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Full text of "CRC Encyclopedia Of Mathematics"

CRC Concise Encyclopedia 

MAmEMAfJCS 



CRC Concise Encyclopedia 

MAfflEMAffG 



Eric W. Weisstein 




CRC Press 
Boca Raton London New York Washington, D.C. 



Library of Congress Cataloging-in-Publication Data 

Weisstein, Eric W. 

The CRC concise encyclopedia of mathematics / Eric W. Weisstein. 
p. cm. 
Includes bibliographical references and index. 
ISBN 0-8493-9640-9 (alk. paper) 
1. Mathematics- -Encyclopedias. I. Title. 
QA5.W45 1998 

510'.3— DC21 98-22385 

CIP 

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources 
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Printed on acid-free paper 



Introduction 



The CRC Concise Encyclopedia of Mathematics is a compendium of mathematical definitions, formulas, 
figures, tabulations, and references. It is written in an informal style intended to make it accessible to a broad 
spectrum of readers with a wide range of mathematical backgrounds and interests. Although mathematics is 
a fascinating subject, it all too frequently is clothed in specialized jargon and dry formal exposition that make 
many interesting and useful mathematical results inaccessible to laypeople. This problem is often further 
compounded by the difficulty in locating concrete and easily understood examples. To give perspective to 
a subject, I find it helpful to learn why it is useful, how it is connected to other areas of mathematics and 
science, and how it is actually implemented. While a picture may be worth a thousand words, explicit 
examples are worth at least a few hundred! This work attempts to provide enough details to give the reader 
a flavor for a subject without getting lost in minutiae. While absolute rigor may suffer somewhat, I hope 
the improvement in usefulness and readability will more than make up for the deficiencies of this approach. 

The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs 
from existing dictionaries of mathematics in a number of important ways. First, the entire text and all 
the equations and figures are available in searchable electronic form on CD-ROM. Second, the entries are 
extensively cross-linked and cross-referenced, not only to related entries but also to many external sites 
on the Internet. This makes locating information very convenient. It also provides a highly efficient way 
to "navigate" from one related concept to another, a feature that is especially powerful in the electronic 
version. Standard mathematical references, combined with a few popular ones, are also given at the end of 
most entries to facilitate additional reading and exploration. In the interests of offering abundant examples, 
this work also contains a large number of explicit formulas and derivations, providing a ready place to locate 
a particular formula, as well as including the framework for understanding where it comes from. 

The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g., 
Borowski and Borwein's HarperCollins Dictionary of Mathematics and Jeans and Jeans' Mathematics Dictio- 
nary). At the same time, the descriptions are more accessible than in "technical" mathematical encyclopedias 
(e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of Mathematics), 
While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate, 
research scientist, or recreational mathematician. In this work, the most useful, interesting, and entertaining 
(at least to my mind) aspects of topics are discussed in addition to their technical definitions. For example, 
in my entry for pi (71-), the definition in terms of the diameter and circumference of a circle is supplemented 
by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. These 
formulas are comprehensible to readers with only minimal mathematical background, and are interesting to 
both those with and without formal mathematics training. However, they have not previously been collected 
in a single convenient location. For this reason, I hope that, in addition to serving as a reference source, this 
work has some of the same flavor and appeal of Martin Gardner's delightful Scientific American columns. 

Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked 
up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections 
weave their way through the physical sciences. It frequently transpires that some piece of recently acquired 
knowledge turns out to be just what I need to solve some apparently unrelated problem. I have therefore 
developed the habit of picking up and storing away odd bits of information for future use. This work has 
provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. I 
have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I 
already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for 
people who are interested in the subjects listed here but who, like me, are not necessarily experts. 

The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and 
continuing during graduate school. The initial document was written in Microsoft Word® on a Mac Plus® 
computer, and had reached about 200 pages by the time I started graduate school in 1990. When Andrew 
Treverrow made his OzTgX program available for the Mac, I began the task of converting all my documents 
to T^X, resulting in a vast improvement in readability. While undertaking the Word to T^}K conversion, I also 
began cross-referencing entries, anticipating that eventually I would be able to convert the entire document 



to hypertext. This hope was realized beginning in 1995, when the Internet explosion was in full swing and 
I learned of Nikos Drakos's excellent I^X to HTML converter, I£TgX2HTML. After some additional effort, 
I was able to post an HTML version of my encyclopedia to the World Wide Web, currently located at 
www . astro . Virginia . edu/ - eww6n/math/. 

The selection of topics included in this compendium is not based on any fixed set of criteria, but rather 
reflects my own random walk through mathematics. In truth, there is no good way of selecting topics in such 
a work. The mathematician James Sylvester may have summed up the situation most aptly. According to 
Sylvester (as quoted in the introduction to Ian Stewart's book From Here to Infinity), "Mathematics is not 
a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to 
ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited 
number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; 
it is not a continent or an ocean, whose area can be mapped out and its "contour defined; it is as limitless as 
that space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are 
forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within 
assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life." 

Several of Sylvester's points apply particularly to this undertaking. As he points out, mathematics itself 
cannot be confined to the pages of a book. The results of mathematics, however, are shared and passed 
on primarily through the printed (and now electronic) medium. While there is no danger of mathematical 
results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical 
results simply because they are not aware of them. Not only does collecting many results in one place provide 
a single starting point for mathematical exploration, but it should also lessen the aggravation of encountering 
explanations for new concepts which themselves use unfamiliar terminology. In this work, the reader is only 
a cross-reference (or a mouse click) away from the necessary background material. As to Sylvester's second 
point, the very fact that the quantity of mathematics is so great means that any attempt to catalog it 
with any degree of completeness is doomed to failure. This certainly does not mean that it's not worth 
trying. Strangely, except for relatively small works usually on particular subjects, there do not appear to 
have been any substantial attempts to collect and display in a place of prominence the treasure trove of 
mathematical results that have been discovered (invented?) over the years (one notable exception being 
Sloane and Plouffe's Encyclopedia of Integer Sequences), This work, the product of the "gazing" of a single 
astronomer, attempts to fill that omission. 

Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, 
neither table of contents nor index are included. In many cases, a particular entry of interest can be located 
from a cross-reference (indicated in SMALL CAPS TYPEFACE in the text) in a related article. In addition, 
most articles are followed by a "see also" list of related entries for quick navigation. This can be particularly 
useful if you are looking for a specific entry (say, "Zeno's Paradoxes"), but have forgotten the exact name. 
By examining the "see also" list at bottom of the entry for "Paradox," you will likely recognize Zeno's name 
and thus quickly locate the desired entry. 

The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning 
with a numeral are ordered by increasing value and appear before the first entry for "A." In multiple-word 
entries containing a space or dash, the space or dash is treated as a character which precedes "a," so entries 
appear in the following order: "Sum," "Sum P. . . ," "Sum-P. . . ," and "Summary." One exception is that 
in a series of entries where a trailing "s" appears in some and not others, the trailing "s" is ignored in the 
alphabetization. Therefore, entries involving Euclid would be alphabetized as follows: "Euclid's Axioms," 
"Euclid Number," "Euclidean Algorithm." Because of the non-standard nomenclature that ensues from 
naming mathematical results after their discoverers, an important result such as the "Pythagorean Theorem" 
is written variously as "Pythagoras 's Theorem," the "Pythagoras Theorem," etc. In this encyclopedia, I have 
endeavored to use the most widely accepted form. I have also tried to consistently give entry titles in the 
singular (e.g., "Knot" instead of "Knots"). 

In cases where the same word is applied in different contexts, the context is indicated in parentheses or 
appended to the end. Examples of the first type are "Crossing Number (Graph)" and "Crossing Number 
(Link)." Examples of the second type are "Convergent Sequence" and "Convergent Series." In the case of 
an entry like "Euler Theorem," which may describe one of three or four different formulas, I have taken the 
liberty of adding descriptive words ("Euler's Something Theorem") to all variations, or kept the standard 



name for the most commonly used variant and added descriptive words for the others. In cases where specific 
examples are derived from a general concept, em dashes ( — ) are used (for example, "Fourier Series," "Fourier 
Series — Power Series," "Fourier Series — Square Wave," "Fourier Series — Triangle"). The decision to put a 
possessive 's at the end of a name or to use a lone trailing apostrophe is based on whether the final "s" 
is pronounced. "Gauss's Theorem" is therefore written out, whereas "Archimedes' Recurrence Formula" is 
not. Finally, given the absence of a definitive stylistic convention, plurals of numerals are written without 
an apostrophe (e.g., 1990s instead of 1990's). 

In an endeavor of this magnitude, errors and typographical mistakes are inevitable. The blame for these 
lies with me alone. Although the current length makes extensive additions in a printed version problematic, 
I plan to continue updating, correcting, and improving the work. 

Eric Weisstein 

Charlottesville, Virginia 
August 8, 1998 



Acknowledgments 



Although I alone have compiled and typeset this work, many people have contributed indirectly and 
directly to its creation. I have not yet had the good fortune to meet Donald Knuth of Stanford University, 
but he is unquestionably the person most directly responsible for making this work possible. Before his 
mathematical typesetting program TfeX, it would have been impossible for a single individual to compile such 
a work as this. Had Prof. Bateman owned a personal computer equipped with T£jX, perhaps his shoe box of 
notes would not have had to await the labors of Erdelyi, Magnus, and Oberhettinger to become a three- volume 
work on mathematical functions. Andrew Trevorrow's shareware implementation of I^X for the Macintosh, 
OzI]eX (www.kagi.com/authors/akt/oztex.html), was also of fundamental importance. Nikos Drakos and 
Ross Moore have provided another building block for this work by developing the IM]gX2HTML program 
(www-dsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html), which has allowed me to 
easily maintain and update an on-line version of the encyclopedia long before it existed in book form. 

I would like to thank Steven Finch of MathSoft, Inc., for his interesting on-line essays about mathemat- 
ical constants (www.mathsoft.com/asolve/constant/constant.html), and also for his kind permission to 
reproduce excerpts from some of these essays. I hope that Steven will someday publish his detailed essays 
in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed 
and on-line (www.research.att.com/-njas/sequences/) versions of the Encyclopedia of Integer Sequences, 
an immensely valuable compilation of useful information which represents a truly mind-boggling investment 
of labor. 

Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript 
and providing a number of helpful suggestions and additions. Thanks also to algebraic topologist Ryan Bud- 
ney for sharing some of his expertise, to Charles Walkden for his helpful comments about dynamical systems 
theory, and to Lambros Lambrou for his contributions. Thanks to David W. Wilson for a number of helpful 
comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permis- 
sion to reproduce their beautiful knot and link diagrams. Thanks to Gavin Theobald for providing diagrams 
of his masterful polygonal dissections. Thanks to Wolfram Research, not only for creating an indispensable 
mathematical tool in Mathematica® , but also for permission to include figures from the Mathematical book 
and MathSource repository for the braid, conical spiral, double helix, Enneper's surfaces, Hadamard matrix, 
helicoid, helix, Henneberg's minimal surface, hyperbolic polyhedra, Klein bottle, Maeder's "owl" minimal 
surface, Penrose tiles, polyhedron, and Scherk's minimal surfaces entries. 

Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading 
the entire document for syntax. Thanks also to Bob Stern, my executive editor from CRC Press, for 
his encouragement, and to Mimi Williams of CRC Press for her careful reading of the manuscript for 
typographical and formatting errors. As this encyclopedia's entry on Proofreading Mistakes shows, the 
number of mistakes that are expected to remain after three independent proofreadings is much lower than 
the original number, but unfortunately still nonzero. Many thanks to the library staff at the University of 
Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I 
would like to thank the hundreds of people who took the time to e-mail me comments and suggestions while 
this work was in its formative stages. Your continued comments and feedback are very welcome. 







10 



Numerals 





see Zero 



The number one (1) is the first Positive Integer. It 
is an Odd Number. Although the number 1 used to be 
considered a PRIME Number, it requires special treat- 
ment in so many definitions and applications involving 
primes greater than or equal to 2 that it is usually placed 
into a class of its own. The number 1 is sometimes also 
called "unity," so the nth roots of 1 are often called the 
nth Roots of Unity. Fractions having 1 as a Nu- 
merator are called Unit Fractions. If only one root, 
solution, etc., exists to a given problem, the solution is 
called Unique. 

The Generating Function have all Coefficients 1 
is given by 

1 ii ,2.3.4. 

1 + x + x -\- x + x + 



l~x 



see also 2, 3, Exactly One, Root of Unity, Unique, 
Unit Fraction, Zero 



The number two (2) is the second POSITIVE INTEGER 
and the first PRIME NUMBER. It is Even, and is the only 
Even Prime (the Primes other than 2 are called the 
Odd Primes). The number 2 is also equal to its Fac- 
torial since 2! = 2. A quantity taken to the Power 2 
is said to be SQUARED. The number of times k a given 
BINARY number & n --*&2&i&o is divisible by 2 is given 
by the position of the first 6^ = 1, counting from the 
right. For example, 12 = 1100 is divisible by 2 twice, 
and 13 = 1101 is divisible by 2 times. 
see also 1, BINARY, 3, SQUARED, ZERO 

2x mod 1 Map 

Let xo be a Real Number in the Closed Interval 
[0, 1], and generate a SEQUENCE using the MAP 



Xn+i = 2x n (mod 1). 



(i) 



Then the number of periodic Orbits of period p (for p 
Prime) is given by 



N„ 



2 p -2 
V 



(2) 



Since a typical Orbit visits each point with equal prob- 
ability, the Natural Invariant is given by 



P {x) = 1. 



(3). 



see also Tent Map 



References 

Ott, E. Chaos in Dynamical Systems. Cambridge: Cam- 
bridge University Press, pp. 26-31, 1993. 



3 is the only INTEGER which is the sum of the preceding 
Positive Integers (1 + 2 = 3) and the only number 
which is the sum of the FACTORIALS of the preceding 
Positive Integers (1! + 2! = 3). It is also the first 
Odd Prime. A quantity taken to the Power 3 is said 
to be Cubed. 

see also 1, 2, 3^ + 1 Mapping, Cubed, Period Three 
Theorem, Super-3 Number, Ternary, Three- 
Colorable, Zero 

3x + 1 Mapping 

see Collatz Problem 

10 

The number 10 (ten) is the basis for the DECIMAL sys- 
tem of notation. In this system, each "decimal place" 
consists of a DIGIT 0-9 arranged such that each Digit 
is multiplied by a POWER of 10, decreasing from left to 
right, and with a decimal place indicating the 10° = Is 
place. For example, the number 1234.56 specifies 

Ixl0 3 +2xl0 2 +3xl0 1 +4xl0° + 5xl0~ 1 +6xl0~ 2 . 

The decimal places to the left of the decimal point 
are 1, 10, 100, 1000, 10000, 10000, 100000, 10000000, 
100000000, ... (Sloane's A011557), called one, ten, 
HUNDRED, THOUSAND, ten thousand, hundred thou- 
sand, Million, 10 million, 100 million, and so on. The 
names of subsequent decimal places for Large Num- 
bers differ depending on country. 

Any Power of 10 which can be written as the PRODUCT 
of two numbers not containing 0s must be of the form 
2 n • 5 n — 10 n for n an INTEGER such that neither 2 n nor 
5 n contains any ZEROS. The largest known such number 



10 33 - 2 33 * 5 33 
= 8, 589, 934, 592 ■ 116, 415, 321, 826, 934, 814, 453, 125. 

A complete list of known such numbers is 

10 1 = 2 1 

10 2 = 2 2 



10 4 



10' 
10 9 

10 18 
10 33 



: 2 9 * 5 9 
: 2 18 ■ 5 1 
2 33 • 5 3 



(Madachy 1979). Since all POWERS of 2 with exponents 
n < 4.6 X 10 7 contain at least one ZERO (M. Cook), no 



12 



18-Point Problem 



other POWER of ten less than 46 million can be written 
as the PRODUCT of two numbers not containing Os. 

see also Billion, Decimal, Hundred, Large Num- 
ber, Milliard, Million, Thousand, Trillion, Zero 

References 

Madachy, J. S. Madachy J s Mathematical Recreations. New 

York: Dover, pp. 127-128, 1979. 
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, 

p. 135, 1995. 
Sloane, N. J. A. Sequence A011557 in "An On-Line Version 

of the Encyclopedia of Integer Sequences." 

12 

One Dozen, or a twelfth of a Gross. 

see also DOZEN, GROSS 

13 

A Number traditionally associated with bad luck. A 
so-called Baker's Dozen is equal to 13. Fear of the 
number 13 is called Triskaidekaphobia. 

see also Baker's Dozen, Friday the Thirteenth, 
Triskaidekaphobia 

15 

see 15 Puzzle, Fifteen Theorem 

15 Puzzle 



2 


1 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 





A puzzle introduced by Sam Loyd in 1878. It consists of 
15 squares numbered from 1 to 15 which are placed in a 
4x4 box leaving one position out of the 16 empty. The 
goal is to rearrange the squares from a given arbitrary 
starting arrangement by sliding them one at a time into 
the configuration shown above. For some initial arrange- 
ments, this rearrangement is possible, but for others, it 
is not. 

To address the solubility of a given initial arrangement, 
proceed as follows. If the SQUARE containing the num- 
ber i appears "before" (reading the squares in the box 
from left to right and top to bottom) n numbers which 
are less than £, then call it an inversion of order n, and 
denote it rii. Then define 



N — X^ n * = 5Z n *' 



where the sum need run only from 2 to 15 rather than 
1 to 15 since there are no numbers less than 1 (so n\ 
must equal 0). If AT is EVEN, the position is possible, 
otherwise it is not. This can be formally proved using 
Alternating Groups. For example, in the following 
arrangement 



ri2 = 1 (2 precedes 1) and all other rii = 0, so N — 1 
and the puzzle cannot be solved. 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, pp. 312- 
316, 1987. 

Bogomolny, A. "Sam Loyd's Fifteen." http://www.cut— the- 
knot.com/pythagoras/fifteen.html. 

Bogomolny, A. "Sam Loyd's Fifteen [History]." http://www. 
cut-the-knot .com/pythagoras/historyl5.html. 

Johnson, W. W. "Notes on the '15 Puzzle. I.'" Amer. J. 
Math. 2, 397-399, 1879. 

Kasner, E. and Newman, J. R. Mathematics and the Imagi- 
nation. Redmond, WA: Tempus Books, pp. 177-180, 1989. 

Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical 
Recreations. New York: W. W. Norton, pp. 302-308, 1942. 

Story, W. E. "Notes on the '15 Puzzle. II.*" Amer. J. Math. 
2, 399-404, 1879. 

16-Cell 

A finite regular 4-D POLYTOPE with SCHLAFLI SYMBOL 
{3, 3, 4} and Vertices which are the PERMUTATIONS 
of (±1, 0, 0, 0). 

see also 24-Cell, 120-Cell, 600-Cell, Cell, Poly- 
tope 

17 

17 is a FERMAT PRIME which means that the 17-sided 
Regular Polygon (the Heptadecagon) is Con- 

STRUCTIBLE using COMPASS and STRAIGHTEDGE (as 

proved by Gauss). 

see also CONSTRUCTIBLE POLYGON , FERMAT PRIME, 

HEPTADECAGON 

References 

Carr, M. "Snow White and the Seven(teen) Dwarfs." 

http:// www . math . harvard . edu / - hmb / issue2.1 / 

SEVENTEEN/seventeen.html. 
Fischer, R. "Facts About the Number 17." http: //tempo, 

harvard . edu / - rf ischer / hcssim / 17_f acts / kelly / 

kelly.html. 
Lefevre, V. "Properties of 17." http://www.ens-lyon.fr/ 

-vlefevre/dl7_eng.html. 
Shell Centre for Mathematical Education. "Number 

17." http : //acorn . educ . nott ingham . ac . uk/ShellCent/ 

Number /Num 17 .html. 

18-Point Problem 

Place a point somewhere on a Line Segment. Now 
place a second point and number it 2 so that each of the 
points is in a different half of the Line SEGMENT. Con- 
tinue, placing every ATth point so that all N points are 
on different (l/iV)th of the Line Segment. Formally, 
for a given N y does there exist a sequence of real num- 
bers xi t X2, • • • , #jv such that for every n £ {1, . - . , N} 
and every k £ {1, . . . , n}, the inequality 

fc- 1 ^ k 

— < Xi < - 

n n 



24-Cell 



196-Algorithm 



holds for some i € {l,...,n}? Surprisingly, it is only 
possible to place 17 points in this manner (Berlekamp 
and Graham 1970, Warmus 1976). 

Steinhaus (1979) gives a 14-point solution (0.06, 0.55, 
0.77, 0.39, 0.96, 0.28, 0.64", 0.13, 0.88, 0.48, 0.19, 0.71, 
0.35, 0.82), and Warmus (1976) gives the 17-point solu- 
tion 

| < a* < ■&> f < X2 < £, jf < x 3 < 1, £ < x 4 < ^, 

IT < ** < IS- H < ** < h 1 < ^ < £, if < ** < h 
I <x 9 < ±,$ <x 10 < *,± <zu < £, 

17 < ^12 < 12 > 2 — Xl2 < 17' U — Xl4 < 17' 

13 ^ ^ ^ 4 5 ^ _ ^ 6 10 ^ ^ ^ 11 

Warmus (1976) states that there are 768 patterns of 17- 
point solutions (counting reversals as equivalent). 
see also Discrepancy Theorem, Point Picking 

References 

Berlekamp, E. R. and Graham, R. L. "Irregularities in the 
Distributions of Finite Sequences." J. Number Th. 2, 152- 
161, 1970. 

Gardner, M. The Last Recreations: Hydras, Eggs, and Other 
Mathematical Mystifications. New York: Springer- Verlag, 
pp. 34-36, 1997. 

Steinhaus, H. "Distribution on Numbers" and "Generaliza- 
tion." Problems 6 and 7 in One Hundred Problems in 
Elementary Mathematics. New York: Dover, pp. 12-13, 
1979. 

Warmus, M. "A Supplementary Note on the Irregularities of 
Distributions." J. Number Th. 8, 260-263, 1976. 

24-Cell 

A finite regular 4-D Polytope with SCHLAFLI Symbol 
{3,4,3}. Coxeter (1969) gives a list of the VERTEX po- 
sitions. The Even coefficients of the D 4 lattice are 1, 
24, 24, 96, ... (Sloane's A004011), and the 24 shortest 
vectors in this lattice form the 24-cell (Coxeter 1973, 
Conway and Sloane 1993, Sloane and Plouffe 1995). 
see also 16-Cell, 120-Cell, 600-Cell, Cell, Poly- 
TOPE 

References 

Conway, J. H. and Sloane, N. J. A. Sphere- Packings, Lattices 
and Groups, 2nd ed. New York: Springer- Verlag, 1993. 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 
York: Wiley, p. 404, 1969. 

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: 
Dover, 1973. 

Sloane, N. J. A. Sequences A004011/M5140 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 

42 

According to Adams, 42 is the ultimate answer to life, 
the universe, and everything, although it is left as an 
exercise to the reader to determine the actual question 
leading to this result. 

References 

Adams, D. The Hitchhiker's Guide to the Galaxy. New York: 
Ballantine Books, 1997. 



72 Rule 

see Rule of 72 

120-Cell 

A finite regular 4-D Polytope with Schlafli Symbol 
{5,3,3} (Coxeter 1969). 

see also 16-Cell, 24-Cell, 600-Cell, Cell, Poly- 
tope 

Preferences 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 
York: Wiley, p. 404, 1969. 

144 

A Dozen Dozen, also called a Gross. 144 is a Square 
Number and a Sum-Product Number. 

see also Dozen 

196-Algorithm 

Take any POSITIVE INTEGER of two DIGITS or more, re- 
verse the DIGITS, and add to the original number. Now 
repeat the procedure with the SUM so obtained. This 
procedure quickly produces PALINDROMIC NUMBERS for 
most INTEGERS. For example, starting with the num- 
ber 5280 produces (5280, 6105, 11121, 23232). The end 
results of applying the algorithm to 1, 2, 3, ... are 1, 2, 
3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, 
... (Sloane's A033865). The value for 89 is especially 
large, being 8813200023188. 

The first few numbers not known to produce PALIN- 
DROMES are 196, 887, 1675, 7436, 13783, . . . (Sloane's 
A006960), which are simply the numbers obtained by 
iteratively applying the algorithm to the number 196. 
This number therefore lends itself to the name of the 
Algorithm. 

The number of terms a(n) in the iteration sequence re- 
quired to produce a Palindromic Number from n (i.e., 
a(n) = 1 for a PALINDROMIC NUMBER, a(n) = 2 if a 
Palindromic Number is produced after a single iter- 
ation of the 196-algorithm, etc.) for n = 1, 2, . . . are 

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 

2, 2, 1, ... (Sloane's A030547). The smallest numbers 
which require n = 0, 1, 2, . . . iterations to reach a palin- 
drome are 0, 10, 19, 59, 69, 166, 79, 188, . . . (Sloane's 
A023109). 

see also Additive Persistence, Digitadition, Mul- 
tiplicative Persistence, Palindromic Number, 
Palindromic Number Conjecture, RATS Se- 
quence, Recurring Digital Invariant 

References 

Gardner, M. Mathematical Circus: More Puzzles, Games, 
Paradoxes and Other Mathematical Entertainments from 
Scientific American. New York: Knopf, pp. 242-245, 1979. 

Gruenberger, F. "How to Handle Numbers with Thousands 
of Digits, and Why One Might Want to." Sci. Amer. 250, 
19-26, Apr. 1984. 

Sloane, N. J. A. Sequences A023109, A030547, A033865, and 
A006960/M5410 in "An On-Line Version of the Encyclo- 
pedia of Integer Sequences." 



239 



65537-gon 



239 

Some interesting properties (as well as a few arcane ones 
not reiterated here) of the number 239 are discussed in 
Beeler et al. (1972, Item 63). 239 appears in Machin's 
Formula 

| 7 r = 4tan(|)-tan- 1 (^), 



which is related to the fact that 



2 * 13 - 1 



239 2 , 



which is why 239/169 is the 7th CONVERGENT of y/2 . 
Another pair of INVERSE TANGENT FORMULAS involv- 
ing 239 is 

tan" 1 ^) = tan" 1 ^) - tan" 1 ^) 



= tan x (^)+tan l (^). 



239 needs 4 SQUARES (the maximum) to express it, 9 
Cubes (the maximum, shared only with 23) to express 
it, and 19 fourth POWERS (the maximum) to express it 
(see Waring'S Problem). However, 239 doesn't need 
the maximum number of fifth POWERS (Beeler et al 
1972, Item 63). 

References 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM-239, Feb. 1972. 

257-gon 

257 is a FERMAT PRIME, and the 257-gon is there- 
fore a Constructible Polygon using Compass and 
Straightedge, as proved by Gauss. An illustration 
of the 257-gon is not included here, since its 257 seg- 
ments so closely resemble a Circle. Richelot and 
Schwendenwein found constructions for the 257-gon in 
1832 (Coxeter 1969). De Temple (1991) gives a con- 
struction using 150 Circles (24 of which are Car- 
lyle Circles) which has Geometrography symbol 
945i + 475 2 + 275Ci + 0C 2 + 150C 3 and Simplicity 
566. 
see also 65537-GON, CONSTRUCTIBLE POLYGON, Fer- 

mat Prime, Heptadecagon, Pentagon 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 
York: Wiley, 1969. 

De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- 
ity of Polygonal Constructions." Amer. Math. Monthly 98, 
97-108, 1991. 

Dixon, R. Mathographics. New York: Dover, p. 53, 1991. 

Rademacher, H. Lectures on Elementary Number Theory. 
New York: Blaisdell, 1964. 



600-Cell 

A finite regular 4-D Polytope with Schlafli Symbol 
{3,3,5}. For Vertices, see Coxeter (1969). 
see also 16-Cell, 24-Cell, 120-Cell, Cell, Poly- 
tope 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 
York: Wiley, p. 404, 1969. 

666 

A number known as the Beast Number appearing in 
the Bible and ascribed various numerological properties. 

see also Apocalyptic Number, Beast Number, Le- 
viathan Number 

References 

Hardy, G. H. A Mathematician's Apology, reprinted with a 
foreword by C. P. Snow. New York: Cambridge University 
Press, p. 96, 1993. 

2187 

The digits in the number 2187 form the two VAMPIRE 

NUMBERS: 21 x 87 = 1827 and 2187 = 27 x 81. 

References 

Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19, 
26-29, Spring 1997. 

65537-gon 

65537 is the largest known Fermat Prime, and the 
65537-gon is therefore a CONSTRUCTIBLE POLYGON us- 
ing Compass and Straightedge, as proved by Gauss. 
The 65537-gon has so many sides that it is, for all in- 
tents and purposes, indistinguishable from a CIRCLE us- 
ing any reasonable printing or display methods. Her- 
mes spent 10 years on the construction of the 65537-gon 
at Gottingen around 1900 (Coxeter 1969). De Temple 
(1991) notes that a Geometric Construction can be 
done using 1332 or fewer Carlyle Circles. 

see also 257-GON, CONSTRUCTIBLE POLYGON, HEP- 
TADECAGON, Pentagon 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 
York: Wiley, 1969. 

De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- 
ity of Polygonal Constructions." Amer. Math. Monthly 98, 
97-108, 1991. 

Dixon, R. Mathographics. New York: Dover, p. 53, 1991. 



A-Integrable 

A 

A-Integrable 

A generalization of the Lebesgue INTEGRAL. A MEA- 
SURABLE Function f(x) is called A-integrable over the 
Closed Interval [a, b] if 

m{x:\f(x)\>n} = 0(n- 1 ), (1) 

where m is the LEBESGUE MEASURE, and 



lim / 



[f(x)] n dx 



(2) 



exists, where 



tf(xW -IfW if 1/0*01 <" |« 

l/(*)J»-| if|/( x )|>„. W 



References 

Titmarsch, E. G. "On Conjugate Functions." Proc. London 
Math. Soc. 29, 49-80, 1928. 

A- Sequence 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

An Infinite Sequence of Positive Integers ai sat- 
isfying 

1 < ai < a-2. < az < ■ • . (1) 

is an A-sequence if no a^ is the SUM of two or more 
distinct earlier terms (Guy 1994). Erdos (1962) proved 

oo 

S{A) = sup Y^ ~ < 103 - ( 2 ) 

all A sequences , a k 

Any A-sequence satisfies the Chi Inequality (Levine 
and O'Sullivan 1977), which gives 5(A) < 3.9998. Ab- 
bott (1987) and Zhang (1992) have given a bound from 
below, so the best result to date is 



AAS Theorem 



Erdos, P. "Remarks on Number Theory III. Some Problems 

in Additive Number Theory." Mat. Lapok 13, 28-38, 1962. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/erdos/erdos.html. 
Guy, R. K. "B 2 -Sequences." §E28 in Unsolved Problems 

in Number Theory, 2nd ed. New York: Springer- Verlag, 

pp. 228-229, 1994. 
Levine, E. and O'Sullivan, J. "An Upper Estimate for the 

Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34, 

9-24, 1977. 
Zhang, Z. X. "A Sum-Free Sequence with Larger Reciprocal 

Sum." Unpublished manuscript, 1992. 

AAA Theorem 




Specifying three ANGLES A, B, and C does not uniquely 
define a Triangle, but any two TRIANGLES with the 
same Angles are SIMILAR. Specifying two ANGLES of 
a TRIANGLE automatically gives the third since the sum 
of Angles in a Triangle sums to 180° (it Radians), 
i.e., 

C = tt-A-B. 

see also AAS Theorem, ASA Theorem, ASS Theo- 
rem, SAS Theorem, SSS Theorem, Triangle 

AAS Theorem 




Specifying two angles A and B and a side a uniquely 
determines a TRIANGLE with AREA 



K ■ 



a 2 sin B sin C a 2 sin B sin(7r — A — B) 



2 sin A 2 sin A 

The third angle is given by 

C = ir - A- B, 



(1) 



(2) 



2.0649 < 5(A) < 3.9998. 



(3) 



since the sum of angles of a Triangle is 180° (n Ra- 
dians). Solving the Law of Sines 



Levine and O'Sullivan (1977) conjectured that the sum 
of Reciprocals of an A-sequence satisfies 



oo 



(4) 



where %% are given by the Levine-O'Sullivan Greedy 

Algorithm. 

see also B 2 -Sequence, Mian-Chowla Sequence 

References 

Abbott, H. L. "On Sum-Free Sequences." Acta Arith. 48, 
93-96, 1987. 



for b gives 



Finally, 



sin A sin B 



sinB 
b = a—r 



sin A 



(3) 
(4) 



c = b cos A + a cos B = a(sin B cot A -f cos B) (5) 
= a sin B(cot A -f cot B) . (6) 

see also AAA Theorem, ASA Theorem, ASS Theo- 
rem, SAS Theorem, SSS Theorem, Triangle 



6 



Abacus 



AbeVs Functional Equation 



Abacus 

A mechanical counting device consisting of a frame hold- 
ing a series of parallel rods on each of which beads are 
strung. Each bead represents a counting unit, and each 
rod a place value. The primary purpose of the abacus 
is not to perform actual computations, but to provide 
a quick means of storing numbers during a calculation. 
Abaci were used by the Japanese and Chinese, as well 
as the Romans. 
see also Roman Numeral, Slide Rule 

References 

Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal 
Fractions." A History of Mathematics, 2nd ed. New York: 
Wiley, pp. 199-201, 1991. 

Fernandes, L. "The Abacus: The Art of Calculating with 
Beads." http : //www . ee . ryerson . ca : 8080/-elf /abacus. 

Gardner, M. "The Abacus." Ch. 18 in Mathematical Circus: 
More Puzzles, Games, Paradoxes and Other Mathemati- 
cal Entertainments from Scientific American. New York: 
Knopf, pp. 232-241, 1979. 

Pappas, T. "The Abacus." The Joy of Mathematics. San 
Carlos, CA: Wide World Publ./Tetra, p. 209, 1989. 

Smith, D. E. "Mechanical Aids to Calculation: The Abacus." 
Ch. 3 §1 in History of Mathematics, Vol. 2. New York: 
Dover, pp. 156-196, 1958. 

abc Conjecture 

A Conjecture due to J. Oesterle and D. W. Masser. 

It states that, for any INFINITESIMAL e > 0, there exists 
a Constant C e such that for any three Relatively 
Prime Integers a, 6, c satisfying 

a 4- b = c, 
the Inequality 

max{|a|,|6|,|c|}<a JJ p 1+e 

p\abc 

holds, where p\abc indicates that the PRODUCT is over 
Primes p which Divide the Product abc. If this 
Conjecture were true, it would imply Fermat's 
Last Theorem for sufficiently large Powers (Goldfeld 
1996). This is related to the fact that the abc conjecture 
implies that there are at least C In x WlEFERlCH PRIMES 
< x for some constant C (Silverman 1988, Vardi 1991). 

see also Fermat's Last Theorem, Mason's Theo- 
rem, Wieferich Prime 

References 

Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. 
Math. Monthly 101, 3-14, 1994. 

Goldfeld, D. "Beyond the Last Theorem." The Sciences, 34- 
40, March/April 1996. 

Guy, R. K, Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, pp. 75-76, 1994. 

Silverman, J. " Wieferich's Criterion and the abc Conjecture." 
J. Number Th. 30, 226-237, 1988. 

Vardi, I. Computational Recreations in Mathematica. Read- 
ing, MA: Addison- Wesley, p. 66, 1991. 



Abelian 

see Abelian Category, Abelian Differential, 
Abelian Function, Abelian Group, Abelian In- 
tegral, Abelian Variety, Commutative 

Abelian Category 

An Abelian category is an abstract mathematical CAT- 
EGORY which displays some of the characteristic prop- 
erties of the Category of all Abelian Groups. 

see also Abelian Group, Category 

Abel's Curve Theorem 

The sum of the values of an INTEGRAL of the "first" or 
"second" sort 

f XltV1 Pdr [*n,vn pd 

/ ^ + - + J ^ = F ^ 



and 



P(xi,2/i) dxx P(xn,Vn) dx N 



Q(rci,yi) dz 



Q(xn,Vn) dz 



dF 

dz ' 



from a FIXED Point to the points of intersection with a 
curve depending rationally upon any number of param- 
eters is a Rational Function of those parameters. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 277, 1959. 

Abelian Differential 

An Abelian differential is an ANALYTIC or MEROMOR- 
phic Differential on a Compact or closed Riemann 

Surface. 

Abelian Function 

An Inverse Function of an Abelian Integral. 

Abelian functions have two variables and four periods. 
They are a generalization of ELLIPTIC FUNCTIONS, and 
are also called Hyperelliptic Functions. 

see also Abelian Integral, Elliptic Function 

References 

Baker, H. F. Abelian Functions: Abel's Theorem and the Al- 
lied Theory, Including the Theory of the Theta Functions. 
New York: Cambridge University Press, 1995. 

Baker, H. F. An Introduction to the Theory of Multiply Pe- 
riodic Functions. London: Cambridge University Press, 
1907. 

Abel's Functional Equation 

Let Li2(x) denote the DlLOGARITHM, defined by 



— n 



Abelian Group 

then 

Li 2 (a) + Li 2 (y) + lA 2 {xy) + 
+ 



(1-2/) 






see a/50 DlLOGARITHM, POLYLOGARITHM, RlEMANN 

Zeta Function 

Abelian Group 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

A Group for which the elements Commute (i.e., AB = 
BA for all elements A and B) is called an Abelian group. 
All Cyclic Groups are Abelian, but an Abelian group 
is not necessarily CYCLIC. All SUBGROUPS of an Abelian 
group are NORMAL. In an Abelian group, each element 
is in a CONJUGACY CLASS by itself, and the CHARACTER 
TABLE involves POWERS of a single element known as a 
Generator. 

No general formula is known for giving the number 
of nonisomorphic Finite GROUPS of a given ORDER. 
However, the number of nonisomorphic Abelian FINITE 
Groups a(n) of any given Order n is given by writing 
n as 

n = Y[pi"\ (1) 

i 

where the pt are distinct PRIME FACTORS, then 



a(n) =Y[P( ai ), 



(2) 



where P is the Partition Function. This gives 1,1, 
1, 2, 1, 1, 1, 3, 2, . . . (Sloane's A000688). The smallest 
orders for which n = 1, 2, 3, ... nonisomorphic Abelian 
groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 
64, 1800, 0, 288, 128, ... (Sloane's A046056), where 
denotes an impossible number (i.e., not a product of 
partition numbers) of nonisomorphic Abelian, groups. 
The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34, 
37, 38, 39, 41, 43, 46, ... (Sloane's A046064). The 
incrementally largest numbers of Abelian groups as a 
function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 
77, 101, . . . (Sloane's A046054), which occur for orders 
1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 
... (Sloane's A046055). 

The Kronecker Decomposition Theorem states 
that every Finite Abelian group can be written as a Di- 
rect Product of Cyclic Groups of Prime Power 
Orders. If the Orders of a Finite Group is a Prime 

p, then there exists a single Abelian group of order p 
(denoted Z p ) and no non-Abelian groups. If the Or- 
ders is a prime squared p 2 , then there are two Abelian 
groups (denoted Z p 2 and Z p & Z p . If the Orders is 



Abelian Group 7 

a prime cubed p 3 , then there are three Abelian groups 
(denoted Z p <g> Z p (g> Z p , Z p % Z p 2, and Z p a), and five 
groups total. If the order is a PRODUCT of two primes 
p and q, then there exists exactly one Abelian group of 
order pq (denoted Z p ® Z q ). 

Another interesting result is that if a(n) denotes the 
number of nonisomorphic Abelian groups of ORDER n, 

then 



^a(n)n- s = CWC(2s)C(3 S )- 



(3) 



n=l 



where ((s) is the Riemann Zeta Function. Srinivasan 
(1973) has also shown that 

N 

Y, a (n) = A 1 N+A 2 N 1/2 +A 3 N 1/3 +O[x 105/407 (]nx) 2 ], 

n=l 

(4) 
where 

( 2.294856591... for k = 1 
Ak = n^(i) = \ - 14 -6475663... for k = 2 (5) 
j=i V } { 118.6924619 ... for k = 3, 

and ( is again the Riemann Zeta Function. [Richert 
(1952) incorrectly gave As = 114.] DeKoninck and Ivic 
(1980) showed that 



^J-^BN + Oi^ilnN)- 1 ' 2 }, 



i(n) 



(6) 



where 



nKE 



P(k - 2) P(k) 






0.752 . . 



(7) 

is a product over Primes. Bounds for the number of 
nonisomorphic non-Abelian groups are given by Neu- 
mann (1969) and Pyber (1993). 

see also Finite Group, Group Theory, Kronecker 
Decomposition Theorem, Partition Function P, 
Ring 

References 

DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func- 
tions: Asymptotic Formulae for Sums of Reciprocals of 
Arithmetical Functions and Related Fields. Amsterdam, 
Netherlands: North- Holland, 1980. 

Erdos, P. and Szekeres, G. "Uber die Anzahl abelscher Grup- 
pen gegebener Ordnung und iiber ein verwandtes zahlen- 
theoretisches Problem." Acta Sci. Math. (Szeged) 7, 95- 
102, 1935. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/abel/abel.html. 

Kendall, D. G. and Rankin, R. A. "On the Number of Abelian 
Groups of a Given Order." Quart J. Oxford 18, 197-208, 
1947. 

Kolesnik, G. "On the Number of Abelian Groups of a Given 
Order." J. Reine Angew. Math. 329, 164-175, 1981. 



8 



Abel's Identity 



Abel's Irreducibility Theorem 



Neumann, P. M. "An Enumeration Theorem for Finite 

Groups." Quart J. Math. Ser. 2 20, 395-401, 1969. 
Pyber, L. "Enumerating Finite Groups of Given Order." 

Ann. Math. 137, 203-220, 1993. 
Richert, H.-E. "Uber die Anzahl abelscher Gruppen 

gegebener Ordnung L" Math. Zeitschr. 56, 21-32, 1952. 
Sloane, N. J. A. Sequence A000688/M0064 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 
Srinivasan, B. R. "On the Number of Abelian Groups of a 

Given Order." Acta Arith. 23, 195-205, 1973. 

Abel's Identity 

Given a homogeneous linear SECOND-ORDER ORDI- 
NARY Differential Equation, 



y" + P(x)y' + Q(x)y = 0, 



(1) 



call the two linearly independent solutions yi(x) and 
y 2 (as). Then 

y'l{x) + P{x)y , l {x) + Q{x)y 1 = ^ (2) 

y' 2 ' (x) + P(x)y' 2 (x) + Q(x)y 2 = 0. (3) 

Now, take yi x (3) - y 2 x (2), 

yilvZ + P(x)y2 + Q(x)y 2 ] 

-V2[yi+P(x)y' 1 +Q(x)y l ]=Q (4) 

(yiy% -y2y")+P(yiy2-yiy2)+Q(yiy2-yiy2) = (5) 

(2/12/2 - 2/22/") + P(2/i2/2 - 2/i2/2) = 0. (6) 

Now, use the definition of the Wronskian and take its 
Derivative, 

W = y t y 2 -2/12/2 (7) 

W = (y[y 2 + yiyi) - (yiyi + 2/12/2) 

= 2/12/2-2/1^2. (8) 



Plugging W and W into (6) gives 
W' 4- PW = 0. 
This can be rearranged to yield 

w = - p ^ dx 

which can then be directly integrated to 
lnl^ = -Ci / P(x)dx, 



(9) 



(10) 



(11) 



where In as is the Natural Logarithm. A second in- 
tegration then yields AbePs identity 

W(x)=C 2 e~f P(x)dx , (12) 

where C\ is a constant of integration and C 2 = e Cl . 
see alsa Ordinary Differential Equation — Sec- 
ond-Order 

References 

Boyce, W. E. and DiPrima, R. C. Elementary Differential 
Equations and Boundary Value Problems, J^th ed. New 
York: Wiley, pp. 118, 262, 277, and 355, 1986. 



Abel's Impossibility Theorem 

In general, Polynomial equations higher than fourth 
degree are incapable of algebraic solution in terms of 
a finite number of Additions, Multiplications, and 
Root extractions. 

see also Cubic Equation, Galois's Theorem, Poly- 
nomial, Quadratic Equation, Quartic Equation, 
Quintic Equation 

References 

Abel, N. H, "Demonstration de l'impossibilite de la resolution 

algebraique des equations generates qui depassent le qua- 

trieme degre." Crelle's J. 1, 1826. 

Abel's Inequality 

Let {f n } and {a n } be Sequences with f n > fn+i > 
for n = 1, 2, . . . , then 



/ ^CLnfn 



<Ah, 



where 



A = max{|ai|, |ai + a 2 \ , . . - , |ai + a 2 + . . . 4- a m |}. 



Abelian Integral 

An Integral of the form 



Jo 



dt 



where R(t) is a POLYNOMIAL of degree > 4. They are 
also called Hyperelliptic Integrals. 

see also Abelian Function, Elliptic Integral 

Abel's Irreducibility Theorem 

If one ROOT of the equation f(x) = 0, which is irre- 
ducible over a Field K, is also a ROOT of the equation 
F(x) = in K, then all the ROOTS of the irreducible 
equation f(x) = are ROOTS of F(x) = 0. Equivalently, 
F(x) can be divided by f(x) without a Remainder, 

F(x) = f{x)F 1 (x) i 

where Fi(x) is also a POLYNOMIAL over K. 

see also ABEL'S LEMMA, KRONECKER'S POLYNOMIAL 

Theorem, Schoenemann's Theorem 

References 

Abel, N. H. "Memoir sur une classe particuliere d'equations 

resolubles algebraiquement." Crelle's J. 4, 1829. 
Dorrie, H. 100 Great Problems of Elementary Mathematics: 

Their History and Solutions. New York: Dover, p. 120, 

1965. 



Abel's Lemma 



Abhyankar's Conjecture 9 



Abel's Lemma 
The pure equation 

x p = C 

of PRIME degree p is irreducible over a FIELD when C 
is a number of the FIELD but not the pth Power of an 
element of the Field. 

see also Abel's Irreducibility Theorem, Gauss's 
Polynomial Theorem, Kronecker's Polynomial 
Theorem, Schoenemann's Theorem 

References 

Dorrie, H. 100 Great Problems of Elementary Mathematics: 

Their History and Solutions. New York: Dover, p. 118, 

1965. 

Abel's Test 

see Abel's Uniform Convergence Test 

Abel's Theorem 

Given a Taylor Series 



F(z) = J2CnZ n = ^Tc n r n e i " 



(1) 



71=0 



n=0 



where the COMPLEX NUMBER z has been written in the 
polar form z = re t& , examine the REAL and IMAGINARY 
Parts 



u(r,8) = ^Tc n r n cos(n6) 

n=0 

oo 

v(r,9) = ^2c n r n sin(n0). 



(2) 



(3) 



Abel's theorem states that, if u(l,9) and v(l,0) are 
Convergent, then 



u{l,0)+iv{\,9) = lim f(re iB ). 



(4) 



Stated in words, Abel's theorem guarantees that, if a 
Real Power Series Converges for some Positive 
value of the argument, the Domain of Uniform Con- 
vergence extends at least up to and including this 
point. Furthermore, the continuity of the sum function 
extends at least up to and including this point. 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, p. 773, 1985. 

Abel Transform 

The following INTEGRAL Transform relationship, 
known as the Abel transform, exists between two func- 
tions f(x) and g(t) for < a < 1, 



a(t \ = sin(7TQ) d f l f(x) d 

yK > tt dtj {x-ty 



_ sin(7ra) 

7V 



u: 



dx 
dx 



df dx | /(0) 

dx{t-xY~ a t 1 -" 



(1) 
(2) 
(3) 



The Abel transform is used in calculating the radial 
mass distribution of galaxies and inverting planetary ra- 
dio occultation data to obtain atmospheric information. 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 875-876, 1985. 

Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, 
NJ: Princeton University Press, p. 651, 1987. 

Bracewell, R. The Fourier Transform and Its Applications. 
New York: McGraw-Hill, pp. 262-266, 1965. 

Abel's Uniform Convergence Test 

Let {u n (x)} be a Sequence of functions. If 

1. u n (x) can be written u n (x) — a n f n (x) 1 

2. ^a n is Convergent, 

3. fn(x) is a Monotonic Decreasing Sequence 
(i.e., fn+i(x) < f n (x)) for all n, and 

4. f n (x) is Bounded in some region (i.e., < f n (x) < 
M for all x e [a, b]) 

then, for all x e [a, 6], the Series Yl Un ( x ) Converges 
Uniformly. 

see also CONVERGENCE TESTS 

References 

Bromwich, T. J. Pa and MacRobert, T. M. An Introduc- 
tion to the Theory of Infinite Series, 3rd ed. New York: 
Chelsea, p. 59, 1991. 

Whittaker, E. T. and Watson, G. N. A Course in Modern 
Analysis, 4th ed. Cambridge, England: Cambridge Uni- 
versity Press, p. 17, 1990. 

Abelian Variety 

An Abelian variety is an algebraic GROUP which is a 
complete Algebraic Variety. An Abelian variety of 
Dimension 1 is an Elliptic Curve. 

see also Albanese Variety 

References 

Murty, V. K. Introduction to Abelian Varieties. Providence, 
RI: Amer. Math, Soc, 1993. 

Abhyankar's Conjecture 

For a Finite Group G, let p(G) be the Subgroup gen- 
erated by all the Sylow p-SuBGROUPS of G. If X is a 
projective curve in characteristic p > 0, and if xq, ...,xt 
are points of X (for t > 0), then a NECESSARY and SUF- 
FICIENT condition that G occur as the GALOIS GROUP 
of a finite covering Y of X, branched only at the points 
a;o, . .., x ti is that the Quotient GROUP G/p{G) has 
2g + 1 generators. 

Raynaud (1994) solved the Abhyankar problem in the 
crucial case of the affine line (i.e., the projective line 
with a point deleted), and Harbater (1994) proved the 
full Abhyankar conjecture by building upon this special 
solution. 

see also FINITE GROUP, GALOIS GROUP, QUOTIENT 
Group, Sylow p-Subgroup 



10 



Ablowitz-Ramani-Segur Conjecture 



Absolute Square 



References 

Abhyankar, S. "Coverings of Algebraic Curves." Airier. J. 

Math. 79, 825-856, 1957. 
American Mathematical Society. "Notices of the AMS, April 

1995, 1995 Prank Nelson Cole Prize in Algebra." http:// 

www. ams . org/notices/199504/prize-cole .html. 
Harbater, D. "Abhyankar's Conjecture on Galois Groups 

Over Curves." Invent. Math. 117, 1-25, 1994. 
Raynaud, M. "Revetements de la droite affine en car- 

acteristique p > et conjecture d' Abhyankar." Invent. 

Math. 116, 425-462, 1994. 

Ablowitz-Ramani-Segur Conjecture 

The Ablowitz-Ramani-Segur conjecture states that a 
nonlinear Partial Differential Equation is solv- 
able by the Inverse Scattering Method only if ev- 
ery nonlinear Ordinary Differential Equation ob- 
tained by exact reduction has the Painleve Property. 

see also Inverse Scattering Method 

References 

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: 
An Introduction. New York: Wiley, p. 351, 1989. 

Abscissa 

The x- (horizontal) axis of a Graph. 

see also Axis, Ordinate, Real Line, a;- Axis, y-Axis, 
z-Axis 

Absolute Convergence 

A Series J^ n u n is said to Converge absolutely if the 
Series J^ |u n | Converges, where |u n | denotes the 
Absolute Value. If a Series is absolutely convergent, 
then the sum is independent of the order in which terms 
are summed. Furthermore, if the SERIES is multiplied by 
another absolutely convergent series, the product series 
will also converge absolutely. 

see also Conditional Convergence, Convergent 
Series, Riemann Series Theorem 

References 

Bromwich, T. J. Pa and MacRobert, T. M. "Absolute Con- 
vergence." Ch. 4 in An Introduction to the Theory of In- 
finite Series, 3rd ed. New York: Chelsea, pp. 69-77, 1991. 

Absolute Deviation 

Let u denote the Mean of a Set of quantities m, then 
the absolute deviation is denned by 

Aui = \m — u\. 



Absolute Error 

The Difference between the measured or inferred 
value of a quantity xq and its actual value x, given by 

Ax = Xq — x 

(sometimes with the ABSOLUTE VALUE taken) is called 
the absolute error. The absolute error of the Sum or 
Difference of a number of quantities is less than or 
equal to the SUM of their absolute errors. 

see also Error Propagation, Percentage Error, 
Relative Error 

References 

Abramowitz, M. and Stegun, C A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 14, 1972. 

Absolute Geometry 

Geometry which depends only on the first four of Eu- 
clid's Postulates and not on the Parallel Postu- 
late. Euclid himself used only the first four postulates 
for the first 28 propositions of the Elements, but was 
forced to invoke the PARALLEL POSTULATE on the 29th. 

see also Affine Geometry, Elements, Euclid's Pos- 
tulates, Geometry, Ordered Geometry, Paral- 
lel Postulate 

References 

Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden 
Braid. New York: Vintage Books, pp. 90-91, 1989. 

Absolute Pseudoprime 

see Carmichael Number 

Absolute Square 

Also known as the squared NORM. The absolute square 
of a Complex Number z is written \z\ 2 and is defined 



zz , 



(1) 



where z* denotes the COMPLEX CONJUGATE of z. For 
a Real Number, (1) simplifies to 



I i2 2 

\z\ = Z . 



(2) 



If the Complex Number is written z — x + iy, then 
the absolute square can be written 



see also Deviation, Mean Deviation, Signed Devi- 
ation, Standard Deviation 



k + w\ 2 = x +y 2 > 



(3) 



An important identity involving the absolute square is 
given by 



a ± be' ld | 2 = (a ± be' ld ){a ± be ld ) 

- a 2 -h b 2 ± ab(e i5 + e~ i5 ) 

— a + b 2 ± 2ab cos S. 



(4) 



Absolute Value 

If a = 1, then (4) becomes 



Abundance 



11 



|l±&e~ ilS | 2 = l + b 2 ±2bcos8 



= l + & 2 ±26[l-2sin 2 (f£)] 
= l±26 + & 2 =F46sin 2 (^) 
- (l±&) 2 q= 4&sin 2 (^). 



(5) 



If a = 1, and 6=1, then 

|1 - e~ iS \ 2 = (1 - l) 2 + 4 ■ lsin 2 (!<5) = 4sin 2 (±<5). (6) 
Finally, 



u^i+e** 3 ! 2 : 



l) I e -i(<t>2-4>i) 



- 2 + e n < 

=:2 + 2cos(02-<^i) = 2[l + cos(^ 2 -0i)] 



= 4 COS (02 - 0l). 



(7) 



Absolute Value 




The absolute value of a REAL Number x is denoted \x\ 
and given by 

, , f x f -x for x < 

|x|=x 8 gn(*) = | a . forx ^ 0j 

where SGN is the sign function. 

The same notation is used to denote the M ODULUS of 
a Complex Number z — x + iy, \z\ = y/x 2 + t/ 2 , a 
p-ADlC absolute value, or a general Valuation. The 
Norm of a Vector x is also denoted |x|, although ||x|| 
is more commonly used. 

Other Notations similar to the absolute value are the 
Floor Function [zj, Nint function [x], and Ceiling 
Function [af|. 

see also Absolute Square, Ceiling Function, 
Floor Function, Modulus (Complex Number), 
Nint, Sgn, Triangle Function, Valuation 



Absolutely Continuous 

Let // be a Positive Measure on a Sigma Algebra 
M and let A be an arbitrary (real or complex) MEASURE 
on M. Then A is absolutely continuous with respect to 
//, written A < /z, if X(E) = for every E e M for 
which fj.(E) = 0. 
see also Concentrated, Mutually Singular 

References 

Rudin, W. Functional Analysis. New York: McGraw-Hill, 
pp. 121-125, 1991. 

Absorption Law 

The law appearing in the definition of a Boolean Al- 
gebra which states 

a A (a V b) = a V (a A b) = a 

for binary operators V and A (which most commonly are 
logical OR and logical And). 

see also BOOLEAN ALGEBRA, LATTICE 

References 

BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra, 
3rd ed. New York: Macmillian, p. 317, 1965. 

Abstraction Operator 

see Lambda Calculus 

Abundance 

The abundance of a number n is the quantity 

A(n) = o~(n) — 2n, 

where <x(n) is the DIVISOR FUNCTION. Kravitz has con- 
jectured that no numbers exist whose abundance is an 
Odd Square (Guy 1994). 

The following table lists special classifications given to 
a number n based on the value of A(n). 

A(n) Number 



< deficient number 

— 1 almost perfect number 

perfect number 

1 quasiperfect number 
> abundant number 

see also DEFICIENCY 

References 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, pp. 45-46, 1994. 



12 



Abundant Number 



Acceleration 



Abundant Number 

An abundant number is an INTEGER n which is not a 

Perfect Number and for which 



s(n) = <r(n) ~ n > n, 



(1) 



where <r(n) is the DIVISOR FUNCTION. The quantity 
cr(n) — 2n is sometimes called the ABUNDANCE. The 
first few abundant numbers are 12, 18, 20, 24, 30, 36, . . . 
(Sloane's A005101). Abundant numbers are sometimes 
called Excessive Numbers. 

There are only 21 abundant numbers less than 100, and 
they are all Even. The first Odd abundant number is 

945 = 3 3 -7-5. (2) 

That 945 is abundant can be seen by computing 

s(945) = 975 > 945. (3) 

Any multiple of a PERFECT NUMBER or an abundant 
number is also abundant. Every number greater than 
20161 can be expressed as a sum of two abundant num- 
bers. 

Define the density function 

\{n : <x(n) > xn}\ 



A(x) = lim 



(4) 



for a POSITIVE Real Number x, then Davenport (1933) 
proved that A(x) exists and is continuous for all x, 
and Erdos (1934) gave a simplified proof (Finch). Wall 
(1971) and Wall et at. (1977) showed that 



0.2441 < A(2) < 0.2909, 
and Deleglise showed that 

0.2474 < A(2) < 0.2480. 



(5) 



(6) 



A number which is abundant but for which all its 
Proper Divisors are Deficient is called a Primitive 
Abundant Number (Guy 1994, p. 46). 

see also Aliquot Sequence, Deficient Number, 
Highly Abundant Number, Multiamicable Num- 
bers, Perfect Number, Practical Number, Prim- 
itive Abundant Number, Weird Number 

References 

Deleglise, M. "Encadrement de la densite des nombres abon- 

dants." Submitted. 
Dickson, L. E. History of the Theory of Numbers, Vol. 1: 

Divisibility and Primality. New York: Chelsea, pp. 3—33, 

1952. 
Erdos, P. "On the Density of the Abundant Numbers." J. 

London Math. Soc. 9, 278-282, 1934. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/abund/abund* html. 
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer- Verlag, pp. 45-46, 1994. 



Singh, S. FermaVs Enigma: The Epic Quest to Solve 

the World's Greatest Mathematical Problem. New York: 

Walker, pp. 11 and 13, 1997. 
Sloane, N. J. A. Sequence A005101/M4825 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 
Wall, C. R. "Density Bounds for the Sum of Divisors Func- 
tion." In The Theory of Arithmetic Functions (Ed. 

A. A. Gioia and D. L. Goldsmith). New York: Springer- 

Verlag, pp. 283-287, 1971. 
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density 

Bounds for the Sum of Divisors Function." Math. Comput. 

26, 773-777, 1972. 
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density 

Bounds for the Sum of Divisors Function." Math. Comput. 

31, 616, 1977. 

Acceleration 

Let a particle travel a distance s(t) as a function of time 
t (here, s can be thought of as the ARC LENGTH of 
the curve traced out by the particle). The SPEED (the 
Scalar Norm of the Vector Velocity) is then given 

§=V(§r + (§)' + (s' <" 

The acceleration is defined as the time DERIVATIVE of 
the Velocity, so the SCALAR acceleration is given by 



dv 
di 


(2) 


d 2 s 
dt 2 


(3) 


dx d 2 x _i_ dji d 2 y , dz d 2 z 
dt dt 7 " ~*~ dt df2 "T" dt di? 


(4) 


jm 2 +m 2 +m* 


dx d 2 x dy d 2 y dz d 2 z 
ds~dF + dsdi? + ds~d¥ 


(5) 


dr d 2 r 

ds ' dt 2 ' 


(6) 



The Vector acceleration is given by 



dv 
dt 



dfr d's~ fdsY <T 

d* = dt*- T+K {di) N - 



(7) 



where T is the UNIT TANGENT VECTOR, k the CURVA- 
TURE, s the Arc Length, and N the Unit Normal 
Vector. 

Let a particle move along a straight LINE so that the 
positions at times £i, £2, and £3 are si, 52, and S3, re- 
spectively. Then the particle is uniformly accelerated 
with acceleration a Iff 



a = 2 



($2 - S 3 )h + (33 - Si)t2 + (Si - 52)^3 
(tl - t 2 )(t2 ~ t 3 )(t 3 - ti) 



(8) 



is a constant (Klamkin 1995, 1996). 



Accidental Cancellation 



Ackermann Function 



13 



Consider the measurement of acceleration in a rotating 
reference frame. Apply the ROTATION OPERATOR 



- f d \ 



■■-( 



\. Ctt / body 



+ u;x 



(9) 



twice to the RADIUS VECTOR r and suppress the body 
notation, 



R 2 r 



(^ +WX )(S +WXr ) 



d 2 r d ( . dr 

d 2 r dr du: dv 

— — + u> x — +r x — - +u> x — 

dt 2 dt dt dt 



+ u?x (u; x r). 



(10) 



Grouping terms and using the definitions of the VELOC- 
ITY v = dr/dt and Angular Velocity a = du/dt 
give the expression 



9-space 



dt 2 



+ 2u? x v + u; x (u> x r) 4- r x ex. (11) 



Now, we can identify the expression as consisting of 
three terms 



= d*r 
a b ody - df2 , 

aCoriolis = 2u? X V, 
a ce ntrifugal = <*> X (u> X I*) , 



(12) 
(13) 
(14) 



a "body" acceleration, centrifugal acceleration, and 
Coriolis acceleration. Using these definitions finally 
gives 

&space = <*body "r ^Coriolis ~~r ^centrifugal + T X Of, (15) 

where the fourth term will vanish in a uniformly ro- 
tating frame of reference (i.e., ex = 0). The centrifugal 
acceleration is familiar to riders of merry ^j-rounds, and 
the Coriolis acceleration is responsible for the motions 
of hurricanes on Earth and necessitates large trajectory 
corrections for intercontinfv: L al ballistic missiles. 

see also Angular Acceleration, Arc Length, 
Jerk, Velocity 

References 

Klamkin, M. S. "Problem 1481." Math. Mag. 68, 307, 1995. 
Klamkin, M. S. "A Characteristic of Constant Acceleration." 
Solution to Problem 1481. Math. Mag. 69, 308, 1996. 

Accidental Cancellation 

see Anomalous Cancellation 



Accumulation Point 

An accumulation point is a Point which is the limit 
of a Sequence, also called a Limit Point. For some 
Maps, periodic orbits give way to Chaotic ones beyond 
a point known as the accumulation point. 

see also Chaos, Logistic Map, Mode Locking, Pe- 
riod Doubling 

Achilles and the Tortoise Paradox 

see Zeno's Paradoxes 

Ackermann Function 

The Ackermann function is the simplest example of a 
well-defined TOTAL FUNCTION which is COMPUTABLE 
but not Primitive Recursive, providing a counterex- 
ample to the belief in the early 1900s that every COM- 
PUTABLE Function was also Primitive Recursive 
(Dotzel 1991). It grows faster than an exponential func- 
tion, or even a multiple exponential function. The Ack- 
ermann function A(x } y) is defined by 



(y+l if x = 

A(x,y)= I A(x-l,l) if 2/ — 

[ A{x — 1, A(x, y — 1)) otherwise. 

Special values for Integer x include 



(i) 



A(0,y) = y + 1 


(2) 


A(l,y) = y + 2 


(3) 


A(2,y) = 2y + 3 


(4) 


A(3,y) = 2"+ 3 - 3 


(5) 


.4(4,2/) = 2^-3. 


(6) 


V+3 





Expressions of the latter form are sometimes called 
Power Towers. A(0,y) follows trivially from the def- 
inition. A(l,y) can be derived as follows, 

A(l,y) = A(0,A(l,y- 1)) = A(l,y- 1) + 1 

= A(0,A(l,y- 2)) + 1 = A(l,y- 2) + 2 
= . . . = .4(1, 0) + y = A(0, l) + y = y + 2. 



(7) 



A(2,y) has a similar derivation, 



A(2,y) = A(l,A(2,y-l)) = A(2,y-.l) + 2 

= i4(l ) A(2,y-2))+2 = i4(2 I y-2) + 4 = ... 

= A(2, 0) + 2y = A(l, 1) + 2y = 2y + 3. (8) 

Buck (1963) defines a related function using the same 
fundamental Recurrence Relation (with arguments 
flipped from Buck's convention) 



F(x,y) = F(x-l,F(x t y-l)), 



(9) 



14 



Ackermann Number 



Acute Triangle 



but with the slightly different boundary values 

^(0, y) = V + 1 
**(1,0) = 2 
F(2,0) = 
F(x,0) = 1 for x 



= 3,4, 



Buck's recurrence gives 



F(l,») = 2 + i/ 
F(2,y) = 2y 
f(3,y) = 2» 

.2 

F(4,j,) = 2 2 . 



(10) 

(11) 
(12) 

(13) 



(14) 
(15) 
(16) 

(17) 



Taking F(4,n) gives the sequence 1, 2, 4, 16, 65536, 
2 65536 , .... Defining ip(x) = F(x, x) for x = 0, 1, ... 

.2 

then gives 1, 3, 4, 8, 65536, 2 2 ' , . . . (Sloane's A001695), 



where m = 2 2 , a truly huge number! 

65536 

see a/50 Ackermann Number, Computable Func- 
tion, Goodstein Sequence, Power Tower, Primi- 
tive Recursive Function, TAK Function, Total 
Function 

References 

Buck, R. C. "Mathematical Induction and Recursive Defini- 
tions." Amer. Math. Monthly 70, 128-135, 1963. 

Dotzel, G. "A Function to End All Functions." Algorithm: 
Recreational Programming 2.4, 16-17, 1991. 

Kleene, S. C. Introduction to Metamathematics. New York: 
Elsevier, 1971. 

Peter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 
1951. 

Reingold, E. H. and Shen, X. "More Nearly Optimal Algo- 
rithms for Unbounded Searching, Part I: The Finite Case." 
SIAM J. Corn-put. 20, 156-183, 1991. 

Rose, H. E. Subrecursion, Functions, and Hierarchies. New 
York: Clarendon Press, 1988. 

Sloane, N. J. A. Sequence A001695/M2352 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Smith, H. J. "Ackermann's Function." http://www.netcom. 
com/-hj smith/Ackerman . html. 

Spencer, J. "Large Numbers and Unprovable Theorems." 
Amer. Math. Monthly 90, 669-675, 1983. 

Tarjan, R. E. Data Structures and Network Algorithms. 
Philadelphia PA: SIAM, 1983. 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addison- Wesley, pp. 11, 227, and 232, 
1991. 

Ackermann Number 

A number of the form n t • • • T™> where Arrow Nota- 

n 

TION has been used. The first few Ackermann numbers 

.3 

are 1 t 1 = 1, 2 tt 2 = 4, and 3 ttt 3 = 3 3 

7,625,597,484,987 



see also Ackermann Function, Arrow Notation, 
Power Tower 

References 

Ackermann, W. "Zum hilbertschen Aufbau der reellen 
Zahlen." Math. Ann. 99, 118-133, 1928. 

Conway, J. H. and Guy, R. K, The Book of Numbers. New 
York: Springer-Verlag, pp. 60-61, 1996. 

Crandall, R. E. "The Challenge of Large Numbers." Sci. 
Amer. 276, 74-79, Feb. 1997. 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addison- Wesley, pp. 11, 227, and 232, 
1991. 

Acnode 

Another name for an ISOLATED POINT. 

see also Crunode, Spinode, Tacnode 

Acoptic Polyhedron 

A term invented by B. Griinbaum in an attempt to pro- 
mote concrete and precise POLYHEDRON terminology. 
The word "coptic" derives from the Greek for "to cut," 
and acoptic polyhedra are defined as POLYHEDRA for 
which the FACES do not intersect (cut) themselves, mak- 
ing them 2-Manifolds. 
see also Honeycomb, Nolid, Polyhedron, Sponge 

Action 

Let M(X) denote the GROUP of all invertible MAPS 
X -> X and let G be any GROUP. A HOMOMORPHISM 
6 :G -> M(X) is called an action of G on X. Therefore, 
6 satisfies 

1. For each g € G, 6(g) is a Map X -> X : x \-> 0(g)x, 

2. 0(gh)x = 6{g)(O(h)x), 

3. 0(e) a; = x, where e is the group identity in G, 

4. 0(g- 1 )x = 6(g)- 1 x. 

see also CASCADE, FLOW, SEMIFLOW 

Acute Angle 

An Angle of less than 7r/2 Radians (90°) is called an 
acute angle. 

see also ANGLE, OBTUSE ANGLE, RIGHT ANGLE, 

Straight Angle 
Acute Triangle 




A Triangle in which all three Angles are Acute An- 
gles. A Triangle which is neither acute nor a RIGHT 
Triangle (i.e., it has an Obtuse Angle) is called an 
Obtuse Triangle. A Square can be dissected into as 
few as 8 acute triangles. 

see also Obtuse Triangle, Right Triangle 



Adams-Bashforth-Moulton Method 



Addition Chain 



15 



Adams-Bashforth-Moulton Method 

see Adams' Method 

Adams' Method 

Adams' method is a numerical METHOD for solving 
linear First-Order Ordinary Differential Equa- 
tions of the form 



dy 
dx 



f{x>y)- 



Let 



: 3?n + l X n 



(i) 



(2) 



be the step interval, and consider the Maclaurin Se- 
ries of y about x n , 



y n +i = y n + ( -T-) ( x ~ x n) 



(x - x n ) 2 + . 



V dx J n + 1 \dxj n \ dx 2 J 



(3) 



(4) 



Here, the Derivatives of y are given by the Backward 
Differences 



\dx/ n Xn+i ~ X 



3/n+i - y n 



h 



(5) 
(6) 

(7) 



etc. Note that by (1), q n is just the value of f{x ni y n ). 

For first-order interpolation, the method proceeds by 
iterating the expression 



2/n+i = yn + q n h 



(8) 



where q n = /(x n ,2/n). The method can then be ex- 
tended to arbitrary order using the finite difference in- 
tegration formula from Beyer (1987) 



/* 

Jo 



/ p ^=(l+IV+£,V 2 + fV 3 



,251 V 4 + J95_V 5 
~720 v ~ 288 v 



19087 V7 6 



V° + ...)/p (9) 



to obtain 



2/n+i -y n = h(q n + \ Vq n -i + ^ V 2 q n -2 + f V 3 g n - 



12 
95 
288 



+ ffivV-4 + ^V 5 g n _5 + ...)■ (10) 



Note that von Karman and Biot (1940) confusingly use 
the symbol normally used for FORWARD DIFFERENCES 
A to denote BACKWARD DIFFERENCES V. 



see also Gill's Method, Milne's Method, Predic- 

TOR-CORRECTOR METHODS, RUNGE-KUTTA METHOD 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 896, 1972. 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, p. 455, 1987. 

Karman, T. von and Biot, M. A. Mathematical Methods in 
Engineering: An Introduction to the Mathematical Treat- 
ment of Engineering Problems. New York: McGraw-Hill, 
pp. 14-20, 1940. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 741, 1992. 

Addend 

A quantity to be Added to another, also called a Sum- 
MAND. For example, in the expression a + 6 + c, a, 6, and 
c are all addends. The first of several addends, or "the 
one to which the others are added" (a in the previous 
example), is sometimes called the AUGEND. 

see also Addition, Augend, Plus, Radicand 



Addition 



i i - 
15 8- 

J- 249 * 
407- 



Y carries 
(-addend 1 
Y addend 2 
hsum 



The combining of two or more quantities using the PLUS 
operator. The individual numbers being combined are 
called ADDENDS, and the total is called the Sum. The 
first of several ADDENDS, or "the one to which the oth- 
ers are added," is sometimes called the AUGEND. The 
opposite of addition is SUBTRACTION. 

While the usual form of adding two n-digit INTEGERS 
(which consists of summing over the columns right to 
left and "Carrying" a 1 to the next column if the sum 
exceeds 9) requires n operations (plus carries), two n- 
digit INTEGERS can be added in about 21gn steps by 
n processors using carry-lookahead addition (McGeoch 
1993). Here, lgx is the Lg function, the LOGARITHM to 
the base 2. 

see also Addend, Amenable Number, Augend, 
Carry, Difference, Division, Multiplication, 
Plus, Subtraction, Sum 

References 

McGeoch, C. C. "Parallel Addition." Amer. Math. Monthly 
100, 867-871, 1993. 

Addition Chain 

An addition chain for a number n is a SEQUENCE 1 = 
ao < ai < . . . < a T = n, such that each member after ao 
is the SUM of the two earlier (not necessarily distinct) 
ones. The number r is called the length of the addition 
chain. For example, 

1,1 + 1 = 2,2 + 2 = 4,4 + 2 = 6,6 + 2 = 8,8 + 6 = 14 



16 Addition-Multiplication Magic Square 



Adele Group 



is an addition chain for 14 of length r = 5 (Guy 1994). 

see also BRAUER CHAIN, HANSEN CHAIN, SCHOLZ CON- 
JECTURE 

References 

Guy, R. K. "Addition Chains. Brauer Chains. Hansen 
Chains." §C6 in Unsolved Problems in Number Theory, 
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. 

Addition-Multiplication Magic Square 



46 


81 


117 


102 


15 


76 


200 


203 


19 


60 


232 


175 


54 


69 


153 


78 


216 


161 


17 


52 


171 


90 


58 


75 


135 


114 


50 


87 


184 


189 


13 


68 


150 


261 


45 


38 


91 


136 


92 


27 


119 


104 


108 


23 


174 


225 


57 


30 


116 


25 


133 


120 


51 


26 


162 


207 


39 


34 


138 


243 


100 


29 


105 


152 



102207290 38 



115216171 



102207290 3 



115216171 



A square which is simultaneously a MAGIC SQUARE and 
Multiplication Magic Square. The three squares 
shown above (the top square has order eight and the 
bottom two have order nine) have addition MAGIC CON- 
STANTS (840, 848, 1200) and multiplicative magic con- 
stants (2,058,068,231,856,000; 5,804,807,833,440,000; 
1,619,541,385,529,760,000), respectively (Hunter and 
Madachy 1975, Madachy 1979). 

see also MAGIC SQUARE 

References 

Hunter, J, A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 

in Mathematical Diversions. New York: Dover, pp, 30-31, 

1975. 
Madachy, J. S. Madachy 's Mathematical Recreations. New 

York: Dover, pp. 89-91, 1979. 

Additive Persistence 

Consider the process of taking a number, adding its DIG- 
ITS, then adding the DIGITS of number derived from it, 
etc., until the remaining number has only one DIGIT. 
The number of additions required to obtain a single 
DIGIT from a number n is called the additive persis- 
tence of n, and the DIGIT obtained is called the DIGITAL 
Root of n. 

For example, the sequence obtained from the starting 
number 9876 is (9876, 30, 3), so 9876 has an additive 
persistence of 2 and a DIGITAL ROOT of 3. The ad- 
ditive persistences of the first few positive integers are 
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 



. . . (Sloane's A031286). The smallest numbers of ad- 
ditive persistence n for n = 0, 1, . . . are 0, 10, 19, 
199, 19999999999999999999999, . . . (Sloane's A006050). 
There is no number < 10 5 ° with additive persistence 
greater than 11. 

It is conjectured that the maximum number lacking the 
DIGIT 1 with persistence 11 is 

77777733332222222222222222222 

There is a stronger conjecture that there is a maximum 
number lacking the DIGIT 1 for each persistence > 2. 

The maximum additive persistence in base 2 is 1. It is 
conjectured that all powers of 2 > 2 15 contain a in base 
3, which would imply that the maximum persistence in 
base 3 is 3 (Guy, 1994). 

see also Digitadition, Digital Root, Multiplica- 
tive Persistence, Narcissistic Number, Recur- 
ring Digital Invariant 

References 

Guy, R. K. "The Persistence of a Number." §F25 in Unsolved 
Problems in Number Theory, 2nd ed. New York: Springer- 
Verlag, pp. 262-263, 1994. 

Hinden, H. J. "The Additive Persistence of a Number." J. 
Recr. Math. 7, 134-135, 1974. 

Sloane, N. J. A. Sequences A031286 and A006050/M4683 in 
"An On-Line Version of the Encyclopedia of Integer Se- 
quences." 

Sloane, N. J. A. "The Persistence of a Number." J. Recr. 
Math. 6, 97-98, 1973. 

Adele 

An element of an Adele GROUP, sometimes called a 
Repartition in older literature. Adeles arise in both 
Number Fields and Function Fields. The adeles of 
a Number Field are the additive Subgroups of all ele- 
ments in Yl kvi where v is the PLACE, whose ABSOLUTE 
Value is < 1 at all but finitely many i/s. 

Let F be a Function Field of algebraic functions of 

one variable. Then a MAP r which assigns to every 

PLACE P of F an element r(P) of F such that there are 

only a finite number of PLACES P for which v P (r(P)) < 

0. 

see also Idele 

References 

Chevalley, C. C. Introduction to the Theory of Algebraic 
Functions of One Variable. Providence, RI: Amer. Math. 
Soc, p. 25, 1951. 

Knapp, A. W. "Group Representations and Harmonic Anal- 
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. 

Adele Group 

The restricted topological Direct Product of the 
GROUP Gk v with distinct invariant open subgroups Go v , 



References 

Weil, A. Adeles and Algebraic Groups. 
Princeton University Press, 1961. 



Princeton, NJ: 



Adem Relations 



Adjoint Operator 17 



Adem Relations 

Relations in the definition of a Steenrod Algebra 
which state that, for i < 2j, 



L*J 



j - k - l\ i+j-k 



Sq* o Sq*(x) = Y.[ 3 i- 2k ' W +J ~" ° S <^' 

where fog denotes function COMPOSITION and |_*J is 
the Floor Function. 

see also STEENROD ALGEBRA 

Adequate Knot 

A class of Knots containing the class of Alternating 
Knots. Let c(K) be the CROSSING Number. Then for 
KNOT Sum Ki#K 2 which is an adequate knot, 

c(K 1 #K 2 )^c(Ki) + c(K2). 

This relationship is postulated to hold true for all 

Knots. 

see also Alternating Knot, Crossing Number 

(Link) 

Adiabatic Invariant 

A property of motion which is conserved to exponential 
accuracy in the small parameter representing the typical 
rate of change of the gross properties of the body. 
see also ALGEBRAIC INVARIANT, LYAPUNOV CHARAC- 
TERISTIC Number 

Adjacency Matrix 

The adjacency matrix of a simple Graph is a Matrix 
with rows and columns labelled by VERTICES, with a 1 
or in position (vi,Vj) according to whether Vi and Vj 
are ADJACENT or not. 
see also INCIDENCE MATRIX 



References 

Chartrand, G. Introductory Graph Theory. 
Dover, p. 218, 1985. 



New York: 



Adjacency Relation 

The Set E of Edges of a Graph (V,E), being a set 
of unordered pairs of elements of V, constitutes a RE- 
LATION on V. Formally, an adjacency relation is any 
Relation which is Irreflexive and Symmetric. 
see also Irreflexive, Relation, Symmetric 

Adjacent Fraction 

Two FRACTIONS are said to be adjacent if their differ- 
ence has a unit NUMERATOR. For example, 1/3 and 1/4 
are adjacent since 1/3 - 1/4 = 1/12, but 1/2 and 1/5 
are not since 1/2 — 1/5 = 3/10. Adjacent fractions can 
be adjacent in a Farey SEQUENCE. 
see also FAREY SEQUENCE, FORD CIRCLE, FRACTION, 

Numerator 

References 

Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, 
p. 119, 1995. 



Adjacent Value 

The value nearest to but still inside an inner FENCE. 

References 

Tukey, J. W. Explanatory Data Analysis. Reading, MA: 
Addison- Wesley, p. 667, 1977. 

Adjacent Vertices 

In a GRAPH G, two VERTICES are adjacent if they are 
joined by an EDGE. 

Adjoint Curve 

A curve which has at least multiplicity Vi — 1 at each 
point where a given curve (having only ordinary singu- 
lar points and cusps) has a multiplicity vi is called the 
adjoint to the given curve. When the adjoint curve is of 
order n — 3, it is called a special adjoint curve. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 30, 1959. 

Adjoint Matrix 

The adjoint matrix, sometimes also called the Ad JU- 
GATE Matrix, is defined by 



a* = (A T r, 



(i) 



where the ADJOINT OPERATOR is denoted * and T de- 
notes the Transpose. If a Matrix is Self- Adjoint, 
it is said to be HERMITIAN. The adjoint matrix of a 
Matrix product is given by 

(oft)^. = [(a6) T ]*, . (2) 

Using the property of transpose products that 

[(a&) T ];, = (6 T a-% = (&<&■)• = (b T ): k (a T y kj 

= b lAj = ( fot «% > ( 3 ) 



it follows that 



(AB) f = BW. 



(4) 



Adjoint Operator 

Given a Second-Order Ordinary Differential 
Equation 

- , . du du t v 

Cu(x) - p — + Pl — + P2 u, (1) 

where pi = Pi(x) and u = u(x), the adjoint operator & 
is defined by 



d 



" ^ (PoU) " di^ PlU) +PaU 

d 2 u f t ,du ( „ , , 
-P°ZT^ + ( 2 Po -pi)^~ + (po -pi +P2)U. 



'dx 2 



dx 



(2) 



18 Adjugate Matrix 



Affine Hull 



Write the two Linearly Independent solutions as 
t/i (x) and 2/2 (#)■ Then the adjoint operator can also 
be written 



?../ 



(y 2 Cyi ~yi£y 2 )dx = 



— {yi 2/2 - 2/13/2 ) 

Po 



(3) 

see a/50 Self-Adjoint Operator, Sturm-Liouville 
Theory 

Adjugate Matrix 

see Adjoint Matrix 

Adjunction 

If a is an element of a Field F over the PRIME Field 
P, then the set of all RATIONAL FUNCTIONS of a with 
Coefficients in P is a Field derived from P by ad- 
junction of a. 

Adleman-Pomerance-Rumely Primality Test 

A modified Miller's Primality Test which gives a 
guarantee of Primality or COMPOSITENESS. The Al- 
gorithm's running time for a number N has been 
provedtobeasO((lniV) clnlnlnJV ) for some c> 0. It was 
simplified by Cohen and Lenstra (1984), implemented by 
Cohen and Lenstra (1987), and subsequently optimized 
by Bosma and van der Hulst (1990). 

References 

Adleman, L. M.; Pomerance, C; and Rumely, R. S. "On 
Distinguishing Prime Numbers from Composite Number." 
Ann. Math. 117, 173-206, 1983. 

Bosma, W. and van der Hulst, M.-P. "Faster Primality Test- 
ing." In Advances in Cryptology, Proc. Eurocrypt '89, 
Houthalen, April 10-13, 1989 (Ed. J.-J. Quisquater). New- 
York: Springer- Verlag, 652-656, 1990. 

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; 
and Tuckerman, B. Factorizations of b n ± 1, b — 2, 
3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, 
RI: Amer. Math. Soc, pp. lxxxiv-lxxxv, 1988. 

Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi 
Sums." Math. Comput. 42, 297-330, 1984. 

Cohen, H. and Lenstra, A. K. "Implementation of a New 
Primality Test." Math. Comput 48, 103-121, 1987. 

Mihailescu, P. "A Primality Test Using Cyclotomic Exten- 
sions." In Applied Algebra, Algebraic Algorithms and 
Error- Correcting Codes (Proc. AAECC-6, Rome, July 
1988). New York: Springer- Verlag, pp. 310-323, 1989. 

Adleman- Rumely Primality Test 

see Adleman-Pomerance-Rumely Primality Test 

Admissible 

A string or word is said to be admissible if that word 
appears in a given SEQUENCE. For example, in the SE- 
QUENCE aabaabaabaabaab . . ., a, aa, baab are all admis- 
sible, but bb is inadmissible. 

see also BLOCK GROWTH 



Affine Complex Plane 

The set A 2 of all ordered pairs of COMPLEX NUMBERS. 
see also Affine Connection, Affine Equation, 
Affine Geometry, Affine Group, Affine Hull, 
Affine Plane, Affine Space, Affine Transforma- 
tion, Affinity, Complex Plane, Complex Projec- 
tive Plane 

Affine Connection 

see Connection Coefficient 

Affine Equation 

A nonhomogeneous Linear Equation or system of 
nonhomogeneous LINEAR EQUATIONS is said to be 
affine. 

see also AFFINE COMPLEX PLANE, AFFINE CONNEC- 
TION, Affine Geometry, Affine Group, Affine 
Hull, Affine Plane, Affine Space, Affine Trans- 
formation, Affinity 

Affine Geometry 

A GEOMETRY in which properties are preserved by PAR- 
ALLEL Projection from one Plane to another. In an 
affine geometry, the third and fourth of Euclid's Pos- 
tulates become meaningless. This type of GEOMETRY 
was first studied by Euler. 

see also ABSOLUTE GEOMETRY, AFFINE COMPLEX 

Plane, Affine Connection, Affine Equation, 
Affine Group, Affine Hull, Affine Plane, Affine 
Space, Affine Transformation, Affinity, Or- 
dered Geometry 

References 

Birkhoff, G. and Mac Lane, S. "Affine Geometry." §9.13 in A 

Survey of Modern Algebra, 3rd ed. New York: Macmillan, 

pp. 268-275, 1965. 

Affine Group 

The set of all nonsingular Affine TRANSFORMATIONS 
of a Translation in Space constitutes a Group known 
as the affine group. The affine group contains the full 
linear group and the group of TRANSLATIONS as SUB- 
GROUPS. 

see also AFFINE COMPLEX PLANE, AFFINE CONNEC- 
TION, Affine Equation, Affine Geometry, Affine 
Hull, Affine Plane, Affine Space, Affine Trans- 
formation, Affinity 

References 

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 
3rd ed. New York: Macmillan, p. 237, 1965. 

Affine Hull 

The IDEAL generated by a SET in a VECTOR SPACE. 

see also Affine Complex Plane, Affine Connec- 
tion, Affine Equation, Affine Geometry, Affine 
Group, Affine Plane, Affine Space, Affine 
Transformation, Affinity, Convex Hull, Hull 



AfRne Plane 



Affine Transformation 



19 



Affine Plane 

A 2-D Affine Geometry constructed over a Finite 
Field. For a Field F of size n, the affine plane consists 
of the set of points which are ordered pairs of elements in 
F and a set of lines which are themselves a set of points. 
Adding a Point at Infinity and Line at Infinity 
allows a Projective Plane to be constructed from an 
affine plane. An affine plane of order n is a BLOCK 
DESIGN of the form (n 2 , n, 1). An affine plane of order 
n exists Iff a PROJECTIVE PLANE of order n exists. 

see also Affine Complex Plane, Affine Connec- 
tion, Affine Equation, Affine Geometry, Affine 
Group, Affine Hull, Affine Space, Affine Trans- 
formation, Affinity, Projective Plane 



References 

Lindner, C. C. and Rodger, C. A. Design Theory. 
Raton, FL: CRC Press, 1997. 



Boca 



Affine Scheme 

A technical mathematical object defined as the SPEC- 
TRUM ct(A) of a set of Prime Ideals of a commutative 
RING A regarded as a local ringed space with a structure 

sheaf. 

see also SCHEME 

References 

Iyanaga, S. and Kawada, Y. (Eds.). "Schemes." §18E in En- 
cyclopedic Dictionary of Mathematics. Cambridge, MA: 
MIT Press, p. 69, 1980. 

Affine Space 

Let V be a VECTOR Space over a FIELD K, and let A 
be a nonempty SET. Now define addition p -f a € A for 
any VECTOR a E V and element p e A subject to the 
conditions 

1. p + 0=p, 

2. (p + a)+b = p+(a + b), 

3. For any q G A, there EXISTS a unique VECTOR a 6 V 
such that q = p + a. 

Here, a, b £ V. Note that (1) is implied by (2) and (3). 
Then A is an affine space and K is called the COEFFI- 
CIENT Field. 

In an affine space, it is possible to fix a point and co- 
ordinate axis such that every point in the SPACE can 
be represented as an n-tuple of its coordinates. Every 
ordered pair of points A and B in an affine space is then 
associated with a VECTOR AB. 

see also Affine Complex Plane, Affine Connec- 
tion, Affine Equation, Affine Geometry, Affine 
Group, Affine Hull, Affine Plane, Affine Space, 
Affine Transformation, Affinity 



Affine Transformation 

Any Transformation preserving Collinearity (i.e., 
all points lying on a Line initially still lie on a Line 
after TRANSFORMATION). An affine transformation is 
also called an AFFINITY. An affine transformation of 

R n is a Map F : R n -> W 1 of the form 



F(p) = Ap + q 



(1) 



for all p € M n , where A is a linear transformation of 
W 1 . If det(A) = 1, the transformation is Orientation- 
Preserving; if det(A) = -1, it is Orientation- 
Reversing. 

Dilation (Contraction, Homothecy), Expansion, 
Reflection, Rotation, and Translation are all 
affine transformations, as are their combinations. A par- 
ticular example combining ROTATION and EXPANSION is 
the rotation-enlargement transformation 



V 


= s 




= s 



cos a 
— sin a 



sin a 
cos a 



x — Xo 

y-yo 



cos a(x — Xo) + sin a(y — yo) 
— sina(x — Xo) + cos a(y — yo) 



(2) 



Separating the equations, 



x — (s cos a)x + (s sin a)y — s(xo cos a + yo sin a) (3) 
y = (— s sin a)x + (5 cos a)y + s(xq sin a — yo cos a). 

(4) 



This can be also written as 



where 



x = ax + by + c 
y = bx + ay + d, 



a = s cos a 
b = —3 sin a. 



The scale factor 5 is then defined by 



8= \/a 2 +6 2 , 
and the rotation Angle by 

■'(-!)■ 



a = tan 



(5) 
(6) 



(7) 
(8) 



(9) 



(10) 



see also Affine Complex Plane, Affine Connec- 
tion, Affine Equation, Affine Geometry, Affine 
Group, Affine Hull, Affine Plane, Affine Space, 
Affine Transformation, Affinity, Equiaffinity, 
Euclidean Motion 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, p. 105, 1993. 



20 Affinity 

Affinity 

see AFFINE TRANSFORMATION 

Affix 

In the archaic terminology of Whittaker and Watson 
(1990), the Complex Number z representing x + iy. 

References 

Whittaker, E. T. and Watson, G. N. A Course in Modem 
Analysis, ^th ed. Cambridge, England: Cambridge Uni- 
versity Press, 1990. 

Aggregate 

An archaic word for infinite SETS such as those consid- 
ered by Georg Cantor. 

see also Class (Set), Set 

AGM 

see Arithmetic-Geometric Mean 

Agnesi's Witch 

see Witch of Agnesi 

Agnesienne 

see Witch of Agnesi 

Agonic Lines 

see Skew Lines 

Ahlfors-Bers Theorem 

The Riemann's Moduli Space gives the solution to 
Riemann's Moduli Problem, which requires an An- 
alytic parameterization of the compact RlEMANN SUR- 
FACES in a fixed HOMEOMORPHISM. 

Airy Differential Equation 

Some authors define a general Airy differential equation 
as 

y" ± k xy — 0. (1) 

This equation can be solved by series solution using the 
expansions 



y = ^a n z n (2) 

71 = 

OO CO 

/ V^ n-1 V^ "-1 

y = > na n x = y ^na n x 

n=0 n=l 

OO 

= ^^(n + l)a n +ix n (3) 

TX-0 

OO OO 

y" — /.( n + l)na n +ix n ~~ = 2_^^ n ~*~ l) na n+i# n ~ 

n=0 n=l 

oo 

= J^(n + 2)(n + l)a n+2 x n . (4) 



Airy Differential Equation 

Specializing to the "conventional" Airy differential equa- 
tion occurs by taking the Minus Sign and setting 
k 2 = 1. Then plug (4) into 

y" -xy = (5) 

to obtain 

OO oo 

^(n + 2)(n + l)a n+2 x n - x ^ a ^ = ° ( 6 ) 

n=0 n—0 

OO oo 

^(n + 2)(n + l)a n+2 z n -^a n :r n+1 =0 (7) 

Tl = Tl = 

OO oo 

2a 2 + ^(n + 2)(n + l)a n+2 z n - ^T ^-ix n = (8) 

n=l n— 1 

OO 

2a 2 + J^[(n + 2)(n + l)a n+2 - a n _i]a; n = 0. (9) 

n = l 

In order for this equality to hold for all #, each term 
must separately be 0. Therefore, 

a 2 = (10) 

(n + 2)(n + l)a n+2 = a n _i. (11) 

Starting with the n = 3 term and using the above RE- 
CURRENCE Relation, we obtain 

5-4a 5 = 20a 5 = a 2 = 0. (12) 

Continuing, it follows by INDUCTION that 

a 2 = a$ = ag = an = . . . a3n-i = (13) 

for n = 1, 2, Now examine terms of the form £3^. 

(14) 



a 3 = 

ae = 



ao 
3^2 

^3 = 

6-5 ~ (6-5)(3-2) 

a& ao 



ao 



(15) 
(16) 



9-8 (9-8)(6-5)(3-2)' 
Again by INDUCTION, 

_ _ao 

0,371 " f(3n)(3n - l)][(3n - 3)(3n - 4)] • ■ • [6 * 5] [3 ■ 2] 

(17) 
for n = 1, 2, Finally, look at terms of the form 

a3n+l, 



a^ 
a 7 



ai 

4-3 

a4 

7^6 = (7-6)(4-3) 
ar 01 



ai 



10-9 (10-9)(7-6)(4-3)" 



(18) 
(19) 

(20) 



Airy-Fock Functions 
By Induction, 

d3n+l = 



0,1 



[(3n + l)(3n)][(3n - 2)(3n - 3)] • * - [7 ■ 6] [4 ■ 3] 

(21) 
for n = 1, 2, The general solution is therefore 



y = a>o 



+ ai 



n=l 

oo 



(3n)(3n - l)(3n - 3)(3n - 4) • • • 3 ■ 2 



(3n + l)(3n)(3n - 2)(3n - 3) ■ ■ ■ 4 • 3 



(22) 



For a general k 2 with a MINUS SIGN, equation (1) is 

y" - k 2 xy = 0, (23) 

and the solution is 

y(x) = fvS [A/_ 1/3 (§W /2 ) - S/ 1/3 (f fcx 3 / 2 )] , 

(24) 
where I is a Modified Bessel Function of the 
First Kind. This is usually expressed in terms of the 
Airy Functions Ai(#) and Bi(#) 

y(x) = A' Ai{k 2/3 x) + B' Bi(fc 2/3 x). (25) 

If the Plus Sign is present instead, then 



y +k xy = 



(26) 



and the solutions are 



y(x) = \& [AJ. 1/3 (\kx z ? 2 ) + BJ 1/Z (f kx^ 2 )] , 

(27) 
where J(z) is a Bessel Function of the First Kind. 

see also Airy-Fock Functions, Airy Functions, 
Bessel Function of the First Kind, Modified 
Bessel Function of the First Kind 

Airy-Fock Functions 

The three Airy-Fock functions are 



v{z) = ~y/irAi(z) 
wi(z) = 2e l7T/6 u(ujz) 
W2(z) = 2e~ t7r/ v(uj~ z) 



(i) 

(2) 
(3) 



where Ai(z) is an Airy Function. These functions 
satisfy 

v{z) = ^W-^W ( 4) 

[w 1 {z)]*=w 2 {z*), (5) 

where z* is the COMPLEX CONJUGATE of z. 
see also AlRY FUNCTIONS 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, p. 65, 1988. 



Airy Functions 21 

Airy Functions 

Watson's (1966, pp. 188-190) definition of an Airy func- 
tion is the solution to the Airy Differential EQUA- 
TION 

$" ±k 2 $x = (1) 

which is Finite at the Origin, where <£' denotes the 
Derivative d$/dx, k 2 — 1/3, and either Sign is per- 
mitted. Call these solutions (l/7r)$(±fc 2 ,:c), then 



£•<*»'*> =jf 



cos(£ 3 ± xi) dt 



(2) 



*(§;*) = Wf 
*(-§;*) = Wf 



(2x 3/2 \ (2x 3/2 \ 



(3) 



r . 2x 3 »\ T fix*'* 



3 3 / 2 



3 3 / 2 



(4) 



where J(z) is a Bessel Function of the First Kind 

and I(z) is a MODIFIED BESSEL FUNCTION OF THE 
First Kind. Using the identity 



K n (x) 



TV I-n(x) - I n (x) 

2 sin(n7r) 



(5) 



where K{z) is a MODIFIED BESSEL FUNCTION OF THE 
Second Kind, the second case can be re-expressed 

(8) 



1 /Fir f 2 * 3/2 ^ 




A more commonly used definition of Airy functions is 
given by Abramowitz and Stegun (1972, pp. 446-447) 
and illustrated above. This definition identifies the 
Ai(x) and Bi(a?) functions as the two LINEARLY INDE- 
PENDENT solutions to (1) with k 2 = 1 and a MINUS 
Sign, 

y -yz^o. (9) 



22 Airy Functions 

The solutions are then written 

y(z) = AAi(z) + BBi(z) 7 
where 



(10) 



Ai(*) = -*(-l,z) 

= |^[/_ 1/3 (I^ /2 )-/ 1/ 3(Iz 3/2 )] 

= ^^/3(I^ /2 ) ("J 

Bi(z) = y|[7_ a/ 3(fz 3/2 ) + / 1 /3(!/ /2 )]. (12) 

In the above plot, Ai(z) is the solid curve and Bi(z) is 
dashed. For zero argument, 



Ai(0) 



3 -2/3 



(13) 



where T(z) is the GAMMA FUNCTION. This means that 
Watson's expression becomes 

/»oo 

(3a)- 1/3 7rAi(±(3a)- 1/3 z)= / cos(at 3 ±xt)dt. (14) 

Jo 

A generalization has been constructed by Hardy. 

The Asymptotic Series of Ai(z) has a different form 
in different QUADRANTS of the COMPLEX PLANE, a fact 
known as the STOKES PHENOMENON, Functions related 
to the Airy functions have been defined as 



Gi(z) 



HiW 



* Jo 



t + zt) dt 



(15) 



exp(-f* 3 +2t)<ft. (16) 



see also AlRY-FoCK FUNCTIONS 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Airy Functions." 
§10.4 in Handbook of Mathematical Functions with Formu- 
las, Graphs, and Mathematical Tables, 9th printing. New 
York: Dover, pp. 446-452, 1972. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Bessel Functions of Fractional Order, Airy 
Functions, Spherical Bessel Functions." §6.7 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 234-245, 1992. 

Spanier, J. and Oldham, K. B. "The Airy Functions Ai(x) 
and Bi(x)." Ch. 56 in An Atlas of Functions. Washington, 
DC: Hemisphere, pp. 555-562, 1987. 

Watson, G. N. A Treatise on the Theory of Bessel Functions, 
2nded. Cambridge, England: Cambridge University Press, 
1966. 



Aitken's 5 2 Process 

Airy Projection 

A Map Projection. The inverse equations for <j> are 
computed by iteration. Let the ANGLE of the projection 
plane be 0&. Define 



for 9 b 

a— < ln[2 cos( -^ it -e b )] 



I* 



t—y otherwise. 

tan[f(|ir-0 b )] 



(1) 



For proper convergence, let Xi = 7r/6 and compute the 
initial point by checking 



Xi = 



exp[-(^fx 2 + y 2 +atanxi)tan#i] . (2) 



As long as x» > 1, take x i+ \ = Xi/2 and iterate again. 
The first value for which Xi < 1 is then the starting 
point. Then compute 

Xi = cos' 1 {exp[-(^/x 2 ~+y 2 -{- atanxi) ta,nxi]} (3) 

until the change in xi between evaluations is smaller 
than the acceptable tolerance. The (inverse) equations 
are then given by 



^7T - 2Xi 



- tan 



-(-;) 



(4) 
(5) 



Aitken's 5 2 Process 

An Algorithm which extrapolates the partial sums s n 
of a Series J^ a n whose Convergence is approxi- 
mately geometric and accelerates its rate of CONVER- 
GENCE. The extrapolated partial sum is given by 



Sn = S n +1 



(S n +1 — S n ) 
S n +1 — 2s n + Sn-1 



see also EULER'S SERIES TRANSFORMATION 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 18, 1972. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 160, 1992. 



Aitken Interpolation 



Albers Equal-Area Conic Projection 23 



Aitken Interpolation 

An algorithm similar to Neville's Algorithm for con- 
structing the Lagrange Interpolating Polynom- 
ial. Let f(x\xo, x\, . . • , Xk) be the unique POLYNOMIAL 
of kth ORDER coinciding with f(x) at xq, . . . , Xfc. Then 



f(x\xo,Xi) = 

f(x\x Qy x 2 ) = 

f(x\xo>x ly x 2 ) = 

f(x\x 0i x 1 ,x 2 ,X3) = 



1 



Xl 


- Xo 




1 


X2 


— Xo 




1 


X 2 


- x± 




1 



/o 


Xo 


— X 


A 


Xl 


— X 


/o 


Xo 


— X 


A 


X 2 


— X 



X 3 — X2 



/(x|x ,xi) Xi - X 
/(x|x 0) x 2 ) x 2 - x 

/(x|x ,Xi,X 2 ) X 2 - X 
/(x|x ,Xi,X 3 ) X 3 - X 



see a/so LAGRANGE INTERPOLATING POLYNOMIAL 

References 

Abramowitz, M. and Stegun, C. A. (Eds.), Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 879, 1972. 

Acton, F. S. Numerical Methods That Work, 2nd printing. 
Washington, DC: Math. Assoc. Amer., pp. 93-94, 1990. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 102, 1992. 



Ajima-Malfatti Points 




The lines connecting the vertices and corresponding 
circle-circle intersections in Malfatti's Tangent Tri- 
angle Problem coincide in a point Y called the first 
Ajima-Malfatti point (Kimberling and MacDonald 1990, 
Kimberling 1994). Similarly, letting A", £", and C" be 
the excenters of ABC, then the lines A 1 A", B'B", and 
C'C" are coincident in another point called the second 
Ajima-Malfatti point. The points are sometimes simply 
called the Malfatti Points (Kimberling 1994). 

References 

Kimberling, C. "Central Points and Central Lines in the 

Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 
Kimberling, C. "1st and 2nd Ajima-Malfatti Points." 

http://vvw . evansville . edu/ -ck6/ tcenters/ recent / 

ajmalf.html. 
Kimberling, C. and MacDonald, I. G. "Problem E 3251 and 

Solution. " Amer. Math. Monthly 97, 612-613, 1990. 



Albanese Variety 

An Abelian Variety which is canonically attached to 
an Algebraic Variety which is the solution to a cer- 
tain universal problem. The Albanese variety is dual to 
the Picard Variety. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, pp. 67-68, 1988. 

Albers Conic Projection 

see Albers Equal- Area Conic Projection 

Albers Equal- Area Conic Projection 




Let <fro be the Latitude for the origin of the Cartesian 
Coordinates and Ao its Longitude. Let 0i and <j>2 
be the standard parallels. Then 



x = p sin v 


11) 


y = po - pcosO, 


(2) 


where 




\JC — In sin 


(3) 


e = n(X- Ao) 


(4) 


yJC — 2nsin<^o 
po = 

n 


(5) 


C = cos 2 0i + 2n sin 0i 


(6) 


n = ~ (sin 0i + sin 02 ) . 


(7) 


The inverse FORMULAS are 






(8) 


A = A + -, 


(9) 



where 



P= \A 2 + (po - y) 2 



= tan 



x 



po-y 



(10) 

(ii) 



References 

Snyder, J. P, Map Projections — A Working Manual. U. S. 
Geological Survey Professional Paper 1395. Washington, 
DC: U. S. Government Printing Office, pp. 98-103, 1987. 



24 Alcuin's Sequence 



Alexander- Conway Polynomial 



Alcuin's Sequence 

The Integer Sequence 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 
7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, . . . 
(Sloane's A005044) given by the COEFFICIENTS of the 
Maclaurin Series for 1/(1 - x 2 )(l - x 3 )(l - x 4 ). The 
number of different TRIANGLES which have INTEGRAL 
sides and Perimeter n is given by 



T(n) = P 3 (n) - J2 P2 W 

l<j<ln/2\ 

[si - lij m 



48 j 



for n even 
for n odd, 



(1) 
(2) 

(3) 



where P2(n) and Ps{n) are PARTITION FUNCTIONS, with 
Pk{n) giving the number of ways of writing n as a sum of 
k terms, [x] is the NiNT function, and |_^J is the FLOOR 
Function (Jordan et al 1979, Andrews 1979, Hons- 
berger 1985). Strangely enough, T(n) for n = 3, 4, . . . 
is precisely Alcuin's sequence. 

see also PARTITION FUNCTION P, TRIANGLE 

References 

Andrews, G. "A Note on Partitions and Triangles with Inte- 
ger Sides." Amer. Math. Monthly 86, 477, 1979. 

Honsberger, R. Mathematical Gems III. Washington, DC: 
Math. Assoc. Amer., pp. 39-47, 1985. 

Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with 
Integer Sides." Amer. Math. Monthly 86, 686-689, 1979. 

Sloane, N. J. A. Sequence A005044/M0146 in 'An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Aleksandrov-Cech Cohomology 

A theory which satisfies all the ElLENBERG-STEENROD 
Axioms with the possible exception of the LONG EX- 
ACT Sequence of a Pair Axiom, as well as a certain 
additional continuity CONDITION. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, p. 68, 1988. 

Aleksandrov's Uniqueness Theorem 

A convex body in Euclidean n-space that is centrally 
symmetric with center at the ORIGIN is determined 
among all such bodies by its brightness function (the 
Volume of each projection). 

see also TOMOGRAPHY 

References 

Gardner, R. J. "Geometric Tomography." Not. Amer. Math. 
Soc. 42, 422-429, 1995. 



Aleph 

The Set Theory symbol (N) for the Cardinality of 
an Infinite Set. 

see also Aleph-0 (N ), Aleph-1 (Ni), Countable 
Set, Countably Infinite Set, Finite, Infinite, 
Transfinite Number, Uncountably Infinite Set 

Aleph-0 (N ) 

The Set Theory symbol for a Set having the same 
Cardinal Number as the "small" Infinite Set of In- 
tegers. The Algebraic Numbers also belong to N . 
Rather surprising properties satisfied by N include 



N r = No 

rN = N 
N + / = N , 
where / is any FINITE SET. However, 

No* = C, 



(1) 

(2) 
(3) 



(4) 



where C is the CONTINUUM. 

see also ALEPH-1, CARDINAL NUMBER, CONTINUUM, 

Continuum Hypothesis, Countably Infinite Set, 
Finite, Infinite, Transfinite Number, Uncount- 
ably Infinite Set 

Aleph-1 (Ni) 

The Set Theory symbol for the smallest Infinite Set 
larger than Alpha-0 (N ). The CONTINUUM HYPOTH- 
ESIS asserts that Ni = c, where c is the CARDINALITY 
of the "large" Infinite Set of Real Numbers (called 
the CONTINUUM in Set Theory). However, the truth 
of the Continuum Hypothesis depends on the version 
of Set Theory you are using and so is Undecidable. 

Curiously enough, n-D SPACE has the same number of 
points (c) as 1-D Space, or any Finite Interval of 1- 
D Space (a Line Segment), as was first recognized by 
Georg Cantor. 

see also Aleph-0 (N ), Continuum, Continuum Hy- 
pothesis, Countably Infinite Set, Finite, Infi- 
nite, Transfinite Number, Uncountably Infinite 
Set 

Alethic 

A term in LOGIC meaning pertaining to TRUTH and 

Falsehood. 

see also False, Predicate, True 

Alexander- Conway Polynomial 

see Conway Polynomial 



Alexander's Horned Sphere 
Alexander's Horned Sphere 



Alexander Matrix 



25 




The above solid, composed of a countable UNION of 
Compact Sets, is called Alexander's horned sphere. 
It is Homeomorphic with the BALL B 3 , and its bound- 
ary is therefore a SPHERE. It is therefore an example of 
a wild embedding in E 3 . The outer complement of the 
solid is not SIMPLY CONNECTED, and its fundamental 
GROUP is not finitely generated. Furthermore, the set 
of nonlocally flat ("bad") points of Alexander's horned 
sphere is a Cantor Set. 

The complement in K of the bad points for Alexan- 
der's horned sphere is SIMPLY CONNECTED, making it 
inequivalent to Antoine'S Horned Sphere. Alexan- 
der's horned sphere has an uncountable infinity of Wild 
POINTS, which are the limits of the sequences of the 
horned sphere's branch points (roughly, the "ends" of 
the horns), since any NEIGHBORHOOD of a limit con- 
tains a horned complex. 

A humorous drawing by Simon Prazer (Guy 1983, 
Schroeder 1991, Albers 1994) depicts mathematician 
John H. Conway with Alexander's horned sphere grow- 
ing from his head. 




see also Antoine's Horned Sphere 

References 

Albers, D. J. Illustration accompanying "The Game of 

'Life'." Math Horizons, p. 9, Spring 1994. 
Guy, R. "Conway's Prime Producing Machine." Math. Mag. 

56, 26-33, 1983. 
Hocking, J. G. and Young, G. S. Topology. New York: Dover, 

1988. 
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 

Perish Press, pp. 80-81, 1976. 



Schroeder, M. Fractals, Chaos, Power Law: Minutes from 
an Infinite Paradise. New York: W. H. Freeman, p. 58, 
1991. 

Alexander Ideal 

The order IDEAL in A, the RING of integral LAURENT 

Polynomials, associated with an Alexander Matrix 

for a Knot K. Any generator of a principal Alexander 
ideal is called an Alexander Polynomial. Because 
the Alexander Invariant of a Tame Knot in S 3 
has a Square presentation Matrix, its Alexander ideal 
is Principal and it has an Alexander Polynomial 
A(t). 

see also Alexander Invariant, Alexander Matrix, 
Alexander Polynomial 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 206-207, 1976. 

Alexander Invariant 

The Alexander invariant i7* (X) of a Knot K is the HO- 
MOLOGY of the Infinite cyclic cover of the complement 
of K, considered as a MODULE over A, the RING of inte- 
gral Laurent Polynomials. The Alexander invariant 
for a classical Tame Knot is finitely presentable, and 
only Hi is significant. 

For any KNOT K n in § n+ whose complement has the 
homotopy type of a FINITE COMPLEX, the Alexander 
invariant is finitely generated and therefore finitely pre- 
sentable. Because the Alexander invariant of a Tame 
Knot in S 3 has a Square presentation Matrix, its 
Alexander Ideal is Principal and it has an Alex- 
ander Polynomial denoted A(t). 

see also Alexander Ideal, Alexander Matrix, Al- 
exander Polynomial 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 206-207, 1976. 

Alexander Matrix 

A presentation matrix for the Alexander Invariant 
Hi(X) of a Knot K. If V is a Seifert Matrix for 
a Tame Knot K in S 3 , then V T - tV and V - tV T 



are Alexander matrices for K, 
Matrix Transpose. 



where V denotes the 



see also Alexander Ideal, Alexander Invariant, 
Alexander Polynomial, Seifert Matrix 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 206-207, 1976. 



26 Alexander Polynomial 



Alexander Polynomial 



Alexander Polynomial 

A Polynomial invariant of a Knot discovered in 1923 
by J. W. Alexander (Alexander 1928). In technical lan- 
guage, the Alexander polynomial arises from the HO- 
MOLOGY of the infinitely cyclic cover of a Knot's com- 
plement. Any generator of a PRINCIPAL ALEXANDER 
Ideal is called an Alexander polynomial (Rolfsen 1976). 
Because the Alexander Invariant of a Tame Knot 
in S 3 has a Square presentation Matrix, its Alex- 
ander Ideal is Principal and it has an Alexander 
polynomial denoted A(i). 

Let * be the MATRIX PRODUCT of BRAID WORDS of a 
Knot, then 



det(l - V) 

1 + *+...+ t*- 



= Az 



(1) 



where Az, is the Alexander polynomial and det is the 
Determinant. The Alexander polynomial of a Tame 
Knot in S 3 satisfies 



A(t) = det(V T -tV), 



(2) 



where V is a Seifert Matrix, det is the Determi- 
nant, and V T denotes the Matrix TRANSPOSE. The 
Alexander polynomial also satisfies 



A(l) = ±l. 



(3) 



The Alexander polynomial of a splittable link is always 
0. Surprisingly, there are known examples of nontrivial 
Knots with Alexander polynomial 1. An example is 
the (-3,5,7) Pretzel Knot. 

The Alexander polynomial remained the only known 
Knot Polynomial until the Jones Polynomial was 
discovered in 1984. Unlike the Alexander polynomial, 
the more powerful JONES POLYNOMIAL does, in most 
cases, distinguish HANDEDNESS. A normalized form of 
the Alexander polynomial symmetric in t and £ _1 and 
satisfying 

A(unknot) = 1 (4) 

was formulated by J. H. Conway and is sometimes de- 
noted Vl • The Notation [a 4- b + c + . . . is an abbrevi- 
ation for the Conway-normalized Alexander polynomial 
of a Knot 



a + b(x + x ) + c(x + x ) + . 



(5) 



For a description of the NOTATION for Links, see Rolf- 
sen (1976, p. 389). Examples of the Conway-Alexander 
polynomials for common KNOTS include 



Vtk 
Vfek 

VsSK 



[1-1 = 

[3-1 = 

[l - i + : 



-x" 1 + 1 



_1 +3-x 



(6) 
(7) 



_1 + l-a: + x 2 (8) 



for the Trefoil Knot, Figure-of-Eight Knot, and 
Solomon's Seal Knot, respectively. Multiplying 
through to clear the NEGATIVE POWERS gives the usual 
Alexander polynomial, where the final SIGN is deter- 
mined by convention. 



\, 



\ 



)( 



s 



s 



u 



L 



J + M) 

Let an Alexander polynomial be denoted A, then there 
exists a Skein Relationship (discovered by J. H. Con- 
way) 

A L+ (t)-A L _(t) + (t- 1/2 -t 1/2 )A Lo (t) = (9) 

corresponding to the above Link Diagrams (Adams 
1994). A slightly different Skein RELATIONSHIP con- 
vention used by Doll and Hoste (1991) is 



V i+ -V £ _ =zV Lo . (10) 



These relations allow Alexander polynomials to be con- 
structed for arbitrary knots by building them up as a 
sequence of over- and undercrossings. 

For a Knot, 

* , n _fl(mod8) ifArf(tf) = 0. (n) 

Ak(-1)= j 5(modg) ifArf(K) = 1) (11) 

where Arf is the Arf Invariant (Jones 1985). If K is 
a Knot and 

|A*(i)|>3, (12) 

then K cannot be represented as a closed 3-BRAID. Also, 
if 

A K (e 27ri/5 )> f, (13) 

then K cannot be represented as a closed 4-braid (Jones 
1985). 

The HOMFLY POLYNOMIAL P{a, z) generalizes the Al- 
exander polynomial (as well at the JONES POLYNOMIAL) 
with 

V(z) = P{l t z) (14) 

(Doll and Hoste 1991). 

Rolfsen (1976) gives a tabulation of Alexander polyno- 
mials for Knots up to 10 Crossings and Links up to 
9 Crossings. 

see also Braid Group, Jones Polynomial, Knot, 
Knot Determinant, Link, Skein Relationship 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 

to the Mathematical Theory of Knots. New York: W. H. 

Freeman, pp. 165-169, 1994. 
Alexander, J. W. "Topological Invariants of Knots and 

Links." Trans. Amer. Math. Soc. 30, 275-306, 1928. 



Alexander-Spanier Cohomology 



Algebra 27 



Alexander, J. W. "A Lemma on a System of Knotted 
Curves." Proc. Nat. Acad. Set. USA 9, 93-95, 1923, 

Doll, H. and Hoste, J. "A Tabulation of Oriented Links." 
Math. Comput. 57, 747-761, 1991. 

Jones, V. "A Polynomial Invariant for Knots via von Neu- 
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 
1985. 

Rolfsen, D. "Table of Knots and Links." Appendix C in 
Knots and Links. Wilmington, DE: Publish or Perish 
Press, pp. 280-287, 1976. 

Stoimenow, A. "Alexander Polynomials." http://www. 
informatik.hu-berlin.de/-stoimeno/ptab/alO.html. 

Stoimenow, A. "Conway Polynomials." http://www. 

informatik.hu-berlin.de/-stoimeno/ptab/clO.html. 

Alexander-Spanier Cohomology 

A fundamental result of DE RHAM COHOMOLOGY 

is that the fcth de Rham Cohomology Vector 
Space of a Manifold M is canonically isomorphic 
to the Alexander-Spanier cohomology VECTOR SPACE 
H k (M;M) (also called cohomology with compact sup- 
port). In the case that M is Compact, Alexander- 
Spanier cohomology is exactly "singular" COHOMOL- 
OGY. 

Alexander's Theorem 

Any Link can be represented by a closed Braid. 

Algebra 

The branch of mathematics dealing with GROUP The- 
ory and Coding Theory which studies number sys- 
tems and operations within them. The word "algebra" 
is a distortion of the Arabic title of a treatise by Al- 
Khwarizmi about algebraic methods. Note that mathe- 
maticians refer to the "school algebra" generally taught 
in middle and high school as "Arithmetic," reserving 
the word "algebra" for the more advanced aspects of the 
subject. 

Formally, an algebra is a Vector Space V, over a 
Field F with a Multiplication which turns it into 
a RING defined such that, if / 6 F and x, y G V, then 

f{*y) = (fx)y = x(fy)- 

In addition to the usual algebra of Real Numbers, 
there are as 1151 additional Consistent algebras which 
can be formulated by weakening the FIELD AXIOMS, at 
least 200 of which have been rigorously proven to be 
self-CONSlSTENT (Bell 1945). 

Algebras which have been investigated and found to be 
of interest are usually named after one or more of their 
investigators. This practice leads to exotic-sounding 
(but unenlightening) names which algebraists frequently 
use with minimal or nonexistent explanation. 

see also ALTERNATE ALGEBRA, ALTERNATING ALGE- 
BRA, i?*-ALGEBRA, BANACH ALGEBRA, BOOLEAN AL- 
GEBRA, Borel Sigma Algebra, C*-Algebra, Cay- 
ley Algebra, Clifford Algebra, Commutative 



Algebra, Exterior Algebra, Fundamental The- 
orem of Algebra, Graded Algebra, Grassmann 
Algebra, Hecke Algebra, Heyting Algebra, Ho- 
mological Algebra, Hopf Algebra, Jordan Al- 
gebra, Lie Algebra, Linear Algebra, Measure 
Algebra, Nonassociative Algebra, Quaternion, 
Robbins Algebra, Schur Algebra, Semisimple Al- 
gebra, Sigma Algebra, Simple Algebra, Steen- 
rod Algebra, von Neumann Algebra 

References 

Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. 

Bell, E. T. The Development of Mathematics, 2nd ed. New 
York: McGraw-Hill, pp. 35-36, 1945. 

Bhattacharya, P. B,; Jain, S. K.; and Nagpu, S. R. (Eds.). 
Basic Algebra, 2nd ed. New York: Cambridge University 
Press, 1994. 

BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra, 
5th ed. New York: Macmillan, 1996. 

Brown, K. S. "Algebra." http://www.seanet.com/-ksbrown/ 
ialgebra.htm. 

Cardano, G. Ars Magna or The Rules of Algebra. New York: 
Dover, 1993. 

Chevalley, C. C. Introduction to the Theory of Algebraic 
Functions of One Variable. Providence, RI: Amer. Math. 
Soc, 1951. 

Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover, 
1961. 

Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: 
University of Chicago Press, 1923. 

Dickson, L. E. Modern Algebraic Theories. Chicago, IL: 
H. Sanborn, 1926. 

Edwards, H. M. Galois Theory, corrected 2nd printing. New 
York: Springer- Verlag, 1993. 

Euler, L. Elements of Algebra. New York: Springer- Verlag, 
1984. 

Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lex- 
ington, MA: D. C. Heath, 1994. 

Grove, L. Algebra. New York: Academic Press, 1983. 

Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to El- 
ementary Algebra for Schools. London: Macmillan, 1960. 

Harrison, M. A. "The Number of Isomorphism Types of Fi- 
nite Algebras." Proc. Amer. Math. Soc. 17, 735-737, 
1966. 

Herstein, I. N. Noncommutative Rings. Washington, DC: 
Math. Assoc. Amer., 1996. 

Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 
1975. 

Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. 
Freeman, 1989. 

Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: Uni- 
versity of Chicago Press, 1995. 

Lang, S. Undergraduate Algebra, 2nd ed. New York: 
Springer- Verlag, 1990. 

Pedersen, J. "Catalogue of Algebraic Systems." http:// 
tarski.math.usf .edu/algctlg/. 

Uspensky, J. V. Theory of Equations. New York: McGraw- 
Hill, 1948. 

van der Waerden, B. L. Algebra, Vol. 2. New York: Springer- 
Verlag, 1991. 

van der Waerden, B. L. Geometry and Algebra in Ancient 
Civilizations. New York: Springer- Verlag, 1983. 

van der Waerden, B. L. A History of Algebra: From Al- 
Khwarizmi to Emmy Noether. New York: Springer- Verlag, 
1985. 

Varadarajan, V. S. Algebra in Ancient and Modern Times. 
Providence, RI: Amer. Math. Soc, 1998. 



28 Algebraic Closure 



Algebraic Invariant 



Algebraic Closure 

The algebraic closure of a Field K is the "smallest" 
Field containing K which is algebraically closed. For 
example, the FIELD of COMPLEX NUMBERS C is the 
algebraic closure of the Field of Reals R. 

Algebraic Coding Theory 

see Coding Theory 

Algebraic Curve 

An algebraic curve over a Field K is an equation 
f(X,Y) = 0, where f{X,Y) is a POLYNOMIAL in X and 
Y with Coefficients in K. A nonsingular algebraic 
curve is an algebraic curve over K which has no SIN- 
GULAR Points over K. A point on an algebraic curve 
is simply a solution of the equation of the curve. A K- 
Rational Point is a point (X, Y) on the curve, where 
X and Y are in the FIELD K. 

see also Algebraic Geometry, Algebraic Variety, 
Curve 



References 

Griffiths, P. A. Introduction to Algebraic Curves. 
dence, RI: Amer. Math. Soc, 1989. 



Provi- 



Algebraic Function 

A function which can be constructed using only a finite 
number of ELEMENTARY FUNCTIONS together with the 
Inverses of functions capable of being so constructed. 

see also Elementary Function, Transcendental 
Function 

Algebraic Function Field 

A finite extension K = Z(z)(w) of the Field C(z) of 
Rational Functions in the indeterminate z, i.e., w is 
a Root of a Polynomial a +aia + a 2 a 2 + . . . + a n a: n , 
where a; € C(z). 

see also Algebraic Number Field, Riemann Sur- 
face 

Algebraic Geometry 

The Study of ALGEBRAIC CURVES, ALGEBRAIC VARI- 
ETIES, and their generalization to n-D. 

see also Algebraic Curve, Algebraic Variety, 
Commutative Algebra, Differential Geometry, 
Geometry, Plane Curve, Space Curve 

References 

Abhyankar, S. S. Algebraic Geometry for Scientists and En- 
gineers. Providence, RI: Amer. Math. Soc, 1990. 

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and 
Algorithms: An Introduction to Algebraic Geometry and 
Commutative Algebra, 2nd ed. New York: Springer- 
Verlag, 1996. 

Eisenbud, D. Commutative Algebra with a View Toward Al- 
gebraic Geometry. New York: Springer- Verlag, 1995. 

Griffiths, P. and Harris, J. Principles of Algebraic Geometry. 
New York: Wiley, 1978. 

Hartshorne, R. Algebraic Geometry, rev. ed. New York: 
Springer- Verlag, 1997. 



Lang, S. Introduction to Algebraic Geometry. New York: 
Interscience, 1958. 

Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, 
Vol. 1. Cambridge, England: Cambridge University Press, 
1994. 

Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, 
Vol. 2. Cambridge, England: Cambridge University Press, 
1994. 

Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, 
Vol. 3. Cambridge, England: Cambridge University Press, 
1994. 

Seidenberg, A. (Ed.). Studies in Algebraic Geometry. Wash- 
ington, DC: Math. Assoc. Amer., 1980. 

Weil, A. Foundations of Algebraic Geometry, enl. ed. Prov- 
idence, RI: Amer. Math. Soc, 1962. 

Algebraic Integer 

If r is a Root of the Polynomial equation 

x n + a n -ix n ~ + . . . + aiz + ao = 0, 

where the a^s are INTEGERS and r satisfies no similar 
equation of degree < n, then r is an algebraic INTEGER 
of degree n. An algebraic INTEGER is a special case of 
an Algebraic Number, for which the leading Coef- 
ficient a n need not equal 1. RADICAL INTEGERS are a 
subring of the ALGEBRAIC INTEGERS. 

A Sum or Product of algebraic integers is again an al- 
gebraic integer. However, Abel's IMPOSSIBILITY THE- 
OREM shows that there are algebraic integers of degree 
> 5 which are not expressible in terms of ADDITION, 

Subtraction, Multiplication, Division, and the ex- 
traction of Roots on Real Numbers. 

The Gaussian Integer are are algebraic integers of 
-1 ), since a + bi are roots of 



z 2 - 2az + a 2 + b 2 = 0. 



see also Algebraic Number, Euclidean Number, 
Radical Integer 

References 

Hancock, H. Foundations of the Theory of Algebraic Num- 
bers, Vol. 1: Introduction to the General Theory. New 
York: Macmillan, 1931. 

Hancock, H. Foundations of the Theory of Algebraic Num- 
bers, Vol. 2: The General Theory. New York: Macmillan, 
1932. 

Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num- 
ber Theory. Cambridge, England: Cambridge University- 
Press, 1989. 

Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in 
Action. New York: W. H. Freeman, pp. 347-353, 1991. 

Algebraic Invariant 

A quantity such as a Discriminant which remains un- 
changed under a given class of algebraic transforma- 
tions. Such invariants were originally called HYPERDE- 
TERMINANTS by Cayley. 

see also DISCRIMINANT (POLYNOMIAL), INVARIANT, 

Quadratic Invariant 



Algebraic Knot 



Algebraic Tangle 29 



References 

Grace, J. H. and Young, A. The Algebra of Invariants. New 
York: Chelsea, 1965. 

Gurevich, G. B. Foundations of the Theory of Algebraic In- 
variants. Groningen, Netherlands: P. NoordhofF, 1964. 

Hermann, R. and Ackerman, M. Hilbert's Invariant Theory 
Papers.rookline, MA: Math Sci Press, 1978. 

Hilbert, D. Theory of Algebraic Invariants. Cambridge, Eng- 
land: Cambridge University Press, 1993. 

Mumford, D.; Fogarty, J.; and Kirwan, F. Geometric Invari- 
ant Theory, 3rd enl. ed. New York: Springer- Verlag, 1994. 

Algebraic Knot 

A single component ALGEBRAIC LINK. 
see also Algebraic Link, Knot, Link 

Algebraic Link 

A class of fibered knots and links which arises in Al- 
gebraic Geometry. An algebraic link is formed by 
connecting the NW and NE strings and the SW and SE 
strings of an ALGEBRAIC Tangle (Adams 1994). 

see also Algebraic Tangle, Fibration, Tangle 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 

to the Mathematical Theory of Knots. New York: W. H. 

Freeman, pp. 48-49, 1994. 
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 

Perish Press, p. 335, 1976. 

Algebraic Number 

If r is a ROOT of the POLYNOMIAL equation 



a$x -\- a±x 



. . + a n - 1 x -\- a n = 0, 



(i) 



where the a^s are Integers and r satisfies no similar 
equation of degree < n, then r is an algebraic number of 
degree n. If r is an algebraic number and ao = 1, then 
it is called an ALGEBRAIC INTEGER. It is also true that 
if the c;s in 



CQX + ClX n + . . . + Cn-lX + C n - 



(2) 



are algebraic numbers, then any ROOT of this equation 
is also an algebraic number. 

If a is an algebraic number of degree n satisfying the 
Polynomial 



a(x — a)(x — j3)(x — 7) ■ 



(3) 



then there are n — 1 other algebraic numbers (3, 7, ... 
called the conjugates of ex. Furthermore, if a satisfies 
any other algebraic equation, then its conjugates also 
satisfy the same equation (Conway and Guy 1996). 

Any number which is not algebraic is said to be TRANS- 
CENDENTAL. 
see also ALGEBRAIC INTEGER, EUCLIDEAN NUMBER, 

Hermite-Lindemann Theorem, Radical Integer, 
Semialgebraic Number, Transcendental Number 



References 

Conway, J. H. and Guy, R. K. "Algebraic Numbers." In The 
Book of Numbers. New York: Springer- Verlag, pp. 189— 
190, 1996. 

Courant, R. and Robbing, H. "Algebraic and Transcendental 
Numbers." §2.6 in What is Mathematics?: An Elementary 
Approach to Ideas and Methods, 2nd ed. Oxford, England: 
Oxford University Press, pp. 103-107, 1996. 

Hancock, H. Foundations of the Theory of Algebraic Num- 
bers. Vol. 1: Introduction to the General Theory. New 
York: Macmillan, 1931. 

Hancock, H. Foundations of the Theory of Algebraic Num- 
bers. Vol. 2: The General Theory. New York: Macmillan, 
1932. 

Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in 
Action. New York: W. H. Freeman, pp. 347-353, 1991. 

Algebraic Number Field 

see Number Field 

Algebraic Surface 

The set of ROOTS of a POLYNOMIAL f(x,y,z) = 0. An 
algebraic surface is said to be of degree n = max(i + J + 
fc), where n is the maximum sum of powers of all terms 
amX lrn y jrn z krn . The following table lists the names of 
algebraic surfaces of a given degree. 



Order 


Surface 


3 


cubic surface 


4 


quartic surface 


5 


quintic surface 


6 


sextic surface 


7 


heptic surface 


8 


octic surface 


9 


nonic surface 


10 


decic surface 



see also Barth Decic, Barth Sextic, Boy Surface, 
Cayley Cubic, Chair, Clebsch Diagonal Cubic, 
Cushion, Dervish, Endrass Octic, Heart Surface, 
Kummer Surface, Order (Algebraic Surface), 
Roman Surface, Surface, Togliatti Surface 

References 

Fischer, G. (Ed.). Mathematical Models from the Collections 

of Universities and Museums. Braunschweig, Germany: 

Vieweg, p. 7, 1986. 

Algebraic Tangle 

Any Tangle obtained by Additions and Multiplica- 
tions of rational TANGLES (Adams 1994). 

see also Algebraic Link 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 

to the Mathematical Theory of Knots. New York: W. H. 

Freeman, pp. 41-51, 1994. 



30 Algebraic Topology 



Algorithm 



Algebraic Topology 

The study of intrinsic qualitative aspects of spatial 
objects (e.g., SURFACES, SPHERES, TORI, CIRCLES, 
Knots, Links, configuration spaces, etc.) that re- 
main invariant under both-directions continuous ONE- 
TO-One (HOMEOMORPHIC) transformations. The dis- 
cipline of algebraic topology is popularly known as 
"Rubber-Sheet Geometry" and can also be viewed 
as the study of Disconnectivities. Algebraic topology 
has a great deal of mathematical machinery for studying 
different kinds of HOLE structures, and it gets the prefix 
"algebraic" since many Hole structures are represented 
best by algebraic objects like GROUPS and RINGS. 

A technical way of saying this is that algebraic topol- 
ogy is concerned with FUNCTORS from the topological 
Category of Groups and Homomorphisms. Here, 
the FUNCTORS are a kind of filter, and given an "input" 
SPACE, they spit out something else in return. The re- 
turned object (usually a Group or Ring) is then a rep- 
resentation of the HOLE structure of the SPACE, in the 
sense that this algebraic object is a vestige of what the 
original SPACE was like (i.e., much information is lost, 
but some sort of "shadow" of the SPACE is retained — 
just enough of a shadow to understand some aspect of its 
HOLE-structure, but no more). The idea is that FUNC- 
TORS give much simpler objects to deal with. Because 
SPACES by themselves are very complicated, they are 
unmanageable without looking at particular aspects. 

COMBINATORIAL TOPOLOGY is a special type of alge- 
braic topology that uses COMBINATORIAL methods. 

see also CATEGORY, COMBINATORIAL TOPOLOGY, DIF- 
FERENTIAL TOPOLOGY, FUNCTOR, HOMOTOPY THE- 
ORY 

References 

Dieudonne, J. A History of Algebraic and Differential Topol- 
ogy: 1900-1960. Boston, MA: Birkhauser, 1989. 

Algebraic Variety 

A generalization to n-D of ALGEBRAIC CURVES. More 
technically, an algebraic variety is a reduced SCHEME of 
Finite type over a Field K. An algebraic variety V is 
defined as the Set of points in the Reals W 1 (or the 
Complex Numbers C n ) satisfying a system of Poly- 
nomial equations fi(xi, . . . , x n ) = for i = 1, 2, 

According to the Hilbert Basis Theorem, a Finite 
number of equations suffices. 

see also Abelian Variety, Albanese Variety, 
Brauer-Severi Variety, Chow Variety, Picard 
Variety 

References 

Ciliberto, C; Laura, E.; and Somese, A. J. (Eds.). Classifica- 
tion of Algebraic Varieties. Providence, RI: Amer. Math. 
Soc, 1994. 



Algebroidal Function 

An Analytic Function f(z) satisfying the irreducible 
algebraic equation 

A (z)f k + Ai(z)/*- 1 + . . . + A k (z) = 

with single- valued MEROMORPHIC functions Aj(z) in a 
Complex Domain G is called a fc-algebroidal function 
in G. 

References 

Iyanaga, S. and Kawada, Y. (Eds.). "Algebroidal Functions." 
§19 in Encyclopedic Dictionary of Mathematics. Cam- 
bridge, MA: MIT Press, pp. 86-88, 1980. 

Algorithm 

A specific set of instructions for carrying out a proce- 
dure or solving a problem, usually with the requirement 
that the procedure terminate at some point. Specific 
algorithms sometimes also go by the name Method, 
Procedure, or Technique. The word "algorithm" is 
a distortion of Al-Khwarizmi, an Arab mathematician 
who wrote an influential treatise about algebraic meth- 
ods. 

see also 196- ALGORITHM, ALGORITHMIC COMPLEXITY, 

Archimedes Algorithm, Bhaskara-Brouckner 
Algorithm, Borchardt-Pfaff Algorithm, Bre- 
laz's Heuristic Algorithm, Buchberger's Algo- 
rithm, Bulirsch-Stoer Algorithm, Bumping Al- 
gorithm, CLEAN Algorithm, Computable Func- 
tion, Continued Fraction Factorization Algo- 
rithm, Decision Problem, Dijkstra's Algorithm, 
Euclidean Algorithm, Ferguson-Forcade Al- 
gorithm, Fermat's Algorithm, Floyd's Algo- 
rithm, Gaussian Approximation Algorithm, Ge- 
netic Algorithm, Gosper's Algorithm, Greedy 
Algorithm, Hasse's Algorithm, HJLS Algo- 
rithm, Jacobi Algorithm, Kruskal's Algorithm, 
Levine-O 'Sullivan Greedy Algorithm, LLL Al- 
gorithm, Markov Algorithm, Miller's Algo- 
rithm, Neville's Algorithm, Newton's Method, 
Prime Factorization Algorithms, Primitive Re- 
cursive Function, Program, PSLQ Algorithm, 
PSOS Algorithm, Quotient-Difference Algo- 
rithm, Risch Algorithm, Schrage's Algorithm, 
Shanks' Algorithm, Spigot Algorithm, Syracuse 
Algorithm, Total Function, Turing Machine, 
Zassenhaus-Berlekamp Algorithm, Zeilberger's 
Algorithm 

References 

Aho, A. V.; Hopcroft, J. E.; and Ullman, J.D. The De- 
sign and Analysis of Computer Algorithms. Reading, MA: 
Addison- Wesley, 1974. 

Baase, S. Computer Algorithms. Reading, MA: Addison- 
Wesley, 1988. 

Brassard, G. and Bratley, P. Fundamentals of Algorithmics. 
Englewood Cliffs, NJ: Prentice-Hall, 1995. 

Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduc- 
tion to Algorithms. Cambridge, MA: MIT Press, 1990. 



Algorithmic Complexity 



Aliquant Divisor 31 



Greene, D. H. and Knuth, D. E. Mathematics for the Analysis 

of Algorithms, 3rd ed. Boston: Birkhauser, 1990. 
Harel, D. Algorithmics: The Spirit of Computing, 2nd ed. 

Reading, MA: Addison- Wesley, 1992. 
Knuth, D. E. The Art of Computer Programming, Vol. 1: 

Fundamental Algorithms, 2nd ed. Reading, MA: Addison- 

Wesley, 1973. 
Knuth, D. E. The Art of Computer Programming, Vol. 2: 

Seminumerical Algorithms, 2nd ed. Reading, MA: 

Addison- Wesley, 1981. 
Knuth, D. E. The Art of Computer Programming, Vol. 3: 

Sorting and Searching, 2nd ed. Reading, MA: Addison- 

Wesley, 1973. 
Kozen, D. C. Design and Analysis and Algorithms. New 

York: Springer- Verlag, 1991. 
Shen, A. Algorithms and Programming. Boston: Birkhauser, 

1996. 
Skiena, S. S. The Algorithm Design Manual. New York: 

Springer- Verlag, 1997. 
Wilf, H. Algorithms and Complexity. Englewood Cliffs, NJ: 

Prentice Hall, 1986. http://www.cis.upenn.edu/-wilf/. 



References 

Dorrie, H. "Alhazen's Billiard Problem." §41 in 100 Great 
Problems of Elementary Mathematics: Their History and 
Solutions. New York: Dover, pp. 197-200, 1965. 

Hogendijk, J. P. "Al-Mutaman's Simplified Lemmas for Solv- 
ing 'Alhazen's Problem'." From Baghdad to Barcelona/De 
Bagdad a Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59- 
101, Anu. Filol. Univ. Bare, XIX B-2, Univ. Barcelona, 
Barcelona, 1996. 

Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tid~ 
skr. 18, 5-35, 1970. 

Neumann, P. Submitted to Amer. Math. Monthly. 

Riede, H. "Reflexion am Kugelspiegel. Oder: das Problem 
des Alhazen." Praxis Math. 31, 65-70, 1989. 

Sabra, A. I. "ibn al-Haytham's Lemmas for Solving 'Al- 
hazen's Problem'." Arch. Hist Exact Sci. 26, 299-324, 
1982. 



Alhazen's Problem 

see Alhazen's Billiard Problem 



Algorithmic Complexity 

see Bit Complexity, Kolmogorov Complexity 

Alhazen's Billiard Problem 

In a given Circle, find an Isosceles Triangle whose 
Legs pass through two given Points inside the Circle. 
This can be restated as: from two POINTS in the Plane 
of a Circle, draw Lines meeting at the Point of the 
Circumference and making equal Angles with the 
Normal at that Point. 

The problem is called the billiard problem because it cor- 
responds to finding the POINT on the edge of a circular 
"BILLIARD" table at which a cue ball at a given POINT 
must be aimed in order to carom once off the edge of the 
table and strike another ball at a second given Point. 
The solution leads to a BIQUADRATIC EQUATION of the 
form 



H{x 2 



V ) 



2Kxy + {x 2 -r y 2 ){hy - kx) = 0. 



The problem is equivalent to the determination of the 
point on a spherical mirror where a ray of light will re- 
flect in order to pass from a given source to an observer. 
It is also equivalent to the problem of finding, given two 
points and a Circle such that the points are both inside 
or outside the Circle, the Ellipse whose Foci are the 
two points and which is tangent to the given CIRCLE. 

The problem was first formulated by Ptolemy in 150 
AD, and was named after the Arab scholar Alhazen, 
who discussed it in his work on optics. It was not until 
1997 that Neumann proved the problem to be insoluble 
using a COMPASS and RULER construction because the 
solution requires extraction of a CUBE ROOT, This is 
the same reason that the CUBE DUPLICATION problem 
is insoluble. 

see also Billiards, Billiard Table Problem, Cube 
Duplication 



Alias' Paradox 

Choose between the following two alternatives: 

1. 90% chance of an unknown amount x and a 10% 
chance of $1 million, or 

2. 89% chance of the same unknown amount x, 10% 
chance of $2.5 million, and 1% chance of nothing. 

The Paradox is to determine which choice has the 
larger expectation value, 0.9x + $100,000 or 0.89:r -f 
$250,000. However, the best choice depends on the un- 
known amount, even though it is the same in both cases! 
This appears to violate the INDEPENDENCE Axiom. 
see also Independence Axiom, Monty Hall Prob- 
lem, Newcomb's Paradox 

Aliasing 

Given a power spectrum (a plot of power vs. frequency), 
aliasing is a false translation of power falling in some fre- 
quency range ( — / c ,/ c ) outside the range. Aliasing can 
be caused by discrete sampling below the NYQUIST FRE- 
QUENCY. The sidelobcs of any INSTRUMENT FUNCTION 
(including the simple SlNC SQUARED function obtained 
simply from FINITE sampling) are also a form of alias- 
ing. Although sidelobe contribution at large offsets can 
be minimized with the use of an APODIZATION FUNC- 
TION, the tradeoff is a widening of the response (i.e., a 
lowering of the resolution). 

see also Apodization Function, Nyquist Fre- 
quency 

Aliquant Divisor 

A number which does not DIVIDE another exactly. For 
instance, 4 and 5 are aliquant divisors of 6. A num- 
ber which is not an aliquant divisor (i.e., one that does 
Divide another exactly) is said to be an Aliquot Di- 
visor. 

see also ALIQUOT DIVISOR, DIVISOR, PROPER DIVISOR 



32 Aliquot Cycle 



Allegory 



Aliquot Cycle 

see Sociable Numbers 

Aliquot Divisor 

A number which DIVIDES another exactly. For instance, 
1, 2, 3, and 6 are aliquot divisors of 6, A number which 
is not an aliquot divisor is said to be an ALIQUANT DI- 
VISOR. The term "aliquot" is frequently used to specif- 
ically mean a PROPER DIVISOR, i.e., a DIVISOR of a 
number other than the number itself. 

see also ALIQUANT DIVISOR, DIVISOR, PROPER DIVI- 
SOR 

Aliquot Sequence 

Let 

s(n) = cr(n) — n, 

where a(n) is the DIVISOR FUNCTION and s(n) is the 
Restricted Divisor Function. Then the Sequence 
of numbers 

s°(n) = n, s 1 (n) = s(n), s (n) — s(s(n)), . . . 

is called an aliquot sequence. If the SEQUENCE for a 
given n is bounded, it either ends at s(l) = or becomes 
periodic. 

1. If the Sequence reaches a constant, the constant is 
known as a PERFECT NUMBER. 

2. If the SEQUENCE reaches an alternating pair, it is 
called an AMICABLE PAIR. 

3. If, after k iterations, the SEQUENCE yields a cycle 
of minimum length t of the form s fc+1 (n), s fc+2 (n), 
..., s k+t (n), then these numbers form a group of 

Sociable Numbers of order t. 

It has not been proven that all aliquot sequences eventu- 
ally terminate and become period. The smallest number 
whose fate is not known is 276, which has been computed 
up to s 487 (276) (Guy 1994). 

see also 196-Algorithm, Additive Persistence, 
Amicable Numbers, Multiamicable Numbers, 
Multiperfect Number, Multiplicative Persis- 
tence, Perfect Number, Sociable Numbers, Uni- 
tary Aliquot Sequence 

References 

Guy, R. K. "Aliquot Sequences." §B6 in Unsolved Problems 
in Number Theory, 2nd ed. New York: Springer- Verlag, 
pp. 60-62, 1994. 

Guy, R. K. and Selfridge, J. L. "What Drives Aliquot Se- 
quences." Math. Corn-put. 29, 101-107, 1975. 

Sloane, N. J. A. Sequences A003023/M0062 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 

All-Poles Model 

see Maximum Entropy Method 



Alladi-Grinstead Constant 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

Let N(n) be the number of ways in which the Facto- 
rial n! can be decomposed into n Factors of the form 
Pk bk arranged in nondecreasing order. Also define 



m(n) = max(pi 1 ), 



(1) 



i.e., m(n) is the Least Prime Factor raised to its 
appropriate POWER in the factorization. Then define 



a(n) = 



lnm(n) 
Inn 



(2) 



where ln(x) is the NATURAL LOGARITHM. For instance, 



2 • 2 ■ 2 2 ■ 5 • 7 • 3 4 
23-5-7-2 3 -3 3 
2 • 5 - 7 ■ 2 3 • 3 2 • 3 2 



9! = 2 


2 


2 


= 2 


2 


2 


= 2 


2 


2 


= 2 


2 


2 


= 2 


2 


2- 


= 2 


2 


2- 


= 2 


2 


3- 


= 2 


2 


3- 


= .2 


3 


3- 


= 2 


3 


3 


= 2 


3 


3- 


= 3 


3 


3- 



2 2 • 2 2 



5 * 7 • 3 2 • 3 2 



3 • 3 • 5 • 7 ■ 3 2 • 2 4 
3 ■ 2 2 • 5 • 7 • 2 3 ■ 3 2 
3 • 3 ■ 3 • 5 • 7 • 2 5 
2 2 • 2 2 • 2 2 ■ 5 • 7 ■ 3 2 
3-3-2 2 .5-7-2 4 



3-3-5 



7 • 2 3 - 2 3 



a(9) = 



3-2 2 -2 2 .5-7-2 3 , 



In 3 In 3 1 



In 9 21n3 2 



For large n, 



lim a(n) = e c_1 = 0.809394020534 . . . , 

n— kx> 



where 



-£WA)- 



(3) 
(4) 

(5) 
(6) 



References 

Alladi, K. and Grinstead, C. "On the Decomposition of n! 

into Prime Powers." J. Number Th, 9, 452-458, 1977. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsof t . c om/ as olve/ const ant /aldgrns/aldgrns .html. 
Guy, R. K. "Factorial n as the Product of n Large Factors." 

§B22 in Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer- Verlag, p. 79, 1994. 

Allegory 

A technical mathematical object which bears the same 
resemblance to binary relations as CATEGORIES do to 
Functions and Sets. 

see also CATEGORY 
References 

Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- 
dam, Netherlands: North-Holland, 1990. 



Allometric 



Almost Integer 33 



Allometric 

Mathematical growth in which one population grows at 
a rate PROPORTIONAL to the POWER of another popu- 
lation. 

References 

Cofrey, W. J. Geography Towards a General Spatial Systems 
Approach. London: Routledge, Chapman & Hall, 1981, 

Almost All 

Given a property P, if P{x) ~ x as x — > oo (so the num- 
ber of numbers less than x not satisfying the property 
P is o(x)), then P is said to hold true for almost all 
numbers. For example, almost all positive integers are 
Composite Numbers (which is not in conflict with the 
second of Euclid's Theorems that there are an infinite 
number of PRIMES). 
see also For All, Normal Order 

References 

Hardy, G. H. and Wright, E. M. An Introduction to the The- 
ory of Numbers, 5th ed. Oxford, England: Clarendon 
Press, p. 8, 1979. 

Almost Alternating Knot 

An Almost Alternating Link with a single compo- 
nent. 

Almost Alternating Link 

Call a projection of a LINK an almost alternating pro- 
jection if one crossing change in the projection makes it 
an alternating projection. Then an almost alternating 
link is a Link with an almost alternating projection, but 
no alternating projection. Every ALTERNATING KNOT 
has an almost alternating projection. A PRIME KNOT 
which is almost alternating is either a Torus Knot or 
a Hyperbolic Knot. Therefore, no Satellite Knot 
is an almost alternating knot. 

All nonalternating 9-crossing PRIME KNOTS are almost 
alternating. Of the 393 nonalternating with 11 or fewer 
crossings, all but five are known to be nonalternating (3 
of these have 11 crossings). The fate of the remaining 
five is not known. The (2,qr), (3,4), and (3,5)-TORUS 
KNOTS are almost alternating. 

see also Alternating Knot, Link 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 

to the Mathematical Theory of Knots. New York: W. H. 

Freeman, pp. 139-146, 1994. 

Almost Everywhere 

A property of X is said to hold almost everywhere if 
the SET of points in X where this property fails has 
Measure 0. 

see also MEASURE 
References 

Sansone, G. Orthogonal Functions, rev. English ed. New 
York: Dover, p. 1, 1991. 



Almost Integer 

A number which is very close to an INTEGER. One sur- 
prising example involving both e and Pi is 



7T = 19.999099979. 



(1) 



which can also be written as 



(tt + 20)* = -0.9999999992 - 0.0000388927i & -1 (2) 

cos(ln(7r + 20)) « -0.9999999992. (3) 

Applying Cosine a few more times gives 

COs(7T COS(7T COs(ln(7T + 20)))) 

« -1 + 3.9321609261 x 10" 35 . (4) 

This curious near-identity was apparently noticed al- 
most simultaneously around 1988 by N. J. A. Sloane, 
J. H. Conway, and S. Plouffe, but no satisfying explana- 
tion as to "why" it has been true has yet been discov- 
ered. 

An interesting near-identity is given by 

i[cos(^) + cosh(^) + 2cos(^^)cosh(^V / 2)] 

= 1 + 2.480... x 10" 13 (5) 

(W. Dubuque). Other remarkable near-identities are 
given by 

5(1 + we )[ g !)]2 =l + 4.5422 -x 10- (6) 

where T(z) is the Gamma FUNCTION (S. Plouffe), and 



e 6 - 7v 4 - tt 5 = 0.000017673 ... (7) 



(D. Wilson). 

A whole class of IRRATIONAL "almost integers" can be 
found using the theory of MODULAR FUNCTIONS, and a 
few rather spectacular examples are given by Ramanu- 
jan (1913-14). Such approximations were also stud- 
ied by Hermite (1859), Kronecker (1863), and Smith 
(1965). They can be generated using some amazing (and 
very deep) properties of the j-FUNCTlON. Some of the 
numbers which are closest approximations to INTEGERS 
are e*^ 1 ^ (sometimes known as the R A MANU J AN Con- 
stant and which corresponds to the field Q(V"163) 
which has Class Number 1 and is the Imaginary 
quadratic field of maximal discriminant), e 22 , e 71 " 37 , 
and e"^, the latter three of which have Class Num- 
ber 2 and are due to Ramanujan (Berndt 1994, Wald- 
schmidt 1988). 



34 Almost Integer 



Almost Prime 



The properties of the j-FUNCTlON also give rise to the 
spectacular identity 



ln(640320 3 + 744) 



163 + 2.32167... x 10" 



(8) 



(Le Lionnais 1983, p. 152). 

The list below gives numbers of the form x = e 71 "^ for 
n < 1000 for which \x] - x < 0.01. 



e^: 
e - 

e = 

jt-v/25 

e = 

nVTf 

e = 

e = 

e : 

e = 

e = 

e = 

tvvT49 

e 

ttvT63 



2,197.990 869 543... 
= 422, 150.997 675 680. . . 
= 614,551.992 885619... 
= 2,508,951.998 257 553. . . 
= 6,635,623.999 341134... 
= 199, 148, 647.999 978 046 551 .. . 
= 884, 736, 743.999 777 466 .. . 
= 24, 591, 257, 751.999 999 822 213 .. . 
= 30, 197, 683, 486.993 182 260 .. . 
= 147, 197, 952, 743.999 998 662 454 .. . 
= 54,551,812,208.999917467 885... 
= 45, 116, 546, 012, 289, 599.991 830 287 . . . 
= 262, 537, 412, 640, 768, 743.999 999 999 999 250 072 . 
= 1, 418, 556, 986, 635, 586, 485.996 179 355 .. . 
= 604, 729, 957, 825, 300, 084, 759.999 992 171 526 .. . 
= 19, 683, 091, 854, 079, 461, 001, 445.992 737 040 .. . 

= 4, 309, 793, 301, 730, 386, 363, 005, 719.996 011 651 . 

= 639, 355, 180, 631, 208, 421, • • • 

■ ■ - 212, 174, 016.997 669 832 . 

= 14, 871, 070, 263, 238, 043, 663, 567, • - • 

• • • 627, 879, 007.999 848 726 . 
= 288, 099, 755, 064, 053, 264, 917, 867, • - • 

•■• 975, 825, 573. 993 898 311. 
= 28, 994, 858, 898, 043, 231, 996, 779, - • - 

■ • ■ 771, 804, 797, 161.992 372 939 . 
= 3, 842, 614, 373, 539, 548, 891, 490, • • - 

' • ■ • 294, 277, 805, 829, 192.999 987 249 . 
= 223, 070, 667, 213, 077, 889, 794, 379, - - - 

- - ■ 623, 183, 838, 336, 437.992 055 118 . 
= 249, 433, 117, 287, 892, 229, 255, 125, • ■ • 

• • • 388, 685, 911, 710, 805.996 097 323 . 
= 365, 698, 321, 891, 389, 219, 219, 142, ■ ■ - 

■ • - 531, 076, 638, 716, 362, 775.998 259 747 . 
= 6, 954, 830, 200, 814, 801, 770, 418, 837, - - ■ 

940, 281, 460, 320, 666, 108.994 649 611 . . 



Gosper noted that the expression 



differs from an Integer by a mere 10 
see also Class Number, j-Function, Pi 

References 

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: 

Springer- Verlag, pp. 90-91, 1994. 
Hermite, C. "Sur la theorie des equations modulaires." C. 

R. Acad. Sci. (Paris) 48, 1079-1084 and 1095-1102, 1859. 
Hermite, C. "Sur la theorie des equations modulaires." C. R. 

Acad. Sci. (Paris) 49, 16-24, 110-118, and 141-144, 1859. 
Kronecker, L. "Uber die Klassenzahl der aus Werzeln der Ein- 

heit gebildeten komplexen Zahlen." Monatsber. K. Preuss. 

Akad. Wiss. Berlin, 340-345. 1863. 
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 

1983. 
Ramanujan, S. "Modular Equations and Approximations to 

7T." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914. 
Smith, H. J, S. Report on the Theory of Numbers. New York: 

Chelsea, 1965. 
Waldschmidt, M. "Some Transcendental Aspects of Ramanu- 

jan's Work." In Ramanujan Revisited: Proceedings of the 

Centenary Conference (Ed. G. E« Andrews, B. C. Berndt, 

and R. A. Rankin). New York: Academic Press, pp. 57-76, 

1988. 

Almost Perfect Number 

A number n for which the DIVISOR FUNCTION satisfies 
cr(n) = 2n — 1 is called almost perfect. The only known 
almost perfect numbers are the POWERS of 2, namely 
1, 2, 4, 8, 16, 32, ... (Sloane's A000079). Singh (1997) 
calls almost perfect numbers SLIGHTLY DEFECTIVE. 

see also QuASIPERFECT NUMBER 

References 

Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, 
Harmonic, Weird, Multiperfect and Hyperperfect Num- 
bers." §B2 in Unsolved Problems in Number Theory, 2nd 
ed. New York: Springer- Verlag, pp. 16 and 45—53, 1994. 

Singh, S. Fermat's Enigma: The Epic Quest to Solve 
the World's Greatest Mathematical Problem. New York: 
Walker, p. 13, 1997. 

Sloane, N. J. A. Sequence A000079/M1129 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Almost Prime 

A number n with prime factorization 



=n»- 



■ 2625374126407G8744e 



-7TV163 



196884e 



-27TN/163 



+103378831900730205293632e~ 37rv/I ^. (9) 



is called ^-almost prime when the sum of the POWERS 
J^^ l di = k. The set of fc-almost primes is denoted Ph. 

The Primes correspond to the "1-almost prime" num- 
bers 2, 3, 5, 7, 11, . . . (Sloane's A000040). The 2-almost 
prime numbers correspond to SEMIPRIMES 4, 6, 9, 10, 
14, 15, 21, 22, ... (Sloane's A001358). The first few 
3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 
45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... 
(Sloane's A014612). The first few 4-almost primes are 
16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, . . . (Sloane's 
A014613). The first few 5-almost primes are 32, 48, 72, 
80, ... (Sloane's A014614). 



Alpha 



Alternate Algebra 35 



see also Chen's Theorem, Prime Number, Semi- 
prime 

References 

Sloane, N. J. A. Sequences A014612, A014613, A014614, 
A000040/M0652, and A001358/M3274 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Alpha 

A financial measure giving the difference between a 
fund's actual return and its expected level of perfor- 
mance, given its level of risk (as measured by Beta). 
A POSITIVE alpha indicates that a fund has performed 
better than expected based on its Beta, whereas a Neg- 
ative alpha indicates poorer performance 

see also Beta, Sharpe Ratio 



Alphamagic Square 

A Magic Square for which the number of letters in 
the word for each number generates another MAGIC 
Square. This definition depends, of course, on the lan- 
guage being used. In English, for example, 



5 


22 


18 


4 


9 


8 


28 


15 


2 


11 


7 


3 


12 


8 


25 


6 


5 


10 



where the MAGIC SQUARE on the right corresponds to 
the number of letters in 



five 

twenty-eight 

twelve 



twenty-two eighteen 
fifteen two 

eight twenty-five 



Alpha Function 




a n (z) = / t n e~ zt dt = n\z- (n+1) e- z ^ 



k\ 



The alpha function satisfies the Recurrence Rela- 
tion 

za n (z) = e~ z + na n -i(z). 

see also BETA FUNCTION (Exponential) 

Alpha Value 

An alpha value is a number < a < 1 such that P(z > 
^observed) < « is considered "Significant," where P is 
a P- Value. 

see also Confidence Interval, P- Value, Signifi- 
cance 

Alphabet 

A Set (usually of letters) from which a Subset is drawn. 
A sequence of letters is called a WORD, and a set of 
Words is called a Code. 
see also CODE, WORD 



References 

Sallows, L. C. F. "Alphamagic Squares." Abacus 4, 28-45, 

1986. 
Sallows, L. C. F. "Alphamagic Squares. 2." Abacus 4, 20-29 

and 43, 1987. 
Sallows, L. C. F. "Alpha Magic Squares." In The Lighter 

Side of Mathematics (Ed. R. K. Guy and R, E. Woodrow). 

Washington, DC: Math. Assoc. Amer., 1994. 

Alphametic 

A CRYPTARITHM in which the letters used to represent 
distinct DIGITS are derived from related words or mean- 
ingful phrases. The term was coined by Hunter in 1955 
(Madachy 1979, p. 178). 

References 

Brooke, M. One Hundred & Fifty Puzzles in Crypt- 
Arithmetic. New York: Dover, 1963. 

Hunter, J. A. H. and Madachy, J. S. "Alphametics and the 
Like." Ch. 9 in Mathematical Diversions, New York: 
Dover, pp. 90-95, 1975. 

Madachy, J. S. "Alphametics." Ch. 7 in Madachy p s Mathe- 
matical Recreations. New York: Dover, pp. 178-200 1979. 

Alternate Algebra 

Let A denote an R-Algebra, so that A is a Vector 
Space over R and 

AxA^A (1) 

(x,y) \->x-y. (2) 

Then A is said to be alternate if, for all x,y £ A, 

(x-y)-y-x-(yy) (3) 

(x-x)-y = x-(x-y). (4) 

Here, VECTOR MULTIPLICATION x • y is assumed to be 
Bilinear. 

References 

Finch, S. "Zero Structures in Real Algebras." http://www. 

raathsof t . com/asolve/zerodiv/zerodiv .html. 
Schafer, R. D. An Introduction to Non- Associative Algebras. 

New York: Dover, 1995. 



36 Alternating Algebra 



Alternating Permutation 



Alternating Algebra 

see Exterior Algebra 

Alternating Group 

Even Permutation Groups A n which are Normal 
Subgroups of the Permutation Group of Order 
n!/2. They are Finite analogs of the families of sim- 
ple Lie GROUPS. The lowest order alternating group is 
60. Alternating groups with n > 5 are non-ABELIAN 
Simple Groups. The number of conjugacy classes in 
the alternating groups A n for n = 2, 3, . . . are 1, 3, 4, 
5, 7, 9, ... (Sloane's A000702). 

see also 15 Puzzle, Finite Group, Group, Lie 
Group, Simple Group, Symmetric Group 

References 

Sloane, N. J, A. Sequence A000702/M2307 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Wilson, R. A. "ATLAS of Finite Group Representation." 
http://for.mat.bham.ac.nk/atlas#alt. 

Alternating Knot 

An alternating knot is a KNOT which possesses a knot 
diagram in which crossings alternate between under- and 
overpasses. Not all knot diagrams of alternating knots 
need be alternating diagrams. 

The Trefoil Knot and Figure-of-Eight Knot are 
alternating knots. One of Tait's Knot Conjectures 
states that the number of crossings is the same for 
any diagram of a reduced alternating knot. Further- 
more, a reduced alternating projection of a knot has 
the least number of crossings for any projection of that 
knot. Both of these facts were proved true by Kauffman 
(1988), Thistlethwaite (1987), and Murasugi (1987). 

If K has a reduced alternating projection of n crossings, 
then the Span of K is An. Let c(K) be the Crossing 
Number. Then an alternating knot K±#K 2 (a Knot 
Sum) satisfies 



Erdener, K. and Flynn, R. "Rolfsen's Table of all Alter- 
nating Diagrams through 9 Crossings." ftp://chs.cusd. 
claremont . e du/pub/knot /Rolf sen_t able .final. 

Kauffman, L. "New Invariants in the Theory of Knots." 
Amer. Math. Monthly 95, 195-242, 1988. 

Murasugi, K. "Jones Polynomials and Classical Conjectures 
in Knot Theory." Topology 26, 297-307, 1987. 

Sloane, N. J. A. Sequence A002864/M0847 in "An On-Line 
Version of the Encyclopedia of Integer Sequences," 

Thistlethwaite, M. "A Spanning Tree Expansion for the Jones 
Polynomial." Topology 26, 297-309, 1987. 

Alternating Knot Diagram 

A Knot Diagram which has alternating under- and 
overcrossings as the KNOT projection is traversed. The 
first KNOT which does not have an alternating diagram 
has 8 crossings. 

Alternating Link 

A Link which has a Link Diagram with alternating 
underpasses and overpasses. 

see also Almost Alternating Link 

References 

Menasco, W. and Thistlethwaite, M. "The Classification of 
Alternating Links." Ann. Math. 138, 113-171, 1993. 

Alternating Permutation 

An arrangement of the elements ci, ..., c n such that 
no element a has a magnitude between a-\ and Ci + i is 
called an alternating (or Zigzag) permutation. The de- 
termination of the number of alternating permutations 
for the set of the first n INTEGERS {1, 2, ... , n} is known 
as Andre's Problem. An example of an alternating 
permutation is (1, 3, 2, 5, 4). 

As many alternating permutations among n elements 
begin by rising as by falling. The magnitude of the c n s 
does not matter; only the number of them. Let the 
number of alternating permutations be given by Z n = 
2A n . This quantity can then be computed from 



In fact, this is true as well for the larger class of Ade- 
quate KNOTS and postulated for all KNOTS. The num- 
ber of Prime alternating knots of n crossing for n = 1, 
2, . . . are 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, . . . (Sloane's 
A002864). 

see also ADEQUATE KNOT, ALMOST ALTERNATING 

Link, Alternating Link, Flyping Conjecture 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 
to the Mathematical Theory of Knots. New York: W. H. 
Freeman, pp. 159-164, 1994. 

Arnold, B.; Au, M.; Candy, C; Erdener, K.; Fan, J.; Flynn, 
R.; Muir, J.; Wu, D.; and Hoste, J. "Tabulating Alter- 
nating Knots through 14 Crossings." ftp://chs.cusd. 
claremont.edu/pub/knot/paper.TeX.txt and ftp://chs. 
cusd. claremont ,edu/pub/knot/AltKnots/. 



2na n 



J2 ar 



(1) 



where r and s pass through all INTEGRAL numbers such 
that 

r + 5==n _l ) (2) 



ao = a\ = 1, and 



A n = n\a n . 



(3) 



The numbers A n are sometimes called the EULER 
Zigzag Numbers, and the first few are given by 1, 1, 
1, 2, 5, 16, 61, 272, ... (Sloane's A000111). The Odd- 
numbered A n s are called Euler Numbers, Secant 
Numbers, or Zig Numbers, and the EvEN-numbered 
ones are sometimes called TANGENT NUMBERS or ZAG 
Numbers. 



Alternating Series 



Altitude 37 



Curiously enough, the SECANT and TANGENT MAC- 
LAURIN SERIES can be written in terms of the A n s as 



X X 

sec x = A + A 2 — - + A 4 — + . . 
2! 4! 

X X 

tan x = AiX + A 3 — - + A 5 — - + . 
o! 5! 



(4) 
(5) 



or combining them, 



sec x + tan x 

t 2 r 3 r 4 



A x 

-A 5 - + .. 



(6) 



see also Entringer Number, Euler Number, Eu- 
ler Zigzag Number, Secant Number, Seidel- 
Entringer-Arnold Triangle, Tangent Number 

References 

Andre, D. "Developments de seccc et tan a?." C. R. Acad. 
Sci. Paris 88, 965-967, 1879. 

Andre, D. "Memoire sur le permutations alternees." J. Math. 
7, 167-184, 1881. 

Arnold, V. I. "Bernoulli-Euler Updown Numbers Associ- 
ated with Function Singularities, Their Combinatorics and 
Arithmetics." Duke Math. J. 63, 537-555, 1991. 

Arnold, V. I. "Snake Calculus and Combinatorics of Ber- 
noulli, Euler, and Springer Numbers for Coxeter Groups." 
Russian Math. Surveys 47, 3-45, 1992. 

Bauslaugh, B. and Ruskey, F. "Generating Alternating Per- 
mutations Lexicographically." BIT 30, 17-26, 1990. 

Conway, J. H. and Guy, R. K. In The Book of Numbers. New 
York: Springer- Verlag, pp. 110-111, 1996. 

Dorrie, H. "Andre's Deviation of the Secant and Tangent 
Series." §16 in 100 Great Problems of Elementary Math- 
ematics: Their History and Solutions. New York: Dover, 
pp. 64-69, 1965. 

Honsberger, R. Mathematical Gems III. Washington, DC: 
Math. Assoc. Amer., pp. 69-75, 1985. 

Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, 
Euler, and Bernoulli Numbers." Math. Comput. 21, 663- 
688, 1967. 

Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op- 
eration on Sequences: The Boustrophedon Transform." J. 
Combin. Th. Ser. A 76, 44-54, 1996. 

Ruskey, F. "Information of Alternating Permutations." 

http:// sue . esc . uvic . ca / - cos / inf / perm / 
Alternat ing . html. 

Sloane, N. J. A. Sequence A000111/M1492 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Alternating Series 

A Series of the form 



k=l 

00 

D-d 



a k 



ajt. 



see also SERIES 



References 

Arfken, G. "Alternating Series." §5.3 in Mathematical Meth- 
ods for Physicists, 3rd ed. Orlando, FL: Academic Press, 
pp. 293-294, 1985. 

Bromwich, T. J. Pa and MacRobert, T. M. "Alternating Se- 
ries." §19 in An Introduction to the Theory of Infinite 
Series, 3rd ed. New York: Chelsea, pp. 55-57, 1991. 

Pinsky, M. A. "Averaging an Alternating Series." Math. 
Mag. 51, 235-237, 1978. 

Alternating Series Test 

Also known as the Leibniz Criterion. An Alternat- 
ing Series Converges if a± > a 2 > . . . and 



lim ak = 0. 



see also CONVERGENCE TESTS 

Alternative Link 

A category of Link encompassing both ALTERNATING 
Knots and Torus Knots. 

see also Alternating Knot, Link, Torus Knot 
References 

Kauffman, L. "Combinatorics and Knot Theory." Contemp. 
Math. 20, 181-200, 1983. 

Altitude 




A r H 3 A 2 

The altitudes of a TRIANGLE are the Cevians AiHi 
which are Perpendicular to the Legs AjAk opposite 
Ai. They have lengths hi = AiHi given by 

hi = at+i sinai+2 = ^+2 sinaii+i 



hi = 



2^/s(s — ai)(s — 0,2) {s — as) 



where s is the Semiperimeter and a% 
interesting FORMULA is 

hihzhz = 2sA 



AiA k 



(1) 

(2) 
Another 

(3) 



(Johnson 1929, p. 191), where A is the Area of the Tri- 
angle. The three altitudes of any TRIANGLE are CON- 
CURRENT at the ORTHOCENTER H. This fundamental 
fact did not appear anywhere in Euclid's Elements. 

Other formulas satisfied by the altitude include 



_1_ 1_ l_ _ 1 
h\ h? /13 v 



(4) 



38 Alysoid 



Amicable Numbers 





1 


= 


1 

h~ 2 


+ 


1 


1 

hx~ 


1 

r 2 


+ 


1 


= 


1 

r 


1 


2 

" hx 



(5) 



(6) 



where r is the INRADIUS and n are the Exradii (John- 
son 1929, p. 189). In addition, 

HA 1 • HHi = HA 2 • HH 2 = HA Z . HH 3 (7) 

Jf Ai • HHi = |(ai 2 + a 2 2 + a 3 2 ) - 4# 2 , (8) 
where R is the ClRCUMRADlUS. 




The points Ai, A 3 , #i, and H 3 (and their permuta- 
tions with respect to indices) all lie on a Circle, as 
do the points A3, Hz, H, and Hi (and their permuta- 
tions with respect to indices). TRIANGLES AA1A2A3 
and AA\H 2 H 3 are inversely similar. 

The triangle H±H 2 H 3 has the minimum PERIMETER 
of any TRIANGLE inscribed in a given Acute TRIAN- 
GLE (Johnson 1929, pp. 161-165). The PERIMETER of 
AHxH 2 H 3 is 2A/R (Johnson 1929, p. 191). Additional 
properties involving the Feet of the altitudes are given 
by Johnson (1929, pp. 261-262). 

see also Cevian, Foot, Orthocenter, Perpendicu- 
lar, Perpendicular Foot 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 

Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40, 

1967. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, 1929. 

Alysoid 

see Catenary 

Ambient Isotopy 

An ambient isotopy from an embedding of a MANIFOLD 
M in N to another is a Homotopy of self Diffeomor- 
phisms (or Isomorphisms, or piecewise-linear transfor- 
mations, etc.) of JV, starting at the IDENTITY Map, such 
that the "last" DlFFEOMORPHISM compounded with the 
first embedding of M is the second embedding of M. 
In other words, an ambient isotopy is like an Isotopy 
except that instead of distorting the embedding, the 
whole ambient SPACE is being stretched and distorted 
and the embedding is just "coming along for the ride." 



For Smooth Manifolds, a Map is Isotopic Iff it is 
ambiently isotopic. 

For KNOTS, the equivalence of MANIFOLDS under con- 
tinuous deformation is independent of the embedding 
Space. Knots of opposite Chirality have ambient 
isotopy, but not REGULAR ISOTOPY. 
see also ISOTOPY, REGULAR ISOTOPY 

References 

Hirsch, M. W. Differential Topology. New York: Springer- 
Verlag, 1988. 

Ambiguous 

An expression is said to be ambiguous (or poorly de- 
fined) if its definition does not assign it a unique inter- 
pretation or value. An expression which is not ambigu- 
ous is said to be Well-Defined. 

see also Well-Defined 

Ambrose-Kakutani Theorem 

For every ergodic Flow on a nonatomic PROBABILITY 
Space, there is a Measurable Set intersecting almost 
every orbit in a discrete set. 

Amenable Number 

A number n which can be built up from INTEGERS ax, 

a 2 , . . . , afc by either ADDITION or MULTIPLICATION such 

that 

k k 

/ a i — \\ a i — n - 
i=x i=X 

The numbers {ai, . . . , a n } in the Sum are simply a Par- 
tition of n. The first few amenable numbers are 

2+2=2x2=4 
1+2+3= 1x2x3=6 

1+1+2+4=1x1x2x4=8 
1 + 1 + 2 + 2 + 2 = 1x1x2x2x2 = 8. 

In fact, all COMPOSITE NUMBERS are amenable. 

See also COMPOSITE NUMBER, PARTITION, SUM 

References 

Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102, 
463, 1995. 

Amicable Numbers 

see Amicable Pair, Amicable Quadruple, Amica- 
ble Triple, Multiamicable Numbers 



Amicable Pair 



Amicable Pair 39 



Amicable Pair 

An amicable pair consists of two Integers m,n for 
which the sum of PROPER DIVISORS (the DIVISORS ex- 
cluding the number itself) of one number equals the 
other. Amicable pairs are occasionally called FRIENDLY 
Pairs, although this nomenclature is to be discouraged 
since FRIENDLY PAIRS are defined by a different, if re- 
lated, criterion. Symbolically, amicable pairs satisfy 



s(m) — n 
s(n) = m, 



(i) 

(2) 



where s(n) is the RESTRICTED Divisor FUNCTION or, 
equivalently, 

cr(m) = cr(n) = s(m) + s(n) = m -f n, (3) 

where <x(n) is the DIVISOR FUNCTION. The smallest 
amicable pair is (220, 284) which has factorizations 



220= 11-5-2^ 
284 = 71 • 2 2 

giving RESTRICTED DIVISOR FUNCTIONS 

s(220) = ^{1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} 

= 284 
S (284) = ^{1,2,4,71,142} 

= 220. 



(4) 
(5) 



The quantity 



<r{m) = cr(n) — s(m) + s(n). 



(6) 
(7) 

(8) 



in this case, 220 + 284 = 504, is called the Pair Sum. 

In 1636, Fermat found the pair (17296, 18416) and in 
1638, Descartes found (9363584, 9437056). By 1747, 
Euler had found 30 pairs, a number which he later ex- 
tended to 60. There were 390 known as of 1946 (Scott 
1946). There are a total of 236 amicable pairs below 
10 8 (Cohen 1970), 1427 below 10 10 (te RhI • 1 ^6), 3340 
less than 10 11 (Moews and Moew? 1 r "3), J' ,ess than 
2.01 x 10 11 (Moews and Moe^ _., < .d 5001 .ess than 
ft* 3.06 x 10 11 (Moews and Moews). 

The first few amicable pairs are (2, 0, 284), (1184, 1210), 
(2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), 
(12285, 14595), (17296 : u -116), (63020, 76084), ... 
(Sloane's A002025 and AQ02046). An exhaustive tab- 
ulation is maintained by D. Moevvo. 

Let an amicable pair be denoted (m, n) with m < n. 
(m,n) is called a regular amicable pair of type (i, j) if 



(m,n) = (gM,gN), 



(9) 



where g = GCD(m,n) is the Greatest Common DI- 
VISOR, 

GCD( 5 ,M) = GCD{g,N) = 1, (10) 

M and N are SQUAREFREE, then the number of Prime 
factors of M and N are i and j. Pairs which are not 
regular are called irregular or exotic (te Riele 1986). 
There are no regular pairs of type (l,j) for j > 1. If 
m = (mod 6) and 



n = cr(m) — m 



(ii) 



is Even, then (m,n) cannot be an amicable pair (Lee 
1969). The minimal and maximal values of m/n found 
by te Riele (1986) were 



938304290/1344480478 = 0.697893577. . . (12) 



and 



4000783984/4001351168 = 0.9998582519 .... (13) 

te Riele (1986) also found 37 pairs of amicable pairs hav- 
ing the same Pair Sum. The first such pair is (609928, 
686072) and (643336, 652664), which has the Pair Sum 



a(m) = cr(n) = m + n = 1,296,000. 



(14) 



te Riele (1986) found no amicable n-tuples having the 
same Pair Sum for n > 2. However, Moews and 
Moews found a triple in 1993, and te Riele found 
a quadruple in 1995. In November 1997, a quin- 
tuple and sextuple were discovered. The sextuple 
is (1953433861918, 2216492794082), (1968039941816, 
2201886714184), (1981957651366, 2187969004634), 
(1993501042130, 2176425613870), (2046897812505, 
2123028843495), (2068113162038, 2101813493962), all 
having PAIR SUM 4169926656000. Amazingly, the sex- 
tuple is smaller than any known quadruple or quintuple, 
and is likely smaller than any quintuple. 

On October 4, 1997, Mariano Garcia found the largest 
known amicable pair, each of whose members has 4829 
Digits. The new pair is 

N x = CM[(P + Q)P 89 - 1] (15) 

N 2 - CQ[(P ~ M)P S9 - 1], (16) 

where 



C = 2 1X P 89 (17) 

M = 287155430510003638403359267 (18) 
P = 574451143340278962374313859 (19) 
Q = 136272576607912041393307632916794623. 

(20) 

P, Q, (P + Q)P 89 - 1, and (P - M)P 89 - 1 are Prime. 



40 



Amicable Pair 



Amicable Triple 



Pomerance (1981) has proved that 

[amicable numbers < n] < ne~^ n ^ J 



(21) 



for large enough n (Guy 1994). No nonfinite lower 
bound has been proven. 

see also Amicable Quadruple, Amicable Triple, 
Augmented Amicable Pair, Breeder, Crowd, Eu- 
ler's Rule, Friendly Pair, Multiamicable Num- 
bers, Pair Sum, Quasiamicable Pair, Sociable 
Numbers, Unitary Amicable Pair 

References 

Alanen, J.; Ore, 0.; and Stemple, J. "Systematic Computa- 
tions on Amicable Numbers." Math. Comput. 21, 242— 
245, 1967. 

Battiato, S. and Borho, W. "Are there Odd Amicable Num- 
bers not Divisible by Three?" Math. Comput. 50, 633- 
637, 1988. 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 62 in 
HAKMEM. Cambridge, MA: MIT Artificial Intelligence 
Laboratory, Memo AIM-239, Feb. 1972. 

Borho, W. and Hoffmann, H. "Breeding Amicable Numbers 
in Abundance." Math. Comput 46, 281-293, 1986. 

Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers 
and Their Distribution." Math. Comput. 24, 431-432, 
1970. 

Cohen, H. "On Amicable and Sociable Numbers." Math. 
Comput. 24, 423-429, 1970. 

Costello, P. "Amicable Pairs of Euler's First Form." J. Rec. 
Math. 10, 183-189, 1977-1978. 

Costello, P. "Amicable Pairs of the Form (i,l)." Math. Com- 
put. 56, 859-865, 1991. 

Dickson, L. E. History of the Theory of Numbers, Vol. 1: 
Divisibility and Primality. New York: Chelsea, pp. 38-50, 
1952. 

Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4, 
108-111, 1955-1956. 

Erdos, P. "On Asymptotic Properties of Aliquot Sequences." 
Math. Comput. 30, 641-645, 1976. 

Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math- 
ematical Magic Show: More Puzzles, Games, Diversions, 
Illusions and Other Mathematical Sleight-of~Mind from 
Scientific American. New York: Vintage, pp. 160-171, 
1978. 

Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems 
in Number Theory, 2nd ed. New York: Springer- Verlag, 
pp. 55-59, 1994. 

Lee, E. J. "Amicable Numbers and the Bilinear Diophantine 
Equation." Math. Comput. 22, 181-197, 1968. 

Lee, E. J. "On Divisibility of the Sums of Even Amicable 
Pairs." Math. Comput. 23, 545-548, 1969. 

Lee, E. J. and Madachy, J. S. "The History and Discovery of 
Amicable Numbers, 1." J. Rec. Math. 5, 77-93, 1972. 

Lee, E. J. and Madachy, J. S. "The History and Discovery of 
Amicable Numbers, II." J. Rec. Math. 5, 153-173, 1972. 

Lee, E. J. and Madachy, J. S. "The History and Discovery of 
Amicable Numbers, HI." J. Rec. Math. 5, 231-249, 1972. 

Madachy, J. S. Madachy's Mathematical Recreations. New 
York: Dover, pp. 145 and 155-156, 1979. 

Moews, D. and Moews, P. C. "A Search for Aliquot Cycles 
and Amicable Pairs." Math. Comput. 61, 935-938, 1993. 

Moews, D. and Moews, P. C. "A List of Amicable Pairs Below 
2.01 x 10 u ." Rev. Jan. 8, 1993. http://xraysgi.ims. 
uconn . edu : 8080/amicable . txt . 

Moews, D. and Moews, P. C. "A List of the First 5001 Am- 
icable Pairs." Rev. Jan. 7, 1996. http://xraysgi.ims. 
uconn.edu: 8080/amicable2. txt. 



Ore, 0. Number Theory and Its History. New York: Dover, 
pp. 96-100, 1988. 

Pedersen, J. M. "Known Amicable Pairs." http://www. 
vejlehs.dk/staff/jmp/aliquot/knwnap.htm. 

Pomerance, C. "On the Distribution of Amicable Numbers." 
J. reine angew. Math. 293/294, 217-222, 1977. 

Pomerance, C. "On the Distribution of Amicable Numbers, 
II." J. reine angew. Math. 325, 182-188, 1981. 

Scott, E. B. E. "Amicable Numbers." Scripta Math. 12, 
61-72, 1946. 

Sloane, N. J. A. Sequences A002025/M5414 and A002046/ 
M5435 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

te Riele, H. J. J. "On Generating New Amicable Pairs from 
Given Amicable Pairs." Math. Comput. 42, 219-223, 
1984. 

te Riele, H. J. J. "Computation of All the Amicable Pairs 
Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35, 
1986. 

te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.; 
and Lee, E. J. "Table of Amicable Pairs Between 10 x and 
10 52 ." Centrum voor Wiskunde en Informatica, Note NM- 
N8603. Amsterdam: Stichting Math. Centrum, 1986. 

te Riele, H. J. J. "A New Method for Finding Amicable 
Pairs." In Mathematics of Computation 1943-1993: A 
Half-Century of Computational Mathematics (Vancouver, 
BC, August 9-13, 1993) (Ed. W. Gautschi). Providence, 
Rl: Amer. Math. Soc, pp. 577-581, 1994. 
$$ Weisstein, E. W. "Sociable and Amicable Num- 
bers." http : //www . astro . Virginia, edu/ -eww6n/math/ 
notebooks/Sociable .m. 

Amicable Quadruple 

An amicable quadruple as a QUADRUPLE (a, b, c, d) such 
that 

a(a) = a(b) — a(c) — cr(d) — a + b + c + d, 



where cr(n) is the DIVISOR FUNCTION. 

References 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, p. 59, 1994. 



Amicable Triple 

Dickson (1913, 1952) defined an amicable triple to be a 

TRIPLE of three numbers (Z,m, n) such that 

s(/) = m + n 
${m) = I + n 
s(n) = / + m, 

where s(n) is the Restricted Divisor Function 
(Madachy 1979). Dickson (1913, 1952) found eight sets 
of amicable triples with two equal numbers, and two 
sets with distinct numbers. The latter are (123228768, 
103340640, 124015008), for which 

s(12322876) = 103340640 + 124015008 = 227355648 
s(103340640) = 123228768 + 124015008 = 24724377 
5(124015008) = 123228768 + 10334064 = 226569408, 



Amortization 



Amplitude 41 



and (1945330728960, 2324196638720, 2615631953920), 
for which 

s(1945330728960) = 2324196638720+2615631953920 

= 4939828592640 
s(2324196638720) = 1945330728960 + 2615631953920 

= 4560962682880 
5(2615631953920) = 1945330728960 + 2324196638720 

= 4269527367680. 



A second definition (Guy 1994) defines an amicable 
triple as a TRIPLE (a, &, c) such that 

a (a) = a(b) — o~(c) = a + b + c, 

where a(n) is the DIVISOR FUNCTION. An example is 

(2 2 3 2 5- 11, 2 5 3 2 7, 2 2 3 2 71). 

see also Amicable Pair, Amicable Quadruple 

References 

Dickson, L. E. "Amicable Number Triples." Amer. Math. 
Monthly 20, 84-92, 1913. 

Dickson, L. E. History of the Theory of Numbers, Vol. 1: 
Divisibility and Primality. New York: Chelsea, p. 50, 1952. 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, p. 59, 1994. 

Madachy, J. S. Madachy's Mathematical Recreations. New 
York: Dover, p. 156, 1979. 

Mason, T. E. "On Amicable Numbers and Their Generaliza- 
tions." Amer, Math. Monthly 28, 195-200, 1921. 
$$ Weisstein, E. W. "Sociable and Amicable Num- 
bers." http : //www . astro . Virginia . edu/~eww6n/math/ 
notebooks/Sociable .m. 

Amortization 

The payment of a debt plus accrued INTEREST by regu- 
lar payments. 



Ampersand Curve 




The Plane CURVE with Cartesian equation 



(y 2 - x 2 ){x - l)(2a> - 3) = 4(z 2 + y 2 - 2x) 2 . 



References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., p. 72, 1989, 



Amphichiral 

An object is amphichiral (also called Reflexible) if it 
is superposable with its Mirror Image (i.e., its image 
in a plane mirror). 

see also Amphichiral Knot, Chiral, Disymmetric, 
Handedness, Mirror Image 

Amphichiral Knot 

An amphichiral knot is a Knot which is capable of be- 
ing continuously deformed into its own MIRROR IMAGE. 
The amphichiral knots having ten or fewer crossings are 
04 O oi (Figure-of-Eight Knot), O6003, O8003, O8009, 
08oi2j 08oi7j O8018) 10oi7,10o33, IO037, IO043, 10o45, 
10o79, IO081, IO088, IO099, IO109, IO115, IO118, and IO123 
(Jones 1985). The HOMFLY Polynomial is good at 
identifying amphichiral knots, but sometimes fails to 
identify knots which are not. No complete invariant (an 
invariant which always definitively determines if a Knot 
is Amphichiral) is known. 

Let 6+ be the Sum of Positive exponents, and 6_ the 
Sum of Negative exponents in the Braid Group B n . 
If 

b + - 3b- - n + 1 > 0, 

then the Knot corresponding to the closed BRAID b is 
not amphichiral (Jones 1985), 

see also Amphichiral, Braid Group, Invertible 
Knot, Mirror Image 

References 

Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 
pp. 311-319, 1985. 

Jones, V. "A Polynomial Invariant for Knots via von Neu- 
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 
1985. 

Jones, V. "Hecke Algebra Representations of Braid Groups 
and Link Polynomials." Ann. Math. 126, 335-388, 1987. 

Amplitude 

The variable <j> used in ELLIPTIC FUNCTIONS and EL- 
LIPTIC Integrals, which can be defined by 



= / dnudu, 



where dn(u) is a JACOBI ELLIPTIC FUNCTION. The term 
"amplitude" is also used to refer to the maximum offset 
of a function from its baseline level. 

see also Argument (Elliptic Integral), Charac- 
teristic (Elliptic Integral), Delta Amplitude, 
Elliptic Function, Elliptic Integral, Jacobi El- 
liptic Functions, Modular Angle, Modulus (El- 
liptic Integral), Nome, Parameter 

References 



Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 590, 1972. 

Fischer, G. (Ed.). Plate 132 in Mathematische Mod- 
elle/ Mathematical Models, Bildband/ Photograph Volume. 
Braunschweig, Germany: Vieweg, p. 129, 1986. 



42 Anallagmatic Curve 



Anchor 



Anallagmatic Curve 

A curve which is invariant under" Inversion. Exam- 
ples include the Cardioid, Cartesian Ovals, Cassini 

Ovals, Limaqon, Strophoid, and Maclaurin Tri- 

SECTRIX. 

Anallagmatic Pavement 

see Hadamard Matrix 

Analogy 

Inference of the Truth of an unknown result obtained 
by noting its similarity to a result already known to be 
TRUE. In the hands of a skilled mathematician, anal- 
ogy can be a very powerful tool for suggesting new and 
extending old results. However, subtleties can render re- 
sults obtained by analogy incorrect, so rigorous PROOF 
is still needed. 

see also INDUCTION 

Analysis 

The study of how continuous mathematical structures 

(Functions) vary around the Neighborhood of a 
point on a Surface. Analysis includes Calculus, Dif- 
ferential Equations, etc. 

see also Analysis Situs, Calculus, Complex Anal- 
ysis, Functional Analysis, Nonstandard Analy- 
sis, Real Analysis 

References 

Bottazzini, U. The "Higher Calculus": A History of Real and 

Complex Analysis from Euler to Weierstraft. New York: 

Springer-Verlag, 1986. 
Bressoud, D. M. A Radical Approach to Real Analysis. 

Washington, DC: Math. Assoc. Amer., 1994. 
Ehrlich, P. Real Numbers, Generalization of the Reals, & 

Theories of Continua. Norwell, MA: Kluwer, 1994. 
Hairer, E. and Wanner, G. Analysis by Its History. New 

York: Springer-Verlag, 1996. 
Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 

1988. 
Wheeden, R. L. and Zygmund, A. Measure and Integral: An 

Introduction to Real Analysis. New York: Dekker, 1977. 
Whittaker, E. T. and Watson, G. N. A Course in Modern 

Analysis, J^th ed. Cambridge, England: Cambridge Uni- 
versity Press, 1990. 



Analytic Function 

A Function in the Complex Numbers C is analy- 
tic on a region R if it is COMPLEX DlFFERENTIABLE 
at every point in R. The terms HOLOMORPHIC FUNC- 
TION and Regular Function are sometimes used in- 
terchangeably with "analytic function." If a Function 
is analytic, it is infinitely DlFFERENTIABLE. 

see also BERGMAN SPACE, COMPLEX DlFFERENTIABLE, 
DlFFERENTIABLE, PSEUDOANALYTIC FUNCTION, SEMI- 
ANALYTIC, SUBANALYTIC 

References 

Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2 
in Methods of Theoretical Physics, Part I. New York: 
McGraw-Hill, pp. 356-374, 1953. 

Analytic Geometry 

The study of the GEOMETRY of figures by algebraic rep- 
resentation and manipulation of equations describing 
their positions, configurations, and separations. Ana- 
lytic geometry is also called Coordinate Geometry 
since the objects are described as n-tuples of points 
(where n = 2 in the PLANE and 3 in Space) in some 
Coordinate System. 

see also Argand Diagram, Cartesian Coordinates, 
Complex Plane, Geometry, Plane, Quadrant, 
Space, x-Axis, y-Axis, z-Axis 

References 

Courant, R. and Robbins, H. "Remarks on Analytic Geome- 
try." §2.3 in What is Mathematics?: An Elementary Ap- 
proach to Ideas and Methods, 2nd ed. Oxford, England: 
Oxford University Press, pp. 72-77, 1996. 

Analytic Set 

A Definable Set, also called a Souslin Set. 

see also COANALYTIC SET, SOUSLIN Set 

Anarboricity 

Given a Graph G, the anarboricity is the maximum 
number of line- disjoint nonacyclic SUBGRAPHS whose 
UNION is G. 

see also ARBORICITY 



Analysis Situs 

An archaic name for TOPOLOGY. 

Analytic Continuation 

A process of extending the region in which a COMPLEX 
FUNCTION is defined. 

see also Monodromy Theorem, Permanence of Al- 
gebraic Form, Permanence of Mathematical Re- 
lations Principle 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 378-380, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 389-390 and 392- 
398, 1953. 



Anchor 

An anchor is the Bundle Map p from a Vector Bun- 
dle A to the Tangent Bundle TB satisfying 

1. [p(X),p(Y)] = p([X,r])and 

2. [x,0y] = 0[x,y] + ( P (x).0)y, 

where X and Y are smooth sections of A, <j> is a 
smooth function of B, and the bracket is the "Jacobi-Lie 
bracket" of a VECTOR FIELD. 

see also Lie Algebroid 

References 

Weinstein, A. "Groupoids: Unifying Internal and External 
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996. 



Anchor Ring 



Andrews-Schur Identity 43 



Anchor Ring 

An archaic name for the TORUS. 

References 

Eisenhart, h. P. A Treatise on the Differential Geometry of 

Curves and Surfaces. New York: Dover, p. 314, 1960. 
Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, 

p. 239, 1977. 
Whittaker, E. T. A Treatise on the Analytical Dynamics of 

Particles & Rigid Bodies, J^th ed. Cambridge, England: 

Cambridge University Press, p. 21, 1959. 

And 

A term (PREDICATE) in LOGIC which yields TRUE if one 
or more conditions are TRUE, and FALSE if any condi- 
tion is False. A AND B is denoted Ak,B, A A B, or 
simply AB. The Binary AND operator has the follow- 
ing Truth Table: 



A 


B 


AAB 


F 


F 


F 


F 


T 


F 


T 


F 


F 


T 


T 


T 



A PRODUCT of ANDs (the AND of n conditions) is 
called a CONJUNCTION, and is denoted 



A*- 



Andre's Reflection Method 

A technique used by Andre (1887) to provide an elegant 
solution to the BALLOT PROBLEM (Hilton and Pederson 
1991). 

References 

Andre, D. "Solution directe du probleme resohi par 
M, Bertrand." Comptes Rendus Acad. Sci. Paris 105, 
436-437, 1887. 

Comtet, L. Advanced Combinatorics. Dordrecht, Nether- 
lands: Reidel, p. 22, 1974. 

Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener- 
alization, and Their Uses." Math. Intel. 13, 64-75, 1991. 

Vardi, I. Computational Recreations in Mathematica. Read- 
ing, MA: Addison- Wesley, p. 185, 1991. 

Andrew's Sine 

The function 



*(*)■■ 



{sin 
o, 



(f) 



< C7T 
> C7T 



which occurs in estimation theory. 
see also SlNE 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 697, 1992. 



Two binary numbers can have the operation AND per- 
formed bitwise with 1 representing TRUE and FALSE. 
Some computer languages denote this operation on A, 
B, and C as A&&B&&C or logand(A,B,C). 

see also BINARY OPERATOR, INTERSECTION, NOT, OR, 

Predicate, Truth Table, XOR 

Anderson-Darling Statistic 

A statistic defined to improve the Kolmogorov- 
SMIRNOV TEST in the TAIL of a distribution. 

see also Kolmogorov-Smirnov Test, Kuiper 
Statistic 

References 

Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 621, 1992. 

Andre's Problem 

The determination of the number of ALTERNATING PER- 
MUTATIONS having elements {1, 2, . . . , n} 

see also ALTERNATING PERMUTATION 



Andrews Cube 

see Semiperfect Magic Cube 

Andrews- Curtis Link 

The Link of 2-spheres in M 4 obtained by Spinning in- 
tertwined arcs. The link consists of a knotted 2-sphere 
and a Spun Trefoil Knot. 
see also Spun Knot, Trefoil Knot 



References 

Rolfsen, D. Knots and Links. 
Perish Press, p. 94, 1976. 



Wilmington, DE: Publish or 



Andrews-Schur Identity 



£« fc2+a 



fc=0 



2n — k + a 
k 



_ V~^ 10fc 2 + (4a-l)fc 



2n + 2a + 2 

n — 5k 



[lOfc + 2a + 2] 
[2n -r 2a + 2] ' 



(1) 



44 Andrica's Conjecture 



Anger Function 



where [x] is a GAUSSIAN POLYNOMIAL. It is a POLY- 
NOMIAL identity for a = 0, 1 which implies the Ro.GERS- 
Ramanujan Identities by taking n -t oo and apply- 
ing the Jacobi Triple Product identity. A variant of 
this equation is 



£ - 

fc=-|_a/2j 



k 2 +2ak 



n 4- k + a 
n — k 



|n/5j 
-L(n+2a+2)/5j 



15fc 2 +(6a+l)fc 



2n + 2a + 2 
5-5/z 

[10A; + 2a + 2] 
[2n 4- 2a + 2] ' 



(2) 



where the symbol [xj in the Sum limits is the Floor 
Function (Paule 1994). The Reciprocal of the iden- 
tity is 

00 k 2 +2ak 

Z^ in- 



(kq) 



2fc+a 



11(1 -q- 

3 = 



1 



2j + l)(1 _ g20j+4a+4)(l _ g20j-4a+16) 



(3) 



for a = 0, 1 (Paule 1994). For g = 1, (1) and (2) become 



£ 

-La/2j 



n + A; -J- a 
n — k 



[n/5j 

£ 

-|_(n+2a+2)/5j 



2n + 2a + 2\ 5fc + a + 1 



n — 5A; 



n + a + 1 



(4) 



References 

Andrews, G. E. "A Polynomial Identity which Implies the 
Rogers-Ramanujan Identities." Scripta Math. 28, 297— 
305, 1970. 

Paule, P. "Short and Easy Computer Proofs of the Rogers- 
Ramanujan Identities and of Identities of Similar Type." 
Electronic J. Combinatorics 1, RIO, 1-9, 1994. http:// 
www. combinatorics . org/Volume JYvolumel .html#R10. 

Andrica's Conjecture 




100 200 300 400 500 

Andrica's conjecture states that, for p n the nth PRIME 
Number, the Inequality 

A n = ^/Pn+l — \/Pn < 1 



holds, where the discrete function A n is plotted above. 
The largest value among the first 1000 PRIMES is for 
n = 4, giving y/u. - \ft « 0.670873. Since the Andrica 
function falls asymptotically as n increases so a PRIME 
Gap of increasing size is needed at large n, it seems 
likely the CONJECTURE is true. However, it has not yet 
been proven. 




100 200 300 400 500 

An bears a strong resemblance to the PRIME DIFFER- 
ENCE Function, plotted above, the first few values of 
which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, . . . (Sloane's 
A001223). 

see also Brocard's Conjecture, Good Prime, For- 
tunate Prime, Polya Conjecture, Prime Differ- 
ence Function, Twin Peaks 

References 

Golomb, S. W. "Problem E2506: Limits of Differences of 
Square Roots." Amer. Math. Monthly 83, 60-61, 1976. 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, p. 21, 1994. 

Rivera, C. "Problems & Puzzles (Conjectures): An- 
drica's Conjecture." http://www.sci.net.mx/-crivera/ 
ppp/conj _008 . htm. 

Sloane, N. J. A. Sequence A001223/M0296 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Anger Function 

A generalization of the Bessel Function OF the 
First Kind defined by 



Mz) 



-tf 



cos(v9 — zsinO) dQ. 



If v is an INTEGER n, then J n (z) = J n (z), where J n (z) 
is a Bessel Function of the First Kind. Anger's 
original function had an upper limit of 27T, but the cur- 
rent Notation was standardized by Watson (1966). 

see also BESSEL FUNCTION, MODIFIED STRUVE FUNC- 
TION, Parabolic Cylinder Function, Struve 
Function, Weber Functions 

References 

Abramowitz, M. and Stegun, C A. (Eds.). "Anger and We- 
ber Functions." §12.3 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 498-499, 1972. 

Watson, G. N. A Treatise on the Theory of Bessel Functions, 
2nd ed. Cambridge, England: Cambridge University Press, 
1966. 



Angle 

Angle 



Angle Bracket 45 




Given two intersecting Lines or Line Segments, the 
amount of ROTATION about the point of intersection 
(the Vertex) required to bring one into correspondence 
with the other is called the angle 6 between them. An- 
gles are usually measured in Degrees (denoted °), Ra- 
dians (denoted rad, or without a unit), or sometimes 
Gradians (denoted grad). 

One full rotation in these three measures corresponds to 
360°, 27r rad, or 400 grad. Half a full ROTATION is called 
a Straight Angle, and a Quarter of a full rotation 
is called a RIGHT ANGLE. An angle less than a RIGHT 
Angle is called an Acute Angle, and an angle greater 
than a Right Angle is called an Obtuse Angle. 

The use of Degrees to measure angles harks back to 
the Babylonians, whose SEXAGESIMAL number system 
was based on the number 60. 360° likely arises from the 
Babylonian year, which was composed of 360 days (12 
months of 30 days each). The DEGREE is further divided 

into 60 Arc Minutes, and an Arc Minute into 60 
Arc Seconds. A more natural measure of an angle is 
the Radian. It has the property that the Arc Length 
around a CIRCLE is simply given by the radian angle 
measure times the Circle Radius. The Radian is also 
the most useful angle measure in CALCULUS because the 
Derivative of Trigonometric functions such as 



dx 

does not require the insertion of multiplicative constants 
like 7r/180. GRADIANS are sometimes used in surveying 
(they have the nice property that a Right Angle is ex- 
actly 100 Gradians), but are encountered infrequently, 
if at all, in mathematics. 

The concept of an angle can be generalized from the 
Circle to the Sphere. The fraction of a Sphere sub- 
tended by an object is measured in StERADIANS, with 
the entire Sphere corresponding to 4n Steradians. 

A ruled Semicircle used for measuring and drawing 
angles is called a Protractor. A Compass can also 
be used to draw circular ARCS of some angular extent. 
see also Acute Angle, Arc Minute, Arc Second, 
Central Angle, Complementary Angle, Degree, 
Dihedral Angle, Directed Angle, Euler Angles, 
Gradian, Horn Angle, Inscribed Angle, Oblique 
Angle, Obtuse Angle, Perigon, Protractor, 
Radian, Right Angle, Solid Angle, Steradian, 
Straight Angle, Subtend, Supplementary Angle, 
Vertex Angle 



References 

Dixon, R. Mathographics. 
1991. 



Angle Bisector 

interior angle 
bisector 




exterior angle 
^ bisection 



The (interior) bisector of an Angle is the LINE or Line 
Segment which cuts it into two equal Angles on the 
same "side" as the Angle. 




Ai h A 2 

The length of the bisector of Angle A± in the above 
Triangle AA!A 2 A 3 is given by 

,, 2 



ti 



a 2 a% 



ax 



(a 2 +a 3 ) 2 



where U = A& and a\ = AjA^. The angle bisectors 
meet at the Incenter J, which has Trilinear Coor- 
dinates 1:1:1. 

see also Angle Bisector Theorem, Cyclic Quad- 
rangle, Exterior Angle Bisector, Isodynamic 
Points, Orthocentric System, Steiner-Lehmus 
Theorem, Trisection 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 
Washington, DC: Math. Assoc. Amer., pp. 9-10, 1967. 

Dixon, R. Mathographics. New York: Dover, p. 19, 1991. 

Mackay, J. S. "Properties Concerned with the Angular Bi- 
sectors of a Triangle." Proc. Edinburgh Math. Soc. 13, 
37-102, 1895. 

Angle Bisector Theorem 

The Angle Bisector of an Angle in a Triangle di- 
vides the opposite side in the same RATIO as the sides 
adjacent to the ANGLE. 

Angle Bracket 

The combination of a Bra and Ket (bra+ket = 
bracket) which represents the INNER PRODUCT of two 
functions or vectors, 



(f\9) 



(V|W) : 



f(x)g(x)dx 



New York: Dover, pp. 99-100, 



By itself, the Bra is a Covariant 1- Vector, and the 
Ket is a Covariant One-Form. These terms are com- 
monly used in quantum mechanics. 
see also Bra, Differential &-Form, Ket, One-Form 



46 Angle of Parallelism 



Annulus Conjecture 



Angle of Parallelism 

P 




Yl(x) 



A C D B 

Given a point P and a Line AB, draw the PERPENDIC- 
ULAR through P and call it PC, Let PD be any other 
line from P which meets CB in D. In a Hyperbolic 
Geometry, as D moves off to infinity along CB, then 
the line PD approaches the limiting line PE, which is 
said to be parallel to CB at P. The angle LCPE which 
PE makes with PC is then called the angle of paral- 
lelism for perpendicular distance x, and is given by 



n(x)-2tan- 1 (e- x ). 



This is known as Lobachevsky's FORMULA. 

see also Hyperbolic Geometry, Lobachevsky's 
Formula 

References 

Manning, H. P. Introductory Non-Euclidean Geometry. New 
York: Dover, pp. 31-32 and 58, 1963. 



Angle Trisection 

see Trisection 

Angular Acceleration 

The angular acceleration ct is defined as the time DE- 
RIVATIVE of the Angular Velocity u>, 



a ~ 



~dt 



d 2 6 „ _ a 
di 2 *' r' 



see also Acceleration, Angular Distance, Angu- 
lar Velocity 

Angular Defect 

The Difference between the Sum of face Angles Ai 
at a Vertex of a Polyhedron and 27r, 



5 = 2ir-^2Ai. 



see also Descartes Total Angular Defect, Jump 
Angle 



Angular Velocity 

The angular velocity U) is the time DERIVATIVE of the 
Angular Distance with direction z Perpendicu- 
lar to the plane of angular motion, 



d0„ v 

io = — z = — . 
dt r 



see also ANGULAR ACCELERATION, ANGULAR DIS- 
TANCE 

Anharmonic Ratio 

see Cross-Ratio 

Anisohedral Tiling 

A fc-anisohedral tiling is a tiling which permits no n- 
ISOHEDRAL TILING with n < k. 

References 

Berglund, J. "Is There a A;-Anisohedral Tile for k > 5?" 

Amer. Math. Monthly 100, 585-588, 1993. 
Klee, V. and Wagon, S. Old and New Unsolved Problems in 

Plane Geometry and Number Theory. Washington, DC: 

Math. Assoc. Amer., 1991. 

Annihilator 

The term annihilator is used in several different ways in 
various aspects of mathematics. It is most commonly 
used to mean the SET of all functions satisfying a given 
set of conditions which is zero on every member of a 

given SET. 

Annulus 

The region in common to two concentric CIRCLES of 
RADII a and b. The AREA of an annulus is 



Aannulus = ?t(& — CL ). 



An interesting identity is as follows. In the figure, 




the AREA of the shaded region A is given by 
A = d + C 2 . 



Angular Distance 

The angular distance traveled around a CIRCLE is the 
number of RADIANS the path subtends, 



0= 7^2tt= -. 

27TT r 



see also CHORD, CIRCLE, CONCENTRIC CIRCLES, LUNE 

(Plane), Spherical Shell 

References 

Pappas, T, "The Amazing Trick," The Joy of Mathematics. 
San Carlos, CA: Wide World Publ./Tetra, p. 69, 1989. 



see also ANGULAR ACCELERATION, ANGULAR VELOC- 
ITY 



Annulus Conjecture 

see Annulus Theorem 



Annulus Theorem 



Anosov Flow 47 



Annulus Theorem 

Let Ki and K^ be disjoint bicollared knots in W n+ or 
S and let U denote the open region between them. 
Then the closure of U is a closed annulus S n x [0,1]. 
Except for the case n = 3, the theorem was proved by 
Kirby (1969). 

References 

Kirby, R. C. "Stable Homeomorphisms and the Annulus Con- 
jecture." Ann. Math. 89, 575-582, 1969. 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, p. 38, 1976. 

Anomalous Cancellation 

The simplification of a FRACTION a/b which gives a cor- 
rect answer by "canceling" DIGITS of a and b. There 
are only four such cases for NUMERATOR and DENOM- 
INATORS of two Digits in base 10: 64/16 = 4/1 = 4, 
98/49 ^ 8/4 = 2, 95/19 = 5/1 = 5, and 65/26 = 5/2 
(Boas 1979). 

The concept of anomalous cancellation can be extended 
to arbitrary bases. PRIME bases have no solutions, but 
there is a solution corresponding to each PROPER DIVI- 
SOR of a Composite b. When b - 1 is Prime, this type 

of solution is the only one. For base 4, for example, 
the only solution is 324/ 134 = 24. Boas gives a table of 
solutions for b < 39. The number of solutions is EVEN 
unless b is an EVEN SQUARE. 



6 


N 


b 


N 


4 


1 


26 


4 


6 


2 


27 


6 


8 


2 


28 


10 


9 


2 


30 


6 


10 


4 


32 


4 


12 


4 


34 


6 


14 


2 


35 


6 


15 


6 


36 


21 


16 


7 


38 


2 


18 


4 


39 


6 


20 


4 






21 


10 






22 


6 






24 


6 







see also Fraction, Printer's Errors, Reduced 
Fraction 

References 

Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathemat- 
ical Plums (Ed. R. Honsberger). Washington, DC: Math. 
Assoc. Amer., pp. 113-129, 1979. 

Ogilvy, C. S. and Anderson, J. T. Excursions in Number 
Theory. New York: Dover, pp. 86-87, 1988. 

Anomalous Number 
see Benford's Law 



Anonymous 

A term in SOCIAL CHOICE Theory meaning invariance 

of a result under permutation of voters. 

see also Dual Voting, Monotonic Voting 

Anosov Automorphism 

A Hyperbolic linear map R n -» R n with Integer en- 
tries in the transformation Matrix and Determinant 
±1 is an Anosov Diffeomorphism of the n-ToRUS, 
called an Anosov automorphism (or HYPERBOLIC AU- 
TOMORPHISM). Here, the term automorphism is used in 
the Group Theory sense. 

Anosov Diffeomorphism 

An Anosov diffeomorphism is a C x DIFFEOMORPHISM <f> 
such that the Manifold M is Hyperbolic with respect 
to (j>. Very few classes of Anosov diffeomorphisms are 
known. The best known is ARNOLD'S Cat Map. 

A Hyperbolic linear map W 1 — > W 1 with Integer 
entries in the transformation Matrix and Determi- 
nant ±1 is an Anosov diffeomorphism of the n-TORUS. 
Not every MANIFOLD admits an Anosov diffeomorphism. 
Anosov diffeomorphisms are EXPANSIVE, and there are 
no Anosov diffeomorphisms on the CIRCLE. 

It is conjectured that if <f> : M —> M is an Anosov dif- 
feomorphism on a Compact Riemannian Manifold 
and the Nonwandering Set Q(<f>) of <f> is M, then <f> 

is TOPOLOGICALLY CONJUGATE to a FlNITE-TO-ONE 

Factor of an Anosov Automorphism of a Nilman- 

ifold. It has been proved that any Anosov diffeomor- 
phism on the n-TORUS is TOPOLOGICALLY CONJUGATE 
to an ANOSOV AUTOMORPHISM, and also that Anosov 
diffeomorphisms are C 1 STRUCTURALLY STABLE. 

see also ANOSOV AUTOMORPHISM, AXIOM A DIFFEO- 
MORPHISM, Dynamical System 

References 

Anosov, D. V. "Geodesic Flow on Closed Riemannian Man- 
ifolds with Negative Curvature." Proc. Steklov Inst, 
A. M. S. 1969. 

Smale, S. "Differentiable Dynamical Systems." Bull. Amer. 
Math. Soc. 73, 747-817, 1967. 

Anosov Flow 

A Flow defined analogously to the Anosov Diffeo- 
morphism, except that instead of splitting the TAN- 
GENT BUNDLE into two invariant sub-BUNDLES, they 
are split into three (one exponentially contracting, one 
expanding, and one which is 1-dimensional and tangen- 
tial to the flow direction). 

see also DYNAMICAL SYSTEM 



48 



Anosov Map 



Anticlastic 



Anosov Map 

An important example of a ANOSOV DlFFEOMORPHISM. 



Xn+l 


= 


2 l" 
1 1 







where x n +i,y n +i are computed mod 1. 
see also ARNOLD'S CAT MAP 

ANOVA 

"Analysis of Variance." A Statistical Test for het- 
erogeneity of Means by analysis of group VARIANCES. 
To apply the test, assume random sampling of a vari- 
ate y with equal VARIANCES, independent errors, and a 
Normal Distribution. Let n be the number of Repli- 
cates (sets of identical observations) within each of K 
FACTOR LEVELS (treatment groups), and y^ be the jth 
observation within FACTOR LEVEL i. Also assume that 
the ANOVA is "balanced" by restricting n to be the 
same for each Factor Level. 

Now define the sum of square terms 



k n 



P\2 



SST = £) £(j/ - J) 



(1) 



\ 2 / u „ v 2 



i=l j = l 
k 



k n 



"*-:E E« -eIE* (3) 



k n 



v j=rl j=l 



.-\2 



i=l j = l 

= SST - SSA, 



(4) 
(5) 



which are the total, treatment, and error sums of 
squares. Here, yi is the mean of observations within 
FACTOR Level i, and y is the "group" mean (i.e., mean 
of means). Compute the entries in the following table, 
obtaining the P- Value corresponding to the calculated 
F- Ratio of the mean squared values 



F = 



MSA 
MSE* 



(6) 



Category SS ° Freedom Mean Squared F- Ratio 
treatment SSA K-l MSA = |P^ §g 

error SSE K(n - 1) MSE = ^^ 



total 



SST Kn - 1 



MST=J^r_ 



If the P- VALUE is small, reject the NULL HYPOTHESIS 
that all Means are the same for the different groups. 

see also Factor Level, Replicate, Variance 



Anthropomorphic Polygon 

A Simple Polygon with precisely two Ears and one 

Mouth. 

References 

Toussaint, G. "Anthropomorphic Polygons." Amer. Math. 
Monthly 122, 31-35, 1991. 

Anthyphairetic Ratio 

An archaic word for a Continued Fraction. 

References 

Fowler, D. H. The Mathematics of Plato's Academy: A New 
Reconstruction. New York: Oxford University Press, 1987. 

Antiautomorphism 

If a Map / : G -> G* from a Group G to a Group G' 
satisfies f(ab) = f(a)f(b) for all a, 6 £ G, then / is said 
to be an antiautomorphism. 

see also AUTOMORPHISM 

Anticevian Triangle 

Given a center a : /3 : 7, the anticevian triangle is 
defined as the TRIANGLE with VERTICES -a : /3 : 7, 
a : -0 : 7, and a : f3 : -7. If A'B'C is the CEVIAN 
TRIANGLE of X and A"B"G" is an anticevian trian- 
gle, then X and A" are HARMONIC CONJUGATE POINTS 
with respect to A and A 1 . 

see also Cevian Triangle 

References 

Kimberling, C. "Central Points and Central Lines in the 
Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 

Antichain 

Let P be a finite PARTIALLY ORDERED Set. An an- 
tichain in P is a set of pairwise incomparable elements 
(a family of SUBSETS such that, for any two members, 
one is not the Subset of another). The WIDTH of P is 
the maximum CARDINALITY of an ANTICHAIN in P. For 
a Partial Order, the size of the longest Antichain 
is called the Width. 

see also Chain, Dilworth's Lemma, Partially Or- 
dered Set, Width (Partial Order) 

References 

Sloane, N. J. A. Sequence A006826/M2469 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Anticlastic 

When the Gaussian Curvature K is everywhere Neg- 
ative, a SURFACE is called anticlastic and is saddle- 
shaped. A Surface on which K is everywhere Posi- 
tive is called Synclastic. A point at which the Gaus- 
sian Curvature is Negative is called a Hyperbolic 
Point. 

see also Elliptic Point, Gaussian Quadrature, 
Hyperbolic Point, Parabolic Point, Planar 

Point, Synclastic 



Anticommutative 



Antimagic Graph 49 



Anticommutative 

An Operator * for which a * b = —6 * a is said to be 

anticommutative. 

see also Commutative 

Anticommutator 

For Operators A and B, the anticommutator is defined 
by 

{i,B} = AB + Si. 

see a/50 Commutator, Jordan Algebra 
Anticomplementary Triangle 




A Triangle AA'B'C* which has a given Triangle 
AABC as its Medial Triangle. The Trilinear Co- 
ordinates of the anticomplementary triangle are 



-1 L-i ^-1 
-a : : c 



A' 

B = a : —0 : c 

s^r -1 7-1 -1 

C = a :b : — c . 



see ateo MEDIAL TRIANGLE 

Antiderivative 

see Integral 



Antihomologous Points 

Two points which are COLLINEAR with respect to 
a Similitude Center but are not Homologous 
Points. Four interesting theorems from Johnson (1929) 
follow. 

1. Two pairs of antihomologous points form inversely 
similar triangles with the HoMOTHETIC CENTER. 

2. The Product of distances from a HOMOTHETIC 
Center to two antihomologous points is a constant. 

3. Any two pairs of points which are antihomologous 
with respect to a Similitude Center lie on a Cir- 
cle. 

4. The tangents to two CIRCLES at antihomologous 
points make equal ANGLES with the LINE through 
the points. 

see also HOMOLOGOUS POINTS, HOMOTHETIC CENTER, 

Similitude Center 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 19-21, 1929. 

Antilaplacian 

The antilaplacian of u with respect to a? is a function 
whose LAPLACIAN with respect to x equals u. The an- 
tilaplacian is never unique. 

see also LAPLACIAN 

Antilinear Operator 

An antilinear OPERATOR satisfies the following two 
properties: 

A[h(x) + f 2 (x)] = Ah{x) + Af 2 (x) 
Acf(x) = c*Af(x), 



Antidifferentiation 

see INTEGRATION 



where c* is the Complex Conjugate of c. 

see also LINEAR OPERATOR 



Antigonal Points 

B 




Given LAXB + IAYB — n RADIANS in the above fig- 
ure, then X and Y are said to be antigonal points with 
respect to A and B. 

Antihomography 

A ClRCLE-preserving TRANSFORMATION composed of 

an Odd number of Inversions. 
see also HOMOGRAPHY 



Antilogarithm 

The Inverse Function of the Logarithm, defined 
such that 

log 6 (antilog 6 z) = z = antilogy (log b z). 

The antilogarithm in base b of z is therefore b z . 
see also Cologarithm, Logarithm, Power 

Antimagic Graph 

A GRAPH with e EDGES labeled with distinct elements 
{l,2,...,e}so that the Sum of the EDGE labels at each 
VERTEX differ. 
see also MAGIC GRAPH 

References 

Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A 
Comprehensive Introduction. San Diego, CA: Academic 
Press, 1990. 



50 Antimagic Square 



Antipedal Triangle 



Antimagic Square 



15 


2 


12 


4 


1 


14 


10 


5 


8 


9 


3 


16 


11 


13 


6 


7 



21 


18 


6 


17 


4 


7 


3 


13 


16 


24 


5 


20 


23 


11 


1 


15 


8 


19 


2 


25 


14 


12 


9 


22 


10 



10 


25 


32 


13 


16 


9 


22 


7 


3 


24 


21 


30 


20 


27 


18 


26 


11 


6 


1 


31 


23 


33 


17 


8 


19 


5 


36 


12 


15 


29 


34 


14 


2 


4 


35 


28 



14 


3 


34 


21 


47 


29 


22 


43 


16 


13 


25 


6 


26 


44 


30 


48 


24 


8 


12 


9 


45 


10 


5 


11 


38 


49 


46 


19 


4 


41 


37 


36 


33 


27 


1 


39 


17 


40 


20 


7 


35 


23 


31 


42 


18 


32 


28 


2 


15 



49 


16 


50 


10 


19 


28 


24 


56 


42 


43 


11 


15 


44 


38 


55 


5 


25 


21 


48 


46 


9 


37 


6 


63 


29 


47 


8 


40 


51 


30 


52 


1 


45 


22 


54 


23 


20 


34 


2 


62 


14 


59 


18 


33 


41 


26 


61 


13 


36 


12 


58 


32 


27 


64 


3 


35 


17 


39 


7 


57 


53 


4 


60 


31 



52 


19 


81 


22 


29 


15 


42 


31 


76 


61 


10 


67 


23 


54 


79 


25 


33 


16 


57 


9 


71 


24 


38 


1 


51 


47 


75 


26 


78 


7 


69 


66 


77 


13 


27 


12 


39 


21 


74 


20 


37 


17 


49 


55 


64 


8 


65 


4 


62 


50 


34 


73 


41 


40 


56 


68 


2 


63 


14 


72 


35 


44 


6 


53 


30 


60 


32 


36 


3 


46 


43 


58 


11 


70 


5 


59 


48 


80 


28 


45 


18 



An antimagic square is an n x n ARRAY of integers from 
1 to n 2 such that each row, column, and main diago- 
nal produces a different sum such that these sums form 
a Sequence of consecutive integers. It is therefore a 
special case of a HETEROSQUARE. 

Antimagic squares of orders one and two are impossi- 
ble, and it is believed that there are also no antimagic 
squares of order three. There are 18 families of an- 
timagic squares of order four. Antimagic squares of or- 
ders 4-9 are illustrated above (Madachy 1979). 

see also HETEROSQUARE, MAGIC SQUARE, TALISMAN 

Square 



References 



Disc. 



Abe, G. "Unsolved Problems on Magic Squares." 

Math. 127, 3-13, 1994. 
Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in 

Madachy 's Mathematical Recreations. New York: Dover, 

pp. 103-113, 1979. 
# Weisstein, E. W. "Magic Squares." http: //www. astro. 

Virginia, edu/~eww6n/math/notebooks/MagicSquares .m. 

Antimorph 

A number which can be represented both in the form 
xo 2 — Dyo 2 and in the form Dx\ 2 — y\ 2 . This is only 
possible when the PELL EQUATION 



2 n 2 

x — Dy 



Antinomy 

A Paradox or contradiction. 

Antiparallel 




A pair of LINES B\ , B2 which make the same ANGLES 
but in opposite order with two other given LINES A\ and 
A2, as in the above diagram, are said to be antiparallel 
to A\ and A2. 

see also HYPERPARALLEL, PARALLEL 

References 

Phillips, A. W. and Fisher, I. Elements of Geometry. New 
York: American Book Co., 1896. 



Antipedal Triangle 




The antipedal triangle A of a given TRIANGLE T is the 
Triangle of which T is the Pedal Triangle. For 
a Triangle with Trilinear Coordinates a : j3 : 7 
and Angles A, B, and C, the antipedal triangle has 
Vertices with Trilinear Coordinates 



is solvable. Then 

x 2 - Dy 2 = ~(x - Dy 2 )(x n 2 - Dy n 2 ) 

= D(x y n - y x n ) 2 - {x x n - Dy y n ) 2 . 

see also Idoneal Number, Polymorph 

References 

Beiler, A. H. Recreations in the Theory of Numbers: The 

Queen of Mathematical Entertains. New York: Dover, 

1964. 

Antimorphic Number 

see Antimorph 



— (/? + a cos C) (7 + a cos i?) : (7 + aicosI?)(a + /?cosC) : 

(0 + a cos C) (a + 7 cos B) 

(7 + cos A)(/3 + a cos C) : -(7 + ^cos A)(a + 0cosC) : 

(a + cos C) (0 + 7 cos A) 

(0 + 7 cos A) (7 + acosi?) : (a + 7 cos B) (7 + ficosA) : 

— (a + jcosB)(0 + 7 cos A). 

The Isogonal Conjugate of the Antipedal Trian- 
gle of a given TRIANGLE is HOMOTHETIC with the orig- 
inal Triangle. Furthermore, the Product of their 
Areas equals the Square of the Area of the original 
Triangle (Gallatly 1913). 
see also Pedal Triangle 



Antipersistent Process 



Antisymmetric Matrix 51 



References 

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. 
London: Hodgson, pp. 56-58, 1913. 

Antipersistent Process 

A Fractal Process for which H < 1/2, so r < 0. 

see also Persistent Process 

Antipodal Map 

The Map which takes points on the surface of a Sphere 
S 2 to their ANTIPODAL POINTS. 

Antipodal Points 

Two points are antipodal (i.e., each is the ANTIPODE of 
the other) if they are diametrically opposite. Examples 
include endpoints of a Line SEGMENT, or poles of a 
Sphere. Given a point on a Sphere with Latitude S 
and Longitude A, the antipodal point has Latitude 
~6 and LONGITUDE A ± 180° (where the sign is taken 
so that the result is between —180° and +180°). 

see also Antipode, Diameter, Great Circle, 
Sphere 

Antipode 

Given a point A, the point B which is the ANTIPODAL 
Point of A is said to be the antipode of A. 

see also ANTIPODAL POINTS 
Antiprism 







Antiquity 

see Geometric Problems of Antiquity 

Antisnowflake 

see Koch Antisnowflake 

Antisquare Number 

A number of the form p a • A is said to be an antisquare 
if it fails to be a Square Number for the two reasons 
that a is ODD and A is a nonsquare modulo p. 

see also Square Number 

Antisymmetric 

A quantity which changes Sign when indices are re- 
versed. For example, Aij = a, — aj is antisymmetric 
since Aij = —Aji. 

see also ANTISYMMETRIC MATRIX, ANTISYMMETRIC 

Tensor, Symmetric 

Antisymmetric Matrix 

An antisymmetric matrix is a MATRIX which satisfies 
the identity 



A=-A* 



(i) 



where A T is the MATRIX TRANSPOSE. In component 
notation, this becomes 



an = —a-* 



Letting k = i = j, the requirement becomes 



cikk — —a-kkj 



(2) 



(3) 







A Semiregular Polyhedron constructed with 2 n- 
gons and 2n TRIANGLES. The 3-antiprism is simply the 
Octahedron. The Duals are the Trapezohedra. 
The Surface Area of a n-gonal antiprism is 

-2[|na 2 cot(^)]+2n(|v / 3a 2 ) 
cot(£)+V3\ 



= \ na2 



see also Octahedron, Prism, Prismoid, Trapezohe- 

DRON 

References 

Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in 

Mathematical Recreations and Essays, 13ili ed. New York; 

Dover, p, 130, 1987. 
Cromwell, P. R. Polyhedra. New York: Cambridge University 

Press, pp. 85-86, 1997. 
Weisstein, E. W. "Prisms and Antiprisms." http://www. 

astro .virginia.edu/-eww6n/math/notebooks/Pr ism. m. 



so an antisymmetric matrix must have zeros on its diag- 
onal. The general 3x3 antisymmetric matrix is of the 
form 

ai2 ai3~ 

-aw a 2 3 • (4) 

. — ai3 — G&23 

Applying A" 1 to both sides of the antisymmetry condi- 
tion gives 

-A^A 1 = I. (5) 

Any SQUARE MATRIX can be expressed as the sum of 
symmetric and antisymmetric parts. Write 



A=i(A + A T ) + f(A-A T ). 



an 
a>2i 



ai2 

«22 



0,2n 



a n i a n 2 



(6) 



(7) 



52 Antisymmetric Relation 



Apeirogon 





an 


a2i 


a n i 




A T = 


ai2 


a22 " * 


0>n2 


) 




_ain 


a2n • • ■ 


Q>nn _ 




2an 


ai2 + C121 




flln + «nl 


0,12 + &21 


2a22 




fl2n + «n2 


_ain + 


a-Tii 


&2n + a„2 




Z(l nn 



(8) 



A + A T = 



which is symmetric, and 

A-A T = 

ai2 - fltei 

-(ai2 - a2i) 

-(flln — flnl) — (tl2n — ^n2) 



(9) 



Oln - Q>nl 

din — &n2 



(10) 



which is antisymmetric. 

see ateo Skew Symmetric Matrix, Symmetric Ma- 
trix 

Antisymmetric Relation 

A RELATION R on a SET S is antisymmetric provided 
that distinct elements are never both related to one an- 
other. In other words xRy and yRx together imply that 
x~y. 

Antisymmetric Tensor 

An antisymmetric tensor is denned as a TENSOR for 
which 

A mn = _ A r,m t ^ 

Any Tensor can be written as a sum of Symmetric 
and antisymmetric parts as 

The antisymmetric part is sometimes denoted using the 
special notation 



A [ab] = U A ab _ A bay 



For a general TENSOR, 



(3) 



(4) 



permutations 



where e ai -a. n is the Levi-Civita Symbol, a.k.a. the 

Permutation Symbol. 

see also Symmetric Tensor 



Antoine's Horned Sphere 

A topological 2-sphere in 3-space whose exterior is not 
Simply Connected. The outer complement of An- 
toine's horned sphere is not Simply Connected. Fur- 
thermore, the group of the outer complement is not 
even finitely generated. Antoine's horned sphere is in- 
equivalent to Alexander's Horned Sphere since the 
complement in E 3 of the bad points for Alexander's 
Horned Sphere is Simply Connected. 
see also Alexander's Horned Sphere 

References 

Alexander, J. W. "An Example of a Simply-Connected Sur- 
face Bounding a Region which is not Simply-Connected." 
Proc. Nat Acad. Sci. 10, 8-10, 1924. 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 76-79, 1976. 

Antoine's Necklace 




Construct a chain C of 2n components in a solid TORUS 
V. Now form a chain C± of 2n solid tori in V, where 

ir x (V - Ci) <* iri(V - C) 

via inclusion. In each component of Ci, construct a 
smaller chain of solid tori embedded in that component. 
Denote the union of these smaller solid tori C^. Con- 
tinue this process a countable number of times, then the 
intersection 



A=f|C 



which is a nonempty compact SUBSET of IR. is called 
Antoine's necklace. Antoine's necklace is HOMEOMOR- 
PHIC with the CANTOR SET. 
see also ALEXANDER'S HORNED SPHERE, NECKLACE 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 73-74, 1976, 

Apeirogon 

The Regular Polygon essentially equivalent to the 
CIRCLE having an infinite number of sides and denoted 
with Schlafli Symbol {oo}. 

see also CIRCLE, REGULAR POLYGON 

References 

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: 
Dover, 1973. 

Schwartzman, S. The Words of Mathematics: An Etymolog- 
ical Dictionary of Mathematical Terms Used in English. 
Washington, DC: Math. Assoc. Amer., 1994. 



Apery 's Constant 



Apery's Constant 53 



Apery's Constant 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 



Apery's constant is defined by 

C(3) = 1.2020569... 



(1) 



(Sloane's A002117) where f(z) is the RlEMANN Zeta 
Function. Apery (1979) proved that £(3) is Irra- 
tional, although it is not known if it is TRANSCEN- 
DENTAL. The Continued Fraction for £(3) is [1, 4, 1, 
18, 1, 1, 1, 4, 1, ...] (Sloane's A013631). The positions 
at which the numbers 1, 2, . . . occur in the continued 
fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, ... . 

Sums related to £(3) are 

c(3) _ 5 f^ (-1)- 1 ,.5f (-ir'(M)» 

(used by Apery), and 

oo 



(2fc + 1)3 

2tt 3 
(3* + l)» " 81^/3 ' 27 



OO 

^ (3& + 1) 

oo 3 

2-> (4/k + l) 3 = 64 + " ^ 3 ) 



+ H C(3) (4) 



(5) 



OO 

^ (6Jfe + 1 



+ iC(3), (6) 



(6fc+l)3 36^3 2 

where X(z) is the Dirichlet Lambda Function. The 
above equations are special cases of a general result due 
to Ramanujan (Berndt 1985). Apery's proof relied on 
showing that the sum 



<»>-£©"Cr)". o 



where (£) is a Binomial Coefficient, satisfies the Re- 
currence Relation 

(n + l) 3 a(n + 1) - (34n 3 + bin 2 + 27n + 5)a(n) 

+n 3 a(n-l) = (8) 

(van der Poorten 1979, Zeilberger 1991). 
Apery's constant is also given by 



Sn,\ 



«3)=x; 2 T i ' 

*-*t nln 



(9) 



where 5 n , m is a Stirling Number of the First Kind. 
This can be rewritten as 



E§= 2 « 3 )' 



(10) 



where H n is the nth HARMONIC NUMBER, Yet another 
expression for £(3) is 



««-*£;?(•♦*+••■ + ;) 



(11) 



(Castellanos 1988). 
Integrals for C(3) include 



CO) 



i r e 

2io c'-l 



cK 



\W 



r ir/4 



= ^ | j7r J ln2 + 2 / a: In (sin a;) da; 



(12) 
(13) 



Gosper (1990) gave 






30& - 11 



4 £? (2* -!)*»(?)' 



(14) 



A Continued Fraction involving Apery's constant is 

JL = 5 _J^ t_ rf 

C(3) 117- 535- ' ' * 34n 3 + 51n 2 + 27n + 5- * * ' 

(15) 
(Apery 1979, Le Lionnais 1983). Amdeberhan (1996) 
used Wilf- Zeilberger Pairs (F,G) with 



F{n,k) 



_ (-l) k k\ 2 (sn-k-l)\ 
(sn + & + l)!(fc + l) 



(16) 



s — 1 to obtain 



c(3) = §f;(-ir- 1 «i ? . (it) 



For 5 = 2, 



oo „ 

ffl)- 1 ^ nn-i 56n 2 -32 + 5 1 



and for s = 3, 



(-i) n 



((3) = V { ~ X) 

^72( 4n )( 3n ) 



6120n + 5265n 4 + 13761n 2 + 13878n 3 + 1040 
(4n-fl)(4n + 3)(n+l)(3n+l) 2 (3n + 2) 2 ^ > 



54 Apery's Constant 



Apoapsis 



(Amdeberhan 1996). The corresponding G(n,k) for s = 
1 and 2 are 

G(n ' fc) -(n + fc + l)!(n+l)' (20) 

and 

<3(n,fc) = 

(-l) fc fci 2 (2n - fe)!(3 + 4n)(4n 2 + 6n + k + 3) 
2(2n-hA; + 2)!(n + l) 2 (2n-hl) 2 

Gosper (1996) expressed C(3) as the MATRIX PRODUCT 

N 



(21) 



lim TTM n = 



C(3) 
1 



(22) 



where 
M n = 

" (n + l) 4 24570Tt 4 + 64161n 3 +62152n 2 +26427n.+4154 

4096(n+f)2(n+J)2 31104(n+|)(n+±)(n+§) 

1 

(23) 

which gives 12 bits per term. The first few terms are 

(24) 

(25) 

(26) 

which gives 

C( 3 ) * IllZlVwl = 1-20205690315732 .... (27) 

Given three INTEGERS chosen at random, the probabil- 
ity that no common factor will divide them all is 





r i 


2077 " 

1728 

1 




Mi = 


19600 





M 2 = 


1 

9801 




7561 " 

4320 

1 






r ° 


50501 

20160 

1 


-1 


M 3 = 


67600 







[CO)]' 



1.202 -1 =0.832.. 



(28) 



B. Haible and T. Papanikolaou computed £(3) to 
1,000,000 Digits using a Wilf-Zeilberger Pair iden- 
tity with 

_ fc n! 6 (2n-fc-l)!fc! 3 

*(n,k)-( 1) 2(n + A . + 1)!2(2rl )!3> W 

5 = 1, and t = 1, giving the rapidly converging 



,vn-V\ 1 ^ rc! 1 °(205n 2 + 250n + 77) 
QW-Z^l- 1 ) 64(2n+l)!« 



(Amdeberhan and Zeilberger 1997). The record as of 

Aug. 1998 was 64 million digits (Plouffe). 

see also Riemann Zeta Function, Wilf-Zeilberger 

Pair 

References 

Amdeberhan, T. "Faster and Faster Convergent Se- 
ries for C(3)-" Electronic J. Combinatorics 3, R13, 
1—2, 1996. http: //www. combinatorics. org/Volume^/ 
volume3 ,html#R13. 

Amdeberhan, T. and Zeilberger, D. "Hypergeometric Se- 
ries Acceleration via the WZ Method." Electronic J. 
Combinatorics 4, No. 2, R3, 1-3, 1997. http: //www. 
combinatorics . org/Volume_4/wilf toe .html#R03. Also 
available at http : //www . math . temple . edu/~zeilberg/ 
mamarim/mamarimhtml/accel . html. 

Apery, R. "Irrationalite de £(2) et C(3)." Asterisque 61, 11- 
13, 1979. 

Berndt, B. C. Ramanujan's Notebooks: Part J. New York: 
Springer- Verlag, 1985. 

Beukers, F. "A Note on the Irrationality of C(3)." Bull. Lon- 
don Math. Soc. 11, 268-272, 1979. 

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in 
Analytic Number Theory and Computational Complexity. 
New York: Wiley, 1987. 

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 
61, 67-98, 1988. 

Conway, J. H. and Guy, R. K. "The Great Enigma." In The 
Book of Numbers. New York: Springer- Verlag, pp. 261— 
262, 1996. 

Ewell, J. A. "A New Series Representation for C(3)." Amer. 
Math. Monthly 97, 219-220, 1990. 

Finch, S. "Favorite Mathematical Constants." http: //www. 
mathsoft.com/asolve/constant/apery/apery.html. 

Gosper, R. W. "Strip Mining in the Abandoned Orefields 
of Nineteenth Century Mathematics." In Computers in 
Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). 
New York: Marcel Dekker, 1990. 

Haible, B. and Papanikolaou, T. "Fast Multiprecision Eval- 
uation of Series of Rational Numbers." Technical Report 
TI-97-7. Darmstadt, Germany: Darmstadt University of 
Technology, Apr. 1997. 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 36, 1983. 

Plouffe, S. "Plouffe's Inverter: Table of Current Records for 
the Computation of Constants." http://lacim.uqam.ca/ 
pi/records. html. 

Plouffe, S. "32,000,279 Digits of Zeta(3)." http://lacim. 
uqam.ca/piDATA/Zet a3.txt. 

Sloane, N. J. A. Sequences A013631 and A002117/M0020 in 
"An On-Line Version of the Encyclopedia of Integer Se- 
quences." 

van der Poorten, A. "A Proof that Euler Missed. . . Apery's 
Proof of the Irrationality of £(3)." Math. Intel. 1,196-203, 
1979. 

Zeilberger, D. "The Method of Creative Telescoping." J. 
Symb. Comput. 11, 195-204, 1991. 

Apoapsis 




(30) 



The greatest radial distance of an Ellipse as measured 
from a FOCUS. Taking v = n in the equation of an 

Ellipse 

a(l-e 2 ) 

r = 

1 + e cos v 



Apocalypse Number 



Apodization Function 55 



gives the apoapsis distance 

r+ =a(l + e). 

Apoapsis for an orbit around the Earth is called apogee, 
and apoapsis for an orbit around the Sun is called aphe- 
lion. 

see also Eccentricity, Ellipse, Focus, Periapsis 

Apocalypse Number 

A number having 666 Digits (where 666 is the Beast 
Number) is called an apocalypse number. The FI- 
BONACCI NUMBER F3184 is an apocalypse number. 
see also Beast Number, Leviathan Number 

References 

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97- 
102, 1995. 

Apocalyptic Number 

A number of the form 2 n which contains the digits 666 
(the Beast Number) is called an Apocalyptic Num- 
ber. 2 157 is an apocalyptic number. The first few such 
powers are 157, 192, 218, 220, . . . (Sloane's A007356). 

see also Apocalypse Number, Leviathan Number 

References 

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97- 
102, 1995. 

Sloane, N. J. A. Sequences A007356/M5405 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 

Apodization 

The application of an APODIZATION FUNCTION. 

Apodization Function 

A function (also called a Tapering Function) used to 

bring an interferogram smoothly down to zero at the 
edges of the sampled region. This suppresses sidelobes 
which would otherwise be produced, but at the expense 
of widening the lines and therefore decreasing the reso- 
lution. 

The following are apodization functions for symmetrical 
(2-sided) interferograms, together with the Instrument 
Functions (or Apparatus Functions) they produce 
and a blowup of the Instrument Function sidelobes. 
The Instrument Function I(k) corresponding to a 
given apodization function A(x) can be computed by 
taking the finite FOURIER COSINE TRANSFORM, 



Apodization Function 



Instrument Function 

1.25 



Instrument Function Sidelobes 



I(k) 



/a 
■a 



cos(27r kx)A(x) dx. 



(1) 



Bartlett 



Connes 




Harming 



Uniform 



Welch 



-1 -0.5 0,5 1 



Type 


Apodization Function 


Instrument Function 


Bartlett 




1 _ i£i 


a sinc 2 (7r/:a) 


Blackman 




B A (x) 


Bi(fc) 


Connes 




o-sr 


8aV27r (2ir*a)«/» 


Cosine 




c °s(ff) 


4aca B (2jrafc) 

TT(l-160 2 fc 2 ) 


Gaussian 




e -*V(2» a ) 


2j a cos(27rfca:)e- l3/(2 ' 2) dx 


Hamming 




Hm.A{x) 


Hmj{k) 


Hanning 




Hn A (x) 


Hnt(k) 


Uniform 




1 


2a sine (27r/ea) 


Welch 




1-S 


W:(k) 



where 
B A (x) = 
Bj(k) = 

Hm A (x) = 
Hrm(k) = 



(1ZX \ / 2-7TX \ 
— J +0.08 cos f J 

a(0.84 - 0.36a 2 k 2 -2.17 x 1Q- X9 a 4 fc 4 ) sinc(27rafc) 



(2) 



(l-a 2 A: 2 )(l-4a 2 fc 2 ) 

0.54 + 0.46 cos (—) 

a(1.08 - 0.64a 2 fc 2 ) sinc(27rafe) 
" l-4a 2 fc 2 "" 



(3) 
(4) 

(5) 



56 Apodization Function 



Apollonius Circles 



Hn A (x) = cos 2 I — ) 



1 + cos 



(?) 



Hrnik) 



1 
" 2 

a sine (2irak) 
' l-4a 2 fc 2 
= a[sinc(27rfca) + ^ sinc(27rA;a — 7r) 

+ ^ sinc(27rA;a + 7r)] 



W}(fc) =a2V2?r 



J3/2 



(27rA;a) 



(27rfca) 3 / 2 
sin(27rfca) — 2nak cos(2-7rafc) 
2a 3 fe 3 7T 3 ' 



(6) 
(7) 
(8) 

(9) 
(10) 

(11) 



Type 


IF FWHM 


IF Peak 


Peak (-) S.L. 
Peak 


Peak (+) S.L. 
Peak 


Bartlett 


1.77179 


1 


0.00000000 


0.0471904 


Blackmail 


2.29880 


0.84 


-0.00106724 


0.00124325 


Cormes 


1.90416 


16 
15 


-0.0411049 


0.0128926 


Cosine 


1.63941 


4. 


-0.0708048 


0.0292720 


Gaussian 


— 


1 


— 


— 


Hamming 


1.81522 


1.08 


-0.00689132 


0.00734934 


Hanning 


2.00000 


1 


-0.0267076 


0.00843441 


Uniform 


1.20671 


2 


-0.217234 


0.128375 


Welch 


1.59044 


4 
3 


-0.0861713 


0.356044 



A general symmetric apodization function A(x) can be 
written as a FOURIER SERIES 

oo 

a n cosl— -J, (12) 

n=l 

where the COEFFICIENTS satisfy 

oo 

a + 2^a„ = 1. (13) 

n = l 

The corresponding apparatus function is 
I(t) = J A{x)e~ 2 ' Ktkx dx = 26Ja sinc(27r£;&) 

oo 

+ y^[sinc(27rA:& + mr) + sinc(27rA;6 - nir)] |. (14) 

n=l 

To obtain an APODIZATION FUNCTION with zero at ka = 
3/4, use 

ao sinc(|7r) + ai[sinc(|7r) + sinc(^7r) = 0. (15) 

Plugging in (13), 

-d-^ +*(£ + £) 

= -|(l-2o 1 ) + oi(i + l) = (16) 



ai = 



ao 



- 5 

3 ° _ 5 




| + | 6-3 + 2-5 28 
1 n„ 28 — 2 • 5 18 

1 lai = 9* ~" 28 - 


9 
14' 



(18) 
(19) 



The Hamming Function is close to the requirement 
that the Apparatus Function goes to at ka — 5/4, 
giving 



a = § « 0.5435 



ai 



21 
92 



0.2283. 



(20) 
(21) 



The Blackman Function is chosen so that the Appa- 
ratus Function goes to at ka — 5/4 and 9/4, giving 



ao 



ai = 



a 2 = 



3969 ,. 
9304 n 
1155 „ 
4652 " 

715 
18608 



0.4266 


(22) 


0.2483 


(23) 


i 0.0384. 


(24) 



ai(! + !) = 



^5 ' 3^ 3 



(IT) 



see also Bartlett Function, Blackman Function, 
Connes Function, Cosine Apodization Function, 
Full Width at Half Maximum, Gaussian Func- 
tion, Hamming Function, Hann Function, Han- 
ning Function, Mertz Apodization Function, 
Parzen Apodization Function, Uniform Apodiza- 
tion Function, Welch Apodization Function 

References 

Ball, J. A. "The Spectral Resolution in a Correlator Sys- 
tem" §4,3.5 in Methods of Experimental Physics 12C (Ed. 
M. L. Meeks). New York: Academic Press, pp. 55-57, 
1976. 

Blackman, R. B. and Tukey, J. W. "Particular Pairs of Win- 
dows." In The Measurement of Power Spectra, From 
the Point of View of Communications Engineering. New 
York: Dover, pp. 95-101, 1959. 

Brault, J. W. "Fourier Transform Spectrometry." In High 
Resolution in Astronomy: 15th Advanced Course of 
the Swiss Society of Astronomy and Astrophysics (Ed. 
A. Benz, M. Huber, and M. Mayor), Geneva Observatory, 
Sauverny, Switzerland, pp. 31-32, 1985. 

Harris, F. J. "On the Use of Windows for Harmonic Analysis 
with the Discrete Fourier Transform." Proc. IEEE 66, 51- 
83, 1978. 

Norton, R. H. and Beer, R. "New Apodizing Functions for 
Fourier Spectroscopy." J. Opt. Soc. Amer. 66, 259-264, 
1976. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 547-548, 1992. 

Schnopper, H. W. and Thompson, R. I. "Fourier Spectrom- 
eters." In Methods of Experimental Physics 12 A (Ed. 
M. L. Meeks). New York: Academic Press, pp. 491-529, 
1974. 

Apollonius Circles 

There are two completely different definitions of the so- 
called Apollonius circles: 

1 . The set of all points whose distances from two fixed 
points are in a constant ratio 1 : \i (Ogilvy 1990). 



Apollonius Point 



Apollonius 3 Problem 57 



2. The eight CIRCLES (two of which are nondegener- 
ate) which solve APOLLONIUS ' PROBLEM for three 
Circles. 

Given one side of a Triangle and the ratio of the 
lengths of the other two sides, the LOCUS of the third 
VERTEX is the Apollonius circle (of the first type) whose 
Center is on the extension of the given side. For a given 
Triangle, there are three circles of Apollonius. 

Denote the three Apollonius circles (of the first type) 
of a Triangle by &i, fo, and £3, and their centers Li, 
L 2) and L 3 . The center L\ is the intersection of the side 
A2A3 with the tangent to the ClRCUMCIRCLE at A\. 
L\ is also the pole of the SYMMEDIAN POINT K with 
respect to ClRCUMCIRCLE. The centers Li, Z/ 2 , and Lz 
are COLLINEAR on the POLAR of K with regard to its 
ClRCUMCIRCLE, called the Lemoine Line. The circle of 
Apollonius ki is also the locus of a point whose Pedal 
Triangle is Isosceles such that P1P2 = P1P3. 




Let U and V be points on the side line BC of a TRI- 
ANGLE AABC met by the interior and exterior ANGLE 
Bisectors of Angles A. The Circle with Diame- 
ter UV is called the A-Apollonian circle. Similarly, 
construct the B- and C-Apollonian circles. The Apol- 
lonian circles pass through the VERTICES A, £?, and C, 
and through the two ISODYNAMIC POINTS S and S' . 
The Vertices of the D-Triangle lie on the respective 
Apollonius circles. 

see also Apollonius' Problem, Apollonius Pursuit 
Problem, Casey's Theorem, Hart's Theorem, Iso- 
dynamic Points, Soddy Circles 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 40 and 294-299, 1929. 

Ogilvy, C. S. Excursions in Geometry. New York: Dover, 
pp. 14-23, 1990. 

Apollonius Point 

Consider the Excircles Fa, T b , and Tc of a Trian- 
gle, and the CIRCLE T internally TANGENT to all three. 
Denote the contact point of T and Fa by A f , etc. Then 



the Lines AA\ BB f , and CC' CONCUR in this point. It 
has Triangle Center Function 

a = sin 2 ,4 cos 2 [§(£-<?)]. 



References 

Kiinherling, C. "Apollonius Point." http://vvv. 

evansville . edu/~ck6/t centers/re cent /apollon. html. 
Kimberling, C. "Central Points and Central Lines in the 

Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 
Kimberling, C; Iwata, S.; and Hidetosi, F. "Problem 1091 

and Solution." Crux Math. 13, 128-129 and 217-218, 

1987. 



Apollonius' Problem 




•3 ^€J •£) 



© #;(•} 



^ 



Given three objects, each of which may be a Point, 
Line, or Circle, draw a Circle that is Tangent to 
each. There are a total of ten cases. The two easi- 
est involve three points or three LINES, and the hardest 
involves three CIRCLES. Euclid solved the two easiest 
cases in his Elements, and the others (with the exception 
of the three CIRCLE problem), appeared in the Tangen- 
cies of Apollonius which was, however, lost. The general 
problem is, in principle, solvable by STRAIGHTEDGE and 

Compass alone. 



58 



Apollonius 7 Problem 



Apollonius Pursuit Problem 




The three-ClRCLE problem was solved by Viete (Boyer 
1968), and the solutions are called Apollonius Cir- 
cles. There are eight total solutions. The simplest 
solution is obtained by solving the three simultaneous 
quadratic equations 

(x - x x f + (y - Vl ) 2 - (r ± n) 2 = (1) 

(x - x 2 f + (y - y 2 f - (r ± r 2 f = (2) 

(x - x z f + (y - y 3 ) 2 - (r ± r 3 ) 2 - (3) 

in the three unknowns x, y y r for the eight triplets of 
signs (Courant and Robbins 1996). Expanding the equa- 
tions gives 

OOO O ^ 

(x + y -r )-2xXi-2yyi±2rri+(xi +yi -n ) = 

(4) 
for i — 1, 2, 3. Since the first term is the same for each 
equation, taking (2) — (1) and (3) — (1) gives 



where 



ax 4- by + cr = d 




(5) 


ax + by + cr=-d, 




(6) 


a = 2(a?i — x 2 ) 




(7) 


b= 2(yi -y 2 ) 




(8) 


c = q=2(ri - r 2 ) 




(9) 


1/2. 2 2\/2, 2 

a = (x 2 +2/2 - r-2 ) - (xi + yi - 


-n 2 ) 


(10) 



and similarly for a , 6' , c and d' (where the 2 subscripts 
are replaced by 3s). Solving these two simultaneous lin- 
ear equations gives 



b'd - bd! - b'cr + bc'r 

ab 1 - ba ! 
—ad + ad' + o! cr — ac'r 
ab' -a'b ' 



(11) 
(12) 



which can then be plugged back into the QUADRATIC 
Equation (1) and solved using the Quadratic For- 
mula. 

Perhaps the most elegant solution is due to Gergonne. 
It proceeds by locating the six HOMOTHETIC CENTERS 
(three internal and three external) of the three given 
CIRCLES. These lie three by three on four lines (illus- 
trated above). Determine the Poles of one of these 
with respect to each of the three CIRCLES and connect 
the Poles with the Radical Center of the Circles. 
If the connectors meet, then the three pairs of intersec- 
tions are the points of tangency of two of the eight circles 
(Johnson 1929, Dorrie 1965). To determine which two 
of the eight Apollonius circles are produced by the three 
pairs, simply take the two which intersect the original 
three CIRCLES only in a single point of tangency. The 
procedure, when repeated, gives the other three pairs of 
Circles. 

If the three CIRCLES are mutually tangent, then the 
eight solutions collapse to two, known as the Soddy 
Circles. 

see also Apollonius Pursuit Problem, Bend (Cur- 
vature), Casey's Theorem, Descartes Circle 
Theorem, Four Coins Problem, Hart's Theorem, 
Soddy Circles 

References 

Boyer, C. B. A History of Mathematics. New York: Wiley, 

p. 159, 1968. 
Courant, R. and Robbins, H. "Apollonius' Problem." §3.3 in 

What is Mathematics? : An Elementary Approach to Ideas 

and Methods , 2nd ed. Oxford, England: Oxford University 

Press, pp. 117 and 125-127, 1996. 
Dorrie, H. "The Tangency Problem of Apollonius." §32 in 

100 Great Problems of Elementary Mathematics: Their 

History and Solutions. New York: Dover, pp. 154-160, 

1965. 
Gauss, C. F. Werke, Vol. 4. New York: George Olms, p. 399, 

1981. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 118-121, 1929. 
Ogilvy, C. S. Excursions in Geometry. New York: Dover, 

pp. 48-51, 1990. 
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide 

World Publ./Tetra, p. 151, 1989. 
Simon, M. Uber die Entwicklung der Element argeometrie im 

XIX Jahrhundert. Berlin, pp. 97-105, 1906. 
^ Weisstein, E. W. "Plane Geometry." http: //www. astro. 

Virginia . edu/-eww6n/math/notebooks/PlaneGeometry . m. 

Apollonius Pursuit Problem 

Given a ship with a known constant direction and speed 
v 1 what course should be taken by a chase ship in pur- 
suit (traveling at speed V) in order to intersect the other 
ship in as short a time as possible? The problem can be 
solved by finding all points which can be simultaneously 
reached by both ships, which is an APOLLONIUS CIRCLE 
with fi = v/V. If the CIRCLE cuts the path of the pur- 
sued ship, the intersection is the point towards which 



Apollonius Theorem 



Appell Transformation 59 



the pursuit ship should steer. If the CIRCLE does not 
cut the path, then it cannot be caught. 
see also Apollonius Circles, Apollonius' Prob- 
lem, Pursuit Curve 

References 

Ogilvy, C. S. Solved by M. S. Klamkin. "A Slow Ship In- 
tercepting a Fast Ship." Problem E991. Amer. Math. 
Monthly 59, 408, 1952. 

Ogilvy, C. S. Excursions in Geometry. New York: Dover, 
p. 17, 1990. 

Steinhaus, H. Mathematical Snapshots, 3rd American ed. 
New York: Oxford University Press, pp. 126-138, 1983. 

Apollonius Theorem 




ma 2 2 + na 3 2 = (m + n)AiP 2 + mPA 3 2 + nPA 2 2 . 



Apothem 




Given a CIRCLE, the PERPENDICULAR distance a from 
the Midpoint of a Chord to the Circle's center is 
called the apothem. It is also equal to the RADIUS r 
minus the SAGITTA s, 

a — r — s. 
see also Chord, Radius, Sagitta, Sector, Segment 

Apparatus Function 

see Instrument Function 

Appell Hypergeometric Function 

A formal extension of the Hypergeometric Function 
to two variables, resulting in four kinds of functions (Ap- 
pell 1925), 



oo oo 



F 1 (a;/3,/3'; 7 ;x,y) = ^^ 



(a)™+«GS)m(/3')« 



m = n = 

oo oo 



-x y 



x y 



m = n = 

F 3 (a,a ;/3,/3 i7i *,y) = ^ JL m!n !( 7 ) m+ „ 

m = Q ti-0 

oo oo 

77 / a < \ V^ V^ ( Q )m + n(^)m + r 1 ^ mj , 



Appell defined the functions in 1880, and Picard showed 
in 1881 that they may all be expressed by INTEGRALS 
of the form 



/' 

Jo 



u a (l - uf{l - xuy(l - yu) S du. 



References 

Appell, P. "Sur les fonctions hypergeometriques de plusieurs 
variables." In Memoir. Sci. Math. Paris: Gauthier-Villars, 
1925. 

Bailey, W. N. Generalised Hypergeometric Series. Cam- 
bridge, England: Cambridge University Press, p. 73, 1935. 

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary 
of Mathematics. Cambridge, MA: MIT Press, p. 1461, 
1980. 

Appell Polynomial 

A type of Polynomial which includes the Bernoulli 
Polynomial, Hermite Polynomial, and Laguerre 
POLYNOMIAL as special cases. The series of POLYNOMI- 
ALS {A n (z)}™ =0 is defined by 



where 



A(t)e** = ^TA n (z)t n , 



A(t) = ^2a k t k 



is a formal POWER series with k = 0, 1, . . . and ao ^ 0. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, pp. 209-210, 1988. 



Appell Transformation 

A HOMOGRAPHIC transformation 



ax + by -\- c 
a"x + b"y + c 

ax + b'y + c' 
a n x + b"y + c" 

with t\ substituted for t according to 



X! — 



yi 



kdti 



dt 



{a"x + b"y + c") 2 ' 



References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, pp. 210-211, 1988. 



m = ti=:0 



m!n!(7) m (7') 71 




A Surface of Revolution defined by Kepler. It con- 
sists of more than half of a circular ARC rotated about 
an axis passing through the endpoints of the Arc. The 
equations of the upper and lower boundaries in the x-z 
Plane are 

z± = ± V / R 2 -(x-r) 2 

for R > r and x E [— (r + R), r + R]. It is the outside 
surface of a Spindle TORUS. 

see also Bubble, Lemon, Sphere-Sphere Intersec- 
tion, Spindle Torus 

Approximately Equal 

If two quantities A and B are approximately equal, this 
is written A « B. 

see also Defined, Equal 

Approximation Theory 

The mathematical study of how given quantities can be 
approximated by other (usually simpler) ones under ap- 
propriate conditions. Approximation theory also stud- 
ies the size and properties of the ERROR introduced by 
approximation. Approximations are often obtained by 
POWER SERIES expansions in which the higher order 
terms are dropped. 
see also LAGRANGE REMAINDER 

References 

Achieser, N. I. and Hyman, C. J. Theory of Approximation. 
New York: Dover, 1993. 

Akheizer, N. I. Theory of Approximation. New York: Dover, 
1992. 

Cheney, E. W. Introduction to Approximation Theory. New 
York: McGraw-Hill, 1966. 

Golomb, M. Lectures on Theory of Approximation. Argonne, 
IL: Argonne National Laboratory, 1962. 

Jackson, D, The Theory of Approximation. New York: 
Amer. Math. Soc, 1930. 

Natanson, I. P. Constructive Function Theory, Vol. 1: Uni- 
form Approximation. New York: Ungar, 1964. 

Petrushev, P. P. and Popov, V. A. Rational Approximation of 
Real Functions. New York: Cambridge University Press, 
1987. 

Rivlin, T. J. An Introduction to the Approximation of Func- 
tions. New York: Dover, 1981. 

Timan, A. F. Theory of Approximation of Functions of a 
Real Variable. New York: Dover, 1994. 



Arbelos 

Arakelov Theory 

A formal mathematical theory which introduces "com- 
ponents at infinity" by defining a new type of divisor 
class group of Integers of a Number Field. The di- 
visor class group is called an "arithmetic surface." 

see also ARITHMETIC GEOMETRY 

Arbelos 




The term "arbelos" means SHOEMAKER'S KNIFE in 
Greek, and this term is applied to the shaded AREA 
in the above figure which resembles the blade of a knife 
used by ancient cobblers (Gardner 1979). Archimedes 
himself is believed to have been the first mathematician 
to study the mathematical properties of this figure. The 
position of the central notch is arbitrary and can be lo- 
cated anywhere along the DIAMETER. 

The arbelos satisfies a number of unexpected identities 
(Gardner 1979). 

1. Call the radii of the left and right SEMICIRCLES a 
and 6, respectively, with a + b = R. Then the arc 
length along the bottom of the arbelos is 

L = 27va + 2tt6 = 2?r(a + b) = 2tvR, 

so the arc lengths along the top and bottom of the 
arbelos are the same. 




2. Draw the PERPENDICULAR BD from the tangent of 
the two Semicircles to the edge of the large Cir- 
cle. Then the Area of the arbelos is the same as 
the Area of the Circle with Diameter BD. 

3. The CIRCLES C\ and C2 inscribed on each half of 
BD on the arbelos (called ARCHIMEDES' CIRCLES) 
each have DIAMETER (AB)(BC)/(AC). Further- 
more, the smallest ClRCUMCIRCLE of these two cir- 
cles has an area equal to that of the arbelos. 

4. The line tangent to the semicircles AB and BC con- 
tains the point E and F which lie on the lines AD 
and CD, respectively. Furthermore, BD and EF bi- 
sect each other, and the points B, D, E, and F are 
CONCYCLIC. 



Arbelos 



Arc Length 61 




5. In addition to the ARCHIMEDES' CIRCLES C± and C 2 
in the arbelos figure, there is a third circle Cz called 
the Bankoff Circle which is congruent to these 
two. 




6. Construct a chain of TANGENT CIRCLES starting 

with the Circle Tangent to the two small ones 
and large one. The centers of the CIRCLES lie on 
an Ellipse, and the Diameter of the nth Cir- 
cle C n is (l/n)th Perpendicular distance to the 
base of the Semicircle. This result is most eas- 
ily proven using INVERSION, but was known to Pap- 
pus, who referred to it as an ancient theorem (Hood 
1961, Cadwell 1966, Gardner 1979, Bankoff 1981). If 
r = AB/AC, then the radius of the nth circle in the 
Pappus Chain is 

_ (1 — r)r 



n 2[n 2 (l-r) 2 +r]" 

This general result simplifies to r n = 1/(6 -f n 2 ) for 
r = 2/3 (Gardner 1979). Further special cases when 
AC = 1 + AB are considered by Gaba (1940). 

If B divides AC in the GOLDEN RATIO 0, then the 
circles in the chain satisfy a number of other special 
properties (Bankoff 1955). 




see also Archimedes' Circles, Bankoff Circle, 
Coxeter's Loxodromic Sequence of Tangent 



Circles, Golden Ratio, Inversion, Pappus Chain, 
Steiner Chain 

References 

Bankoff, L. "The Fibonacci Arbelos." Scripta Math. 20, 

218, 1954. 
Bankoff, L. "The Golden Arbelos." Scripta Math. 21, 70-76, 

1955. 
Bankoff, L. "Are the Twin Circles of Archimedes Really 

Twins?" Math. Mag. 47, 214-218, 1974, 
Bankoff, L. "How Did Pappus Do It?" In The Mathematical 

Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, 

and Schmidt, pp. 112-118, 1981. 
Bankoff, L. "The Marvelous Arbelos." In The Lighter Side of 

Mathematics (Ed. R. K. Guy and R. E. Woodrow). Wash- 
ington, DC: Math. Assoc. Amer., 1994. 
Cadwell, J. H. Topics in Recreational Mathematics. Cam- 
bridge, England: Cambridge University Press, 1966. 
Gaba, M. G. "On a Generalization of the Arbelos." Amer. 

Math. Monthly 47, 19-24, 1940. 
Gardner, M. "Mathematical Games: The Diverse Pleasures 

of Circles that Are Tangent to One Another." Sci. Amer. 

240, 18-28, Jan. 1979. 
Heath, T. L. The Works of Archimedes with the Method of 

Archimedes. New York: Dover, 1953. 
Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134- 

137, 1961. 
Johnson, R. A. Modern Geometry; An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 116-117, 1929. 
Ogilvy, C S. Excursions in Geometry. New York: Dover, 

pp. 54-55, 1990. 

Arborescence 

A Digraph is called an arborescence if, from a given 
node x known as the ROOT, there is exactly one ele- 
mentary path from x to every other node y. 

see also Arboricity 

Arboricity 

Given a GRAPH G, the arboricity is the MINIMUM num- 
ber of line-disjoint acyclic SUBGRAPHS whose UNION is 
G. 
see also ANARBORICITY 

Arc 

In general, any smooth curve joining two points. In 
particular, any portion (other than the entire curve) of 
a Circle or Ellipse. 

see also APPLE, ClRCLE-ClRCLE INTERSECTION, FlVE 

Disks Problem, Flower of Life, Lemon, Lens, 
Piecewise Circular Curve, Reuleaux Polygon, 
Reuleaux Triangle, Salinon, Seed of Life, Tri- 
angle Arcs, Venn Diagram, Yin- Yang 

Arc Length 

Arc length is defined as the length along a curve, 



J a 



\d£\. 



(1) 



Defining the line element ds 2 = \d£\ 2 , parameterizing 
the curve in terms of a parameter t, and noting that 



62 Arc Minute 



Archimedes Algorithm 



ds/dt is simply the magnitude of the VELOCITY with 
which the end of the Radius Vector r moves gives 



= / ds = I ft dt= I |r ' ( * )|dt - (2) 



In Polar Coordinates, 



d£ = rdr + r§d6= (^-r + rd\ dd, (3) 



so 



ds=\de\ = X /r*+(j£) d0 



In Cartesian Coordinates, 
di = x± + yy 



Therefore, if the curve is written 

r(x) = xx-\- f(x)y, 
then 



J a 



* = / x/l + f' 2 {x)dx. 

If the curve is instead written 

r(t) = x(t)x + y(t)y t 
then 



J a 



(4) 



=J m =C^ 2+ (%) 2de - (5) 



(6) 



ds= ^dx 2 + dy 2 = A/(£) +ldx. (7) 



(8) 



(9) 



(10) 



'= I ^x"(t) + y*(t)dt. (11) 

J a 

Or, in three dimensions, 

r(t) = x(t)x + y(t)y + z(t)z, (12) 



(t)+y' 2 {t) + z' 2 (t)dt. (13) 



see also Curvature, Geodesic, Normal Vector, 
Radius of Curvature, Radius of Torsion, Speed, 
Surface Area, Tangential Angle, Tangent Vec- 
tor, Torsion (Differential Geometry), Veloc- 
ity 

Arc Minute 

A unit of Angular measure equal to 60 Arc Seconds, 
or 1/60 of a DEGREE. The arc minute is denoted ' (not 
to be confused with the symbol for feet). 



Arc Second 

A unit of Angular measure equal to 1/60 of an Arc 
MINUTE, or 1/3600 of a DEGREE. The arc second is de- 
noted " (not to be confused with the symbol for inches). 

Arccosecant 

see Inverse Cosecant 

Arccosine 

see Inverse Cosine 

Arccotangent 

see Inverse Cotangent 

Arch 




A 4-POLYHEX. 

References 

Gardner, M. Mathematical Magic Show: More Puzzles, 
Games, Diversions, Illusions and Other Mathematical 
Sleight- of- Mind from Scientific American. New York: 
Vintage, p. 147, 1978. 

Archimedes Algorithm 

Successive application of ARCHIMEDES' RECURRENCE 
FORMULA gives the Archimedes algorithm, which can 
be used to provide successive approximations to it (Pi). 
The algorithm is also called the Borchardt-Pfaff Al- 
gorithm. Archimedes obtained the first rigorous ap- 
proximation of TV by Circumscribing and Inscribing 
n = 6 • 2 fe -gons on a CIRCLE. Prom ARCHIMEDES' RE- 
CURRENCE Formula, the Circumferences a and b of 
the circumscribed and inscribed POLYGONS are 



a(n) = 2ntan ( — ) 
b(n) = 2nsin ( — ) , 



(i) 

(2) 



where 



b(n) < C = 27rr = 2tt • 1 = 2tt < a(n). (3) 

For a Hexagon, n = 6 and 

a = a(6) = 4\/3 (4) 



feo = 6(6) = 6, 



(5) 



where a^ = a(6-2 k ). The first iteration of ARCHIMEDES' 
Recurrence Formula then gives 

2-6 -4^3 24^ nA , n /-, 

ffll = 7Tvr = ^vi = 24(2 -^ ) (6) 

h = yj 24(2 - V3) • 6 = 12\/2- \/3 

-6(v / 6-v / 2). (7) 



Archimedes 7 Axiom 



Archimedes' Cattle Problem 



63 



Additional iterations do not have simple closed forms, 
but the numerical approximations for k = 0, 1, 2, 3, 4 
(corresponding to 6-, 12-, 24-, 48-, and 96-gons) are 



3.00000 < TV < 3.46410 
3.10583 <tt< 3.21539 
3.13263 < 7T < 3.15966 
3.13935 < TV < 3.14609 
3.14103 < 7T < 3.14271. 



(8) 

(9) 

(10) 

(11) 
(12) 



By taking k = 4 (a 96-gon) and using strict inequalities 
to convert irrational bounds to rational bounds at each 
step, Archimedes obtained the slightly looser result 



^ =3.14084... <tt < f 



: 3.14285. 



(13) 



References 

Miel, G. "Of Calculations Past and Present: The Archimed- 
ean Algorithm." Amer. Math. Monthly 90, 17-35, 1983. 

Phillips, G. M. "Archimedes in the Complex Plane." Amer. 
Math. Monthly 91, 108-114, 1984. 

Archimedes' Axiom 

An Axiom actually attributed to Eudoxus (Boyer 1968) 
which states that 

a/6 = c/d 

IFF the appropriate one of following conditions is satis- 
fied for Integers m and n: 

1. If ma < nb, then mc < md. 

2. If ma — rid, then mc = nd. 

3. If ma > nd, then mc > nd. 

Archimedes' Lemma is sometimes also known as Arch- 
imedes' axiom. 

References 

Boyer, C. B. A History of Mathematics. New York: Wiley, 
p. 99, 1968. 

Archimedes' Cattle Problem 

Also called the Bovinum PROBLEMA. It is stated as 
follows: "The sun god had a herd of cattle consisting of 
bulls and cows, one part of which was white, a second 
black, a third spotted, and a fourth brown. Among the 
bulls, the number of white ones was one half plus one 
third the number of the black greater than the brown; 
the number of the black, one quarter plus one fifth the 
number of the spotted greater than the brown; the num- 
ber of the spotted, one sixth and one seventh the number 
of the white greater than the brown. Among the cows, 
the number of white ones was one third plus one quarter 
of the total black cattle; the number of the black, one 
quarter plus one fifth the total of the spotted cattle; the 
number of spotted, one fifth plus one sixth the total of 
the brown cattle; the number of the brown, one sixth 
plus one seventh the total of the white cattle. What 
was the composition of the herd?" 



Solution consists of solving the simultaneous DlOPHAN- 

tine Equations in Integers W, X, Y, Z (the number 

of white, black, spotted, and brown bulls) and w y x y y, z 
(the number of white, black, spotted, and brown cows), 



w ^ \x + z 



_9_ 
20 J 

42 ' 
_7_ 
12 v 



Y + Z 
W + Z 
(X + x) 
x =±(Y + y) 

(W + w). 



_ 13 



(i) 

(2) 
(3) 
(4) 
(5) 
(6) 
(7) 



The smallest solution in INTEGERS is 



W = 10,366,482 


(8) 


X = 7,460,514 


(9) 


Y = 7,358,060 


(10) 


Z = 4,149,387 


(11) 


w= 7,206,360 


(12) 


x = 4,893,246 


(13) 


y= 3,515,820 


(14) 


z = 5,439,213. 


(15) 



A more complicated version of the problem requires that 
W+X be a Square Number and Y+Z a Triangular 
Number. The solution to this Problem are numbers 
with 206544 or 206545 digits. 

References 

Amthor, A. and Krumbiegel B. "Das Problema bovinum des 
Archimedes." Z. Math. Phys. 25, 121-171, 1880. 

Archibald, R. C. "Cattle Problem of Archimedes." Amer. 
Math. Monthly 25, 411-414, 1918. 

Beiler, A. H. Recreations in the Theory of Numbers: The 
Queen of Mathematics Entertains. New York: Dover, 
pp. 249-252, 1966. 

Bell, A. H. "Solution to the Celebrated Indeterminate Equa- 
tion x 2 - ny 2 = 1." Amer. Math. Monthly 1, 240, 1894. 

Bell, A. H. "'Cattle Problem.' By Archimedes 251 BC." 
Amer. Math. Monthly 2, 140, 1895. 

Bell, A. H. "Cattle Problem of Archimedes." Math. Mag. 1, 
163, 1882-1884. 

Calkins, K. G. "Archimedes' Problema Bovinum." http:// 
www. andrews . edu/-calkins/cattle .html. 

Dorrie, H. "Archimedes' Problema Bovinum" §1 in 100 
Great Problems of Elementary Mathematics: Their His- 
tory and Solutions. New York: Dover, pp. 3-7, 1965. 

Grosjean, C. C. and de Meyer, H. E. "A New Contribution 
to the Mathematical Study of the Cattle-Problem of Arch- 
imedes." In Constantin Caratheodory: An International 
Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ: 
World Scientific, pp. 404-453, 1991. 

Merriman, M. "Cattle Problem of Archimedes." Pop. Sci. 
Monthly 67, 660, 1905. 

Rorres, C. "The Cattle Problem." http: //www. mcs.drexel. 
edu/-crorres/Archimedes/Cattle/Statement .html. 

Vardi, I. "Archimedes' Cattle Problem." Amer. Math. 
Monthly 105, 305-319, 1998. 



64 Archimedes' Circles 

Archimedes' Circles 




Draw the PERPENDICULAR LINE from the intersection 
of the two small SEMICIRCLES in the ARBELOS. The 
two Circles Ci and Ci Tangent to this line, the large 
SEMICIRCLE, and each of the two SEMICIRCLES are then 
congruent and known as Archimedes' circles. 

see also ARBELOS, BANKOFF CIRCLE, SEMICIRCLE 

Archimedes' Constant 

see Pi 

Archimedes' Hat-Box Theorem 

Enclose a Sphere in a Cylinder and slice Perpen- 
dicularly to the Cylinder's axis. Then the Surface 
Area of the of Sphere slice is equal to the Surface 
Area of the Cylinder slice. 

Archimedes' Lemma 

Also known as the continuity axiom, this Lemma sur- 
vives in the writings of Eudoxus (Boyer 1968). It states 
that, given two magnitudes having a ratio, one can find 
a multiple of either which will exceed the other. This 
principle was the basis for the EXHAUSTION METHOD 
which Archimedes invented to solve problems of Area 
and Volume. 
see also Continuity Axioms 

References 

Boyer, C. B. A History of Mathematics. New York: Wiley, 
p. 100, 1968. 

Archimedes' Midpoint Theorem 




Let M be the Midpoint of the Arc AMB. Pick C 
at random and pick D such that MD _L AC (where J_ 
denotes PERPENDICULAR). Then 

AD = DC + BC. 



see also MIDPOINT 

References 

Honsberger, R. More Mathematical Morsels. 
DC: Math. Assoc. Amer., pp. 31-32, 1991. 



Washington, 



Archimedes 7 Recurrence Formula 

Archimedes' Postulate 

see Archimedes' Lemma 

Archimedes' Problem 

Cut a Sphere by a Plane in such a way that the VOL- 
UMES of the Spherical Segments have a given Ratio. 

see also SPHERICAL SEGMENT 
Archimedes' Recurrence Formula 





Let a n and b n be the Perimeters of the Circum- 
scribed and Inscribed n-gon and a2 n and fen the 
Perimeters of the Circumscribed and Inscribed 2n- 

gon. Then 



di 



2a n b n 



a n + b n 

&2n = V d2nK • 



(1) 

(2) 



The first follows from the fact that side lengths of the 
Polygons on a Circle of Radius r = 1 are 



SR 



2 tan 

2 sin 



CD 



But 



a n = 2ntan ( — ) 

b n — 2nsin f — 1 . 

2an b n 2-2ntan(i)-2nsin(^) 



(3) 
(4) 

(5) 
(6) 



a n +b n 2ntan(^) +2nsin(^) 
tan(S) sin (?) 



An 



tan(^)+sin(^)- 



Using the identity 



tan(|x) = 



tan x sin x 
tan x + sin x 



then gives 



2a n frn 
a n + b n 



— 4ntan 



(- 

V2n 



2n) 



&2n 



(7) 



(8) 



(9) 



Archimedean Solid 

The second follows from 



Archimedean Solid 



65 



\Zd2nbn = W4ntan ( — j • 2nsin (-) (10) 



Using the identity 

since = 2sin(|x) cos(|x) 
gives 



(ii) 



y^X = 2n v /2tan (£) - 2 sin (£) cos ( j) 

= 4n v sin2 (£) = 4nsin (£) = b2n - (12) 

Successive application gives the Archimedes Algo- 
rithm, which can be used to provide successive approx- 
imations to Pi (it). 
see also ARCHIMEDES ALGORITHM, Pi 

References 

Dorrie, H. 100 Great Problems of Elementary Mathematics: 

Their History and Solutions. New York: Dover, p. 186, 

1965. 

Archimedean Solid 

The Archimedean solids are convex Polyhedra which 
have a similar arrangement of nonintersecting regu- 
lar plane Convex Polygons of two or more differ- 
ent types about each VERTEX with all sides the same 
length. The Archimedean solids are distinguished from 

the Prisms, Antiprisms, and Elongated Square 

GYROBICUPOLA by their symmetry group: the Arch- 
imedean solids have a spherical symmetry, while the 
others have "dihedral" symmetry. The Archimedean 
solids are sometimes also referred to as the SEMIREG- 
ular Polyhedra. 

Pugh (1976, p. 25) points out the Archimedean solids 
are all capable of being circumscribed by a regular Tet- 
rahedron so that four of their faces lie on the faces 
of that Tetrahedron. A method of constructing the 
Archimedean solids using a method known as "expan- 
sion" has been enumerated by Stott (Stott 1910; Ball 
and Coxeter 1987, pp. 139-140). 

Let the cyclic sequence S = (pi,P2, . . . ,p q ) represent the 
degrees of the faces surrounding a vertex (i.e., S is a list 
of the number of sides of all polygons surrounding any 
vertex). Then the definition of an Archimedean solid 
requires that the sequence must be the same for each 
vertex to within ROTATION and REFLECTION. Walsh 
(1972) demonstrates that S represents the degrees of the 
faces surrounding each vertex of a semiregular convex 
polyhedron or TESSELLATION of the plane IFF 

1. q > 3 and every member of S is at least 3, 

2. ^2? =1 ~ > \q — 1, with equality in the case of a 
plane TESSELLATION, and 



3. for every ODD NUMBER p £ 5, S contains a subse- 
quence (6, p, 6). 

Condition (1) simply says that the figure consists of two 
or more polygons, each having at least three sides. Con- 
dition (2) requires that the sum of interior angles at a 
vertex must be equal to a full rotation for the figure to 
lie in the plane, and less than a full rotation for a solid 
figure to be convex. 

The usual way of enumerating the semiregular polyhe- 
dra is to eliminate solutions of conditions (1) and (2) 
using several classes of arguments and then prove that 
the solutions left are, in fact, semiregular (Kepler 1864, 
pp. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940, 
p. 394; Coxeter et al. 1954; Lines 1965, pp. 202-203; 
Walsh 1972). The following table gives all possible reg- 
ular and semiregular polyhedra and tessellations. In 
the table, 'P } denotes PLATONIC SOLID, 'M' denotes a 
PRISM or ANTIPRISM, 'A' denotes an Archimedean solid, 
and 'T' a plane tessellation. 



Fg. Solid 



Schlafli 



3,3) 

4,4) 

6,6) 

8,8) 

10, 10) 

12, 12) 

4,n) 

4, 4) 

6,6) 

6,8) 

6,10) 

6,12) 

8, 8) 

5,5) 

6,6) 

6,6) 

3, 3, n) 

3, 3, 3) 

4, 3, 4) 

5, 3, 5) 

6, 3, 6) 
4, 4, 4) 
4, 5, 4) 
4, 6, 4) 
4, 4, 4) 
o, o, o, 

3, 3, 3, 

o, o, o, 

o, o, o, 

3, 3, 4, 

3, 4, 3, 

3, 3, 3, 



P tetrahedron {3j3} 

M triangular prism t{2,3} 

A truncated tetrahedron t{3, 3} 

A truncated cube t{4, 3} 

A truncated dodecahedron t{5,3} 

T (plane tessellation) t{6,3} 

M n-gonal Prism t{2,n} 

P cube {4, 3} 

A truncated octahedron t{3,4} 

A great rhombicuboct. 

A great rhombicosidodec. 

T (plane tessellation) 

T (plane tessellation) 

P dodecahedron 

A truncated icosahedron 

T (plane tessellation) 

M n-gonal antiprism 

P octahedron 

A cuboctahedron 

A icosidodecahedron 

T (plane tessellation) 

A small rhombicuboct. 

A small rhombicosidodec. 

T (plane tessellation) 

T (plane tessellation) 

P icosahedron 

A snub cube 

A snub dodecahedron 

T (plane tessellation) 

T (plane tessellation) — 

T (plane tessellation) s 1 4 J 

T (plane tessellation) {3,6} 



As shown in the above table, there are exactly 13 Ar- 
chimedean solids (Walsh 1972, Ball and Coxeter 1987). 



66 



Archimedean Solid 



Archimedean Solid 



They are called the CUBOCTAHEDRON, GREAT RHOMB- 
ICOSIDODECAHEDRON, GREAT RHOMBICUBOCTAHE- 
DRON, ICOSIDODECAHEDRON, SMALL RHOMBICOSIDO- 
DECAHEDRON, SMALL RHOMBICUBOCTAHEDRON, SNUB 

Cube, Snub Dodecahedron, Truncated Cube, 
Truncated Dodecahedron, Truncated Icosahe- 
dron (soccer ball), Truncated Octahedron, and 
Truncated Tetrahedron. The Archimedean solids 
satisfy 

(27T- <t)V — 4tt, 

where a is the sum of face- angles at a vertex and V is 
the number of vertices (Steinitz and Rademacher 1934, 
Ball and Coxeter 1987). 

Here are the Archimedean solids shown in alphabetical 
order (left to right, then continuing to the next row). 





ry 


\ / 


^m 


n A , 










Li 


/^ 


aM 


LV 





The following table lists the symbol and number of faces 
of each type for the Archimedean solids (Wenninger 
1989, p. 9). 



Solid 


Schlafli 


Wythoff 


C&R 


cuboctahedron 


i 3 \ 

X 4 1 


2 | 34 


(3.4) 2 


great rhombicosidodecahedron 


*{*} 


2 3 5 | 




great rhombicuboctahedron 


*{:} 


234 | 




icosidodecahedron 


/ 3 \ 
1 5 J 


2 | 3 5 


(3-5) 2 


small rhombicosidodecahedron 


'it) 


3 5 | 2 


3.4.5.4 


small rhombicuboctahedron 


r l:l 


3 4)2 


3.4 3 


snub cube 


s i:i 


| 2 3 4 


3 4 .4 


snub dodecahedron 


*{*} 


| 2 3 5 


3 4 .5 


truncated cube 


t{4,3} 


2 3 | 4 


3.8 2 


truncated dodecahedron 


t{5,3} 


23[5 


3.10 2 


truncated icosahedron 


t{3,5} 


2 5 | 3 


5.6 2 


truncated octahedron 


t{3,4} 


2 4 | 3 


4.6 2 


truncated tetrahedron 


t{3,3} 


23 | 3 


3.6 2 



Solid 


V 


e 


h 


h 


A 


h 


h 


/io 


cuboctahedron 


12 


24 


8 


6 










great rhombicos. 


120 


180 




30 




2G 




12 


great rhombicub. 


48 


72 




12 




8 


6 




icosidodecahedron 


30 


60 


20 




12 








small rhombicos. 


60 


120 


20 


30 


12 








small rhombicub. 


24 


48 


8 


18 










snub cube 


24 


60 


32 


6 










snub dodecahedron 


60 


150 


80 




12 








trunc. cube 


24 


36 


8 








6 




trunc. dodec. 


60 


90 


20 










12 


trunc. icosahedron 


60 


90 






12 


20 






trunc. octahedron 


24 


36 




6 




8 






trunc. tetrahedron 


12 


18 


4 






4 







Let r be the INRADIUS, p the MIDRADIUS, and R the 
ClRCUMRADIUS. The following tables give the analytic 
and numerical values of r, p, and R for the Archimedean 
solids with EDGES of unit length. 



Solid 


r 


cuboctahedron 

great rhombicosidodecahedron 

great rhombicuboctahedron 

icosidodecahedron 

small rhombicosidodecahedron 

small rhombicuboctahedron 
snub cube 
snub dodecahedron 
truncated cube 

truncated dodecahedron 
truncated icosahedron 
truncated octahedron 
truncated tetrahedron 


3 

4 


aii (105 + 6^5 )\/31 4- 12 VE 


£(14 + >/2)\/l3 + 6^ 

±(5 + 3^5) 

^(15 + 2^)^11 + 4^5 


^r(6 + v / 2)V /s + 2 v / 2 
* 
* 


£(5 + 2^)^7 + 4^ 

4§s (17V2 + 3</l0 ) ^37 + ISn/5 

? f^(21 + Vo")V /58 + 18 v / 5 

£v^2 



Archimedean Solid 



Archimedean Solid 67 



Solid 


P 


"i2 


cuboctahedron 

great rhombicosidodecahedron 


ivs. 


1 


1^/30 + 12^ 


1^31 + 12 V5 


great rhombicuboctahedron 


IA/12 + 6X/2 


I ^13 + 6x72 


icosidodecahedron 

small rhombicosidodecahedron 


§\/ 5 + 2 >/5 


|- V / 11 + 4 ^ 


\y/io + ±y/z 


small rhombicuboctahedron 


J-/4 + 2V2 


| ^5 + 2y/2 


snub cube 


* 


* 


snub dodecahedron 
truncated cube 


* 
i(2 + v/2) 


* 


i ^/V + 4V5 


truncated dodecahedron 


^(5 + 3^) 


^V /t4 + 3 °v / 5 


truncated icosahedron 


ia + V5) 


JV/58+18X/5 


truncated octahedron 


3 

2 


Iv'lO 


truncated tetrahedron 


f\/2 


IV22 



*The complicated analytic expressions for the ClRCUM- 
RADII of these solids are given in the entries for the SNUB 
Cube and Snub Dodecahedron. 



Solid 


r 


P 


R 


cuboctahedron 


0.75 


0.86603 


1 


great rhombicosidodecahedron 


3.73665 


3.76938 


3.80239 


great rhombicuboctahedron 


2.20974 


2.26303 


2.31761 


icosidodecahedron 


1.46353 


1.53884 


1.61803 


small rhombicosidodecahedron 


2.12099 


2.17625 


2.23295 


small rhombicuboctahedron 


1.22026 


1.30656 


1.39897 


snub cube 


1.15766 


1.24722 


1.34371 


snub dodecahedron 


2.03987 


2.09705 


2.15583 


truncated cube 


1.63828 


1.70711 


1.77882 


truncated dodecahedron 


2.88526 


2.92705 


2.96945 


truncated icosahedron 


2.37713 


2.42705 


2.47802 


truncated octahedron 


1.42302 


1.5 


1.58114 


truncated tetrahedron 


0.95940 


1.06066 


1.17260 



The Duals of the Archimedean solids, sometimes called 
the Catalan Solids, are given in the following table. 



Archimedean Solid 



Dual 



rhombicosidodecahedron 
small rhombicuboctahedron 
great rhombicuboctahedron 
great rhombicosidodecahedron 
truncated icosahedron 

snub dodecahedron (laevo) 

snub cube (laevo) 

cuboctahedron 
icosidodecahedron 
truncated octahedron 
truncated dodecahedron 
truncated cube 
truncated tetrahedron 



deltoidal hexecontahedron 
deltoidal icositetrahedron 
disdyakis dodecahedron 
disdyakis triacontahedron 
pentakis dodecahedron 
pentagonal hexecontahedron 

(dextro) 
pentagonal icositetrahedron 

(dextro) 
rhombic dodecahedron 
rhombic triacontahedron 
tctrakis hexahedron 
triakis icosahedron 
triakis octahedron 
triakis tetrahedron 



Here are the Archimedean DUALS (Holden 1971, Pearce 
1978) displayed in alphabetical order (left to right, then 
continuing to the next row). 




Here are the Archimedean solids paired with their DU- 
ALS. 




The Archimedean solids and their DUALS are all 
Canonical Polyiiedra. 

see also Archimedean Solid Stellation, Cata- 
lan Solid, Deltahedron, Johnson Solid, Kepler- 
Poinsot Solid, Platonic Solid, Semiregular 
Polyhedron, Uniform Polyhedron 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 136, 
1987. 

Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). 
Fundamentals of Mathematics, Vol. 2. Cambridge, MA: 
MIT Press, pp. 269-286, 1974. 

Catalan, E. "Memoire sur la Theorie des Polyedres." J. 
I'Ecole Polytechnique (Paris) 41, 1-71, 1865. 

Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six 
and Seven Dimensions." Proc. Cambridge Phil Soc. 24, 
1-9, 1928. 

Coxeter, H. S. M. "Regular and Semi- Regular Polytopes I." 
Math. Z. 46, 380-407, 1940. 

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: 
Dover, 1973. 

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, 
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon- 
don Ser. A 246, 401-450, 1954. 

Critchlow, K. Order in Space: A Design Source Book. New 
York: Viking Press, 1970. 



68 



Archimedean Solid Stellation 



Archimedes' Spiral 



Cromwell, P. R. Polyhedra. New York: Cambridge University 

Press, pp. 79-86, 1997. 
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 

Stradbroke, England: Tarquin Pub., 1989. 
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 

p. 54, 1991. 
Kepler, J. "Harmonice Mundi." Opera Omnia, Vol. 5. 

Frankfurt, pp. 75-334, 1864. 
Kraitchik, M. Mathematical Recreations. New York: 

W. W. Norton, pp. 199-207, 1942. 
Le, Ha. "Archimedean Solids." http : //daisy, uwaterloo. 

ca/~hqle/archimedean.htnil. 
Pearce, P. Structure in Nature is a Strategy for Design. Cam- 
bridge, MA: MIT Press, pp. 34-35, 1978. 
Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer- 
sity of California Press, p. 25, 1976. 
Rawles, B. A. "Platonic and Archimedean Solids — Faces, 

Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios." 

http://www.intent.com/sg/polyhedra.html. 
Rorres, C. "Archimedean Solids: Pappus." http://www.mcs. 

drexel.edu/-crorres/Archimedes/Solids/Pappus.html. 
Steinitz, E. and Rademacher, H. Vorlesungen uber die The- 

orie der Polyheder. Berlin, p. 11, 1934. 
Stott, A. B. Verhandelingen der Konniklijke Akad. Weten- 

schappen, Amsterdam 11, 1910. 
Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods 

of Semi- Regular Polyhedra." Geometriae Dedicata 1, 117- 

123, 1972. 
Wenninger, M. J. Polyhedron Models. New York: Cambridge 

University Press, 1989. 

Archimedean Solid Stellation 

A large class of Polyhedra which includes the Do- 

DECADODECAHEDRON and GREAT ICOSIDODECAHE- 
DRON. No complete enumeration (even with restrictive 
uniqueness conditions) has been worked out. 

References 

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, 
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon- 
don Ser. A 246, 401-450, 1954. 

Wenninger, M. J. Polyhedron Models. New York: Cambridge 
University Press, pp. 66-72, 1989. 

Archimedean Spiral 

A Spiral with Polar equation 

r = a0 1/7n , 



see also Archimedes' Spiral, Daisy, Fermat's Spi- 
ral, Hyperbolic Spiral, Lituus, Spiral 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993. 

Lauweirer, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 59- 
60, 1991. 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 186 and 189, 1972. 

Lee, X. "Archimedean Spiral." http://www.best.com/-xah/ 
Special Plane Curves _ dir / Archimedean Spiral _ dir / 
archimedeanSpiral .html. 

Lockwood, E. H. A Book of Curves. Cambridge, England: 
Cambridge University Press, p. 175, 1967. 

MacTutor History of Mathematics Archive. "Spiral of Arch- 
imedes." http: // www - groups . dcs . st - and .ac.uk/ 
-history/Curves/Spiral.html. 

Pappas, T. "The Spiral of Archimedes." The Joy of Mathe- 
matics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 
1989. 

Archimedean Spiral Inverse Curve 

The Inverse Curve of the Archimedean Spiral 



1/77 



aO 



with Inversion Center at the origin and inversion Ra- 
dius k is the Archimedean Spiral 

r = ka6 l/m . 



Archimedes' Spiral 




An Archimedean Spiral with Polar equation 



where r is the radial distance, 6 is the polar angle, and m 
is a constant which determines how tightly the spiral is 
"wrapped." The Curvature of an Archimedean spiral 
is given by 

_ n(9 1 - 1 / n (l + n + n 2 l9 2 ) 
K ~ a(l + n 2 2 ) 3 / 2 

Various special cases are given in the following table. 



Name 



lituus 

hyperbolic spiral 
Archimedes' spiral 
Fermat's spiral 



m 



-2 
-1 

1 
2 



This spiral was studied by Conon, and later by Archi- 
medes in On Spirals about 225 BC. Archimedes was able 
to work out the lengths of various tangents to the spiral. 

Archimedes' spiral can be used for COMPASS and 
Straightedge division of an Angle into n parts (in- 
cluding Angle Trisection) and can also be used for 
Circle Squaring. In addition, the curve can be used 
as a cam to convert uniform circular motion into uni- 
form linear motion. The cam consists of one arch of the 
spiral above the cc-AxiS together with its reflection in 
the z-AxiS. Rotating this with uniform angular veloc- 
ity about its center will result in uniform linear motion 
of the point where it crosses the y-AxiS. 



Archimedes' Spiral Inverse 



Area-Preserving Map 69 



see also ARCHIMEDEAN SPIRAL 

References 

Gardner, M. The Unexpected Hanging and Other Mathemat- 
ical Diversions. Chicago, IL: Chicago University Press, 
pp. 106-107, 1991. 

Gray, A- Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993. 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 186-187, 1972. 

Lockwood, E. H. A Book of Curves. Cambridge, England: 
Cambridge University Press, pp. 173 164, 1967. 

Archimedes' Spiral Inverse 

Taking the ORIGIN as the INVERSION CENTER, ARCHI- 
MEDES' Spiral r = aO inverts to the Hyperbolic Spi- 
ral r = a/6. 

Archimedean Valuation 

A Valuation for which |a;| < 1 Implies |1 + z| < C for 
the constant C — 1 (independent of x). Such a VALUA- 
TION does not satisfy the strong TRIANGLE INEQUALITY 

\x + y\< maxO|,|y|). 



Arcsecant 

see Inverse Secant 

Arcsine 

see Inverse Sine 



Calculus and, in particular, the Integral, are power- 
ful tools for computing the AREA between a curve f(x) 
and the x-Axis over an INTERVAL [a, 6], giving 



A = f f(x) 

J a 



c)dx. (6) 

The Area of a Polar curve with equation r = r(8) is 
A= | fr 2 dO. (7) 

Written in CARTESIAN COORDINATES, this becomes 

*-j/(-2-'i)* (8 > 



-\! 



(xdy — ydx). 



(9) 



For the AREA of special surfaces or regions, see the en- 
try for that region. The generalization of AREA to 3-D 
is called Volume, and to higher Dimensions is called 
Content. 

see also ARC LENGTH, AREA ELEMENT, CONTENT, 

Surface Area, Volume 

References 

Gray, A. "The Intuitive Idea of Area on a Surface." §13.2 

in Modern Differential Geometry of Curves and Surfaces. 

Boca Raton, FL: CRC Press, pp. 259-260, 1993. 



Arctangent 

see Inverse Tangent 

Area 

The Area of a Surface is the amount of material 
needed to "cover" it completely. The AREA of a Trian- 
gle is given by 

A A = \lh, (1) 

where I is the base length and h is the height, or by 
Heron's Formula 



Aa = a/ s(s — a)(s — b)($ — c), 



(2) 



where the side lengths are a, b, and c and s the 
Semiperimeter. The Area of a Rectangle is given 
by 



^rectangle 



a6, 



(3) 



where the sides are length a and b. This gives the special 
case of 

^square = & yQ) 

for the Square. The Area of a regular Polygon with 
n sides and side length s is given by 



-^Ti-gon — 4^^ COt I 1 



(5) 



Area Element 

The area element for a Surface with Riemannian 
Metric 

ds 2 = Edu 2 + 2Fdudv + Gdv 2 



dA = y^EG - F 2 du A dv, 

where du A dv is the WEDGE PRODUCT. 

see also Area, Line Element, Riemannian Metric, 

Volume Element 

References 

Gray, A. "The Intuitive Idea of Area on a Surface." §13.2 
in Modern Differential Geometry of Curves and Surfaces, 
Boca Raton, FL; CRC Press, pp. 259-260, 1993. 

Area-Preserving Map 

A Map F from R n to W 1 is AREA-preserving if 

m{F(A)) = m(A) 

for every subregion A of M n , where m(A) is the n- 
D Measure of A. A linear transformation is AREA- 
preserving if its corresponding DETERMINANT is equal 
to 1. 
see also Conformal Map, Symplectic Map 



70 Area Principle 

Area Principle 





The "AREA principle" states that 

|i4iP| _ \A,BC\ 



\A 2 P\ \A 2 BC\' 
This can also be written in the form 



[ AiP -1 = \AiBCl 
IA 2 P\ [A2BCI ' 



where 



AB 



CD 



(1) 



(2) 



(3) 



is the ratio of the lengths [A, B] and [C, D] for AB\\CD 
with a PLUS or MINUS SIGN depending on if these seg- 
ments have the same or opposite directions, and 



ABC 1 



DEFGl 



(4) 



is the Ratio of signed Areas of the Triangles. 

Griinbaum and Shepard show that Ceva'S THEOREM, 
Hoehn's Theorem, and Menelaus' Theorem are the 
consequences of this result. 

see also Ceva's Theorem, Hoehn's Theorem, Men- 
elaus' Theorem, Self-Transversality Theorem 

References 

Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the 
Area Principle." Math. Mag. 68, 254-268, 1995. 

Areal Coordinates 

Trilinear Coordinates normalized so that 

tl +*2+*3 = 1. 

When so normalized, they become the AREAS of the 
Triangles PAiA 2 , PAiA$, and PA 2 A 3 , where P is 
the point whose coordinates have been specified. 

Arf Invariant 

A LINK invariant which always has the value or 1. 
A Knot has Arf Invariant if the Knot is "pass 
equivalent" to the UNKNOT and 1 if it is pass equiv- 
alent to the Trefoil Knot. If iC+, if_, and L are 
projections which are identical outside the region of the 
crossing diagram, and K+ and K- are Knots while L 
is a 2-component LINK with a nonintersecting crossing 



Argoh's Conjecture 

diagram where the two left and right strands belong to 
the different LINKS, then 



a{K+)=a(K-) + l{L u L 2 ), 



(1) 



where I is the Linking Number of L\ and L 2 - The 
Arf invariant can be determined from the ALEXANDER 
Polynomial or Jones Polynomial for a Knot. For 
A K the Alexander Polynomial of K, the Arf invari- 
ant is given by 



a*(-i; 



■{i 



(mod 8) 
5 (mod 8) 



if Arf(K) = 
if Arf(J0 = 1 



(2) 



(Jones 1985). For the Jones Polynomial W K of a 
Knot K , 

Arf(K) = W K (i) (3) 

(Jones 1985), where i is the Imaginary Number. 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 
to the Mathematical Theory of Knots. New York: W. H. 
Freeman, pp. 223-231, 1994. 

Jones, V. "A Polynomial Invariant for Knots via von Neu- 
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 
1985. 
# Weisstein, E. W. "Knots." http://www. astro. Virginia. 
edu/-eww6n/math/notebooks/Knots.m. 

Argand Diagram 

A plot of Complex Numbers as points 

z = x + iy 

using the x-AxiS as the Real axis and y-AxiS as the 
Imaginary axis. This is also called the Complex 
Plane or Argand Plane. 

Argand Plane 

see Argand Diagram 

Argon's Conjecture 

Let B k be the fcth BERNOULLI NUMBER. Then does 

nBn-i = —1 (mod n) 

Iff n is Prime? For example, for n = 1, 2, . . . , nB n -i 
(mod n) is 0, -1, -1, 0, -1, 0, -1, 0, -3, 0, -1, .... 
There are no counterexamples less than n = 5, 600. Any 
counterexample to Argon's conjecture would be a con- 
tradiction to Giuga's Conjecture, and vice versa. 

see also BERNOULLI NUMBER, GlUGA'S CONJECTURE 
References 

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- 
sohn, R. "Giuga's Conjecture on Primality." Amer. Math. 
Monthly 103, 40-50, 1996. 



Argument Addition Relation 



Aristotle's Wheel Paradox 



71 



Argument Addition Relation 

A mathematical relationship relating f(x + y) to f(x) 

and f(y). 

see also ARGUMENT MULTIPLICATION RELATION, 

Recurrence Relation, Reflection Relation, 
Translation Relation 



Argument (Complex Number) 

A Complex Number z may be represented as 

z = x + iy = \z\e ld , 



(i) 



where \z\ is called the Modulus of z, and is called the 
argument 

wg(x + iy) = tern' 1 (^j. (2) 

Therefore, 

arg(^) = argGzlc^Me"") = oxg(e ie 'e i&v> ) 

= arg[e i( ^ + ^ } ] = arg(z) + arg(u/). (3) 



Extending this procedure gives 

arg(z n ) = narg(z). 



(4) 



The argument of a COMPLEX NUMBER is sometimes 
called the PHASE. 

see also Affix, Complex Number, de Moivre's 
Identity, Euler Formula, Modulus (Complex 
Number), Phase, Phasor 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 16, 1972. 

Argument (Elliptic Integral) 

Given an Amplitude <f> in an Elliptic Integral, the 
argument u is defined by the relation 

4> = am u. 
see also Amplitude, Elliptic Integral 

Argument (Function) 

An argument of a FUNCTION /(xi,...,x n ) is one of 
the n parameters on which the function's value de- 
pends. For example, the SINE since is a one-argument 
function, the BINOMIAL Coefficient (™) is a two- 
argument function, and the Hypergeometric Func- 
tion 2 Fi (a, b; c; z) is a four-argument function. 

Argument Multiplication Relation 

A mathematical relationship relating f(nx) to f(x) for 
Integer n. 

see also Argument Addition Relation, Recur- 
rence Relation, Reflection Relation, Transla- 
tion Relation 



Argument Principle 

If f(z) is MEROMORPHIC in a region R enclosed by a 
curve 7, let N be the number of COMPLEX ROOTS of 
f(z) in 7, and P be the number of POLES in 7, then 

J(z)dz 
2ttz 



N 



J_ [ f'(z)d. 
2iriL f(z) 



Defining w = f(z) and a = / (7) gives 

1 f dw 
2-xi I w 



N ■ 



see also VARIATION OF ARGUMENT 

References 

Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Ar- 
gument Principle for Harmonic Functions." Math. Mag. 
103, 411-415, 1996. 

Argument Variation 

see Variation of Argument 

Aristotle's Wheel Paradox 



<a=® 



A PARADOX mentioned in the Greek work Mechanica, 
dubiously attributed to Aristotle. Consider the above 
diagram depicting a wheel consisting of two concen- 
tric Circles of different Diameters (a wheel within 
a wheel). There is a 1:1 correspondence of points on 
the large CIRCLE with points on the small CIRCLE, so 
the wheel should travel the same distance regardless of 
whether it is rolled from left to right on the top straight 
line or on the bottom one. This seems to imply that 
the two Circumferences of different sized Circles 
are equal, which is impossible. 

The fallacy lies in the assumption that a 1:1 correspon- 
dence of points means that two curves must have the 
same length. In fact, the CARDINALITIES of points in 
a Line Segment of any length (or even an Infinite 
Line, a Plane, a 3-D Space, or an infinite dimensional 
Euclidean Space) are all the same: Hi (Aleph-1), so 
the points of any of these can be put in a One-TO-One 
correspondence with those of any other. 
see also ZENO'S PARADOXES 

References 

Ballew, D. "The Wheel of Aristotle." Math. Teacher 65, 
507-509, 1972. 

Costabel, P. "The Wheel of Aristotle and French Considera- 
tion of Galileo's Arguments." Math. Teacher 61, 527-534, 
1968. 

Drabkin, I. "Aristotle's Wheel: Notes on the History of the 
Paradox." Osiris 9, 162-198, 1950. 

Gardner, M. Wheels, Life, and other Mathematical Amuse- 
ments. New York: W. H. Freeman, pp. 2-4, 1983. 

Pappas, T. "The Wheel of Paradox Aristotle." The Joy of 
Mathematics. San Carlos, CA: Wide World Publ./Tetra, 
p. 202, 1989. 

vos Savant, M. The World's Most Famous Math Problem. 
New York: St. Martin's Press, pp. 48-50, 1993. 



72 



Arithmetic 



Arithmetic 

The branch of mathematics dealing with Integers 
or, more generally, numerical computation. Arithmeti- 
cal operations include Addition, Congruence cal- 
culation, Division, Factorization, Multiplication, 
Power computation, Root extraction, and SUBTRAC- 
TION. 

The Fundamental Theorem of Arithmetic, also 
called the Unique Factorization Theorem, states 
that any Positive Integer can be represented in ex- 
actly one way as a PRODUCT of PRIMES. 

The Lowenheimer-Skolem Theorem, which is a fun- 
damental result in Model Theory, establishes the ex- 
istence of "nonstandard" models of arithmetic. 
see also Algebra, Calculus, Fundamental The- 
orem of Arithmetic, Group Theory, Higher 
Arithmetic, Linear Algebra, Lowenheimer- 
Skolem Theorem, Model Theory, Number The- 
ory, Trigonometry 

References 

Karpinski, L. C. The History of Arithmetic. Chicago, IL: 

Rand, McNally, & Co., 1925. 
Maxfield, J. E. and Maxfield, M. W. Abstract Algebra and 

Solution by Radicals. Philadelphia, PA: Saunders, 1992. 
Thompson, J. E. Arithmetic for the Practical Man. New 

York: Van Nostrand Reinhold, 1973. 

Arithmetic-Geometric Mean 

The arithmetic-geometric mean (AGM) M(a, b) of two 
numbers a and b is defined by starting with clq = a and 
bo = &, then iterating 



CLn + l = 2 ( a ™ + kn) 



b n + l = yCLnbn 



(1) 

(2) 



until a n = b n . a n and b n converge towards each other 
since 



a n +i - b n +i = \{a n -\- b n ) - ydnb n 



a n — 2\/a n b n + b n 



(3) 



But "s/Sn < V^"» SO 



2b n < 2^a n b n . (4) 

Now, add a n — b n — 2y/a n b n to each side 

a n + b n — 2\/a n b n < a n — b n > (5) 



CLn + l - b n + l < 2^ an ~ bn)- 



(6) 



The AGM is very useful in computing the values of 
complete Elliptic Integrals and can also be used 
for finding the INVERSE TANGENT. The special value 
l/M(l,\/2) is called Gauss's Constant. 



Arithmetic- Geometric Mean 

The AGM has the properties 

AM(a,6) = M(Aa,A6) (7) 

M(a,6) = M(£(a + 6),>/S) (8) 

M(l, V 1 - x 2 ) = M (! + s, 1 ~ x) (9) 

The Legendre form is given by 

Af(l,x) = JJi(l + fc„), (11) 

where ko = x and 

Solutions to the differential equation 



(x 3 -z)^| + (3x 2 - l)^-+xy = (13) 

ax* ax 



are given by [M(l + x, 1 - x)] 1 and[M(l,x)] 1 . 

A generalization of the Arithmetic-Geometric 
Mean is 



f°° x p ~ 2 dx 



(14) 



which is related to solutions of the differential equation 

x(l-x p )Y" + [l-fr+l)x p ]Y'-(p-l)x p - 1 Y = 0. (15) 

When p = 2 or p = 3, there is a modular transformation 
for the solutions of (15) that are bounded as x — > 0. Let- 
ting J p (x) be one of these solutions, the transformation 
takes the form 

J p (\) = M J p (z), (16) 



where 



A 



1-u 



and 



1 + (p - l)u 
1 + (p - l)u 



x p + u p = 1. 



(17) 
(18) 

(19) 



The case p = 2 gives the Arithmetic- Geometric 
Mean, and p = 3 gives a cubic relative discussed by 
Borwein and Borwein (1990, 1991) and Borwein (1996) 
in which, for a, b > and I(a, b) defined by 



/(a, 6) 



Jo V& + 



tdt 



*3)(6 3 +f 3 ) 2 ] 1 / 3 ' 



(20) 



Arithmetic Geometry 

Iia , b) = l(^,[^+ab + b 2 )]). (21) 

For iteration with ao = a and bo = 6 and 



a n +i 



a n + 26 n 



fcn + l = — (dn + fln&Ti + b n ) 



lim a n = lim b n 



Hhi) 

I(a,b)- 



(22) 
(23) 

(24) 



Modular transformations are known when p = 4 and 

p = 6, but they do not give identities for p = 6 (Borwein 

1996). 

see also Arithmetic-Harmonic Mean 

References 

Abramowitz, M. and Stegun, C. A. (Eds.), "The Process 
of the Arithmetic-Geometric Mean." §17.6 in Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
pp. 571 ad 598-599, 1972. 

Borwein, J. M. Problem 10281. "A Cubic Relative of the 
AGM." Amer, Math. Monthly 103, 181-183, 1996. 

Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic It- 
eration." In Computational Method & Function Theory: 
Proc. Conference Held in Valparaiso, Chile, March 13- 
18, i9SP0387527680 (Ed. A. Dold, B. Eckmann, F. Tak- 
ens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C, and 
R, S. Varga). New York: Springer- Vcrlag, 1990. 

Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of 
Jacobi's Identity and the AGM." Trans. Amer. Math. Soc. 
323, 691-701, 1991. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 906-907, 1992. 

Arithmetic Geometry 

A vaguely defined branch of mathematics dealing with 

Varieties, the Mordell Conjecture, Arakelov 
Theory, and Elliptic Curves. 

References 

Cornell, G. and Silverman, J. H. (Eds.). Arithmetic Geome- 
try. New York: Springer- Verlag, 1986. 

Lorenzini, D. An Invitation to Arithmetic Geometry. Provi- 
dence, RI: Amer. Math. Soc, 1996. 

Arithmetic-Harmonic Mean 

Let 



a n+ x = \{a n + b n ) 



b n 



+i 



a n + b n 



Then 

A(ao,bo) = lim a n = lim b n 

n — ► oo n— »-oo 

which is just the GEOMETRIC MEAN. 



(1) 

(2) 
'aobo, (3) 



Arithmetic Mean 73 

Arithmetic-Logarithmic-Geometric Mean 
Inequality 

a + b b — a i—r 

—— > — — : — > Vab. 
2 In o — In a 

see also Napier's Inequality 

References 

Nelson, R. B. "Proof without Words: The Arithmetic- 
Logarithmic-Geometric Mean Inequality." Math. Mag. 
68, 305, 1995. 

Arithmetic Mean 

For a Continuous Distribution function, the arith- 
metic mean of the population, denoted /*, x, {x) t or 
A(x) t is given by 



-/. 



H=(f(x))= / P(x)f(x)dx, 



(1) 



where (x) is the EXPECTATION VALUE. For a DISCRETE 

Distribution, 

„ = </(*)> ss E ^°/ (a " )/( ; n) = 5><*.)/<*.). 

l^n = Q F V Xn ) n=0 

(2) 

The population mean satisfies 

{f(x)+g(x)) = {f(x)) + (g(x)} (3) 

<c/(x))=c </(*)>, (4) 



and 



{f(x)g(y)) = </(*)> (g(y)) 



(5) 



if x and y are INDEPENDENT STATISTICS. The "sample 
mean," which is the mean estimated from a statistical 
sample, is an UNBIASED ESTIMATOR for the population 
mean. 

For small samples, the mean is more efficient than the 
Median and approximately tt/2 less (Kenney and Keep- 
ing 1962, p. 211). A general expression which often holds 
approximately is 

mean — mode « 3(mean — median). (6) 

Given a set of samples {a;*}, the arithmetic mean is 

N 

A(x) =x = hee{x) = ^^2 Xi ' W 

Hoehn and Niven (1985) show that * 

A(a± +c,a 2 +c, . ..,a n +c) = c + j4(ai,a2,...,a n ) (8) 

for any POSITIVE constant c. The arithmetic mean sat- 
isfies 

(9) 



74 Arithmetic Mean 



Arithmetic Progression 



where G is the Geometric Mean and H is the Har- 
monic Mean (Hardy et al. 1952; Mitrinovic 1970; Beck- 
enbach and Bellman 1983; Bullen et ah 1988; Mitrinovic 
et al. 1993; Alzer 1996). This can be shown as follows. 
For a, b > 0, 







P--^Y> 



1 2 1 rt 

1 1^2 

- + r > -7= 



a ~ b 

H>G, 



(10) 

(11) 
(12) 

(13) 
(14) 



with equality Iff b = a. To show the second part of the 
inequality, 

{yfa-Vbf = a-2\/a6 + &> (15) 



<> + b 



> Vab 



2 

A> H< 



(16) 

(17) 

with equality Iff a = b. Combining (14) and (17) then 
gives (9). 

Given n independent random GAUSSIAN DISTRIBUTED 
variates #», each with population mean fii = \i and 
Variance <n 2 = a 2 , 









= ^E^=^)=/i, (19) 

Z = l 

so the sample mean is an Unbiased Estimator of 
population mean. However, the distribution of x de- 
pends on the sample size. For large samples, x is ap- 
proximately Normal. For small samples, Student's 
^-Distribution should be used. 

The Variance of the population mean is independent 
of the distribution. 

var(z) = var I - > Xi\ = —— var > x { 

n N 2 



From /c-Statistics for a GAUSSIAN DISTRIBUTION, the 
Unbiased Estimator for the Variance is given by 



N 



N - 1 



where 






var (a:) = 



JV-1 



The Square Root of this, 

s 

is called the Standard Error. 

var(x) = (% 2 ) ~~ (^) 2 > 



(21) 



(22) 



(23) 



(24) 



(25) 



(x 2 )=var(x) + (a-) 2 = ^+/A (26) 

see also Arithmetic-Geometric Mean, Arith- 
metic-Harmonic Mean, Carleman's Inequal- 
ity, Cumulant, Generalized Mean, Geomet- 
ric Mean, Harmonic Mean, Harmonic-Geometric 
Mean, Kurtosis, Mean, Mean Deviation, Median 
(Statistics), Mode, Moment, Quadratic Mean, 
Root-Mean-Square, Sample Variance, Skewness, 
Standard Deviation, Trimean, Variance 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

p. 10, 1972. 
Alzer, H. "A Proof of the Arithmetic Mean-Geometric Mean 

Inequality." Amer. Math. Monthly 103, 585, 1996. 
Beckenbach, E. F. and Bellman, R. Inequalities. New York: 

Springer- Verlag, 1983. 
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 

Boca Raton, FL: CRC Press, p. 471, 1987. 
Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means & 

Their Inequalities. Dordrecht, Netherlands: Reidel, 1988. 
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities. 

Cambridge, England: Cambridge University Press, 1952. 
Hoehn, L. and Niven, I. "Averages on the Move." Math. 

Mag. 58, 151-156, 1985. 
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, 

Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. 
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical 

and New Inequalities in Analysis. Dordrecht, Netherlands: 

Kluwer, 1993. 
Vasic, P. M. and Mitrinovic, D. S. Analytic Inequalities. New 

York: Springer- Verlag, 1970. 

Arithmetic Progression 

see Arithmetic Series 



(20) 



Arithmetic Sequence 



Arnold's Cat Map 75 



Arithmetic Sequence 

A Sequence of n numbers {do 4- kd} 7 ^ such that the 
differences between successive terms is a constant d. 

see also ARITHMETIC SERIES, SEQUENCE 

Arithmetic Series 

An arithmetic series is the Sum of a SEQUENCE {a^}, 
k = 1, 2, ..., in which each term is computed from 
the previous one by adding (or subtracting) a constant. 
Therefore, for k > 1, 



a>k = a,k-i + d = afc-2 + 2d • 



:ai+d(fc-l). (1) 



The sum of the sequence of the first n terms is then 
given by 

n n 

S n = ]Ta fc =J^[ai + (* - l)d] 

k=l k=l 

n n 

= nai + d^ik — 1) = noi + d /(& - 1) 

fc = l k = 2 

n-l 

= nai -\- dj k (2) 



Using the SUM identity 



]T = §n(n+l) 



(3) 



then gives 

S n = nai + \d(n - 1) = \ n[2ai + d(n - 1)]. (4) 
Note, however, that 

ai + a n = ai + [a\ + d(n — 1)] — 2ai + d(n — 1), (5) 



5 n = \n{a\ +a n ), 



(6) 



or n times the AVERAGE of the first and last terms! 
This is the trick Gauss used as a schoolboy to solve 
the problem of summing the INTEGERS from 1 to 100 
given as busy-work by his teacher. While his classmates 
toiled away doing the ADDITION longhand, Gauss wrote 
a single number, the correct answer 



|(100)(1 + 100) = 50 ■ 101 = 5050 



(7) 



on his slate. When the answers were examined, Gauss's 
proved to be the only correct one. 

see also Arithmetic Sequence, Geometric Series, 
Harmonic Series, Prime Arithmetic Progression 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 



Mathematical Tables, 9th printing. New York: Dover, 
p. 10, 1972. 

Beyer, W. H. (Ed.), CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. 

Courant, R. and Robbins, H. "The Arithmetical Progres- 
sion." §1.2.2 in What is Mathematics?: An Elementary 
Approach to Ideas and Methods, 2nd ed. Oxford, England: 
Oxford University Press, pp. 12-13, 1996. 

Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide 
World Publ./Tetra, p. 164, 1989. 

Armstrong Number 

The n-digit numbers equal to sum of nth powers of their 
digits (a finite sequence), also called PLUS PERFECT 
NUMBERS. They first few are given by 1, 2, 3, 4, 5, 
6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 
... (Sloane's A005188). 

see also NARCISSISTIC NUMBER 

References 

Sloane, N. J. A. Sequence A005188/M0488 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Arnold's Cat Map 

The best known example of an ANOSOV DlFFEOMOR- 
PHISM. It is given by the TRANSFORMATION 



(i) 



where x n +i and y n +i are computed mod 1. The Arnold 
cat mapping is non-Hamiltonian, nonanalytic, and mix- 
ing. However, it is Area- Preserving since the Deter- 
minant is 1. The Lyapunov Characteristic Expo- 
nents are given by 



Xn+l 


= 


"l l" 

1 2 




x n 
y n _ 



l-a 1 

1 2-<T 



3(7 + 1 = 0, 



(2) 



ct± = |(3±v / 5). (3) 

The Eigenvectors are found by plugging <r± into the 
Matrix Equation 



1 



1 

2 - cr± 



(4) 



For <r+, the solution is 

y=\{l + ^)x = 4>x, (5) 

where <j> is the GOLDEN RATIO, so the unstable (normal- 
ized) Eigenvector is 



^+ = ^\ / 50-10v / 5 



1(1 + V5) 



Similarly, for <j- , the solution is 

y = -±(V5-l)x~(/>- 1 x y 
so the stable (normalized) Eigenvector is 



£_ = ^\/50 + 10v / 5 
see also Anosov Map 



1(1 -v/5) 



(6) 



(7) 



(8) 



76 



Arnold Diffusion 



Array 



Arnold Diffusion 

The nonconservation of ADIABATIC INVARIANTS which 
arises in systems with three or more DEGREES OF FREE- 
DOM. 

Arnold Tongue 

Consider the Circle Map. If K is Nonzero, then 
the motion is periodic in some FINITE region surround- 
ing each rational Q. This execution of periodic motion 
in response to an irrational forcing is known as MODE 
LOCKING. If a plot is made of K versus Q with the re- 
gions of periodic MODE-LOCKED parameter space plot- 
ted around rational ft values (the WINDING Numbers), 
then the regions are seen to widen upward from at 
K = to some FINITE width at K = 1. The region 
surrounding each RATIONAL NUMBER is known as an 
Arnold Tongue. 

At K — 0, the Arnold tongues are an isolated set of 
MEASURE zero. At K = 1, they form a general CAN- 
TOR Set of dimension d w 0.8700. In general, an Arnold 
tongue is defined as a resonance zone emanating out 
from RATIONAL NUMBERS in a two-dimensional param- 
eter space of variables. 
see also Circle Map 

Aronhold Process 

The process used to generate an expression for a covari- 
ant in the first degree of any one of the equivalent sets 

of Coefficients for a curve. 

see also Clebsch-Aronhold Notation, Joachims- 

thal's Equation 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- 
York: Dover, p. 74, 1959. 

Aronson's Sequence 

The sequence whose definition is: "t is the first, fourth, 
eleventh, . . . letter of this sentence." The first few val- 
ues are 1, 4, 11, 16, 24, 29, 33, 35, 39, ... (Sloane's 
A005224). 

References 

Hofstadter, D. R. Metamagical Themas: Questing of Mind 

and Pattern. New York: BasicBooks, p. 44, 1985. 
Sloane, N. J. A. Sequence A005224/M3406 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Arrangement 

In general, an arrangement of objects is simply a group- 
ing of them. The number of "arrangements" of n items 
is given either by a COMBINATION (order is ignored) or 
Permutation (order is significant). 

The division of Space into cells by a collection of Hy- 
PERPLANES is also called an arrangement. 

see also COMBINATION, CUTTING, HYPERPLANE, OR- 
DERING, Permutation 



Arrangement Number 

see Permutation 

Array 

An array is a "list of lists" with the length of each 
level of list the same. The size (sometimes called the 
"shape") of a d-dimensional array is then indicated as 
m x n x • • • x p. The most common type of array en- 

d 
countered is the 2-D m x n rectangular array having m 

columns and n rows. If m = n, a square array results. 
Sometimes, the order of the elements in an array is sig- 
nificant (as in a MATRIX), whereas at other times, arrays 
which are equivalent modulo reflections (and rotations, 
in the case of a square array) are considered identical 
(as in a MAGIC SQUARE or PRIME ARRAY). 

In order to exhaustively list the number of distinct ar- 
rays of a given shape with each element being one of k 
possible choices, the naive algorithm of running through 
each case and checking to see whether it's equivalent to 
an earlier one is already just about as efficient as can 
be. The running time must be at least the number of 
answers, and this is so close to k mn '" p that the difference 
isn't significant. 

However, finding the number of possible arrays of a given 
shape is much easier, and an exact formula can be ob- 
tained using the POLYA ENUMERATION THEOREM. For 
the simple case of an m x n array, even this proves un- 
necessary since there are only a few possible symmetry 
types, allowing the possibilities to be counted explicitly. 
For example, consider the case of m and n EVEN and 
distinct, so only reflections need be included. To take a 
specific case, let m = 6 and n = 4 so the array looks like 



a 


b 


c 


1 


d 


e 


f 


9 


h 


i 


1 

+ 

1 


3 


k 


I 


m 


n 





V 


Q 


r 


s 


t 


u 


1 


V 


w 


X, 



where each a, 6, . . . , x can take a value from 1 to k. The 
total number of possible arrangements is k 24 (k mn in 
general). The number of arrangements which are equiv- 
alent to their left-right mirror images is k 1 (in general, 
k mn/2 ), as is the number equal to their up-down mirror 
images, or their rotations through 180°. There are also 
k Q arrangements (in general, fc mn/4 ) with full symmetry. 

In general, it is therefore true that 



jL7TiTl/4 

j^mn/2 _ fcmn/4 
femn/2 _ pn/4 
femn/2 _ femn/4 



with full symmetry 
with only left-right reflection 
with only up-down reflection 
with only 180° rotation, 



so there are 



3k 



ran/2 , rw mn./4 



Arrow Notation 



Artin Braid Group 77 



arrangements with no symmetry. Now dividing by the 
number of images of each type, the result, for m -£ n 
with m, n EVEN, is 



N(m,n,k) = |A; mn + (|)(3)(fc mn/2 - A; mn/4 ) 

mn i 
+ 1*" 



4. \(k mn -3k mn/2 + 2k mn/4 ) 



4\ 
lfcrnn _,_ 3 j.mn/2 _,_ lj.mn/4 



+ ifc" 



The number is therefore of order C>(fc mn /4), with "cor- 
rection" terms of much smaller order. 

see also Antimagic Square, Euler Square, 
Kirkman's Schoolgirl Problem, Latin Rect- 
angle, Latin Square, Magic Square, Matrix, 
Mrs. Perkins' Quilt, Multiplication Table, Or- 
thogonal Array, Perfect Square, Prime Array, 
Quotient-Difference Table, Room Square, Sto- 
larsky Array, Truth Table, Wythoff Array 

Arrow Notation 

A Notation invented by Knuth (1976) to represent 
Large Numbers in which evaluation proceeds from the 
right (Conway and Guy 1996, p. 60). 



m t n 



m * m- - -m 



n 

m ttt n m tt m tt ■ ' ' tt m 

n 

For example, 

m t n ~ m n 

m"[ J \-2 = m J [m~m^m = rn 71 

2 

m tt 3 = m t rn t m = m t ( m t rn) 

v v / 

3 

= m|m m = m mm 
m ttt 2 = mtt^ = ^tt^ = mm 

m ttt 3 = m tt rn tt ™ = m tt ™ m 

v v ' ' ^~ 

3 m 

= m t • • * t rn — m m 



(1) 
(2) 

(3) 
(4) 



(5) 



m tt n is sometimes called a Power Tower. The 
values nt • * • t n are called ACKERMANN NUMBERS. 



see also Ackermann Number, Chained Arrow No- 
tation, Down Arrow Notation, Large Number, 
Power Tower, Steinhaus-Moser Notation 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New- 
York: Springer- Verlag, pp. 59-62, 1996. 

Guy, R. K. and Selfridge, J. L. "The Nesting and Roost- 
ing Habits of the Laddered Parenthesis." Amer. Math. 
Monthly 80, 868-876, 1973. 

Knuth, D. E. "Mathematics and Computer Science: Coping 
with Finiteness. Advances in Our Ability to Compute are 
Bringing Us Substantially Closer to Ultimate Limitations." 
Science 194, 1235-1242, 1976. 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addison- Wesley, pp. 11 and 226-229, 1991. 

Arrow's Paradox 

Perfect democratic voting is, not just in practice but in 
principle, impossible. 

References 

Gardner, M. Time Travel and Other Mathematical Bewilder- 
ments. New York: W. H. Freeman, p. 56, 1988. 

Arrowhead Curve 

see Sierpinski Arrowhead Curve 

Art Gallery Theorem 

Also called Chvatal's Art Gallery Theorem. If 
the walls of an art gallery are made up of n straight 
Lines Segments, then the entire gallery can always be 
supervised by [n/3\ watchmen placed in corners, where 
[x\ is the Floor Function. This theorem was proved 
by V. Chvatal in 1973. It is conjectured that an art 
gallery with n walls and h HOLES requires [(n + h)/3j 
watchmen. 

see also Illumination Problem 

References 

Honsberger, R. "Chvatal's Art Gallery Theorem." Ch. 11 
in Mathematical Gems II. Washington, DC: Math. Assoc. 
Amer., pp. 104-110, 1976. 

O'Ronrke, J. Art Gallery Theorems and Algorithms. New- 
York: Oxford University Press, 1987. 

Stewart, I. "How Many Guards in the Gallery?" Sci. Amer. 
270, 118-120, May 1994. 

Tucker, A. "The Art Gallery Problem." Math Horizons, 
pp. 24-26, Spring 1994. 

Wagon, S. "The Art Gallery Theorem." §10.3 in Mathema- 
tica in Action. New York: W. H. Freeman, pp. 333-345, 
1991. 

Articulation Vertex 

A VERTEX whose removal will disconnect a GRAPH, also 
called a Cut- Vertex. 

see also Bridge (Graph) 

References 

Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introduc- 
tory Graph Theory. New York: Dover, pp. 45—49, 1985. 

Artin Braid Group 

see Braid Group 



78 Artin's Conjecture 



Artistic Series 



Artin's Conjecture 

There are at least two statements which go by the name 
of Artin's conjecture. The first is the RlEMANN HY- 
POTHESIS. The second states that every INTEGER not 
equal to —1 or a SQUARE NUMBER is a primitive root 
modulo p for infinitely many p and proposes a density 
for the set of such p which are always rational multi- 
ples of a constant known as ARTIN'S CONSTANT. There 
is an analogous theorem for functions instead of num- 
bers which has been proved by Billharz (Shanks 1993, 
p. 147). 

see also ARTIN'S CONSTANT, RlEMANN HYPOTHESIS 

References 

Shanks, D. Solved and Unsolved Problems in Number Theory, 
4th ed. New York: Chelsea, pp. 31, 80-83, and 147, 1993. 

Artin's Constant 

If n / -1 and n is not a PERFECT SQUARE, then Artin 
conjectured that the SET S(n) of all PRIMES for which n 
is a PRIMITIVE ROOT is infinite. Under the assumption 
of the Extended Riemann Hypothesis, Artin's con- 
jecture was solved in 1967 by C. Hooley. If, in addition, 
n is not an rth POWER for any r > 1, then Artin con- 
jectured that the density of S(n) relative to the Primes 
is CArtin (independent of the choice of n) , where 



CAn 



n 



1- 



1 



<?(<?-!) 



= 0.3739558136.. 



and the PRODUCT is over Primes. The significance of 
this constant is more easily seen by describing it as the 
fraction of PRIMES p for which 1/p has a maximal DEC- 
IMAL Expansion (Conway and Guy 1996). 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 

York: Springer-Verlag, p. 169, 1996. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/artin/artin.html. 
Hooley, C. "On Artin's Conjecture." J. reine angew. Math. 

225, 209-220, 1967. 
Ireland, K. and Rosen, M. A Classical Introduction to Mod- 
ern Number Theory, 2nd ed. New York: Springer-Verlag, 

1990. 
Ribenboim, P. The Book of Prime Number Records. New 

York: Springer-Verlag, 1989. 
Shanks, D. Solved and Unsolved Problems in Number Theory, 

4th ed. New York: Chelsea, pp. 80-83, 1993. 
Wrench, J. W. "Evaluation of Artin's Constant and the Twin 

Prime Constant." Math. Comput. 15, 396-398, 1961. 

Artin L- Function 

An Artin L-function over the Rationals Q encodes in 
a Generating Function information about how an 
irreducible monic POLYNOMIAL over Z factors when re- 
duced modulo each PRIME. For the POLYNOMIAL x 2 + l, 
the Artin L-function is 



L(s,Q(i)/Q,sgn): 



n itt^ 



where ( — 1/p) is a Legendre Symbol, which is equiv- 
alent to the Euler L-FUNCTION. The definition over 
arbitrary POLYNOMIALS generalizes the above expres- 
sion. 
see also Langlands Reciprocity 

References 

Knapp, A. W. "Group Representations and Harmonic Anal- 
ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996. 

Artin Reciprocity 

see Artin's Reciprocity Theorem 

Artin's Reciprocity Theorem 

A general RECIPROCITY Theorem for all orders. If R 
is a NUMBER FIELD and R f a finite integral extension, 
then there is a SURJECTION from the group of fractional 
IDEALS prime to the discriminant, given by the Artin 
symbol. For some cycle c, the kernel of this SURJECTION 
contains each Principal fractional Ideal generated by 
an element congruent to 1 mod c. 

see also LANGLANDS PROGRAM 

Artinian Group 

A GROUP in which any decreasing CHAIN of distinct 
Subgroups terminates after a Finite number. 

Artinian Ring 

A noncommutative Semisimple RING satisfying the 

"descending chain condition." 

see also GORENSTEIN RING, SEMISIMPLE RING 

References 

Artin, E. "Zur Theorie der hyperkomplexer Zahlen." Hamb. 

Abh. 5, 251-260, 1928. 
Artin, E. "Zur Arithmetik hyperkomplexer Zahlen." Hamb. 

Abh. 5, 261-289, 1928. 

Artistic Series 

A Series is called artistic if every three consecutive 
terms have a common three-way ratio 



-P[ai,ai+i,a; + 2] 



(ai + aj+i + ai+2)ai+i 



aiOi+2 



A Series is also artistic Iff its BIAS is a constant. A 
Geometric Series with Ratio r > is an artistic 

series with 

P=i + l + r>3. 

r 

see also BIAS (SERIES), GEOMETRIC SERIES, MELODIC 

Series 

References 

Duffin, R. J. "On Seeing Progressions of Constant Cross Ra- 
tio." Amer. Math. Monthly 100, 38-47, 1993. 



p odd prime 



(?)*" 



ASA Theorem 

ASA Theorem 




Specifying two adjacent ANGLES A and B and the side 
between them c uniquely determines a Triangle with 
Area 



K = 



2(coti4 + cot£)" 
The angle C is given in terms of A and B by 

C = n-A-B, 



(1) 



(2) 



and the sides a and b can be determined by using the 
Law of Sines 



to obtain 



sin A sin B sin C 



sin A 



sin(7r — A — B) 

sinB 
sin(7r — A — B) 



(3) 

(4) 
(5) 



see also AAA Theorem, AAS Theorem, ASS Theo- 
rem, SAS Theorem, SSS Theorem, Triangle 

Aschbacher's Component Theorem 

Suppose that E(G) (the commuting product of all com- 
ponents of G) is SIMPLE and G contains a SEMISIM- 
ple Involution. Then there is some Semisimple 
Involution x such that C G (x) has a Normal Sub- 
group K which is either QUASISIMPLE or ISOMORPHIC 
to + (4,q)' and such that Q — C G {K) is Tightly Em- 
bedded. 

see also Involution (Group), Isomorphic Groups, 
Normal Subgroup, Quasisimple Group, Simple 
Group, Tightly Embedded 

ASS Theorem 




c c c 

Specifying two adjacent side lengths a and b of a TRIAN- 
GLE (taking a > b) and one ACUTE ANGLE A opposite 
a does not, in general, uniquely determine a triangle. 
If sin A < a/cy there are two possible TRIANGLES satis- 
fying the given conditions. If sin A = a/c, there is one 
possible Triangle. If sin A > a/c, there are no possible 
TRIANGLES. Remember: don't try to prove congruence 
with the ASS theorem or you will make make an ASS 
out of yourself. 

see also AAA Theorem, AAS Theorem, SAS Theo- 
rem, SSS Theorem, Triangle 



Associative Magic Square 79 

Associative 

In simple terms, let x, y, and z be members of an Al- 
gebra. Then the Algebra is said to be associative 
if 

x - (y - z) = (x - y) • z, (1) 

where • denotes MULTIPLICATION. More formally, let A 
denote an IR-algebra, so that A is a VECTOR SPACE over 

Rand 

Ax A-+ A (2) 



(x,y) \->x-y. 



(3) 



Then A is said to be m-associative if there exists an m-D 
Subspace S of A such that 



(y-x)-z = y-(x-z) 



(4) 



for all y,z € A and x € S. Here, VECTOR MULTIPLI- 
CATION x • y is assumed to be Bilinear. An n-D n- 
associative ALGEBRA is simply said to be "associative." 

see also COMMUTATIVE, DISTRIBUTIVE 
References 

Finch, S. "Zero Structures in Real Algebras." http://www. 
mathsoft.com/asolve/zerodiv/zerodiv.html. 

Associative Magic Square 



1 


15 


24 


8 


17 


23 


7 


16 


5 


14 


20 


4 


13 


22 


6 


12 


21 


10 


19 


3 


9 


18 


2 


11 


25 



An n x n Magic Square for which every pair of num- 
bers symmetrically opposite the center sum to n 2 + 1. 
The Lo Shu is associative but not PANMAGIC. Order 
four squares can be PANMAGIC or associative, but not 
both. Order five squares are the smallest which can be 
both associative and PANMAGIC, and 16 distinct asso- 
ciative PANMAGIC Squares exist, one of which is illus- 
trated above (Gardner 1988). 

see also Magic Square, Panmagic Square 

References 

Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time 
Travel and Other Mathematical Bewilderments. New 
York: W. H, Freeman, 1988. 



80 Astroid 

Astroid 




A 4-cusped HYPOCYCLOID which is sometimes also 
called a Tetracuspid, Cubocycloid, or Paracycle. 
The parametric equations of the astroid can be obtained 
by plugging in n = a/b = 4 or 4/3 into the equations for 
a general HYPOCYCLOID, giving 

x = 3bcos(j> + 6cos(30) = 46 cos 3 <j> — a cos 3 <j) (1) 
y — 36 sin ^ — bs'm(3<f>) = 46 sin <j) = asin <j>. (2) 



In Cartesian Coordinates, 



2/3 . 2/3 2/3 



(3) 



In Pedal Coordinates with the Pedal Point at the 
center, the equation is 



2 , o 2 2 

r + op — a . 



(4) 






v J 




I J 


r ^ 




n 






The Arc Length, Curvature, and Tangential An- 
gle are 



s(t) 



/ |sin(2*')|d*' 
Jo 



f sin 2 t (5) 



K(t) = -|csc(2t) 

4>(t) = -t. 



(6) 
(7) 



As usual, care must be taken in the evaluation of s(t) for 
t > it/ 2. Since (5) comes from an integral involving the 
Absolute Value of a function, it must be monotonic 
increasing. Each QUADRANT can be treated correctly 
by defining 

'»=[fj+l. (8) 

where [x] is the FLOOR FUNCTION, giving the formula 

S (t) = (-l) 1+ l" < mod 2 » | sin 2 t + 3 [|nj . (9) 

The overall Arc Length of the astroid can be com- 
puted from the general HYPOCYCLOID formula 



&a(n - 1) 



(10) 



with n = 4, 



54 = 6a. 



Astroid 



(ll) 



The Area is given by 



An = ("-D("-2) ro . 



with n = 4, 



I- 2 



(12) 



(13) 



The Evolute of an Ellipse is a stretched Hypocy- 
CLOID. The gradient of the TANGENT T from the point 
with parameter p is — tan p. The equation of this TAN- 
GENT T is 



xsinp + ycosp = |asin(2p) 



(14) 



(MacTutor Archive). Let T cut the z-Axis and the y- 
Axis at X and Y, respectively. Then the length XY is 
a constant and is equal to a. 



t 

L 




The astroid can also be formed as the ENVELOPE pro- 
duced when a Line Segment is moved with each end 
on one of a pair of PERPENDICULAR axes (e.g., it is the 
curve enveloped by a ladder sliding against a wall or a 
garage door with the top corner moving along a verti- 
cal track; left figure above). The astroid is therefore 
a GLISSETTE. To see this, note that for a ladder of 
length L, the points of contact with the wall and floor 
are (xo,0) and (0, y/L 2 — xq 1 ), respectively. The equa- 
tion of the Line made by the ladder with its foot at 
(xojO) is therefore 



VL 2 - xq 2 , . 

y — = (x - xo) 



-xq 



(15) 



which can be written 



U{x,y,xo) = y + — (x-x ). (16) 

Xo 

The equation of the Envelope is given by the simulta- 
neous solution of 



U(x,y,x ) = y+ V L xn X ° (x-xo) = 







ax o *oV^ 2 -*o 2 



-0, 



which is 



Xq 



I? 

(L 2 - xq 2 ) 3 / 2 
L 2 



(17) 

(18) 
(19) 



Astroid 






Noting that 








~ 2 

2/3 _ ^0 

~ L 4 /3 


(20) 




r 2 2 
2/3 _ -k — x 

y ~~ TAlZ 


(21) 



Astroid Involute 



81 



allows this to be written implicitly as 



x 2/3 +y 2/3 = L 2/3 , 



the equation of the astroid, as promised. 



A 



/~\ 



slotted 
track 



^ 




(22) 



The related problem obtained by having the "garage 
door" of length L with an "extension" of length AL 
move up and down a slotted track also gives a surprising 
answer. In this case, the position of the "extended" end 
for the foot of the door at horizontal position xo and 
Angle 6 is given by 



x = — ALcosO 



y = \/L 2 - xo 2 + ALsin0. 



(23) 
(24) 




The astroid is also the Envelope of the family of El- 
lipses 



y 



(1 - c)'< 



-1 = 0, 



(30) 



illustrated above. 

see also Deltoid, Ellipse Envelope, Lame Curve, 
Nephroid, Ranunculoid 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 172-175, 1972. 
Lee, X. "Astroid." http://www.best.com/-xah/Special 

PlaneCurves_dir/Astroid_dir/astroid.html. 
Lockwood, E. H. "The Astroid." Ch. 6 in A Book of Curves. 

Cambridge, England: Cambridge University Press, pp. 52- 

61, 1967. 
MacTutor History of Mathematics Archive. "Astroid." 

http: //www-groups .dcs .st-and.ac.uk/-history/Curves 

/Astroid. html. 
Yates, R. C. "Astroid." A Handbook on Curves and Their 

Properties. Ann Arbor, Ml: J. W. Edwards, pp. 1-3, 1952. 



Using 
then gives 



xq = L cos 



AL 



x = —xo 



= yz^^( 1 + ^) 



(25) 

(26) 
(27) 



Astroid Evolute 



Solving (26) for xo, plugging into (27) and squaring then 
gives 



(Ai) 2 
Rearranging produces the equation 



2 f2 Z/V / ALV 



y 



(AL) 2 (L + AL) 2 



(28) 



(29) 



the equation of a (QUADRANT of an) ELLIPSE with 
Semimajor and Semiminor Axes of lengths AL and 

L + AL. 




A Hypocycloid Evolute for n = 4 is another As- 
troid scaled by a factor n/(n — 2) = 4/2 = 2 and 
rotated 1/(2 • 4) = 1/8 of a turn. 

Astroid Involute 





V 


/ 


\ 


/ 


\ 


/ 


\ 


/V- 


--7V 


y \ 


/ ^ 






-- 


^, 


'^i- 


4."" 


\ 
\ 

\ 


/ 
/ 

/ 
/ 
/ 



A Hypocycloid Involute for n = 4 is another As- 
troid scaled by a factor (n — 2)/2 = 2/4 = 1/2 and 
rotated 1/(2 • 4) = 1/8 of a turn. 



82 Astroid Pedal Curve 

Astroid Pedal Curve 



Asymptotic Curve 




The Pedal Curve of an Astroid with Pedal Point 

at the center is a QUADRIFOLIUM. 
Astroid Radial Curve 




The QUADRIFOLIUM 



x = xo + 3a cos t — 3a cos(3£) 
y = y -\- 3a sin t + 3a sin(3t) . 



Astroidal Ellipsoid 

The surface which is the inverse of the ELLIPSOID in the 
sense that it "goes in" where the ELLIPSOID "goes out." 
It. is given by the parametric equations 

x = (acosticosv) 
y = (b sin u cost;) 3 
z — (csinv) 3 

for u € [— 7r/2,7r/2] and v G [— 7r,7r]. The special case 
a = b = c = 1 corresponds to the HYPERBOLIC OCTA- 
HEDRON. 
see also Ellipsoid, Hyperbolic Octahedron 

References 

Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/ 
people/nfytn/ asttxt.htm. 

Asymptosy 

Asymptotic behavior. A useful yet endangered word, 

found rarely outside the captivity of the Oxford English 

Dictionary. 

see also ASYMPTOTE, ASYMPTOTIC 



Asymptote 




asymptotes 
A curve approaching a given curve arbitrarily closely, as 
illustrated in the above diagram. 
see also ASYMPTOSY, ASYMPTOTIC, ASYMPTOTIC 

Curve 

References 

Giblin, P. J. "What is an Asymptote?" Math. Gaz. 56, 
274-284, 1972. 

Asymptotic 

Approaching a value or curve arbitrarily closely (i.e., 
as some sort of Limit is taken). A Curve A which is 
asymptotic to given CURVE C is called the ASYMPTOTE 
of C. 
see also ASYMPTOSY, ASYMPTOTE, ASYMPTOTIC 

Curve, Asymptotic Direction, Asymptotic Se- 
ries, Limit 

Asymptotic Curve 

Given a Regular Surface M, an asymptotic curve 
is formally defined as a curve x(i) on M such that the 
Normal Curvature is in the direction x'(t) for all 
t in the domain of x. The differential equation for the 
parametric representation of an asymptotic curve is 



eu -\-2fuv + gv = 0, 



(i) 



where e, /, and g are second FUNDAMENTAL FORMS. 
The differential equation for asymptotic curves on a 
Monge Patch (u,v,h(u t v)) is 



h uu u + 2h U uU v + h vv v = 0, 



and on a polar patch (r cos0,rsin#, h(r)) is 
ti'(r)r ,2 +ti{r)rd' 2 =0. 



(2) 



(3) 



The images below show asymptotic curves for the EL- 
LIPTIC Helicoid, Funnel, Hyperbolic Paraboloid, 
and Monkey Saddle. 




Asymptotic Direction 



Atiyah-Singer Index Theorem 83 



see also RULED SURFACE 

References 

Gray, A. "Asymptotic Curves," "Examples of Asymp- 
totic Curves," "Using Mathematica to Find Asymptotic 
Curves." §16.1, 16.2, and 16.3 in Modern Differential Ge- 
ometry of Curves and Surfaces. Boca Raton, FL: CRC 
Press, pp, 320-331, 1993. 

Asymptotic Direction 

An asymptotic direction at a point p of a REGULAR 
Surface M e M 3 is a direction in which the NORMAL 
Curvature of M vanishes. 

1. There are no asymptotic directions at an Elliptic 

Point. 

2. There are exactly two asymptotic directions at a HY- 
PERBOLIC Point. 

3. There is exactly one asymptotic direction at a PAR- 
ABOLIC Point. 

4. Every direction is asymptotic at a Planar Point. 

see also ASYMPTOTIC CURVE 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces.Boca, Raton, FL: CRC Press, pp. 270 and 320, 1993. 

Asymptotic Notation 

Let n be a integer variable which tends to infinity and let 
xbea continuous variable tending to some limit. Also, 
let 4>(n) or (j){x) be a positive function and f(n) or f{x) 
any function. Then Hardy and Wright (1979) define 

1. / = 0{(j>) to mean that |/| < A<f> for some constant 
A and all values of n and x y 

2. f = o(<j>) to mean that f/<j> — y 0, 

3. / ~ <j> to mean that f /<j> — > 1, 

4. / -< <j> to mean the same as / = o((f>) 7 

5. f y </> to mean f/<j> — > oo, and 

6. / x <fc to mean Ai<j> < / < A 2 for some positive 
constants A± and A 2 . 

f = o(<j>) implies and is stronger than / = 0(<}>). 

References 

Hardy, G. H. and Wright, E. M. "Some Notation." §1.6 in 
An Introduction to the Theory of Numbers, 5th ed. Oxford, 
England: Clarendon Press, pp. 7-8, 1979. 

Asymptotic Series 

An asymptotic series is a SERIES EXPANSION of a FUNC- 
TION in a variable x which may converge or diverge 
(Erdelyi 1987, p. 1), but whose partial sums can be made 
an arbitrarily good approximation to a given function 
for large enough x. To form an asymptotic series R(x) 
of /(#), written 

/(*) ~ R(x), (1) 



where 



c / \ — i ai i a2 i i an 

S n (x) = a H h -=■ + ... + — -■ 



(3) 



The asymptotic series is defined to have the properties 



lim x n R n (x) = for fixed n 



(4) 



lim x n R n (x) = oo for fixed x. (5) 



Therefore, 



f(x) « 22 anX 



(6) 



in the limit x — > oo. If a function has an asymptotic 
expansion, the expansion is unique. The symbol ~ is 
also used to mean directly Similar. 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 15, 1972. 

Arfken, G. "Asymptotic of Semiconvergent Series." §5.10 in 
Mathematical Methods for Physicists, 3rd ed. Orlando, 
FL: Academic Press, pp. 339-346, 1985. 

Bleistein, N. and Handelsman, R. A. Asymptotic Expansions 
of Integrals. New York: Dover, 1986. 

Copson, E. T. Asymptotic Expansions. Cambridge, England: 
Cambridge University Press, 1965. 

de Bruijn, N. G. Asymptotic Methods in Analysis, 2nd ed. 
New York: Dover, 1982. 

Dingle, R. B. Asymptotic Expansions: Their Derivation and 
Interpretation. London: Academic Press, 1973. 

Erdelyi, A. Asymptotic Expansions. New York: Dover, 1987. 

Morse, P. M. and Feshbach, H. "Asymptotic Series; Method 
of Steepest Descent." §4.6 in Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 434-443, 1953. 

Olver, F. W. J. Asymptotics and Special Functions. New 
York: Academic Press, 1974. 

Wasow, W. R. Asymptotic Expansions for Ordinary Differ- 
ential Equations. New York: Dover, 1987. 

Atiyah-Singer Index Theorem 

A theorem which states that the analytic and topological 
"indices" are equal for any elliptic differential operator 
on an n-D Compact Differentiable C°° boundary- 
less Manifold. 

see also Compact Manifold, Differentiable Man- 
ifold 

References 

Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Op- 
erators on Compact Manifolds." Bull. Amer. Math. Soc. 
69, 322-433, 1963. 

Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Oper- 
ators I, II, III." Ann. Math. 87, 484-604, 1968. 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, p. 4, 1996. 



take 



x n R n (x)=x n [f(x)-S n (x)], 



(2) 



84 



Atkin-Goldwasser-Kilian-Morain Certificate 



Augmented Amicable Pair 



Atkin-Goldwasser-Kilian-Morain Certificate 

A recursive PRIMALITY CERTIFICATE for a PRIME p. 
The certificate consists of a list of 

1. A point on an ELLIPTIC CURVE C 

y 2 - x 3 + 92X + p 3 (mod p) 

for some numbers £2 and #3- 

2. A Prime g with q > (p 1 ^ 4 + l) 2 , such that for 
some other number k and m = kq with k ^ 1, 
mC{X)y,g2 ) g$ ) p) is the identity on the curve, but 
kC(x,y,g2 ) gz,p) is not the identity. This guaran- 
tees PRIMALITY of p by a theorem of Goldwasser 
and Kilian (1986). 

3. Each q has its recursive certificate following it. So if 
the smallest q is known to be PRIME, all the numbers 
are certified PRIME up the chain. 

A Pratt Certificate is quicker to generate for 
small numbers. The Mathematica® (Wolfram Re- 
search, Champaign, IL) task ProvablePrime [n] there- 
fore generates an Atkin-Goldwasser-Kilian-Morain cer- 
tificate only for numbers above a certain limit (10 10 by 
default), and a Pratt CERTIFICATE for smaller num- 
bers. 

see also Elliptic Curve Primality Proving, Ellip- 
tic Pseudoprime, Pratt Certificate, Primality 
Certificate, Witness 

References 

Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal- 
ity Proving." Math. Comput. 61, 29-68, 1993. 

Bressoud, D, M. Factorization and Prime Testing. New 
York: Springer- Verlag, 1989. 

Goldwasser, S. and Kilian, J. "Almost All Primes Can Be 
Quickly Certified." Proc. 18th STOC. pp. 316-329, 1986. 

Morain, F. "Implementation of the Atkin-Goldwasser-Kilian 
Primality Testing Algorithm." Rapport de Recherche 911, 
INRIA, Octobre 1988. 

Schoof, R. "Elliptic Curves over Finite Fields and the Com- 
putation of Square Roots mod p." Math. Comput. 44, 
483-494, 1985. 

Wunderlich, M. C "A Performance Analysis of a Simple 
Prime-Testing Algorithm." Math. Comput. 40, 709-714, 
1983. 

Atomic Statement 

In LOGIC, a statement which cannot be broken down 
into smaller statements. 

Attraction Basin 

see Basin of Attraction 

Attractor 

An attractor is a Set of states (points in the Phase 
Space), invariant under the dynamics, towards which 
neighboring states in a given Basin of Attraction 
asymptotically approach in the course of dynamic evo- 
lution. An attractor is denned as the smallest unit which 
cannot be itself decomposed into two or more attractors 



with distinct BASINS OF ATTRACTION. This restriction 
is necessary since a Dynamical System may have mul- 
tiple attractors, each with its own Basin OF Attrac- 
tion. 

Conservative systems do not have attractors, since the 
motion is periodic. For dissipative Dynamical Sys- 
tems, however, volumes shrink exponentially so attrac- 
tors have volume in n-D phase space. 

A stable FIXED Point surrounded by a dissipative re- 
gion is an attractor known as a SINK. Regular attractors 
(corresponding to Lyapunov Characteristic Ex- 
ponents) act as Limit Cycles, in which trajectories 
circle around a limiting trajectory which they asymp- 
totically approach, but never reach. STRANGE ATTRAC- 
TORS are bounded regions of PHASE SPACE (correspond- 
ing to Positive Lyapunov Characteristic Expo- 
nents) having zero MEASURE in the embedding PHASE 
Space and a Fractal Dimension. Trajectories within 
a Strange Attractor appear to skip around ran- 
domly. 

see also Barnsley's Fern, Basin of Attraction, 
Chaos Game, Fractal Dimension, Limit Cycle, 
Lyapunov Characteristic Exponent, Measure, 
Sink (Map), Strange Attractor 

Auction 

A type of sale in which members of a group of buyers 
offer ever increasing amounts. The bidder making the 
last bid (for which no higher bid is subsequently made 
within a specified time limit: "going once, going twice, 
sold") must then purchase the item in question at this 
price. Variants of simple bidding are also possible, as in 
a Vickery Auction. 

see also Vickery Auction 

Augend 

The first of several Addends, or "the one to which 
the others are added," is sometimes called the augend. 
Therefore, while a, 6, and c are ADDENDS in a + 6 -J- c, 
a is the augend. 

see also ADDEND, ADDITION 

Augmented Amicable Pair 

A Pair of numbers m and n such that 

a(m) — cr(n) = m + n — 1, 

where a{m) is the DIVISOR FUNCTION. Beck and Najar 
(1977) found 11 augmented amicable pairs. 

see also Amicable Pair, Divisor Function, Quasi- 
amicable Pair 

References 

Beck, W. E. and Najar, R. M. "More Reduced Amicable 

Pairs." Fib. Quart. 15, 331-332, 1977. 
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer- Verlag, p. 59, 1994. 



Augmented Dodecahedron 



Authalic Latitude 



85 



Augmented Dodecahedron 

see Johnson Solid 

Augmented Hexagonal Prism 

see Johnson Solid 



where h = 2k — 1 and 



L 2hi M 2h = 2 h + lT2 (7) 

L 3hi M 3h = 3 h + lT3 k (8) 

L 6h , M 5h = 5 2/l + 3 • S' 1 + 1 =F 5* (5* + 1). (9) 



Augmented Pentagonal Prism 

see Johnson Solid 

Augmented Polyhedron 

A Uniform Polyhedron with one or more other solids 
adjoined. 

Augmented Sphenocorona 

see Johnson Solid 

Augmented Triangular Prism 

see Johnson Solid 

Augmented Tridiminished Icosahedron 

see Johnson Solid 

Augmented Truncated Cube 

see Johnson Solid 

Augmented Truncated Dodecahedron 

see Johnson Solid 

Augmented Truncated Tetrahedron 

see Johnson Solid 

Aureum Theorema 

Gauss's name for the QUADRATIC RECIPROCITY THE- 
OREM. 

Aurifeuillean Factorization 

A factorization of the form 

2 4n + 2 + x = ^2n + l _ ^ + 1 + 1 )( 2 2n+1 + 2 n + 1 + 1). (1) 

The factorization for n — 14 was discovered by Au- 
rifeuille, and the general form was subsequently discov- 
ered by Lucas. The large factors are sometimes written 
as L and M as follows 

2 4fc-2 + : = ^ 2 fe-i _ 2 fc + 1 )( 2 2 *- 1 + 2 k + 1) (2) 

3 6fc-3 + x = ^2*-i + 1 j( 3 2fc-i _ 3 fc + i)^ 2 *" 1 + 3 fc + 1), 



(3) 



see also GAUSS'S FORMULA 

References 

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; WagstafT, S. S. Jr.; 
and Tuckerman, B. Factorizations of b n ± 1, b = 2, 
3, 5j 6, 7j 10, 11, 12 Up to High Powers, rev. ed. Providence, 
RI: Amer. Math. Soc, pp. lxviii-lxxii, 1988. 

WagstafT, S. S. Jr. "Aurifeullian Factorizations and the Pe- 
riod of the Bell Numbers Modulo a Prime." Math. Corn- 
put. 65, 383-391, 1996. 

Ausdehnungslehre 

see Exterior Algebra 

Authalic Latitude 

An Auxiliary Latitude which gives a Sphere equal 
Surface Area relative to an Ellipsoid. The authalic 
latitude is defined by 



(i) 







/^sin- 1 -*- 


where 




Q=(l-e 2 ) 


- 


sin0 1 ^ 


1- 


- e 2 sin 2 4> 2e 



Li n f 1 ~ esin( A 
le \ l-\-es'm<j)J 



. (2) 



and q p is q evaluated at the north pole (0 = 90°). Let R q 
be the Radius of the Sphere having the same Surface 
Area as the Ellipsoid, then 



Rq 



V 2 



(3) 



The series for j3 is 



/3 = ^-Ge 2 + ^e 4 + a i e 6 + ...)sin(2<A) 
+ (^ e4 + lio e6 + ---)sin(4^) 
-(4lfoe 6 + ...)sin(60) + .... 



(4) 



The inverse FORMULA is found from 



A<f> = 



(l-e 2 sin 2 0) 2 
2cos0 



sin<j) 



1-e 2 

+ 



1 — e 2 sin <j> 



which can be written 



1^/l-esin.A 

le VI -f esin<j> J 



(5) 



2 in + 1 = L 2h M 2h 

3 3h + 1 = (3 h + l)L 3h M 3h 



(4) 
(5) 
(6) 



where 



q = q p sm/3 



(6) 



86 Autocorrelation 

and (f>o — s'm~ 1 (q/2). This can be written in series form 



+ (^e* + ^ e 6 + ...)sin(4/3) 



(7) 



see a/so LATITUDE 

References 

Adams, O. S. "Latitude Developments Connected with 
Geodesy and Cartography with Tables, Including a Table 
for Lambert Equal-Area Meridional Projections." Spec. 
Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. 

Snyder, J. P. Map Projections — A Working Manual. U. S. 
Geological Survey Professional Paper 1395. Washington, 
DC: U. S. Government Printing Office, p. 16, 1987. 

Autocorrelation 

The autocorrelation function is denned by 



F 



C f {t) = f*f = f(-t)*nt)= I r(T)f(t + r)dr, 

(1) 

where * denotes CONVOLUTION and • denotes CROSS- 
CORRELATION. A finite autocorrelation is given by 

Cf(r) = ([y(t)-y][y(t + r)-y]) (2) 

pT/2 

= lim / [y(t)-y][y(t + r)-y]dt. (3) 



If / is a Real Function, 

/* = /, 

and an Even Function so that 

f(-r) = / (r), 
then 



(4) 



(5) 
(6) 



Cf(t)= I f(r)f(t + r)dr. 

J — OO 

But let t' = — r, so dr ~ —dr, then 

/» — OO 

Cf{t)= f(-r)f(t-r)(-dr) 

J OO 

OO 

f(-r)f(t-r)dr 
f(r)f(t-r)dr = f*f. (7) 



-F 

-F 



The autocorrelation discards phase information, return- 
ing only the POWER. It is therefore not reversible. 

There is also a somewhat surprising and extremely im- 
portant relationship between the autocorrelation and 



Autocorrelation 

the Fourier Transform known as the Wiener- 
Khintchine Theorem. Let FF[f{x)] — F(fc), and F* 
denote the COMPLEX CONJUGATE of F, then the FOUR- 
IER Transform of the Absolute Square of F(k) is 
given by 



n\F(k)\'}= r r(r)f(r + x)dr. (8) 

t/-oo 

The autocorrelation is a Hermitian Operator since 
Cf(-t) = C f *(t). /*/ is Maximum at the Origin. In 
other words, 

/oo /»oo 

f(u)f(u + x)du< / f 2 (u)du. (9) 

•oo J — oo 

To see this, let e be a Real Number. Then 

/oo 
[f{u) + ef(u + x)] 2 du>Q (10) 

■oo 

/oo /»oo 

f(u)du + 2e l f{u)f(u + x)du 
-oo J —oo 

/oo 
f 2 (u + x)du> (11) 
■oo 



+e 



/ f 2 (u)du + 2e / 

J — oo J — 



) du + 2e / f{u)f(u + x) du 



+e 



/oo 
-oo 



) du > 0. (12) 



Define 



/oo 
f(u)du (13) 

■oo 
/oo 
f(u)f(u + x)du. (14) 

■oo 

Then plugging into above, we have ae 2 +be-\-c > 0. This 
Quadratic Equation does not have any Real Root, 
so b 2 — 4ac < 0, i.e., 6/2 < a. It follows that 



F 



f(u)f(u + x) du < 



/oo 
f{u)du, 
-OO 



(15) 



with the equality at x — 0. This proves that / * / is 
Maximum at the Origin. 

see also CONVOLUTION, CROSS-CORRELATION, QUAN- 
TIZATION Efficiency, Wiener-Khintchine Theo- 
rem 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Correlation and Autocorrelation Using the 
FFT." §13.2 in Numerical Recipes in FORTRAN: The Art 
of Scientific Computing, 2nd ed. Cambridge, England: 
Cambridge University Press, pp. 538-539, 1992. 



Automorphic Function 

Automorphic Function 

An automorphic function f(z) of a COMPLEX variable 
z is one which is analytic (except for POLES) in a do- 
main D and which is invariant under a DENUMERABLY 
Infinite group of Linear Fractional Transforma- 
tions (also known as MOBIUS TRANSFORMATIONS) 

, az + 6 

z = -. 

cz + a 

Automorphic functions are generalizations of TRIGONO- 
METRIC Functions and Elliptic Functions. 
see also Modular Function, Mobius Transforma- 
tions, Zeta Fuchsian 

Automorphic Number 

A number k such that nk 2 has its last digits equal to 
k is called n-automorphic. For example, 1 • 5 2 = 25 
and 1 ■ 6 2 = 36 are 1-automorphic and 2 ■ 8 2 — 128 
and 2 • 88 2 = 15488 are 2-automorphic. de Guerre and 
Fairbairn (1968) give a history of automorphic numbers. 

The first few 1-automorphic numbers are 1, 5, 6, 25, 
76, 376, 625, 9376, 90625, . . . (Sloane's A003226, Wells 
1986, p. 130). There are two 1-automorphic numbers 
with a given number of digits, one ending in 5 and one in 
6 (except that the 1-digit automorphic numbers include 
1), and each of these contains the previous number with 
a digit prepended. Using this fact, it is possible to con- 
struct automorphic numbers having more than 25,000 
digits (Madachy 1979). The first few 1-automorphic 
numbers ending with 5 are 5, 25, 625, 0625, 90625, . . . 
(Sloane's A007185), and the first few ending with 6 are 
6, 76, 376, 9376, 09376, . . . (Sloane's A016090). The 1- 
automorphic numbers a(n) ending in 5 are IDEMPOTENT 
(mod 10") since 

[a(n)] 2 = a(n) (mod 10 n ) 

(Sloane and Plouffe 1995). 

The following table gives the 10-digit n-automorphic 
numbers. 
n n-Automorphic Numbers Sloane 



1 0000000001, 8212890625, 1787109376 

2 0893554688 

3 6666666667, 7262369792, 9404296875 

4 0446777344 

5 3642578125 

6 3631184896 

7 7142857143, 4548984375, 1683872768 

8 0223388672 

9 5754123264, 3134765625, 8888888889 



— , A007185, A016090 

A030984 

— , A030985, A030986 

A030987 

A030988 

A030989 

A030990, A030991, 

A030992 
A030993 
A030994, A030995, — 



see also IDEMPOTENT, NARCISSISTIC NUMBER, NUM- 
BER Pyramid, Trimorphic Number 

References 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 59 in 

HAKMEM. Cambridge, MA: MIT Artificial Intelligence 

Laboratory, Memo AIM-239, Feb. 1972. 



Autoregressive Model 87 

Fairbairn, R. A. "More on Automorphic Numbers." J. Recr. 
Math. 2, 170-174, 1969. 

Fairbairn, R. A. Erratum to "More on Automorphic Num- 
bers." J. Recr. Math. 2, 245, 1969. 

de Guerre, V. and Fairbairn, R. A. "Automorphic Numbers." 
J. Recr. Math. 1, 173-179, 1968. 

Hunter, J. A. H. "Two Very Special Numbers." Fib. Quart 
2, 230, 1964. 

Hunter, J. A. H. "Some Polyautomorphic Numbers." J. Recr. 
Math. 5, 27, 1972. 

Kraitchik, M. "Automorphic Numbers." §3.8 in Mathemat- 
ical Recreations. New York: W. W. Norton, pp. 77-78, 
1942. 

Madachy, J. S. Madachy 's Mathematical Recreations. New 
York: Dover, pp. 34-54 and 175-176, 1979. 

Sloane, N. J. A. Sequences A016090, A003226/M3752, and 
A007185/M3940 in "An On-Line Version of the Encyclo- 
pedia of Integer Sequences." 

Wells, D. The Penguin Dictionary of Curious and Interesting 
Numbers. Middlesex: Penguin Books, pp. 171, 178, 191- 
192, 1986. 

Automorphism 

An Isomorphism of a system of objects onto itself. 
see also ANOSOV AUTOMORPHISM 

Automorphism Group 

The GROUP of functions from an object G to itself which 
preserve the structure of the object, denoted Aut(G). 
The automorphism group of a GROUP preserves the 
Multiplication table, the automorphism group of a 
Graph the Incidence Matrices, and that of a Field 
the Addition and Multiplication tables, 
see also Outer Automorphism Group 

Autonomous 

A differential equation or system of ORDINARY DIFFER- 
ENTIAL EQUATIONS is said to be autonomous if it does 
not explicitly contain the independent variable (usu- 
ally denoted i). A second-order autonomous differen- 
tial equation is of the form F{y,y \y") — 0, where 
y = dy/dt = v. By the CHAIN RULE, y" can be ex- 
pressed as 



y 



dv 
It 



dv dy __ dv 
dy dt dy 



For an autonomous ODE, the solution is independent of 
the time at which the initial conditions are applied. This 
means that all particles pass through a given point in 
phase space. A nonautonomous system of n first-order 
ODEs can be written as an autonomous system of n + 1 
ODEs by letting t = x n +i and increasing the dimension 
of the system by 1 by adding the equation 



dx 



Tl + l 



dt 



1. 



Autoregressive Model 

see Maximum Entropy Method 



88 Auxiliary Circle 



Axiom A Flow 



Auxiliary Circle 

The ClRCUMCIRCLE of an ELLIPSE, i.e., the CIRCLE 
whose center corresponds with that of the ELLIPSE and 
whose Radius is equal to the Ellipse's Semimajor 
Axis. 

see also CIRCLE, ECCENTRIC ANGLE, ELLIPSE 

Auxiliary Latitude 

see Authalic Latitude, Conformal Latitude, 
Geocentric Latitude, Isometric Latitude, Lat- 
itude, Parametric Latitude, Rectifying Lati- 
tude, Reduced Latitude 

Auxiliary Triangle 

see Medial Triangle 

Average 

see Mean 

Average Absolute Deviation 

N 

a= — ^\xi- fi\ = (\xi-n\). 

i=l 
see also ABSOLUTE DEVIATION, DEVIATION, STANDARD 

Deviation, Variance 

Average Function 

If / is Continuous on a Closed Interval [a, 6], then 
there is at least one number x* in [a, 6] such that 



/ 

J a 



f(x)dx = f(x*)(b- a). 



The average value of the FUNCTION (/) on this interval 
is then given by f(x*). 

see Mean- Value Theorem 



Average Seek Time 

see POINT-POINT DlSTANCE- 



-1-D 



Ax-Kochen Isomorphism Theorem 

Let P be the Set of PRIMES, and let Q p and Z p (t) be the 
Fields of p-ADic Numbers and formal Power series 
over Z p = (0, 1, ... ,p — 1). Further, suppose that D is a 
"nonprincipal maximal filter" on P. Then Y[ GP Q p /D 
and Y[ ep Z p (t)/D are ISOMORPHIC. 

see also Hyperreal Number, Nonstandard Analy- 
sis 

Axial Vector 

see PSEUDOVECTOR 



Axiom 

A Proposition regarded as self-evidently True with- 
out Proof. The word "axiom" is a slightly archaic syn- 
onym for Postulate. Compare Conjecture or Hy- 
pothesis, both of which connote apparently TRUE but 
not self- evident statements. 

see also ARCHIMEDES' AXIOM, AXIOM OF CHOICE, AX- 
IOMATIC System, Cantor-Dedekind Axiom, Con- 
gruence Axioms, Conjecture, Continuity Ax- 
ioms, Countable Additivity Probability Axiom, 
Dedekind's Axiom, Dimension Axiom, Eilenberg- 
Steenrod Axioms, Euclid's Axioms, Excision Ax- 
iom, Fano's Axiom, Field Axioms, Hausdorff Ax- 
ioms, Hilbert's Axioms, Homotopy Axiom, In- 
accessible Cardinals Axiom, Incidence Axioms, 
Independence Axiom, Induction Axiom, Law, 
Lemma, Long Exact Sequence of a Pair Axiom, 
Ordering Axioms, Parallel Axiom, Pasch's Ax- 
iom, Peano's Axioms, Playfair's Axiom, Porism, 
Postulate, Probability Axioms, Proclus' Axiom, 
Rule, T2-Separation Axiom, Theorem, Zermelo's 
Axiom of Choice, Zermelo-Fraenkel Axioms 

Axiom A Diffeomorphism 

Let 4> : M -¥ M be a C 1 Diffeomorphism on a com- 
pact Riemannian Manifold M. Then <f> satisfies Ax- 
iom A if the Nonwandering set Q(4>) of is hyperbolic 
and the Periodic Points of <j> are Dense in Q(<f>). Al- 
though it was conjectured that the first of these condi- 
tions implies the second, they were shown to be indepen- 
dent in or around 1977. Examples include the AN0S0V 
Diffeomorphisms and Smale Horseshoe Map. 

In some cases, Axiom A can be replaced by the condi- 
tion that the DIFFEOMORPHISM is a hyperbolic diffeo- 
morphism on a hyperbolic set (Bowen 1975, Parry and 
Pollicott 1990). 

see also Anosov Diffeomorphism, Axiom A Flow, 
Diffeomorphism, Dynamical System, Riemannian 
Manifold, Smale Horseshoe Map 

References 

Bowen, R. Equilibrium States and the Ergodic Theory of 
Anosov Diffeomorphisms. New York: Springer- Verlag, 
1975. 

Ott, E. Chaos in Dynamical Systems. New York: Cambridge 
University Press, p. 143, 1993. 

Parry, W. and Pollicott, M. "Zeta Functions and the Peri- 
odic Orbit Structure of Hyperbolic Dynamics," Asterisque 
No. 187-188, 1990. 

Smale, S. "Different iable Dynamical Systems." Bull Amer. 
Math. Soc. 73, 747-817, 1967. 

Axiom A Flow 

A Flow defined analogously to the Axiom A Diffeo- 
morphism, except that instead of splitting the Tan- 
gent Bundle into two invariant sub-BUNDLES, they 
are split into three (one exponentially contracting, one 
expanding, and one which is 1-dimensional and tangen- 
tial to the flow direction). 
see also DYNAMICAL SYSTEM 



Axiom of Choice 



Azimuthal Projection 89 



Axiom of Choice 

An important and fundamental result in Set Theory 
sometimes called Zermelo'S Axiom of Choice. It was 
formulated by Zermelo in 1904 and states that, given any 
Set of mutually exclusive nonempty SETS, there exists 
at least one Set that contains exactly one element in 
common with each of the nonempty SETS. 

It is related to HlLBERT'S PROBLEM IB, and was proved 
to be consistent with other Axioms in Set Theory in 
1940 by GodeL In 1963, Cohen demonstrated that the 
axiom of choice is independent of the other Axioms in 
Cantorian Set Theory, so the Axiom cannot be proved 
within the system (Boyer and Merzbacher 1991, p. 610). 
see also Hilbert's Problems, Set Theory, Well- 
Ordered Set, Zermelo-Fraenkel Axioms, Zorn's 
Lemma 

References 

Boyer, C. B. and Merzbacher, U. C, A History of Mathemat- 
ics, 2nd ed. New York: Wiley, 1991. 

Cohen, P. J, "The Independence of the Continuum Hypoth- 
esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963. 

Cohen, P. J. "The Independence of the Continuum Hypothe- 
sis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964. 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, pp. 274-276, 1996. 

Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Devel- 
opment, and Influence. New York: Springer- Verlag, 1982. 

Axiomatic Set Theory 

A version of Set Theory in which axioms are taken 
as uninterpreted rather than as formalizations of pre- 
existing truths. 

see also Naive Set Theory, Set Theory 

Axiomatic System 

A logical system which possesses an explicitly stated 
Set of Axioms from which Theorems can be derived. 

see also Complete Axiomatic Theory, Consis- 
tency, Model Theory, Theorem 

Axis 

A LINE with respect to which a curve or figure is drawn, 
measured, rotated, etc. The term is also used to refer 

to a Line Segment through a Range (Woods 1961). 
see also Abscissa, Ordinate, cc-AxiS, y-AxiS, z-Axis 

References 

Woods, F. S. Higher Geometry: An Introduction to Advanced 
Methods in Analytic Geometry. New York: Dover, p. 8, 
1961. 

Axonometry 

A Method for mapping 3-D figures onto the Plane. 

see also CROSS-SECTION, Map Projection, Pohlke's 
Theorem, Projection, Stereology 

References 

Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York: 
Dover, p. 313, 1973. 



Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, pp. 322-323, 1988. 

Azimuthal Equidistant Projection 



'^rk 




An Azimuthal Projection which is neither equal- 
Area nor CONFORMAL. Let <p± and Ao be the LATI- 
TUDE and LONGITUDE of the center of the projection, 
then the transformation equations are given by 



x - fc'cos0sin(A - Ao) 

y = fc'jcos^i sin0 — sin<^i cos<£cos(A — Ao)]. 



Here, 



and 



k' 



c 
sine 



(i) 

(2) 
(3) 



cose — sin 0i sin0 + cos^i cos0cos(A — Ao), (4) 

where c is the angular distance from the center. The 
inverse FORMULAS are 



-( 



= sin I cose sin 0i + 



y sin c cos <f>. 



l ) (5) 



and 



( A + tan" 1 ( -r xsinc . . . 

u V ccos <pi cos c — y sin q>\ sin c 

for 0! ^ ±90° 
Ao+tan-^-l) 

for 0! = 90° 
Ao+tan- 1 ^), 

for 0i = -90°, 



) 



(6) 



with the angular distance from the center given by 

c = V^ + y 2 . (7) 

References 

Snyder, J. P. Map Projections — A Working Manual U. S. 
Geological Survey Professional Paper 1395. Washington, 
DC: U. S. Government Printing Office, pp. 191-202, 1987. 

Azimuthal Projection 

see Azimuthal Equidistant Projection, Lam- 
bert Azimuthal Equal-Area Projection, Ortho- 
graphic Projection, Stereographic Projection 



B* -Algebra 

B 



B-Spline 91 



E*-Algebra 

A Banach Algebra with an Antiautomorphic In- 
volution * which satisfies 



(5) 



A C*-Algebra is a special type of i?*-algebra. 
see also Banach Algebra, C*-Algebra 

i?2- Sequence 

N. B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

Also called a Sidon Sequence. An Infinite Se- 
quence of Positive Integers 



1 < h < 6 2 < h < 



such that all pairwise sums 



bi + bj 



(i) 



(2) 



for i < j are distinct (Guy 1994). An example is 1, 2, 4, 
8, 13, 21, 31, 45, 66, 81, . . . (Sloane's A005282). 

Zhang (1993, 1994) showed that 



S(B2) = sup V — > 2.1597. 

all B2 sequences ~~j ®k 



(3) 



The definition can be extended to B n -sequences (Guy 
1994). 

see also ^-Sequence, Mian-Chowla Sequence 

References 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsof t . com/asolve/constant/erdos/erdos .html. 

Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with 
Distinct Sums," and "^-Sequences," and B 2 -Sequences 
Formed by the Greedy Algorithm." §C9, Cll, E28, and 
E32 in Unsolved Problems in Number Theory, 2nd ed. New 
York: Springer- Verlag, pp. 115-118, 121-123, 228-229, and 
232-233, 1994. 

Sloane, N. J. A. Sequence A005282/M1094 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum." 
Math. Comput. 60, 835-839, 1993. 

Zhang, Z. X. "Finding Finite B2-Sequences with Larger m — 
a™ 1 ' 2 ." Math. Comput. 63, 403-414, 1994. 



B p - Theorem 

If Op' (G) = 1 and if a? is a p-element of G, then 

L p ,(C g (x)<E(Cg(x)), 
where L p > is the p-LAYER. 



X = X 


(i) 

(2) 


B 


-Spline 

Po®t 


x* +y* = {x + y)* 


(3) 






(ex)* = ex* 
satisfies 


(4) 




• 

Pi 




A generalization of the Bezier Curve. Let a vector 
known as the KNOT VECTOR be defined 



T = {£o,£ij • ■ , tm}i 



(i) 



where T is a nondecreasing SEQUENCE with U 6 [0, 1], 
and define control points Po, . . . , Pn- Define the degree 
as 

p = m — n — 1. (2) 

The "knots" £ p +i, ..., tm- P -i are called Internal 
Knots. 

Define the basis functions as 



at / ,\ _ f 1 if ti < t < ti+i and U < tt+i 
1 otherwise 

ti + v+l — t 



N ilP (t) 



t-U 

ti-\-p Ci 



(3) 



ii+p+1 — *i+l 



Then the curve defined by 



C(t) = £p<M,p(t) 



(4) 



(5) 



is a B-spline. Specific types include the nonperiodic B- 
spline (first p + 1 knots equal and last p + 1 equal to 
1) and uniform B-spline (INTERNAL KNOTS are equally 
spaced). A B-Spline with no INTERNAL KNOTS is a 
Bezier Curve. 

The degree of a B-spline is independent of the number of 
control points, so a low order can always be maintained 
for purposes of numerical stability. Also, a curve is p — k 
times differentiate at a point where k duplicate knot 
values occur. The knot values determine the extent of 
the control of the control points. 

A nonperiodic B-spline is a B-spline whose first p + 1 
knots are equal to and last p -f 1 knots are equal to 
1. A uniform B-spline is a B-spline whose INTERNAL 
Knots are equally spaced. 

see also Bezier Curve, NURBS Curve 



92 



B-Tree 



Backtracking 



B-Tree 

B-trees were introduced by Bayer (1972) and Mc- 
Creight. They are a special m-ary balanced tree used in 
databases because their structure allows records to be 
inserted, deleted, and retrieved with guaranteed worst- 
case performance. An n-node £?-tree has height C(lg2), 
where Lg is the LOGARITHM to base 2. The Apple® 
Macintosh® (Apple Computer, Cupertino, CA) HFS fil- 
ing system uses B-trees to store disk directories (Bene- 
dict 1995). A B-tree satisfies the following properties: 

1. The Root is either a Leaf (Tree) or has at least 
two Children, 

2. Each node (except the ROOT and LEAVES) has be- 
tween \m/2\ and m Children, where \x\ is the 
Ceiling Function. 

3. Each path from the Root to a Leaf (Tree) has the 
same length. 

Every 2-3 Tree is a B-tree of order 3. The number of 
B-trees of order n = 1, 2, . . . are 0, 1, 1, 1, 2, 2, 3, 4, 5, 
8, 14, 23, 32, 43, 63, . . . (Ruskey, Sloane's A014535). 
see also Red-Black Tree 

References 

Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc- 
tures and Algorithms. Reading, MA: Addison-Wesley, 
pp. 369-374, 1987. 

Benedict, B. Using Norton Utilities for the Macintosh. Indi- 
anapolis, IN: Que, pp. B-17-B-33, 1995. 

Beyer, R. "Symmetric Binary jB-Trees: Data Structures and 
Maintenance Algorithms." Acta Informat. 1, 290-306, 
1972. 

Ruskey, F. "Information on B-Trees." http://sue.csc.uvic 
. ca/~cos/inf /tree/BTrees .html. 

Sloane, N. J. A. Sequence A014535 in "An On-Line Version 
of the Encyclopedia of Integer Sequences." 

Baby Monster Group 

Also known as FISCHER'S BABY MONSTER GROUP. The 

Sporadic Group B. It has Order 

2 4i . 3 i3 . 5 6 . 7 2 . ii . 13 . 17 . 19 . 23 • 31 ■ 47. 

see also MONSTER GROUP 

References 

Wilson, R. A. "ATLAS of Finite Group Representation." 
http : //for . mat . bham . ac .uk/atlas/BM . html. 

BAC-CAB Identity 

The Vector Triple Product identity 

A x (B x C) = B(A ■ C) - C(A • B). 
This identity can be generalized to n-D 



a 2 x ■ • • x a n _i x (bi x ■ • • x b n _i) 
h x 
a 2 ■ In 

= (-*) 

a n _i ■ bi 
See also LAGRANGE'S IDENTITY 



b n -i 
a 2 • b n _i 

&n-i " b n _i 



BAC-CAB Rule 

see BAC-CAB IDENTITY 

Bachelier Function 

see Brown Function 

Bachet's Conjecture 

see Lagrange's Four-Square Theorem 

Bachet Equation 

The Diophantine Equation 



x 2 + k = y 3 , 



which is also an Elliptic Curve. The general equation 
is still the focus of ongoing study. 

Backhouse's Constant 

Let P(x) be defined as the POWER series whose nth term 
has a Coefficient equal to the nth Prime, 

oo 

P(x) = Y^PhX k = l + 2z + 3z 2 + 5z 3 + 7z 4 -hllz 5 + ..., 
and let Q(x) be defined by 

on 
1 



Q(*) = 



P(x) 



y^qkX h . 
k=o 



Then N. Backhouse conjectured that 



lim 

n—t-oc 



<7n+l 



q n 



1.456074948582689671399595351116. . . . 



The constant was subsequently shown to exist by P. Fla- 
jolet. 

References 

Finch, S. "Favorite Mathematical Constants." http: //www. 

mathsof t . com/asolve/constant/backhous/ 

backhous .html. 

Backlund Transformation 

A method for solving classes of nonlinear Partial Dif- 
ferential Equations. 

see also INVERSE SCATTERING METHOD 

References 

Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and 

Chaos. Cambridge, England: Cambridge University Press, 

p. 196, 1990. 
Miura, R. M. (Ed.) Backlund Transformations, the Inverse 

Scattering Method, Solitons, and Their Applications. New 

York: Springer- Verlag, 1974. 

Backtracking 

A method of drawing FRACTALS by appropriate num- 
bering of the corresponding tree diagram which does not 
require storage of intermediate results. 



Backus-Gilbert Method 



Baire Category Theorem 93 



Backus-Gilbert Method 

A method which can be used to solve some classes of 
INTEGRAL EQUATIONS and is especially useful in im- 
plementing certain types of data inversion. It has been 
applied to invert seismic data to obtain density profiles 
in the Earth. 

References 

Backus, G. and Gilbert, F. "The Resolving Power of Growth 

Earth Data." Geophys. J. Roy. Astron. Soc. 16, 169-205, 

1968. 
Backus, G. E. and Gilbert, F. "Uniqueness in the Inversion 

of Inaccurate Gross Earth Data." Phil Trans. Roy. Soc. 

London Ser. A 266, 123-192, 1970. 
Loredo, T. J. and Epstein, R. I. "Analyzing Gamma-Ray 

Burst Spectral Data." Astrophys. J. 336, 896-919, 1989. 
Parker, R. L. "Understanding Inverse Theory." Ann. Rev. 

Earth Planet Sci. 5, 35-64, 1977. 
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 

ling, W. T. "Backus-Gilbert Method." §18.6 in Numerical 

Recipes in FORTRAN: The Art of Scientific Computing, 

2nd ed. Cambridge, England: Cambridge University Press, 

pp. 806-809, 1992. 

Backward Difference 

The backward difference is a Finite Difference de- 
fined by 

Vp = V/ p s/p-/p_i. (1) 

Higher order differences are obtained by repeated oper- 
ations of the backward difference operator, so 

Vp = V(Vp) = V(/ p - /„_!) = V/ p - V/,_i (2) 
= {fp ~ fp-i) ~ (fp-i ~ /p-z) 

= fp~ 2 /p-l + fp-2 



(3) 



In general, 



v5 = vv, = £(-ir(*W* +m > 



(4) 



where (^) is a BINOMIAL COEFFICIENT. 

Newton's Backward Difference Formula ex- 
presses f p as the sum of the nth backward differences 

/ P = /o+pVo + ^p(p + l)V? + J T p(p + l)(p + 2)Vg + ..., 

(5) 
where Vq is the first nth difference computed from the 
difference table. 

see also Adams' Method, Difference Equation, 
Divided Difference, Finite Difference, For- 
ward Difference, Newton's Backward Differ- 
ence Formula, Reciprocal Difference 

References 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, pp. 429 and 433, 1987. 



Bader-Deuflhard Method 

A generalization of the Bulirsch-Stoer Algorithm 
for solving Ordinary Differential Equations. 

References 

Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point 
Rule for Stiff Systems of Ordinary Differential Equations." 
Numer. Math. 41, 373-398, 1983. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 730, 1992. 

Baguenaudier 

A Puzzle involving disentangling a set of rings from a 
looped double rod (also called CHINESE Rings). The 
minimum number of moves needed for n rings is 



§(2 n+1 -2) 
|(2 n+1 -l) 



n even 
n odd. 



By simultaneously moving the two end rings, the num- 
ber of moves can be reduced to 

f 2 n_1 -In even 
I 2 71 " 1 n odd. 

The solution of the baguenaudier is intimately related 
to the theory of GRAY CODES. 

References 

Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Pro- 
duce a Chinese Monster." Quantum 6, 61-65 (Mar.) and 
58-59 (Apr.), 1996. 

Gardner, M. "The Binary Gray Code." In Knotted Dough- 
nuts and Other Mathematical Entertainments. New York: 
W. H. Freeman, pp. 15-17, 1986. 

Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical 
Recreations. New York: W. W. Norton, pp. 89-91, 1942. 

Steinhaus, H. Mathematical Snapshots, 3rd American ed. 
New York: Oxford University Press, p. 268, 1983. 

Bailey's Method 

see Lambert's Method 

Bailey's Theorem 

Let T(z) be the GAMMA FUNCTION, then 



r(m+|) 



V(m) 



[i (-Y — — (-Y—!— 



I>+§) 



V(n) 



1 /iy_j_ /i-3\ 2 1 

n + \2J n-M + \2.4y n + 2 + ' 



Baire Category Theorem 

A nonempty complete Metric Space cannot be repre- 
sented as the Union of a Countable family of nowhere 
Dense Subsets. 



94 Baire Space 



Ball Triangle Picking 



Baire Space 

A Topological Space X in which each Subset of X 
of the "first category" has an empty interior. A TOPO- 
LOGICAL Space which is Homeomorphic to a complete 
Metric Space is a Baire space. 

Bairstow's Method 

A procedure for rinding the quadratic factors for the 
Complex Conjugate Roots of a Polynomial P(x) 
with Real Coefficients. 

[x — (a + ib)][x - (a — ib)] 

= x 2 + 2ax + (a 2 + b 2 ) = x 2 + Bx + C. (1) 

Now write the original POLYNOMIAL as 

P(x) = (x 2 +Bx + C)Q{x) + Rx + S (2) 

R(B + SB,C + 6C)KR(B,C) + ^dB+^dC (3) 



dB 



8C 



S(B + 5B,C + 5C)*d(B,C) + ^dB+^dC (4) 

£ — <.- + «.♦* ,« + «, + g5 + » ( .> 

. QW = ( I . + B , + C )g + g + f (6) 

"*«M = <** + «* + C >I + 1 + !' (8) 

Now use the 2-D Newton's Method to find the simul- 
taneous solutions. 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in C: The Art of Scientific 
Computing. Cambridge, England: Cambridge University 
Press, pp. 277 and 283-284, 1989. 



Baker's Dozen 

The number 13. 
see also 13, DOZEN 



Baker's Map 

The Map 



X n +1 = 2^£ n , 



(1) 



where x is computed modulo 1. A generalized Baker's 
map can be defined as 



Vn < a 



(2) 



Xn+1 -\(1-X b ) + X b x n y n >a 



where (3 = 1 — a, A + A 6 < 1, and x and y are computed 
mod 1. The q = 1 g-DlMENSION is 

aln(±)+/31n(|) 
D 1 = 1 + Va) )*' ■ (4) 



' ta (£)+*»»(*)' 



If A a = A&, then the general g-DlMENSION is 
1 In (a q +f3 q ) 



D q = l + 



q — 1 In A 



(5) 



References 

Lichtenberg, A. and Lieberman, M. Regular and Stochastic 
Motion. New York: Springer- Verlag, p. 60, 1983. 

Ott, E. Chaos in Dynamical Systems. Cambridge, England: 
Cambridge University Press, pp. 81-82, 1993. 

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. 
New York: Wiley, p. 32, 1990. 

Balanced ANOVA 

An ANOVA in which the number of REPLICATES (sets 
of identical observations) is restricted to be the same for 
each Factor Level (treatment group). 
see also ANOVA 

Balanced Incomplete Block Design 

see Block Design 

Ball 

The n-ball, denoted B n , is the interior of a SPHERE 
S™" 1 , and sometimes also called the n-DlSK. (Al- 
though physicists often use the term "SPHERE" to mean 
the solid ball, mathematicians definitely do not!) Let 
Vol(B n ) denote the volume of an n-D ball of RADIUS r. 
Then 

oo 

Y^ Vol(S n ) = e^ 2 [1 + erf (r^ )], 

where erf(x) is the ERF function. 
see also Alexander's Horned Sphere, Banach- 
Tarski Paradox, Bing's Theorem, Bishop's In- 
equality, Bounded, Disk, Hypersphere, Sphere, 
Wild Point 

References 

Preden, E. Problem 10207. "Summing a Series of Volumes." 
Amer. Math. Monthly 100, 882, 1993. 

Ball Triangle Picking 

The determination of the probability for obtaining an 
Obtuse Triangle by picking 3 points at random in 
the unit Disk was generalized by Hall (1982) to the n- 
D Ball. Buchta (1986) subsequently gave closed form 



Ballantine 



Banach Measure 



95 



evaluations for Hall's integrals, with the first few solu- 
tions being 

9 4 
P 2 = - - — « 0.72 

8 7V d 

P 4 « 0.39 
P 5 « 0.29. 

The case P^ corresponds to the usual DISK case. 

see also Cube Triangle Picking, Obtuse Triangle 

References 

Buchta, C. "A Note on the Volume of a Random Polytope in 
a Tetrahedron." III. J. Math. 30, 653-659, 1986. 

Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 
19, 712-715, 1982. 

Ballantine 

see Borromean Rings 

Ballieu's Theorem 

For any set fi = (^1,^2, ■ ■ ■ ,fi n ) of POSITIVE numbers 
with ^o = and 



M M = max 



flk + {ln\b n -k\ 



0<k<n-l /ifc + 1 

Then all the EIGENVALUES A satisfying P(X) = 0, where 

P{\) is the Characteristic Polynomial, lie on the 
Disk \z\ < M M . 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1119, 1979. 

Ballot Problem 

Suppose A and B are candidates for office and there are 
2n voters, n voting for A and n for B, In how many ways 
can the ballots be counted so that A is always ahead of 
or tied with B1 The solution is a CATALAN NUMBER 

A related problem also called "the" ballot problem is to 
let A receive a votes and B b votes with a > b. This ver- 
sion of the ballot problem then asks for the probability 
that A stays ahead of B as the votes are counted (Vardi 
1991). The solution is (a — b)/(a + 6), as first shown 
by M. Bertrand (Hilton and Pedersen 1991). Another 
elegant solution was provided by Andre (1887) using the 
so-called Andre's Reflection Method. 

The problem can also be generalized (Hilton and Ped- 
ersen 1991). Furthermore, the TAK FUNCTION is con- 
nected with the ballot problem (Vardi 1991). 
see also Andre's Reflection Method, Catalan 
Number, TAK Function 



References 

Andre, D. "Solution directe du probleme resolu par 
M. Bertrand." Comptes Rendus Acad. Sci. Paris 105, 
436-437, 1887. 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 49, 1987. 

Carlitz, L. "Solution of Certain Recurrences." SIAM J. Appl. 
Math. 17, 251-259, 1969. 

Comtet, L. Advanced Combinatorics. Dordrecht, Nether- 
lands: Reidel, p. 22, 1974. 

Feller, W. An Introduction to Probability Theory and Its Ap- 
plications, Vol. 1, 3rd ed. New York: Wiley, pp. 67-97, 
1968. 

Hilton, P. and Pedersen, J. "The Ballot Problem and Cata- 
lan Numbers." Nieuw Archief voor Wiskunde 8, 209-216, 
1990. 

Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener- 
alization, and Their Uses." Math. Intel. 13, 64-75, 1991. 

Kraitchik, M. "The Ballot-Box Problem." §6.13 in Mathe- 
matical Recreations. New York: W. W. Norton, p. 132, 
1942. 

Motzkin, T. "Relations Between Hypersurface Cross Ratios, 
and a Combinatorial Formula for Partitions of a Polygon, 
for Permanent Preponderance, and for Non-Associative 
Products." Bull Amer. Math. Soc. 54, 352-360, 1948. 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addison- Wesley, pp. 185-187, 1991. 

Banach Algebra 

An Algebra A over a Field F with a Norm that 
makes A into a COMPLETE METRIC SPACE, and there- 
fore, a Banach Space. F is frequently taken to be the 
Complex Numbers in order to assure that the Spec- 
trum fully characterizes an Operator (i.e., the spec- 
tral theorems for normal or compact normal operators 
do not, in general, hold in the Spectrum over the Real 
Numbers). 

see also £?*-Algebra 

Banach Fixed Point Theorem 

Let / be a contraction mapping from a closed SUBSET 
F of a Banach Space E into F. Then there exists a 
unique z £ F such that f(z) = z. 

see also FIXED POINT THEOREM 

References 

Debnath, L. and Mikusiriski, P. Introduction to Hilbert 

Spaces with Applications. San Diego, CA: Academic Press, 

1990. 

Banach-Hausdorff-Tarski Paradox 

see Banach- Tarski Paradox 

Banach Measure 

An "Area" which can be defined for every set — even 
those without a true geometric AREA — which is rigid 
and finitely additive. 



96 



Banach Space 



Baibiefs Theorem 



Banach Space 

A normed linear Space which is Complete in the norm- 
determined Metric. A Hilbert Space is always a Ba- 
nach space, but the converse need not hold. 
see also Besov Space, Hilbert Space, Schauder 
Fixed Point Theorem 

Banach-Steinhaus Theorem 

see Uniform Boundedness Principle 

Banach- Tarski Paradox 

First stated in 1924, this theorem demonstrates that it 
is possible to dissect a Ball into six pieces which can 
be reassembled by rigid motions to form two balls of 
the same size as the original. The number of pieces was 
subsequently reduced to five. However, the pieces are 
extremely complicated. A generalization of this theo- 
rem is that any two bodies in R which do not extend 
to infinity and each containing a ball of arbitrary size 
can be dissected into each other (they are are EQUIDE- 
composable). 

References 

Stromberg, K. "The Banach- Tarski Paradox." Amer. Math. 
Monthly 86, 3, 1979. 

Wagon, S. The Banach-Tarski Paradox. New York: Cam- 
bridge University Press, 1993. 

Bang's Theorem 

The lines drawn to the Vertices of a face of a Tetra- 
hedron from the point of contact of the FACE with the 
INSPHERE form three ANGLES at the point of contact 
which are the same three ANGLES in each FACE. 

References 

Brown, B. H. "Theorem of Bang. Isosceles Tetrahedra." 

Amer. Math. Monthly 33, 224-226, 1926. 
Honsberger, R. Mathematical Gems II. Washington, DC: 

Math. Assoc. Amer., p. 93, 1976. 

Bankoff Circle 




References 

Bankoff, L. "Are the Twin Circles of Archimedes Really 

Twins?" Math. Mag. 47, 214-218, 1974. 
Gardner, M. "Mathematical Games: The Diverse Pleasures 

of Circles that Are Tangent to One Another." Sci. Amer. 

240, 18-28, Jan. 1979. 

Banzhaf Power Index 

The number of ways in which a group of n with weights 
X^r=i Wi = 1 can cnan g e a losing coalition (one with 
^2 w i < 1/2) to a winning one, or vice versa. It was 
proposed by the lawyer J. F. Banzhaf in 1965. 

References 

Paulos, J. A. A Mathematician Reads the Newspaper. New 
York: BasicBooks, pp. 9-10, 1995. 

Bar (Edge) 

The term in rigidity theory for the EDGES of a GRAPH. 

see also Configuration, Framework 
Bar Polyhex 




A Polyhex consisting of Hexagons arranged along a 

line. 

see also Bar Polyiamond 

References 

Gardner, M. Mathematical Magic Show: More Puzzles, 
Games, Diversions, Illusions and Other Mathematical 
Sleight- of- Mind from Scientific American. New York: 
Vintage, p. 147, 1978. 

Bar Polyiamond 



In addition to the ARCHIMEDES' CIRCLES d and C 2 in 
the Arbelos figure, there is a third circle C3 congruent 
to these two as illustrated in the above figure. 

see also ARBELOS 



A Polyiamond consisting of Equilateral Triangles 

arranged along a line. 
see also Bar Polyhex 

References 

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, 
and Packings, 2nd ed. Princeton, NJ: Princeton University 
Press, p. 92, 1994. 

Barber Paradox 

A man of Seville is shaved by the Barber of Seville IFF 
the man does not shave himself. Does the barber shave 
himself? Proposed by Bertrand Russell. 

Barbier's Theorem 

All Curves of Constant Width of width w have the 
same Perimeter ttw. 



Bare Angle Center 



Barth Decic 



97 



Bare Angle Center 

The Triangle Center with Triangle Center 
Function 

a = A. 

References 

Kimberling, C. "Major Centers of Triangles." Amer. Math. 
Monthly 104, 431-438, 1997. 

Barnes G- Function 

see G-Function 

Barnes' Lemma 

If a Contour in the Complex Plane is curved such 
that it separates the increasing and decreasing sequences 
of Poles, then 



2-m . v 

</ —too 



+ s)r(0 + s)T('y-s)r(5-s)ds 

= T(a + 7)r(a + 6)r{(3 + j)T{p + 6) 
r(a + /3 + 7 + <5) 



where T(z) is the Gamma Function. 

Barnes- Wall Lattice 

A lattice which can be constructed from the LEECH LAT- 
TICE A 2 4- 

see also Coxeter-Todd Lattice, Lattice Point, 
Leech Lattice 

References 

Barnes, E. S. and Wall, G. E. "Some Extreme Forms Denned 
in Terms of Abelian Groups." J. Austral Math. Soc. 1, 
47-63, 1959. 

Conway, J. H. and Sloane, N. J, A, "The 16- Dimensional 
Barnes- Wall Lattice Ai 6 ." §4.10 in Sphere Packings, Lat- 
tices, and Groups, 2nd ed. New York: Springer- Verlag, 
pp. 127-129, 1993, 



Barnsley's Fern 



^■,;f^' 



*^7&g$^~~ 



0.85 0.04" 




X 


+ 


"o.oo" 


(1) 


-0.04 0.85 




_y 


1.60 


-0.15 0.28" 




X 


+ 


"o.oo" 


(2) 


0.26 0.24 




y '. 


0.44 


0.20 -0.26' 




X 


+ 


"o.oo" 


(3) 


0.23 0.22 




y . 


1.60 


0.00 0.00 " 




X 




(4) 


0.00 0.16 




y 









The Attractor of the Iterated Function System 
given by the set of "fern functions" 



h(x,y) = 

fs(x,y) = 
U(x,y) = 



(Barnsley 1993, p. 86; Wagon 1991). These Affine 
Transformations are contractions. The tip of the 
fern (which resembles the black spleehwort variety of 
fern) is the fixed point of /i , and the tips of the lowest 
two branches are the images of the main tip under J2 
and f z (Wagon 1991). 

see also Dynamical System, Fractal, Iterated 
Function System 

References 

Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Aca- 
demic Press, pp. 86, 90, 102 and Plate 2, 1993. 

Gleick, J. Chaos: Making a New Science. New York: Pen- 
guin Books, p. 238, 1988. 

Wagon, S. "Biasing the Chaos Game: Barnslej^s Fern." §5.3 
in Mathematica in Action. New York: W. H. Freeman, 
pp. 156-163, 1991. 

Barrier 

A number n is called a barrier of a number-theoretic 
function f(m) if, for all m < n, m + f(m) < n. Neither 
the Totient Function <p(n) nor the Divisor Func- 
tion o-(n) has barriers. 

References 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, pp. 64-65, 1994. 



Barth Decic 



\ 1 7 / . 




98 



Barth Sextic 



Bartlett Function 



The Barth decic is a Decic Surface in complex three- 
dimensional projective space having the maximum pos- 
sible number of ORDINARY DOUBLE POINTS (345). It is 
given by the implicit equation 



■A 2 ) 



■2yV) 



x( :c 4 + y 4 + z 4 -2 2; V- 
+(3 + 50)(;r 2 +y 2 +z 2 -w 2 ) 2 [x 2 4-y 2 \z 2 -(2-0)u? 2 ]V 

= 0, 

where is the Golden Mean and w is a parameter 
(EndraB, Nordstrand), taken as w = 1 in the above plot. 
The Barth decic is invariant under the ICOSAHEDRAL 
Group. 

see also ALGEBRAIC SURFACE, BARTH SEXTIC, DECIC 

Surface, Ordinary Double Point 

References 

Barth, W. "Two Projective Surfaces with Many Nodes Ad- 
mitting the Symmetries of the Icosahedron." J. Alg. Geom. 
5, 173-186, 1996. 

Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV- 
Mitteilungen 4, 17-20, 4/1995. 

Endrafi, S. "Barth's Decic." http://www.mathematik.uni- 
mainz . de/AlgebraischeGeometrie/docs/ 
Ebarthdecic . shtml. 

Nordstrand, T. "Batch Decic." http://www.uib.no/people/ 
nf ytn/bdectxt .htm. 

Barth Sextic 




The Barth-sextic is a SEXTIC SURFACE in complex 
three-dimensional projective space having the maximum 
possible number of ORDINARY DOUBLE POINTS (65). It 
is given by the implicit equation 



A{4> 2 x 2 -y 2 ){4> 2 y 2 ~z'){<t>-z- -x 



2 2 2w ,2 2 2x 

-(1 + 2<P)(x 2 + y 2 + z 2 - w 2 ) 2 w 2 



0. 



where 4> is the GOLDEN Mean, and w is a parameter 
(Endrafi, Nordstrand), taken as w — 1 in the above plot. 
The Barth sextic is invariant under the ICOSAHEDRAL 
Group. Under the map 

/ \ v / 2 2 2 2\ 

(x,y,z,w) -+ (x ,y ,z 9 w ), 



the surface is the eightfold cover of the Cayley Cubic 

(Endrafi). 

see also ALGEBRAIC SURFACE, BARTH DECIC, CAYLEY 

Cubic, Ordinary Double Point, Sextic Surface 

References 

Barth, W. "Two Projective Surfaces with Many Nodes Ad- 
mitting the Symmetries of the Icosahedron." J. Alg. Geom. 
5, 173-186, 1996. 

Endrafl, S. "Flachen mit vielen Doppelpunkten." DMV- 
Mitteilungen 4, 17-20, 4/1995. 

Endrafl, S. "Barth's Sextic." http://www.mathematik.uni- 
mainz.de/AlgebraischeGeometrie/docs/ 
Ebarthsextic . shtml. 

Nordstrand, T. "Barth Sextic." http://www.uib.no/people/ 
nf ytn/sexttxt .htm. 

Bartlett Function 




o.is 



o.c 
o.c 
oflc 



J '-0725 
-1 -0.5 ' 075 1 -0.5 

The Apodization Function 



L 



f{x) = 1 



(1) 



which is a generalization of the one-argument TRIANGLE 
Function. Its Full Width at Half Maximum is a. 
It has Instrument Function 

I(x) = ^ e~ 2 * ikx (l - M) dx 

v —a 

+ fe- J,iJ, (l-j)<b. (2) 

Letting x' = —x in the first part therefore gives 
f° e- 2 " ikx (l + |) dx = I e Mk *' (l - ^\ (-dx') 

Rewriting (2) using (3) gives 



(3) 



7-/ \ / 2irikx . — 2-rrikx\ ( -, % \ 

I(x) = (e +e H aj 



dx 



= 2 / cos(27rfcz) (l - -J dx. 



(4) 



Integrating the first part and using the integral 



/ 



x cos(bx) dx — — cos(6;c) + — sin(for) (5) 

b 1 b 



Barycentric Coordinates 

for the second part gives 
sin(27rA;a;) 



I(x) = 2 



2irk 



[s\n(2Trk 
2™fe~ 



= 2 { l" sin ( 27rfc a ) __ 



cos(27rfca) — 1 asm.{2nka) 



47T 2 fc 2 



27r 2 a/c 2 
: a sine (7rka), 



[cos(27r&a) — 1] = a 



2ttA; 
sin 2 (7rfca) 



7r 2 k 2 a 2 



(6) 



where sine x is the SlNC FUNCTION. The peak (in units 
of a) is 1. The function I(x) is always positive, so there 
are no Negative sidelobes. The extrema are given by 
letting j3 = nka and solving 



d ( sin j3 



2 sin/9sin/3-/3cos/9 . 



P 



P 



sin/3(sin/?-/?cos/3) = 

sin/3-/3cos/3 = 

tan/3 = /3. 



(8) 

(9) 

(10) 



Solving this numerically gives j3 = 4.49341 for the first 
maximum, and the peak POSITIVE sidelobe is 0.047190. 
The full width at half maximum is given by setting x = 
nka and solving 

sine x = | (11) 

for #1/2, yielding 

Ei/2 = 7rfci /2 a = 1.39156. (12) 

Therefore, with L = 2a, 



FWHM = 2fei /2 = 



0.885895 1.77179 



a 



(13) 



see a/so APODIZATION FUNCTION, PARZEN ApODIZA- 

tion Function, Triangle Function 

References 

Bartlett, M. S. "Periodogram Analysis and Continuous Spec- 
tra." Biometrika 37, 1-16, 1950. 

Barycentric Coordinates 

Also known as HOMOGENEOUS COORDINATES or TRI- 
linear Coordinates. 

see Trilinear Coordinates 

Base Curve 

see Directrix (Ruled Surface) 



Base (Number) 99 

Base (Logarithm) 

The number used to define a LOGARITHM, which is then 
written log 6 . The symbol logo; is an abbreviation for 
log 10 x, In as for log e x (the Natural Logarithm), and 
lga: for log 2 x. 

see also e, Lg, Ln, Logarithm, Napierian Loga- 
rithm, Natural Logarithm 

Base (Neighborhood System) 

A base for a neighborhood system of a point x is a col- 
lection N of Open Sets such that x belongs to every 
member of iV, and any Open Set containing x also con- 
tains a member of N as a Subset. 

Base (Number) 

A Real Number x can be represented using any Inte- 
ger number b as a base (sometimes also called a RADIX 
or SCALE). The choice of a base yields to a representa- 
tion of numbers known as a Number System. In base 
6, the DIGITS 0, 1, . . . , b - 1 are used (where, by con- 
vention, for bases larger than 10, the symbols A, B, C, 
. . . are generally used as symbols representing the DEC- 
IMAL numbers 10, 11, 12, . . . ). 



Base 


Name 


2 


binary 


3 


ternary 


4 


quaternary 


5 


quinary 


6 


senary 


7 


septenary 


8 


octal 


9 


nonary 


10 


decimal 


11 


undenary 


12 


duodecimal 


16 


hexadecimal 


20 


vigesimal 


60 


sexagesimal 



Let the base b representation of a number x be written 



(a n Cin-i ... ao- a_i . . .)*,, 



(1) 



(e.g., 123.456io), then the index of the leading DIGIT 
needed to represent the number is 



n = |k>g 6 x\ , 



(2) 



where \_x\ is the FLOOR FUNCTION. Now, recursively 
compute the successive Digits 



ai = L?J • 



where r n = x and 



n-! = n 



(lib 1 



(3) 



(4) 



100 



Base Space 



Basis 



for i = n, n — 1, . . . , 1,0, This gives the base b 

representation of x. Note that if x is an Integer, then 
i need only run through 0, and that if x has a fractional 
part, then the expansion may or may not terminate. 
For example, the HEXADECIMAL representation of 0.1 
(which terminates in DECIMAL notation) is the infinite 
expression 0.19999. . .h- 

Some number systems use a mixture of bases for count- 
ing. Examples include the Mayan calendar and the old 
British monetary system (in which ha'pennies, pennies, 
threepence, sixpence, shillings, half crowns, pounds, and 
guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240, 
and 252, respectively). 

Knuth has considered using TRANSCENDENTAL bases. 
This leads to some rather unfamiliar results, such as 
equating -k to 1 in "base 7r," 7r = I*.. 

see also Binary, Decimal, Hereditary Represen- 
tation, Hexadecimal, Octal, Quaternary, Sexa- 
gesimal, Ternary, Vigesimal 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 28, 1972. 

Bogomolny, A. "Base Converter." http : //www . cut-the- 
knot . com/binary .html. 

Lauwerier, II. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 6-11, 
1991. 
i$ Weisstein, E. W. "Bases." http: //www. astro. Virginia. 
edu/~eww6n/math/notebooks/Bases.m. 

Base Space 

The Space B of a Fiber Bundle given by the Map 
/ : E -> B, where E is the Total Space of the Fiber 
Bundle. 

see also FIBER BUNDLE, TOTAL SPACE 

Baseball 

The numbers 3 and 4 appear prominently in the game 
of baseball. There are 3*3 = 9 innings in a game, and 
three strikes are an out. However, 4 balls are needed for 
a walk. The number of bases can either be regarded as 
3 (excluding HOME Plate) or 4 (including it). 

see Baseball Cover, Home Plate 



A pair of identical plane regions (mirror symmetric 
about two perpendicular lines through the center) which 
can be stitched together to form a baseball (or tennis 
ball). A baseball has a CIRCUMFERENCE of 9 1/8 inches. 
The practical consideration of separating the regions far 
enough to allow the pitcher a good grip requires that 
the "neck" distance be about 1 3/16 inches. The base- 
ball cover was invented by Elias Drake as a boy in the 
1840s. (Thompson's attribution of the current design 
to trial and error development by C. H. Jackson in the 
1860s is apparently unsubstantiated, as discovered by 
George Bart.) 

One way to produce a baseball cover is to draw the re- 
gions on a Sphere, then cut them out. However, it is 
difficult to produce two identical regions in this man- 
ner. Thompson (1996) gives mathematical expressions 
giving baseball cover curves both in the plane and in 
3-D. J. H. Conway has humorously proposed the follow- 
ing "baseball curve conjecture:" no two definitions of 
"the" baseball curve will give the same answer unless 
their equivalence was obvious from the start. 

see also Baseball, Home Plate, Tennis Ball The- 
orem, Yin- Yang 

References 

Thompson, R. B. "Designing a Baseball Cover. 1860's: Pa- 
tience, Trial, and Error. 1990's: Geometry, Calculus, 
and Computation," http://www.mathsoft.com/asolve/ 
baseball/baseball. html. Rev. March 5, 1996. 

Basin of Attraction 

The set of points in the space of system variables such 
that initial conditions chosen in this set dynamically 
evolve to a particular Attractor. 

see also Wada Basin 

Basis 

A (vector) basis is any Set of n LINEARLY INDEPEN- 
DENT Vectors capable of generating an n-dimensional 
SUBSPACE of R n . Given a IlYPERPLANE defined by 

xi + x 2 + X3 4- x 4 + x$ = 0, 

a basis is found by solving for Xi in terms of #2, #3, 2:4, 
and £5. Carrying out this procedure, 



Baseball Cover 



Xi 



-X2 — X3 — X4 — £5, 




~Xi~ 




--1- 




--1- 




--1- 




--1- 


X2 




1 

















X3 


= x 2 





+£3 


1 


~\-X4 





-\-x 5 





X4 














1 







-335- 




. . 




- - 




. . 




. 1 - 



Basis Theorem 



B ayes' Formula 101 



and the above VECTOR form an (unnormalized) BASIS. 
Given a MATRIX A with an orthonormal basis, the MA- 
TRIX corresponding to a new basis, expressed in terms 
of the original xi , . . . , x n is 



A' = [Axi 



Ax n ]. 



see also Bilinear Basis, Modular System Basis, 
Orthonormal Basis, Topological Basis 

Basis Theorem 

see Hilbert Basis Theorem 

Basler Problem 

The problem of analytically finding the value of C(2), 
where £ is the Riemann Zeta Function. 

References 

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 
61, 67-98, 1988. 

Basset Function 

see Modified Bessel Function of the Second 
Kind 

Batch 

A set of values of similar meaning obtained in any man- 
ner. 

References 

Tukey, J. W. Explanatory Data Analysis. Reading, MA: 
Addison-Wesley, p. 667, 1977. 

Bateman Function 

Mx) - r(i < +V ( "^'°' fa) 

for x > 0, where U is a Confluent Hypergeometric 

Function of the Second Kind. 

see also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL 

Equation, Hypergeometric Function 

Batrachion 

A class of CURVE defined at Integer values which hops 
from one value to another. Their name derives from the 
word batrachion, which means "frog- like." Many ba- 
trachions are FRACTAL. Examples include the BLANC- 
MANGE Function, Hofstadter-Conway $10,000 Se- 
quence, Hofstadter's Q-Sequence, and Mallow's 
Sequence. 

References 

Pickover, C. A. "The Crying of Fractal Batrachion 1,489." 

Ch. 25 in Keys to Infinity. New York: W. H. Freeman, 

pp. 183-191, 1995. 



Bauer's Identical Congruence 

Let t(m) denote the set of the </>(m) numbers less than 
and Relatively Prime to m, where <f>(n) is the To- 
tient Function. Define 



f m {x)= n (*-*)• 



(i) 



t(m) 



A theorem of Lagrange states that 

f m {x) = x Hm) -1 (mod to). (2) 

This can be generalized as follows. Let p be an ODD 
Prime Divisor of m and p a the highest Power which 
divides to, then 

f m (x) = (x*- 1 - l)*^)/^- 1 ) (mod p») (3) 

and, in particular, 

/„.(*) = (a*" 1 -l)*" -1 (mod/). (4) 

Furthermore, if to > 2 is EVEN and 2 a is the highest 
POWER of 2 that divides m, then 

/ m (a:) = (a: 2 -l)* (m)/2 (mod 2 a ) (5) 

and, in particular, 

f 2a ( x ) = ( x 2 -l) 2a ~ 2 (mod2 a ). (6) 

see also Leudesdorf Theorem 

References 

Hardy, G. H. and Wright, E. M. "Bauer's Identical Congru- 
ence." §8.5 in An Introduction to the Theory of Numbers, 
5th ed. Oxford, England: Clarendon Press, pp. 98-100, 
1979. 

Bauer's Theorem 

see Bauer's Identical Congruence 

Bauspiel 

A construction for the RHOMBIC DODECAHEDRON. 

References 

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: 
Dover, pp. 26 and 50, 1973. 

Bayes' Formula 

Let A and Bj be Sets. Conditional Probability 
requires that 

P(AC\B j )=P(A)P(B j \Al (1) 

where n denotes INTERSECTION ("and"), and also that 

P(A n Bj) = P(Bj n A) - P{Bj)P{A\Bj) (2) 



102 Bayes' Theorem 

and 

P{B j nA)=P{B j )P{A\B j ). (3) 

Since (2) and (3) must be equal, 

P(AnB j ) = P(B j nA). (4) 

Prom (2) and (3), 

P(AnB j ) = P(B j )P(A\B j ). (5) 

Equating (5) with (2) gives 

P(A)P(B j \A) = P(B i )P(A\B j ), (6) 



so 



P(Bj\A) 



PjB^PjAlBj) 
P(A) ■ 



(7) 



Now, let 



S=U^> (8) 

i=l 

so Ai is an event is S and A» O Aj = for i ^ j, then 

/ N \ JV 

A = A n 5 - A n ( (J ^ J = (J (A n Ai) (9) 



\ N 



P(A) = Pl\J(AnA i )\=Y i P(AnA i ). (10) 
Prom (5), this becomes 

N 

P(A) = Y,P(Ai)P(E\Ai), (11) 

i=l 
SO 

P{Ai)P(A\Ai) 



P(Ai\A) N 

£ P(Ai)P(A\Ai) 

3 = 1 



(12) 



5ee also CONDITIONAL PROBABILITY, INDEPENDENT 

Statistics 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, p. 810, 1992. 

Bayes' Theorem 

see Bayes' Formula 



Beam Detector 

Bayesian Analysis 

A statistical procedure which endeavors to estimate pa- 
rameters of an underlying distribution based on the ob- 
served distribution. Begin with a "PRIOR Distribu- 
tion" which may be based on anything, including an 
assessment of the relative likelihoods of parameters or 
the results of non-Bayesian observations. In practice, it 
is common to assume a UNIFORM DISTRIBUTION over 
the appropriate range of values for the PRIOR Distri- 
bution. 

Given the Prior Distribution, collect data to obtain 
the observed distribution. Then calculate the LIKELI- 
HOOD of the observed distribution as a function of pa- 
rameter values, multiply this likelihood function by the 
PRIOR Distribution, and normalize to obtain a unit 
probability over all possible values. This is called the 
Posterior Distribution. The Mode of the distribu- 
tion is then the parameter estimate, and "probability 
intervals" (the Bayesian analog of Confidence Inter- 
vals) can be calculated using the standard procedure. 
Bayesian analysis is somewhat controversial because the 
validity of the result depends on how valid the PRIOR 
DISTRIBUTION is, and this cannot be assessed statisti- 
cally. 

see also Maximum Likelihood, Prior Distribution, 
Uniform Distribution 

References 

Hoel, P. G.; Port, S. C; and Stone, C. J. Introduction to 
Statistical Theory. New York: Houghton Mifflin, pp. 36- 
42, 1971. 

Iversen, G. R. Bayesian Statistical Inference. Thousand 
Oaks, CA: Sage Pub., 1984. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 799-806, 1992. 

Sivia, D. S. Data Analysis: A Bayesian Tutorial. New York: 
Oxford University Press, 1996. 

Bays' Shuffle 

A shuffling algorithm used in a class of RANDOM NUM- 
BER generators. 

References 

Knuth, D. E. §3.2 and 3.3 in The Art of Computer Program- 
ming, Vol. 2: Seminumerical Algorithms, 2nd ed. Read- 
ing, MA: Addison-Wesley, 1981. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 270-271, 1992. 

Beam Detector 

N. B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 



Bean Curve 




A "beam detector" for a given curve C is defined as 
a curve (or set of curves) through which every Line 
tangent to or intersecting C passes. The shortest 1- 
arc beam detector, illustrated in the upper left figure, 
has length L\ — n + 2. The shortest known 2-arc beam 
detector, illustrated in the right figure, has angles 



Ox « 1.286 rad 
6 2 « 1.191 rad, 



(1) 
(2) 



given by solving the simultaneous equations 

2 cos <9i -sin(§0 2 ) = (3) 

tan(§0i)cos(f 2 ) + sm{±0 2 )[sec 2 {±6 2 ) + 1] = 2. (4) 
The corresponding length is 

L 2 =2tt-26>i -0 2 + 2tan(§0i)+sec(|0 2 ) 

- cos(§<9 2 )+tan(§6>i) sin(±<9 2 ) = 4.8189264563. . . . (5) 

A more complicated expression gives the shortest known 
3-arc length L 3 = 4.799891547. . .. Finch defines 



L = inf L n 

n>l 



(6) 



as the beam detection constant, or the Trench Dig- 
gers' Constant. It is known that L>n. 

References 

Croft, H, T.; Falconer, K, J.; and Guy, R. K. §A30 in Un- 
solved Problems in Geometry. New York: Springer- Verlag, 

1991. 
Faber, V.; Mycielski, J.; and Pedersen, P. "On the Shortest 

Curve which Meets All Lines which Meet a Circle." Ann. 

Polon. Math. 44, 249-266, 1984. 
Faber, V. and Mycielski, J. "The Shortest Curve that Meets 

All Lines that Meet a Convex Body." Amer. Math. 

Monthly 93, 796-801, 1986. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/beam/beam.html. 
Makai, E. "On a Dual of Tarski's Plank Problem." In 

Diskrete Geometric 2 Kolloq., Inst. Math. Univ. Salzburg, 

127-132, 1980. 
Stewart, L "The Great Drain Robbery." Sci. Amer., 206- 

207, 106, and 125, Sept. 1995, Dec. 1995, and Feb. 1996. 

Bean Curve 




Beast Number 103 

The Plane Curve given by the Cartesian equation 
x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 ). 

References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., 1989. 

Beast Number 

The occult "number of the beast" associated in the Bible 
with the Antichrist. It has figured in many numerolog- 
ical studies. It is mentioned in Revelation 13:13: "Here 
is wisdom. Let him that hath understanding count the 
number of the beast: for it is the number of a man; and 
his number is 666." 

The beast number has several interesting properties 
which numerologists may find particularly interesting 
(Keith 1982-83). In particular, the beast number is 
equal to the sum of the squares of the first 7 PRIMES 

2 2 + 3 2 -h 5 2 + 7 2 + ll 2 + 13 2 + 17 2 = 666, (1) 

satisfies the identity 

0(666) = 6-6-6, (2) 

where 4> is the Totient Function, as well as the sum 



^2 = 666. 



(3) 



The number 666 is a sum and difference of the first three 
6th Powers, 

666 = l 6 - 2 6 + 3 6 (4) 

(Keith). Another curious identity is that there are ex- 
actly two ways to insert "+" signs into the sequence 
123456789 to make the sum 666, and exactly one way 
for the sequence 987654321, 

666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 

(5) 
666 = 9 + 87 + 6 + 543 + 21 (6) 

(Keith). 666 is a Repdigit, and is also a Triangular 

Number 

T 6 . 6 = T 36 = 666. (7) 

In fact, it is the largest Repdigit Triangular Num- 
ber (Bellew and Weger 1975-76). 666 is also a Smith 
Number. The first 144 Digits of n - 3, where n is Pi, 
add to 666. In addition 144 = (6 + 6) x (6 + 6) (Blatner 
1997). 

A number of the form 2 1 which contains the digits of the 
beast number "666" is called an Apocalyptic Num- 
ber, and a number having 666 digits is called an APOC- 
ALYPSE Number. 



104 Beatty Sequence 



Bei 



see also Apocalypse Number, Apocalyptic Num- 
ber, Bimonster, Monster Group 

References 

Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num- 
bers." J. Recr. Math. 8, 96-97, 1975-76. 
Blatner, D. The Joy of Pi. New York: Walker, back jacket, 

1997. 
Castellanos, D. "The Ubiquitous tt." Math. Mag. 61, 153- 

154, 1988. 
Hardy, G. H. A Mathematician's Apology, reprinted with a 

foreword by C. P. Snow. New York: Cambridge University 

Press, p, 96, 1993. 
Keith, M. "The Number of the Beast." http://users.aol. 

com/s6sj7gt/mike666.htm. 
Keith, M. "The Number 666." J. Recr. Math. 15, 85-87, 

1982-1983. 




Bee 



A 4-P0LYHEX. 

References 

Gardner, M. Mathematical Magic Show: More Puzzles, 
Games, Diversions, Illusions and Other Mathematical 
Sleight- of -Mind from Scientific American. New York: 
Vintage, p. 147, 1978. 

Behrens-Fisher Test 

see Fisher-Behrens Problem 



Beatty Sequence 

The Beatty sequence is a Spectrum Sequence with an 
Irrational base. In other words, the Beatty sequence 
corresponding to an Irrational Number 6 is given by 
[0J, [20 \, [30J, . . . , where \_x\ is the Floor Function. 
If a and f3 are Positive Irrational Numbers such 

that 

1 1 , 

a p 

then the Beatty sequences [a J , [2aJ , . . . and [f3\ , \_W\ > 
. . . together contain all the POSITIVE INTEGERS without 
repetition. 

References 

Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the 
Return of Dr. Matrix, reissue ed. New York: W. H. Free- 
man, p. 21, 1989. 

Graham, R. L.; Lin, S.; and Lin, C.-S. "Spectra of Numbers." 
Math. Mag. 51, 174-176, 1978. 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, p. 227, 1994. 

Sloane, N. J. A. A Handbook of Integer Sequences. Boston, 
MA: Academic Press, pp. 29-30, 1973. 

Beauzamy and Degot's Identity 

For P, Q, R, and S POLYNOMIALS in n variables 



[PQ,RS]= J^ 



A 



ii,...,t n >0 



iil---i n ] - 



vhere 



A=[Rl i i>-"M(D li ...,D n )Q(x u ... i x n ) 

XP (il, - ,iB) (ft 2?n)5(Xl,.. M In)] 

Di = d/dxi is the Differential Operator, [X,Y] is 
the Bombieri Inner Product, and 

p(ti,...,i™) =D i 1 1 -.-D i r TP. 



Behrraann Cylindrical Equal- Area 
Projection 

A Cylindrical Area-Preserving projection which 

uses 30° N as the no-distortion parallel. 

References 

Dana, P. H. "Map Projections." http://www.utexas.edu/ 
depts/grg/gcraft/notes/mapproj/mapproj ,html, 

Bei 






I Bei z| 



10000 
.10 5000 




-1000UE^^^^^/5 -500 

Re[z] ^i^-lO Re[z]" 5 ^10 

The Imaginary Part of 

J„(xe 3vi/4 ) = ber„(a;) +ibei„(x). (1) 

The special case v = gives 

Jo(iVix) = ber(rc) + ibei(sc), (2) 

where J Q (z) is the zeroth order BESSEL FUNCTION OF 
the First Kind. 



bei (x) = ^ [(2n) , ]2 



(3) 



see also Reznik's Identity 



see also Ber, Bessel Function, Kei, Kelvin Func- 
tions, Ker 



Bell Curve 



Bell Number 105 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- 
tions." §9.9 in Handbook of Mathematical Functions with 
Formulas, Graphs, and Mathematical Tables, 9th printing. 
New York: Dover, pp. 379-381, 1972. 

Spanier, J. and Oldham, K. B. "The Kelvin Functions." 
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- 
sphere, pp. 543-554, 1987. 

Bell Curve 

see Gaussian Distribution, Normal Distribution 

Bell Number 

The number of ways a Set of n elements can be PARTI- 
TIONED into nonempty Subsets is called a Bell Num- 
ber and is denoted B n . For example, there are five 
ways the numbers {1, 2, 3} can be partitioned: {{1}, 
{2}, {3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{1}, {2, 3}}, 
and {{1, 2, 3}}, so B 3 = 5. B = 1 and the first few 
Bell numbers for n = 1, 2, ... are 1, 2, 5, 15, 52, 203, 
877, 4140, 21147, 115975, ... (Sloane's A000110). Bell 
numbers are closely related to CATALAN NUMBERS. 

The diagram below shows the constructions giving B 3 = 
5 and B± = 15, with line segments representing elements 
in the same Subset and dots representing subsets con- 
taining a single element (Dickau). 



B, 



The Integers B n can be defined by the sum 




fc=i ^ J 

= {l} isa 



(i) 



where s£ fc) = i '," *> is a STIRLING NUMBER OF THE 
SECOND Kind, or by the generating function 



B„ 



6 = 2^ 



(2) 



The Bell numbers can also be generated using the BELL 
Triangle, using the Recurrence Relation 



Jn+l 






(3) 



where (£) is a Binomial Coefficient, or using the 
formula of Comtet (1974) 



B n 



-E 



m 



(4) 



where \x] denotes the Ceiling Function. 



The Bell number B n is also equal to n (l), where <t> n (x) 
is a Bell Polynomial. Dobinski's Formula gives 
the nth Bell number 



oo 






(5) 



Lovasz (1993) showed that this formula gives the asymp- 
totic limit 



-1/2 



[A(n)] 



n+l/2 A(n)-n-l 



where A(n) is defined implicitly by the equation 

A(n)log[A(n)] = n. 
A variation of DOBINSKI'S FORMULA gives 

- -«■ ( _ 1)S 



B * = E 5- E 



(6) 



(?) 



(8) 



for 1 < k < n (Pitman 1997). de Bruijn (1958) gave the 
asymptotic formula 



InBn , , , In Inn 1 

= lnn — Inlnn — 1 + — h - — 

n Inn Inn 

WlnlnnX 2 

^2 V Inn / 



In Inn 



(Inn) 2 



Touchard's Congruence states 

B p+k = B k + B k+1 (mod p) , 



(9) 



(10) 



when p is Prime. The only PRIME Bell numbers for 
n < 1000 are B 2 , B 3i B 7 , B 13 , B 42 , and £55. The Bell 
numbers also have the curious property that 



Bq B\ 
B\ £2 



B n 



?n + l 



B 2 
B 3 



B n ^ 



B n 

B n +i 

B 2n 



J[n\ (11) 



(Lenard 1986). 

see also Bell Polynomial, Bell Triangle, Dobin- 
ski's Formula, Stirling Number of the Second 
Kind, Touchard's Congruence 



106 Bell Polynomial 



Beltrami Differential Equation 



References 

Bell, E. T. "Exponential Numbers." Amer. Math. Monthly 
41, 411-419, 1934. 

Comtet, L. Advanced Combinatorics. Dordrecht, Nether- 
lands: Reidel, 1974. 

Conway, J. H. and Guy, R. K. In The Book of Numbers. New 
York: Springer- Verlag, pp. 91-94, 1996. 

de Bruijn, N. G. Asymptotic Methods in Analysis. New York: 
Dover, pp. 102-109, 1958. 

Dickau, R. M. "Bell Number Diagrams." http:// forum . 
swarthmore.edu/advanced/robertd/bell.html. 

Gardner, M. "The Tinkly Temple Bells." Ch. 2 in Fractal 
Music, HyperCards, and More Mathematical Recreations 
from Scientific American Magazine. New York: W. H. 
Freeman, 1992. 

Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- 
phy of Two Special Number Sequences, 6th ed. Morgan- 
town, WV: Math Monongliae, 1985. 

Lenard, A. In Fractal Music, HyperCards, and More Math- 
ematical Recreations from Scientific American Magazine. 
(M. Gardner). New York: W. H. Freeman, pp. 35-36, 
1992. 

Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p, 
of First Order Bell Exponential Integrals." Math. Comput. 
16, 416-423, 1962. 

Lovasz, L. Combinatorial Problems and Exercises, 2nd ed. 
Amsterdam, Netherlands: North-Holland, 1993. 

Pitman, J. "Some Probabilistic Aspects of Set Partitions." 
Amer. Math. Monthly 104, 201-209, 1997. 

Rota, G.-C. "The Number of Partitions of a Set." Amer. 
Math. Monthly 71, 498-504, 1964. 

Sloane, N. J. A. Sequence A000110/M1484 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Bell Polynomial 




0.2 0.4 0.6 0.8 1 

Two different GENERATING FUNCTIONS for the Bell 
polynomials for n > are given by 



<t> n {x) =e x ^ 



k n ~ 1 x k 



The Bell polynomials are denned such that <f> n (l) = B nj 
where B n is a Bell NUMBER. The first few Bell poly- 
nomials are 



4>o(x 
<pi(x 

4>2(x 

fo{x 
(J>a{x 

<p 6 (x 



= 1 
= X 

= x + x 2 

- x + 3z 2 + x 3 

= x + 7x 2 + 6x 3 + x 4 

= x 4- 15x 2 + 25a; 3 + 10z 4 + x 5 

= x + Six 2 + 90x 3 + 65z 4 + 15a; 5 + x 6 . 



see also Bell Number 

References 

Bell, E. T. "Exponential Polynomials." 
258-277, 1934. 



Ann. Math. 35, 



Bell Triangle 

12 5 15 52 203 877 ... 

1 3 10 37 151 674 \ 

2 7 27 114 523 \ 

5 20 87 409 \ 

15 67 322 \ 

52 255 •■. 

203 ■-. 

A triangle of numbers which allow the Bell Numbers 
to be computed using the Recurrence Relation 



= Va 



B n+1 = 2^B k { n k 

k-o 



see also Bell Number, Clark's Triangle, Leibniz 
Harmonic Triangle, Number Triangle, Pascal's 
Triangle, Seidel-Entringer-Arnold Triangle 

Bellows Conjecture 

see Flexible Polyhedron 

Beltrami Differential Equation 

For a measurable function /z, the Beltrami differential , 
equation is given by 



n~ 1 s v 

4> n (x) = x^2 [ k-1 j^" 1 ^)' 

where (£) is a Binomial Coefficient. 



where f z is a PARTIAL DERIVATIVE and z* denotes the 

Complex Conjugate of z. 

see also QUASICONFORMAL MAP 

References 

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary 
of Mathematics. Cambridge, MA: MIT Press, p. 1087, 
1980. 



Beltrami Field 



Ben ford's Law 107 



Beltrami Field 

A Vector Field u satisfying the vector identity 

u x (V x u) = 

where A x B is the CROSS Product and V x A is the 

CURL is said to be a Beltrami field. 

see also DlVERGENCELESS FIELD, IRROTATIONAL 

Field, Solenoidal Field 

Beltrami Identity 

An identity in CALCULUS OF VARIATIONS discovered in 
1868 by Beltrami. The Euler-Lagrange Differen- 
tial Equation is 



d£__d_ 
dy dx 



(&)-* 



Now, examine the DERIVATIVE of x 

— ~ l/x T" n t/xx ~r • 

ax oy oy x ox 



Solving for the df /dy term gives 



dy 1 



dx dy x 



0/ b 

dx' 



Now, multiplying (1) by y x gives 



(i) 



(2) 



(3) 



(4) 



(5) 



(6) 



This form is especially useful if f x = 0, since in that case 



0/ _ d_ 
oy ax 



dy* J 



Substituting (3) into (4) then gives 



dx 


dy x Vxz dx 


x dx \dy x 




dx dx \ 


- y *dyZ) = 



dx 
which immediately gives 

/ 



dy x 



= 0, 



dy x 



(7) 



(8) 



where C is a constant of integration. 

The Beltrami identity greatly simplifies the solution for 
the minimal AREA SURFACE OF REVOLUTION about 
a given axis between two specified points. It also al- 
lows straightforward solution of the BRACHISTOCHRONE 
Problem. 

see also Brachistochrone Problem, Calculus of 
Variations, Euler-Lagrange Differential Equa- 
tion, Surface of Revolution 



Bend (Curvature) 

Given four mutually tangent circles, their bends are de- 
fined as the signed CURVATURES of the CIRCLES. If the 
contacts are all external, the signs are all taken as Pos- 
itive, whereas if one circle surrounds the other three, 
the sign of this circle is taken as NEGATIVE (Coxeter 
1969). 

see also Curvature, Descartes Circle Theorem, 
Soddy Circles 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New- 
York: Wiley, pp. 13-14, 1969. 

Bend (Knot) 

A Knot used to join the ends of two ropes together to 
form a longer length. 

References 

Owen, P. Knots. Philadelphia, PA: Courage, p. 49, 1993. 

Benford's Law 

Also called the FIRST DIGIT LAW, FIRST DIGIT PHE- 
NOMENON, or Leading Digit Phenomenon. In list- 
ings, tables of statistics, etc., the DIGIT 1 tends to oc- 
cur with Probability ~ 30%, much greater than the 
expected 10%. This can be observed, for instance, by 
examining tables of LOGARITHMS and noting that the 
first pages are much more worn and smudged than later 
pages. The table below, taken from Benford (1938), 
shows the distribution of first digits taken from several 
disparate sources. Of the 54 million real constants in 
Plouffe's "Inverse Symbolic Calculator" database, 30% 
begin with the Digit 1. 



Title 




First Digit 




# 




12 3 


4 


5 6 7 8 


9 




Rivers, Area 


31.0 16.4 10.7 11.3 


7.2 8.6 5.5 4.2 


5.1 


335 


Population 


33.9 20.4 14.2 


8.1 


7.2 6.2 4.1 3.7 


2.2 3259 


Constants 


41.3 14.4 4.8 


8.6 10.6 5.8 1.0 2.9 10.6 


104 


Newspapers 


30.0 18.0 12.0 


10.0 


8.0 6.0 6.0 5.0 


5.0 


100 


Specific Heat 


24.0 18.4 16.2 


14.6 10.6 4.1 3.2 4.8 


4.1 


1389 


Pressure 


29.6 18.3 12.8 


9.8 


8.3 6.4 5.7 4.4 


4.7 


703 


H.P. Lost 


30.0 18.4 11.9 


10.8 


8.1 7.0 5.1 5.1 


3.6 


690 


Mol. Wgt. 


26.7 25.2 15.4 


10.8 


6.7 5.1 4.1 2.8 


3.2 1800 


Drainage 


27.1 23.9 13.8 


12.6 


8.2 5.0 5.0 2.5 


1.9 


159 


Atomic Wgt. 


47.2 18.7 5.5 


4.4 


6.6 4.4 3.3 4.4 


5.5 


91 


n" 1 , sfn 


25.7 20.3 9.7 


6.8 


6.6 6.8 7.2 8.0 


8.9 5000 


Design 


26.8 14.8 14.3 


7.5 


8.3 8.4 7.0 7.3 


5.6 


560 


Reader's Dig. 


33.4 18.5 12.4 


7.5 


7.1 6.5 5.5 4.9 


4.2 


308 


Cost Data 


32.4 18.8 10.1 


10.1 


9.8 5.5 4.7 5.5 


3.1 


741 


X-Ray Volts 


27.9 17.5 14.4 


9.0 


8.1 7.4 5.1 5.8 


4.8 


707 


Am. League 


32.7 17.6 12.6 


9.8 


7.4 6.4 4.9 5.6 


3.0 


1458 


Blackbody 


31.0 17.3 14.1 


8.7 


6.6 7.0 5.2 4.7 


5.4 1165 


Addresses 


28.9 19.2 12.6 


8.8 


8.5 6.4 5.6 5.0 


5.0 


342 


n 1 , n 2 - • - n\ 


25.3 16.0 12.0 10.0 


8.5 8.8 6.8 7.1 


5.5 


900 


Death Rate 


27.0 18.6 15.7 


9.4 


6.7 6.5 7.2 4.8 


4.1 


418 


Average 


30.6 18.5 12.4 


9.4 


8.0 6.4 5.1 4.9 


4.7 1011 


Prob. Error 


0.8 0.4 0.4 


0.3 


0.2 0.2 0.2 0.2 


0.3 





108 



Benham's Wheel 



Benson's Formula 



In fact, the first SIGNIFICANT DIGIT seems to follow a 

Logarithmic Distribution, with 

P(n) « log(n + 1) - logn 

for n — 1, . . . , 9. One explanation uses Central Limit- 
like theorems for the MANTISSAS of random variables 
under Multiplication. As the number of variables in- 
creases, the density function approaches that of a LOG- 
ARITHMIC DISTRIBUTION. 

References 

Benford, F. "The Law of Anomalous Numbers." Proc. Amer. 
Phil Soc. 78, 551-572, 1938. 

Boyle, J. "An Application of Fourier Series to the Most Sig- 
nificant Digit Problem." Amer. Math. Monthly 101, 879™ 
886, 1994. 

Hill, T. P. "Base-Invariance Implies Benford 's Law." Proc. 
Amer. Math. Soc. 12, 887-895, 1995. 

Hill, T. P. "The Significant-Digit Phenomenon." Amer. 
Math. Monthly 102, 322-327, 1995. 

Hill, T. P. "A Statistical Derivation of the Significant-Digit 
Law." Stat Sci. 10, 354-363, 1996. 

Hill, T. P. "The First Digit Phenomenon." Amer. Sci. 86, 
358-363, 1998. 

Ley, E. "On the Peculiar Distribution of the U.S. Stock In- 
dices Digits." Amer. Stat. 50, 311-313, 1996. 

Newcomb, S. "Note on the Frequency of the Use of Digits in 
Natural Numbers." Amer. J. Math. 4, 39-40, 1881. 

Nigrini, M. "A Taxpayer Compliance Application of Ben- 
ford's Law." J. Amer. Tax. Assoc. 18, 72-91, 1996. 

Plouffe, S. "Inverse Symbolic Calculator." http://www.cecm. 
sfu.ca/projects/ISC/. 

Raimi, R. A. "The Peculiar Distribution of First Digits." Sci. 
Amer. 221, 109-119, Dec. 1969. 

Raimi, R. A. "The First Digit Phenomenon." Amer. Math, 
Monthly 83, 521-538, 1976. 

Benham's Wheel 




An optical ILLUSION consisting of a spinnable top 
marked in black with the pattern shown above. When 
the wheel is spun (especially slowly), the black broken 
lines appear as green, blue, and red colored bands! 

References 

Cohen, J. and Gordon, D. A. "The Prevost-Fechner-Benham 
Subjective Colors." Psycholog. Bull. 46, 97-136, 1949. 

Festinger, L.; Allyn, M. R.; and White, C. W. "The Percep- 
tion of Color with Achromatic Stimulation." Vision Res. 
11, 591-612, 1971. 

Fineman, M. The Nature of Visual Illusion. New York: 
Dover, pp. 148-151, 1996. 

Trolland, T. L. "The Enigma of Color Vision." Amer. J. 
Physiology 2, 23-48, 1921. 



Bennequin's Conjecture 

A BRAID with M strands and R components with P 
positive crossings and N negative crossings satisfies 

\P - N\ < 2U + M - R < P + iV, 

where U is the UNKNOTTING NUMBER. While the 
second part of the Inequality was already known to 
be true (Boileau and Weber, 1983, 1984) at the time 
the conjecture was proposed, the proof of the entire 
conjecture was completed using results of Kronheimer 
and Mrowka on MlLNOR'S CONJECTURE (and, indepen- 
dently, using Menasco's Theorem). 

see also Braid, Menasco's Theorem, Milnor's Con- 
jecture, Unknotting Number 

References 

Bennequin, D. "L'instanton gordien (d'apres P. B. Kron- 
heimer et T. S. Mrowka)." Asterisque 216, 233-277, 1993. 

Birman, J. S. and Menasco, W. W. "Studying Links via 
Closed Braids. II. On a Theorem of Bennequin." Topology 
Appl. 40, 71-82, 1991. 

Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le 
nombre gordien des nceuds algebriques." Enseign. Math. 
30, 173-222, 1984. 

Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le 
nombre gordien des nceuds algebriques." In Knots, Braids 
and Singularities (Plans- sur- Bex, 1982). Geneva, Switzer- 
land: Monograph. Enseign. Math. Vol. 31, pp. 49-98, 
1983. 

Cipra, B. What's Happening in the Mathematical Sciences, 
Vol. 2. Providence, RI: Amer. Math. Soc, pp. 8-13, 1994. 

Kronheimer, P. B. "The Genus-Minimizing Property of Al- 
gebraic Curves." Bull. Amer. Math. Soc. 29, 63-69, 1993. 

Kronheimer, P. B. and Mrowka, T. S, "Gauge Theory for 
Embedded Surfaces. I." Topology 32, 773-826, 1993. 

Kronheimer, P. B. and Mrowka, T. S. "Recurrence Relations 
and Asymptotics for Four-Manifold Invariants." Bull. 
Amer. Math. Soc. 30, 215-221, 1994. 

Menasco, W. W. "The Bennequin-Milnor Unknotting Con- 
jectures." C. R. Acad. Sci. Paris Ser. I Math. 318, 831- 
836, 1994, 

Benson's Formula 

An equation for a LATTICE SUM with n = 3 

i+i+fe+l 



i, j,k= — oo V J 



= 12?r ^ sech 2 (!7iVm 2 +n 2 ). 

m, n=l, 3, ... 

Here, the prime denotes that summation over (0, 0, 0) is 
excluded. The sum is numerically equal to —1.74756 . . ., 
a value known as "the" MADELUNG CONSTANT. 

see also MADELUNG CONSTANTS 

References 

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in 

Analytic Number Theory and Computational Complexity. 

New York: Wiley, p. 301, 1987. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/mdlung/mdlTing.html. 



Ber 

Ber 





| Ber z | 




The Real Part of 

J„(xe 3ni/4 ) = beT v (x)+ibei v (x). 
The special case v = gives 

Jo(iV^x) = ber(:r) + zbei(z), 



(1) 



(2) 



where J is the zeroth order BESSEL FUNCTION OF THE 
First Kind. 

i 2+4n 



ber( !B ) = ^ [(2n + 1)!] 2 ■ 



(3) 



see a/so Bei, Bessel Function, Kei, Kelvin Func- 
tions, Ker 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- 
tions." §9.9 in Handbook of Mathematical Functions with 
Formulas, Graphs, and Mathematical Tables, 9th printing. 
New York: Dover, pp. 379-381, 1972. 

Spanier, J. and Oldham, K. B. "The Kelvin Functions." 
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- 
sphere, pp. 543-554, 1987. 

Beraha Constants 

The nth Beraha constant is given by 

'2tt\ 



Be„ = 2 + 2 cos 



(!)- 



The first few are 

Bei =4 

Be 2 = 

Be 3 = 1 

Be 4 = 2 

Be 5 = |(3 + \/5)« 2.618 

Be 6 = 3 

Be 7 = 2 + 2cos(|7r) « 3.247.... 

They appear to be ROOTS of the CHROMATIC POLY- 
NOMIALS of planar triangular GRAPHS. Be 4 is 0+1, 

where <p is the Golden Ratio, and Be 7 is the Silver 
Constant. 

References 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 143, 1983. 



Bernoulli Differential Equation 109 

Berger-Kazdan Comparison Theorem 

Let M be a compact n-D Manifold with Injectivity 
radius inj(M). Then 



Vol(M) > 



qnj(M) 



with equality IFF M is ISOMETRIC to the standard round 
Sphere S n with Radius inj(M), where c n {r) is the 
Volume of the standard u-Hypersphere of Radius 
r. 

see also Blaschke Conjecture, Hypersphere, In- 

jective, Isometry 

References 

Chavel, I. Riemannian Geometry: A Modern Introduction. 
New York: Cambridge University Press, 1994. 

Bergman Kernel 

A Bergman kernel is a function of a COMPLEX VARI- 
ABLE with the "reproducing kernel" property defined 
for any Domain in which there exist NONZERO Ana- 
lytic Functions of class L 2 (D) with respect to the 

Lebesgue Measure dV. 

References 

Hazewinkel, M. (Managing Ed,). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, pp. 356-357, 1988. 

Bergman Space 

Let G be an open subset of the COMPLEX PLANE C, and 
let Ll(G) denote the collection of all Analytic Func- 
tions / : G — y C whose Modulus is square integrable 
with respect to Area measure. Then L 2 a {G), sometimes 
also denoted A 2 (G), is called the Bergman space for G. 
Thus, the Bergman space consists of all the ANALYTIC 
FUNCTIONS in L 2 (G). The Bergman space can also be 
generalized to L?(G), where < p < oo. 

Bernoulli Differential Equation 



-£ + p(x)y = q(x)y n . 
Let v ~ y 1 " 71 for n ^ 1, then 

dv , . - n dy 

— — (l - n)y — . 
dx v }y dx 

Rewriting (1) gives 

y~ n f; = q(x) - p{x)y'~ n = q(x) - vp(x). 



Plugging (3) into (2), 



dv 



— = (I - n)[q(x) - vp(x)]. 



(1) 



(2) 



(3) 



(4) 



110 



Bernoulli Distribution 



Bernoulli Function 



Now, this is a linear First-Order Ordinary Differ- 
ential Equation of the form 



^+vP(x) = Q{x), 



(5) 



where P(x) = (l-n)p(x) and Q(x) = (l-n)q(x). It can 
therefore be solved analytically using an Integrating 
Factor 



/ 



Jri'U 



c Q(x)dx + C 



J P(x)dx 

_ (1 - n) / e (1 - n) / pW dx g(x) dx + C 

(1-n) J p(x) dx 



(6) 



where C is a constant of integration. If n = 1, then 
equation (1) becomes 



dy 
dx 



= y(q-p) 



— = {q-p)dx 



(?) 
(8) 



y = C 2 ef [q{x) - p{x)]dx . (9) 

The general solution is then, with C\ and C2 constants, 

l/(l-n) 



y= < 



'(l-nlp 1 - 10 /* 



a;) da; 



4(3;) das+Ci 



(1 



r^j 



p(ar) da: 



for n ^ 1 
C e/ te(s)-p(x)]dx 

for n = 1. 



(10) 



Bernoulli Distribution 

A Distribution given by 



p M = {l 



q = 1 — p for n = 
for n — 1 



— p n (l— p) n for n = 0,1. 



(i) 

(2) 



The distribution of heads and tails in COIN TOSSING is 
a Bernoulli distribution with p = q — 1/2. The GENER- 
ATING FUNCTION of the Bernoulli distribution is 

1 

M « = (*'"> = E ^w - p) 1_n = e °( 1 - p) + e ^ 

(3) 

SO 



M(t) = (l-p)+pe t 


(4) 


M'{t) = pe 


(5) 


M"(t)=pe t 


(6) 


M (n) (t)=pe\ 


(7) 



and the Moments about are 

Ml=M = M'(0)=p (8) 

& = M"(0) = p (9) 

M ; = M ( " ) (0)=p. (10) 

The Moments about the Mean are 

P2 = p' 2 -(pi) 2 =P-P 2 =p(l-p) (11) 

p.3=p 3 - 3^2Pi + 2(p' 1 ) 3 = p - 3p 2 + 2p 3 

= p(l-p)(l-2p) (12) 

P4 = A»4 - 4/i3pi + 6^2 (m!) 2 - 3(pi) 4 



= p - 4p 2 + 6p 3 - 3p 4 
= p(l-p)(3p 2 -3p+l). 



(13) 



The Mean, Variance, Skewness, and Kurtosis are 
then 



P = Pi = P 
cr 2 - p.2 = p(l - p) 

_fi 3 _ p(l-p)(l-2p) 

71 <T 3 [p(l - p)]3/ 2 

_ l-2p 

H4 p(l-2p)(2p 2 -2p+l) 

72 = —t - 3 = 



P 2 (l-P) 2 



6p 2 - 6p + 1 



p(l-p) 
To find an estimator for a population mean, 

V^ ( N 



(14) 
(15) 



(16) 



(17) 



■0) 



JVp=0 v / 



Np=l 



= e[e + (i-8)] N - 1 = e, 



(18) 



so (p) is an Unbiased Estimator for 9, The probabil- 
ity of Np successes in N trials is then 



N 
Np 



e Np (i-o) Nq , 



(19) 



where 



__ [number of successes] _ n 

p- x =77- _ (20) 



see also BINOMIAL DISTRIBUTION 

Bernoulli Function 

see Bernoulli Polynomial 



Bernoulli Inequality 
Bernoulli Inequality 

(l + x) n > 1 + nx, 



(1) 



where x£l> — 1^0, n€Z> 1. This inequality can 
be proven by taking a MACLAURIN SERIES of (1 + x) n , 



Bernoulli Number 111 

B n Bernoulli numbers may be calculated from the inte- 
gral 

(3) 



Bn=4n L **=r 



and analytically from 



(l+x) n = l+n^+|n(n-l)x +|n(n-l)(n-2)a; +.... 

(2) 
Since the series terminates after a finite number of terms 
for INTEGRAL n, the Bernoulli inequality for x > is 
obtained by truncating after the first-order term. When 
— 1 < x < 0, slightly more finesse is needed. In this case, 
let y = \x\ = — cc > so that < y < 1, and take 

(l-y) n = l-ny+in(n-l)y 2 - in(n-l)(n-2)y 3 + . . . . 

(3) 
Since each Power of y multiplies by a number < 1 and 
since the ABSOLUTE VALUE of the COEFFICIENT of each 
subsequent term is smaller than the last, it follows that 
the sum of the third order and subsequent terms is a 
Positive number. Therefore, 



(i - vT > i 



ny, 



(4) 



(1 -f x) n > 1 + nx, for - 1 < x < 0, (5) 

completing the proof of the INEQUALITY over all ranges 
of parameters. 



Bernoulli Lemniscate 

see Lemniscate 

Bernoulli Number 

There are two definitions for the Bernoulli numbers. The 
older one, no longer in widespread use, defines the Ber- 
noulli numbers B* by the equations 



-12 *-*> 



n — 1 r>* ™2n 



i-rr^B^x 

(2n)! 



B{x 2 B$x A Btx [ 



2! 



+ 



4! 6! 



-f ... (1) 



for \x\ < 27r, or 



2(2n)! v . 



p=i 



2(2n)! 
(2tt) 2 " 



C(2r 



(4) 



where ((z) is the RlEMANN Zeta 



Function. 

The first few Bernoulli numbers B* are 



b; 


= 


i 

6 


b; 


= 


1 
30 


b; 


= 


1 
42 


bx 


= 


1 
30 


b; 


= 


5 
66 


bi 


= 


691 
2,730 


b; 


= 


7 
6 


B' 8 


= 


3,617 
510 


b; 


= 


43,867 
798 


^10 


= 


174,611 
330 


*n 


= 


854,513 
138 



Bernoulli numbers defined by the modern definition are 
denoted B n and also called "EVEN-index" Bernoulli 
numbers. These are the Bernoulli numbers returned by 
the Mathematical (Wolfram Research, Champaign, IL) 
function BernoulliB[n] . These Bernoulli numbers are 
a superset of the archaic ones B n since 



r 1 



B n 



for n = 
for n = 1 



(-l)^/ 2 )- 1 ^;^ for n even 
< for n odd. 

The B n can be defined by the identity 

B n x n 



(5) 






(6) 



, x (x\ ^ B n x 2r 
'- 2 COt (2J S T,~§M 



2! 



+ 



B* 2 x A 
4! 



+ 



D* ™6 
-P3^ 

6! 



+ ... (2) 



for \x\ < 7T (Whittaker and Watson 1990, p. 125). Grad- 
shteyn and Ryzhik (1979) denote these numbers B n , 
while Bernoulli numbers defined by the newer (National 
Bureau of Standards) definition are denoted B, The 



These relationships can be derived using the generating 
function 



F(*,t) = £*££, 



(7) 



which converges uniformly for \t\ < 2tt and all x (Castel- 
lanos 1988). Taking the partial derivative gives 

dF(x,t) _ A B n ^(x)t n _ + ^ B n {x)t n 



dx 



Z— < ( n - i)! Z-, n \ 



(8) 



112 Bernoulli Number 

The solution to this differential equation is 

F(x,t) = T(t)e xt , 
so integrating gives 

/ F(x,t)dx = T(t) / e xt dx = T{t)^—- - 
./o Jo l 

00 *«- r 1 

n = l * / ° 



(9) 



(a;) da? 



1 + 



te 



_ 1 ~ 2^ n : 






(a;)da; = 1 (10) 



(11) 



(Castellanos 1988). Setting x = and adding t/2 to 
both sides then gives 



B2nt 



itcoth(It) = ^ 

n—O 

Letting t = 2ix then gives 

00 . 2 

xcotx = ^(-i)"^*^ 

n=0 



(12) 



2a 2 
(2n)! 



(13) 



for x 6 [— 7r,7r], The Bernoulli numbers may also be 
calculated from the integral 



n! f z dz 
n=r 2^7 ^TT^+T' 



(14) 



(15) 



or from 

Bn= \ dn x ' 

[dx n e x — 1_ 
The Bernoulli numbers satisfy the identity 



*t>H*r)*- + - + (*i> +fl —- 

(16) 

where (£) is a BINOMIAL COEFFICIENT. An asymptotic 
Formula is 

lim \B 2n \ ~4,^{ — \ U . (17) 

n-voo \7re/ 

Bernoulli numbers appear in expressions of the form 

X^fe = i k P y wnere V — I? 2, Bernoulli numbers also 

appear in the series expansions of functions involving 
tanx, cotx, csccc, ln|sinx|, ln|cosa?|, ln|tanx|, tanhx, 



Bernoulli Number 

cothx, and cschx. An analytic solution exists for EVEN 
orders, 



B 2 



(-l)- 1 2(2n)! ^ -2n _ (-l)- 1 2(2n)! 



(2») 



ir) 2 n ^—~f 



P 



p=i 



(2w) 2n 



: C(2n) 

(18) 

for n = 1, 2, ..., where ((2n) is the RlEMANN ZETA 
FUNCTION. Another intimate connection with the RlE- 
MANN Zeta Function is provided by the identity 



£ n = (-l) n+1 nC(l-n). 



(19) 



The Denominator of B 2k is given by the von Staudt- 

Clausen Theorem 



2fc + l 



denom(B 2 fc) = fj P> 



(20) 



p prime 
(p-l)|2fc 



which also implies that the DENOMINATOR of B 2 k is 
Squarefree (Hardy and Wright 1979). Another curi- 
ous property is that the fraction part of B n in DECIMAL 
has a Decimal Period which divides n, and there is a 
single digit before that period (Conway 1996). 



B = 


1 


B 1 = 


1 
2 


B 2 = 


1 

6 


£4 = 


1 
30 


B<> = 


1 
42 


B 8 = 


1 
30 


3io = 


5 
66 



B12 = — 

B14 = 6 

Big = — 



691 
2,730 



798 
174,611 



518 
#20 



D 854,513 

^22 - i3 8 



(Sloane's A000367 and A002445). In addition, 



B2n+1 — 



(21) 



for n = 1, 2, 



Bernoulli first used the Bernoulli numbers while com- 
puting X)fc=i ^ P - l* e used the property of the FlGURATE 
Number Triangle that 



£< 



(n + l)a n 

i + i 



(22) 



Bernoulli Number 



Bernoulli Polynomial 113 



along with a form for a n j which he derived inductively 
to compute the sums up to n = 10 (Boyer 1968, p. 85). 
For p € Z > 0, the sum is given by 



where the NOTATION B^ means the quantity in ques- 
tion is raised to the appropriate POWER fc, and all terms 
of the form B™ are replaced with the corresponding Ber- 
noulli numbers B m . Written explicitly in terms of a sum 
of Powers, 



I> = 



B kP l 



fc!(p-fc + l)! 



j-Hl 



(24) 



Plouffe, S. "Plouffe's Inverter: Table of Current Records for 
the Computation of Constants." http://lacim.uqam.ca/ 
pi/records .html. 

Ramanujan, S. "Some Properties of Bernoulli's Numbers." 
J. Indian Math. Soc. 3, 219-234, 1911. 

Sloane, N. J. A. Sequences A000367/M4039 and A002445/ 
M4189 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Spanier, J. and Oldham, K. B. "The Bernoulli Numbers, 
B n ." Ch. 4 in An Atlas of Functions. Washington, DC: 
Hemisphere, pp. 35-38, 1987, 

Wagstaff, S. S. Jr. "Ramanujan's Paper on Bernoulli Num- 
bers." J. Indian Math. Soc. 45, 49-65, 1981. 

Whit taker, E. T. and Watson, G. N. A Course in Modern 
Analysis, 4th ed. Cambridge, England: Cambridge Uni- 
versity Press, 1990. 

Bernoulli's Paradox 

Suppose the Harmonic Series converges to h: 



It is also true that the COEFFICIENTS of the terms in 
such an expansion sum to 1 (which Bernoulli stated 
without proof). Ramanujan gave a number of curi- 
ous infinite sum identities involving Bernoulli numbers 
(Berndt 1994). 

G. J. Fee and S. Plouffe have computed #200,000? which 
has ~ 800,000 Digits (Plouffe). Plouffe and collabora- 
tors have also calculated B n for n up to 72,000. 

see also Argoh's Conjecture, Bernoulli Func- 
tion, Bernoulli Polynomial, Debye Functions, 
Euler-Maclaurin Integration Formulas, Euler 
Number, Figurate Number Triangle, Genocchi 
Number, Pascal's Triangle, Riemann Zeta Func- 
tion, von Staudt-Clausen Theorem 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli 
and Euler Polynomials and the Euler-Maclaurin Formula." 
§23.1 in Handbook of Mathematical Functions with Formu- 
las, Graphs, and Mathematical Tables, 9th printing. New 
York: Dover, pp. 804-806, 1972. 

Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula." 
§5.9 in Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 327-338, 1985. 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 71, 1987. 

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: 
Springer- Verlag, pp. 81-85, 1994. 

Boyer, C. B. A History of Mathematics. New York: Wiley, 
1968. 

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 
61, 67-98, 1988. 

Conway, J. H. and Guy, R. K. In The Book of Numbers. New 
York: Springer- Verlag, pp. 107-110, 1996. 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, 1980. 

Hardy, G. H. and Wright, W. M. An Introduction to the The- 
ory of Numbers, 5th ed. Oxford, England: Oxford Univer- 
sity Press, pp. 91-93, 1979. 

Ireland, K. and Rosen, M. "Bernoulli Numbers." Ch. 15 in 
A Classical Introduction to Modern Number Theory, 2nd 
ed. New York: Springer- Verlag, pp. 228-248, 1990. 

Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, 
Euler, and Bernoulli Numbers." Math. Comput. 21, 663- 
688, 1967. 



00 



Then rearranging the terms in the sum gives 
h-l = h, 

which is a contradiction. 

References 

Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3 

in Mathematical Plums (Ed. R. Honsberger). Washington, 

DC: Math. Assoc. Amer., pp. 39-40, 1979. 

Bernoulli Polynomial 




There are two definitions of Bernoulli polynomials in 
use. The nth Bernoulli polynomial is denoted here by 
B n (x)i and the archaic Bernoulli polynomial by -B*(x). 
These definitions correspond to the BERNOULLI NUM- 
BERS evaluated at 0, 



B n = B n (0) 

b: = s;(o). 



They also satisfy 



and 



B„(l) = (-l) n B n (0) 
B n (l-x) = (-l) n B n (x) 



(1) 

(2) 



(3) 
(4) 



114 Bernoulli Polynomial 



Bernstein's Constant 



(Lehmer 1988). The first few Bernoulli POLYNOMIALS 
are 

B (x) = l 
B!(x) = x- \ 
' B 2 (x) = x 2 -i+ | 
B 3 (x) = x 3 - §z 2 + \x 
B A {x) = x 4 -2x z + x 2 - ^ 
B 5 (x) = x 5 -%x 4 + lx 3 -±x 
B 6 (x) = x 6 - 3x 5 + f x 4 ~ \x 2 + ^. 

Bernoulli (1713) defined the POLYNOMIALS in terms of 
sums of the Powers of consecutive integers, 



fc=0 



&"- 1 = -[B n {m) - B„(0)]. 



(5) 



Euler (1738) gave the Bernoulli POLYNOMIALS B n (x) in 
terms of the generating function 



e 4 - 1 ^-^ n\ 



They satisfy recurrence relation 
dB n 

T = nB - l(l) 

(Appell 1882), and obey the identity 
B n (x) = (B + x) n , 



(6) 



(7) 



(8) 



where B k is interpreted here as Bk(x). Hurwitz gave 
the Fourier Series 



B n {x) 



(2«) 



- ^ A-V"^ (9) 



for < x < 1, and Raabe (1851) found 

m-l 

~ 1Z B " ( X + ) = m " n5 "( mX )' ( 10 ) 

fc=0 

A sum identity involving the Bernoulli POLYNOMIALS is 



f2(™)B k (a)B m - k (0) 



= _( m -l)B m (a + /3)+m(a + /3-l)B m _i(a + /3) (11) 

for an INTEGER m and arbitrary REAL NUMBERS a and 
P. 

see also Bernoulli Number, Euler-Maclaurin In- 
tegration Formulas, Euler Polynomial 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli 
and Euler Polynomials and the Euler-Maclaurin Formula." 
§23.1 in Handbook of Mathematical Functions with Formu- 
las, Graphs, and Mathematical Tables, 9th printing. New 
York: Dover, pp. 804-806, 1972. 

Appell, P. E. "Sur une classe de polynomes." Annales d'Ecole 
Normal Superieur, Ser. 2 9, 119-144, 1882. 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, p. 330, 1985. 

Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. 
Published posthumously. 

Euler, L. "Methodus generalis summandi progressiones." 
Comment. Acad. Set. Petropol. 6, 68-97, 1738. 

Lehmer, D. H. "A New Approach to Bernoulli Polynomials." 
Amer. Math. Monthly. 95, 905-911, 1988. 

Lucas, E. Ch. 14 in Theorie des Nombres. Paris, 1891. 

Raabe, J. L. "Zuruckfiihrung einiger Summen und bes- 
timmten Integrale auf die Jakob Bernoullische Function." 
J. reine angew. Math. 42, 348-376, 1851. 

Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial 
B n (x)" Ch. 19 in An Atlas of Functions. Washington, 
DC: Hemisphere, pp. 167-173, 1987. 

Bernoulli's Theorem 

see Weak Law of Large Numbers 

Bernoulli Trial 

An experiment in which s TRIALS are made of an event, 
with probability p of success in any given TRIAL. 

Bernstein-Bezier Curve 

see Bezier Curve 

Bernstein's Constant 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

Let E n (f) be the error of the best uniform approxima- 
tion to a Real function f(x) on the Interval [—1,1] 
by Real Polynomials of degree at most n. If 

«(*) = M> (i) 

then Bernstein showed that 

0.267... < lim 2nE 2n {a) < 0.286. (2) 

n— >oo 

He conjectured that the lower limit {(5) was f3 — 
1/(2^/7?). However, this was disproven by Varga and 
Carpenter (1987) and Varga (1990), who computed 



/? = 0.2801694990.... 



(3) 



For rational approximations p(x)/q(x) for p and q of 
degree m and n, D. J. Newman (1964) proved 



i e _ 9v ^ < Enn ( a) < 3e -^ 



(4) 



Bernstein's Inequality 



Bernstein-Szego Polynomials 115 



for n > 4. Gonchar (1967) and Bulanov (1975) improved 
the lower bound to 



-7rVn+T 



< K,„(a) < 3e~^\ 



(5) 



Vjacheslavo (1975) proved the existence of POSITIVE 
constants m and M such that 



m<e Vy/K E^ n [pL) <M 



(6) 



(Petrushev 1987, pp. 105-106). Varga et al (1993) con- 
jectured and Stahl (1993) proved that 



lim e 2n i?2Ti,2n,(a) = 8. 

n—too 



(7) 



Bernstein Minimal Surface Theorem 

If a Minimal Surface is given by the equation z = 
f(x, y) and / has CONTINUOUS first and second PARTIAL 

Derivatives for all Real x and y, then / is a Plane. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, p. 369, 1988. 

Bernstein Polynomial 

The Polynomials defined by 



B itn (t)= ('.') **(!-*)* 



References 

Bulanov, A. P. "Asymptotics for the Best Rational Approxi- 
mation of the Function Sign a." Mat. Sbornik 96, 171-178, 
1975. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsof t . com/asolve/constant/brnstn/brnstn.html. 

Gonchar, A. A. "Estimates for the Growth of Rational Func- 
tions and their Applications." Mat. Sbornik 72, 489-503, 
1967. 

Newman, D. J. "Rational Approximation to |x|." Michigan 
Math. J. 11, 11-14, 1964. 

Petrushev, P. P. and Popov, V. A. Rational Approximation of 
Real Functions. New York: Cambridge University Press, 
1987. 

Stahl, H. "Best Uniform Rational Approximation of \x\ on 
[-1,1]." Russian Acad. Sci. Sb. Math. 76, 461-487, 1993. 

Varga, R. S. Scientific Computations on Mathematical Prob- 
lems and Conjectures. Philadelphia, PA: SIAM, 1990. 

Varga, R. S. and Carpenter, A. J. "On a Conjecture of 
S. Bernstein in Approximation Theory." Math. USSR 
Sbornik 57, 547-560, 1987. 

Varga, R. S.; Rut tan, A.; and Carpenter, A. J. "Numerical 
Results on Best Uniform Rational Approximations to |x| 
on [-1,+1]. Math. USSR Sbornik 74, 271-290, 1993. 

Vjacheslavo, N. S. "On the Uniform Approximation of \x\ by 
Rational Functions." Dokl Akad. Nauk SSSR 220, 512- 
515, 1975. 

Bernstein's Inequality 

Let P be a POLYNOMIAL of degree n with derivative P' . 
Then 

HP'lloo <n||P||oo, 



where (™) is a BINOMIAL COEFFICIENT. The Bernstein 
polynomials of degree n form a basis for the POWER 
Polynomials of degree n. 

see also Bezier Curve 

Bernstein's Polynomial Theorem 

If g(9) is a trigonometric POLYNOMIAL of degree m sat- 
isfying the condition \g(0) \ < 1 where 6 is arbitrary and 
real, then g'{9) < m. 

References 

Szego, G. Orthogonal Polynomials, ^.th ed. Providence, RI: 
Amer. Math. Soc, p. 5, 1975. 

Bernstein-Szego Polynomials 

The POLYNOMIALS on the interval [-1,1] associated 
with the Weight Functions 

w{x) — (1 - z 2 ) _1/ 
w(x) = (1 - x 2 ) 1/2 

w(x) - 



1 + x* 



also called BERNSTEIN POLYNOMIALS. 

References 

Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: 
Amer. Math. Soc, pp. 31-33, 1975. 



where 



|F||oo = mK|PW|. 



116 Berry-Osseen Inequality 



Bertrand's Problem 



Berry-Osseen Inequality 

Gives an estimate of the deviation of a DISTRIBUTION 
Function as a Sum of independent Random Vari- 
ables with a Normal Distribution. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, p. 369, 1988. 

Berry Paradox 

There are several versions of the Berry paradox, the 
original version of which was published by Bertrand 
Russell and attributed to Oxford University librarian 
Mr. G. Berry. In one form, the paradox notes that the 
number "one million, one hundred thousand, one hun- 
dred and twenty one" can be named by the description: 
"the first number not nameable in under ten words." 
However, this latter expression has only nine words, so 
the number can be named in under ten words, so there 
is an inconsistency in naming it in this manner! 



References 

Chaitin, G. J. "The Berry Paradox." 
1995. 



Complexity 1, 26-30, 



Bertelsen's Number 

An erroneous value of 7r(10 9 ), where tt(x) is the PRIME 
Counting Function. Bertelsen's value of 50,847,478 
is 56 lower than the correct value of 50,847,534. 

References 

Brown, K. S. "Bertelsen's Number." http://www.seanet . 
com/-ksbrown/kmath049.htm. 

Bertini's Theorem 

The general curve of a system which is LINEARLY IN- 
DEPENDENT on a certain number of given irreducible 
curves will not have a singular point which is not fixed 
for all the curves of the system. 



References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves 
York: Dover, p. 115, 1959. 



New 



Bertrand Curves 

Two curves which, at any point, have a common princi- 
pal Normal Vector are called Bertrand curves. The 
product of the TORSIONS of Bertrand curves is a con- 
stant. 

Bertrand's Paradox 

see Bertrand's Problem 



Bertrand's Postulate 

If n > 3, there is always at least one PRIME between n 
and 2n — 2. Equivalently, if n > 1, then there is always 
at least one PRIME between n and 2n, It was proved 
in 1850-51 by Chebyshev, and is therefore sometimes 
known as Chebyshev's Theorem. An elegant proof 
was later given by Erdos. An extension of this result is 
that if n > k, then there is a number containing a Prime 
divisor > k in the sequence n, n + 1, . . . , n + k — 1. (The 
case n = k + 1 then corresponds to Bertrand's postu- 
late.) This was first proved by Sylvester, independently 
by Schur, and a simple proof was given by Erdos. 

A related problem is to find the least value of 8 so that 
there exists at least one PRIME between n and n + O(n ) 
for sufficiently large n (Berndt 1994). The smallest 
known value is 9 = 6/11 -f e (Lou and Yao 1992). 
see also Choquet Theory, de Polignac's Conjec- 
ture, Prime Number 

References 

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: 
Springer- Verlag, p. 135, 1994. 

Erdos, P. "Ramanujan and I." In Proceedings of the Inter- 
national Ramanujan Centenary Conference held at Anna 
University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New 
York: Springer- Verlag, pp. 1-20, 1989. 

Lou, S. and Yau, Q. "A Chebyshev's Type of Prime Number 
Theorem in a Short Interval (II)." Hardy- Ramanujan J. 
15, 1-33, 1992. 

Bertrand's Problem 

What is the Probability that a Chord drawn at Ran- 
dom on a Circle of Radius r has length > r? The an- 
swer, it turns out, depends on the interpretation of "two 
points drawn at RANDOM." In the usual interpretation 
that Angles #i and 6i are picked at Random on the 
Circumference, 

t, 7r " f 2 
P= *-=-• 

7T 3 

However, if a point is instead placed at RANDOM on a 
Radius of the Circle and a Chord drawn Perpen- 
dicular to it, 

r 2 

The latter interpretation is more satisfactory in the 
sense that the result remains the same for a rotated CIR- 
CLE, a slightly smaller CIRCLE INSCRIBED in the first, 
or for a CIRCLE of the same size but with its center 
slightly offset. Jaynes (1983) shows that the interpre- 
tation of "Random" as a continuous Uniform Distri- 
bution over the RADIUS is the only one possessing all 
these three invariances. 

References 

Bogomolny, A. "Bertrand's Paradox." http: //www. cut-the- 
knot . com/bertrand.html. 

Jaynes, E. T. Papers on Probability, Statistics, and Statisti- 
cal Physics. Dordrecht, Netherlands: Reidel, 1983. 

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42- 
45, 1995. 



Bertrand's Test 

Bertrand's Test 

A Convergence Test also called de Morgan's and 
Bertrand's Test. If the ratio of terms of a Series 
{flnj^Li can be written in the form 



an 



1 



= 1 + - + 



Pn 



n n In n ' 



then the series converges if lim n ->oo pn > 1 and diverges 
if lim n _^oo/0n < 1, where lim w ->oo is the Lower Limit 
and lim n _>.oo is the Upper Limit. 
see also Rummer's Test 

References 

Bromwich, T. J. Pa and MacRobert, T. M. An Introduc- 
tion to the Theory of Infinite Series, 3rd ed. New York: 
Chelsea, p. 40, 1991. 

Bertrand's Theorem 

see Bertrand's Postulate 



Bessel Differential Equation 
Bessel Differential Equation 

m?)y = 0. 
Equivalently, dividing through by z 2 , 



2d 2 y dy 2 „^ 



117 



(i) 



The solutions to this equation define the BESSEL FUNC- 
TIONS. The equation has a regular SINGULARITY at 
and an irregular SINGULARITY at oo. 

A transformed version of the Bessel differential equation 
given by Bowman (1958) is 



* 2 § + (2p+l)sg + (aV r + /? 2 )y = 0. (3) 



The solution is 



Besov Space 

A type of abstract Space which occurs in Spline and 
Rational Function approximations. The Besov space 
Bp yQ is a complete quasinormed space which is a Ba- 
NACH Space when 1 < p, q < oo (Petrushev and Popov 
1987). 

References 

Bergh, J. and Lofstrom, J. Interpolation Spaces. New York: 
Springer- Verlag, 1976. 

Peetre, J. New Thoughts on Besov Spaces. Durham, NC: 
Duke University Press, 1976. 

Petrushev, P. P. and Popov, V. A. "Besov Spaces." §7.2 
in Rational Approximation of Real Functions. New York: 
Cambridge University Press, pp. 201-203, 1987. 

Triebel, H. Interpolation Theory, Function Spaces, Differen- 
tial Operators. New York: Elsevier, 1978. 

Bessel's Correction 

The factor (N — 1)/N in the relationship between the 
Variance a and the Expectation Values of the Sam- 
ple Variance, 



y = x p 



I 2\ N-l 2 



s 2 = (x 1 ) - (x) 2 . 



N lSl 2 +N 2 s 2 2 
Ni+N 2 -2 ' 



see also Sample Variance, Variance 



where 



For two samples, 



(i) 

(2) 
(3) 



c 1 J q/r (^-)+c 2 r g/r (^) 



where 



q = vV - P\ 



(4) 



(5) 



J and Y are the Bessel Functions of the First and 
SECOND KINDS, and C\ and Ci are constants. Another 
form is given by letting y = x a J n (/3x' y ) i tj — yx~ a , and 
£ = 0x 7 (Bowman 1958, p. 117), then 

(6) 



The solution is 



= f x a [AJ n {(3x' r ) + BYniPx 1 )] for integral n 
V \ AJniffx 7 ) + BJ-niPx 1 )] for nonintegral u. 

(?) 
see also AlRY FUNCTIONS, ANGER FUNCTION, Bei, 

Ber, Bessel Function, Bourget's Hypothesis, 
Catalan Integrals, Cylindrical Function, Dini 
Expansion, Hankel Function, Hankel's Integral, 
Hemispherical Function, Kapteyn Series, Lip- 
schitz's Integral, Lommel Differential Equa- 
tion, Lommel Function, Lommel's Integrals, 
Neumann Series (Bessel Function), Parseval's 
Integral, Poisson Integral, Ramanujan's Inte- 
gral, Riccati Differential Equation, Sonine's 
Integral, Struve Function, Weber Functions, 
Weber's Discontinuous Integrals 

References 

Bowman, F. Introduction to Bessel Functions. New York: 
Dover, 1958. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, p. 550, 1953. 



118 BesseVs Finite Difference Formula 



Bessel Function of the First Kind 



Bessel's Finite Difference Formula 

An Interpolation formula also sometimes known as 



The Bessel functions are more frequently defined as so- 
lutions to the Differential Equation 



f P = fo+ pSi/2 + B 2 {Sl + <$i ) + B 3 8l /2 

+ B 4 05$ + tf) + B 5 *? /a + ... ) (1) 



for p e [0, 1], where 6 is the Central Difference and 

Bin = ^Gln = g ("^2n + i*2n) (2) 

B2n + 1 = G2n + 1 ~ 2^ 2n ~ 2 (^ 2ri ~ ^2n) (**) 

£?2n = ^2n — G 2n +1 = Bin — #2n + l (4) 

F 2 n = t?2n+l = B 2n + #2n+l> (5) 

where Gk are the COEFFICIENTS from GAUSS'S BACK- 
WARD Formula and Gauss's Forward Formula and 
E k and Fk are the Coefficients from Everett's FOR- 
MULA. The i?fcS also satisfy 



B 2n {p) = B 2n (q) 
B 2n+X {p) = -B 2n +i(q), 



for 



(6) 
(7) 

(8) 



q = l-p. 

see also Everett's Formula 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

p. 880, 1972. 
Acton, F. S. Numerical Methods That Work, 2nd printing, 

Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990. 
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 

Boca Raton, FL: CRC Press, p. 433, 1987. 

Bessel's First Integral 

i r 

J n (x) = — / cos(n# - xsinO) d8, 
77 Jo 

where J n (x) is a BESSEL FUNCTION OF THE FIRST 

Kind. 

Bessel's Formula 

see Bessel's Finite Difference Formula, Bes- 
sel's Interpolation Formula, Bessel's Statisti- 
cal Formula 

Bessel Function 

A function Z(x) defined by the RECURRENCE RELA- 
TIONS 

Zm + l + Z m — 1 — Zm 



and 



&m+l — ^m-1 



Zm~l — —2 



dx 



2d 2 y dy 2 

X dx^ +X dx- + {x 



m )y — 0. 



There are two classes of solution, called the BESSEL 
Function of the First Kind J and Bessel Func- 
tion of the Second Kind Y. (A Bessel Function 
OF THE THIRD Kind is a special combination of the first 
and second kinds.) Several related functions are also de- 
fined by slightly modifying the defining equations. 

see also Bessel Function of the First Kind, 
Bessel Function of the Second Kind, Bessel 
Function of the Third Kind, Cylinder Func- 
tion, Hemicylindrical Function, Modified Bes- 
sel Function of the First Kind, Modified Bessel 
Function of the Second Kind, Spherical Bessel 
Function of the First Kind, Spherical Bessel 
Function of the Second Kind 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions 
of Integer Order," "Bessel Functions of Fractional Order," 
and "Integrals of Bessel Functions." Chs. 9-11 in Hand- 
book of Mathematical Functions with Formulas, Graphs, 
and Mathematical Tables, 9th printing. New York: Dover, 
pp. 355-389, 435-456, and 480-491, 1972. 

Arfken, G. "Bessel Functions." Ch. 11 in Mathematical 
Methods for Physicists, 3rd ed. Orlando, FL: Academic 
Press, pp. 573-636, 1985. 

Bickley, W. G. Bessel Functions and Formulae. Cambridge, 
England: Cambridge University Press, 1957. 

Bowman, F. Introduction to Bessel Functions. New York: 
Dover, 1958. 

Gray, A. and Matthews, G. B. A Treatise on Bessel Func- 
tions and Their Applications to Physics, 2nd ed. New 
York: Dover, 1966. 

Luke, Y. L. Integrals of Bessel Functions. New York: 
McGraw-Hill, 1962. 

McLachlan, N. W. Bessel Functions for Engineers, 2nd ed. 
with corrections. Oxford, England: Clarendon Press, 1961. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Bessel Functions of Integral Order" and 
"Bessel Functions of Fractional Order, Airy Functions, 
Spherical Bessel Functions." §6.5 and 6.7 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 223-229 and 234-245, 1992. 

Watson, G. N. A Treatise on the Theory of Bessel Functions, 
2nd ed. Cambridge, England: Cambridge University Press, 
1966. 

Bessel Function of the First Kind 




-0.2 



Bessel Function of the First Kind 



Bessel Function of the First Kind 119 



The Bessel functions of the first kind J n {x) are defined as 
the solutions to the BESSEL DIFFERENTIAL EQUATION 

which are nonsingular at the origin. They are some- 
times also called Cylinder Functions or Cylindri- 
cal Harmonics. The above plot shows J n (x) for n = 1, 

2,..., 5. 

To solve the differential equation, apply FROBENIUS 
METHOD using a series solution of the form 



First, look at the special case m = —1/2, then (9) be- 
comes 

oo 

^[a n n(n-l) + a n _ 2 ]x m +" = 0, (10) 



n(n — 1) 
Now let n = 2/, where / = 1, 2, 



fln-2. 



(11) 



&21 



1 



2/(2/ - 1) 



0*21-2 



(-i)' 



y = x k ^ a n x n = JT a n x n+k . 



(2) 



n=0 n=0 

Plugging into (1) yields 

oo 

x 2 ^{k + n)(k + n~ l)a n x k+n - 2 

n—0 

oo oo 

+X Y^(k + n)OnX h+n - 1 +X 2 J2 a nX k+n 
n~ n — 

oo 

-m 2 ^2a n x n + k = (3) 



[2/(2/ - 1)][2(Z - 1)(2Z - 3)] ■ * - [2 - 1 • 1] 

7«0, 



do 



2 l l\(2l~l)\\ 
which, using the identity 2 l l\(2l - 1)!! = (2/)!, gives 

a 2/ = /rtlXI Q0> 



(12) 



(20! °' 



(13) 



Similarly, letting n = 21 + 1 



fl2i+i = — 



1 



(2/ + l)(2/) 



«2i-l 



(-1)' 



[2/(2/ + 1)][2(I - 1)(2/ - 1)] ... [2 • 1 • 3][1] au 

(14) 



^(fc + n)(fc + n - l)a n x fc+n + ^(fc + n)a n x fe+n which, using the identity 2 l l\(2l + 1)!! = (2/ + 1)!, gives 



]Ta n - 2 x k+n -m 2 J2a n x n+k = 0. (4) 



n=2 n=0 



The INDICIAL EQUATION, obtained by setting n = 0, is 

a [fc(£; - 1) + k - m 2 ] = a (k 2 - m 2 ) = 0. (5) 

Since ao is defined as the first NONZERO term, k 2 —m 2 = 
0, so k = ±ra. Now, if k ~ m, 

oo 

^[(m + n)(m + n - 1) + (m + n) - m 2 }a n x 7n+n 

n~0 

oo 

+ Y. an ~ 2^ m+n = (6) 

n^2 

oo oo 

£][(m + n) 2 - m 2 ]a n x m+n + ^ a„_ 2 x m+n = (7) 

n=0 n = 2 

oo oo 

^ n(2m + n)a n i m+ " + J] a„. 2 2 ra+n = (8) 

n — n=2 

OO 

ai(2m + 1) 4- ^[a n n(2m + n) + a„_ 2 ]a; m+n = 0. (9) 



(-1) 1 (-l) z 

a2/+1 " 2^/!(2/ + l)!! ai = (27TI)! ai ' (15) 

Plugging back into (2) with k = m = —1/2 gives 



2/ = x 1/2 N a n a; n 

t=0 

oo oo 

2. a n x n + N^ a n x n 

_n~l,3,5,... n-0,2,4 

oo oo 

E2J . V~^ 
CL21X + J> ^21 + lX 



-1/2 



-1/2 



n = 0,2,4,... 
2J + 1 



-1/2 



J = 1=0 



(-1) 



2^ ( 2 n! X +ai 2^(2Z + l)! 



;=o ■ ' z=o 



(20! 

-1/2/ , • \ 

= x ' (ao cos a; + a\ since). 



(2/ + 1) 



(16) 



The BESSEL FUNCTIONS of order ±1/2 are therefore de- 
fined as 



(17) 
(18) 



J-i/ 2 (x) =4/ — cosa; 

' U 7TX 



Ji/ 2 (x) = 4/ — sinx, 

17 7TZ 



120 Bessel Function of the First Kind 

so the general solution for m = ±1/2 is 

y = a' J- 1/2 {x) + a 1 J 1/2 (x). (19) 

Now, consider a general m ^ —1/2. Equation (9) re- 
quires 

ai(2m+l)=0 (20) 

[a n n(2m + n) + a n _ 2 ]z m+n = (21) 



for n = 2, 3, . . . , so 

ai =0 



n(2m + n 



■fln-2 



(22) 
(23) 



for n = 2, 3, Let n = 2Z + 1, where Z = 1, 2, . . . , 

then 



«2/ + l 



1 



Tfl2Z-l 



(2Z + l)[2(m + l) + l] 
= ... = /(n,m)ai =0, (24) 



where f(n,m) is the function of Z and m obtained by 
iterating the recursion relationship down to a\ . Now let 
n = 2Z, where Z = 1, 2, . . . , so 

1 1 
a 2* = ~~ 77777; r~^ a 2/-2 = —777 — 77^-2 



2l{2m + 2l) " 4Z(m + Z) 

tn 

[4Z(m + Z)][4(Z - l)(m + Z - 1)] • • ■ [4 • (m + 1)] 



ao- 



(25) 



Plugging back into (9), 

a n x = > a n x + y a n x 

n = n = l,3,5,... n = 0,2,4,,,, 

E2I + m + l . \~^ 2I + m 

G 2 i-M^ + > a 2lX 

1=0 (=0 

„ v^ (_z}Y « + ™ 

= tin 7 X 

Z^ [4i(m + l)][4{l - l)(m + I - 1)] • • . [4 • (m + 1)] 
1=0 

[(-l) f m(m- l)---l]x 2t+m 

[4/(m + i)][4(i - l)(m + i - 1)] • ■ ■ [m(m - 1) ■ • • 1] 



Bessel Function of the First Kind 

Returning to equation (5) and examining the case k — 

— m, 

00 
ai(l-2m) + ^[a n ra(ra-2m) + a n _ 2 ]a; n ~ m = 0. (29) 



However, the sign of m is arbitrary, so the solutions must 
be the same for +ra and — m. We are therefore free to 

replace — m with — |m|, so 

00 
oi(l + 2|m|) + ^[o n n(n + 2|m|) + a n _ 2 ]x |m|+n = 0, 

n = 2 

. (30) 

and we obtain the same solutions as before, but with m 
replaced by \m\. 



*J<m\X) '■ 



v^oo (-1)' 2Z+|m| f or | rn |^_I 



for m = — | 
for m = |. 



(31) 

We can relate J m and J_ m (when m is an Integer) by 
writing 



( — lV 

1=0 v ' 



(32) 



Now let 1 = 1' + m. Then 



J-m(x) = ^ 



(-1) 



Z' + m 



Z' + m=0 

-1 



2 2 <'+™(Z' + m)!Z! 
(-1)''+™ 



2l'+m 



V I- 1 ) 2Z'+m 

Z^ 2 2 <'+™Z'!(Z'+m)! 

l' = — m 



+ 2-1 2«'+"Z'!(Z'+m)! a;2 ' +m ' (33) 
i'=o v 

But Z'! = oo for Z' = -m, ...,-1, so the Denomina- 
tor is infinite and the terms on the right are zero. We 
therefore have 



--Ej^--£?&- w '-<*>- Es.J^fe'" 4 "-'- 1 '"^ 



\(m + l)\ ~ u ^ 2 2 <Z!(m + Z) 

Z=0 ' 1=0 / 



Now define 



OO ; 

Jm(x) = Jj 2 2 <+™Z!(m + Z)! x2 ' +m ' (27) 



where the factorials can be generalized to Gamma 
FUNCTIONS for nonintegral m. The above equation then 
becomes 



(34) 

Note that the Bessel Differential Equation is 
second-order, so there must be two linearly independent 
solutions. We have found both only for \m\ = 1/2. For 
a general nonintegral order, the independent solutions 
are J m and J~ m . When m is an INTEGER, the general 
(real) solution is of the form 



Z m = C 1 J m (x) + C 2 Y rn (x), 



(35) 



y = a 2 m m\J m (x) — a' J m (x). 



(28) 



Bessel Function of the First Kind 



Bessel Function of the First Kind 121 



where J m is a Bessel function of the first kind, F m 
(a.k.a. iV m ) is the BESSEL FUNCTION OF THE SECOND 
Kind (a.k.a. Neumann Function or Weber Func- 
tion), and C\ and C 2 are constants. Complex solutions 
are given by the Hankel Functions (a.k.a. Bessel 
Functions of the Third Kind). 

The Bessel functions are ORTHOGONAL in [0, 1] with re- 
spect to the weight factor x. Except when 2n is a NEG- 
ATIVE Integer, 



Jrn(z) 



-1/2 



2 2m+l/2 i m + l/2 r ( m+1 ^ 



Mo im (2iz) ) (36) 



where T(x) is the Gamma Function and M , m is a 
Whittaker Function. 

In terms of a Confluent Hypergeometric Func- 
tion of the First Kind, the Bessel function is written 



Mz) 



^fryo^^ + i;-^ 2 )- (37) 



A derivative identity for expressing higher order Bessel 
functions in terms of Jo(x) is 



Jn(x) — i n T n li-j-) Jo( 



(38) 



where T n (x) is a Chebyshev Polynomial of the 
First Kind. Asymptotic forms for the Bessel functions 
are 



J - {x) * fd+T) (!) 



for x <^ 1 and 

J m (x) : 



/ ran tt\ 

x 

V 2 4/ 



for x ^> 1. A derivative identity is 
d 



dx 



[x^Jmix)] = X^Jm-lix). 



An integral identity is 

uJo(u)du —uJ\{u). 



F 

Jo 



Some sum identities are 

1 = [Jo(x)] 2 + 2[J 1 {x)f + 2[J 2 (x)] 2 + , 

1 = J (x) + 2J 2 {x) + 2J A {x) + . . 
and the Jacobi-Anger Expansion 



% J n (z)e 



(39) 
(40) 

(41) 
(42) 

(43) 
(44) 

(45) 



which can also be written 

00 
e tzcose = J (z) + 2^2i n J n (z)cos(n8). (46) 

n=l 

The Bessel function addition theorem states 

00 

My + z) = ^ J™{y) J n-m{z). (47) 



m=-oo 



ROOTS of the FUNCTION J n (x) are given in the following 
table. 



zero 


J Q (x) 


Ji(x) 


J 2 {x) 


Mx) 


Mx) 


J*(x) 


1 


2.4048 


3.8317 


5.1336 


6.3802 


7.5883 


8.7715 


2 


5.5201 


7.0156 


8.4172 


9.7610 


11.0647 


12.3386 


3 


8.6537 


10.1735 


11.6198 


13.0152 


14.3725 


15.7002 


4 


11.7915 


13.3237 


14.7960 


16.2235 


17.6160 


18.9801 


5 


14.9309 


16.4706 


17.9598 


19.4094 


20,8269 


22.2178 



Let x n be the nth ROOT of the Bessel function Jo(#), 
then 



Y — 



71 = 1 

(Le Lionnais 1983). 



2"n*J§y£n} 



= 0.38479... 



(48) 



The Roots of its Derivatives are given in the following 
table. 



zero 


Jo'(x) 


•V(z) 


•V(s) 


J 3 '(x) 


J 4 '(x) 


J 5 '(x) 


1 


3.8317 


1.8412 


3.0542 


4.2012 


5,3175 


6.4156 


2 


7.0156 


5.3314 


6.7061 


8.0152 


9.2824 


10.5199 


3 


10.1735 


8.5363 


9.9695 


11.3459 


12.6819 


13.9872 


4 


13.3237 


11.7060 


13.1704 


14.5858 


15.9641 


17.3128 


5 


16.4706 


14.8636 


16.3475 


17.7887 


19.1960 


20.5755 



Various integrals can be expressed in terms of Bessel 
functions 



1 f 2 " 

* w - s y «•• 



' cos <j> d(f> 

i 

i r 

J n (z) = — / cos(z sin — n6) d8 , 
n Jo 

which is BESSEL'S FIRST INTEGRAL, 



(49) 
(50) 



.-71 f* 

./«(*) = —/ e izcose cos(n9)d0 (51) 

w Jo 



Jn{z) 



JL_ [ 2 \i 



V"* d<t> 



z cos <p in.1 



(52) 



J, . . . , 



for n = 1, 2 

2 x 



J»W 



7r (2m 
for n = I, 2, . . . , 






sin n u cos(x cos u) du (53) 



71— — OO 



T f~\ 1 I (x/2)(z-l/z) -71-1 , 

Jtl(x) = - — ; / e K ' A ' } z dz 

2tvi J 



(54) 



122 Bessel Function Fourier Expansion 



Bessel Function of the Second Kind 



for n > —1/2. Integrals involving J\(x) include 



(Bowman 1958, p. 108), so 



/ J\ (x) dx = 1 
Jo 


(55) 


r[¥ 


dx = h 


(56) 


cm 


xdx = — . 
2 


(57) 



see also BESSEL FUNCTION OF THE SECOND KIND, DE- 

bye's Asymptotic Representation, Dixon-Ferrar 
Formula, Hansen-Bessel Formula, Kapteyn Se- 
ries, Kneser-Sommerfeld Formula, Mehler's 
Bessel Function Formula, Nicholson's Formula, 
Poisson's Bessel Function Formula, Schlafli's 
Formula, Schlomilch's Series, Sommerfeld's 
Formula, Sonine-Schafheitlin Formula, Wat- 
son's Formula, Watson-Nicholson Formula, We- 
ber's Discontinuous Integrals, Weber's For- 
mula, Weber-Sonine Formula, Weyrich's For- 
mula 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func- 
tions J and V." §9.1 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 358-364, 1972. 

Arfken, G. "Bessel Functions of the First Kind, J„(;r)" and 
"Orthogonality." §11.1 and 11,2 in Mathematical Meth- 
ods for Physicists, 3rd ed. Orlando, FL: Academic Press, 
pp. 573-591 and 591-596, 1985. 

Lehmer, D. H. "Arithmetical Periodicities of Bessel Func- 
tions." Ann. Math. 33, 143-150, 1932. 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 25, 1983. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 619-622, 1953. 

Spanier, J. and Oldham, K. B. "The Bessel Coefficients Jq(x) 
and Ji(x)" and "The Bessel Function J u (x)." Chs. 52-53 
in An Atlas of Functions. Washington, DC: Hemisphere, 
pp. 509-520 and 521-532, 1987. 

Bessel Function Fourier Expansion 

Let n > 1/2 and e*i, a 2 , ...be the POSITIVE ROOTS 
of J n (x) — 0. An expansion of a function in the inter- 
val (0,1) in terms of BESSEL FUNCTIONS OF THE FIRST 

Kind 



f( x ) = /]A r J n (xa r ), 



(i) 



has Coefficients found as follows: 

/ xf(x)J n (xai)dx = y^A r / xJ n (xa r )Jn(xai)dx. 
Jo r=1 Jo 

(2) 
But Orthogonality of Bessel Function Roots 
gives 



/' 

Jo 



xJ n (xai)J n (xa r )dx = ^Sl^Jn + 1 (&r) (3) 



ol °° 

/ xf(x)J n (xai)dx - \ }^ Ar5i, r J n +i 2 (xa r ) 

J° r=l 



I A. T . . 2 (^,.\ U) 



= ^AiJ n+1 (on), 



and the COEFFICIENTS are given by 
2 



A t = 



J n+ i 2 (ai) 



f 

Jo 



xf(x)Jn(xai)dx. (5) 



References 

Bowman, F. Introduction to Bessel Functions. New York: 
Dover, 1958. 



Bessel Function of the Second Kind 




A Bessel function of the second kind Y n (x) is a solution 
to the Bessel Differential Equation which is sin- 
gular at the origin. Bessel functions of the second kind 
are also called Neumann Functions or Weber Func- 
tions. The above plot shows Y n {x) for n = 1, 2, . . . , 
5. 

Let v = Jm{x) be the first solution and u be the 
Other one (since the BESSEL DIFFERENTIAL EQUATION 

is second-order, there are two Linearly Independent 
solutions). Then 



xu + u + xu = 

XV + V + XV = 0. 

Take v x (1) - u x (2), 

x{u v — uv ) -\- u v — uv =0 

— \x(uv — uv')] = 0, 
ax 



(i) 

(2) 

(3) 
(4) 



so x(uv — uv) = B, where B is a constant. Divide by 
xv 2 , 

uv — uv _ d /u\ _ B ( . 

v 2 dx \v ) xv 2 



V 



f- 

J & 



,2* 



(6) 



Bessel Function of the Third Kind 

Rearranging and using v = J m (x) gives 



u = AJm(x) + BJ m (x) 



I. 



dx 



XJrn \X~) 

= A , J m (x)-{-B'Y rn (x), (7) 

where the Bessel function of the second kind is denned 

by 



Y m (x) 



J m (x) cos(mir) — J_ m (x) 
sin(m7r) 



* Z. 2«+»*!(m + *)! [ 2 ln 1 2 j + 27 " bm+k ~ bk 



1 v^ x~ m+2k (m -k-l)\ 



--J2 



2-m4-2fcfc| 



(8) 



m = 0, 1, 2, . . . , 7 is the Euler-Mascheroni Con- 
stant, and 



Jo k = 0, 



(9) 



The function is given by 

Y n (z) = - / sin(z sin d-n0)d0 
* Jo 



I!?-" 



— nt ( ,\ni —z sinh t 



+ e~ nt (-l) n ]e 



dt, (10) 



Asymptotic equations are 



m() ~l-^(f) m m^0,x«l (U) 

rmW = V^ sm r T"4J * >>x ' (12) 

where r(z) is a Gamma Function. 

see also Bessel Function of the First Kind, Bour- 

get's Hypothesis, Hankel Function 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func- 
tions J and Y. n §9.1 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 358-364, 1972. 

Arfken, G. "Neumann Functions, Bessel Functions of the Sec- 
ond Kind, N v (x). n §11.3 in Mathematical Methods for 
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596- 
604, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 625-627, 1953. 

Spanier, J. and Oldham, K. B. "The Neumann Function 
Y u (x)" Ch. 54 in An Atlas of Functions. Washington, 
DC: Hemisphere, pp. 533-542, 1987. 

Bessel Function of the Third Kind 

see Hankel Function 



Bessel's Inequality 123 

Bessel's Inequality 

If f(x) is piecewise CONTINUOUS and has a general 
Fourier Series 

5^Mi(x) (1) 

i 

with Weighting Function w(x), it must be true that 



/ 



/O) - y^ai<fc(aQ 



w(x)dx > (2) 



+ 5^a< 2 <t>i 2 (x)w{x)dx>0. (3) 
i J 

But the Coefficient of the generalized Fourier Se- 
ries is given by 

a m = / f(x)<f> m (x)w(x)dx, (4) 

so 

/ f 2 (x)w(x)dx-2^2ai 2 -h^di 2 > (5) 

i i 

f{x)w(x)dx>Y^ai 2 - (6) 

i 

Equation (6) is an inequality if the functions <j>i are not 
Complete. If they are Complete, then the inequality 
(2) becomes an equality, so (6) becomes an equality and 
is known as PARSEVAL's THEOREM. If f(x) has a simple 
Fourier Series expansion with Coefficients a , ai, 
. . . , a n and &i, . . . , b ni then 

ia 2 + ^(a fc 2 +6 fc 2 )<- / [f(x)] 2 dx. (7) 

fc = l n J—* 

The inequality can also be derived from SCHWARZ'S IN- 
EQUALITY 

I (f\g) I 2 < {/I/} (g\g) (8) 

by expanding g in a superposition of ElGENFUNCTlONS 

0f/,S= Yji a ifc- Then 



(/|5) = X)°* </!/*> ^Z) fli - 



(9) 



(f\g) r < 



Y< ai 



= 5> 4 a«' < </|/> <s| S ) . (10) 



124 BesseVs Interpolation Formula 

If g is normalized, then (g\g) = 1 and 

</!/>> 5> t a t *. (11) 



see also Schwarz's Inequality, Triangle Inequal- 
ity 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 526-527, 1985. 

Gradshteyn, I. S. and Ryzhik, L M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1102, 1980. 

BessePs Interpolation Formula 

see Bessel's Finite Difference Formula 

Bessel Polynomial 

see Bessel Function 

Bessel's Second Integral 

see POISSON INTEGRAL 



Bessel's Statistical Formula 

W — UJ W — U) 



•'** t lzL 



(i) 



(wi-w) 2 



N(N-l) 



where 



w = X\ — X2 

u = M(i) - M(2) 

N = N 1 + N 2 . 



(2) 
(3) 
(4) 



Beta 

A financial measure of a fund's sensitivity to market 
movements which measures the relationship between a 
fund's excess return over Treasury Bills and the excess 
return of a benchmark index (which, by definition, has 
j3 = 1). A fund with a beta of (3 has performed r — 
(j3 - 1) x 100% better (or \r\ worse if r < 0) than its 
benchmark index (after deducting the T-bill rate) in up 
markets and \r\ worse (or \r\ better if r < 0) in down 
markets. 

see also Alpha, Sharpe Ratio 
Beta Distribution 



/^ "\ (a.6) = (l. 1) 


Q 


/ //ia.G) = {l, 1) 


<2.3)-\ 
/(J. 2} ^ 



Beta Distribution 

A general type of statistical DISTRIBUTION which is re- 
lated to the Gamma Distribution. Beta distributions 
have two free parameters, which are labeled according 
to one of two notational conventions. The usual defini- 
tion calls these a and /?, and the other uses /?' = j3 — 1 
and d = a - 1 (Beyer 1987, p. 534). The above plots 
are for (a,/3) = (1,1) [solid], (1, 2) [dotted], and (2, 3) 
[dashed]. The probability function P(x) and DISTRIBU- 
TION Function D(x) are given by 



P{x) 



0-l„a-l 



(l-xf- l x' 
B(a,0) 



T(a)T((3) 
D{x) = I(x; a, 6), 



(l-xf^x 



/3-1 a-1 



(1) 

(2) 



where B(a,b) is the BETA FUNCTION, J(x;a,6) is the 
Regularized Beta Function, and < x < 1 where 
a, f3 > 0. The distribution is normalized since 



Jo 



P(x) dx : 



r(a)r(/3) 

r(a + /3) 



Jo 



(l-xf^dx 
B(a,0) = l. (3) 



T(a)T(/3) 
The Characteristic Function is 

</>(*) = ^faa + bjit) 
The Moments are given by 

: + /3)r(a + r 



P 1 T(rv 

M r = (a- fi) r dx= ~^— 

Jo r ( a 



+ /3 + r)r(a) 



(4) 



(5) 



The Mean is 



M -r(a)r ( ^y (1 x) 



T(a + P) 



B(a + l,f3) 



r(a + y3)r(a + l)r(/3) _ a 



r(a)r(/?)r(a + /? + i) a + /?' 

and the Variance, SKEWNESS, and KURTOSIS are 

2__ a/3 

a {a + f3) 2 {a + (3 + l) 

_ 2(yff- y^)( % /S+V^)Vl + q + /? 

71 ~ V^p(a + /3 + 2) 

_ 6(a 2 + a 3 - Aaj3 - 2a 2 (3 + (3 2 - 2af3 2 + /3 3 ) 

72 ~ a/3(a + /3 + 2)(a + /3 + 3) 



(6) 

(7) 
(8) 

(9) 



The Mode of a variate distributed as /3(a,/3) is 
- - Q ~ 1 



(10) 



Beta Function 

In "normal" form, the distribution is written 

and the MEAN, VARIANCE, SKEWNESS, and KURTOSIS 
are 



A* = 



a + /3 



2 OL0 

a = 



7i = 



72 



(a + /?) 2 (l + a + /?) 
_ 2(Vo:-^)( v /a + v / g)Vl + tt + /3 
V^(a + /? + 2) 
3(1 + a + /?)(2a 2 - 2a/? + a 2 /3 + 2/3 2 + a/3 2 ) 



(12) 
(13) 

(14) 



a/?(a + /3 + 2)(a + /? + 3) 



(15) 



see a/so GAMMA DISTRIBUTION 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

pp. 944-945, 1972. 
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 

Boca Raton, FL: CRC Press, pp. 534-535, 1987. 

Beta Function 

The beta function is the name used by Legendre and 
Whittaker and Watson (1990) for the Eulerian Inte- 
gral of the Second Kind. To derive the integral 
representation of the beta function, write the product 
of two Factorials as 



POO no 

i\n\= I e- u u m du I 
Jo Jo 



Now, let u = a? 2 , v = y 2 , so 



e v v n dv. (1) 



dy 



/oo />oo 

e- x2 x 2rn+1 dx e~ y2 y 2n+ \ 

/oo poo 
/ e-^ + ^x 2m+1 y 2m+1 dxdy. (2) 
-oo «/ — oo 

Transforming to POLAR COORDINATES with x ~ rcos9, 
y — r sin 6 



pTT/2 pO 

n! = 4/ / e~ r '(r cos dY m+1 (r sin 6) 2n+1 rdrc 

Jo Jo 

poo /' 7r /2 

A I -r 2 2m+2n+3 i / 2ro+l n ■ 2n+l n jq 

■ 4 / e r dr cos v sin ^ v dv 

Jo Jo 

tt/2 



2(m + 



n + 1)! / 
Jo 



cos" m+1 sin n+1 (9 d0. (3) 



Beta Function 125 

The beta function is then defined by 



B(m + l,n + l) = 5(71 + 1,771+ 1) 

/.tt/2 



= 2 / cos 2 ™ +1 flsin 2 " +1 ^= / m]n ' „ . 
Jo (m + n + 1)! 



(4) 



Rewriting the arguments, 



B( P a) - r &™ - (P-D'(g-l)! (5) 

The general trigonometric form is 

o 



/•tt/2 

/ sin n a;cos m ;rdx = \B(n+ |,m+ |). (6) 



Equation (6) can be transformed to an integral over 
Polynomials by letting u = cos 2 0, 



— = u (1-u) du. 
' n ) Jo 



B(m,n) 



T(m)T(r, 
r(m + i 



du 
(7) 
(8) 



To put it in a form which can be used to derive the 
Legendre Duplication Formula, let x = y/u, so 
u = x and du — 2x dx, and 

B(m y n)= / x 2irn ~ 1) (l-x 2 ) n - 1 {2xdx) 
Jo 

-'f- 

Jo 



2m — 1/-. 2\n~l 



(l-x^^dx. (9) 



To put it in a form which can be used to develop integral 
representations of the Bessel Functions and Hyper- 
geometric Function, let u = x/(l + x), so 



£(m + l,n + l)= H , ""> 



(10) 



Various identities can be derived using the GAUSS MUL- 
TIPLICATION Formula 



B(np, nq) 



T(np)T(nq) 



T[n(p + q)} 
_ - nq B(p,q)B(p+ l, t )- B(p+ 2=1, q) 

B(q,q)B(2q,q)---B([n-l]q,q) ' ( > 

Additional identities include 

B(va ^) = r(p)F(9 + 1) = g T(p + l)r(q) 
(P ' q+ ' T(p + q + l) p r(\p+l]q) 

= |s(p+l, ff ) (12) 



B(p,q) = B(p+l,q) + B(p,q+l) (13) 



126 Beta Function (Exponential) 



B{p,q+1) 



P + Q 
If n is a Positive Integer, then 



B(p,q). 



(14) 



1 * 2 • • • 71 ,„ ^ x 

B(p, n + 1 = , . x (15) 

p(p + 1) • • • (p + n) 



S(P,p)5(P+iP+5) = 



(16) 



2'^ ' 2> 2 4 P" X p 

5(p + <?)#(p + 9, r) = £(<?, r)B(q + r,p). (17) 

A generalization of the beta function is the incomplete 
beta function 



B(t;x,y)= r«"- 1 (l-u)*- 1 
Jo 



+P |1 , i-w. 



\ X X + 1 



(l-y)---(n-y), w 



n!(rr + n) 



r + . . • 



(18) 



see aZso Central Beta Function, Dirichlet In- 
tegrals, Gamma Function, Regularized Beta 

Function 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Beta Function" 
and "Incomplete Beta Function." §6.2 and 6.6 in Hand- 
book of Mathematical Functions with Formulas, Graphs, 
and Mathematical Tables, 9th printing. New York: Dover, 
pp. 258 and 263, 1972. 

Arfken, G. "The Beta Function." §10.4 in Mathematical 
Methods for Physicists, 3rd ed. Orlando, FL: Academic 
Press, pp. 560-565, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part L New York: McGraw-Hill, p. 425, 1953. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Gamma Function, Beta Function, Facto- 
rials, Binomial Coefficients" and "Incomplete Beta Func- 
tion, Student's Distribution, F-Distribution, Cumulative 
Binomial Distribution." §6.1 and 6.2 in Numerical Recipes 
in FORTRAN: The Art of Scientific Computing, 2nd 
ed. Cambridge, England: Cambridge University Press, 
pp. 206-209 and 219-223, 1992. 

Spanier, J. and Oldham, K. B. "The Incomplete Beta Func- 
tion B(v\mx)" Ch, 58 in An Atlas of Functions. Wash- 
ington, DC: Hemisphere, pp. 573-580, 1987. 

Whittaker, E. T. and Watson, G. N. A Course of Modern 
Analysis, 4th ed. Cambridge, England: Cambridge Uni- 
versity Press, 1990. 

Beta Function (Exponential) 




Betti Group 

Another "Beta Function" defined in terms of an in- 
tegral is the "exponential" beta function, given by 



/?»(*) 



5 />-" 



dt 



i!*-< n+1 > 



'£ 



(-i)* 



L fc=o 



fc! 



2-r fc! 



fc=0 



(1) 



(2) 



The exponential beta function satisfies the Recur- 
rence Relation 



z(3 n (z) = (-l) n e z - e- z +n(3 n ^(z). 

The first few integral values are 
2 sinh z 






2 (sinh z — z cosh z) 



_ , , 2(2 + z 2 ) sinh z - 4z cosh z 
02(a) = ^ . 



see also ALPHA FUNCTION 

Beta Prime Distribution 

A distribution with probability function 



(3) 

(4) 
(5) 
(6) 



P{x) = 



x a - 1 (l + xy 
B(a,l3) 



-OL-P 



where B is a Beta Function. The Mode of a variate 
distributed as (3 f (a,(3) is 

. a-1 



+ 1' 



If x is a f (a,0) variate, then 1/x is a j9'(/3,a) variate. 
If x is a j3(a,/3) variate, then (1 - x)/x and x/(l — x) 
are 0\0 ) ct) and 0'{a,0) variates. If x and y are 7( a i) 
and 7(0:2) variates, then x/y is a /?' (0:1,0:2) variate. If 
x 2 /2 and y 2 /2 are 7(1/2) variates, then z 2 = (x/y) 2 is 
a £'(1/2, 1/2) variate. 

Bethe Lattice 

see Cayley Tree 

Betrothed Numbers 

see QUASIAMICABLE PAIR 

Betti Group 

The free part of the Homology Group with a domain 
of Coefficients in the Group of Integers (if this 
Homology Group is finitely generated). 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, p. 380, 1988. 



Betti Number 



Bhargava's Theorem 127 



Betti Number 

Betti numbers are topological objects which were proved 
to be invariants by Poincare, and used by him to ex- 
tend the Polyhedral Formula to higher dimensional 
spaces. The nth Betti number is the rank of the nth 
Homology Group. Let p r be the Rank of the Ho- 
mology Group H r of a Topological Space K. For 
a closed, orientable surface of GENUS g, the Betti num- 
bers are po = 1, Pi = 2#, and p 2 = I. For a nonori- 
entable surface with k CROSS-CAPS, the Betti numbers 
are po = 1, Pi = fc - 1, and p<z = 0. 
see also Euler Characteristic, Poincare Duality 

Bezier Curve 



the fact that moving a single control point changes the 
global shape of the curve. The former is sometimes 
avoided by smoothly patching together low-order Bezier 
curves. A generalization of the Bezier curve is the B- 

Spline. 

see also B-Spline, NURBS Curve 

Bezier Spline 

see Bezier Curve, Spline 

Bezout Numbers 

Integers (A,//) for a and b such that 

Aa + fib = GCD(a,6). 







Given a set of n control points, the corresponding Bezier 
curve (or BernSTEIN-Bezier Curve) is given by 



C(t) = 5^P.B i>n (t), 



where Bi n (t) is a Bernstein Polynomial and t € 

[0,1]- 

A "rational" Bezier curve is defined by 



C(*) = 



jy; =0 B itP (t)wii>i 



where p is the order, B itP are the BERNSTEIN POLYNO- 
MIALS, Pi are control points, and the weight Wi of Pi is 
the last ordinate of the homogeneous point P™. These 
curves are closed under perspective transformations, and 
can represent CONIC SECTIONS exactly. 

The Bezier curve always passes through the first and 
last control points and lies within the CONVEX Hull of 
the control points. The curve is tangent to Pi — Po and 
P n -P n _i at the endpoints. The "variation diminishing 
property" of these curves is that no line can have more 
intersections with a Bezier curve than with the curve 
obtained by joining consecutive points with straight line 
segments. A desirable property of these curves is that 
the curve can be translated and rotated by performing 
these operations on the control points. 

Undesirable properties of Bezier curves are their numer- 
ical instability for large numbers of control points, and 



For Integers ai, . . . , a n , the Bezout numbers are a set 
of numbers k\ , . . . , k n such that 

k\a\ + k-2<i2 + . . . + k n a n = d, 

where d is the Greatest Common Divisor of ai, . . . , 

a n . 

see also GREATEST COMMON DIVISOR 

Bezout's Theorem 

In general, two algebraic curves of degrees m and n in- 
tersect inm-n points and cannot meet in more than m-n 
points unless they have a component in common (i.e., 
the equations defining them have a common factor). 
This can also be stated: if P and Q are two POLYNOMI- 
ALS with no roots in common, then there exist two other 
Polynomials A and B such that AP + BQ = 1. Simi- 
larly, given N Polynomial equations of degrees m, ri2, 
. . . tin in N variables, there are in general niti2 • • • tin 
common solutions. 

see also POLYNOMIAL 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 10, 1959. 

Bhargava's Theorem 

Let the nth composition of a function f(x) be denoted 
/ (n) (x), such that / (0) (z) = x and / Cl) (z) = f(x). De- 
note / o g(x) = f(g(x)), and define 



Let 



u = (a, 6, c) 

\u\ = a-h b + c 
u\\ = a 4 + 6 4 + c 4 , 



(2) 
(3) 
(4) 



128 Bhaskara-Brouckner Algorithm 

and 

/(«) = (/i(«),/ a («),/3(t*)) (5) 

= (a(b - c), b(c - a),c(a - &)) (6) 

S(w) = (5i( u )»P2H,53(«)) 

= (^a 2 6,^a& 2 ,3a&c) . (7) 

Then if |u| = 0, 



||/ (m) o 5 (n) (tx)|| = 2(a6 + 6c + ca) 2m+l3 " 

= llff (n) o/ (m) (u)|| ) 



(8) 



where 771, n E {0, 1, ...} and composition is done in 

terms of components. 

see also DlOPHANTINE EQUATION — QUARTIC, FORD'S 

Theorem 

References 

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: 

Springer- Verlag, pp. 97-100, 1994. 
Bhargava, S. "On a Family of Ramanujan's Formulas for 

Sums of Fourth Powers." Ganita 43, 63-67, 1992. 

Bhaskara-Brouckner Algorithm 

see Square Root 

Bi-Connected Component 

A maximal SUBGRAPH of an undirected graph such that 

any two edges in the SUBGRAPH lie on a common simple 

cycle. 

see also Strongly Connected Component 

Bianchi Identities 

The Riemann Tensor is defined by 



-IJLf 



dx K dxv- 



2 q2 q2 

9^u a g\ K a g^ K 



dx K dx x dx^dx u dx u dx x 



Permuting 1/, «, and 77 (Weinberg 1972, pp. 146-147) 
gives the Bianchi identities 

see also BlANCHI IDENTITIES (CONTRACTED), RlE- 

mann Tensor 

References 

Weinberg, S. Gravitation and Cosmology: Principles and 

Applications of the General Theory of Relativity. New 

York: Wiley, 1972. 



BIBD 

Bianchi Identities (Contracted) 

Contracting A with v in the Bianchi Identities 



gives 



(2) 



Contracting again, 

R-n — R n\ii ~ R n-,v — 0, (3) 



or 



{R% - i<J%fi) ;M = 0, 



(fl"" - \!TR);* = 0. 



(4) 
(5) 



Bias (Estimator) 

The bias of an ESTIMATOR 9 is defined as 

b0) = (e) - e. 

It is therefore true that 

6 -6 = (8- (§)) + ((6) -$) = (0 - (§)) + B(0). 

An Estimator for which B = is said to be Unbiased. 
see also ESTIMATOR, UNBIASED 

Bias (Series) 

The bias of a Series is defined as 



Q[ai, at+i,a»+2] '■■ 






A Series is Geometric Iff Q = 0. A Series is Artis- 
tic Iff the bias is constant. 

see also Artistic Series, Geometric Series 

References 

Duffin, R. J. "On Seeing Progressions of Constant Cross Ra- 
tio." Amer. Math. Monthly 100, 38-47, 1993. 

Biased 

An Estimator which exhibits Bias. 

Biaugmented Pentagonal Prism 

see Johnson Solid 

Biaugmented Triangular Prism 

see Johnson Solid 

Biaugmented Truncated Cube 

see Johnson Solid 



BIBD 

see Block Design 



Bicentric Polygon 

Dicentric Polygon 



Bicorn 



129 




A Polygon which has both a Circumcircle and an 
INCIRCLE, both of which touch all VERTICES. All TRI- 
ANGLES are bicentric with 



R 2 -s 2 = 2Rr, 



(1) 



where R is the ClRCUMRADlUS, r is the Inradius, and s 
is the separation of centers. In 1798, N. Puss character- 
ized bicentric POLYGONS of n = 4, 5, 6, 7, and 8 sides. 
For bicentric QUADRILATERALS (FUSS'S PROBLEM), the 

Circles satisfy 



2r 2 (R 2 ~s 2 ) 
(Dorrie 1965) and 

Vabcd 



(R 2 -s 2 ) 2 -4r 2 s 2 



1 {ac + bd)(ad + bc)(ab + cd) 



4 V abed 

(Beyer 1987). In addition, 

1 1 

+ 



{R-s) 2 {R + s 



and 



a + c = b + d. 
The Area of a bicentric quadrilateral is 

A = vabed. 



(2) 

(3) 
(4) 

(5) 
(6) 
(7) 



If the circles permit successive tangents around the In- 
CIRCLE which close the POLYGON for one starting point 
on the CIRCUMCIRCLE, then they do so for all points on 
the Circumcircle. 

see also PONCELET'S CLOSURE THEOREM 

References 

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. 

Dorrie, H. "Fuss' Problem of the Chord- Tangent Quadrilat- 
eral," §39 in 100 Great Problems of Elementary Mathe- 
matics: Their History and Solutions. New York: Dover, 
pp. 188-193, 1965. 



Bicentric Quadrilateral 

A 4-sided Bicentric Polygon, also called a Cyclic- 
Inscriptable Quadrilateral. 

References 

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. 

Bichromatic Graph 

A Graph with Edges of two possible "colors," usually 
identified as red and blue. For a bichromatic graph with 
R red EDGES and B blue Edges, 

R + B>2. 

see also Blue-Empty Graph, Extremal Coloring, 
Extremal Graph, Monochromatic Forced Tri- 
angle, Ramsey Number 

Bicollared 

A SUBSET X C Y is said to be bicollared in Y if there 
exists an embedding 6 : X x [-1, 1] -> Y such that 
b(x, 0) = x when x £ X. The MAP 6 or its image is then 
said to be the bicollar. 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 34-35, 1976. 

Bicorn 




The bicorn is the name of a collection of QUARTIC 
CURVES studied by Sylvester in 1864 and Cayley in 1867 
(MacTutor Archive). The bicorn is given by the para- 
metric equations 



V 



asint 

a cos 2 t(2 + cost) 
3 sin 2 t 



The graph is similar to that of the COCKED HAT CURVE. 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New- 
York: Dover, pp. 147-149, 1972. 

MacTutor History of Mathematics Archive. "Bicorn." http: 
// www - groups . des . st - and .ac.uk/ -history / Curves / 
Bicorn.html. 



130 Bicubic Spline 



Bieberbach Conjecture 



Bicubic Spline 

A bicubic spline is a special case of bicubic interpolation 
which uses an interpolation function of the form 

4 4 

t=l j = l 
4 4 



Bidiakis Cube 



J- 2 



4 4 

y X2 (xi,x 2 ) = 5^ 5^0" - l)cijt"~V 

4 4 

t=l J=l 

where Cij are constants and u and £ are parameters rang- 
ing from to 1. For a bicubic spline, however, the partial 
derivatives at the grid points are determined globally by 
1-D Splines. 
see also B-Spline, Spline 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 118-122, 1992. 

Bicupola 

Two adjoined CUPOLAS. 

see also Cupola, Elongated Gyrobicupola, Elon- 
gated Orthobicupola, Gyrobicupola, Orthobi- 

CUPOLA 

Bicuspid Curve 




The Plane Curve given by the Cartesian equation 



(x 2 - a 2 )(x - a) 2 + (y 2 - a 2 ) 2 = 0. 



Bicylinder 

see Steinmetz Solid 



f^ 





The 12- Vertex graph consisting of a Cube in which two 
opposite faces (say, top and bottom) have edges drawn 
across them which connect the centers of opposite sides 
of the faces in such a way that the orientation of the 
edges added on top and bottom are PERPENDICULAR to 
each other. 

see also Bislit Cube, Cube, Cubical Graph 

Bieberbach Conjecture 

The nth. Coefficient in the Power series of a Univa- 
lent Function should be no greater than n. In other 
words, if 

f(z) = a + aiz 4- a 2 z 2 + . . . + a n z n + ... 

is a conformal transformation of a unit disk on any do- 
main, then|a n | < n|ai|. In more technical terms, "ge- 
ometric extremality implies metric extremality." The 
conjecture had been proven for the first six terms (the 
cases n = 2, 3, and 4 were done by Bieberbach, Lowner, 
and Sniffer and Garbedjian, respectively), was known 
to be false for only a finite number of indices (Hayman 
1954), and true for a convex or symmetric domain (Le 
Lionnais 1983). The general case was proved by Louis 
de Branges (1985). De Branges proved the MlLlN CON- 
JECTURE, which established the ROBERTSON CONJEC- 
TURE, which in turn established the Bieberbach conjec- 
ture (Stewart 1996). 

References 

de Branges, L. "A Proof of the Bieberbach Conjecture." Acta 

Math. 154, 137-152, 1985. 
Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge, 

England: Cambridge University Press, 1994. 
Hayman, W. K. and Stewart, F. M. "Real Inequalities with 

Applications to Function Theory." Proc. Cambridge Phil. 

Soc. 50, 250-260, 1954. 
Kazarinoff, N. D. "Special Functions and the Bieberbach 

Conjecture." Amer. Math. Monthly 95, 689-696, 1988. 
Korevaar, J. "Ludwig Bieberbach's Conjecture and its 

Proof." Amer. Math. Monthly 93, 505-513, 1986. 
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 

p. 53, 1983. 
Pederson, R. N. "A Proof of the Bieberbach Conjecture for 

the Sixth Coefficient." Arch. Rational Mech. Anal. 31, 

331-351, 1968/1969. 
Pederson, R. and SchifFer, M. "A Proof of the Bieberbach 

Conjecture for the Fifth Coefficient." Arch. Rational 

Mech. Anal. 45, 161-193, 1972. 
Stewart, I. "The Bieberbach Conjecture." In From Here to 

Infinity: A Guide to Today's Mathematics. Oxford, Eng- 
land: Oxford University Press, pp. 164-166, 1996. 



Bienayme-Chebyshev Inequality 



Biharmonic Equation 131 



Bienayme-Chebyshev Inequality 

see Chebyshev Inequality 

Bifoliate 




The Plane Curve given by the Cartesian equation 



x A + y = 2axy . 



References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., p. 72, 1989. 

Bifolium 




A Folium with 6 = 0. The bifolium is the Pedal 
Curve of the Deltoid, where the Pedal Point is the 
Midpoint of one of the three curved sides. The Carte- 
sian equation is 

(x 2 +y 2 ) 2 =4axy 2 

and the POLAR equation is 

r = 4a sin 2 OcosO. 

see also FOLIUM, QuADRIFOLIUM, TRIFOLIUM 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 152-153, 1972. 
MacTutor History of Mathematics Archive. "Double 

Folium." http : // www - groups . dcs . st - and .ac.uk/ 

-history/Curves/Double .html. 

Bifurcation 

A period doubling, quadrupling, etc., that accompanies 
the onset of Chaos. It represents the sudden appear- 
ance of a qualitatively different solution for a nonlin- 
ear system as some parameter is varied. Bifurcations 
come in four basic varieties: FLIP BIFURCATION, FOLD 
Bifurcation, Pitchfork Bifurcation, and Trans- 
critical Bifurcation (Rasband 1990). 

see also CODIMENSION, FEIGENBAUM CONSTANT, 

Feigenbaum Function, Flip Bifurcation, Hopf 



Bifurcation, Logistic Map, Period Doubling, 
Pitchfork Bifurcation, Tangent Bifurcation, 
Transcritical Bifurcation 

References 

Guckenheimer, J. and Holmes, P. "Local Bifurcations." Ch. 3 
in Nonlinear Oscillations, Dynamical Systems, and Bifur- 
cations of Vector Fields, 2nd pr., rev. corr. New York: 
Springer- Verlag, pp. 117-165, 1983. 

Lichtenberg, A. J. and Lieberman, M. A. "Bifurcation Phe- 
nomena and Transition to Chaos in Dissipative Systems." 
Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New- 
York: Springer- Verlag, pp. 457-569, 1992. 

Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4 
in Chaotic Dynamics of Nonlinear Systems. New York: 
Wiley, pp. 25-31, 1990. 

Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to 
Applied Nonlinear Dynamical Systems and Chaos. New 
York: Springer- Verlag, pp. 253-419, 1990. 

Bifurcation Theory 

The study of the nature and properties of BIFURCA- 
TIONS. 

see also CHAOS, DYNAMICAL SYSTEM 



Digraph 

see Bipartite Graph 

Bigyrate Diminished 
Rhombicosidodecahedron 

see Johnson Solid 



Biharmonic Equation 

The differential equation obtained by applying the Bi- 
harmonic Operator and setting to zero. 

vV = o. (i) 

In Cartesian Coordinates, the biharmonic equation 



V 2 (V 2 )0 

dx 2 + dy 2 + dz 2 ) \dx 2 + dy 2 + dz 2 J * 
-4 + ^-t + -Tnr +^-- 



dx 4 dy 4 dz 4 dx 2 dy 2 
0. 



d A (j> n d 4 <f> 



(2) 



dy 2 dz 2 dx 2 dz 2 
In Polar Coordinates (Kaplan 1984, p. 148) 
2 12 

V (p = (prrrr H 2^ rr9$ ~* 4^0090 H 4>rrr 

2 14 1 

~<t>rdd ~4>rr + ~7<l>e0 + ~^4>r = 0. (3) 



132 Biharmonic Operator 



Billiards 



For a radial function </>(r), the biharmonic equation be- 
comes 

Id f d [1 d 



r dr \ dr [ r dr V dr J J J 



2 11 

Vr-rr + ~<firrr ~ ^<t>rr + -3 0r = 0. (4) 



Writing the inhomogeneous equation as 

V 4 = 64/3, 
we have 



M rdr = d{r±\ 1 -±(r^)]} 

I dr lr dr \ dr / J J 



2 Vlnr- |r 2 



to obtain 



# 



(5) 



dr L r dr V dr / J 

r dr \ dr J 

(16j3r 3 + Cir Inr + C 2 r) dr = d {r*j-\ . (10) 

Now use 

/ r In r dr = \ 



(6) 

(7) 
(8) 
(9) 



(11) 



4/3r 4 + d(±r 2 lnr - \r 2 ) + §C 2 r 2 + ^ 3 = r^ (12) 
(4/3r 3 + C> In r + C 2 r+— \ dr = d<f> (13) 

</>(r)=/?r 4 -f C[ (|r 2 lnr- \r 2 ) 

+ §C 2 r 2 + C 3 lnr + C 4 
= /?r 4 + or 2 + 6 4- (cr 2 + d) In (?-) . (14) 

The homogeneous biharmonic equation can be separated 
and solved in 2-D Bipolar Coordinates. 

References 

Kaplan, W. Advanced Calculus, ^th ed. Reading, MA: 
Addison-Wesley, 1991. 



Biharmonic Operator 

Also known as the BlLAPLAClAN. 



In n-D space, 



V 4 = (V 2 ) 2 . 



, 4 /'1\ _ 3(15 -8n + n 2 ) 



(;)- 



Bijection 

A transformation which is One-TO-One and ONTO. 

see also One-to-One, Onto, Permutation 



Bilaplacian 

see Biharmonic Operator 

Bilinear 

A function of two variables is bilinear if it is linear with 
respect to each of its variables. The simplest example is 
f(x,y) =xy. 

Bilinear Basis 

A bilinear basis is a BASIS, which satisfies the conditions 

(ax + by) • z = a(x * z) + 6(y • z) 
z • (ax 4- by) = a(z • x) + 6(z • y). 
see also Basis 

Billiard Table Problem 

Given a billiard table with only corner pockets and sides 
of Integer lengths m and n, a ball sent at a 45° angle 
from a corner will be pocketed in a corner after m+n-2 
bounces. 

see also Alhazen's Billiard Problem, Billiards 
Billiards 

The game of billiards is played on a RECTANGULAR table 
(known as a billiard table) upon which balls are placed. 
One ball (the "cue ball") is then struck with the end 
of a "cue" stick, causing it to bounce into other balls 
and Reflect off the sides of the table. Real billiards 
can involve spinning the ball so that it does not travel 
in a straight LINE, but the mathematical study of bil- 
liards generally consists of REFLECTIONS in which the 
reflection and incidence angles are the same. However, 
strange table shapes such as CIRCLES and Ellipses are 
often considered. Many interesting problems can arise. 

For example, Alhazen's BILLIARD PROBLEM seeks to 
find the point at the edge of a circular "billiards" table 
at which a cue ball at a given point must be aimed in 
order to carom once off the edge of the table and strike 
another ball at a second given point. It was not until 
1997 that Neumann proved that the problem is insoluble 
using a COMPASS and RULER construction. 

On an ELLIPTICAL billiard table, the ENVELOPE of a 
trajectory is a smaller ELLIPSE, a HYPERBOLA, a LINE 
through the FOCI of the ELLIPSE, or periodic curve (e.g., 
DlAMOND-shape) (Wagon 1991). 

see also Alhazen's Billiard Problem, Billiard Ta- 
ble Problem, Reflection Property 



see also Biharmonic Equation 



Billion 



Binary 133 



References 

Davis, D.; Ewing, C; He, Z.; and Shen, T. "The 
Billiards Simulation." http : //serendip .brynmawr . edu/ 
chao s /home . html . 

Dullin, H. R.; Richter, RH.; and Wittek, A. "A Two- 
Parameter Study of the Extent of Chaos in a Billiard Sys- 
tem." Chaos 6, 43-58, 1996. 

Madachy, J. S. "Bouncing Billiard Balls." In Madachy's 
Mathematical Recreations. New York: Dover, pp. 231— 
241, 1979. 

Neumann, P. Submitted to Amer. Math. Monthly. 

Pappas, T. "Mathematics of the Billiard Table." The Joy of 
Mathematics. San Carlos, CA: Wide World Publ./Tetra, 
p. 43, 1989. 

Peterson, I. "Billiards in the Round." http : //www . 
sciencenews.org/sn_arc97/3-l_97/mathland.htm. 

Wagon, S. "Billiard Paths on Elliptical Tables." §10.2 in 
Mathematica in Action. New York: W. H. Freeman, 
pp. 330-333, 1991. 

Billion 

The word billion denotes different numbers in American 
and British usage. In the American system, one billion 
equals 10 9 . In the British, French, and German systems, 
one billion equals 10 12 . 

see also LARGE NUMBER, MILLIARD, MILLION, TRIL- 
LION 

Bilunabirotunda 

see Johnson Solid 

Bimagic Square 



16 


41 


36 


5 


27 


62 


55 


18 


26 


63 


54 


19 


13 


44 


33 


8 


1 


40 


45 


12 


22 


51 


58 


31 


23 


50 


59 


30 


4 


37 


48 


9 


38 


3 


10 


47 


49 


24 


29 


60 


52 


21 


32 


57 


39 


2 


11 


46 


43 


14 


7 


34 


64 


25 


20 


53 


61 


28 


17 


56 


42 


15 


6 


35 



If replacing each number by its square in a MAGIC 
Square produces another Magic Square, the square 
is said to be a bimagic square. The first bimagic square 
(shown above) has order 8 with magic constant 260 for 
addition and 11,180 after squaring. Bimagic squares 
are also called Doubly Magic Squares, and are 2- 
Multimagic Squares. 

see also MAGIC SQUARE, MULTIMAGIC SQUARE, 

Trimagic Square 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 212, 
1987. 

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 
in Mathematical Diversions. New York: Dover, p. 31, 
1975. 

Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical 
Recreations. New York: W. W. Norton, pp. 176-178, 1942. 




M BC 



A' M AB 

A Line Segment joining the Midpoints of opposite 

sides of a QUADRILATERAL. 

see also Median (Triangle), Varignon's Theorem 

Bimodal Distribution 

A Distribution having two separated peaks. 

see also Unimodal Distribution 

Bimonster 

The wreathed product of the Monster Group by Z 2 . 
The bimonster is a quotient of the Coxeter Group 
with the following Coxeter-Dynkin Diagram. 




This had been conjectured by Conway, but was proven 
around 1990 by Ivanov and Norton. If the parameters 
p,<?, r in Coxeter's NOTATION [3 F,q>r ] are written side 
by side, the bimonster can be denoted by the BEAST 
Number 666. 

Bin 

An interval into which a given data point does or does 
not fall. 

see also HISTOGRAM 

Binary 

The BASE 2 method of counting in which only the digits 
and 1 are used. In this Base, the number 1011 equals 
l-2° + l-2 + 0-2 2 + l-2 3 = 11. This Base is used in com- 
puters, since all numbers can be simply represented as 
a string of electrically pulsed ons and offs. A NEGATIVE 
— n is most commonly represented as the complement of 
the Positive number n - 1, so -11 = 00001011 2 would 
be written as the complement of 10 — OOOOIOIO2, or 
11110101. This allows addition to be carried out with 
the usual carrying and the left-most digit discarded, so 
17 — 11 = 6 gives 

00010001 17 
11110101 -11 
00000110 6 



134 Binary Bracketing 



Binary Tree 



The number of times k a given binary number 
b n ■ • -&2&1&0 is divisible by 2 is given by the position 
of the first bk = 1 counting from the right. For exam- 
ple, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is 
divisible by 2 times. 

Unfortunately, the storage of binary numbers in com- 
puters is not entirely standardized. Because computers 
store information in 8-bit bytes (where a bit is a sin- 
gle binary digit), depending on the "word size" of the 
machine, numbers requiring more than 8 bits must be 
stored in multiple bytes. The usual F0RTRAN77 integer 
size is 4 bytes long. However, a number represented as 
(bytel byte2 byte3 byte4) in a VAX would be read and 
interpreted as (byte4 byte3 byte2 bytel) on a Sun. The 
situation is even worse for floating point (real) num- 
bers, which are represented in binary as a MANTISSA 
and Characteristic, and worse still for long (8-byte) 
reals! 

Binary multiplication of single bit numbers (0 or 1) is 
equivalent to the AND operation, as can be seen in the 
following Multiplication Table. 



X 


1 



1 



1 



see also Base (Number), Decimal, Hexadecimal, 
Octal, Quaternary, Ternary 

References 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 6-9, 
1991. 

Pappas, T. "Computers, Counting, & Electricity." The Joy 
of Mathematics. San Carlos, CA: Wide World Publ./ 
Tetra, pp. 24-25, 1989. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Error, Accuracy, and Stability" and "Diag- 
nosing Machine Parameters." §1.2 and §20.1 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 18-21, 276, and 881-886, 1992. 
^ Weisstein, E. W. "Bases." http: //www. astro. Virginia. 
edu/~eww6n/math/notebooks/Bases.m. 

Binary Bracketing 

A binary bracketing is a BRACKETING built up entirely 
of binary operations. The number of binary bracket ings 
of n letters (Catalan's Problem) are given by the 
Catalan Numbers C n _i, where 



C n = 



n + 1 



2n\ _ 1 (2ra)! _ 
n ) n+ 1 n! 2 



(2n)! 



(n+l)!n! 



where ( 2 ™) denotes a Binomial Coefficient and n\ 
is the usual FACTORIAL, as first shown by Catalan in 
1838, For example, for the four letters a, 6, c, and d 
there are five possibilities: ({ab)c)d, (a(6c))d, (a&)(cd), 
a((bc)d), and a(6(cd)), written in shorthand as {(xx)x)x } 
(x(xx))x, (xx)(xx), x((xx)x), and x(x(xx)). 



see also BRACKETING, CATALAN NUMBER, CATALAN'S 

Problem 

References 

Schroder, E. "Vier combinatorische Probleme." Z. Math. 
Physik 15, 361-376, 1870. 

Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 

Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough." 
Amer. Math. Monthly 104, 344-350, 1997. 

Binary Operator 

An Operator which takes two mathematical objects 
as input and returns a value is called a binary operator. 
Binary operators are called compositions by Rosenfeld 
(1968). Sets possessing a binary multiplication opera- 
tion include the Group, Groupoid, Monoid, Quasi- 
group, and Semigroup. Sets possessing both a bi- 
nary multiplication and a binary addition operation in- 
clude the Division Algebra, Field, Ring, Ringoid, 
Semiring, and Unit Ring. 

see also AND, BOOLEAN ALGEBRA, CLOSURE, DIVI- 
SION Algebra, Field, Group, Groupoid, Monoid, 
Operator, Or, Monoid, Not, Quasigroup, Ring, 
Ringoid, Semigroup, Semiring, XOR, Unit Ring 

References 

Rosenfeld, A. An Introduction to Algebraic Structures. New 
York: Holden-Day, 1968. 

Binary Quadratic Form 

A 2-variable QUADRATIC FORM of the form 



Q(x, y) = aux 2 + 2a\ixy + a 2 2V . 



see also QUADRATIC FORM, QUADRATIC INVARIANT 

Binary Remainder Method 

An Algorithm for computing a Unit Fraction 

(Stewart 1992). 

References 

Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer. 
266, 122-124, June 1992. 

Binary Tree 

A Tree with two Branches at each Fork and with 
one or two Leaves at the end of each Branch. (This 
definition corresponds to what is sometimes known as 
an "extended" binary tree.) The height of a binary tree 
is the number of levels within the TREE. For a binary 
tree of height H with n nodes, 

H < n < 2 H - 1, 



Binet Forms 



Binomial Coefficient 135 



These extremes correspond to a balanced tree (each 
node except the Leaves has a left and right Child, 
arid all LEAVES are at the same level) and a degenerate 
tree (each node has only one outgoing BRANCH), respec- 
tively. For a search of data organized into a binary tree, 
the number of search steps S(n) needed to find an item 
is bounded by 

lgn < S(n) < n. 

Partial balancing of an arbitrary tree into a so-called 
AVL binary search tree can improve search speed. 

The number of binary trees with n internal nodes is 
the Catalan Number C n (Sloane's A000108), and the 
number of binary trees of height b is given by Sloane's 
A001699. 

see also S-Tree, Quadtree, Quaternary Tree, 
Red-Black Tree, Stern-Brocot Tree, Weakly 
Binary Tree 

References 

Lucas, J.; Roelants van Baronaigien, D.; and Ruskey, F. 
"Generating Binary Trees by Rotations." J. Algorithms 
15, 343-366, 1993. 

Ranum, D. L. "On Some Applications of Fibonacci Num- 
bers." Amer. Math. Monthly 102, 640-645, 1995. 

Ruskey, F. "Information on Binary Trees." http://sue.csc 
,uvic.ca/~cos/inf/tree/BinaryTrees.html. 

Ruskey, F. and Proskurowski, A. "Generating Binary Trees 
by Transpositions." J. Algorithms 11, 68-84, 1990. 

Sloane, N. J. A. Sequences A000108/M1459 and A001699/ 
M3087 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Binet Forms 

The two Recurrence Sequences 



U n — mUn-l + U n ~2 

V n =mV n - 1 + V n - 2 



(1) 
(2) 



with Uo = 0, Ui = 1 and Vo = 2, V\ — m, can be solved 
for the individual U n and V n . They are given by 



" ~ P (3) 

(4) 

(5) 
(6) 

(7) 

(8) 

Binet' S Formula is a special case of the Binet form 
for U n corresponding to m = 1. 
see also Fibonacci Q-Matrix 





yJ-n — 


A 




V n = 


= a n + ^, 


where 


A = 






\/m 2 + 4 




a = 


771+ A 

2 




= 


m - A 
2 


A useful related 


identity 


is 




Un-l+Un+l = Vn- 



Binet 's Formula 

A special case of the U n Binet Form with m 
corresponding to the nth FIBONACCI NUMBER, 

_ _ (1 + V5)"-(1-a/5) b 



0, 



2 n VE 

It was derived by Binet in 1843, although the result 
was known to Euler and Daniel Bernoulli more than a 
century earlier. 

see also Binet Forms, Fibonacci Number 

Bing's Theorem 

If M 3 is a closed oriented connected 3-MANIFOLD such 
that every simple closed curve in M lies interior to a 
BALL in Af , then M is HOMEOMORPHIC with the Hy- 
persphere, S 3 . 

see also Ball, Hypersphere 

References 

Bing, R. H. "Necessary and Sufficient Conditions that a 3- 

Manifold be S 3 ." Ann. Math. 68, 17-37, 1958. 
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 

Perish Press, pp. 251-257, 1976. 

Binomial 

A Polynomial with 2 terms. 

see also Monomial, Polynomial, Trinomial 

Binomial Coefficient 

The number of ways of picking n unordered outcomes 
from N possibilities. Also known as a COMBINATION. 
The binomial coefficients form the rows of PASCAL'S 
Triangle. The symbols N C n and 



n) = (N - n)\n\ 



(1) 



are used, where the latter is sometimes known as N 
CHOOSE n. The number of LATTICE PATHS from the 
Origin (0, 0) to a point (a, b) is the Binomial Coeffi- 
cient (°+ 6 ) (Hilton and Pedersen 1991). 

For Positive integer n, the Binomial Theorem gives 



i.— n V / 



(2) 



The Finite Difference analog of this identity is 
known as the Chu-Vandermonde Identity. A sim- 
ilar formula holds for Negative Integral n, 






~ n \ k-Tl-k 

,x a 

k 



(3) 



A general identity is given by 
(a + b) n 



= Eu)( a -* : ) i " 1 ( 6 +j' c ) B "' w 



136 Binomial Coefficient 

(Prudnikov et ol. 1986), which gives the BINOMIAL THE- 
OREM as a special case with c = 0. 

The binomial coefficients satisfy the identities: 



n 


n 
k 

n + 1 
A: 



„-»-<-<":"' 



n \ I n 



(5) 
(6) 
(7) 



Sums of powers include 



fc=0 x y 

E<-')"(I)=» 

k=0 x 7 

t(;V -(•+')• 



(8) 

(9) 
(10) 



(the Binomial Theorem), and 

^2n + s 



£ 



a; n = 2^1(1(5 4- 1), §(* + 2); s + 1, 4x) 



2 s 



(\/l - 4x + l)Vl - 4a; 



, (11) 



where 2F1 (a, 6; c;z) is a Hypergeometric Function 
(Abramowitz and Stegun 1972, p. 555; Graham et al. 
1994, p. 203). For NONNEGATIVE INTEGERS n and r 
with r < n + 1, 



££?(;) [D-*G")"- fl ' 

fc^o v 7 L j=o v y 

+Et- 1 ) i (")( n+l - r -j) n - fc 



Taking n = 2r — 1 gives 

r-l 



n!. (12) 






-fc _ i 



in!. (13) 



Another identity is 



E ( n £ k ) [xn+1(1 - x)fc + (1 - x)n+1 ^ = 1 w 



Binomial Coefficient 
Recurrence Relations of the sums 



EI 



— . <15) 

are given by 

2si(n)-si(n + l) = (16) 

-2(2n + l)s 2 (n) + (n + l)s 2 (n) = (17) 

-8(n + l) 2 s 3 (n) + (-16 - 21n - 7n 2 )s 3 (n + 1) 

+(n + 2) 2 53 (n + 2) = (18) 

-4(n + l)(4n + 3)(4n + 5)s 4 (n) 

-2(2n + 3)(3n 2 + 9n + 7)s 4 (n + 1) 

+(n + 2) 3 s 4 (n + 2) = 0. (19) 

This sequence for S3 cannot be expressed as a fixed 
number of hypergeometric terms (Petkovsek et a/. 1996, 
p. 160). 

A fascinating series of identities involving binomial co- 
efficients times small powers are 



00 
£ 72^Y = 27 ( 27r ^ + 9 ) = 0.7363998587 . . . 

n=l V n ) 

00 

E— 1_ = Ittv^ = 0.6045997881 . . . 
n( 2n ) 9 

n=l \n J 

n=l V n 7 

°° 1 

/ ^ 4(2n\ 36 ^W 3240 /l 

71=1 V n / 



(Comtet 1974, p. 89) and 

- ( _ 1} n-i 



£ 



8 a 



= I C(3), 



(20) 

(21) 
(22) 
(23) 

(24) 



where ((z) is the Riemann Zeta Function (Le Lion- 
nais 1983, pp. 29, 30, 41, 36, and 35; Guy 1994, p. 257). 

As shown by Kummer in 1852, the exact Power of p 
dividing ( a ^ b ) is equal to 



eo + ei + . . . + e £ , 



(25) 



(Beeler et al 1972, Item 42). 



where this is the number of carries in performing the 
addition of a and b written in base b (Graham et al. 
1989, Exercise 5.36; Ribenboim 1989; Vardi 1991, p. 68). 
Kummer's result can also be stated in the form that the 



Binomial Coefficient 



Binomial Coefficient 



137 



exponent of a Prime p dividing (j^j is given by the 
number of integers j > for which 



frac(ra/p J ) > frac (n/p 3 ). 



(26) 



where frac(cc) denotes the FRACTIONAL PART of x. This 
inequality may be reduced to the study of the exponen- 
tial sums ^2 n A(n)e(x/n), where A(n) is the MANGOLDT 
FUNCTION. Estimates of these sums are given by Jutila 
(1974, 1975), but recent improvements have been made 
by Granville and Ramare (1996). 

R. W. Gosper showed that 

/( n ) = (l(n~-l)) ~ (- 1 ) < "" 1)/2 ( mod ") (27) 

for all Primes, and conjectured that it holds only for 
Primes. This was disproved when Skiena (1990) found 
it also holds for the Composite Number n = 3xllx 
179. Vardi (1991, p. 63) subsequently showed that n = 
p 2 is a solution whenever p is a Wieferich Prime and 
that if n = p k with k > 3 is a solution, then so is n = 
p k ~ 1 . This allowed him to show that the only solutions 
for Composite n < 1.3xl0 7 are 5907, 1093 2 , and 3511 2 , 
where 1093 and 3511 are Wieferich PRIMES. 

Consider the binomial coefficients ( n ~ )•> the first few 
of which are 1, 3, 10, 35, 126, ... (Sloane's A001700). 
The Generating Function is 



Vl-4o; 



: x + 3x 2 + 10x 3 + 35x 4 + . 



(28) 



These numbers are SQUAREFREE only for n = 2, 3, 4, 
6, 9, 10, 12, 36, . . . (Sloane's A046097), with no others 
less than n = 10, 000. Erdos showed that the binomial 
coefficient (™) is never a Power of an Integer for n > 
3 where A; ^ 0, 1, n— 1, and n (Le Lionnais 1983, p. 48). 

The binomial coefficients (| n / 2 |) are called CENTRAL 

Binomial Coefficients, where |xj is the Floor 
Function, although the subset of coefficients ( 2 ™) is 
sometimes also given this name. Erdos and Graham 
(1980, p. 71) conjectured that the Central Binomial 
Coefficient ( 2 ^) is never Squarefree for n > 4, and 
this is sometimes known as the Erdos SQUAREFREE 
Conjecture. Sarkozy's Theorem (Sarkozy 1985) 
provides a partial solution which states that the BINO- 
MIAL Coefficient ( 2 ^) is never Squarefree for all 
sufficiently large n > no (Vardi 1991). Granville and 
Ramare (1996) proved that the only SQUAREFREE val- 
ues are n = 2 and 4. Sander (1992) subsequently showed 
that ( 2n ^ d ) are also never SQUAREFREE for sufficiently 
large n as long as d is not "too big." 

For p, qr, and r distinct PRIMES, then the above function 
satisfies 

f(pqr)f(p)f(q)f(r) = f {pq) f (pr)p(qr) (mod pqr) 

(29) 



(Vardi 1991, p. 66). 

The binomial coefficient (™) mod 2 can be computed 
using the XOR operation n XOR m, making Pascal's 
Triangle mod 2 very easy to construct. 




The binomial coefficient "function" can be defined as 



C{z,y) 



y\(x - y)\ 



(30) 



(Fowler 1996), shown above. It has a very complicated 
Graph for Negative x and y which is difficult to render 
using standard plotting programs. 

see also BALLOT PROBLEM, BINOMIAL DISTRIBU- 
TION, Binomial Theorem, Central Binomial Co- 
efficient, Chu-Vandermonde Identity, Combi- 
nation, Deficiency, Erdos Squarefree Conjec- 
ture, Gaussian Coefficient, Gaussian Polynom- 
ial, Kings Problem, Multinomial Coefficient, 
Permutation, Roman Coefficient, Sarkozy's 
Theorem, Strehl Identity, Wolstenholme's The- 
orem 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Binomial Co- 
efficients. " §24.1.1 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 10 and 822-823, 1972. 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM-239, Feb. 1972. 

Comtet, L. Advanced Combinatorics. Amsterdam, Nether- 
lands: Kluwer, 1974. 

Conway, J. H. and Guy, R. K. In The Book of Numbers. New 
York: Springer- Verlag, pp. 66-74, 1996. 

Erdos, P.; Graham, R. L.; Nathanson, M. B.; and Jia, X. Old 
and New Problems and Results in Combinatorial Number 
Theory. New York: Springer- Verlag, 1998, 

Fowler, D. "The Binomial Coefficient Function." Amer. 
Math. Monthly 103, 1-17, 1996. 

Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial 
Coefficients." Ch. 5 in Concrete Mathematics: A Foun- 
dation for Computer Science. Reading, MA: Addison- 
Wesley, pp. 153-242, 1990. 

Granville, A. and Ramare, O. "Explicit Bounds on Exponen- 
tial Sums and the Scarcity of Squarefree Binomial Coeffi- 
cients." Mathematika 43, 73-107, 1996. 



138 



Binomial Distribution 



Binomial Distribution 



Guy, R. K. "Binomial Coefficients," "Largest Divisor of a 
Binomial Coefficient," and "Series Associated with the £- 
Function." §B31, B33, and F17 in Unsolved Problems 
in Number Theory, 2nd ed. New York: Springer- Verlag, 
pp. 84-85, 87-89, and 257-258, 1994. 

Harborth, H. "Number of Odd Binomial Coefficients." Not. 
Amer. Math. Soc. 23, 4, 1976. 

Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener- 
alization, and Their Uses." Math. Intel 13, 64-75, 1991. 

Jutila, M. "On Numbers with a Large Prime Factor." J. 
Indian Math. Soc. 37, 43-53, 1973. 

Jutila, M. "On Numbers with a Large Prime Factor. II," J. 
Indian Math. Soc. 38, 125-130, 1974. 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
1983. 

Ogilvy, C. S. "The Binomial Coefficients." Amer. Math. 
Monthly 57, 551-552, 1950. 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, 1996. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Gamma Function, Beta Function, Factorials, 
Binomial Coefficients." §6.1 in Numerical Recipes in FOR- 
TRAN: The Art of Scientific Computing, Qnd ed. Cam- 
bridge, England: Cambridge University Press, pp. 206— 
209, 1992. 

Prudnikov, A. P.; Marichev, O. I.; and Brychkow, Yu. A. 
Formula 41 in Integrals and Series, Vol. 1: Elementary 
Functions. Newark, NJ: Gordon & Breach, p. 611, 1986. 

Ribenboim, P. The Book of Prime Number Records, 2nd ed. 
New York: Springer- Verlag, pp. 23-24, 1989. 

Riordan, J. "Inverse Relations and Combinatorial Identities." 
Amer. Math. Monthly 71, 485-498, 1964. 

Sander, J. W. "On Prime Divisors of Binomial Coefficients." 
Bull. London Math. Soc. 24,140-142, 1992. 

Sarkozy, A. "On the Divisors of Binomial Coefficients, I." J. 
Number Th. 20, 70-80, 1985. 

Skiena, S, Implementing Discrete Mathematics: Combina- 
torics and Graph Theory with Mathematica. Reading, 
MA: Addison- Wesley, p. 262, 1990. 

Sloane, N. J. A. Sequences A046097 and A001700/M2848 in 
"An On-Line Version of the Encyclopedia of Integer Se- 
quences." 

Spanier, J. and Oldham, K. B. "The Binomial Coefficients 
(^)." Ch. 6 in An Atlas of Functions. Washington, DC: 
Hemisphere, pp. 43-52, 1987. 

Sved, M. "Counting and Recounting." Math. Intel 5, 21-26, 
1983. 

Vardi, I. "Application to Binomial Coefficients," "Binomial 
Coefficients," "A Class of Solutions," "Computing Bino- 
mial Coefficients," and "Binomials Modulo and Integer." 
§2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations 
in Mathematica. Redwood City, CA: Addison- Wesley, 
pp. 25-28 and 63-71, 1991. 

Wolfram, S. "Geometry of Binomial Coefficients." Amer. 
Math. Monthly 91, 566-571, 1984. 

Binomial Distribution 



The probability of n successes in N BERNOULLI TRIALS 
is 




n N — n 

P Q 



(1) 

The probability of obtaining more successes than the n 
observed is 



*=E 



k = n + l 



N ^p k (l-p) N - k =I p (n + l,N-N), (2) 



where 



Ix{a,b) 



B{x\ a y b) 
B(a,b) ' 



(3) 



B(a,b) is the Beta Function, and B(x\a,b) is the 
incomplete BETA FUNCTION. The CHARACTERISTIC 

Function is 

<f>(t) = ( q + pe it )\ (4) 

The Moment-Generating Function M for the dis- 
tribution is 

N y v 

M(t) = <e tn > = XX n ( ^ V<? N " n 

n=0 ^ ' 

= E(?V)"(i-p) n -" 



= \pe t + (l-p)] N 
At (t) = Nfre* + (1 - p)] JV - 1 (pe') 
M"it) = NiN - l)[pc* + (1 - p)]"- V) 2 
+ N\pe t + il-p)} N - 1 ipe t ). 

The Mean is 

H = M'(0) = Nip + 1 - p)p = Np. 

The Moments about are 



(5) 
(6) 

(7) 



(8) 



/*!=/* = Np (9) 

l& = Np(l-p + Np) (10) 

^ = Np(l - 3p + 3Np + 2p 2 - 3NP 2 + N 2 p 2 ) (11) 
/4 = Np(l - 7p + 7Np + 12p 2 - ISNp 2 + 6iVV 
- 6p 3 + HATp 3 - 6iVV + A^p 3 ), (12) 

so the Moments about the Mean are 



M2 = a 2 = [N(N - l)p 2 + Np] - {Np) 2 
= N 2 p 2 - Np 2 +Np- N 2 p 2 
= Np(l -p) = Npq 



(13) 
(14) 



Pz = P3 ~ 3p f 2Pi + 2{pif 

= Np(l-p)(l-2p) 
li4 = fJ,4- 4/4/ii + 6/i2(/ii) 2 - 3(/ii) 4 

= Np(l - p)[3p 2 (2 - N) + 3p(N - 2) + 1]. (15) 



Binomial Distribution 

The SKEWNESS and KURTOSIS are 

/is = Np(l-p){l-2p) 

<7 3 [iV>(l-p)] 3 /2 



7i 



l-2p 



q-p 



y/Np(l-p) y/Npq 



7 2 = ^-3: 
cr 4 



6p — 6p + 1 1 — 6pg 



iVp(l-p) 



A^pg 



(16) 



(17) 



An approximation to the Bernoulli distribution for large 
N can be obtained by expanding about the value n 
where P(n) is a maximum, i.e., where dP/dn = 0. Since 
the Logarithm function is Monotonic, we can instead 
choose to expand the LOGARITHM. Let n = h + to, then 



ln[F(n)]-ln[P(n)] + B 1 7?+|B2T? 2 + |jS 3 7 ? 3 + ..., (18) 

where 

(19) 



B k = 



d k \n[P(n)] 



dn k 



But we are expanding about the maximum, so, by defi 
nition, 

~dln[P(n)] 



Bi 



dn 



= 0. 



(20) 



This also means that B2 is negative, so we can write 

B 2 = — 1B 2 1 . Now, taking the LOGARITHM of (1) gives 

ln[P(n)] = lnNl-\nn\-ln(N-n)\ + nlnp+(N-n)\nq. 

(21) 
For large n and N — n we can use STIRLING'S APPROX- 
IMATION 

ln(n!) « toIxito-to, (22) 



so 



d[ln(w!)] 
dn 
d[\n(N-n)\] 
dn 



« (Inn -I- 1) - 1 = lnn 
d 



(23) 



dn 



[{N - n) \vv{N -n)-(N - 



= -ln(7V-n), 



(24) 



and 



dln{P(n)} ^ _ lnn + ln(JV _ w) + lnp _ lnq / 25 v 

dn 



To find n, set this expression to and solve for ra, 

(26) 

N — hp 



h (^)=' 



1 
n q 

(N — n)p = hq 

n(q + p) = h = Np, 



(27) 

(28) 
(29) 



Binomial Distribution 139 

since p + q — 1. We can now find the terms in the 
expansion 



B 2 



d 2 \n[P(n)] 



dn 2 



1 1 



h N ~ h 



1 1 _ _ J_ (\ l\ 

Np N(l-p)~ N\p + q) 



i (p + q 



B 3 = 



N \ pq 

d*\n[P(n)] 
dn 3 

1 1 



1 



1 



Npq N(l-p) 

1 1 



(30) 



h 2 (N - h) 2 



„2 2 



N 2 p 2 N 2 q 2 N 2 p 2 q 2 
(l-2p + p 2 )-p 2 _ 



l-2p 



B 4 = 



N 2 p 2 (l-p) 2 N 2 p 2 {l-p) 2 

d 4 \n[P(n)] 



(31) 



dn 4 



h 3 (n — h) 3 



-2 



3 



_J_ _J_\ = 2(P 3 +Q 
N 3 p 3 N 3 q 3 J N 3 p 3 q< 

= 2(p 2 -pq + q 2 ) 
N z p 3 q 3 

= 2\p 2 -p{l-p) + {l-2p + p 2 )] 
N 3 p 3 (l-p 3 ) 

= 2(3p 2 -3p+l) 
N 3 p 3 (l-p 3 ) ' 

Now, treating the distribution as continuous, 



(32) 



■W p /»oo 

lim y^P(n)^ P(n)dn= / P(h + to) dn - 1. 

(33) 
Since each term is of order 1/iV ~ 1/<t 2 smaller than the 
previous, we can ignore terms higher than B 2 , so 



P(n) = P(n)e- |B2| " 2/2 . 
The probability must be normalized, so 



(34) 



J~ P(fi)e-W 2 '*dT, = P(n)^=l, (35) 



and 



P(n) 



\Bl\ -\B 2 \(n-n) 2 /2 

2tt 



yj2nNpq 



exp 



Defining a 2 = 2Npq, 



P(n) 



(7V27T 



: exp 



(n - iVp) 2 
2iVpg 



(to - n) 2 
2a 2 



(36) 



(37) 



140 Binomial Expansion 



Binomial Series 



which is a GAUSSIAN DISTRIBUTION. For p < 1, a 
different approximation procedure shows that the bi- 
nomial distribution approaches the PoiSSON DISTRIBU- 
TION. The first Cumulant is 



m = np, 



(38) 



and subsequent Cumulants are given by the RECUR- 
RENCE Relation 



kv+i = pq 



dp ' 



(39) 



Let x and y be independent binomial Random Vari- 
ables characterized by parameters n,p and m,p. The 
Conditional Probability of x given that x + y = k 
is 



P(a; = i|a; + y — k) 



P(x = i y x + y = k) 



P{x + y = k) 
P(x = i,y = k-i) _ P(x = i)P(y = k-i) 
P(x-\-y = k) ~ P{x + y = k) 

( n t m )p fc (i-p) n+m - fc 






(40) 



Note that this is a Hypergeometric Distribution! 

see also de Moivre-Laplace Theorem, Hypergeo- 
metric Distribution, Negative Binomial Distri- 
bution 

References 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, p. 531, 1987, 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Incomplete Beta Function, Student's Distribu- 
tion, F-Distribution, Cumulative Binomial Distribution." 
§6.2 in Numerical Recipes in FORTRAN: The Art of Sci- 
entific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 219-223, 1992. 

Spiegel, M. R. Theory and Problems of Probability and 
Statistics. New York: McGraw-Hill, p. 108-109, 1992. 

Binomial Expansion 

see Binomial Series 

Binomial Formula 

see Binomial Series, Binomial Theorem 

Binomial Number 

A number of the form a n ± b n , where a, 6, and n are 
Integers. They can be factored algebraically 

a n -& n = (a-6)(a ri ~ 1 +a Tl - 2 b + ... + a6 n " 2 +b n - 1 ) (1) 

a n + b n -(a + 6)(a n - 1 -a n - 2 6 + ...-ab n - 2 +6 n - 1 ) (2) 



a nm - b nTn = (a m - 6 m )[a m(n ~ 1) + a m(n ' 2) 6 m 

+ ... + 6 m(n_1) ]. (3) 

In 1770, Euler proved that if (a, b) = 1, then every FAC- 
TOR of 

o a "+6 jn (4) 

is either 2 or of the form 2 n+1 K + 1. If p and q are 
Primes, then 



a pq -l)(a-l) 
(aP-l)(a«-l] 



- 1 



(5) 



is Divisible by every Prime Factor of a p 1 not divid- 
ing a q — 1. 
see also CUNNINGHAM NUMBER, FERMAT NUMBER, 

Mersenne Number, Riesel Number, Sierpinski 
Number of the Second Kind 

References 

Guy, R. K. "When Does 2 a - 2 b Divide n a - n 6 ." §B47 in 

Unsolved Problems in Number Theory, 2nd ed. New York: 

Springer- Verlag, p. 102, 1994. 
Qi, S and Ming-Zhi, Z. "Pairs where 2 a - a b Divides n a - n h 

for All n." Proc. Amer. Math. Soc. 93, 218-220, 1985. 
Schinzel, A. "On Primitive Prime Factors of a n — 6 n ." Proc. 

Cambridge Phil Soc. 58, 555-562, 1962. 

Binomial Series 

For Id < 1, 



(i + x y 



- £(:)■* 

fc=o v 7 



(i) 



= i + 



;x + 



l!(n-l)! (n-2)!2! 



" ! x= + ...(3) 



n(n — 1) o 
l + nz+ -^ — -x 2 + . 



(4) 



The binomial series also has the CONTINUED FRACTION 
representation 



(1 + *)" = 



-. (5) 



1 + 



l-(l + n) 



1-2 



1 + 



1 ■(!-") , 
2-3 



1 + 



2(2 + n) 
3-4 



1+- 



2(2 -n) , 

4-5 



3(3 + n) 
5-6 
1 + ... 



1 + 



Binomial Theorem 



Biotic Potential 



141 



see also Binomial Theorem, Multinomial Series, 
Negative Binomial Series 

References 

Abramowitz, M. and Stegun, C, A, (Eds,). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
pp. 14-15, 1972. 

Pappas, T. "Pascal's Triangle, the Fibonacci Sequence &; Bi- 
nomial Formula." The Joy of Mathematics. San Carlos, 
CA: Wide World Publ./Tetra, pp. 40-41, 1989. 

Binomial Theorem 

The theorem that, for INTEGRAL POSITIVE n, 



Z_/ kun - 



A;=0 



(n-k)\ 



^r 



k=0 



the so-called Binomial Series, where (™) are Bino- 
mial Coefficients. The theorem was known for the 
case n = 2 by Euclid around 300 BC, and stated in its 
modern form by Pascal in 1665. Newton (1676) showed 
that a similar formula (with Infinite upper limit) holds 
for Negative Integral n, 



(* + a)-» = £; ( fc n y a - 



the so-called Negative Binomial Series, which con- 
verges for |x| > \a\. 

see also BINOMIAL COEFFICIENT, BINOMIAL SERIES, 

Cauchy Binomial Theorem, Chu-Vandermonde 
Identity, Logarithmic Binomial Formula, Nega- 
tive Binomial Series, <?-Binomial Theorem, Ran- 
dom Walk 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 10, 1972. 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 307-308, 1985. 

Conway, J. H. and Guy, R. K. "Choice Numbers Are Bino- 
mial Coefficients." In The Book of Numbers. New York: 
Springer- Verlag, pp. 72-74, 1996. 

Coolidge, J. L. "The Story of the Binomial Theorem," Amer. 
Math. Monthly 56, 147-157, 1949. 

Courant, R. and Robbins, H. "The Binomial Theorem." §1.6 
in What is Mathematics?: An Elementary Approach to 
Ideas and Methods, 2nd ed. Oxford, England: Oxford Uni- 
versity Press, pp. 16-18, 1996. 

Binomial Triangle 

see Pascal's Triangle 



Binormal Developable 

A Ruled Surface M is said to be a binormal de- 
velopable of a curve y if M can be parameterized by 
x(«,v) = y(u)+t;B(u), where B is the BINORMAL VEC- 
TOR. 

see also NORMAL DEVELOPABLE, TANGENT DEVEL- 
OPABLE 

References 

Gray, A. "Developables." §17.6 in Modern Differential Ge- 
ometry of Curves and Surfaces. Boca Raton, FL: CRC 
Press, pp. 352-354, 1993. 

Binormal Vector 



:TxN 
r' x r" 

' |r' xr'T 



(1) 
(2) 



where the unit TANGENT VECTOR T and unit "princi- 
pal" NORMAL VECTOR N are defined by 



t - r'(s) 



N: 



|r'( S )| 
\t"(s)\ 



(3) 
(4) 



Here, r is the Radius Vector, s is the Arc Length, r 
is the TORSION, and « is the Curvature. The binormal 
vector satisfies the remarkable identity 



[B,B,B1 



ds 



(") 



(5) 



see also Frenet Formulas, Normal Vector, Tan- 
gent Vector 

References 

Kreyszig, E. "Binormal. Moving Trihedron of a Curve." §13 

in Differential Geometry. New York: Dover, p. 36—37, 

1991. 

Bioche's Theorem 

If two complementary PLUCKER CHARACTERISTICS are 
equal, then each characteristic is equal to its comple- 
ment except in four cases where the sum of order and 
class is 9. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- 
York: Dover, p. 101, 1959. 

Biotic Potential 

see Logistic Equation 



142 Bipartite Graph 

Bipartite Graph 




A set of VERTICES decomposed into two disjoint sets 
such that no two VERTICES within the same set are 
adjacent. A bigraph is a special case of a &- Partite 
Graph with k = 2. 

see also Complete Bipartite Graph, /c-Partite 
Graph, Konig-Egevary Theorem 

References 

Chartrand, G. Introductory Graph Theory. New York: 

Dover, p. 116, 1985. 
Saaty, T. L. and Kainen, P. C. The Four- Color Problem: 

Assaults and Conquest, New York: Dover, p. 12, 1986. 

Biplanar Double Point 

see Isolated Singularity 

Bipolar Coordinates 

Bipolar coordinates are a 2-D system of coordinates. 
There are two commonly defined types of bipolar co- 
ordinates, the first of which is defined by 



a sinh v 



y = 



cosh v — cos u 

as'mu 
cosh v — cos u ' 



(i) 

(2) 



where u € [0,27r), v G (—00,00). The following identi- 
ties show that curves of constant u and v are CIRCLES 
in ay-space. 



x 2 + {y — a cot u) 2 — a 2 esc 2 u 



(x — a coth v) 2 + y = a 2 csch 2 v. 



The Scale Factors 


are 








h u - 


a 








coshi; — 


cosu 




h v - 


a 








coshv — 


cosu 




The Laplacian is 










^2 _ (coshi; 


\2 

— cos u) 

a 2 


( d 2 

\du 2 


+ 


d 2 

dv 2 



(3) 
(4) 

(5) 
(6) 

(7) 



Laplace's Equation is separable. 



Bipolar Cylindrical Coordinates 

Two-center bipolar coordinates are two coordinates giv- 
ing the distances from two fixed centers r\ and V2 , some- 
times denoted r and r'. For two-center bipolar coordi- 
nates with centers at (±c, 0), 



ri 2 = ( x + c) 2 +y 2 

2 / \2 , 2 

r 2 ■ = (x - c) + y . 
Combining (8) and (9) gives 



2 2 A 

ri — ri = 4cx. 



(8) 
(9) 



(10) 



Solving for CARTESIAN COORDINATES x and y gives 



* 2 „ 2 

Ti — 7*2 

4c 



(11) 



y = ±^y/l6c 2 n 2 - (n 2 - r 2 2 + 4c 2 ). (12) 



Solving for POLAR COORDINATES gives 



ri 2 + r 2 2 -2c 2 



8 — tan 



8c 2 (n 2 +r 2 2 -2c 2 ) 



(13) 
(14) 



References 

Lockwood, E. H. "Bipolar Coordinates." Ch. 25 in A Book 

of Curves. Cambridge, England: Cambridge University 

Press, pp. 186-190, 1967. 



Bipolar Cylindrical Coordinates 




A set of Curvilinear Coordinates defined by 
a sinh v 



cosh v — cos u 

asinu 
cosh v — cos u 

z = z, 



2/ = 



(1) 

(2) 
(3) 



where u 6 [0,27r), v £ (-00,00), and z e (—00,00). 
There are several notational conventions, and whereas 
(u,v,z) is used in this work, Arfken (1970) prefers 



Biprism 



Biquadratic Number 143 



(77, £, z). The following identities show that curves of 
constant u and v are CIRCLES in xy- space. 



2 , / x \2 2 2 

x -\- (y — a cot u) = a esc it 

(x — acothv) + y =a csch v. 
The Scale Factors are 

a 



h u = 
h v = 



cosh v — cos u 
a 

cosh v — cos u 

1. 



The Laplacian is 



2 (cosh v — cos u) 2 ( d 2 d 2 



(4) 
(5) 

(6) 

(7) 
(8) 



d 2 



Laplace's Equation is not separable in Bipolar 
Cylindrical Coordinates, but it is in 2-D Bipolar 
Coordinates. 

References 

Arfken, G. "Bipolar Coordinates (£, 77, z)." §2.9 in Math- 
ematical Methods for Physicists, 2nd ed. Orlando, FL: 
Academic Press, pp. 97-102, 1970. 

Biprism 

Two slant triangular PRISMS fused together. 

see also Prism, Schmitt-Conway Biprism 

Bipyramid 

see Dipyramid 

Biquadratefree 



60 



40 



20 40 60 80 100 

A number is said to be biquadratefree if its Prime de- 
composition contains no tripled factors. All PRIMES are 
therefore trivially biquadratefree. The biquadratefree 
numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 
15, 17, . . . (Sloane's A046100). The biquadrateful num- 
bers (i.e., those that contain at least one biquadrate) 
are 16, 32, 48, 64, 80, 81, 96, ... (Sloane's A046101). 
The number of biquadratefree numbers less than 10, 100, 
1000, ... are 10, 93, 925, 9240, 92395, 923939, . . . , and 
their asymptotic density is 1/C(4) = 90/tt 4 « 0.923938, 
where C(n) is the Riemann Zeta Function. 



see also Cubefree, Prime Number, Riemann Zeta 
Function, Squarefree 

References 

Sloane, N. J. A. Sequences A046100 and A046101 in "An On- 
Line Version of the Encyclopedia of Integer Sequences." 

Biquadratic Equation 

see Quartic Equation 

Biquadratic Number 

A biquadratic number is a fourth POWER, n 4 . The first 
few biquadratic numbers are 1, 16, 81, 256, 625, ... 
(Sloane's A000583). The minimum number of squares 
needed to represent the numbers 1, 2, 3, . . . are 1, 2, 3, 
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, . . . 
(Sloane's A002377), and the number of distinct ways to 
represent the numbers 1, 2, 3, . . . in terms of biquadratic 
numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 

2, 2, 2, A brute-force algorithm for enumerating the 

biquadratic permutations of n is repeated application of 
the Greedy Algorithm. 

Every POSITIVE integer is expressible as a SUM of (at 
most) 5(4) = 19 biquadratic numbers (WARING'S PROB- 
LEM). Davenport (1939) showed that G(4) = 16, mean- 
ing that all sufficiently large integers require only 16 
biquadratic numbers. The following table gives the first 
few numbers which require 1, 2, 3, . . . , 19 biquadratic 
numbers to represent them as a sum, with the sequences 
for 17, 18, and 19 being finite. 

# Sloane Numbers 

1, 16, 81, 256, 625, 1296, 2401, 4096, ... 

2, 17, 32, 82, 97, 162, 257, 272, . . . 

3, 18, 33, 48, 83, 98, 113, 163, ... 

4, 19, 34, 49, 64, 84, 99, 114, 129, . . 

5, 20, 35, 50, 65, 80, 85, 100, 115, .. 

6, 21, 36, 51, 66, 86, 96, 101, 116, .. 

7, 22, 37, 52, 67, 87, 102, 112, 117, . 

8, 23, 38, 53, 68, 88, 103, 118, 128, . 

9, 24, 39, 54, 69, 89, 104, 119, 134, . 

10, 25, 40, 55, 70, 90, 105, 120, 135, 

11, 26, 41, 56, 71, 91, 106, 121, 136, 

12, 27, 42, 57, 72, 92, 107, 122, 137, 



1 


000290 


2 


003336 


3 


003337 


4 


003338 


5 


003339 


6 


003340 


7 


003341 


8 


003342 


9 


003343 


10 


003344 


11 


003345 


12 


003346 



The following table gives the numbers which can be rep- 
resented in n different ways as a sum of k biquadrates. 

k n Sloane Numbers 

1 1 000290 1, 16, 81, 256, 625, 1296, 2401, 4096, . . . 
2 2 635318657, 3262811042, 8657437697, ... 

The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 
15, 18, 19, 20, 21, ... (Sloane's A046039) cannot be 
represented using distinct biquadrates. 

see also CUBIC NUMBER, SQUARE NUMBER, WARING'S 

Problem 

References 

Davenport, H. "On Waring's Problem for Fourth Powers." 
Ann. Math. 40, 731-747, 1939. 



144 Biquadratic Reciprocity Theorem 

Biquadratic Reciprocity Theorem 



x = q (mod p) . 



(i) 



This was solved by Gauss using the GAUSSIAN INTEGERS 
as 



(J).®.-'-"""*'- 



)/4][(JV(<r)-l)/4] 



(2) 



'4 \TV / 4 

where n and a are distinct GAUSSIAN INTEGER PRIMES, 



N(a + hi) = yja? + b 2 
and N is the norm. 



(3) 



■{ 



1 if x 4 = a (mod 7r) is solvable 

— l,i, or — i otherwise, 



(4) 



where solvable means solvable in terms of Gaussian In- 
tegers. 

see also RECIPROCITY THEOREM 

Biquaternion 

A Quaternion with Complex coefficients. The Alge- 
bra of biquaternions is isomorphic to a full matrix ring 
over the complex number field (van der Waerden 1985). 
see also Quaternion 

References 

Clifford, W. K. "Preliminary Sketch of Biquaternions." Proc. 

London Math. Soc. 4, 381-395, 1873. 
Hamilton, W. R. Lectures on Quaternions: Containing a 

Systematic Statement of a New Mathematical Method. 

Dublin: Hodges and Smith, 1853. 
Study, E. "Von den Bewegung und Umlegungen." Math. 

Ann. 39, 441-566, 1891. 
van der Waerden, B. L. A History of Algebra from al- 

Khwarizmi to Emmy Noether. New York: Springer- Verlag, 

pp. 188-189, 1985. 

Birational Transformation 

A transformation in which coordinates in two SPACES 
are expressed rationally in terms of those in another. 

see also Riemann Curve Theorem, Weber's Theo- 
rem 

Birch Conjecture 

see Swinnerton-Dyer Conjecture 

Birch-Swinnerton-Dyer Conjecture 

see Swinnerton-Dyer Conjecture 



Birthday Attack 

Birkhoff 's Ergodic Theorem 

Let T be an ergodic ENDOMORPHISM of the PROBABIL- 
ITY SPACE X and let / : X -t R be a real-valued MEA- 
SURABLE Function. Then for Almost Every x € X, 
we have 



-^TfoF j (x)^ If dm 



as n — v oo. To illustrate this, take / to be the charac- 
teristic function of some Subset A of X so that 



/(*)={; 



if xe A 
if x £ A. 



The left-hand side of (-1) just says how often the or- 
bit of x (that is, the points x, Tx, T 2 x, . . . ) lies in 
A, and the right-hand side is just the MEASURE of A. 
Thus, for an ergodic ENDOMORPHISM, "space-averages 
= time- averages almost everywhere." Moreover, if T is 
continuous and uniquely ergodic with BOREL PROBA- 
BILITY MEASURE m and / is continuous, then we can 
replace the Almost Everywhere convergence in (-1) 
to everywhere. 

Birotunda 

Two adjoined ROTUNDAS. 

see also BlLUNABIROTUNDA, CUPOLAROTUNDA, ELON- 
GATED Gyrocupolarotunda, Elongated Ortho- 

CUPOLAROTUNDA, ELONGATED ORTHOBIROTUNDA, 

Gyrocupolarotunda, Gyroelongated Rotunda, 

ORTHOBIROTUNDA, TRIANGULAR HEBESPHENOROTUN- 
DA 

Birthday Attack 

Birthday attacks are a class of brute-force techniques 
used in an attempt to solve a class of cryptographic 
hash function problems. These methods take advantage 
of functions which, when supplied with a random in- 
put, return one of k equally likely values. By repeatedly 
evaluating the function for different inputs, the same 
output is expected to be obtained after about 1.2\/fc 
evaluations. 
see also Birthday Problem 

References 

RSA Laboratories. "Question 95. What is a Birthday At- 
tack." http : //www . rsa . com/rsalabs/newf aq/q95 . html. 
"Question 96. How Does the Length of a Hash Value 
Affect Security?" http : //www . rsa . com/r salabs/newf aq/ 
q96.html. 

van Oorschot, P. and Wiener, M. "A Known Plaintext At- 
tack on Two-Key Triple Encryption." In Advances in 
Cryptology — Eurocrypt '90. New York: Springer- Verlag, 
pp. 366-377, 1991. 

Yuval, G. "How to Swindle Rabin." Cryptologia 3, 187-189, 
Jul. 1979. 



Birthday Problem 



Birthday Problem 145 



Birthday Problem 

Consider the probability Qi(n, d) that no two people out 
of a group of n will have matching birthdays out of d 
equally possible birthdays. Start with an arbitrary per- 
son's birthday, then note that the probability that the 
second person's birthday is different is (d — l)/d, that 
the third person's birthday is different from the first two 
is [(d — l)/d][(d — 2)/d], and so on, up through the nth 
person. Explicitly, 



Qi(n,d) = 



Id- 2 d-(n-l) 



d d d 

_ (d-l)(d-2)---[d-(n-l)] 
d n 
But this can be written in terms of FACTORIALS as 

dl 



Qi(n,d) 



(d-n)\d> 



71 ' 



(1) 



(2) 



so the probability P 2 (n, 365) that two people out of a 
group of n do have the same birthday is therefore 



P 2 (n,d) = 1-Qi(n,d) = 1 



d\ 



(d-n)\d n ' 



(3) 



If 365-day years have been assumed, i.e., the existence of 
leap days is ignored, then the number of people needed 
for there to be at least a 50% chance that two share 
birthdays is the smallest n such that p2(^, 365) > 1/2. 
This is given by n — 23, since 

P 2 (23,365) = 

38093904702297390785243708291056390518886454060947061 
75091883268515350125426207425223147563269805908203125 

« 0.507297. (4) 

The number of people needed to obtain Pzin, 365) > 1/2 
for n = 1, 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, . . . 
(Sloane's A033810). 

The probability P2(n,d) can be estimated as 

P 2 (n,d)^l-e- n(n " 1)/2d (5) 

— ('-sr- <■> 

where the latter has error 

C < — 7~7TK (7) 



6(d-n + l) 2 



(Sayrafiezadeh 1994). 




In general, let Qi(n,d) denote the probability that a 
birthday is shared by exactly i (and no more) people 
out of a group of n people. Then the probability that a 
birthday is shared by k or more people is given by 



P k {n,d) = l-^Qi(n,d). 
Qi can be computed explicitly as 



(8) 



L«/2J 



~ ~cF 2^i 2H\(n 



dl 



(-l) n 



(n~2i)l(d- n + i)\ 



(9) 



where (™) is a BINOMIAL COEFFICIENT, T(n) is a 

Gamma Function, and Pj[ x \x) is an Ultraspheri- 
cal Polynomial. This gives the explicit formula for 
P^n^d) as 

Pz(n 7 d) = 1 - Qi(n,d) - Q 2 (n,d) 

(-1)^(71+ l)Pi~ d) (2-^) 
" ^ 2 n / 2 d n * K J 

Qz{n,d) cannot be computed in entirely closed form, 
but a partially reduced form is 



Qz{n,d) = 



r(d+i) 

d n 



(-irF(f)-F(-f) 

T(d-n + l) 



+(-i) r(i + n) ^ r(d-i + i)r(i + i) 



(ii) 



where 

F = F(n,d, a) = 1-3^2 



■ i(l_„),l(2-n),-I 
i(d-n+l) i(d-n + 2)'' 



(12) 



and 3^2 (a, 6, c; d, e; z) is a GENERALIZED HYPERGEO- 
metric Function. 

In general, Qk(n,d) can be computed using the RECUR- 
RENCE Relation 



[n/kj 

Qk(n,d) — y^ 



n!rf! 



d ik i\(k\y(n-ik)\(d-i)\ 



x^2Qj{n-k,d-i) 



; Jd-iy 



j=i 



(Jn — ik 



(13) 



146 Birthday Problem 



Bisection Procedure 



(Finch). However, the time to compute this recursive 
function grows exponentially with k and so rapidly be- 
comes unwieldy. The minimal number of people to give 
a 50% probability of having at least n coincident birth- 
days is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 
1385, 1596, 1813, . . . (Sloane's A014088; Diaconis and 
Mosteller 1989). 

A good approximation to the number of people n such 
that p = Pk(n,d) is some given value can given by solv- 
ing the equation 



ne 



-n/(dk) 



d* _1 fc!ln 



1 



1- 



d(fc + l) 



i/fc 



(14) 

for n and taking [n], where [n] is the CEILING Func- 
tion (Diaconis and Mosteller 1989). For p = 0.5 and 
k — 1, 2, 3, ... , this formula gives n = 1, 23, 88, 187, 
313, 459, 722, 797, 983, 1179, 1382, 1592, 1809, ..., 
which differ from the true values by from to 4. A 
much simpler but also poorer approximation for n such 
that p — 0.5 for k < 20 is given by 



n = 47(fe-1.5)' 



3/2 



(15) 



(Diaconis and Mosteller 1989), which gives 86, 185, 307, 
448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for k = 3, 
4,.... 

The "almost" birthday problem, which asks the number 
of people needed such that two have a birthday within 
a day of each other, was considered by Abramson and 
Moser (1970), who showed that 14 people suffice. An ap- 
proximation for the minimum number of people needed 
to get a 50-50 chance that two have a match within k 
days out of d possible is given by 



n(k y d) = 1.2 



d 



2k + 1 



(16) 



(Sevast'yanov 1972, Diaconis and Mosteller 1989). 

see also Birthday Attack, Coincidence, Small 
World Problem 

References 

Abramson, M. and Moser, W. O. J. "More Birthday Sur- 
prises." Amer. Math. Monthly 77, 856-858, 1970. 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, pp. 45-46, 
1987. 

Bloom, D. M. "A Birthday Problem." Amer. Math. Monthly 
80, 1141-1142, 1973. 

Bogomolny, A. "Coincidence." http://www.cut-* the-knot . 
com/do_you_know/coincidence.html. 

Clevenson, M. L. and Watkins, W. "Majorization and the 
Birthday Inequality." Math. Mag. 64, 183-188, 1991. 

Diaconis, P. and Mosteller, F. "Methods of Studying Coinci- 
dences." J. Amer. Statist. Assoc. 84, 853-861, 1989. 

Feller, W. An Introduction to Probability Theory and Its Ap- 
plications, Vol. 1, 3rd ed. New York: Wiley, pp. 31-32, 
1968. 



Finch, S. "Puzzle #28 [June 1997]: Coincident Birthdays." 
http: //www. maths oft . com/mathcad/library /puzzle/ 
soln28/soln28.html. 

Gehan, E. A. "Note on the 'Birthday Problem.'" Amer. Stat. 
22, 28, Apr. 1968. 

Heuer, G. A. "Estimation in a Certain Probability Problem." 
Amer. Math. Monthly 66, 704-706, 1959. 

Hocking, R. L. and Schwertman, N. C. "An Extension of the 
Birthday Problem to Exactly k Matches." College Math. 
J. 17, 315-321, 1986. 

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- 
sions. New York: Dover, pp. 102-103, 1975. 

Klamkin, M. S. and Newman, D. J. "Extensions of the Birth- 
day Surprise." J. Combin. Th. 3, 279-282, 1967. 

Levin, B. "A Representation for Multinomial Cumulative 
Distribution Functions." Ann. Statistics 9, 1123-1126, 
1981. 

McKinney, E. H. "Generalized Birthday Problem." Amer. 
Math. Monthly 73, 385-387, 1966. 

Mises, R. von. "Uber Aufteilungs — und Besetzungs- 

Wahrscheinlichkeiten." Revue de la Faculte des Sci- 
ences de VUniversite d'Istanbul, N. S. 4, 145—163, 1939. 
Reprinted in Selected Papers of Richard von Mises, Vol. 2 
(Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. Szego, 
and G. BirkhofF), Providence, RI: Amer. Math. Soc, 
pp. 313-334, 1964. 

Riesel, H. Prime Numbers and Computer Methods for Fac- 
torization, 2nd ed. Boston, MA: Birkhauser, pp. 179-180, 
1994. 

Sayrafiezadeh, M. "The Birthday Problem Revisited." Math. 
Mag. 67, 220-223, 1994. 

Sevast'yanov, B. A. "Poisson Limit Law for a Scheme of Sums 
of Dependent Random Variables." Th. Prob. Appl. 17, 
695-699, 1972. 

Sloane, N. J. A. Sequences A014088 and A033810 in "An On- 
Line Version of the Encyclopedia of Integer Sequences." 

Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96, 
June 1998. 

Tesler, L. "Not a Coincidence!" http://www.nomodes.com/ 
coincidence .html. 

Bisected Perimeter Point 

see Nagel Point 

Bisection Procedure 

Given an interval [a, &], let a n and b n be the endpoints 
at the nth iteration and r n be the nth approximate solu- 
tion. Then, the number of iterations required to obtain 
an error smaller than e is found as follows. 



1 






(1) 



(2) 



\r n -r\<±{b n - a n ) = 2~ n (b - a) < e (3) 



— n In 2 < In e — ln(6 — a), 
ln(6 — a) — In e 



n > 



In 2 



(4) 
(5) 



so 

see also ROOT 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 964-965, 1985. 



Bisector 



Bislit Cube 



147 



Press, W. H.; Fiannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Bracketing and Bisection." §9.1 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 343-347, 1992. 

Bisector 

Bisection is the division of a given curve or figure into 
two equal parts (halves). 

see also Angle Bisector, Bisection Procedure, 
Exterior Angle Bisector, Half, Hemisphere, 
Line Bisector, Perpendicular Bisector, Trisec- 

TION 

Bishop's Inequality 

Let V{r) be the volume of a BALL of radius r in a com- 
plete 7l-D RlEMANNIAN MANIFOLD with RlCCI CURVA- 
TURE > (n - 1)k. Then V(r) > V K (r), where V K is 
the volume of a Ball in a space having constant Sec- 
tional Curvature. In addition, if equality holds for 
some Ball, then this Ball is Isometric to the Ball 
of radius r in the space of constant SECTIONAL CURVA- 
TURE K. 

References 

Chavel, I. Riemannian Geometry: A Modern Introduction. 
New York: Cambridge University Press, 1994. 

Bishops Problem 



B 
















B 














B 


B 














B 


B 














B 


B 














B 


B 














B 


B 














B 


B 

















Find the maximum number of bishops B(n) which can 
be placed on an n x n Chessboard such that no two 
attack each other. The answer is 2n — 2 (Dudeney 1970, 
Madachy 1979), giving the sequence 2, 4, 6, 8, . . . (the 
Even Numbers) for n = 2, 3, One maximal so- 
lution for n = 8 is illustrated above. The number of 
distinct maximal arrangements of bishops for n — 1, 2, 
... are 1, 4, 26, 260, 3368, . . . (Sloane's A002465). The 
number of rotationally and reflectively distinct solutions 
on an n x n board for n > 2 is 



B(n) 



/ 2 (n-4)/2 [2 (n-2)/2 + y ^ n ey( . Q 
| 2 (n-3)/2 [2 („-3)/2 + 1 ] fornodd 



where |nj is the FLOOR FUNCTION, giving the sequence 
for n = 1, 2, . . . as 1, 1, 2, 3, 6, 10, 20, 36, . . . (Sloane's 

A005418). 









B 
















B 
















B 
















B 
















B 
















B 
















B 
















B 











The minimum number of bishops needed to occupy or 
attack all squares on an n x n Chessboard is n, ar- 
ranged as illustrated above. 

see also Chess, Kings Problem, Knights Problem, 
Queens Problem, Rooks Problem 

References 

Ahrens, W. Mathematische Unterhaltungen und Spiele, 

Vol 1, 3rd ed. Leipzig, Germany: Teubner, p. 271, 1921. 
Dudeney, H. E. "Bishops — Unguarded" and "Bishops — 

Guarded." §297 and 298 in Amusements in Mathematics. 

New York: Dover, pp. 88-89, 1970. 
Guy, R. K. "The n Queens Problem." §C18 in Unsolved 

Problems in Number Theory, 2nd ed. New York: Springer- 

Verlag, pp. 133-135, 1994. 
Madachy, J. Madachy's Mathematical Recreations. New 

York: Dover, pp. 36-46, 1979. 
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 74- 

75, 1995. 
Sloane, N. J. A. Sequences A002465/M3616 and A005418/ 

M0771 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Bislit Cube 






The 8- Vertex graph consisting of a Cube in which two 
opposite faces have DIAGONALS oriented PERPENDICU- 
LAR to each other. 

see also Bidiakis Cube, Cube, Cubical Graph 



(Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 
1995). An equivalent formula is 



J B(n) = 2 n - 3 + 2 L( ' l - 1)/2J - 1 , 



148 Bispherical Coordinates 

Bispherical Coordinates 




A system of CURVILINEAR COORDINATES defined by 
a sin £ cos <fi 



y- 



cosh 77 — cos £ 

a sin £ sin <\> 
cosh 77 — cos £ 

a sinh 77 
cosh T] — cos £ 

The Scale Factors are 

h a 

h v 



cos 77 — cos £ 

a 



The Laplacian is 



2 _ / — cos u co ^ 2 u ' 
\ cosh t> 



cosh 77 — cos £ 

asin£ 
cosh 77 — cos £ 



+ 3 cosh v cot u 



a) 

(2) 

(3) 

(4) 
(5) 
(6) 



-3 cosh 2 v cot u esc u + cosh vcsc u 
cosh v — cos ti 



d(j> 2 



+ (cosu — cosh v) sinh v~ — h (cosh v - cosu) ■^-^ 



<% 



0v 2 



a 



+ (cosh v — cos ti) (cosh v cot w — sin u — cos u cot u) — 



+(cosh 2 i; — cos u) -^—7 . 
ou 2 



(7) 



In bispherical coordinates, LAPLACE'S EQUATION is sep- 
arable, but the Helmholtz Differential Equation 
is not. 

see also Laplace's Equation— Bispherical Coor- 
dinates, Toroidal Coordinates 

References 

Arfken, G. "Bispherical Coordinates (£, 77, <£)." §2.14 in 
Mathematical Methods for Physicists, 2nd ed. Orlando, 
FL: Academic Press, pp. 115-117, 1970. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 665-666, 1953. 



Black-Scholes Theory 

Bit Complexity 

The number of single operations (of ADDITION, SUB- 
TRACTION, and Multiplication) required to complete 
an algorithm. 
see also STRASSEN FORMULAS 

References 

Borodin, A. and Munro, I. The Computational Complexity 

of Algebraic and Numeric Problems. New York: American 

Elsevier, 1975. 



Bitangent 



bitangent 




A Line which is Tangent to a curve at two distinct 
points. 

see also Klein's Equation, Plucker Characteris- 
tics, Secant Line, Solomon's Seal Lines, Tangent 

Line 

Bivariate Distribution 

see Gaussian Bivariate Distribution 

Bivector 

An antisymmetric TENSOR of second Rank (a.k.a. 2- 
form) . 

X = X ah u) a A u) , 

where A is the Wedge Product (or Outer Prod- 
uct). 

Biweight 

see TUKEY'S BIWEIGHT 

Black-Scholes Theory 

The theory underlying financial derivatives which in- 
volves "stochastic calculus" and assumes an uncor- 
rected Log Normal Distribution of continuously 
varying prices. A simplified "binomial" version of the 
theory was subsequently developed by Sharpe et al. 
(1995) and Cox et al (1979). It reproduces many re- 
sults of the full-blown theory, and allows approximation 
of options for which analytic solutions are not known 
(Price 1996). 
see also Garman-Kohlhagen Formula 

References 

Black, F. and Scholes, M. S. "The Pricing of Options and 
Corporate Liabilities." J. Political Econ. 81, 637-659, 
1973. 

Cox, J. C; Ross, A.; and Rubenstein, M. "Option Pricing: A 
Simplified Approach." J. Financial Economics 7, 229-263, 
1979. 

Price, J. F. "Optional Mathematics is Not Optional." Not. 
Amer. Math. Soc. 43, 964-971, 1996. 

Sharpe, W. F.; Alexander, G. J,; and Bailey, J. V. Invest- 
ments, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. 



Black Spleenwort Fern 

Black Spleenwort Fern 

see BARNSLEY'S FERN 

Blackman Function 



Blecksmith-Brillhart- Gerst Theorem 



149 




-1 -0.5 0.5 1 -0.5 

An Apodization Function given by 



A(x) = 0.42 + 0.5 cos 



(?) 



+ 0.08 cos 



(^) 



a) 



Its Full Width at Half Maximum is 0.810957a. The 
Apparatus Function is 



I(k) = 

a(0.84 - 0.36a 2 fc 2 - 2.17 x 10~ x Vfe 4 ) sin(27raA:) 
(l-a 2 fc 2 )(l-4a 2 A; 2 ) 



The Coefficients are approximations to 



ao 



ai 



a 2 = 



3969 
9304 
1155 
4652 
715 

18608' 



(2) 

(3) 
(4) 
(5) 



which would have produced zeros of I(k) at k = (7/4)a 
and k = (9/4)a. 

see also APODIZATION FUNCTION 

References 

Blackman, R. B. and Tukey, J, W. "Particular Pairs of Win- 
dows." In The Measurement of Power Spectra, From 
the Point of View of Communications Engineering. New 
York: Dover, pp. 98-99, 1959. 

Blancmange Function 




A Continuous Function which is nowhere Differ- 

ENTIABLE. The iterations towards the continuous func- 
tion are Batrachions resembling the Hofstadter- 
Conway $10,000 Sequence. The first six iterations 
are illustrated below. The dth iteration contains TV + 1 



points, where TV = 2 d , and can be obtained by setting 
6(0) = b(N) = 0, letting 

b{m + 2 71 " 1 ) = 2 n + \[b{m) + b{m + 2 n )], 

and looping over n = d to 1 by steps of —1 and m = 
to TV- 1 by steps of 2 n . 




Peitgen and Saupe (1988) refer to this curve as the Tak- 
agi Fractal Curve. 

see also HOFSTADTER-CONWAY $10,000 SEQUENCE, 

Weierstrak Function 

References 

Dixon, R. Mathographics. New York: Dover, pp. 175-176 
and 210, 1991. 

Peitgen, H.-O. and Saupe, D. (Eds.). "Midpoint Displace- 
ment and Systematic Fractals: The Takagi Fractal Curve, 
Its Kin, and the Related Systems." §A.1.2 in The Science 
of Fractal Images. New York: Springer- Verlag, pp. 246- 
248, 1988. 

Takagi, T. "A Simple Example of the Continuous Function 
without Derivative." Proc. Phys. Math. Japanl y 176-177, 
1903. 

Tall, D. O. "The Blancmange Function, Continuous Every- 
where but DifTerentiable Nowhere." Math. Gaz. 66,11-22, 
1982. 

Tall, D. "The Gradient of a Graph." Math. Teaching 111, 
48-52, 1985. 

Blaschke Conjecture 

The only WlEDERSEHEN MANIFOLDS are the standard 
round spheres. The conjecture has been proven by com- 
bining the Berger-Kazdan Comparison Theorem 
with A. Weinstein's results for n Even and C. T. Yang's 
for n Odd. 

References 

Chavel, I. Riemannian Geometry: A Modern Introduction. 
New York: Cambridge University Press, 1994. 

Blaschke's Theorem 

A convex planar domain in which the minimal length is 
> 1 always contains a Circle of Radius 1/3. 

References 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 25, 1983. 

Blecksmith-Brillhart-Gerst Theorem 

A generalization of Schroter'S FORMULA. 

References 

Berndt, B. C Ramanujan's Notebooks, Part III. New York: 
Springer- Verlag, p. 73, 1985. 




BlichfeldVs Lemma 



Blichfeldt's Lemma 

see Blichfeldt's Theorem 

Blichfeldt's Theorem 

Published in 1914 by Hans Blichfeldt. It states that any 
bounded planar region with POSITIVE AREA > A placed 
in any position of the UNIT SQUARE LATTICE can be 
Translated so that the number of Lattice Points 
inside the region will be at least A + 1. The theorem 
can be generalized to n-D. 

BLM/Ho Polynomial 

A 1-variable unoriented Knot Polynomial Q(x). It 
satisfies 

Qunknot = 1 (l) 

and the SKEIN RELATIONSHIP 

Ql^+Ql^ =x(Q Lq + Q Lqo ). (2) 

It also satisfies 

Qlx#l 2 = Ql y Ql 2 , (3) 

where # is the KNOT Sum and 

Ql*=Ql> (4) 

where L* is the Mirror Image of L. The BLM/Ho 
polynomials of Mutant KNOTS are also identical. 
Brandt et al. (1986) give a number of interesting prop- 
erties. For any Link L with > 2 components, Ql — 1 is 
divisible by 2 (x — 1). If L has c components, then the 
lowest POWER of x in Ql(x) is 1 — c, and 



lim x c 



lim (-m) c - 1 P L (£,m) J (5) 

(^m)-4(l,0) n V ' 

where P L is the HOMFLY Polynomial. Also, the de- 
gree of Ql is less than the Crossing Number of L. If 
L is a 2-Bridge Knot, then 

Q L (z) = 2z~' 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ), (6) 

where z = -t - r -1 (Kanenobu and Sumi 1993). 
The Polynomial was subsequently extended to the 2- 
variable Kauffman Polynomial F(a i z) y which satis- 
fies 

Q(x) = F{l,x). (7) 

Brandt et al. (1986) give a listing of Q POLYNOMIALS 
for KNOTS up to 8 crossings and links up to 6 crossings. 

References 

Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. "A 
Polynomial Invariant for Unoriented Knots and Links." In- 
vent Math. 84, 563-573, 1986. 

Ho, C. F. "A New Polynomial for Knots and Links — 
Preliminary Report." Abstracts Amer. Math. Soc. 6, 300, 
1985. 

Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- 
Bridge Knots through 22-Crossings." Math. Comput. 60, 
771-778 and S17-S28, 1993. 

Stoimenow, A. "Brandt-Lickorish-Millett-Ho Polynomi- 
als." http: //www, informatik.hu-berlin.de/-stoimeno/ 
ptab/blmhlO . html. 
^ Weisstein, E. W. "Knots." http: //www. astro. Virginia, 
edu/ - eww6n/math/not ebooks/Knot s . m. 



Block Design 

Bloch Constant 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

Let F be the set of Complex analytic functions / de- 
fined on an open region containing the closure of the 
unit disk D = {z : \z\ < 1} satisfying /(0) = and 
df/dz(Q) = 1. For each / in F, let b(f) be the SUPRE- 
MUM of all numbers r such that there is a disk S in D on 
which / is ONE-TO-ONE and such that f(S) contains a 
disk of radius r. In 1925, Bloch (Conway 1978) showed 
that b(f) > 1/72. Define Bloch's constant by 

B = mi{btf):f£F}. 

Ahlfors and Grunsky (1937) derived 
0.433012701...= \VZ<B 

i r(i)r(i§) 



4 
< 



v / nm r (?) 



f^- < 0.4718617. 



They also conjectured that the upper limit is actually 
the value of B, 



1 



r(j)r(M) 



v/TTv! r (i) 



iV 



= 0.4718617X 



/ 



V 



^ 



°4? 



(Le Lionnais 1983). 

see also Landau Constant 

References 

Conway, J. B. Functions of One Complex Variable, 2nd ed. 

New York: Springer- Verlag, 1989. 
Finch, S, "Favorite Mathematical Constants." http: //www. 

mathsof t . com/asolve/constant/bloch/bloch.html. 
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 

p. 25, 1983. 
Minda, CD. "Bloch Constants." J. d Analyse Math. 41, 

54-84, 1982. 

BIoch-Landau Constant 

see Landau Constant 

Block 

see also Block Design, Square Polyomino 

Block Design 

An incidence system (v, fc, A, r, 6) in which a set X 
of v points is partitioned into a family A of b subsets 
(blocks) in such a way that any two points determine A 
blocks, there are k points in each block, and each point 
is contained in r different blocks. It is also generally 
required that k < v , which is where the "incomplete" 
comes from in the formal term most often encountered 



Block Design 



Blow-Up 151 



for block designs, Balanced Incomplete Block De- 
signs (BIBD). The five parameters are not independent, 
but satisfy the two relations 



bk 



X(v~ 1) = r(fc-l). 



(1) 
(2) 



A BIBD is therefore commonly written as simply (v, &, 
A), since b and r are given in terms of u, k, and A by 



v(v - 1)A 
k(k - 1) 



(3) 
(4) 



A BIBD is called SYMMETRIC if b = v (or, equivalently, 
r = k). 

Writing X = {^}Li and A — {Aj} b j=1 , then the IN- 
CIDENCE Matrix of the BIBD is given by the v x b 
Matrix M defined by 



1J I othe 



GA 
otherwise. 



This matrix satisfies the equation 

MM T = (r-A)l + AJ, 



(5) 



(6) 



where I is a v x v IDENTITY MATRIX and J is a v x v 
matrix of Is (Dinitz and Stinson 1992). 

Examples of BIBDs are given in the following table. 



Block Design 



(v, K A) 



affine plane (n , n, 1) 

Fano plane (7, 3, 1)) 

Hadamard design symmetric (An + 3, 2n -f- 1, n) 

projective plane symmetric (n 2 + n -j- 1, n + 1, 1) 

Steiner triple system (v, 3, 1) 

unital (g 3 + 1, q+ 1, 1) 

see also Affine Plane, Design, Fano Plane, Hada- 
mard Design, Parallel Class, Projective Plane, 
Resolution, Resolvable, Steiner Triple System, 
Symmetric Block Design, Unital 

References 

Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to 
Design Theory." Ch. 1 in Contemporary Design Theory: A 
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). 
New York: Wiley, pp. 1-12, 1992. 

Ryser, H. J. "The {b,v,r, k, A)-Configuration." §8.1 in Com- 
binatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 
pp. 96-102, 1963. 



Block Growth 

Let (rco£i£2 • • •) be a sequence over a finite ALPHABET 
A (all the entries are elements of A). Define the block 
growth function B(n) of a sequence to be the number 
of Admissible words of length n. For example, in the 
sequence aabaabaabaabaab. . . , the following words are 
Admissible 

Length Admissible Words 



a, b 

aa, a&, ba 
aab, aba, baa 
aaba, abaa, baab 



so B(l) = 2, B(2) = 3, 5(3) = 3, B(4) = 3, and so 
on. Notice that B(n) < B(n + 1), so the block growth 
function is always nondecreasing. This is because any 
Admissible word of length n can be extended right- 
wards to produce an Admissible word of length n + 1. 
Moreover, suppose B(n) = B(n + 1) for some n. Then 
each admissible word of length n extends to a unique 

Admissible word of length n + 1. 

For a SEQUENCE in which each substring of length n 
uniquely determines the next symbol in the SEQUENCE, 
there are only finitely many strings of length n, so the 
process must eventually cycle and the SEQUENCE must 
be eventually periodic. This gives us the following the- 
orems: 

1. If the Sequence is eventually periodic, with least 
period p, then B(n) is strictly increasing until it 
reaches p, and B(n) is constant thereafter. 

2. If the Sequence is not eventually periodic, then 
B(n) is strictly increasing and so B(n) > n + 1 for all 
n. If a Sequence has the property that B(n) = n+1 
for all n, then it is said to have minimal block growth, 
and the Sequence is called a Sturmian Sequence. 

The block growth is also called the GROWTH FUNCTION 
or the Complexity of a Sequence. 

Block Matrix 

A square Diagonal Matrix in which the diagonal ele- 
ments are Square Matrices of any size (possibly even 
lxl), and the off-diagonal elements are 0. 

Block (Set) 

One of the disjoint Subsets making up a Set Parti- 
tion. A block containing n elements is called an n- 
block. The partitioning of sets into blocks can be de- 
noted using a RESTRICTED GROWTH STRING. 
see also Block Design, Restricted Growth 
String, Set Partition 

Blow-Up 

A common mechanism which generates SINGULARITIES 
from smooth initial conditions. 



152 Blue-Empty Coloring 



Bohemian Dome 



Blue-Empty Coloring 

see Blue-Empty Graph 

Blue-Empty Graph 

An Extremal Graph in which the forced Trian- 
gles are all the same color. Call R the number of 
red Monochromatic Forced Triangles and B the 
number of blue Monochromatic Forced Triangles, 
then a blue-empty graph is an Extremal Graph with 
B = 0. For Even n, a blue-empty graph can be 
achieved by coloring red two Complete SUBGRAPHS 
of n/2 points (the RED Net method). There is no blue- 
empty coloring for Odd n except for n = 7 (Lorden 
1962). 

see also Complete Graph, Extremal Graph, 
Monochromatic Forced Triangle, Red Net 

References 

Lorden, G. "Blue-Empty Chromatic Graphs." Amer. Math. 

Monthly 69, 114-120, 1962. 
Sauve, L. "On Chromatic Graphs." Amer. Math. Monthly 

68, 107-111, 1961. 

Board 

A subset of d x d, where d = {1, 2, . . . , d}. 

see also Rook Number 

Boatman's Knot 

see Clove Hitch 

Bochner Identity 

For a smooth Harmonic Map u : M -► TV, 

A(|Vu| 2 ) = \V(du)\ 2 + {RicM Vu,Vu) 

- (Riem N (u)(Vu, Vu)Vu, Vu> , 

where V is the GRADIENT, Ric is the RlCCl TENSOR, 
and Riem is the Riemann Tensor. 

References 

Eels, J. and Lemaire, L. "A Report on Harmonic Maps." 
Bull. London Math. Soc. 10, 1-68, 1978. 

Bochner's Theorem 

Among the continuous functions on R n , the POSITIVE 
Definite Functions are those functions which are the 
Fourier Transforms of finite measures. 

Bode's Rule 



J XI 



f{x) dx = ^/i(7/i + 32/ 2 + 12/ 3 + 32/ 4 + 7/5) 

-sfeW'K). 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 886, 1972. 

Bogdanov Map 

A 2-D MAP which is conjugate to the Henon Map in 
its nondissipative limit. It is given by 

x = x + y 

y' =y + ey + kx(x - l) + [ixy. 

see also Henon Map 

References 

Arrowsmith, D. K.; Cartwright, J. H. E.; Lansbury, A. N.; 
and Place, C. M. "The Bogdanov Map: Bifurcations, Mode 
Locking, and Chaos in a Dissipative System." Int. J. Bi- 
furcation Chaos 3, 803-842, 1993. 

Bogdanov, R. "Bifurcations of a Limit Cycle for a Family 
of Vector Fields on the Plane." Selecta Math. Soviet 1, 
373-388, 1981. 

Bogomolov-Miyaoka-Yau Inequality 

Relates invariants of a curve defined over the INTEGERS. 
If this inequality were proven true, then FERMAT'S Last 
THEOREM would follow for sufficiently large exponents. 
Miyaoka claimed to have proven this inequality in 1988, 
but the proof contained an error. 
see also FERMAT'S LAST THEOREM 

References 

Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. 
Math. Monthly 101, 3-14, 1994. 

Bohemian Dome 




see also Hardy's Rule, Newton-Cotes Formulas, 
Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal 
Rule, Weddle's Rule 



A Quartic Surface which can be constructed as fol- 
lows. Given a CIRCLE C and PLANE E PERPENDICULAR 
to the Plane of C, move a second Circle K of the 
same Radius as C through space so that its Center 
always lies on C and it remains PARALLEL to E. Then 
K sweeps out the Bohemian dome. It can be given by 
the parametric equations 

x = a cos u 

y = b cos v + a sin u 

z — csinv 

where u, v 6 [0, 27r). In the above plot, a = 0.5, b = 1.5, 

and c = 1. 

see also Quartic Surface 



Bohr-Favard Inequalities 



Bombieri Norm 



153 



References 

Fischer, G. (Ed.). Mathematical Models from the Collections 

of Universities and Museums. Braunschweig, Germany: 

Vieweg, pp. 19-20, 1986. 
Fischer, G. (Ed.). Plate 50 in Mathematische Mod- 

elle/ Mathematical Models, Bildband/ Photograph Volume. 

Braunschweig, Germany: Vieweg, p. 50, 1986. 
Nordstrand, T. "Bohemian Dome." http://www.uib.no/ 

people/nf ytn/bodtxt .htm. 

Bohr-Favard Inequalities 

If / has no spectrum in [—A, A], then 



saii'i 



(Bohr 1935). A related inequality states that if Ak is 
the class of functions such that 

/(*) = /(* + 2*), /(*),/'(*),... ./^(a:) 

are absolutely continuous and f w f(x) dx = 0, then 

4 



1 _1)^ 
S ^2^ ( 2 ^+ l)M-i N/ wi 



(Northcott 1939). Further, for each value of k, there is 
always a function f(x) belonging to Ak and not identi- 
cally zero, for which the above inequality becomes an in- 
equality (Favard 1936). These inequalities are discussed 
in Mitrinovic et al. (1991). 

References 

Bohr, H. "Ein allgemeiner Satz iiber die Integration eines 
trigonometrischen Polynoms." Prace Matem.-Fiz. 43, 
1935. 

Favard, J. "Application de la formule soiiimaloire d'Euler 
a la demonstration de quelques proprietes extremales des 
integrale des fonctions periodiques ou presqueperiodiqu.es." 
Mat Tidsskr. B, 81-94, 1936. [Reviewed in Zentralblatt f. 
Math. 16, 58-59, 1939.] 

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities 
Involving Functions and Their Integrals and Derivatives. 
Dordrecht, Netherlands: Kluwer, pp. 71-72, 1991. 

Northcott, D. G. "Some Inequalities Between Periodic Func- 
tions and Their Derivatives." J. London Math. Soc. 14, 
198-202, 1939. 

Tikhomirov, V. M. "Approximation Theory." In Analysis 
II (Ed. R. V. Gamrelidze). New York: Springer- Verlag, 
pp." 93-255, 1990. 

Bolyai-Gerwein Theorem 

see Wallace-Bolyai-Gerwein Theorem 

Bolza Problem 

Given the functional 

U= /(2/l,---,S/n;3/l\...,2/n') d * 

Jt 

+G(yi , . . . , 2Mr; 2/11, • ■ ■ , 2/m), 



find in a class of arcs satisfying p differential and q finite 
equations 

<M3/i,-..,2/n;3/i',...,3/n') = ° for a = l,...,p 
VV3(yi»--->yn) = for = l,...,g 
as well as the r equations on the endpoints 
X7(yio,---)2/nr;3/ii,...,2/ni) = for 7 = 1, . . . , r, 

one which renders U a minimum. 

References 

Goldstine, H. H, A History of the Calculus of Variations from 
the 17th through the 19th Century. New York: Springer- 
Verlag, p. 374, 1980. 

Bolzano Theorem 

see Bolzano- WeierstraB Theorem 

Bolzano- Weierstrafl Theorem 

Every Bounded infinite set in W 1, has an ACCUMULA- 
TION Point. For n = 1, the theorem can be stated as 
follows: If a Set in a METRIC SPACE, finite-dimensional 
Euclidean Space, or First-Countable Space has 
infinitely many members within a finite interval x 6 
[a, 6], then it has at least one Limit Point x such that 
x e [a, &]. The theorem can be used to prove the Inter- 
mediate Value Theorem. 

Bombieri's Inequality 

For Homogeneous Polynomials P and Q of degree 
m and n, then 



[P ■ Qh > 



tM. 



(m + n)\ 



-jiPhlQb, 



where [P • Q] 2 is the BOMBIERI Norm. If m = n, this 
becomes 

[P'Qh>[P]2[Q]2. 

see also Beauzamy and Degot's Identity, Reznik's 
Identity 

Bombieri Inner Product 

For Homogeneous Polynomials P and Q of degree 
n, 

[P,Q]= J2 C*! 1 "-^ 1 )^,..^^!,..^)- 
ii,...,i„>0 



Bombieri Norm 

For Homogeneous Polynomials P of degree m, 



mV 



[P], = y/frF]=\ J2 S |a ' 
y |a|=m 

see also POLYNOMIAL BAR NORM 



154 BombievVs Theorem 



Bonne Projection 



Bombieri's Theorem 

Define 



E(x;q,a) = ip(x\q,a) - 



<KqV 



where 



■tP(x;q,a)= ^ A(n) 



(1) 
(2) 



n<x 
n = a (mod g) 



(Davenport 1980, p. 121), A(n) is the MANGOLDT 
Function, and <j>(q) is the Totient Function. Now 
define 

E(x;q)= max \E(x\q y a)\ (3) 

(a,q°) = l 

where the sum is over a RELATIVELY PRIME to q, 
(a,g) = 1, and 

E*(x,q) = ma,xE{y,q). (4) 

y<x 

Bombieri's theorem then says that for A > fixed, 

^E*(x,q) « ^Q{\nx)\ (5) 

q<Q 

provided that ^(lnx)" 4 < Q < \fx. 

References 

Bombieri, E. "On the Large Sieve." Mathematika 12, 201- 
225, 1965. 

Davenport, H. "Bombieri's Theorem." Ch. 28 in Multiplica- 
tive Number Theory, 2nd ed. New York: Springer- Verlag, 
pp. 161-168, 1980. 

Bond Percolation 




bond percolation site percolation 

A Percolation which considers the lattice edges as the 
relevant entities (left figure). 
see also Percolation Theory, Site Percolation 

Bonferroni Correction 

The Bonferroni correction is a multiple-comparison cor- 
rection used when several independent STATISTICAL 
TESTS are being performed simultaneously (since while 
a given Alpha Value a may be appropriate for each 
individual comparison, it is not for the set of all com- 
parisons). In order to avoid a lot of spurious positives, 
the Alpha Value needs to be lowered to account for 
the number of comparisons being performed. 

The simplest and most conservative approach is the 
Bonferroni correction, which sets the ALPHA VALUE for 
the entire set of n comparisons equal to a by taking the 



Alpha Value for each comparison equal to cx/n. Ex- 
plicitly, given n tests Ti for hypotheses Hi (1 < i < n) 
under the assumption Ho that all hypotheses Hi are 
false, and if the individual test critical values are < a/n, 
then the experiment-wide critical value is < a. In equa- 
tion form, if 

P(Ti passes \H Q ) < - 
n 

for 1 < i < ra, then 

P(some Ti passes \H ) < a, 

which follows from BONFERRONl'S INEQUALITY. 

Another correction instead uses 1 — (1— a) 1 / 71 . While this 
choice is applicable for two-sided hypotheses, multivari- 
ate normal statistics, and positive orthant dependent 
statistics, it is not, in general, correct (Shaffer 1995). 
see also ALPHA VALUE, HYPOTHESIS TESTING, STATIS- 
TICAL Test 

References 

Bonferroni, C. E. "II calcolo delle assicurazioni su gruppi di 
teste." In Studi in Onore del Professore Salvatore Ortu 
Carboni. Rome: Italy, pp. 13-60, 1935. 

Bonferroni, C. E. "Teoria statistica delle classi e calcolo delle 
probabilita." Pubblicazioni del R Istituto Superiore di 
Scienze Economiche e Commerciali di Firenze 8, 3-62, 
1936. 

Dewey, M. "Carlo Emilio Bonferroni: Life and Works." 
http://www.nottingham.ac.uk/-mh2md/life.html. 

Miller, R. G. Jr. Simultaneous Statistical Inference. New 
York: Springer- Verlag, 1991. 

Perneger, T. V. "What's Wrong with Bonferroni Adjust- 
ments." Brit Med. J. 316, 1236-1238, 1998. 

Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev. 
Psych. 46, 561-584, 1995. 

Bonferroni's Inequality 

Let P(Ei) be the probability that £?» is true, and 
P(U" =1 ^i) be the probability that E u E 2j ..., E n 
are all true. Then 



fU 1 *") *!><*>• 



Bonferroni Test 

see Bonferroni Correction 

Bonne Projection 




Book Stacking Problem 



Boolean Algebra 155 



A Map Projection which resembles the shape of a 
heart. Let (pi be the standard parallel and Ao the central 
meridian. Then 



where 



x = p sin E 

y — R cot 0i — p cos R ) 



p = cot (pi + 0i — <(> 

(A- Aq)cos0 



The inverse FORMULAS are 
<p = cot 01 + (f>i — p 



A = A + 



COS0 



■ tan 



-l ( x 

\ cot (pi -y 



where 



p = ±\/x 2 + (cot 0i -y) 2 . 



(1) 
(2) 



(3) 
(4) 

(5) 
(6) 

(7) 



References 

Snyder, J. P. Map Projections — A Working Manual. U. S. 
Geological Survey Professional Paper 1395. Washington, 
DC: U. S. Government Printing Office, pp. 138-140, 1987. 



(Sloane's A001008 and A002805). 

In order to find the number of stacked books required to 
obtain d book-lengths of overhang, solve the d n equation 
for d, and take the Ceiling Function. For n = 1, 2, . . . 
book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 
675214, 4989191, 36865412, 272400600, ... (Sloane's 
A014537) books are needed. 

References 

Dickau, R. M. "The Book-Stacking Problem." http://wwv. 
prairienet.org/-pops/BookStacking.html. 

Eisner, L. "Leaning Tower of the Physical Review." Amer. 
J. Phys. 27, 121, 1959. 

Gardner, M. Martin Gardner's Sixth Book of Mathematical 
Games from Scientific American. New York: Scribner's, 
p. 167, 1971. 

Graham, R. L.; Knuth, D, E.; and Patashnik, O. Concrete 
Mathematics: A Foundation for Computer Science. Read- 
ing, MA: Addison- Wesley, pp. 272-274, 1990. 

Johnson, P. B. "Leaning Tower of Lire." Amer. J. Phys. 23, 
240, 1955. 

Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 1, 322, 1953. 

Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 2, 411, 1954. 

Sloane, N. J. A. Sequences A014537, A001008/M2885, and 
A002805/M1589 in "An On-Line Version of the Encyclo- 
pedia of Integer Sequences." 

Boole's Inequality 



Book Stacking Problem 




How far can a stack of n books protrude over the edge 
of a table without the stack falling over? It turns out 
that the maximum overhang possible d n for n books (in 
terms of book lengths) is half the nth partial sum of the 
Harmonic Series, given explicitly by 



d n 



n 



where <&(z) is the DiGAMMA FUNCTION and 7 is the 
Euler-Mascheroni Constant. The first few values 
are 



di = -= 0.5 



3 

4 
— 11 



0.75 



d 3 = i| « 0.91667 



A — 25 
rf 4 - 24 



p U £ 0^E p ^)- 



1.04167, 



If Ei and Ej are Mutually Exclusive for all i and j, 
then the INEQUALITY becomes an equality. 

Boolean Algebra 

A mathematical object which is similar to a BOOLEAN 
RING, but which uses the meet and join operators in- 
stead of the usual addition and multiplication operators. 
A Boolean algebra is a set B of elements a, 6, ... with 
Binary Operators + and * such that 

la. If a and b are in the set S, then a + b is in the set 
B. 

lb. If a and b are in the set B, then a • b is in the set 
B. 

2a. There is an element Z (zero) such that a + Z = a 
for every element a. 

2b. There is an element U (unity) such that a • U = a 

for every element a. 
3a. a + 6 = b + a 
3b. a - b = b ■ a 
4a. a + 6 ■ c = (a + b) (a + c) 
4b. a ■ (b-\- c) — a - b-\- a ■ c 

5. For every element a there is an element a such that 
a + a' — U and a ■ a' = Z. 

6. There are are least two distinct elements in the set 
B. 

(Bell 1937, p. 444). 



156 Boolean Algebra 



Boolean Ring 



In more modern terms, a Boolean algebra is a Set B of 
elements a, 6, ... with the following properties: 

1. B has two binary operations, A (Wedge) and V 
(Vee), which satisfy the IDEMPOTENT laws 

aAa = a\/a = a, 

the Commutative laws 

a A b — b A a 

aVb^bV a, 
and the Associative laws 

a A (b A c) = (a A b) A c 

aV(6Vc) = (aVb) V c. 

2. The operations satisfy the ABSORPTION LAW 

a A (a V b) = a V (a A 6) = a. 

3. The operations are mutually distributive 

a A (6Vc) = (a A 6) V (a Ac) 

a V (6 A c) = (a V 6) A (a V c). 

4. I? contains universal bounds 0,/ which satisfy 

OAa = 

O Va = a 
/ A a = a 

/Vfl = J. 

5. B has a unary operation a —± a' of complementation 
which obeys the laws 

a A a = O 

aV a = I 

(Birkhoff and Mac Lane 1965). Under intersection, 
union, and complement, the subsets of any set I form a 
Boolean algebra. 

Huntington (1933a, b) presented the following basis for 
Boolean algebra, 

1. Commutivity. x + y = y + x. 

2. Associativity, (x + y) + z = x + (y + z). 

3. Huntington Equation. n(n(x) + y) + n(n(a;) + 
n(y)) = x. 

H. Robbins then conjectured that the Huntington 
Equation could be replaced with the simpler Robbins 
Equation, 

n(n(x + y) + n(x + n(j/))) = x. 



The Algebra defined by commutivity, associativity, 
and the Robbins EQUATION is called ROBBINS ALGE- 
BRA. Computer theorem proving demonstrated that ev- 
ery Robbins Algebra satisfies the second Winkler 
Condition, from which it follows immediately that all 
Robbins Algebras are Boolean. 

References 

Bell, E. T. Men of Mathematics. New York: Simon and 

Schuster, 1986. 
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 

3rd ed. New York: Macmillian, p. 317, 1965. 
Halmos, P. Lectures on Boolean Algebras. Princeton, NJ: 

Van Nostrand, 1963. 
Huntington, E. V. "New Sets of Independent Postulates for 

the Algebra of Logic." Trans. Amer. Math. Soc. 35, 274- 

304, 1933a. 
Huntington, E. V. "Boolean Algebras: A Correction." Trans. 

Amer. Math. Soc. 35, 557-558, 1933. 
McCune, W. "Robbins Algebras are Boolean." http://www. 

mcs.anl.gov/-mccune/papers/robbins/. 

Boolean Connective 

One of the Logic operators And A, Or V, and Not ->. 

see also QUANTIFIER 

Boolean Function 

A Boolean function in n variables is a function 

J\Xi , . . . , x n J, 

where each Xi can be or 1 and / is or 1. Determining 
the number of monotone Boolean functions of n vari- 
ables is known as Dedekind'S Problem. The number 
of monotonic increasing Boolean functions of n variables 
is given by 2, 3, 6, 20, 168, 7581, 7828354, . . . (Sloane's 
A000372, Beeler et al. 1972, Item 17). The number of 
inequivalent monotone Boolean functions of n variables 
is given by 2, 3, 5, 10, 30, . . . (Sloane's A003182). 

Let M(n, k) denote the number of distinct monotone 
Boolean functions of n variables with k mincuts. Then 

M(n,0) = 1 

M(n,l)-2 n 

M(n, 2) = 2 n " 1 (2 n - 1) - 3 n + 2 n 

M(n,3) = |(2 n )(2 n - l)(2 n - 2) - 6 n + 5" + 4 n - 3 n . 



References 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM-239, Feb. 1972. 

Sloane, N, J. A. Sequences A003182/M0729 and A000372/ 
M0817 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Boolean Ring 

A Ring with a unit element in which every element is 

IDEMPOTENT. 

see also BOOLEAN ALGEBRA 



Borchardt-Pfaff Algorithm 



Borel Probability Measure 157 



Borchardt-Pfaff Algorithm 

see Archimedes Algorithm 

Border Square 



40 


1 


2 


3 


42 


41 


46 


38 


31 


13 


14 


32 


35 


12 


39 


30 


26 


21 


28 


20 


11 


43 


33 


27 


25 


23 


17 


7 


6 


16 


22 


29 


24 


34 


44 


5 


15 


37 


36 


18 


19 


45 


4 


49 


48 


47 


8 


9 


10 



31 


13 


14 


32 


35 


30 


26 


21 


28 


20 


33 


27 


25 


23 


17 


16 


22 


29 


24 


34 


15 


37 


36 


18 


19 



26 


21 


28 


27 


25 


23 


22 


29 


24 



A MAGIC SQUARE that remains magic when its bor- 
der is removed. A nested magic square remains magic 
after the border is successively removed one ring at a 
time. An example of a nested magic square is the order 
7 square illustrated above (i.e., the order 7, 5, and 3 
squares obtained from it are all magic). 

see also MAGIC SQUARE 

References 

Kraitchik, M. "Border Squares." §7.7 in Mathematical Recre- 
ations. New York: W. W. Norton, pp. 167-170, 1942. 

Bordism 

A relation between Compact boundaryless Manifolds 
(also called closed Manifolds). Two closed Mani- 
folds are bordant IFF their disjoint union is the bound- 
ary of a compact (n+l)-MANlFOLD. Roughly, two Man- 
ifolds are bordant if together they form the boundary 
of a Manifold. The word bordism is now used in place 
of the original term COBORDISM. 

References 

Budney, R. "The Bordism Project." http: //math. Cornell. 
eduArybu/bordism/bordism.html. 

Bordism Group 

There are bordism groups, also called Cobordism 
Groups or Cobordism Rings, and there are singu- 
lar bordism groups. The bordism groups give a frame- 
work for getting a grip on the question, "When is a 
compact boundaryless MANIFOLD the boundary of an- 
other Manifold?" The answer is, precisely when all of 
its Stiefel- Whitney Classes are zero. Singular bor- 
dism groups give insight into STEENROD's REALIZATION 
PROBLEM: "When can homology classes be realized as 
the image of fundamental classes of manifolds?" That 
answer is known, too. 

The machinery of the bordism group winds up being 
important for HOMOTOPY THEORY as well. 

References 

Budney, R. "The Bordism Project." http: //math. Cornell. 
edu/-rybu/bordism/bordism.html. 



Borel-Cantelli Lemma 

Let {^4.n}£Lo De a Sequence of events occurring with a 
certain probability distribution, and let A be the event 
consisting of the occurrence of a finite number of events 
A ni n = 1, Then if 



then 



^2p(A n ) < oo, 



P(A) = 1. 



References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet u Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, pp. 435-436, 1988. 

Borel Determinacy Theorem 

Let T be a tree defined on a metric over a set of paths 
such that the distance between paths p and q is 1/n, 
where n is the number of nodes shared by p and q. Let 
A be a Borel set of paths in the topology induced by this 
metric. Suppose two players play a game by choosing a 
path down the tree, so that they alternate and each time 
choose an immediate successor of the previously chosen 
point. The first player wins if the chosen path is in A, 
Then one of the players has a winning STRATEGY in this 
Game. 

see also Game Theory, Strategy 

BorePs Expansion 

Let <p(t) = Xl^lo ^nt 71 ^ e any function for which the 
integral 

/>oo 

I(x) = / e- tx t v 4>{t) dt 
Jo 

converges. Then the expansion 



I(x) 



XP+ 



■^l[Ao + ( P + iy- 



+ (p+l)(p + 2)^ + ... 



where F(z) is the Gamma Function, is usually an 
Asymptotic Series for I(x). 

Borel Measure 

If F is the Borel Sigma Algebra on some Topolog- 
ical Space, then a Measure m : F -+ R is said to be 
a Borel measure (or BOREL PROBABILITY MEASURE). 
For a Borel measure, all continuous functions are MEA- 
SURABLE. 



Borel Probability Measure 

see BOREL MEASURE 



158 



Borel Set 



Borwein Conjectures 



Borel Set 

A Definable Set derived from the Real Line by re- 
moving a Finite number of intervals. Borel sets are 
measurable and constitute a special type of Sigma Al- 
gebra called a BOREL SIGMA ALGEBRA. 

see also Standard Space 

Borel Sigma Algebra 

A Sigma Algebra which is related to the Topology 
of a Set, The Borel sigma-algebxa is defined to be 
the Sigma Algebra generated by the Open Sets (or 
equivalently, by the CLOSED Sets). 

see also Borel MEASURE 

Borel Space 

A Set equipped with a Sigma Algebra of Subsets. 

Borromean Rings 




Three mutually interlocked rings named after the Italian 
Renaissance family who used them on their coat of arms. 
No two rings are linked, so if one of the rings is cut, all 
three rings fall apart. They are given the Link symbol 
O603, and are also called the Ballantine. The Bor- 
romean rings have BRAID WORD c^ -1 o- 2 o'i~ 1 <J 2 ai _1 &2 
and are also the simplest Brunnian Link. 

References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., pp. 58-59, 1989. 

Gardner, M. The Unexpected Hanging and Other Mathemat- 
ical Diversions. Chicago, IL: University of Chicago Press, 
1991. 

Jablan, S. "Borromean Triangles." http:/ /members. tripod, 
com/ -modularity/links .htm. 

Pappas, T. "Trinity of Rings— A Topological Model." The 
Joy of Mathematics. San Carlos, CA: Wide World Publ./ 
Tetra, p. 31, 1989. 



Borrow 



1 2 3 
-78 




borrows 



4 4 5 

The procedure used in SUBTRACTION to "borrow" 10 
from the next higher Digit column in order to obtain a 
Positive Difference in the column in question. 

see also Carry 



Borsuk's Conjecture 

Borsuk conjectured that it is possible to cut an n-D 
shape of DIAMETER 1 into n + 1 pieces each with di- 
ameter smaller than the original. It is true for n = 2, 
3 and when the boundary is "smooth." However, the 
minimum number of pieces required has been shown to 
increase as ~ 1.1^. Since 1.1^ > n + 1 at n = 9162, 
the conjecture becomes false at high dimensions. In fact, 
the limit has been pushed back to ~ 2000. 

see also DIAMETER (GENERAL), KELLER'S CONJEC- 
TURE, Lebesgue Minimal Problem 

References 

Borsuk, K. "Uber die Zerlegung einer Euklidischen n- 

dimensionalen Vollkugel in n Mengen." Verh. Internat. 

Math.-Kongr. Zurich 2, 192, 1932. 
Borsuk, K. "Drei Satze iiber die n-dimensionale euklidische 

Sphare." Fund. Math. 20, 177-190, 1933. 
Cipra, B. "If You Can't See It, Don't Believe It. . . ." Science 

259, 26-27, 1993. 
Cipra, B. What's Happening in the Mathematical Sciences, 

Vol. 1. Providence, RI: Amer. Math, Soc, pp. 21-25, 1993, 
Grunbaum, B. "Borsuk's Problem and Related Questions." 

In Convexity, Proceedings of the Seventh Symposium in 

Pure Mathematics of the American Mathematical Society, 

Held at the University of Washington, Seattle, June 13- 

15, 1961. Providence, RI: Amer. Math. Soc, pp. 271-284, 

1963. 
Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture." 

Bull. Amer. Math. Soc. 329, 60-62, 1993. Listernik, L. 

and Schnirelmann, L. Topological Methods in Variational 

Problems. Moscow, 1930. 

Borwein Conjectures 

Use the definition of the q- Series 



{a\q)n = JJ(l-ag') 



j=o 



and define 



N 
M 



[q \q)m 

(Q\Q)m 



(1) 



(2) 



Then P. Borwein has conjectured that (1) the Polyno- 
mials A n (q), B n (q), and C n (q) defined by 

(q\ </ 3 W<Z 2 ; qX = A n (q 3 ) - qB n (q 3 ) - q 2 C n (q 3 ) (3) 

have NONNEGATIVE COEFFICIENTS, (2) the POLYNOMI- 
ALS A* n {q), B*{q), and C*(q) defined by 

(q;qX(q 2 ;q 3 )l = A:(q S )- q B:(q S )~ q 2 C:(q 3 ) (4) 

have Nonnegative Coefficients, (3) the Polynomi- 
als A* n {q), B*{q), C*(q), D*(q), and E* n (q) defined by 



(9; 5 )n(q ;q )n(q ;q)n(q;q)n- 
AUq 5 )-qB* n {f , )-q 2 C* n {qS)-q 3 Dl{q 5 )-q 4 EUq 5 ) (5) 



Bouligand Dimension 



Boundary Point 159 



have NONNEGATIVE COEFFICIENTS, (4) the POLYNOMI- 
ALS Al l (m i n,t,q) 1 £* (m,n,£, g), and C^m^n^t^q) de- 
fined by- 



Bound Variable 

An occurrence of a variable in a LOGIC which is not 
Free. 



(?; q 3 )m(q 2 ; q Z )m{zq\ q 3 ) n {zq 2 ; q 3 ) n 

2m 

= > z [A* (m,n, £, q ) — qB* (m, n,t,q ) 

t=Q 

-q 2 C\m,n,t,q 3 )} (6) 

have Nonnegative Coefficients, (5) for k Odd and 
1 < a < k/2, consider the expansion 



(q a ;q k U(q k - a ;q k )n 



(fc-D/2 

E 

t/=(l-fc)/2 



(_ 1 )^M- 2 +-)/2-a,^ ( ^ ) (7) 



with 

oo 
_ V^ f-lY 3(k 2 j + 2ku + k-2a)/2 



m 4- n 
m + v + kj 



(8) 



then if a is Relatively Prime to k and m = n, the CO- 
EFFICIENTS of F^qr) are NONNEGATIVE, and (6) given 
a J rf3< 2'K and — K + /? < n — m < K — a, consider 

G(a,0,K;q) = ^(_i)V 1JC(a+w+lf(a+/9)1/a 



ra + n 

171+ Kj 



, (9) 



the Generating Function for partitions inside an mx 
n rectangle with hook difference conditions specified by 
a, /?, and if. Let a and /? be POSITIVE RATIONAL 
Numbers and K > 1 an Integer such that aK and 
/3Jf are integers. Then if 1 < a + < 2K-1 (with strict 
inequalities for K = 2) and —if + /3<n — m < K — a, 
then G(a,j3,K;q) has NONNEGATIVE COEFFICIENTS, 
see ateo ^-SERIES 

References 

Andrews, G. E. ei al. "Partitions with Prescribed Hook Dif- 
ferences." Europ. J. Combin. 8, 341-350, 1987. 

Bressoud, D. M. "The Borwein Conjecture and Partitions 
with Prescribed Hook Differences. " Electronic J. Com- 
binatorics 3, No. 2, R4, 1-14, 1996. http://www. 
combinatorics. org/Volume^3/volume3_2.html#R4. 

Bouligand Dimension 

see MlNKOWSKI-BOULIGAND DIMENSION 

Bound 

see Greatest Lower Bound, Infimum, Least Up- 
per Bound, Supremum 



Boundary 

The set of points, known as Boundary Points, which 
are members of the CLOSURE of a given set 5 and the 
CLOSURE of its complement set. The boundary is some- 
times called the FRONTIER. 
see also SURGERY 

Boundary Conditions 

There are several types of boundary conditions com- 
monly encountered in the solution of PARTIAL DIFFER- 
ENTIAL Equations. 

1. Dirichlet Boundary Conditions specify the 
value of the function on a surface T = /(r,£). 

2. Neumann Boundary Conditions specify the nor- 
mal derivative of the function on a surface, 



dT 
dn 



_=fi-vr = /(r, y ). 



3. Cauchy Boundary Conditions specify a weighted 
average of first and second kinds. 

4. Robin Boundary Conditions. For an elliptic par- 
tial differential equation in a region Q, Robin bound- 
ary conditions specify the sum of au and the normal 
derivative of u = / at all points of the boundary of 
Q } with a and / being prescribed. 

see also BOUNDARY VALUE PROBLEM, DlRICHLET 
Boundary Conditions, Initial Value Problem, 
Neumann Boundary Conditions, Partial Differ- 
ential Equation, Robin Boundary Conditions 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 502-504, 1985. 

Morse, P. M. and Feshbach, H. "Boundary Conditions and 
Eigcnfunctions." Ch. 6 in Methods of Theoretical Physics, 
Part L New York: McGraw-Hill, pp. 495-498 and 676-790, 
1953. 

Boundary Map 

The Map H n {X, A) -► H n - 1 (A) appearing in the Long 
Exact Sequence of a Pair Axiom. 

see also Long Exact Sequence of a Pair Axiom 

Boundary Point 

A point which is a member of the Closure of a given 
set S and the CLOSURE of its complement set. If A is a 
subset of M n , then a point x € M. n is a boundary point 
of A if every NEIGHBORHOOD of x contains at least one 
point in A and at least one point not in A. 
see also BOUNDARY 



160 



Boundary Set 



Boustrophedon Transform 



Boundary Set 

A (symmetrical) boundary set of RADIUS r and center 
xq is the set of all points x such that 



Bourget Function 



x- x = r. 



Let xo be the ORIGIN. In IR , the boundary set is then 

\ the 



-r. In 



the pair of points x — r and x 
boundary set is a CIRCLE. In R 
is a Sphere. 

see also Circle, Disk, Open Set, Sphere 



the boundary set 



Boundary Value Problem 

A boundary value problem is a problem, typically an 
Ordinary Differential Equation or a Partial 
Differential Equation, which has values assigned 
on the physical boundary of the Domain in which the 
problem is specified. For example, 



u(O t t) 



V 2 u = f 



m 



f*(0,t)=u 2 



in Q 
on dQ 
on dQ, 



where dCl denotes the boundary of O, is a boundary 

problem. 

see also Boundary Conditions, Initial Value 

Problem 

References 

Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Compu- 
tational Differential Equations. Lund: Studentlitteratur, 
1996. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Two Point Boundary Value Problems." Ch. 17 
in Numerical Recipes in FORTRAN: The Art of Scientific 
Computing, 2nd ed. Cambridge, England: Cambridge Uni- 
versity Press, pp. 745-778, 1992. 

Bounded 

A Set in a Metric Space (X,d) is bounded if it has 
a FINITE diameter, i.e., there is an R < oo such that 
d(#, y) < R for all x, y € X. A Set in W 1 is bounded if 
it is contained inside some Ball x\ 2 + . . . + x n 2 < R 2 
of Finite Radius R (Adams 1994). 
see also Bound, Finite 

References 

Adams, R. A. Calculus: A Complete Course, Reading, MA: 
Addison- Wesley, p. 707, 1994. 

Bounded Variation 

A Function f(x) is said to have bounded variation if, 
over the Closed Interval x e [a, b], there exists an M 
such that 

\f(xi)-f(a)\ + \f(x2)-f(x 1 )\ + . . .+ |/(6)-/(x„_i)| < M 

for all a < xi < X2 < ■ . . < x n -i < b. 



J -*u-hf t ""( ,+ \Y"*[i'('-\)]* 



* Jo 



(2 cos d) k cos(n(9 - z sin 6) d0. 



see also Bessel Function of the First Kind 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- 
lands: Reidel, p. 465, 1988. 

Bourget's Hypothesis 

When n is an INTEGER > 0, then J n (z) and J n +m(z) 
have no common zeros other than at z = for m an 
Integer > 1, where J n (z) is a Bessel Function of 
THE First Kind. The theorem has been proved true 
for m=l 2, 3, and 4. 

References 

Watson, G. N. A Treatise on the Theory of Bessel Functions, 
2nd ed. Cambridge, England: Cambridge University Press, 
1966. 

Boustrophedon Transform 

The boustrophedon ( "ox-plowing" ) transform b of a se- 
quence a is given by 



bn = 7 7 \dkEn-k 

k=o v / 

— ±(-')-(0 

fc=0 x ' 



bkEn~k 



(1) 

(2) 



for n > 0, where E n is a Secant Number or Tangent 
Number defined by 



Ex n 
E n — 7 = sec X + 



tanz. 



(3) 



The exponential generating functions of a and b are 
related by 

B(x) = (sec a? + tanz)^4(#), (4) 

where the exponential generating function is defined by 



A(x) = Y,An 



x 



(5) 



see also ALTERNATING PERMUTATION, ENTRINGER 

Number, Secant Number, Seidel-Entringer- 
Arnold Triangle, Tangent Number 

References 

Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op- 
eration on Sequences: The Boustrophedon Transform." J. 
Combin. Th. Ser. A 76, 44-54, 1996. 



Bovinum Problema 



Box Fractal 161 



Bovinum Problema 

see Archimedes' Cattle Problem 

Bow 




4 2 3 

x = x y — y . 



References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., p. 72, 1989. 

Bowditch Curve 

see Lissajous Curve 

Bowley Index 

The statistical Index 

where P L is Laspeyres' Index and P P is Paasche's 
Index. 

see also INDEX 

References 

Kenney, J. F, and Keeping, E. S. Mathematics of Statistics, 
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962. 

Bowley Skewness 

Also known as Quartile Skewness Coefficient, 

(Q 3 - Q 2 ) - (<?2 - <2i) _ Qi - 2Q 2 + Q 3 



Qz-Qi) 



Qz-Qi 



where the Qs denote the Interquartile Ranges. 
see also SKEWNESS 

Bowling 

Bowling is a game played by rolling a heavy ball down 
a long narrow track and attempting to knock down ten 
pins arranged in the form of a TRIANGLE with its vertex 
oriented towards the bowler. The number 10 is, in fact, 
the Triangular Number T 4 = 4(4 4- l)/2 = 10. 

Two "bowls" are allowed per "frame." If all the pins are 
knocked down in the two bowls, the score for that frame 
is the number of pins knocked down. If some or none of 
the pins are knocked down on the first bowl, then all the 
pins knocked down on the second, it is called a "spare," 
and the number of points tallied is 10 plus the number 
of pins knocked down on the bowl of the next frame. 
If all of the pins are knocked down on the first bowl, 
the number of points tallied is 10 plus the number of 



pins knocked down on the next two bowls. Ten frames 
are bowled, unless the last frame is a strike or spare, in 
which case an additional bowl is awarded. 

The maximum number of points possible, corresponding 
to knocking down all 10 pins on every bowl, is 300. 

References 

Cooper, C N. and Kennedy, R. E. "A Generating Function 
for the Distribution of the Scores of All Possible Bowl- 
ing Games." In The Lighter Side of Mathematics (Ed. 
R. K. Guy and R. E. Woodrow). Washington, DC: Math. 
Assoc. Amer., 1994. 

Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling 
Score Awful?" In The Lighter Side of Mathematics (Ed. 
R. K. Guy and R. E. Woodrow). Washington, DC: Math. 
Assoc. Amer., 1994. 

Box 

see Cuboid 



Box-and- Whisker Plot 



X 



T 



i 



A HlSTOGRAM-like method of displaying data invented 
by J. Tukey (1977). Draw a box with ends at the QUAR- 
TILES Qi and Q 3 . Draw the MEDIAN as a horizontal 
line in the box. Extend the "whiskers" to the farthest 
points. For every point that is more than 3/2 times the 
Interquartile Range from the end of a box, draw a 
dot on the corresponding top or bottom of the whisker. 
If two dots have the same value, draw them side by side. 



References 

Tukey, J. W. Explanatory Data Analysis. 
Addison- Wesley, pp. 39-41, 1977. 

Box Counting Dimension 

see Capacity Dimension 

Box Fractal 



Reading, MA: 




A Fractal which can be constructed using String 
Rewriting by creating a matrix with 3 times as 
many entries as the current matrix using the rules 



line 1 
line 2 
line 3 



11 jkii — S 11 sk " " 



'■_>11 " 



162 



Box-Muller Transformation 



Boy Surface 



Let N n be the number of black boxes, L n the length of 
a side of a white box, and A n the fractional AREA of 
black boxes after the nth iteration. 



N„=5 n 

Ln = (!)"= 3"" 

The Capacity Dimension is therefore 



(i) 

(2) 
(3) 



lniV n 



d cap = - lim lnL 



lim 



ln(5") 



n— >oo 

In 5 
m~3 



n-voo ln(3- n ) 
1.464973521.... 



(4) 



Boxcar Function 



y = c[H(x-o)-H{x-b)], 

where H is the Heaviside Step Function. 

References 

von Seggern, D. CRC Standard Curves and Surfaces. Boca 
Raton, FL: CRC Press, p. 324, 1993. 



see also Cantor Dust, Sierpinski Carpet, Sierpin- 
ski Sieve 

References 

$ Weisstein, E. W. "Fractals." http: //www. astro. Virginia. 
edu/~evw6n/math/notebooks/Fractal.m. 

Box-Muller Transformation 

A transformation which transforms from a 2-D contin- 
uous Uniform Distribution to a 2-D Gaussian Bi- 
variate Distribution (or Complex Gaussian Dis- 
tribution). If xi and X2 are uniformly and indepen- 
dently distributed between and 1, then z\ and z 2 as de- 
fined below have a Gaussian Distribution with Mean 
li = and Variance <t 2 = 1. 



z\ — y — 21na;i cos(27ra;2) 
Z2 = v — 21n#i sin(27ra;2)- 

This can be verified by solving for x\ and x 2 , 

-( Zl 2 + Z2 2 )/2 



x x 

X 2 



2tt 



■ tan 



■■(!)• 



(1) 
(2) 

(3) 
(4) 



Taking the Jacobian yields 
d(xi,x 2 ) 



d(z u z 2 ) 



d%i dxi 

dz± dz 2 

dx 2 dx 2 

£zi dz 2 



1 -Z! 2 /2 

— e * ' 



2tt 



\/2^ 



^ 2 2 /2 



(5) 



Box-Packing Theorem 

The number of "prime" boxes is always finite, where a 
set of boxes is prime if it cannot be built up from one 
or more given configurations of boxes. 
see also Conway Puzzle, Cuboid, de Bruijn's Theo- 
rem, Klarner's Theorem, Slothouber-Graatsma 
Puzzle 

References 

Honsberger, R. Mathematical Gems II. Washington, DC: 
Math. Assoc. Amer., p. 74, 1976. 



Boxcars 

A roll of two 6s (the highest roll possible) on a pair of 
6-sided DICE, The probability of rolling boxcars is 1/36, 
or 2.777...%. 

see also Dice, Double Sixes, Snake Eyes 

Boy Surface 

A Nonorientable Surface which is one of the three 
possible Surfaces obtained by sewing a Mobius Strip 
to the edge of a DISK. The other two are the CROSS- 
Cap and ROMAN SURFACE. The Boy surface is a model 
of the Projective Plane without singularities and is 
a Sextic Surface. 

The Boy surface can be described using the general 
method for NONORIENTABLE SURFACES, but this was 

not known until the analytic equations were found by 
Apery (1986). Based on the fact that it had been proven 
impossible to describe the surface using quadratic poly- 
nomials, Hopf had conjectured that quartic polynomials 
were also insufficient (Pinkall 1986). Apery's IMMER- 
SION proved this conjecture wrong, giving the equations 
explicitly in terms of the standard form for a NONORI- 
ENTABLE Surface, 

/i(*,y,s) = |[(2x 2 - y 2 - z 2 )(x 2 + y 2 + z 2 ) 
+ 2yz(y — z ) + zx(x — z ) 
+ xy(y 2 -x 2 )] (1) 

Mx,y,z) = \s/Z[{y 2 - z 2 )(x 2 + y 2 + z 2 ) 

+ zx{z 2 - x 2 ) + xy(y 2 - x 2 )} (2) 

f 3 (x, y,z) = i(x + y + z)[(x + y + z) 3 

+ A(y-x)(z-y)(x-z)]. (3) 






Boy Surface 

Plugging in 



x = cos u sin v 
y = sin u sin v 

Z = COS V 



(4) 
(5) 
(6) 



and letting u G [0, tv] and v € [0, 7r] then gives the Boy 
surface, three views of which are shown above. 

The K. parameterization can also be written as 



V = 



V2cos 2 vcos(2it) + cosusin(2t;) 
2- v / 2sin(3u)sin(2v) 
__ \/2cos 2 vsin(2u) + cos^sin(2i;) 
2- v / 2sin(3u)sin(2i;) 
3 cos 2 v 



2- V2sin(3u)sin(2t;) 
(Nordstrand) for u 6 [-7r/2,7r/2] and i; G [0,7r]. 



(7) 
(8) 
(9) 






Three views of the surface obtained using this parame- 
terization are shown above. 

In fact, a HOMOTOPY (smooth deformation) between 
the Roman Surface and Boy surface is given by the 
equations 



x(u,v) = 
y{u,v) = 

Z(U y V) = 



\[2 cos(2n) cos 2 v + cos u sin(2v) 
2 — a\/2 sin(3ti) sin(2t;) 

\/2sin(2u) cos 2 v — sinusin(2t>) 
2-aA/2sin(3tx)sin(2i;) 
3 cos 2 v 

2 — a\/2 sin(3u) sin(2v) 



(10) 

(11) 

(12) 



as a varies from to 1, where a — corresponds to the 
Roman Surface and a = 1 to the Boy surface (Wang), 
shown below. 









Boy Surface 163 

In K. , the parametric representation is 

xq = 3[(u + v +w )(u + v ) — V2vw(3u — v )] 

(13) 
X! = V2(u 2 + v 2 )(u 2 - v 2 + v^uty) (14) 

a?2 = V2(u 2 + v 2 )(2wu - V2vw) (15) 

X3 = 3(u 2 + v 2 ) 2 , (16) 

and the algebraic equation is 

64(x — £3) 3 #3 3 — 48(x — ^3) 2 ^3 2 (32;i 2 + Sx2 2 + 2x 3 2 ) 
+12(z - x 3 )x 3 [27(x 1 2 + z 2 2 ) 2 - 24z 3 2 (zi 2 + x 2 2 ) 
+36^3:2^3 (x2 2 — 3cci 2 ) + X3 4 ] 
+(9zi 2 +92 2 2 - 2x 3 2 ) 

x[-81(^i 2 + x 2 2 ) 2 - 72x 3 2 (xi 2 + x 2 2 ) 
+10%V2x 1 x 3 {x 1 2 - 3z 2 2 ) + 4z 3 4 ] = (17) 



(Apery 1986). Letting 



Xq — 1 

Xi = X 

x 2 =y 

X3 = z 



(18) 
(19) 
(20) 
(21) 



gives another version of the surface in M . 

see also Cross-Cap, Immersion, Mobius Strip, 
nonorientable surface, real projective plane, 
Roman Surface, Sextic Surface 

References 

Apery, F. "The Boy Surface." Adv. Math. 61, 185-266, 1986. 

Boy, W. "Uber die Curvatura Integra und die Topologie 

geschlossener Flachen." Math. Ann 57, 151-184, 1903. 
Brehm, U. "How to Build Minimal Polyhedral Models of the 

Boy Surface." Math. Intell. 12, 51-56, 1990. 
Carter, J. S. "On Generalizing Boy Surface — Constructing a 

Generator of the 3rd Stable Stem." Trans. Amer. Math. 

Soc. 298, 103-122, 1986. 
Fischer, G. (Ed.). Plates 115-120 in Mathematische Mod- 

elle/ Mathematical Models, Bildband/ Photograph Volume. 

Braunschweig, Germany: Vieweg, pp. 110-115, 1986. 
Geometry Center. "Boy's Surface." http://www.geom.umn. 

edu/zoo/toptype/pplane/boy/. 
Hilbert, D. and Cohn-Vossen, S. §46—47 in Geometry and the 

Imagination. New York: Chelsea, 1952. 
Nordstrand, T. "Boy's Surface." http : //www . uib . no/ 

people/nf ytn/boytxt . htm. 
Petit, J .-P. and Souriau, J. "Une representation analytique 

de la surface de Boy." C. R. Acad. Sci. Paris Sir. 1 Math 

293, 269-272, 1981. 
Pinkall, U. Mathematical Models from the Collections of Uni- 
versities and Museums (Ed. G. Fischer). Braunschweig, 

Germany: Vieweg, pp. 64-65, 1986. 
Stewart, I. Game, Set and Math. New York: Viking Penguin, 

1991. 
Wang, P. "Renderings." http: //www.ugcs . caltech.edu/ 

-pet erw/portf olio/renderings/. 



164 



Bra 



Brachistochrone Problem 



Bra 

A (COVARIANT) 1-VECTOR denoted (V>|- The bra is 
Dual to the Contravariant Ket, denoted \ip). Taken 
together, the bra and KET form an ANGLE BRACKET 
(bra+ket = bracket). The bra is commonly encountered 
in quantum mechanics. 

see also Angle Bracket, Bracket Product, Co- 
variant Vector, Differential /.-Form, Ket, One- 
Form 

Brachistochrone Problem 

Find the shape of the CURVE down which a bead sliding 
from rest and Accelerated by gravity will slip (with- 
out friction) from one point to another in the least time. 
This was one of the earliest problems posed in the CAL- 
CULUS of Variations. The solution, a segment of a 
Cycloid, was found by Leibniz, L'Hospital, Newton, 
and the two Bernoullis. 

The time to travel from a point Pi to another point Pi 
is given by the INTEGRAL 



= C - 



(i) 



The VELOCITY at any point is given by a simple appli- 
cation of energy conservation equating kinetic energy to 
gravitational potential energy, 



1 2 

2 mv 



mgy, 



v = y/2gy. 
Plugging this into (1) then gives 



tl2 



i: 



a/i + y' 2 
s/5gy 



dx ■ 



i: 



l + y' 2 
tgy 



dx. 



The function to be varied is thus 

f = (l + y ,2 ) 1/2 (2gy)-^. 



(2) 



(3) 



(4) 



(5) 



subtracting y'{df/dy') from /, and simplifying then 
gives 

C. (9) 



V^gy^i + y' 2 

Squaring both sides and rearranging slightly results in 



1 + 



[dx) 



2gC* 



(10) 



where the square of the old constant C has been ex- 
pressed in terms of a new (POSITIVE) constant k 2 . This 
equation is solved by the parametric equations 



x - 

y 



\k 2 {e-s\n9) 

§fc 2 (l-cos6>), 



(11) 

(12) 



which are — lo and behold — the equations of a CYCLOID. 

If kinetic friction is included, the problem can also be 
solved analytically, although the solution is significantly 
messier. In that case, terms corresponding to the normal 
component of weight and the normal component of the 
Acceleration (present because of path Curvature) 
must be included. Including both terms requires a con- 
strained variational technique (Ashby et al. 1975), but 
including the normal component of weight only gives an 
elementary solution. The Tangent and Normal Vec- 
tors are 



(13) 
(14) 



T = 


dx „ 
ds 


dy- 

ds 


N = 


dy ~ 
ds 


dx ^ 



gravity and friction are then 



• gravity 



: mgy 



dx r 



Ff r i c tion = ~M( F gravityN)T = - flTTig — T, 



and the components along the curve are 



(15) 
(16) 



To proceed, one would normally have to apply the full- 
blown Euler-Lagrange Differential Equation 



21 
dy 



dx \dy'J 



0. 



(6) 



However, the function f{y,y' } x) is particularly nice 
since x does not appear explicitly. Therefore, df /dx = 
0, and we can immediately use the Beltrami Identity 



>-<%-<>■ 



Computing 



8y' 



y {l + y 



/2\-l/2 



(2gy) 



-1/2 



(7) 



(8) 



-T gravity J- : 
■T friction -L 



dy 

m9 dS 



-fj,mg 



dx 
ds ' 



so Newton's Second Law gives 



dv 



dy 



m— — mg- 1 
dt ds 



limg 



dx 
ds 



But 



dv dv 

— = v — 
at as 



1 d 2 ^ 
2dS^ V) 



\v 2 = g{y - fix) 
v = y/2g{y - fix), 



(17) 
(18) 

(19) 

(20) 

(21) 
(22) 



Bracket 



Bracketing 165 



-Jyi 



+ (y') 2 



dx. 



(23) 
2< ? (y - fix) ~' v 

Using the Euler-Lagrange Differential Equation 
gives 

[i + y 2 ](i + aV) + 2(2/ - ^)y" - 0. (24) 

This can be reduced to 

i + (y') 2 _ c 



Now letting 



the solution is 



(1 + /X2/') 2 y- iix' 



y'=cot(±0), 



(25) 
(26) 



a; = ffc 2 [(0-sm0)+Ai(l-cos6O] (27) 

y = |A; 2 [(1 - cos<9) + ^(<9 + sin0)]. (28) 

see also Cycloid, Tautochrone Problem 
References 

Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. "Bra- 
chistochrone with Coulomb Friction." Amer. J. Phys. 43, 
902-905, 1975. 

Haws, L. and Kiser, T. "Exploring the Brachistochrone Prob- 
lem." Amer. Math. Monthly 102, 328-336, 1995. 

Wagon, S. Mathematica in Action. New York: W. H. Free- 
man, pp. 60-66 and 385-389, 1991. 

Bracket 

see Angle Bracket, Bra, Bracket Polynomial, 
Bracket Product, Iverson Bracket, Ket, La- 
grange Bracket, Poisson Bracket 

Bracket Polynomial 

A one- variable KNOT POLYNOMIAL related to the JONES 
Polynomial. The bracket polynomial, however, is not 
a topological invariant, since it is changed by type I REI- 
demeister Moves. However, the Span of the bracket 
polynomial is a knot invariant. The bracket polynom- 
ial is occasionally given the grandiose name REGULAR 
Isotopy Invariant. It is defined by 



<L)(A,*,d) = ^<2W 



Ikll 



(1) 



where A and B are the "splitting variables," a runs 
through all "states" of L obtained by Splitting the 
LINK, (L\a) is the product of "splitting labels" corre- 
sponding to cr, and 

\W\\ = N L -1, (2) 

where JV& is the number of loops in er. Letting 



-l 



B = A 

d^-A 2 -A' 2 



(3) 
(4) 



gives a Knot Polynomial which is invariant under 
Regular Isotopy, and normalizing gives the Kauff- 
man Polynomial X which is invariant under Ambient 
Isotopy. The bracket Polynomial of the Unknot is 
1. The bracket Polynomial of the Mirror Image K* 
is the same as for K but with A replaced by A -1 . In 
terms of the one-variable KAUFFMAN POLYNOMIAL X, 
the two-variable KAUFFMAN POLYNOMIAL F and the 

Jones Polynomial V\ 



X(A) 
(L) (A) 



( 



-A*y 



-«.(£) 



(L), 



F(-A 3 ,A + A- 1 ) 



(5) 
(6) 

(7) 



<L> (A) = V{A~% 

where w(L) is the WRITHE of L. 

see also SQUARE BRACKET POLYNOMIAL 



References 

Adams, C. C. The Knot Book: An Elementary Introduction 
to the Mathematical Theory of Knots. New York: W. H. 
Freeman, pp. 148-155, 1994. 

Kauffman, L. "New Invariants in the Theory of Knots." 
Amer. Math. Monthly 95, 195-242, 1988. 

Kauffman, L. Knots and Physics. Teaneck, NJ: World Sci- 
entific, pp. 26-29, 1991. 
i$ Weisstein, E. W. "Knots and Links." http: //www. astro. 
virginia.edu/~eww6n/math/notebooks/Knots .m. 

Bracket Product 

The Inner Product in an Li Space represented by an 
Angle Bracket. 

see also Angle Bracket, Bra, Ket, L 2 Space, One- 
Form 

Bracketing 

Take x itself to be a bracketing, then recursively de- 
fine a bracketing as a sequence B = (jBi, . . . , Bk) where 
k > 2 and each Bi is a bracketing. A bracketing can be 
represented as a parenthesized string of xs, with paren- 
theses removed from any single letter x for clarity of 
notation (Stanley 1997). Bracketings built up of binary 
operations only are called BINARY BRACKETINGS. For 
example, four letters have 11 possible bracketings: 

xxxx (xx)xx x(xx)x xx(xx) 

(xxx)x x(xxx) ((xx)x)x (x(xx))x 
{xx)(xx) x((xx)x) x(x(xx)), 

the last five of which are binary. 

The number of bracketings on n letters is given by the 
Generating Function 

\(l + x- y/l ~6x + x 2 ) = x + x 2 + 3x 3 + llx 4 + 45x 5 

(Schroder 1870, Stanley 1997) and the RECURRENCE 
Relation 

_ 3(2n — 3)s n -i — (n — 3)s n -2 



166 



Bradley's Theorem 



Brahmagupta Matrix 



(Sloane), giving the sequence for s n as 1, 1, 3, 11, 45, 
197, 903, . . . (Sloane's A001003). The numbers are also 
given by 

s n = ^ s(ii) • • - s(i k ) 

for n > 2 (Stanley 1997). 

The first PLUTARCH NUMBER 103,049 is equal to $io 
(Stanley 1997), suggesting that Plutarch's problem of 
ten compound propositions is equivalent to the number 
of bracketings. In addition, Plutarch's second number 
310,954 is given by (sio + sn)/2 = 310,954 (Habsieger 
et al. 1998). 
see also Binary Bracketing, Plutarch Numbers 

References 

Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second 

Number of Plutarch." Amer. Math. Monthly 105, 446, 

1998. 
Schroder, E. "Vier combinatorische Probleme." Z. Math. 

Physik 15, 361-376, 1870. 
Sloane, N. J. A. Sequence A001003/M2898 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 
Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough." 

Amer. Math. Monthly 104, 344-350, 1997. 

Bradley's Theorem 

Let 

S(a,(3,m;z) = 

y> T(m + j(z + l))rpg + 1 + jz) (a) + j 

m 2^ T{m + jz + l)r(a + + 1 + j(z + 1)) j! 

j = 



and a be a Negative Integer. Then 

T(/3 + 1 - m) 



S{a,(3,m\z) = 



r(a + /3 + l-m)' 



where T(z) is the GAMMA FUNCTION. 

References 

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: 

Springer- Verlag, pp. 346-348, 1994. 
Bradley, D. "On a Claim by Ramanujan about Certain Hy- 

pergeometric Series." Proc. Amer. Math. Soc. 121, 1145- 

1149, 1994. 



Brahmagupta's Formula 

For a Quadrilateral with sides of length a, 6, c, and 
d, the Area K is given by 



K : 



J(s - a)(s - b)(s - c)(s - d) - abcdcos 2 [\{A + B)], 

(1) 



where 



s= |(a + 6 + c + d) 



(2) 



is the Semiperimeter, A is the Angle between a and 
d, and B is the Angle between b and c. For a Cyclic 



Quadrilateral (i.e., a Quadrilateral inscribed in 

a Circle), A + B — 7r, so 



K = ^/(s-a)(s-b){s-c){s-d) (3) 



y/(bc + ad)(ac + bd)(ab -f- cd) 



4R 



(4) 



where R is the RADIUS of the CiRCUMClRCLE. If the 
Quadrilateral is Inscribed in one Circle and Cir- 
cumscribed on another, then the Area Formula sim- 
plifies to 

K = \fabc~d. (5) 

see also Bretschneider's Formula, Heron's For- 
mula 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 

Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 81-82, 1929. 

Brahmagupta Identity 

Let 

0=\B\^x 2 -ty\ 

where B is the Brahmagupta Matrix, then 

det[B(x u yi)B(x 2 ,y2)] = det[B(x u yi )] det[B(x 2 , y 2 )] 
= Pifo. 



References 

Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. 
Quart. 34, 30-39, 1996. 



Brahmagupta Matrix 

5(z,y) = 



x y 

±ty ±x 



It satisfies 

B(x!,yi)B(x 27 y2) = B{xxx 2 ±tyiy2,x 1 y 2 ±2/1X2). 
Powers of the matrix are defined by 



B n = 



X 


y 


n 


X<n 


Vn 


ty 


X 




ty n 


Xn 



= B n . 



The x n and y n are called BRAHMAGUPTA POLYNOMI- 
ALS. The Brahmagupta matrices can be extended to 

Negative Integers 

n-n _ x y _ *-n y-n _ d 



X 


y 


— n 


X — n 


y- 


ty 


X 




ty-n 


X- 



see also Brahmagupta Identity 

References 

Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. 
Quart. 34, 30-39, 1996. 



Brahmagupta Polynomial 



Braid Group 167 



Brahmagupta Polynomial 

One of the POLYNOMIALS obtained by taking POWERS 
of the Brahmagupta Matrix. They satisfy the recur- 
rence relation 



x n +! = xx n + tyy n 
y n+1 = xy n + yx n . 



(1) 
(2) 



A list of many others is given by Suryanarayan (1996). 
Explicitly, 



X +t 



(;)«-v+« a (j)«-- 4 » 4 +... (3) 



n-l . .i n \ n-3 3 . .2l n \ n-5 5 . 

rix y + t[\x y +t i ]x y + 



The Brahmagupta POLYNOMIALS satisfy 



dx 

dx n 
dy 



nx n -i 



dyn 

dy 

,dy n 

t—- = ntyn-L 
dy 



(4) 

(5) 
(6) 



The first few POLYNOMIALS are 

xo = 

xi = x 

x 2 — x 2 + ty 2 

xz = x 3 + 3txy 2 

4 . «, 2 2 , ,2 4 

X4 = x + otx y +t y 
and 

yo-o 

2/1=2/ 

y 2 = 2xy 

2/3 = 3x 2 y + ty 3 

2/4 = ^x z y -\- Atxy z . 

Taking x = i/ = 1 and £ = 2 gives j/„ equal to the PELL 
Numbers and x n equal to half the Pell-Lucas num- 
bers. The Brahmagupta POLYNOMIALS are related to 
the Morgan- Voyce Polynomials, but the relation- 
ship given by Suryanarayan (1996) is incorrect. 

References 

Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. 
Quart. 34, 30-39, 1996. 

Brahmagupta's Problem 

Solve the PELL EQUATION 

x 2 - 92y 2 = 1 

in Integers. The smallest solution is x = 1151, y = 

120. 

see also Diophantine Equation, Pell Equation 



Braid 

An intertwining of strings attached to top and bottom 
"bars" such that each string never "turns back up." In 
other words, the path of a braid in something that a 
falling object could trace out if acted upon only by grav- 
ity and horizontal forces. 

see also Braid GROUP 
References 

Christy, J. "Braids." http://www.mathsource.com/cgi-bin 
/MathSource/Applications/Mathematics/0202-228. 

Braid Group 

Also called Artin Braid Groups. Consider n strings, 
each oriented vertically from a lower to an upper "bar." 
If this is the least number of strings needed to make a 
closed braid representation of a LINK, n is called the 
Braid Index. Now enumerate the possible braids in a 
group, denoted B n . A general n-braid is constructed by 
iteratively applying the <Tj (i = 1, . . . ,n — 1) operator, 
which switches the lower endpoints of the ith and (i + 
l)th strings — keeping the upper endpoints fixed — with 
the (i + l)th string brought above the ith string. If the 
(i + l)th string passes below the zth string, it is denoted 



1 2 




Topological equivalence for different representations of 
a BRAID Word JJ o~i and J^ a^ is guaranteed by the 

conditions 



CTiCTj — <Tj<Ti 

/ it 

O'iO'i + iO'i — 0'i-\-\(T i <Ti + i 



for \i-j\ >2 
for all i 



as first proved by E. Artin. Any n-braid is expressed as 
a Braid Word, e.g., G^aicr^a^ a\ is a Braid Word 
for the braid group #3 . When the opposite ends of the 
braids are connected by nonintersecting lines, KNOTS 
are formed which are identified by their braid group and 

Braid Word. The Burau Representation gives a 

matrix representation of the braid groups. 

References 

Birman, J. S. "Braids, Links, and the Mapping Class 
Groups." Ann. Math. Studies, No. 82. Princeton, NJ: 
Princeton University Press, 1976. 

Birman, J. S. "Recent Developments in Braid and Link The- 
ory." Math. Intell. 13, 52-60, 1991. 

Christy, J. "Braids." http://www.mathsource.com/cgi-bin 
/MathSource/Applications/Mathematics/0202-228. 

Jones, V. F. R. "Hecke Algebra Representations of Braid 
Groups and Link Polynomials." Ann. Math. 126, 335- 
388, 1987. 
^ Weisstein, E. W. "Knots and Links." http: //www. astro. 
Virginia. edu/-eww6n/math/notebooks/Knots .m. 



168 



Braid Index 



Branch Point 



Braid Index 

The least number of strings needed to make a closed 
braid representation of a LINK. The braid index is equal 
to the least number of Seifert Circles in any projec- 
tion of a Knot (Yamada 1987). Also, for a nonsplit- 
table Link with Crossing Number c(L) and braid in- 
dex i{L) y 

c(L) > 2[i(L) - 1] 

(Ohyama 1993). Let E be the largest and e the small- 
est Power of £ in the HOMFLY Polynomial of an 
oriented LINK, and i be the braid index. Then the 
Morton-Franks- Williams Inequality holds, 

i>\{E-e) + l 

(Franks and Williams 1987). The inequality is sharp for 
all Prime Knots up to 10 crossings with the exceptions 
of 09 42, 09 49, IO132, IO150, and 10i 5 6- 

References 

Franks, J. and Williams, R. F. "Braids and the Jones Poly- 
nomial." Trans. Amer. Math. Soc. 303, 97-108, 1987. 

Jones, V. F. R. "Hecke Algebra Representations of Braid 
Groups and Link Polynomials." Ann. Math. 126, 335- 
388, 1987. 

Ohyama, Y. "On the Minimal Crossing Number and the Brad 
Index of Links," Canad. J. Math. 45, 117-131, 1993. 

Yamada, S. "The Minimal Number of Seifert Circles Equals 
the Braid Index of a Link." Invent. Math. 89, 347-356, 
1987. 

Braid Word 

Any n-braid is expressed as a braid word, e.g., 
o-i^osa^ <y\ is a braid word for the Braid Group S3. 
By Alexander's Theorem, any LINK is representable 
by a closed braid, but there is no general procedure for 
reducing a braid word to its simplest form. However, 
Markov's Theorem gives a procedure for identifying 
different braid words which represent the same LINK. 

Let 6+ be the sum of Positive exponents, and 6_ the 
sum of Negative exponents in the Braid Group B n . 
If 

b + - 36_ - n+ 1 > 0, 

then the closed braid b is not AMPHICHIRAL (Jones 
1985). 

see also Braid GROUP 
References 

Jones, V. F. R. "A Polynomial Invariant for Knots via von 

Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 

1985. 
Jones, V. F. R. "Hecke Algebra Representations of Braid 

Groups and Link Polynomials." Ann. Math. 126, 335- 

388, 1987. 



Braikenridge-Maclaurin Construction 

The converse of PASCAL'S THEOREM. Let Ai, B 2i Ci, 
A 2 , and Br be the five points on a Conic. Then the 
Conic is the Locus of the point 

C 2 =Ax{z> dA 2 ) ■ B x (z • C1B2), 

where z is a line through the point AiB 2 • B\A 2 . 
see also PASCAL'S THEOREM 

Branch 

The segments of a TREE between the points of connec- 
tion (Forks). 

see also FORK, LEAF (TREE) 



Branch Cut 



|Sqrt z| 




A line in the COMPLEX PLANE across which a FUNCTION 
is discontinuous. 



function 


branch cut(s) 


cos -1 z 


(— 00, — 1) and (l,oo) 




cosh -1 


(-oo,l) 




cot -1 z 


(-i,i) 




coth" 1 


[-1,1] 




esc -1 z 


(-1,1) 




csch -1 


(-m) 




In z 


(-oo,0] 




sec" 1 z 


(-1,1) 




sech -1 


(oo,0] and (1, 00) 




sin - z 


(— 00,— 1) and (l,oo) 




sinh -1 


(—200, —i) and (2,200) 




v/i 


(-oo,0) 




tan x z 


(-ioo, -i) and (2,200) 




tanh -1 


( — 00, —1] and [1, 00) 




z n ,n<£Z 


(-oo,0) for R[n] < 0; (- 


-oo,0] for R[n] > 



see also Branch Point 

References 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 399-401, 1953. 

Branch Line 

see Branch Cut 

Branch Point 

An argument at which identical points in the COMPLEX 
PLANE are mapped to different points. For example, 

consider 



Brauer Chain 



Breeder 



169 



Then f(e oi ) = /(l) = 1, but f(e 27ri ) = e 2 ™, despite 
the fact that e i0 = e 2ni . Pinch Points are also called 
branch points. 

see also BRANCH CUT, PlNCH POINT 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 397-399, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 391-392 and 399- 
401, 1953. 

Brauer Chain 

A Brauer chain is an ADDITION CHAIN in which each 
member uses the previous member as a summand. A 
number n for which a shortest chain exists which is a 
Brauer chain is called a BRAUER NUMBER. 

see also Addition Chain, Brauer Number, Hansen 
Chain 

References 

Guy, R. K. "Addition Chains, Brauer Chains. Hansen 
Chains." §C6 in Unsolved Problems in Number Theory, 
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. 

Brauer Group 

The GROUP of classes of finite dimensional central sim- 
ple Algebras over k with respect to a certain equiva- 
lence. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, p. 479, 1988. 

Brauer Number 

A number n for which a shortest chain exists which is 
a BRAUER Chain is called a Brauer number. There are 
infinitely many non-Brauer numbers. 

see also Brauer Chain, Hansen Number 

References 

Guy, R. K. "Addition Chains. Brauer Chains. Hansen 
Chains." §C6 in Unsolved Problems in Number Theory, 
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. 

Brauer- Severi Variety 

An Algebraic Variety over a Field K that becomes 
Isomorphic to a Projective Space. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, pp. 480-481, 1988. 



Brauer's Theorem 

If, in the Gersgorin Circle Theorem for a given m, 

for all j f^ m, then exactly one EIGENVALUE of A lies in 
the Disk F m . 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1121, 1979. 

Braun's Conjecture 

Let B = {&i,& 2 ,...} be an Infinite Abelian Semi- 
group with linear order &i < & 2 < . . . such that &i is the 
unit element and a < b Implies ac < be for a,b,c 6 B. 
Define a Mobius Function jj, on B by /x(6i) = 1 and 



Yl ^ = ° 



b d \b n 



for n = 2, 3, Further suppose that /x(6 n ) = M n ) 

(the true MOBIUS FUNCTION) for all n > 1. Then 
Braun's conjecture states that 

for all m,n> 1. 

see also MOBIUS PROBLEM 

References 

Flath, A. and Zulauf, A. "Does the Mobius Function Deter- 
mine Multiplicative Arithmetic?" Amer. Math. Monthly 
102, 354-256, 1995. 

Breeder 

A pair of POSITIVE INTEGERS (ai,a 2 ) such that the 
equations 

a\ 4- a 2 x = cr(a\) — a(a 2 )(x 4- 1) 

have a POSITIVE INTEGER solution x, where a(n) is the 
DIVISOR FUNCTION. If x is Prime, then (ai,a 2 x) is an 
Amicable Pair (te Riele 1986). (ai,a 2 ) is a "special" 
breeder if 

a± = au 

a 2 = a, 

where a and u are Relatively Prime, (a, u) — 1. If 
regular amicable pairs of type (i,l) with i > 2 are of 
the form (au,ap) with p PRIME, then (au,a) are special 
breeders (te Riele 1986). 

References 

te Riele, H. J. J. "Computation of All the Amicable Pairs 

Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35, 

1986. 



170 Brelaz's Heuristic Algorithm 



Bretschneider's Formula 



Brelaz's Heuristic Algorithm 

An Algorithm which can be used to find a good, but 
not necessarily minimal, EDGE or VERTEX coloring for 

a Graph. 

see also Chromatic Number 

Brent's Factorization Method 

A modification of the POLLARD p FACTORIZATION 
Method which uses 



Xi+i = Xi — c (mod n). 



References 

Brent, R. "An Improved Monte Carlo Factorization Algo- 
rithm." Nordisk Tidskrift for Informationsbehandlung 
(BIT) 20, 176-184, 1980. 

Brent's Method 

A RoOT-finding ALGORITHM which combines root 

bracketing, bisection, and Inverse Quadratic In- 
terpolation. It is sometimes known as the VAN 
Wijngaarden-Deker-Brent Method. 

Brent's method uses a LAGRANGE INTERPOLATING 
Polynomial of degree 2. Brent (1973) claims that this 
method will always converge as long as the values of the 
function are computable within a given region contain- 
ing a ROOT. Given three points asi, x 2 , and £3, Brent's 
method fits x as a quadratic function of y, then uses the 
interpolation formula 



[y-f(*i)][y-f{ x 2)] x 3 



[/(**) 



+ 



f(x 1 )][f(x 3 )-f(x 2 )} 

[y- /Qg2)][y- f(x s )]xi 

[f(x 1 )-f(x 2 )][f(x 1 )-f(x 3 )] 

[y- f(x3)][y- f(xi)]x 2 



+ 



[f( X 2)-f(x 3 )][f(x 2 )-f(x 1 )Y 



(1) 



Subsequent root estimates are obtained by setting y = 0, 
giving 



, P 



(2) 



where 



P = S[R(R - T)(x 3 - x 2 ) - (1 - R)(x 2 - zi)] (3) 
Q = (T-1)(R-1)(S-1) (4) 



with 



R = 



f(X2) 



/(*s) 

a - /(*») 

" f(xi) 

T _/(*l) 



(5) 
(6) 
(7) 



References 

Brent, R. P. Ch. 3-4 in Algorithms for Minimization Without 
Derivatives. Englewood Cliffs, NJ: Prentice- Hall, 1973. 

Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. §7.2 in 
Computer Methods for Mathematical Computations. En- 
glewood Cliffs, NJ: Prentice-Hall, 1977. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Van Wijngaarden-Dekker-Brent Method." 
§9.3 in Numerical Recipes in FORTRAN: The Art of Sci- 
entific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 352—355, 1992. 

Brent- Salamin Formula 

A formula which uses the Arithmetic-Geometric 
MEAN to compute Pi. It has quadratic convergence 
and is also called the Gauss-Salamin Formula and 
Salamin Formula. Let 



CLn + l = 2 {,Q>n + O n ) 


(i) 


&n+l = ydnbn 


(2) 


C n +i = 2 ( a n — b n ) 


(3) 


A — 2 h 2 

0, n = €L n On y 


(4) 



and define the initial conditions to be ao = 1, &o = 
l/\/2- Then iterating a„ and 6„ gives the ARITHMETIC- 
GEOMETRIC MEAN, and it is given by 



4[M(1,2- 1 / 2 )] 2 

4[M(l,2- 1 / 2 )] 2 
l-£~i2 i+ V 



(5) 



(6) 



King (1924) showed that this formula and the LEGEN- 
DRE RELATION are equivalent and that either may be 
derived from the other. 

see also Arithmetic-Geometric Mean, Pi 

References 

Borwein, J. M. and Borwein, P. B, Pi & the AGM: A Study in 

Analytic Number Theory and Computational Complexity. 

New York: Wiley, pp. 48-51, 1987. 
Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 

61, 148-163, 1988. 
King, L. V. On the Direct Numerical Calculation of Elliptic 

Functions and Integrals. Cambridge, England: Cambridge 

University Press, 1924. 
Lord, N. J. "Recent Calculations of n: The Gauss-Salamin 

Algorithm." Math. Gaz. 76, 231-242, 1992. 
Salamin, E. "Computation of n Using Arithmetic-Geometric 

Mean." Math. Comput. 30, 565-570, 1976. 

Bretschneider's Formula 

Given a general QUADRILATERAL with sides of lengths 
a, 6, c, and d (Beyer 1987), the Area is given by 



(Press et al. 1992). 



^quadrilateral = \ ^4p 2 q 2 - (b 2 + d 2 - d 2 - C 2 ) 2 , 

where p and q are the diagonal lengths. 

see also BRAHMAGUPTA'S FORMULA, HERON'S FOR- 
MULA 

References 

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. 



Brianchon Point 



Bridge (Graph) 171 



Brianchon Point 

The point of CONCURRENCE of the joins of the VER- 
TICES of a Triangle and the points of contact of a 
Conic Section Inscribed in the Triangle. A Conic 
Inscribed in a Triangle has an equation of the form 



the chance that one of four players will receive a hand 
of a single suit is 



39,688,347,497 



/ 9 h 

- + - + - 

U V w 



o, 



an it.s HrianrTinn -nrnnt Viae Trttttmrar nnrmnTM atpq 



There are special names for specific types of hands. A 
ten, jack, queen, king, or ace is called an "honor." Get- 



suits and the ace, king, and queen, and jack of the re- 
maining suit is called 13 top honors. Getting all cards of 
the same suit is called a 13-card suit. Getting 12 cards 
of same suit with ace high and the 13th card not an 
ace is called 2-card suit, ace high. Getting no honors is 
called a Yarborough. 

The probabilities of being dealt 13-card bridge hands 
of a given type are given below. As usual, for a hand 
with probability P, the Odds against being dealt it are 
(1/P) -1:1. 



(1//, l/g,l/h). For Kiepert's Parabola, the Bran- 
chion point has TRIANGLE CENTER FUNCTION 



a(b 2 - 
which is the Steiner Point. 



2). 



Brianchon's Theorem 

The Dual of Pascal's Theorem. It states that, given 
a 6-sided Polygon Circumscribed on a Conic SEC- 
TION, the lines joining opposite VERTICES (DIAGONALS) 
meet in a single point. 




see also DUALITY PRINCIPLE, PASCAL'S THEOREM 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 

Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967. 
Ogilvy, C. S. Excursions in Geometry. New York: Dover, 

p. 110, 1990. 

Brick 

see Euler Brick, Harmonic Brick, Rectangular 
Parallelepiped 



Bride's Chair 

One name for the figure used by Euclid to prove the 
Pythagorean Theorem. 

see also Peacock's Tail, Windmill 

Bridge Card Game 

Bridge is a CARD game played with a normal deck of 52 

cards. The number of possible distinct 13-card hands is 



N = 



635,013,559,600. 



where (£) is a Binomial Coefficient. While the 

chances of being dealt a hand of 13 CARDS (out of 52) 
of the same suit are 

4 1 



Hand 




Exact 


Probability 


13 top honors 


high 


4 
N 

4 
N 

4-12-36 
N 

(S) 

N 

ill 

mm 

AT 


i 


158,753,389,900 
1 


12-card suit, ace 
Yarborough 
four aces 
nine honors 


158,753,389,900 
4 


1,469,938,795 

5,394 
9,860,459 

11 
4,165 

888,212 
93,384,347 




Hand 


Probability 


Odds 


13 top honors 

13-card suit 

12-card suit, ace high 

Yarborough 

four aces 

nine honors 


6.30 
6,30 

2.72 
5.47 
2.64 
9.51 


x 10~ 12 
x 10" 12 
x 10" 9 
x 10" 4 
x 10~ 3 
x 10" 3 


158,753,389,899:1 

158,753,389,899:1 

367,484,697.8:1 

1,827.0:1 

377.6:1 

104.1:1 


see also CARDS, POKER 
References 







Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, pp. 48-49, 
1987. 

Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recre- 
ations. New York: W. W. Norton, pp. 119-121, 1942. 

Bridge (Graph) 

The bridges of a Graph are the Edges whose removal 
disconnects the Graph. 

see also Articulation Vertex 

References 

Chartrand, G. "Cut- Vertices and Bridges." §2.4 in Introduc- 
tory Graph Theory. New York: Dover, pp. 45-49, 1985. 



(«) 158,753,389,900' 



172 Bridge Index 



Bring Quintic Form 



Bridge Index 

A numerical KNOT invariant. For a TAME KNOT K, the 
bridge index is the least BRIDGE NUMBER of all planar 
representations of the Knot. The bridge index of the 
Unknot is defined as 1. 

see also Bridge Number, Crookedness 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 

Perish Press, p. 114, 1976. 
Schubert, H. "Uber eine numerische Knotteninvariante." 

Math. Z. 61, 245-288, 1954. 

Bridge of Konigsberg 

see Konigsberg Bridge Problem 



Bridge Number 

The least number of unknotted arcs lying above the 
plane in any projection. The knot 05os has bridge num- 
ber 2. Such knots are called 2-BRIDGE KNOTS. There is 
a one-to-one correspondence between 2-Bridge KNOTS 
and rational knots. The knot O8010 is a 3-bridge knot. A 
knot with bridge number b is an n-EMBEDDABLE KNOT 
where n < b. 
see also BRIDGE INDEX 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 

to the Mathematical Theory of Knots. New York: W. H. 

Freeman, pp. 64-67, 1994. 
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 

Perish Press, p. 115, 1976. 



Bridge Knot 

An n-bridge knot is a knot with BRIDGE Number n. 
The set of 2-bridge knots is identical to the set of rational 
knots. If L is a 2-Bridge Knot, then the BLM/Ho 
Polynomial Q and Jones Polynomial V satisfy 

Q L (z) = 2z- 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ), 

where z = — t — i" 1 (Kanenobu and Sumi 1993). Ka- 
nenobu and Sumi also give a table containing the num- 
ber of distinct 2-bridge knots of n crossings for n — 10 
to 22, both not counting and counting MIRROR IMAGES 
as distinct. 



n 


K n 


K n + K n 


3 








4 








5 






6 






7 






8 






9 






10 


45 


85 


11 


91 


182 


12 


176 


341 


13 


352 


704 


14 


693 


1365 


15 


1387 


2774 


16 


2752 


5461 


17 


5504 


11008 


18 


10965 


21845 


19 


21931 


43862 


20 


43776 


87381 


21 


87552 


175104 


22 


174933 


349525 



References 

Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- 
Bridge Knots through 22-Crossings." Math. Comput. 60, 
771-778 and S17-S28, 1993. 

Schubert, H. "Knotten mit zwei Briicken." Math. Z. 65, 
133-170, 1956. 



Brill-Noether Theorem 

If the total group of the canonical series is divided into 
two parts, the difference between the number of points 
in each part and the double of the dimension of the 
complete series to which it belongs is the same. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 263, 1959. 

Bring-Jerrard Quintic Form 

A TSCHIRNHAUSEN TRANSFORMATION can be used to 

algebraically transform a general Quintic Equation 
to the form 



z + C\Z + Co == 0. 



(1) 



y + b 2 y 2 + hy + b -0 



In practice, the general quintic is first reduced to the 

Principal Quintic Form 

(2) 

before the transformation is done. Then, we require that 
the sum of the third POWERS of the ROOTS vanishes, 
so ss(yj) = 0. We assume that the ROOTS Zi of the 
Bring-Jerrard quintic are related to the ROOTS yi of the 
Principal Quintic Form by 

Zi = ayi 4 + j3yi 3 + jyi 2 + 6yi + e. (3) 

In a similar manner to the Principal Quintic Form 
transformation, we can express the COEFFICIENTS Cj in 
terms of the bj . 

see also Bring Quintic Form, Principal Quintic 
Form, Quintic Equation 

Bring Quintic Form 

A TSCHIRNHAUSEN Transformation can be used to 
take a general Quintic Equation to the form 



x — x — a : 



0, 



where a may be Complex. 

see also Bring-Jerrard Quintic Form, Quintic 

Equation 

References 

Ruppert, W. M. "On the Bring Normal Form of a Quintic in 
Characteristic 5." Arch. Math. 58, 44-46, 1992. 



Brioschi Formula 



Brocard Angle 173 



Brioschi Formula 

For a curve with METRIC 



Brocard Angle 



ds 2 = E du + F dudv + G dv 2 , 



(1) 



where E, F, and G is the first FUNDAMENTAL FORM, 
the Gaussian Curvature is 

Mi + M 2 /0 v 



where 



Mi = 



M 2 





-F 2 


r 








2 U1; ~t" -^tit; 2 uu 


2 Eu 


F u 


- ^E v 




i*V — 2^1* 




E 




F 




2^« 




F 




G 












(3) 


2 ^v 2 "" 










§£ v £ F 


i 






(4) 


\G U F G 













which can also be written 



K = 



d_ (j_dVG\ d_ ( i d^E\ 



r EG [du \^E du J dv \^Q dv J _ 

d ( G u \ 3 ( E v \ 
du \y/EGj dv \<JEGJ 



2VEG 



(5) 
(6) 



see also Fundamental Forms, Gaussian Curvature 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, pp. 392-393, 1993. 

Briot-Bouquet Equation 

An Ordinary Differential Equation of the form 

where m is a Positive Integer, / is Analytic at x ~ 
y = 0, /(0,0) = 0, and /i(0, 0)^0. 

References 

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- 
ematics: An Updated and Annotated Translation of the 
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- 
lands: Reidel, pp. 481-482, 1988. 




A C 

Define the first Brocard Point as the interior point Q 
of a Triangle for which the Angles ICIAB, IQBC, 
and iVtCA are equal. Similarly, define the second BRO- 
CARD Point as the interior point Cl' for which the AN- 
GLES IQ'AC, /-0,'CB, and IQ'BA are equal. Then the 
Angles in both cases are equal, and this angle is called 
the Brocard angle, denoted u). 

The Brocard angle u> of a Triangle AABC is given by 
the formulas 



cot u) = cot A 4- cot B + cot C 

1 + cos ai cos ct2 cos az 

sin ct\ sin 0:2 sin otz 

_ sin 2 ai + sin 2 c*2 + sin 2 0:3 

2sinai sina2 sin 0:3 
_ ai sin ai + 02 sin 0:2 + az sin a<3 

a± cos a± + a2 cos 0:2 + &z cos a3 

2 

; a2 

2A 



2 2,2,2 

csc w = csc a± + csc a.2 + esc otz 



-s/ai 2 a2 2 + a2 2 a 3 2 + a 3 2 ai 2 



(i) 

(2) 
(3) 
(4) 

(5) 

(6) 
(7) 



where A is the Triangle Area, A, B, and C are An- 
gles, and a, b, and c are side lengths. 

If an Angle a of a Triangle is given, the maximum 
possible Brocard angle is given by 

coto; = § tan(ia) + 5COs(|a). (8) 

Let a Triangle have Angles A, B, and C. Then 



sin A sin B sin C < kABC, 



(9) 



where 



k=[^) (10) 



(Le Lionnais 1983). This can be used to prove that 

8a; 3 < ABC (11) 

(Abi-Khuzam 1974). 



174 



Brocard Axis 



Brocard Line 



see also BROCARD CIRCLE, BROCARD LINE, EQUI- 

Brocard Center, Fermat Point, Isogonic Cen- 
ters 

References 

Abi-Khuzam, F. "Proof of YfFs Conjecture on the Brocard 
Angle of a Triangle." Elem. Math. 29, 141-142, 1974. 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 263-286 and 289-294, 1929. 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 28, 1983. 

Brocard Axis 

The Line KO passing through the Lemoine Point K 
and Circumcenter O of a Triangle. The distance 
OK is called the Brocard Diameter. The Brocard 
axis is Perpendicular to the Lemoine Axis and is 
the Isogonal Conjugate of Kiepert's Hyperbola. 
It has equations 

sin(£ - C)a + sin(C - A)f3 + sin(A - B)j = 

bc(b 2 - c 2 )a + ca(c 2 - a 2 )p + ab(a 2 - 6 2 ) 7 = 0. 

The Lemoine Point, Circumcenter, Isodynamic 
Points, and BROCARD Midpoint all lie along the Bro- 
card axis. Note that the Brocard axis is not equivalent 
to the Brocard Line. 

see also Brocard Circle, Brocard Diameter, Bro- 
card Line 

Brocard Circle 




The CIRCLE passing through the first and second Bro- 
card Points ft and ft', the Lemoine Point K, and 
the Circumcenter O of a given Triangle. The Bro- 
card Points ft and ft' are symmetrical about the Line 



KO' 



which is called the Brocard Line. The Line 



Segment KO is called the Brocard Diameter, and 
it has length 



OK : 



on 

COS UJ 



R^Jl -4sin 2 cj 



cos a; 



where R is the ClRCUMRADlUS and u> is the BROCARD 
Angle. The distance between either of the Brocard 
Points and the Lemoine Point is 

OK = TVK = Tld tan a;. 



see also Brocard Angle, Brocard Diameter, Bro- 
card Points 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, p. 272, 1929. 

Brocard's Conjecture 

7r(pn+i 2 ) -7r(Pn 2 ) > 4 
for n > 2 where tt is the Prime Counting Function. 

see also ANDRICA'S CONJECTURE 

Brocard Diameter 

The Line Segment KO joining the Lemoine Point K 
and Circumcenter O of a given Triangle. It is the 
Diameter of the Triangle's Brocard Circle, and 
lies along the BROCARD Axis. The Brocard diameter 

has length 



— — On R\/l - 4 sin 2 w 
OK = — , 

COS UJ COS U) 

where ft is the first Brocard Point, R is the Circum- 
RADIUS, and w is the Brocard Angle. 
see also Brocard Axis, Brocard Circle, Brocard 
Line, Brocard Points 

Brocard Line 




^3 "3 

A Line from any of the Vertices Ai of a Triangle 
to the first ft or second ft' BROCARD POINT, Let the 
Angle at a Vertex A» also be denoted A i} and denote 
the intersections of A±Q and Aifl' with A2A3 as Wi and 
W2. Then the ANGLES involving these points are 



LA&Wz^Ax 



(1) 



IW Z QA 2 = A 3 (2) 

LA 2 £IW 1 =A 2 . (3) 

Distances involving the points Wi and W[ are given by 

a 3 



,4 2 ft 



sin A2 



(4) 



Brocard Midpoint 



Brocard Points 175 



A 2 Q 

A 3 n 


_ a 3 2 _ sin(^4 3 - ll>) 
aia2 sin a; 


W 3 Ai _ 

W3A2 


a2 sin u) (0,2 
ai sin(A3 — uj) \a% 



(5) 



(6) 



where uj is the Brocard Angle (Johnson 1929, 
pp. 267-268). 

The Brocard line, MEDIAN M, and LEMOINE POINT K 
are concurrent, with A1Q1, A2K ', and A3M meeting at 
a point P. Similarly, AiQ' , A2M, and A3K meet at 
a point which is the ISOGONAL CONJUGATE point of P 
(Johnson 1929, pp. 268-269). 

see also Brocard Axis, Brocard Diameter, Bro- 
card Points, Isogonal Conjugate, Lemoine 
Point, Median (Triangle) 

References 

Johnson, It. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 263-286, 1929. 

Brocard Midpoint 

The Midpoint of the Brocard Points. It has Tri- 
angle Center Function 

a = a(b + c ) — sin(^4 -f- a;), 

where uj is the Brocard Angle. It lies on the Bro- 
card Axis. 

References 

Kimberling, C. "Central Points and Central Lines in the 
Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 

Brocard Points 




A C 

The first Brocard point is the interior point H (or n 
or Zx) of a Triangle for which the Angles IQAB, 
ZfiBC, and IQCA are equal. The second Brocard point 
is the interior point fi' (or T2 or Z2) for which the An- 
gles IQ'AC, IQ'CB, and Itt'BA are equal. The AN- 
GLES in both cases are equal to the BROCARD ANGLE 



uj = IttAB = IttBC = mCA 

= in' ac = m'CB - iq'ba. 

The first two Brocard points are ISOGONAL Conju- 
gates (Johnson 1929, p. 266). 




Let Cbc be the CIRCLE which passes through the ver- 
tices B and C and is TANGENT to the line AC at C, and 
similarly for Cab and Cbc- Then the CIRCLES Cab, 
Cbc, and Cac intersect in the first Brocard point Q. 
Similarly, let C' BC be the CIRCLE which passes through 
the vertices B and C and is TANGENT to the line AB at 
B, and similarly for C' AB and C' AC . Then the CIRCLES 
C A b j C'bC) anc * Cac intersect in the second Brocard 
points £V (Johnson 1929, pp. 264-265). 





a c a c 

The Pedal Triangles of Q and 0! are congruent, 
and Similar to the Triangle AABC (Johnson 1929, 
p. 269). Lengths involving the Brocard points include 



OQ = OW = R\/l-4sm 2 uj 
nO' = 2Rs\xiu\/l -4sin 2 u>. 



(i) 

(2) 



Brocard's third point is related to a given TRIANGLE by 
the Triangle Center Function 



(3) 



(Casey 1893, Kimberling 1994). The third Brocard 
point Q" (or r 3 or Z z ) is COLLINEAR with the SPIEKER 

Center and the Isotomic Conjugate Point of its 
Triangle's Incenter. 

see also Brocard Angle, Brocard Midpoint, Equi- 
Brocard Center, Yff Points 

References 

Casey, J. A Treatise on the Analytical Geometry of the Point, 
Line, Circle, and Conic Sections, Containing an Account 
of Its Most Recent Extensions, with Numerous Examples, 
2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 
1893. 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 263-286, 1929. 

Kimberling, C "Central Points and Central Lines in the 
Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 

Stroeker, R. J. "Brocard Points, Circulant Matrices, and 
Descartes' Folium." Math. Mag. 61, 172-187, 1988. 



176 



Brocard's Problem 



Brown Function 



Brocard's Problem 

Find the values of n for which n! + 1 is a SQUARE NUM- 
BER m 2 , where n! is the FACTORIAL (Brocard 1876, 
1885). The only known solutions are n = 4, 5, and 
7, and there are no other solutions < 1027. The pairs of 
numbers (m,n) are called Brown NUMBERS. 

see also BROWN NUMBERS, FACTORIAL, SQUARE NUM- 
BER 

References 

Brocard, H. Question 166. Nouv. Corres. Math. 2, 287, 

1876. 
Brocard, H. Question 1532. Nouv. Ann. Math. 4, 391, 1885. 
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer- Verlag, p. 193, 1994. 

Brocard Triangles 

Let the point of intersection of A 2 ^l and Azfl' be Bi, 
where Q and fl f are the Brocard Points, and similarly 
define B2 and £3. B1B2BZ is the first Brocard trian- 
gle, and is inversely similar to A1A2A3. It is inscribed 
in the BROCARD CIRCLE drawn with OK as the DIAM- 
ETER. The triangles B1A2A3, £ 2 A 3 Ai, and B3A1A2 
are ISOSCELES TRIANGLES with base angles lj, where u; 
is the Brocard Angle. The sum of the areas of the 
Isosceles Triangles is A, the Area of Triangle 
A1A2A3. The first Brocard triangle is in perspective 
with the given TRIANGLE, with AtB^ A 2 B 2 , and A3B3 
Concurrent. The Median Point of the first Brocard 
triangle is the MEDIAN POINT M of the original triangle. 
The Brocard triangles are in perspective at M. 

Let ci, c 2 , and c 3 and ci, c 2 , and c 3 be the CIRCLES 
intersecting in the Brocard Points Q and Q' , respec- 
tively. Let the two circles c\ and c[ tangent at A\ to 
A1A2 and A\A$, and passing respectively through As 
and A 2 , meet again at C\. The triangle C1C2C3 is the 
second Brocard triangle. Each Vertex of the second 
Brocard triangle lies on the second Brocard Circle. 

The two Brocard triangles arc in perspective at M. 

see also Steiner Points, Tarry Point 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 277-281, 1929. 

Bromwich Integral 

The inverse of the Laplace Transform, given by 



2iri I 

J -y — to 



'7—100 



where 7 is a vertical Contour in the Complex Plane 
chosen so that all singularities of f(s) are to the left of 
it. 

References 

Arfken, G. "Inverse Laplace Transformation." §15.12 in 

Mathematical Methods for Physicists, 3rd ed. Orlando, 

FL: Academic Press, pp. 853-861, 1985. 



Brothers 

A Pair of consecutive numbers. 

see also Pair, Smith Brothers, Twins 



Brouwer Fixed Point Theorem 

Any continuous FUNCTION G : D n -> D n has a FIXED 
Point, where 

£> n = {x€M n :xi 2 + ... + a;„ 2 <1} 

is the unit n-BALL. 

see also FIXED POINT THEOREM 

References 

Milnor, J. W. Topology from the Differentiate Viewpoint. 
Princeton, NJ: Princeton University Press, p. 14, 1965. 

Browkin's Theorem 

For every Positive Integer n, there exists a Square 
in the plane with exactly n Lattice Points in its inte- 
rior. This was extended by Schinzel and Kulikowski to 
all plane figures of a given shape. The generalization of 
the Square in 2-D to the Cube in 3-D was also proved 
by Browkin. 
see also Cube, Schinzel's Theorem, Square 

References 

Honsberger, R. Mathematical Gems I. Washington, DC: 
Math. Assoc. Amer., pp. 121-125, 1973. 

Brown's Criterion 

A Sequence {^} of nondecreasing Positive Integers 
is Complete Iff 

1. 1/1 = 1. 

2. For all k = 2, 3, . . . , 

S k -1 = v\ + ^2 + . . • + ffc-l > Vk - 1. 

A corollary states that a Sequence for which v\ = 1 
and v>k+i < 2vk is COMPLETE (Honsberger 1985). 
see also COMPLETE SEQUENCE 

References 

Brown, J. L. Jr. "Notes on Complete Sequences of Integers." 

Amer. Math. Monthly, 557-560, 1961. 
Honsberger, R. Mathematical Gems III. Washington, DC: 

Math. Assoc. Amer., pp. 123-130, 1985. 

Brown Function 

For a Fractal Process with values y(t — At) and y(t+ 
At) j the correlation between these two values is given by 
the Brown function 



1, 



also known as the Bachelier Function, Levy Func- 
tion, or Wiener Function. 



Brown Numbers 



Brun's Constant 



177 



Brown Numbers 

Brown numbers are Pairs (m, n) of Integers satisfying 

the condition of Brocard's Problem, i.e., such that 

n! + 1 = m 

where n! is the FACTORIAL and m 2 is a SQUARE Num- 
ber. Only three such Pairs of numbers are known: 
(5,4), (11,5), (71,7), and Erdos conjectured that these 
are the only three such Pairs. Le Lionnais (1983) points 
out that there are 3 numbers less than 200,000 for which 

(n-l)! + l = (mod n 2 ) , 

namely 5, 13, and 563. 

see also Brocard's Problem, Factorial, Square 
Number 

References 

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer- Verlag, p. 193, 1994. 
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 

p. 56, 1983. 
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, 

p. 170, 1995. 

Broyden's Method 

An extension of the secant method of root finding to 
higher dimensions. 

References 

Broyden, C. G. "A Class of Methods for Solving Nonlinear 
Simultaneous Equations." Math. Comput. 19, 577-593, 
1965. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 382-385, 1992. 

Bruck-Ryser-Chowla Theorem 

If n = 1, 2 (mod 4), and the SQUAREFREE part of n is di- 
visible by a Prime p = 3 (mod 4), then no Difference 
Set of ORDER n exists. Equivalently, if a PROJECTIVE 
PLANE of order n exists, and n — 1 or 2 (mod 4), then 
n is the sum of two SQUARES. 

Dinitz and Stinson (1992) give the theorem in the fol- 
lowing form. If a symmetric (v, k, A)-BLOCK DESIGN 
exists, then 

1. If v is Even, then k - A is a Square Number, 

2. If v is Odd, the the Diophantine Equation 



x 2 ^(k-\)y 2 + (-l) 



(f-l)/2 



\z z 



has a solution in integers, not all of which are 0. 

see also Block Design, Fisher's Block Design In- 
equality 

References 

Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to 
Design Theory." Ch. 1 in Contemporary Design Theory: A 



Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). 

New York: Wiley, pp. 1-12, 1992. 
Gordon, D. M. "The Prime Power Conjecture is True 

for n < 2,000,000." Electronic J. Combinatorics 1, 

R6, 1-7, 1994. http://www.combinatorics.org/Volume_l/ 

volume 1 ,html#R6. 
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: 

Math. Assoc. Amer., 1963. 

Bruck-Ryser Theorem 

see BRUCK-RYSER-CHOWLA Theorem 

Brun's Constant 

The number obtained by adding the reciprocals of the 

Twin Primes, 

(1) 
By Brun's Theorem, the constant converges to a def- 
inite number as p — > oo. Any finite sum underesti- 
mates B. Shanks and Wrench (1974) used all the Twin 
PRIMES among the first 2 million numbers. Brent (1976) 
calculated all Twin Primes up to 100 billion and ob- 
tained (Ribenboim 1989, p. 146) 



B « 1.90216054, 



(2) 



assuming the truth of the first HARDY-LlTTLEWOOD 
Conjecture. Using Twin Primes up to 10 14 , Nicely 
(1996) obtained 



B^ 1.9021605778 ±2.1 x 10 



-9 



(3) 



(Cipra 1995, 1996), in the process discovering a bug in 
Intel's® Pentium™ microprocessor. The value given by 
Le Lionnais (1983) is incorrect. 

see also Twin Primes, Twin Prime Conjecture, 
Twin Primes Constant 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 64, 1987. 

Brent, R. P. "Tables Concerning Irregularities in the Distri- 
bution of Primes and Twin Primes Up to 10 11 ." Math. 
Comput 30, 379, 1976. 

Cipra, B. "How Number Theory Got the Best of the Pentium 
Chip." Science 267, 175, 1995. 

Cipra, B. "Divide and Conquer." What's Happening in the 
Mathematical Sciences, 1995-1996, Vol 3. Providence, 
RI: Amer. Math. Soc, pp. 38-47, 1996. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/brun/brun.html. 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 41, 1983. 

Nicely, T. "Enumeration to 10 14 of the Twin Primes and 
Brun's Constant." Virginia J. Sci. 46, 195-204, 1996. 

Ribenboim, P. The Book of Prime Number Records, 2nd ed. 
New York: Springer- Verlag, 1989. 

Shanks, D. and Wrench, J. W. "Brun's Constant." Math. 
Comput 28, 293-299, 1974. 

Wolf, M. "Generalized Brun's Constants." http://www.ift. 
uni.wroc.pl/-mwolf/. 



178 Brunn-Minkowski Inequality 



Buffon's Needle Problem 



Brunn-Minkowski Inequality 

The nth root of the Content of the set sum of two sets 
in Euclidean n-space is greater than or equal to the sum 
of the nth roots of the Contents of the individual sets. 
see also TOMOGRAPHY 

References 

Cover, T. M. "The Entropy Power Inequality and the Brunn- 
Minkowski Inequality" §5.10 in In Open Problems in Com- 
munications and Computation. (Ed. T. M. Cover and 
B. Gopinath). New York: Springer- Verlag, p. 172, 1987. 

Schneider, R. Convex Bodies: The Brunn-Minkowski The- 
ory. Cambridge, England: Cambridge University Press, 
1993. 

Brun's Sum 

see Brun's Constant 

Brun's Theorem 

The series producing Brun's Constant Converges 
even if there are an infinite number of TWIN PRIMES. 
Proved in 1919 by V. Brun. 

Brunnian Link 

A Brunnian link is a set of n linked loops such that 
each proper sublink is trivial, so that the removal of any 
component leaves a set of trivial unlinked Unknots. 
The Borromean Rings are the simplest example and 
have n = 3. 
see also Borromean Rings 

References 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, 1976. 

Brute Force Factorization 

see Direct Search Factorization 

Bubble 

A bubble is a MINIMAL SURFACE of the type that is 
formed by soap film. The simplest bubble is a single 
Sphere. More complicated forms occur when multi- 
ple bubbles are joined together. Two outstanding prob- 
lems involving bubbles are to find the arrangements with 
the smallest Perimeter (planar problem) or Surface 
Area (Area problem) which enclose and separate n 
given unit areas or volumes in the plane or in space. 
For n — 2, the problems are called the DOUBLE BUB- 
BLE CONJECTURE and the solution to both problems is 
known to be the DOUBLE Bubble. 



see also Double Bubble, Minimal 
Plateau's Laws, Plateau's Problem 



Surface, 



References 

Morgan, F. "Mathematicians, Including Undergraduates, 

Look at Soap Bubbles." Amer. Math. Monthly 101, 343- 

351, 1994. 
Pappas, T. "Mathematics & Soap Bubbles." The Joy of 

Mathematics. San Carlos, CA: Wide World Publ./Tetra, 

p. 219, 1989. 



Buchberger's Algorithm 

The algorithm for the construction of a GROBNER BASIS 
from an arbitrary ideal basis. 

see also GROBNER BASIS 

References 

Becker, T. and Weispfenning, V. Grobner Bases: A Com- 
putational Approach to Commutative Algebra. New York: 
Springer- Verlag, pp. 213-214, 1993. 

Buchberger, B. "Theoretical Basis for the Reduction of Poly- 
nomials to Canonical Forms." SIGSAM Bull 39, 19-24, 
Aug. 1976. 

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and 
Algorithms: An Introduction to Algebraic Geometry and 
Commutative Algebra, 2nd ed. New York: Springer- 
Verlag, 1996. 

Buckminster Fuller Dome 

see Geodesic Dome 

Buffon-Laplace Needle Problem 







-4 1 X 


t v 


/ v 


£~ 


/ 








, ^ 


r ,% 


^ i 


^ + 





b 

h 

Find the probability P(£, a, b) that a needle of length £ 
will land on a line, given a floor with a grid of equally 
spaced Parallel Lines distances a and b apart, with 
£ > a,b. 

2£(a + b)-P 



P(*,a,6) = 



-nab 



see also BUFFON'S NEEDLE PROBLEM 

BufFon's Needle Problem 



/ 



^ 



/ 



/ 



Bulirsch-Stoer Algorithm 



Burau Representation 179 



Find the probability P(£>d) that a needle of length £ 
will land on a line, given a floor with equally spaced 
Parallel Lines a distance d apart. 



P&d) 



-f 

Jo 



£\cosO\ dd _ t 



= -[8in*] ' 



27r 2nd 

- *L 

ird 



/.tt/2 

7 ' 

Jo 



cos 8 dO 



Several attempts have been made to experimentally de- 
termine 7r by needle- tossing. For a discussion of the 
relevant statistics and a critical analysis of one of the 
more accurate (and least believable) needle-tossings, see 
Badger (1994). 

see also Buffon-Laplace Needle Problem 

References 

Badger, L. "Lazzarini's Lucky Approximation of 7r." Math. 

Mag. 67, 83-91, 1994. 
Dorrie, H. "Buffon's Needle Problem." §18 in 100 Great 

Problems of Elementary Mathematics: Their History and 

Solutions. New York: Dover, pp. 73-77, 1965. 
Kraitchik, M. "The Needle Problem." §6.14 in Mathematical 

Recreations. New York: W. W. Norton, p. 132, 1942. 
Wegert, E. and Trefethen, L, N. "Prom the Buffon Needle 

Problem to the Kreiss Matrix Theorem." Amer. Math. 

Monthly 101, 132-139, 1994. 

Bulirsch-Stoer Algorithm 

An algorithm which finds RATIONAL FUNCTION extrap- 
olations of the form 



Ri(i + l)---(i+m) 



Py(x) __ po + p\x + . . . +p^x M 
P„(x) qo + qix + . . . + q u x v 



and can be used in the solution of Ordinary Differ- 
ential Equations. 

References 

Bulirsch, R. and Stoer, J. §2.2 in Introduction to Numerical 
Analysis. New York: Springer- Verlag, 1991. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Richardson Extrapolation and the Bulirsch- 
Stoer Method." §16.4 in Numerical Recipes in FORTRAN: 
The Art of Scientific Computing, 2nd ed. Cambridge, Eng- 
land: Cambridge University Press, pp. 718-725, 1992. 

Bullet Nose 



A plane curve with implicit equation 



x 1 y 2 



(1) 



The Curvature is 



x = a cost 
y = b cot t. 

Sab cot t esc t 



(6 2 csc 4 i + a 2 sin 2 i) 3 / 2 
and the TANGENTIAL ANGLE is 



■ = tan 



_i /bcsc 3 A 



(2) 
(3) 

(4) 
(5) 



References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 127-129, 1972. 

Bumping Algorithm 

Given a Permutation {pi,f>2, ■ ■ • ,Vn) of {1, . . . , n}, 
the bumping algorithm constructs a standard YOUNG 
Tableau by inserting the pi one by one into an already 
constructed YOUNG TABLEAU. To apply the bump- 
ing algorithm, start with {{pi}}, which is a YOUNG 
TABLEAU. If p\ through pk have already been inserted, 
then in order to insert pfc+i, start with the first line of 
the already constructed YOUNG TABLEAU and search 
for the first element of this line which is greater than 
Pk+i- If there is no such element, append Pk+\ to the 
first line and stop. If there is such an element (say, p p ), 
exchange p p for pjt+i, search the second line using p p , 
and so on. 

see also YOUNG TABLEAU 

References 

Skiena, S. Implementing Discrete Mathematics: Combina- 
torics and Graph Theory with Mathematica. Reading, 
MA: Addison- Wesley, 1990. 

Bundle 

see Fiber Bundle 

Burau Representation 

Gives a Matrix representation b* of a Braid Group 
in terms of (n - 1) x (n - 1) Matrices. A -t always 

appears in the (i,i) position. 



bi = 



-too 

-1 1 
1 



(1) 



bi = 



In parametric form, 



1 


•• 


• 


• 





■ ■ -t 


• 


• 





•• -t 


o . 


• 





• ■ -1 


1 ■ 


• 





'. 










•• 


• 


• 1 



(2) 



180 Burkhardt Quartic 



Burnside Problem 



b n _ 





rl 


• 


• 










1 ■ 


• 





1 — 















■ 


• 


— 




Lo 


• 


• 


— 



(3) 



Let * be the Matrix Product of Braid Words, then 

det(l - 9) _ , , 

l + t + . .. + t»-i " AL ' (4) 

where A L is the ALEXANDER POLYNOMIAL and det is 
the Determinant. 

References 

Burau, W. "Uber Zopfgruppen und gleichsinnig verdrilte Ver- 

kettungen." Abh. Math. Sem. Hanischen Univ. 11, 171- 

178, 1936. 
Jones, V. "Hecke Algebra Representation of Braid Groups 

and Link Polynomials." Ann. Math. 126, 335-388, 1987. 

Burkhardt Quartic 

The Variety which is an invariant of degree four and 
is given by the equation 



yt 



2/0(2/? 



3 , 3 

■ 2/2 + yz ■ 



■2/1) + 32/12/22/32/4 = 0. 



References 

Burkhardt, H. "Untersuchungen aus dem Gebiet der hyperel- 
liptischen Modulfunctionen. II." Math. Ann. 38, 161-224, 
1890. 

Burkhardt, H. "Untersuchungen aus dem Gebiet der hyper- 
elliptischen Modulfunctionen. III." Math. Ann. 40, 313- 
343, 1892. 

Hunt, B. "The Burkhardt Quartic." Ch. 5 in The Geom- 
etry of Some Special Arithmetic Quotients. New York: 
Springer- Verlag, pp. 168-221, 1996. 

Burnside's Conjecture 

Every non-ABELIAN SIMPLE GROUP has EVEN ORDER. 

see also Abelian Group, Simple Group 

Burnside's Lemma 

Let J be a Finite Group and the image R(J) be a 
representation which is a HOMEOMORPHISM of J into a 
Permutation Group S(X), where S(X) is the Group 
of all permutations of a Set X. Define the orbits o£R(J) 
as the equivalence classes under x ~ y, which is true if 
there is some permutation p in R( J) such that p(x) = y. 
Define the fixed points of p as the elements x of X for 
which p(x) = x. Then the AVERAGE number of FIXED 
POINTS of permutations in R(J) is equal to the number 
of orbits of R(J). 

The LEMMA was apparently known by Cauchy (1845) in 
obscure form and Frobenius (1887) prior to Burnside's 
(1900) rediscovery. It was subsequently extended and 
refined by Polya (1937) for applications in COMBINATO- 
RIAL counting problems. In this form, it is known as 

Polya Enumeration Theorem. 

References 

Polya, G. "Kombinatorische Anzahlbestimmungen fur Grup- 

pen, Graphen, und chemische Verbindungen." Acta Math. 

68, 145-254, 1937. 



Burnside Problem 

A problem originating with W. Burnside (1902), who 
wrote, "A still undecided point in the theory of dis- 
continuous groups is whether the Order of a Group 
may be not finite, while the order of every operation 
it contains is finite." This question would now be 
phrased as "Can a finitely generated group be infinite 
while every element in the group has finite order?" 
(Vaughan-Lee 1990). This question was answered by 
Golod (1964) when he constructed finitely generated in- 
finite p-GROUPS. These GROUPS, however, do not have 
a finite exponent. 

Let F r be the Free Group of Rank r and let N be 
the Subgroup generated by the set of nth POWERS 
{g n \g e F r }. Then TV is a normal subgroup of F r . We 
define B(r, n) = F r /N to be the QUOTIENT GROUP. We 
call B(r,n) the r-generator Burnside group of exponent 
n. It is the largest r-generator group of exponent n, in 
the sense that every other such group is a HOMEOMOR- 
PHIC image of B(r, n). The Burnside problem is usually 
stated as: "For which values of r and n is £(r,n) a 
Finite Group?" 

An answer is known for the following values. For r = 1, 
5(1,77) is a Cyclic Group of Order n. For n = 2, 
B(r, 2) is an elementary Abelian 2-group of Order 2 n , 
For n = 3, B(r, 3) was proved to be finite by Burnside. 
The ORDER of the B(r,3) groups was established by 
Levi and van der Waerden (1933), namely 3 a where 



:r + 



(1) 



where (™) is a Binomial COEFFICIENT. For n = 4, 
B(r> 4) was proved to be finite by Sanov (1940). Groups 
of exponent four turn out to be the most complicated 
for which a POSITIVE solution is known. The precise 
nilpotency class and derived length are known, as are 
bounds for the ORDER. For example, 



|S(2,4)| = 2 12 
|B(3,4)| = 2 69 
|S(4,4)| = 2 422 
|B(5,4)|=2 2728 



(2) 
(3) 
(4) 
(5) 



while for larger values of r the exact value is not yet 
known. For n = 6, B(r,6) was proved to be finite by 
Hall (1958) with ORDER 2 a 3 6 , where 



a = 1 + (r - 1)3 C 
6 = l + (r-l)2 r 



c = r + 



+ 



(6) 
(7) 

(8) 



No other Burnside groups are known to be finite. On 
the other hand, for r > 2 and n > 665, with n ODD, 



Busemann-Petty Problem 

B(r,n) is infinite (Novikov and Adjan 1968). There is a 
similar fact for r > 2 and n a large Power of 2. 

E. Zelmanov was awarded a Fields Medal in 1994 for 
his solution of the "restricted" Burnside problem. 

see also FREE GROUP 

References 

Burnside, W. "On an Unsettled Question in the Theory of 

Discontinuous Groups." Quart. J. Pure Appl. Math. 33, 

230-238, 1902. 
Golod, E. S. "On Nil-Algebras and Residually Finite p- 

Groups." Isv. Akad. Nauk SSSR Ser. Mat. 28, 273-276, 

1964. 
Hall, M. "Solution of the Burnside Problem for Exponent 

Six." Ill J. Math. 2, 764-786, 1958. „ 
Levi, F. and van der Waerden, B. L. "Uber eine besondere 

Klasse von Gruppen." Abh. Math. Sem. Univ. Hamburg 

9, 154-158, 1933. 
Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I, 

II, III." Izv. Akad. Nauk SSSR Ser. Mat 32, 212-244, 

251-524, and 709-731, 1968. 
Sanov, I. N. "Solution of Burnside's problem for exponent 

four." Leningrad State Univ. Ann. Math. Ser. 10, 166— 

170, 1940. 
Vaughan-Lee, M. The Restricted Burnside Problem, 2nd ed. 

New York: Clarendon Press, 1993. 

Busemann-Petty Problem 

If the section function of a centered convex body in Eu- 
clidean n-space (n > 3) is smaller than that of another 
such body, is its volume also smaller? 

References 

Gardner, R. J. "Geometric Tomography." Not. Amer. Math. 
Soc. 42, 422-429, 1995. 

Busy Beaver 

A busy beaver is an n-state, 2-symbol, 5-tuple Turing 
MACHINE which writes the maximum possible number 
BB(n) of Is on an initially blank tape before halting. 
For n = 0, 1, 2, ... , BB(n) is given by 0, 1, 4, 6, 13, 

> 4098, > 136612, The busy beaver sequence is 

also known as Rado's Sigma Function. 

see also HALTING PROBLEM, TURING MACHINE 

References 

Chaitin, G. J. "Computing the Busy Beaver Function." §4.4 

in Open Problems in Communication and Computation 

(Ed. T. M. Cover and B. Gopinath). New York: Springer- 

Verlag, pp. 108-112, 1987. 
Dewdney, A. K. "A Computer Trap for the Busy Beaver, 

the Hardest- Working Turing Machine." Sci. Amer. 251, 

19-23, Aug. 1984. 
Marxen, H. and Buntrock, J. "Attacking the Busy Beaver 5." 

Bull. EATCS40, 247-251, Feb. 1990. 
Sloane, N. J. A. Sequence A028444 in "An On-Line Version 

of the Encyclopedia of Integer Sequences." 



Butterfly Fractal 181 
Butterfly Catastrophe 




A Catastrophe which can occur for four control fac- 
tors and one behavior axis. The equations 

x = c(Sat 3 + 24t 5 ) 
y = c(-6ai 2 - 15t 4 ) 

display such a catastrophe (von Seggern 1993). 

References 

von Seggern, D. CRC Standard Curves and Surfaces. Boca 
Raton, FL: CRC Press, p. 94, 1993. 

Butterfly Curve 




A Plane Curve given by the implicit equation 
y =(x -x ). 

see also DUMBBELL CURVE, EIGHT CURVE, PIRIFORM 

References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., p. 72, 1989. 

Butterfly Effect 

Due to nonlinearities in weather processes, a butterfly 
flapping its wings in Tahiti can, in theory, produce a 
tornado in Kansas. This strong dependence of outcomes 
on very slightly differing initial conditions is a hallmark 
of the mathematical behavior known as CHAOS. 

see also Chaos, Lorenz System 
Butterfly Fractal 




The FRACTAL-like curve generated by the 2-D function 
(z 2 -y 2 )sin(^) 



ffay) = 



x 2 +y 2 



182 Butterfly Polyiamond Butterfly Theorem 

Butterfly Polyiamond 




A 6-POLYIAMOND. 

References 

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, 
and Packings, 2nd ed. Princeton, NJ: Princeton University 
Press, p. 92, 1994. 

Butterfly Theorem 

A 




Given a Chord PQ of a Circle, draw any other two 
CHORDS AB and CD passing through its MIDPOINT. 
Call the points where AD and BC meet PQ X and Y. 
Then M is the Midpoint of XY. 

see also CHORD, CIRCLE, MIDPOINT 
References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited, 
Washington, DC: Math. Assoc. Amer., pp. 45-46, 1967. 



Cake Cutting 183 



C 



C-Table 

see C-Determinant 



The Field of Complex Numbers, denoted C. 
see also C\ Complex Number, I, N, Q, R, Z 

C* 

The Riemann Sphere C U {oo}, 

see also C, Complex Number, Q, R, Riemann 
Sphere, Z 

C*-Algebra 

A special type of B* -Algebra in which the Involu- 
tion is the Adjoint Operator in a Hilbert Space. 

see also £*-ALGEBRA, fc-THEORY 

References 

Davidson, K. R. C* -Algebras by Example. Providence, RI: 
Amer. Math. Soc, 1996. 

C- Curve 

see Levy Fractal 



Cable Knot 

Let Ki be a Torus Knot. Then the Satellite Knot 
with Companion Knot K 2 is a cable knot on K 2 . 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 
to the Mathematical Theory of Knots. New York: W. H. 
Freeman, p. 118, 1994. 

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or 
Perish Press, pp. 112 and 283, 1976. 



Cactus Fractal 



•m 



A Mandelbrot SET-like Fractal obtained by iterat- 
ing the map 

Zn+l = Z n + (ZQ — l)z n — Zq - 



C-Determinant 

A Determinant appearing in Pade Approximant 
identities: 



a 



s + l <Xr-s+2 



a r +\ 



Gr+s-1 



see also Pade APPROXIMANT 

C-Matrix 

Any Symmetric Matrix (A t = A) or Skew Symmet- 
ric Matrix (A t = -A) C™ with diagonal elements 
and others ±1 satisfying 

CC T = (n-l)l, 

where I is the IDENTITY MATRIX, is known as a C- 
matrix (Ball and Coxeter 1987), Examples include 



c 4 = 






+ 


+ 


+ 






- 





- 


+ 






- 


+ 





- 






- 


- 


+ 


0_ 









+ 


+ 


+ 


+ 


+ 


+ 





+ 


- 


- 


+ 


+ 


+ 





+ 


+ 


__ 


+ 


- 


+ 





+ 


- 


+ 


- 


- 


+ 





+ 


+ 


+ 


- 


- 


+ 






c 6 = 



References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, pp. 308- 
309, 1987. 



see also FRACTAL, JULIA SET, MANDELBROT SET 

Cake Cutting 

It is always possible to "fairly" divide a cake among n 
people using only vertical cuts. Furthermore, it is pos- 
sible to cut and divide a cake such that each person 
believes that everyone has received 1/n of the cake ac- 
cording to his own measure. Finally, if there is some 
piece on which two people disagree, then there is a way 
of partitioning and dividing a cake such that each par- 
ticipant believes that he has obtained more than 1/n of 
the cake according to his own measure. 

Ignoring the height of the cake, the cake-cutting problem 
is really a question of fairly dividing a CIRCLE into n 
equal Area pieces using cuts in its plane. One method 
of proving fair cake cutting to always be possible relies 
on the Frobenius-Konig Theorem. 

see also CIRCLE CUTTING, CYLINDER CUTTING, EN- 
VYFREE, FROBENIUS-KONIG THEOREM, HAM SAND- 
WICH Theorem, Pancake Theorem, Pizza Theo- 
rem, Square Cutting, Torus Cutting 

References 

Brams, S. J. and Taylor, A. D. "An Envy-Free Cake Division 
Protocol." Amer. Math. Monthly 102, 9-19, 1995. 

Brams, S. J. and Taylor, A. D. Fair Division: From Cake- 
Cutting to Dispute Resolution. New York: Cambridge Uni- 
versity Press, 1996. 

Dubbins, L. and Spanier, E. "How to Cut a Cake Fairly." 
Amer. Math. Monthly 68, 1-17, 1961. 

Gale, D. "Dividing a Cake." Math. Intel. 15, 50, 1993. 

Jones, M. L. "A Note on a Cake Cutting Algorithm of Banach 
and Knaster." Amer. Math. Monthly 104, 353-355, 1997. 

Rebman, K. "How to Get (At Least) a Fair Share of the 
Cake." In Mathematical Plums (Ed. R. Honsberger). 
Washington, DC: Math. Assoc. Amer., pp. 22-37, 1979. 



184 



Cal 



Calculus of Variations 



rsi 



Steinhaus, H. "Sur la division progmatique." Ekonometrika 

(Supp.) 17, 315-319, 1949. 
Stromquist, W. "How to Cut a Cake Fairly." Amer. Math. 

Monthly 87, 640-644, 1980. 

Cal 

see Walsh Function 

Calabi's Triangle 



and Integrals 



/ 



f(x) dx, 





Equilateral Triangle Calabi's Triangle 

The one TRIANGLE in addition to the EQUILATERAL 
Triangle for which the largest inscribed Square 
can be inscribed in three different ways. The ra- 
tio of the sides to that of the base is given by x = 
1.55138752455. . . (Sloane's A046095), where 



11 



_ 1 (-23 + 3zy / 237) 1/3 

X ~ 3 + 3-2 2 /3 + 3[ 2 (-23 + 3iv / 237)] 1 / 3 



is the largest POSITIVE ROOT of 



2x 3 - 2x 2 - 3z + 2 = 0, 



which has CONTINUED FRACTION [1, 1, 1, 4, 2, 1, 2, 1, 
5, 2, 1, 3, 1, 1, 390, . . .] (Sloane's A046096). 

see also GRAHAM'S BIGGEST LITTLE HEXAGON 

References 

Conway, J. H. and Guy, R. K. "Calabi's Triangle." In The 

Book of Numbers. New York: Springer- Verlag, p. 206, 

1996, 
Sloane, N. J. A. Sequences A046095 and A046096 in "An On- 

Line Version of the Encyclopedia of Integer Sequences." 

Calabi-Yau Space 

A structure into which the 6 extra Dimensions of 10-D 
string theory curl up. 

Calculus 

In general, "a" calculus is an abstract theory developed 
in a purely formal way. 

"The" calculus, more properly called ANALYSIS (or 
Real Analysis or, in older literature, Infinitesimal 
Analysis) is the branch of mathematics studying the 
rate of change of quantities (which can be interpreted as 
Slopes of curves) and the length, Area, and Volume 
of objects. The CALCULUS is sometimes divided into 
Differential and Integral Calculus, concerned 
with Derivatives 



respectively. 

While ideas related to calculus had been known for some 
time (Archimedes' Exhaustion Method was a form 
of calculus), it was not until the independent work of 
Newton and Leibniz that the modern elegant tools and 
ideas of calculus were developed. Even so, many years 
elapsed until the subject was put on a mathematically 
rigorous footing by mathematicians such as Weierstraft. 
see also Arc Length, Area, Calculus of Vari- 
ations, Change of Variables Theorem, De- 
rivative, Differential Calculus, Ellipsoidal 
Calculus, Extensions Calculus, Fluent, Flux- 
ion, Fractional Calculus, Functional Calculus, 
Fundamental Theorems of Calculus, Heaviside 
Calculus, Integral, Integral Calculus, Jaco- 
bian, Lambda Calculus, Kirby Calculus, Malli- 
avin Calculus, Predicate Calculus, Proposi- 
tional Calculus, Slope, Tensor Calculus, Um- 
bral Calculus, Volume 

References 

Anton, H. Calculus with Analytic Geometry, 5th ed. New 
York: Wiley, 1995. 

Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Cal- 
culus, with an Introduction to Linear Algebra. Waltham, 
MA: Blaisdell, 1967. 

Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Cal- 
culus and Linear Algebra, with Applications to Differential 
Equations and Probability. Waltham, MA: Blaisdell, 1969. 

Apostol, T. M. A Century of Calculus, 2 vols. Pt. 1: 1894~ 
1968. Pt. 2: 1969-1991. Washington, DC: Math. Assoc. 
Amer., 1992. 

Ayres, F. Jr. and Mendelson, E. Schaum's Outline of Theory 
and Problems of Differential and Integral Calculus, 3rd ed. 
New York: McGraw-Hill, 1990. 

Borden, R. S, A Course in Advanced Calculus. New York: 
Dover, 1998. 

Boyer, C B. A History of the Calculus and Its Conceptual 
Development. New York: Dover, 1989. 

Brown, K. S. "Calculus and Differential Equations." http:// 
www. seanet . com/-ksbrown/icalculu.htm. 

Courant, R. and John, F. Introduction to Calculus and Anal- 
ysis, Vol. 1. New York: Springer- Verlag, 1990. 

Courant, R. and John, F. Introduction to Calculus and Anal- 
ysis, Vol. 2. New York: Springer- Verlag, 1990. 

Hahn, A. Basic Calculus: From Archimedes to Newton to Its 
Role in Science. New York: Springer- Verlag, 1998. 

Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: 
Addison- Wesley, 1992. 

Marsden, J. E. and Tromba, A. J. Vector Calculus, ^i/i ed. 
New York: W. H. Freeman, 1996. 

Strang, G. Calculus. Wellesley, MA: Wellesley-Cambridge 
Press, 1991. 

Calculus of Variations 

A branch of mathematics which is a sort of general- 
ization of CALCULUS. Calculus of variations seeks to 
find the path, curve, surface, etc., for which a given 
Function has a Stationary Value (which, in physical 



Calcus 

problems, is usually a Minimum or Maximum). Mathe- 
matically, this involves finding STATIONARY VALUES of 
integrals of the form 



'= / /(y.y, 



x) dx. 



(i) 



J has an extremum only if the Euler-Lagrange Dif- 
ferential Equation is satisfied, i.e., if 



dy 



dx \dyj 



(2) 



The Fundamental Lemma of Calculus of Varia- 
tions states that, if 



t/ a 



M(x)h(x)dx = 



(3) 



for all h(x) with CONTINUOUS second PARTIAL DERIVA- 
TIVES, then 

M(x) = (4) 

on (a, 6). 

see also BELTRAMI IDENTITY, BOLZA PROBLEM, 

Brachistochrone Problem, Catenary, Enve- 
lope Theorem, Euler-Lagrange Differential 
Equation, Isoperimetric Problem, Isovolume 
Problem, Lindelof's Theorem, Plateau's Prob- 
lem, Point-Point Distance — 2-D, Point-Point 
Distance— 3-D, Roulette, Skew Quadrilateral, 
Sphere with Tunnel, Unduloid, WeierstraB- 
Erdman Corner Condition 

References 

Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical 

Methods for Physicists, 3rd ed. Orlando, FL: Academic 

Press, pp. 925-962, 1985. 
Bliss, G. A. Calculus of Variations. Chicago, IL: Open 

Court, 1925. 
Forsyth, A. R. Calculus of Variations. New York: Dover, 

1960. 
Fox, C An Introduction to the Calculus of Variations. New- 
York: Dover, 1988. 
Isenberg, C The Science of Soap Films and Soap Bubbles. 

New York: Dover, 1992. 
Menger, K. "What is the Calculus of Variations and What 

are Its Applications?" In The World of Mathematics (Ed. 

K. Newman). Redmond, WA: Microsoft Press, pp. 886- 

890, 1988. 
Sagan, H. Introduction to the Calculus of Variations. New 

York: Dover, 1992. 
Todhunter, I. History of the Calculus of Variations During 

the Nineteenth Century. New York: Chelsea, 1962. 
Weinstock, R. Calculus of Variations, with Applications to 

Physics and Engineering. New York: Dover, 1974. 



Calcus 



1 calcus = 



see also Half, Quarter, Scruple, Uncia, Unit 
Fraction 



Cancellation Law 185 
Calderon's Formula 

/oo /*oo 
/ (f,tp a ' b )i> a - b (x)a.- 2 dadb, 
-oo J — CO 



where 



r' b (x) = \a\-^(^.). 



This result was originally derived using HARMONIC 
Analysis, but also follows from a Wavelets viewpoint. 

Caliban Puzzle 

A puzzle in LOGIC in which one or more facts must be 
inferred from a set of given facts. 

Calvary Cross 






see also CROSS 

Cameron's Sum-Free Set Constant 

A set of POSITIVE INTEGERS S is sum-free if the equa- 
tion x 4- y = z has no solutions x, y, z 6 S. The proba- 
bility that a random sum-free set S consists entirely of 
Odd Integers satisfies 

0.21759 < c < 0.21862. 



References 

Cameron, P. J. "Cyclic Automorphisms of a Countable 
Graph and Random Sum-Free Sets." Graphs and Com- 
binatorics 1, 129-135, 1985. 

Cameron, P. J. "Portrait of a Typical Sum- Free Set." In 
Surveys in Combinatorics 1987 (Ed. C. Whitehead). New 
York: Cambridge University Press, 13-42, 1987. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/cameron/cameron.html. 

Cancellation 

see Anomalous Cancellation 

Cancellation Law 

If be = bd (mod a) and (6, a) — 1 (i.e., a and b are 
Relatively Prime), then c~ d (mod a). 

see also CONGRUENCE 

References 

Courant, R. and Robbins, H. What is Mathematics?: An El- 
ementary Approach to Ideas and Methods, 2nd ed. Oxford, 
England: Oxford University Press, p. 36, 1996. 

Shanks, D. Solved and Unsolved Problems in Number Theory, 
4th ed. New York: Chelsea, p. 56, 1993. 



186 



Cannonball Problem 



Cantor Dust 



Cannonball Problem 

Find a way to stack a SQUARE of cannonballs laid out on 
the ground into a Square Pyramid (i.e., find a Square 
Number which is also Square Pyramidal). This cor- 
responds to solving the DlOPHANTINE EQUATION 



Cantor-Dedekind Axiom 

The points on a line can be put into a One-to-One 
correspondence with the REAL NUMBERS. 

see also Cardinal Number, Continuum Hypothe- 
sis, Dedekind Cut 



£V = I*(1 + *)(! + 2*) 



N 2 



for some pyramid height k. The only solution is k = 24, 
N = 70, corresponding to 4900 cannonballs (Ball and 
Coxeter 1987, Dickson 1952), as conjectured by Lucas 
(1875, 1876) and proved by Watson (1918). 
see also Sphere Packing, Square Number, Square 
Pyramid, Square Pyramidal Number 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, p. 59, 1987. 

Dickson, L. E. History of the Theory of Numbers, Vol. 2: 
Diophantine Analysis. New York: Chelsea, p. 25, 1952. 

Lucas, E. Question 1180. Nouvelles Ann. Math. Ser. 2 14, 
336, 1875. 

Lucas, E. Solution de Question 1180. Nouvelles Ann. Math. 
Ser. 2 15, 429-432, 1876. 

Ogilvy, C. S. and Anderson, J. T. Excursions in Number 
Theory. New York: Dover, pp. 77 and 152, 1988. 

Pappas, T. "Cannon Balls & Pyramids." The Joy of Math- 
ematics. San Carlos, CA: Wide World Publ./Tetra, p. 93, 
1989. 

Watson, G. N. "The Problem of the Square Pyramid." Mes- 
senger. Math. 48, 1-22, 1918. 

Canonical Form 

A clear-cut way of describing every object in a class in 
a One-to-One manner. 

see also Normal Form, One-to-One 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, p. 7, 1996. 

Canonical Polyhedron 

A Polyhedron is said to be canonical if all its Edges 
touch a Sphere and the center of gravity of their contact 
points is the center of that Sphere. Each combinato- 
rial type of (GENUS zero) polyhedron contains just one 
canonical version. The ARCHIMEDEAN SOLIDS and their 
DUALS are all canonical. 



References 

Conway, J. H. "Re: polyhedra database." 
geometry. forum newsgroup, Aug. 31, 1995. 

Canonical Transformation 

see Symplectic Diffeomorphism 

Cantor Comb 
see Cantor Set 



Posting to 



Cantor Diagonal Slash 

A clever and rather abstract technique used by Georg 
Cantor to show that the Integers and Reals cannot be 
put into a One-to-One correspondence (i.e., the INFIN- 
ITY of Real Numbers is "larger" than the Infinity of 
INTEGERS), It proceeds by constructing a new member 
S' of a Set from already known members S by arrang- 
ing its nth term to differ from the nth term of the nth 
member of S. The tricky part is that this is done in 
such a way that the Set including the new member has 
a larger CARDINALITY than the original SET S. 

see also Cardinality, Continuum Hypothesis, De- 

NUMERABLE SET 

References 

Courant, R. and Robbins, H. What is Mathematics?: An El- 
ementary Approach to Ideas and Methods, 2nd ed. Oxford, 
England: Oxford University Press, pp. 81-83, 1996. 

Penrose, R. The Emperor's New Mind: Concerning Comput- 
ers, Minds, and the Laws of Physics. Oxford, England: 
Oxford University Press, pp. 84-85, 1989. 

Cantor Dust 




A Fractal which can be constructed using String Re- 
writing by creating a matrix three times the size of the 
current matrix using the rules 

line 1: "*"->"* *",'' "->" " 
line 2: "*"->" "," *'->" 
line 3: "*»->"* *",» »->" 

Let N n be the number of black boxes, L n the length of 
a side of a white box, and A n the fractional Area of 
black boxes after the nth iteration. 



iVn-5 71 

A n = L n 2 N n = ($) n . 
The Capacity Dimension is therefore 



(1) 
(2) 
(3) 



ln(5 n ) 



r lniV n 
= - hm - — — = - hm /0 _ . 

n-J-oo III L n n->-oo Ul(cJ n ) 



In 5 
ln3 



1.464973521. 



(4) 



see also Box FRACTAL, SlERPINSKI CARPET, SlERPIN- 

ski Sieve 



Cantor's Equation 



Cantor Square Fractal 187 



References 

Dickau, R. M. "Cantor Dust." http://f orum . swarthmore . 

edu/advanced/robertd/cantor .html. 
Ott, E. Chaos in Dynamical Systems. New York: Cambridge 

University Press, pp. 103-104, 1993. 
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia. 

edu/~eww6n/math/notebooks/Fractal.m. 

Cantor's Equation 



Cantor Set 

The Cantor set (Too) is given by taking the interval [0,1] 
(set To), removing the middle third (Ti), removing the 
middle third of each of the two remaining pieces (T2), 
and continuing this procedure ad infinitum. It is there- 
fore the set of points in the INTERVAL [0,1] whose ternary 
expansions do not contain 1, illustrated below. 



where uj is an Ordinal Number and e is an Inacces- 
sible Cardinal, 

see also INACCESSIBLE CARDINAL, ORDINAL NUMBER 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, p. 274, 1996. 

Cantor Function 

The function whose values are 



2 V 2 



Cm-l . 2 
Orn — 1 Orn 



for any number between 



This produces the Set of Real Numbers {x} such that 

(i) 



Cl C n 

X= 3+--- + F + ---' 



where c n may equal or 2 for each n. This is an infinite, 
Perfect Set. The total length of the Line Segments 

in the nth iteration is 



*•-(!)"■ 



(2) 



and the number of LINE SEGMENTS is N n = 2 n , so the 
length of each element is 



tn - N~ (3) 



(3) 



Cl Cm-l 

"3 * ' ' 3™" 1 



and 



Cl 



+ ■ 



Cm-l _2_ 



Chalice (1991) shows that any real- values function F(x) 
on [0, 1] which is MONOTONE INCREASING and satisfies 

1. F(0) = 0, 

2. F(x/S) = F{x)/2, 

3. F(l-x) = 1-F(x) 
is the Cantor function. 

see also CANTOR SET, DEVIL'S STAIRCASE 

References 

Chalice, D. R. "A Characterization of the Cantor Function." 
Amer. Math. Monthly 98, 255-258, 1991. 

Wagon, S. "The Cantor Function" and "Complex Cantor 
Sets." §4.2 and 5.1 in Mathematica in Action. New York: 
W. H. Freeman, pp. 102-108 and 143-149, 1991. 

Cantor's Paradox 

The Set of all Sets is its own Power Set. Therefore, 
the Cardinality of the Set of all Sets must be bigger 
than itself. 
see also CANTOR'S THEOREM, POWER SET 



and the Capacity DIMENSION is 

In AT 



lim _ 
€-►0+ me 



lim 



nln2 



00 — nln3 



In 2 

In 3 



0.630929... 



(4) 



The Cantor set is nowhere Dense, so it has LEBESGUE 

MEASURE 0. 

A general Cantor set is a CLOSED SET consisting en- 
tirely of BOUNDARY POINTS. Such sets are UNCOUNT- 
ABLE and may have or POSITIVE LEBESGUE MEA- 
SURE. The Cantor set is the only totally disconnected, 
perfect, Compact Metric Space up to a Homeomor- 
PHISM (Willard 1970). 

see also Alexander's Horned Sphere, Antoine's 
Necklace, Cantor Function 

References 

Boas, R. P. Jr. A Primer of Real Functions. Washington, 
DC: Amer. Math. Soc, 1996. 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princetqn, NJ: Princeton University Press, pp. 15- 
20, 1991. 

Willard, S. §30.4 in General Topology. Reading, MA: 
Addison- Wesley, 1970. 

Cantor Square Fractal 



188 



Cantor's Theorem 



Cardano's Formula 



A Fractal which can be constructed using String Re- 
writing by creating a matrix three times the size of the 
current matrix using the rules 

line 1: "*"->"***"," "->" " 

line 2: "*"->"* *"," "->" " 

line 3: "*"->"***",» "->" " 

The first few steps are illustrated above. 

The size of the unit element after the nth iteration is 



L n 



G)" 



and the number of elements is given by the RECUR- 
RENCE Relation 

N n = 4JV n _i + 5(9 n ) 

where Ni = 5, and the first few numbers of elements are 
5, 65, 665, 6305, Expanding out gives 



N n 



5 \p 4 n-fc g fc-l =9 n_ 4 n_ 



fc=0 



The Capacity Dimension is therefore 



liml^-lim^ 9 "- 4 ") 



th-oo In L n 
ln(9 n 



n-+oo ln(3- n ) 



n-^oo ln(3" n ) 
ln9 _ 21n3 _ 
ln3 "" In 3 ~ 



2. 



Since the DIMENSION of the filled part is 2 (i.e., the 
SQUARE is completely filled), Cantor's square fractal is 
not a true FRACTAL. 

see also Box Fractal, Cantor Dust 

References 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 82- 
83, 1991. 
^ Weisstein, E. W. "Fractals." http://www. astro. Virginia. 
edu/-eww6n/math/notebooks/Fractal.m. 

Cantor's Theorem 

The Cardinal Number of any set is lower than the 
Cardinal Number of the set of all its subsets. A 
Corollary is that there is no highest N (Aleph). 

see also Cantor's Paradox 

Cap 

see Cross-Cap, Spherical Cap 

Capacity 

see Transfinite Diameter 



Capacity Dimension 

A Dimension also called the Fractal Dimen- 
sion, Hausdorff Dimension, and Hausdorff- 
Besicovitch Dimension in which nonintegral values 
are permitted. Objects whose capacity dimension is dif- 
ferent from their TOPOLOGICAL Dimension are called 
Fractals. The capacity dimension of a compact Met- 
ric Space X is a Real Number capacity such that if 
n(e) denotes the minimum number of open sets of diam- 
eter less than or equal to e, then n(e) is proportional to 
e~ D as e — > 0. Explicitly, 



-^capacity 



,. miV 

hm 

€-►0+ hie 



(if the limit exists), where N is the number of elements 
forming a finite Cover of the relevant Metric SPACE 
and e is a bound on the diameter of the sets involved 
(informally, e is the size of each element used to cover 
the set, which is taken to to approach 0). If each ele- 
ment of a Fractal is equally likely to be visited, then 

^capacity = ^information, where ^information is the INFOR- 
MATION Dimension. The capacity dimension satisfies 

^correlation S: ^information S: ^capacity 

where correlation is the Correlation Dimension, and 
is conjectured to be equal to the LYAPUNOV DIMENSION. 

see also CORRELATION EXPONENT, DIMENSION, HAUS- 
DORFF Dimension, Kaplan- Yorke Dimension 

References 

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear 
Dynamics: Analytical, Computational, and Experimental 
Methods. New York: Wiley, pp. 538-541, 1995. 

Peitgen, H.-O. and Richter, D. H. The Beauty of Frac- 
tals: Images of Complex Dynamical Systems. New York: 
Springer- Verlag, 1986. 

Wheeden, R. L. and Zygmund, A. Measure and Integral: An 
Introduction to Real Analysis. New York: M. Dekker, 
1977. 

Caratheodory Derivative 

A function / is Caratheodory differentiate at a if there 
exists a function which is CONTINUOUS at a such that 

f(x) -/(a) = <t>(x)(x-a). 

Every function which is Caratheodory differentiable is 
also FRECHET DIFFERENTIABLE. 

see also Derivative, Frechet Derivative 

Caratheodory's Fundamental Theorem 

Each point in the CONVEX Hull of a set S in R n is in 
the convex combination of n + 1 or fewer points of 5. 

see also Convex Hull, Helly's Theorem 



Cardano's Formula 

see Cubic Equation 



Cardinal Number 



Cardioid 189 



Cardinal Number 

In informal usage, a cardinal number is a number used 
in counting (a Counting Number), such as 1, 2, 3, 

Formally, a cardinal number is a type of number defined 
in such a way that any method of counting SETS using it 
gives the same result. (This is not true for the ORDINAL 
Numbers.) In fact, the cardinal numbers are obtained 
by collecting all ORDINAL NUMBERS which are obtain- 
able by counting a given set. A set has No (ALEPH-0) 
members if it can be put into a One-TO-One correspon- 
dence with the finite ORDINAL NUMBERS. 

Two sets are said to have the same cardinal number if 
all the elements in the sets can be paired off One-to- 
One. An Inaccessible Cardinal cannot be expressed 
in terms of a smaller number of smaller cardinals. 

see also Aleph, Aleph-0 (Ho), Aleph-1 (Hi), Can- 
tor-Dedekind Axiom, Cantor Diagonal Slash, 
Conttnuum, Continuum Hypothesis, Equipol- 
lent, Inaccessible Cardinals Axiom, Infinity, 
Ordinal Number, Power Set, Surreal Number, 
Uncountable Set 

References 

Cantor, G. Uber unendliche, lineare Punktmannigfaltig- 

keiten, Arbeiten zur Mengenlehre aus dem Jahren 1872- 

1884. Leipzig, Germany: Teubner, 1884. 
Conway, J. H. and Guy, R. K. "Cardinal Numbers." In The 

Book of Numbers. New York: Springer- Verlag, pp. 277- 

282, 1996. 
Courant, R. and Robbins, H. "Cantor's 'Cardinal Numbers.'" 

§2.4.3 in What is Mathematics?: An Elementary Approach 

to Ideas and Methods, 2nd ed. Oxford, England: Oxford 

University Press, pp. 83-86, 1996. 

Cardinality 

see Cardinal Number 

Cardioid 



and the parametric equations 




The curve given by the POLAR equation 
r = a(l + cos#), 

sometimes also written 

r = 26(1 + cos 0), 

where b = a/2, the Cartestan equation 

/ 2 . 2 n2 2/ 2 . 2\ 

[x + y -ax) — a (x +y ), 



(1) 



(2) 



(3) 



x = acost(l + cost) 
y = asini(l + cost). 



(4) 
(5) 



The cardioid is a degenerate case of the LlMA<JON. It is 
also a 1-CuSPED EPICYCLOID (with r = R) and is the 
CAUSTIC formed by rays originating at a point on the 
circumference of a CIRCLE and reflected by the Circle. 

The name cardioid was first used by de Castillon in 
Philosophical Transactions of the Royal Society in 1741. 
Its Arc Length was found by La Hire in 1708. There 
are exactly three PARALLEL TANGENTS to the cardioid 
with any given gradient. Also, the TANGENTS at the 
ends of any Chord through the Cusp point are at 
Right Angles. The length of any Chord through the 
Cusp point is 2a. 




The cardioid may also be generated as follows. Draw 
a CIRCLE C and fix a point A on it. Now draw a set 
of Circles centered on the Circumference of C and 
passing through A. The ENVELOPE of these Circles 
is then a cardioid (Pedoe 1995). Let the CIRCLE C be 
centered at the origin and have RADIUS 1, and let the 
fixed point be A — (1, 0). Then the RADIUS of a CIRCLE 
centered at an ANGLE 9 from (1, 0) is 



r 2 = (0-cos(9) 2 + (l-sin(9) 2 
= cos 2 0+l-2sin0 + sin 2 
= 2(1- sin 0). 



(6) 




J 


^ 




The Arc Length, Curvature, and Tangential An- 
gle are 



/' 

Jo 



2|cos(!i)|dt = 4asin(i0) 



3|sec(i0)| 



4o 



(7) 

(8) 
(9) 



As usual, care must be taken in the evaluation of s(t) 
for t > n. Since (7) comes from an integral involving the 



190 



Cardioid Caustic 



Cards 



ABSOLUTE Value of a function, it must be monotonic 
increasing- Each Quadrant can be treated correctly 
by defining 

+ 1, (10) 



Cardioid Evolute 



l_7T 



where [a; J is the FLOOR FUNCTION, giving the formula 
s(t) = (~l) 1+[n (mod 2)] 4sin(|i) + 8 Ll n J ' ( U > 



The Perimeter of the curve is 

/»2tt 



/ 

Jo 



|2acos(|i9)|d0 = 4a 



/ cos (| 
Jo 



9)dB 



/•7r/2 / 1 t/2 

= 4a / cos <j>(2 d<$>) — 8a / cos (j)d<fi 
Jo Jo 

-8a[sin0]o /2 = 8a. (12) 



The Area is 

/•27T 



A= \ I r 2 d6=\a I (1 + 2cos<9 + cos 2 6) dO 
Jo Jo 

= 2 a / 

Jo 



{1 + 2 cos + | [1 + cos(26>)]} d0 



/»27T 

= |a 2 / [§ + 2cos(9+|cos(26>)]dl9 
Jo 



= \A¥ + 2sin # + \ sin^lo" = 



2tt _ 3 2 



(13) 



see also Circle, Cissoid, Conchoid, Equiangular 

Spiral, Lemniscate, LiMAgoN, Mandelbrot Set 

References 

Gray, A. "Cardioids." §3.3 in Modern Differential Geometry 

of Curves and Surfaces. Boca Raton, FL: CRC Press, 

pp. 41-42, 1993. 
Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 118-121, 1972. 
Lee, X. "Cardioid." http : //www .best . com/ ~xah/ Special 

PlaneCurves_dir/Cardioid_dir/cardioid.html. 
Lee, X. "Cardioid." http://www.best.com/-xah/Special 

PlaneCurves_dir/Cardioid_dir/cardioidGG.html. 
Lockwood, E. H. "The Cardioid." Ch. 4 in A Book of Curves. 

Cambridge, England: Cambridge University Press, pp. 34- 

43, 1967. 
MacTutor History of Mathematics Archive. "Cardioid." 

http : //www-groups . dcs . st-and. ac .uk/ -history/Curves 

/Cardioid. html. 
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- 
ton, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995. 
Yates, R. C. "The Cardioid." Math. Teacher 52, 10-14, 1959. 
Yates, R. C. "Cardioid." A Handbook on Curves and Their 

Properties. Ann Arbor, Ml: J. W. Edwards, pp. 4-7, 1952. 

Cardioid Caustic 

The Catacaustic of a Cardioid for a Radiant Point 
at the Cusp is a Nephroid. The Catacaustic for 
Parallel rays crossing a Circle is a Cardioid. 





y^ 


~-^ 






/ 




V 


/ 






\ 


/ 






\ 


/ 
\ 
\ 


/^~^\ 




\ 

\ 
\ 


\ 


f \ 




\ 


V 


\ I 




1 


/ 
\ 


u 




; 

/ 
/ 
/ 


\ 






/ 


\ 






/ 








/ 




N. 




/ 












*"■' 


**" 





x = -a + |a cos 0(1 — cos#) 
y = |asin#(l — cos#). 

This is a mirror-image Cardioid with a = a/3. 

Cardioid Inverse Curve 

If the Cusp of the cardioid is taken as the Inversion 
Center, the cardioid inverts to a Parabola. 

Cardioid Involute 




x — 2a + 3a cos 9(1 — cos 0) 
y = 3a sin 0(1 — cos#). 

This is a mirror-image CARDIOID with a 1 = 3a. 

Cardioid Pedal Curve 



/ / 
/ / 


y 


- 


- 


NX 


V \ 










/ / 










\ 
\ \ 










\ \ 








// 






- 















The Pedal Curve of the Cardioid where the Pedal 
Point is the Cusp is Cayley's Sextic. 

Cards 

Cards are a set of n rectangular pieces of cardboard 
with markings on one side and a uniform pattern on the 
other. The collection of all cards is called a "deck," and 
a normal deck of cards consists of 52 cards of four dif- 
ferent "suits." The suits are called clubs (Jt), diamonds 
(<0>), hearts (\?), and spades (♦). Spades and clubs are 



Carleman's Inequality 

colored black, while hearts and diamonds are colored 
red. The cards of each suit are numbered 1 through 13, 
where the special terms ace (1), jack (11), queen (12), 
and king (13) are used instead of numbers 1 and 11-13. 

The randomization of the order of cards in a deck is 
called Shuffling. Cards are used in many gambling 
games (such as POKER), and the investigation of the 
probabilities of various outcomes in card games was one 
of the original motivations for the development of mod- 
ern Probability theory. 

see also Bridge Card Game, Clock Solitaire, 
Coin, Coin Tossing, Dice, Poker, Shuffle 

Carleman's Inequality- 
Let {a,i}™ =1 be a Set of Positive numbers. Then the 
Geometric Mean and Arithmetic Mean satisfy 

n n 

^J(aia 2 • • • a;) 1/j < - ^J a». 

Here, the constant e is the best possible, in the sense 
that counterexamples can be constructed for any stricter 
Inequality which uses a smaller constant. 

see also Arithmetic Mean, e, Geometric Mean 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1094, 1979. 

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 249-250, 1988. 

Carlson-Levin Constant 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 

Assume that / is a Nonnegative Real function on 
[0, oo) and that the two integrals 



Carlyle Circle 191 



/ 

Jo 



x P ~ [f(x)] p dx 



*'- 1+M [/(aO]' dx 



(1) 



(2) 



exist and are FINITE. If p = q — 2 and A = /x = 1, 
Carlson (1934) determined 



/ f(x)dx< \M / 

(I 



[f(x)] 2 dx 



1/4 



x / x*[f{x)Ydx\ (3) 



1/4 



and showed that ^pK is the best constant (in the sense 
that counterexamples can be constructed for any stricter 



INEQUALITY which uses a smaller constant). For the 
general case 

/ f(x)dx<cl x p - 1 - x [f(x)] p dx\ 



C 9 - 1+ "[/(x)]* dx 

and Levin (1948) showed that the best constant 

r(;)r(i) 



(4) 



(pa)*(qty 



(A + / «)r(4±i) 



where 



t = 



ppL + qX 
A 



pfi + q\ 

a = 1 — s — t 

and T(z) is the GAMMA FUNCTION. 



(5) 



(6) 

(7) 
(8) 



References 

Beckenbach, E. F.; and Bellman, R. Inequalities. New York: 

Springer- Verlag, 1983. 
Boas, R. P. Jr. Review of Levin, V. I. "Exact Constants 

in Inequalities of the Carlson Type." Math. Rev. 9, 415, 

1948. 
Finch, S. "Favorite Mathematical Constants." http://www. 

mathsoft.com/asolve/constant/crlslvn/crlslvn.htnil. 
Levin, V. L "Exact Constants in Inequalities of the Carlson 

Type." Doklady Akad. Nauk. SSSR (N. S.) 59, 635-638, 

1948. English review in Boas (1948). 
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities 

Involving Functions and Their Integrals and Derivatives. 

Kluwer, 1991. 

Carlson's Theorem 

If f(z) is regular and of the form <9(e fc '*') where k < tt, 
for K[z] > 0, and if f(z) = for z = 0, 1, . . . , then f(z) 
is identically zero. 

see also Generalized Hypergeometric Function 

References 

Bailey, W. N. "Carlson's Theorem." §5.3 in Generalised 
Hypergeometric Series. Cambridge, England: Cambridge 
University Press, pp. 36—40, 1935. 

Carlyle Circle 

n 
A = (1,0) 




B = {s,p) 



Y=(0,p+l) 

C=(0,p) 



H 2 S = (j, 0) 



Consider a Quadratic Equation x 2 -sx+p = where 
s and p denote signed lengths. The CIRCLE which has 



192 



Carmichael Condition 



Carmichael Number 



the points A = (0,1) and B — (s,p) as a DIAMETER 
is then called the Carlyle circle C S>P of the equation. 
The Center of C SjP is then at the Midpoint of AB, 
M = (s/2,(l +p)/2), which is also the Midpoint of 
S = (s, 0) and Y = (0, 1 + p). Call the points at which 
C SiP crosses the x-AxiS Hi = (2:1,0) and #2 = (#2,0) 
(with x\ > X2)> Then 



s = Xi -\- X2 

p = X1X2 

(# — x\)(x — X2) = x 2 — sx + p, 

so xi and X2 are the ROOTS of the quadratic equation. 
see also 257-gon, 65537-gon, Heptadecagon, Pen- 
tagon 

References 

De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- 
ity of Polygonal Constructions." Amer. Math. Monthly 98, 
97-108, 1991. 

Eves, H. An Introduction to the History of Mathematics, 6th 
ed. Philadelphia, PA: Saunders, 1990. 

Leslie, J. Elements of Geometry and Plane Trigonome- 
try with an Appendix and Very Copious Notes and Il- 
lustrations, J^th ed., improved and exp. Edinburgh: 
W. & G. Tait, 1820. 

Carmichael Condition 

A number n satisfies the Carmichael condition IFF (p — 
l)\(n/p - 1) for all PRIME DIVISORS p of n. This is 
equivalent to the condition (p - l)\(n - 1) for all Prime 
Divisors pofn. 

see also Carmichael Number 

References 

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- 

sohn, R. "Giuga's Conjecture on Primality." Amer. Math. 

Monthly 103, 40-50, 1996. 

Carmichael's Conjecture 

CarmichaeFs conjecture asserts that there are an In- 
finite number of Carmichael Numbers. This was 
proven by Alford et al. (1994). 

see also CARMICHAEL NUMBER, CARMICHAEL'S TO- 
tient Function Conjecture 

References 

Alford, W. R.; Granville, A.; and Pomerance, C. "There Are 

Infinitely Many Carmichael Numbers." Ann. Math. 139, 

703-722, 1994. 
Cipra, B. What's Happening in the Mathematical Sciences, 

Vol 1. Providence, RI: Amer. Math. Soc, 1993. 
Guy, R. K. "Carmichael's Conjecture." §B39 in Unsolved 

Problems in Number Theory, 2nd ed. New York: Springer- 

Verlag, p. 94, 1994. 
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The 

Pseudoprimesto25-10 9 ." Math. Comput. 35,1003-1026, 

1980. 
Ribenboim, P. The Book of Prime Number Records, 2nd ed. 

New York: Springer- Verlag, pp. 29-31, 1989. 
Schlafly, A. and Wagon, S. "Carmichael's Conjecture on the 

Euler Function is Valid Below lO 10 - 000 - 000 ." Math. Com- 
put. 63, 415-419, 1994. 



Carmichael Function 

A(n) is the LEAST COMMON MULTIPLE (LCM) of all the 
Factors of the Totient Function <j>(n), except that 
if 8|n, then 2 a ~ 2 is a FACTOR instead of 2 a ~ 1 . 



\{n) = < 



0(n) 

for n = p a ,p = 2 and a < 2, or p > 3 
\<t>{n) 

for n = 2 a and a > 3 
LCM[X(jH ai )]i 

for n = YiiPi ai 



Some special values are 



for r > 3, and 



A(l) = 1 
A(2) = 1 
A(4) = 2 
A(2 r ) - 2 r ~ 2 

X(p r ) = 4>tf) 



for p an ODD PRIME and r > 1. The ORDER of a (mod 
n) is at most A(n) (Ribenboim 1989). The values of A(n) 
for the first few n are 1, 1, 2, 2, 4, 2, 6, 4, 10, 2, 12, . . . 
(Sloane's A011773). 
see also MODULO MULTIPLICATION GROUP 

References 

Ribenboim, P. The Book of Prime Number Records, 2nd ed. 
New York: Springer- Verlag, p. 27, 1989. 

Riesel, H. "Carmichael's Function." Prime Numbers and 
Computer Methods for Factorization, 2nd ed. Boston, 
MA: Birkhauser, pp. 273-275, 1994. 

Sloane, N. J. A. Sequence A011773 in "An On-Line Version 
of the Encyclopedia of Integer Sequences." 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addison- Wesley, p. 226, 1991. 

Carmichael Number 

A Carmichael number is an Odd Composite Number 
n which satisfies Fermat's Little Theorem 

a n_1 -1 = (mod n) 

for every choice of a satisfying (a,n) = 1 (i.e., a and 
n are Relatively Prime) with 1 < a < n. A Car- 
michael number is therefore a PSEUDOPRIMES to any 
base. Carmichael numbers therefore cannot be found 
to be Composite using Fermat's Little Theorem. 
However, if (a,n) ^ 1, the congruence of Fermat's Lit- 
tle Theorem is sometimes Nonzero, thus identifying 
a Carmichael number n as COMPOSITES, 

Carmichael numbers are sometimes called ABSOLUTE 
PSEUDOPRIMES and also satisfy KORSELT'S CRITERION. 
R. D. Carmichael first noted the existence of such num- 
bers in 1910, computed 15 examples, and conjectured 
that there were infinitely many (a fact finally proved by 
Alford et al. 1994). 



Carmichael Number 



CarmichaeFs Totient Function Conjecture 193 



The first few Carmichael numbers are 561, 1105, 1729, 
2465, 2821, 6601, 8911, 10585, 15841, 29341, ... 
(Sloane's A002997). Carmichael numbers have at least 
three PRIME FACTORS. For Carmichael numbers with 
exactly three PRIME FACTORS, once one of the PRIMES 
has been specified, there are only a finite number of Car- 
michael numbers which can be constructed. Numbers of 
the form (6fc + l)(12fc + l)(18fc + l) are Carmichael num- 
bers if each of the factors is Prime (Korselt 1899, Ore 
1988, Guy 1994). This can be seen since for 



N = (6fc+l)(12fc+l)(18fc+l) 



1296fc 3 +396/c 2 +36£;+l, 



N - 1 is a multiple of 36k and the LEAST COMMON 
Multiple of 6fc, 12fc, and 18k is 36fc, so a^" 1 = 1 
modulo each of the PRIMES 6A; + 1, 12k + 1, and lSk + 
1, hence a N ~ x = 1 modulo their product. The first 
few such Carmichael numbers correspond to k = 1, 6, 
35, 45, 51, 55, 56, ... and are 1729, 294409, 56052361, 
118901521, ... (Sloane's A046025). The largest known 
Carmichael number of this form was found by H. Dubner 
in 1996 and has 1025 digits. 

The smallest Carmichael numbers having 3, 4, ... fac- 
tors are 561 = 3 x 11 x 17, 41041 = 7 x 11 x 13 x 41, 
825265, 321197185, ... (Sloane's A006931). In total, 
there are only 43 Carmichael numbers < 10 6 , 2163 
< 2.5 x 10 10 , 105,212 < 10 15 , and 246,683 < 10 16 (Pinch 
1993). Let C(n) denote the number of Carmichael num- 
bers less than n. Then, for sufficiently large n (n ~ 10 7 
from numerical evidence), 



C(n) 



2/7 



(Alford et al. 1994). 

The Carmichael numbers have the following properties: 

1. If a PRIME p divides the Carmichael number 
n, then n = 1 (mod p — 1) implies that n = 
p (mod p(p — 1)). 

2. Every Carmichael number is SQUAREFREE. 

3. An Odd Composite Squarefree number n is a 

Carmichael number Iff n divides the DENOMINATOR 

of the Bernoulli Number B n -\. 

see also CARMICHAEL CONDITION, PSEUDOPRIME 

References 

Alford, W. R.; Granville, A.; and Pomerance, C. "There are 

Infinitely Many Carmichael Numbers." Ann. Math. 139, 

703-722, 1994. 
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 

Boca Raton, FL: CRC Press, p. 87, 1987. 
Guy, R. K. "Carmichael Numbers." §A13 in Unsolved Prob- 
lems in Number Theory, 2nd ed. New York: Springer- 

Verlag, pp. 30-32, 1994. 
Korselt, A. "Probleme chinois." L 'intermediate math. 6, 

143-143, 1899. 
Ore, 0. Number Theory and Its History. New York: Dover, 

1988. 
Pinch, R. G. E. "The Carmichael Numbers up to 10 15 ." 

Math. Comput. 55, 381-391, 1993. 



Pinch, R. G. E. ftp:// emu . pmms . cam .ac.uk/ pub / 
Carmichael/. 

Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The 
Pseudoprimesto25'10 9 ." Math. Corn-put 35, 1003-1026, 
1980. 

Riesel, H. Prime Numbers and Computer Methods for Fac- 
torization, 2nd ed. Basel: Birkhauser, pp. 89-90 and 94- 
95, 1994. 

Shanks, D. Solved and Unsolved Problems in Number Theory, 
4th ed. New York: Chelsea, p'. 116, 1993. 

Sloane, N. J. A. Sequences A002997/M5462 and A006931/ 
M5463 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Carmichael Sequence 

A Finite, Increasing Sequence of Integers {a ly 
. . . , a m } such that 

(en - l)|(ai •■ -ai-i) 

for i = 1, . . . , ?n, where m\n indicates that m DIVIDES n. 
A Carmichael sequence has exclusive EVEN or Odd ele- 
ments. There are infinitely many Carmichael sequences 
for every order. 

see also GlUGA SEQUENCE 

References 

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- 

sohn, R. "Giuga's Conjecture on Primality." ^4mer. Math. 

Monthly 103, 40-50, 1996. 

CarmichaePs Theorem 

If a and n are RELATIVELY PRIME so that the GREATEST 
Common Denominator GCD(a,n) = 1, then 

a ^ = 1 (mod n) , 

where A is the Carmichael Function. 

CarmichaePs Totient Function Conjecture 

It is thought that the Totient Valence Function 
AT (m) > 2 (i.e., the TOTIENT VALENCE FUNCTION 
never takes the value 1). This assertion is called Car- 
michael's totient function conjecture and is equivalent 
to the statement that there exists an m ^ n such 
that <t>{n) = <p(m) (Ribenboim 1996, pp. 39-40). Any 
counterexample to the conjecture must have more than 
10,000 DIGITS (Conway and Guy 1996). Recently, 
the conjecture was reportedly proven by F. Saidak in 
November, 1997 with a proof short enough to fit on a 
postcard. 

see also Totient Function, Totient Valence 
Function 

References 

Conway, J. H, and Guy, R. K. The Book of Numbers. New 

York: Springer- Verlag, p. 155, 1996. 
Ribenboim, P. The New Book of Prime Number Records. 

New York: Springer- Verlag, 1996. 



194 Carnot's Polygon Theorem 



Cartan Torsion Coefficient 



Carnot's Polygon Theorem 

If Pi, P2, . • ■ , are the VERTICES of a finite POLYGON 
with no "minimal sides" and the side PiPj meets a curve 
in the POINTS Piji and Pj-,2, then 



Ui^ P ^Ui P 2P23i--Ui P ^ P ^ 



= 1, 



where AB denotes the DISTANCE from POINT A to B. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 190, 1959. 

Carnot's Theorem 

Given any TRIANGLE A 1 A2A$ i the signed sum of PER- 
PENDICULAR distances from the C IRC UM CENTER O to 
the sides is 

OOi + OO2 + OO3 = R + r, 

where r is the INRADIUS and R is the ClRCUMRADIUS. 
The sign of the distance is chosen to be POSITIVE IFF 
the entire segment OOi lies outside the TRIANGLE. 
see also JAPANESE TRIANGULATION THEOREM 

References 

Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: 

Allyn and Bacon, pp. 256 and 262, 1972. 
Honsbergcr, R. Mathematical Gems III. Washington, DC: 

Math. Assoc. Amer., p. 25, 1985. 

Carotid-Kundalini Fractal 

Fractal Valley Gaussian Mtn. Oscillation Land 



0.5 



-1 



, ,11:., 



111 



//>;'); V 



. Mmmm 






I 






'''■-jffi/i 

x m 



\ i 



m 



0.5 



A fractal-like structure is produced for x < by super- 
posing plots of Carotid-Kundalini Functions CK n 
of different orders n. The region — 1 < x < is called 
FRACTAL LAND by Pickover (1995), the central region 
the Gaussian Mountain Range, and the region x > 
Oscillation Land. The plot above shows n — 1 to 25. 
Gaps in FRACTAL LAND occur whenever 



cos(27rr/<?) for r = 0, 1, ..., [q/2\, where \z\ is the 

Ceiling Function and L^J is the Floor Function. 

References 

Pickover, C. A. "Are Infinite Carotid-Kundalini Functions 

Fractal?" Ch. 24 in Keys to Infinity. New York: W. H. 

Freeman, pp. 179-181, 1995. 

Carotid-Kundalini Function 

The Function given by 

CK n (x) = cos(nxcos _1 x), 

where n is an Integer and — 1 < x < 1. 
see also Carotid-Kundalini Fractal 



Carry 



l 1 
1 5 8- 

H 249 - 
407- 



-carries 

- addend 1 

- addend 2 
-sum 



The operating of shifting the leading DIGITS of an AD- 
DITION into the next column to the left when the Sum of 
that column exceeds a single DIGIT (i.e., 9 in base 10). 
see also ADDEND, ADDITION, BORROW 

Carrying Capacity 

see Logistic Growth Curve 

Cartan Matrix 

A Matrix used in the presentation of a Lie Algebra. 

References 

Jacobson, N. Lie Algebras. New York: Dover, p. 121, 1979. 

Cartan Relation 

The relationship Sq*(x ^ y) = Z j+k =iSq j (x) -- Sq k {y) 
encountered in the definition of the Steenrod Alge- 
bra. 

Cartan Subgroup 

A type of maximal Abelian SUBGROUP. 

References 

Knapp, A. W. "Group Representations and Harmonic Anal- 
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. 

Cartan Torsion Coefficient 

The Antisymmetric parts of the Connection Coef- 
ficient r A u „. 



-i p 

xcos X = 2it- 
Q 

for p and q RELATIVELY PRIME INTEGERS. At such 
points #, the functions assume the \(q + l)/2] values 



Cartesian Coordinates 



Cartesian Ovals 195 



Cartesian Coordinates 

2-axis 

A 



The Gradient of the Divergence is 




y-axis 
Cartesian coordinates are rectilinear 2-D or 3-D coordi- 
nates (and therefore a special case of CURVILINEAR CO- 
ORDINATES) which are also called Rectangular Co- 
ordinates. The three axes of 3-D Cartesian coordi- 
nates, conventionally denoted the a>, y-, and z-Axes (a 
Notation due to Descartes) are chosen to be linear and 
mutually PERPENDICULAR. In 3-D, the coordinates x, 
y, and z may lie anywhere in the INTERVAL ( — 00,00). 

The Scale Factors of Cartesian coordinates are all 
unity, hi = l. The Line Element is given by 



ds — dx x + dy y + dz z, 
and the Volume Element by 

dV = dx dy dz. 
The Gradient has a particularly simple form, 

„J?_ ,d_ ^d_ 
dx dy dz ' 



as does the Laplacian 



dx 2 dy 2 dz 2 * 



(i) 



(2) 



(3) 



(4) 



The Laplacian is 

V 2 F = V-(VF) 



d 2 F d 2 F 
dx 2 dy 2 



d 2 F 
dz 2 



+ y 

+ z 

The Divergence is 
V-F - 
and the CURL is 



d 2 F x d 2 F 2 
dx 2 + 

d 2 F v 



dx 2 

d 2 F z 
dx 2 



dy 2 

d 2 F y 
dy 2 



d 2 F x 
dz 2 

d 2 F, 



+ 



+ 



d^F z 
dy 2 



+ 



dz 2 

d 2 F z 
dz 2 



dF x 
dx 



dF v . 8F X 



dy 



+ 



dz 



(5) 



(6) 



V x F : 



x 

_d_ 
dx 
F x 



y 
a 



+ 



(dF z 
\ dy 

y dx dy 



z 

d_ 
dz 
F z 

dFy 

dz 
dF Q 



x + 



( dF x 
V dz 



V(V-u) 



a ( du_x_ 1 du y 1 <t 
x "I" dy ~r c 





du x 1 9uy_ du z \ 
dx ~T~ dy ^~ dz J 



dy 



o 1 du x 1 dv-y 1 
*~ l dx ~*~ dy ^ 



r A. 
% 
dy 
_d_ 
dz 



du x du v du z 

___ _j * _j 

dx dy dz 



(8) 



Laplace's Equation is separable in Cartesian coordi- 
nates. 

see also COORDINATES, HELMHOLTZ DIFFERENTIAL 

Equation— Cartesian Coordinates 

References 

Arfken, G. "Special Coordinate Systems— Rectangular 
Cartesian Coordinates." §2.3 in Mathematical Methods for 
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94- 
95, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, p. 656, 1953. 

Cartesian Ovals 




A curve consisting of two ovals which was first studied 
by Descartes in 1637. It is the locus of a point P whose 
distances from two FOCI F\ and F2 in two-center BIPO- 
LAR Coordinates satisfy 



mr ± nr = k, 



(i) 



where m,n are Positive Integers, A; is a Positive 
real, and r and r are the distances from F\ and F2. If 
m = n, the oval becomes an an ELLIPSE. In CARTESIAN 
Coordinates, the Cartesian ovals can be written 



iy/(x - a) 2 + y 2 + ny/(x + a) 2 + 1 



(2) 



/ 2 , 2 . 2w 2 2\ / 2 . 2\ 7 2 

(x -\- y + a ){m — n ) — 2ax{m + n ) — k 

= -2n^{x + a) 2 + y 2 , (3) 

[(m 2 - n 2 )(x 2 + y 2 + a 2 ) - 2ax(m 2 + n 2 )] 2 

= 2(m 2 + n 2 )(n 2 + y 2 + a 2 ) - 4ax(m 2 - n 2 ) - A; 2 . (4) 



(5) 
(6) 



dF z \ „ 


Now define 




(7) 




,22 
— 771 — n 

_ 2 . 2 

c = m +n , 



196 



Cartesian Product 



Cassini Ovals 



and set a = 1. Then 

[b(x 2 +y 2 )-2cx + bf +Abx + k 2 -2c = 2c(x 2 +y 2 ). (7) 
If c is the distance between Fi and F2, and the equation 
r 4- mr = a (8) 

is used instead, an alternate form is 

[(l-m 2 )(x 2 +y 2 )+2m 2 c'x+a' 2 -m 2 c 12 } 2 = 4a' 2 (x 2 +y 2 ). 

(9) 

The curves possess three Foci. If m — 1, one Cartesian 

oval is a central CONIC, while if m = a/c % then the curve 
is a LlMAgON and the inside oval touches the outside 
one. Cartesian ovals are ANALLAGMATIC CURVES. 

References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 

Stradbroke, England: Tarquin Pub., p. 35, 1989. 
Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 155-157, 1972. 
Lockwood, E. H. A Book of Curves. Cambridge, England: 

Cambridge University Press, p. 188, 1967. 
MacTutor History of Mathematics Archive. "Cartesian 

Oval." http : //www-groups . dcs . st -and . ac . uk/ -history/ 

Curves /Cart esian.html. 

Cartesian Product 

see Direct Product (Set) 

Cartesian Trident 

see Trident of Descartes 

Cartography 

The study of Map Projections and the making of ge- 
ographical maps. 

see also Map Projection 

Cascade 

A Z-Action or N- Action. A cascade and a single Map 
X — ¥ X are essentially the same, but the term "cascade" 
is preferred by many Russian authors. 
see also Action, Flow 

Casey's Theorem 

Four Circles are Tangent to a fifth Circle or a 
straight Line Iff 

£12^34 i £13^42 db £14^23 = 0, 

where Uj is a common TANGENT to CIRCLES i and j. 
see also PURSER'S THEOREM 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 121-127, 1929. 



Casimir Operator 

An Operator 






on a representation R of a LIE ALGEBRA. 

References 

Jacobson, N. Lie Algebras. New York: Dover, p. 78, 1979. 

Cassini Ellipses 

see Cassini Ovals 

Cassini's Identity 

For F n the nth FIBONACCI NUMBER, 

Fn~iF n +i — F n — (— l) n . 

see also Fibonacci Number 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, p. 12, 1996. 

Cassini Ovals 



The curves, also called CASSINI ELLIPSES, described by 
a point such that the product of its distances from two 
fixed points a distance 2a apart is a constant b . The 
shape of the curve depends on b/a. If a < 6, the curve 
is a single loop with an Oval (left figure above) or dog 
bone (second figure) shape. The case a = b produces 
a Lemniscate (third figure). If a > b, then the curve 
consists of two loops (right figure). The curve was first 
investigated by Cassini in 1680 when he was studying 
the relative motions of the Earth and the Sun. Cassini 
believed that the Sun traveled around the Earth on one 
of these ovals, with the Earth at one FOCUS of the oval. 

Cassini ovals are Anallagmatic Curves. The Cassini 
ovals are defined in two-center Bipolar Coordinates 
by the equation 



T\T2 = b , 



(1) 



with the origin at a FOCUS. Even more incredible curves 
are produced by the locus of a point the product of 
whose distances from 3 or more fixed points is a con- 
stant. 

The Cassini ovals have the CARTESIAN equation 

[(x-a) 2 +y 2 ][(x + a) 2 +2/ 2 ] = 6 4 (2) 

or the equivalent form 



(x 4- y + a ) — 4a x = b (3) 



Cassini Ovals 



Cassini Surface 197 



and the polar equation 



Cassini Projection 



4 . 4 

r 4- a 



2aVcos(2(9) = & 4 . 



(4) 



Solving for r 2 using the QUADRATIC Equation gives 



2 

r = 



2a 2 cos(2(9) + ^a 4 cos 2 (20) - 4(a 4 - b 4 ) 



= a 2 003(20) + V / a 4 cos 2 (2(9) + 6 4 -a 4 
= a 2 cos(20) v/a 4 [cos 2 (20) - 1] + fe 4 
= a 2 cos(20) + ^b 4 - a 4 sin 2 (20) 



cos(20) + J(-} -sin 2 (20) 



(5) 



If a < 6, the curve has Area 

A= L r i de = 2 (l) f r 2 c 

J-tv/4 



a J +6^(- ), (6) 



where the integral has been done over half the curve 
and then multiplied by two and E(x) is the complete 
Elliptic Integral of the Second Kind. If a = 6, 
the curve becomes 

r 2 = a 2 |cos(20) + >/l-sin 2 0l = 2a 2 cos(2<9), (7) 

which is a Lemniscate having Area 

A = 2a 2 (8) 

(two loops of a curve y/2 the linear scale of the usual 
lemniscate r 2 — a 2 cos(2#), which has area A = a 2 /2 
for each loop). If a > 6, the curve becomes two disjoint 
ovals with equations 



r = ±aJ cos(20) ± J (~) -sin 2 (20), (9) 

where £ [— 0o,9q] and 



A — 1 * "I 

t/o = f sin 



&' 



(10) 



see a/so Cassini Surface, Lemniscate, Mandelbrot 
Set, Oval 

References 

Gray, A. "Cassinian Ovals." §4.2 in Modern Differential Ge- 
ometry of Curves and Surfaces. Boca Raton, FL: CRC 

Press, pp. 63-65, 1993. 
Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 153-155, 1972. 
Lee, X. "Cassinian Oval," http : // www . best . com / - xah / 

SpecialPlane Curves _dir/CassinianOval_dir/ cassinian 

Oval.html. 
Lockwood, E. H. A Book of Curves. Cambridge, England: 

Cambridge University Press, pp. 187-188, 1967. 
MacTutor History of Mathematics Archive. "Cassinian 

Ovals." http: //www-groups .dcs .st-and.ac .uk/ -history 

/Curves/Cassinian.html. 
Yates, R. C. "Cassinian Curves." A Handbook on Curves 

and Their Properties. Ann Arbor, MI: J. W. Edwards, 

pp. 8-11, 1952. 




iCTION. 






x — sin - B 


(i) 


y = tan - 


tan<£ 


(2) 


cos(A — Ao) 



where 

B = cos</>sin(A - Ao). 

The inverse FORMULAS are 

<t> = sin -1 (sin D cos x) 

-l ( tan x \ 

A = Ao + tan ( — 1 , 

V cos D J 

where 

D = y + <f> . 



(3) 

(4) 
(5) 

(6) 



References 

Snyder, J. P. Map Projections — A Working Manual. U. S. 
Geological Survey Professional Paper 1395. Washington, 
DC: U. S. Government Printing Office, pp. 92-95, 1987. 

Cassini Surface 




The QUARTIC SURFACE obtained by replacing the con- 
stant c in the equation of the CASSINI OVALS 



{(x-a) 2 +y 2 ][(x + af + y 2 ] = c 2 
by c = z 2 , obtaining 

[(x-a) 2 +y 2 }[(x + a) 2 +y 2 ] = z 4 . 
As can be seen by letting y = to obtain 

/ 2 2\2 4 

(x — a ) — z 



2.2 2 

x + z = a , 



(i) 

(2) 

(3) 
(4) 



198 



Castillon's Problem 



Catalan's Conjecture 



the intersection of the surface with the y — PLANE is 
a Circle of Radius a. 

References 

Fischer, G. (Ed.). Mathematical Models from the Collections 

of Universities and Museums. Braunschweig, Germany: 

Vieweg, p. 20, 1986. 
Fischer, G. (Ed.). Plate 51 in Mathematische Mod- 

elle/ Mathematical Models, Bildband/ Photograph Volume. 

Braunschweig, Germany: Vieweg, p. 51, 1986. 

Castillon's Problem 




Inscribe a TRIANGLE in a CIRCLE such that the sides of 
the Triangle pass through three given Points A, B, 
and C> 

References 

Dorrie, H. "Castillon's Problem." §29 in 100 Great Problems 

of Elementary Mathematics: Their History and Solutions. 

New York: Dover, pp. 144-147, 1965. 

Casting Out Nines 

An elementary check of a Multiplication which makes 
use of the CONGRUENCE 10 n = 1 (mod 9) for n > 2. 
Prom this CONGRUENCE, a MULTIPLICATION ab — c 
must give 

a = > a,i = a* 



bi = b* 



C = 2~J Ci — c* , 

so ab = a*b* must be = c* (mod 9). Casting out nines 
is sometimes also called "the Hindu Check." 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, pp. 28-29, 1996. 

Cat Map 

see Arnold's Cat Map 

Catacaustic 

The curve which is the ENVELOPE of reflected rays. 



Curve 


Source 


Catacaustic 


cardioid 


cusp 


nephroid 


circle 


not on circumf. 


lima^on 


circle 


on circumf. 


cardioid 


circle 


point at oo 


nephroid 


cissoid of Diocles 


focus 


cardioid 


1 arch of a cycloid 


rays _L axis 


2 arches of a cycloid 


deltoid 


point at oo 


astroid 


In x 


rays || axis 


catenary 


logarithmic spiral 


origin 


equal logarithmic spiral 


parabola 


rays _L axis 


Tschirnhausen cubic 


quadrifolium 


center 


astroid 


Tschirnhausen cubic 


focus 


semicubical parabola 


see also CAUSTIC, 


DlACAUSTIC 




References 







Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 60 and 207, 1972. 

Catalan's Conjecture 

8 and 9 (2 3 and 3 2 ) are the only consecutive POWERS 
(excluding and 1), i.e., the only solution to Cata- 
lan's Diophantine PROBLEM. Solutions to this prob- 
lem (Catalan's Diophantine Problem) are equiva- 
lent to solving the simultaneous Diophantine Equa- 
tions 



X 2 - Y s = 1 
X 3 -Y 2 = 1. 



This Conjecture has not yet been proved or refuted, 
although it has been shown to be decidable in a Fi- 
nite (but more than astronomical) number of steps. 
In particular, if n and n H- 1 are POWERS, then n < 
exp exp exp exp 730 (Guy 1994, p. 155), which follows 
from R. Tijdeman's proof that there can be only a FI- 
NITE number of exceptions should the CONJECTURE not 
hold. 

Hyyro and Makowski proved that there do not exist 
three consecutive POWERS (Ribenboim 1996), and it is 
also known that 8 and 9 are the only consecutive CUBIC 

and Square Numbers (in either order). 

see also Catalan's Diophantine Problem 

References 

Guy, R. K. "Difference of Two Power." §D9 in Unsolved 

Problems in Number Theory, 2nd ed. New York: Springer- 

Verlag, pp. 155-157, 1994. 
Ribenboim, P. Catalan's Conjecture. Boston, MA: Academic 

Press, 1994. 
Ribenboim, P. "Catalan's Conjecture." Amer. Math. 

Monthly 103, 529-538, 1996. 
Ribenboim, P. "Consecutive Powers." Expositiones Mathe- 

maticae 2, 193-221, 1984. 



Catalan's Constant 



Catalan's Constant 



199 



Catalan's Constant 

A constant which appears in estimates of combinatorial 
functions. It is usually denoted K, /3(2), or G. It is not 
known if K is IRRATIONAL. Numerically, 



K = 0.915965594177... 



(1) 



(Sloane's A006752). The CONTINUED FRACTION for K 
is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (Sloane's A014538). 
K can be given analytically by the following expressions, 



K = /3(2) 



(-l) fe _ J__jL 1 
(2fc-fl) 2 ~ l 2 3 2 + 5 2 + " 



(2) 
(3) 



= 1 + 

71 = 1 



oo oo 

^ (4n + l) 2 ~ 9 ~ ^ (4n + 3) 2 ^ 



I 

Jo 



(4 
tan -1 xdx 



l 



In xdx 



(5) 
(6) 



where (3(z) is the Dirichlet Beta Function. In terms 
of the POLYGAMMA FUNCTION *i(as), 

*=£*iU)-£Mi) (7) 

= ^*i(A) + ^*i(A)-> 2 (8) 

= i* 1 (l)-i* 1 (|)-i^. (9) 



Applying CONVERGENCE IMPROVEMENT to (3) gives 



^=^E( TO + 1 )^C(m + 2), (10) 



where ((z) is the Riemann Zeta Function and the 
identity 



1 1__ _ ^ 3 m -l 

(l-3^) 2 (I-*) 2 ~ 2^ TO + 1 > 4 „ 






has been used (Flajolet and Vardi 1996). The Flajolet 
and Vardi algorithm also gives 



K - - 1 - n (i - ±-\ W-l 

V2 11 \ 2»V/3(2*) 



k^i^/i 2 ^ 1 ) 



(12) 



where f3(z) is the Dirichlet Beta Function. Glaisher 
(1913) gave 



*-i-E 



nC(2n + l) 



16 n 



(13) 



(Vardi 1991, p. 159). W. Gosper used the related FOR- 
MULA 



K = 



where 



V2 



*(2) - 1 



n 



-il/(2 fe+1 ) 



*(m) 



-*(2 fc ) -1 

K-m^rn _ l)4 m - 1 S m ' 



(14) 
(15) 



where B n is a Bernoulli Number and ip(x) is a Poly- 
gamma Function (Finch). The Catalan constant may 
also be defined by 



Jo 



K{k) dk, 



(16) 



where K(k) (not to be confused with Catalan's constant 
itself, denoted K) is a complete Elliptic Integral of 
the First Kind. 



K = 



7rln2 
8 



+£ 



at 



2L(i+l)/2Ji2> 



where 



{o<} = {1,1,1,0,-1,-1,-1,0} 



(17) 



(18) 



is given by the periodic sequence obtained by appending 
copies of {1, 1, 1, 0, — 1, — 1, — 1, 0} (in other words, 
en = a[(t-i) (mod 8)]+i for i > 8) and [x\ is the FLOOR 
Function (Nielsen 1909). 

see also Dirichlet Beta Function 

References 

Abramowitz, M, and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
pp. 807-808, 1972. 

Adamchik, V. "32 Representations for Catalan's Con- 
stant." http://www.wolfram.com/-victor/articles/ 
catalan/catalan.html. 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 551—552, 1985. 

Fee, G. J. "Computation of Catalan's Constant using Ra- 
in anuj an' s Formula." ISAAC '90. Proc. Internal. Symp. 
Symbolic Algebraic Cornp., Aug. 1990. Reading, MA: 
Addison-Wesley, 1990. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsof t . com/ asolve/constant/catalan/catalan. html. 

Flajolet, P. and Vardi, I. "Zeta Function Expan- 
sions of Classical Constants." Unpublished manu- 
script. 1996. http://pauillac.inria.fr/algo/flajolet/ 
Publications/landau. ps. 

Glaisher, J. W. L. "Numerical Values of the Series 1 - 1/3" + 
1/5" - 1/7" + 1/9" - &c for n = 2, 4, 6." Messenger Math. 
42, 35-58, 1913. 

Gosper, R. W. "A Calculus of Series Rearrangements." In 
Algorithms and Complexity: New Directions and Recent 
Results (Ed. J. F. Traub). New York: Academic Press, 
1976. 

Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany: 
Halle, pp. 105 and 151, 1909. 



200 Catalan's Diophantine Problem 



Catalan Number 



Plouffe, S. "PloufiVs Inverter: Table of Current Records for 
the Computation of Constants." http://lacim.uqam.ca/ 
pi/records .html. 

Sloane, N. J. A. Sequences A014538 and A006752/M4593 in 
"An On-Line Version of the Encyclopedia of Integer Se- 
quences." 

Srivastava, H. M. and Miller, E. A. "A Simple Reducible 
Case of Double Hypergeometric Series involving Catalan's 
Constant and Riemann's Zeta Function." Int. J. Math. 
Educ. Sci. Technol. 21, 375-377, 1990. 

Vardi, I. Computational Recreations in Mathematica. Read- 
ing, MA: Addison-Wesley, p. 159, 1991. 

Yang, S. "Some Properties of Catalan's Constant G." Int. J. 
Math. Educ. Sci. Technol 23, 549-556, 1992. 

Catalan's Diophantine Problem 

Find consecutive POWERS, i.e., solutions to 

b d -, 
a — c = 1, 

excluding and 1. CATALAN'S CONJECTURE is that the 
only solution is 3 2 - 2 3 = 1, so 8 and 9 (2 3 and 3 2 ) are 
the only consecutive POWERS (again excluding and 1). 

see also CATALAN'S CONJECTURE 

References 

Cassels, J. W. S. "On the Equation a x - 6^ = 1. II." Proc. 

Cambridge Phil Soc. 56, 97-103, 1960. 
Inkeri, K. "On Catalan's Problem." Acta Arith. 9, 285-290, 

1964. 

Catalan Integrals 

Special cases of general FORMULAS due to Bessel. 

Jo(\A 2 -2/ 2 ) = - / e ycosd cos(z sin 0)d6, 
77 Jo 

where J is a BESSEL FUNCTION OF THE FIRST KIND. 
Now, let z = 1 — z' and y = 1 + z' . Then 

Jo(2iv^) = - / e (1+z)cos6 cos[(l-z)sm0]d6. 
n Jo 



Catalan Number 

The Catalan numbers are an INTEGER SEQUENCE {C n } 
which appears in TREE enumeration problems of the 
type, "In how many ways can a regular n-gon be di- 
vided into n — 2 TRIANGLES if different orientations 
are counted separately?" (EULER'S POLYGON DIVI- 
SION Problem). The solution is the Catalan number 
Cn-2 (Dorrie 1965, Honsberger 1973), as graphically il- 
lustrated below (Dickau). 




The first few Catalan numbers are 1, 2, 5, 14, 42, 132, 
429, 1430, 4862, 16796, ... (Sloane's A000108). The 
only Odd Catalan numbers are those of the form c 2 fc_i, 
and the last DIGIT is five for k = 9 to 15. The only 
PRIME Catalan numbers for n < 2 15 - 1 are C 2 = 2 and 
C 3 = 5. 

The Catalan numbers turn up in many other related 
types of problems. For instance, the Catalan number 
C n -i gives the number of BINARY BRACKETINGS of n 
letters (CATALAN'S Problem). The Catalan numbers 
also give the solution to the Ballot PROBLEM, the 
number of trivalent Planted Planar Trees (Dickau), 



^J^O^^ 



the number of states possible in an n-FLEXAGON, the 
number of different diagonals possible in a FRIEZE PAT- 
TERN with n+1 rows, the number of ways of forming 
an n-fold exponential, the number of rooted planar bi- 
nary trees with n internal nodes, the number of rooted 
plane bushes with n EDGES, the number of extended 
Binary Trees with n internal nodes, the number of 
mountains which can be drawn with n upstrokes and 
n downstrokes, the number of noncrossing handshakes 
possible across a round table between n pairs of peo- 
ple (Conway and Guy 1996), and the number of SE- 
QUENCES with NONNEGATIVE PARTIAL SUMS which can 
be formed from n Is and n -Is (Bailey 1996, Buraldi 
1992)! 

An explicit formula for C n is given by 

'2n\ _ _^_ (2n)! _ (2n)! 

n 



C n — 



1 
n+1 



n + 1 n! 2 (n + l)!n!' 



(1) 



■&mQm<s><^ 



where ( 2 ™) denotes a BINOMIAL COEFFICIENT and n\ is 
the usual Factorial. A Recurrence Relation for 
C n is obtained from 

Cn+i (2n + 2)! (n+l)(n!) 2 

C n (n + 2)[(n+l)!] 2 (2n)! 
__ (2n + 2)(2n + l)(n + l) 
(n + 2)(n-f-l) 2 

_ 2(2n + l)(n + l) 2 _ 2(2n+l) 
(n+l) 2 (n + 2) ~ n + 2 ' 



(2) 



Catalan Number 



Catalan Number 201 



_ 2(2n + l) 

t-'n+l — T~^ ^n* 



n + 2 



Other forms include 



C n — 



2-6-10---(4n-2) 

(n + 1)! 
2 n (2n~l)!! 

(n + 1)! 

(2n)! 



n!(n+l)f 



(3) 

(4) 
(5) 
(6) 



Segner's Recurrence Formula, given by Segner in 

1758, gives the solution to Euler's POLYGON DIVISION 
Problem 

E n = E^En-x + EsE n -2 + . . . + E n -iE2. (7) 

With Ei = E 2 = 1, the above RECURRENCE RELATION 
gives the Catalan number C n _2 = Z2 n . 

The Generating Function for the Catalan numbers 
is given by 



1 VI 4x = Y CnX " = i + x + 2x 2 + bx s + .... (8) 

n=0 

The asymptotic form for the Catalan numbers is 



C k 



v^FP/2 



(9) 



(Vardi 1991, Graham et al. 1994). 

A generalization of the Catalan numbers is defined by 



if pk \_ 1 (pk 



(10) 



for k > 1 (Klarner 1970, Hilton and Pederson 1991). 
The usual Catalan numbers Ck = 2<ih are a special case 
with j) —2. p dk gives the number of p-ary TREES with k 
source-nodes, the number of ways of associating k appli- 
cations of a given p-ary OPERATOR, the number of ways 
of dividing a convex POLYGON into k disjoint (p + 1)- 
gons with nonintersecting DIAGONALS, and the number 
of p-GoOD PATHS from (0, -1) to (]fe, (p-l)k-l) (Hilton 
and Pederson 1991). 

A further generalization is obtained as follows. Let p 
be an INTEGER > 1, let P k = (k,(p - l)k - 1) with 
k > 0, and q < p - 1. Then define p d q o = 1 and let p d q k 
be the number of p-GoOD PATHS from (1, q — 1) to Pk 
(Hilton and Pederson 1991). Formulas for p d q i include 
the generalized JONAH FORMULA 



k 

z = l 



- pi 



(11) 



and the explicit formula 

p^qk 



p-q (pk - q\ 

ok — qyk ~ 1 J 



A Recurrence Relation is given by 



pd q k - 2_^ 



p&p — T,i P^Q-^Tyj 



(12) 



(13) 



k + 1 



where i,j, r > 1, k > 1, q < p — r, and i 4- j 
(Hilton and Pederson 1991). 

see also BALLOT PROBLEM, BINARY BRACKETING, 
Binary Tree, Catalan's Problem, Catalan's 
Triangle, Delannoy Number, Euler's Polygon 
Division Problem, Flexagon, Frieze Pattern, 
Motzkin Number, p-Good Path, Planted Planar 
Tree, Schroder Number, Super Catalan Number 

References 

Alter, R. "Some Remarks and Results on Catalan Numbers." 
Proc. 2nd Louisiana Conf. Comb., Graph Th., and Corn- 
put, 109-132, 1971. 

Alter, R. and Kubota, K. K. "Prime and Prime Power Divis- 
ibility of Catalan Numbers." J. Combin. Th. 15,243-256, 
1973. 

Bailey, D. F. "Counting Arrangements of l's and — l's." 
Math. Mag. 69, 128-131, 1996. 

Brualdi, R. A. Introductory Combinatorics, 3rd ed. New 
York: Elsevier, 1997. 

Campbell, D. "The Computation of Catalan Numbers." 
Math. Mag. 57, 195-208, 1984, 

Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration 
of Certain Sets of Planted Trees." J. Combin. Th. Ser. B 
18, 209-221, 1975. 

Chu, W. "A New Combinatorial Interpretation for General- 
ized Catalan Numbers." Disc. Math. 65, 91-94, 1987. 

Conway, J. H. and Guy, R. K. In The Book of Numbers. New- 
York: Springer- Verlag, pp. 96-106, 1996. 

Dershowitz, N. and Zaks, S. "Enumeration of Ordered Trees." 
Disc, Math. 31, 9-28, 1980. 

Dickau, R. M. "Catalan Numbers." http: //forum. 

swarthmore.edu/advanced/robertd/catalan.html. 

Dorrie, H. "Euler's Problem of Polygon Division." §7 in 100 
Great Problems of Elementary Mathematics: Their His- 
tory and Solutions. New York: Dover, pp. 21-27, 1965. 

Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How 
Likely is a Function to be Convex?" Math. Mag. 61, 211— 
219, 1988. 

Gardner, M. "Catalan Numbers." Ch. 20 in Time Travel and 
Other Mathematical Bewilderments. New York: W. H. 
Freeman, 1988. 

Gardner, M. "Catalan Numbers: An Integer Sequence that 
Materializes in Unexpected Places." Sci. Amer. 234, 120- 
125, June 1976. 

Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- 
phy of Two Special Number Sequences, 6th ed. Morgan- 
town, WV: Math Monongliae, 1985. 

Graham, R. L.; Knuth, D. E.; and Patashnik, 0. Exercise 
9.8 in Concrete Mathematics: A Foundation for Computer 
Science, 2nd ed. Reading, MA: Addison- Wesley, 1994. 

Guy, R. K. "Dissecting a Polygon Into Triangles." Bull. 
Malayan Math. Soc. 5, 57-60, 1958. 

Hilton, P. and Pederson, J. "Catalan Numbers, Their Gen- 
eralization, and Their Uses." Math. Int. 13, 64-75, 1991. 

Honsberger, R. Mathematical Gems I. Washington, DC: 
Math. Assoc. Amer., pp. 130-134, 1973. 



202 



Catalan's Problem 



Catalan's Surface 



Honsberger, R. Mathematical Gems III. Washington, DC: 
Math. Assoc. Amer., pp. 146-150, 1985, 

Klarner, D. A. "Correspondences Between Plane Trees and 
Binary Sequences." J. Comb. Th. 9, 401-411, 1970. 

Rogers, D. G. "Pascal Triangles, Catalan Numbers and Re- 
newal Arrays." Disc. Math. 22, 301-310, 1978. 

Sands, A. D. "On Generalized Catalan Numbers." Disc. 
Math. 21, 218-221, 1978. 

Singmaster, D. "An Elementary Evaluation of the Catalan 
Numbers." Amer. Math. Monthly 85, 366-368, 1978. 

Sloane, N. J. A. A Handbook of Integer Sequences. Boston, 
MA: Academic Press, pp. 18-20, 1973. 

Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 

Vardi, I. Computational Recreations in Mathematica. Red- 
wood City, CA: Addis on- Wesley, pp. 187-188 and 198-199, 
1991. 

Wells, D. G. The Penguin Dictionary of Curious and Inter- 
esting Numbers. London: Penguin, pp. 121-122, 1986. 

Catalan's Problem 

The problem of finding the number of different ways in 
which a PRODUCT of n different ordered FACTORS can be 
calculated by pairs (i.e., the number of BINARY Brack- 
ETINGS of n letters). For example, for the four FAC- 
TORS a, 6, c, and d } there are five possibilities: ((ab)c)d, 
(a(bc))d, (ab)(cd), a((bc)d) y and a(b(cd)). The solution 
was given by Catalan in 1838 as 



c: = 



2 ■ 6 • 10 • (4n - 6) 



r\ 



C' 



and is equal to the CATALAN NUMBER C n -i 

see also Binary Bracketing, Catalan's Diophan- 
tine Problem, Euler's Polygon Division Problem 

References 

Dorrie, H. 100 Great Problems of Elementary Mathematics: 
Their History and Solutions. New York: Dover, p. 23, 
1965. 

Catalan Solid 

The Dual Polyhedra of the Archimedean Solids, 
given in the following table. 



Archimedean Solid 



Dual 



rhombicosidodecahedron 
small rhombicuboctahedron 
great rhombicuboctahedron 
great rhombicosidodecahedron 
truncated icosahedron 
snub dodecahedron 

(laevo) 
snub cube 

(laevo) 
cuboctahedron 
icosidodecahedron 
truncated octahedron 
truncated dodecahedron 
truncated cube 
truncated tetrahedron 



deltoidal hexecontahedron 
deltoidal icositetrahedron 
disdyakis dodecahedron 
disdyakis triacontahedron 
pentakis dodecahedron 
pentagonal hexecontahedron 

(dextro) 
pentagonal icositetrahedron 

(dextro) 
rhombic dodecahedron 
rhombic triacontahedron 
tetrakis hexahedron 
triakis icosahedron 
triakis octahedron 
triakis tetrahedron 



Here are the Archimedean DUALS (Holden 1971, 
Pearce 1978) displayed in alphabetical order (left to 
right, then continuing to the next row). 




Here are the Archimedean solids paired with the corre- 
sponding Catalan solids. 



O 



© Q 



© € 



© w 



see also Archimedean Solid, Dual Polyhedron, 
Semiregular Polyhedron 

References 

Catalan, E. "Memoire sur la Theorie des Polyedres." J. 

I'Ecole Polytechnique (Paris) 41, 1-71, 1865. 
Holden, A. Shapes, Space, and Symmetry. New York: Dover, 

1991. 

Catalan's Surface 




A Minimal Surface given by the parametric equations 



x(u, v) = u — sin u cosh v 
y(u, v) = 1 — cos u cosh v 
z(u,v) = 4sin(|w)sinh(|u) 



(i) 

(2) 
(3) 



Catalan's Triangle 



Categorical Variable 203 



(Gray 1993), or 



x(r, <j>) = asin(2</>) — 2a<fi + \o>v 2 cos(2<fi) 
y(r, <j>) = — acos(2<p) — ~av 2 cos(2(p) 
z(r,(fi) = 2avsin0, 



where 



-r + 



(4) 
(5) 
(6) 

(?) 



(do Carmo 1986). 



References 

Catalan, E. "Memoir sur les surfaces dont les rayons de 
courburem en chaque point, sont egaux et des signes con- 
traires." C. R. Acad. Sci. Paris 41, 1019-1023, 1855. 

do Carmo, M. P. "Catalan's Surface" §3.5D in Mathemati- 
cal Models from the Collections of Universities and Muse- 
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, 
pp. 45-46, 1986. 

Fischer, G. (Ed.). Plates 94-95 in Mathematische Mod- 
elle/ Mathematical Models, Bildband/ Photograph Volume. 
Braunschweig, Germany: Vieweg, pp. 90-91, 1986. 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, pp, 448-449, 1993. 

Catalan's Triangle 

A triangle of numbers with entries given by 

(n + m)\{n — m + 1) 
Cnrn= m!(n+l)! 

for < m < n, where each element is equal to the one 
above plus the one to the left. Furthermore, the sum 
of each row is equal to the last element of the next row 
and also equal to the CATALAN NUMBER C n . 



5 

14 14 



14 28 42 42 

20 48 90 132 132 



(Sloane's A009766). 

see also Bell Triangle, Clark's Triangle, Eu- 
ler's Triangle, Leibniz Harmonic Triangle, Num- 
ber Triangle, Pascal's Triangle, Prime Trian- 
gle, Seidel-Entringer-Arnold Triangle 

References 

Sloane, N. J. A. Sequence A009766 in "An On-Line Version 
of the Encyclopedia of Integer Sequences." 

Catalan's Trisectrix 

see TSCHIRNHAUSEN CUBIC 



Catastrophe 

see Butterfly Catastrophe, Catastrophe The- 
ory, Cusp Catastrophe, Elliptic Umbilic Catas- 
trophe, Fold Catastrophe, Hyperbolic Umbilic 
Catastrophe, Parabolic Umbilic Catastrophe, 
Swallowtail Catastrophe 

Catastrophe Theory 

Catastrophe theory studies how the qualitative nature 
of equation solutions depends on the parameters that 
appear in the equations. Subspecializations include bi- 
furcation theory, nonequilibrium thermodynamics, sin- 
gularity theory, synergetics, and topological dynamics. 
For any system that seeks to minimize a function, only 
seven different local forms of catastrophe "typically" oc- 
cur for four or fewer variables: (1) FOLD CATASTROPHE, 
(2) Cusp Catastrophe, (3) Swallowtail Catastro- 
phe, (4) Butterfly Catastrophe, (5) Elliptic Um- 
bilic Catastrophe, (6) Hyperbolic Umbilic Catas- 
trophe, (7) Parabolic Umbilic Catastrophe. 

More specifically, for any system with fewer than five 
control factors and fewer than three behavior axes, these 
are the only seven catastrophes possible. The following 
tables gives the possible catastrophes as a function of 
control factors and behavior axes (Goetz). 



Control 
Factors 



1 Behavior 

Axis 



2 Behavior 
Axes 



fold 
cusp 

swallowtail 
butterfly- 



hyperbolic umbilic, elliptic umbilic 
parabolic umbilic 



References 

Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin: Springer- 
Verlag, 1992. 

Gilmore, R. Catastrophe Theory for Scientists and Engi- 
neers. New York: Dover, 1993. 

Goetz, P. "Phil's Good Enough Complexity Dictionary." 
http ; //www . cs .but f alo . edu/~goetz/dict .html. 

Saunders, P. T. An Introduction to Catastrophe Theory. 
Cambridge, England: Cambridge University Press, 1980. 

Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, 
England: Oxford University Press, p. 211, 1987. 

Thorn, R. Structural Stability and Morphogenesis: An Out- 
line of a General Theory of Models. Reading, MA: Read- 
ing, MA: Addison- Wesley, 1993. 

Thompson, J. M. T. Instabilities and Catastrophes in Science 
and Engineering. New York: Wiley, 1982. 

Woodcock, A. E. R. and Davis, M. Catastrophe Theory. New 
York: E. P. Dutton, 1978. 

Zeeman, E. C. Catastrophe Theory — Selected Papers 1972- 
1977. Reading, MA: Addis on- Wesley, 1977. 

Categorical Game 

A Game in which no draw is possible. 

Categorical Variable 

A variable which belongs to exactly one of a finite num- 
ber of Categories. 



204 Category 



Catenary 



Category 

A category consists of two things: an OBJECT and a 
MORPHISM (sometimes called an "arrow"). An OB- 
JECT is some mathematical structure (e.g., a GROUP, 
Vector Space, or Differentiable Manifold) and a 
Morphism is a Map between two Objects. The Mor- 
PHISMS are then required to satisfy some fairly natural 
conditions; for instance, the IDENTITY MAP between 
any object and itself is always a Morphism, and the 
composition of two MORPHISMS (if defined) is always a 
Morphism. 

One usually requires the MORPHISMS to preserve the 
mathematical structure of the objects. So if the objects 
are all groups, a good choice for a MORPHISM would be 
a group HOMOMORPHISM. Similarly, for vector spaces, 
one would choose linear maps, and for differentiable 
manifolds, one would choose differentiable maps. 

In the category of TOPOLOGICAL SPACES, homomor- 
phisms are usually continuous maps between topologi- 
cal spaces. However, there are also other category struc- 
tures having TOPOLOGICAL SPACES as objects, but they 
are not nearly as important as the "standard" category 
of Topological Spaces and continuous maps. 

see also Abelian Category, Allegory, Eilenberg- 
Steenrod Axioms, Groupoid, Holonomy, Logos, 
monodromy, topos 

References 

Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- 
dam, Netherlands: North-Holland, 1990. 

Category Theory 

The branch of mathematics which formalizes a number 
of algebraic properties of collections of transformations 
between mathematical objects (such as binary relations, 
groups, sets, topological spaces, etc.) of the same type, 
subject to the constraint that the collections contain the 
identity mapping and are closed with respect to compo- 
sitions of mappings. The objects studied in category 
theory are called CATEGORIES. 

see also CATEGORY 
Catenary 



The curve a hanging flexible wire or chain assumes when 
supported at its ends and acted upon by a uniform grav- 
itational force. The word catenary is derived from the 
Latin word for "chain." In 1669, Jungius disproved 
Galileo's claim that the curve of a chain hanging un- 
der gravity would be a PARABOLA (MacTutor Archive). 
The curve is also called the ALYSOID and CHAINETTE. 
The equation was obtained by Leibniz, Huygens, and 
Johann Bernoulli in 1691 in response to a challenge by 
Jakob Bernoulli. 



Huygens was the first to use the term catenary in a letter 
to Leibniz in 1690, and David Gregory wrote a treatise 
on the catenary in 1690 (MacTutor Archive). If you roll 
a PARABOLA along a straight line, its FOCUS traces out 
a catenary. As proved by Euler in 1744, the catenary is 
also the curve which, when rotated, gives the surface of 
minimum SURFACE Area (the Catenoid) for the given 
bounding CIRCLE. 

The Cartesian equation for the catenary is given by 
y =l a (e x/a + e- K/a ) = acoshg), (1) 

and the Cesaro Equation is 

{s 2 +a 2 )K=-a. (2) 

The catenary gives the shape of the road over which a 
regular polygonal "wheel" can travel smoothly. For a 
regular n-gon, the corresponding catenary is 



where 



y = -Acosh I — j , 



A = R cos 



(3) 
(4) 




The Arc Length, Curvature, and Tangential An- 
gle are 

s = asinh ( — ) , (5) 

n=--sedi 2 (-) y (6) 

a \a/ 

<f>= -2 tan" 1 [tanh (^-)1 * (?) 

The slope is proportional to the Arc Length as mea- 
sured from the center of symmetry. 
see also Calculus of Variations, Catenoid, Linde- 
lof's Theorem, Surface of Revolution 

References 

Geometry Center. "The Catenary." http://www.geom.umn. 

edu/zoo/diffgeom/surf space/catenoid/catenary.html. 
Gray, A. "The E volute of a Tract rix is a Catenary." §5.3 

in Modern Differential Geometry of Curves and Surfaces. 

Boca Raton, FL: CRC Press, pp. 80-81, 1993. 
Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 195 and 199-200, 1972. 
Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A 

Book of Curves. Cambridge, England: Cambridge Univer- 
sity Press, pp. 118-124, 1967, 
MacTutor History of Mathematics Archive. "Catenary." 

http : //www-groups . dcs . st-and . ac . uk/ -history/Curves 

/Catenary .html. 
Pappas, T. "The Catenary & the Parabolic Curves." The 

Joy of Mathematics. San Carlos, CA: Wide World Publ./ 

Tetra, p. 34, 1989. 
Yates, R. C. "Catenary." A Handbook on Curves and Their 

Properties. Ann Arbor, MI: J. W. Edwards, pp. 12-14, 

1952. 



Catenary Evolute 
Catenary Evolute 




x = a[x — \ sinh(2t)] 
y = 2a cosh t. 



Catenary Involute 
\ 
\ 



\ 



/ 



y 



/ 




The parametric equation for a Catenary is 



dx 

dt 
dr 
dt 



1 
sinh 2 



ayl + sinh 2 t = acoshi 



and 



dr 

rpi dt 

i dt | 



secht 
tanhi 



(1) 



(2) 



(3) 



(4) 



ds 2 = \dr 2 \ = a 2 (I + sinh 2 t) dt 2 = a 2 cosh 2 <ft 2 (5) 



dt 



a cosh i. 



Therefore, 



-•/ 



cosh tdt = a sinh £ 



and the equation of the INVOLUTE is 

x = a(t — tanht) 
y — asechi. 

This curve is called a TRACTRIX. 



(6) 



(7) 



(8) 
(9) 



Catenoid 205 



Catenary Radial Curve 



\ 


/ 


\ 


/ 




/ ^ 


"^^v^ 


S^^***'^ 


^^^_ 


^^^^ 



The Kampyle of Eudoxus. 
Catenoid 




A Catenary of Revolution. The catenoid and Plane 
are the only SURFACES OF Revolution which are also 
Minimal Surfaces. The catenoid can be given by the 
parametric equations 



x = ccosh 



cosu 



y = c cosh ( - J sin u 



(i) 

(2) 

(3) 



where u G [0, 2w). The differentials are 

dx — sinh ( - j cos u dv - cosh ( - J sin u du (4) 

dy = sinh I - J sin u dv -f cosh [ - j cos u du (5) 

dz = du, (6) 

so the Line Element is 

ds 2 = dx 2 + dy 2 + dz 2 

= [sinh 2 Q) + l] dv 2 + cosh 2 Q) du * 

= cosh 2 f^\ dv 2 + cosh 2 (-) du 2 . (7) 

The Principal Curvatures are 



Kl — — sech 2 f - j 



K2 — - sech 2 ( - ) • 
The Mean Curvature of the catenoid is 



(8) 
(9) 

(10) 



206 



Caterpillar Graph 



Cauchy Distribution 



and the GAUSSIAN CURVATURE is 






(i) 



(ii) 









The HELICOID can be continuously deformed into a 
catenoid with c = 1 by the transformation 

x(u, v) = cos a sinh v sin u + sin a cosh v cos u (12) 
y(u, v) = — cos a sinh v cos u -f sin a cosh f sin u (13) 
z(?z, u) = u cos a + v sin a, (14) 

where a = corresponds to a HELICOID and a = n/2 
to a catenoid. 

see also CATENARY, COSTA MINIMAL SURFACE, HELI- 
COID, Minimal Surface, Surface of Revolution 

References 

do Carmo, M. P. "The Catenoid." §3.5A in Mathematical 
Models from the Collections of Universities and Museums 
(Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 
1986. 

Fischer, G. (Ed.). Plate 90 in Mathematische Modelle/ 
Mathematical Models, Bildband/ Photograph Volume. 
Braunschweig, Germany: Vieweg, p. 86, 1986. 

Geometry Center. "The Catenoid." http://www.geom.umn, 
edu/zoo/diffgeom/surf space/catenoid/. 

Gray, A. "The Catenoid." §18.4 Modern Differential Geom- 
etry of Curves and Surfaces. Boca Raton, FL: CRC Press, 
pp. 367-369, 1993. 

Meusnier, J. B. "Memoire sur la courbure des surfaces." 
Mem. des savans etrangers 10 (lu 1776), 477-510, 1785. 

Caterpillar Graph 

A TREE with every NODE on a central stalk or only one 
EDGE away from the stalk. 

References 

Gardner, M. Wheels, Life, and other Mathematical Amuse- 
ments. New York: W. H. Freeman, p. 160, 1983. 

Cattle Problem of Archimedes 

see Archimedes' Cattle Problem 

Cauchy Binomial Theorem 



V^ y m q m(m+l)/2 ( tl 
m=0 ^ 



J[(l + yq k ), 



where ( n ) is a Gaussian Coefficient. 

\m/ q 

see also g-BlNOMIAL THEOREM 



Cauchy Boundary Conditions 

Boundary Conditions of a Partial Differential 
Equation which are a weighted Average of Dirich- 
let Boundary Conditions (which specify the value 
of the function on a surface) and Neumann Boundary 
CONDITIONS (which specify the normal derivative of the 
function on a surface). 

see also Boundary Conditions, Cauchy Prob- 
lem, Dirichlet Boundary Conditions, Neumann 
Boundary Conditions 

References 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 678-679, 1953. 

Cauchy's Cosine Integral Formula 



/.tt/2 

/ ' 

J-rr/2 



a + u-2 ni0(v.-v+2£) 



dO 



7rV(fl + V ~ 1) 



2*+"- 2 r( M + 0rV-0' 

where F(z) is the GAMMA Function. 

Cauchy Criterion 

A Necessary and Sufficient condition for a Se- 
quence Si to CONVERGE. The Cauchy criterion is sat- 
isfied when, for all e > 0, there is a fixed number N such 
that \Sj - Si\ < e for all i,j > N. 

Cauchy Distribution 




The Cauchy distribution, also called the Lorentzian 
Distribution, describes resonance behavior. It also de- 
scribes the distribution of horizontal distances at which 
a Line Segment tilted at a random Angle cuts the 
x-AxiS. Let 6 represent the ANGLE that a line, with 
fixed point of rotation, makes with the vertical axis, as 
shown above. Then 



tan# : 



b 
6 = tan~ 



■(?) 



dx 



bdx 



1 + fJ 6 b 2 -rx 2 ' 

so the distribution of ANGLE is given by 
<W_ _ 1 bdx 

7T 7T b 2 + X 2 ' 



(i) 

(2) 
(3) 

(4) 



Cauchy Distribution 



This is normalized over all angles, since 



/7T/2 
■tt/2 



d9 



= 1 



(5) 



and 



f 

J — c 



i feds _ i [-,/nr 

■K b 2 +X 2 7T L VX/J _oo 

= i[i w -(-i ff )] = l. 



(6) 




The general Cauchy distribution and its cumulative dis- 
tribution can be written as 



P(x) 



2 X 



7r(x- M ) 2 + (|r)2 



.(*)=I + i tan -l(^) 



(7) 
(8) 



where T is the FULL WIDTH AT HALF MAXIMUM (r = 
26 in the above example) and /x is the MEAN (/x — in 
the above example). The Characteristic Function 
is 



<m 



dx 



~ 7T / 1 

t/ — oo 



_ e -i M t-r|ti/2^ 



1 + x 2 
cos(Tta/2) 



+ (r^/2) 2 



dz 



The Moments are given by 

2 



\i2 = cr = oo 

for ji = 



M3 



.oo for fi / 



/44 = oo, 



(9) 

(10) 

(11) 
(12) 



and the STANDARD DEVIATION, SKEWNESS, and KUR- 
TOSIS by 



_ f for fj, = 

71 ~ I oo for /x # 

72 = oo. 



(13) 
(14) 
(15) 



If X and Y are variates with a NORMAL DISTRIBUTION, 
then Z = X/Y has a Cauchy distribution with MEAN 
fi — and full width 






(16) 



Cauchy Inequality 207 

see a/so Gaussian Distribution, Normal Distribu- 
tion 

References 

Spiegel, M. R, Theory and Problems of Probability and 
Statistics. New York: McGraw-Hill, pp. 114-115, 1992. 

Cauchy Equation 

see Euler Equation 

Cauchy's Formula 

The Geometric Mean is smaller than the Arith- 
metic Mean, 



1/JV 



n~) <%=■ 



Cauchy Functional Equation 

The fifth of HlLBERT'S PROBLEMS is a generalization of 
this equation. 

Cauchy-Hadamard Theorem 

The Radius of Convergence of the Taylor Series 

ao + cl\z + aiz + . . . 

is 

1 



r = 



lim (Kl) 1 /" 

n— too 
see also RADIUS OF CONVERGENCE, TAYLOR SERIES 

Cauchy Inequality 

A special case of the HOLDER SUM INEQUALITY with 



y ^flfc&fc 



E- 2 E»* 2 • w 



Ok I < ^^ 

. k=l / \ k-1 / \ k=l 



where equality holds for ak = cbk- In 2-D, it becomes 

(2) 

It can be proven by writing 



(a 2 +6 2 )(c 2 + a 2 ) > {ac + bdf. 



Y^iatx + bi) 2 = f> 2 (x+ ^-) 2 = 0. (3) 

i=l i=l 

If bi/di is a constant c, then x = — c. If it is not a 
constant, then all terms cannot simultaneously vanish 
for REAL x, so the solution is COMPLEX and can be 
found using the QUADRATIC EQUATION 



2j2a i b i ±^4&a i b i ) -4^a, 2 ^6 i 2 
2J>^ 



• (4) 



208 Cauchy Integral Formula 

In order for this to be COMPLEX, it must be true that 



$>* <£«.'£« 



(5) 



with equality when hi /at is a constant. The VECTOR 
derivation is much simpler, 



(a-b) 2 = aV cos 2 6 < ab 2 , 



yhere 



= E^ 2 



2 _ V^ 2 

a = a • a — x 



(6) 
(7) 



and similarly for b. 

see also Chebyshev Inequality, Holder Sum In- 
equality 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 11, 1972. 

Cauchy Integral Formula 




r Yo Y r 

Given a Contour Integral of the form 



/ 



/(*) dz 

Z — Zo 



(1) 



define a path 70 as an infinitesimal CIRCLE around the 
point zo (the dot in the above illustration). Define the 
path 7 r as an arbitrary loop with a cut line (on which 
the forward and reverse contributions cancel each other 
out) so as to go around zq. 



The total path is then 



7 = 7o + It 



(2) 



tm±=tm*+tixr=L. (3 ) 

L z ~ z ° L a z ~ z o L r ~ - 



Z - Zq 



Prom the Cauchy Integral Theorem, the Contour 
Integral along any path not enclosing a Pole is 0. 
Therefore, the first term in the above equation is since 
70 does not enclose the Pole, and we are left with 



r Hz)dz = r f_(z)dz 



Cauchy Integral Formula 



Now, let z = z + re iB , so dz = ire w d9. Then 
f fWdz = f 

A z ~ Zo A, 

-I 



f{Zo + r / ) ire ig d0 

re™ 



f{zo + re ie )id9. 



(5) 



But we are free to allow the radius r to shrink to 0, so 
f Hz)dz = lim f f f ZQ + re ™\ id0 = f f( ZQ )idO 

/ Z - ZQ r->Q / / 

= if(zo) [ dd = 2<Kif(z ), (6) 

J It 



and 



/(*>) 



2 ™L 



f(z) dz 
z — Zq' 



(7) 



If multiple loops are made around the POLE, then equa- 
tion (7) becomes 



t/ 7 



)dz 



(8) 



where 71(7, z ) is the WINDING NUMBER. 

A similar formula holds for the derivatives of f(z), 

f(zo) = i im n«+h)-m 

h^t-0 h 

= ]im J-([ f^ dz - f M*z\ 

/i^o 2izih \Jz — zo-~h J z — zo I 

_ y 1 f f(z)[(z - z ) - (z - zo - h)} dz 

h^o 2nih j 



lim 
h 



im — — - / 

■-+0 27Vih I 

2 ™ 7 7 ( z - > 



(z - zo - h)(z - zo) 
hf(z) dz 



(z — zo — h)(z — zq) 



Iterating again, 



™-ht£ 



z) dz 



zo) 3 



(9) 



(10) 



Continuing the process and adding the WINDING Num- 
ber n, 



see also Morera's Theorem 

References 

Arfken, G. "Cauchy's Integral Formula." §6.4 in Mathemati- 
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic 
Press, pp. 371-376, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 367-372, 1953. 



Cauchy Integral Test 



Cauchy Ratio Test 209 



Cauchy Integral Test 

see Integral Test 

Cauchy Integral Theorem 

If / is continuous and finite on a simply connected region 
R and has only finitely many points of nondifferentia- 
bility in i£, then 



£ 



f(z)dz = 



(1) 



for any closed CONTOUR 7 completely contained in R. 
Writing z as 

z = x + iy (2) 

and f(z) as 

f(z)=u + iv (3) 

then gives 

(p f(z) dz — \ (u + iv)(dx + idy) 

= / udx -vdy + i / vdx + udy. (4) 
Prom Green's Theorem, 



J f(x J y)dx-g(x J y)dy=- fj (f| + fj) <**<fo 

/ f(x,y)dx+g{x,y)dy^ // ( 
so (4) becomes 



<9x % 



(5) 
dxdy (6) 



-h//(£-£')«M* (7) 



But the Cauchy-Riemann Equations require that 

du _ <9v 
dx dy 

du dv 

dy dx ' 



(8) 
(9) 



£ 



f(z)dz = 0, 



Q. E. D. 

For a Multiply Connected region, 



f f(z)dz= f f(z)dz. 



(10) 



(11) 



see also Cauchy Integral Theorem, Morera's 
Theorem, Residue Theorem (Complex Analysis) 

References 

Arfken, G. "Cauchy's Integral Theorem." §6.3 in Mathemati- 
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic 
Press, pp. 365-371, 1985. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part I. New York: McGraw-Hill, pp. 363-367, 1953. 



Cauchy-Kovalevskaya Theorem 

The theorem which proves the existence and uniqueness 
of solutions to the Cauchy Problem. 
see also Cauchy Problem 

Cauchy-Lagrange Identity 



(ax 2 + a 2 2 + ■ • • + an 2 )(&i 2 + b 2 2 + . . . + b n 2 ) 
= (aib 2 - a 2 h) 2 + (ai& 3 - a3&i) 2 + • • • 

+(a n -i&n - a n &n-i) • 

From this identity, the n-D Cauchy Inequality fol- 
lows. 

Cauchy-Maclaurin Theorem 

see Maclaurin-Cauchy Theorem 

Cauchy Mean Theorem 

For numbers > 0, the Geometric Mean < the Arith- 
metic Mean. 

Cauchy Principal Value 



fix) dx = lim 

-00 



/ f(x)dx 

J-R 



I 

J a 



PV I f{x)dx = lim 



I f{x)dx+ / f(x)dx 

J a J c+e 



where e > and a < c <b. 

References 

Arfken, G. Mathematical Methods for Physicists f 3rd ed. Or- 
lando, FL: Academic Press, pp. 401-403, 1985. 

Sansone, G. Orthogonal Functions, rev. English ed. New 
York: Dover, p. 158, 1991. 

Cauchy Problem 

Tf f(x,y) is an ANALYTIC FUNCTION in a NEIGHBOR- 
HOOD of the point (xo,yo) (i.e., it can be expanded in 
a series of Nonnegative Integer Powers of (x - x ) 
and (y — yo)), find a solution y(x) of the DIFFERENTIAL 
Equation 

dy 



dx 



/(*), 



with initial conditions y = yo and x = xq. The existence 
and uniqueness of the solution were proven by Cauchy 
and Kovalevskaya in the Cauchy-Kovalevskaya The- 
orem. The Cauchy problem amounts to determining 
the shape of the boundary and type of equation which 
yield unique and reasonable solutions for the CAUCHY 
Boundary Conditions. 

see also Cauchy Boundary Conditions 

Cauchy Ratio Test 

see Ratio Test 



210 Cauchy Remainder Form 



Cauchy Root Test 



Cauchy Remainder Form 

The remainder of n terms of a TAYLOR Series is given 

by 

(x-c) n_1 (a;-a) 



Rn — 



where a < c < x. 



(n-l)! 



r'(c), 



Cauchy- Riemann Equations 
Let 

f(x,y) = u(x,y) + iv(x,y) y 



where 



z = x + iy, 



(1) 
(2) 



These are known as the Cauchy- Riemann equations. 
They lead to the condition 



d 2 u 



d 2 v 



dxdy dxdy 



(14) 



The Cauchy-Riemann equations may be concisely writ- 
ten as 



(du .dv\ . ( du .dv\ 
\dx dx) \dy dy J 



df _ df df _ (du ( . dv \ t . ( du t . dv 
)x dy 

du dv 



dx) \dy 
. . du dv\ 

+ * -^- + -^- =0. 



dz* dx dy \dx dx) \dy dy 

du dv 
dx dy J ' " \dy dx 



(15) 



dz = dx -\- i dy. 



(3) 



The total derivative of / with respect to z may then be 
computed as follows. 



(4) 
(5) 



x = z - ty, 



dy __ 1 



dz 
dx 
dz 



and 



In terms of u and t>, (8) becomes 



df __ / du .dv\ . I du .dv 
\dx dx) \dy dy 



dz \dx dx j 



(6) 
(7) 



V = dldx + dldy = dl_ i dl 

dz dx dz dy dz dx dy' 



(du ,dv\ ( .du dv\ ,„, 

= U + ^) + (-^ + ^J- (9) 

Along the real, or as- Axis, df /dy = 0, so 

df _ du .dv . . 

dz dx dx 

Along the imaginary, or y-axis, df /dx = 0, so 

df _ .du dv . . 

dz dy dy * 

If / is Complex Differentiable, then the value of the 
derivative must be the same for a given dz, regardless of 
its orientation. Therefore, (10) must equal (11), which 
requires that 



and 



dv du 

dx dy' 



(13) 



In Polar Coordinates, 

f(re ie ) = R(r,0)e i@(r ' e \ 
so the Cauchy-Riemann equations become 



dR 

dr 

IdR 



RdQ 

r d6 



— = -*$©. 

r dd dr 



(16) 

(17) 
(18) 



If u and v satisfy the Cauchy-Riemann equations, they 
also satisfy Laplace's Equation in 2-D, since 



d^u d?u 

dx 2 dy 2 



d_ (dv 

dx \dy 



)+ £(-£)- " 9 > 

d 2 v d 2 v _ d ( du\ d (du\_ { . 

dx 2 dy 2 dx \ dy J dy \dx) 

By picking an arbitrary f(z), solutions can be found 
which automatically satisfy the Cauchy-Riemann equa- 
tions and Laplace's Equation. This fact is used to 
find so-called Conformal Solutions to physical prob- 
lems involving scalar potentials such as fluid flow and 
electrostatics. 

see also Cauchy Integral Theorem, Conformal 
Solution, Monogenic Function, Polygenic Func- 
tion 

References 

Abramowitz, M . and Stegun, C . A . (Eds . ) . Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 17, 1972. 

Arfken, G. "Cauchy-Riemann Conditions." §6.2 in Mathe- 
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca- 
demic Press, pp. 3560-365, 1985. 



Cauchy's Rigidity Theorem 

see Rigidity Theorem 

Cauchy Root Test 

see Root Test 



Cauchy-Schwarz Integral Inequality 



Cayley Cubic 211 



Cauchy-Schwarz Integral Inequality 

Let f(x) and g(x) by any two Real integrable functions 
of [a, 6], then 



'/"■ 



x)g(x) dx 



< 



nb "I r pb 

I f 2 (x)dx / g 2 (x)dx 

yd J Lv a 



with equality IFF f(x) = kg(x) with k real. 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1099, 1993. 

Cauchy-Schwarz Sum Inequality 

|a-b|<|ailb|. 



E 

, fe = l 



akbk 




Equality holds IFF the sequences ai, a2, ... and &i, 62, 

. . . are proportional. 

see also Fibonacci Identity 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1092, 1979. 

Cauchy Sequence 

A Sequence ai, 02, . . . such that the Metric d(a m , a n ) 
satisfies 

lim d(a m , a n ) = 0. 

min(m,n)— ^00 

Cauchy sequences in the rationals do not necessarily 
Converge, but they do Converge in the Reals. 

Real Numbers can be defined using either Dedekind 
Cuts or Cauchy sequences. 

see also Dedekind Cut 

Cauchy Test 

see Ratio Test 

Caustic 

The curve which is the ENVELOPE of reflected (CAT- 
ACAUSTIC) or refracted (DIACAUSTIC) rays of a given 
curve for a light source at a given point (known as the 
Radiant Point). The caustic is the Evolute of the 
Orthotomic. 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York; Dover, p. 60, 1972. 
Lee, X. "Caustics." http://www.best.com/-xah/Special 

PlaneCurves_dir/Caustics-dir/caustics.html. 
Lockwood, E. H. "Caustic Curves." Ch. 24 in A Book 

of Curves. Cambridge, England: Cambridge University 

Press, pp. 182-185, 1967. 
Yates, R. C. "Caustics." A Handbook on Curves and Their 

Properties. Ann Arbor, MI: J. W. Edwards, pp. 15-20, 

1952. 



Cavalieri's Principle 

1. If the lengths of every one-dimensional slice are equal 
for two regions, then the regions have equal Areas. 

2. If the AREAS of every two-dimensional slice (CROSS- 
Section) are equal for two SOLIDS, then the SOLIDS 
have equal Volumes. 

see also Cross-Section, Pappus's Centroid Theo- 
rem 

References 

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 126 and 132, 
1987. 

Cayley Algebra 

The only Nonassociative Division Algebra with 
REAL SCALARS. There is an 8-square identity corre- 
sponding to this algebra. The elements of a Cayley al- 
gebra are called CAYLEY NUMBERS or OCTONIONS. 

References 

Kurosh, A. G. General Algebra. New York: Chelsea, pp. 226- 
28, 1963. 

Cayley-Bacharach Theorem 

Let Xi, X 2 C P 2 be CUBIC plane curves meeting in nine 
points pi, . . . , pq. If X C P 2 is any CUBIC containing 
Pi, ■ - ■ , Ps, then X contains pg as well. It is related to 
GORENSTEIN RINGS, and is a generalization of PAPPUS'S 
Hexagon Theorem and Pascal's Theorem, 

References 

Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach 

Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 

295-324, 1996. 



Cayley Cubic 



* 



**4 



A Cubic Ruled Surface (Fischer 1986) in which the 
director line meets the director CONIC SECTION. Cay- 
ley's surface is the unique cubic surface having four OR- 
DINARY Double Points (Hunt), the maximum possible 
for Cubic Surface (EndraB). The Cayley cubic is in- 
variant under the TETRAHEDRAL GROUP and contains 
exactly nine lines, six of which connect the four nodes 
pairwise and the other three of which are coplanar (En- 
draB). 

If the Ordinary Double Points in projective 3-space 

are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 
0, 0, 1), then the equation of the surface in projective 
coordinates is 



1 1 1 1 

— + — + — + — =0 

Xq X\ X2 X3 



212 



Cay ley Cubic 



Cayley -Hamilton Theorem 



(Hunt). Denning "affine" coordinates with plane at in- 
finity v — Xq 4- x\ + X2 + 2^3 and 



Xq 

v 
v 

X 2 



then gives the equation 

-b(x 2 y+x 2 z+y 2 x+y 2 z+z 2 y+z 2 x)+2(xy+xz+yz) = 

plotted in the left figure above (Hunt). The slightly 
different form 



4(x 3 + y + z 3 + w ) - (x + y'+ z + • 







is given by Endrafi which, when rewritten in Tetrahe- 
dral Coordinates, becomes 



x + y — xz + yz-\-z — 1 = 0, 



plotted in the right figure above. 




The Hessian of the Cayley cubic is given by 

= Z 2 (xia:2 + X 1 X 3 + X2Xz) + X l (x X2 + X0X3 + Z2Z3) 
+xl(x Xi + XqX 3 + X1X3) + xI(xqX! + X X 2 + X1X2). 

in homogeneous coordinates xq, #1, x 2 , and X3. Taking 
the plane at infinity as v = 5(:ro + x\ + #2 + 2xz)j2 and 
setting a;, y, and 2 as above gives the equation 

25[x 3 (y+z)+y 3 (x+z)+z 3 {x+y)]+b0(x 2 y 2 +x 2 z 2 +y 2 z 2 ) 
— 125(x 2 yz + y xz-\-z xy)-{-60xyz — 4(xy-{-xz-\-yz) = 0, 

plotted above (Hunt). The Hessian of the Cayley cubic 
has 14 ORDINARY Double Points, four more than a 
the general Hessian of a smooth CUBIC SURFACE (Hunt). 

References 

Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV- 
Mitteilungen 4, 17-20, Apr. 1995. 



Endrafi, S. "The Cayley Cubic." http://www.mathematik. 

uni-mainz . de/AlgebraischeGeometrie/docs/ 

Ecayley.shtml. 
Fischer, G. (Ed.). Mathematical Models from the Collections 

of Universities and Museums. Braunschweig, Germany: 

Vieweg, p. 14, 1986. 
Fischer, G. (Ed.). Plate 33 in Mathematische Mod- 

elle/ Mathematical Models, Bildband/ Photograph Volume. 

Braunschweig, Germany: Vieweg, p. 33, 1986. 
Hunt, B. "Algebraic Surfaces." http://www.mathematik. 

uni-kl . de/-wwwagag/Galerie . html. 
Hunt, B. The Geometry of Some Special Arithmetic Quo- 
tients. New York: Springer- Verlag, pp. 115-122, 1996. 
Nordstrand, T. "The Cayley Cubic." http://www.uib.no/ 

people/nfytn/cleytxt.htm. 

Cayley Graph 

The representation of a GROUP as a network of directed 
segments, where the vertices correspond to elements and 
the segments to multiplication by group generators and 

their inverses. 

see also Cayley Tree 

References 

Grossman, I. and Magnus, W. Groups and Their Graphs. 
New York: Random House, p. 45, 1964. 

Cayley's Group Theorem 

Every Ftntte GROUP of order n can be represented as 
a Permutation Group on n letters, as first proved by 
Cayley in 1878 (Rotman 1995). 

see also Finite Group, Permutation Group 
References 

Rotman, J, J. An Introduction to the Theory of Groups, J^th 
ed. New York: Springer- Verlag, p. 52, 1995. 

Cayley-Hamilton Theorem 

Given 



a>\\ ~ 


X 


ai2 






aim 








0,21 




&22 — 


X 




ft2m 








dml 




dm2 






a>mrn X 
















— X ~T~ Cjn — \X 


771—1 , 


• + c , 


(1) 



then 



A m + c m - 1 A m - 1 + ... + c l = 0, 



(2) 



where I is the Identity Matrix. Cayley verified this 
identity for m = 2 and 3 and postulated that it was true 
for all m. For m = 2, direct verification gives 



a — x b 
c d — x 



= (a — x)(d — x) — be 



— x 2 — (a + d)x + {ad — be) = x 2 + c\x + C2 (3) 



Cayley's Hypergeometric Function Theorem 



Cayley-Klein Parameters 213 



A = 

A 2 = 

-{a + d)A = 
(ad — be) I = 



a b 








c d 




a b 




a b 




c d 




c d 




a 2 + be ab + bd 


ac + cd 


be + d 2 



—a — ad —ab — 


bd' 


—ac — dc —ad — d 2 


ad — be 




ad — be 


) 


\-(ad-bc)\ = 


"o o" 





(4) 

(5) 
(6) 
(7) 

(8) 



The Cayley-Hamilton theorem states that a n x n MA- 
TRIX A is annihilated by its Characteristic Poly- 
nomial det(xl — A), which is monic of degree n. 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed, San Diego, CA: Academic 
Press, p. 1117, 1979. 

Segercrantz, J. "Improving the Cayley-Hamilton Equation 
for Low-Rank Transformations." Amer. Math. Monthly 
99, 42-44, 1992. 

Cayley's Hypergeometric Function Theorem 

If 

oo 

(1 - z) a+h ~ c 2 Fi (2a, 26; 2c; z) = VJ a n z n , 

n = 

then 



2 Fi(a,6;c+ \- 1 z) 2 F 1 (c- a,c- b;e\\z) 

oo 

= E 



( c )" a ,» 



where 2 Fi (a, b; c; z) is a HYPERGEOMETRIC FUNCTION. 
see also Hypergeometric Function 

Cayley-Klein Parameters 

The parameters a, f3, 7, and S which, like the three 
Euler Angles, provide a way to uniquely characterize 
the orientation of a solid body. These parameters satisfy 
the identities 



and 



aa* + 77* = 1 
aa* + 00* = 1 
00* + SS* = 1 
a*/? + 7*5 = 
a5 — /?7 = 1 



/3 = -7* 
5 = a*, 



(i) 

(2) 
(3) 

(4) 
(5) 



(6) 
(7) 



where z* denotes the COMPLEX CONJUGATE. In terms 
of the EULER ANGLES 8, </>, and tj>, the Cayley-Klein 
parameters are given by 



a = e *(*+*)/ a OOB(i#) 

s i(V>-«)/2, 

d = c -(*+*)/ a cos(itf) 



/3 = te* l *- w/:, 8in(ie) 
■y = le " v ' r ' r " J sin(^) 



(8) 

(9) 

(10) 

(11) 



(Goldstein 1960, p. 155). 

The transformation matrix is given in terms of the 
Cayley-Klein parameters by 



A = 

I (a 2 - 7 2 + S 2 - (3 2 ) |i( 7 2 - a 2 + S 2 - /3 2 ) 7* - ct{3 

\i{a 2 + 7 2 - P 2 - 6 2 ) i (a 2 + 7 2 + ^ 2 + <* 2 ) -i(a/3 + 7 tf) 



/3£ — a7 



(Goldstein 1960, p. 153). 



i(ay + p8) 



a<5 + /37 



(12) 



The Cayley-Klein parameters may be viewed as param- 
eters of a matrix (denoted Q for its close relationship 

with Quaternions) 



Q = 



a 

7 6 



which characterizes the transformations 

u = au + 0v 



(13) 



(14) 
(15) 



of a linear space having complex axes. This matrix sat- 
isfies 

Q f Q = 0(^ = 1, (16) 

where I is the IDENTITY MATRIX and A f the MATRIX 
Transpose, as well as 



iQriQi = i. 



(17) 



In terms of the Euler Parameters a and the Pauli 
MATRICES cr iy the Q-matrix can be written as 



Q = e l + z(ei<n + e 2 a 2 + e 3 cr 3 ) 



(18) 



(Goldstein 1980, p. 156). 

see also EULER ANGLES, EULER PARAMETERS, PAULI 

Matrices, Quaternion 

References 

Goldstein, H. "The Cayley-Klein Parameters and Related 
Quantities." §4-5 in Classical Mechanics, 2nd ed. Read- 
ing, MA: Addison- Wesley, pp. 148-158, 1980. 



214 Cayley-Klein-Hilbert Metric 



Cayley's Sextic Evolute 



Cayley-Klein-Hilbert Metric 

The METRIC of Felix Klein's model for HYPERBOLIC 
Geometry, 



9ii 



912 



922 



a 2 (l-x 2 2 ) 

(1-Z! 2 -Z 2 2 ) 2 

a X\X2 

(1-Zl 2 ~X 2 2 ) 2 

a 2 (l-X! 2 ) 



(1-xi 2 -X2 2 ) 2 ' 

see also HYPERBOLIC GEOMETRY 

Cayley Number 

There are two completely different definitions of Cayley 
numbers. The first type Cayley numbers is one of the 
eight elements in a Cayley Algebra, also known as 
an OCTONION. A typical Cayley number is of the form 

a + bio + ci\ + dii + ei 3 + fU + gh + hi Qi 

where each of the triples (10,11,13), (n,^,^), (22,^3,25), 
(z3,i4)*6)) (i4,*5,*o)» (*5»*6j*i), (*e,*o,«2) behaves like 
the QUATERNIONS (i,j,k). Cayley numbers are not AS- 
SOCIATIVE. They have been used in the study of 7- and 
8-D space, and a general rotation in 8-D space can be 
written 

x ' -> {{{{{( xc i)c2)c3)c 4 )c 5 )cq)c 7 . 



The second type of Cayley number is a quantity which 
describes a Del Pezzo Surface. 

see also Complex Number, Del Pezzo Surface, 
Quaternion, Real Number 

References 

Conway, J. H. and Guy, R. K. "Cayley Numbers." In The 
Book of Numbers. New York: Springer- Ver lag, pp. 234- 
235, 1996. 

Okubo, S. Introduction to Octonion and Other Non- 
Associative Algebras in Physics. New York: Cambridge 
University Press, 1995. 

Cayley's Ruled Surface 

see Cayley Cubic 



Cayley's Sextic 




A plane curve discovered by Maclaurin but first studied 
in detail by Cayley. The name Cayley's sextic is due 
to R. C. Archibald, who attempted to classify curves in 
a paper published in Strasbourg in 1900 (MacTutor Ar- 
chive). Cayley's sextic is given in POLAR COORDINATES 

by 

r = acos 3 (|0), (1) 



or 



r = 4&cos 3 (§0), (2) 



where b = a/4. In the latter case, the CARTESIAN equa- 
tion is 



4(x 2 + y 2 - bxf - 27a 2 (x 2 + y 2 ) 2 . 

The parametric equations are 

x(t) = 4a cos 4 (I t) (2 cost - 1) 
y(t) =4acos 3 (|t)sin(|t). 



(3) 



(4) 
(5) 




JV_ 




The Arc Length, Curvature, and Tangential An- 
gle are 



s(t) = 3(i + sini), 
K(i) = !sec 2 (£t), 
<f>(t) = 2t. 



(6) 
(7) 

(8) 



References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 178 and 180, 1972. 

MacTutor History of Mathematics Archive. "Cayley's Sex- 
tic." http: //www-groups . dcs . st-and. ac . uk/ -history/ 
Curves/Cayleys.html. 



Cayley's Sextic Evolute 



/ 


\ 


/ 


\ 


1 


\ 


\ 


\ 


\ 


'"""N ^ 


\ I 


A \ 


\ A 


) 1 


S^T 


s < 


/ 


) j 


/ \ 


^J i 


1 


^-^ 1 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 




y 




•^ y 



The Evolute of Cayley's sextic is 

x=\a + ^a[3cos(|t) - cos(2<)] 
y=^a[3sin(|t)-sin(2t)] ) 

which is a Nephroid. 



Cayley Tree 



Cellular Automaton 



215 



Cayley Tree 

A Tree in which each NODE has a constant number of 
branches. The PERCOLATION THRESHOLD for a Cayley 
tree having z branches is 

1 



Pc 



see also CAYLEY GRAPH 



1" 



Cayleyian Curve 

The Envelope of the lines connecting correspond- 
ing points on the JACOBIAN CURVE and STEINERIAN 
CURVE. The Cayleyian curve of a net of curves of or- 
der n has the same Genus (Curve) as the JACOBIAN 
Curve and Steinerian Curve and, in general, the 
class 3n(n— 1). 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 150, 1959. 

Cech Cohomology 

The direct limit of the COHOMOLOGY groups with CO- 
EFFICIENTS in an ABELIAN GROUP of certain coverings 
of a Topological Space. 

Ceiling Function 



1**1 Ceiling 

[x] Nint (Round) 



|jc| Floor 





-4 




-2 

Jj_: 

i 






JT 


i 
i 




JT 


i 




L 


i 

_ j 







u 



' -2 




The function \x] which gives the smallest INTEGER > as, 
shown as the thick curve in the above plot. Schroeder 
(1991) calls the ceiling function symbols the "Gallows" 
because of the similarity in appearance to the structure 
used for hangings. The name and symbol for the ceiling 
function were coined by K. E. Iverson (Graham et al. 
1990). It can be implemented as ceil(x)=-int (-x), 
where int(x) is the INTEGER PART of x. 

set also Floor Function, Integer Part, Nint 

References 

Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer 
Functions." Ch. 3 in Concrete Mathematics: A Foun- 
dation for Computer Science. Reading, MA: Addison- 
Wesley, pp. 67-101, 1990. 

Iverson, K. E. A Programming Language. New York: Wiley, 
p. 12, 1962. 

Schroeder, M. Fractals, Chaos, Power Laws: Minutes from 
an Infinite Paradise. New York: W. H. Freeman, p. 57, 
1991. 



Cell 

A finite regular POLYTOPE. 

see also 16-Cell, 24-Cell, 120-Cell, 600-Cell 

Cellular Automaton 

A grid (possibly 1-D) of cells which evolves according to 
a set of rules based on the states of surrounding cells, 
von Neumann was one of the first people to consider 
such a model, and incorporated a cellular model into 
his "universal constructor." von Neumann proved that 
an automaton consisting of cells with four orthogonal 
neighbors and 29 possible states would be capable of 
simulating a TURING MACHINE for some configuration 
of about 200,000 cells (Gardner 1983, p. 227). 

l-D automata are called "elementary" and are repre- 
sented by a row of pixels with states either or 1. 
These can be represented with an 8-bit binary num- 
ber, as shown by Stephen Wolfram. Wolfram further 
restricted the number from 2 8 = 256 to 32 by requiring 
certain symmetry conditions. 

The most well-known cellular automaton is Conway's 
game of Life, popularized in Martin Gardner's Scien- 
tific American columns. Although the computation of 
successive Life generations was originally done by hand, 
the computer revolution soon arrived and allowed more 
extensive patterns to be studied and propagated. 

see Life, Langton's Ant 
References 

Adami, C. Artificial Life. Cambridge, MA: MIT Press, 1998. 

Buchi, J. R. and Siefkes, D. (Eds.). Finite Automata, Their 
Algebras and Grammars: Towards a Theory of Formal Ex- 
pressions. New York: Springer- Verlag, 1989. 

Burks, A. W. (Ed.). Essays on Cellular Automata. Urbana- 
Champaign, IL: University of Illinois Press, 1970. 

Cipra, B. "Cellular Automata Offer New Outlook on Life, the 
Universe, and Everything." In What's Happening in the 
Mathematical Sciences, 1995-1996, Vol 3. Providence, 
RI: Amer. Math. Soc, pp. 70-81, 1996. 

Dewdney, A. K. The Armchair Universe: An Exploration of 
Computer Worlds. New York: W. H. Freeman, 1988. 

Gardner, M. "The Game of Life, Parts I— III." Chs. 20-22 in 
Wheels, Life, and Other Mathematical Amusements. New 
York: W. H. Freeman, pp. 219 and 222, 1983. 

Gutowitz, H. (Ed.). Cellular Automata: Theory and Exper- 
iment. Cambridge, MA: MIT Press, 1991. 

Levy, S. Artificial Life: A Report from the Frontier Where 
Computers Meet Biology. New York: Vintage, 1993. 

Martin, O.; Odlyzko, A.; and Wolfram, S. "Algebraic Aspects 
of Cellular Automata." Communications in Mathematical 
Physics 93, 219-258, 1984. 

Mcintosh, H. V. "Cellular Automata." http://www.es. 
cinvestav.mx/mcintosh/cellular.html. 

Preston, K. Jr. and Duff, M. J. B. Modern Cellular Au- 
tomata: Theory and Applications. New York: Plenum, 
1985. 

Sigmund, K. Games of Life: Explorations in Ecology, Evo- 
lution and Behaviour. New York: Penguin, 1995. 

Sloane, N. J. A. Sequences A006977/M2497 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- 
clopedia of Integer Sequences. San Diego: Academic Press, 
1995. 



216 Cellular Space 



Toffoli, T. and Margolus, N. Cellular Automata Machines: 
A New Environment for Modeling. Cambridge, MA: MIT 
Press, 1987. 

Wolfram, S. "Statistical Mechanics of Cellular Automata." 
Rev. Mod. Phys. 55, 601-644, 1983. 

Wolfram, S. (Ed.). Theory and Application of Cellular Au- 
tomata. Reading, MA: Addis on- Wesley, 1986. 

Wolfram, S. Cellular Automata and Complexity: Collected 
Papers. Reading, MA: Addison- Wesley, 1994. 

Wuensche, A. and Lesser, M. The Global Dynamics of Cel- 
lular Automata: An Atlas of Basin of Attraction Fields 
of One- Dimensional Cellular Automata. Reading, MA: 
Addison- Wesley, 1992. 

Cellular Space 

A Hausdorff Space which has the structure of a so- 
called CW-COMPLEX. 

Center 

A special POINT which usually has some symmetric 
placement with respect to points on a curve or in a 
SOLID. The center of a CIRCLE is equidistant from all 
points on the CIRCLE and is the intersection of any two 
distinct DIAMETERS. The same holds true for the center 
of a Sphere. 
see also Center (Group), Center of Mass, Cir- 

CUMCENTER, CURVATURE CENTER, ELLIPSEj EQUI- 

Brocard Center, Excenter, Homothetic Cen- 
ter, Incenter, Inversion Center, Isogonic Cen- 
ters, Major Triangle Center, Nine-Point Cen- 
ter, Orthocenter, Perspective Center, Point, 
Radical Center, Similitude Center, Sphere, 
Spieker Center, Taylor Center, Triangle Cen- 
ter, Triangle Center Function, Yff Center of 
Congruence 

Center Function 

see Triangle Center Function 

Center of Gravity 

see Center of Mass 

Center (Group) 

The center of a GROUP is the set of elements which 
commute with every member of the GROUP. It is equal 
to the intersection of the Centralizers of the Group 
elements. 

see also ISOCLINIC GROUPS, NlLPOTENT GROUP 

Center of Mass 

see Centroid (Geometric) 



Centered Pentagonal Number 

Centered Cube Number 




A Figurate Number of the form, 

CCub n = n +(n- l) 3 = (2n - l)(n 2 - n + 1). 

The first few are 1, 9, 35, 91, 189, 341, ... (Sloane's 
A005898). The Generating Function for the cen- 
tered cube numbers is 

x(x 3 + 5z 2 + 5x + 1) n 2 * ^ 4 

-^ -, -^j [ — - = x + 9x 2 + 35z + 91a? 4 + . . . . 

(x- l) 4 

see also Cubic Number 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 

York: Springer- Verlag, p. 51, 1996. 
Sloane, N. J. A. Sequence A005898/M4616 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Centered Hexagonal Number 

see Hex Number 

Centered Pentagonal Number 




A Centered Polygonal Number consisting of a cen- 
tral dot with five dots around it, and then additional 
dots in the gaps between adjacent dots. The general 
term is (5n 2 - 5n + 2)/2, and the first few such num- 
bers are 1, 6, 16, 31, 51, 76, ... (Sloane's A005891). 
The Generating Function of the centered pentago- 
nal numbers is 



x(x 2 + Sx + 1) 
(z-1) 3 



x + 6x 2 + 16z 3 + 31z 4 + . . . . 



see also CENTERED SQUARE NUMBER, CENTERED TRI- 
ANGULAR Number 

References 

Sloane, N. J. A. Sequence A005891/M4112 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 



Centered Polygonal Number 
Centered Polygonal Number 




N, / ^"*--~- / ^ — • — • — d 



A Figurate Number in which layers of Polygons are 
drawn centered about a point instead of with the point 
at a Vertex. 

see also Centered Pentagonal Number, Centered 
Square Number, Centered Triangular Number 

References 

Sloane, N. J. A. and Plouffe, S. Extended entry for sequence 
M3826 in The Encyclopedia of Integer Sequences. San 
Diego, CA: Academic Press, 1995. 

Centered Square Number 




A Centered Polygonal Number consisting of a cen- 
tral dot with four dots around it, and then additional 
dots in the gaps between adjacent dots. The general 
term is n 2 + (n — l) 2 , and the first few such numbers 
are 1, 5, 13, 25, 41, ... (Sloane's A001844). Centered 
square numbers are the sum of two consecutive SQUARE 
Numbers and are congruent to 1 (mod 4). The Gen- 
erating Function giving the centered square numbers 
is 

(1 — x) 6 

see also Centered Pentagonal Number, Centered 
Polygonal Number, Centered Triangular Num- 
ber, Square Number 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 

York: Springer- Verlag, p. 41, 1996. 
Sloane, N. J. A. Sequence A001844/M3826 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Centered Triangular Number 




A Centered Polygonal Number consisting of a cen- 
tral dot with three dots around it, and then additional 



Central Beta Function 217 

dots in the gaps between adjacent dots. The general 
term is (3n — 3n + 2)/2, and the first few such numbers 
are 1, 4, 10, 19, 31, 46, 64, . . . (Sloane's A005448). The 
Generating Function giving the centered triangular 
numbers is 

x{x' + x + l) =x + 4x * + 10x > + 19x * + .... 
(1 — X) 6 

see also CENTERED PENTAGONAL NUMBER, CENTERED 

Square Number 

References 

Sloane, N. J. A. Sequence A005448/M3378 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Centillion 

In the American system, 10 303 . 

see also Large Number 
Central Angle 




An Angle having its Vertex at a Circle's center 
which is formed by two points on the CIRCLE'S Cir- 
cumference. For angles with the same endpoints, 

C = 29 i, 

where 0; is the INSCRIBED ANGLE. 

References 

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- 
ton, DC: Math. Assoc. Amer., pp. xxi— xxii, 1995. 

Central Beta Function 

10r 




;im[zj 




;im[z] 




MzH*? 



The central beta function is defined by 
f3(p) = B(p,p), 



(1) 



218 Central Binomial Coefficient 



Central Conic 



where B(p,q) is the BETA FUNCTION. It satisfies the 
identities 



^(p) = 2 1 - ap B(p > i) 



(2) 

= 2 1 - 2p cos(7rp)B(f-p,p) (3) 

1 t p dt 



_ 2 T-r n(n + 2p) 

V *1 (n + p)(n + p)" 



(4) 
(5) 



With p = 1/2, the latter gives the WALLIS FORMULA. 
When p = a/b, 



b/3(a/b) = 2 1 - 2a/b J(a,b), 



where 



a 



a,b)= f 

Jo 



1 1*- 1 dt 



The central beta function satisfies 

(2 -\- 4x)0(l -\- x) = x0(x) 

(1 - 2x)j8(l - x)f3(x) = 27rcot(7nr) 
P(\ - x) = 2 4x_1 t<m(7rx)/3(x) 



(6) 



(7) 



(8) 

(9) 
(10) 



P(x)0(x + |) = 2 4 * +1 7r/?(2z)/3(2; C + §). (11) 

For p an Odd Positive Integer, the central beta func- 
tion satisfies the identity 



^ )= vP n -^ 

V fc=l fc=0 



n>(" + i;J- < 12 > 



see a/so BETA FUNCTION, REGULARIZED BETA FUNC- 
TION 

References 

Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua- 
tion of the Gamma Function at Rational Values of Small 
Denominators." IMA J. Numerical Analysis 12, 519—526, 
1992. 

Central Binomial Coefficient 

The nth central binomial coefficient is defined as ( i n / 2 i ) > 
where (™) is a BINOMIAL COEFFICIENT and [n\ is the 
Floor Function. The first few values are 1, 2, 3, 6, 10, 
20, 35, 70, 126, 252, . . . (Sloane's A001405). The central 
binomial coefficients have GENERATING FUNCTION 



2(2# 3 - x 2 ) 

The central binomial coefficients are SQUAREFREE only 
for n = 1, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, . . . (Sloane's 
A046098), with no others less than 1500. 



The above coefficients are a superset of the alternative 
"central" binomial coefficients 



CD- 



(2n)! 

(n!) 2 ' 



which have GENERATING FUNCTION 



v 7 ! - 4z 



: 1 + 2x + 6x 2 + 20z 3 + 70x 4 + . . . . 



The first few values are 2, 6, 20, 70, 252, 924, 3432, 
12870, 48620, 184756, ... (Sloane's A000984). 

Erdos and Graham (1980, p. 71) conjectured that 
the central binomial coefficient ( 2 ^) is never SQUARE- 
FREE for n > 4, and this is sometimes known as the 
Erdos Squarefree Conjecture. Sarkozy's The- 
orem (Sarkozy 1985) provides a partial solution which 
states that the BINOMIAL COEFFICIENT ( 2 ") is never 
Squarefree for all sufficiently large n > no (Vardi 
1991). Granville and Ramare (1996) proved that the 
only Squarefree values are n — 2 and 4. Sander 
(1992) subsequently showed that ( 2n T f d ) are also never 
SQUAREFREE for sufficiently large n as long as d is not 

"too big." 

see also BINOMIAL COEFFICIENT, CENTRAL TRINO- 
MIAL Coefficient, Erdos Squarefree Conjec- 
ture, Sarkozy's Theorem, Quota System 

References 

Granville, A. and Ramare, O. "Explicit Bounds on Exponen- 
tial Sums and the Scarcity of Squarefree Binomial Coeffi- 
cients." Mathematika 43, 73-107, 1996. 

Sander, J. W. "On Prime Divisors of Binomial Coefficients." 
Bull London Math. Soc. 24, 140-142, 1992. 

Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. 
Number Th. 20, 70-80, 1985. 

Sloane, N. J. A. Sequences A046098, A000984/M1645, and 
A001405/M0769 in "An On-Line Version of the Encyclo- 
pedia of Integer Sequences." 

Vardi, I. "Application to Binomial Coefficients," "Binomial 
Coefficients," "A Class of Solutions," "Computing Bino- 
mial Coefficients," and "Binomials Modulo and Integer." 
§2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations 
in Mathematica. Redwood City, CA: Addison- Wesley, 
pp. 25-28 and 63-71, 1991. 

Central Conic 

An Ellipse or Hyperbola. 

see also CONIC Section 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited, 
Washington, DC: Math. Assoc. Amer., pp. 146-150, 1967. 

Ogilvy, C. S. Excursions in Geometry. New York: Dover, 
p. 77, 1990. 



Central Difference 



Central Limit Theorem 219 



Central Difference 

The central difference for a function tabulated at equal 
intervals fi is defined by 

^(/n+l/2) = <Wi/ 2 = $n + l/2 = /n+1 - fn- (1) 

Higher order differences may be computed for Even and 
Odd powers, 



2fc / \ 

C +1 /2 =£(-1)' 2 f/n+ fc 

2fc+l / \ 



(2) 
+fc+i-j- (3) 



see a/so Backward Difference, Divided Differ- 
ence, Forward Difference 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Differences." 
§25.1 in Handbook of Mathematical Functions with Formu- 
las, Graphs, and Mathematical Tables, 9th printing. New 
York: Dover, pp. 877-878, 1972. 

Central Limit Theorem 

Let x\ , X2 , . . . , xn be a set of AT INDEPENDENT random 
variates and each Xi have an arbitrary probability distri- 
bution P(a?i, . . . , xn) with MEAN fii and a finite VARI- 
ANCE cr^ 2 . Then the normal form variate 



A norra — 



v^ 



(1) 



Vi* 



has a limiting distribution which is NORMAL (GAUS- 
SIAN) with Mean \l = and Variance a 2 ~ 1. If 
conversion to normal form is not performed, then the 
variate 



X 



^5> 



(2) 



is Normally Distributed with fi x = \x x and a x = 
o~ x /y/N. To prove this, consider the Inverse Fourier 

Transform of Px{}). 

/oo 
e 2 * ifX p(X)dX 
-OO 



J —c 



sr ( 27ri 



n=Q 



•J —oo 



p{X) dx 



(2^/) n /„Xn 



E^<*> 



(3) 



Now write 

(X n ) = (AT n (xi +X2 + ... + x N ) n ) 

/OO 
N~ n (xi + . .. + xn) u p(xi) - • -p(xN)dxi ---cIxn, 
•oo 

(4) 



so we have 



(2«/)» 



*■ — ' n 



n = «/-oo 

/*°° y^ r 27rz/(x 1 + ... + x JV ) l" 1 

<J — oo _ rt 



+ .., + x^) n 

x p(xi) • • -p(x N ) dxi ■ • ■ dxj\ 



x p(#i) • * -p(xjyf) dx\ • • 'dxj\ 



/oo 



pix^dx! 



F 

V — c 



w p(xn) dxjsr 



p(x) dx 



}' 



= / 6 a-</-/JVp(a.) da . 

= / p(x)dx-\ / xp(x) dx 

L 1 ' — oo <J — oo 



Now expand 



ln(l + x) = x-\x 2 + \x 3 + ... 



(5) 



(6) 



w exp < AT 



N {X) 2N* \ X I 



+ ^<*> 2 + <^- 3 ) 



: exp 



J exp 



(2nf) 2 ((x 2 ) - (x) 2 ) 



2iviffi x 



(27T/)V, 2 

2N 



(7) 



220 



Central Limit Theorem 



Centroid (Geometric) 



Hx = (x) 

a 2 = (x 2 ) - (x) 2 
Taking the FOURIER TRANSFORM, 



(8) 
(9) 



/OO 
e -wr-i[P x (f)]df 
-oo 

= f°° e 2^if(^ x -x)-(2^f) 2 a x 2 /2N d , ^ 

J — oo 



This is of the form 



/CO 
iaf - 
e 
-CO 



bf 



df, 



(11) 



where a = 2iz(ti x — x) and 6 = (27ro~ x ) 2 /2N. But, from 
Abramowitz and Stegun (1972, p. 302, equation 7.4.6), 



/CO 
e iaf- 
-oo 



bf 2 



df = e 



-a 2 /ib /W 



(12) 



Therefore, 



7T J -[27T(fl x -X)} 2 

exp ' — — 



2AT 

27TJV 
47T 2 <7 X 2 



exp 



4 (2™*) 2 f 

^ 2AT J 

4tt 2 (^ -x) 2 2iV 
4 • 47T 2 cr x 2 



ViV 



But ax = <7 x /VN and //x = Man so 



P x = \ c -(mx-^) 2 ^x 2 



(rx\/27r 



(13) 



(14) 



Central Trinomial Coefficient 

The nth central binomial coefficient is denned as the co- 
efficient of x n in the expansion of (l-\-x-\-x 2 ) n . The first 
few are 1, 3, 7, 19, 51, 141, 393, . . . (Sloane's A002426). 
This sequence cannot be expressed as a fixed number 
of hypergeometric terms (Petkovsek et al. 1996, p. 160). 
The Generating Function is given by 



/(*) = 



1 



^(l + z)(l-3x) 



= 1 + x + 3z 2 + 7x 3 + . . . . 



see also Central Binomial Coefficient 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, 1996. 

Sloane, N. J. A. Sequence A002426/M2673 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Centralizer 

The centralizer of a Finite non-ABELiAN Simple 
Group G is an element z of order 2 such that 

C G (z) = {geG:gz = zg}. 

see also Center (Group), Normauzer 
Centrode 

C = rT + kB, 
where r is the TORSION, k is the CURVATURE, T is the 

Tangent Vector, and B is the Binormal Vector. 

Centroid (Function) 

By analogy with the GEOMETRIC CENTROID, the cen- 
troid of an arbitrary function f(x) is defined as 



{x} = 



IZo f( x ) dx 



The "fuzzy" central limit theorem says that data which 
are influenced by many small and unrelated random ef- 
fects are approximately NORMALLY DISTRIBUTED. 

see also LlNDEBERG Condition, Lindeberg-Feller 

Central Limit Theorem, Lyapunov Condition 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

1972, 
Spiegel, M. R. Theory and Problems of Probability and 

Statistics. New York: McGraw-Hill, pp. 112-113, 1992. 
Zabell, S. L. "Alan Turing and the Central Limit Theorem." 

Amer. Math. Monthly 102, 483-494, 1995. 



References 

Bracewell, R. The Fourier Transform and Its Applications. 
New York: McGraw-Hill, pp. 139-140 and 156, 1965. 



Centroid (Geometric) 

The Center of Mass of a 2-D planar Lamina or a 
3-D solid. The mass of a LAMINA with surface density 
function o~(x,y) is 



M-- 



: //' (x ' 



y)dA. 



(1) 



The coordinates of the centroid (also called the CENTER 
of Gravity) are 



ff xo~(x,y) dA 



M 



(2) 



Centroid (Orthocentric System) 

Jfya(x,y)dA 



y 



M 



(3) 



The centroids of several common laminas along the non- 
symmetrical axis are summarized in the following table. 



Figure 



y 



parabolic segment |/t 

3tt 



semicircle 



In 3-D , the mass of a solid with density function 
p(x,y,z) is 



Iff**'* 



M= I I I p(x,y,z)dV, (4) 

and the coordinates of the center of mass are 
_ _ fffxp(x,y,z)dV 



M 

JJfyp(x,y,z)dV 

M 

JfJzp(x,y,z)dV 

M 



(5) 

(6) 
(7) 



Figure 



cone ^ h 

conical frustum ^Y^t 3 ^ 

hemisphere 

paraboloid 

pyramid 



4(R 1 2 +R 1 R 2 +R2 2 ) 



\h 



see also Pappus's Centroid Theorem 

References 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, p. 132, 1987. 

McLean, W. G. and Nelson, E. W. "First Moments and Cen- 
troids." Ch. 9 in Schaum's Outline of Theory and Prob- 
lems of Engineering Mechanics: Statics and Dynamics, 
4th ed. New York: McGraw-Hill, pp. 134-162, 1988. 

Centroid (Orthocentric System) 

The centroid of the four points constituting an ORTHO- 
CENTRIC System is the center of the common Nine- 
Point Circle (Johnson 1929, p. 249). This fact auto- 
matically guarantees that the centroid of the Incenter 
and Excenters of a Triangle is located at the Cir- 
cumcenter. 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, 1929. 



Centroid (Triangle) 221 

Centroid (Triangle) 

The centroid (Center of Mass) of the Vertices of 
a Triangle is the point M (or G) of intersection of 
the Triangle's three Medians, also called the Median 
Point (Johnson 1929, p. 249). The centroid is always 
in the interior of the TRIANGLE, and has TRILINEAR 
Coordinates 



csc A : esc B : esc C. 
If the sides of a TRIANGLE are divided so that 



A 2 Pi A3P2 A ± P 2 



PiA 3 P 2 A X P 3 A 2 



P 

9 



(2) 



(3) 



the centroid of the TRIANGLE AP1P2P3 is M (Johnson 
1929, p. 250). 

Pick an interior point X. The TRIANGLES BXC, CXA, 
and AXB have equal areas IFF X corresponds to the 
centroid. The centroid is located one third of the way 
from each Vertex to the Midpoint of the opposite side. 
Each median divides the triangle into two equal areas; 
all the medians together divide it into six equal parts, 
and the lines from the Median Point to the Vertices 
divide the whole into three equivalent TRIANGLES. In 
general, for any line in the plane of a Triangle ABC, 



d= l{d A + d B + d c ), 



(4) 



where d } d A , ds, and dc are the distances from the cen- 
troid and Vertices to the line. A Triangle will bal- 
ance at the centroid, and along any line passing through 
the centroid. The Trilinear Polar of the centroid is 
called the Lemoine Axis. The Perpendiculars from 
the centroid are proportional to s^ -1 , 

CL1P2 = CL2P2 = dtps - § A, (5) 

where A is the Area of the Triangle. Let P be an 
arbitrary point, the Vertices be Ai, A 2) and A 3) and 
the centroid M. Then 



PA X +PA 2 +PA 3 = MA! +Mi 2 +MA Z +3PM . 

(6) 
If O is the ClRCUMCENTER of the triangle's centroid, 
then 

OM 2 =,R 2 -|(a 2 + 6 2 +c 2 ). (7) 

The centroid lies on the EULER LINE. 

The centroid of the PERIMETER of a TRIANGLE is the 
triangle's Spieker Center (Johnson 1929, p. 249). 

see also ClRCUMCENTER, EULER LlNE, EXMEDIAN 

Point, Incenter, Orthocenter 

References 

Carr, G. S. Formulas and Theorems in Pure Mathematics, 
2nd ed. New York: Chelsea, p. 622, 1970. 



222 



Certificate of Compositeness 



Ceva's Theorem 



Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 

Washington, DC: Math. Assoc. Amer., p. 7, 1967. 
Dixon, R. Mathographics. New York: Dover, pp. 55-57, 1991. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 173-176 and 249, 1929. 
Kimberling, C. "Central Points and Central Lines in the 

Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 
Kimberling, C. "Centroid." http : //www . evansville . edu/ 

-ck6/tcenters/class/centroid.html. 

Certificate of Compositeness 

see Compositeness Certificate 

Certificate of Primality 

see Primality Certificate 

Cesaro Equation 

An Intrinsic Equation which expresses a curve in 
terms of its ARC LENGTH s and RADIUS OF CURVA- 
TURE R (or equivalently, the CURVATURE k). 

see also Arc Length, Intrinsic Equation, Natural 
Equation, Radius of Curvature, Whewell Equa- 
tion 

References 

Yates, R. C. "Intrinsic Equations." A Handbook on Curves 

and Their Properties. Ann Arbor, MI: J. W. Edwards, 

pp. 123-126, 1952. 

Cesaro Fractal 




A Fractal also known as the Torn Square Frac- 
tal. The base curves and motifs for the two fractals 
illustrated above are show below. 




see also Fractal, Koch Snowflake 

References 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, p. 43, 
1991. 

Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide 
World Publ./Tetra, p. 79, 1989. 
^ Weisstein, E. W. "Fractals." http://www. astro. Virginia. 
edu/-eww6n/math/notebooks/Fractal.m. 



Cesaro Mean 

see FEJES TOTH'S INTEGRAL 

Ceva's Theorem 




Given a Triangle with Vertices A, £?, and C and 
points along the sides D, E, and F, a NECESSARY and 
Sufficient condition for the Cevians AD, BE, and 
CF to be Concurrent (intersect in a single point) is 
that 

BDCE-AF^DCEA- FB. (1) 

Let P = [Vi, . . . , V^] be an arbitrary n-gon, C a given 
point, and k a Positive Integer such that 1 < k < 
n/2. For i = 1, . . . , n, let Wi be the intersection of the 
lines CVi and Vi-kV i+ k, then 



n 



Vi-kWi 



WtVi 



i+k 



= 1. 



Here, AB\\CD and 



AB 



VCD\ 



(2) 



(3) 



is the Ratio of the lengths [A, B] and [C, D] with a plus 
or minus sign depending on whether these segments have 
the same or opposite directions (Grunbaum and Shepard 
1995). 

Another form of the theorem is that three Concurrent 
lines from the Vertices of a Triangle divide the op- 
posite sides in such fashion that the product of three 
nonadjacent segments equals the product of the other 
three (Johnson 1929, p. 147). 
see also Hoehn's Theorem, Menelaus' Theorem 

References 

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 
28th ed. Boca Raton, FL: CRC Press, p. 122, 1987. 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 
Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967. 

Grunbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the 
Area Principle." Math. Mag. 68, 254-268, 1995. 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 145-151, 1929. 

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- 
ton, DC: Math. Assoc. Amer., p. xx, 1995. 



Cevian 
Cevian 




A line segment which joins a Vertex of a Triangle 
with a point on the opposite side (or its extension). In 
the above figure, 

6 sin a 



sin(7 + a') 



References 

Thebault, V. "On the Cevians of a Triangle." Amer. Math. 
Monthly 60, 167-173, 1953. 

Cevian Conjugate Point 

see ISOTOMIC CONJUGATE POINT 

Cevian Transform 

Vandeghen's (1965) name for the transformation taking 
points to their ISOTOMIC CONJUGATE POINTS. 

see also Isotomic Conjugate Point 

References 

Vandeghen, A. "Some Remarks on the Isogonal and Cevian 
Transforms. Alignments of Remarkable Points of a Trian- 
gle." Amer. Math. Monthly 72, 1091-1094, 1965. 

Cevian Triangle 




Given a center a : (3 : 7, the cevian triangle is defined 
as that with VERTICES : : 7, a : : 7, and a : 
P : 0. If A'B'C is the CEVIAN TRIANGLE of X and 
A"B"C" is the Anticevian Triangle, then X and 
A" are Harmonic Conjugate Points with respect to 
A and A', 
see also Anticevian Triangle 



Chain Rule 223 

Chain 

Let P be a finite Partially Ordered Set. A chain 
in P is a set of pairwise comparable elements (i.e., a 
Totally Ordered subset). The Width of P is the 
maximum CARDINALITY of an Antichain in P. For a 
Partial Order, the size of the longest Chain is called 
the Width. 

see also Addition Chain, Antichain, Brauer Chain, 
Chain (Graph), Dilworth's Lemma, Hansen Chain 

Chain Fraction 

see Continued Fraction 

Chain (Graph) 

A chain of a GRAPH is a SEQUENCE {x u z 2 , . . ■ , x n } such 
that {x u x 2 ), (052,2:3), .--, (z„_i,a:n) are EDGES of the 
Graph. 

Chain Rule 

If g(x) is DlFFERENTlABLE at the point x and f(x) is 
DlFFERENTIABLE at the point g(x), then / o g is DlF- 
FERENTlABLE at x. Furthermore, let y = f(g(x)) and 

u = g(x), then 

dy _ dy du 

dx du dx 



(i) 



There are a number of related results which also go un- 
der the name of "chain rules." For example, if z — 
f(x,y), x = g{t), and y - h(t), then 



dz _ dz dx 
dt dx dt 



dz dy 

dy dt ' 



(2) 



The "general" chain rule applies to two sets of functions 



yi 



/1 (ui,..., «p) 



and 



:(3) 

y m - fm{ui i ... J Up) 



U± = £l(25i, . .. ,X n ) 



:(4) 

U P = 0p(#l»- • • j^n)- 

Defining the m X n JACOBI MATRIX by 



dyi 
dx. 





dvi . 
dx 2 


dyi 
dx n 


dxi 


dy m 
8x2 


dXn 



(5) 



and similarly for (dyi/duj) and (diii/dxj) then gives 



dyi 
dx. 



-(£)(£)■ m 



224 



Chained Arrow Notation 



Champernowne Constant 



In differential form, this becomes 

dpi du p 
du p dxi 



d _ | dy^diH 
* dui dxi 



+ ^^L ]dxi 



I dmdu± ^dup\ ^ 

du\ &X2 ' ' ' du p 8x2 J 

(Kaplan 1984). 

see also Derivative, Jacobian, Power Rule, Prod- 
uct Rule 

References 

Anton, H. Calculus with Analytic Geometry, 2nd ed. New 

York: Wiley, p. 165, 1984. 
Kaplan, W. "Derivatives and Differentials of Composite 

Functions" and "The General Chain Rule." §2.8 and 2.9 

in Advanced Calculus, 3rd ed. Reading, MA: Addison- 

Wesley, pp. 101-105 and 106-110, 1984. 

Chained Arrow Notation 

A Notation which generalizes Arrow Notation and 
is defined as 

a\-<*"\h = a^b^>c. 



see also Arrow Notation 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, p. 61, 1996. 

Chainette 

see Catenary 

Chair 



Chaitin's Constant 

An Irrational Number Q which gives the probability 
that for any set of instructions, a Universal Turing 
MACHINE will halt. The digits in are random and 
cannot be computed ahead of time. 

see also Halting Problem, Turing Machine, Uni- 
versal Turing Machine 

References 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/chaitin/chaitin.html. 

Gardner, M. "The Random Number Bids Fair to Hold 
the Mysteries of the Universe." Set. Amer. 241, 20-34, 
Nov. 1979. 

Gardner, M. "Chaitin's Omega." Ch. 21 in Fractal Music, 
HyperCards, and More Mathematical Recreations from Sci- 
entific American Magazine. New York: W. H. Freeman, 
1992. 

Kobayashi, K. "Sigma(N)0-Complete Properties of Pro- 
grams and Lartin-Lof Randomness." Information Proc. 
Let 46, 37-42, 1993. 

Chaitin's Number 

see Chaitin's Constant 

Chaitin's Omega 

see Chaitin's Constant 

Champernowne Constant 

Champernowne's number 0.1234567891011. . . (Sloane's 
A033307) is the decimal obtained by concatenating the 
Positive Integers. It is Normal in base 10. In 1961, 
Mahler showed it to also be TRANSCENDENTAL. 

The Continued Fraction of the Champernowne con- 
stant is [0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 
1, 15, 




A Surface with tetrahedral symmetry which, according 
to Nordstrand, looks like an inflatable chair from the 
1970s. It is given by the implicit equation 

(x 2 +y 2 + z 2 -ak 2 ) 2 -b[(z-k) 2 -2x 2 ][{z + k) 2 ~2y 2 ] = 0. 

see also Bride's Chair 

References 

Nordstrand, T. "Chair." http://www.uib.no/people/nfytn/ 
chairtxt.htm. 



457540111391031076483646628242956118599603939- • • 
710457555000662004393090262659256314937953207- - • 
747128656313864120937550355209460718308998457* • * 
5801469863148833592141783010987, 

6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 
58, 8, 54, ...] (Sloane's A030167). The next term of 
the Continued Fraction is huge, having 2504 digits. 
In fact, the coefficients eventually become unbounded, 
making the continued fraction difficult to calculate for 
too many more terms. Large terms greater than 10 5 oc- 
cur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and 
have 6, 166, 2504, 140, 33102, 109, 2468, 136, . . . digits 
(Plouffe). Interestingly, the Copeland-Erdos Con- 
stant, which is the decimal obtained by concatenating 
the Primes, has a well-behaved Continued Fraction 
which does not show the "large term" phenomenon. 
see also COPELAND-ERDOS CONSTANT, SMARANDACHE 

Sequences 



Change of Variables Theorem 



Chaos 225 



References 

Champernowne, D. G. "The Construction of Decimals Nor- 
mal in the Scale of Ten." J. London Math. Soc. 8, 1933. 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/cntfrc/cntfrc.html. 

Sloane, N. J. A. Sequences A030167 and A033307 in "An On- 
Line Version of the Encyclopedia of Integer Sequences." 

Change of Variables Theorem 

A theorem which effectively describes how lengths, ar- 
eas, volumes, and generalized n-dimensional volumes 

(Contents) are distorted by Differentiable Func- 
tions. In particular, the change of variables theorem 
reduces the whole problem of figuring out the distortion 
of the content to understanding the infinitesimal dis- 
tortion, i.e., the distortion of the DERIVATIVE (a linear 
Map), which is given by the linear Map's Determi- 
nant. So / : R n -► W 1 is an Area-Preserving linear 
MAP Iff |det(/)| = 1, and in more generality, if S is 
any subset of MJ 1 , the CONTENT of its image is given by 
I det(/)| times the CONTENT of the original. The change 
of variables theorem takes this infinitesimal knowledge, 
and applies CALCULUS by breaking up the DOMAIN into 
small pieces and adds up the change in AREA, bit by 
bit. 

The change of variable formula persists to the general- 
ity of Differential Forms on Manifolds, giving the 
formula 



/ (/*w) = f (u 
Jm Jw 



under the conditions that M and W are compact con- 
nected oriented MANIFOLDS with nonempty boundaries, 
/ : M — > W is a smooth map which is an orientation- 
preserving DlFFEOMORPHISM of the boundaries. 

In 2-D, the explicit statement of the theorem is 



/. 



f(x,y)dxdy 



-L 



f[x(u,v),y(u,v)] 



d(x,y) 



d(u,v) 



dudv 



and in 3-D, it is 



/ 



/(a;, y, z) dx dy dz 



■ I f[x(u, v,w), y(u, v, w) } z(u, u, 

J R* 



W)] 



d(x,y,z) 



du dv dw , 



d(u, v, w) 
where R = f(R*) is the image of the original region R* , 



d(u,v,w) 



is the JACOBIAN, and / is a global orientation-preserving 
DlFFEOMORPHISM of R and R* (which are open subsets 
ofM n ). 



The change of variables theorem is a simple consequence 
of the Curl Theorem and a little de Rham Cohomol- 
OGY. The generalization to n-D requires no additional 
assumptions other than the regularity conditions on the 
boundary. 

see also Implicit Function Theorem, Jacobian 

References 

Kaplan, W. "Change of Variables in Integrals." §4.6 in Ad- 
vanced Calculus, 3rd ed. Reading, MA: Addison- Wesley, 
pp. 238-245, 1984. 

Chaos 

A Dynamical System is chaotic if it 

1. Has a Dense collection of points with periodic or- 
bits, 

2. Is sensitive to the initial condition of the system (so 
that initially nearby points can evolve quickly into 
very different states), and 

3. Is TOPOLOGICALLY TRANSITIVE. 

Chaotic systems exhibit irregular, unpredictable behav- 
ior (the Butterfly Effect). The boundary between 
linear and chaotic behavior is characterized by PERIOD 
DOUBLING, following by quadrupling, etc. 

An example of a simple physical system which displays 
chaotic behavior is the motion of a magnetic pendulum 
over a plane containing two or more attractive magnets. 
The magnet over which the pendulum ultimately comes 
to rest (due to frictional damping) is highly dependent 
on the starting position and velocity of the pendulum 
(Dickau). Another such system is a double pendulum (a 
pendulum with another pendulum attached to its end). 

see also Accumulation Point, Attractor, Basin 
of Attraction, Butterfly Effect, Chaos Game, 
Feigenbaum Constant, Fractal Dimension, Gin- 
gerbreadman Map, Henon-Heiles Equation, 
Henon Map, Limit Cycle, Logistic Equation, Lya- 
punov Characteristic Exponent, Period Three 
Theorem, Phase Space, Quantum Chaos, Reso- 
nance Overlap Method, Sarkovskii's Theorem, 
Shadowing Theorem, Sink (Map), Strange At- 
tractor 

References 

Bai-Lin, H. Chaos. Singapore: World Scientific, 1984. 

Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Intro- 
duction, 2nd ed. Cambridge: Cambridge University Press, 
1996. 

Cvitanovic, P. Universality in Chaos: A Reprint Selection, 
2nd ed. Bristol: Adam Hilger, 1989. 

Dickau, R. M. "Magnetic Pendulum." http:// forum . 
swarthmore . edu / advanced / robertd / magnetic 
pendulum . html . 

Drazin, P. G. Nonlinear Systems. Cambridge, England: 
Cambridge University Press, 1992. 

Field, M. and Golubitsky, M. Symmetry in Chaos: A Search 
for Pattern in Mathematics, Art and Nature. Oxford, 
England: Oxford University Press, 1992. 

Gleick, J. Chaos: Making a New Science. New York: Pen- 
guin, 1988. 



226 



Chaos Game 



Character Table 



Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, 
Dynamical Systems, and Bifurcations of Vector Fields, 3rd 
ed. New York: Springer- Verlag, 1997. 

Lichtenberg, A. and Lieberman, M. Regular and Stochastic 
Motion, 2nd ed. New York: Springer- Verlag, 1994. 

Lorenz, E. N. The Essence of Chaos. Seattle, WA: University 
of Washington Press, 1996. 

Ott, E, Chaos in Dynamical Systems. New York: Cambridge 
University Press, 1993. 

Ott, E.; Sauer, T.; and Yorke, J. A. Coping with Chaos: 
Analysis of Chaotic Data and the Exploitation of Chaotic 
Systems. New York: Wiley, 1994. 

Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac- 
tals: New Frontiers of Science. New York: Sprhiger- 
Verlag, 1992. 

Poon, L. "Chaos at Maryland." http://www-chaos.umd.edu. 

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. 
New York: Wiley, 1990. 

Strogatz, S. H. Nonlinear Dynamics and Chaos, with Appli- 
cations to Physics, Biology, Chemistry, and Engineering. 
Reading, MA: Addis on- Wesley, 1994. 

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: 
An Introduction. New York: Wiley, 1989, 

Tufillaro, N.; Abbott, T. R.; and Reilly, J. An Experimental 
Approach to Nonlinear Dynamics and Chaos. Redwood 
City, CA: Addison-Wesley, 1992. 

Wiggins, S. Global Bifurcations and Chaos: Analytical Meth- 
ods. New York: Springer- Verlag, 1988. 

Wiggins, S. Introduction to Applied Nonlinear Dynamical 
Systems and Chaos. New York: Springer- Verlag, 1990. 

Chaos Game 

Pick a point at random inside a regular n-gon. Then 
draw the next point a fraction r of the distance between 
it and a Vertex picked at random. Continue the pro- 
cess (after throwing out the first few points). The result 
of this "chaos game" is sometimes, but not always, a 
Fractal. The case (n,r) = (4,1/2) gives the interior 
of a SQUARE with all points visited with equal probabil- 
ity. 












<Tfc 
******** 



A A 






4% 



Ah; 

f\ A. 



A 






******** 



(3,1/2) 

&& pig 



(5,1/3) 



& 



hi 

(5,3/8) 
see a/so Barnsley's Fern 




(6,1/3) 



References 

Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. 
Boston, MA: Academic Press, 1993. 

Dickau, R. M. "The Chaos Game." http:// forum . 
swarthmore.edu/advanced/robertd/chaos_game.html. 

Wagon, S. Mathematica in Action. New York: W. H. Free- 
man, pp. 149-163, 1991. 
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia. 
edu/-eww6n/math/notebooks/Fractal.m. 

Character (Group) 

The Group Theory term for what is known to physi- 
cists as the Trace. All members of the same Conju- 
GACY Class in the same representation have the same 
character. Members of other Conjugacy Classes may 
also have the same character, however. An (abstract) 
Group can be uniquely identified by a listing of the 
characters of its various representations, known as a 
Character Table. Some of the Schonflies Sym- 
bols denote different sets of symmetry operations but 
correspond to the same abstract GROUP and so have the 
same Character Tables. 

Character (Multiplicative) 

A continuous HOMEOMORPHISM of a GROUP into the 

Nonzero Complex Numbers. A multiplicative char- 
acter w gives a REPRESENTATION on the 1-D SPACE C 

of Complex Numbers, where the Representation ac- 
tion by g 6 G is multiplication by uj(g). A multiplicative 
character is UNITARY if it has ABSOLUTE VALUE 1 ev- 
erywhere. 

References 

Knapp, A. W. "Group Representations and Harmonic Anal- 
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. 

Character (Number Theory) 

A number theoretic function Xk(n) for POSITIVE integral 

n is a character modulo k if 

X*(l) = l 
Xk{n) = Xk(n + k) 
Xk(m)xk(n) = Xk(mn) 



for all m^n, and 



X*(») = 0- 



if (fc,n) ^ 1. Xk can only assume values which are (j>{k) 

Roots of Unity, where <j> is the Totient Function. 

see also DlRlCHLET L-SERIES 
Character Table 



C x 


E 








A 


1 








C 8 


E 


CTh 






A 
B 


1 
1 


1 
-1 


-3 j JXx , •t* j y 


2 2 2 

x ,y ,z 
yz,xz 


xy 



Character Table 



Character Table 227 



a 


E 


i 






A 9 
A u 


1 

1 


1 

-1 


x,y,z 


x 2 ,y 2 ,z 2 ,xy,xz 


yz 














c 2 


E 


c 2 








A 
B 


1 
1 


1 
-1 


z,R z 
x,y,R x ,R y 


x'\y\z'\xy 
yz,xz 





C 3 


E C3 Cz 




e = exp(27rz/3) 


A 
E 


111 

{! I- f } 


z,R z 

{x,y)(R x ,R y ) 


222 
x ,y ,z ,xy 

(x 2 -y 2 ,xy){yz,xz) 



c 4 


E O3 C 2 C4 






A 
B 

E 


1111 
1-1 1-1 

ri i -1 -n 

ll-i 1 i) 


z,R z 

(x,y)(R x ,R y ) 


2,22 

x +y ,z 
x 2 -y 2 ,xy 

(yz,xz) 



D 6 


E 2C 6 


2O3 O2 3Gj 3G 2 






A, 


1 1 


1111 




x 2 +y\z 2 


A 2 


1 1 


1 1-1-1 


z, R z 




B 1 


1 -1 


1-1 1-1 






B 2 


1 -1 


1-1-1 1 


(x^yXR^Ry) 




E 1 


2 1 


-1-2 




(xz,yz) 


E 2 


2 -1 


-12 




(x 2 -y 2 ,xy) 



C2v 


E C 2 


cr v (xz) 


°'v{yz) 






A 1 


1 1 


1 


1 


z 


2 2 
x ,y 


z 2 


A 2 


1 1 


-1 


-1 


Rz 


xy 




3i 


1 -1 


1 


-1 


X, ity 


xz 




B 2 


1 -1 


-1 


1 


y,Rx 


yz 





c$ v 



Ai 
A 2 
E 



E 2 C3 3<x v 



1 1 1 
1 1 -1 
2-10 



z 
Rz 

(x,y)(R x ,R y ) 



~^2~, 2 2~~ 

x +y ,z 



(x 2 -y 2 ,xy)(xz,yz) 



c & 


E C 5 C 5 2 


c 5 3 


c 5 4 




e = exp(27ri/5) 


A 


11 1 


1 


1 


2,H, 


2,22 
a; 4- y ,z 


E, 


fie e 2 
tl e * e 2 ' 


e 2 * 
e 2 


r} 


(x,*/)^,^) 


(yz, xz) 


E 2 


(1 £ 2 e* 

ll e 2 * e 


e 

£* 


?} 




(x 2 — y 2 ,xy) 



c. 


E 


c 6 


c 3 


C 2 Gz 


<V 




£ = exp(27rt/6) 


A 


1 


1 


1 


1 1 


1 


z,R x 


2 1 2 2 
x + y ,z 


B 


1 


_i 


1 


-1 1 


-1 






Ei 


(I 


£ 


— £* 

— e 


-1 -£ 
~1 ~<T 


I'} 


(s,y) 

(R x , Ry) 


(f,^) 


E 2 


(i 


— £ 

— £* 


— £ 

— £* 


1 -£* 

1 -£ 


-:■} 




(x 2 - y 2 , xy) 



Z>2 


E C 2 (z) C 2 {y) C 2 (x) 






A 1 


1111 




2,22 

x +y ,z 


B 1 


1 1-1-1 


z,R z 


xy 


B 2 


1-1 1-1 


y,Ry 


xz 


B 3 


1-1-1 1 


z,R z 


yz 



D 3 



A 1 
A 2 

E 



E 2C 3 3C 2 



111 

1 1 -1 

2 -1 



z,R z 

(x,y)(R x ,R y ) 



ar -\-y,z z 

xy 

(x 2 -y 2 ,xy){xz,yz) 



D 4 


E 2C4 C 2 2C 2 2C2 






Ai 


11111 




2,22 

x +y ,z 


A 2 


1 11-1-1 


z,R z 




Bi 


1-11 1-1 




2 2 
x -y 


B 2 


1-11-1 1 




xy 


E 


2 0-20 


(x,y)(R x ,R y ) 


(xz,yz) 



D 5 


E 2C 5 


2C 5 2 


5C 2 






A x 
Bi 
B 2 
£3 


1 1 

1 1 

2 2 cos 72° 
2 2 cos 144° 


1 

1 
2 cos 144° 
2 cos 72° 


1 

-1 




z,R z 

(x,y)(R x ,R y ) 


x 2 ^y 2 ,z 2 

{xz,yz) 

{x 2 -y 2 ,xy) 



Cav 


E 


2C4 O2 2(T V 2(Td 






A 2 
B 1 
B 2 
E 


1 
1 
1 

1 
2 


1111 

1 1-1-1 

-11 1-1 

-11-1 1 

0-200 


z 
Rz 

(x,y)(R x ,R y ) 


2,22 

x z +y,z z 

2 2 
x -y 

xy 

(xz,yz) 



c 5v 


E 2C 5 


2C 5 2 


5<7v 






A x 
Bi 
B 2 
B3 


1 1 

1 1 

2 2 cos 72° 
2 2 cos 144° 


1 

1 
2 cos 144° 
2 cos 72° 


1 

-1 






z 

R z 

(x,y)(R x ,R y ) 


x 2 +y 2 ,z 2 

(xz.yz) 

(x 2 -y 2 ,xy) 



c, v 


E 


2C 6 


2C 3 


C 2 3cr v 


3cr d 






A! 


1 






1 1 


1 


z 


* 2 \y\z 2 


A 2 


1 






1 -1 


-1 


Rz 




B l 


1 


-1 




-1 1 


-1 






B 2 


1 


-1 




-1 _i 


1 






E l 


2 




-1 


-2 





(x,y)(R x ,R y ) 


(xz,yz) 


E 2 


2 


-1 


-1 


2 







(x 2 - y 2 ,xy) 



c». 


£7 


Coo* • 


oocr„ 






A x = S + 


1 


1 


1 


z 


x 2 +y 2 ,z 2 


A 2 = E" 


1 


1 


.. -1 


Rz 




E x = n 


2 


2 cos <£ 





(x,y);(R x ,R y ) 


(xz,yz) 


£? 2 = A 


2 


2 cos 2* 







(x 2 - y 2 ,xy) 


S 3 =* 


2 


2 cos 3* 










References 

Bishop, D. M. "Character Tables." Appendix 1 in Group 
Theory and Chemistry. New York: Dover, pp. 279—288, 
1993. 

Cotton, F. A. Chemical Applications of Group Theory, 3rd 
ed. New York: Wiley, 1990. 

Iyanaga, S. and Kawada, Y. (Eds.). "Characters of Finite 
Groups." Appendix B, Table 5 in Encyclopedic Dictionary 
of Mathematics. Cambridge, MA: MIT Press, pp. 1496- 
1503, 1980. 



228 



Characteristic Class 



Characteristic (Field) 



Characteristic Class 

Characteristic classes are Cohomology classes in the 
Base Space of a Vector Bundle, defined through 
Obstruction theory, which are (perhaps partial) ob- 
structions to the existence of k everywhere linearly 
independent vector Fields on the Vector Bundle. 
The most common examples of characteristic classes 
are the Chern, Pontryagin, and Stiefel- Whitney 
Classes. 

Characteristic (Elliptic Integral) 

A parameter n used to specify an ELLIPTIC INTEGRAL 
of the Third Kind. 

see also AMPLITUDE, ELLIPTIC INTEGRAL, MODULAR 

Angle, Modulus (Elliptic Integral), Nome, Pa- 
rameter 

References 

Abramowitz, M. and Stegun, C. A, (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 590, 1972. 

Characteristic Equation 

The equation which is solved to find a Matrix's Eigen- 
values, also called the CHARACTERISTIC POLYNOMIAL. 
Given a 2 x 2 system of equations with MATRIX 



M 



the Matrix Equation is 



a b 
c d 



a b 
c d 




X 


= t 


X 

y_ 



which can be rewritten 



(i) 



(2) 



(3) 



(4) 



which contradicts our ability to pick arbitrary x and y. 
Therefore, M has no inverse, so its Determinant is 0. 
This gives the characteristic equation 



a — t b 
c d — t 



= t 



M can have no Matrix Inverse, since otherwise 



X 


= M" 1 


"o" 




= 


V 





a — t b 
c d — t 



= 0, 



(5) 



where | A| denotes the Determinant of A. For a general 
k x k Matrix 



(6) 



an 


ai2 ■ 


- • aifc 


021 


^22 • 


. . Q>2k 


afci 


&k2 . 


•• a>kh 



the characteristic equation is 



an — t a 12 

0,21 CL22 — t 



CLkl 



ak2 



aifc 
a2fc 

&kk — t 



(7) 



see also Ballieu's Theorem, Cayley-Hamilton 
Theorem, Parodi's Theorem, Routh-Hurwitz 
Theorem 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, pp. 1117-1119, 1979. 

Characteristic (Euler) 

see Euler Characteristic 

Characteristic Factor 

A characteristic factor is a factor in a particular fac- 
torization of the Totient Function <j>(n) such that 
the product of characteristic factors gives the represen- 
tation of a corresponding abstract Group as a Direct 
PRODUCT. By computing the characteristic factors, any 
Abelian Group can be expressed as a Direct Prod- 
uct of Cyclic Subgroups, for example, Z 2 ® Z 4 or 
Z2® Z2® Z 2 . There is a simple algorithm for determining 
the characteristic factors of Modulo Multiplication 
Groups. 

see also Cyclic Group, Direct Product (Group), 
Modulo Multiplication Group, Totient Func- 
tion 

References 

Shanks, D. Solved and Unsolved Problems in Number Theory, 
4th ed. New York: Chelsea, p. 94, 1993. 

Characteristic (Field) 

For a FIELD K with multiplicative identity 1, consider 
the numbers 2 = 1 + 1, 3 = 1 + 1 + 1,4 = 1 + 1 + 1 + 1, 
etc. Either these numbers are all different, in which 
case we say that K has characteristic 0, or two of them 
will be equal. In this case, it is straightforward to show 
that, for some number p, we have 1 + 1 + .. . + 1 = 0. 

p times 

If p is chosen to be as small as possible, then p will 
be a Prime, and we say that K has characteristic p. 
The Fields Q, E, C, and the /?-adic Numbers Q p 
have characteristic 0. For p a Prime, the Galois Field 
GF(p n ) has characteristic p. 

If H is a Subfield of K, then H and K have the same 
characteristic. 

see also Field, Subfield 



Characteristic Function 



Chasles's Polars Theorem 229 



Characteristic Function 

The characteristic function <j>(t) is defined as the Four- 
ier Transform of the Probability Density Func- 
tion, 

/CO 
e iix P{x)dx (1) 

■oo 

/OO /"OO 

P(x)dx + it / xP(x)dx 
■oo J — OO 

/OO 
x 2 P(z)dx + ... (2) 

OO 



= 1 + ii/i'i - ^2 - ^f« 3 /*3 + ^Vi + . . . , (4) 

where fi f n (sometimes also denoted i/ n ) is the nth MO- 
MENT about and {j! = 1. The characteristic function 
can therefore be used to generate MOMENTS about 0, 



or the Cumulants « n , 

OO 

z — ' n! 



(5) 



(6) 



A Distribution is not uniquely specified by its Mo- 
ments, but is uniquely specified by its characteristic 
function. 

see also Cumulant, Moment, Moment-Generating 
Function, Probability Density Function 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 928, 1972. 

Kenney, J. F. and Keeping, E. S. "Moment-Generating and 
Characteristic Functions," "Some Examples of Moment- 
Generating Functions," and "Uniqueness Theorem for 
Characteristic Functions." §4.6—4.8 in Mathematics of 
Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 
pp. 72-77, 1951. 

Characteristic (Partial Differential 
Equation) 

Paths in a 2-D plane used to transform Partial Dif- 
ferential Equations into systems of Ordinary Dif- 
ferential EQUATIONS. They were invented by Rie- 
mann. For an example of the use of characteristics, con- 
sider the equation 

Ut - 6uu x = 0. 

Now let u(s) = u(x(s))t(s)). Since 



it follows that dt/ds = 1, dx/ds = — 6u, and du/ds = 
0. Integrating gives t(s) = s, x(s) — -6su (x) J and 
u(s) = uo(x) 7 where the constants of integration are 
and Uq(x) = u(x, 0). 

Characteristic Polynomial 

The expanded form of the CHARACTERISTIC EQUATION. 

det(al - A), 

where A is an n x n MATRIX and I is the IDENTITY 
Matrix. 

see also Cayley-Hamilton Theorem 

Characteristic (Real Number) 

For a Real Number x, [^J = int(x) is called the char- 
acteristic. Here, [x\ is the FLOOR FUNCTION. 

see also MANTISSA, SCIENTIFIC NOTATION 

Charlier's Check 

A check which can be used to verify correct computation 

of Moments. 

Chasles-Cayley-Brill Formula 

The number of coincidences of a (i/, i/') correspondence 
of value 7 on a curve of Genus p is given by 

v + v + 2^7. 

see also Zeuthen's Theorem 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 129, 1959. 

Chasles's Contact Theorem 

If a one-parameter family of curves has index N and 
class M, the number tangent to a curve of order m and 
class mi in general position is 

mi TV -hm M. 



References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 436, 1959. 



Chasles's Polars Theorem 

If the Trilinear Polars of the Vertices of a Tri- 
angle are distinct from the respectively opposite sides, 
they meet the sides in three Collinear points. 

see also COLLINEAR, TRIANGLE, TRILINEAR POLAR 



du 
ds 



dx dt 

~ru x + -j-u t , 
ds ds 



230 



Chasles's Theorem 



Chasles's Theorem 

If two projective PENCILS of curves of orders n and n' 
have no common curve, the LOCUS of the intersections of 
corresponding curves of the two is a curve of order n + n f 
through all the centers of either PENCIL. Conversely, if 
a curve of order n + n 1 contains all centers of a PENCIL 
of order n to the multiplicity demanded by Noether'S 
Fundamental Theorem, then it is the Locus of the 
intersections of corresponding curves of this PENCIL and 
one of order n projective therewith. 
see also Noether's Fundamental Theorem, Pencil 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 33, 1959. 

Chebyshev Approximation Formula 

Using a Chebyshev Polynomial of the First Kind 
T\ define 



Cj = ^^f{x k )Tj{x k ) 



k=i 

N 



= NZ^ f cos {^v— / cos { 

k=l L y J J ^ 



"*i(*-§) 



jv 



Then 



f{x)K^c k T k (x)-\c . 



It is exact for the TV zeros of T N (x). This type of ap- 
proximation is important because, when truncated, the 
error is spread smoothly over [—1,1]. The Chebyshev 
approximation formula is very close to the MlNIMAX 
Polynomial. 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and 
Vetterling, W. T. "Chebyshev Approximation," "Deriva- 
tives or Integrals of a Chebyshev- Approximated Function," 
and "Polynomial Approximation from Chebyshev Coeffi- 
cients." §5.8, 5.9, and 5.10 in Numerical Recipes in FOR- 
TRAN: The Art of Scientific Computing, 2nd ed. Cam- 
bridge, England: Cambridge University Press, pp. 184- 
188, 189-190, and 191-192, 1992. 

Chebyshev Constants 

N.B. A detailed on-line essay by S. Finch was the start- 
ing point for this entry. 



The constants 



where 



inf sup \e x ~ r(x)\, 
reRm, n x >o 



r(x) = 



P(s) 
q{xY 



p and q are mth and nth order POLYNOMIALS, and R mt n 
is the set all RATIONAL FUNCTIONS with REAL coeffi- 
cients. 



Chebyshev Differential Equation 

see also One-Ninth Constant, Rational Function 

References 

Finch, S. "Favorite Mathematical Constants." http://www. 
mathsoft.com/asolve/constant/onenin/onenin.html. 

Petrushev, P. P. and Popov, V. A. Rational Approximation of 
Real Functions. New York: Cambridge University Press, 
1987. 

Varga, R. S. Scientific Computations on Mathematical Prob- 
lems and Conjectures. Philadelphia, PA: SIAM, 1990. 

Philadelphia, PA: SIAM, 1990. 

Chebyshev Deviation 



max {|/(x) - p(x)\w(x)}. 

a<x<b 



References 

Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI: 
Amer. Math. Soc, p. 41, 1975. 

Chebyshev Differential Equation 

( 1 -^)S-S+-'» = o W 

for | x | < 1. The Chebyshev differential equation has reg- 
ular Singularities at -1, 1, and oo. It can be solved 
by series solution using the expansions 



y = ^2a n x n (2) 

OO oo 

y = \ na n x n ~ = y na n x n ~ 

n=0 n=l 

oo 

= J^(™ + l)a n +ix n (3) 

71 = 

oo oo 

y" = ^(n + l)na ri+ ix n " 1 = ^(n + ^na^ix 71 ' 1 

n=0 n=l 

oo 

= ^(n + 2)(n + l)a n+2 x n . (4) 

71 = 

Now, plug (2-4) into the original equation (1) to obtain 

oo 

(1 - x 2 ) ^(n + 2)(n + l)a n+2 x n 

n-0 

oo oo 

-x "^(n + l)n n+1 x n + rn ^ a n x n = (5) 

n=0 n=0 

oo oo 

^(n + 2)(n + l)a n+2 x n - ^(n + 2)(n + l)a n + 2 x n+2 

n=0 n=0 

oo oo 

- J](n+l)a n+1 x n+1 +m 2 ^a„x n = (6) 



Chebyshev Differential Equation 

OO CO 

VVn + 2)(n + l)a n+ 2X n - V^ n(n - l)a n x n+2 

n=0 n=2 

OO OO 

— > na n x n -\- m /, a nX n = (?) 



2 2 

2 * la2 + 3 • 2a%x — 1 • ax + m ao + m aiz 



+ y^[(" + 2)(n + l)a n+2 - n(n - l)a„ 

— na n + m an]/ = (8) 



n=2 



(2a 2 4- m 2 a ) + [(m 2 - l)ai + 6a 3 ]a 



+ ^[(n + 2)(n + l)a n+2 + (m 2 - nVl^ = °> ( 9 ) 



2a 2 +771 ao = 
(m 2 — l)ai + 6a3 = 



a n +2 



2 2 

n — m 



for n = 2, 3, . 



(10) 
(11) 
(12) 



(n + l)(n + 2) 
The first two are special cases of the third, so the general 



recurrence relation is 
n 2 — m 



for n = 0, 1, 



(n+l)(n + 2) 
Prom this, we obtain for the EVEN COEFFICIENTS 



a 2 = -|m 2 ao 



a4 



a2n 



-a 2 = 



(2 2 - m 2 )(-m 2 ) 



ao 



3*4 ~* 1-2*3*4 

[(2n) 2 - m 2 ][(2n - 2) 2 - m 2 } • • • [-m 2 ] 
(2n)! 



ao, 



(13) 

(14) 
(15) 

(16) 



and for the Odd Coefficients 



So the general solution is 

[A,* _ m 2 ][(k - 2) 2 -m 2 ]---[-m 2 ] r 



V = a 



1 + 



E 



z + 



E 



[{k - 2) 2 - m 2 ][(Jfe - 2) 2 - m 2 ] ■ • • [I 2 - m 2 ] 

3 

fc! 



Chebyshev- Gauss Quadrature 231 



If n is Even, then y\ terminates and is a Polynomial 
solution, whereas if n is ODD, then y 2 terminates and 
is a Polynomial solution. The Polynomial solutions 
defined here are known as CHEBYSHEV POLYNOMIALS 
of the First Kind. The definition of the Chebyshev 
Polynomial of the Second Kind gives a similar, but 
distinct, recurrence relation 

, (n+ l) 2 - m 2 , , . 

fln+2 = ; , w .^ n for n = 0, 1, . . . . (21) 
(n + 2)(n + 3) 

Chebyshev Function 

0(z) = ^lnp, 

p<a: 

where the sum is over PRIMES p, so 



hm -^-r = 1. 



Chebyshev-Gauss Quadrature 

Also called Chebyshev Quadrature. A Gaussian 
Quadrature over the interval [—1,1] with Weight- 
ing Function W(x) = l/\/i - z 2 - The Abscissas for 

quadrature order n are given by the roots of the CHEBY- 
SHEV Polynomial of the First Kind T n (x), which 

occur symmetrically about 0. The WEIGHTS are 



Wi ■ 



A n +l7n 



A n 



7n-l 



' A n Tk(xi)T n +i(xi) A n -! T n - l (x i )T n (x t )' 

(1) 
where A n is the COEFFICIENT of x n in T n (x). For HER- 
mite Polynomials, 



1-m 2 
o 


(17) 


Additionally, 


3 2 -m 2 (3 2 -m 2 )(l 2 -m 2 ) 
a 5 = 4 5 a 3 = 5 , 


(18) 


so 


[(2n - l) 2 - m 2 ][{2n - 3) 2 - m 2 ] ■ ■ ■ [l 2 - 


-m 2 ] 


Since 


a ' n - L ~ (2n + l)! 


ai- 




(19) 





A n = 2 

A n+1 
A n 

In = 



= 2. 



|tt, 



WJi = 



T n+1 (xi)T n (xi)' 



T n {x) = cos(ncos x), 
the ABSCISSAS are given explicitly by 

(2i- 1)tt" 



Since 



Xi = cos 



T' n {Xi) = 



In 



(~1)' +1 » 



(20) 



T„ + i(o;i) = (-l)'sinai, 



(2) 

(3) 

(4) 
(5) 

(6) 
(7) 

(8) 
(9) 



232 Chebyshev Inequality 

where 



on = 



(2i - 1)tt 
2n ' 



all the Weights are 



Wi 



(10) 



(11) 



The explicit Formula is then 
f(x)dx 



i: 



vr 



Zt'hF^)]*^'™®- < 12 > 



11^ 



2 ±0.707107 1.5708 

3 1.0472 
±0.866025 1.0472 

4 ±0.382683 0.785398 
±0.92388 0.785398 

5 0.628319 
±0.587785 0.628319 
±0.951057 0.628319 



References 

Hildebrand, F. B. Introduction to Numerical Analysis. New 
York: McGraw-Hill, pp. 330-331, 1956. 



Chebyshev Inequality 

Apply Markov's Inequality with a = k 2 to obtain 



P[{x-fxf >k 2 } < 



((x-nf) _a 2 



k 2 



= h- (^ 



Therefore, if a RANDOM Variable x has a finite Mean 
H and finite VARIANCE <r 2 , then V ft > 0, 



P(\x - fi\ > ft) < -^ 



P(\x - fi\> ka) < 



(2) 
(3) 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 11, 1972. 



Chebyshev Integral 

x p (l-x) q dx. 



/■ 



Chebyshev Polynomial 

Chebyshev Integral Inequality 

/ fi(x)dx I f 2 (x)dx--- I f n (x)dx 

«/ a J a J a 



<{b- 



J a 



f(xi)f(x 2 )"-f n (x)dx t 



where /i, / 2 , . . . , f n are NONNEGATIVE integrable func- 
tions on [a, 6] which are monotonic increasing or decreas- 
ing. 

References 

Gradshteyn, IS. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1092, 1979. 

Chebyshev Phenomenon 
see Prime Quadratic Effect 

Chebyshev Polynomial of the First Kind 



0.5 



-0.5 



A set of Orthogonal Polynomials defined as the so- 
lutions to the Chebyshev Differential Equation 
and denoted T n (x). They are used as an approxima- 
tion to a Least Squares Fit, and are a special case 
of the Ultraspherical Polynomial with a = 0. The 
Chebyshev polynomials of the first kind T n (x) are illus- 
trated above for x £ [0, 1] and n = 1, 2, . . . , 5. 

The Chebyshev polynomials of the first kind can be ob- 
tained from the generating functions 




9i(t>n) 



\-t z 



1 - 2xt + t 2 



= T {x) + 2j2T n (x)t n (1) 



and 



9*(t,z)= , \j\^ =Y. T ^ tn ( 2 ) 



l-2xt + t 2 



n=0 



for \x\ < 1 and \t\ < 1 (Beeler et al 1972, Item 15). 
(A closely related Generating Function is the basis 
for the definition of Chebyshev Polynomial of the 
Second Kind.) They are normalized such that T„(l) = 
1. They can also be written 






(3) 



Chebyshev Polynomial 



Chebyshev Polynomial 233 



or in terms of a DETERMINANT 



X 


1 





■ 


•• 





1 


2x 


1 


■ 


■■ 








1 


2x 


1 ■ 


■■ 











1 


2x ■ 


■• 














* 


•• 1 


2x 



(4) 



In closed form, 



L«/2J / v 

T n (x) = cosmos" 1 z) = ^ I £)* n ~ 2m (* 2 " 1)™ 

m=0 ^ ' 

(5) 
where (™) is a BINOMIAL COEFFICIENT and \_x\ is the 
Floor Function. Therefore, zeros occur when 



*(*-§) 



for k — 1, 2, . . . , n. Extrema occur for 
X — cos I — J , 



(6) 



(7) 



where k = 0, 1, . . . , n. At maximum, T n (x) = 1, and 
at minimum, T n (x) = -1. The Chebyshev POLYNOMI- 
ALS are Orthonormal with respect to the Weighting 

Function (1 - x 2 )~ 1/2 



/', 



T m (x)T n {x)dx 
Vl-x 2 



{I 



ir8 n m for m ^ 0, n ^ 
for m = n = 0, 



(8) 

where £ m n is the KRONECKER DELTA. Chebyshev poly- 
nomials of the first kind satisfy the additional discrete 

identity 

m s 

-£-' m for % = 7 = 0, 

where Xk for fc = 1, . . . , m are the m zeros of T m (x). 
They also satisfy the Recurrence Relations 

T n+1 (x) = 2xT n (x) - T n _i(x) (10) 

T n+ i(a:) - xT„(x) - ^/(l- x 2){l-[T n (x)}2} (11) 

for n > 1. They have a Complex integral representa- 
tion 

Tn{x) = 4ri I l-2 X z + z> (12) 

and a Rodrigues representation 



Using a FAST FIBONACCI TRANSFORM with multiplica- 
tion law 

(A, B)(C, D) = (AD + BC + 2xAC, BD - AC) (14) 

gives 

(T n+ i(aO,-T n (aO) = (Ti(aO,-T (aO)(l,0) n . (15) 

Using Gram-Schmidt Orthonormalization in the 
range (-1,1) with Weighting Function (1-x 2 ) c ~ 1/2) 
gives 



Po(x) = 
pi(x) = 



p 2 {x) = 



/^^(1-x 2 )- 1 / 2 ^ 
/^(l-a: 2 )- 1 ^^ 

[-(l-* a ) 1/3 ]li =g ' 
[sin 1 a:]l: 1 

/^(l-x 2 )- 1 / 2 ^ 
/^^(l-o: 2 )- 1 / 2 ^ 

f\(l - x 2 )- 1 / 2 dx 



(16) 



(17) 



X - 



• 1 



= [x — 0]x — - = x — h, 
etc. Normalizing such that T n (l) = 1 gives 

T (x) = 1 

Tx(x) = x 

T 2 (x) = 2x 2 -1 

T 3 (x) = 4x 3 -Sx 

T 4 (x) = 8x 4 -8x 2 + l 

Ts(x) = 16z 5 -20z 3 + 5z 

T 6 (x) = 32z 6 - 48a; 4 + 18x 2 - 1. 



(18) 



The Chebyshev polynomial of the first kind is related 
to the Bessel Function of the First Kind J„(x) 
and Modified Bessel Function of the First Kind 
I n {x) by the relations 

J n (x) = i n T n (i-j^j Jo(x) (19) 

I n {x)=T n (J^)lo(x). (20) 

Letting x = cos 8 allows the Chebyshev polynomials of 
the first kind to be written as 

T n (x) = cos(rz0) = cos(ncos~ x). (21) 



234 Chebyshev Polynomial 



Chebyshev Polynomial 



The second linearly dependent solution to the trans- 
formed differential equation 



d T n t 2 



d9 2 



+ ri T n = 



(22) 



is then given by 

V n (x) = sin(n#) = sin(ncos~ a;), (23) 



which can also be written 



V n (x) = Vl-X 2 C/„-i(x), 



(24) 



where U n is a Chebyshev Polynomial of the Sec- 
ond Kind. Note that V n (x) is therefore not a Poly- 
nomial. 



The Polynomial 



x n - 2 L - n T n (x) 



(25) 



(of degree n — 2) is the POLYNOMIAL of degree < n which 
stays closest to x n in the interval (—1,1). The maximum 
deviation is 2 1 ~ n at the n -+- 1 points where 









(26) 



for k = 0, 1, . . . , n (Beeler et al. 1972, Item 15). 

see also Chebyshev Approximation Formula, 

Chebyshev Polynomial of the Second Kind 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal 
Polynomials." Ch. 22 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 771-802, 1972. 

Arfken, G. "Chebyshev (TschebyschefF) Polynomials" and 
"Chebyshev Polynomials — Numerical Applications." §13.3 
and 13.4 in Mathematical Methods for Physicists, 3rd ed. 
Orlando, FL: Academic Press, pp. 731-748, 1985. 

Beeler, M.; Gosper, R. W.; and Schroeppel, R HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM 239, Feb. 1972. 

Iyanaga, S. and Kawada, Y. (Eds.). "Cebysev (Tscheby- 
schefF) Polynomials." Appendix A, Table 20.11 in Encyclo- 
pedic Dictionary of Mathematics. Cambridge, MA: MIT 
Press, pp. 1478-1479, 1980. 

Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 
1990. 

Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi- 
als T n (x) and U n (x)" Ch. 22 in An Atlas of Functions. 
Washington, DC: Hemisphere, pp. 193-207, 1987. 

Chebyshev Polynomial of the Second Kind 




A modified set of Chebyshev Polynomials defined by a 
slightly different GENERATING FUNCTION. Used to de- 
velop four- dimensional SPHERICAL HARMONICS in an- 
gular momentum theory. They are also a special case 
of the Ultraspherical Polynomial with a = 1. The 
Chebyshev polynomials of the second kind U n (x) are 
illustrated above for x 6 [0, 1] and n— 1, 2, ..., 5. 

The defining GENERATING FUNCTION of the Chebyshev 
polynomials of the second kind is 



g<2(t,x) = 



1 



1 - 2xt + t 2 



Y,Un(x)t n (1) 



for \x\ < 1 and \t\ < 1. To see the relationship to 

a Chebyshev Polynomial of the First Kind (T), 
take dg/Ot, 

^ = -(1 - 2xt + t 2 )~\~2x + 2t) 
- 2(t - x){l - 2xt -\- 1 2 )~ 2 

oo 

= \ nC/n(x)£ n-1 . 

n— 

Multiply (2) by t, 

oo 

{2t 2 -2xt){l-2xt-rt 2 )~ 2 = ^nU n {x)t n 

n=0 

and take (3) -(2), 

{2t 2 - 2tx) - (1 - 2xt + t 2 ) _ t 2 - 1 



(2) 



(3) 



(l-2xt + t 2 ) 2 



{l-2xt + t) 2 

oo 

= 5> -!)£/„(*)*"• (4) 



The Rodrigues representation is 



Un{x) = 



(-i)> + iysF 



2\n+l/2i 



[(1 _ X *)W} 



2»+ 1 (n+ |)!(1 -x 2 y/*dx n 
The polynomials can also be written 

u n {x)= X)(-ir( n /)(2xr- a - 

rv 2 i / x 

^ \2m + l/ v } 



(5) 



(6) 



where [a; J is the Floor Function and \x] is the Ceil- 
ing Function, or in terms of a Determinant 



U n 



2x 1 

2x 1 

1 2x 1 











1 2x 



(7) 



Chebyshev Quadrature 



Chebyshev Quadrature 235 



The first few POLYNOMIALS are 



U (x) 


= 1 


Ui(x) 


= 2x 


U 2 {x) 


= 4x 2 - 1 


U 3 (x) 


= 8x 3 - 4x 


Ut{x) 


= 16z 4 - 12z 2 + 1 


U 5 (x) 


= 32a; 5 - 32a; 3 + 6a; 


U 6 (x) 


= 64a; 6 - 80a; 4 + 24a; 2 - 1 



Letting x = cos 6 allows the Chebyshev polynomials of 
the second kind to be written as 



U n (x) = 



sin[(ra+l)fl] 
sin# 



(8) 



The second linearly dependent solution to the trans- 
formed differential equation is then given by 



W n (x) 



cos[(n+l)fl] 
sin# 



which can also be written 

W n (x) = {l-x 2 )- 1/2 T n + 1 (x), 



(9) 



(10) 



where T n is a CHEBYSHEV POLYNOMIAL OF THE FIRST 

Kind. Note that W n (x) is therefore not a Polynomial. 
see also Chebyshev Approximation Formula, 
Chebyshev Polynomial of the First Kind, Ultra- 
spherical Polynomial 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal 
Polynomials." Ch. 22 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 771-802, 1972. 

Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and 
"Chebyshev Polynomials — Numerical Applications." §13.3 
and 13.4 in Mathematical Methods for Physicists, 3rd ed. 
Orlando, FL: Academic Press, pp. 731-748, 1985. 

Rivlin, T. J. Chebyshev Polynomials, New York: Wiley, 
1990. 

Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi- 
als T n (x) and U n [x). n Ch. 22 in An Atlas of Functions. 
Washington, DC: Hemisphere, pp. 193-207, 1987. 

Chebyshev Quadrature 

A Gaussian QuADRATURE-like Formula for numeri- 
cal estimation of integrals. It uses Weighting Func- 
tion W(x) = 1 in the interval [-1, 1] and forces all the 
weights to be equal. The general FORMULA is 



/; 



f(x)dx = - \ }{xi). 

n *■ — ^ 



The ABSCISSAS are found by taking terms up to y n in 
the MACLAURIN SERIES of 



Sn(y) = exp < 



| in -2 + ln(l-y)(l-i) 
+ ln(l + y) 



H)]} 



and then defining 



G n (x) = x n s n (-) 



The ROOTS o£G n (x) then give the ABSCISSAS. The first 
few values are 

G (x) = 1 

G\{x) = x 

G 2 (x) = l(3x 2 ~l) 

G s {x) = l(2x 3 -x) 

G*(x) = ^(45z 4 -30:£ 2 + i) 

G s (x) = ^(72a; 5 - 60x 3 + 7x) 

G G {x) = ^(105x 6 - 105x 4 + 21z 2 - 1) 



Gr{x) 
G 8 (x) 

G 9 (x) = 



j^ (6480a; 7 - 7560a; 5 + 2142a; 3 - 149a;) 



56700x 6 + 20790a; 4 



6480 

42k (42525a; 8 
- 2220a; 2 - 43) 
22^ (22400a; 9 - 33600x 7 + 15120a; 5 



2280a; 3 + 53a;). 



Because the ROOTS are all REAL for n < 7 and n = 9 
only (Hildebrand 1956), these are the only permissible 
orders for Chebyshev quadrature. The error term is 



_ I c n (n+1)! n 
n ~) c f {n+2) U) _ 

I ° Tl (n+2)! U 



odd 

even. 



where 



{J_ xG n (x)dx n odd 
I-i x 2 Gn{x)dx n even. 

The first few values of c n are 2/3, 8/45, 1/15, 32/945, 
13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) 
gives abscissas up to n = 7 and Hildebrand (1956) up 
to n = 9. 



236 Chebyshev-Radau Quadrature 



Chebyshev's Theorem 



cally for small n. 



n 


x» 


2 


±0.57735 


3 







±0.707107 


4 


±0.187592 




±0.794654 


5 







±0.374541 




±0.832497 


6 


±0.266635 




±0.422519 




±0.866247 


7 







±0.323912 




±0.529657 




±0.883862 


9 







±0.167906 




±0.528762 




±0.601019 




±0.911589 


d w 


eights can be 


n 


Xi 


2 


±|V3 


3 




±|V2 


4 
5 


i ■ 1 y/h-2 

± V sVs 





±\^-^F 




.1 /s+x/TT 

=C 2 V 3 



see a/so Chebyshev Quadrature, Lobatto Quad- 
rature 

References 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 

Boca Raton, FL: CRC Press, p. 466, 1987. 
Hildebrand, F. B. Introduction to Numerical Analysis. New 

York: McGraw-Hill, pp. 345-351, 1956. 

Chebyshev-Radau Quadrature 

A Gaussian QuADRATURE-like Formula over the in- 
terval [-1, 1] which has Weighting Function W(x) = 
x. The general FORMULA is 

/l " 

xf(x)dx = ^Wilfixt) - f(-Xi)]. 
1 i=i 



n 


Xi 


Wi 


1 


0.7745967 


0.4303315 


2 


0.5002990 


0.2393715 




0.8922365 


0.2393715 


3 


0.4429861 


0.1599145 




0.7121545 


0.1599145 




0.9293066 


0.1599145 


4 


0.3549416 


0.1223363 




0.6433097 


0.1223363 




0.7783202 


0.1223363 




0.9481574 


0.1223363 


References 







Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, p. 466, 1987. 

Chebyshev Sum Inequality 

If 

Cb\ > 0,2 > . • • > 0,-n 



h >b 2 >...>6n, 



then 



n z2 akbk - ( Z-s ak } [ z2^ k J ' 
k^i \ fc=i / \ k=i / 

This is true for any distribution. 

see also CAUCHY INEQUALITY, HOLDER SUM INEQUAL- 
ITY 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1092, 1979. 

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 43-44, 1988. 

Chebyshev-Sylvester Constant 

In 1891, Chebyshev and Sylvester showed that for suf- 
ficiently large x, there exists at least one prime number 

p satisfying 

x < p < (1 + a)x, 

where a = 0.092.... Since the PRIME NUMBER THE- 
OREM shows the above inequality is true for all a > 
for sufficiently large x t this constant is only of historical 

interest. 

References 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 
p. 22, 1983. 

Chebyshev's Theorem 

see Bertrand's Postulate 



Checker-Jumping Problem 



Chern Number 237 



Checker-Jumping Problem 

Seeks the minimum number of checkers placed on a 
board required to allow pieces to move by a sequence of 
horizontal or vertical jumps (removing the piece jumped 
over) n rows beyond the forward-most initial checker. 
The first few cases are 2, 4, 8, 20. It is, however, impos- 
sible to reach level 5. 

References 

Honsberger, R. Mathematical Gems II. Washington, DC: 
Math. Assoc. Amer., pp. 23-28, 1976. 

Checkerboard 

see Chessboard 

Checkers 

Beeler et al. (1972, Item 93) estimated that there are 
about 10 12 possible positions. However, this disagrees 
with the estimate of Jon Schaeffer of 5 x 10 20 plausible 
positions, with 10 18 reachable under the rules of the 
game. Because "solving" checkers may require only the 
Square Root of the number of positions in the search 
space (i.e., 10 9 ), so there is hope that some day checkers 
may be solved (i.e., it may be possible to guarantee a 
win for the first player to move before the game is even 
started; Dubuque 1996). 

Depending on how they are counted, the number of Eu- 
LERIAN CIRCUITS on an n x n checkerboard are either 
1, 40, 793, 12800, 193721, ... (Sloane's A006240) or 1, 
13, 108, 793, 5611, 39312, . . . (Sloane's A006239). 
see also Checkerboard, Checker-Jumping Prob- 
lem 

References 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM-239, Feb. 1972. 

Dubuque, W. "Re: number of legal chess positions." math- 
fun@cs.arizona.edu posting, Aug 15, 1996. 

Kraitchik, M. "Chess and Checkers" and "Checkers 
(Draughts)." §12.1.1 and 12.1.10 in Mathematical Recre- 
ations. New York: W. W. Norton, pp. 267-276 and 284- 
287, 1942. 

Schaeffer, J. One Jump Ahead: Challenging Human 
Supremacy in Checkers. New York: Springer- Verlag, 1997. 

Sloane, N. J. A. Sequences A006239/M4909 and A006240/ 
M5271 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 

Checksum 

A sum of the digits in a given transmission modulo some 
number. The simplest form of checksum is a parity bit 
appended on to 7-bit numbers (e.g., ASCII characters) 
such that the total number of Is is always EVEN ("even 
parity") or Odd ("odd parity"). A significantly more 
sophisticated checksum is the CYCLIC REDUNDANCY 
Check (or CRC), which is based on the algebra of poly- 
nomials over the integers (mod 2). It is substantially 
more reliable in detecting transmission errors, and is 
one common error- checking protocol used in modems. 



see also Cyclic Redundancy Check, Error- 
Correcting Code 

References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Cyclic Redundancy and Other Checksums." 
Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of 
Scientific Computing, 2nd ed. Cambridge, England: Cam- 
bridge University Press, pp. 888-895, 1992. 

Cheeger's Finiteness Theorem 

Consider the set of compact n-RlEMANNlAN MANIFOLDS 
M with diameter(M) < d, Volume(M) > V, and \K\ < 
k where k is the Sectional Curvature. Then there 
is a bound on the number of DlFFEOMORPHlSMS classes 
of this set in terms of the constants n, d, V, and k. 

References 

Chavel, I. Riemannian Geometry: A Modern Introduction. 
New York: Cambridge University Press, 1994. 

Chefalo Knot 

A fake KNOT created by tying a SQUARE Knot, then 
looping one end twice through the KNOT such that when 
both ends are pulled, the KNOT vanishes. 

Chen's Theorem 

Every "large" EVEN INTEGER may be written as 2n = 
p -J- m where p is a Prime and m 6 P2 is the Set of 
Semiprimes (i.e., 2-Almost Primes). 

see also ALMOST PRIME, PRIME NUMBER, SEMIPRIME 

References 

Rivera, C "Problems & Puzzles (Conjectures): Chen's 

Conjecture." http://www.sci.net .mx/-crivera/ppp/ 

conj_002.htm. 

Chern Class 

A Gadget defined for Complex Vector Bundles. 
The Chern classes of a Complex Manifold are the 
Chern classes of its Tangent Bundle. The ith Chern 
class is an OBSTRUCTION to the existence of (n — i + 
1) everywhere COMPLEX linearly independent VECTOR 
Fields on that Vector Bundle. The zth Chern class 
is in the (2z)th cohomology group of the base SPACE. 
see also OBSTRUCTION, PONTRYAGIN CLASS, STIEFEL- 

Whitney Class 

Chern Number 

The Chern number is defined in terms of the Chern 
Class of a Manifold as follows. For any collection 
Chern Classes such that their cup product has the 
same Dimension as the Manifold, this cup product 
can be evaluated on the Manifold's Fundamental 
CLASS. The resulting number is called the Chern num- 
ber for that combination of Chern classes. The most 
important aspect of Chern numbers is that they are 
COBORDISM invariant. 

see also Pontryagin Number, Stiefel-Whitney 
Number 



238 



Chemoff Face 



Chess 



Chernoff Face 

A way to display n variables on a 2-D surface. For in- 
stance, let x be eyebrow slant, y be eye size, z be nose 
length, etc. 

References 

Gonick, L. and Smith, W. The Cartoon Guide to Statistics. 
New York: Harper Perennial, p. 212, 1993. 

Chess 

Chess is a game played on an 8x8 board, called a CHESS- 
BOARD, of alternating black and white squares. Pieces 
with different types of allowed moves are placed on the 
board, a set of black pieces in the first two rows and 
a set of white pieces in the last two rows. The pieces 
are called the bishop (2), king (1), knight (2), pawn (8), 
queen (1), and rook (2). The object of the game is to 
capture the opponent's king. It is believed that chess 
was played in India as early as the sixth century AD. 

In a game of 40 moves, the number of possible board 
positions is at least 10 120 according to Peterson (1996). 
However, this value does not agree with the 10 pos- 
sible positions given by Beeler et al. (1972, Item 95). 
This value was obtained by estimating the number of 
pawn positions (in the no-captures situation, this is 15 ), 
times all pieces in all positions, dividing by 2 for each 
of the (rook, knight) which are interchangeable, divid- 
ing by 2 for each pair of bishops (since half the posi- 
tions will have the bishops on the same color squares). 
There are more positions with one or two captures, since 
the pawns can then switch columns (Schroeppel 1996). 
Shannon (1950) gave the value 



P(40) : 



64! 



32!(8!) 2 (2!) 6 



10 4 



The number of chess games which end in exactly n plies 
(including games that mate in fewer than n plies) for 
n = 1, 2, 3, . . . are 20, 400, 8902, 197742, 4897256, 
119060679, 3195913043, ... (K. Thompson, Sloane's 
A007545). Rex Stout's fictional detective Nero Wolfe 
quotes the number of possible games after ten moves as 
follows: "Wolfe grunted. One hundred and sixty-nine 
million, five hundred and eighteen thousand, eight hun- 
dred and twenty-nine followed by twenty-one ciphers. 
The number of ways the first ten moves, both sides, 
may be played" (Stout 1983). The number of chess 
positions after n moves for n — 1, 2, . , . are 20, 400, 
5362, 71852, 809896?, 9132484?, . . . (Schwarzkopf 1994, 
Sloane's A019319). 

Cunningham (1889) incorrectly found 197,299 games 
and 71,782 positions after the fourth move. C. Flye 
St. Marie was the first to find the correct number of po- 
sitions after four moves: 71,852. Dawson (1946) gives 
the source as Intermediare des Mathematiques (1895), 
but K. Fabel writes that Flye St. Marie corrected the 
number 71,870 (which he found in 1895) to 71,852 in 



1903. The history of the determination of the chess se- 
quences is discussed in Schwarzkopf (1994). 

Two problems in recreational mathematics ask 

1. How many pieces of a given type can be placed on a 
Chessboard without any two attacking. 

2. What is the smallest number of pieces needed to oc- 
cupy or attack every square. 

The answers are given in the following table (Madachy 
1979). 



Piece 


Max. 


Min. 


bishops 


14 


8 


kings 


16 


9 


knights 


32 


12 


queens 


8 


5 


rooks 


8 


8 



see also BISHOPS PROBLEM, CHECKERBOARD, CHECK- 
ERS, Fairy Chess, Go, Gomory's Theorem, Hard 
Hexagon Entropy Constant, Kings Problem, 
Knight's Tour, Magic Tour, Queens Problem, 
Rooks Problem, Tour 

References 

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- 
ations and Essays, 13th ed. New York: Dover, pp. 124- 
127, 1987. 

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 
Cambridge, MA: MIT Artificial Intelligence Laboratory, 
Memo AIM-239, Feb. 1972. 

Dawson, T. R. "A Surprise Correction." The Fairy Chess 
Review 6, 44, 1946. 

Dickins, A. "A Guide to Fairy Chess." p. 28, 1967/1969/ 
1971. 

Dudeney, H. E. "Chessboard Problems," Amusements in 
Mathematics. New York: Dover, pp. 84-109, 1970. 

Fabel, K. "Nusse." Die Schwalbe 84, 196, 1934. 

Fabel, K. "Weihnachtsniisse." Die Schwalbe 190, 97, 1947. 

Fabel, K. "Weihnachtsniisse." Die Schwalbe 195, 14, 1948. 

Fabel, K. "Eroffnungen." Am Rande des Schachbretts, 34— 
35, 1947. 

Fabel, K. "Die ersten Schritte." Rund um das Schachbrett, 
107-109, 1955. 

Fabel, K. "Eroffnungen." Schach und Zahl 8, 1966/1971. 

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- 
sions. New York: Dover, pp. 86-89, 1975. 

Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathemati- 
cal Recreations. New York: W. W. Norton, pp. 267-276, 
1942. 

Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in 
Madachy 's Mathematical Recreations. New York: Dover, 
pp. 34-54, 1979. 

Peterson, I. "The Soul of a Chess Machine: Lessons Learned 
from a Contest Pitting Man Against Computer." Sci. 
News 149, 200-201, Mar. 30, 1996. 

Petkovic, M. Mathematics and Chess. New York: Dover, 
1997. 

Schroeppel, R. "Reprise: Number of legal chess positions." 
tech-news@cs.arizona.edu posting, Aug. 18, 1996. 

Schwarzkopf, B. "Die ersten Ziige." Problemkiste, 142—143, 
No. 92, Apr. 1994. 

Shannon, C. "Programming a Computer for Playing Chess." 
Phil. Mag. 41, 256-275, 1950. 

Sloane, N. J. A. Sequences A019319 and A007545/M5100 in 
"An On-Line Version of the Encyclopedia of Integer Se- 
quences." 



Chessboard 



Chi Distribution 239 



Stout, R. "Gambit." In Seven Complete Nero Wolfe Novels. 
New York: Avenic Books, p. 475, 1983. 

Chessboard 




A board containing 8x8 squares alternating in color 
between black and white on which the game of Chess is 
played. The checkerboard is identical to the chessboard 
except that chess's black and white squares are colored 
red and white in CHECKERS. It is impossible to cover a 
chessboard from which two opposite corners have been 
removed with DOMINOES. 

see also Checkers, Chess, Domino, Gomory's The- 
orem, Wheat and Chessboard Problem 

References 

Pappas, T. "The Checkerboard." The Joy of Mathematics. 
San Carlos, CA: Wide World Publ./Tetra, pp. 136 and 232, 
1989. 

Chevalley Groups 

Finite Simple Groups of Lie-Type. They include 
four families of linear SIMPLE GROUPS: PSL(n,q), 
PSU(n,q), PSp(2n,q), or PQ € (n,q). 

see also Twisted Chevalley Groups 

References 

Wilson, R. A. "ATLAS of Finite Group Representation." 
http : //f or . mat . bham . ac . uk/atlas#chev. 

Chevalley's Theorem 

Let f{x) be a member of a Finite Field 
F[xx, #2, . . • jX n ] and suppose /(0,0,...,0) = and n 
is greater than the degree of /, then / has at least two 
zeros in A n {F). 

References 

Chevalley, C "Demonstration d'une hypothese de M. Artin." 
Abhand. Math. Sem. Hamburg 11, 73-75, 1936. 

Ireland, K. and Rosen, M. "Chevalley's Theorem." §10.2 in 
A Classical Introduction to Modern Number Theory, 2nd 
ed. New York: Springer- Verlag, pp. 143-144, 1990. 




Chevron 



A 6-Polyiamond. 
References 

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, 
and Packings, 2nd ed. Princeton, NJ: Princeton University 
Press, p. 92, 1994. 

Chi 




^, ./ s , f Z cosht — 1 . ■ 

Chi(jz) = 7 + In z + / dt } 

Jo ^ 

where 7 is the Euler-Mascheroni Constant. The 
function is given by the Mathematica® (Wolfram Re- 
search, Champaign, IL) command CoshlntegralEz] . 

see also Cosine Integral, Shi, Sine Integral 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co- 
sine Integrals." §5.2 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 231-233, 1972. 

Chi Distribution 

The probability density function and cumulative distri- 
bution function are 



Pn{x) 



2 l-n/2 x n~l e -x 2 /2 



D n {x) = Q{\n,\x 2 ), 

where Q is the Regularized Gamma Function. 

v^r(i(n + i)) 
M= r(in) 

2 ^ 2[r(in)r(l + ln)-r 2 (f(n+l))] 

= 2T*{\{n + 1)) - 3r(|n)r(§(n + l))r(l + \n) 
71 [r(In)r(l + in)-r»(i(n + l))] 8 / a 



(1) 

(2) 

(3) 
(4) 



240 



72 = 



Chi Inequality 



[r(in)r(l + in)-H(I(„ + l))]3/2 

-3r*(i(n + 1)) + er(§n) + r 2 (|(n + i))r(i + i n ) 



(5) 



[r(|n)r(^)-r»(i(n + i))]» 
-AT*C-n)T(\(n + i))r(*±=) + r»(±n)r(±±*) 



[r(^)r(^)-r 2 (i(n + i))] 2 



(6) 



where m is the MEAN, <r 2 the VARIANCE, 71 the Skew- 
ness, and 72 the Kurtosis. For n = 1, the x distribu- 
tion is a Half-Normal Distribution with = 1. For 
n = 2, it is a Rayleigh Distribution with a = 1. 

see a/50 Chi-Squared Distribution, Half-Normal 
Distribution, Rayleigh Distribution 

Chi Inequality 

The inequality 

(j + l)aj -V ca> (j + l)i, 



which is satisfied by all ^-SEQUENCES. 

References 

Levine, E. and O'Sullivan, J. "An Upper Estimate for the 
Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34, 
9-24, 1977. 



Chi-Squared Distribution 

A x 2 distribution is a Gamma Distribution with = 2 
and a = r/2, where r is the number of DEGREES OF 
Freedom. If Y» have Normal Independent distribu- 
tions with MEAN and VARIANCE 1, then 



-£* 2 



(i) 



is distributed as x* witn n DEGREES OF FREEDOM. If 
Xi 2 are independently distributed according to a x 2 dis- 
tribution with m, 712, . . . , n*. DEGREES OF FREEDOM, 
then 






Xj 



(2) 



is distributed according to x with n = X] n =i n J DE- 
GREES of Freedom. 



P n (x) = \ r(|r)2-/2 - (3) 

for x < 0. 



The cumulative distribution function is then 



_ , a , f x t^e-^dt 



Chi-Squared Distribution 

where P(a, z) is a REGULARIZED GAMMA FUNCTION. 
The Confidence Intervals can be found by finding 
the value of x for which D n (x) equals a given value. 

The Moment-Generating Function of the x 2 distri- 
bution is 



M(t)-- 


= (1 


- 2t)~ T/2 






(5) 


R(t) = 


Eblj 


M(t) = - 


§rln(l- 


-2t) 


(6) 


R'(t) -- 


1- 


r 

-2t 






(7) 


R"(t) = 




2r 






(8) 


(1 


-2ty 




M 


= R'(0) = 


= r 




(9) 




2 


= R"(0)-- 


= 2r 




(10) 




71 


12 






(11) 




72 








(12) 



The nth Moment about zero for a distribution with n 
Degrees of Freedom is 

m' n = 2- r( ' 1 1 ^ = r(r + 2) ■ ■ ■ (r + 2n - 2), (13) 

and the moments about the MEAN are 

fJL2 = 2r (14) 

A*3 = 8r (15) 

p 4 = 12n 2 + 48n. (16) 

The nth CUMULANT is 



« n = 2 n r(n)(|r) = 2 n - x (n - l)!r, (17) 



The Moment-Generating Function is 

-r/2 






9 *\/2A 



-r/2 



As r* — ► 00, 
so for large r, 



lim M(t) = e* 2/2 , 



r/2 



^i/E 



(x< - /J,) 2 



<Ti' 



(18) 
(19) 

(20) 



Chi-Squared Distribution 



Chi-Squared Test 241 



is approximately a Gaussian Distribution with 
MEAN y/2r and VARIANCE <t 2 = 1. Fisher showed that 



X 2 ~r 
V27--1 



(21) 



is an improved estimate for moderate r. Wilson and 
Hilferty showed that 



1/3 



(22) 



is a nearly GAUSSIAN DISTRIBUTION with MEAN \i = 
1 - 2/(9r) and VARIANCE a 2 = 2/(9r). 

In a Gaussian Distribution, 

P(x) dx = ~^=e~ (x ~ » )2/2(r2 dx, (23) 



let 



Then 



so 



But 



z = (x — fi) I a . 



dx = — -=dz. 

2v^ 



P(z)dz = 2P(x)dx, 



r(f)2V2 



\/27r 



(24) 



dz ^2(x-^ dx= 2^z dx 

(T z (7 



(26) 
(27) 



P(x) dx = 2 —^—e-^ 2 dz = -^=e~ z/2 dz. (28) 

This is a \ 2 distribution with r = 1, since 

1/2-1 -z/2 1/2-1/2 
P(z) ^ = e d* = -L dz. (29) 



oFi is the Confluent Hypergeometric Limit Func- 
tion and T is the GAMMA FUNCTION. The Mean, 
Variance, Skewness, and Kurtosis are 



\i = A + n 

2 



7i 



72 



2(2A + n) 

2y / 2(3A + n) 
(2A + n)3/2 

12(4A + n) 
(2A + n) 2 * 



(34) 
(35) 

(36) 
(37) 



see also Chi Distribution, Snedecor's F-Distribu- 
tion 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
pp. 940-943, 1972. 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, p. 535, 1987. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Incomplete Gamma Function, Error Func- 
tion, Chi-Square Probability Function, Cumulative Poisson 
Function." §6.2 in Numerical Recipes in FORTRAN: The 
Art of Scientific Computing, 2nd ed. Cambridge, England: 
Cambridge University Press, pp. 209-214, 1992. 

Spiegel, M . R. Theory and Problems of Probability and 
Statistics. New York: McGraw-Hill, pp. 115-116, 1992. 

Chi-Squared Test 

Let the probabilities of various classes in a distribution 
be pi , p2 , . . . , Pk • The expected frequency 



£ 



(mi - Npj) 2 

N Pi 



is a measure of the deviation of a sample from expecta- 
tion. Karl Pearson proved that the limiting distribution 
of \s 2 is x 2 (Kenney and Keeping 1951, pp. 114-116). 



If Xi are independent variates with a NORMAL DISTRI- 
BUTION having MEANS \i{ and VARIANCES a 2 for i = 1, 
. . . , n, then 



i 2 _ v^ (Xi -in) 



= £ 



2 A ~ £^ 2<7i 2 

1=1 



(30) 



is a Gamma Distribution variate with a = n/2, 

r( ? n) 

(31) 

The noncentral chi-squared distribution is given by 

P(x) = 2-" /2 e - (A+l)/2 x n/2 - 1 F(in, f Ax), (32) 

where 



F(a,z) = 



oFi(;a;z) 
T(a) ' 



(33) 



Pr(* 2 >X* 2 )= f^ f(x 2 )d( X 2 ) 

Jxs 2 

2\M)/2 



= 1 

~ 2 
= 1 



f 



(*) 



,V/3 



r(ft=i) 



d(x 2 ) 



= 1-1 



Xs 



k-3 



V^^T)' 2 



where I(x i n) is PEARSON'S FUNCTION. There are some 
subtleties involved in using the x 2 test to fit curves (Ken- 
ney and Keeping 1951, pp. 118-119). 

When fitting a one-parameter solution using x 2 > the 
best-fit parameter value can be found by calculating % 2 



242 



Child 



Choose 



at three points, plotting against the parameter values of 
these points, then finding the minimum of a PARABOLA 
fit through the points (Cuzzi 1972, pp. 162-168). 

References 

Cuzzi, J. The Subsurface Nature of Mercury and Mars from 
Thermal Microwave Emission. Ph.D. Thesis. Pasadena, 
CA: California Institute of Technology, 1972. 

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, 
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. 

Child 

A node which is one EDGE further away from a given 
Edge in a Rooted Tree. 

see also Root (Tree), Rooted Tree, Sibling 

Chinese Hypothesis 

A Prime p always satisfies the condition that 2 P — 2 
is divisible by p. However, this condition is not true 
exclusively for PRIME (e.g., 2 341 — 2 is divisible by 341 = 
11*31). Composite Numbers n (such as 341) for which 
2 n - 2 is divisible by n are called Poulet Numbers, 
and are a special class of Fermat Pseudoprimes. The 
Chinese hypothesis is a special case of FERMAT's LITTLE 
Theorem. 

see also Carmichael Number, Euler's Theorem, 
Fermat's Little Theorem, Fermat Pseudoprime, 
Poulet Number, Pseudoprime 

References 

Shanks, D. Solved and Unsolved Problems in Number Theory, 
4th ed. New York: Chelsea, pp. 19-20, 1993. 

Chinese Remainder Theorem 

Let r and s be Positive Integers which are Rela- 
tively Prime and let a and b be any two Integers. 
Then there is an INTEGER N such that 



and the 6, are determined from 



M 



and 



N = a (mod r) 



N = b (mod 5) . 



(i) 



(2) 



Moreover, iV is uniquely determined modulo rs. An 
equivalent statement is that if (r,s) = 1, then every 
pair of Residue Classes modulo r and s corresponds 
to a simple RESIDUE CLASS modulo rs. 

The theorem can also be generalized as follows. Given 
a set of simultaneous CONGRUENCES 

x = a,i (mod rrii) (3) 

for i — 1, . . . , r and for which the rrti are pairwise Rela- 
tively Prime, the solution of the set of Congruences 

is 

x = aibi (- . . . -h a r b r (mod M), (4) 

mi m r 



bi — = 1 (mod rrii). 

TTli 



(6) 



where 



M = m\m2 - - *rn r 



(5) 



References 

Ireland, K. and Rosen, M. "The Chinese Remainder Theo- 
rem." §3.4 in A Classical Introduction to Modern Number 
Theory, 2nd ed. New York: Springer- Verlag, pp. 34-38, 
1990. 

Uspensky, J. V. and Heaslet, M. A. Elementary Number The- 
ory. New York: McGraw-Hill, pp. 189-191, 1939. 

Wagon, S. "The Chinese Remainder Theorem." §8.4 in Math- 
ematica in Action. New York: W. H. Freeman, pp. 260- 
263, 1991. 

Chinese Rings 

see Baguenaudier 

Chiral 

Having forms of different HANDEDNESS which are not 

mirror-symmetric. 

see also Disymmetric, Enantiomer, Handedness, 

Mirror Image, Reflexible 

Choice Axiom 

see Axiom of Choice 

Choice Number 
see Combination 

Cholesky Decomposition 

Given a symmetric POSITIVE DEFINITE MATRIX A, the 
Cholesky decomposition is an upper TRIANGULAR MA- 
TRIX U such that 

A-U T U. 

see also LU Decomposition, QR Decomposition 

References 

Nash, J. C. "The Choleski Decomposition." Ch. 7 in Com- 
pact Numerical Methods for Computers: Linear Algebra 
and Function Minimisation, 2nd ed. Bristol, England: 
Adam Hilger, pp. 84-93, 1990. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Cholesky Decomposition." §2.9 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 89-91, 1992. 

Choose 

An alternative term for a BINOMIAL COEFFICIENT, in 
which C?} is read as "n choose k" R. K. Guy suggested 
this pronunciation around 1950, when the notations n C r 
and n C r were commonly used. Leo Moser liked the pro- 
nunciation and he and others spread it around. It got 
the final seal of approval from Donald Knuth when he 
incorporated it into the TeX mathematical typesetting 
language as {n\choose k}. 



Choquet Theory 



Chow Coordinates 



243 



Choquet Theory 

Erdos proved that there exist at least one Prime of the 
form Ak + 1 and at least one Prime of the form 4k -f 3 
between n and 2n for all n > 6. 

see also Equinumerous, Prime Number 



Chord 



chord^ 




The Line Segment joining two points on a curve. The 
term is often used to describe a LINE Segment whose 
ends lie on a CIRCLE. In the above figure, r is the RA- 
DIUS of the CIRCLE, a is called the Apothem, and s the 
Sagitta. 

s s_ 




The shaded region in the left figure is called a Sector, 
and the shaded region in the right figure is called a SEG- 
MENT. 

All ANGLES inscribed in a Circle and subtended by 
the same chord are equal. The converse is also true: 
The LOCUS of all points from which a given segment 
subtends equal ANGLES is a CIRCLE. 




Let a Circle of Radius R have a Chord at distance r. 
The Area enclosed by the Chord, shown as the shaded 
region in the above figure, is then 



f , v / J? 2„ 7 .2 

A = 2 / x(y) dy. 



Jo 



But 



y 2 + (r + x) 2 = R 2 , 



x(y) = \/R 2 - y 2 - r 



(1) 

(2) 
(3) 



and 



A = 2 



/ (y/R 2 -y 2 

Jo 



r)dy 



y^R 2 -y 2 +R 2 tan" 1 



■i^ 



2ry 



■.ry/B? ™r 2 + J^ 2 tan" 1 



sfR? 



:i)'- 



= i^tan" 1 



(f) : 



- r^R 2 - \ 



2r^R? - r 2 

(4) 



Checking the limits, when r = R, A = and when 

r->0, 

A=\kR\ (5) 

see also Annulus, Apothem, Bertrand's Problem, 
Concentric Circles, Radius, Sagitta, Sector, 
Segment 

Chordal 

see Radical Axis 

Chordal Theorem 





The LOCUS of the point at which two given CIRCLES 
possess the same POWER is a straight line PERPENDIC- 
ULAR to the line joining the MIDPOINTS of the CIRCLE 
and is known as the chordal (or RADICAL Axis) of the 
two Circles. 

References 

Dorrie, H. 100 Great Problems of Elementary Mathematics: 
Their History and Solutions. New York: Dover, p. 153, 
1965. 

Chow Coordinates 

A generalization of GRASSMANN COORDINATES to m-D 
varieties of degree d in P n , where P n is an n-D pro- 
jective space. To define the Chow coordinates, take 
the intersection of a m-D VARIETY Z of degree d by 
an (n - m)-D SUBSPACE U of P n . Then the coordi- 
nates of the d points of intersection are algebraic func- 
tions of the Grassmann Coordinates of U, and by 
taking a symmetric function of the algebraic functions, 
a hHOMOGENEOUS POLYNOMIAL known as the Chow 
form of Z is obtained. The Chow coordinates are then 



244 Chow Ring 

the Coefficients of the Chow form. Chow coordinates 
can generate the smallest field of definition of a divisor. 

References 

Chow, W.-L. and van der Waerden., B. L. "Zur algebraische 

Geometrie IX." Math. Ann. 113, 692-704, 1937. 
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and 

Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 

43, 1117-1124, 1996. 

Chow Ring 

The intersection product for classes of rational equiva- 
lence between cycles on an Algebraic Variety. 

References 

Chow, W.-L. "On Equivalence Classes of Cycles in an Alge- 
braic Variety." Ann. Math. 64, 450-479, 1956. 

Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and 
Igusa, J.-L "Wei-Liang Chow." Not. Amer. Math. Soc. 
43, 1117-1124, 1996. 

Chow Variety 

The set C n ,m,d of all rn-D varieties of degree d in an n-D 
projective space P n into an M-D projective space P M . 

References 

Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and 
Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 
43, 1117-1124, 1996. 

Christoffel-Darboux Formula 

For three consecutive ORTHOGONAL POLYNOMIALS 

Pn(x) = (A n X + B n )p n -lX ~ C n p n -2(x) (l) 

for n = 2, 3, . . . , where A n > 0, B n , and C n > are 
constants. Denoting the highest Coefficient of p n (x) 
by fc n , 



A n = 



kn-l 
•A-n rCn^n — 2 

A n -i kn-i 2 



(2) 
(3) 



Then 



Po(x)po{y) 4- . . -+p n (x)p n {y) 

= k n Pn + l(x)p n (y) - Pn(x)p n + l(y) 

kn+x x-y 

In the special case of x = y, (4) gives 



(4) 



\P0(X)} 2 + . . . + \p n (x)] 

k 



kn+l 



\Pn+l{x)Pn{x) ~ P n ( X )Pn+l(x)}. (5) 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

p. 785, 1972. 
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI: 

Amer. Math. Soc, pp. 42-44, 1975. 



Christoffel Number 
Christoffel-Darboux Identity 

<f> k (x)(f) k (y) _ <p m +l(x)(f>m(y) - (t>m{x)<f>m+l{y) 



a m ^ m {x -y), 



k=0 "" ' x 

(1) 

where 4>k{x) are ORTHOGONAL POLYNOMIALS with 
Weighting Function W(x) y 

7m= J[cj>r n {x)fW{x)dx, (2) 



and 



Q>k — 



Z±±i 
A k 



(3) 



where A k is the COEFFICIENT of x k in <f>k(x). 

References 

Hildebrand, F. B. Introduction to Numerical Analysis. New 
. York: McGraw-Hill, p. 322, 1956. 

Christoffel Formula 

Let {p n {x)} be orthogonal Polynomials associated 
with the distribution da(x) on the interval [a, 6]. Also 
let 

p = c(x — Xi)(x - X2) ' ' • (x — Xi) 

(for c ^ 0) be a Polynomial of order I which is 
NONNEGATIVE in this interval. Then the orthogonal 
Polynomials {q(x)} associated with the distribution 
p(x) da(x) can be represented in terms of the POLYNO- 
MIALS p n {x) as 



p{x)q n {x) = 



Pn(x) p n + l(x) 
Pn(xi) Pn + l(xi) 

Pn(Xl) Pn+l{xi) 



Pn+l{x) 
Pn+l(xi) 

Pn+l{xi) 



In the case of a zero x k of multiplicity m > 1, we replace 
the corresponding rows by the derivatives of order 0, 1, 
2, . . . , m - 1 of the POLYNOMIALS p n (xi), . . . , p n +l{xi) 
at x — — x k . 

References 

Szego, G. Orthogonal Polynomials, J^.th ed. Providence, RI: 
Amer. Math. Soc, pp. 29-30, 1975. 

Christoffel Number 

One of the quantities Xi appearing in the GAUSS-JACOBI 

Mechanical Quadrature. They satisfy 



Ai + A 2 + 



. . . + A„ = / 

J a 



da(x) = a{b) - a(a) (1) 



Christoffel Symbol of the First Kind 



Christoffel Symbol of the Second Kind 245 



and are given by 



J a [Pn(x v )(X - X, 



A„ = 



&n 



+ 1 



1 



k n Pn+l(Xv)Pn(Xv) 
k n 1 



da(x) (2) 

(3) 



(4) 
(5) 



k n -\ p n -r{xu)Pk{x u ) 

where A; n is the higher COEFFICIENT of p n (x). 

References 

Szego, G. Orthogonal Polynomials, ^th ed. Providence, RI: 
Amer. Math. Soc, pp. 47-48, 1975. 

Christoffel Symbol of the First Kind 

Variously denoted [ij,k], [\ J ], r obc , or {ab,c}. 



[ij, k] : 



(i) 

where p mfc is the METRIC TENSOR and 



But 



df 



dq k ~ dq* [€i ' 6j) " a 9 * ' ej ei ■ a<? fc 

= [»M + b"M, (3) 

so 

[ab,c]= \{9ac,b+ 9bc,a- 9ab,c)' (4) 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 160-167, 1985. 

Christoffel Symbol of the Second Kind 

Variously denoted { . m . } or rg. 



-*m C76i kmr • • j i 



_ 1 fcm ( <9ffi 

2 5 \ dgJ 



rJ d^ dq k 



(1) 



where rjj is a CONNECTION COEFFICIENT and {6c, d} 
is a Christoffel Symbol of the First Kind. 



\ b a c j =9ad{bc,d}. 



(2) 



The Christoffel symbols are given in terms of the first 
Fundamental Form E, F, and G by 



r 12 



1 _ GE U - 2FF U + FE V 
2(EG-F 2 ) 

GE V — FG U 

2{EG - F 2 ) 

2GF V — GG U — FG V 
2(EG - F 2 ) 

2£F U - EE V - FE U 
2(EG-F 2 ) 

EG U — FE V 



r*22 



r 2 - 

1 11 — 



r 2 - 
1 12 — 



2(£G-F 2 ) 



■p2 SGd — 2FF V + FG U 

1 22 — 



2(£G - F 2 ) 



(3) 
(4) 
(5) 
(6) 
(7) 
(8) 



and T^ = T\ 2 and T^ = r? 3 . If F = 0, the Christoffel 
symbols of the second kind simplify to 



(9) 
(10) 

(11) 
(12) 

(13) 

(14) 



(Gray 1993). 

The following relationships hold between the Christoffel 
symbols of the second kind and coefficients of the first 

Fundamental Form, 



r 1 

1 ii 


= 


E u 
2E 


r 1 

1 12 


= 


E v 
2E 


r 2 2 


= 


G u 
2E 


r 2 

-L 11 


= 


E v 
2G 


r 2 

1 12 


= 


G u 
2G 


r 2 

1 22 


= 


G v 
2G 



T\ 1 E + T\ 1 F=\E U 
T 12 E + T 12 F — ^E v 

^22^ + 1^22^ ~ -Pw — 2^* u 

^nF + T 1X G = F u — -E v 

r"l2-^ + ^12^? = ^G u 
1^22^ + T22G = oG v 



(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 



Fii + T? 2 = (In y/EG - F=> )„ 
ria + Im = (In yjEG - F* ). 

(Gray 1993). 

For a surface given in Monge'S Form 2 = F(x,y), 

r k - = ZijZk C2S^ 



see also Christoffel Symbol of the First Kind, 
Connection Coefficient, Gauss Equations 



246 



Chromatic Number 



ci 



References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 160-167, 1985. 

Gray, A. "Christoffel Symbols." §20.3 in Modern Differential 
Geometry of Curves and Surfaces. Boca Raton, FL: CRC 
Press, pp. 397-400, 1993. 

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- 
ics, Part L New York: McGraw-Hill, pp. 47-48, 1953. 

Chromatic Number 

The fewest number of colors j(G) necessary to color a 
Graph or surface. The chromatic number of a surface 
of GENUS g is given by the HEAWOOD CONJECTURE, 



l(9)= §(7+7485 + 1) 



where [x\ is the Floor Function. j(g) is sometimes 
also denoted x(p)- For g = 0, 1, ... , the first few values 
of x(9) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 
16, ... (Sloane's A000934). 

The fewest number of colors necessary to color each 
Edge of a Graph so that no two Edges incident on the 
same Vertex have the same color is called the "Edge 
chromatic number." 

see also Brelaz's Heuristic Algorithm, Chro- 
matic Polynomial, Edge-Coloring, Euler Char- 
acteristic, Heawood Conjecture, Map Color- 
ing, Torus Coloring 

References 

Chartrand, G. "A Scheduling Problem: An Introduction to 

Chromatic Numbers." §9.2 in Introductory Graph Theory. 

New York: Dover, pp. 202-209, 1985. 
Eppstein, D. "The Chromatic Number of the Plane." 

http:// www . ics . uci . edu / - eppstein / junkyard / 

plane-color/. 
Sloane, N. J. A. Sequence A000934/M3292 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Chromatic Polynomial 

A Polynomial P(z) of a graph g which counts the 
number of ways to color g with exactly z colors. Tutte 
(1970) showed that the chromatic POLYNOMIALS of pla- 
nar triangular graphs possess a ROOT close to <j> 2 = 
2.618033 . . ., where <j> is the GOLDEN Mean. More pre- 
cisely, if n is the number of VERTICES of G, then 

(Le Lionnais 1983). 

References 

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 

p. 46, 1983. 
Tutte, W. T. "On Chromatic Polynomials and the Golden 

Ratio." J. Corabin. Th. 9, 289-296, 1970. 



Chu Space 

A Chu space is a binary relation from a Set A to an 
antiset X which is defined as a Set which transforms 
via converse functions. 

References 

Stanford Concurrency Group. "Guide to Papers on Chu 
Spaces." http : //boole . Stanford . edu/ chuguide .html. 

Chu-Vandermonde Identity 

(x + a) n = Y^ Uj(a)fcO*On-fc 

where (™) is a Binomial Coefficient and (a) n = 
a(a - 1) • • • (a - n + 1) is the Pochhammer Symbol. A 
special case gives the identity 



max(fe,n) 

£ 

( = 



m 
k-l 



i)-\ k )■ 



see also BINOMIAL THEOREM, UMBRAL CALCULUS 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, pp. 130 and 181-182, 1996. 

Church's Theorem 

No decision procedure exists for Arithmetic. 

Church's Thesis 

see Church-Turing Thesis 

Church- Turing Thesis 

The Turing Machine concept defines what is meant 
mathematically by an algorithmic procedure. Stated 
another way, a function / is effectively COMPUTABLE 
Iff it can be computed by a Turing Machine. 

see also ALGORITHM, COMPUTABLE FUNCTION, TUR- 
ING Machine 

References 

Penrose, R. The Emperor's New Mind: Concerning Comput- 
ers, Minds, and the Laws of Physics. Oxford, England: 
Oxford University Press, pp. 47-49, 1989. 

Chvatal's Art Gallery Theorem 

see Art Gallery Theorem 

Chvatal's Theorem 

Let the Graph G have Vertices with Valences di < 
. . . < d m . If for every i < n/2 we have either efc > i + 1 
or d n -i > n - 2, then the Graph is Hamiltonian. 



Chu Identity 

see Chu-Vandermonde Identity 



ci 

see Cosine Integral 



a 



Circle 247 



Ci 

see Cosine Integral 

Cigarettes 

It is possible to place 7 cigarettes in such a way that 
each touches the other if l/d > 7\/3/2 (Gardner 1959, 
p. 115). 

References 

Gardner, M. The Scientific American Book of Mathematical 

Puzzles & Diversions. New York: Simon and Schuster, 

1959. 

Cin 

see Cosine Integral 

Circle 







A circle is the set of points equidistant from a given 
point O. The distance r from the Center is called the 
Radius, and the point O is called the Center. Twice 
the Radius is known as the Diameter d = 2r. The 
Perimeter C of a circle is called the Circumference, 
and is given by 

C = ird = 2tt7\ (1) 

The circle is a Conic SECTION obtained by the intersec- 
tion of a Cone with a Plane Perpendicular to the 
Cone's symmetry axis. A circle is the degenerate case 
of an Ellipse with equal semimajor and semiminor axes 
(i.e., with ECCENTRICITY 0). The interior of a circle is 
called a Disk. The generalization of a circle to 3-D is 
called a SPHERE, and to n-D for n > 4 a HYPERSPHERE. 

The region of intersection of two circles is called a LENS. 
The region of intersection of three symmetrically placed 
circles (as in a VENN DIAGRAM), in the special case of 
the center of each being located at the intersection of 
the other two, is called a Reuleaux Triangle. 

The parametric equations for a circle of RADIUS a are 



x — a cos t 
y = a sin t. 

For a body moving uniformly around the circle, 



X 

t 

y 



-asint 
a cost, 



and 



x = —a cost 
y" = —asint. 



(2) 
(3) 



(4) 
(5) 



(6) 
(7) 



When normalized, the former gives the equation for the 
unit Tangent Vector of the circle, (-sint,cost). The 
circle can also be parameterized by the rational func- 
tions 



x = 



y- 



2t 

1 + t 2 ' 



(8) 
(9) 



but an Elliptic Curve cannot. The following plots 
show a sequence of NORMAL and TANGENT VECTORS 
for the circle. 




The Arc Length s, Curvature k, and Tangential 
ANGLE <j> of the circle are 



s(t) = ds= \/x f2 + y' 2 dt = at (10) 

(j>(t) = I K(t)dt= -. (12) 

The Cesaro Equation is 

K=~. (13) 

a 

In POLAR COORDINATES, the equation of the circle has 
a particularly simple form. 

r = a (14) 

is a circle of RADIUS a centered at Origin, 

r = 2acos9 (15) 

is circle of RADIUS a centered at (a, 0), and 

r = 2asm6 (16) 



248 



Circle 



Circle 



is a circle of RADIUS a centered on (0, a). In CARTE- 
SIAN Coordinates, the equation of a circle of Radius 

a centered on (xo,2/o) is 



(x - x ) 2 + (y-yo) 2 



(17) 



In Pedal Coordinates with the Pedal Point at the 
center, the equation is 

pa = r 2 . (18) 

The circle having P1P2 as a diameter is given by 

(x - xi)(x - x 2 ) + (2/ - yi){y - 2/2) = 0. (19) 

The equation of a circle passing through the three points 
(xi,yi) for i = 1, 2, 3 (the Circumcircle of the Tri- 
angle determined by the points) is 



(20) 



The Center and Radius of this circle can be identified 
by assigning coefficients of a Quadratic Curve 



2 , 2 

x +y 


X 


y 


1 


2 1 2 
xi +2/1 


Xi 


2/1 


1 


2 1 2 

x 2 +t/2 


X 2 


2/2 


1 


2 , 2 

XZ +J/3 


xz 


2/3 


1 



ax 2 + cy 2 + dx + ey + / = 0, 



(21) 



where a — c and 6 = (since there is no xy cross term) . 
Completing the Square gives 



The Center can then be identified as 



Xq 

2/o 



2a 
e 

2a 



and the Radius as 



where 



d 2 + e 2 / 
a 



4a 2 



(23) 
(24) 

(25) 



e = 



xi 2/1 I 








#2 2/2 1 


(26) 


xz 2/3 1 






#i 2 +2/i 2 2/i 1 






Z2 2 +2/2 2 2/2 1 


(27) 




£3 2 +2/3 2 2/3 1 




zi 2 +2/i 2 X! 1 




Z2 2 +2/2 2 £ 2 1 


(28) 


Xz 2 + 2/3 2 #3 1 






#i 2 +2/1 2 asi 2/i 






Z2 2 +2/2 2 Z 2 2/2 


(29) 




#3 2 +2/3^ 


! xz 


2/3 





Four or more points which lie on a circle are said to be 
Concyclic. Three points are trivially concyclic since 
three noncollinear points determine a circle. 

The ClRCUMFERENCE-to-DlAMETER ratio C/d for a cir- 
cle is constant as the size of the circle is changed (as 
it must be since scaling a plane figure by a factor s in- 
creases its Perimeter by s), and d also scales by s. This 
ratio is denoted -k (Pi), and has been proved Transcen- 
dental. With d the Diameter and r the Radius, 



C == 7rd = 27r?\ 



(30) 



Knowing C/d, we can then compute the Area of the 
circle either geometrically or using CALCULUS. From 

Calculus, 



A = 



p1t\ nr 

Jo Jo 



rdr = (27r)(^r ) = irr 



(31) 



Now for a few geometrical derivations. Using concentric 
strips, we have 




As the number of strips increases to infinity, we are left 
with a Triangle on the right, so 



A = \{2nr)r = nr . 



(32) 



This derivation was first recorded by Archimedes in 
Measurement of a Circle (ca. 225 BC). If we cut the 
circle instead into wedges, 

^ *+ nr ► 




As the number of wedges increases to infinity, we are 
left with a RECTANGLE, so 



(-Kr)r = nr . 



(33) 



see also Arc, Blaschke's Theorem, Brahmagupta's 
Formula, Brocard Circle, Casey's Theorem, 
Chord, Circumcircle, Circumference, Clif- 
ford's Circle Theorem, Closed Disk, Concentric 
Circles, Cosine Circle, Cotes Circle Property, 
Diameter, Disk, Droz-Farny Circles, Euler Tri- 
angle Formula, Excircle, Feuerbach's Theorem, 



Circles-and-Squares Fractal 



Circle-Circle Intersection 



249 



Five Disks Problem, Flower of Life, Ford Cir- 
cle, Fuhrmann Circle, Gersgorin Circle Theo- 
rem, Hopf Circle, Incircle, Inversive Distance, 
Johnson Circle, Kinney's Set, Lemoine Circle, 
Lens, Magic Circles, Malfatti Circles, McCay 
Circle, Midcircle, Monge's Theorem, Moser's 
Circle Problem, Neuberg Circles, Nine-Point 
Circle, Open Disk, P-Circle, Parry Circle, Pi, 
Polar Circle, Power (Circle), Prime Circle, 
Ptolemy's Theorem, Purser's Theorem, Radi- 
cal Axis, Radius, Reuleaux Triangle, Seed of 
Life, Seifert Circle, Semicircle, Soddy Circles, 
Sphere, Taylor Circle, Triangle Inscribing in 
a Circle, Triplicate-Ratio Circle, Tucker Cir- 
cles, Unit Circle, Venn Diagram, Villarceau 
Circles, Yin- Yang 

References 

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. 
Boca Raton, FL: CRC Press, pp. 125 and 197, 1987. 

Casey, J. "The Circle." Ch. 3 in A Treatise on the Analyt- 
ical Geometry of the Point, Line, Circle, and Conic Sec- 
tions, Containing an Account of Its Most Recent Exten- 
sions, with Numerous Examples, 2nd ed., rev. enl. Dublin: 
Hodges, Figgis, & Co., pp. 96-150, 1893. 

Courant, R. and Robbins, H. What is Mathematics?: An El- 
ementary Approach to Ideas and Methods, 2nd ed. Oxford, 
England: Oxford University Press, pp. 74-75, 1996. 

Dunham, W. "Archimedes' Determination of Circular Area." 
Ch. 4 in Journey Through Genius: The Great Theorems 
of Mathematics. New York: Wiley, pp. 84-112, 1990. 

Eppstein, D. "Circles and Spheres." http://www. ics . uci . 
edu/*eppstein/ junkyard/sphere. html. 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 65-66, 1972. 

MacTutor History of Mathematics Archive. "Circle." http: 
//www -groups . dcs . st -and .ac.uk/ -history /Curves/ 
Circle.html. 

Pappas, T. "Infinity & the Circle" and "Japanese Calculus." 
The Joy of Mathematics. San Carlos, CA: Wide World 
Publ./Tetra, pp. 68 and 139, 1989. 

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- 
ton, DC: Math. Assoc. Amer., 1995. 

Yates, R. C "The Circle." A Handbook on Curves and Their 
Properties. Ann Arbor, MI: J. W. Edwards, pp. 21-25, 
1952. 

Circles-and-Squares Fractal 

m 




A FRACTAL produced by iteration of the equation 

Zn+i = z n (mod m) 
which results in a M0IRE-Iike pattern. 

see also FRACTAL, M0IRE PATTERN 



Circle Caustic 

Consider a point light source located at a point (//,0). 
The CATACAUSTIC of a unit CIRCLE for the light at fi = 

oo is the Nephroid 



x = ~ [3 cost - cos(3£)] 
y = \ [3 sin t — sin(3i)]. 



a) 

(2) 



The CATACAUSTIC for the light at a finite distance fx > 1 
is the curve 



V : 



fi(l — 3/j, cos t + 2fi cos 3 t) 
-(l-\-2fi 2 )-r3ficost 
2fi 2 sin 3 t 



1 + 2// 2 — 3/xcost ' 



(3) 
(4) 



and for the light on the CIRCUMFERENCE of the CIRCLE 
{i — 1 is the CARDIOID 



x = | cos t(l + cos t) - | 
y — | sini(l + cost). 



(5) 
(6) 



If the point is inside the circle, the catacaustic is a dis- 
continuous two-part curve. These four cases are illus- 
trated below. 




The CATACAUSTIC for PARALLEL rays crossing a CIRCLE 
is a Cardioid. 

see also CATACAUSTIC, CAUSTIC 
Circle-Circle Intersection 




Let two Circles of Radii R and r and centered at (0, 0) 
and (d, 0) intersect in a LENS-shaped region. The equa- 
tions of the two circles are 



2,2 D 2 

x +y — R 



(x - df +y 2 = r 2 



(1) 
(2) 



250 Circle-Circle Intersection 

Combining (1) and (2) gives 

(x-d) 2 + (R 2 -x 2 ) = r 2 . 
Multiplying through and rearranging gives 



x 2 - 2dx + d 2 - x 2 = r 2 - R 2 . 



Solving for x results in 



d 2 - r 2 + R 2 
2d 



(3) 



(4) 



(5) 



The line connecting the cusps of the LENS therefore has 
half-length given by plugging x back in to obtain 



2 D 2 2 D 2 / d - r + R 
y = R — x = R 



2d 



Ad 2 R 2 -{d 2 -r 2 +R 2 ) 2 
Ad? 



(6) 



giving a length of 



a= ^V 4 ^ 1 * 2 ~ ( d2 ~ r2 + R2 ) 2 

= h(-d + r-R)(-d-r + R) 

a 

x [(-d + r + R){d + r + R)] 1/2 . (7) 

This same formulation applies directly to the SPHERE- 
Sphere Intersection problem. 

To find the AREA of the asymmetric "Lens" in which 
the Circles intersect, simply use the formula for the 
circular SEGMENT of radius i^'and triangular height d' 

A{R!,d') = i^cos" 1 f^\ -d'^R' 2 -d<* (8) 

twice, one for each half of the "Lens." Noting that the 
heights of the two segment triangles are 



di = x ■ 



d 2 -r 2 + R 2 



dz = d — x ■■ 



2d 
d 2 +r 2 - R 2 
2d 



(9) 
(10) 



The result is 



A = A(Ri,d 1 )+A(R 2 ,d 2 ) 
_i (d 2 + r 2 -R 2 



2 

r cos 



2dr 



+ R* cos 



/ d 2 +E 2 -r 2 \ 
^ 2dR ) 



- \^{d - r - R)(d + r - R){d - r + R)(d + r + R). 

(11) 



Circle Cutting 

The limiting cases of this expression can be checked to 
give when d — R + r and 

A = 2R 2 cos" 1 (^) - \d\/AR? - d? (12) 



= 2A{\d,R) 



(13) 



when r = i2, as expected. In order for half the area of 

two Unit Disks (R = 1) to overlap, set A = irR 2 /2 = 
7r/2 in the above equation 



|tt = 2cos~ l (±d) - \d^J\ - d? (14) 

and solve numerically, yielding d w 0.807946. 
see also Lens, Segment, Sphere-Sphere Intersec- 
tion 

Circle Cutting 




2 4 7 11 

Determining the maximum number of pieces in which 
it is possible to divide a CIRCLE for a given number of 
cuts is called the circle cutting, or sometimes PANCAKE 
Cutting, problem. The minimum number is always 
n + 1, where n is the number of cuts, and it is always 
possible to obtain any number of pieces between the 
minimum and maximum. The first cut creates 2 regions, 
and the nth cut creates n new regions, so 



/(l) = 2 


(1) 


/(2) = 2 + /(l) 


(2) 


/(n) = n+/(n-l). 


(3) 



Therefore, 

f(n) = n+[(n-l) + f(n-2)} 

n 

= n + (n-l) + ... + 2 + /(l) = J^ k fW 

fc-2 
n 

= ^fc-l + /(l)-in(n+l)-l + 2 

k = l 

= §(n 2 +n + 2). (4) 

Evaluating for n = 1, 2, . . . gives 2, 4, 7, 11, 16, 22, . . . 
(Sloane's A000124). 



OO 





12 4 8 

A related problem, sometimes called Moser's CIRCLE 
PROBLEM, is to find the number of pieces into which 
a Circle is divided if n points on its Circumference 



Circle Evolute 



Circle Involute 251 



are joined by Chords with no three Concurrent. The 
answer is 



»<»>=(:)+©+> 



= 5j(n 4 - 6n 3 + 23n 2 - 18n + 24), 



(5) 
(6) 



(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 
1996, Noy 1996), where (£) is a Binomial Coeffi- 
cient. The first few values are 1, 2, 4, 8, 16, 31, 57, 
99, 163, 256, ... (Sloane's A000127). This sequence 
and problem are an example of the danger in making 
assumptions based on limited trials. While the series 
starts off like 2 n ~ 1 , it begins differing from this GEO- 
METRIC Series at n = 6. 

see also Cake Cutting, Cylinder Cutting, Ham 
Sandwich Theorem, Pancake Theorem, Pizza 
Theorem, Square Cutting, Torus Cutting 

References 

Conway, J. H. and Guy, R. K. "How Many Regions." In The 

Book of Numbers. New York: Springer- Verlag, pp. 76-79, 

1996. 
Guy, R. K. "The Strong Law of Small Numbers." Amer. 

Math. Monthly 95, 697-712, 1988. 
Noy, M. "A Short Solution of a Problem in Combinatorial 

Geometry." Math. Mag. 69, 52-53, 1996. 
Sloane, N. J. A. Sequences A000124/M1041 and A000127/ 

M1119 in "An On-Line Version of the Encyclopedia of In- 
teger Sequences." 
Yaglom, A. M. and Yaglom, I. M. Problem 47. Challenging 

Mathematical Problems with Elementary Solutions, Vol. 1. 

New York: Dover, 1987. 

Circle Evolute 



x = cos t x = — sin t x ~ 


— cost 


(i) 


y = sin t y = cos t y = - 


- sin t, 


(2) 


so the Radius of Curvature is 






^_(x' 2 +y' 2 ) 3/2 

y" x' — x"y' 






(sin 2 t + cos 2 t) 3/2 


— i 


i"*t 



(— sint)(— sint) — (— cost) cost 
and the TANGENT VECTOR is 



— sint 
cost 



Therefore, 



cos r —T • x = — sin t 
sin r ~T • y = cos t, 



(4) 



(5) 
(6) 



and the EVOLUTE degenerates to a POINT at the ORI- 
GIN. 

see also CIRCLE INVOLUTE 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, p. 77, 1993. 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 55- 
59, 1991. 

Circle Inscribing 

If r is the Radius of a Circle inscribed in a Right 
Triangle with sides a and b and Hypotenuse c, then 

r = i( a + 6-c). 
see Inscribed, Polygon 

Circle Involute 

First studied by Huygens when he was considering clocks 
without pendula for use on ships at sea. He used the cir- 
cle involute in his first pendulum clock in an attempt to 
force the pendulum to swing in the path of a CYCLOID. 




For a Circle with a = 1, the parametric equations of 
the circle and their derivatives are given by 

x = cost x =— sint x =— cost (1) 



y — sin t y = cos t 
The Tangent Vector is 



- sin t. 



T = 



— sint 
cost 



and the Arc LENGTH along the circle is 
so the involute is given by 



(2) 



(3) 



(4) 



n = r - sT = 



cost 

sint 

j 


-t 


— sint 
cost 


= 


cos t + t sin t 
sin t — t cos t 



(5) 



£(t) = x — R sin r — cos t — 1 • cos t = (7) 

>q(t) = y + Rcosr = sint + 1 * (-sint) = 0, (8) 



x = a(cost -f tsint) 
y = a(sint — tcost). 



(6) 
(7) 



252 



Circle Involute Pedal Curve 



Circle Lattice Points 





The Arc Length, Curvature, and Tangential An- 
gle are 



J ds= / ^x' 2 + y' 2 dt = \ 



1 

K = 

= i. 
The Cesaro Equation is 



Vas' 



at 2 (8) 

(9) 
(10) 



(11) 



see also Circle, Circle Evolute, Ellipse Involute, 
Involute 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, p. 83, 1993. 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 190-191, 1972. 

MacTutor History of Mathematics Archive. "Involute of a 
Circle." http://www-groups.dcs.st-and.ac.uk/-history 
/Curves/Involute. html. 

Circle Involute Pedal Curve 




The Pedal Curve of Circle Involute 

/ = cos t + t sin t 
g = sin t — t cos t 

with the center as the PEDAL Point is the ARCHIME- 
DES' Spiral 

x ~ t sin t 
y = — tcost. 



Circle Lattice Points 

For every POSITIVE INTEGER n, there exists a CIRCLE 
which contains exactly n lattice points in its interior. 
H. Steinhaus proved that for every POSITIVE INTEGER 
n, there exists a Circle of Area n which contains ex- 
actly n lattice points in its interior. 



Schinzel's Theorem shows that for every Positive 
Integer n, there exists a Circle in the Plane hav- 
ing exactly n LATTICE POINTS on its CIRCUMFERENCE. 
The theorem also explicitly identifies such "Schinzel 
Circles" as 



{x 

(x 



l) 2 + y 2 



1 cfc-] 
4 5 
1 r2fc 
9 5 



for n = 2k 
for n = 2fc + 1. 



(1) 



Note, however, that these solutions do not necessarily 
have the smallest possible RADIUS, For example, while 
the Schinzel Circle centered at (1/3, 0) and with 
RADIUS 625/3 has nine lattice points on its CIRCUM- 
FERENCE, so does the CIRCLE centered at (1/3, 0) with 
Radius 65/3. 

Let r be the smallest INTEGER RADIUS of a CIRCLE cen- 
tered at the Origin (0, 0) with L(r) Lattice Points. 
In order to find the number of lattice points of the Cir- 
cle, it is only necessary to find the number in the first 
octant, i.e., those with < y < [r/v^J , where [z\ is the 
Floor Function. Calling this N(r% then for r > 1, 
L(r) = 8N(r) - 4, so L(r) = 4 (mod 8). The multipli- 
cation by eight counts all octants, and the subtraction 
by four eliminates points on the axes which the multi- 
plication counts twice. (Since ^/2 is IRRATIONAL, the 
MIDPOINT of a are is never a LATTICE POINT.) 

Gauss's Circle Problem asks for the number of lat- 
tice points within a CIRCLE of RADIUS r 



N(r) = 1 + 4 [rj + 4 ^ ^r 2 - i 2 . 



Gauss showed that 



where 



N(r) = nr 2 + E(r), 
\E(r)\ < 2V2nr. 



(2) 

(3) 
(4) 




i 



The number of lattice points on the CIRCUMFERENCE of 
circles centered at (0, 0) with radii 0, 1, 2, . . . are 1, 4, 4, 
4, 4, 12, 4, 4, 4, 4, 12, 4, 4, . . . (Sloane's A046109). The 
following table gives the smallest RADIUS r < 111,000 
for a circle centered at (0, 0) having a given number of 
LATTICE POINTS L(r). Note that the high water mark 
radii are always multiples of five. 



Circle Lattice Points 



Circle Map 253 



L(r) 


r 


1 





4 


1 


12 


5 


20 


25 


28 


125 


36 


65 


44 


3,125 


52 


15,625 


60 


325 


68 


< 390,625 


76 


< 1,953,125 


84 


1,625 


92 


< 48,828,125 


100 


4,225 


108 


1,105 


132 


40,625 


140 


21,125 


180 


5,525 


252 


27,625 


300 


71,825 


324 


32,045 




* 




If the CIRCLE is instead centered at (1/2, 0), then the 
Circles of Radii 1/2, 3/2, 5/2, . . . have 2, 2, 6, 2, 2, 

2, 6, 6, 6, 2, 2, 2, 10, 2, . . . (Sloane's A046110) on their 
Circumferences. If the Circle is instead centered 
at (1/3, 0), then the number of lattice points on the 
Circumference of the Circles of Radius 1/3, 2/3, 
4/3, 5/3, 7/3, 8/3, ... are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 

3, 1, 3, 1, 1, 5, 3, . . . (Sloane's A046111). 

Let 

1. a n be the RADIUS of the CIRCLE centered at (0, 0) 
having 8n + 4 lattice points on its CIRCUMFERENCE, 

2. b n /2 be the RADIUS of the Circle centered at (1/2, 
0) having 4n + 2 lattice points on its CIRCUMFER- 
ENCE, 

3. c n /3 be the Radius of Circle centered at (1/3, 0) 
having 2n + 1 lattice points on its CIRCUMFERENCE. 

Then the sequences {a n }, {&n}, and {c n } are equal, with 
the exception that b n — if 2|n and c n = if 3|n. How- 
ever, the sequences of smallest radii having the above 
numbers of lattice points are equal in the three cases 



and given by 1, 5, 25, 125, 65, 3125, 15625, 325, ... 
(Sloane's A046112). 

Kulikowski's Theorem states that for every Posi- 
tive Integer n, there exists a 3-D Sphere which has 
exactly n Lattice Points on its surface. The Sphere 
is given by the equation 



(x-a) 2 + {y-b) 2 + (z-^) 2 



: C + 2, 



where a and b are the coordinates of the center of the 
so-called Schinzel Circle and c is its Radius (Hons- 

berger 1973). 

see also CIRCLE, CIRCUMFERENCE, GAUSS'S CIRCLE 

Problem, Kulikowski's Theorem, Lattice Point, 
Schinzel Circle, Sciiinzel's Theorem 

References 

Honsberger, R. "Circles, Squares, and Lattice Points." 

Ch. 11 in Mathematical Gems I. Washington, DC: Math. 

Assoc. Amer., pp. 117-127, 1973. 
Kulikowski, T. "Sur l'existence d'une sphere passant par un 

nombre donne aux coordonnees entieres." L'Enseignement 

Math. Ser. 2 5, 89-90, 1959. 
Schinzel, A. "Sur l'existence d'un cercle passant par un 

nombre donne de points aux coordonnees entieres." 

L'Enseignement Math. Ser. 2 4, 71-72, 1958. 
Sierpiiiski, W. "Sur quelques problemes concernant les points 

aux coordonnees entieres." L'Enseignement Math. Ser. 2 

4, 25-31, 1958. 
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant 

les ensembles de points sur le plan." Fund. Math. 46, 

191-194, 1959. 
Sierpinski, W. A Selection of Problems in the Theory of 

Numbers. New York: Pergamon Press, 1964. 
# Weisstein, E. W. "Circle Lattice Points." http:// www . 

astro . Virginia . edu/ -eww6n/ math /notebooks /Circle 

LatticePoints .m. 

Circle Lattice Theorem 

see Gauss's Circle Problem 

Circle Map 

A 1-D Map which maps a CIRCLE onto itself 

0n+i = n + Q-^- sin(27r0„), (1) 

where # n +i is computed mod 1. Note that the circle map 
has two parameters: Q and K. Q can be interpreted as 
an externally applied frequency, and K as a strength of 
nonlinearity. The 1-D JACOBIAN is 



d9, 



n+l 



d0 n 



l-ii:cos(27r(9n), 



(2) 



so the circle map is not Area-Preserving. It is related 

to the Standard Map 



/n+l = Jn + — sin(27r0 n ) 

@n + l — n + /n + l, 



(3) 
(4) 



254 



Circle Method 



for / and computed mod 1. Writing 8 n +i as 

n+ i = n + /„ + ^- sin(27rl9 n ) (5) 

gives the circle map with I n = Q, and K = —K. The 
unperturbed circle map has the form 

0n + l=0n+fi. (6) 

If fi is RATIONAL, then it is known as the map WINDING 
Number, defined by 






(7) 



and implies a periodic trajectory, since n will return 
to the same point (at most) every q ORBITS. If Q is 
Irrational, then the motion is quasiperiodic. If K is 
NONZERO, then the motion may be periodic in some 
finite region surrounding each RATIONAL Q. This exe- 
cution of periodic motion in response to an IRRATIONAL 
forcing is known as Mode Locking. 

If a plot is made of K vs. Q with the regions of pe- 
riodic MODE-LOCKED parameter space plotted around 
Rational Q values (Winding Numbers), then the re- 
gions are seen to widen upward from at K = to some 
finite width at K = 1. The region surrounding each Ra- 
tional Number is known as an Arnold Tongue. At 
K = 0, the Arnold Tongues are an isolated set of 
Measure zero. At K = 1, they form a Cantor Set 
of Dimension d « 0.08700. For K > 1, the tongues 
overlap, and the circle map becomes noninvertible. The 
circle map has a Feigenbaum Constant 



6= lim 

n—¥oo U n + 1 



On — On-1 
n 



2.833. 



(8) 



see also Arnold Tongue, Devil's Staircase, Mode 
Locking, Winding Number (Map) 

Circle Method 

see Partition Function P 

Circle Negative Pedal Curve 

The Negative Pedal Curve of a circle is an Ellipse 
if the Pedal Point is inside the Circle, and a Hy- 
perbola if the Pedal Point is outside the Circle. 

Circle Notation 

A Notation for Large Numbers due to Steinhaus 
(1983) in which is defined in terms of STEINHAUS- 
Moser Notation as n in n SQUARES. The particular 
number known as the MEGA is then defined as follows. 



©-E 



A-\A 



4 4 



256 



see also Mega, Megistron, Steinhaus-Moser No- 
tation 

References 

Steinhaus, H. Mathematical Snapshots, 3rd American ed. 
New York: Oxford University Press, pp. 28-29, 1983. 



Circle Packing 



Circle Order 

A Poset P is a circle order if it is Isomorphic to a Set 
of Disks ordered by containment. 

see also ISOMORPHIC POSETS, PARTIALLY ORDERED 

Set 

Circle Orthotomic 




The Orthotomic of the Circle represented by 

X = cos t 
y = sin t 

with a source at (x, y) is 



(1) 
(2) 



x = x cos(2£) - y sin(2t) + 2 sin t (3) 

y = ~x sin(2i) - y cos(2t) + 2 cos t. (4) 



Circle Packing 




The densest packing of spheres in the PLANE is the 
hexagonal lattice of the bee's honeycomb (illustrated 
above), which has a Packing Density of 



2\/3 



= 0.9068996821.. 



Gauss proved that the hexagonal lattice is the densest 
plane lattice packing, and in 1940, L. Fejes Toth proved 
that the hexagonal lattice is indeed the densest of all 
possible plane packings. 

Solutions for the smallest diameter CIRCLES into which 
n Unit Circles can be packed have been proved op- 
timal for n = 1 through 10 (Kravitz 1967). The best 
known results are summarized in the following table. 



Circle Packing 



Circle-Point Midpoint Theorem 255 



n 


d exact 


d approx. 


1 
2 
3 
4 

5 
6 

7 
8 

9 

10 
11 
12 


1 
2 

l+fx/3 
1 + V2 


1.00000 
2.00000 
2.15470... 
2.41421... 

2.70130... 
3.00000 
3.00000 
3.30476... 

3.61312... 

3.82... 

4.02... 


1 + \/2(l + l/\/5) 

3 

3 

1 + csc(tt/7) 

1 + ^/2(2 + ^/2) 



For Circle packing inside a Square, proofs are known 
only for n = 1 to 9. 



n 


d exact 


d approx. 


1 


1 


1.000 


2 




0.58... 


3 




0.500... 


4 


i 

2 


0.500 


5 




0.41... 


6 




0.37. . . 


7 




0.348... 


8 




0.341... 


9 


1 
3 


0.333. . . 


10 




0.148204... 



The smallest Square into which two Unit Circles, 

one of which is split into two pieces by a chord, can be 
packed is not known (Goldberg 1968, Ogilvy 1990). 

see also Hypersphere Packing, Malfatti's Right 
Triangle Problem, Mergelyan-Wesler Theorem, 
Sphere Packing 

References 

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, 
and Groups, 2nd ed. New York: Springer- Verlag, 1992. 

Eppstein, D. "Covering and Packing." http://www.ics.uci 
. edu/-eppstein/ junkyard/cover, html. 

Folkman, J. H. and Graham, R. "A Packing Inequality for 
Compact Convex Subsets of the Plane." Canad. Math, 
Bull. 12, 745-752, 1969. 

Gardner, M. "Mathematical Games: The Diverse Pleasures 
of Circles that Are Tangent to One Another." ScL Amer. 
240, 18-28, Jan. 1979. 

Gardner, M. "Tangent Circles." Ch. 10 in Fractal Music, 
HyperCards, and More Mathematical Recreations from Sci- 
entific American Magazine. New York: W. H. Freeman, 
1992. 

Goldberg, M. "Problem E1924." Amer. Math. Monthly 75, 
195, 1968. 

Goldberg, M. "The Packing of Equal Circles in a Square." 
Math. Mag. 43, 24-30, 1970. 

Goldberg, M. "Packing of 14, 16, 17, and 20 Circles in a 
Circle." Math. Mag. 44, 134-139, 1971. 

Graham, R. L. and Luboachevsky, B, D, "Repeated Patterns 
of Dense Packings of Equal Disks in a Square." Elec- 
tronic J. Combinatorics 3, R16, 1-17, 1996. http://www. 
combinatorics. org/Volume^3/volume3.html#R16. 

Kravitz, S. "Packing Cylinders into Cylindrical Containers." 
Math. Mag. 40, 65-70, 1967. 



McCaughan, F, "Circle Packings." http://www.pmms.cam. 

ac, uk/ -gj ml i/cpacking/ info. html. 
Molland, M. and Payan, Charles. "A Better Packing of Ten 

Equal Circles in a Square." Discrete Math. 84, 303-305, 

1990. 
Ogilvy, C. S. Excursions in Geometry. New York: Dover, 

p. 145, 1990. 
Reis, G. E. "Dense Packing of Equal Circle within a Circle." 

Math. Mag. 48, 33-37, 1975. 
Schaer, J. "The Densest Packing of Nine Circles in a Square." 

Can. Math. Bui. 8, 273-277, 1965. 
Schaer, J. "The Densest Packing of Ten Equal Circles in a 

Square." Math. Mag. 44, 139-140, 1971. 
Valette, G. "A Better Packing of Ten Equal Circles in a 

Square." Discrete Math. 76, 57-59, 1989. 

Circle Pedal Curve 















/ s* 




\ 1 


/ / 






/ / 




\^\ 


/ / 




\ \ 


/ 
1/ 




1 


y 




1 / 
/ / 
/ / 
/ / 




^ = ^^; 


— " ^s^ 



The Pedal Curve of a Circle is a Cardioid if the 
Pedal Point is taken on the Circumference, 




and otherwise a LlMAQON. 
Circle-Point Midpoint Theorem 




Taking the locus of MIDPOINTS from a fixed point to a 
circle of radius r results in a circle of radius r/2. This 
follows trivially from 



r(0) 



—x 



+K 


rcosS 

rsinO 


- 


—x 



~r cos9 — \x 




- 


\ sin 











References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, p. 17, 1929. 



256 Circle Radial Curve 

Circle Radial Curve 




The Radial Curve of a unit Circle from a Radial 
Point (x, 0) is another Circle with parametric equa- 
tions 

x(t) = x — cost 
y(i) = — sint. 



Circle Squaring 

Construct a SQUARE equal in Area to a CIRCLE using 
only a STRAIGHTEDGE and COMPASS. This was one of 
the three Geometric Problems of Antiquity, and 
was perhaps first attempted by Anaxagoras. It was fi- 
nally proved to be an impossible problem when Pi was 
proven to be TRANSCENDENTAL by Lindemann in 1882. 

However, approximations to circle squaring are given 
by constructing lengths close to tt = 3.1415926.... 
Ramanujan (1913-14) and Olds (1963) give geomet- 
ric constructions for 355/113 = 3.1415929.... Gard- 
ner (1966, pp. 92-93) gives a geometric construc- 
tion for 3+ 16/113 = 3.1415929.... Dixon (1991) 
gives constructions for 6/5(1 + <fi) = 3.141640... and 
y / 4+[3-tan(30°)] = 3.141533 . . .. 

While the circle cannot be squared in EUCLIDEAN 

Space, it can in Gauss-Bolyai-Lobachevsky Space 

(Gray 1989). 

see also GEOMETRIC CONSTRUCTION, QUADRATURE, 

Squaring 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, pp. 190-191, 1996. 

Dixon, R. M athographics. New York: Dover, pp. 44-49 and 
52-53, 1991. 

Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1 
in Journey Through Genius: The Great Theorems of 
Mathematics. New York: Wiley, pp. 20-26, 1990. 

Gardner, M. "The Transcendental Number Pi." Ch. 8 in 
Martin Gardner's New Mathematical Diversions from Sci- 
entific American. New York: Simon and Schuster, 1966. 

Gray, J. Ideas of Space. Oxford, England: Oxford University 
Press, 1989. 

Meyers, L. F. "Update on William Wernick's 'Triangle Con- 
structions with Three Located Points,"' Math. Mag. 69, 
46-49, 1996. 

Olds, C. D. Continued Fractions. New York: Random House, 
pp. 59-60, 1963. 

Ramanujan, S. "Modular Equations and Approximations to 
7T." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914, 



Circle Tangents 

Circle Strophoid 

The Strophoid of a Circle with pole at the center 
and fixed point on the CIRCUMFERENCE is a FREETH'S 

Nephroid. 

Circle Tangents 

There are four CIRCLES that touch all the sides of a 
given TRIANGLE. These are all touched by the CIRCLE 
through the intersection of the ANGLE BISECTORS of 
the Triangle, known as the Nine-Point Circle. 




Given the above figure, GE — FH, since 

AB = AG 4- GB = GE + GF = GE + {GE + EF) 

= 2G + EF 
CD = CH + HD = EH + FH = FH + (FH + EF) 

= EF + 2FH. 

Because AB = CD, it follows that GE = FH. 




The line tangent to a CIRCLE of RADIUS a centered at 

(a,y) 

x — x + a cos t 
V — V + o, sin t 

through (0,0) can be found by solving the equation 



x + a cos t 
y 4- a sin t 



a cost 
a sint 



giving 



t — db cos 



—ax db y\/x 2 -\- y 2 — a 2 



x 2 + y 2 



Circuit 

Two of these four solutions give tangent lines, as illus- 
trated above. 
see also KISSING CIRCLES PROBLEM, MlQUEL POINT, 

Monge's Problem, Pedal Circle, Tangent Line, 
Triangle 

References 

Dixon, R. Mathographics. New York: Dover, p. 21, 1991. 
Honsberger, R. More Mathematical Morsels. Washington, 
DC: Math. Assoc. Amer., pp. 4-5, 1991. 

Circuit 

see Cycle (Graph) 

Circuit Rank 

Also known as the Cyclomatic Number. The circuit 
rank is the smallest number of EDGES 7 which must be 
removed from a GRAPH of N EDGES and n nodes such 
that no Circuit remains. 

7 = N - n + 1. 



Circulant Determinant 

Gradshteyn and Ryzhik (1970) define circulants by 

Xn 
X n -1 
Xn-2 



Circular Functions 



257 



Xl 


X 2 


X3 


Xn 


Xl 


x 2 


m-1 


X n 


Xl 



X2 Xz X4 



Xl 



= Y\( Xl + X 2ti>j +X3Wj 2 + .- ■ +Xn(Jj n ), (1) 



i=i 



where u>j is the nth ROOT OF Unity. The second-order 
circulant determinant is 



Xl X2 
X2 Xi 

and the third order is 



Xl X2 Xz 
Xz Xi X2 
X2 Xz Xi 



= (xi -\-x 2 )(xi - x 2 ), 



(2) 



= (xi + x 2 + X3)(asi + ujx 2 + oj xz){xi + OJ X2 + UJXz), 

(3) 
where u) and u 2 are the COMPLEX CUBE ROOTS of 

Unity. 

The Eigenvalues A of the corresponding n x n circulant 
matrix are 



\j = xi -f- X20JJ 4- Xz^j + . . . + x n ujj n 



see also CIRCULANT MATRIX 



(4) 



References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, pp. 1111-1112, 1979. 

Vardi, I. Computational Recreations in Mathematica. Read- 
ing, MA: Addison-Wesley, p. 114, 1991. 



Circulant Graph 

A Graph of n Vertices in which the zth Vertex is 
adjacent to the (i + j)th and (i - j)th Vertices for 
each j in a list I. 



Circulant Matrix 

An n x n MATRIX C defined as follows, 

1 (?) G) - UO 

L (?) (?) (?)■•• i 

c = n[(i+u,,r-i], 

3 = 



where u;o = 1, cji, ..., u) n -i are the nth ROOTS 
OF UNITY. Circulant matrices are examples of LATIN 
Squares. 

see also CIRCULANT DETERMINANT 

References 

Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 
1994. 

Stroeker, R. J. "Brocard Points, Circulant Matrices, and 
Descartes' Folium." Math. Mag. 61, 172-187, 1988. 

Vardi, I. Computational Recreations in Mathematica. Read- 
ing, MA: Addison-Wesley, p. 114, 1991. 

Circular Cylindrical Coordinates 

see Cylindrical Coordinates 

Circular Functions 

The functions describing the horizontal and vertical po- 
sitions of a point on a Circle as a function of Angle 
(COSINE and Sine) and those functions derived from 
them: 



cot a; = 



tana; = 



tana; 

1 
sinx 

1 
cos a; 
sinx 



(i) 

(2) 
(3) 
(4) 



The study of circular functions is called TRIGONOME- 
TRY. 

see also COSECANT, COSINE, COTANGENT, ELLIPTIC 

Function, Generalized Hyperbolic Functions, 
Hyperbolic Functions, Secant, Sine, Tangent, 
Trigonometry 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- 
tions." §4.3 in Handbook of Mathematical Functions with 
Formulas, Graphs, and Mathematical Tables, 9th printing. 
New York: Dover, pp. 71-79, 1972. 



258 



Circular Permutation 



Circumcircle 



Circular Permutation 

The number of ways to arrange n distinct objects along 
a Circle is 

P n = (n- 1)1 

The number is (n - 1)! instead of the usual FACTORIAL 
n! since all Cyclic Permutations of objects are equiv- 
alent because the CIRCLE can be rotated. 

see also Permutation, Prime Circle 
Circumcenter 




The center O of a TRIANGLE'S CIRCUMCIRCLE. It can 
be found as the intersection of the PERPENDICULAR BI- 
SECTORS. If the Triangle is Acute, the circumcenter 
is in the interior of the TRIANGLE. In a RIGHT TRI- 
ANGLE, the circumcenter is the Midpoint of the Hy- 
potenuse. 



OOi + OQ 2 + OOz =R + r, 



(1) 



where Oi are the MIDPOINTS of sides Ai, R is the 
Circumradius, and r is the INRADIUS (Johnson 1929, 
p. 190), The Trilinear Coordinates of the circum- 
center are 

cos A : cos B : cos C, (2) 



and the exact trilinears are therefore 

R cos A : R cos B : R cos C. 
The Areal Coordinates are 

(^acotA, \bcotB, |ccotC). 



(3) 



(4) 



The distance b etween the Incenter and circumcenter 
is ^R(R — 2r). Given an interior point, the distances 
to the Vertices are equal Iff this point is the circum- 
center. It lies on the BROCARD AXIS. 




The circumcenter O and ORTHOCENTER H are ISOGO- 

nal Conjugates. 




The Orthocenter H of the Pedal Triangle 

AO1O2O3 formed by the CIRCUMCENTER O concurs 
with the circumcenter O itself, as illustrated above. The 
circumcenter also lies on the EULER LINE. 
see also Brocard Diameter, Carnot's Theorem, 
Centroid (Triangle), Circle, Euler Line, Incen- 
ter, Orthocenter 

References 

Carr, G. S. Formulas and Theorems in Pure Mathematics, 

2nd ed. New York: Chelsea, p, 623, 1970. 
Dixon, R. Mathographics. New York: Dover, p. 55, 1991. 
Eppstein, D. "Circumcenters of Triangles." http://www.ics 

.uci.edu/-eppstein/junkyard/circumcenter.htnil. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, 1929. 
Kimberling, C. "Central Points and Central Lines in the 

Plane of a Triangle." Math. Mag. 67, 163-187, 1994. 
Kimberling, C. "Circumcenter." http://vvv.evansville, 

edu/-ck6/tcenters/class/ccenter.html. 

Circumcircle 




Circumcircle 



Circumradius 259 



A Triangle's circumscribed circle. Its center O is 
called the Circumcenter, and its Radius R the Cir- 
cumradius. The circumcircle can be specified using 
Trilinear Coordinates as 



Pya. + yab + a/3c = 0. 



(i) 



The Steiner Point S and Tarry Point T lie on the 
circumcircle. 

A Geometric Construction for the circumcircle is 
given by Pedoe (1995, pp. xii-xiii). The equation for the 
circumcircle of the Triangle with Vertices (zu, yi) for 
i = 1, 2, 3 is 



2 , 2 

x + y 


X 


y 


1 


2 i 2 

xi +2/i 


X\ 


yi 


1 


2 , 2 
X 2 +V2 


X2 


2/2 


1 


2 , 2 

xz +2/3 


xz 


2/3 


1 



= 0. 



Expanding the DETERMINANT, 

a(x 2 + y 2 ) + 2dx + 2/y + 5 = 0, 
where 



(2) 



(3) 





Xi 


yi l 




a — 


X 2 2/2 1 
X3 2/3 1 




d=- 


1 
2 


xi 2 +2/ 
# 2 2 +2/ 
Z3 2 +2/ 


2 
1 

2 
2 

2 
3 



yi 

2/2 
2/3 





Xi 2 +2/i 2 


Xi 


J 2 


x 2 2 + 2/2 2 


x 2 




2 i 2 
Xz +2/3 


xz 




2 , 2 
Xl +2/1 


Xi 


9 = ~ 


2 , 2 
#2 +2/2 


x 2 




2 , 2 
Z3 +2/3 


xz 


COMPLETING THE SQUARE gives 


a { x+ lY +a (" + z?- 


a 


which is a CIRCLE o 


: the form 





1 
1 

1 

2/1 

2/2 

2/3 



(x - zo) 2 + (y- yo) 2 = r 2 , 



with ClRCUMCENTER 



Xq 



yo 



a 

./ 

a 



and Circumradius 



P±&_9 

a 2 a 



(4) 
(5) 
(6) 
(?) 

+ 5 = (8) 

(9) 

(10) 
(11) 

(12) 



see also CIRCLE, ClRCUMCENTER, CIRCUMRADIUS, EX- 
CIRCLE, INCIRCLE, PARRY POINT, PURSER'S THEOREM, 

Steiner Points, Tarry Point 
References 

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- 
ton, DC: Math. Assoc. Amer., 1995. 



Circumference 

The Perimeter of a Circle. For Radius r or Diam- 
eter d = 2r, 

C = 27vr = ltd, 

where tv is Pi. 

see also Circle, Diameter, Perimeter, Pi, Radius 

Circuminscribed 

Given two closed curves, the circuminscribed curve is 
simultaneously INSCRIBED in the outer one and CIR- 
CUMSCRIBED on the inner one. 

see also Poncelet's Closure Theorem 

Circumradius 

The radius of a TRIANGLE'S CIRCUMCIRCLE or of a 
Polyhedron's Circumsphere, denoted R. For a Tri- 
angle, 



R = 



abc 



y/(a + b + c)(b + c - a)(c + a - b)(a + b - c) 

(1) 
where the side lengths of the TRIANGLE are a, 6, and c. 




This equation can also be expressed in terms of the 
Radii of the three mutually tangent Circles centered 
at the Triangle's Vertices. Relabeling the diagram 
for the SODDY CIRCLES with VERTICES Oi, O2, and 3 
and the radii 7*1, r 2 , and rz, and using 



a = T\ + V2 
b = V2 + 7"3 

c — r\-\-rz 



(2) 
(3) 
(4) 



then gives 



R = (n +r 2 )(n + r 3 )(r 2 +r 3 ) 



4^/Vir 2 r3(ri + r 2 + rz) 



If O is the ClRCUMCENTER and M is the triangle Cen- 
TROID, then 



OM 2 =R 2 - §(a 2 + 6 2 + c 2 ). 



Rr = 



Q1Q2Q3 
As 



(6) 
(?) 



260 



Circumscribed 



Cissoid of Diodes 



COS CKi + COS Ct2 + cos 0:3 — 1 + 



R 



v = 2R cos ai cos 0:2 cos a$ 
ai 2 + a 2 2 + a 3 2 = 4r# + 8iZ 2 



(8) 

(9) 
(10) 



(Johnson 1929, pp. 189-191). Let d be the_distance 
between INRADIUS r and circumradius R, d = rR. Then 



= 2Rr 
1 1 



R- d R+d 



(11) 



(12) 



(Mackay 1886-87). These and many other identities are 
given in Johnson (1929, pp. 186-190). 

For an ARCHIMEDEAN SOLID, expressing the circumra- 
dius in terms of the INRADIUS r and MlDRADIUS p gives 



tf =±(r + xA 2 +a 2 ) 



s> 



(13) 
(14) 



for an Archimedean Solid. 

see also Carnot's Theorem, Circumcircle, Cir- 

CUMSPHERE 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, 1929. 
Mackay, J. S. "Historical Notes on a Geometrical Theorem 

and its Developments [18th Century]." Proc. Edinburgh 

Math. Soc. 5, 62-78, 1886-1887. 

Circumscribed 

A geometric figure which touches only the VERTICES (or 
other extremities) of another figure. 

see also ClRCUMCENTER, CIRCUMCIRCLE, ClRCUMIN- 

scribed, Circumradius, Inscribed 

Circumsphere 

A Sphere circumscribed in a given solid. Its radius is 
called the CIRCUMRADIUS. 
see also Insphere 



Cis 



Cis x = e 1 ' 



■ cosx 4- i since. 



Cissoid 

Given two curves C\ and C2 and a fixed point O, let a 
line from O cut C at Q and C at R. Then the LOCUS of 
a point P such that OP = QR is the cissoid. The word 
cissoid means "ivy shaped." 



Curve 1 Curve 2 



Pole 



Cissoid 



line 
line 

circle 
circle 

circle 
circle 
circle 



parallel line 
circle 

tangent line 
tangent line 

radial line 
concentric circle 
same circle 



any point 
center 

on C 

on C opp. 
tangent 
on C 
center 

(0A0) 



line 

conchoid of 
Nicomedes 
oblique cissoid 
cissoid of Diocles 

strophoid 

circle 

lemniscate 



see also Cissoid of Diocles 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 53-56 and 205, 1972. 
Lee, X. "Cissoid." http : //www . best . com/~xah/Special 

PlaneCurves^dir/Cissoid_dir/c issoid.html. 
Lockwood, E. H. "Cissoids." Ch. 15 in A Book of 

Curves. Cambridge, England: Cambridge University 

Press, pp. 130-133, 1967. 
Yates, R. C. "Cissoid." A Handbook on Curves and Their 

Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30, 

1952. 



Cissoid of Diocles 




A curve invented by Diocles in about 180 BC in con- 
nection with his attempt to duplicate the cube by geo- 
metrical methods. The name "cissoid" first appears in 
the work of Geminus about 100 years later. Fermat and 
Roberval constructed the tangent in 1634. Huygens and 
Wallis found, in 1658, that the Area between the curve 
and its asymptote was 3a (MacTutor Archive). From a 
given point there are either one or three TANGENTS to 
the cissoid. 

Given an origin O and a point P on the curve, let S be 
the point where the extension of the line OP intersects 
the line x — 2a and R be the intersection of the CIRCLE 
of RADIUS a and center (a, 0) with the extension of OP. 
Then the cissoid of Diocles is the curve which satisfies 
OP = RS. 



Cissoid of Diodes 



Clark's Triangle 261 



The cissoid of Diodes is the Roulette of the Vertex 
of a Parabola rolling on an equal Parabola. Newton 
gave a method of drawing the cissoid of Diocles using 
two line segments of equal length at RIGHT ANGLES. If 
they are moved so that one line always passes through a 
fixed point and the end of the other line segment slides 
along a straight line, then the MIDPOINT of the sliding 
line segment traces out a cissoid of Diocles. 

The cissoid of Diocles is given by the parametric equa- 
tions 



x = 2a sin 6 
_ 2a sin 3 
V ~ cos 6 

Converting these to POLAR COORDINATES gives 

sin 6 ' 



(1) 
(2) 



2 2.2 

r = x + y 



: 4a 2 [ sin 4 + 



cos 2 (9, 
: 4a 2 sin 4 0(1 + tan 2 6) = 4a 2 sin 4 6 sec 2 0, (3) 



so 

r = 2a sin 2 sec = 2a sin 6 tan 0. 

In Cartesian Coordinates, 

,3 Qrt 3 • 6/1 



(4) 



X 



2a -x 2a — 2a sin 2 

. 2 sin 6 2 
= 4a — = y . 



= 4a* 



sin 



1 - sin 2 8 



s 2 9 



An equivalent form is 



x(x 2 -\-y 2 ) = 2ay . 



Using the alternative parametric form 



*(*) = 

y(t) 



2at 2 
1 + i 2 
2at 3 



1 + t 2 
(Gray 1993), gives the Curvature as 

*<*)= a \t\{t* + 4)3/2- 



(5) 
(6) 

(7) 
(8) 

(9) 



References 

Gray, A. "The Cissoid of Diocles." §3.4 in Modern Differ- 
ential Geometry of Curves and Surf 'aces. Roca Raton, FL: 
CRC Press, pp. 43-46, 1993. 

Lawrence, J. D. A Catalog of Special Plane Curves. New 
York: Dover, pp. 98-100, 1972. 

Lee, X. "Cissoid of Diocles." http://www.best.com/-xah/ 
SpecialPlaneCurvesjdir/CissoidOf Diocles jdir/cissoid 
OfDiocles.html. 

Lockwood, E. H. A Book of Curves. Cambridge, England: 
Cambridge University Press, pp. 130-133, 1967. 

MacTutor History of Mathematics Archive. "Cissoid of Dio- 
cles." http: //www-groups . dcs . st-and.ac.uk/-history/ 
Curves/Cissoid.html. 

Yates, R. C. "Cissoid." A Handbook on Curves and Their 
Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30, 
1952. 



Cissoid of Diocles Caustic 

The Caustic of the cissoid where the Radiant Point 
is taken as (8a, 0) is a CARDIOID. 

Cissoid of Diocles Inverse Curve 

If the cusp of the CISSOID OF DIOCLES is taken as 
the Inversion Center, then the cissoid inverts to a 
PARABOLA. 

Cissoid of Diocles Pedal Curve 





\ 

\ 
\ 

The Pedal Curve of the cissoid, when the Pedal 
Point is on the axis beyond the Asymptote at a dis- 
tance from the cusp which is four times that of the 
Asymptote is a Cardioid. 

Clairaut's Differential Equation 

- x ^y + f f^M.\ 
dx V dx ) 

or 

y = px + f(p), 

where / is a Function of one variable and p = dy/dx. 

The general solution is y — ex + /(c). The singular 

solution ENVELOPES are x — ~f f (c) and y = f(c) - 

cf'(c). 

see also d'Alembert's Equation 

References 

Boyer, C B. A History of Mathematics. New York: Wiley, 
p. 494, 1968. 

Clarity 

The Ratio of a measure of the size of a "fit" to the size 
of a "residual." 

References 

Tukey, J. W. Explanatory Data Analysis. Reading, MA: 
Addison- Wesley, p. 667, 1977. 



Clark's Triangle 



(m-l)3 



12 7 1 t> 



18 19 8 1 

24 37 27 9 1 

30 61 64 36 10 1 

36 91 125 100 46 11 1 



// 



262 Clark's Triangle 

A Number Triangle created by setting the Vertex 
equal to 0, filling one diagonal with Is, the other diag- 
onal with multiples of an INTEGER /, and rilling in the 
remaining entries by summing the elements on either 
side from one row above. Call the first column n = 
and the last column m = nso that 



c(m, 0) = fm 
c(rri) m) = 1, 



(1) 
(2) 



then use the Recurrence Relation 

c(m, n) = c(m — 1, n — 1) + c(m — 1, n) (3) 

to compute the rest of the entries. For n = 1, we have 
c(m, 1) = c(m -1,0) + c(m - 1, 1) (4) 

c(m, 1) - c(m - 1, 1) = c(m -1,0) = f(m - 1). (5) 

For arbitrary m, the value can be computed by Sum- 
ming this Recurrence, 

c(m, 1) = / j J2 k I + X = l/ m ( m - 1) + 1. (6) 

Now, for n = 2 we have 

c(m, 2) = c(m - 1, 1) + c(m - 1, 2) (7) 

c(m,2)-c(m-l,2) = c(m-l,l) = |/(m-l)m+l, (8) 
so Summing the Recurrence gives 

c(m, 2) = 5}±/*(* - 1) + 1] = ]T(§/fc 2 " 3** + X ) 



fc=i 



fc=i 



= \f[\m{m + l)(2m + 1)] - \f[\m{m + 1)] + m 

= ±(m-l)(/m 2 -2/m + 6). (9) 

Similarly, for n = 3 we have 

c(m, 3) - c(m -1,3) = c(m - 1, 2) 

= |/m 3 -/m 2 + (^/ + l)m-(/ + 2). (10) 

Taking the Sum, 

m 

c(m,3) = ^ i/fc 3 - /fc 2 + (ff + l)k - (/ + 2). (11) 

fc = 2 

Evaluating the Sum gives 

c(m,3) = ^(m- l)(m-2)(/m 2 -3/m+12). (12) 



Ciass Number 

So far, this has just been relatively boring Algebra. 
But the amazing part is that if / = 6 is chosen as the 
Integer, then c(m, 2) and c(tm, 3) simplify to 

c(m, 2) = \{m - l)(6m 2 - 12m + 6) 

-(m-1) 3 (13) 

c(m,3)=|(m-l) 2 (m-2) 2 , (14) 

which are consecutive Cubes (m — l) 3 and nonconsecu- 
tive Squares n 2 = [(m - l)(m - 2)/2] 2 . 

see a/so Bell Triangle, Catalan's Triangle, 
Euler's Triangle, Leibniz Harmonic Triangle, 
Number Triangle, Pascal's Triangle, Seidel- 
Entringer-Arnold Triangle, Sum 

References 

Clark, J. E. "Clark's Triangle." Math. Student 26, No. 2, 
p. 4, Nov. 1978. 



Class 

see Characteristic Class, Class Interval, Class 
(Multiply Perfect Number), Class Number, 
Class (Set), Conjugacy Class 

Class (Group) 

see Conjugacy Class 

Class Interval 

The constant bin size in a HISTOGRAM, 

see also Sheppard's Correction 

Class (Map) 

A Map u : R n -► R n from a Domain G is called a map 
of class C r if each component of 

u(x) - (ui(zi,...,Xn),...,u m (a;i J ...,x„)) 

is of class C r (0 < r < 00 or r — w) in G, where C d 
denotes a continuous function which is differentiable d 
times. 

Class (Multiply Perfect Number) 

The number k in the expression s(n) — kn for a Mul- 
tiply Perfect Number is called its class. 

Class Number 

For any IDEAL 7, there is an IDEAL 7* such that 



Hi = z, 



(1) 



where z is a Principal IDEAL, (i.e., an IDEAL of rank 
1). Moreover, there is a finite list of ideals h such that 
this equation may be satisfied for every I. The size 
of this list is known as the class number. When the 
class number is 1, the Ring corresponding to a given 
IDEAL has unique factorization and, in a sense, the class 



Class Number 



Class Number 



263 



number is a measure of the failure of unique factorization 
in the original number ring. 

A finite series giving exactly the class number of a Ring 
is known as a CLASS NUMBER FORMULA. A CLASS 
Number Formula is known for the full ring of cyclo- 
tomic integers, as well as for any subring of the cyclo- 
tomic integers. Finding the class number is a computa- 
tionally difficult problem. 

Let h(d) denote the class number of a quadratic ring, 
corresponding to the Binary Quadratic Form 



ax + bxy + cy , 



with Discriminant 



d = b — 4ac. 



(2) 



(3) 



Then the class number h(d) for DISCRIMINANT d gives 
the number of possible factorizations of ax 2 + bxy + cy 2 
in the QUADRATIC Field Q(y/d). Here, the factors are 
of the form x 4- yVd, with x and y half INTEGERS. 

Some fairly sophisticated mathematics shows that the 
class number for discriminant d can be given by the 

Class Number Formula 

,, f-^E^VWlnsin(^) ford>0 /x 



mElt\d\r)r 



for d < 0, 



where (d\r) is the Kronecker Symbol, 77(d) is the 
Fundamental Unit, w(d) is the number of substitu- 
tions which leave the Binary Quadratic Form un- 
changed 

( 6 for d = -3 
w(d) ^<4 for d = -4 (5) 

[ 2 otherwise, 

and the sums are taken over all terms where the Kron- 
ecker SYMBOL is defined (Cohn 1980). The class num- 
ber for d > can also be written 



^M-) = TJ Bin -(-|r)^^ 



(6) 



for d > 0, where the PRODUCT is taken over terms for 
which the Kronecker Symbol is defined. 

The class number is related to the DlRlCHLET L-Series 

by 

L„(l) 



h(d) = 



K{d) 



(7) 



where /c(d) is the DlRlCHLET STRUCTURE CONSTANT. 
Wagner (1996) shows that class number h(—d) satisfies 

the Inequality 



-»^(>-M) 



lnd, 



(8) 



for -d < 0, where [x] is the Floor Function, the 
product is over PRIMES dividing d, and the * indicates 
that the Greatest Prime Factor of d is omitted from 
the product. 

The Mathematica® (Wolfram Research, Champaign, 
IL) function NumberTheory'NumberTheoryFunct ions' 
ClassNumber [n] gives the class number h{d) for d a 
Negative Squarefree number of the form 4k -f 1, 

Gauss's Class Number Problem asks to determine 
a complete list of fundamental DISCRIMINANTS — d such 
that the CLASS Number is given by h(—d) = m for 
a given m. This problem has been solved for n < 7 
and Odd n < 23. Gauss conjectured that the class 
number h(—d) of an IMAGINARY quadratic field with 
Discriminant —d tends to infinity with d, an assertion 
now known as Gauss's Class Number Conjecture. 

The discriminants d having h(~d) = 1, 2, 3, 4, 5, ... 
are Sloane's A014602 (Cohen 1993, p. 229; Cox 1997, 
p. 271), Sloane's A014603 (Cohen 1993, p. 229), Sloane's 
A006203 (Cohen 1993, p. 504), Sloane's A013658 (Co- 
hen 1993, p. 229), Sloane's A046002, Sloane's A046003, 
The complete set of negative discriminants hav- 
ing class numbers 1-5 and Odd 7-23 are known. Buell 
(1977) gives the smallest and largest fundamental class 
numbers for d < 4, 000, 000, partitioned into EVEN dis- 
criminants, discriminants 1 (mod 8), and discriminants 
5 (mod 8). Arno et al. (1993) give complete lists of val- 
ues of d with h{-d) = k for ODD k = 5, 7, 9, . . . , 23. 
Wagner gives complete lists of values for k = 5, 6, and 
7. 

Lists of NEGATIVE discriminants co rrespon ding to 
Imaginary Quadratic Fields Q(y/—d(n) ) having 
small class numbers h{—d) are given in the table below. 
In the table, N is the number of "fundamental" values 
of — d with a given class number h{—d)^ where "funda- 
mental" means that — d is not divisible by any SQUARE 
Number s 2 such that h(—d/s 2 ) < h(—d). For example, 
although h(— 63) = 2, —63 is not a fundamental dis- 
criminant since 63 = 3 2 • 7 and h(-63/3 2 ) = h(-7) = 
1 < h(-63). Even values 8 < h(-d) < 18 have been 
computed by Weisstein. The number of negative dis- 
criminants having class number 1, 2, 3, . . . are 9, 18, 
16, 54, 25, 51, 31, ... (Sloane's A046125). The largest 
negative discriminants having class numbers 1, 2, 3, . . . 
are 163, 427, 907, 1555, 2683, . . . (Sloane's A038552). 

The following table lists the numbers with small class 
numbers < 11. Lists including larger class numbers are 
given by Weisstein. 

h(-d) N d 

1 9 3, 4, 7, 8, 11, 19, 43, 67, 163 

2 18 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 

123, 148, 187, 232, 235, 267, 403, 427 

3 16 23, 31, 59, 83, 107, 139, 211, 283, 307, 

331, 379, 499, 547, 643, 883, 907 



264 



Class Number 



Class Number 



h(-d) N d 



4 54 39, 55, 56, 68, 84, 120, 132, 136, 155, 

168, 184, 195, 203, 219, 228, 259, 280, 
291, 292, 312, 323, 328, 340, 355, 372, 
388, 408, 435, 483, 520, 532, 555, 568, 
595, 627, 667, 708, 715, 723, 760, 763, 
772, 795, 955, 1003, 1012, 1027, 1227, 
1243, 1387, 1411, 1435, 1507, 1555 

5 25 47, 79, 103, 127, 131, 179, 227, 347, 443, 

523, 571, 619, 683, 691, 739, 787, 947, 
1051, 1123, 1723, 1747, 1867, 2203, 2347, 
2683 

6 51 87, 104, 116, 152, 212, 244, 247, 339, 

411, 424, 436, 451, 472, 515, 628, 707, 
771, 808, 835, 843, 856, 1048, 1059, 1099, 
1108, 1147, 1192, 1203, 1219, 1267, 1315, 
1347, 1363, 1432, 1563, 1588, 1603, 1843, 
1915, 1963, 2227, 2283, 2443, 2515, 2563, 
2787, 2923, 3235, 3427, 3523, 3763 

7 31 71, 151, 223, 251, 463, 467, 487, 587, 

811, 827, 859, 1163, 1171, 1483, 1523, 
1627, 1787, 1987, 2011, 2083, 2179, 2251, 
2467, 2707, 3019, 3067, 3187, 3907, 4603, 
5107, 5923 

8 131 95, 111, 164, 183, 248, 260, 264, 276, 

295, 299, 308, 371, 376, 395, 420, 452, 
456, 548, 552, 564, 579, 580, 583, 616, 
632, 651, 660, 712, 820, 840, 852, 868, 
904, 915, 939, 952, 979, 987, 995, 1032, 
1043, 1060, 1092, 1128, 1131, 1155, 
1195, 1204, 1240, 1252, 1288, 1299, 1320, 
1339, 1348, 1380, 1428, 1443, 1528, 1540, 
1635, 1651, 1659, 1672, 1731, 1752, 1768, 
1771, 1780, 1795, 1803, 1828, 1848, 1864, 
1912, 1939, 1947, 1992, 1995, 2020, 2035, 
2059, 2067, 2139, 2163, 2212, 2248, 2307, 
2308, 2323, 2392, 2395, 2419, 2451, 2587, 
2611, 2632, 2667, 2715, 2755, 2788, 2827, 
2947, 2968, 2995, 3003, 3172, 3243, 3315, 
3355, 3403, 3448, 3507, 3595, 3787, 3883, 
3963, 4123, 4195, 4267, 4323, 4387, 4747, 
4843, 4867, 5083, 5467, 5587, 5707, 5947, 
6307 

9 34 199, 367, 419, 491, 563, 823, 1087, 1187, 

1291, 1423, 1579, 2003, 2803, 3163, 3259, 
3307, 3547, 3643, 4027, 4243, 4363, 4483, 
4723, 4987, 5443, 6043, 6427, 6763, 6883, 
7723, 8563, 8803, 9067, 10627 

10 87 119, 143, 159, 296, 303, 319, 344, 415, 
488, 611, 635, 664, 699, 724, 779, 788, 
803, 851, 872, 916, 923, 1115, 1268, 
1384, 1492, 1576, 1643, 1684, 1688, 1707, 

1779, 1819, 1835, 1891, 1923, 2152, 2164, 



h(~d) N d 



2363, 2452, 2643, 2776, 2836, 2899, 3028, 
3091, 3139, 3147, 3291, 3412, 3508, 3635, 
3667, 3683, 3811, 3859, 3928, 4083, 4227, 
4372, 4435, 4579, 4627, 4852, 4915, 5131, 
5163, 5272, 5515, 5611, 5667, 5803, 6115, 
6259, 6403, 6667, 7123, 7363, 7387, 7435, 
7483, 7627, 8227, 8947, 9307, 10147, 
10483, 13843 
11 41 167, 271, 659, 967, 1283, 1303, 1307, 

1459, 1531, 1699, 2027, 2267, 2539, 2731, 
2851, 2971, 3203, 3347, 3499, 3739, 3931, 
4051, 5179, 5683, 6163, 6547, 7027, 7507, 
7603, 7867, 8443, 9283, 9403, 9643, 9787, 
10987, 13003, 13267, 14107, 14683, 15667 

The table below gives lists of Positive fundamental 
discriminants d having small class numbers h(d), cor- 
responding to Real quadratic fields. All Positive 
SQUAREFREE values of d < 97 (for which the KRON- 
ECKER SYMBOL is defined) are included. 



h(d) d 



1 5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77 

2 65 

The POSITIVE d for which h(d) = 1 is given by Sloane's 
A014539. 

see also Class Number Formula, Dirichlet L- 
Series, Discriminant (Binary Quadratic Form), 
Gauss's Class Number Conjecture, Gauss's 
Class Number Problem, Heegner Number, Ideal, 

j-FUNCTION 

References 

Arno, S. "The Imaginary Quadratic Fields of Class Number 
4." Acta Arith. 40, 321-334, 1992. 

Arno, S.; Robinson, M. L«; and Wheeler, F. S. "Imaginary 
Quadratic Fields with Small Odd Class Number." http:// 
www.math.uiuc . edu/Algebraic -Number-Theory/ 0009/. 

Buell, D. A. "Small Class Numbers and Extreme Values of 
//-Functions of Quadratic Fields." Math. Comput. 139, 
786-796, 1977. 

Cohen, H. A Course in Computational Algebraic Number 
Theory. New York: Springer- Verlag, 1993. 

Cohn, H. Advanced Number Theory. New York: Dover, 
pp. 163 and 234, 1980. 

Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field 
Theory and Complex Multiplication. New York: Wiley, 
1997. 

Davenport, H. "Dirichlet's Class Number Formula." Ch. 6 
in Multiplicative Number Theory, 2nd ed. New York: 
Springer- Verlag, pp. 43-53, 1980. 

Iyanaga, S. and Kawada, Y. (Eds.). "Class Numbers of Al- 
gebraic Number Fields." Appendix B, Table 4 in Encyclo- 
pedic Dictionary of Mathematics. Cambridge, MA: MIT 
Press, pp. 1494-1496, 1980. 

Montgomery, H. and Weinberger, P. "Notes on Small Class 
Numbers." Acta. Arith. 24, 529-542, 1974. 

Sloane, N. J. A. Sequences A014539, A038552, A046125, and 
A003657/M2332 in "An On-Line Version of the Encyclo- 
pedia of Integer Sequences." 



Class Number Formula 



Clausen Formula 265 



Stark, H. M. "A Complete Determination of the Complex 
Quadratic Fields of Class Number One." Michigan Math. 
J. 14, 1-27, 1967. 

Stark, H, M. "On Complex Quadratic Fields with Class Num- 
ber Two." Math. Comput. 29, 289-302, 1975. 

Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 
785-800, 1996. 
# Weisstein, E. W. "Class Numbers." http: //www. astro . 
Virginia. edu/~eww6n/math/notebooks/ClassNumbers .m. 

Class Number Formula 

A class number formula is a finite series giving exactly 
the Class Number of a Ring. For a Ring of quadratic 
integers, the class number is denoted h(d) y where d is the 
discriminant. A class number formula is known for the 
full ring of cyclotomic integers, as well as for any subring 
of the cyclotomic integers. This formula includes the 
quadratic case as well as many cubic and higher-order 
rings. 

see also Class Number 

Class Representative 

A set of class representatives is a SUBSET of X which 
contains exactly one element from each Equivalence 
Class. 

Class (Set) 

A class is a special kind of Set invented to get around 
RUSSELL'S PARADOX while retaining the arbitrary cri- 
teria for membership which leads to difficulty for Sets. 
The members of classes are Sets, but it is possible to 
have the class C of "all Sets which are not members of 
themselves" without producing a paradox (since C is a 
proper class (and not a Set), it is not a candidate for 
membership in C). 

see also Aggregate, Russell's Paradox, Set 

Classical Groups 

The four following types of GROUPS, 

1. Linear Groups, 

2. Orthogonal Groups, 

3. Symplectic Groups, and 

4. Unitary Groups, 

which were studied before more exotic types of groups 
(such as the SPORADIC GROUPS) were discovered. 

see also GROUP, LINEAR GROUP, ORTHOGONAL 
Group, Symplectic Group, Unitary Group 

Classification 

The classification of a collection of objects generally 
means that a list has been constructed with exactly one 
member from each ISOMORPHISM type among the ob- 
jects, and that tools and techniques can effectively be 
used to identify any combinatorially given object with 
its unique representative in the list. Examples of math- 
ematical objects which have been classified include the 
finite Simple Groups and 2-Manifolds but not, for 
example, Knots. 



Classification Theorem 

The classification theorem of FINITE Simple GROUPS, 
also known as the ENORMOUS THEOREM, which states 
that the Finite Simple Groups can be classified com- 
pletely into 

1. Cyclic Groups Z p of Prime Order, 

2. Alternating Groups A n of degree at least five, 

3. Lie-Type Chevalley Groups PSL(n,q), 
PSU(n,q), PsP(2n,g), and Pft € (n,g), 

4. Lie-Type (Twisted Chevalley Groups or the 
Tits Group) s D 4 (q) y E Q (q) y E 7 (q), E s (q), F 4 (g), 
2 F 4 (2*% G 2 (q), 2 G 2 (3 n ), 2 B(2 n ), 

5. Sporadic Groups Mu, M i2 , M 22 , M23, M 24 , Ji = 
HJ, Suz, HS, McL, Co 3 , Co 2 , C01, He, Fi 22} ^'23, 
Fi' 24 , HN, Th, B, M, J u OW, J 3 , Ly, Ru, J 4 . 

The "Proof" of this theorem is spread throughout the 
mathematical literature and is estimated to be approx- 
imately 15,000 pages in length. 

see also FINITE GROUP, GROUP, j-FUNCTION, SIMPLE 

Group 

References 

Cartwright, M. "Ten Thousand Pages to Prove Simplicity." 
New Scientist 109, 26-30, 1985. 

Cipra, B. "Are Group Theorists Simpleminded?" What's 
Happening in the Mathematical Sciences, 1995-1996, 
Vol 3. Providence, RJ: Amer. Math. Soc, pp. 82-99, 1996. 

Cipra, B. "Slimming an Outsized Theorem." Science 267, 
794-795, 1995. 

Gorenstein, D. "The Enormous Theorem," Set Amer, 253, 
104-115, Dec. 1985. 

Solomon, R. "On Finite Simple Groups and Their Classifica- 
tion." Not Amer. Math. Soc. 42, 231-239, 1995. 



Clausen Formula 

Clausen's 4^3 identity 



/ 9 



(2a)| d |(a + %|(26)| d | 
(2a + 2b)\d\a\ d \b\d\ 



holds for a + b + c- d= 1/2, e = a + 6 + 1/2, a + / = 
d+l = 6 + p, da nonpositive integer, and (a) n is the 
POCHHAMMER Symbol (Petkovsek tt al. 1996). 

Another identity ascribed to Clausen which in- 
volves the Hypergeometric Function 2 i*i(a, b\c\z) 

and the GENERALIZED HYPERGEOMETRIC FUNCTION 

3F2 (a, 6, c; d, e; z) is given by 



a, 6 
a + b+k'' X 



= 3-^2 



(• 



2a, a + b, 2b 
+ 6+|,2a + 26 ;:C 



see also GENERALIZED HYPERGEOMETRIC FUNCTION, 
HYPERGEOMETRIC FUNCTION 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K, Peters, pp. 43 and 127, 1996. 



266 Clausen Function 

Clausen Function 




sin(kx) 



*.(*>- £n£ 



C n {x) = J2 



cos(kx) 



(i) 

(2) 



and write 



cl " (x) = \c„(x)=Er=i ££ ^ 1 "° dd - 

Then the Clausen function Cl n (x) can be given symbol- 
ically in terms of the Polylogarithm as 



/ii[Li n (e-")-Li n (e-)] r. 
Ol nW - | i [Lin(e -i*) + Li n (e-)] n 



even 
odd. 



For n = l, the function takes on the special form 

Cli(x) = Ci(x) = -ln|2sin(|x)| 
and for n = 2, it becomes Clausen's Integral 

Cl 2 (a:) - S 2 (x) = - / ln[2sin(ft)]dt. 



(4) 



(5) 



The symbolic sums of opposite parity are summable 
symbolically, and the first few are given by 



i~ 2 



1_ 4 
48^ 



C 2 (ac) = ±tt - ±ttx+±x 

C 4 (z) = ^ - T^ 2 ^ 2 + T2 7 ™ 3 ~ -h* 

5i(x)=§(7T-x) 



(6) 
(7) 
(8) 
(9) 



5 5 (x) = i7r 4 x-^7rV + ^7rx 4 -^x 5 (10) 
for < x < 27r (Abramowitz and Stegun 1972). 

see also CLAUSEN'S INTEGRAL, POLYGAMMA FUNC- 
TION, Polylogarithm 



CLEAN Algorithm 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Clausen's Inte- 
gral and Related Summations" §27.8 in Handbook of Math- 
ematical Functions with Formulas, Graphs, and Mathe- 
matical Tables, 9th printing. New York: Dover, pp. 1005- 
1006, 1972. 

Arfken, G. Mathematical Methods {or Physicists, 3rd ed. Or- 
lando, FL: Academic Press, p. 783, 1985. 

Clausen, R. "Uber die Zerlegung reeller gebrochener . Funk- 
tionen." J. reine angew. Math. 8, 298-300, 1832. 

Grosjean, C. C. "Formulae Concerning the Computation of 
the Clausen Integral Cl 2 (a)." J. Comput. Appl. Math. 11, 
331-342, 1984. 

Jolley, L. B. W. Summation of Series. London: Chapman, 
1925. 

Wheelon, A. D. A Short Table of Summable Series. Report 
No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 
1953. 



Clausen's Integral 



0.5 




-1- 



The Clausen Function 



C1 2 (0) = - / \n[2sm(lt)]dt 

t/0 



see also CLAUSEN FUNCTION 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 

of Mathematical Functions with Formulas, Graphs, and 

Mathematical Tables, 9th printing. New York: Dover, 

pp. 1005-1006, 1972. 
Ashour, A. and Sabri, A. "Tabulation of the Function ip(0) = 

V°° -i£l»£i.» Math. Tables Aids Comp. 10, 54 and 57- 

65, 1956. 
Clausen, R. "Uber die Zerlegung reeller gebrochener Funk- 

tionen." J. reine angew. Math. 8, 298-300, 1832. 

CLEAN Algorithm 

An iterative algorithm which DECONVOLVES a sampling 
function (the "Dirty Beam") from an observed bright- 
ness ("DIRTY Map") of a radio source. This algorithm 
is of fundamental importance in radio astronomy, where 
it is used to create images of astronomical sources which 
are observed using arrays of radio telescopes ( "synthesis 
imaging"). As a result of the algorithm's importance to 
synthesis imaging, a great deal of effort has gone into 
optimizing and adjusting the ALGORITHM. CLEAN is a 
nonlinear algorithm, since linear DECONVOLUTION algo- 
rithms such as Wiener Filtering and inverse filtering 



CLEAN Algorithm 



CLEAN Algorithm 267 



are inapplicable to applications with invisible distribu- 
tions (i.e., incomplete sampling of the spatial frequency 
plane) such as map obtained in synthesis imaging. 

The basic CLEAN method was developed by Hogbom 
(1974). It was originally designed for point sources, but 
it has been found to work well for extended sources 
as well when given a reasonable starting model. The 
Hogbom CLEAN constructs discrete approximations I n 
to the CLEAN Map in the (£,77) plane from the CON- 
VOLUTION equation 



b' *I = /', 



(1) 



where b' is the Dirty Beam, I' is the Dirty Map (both 
in the (£>r?) Plane), and f*g denotes a Convolution. 

The CLEAN algorithm starts with an initial approxi- 
mation Jo = 0. At the nth iteration, it then searches for 
the largest value in the residual map 



I n ^ I' - b' * I n -1. 



(2) 



A Delta Function is then centered at the location of 
the largest residual flux and given an amplitude /x (the 
so-called "Loop Gain") times this value. An antenna's 
response to the Delta FUNCTION, the DlRTY Beam, is 
then subtracted from I n -i to yield I n . Iteration con- 
tinues until a specified iteration limit N is reached, or 
until the peak residual or Root-Mean-Square resid- 
ual decreases to some level. The resulting final map is 
denoted In, and the position of each Delta Function 
is saved in a "CLEAN component" table in the CLEAN 
Map file. At the point where component subtraction is 
stopped, it is assumed that the residual brightness dis- 
tribution consists mainly of NOISE. 

To diminish high spatial frequency features which may 
be spuriously extrapolated from the measured data, 
each CLEAN component is convolved with the so-called 
CLEAN Beam 6, which is simply a suitably smoothed 
version of the sampling function ("Dirty Beam"). Usu- 
ally, a Gaussian is used. A good CLEAN Beam should: 

1. Have a unity FOURIER TRANSFORM inside the sam- 
pled region of (u, v) space, 

2. Have a FOURIER TRANSFORM which tends to out- 
side the sampled (u, v) region as quickly as possible, 
and 

3. Not have any effects produced by NEGATIVE side- 
lobes larger than the NOISE level. 

A CLEAN Map is produced when the final residual 
map is added to the the approximate solution, 



[clean map] = In * b -\- [I — b * In] 



in order to include the NOISE. 



(3) 



CLEAN will always converge to one (of possibly many) 
solutions if the following three conditions are satisfied 
(Schwarz 1978): 



1. The beam must be symmetric. 

2. The Fourier Transform of the Dirty Beam is 
NONNEGATIVE (positive definite or positive semidef- 
inite). 

3. There must be no spatial frequencies present in the 
dirty image which are not also present in the Dirty 
Beam. 

These conditions are almost always satisfied in practice. 
If the number of CLEAN components does not exceed 
the number of independent (u,v) points, CLEAN con- 
verges to a solution which is the least squares fit of the 
Fourier Transforms of the Delta Function com- 
ponents to the measured visibility (Thompson et al. 
1986, p. 347). Schwarz claims that the CLEAN algo- 
rithm is equivalent to a least squares fitting of cosine 
and sine parts in the (u, v) plane of the visibility data. 
Schwab has produced a NOISE analysis of the CLEAN 
algorithm in the case of least squares minimization of 
a noiseless image which involves am N x M MATRIX. 
However, no NOISE analysis has been performed for a 
Real image. 

Poor modulation of short spacings results in an under- 
estimation of the flux, which is manifested in a bowl of 
negative surface brightness surrounding an object. Pro- 
viding an estimate of the "zero spacing" flux (the to- 
tal flux of the source, which cannot be directly mea- 
sured by an interferometer) can considerably reduce 
this effect. Modulations or stripes can occur at spa- 
tial frequencies corresponding to undersampled parts 
of the (u,v) plane. This can result in a golf ball-like 
mottling for disk sources such as planets, or a corru- 
gated pattern of parallel lines of peaks and troughs 
("stripes"). A more accurate model can be used to sup- 
press the "golf ball" modulations, but may not elimi- 
nate the corrugations. A tapering function which de- 
emphasizes data near (u, v) = (0,0) can also be used. 
Stripes can sometimes be eliminated using the Cornwell 
smoothness-stabilized CLEAN (a.k.a. Prussian helmet 
algorithm; Thompson et al 1986). CLEANing part way, 
then restarting the CLEAN also seems to eliminate the 
stripes, although this fact is more disturbing than reas- 
suring. Stability the the CLEAN algorithm is discussed 
by Tan (1986). 

In order to CLEAN a map of a given dimension, it is nec- 
essary to have a beam pattern twice as large so a point 
source can be subtracted from any point in the map. 
Because the CLEAN algorithm uses a Fast FOURIER 
Transform, the size must also be a Power of 2. 

There are many variants of the basic Hogbom CLEAN 
which extend the method to achieve greater speed and 
produce more realistic maps. Alternate nonlinear De- 
convolution methods, such as the Maximum En- 
tropy Method, may also be used, but are gener- 
ally slower than the CLEAN technique. The Astro- 
nomical Image Processing Software (AIPS) of the Na- 
tional Radio Astronomical Observatory includes 2-D 



268 CLEAN Algorithm 



CLEAN Algorithm 



DECONVOLUTION algorithms in the tasks DCONV and 
UVMAP. Among the variants of the basic Hogbom CLEAN 
are Clark, Cornwell smoothness stabilized (Prussian 
helmet), Cotton-Schwab, Gerchberg-Saxton (Fienup), 
Steer, Steer-Dewdney-Ito, and van Cittert iteration. 

In the Clark (1980) modification, CLEAN picks out only 
the largest residual points, and subtracts approximate 
point source responses in the (£,77) plane during minor 
(Hogbom CLEAN) cycles. It only occasionally (dur- 
ing major cycles) computes the full /„, residual map by 
subtracting the identified point source responses in the 
(ujv) plane using a Fast Fourier Transform for the 
Convolution. The Algorithm then returns to a mi- 
nor cycle. This algorithm modifies the Hogbom method 
to take advantage of the array processor (although it also 
works without one). It is therefore a factor of 2-10 faster 
than the simple Hogbom routine. It is implemented as 
the AIPS task APCLN. 

The Cornwell smoothness stabilized variant was devel- 
oped because, when dealing with two-dimensional ex- 
tended structures, CLEAN can produce artifacts in the 
form of low-level high frequency stripes running through 
the brighter structure. These stripes derive from poor 
interpolations into unsampled or poorly sampled re- 
gions of the (u, v) plane. When dealing with quasi-one- 
dimensional sources (i.e., jets), the artifacts resemble 
knots (which may not be so readily recognized as spuri- 
ous). APCLN can invoke a modification of CLEAN that 
is intended to bias it toward generating smoother solu- 
tions to the deconvolution problem while preserving the 
requirement that the transform of the CLEAN compo- 
nents list fits the data. The mechanism for introducing 
this bias is the addition to the Dirty Beam of a Delta 
FUNCTION (or "spike") of small amplitude (PHAT) while 
searching for the CLEAN components. The beam used 
for the deconvolution resembles the helmet worn by Ger- 
man military officers in World War I, hence the name 
"Prussian helmet" CLEAN. 

The theory underlying the Cornwell smoothness stabi- 
lized algorithm is given by Cornwell (1982, 1983), where 
it is described as the smoothness stabilized CLEAN. It 
is implemented in the AIPS tasks APCLN and MX. The 
spike performs a NEGATIVE feedback into the dirty im- 
age, thus suppressing features not required by the data. 
Spike heights of a few percent and lower than usual loop 
gains are usually needed. Also according to the MX doc- 
umentation, 



PHAT ; 



(noise) 1 

2(signal) 2 ~ 2(SNR) 2 



Unfortunately, the addition of a Prussian helmet gen- 
erally has "limited success," so resorting to another de- 
convolution method such as the MAXIMUM ENTROPY 
METHOD is sometimes required. 



The Cotton-Schwab uses the Clark method, but the 
major cycle subtractions of CLEAN components are 
performed on ungridded visibility data. The Cotton- 
Schwab technique is often faster than the Clark variant. 
It is also capable of including the w baseline term, thus 
removing distortions from noncoplanar baselines. It is 
often faster than the Clark method. The Cotton-Schwab 
technique is implemented as the AIPS task MX. 

The Gerchberg-Saxton variant, also called the Fienup 
variant, is a technique originally introduced for solv- 
ing the phase problem in electron microscopy. It was 
subsequently adapted for visibility amplitude measure- 
ments only. A Gerchberg-Saxton map is constrained to 
be Nonzero, and positive. Data and image plane con- 
straints are imposed alternately while transforming to 
and from the image plane. If the boxes to CLEAN are 
chosen to surround the source snugly, then the algorithm 
will converge faster and will have more chance of finding 
a unique image. The algorithm is slow, but should be 
comparable to the Clark technique (APCLN) if the map 
contains many picture elements. However, the resolu- 
tion is data dependent and varies across the map. It is 
implemented as the AIPS task APGS (Pearson 1984). 

The Steer variant is a modification of the Clark variant 
(Cornwell 1982). It is slow, but should be comparable 
to the Clark algorithm if the map contains many pic- 
ture elements. The algorithm used in the program is 
due to David Steer. The principle is similar to Barry 
Clark's CLEAN except that in the minor cycle only 
points above the (trim level) x (peak in the residual map) 
are selected. In the major cycle these are removed us- 
ing a Fast Fourier Transform. If boxes are chosen 
to surround the source snugly, then the algorithm will 
converge faster and will have more chance of finding a 
unique image. It is implemented in AIPS as the exper- 
imental program STEER and as the Steer-Dewdney-Ito 
variant combined with the Clark algorithm as SDCLN. 

The Steer-Dewdney-Ito variant is similar to the Clark 
variant, but the components are taken as all pixels 
having residual flux greater than a cutoff value times 
the current peak residual. This method should avoid 
the "ripples" produced by the standard CLEAN on ex- 
tended emission. The AIPS task SDCLN does an AP- 
based CLEAN of the the Clark type, but differs from 
APCLN in that it offers the option to switch to the Steer- 
Dewdney-Ito method. 

Finally, van Cittert iteration consists of two steps: 

1. Estimate a correction to add to the current map es- 
timate by multiplying the residuals by some weight. 
In the classical van Cittert algorithm, this weight is 
a constant, where as in CLEAN the weight is zero 
everywhere except at the peak of the residuals. 

2. Add the step to the current estimate, and subtract 
the estimate, convolved with the DIRTY BEAM, from 
the residuals. 



CLEAN Beam 



Clebsch Diagonal Cubic 269 



Though it is a simple algorithm, it works well (if slowly) 
for cases where the DlRTY BEAM is positive semidefmite 
(as it is in astronomical observations). The basic idea is 
that the DlRTY MAP is a reasonably good estimate of 
the deconvolved map. The different iterations vary only 
in the weight to apply to each residual in determining 
the correction step, van Cittert iteration is implemented 
as the AIPS task APVC, which is a rather experimental 
and ad hoc procedure. In some limiting cases, it reduces 
to the standard CLEAN algorithm (though it would be 
unpractically slow). 

see also CLEAN Beam, CLEAN Map, Dirty Beam, 
Dirty Map 

References 

Christiansen, W. N. and Hogbom, J. A. Radiotelescopes, 2nd 
ed. Cambridge, England: Cambridge University Press, 
pp. 214-216, 1985, 

Clark, B, G. "An Efficient Implementation of the Algorithm 
'CLEAN'." Astron. Astrophys, 89, 377-378, 1980. 

Cornwell, T. J. "Can CLEAN be Improved?" VLA Scientific 
Memorandum No. 141, 1982. 

Cornwell, T\ J. "Image Restoration (and the CLEAN Tech- 
nique)." Lecture 9. NRAO VLA Workshop on Synthesis 
Mapping, p. 113, 1982, 

Cornwell, T. J. "A Method of Stabilizing the CLEAN Algo- 
rithm." Astron. Astrophys. 121, 281-285, 1983. 

Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Syn- 
thesis Imaging in Radio Astronomy: Third NRAO Sum- 
mer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and 
A. H. Bridle). San Francisco, CA: Astronomical Society of 
the Pacific, pp. 178-179, 1989. 

Hogbom, J. A. "Aperture Synthesis with a Non-Regular Dis- 
tribution of Interferometric Baselines." Astron. Astrophys. 
Supp. 15, 417-426, 1974. 

National Radio Astronomical Observatory. Astronomical Im- 
age Processing Software (AIPS) software package. APCLN, 
MX, and UVMAP tasks. 

Pearson, T. J. and Readhead, A. C. S. "Image Formation by 
Self-Calibration in Radio Astronomy." Ann. Rev. Astron. 
Astrophys. 22, 97-130, 1984. 

Schwarz, U. J. "Mathematical-Statistical Description of the 
Iterative Beam Removing Technique (Method CLEAN)." 
Astron. Astrophys. 65, 345-356, 1978. 

Tan, S. M. "An Analysis of the Properties of CLEAN and 
Smoothness Stabilized CLEAN — Some Warnings." Mon. 
Not. Royal Astron. Soc. 220, 971-1001, 1986. 

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. 
Inter jerometry and Synthesis in Radio Astronomy. New 
York: Wiley, p. 348, 1986. 

CLEAN Beam 

An Elliptical Gaussian fit to the Dirty Beam in 

order to remove sidelobes. The CLEAN beam is con- 
volved with the final CLEAN iteration to diminish spu- 
rious high spatial frequencies. 

see also CLEAN Algorithm, CLEAN Map, Decon- 
volution, Dirty Beam, Dirty Map 

CLEAN Map 

The deconvolved map extracted from a finitely sampled 

Dirty Map by the CLEAN Algorithm, Maximum 
Entropy Method, or any other Deconvolution pro- 
cedure. 



see also CLEAN Algorithm, CLEAN Beam, Decon- 
volution, Dirty Beam, Dirty Map 

Clebsch- Aronhold Notation 

A notation used to describe curves. The fundamen- 
tal principle of Clebsch-Aronhold notation states that 
if each of a number of forms be replaced by a POWER of 
a linear form in the same number of variables equal to 
the order of the given form, and if a sufficient number 
of equivalent symbols are introduced by the ARONHOLD 
Process so that no actual Coefficient appears except 
to the first degree, then every identical relation holding 
for the new specialized forms holds for the general ones. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New 
York: Dover, p. 79, 1959. 

Clebsch Diagonal Cubic 




A Cubic Algebraic Surface given by the equation 

xo 3 + xi S + x 2 3 + x 3 3 + X4 3 = 0, (1) 

with the added constraint 



xo + Xi + X2 + £3 + X4 



0. 



(2) 



The implicit equation obtained by taking the plane at 
infinity as xq + x\ + x 2 + x$/2 is 

81(x -hy -\-z ) — 189(x y-\-x z + y x-\-y z + z x + z y) 
+54xyz + 126(xy + xz + yz) - 9(x 2 + y 2 + z 2 ) 

-9(x + y + z) + 1 = (3) 

(Hunt, Nordstrand). On Clebsch's diagonal surface, 
all 27 of the complex lines (Solomon's Seal Lines) 
present on a general smooth CUBIC SURFACE are real. 
In addition, there are 10 points on the surface where 3 
of the 27 lines meet. These points are called ECKARDT 
POINTS (Fischer 1986, Hunt), and the Clebsch diago- 
nal surface is the unique CUBIC SURFACE containing 10 
such points (Hunt). 

If one of the variables describing Clebsch's diagonal sur- 
face is dropped, leaving the equations 



xq 3 + xi 3 + x 2 3 + #3 3 = 0, 



(4) 



270 Clebsch-Gordon Coefficient 

x + xi + x 2 + xz = 0, (5) 

the equations degenerate into two intersecting Planes 
given by the equation 

{x + y)(x + z){y + z) = Q. (6) 

see also Cubic Surface, Eckardt Point 

References 

Fischer, G. (Ed.). Mathematical Models from the Collections 
of Universities and Museums. Braunschweig, Germany: 
Vieweg, pp. 9-11, 1986. 

Fischer, G. (Ed.). Plates 10-12 in Mathematische Mod- 
elle/ Mathematical Models, Bildband/ Photograph Volume. 
Braunschweig, Germany: Vieweg, pp. 13-15, 1986. 

Hunt, B. The Geometry of Some Special Arithmetic Quo- 
tients. New York: Springer- Verlag, pp. 122-128, 1996. 

Nordstrand, T. "Clebsch Diagonal Surface." http://www. 
uib , no/people/nf ytn/clebtxt . htm. 

Clebsch-Gordon Coefficient 

A mathematical symbol used to integrate products of 
three SPHERICAL HARMONICS. Clebsch-Gordon coeffi- 
cients commonly arise in applications involving the ad- 
dition of angular momentum in quantum mechanics. If 
products of more than three SPHERICAL HARMONICS 
are desired, then a generalization known as WlGNER 
6J-SYMBOLS or WlGNER 9?'-Symb0LS is used. The 
Clebsch-Gordon coefficients are written 

C J mim2 = UiJ2mim 2 \jiJ2Jm) (1) 

and are denned by 

^jm = 2_^ Cm 1 m 2 ^m 1 m 2 , ( 2 ) 

M=Mi+M 2 

where J = Ji 4- J 2 - The Clebsch-Gordon coefficients 
are sometimes expressed using the related RACAH V- 
COEFFICIENTS 



V(jiJ2J;m 1 7n 2 7n) 



(3) 



or Wigner 3 j- Symbols. Connections among the three 
are 



(jiJ2mim2\jiJ2m) 
(jiJ2m 1 m 2 \jiJ2Jm) 



3i 

mi 



32 
7712 



(4) 



V(ji32J;rn 1 m 2 m) = (-1)" 



-h+32+3 I 3i 32 3i 
m 2 mi m 2 



Clenshaw Recurrence Formula 

They have the symmetry 

(jiJ2mim 2 \jij 2 jm) = (-iyi+w (j 2 j 1 Tn 2 m 1 \j 2 jijm), 

(7) 
and obey the orthogonality relationships 

"y y j (jiJ2Tn 1 m2\jiJ2Jm)(jiJ2Jm\j 1 J2Tn' 1 tn' 2 ) 



= S, 



>6„ 



Tn, l TTl i Tri 2 Tn { 



(8) 



^ (ji J2mim 2 \jiJ2Jm)(jiJ2J'm'\jiJ2mim2) 

see also Racah ^-Coefficient, Racah ^-Coef- 
ficient, Wigner 3j-Symbol, Wigner 6j-Symbol, 

WlGNER 9J-SYMBOL 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition 
Coefficients." §27.9 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 1006-1010, 1972. 

Cohen- Tannoudji, C; Diu, B.; and Laloe, F. "Clebsch- 
Gordon Coefficients." Complement B x in Quantum Me- 
chanics, Vol 2. New York: Wiley, pp. 1035-1047, 1977. 

Condon, E. U. and Shortley, G. §3.6-3.14 in The Theory of 
Atomic Spectra. Cambridge, England: Cambridge Univer- 
sity Press, pp. 56-78, 1951. 

Fano, U. and Fano, L. Basic Physics of Atoms and Molecules. 
New York: Wiley, p. 240, 1959. 

Messiah, A. "Clebsch-Gordon (C.-G.) Coefficients and 'Sf 
Symbols." Appendix C.I in Quantum Mechanics, Vol. 2. 
Amsterdam, Netherlands: North-Holland, pp. 1054-1060, 
1962. 

Shore, B. W. and Menzel, D. H. "Coupling and Clebsch- 
Gordon Coefficients." §6.2 in Principles of Atomic Spectra. 
New York: Wiley, pp. 268-276, 1968. 

Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec- 
tra and Radiative Transitions, 2nd ed. Berlin: Springer- 
Verlag, 1992. 

Clement Matrix 

see Kac Matrix 

Clenshaw Recurrence Formula 

The downward Clenshaw recurrence formula evaluates a 
sum of products of indexed COEFFICIENTS by functions 
which obey a recurrence relation. If 



f( X ) = Y, ckFk ^ 



fc-0 



(-l) i+ "V2j + lV{jij 2 j;mim 2 - m) (5) and 



F n +i(x) = a(n,x)F n (x) + f3(n,x)F n -i(x), 



(6) 



Cliff Random Number Generator 

where the CfcS are known, then define 

VN+2 = Vn+i = 

y k = a(/c, x)y k +i + 0{k + 1, x)y k+2 + c k 

for k ~ N, N - 1, . . . and solve backwards to obtain y 2 
and yi. 

Cfc = J/* - a(fe, ^)y fc+ i - /?(fc + 1, x)y fc +2 

N 

f(x) = ^2c k F k (x) 

fc=0 

- coFo(x) + [t/i - a(l,x)y 2 - /3(2,x)y 3 ]F 1 (x) 
+ [y 2 - a(2,x)y<i - (3(3,x)y4]F 2 (x) 
+ [ys - a(3,x)y 4 - j3(4,x)y5]F s {x) 
+ [y 4 - a(4, a) 3/5 - /3(5, x)y 6 ]i ? 4(x) + . . . 

= c Fo(x) + yi Fi (a:) + y 2 [F 2 (x) - a(l, z)Pi(z)] 
+ ys[F 3 (x) - a(2, z)P 2 (:r) - 0(2, a)] 
+ 2/ 4 [F 4 (x) - a(3,z)F 3 (x) - 0(3, x)] + . . . 

= c Fo(x) + y2[{a(l,x)F 1 ( : r)+/?(l,x)Fo( : r)} 
~a{l t x)F 1 (x)]+yiF 1 (x) 

= c F Q {x) + yiFi(a) + 0(l,x)F o (x)y 2 . 

The upward Clenshaw recurrence formula is 

y-2 = y-i = 

_ 1 

y *~/?(fc+l,x) 

for fe = 0, 1,..., N - 1. 



[y fc _ 2 - a(k 1 x)y k -i - c k ] 



f(x) = c N F N {x) - P(N ) x)F N -i(x)y N -i - F N (x)y N - 2 . 



References 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- 
ling, W. T. "Recurrence Relations and Clenshaw's Recur- 
rence Formula." §5.5 in Numerical Recipes in FORTRAN: 
The Art of Scientific Computing, 2nd ed. Cambridge, Eng- 
land: Cambridge University Press, pp. 172-178, 1992. 

Cliff Random Number Generator 

A Random Number generator produced by iterating 

X n+ i = 1 100 In X„ (mod 1)| 

for a Seed X = 0.1. This simple generator passes 
the NOISE SPHERE test for randomness by showing no 
structure. 

see also RANDOM NUMBER, SEED 

References 

Pickover, C. A. "Computers, Randomness, Mind, and In- 
finity." Ch. 31 in Keys to Infinity. New York: W. H. 
Freeman, pp. 233-247, 1995. 



Clique Number 271 

Clifford Algebra 

Let V be an n-D linear Space over a Field K, and let Q 
be a Quadratic Form on V. A Clifford algebra is then 
defined over the T{V)/I(Q), where T(V) is the tensor 
algebra over V and I is a particular Ideal of T(V). 

References 

Iyanaga, S. and Kawada, Y. (Eds.). "Clifford Algebras." §64 
in Encyclopedic Dictionary of Mathematics. Cambridge, 
MA: MIT Press, pp. 220-222, 1980. 

Lounesto, P. "Counterexamples to Theorems Published and 
Proved in Recent Literature on Clifford Algebras, Spinors, 
Spin Groups, and the Exterior Algebra." http://www.hit. 
f i/~lounesto/counterexamples .htm. 

Clifford's Circle Theorem 

Let Ci, <7 2 , C 3 , and C 4 be four CIRCLES of GENERAL 
POSITION through a point P. Let Pij be the second 
intersection of the CIRCLES C» and Cj. Let dj k be 
the Circle PijP ik Pjk- Then the four Circles P234, 
Pi34, P124, and P123 all pass through the point P1234. 
Similarly, let C 5 be a fifth CIRCLE through P. Then the 
five points P2345, P1345, P1245, A235 and P1234 all lie on 
one Circle C12345. And so on. 
see also CIRCLE, Cox's THEOREM 

Clifford's Curve Theorem 

The dimension of a special series can never exceed half 
its order. 

References 

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- 
York: Dover, p. 263, 1959. 

Clique 

In a Graph of N Vertices, a subset of pairwise ad- 
jacent Vertices is known as a clique. A clique is a 
fully connected subgraph of a given graph. The prob- 
lem of finding the size of a clique for a given GRAPH is 
an NP-Complete Problem. The number of graphs on 
n nodes having 3 cliques are 0, 0, 1, 4, 12, 31, 67, ... 
(Sloane's A005289). 

see also Clique Number, Maximum Clique Prob- 
lem, Ramsey Number, Turan's Theorem 

References 

Sloane, N. J. A. Sequence A005289/M3440 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Clique Number 

The number of VERTICES in the largest CLIQUE of G, 
denoted u)(G). For an arbitrary GRAPH, 



^— ' n-di 



where di is the DEGREE of VERTEX i. 
References 

Aigner, M. "Turan's Graph Theorem." Amer. Math. 
Monthly 102, 808-816, 1995. 



272 



Clock Solitaire 



Closure 



Clock Solitaire 

A solitaire game played with Cards. The chance of 
winning is 1/13, and the AVERAGE number of CARDS 
turned up is 42.4. 

References 

Gardner, M. Mathematical Magic Show: More Puzzles, 
Games, Diversions, Illusions and Other Mathematical 
Sleight- of- Mind from Scientific American. New York: 
Vintage, pp. 244-247, 1978. 

Close Packing 

see Sphere Packing 

Closed Curve 





closed curves open curves 

A CURVE with no endpoints which completely encloses 
an AREA. A closed curve is formally denned as the con- 
tinuous Image of a Closed Set. 

see also SIMPLE CURVE 

Closed Curve Problem 

Find Necessary and Sufficient conditions that de- 
termine when the integral curve of two periodic func- 
tions k(s) and t(s) with the same period L is a CLOSED 
Curve. 

Closed Disk 

An n-D closed disk of Radius r is the collection of points 
of distance < r from a fixed point in EUCLIDEAN n- 
space. 

see also Disk, Open Disk 

Closed Form 

A discrete FUNCTION A(n,k) is called closed form (or 
sometimes "hypergeometric" ) in two variables if the ra- 
tios A(n-rl,k)/A(n, k) and A(n,k-\-l)/A(n i k) are both 

Rational Functions. A pair of closed form functions 
(F, G) is said to be a Wilf-Zeilberger Pair if 

F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k). 

see also Rational Function, Wilf-Zeilberger Pair 

References 

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- 
ley, MA: A. K. Peters, p. 141, 1996. 

Zeilberger, D. "Closed Form (Pun Intended!)." Contempo- 
rary Math. 143, 579-607, 1993. 



Closed Graph Theorem 

A linear Operator between two Banach Spaces is 
continuous IFF it has a "closed" GRAPH. 

see also Banach SPACE 

References 

Zeidler, E. Applied Functional Analysis: Applications to 
Mathematical Physics. New York: Springer- Verlag, 1995. 

Closed Interval 

An Interval which includes its Limit Points. If the 
endpoints of the interval are Finite numbers a and b, 
then the Interval is denoted [a, 6]. If one of the end- 
points is ±oo, then the interval still contains all of its 
Limit Points, so [a, oo) and ( — 00,6] are also closed 
intervals. 

see also Half-Closed Interval, Open Interval 

Closed Set 

There are several equivalent definitions of a closed Set. 
A Set S is closed if 

1. The Complement of S is an Open Set, 

2. S is its own CLOSURE, 

3. Sequences/nets/filters in S which converge do so 
within 5, 

4. Every point outside S has a NEIGHBORHOOD disjoint 
from 5. 

The Point-Set Topological definition of a closed set 
is a set which contains all of its Limit POINTS. There- 
fore, a closed set C is one for which, whatever point x 
is picked outside of C, x can always be isolated in some 
Open Set which doesn't touch C. 

see also CLOSED INTERVAL 

Closure 

A Set S and a Binary Operator * are said to ex- 
hibit closure if applying the Binary Operator to two 
elements S returns a value which is itself a member of 
S. 

The term "closure" is also used to refer to a "closed" 
version of a given set. The closure of a Set can be 
denned in several equivalent ways, including 

1. The Set plus its Limit Points, also called "bound- 
ary" points, the union of which is also called the 
"frontier," 

2. The unique smallest CLOSED Set containing the 
given Set, 

3. The Complement of the interior of the Comple- 
ment of the set, 

4. The collection of all points such that every NEIGH- 
BORHOOD of them intersects the original Set in a 
nonempty SET. 

In topologies where the T2-Separation Axiom is as- 
sumed, the closure of a finite Set S is S itself. 



Clothoid 



Cobordism 



273 



see also Binary Operator, Existential Closure, 
Reflexive Closure, Tight Closure, Transitive 
Closure 

Clothoid 

see also CORNU SPIRAL 

Clove Hitch 




A Hitch also called the Boatman's Knot or Peg 
Knot. 

References 

Owen, P. Knots. Philadelphia, PA: Courage, pp. 24-27, 1993. 

Clump 

see Run 

Cluster 

Given a lattice, a cluster is a group of filled cells which 
are all connected to their neighbors vertically or hori- 
zontally. 

see also Cluster Perimeter, Percolation Theory, 
s-Cluster, s-Run 

References 

StaufFer, D. and Aharony, A. Introduction to Percolation 
Theory, 2nd ed. London: Taylor & Francis, 1992. 

Cluster Perimeter 

The number of empty neighbors of a CLUSTER. 

see also PERIMETER POLYNOMIAL 

Coanalytic Set 

A Definable Set which is the complement of an An- 
alytic Set. 
see also Analytic Set 

Coastline Paradox 

Determining the length of a country's coastline is not 
as simple as it first appears, as first considered by 
L. F. Richardson (1881-1953). In fact, the answer de- 
pends on the length of the RULER you use for the mea- 
surements. A shorter RULER measures more of the sin- 
uosity of bays and inlets than a larger one, so the esti- 
mated length continues to increase as the Ruler length 
decreases. 

In fact, a coastline is an example of a Fractal, and 
plotting the length of the Ruler versus the measured 
length of the coastline on a log-log plot gives a straight 
line, the slope of which is the FRACTAL DIMENSION of 
the coastline (and will be a number between 1 and 2). 



References 

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- 
ures. Princeton, NJ: Princeton University Press, pp. 29- 
31, 1991. 

Coates- Wiles Theorem 

In 1976, Coates and Wiles showed that Elliptic 
Curves with Complex Multiplication having an in- 
finite number of solutions have //-functions which are 
zero at the relevant fixed point. This is a special case of 
the Swinnerton-Dyer Conjecture. 

References 

Cipra, B. "Fermat Prover Points to Next Challenges." Sci- 
ence 271, 1668-1669, 1996. 

Coaxal Circles 




Circles which share a Radical Line with a given cir- 
cle are said to be coaxal. The centers of coaxal circles 
are COLLINEAR. It is possible to combine the two types 
of coaxal systems illustrated above such that the sets 
are orthogonal. 

see also Circle, Coaxaloid System, Gauss- 
Bodenmiller Theorem, Radical Line 

References 

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. 

Washington, DC: Math. Assoc. Amer., pp. 35-36 and 122, 

1967. 
Dixon, R. Mathographics. New York: Dover, pp, 68-72, 1991. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 34-37, 199, and 279, 1929. 

Coaxal System 

A system of COAXAL CIRCLES. 

Coaxaloid System 

A system of circles obtained by multiplying each Radius 
in a Coaxal System by a constant. 

References 

Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 276-277, 1929. 

Cobordant Manifold 

Two open MANIFOLDS M and M' are cobordant if there 
exists a MANIFOLD with boundary W n+1 such that an 
acceptable restrictive relationship holds. 

see also COBORDISM, /i-COBORDISM THEOREM, MORSE 

Theory 

Cobordism 

see Bordism, /i-Cobordism 



274 Cobordism Group 



Code 



Cobordism Group 

see Bordism Group 

Cobordism Ring 

see Bordism Group 



with Inversion Center at the Origin and inversion 
radius k is the QuADRATRIX OF HlPPIAS. 



x = kt cot 
y = kt. 



(2) 
(3) 



Cochleoid 




The cochleoid, whose name means "snail-form" in Latin, 
was first discussed by J. Peck in 1700 (MacTutor Ar- 
chive). The points of contact of PARALLEL TANGENTS 
to the cochleoid lie on a Strophoid. 



In Polar Coordinates, 

asin# 

In Cartesian Coordinates, 

(x 2 + 2/ 2 )tan- 1 (|) 

The Curvature is 

_ 2y / 2l9 3 [2l9-sin(2fl)] 



ay. 



[1 + 20 2 - cos(2(9) - 2(9 sin(2<9)] 3 / 2 ' 



(1) 



(2) 



(3) 



see also QUADRATRIX OF HlPPIAS 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 192 and 196, 1972. 
MacTutor History of Mathematics Archive. "Cochleoid." 

http: //www-groups . dcs . st-and.ac . uk/ -history /Curves 

/Cochleoid. html. 



Cochleoid Inverse Curve 




The Inverse Curve of the Cochleoid 



Cochloid 

see Conchoid of Nicomedes 

Cochran's Theorem 

The converse of FISHER'S THEOREM. 

Cocked Hat Curve 




The Plane Curve 

(x 2 + 2ay - a 2 ) 2 = y 2 (a 2 - x 2 ), 

which is similar to the BlCORN. 

References 

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. 
Stradbroke, England: Tarquin Pub., p. 72, 1989. 



Cocktail Party Graph 




(i) 



A Graph consisting of two rows of paired nodes in which 
all nodes but the paired ones are connected with an 
Edge. It is the complement of the Ladder Graph. 

Coconut 

see Monkey and Coconut Problem 

Codazzi Equations 

see MAINARDI-CODAZZI EQUATIONS 

Code 

A code is a set of n-tuples of elements ("WORDS") taken 
from an ALPHABET. 

see also Alphabet, Coding Theory, Encoding, 
Error-Correcting Code, Gray Code, Huffman 
Coding, ISBN, Linear Code, Word 



Codimension 



Coin 



275 



Codimension 

The minimum number of parameters needed to fully de- 
scribe all possible behaviors near a nonstructurally sta- 
ble element. 
see also BIFURCATION 

Coding Theory 

Coding theory, sometimes called ALGEBRAIC CODING 
THEORY, deals with the design of ERROR-CORRECTING 
CODES for the reliable transmission of information 
across noisy channels. It makes use of classical and 
modern algebraic techniques involving Finite Fields, 
Group Theory, and polynomial algebra. It has con- 
nections with other areas of DISCRETE MATHEMATICS, 
especially Number Theory and the theory of experi- 
mental designs. 

see also Encoding, Error-Correcting Code, Ga- 
lois Field, Hadamard Matrix 

References 

Alexander, B. "At the Dawn of the Theory of Codes." Math. 
Intel 15, 20-26, 1993. 

Golomb, S. W.; Peile, R. E.; and Scholtz, R. A. Basic Con- 
cepts in Information Theory and Coding: The Adventures 
of Secret Agent 00111. New York: Plenum, 1994. 

Humphreys, O. F. and Prest, M. Y. Numbers, Groups, and 
Codes. New York: Cambridge University Press, 1990. 

MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- 
Correcting Codes. New York: Elsevier, 1978. 

Roman, S. Coding and Information Theory. New York: 
Springer- Verlag, 1992. 

Coefficient 

A multiplicative factor (usually indexed) such as one of 
the constants ai in the Polynomial a n x n + a n -i£ n_1 4- 
. . . + aix 2 -f- a\x + a . 

see also Binomial Coefficient, Cartan Tor- 
sion Coefficient, Central Binomial Coeffi- 
cient, Clebsch-Gordon Coefficient, Coeffi- 
cient Field, Commutation Coefficient, Con- 
nection Coefficient, Correlation Coefficient, 
Cross-Correlation Coefficient, Excess Coef- 
ficient, Gaussian Coefficient, Lagrangian Co- 
efficient, Multinomial Coefficient, Pearson's 
Skewness Coefficients, Product-Moment Co- 
efficient of Correlation, Quartile Skewness 
Coefficient, Quartile Variation Coefficient, 
Racah V-Coefficient, Racah ^-Coefficient, Re- 
gression Coefficient, Roman Coefficient, Tri- 
angle Coefficient, Undetermined Coefficients 
Method, Variation Coefficient 



Coercive Functional 

A bilinear FUNCTIONAL <j> on a normed SPACE E is called 
coercive (or sometimes Elliptic) if there exists a POS- 
ITIVE constant K such that 

<i>(x,x)>K\\x\\ 2 

for all x £ E. 

see also Lax-Milgram Theorem 

References 

Debnath, L. and Mikusinski, P. Introduction to Hilbert 

Spaces with Applications. San Diego, CA: Academic Press, 

1990. 

Cofactor 

The Minor of a Determinant is another Determi- 
nant |C| formed by omitting the ith row and jth col- 
umn of the original DETERMINANT |M|. 

dj = (-l) i+J 'oiM y . 
see also Determinant Expansion by Minors, Minor 

Cohen-Kung Theorem 

Guarantees that the trajectory of Langton's Ant is 
unbounded. 

Cohomology 

Cohomology is an invariant of a TOPOLOGICAL SPACE, 
formally "dual" to HOMOLOGY, and so it detects "holes'* 
in a SPACE. Cohomology has more algebraic structure 
than Homology, making it into a graded ring (multi- 
plication given by "cup product"), whereas HOMOLOGY 
is just a graded Abelian Group invariant of a Space. 

A generalized homology or cohomology theory must sat- 
isfy all of the Eilenberg-Steenrod Axioms with the 
exception of the dimension axiom. 

see also Aleksandrov-Cech Cohomology, Alexan- 
der-Spanier Cohomology, Cech Cohomology, de 
Rham Cohomology, Homology (Topology) 

Cohomotopy Group 

Cohomotopy groups are similar to HOMOTOPY GROUPS. 
A cohomotopy group is a Group related to the Homo- 
topy classes of Maps from a Space X into a Sphere 

see also HOMOTOPY GROUP 



Coefficient Field 

Let V be a Vector Space over a Field K, and let A be 
a nonempty Set. For an appropriately defined Affine 
Space A, K is called the Coefficient field. 



Coin 

A flat disk which acts as a two-sided Die. 

see Bernoulli Trial, Cards, Coin Paradox, Coin 

Tossing, Dice, Feller's Coin-Tossing Constants, 
Four Coins Problem, Gambler's Ruin 

References 

Brooke, M. Fun for the Money. New York: Scribner's, 1963. 



276 Coin Flipping 



Coin Tossing 



Coin Flipping 

see Coin Tossing 

Coin Paradox 




After a half rotation of the coin on the left around the 
central coin (of the same RADIUS), the coin undergoes 
a complete rotation. 

References 

Pappas, T. "The Coin Paradox." The Joy of Mathematics. 
San Carlos, CA: Wide World Publ./Tetra, p. 220, 1989. 

Coin Problem 

Let there be n > 2 INTEGERS < a\ < . . . < a n with 
(ai,a 2 ,...,a n ) = 1 (all Relatively Prime). For large 
enough N = X^-i a i x ii there is a solution in NoNNEG- 
ATIVE INTEGERS xi. The greatest N — g(ai,a 2 , ...a n ) 
for which there is no solution is called the coin problem. 
Sylvester showed 

g(ai,a 2 ) - {a\ - l)(o 3 - 1) - 1, 

and an explicit solution is known for n — 3, but no 
closed form solution is known for larger N. 

References 

Guy, R. K. "The Money- Changing Problem." §C7 in Un- 
solved Problems in Number Theory, 2nd ed. New York: 
Springer- Verlag, pp. 113-114, 1994. 

Coin Tossing 

An idealized coin consists of a circular disk of zero thick- 
ness which, when thrown in the air and allowed to fall, 
will rest with either side face up ("heads" H or "tails" T) 
with equal probability. A coin is therefore a two-sided 
Die. A coin toss corresponds to a Bernoulli Distri- 
bution with p = 1/2. Despite slight differences between 
the sides and NONZERO thickness of actual coins, the 
distribution of their tosses makes a good approximation 
to a p = 1/2 Bernoulli Distribution. 

There are, however, some rather counterintuitive prop- 
erties of coin tossing. For example, it is twice as likely 
that the triple TTH will be encountered before THT 
than after it, and three times as likely that THH will 
precede HTT. Furthermore, it is six times as likely that 
HTT will be the first of HTT, TTH, and TTT to oc- 
cur (Honsberger 1979). More amazingly still, spinning 
a penny instead of tossing it results in heads only about 
30% of the time (Paulos 1995). 

Let w(n) be the probability that no RUN of three consec- 
utive heads appears in n independent tosses of a Coin. 
The following table gives the first few values of w{n). 



n 


w(n) 





1 


1 


1 


2 


1 


3 


7 
8 


4 


13 




16 


5 


3 

4 



Feller (1968, pp. 278-279) proved that 



lim w(n)a 



n + l 



■0, 



(1) 



vhere 



a = f [(136 + 24v / 33) 1/3 - 8(136 + 24v / 33)~ 1/3 - 2] 



- 1.087378025. 



and 



= ^ — — = 1.236839845 . . 
4 — 3a 



(2) 



(3) 



The corresponding constants for a RUN of k > 1 heads 
are a*, the smallest Positive Root of 



and 



i -* + (!*) 



k = 



k + 1 



o, 



k + 1 — kak 



(4) 



(5) 



These arc modified for unfair coins with P(H) = p and 
P(T) = q = 1 - p to a' k , the smallest Positive Root 
of 

l-z + <2pV +1 -0, (6) 



and 



& = 



P&k 



(7) 



(k + 1 -ka' k )p 
(Feller 1968, pp. 322-325). 

see also BERNOULLI DISTRIBUTION, CARDS, COIN, 

Dice, Gambler's Ruin, Martingale, Run, Saint 
Petersburg Paradox 

References 

Feller, W. An Introduction to Probability Theory and Its Ap- 
plication, Vol. 1, 3rd ed. New York: Wiley, 1968. 

Finch, S. u Favorite Mathematical Constants.' 1 http://www. 
mathsoft.com/asolve/constant/feller/feller.htnil. 

Ford, J. "How Random is a Coin Toss?" Physics Today 36, 
40-47, 1983. 

Honsberger, R. "Some Surprises in Probability." Ch. 5 in 
Mathematical Plums (Ed. R. Honsberger). Washington, 
DC: Math. Assoc. Amer., pp. 100-103, 1979. 

Keller, J. B. "The Probability of Heads." Amer. Math. 
Monthly 93, 191-197, 1986. 

Paulos, J. A. A Mathematician Reads the Newspaper. New 
York: BasicBooks, p. 75, 1995. 

Peterson, I. Islands of Truth: A Mathematical Mystery 
Cruise. New York: W. H. Freeman, pp. 238-239, 1990. 

Spencer, J. "Combinatorics by Coin Flipping." Coll. Math. 
J., 17, 407-412, 1986. 



Coincidence 



Collatz Problem 



277 



Coincidence 

A coincidence is a surprising concurrence of events, per- 
ceived as meaningfully related, with no apparent causal 
connection (Diaconis and Mosteller 1989). 

see also Birthday Problem, Law of Truly Large 
Numbers, Odds, Probability, Random Number 

References 

Bogomolny, A. "Coincidence." http://www.cut— the-knot . 
com/ do_you_know/coincidence. html. 

Falk, R. "On Coincidences." Skeptical Inquirer 6, 18—31, 
1981-82. 

Falk, R. "The Judgment of Coincidences: Mine Versus 
Yours." Amer. J. Psych. 102, 477-493, 1989. 

Falk, R. and MacGregor, D. "The Surprisingness of Coinci- 
dences." In Analysing and Aiding Decision Processes (Ed. 
P. Humphreys, O. Svenson, and A. Vari). New York: El- 
sevier, pp. 489-502, 1984. 

Diaconis, P. and Mosteller, F. "Methods of Studying Coinci- 
dences." J. Amer. Statist. Assoc. 84, 853-861, 1989. 

Jung, C. G. Synchronicity: An Acausal Connecting Princi- 
ple. Princeton, NJ: Princeton University Press, 1973. 

Kammerer, P. Das Gesetz der Serie: Eine Lehre von 
den Wiederholungen im Lebens — und im Weltgeschehen. 
Stuttgart, Germany: Deutsche Verlags-Anstahlt, 1919. 

Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96, 
June 1998. 

Colatitude 

The polar angle on a SPHERE measured from the North 
Pole instead of the equator. The angle <j> in SPHERICAL 

Coordinates is the Colatitude. It is related to the 
Latitude 5 by <p = 90° - S. 

see also LATITUDE, LONGITUDE, SPHERICAL COORDI- 
NATES 

Colinear 

see COLLINEAR 

Collatz Problem 

A problem posed by L. Collatz in 1937, also called the 
3x + 1 Mapping, Hasse's Algorithm, Kakutani's 
Problem, Syracuse Algorithm, Syracuse Prob- 
lem, Thwaites Conjecture, and Ulam's Problem 
(Lagarias 1985). Thwaites (1996) has offered a £1000 
reward for resolving the Conjecture. Let n be an In- 
teger. Then the Collatz problem asks if iterating 



fin) 



i 1 
I 3 



3n+l 



for n even 
for n odd 



(i) 



always returns to 1 for POSITIVE n. This question 
has been tested and found to be true for all numbers 
< 5.6 x 10 13 (Leavens and Vermeulen 1992), and more 
recently, 10 15 (Vardi 1991, p. 129). The members of 
the SEQUENCE produced by the Collatz are sometimes 
known as Hailstone NUMBERS. Because of the dif- 
ficulty in solving this problem, Erdos commented that 
"mathematics is not yet ready for such problems" (La- 
garias 1985). If NEGATIVE numbers are included, there 
are four known cycles (excluding the trivial cycle): (4, 



2, 1), (-2, -1), (-5, -7, -10), and (-17, -25, -37, 
-55, -82, -41, -61, -91, -136, -68, -34). The num- 
ber of tripling steps needed to reach 1 for n = 1, 2, ... 
are 0, 0, 2, 0, 1, 2, 5, 0, 6, . . . (Sloane's A006667). 

The Collatz problem was modified by Terras (1976, 
1979), who asked if iterating 



T(x) 



-{I 



X 

(Sx + 1) 



for x even 
for x odd 



(2) 



always returns to 1. If NEGATIVE numbers are included, 
there are 4 known cycles: (1, 2), (-1), (-5, -7, -10), 
and (-17, -25, -37, -55, -82, -41, -61, -91, -136, 
—68, —34). It is a special case of the "generalized Collatz 
problem" with d = 2, mo = 1, mi = 3, ro — 0, and 
ri = -1. Terras (1976, 1979) also proved that the set 
of Integers Sk = {n : n has stopping time < k} has a 
limiting asymptotic density F(h), so the limit 



F(k)= lim -, 

a:-»oo X 



(3) 



for {n : n < x and cr(n) < k} exists. Furthermore, 
F(k) — >- 1 as k -4 oo, so almost all INTEGERS have a 
finite stopping time. Finally, for all k > 1, 



1 - F(k) 



lim - < 2 

£->00 X 



-T]k 



where 



7] = 1-H(0) = 0.05004... 
H (x) = —x lg x — (1 — x) lg(l — x) 

"Si 



(4) 



(5) 
(6) 

(7) 



(Lagarias 1985). 

Conway proved that the original Collatz problem has 
no nontrivial cycles of length < 400. Lagarias (1985) 
showed that there are no nontrivial cycles with length 
< 275,000. Conway (1972) also proved that Collatz- 
type problems can be formally Undecidable. 

A generalization of the COLLATZ PROBLEM lets d > 2 be 
a Positive Integer and mo, . . . , md-i be Nonzero 
Integers. Also let r»eZ satisfy 



n = irfii (mod d) . 



Then 



T(x) = 



mix — Ti 



(8) 



(9) 



for x = i (mod d) defines a generalized Collatz mapping. 
An equivalent form is 



w-L'r 



+ x t 



(10) 



278 



Collatz Problem 



Collineation 



for x = i (mod d) where Xo, . . . , Xd-\ are INTEGERS 
and [r\ is the FLOOR FUNCTION. The problem is con- 
nected with Ergodic Theory and Markov Chains 
(Matthews 1995). Matthews (1995) obtained the fol- 
lowing table for the mapping 



Tk(x) 



\i(3x 



for x = (mod 2) 
+ k) for x = 1 (mod 2), 



(11) 



where k = T*\ 



k 


# Cycles 


Max. 


Cycle Length 





5 




27 


1 


10 




34 


2 


13 




118 


3 


17 




118 


4 


19 




118 


5 


21 




165 


6 


23 




433 



Matthews and Watts (1984) proposed the following con- 
jectures. 

1. If | mo ■ ■ -rrid-il < d d , then all trajectories {T K (n)} 
for n € Z eventually cycle. 

2. If |mo---md-i| > <2 d , then almost all trajectories 
{T K (n)} for n € Z are divergent, except for an ex- 
ceptional set of Integers n satisfying 

#{n £S\-X<n<X} = o(X). 

3. The number of cycles is finite. 

4. If the trajectory {T K (n)} for n 6 Z is not eventually 
cyclic, then the iterates are uniformly distribution 
mod d a for each a > 1, with 



1 



lim 

iv^oo AT+ 1 



card{if < N\T K (n) = j (mod d a )} 



(12) 



for < j < d a - 1. 
Matthews believes that the map 



T(x) 



"{i 

v 3 



7a: + 3 

(7a: + 2) 
3^-2) 



for x = (mod 3) 
for x = 1 (mod 3) 
for x = 2 (mod 3) 



(13) 



will either reach (mod 3) or will enter one of the cycles 
( — 1) or (-2,-4), and offers a $100 (Australian?) prize 
for a proof. 

see also HAILSTONE Number 

References 

Applegate, D. and Lagarias, J. C. "Density Bounds for the 

3z + 1 Problem 1. Tree-Search Method." Math. Comput 

64, 411-426, 1995. 
Applegate, D. and Lagarias, J. C. "Density Bounds for the 

Sx + 1 Problem 2. Krasikov Inequalities." Math. Comput. 

64, 427-438, 1995. 



Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. 

Cambridge, MA: MIT Artificial Intelligence Laboratory, 

Memo AIM-239, Feb. 1972. 
Burckel, S. "Functional Equations Associated with Congru- 

ential Functions." Theor. Comp. Set. 123, 397-406, 1994. 
Conway, J. H. "Unpredictable Iterations." Proc. 1972 Num- 
ber Th. Conf., University of Colorado, Boulder, Colorado, 

pp. 49-52, 1972. 
Crandall, R. "On the ( 3z + 1' Problem." Math. Comput 32, 

1281-1292, 1978. 
Everett, C. "Iteration of the Number Theoretic Function 

f(2n) = n, f(2n + 1) = f(3n + 2)." Adv. Math. 25, 

42-45, 1977. 
Guy, R. K. "Collatz's Sequence." §E16 in Unsolved Problems 

in Number Theory, 2nd ed. New York: Springer- Verlag, 

pp. 215-218, 1994. 
Lagarias, J. C. "The 3x + l Problem and Its Generalizations." 

Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm, 

sfu. ca/organics/papers/lagarias/. 
Leavens, G. T. and Vermeulen, M. "3x + l Search Programs." 

Comput. Math. Appl. 24, 79-99, 1992. 
Matthews, K. R. "The Generalized 3x+l Mapping." http:// 

www.maths.uq.oz.au/-krm/survey.dvi. Rev. Sept. 10, 

1995. 
Matthews, K. R. "A Generalized 3z + 1 Conjecture." [$100 

Reward for a Proof.] ftp://www.maths.uq.edu.au/pub/ 

krm/gnubc/challenge. 
Matthews, K. R. and Watts, A. M. "A Generalization of 

Hasses's Generalization of the Syracuse Algorithm." Acta 

Arith. 43, 167-175, 1984. 
Sloane, N. J. A. Sequence A006667/M0019 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 
Terras, R. "A Stopping Time Problem on the Positive Inte- 
gers." Acta Arith. 30, 241-252, 1976. 
Terras, R. "On the Existence of a Density." Acta Arith. 35, 

101-102, 1979. 
Thwaites, B. "Two Conjectures, or How to win £1100." 

Math.Gaz. 80, 35-36, 1996. 
Vardi, I. "The 3# + 1 Problem." Ch. 7 in Computational 

Recreations in Mathematica. Redwood City, CA: Addison- 

Wesley, pp. 129-137, 1991. 

Collinear 




Three or more points Pi, P2, P3, . .., are said to be 
collinear if they lie on a single straight LINE L. (Two 
points are always collinear.) This will be true IFF the 
ratios of distances satisfy 

X2 - xi : y 2 - yi : Z2 - zi = x 3 - xi : y 3 — yi : zs - zi. 

Two points are trivially collinear since two points deter- 
mine a Line. 

see also Concyclic, Directed Angle, N-Cluster, 

Sylvester's Line Problem 

Collineation 

A transformation of the plane which transforms COL- 
LINEAR points into COLLINEAR points. A projective 
collineation transforms every 1-D form projectively, and 
a perspective collineation is a collineation which leaves 
all lines through a point and points through a line invari- 
ant. In an ELATION, the center and axis are incident; in 



Cologarithm 



Combination 279 



a HOMOLOGY they are not. For further discussion, see 
Coxeter (1969, p. 248). 

see also Affinity, Correlation, Elation, Equi- 
affinity, Homology (Geometry), Perspective 
Collineation, Projective Collineation 

References 

Coxeter, H. S. M. "Collineations and Correlations." §14.6 
in Introduction to Geometry, 2nd ed. New York: Wiley, 
pp. 247-251, 1969. 

Cologarithm 

The Logarithm of the Reciprocal of a number, equal 
to the Negative of the Logarithm of the number it- 
self, 

colog x = log ( — J — — log x. 

see also Antilogarithm, Logarithm 

Colon Product 

Let AB and CD be Dyads. Their colon product is 
defined by 

AB : CD = C AB D = (A C)(B D). 



Colorable 

Color each segment of a KNOT DIAGRAM using one of 
three colors. If 

1. at any crossing, either the colors are all different or 
all the same, and 

2. at least two colors are used, 

then a KNOT is said to be colorable (or more specif- 
ically, Three- Colorable). Color ability is invariant 
under REIDEMEISTER Moves, and can be generalized. 
For instance, for five colors 0, 1, 2, 3, and 4, a KNOT is 
five-colorable if 

1. at any crossing, three segments meet. If the overpass 
is numbered a and the two underpasses B and C, 
then 2a = b -f c (mod 5), and 

2. at least two colors are used. 

Colorability cannot alway distinguish HANDEDNESS. 
For instance, three-colorability can distinguish the mir- 
ror images of the TREFOIL KNOT but not the FlGURE- 
OF-ElGHT KNOT. Five-colorability, on the other hand, 
distinguishes the MIRROR Images of the FlGURE-OF- 
Eight Knot but not the Trefoil Knot. 

see also Coloring, Three-Colorable 

Coloring 

A coloring of plane regions, Link segments, etc., is an 
assignment of a distinct labelling (which could be a 
number, letter, color, etc.) to each component. Col- 
oring problems generally involve TOPOLOGICAL consid- 
erations (i.e., they depend on the abstract study of the 
arrangement of objects), and theorems about colorings, 



such as the famous Four-Color THEOREM, can be ex- 
tremely difficult to prove. 
see also COLORABLE, EDGE-COLORING, FOUR-COLOR 

Theorem, ^-Coloring, Polyhedron Coloring, 

Six-Color Theorem, Three-Colorable, Vertex 
Coloring 

References 

Eppstein, D. "Coloring," http://vvv . ics . uci . edu / - 

eppstein/ junkyard/color. html. 
Saaty, T. L. and Kainen, P. C The Four-Color Problem: 

Assaults and Conquest. New York: Dover, 1986. 

Columbian Number 

see Self Number 

Colunar Triangle 

Given a SCHWARZ TRIANGLE (p q r), replacing each 
Vertex with its antipodes gives the three colunar 
Spherical Triangles 



(p q r'),(p q r f ),(p q r), 



where 



P P 

q q' 

r r 
see also SCHWARZ TRIANGLE, SPHERICAL TRIANGLE 

References 

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: 
Dover, p. 112, 1973. 



Comb Function 
see Shah Function 

Combination 

The number of ways of picking r unordered outcomes 
from n possibilities. Also known as the Binomial Co- 
efficient or Choice Number and read "n choose r." 



t.Ct. = 



rl(n - 



where n\ is a FACTORIAL. 

see also Binomial Coefficient, Derangement, Fac- 
torial, Permutation, Subfactorial 

References 

Conway, J. H. and Guy, R. K. "Choice Numbers." In The 

Book of Numbers. New York: Springer- Verlag, pp. 67-68, 

1996. 
Ruskey, F. "Information on Combinations of a Set." 

http://sue . esc . uvic . ca/~cos/inf /comb/Combinations 

Info.html. 



280 



Combination Lock 



Combinatorics 



Combination Lock 

Let a combination of n buttons be a SEQUENCE of dis- 
joint nonempty Subsets of the Set {1, 2, . . . , n}. If 
the number of possible combinations is denoted a n , then 
a n satisfies the RECURRENCE RELATION 



i— n \ / 



with ao = 1. This can also be written 






2 / , 2 k ' 
k=0 



(1) 



(2) 



where the definition 0° = 1 has been used. Furthermore, 



a n = 2^i n ,fe2 n = ^^^4n,fc2 , 



(3) 



fc = l 



where A n ,k are EULERIAN NUMBERS. In terms of the 

Stirling Numbers of the Second Kind s{n,k), 



a n = \, k\s(n,k). 

k = l 



a n can also be given in closed form as 
a n — 2 Ll -n(2)> 



(4) 



(5) 



where Li n (z) is the POLYLOGARITHM. The first few 
values of a n for n = 1, 2, ... are 1, 3, 13, 75, 541, 
4683, 47293, 545835, 7087261, 102247563, ... (Sloane's 
A000670). 

The quantity 



b n = 



satisfies the inequality 

1 



2(ln2) n 



<b n < 



(ln2) n * 



(6) 



(7) 



References 

Sloane, N. J. A. Sequence A000670/M2952 in "An On-Line 
Version of the Encyclopedia of Integer Sequences." 

Velleman, D. J. and Call, G. S. "Permutations and Combi- 
nation Locks." Math. Mag. 68, 243-253, 1995. 

Combinatorial Species 

see Species 

Combinatorial Topology 

Combinatorial topology is a special type of Algebraic 
Topology that uses Combinatorial methods. For 
example, Simplicial Homology is a combinatorial 
construction in ALGEBRAIC TOPOLOGY, so it belongs 
to combinatorial topology. 

see also ALGEBRAIC TOPOLOGY, SlMPLICIAL HOMO- 
LOGY, Topology 



Combinatorics 

The branch of mathematics studying the enumeration, 
combination, and permutation of sets of elements and 
the mathematical relations which characterize these 
properties. 

see also Antichain, Chain, Dilworth's Lemma, 
Diversity Condition, Erdos-Szekeres Theo- 
rem, Inclusion-Exclusion Principle, Kirkman's 
Schoolgirl Problem, Kirkman Triple System, 
Length (Partial Order), Partial Order, Pigeon- 
hole Principle, Ramsey's Theorem, Schroder- 
Bernstein Theorem, Schur's Lemma, Sperner's 
Theorem, Total Order, van der Waerden's The- 
orem, Width (Partial Order) 

References 

Abramowitz, M. and Stegun, C A. (Eds.). "Combinatorial 
Analysis." Ch. 24 in Handbook of Mathematical Func- 
tions with Formulas, Graphs, and Mathematical Tables, 
9th printing. New York: Dover, pp. 821-8827, 1972. 

Aigner, M. Combinatorial Theory. New York: Springer- 
Verlag, 1997. 

Bellman, R. and Hall, M. Combinatorial Analysis. Amer. 
Math. Soc, 1979. 

Biggs, N. L. "The Roots of Combinatorics." Historia Math- 
ematica 6, 109-136, 1979. 

Bose, R. C. and Manvel, B. Introduction to Combinatorial 
Theory. New York: Wiley, 1984. 

Brown, K. S. "Combinatorics." http://www.seanet.com/ 
-ksbrown/icombina.htm. 

Cameron, P. J. Combinatorics: Topics, Techniques, Algo- 
rithms. New York: Cambridge University Press, 1994. 

Cohen, D. Basic Techniques of Combinatorial Theory. New 
York: Wiley, 1978. 

Cohen, D. E. Combinatorial Group Theory: A Topological 
Approach. New York: Cambridge University Press, 1989. 

Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- 
natorial Designs. Boca Raton, FL: CRC Press, 1996. 

Comtet, L. Advanced Combinatorics. Dordrecht, Nether- 
lands: Reidel, 1974. 

Coolsaet, K. "Index of Combinatorial Objects." http://www. 
hogent.be/~kc/ico/. 

Dinitz, J. H. and Stinson, D. R. (Eds.). Contemporary De- 
sign Theory: A Collection of Surveys. New York: Wiley, 
1992. 

Electronic Journal of Combinatorics. http : //www . 

combinatorics.org/previousjvolumes.html. 

Eppstein, D. "Combinatorial Geometry." http://www.ics. 
uci.edu/-eppstein/junkyard/combinatorial.html. 

Erickson, M. J. Introduction to Combinatorics. New York: 
Wiley, 1996. 

Fields, J. "On-Line Dictionary of Combinatorics." http:// 
math.uic.edu/-fields/dic/. 

Godsil, C. D. "Problems in Algebraic Combinatorics." Elec- 
tronic J. Combinatorics 2, Fl, 1-20, 1995. http: //www. 
combinatorics. org/Volume_2/volume2.html#Fl. 

Graham, R. L.; Grotschel, M.; and Lovasz, L. (Eds.). Hand- 
book of Combinatorics, 2 vols. Cambridge, MA: MIT 
Press, 1996. 

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete 
Mathematics: A Foundation for Computer Science, 2nd 
ed. Reading, MA: Add is on- Wesley, 1994. 

Hall, M. Jr. Combinatorial Theory, 2nd ed. New York: Wi- 
ley, 1986. 

Knuth, D. E. (Ed.). Stable Marriage and Its Relation to 
Other Combinatorial Problems. Providence, RI: Amer. 
Math. Soc, 1997. 



Comma Derivative 



Commutation Coefficient 281 



Kucera, L. Combinatorial Algorithms. Bristol, England: 
Adam Hilger, 1989. 

Liu, C. L. Introduction to Combinatorial Mathematics. New- 
York: McGraw-Hill, 1968. 

MacMahon, P. A. Combinatory Analysis. New York: 
Chelsea, 1960. 

Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for 
Computers and Calculators, 2nd ed. New York: Academic 
Press, 1978. 

Riordan, J. Combinatorial Identities, reprint ed. with correc- 
tions. Huntington, NY: Krieger, 1979. 

Riordan, J. An Introduction to Combinatorial Analysis. New 
York: Wiley, 1980. 

Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: 
Prentice-Hall, 1984. 

Rota, G.-C. (Ed.). Studies in Combinatorics. Providence, 
RI: Math. Assoc. Amer., 1978. 

Ruskey, F. "The (Combinatorial) Object Server." http:// 
sue.csc.uvic.ca/-cos. 

Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: 
Math. Assoc. Amer., 1963. 

Skiena, S. S. Implementing Discrete Mathematics: Combi- 
natorics and Graph Theory with Mathematica. Reading, 
MA: Addison- Wesley, 1990. 

Sloane, N. J. A. "An On-Line Version of the Encyclopedia 
of Integer Sequences." http://www.research.att.com/ 
-njas/sequences/eisonline.html. 

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer 
Sequences. San Diego, CA: Academic Press, 1995. 

Street, A. P. and Wallis, W. D. Combinatorial Theory: An 
Introduction. Winnipeg, Manitoba: Charles Babbage Re- 
search Center, 1977. 

Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley, 
1995. 

van Lint, J. H. and Wilson, R. M. A Course in Combina- 
torics. New York: Cambridge University Press, 1992. 

Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- 
phia, PA: SIAM, 1989. 

Comma Derivative 






9k dx k 

see also COVARIANT DERIVATIVE, SEMICOLON DERIV- 
ATIVE 

Comma of Didymus 

The musical interval by which four fifths exceed a sev- 
enteenth (i.e., two octaves and a major third), 



(I) 

2 2(|) 2^.5 



81 

80 



1.0125, 



also called a Syntonic Comma. 

see also COMMA OF PYTHAGORAS, DlESIS, SCHISMA 



Comma of Pythagoras 

The musical interval by which twelve fifths exceed seven 
octaves, 



ill 
2 7 



3^ 

2 19 



531441 

524288 



1.013643265. 



Successive CONTINUED FRACTION CONVERGENTS to 
log 2/ log (3/2) give increasingly close approximations 
m/n of m fifths by n octaves as 1, 2, 5/3, 12/7, 41/24, 
53/31, 306/179, 665/389, ... (Sloane's A005664 and 
A046102; Jeans 1968, p. 188), shown in bold in the ta- 
ble below. All near-equalities of m fifths and n octaves 
having 



R. 



(§r 



2^ Om+n 

with \R — 1| < 0.02 are given in the following table. 



m 


n 


Ratio 


m 


n 


Ratio 


12 


7 


1.013643265 


265 


155 


1.010495356 


41 


24 


0.9886025477 


294 


172 


0.9855324037 


53 


31 


1.002090314 


306 


179 


0.9989782832 


65 


38 


1.015762098 


318 


186 


1.012607608 


94 


55 


0.9906690375 


347 


203 


0.9875924759 


106 


62 


1.004184997 


359 


210 


1.001066462 


118 


69 


1.017885359 


371 


217 


1.014724276 


147 


86 


0.9927398469 


400 


234 


0.9896568543 


159 


93 


1.006284059 


412 


241 


1.003159005 


188 


110 


0.9814251419 


424 


248 


1.016845369 


200 


117 


0.994814985 


453 


265 


0.9917255479 


212 


124 


1.008387509 


465 


272 


1.005255922 


241 


141 


0.9834766286 


477 


279 


1.018970895 


253 


148 


0.9968944607 


494 


289 


0.9804224033 



see also COMMA OF DlDYMUS, DlESIS, SCHISMA 

References 

Conway, J. H. and Guy, R. K. The Book of Numbers. New 
York: Springer- Verlag, p. 257, 1995. 

Guy, R. K. "Small Differences Between Powers of 2 and 3." 
§F23 in Unsolved Problems in Number Theory, 2nd ed. 
New York: Springer- Verlag, p. 261, 1994. 

Sloane, N. J. A. Sequences A005664 and A046102 in "An On- 
Line Version of the Encyclopedia of Integer Sequences." 

Common Cycloid 

see Cycloid 

Common Residue 

The value of fr, where a = b (mod m), taken to be NON- 
NEGATIVE and smaller than m. 

see also Minimal Residue, Residue (Congruence) 

Commutation Coefficient 

A coefficient which gives the difference between partial 
derivatives of two coordinates with respect to the other 
coordinate, 

c ap^ — [^cn^a] = V^e/3 - V^e a . 

see also CONNECTION COEFFICIENT 



282 



Commutative 



Compactness Theorem 



Commutative 

Let A denote an M- algebra, so that A is a VECTOR 
Space over R and 

A x A ->■ A 

(x,y) M- x-y. 

Now define 

Z = {x e a ; x • y foi some y 6 A / 0}, 

where € Z. An ASSOCIATIVE R-algebra is commuta- 
tive if x • y = y * x for all x, y € A. Similarly, a Ring is 
commutative if the MULTIPLICATION operation is com- 
mutative, and a LIE ALGEBRA is commutative if the 
Commutator [A, B] is for every A and B in the LIE 
Algebra. 
see also Abelian, Associative, Transitive 

References 

Finch, S. "Zero Structures in Real Algebras." http://www. 
mathsoft.com/asolve/zerodiv/zerodiv.html. 

MacDonald, I. G. and Atiyah, M. F. Introduction to Com- 
mutative Algebra. Reading, MA: Addison- Wesley, 1969. 

Commutative Algebra 

An Algebra in which the + operators and x are Com- 
mutative. 
see also Algebraic Geometry, Grobner Basis 

References 

MacDonald, I. G. and Atiyah, M. F. Introduction to Com- 
mutative Algebra. Reading, MA: Addison-Wesley, 1969. 

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and 
Algorithms: An Introduction to Algebraic Geometry and 
Commutative Algebra, 2nd ed. New York: Springer- 
Verlag, 1996. 

Samuel, P. and Zariski, O, Commutative Algebra, Vol. 2. 
New York: Springer- Verlag, 1997. 

Commutator 

Let A, £, . . .be Operators. Then the commutator of 
A and B is defined as 

[A,B] = AB-BA. (1) 

Let a, 6, ... be constants. Identities include 

[/(*),*] = (2) 

[A,A]=0 (3) 

[A,B] = -[B,A] (4) 

[A,BC] = [A,B]C + B[A,C] (5) 

[AB, C] = [A, C]B + A[B, C] (6) 

[a + A,b + B] = [A,B] (7) 

[A + B,C + D} = [A,C] + [A,D] + [B,C] + [B,D]. 

(8) 

The commutator can be interpreted as the "infinitesi- 
mal" of the commutator of a Lie Group. 

Let A and B be Tensors. Then 

[A,B]=X? a B-VbA. (9) 

see also Anticommutator, Jacobi Identities 



Compact Group 

If the parameters of a LIE GROUP vary over a CLOSED 
Interval, the GROUP is compact. Every representation 
of a compact group is equivalent to a UNITARY repre- 
sentation. 

Compact Manifold 

A Manifold which can be "charted" with finitely many 

Euclidean Space charts. The Circle is the only com- 
pact l-D Manifold. The Sphere and n-ToRUS are 
the only compact 2-D MANIFOLDS. It is an open ques- 
tion if the known compact MANIFOLDS in 3-D are com- 
plete, and it is not even known what a complete list in 
4-D should look like. The following terse table there- 
fore summarizes current knowledge about the number 
of compact manifolds N(D) of D dimensions. 

D N(D) 



see also Tychonof Compactness Theorem 

Compact Set 

The Set S is compact if, from any Sequence of ele- 
ments Xi, X 2y ...of S, a subsequence can always be 
extracted which tends to some limit element X of S. 
Compact sets are therefore closed and bounded. 

Compact Space 

A Topological Space is compact if every open cover 

of X has a finite subcover. In other words, if X is the 
union of a family of open sets, there is a finite subfamily 
whose union is X. A subset A of a Topological Space 
X is compact if it is compact as a TOPOLOGICAL Space 
with the relative topology (i.e., every family of open 
sets of X whose union contains A has a finite subfamily 
whose union contains A). 

Compact Surface 

A surface with a finite number of TRIANGLES in its TRI- 
angulation. The Sphere and TORUS are compact, 
but the PLANE and TORUS minus a Disk are not. 

Compactness Theorem 

Inside a Ball B in R 3 , 

{rectifiable currents 5 in BL Area S < c, 

length dS < c} 

is compact under the Flat Norm. 

References 

Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103, 
369-376, 1996. 



Companion Knot 



Complete Axiomatic Theory 283 



Companion Knot 

Let Ki be a knot inside a TORUS. Now knot the TORUS 
in the shape of a second knot (called the companion 
knot) K2. Then the new knot resulting from K\ is called 
the Satellite Knot K 3 . 

References 

Adams, C. C. The Knot Book: An Elementary Introduction 
to the Mathematical Theory of Knots, New York: W. H. 
Freeman, pp. 115-118, 1994. 

Comparability Graph 

The comparability graph of a POSET P = (X, <) is the 
Graph with vertex set X for which vertices x and y are 
adjacent IFF either x < y or y < x in P. 

see also INTERVAL GRAPH, PARTIALLY ORDERED SET 

Comparison Test 

Let J2 ak and J2^ k be a Series with Positive terms 
and suppose a\ < &i, 02 < ta, 

1. If the bigger series CONVERGES, then the smaller 
series also Converges. 

2. If the smaller series DIVERGES, then the bigger series 
also Diverges. 

see also Convergence Tests 

References 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 280-281, 1985. 

Compass 

A tool with two arms joined at their ends which can 
be used to draw Circles. In Geometric Construc- 
tions, the classical Greek rules stipulate that the com- 
pass cannot be used to mark off distances, so it must 
"collapse" whenever one of its arms is removed from 
the page. This results in significant complication in the 
complexity of GEOMETRIC CONSTRUCTIONS, 

see also Constructible Polygon, Geometric Con- 
struction, Geometrography, Mascheroni Con- 
struction, Plane Geometry, Polygon, Poncelet- 
Steiner Theorem, Ruler, Simplicity, Steiner 
Construction, Straightedge 

References 

Dixon, R. "Compass Drawings." Ch. 1 in Mathographics. 
New York: Dover, pp. 1-78, 1991. 

Compatible 

Let 1 1 A 1 1 be the MATRIX NORM associated with the MA- 
TRIX A and ||x|| be the Vector Norm associated with 
a Vector x. Let the product Ax be defined, then ||A|| 
and ||x|| are said to be compatible if 

l|Ax||<||A||||x||. 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1115, 1980. 



Complement Graph 

The complement Graph G of G has the same Vertices 
as G but contains precisely those two-element SUBSETS 
which are not in G. 

Complement Knot 

see Knot Complement 

Complement Set 

Given a set S with a subset F, the complement of E is 
defined as 

E' = {F:FeS,F^E}. (1) 



If E = 5, then 



E' = S' = 0, 



(2) 



where is the EMPTY SET. Given a single Set, the 
second Probability Axiom gives 



1 = P(S) = P(EUE'). 

Using the fact that E n E f = 0, 

1 = P(E) + P(E') 
P(E') = 1-P(E). 



(3) 

(4) 
(5) 



This demonstrates that 

P(S') = P{0) = 1 - P(S) = 1-1-0. (6) 

Given two Sets, 

P(E O F') = P(E) - P(E F) (7) 

P(E' r\F f ) = l- P(E) - P{F) + P(E O F). (8) 



Complementary Angle 

Two ANGLES a and 7r/2 - a are said to be complemen- 
tary. 

see also ANGLE, SUPPLEMENTARY ANGLE 

Complete 

see Complete Axiomatic Theory, Complete Bi- 
graph, Complete Functions, Complete Graph, 
Complete Quadrangle, Complete Quadrilat- 
eral, Complete Sequence, Complete Space, 
Completeness Property, Weakly Complete Se- 
quence 

Complete Axiomatic Theory 

An axiomatic theory (such as a Geometry) is said to be 
complete if each valid statement in the theory is capable 
of being proven true or false. 

see also CONSISTENCY 



284 Complete Bigraph 



Complete Graph 



Complete Bigraph 

see Complete Bipartite Graph 

Complete Bipartite Graph 



Complete Graph 





A Bipartite Graph (i.e., a set of Vertices decom- 
posed into two disjoint sets such that there are no two 
VERTICES within the same set are adjacent) such that 
every pair of VERTICES in the two sets are adjacent. If 
there are p and q VERTICES in the two sets, the complete 
bipartite graph (sometimes also called a COMPLETE Bl- 
GRAPH) is denoted K p , q . The above figures show K^^ 
and i^2,5* 

see also Bipartite Graph, Complete Graph, 
Complete ^-Partite Graph, ^-Partite Graph, 
Thomassen Graph, Utility Graph 

References 

Saaty, T. L. and Kainen, P. C. The Four-Color Problem; 
Assaults and Conquest. New York: Dover, p. 12, 1986. 

Complete Functions 

A set of Orthonormal Functions </> n (x) is termed 
complete in the CLOSED INTERVAL x € [a, b] if, for every 
piecewise CONTINUOUS Function f(x) in the interval, 
the minimum square error 

E n = ||/-(ci0i + ... + c n n )|| 2 

(where || denotes the Norm) converges to zero as n be- 
comes infinite. Symbolically, a set of functions is com- 
plete if 



lim 

771— »-00 



f 



f(x) - y^an4>n(x) 



n=Q 



w(x) dx — 0, 



where w(x) is a Weighting Function and the above 
is a Lebesgue Integral. 

see also BESSEL'S INEQUALITY, HlLBERT SPACE 

References 

Arfken, G. "Completeness of Eigenfunctions." §9.4 in Mathe- 
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca- 
demic Press, pp. 523-538, 1985. 




A Graph in which each pair of VERTICES is connected 
by an EDGE. The complete graph with n VERTICES is 
denoted K n . In older literature, complete GRAPHS are 
called UNIVERSAL GRAPHS. 

K 4 is the Tetrahedral Graph and is therefore a PLA- 
NAR GRAPH. K$ is nonplanar. Conway and Gordon 
(1983) proved that every embedding of K G is INTRINSI- 
CALLY Linked with at least one pair of linked triangles. 
They also showed that any embedding of Kj contains a 
knotted Hamiltonian Cycle. 

The number of Edges in K v is v(v — l)/2, and the 
Genus is (v — 3)(v — 4)/12 for v > 3. The number of dis- 
tinct variations for K n (GRAPHS which cannot be trans- 
formed into each other without passing nodes through 
an EDGE or another node) for n — 1, 2, . . . are 1, 1, 1, 

1, 1, 1, 6, 3, 411, 37, The Adjacency Matrix of 

the complete graph takes the particularly simple form 
of all Is with Os on the diagonal. 

It is not known in general if a set of Trees with 1,2,..., 
n — 1 Edges can always be packed into K n . However, 
if the choice of TREES is restricted to either the path or 
star from each family, then the packing can always be 
done (Zaks and Liu 1977, Honsberger 1985). 

References 

Chartrand, G. Introductory Graph Theory. New York: 
Dover, pp. 29-30, 1985. 

Conway, J. H. and Gordon, C. M. "Knots and Links in Spatial 
Graphs." J. Graph Th. 7, 445-453, 1983. 

Honsberger, R. Mathematical Gems III. Washington, DC: 
Math. Assoc. Amer., pp. 60-63, 1985. 

Saaty, T. L. and Kainen, P. C. The Four-Color Problem,: 
Assaults and Conquest. New York: Dover, p. 12, 1986. 

Zaks, S. and Liu, C. L. "Decomposition of Graphs into 
Trees." Proc. Eighth Southeastern Conference on Com- 
binatorics, Graph Theory, and Computing, pp. 643-654, 
1977. 



Complete k-Partite Graph 

Complete fc-Partite Graph 




A A;-Partite Graph (i.e., a set of Vertices decom- 
posed into k disjoint sets such that no two VERTICES 
within the same set are adjacent) such that every pair 
of Vertices in the k sets are adjacent. If there are 
p, q, . . . , r Vertices in the k sets, the complete bi- 
partite graph is denoted i^ P) ^,...,r- The above figure 

Shows 1^2,3,5- 

see also COMPLETE GRAPH, COMPLETE fc-PARTITE 

Graph, ^-Partite Graph 

References 

Saaty, T. L. and Kainen, P. C. The Four-Color Problem: 
Assaults and Conquest, New York: Dover, p. 12, 1986. 

Complete Metric Space 

A complete metric space is a METRIC SPACE in which 
every CAUCHY SEQUENCE is CONVERGENT. Examples 
include the Real Numbers with the usual metric and 
the p-ADic Numbers. 

Complete Permutation 

see Derangement 

Complete Quadrangle 

If the four points making up a Quadrilateral are 
joined pairwise by six distinct lines, a figure known as 
a complete quadrangle results. Note that a complete 
quadrilateral is defined differently from a COMPLETE 
Quadrangle. 

The midpoints of the sides of any complete quadrangle 
and the three diagonal points all lie on a CONIC known 

as the Nine-Point Conic If it is an Orthocentric 
Quadrilateral, the Conic reduces to a Circle. The 
Orthocenters of the four Triangles of a complete 
quadrangle are COLLINEAR on the RADICAL Line of the 
Circles on the diameters of a Quadrilateral. 

see also Complete Quadrangle, Ptolemy's Theo- 
rem 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 

York: Wiley, pp. 230-231, 1969. 
Demir, H. "The Compleat [sic] Cyclic Quadrilateral." Amer. 

Math. Monthly 79, 777-778, 1972. 



Complete Sequence 285 

Johnson, R. A. Modern Geometry: An Elementary Treatise 
on the Geometry of the Triangle and the Circle. Boston, 
MA: Houghton Mifflin, pp. 61-62, 1929. 

Ogilvy, C. S. Excursions in Geometry. New York: Dover, 
pp. 101-104, 1990. 

Complete Quadrilateral 

The figure determined by four lines and their six points 
of intersection (Johnson 1929, pp. 61-62). Note that 
this is different from a COMPLETE QUADRANGLE. The 
midpoints of the diagonals of a complete quadrilateral 
are COLLINEAR (Johnson 1929, pp. 152-153). 

A theorem due to Steiner (Mention 1862, Johnson 1929, 
Steiner 1971) states that in a complete quadrilateral, the 
bisectors of angles are CONCURRENT at 16 points which 
are the incenters and EXCENTERS of the four TRIAN- 
GLES. Furthermore, these points are the intersections of 
two sets of four CIRCLES each of which is a member of 
a conjugate coaxal system. The axes of these systems 
intersect at the point common to the ClRCUMCIRCLES 
of the quadrilateral. 

see also COMPLETE QUADRANGLE, GAUSS-BODENMIL- 

ler Theorem, Polar Circle 

References 

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New 

York: Wiley, pp. 230-231, 1969. 
Johnson, R. A. Modern Geometry: An Elementary Treatise 

on the Geometry of the Triangle and the Circle. Boston, 

MA: Houghton Mifflin, pp. 61-62, 149, 152-153, and 255- 

256, 1929. 
Mention, M. J. "Demonstration d'un Theoreme de 

M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 16-20, 1862. 
Mention, M. J. "Demonstration d'un Theoreme de 

M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 65-67, 1862. 
Steiner, J. Gesammelte Werke, 2nd ed, Vol. 1. New York: 

Chelsea, p. 223, 1971. 

Complete Residue System 

A set of numbers clq, cli, ..., a m -i (mod m) form a 
complete set of residues if they satisfy 

ai = i (mod m) 

for i = 0, 1, . . . , m — 1. In other words, a complete 
system of residues is formed by a base and a modulus if 
the residues r; in b l = Vi (mod m) for i = 1, . . . , m - 1 
run through the values 1, 2, ..., m — 1. 

see also Haupt-Exponent 

Complete Sequence 

A Sequence of numbers V — {u n } is complete if every 
Positive Integer n is the sum of some subsequence of 
V, i.e., there exist a; = or 1 such that 



/ v aM 



(Honsberger 1985, pp. 123-126). The Fibonacci Num- 
bers are complete. In fact, dropping one number still 



286 



Complete Space 



Complex Analysis 



leaves a complete sequence, although dropping two num- 
bers does not (Honsberger 1985, pp. 123 and 126). The 
Sequence of Primes with the element {1} prepended, 

{1,2,3,5,7,11,13,17,19,23,...} 

is complete, even if any number of Primes each > 7 are 
dropped, as long as the dropped terms do not include 
two consecutive PRIMES (Honsberger 1985, pp. 127— 

128). This is a consequence of BERTRAND'S POSTU- 
LATE. 

see also Bertrand's Postulate, Brown's Cri- 
terion, Fibonacci Dual Theorem, Greedy Al- 
gorithm, Weakly Complete Sequence, Zeck- 
endorf's Theorem 

References 

Brown, J. L. Jr. "Unique Representations of Integers as Sums 
of Distinct Lucas Numbers." Fib. Quart. 7,243-252,1969. 

Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for 
Fibonacci Numbers. XIL" Fib. Quart. 11, 317-331, 1973. 

Honsberger, R. Mathematical Gems III. Washington, DC: 
Math. Assoc. Amer., 1985. 

Complete Space 

A Space of Complete Functions. 

see also COMPLETE METRIC SPACE 
Completely Regular Graph 

A POLYHEDRAL Graph is completely regular if the 
Dual Graph is also Regular. There are only five 
types. Let p be the number of EDGES at each node, p* 
the number of EDGES at each node of the DUAL GRAPH, 
V the number of VERTICES, E the number of EDGES, 
and F the number of faces in the Platonic Solid cor- 
responding to the given graph. The following table sum- 
marizes the completely regular graphs. 



Type 


9 


P* 


V 


E 


F 


Tetrahedral 


3 


3 


4 


6 


4 


Cubical 


3 


4 


8 


12 


6 


Dodecahedral 


3 


5 


20 


39 


12 


Octahedral 


4 


3 


6 


12 


8 


Icosahedral 


5 


3 


12 


30 


20 



Completeness Property 

All lengths can be expressed as Real Numbers. 

Completing the Square 

The conversion of an equation of the form ax 2 + bx + c 
to the form 



a { x + ^) 



+ ic -4-al' 



which, defining B = b/2a and C = c — b 2 /4a, simplifies 
to 

a(x + B) 2 + C. 



Complex 

A finite Set of SlMPLEXES such that no two have a 
common point. A 1-D complex is called a GRAPH. 
see also CW-Complex, Simplicial Complex 

Complex Analysis 

The study of Complex NUMBERS, their DERIVATIVES, 
manipulation, and other properties. Complex analysis is 
an extremely powerful tool with an unexpectedly large 
number of practical applications to the solution of phys- 
ical problems. CONTOUR INTEGRATION, for example, 
provides a method of computing difficult INTEGRALS by 
investigating the singularities of the function in regions 
of the Complex Plane near and between the limits of 
integration. 

The most fundamental result of complex analysis is the 
Cauchy-Riemann Equations, which give the condi- 
tions a Function must satisfy in order for a com- 
plex generalization of the Derivative, the so-called 
Complex Derivative, to exist. When the Complex 
Derivative is defined "everywhere," the function is 
said to be ANALYTIC. A single example of the unex- 
pected power of complex analysis is PlCARD'S Theo- 
rem, which states that an Analytic Function as- 
sumes every Complex Number, with possibly one ex- 
ception, infinitely often in any NEIGHBORHOOD of an 
Essential Singularity! 

see also ANALYTIC CONTINUATION, BRANCH CUT, 

Branch Point, Cauchy Integral Formula, Cau- 
chy Integral Theorem, Cauchy Principal Value, 
Cauchy-Riemann Equations, Complex Number, 
Conformal Map, Contour Integration, de 
Moivre's Identity, Euler Formula, Inside- 
Outside Theorem, Jordan's Lemma, Laurent Se- 
ries, Liouville's Conformality Theorem, Mono- 
genic Function, Morera's Theorem, Permanence 
of Algebraic Form, Picard's Theorem, Pole, 
Polygenic Function, Residue (Complex Analy- 
sis) 

References 

Arfken, G. "Functions of a Complex Variable I: Analytic 
Properties, Mapping" and "Functions of a Complex Vari- 
able II: Calculus of Residues." Chs. 6—7 in Mathematical 
Methods for Physicists, 3rd ed. Orlando, FL: Academic 
Press, pp. 352-395 and 396-436, 1985. 

Boas, R. P. Invitation to Complex Analysis. New York: Ran- 
dom House, 1987. 

Churchill, R. V. and Brown, J. W. Complex Variables and 
Applications, 6th ed. New York: McGraw-Hill, 1995. 

Conway, J. B. Functions of One Complex Variable, 2nd ed. 
New York: Springer- Verlag, 1995. 

Forsyth, A. R. Theory of Functions of a Complex Variable, 
3rd ed, Cambridge, England: Cambridge University Press, 
1918. . 

Lang, S. Complex Analysis, 3rd ed. New York: Springer- 
Verlag, 1993. 

Morse, P. M. and Feshbach, H. "Functions of a Complex Vari- 
able" and "Tabulation of Properties of Functions of Com- 
plex Variables." Ch. 4 in Methods of Theoretical Physics, 
Part I. New York: McGraw-Hill, pp. 348-491 and 480-485, 
1953. 



Complex Conjugate 



Complex Number 287 



Complex Conjugate 

The complex conjugate of a Complex Number z = 
a+bi is defined to be z* = a— hi. The complex conjugate 
is Associative, (zi + z 2 )* = zi* + z 2 *, since 

(ai H- M)* + (a 2 + M)* — ai - ibi + a 2 - i&2 

= (ai - ibi) + (a 2 - ib 2 ) 
= (ai+6i)* + (a 2 + b 2 )*, 

and Distributive, (ziz 2 ) m = zi*z 2 *, since 

[(ai + bii)(a 2 + 62*)]* = [( a i a 2 - 6162) + i(ai& 2 + 0261)]* 
= (ai(X2 — &ifr 2 ) — i(ai6 2 + a 2 6i) 
= (ai - z6i)(a 2 - i6 2 ) 
— (ai + i6i)*(a2 + 162)*. 



References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
p. 16, 1972. 

Complex Derivative 

A Derivative of a Complex function, which must sat- 
isfy the Cauchy-Riemann Equations in order to be 
Complex Differentiable. 



see also Cauchy-Riemann Equations, 
Differentiable, Derivative 



Complex 



Complex Differentiable 

If the Cauchy-Riemann Equations are satisfied for a 
function f(x) = u(x) + iv(x) and the PARTIAL DERIVA- 
TIVES of u(x) and v(x) are Continuous, then the Com- 
plex Derivative df/dz exists. 

see also Analytic Function, Cauchy-Riemann 
Equations, Complex Derivative, Pseudoanalytic 
Function 

Complex Function 

A Function whose Range is in the Complex Num- 
bers is said to be a complex function. 

see also Real Function, Scalar Function, Vector 
Function 

Complex Matrix 

A Matrix whose elements may contain Complex Num- 
bers. The Matrix Product of two 2x2 complex 
matrices is given by 



xu + 2/i 1* Z12 + y 12 i 
£21 + V2ii £22 + 2/22* 



uu -\-Vni 


U12 


+ V121 




U21 -\-v21i 


1*22 + ^22^ 




R11 R12 


-H 


111 1 12 


R21 R22 


hi 1 


22 



where 

R11 — u\\x\\ + u 2 ixi 2 — viij/11 — v 2 iyi 2 

Rl2 — Wl2Xll + ^22^12 - V122/11 - U222/12 

R 2 1 = U11X21 + U21X22 - Ul 12/21 - V21J/22 

R 22 = Ui 2 X 2 ± + u 22 x 22 — vi 2 y 2 i — V 222/22 

In = vnxii + ^21X12 + wnyii + U21IJ12 

111 = V12X11 + ^22^12 + U122/11 + ^222/12 
^21 = ^113521 + ^21^22 + U112/21 + ^212/22 
i~22 = V\ 2 X 2 1 + V 22 #22 + ^122/21 + ^222/22- 

see a/so Real Matrix 

Complex Multiplication 

Two Complex Numbers x = a + ib and y = c + id are 
multiplied as follows: 

xy — (a + i&)(c + zd) = ac + ibc + zad — 6d 
= (ac - bd) + i(ad -f 6c). 

However, the multiplication can be carried out using 
only three REAL multiplications, ac, bd, and (a+b)(c-\-d) 
as 

R[(a + ib)(c + id)] = ac - bd 

9f[(a + ifc)(c + id)] = (a + 6)(c + d) - ac - bd. 

Complex multiplication has a special meaning for EL- 
LIPTIC Curves. 

see also Complex Number, Elliptic Curve, Imagi- 
nary Part, Multiplication, Real Part 

References 

Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field 

Theory and Complex Multiplication. New York: Wiley, 

1997. 

Complex Number 

The complex numbers are the Field C of numbers of the 
form x + iy, where x and y are REAL NUMBERS and i is 
the Imaginary Number equal to >/-!• When a single 
letter z - x + iy is used to denote a complex number, it 
is sometimes called an "AFFIX." The FIELD of complex 
numbers includes the Field of Real Numbers as a 
Subfield. 

Through the Euler FORMULA, a complex number 

z = x -f iy (1) 

may be written in "PHASOR" form 

z = \z\ (cos + i sin 6) = \z\e ie . (2) 

Here, \z\ is known as the Modulus and 9 is known as 
the Argument or Phase. The Absolute Square of 



288 Complex Number 



Complex Structure 



z is defined by \z\ 2 — zz* , and the argument may be 
computed from 



Complex Plane 



arg(z) — = tan I — J 



(3) 



de Moivre's Identity relates Powers of complex 
numbers 



z n = |z| n [cos(n#) + zsin(n#)]. 



(4) 



Finally, the Real R(z) and Imaginary Parts $s(z) are 
given by 



»w = i(^+o (5) 

*(*> = ^^ = ~W - O = 5*(** " *)■ ( 6 ) 



2z 



The Powers of complex numbers can be written in 
closed form as follows: 



-0 



ri-2 2 . I n \ n-4 4 

x y + 1 4 p y 



+ i 



>~v 



3 F y +.. 



(7) 



The first few are explicitly 

z 2 = (x 2 - y 2 ) -{- i(2xy) 

z = (x — 3xy ) + i(3x y — y ) 

z 4 = (x 4 - 6x 2 y 2 + y 4 ) 4- i(4z 3 y - 4xy 3 ) 

z 5 = ( x 5 - I0x 3 y 2 + 5zy 4 ) + i{$x A y - 10xV + y 5 ) 



(8) 

(9) 

(10) 



(11) 



(Abramowitz and Stegun 1972). 

see also Absolute Square, Argument (Complex 
Number), Complex Plane, i, Imaginary Number, 
Modulus, Phase, Phasor, Real Number, Surreal 
Number 

References 

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook 
of Mathematical Functions with Formulas, Graphs, and 
Mathematical Tables, 9th printing. New York: Dover, 
pp. 16-17, 1972. 

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- 
lando, FL: Academic Press, pp. 353-357, 1985. 

Courant, R. and Robbins, H. "Complex Numbers." §2.5 in 
What is Mathematics? : An Elementary Approach to Ideas 
and Methods, 2nd ed. Oxford, England: Oxford University- 
Press, pp. 88-103, 1996. 

Morse, P. M. and Feshbach, H. "Complex Numbers and Vari- 
ables." §4.1 in Methods of Theoretical Physics, Part I. New 
York: McGraw-Hill, pp. 349-356, 1953. 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- 
terling, W. T. "Complex Arithmetic." §5.4 in Numerical 
Recipes in FORTRAN: The Art of Scientific Computing, 
2nd ed. Cambridge, England: Cambridge University Press, 
pp. 171-172, 1992. 



Imaginary 




The plane of COMPLEX Numbers spanned by the vec- 
tors 1 and i, where i is the IMAGINARY NUMBER. Every 
Complex Number corresponds to a unique Point in 
the complex plane. The LINE in the plane with i = is 
the Real Line. The complex plane is sometimes called 
the Argand Plane or Gauss Plane, and a plot of 
Complex Numbers in the plane is sometimes called 
an Argand Diagram. 

see also AFFINE COMPLEX PLANE, ARGAND DIAGRAM, 
Argand Plane, Bergman Space, Complex Projec- 
tive Plane 

References 

Courant, R. and Robbins, H. "The Geometric Interpretation 
of Complex Numbers." §5.2 in What is Mathematics?: An 
Elementary Approach to Ideas and Methods, 2nd ed. Ox- 
ford, England: Oxford University Press, pp. 92-97, 1996. 

Complex Projective Plane 

The set P 2 is the set of all Equivalence Classes 
[a, 6,c] of ordered triples (a, 6, c) E C 3 \(0,0,0) under 
the equivalence relation (a, 6, c) ~ (a', &', c') if (a, 6, c) = 
(Aa', A6',Ac') for some Nonzero Complex Number A. 

Complex Representation 

see Phasor 



Complex Structure 

The complex structure of a point x = 
PLANE is defined by the linear MAP J : '. 

J{Xi,X 2 ) - (-Z2,Zl), 



X\ , X2 in the 



and corresponds to a clockwise rotation by rr/2. This 
map satisfies 

J 2 = -I 
(Jx).(Jy) = x.y 
( Jx) • x = 0, 

where / is the IDENTITY MAP. 

More generally, if V is a 2-D Vector SPACE, a linear 
map J : V — > V such that J 2 = — I is called a complex 
structure on7. If V = M. , this collapses to the previous 
definition. 

References 

Gray, A. Modern Differential Geometry of Curves and Sur- 
faces. Boca Raton, FL: CRC Press, pp. 3 and 229, 1993. 



Complexity (Number) 

Complexity (Number) 

The number of Is needed to represent an INTEGER us- 
ing only additions, multiplications, and parentheses are 
called the integer's complexity. For example, 

1 = 1 

2 = 1 + 1 
3=1+1+1 

4=(1 + 1)(1 + 1) = 1 + 1 + 1 + 1 

5 = (1 + 1)(1 + 1) + 1 = 1 + 1 + 1 + 1 + 1 

6 = (1 + 1)(1 + 1 + 1) 

7 = (1 + 1)(1 + 1 + 1) + 1 

8 = (1 + 1)(1 + 1)(1 + 1) 
9=(1 + 1 + 1)(1 + 1 + 1) 

10 = (1 + 1 + 1)(1 + 1 + 1) + 1 
= (1 + 1)(1 + 1 + 1 + 1 + 1) 

So, for the first few n, the complexity is 1, 2, 3, 4, 5, 5, 

6, 6, 6, 7, 8, 7, 8, . . . (Sloane's A005245). 

References 

Guy, R. K. "Expressing Numbers Using Just Ones." §F26 in 

Unsolved Problems in Number Theory, 2nd ed. New York: 

Springer- Verlag, p. 263, 1994. 
Guy, R. K. "Some Suspiciously Simple Sequences." Amer. 

Math. Monthly 93, 186-190, 1986. 
Guy, R. K. "Monthly Unsolved Problems, 1969-1987." 

Amer. Math. Monthly 94, 961-970, 1987. 
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. 

Monthly 96, 903-909, 1989. 
Rawsthorne, D. A. "How Many l's are Needed?" Fib. Quart. 

27, 14-17, 1989. 
Sloane, N. J. A. Sequence A005245/M0457 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Complexity (Sequence) 

see Block Growth 

Complexity Theory 

Divides problems into "easy" and "hard" categories. 
A problem is easy and assigned to the P-Problem 
(Polynomial time) class if the number of steps needed 
to solve it is bounded by some Power of the prob- 
lem's size. A problem is hard and assigned to the NP- 
PROBLEM (nondeterministic POLYNOMIAL time) class if 
the number of steps is not bounded and may grow ex- 
ponentially. 

However, if a solution is known to an NP-Problem, it 
can be reduced to a single period verification. A prob- 
lem is NP-Complete if an Algorithm for solving it 
can be translated into one for solving any other NP- 
Problem. Examples of NP-Complete Problems in- 
clude the Hamiltonian Cycle and Traveling Sales- 
man Problems. Linear Programming, thought to 
be an NP-PROBLEM, was shown to actually be a P- 
PROBLEM by L. Khachian in 1979. It is not known if all 
apparently NP-PROBLEMS are actually P-PROBLEMS. 



Composite Number 289 

see also Bit Complexity, NP-Complete Problem, 
NP-Problem, P-Problem 

References 

Bridges, D. S. Computability. New York: Springer- Verlag, 
1994. 

Brookshear, J. G. Theory of Computation: Formal Lan- 
guages, Automata, and Complexity. Redwood City, CA: 
Benjamin/Cummings, 1989. 

Cooper, S. B.; Slaman, T. A.; and Wainer, S. S. (Eds.). Com- 
putability, Enumerability, Unsolvability: Directions in Re- 
cursion Theory. New York: Cambridge University Press, 
1996. 

Garey, M. R. and Johnson, D. S. Computers and Intractabil- 
ity: A Guide to the Theory of NP- Completeness. New 
York: W. H. Freeman, 1983. 

Goetz, P. "Phil Goetz's Complexity Dictionary." http:// 
www . cs .buf f alo . edu/~goetz/dict .html. 

Hopcroft, J. E. and Ullman, J. D. Introduction to Auto- 
mated Theory, Languages, and Computation. Reading, 
MA: Addison-Wesley, 1979. 

Lewis, H. R. and Papadimitriou, C. H. Elements of the 
Theory of Computation, 2nd ed. Englewood Cliffs, NJ: 
Prentice-Hall, 1997. 

Sudkamp, T. A. Language and Machines: An Introduction 
to the Theory of Computer Science, 2nd ed. Reading, MA: 
Addison-Wesley, 1996. 

Welsh, D. J. A. Complexity: Knots, Colourings and Count- 
ing. New York: Cambridge University Press, 1993. 

Component 

A Group L is a component of H if L is a Quasisimple 

Group which is a Subnormal Subgroup of H, 

see also GROUP, QUASISIMPLE GROUP, SUBGROUP, 

Subnormal 

Composite Knot 

A Knot which is not a Prime Knot. Composite knots 
are special cases of Satellite Knots. 

see also Knot, Prime Knot, Satellite Knot 

Composite Number 

A Positive Integer which is not Prime (i.e., which 
has FACTORS other than 1 and itself). 

A composite number C can always be written as a 
Product in at least two ways (since 1 ■ C is always 
possible). Call these two products 



C = ab = cd. 



(i) 



then it is obviously the case that C\ab (C divides ab). 
Set 

c = mn 1 (2) 

where m is the part of C which divides a, and n the part 
of C which divides n. Then there are p and q such that 



a = mp 
b = nq. 



(3) 
(4) 



290 Composite Runs 



Composition Theorem 



Solving ab = cd for d gives 



ab __ (mp)(nqr) _ 

£j — — — pq t 

c mn 



(5) 



It then follows that 



S = a 2 + b 2 + c 2 + d 2 

2 2. 22. 22. 22 

= m p + n q +m n -\- p q 
= (m 2 + q 2 )(n 2 +p 2 ). 



(6) 



It therefore follows that a 2 + b 2 + c 2 + d 2 is never Prime! 
In fact, the more general result that 



S = a k + 6 fc + c fc -f d k 



(7) 



is never Prime for k an Integer > also holds (Hons- 
berger 1991). 

There are infinitely many integers of the form |_(3/2) Tt J 
and L(4/3) n J which are composite, where [^J is the 
Floor Function (Forman and Shapiro, 1967; Guy 
1994, p. 220). The first few composite |_(3/2) n J occur 
for n = 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 
23, ..., and the the few composite |_(4/3) n J occur for 
n = 5, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... . 
see also Amenable Number, Grimm's Conjecture, 
Highly Composite Number, Prime Factorization 
Prime Gaps, Prime Number 

References 

Forman, W. and Shapiro, H. N. "An Arithmetic Property of 

Certain Rational Powers." Comm. Pure AppL Math. 20, 

561-573, 1967. 
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. 

New York: Springer-Verlag, 1994. 
Honsberger, R. More Mathematical Morsels. Washington, 

DC: Math. Assoc. Amer., pp. 19-20, 1991. 
Sloane, N. J. A. Sequence A002808/M3272 in "An On-Line 

Version of the Encyclopedia of Integer Sequences." 

Composite Runs 

see Prime Gaps 

Compositeness Certificate 

A compositeness certificate is a piece of information 
which guarantees that a given number p is COMPOSITE. 
Possible certificates consist of a Factor of a number 
(which, in general, is much quicker to check by direct 
division than to determine initially), or of the determi- 
nation that either 

a p_1 ^ 1 (modp), 

(i.e., p violates Fermat's Little Theorem), or 

a ^ —1, 1 and a = 1 (mod p) . 

A quantity a satisfying either property is said to be a 
Witness to p's compositeness. 

see also Adleman-Pomerance-Rumely Primality 
Test, Fermat's Little Theorem, Miller's Pri- 
mality Test, Primality Certificate, Witness 



Compositeness Test 

A test which always identifies Prime numbers correctly, 
but may incorrectly identify a Composite Number as 
a Prime. 

see also Primality Test 

Composition 

The combination of two FUNCTIONS to form a single new 
Operator. The composition of two functions / and g 
is denoted fog and is defined by 

/°S = /(#(#)) 

when / and g are both functions of x. 

An operation called composition is also defined on Bi- 
nary Quadratic Forms. For two numbers repre- 
sented by two forms, the product can then be repre- 
sented by the composition. For example, the composi- 
tion of the forms 2x 2 + 15y 2 and 3x 2 + 10y 2 is given by 
6x 2 + 5y 2 , and in this case, the product of 17 and 13 
would be represented as (6 * 36 + 5 - 1 = 221). There 
are several algorithms for computing binary quadratic 
form composition, which is the basis for some factoring 
methods. 

see also ADEM RELATIONS, BINARY OPERATOR, BI- 
NARY Quadratic Form 

Composition Series 

Every Finite GROUP G of order greater than one pos- 
sesses a finite series of SUBGROUPS, called a composition 
series, such that 

/ C H s C . . . C H 2 C H! C G, 

where if;+i is a maximal subgroup of Hi. The Quo- 
tient Groups G/Hi, H 1 /H 2 , ■•-, H 3 -i/H 3 , H s are 
called composition quotient groups. 
see also Finite Group, Jordan-Holder Theorem, 
Quotient Group, Subgroup 

References 

Lomont, J. S. Applications of Finite Groups. New York: 
Dover, p. 26, 1993. 

Composition Theorem 

Let 

Q( X ,y)= X 2 +y 2 . 



Then 



Q(x,y)Q(x,y) = Q{xx - yy \xy + xy), 



{x 2 + y 2 ){x 2 + y 2 ) = (xx - yy 1 ) 2 + (xy + xy) 2 

2 /2 , 2 12 , /2 2 , 2/2 

= x x +y y +x y +x y . 



see also Genus Theorem 



Compound Interest 



Concatenation 291 



Compound Interest 

Let P be the Principal (initial investment), r be the 
annual compounded rate, v n > the "nominal rate," rt be 
the number of times INTEREST is compounded per year 
(i.e., the year is divided into n CONVERSION PERIODS), 
and t be the number of years (the "term"). The INTER- 
EST rate per CONVERSION PERIOD is then 



-■(«) 



(1) 



If interest is compounded n times at an annual rate of r 
(where, for example, 10% corresponds to r = 0.10), then 
the effective rate over 1/n the time (what an investor 
would earn if he did not redeposit his interest after each 
compounding) is 



(i + ' 



Nl/n 



(2) 



The total amount of holdings A after a time t when 
interest is re-invested is then 



A = P[l + —) =P(l + r) nt . (3) 



Note that even if interest is compounded continuously, 
the return is still finite since 



lim (l + -V 

n— >oo \ 71 J 



(4) 



where e is the base of the NATURAL LOGARITHM. 



The time required for a given PRINCIPAL to double (as- 
suming n = l Conversion Period) is given by solving 



2P = P(l + r) t , 



In 2 



ln(l + r)' 



(5) 



(6) 



where Ln is the NATURAL LOGARITHM. This function 
can be approximated by the so-called RULE OF 72: 



0.72 
r 



(?) 



see also e, Interest, Ln, Natural Logarithm, Prin- 
cipal, Rule of 72, Simple Interest 

References 

Kellison, S. G. The Theory of Interest, 2nd ed. Burr Ridge, 

IL: Richard D. Irwin, pp. 14-16, 1991. 
Milanfar, P. "A Persian Folk Method of Figuring Interest." 

Math. Mag. 69, 376, 1996. 

Compound Polyhedron 

see Polyhedron Compound 

Comput ability 

see Complexity Theory 



Computable Function 

Any computable function can be incorporated into a 
Program using while-loops (i.e., "while something is 
true, do something else"). For-loops (which have a fixed 
iteration limit) are a special case of while-loops, so com- 
putable functions could also be coded using a combina- 
tion of for- and while-loops. The ACKERMANN FUNC- 
TION is the simplest example of a well-defined TOTAL 
Function which is computable but not Primitive Re- 
cursive, providing a counterexample to the belief in 
the early 1900s that every computable function was also 
primitive recursive (Dotzel 1991). 

see also Ackermann Function, Church's Thesis, 
Computable Number, Primitive Recursive Func- 
tion, Turing Machine 

References 

Dotzel, G. "A Function to End All Functions." Algorithm: 
Recreational Programming 2, 16—17, 1991. 

Computable Number 

A number which can be computed to any number of 
Digits desired by a Turing Machine. Surprisingly, 
most Irrationals are not computable numbers! 

References 

Penrose, R. The Emperor's New Mind: Concerning Comput- 
ers, Minds, and the Laws of Physics. Oxford, England: 
Oxford University Press, 1989. 

Computational Complexity 

see Complexity Theory 

Concatenated Number Sequences 

see Consecutive Number Sequences 

Concatenation 

The concatenation of two strings a and b is the string ab 
formed by joining a and b. Thus the concatenation of 
the strings "book" and "case" is the string "bookcase". 
The concatenation of two strings a and 6 is often de- 
noted ab, a\\b, or (in Mathematica® (Wolfram Research, 
Champaign, IL) a <> b. Concatenation is an asso- 
ciative operation, so that the concatenation of three or 
more strings, for example abc, abed, etc., is well-defined. 

The concatenation of two or more numbers is the num- 
ber formed by concatenating their numerals. For exam- 
ple, the concatenation of 1, 234, and 5678 is 12345678. 
The value of the result depends on the numeric base, 
which is typically understood from context. 

The formula for the concatenation of numbers p and q 
in base b is 

p\\q=pb lM +q, 



where 



i(«) = |>g fc «J + i 



is the Length of q in base b and [x\ is the Floor 
Function. 



292 



Concave 



Conchoid 



see also CONSECUTIVE NUMBER SEQUENCES, LENGTH 

(Number), Smarandache Sequences 
Concave 





A Set in R is concave if it does not contain all the 
Line Segments connecting any pair of its points. If 
the Set does contain all the Line Segments, it is called 
Convex. 

see also CONNECTED SET, CONVEX FUNCTION, CON- 
VEX Hull, Convex Optimization Theory, Convex 
Polygon, Delaunay Triangulation, Simply Con- 
nected 

Concave Function 

A function f(x) is said to be concave on an interval [a, b] 
if, for any points x\ and X2 in [a, 6], the function —f(x) 
is Convex on that interval. If the second Derivative 
of/ 

/"(*) > 0, 

on an open interval (a, fo) (where f"(x) is the second 
Derivative), then / is concave up on the interval. If 

/"(*) < o 

on the interval, then / is concave down on it. 

References 

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- 
ries, and Products, 5th ed. San Diego, CA: Academic 
Press, p. 1100, 1980. 

Concentrated 

Let fi be a POSITIVE MEASURE on a SlGMA ALGEBRA 
M, and let A be an arbitrary (real or complex) MEASURE 
on M. If there is a Set A € M such that X(E) = 
A(Afl^) for every E G M, then lambda is said to be 
concentrated on A. This is equivalent to requiring that 
X(E) = whenever E n A = 0. 

see also Absolutely Continuous, Mutually Singu- 
lar 

References 

Rudin, W. Functional Analysis. New York: McGraw-Hill, 
p. 121, 1991. 

Concentric 

Two geometric figures are said to be concentric if their 
Centers coincide. The region between two concentric 
Circles is called an Annulus. 

see also Annulus, Concentric Circles, Concyclic, 
Eccentric 



Concentric Circles 

The region between two CONCENTRIC circles of different 
Radii is called an Annulus. 

Given two concentric circles with RADII R and 2R, what 
is the probability that a chord chosen at random from 
the outer circle will cut across the inner circle? Depend- 
ing on how the "random" CHORD is chosen, 1/2, 1/3, or 
1/4 could all be correct answers. 

1. Picking any two points on the outer circle and con- 
necting them gives 1/3. 

2. Picking any random point on a diagonal and then 
picking the Chord that perpendicularly bisects it 
gives 1/2. 

3. Picking any point on the large circle, drawing a line 
to the center, and then drawing the perpendicularly 
bisected CHORD gives 1/4. 

So some care is obviously needed in specifying what is 
meant by "random" in this problem. 

Given an arbitrary Chord BB' to the larger of two 
concentric CIRCLES centered on O, the distance be- 
tween inner and outer intersections is equal on both 
sides (AB = A'B'). To prove this, take the PERPEN- 
DICULAR to BB' passing through O and crossing at P. 
By symmetry, it must be true that PA and PA' are 
equal. Similarly, PB and PB' must be equal. There- 
fore, PB - PA = AB equals PB' - PA' = A'B'. Inci- 
dentally, this is also true for HOMEOIDS, but the proof 
is nontrivial. 




see also Annulus 

Concho-Spiral 

The Space Curve with parametric equations 

u 

r = fj, a 
6 = u 

u 
Z = fJ, C. 

see also CONICAL SPIRAL, SPIRAL 

Conchoid 

A curve whose name means "shell form." Let C be a 
curve and O a fixed point. Let P and P f be points 
on a line from O to C meeting it at Q, where P'Q = 
QP — k, with k a given constant. For example, if C is a 
CIRCLE and O is on C, then the conchoid is a LlMAQON, 
while in the special case that k is the DIAMETER of C, 



Conchoid of de Sluze 



Concordant Form 293 



then the conchoid is a CARDIOID. The equation for a 
parametrically represented curve (f(t),g(t)) with O = 
(x Qi yo) is 



x = f± 

y = g± 



k(f - xp) 



y/(f-x )* + {g-yo) 2 
fe(g ~ 2/Q ) 

A/(/-*o) a + (5-yd) 2 ' 



see a/so CONCHO-SPIRAL, CONCHOID OF DE SLUZE, 

Conchoid of Nicomedes, Conical Spiral, Durer's 
Conchoid 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 49-51, 1972. 
Lee, X. "Conchoid." http : //www. best . com/ -xah/ Special 

PlaneCurves-dir/Conchoid_dir/conchoid.html. 
Lockwood, E. H. "Conchoids." Ch. 14 in A Book of 

Curves. Cambridge, England: Cambridge University 

Press, pp. 126-129, 1967. 
Yates, R. C. "Conchoid." A Handbook on Curves and Their 

Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33, 

1952. 

Conchoid of de Sluze 




A curve first constructed by Rene de Sluze in 1662. In 
Cartesian Coordinates, 

a(x — a){x + y ) = k x , 

and in POLAR COORDINATES, 

k 2 cos 8 „ 

r = h a sec v. 

a 

The above curve has k 2 fa — 1, a = —0.5. 
Conchoid of Nicomedes 




A curve studied by the Greek mathematician Nicomedes 
in about 200 BC, also called the Cochloid. It is the 
LOCUS of points a fixed distance away from a line as 
measured along a line from the FOCUS point (MacTutor 
Archive). Nicomedes recognized the three distinct forms 



seen in this family. This curve was a favorite with 17th 
century mathematicians and could be used to solve the 
problems of CUBE DUPLICATION and ANGLE TRISEC- 
TION. 



In Polar Coordinates, 

r = b + asecO. 
In Cartesian Coordinates, 

/ \2/ 2 . 2\ ,2 2 

{x — a) [x + y ) — b x . 



(i) 



(2) 



The conchoid has x — a as an asymptote and the Area 
between either branch and the ASYMPTOTE is infinite. 
The Area of the loop is 



+ 6 2 cos- 1 (JJ. (3) 

see also CONCHOID 

References 

Lawrence, J. D. A Catalog of Special Plane Curves. New 

York: Dover, pp. 135-139, 1972. 
Lee, X. "Conchoid of Nicomedes." http://www.best.com/ 

-xah/SpecialPlaneCurvesjdir/ConchoidOfNicomedesjdir 

/conchoidOf Nicomedes .html. 
MacTutor History of Mathematics Archive. "Conchoid." 

http : //www-groups . dcs . st-and . ac . uk/ -history/Curves 

/Conchoid. html. 
Pappas, T. "Conchoid of Nicomedes." The Joy of Mathemat- 
ics. San Carlos, CA: Wide World Publ./Tetra, pp. 94-95, 

1989. 
Yates, R. C. "Conchoid." A Handbook on Curves and Their 

Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33, 

1952. 

Concordant Form 

A concordant form is an integer TRIPLE (a, 6, N) where 

fa 2 +6 2 =c 2 
\a 2 +Nb 2 = d 2 , 

with c and d integers. Examples include 

/ 14663 2 + 111384 2 = 112345 2 



14663 2 + 47 • 111384 2 = 763751 2 
1141 2 + 13260 2 = 13309 2 
1141 2 + 53-13260 2 =96541 2 
2873161 2 + 2401080 2 = 3744361 2 
2873161 2 + 83 • 2401080 2 = 22062761 2 



Dickson (1962) states that C. H. Brooks and S. Watson 
found in The Ladies' and Gentlemen's Diary (1857) that 
x 2 + y 2 and x 2 + Ny 2 can be simultaneously squares for 
N < 100 only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 
30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61, 
68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 



294 



Concur 



Conditional Convergence 



99, and 100 (which evidently omits 47, 53, and 83 from 
above). The list of concordant primes less than 1000 
is now complete with the possible exception of the 16 
primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593, 
607, 641, 743, 821, 929, and 971 (Brown). 
see also Congruum 

References 

Brown, K. S. "Concordant Forms." http : //www . seanet . 

com/~ksbrown/kmath286 .htm. 
Dick