CRC Concise Encyclopedia
MAmEMAfJCS
CRC Concise Encyclopedia
MAfflEMAffG
Eric W. Weisstein
CRC Press
Boca Raton London New York Washington, D.C.
Library of Congress CataloginginPublication Data
Weisstein, Eric W.
The CRC concise encyclopedia of mathematics / Eric W. Weisstein.
p. cm.
Includes bibliographical references and index.
ISBN 0849396409 (alk. paper)
1. Mathematics Encyclopedias. I. Title.
QA5.W45 1998
510'.3— DC21 9822385
CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources
are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and
the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,
microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific
permission must be obtained in writing from CRC Press LLC for such copying.
Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are only used for identification and explanation,
without intent to infringe.
© 1999 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 0849396409
Library of Congress Card Number 9822385
Printed in the United States of America 1234567890
Printed on acidfree 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 CDROM. Second, the entries are
extensively crosslinked and crossreferenced, 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 jumpingoff 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 crossreferencing 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 crossreference (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 crossreference (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 multipleword
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. . . ," "SumP. . . ," 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 nonstandard 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
(wwwdsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html), which has allowed me to
easily maintain and update an online version of the encyclopedia long before it existed in book form.
I would like to thank Steven Finch of MathSoft, Inc., for his interesting online 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 online (www.research.att.com/njas/sequences/) versions of the Encyclopedia of Integer Sequences,
an immensely valuable compilation of useful information which represents a truly mindboggling 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 email 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. 2631, 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, Super3 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 09 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
18Point 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. 127128, 1979.
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 135, 1995.
Sloane, N. J. A. Sequence A011557 in "An OnLine 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
socalled 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.
cuttheknot .com/pythagoras/historyl5.html.
Johnson, W. W. "Notes on the '15 Puzzle. I.'" Amer. J.
Math. 2, 397399, 1879.
Kasner, E. and Newman, J. R. Mathematics and the Imagi
nation. Redmond, WA: Tempus Books, pp. 177180, 1989.
Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical
Recreations. New York: W. W. Norton, pp. 302308, 1942.
Story, W. E. "Notes on the '15 Puzzle. II.*" Amer. J. Math.
2, 399404, 1879.
16Cell
A finite regular 4D POLYTOPE with SCHLAFLI SYMBOL
{3, 3, 4} and Vertices which are the PERMUTATIONS
of (±1, 0, 0, 0).
see also 24Cell, 120Cell, 600Cell, Cell, Poly
tope
17
17 is a FERMAT PRIME which means that the 17sided
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.enslyon.fr/
vlefevre/dl7_eng.html.
Shell Centre for Mathematical Education. "Number
17." http : //acorn . educ . nott ingham . ac . uk/ShellCent/
Number /Num 17 .html.
18Point 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
24Cell
196Algorithm
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 14point 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 17point 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. 3436, 1997.
Steinhaus, H. "Distribution on Numbers" and "Generaliza
tion." Problems 6 and 7 in One Hundred Problems in
Elementary Mathematics. New York: Dover, pp. 1213,
1979.
Warmus, M. "A Supplementary Note on the Irregularities of
Distributions." J. Number Th. 8, 260263, 1976.
24Cell
A finite regular 4D 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 24cell (Coxeter 1973,
Conway and Sloane 1993, Sloane and Plouffe 1995).
see also 16Cell, 120Cell, 600Cell, 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 OnLine
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
120Cell
A finite regular 4D Polytope with Schlafli Symbol
{5,3,3} (Coxeter 1969).
see also 16Cell, 24Cell, 600Cell, 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 SumProduct Number.
see also Dozen
196Algorithm
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 196algorithm, 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. 242245, 1979.
Gruenberger, F. "How to Handle Numbers with Thousands
of Digits, and Why One Might Want to." Sci. Amer. 250,
1926, Apr. 1984.
Sloane, N. J. A. Sequences A023109, A030547, A033865, and
A006960/M5410 in "An OnLine Version of the Encyclo
pedia of Integer Sequences."
239
65537gon
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 AIM239, Feb. 1972.
257gon
257 is a FERMAT PRIME, and the 257gon is there
fore a Constructible Polygon using Compass and
Straightedge, as proved by Gauss. An illustration
of the 257gon is not included here, since its 257 seg
ments so closely resemble a Circle. Richelot and
Schwendenwein found constructions for the 257gon 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 65537GON, 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,
97108, 1991.
Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
Rademacher, H. Lectures on Elementary Number Theory.
New York: Blaisdell, 1964.
600Cell
A finite regular 4D Polytope with Schlafli Symbol
{3,3,5}. For Vertices, see Coxeter (1969).
see also 16Cell, 24Cell, 120Cell, 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,
2629, Spring 1997.
65537gon
65537 is the largest known Fermat Prime, and the
65537gon is therefore a CONSTRUCTIBLE POLYGON us
ing Compass and Straightedge, as proved by Gauss.
The 65537gon 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 65537gon
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 257GON, 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,
97108, 1991.
Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
AIntegrable
A
AIntegrable
A generalization of the Lebesgue INTEGRAL. A MEA
SURABLE Function f(x) is called Aintegrable 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, 4980, 1928.
A Sequence
N.B. A detailed online essay by S. Finch was the start
ing point for this entry.
An Infinite Sequence of Positive Integers ai sat
isfying
1 < ai < a2. < az < ■ • . (1)
is an Asequence 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 Asequence 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, 2838, 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. 228229, 1994.
Levine, E. and O'Sullivan, J. "An Upper Estimate for the
Reciprocal Sum of a Sum Free Sequence." Acta Arith. 34,
924, 1977.
Zhang, Z. X. "A SumFree 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 = ttAB.
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 Asequence satisfies
oo
(4)
where %% are given by the LevineO'Sullivan Greedy
Algorithm.
see also B 2 Sequence, MianChowla Sequence
References
Abbott, H. L. "On SumFree Sequences." Acta Arith. 48,
9396, 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. 199201, 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. 232241, 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. 156196, 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, 314, 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. 7576, 1994.
Silverman, J. " Wieferich's Criterion and the abc Conjecture."
J. Number Th. 30, 226237, 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) +
+
(12/)
see a/50 DlLOGARITHM, POLYLOGARITHM, RlEMANN
Zeta Function
Abelian Group
N.B. A detailed online 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 nonAbelian 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 nonAbelian 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, 197208,
1947.
Kolesnik, G. "On the Number of Abelian Groups of a Given
Order." J. Reine Angew. Math. 329, 164175, 1981.
8
Abel's Identity
Abel's Irreducibility Theorem
Neumann, P. M. "An Enumeration Theorem for Finite
Groups." Quart J. Math. Ser. 2 20, 395401, 1969.
Pyber, L. "Enumerating Finite Groups of Given Order."
Ann. Math. 137, 203220, 1993.
Richert, H.E. "Uber die Anzahl abelscher Gruppen
gegebener Ordnung L" Math. Zeitschr. 56, 2132, 1952.
Sloane, N. J. A. Sequence A000688/M0064 in "An OnLine
Version of the Encyclopedia of Integer Sequences."
Srinivasan, B. R. "On the Number of Abelian Groups of a
Given Order." Acta Arith. 23, 195205, 1973.
Abel's Identity
Given a homogeneous linear SECONDORDER 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(yiy2yiy2)+Q(yiy2yiy2) = (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)
= 2/12/22/1^2. (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
ondOrder
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 {xty
_ sin(7ra)
7V
u:
dx
dx
df dx  /(0)
dx{txY~ 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. 875876, 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: McGrawHill, pp. 262266, 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 pSuBGROUPS 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 pSubgroup
10
AblowitzRamaniSegur Conjecture
Absolute Square
References
Abhyankar, S. "Coverings of Algebraic Curves." Airier. J.
Math. 79, 825856, 1957.
American Mathematical Society. "Notices of the AMS, April
1995, 1995 Prank Nelson Cole Prize in Algebra." http://
www. ams . org/notices/199504/prizecole .html.
Harbater, D. "Abhyankar's Conjecture on Galois Groups
Over Curves." Invent. Math. 117, 125, 1994.
Raynaud, M. "Revetements de la droite affine en car
acteristique p > et conjecture d' Abhyankar." Invent.
Math. 116, 425462, 1994.
AblowitzRamaniSegur Conjecture
The AblowitzRamaniSegur 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, yAxis,
zAxis
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. 6977, 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. 9091, 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[l2sin 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>24>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
pADlC 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: McGrawHill,
pp. 121125, 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. 4546, 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 75. (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, 278282, 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. 4546, 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 OnLine
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. 283287, 1971.
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density
Bounds for the Sum of Divisors Function." Math. Comput.
26, 773777, 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
9space
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 ^jrounds, 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
welldefined 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(xl,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,yl)) = A(2,y.l) + 2
= i4(l ) A(2,y2))+2 = i4(2 I y2) + 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(xl,F(x t yl)),
(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, 128135, 1963.
Dotzel, G. "A Function to End All Functions." Algorithm:
Recreational Programming 2.4, 1617, 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. Cornput. 20, 156183, 1991.
Rose, H. E. Subrecursion, Functions, and Hierarchies. New
York: Clarendon Press, 1988.
Sloane, N. J. A. Sequence A001695/M2352 in "An OnLine
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, 669675, 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, 118133, 1928.
Conway, J. H. and Guy, R. K, The Book of Numbers. New
York: SpringerVerlag, pp. 6061, 1996.
Crandall, R. E. "The Challenge of Large Numbers." Sci.
Amer. 276, 7479, 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 2Manifolds.
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
AdamsBashforthMoulton Method
Addition Chain
15
AdamsBashforthMoulton Method
see Adams' Method
Adams' Method
Adams' method is a numerical METHOD for solving
linear FirstOrder 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 firstorder 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
+ ffivV4 + ^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
TORCORRECTOR METHODS, RUNGEKUTTA 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: McGrawHill,
pp. 1420, 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 ndigit 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 carrylookahead 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, 867871, 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 AdditionMultiplication 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. 111113, 1994.
AdditionMultiplication 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, 3031,
1975.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 8991, 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. 262263, 1994.
Hinden, H. J. "The Additive Persistence of a Number." J.
Recr. Math. 7, 134135, 1974.
Sloane, N. J. A. Sequences A031286 and A006050/M4683 in
"An OnLine Version of the Encyclopedia of Integer Se
quences."
Sloane, N. J. A. "The Persistence of a Number." J. Recr.
Math. 6, 9798, 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, 537549, 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+jk
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 SecondOrder 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 SelfAdjoint Operator, SturmLiouville
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.
AdlemanPomeranceRumely 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, 173206, 1983.
Bosma, W. and van der Hulst, M.P. "Faster Primality Test
ing." In Advances in Cryptology, Proc. Eurocrypt '89,
Houthalen, April 1013, 1989 (Ed. J.J. Quisquater). New
York: Springer Verlag, 652656, 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. lxxxivlxxxv, 1988.
Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi
Sums." Math. Comput. 42, 297330, 1984.
Cohen, H. and Lenstra, A. K. "Implementation of a New
Primality Test." Math. Comput 48, 103121, 1987.
Mihailescu, P. "A Primality Test Using Cyclotomic Exten
sions." In Applied Algebra, Algebraic Algorithms and
Error Correcting Codes (Proc. AAECC6, Rome, July
1988). New York: Springer Verlag, pp. 310323, 1989.
Adleman Rumely Primality Test
see AdlemanPomeranceRumely 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. 268275, 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 2D 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 ntuple 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 rotationenlargement transformation
V
= s
= s
cos a
— sin a
sin a
cos a
x — Xo
yyo
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 ArithmeticGeometric Mean
Agnesi's Witch
see Witch of Agnesi
Agnesienne
see Witch of Agnesi
Agonic Lines
see Skew Lines
AhlforsBers 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^ n1 V^ "1
y = > na n x = y ^na n x
n=0 n=l
OO
= ^^(n + l)a n +ix n (3)
TX0
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
54a 5 = 20a 5 = a 2 = 0. (12)
Continuing, it follows by INDUCTION that
a 2 = a$ = ag = an = . . . a3ni = (13)
for n = 1, 2, Now examine terms of the form £3^.
(14)
a 3 =
ae =
ao
3^2
^3 =
65 ~ (65)(32)
a& ao
ao
(15)
(16)
98 (98)(65)(32)'
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
43
a4
7^6 = (76)(43)
ar 01
ai
109 (109)(76)(43)"
(18)
(19)
(20)
AiryFock 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 AiryFock Functions, Airy Functions,
Bessel Function of the First Kind, Modified
Bessel Function of the First Kind
AiryFock Functions
The three AiryFock 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. 188190) 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 In(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 reexpressed
(8)
1 /Fir f 2 * 3/2 ^
A more commonly used definition of Airy functions is
given by Abramowitz and Stegun (1972, pp. 446447)
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 AlRYFoCK 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. 446452, 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. 234245, 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. 555562, 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(ir0 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 + Sn1
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 EqualArea 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
/(xx ,xi) Xi  X
/(xx 0) x 2 ) x 2  x
/(xx ,Xi,X 2 ) X 2  X
/(xx ,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. 9394, 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.
AjimaMalfatti Points
The lines connecting the vertices and corresponding
circlecircle intersections in Malfatti's Tangent Tri
angle Problem coincide in a point Y called the first
AjimaMalfatti 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
AjimaMalfatti 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, 163187, 1994.
Kimberling, C. "1st and 2nd AjimaMalfatti 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, 612613, 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. 6768, 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
poy
(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. 98103, 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. 3947, 1985.
Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with
Integer Sides." Amer. Math. Monthly 86, 686689, 1979.
Sloane, N. J. A. Sequence A005044/M0146 in 'An OnLine
Version of the Encyclopedia of Integer Sequences."
AleksandrovCech Cohomology
A theory which satisfies all the ElLENBERGSTEENROD
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 nspace 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, 422429, 1995.
Aleph
The Set Theory symbol (N) for the Cardinality of
an Infinite Set.
see also Aleph0 (N ), Aleph1 (Ni), Countable
Set, Countably Infinite Set, Finite, Infinite,
Transfinite Number, Uncountably Infinite Set
Aleph0 (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 ALEPH1, CARDINAL NUMBER, CONTINUUM,
Continuum Hypothesis, Countably Infinite Set,
Finite, Infinite, Transfinite Number, Uncount
ably Infinite Set
Aleph1 (Ni)
The Set Theory symbol for the smallest Infinite Set
larger than Alpha0 (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, nD SPACE has the same number of
points (c) as 1D Space, or any Finite Interval of 1
D Space (a Line Segment), as was first recognized by
Georg Cantor.
see also Aleph0 (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, 2633, 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. 8081, 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. 206207, 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. 206207, 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. 206207, 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 Conwaynormalized 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 ConwayAlexander
polynomials for common KNOTS include
Vtk
Vfek
VsSK
[11 =
[31 =
[l  i + :
x" 1 + 1
_1 +3x
(6)
(7)
_1 + la: + x 2 (8)
for the Trefoil Knot, FigureofEight 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 3BRAID. Also,
if
A K (e 27ri/5 )> f, (13)
then K cannot be represented as a closed 4braid (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. 165169, 1994.
Alexander, J. W. "Topological Invariants of Knots and
Links." Trans. Amer. Math. Soc. 30, 275306, 1928.
AlexanderSpanier Cohomology
Algebra 27
Alexander, J. W. "A Lemma on a System of Knotted
Curves." Proc. Nat. Acad. Set. USA 9, 9395, 1923,
Doll, H. and Hoste, J. "A Tabulation of Oriented Links."
Math. Comput. 57, 747761, 1991.
Jones, V. "A Polynomial Invariant for Knots via von Neu
mann Algebras." Bull. Amer. Math. Soc. 12, 103111,
1985.
Rolfsen, D. "Table of Knots and Links." Appendix C in
Knots and Links. Wilmington, DE: Publish or Perish
Press, pp. 280287, 1976.
Stoimenow, A. "Alexander Polynomials." http://www.
informatik.huberlin.de/stoimeno/ptab/alO.html.
Stoimenow, A. "Conway Polynomials." http://www.
informatik.huberlin.de/stoimeno/ptab/clO.html.
AlexanderSpanier 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 AlexanderSpanier 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
selfCONSlSTENT (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 exoticsounding
(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: PrenticeHall, 1991.
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGrawHill, pp. 3536, 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, 735737,
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 nD.
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. 347353, 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. 4849, 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 + . . . + CnlX + 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,
HermiteLindemann 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. 103107, 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. 347353, 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. 4151, 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 bothdirections continuous ONE
TOOne (HOMEOMORPHIC) transformations. The dis
cipline of algebraic topology is popularly known as
"RubberSheet 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
HOLEstructure, 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: 19001960. Boston, MA: Birkhauser, 1989.
Algebraic Variety
A generalization to nD 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,
BrauerSeveri 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 fcalgebroidal function
in G.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Algebroidal Functions."
§19 in Encyclopedic Dictionary of Mathematics. Cam
bridge, MA: MIT Press, pp. 8688, 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 AlKhwarizmi, an Arab mathematician
who wrote an influential treatise about algebraic meth
ods.
see also 196 ALGORITHM, ALGORITHMIC COMPLEXITY,
Archimedes Algorithm, BhaskaraBrouckner
Algorithm, BorchardtPfaff Algorithm, Bre
laz's Heuristic Algorithm, Buchberger's Algo
rithm, BulirschStoer Algorithm, Bumping Al
gorithm, CLEAN Algorithm, Computable Func
tion, Continued Fraction Factorization Algo
rithm, Decision Problem, Dijkstra's Algorithm,
Euclidean Algorithm, FergusonForcade 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,
LevineO '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, QuotientDifference Algo
rithm, Risch Algorithm, Schrage's Algorithm,
Shanks' Algorithm, Spigot Algorithm, Syracuse
Algorithm, Total Function, Turing Machine,
ZassenhausBerlekamp 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: PrenticeHall, 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. 197200, 1965.
Hogendijk, J. P. "AlMutaman'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 B2, Univ. Barcelona,
Barcelona, 1996.
Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tid~
skr. 18, 535, 1970.
Neumann, P. Submitted to Amer. Math. Monthly.
Riede, H. "Reflexion am Kugelspiegel. Oder: das Problem
des Alhazen." Praxis Math. 31, 6570, 1989.
Sabra, A. I. "ibn alHaytham's Lemmas for Solving 'Al
hazen's Problem'." Arch. Hist Exact Sci. 26, 299324,
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 196Algorithm, 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. 6062, 1994.
Guy, R. K. and Selfridge, J. L. "What Drives Aliquot Se
quences." Math. Cornput. 29, 101107, 1975.
Sloane, N. J. A. Sequences A003023/M0062 in "An OnLine
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.
AllPoles Model
see Maximum Entropy Method
AlladiGrinstead Constant
N.B. A detailed online 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
23572 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
332 2 .572 4
335
7 • 2 3  2 3
a(9) =
32 2 2 2 .572 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, 452458, 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: NorthHolland, 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 9crossing 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. 139146, 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 nearidentity 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 nearidentity is given by
i[cos(^) + cosh(^) + 2cos(^^)cosh(^V / 2)]
= 1 + 2.480... x 10" 13 (5)
(W. Dubuque). Other remarkable nearidentities 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 (191314). 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 jFUNCTlON. 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 jFUNCTlON 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 =
jtv/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, jFunction, Pi
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer Verlag, pp. 9091, 1994.
Hermite, C. "Sur la theorie des equations modulaires." C.
R. Acad. Sci. (Paris) 48, 10791084 and 10951102, 1859.
Hermite, C. "Sur la theorie des equations modulaires." C. R.
Acad. Sci. (Paris) 49, 1624, 110118, and 141144, 1859.
Kronecker, L. "Uber die Klassenzahl der aus Werzeln der Ein
heit gebildeten komplexen Zahlen." Monatsber. K. Preuss.
Akad. Wiss. Berlin, 340345. 1863.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure Appl. Math. 45, 350372, 19131914.
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. 5776,
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, QuasiPerfect, 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 OnLine
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 fcalmost primes is denoted Ph.
The Primes correspond to the "1almost prime" num
bers 2, 3, 5, 7, 11, . . . (Sloane's A000040). The 2almost
prime numbers correspond to SEMIPRIMES 4, 6, 9, 10,
14, 15, 21, 22, ... (Sloane's A001358). The first few
3almost 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 4almost primes are
16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, . . . (Sloane's
A014613). The first few 5almost 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 OnLine
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
twentyeight
twelve
twentytwo eighteen
fifteen two
eight twentyfive
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, 2845,
1986.
Sallows, L. C. F. "Alphamagic Squares. 2." Abacus 4, 2029
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. 9095, 1975.
Madachy, J. S. "Alphametics." Ch. 7 in Madachy p s Mathe
matical Recreations. New York: Dover, pp. 178200 1979.
Alternate Algebra
Let A denote an RAlgebra, so that A is a Vector
Space over R and
AxA^A (1)
(x,y) \>xy. (2)
Then A is said to be alternate if, for all x,y £ A,
(xy)yx(yy) (3)
(xx)y = x(xy). (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 nonABELIAN
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 OnLine
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 FigureofEight 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, 195242, 1988.
Murasugi, K. "Jones Polynomials and Classical Conjectures
in Knot Theory." Topology 26, 297307, 1987.
Sloane, N. J. A. Sequence A002864/M0847 in "An OnLine
Version of the Encyclopedia of Integer Sequences,"
Thistlethwaite, M. "A Spanning Tree Expansion for the Jones
Polynomial." Topology 26, 297309, 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, 113171, 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. 159164, 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 EvENnumbered
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
EntringerArnold Triangle, Tangent Number
References
Andre, D. "Developments de seccc et tan a?." C. R. Acad.
Sci. Paris 88, 965967, 1879.
Andre, D. "Memoire sur le permutations alternees." J. Math.
7, 167184, 1881.
Arnold, V. I. "BernoulliEuler Updown Numbers Associ
ated with Function Singularities, Their Combinatorics and
Arithmetics." Duke Math. J. 63, 537555, 1991.
Arnold, V. I. "Snake Calculus and Combinatorics of Ber
noulli, Euler, and Springer Numbers for Coxeter Groups."
Russian Math. Surveys 47, 345, 1992.
Bauslaugh, B. and Ruskey, F. "Generating Alternating Per
mutations Lexicographically." BIT 30, 1726, 1990.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer Verlag, pp. 110111, 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. 6469, 1965.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 6975, 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, 4454, 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 OnLine
Version of the Encyclopedia of Integer Sequences."
Alternating Series
A Series of the form
k=l
00
Dd
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. 293294, 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. 5557, 1991.
Pinsky, M. A. "Averaging an Alternating Series." Math.
Mag. 51, 235237, 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, 181200, 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. 161165). 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. 261262).
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 3640,
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 piecewiselinear 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 WellDefined.
see also WellDefined
AmbroseKakutani 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= 1152^
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 ntuples 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 AIM239, Feb. 1972.
Borho, W. and Hoffmann, H. "Breeding Amicable Numbers
in Abundance." Math. Comput 46, 281293, 1986.
Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers
and Their Distribution." Math. Comput. 24, 431432,
1970.
Cohen, H. "On Amicable and Sociable Numbers." Math.
Comput. 24, 423429, 1970.
Costello, P. "Amicable Pairs of Euler's First Form." J. Rec.
Math. 10, 183189, 19771978.
Costello, P. "Amicable Pairs of the Form (i,l)." Math. Com
put. 56, 859865, 1991.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 3850,
1952.
Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4,
108111, 19551956.
Erdos, P. "On Asymptotic Properties of Aliquot Sequences."
Math. Comput. 30, 641645, 1976.
Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math
ematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleightof~Mind from
Scientific American. New York: Vintage, pp. 160171,
1978.
Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer Verlag,
pp. 5559, 1994.
Lee, E. J. "Amicable Numbers and the Bilinear Diophantine
Equation." Math. Comput. 22, 181197, 1968.
Lee, E. J. "On Divisibility of the Sums of Even Amicable
Pairs." Math. Comput. 23, 545548, 1969.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, 1." J. Rec. Math. 5, 7793, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, II." J. Rec. Math. 5, 153173, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, HI." J. Rec. Math. 5, 231249, 1972.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 145 and 155156, 1979.
Moews, D. and Moews, P. C. "A Search for Aliquot Cycles
and Amicable Pairs." Math. Comput. 61, 935938, 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. 96100, 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, 217222, 1977.
Pomerance, C. "On the Distribution of Amicable Numbers,
II." J. reine angew. Math. 325, 182188, 1981.
Scott, E. B. E. "Amicable Numbers." Scripta Math. 12,
6172, 1946.
Sloane, N. J. A. Sequences A002025/M5414 and A002046/
M5435 in "An OnLine 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, 219223,
1984.
te Riele, H. J. J. "Computation of All the Amicable Pairs
Below 10 10 ." Math. Comput. 47, 361368 and S9S35,
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 19431993: A
HalfCentury of Computational Mathematics (Vancouver,
BC, August 913, 1993) (Ed. W. Gautschi). Providence,
Rl: Amer. Math. Soc, pp. 577581, 1994.
$$ Weisstein, E. W. "Sociable and Amicable Num
bers." http : //www . astro . Virginia, edu/ eww6n/math/
notebooks/Sociable .m.
Amicable Quadruple
An amicable quadruple as a QUADRUPLE (a, b, c, d) such
that
a(a) = a(b) — a(c) — cr(d) — a + b + c + d,
where cr(n) is the DIVISOR FUNCTION.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer Verlag, p. 59, 1994.
Amicable Triple
Dickson (1913, 1952) defined an amicable triple to be a
TRIPLE of three numbers (Z,m, n) such that
s(/) = m + n
${m) = I + n
s(n) = / + m,
where s(n) is the Restricted Divisor Function
(Madachy 1979). Dickson (1913, 1952) found eight sets
of amicable triples with two equal numbers, and two
sets with distinct numbers. The latter are (123228768,
103340640, 124015008), for which
s(12322876) = 103340640 + 124015008 = 227355648
s(103340640) = 123228768 + 124015008 = 24724377
5(124015008) = 123228768 + 10334064 = 226569408,
Amortization
Amplitude 41
and (1945330728960, 2324196638720, 2615631953920),
for which
s(1945330728960) = 2324196638720+2615631953920
= 4939828592640
s(2324196638720) = 1945330728960 + 2615631953920
= 4560962682880
5(2615631953920) = 1945330728960 + 2324196638720
= 4269527367680.
A second definition (Guy 1994) defines an amicable
triple as a TRIPLE (a, &, c) such that
a (a) = a(b) — o~(c) = a + b + c,
where a(n) is the DIVISOR FUNCTION. An example is
(2 2 3 2 5 11, 2 5 3 2 7, 2 2 3 2 71).
see also Amicable Pair, Amicable Quadruple
References
Dickson, L. E. "Amicable Number Triples." Amer. Math.
Monthly 20, 8492, 1913.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, p. 50, 1952.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer Verlag, p. 59, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 156, 1979.
Mason, T. E. "On Amicable Numbers and Their Generaliza
tions." Amer, Math. Monthly 28, 195200, 1921.
$$ 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 (FigureofEight 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. 311319, 1985.
Jones, V. "A Polynomial Invariant for Knots via von Neu
mann Algebras." Bull. Amer. Math. Soc. 12, 103111,
1985.
Jones, V. "Hecke Algebra Representations of Braid Groups
and Link Polynomials." Ann. Math. 126, 335388, 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:
SpringerVerlag, 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: SpringerVerlag, 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:
McGrawHill, pp. 356374, 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 ntuples 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, xAxis, yAxis, zAxis
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. 7277, 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. 378380, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part I. New York: McGrawHill, pp. 389390 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 "JacobiLie
bracket" of a VECTOR FIELD.
see also Lie Algebroid
References
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744752, 1996.
Anchor Ring
AndrewsSchur 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,
436437, 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, 6475, 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
AndersonDarling Statistic
A statistic defined to improve the Kolmogorov
SMIRNOV TEST in the TAIL of a distribution.
see also KolmogorovSmirnov 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 2spheres in M 4 obtained by Spinning in
tertwined arcs. The link consists of a knotted 2sphere
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
AndrewsSchur Identity
£« fc2+a
fc=0
2n — k + a
k
_ V~^ 10fc 2 + (4al)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
55/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 _ g20j4a+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
RogersRamanujan 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, 19, 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, 6061, 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 OnLine
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. 498499, 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, SteinerLehmus
Theorem, Trisection
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 910, 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,
37102, 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)
(VW) :
f(x)g(x)dx
New York: Dover, pp. 99100,
By itself, the Bra is a Covariant 1 Vector, and the
Ket is a Covariant OneForm. These terms are com
monly used in quantum mechanics.
see also Bra, Differential &Form, Ket, OneForm
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 NonEuclidean Geometry. New
York: Dover, pp. 3132 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 CrossRatio
Anisohedral Tiling
A fcanisohedral 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, 585588, 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, 575582, 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. 113129, 1979.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 8687, 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 nToRUS,
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 nTORUS.
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 FlNITETOONE
Factor of an Anosov Automorphism of a Nilman
ifold. It has been proved that any Anosov diffeomor
phism on the nTORUS 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, 747817, 1967.
Anosov Flow
A Flow defined analogously to the Anosov Diffeo
morphism, except that instead of splitting the TAN
GENT BUNDLE into two invariant subBUNDLES, they
are split into three (one exponentially contracting, one
expanding, and one which is 1dimensional 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 Kl 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, 3135, 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, 163187, 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 OnLine
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 Li ^1
a : : c
A'
B = a : —0 : c
s^r 1 71 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. 1921, 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 ClRCLEpreserving 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 49 are illustrated above (Madachy 1979).
see also HETEROSQUARE, MAGIC SQUARE, TALISMAN
Square
References
Disc.
Abe, G. "Unsolved Problems on Magic Squares."
Math. 127, 313, 1994.
Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 103113, 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. 5658, 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 — —akkj
(2)
(3)
A Semiregular Polyhedron constructed with 2 n
gons and 2n TRIANGLES. The 3antiprism is simply the
Octahedron. The Duals are the Trapezohedra.
The Surface Area of a ngonal 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. 8586, 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(AA 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 +
aTii
&2n + a„2
Z(l nn
(8)
A + A T =
which is symmetric, and
AA 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 LeviCivita Symbol, a.k.a. the
Permutation Symbol.
see also Symmetric Tensor
Antoine's Horned Sphere
A topological 2sphere in 3space 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 SimplyConnected Sur
face Bounding a Region which is not SimplyConnected."
Proc. Nat Acad. Sci. 10, 810, 1924.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 7679, 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=fC
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. 7374, 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 online 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(nl) = (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 (snkl)\
(sn + & + l)!(fc + l)
(16)
s — 1 to obtain
c(3) = §f;(ir 1 «i ? . (it)
For 5 = 2,
oo „
ffl) 1 ^ nni 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
(4nfl)(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(2nhA; + 2)!(n + l) 2 (2nhl) 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 = 120205690315732 .... (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 WilfZeilberger Pair iden
tity with
_ fc n! 6 (2nfcl)!fc! 3
*(n,k)( 1) 2(n + A . + 1)!2(2rl )!3> W
5 = 1, and t = 1, giving the rapidly converging
,vnV\ 1 ^ rc! 1 °(205n 2 + 250n + 77)
QWZ^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, WilfZeilberger
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, 13, 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, 268272, 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, 6798, 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, 219220, 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
TI977. 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 OnLine 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. 1,196203,
1979.
Zeilberger, D. "The Method of Creative Telescoping." J.
Symb. Comput. 11, 195204, 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(le 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 OnLine
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
(2sided) 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
osr
8aV27r (2ir*a)«/»
Cosine
c °s(ff)
4aca B (2jrafc)
TT(l160 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
1S
W:(k)
where
B A (x) =
Bj(k) =
Hm A (x) =
Hrm(k) =
(1ZX \ / 27TX \
— 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)
(la 2 A: 2 )(l4a 2 fc 2 )
0.54 + 0.46 cos (—)
a(1.08  0.64a 2 fc 2 ) sinc(27rafe)
" l4a 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)
' l4a 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(27rafc)
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^ +*(£ + £)
= (l2o 1 ) + oi(i + l) = (16)
ai =
ao
 5
3 ° _ 5
 +  63 + 25 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. 5557,
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. 95101, 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. 3132, 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, 259264,
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. 547548, 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. 491529,
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 AApollonian circle. Similarly,
construct the B and CApollonian 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 DTriangle 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 294299, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 1423, 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, 163187, 1994.
Kimberling, C; Iwata, S.; and Hidetosi, F. "Problem 1091
and Solution." Crux Math. 13, 128129 and 217218,
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 threeClRCLE 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 )2xXi2yyi±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  r2 )  (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 125127, 1996.
Dorrie, H. "The Tangency Problem of Apollonius." §32 in
100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 154160,
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. 118121, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 4851, 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. 97105, 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. 126138, 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 ti0
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: GauthierVillars,
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. 209210, 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. 210211, 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 xz
Plane are
z± = ± V / R 2 (xr) 2
for R > r and x E [— (r + R), r + R]. It is the outside
surface of a Spindle TORUS.
see also Bubble, Lemon, SphereSphere 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: McGrawHill, 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 (lr) 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, 7076,
1955.
Bankoff, L. "Are the Twin Circles of Archimedes Really
Twins?" Math. Mag. 47, 214218, 1974,
Bankoff, L. "How Did Pappus Do It?" In The Mathematical
Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber,
and Schmidt, pp. 112118, 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, 1924, 1940.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 1828, 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. 116117, 1929.
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 5455, 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 linedisjoint 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, ClRCLEClRCLE 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 4POLYHEX.
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 BorchardtPfaff 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(62 k ). The first iteration of ARCHIMEDES'
Recurrence Formula then gives
26 4^3 24^ nA , n /,
ffll = 7Tvr = ^vi = 24(2 ^ ) (6)
h = yj 24(2  V3) • 6 = 12\/2 \/3
6(v / 6v / 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 96gons) 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 96gon) 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, 1735, 1983.
Phillips, G. M. "Archimedes in the Complex Plane." Amer.
Math. Monthly 91, 108114, 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, 121171, 1880.
Archibald, R. C. "Cattle Problem of Archimedes." Amer.
Math. Monthly 25, 411414, 1918.
Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains. New York: Dover,
pp. 249252, 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, 18821884.
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. 37, 1965.
Grosjean, C. C. and de Meyer, H. E. "A New Contribution
to the Mathematical Study of the CattleProblem of Arch
imedes." In Constantin Caratheodory: An International
Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ:
World Scientific, pp. 404453, 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, 305319, 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' HatBox 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. 3132, 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 ngon 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 22ntan(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. 139140).
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. 116126; Catalan 1865, pp. 2532; Coxeter 1940,
p. 394; Coxeter et al. 1954; Lines 1965, pp. 202203;
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 ngonal 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 ngonal 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
(35) 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. 269286, 1974.
Catalan, E. "Memoire sur la Theorie des Polyedres." J.
I'Ecole Polytechnique (Paris) 41, 171, 1865.
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six
and Seven Dimensions." Proc. Cambridge Phil Soc. 24,
19, 1928.
Coxeter, H. S. M. "Regular and Semi Regular Polytopes I."
Math. Z. 46, 380407, 1940.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Coxeter, H. S. M.; LonguetHiggins, M. S.; and Miller,
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon
don Ser. A 246, 401450, 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. 7986, 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. 75334, 1864.
Kraitchik, M. Mathematical Recreations. New York:
W. W. Norton, pp. 199207, 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. 3435, 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.; LonguetHiggins, M. S.; and Miller,
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon
don Ser. A 246, 401450, 1954.
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, pp. 6672, 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. 6970, 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 ccAxiS together with its reflection in
the zAxiS. Rotating this with uniform angular veloc
ity about its center will result in uniform linear motion
of the point where it crosses the yAxiS.
Archimedes' Spiral Inverse
AreaPreserving Map 69
see also ARCHIMEDEAN SPIRAL
References
Gardner, M. The Unexpected Hanging and Other Mathemat
ical Diversions. Chicago, IL: Chicago University Press,
pp. 106107, 1991.
Gray, A Modern Differential Geometry of Curves and Sur
faces. Boca Raton, FL: CRC Press, pp. 6970, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 186187, 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 xAxis 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 3D
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. 259260, 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
^Tigon — 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. 259260, 1993.
AreaPreserving Map
A Map F from R n to W 1 is AREApreserving 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, SelfTransversality Theorem
References
Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254268, 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 2component 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. 223231, 1994.
Jones, V. "A Polynomial Invariant for Knots via von Neu
mann Algebras." Bull. Amer. Math. Soc. 12, 103111,
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 xAxiS as the Real axis and yAxiS 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
nBni = —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, 4050, 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 oneargument
function, the BINOMIAL Coefficient (™) is a two
argument function, and the Hypergeometric Func
tion 2 Fi (a, b; c; z) is a fourargument 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
2xi 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, 411415, 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 3D Space, or an infinite dimensional
Euclidean Space) are all the same: Hi (Aleph1), so
the points of any of these can be put in a OneTOOne
correspondence with those of any other.
see also ZENO'S PARADOXES
References
Ballew, D. "The Wheel of Aristotle." Math. Teacher 65,
507509, 1972.
Costabel, P. "The Wheel of Aristotle and French Considera
tion of Galileo's Arguments." Math. Teacher 61, 527534,
1968.
Drabkin, I. "Aristotle's Wheel: Notes on the History of the
Paradox." Osiris 9, 162198, 1950.
Gardner, M. Wheels, Life, and other Mathematical Amuse
ments. New York: W. H. Freeman, pp. 24, 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. 4850, 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 LowenheimerSkolem 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.
ArithmeticGeometric Mean
The arithmeticgeometric 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 ArithmeticGeometric
Mean is
f°° x p ~ 2 dx
(14)
which is related to solutions of the differential equation
x(lx p )Y" + [lfr+l)x p ]Y'(pl)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
1u
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 ArithmeticHarmonic Mean
References
Abramowitz, M. and Stegun, C. A. (Eds.), "The Process
of the ArithmeticGeometric Mean." §17.6 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 571 ad 598599, 1972.
Borwein, J. M. Problem 10281. "A Cubic Relative of the
AGM." Amer, Math. Monthly 103, 181183, 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, 691701, 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. 906907, 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.
ArithmeticHarmonic 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
ArithmeticLogarithmicGeometric 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
LogarithmicGeometric 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,
{yfaVbf = a2\/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 /cStatistics for a GAUSSIAN DISTRIBUTION, the
Unbiased Estimator for the Variance is given by
N
N  1
where
var (a:) =
JV1
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 ArithmeticGeometric Mean, Arith
meticHarmonic Mean, Carleman's Inequal
ity, Cumulant, Generalized Mean, Geomet
ric Mean, Harmonic Mean, HarmonicGeometric
Mean, Kurtosis, Mean, Mean Deviation, Median
(Statistics), Mode, Moment, Quadratic Mean,
RootMeanSquare, 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 MeanGeometric 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, 151156, 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,ki + d = afc2 + 2d •
:ai+d(fcl). (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
nl
= 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 busywork 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. 1213, 1996.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 164, 1989.
Armstrong Number
The ndigit 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 OnLine
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 nonHamiltonian, 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 _
la 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
^+ = ^\ / 5010v / 5
1(1 + V5)
Similarly, for <j , the solution is
y = ±(V5l)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 MODELOCKED 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 twodimensional 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 ClebschAronhold 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 OnLine
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 ddimensional array is then indicated as
m x n x • • • x p. The most common type of array en
d
countered is the 2D 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 leftright mirror images is k 1 (in general,
k mn/2 ), as is the number equal to their updown 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 leftright reflection
with only updown 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,
QuotientDifference 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
= mm 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, SteinhausMoser Notation
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer Verlag, pp. 5962, 1996.
Guy, R. K. and Selfridge, J. L. "The Nesting and Roost
ing Habits of the Laddered Parenthesis." Amer. Math.
Monthly 80, 868876, 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, 12351242, 1976.
Vardi, I. Computational Recreations in Mathematica. Red
wood City, CA: Addison Wesley, pp. 11 and 226229, 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. 104110, 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, 118120, May 1994.
Tucker, A. "The Art Gallery Problem." Math Horizons,
pp. 2426, Spring 1994.
Wagon, S. "The Art Gallery Theorem." §10.3 in Mathema
tica in Action. New York: W. H. Freeman, pp. 333345,
1991.
Articulation Vertex
A VERTEX whose removal will disconnect a GRAPH, also
called a Cut Vertex.
see also Bridge (Graph)
References
Chartrand, G. "CutVertices 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, 8083, 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: SpringerVerlag, 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, 209220, 1967.
Ireland, K. and Rosen, M. A Classical Introduction to Mod
ern Number Theory, 2nd ed. New York: SpringerVerlag,
1990.
Ribenboim, P. The Book of Prime Number Records. New
York: SpringerVerlag, 1989.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 8083, 1993.
Wrench, J. W. "Evaluation of Artin's Constant and the Twin
Prime Constant." Math. Comput. 15, 396398, 1961.
Artin L Function
An Artin Lfunction 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 Lfunction is
L(s,Q(i)/Q,sgn):
n itt^
where ( — 1/p) is a Legendre Symbol, which is equiv
alent to the Euler LFUNCTION. 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, 537549, 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, 251260, 1928.
Artin, E. "Zur Arithmetik hyperkomplexer Zahlen." Hamb.
Abh. 5, 261289, 1928.
Artistic Series
A Series is called artistic if every three consecutive
terms have a common threeway 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, 3847, 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 = nAB,
(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 IRalgebra, so that A is a VECTOR SPACE over
Rand
Ax A+ A (2)
(x,y) \>xy.
(3)
Then A is said to be massociative if there exists an mD
Subspace S of A such that
(yx)z = y(xz)
(4)
for all y,z € A and x € S. Here, VECTOR MULTIPLI
CATION x • y is assumed to be Bilinear. An nD 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 4cusped 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 zAxis 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 + — (xx ). (16)
Xo
The equation of the Envelope is given by the simulta
neous solution of
U(x,y,x ) = y+ V L xn X ° (xxo) =
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. 172175, 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: //wwwgroups .dcs .stand.ac.uk/history/Curves
/Astroid. html.
Yates, R. C. "Astroid." A Handbook on Curves and Their
Properties. Ann Arbor, Ml: J. W. Edwards, pp. 13, 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,
274284, 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
AtiyahSinger 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, 320331, 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. 78, 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. 339346, 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: McGrawHill, pp. 434443, 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.
AtiyahSinger Index Theorem
A theorem which states that the analytic and topological
"indices" are equal for any elliptic differential operator
on an nD 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, 322433, 1963.
Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Oper
ators I, II, III." Ann. Math. 87, 484604, 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
AtkinGoldwasserKilianMorain Certificate
Augmented Amicable Pair
AtkinGoldwasserKilianMorain 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 AtkinGoldwasserKilianMorain 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, 2968, 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. 316329, 1986.
Morain, F. "Implementation of the AtkinGoldwasserKilian
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,
483494, 1985.
Wunderlich, M. C "A Performance Analysis of a Simple
PrimeTesting Algorithm." Math. Comput. 40, 709714,
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 nD 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, 331332, 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 4fc2 + : = ^ 2 fei _ 2 fc + 1 )( 2 2 * 1 + 2 k + 1) (2)
3 6fc3 + x = ^2*i + 1 j( 3 2fci _ 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. lxviiilxxii, 1988.
WagstafT, S. S. Jr. "Aurifeullian Factorizations and the Pe
riod of the Bell Numbers Modulo a Prime." Math. Corn
put. 65, 383391, 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=(le 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> =
(le 2 sin 2 0) 2
2cos0
sin<j)
1e 2
+
1 — e 2 sin <j>
which can be written
1^/lesin.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 EqualArea 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(tr)(dr)
J OO
OO
f(r)f(tr)dr
f(r)f(tr)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, CROSSCORRELATION, QUAN
TIZATION Efficiency, WienerKhintchine 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. 538539, 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 nautomorphic. For example, 1 • 5 2 = 25
and 1 ■ 6 2 = 36 are 1automorphic and 2 ■ 8 2 — 128
and 2 • 88 2 = 15488 are 2automorphic. de Guerre and
Fairbairn (1968) give a history of automorphic numbers.
The first few 1automorphic numbers are 1, 5, 6, 25,
76, 376, 625, 9376, 90625, . . . (Sloane's A003226, Wells
1986, p. 130). There are two 1automorphic numbers
with a given number of digits, one ending in 5 and one in
6 (except that the 1digit 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 1automorphic
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 10digit nautomorphic
numbers.
n nAutomorphic 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 AIM239, Feb. 1972.
Autoregressive Model 87
Fairbairn, R. A. "More on Automorphic Numbers." J. Recr.
Math. 2, 170174, 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, 173179, 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. 7778,
1942.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 3454 and 175176, 1979.
Sloane, N. J. A. Sequences A016090, A003226/M3752, and
A007185/M3940 in "An OnLine 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 secondorder 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 firstorder
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\ = (\xin\).
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 POINTPOINT DlSTANCE
1D
AxKochen Isomorphism Theorem
Let P be the Set of PRIMES, and let Q p and Z p (t) be the
Fields of pADic 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 selfevidently 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, CantorDedekind 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, T2Separation Axiom, Theorem, Zermelo's
Axiom of Choice, ZermeloFraenkel 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. 187188, 1990.
Smale, S. "Different iable Dynamical Systems." Bull Amer.
Math. Soc. 73, 747817, 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 subBUNDLES, they
are split into three (one exponentially contracting, one
expanding, and one which is 1dimensional 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, ZermeloFraenkel 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, 11431148, 1963.
Cohen, P. J. "The Independence of the Continuum Hypothe
sis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105110, 1964.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer Verlag, pp. 274276, 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, ccAxiS, yAxiS, zAxis
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 3D figures onto the Plane.
see also CROSSSECTION, 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. 322323, 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. 191202, 1987.
Azimuthal Projection
see Azimuthal Equidistant Projection, Lam
bert Azimuthal EqualArea Projection, Ortho
graphic Projection, Stereographic Projection
B* Algebra
B
BSpline 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 online 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, MianChowla Sequence
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/erdos/erdos .html.
Guy, R. K. "Packing Sums of Pairs," "ThreeSubsets 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. 115118, 121123, 228229, and
232233, 1994.
Sloane, N. J. A. Sequence A005282/M1094 in "An OnLine
Version of the Encyclopedia of Integer Sequences."
Zhang, Z. X. "A B2Sequence with Larger Reciprocal Sum."
Math. Comput. 60, 835839, 1993.
Zhang, Z. X. "Finding Finite B2Sequences with Larger m —
a™ 1 ' 2 ." Math. Comput. 63, 403414, 1994.
B p  Theorem
If Op' (G) = 1 and if a? is a pelement of G, then
L p ,(C g (x)<E(Cg(x)),
where L p > is the pLAYER.
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)
tU
ti\p Ci
(3)
ii+p+1 — *i+l
Then the curve defined by
C(t) = £p<M,p(t)
(4)
(5)
is a Bspline. Specific types include the nonperiodic B
spline (first p + 1 knots equal and last p + 1 equal to
1) and uniform Bspline (INTERNAL KNOTS are equally
spaced). A BSpline with no INTERNAL KNOTS is a
Bezier Curve.
The degree of a Bspline 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 Bspline is a Bspline whose first p + 1
knots are equal to and last p f 1 knots are equal to
1. A uniform Bspline is a Bspline whose INTERNAL
Knots are equally spaced.
see also Bezier Curve, NURBS Curve
92
BTree
Backtracking
BTree
Btrees were introduced by Bayer (1972) and Mc
Creight. They are a special mary balanced tree used in
databases because their structure allows records to be
inserted, deleted, and retrieved with guaranteed worst
case performance. An nnode £?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 Btrees to store disk directories (Bene
dict 1995). A Btree 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 23 Tree is a Btree of order 3. The number of
Btrees 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 RedBlack Tree
References
Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc
tures and Algorithms. Reading, MA: AddisonWesley,
pp. 369374, 1987.
Benedict, B. Using Norton Utilities for the Macintosh. Indi
anapolis, IN: Que, pp. B17B33, 1995.
Beyer, R. "Symmetric Binary jBTrees: Data Structures and
Maintenance Algorithms." Acta Informat. 1, 290306,
1972.
Ruskey, F. "Information on BTrees." http://sue.csc.uvic
. ca/~cos/inf /tree/BTrees .html.
Sloane, N. J. A. Sequence A014535 in "An OnLine 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.
BACCAB Identity
The Vector Triple Product identity
A x (B x C) = B(A ■ C)  C(A • B).
This identity can be generalized to nD
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
&ni " b n _i
BACCAB Rule
see BACCAB IDENTITY
Bachelier Function
see Brown Function
Bachet's Conjecture
see Lagrange's FourSquare 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—toc
<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.
BackusGilbert Method
Baire Category Theorem 93
BackusGilbert 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, 169205,
1968.
Backus, G. E. and Gilbert, F. "Uniqueness in the Inversion
of Inaccurate Gross Earth Data." Phil Trans. Roy. Soc.
London Ser. A 266, 123192, 1970.
Loredo, T. J. and Epstein, R. I. "Analyzing GammaRay
Burst Spectral Data." Astrophys. J. 336, 896919, 1989.
Parker, R. L. "Understanding Inverse Theory." Ann. Rev.
Earth Planet Sci. 5, 3564, 1977.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter
ling, W. T. "BackusGilbert Method." §18.6 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 806809, 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 ~ fpi) ~ (fpi ~ /pz)
= fp~ 2 /pl + fp2
(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.
BaderDeuflhard Method
A generalization of the BulirschStoer Algorithm
for solving Ordinary Differential Equations.
References
Bader, G. and Deuflhard, P. "A SemiImplicit MidPoint
Rule for Stiff Systems of Ordinary Differential Equations."
Numer. Math. 41, 373398, 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, 6165 (Mar.) and
5859 (Apr.), 1996.
Gardner, M. "The Binary Gray Code." In Knotted Dough
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, pp. 1517, 1986.
Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical
Recreations. New York: W. W. Norton, pp. 8991, 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_ /i3\ 2 1
n + \2J nM + \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 2D 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 283284, 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 \(1X 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 gDlMENSION is
aln(±)+/31n()
D 1 = 1 + Va) )*' ■ (4)
' ta (£)+*»»(*)'
If A a = A&, then the general gDlMENSION 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. 8182, 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 nball, denoted B n , is the interior of a SPHERE
S™" 1 , and sometimes also called the nDlSK. (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 nD 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, 653659, 1986.
Hall, G. R. "Acute Triangles in the nBall." J. Appl. Prob.
19, 712715, 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<nl /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
socalled 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,
436437, 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, 251259, 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. 6797,
1968.
Hilton, P. and Pedersen, J. "The Ballot Problem and Cata
lan Numbers." Nieuw Archief voor Wiskunde 8, 209216,
1990.
Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener
alization, and Their Uses." Math. Intel. 13, 6475, 1991.
Kraitchik, M. "The BallotBox 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 NonAssociative
Products." Bull Amer. Math. Soc. 54, 352360, 1948.
Vardi, I. Computational Recreations in Mathematica. Red
wood City, CA: Addison Wesley, pp. 185187, 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.
BanachHausdorffTarski 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
BanachSteinhaus 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 BanachTarski 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, 224226, 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, 214218, 1974.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 1828, 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. 910, 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, 431438, 1997.
Barnes G Function
see GFunction
Barnes' Lemma
If a Contour in the Complex Plane is curved such
that it separates the increasing and decreasing sequences
of Poles, then
2m . v
</ —too
+ s)r(0 + s)T('ys)r(5s)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 CoxeterTodd 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,
4763, 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. 127129, 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. 156163, 1991.
Barrier
A number n is called a barrier of a numbertheoretic
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. 6465, 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 4y 2 \z 2 (20)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, 173186, 1996.
Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV
Mitteilungen 4, 1720, 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 Barthsextic is a SEXTIC SURFACE in complex
threedimensional 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, 173186, 1996.
Endrafl, S. "Flachen mit vielen Doppelpunkten." DMV
Mitteilungen 4, 1720, 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 oneargument 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, (lj)<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 . — 2rrikx\ ( , % \
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, 116, 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 Cini ... 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 . cutthe
knot . com/binary .html.
Lauwerier, II. Fractals: Endlessly Repeated Geometric Fig
ures. Princeton, NJ: Princeton University Press, pp. 611,
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
3D. 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 ndimensional
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, 6798, 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:
AddisonWesley, 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, HofstadterConway $10,000 Se
quence, Hofstadter's QSequence, 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. 183191, 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. 98100,
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 nonBayesian 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. 799806, 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: AddisonWesley, 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. 270271, 1992.
Beam Detector
N. B. A detailed online 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 2arc 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 =2tt26>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
3arc 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, 249266, 1984.
Faber, V. and Mycielski, J. "The Shortest Curve that Meets
All Lines that Meet a Convex Body." Amer. Math.
Monthly 93, 796801, 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,
127132, 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 198283). 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) = 666, (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 197576). 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, 9697, 197576.
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, 8587,
19821983.
Bee
A 4P0LYHEX.
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.
BehrensFisher Test
see FisherBehrens 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, 174176, 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. 2930, 1973.
Beauzamy and Degot's Identity
For P, Q, R, and S POLYNOMIALS in n variables
[PQ,RS]= J^
A
ii,...,t n >0
iili 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 AreaPreserving projection which
uses 30° N as the nodistortion 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. 379381, 1972.
Spanier, J. and Oldham, K. B. "The Kelvin Functions."
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi
sphere, pp. 543554, 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)nl
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, 411419, 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. 9194, 1996.
de Bruijn, N. G. Asymptotic Methods in Analysis. New York:
Dover, pp. 102109, 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. 3536,
1992.
Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p,
of First Order Bell Exponential Integrals." Math. Comput.
16, 416423, 1962.
Lovasz, L. Combinatorial Problems and Exercises, 2nd ed.
Amsterdam, Netherlands: NorthHolland, 1993.
Pitman, J. "Some Probabilistic Aspects of Set Partitions."
Amer. Math. Monthly 104, 201209, 1997.
Rota, G.C. "The Number of Partitions of a Set." Amer.
Math. Monthly 71, 498504, 1964.
Sloane, N. J. A. Sequence A000110/M1484 in "An OnLine
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."
258277, 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
ko
see also Bell Number, Clark's Triangle, Leibniz
Harmonic Triangle, Number Triangle, Pascal's
Triangle, SeidelEntringerArnold 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 [ k1 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 EulerLagrange 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, EulerLagrange 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. 1314, 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
XRay 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, 551572, 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. "BaseInvariance Implies Benford 's Law." Proc.
Amer. Math. Soc. 12, 887895, 1995.
Hill, T. P. "The SignificantDigit Phenomenon." Amer.
Math. Monthly 102, 322327, 1995.
Hill, T. P. "A Statistical Derivation of the SignificantDigit
Law." Stat Sci. 10, 354363, 1996.
Hill, T. P. "The First Digit Phenomenon." Amer. Sci. 86,
358363, 1998.
Ley, E. "On the Peculiar Distribution of the U.S. Stock In
dices Digits." Amer. Stat. 50, 311313, 1996.
Newcomb, S. "Note on the Frequency of the Use of Digits in
Natural Numbers." Amer. J. Math. 4, 3940, 1881.
Nigrini, M. "A Taxpayer Compliance Application of Ben
ford's Law." J. Amer. Tax. Assoc. 18, 7291, 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, 109119, Dec. 1969.
Raimi, R. A. "The First Digit Phenomenon." Amer. Math,
Monthly 83, 521538, 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 PrevostFechnerBenham
Subjective Colors." Psycholog. Bull. 46, 97136, 1949.
Festinger, L.; Allyn, M. R.; and White, C. W. "The Percep
tion of Color with Achromatic Stimulation." Vision Res.
11, 591612, 1971.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. 148151, 1996.
Trolland, T. L. "The Enigma of Color Vision." Amer. J.
Physiology 2, 2348, 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, 233277, 1993.
Birman, J. S. and Menasco, W. W. "Studying Links via
Closed Braids. II. On a Theorem of Bennequin." Topology
Appl. 40, 7182, 1991.
Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le
nombre gordien des nceuds algebriques." Enseign. Math.
30, 173222, 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. 4998,
1983.
Cipra, B. What's Happening in the Mathematical Sciences,
Vol. 2. Providence, RI: Amer. Math. Soc, pp. 813, 1994.
Kronheimer, P. B. "The GenusMinimizing Property of Al
gebraic Curves." Bull. Amer. Math. Soc. 29, 6369, 1993.
Kronheimer, P. B. and Mrowka, T. S, "Gauge Theory for
Embedded Surfaces. I." Topology 32, 773826, 1993.
Kronheimer, P. B. and Mrowka, T. S. "Recurrence Relations
and Asymptotics for FourManifold Invariants." Bull.
Amer. Math. Soc. 30, 215221, 1994.
Menasco, W. W. "The BennequinMilnor 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. 379381, 1972.
Spanier, J. and Oldham, K. B. "The Kelvin Functions."
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi
sphere, pp. 543554, 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
BergerKazdan Comparison Theorem
Let M be a compact nD 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 uHypersphere 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. 356357, 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 FirstOrder Ordinary Differ
ential Equation of the form
^+vP(x) = Q{x),
(5)
where P(x) = (ln)p(x) and Q(x) = (ln)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
(1n) J p(x) dx
(6)
where C is a constant of integration. If n = 1, then
equation (1) becomes
dy
dx
= y(qp)
— = {qp)dx
(?)
(8)
y = C 2 ef [q{x)  p{x)]dx . (9)
The general solution is then, with C\ and C2 constants,
l/(ln)
y= <
'(lnlp 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) = (lp)+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 =PP 2 =p(lp) (11)
p.3=p 3  3^2Pi + 2(p' 1 ) 3 = p  3p 2 + 2p 3
= p(lp)(l2p) (12)
P4 = A»4  4/i3pi + 6^2 (m!) 2  3(pi) 4
= p  4p 2 + 6p 3  3p 4
= p(lp)(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(lp)(l2p)
71 <T 3 [p(l  p)]3/ 2
_ l2p
H4 p(l2p)(2p 2 2p+l)
72 = —t  3 =
P 2 (lP) 2
6p 2  6p + 1
p(lp)
To find an estimator for a population mean,
V^ ( N
(14)
(15)
(16)
(17)
■0)
JVp=0 v /
Np=l
= e[e + (i8)] 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 (io) 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(nl)x +n(nl)(n2)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 firstorder term. When
— 1 < x < 0, slightly more finesse is needed. In this case,
let y = \x\ = — cc > so that < y < 1, and take
(ly) n = lny+in(nl)y 2  in(nl)(n2)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
irr^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 "EVENindex" 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)
nvoo \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, lnsinx, lncosa?, lntanx, 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(ln).
(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
(pl)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!(pfc + l)!
jHl
(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, 219234, 1911.
Sloane, N. J. A. Sequences A000367/M4039 and A002445/
M4189 in "An OnLine 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. 3538, 1987,
Wagstaff, S. S. Jr. "Ramanujan's Paper on Bernoulli Num
bers." J. Indian Math. Soc. 45, 4965, 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,
EulerMaclaurin Integration Formulas, Euler
Number, Figurate Number Triangle, Genocchi
Number, Pascal's Triangle, Riemann Zeta Func
tion, von StaudtClausen Theorem
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli
and Euler Polynomials and the EulerMaclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804806, 1972.
Arfken, G. "Bernoulli Numbers, EulerMaclaurin Formula."
§5.9 in Mathematical Methods for Physicists, 3rd ed. Or
lando, FL: Academic Press, pp. 327338, 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. 8185, 1994.
Boyer, C. B. A History of Mathematics. New York: Wiley,
1968.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 6798, 1988.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer Verlag, pp. 107110, 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. 9193, 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. 228248, 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
hl = 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. 3940, 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 (lx) = (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«)
 ^ AV"^ (9)
for < x < 1, and Raabe (1851) found
ml
~ 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 + /3l)B m _i(a + /3) (11)
for an INTEGER m and arbitrary REAL NUMBERS a and
P.
see also Bernoulli Number, EulerMaclaurin In
tegration Formulas, Euler Polynomial
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli
and Euler Polynomials and the EulerMaclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804806, 1972.
Appell, P. E. "Sur une classe de polynomes." Annales d'Ecole
Normal Superieur, Ser. 2 9, 119144, 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, 6897, 1738.
Lehmer, D. H. "A New Approach to Bernoulli Polynomials."
Amer. Math. Monthly. 95, 905911, 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, 348376, 1851.
Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial
B n (x)" Ch. 19 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 167173, 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.
BernsteinBezier Curve
see Bezier Curve
Bernstein's Constant
N.B. A detailed online 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
BernsteinSzego 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. 105106). 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, 171178,
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, 489503,
1967.
Newman, D. J. "Rational Approximation to x." Michigan
Math. J. 11, 1114, 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, 461487, 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, 547560, 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, 271290, 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 <nPoo,
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.
BernsteinSzego 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. 3133, 1975.
where
Foo = mKPW.
116 BerryOsseen Inequality
Bertrand's Problem
BerryOsseen 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, 2630,
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 185051 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. 120, 1989.
Lou, S. and Yau, Q. "A Chebyshev's Type of Prime Number
Theorem in a Short Interval (II)." Hardy Ramanujan J.
15, 133, 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. cutthe
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. 201203, 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\ Nl 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 ) + BJniPx 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: McGrawHill, 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 = lp.
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. 9091, 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 — ^m1
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. 911 in Hand
book of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover,
pp. 355389, 435456, and 480491, 1972.
Arfken, G. "Bessel Functions." Ch. 11 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 573636, 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:
McGrawHill, 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. 223229 and 234245, 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(nl) + a n _ 2 ]x m +" = 0, (10)
n(n — 1)
Now let n = 2/, where / = 1, 2,
fln2.
(11)
&21
1
2/(2/  1)
0*212
(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/)
«2il
(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,... n0,2,4
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)
Ji/ 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
■fln2
(22)
(23)
for n = 2, 3, Let n = 2Z + 1, where Z = 1, 2, . . . ,
then
«2/ + l
1
Tfl2Zl
(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 iM^ + > 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(l2m) + ^[a n ra(ra2m) + 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 + 2m) + ^[o n n(n + 2m) + 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
Jm(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
+ 21 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
secondorder, 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 lij) 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^Jmlix).
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 JacobiAnger 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 nm{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)(zl/z) 711 ,
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, DixonFerrar
Formula, HansenBessel Formula, Kapteyn Se
ries, KneserSommerfeld Formula, Mehler's
Bessel Function Formula, Nicholson's Formula,
Poisson's Bessel Function Formula, Schlafli's
Formula, Schlomilch's Series, Sommerfeld's
Formula, SonineSchafheitlin Formula, Wat
son's Formula, WatsonNicholson Formula, We
ber's Discontinuous Integrals, Weber's For
mula, WeberSonine 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. 358364, 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. 573591 and 591596, 1985.
Lehmer, D. H. "Arithmetical Periodicities of Bessel Func
tions." Ann. Math. 33, 143150, 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: McGrawHill, pp. 619622, 1953.
Spanier, J. and Oldham, K. B. "The Bessel Coefficients Jq(x)
and Ji(x)" and "The Bessel Function J u (x)." Chs. 5253
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 509520 and 521532, 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 secondorder, 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 kl)\
J2
2m42fcfc
(8)
m = 0, 1, 2, . . . , 7 is the EulerMascheroni Con
stant, and
Jo k = 0,
(9)
The function is given by
Y n (z) =  / sin(z sin dn0)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. 358364, 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: McGrawHill, pp. 625627, 1953.
Spanier, J. and Oldham, K. B. "The Neumann Function
Y u (x)" Ch. 54 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 533542, 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)dx2^2ai 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. 526527, 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)
(wiw) 2
N(Nl)
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 Tbill 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)
0l„al
(lxf l x'
B(a,0)
T(a)T((3)
D{x) = I(x; a, 6),
(lxf^x
/31 a1
(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
(lxf^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. 944945, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 534535, 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 &™  (PD'(gl)! (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 (1u) 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) (lx 2 ) n  1 {2xdx)
Jo
'f
Jo
2m — 1/. 2\n~l
(lx^^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([nl]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 (lu)* 1
Jo
+P 1 , iw.
\ X X + 1
(ly)(ny), 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. 560565, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part L New York: McGrawHill, 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, FDistribution, 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. 206209 and 219223, 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. 573580, 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!
2r 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)
OLP
where B is a Beta Function. The Mode of a variate
distributed as (3 f (a,(3) is
. a1
+ 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 CROSSCAPS, 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 loworder Bezier
curves. A generalization of the Bezier curve is the B
Spline.
see also BSpline, 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 BernSTEINBezier 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\ + k2<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 inmn points and cannot meet in more than mn
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\ = ah b + c
u\\ = a 4 + 6 4 + c 4 ,
(2)
(3)
(4)
128 BhaskaraBrouckner 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. 97100, 1994.
Bhargava, S. "On a Family of Ramanujan's Formulas for
Sums of Fourth Powers." Ganita 43, 6367, 1992.
BhaskaraBrouckner Algorithm
see Square Root
BiConnected 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. 146147)
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,
Rn — 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, 3847, 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
+
{Rs) 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. 188193, 1965.
Bicentric Quadrilateral
A 4sided 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 BlueEmpty 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. 3435, 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. 147149, 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
1D Splines.
see also BSpline, 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. 118122, 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, thena n  < nai. 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, 137152, 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, 250260, 1954.
Kazarinoff, N. D. "Special Functions and the Bieberbach
Conjecture." Amer. Math. Monthly 95, 689696, 1988.
Korevaar, J. "Ludwig Bieberbach's Conjecture and its
Proof." Amer. Math. Monthly 93, 505513, 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,
331351, 1968/1969.
Pederson, R. and SchifFer, M. "A Proof of the Bieberbach
Conjecture for the Fifth Coefficient." Arch. Rational
Mech. Anal. 45, 161193, 1972.
Stewart, I. "The Bieberbach Conjecture." In From Here to
Infinity: A Guide to Today's Mathematics. Oxford, Eng
land: Oxford University Press, pp. 164166, 1996.
BienaymeChebyshev Inequality
Biharmonic Equation 131
BienaymeChebyshev 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. 152153, 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. 117165, 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. 457569, 1992.
Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4
in Chaotic Dynamics of Nonlinear Systems. New York:
Wiley, pp. 2531, 1990.
Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to
Applied Nonlinear Dynamical Systems and Chaos. New
York: Springer Verlag, pp. 253419, 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
Vrrr + ~<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 2D Bipolar Coordinates.
References
Kaplan, W. Advanced Calculus, ^th ed. Reading, MA:
AddisonWesley, 1991.
Biharmonic Operator
Also known as the BlLAPLAClAN.
In nD space,
V 4 = (V 2 ) 2 .
, 4 /'1\ _ 3(15 8n + n 2 )
(;)
Bijection
A transformation which is OneTOOne and ONTO.
see also OnetoOne, 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+n2
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.,
DlAMONDshape) (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, 4358, 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_arc97/3l_97/mathland.htm.
Wagon, S. "Billiard Paths on Elliptical Tables." §10.2 in
Mathematica in Action. New York: W. H. Freeman,
pp. 330333, 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. 176178, 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 CoxeterDynkin 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
l2° + l2 + 02 2 + l2 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 leftmost 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 8bit 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 (8byte)
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. 69,
1991.
Pappas, T. "Computers, Counting, & Electricity." The Joy
of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 2425, 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. 1821, 276, and 881886, 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, 361376, 1870.
Sloane, N. J. A. Sequences A000108/M1459 in "An OnLine
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, 344350, 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: HoldenDay, 1968.
Binary Quadratic Form
A 2variable 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, 122124, 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 socalled
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 STree, Quadtree, Quaternary Tree,
RedBlack Tree, SternBrocot Tree, Weakly
Binary Tree
References
Lucas, J.; Roelants van Baronaigien, D.; and Ruskey, F.
"Generating Binary Trees by Rotations." J. Algorithms
15, 343366, 1993.
Ranum, D. L. "On Some Applications of Fibonacci Num
bers." Amer. Math. Monthly 102, 640645, 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, 6884, 1990.
Sloane, N. J. A. Sequences A000108/M1459 and A001699/
M3087 in "An OnLine Version of the Encyclopedia of In
teger Sequences."
Binet Forms
The two Recurrence Sequences
U n — mUnl + 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 QMatrix
yJn —
A
V n =
= a n + ^,
where
A =
\/m 2 + 4
a =
771+ A
2
=
m  A
2
A useful related
identity
is
Unl+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)"(1a/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 3MANIFOLD 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, 1737, 1958.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 251257, 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 ChuVandermonde Identity. A sim
ilar formula holds for Negative Integral n,
~ n \ kTlk
,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
rl
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} ni
£
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
Vl4o;
: 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, ChuVandermonde 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 822823, 1972.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM239, 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. 6674, 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, 117, 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. 153242, 1990.
Granville, A. and Ramare, O. "Explicit Bounds on Exponen
tial Sums and the Scarcity of Squarefree Binomial Coeffi
cients." Mathematika 43, 73107, 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. 8485, 8789, and 257258, 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, 6475, 1991.
Jutila, M. "On Numbers with a Large Prime Factor." J.
Indian Math. Soc. 37, 4353, 1973.
Jutila, M. "On Numbers with a Large Prime Factor. II," J.
Indian Math. Soc. 38, 125130, 1974.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Ogilvy, C. S. "The Binomial Coefficients." Amer. Math.
Monthly 57, 551552, 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. 2324, 1989.
Riordan, J. "Inverse Relations and Combinatorial Identities."
Amer. Math. Monthly 71, 485498, 1964.
Sander, J. W. "On Prime Divisors of Binomial Coefficients."
Bull. London Math. Soc. 24,140142, 1992.
Sarkozy, A. "On the Divisors of Binomial Coefficients, I." J.
Number Th. 20, 7080, 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 OnLine 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. 4352, 1987.
Sved, M. "Counting and Recounting." Math. Intel 5, 2126,
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. 2528 and 6371, 1991.
Wolfram, S. "Geometry of Binomial Coefficients." Amer.
Math. Monthly 91, 566571, 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 (lp) N  k =I p (n + l,NN), (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 MomentGenerating Function M for the dis
tribution is
N y v
M(t) = <e tn > = XX n ( ^ V<? N " n
n=0 ^ '
= E(?V)"(ip) n "
= \pe t + (lp)] N
At (t) = Nfre* + (1  p)] JV  1 (pe')
M"it) = NiN  l)[pc* + (1  p)]" V) 2
+ N\pe t + ilp)} 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(lp + 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(lp)(l2p)
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(lp){l2p)
<7 3 [iV>(lp)] 3 /2
7i
l2p
qp
y/Np(lp) y/Npq
7 2 = ^3:
cr 4
6p — 6p + 1 1 — 6pg
iVp(lp)
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(Nn)\ + nlnp+(Nn)\nq.
(21)
For large n and N — n we can use STIRLING'S APPROX
IMATION
ln(n!) « toIxitoto, (22)
so
d[ln(w!)]
dn
d[\n(Nn)\]
dn
« (Inn I 1)  1 = lnn
d
(23)
dn
[{N  n) \vv{N n)(N 
= ln(7Vn),
(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(lp)~ N\p + q)
i (p + q
B 3 =
N \ pq
d*\n[P(n)]
dn 3
1 1
1
1
Npq N(lp)
1 1
(30)
h 2 (N  h) 2
„2 2
N 2 p 2 N 2 q 2 N 2 p 2 q 2
(l2p + p 2 )p 2 _
l2p
B 4 =
N 2 p 2 (lp) 2 N 2 p 2 {lp) 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{lp) + {l2p + p 2 )]
N 3 p 3 (lp 3 )
= 2(3p 2 3p+l)
N 3 p 3 (lp 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)eW 2 '*dT, = P(n)^=l, (35)
and
P(n)
\Bl\ \B 2 \(nn) 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; = ia; + y — k)
P(x = i y x + y = k)
P{x + y = k)
P(x = i,y = ki) _ P(x = i)P(y = ki)
P(x\y = k) ~ P{x + y = k)
( n t m )p fc (ip) n+m  fc
(40)
Note that this is a Hypergeometric Distribution!
see also de MoivreLaplace 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, FDistribution, 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. 219223, 1992.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGrawHill, p. 108109, 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 = (a6)(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)(al)
(aPl)(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 MingZhi, Z. "Pairs where 2 a  a b Divides n a  n h
for All n." Proc. Amer. Math. Soc. 93, 218220, 1985.
Schinzel, A. "On Primitive Prime Factors of a n — 6 n ." Proc.
Cambridge Phil Soc. 58, 555562, 1962.
Binomial Series
For Id < 1,
(i + x y
 £(:)■*
fc=o v 7
(i)
= i +
;x +
l!(nl)! (n2)!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)
12
1 +
1 ■(!") ,
23
1 +
2(2 + n)
34
1+
2(2 n) ,
45
3(3 + n)
56
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. 1415, 1972.
Pappas, T. "Pascal's Triangle, the Fibonacci Sequence &; Bi
nomial Formula." The Joy of Mathematics. San Carlos,
CA: Wide World Publ./Tetra, pp. 4041, 1989.
Binomial Theorem
The theorem that, for INTEGRAL POSITIVE n,
Z_/ kun 
A;=0
(nk)\
^r
k=0
the socalled 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 socalled Negative Binomial Series, which con
verges for x > \a\.
see also BINOMIAL COEFFICIENT, BINOMIAL SERIES,
Cauchy Binomial Theorem, ChuVandermonde
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. 307308, 1985.
Conway, J. H. and Guy, R. K. "Choice Numbers Are Bino
mial Coefficients." In The Book of Numbers. New York:
Springer Verlag, pp. 7274, 1996.
Coolidge, J. L. "The Story of the Binomial Theorem," Amer.
Math. Monthly 56, 147157, 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. 1618, 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. 352354, 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, /cPartite
Graph, KonigEgevary 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 2D 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 ayspace.
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
Twocenter bipolar coordinates are two coordinates giv
ing the distances from two fixed centers r\ and V2 , some
times denoted r and r'. For twocenter 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. 186190, 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 2D Bipolar
Coordinates.
References
Arfken, G. "Bipolar Coordinates (£, 77, z)." §2.9 in Math
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 97102, 1970.
Biprism
Two slant triangular PRISMS fused together.
see also Prism, SchmittConway 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 bruteforce 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, 731747, 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, 381395, 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, 441566, 1891.
van der Waerden, B. L. A History of Algebra from al
Khwarizmi to Emmy Noether. New York: Springer Verlag,
pp. 188189, 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 SwinnertonDyer Conjecture
BirchSwinnertonDyer Conjecture
see SwinnertonDyer 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 realvalued 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 lefthand 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 righthand side is just the MEASURE of A.
Thus, for an ergodic ENDOMORPHISM, "spaceaverages
= 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 bruteforce 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 TwoKey Triple Encryption." In Advances in
Cryptology — Eurocrypt '90. New York: Springer Verlag,
pp. 366377, 1991.
Yuval, G. "How to Swindle Rabin." Cryptologia 3, 187189,
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(nl)
d d d
_ (dl)(d2)[d(nl)]
d n
But this can be written in terms of FACTORIALS as
dl
Qi(n,d)
(dn)\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) = 1Qi(n,d) = 1
d\
(dn)\d n '
(3)
If 365day 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)^le n(n " 1)/2d (5)
— ('sr <■>
where the latter has error
C < — 7~7TK (7)
6(dn + 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(dn + l)
+(i) r(i + n) ^ r(di + i)r(i + i)
(ii)
where
F = F(n,d, a) = 13^2
■ i(l_„),l(2n),I
i(dn+l) i(dn + 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(nik)\(di)\
x^2Qj{nk,di)
; Jdiy
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(fe1.5)'
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 5050 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, 856858, 1970.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre
ations and Essays, 13th ed. New York: Dover, pp. 4546,
1987.
Bloom, D. M. "A Birthday Problem." Amer. Math. Monthly
80, 11411142, 1973.
Bogomolny, A. "Coincidence." http://www.cut* theknot .
com/do_you_know/coincidence.html.
Clevenson, M. L. and Watkins, W. "Majorization and the
Birthday Inequality." Math. Mag. 64, 183188, 1991.
Diaconis, P. and Mosteller, F. "Methods of Studying Coinci
dences." J. Amer. Statist. Assoc. 84, 853861, 1989.
Feller, W. An Introduction to Probability Theory and Its Ap
plications, Vol. 1, 3rd ed. New York: Wiley, pp. 3132,
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, 704706, 1959.
Hocking, R. L. and Schwertman, N. C. "An Extension of the
Birthday Problem to Exactly k Matches." College Math.
J. 17, 315321, 1986.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver
sions. New York: Dover, pp. 102103, 1975.
Klamkin, M. S. and Newman, D. J. "Extensions of the Birth
day Surprise." J. Combin. Th. 3, 279282, 1967.
Levin, B. "A Representation for Multinomial Cumulative
Distribution Functions." Ann. Statistics 9, 11231126,
1981.
McKinney, E. H. "Generalized Birthday Problem." Amer.
Math. Monthly 73, 385387, 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. 313334, 1964.
Riesel, H. Prime Numbers and Computer Methods for Fac
torization, 2nd ed. Boston, MA: Birkhauser, pp. 179180,
1994.
Sayrafiezadeh, M. "The Birthday Problem Revisited." Math.
Mag. 67, 220223, 1994.
Sevast'yanov, B. A. "Poisson Limit Law for a Scheme of Sums
of Dependent Random Variables." Th. Prob. Appl. 17,
695699, 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, 9596,
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. 964965, 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. 343347, 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 7lD 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 (n4)/2 [2 (n2)/2 + y ^ n ey( . Q
 2 (n3)/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. 8889, 1970.
Guy, R. K. "The n Queens Problem." §C18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer
Verlag, pp. 133135, 1994.
Madachy, J. Madachy's Mathematical Recreations. New
York: Dover, pp. 3646, 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 OnLine 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. 115117, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part I. New York: McGrawHill, pp. 665666, 1953.
BlackScholes 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
BlackScholes 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 fullblown theory, and allows approximation
of options for which analytic solutions are not known
(Price 1996).
see also GarmanKohlhagen Formula
References
Black, F. and Scholes, M. S. "The Pricing of Options and
Corporate Liabilities." J. Political Econ. 81, 637659,
1973.
Cox, J. C; Ross, A.; and Rubenstein, M. "Option Pricing: A
Simplified Approach." J. Financial Economics 7, 229263,
1979.
Price, J. F. "Optional Mathematics is Not Optional." Not.
Amer. Math. Soc. 43, 964971, 1996.
Sharpe, W. F.; Alexander, G. J,; and Bailey, J. V. Invest
ments, 5th ed. Englewood Cliffs, NJ: PrenticeHall, 1995.
Black Spleenwort Fern
Black Spleenwort Fern
see BARNSLEY'S FERN
Blackman Function
BlecksmithBrillhart 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:)
(la 2 fc 2 )(l4a 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. 9899, 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 HOFSTADTERCONWAY $10,000 SEQUENCE,
Weierstrak Function
References
Dixon, R. Mathographics. New York: Dover, pp. 175176
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 176177,
1903.
Tall, D. O. "The Blancmange Function, Continuous Every
where but DifTerentiable Nowhere." Math. Gaz. 66,1122,
1982.
Tall, D. "The Gradient of a Graph." Math. Teaching 111,
4852, 1985.
Blaschke Conjecture
The only WlEDERSEHEN MANIFOLDS are the standard
round spheres. The conjecture has been proven by com
bining the BergerKazdan 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.
BlecksmithBrillhartGerst 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 nD.
BLM/Ho Polynomial
A 1variable 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 2Bridge 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, 563573, 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 22Crossings." Math. Comput. 60,
771778 and S17S28, 1993.
Stoimenow, A. "BrandtLickorishMillettHo Polynomi
als." http: //www, informatik.huberlin.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 online 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 ONETOONE 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,
5484, 1982.
BIochLandau 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
BlowUp 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(fcl).
(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 = (rA)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. 112, 1992.
Ryser, H. J. "The {b,v,r, k, A)Configuration." §8.1 in Com
binatorial Mathematics. Buffalo, NY: Math. Assoc. Amer.,
pp. 96102, 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 offdiagonal 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
BlowUp
A common mechanism which generates SINGULARITIES
from smooth initial conditions.
152 BlueEmpty Coloring
Bohemian Dome
BlueEmpty Coloring
see BlueEmpty Graph
BlueEmpty 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 blueempty graph is an Extremal Graph with
B = 0. For Even n, a blueempty 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. "BlueEmpty Chromatic Graphs." Amer. Math.
Monthly 69, 114120, 1962.
Sauve, L. "On Chromatic Graphs." Amer. Math. Monthly
68, 107111, 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, 168, 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 2D 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, 803842, 1993.
Bogdanov, R. "Bifurcations of a Limit Cycle for a Family
of Vector Fields on the Plane." Selecta Math. Soviet 1,
373388, 1981.
BogomolovMiyaokaYau 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, 314, 1994.
Bohemian Dome
see also Hardy's Rule, NewtonCotes 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
BohrFavard Inequalities
Bombieri Norm
153
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 1920, 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.
BohrFavard 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)Mi 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, 8194, 1936. [Reviewed in Zentralblatt f.
Math. 16, 5859, 1939.]
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives.
Dordrecht, Netherlands: Kluwer, pp. 7172, 1991.
Northcott, D. G. "Some Inequalities Between Periodic Func
tions and Their Derivatives." J. London Math. Soc. 14,
198202, 1939.
Tikhomirov, V. M. "Approximation Theory." In Analysis
II (Ed. R. V. Gamrelidze). New York: Springer Verlag,
pp." 93255, 1990.
BolyaiGerwein Theorem
see WallaceBolyaiGerwein 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, finitedimensional
Euclidean Space, or FirstCountable 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. 161168, 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 multiplecomparison 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 experimentwide 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 twosided 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. 1360, 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, 362,
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, 12361238, 1998.
Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev.
Psych. 46, 561584, 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. 138140, 1987.
(Sloane's A001008 and A002805).
In order to find the number of stacked books required to
obtain d booklengths of overhang, solve the d n equation
for d, and take the Ceiling Function. For n = 1, 2, . . .
booklengths of overhang, 4, 31, 227, 1674, 12367, 91380,
675214, 4989191, 36865412, 272400600, ... (Sloane's
A014537) books are needed.
References
Dickau, R. M. "The BookStacking 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. 272274, 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 OnLine 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
EulerMascheroni 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, 557558, 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 AIM239, Feb. 1972.
Sloane, N, J. A. Sequences A003182/M0729 and A000372/
M0817 in "An OnLine 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
BorchardtPfaff Algorithm
Borel Probability Measure 157
BorchardtPfaff 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. 167170, 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.
BorelCantelli 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. 435436, 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 sigmaalgebxa 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. 5859, 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 nD
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 ndimensionale euklidische
Sphare." Fund. Math. 20, 177190, 1933.
Cipra, B. "If You Can't See It, Don't Believe It. . . ." Science
259, 2627, 1993.
Cipra, B. What's Happening in the Mathematical Sciences,
Vol. 1. Providence, RI: Amer. Math, Soc, pp. 2125, 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. 271284,
1963.
Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture."
Bull. Amer. Math. Soc. 329, 6062, 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(lag')
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
(fcD/2
E
t/=(lfc)/2
(_ 1 )^M 2 +)/2a,^ ( ^ ) (7)
with
oo
_ V^ flY 3(k 2 j + 2ku + k2a)/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 + < 2K1 (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, 341350, 1987.
Bressoud, D. M. "The Borwein Conjecture and Partitions
with Prescribed Hook Differences. " Electronic J. Com
binatorics 3, No. 2, R4, 114, 1996. http://www.
combinatorics. org/Volume^3/volume3_2.html#R4.
Bouligand Dimension
see MlNKOWSKIBOULIGAND 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
_=fivr = /(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. 502504, 1985.
Morse, P. M. and Feshbach, H. "Boundary Conditions and
Eigcnfunctions." Ch. 6 in Methods of Theoretical Physics,
Part L New York: McGrawHill, pp. 495498 and 676790,
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. 745778, 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 *uhf 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 ( "oxplowing" ) transform b of a se
quence a is given by
bn = 7 7 \dkEnk
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, SeidelEntringer
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, 4454, 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
QzQi)
QzQi
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
Boxand Whisker Plot
X
T
i
A HlSTOGRAMlike 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. 3941, 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
BoxMuller 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
nvoo ln(3 n )
1.464973521....
(4)
Boxcar Function
y = c[H(xo)H{xb)],
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.
BoxMuller Transformation
A transformation which transforms from a 2D contin
uous Uniform Distribution to a 2D 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)
BoxPacking 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, SlothouberGraatsma
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
6sided 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(yx)(zy)(xz)]. (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>)
2aA/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 CrossCap, Immersion, Mobius Strip,
nonorientable surface, real projective plane,
Roman Surface, Sextic Surface
References
Apery, F. "The Boy Surface." Adv. Math. 61, 185266, 1986.
Boy, W. "Uber die Curvatura Integra und die Topologie
geschlossener Flachen." Math. Ann 57, 151184, 1903.
Brehm, U. "How to Build Minimal Polyhedral Models of the
Boy Surface." Math. Intell. 12, 5156, 1990.
Carter, J. S. "On Generalizing Boy Surface — Constructing a
Generator of the 3rd Stable Stem." Trans. Amer. Math.
Soc. 298, 103122, 1986.
Fischer, G. (Ed.). Plates 115120 in Mathematische Mod
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 110115, 1986.
Geometry Center. "Boy's Surface." http://www.geom.umn.
edu/zoo/toptype/pplane/boy/.
Hilbert, D. and CohnVossen, 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, 269272, 1981.
Pinkall, U. Mathematical Models from the Collections of Uni
versities and Museums (Ed. G. Fischer). Braunschweig,
Germany: Vieweg, pp. 6465, 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) 1VECTOR 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 {es\n9)
§fc 2 (lcos6>),
(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 EulerLagrange 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 EulerLagrange 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 [(0sm0)+Ai(lcos6O] (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,
902905, 1975.
Haws, L. and Kiser, T. "Exploring the Brachistochrone Prob
lem." Amer. Math. Monthly 102, 328336, 1995.
Wagon, S. Mathematica in Action. New York: W. H. Free
man, pp. 6066 and 385389, 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 onevariable KAUFFMAN POLYNOMIAL X,
the twovariable 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. 148155, 1994.
Kauffman, L. "New Invariants in the Theory of Knots."
Amer. Math. Monthly 95, 195242, 1988.
Kauffman, L. Knots and Physics. Teaneck, NJ: World Sci
entific, pp. 2629, 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, 361376, 1870.
Sloane, N. J. A. Sequence A001003/M2898 in "An OnLine
Version of the Encyclopedia of Integer Sequences."
Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough."
Amer. Math. Monthly 104, 344350, 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 + lm)'
where T(z) is the GAMMA FUNCTION.
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer Verlag, pp. 346348, 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 = ^/(sa)(sb){sc){sd) (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. 5660, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 8182, 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, 3039, 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
nn _ x y _ *n yn _ d
X
y
— n
X — n
y
ty
X
tyn
X
see also Brahmagupta Identity
References
Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib.
Quart. 34, 3039, 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)
nl . .i n \ n3 3 . .2l n \ n5 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— = ntynL
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
yoo
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 PellLucas 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, 3039, 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/cgibin
/MathSource/Applications/Mathematics/0202228.
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 nbraid 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 \ij\ >2
for all i
as first proved by E. Artin. Any nbraid 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, 5260, 1991.
Christy, J. "Braids." http://www.mathsource.com/cgibin
/MathSource/Applications/Mathematics/0202228.
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
MortonFranks Williams Inequality holds,
i>\{Ee) + 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, 97108, 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, 117131, 1993.
Yamada, S. "The Minimal Number of Seifert Circles Equals
the Braid Index of a Link." Invent. Math. 89, 347356,
1987.
Braid Word
Any nbraid is expressed as a braid word, e.g.,
oi^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, 103111,
1985.
Jones, V. F. R. "Hecke Algebra Representations of Braid
Groups and Link Polynomials." Ann. Math. 126, 335
388, 1987.
BraikenridgeMaclaurin 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: McGrawHill, pp. 399401, 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. 397399, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part I. New York: McGrawHill, pp. 391392 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. 111113, 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 nonBrauer 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. 111113, 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. 480481, 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, 354256, 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, 361368 and S9S35,
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, 176184, 1980.
Brent's Method
A RoOTfinding ALGORITHM which combines root
bracketing, bisection, and Inverse Quadratic In
terpolation. It is sometimes known as the VAN
WijngaardenDekerBrent 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
[yf(*i)][yf{ 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 = (T1)(R1)(S1) (4)
with
R =
f(X2)
/(*s)
a  /(*»)
" f(xi)
T _/(*l)
(5)
(6)
(7)
References
Brent, R. P. Ch. 34 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: PrenticeHall, 1977.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet
terling, W. T. "Van WijngaardenDekkerBrent 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 ArithmeticGeometric
MEAN to compute Pi. It has quadratic convergence
and is also called the GaussSalamin 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 ArithmeticGeometric 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. 4851, 1987.
Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag.
61, 148163, 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 GaussSalamin
Algorithm." Math. Gaz. 76, 231242, 1992.
Salamin, E. "Computation of n Using ArithmeticGeometric
Mean." Math. Comput. 30, 565570, 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 13card suit. Getting 12 cards
of same suit with ace high and the 13th card not an
ace is called 2card suit, ace high. Getting no honors is
called a Yarborough.
The probabilities of being dealt 13card 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 6sided 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. 7779, 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 13card 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
41236
N
(S)
N
ill
mm
AT
i
158,753,389,900
1
12card 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
13card suit
12card 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. 4849,
1987.
Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recre
ations. New York: W. W. Norton, pp. 119121, 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. 4549, 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, 245288, 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 2BRIDGE KNOTS. There is
a onetoone correspondence between 2Bridge KNOTS
and rational knots. The knot O8010 is a 3bridge knot. A
knot with bridge number b is an nEMBEDDABLE 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. 6467, 1994.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 115, 1976.
Bridge Knot
An nbridge knot is a knot with BRIDGE Number n.
The set of 2bridge knots is identical to the set of rational
knots. If L is a 2Bridge 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 2bridge 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 22Crossings." Math. Comput. 60,
771778 and S17S28, 1993.
Schubert, H. "Knotten mit zwei Briicken." Math. Z. 65,
133170, 1956.
BrillNoether 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.
BringJerrard 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
BringJerrard 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 BringJerrard Quintic Form, Quintic
Equation
References
Ruppert, W. M. "On the Bring Normal Form of a Quintic in
Characteristic 5." Arch. Math. 58, 4446, 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. 392393, 1993.
BriotBouquet 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. 481482, 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)
(AbiKhuzam 1974).
174
Brocard Axis
Brocard Line
see also BROCARD CIRCLE, BROCARD LINE, EQUI
Brocard Center, Fermat Point, Isogonic Cen
ters
References
AbiKhuzam, F. "Proof of YfFs Conjecture on the Brocard
Angle of a Triangle." Elem. Math. 29, 141142, 1974.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 263286 and 289294, 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. 267268).
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. 268269).
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. 263286, 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, 163187, 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. 264265).
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\/l4sm 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. 263286, 1929.
Kimberling, C "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163187, 1994.
Stroeker, R. J. "Brocard Points, Circulant Matrices, and
Descartes' Folium." Math. Mag. 61, 172187, 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. 277281, 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. 853861, 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 nBALL.
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 2D to the Cube in 3D was also proved
by Browkin.
see also Cube, Schinzel's Theorem, Square
References
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 121125, 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 + . . • + ffcl > 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, 557560, 1961.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 123130, 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
(nl)! + 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, 577593,
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. 382385, 1992.
BruckRyserChowla 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)
(fl)/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. 112, 1992.
Gordon, D. M. "The Prime Power Conjecture is True
for n < 2,000,000." Electronic J. Combinatorics 1,
R6, 17, 1994. http://www.combinatorics.org/Volume_l/
volume 1 ,html#R6.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., 1963.
BruckRyser Theorem
see BRUCKRYSERCHOWLA 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 HARDYLlTTLEWOOD
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, 19951996, Vol 3. Providence,
RI: Amer. Math. Soc, pp. 3847, 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, 195204, 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, 293299, 1974.
Wolf, M. "Generalized Brun's Constants." http://www.ift.
uni.wroc.pl/mwolf/.
178 BrunnMinkowski Inequality
Buffon's Needle Problem
BrunnMinkowski Inequality
The nth root of the Content of the set sum of two sets
in Euclidean nspace 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 BrunnMinkowski 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. 213214, 1993.
Buchberger, B. "Theoretical Basis for the Reduction of Poly
nomials to Canonical Forms." SIGSAM Bull 39, 1924,
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
BuffonLaplace 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
/
^
/
/
BulirschStoer 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) needletossings, see
Badger (1994).
see also BuffonLaplace Needle Problem
References
Badger, L. "Lazzarini's Lucky Approximation of 7r." Math.
Mag. 67, 8391, 1994.
Dorrie, H. "Buffon's Needle Problem." §18 in 100 Great
Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, pp. 7377, 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, 132139, 1994.
BulirschStoer 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. 718725, 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. 127129, 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, 335388, 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, 161224,
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. 168221, 1996.
Burnside's Conjecture
Every nonABELIAN 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, 145254, 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?"
(VaughanLee 1990). This question was answered by
Golod (1964) when he constructed finitely generated in
finite pGROUPS. 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 rgenerator Burnside group of exponent
n. It is the largest rgenerator 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 2group 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 + (rl)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,
BusemannPetty 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,
230238, 1902.
Golod, E. S. "On NilAlgebras and Residually Finite p
Groups." Isv. Akad. Nauk SSSR Ser. Mat. 28, 273276,
1964.
Hall, M. "Solution of the Burnside Problem for Exponent
Six." Ill J. Math. 2, 764786, 1958. „
Levi, F. and van der Waerden, B. L. "Uber eine besondere
Klasse von Gruppen." Abh. Math. Sem. Univ. Hamburg
9, 154158, 1933.
Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I,
II, III." Izv. Akad. Nauk SSSR Ser. Mat 32, 212244,
251524, and 709731, 1968.
Sanov, I. N. "Solution of Burnside's problem for exponent
four." Leningrad State Univ. Ann. Math. Ser. 10, 166—
170, 1940.
VaughanLee, M. The Restricted Burnside Problem, 2nd ed.
New York: Clarendon Press, 1993.
BusemannPetty Problem
If the section function of a centered convex body in Eu
clidean nspace (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, 422429, 1995.
Busy Beaver
A busy beaver is an nstate, 2symbol, 5tuple 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. 108112, 1987.
Dewdney, A. K. "A Computer Trap for the Busy Beaver,
the Hardest Working Turing Machine." Sci. Amer. 251,
1923, Aug. 1984.
Marxen, H. and Buntrock, J. "Attacking the Busy Beaver 5."
Bull. EATCS40, 247251, Feb. 1990.
Sloane, N. J. A. Sequence A028444 in "An OnLine 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 FRACTALlike curve generated by the 2D function
(z 2 y 2 )sin(^)
ffay) =
x 2 +y 2
182 Butterfly Polyiamond Butterfly Theorem
Butterfly Polyiamond
A 6POLYIAMOND.
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. 4546, 1967.
Cake Cutting 183
C
CTable
see CDeterminant
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, fcTHEORY
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 SETlike Fractal obtained by iterat
ing the map
Zn+l = Z n + (ZQ — l)z n — Zq 
CDeterminant
A Determinant appearing in Pade Approximant
identities:
a
s + l <Xrs+2
a r +\
Gr+s1
see also Pade APPROXIMANT
CMatrix
Any Symmetric Matrix (A t = A) or Skew Symmet
ric Matrix (A t = A) C™ with diagonal elements
and others ±1 satisfying
CC T = (nl)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 cakecutting 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 FrobeniusKonig Theorem.
see also CIRCLE CUTTING, CYLINDER CUTTING, EN
VYFREE, FROBENIUSKONIG THEOREM, HAM SAND
WICH Theorem, Pancake Theorem, Pizza Theo
rem, Square Cutting, Torus Cutting
References
Brams, S. J. and Taylor, A. D. "An EnvyFree Cake Division
Protocol." Amer. Math. Monthly 102, 919, 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, 117, 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, 353355, 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. 2237, 1979.
184
Cal
Calculus of Variations
rsi
Steinhaus, H. "Sur la division progmatique." Ekonometrika
(Supp.) 17, 315319, 1949.
Stromquist, W. "How to Cut a Cake Fairly." Amer. Math.
Monthly 87, 640644, 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 + 32 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."
CalabiYau Space
A structure into which the 6 extra Dimensions of 10D
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: OneVariable Cal
culus, with an Introduction to Linear Algebra. Waltham,
MA: Blaisdell, 1967.
Apostol, T. M. Calculus, 2nd ed., Vol. 2: MultiVariable 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: 19691991. 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: McGrawHill, 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: WellesleyCambridge
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 EulerLagrange 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, EulerLagrange Differential
Equation, Isoperimetric Problem, Isovolume
Problem, Lindelof's Theorem, Plateau's Prob
lem, PointPoint Distance — 2D, PointPoint
Distance— 3D, 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. 925962, 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 SumFree Set Constant
A set of POSITIVE INTEGERS S is sumfree if the equa
tion x 4 y = z has no solutions x, y, z 6 S. The proba
bility that a random sumfree 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 SumFree Sets." Graphs and Com
binatorics 1, 129135, 1985.
Cameron, P. J. "Portrait of a Typical Sum Free Set." In
Surveys in Combinatorics 1987 (Ed. C. Whitehead). New
York: Cambridge University Press, 1342, 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
CantorDedekind Axiom
The points on a line can be put into a OnetoOne
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, 429432, 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, 122, 1918.
Canonical Form
A clearcut way of describing every object in a class in
a OnetoOne manner.
see also Normal Form, OnetoOne
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 OnetoOne 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. 8183, 1996.
Penrose, R. The Emperor's New Mind: Concerning Comput
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, pp. 8485, 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.
iVn5 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 _ .
nJoo 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. 103104, 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
Cml . 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 Cml
"3 * ' ' 3™" 1
and
Cl
+ ■
Cml _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(lx) = 1F(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, 255258, 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. 102108 and 143149, 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 nfc g fcl =9 n_ 4 n_
fc=0
The Capacity Dimension is therefore
liml^lim^ 9 " 4 ")
thoo 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 CrossCap, 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. 538541, 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)(xa).
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 (ALEPH0)
members if it can be put into a OneTOOne 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 Oneto
One. An Inaccessible Cardinal cannot be expressed
in terms of a smaller number of smaller cardinals.
see also Aleph, Aleph0 (Ho), Aleph1 (Hi), Can
torDedekind 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. 8386, 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 1CuSPED 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 = (0cos(9) 2 + (lsin(9) 2
= cos 2 0+l2sin0 + sin 2
= 2(1 sin 0).
(6)
J
^
The Arc Length, Curvature, and Tangential An
gle are
/'
Jo
2cos(!i)dt = 4asin(i0)
3sec(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. 4142, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 118121, 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 : //wwwgroups . dcs . stand. ac .uk/ history/Curves
/Cardioid. html.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing
ton, DC: Math. Assoc. Amer., pp. xxvixxvii, 1995.
Yates, R. C. "The Cardioid." Math. Teacher 52, 1014, 1959.
Yates, R. C. "Cardioid." A Handbook on Curves and Their
Properties. Ann Arbor, Ml: J. W. Edwards, pp. 47, 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 mirrorimage 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 mirrorimage 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 1113.
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. 249250, 1988.
CarlsonLevin Constant
N.B. A detailed online 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, 635638,
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 xAxiS 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 257gon, 65537gon, Heptadecagon, Pen
tagon
References
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97108, 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, 4050, 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,
703722, 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
Pseudoprimesto2510 9 ." Math. Comput. 35,10031026,
1980.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer Verlag, pp. 2931, 1989.
Schlafly, A. and Wagon, S. "Carmichael's Conjecture on the
Euler Function is Valid Below lO 10  000  000 ." Math. Com
put. 63, 415419, 1994.
Carmichael Function
A(n) is the LEAST COMMON MULTIPLE (LCM) of all the
Factors of the Totient Function <j>(n), except that
if 8n, 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. 273275, 1994.
Sloane, N. J. A. Sequence A011773 in "An OnLine 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,
703722, 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. 3032, 1994.
Korselt, A. "Probleme chinois." L 'intermediate math. 6,
143143, 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, 381391, 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. Cornput 35, 10031026,
1980.
Riesel, H. Prime Numbers and Computer Methods for Fac
torization, 2nd ed. Basel: Birkhauser, pp. 8990 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 OnLine 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 •■ aii)
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, 4050, 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. 3940). 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 2P23iUi 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.
CarotidKundalini Fractal
Fractal Valley Gaussian Mtn. Oscillation Land
0.5
1
, ,11:.,
111
//>;'); V
. Mmmm
I
'''■jffi/i
x m
\ i
m
0.5
A fractallike structure is produced for x < by super
posing plots of CarotidKundalini 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 CarotidKundalini Functions
Fractal?" Ch. 24 in Keys to Infinity. New York: W. H.
Freeman, pp. 179181, 1995.
CarotidKundalini Function
The Function given by
CK n (x) = cos(nxcos _1 x),
where n is an Integer and — 1 < x < 1.
see also CarotidKundalini 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, 537549, 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
2axis
A
The Gradient of the Divergence is
yaxis
Cartesian coordinates are rectilinear 2D or 3D coordi
nates (and therefore a special case of CURVILINEAR CO
ORDINATES) which are also called Rectangular Co
ordinates. The three axes of 3D Cartesian coordi
nates, conventionally denoted the a>, y, and zAxes (a
Notation due to Descartes) are chosen to be linear and
mutually PERPENDICULAR. In 3D, 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
VF 
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(Vu)
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 dvy 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: McGrawHill, 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 twocenter 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
[(lm 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. 155157, 1972.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 188, 1967.
MacTutor History of Mathematics Archive. "Cartesian
Oval." http : //wwwgroups . 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 ZAction 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. 121127, 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 twocenter 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
[(xa) 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
Jtv/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) + >/lsin 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. 6365, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 153155, 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. 187188, 1967.
MacTutor History of Mathematics Archive. "Cassinian
Ovals." http: //wwwgroups .dcs .stand.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. 811, 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. 9295, 1987.
Cassini Surface
The QUARTIC SURFACE obtained by replacing the con
stant c in the equation of the CASSINI OVALS
{(xa) 2 +y 2 ][(x + af + y 2 ] = c 2
by c = z 2 , obtaining
[(xa) 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. 144147, 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. 2829, 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. 155157, 1994.
Ribenboim, P. Catalan's Conjecture. Boston, MA: Academic
Press, 1994.
Ribenboim, P. "Catalan's Conjecture." Amer. Math.
Monthly 103, 529538, 1996.
Ribenboim, P. "Consecutive Powers." Expositiones Mathe
maticae 2, 193221, 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
(2fcfl) 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
(l3^) 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  ±\ Wl
V2 11 \ 2»V/3(2*)
k^i^/i 2 ^ 1 )
(12)
where f3(z) is the Dirichlet Beta Function. Glaisher
(1913) gave
*iE
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
Km^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[(ti) (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. 807808, 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:
AddisonWesley, 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, 3558, 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 OnLine 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, 375377, 1990.
Vardi, I. Computational Recreations in Mathematica. Read
ing, MA: AddisonWesley, p. 159, 1991.
Yang, S. "Some Properties of Catalan's Constant G." Int. J.
Math. Educ. Sci. Technol 23, 549556, 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, 97103, 1960.
Inkeri, K. "On Catalan's Problem." Acta Arith. 9, 285290,
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[(lz)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 ngon 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
Cn2 (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 nFLEXAGON, the
number of different diagonals possible in a FRIEZE PAT
TERN with n+1 rows, the number of ways of forming
an nfold 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)(nfl) 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 —
2610(4n2)
(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^Enx + 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 pary TREES with k
sourcenodes, the number of ways of associating k appli
cations of a given pary OPERATOR, the number of ways
of dividing a convex POLYGON into k disjoint (p + 1)
gons with nonintersecting DIAGONALS, and the number
of pGoOD PATHS from (0, 1) to (]fe, (pl)kl) (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 pGoOD 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
pq (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, pGood 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, 109132, 1971.
Alter, R. and Kubota, K. K. "Prime and Prime Power Divis
ibility of Catalan Numbers." J. Combin. Th. 15,243256,
1973.
Bailey, D. F. "Counting Arrangements of l's and — l's."
Math. Mag. 69, 128131, 1996.
Brualdi, R. A. Introductory Combinatorics, 3rd ed. New
York: Elsevier, 1997.
Campbell, D. "The Computation of Catalan Numbers."
Math. Mag. 57, 195208, 1984,
Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration
of Certain Sets of Planted Trees." J. Combin. Th. Ser. B
18, 209221, 1975.
Chu, W. "A New Combinatorial Interpretation for General
ized Catalan Numbers." Disc. Math. 65, 9194, 1987.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer Verlag, pp. 96106, 1996.
Dershowitz, N. and Zaks, S. "Enumeration of Ordered Trees."
Disc, Math. 31, 928, 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. 2127, 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, 5760, 1958.
Hilton, P. and Pederson, J. "Catalan Numbers, Their Gen
eralization, and Their Uses." Math. Int. 13, 6475, 1991.
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 130134, 1973.
202
Catalan's Problem
Catalan's Surface
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 146150, 1985,
Klarner, D. A. "Correspondences Between Plane Trees and
Binary Sequences." J. Comb. Th. 9, 401411, 1970.
Rogers, D. G. "Pascal Triangles, Catalan Numbers and Re
newal Arrays." Disc. Math. 22, 301310, 1978.
Sands, A. D. "On Generalized Catalan Numbers." Disc.
Math. 21, 218221, 1978.
Singmaster, D. "An Elementary Evaluation of the Catalan
Numbers." Amer. Math. Monthly 85, 366368, 1978.
Sloane, N. J. A. A Handbook of Integer Sequences. Boston,
MA: Academic Press, pp. 1820, 1973.
Sloane, N. J. A. Sequences A000108/M1459 in "An OnLine
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. 187188 and 198199,
1991.
Wells, D. G. The Penguin Dictionary of Curious and Inter
esting Numbers. London: Penguin, pp. 121122, 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, 171, 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, 10191023, 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. 4546, 1986.
Fischer, G. (Ed.). Plates 9495 in Mathematische Mod
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 9091, 1986.
Gray, A. Modern Differential Geometry of Curves and Sur
faces. Boca Raton, FL: CRC Press, pp, 448449, 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, SeidelEntringerArnold Triangle
References
Sloane, N. J. A. Sequence A009766 in "An OnLine 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: NorthHolland, 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 ngon, 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. 8081, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 195 and 199200, 1972.
Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A
Book of Curves. Cambridge, England: Cambridge Univer
sity Press, pp. 118124, 1967,
MacTutor History of Mathematics Archive. "Catenary."
http : //wwwgroups . dcs . stand . 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. 1214,
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. 367369, 1993.
Meusnier, J. B. "Memoire sur la courbure des surfaces."
Mem. des savans etrangers 10 (lu 1776), 477510, 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 gBlNOMIAL 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: McGrawHill, pp. 678679, 1953.
Cauchy's Cosine Integral Formula
/.tt/2
/ '
Jrr/2
a + u2 ni0(v.v+2£)
dO
7rV(fl + V ~ 1)
2*+" 2 r( M + 0rV0'
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
xAxiS. 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 trti/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: McGrawHill, pp. 114115, 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.
CauchyHadamard 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 / \ k1 / \ k=l
where equality holds for ak = cbk In 2D, 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,
(ab) 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^t0 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. 371376, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part I. New York: McGrawHill, pp. 367372, 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)dxg(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 CauchyRiemann 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. 365371, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part I. New York: McGrawHill, pp. 363367, 1953.
CauchyKovalevskaya Theorem
The theorem which proves the existence and uniqueness
of solutions to the Cauchy Problem.
see also Cauchy Problem
CauchyLagrange 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 &ni) •
From this identity, the nD Cauchy Inequality fol
lows.
CauchyMaclaurin Theorem
see MaclaurinCauchy Theorem
Cauchy Mean Theorem
For numbers > 0, the Geometric Mean < the Arith
metic Mean.
Cauchy Principal Value
fix) dx = lim
00
/ f(x)dx
JR
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. 401403, 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 CauchyKovalevskaya 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
(xc) n_1 (a;a)
Rn —
where a < c < x.
(nl)!
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 CauchyRiemann 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 yaxis, 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 CauchyRiemann equations become
dR
dr
IdR
RdQ
r d6
— = *$©.
r dd dr
(16)
(17)
(18)
If u and v satisfy the CauchyRiemann equations, they
also satisfy Laplace's Equation in 2D, 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 CauchyRiemann equa
tions and Laplace's Equation. This fact is used to
find socalled 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. "CauchyRiemann Conditions." §6.2 in Mathe
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca
demic Press, pp. 3560365, 1985.
Cauchy's Rigidity Theorem
see Rigidity Theorem
Cauchy Root Test
see Root Test
CauchySchwarz Integral Inequality
Cayley Cubic 211
CauchySchwarz 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.
CauchySchwarz Sum Inequality
ab<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/Causticsdir/caustics.html.
Lockwood, E. H. "Caustic Curves." Ch. 24 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 182185, 1967.
Yates, R. C. "Caustics." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 1520,
1952.
Cavalieri's Principle
1. If the lengths of every onedimensional slice are equal
for two regions, then the regions have equal Areas.
2. If the AREAS of every twodimensional slice (CROSS
Section) are equal for two SOLIDS, then the SOLIDS
have equal Volumes.
see also CrossSection, 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 8square 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.
CayleyBacharach 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. "CayleyBacharach
Theorems and Conjectures." Bull. Amer. Math. Soc. 33,
295324, 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 3space
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, 1720, Apr. 1995.
Endrafi, S. "The Cayley Cubic." http://www.mathematik.
unimainz . 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.
unikl . de/wwwagag/Galerie . html.
Hunt, B. The Geometry of Some Special Arithmetic Quo
tients. New York: Springer Verlag, pp. 115122, 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.
CayleyHamilton 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
CayleyKlein 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
)
\(adbc)\ =
"o o"
(4)
(5)
(6)
(7)
(8)
The CayleyHamilton 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 CayleyHamilton Equation
for LowRank Transformations." Amer. Math. Monthly
99, 4244, 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
CayleyKlein 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 CayleyKlein
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
CayleyKlein 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 CayleyKlein 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 Qmatrix 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 CayleyKlein Parameters and Related
Quantities." §45 in Classical Mechanics, 2nd ed. Read
ing, MA: Addison Wesley, pp. 148158, 1980.
214 CayleyKleinHilbert Metric
Cayley's Sextic Evolute
CayleyKleinHilbert Metric
The METRIC of Felix Klein's model for HYPERBOLIC
Geometry,
9ii
912
922
a 2 (lx 2 2 )
(1Z! 2 Z 2 2 ) 2
a X\X2
(1Zl 2 ~X 2 2 ) 2
a 2 (lX! 2 )
(1xi 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
8D space, and a general rotation in 8D 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: //wwwgroups . dcs . stand. 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. 67101, 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 16Cell, 24Cell, 120Cell, 600Cell
Cellular Automaton
A grid (possibly 1D) 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).
lD automata are called "elementary" and are repre
sented by a row of pixels with states either or 1.
These can be represented with an 8bit 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 wellknown 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, 19951996, Vol 3. Providence,
RI: Amer. Math. Soc, pp. 7081, 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. 2022 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, 219258, 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 OnLine
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, 601644, 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 CWCOMPLEX.
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, NinePoint 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 OnLine
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)
(z1) 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 OnLine
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 OnLine
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 OnLine
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(fp,p) (3)
1 t p dt
_ 2 Tr 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, 73107, 1996.
Sander, J. W. "On Prime Divisors of Binomial Coefficients."
Bull London Math. Soc. 24, 140142, 1992.
Sarkozy, A. "On Divisors of Binomial Coefficients. I." J.
Number Th. 20, 7080, 1985.
Sloane, N. J. A. Sequences A046098, A000984/M1645, and
A001405/M0769 in "An OnLine 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. 2528 and 6371, 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. 146150, 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+ij (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. 877878, 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 wri[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)(l3x)
= 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 OnLine
Version of the Encyclopedia of Integer Sequences."
Centralizer
The centralizer of a Finite nonABELiAN 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, LindebergFeller
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: McGrawHill, pp. 112113, 1992.
Zabell, S. L. "Alan Turing and the Central Limit Theorem."
Amer. Math. Monthly 102, 483494, 1995.
References
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGrawHill, pp. 139140 and 156, 1965.
Centroid (Geometric)
The Center of Mass of a 2D planar Lamina or a
3D 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 3D , 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: McGrawHill, pp. 134162, 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. 5557, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 173176 and 249, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163187, 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. 123126, 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
BDCEAF^DCEA FB. (1)
Let P = [Vi, . . . , V^] be an arbitrary ngon, 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 VikV i+ k, then
n
VikWi
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. 45, 1967.
Grunbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254268, 1995.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 145151, 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, 167173, 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, 10911094, 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. 101105 and 106110, 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, 2034,
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)0Complete Properties of Pro
grams and LartinLof Randomness." Information Proc.
Let 46, 3742, 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[(zk) 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 CopelandErdos Con
stant, which is the decimal obtained by concatenating
the Primes, has a wellbehaved Continued Fraction
which does not show the "large term" phenomenon.
see also COPELANDERDOS 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 ndimensional 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 AreaPreserving 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 2D, 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 3D, 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 orientationpreserving
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 nD 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. 238245, 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, HenonHeiles 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
BaiLin, 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://wwwchaos.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: AddisonWesley, 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 ngon. 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. 149163, 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 1D 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, 537549, 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 LSERIES
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
11 11
ri i 1 n
lli 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
1 111
z, R z
B 1
1 1
11 11
B 2
1 1
111 1
(x^yXR^Ry)
E 1
2 1
12
(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
210
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
1 111
z,R z
xy
B 2
11 11
y,Ry
xz
B 3
111 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 1111
z,R z
Bi
111 11
2 2
x y
B 2
1111 1
xy
E
2 020
(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
1 111
11 11
111 1
0200
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, CayleyHamilton
Theorem, Parodi's Theorem, RouthHurwitz
Theorem
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 11171119, 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, MomentGenerating
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. "MomentGenerating 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. 7277, 1951.
Characteristic (Partial Differential
Equation)
Paths in a 2D 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 CayleyHamilton 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.
ChaslesCayleyBrill 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 oneparameter 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 + ju 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, 189190, and 191192, 1992.
Chebyshev Constants
N.B. A detailed online 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 OneNinth 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 ^)SS+'» = 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 (24) into the original equation (1) to obtain
oo
(1  x 2 ) ^(n + 2)(n + l)a n+2 x n
n0
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 ~* 12*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.
ChebyshevGauss 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
7nl
' 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,
1m 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: McGrawHill, pp. 330331, 1956.
Chebyshev Inequality
Apply Markov's Inequality with a = k 2 to obtain
P[{xfxf >k 2 } <
((xnf) _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 (lx) 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 )
l2xt + 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
Vlx 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 l2 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 GramSchmidt Orthonormalization in the
range (1,1) with Weighting Function (1x 2 ) c ~ 1/2)
gives
Po(x) =
pi(x) =
p 2 {x) =
/^^(1x 2 ) 1 / 2 ^
/^(la: 2 ) 1 ^^
[(l* a ) 1/3 ]li =g '
[sin 1 a:]l: 1
/^(lx 2 ) 1 / 2 ^
/^^(lo: 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 (ij^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) = VlX 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. 771802, 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. 731748, 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. 14781479, 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. 193207, 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)£ n1 .
n—
Multiply (2) by t,
oo
{2t 2 2xt){l2xtrt 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)
(l2xt + t 2 ) 2
{l2xt + 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) = {lx 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. 771802, 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. 731748, 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. 193207, 1987.
Chebyshev Quadrature
A Gaussian QuADRATURElike 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(ly)(li)
+ 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
Ii 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 ChebyshevRadau 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/h2
± 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: McGrawHill, pp. 345351, 1956.
ChebyshevRadau Quadrature
A Gaussian QuADRATURElike 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  ( Zs 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. 4344, 1988.
ChebyshevSylvester 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
CheckerJumping Problem
Chern Number 237
CheckerJumping 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 forwardmost 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. 2328, 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, CheckerJumping Prob
lem
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM239, 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. 267276 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 OnLine 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 7bit 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. 888895, 1992.
Cheeger's Finiteness Theorem
Consider the set of compact nRlEMANNlAN 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., 2Almost 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, StiefelWhitney
Number
238
Chemoff Face
Chess
Chernoff Face
A way to display n variables on a 2D 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 nocaptures 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 sixtynine
million, five hundred and eighteen thousand, eight hun
dred and twentynine followed by twentyone 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 AIM239, 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. 84109, 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,
107109, 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. 8689, 1975.
Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathemati
cal Recreations. New York: W. W. Norton, pp. 267276,
1942.
Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 3454, 1979.
Peterson, I. "The Soul of a Chess Machine: Lessons Learned
from a Contest Pitting Man Against Computer." Sci.
News 149, 200201, Mar. 30, 1996.
Petkovic, M. Mathematics and Chess. New York: Dover,
1997.
Schroeppel, R. "Reprise: Number of legal chess positions."
technews@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, 256275, 1950.
Sloane, N. J. A. Sequences A019319 and A007545/M5100 in
"An OnLine 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 LieType. 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, 7375, 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. 143144, 1990.
Chevron
A 6Polyiamond.
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 EulerMascheroni 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. 231233, 1972.
Chi Distribution
The probability density function and cumulative distri
bution function are
Pn{x)
2 ln/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*Cn)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 HalfNormal Distribution with = 1. For
n = 2, it is a Rayleigh Distribution with a = 1.
see a/50 ChiSquared Distribution, HalfNormal
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,
924, 1977.
ChiSquared 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
ChiSquared 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 MomentGenerating 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 MomentGenerating 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)
ChiSquared Distribution
ChiSquared Test 241
is approximately a Gaussian Distribution with
MEAN y/2r and VARIANCE <t 2 = 1. Fisher showed that
X 2 ~r
V271
(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
1/21 z/2 1/21/2
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 FDistribu
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. 940943, 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, ChiSquare 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. 209214, 1992.
Spiegel, M . R. Theory and Problems of Probability and
Statistics. New York: McGrawHill, pp. 115116, 1992.
ChiSquared 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. 114116).
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 chisquared 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 )
= 11
Xs
k3
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. 118119).
When fitting a oneparameter solution using x 2 > the
bestfit 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. 162168).
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. 1920, 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. 3438,
1990.
Uspensky, J. V. and Heaslet, M. A. Elementary Number The
ory. New York: McGrawHill, pp. 189191, 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
mirrorsymmetric.
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
AU 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. 8493, 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. 8991, 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 mD
varieties of degree d in P n , where P n is an nD pro
jective space. To define the Chow coordinates, take
the intersection of a mD 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, 692704, 1937.
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.I. "WeiLiang Chow." Not. Amer. Math. Soc.
43, 11171124, 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, 450479, 1956.
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.L "WeiLiang Chow." Not. Amer. Math. Soc.
43, 11171124, 1996.
Chow Variety
The set C n ,m,d of all rnD varieties of degree d in an nD
projective space P n into an MD projective space P M .
References
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.I. "WeiLiang Chow." Not. Amer. Math. Soc.
43, 11171124, 1996.
ChristoffelDarboux 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 =
knl
•An rCn^n — 2
A n i kni 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 xy
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. 4244, 1975.
Christoffel Number
ChristoffelDarboux 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: McGrawHill, 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. 2930, 1975.
Christoffel Number
One of the quantities Xi appearing in the GAUSSJACOBI
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. 4748, 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. 160167, 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(EGF 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(EGF 2 )
EG U — FE V
r*22
r 2 
1 11 —
r 2 
1 12 —
2(£GF 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. 160167, 1985.
Gray, A. "Christoffel Symbols." §20.3 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 397400, 1993.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys
ics, Part L New York: McGrawHill, pp. 4748, 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, EdgeColoring, 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. 202209, 1985.
Eppstein, D. "The Chromatic Number of the Plane."
http:// www . ics . uci . edu /  eppstein / junkyard /
planecolor/.
Sloane, N. J. A. Sequence A000934/M3292 in "An OnLine
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, 289296, 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.
ChuVandermonde Identity
(x + a) n = Y^ Uj(a)fcO*Onfc
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
kl
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 181182, 1996.
Church's Theorem
No decision procedure exists for Arithmetic.
Church's Thesis
see ChurchTuring 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. 4749, 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 ChuVandermonde 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 3D is
called a SPHERE, and to nD 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 + (yyo) 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 ClRCUMFERENCEtoDlAMETER 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, DrozFarny Circles, Euler Tri
angle Formula, Excircle, Feuerbach's Theorem,
CirclesandSquares Fractal
CircleCircle 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, NinePoint
Circle, Open Disk, PCircle, 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, TriplicateRatio 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. 96150, 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. 7475, 1996.
Dunham, W. "Archimedes' Determination of Circular Area."
Ch. 4 in Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 84112, 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. 6566, 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. 2125,
1952.
CirclesandSquares Fractal
m
A FRACTAL produced by iteration of the equation
Zn+i = z n (mod m)
which results in a M0IREIike 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 twopart curve. These four cases are illus
trated below.
The CATACAUSTIC for PARALLEL rays crossing a CIRCLE
is a Cardioid.
see also CATACAUSTIC, CAUSTIC
CircleCircle Intersection
Let two Circles of Radii R and r and centered at (0, 0)
and (d, 0) intersect in a LENSshaped 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 CircleCircle Intersection
Combining (1) and (2) gives
(xd) 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
halflength 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 + rR)(dr + 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, SphereSphere 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+/(nl).
(3)
Therefore,
f(n) = n+[(nl) + f(n2)}
n
= n + (nl) + ... + 2 + /(l) = J^ k fW
fc2
n
= ^fcl + /(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. 7679,
1996.
Guy, R. K. "The Strong Law of Small Numbers." Amer.
Math. Monthly 95, 697712, 1988.
Noy, M. "A Short Solution of a Problem in Combinatorial
Geometry." Math. Mag. 69, 5253, 1996.
Sloane, N. J. A. Sequences A000124/M1041 and A000127/
M1119 in "An OnLine 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 + 6c).
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. 190191, 1972.
MacTutor History of Mathematics Archive. "Involute of a
Circle." http://wwwgroups.dcs.stand.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 2n and c n = if 3n. 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 3D Sphere which has
exactly n Lattice Points on its surface. The Sphere
is given by the equation
(xa) 2 + {yb) 2 + (z^) 2
: C + 2,
where a and b are the coordinates of the center of the
socalled 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. 117127, 1973.
Kulikowski, T. "Sur l'existence d'une sphere passant par un
nombre donne aux coordonnees entieres." L'Enseignement
Math. Ser. 2 5, 8990, 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, 7172, 1958.
Sierpiiiski, W. "Sur quelques problemes concernant les points
aux coordonnees entieres." L'Enseignement Math. Ser. 2
4, 2531, 1958.
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant
les ensembles de points sur le plan." Fund. Math. 46,
191194, 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 1D 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 1D JACOBIAN is
d9,
n+l
d0 n
lii:cos(27r(9n),
(2)
so the circle map is not AreaPreserving. 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 MODELOCKED 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 — On1
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, SteinhausMoser No
tation
References
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 2829, 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
CirclePoint 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, MergelyanWesler 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, 745752, 1969.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." ScL Amer.
240, 1828, 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, 2430, 1970.
Goldberg, M. "Packing of 14, 16, 17, and 20 Circles in a
Circle." Math. Mag. 44, 134139, 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, 117, 1996. http://www.
combinatorics. org/Volume^3/volume3.html#R16.
Kravitz, S. "Packing Cylinders into Cylindrical Containers."
Math. Mag. 40, 6570, 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, 303305,
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, 3337, 1975.
Schaer, J. "The Densest Packing of Nine Circles in a Square."
Can. Math. Bui. 8, 273277, 1965.
Schaer, J. "The Densest Packing of Ten Equal Circles in a
Square." Math. Mag. 44, 139140, 1971.
Valette, G. "A Better Packing of Ten Equal Circles in a
Square." Discrete Math. 76, 5759, 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.
CirclePoint 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 (191314) and Olds (1963) give geomet
ric constructions for 355/113 = 3.1415929.... Gard
ner (1966, pp. 9293) 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+[3tan(30°)] = 3.141533 . . ..
While the circle cannot be squared in EUCLIDEAN
Space, it can in GaussBolyaiLobachevsky 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. 190191, 1996.
Dixon, R. M athographics. New York: Dover, pp. 4449 and
5253, 1991.
Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 2026, 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,
4649, 1996.
Olds, C. D. Continued Fractions. New York: Random House,
pp. 5960, 1963.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure. Appl. Math. 45, 350372, 19131914,
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 NinePoint 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. 45, 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
Xn2
Circular Functions
257
Xl
X 2
X3
Xn
Xl
x 2
m1
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 secondorder
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. 11111112, 1979.
Vardi, I. Computational Recreations in Mathematica. Read
ing, MA: AddisonWesley, 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,,ri],
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, 172187, 1988.
Vardi, I. Computational Recreations in Mathematica. Read
ing, MA: AddisonWesley, 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. 7179, 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, 163187, 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. xiixiii). 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. 189191). Let d be the_distance
between INRADIUS r and circumradius R, d = rR. Then
= 2Rr
1 1
R d R+d
(11)
(12)
(Mackay 188687). These and many other identities are
given in Johnson (1929, pp. 186190).
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, 6278, 18861887.
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. 5356 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. 130133, 1967.
Yates, R. C. "Cissoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 2630,
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. 4346, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 98100, 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. 130133, 1967.
MacTutor History of Mathematics Archive. "Cissoid of Dio
cles." http: //wwwgroups . dcs . stand.ac.uk/history/
Curves/Cissoid.html.
Yates, R. C. "Cissoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 2630,
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
(ml)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(ml,2) = c(ml,l) = /(ml)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
= ±(ml)(/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)(m2)(/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)
(m1) 3 (13)
c(m,3)=(ml) 2 (m2) 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
EntringerArnold 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 LSeries
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 15 and Odd 723 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,
jFUNCTION
References
Arno, S. "The Imaginary Quadratic Fields of Class Number
4." Acta Arith. 40, 321334, 1992.
Arno, S.; Robinson, M. L«; and Wheeler, F. S. "Imaginary
Quadratic Fields with Small Odd Class Number." http://
www.math.uiuc . edu/Algebraic NumberTheory/ 0009/.
Buell, D. A. "Small Class Numbers and Extreme Values of
//Functions of Quadratic Fields." Math. Comput. 139,
786796, 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. 4353, 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. 14941496, 1980.
Montgomery, H. and Weinberger, P. "Notes on Small Class
Numbers." Acta. Arith. 24, 529542, 1974.
Sloane, N. J. A. Sequences A014539, A038552, A046125, and
A003657/M2332 in "An OnLine 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, 127, 1967.
Stark, H, M. "On Complex Quadratic Fields with Class Num
ber Two." Math. Comput. 29, 289302, 1975.
Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65,
785800, 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 higherorder
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 2Manifolds 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. LieType Chevalley Groups PSL(n,q),
PSU(n,q), PsP(2n,g), and Pft € (n,g),
4. LieType (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, jFUNCTION, SIMPLE
Group
References
Cartwright, M. "Ten Thousand Pages to Prove Simplicity."
New Scientist 109, 2630, 1985.
Cipra, B. "Are Group Theorists Simpleminded?" What's
Happening in the Mathematical Sciences, 19951996,
Vol 3. Providence, RJ: Amer. Math. Soc, pp. 8299, 1996.
Cipra, B. "Slimming an Outsized Theorem." Science 267,
794795, 1995.
Gorenstein, D. "The Enormous Theorem," Set Amer, 253,
104115, Dec. 1985.
Solomon, R. "On Finite Simple Groups and Their Classifica
tion." Not Amer. Math. Soc. 42, 231239, 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) = ln2sin(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)=§(7Tx)
(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, 298300, 1832.
Grosjean, C. C. "Formulae Concerning the Computation of
the Clausen Integral Cl 2 (a)." J. Comput. Appl. Math. 11,
331342, 1984.
Jolley, L. B. W. Summation of Series. London: Chapman,
1925.
Wheelon, A. D. A Short Table of Summable Series. Report
No. SM14642. 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. 10051006, 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, 298300, 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
socalled "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 RootMeanSquare 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 socalled
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 balllike
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
smoothnessstabilized 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 2D
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), CottonSchwab, GerchbergSaxton (Fienup),
Steer, SteerDewdneyIto, 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 210 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 twodimensional ex
tended structures, CLEAN can produce artifacts in the
form of lowlevel 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 quasione
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 CottonSchwab 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 CottonSchwab
technique is implemented as the AIPS task MX.
The GerchbergSaxton 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 GerchbergSaxton 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 SteerDewdneyIto
variant combined with the Clark algorithm as SDCLN.
The SteerDewdneyIto 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
DewdneyIto 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. 214216, 1985,
Clark, B, G. "An Efficient Implementation of the Algorithm
'CLEAN'." Astron. Astrophys, 89, 377378, 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, 281285, 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. 178179, 1989.
Hogbom, J. A. "Aperture Synthesis with a NonRegular Dis
tribution of Interferometric Baselines." Astron. Astrophys.
Supp. 15, 417426, 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
SelfCalibration in Radio Astronomy." Ann. Rev. Astron.
Astrophys. 22, 97130, 1984.
Schwarz, U. J. "MathematicalStatistical Description of the
Iterative Beam Removing Technique (Method CLEAN)."
Astron. Astrophys. 65, 345356, 1978.
Tan, S. M. "An Analysis of the Properties of CLEAN and
Smoothness Stabilized CLEAN — Some Warnings." Mon.
Not. Royal Astron. Soc. 220, 9711001, 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 ClebschAronhold 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 ClebschGordon 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. 911, 1986.
Fischer, G. (Ed.). Plates 1012 in Mathematische Mod
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 1315, 1986.
Hunt, B. The Geometry of Some Special Arithmetic Quo
tients. New York: Springer Verlag, pp. 122128, 1996.
Nordstrand, T. "Clebsch Diagonal Surface." http://www.
uib , no/people/nf ytn/clebtxt . htm.
ClebschGordon Coefficient
A mathematical symbol used to integrate products of
three SPHERICAL HARMONICS. ClebschGordon 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
6JSYMBOLS or WlGNER 9?'Symb0LS is used. The
ClebschGordon 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 ClebschGordon 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 3jSymbol, Wigner 6jSymbol,
WlGNER 9JSYMBOL
References
Abramowitz, M. and Stegun, C. A. (Eds.). "VectorAddition
Coefficients." §27.9 in Handbook of Mathematical Func
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 10061010, 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. 10351047, 1977.
Condon, E. U. and Shortley, G. §3.63.14 in The Theory of
Atomic Spectra. Cambridge, England: Cambridge Univer
sity Press, pp. 5678, 1951.
Fano, U. and Fano, L. Basic Physics of Atoms and Molecules.
New York: Wiley, p. 240, 1959.
Messiah, A. "ClebschGordon (C.G.) Coefficients and 'Sf
Symbols." Appendix C.I in Quantum Mechanics, Vol. 2.
Amsterdam, Netherlands: NorthHolland, pp. 10541060,
1962.
Shore, B. W. and Menzel, D. H. "Coupling and Clebsch
Gordon Coefficients." §6.2 in Principles of Atomic Spectra.
New York: Wiley, pp. 268276, 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 ^
fc0
(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
y2 = yi =
_ 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. 172178, 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. 233247, 1995.
Clique Number 271
Clifford Algebra
Let V be an nD 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. 220222, 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 NPComplete 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 OnLine
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,
^— ' ndi
where di is the DEGREE of VERTEX i.
References
Aigner, M. "Turan's Graph Theorem." Amer. Math.
Monthly 102, 808816, 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. 244247, 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 nD 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(nrl,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 WilfZeilberger Pair if
F(n + 1, k)  F(n, k) = G(n, k + 1)  G(n, k).
see also Rational Function, WilfZeilberger 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, 579607, 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 HalfClosed 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 PointSet 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 T2Separation 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. 2427, 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,
sCluster, sRun
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 (18811953). 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 loglog 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 SwinnertonDyer Conjecture.
References
Cipra, B. "Fermat Prover Points to Next Challenges." Sci
ence 271, 16681669, 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. 3536 and 122,
1967.
Dixon, R. Mathographics. New York: Dover, pp, 6872, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 3437, 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. 276277, 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, /iCOBORDISM THEOREM, MORSE
Theory
Cobordism
see Bordism, /iCobordism
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 "snailform" 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 [2l9sin(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: //wwwgroups . dcs . stand.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 MAINARDICODAZZI EQUATIONS
Code
A code is a set of ntuples of elements ("WORDS") taken
from an ALPHABET.
see also Alphabet, Coding Theory, Encoding,
ErrorCorrecting 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 ERRORCORRECTING
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, ErrorCorrecting Code, Ga
lois Field, Hadamard Matrix
References
Alexander, B. "At the Dawn of the Theory of Codes." Math.
Intel 15, 2026, 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, ClebschGordon Coefficient, Coeffi
cient Field, Commutation Coefficient, Con
nection Coefficient, Correlation Coefficient,
CrossCorrelation Coefficient, Excess Coef
ficient, Gaussian Coefficient, Lagrangian Co
efficient, Multinomial Coefficient, Pearson's
Skewness Coefficients, ProductMoment Co
efficient of Correlation, Quartile Skewness
Coefficient, Quartile Variation Coefficient,
Racah VCoefficient, 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 LaxMilgram 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
CohenKung 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 EilenbergSteenrod Axioms with the
exception of the dimension axiom.
see also AleksandrovCech Cohomology, Alexan
derSpanier 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 twosided Die.
see Bernoulli Trial, Cards, Coin Paradox, Coin
Tossing, Dice, Feller's CoinTossing 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. 113114, 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 twosided
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. 278279) 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
lz + <2pV +1 0, (6)
and
& =
P&k
(7)
(k + 1 ka' k )p
(Feller 1968, pp. 322325).
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,
4047, 1983.
Honsberger, R. "Some Surprises in Probability." Ch. 5 in
Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., pp. 100103, 1979.
Keller, J. B. "The Probability of Heads." Amer. Math.
Monthly 93, 191197, 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. 238239, 1990.
Spencer, J. "Combinatorics by Coin Flipping." Coll. Math.
J., 17, 407412, 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— theknot .
com/ do_you_know/coincidence. html.
Falk, R. "On Coincidences." Skeptical Inquirer 6, 18—31,
198182.
Falk, R. "The Judgment of Coincidences: Mine Versus
Yours." Amer. J. Psych. 102, 477493, 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. 489502, 1984.
Diaconis, P. and Mosteller, F. "Methods of Studying Coinci
dences." J. Amer. Statist. Assoc. 84, 853861, 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 VerlagsAnstahlt, 1919.
Stewart, I. "What a Coincidence!" Sci. Amer. 278, 9596,
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] = 1H(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, . . . , mdi 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
wL'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 ■ ■ rridil < d d , then all trajectories {T K (n)}
for n € Z eventually cycle.
2. If momdi > <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. TreeSearch Method." Math. Comput
64, 411426, 1995.
Applegate, D. and Lagarias, J. C. "Density Bounds for the
Sx + 1 Problem 2. Krasikov Inequalities." Math. Comput.
64, 427438, 1995.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM239, Feb. 1972.
Burckel, S. "Functional Equations Associated with Congru
ential Functions." Theor. Comp. Set. 123, 397406, 1994.
Conway, J. H. "Unpredictable Iterations." Proc. 1972 Num
ber Th. Conf., University of Colorado, Boulder, Colorado,
pp. 4952, 1972.
Crandall, R. "On the ( 3z + 1' Problem." Math. Comput 32,
12811292, 1978.
Everett, C. "Iteration of the Number Theoretic Function
f(2n) = n, f(2n + 1) = f(3n + 2)." Adv. Math. 25,
4245, 1977.
Guy, R. K. "Collatz's Sequence." §E16 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer Verlag,
pp. 215218, 1994.
Lagarias, J. C. "The 3x + l Problem and Its Generalizations."
Amer. Math. Monthly 92, 323, 1985. http://www.cecm,
sfu. ca/organics/papers/lagarias/.
Leavens, G. T. and Vermeulen, M. "3x + l Search Programs."
Comput. Math. Appl. 24, 7999, 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, 167175, 1984.
Sloane, N. J. A. Sequence A006667/M0019 in "An OnLine
Version of the Encyclopedia of Integer Sequences."
Terras, R. "A Stopping Time Problem on the Positive Inte
gers." Acta Arith. 30, 241252, 1976.
Terras, R. "On the Existence of a Density." Acta Arith. 35,
101102, 1979.
Thwaites, B. "Two Conjectures, or How to win £1100."
Math.Gaz. 80, 3536, 1996.
Vardi, I. "The 3# + 1 Problem." Ch. 7 in Computational
Recreations in Mathematica. Redwood City, CA: Addison
Wesley, pp. 129137, 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, NCluster,
Sylvester's Line Problem
Collineation
A transformation of the plane which transforms COL
LINEAR points into COLLINEAR points. A projective
collineation transforms every 1D 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. 247251, 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
fivecolorable 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, threecolorability can distinguish the mir
ror images of the TREFOIL KNOT but not the FlGURE
OFElGHT KNOT. Fivecolorability, on the other hand,
distinguishes the MIRROR Images of the FlGUREOF
Eight Knot but not the Trefoil Knot.
see also Coloring, ThreeColorable
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 FourColor THEOREM, can be ex
tremely difficult to prove.
see also COLORABLE, EDGECOLORING, FOURCOLOR
Theorem, ^Coloring, Polyhedron Coloring,
SixColor Theorem, ThreeColorable, Vertex
Coloring
References
Eppstein, D. "Coloring," http://vvv . ics . uci . edu / 
eppstein/ junkyard/color. html.
Saaty, T. L. and Kainen, P. C The FourColor 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. 6768,
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 OnLine
Version of the Encyclopedia of Integer Sequences."
Velleman, D. J. and Call, G. S. "Permutations and Combi
nation Locks." Math. Mag. 68, 243253, 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, ErdosSzekeres Theo
rem, InclusionExclusion 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. 8218827, 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, 109136, 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. "OnLine Dictionary of Combinatorics." http://
math.uic.edu/fields/dic/.
Godsil, C. D. "Problems in Algebraic Combinatorics." Elec
tronic J. Combinatorics 2, Fl, 120, 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: McGrawHill, 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:
PrenticeHall, 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 OnLine 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 nearequalities 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 xy.
Now define
Z = {x e a ; x • y foi some y 6 A / 0},
where € Z. An ASSOCIATIVE Ralgebra 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: AddisonWesley, 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] = ABBA. (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 BVbA. (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 lD Manifold. The Sphere and nToRUS are
the only compact 2D MANIFOLDS. It is an open ques
tion if the known compact MANIFOLDS in 3D are com
plete, and it is not even known what a complete list in
4D 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,
369376, 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. 115118, 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. 280281, 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. 178, 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
lAx<Ax.
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 twoelement 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') = 1P(E).
(3)
(4)
(5)
This demonstrates that
P(S') = P{0) = 1  P(S) = 110. (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 FourColor 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. 523538, 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. 2930, 1985.
Conway, J. H. and Gordon, C. M. "Knots and Links in Spatial
Graphs." J. Graph Th. 7, 445453, 1983.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 6063, 1985.
Saaty, T. L. and Kainen, P. C. The FourColor 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. 643654,
1977.
Complete kPartite Graph
Complete fcPartite 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 fcPARTITE
Graph, ^Partite Graph
References
Saaty, T. L. and Kainen, P. C. The FourColor 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 pADic 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 NinePoint 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. 230231, 1969.
Demir, H. "The Compleat [sic] Cyclic Quadrilateral." Amer.
Math. Monthly 79, 777778, 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. 6162, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 101104, 1990.
Complete Quadrilateral
The figure determined by four lines and their six points
of intersection (Johnson 1929, pp. 6162). Note that
this is different from a COMPLETE QUADRANGLE. The
midpoints of the diagonals of a complete quadrilateral
are COLLINEAR (Johnson 1929, pp. 152153).
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, GAUSSBODENMIL
ler Theorem, Polar Circle
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 230231, 1969.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 6162, 149, 152153, and 255
256, 1929.
Mention, M. J. "Demonstration d'un Theoreme de
M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 1620, 1862.
Mention, M. J. "Demonstration d'un Theoreme de
M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 6567, 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 HauptExponent
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. 123126). 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. 7,243252,1969.
Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for
Fibonacci Numbers. XIL" Fib. Quart. 11, 317331, 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 4al'
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 1D complex is called a GRAPH.
see also CWComplex, 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
CauchyRiemann Equations, which give the condi
tions a Function must satisfy in order for a com
plex generalization of the Derivative, the socalled
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,
CauchyRiemann 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. 352395 and 396436, 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: McGrawHill, 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: McGrawHill, pp. 348491 and 480485,
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 CauchyRiemann Equations in order to be
Complex Differentiable.
see also CauchyRiemann Equations,
Differentiable, Derivative
Complex
Complex Differentiable
If the CauchyRiemann 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, CauchyRiemann
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 $s(z) are
given by
»w = i(^+o (5)
*(*> = ^^ = ~W  O = 5*(** " *)■ ( 6 )
2z
The Powers of complex numbers can be written in
closed form as follows:
0
ri2 2 . I n \ n4 4
x y + 1 4 p y
+ i
>~v
3 F y +..
(7)
The first few are explicitly
z 2 = (x 2  y 2 ) { i(2xy)
z = (x — 3xy ) + i(3x y — y )
z 4 = (x 4  6x 2 y 2 + y 4 ) 4 i(4z 3 y  4xy 3 )
z 5 = ( x 5  I0x 3 y 2 + 5zy 4 ) + i{$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. 1617, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or
lando, FL: Academic Press, pp. 353357, 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. 88103, 1996.
Morse, P. M. and Feshbach, H. "Complex Numbers and Vari
ables." §4.1 in Methods of Theoretical Physics, Part I. New
York: McGrawHill, pp. 349356, 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. 171172, 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. 9297, 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 2D 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, 186190, 1986.
Guy, R. K. "Monthly Unsolved Problems, 19691987."
Amer. Math. Monthly 94, 961970, 1987.
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903909, 1989.
Rawsthorne, D. A. "How Many l's are Needed?" Fib. Quart.
27, 1417, 1989.
Sloane, N. J. A. Sequence A005245/M0457 in "An OnLine
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 PProblem
(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 NPProblem, it
can be reduced to a single period verification. A prob
lem is NPComplete if an Algorithm for solving it
can be translated into one for solving any other NP
Problem. Examples of NPComplete Problems in
clude the Hamiltonian Cycle and Traveling Sales
man Problems. Linear Programming, thought to
be an NPPROBLEM, was shown to actually be a P
PROBLEM by L. Khachian in 1979. It is not known if all
apparently NPPROBLEMS are actually PPROBLEMS.
Composite Number 289
see also Bit Complexity, NPComplete Problem,
NPProblem, PProblem
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: AddisonWesley, 1979.
Lewis, H. R. and Papadimitriou, C. H. Elements of the
Theory of Computation, 2nd ed. Englewood Cliffs, NJ:
PrenticeHall, 1997.
Sudkamp, T. A. Language and Machines: An Introduction
to the Theory of Computer Science, 2nd ed. Reading, MA:
AddisonWesley, 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,
561573, 1967.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: SpringerVerlag, 1994.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 1920, 1991.
Sloane, N. J. A. Sequence A002808/M3272 in "An OnLine
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 AdlemanPomeranceRumely 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, JordanHolder 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 reinvested 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 socalled 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. 1416, 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 whileloops (i.e., "while something is
true, do something else"). Forloops (which have a fixed
iteration limit) are a special case of whileloops, so com
putable functions could also be coded using a combina
tion of for and whileloops. The ACKERMANN FUNC
TION is the simplest example of a welldefined 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 welldefined.
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