CRC Concise Encyclopedia
MAmEMAfJCS
CRC Concise Encyclopedia
MAfflEMAffG
Eric W. Weisstein
CRC Press
Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data
Weisstein, Eric W.
The CRC concise encyclopedia of mathematics / Eric W. Weisstein.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-9640-9 (alk. paper)
1. Mathematics- -Encyclopedias. I. Title.
QA5.W45 1998
510'.3— DC21 98-22385
CIP
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Introduction
The CRC Concise Encyclopedia of Mathematics is a compendium of mathematical definitions, formulas,
figures, tabulations, and references. It is written in an informal style intended to make it accessible to a broad
spectrum of readers with a wide range of mathematical backgrounds and interests. Although mathematics is
a fascinating subject, it all too frequently is clothed in specialized jargon and dry formal exposition that make
many interesting and useful mathematical results inaccessible to laypeople. This problem is often further
compounded by the difficulty in locating concrete and easily understood examples. To give perspective to
a subject, I find it helpful to learn why it is useful, how it is connected to other areas of mathematics and
science, and how it is actually implemented. While a picture may be worth a thousand words, explicit
examples are worth at least a few hundred! This work attempts to provide enough details to give the reader
a flavor for a subject without getting lost in minutiae. While absolute rigor may suffer somewhat, I hope
the improvement in usefulness and readability will more than make up for the deficiencies of this approach.
The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs
from existing dictionaries of mathematics in a number of important ways. First, the entire text and all
the equations and figures are available in searchable electronic form on CD-ROM. Second, the entries are
extensively cross-linked and cross-referenced, not only to related entries but also to many external sites
on the Internet. This makes locating information very convenient. It also provides a highly efficient way
to "navigate" from one related concept to another, a feature that is especially powerful in the electronic
version. Standard mathematical references, combined with a few popular ones, are also given at the end of
most entries to facilitate additional reading and exploration. In the interests of offering abundant examples,
this work also contains a large number of explicit formulas and derivations, providing a ready place to locate
a particular formula, as well as including the framework for understanding where it comes from.
The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g.,
Borowski and Borwein's HarperCollins Dictionary of Mathematics and Jeans and Jeans' Mathematics Dictio-
nary). At the same time, the descriptions are more accessible than in "technical" mathematical encyclopedias
(e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of Mathematics),
While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate,
research scientist, or recreational mathematician. In this work, the most useful, interesting, and entertaining
(at least to my mind) aspects of topics are discussed in addition to their technical definitions. For example,
in my entry for pi (71-), the definition in terms of the diameter and circumference of a circle is supplemented
by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. These
formulas are comprehensible to readers with only minimal mathematical background, and are interesting to
both those with and without formal mathematics training. However, they have not previously been collected
in a single convenient location. For this reason, I hope that, in addition to serving as a reference source, this
work has some of the same flavor and appeal of Martin Gardner's delightful Scientific American columns.
Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked
up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections
weave their way through the physical sciences. It frequently transpires that some piece of recently acquired
knowledge turns out to be just what I need to solve some apparently unrelated problem. I have therefore
developed the habit of picking up and storing away odd bits of information for future use. This work has
provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. I
have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I
already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for
people who are interested in the subjects listed here but who, like me, are not necessarily experts.
The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and
continuing during graduate school. The initial document was written in Microsoft Word® on a Mac Plus®
computer, and had reached about 200 pages by the time I started graduate school in 1990. When Andrew
Treverrow made his OzTgX program available for the Mac, I began the task of converting all my documents
to T^X, resulting in a vast improvement in readability. While undertaking the Word to T^}K conversion, I also
began cross-referencing entries, anticipating that eventually I would be able to convert the entire document
to hypertext. This hope was realized beginning in 1995, when the Internet explosion was in full swing and
I learned of Nikos Drakos's excellent I^X to HTML converter, I£TgX2HTML. After some additional effort,
I was able to post an HTML version of my encyclopedia to the World Wide Web, currently located at
www . astro . Virginia . edu/ - eww6n/math/.
The selection of topics included in this compendium is not based on any fixed set of criteria, but rather
reflects my own random walk through mathematics. In truth, there is no good way of selecting topics in such
a work. The mathematician James Sylvester may have summed up the situation most aptly. According to
Sylvester (as quoted in the introduction to Ian Stewart's book From Here to Infinity), "Mathematics is not
a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to
ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited
number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests;
it is not a continent or an ocean, whose area can be mapped out and its "contour defined; it is as limitless as
that space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are
forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within
assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life."
Several of Sylvester's points apply particularly to this undertaking. As he points out, mathematics itself
cannot be confined to the pages of a book. The results of mathematics, however, are shared and passed
on primarily through the printed (and now electronic) medium. While there is no danger of mathematical
results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical
results simply because they are not aware of them. Not only does collecting many results in one place provide
a single starting point for mathematical exploration, but it should also lessen the aggravation of encountering
explanations for new concepts which themselves use unfamiliar terminology. In this work, the reader is only
a cross-reference (or a mouse click) away from the necessary background material. As to Sylvester's second
point, the very fact that the quantity of mathematics is so great means that any attempt to catalog it
with any degree of completeness is doomed to failure. This certainly does not mean that it's not worth
trying. Strangely, except for relatively small works usually on particular subjects, there do not appear to
have been any substantial attempts to collect and display in a place of prominence the treasure trove of
mathematical results that have been discovered (invented?) over the years (one notable exception being
Sloane and Plouffe's Encyclopedia of Integer Sequences), This work, the product of the "gazing" of a single
astronomer, attempts to fill that omission.
Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia,
neither table of contents nor index are included. In many cases, a particular entry of interest can be located
from a cross-reference (indicated in SMALL CAPS TYPEFACE in the text) in a related article. In addition,
most articles are followed by a "see also" list of related entries for quick navigation. This can be particularly
useful if you are looking for a specific entry (say, "Zeno's Paradoxes"), but have forgotten the exact name.
By examining the "see also" list at bottom of the entry for "Paradox," you will likely recognize Zeno's name
and thus quickly locate the desired entry.
The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning
with a numeral are ordered by increasing value and appear before the first entry for "A." In multiple-word
entries containing a space or dash, the space or dash is treated as a character which precedes "a," so entries
appear in the following order: "Sum," "Sum P. . . ," "Sum-P. . . ," and "Summary." One exception is that
in a series of entries where a trailing "s" appears in some and not others, the trailing "s" is ignored in the
alphabetization. Therefore, entries involving Euclid would be alphabetized as follows: "Euclid's Axioms,"
"Euclid Number," "Euclidean Algorithm." Because of the non-standard nomenclature that ensues from
naming mathematical results after their discoverers, an important result such as the "Pythagorean Theorem"
is written variously as "Pythagoras 's Theorem," the "Pythagoras Theorem," etc. In this encyclopedia, I have
endeavored to use the most widely accepted form. I have also tried to consistently give entry titles in the
singular (e.g., "Knot" instead of "Knots").
In cases where the same word is applied in different contexts, the context is indicated in parentheses or
appended to the end. Examples of the first type are "Crossing Number (Graph)" and "Crossing Number
(Link)." Examples of the second type are "Convergent Sequence" and "Convergent Series." In the case of
an entry like "Euler Theorem," which may describe one of three or four different formulas, I have taken the
liberty of adding descriptive words ("Euler's Something Theorem") to all variations, or kept the standard
name for the most commonly used variant and added descriptive words for the others. In cases where specific
examples are derived from a general concept, em dashes ( — ) are used (for example, "Fourier Series," "Fourier
Series — Power Series," "Fourier Series — Square Wave," "Fourier Series — Triangle"). The decision to put a
possessive 's at the end of a name or to use a lone trailing apostrophe is based on whether the final "s"
is pronounced. "Gauss's Theorem" is therefore written out, whereas "Archimedes' Recurrence Formula" is
not. Finally, given the absence of a definitive stylistic convention, plurals of numerals are written without
an apostrophe (e.g., 1990s instead of 1990's).
In an endeavor of this magnitude, errors and typographical mistakes are inevitable. The blame for these
lies with me alone. Although the current length makes extensive additions in a printed version problematic,
I plan to continue updating, correcting, and improving the work.
Eric Weisstein
Charlottesville, Virginia
August 8, 1998
Acknowledgments
Although I alone have compiled and typeset this work, many people have contributed indirectly and
directly to its creation. I have not yet had the good fortune to meet Donald Knuth of Stanford University,
but he is unquestionably the person most directly responsible for making this work possible. Before his
mathematical typesetting program TfeX, it would have been impossible for a single individual to compile such
a work as this. Had Prof. Bateman owned a personal computer equipped with T£jX, perhaps his shoe box of
notes would not have had to await the labors of Erdelyi, Magnus, and Oberhettinger to become a three- volume
work on mathematical functions. Andrew Trevorrow's shareware implementation of I^X for the Macintosh,
OzI]eX (www.kagi.com/authors/akt/oztex.html), was also of fundamental importance. Nikos Drakos and
Ross Moore have provided another building block for this work by developing the IM]gX2HTML program
(www-dsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html), which has allowed me to
easily maintain and update an on-line version of the encyclopedia long before it existed in book form.
I would like to thank Steven Finch of MathSoft, Inc., for his interesting on-line essays about mathemat-
ical constants (www.mathsoft.com/asolve/constant/constant.html), and also for his kind permission to
reproduce excerpts from some of these essays. I hope that Steven will someday publish his detailed essays
in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed
and on-line (www.research.att.com/-njas/sequences/) versions of the Encyclopedia of Integer Sequences,
an immensely valuable compilation of useful information which represents a truly mind-boggling investment
of labor.
Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript
and providing a number of helpful suggestions and additions. Thanks also to algebraic topologist Ryan Bud-
ney for sharing some of his expertise, to Charles Walkden for his helpful comments about dynamical systems
theory, and to Lambros Lambrou for his contributions. Thanks to David W. Wilson for a number of helpful
comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permis-
sion to reproduce their beautiful knot and link diagrams. Thanks to Gavin Theobald for providing diagrams
of his masterful polygonal dissections. Thanks to Wolfram Research, not only for creating an indispensable
mathematical tool in Mathematica® , but also for permission to include figures from the Mathematical book
and MathSource repository for the braid, conical spiral, double helix, Enneper's surfaces, Hadamard matrix,
helicoid, helix, Henneberg's minimal surface, hyperbolic polyhedra, Klein bottle, Maeder's "owl" minimal
surface, Penrose tiles, polyhedron, and Scherk's minimal surfaces entries.
Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading
the entire document for syntax. Thanks also to Bob Stern, my executive editor from CRC Press, for
his encouragement, and to Mimi Williams of CRC Press for her careful reading of the manuscript for
typographical and formatting errors. As this encyclopedia's entry on Proofreading Mistakes shows, the
number of mistakes that are expected to remain after three independent proofreadings is much lower than
the original number, but unfortunately still nonzero. Many thanks to the library staff at the University of
Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I
would like to thank the hundreds of people who took the time to e-mail me comments and suggestions while
this work was in its formative stages. Your continued comments and feedback are very welcome.
10
Numerals
see Zero
The number one (1) is the first Positive Integer. It
is an Odd Number. Although the number 1 used to be
considered a PRIME Number, it requires special treat-
ment in so many definitions and applications involving
primes greater than or equal to 2 that it is usually placed
into a class of its own. The number 1 is sometimes also
called "unity," so the nth roots of 1 are often called the
nth Roots of Unity. Fractions having 1 as a Nu-
merator are called Unit Fractions. If only one root,
solution, etc., exists to a given problem, the solution is
called Unique.
The Generating Function have all Coefficients 1
is given by
1 ii ,2.3.4.
1 + x + x -\- x + x +
l~x
see also 2, 3, Exactly One, Root of Unity, Unique,
Unit Fraction, Zero
The number two (2) is the second POSITIVE INTEGER
and the first PRIME NUMBER. It is Even, and is the only
Even Prime (the Primes other than 2 are called the
Odd Primes). The number 2 is also equal to its Fac-
torial since 2! = 2. A quantity taken to the Power 2
is said to be SQUARED. The number of times k a given
BINARY number & n --*&2&i&o is divisible by 2 is given
by the position of the first 6^ = 1, counting from the
right. For example, 12 = 1100 is divisible by 2 twice,
and 13 = 1101 is divisible by 2 times.
see also 1, BINARY, 3, SQUARED, ZERO
2x mod 1 Map
Let xo be a Real Number in the Closed Interval
[0, 1], and generate a SEQUENCE using the MAP
Xn+i = 2x n (mod 1).
(i)
Then the number of periodic Orbits of period p (for p
Prime) is given by
N„
2 p -2
V
(2)
Since a typical Orbit visits each point with equal prob-
ability, the Natural Invariant is given by
P {x) = 1.
(3).
see also Tent Map
References
Ott, E. Chaos in Dynamical Systems. Cambridge: Cam-
bridge University Press, pp. 26-31, 1993.
3 is the only INTEGER which is the sum of the preceding
Positive Integers (1 + 2 = 3) and the only number
which is the sum of the FACTORIALS of the preceding
Positive Integers (1! + 2! = 3). It is also the first
Odd Prime. A quantity taken to the Power 3 is said
to be Cubed.
see also 1, 2, 3^ + 1 Mapping, Cubed, Period Three
Theorem, Super-3 Number, Ternary, Three-
Colorable, Zero
3x + 1 Mapping
see Collatz Problem
10
The number 10 (ten) is the basis for the DECIMAL sys-
tem of notation. In this system, each "decimal place"
consists of a DIGIT 0-9 arranged such that each Digit
is multiplied by a POWER of 10, decreasing from left to
right, and with a decimal place indicating the 10° = Is
place. For example, the number 1234.56 specifies
Ixl0 3 +2xl0 2 +3xl0 1 +4xl0° + 5xl0~ 1 +6xl0~ 2 .
The decimal places to the left of the decimal point
are 1, 10, 100, 1000, 10000, 10000, 100000, 10000000,
100000000, ... (Sloane's A011557), called one, ten,
HUNDRED, THOUSAND, ten thousand, hundred thou-
sand, Million, 10 million, 100 million, and so on. The
names of subsequent decimal places for Large Num-
bers differ depending on country.
Any Power of 10 which can be written as the PRODUCT
of two numbers not containing 0s must be of the form
2 n • 5 n — 10 n for n an INTEGER such that neither 2 n nor
5 n contains any ZEROS. The largest known such number
10 33 - 2 33 * 5 33
= 8, 589, 934, 592 ■ 116, 415, 321, 826, 934, 814, 453, 125.
A complete list of known such numbers is
10 1 = 2 1
10 2 = 2 2
10 4
10'
10 9
10 18
10 33
: 2 9 * 5 9
: 2 18 ■ 5 1
2 33 • 5 3
(Madachy 1979). Since all POWERS of 2 with exponents
n < 4.6 X 10 7 contain at least one ZERO (M. Cook), no
12
18-Point Problem
other POWER of ten less than 46 million can be written
as the PRODUCT of two numbers not containing Os.
see also Billion, Decimal, Hundred, Large Num-
ber, Milliard, Million, Thousand, Trillion, Zero
References
Madachy, J. S. Madachy J s Mathematical Recreations. New
York: Dover, pp. 127-128, 1979.
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 135, 1995.
Sloane, N. J. A. Sequence A011557 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
12
One Dozen, or a twelfth of a Gross.
see also DOZEN, GROSS
13
A Number traditionally associated with bad luck. A
so-called Baker's Dozen is equal to 13. Fear of the
number 13 is called Triskaidekaphobia.
see also Baker's Dozen, Friday the Thirteenth,
Triskaidekaphobia
15
see 15 Puzzle, Fifteen Theorem
15 Puzzle
2
1
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A puzzle introduced by Sam Loyd in 1878. It consists of
15 squares numbered from 1 to 15 which are placed in a
4x4 box leaving one position out of the 16 empty. The
goal is to rearrange the squares from a given arbitrary
starting arrangement by sliding them one at a time into
the configuration shown above. For some initial arrange-
ments, this rearrangement is possible, but for others, it
is not.
To address the solubility of a given initial arrangement,
proceed as follows. If the SQUARE containing the num-
ber i appears "before" (reading the squares in the box
from left to right and top to bottom) n numbers which
are less than £, then call it an inversion of order n, and
denote it rii. Then define
N — X^ n * = 5Z n *'
where the sum need run only from 2 to 15 rather than
1 to 15 since there are no numbers less than 1 (so n\
must equal 0). If AT is EVEN, the position is possible,
otherwise it is not. This can be formally proved using
Alternating Groups. For example, in the following
arrangement
ri2 = 1 (2 precedes 1) and all other rii = 0, so N — 1
and the puzzle cannot be solved.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 312-
316, 1987.
Bogomolny, A. "Sam Loyd's Fifteen." http://www.cut— the-
knot.com/pythagoras/fifteen.html.
Bogomolny, A. "Sam Loyd's Fifteen [History]." http://www.
cut-the-knot .com/pythagoras/historyl5.html.
Johnson, W. W. "Notes on the '15 Puzzle. I.'" Amer. J.
Math. 2, 397-399, 1879.
Kasner, E. and Newman, J. R. Mathematics and the Imagi-
nation. Redmond, WA: Tempus Books, pp. 177-180, 1989.
Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical
Recreations. New York: W. W. Norton, pp. 302-308, 1942.
Story, W. E. "Notes on the '15 Puzzle. II.*" Amer. J. Math.
2, 399-404, 1879.
16-Cell
A finite regular 4-D POLYTOPE with SCHLAFLI SYMBOL
{3, 3, 4} and Vertices which are the PERMUTATIONS
of (±1, 0, 0, 0).
see also 24-Cell, 120-Cell, 600-Cell, Cell, Poly-
tope
17
17 is a FERMAT PRIME which means that the 17-sided
Regular Polygon (the Heptadecagon) is Con-
STRUCTIBLE using COMPASS and STRAIGHTEDGE (as
proved by Gauss).
see also CONSTRUCTIBLE POLYGON , FERMAT PRIME,
HEPTADECAGON
References
Carr, M. "Snow White and the Seven(teen) Dwarfs."
http:// www . math . harvard . edu / - hmb / issue2.1 /
SEVENTEEN/seventeen.html.
Fischer, R. "Facts About the Number 17." http: //tempo,
harvard . edu / - rf ischer / hcssim / 17_f acts / kelly /
kelly.html.
Lefevre, V. "Properties of 17." http://www.ens-lyon.fr/
-vlefevre/dl7_eng.html.
Shell Centre for Mathematical Education. "Number
17." http : //acorn . educ . nott ingham . ac . uk/ShellCent/
Number /Num 17 .html.
18-Point Problem
Place a point somewhere on a Line Segment. Now
place a second point and number it 2 so that each of the
points is in a different half of the Line SEGMENT. Con-
tinue, placing every ATth point so that all N points are
on different (l/iV)th of the Line Segment. Formally,
for a given N y does there exist a sequence of real num-
bers xi t X2, • • • , #jv such that for every n £ {1, . - . , N}
and every k £ {1, . . . , n}, the inequality
fc- 1 ^ k
— < Xi < -
n n
24-Cell
196-Algorithm
holds for some i € {l,...,n}? Surprisingly, it is only
possible to place 17 points in this manner (Berlekamp
and Graham 1970, Warmus 1976).
Steinhaus (1979) gives a 14-point solution (0.06, 0.55,
0.77, 0.39, 0.96, 0.28, 0.64", 0.13, 0.88, 0.48, 0.19, 0.71,
0.35, 0.82), and Warmus (1976) gives the 17-point solu-
tion
| < a* < ■&> f < X2 < £, jf < x 3 < 1, £ < x 4 < ^,
IT < ** < IS- H < ** < h 1 < ^ < £, if < ** < h
I <x 9 < ±,$ <x 10 < *,± <zu < £,
17 < ^12 < 12 > 2 — Xl2 < 17' U — Xl4 < 17'
13 ^ ^ ^ 4 5 ^ _ ^ 6 10 ^ ^ ^ 11
Warmus (1976) states that there are 768 patterns of 17-
point solutions (counting reversals as equivalent).
see also Discrepancy Theorem, Point Picking
References
Berlekamp, E. R. and Graham, R. L. "Irregularities in the
Distributions of Finite Sequences." J. Number Th. 2, 152-
161, 1970.
Gardner, M. The Last Recreations: Hydras, Eggs, and Other
Mathematical Mystifications. New York: Springer- Verlag,
pp. 34-36, 1997.
Steinhaus, H. "Distribution on Numbers" and "Generaliza-
tion." Problems 6 and 7 in One Hundred Problems in
Elementary Mathematics. New York: Dover, pp. 12-13,
1979.
Warmus, M. "A Supplementary Note on the Irregularities of
Distributions." J. Number Th. 8, 260-263, 1976.
24-Cell
A finite regular 4-D Polytope with SCHLAFLI Symbol
{3,4,3}. Coxeter (1969) gives a list of the VERTEX po-
sitions. The Even coefficients of the D 4 lattice are 1,
24, 24, 96, ... (Sloane's A004011), and the 24 shortest
vectors in this lattice form the 24-cell (Coxeter 1973,
Conway and Sloane 1993, Sloane and Plouffe 1995).
see also 16-Cell, 120-Cell, 600-Cell, Cell, Poly-
TOPE
References
Conway, J. H. and Sloane, N. J. A. Sphere- Packings, Lattices
and Groups, 2nd ed. New York: Springer- Verlag, 1993.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 404, 1969.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Sloane, N. J. A. Sequences A004011/M5140 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
42
According to Adams, 42 is the ultimate answer to life,
the universe, and everything, although it is left as an
exercise to the reader to determine the actual question
leading to this result.
References
Adams, D. The Hitchhiker's Guide to the Galaxy. New York:
Ballantine Books, 1997.
72 Rule
see Rule of 72
120-Cell
A finite regular 4-D Polytope with Schlafli Symbol
{5,3,3} (Coxeter 1969).
see also 16-Cell, 24-Cell, 600-Cell, Cell, Poly-
tope
Preferences
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 404, 1969.
144
A Dozen Dozen, also called a Gross. 144 is a Square
Number and a Sum-Product Number.
see also Dozen
196-Algorithm
Take any POSITIVE INTEGER of two DIGITS or more, re-
verse the DIGITS, and add to the original number. Now
repeat the procedure with the SUM so obtained. This
procedure quickly produces PALINDROMIC NUMBERS for
most INTEGERS. For example, starting with the num-
ber 5280 produces (5280, 6105, 11121, 23232). The end
results of applying the algorithm to 1, 2, 3, ... are 1, 2,
3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121,
... (Sloane's A033865). The value for 89 is especially
large, being 8813200023188.
The first few numbers not known to produce PALIN-
DROMES are 196, 887, 1675, 7436, 13783, . . . (Sloane's
A006960), which are simply the numbers obtained by
iteratively applying the algorithm to the number 196.
This number therefore lends itself to the name of the
Algorithm.
The number of terms a(n) in the iteration sequence re-
quired to produce a Palindromic Number from n (i.e.,
a(n) = 1 for a PALINDROMIC NUMBER, a(n) = 2 if a
Palindromic Number is produced after a single iter-
ation of the 196-algorithm, etc.) for n = 1, 2, . . . are
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3,
2, 2, 1, ... (Sloane's A030547). The smallest numbers
which require n = 0, 1, 2, . . . iterations to reach a palin-
drome are 0, 10, 19, 59, 69, 166, 79, 188, . . . (Sloane's
A023109).
see also Additive Persistence, Digitadition, Mul-
tiplicative Persistence, Palindromic Number,
Palindromic Number Conjecture, RATS Se-
quence, Recurring Digital Invariant
References
Gardner, M. Mathematical Circus: More Puzzles, Games,
Paradoxes and Other Mathematical Entertainments from
Scientific American. New York: Knopf, pp. 242-245, 1979.
Gruenberger, F. "How to Handle Numbers with Thousands
of Digits, and Why One Might Want to." Sci. Amer. 250,
19-26, Apr. 1984.
Sloane, N. J. A. Sequences A023109, A030547, A033865, and
A006960/M5410 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
239
65537-gon
239
Some interesting properties (as well as a few arcane ones
not reiterated here) of the number 239 are discussed in
Beeler et al. (1972, Item 63). 239 appears in Machin's
Formula
| 7 r = 4tan(|)-tan- 1 (^),
which is related to the fact that
2 * 13 - 1
239 2 ,
which is why 239/169 is the 7th CONVERGENT of y/2 .
Another pair of INVERSE TANGENT FORMULAS involv-
ing 239 is
tan" 1 ^) = tan" 1 ^) - tan" 1 ^)
= tan x (^)+tan l (^).
239 needs 4 SQUARES (the maximum) to express it, 9
Cubes (the maximum, shared only with 23) to express
it, and 19 fourth POWERS (the maximum) to express it
(see Waring'S Problem). However, 239 doesn't need
the maximum number of fifth POWERS (Beeler et al
1972, Item 63).
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
257-gon
257 is a FERMAT PRIME, and the 257-gon is there-
fore a Constructible Polygon using Compass and
Straightedge, as proved by Gauss. An illustration
of the 257-gon is not included here, since its 257 seg-
ments so closely resemble a Circle. Richelot and
Schwendenwein found constructions for the 257-gon in
1832 (Coxeter 1969). De Temple (1991) gives a con-
struction using 150 Circles (24 of which are Car-
lyle Circles) which has Geometrography symbol
945i + 475 2 + 275Ci + 0C 2 + 150C 3 and Simplicity
566.
see also 65537-GON, CONSTRUCTIBLE POLYGON, Fer-
mat Prime, Heptadecagon, Pentagon
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.
Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
Rademacher, H. Lectures on Elementary Number Theory.
New York: Blaisdell, 1964.
600-Cell
A finite regular 4-D Polytope with Schlafli Symbol
{3,3,5}. For Vertices, see Coxeter (1969).
see also 16-Cell, 24-Cell, 120-Cell, Cell, Poly-
tope
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 404, 1969.
666
A number known as the Beast Number appearing in
the Bible and ascribed various numerological properties.
see also Apocalyptic Number, Beast Number, Le-
viathan Number
References
Hardy, G. H. A Mathematician's Apology, reprinted with a
foreword by C. P. Snow. New York: Cambridge University
Press, p. 96, 1993.
2187
The digits in the number 2187 form the two VAMPIRE
NUMBERS: 21 x 87 = 1827 and 2187 = 27 x 81.
References
Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19,
26-29, Spring 1997.
65537-gon
65537 is the largest known Fermat Prime, and the
65537-gon is therefore a CONSTRUCTIBLE POLYGON us-
ing Compass and Straightedge, as proved by Gauss.
The 65537-gon has so many sides that it is, for all in-
tents and purposes, indistinguishable from a CIRCLE us-
ing any reasonable printing or display methods. Her-
mes spent 10 years on the construction of the 65537-gon
at Gottingen around 1900 (Coxeter 1969). De Temple
(1991) notes that a Geometric Construction can be
done using 1332 or fewer Carlyle Circles.
see also 257-GON, CONSTRUCTIBLE POLYGON, HEP-
TADECAGON, Pentagon
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.
Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
A-Integrable
A
A-Integrable
A generalization of the Lebesgue INTEGRAL. A MEA-
SURABLE Function f(x) is called A-integrable over the
Closed Interval [a, b] if
m{x:\f(x)\>n} = 0(n- 1 ), (1)
where m is the LEBESGUE MEASURE, and
lim /
[f(x)] n dx
(2)
exists, where
tf(xW -IfW if 1/0*01 <" |«
l/(*)J»-| if|/( x )|>„. W
References
Titmarsch, E. G. "On Conjugate Functions." Proc. London
Math. Soc. 29, 49-80, 1928.
A- Sequence
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
An Infinite Sequence of Positive Integers ai sat-
isfying
1 < ai < a-2. < az < ■ • . (1)
is an A-sequence if no a^ is the SUM of two or more
distinct earlier terms (Guy 1994). Erdos (1962) proved
oo
S{A) = sup Y^ ~ < 103 - ( 2 )
all A sequences , a k
Any A-sequence satisfies the Chi Inequality (Levine
and O'Sullivan 1977), which gives 5(A) < 3.9998. Ab-
bott (1987) and Zhang (1992) have given a bound from
below, so the best result to date is
AAS Theorem
Erdos, P. "Remarks on Number Theory III. Some Problems
in Additive Number Theory." Mat. Lapok 13, 28-38, 1962.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/erdos/erdos.html.
Guy, R. K. "B 2 -Sequences." §E28 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 228-229, 1994.
Levine, E. and O'Sullivan, J. "An Upper Estimate for the
Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34,
9-24, 1977.
Zhang, Z. X. "A Sum-Free Sequence with Larger Reciprocal
Sum." Unpublished manuscript, 1992.
AAA Theorem
Specifying three ANGLES A, B, and C does not uniquely
define a Triangle, but any two TRIANGLES with the
same Angles are SIMILAR. Specifying two ANGLES of
a TRIANGLE automatically gives the third since the sum
of Angles in a Triangle sums to 180° (it Radians),
i.e.,
C = tt-A-B.
see also AAS Theorem, ASA Theorem, ASS Theo-
rem, SAS Theorem, SSS Theorem, Triangle
AAS Theorem
Specifying two angles A and B and a side a uniquely
determines a TRIANGLE with AREA
K ■
a 2 sin B sin C a 2 sin B sin(7r — A — B)
2 sin A 2 sin A
The third angle is given by
C = ir - A- B,
(1)
(2)
2.0649 < 5(A) < 3.9998.
(3)
since the sum of angles of a Triangle is 180° (n Ra-
dians). Solving the Law of Sines
Levine and O'Sullivan (1977) conjectured that the sum
of Reciprocals of an A-sequence satisfies
oo
(4)
where %% are given by the Levine-O'Sullivan Greedy
Algorithm.
see also B 2 -Sequence, Mian-Chowla Sequence
References
Abbott, H. L. "On Sum-Free Sequences." Acta Arith. 48,
93-96, 1987.
for b gives
Finally,
sin A sin B
sinB
b = a—r
sin A
(3)
(4)
c = b cos A + a cos B = a(sin B cot A -f cos B) (5)
= a sin B(cot A -f cot B) . (6)
see also AAA Theorem, ASA Theorem, ASS Theo-
rem, SAS Theorem, SSS Theorem, Triangle
6
Abacus
AbeVs Functional Equation
Abacus
A mechanical counting device consisting of a frame hold-
ing a series of parallel rods on each of which beads are
strung. Each bead represents a counting unit, and each
rod a place value. The primary purpose of the abacus
is not to perform actual computations, but to provide
a quick means of storing numbers during a calculation.
Abaci were used by the Japanese and Chinese, as well
as the Romans.
see also Roman Numeral, Slide Rule
References
Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal
Fractions." A History of Mathematics, 2nd ed. New York:
Wiley, pp. 199-201, 1991.
Fernandes, L. "The Abacus: The Art of Calculating with
Beads." http : //www . ee . ryerson . ca : 8080/-elf /abacus.
Gardner, M. "The Abacus." Ch. 18 in Mathematical Circus:
More Puzzles, Games, Paradoxes and Other Mathemati-
cal Entertainments from Scientific American. New York:
Knopf, pp. 232-241, 1979.
Pappas, T. "The Abacus." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, p. 209, 1989.
Smith, D. E. "Mechanical Aids to Calculation: The Abacus."
Ch. 3 §1 in History of Mathematics, Vol. 2. New York:
Dover, pp. 156-196, 1958.
abc Conjecture
A Conjecture due to J. Oesterle and D. W. Masser.
It states that, for any INFINITESIMAL e > 0, there exists
a Constant C e such that for any three Relatively
Prime Integers a, 6, c satisfying
a 4- b = c,
the Inequality
max{|a|,|6|,|c|}<a JJ p 1+e
p\abc
holds, where p\abc indicates that the PRODUCT is over
Primes p which Divide the Product abc. If this
Conjecture were true, it would imply Fermat's
Last Theorem for sufficiently large Powers (Goldfeld
1996). This is related to the fact that the abc conjecture
implies that there are at least C In x WlEFERlCH PRIMES
< x for some constant C (Silverman 1988, Vardi 1991).
see also Fermat's Last Theorem, Mason's Theo-
rem, Wieferich Prime
References
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
Goldfeld, D. "Beyond the Last Theorem." The Sciences, 34-
40, March/April 1996.
Guy, R. K, Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 75-76, 1994.
Silverman, J. " Wieferich's Criterion and the abc Conjecture."
J. Number Th. 30, 226-237, 1988.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 66, 1991.
Abelian
see Abelian Category, Abelian Differential,
Abelian Function, Abelian Group, Abelian In-
tegral, Abelian Variety, Commutative
Abelian Category
An Abelian category is an abstract mathematical CAT-
EGORY which displays some of the characteristic prop-
erties of the Category of all Abelian Groups.
see also Abelian Group, Category
Abel's Curve Theorem
The sum of the values of an INTEGRAL of the "first" or
"second" sort
f XltV1 Pdr [*n,vn pd
/ ^ + - + J ^ = F ^
and
P(xi,2/i) dxx P(xn,Vn) dx N
Q(rci,yi) dz
Q(xn,Vn) dz
dF
dz '
from a FIXED Point to the points of intersection with a
curve depending rationally upon any number of param-
eters is a Rational Function of those parameters.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 277, 1959.
Abelian Differential
An Abelian differential is an ANALYTIC or MEROMOR-
phic Differential on a Compact or closed Riemann
Surface.
Abelian Function
An Inverse Function of an Abelian Integral.
Abelian functions have two variables and four periods.
They are a generalization of ELLIPTIC FUNCTIONS, and
are also called Hyperelliptic Functions.
see also Abelian Integral, Elliptic Function
References
Baker, H. F. Abelian Functions: Abel's Theorem and the Al-
lied Theory, Including the Theory of the Theta Functions.
New York: Cambridge University Press, 1995.
Baker, H. F. An Introduction to the Theory of Multiply Pe-
riodic Functions. London: Cambridge University Press,
1907.
Abel's Functional Equation
Let Li2(x) denote the DlLOGARITHM, defined by
— n
Abelian Group
then
Li 2 (a) + Li 2 (y) + lA 2 {xy) +
+
(1-2/)
see a/50 DlLOGARITHM, POLYLOGARITHM, RlEMANN
Zeta Function
Abelian Group
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
A Group for which the elements Commute (i.e., AB =
BA for all elements A and B) is called an Abelian group.
All Cyclic Groups are Abelian, but an Abelian group
is not necessarily CYCLIC. All SUBGROUPS of an Abelian
group are NORMAL. In an Abelian group, each element
is in a CONJUGACY CLASS by itself, and the CHARACTER
TABLE involves POWERS of a single element known as a
Generator.
No general formula is known for giving the number
of nonisomorphic Finite GROUPS of a given ORDER.
However, the number of nonisomorphic Abelian FINITE
Groups a(n) of any given Order n is given by writing
n as
n = Y[pi"\ (1)
i
where the pt are distinct PRIME FACTORS, then
a(n) =Y[P( ai ),
(2)
where P is the Partition Function. This gives 1,1,
1, 2, 1, 1, 1, 3, 2, . . . (Sloane's A000688). The smallest
orders for which n = 1, 2, 3, ... nonisomorphic Abelian
groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144,
64, 1800, 0, 288, 128, ... (Sloane's A046056), where
denotes an impossible number (i.e., not a product of
partition numbers) of nonisomorphic Abelian, groups.
The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34,
37, 38, 39, 41, 43, 46, ... (Sloane's A046064). The
incrementally largest numbers of Abelian groups as a
function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56,
77, 101, . . . (Sloane's A046054), which occur for orders
1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192,
... (Sloane's A046055).
The Kronecker Decomposition Theorem states
that every Finite Abelian group can be written as a Di-
rect Product of Cyclic Groups of Prime Power
Orders. If the Orders of a Finite Group is a Prime
p, then there exists a single Abelian group of order p
(denoted Z p ) and no non-Abelian groups. If the Or-
ders is a prime squared p 2 , then there are two Abelian
groups (denoted Z p 2 and Z p & Z p . If the Orders is
Abelian Group 7
a prime cubed p 3 , then there are three Abelian groups
(denoted Z p <g> Z p (g> Z p , Z p % Z p 2, and Z p a), and five
groups total. If the order is a PRODUCT of two primes
p and q, then there exists exactly one Abelian group of
order pq (denoted Z p ® Z q ).
Another interesting result is that if a(n) denotes the
number of nonisomorphic Abelian groups of ORDER n,
then
^a(n)n- s = CWC(2s)C(3 S )-
(3)
n=l
where ((s) is the Riemann Zeta Function. Srinivasan
(1973) has also shown that
N
Y, a (n) = A 1 N+A 2 N 1/2 +A 3 N 1/3 +O[x 105/407 (]nx) 2 ],
n=l
(4)
where
( 2.294856591... for k = 1
Ak = n^(i) = \ - 14 -6475663... for k = 2 (5)
j=i V } { 118.6924619 ... for k = 3,
and ( is again the Riemann Zeta Function. [Richert
(1952) incorrectly gave As = 114.] DeKoninck and Ivic
(1980) showed that
^J-^BN + Oi^ilnN)- 1 ' 2 },
i(n)
(6)
where
nKE
P(k - 2) P(k)
0.752 . .
(7)
is a product over Primes. Bounds for the number of
nonisomorphic non-Abelian groups are given by Neu-
mann (1969) and Pyber (1993).
see also Finite Group, Group Theory, Kronecker
Decomposition Theorem, Partition Function P,
Ring
References
DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func-
tions: Asymptotic Formulae for Sums of Reciprocals of
Arithmetical Functions and Related Fields. Amsterdam,
Netherlands: North- Holland, 1980.
Erdos, P. and Szekeres, G. "Uber die Anzahl abelscher Grup-
pen gegebener Ordnung und iiber ein verwandtes zahlen-
theoretisches Problem." Acta Sci. Math. (Szeged) 7, 95-
102, 1935.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/abel/abel.html.
Kendall, D. G. and Rankin, R. A. "On the Number of Abelian
Groups of a Given Order." Quart J. Oxford 18, 197-208,
1947.
Kolesnik, G. "On the Number of Abelian Groups of a Given
Order." J. Reine Angew. Math. 329, 164-175, 1981.
8
Abel's Identity
Abel's Irreducibility Theorem
Neumann, P. M. "An Enumeration Theorem for Finite
Groups." Quart J. Math. Ser. 2 20, 395-401, 1969.
Pyber, L. "Enumerating Finite Groups of Given Order."
Ann. Math. 137, 203-220, 1993.
Richert, H.-E. "Uber die Anzahl abelscher Gruppen
gegebener Ordnung L" Math. Zeitschr. 56, 21-32, 1952.
Sloane, N. J. A. Sequence A000688/M0064 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Srinivasan, B. R. "On the Number of Abelian Groups of a
Given Order." Acta Arith. 23, 195-205, 1973.
Abel's Identity
Given a homogeneous linear SECOND-ORDER ORDI-
NARY Differential Equation,
y" + P(x)y' + Q(x)y = 0,
(1)
call the two linearly independent solutions yi(x) and
y 2 (as). Then
y'l{x) + P{x)y , l {x) + Q{x)y 1 = ^ (2)
y' 2 ' (x) + P(x)y' 2 (x) + Q(x)y 2 = 0. (3)
Now, take yi x (3) - y 2 x (2),
yilvZ + P(x)y2 + Q(x)y 2 ]
-V2[yi+P(x)y' 1 +Q(x)y l ]=Q (4)
(yiy% -y2y")+P(yiy2-yiy2)+Q(yiy2-yiy2) = (5)
(2/12/2 - 2/22/") + P(2/i2/2 - 2/i2/2) = 0. (6)
Now, use the definition of the Wronskian and take its
Derivative,
W = y t y 2 -2/12/2 (7)
W = (y[y 2 + yiyi) - (yiyi + 2/12/2)
= 2/12/2-2/1^2. (8)
Plugging W and W into (6) gives
W' 4- PW = 0.
This can be rearranged to yield
w = - p ^ dx
which can then be directly integrated to
lnl^ = -Ci / P(x)dx,
(9)
(10)
(11)
where In as is the Natural Logarithm. A second in-
tegration then yields AbePs identity
W(x)=C 2 e~f P(x)dx , (12)
where C\ is a constant of integration and C 2 = e Cl .
see alsa Ordinary Differential Equation — Sec-
ond-Order
References
Boyce, W. E. and DiPrima, R. C. Elementary Differential
Equations and Boundary Value Problems, J^th ed. New
York: Wiley, pp. 118, 262, 277, and 355, 1986.
Abel's Impossibility Theorem
In general, Polynomial equations higher than fourth
degree are incapable of algebraic solution in terms of
a finite number of Additions, Multiplications, and
Root extractions.
see also Cubic Equation, Galois's Theorem, Poly-
nomial, Quadratic Equation, Quartic Equation,
Quintic Equation
References
Abel, N. H, "Demonstration de l'impossibilite de la resolution
algebraique des equations generates qui depassent le qua-
trieme degre." Crelle's J. 1, 1826.
Abel's Inequality
Let {f n } and {a n } be Sequences with f n > fn+i >
for n = 1, 2, . . . , then
/ ^CLnfn
<Ah,
where
A = max{|ai|, |ai + a 2 \ , . . - , |ai + a 2 + . . . 4- a m |}.
Abelian Integral
An Integral of the form
Jo
dt
where R(t) is a POLYNOMIAL of degree > 4. They are
also called Hyperelliptic Integrals.
see also Abelian Function, Elliptic Integral
Abel's Irreducibility Theorem
If one ROOT of the equation f(x) = 0, which is irre-
ducible over a Field K, is also a ROOT of the equation
F(x) = in K, then all the ROOTS of the irreducible
equation f(x) = are ROOTS of F(x) = 0. Equivalently,
F(x) can be divided by f(x) without a Remainder,
F(x) = f{x)F 1 (x) i
where Fi(x) is also a POLYNOMIAL over K.
see also ABEL'S LEMMA, KRONECKER'S POLYNOMIAL
Theorem, Schoenemann's Theorem
References
Abel, N. H. "Memoir sur une classe particuliere d'equations
resolubles algebraiquement." Crelle's J. 4, 1829.
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 120,
1965.
Abel's Lemma
Abhyankar's Conjecture 9
Abel's Lemma
The pure equation
x p = C
of PRIME degree p is irreducible over a FIELD when C
is a number of the FIELD but not the pth Power of an
element of the Field.
see also Abel's Irreducibility Theorem, Gauss's
Polynomial Theorem, Kronecker's Polynomial
Theorem, Schoenemann's Theorem
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 118,
1965.
Abel's Test
see Abel's Uniform Convergence Test
Abel's Theorem
Given a Taylor Series
F(z) = J2CnZ n = ^Tc n r n e i "
(1)
71=0
n=0
where the COMPLEX NUMBER z has been written in the
polar form z = re t& , examine the REAL and IMAGINARY
Parts
u(r,8) = ^Tc n r n cos(n6)
n=0
oo
v(r,9) = ^2c n r n sin(n0).
(2)
(3)
Abel's theorem states that, if u(l,9) and v(l,0) are
Convergent, then
u{l,0)+iv{\,9) = lim f(re iB ).
(4)
Stated in words, Abel's theorem guarantees that, if a
Real Power Series Converges for some Positive
value of the argument, the Domain of Uniform Con-
vergence extends at least up to and including this
point. Furthermore, the continuity of the sum function
extends at least up to and including this point.
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 773, 1985.
Abel Transform
The following INTEGRAL Transform relationship,
known as the Abel transform, exists between two func-
tions f(x) and g(t) for < a < 1,
a(t \ = sin(7TQ) d f l f(x) d
yK > tt dtj {x-ty
_ sin(7ra)
7V
u:
dx
dx
df dx | /(0)
dx{t-xY~ a t 1 -"
(1)
(2)
(3)
The Abel transform is used in calculating the radial
mass distribution of galaxies and inverting planetary ra-
dio occultation data to obtain atmospheric information.
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 875-876, 1985.
Binney, J. and Tremaine, S. Galactic Dynamics. Princeton,
NJ: Princeton University Press, p. 651, 1987.
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 262-266, 1965.
Abel's Uniform Convergence Test
Let {u n (x)} be a Sequence of functions. If
1. u n (x) can be written u n (x) — a n f n (x) 1
2. ^a n is Convergent,
3. fn(x) is a Monotonic Decreasing Sequence
(i.e., fn+i(x) < f n (x)) for all n, and
4. f n (x) is Bounded in some region (i.e., < f n (x) <
M for all x e [a, b])
then, for all x e [a, 6], the Series Yl Un ( x ) Converges
Uniformly.
see also CONVERGENCE TESTS
References
Bromwich, T. J. Pa and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 59, 1991.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, p. 17, 1990.
Abelian Variety
An Abelian variety is an algebraic GROUP which is a
complete Algebraic Variety. An Abelian variety of
Dimension 1 is an Elliptic Curve.
see also Albanese Variety
References
Murty, V. K. Introduction to Abelian Varieties. Providence,
RI: Amer. Math, Soc, 1993.
Abhyankar's Conjecture
For a Finite Group G, let p(G) be the Subgroup gen-
erated by all the Sylow p-SuBGROUPS of G. If X is a
projective curve in characteristic p > 0, and if xq, ...,xt
are points of X (for t > 0), then a NECESSARY and SUF-
FICIENT condition that G occur as the GALOIS GROUP
of a finite covering Y of X, branched only at the points
a;o, . .., x ti is that the Quotient GROUP G/p{G) has
2g + 1 generators.
Raynaud (1994) solved the Abhyankar problem in the
crucial case of the affine line (i.e., the projective line
with a point deleted), and Harbater (1994) proved the
full Abhyankar conjecture by building upon this special
solution.
see also FINITE GROUP, GALOIS GROUP, QUOTIENT
Group, Sylow p-Subgroup
10
Ablowitz-Ramani-Segur Conjecture
Absolute Square
References
Abhyankar, S. "Coverings of Algebraic Curves." Airier. J.
Math. 79, 825-856, 1957.
American Mathematical Society. "Notices of the AMS, April
1995, 1995 Prank Nelson Cole Prize in Algebra." http://
www. ams . org/notices/199504/prize-cole .html.
Harbater, D. "Abhyankar's Conjecture on Galois Groups
Over Curves." Invent. Math. 117, 1-25, 1994.
Raynaud, M. "Revetements de la droite affine en car-
acteristique p > et conjecture d' Abhyankar." Invent.
Math. 116, 425-462, 1994.
Ablowitz-Ramani-Segur Conjecture
The Ablowitz-Ramani-Segur conjecture states that a
nonlinear Partial Differential Equation is solv-
able by the Inverse Scattering Method only if ev-
ery nonlinear Ordinary Differential Equation ob-
tained by exact reduction has the Painleve Property.
see also Inverse Scattering Method
References
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, p. 351, 1989.
Abscissa
The x- (horizontal) axis of a Graph.
see also Axis, Ordinate, Real Line, a;- Axis, y-Axis,
z-Axis
Absolute Convergence
A Series J^ n u n is said to Converge absolutely if the
Series J^ |u n | Converges, where |u n | denotes the
Absolute Value. If a Series is absolutely convergent,
then the sum is independent of the order in which terms
are summed. Furthermore, if the SERIES is multiplied by
another absolutely convergent series, the product series
will also converge absolutely.
see also Conditional Convergence, Convergent
Series, Riemann Series Theorem
References
Bromwich, T. J. Pa and MacRobert, T. M. "Absolute Con-
vergence." Ch. 4 in An Introduction to the Theory of In-
finite Series, 3rd ed. New York: Chelsea, pp. 69-77, 1991.
Absolute Deviation
Let u denote the Mean of a Set of quantities m, then
the absolute deviation is denned by
Aui = \m — u\.
Absolute Error
The Difference between the measured or inferred
value of a quantity xq and its actual value x, given by
Ax = Xq — x
(sometimes with the ABSOLUTE VALUE taken) is called
the absolute error. The absolute error of the Sum or
Difference of a number of quantities is less than or
equal to the SUM of their absolute errors.
see also Error Propagation, Percentage Error,
Relative Error
References
Abramowitz, M. and Stegun, C A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 14, 1972.
Absolute Geometry
Geometry which depends only on the first four of Eu-
clid's Postulates and not on the Parallel Postu-
late. Euclid himself used only the first four postulates
for the first 28 propositions of the Elements, but was
forced to invoke the PARALLEL POSTULATE on the 29th.
see also Affine Geometry, Elements, Euclid's Pos-
tulates, Geometry, Ordered Geometry, Paral-
lel Postulate
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 90-91, 1989.
Absolute Pseudoprime
see Carmichael Number
Absolute Square
Also known as the squared NORM. The absolute square
of a Complex Number z is written \z\ 2 and is defined
zz ,
(1)
where z* denotes the COMPLEX CONJUGATE of z. For
a Real Number, (1) simplifies to
I i2 2
\z\ = Z .
(2)
If the Complex Number is written z — x + iy, then
the absolute square can be written
see also Deviation, Mean Deviation, Signed Devi-
ation, Standard Deviation
k + w\ 2 = x +y 2 >
(3)
An important identity involving the absolute square is
given by
a ± be' ld | 2 = (a ± be' ld ){a ± be ld )
- a 2 -h b 2 ± ab(e i5 + e~ i5 )
— a + b 2 ± 2ab cos S.
(4)
Absolute Value
If a = 1, then (4) becomes
Abundance
11
|l±&e~ ilS | 2 = l + b 2 ±2bcos8
= l + & 2 ±26[l-2sin 2 (f£)]
= l±26 + & 2 =F46sin 2 (^)
- (l±&) 2 q= 4&sin 2 (^).
(5)
If a = 1, and 6=1, then
|1 - e~ iS \ 2 = (1 - l) 2 + 4 ■ lsin 2 (!<5) = 4sin 2 (±<5). (6)
Finally,
u^i+e** 3 ! 2 :
l) I e -i(<t>2-4>i)
- 2 + e n <
=:2 + 2cos(02-<^i) = 2[l + cos(^ 2 -0i)]
= 4 COS (02 - 0l).
(7)
Absolute Value
The absolute value of a REAL Number x is denoted \x\
and given by
, , f x f -x for x <
|x|=x 8 gn(*) = | a . forx ^ 0j
where SGN is the sign function.
The same notation is used to denote the M ODULUS of
a Complex Number z — x + iy, \z\ = y/x 2 + t/ 2 , a
p-ADlC absolute value, or a general Valuation. The
Norm of a Vector x is also denoted |x|, although ||x||
is more commonly used.
Other Notations similar to the absolute value are the
Floor Function [zj, Nint function [x], and Ceiling
Function [af|.
see also Absolute Square, Ceiling Function,
Floor Function, Modulus (Complex Number),
Nint, Sgn, Triangle Function, Valuation
Absolutely Continuous
Let // be a Positive Measure on a Sigma Algebra
M and let A be an arbitrary (real or complex) MEASURE
on M. Then A is absolutely continuous with respect to
//, written A < /z, if X(E) = for every E e M for
which fj.(E) = 0.
see also Concentrated, Mutually Singular
References
Rudin, W. Functional Analysis. New York: McGraw-Hill,
pp. 121-125, 1991.
Absorption Law
The law appearing in the definition of a Boolean Al-
gebra which states
a A (a V b) = a V (a A b) = a
for binary operators V and A (which most commonly are
logical OR and logical And).
see also BOOLEAN ALGEBRA, LATTICE
References
BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillian, p. 317, 1965.
Abstraction Operator
see Lambda Calculus
Abundance
The abundance of a number n is the quantity
A(n) = o~(n) — 2n,
where <x(n) is the DIVISOR FUNCTION. Kravitz has con-
jectured that no numbers exist whose abundance is an
Odd Square (Guy 1994).
The following table lists special classifications given to
a number n based on the value of A(n).
A(n) Number
< deficient number
— 1 almost perfect number
perfect number
1 quasiperfect number
> abundant number
see also DEFICIENCY
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 45-46, 1994.
12
Abundant Number
Acceleration
Abundant Number
An abundant number is an INTEGER n which is not a
Perfect Number and for which
s(n) = <r(n) ~ n > n,
(1)
where <r(n) is the DIVISOR FUNCTION. The quantity
cr(n) — 2n is sometimes called the ABUNDANCE. The
first few abundant numbers are 12, 18, 20, 24, 30, 36, . . .
(Sloane's A005101). Abundant numbers are sometimes
called Excessive Numbers.
There are only 21 abundant numbers less than 100, and
they are all Even. The first Odd abundant number is
945 = 3 3 -7-5. (2)
That 945 is abundant can be seen by computing
s(945) = 975 > 945. (3)
Any multiple of a PERFECT NUMBER or an abundant
number is also abundant. Every number greater than
20161 can be expressed as a sum of two abundant num-
bers.
Define the density function
\{n : <x(n) > xn}\
A(x) = lim
(4)
for a POSITIVE Real Number x, then Davenport (1933)
proved that A(x) exists and is continuous for all x,
and Erdos (1934) gave a simplified proof (Finch). Wall
(1971) and Wall et at. (1977) showed that
0.2441 < A(2) < 0.2909,
and Deleglise showed that
0.2474 < A(2) < 0.2480.
(5)
(6)
A number which is abundant but for which all its
Proper Divisors are Deficient is called a Primitive
Abundant Number (Guy 1994, p. 46).
see also Aliquot Sequence, Deficient Number,
Highly Abundant Number, Multiamicable Num-
bers, Perfect Number, Practical Number, Prim-
itive Abundant Number, Weird Number
References
Deleglise, M. "Encadrement de la densite des nombres abon-
dants." Submitted.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 3—33,
1952.
Erdos, P. "On the Density of the Abundant Numbers." J.
London Math. Soc. 9, 278-282, 1934.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/abund/abund* html.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 45-46, 1994.
Singh, S. FermaVs Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem. New York:
Walker, pp. 11 and 13, 1997.
Sloane, N. J. A. Sequence A005101/M4825 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Wall, C. R. "Density Bounds for the Sum of Divisors Func-
tion." In The Theory of Arithmetic Functions (Ed.
A. A. Gioia and D. L. Goldsmith). New York: Springer-
Verlag, pp. 283-287, 1971.
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density
Bounds for the Sum of Divisors Function." Math. Comput.
26, 773-777, 1972.
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density
Bounds for the Sum of Divisors Function." Math. Comput.
31, 616, 1977.
Acceleration
Let a particle travel a distance s(t) as a function of time
t (here, s can be thought of as the ARC LENGTH of
the curve traced out by the particle). The SPEED (the
Scalar Norm of the Vector Velocity) is then given
§=V(§r + (§)' + (s' <"
The acceleration is defined as the time DERIVATIVE of
the Velocity, so the SCALAR acceleration is given by
dv
di
(2)
d 2 s
dt 2
(3)
dx d 2 x _i_ dji d 2 y , dz d 2 z
dt dt 7 " ~*~ dt df2 "T" dt di?
(4)
jm 2 +m 2 +m*
dx d 2 x dy d 2 y dz d 2 z
ds~dF + dsdi? + ds~d¥
(5)
dr d 2 r
ds ' dt 2 '
(6)
The Vector acceleration is given by
dv
dt
dfr d's~ fdsY <T
d* = dt*- T+K {di) N -
(7)
where T is the UNIT TANGENT VECTOR, k the CURVA-
TURE, s the Arc Length, and N the Unit Normal
Vector.
Let a particle move along a straight LINE so that the
positions at times £i, £2, and £3 are si, 52, and S3, re-
spectively. Then the particle is uniformly accelerated
with acceleration a Iff
a = 2
($2 - S 3 )h + (33 - Si)t2 + (Si - 52)^3
(tl - t 2 )(t2 ~ t 3 )(t 3 - ti)
(8)
is a constant (Klamkin 1995, 1996).
Accidental Cancellation
Ackermann Function
13
Consider the measurement of acceleration in a rotating
reference frame. Apply the ROTATION OPERATOR
- f d \
■■-(
\. Ctt / body
+ u;x
(9)
twice to the RADIUS VECTOR r and suppress the body
notation,
R 2 r
(^ +WX )(S +WXr )
d 2 r d ( . dr
d 2 r dr du: dv
— — + u> x — +r x — - +u> x —
dt 2 dt dt dt
+ u?x (u; x r).
(10)
Grouping terms and using the definitions of the VELOC-
ITY v = dr/dt and Angular Velocity a = du/dt
give the expression
9-space
dt 2
+ 2u? x v + u; x (u> x r) 4- r x ex. (11)
Now, we can identify the expression as consisting of
three terms
= d*r
a b ody - df2 ,
aCoriolis = 2u? X V,
a ce ntrifugal = <*> X (u> X I*) ,
(12)
(13)
(14)
a "body" acceleration, centrifugal acceleration, and
Coriolis acceleration. Using these definitions finally
gives
&space = <*body "r ^Coriolis ~~r ^centrifugal + T X Of, (15)
where the fourth term will vanish in a uniformly ro-
tating frame of reference (i.e., ex = 0). The centrifugal
acceleration is familiar to riders of merry ^j-rounds, and
the Coriolis acceleration is responsible for the motions
of hurricanes on Earth and necessitates large trajectory
corrections for intercontinfv: L al ballistic missiles.
see also Angular Acceleration, Arc Length,
Jerk, Velocity
References
Klamkin, M. S. "Problem 1481." Math. Mag. 68, 307, 1995.
Klamkin, M. S. "A Characteristic of Constant Acceleration."
Solution to Problem 1481. Math. Mag. 69, 308, 1996.
Accidental Cancellation
see Anomalous Cancellation
Accumulation Point
An accumulation point is a Point which is the limit
of a Sequence, also called a Limit Point. For some
Maps, periodic orbits give way to Chaotic ones beyond
a point known as the accumulation point.
see also Chaos, Logistic Map, Mode Locking, Pe-
riod Doubling
Achilles and the Tortoise Paradox
see Zeno's Paradoxes
Ackermann Function
The Ackermann function is the simplest example of a
well-defined TOTAL FUNCTION which is COMPUTABLE
but not Primitive Recursive, providing a counterex-
ample to the belief in the early 1900s that every COM-
PUTABLE Function was also Primitive Recursive
(Dotzel 1991). It grows faster than an exponential func-
tion, or even a multiple exponential function. The Ack-
ermann function A(x } y) is defined by
(y+l if x =
A(x,y)= I A(x-l,l) if 2/ —
[ A{x — 1, A(x, y — 1)) otherwise.
Special values for Integer x include
(i)
A(0,y) = y + 1
(2)
A(l,y) = y + 2
(3)
A(2,y) = 2y + 3
(4)
A(3,y) = 2"+ 3 - 3
(5)
.4(4,2/) = 2^-3.
(6)
V+3
Expressions of the latter form are sometimes called
Power Towers. A(0,y) follows trivially from the def-
inition. A(l,y) can be derived as follows,
A(l,y) = A(0,A(l,y- 1)) = A(l,y- 1) + 1
= A(0,A(l,y- 2)) + 1 = A(l,y- 2) + 2
= . . . = .4(1, 0) + y = A(0, l) + y = y + 2.
(7)
A(2,y) has a similar derivation,
A(2,y) = A(l,A(2,y-l)) = A(2,y-.l) + 2
= i4(l ) A(2,y-2))+2 = i4(2 I y-2) + 4 = ...
= A(2, 0) + 2y = A(l, 1) + 2y = 2y + 3. (8)
Buck (1963) defines a related function using the same
fundamental Recurrence Relation (with arguments
flipped from Buck's convention)
F(x,y) = F(x-l,F(x t y-l)),
(9)
14
Ackermann Number
Acute Triangle
but with the slightly different boundary values
^(0, y) = V + 1
**(1,0) = 2
F(2,0) =
F(x,0) = 1 for x
= 3,4,
Buck's recurrence gives
F(l,») = 2 + i/
F(2,y) = 2y
f(3,y) = 2»
.2
F(4,j,) = 2 2 .
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
Taking F(4,n) gives the sequence 1, 2, 4, 16, 65536,
2 65536 , .... Defining ip(x) = F(x, x) for x = 0, 1, ...
.2
then gives 1, 3, 4, 8, 65536, 2 2 ' , . . . (Sloane's A001695),
where m = 2 2 , a truly huge number!
65536
see a/50 Ackermann Number, Computable Func-
tion, Goodstein Sequence, Power Tower, Primi-
tive Recursive Function, TAK Function, Total
Function
References
Buck, R. C. "Mathematical Induction and Recursive Defini-
tions." Amer. Math. Monthly 70, 128-135, 1963.
Dotzel, G. "A Function to End All Functions." Algorithm:
Recreational Programming 2.4, 16-17, 1991.
Kleene, S. C. Introduction to Metamathematics. New York:
Elsevier, 1971.
Peter, R. Rekursive Funktionen. Budapest: Akad. Kiado,
1951.
Reingold, E. H. and Shen, X. "More Nearly Optimal Algo-
rithms for Unbounded Searching, Part I: The Finite Case."
SIAM J. Corn-put. 20, 156-183, 1991.
Rose, H. E. Subrecursion, Functions, and Hierarchies. New
York: Clarendon Press, 1988.
Sloane, N. J. A. Sequence A001695/M2352 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Smith, H. J. "Ackermann's Function." http://www.netcom.
com/-hj smith/Ackerman . html.
Spencer, J. "Large Numbers and Unprovable Theorems."
Amer. Math. Monthly 90, 669-675, 1983.
Tarjan, R. E. Data Structures and Network Algorithms.
Philadelphia PA: SIAM, 1983.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 11, 227, and 232,
1991.
Ackermann Number
A number of the form n t • • • T™> where Arrow Nota-
n
TION has been used. The first few Ackermann numbers
.3
are 1 t 1 = 1, 2 tt 2 = 4, and 3 ttt 3 = 3 3
7,625,597,484,987
see also Ackermann Function, Arrow Notation,
Power Tower
References
Ackermann, W. "Zum hilbertschen Aufbau der reellen
Zahlen." Math. Ann. 99, 118-133, 1928.
Conway, J. H. and Guy, R. K, The Book of Numbers. New
York: Springer-Verlag, pp. 60-61, 1996.
Crandall, R. E. "The Challenge of Large Numbers." Sci.
Amer. 276, 74-79, Feb. 1997.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 11, 227, and 232,
1991.
Acnode
Another name for an ISOLATED POINT.
see also Crunode, Spinode, Tacnode
Acoptic Polyhedron
A term invented by B. Griinbaum in an attempt to pro-
mote concrete and precise POLYHEDRON terminology.
The word "coptic" derives from the Greek for "to cut,"
and acoptic polyhedra are defined as POLYHEDRA for
which the FACES do not intersect (cut) themselves, mak-
ing them 2-Manifolds.
see also Honeycomb, Nolid, Polyhedron, Sponge
Action
Let M(X) denote the GROUP of all invertible MAPS
X -> X and let G be any GROUP. A HOMOMORPHISM
6 :G -> M(X) is called an action of G on X. Therefore,
6 satisfies
1. For each g € G, 6(g) is a Map X -> X : x \-> 0(g)x,
2. 0(gh)x = 6{g)(O(h)x),
3. 0(e) a; = x, where e is the group identity in G,
4. 0(g- 1 )x = 6(g)- 1 x.
see also CASCADE, FLOW, SEMIFLOW
Acute Angle
An Angle of less than 7r/2 Radians (90°) is called an
acute angle.
see also ANGLE, OBTUSE ANGLE, RIGHT ANGLE,
Straight Angle
Acute Triangle
A Triangle in which all three Angles are Acute An-
gles. A Triangle which is neither acute nor a RIGHT
Triangle (i.e., it has an Obtuse Angle) is called an
Obtuse Triangle. A Square can be dissected into as
few as 8 acute triangles.
see also Obtuse Triangle, Right Triangle
Adams-Bashforth-Moulton Method
Addition Chain
15
Adams-Bashforth-Moulton Method
see Adams' Method
Adams' Method
Adams' method is a numerical METHOD for solving
linear First-Order Ordinary Differential Equa-
tions of the form
dy
dx
f{x>y)-
Let
: 3?n + l X n
(i)
(2)
be the step interval, and consider the Maclaurin Se-
ries of y about x n ,
y n +i = y n + ( -T-) ( x ~ x n)
(x - x n ) 2 + .
V dx J n + 1 \dxj n \ dx 2 J
(3)
(4)
Here, the Derivatives of y are given by the Backward
Differences
\dx/ n Xn+i ~ X
3/n+i - y n
h
(5)
(6)
(7)
etc. Note that by (1), q n is just the value of f{x ni y n ).
For first-order interpolation, the method proceeds by
iterating the expression
2/n+i = yn + q n h
(8)
where q n = /(x n ,2/n). The method can then be ex-
tended to arbitrary order using the finite difference in-
tegration formula from Beyer (1987)
/*
Jo
/ p ^=(l+IV+£,V 2 + fV 3
,251 V 4 + J95_V 5
~720 v ~ 288 v
19087 V7 6
V° + ...)/p (9)
to obtain
2/n+i -y n = h(q n + \ Vq n -i + ^ V 2 q n -2 + f V 3 g n -
12
95
288
+ ffivV-4 + ^V 5 g n _5 + ...)■ (10)
Note that von Karman and Biot (1940) confusingly use
the symbol normally used for FORWARD DIFFERENCES
A to denote BACKWARD DIFFERENCES V.
see also Gill's Method, Milne's Method, Predic-
TOR-CORRECTOR METHODS, RUNGE-KUTTA METHOD
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 896, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 455, 1987.
Karman, T. von and Biot, M. A. Mathematical Methods in
Engineering: An Introduction to the Mathematical Treat-
ment of Engineering Problems. New York: McGraw-Hill,
pp. 14-20, 1940.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 741, 1992.
Addend
A quantity to be Added to another, also called a Sum-
MAND. For example, in the expression a + 6 + c, a, 6, and
c are all addends. The first of several addends, or "the
one to which the others are added" (a in the previous
example), is sometimes called the AUGEND.
see also Addition, Augend, Plus, Radicand
Addition
i i -
15 8-
J- 249 *
407-
Y carries
(-addend 1
Y addend 2
hsum
The combining of two or more quantities using the PLUS
operator. The individual numbers being combined are
called ADDENDS, and the total is called the Sum. The
first of several ADDENDS, or "the one to which the oth-
ers are added," is sometimes called the AUGEND. The
opposite of addition is SUBTRACTION.
While the usual form of adding two n-digit INTEGERS
(which consists of summing over the columns right to
left and "Carrying" a 1 to the next column if the sum
exceeds 9) requires n operations (plus carries), two n-
digit INTEGERS can be added in about 21gn steps by
n processors using carry-lookahead addition (McGeoch
1993). Here, lgx is the Lg function, the LOGARITHM to
the base 2.
see also Addend, Amenable Number, Augend,
Carry, Difference, Division, Multiplication,
Plus, Subtraction, Sum
References
McGeoch, C. C. "Parallel Addition." Amer. Math. Monthly
100, 867-871, 1993.
Addition Chain
An addition chain for a number n is a SEQUENCE 1 =
ao < ai < . . . < a T = n, such that each member after ao
is the SUM of the two earlier (not necessarily distinct)
ones. The number r is called the length of the addition
chain. For example,
1,1 + 1 = 2,2 + 2 = 4,4 + 2 = 6,6 + 2 = 8,8 + 6 = 14
16 Addition-Multiplication Magic Square
Adele Group
is an addition chain for 14 of length r = 5 (Guy 1994).
see also BRAUER CHAIN, HANSEN CHAIN, SCHOLZ CON-
JECTURE
References
Guy, R. K. "Addition Chains. Brauer Chains. Hansen
Chains." §C6 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994.
Addition-Multiplication Magic Square
46
81
117
102
15
76
200
203
19
60
232
175
54
69
153
78
216
161
17
52
171
90
58
75
135
114
50
87
184
189
13
68
150
261
45
38
91
136
92
27
119
104
108
23
174
225
57
30
116
25
133
120
51
26
162
207
39
34
138
243
100
29
105
152
102207290 38
115216171
102207290 3
115216171
A square which is simultaneously a MAGIC SQUARE and
Multiplication Magic Square. The three squares
shown above (the top square has order eight and the
bottom two have order nine) have addition MAGIC CON-
STANTS (840, 848, 1200) and multiplicative magic con-
stants (2,058,068,231,856,000; 5,804,807,833,440,000;
1,619,541,385,529,760,000), respectively (Hunter and
Madachy 1975, Madachy 1979).
see also MAGIC SQUARE
References
Hunter, J, A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3
in Mathematical Diversions. New York: Dover, pp, 30-31,
1975.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 89-91, 1979.
Additive Persistence
Consider the process of taking a number, adding its DIG-
ITS, then adding the DIGITS of number derived from it,
etc., until the remaining number has only one DIGIT.
The number of additions required to obtain a single
DIGIT from a number n is called the additive persis-
tence of n, and the DIGIT obtained is called the DIGITAL
Root of n.
For example, the sequence obtained from the starting
number 9876 is (9876, 30, 3), so 9876 has an additive
persistence of 2 and a DIGITAL ROOT of 3. The ad-
ditive persistences of the first few positive integers are
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1,
. . . (Sloane's A031286). The smallest numbers of ad-
ditive persistence n for n = 0, 1, . . . are 0, 10, 19,
199, 19999999999999999999999, . . . (Sloane's A006050).
There is no number < 10 5 ° with additive persistence
greater than 11.
It is conjectured that the maximum number lacking the
DIGIT 1 with persistence 11 is
77777733332222222222222222222
There is a stronger conjecture that there is a maximum
number lacking the DIGIT 1 for each persistence > 2.
The maximum additive persistence in base 2 is 1. It is
conjectured that all powers of 2 > 2 15 contain a in base
3, which would imply that the maximum persistence in
base 3 is 3 (Guy, 1994).
see also Digitadition, Digital Root, Multiplica-
tive Persistence, Narcissistic Number, Recur-
ring Digital Invariant
References
Guy, R. K. "The Persistence of a Number." §F25 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 262-263, 1994.
Hinden, H. J. "The Additive Persistence of a Number." J.
Recr. Math. 7, 134-135, 1974.
Sloane, N. J. A. Sequences A031286 and A006050/M4683 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Sloane, N. J. A. "The Persistence of a Number." J. Recr.
Math. 6, 97-98, 1973.
Adele
An element of an Adele GROUP, sometimes called a
Repartition in older literature. Adeles arise in both
Number Fields and Function Fields. The adeles of
a Number Field are the additive Subgroups of all ele-
ments in Yl kvi where v is the PLACE, whose ABSOLUTE
Value is < 1 at all but finitely many i/s.
Let F be a Function Field of algebraic functions of
one variable. Then a MAP r which assigns to every
PLACE P of F an element r(P) of F such that there are
only a finite number of PLACES P for which v P (r(P)) <
0.
see also Idele
References
Chevalley, C. C. Introduction to the Theory of Algebraic
Functions of One Variable. Providence, RI: Amer. Math.
Soc, p. 25, 1951.
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Adele Group
The restricted topological Direct Product of the
GROUP Gk v with distinct invariant open subgroups Go v ,
References
Weil, A. Adeles and Algebraic Groups.
Princeton University Press, 1961.
Princeton, NJ:
Adem Relations
Adjoint Operator 17
Adem Relations
Relations in the definition of a Steenrod Algebra
which state that, for i < 2j,
L*J
j - k - l\ i+j-k
Sq* o Sq*(x) = Y.[ 3 i- 2k ' W +J ~" ° S <^'
where fog denotes function COMPOSITION and |_*J is
the Floor Function.
see also STEENROD ALGEBRA
Adequate Knot
A class of Knots containing the class of Alternating
Knots. Let c(K) be the CROSSING Number. Then for
KNOT Sum Ki#K 2 which is an adequate knot,
c(K 1 #K 2 )^c(Ki) + c(K2).
This relationship is postulated to hold true for all
Knots.
see also Alternating Knot, Crossing Number
(Link)
Adiabatic Invariant
A property of motion which is conserved to exponential
accuracy in the small parameter representing the typical
rate of change of the gross properties of the body.
see also ALGEBRAIC INVARIANT, LYAPUNOV CHARAC-
TERISTIC Number
Adjacency Matrix
The adjacency matrix of a simple Graph is a Matrix
with rows and columns labelled by VERTICES, with a 1
or in position (vi,Vj) according to whether Vi and Vj
are ADJACENT or not.
see also INCIDENCE MATRIX
References
Chartrand, G. Introductory Graph Theory.
Dover, p. 218, 1985.
New York:
Adjacency Relation
The Set E of Edges of a Graph (V,E), being a set
of unordered pairs of elements of V, constitutes a RE-
LATION on V. Formally, an adjacency relation is any
Relation which is Irreflexive and Symmetric.
see also Irreflexive, Relation, Symmetric
Adjacent Fraction
Two FRACTIONS are said to be adjacent if their differ-
ence has a unit NUMERATOR. For example, 1/3 and 1/4
are adjacent since 1/3 - 1/4 = 1/12, but 1/2 and 1/5
are not since 1/2 — 1/5 = 3/10. Adjacent fractions can
be adjacent in a Farey SEQUENCE.
see also FAREY SEQUENCE, FORD CIRCLE, FRACTION,
Numerator
References
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 119, 1995.
Adjacent Value
The value nearest to but still inside an inner FENCE.
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 667, 1977.
Adjacent Vertices
In a GRAPH G, two VERTICES are adjacent if they are
joined by an EDGE.
Adjoint Curve
A curve which has at least multiplicity Vi — 1 at each
point where a given curve (having only ordinary singu-
lar points and cusps) has a multiplicity vi is called the
adjoint to the given curve. When the adjoint curve is of
order n — 3, it is called a special adjoint curve.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 30, 1959.
Adjoint Matrix
The adjoint matrix, sometimes also called the Ad JU-
GATE Matrix, is defined by
a* = (A T r,
(i)
where the ADJOINT OPERATOR is denoted * and T de-
notes the Transpose. If a Matrix is Self- Adjoint,
it is said to be HERMITIAN. The adjoint matrix of a
Matrix product is given by
(oft)^. = [(a6) T ]*, . (2)
Using the property of transpose products that
[(a&) T ];, = (6 T a-% = (&<&■)• = (b T ): k (a T y kj
= b lAj = ( fot «% > ( 3 )
it follows that
(AB) f = BW.
(4)
Adjoint Operator
Given a Second-Order Ordinary Differential
Equation
- , . du du t v
Cu(x) - p — + Pl — + P2 u, (1)
where pi = Pi(x) and u = u(x), the adjoint operator &
is defined by
d
" ^ (PoU) " di^ PlU) +PaU
d 2 u f t ,du ( „ , ,
-P°ZT^ + ( 2 Po -pi)^~ + (po -pi +P2)U.
'dx 2
dx
(2)
18 Adjugate Matrix
Affine Hull
Write the two Linearly Independent solutions as
t/i (x) and 2/2 (#)■ Then the adjoint operator can also
be written
?../
(y 2 Cyi ~yi£y 2 )dx =
— {yi 2/2 - 2/13/2 )
Po
(3)
see a/50 Self-Adjoint Operator, Sturm-Liouville
Theory
Adjugate Matrix
see Adjoint Matrix
Adjunction
If a is an element of a Field F over the PRIME Field
P, then the set of all RATIONAL FUNCTIONS of a with
Coefficients in P is a Field derived from P by ad-
junction of a.
Adleman-Pomerance-Rumely Primality Test
A modified Miller's Primality Test which gives a
guarantee of Primality or COMPOSITENESS. The Al-
gorithm's running time for a number N has been
provedtobeasO((lniV) clnlnlnJV ) for some c> 0. It was
simplified by Cohen and Lenstra (1984), implemented by
Cohen and Lenstra (1987), and subsequently optimized
by Bosma and van der Hulst (1990).
References
Adleman, L. M.; Pomerance, C; and Rumely, R. S. "On
Distinguishing Prime Numbers from Composite Number."
Ann. Math. 117, 173-206, 1983.
Bosma, W. and van der Hulst, M.-P. "Faster Primality Test-
ing." In Advances in Cryptology, Proc. Eurocrypt '89,
Houthalen, April 10-13, 1989 (Ed. J.-J. Quisquater). New-
York: Springer- Verlag, 652-656, 1990.
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.;
and Tuckerman, B. Factorizations of b n ± 1, b — 2,
3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence,
RI: Amer. Math. Soc, pp. lxxxiv-lxxxv, 1988.
Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi
Sums." Math. Comput. 42, 297-330, 1984.
Cohen, H. and Lenstra, A. K. "Implementation of a New
Primality Test." Math. Comput 48, 103-121, 1987.
Mihailescu, P. "A Primality Test Using Cyclotomic Exten-
sions." In Applied Algebra, Algebraic Algorithms and
Error- Correcting Codes (Proc. AAECC-6, Rome, July
1988). New York: Springer- Verlag, pp. 310-323, 1989.
Adleman- Rumely Primality Test
see Adleman-Pomerance-Rumely Primality Test
Admissible
A string or word is said to be admissible if that word
appears in a given SEQUENCE. For example, in the SE-
QUENCE aabaabaabaabaab . . ., a, aa, baab are all admis-
sible, but bb is inadmissible.
see also BLOCK GROWTH
Affine Complex Plane
The set A 2 of all ordered pairs of COMPLEX NUMBERS.
see also Affine Connection, Affine Equation,
Affine Geometry, Affine Group, Affine Hull,
Affine Plane, Affine Space, Affine Transforma-
tion, Affinity, Complex Plane, Complex Projec-
tive Plane
Affine Connection
see Connection Coefficient
Affine Equation
A nonhomogeneous Linear Equation or system of
nonhomogeneous LINEAR EQUATIONS is said to be
affine.
see also AFFINE COMPLEX PLANE, AFFINE CONNEC-
TION, Affine Geometry, Affine Group, Affine
Hull, Affine Plane, Affine Space, Affine Trans-
formation, Affinity
Affine Geometry
A GEOMETRY in which properties are preserved by PAR-
ALLEL Projection from one Plane to another. In an
affine geometry, the third and fourth of Euclid's Pos-
tulates become meaningless. This type of GEOMETRY
was first studied by Euler.
see also ABSOLUTE GEOMETRY, AFFINE COMPLEX
Plane, Affine Connection, Affine Equation,
Affine Group, Affine Hull, Affine Plane, Affine
Space, Affine Transformation, Affinity, Or-
dered Geometry
References
Birkhoff, G. and Mac Lane, S. "Affine Geometry." §9.13 in A
Survey of Modern Algebra, 3rd ed. New York: Macmillan,
pp. 268-275, 1965.
Affine Group
The set of all nonsingular Affine TRANSFORMATIONS
of a Translation in Space constitutes a Group known
as the affine group. The affine group contains the full
linear group and the group of TRANSLATIONS as SUB-
GROUPS.
see also AFFINE COMPLEX PLANE, AFFINE CONNEC-
TION, Affine Equation, Affine Geometry, Affine
Hull, Affine Plane, Affine Space, Affine Trans-
formation, Affinity
References
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, p. 237, 1965.
Affine Hull
The IDEAL generated by a SET in a VECTOR SPACE.
see also Affine Complex Plane, Affine Connec-
tion, Affine Equation, Affine Geometry, Affine
Group, Affine Plane, Affine Space, Affine
Transformation, Affinity, Convex Hull, Hull
AfRne Plane
Affine Transformation
19
Affine Plane
A 2-D Affine Geometry constructed over a Finite
Field. For a Field F of size n, the affine plane consists
of the set of points which are ordered pairs of elements in
F and a set of lines which are themselves a set of points.
Adding a Point at Infinity and Line at Infinity
allows a Projective Plane to be constructed from an
affine plane. An affine plane of order n is a BLOCK
DESIGN of the form (n 2 , n, 1). An affine plane of order
n exists Iff a PROJECTIVE PLANE of order n exists.
see also Affine Complex Plane, Affine Connec-
tion, Affine Equation, Affine Geometry, Affine
Group, Affine Hull, Affine Space, Affine Trans-
formation, Affinity, Projective Plane
References
Lindner, C. C. and Rodger, C. A. Design Theory.
Raton, FL: CRC Press, 1997.
Boca
Affine Scheme
A technical mathematical object defined as the SPEC-
TRUM ct(A) of a set of Prime Ideals of a commutative
RING A regarded as a local ringed space with a structure
sheaf.
see also SCHEME
References
Iyanaga, S. and Kawada, Y. (Eds.). "Schemes." §18E in En-
cyclopedic Dictionary of Mathematics. Cambridge, MA:
MIT Press, p. 69, 1980.
Affine Space
Let V be a VECTOR Space over a FIELD K, and let A
be a nonempty SET. Now define addition p -f a € A for
any VECTOR a E V and element p e A subject to the
conditions
1. p + 0=p,
2. (p + a)+b = p+(a + b),
3. For any q G A, there EXISTS a unique VECTOR a 6 V
such that q = p + a.
Here, a, b £ V. Note that (1) is implied by (2) and (3).
Then A is an affine space and K is called the COEFFI-
CIENT Field.
In an affine space, it is possible to fix a point and co-
ordinate axis such that every point in the SPACE can
be represented as an n-tuple of its coordinates. Every
ordered pair of points A and B in an affine space is then
associated with a VECTOR AB.
see also Affine Complex Plane, Affine Connec-
tion, Affine Equation, Affine Geometry, Affine
Group, Affine Hull, Affine Plane, Affine Space,
Affine Transformation, Affinity
Affine Transformation
Any Transformation preserving Collinearity (i.e.,
all points lying on a Line initially still lie on a Line
after TRANSFORMATION). An affine transformation is
also called an AFFINITY. An affine transformation of
R n is a Map F : R n -> W 1 of the form
F(p) = Ap + q
(1)
for all p € M n , where A is a linear transformation of
W 1 . If det(A) = 1, the transformation is Orientation-
Preserving; if det(A) = -1, it is Orientation-
Reversing.
Dilation (Contraction, Homothecy), Expansion,
Reflection, Rotation, and Translation are all
affine transformations, as are their combinations. A par-
ticular example combining ROTATION and EXPANSION is
the rotation-enlargement transformation
V
= s
= s
cos a
— sin a
sin a
cos a
x — Xo
y-yo
cos a(x — Xo) + sin a(y — yo)
— sina(x — Xo) + cos a(y — yo)
(2)
Separating the equations,
x — (s cos a)x + (s sin a)y — s(xo cos a + yo sin a) (3)
y = (— s sin a)x + (5 cos a)y + s(xq sin a — yo cos a).
(4)
This can be also written as
where
x = ax + by + c
y = bx + ay + d,
a = s cos a
b = —3 sin a.
The scale factor 5 is then defined by
8= \/a 2 +6 2 ,
and the rotation Angle by
■'(-!)■
a = tan
(5)
(6)
(7)
(8)
(9)
(10)
see also Affine Complex Plane, Affine Connec-
tion, Affine Equation, Affine Geometry, Affine
Group, Affine Hull, Affine Plane, Affine Space,
Affine Transformation, Affinity, Equiaffinity,
Euclidean Motion
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 105, 1993.
20 Affinity
Affinity
see AFFINE TRANSFORMATION
Affix
In the archaic terminology of Whittaker and Watson
(1990), the Complex Number z representing x + iy.
References
Whittaker, E. T. and Watson, G. N. A Course in Modem
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Aggregate
An archaic word for infinite SETS such as those consid-
ered by Georg Cantor.
see also Class (Set), Set
AGM
see Arithmetic-Geometric Mean
Agnesi's Witch
see Witch of Agnesi
Agnesienne
see Witch of Agnesi
Agonic Lines
see Skew Lines
Ahlfors-Bers Theorem
The Riemann's Moduli Space gives the solution to
Riemann's Moduli Problem, which requires an An-
alytic parameterization of the compact RlEMANN SUR-
FACES in a fixed HOMEOMORPHISM.
Airy Differential Equation
Some authors define a general Airy differential equation
as
y" ± k xy — 0. (1)
This equation can be solved by series solution using the
expansions
y = ^a n z n (2)
71 =
OO CO
/ V^ n-1 V^ "-1
y = > na n x = y ^na n x
n=0 n=l
OO
= ^^(n + l)a n +ix n (3)
TX-0
OO OO
y" — /.( n + l)na n +ix n ~~ = 2_^^ n ~*~ l) na n+i# n ~
n=0 n=l
oo
= J^(n + 2)(n + l)a n+2 x n . (4)
Airy Differential Equation
Specializing to the "conventional" Airy differential equa-
tion occurs by taking the Minus Sign and setting
k 2 = 1. Then plug (4) into
y" -xy = (5)
to obtain
OO oo
^(n + 2)(n + l)a n+2 x n - x ^ a ^ = ° ( 6 )
n=0 n—0
OO oo
^(n + 2)(n + l)a n+2 z n -^a n :r n+1 =0 (7)
Tl = Tl =
OO oo
2a 2 + ^(n + 2)(n + l)a n+2 z n - ^T ^-ix n = (8)
n=l n— 1
OO
2a 2 + J^[(n + 2)(n + l)a n+2 - a n _i]a; n = 0. (9)
n = l
In order for this equality to hold for all #, each term
must separately be 0. Therefore,
a 2 = (10)
(n + 2)(n + l)a n+2 = a n _i. (11)
Starting with the n = 3 term and using the above RE-
CURRENCE Relation, we obtain
5-4a 5 = 20a 5 = a 2 = 0. (12)
Continuing, it follows by INDUCTION that
a 2 = a$ = ag = an = . . . a3n-i = (13)
for n = 1, 2, Now examine terms of the form £3^.
(14)
a 3 =
ae =
ao
3^2
^3 =
6-5 ~ (6-5)(3-2)
a& ao
ao
(15)
(16)
9-8 (9-8)(6-5)(3-2)'
Again by INDUCTION,
_ _ao
0,371 " f(3n)(3n - l)][(3n - 3)(3n - 4)] • ■ • [6 * 5] [3 ■ 2]
(17)
for n = 1, 2, Finally, look at terms of the form
a3n+l,
a^
a 7
ai
4-3
a4
7^6 = (7-6)(4-3)
ar 01
ai
10-9 (10-9)(7-6)(4-3)"
(18)
(19)
(20)
Airy-Fock Functions
By Induction,
d3n+l =
0,1
[(3n + l)(3n)][(3n - 2)(3n - 3)] • * - [7 ■ 6] [4 ■ 3]
(21)
for n = 1, 2, The general solution is therefore
y = a>o
+ ai
n=l
oo
(3n)(3n - l)(3n - 3)(3n - 4) • • • 3 ■ 2
(3n + l)(3n)(3n - 2)(3n - 3) ■ ■ ■ 4 • 3
(22)
For a general k 2 with a MINUS SIGN, equation (1) is
y" - k 2 xy = 0, (23)
and the solution is
y(x) = fvS [A/_ 1/3 (§W /2 ) - S/ 1/3 (f fcx 3 / 2 )] ,
(24)
where I is a Modified Bessel Function of the
First Kind. This is usually expressed in terms of the
Airy Functions Ai(#) and Bi(#)
y(x) = A' Ai{k 2/3 x) + B' Bi(fc 2/3 x). (25)
If the Plus Sign is present instead, then
y +k xy =
(26)
and the solutions are
y(x) = \& [AJ. 1/3 (\kx z ? 2 ) + BJ 1/Z (f kx^ 2 )] ,
(27)
where J(z) is a Bessel Function of the First Kind.
see also Airy-Fock Functions, Airy Functions,
Bessel Function of the First Kind, Modified
Bessel Function of the First Kind
Airy-Fock Functions
The three Airy-Fock functions are
v{z) = ~y/irAi(z)
wi(z) = 2e l7T/6 u(ujz)
W2(z) = 2e~ t7r/ v(uj~ z)
(i)
(2)
(3)
where Ai(z) is an Airy Function. These functions
satisfy
v{z) = ^W-^W ( 4)
[w 1 {z)]*=w 2 {z*), (5)
where z* is the COMPLEX CONJUGATE of z.
see also AlRY FUNCTIONS
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, p. 65, 1988.
Airy Functions 21
Airy Functions
Watson's (1966, pp. 188-190) definition of an Airy func-
tion is the solution to the Airy Differential EQUA-
TION
$" ±k 2 $x = (1)
which is Finite at the Origin, where <£' denotes the
Derivative d$/dx, k 2 — 1/3, and either Sign is per-
mitted. Call these solutions (l/7r)$(±fc 2 ,:c), then
£•<*»'*> =jf
cos(£ 3 ± xi) dt
(2)
*(§;*) = Wf
*(-§;*) = Wf
(2x 3/2 \ (2x 3/2 \
(3)
r . 2x 3 »\ T fix*'*
3 3 / 2
3 3 / 2
(4)
where J(z) is a Bessel Function of the First Kind
and I(z) is a MODIFIED BESSEL FUNCTION OF THE
First Kind. Using the identity
K n (x)
TV I-n(x) - I n (x)
2 sin(n7r)
(5)
where K{z) is a MODIFIED BESSEL FUNCTION OF THE
Second Kind, the second case can be re-expressed
(8)
1 /Fir f 2 * 3/2 ^
A more commonly used definition of Airy functions is
given by Abramowitz and Stegun (1972, pp. 446-447)
and illustrated above. This definition identifies the
Ai(x) and Bi(a?) functions as the two LINEARLY INDE-
PENDENT solutions to (1) with k 2 = 1 and a MINUS
Sign,
y -yz^o. (9)
22 Airy Functions
The solutions are then written
y(z) = AAi(z) + BBi(z) 7
where
(10)
Ai(*) = -*(-l,z)
= |^[/_ 1/3 (I^ /2 )-/ 1/ 3(Iz 3/2 )]
= ^^/3(I^ /2 ) ("J
Bi(z) = y|[7_ a/ 3(fz 3/2 ) + / 1 /3(!/ /2 )]. (12)
In the above plot, Ai(z) is the solid curve and Bi(z) is
dashed. For zero argument,
Ai(0)
3 -2/3
(13)
where T(z) is the GAMMA FUNCTION. This means that
Watson's expression becomes
/»oo
(3a)- 1/3 7rAi(±(3a)- 1/3 z)= / cos(at 3 ±xt)dt. (14)
Jo
A generalization has been constructed by Hardy.
The Asymptotic Series of Ai(z) has a different form
in different QUADRANTS of the COMPLEX PLANE, a fact
known as the STOKES PHENOMENON, Functions related
to the Airy functions have been defined as
Gi(z)
HiW
* Jo
t + zt) dt
(15)
exp(-f* 3 +2t)<ft. (16)
see also AlRY-FoCK FUNCTIONS
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Airy Functions."
§10.4 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 446-452, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Bessel Functions of Fractional Order, Airy
Functions, Spherical Bessel Functions." §6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 234-245, 1992.
Spanier, J. and Oldham, K. B. "The Airy Functions Ai(x)
and Bi(x)." Ch. 56 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 555-562, 1987.
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nded. Cambridge, England: Cambridge University Press,
1966.
Aitken's 5 2 Process
Airy Projection
A Map Projection. The inverse equations for <j> are
computed by iteration. Let the ANGLE of the projection
plane be 0&. Define
for 9 b
a— < ln[2 cos( -^ it -e b )]
I*
t—y otherwise.
tan[f(|ir-0 b )]
(1)
For proper convergence, let Xi = 7r/6 and compute the
initial point by checking
Xi =
exp[-(^fx 2 + y 2 +atanxi)tan#i] . (2)
As long as x» > 1, take x i+ \ = Xi/2 and iterate again.
The first value for which Xi < 1 is then the starting
point. Then compute
Xi = cos' 1 {exp[-(^/x 2 ~+y 2 -{- atanxi) ta,nxi]} (3)
until the change in xi between evaluations is smaller
than the acceptable tolerance. The (inverse) equations
are then given by
^7T - 2Xi
- tan
-(-;)
(4)
(5)
Aitken's 5 2 Process
An Algorithm which extrapolates the partial sums s n
of a Series J^ a n whose Convergence is approxi-
mately geometric and accelerates its rate of CONVER-
GENCE. The extrapolated partial sum is given by
Sn = S n +1
(S n +1 — S n )
S n +1 — 2s n + Sn-1
see also EULER'S SERIES TRANSFORMATION
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 18, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 160, 1992.
Aitken Interpolation
Albers Equal-Area Conic Projection 23
Aitken Interpolation
An algorithm similar to Neville's Algorithm for con-
structing the Lagrange Interpolating Polynom-
ial. Let f(x\xo, x\, . . • , Xk) be the unique POLYNOMIAL
of kth ORDER coinciding with f(x) at xq, . . . , Xfc. Then
f(x\xo,Xi) =
f(x\x Qy x 2 ) =
f(x\xo>x ly x 2 ) =
f(x\x 0i x 1 ,x 2 ,X3) =
1
Xl
- Xo
1
X2
— Xo
1
X 2
- x±
1
/o
Xo
— X
A
Xl
— X
/o
Xo
— X
A
X 2
— X
X 3 — X2
/(x|x ,xi) Xi - X
/(x|x 0) x 2 ) x 2 - x
/(x|x ,Xi,X 2 ) X 2 - X
/(x|x ,Xi,X 3 ) X 3 - X
see a/so LAGRANGE INTERPOLATING POLYNOMIAL
References
Abramowitz, M. and Stegun, C. A. (Eds.), Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 879, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 93-94, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 102, 1992.
Ajima-Malfatti Points
The lines connecting the vertices and corresponding
circle-circle intersections in Malfatti's Tangent Tri-
angle Problem coincide in a point Y called the first
Ajima-Malfatti point (Kimberling and MacDonald 1990,
Kimberling 1994). Similarly, letting A", £", and C" be
the excenters of ABC, then the lines A 1 A", B'B", and
C'C" are coincident in another point called the second
Ajima-Malfatti point. The points are sometimes simply
called the Malfatti Points (Kimberling 1994).
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "1st and 2nd Ajima-Malfatti Points."
http://vvw . evansville . edu/ -ck6/ tcenters/ recent /
ajmalf.html.
Kimberling, C. and MacDonald, I. G. "Problem E 3251 and
Solution. " Amer. Math. Monthly 97, 612-613, 1990.
Albanese Variety
An Abelian Variety which is canonically attached to
an Algebraic Variety which is the solution to a cer-
tain universal problem. The Albanese variety is dual to
the Picard Variety.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, pp. 67-68, 1988.
Albers Conic Projection
see Albers Equal- Area Conic Projection
Albers Equal- Area Conic Projection
Let <fro be the Latitude for the origin of the Cartesian
Coordinates and Ao its Longitude. Let 0i and <j>2
be the standard parallels. Then
x = p sin v
11)
y = po - pcosO,
(2)
where
\JC — In sin
(3)
e = n(X- Ao)
(4)
yJC — 2nsin<^o
po =
n
(5)
C = cos 2 0i + 2n sin 0i
(6)
n = ~ (sin 0i + sin 02 ) .
(7)
The inverse FORMULAS are
(8)
A = A + -,
(9)
where
P= \A 2 + (po - y) 2
= tan
x
po-y
(10)
(ii)
References
Snyder, J. P, Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 98-103, 1987.
24 Alcuin's Sequence
Alexander- Conway Polynomial
Alcuin's Sequence
The Integer Sequence 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4,
7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, . . .
(Sloane's A005044) given by the COEFFICIENTS of the
Maclaurin Series for 1/(1 - x 2 )(l - x 3 )(l - x 4 ). The
number of different TRIANGLES which have INTEGRAL
sides and Perimeter n is given by
T(n) = P 3 (n) - J2 P2 W
l<j<ln/2\
[si - lij m
48 j
for n even
for n odd,
(1)
(2)
(3)
where P2(n) and Ps{n) are PARTITION FUNCTIONS, with
Pk{n) giving the number of ways of writing n as a sum of
k terms, [x] is the NiNT function, and |_^J is the FLOOR
Function (Jordan et al 1979, Andrews 1979, Hons-
berger 1985). Strangely enough, T(n) for n = 3, 4, . . .
is precisely Alcuin's sequence.
see also PARTITION FUNCTION P, TRIANGLE
References
Andrews, G. "A Note on Partitions and Triangles with Inte-
ger Sides." Amer. Math. Monthly 86, 477, 1979.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 39-47, 1985.
Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with
Integer Sides." Amer. Math. Monthly 86, 686-689, 1979.
Sloane, N. J. A. Sequence A005044/M0146 in 'An On-Line
Version of the Encyclopedia of Integer Sequences."
Aleksandrov-Cech Cohomology
A theory which satisfies all the ElLENBERG-STEENROD
Axioms with the possible exception of the LONG EX-
ACT Sequence of a Pair Axiom, as well as a certain
additional continuity CONDITION.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, p. 68, 1988.
Aleksandrov's Uniqueness Theorem
A convex body in Euclidean n-space that is centrally
symmetric with center at the ORIGIN is determined
among all such bodies by its brightness function (the
Volume of each projection).
see also TOMOGRAPHY
References
Gardner, R. J. "Geometric Tomography." Not. Amer. Math.
Soc. 42, 422-429, 1995.
Aleph
The Set Theory symbol (N) for the Cardinality of
an Infinite Set.
see also Aleph-0 (N ), Aleph-1 (Ni), Countable
Set, Countably Infinite Set, Finite, Infinite,
Transfinite Number, Uncountably Infinite Set
Aleph-0 (N )
The Set Theory symbol for a Set having the same
Cardinal Number as the "small" Infinite Set of In-
tegers. The Algebraic Numbers also belong to N .
Rather surprising properties satisfied by N include
N r = No
rN = N
N + / = N ,
where / is any FINITE SET. However,
No* = C,
(1)
(2)
(3)
(4)
where C is the CONTINUUM.
see also ALEPH-1, CARDINAL NUMBER, CONTINUUM,
Continuum Hypothesis, Countably Infinite Set,
Finite, Infinite, Transfinite Number, Uncount-
ably Infinite Set
Aleph-1 (Ni)
The Set Theory symbol for the smallest Infinite Set
larger than Alpha-0 (N ). The CONTINUUM HYPOTH-
ESIS asserts that Ni = c, where c is the CARDINALITY
of the "large" Infinite Set of Real Numbers (called
the CONTINUUM in Set Theory). However, the truth
of the Continuum Hypothesis depends on the version
of Set Theory you are using and so is Undecidable.
Curiously enough, n-D SPACE has the same number of
points (c) as 1-D Space, or any Finite Interval of 1-
D Space (a Line Segment), as was first recognized by
Georg Cantor.
see also Aleph-0 (N ), Continuum, Continuum Hy-
pothesis, Countably Infinite Set, Finite, Infi-
nite, Transfinite Number, Uncountably Infinite
Set
Alethic
A term in LOGIC meaning pertaining to TRUTH and
Falsehood.
see also False, Predicate, True
Alexander- Conway Polynomial
see Conway Polynomial
Alexander's Horned Sphere
Alexander's Horned Sphere
Alexander Matrix
25
The above solid, composed of a countable UNION of
Compact Sets, is called Alexander's horned sphere.
It is Homeomorphic with the BALL B 3 , and its bound-
ary is therefore a SPHERE. It is therefore an example of
a wild embedding in E 3 . The outer complement of the
solid is not SIMPLY CONNECTED, and its fundamental
GROUP is not finitely generated. Furthermore, the set
of nonlocally flat ("bad") points of Alexander's horned
sphere is a Cantor Set.
The complement in K of the bad points for Alexan-
der's horned sphere is SIMPLY CONNECTED, making it
inequivalent to Antoine'S Horned Sphere. Alexan-
der's horned sphere has an uncountable infinity of Wild
POINTS, which are the limits of the sequences of the
horned sphere's branch points (roughly, the "ends" of
the horns), since any NEIGHBORHOOD of a limit con-
tains a horned complex.
A humorous drawing by Simon Prazer (Guy 1983,
Schroeder 1991, Albers 1994) depicts mathematician
John H. Conway with Alexander's horned sphere grow-
ing from his head.
see also Antoine's Horned Sphere
References
Albers, D. J. Illustration accompanying "The Game of
'Life'." Math Horizons, p. 9, Spring 1994.
Guy, R. "Conway's Prime Producing Machine." Math. Mag.
56, 26-33, 1983.
Hocking, J. G. and Young, G. S. Topology. New York: Dover,
1988.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 80-81, 1976.
Schroeder, M. Fractals, Chaos, Power Law: Minutes from
an Infinite Paradise. New York: W. H. Freeman, p. 58,
1991.
Alexander Ideal
The order IDEAL in A, the RING of integral LAURENT
Polynomials, associated with an Alexander Matrix
for a Knot K. Any generator of a principal Alexander
ideal is called an Alexander Polynomial. Because
the Alexander Invariant of a Tame Knot in S 3
has a Square presentation Matrix, its Alexander ideal
is Principal and it has an Alexander Polynomial
A(t).
see also Alexander Invariant, Alexander Matrix,
Alexander Polynomial
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 206-207, 1976.
Alexander Invariant
The Alexander invariant i7* (X) of a Knot K is the HO-
MOLOGY of the Infinite cyclic cover of the complement
of K, considered as a MODULE over A, the RING of inte-
gral Laurent Polynomials. The Alexander invariant
for a classical Tame Knot is finitely presentable, and
only Hi is significant.
For any KNOT K n in § n+ whose complement has the
homotopy type of a FINITE COMPLEX, the Alexander
invariant is finitely generated and therefore finitely pre-
sentable. Because the Alexander invariant of a Tame
Knot in S 3 has a Square presentation Matrix, its
Alexander Ideal is Principal and it has an Alex-
ander Polynomial denoted A(t).
see also Alexander Ideal, Alexander Matrix, Al-
exander Polynomial
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 206-207, 1976.
Alexander Matrix
A presentation matrix for the Alexander Invariant
Hi(X) of a Knot K. If V is a Seifert Matrix for
a Tame Knot K in S 3 , then V T - tV and V - tV T
are Alexander matrices for K,
Matrix Transpose.
where V denotes the
see also Alexander Ideal, Alexander Invariant,
Alexander Polynomial, Seifert Matrix
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 206-207, 1976.
26 Alexander Polynomial
Alexander Polynomial
Alexander Polynomial
A Polynomial invariant of a Knot discovered in 1923
by J. W. Alexander (Alexander 1928). In technical lan-
guage, the Alexander polynomial arises from the HO-
MOLOGY of the infinitely cyclic cover of a Knot's com-
plement. Any generator of a PRINCIPAL ALEXANDER
Ideal is called an Alexander polynomial (Rolfsen 1976).
Because the Alexander Invariant of a Tame Knot
in S 3 has a Square presentation Matrix, its Alex-
ander Ideal is Principal and it has an Alexander
polynomial denoted A(i).
Let * be the MATRIX PRODUCT of BRAID WORDS of a
Knot, then
det(l - V)
1 + *+...+ t*-
= Az
(1)
where Az, is the Alexander polynomial and det is the
Determinant. The Alexander polynomial of a Tame
Knot in S 3 satisfies
A(t) = det(V T -tV),
(2)
where V is a Seifert Matrix, det is the Determi-
nant, and V T denotes the Matrix TRANSPOSE. The
Alexander polynomial also satisfies
A(l) = ±l.
(3)
The Alexander polynomial of a splittable link is always
0. Surprisingly, there are known examples of nontrivial
Knots with Alexander polynomial 1. An example is
the (-3,5,7) Pretzel Knot.
The Alexander polynomial remained the only known
Knot Polynomial until the Jones Polynomial was
discovered in 1984. Unlike the Alexander polynomial,
the more powerful JONES POLYNOMIAL does, in most
cases, distinguish HANDEDNESS. A normalized form of
the Alexander polynomial symmetric in t and £ _1 and
satisfying
A(unknot) = 1 (4)
was formulated by J. H. Conway and is sometimes de-
noted Vl • The Notation [a 4- b + c + . . . is an abbrevi-
ation for the Conway-normalized Alexander polynomial
of a Knot
a + b(x + x ) + c(x + x ) + .
(5)
For a description of the NOTATION for Links, see Rolf-
sen (1976, p. 389). Examples of the Conway-Alexander
polynomials for common KNOTS include
Vtk
Vfek
VsSK
[1-1 =
[3-1 =
[l - i + :
-x" 1 + 1
_1 +3-x
(6)
(7)
_1 + l-a: + x 2 (8)
for the Trefoil Knot, Figure-of-Eight Knot, and
Solomon's Seal Knot, respectively. Multiplying
through to clear the NEGATIVE POWERS gives the usual
Alexander polynomial, where the final SIGN is deter-
mined by convention.
\,
\
)(
s
s
u
L
J + M)
Let an Alexander polynomial be denoted A, then there
exists a Skein Relationship (discovered by J. H. Con-
way)
A L+ (t)-A L _(t) + (t- 1/2 -t 1/2 )A Lo (t) = (9)
corresponding to the above Link Diagrams (Adams
1994). A slightly different Skein RELATIONSHIP con-
vention used by Doll and Hoste (1991) is
V i+ -V £ _ =zV Lo . (10)
These relations allow Alexander polynomials to be con-
structed for arbitrary knots by building them up as a
sequence of over- and undercrossings.
For a Knot,
* , n _fl(mod8) ifArf(tf) = 0. (n)
Ak(-1)= j 5(modg) ifArf(K) = 1) (11)
where Arf is the Arf Invariant (Jones 1985). If K is
a Knot and
|A*(i)|>3, (12)
then K cannot be represented as a closed 3-BRAID. Also,
if
A K (e 27ri/5 )> f, (13)
then K cannot be represented as a closed 4-braid (Jones
1985).
The HOMFLY POLYNOMIAL P{a, z) generalizes the Al-
exander polynomial (as well at the JONES POLYNOMIAL)
with
V(z) = P{l t z) (14)
(Doll and Hoste 1991).
Rolfsen (1976) gives a tabulation of Alexander polyno-
mials for Knots up to 10 Crossings and Links up to
9 Crossings.
see also Braid Group, Jones Polynomial, Knot,
Knot Determinant, Link, Skein Relationship
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 165-169, 1994.
Alexander, J. W. "Topological Invariants of Knots and
Links." Trans. Amer. Math. Soc. 30, 275-306, 1928.
Alexander-Spanier Cohomology
Algebra 27
Alexander, J. W. "A Lemma on a System of Knotted
Curves." Proc. Nat. Acad. Set. USA 9, 93-95, 1923,
Doll, H. and Hoste, J. "A Tabulation of Oriented Links."
Math. Comput. 57, 747-761, 1991.
Jones, V. "A Polynomial Invariant for Knots via von Neu-
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111,
1985.
Rolfsen, D. "Table of Knots and Links." Appendix C in
Knots and Links. Wilmington, DE: Publish or Perish
Press, pp. 280-287, 1976.
Stoimenow, A. "Alexander Polynomials." http://www.
informatik.hu-berlin.de/-stoimeno/ptab/alO.html.
Stoimenow, A. "Conway Polynomials." http://www.
informatik.hu-berlin.de/-stoimeno/ptab/clO.html.
Alexander-Spanier Cohomology
A fundamental result of DE RHAM COHOMOLOGY
is that the fcth de Rham Cohomology Vector
Space of a Manifold M is canonically isomorphic
to the Alexander-Spanier cohomology VECTOR SPACE
H k (M;M) (also called cohomology with compact sup-
port). In the case that M is Compact, Alexander-
Spanier cohomology is exactly "singular" COHOMOL-
OGY.
Alexander's Theorem
Any Link can be represented by a closed Braid.
Algebra
The branch of mathematics dealing with GROUP The-
ory and Coding Theory which studies number sys-
tems and operations within them. The word "algebra"
is a distortion of the Arabic title of a treatise by Al-
Khwarizmi about algebraic methods. Note that mathe-
maticians refer to the "school algebra" generally taught
in middle and high school as "Arithmetic," reserving
the word "algebra" for the more advanced aspects of the
subject.
Formally, an algebra is a Vector Space V, over a
Field F with a Multiplication which turns it into
a RING defined such that, if / 6 F and x, y G V, then
f{*y) = (fx)y = x(fy)-
In addition to the usual algebra of Real Numbers,
there are as 1151 additional Consistent algebras which
can be formulated by weakening the FIELD AXIOMS, at
least 200 of which have been rigorously proven to be
self-CONSlSTENT (Bell 1945).
Algebras which have been investigated and found to be
of interest are usually named after one or more of their
investigators. This practice leads to exotic-sounding
(but unenlightening) names which algebraists frequently
use with minimal or nonexistent explanation.
see also ALTERNATE ALGEBRA, ALTERNATING ALGE-
BRA, i?*-ALGEBRA, BANACH ALGEBRA, BOOLEAN AL-
GEBRA, Borel Sigma Algebra, C*-Algebra, Cay-
ley Algebra, Clifford Algebra, Commutative
Algebra, Exterior Algebra, Fundamental The-
orem of Algebra, Graded Algebra, Grassmann
Algebra, Hecke Algebra, Heyting Algebra, Ho-
mological Algebra, Hopf Algebra, Jordan Al-
gebra, Lie Algebra, Linear Algebra, Measure
Algebra, Nonassociative Algebra, Quaternion,
Robbins Algebra, Schur Algebra, Semisimple Al-
gebra, Sigma Algebra, Simple Algebra, Steen-
rod Algebra, von Neumann Algebra
References
Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, pp. 35-36, 1945.
Bhattacharya, P. B,; Jain, S. K.; and Nagpu, S. R. (Eds.).
Basic Algebra, 2nd ed. New York: Cambridge University
Press, 1994.
BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra,
5th ed. New York: Macmillan, 1996.
Brown, K. S. "Algebra." http://www.seanet.com/-ksbrown/
ialgebra.htm.
Cardano, G. Ars Magna or The Rules of Algebra. New York:
Dover, 1993.
Chevalley, C. C. Introduction to the Theory of Algebraic
Functions of One Variable. Providence, RI: Amer. Math.
Soc, 1951.
Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover,
1961.
Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL:
University of Chicago Press, 1923.
Dickson, L. E. Modern Algebraic Theories. Chicago, IL:
H. Sanborn, 1926.
Edwards, H. M. Galois Theory, corrected 2nd printing. New
York: Springer- Verlag, 1993.
Euler, L. Elements of Algebra. New York: Springer- Verlag,
1984.
Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lex-
ington, MA: D. C. Heath, 1994.
Grove, L. Algebra. New York: Academic Press, 1983.
Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to El-
ementary Algebra for Schools. London: Macmillan, 1960.
Harrison, M. A. "The Number of Isomorphism Types of Fi-
nite Algebras." Proc. Amer. Math. Soc. 17, 735-737,
1966.
Herstein, I. N. Noncommutative Rings. Washington, DC:
Math. Assoc. Amer., 1996.
Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley,
1975.
Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H.
Freeman, 1989.
Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: Uni-
versity of Chicago Press, 1995.
Lang, S. Undergraduate Algebra, 2nd ed. New York:
Springer- Verlag, 1990.
Pedersen, J. "Catalogue of Algebraic Systems." http://
tarski.math.usf .edu/algctlg/.
Uspensky, J. V. Theory of Equations. New York: McGraw-
Hill, 1948.
van der Waerden, B. L. Algebra, Vol. 2. New York: Springer-
Verlag, 1991.
van der Waerden, B. L. Geometry and Algebra in Ancient
Civilizations. New York: Springer- Verlag, 1983.
van der Waerden, B. L. A History of Algebra: From Al-
Khwarizmi to Emmy Noether. New York: Springer- Verlag,
1985.
Varadarajan, V. S. Algebra in Ancient and Modern Times.
Providence, RI: Amer. Math. Soc, 1998.
28 Algebraic Closure
Algebraic Invariant
Algebraic Closure
The algebraic closure of a Field K is the "smallest"
Field containing K which is algebraically closed. For
example, the FIELD of COMPLEX NUMBERS C is the
algebraic closure of the Field of Reals R.
Algebraic Coding Theory
see Coding Theory
Algebraic Curve
An algebraic curve over a Field K is an equation
f(X,Y) = 0, where f{X,Y) is a POLYNOMIAL in X and
Y with Coefficients in K. A nonsingular algebraic
curve is an algebraic curve over K which has no SIN-
GULAR Points over K. A point on an algebraic curve
is simply a solution of the equation of the curve. A K-
Rational Point is a point (X, Y) on the curve, where
X and Y are in the FIELD K.
see also Algebraic Geometry, Algebraic Variety,
Curve
References
Griffiths, P. A. Introduction to Algebraic Curves.
dence, RI: Amer. Math. Soc, 1989.
Provi-
Algebraic Function
A function which can be constructed using only a finite
number of ELEMENTARY FUNCTIONS together with the
Inverses of functions capable of being so constructed.
see also Elementary Function, Transcendental
Function
Algebraic Function Field
A finite extension K = Z(z)(w) of the Field C(z) of
Rational Functions in the indeterminate z, i.e., w is
a Root of a Polynomial a +aia + a 2 a 2 + . . . + a n a: n ,
where a; € C(z).
see also Algebraic Number Field, Riemann Sur-
face
Algebraic Geometry
The Study of ALGEBRAIC CURVES, ALGEBRAIC VARI-
ETIES, and their generalization to n-D.
see also Algebraic Curve, Algebraic Variety,
Commutative Algebra, Differential Geometry,
Geometry, Plane Curve, Space Curve
References
Abhyankar, S. S. Algebraic Geometry for Scientists and En-
gineers. Providence, RI: Amer. Math. Soc, 1990.
Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and
Algorithms: An Introduction to Algebraic Geometry and
Commutative Algebra, 2nd ed. New York: Springer-
Verlag, 1996.
Eisenbud, D. Commutative Algebra with a View Toward Al-
gebraic Geometry. New York: Springer- Verlag, 1995.
Griffiths, P. and Harris, J. Principles of Algebraic Geometry.
New York: Wiley, 1978.
Hartshorne, R. Algebraic Geometry, rev. ed. New York:
Springer- Verlag, 1997.
Lang, S. Introduction to Algebraic Geometry. New York:
Interscience, 1958.
Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry,
Vol. 1. Cambridge, England: Cambridge University Press,
1994.
Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry,
Vol. 2. Cambridge, England: Cambridge University Press,
1994.
Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry,
Vol. 3. Cambridge, England: Cambridge University Press,
1994.
Seidenberg, A. (Ed.). Studies in Algebraic Geometry. Wash-
ington, DC: Math. Assoc. Amer., 1980.
Weil, A. Foundations of Algebraic Geometry, enl. ed. Prov-
idence, RI: Amer. Math. Soc, 1962.
Algebraic Integer
If r is a Root of the Polynomial equation
x n + a n -ix n ~ + . . . + aiz + ao = 0,
where the a^s are INTEGERS and r satisfies no similar
equation of degree < n, then r is an algebraic INTEGER
of degree n. An algebraic INTEGER is a special case of
an Algebraic Number, for which the leading Coef-
ficient a n need not equal 1. RADICAL INTEGERS are a
subring of the ALGEBRAIC INTEGERS.
A Sum or Product of algebraic integers is again an al-
gebraic integer. However, Abel's IMPOSSIBILITY THE-
OREM shows that there are algebraic integers of degree
> 5 which are not expressible in terms of ADDITION,
Subtraction, Multiplication, Division, and the ex-
traction of Roots on Real Numbers.
The Gaussian Integer are are algebraic integers of
-1 ), since a + bi are roots of
z 2 - 2az + a 2 + b 2 = 0.
see also Algebraic Number, Euclidean Number,
Radical Integer
References
Hancock, H. Foundations of the Theory of Algebraic Num-
bers, Vol. 1: Introduction to the General Theory. New
York: Macmillan, 1931.
Hancock, H. Foundations of the Theory of Algebraic Num-
bers, Vol. 2: The General Theory. New York: Macmillan,
1932.
Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num-
ber Theory. Cambridge, England: Cambridge University-
Press, 1989.
Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in
Action. New York: W. H. Freeman, pp. 347-353, 1991.
Algebraic Invariant
A quantity such as a Discriminant which remains un-
changed under a given class of algebraic transforma-
tions. Such invariants were originally called HYPERDE-
TERMINANTS by Cayley.
see also DISCRIMINANT (POLYNOMIAL), INVARIANT,
Quadratic Invariant
Algebraic Knot
Algebraic Tangle 29
References
Grace, J. H. and Young, A. The Algebra of Invariants. New
York: Chelsea, 1965.
Gurevich, G. B. Foundations of the Theory of Algebraic In-
variants. Groningen, Netherlands: P. NoordhofF, 1964.
Hermann, R. and Ackerman, M. Hilbert's Invariant Theory
Papers.rookline, MA: Math Sci Press, 1978.
Hilbert, D. Theory of Algebraic Invariants. Cambridge, Eng-
land: Cambridge University Press, 1993.
Mumford, D.; Fogarty, J.; and Kirwan, F. Geometric Invari-
ant Theory, 3rd enl. ed. New York: Springer- Verlag, 1994.
Algebraic Knot
A single component ALGEBRAIC LINK.
see also Algebraic Link, Knot, Link
Algebraic Link
A class of fibered knots and links which arises in Al-
gebraic Geometry. An algebraic link is formed by
connecting the NW and NE strings and the SW and SE
strings of an ALGEBRAIC Tangle (Adams 1994).
see also Algebraic Tangle, Fibration, Tangle
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 48-49, 1994.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 335, 1976.
Algebraic Number
If r is a ROOT of the POLYNOMIAL equation
a$x -\- a±x
. . + a n - 1 x -\- a n = 0,
(i)
where the a^s are Integers and r satisfies no similar
equation of degree < n, then r is an algebraic number of
degree n. If r is an algebraic number and ao = 1, then
it is called an ALGEBRAIC INTEGER. It is also true that
if the c;s in
CQX + ClX n + . . . + Cn-lX + C n -
(2)
are algebraic numbers, then any ROOT of this equation
is also an algebraic number.
If a is an algebraic number of degree n satisfying the
Polynomial
a(x — a)(x — j3)(x — 7) ■
(3)
then there are n — 1 other algebraic numbers (3, 7, ...
called the conjugates of ex. Furthermore, if a satisfies
any other algebraic equation, then its conjugates also
satisfy the same equation (Conway and Guy 1996).
Any number which is not algebraic is said to be TRANS-
CENDENTAL.
see also ALGEBRAIC INTEGER, EUCLIDEAN NUMBER,
Hermite-Lindemann Theorem, Radical Integer,
Semialgebraic Number, Transcendental Number
References
Conway, J. H. and Guy, R. K. "Algebraic Numbers." In The
Book of Numbers. New York: Springer- Verlag, pp. 189—
190, 1996.
Courant, R. and Robbing, H. "Algebraic and Transcendental
Numbers." §2.6 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 103-107, 1996.
Hancock, H. Foundations of the Theory of Algebraic Num-
bers. Vol. 1: Introduction to the General Theory. New
York: Macmillan, 1931.
Hancock, H. Foundations of the Theory of Algebraic Num-
bers. Vol. 2: The General Theory. New York: Macmillan,
1932.
Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in
Action. New York: W. H. Freeman, pp. 347-353, 1991.
Algebraic Number Field
see Number Field
Algebraic Surface
The set of ROOTS of a POLYNOMIAL f(x,y,z) = 0. An
algebraic surface is said to be of degree n = max(i + J +
fc), where n is the maximum sum of powers of all terms
amX lrn y jrn z krn . The following table lists the names of
algebraic surfaces of a given degree.
Order
Surface
3
cubic surface
4
quartic surface
5
quintic surface
6
sextic surface
7
heptic surface
8
octic surface
9
nonic surface
10
decic surface
see also Barth Decic, Barth Sextic, Boy Surface,
Cayley Cubic, Chair, Clebsch Diagonal Cubic,
Cushion, Dervish, Endrass Octic, Heart Surface,
Kummer Surface, Order (Algebraic Surface),
Roman Surface, Surface, Togliatti Surface
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 7, 1986.
Algebraic Tangle
Any Tangle obtained by Additions and Multiplica-
tions of rational TANGLES (Adams 1994).
see also Algebraic Link
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 41-51, 1994.
30 Algebraic Topology
Algorithm
Algebraic Topology
The study of intrinsic qualitative aspects of spatial
objects (e.g., SURFACES, SPHERES, TORI, CIRCLES,
Knots, Links, configuration spaces, etc.) that re-
main invariant under both-directions continuous ONE-
TO-One (HOMEOMORPHIC) transformations. The dis-
cipline of algebraic topology is popularly known as
"Rubber-Sheet Geometry" and can also be viewed
as the study of Disconnectivities. Algebraic topology
has a great deal of mathematical machinery for studying
different kinds of HOLE structures, and it gets the prefix
"algebraic" since many Hole structures are represented
best by algebraic objects like GROUPS and RINGS.
A technical way of saying this is that algebraic topol-
ogy is concerned with FUNCTORS from the topological
Category of Groups and Homomorphisms. Here,
the FUNCTORS are a kind of filter, and given an "input"
SPACE, they spit out something else in return. The re-
turned object (usually a Group or Ring) is then a rep-
resentation of the HOLE structure of the SPACE, in the
sense that this algebraic object is a vestige of what the
original SPACE was like (i.e., much information is lost,
but some sort of "shadow" of the SPACE is retained —
just enough of a shadow to understand some aspect of its
HOLE-structure, but no more). The idea is that FUNC-
TORS give much simpler objects to deal with. Because
SPACES by themselves are very complicated, they are
unmanageable without looking at particular aspects.
COMBINATORIAL TOPOLOGY is a special type of alge-
braic topology that uses COMBINATORIAL methods.
see also CATEGORY, COMBINATORIAL TOPOLOGY, DIF-
FERENTIAL TOPOLOGY, FUNCTOR, HOMOTOPY THE-
ORY
References
Dieudonne, J. A History of Algebraic and Differential Topol-
ogy: 1900-1960. Boston, MA: Birkhauser, 1989.
Algebraic Variety
A generalization to n-D of ALGEBRAIC CURVES. More
technically, an algebraic variety is a reduced SCHEME of
Finite type over a Field K. An algebraic variety V is
defined as the Set of points in the Reals W 1 (or the
Complex Numbers C n ) satisfying a system of Poly-
nomial equations fi(xi, . . . , x n ) = for i = 1, 2,
According to the Hilbert Basis Theorem, a Finite
number of equations suffices.
see also Abelian Variety, Albanese Variety,
Brauer-Severi Variety, Chow Variety, Picard
Variety
References
Ciliberto, C; Laura, E.; and Somese, A. J. (Eds.). Classifica-
tion of Algebraic Varieties. Providence, RI: Amer. Math.
Soc, 1994.
Algebroidal Function
An Analytic Function f(z) satisfying the irreducible
algebraic equation
A (z)f k + Ai(z)/*- 1 + . . . + A k (z) =
with single- valued MEROMORPHIC functions Aj(z) in a
Complex Domain G is called a fc-algebroidal function
in G.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Algebroidal Functions."
§19 in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, pp. 86-88, 1980.
Algorithm
A specific set of instructions for carrying out a proce-
dure or solving a problem, usually with the requirement
that the procedure terminate at some point. Specific
algorithms sometimes also go by the name Method,
Procedure, or Technique. The word "algorithm" is
a distortion of Al-Khwarizmi, an Arab mathematician
who wrote an influential treatise about algebraic meth-
ods.
see also 196- ALGORITHM, ALGORITHMIC COMPLEXITY,
Archimedes Algorithm, Bhaskara-Brouckner
Algorithm, Borchardt-Pfaff Algorithm, Bre-
laz's Heuristic Algorithm, Buchberger's Algo-
rithm, Bulirsch-Stoer Algorithm, Bumping Al-
gorithm, CLEAN Algorithm, Computable Func-
tion, Continued Fraction Factorization Algo-
rithm, Decision Problem, Dijkstra's Algorithm,
Euclidean Algorithm, Ferguson-Forcade Al-
gorithm, Fermat's Algorithm, Floyd's Algo-
rithm, Gaussian Approximation Algorithm, Ge-
netic Algorithm, Gosper's Algorithm, Greedy
Algorithm, Hasse's Algorithm, HJLS Algo-
rithm, Jacobi Algorithm, Kruskal's Algorithm,
Levine-O 'Sullivan Greedy Algorithm, LLL Al-
gorithm, Markov Algorithm, Miller's Algo-
rithm, Neville's Algorithm, Newton's Method,
Prime Factorization Algorithms, Primitive Re-
cursive Function, Program, PSLQ Algorithm,
PSOS Algorithm, Quotient-Difference Algo-
rithm, Risch Algorithm, Schrage's Algorithm,
Shanks' Algorithm, Spigot Algorithm, Syracuse
Algorithm, Total Function, Turing Machine,
Zassenhaus-Berlekamp Algorithm, Zeilberger's
Algorithm
References
Aho, A. V.; Hopcroft, J. E.; and Ullman, J.D. The De-
sign and Analysis of Computer Algorithms. Reading, MA:
Addison- Wesley, 1974.
Baase, S. Computer Algorithms. Reading, MA: Addison-
Wesley, 1988.
Brassard, G. and Bratley, P. Fundamentals of Algorithmics.
Englewood Cliffs, NJ: Prentice-Hall, 1995.
Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduc-
tion to Algorithms. Cambridge, MA: MIT Press, 1990.
Algorithmic Complexity
Aliquant Divisor 31
Greene, D. H. and Knuth, D. E. Mathematics for the Analysis
of Algorithms, 3rd ed. Boston: Birkhauser, 1990.
Harel, D. Algorithmics: The Spirit of Computing, 2nd ed.
Reading, MA: Addison- Wesley, 1992.
Knuth, D. E. The Art of Computer Programming, Vol. 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison- Wesley, 1981.
Knuth, D. E. The Art of Computer Programming, Vol. 3:
Sorting and Searching, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Kozen, D. C. Design and Analysis and Algorithms. New
York: Springer- Verlag, 1991.
Shen, A. Algorithms and Programming. Boston: Birkhauser,
1996.
Skiena, S. S. The Algorithm Design Manual. New York:
Springer- Verlag, 1997.
Wilf, H. Algorithms and Complexity. Englewood Cliffs, NJ:
Prentice Hall, 1986. http://www.cis.upenn.edu/-wilf/.
References
Dorrie, H. "Alhazen's Billiard Problem." §41 in 100 Great
Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, pp. 197-200, 1965.
Hogendijk, J. P. "Al-Mutaman's Simplified Lemmas for Solv-
ing 'Alhazen's Problem'." From Baghdad to Barcelona/De
Bagdad a Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59-
101, Anu. Filol. Univ. Bare, XIX B-2, Univ. Barcelona,
Barcelona, 1996.
Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tid~
skr. 18, 5-35, 1970.
Neumann, P. Submitted to Amer. Math. Monthly.
Riede, H. "Reflexion am Kugelspiegel. Oder: das Problem
des Alhazen." Praxis Math. 31, 65-70, 1989.
Sabra, A. I. "ibn al-Haytham's Lemmas for Solving 'Al-
hazen's Problem'." Arch. Hist Exact Sci. 26, 299-324,
1982.
Alhazen's Problem
see Alhazen's Billiard Problem
Algorithmic Complexity
see Bit Complexity, Kolmogorov Complexity
Alhazen's Billiard Problem
In a given Circle, find an Isosceles Triangle whose
Legs pass through two given Points inside the Circle.
This can be restated as: from two POINTS in the Plane
of a Circle, draw Lines meeting at the Point of the
Circumference and making equal Angles with the
Normal at that Point.
The problem is called the billiard problem because it cor-
responds to finding the POINT on the edge of a circular
"BILLIARD" table at which a cue ball at a given POINT
must be aimed in order to carom once off the edge of the
table and strike another ball at a second given Point.
The solution leads to a BIQUADRATIC EQUATION of the
form
H{x 2
V )
2Kxy + {x 2 -r y 2 ){hy - kx) = 0.
The problem is equivalent to the determination of the
point on a spherical mirror where a ray of light will re-
flect in order to pass from a given source to an observer.
It is also equivalent to the problem of finding, given two
points and a Circle such that the points are both inside
or outside the Circle, the Ellipse whose Foci are the
two points and which is tangent to the given CIRCLE.
The problem was first formulated by Ptolemy in 150
AD, and was named after the Arab scholar Alhazen,
who discussed it in his work on optics. It was not until
1997 that Neumann proved the problem to be insoluble
using a COMPASS and RULER construction because the
solution requires extraction of a CUBE ROOT, This is
the same reason that the CUBE DUPLICATION problem
is insoluble.
see also Billiards, Billiard Table Problem, Cube
Duplication
Alias' Paradox
Choose between the following two alternatives:
1. 90% chance of an unknown amount x and a 10%
chance of $1 million, or
2. 89% chance of the same unknown amount x, 10%
chance of $2.5 million, and 1% chance of nothing.
The Paradox is to determine which choice has the
larger expectation value, 0.9x + $100,000 or 0.89:r -f
$250,000. However, the best choice depends on the un-
known amount, even though it is the same in both cases!
This appears to violate the INDEPENDENCE Axiom.
see also Independence Axiom, Monty Hall Prob-
lem, Newcomb's Paradox
Aliasing
Given a power spectrum (a plot of power vs. frequency),
aliasing is a false translation of power falling in some fre-
quency range ( — / c ,/ c ) outside the range. Aliasing can
be caused by discrete sampling below the NYQUIST FRE-
QUENCY. The sidelobcs of any INSTRUMENT FUNCTION
(including the simple SlNC SQUARED function obtained
simply from FINITE sampling) are also a form of alias-
ing. Although sidelobe contribution at large offsets can
be minimized with the use of an APODIZATION FUNC-
TION, the tradeoff is a widening of the response (i.e., a
lowering of the resolution).
see also Apodization Function, Nyquist Fre-
quency
Aliquant Divisor
A number which does not DIVIDE another exactly. For
instance, 4 and 5 are aliquant divisors of 6. A num-
ber which is not an aliquant divisor (i.e., one that does
Divide another exactly) is said to be an Aliquot Di-
visor.
see also ALIQUOT DIVISOR, DIVISOR, PROPER DIVISOR
32 Aliquot Cycle
Allegory
Aliquot Cycle
see Sociable Numbers
Aliquot Divisor
A number which DIVIDES another exactly. For instance,
1, 2, 3, and 6 are aliquot divisors of 6, A number which
is not an aliquot divisor is said to be an ALIQUANT DI-
VISOR. The term "aliquot" is frequently used to specif-
ically mean a PROPER DIVISOR, i.e., a DIVISOR of a
number other than the number itself.
see also ALIQUANT DIVISOR, DIVISOR, PROPER DIVI-
SOR
Aliquot Sequence
Let
s(n) = cr(n) — n,
where a(n) is the DIVISOR FUNCTION and s(n) is the
Restricted Divisor Function. Then the Sequence
of numbers
s°(n) = n, s 1 (n) = s(n), s (n) — s(s(n)), . . .
is called an aliquot sequence. If the SEQUENCE for a
given n is bounded, it either ends at s(l) = or becomes
periodic.
1. If the Sequence reaches a constant, the constant is
known as a PERFECT NUMBER.
2. If the SEQUENCE reaches an alternating pair, it is
called an AMICABLE PAIR.
3. If, after k iterations, the SEQUENCE yields a cycle
of minimum length t of the form s fc+1 (n), s fc+2 (n),
..., s k+t (n), then these numbers form a group of
Sociable Numbers of order t.
It has not been proven that all aliquot sequences eventu-
ally terminate and become period. The smallest number
whose fate is not known is 276, which has been computed
up to s 487 (276) (Guy 1994).
see also 196-Algorithm, Additive Persistence,
Amicable Numbers, Multiamicable Numbers,
Multiperfect Number, Multiplicative Persis-
tence, Perfect Number, Sociable Numbers, Uni-
tary Aliquot Sequence
References
Guy, R. K. "Aliquot Sequences." §B6 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 60-62, 1994.
Guy, R. K. and Selfridge, J. L. "What Drives Aliquot Se-
quences." Math. Corn-put. 29, 101-107, 1975.
Sloane, N. J. A. Sequences A003023/M0062 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
All-Poles Model
see Maximum Entropy Method
Alladi-Grinstead Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let N(n) be the number of ways in which the Facto-
rial n! can be decomposed into n Factors of the form
Pk bk arranged in nondecreasing order. Also define
m(n) = max(pi 1 ),
(1)
i.e., m(n) is the Least Prime Factor raised to its
appropriate POWER in the factorization. Then define
a(n) =
lnm(n)
Inn
(2)
where ln(x) is the NATURAL LOGARITHM. For instance,
2 • 2 ■ 2 2 ■ 5 • 7 • 3 4
23-5-7-2 3 -3 3
2 • 5 - 7 ■ 2 3 • 3 2 • 3 2
9! = 2
2
2
= 2
2
2
= 2
2
2
= 2
2
2
= 2
2
2-
= 2
2
2-
= 2
2
3-
= 2
2
3-
= .2
3
3-
= 2
3
3
= 2
3
3-
= 3
3
3-
2 2 • 2 2
5 * 7 • 3 2 • 3 2
3 • 3 • 5 • 7 ■ 3 2 • 2 4
3 ■ 2 2 • 5 • 7 • 2 3 ■ 3 2
3 • 3 ■ 3 • 5 • 7 • 2 5
2 2 • 2 2 • 2 2 ■ 5 • 7 ■ 3 2
3-3-2 2 .5-7-2 4
3-3-5
7 • 2 3 - 2 3
a(9) =
3-2 2 -2 2 .5-7-2 3 ,
In 3 In 3 1
In 9 21n3 2
For large n,
lim a(n) = e c_1 = 0.809394020534 . . . ,
n— kx>
where
-£WA)-
(3)
(4)
(5)
(6)
References
Alladi, K. and Grinstead, C. "On the Decomposition of n!
into Prime Powers." J. Number Th, 9, 452-458, 1977.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . c om/ as olve/ const ant /aldgrns/aldgrns .html.
Guy, R. K. "Factorial n as the Product of n Large Factors."
§B22 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 79, 1994.
Allegory
A technical mathematical object which bears the same
resemblance to binary relations as CATEGORIES do to
Functions and Sets.
see also CATEGORY
References
Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster-
dam, Netherlands: North-Holland, 1990.
Allometric
Almost Integer 33
Allometric
Mathematical growth in which one population grows at
a rate PROPORTIONAL to the POWER of another popu-
lation.
References
Cofrey, W. J. Geography Towards a General Spatial Systems
Approach. London: Routledge, Chapman & Hall, 1981,
Almost All
Given a property P, if P{x) ~ x as x — > oo (so the num-
ber of numbers less than x not satisfying the property
P is o(x)), then P is said to hold true for almost all
numbers. For example, almost all positive integers are
Composite Numbers (which is not in conflict with the
second of Euclid's Theorems that there are an infinite
number of PRIMES).
see also For All, Normal Order
References
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 8, 1979.
Almost Alternating Knot
An Almost Alternating Link with a single compo-
nent.
Almost Alternating Link
Call a projection of a LINK an almost alternating pro-
jection if one crossing change in the projection makes it
an alternating projection. Then an almost alternating
link is a Link with an almost alternating projection, but
no alternating projection. Every ALTERNATING KNOT
has an almost alternating projection. A PRIME KNOT
which is almost alternating is either a Torus Knot or
a Hyperbolic Knot. Therefore, no Satellite Knot
is an almost alternating knot.
All nonalternating 9-crossing PRIME KNOTS are almost
alternating. Of the 393 nonalternating with 11 or fewer
crossings, all but five are known to be nonalternating (3
of these have 11 crossings). The fate of the remaining
five is not known. The (2,qr), (3,4), and (3,5)-TORUS
KNOTS are almost alternating.
see also Alternating Knot, Link
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 139-146, 1994.
Almost Everywhere
A property of X is said to hold almost everywhere if
the SET of points in X where this property fails has
Measure 0.
see also MEASURE
References
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 1, 1991.
Almost Integer
A number which is very close to an INTEGER. One sur-
prising example involving both e and Pi is
7T = 19.999099979.
(1)
which can also be written as
(tt + 20)* = -0.9999999992 - 0.0000388927i & -1 (2)
cos(ln(7r + 20)) « -0.9999999992. (3)
Applying Cosine a few more times gives
COs(7T COS(7T COs(ln(7T + 20))))
« -1 + 3.9321609261 x 10" 35 . (4)
This curious near-identity was apparently noticed al-
most simultaneously around 1988 by N. J. A. Sloane,
J. H. Conway, and S. Plouffe, but no satisfying explana-
tion as to "why" it has been true has yet been discov-
ered.
An interesting near-identity is given by
i[cos(^) + cosh(^) + 2cos(^^)cosh(^V / 2)]
= 1 + 2.480... x 10" 13 (5)
(W. Dubuque). Other remarkable near-identities are
given by
5(1 + we )[ g !)]2 =l + 4.5422 -x 10- (6)
where T(z) is the Gamma FUNCTION (S. Plouffe), and
e 6 - 7v 4 - tt 5 = 0.000017673 ... (7)
(D. Wilson).
A whole class of IRRATIONAL "almost integers" can be
found using the theory of MODULAR FUNCTIONS, and a
few rather spectacular examples are given by Ramanu-
jan (1913-14). Such approximations were also stud-
ied by Hermite (1859), Kronecker (1863), and Smith
(1965). They can be generated using some amazing (and
very deep) properties of the j-FUNCTlON. Some of the
numbers which are closest approximations to INTEGERS
are e*^ 1 ^ (sometimes known as the R A MANU J AN Con-
stant and which corresponds to the field Q(V"163)
which has Class Number 1 and is the Imaginary
quadratic field of maximal discriminant), e 22 , e 71 " 37 ,
and e"^, the latter three of which have Class Num-
ber 2 and are due to Ramanujan (Berndt 1994, Wald-
schmidt 1988).
34 Almost Integer
Almost Prime
The properties of the j-FUNCTlON also give rise to the
spectacular identity
ln(640320 3 + 744)
163 + 2.32167... x 10"
(8)
(Le Lionnais 1983, p. 152).
The list below gives numbers of the form x = e 71 "^ for
n < 1000 for which \x] - x < 0.01.
e^:
e -
e =
jt-v/25
e =
nVTf
e =
e =
e :
e =
e =
e =
tvvT49
e
ttvT63
2,197.990 869 543...
= 422, 150.997 675 680. . .
= 614,551.992 885619...
= 2,508,951.998 257 553. . .
= 6,635,623.999 341134...
= 199, 148, 647.999 978 046 551 .. .
= 884, 736, 743.999 777 466 .. .
= 24, 591, 257, 751.999 999 822 213 .. .
= 30, 197, 683, 486.993 182 260 .. .
= 147, 197, 952, 743.999 998 662 454 .. .
= 54,551,812,208.999917467 885...
= 45, 116, 546, 012, 289, 599.991 830 287 . . .
= 262, 537, 412, 640, 768, 743.999 999 999 999 250 072 .
= 1, 418, 556, 986, 635, 586, 485.996 179 355 .. .
= 604, 729, 957, 825, 300, 084, 759.999 992 171 526 .. .
= 19, 683, 091, 854, 079, 461, 001, 445.992 737 040 .. .
= 4, 309, 793, 301, 730, 386, 363, 005, 719.996 011 651 .
= 639, 355, 180, 631, 208, 421, • • •
■ ■ - 212, 174, 016.997 669 832 .
= 14, 871, 070, 263, 238, 043, 663, 567, • - •
• • • 627, 879, 007.999 848 726 .
= 288, 099, 755, 064, 053, 264, 917, 867, • - •
•■• 975, 825, 573. 993 898 311.
= 28, 994, 858, 898, 043, 231, 996, 779, - • -
■ • ■ 771, 804, 797, 161.992 372 939 .
= 3, 842, 614, 373, 539, 548, 891, 490, • • -
' • ■ • 294, 277, 805, 829, 192.999 987 249 .
= 223, 070, 667, 213, 077, 889, 794, 379, - - -
- - ■ 623, 183, 838, 336, 437.992 055 118 .
= 249, 433, 117, 287, 892, 229, 255, 125, • ■ •
• • • 388, 685, 911, 710, 805.996 097 323 .
= 365, 698, 321, 891, 389, 219, 219, 142, ■ ■ -
■ • - 531, 076, 638, 716, 362, 775.998 259 747 .
= 6, 954, 830, 200, 814, 801, 770, 418, 837, - - ■
940, 281, 460, 320, 666, 108.994 649 611 . .
Gosper noted that the expression
differs from an Integer by a mere 10
see also Class Number, j-Function, Pi
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 90-91, 1994.
Hermite, C. "Sur la theorie des equations modulaires." C.
R. Acad. Sci. (Paris) 48, 1079-1084 and 1095-1102, 1859.
Hermite, C. "Sur la theorie des equations modulaires." C. R.
Acad. Sci. (Paris) 49, 16-24, 110-118, and 141-144, 1859.
Kronecker, L. "Uber die Klassenzahl der aus Werzeln der Ein-
heit gebildeten komplexen Zahlen." Monatsber. K. Preuss.
Akad. Wiss. Berlin, 340-345. 1863.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.
Smith, H. J, S. Report on the Theory of Numbers. New York:
Chelsea, 1965.
Waldschmidt, M. "Some Transcendental Aspects of Ramanu-
jan's Work." In Ramanujan Revisited: Proceedings of the
Centenary Conference (Ed. G. E« Andrews, B. C. Berndt,
and R. A. Rankin). New York: Academic Press, pp. 57-76,
1988.
Almost Perfect Number
A number n for which the DIVISOR FUNCTION satisfies
cr(n) = 2n — 1 is called almost perfect. The only known
almost perfect numbers are the POWERS of 2, namely
1, 2, 4, 8, 16, 32, ... (Sloane's A000079). Singh (1997)
calls almost perfect numbers SLIGHTLY DEFECTIVE.
see also QuASIPERFECT NUMBER
References
Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect,
Harmonic, Weird, Multiperfect and Hyperperfect Num-
bers." §B2 in Unsolved Problems in Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 16 and 45—53, 1994.
Singh, S. Fermat's Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem. New York:
Walker, p. 13, 1997.
Sloane, N. J. A. Sequence A000079/M1129 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Almost Prime
A number n with prime factorization
=n»-
■ 2625374126407G8744e
-7TV163
196884e
-27TN/163
+103378831900730205293632e~ 37rv/I ^. (9)
is called ^-almost prime when the sum of the POWERS
J^^ l di = k. The set of fc-almost primes is denoted Ph.
The Primes correspond to the "1-almost prime" num-
bers 2, 3, 5, 7, 11, . . . (Sloane's A000040). The 2-almost
prime numbers correspond to SEMIPRIMES 4, 6, 9, 10,
14, 15, 21, 22, ... (Sloane's A001358). The first few
3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44,
45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ...
(Sloane's A014612). The first few 4-almost primes are
16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, . . . (Sloane's
A014613). The first few 5-almost primes are 32, 48, 72,
80, ... (Sloane's A014614).
Alpha
Alternate Algebra 35
see also Chen's Theorem, Prime Number, Semi-
prime
References
Sloane, N. J. A. Sequences A014612, A014613, A014614,
A000040/M0652, and A001358/M3274 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Alpha
A financial measure giving the difference between a
fund's actual return and its expected level of perfor-
mance, given its level of risk (as measured by Beta).
A POSITIVE alpha indicates that a fund has performed
better than expected based on its Beta, whereas a Neg-
ative alpha indicates poorer performance
see also Beta, Sharpe Ratio
Alphamagic Square
A Magic Square for which the number of letters in
the word for each number generates another MAGIC
Square. This definition depends, of course, on the lan-
guage being used. In English, for example,
5
22
18
4
9
8
28
15
2
11
7
3
12
8
25
6
5
10
where the MAGIC SQUARE on the right corresponds to
the number of letters in
five
twenty-eight
twelve
twenty-two eighteen
fifteen two
eight twenty-five
Alpha Function
a n (z) = / t n e~ zt dt = n\z- (n+1) e- z ^
k\
The alpha function satisfies the Recurrence Rela-
tion
za n (z) = e~ z + na n -i(z).
see also BETA FUNCTION (Exponential)
Alpha Value
An alpha value is a number < a < 1 such that P(z >
^observed) < « is considered "Significant," where P is
a P- Value.
see also Confidence Interval, P- Value, Signifi-
cance
Alphabet
A Set (usually of letters) from which a Subset is drawn.
A sequence of letters is called a WORD, and a set of
Words is called a Code.
see also CODE, WORD
References
Sallows, L. C. F. "Alphamagic Squares." Abacus 4, 28-45,
1986.
Sallows, L. C. F. "Alphamagic Squares. 2." Abacus 4, 20-29
and 43, 1987.
Sallows, L. C. F. "Alpha Magic Squares." In The Lighter
Side of Mathematics (Ed. R. K. Guy and R, E. Woodrow).
Washington, DC: Math. Assoc. Amer., 1994.
Alphametic
A CRYPTARITHM in which the letters used to represent
distinct DIGITS are derived from related words or mean-
ingful phrases. The term was coined by Hunter in 1955
(Madachy 1979, p. 178).
References
Brooke, M. One Hundred & Fifty Puzzles in Crypt-
Arithmetic. New York: Dover, 1963.
Hunter, J. A. H. and Madachy, J. S. "Alphametics and the
Like." Ch. 9 in Mathematical Diversions, New York:
Dover, pp. 90-95, 1975.
Madachy, J. S. "Alphametics." Ch. 7 in Madachy p s Mathe-
matical Recreations. New York: Dover, pp. 178-200 1979.
Alternate Algebra
Let A denote an R-Algebra, so that A is a Vector
Space over R and
AxA^A (1)
(x,y) \->x-y. (2)
Then A is said to be alternate if, for all x,y £ A,
(x-y)-y-x-(yy) (3)
(x-x)-y = x-(x-y). (4)
Here, VECTOR MULTIPLICATION x • y is assumed to be
Bilinear.
References
Finch, S. "Zero Structures in Real Algebras." http://www.
raathsof t . com/asolve/zerodiv/zerodiv .html.
Schafer, R. D. An Introduction to Non- Associative Algebras.
New York: Dover, 1995.
36 Alternating Algebra
Alternating Permutation
Alternating Algebra
see Exterior Algebra
Alternating Group
Even Permutation Groups A n which are Normal
Subgroups of the Permutation Group of Order
n!/2. They are Finite analogs of the families of sim-
ple Lie GROUPS. The lowest order alternating group is
60. Alternating groups with n > 5 are non-ABELIAN
Simple Groups. The number of conjugacy classes in
the alternating groups A n for n = 2, 3, . . . are 1, 3, 4,
5, 7, 9, ... (Sloane's A000702).
see also 15 Puzzle, Finite Group, Group, Lie
Group, Simple Group, Symmetric Group
References
Sloane, N. J, A. Sequence A000702/M2307 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.nk/atlas#alt.
Alternating Knot
An alternating knot is a KNOT which possesses a knot
diagram in which crossings alternate between under- and
overpasses. Not all knot diagrams of alternating knots
need be alternating diagrams.
The Trefoil Knot and Figure-of-Eight Knot are
alternating knots. One of Tait's Knot Conjectures
states that the number of crossings is the same for
any diagram of a reduced alternating knot. Further-
more, a reduced alternating projection of a knot has
the least number of crossings for any projection of that
knot. Both of these facts were proved true by Kauffman
(1988), Thistlethwaite (1987), and Murasugi (1987).
If K has a reduced alternating projection of n crossings,
then the Span of K is An. Let c(K) be the Crossing
Number. Then an alternating knot K±#K 2 (a Knot
Sum) satisfies
Erdener, K. and Flynn, R. "Rolfsen's Table of all Alter-
nating Diagrams through 9 Crossings." ftp://chs.cusd.
claremont . e du/pub/knot /Rolf sen_t able .final.
Kauffman, L. "New Invariants in the Theory of Knots."
Amer. Math. Monthly 95, 195-242, 1988.
Murasugi, K. "Jones Polynomials and Classical Conjectures
in Knot Theory." Topology 26, 297-307, 1987.
Sloane, N. J. A. Sequence A002864/M0847 in "An On-Line
Version of the Encyclopedia of Integer Sequences,"
Thistlethwaite, M. "A Spanning Tree Expansion for the Jones
Polynomial." Topology 26, 297-309, 1987.
Alternating Knot Diagram
A Knot Diagram which has alternating under- and
overcrossings as the KNOT projection is traversed. The
first KNOT which does not have an alternating diagram
has 8 crossings.
Alternating Link
A Link which has a Link Diagram with alternating
underpasses and overpasses.
see also Almost Alternating Link
References
Menasco, W. and Thistlethwaite, M. "The Classification of
Alternating Links." Ann. Math. 138, 113-171, 1993.
Alternating Permutation
An arrangement of the elements ci, ..., c n such that
no element a has a magnitude between a-\ and Ci + i is
called an alternating (or Zigzag) permutation. The de-
termination of the number of alternating permutations
for the set of the first n INTEGERS {1, 2, ... , n} is known
as Andre's Problem. An example of an alternating
permutation is (1, 3, 2, 5, 4).
As many alternating permutations among n elements
begin by rising as by falling. The magnitude of the c n s
does not matter; only the number of them. Let the
number of alternating permutations be given by Z n =
2A n . This quantity can then be computed from
In fact, this is true as well for the larger class of Ade-
quate KNOTS and postulated for all KNOTS. The num-
ber of Prime alternating knots of n crossing for n = 1,
2, . . . are 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, . . . (Sloane's
A002864).
see also ADEQUATE KNOT, ALMOST ALTERNATING
Link, Alternating Link, Flyping Conjecture
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 159-164, 1994.
Arnold, B.; Au, M.; Candy, C; Erdener, K.; Fan, J.; Flynn,
R.; Muir, J.; Wu, D.; and Hoste, J. "Tabulating Alter-
nating Knots through 14 Crossings." ftp://chs.cusd.
claremont.edu/pub/knot/paper.TeX.txt and ftp://chs.
cusd. claremont ,edu/pub/knot/AltKnots/.
2na n
J2 ar
(1)
where r and s pass through all INTEGRAL numbers such
that
r + 5==n _l ) (2)
ao = a\ = 1, and
A n = n\a n .
(3)
The numbers A n are sometimes called the EULER
Zigzag Numbers, and the first few are given by 1, 1,
1, 2, 5, 16, 61, 272, ... (Sloane's A000111). The Odd-
numbered A n s are called Euler Numbers, Secant
Numbers, or Zig Numbers, and the EvEN-numbered
ones are sometimes called TANGENT NUMBERS or ZAG
Numbers.
Alternating Series
Altitude 37
Curiously enough, the SECANT and TANGENT MAC-
LAURIN SERIES can be written in terms of the A n s as
X X
sec x = A + A 2 — - + A 4 — + . .
2! 4!
X X
tan x = AiX + A 3 — - + A 5 — - + .
o! 5!
(4)
(5)
or combining them,
sec x + tan x
t 2 r 3 r 4
A x
-A 5 - + ..
(6)
see also Entringer Number, Euler Number, Eu-
ler Zigzag Number, Secant Number, Seidel-
Entringer-Arnold Triangle, Tangent Number
References
Andre, D. "Developments de seccc et tan a?." C. R. Acad.
Sci. Paris 88, 965-967, 1879.
Andre, D. "Memoire sur le permutations alternees." J. Math.
7, 167-184, 1881.
Arnold, V. I. "Bernoulli-Euler Updown Numbers Associ-
ated with Function Singularities, Their Combinatorics and
Arithmetics." Duke Math. J. 63, 537-555, 1991.
Arnold, V. I. "Snake Calculus and Combinatorics of Ber-
noulli, Euler, and Springer Numbers for Coxeter Groups."
Russian Math. Surveys 47, 3-45, 1992.
Bauslaugh, B. and Ruskey, F. "Generating Alternating Per-
mutations Lexicographically." BIT 30, 17-26, 1990.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 110-111, 1996.
Dorrie, H. "Andre's Deviation of the Secant and Tangent
Series." §16 in 100 Great Problems of Elementary Math-
ematics: Their History and Solutions. New York: Dover,
pp. 64-69, 1965.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 69-75, 1985.
Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent,
Euler, and Bernoulli Numbers." Math. Comput. 21, 663-
688, 1967.
Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op-
eration on Sequences: The Boustrophedon Transform." J.
Combin. Th. Ser. A 76, 44-54, 1996.
Ruskey, F. "Information of Alternating Permutations."
http:// sue . esc . uvic . ca / - cos / inf / perm /
Alternat ing . html.
Sloane, N. J. A. Sequence A000111/M1492 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Alternating Series
A Series of the form
k=l
00
D-d
a k
ajt.
see also SERIES
References
Arfken, G. "Alternating Series." §5.3 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 293-294, 1985.
Bromwich, T. J. Pa and MacRobert, T. M. "Alternating Se-
ries." §19 in An Introduction to the Theory of Infinite
Series, 3rd ed. New York: Chelsea, pp. 55-57, 1991.
Pinsky, M. A. "Averaging an Alternating Series." Math.
Mag. 51, 235-237, 1978.
Alternating Series Test
Also known as the Leibniz Criterion. An Alternat-
ing Series Converges if a± > a 2 > . . . and
lim ak = 0.
see also CONVERGENCE TESTS
Alternative Link
A category of Link encompassing both ALTERNATING
Knots and Torus Knots.
see also Alternating Knot, Link, Torus Knot
References
Kauffman, L. "Combinatorics and Knot Theory." Contemp.
Math. 20, 181-200, 1983.
Altitude
A r H 3 A 2
The altitudes of a TRIANGLE are the Cevians AiHi
which are Perpendicular to the Legs AjAk opposite
Ai. They have lengths hi = AiHi given by
hi = at+i sinai+2 = ^+2 sinaii+i
hi =
2^/s(s — ai)(s — 0,2) {s — as)
where s is the Semiperimeter and a%
interesting FORMULA is
hihzhz = 2sA
AiA k
(1)
(2)
Another
(3)
(Johnson 1929, p. 191), where A is the Area of the Tri-
angle. The three altitudes of any TRIANGLE are CON-
CURRENT at the ORTHOCENTER H. This fundamental
fact did not appear anywhere in Euclid's Elements.
Other formulas satisfied by the altitude include
_1_ 1_ l_ _ 1
h\ h? /13 v
(4)
38 Alysoid
Amicable Numbers
1
=
1
h~ 2
+
1
1
hx~
1
r 2
+
1
=
1
r
1
2
" hx
(5)
(6)
where r is the INRADIUS and n are the Exradii (John-
son 1929, p. 189). In addition,
HA 1 • HHi = HA 2 • HH 2 = HA Z . HH 3 (7)
Jf Ai • HHi = |(ai 2 + a 2 2 + a 3 2 ) - 4# 2 , (8)
where R is the ClRCUMRADlUS.
The points Ai, A 3 , #i, and H 3 (and their permuta-
tions with respect to indices) all lie on a Circle, as
do the points A3, Hz, H, and Hi (and their permuta-
tions with respect to indices). TRIANGLES AA1A2A3
and AA\H 2 H 3 are inversely similar.
The triangle H±H 2 H 3 has the minimum PERIMETER
of any TRIANGLE inscribed in a given Acute TRIAN-
GLE (Johnson 1929, pp. 161-165). The PERIMETER of
AHxH 2 H 3 is 2A/R (Johnson 1929, p. 191). Additional
properties involving the Feet of the altitudes are given
by Johnson (1929, pp. 261-262).
see also Cevian, Foot, Orthocenter, Perpendicu-
lar, Perpendicular Foot
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40,
1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Alysoid
see Catenary
Ambient Isotopy
An ambient isotopy from an embedding of a MANIFOLD
M in N to another is a Homotopy of self Diffeomor-
phisms (or Isomorphisms, or piecewise-linear transfor-
mations, etc.) of JV, starting at the IDENTITY Map, such
that the "last" DlFFEOMORPHISM compounded with the
first embedding of M is the second embedding of M.
In other words, an ambient isotopy is like an Isotopy
except that instead of distorting the embedding, the
whole ambient SPACE is being stretched and distorted
and the embedding is just "coming along for the ride."
For Smooth Manifolds, a Map is Isotopic Iff it is
ambiently isotopic.
For KNOTS, the equivalence of MANIFOLDS under con-
tinuous deformation is independent of the embedding
Space. Knots of opposite Chirality have ambient
isotopy, but not REGULAR ISOTOPY.
see also ISOTOPY, REGULAR ISOTOPY
References
Hirsch, M. W. Differential Topology. New York: Springer-
Verlag, 1988.
Ambiguous
An expression is said to be ambiguous (or poorly de-
fined) if its definition does not assign it a unique inter-
pretation or value. An expression which is not ambigu-
ous is said to be Well-Defined.
see also Well-Defined
Ambrose-Kakutani Theorem
For every ergodic Flow on a nonatomic PROBABILITY
Space, there is a Measurable Set intersecting almost
every orbit in a discrete set.
Amenable Number
A number n which can be built up from INTEGERS ax,
a 2 , . . . , afc by either ADDITION or MULTIPLICATION such
that
k k
/ a i — \\ a i — n -
i=x i=X
The numbers {ai, . . . , a n } in the Sum are simply a Par-
tition of n. The first few amenable numbers are
2+2=2x2=4
1+2+3= 1x2x3=6
1+1+2+4=1x1x2x4=8
1 + 1 + 2 + 2 + 2 = 1x1x2x2x2 = 8.
In fact, all COMPOSITE NUMBERS are amenable.
See also COMPOSITE NUMBER, PARTITION, SUM
References
Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102,
463, 1995.
Amicable Numbers
see Amicable Pair, Amicable Quadruple, Amica-
ble Triple, Multiamicable Numbers
Amicable Pair
Amicable Pair 39
Amicable Pair
An amicable pair consists of two Integers m,n for
which the sum of PROPER DIVISORS (the DIVISORS ex-
cluding the number itself) of one number equals the
other. Amicable pairs are occasionally called FRIENDLY
Pairs, although this nomenclature is to be discouraged
since FRIENDLY PAIRS are defined by a different, if re-
lated, criterion. Symbolically, amicable pairs satisfy
s(m) — n
s(n) = m,
(i)
(2)
where s(n) is the RESTRICTED Divisor FUNCTION or,
equivalently,
cr(m) = cr(n) = s(m) + s(n) = m -f n, (3)
where <x(n) is the DIVISOR FUNCTION. The smallest
amicable pair is (220, 284) which has factorizations
220= 11-5-2^
284 = 71 • 2 2
giving RESTRICTED DIVISOR FUNCTIONS
s(220) = ^{1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}
= 284
S (284) = ^{1,2,4,71,142}
= 220.
(4)
(5)
The quantity
<r{m) = cr(n) — s(m) + s(n).
(6)
(7)
(8)
in this case, 220 + 284 = 504, is called the Pair Sum.
In 1636, Fermat found the pair (17296, 18416) and in
1638, Descartes found (9363584, 9437056). By 1747,
Euler had found 30 pairs, a number which he later ex-
tended to 60. There were 390 known as of 1946 (Scott
1946). There are a total of 236 amicable pairs below
10 8 (Cohen 1970), 1427 below 10 10 (te RhI • 1 ^6), 3340
less than 10 11 (Moews and Moew? 1 r "3), J' ,ess than
2.01 x 10 11 (Moews and Moe^ _., < .d 5001 .ess than
ft* 3.06 x 10 11 (Moews and Moews).
The first few amicable pairs are (2, 0, 284), (1184, 1210),
(2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856),
(12285, 14595), (17296 : u -116), (63020, 76084), ...
(Sloane's A002025 and AQ02046). An exhaustive tab-
ulation is maintained by D. Moevvo.
Let an amicable pair be denoted (m, n) with m < n.
(m,n) is called a regular amicable pair of type (i, j) if
(m,n) = (gM,gN),
(9)
where g = GCD(m,n) is the Greatest Common DI-
VISOR,
GCD( 5 ,M) = GCD{g,N) = 1, (10)
M and N are SQUAREFREE, then the number of Prime
factors of M and N are i and j. Pairs which are not
regular are called irregular or exotic (te Riele 1986).
There are no regular pairs of type (l,j) for j > 1. If
m = (mod 6) and
n = cr(m) — m
(ii)
is Even, then (m,n) cannot be an amicable pair (Lee
1969). The minimal and maximal values of m/n found
by te Riele (1986) were
938304290/1344480478 = 0.697893577. . . (12)
and
4000783984/4001351168 = 0.9998582519 .... (13)
te Riele (1986) also found 37 pairs of amicable pairs hav-
ing the same Pair Sum. The first such pair is (609928,
686072) and (643336, 652664), which has the Pair Sum
a(m) = cr(n) = m + n = 1,296,000.
(14)
te Riele (1986) found no amicable n-tuples having the
same Pair Sum for n > 2. However, Moews and
Moews found a triple in 1993, and te Riele found
a quadruple in 1995. In November 1997, a quin-
tuple and sextuple were discovered. The sextuple
is (1953433861918, 2216492794082), (1968039941816,
2201886714184), (1981957651366, 2187969004634),
(1993501042130, 2176425613870), (2046897812505,
2123028843495), (2068113162038, 2101813493962), all
having PAIR SUM 4169926656000. Amazingly, the sex-
tuple is smaller than any known quadruple or quintuple,
and is likely smaller than any quintuple.
On October 4, 1997, Mariano Garcia found the largest
known amicable pair, each of whose members has 4829
Digits. The new pair is
N x = CM[(P + Q)P 89 - 1] (15)
N 2 - CQ[(P ~ M)P S9 - 1], (16)
where
C = 2 1X P 89 (17)
M = 287155430510003638403359267 (18)
P = 574451143340278962374313859 (19)
Q = 136272576607912041393307632916794623.
(20)
P, Q, (P + Q)P 89 - 1, and (P - M)P 89 - 1 are Prime.
40
Amicable Pair
Amicable Triple
Pomerance (1981) has proved that
[amicable numbers < n] < ne~^ n ^ J
(21)
for large enough n (Guy 1994). No nonfinite lower
bound has been proven.
see also Amicable Quadruple, Amicable Triple,
Augmented Amicable Pair, Breeder, Crowd, Eu-
ler's Rule, Friendly Pair, Multiamicable Num-
bers, Pair Sum, Quasiamicable Pair, Sociable
Numbers, Unitary Amicable Pair
References
Alanen, J.; Ore, 0.; and Stemple, J. "Systematic Computa-
tions on Amicable Numbers." Math. Comput. 21, 242—
245, 1967.
Battiato, S. and Borho, W. "Are there Odd Amicable Num-
bers not Divisible by Three?" Math. Comput. 50, 633-
637, 1988.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 62 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, Feb. 1972.
Borho, W. and Hoffmann, H. "Breeding Amicable Numbers
in Abundance." Math. Comput 46, 281-293, 1986.
Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers
and Their Distribution." Math. Comput. 24, 431-432,
1970.
Cohen, H. "On Amicable and Sociable Numbers." Math.
Comput. 24, 423-429, 1970.
Costello, P. "Amicable Pairs of Euler's First Form." J. Rec.
Math. 10, 183-189, 1977-1978.
Costello, P. "Amicable Pairs of the Form (i,l)." Math. Com-
put. 56, 859-865, 1991.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 38-50,
1952.
Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4,
108-111, 1955-1956.
Erdos, P. "On Asymptotic Properties of Aliquot Sequences."
Math. Comput. 30, 641-645, 1976.
Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math-
ematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight-of~Mind from
Scientific American. New York: Vintage, pp. 160-171,
1978.
Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 55-59, 1994.
Lee, E. J. "Amicable Numbers and the Bilinear Diophantine
Equation." Math. Comput. 22, 181-197, 1968.
Lee, E. J. "On Divisibility of the Sums of Even Amicable
Pairs." Math. Comput. 23, 545-548, 1969.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, 1." J. Rec. Math. 5, 77-93, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, II." J. Rec. Math. 5, 153-173, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of
Amicable Numbers, HI." J. Rec. Math. 5, 231-249, 1972.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 145 and 155-156, 1979.
Moews, D. and Moews, P. C. "A Search for Aliquot Cycles
and Amicable Pairs." Math. Comput. 61, 935-938, 1993.
Moews, D. and Moews, P. C. "A List of Amicable Pairs Below
2.01 x 10 u ." Rev. Jan. 8, 1993. http://xraysgi.ims.
uconn . edu : 8080/amicable . txt .
Moews, D. and Moews, P. C. "A List of the First 5001 Am-
icable Pairs." Rev. Jan. 7, 1996. http://xraysgi.ims.
uconn.edu: 8080/amicable2. txt.
Ore, 0. Number Theory and Its History. New York: Dover,
pp. 96-100, 1988.
Pedersen, J. M. "Known Amicable Pairs." http://www.
vejlehs.dk/staff/jmp/aliquot/knwnap.htm.
Pomerance, C. "On the Distribution of Amicable Numbers."
J. reine angew. Math. 293/294, 217-222, 1977.
Pomerance, C. "On the Distribution of Amicable Numbers,
II." J. reine angew. Math. 325, 182-188, 1981.
Scott, E. B. E. "Amicable Numbers." Scripta Math. 12,
61-72, 1946.
Sloane, N. J. A. Sequences A002025/M5414 and A002046/
M5435 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
te Riele, H. J. J. "On Generating New Amicable Pairs from
Given Amicable Pairs." Math. Comput. 42, 219-223,
1984.
te Riele, H. J. J. "Computation of All the Amicable Pairs
Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35,
1986.
te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.;
and Lee, E. J. "Table of Amicable Pairs Between 10 x and
10 52 ." Centrum voor Wiskunde en Informatica, Note NM-
N8603. Amsterdam: Stichting Math. Centrum, 1986.
te Riele, H. J. J. "A New Method for Finding Amicable
Pairs." In Mathematics of Computation 1943-1993: A
Half-Century of Computational Mathematics (Vancouver,
BC, August 9-13, 1993) (Ed. W. Gautschi). Providence,
Rl: Amer. Math. Soc, pp. 577-581, 1994.
$$ Weisstein, E. W. "Sociable and Amicable Num-
bers." http : //www . astro . Virginia, edu/ -eww6n/math/
notebooks/Sociable .m.
Amicable Quadruple
An amicable quadruple as a QUADRUPLE (a, b, c, d) such
that
a(a) = a(b) — a(c) — cr(d) — a + b + c + d,
where cr(n) is the DIVISOR FUNCTION.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 59, 1994.
Amicable Triple
Dickson (1913, 1952) defined an amicable triple to be a
TRIPLE of three numbers (Z,m, n) such that
s(/) = m + n
${m) = I + n
s(n) = / + m,
where s(n) is the Restricted Divisor Function
(Madachy 1979). Dickson (1913, 1952) found eight sets
of amicable triples with two equal numbers, and two
sets with distinct numbers. The latter are (123228768,
103340640, 124015008), for which
s(12322876) = 103340640 + 124015008 = 227355648
s(103340640) = 123228768 + 124015008 = 24724377
5(124015008) = 123228768 + 10334064 = 226569408,
Amortization
Amplitude 41
and (1945330728960, 2324196638720, 2615631953920),
for which
s(1945330728960) = 2324196638720+2615631953920
= 4939828592640
s(2324196638720) = 1945330728960 + 2615631953920
= 4560962682880
5(2615631953920) = 1945330728960 + 2324196638720
= 4269527367680.
A second definition (Guy 1994) defines an amicable
triple as a TRIPLE (a, &, c) such that
a (a) = a(b) — o~(c) = a + b + c,
where a(n) is the DIVISOR FUNCTION. An example is
(2 2 3 2 5- 11, 2 5 3 2 7, 2 2 3 2 71).
see also Amicable Pair, Amicable Quadruple
References
Dickson, L. E. "Amicable Number Triples." Amer. Math.
Monthly 20, 84-92, 1913.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, p. 50, 1952.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 59, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 156, 1979.
Mason, T. E. "On Amicable Numbers and Their Generaliza-
tions." Amer, Math. Monthly 28, 195-200, 1921.
$$ Weisstein, E. W. "Sociable and Amicable Num-
bers." http : //www . astro . Virginia . edu/~eww6n/math/
notebooks/Sociable .m.
Amortization
The payment of a debt plus accrued INTEREST by regu-
lar payments.
Ampersand Curve
The Plane CURVE with Cartesian equation
(y 2 - x 2 ){x - l)(2a> - 3) = 4(z 2 + y 2 - 2x) 2 .
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989,
Amphichiral
An object is amphichiral (also called Reflexible) if it
is superposable with its Mirror Image (i.e., its image
in a plane mirror).
see also Amphichiral Knot, Chiral, Disymmetric,
Handedness, Mirror Image
Amphichiral Knot
An amphichiral knot is a Knot which is capable of be-
ing continuously deformed into its own MIRROR IMAGE.
The amphichiral knots having ten or fewer crossings are
04 O oi (Figure-of-Eight Knot), O6003, O8003, O8009,
08oi2j 08oi7j O8018) 10oi7,10o33, IO037, IO043, 10o45,
10o79, IO081, IO088, IO099, IO109, IO115, IO118, and IO123
(Jones 1985). The HOMFLY Polynomial is good at
identifying amphichiral knots, but sometimes fails to
identify knots which are not. No complete invariant (an
invariant which always definitively determines if a Knot
is Amphichiral) is known.
Let 6+ be the Sum of Positive exponents, and 6_ the
Sum of Negative exponents in the Braid Group B n .
If
b + - 3b- - n + 1 > 0,
then the Knot corresponding to the closed BRAID b is
not amphichiral (Jones 1985),
see also Amphichiral, Braid Group, Invertible
Knot, Mirror Image
References
Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter,
pp. 311-319, 1985.
Jones, V. "A Polynomial Invariant for Knots via von Neu-
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111,
1985.
Jones, V. "Hecke Algebra Representations of Braid Groups
and Link Polynomials." Ann. Math. 126, 335-388, 1987.
Amplitude
The variable <j> used in ELLIPTIC FUNCTIONS and EL-
LIPTIC Integrals, which can be defined by
= / dnudu,
where dn(u) is a JACOBI ELLIPTIC FUNCTION. The term
"amplitude" is also used to refer to the maximum offset
of a function from its baseline level.
see also Argument (Elliptic Integral), Charac-
teristic (Elliptic Integral), Delta Amplitude,
Elliptic Function, Elliptic Integral, Jacobi El-
liptic Functions, Modular Angle, Modulus (El-
liptic Integral), Nome, Parameter
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 590, 1972.
Fischer, G. (Ed.). Plate 132 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 129, 1986.
42 Anallagmatic Curve
Anchor
Anallagmatic Curve
A curve which is invariant under" Inversion. Exam-
ples include the Cardioid, Cartesian Ovals, Cassini
Ovals, Limaqon, Strophoid, and Maclaurin Tri-
SECTRIX.
Anallagmatic Pavement
see Hadamard Matrix
Analogy
Inference of the Truth of an unknown result obtained
by noting its similarity to a result already known to be
TRUE. In the hands of a skilled mathematician, anal-
ogy can be a very powerful tool for suggesting new and
extending old results. However, subtleties can render re-
sults obtained by analogy incorrect, so rigorous PROOF
is still needed.
see also INDUCTION
Analysis
The study of how continuous mathematical structures
(Functions) vary around the Neighborhood of a
point on a Surface. Analysis includes Calculus, Dif-
ferential Equations, etc.
see also Analysis Situs, Calculus, Complex Anal-
ysis, Functional Analysis, Nonstandard Analy-
sis, Real Analysis
References
Bottazzini, U. The "Higher Calculus": A History of Real and
Complex Analysis from Euler to Weierstraft. New York:
Springer-Verlag, 1986.
Bressoud, D. M. A Radical Approach to Real Analysis.
Washington, DC: Math. Assoc. Amer., 1994.
Ehrlich, P. Real Numbers, Generalization of the Reals, &
Theories of Continua. Norwell, MA: Kluwer, 1994.
Hairer, E. and Wanner, G. Analysis by Its History. New
York: Springer-Verlag, 1996.
Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan,
1988.
Wheeden, R. L. and Zygmund, A. Measure and Integral: An
Introduction to Real Analysis. New York: Dekker, 1977.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Analytic Function
A Function in the Complex Numbers C is analy-
tic on a region R if it is COMPLEX DlFFERENTIABLE
at every point in R. The terms HOLOMORPHIC FUNC-
TION and Regular Function are sometimes used in-
terchangeably with "analytic function." If a Function
is analytic, it is infinitely DlFFERENTIABLE.
see also BERGMAN SPACE, COMPLEX DlFFERENTIABLE,
DlFFERENTIABLE, PSEUDOANALYTIC FUNCTION, SEMI-
ANALYTIC, SUBANALYTIC
References
Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2
in Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 356-374, 1953.
Analytic Geometry
The study of the GEOMETRY of figures by algebraic rep-
resentation and manipulation of equations describing
their positions, configurations, and separations. Ana-
lytic geometry is also called Coordinate Geometry
since the objects are described as n-tuples of points
(where n = 2 in the PLANE and 3 in Space) in some
Coordinate System.
see also Argand Diagram, Cartesian Coordinates,
Complex Plane, Geometry, Plane, Quadrant,
Space, x-Axis, y-Axis, z-Axis
References
Courant, R. and Robbins, H. "Remarks on Analytic Geome-
try." §2.3 in What is Mathematics?: An Elementary Ap-
proach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 72-77, 1996.
Analytic Set
A Definable Set, also called a Souslin Set.
see also COANALYTIC SET, SOUSLIN Set
Anarboricity
Given a Graph G, the anarboricity is the maximum
number of line- disjoint nonacyclic SUBGRAPHS whose
UNION is G.
see also ARBORICITY
Analysis Situs
An archaic name for TOPOLOGY.
Analytic Continuation
A process of extending the region in which a COMPLEX
FUNCTION is defined.
see also Monodromy Theorem, Permanence of Al-
gebraic Form, Permanence of Mathematical Re-
lations Principle
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 378-380, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 389-390 and 392-
398, 1953.
Anchor
An anchor is the Bundle Map p from a Vector Bun-
dle A to the Tangent Bundle TB satisfying
1. [p(X),p(Y)] = p([X,r])and
2. [x,0y] = 0[x,y] + ( P (x).0)y,
where X and Y are smooth sections of A, <j> is a
smooth function of B, and the bracket is the "Jacobi-Lie
bracket" of a VECTOR FIELD.
see also Lie Algebroid
References
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.
Anchor Ring
Andrews-Schur Identity 43
Anchor Ring
An archaic name for the TORUS.
References
Eisenhart, h. P. A Treatise on the Differential Geometry of
Curves and Surfaces. New York: Dover, p. 314, 1960.
Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley,
p. 239, 1977.
Whittaker, E. T. A Treatise on the Analytical Dynamics of
Particles & Rigid Bodies, J^th ed. Cambridge, England:
Cambridge University Press, p. 21, 1959.
And
A term (PREDICATE) in LOGIC which yields TRUE if one
or more conditions are TRUE, and FALSE if any condi-
tion is False. A AND B is denoted Ak,B, A A B, or
simply AB. The Binary AND operator has the follow-
ing Truth Table:
A
B
AAB
F
F
F
F
T
F
T
F
F
T
T
T
A PRODUCT of ANDs (the AND of n conditions) is
called a CONJUNCTION, and is denoted
A*-
Andre's Reflection Method
A technique used by Andre (1887) to provide an elegant
solution to the BALLOT PROBLEM (Hilton and Pederson
1991).
References
Andre, D. "Solution directe du probleme resohi par
M, Bertrand." Comptes Rendus Acad. Sci. Paris 105,
436-437, 1887.
Comtet, L. Advanced Combinatorics. Dordrecht, Nether-
lands: Reidel, p. 22, 1974.
Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel. 13, 64-75, 1991.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 185, 1991.
Andrew's Sine
The function
*(*)■■
{sin
o,
(f)
< C7T
> C7T
which occurs in estimation theory.
see also SlNE
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 697, 1992.
Two binary numbers can have the operation AND per-
formed bitwise with 1 representing TRUE and FALSE.
Some computer languages denote this operation on A,
B, and C as A&&B&&C or logand(A,B,C).
see also BINARY OPERATOR, INTERSECTION, NOT, OR,
Predicate, Truth Table, XOR
Anderson-Darling Statistic
A statistic defined to improve the Kolmogorov-
SMIRNOV TEST in the TAIL of a distribution.
see also Kolmogorov-Smirnov Test, Kuiper
Statistic
References
Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 621, 1992.
Andre's Problem
The determination of the number of ALTERNATING PER-
MUTATIONS having elements {1, 2, . . . , n}
see also ALTERNATING PERMUTATION
Andrews Cube
see Semiperfect Magic Cube
Andrews- Curtis Link
The Link of 2-spheres in M 4 obtained by Spinning in-
tertwined arcs. The link consists of a knotted 2-sphere
and a Spun Trefoil Knot.
see also Spun Knot, Trefoil Knot
References
Rolfsen, D. Knots and Links.
Perish Press, p. 94, 1976.
Wilmington, DE: Publish or
Andrews-Schur Identity
£« fc2+a
fc=0
2n — k + a
k
_ V~^ 10fc 2 + (4a-l)fc
2n + 2a + 2
n — 5k
[lOfc + 2a + 2]
[2n -r 2a + 2] '
(1)
44 Andrica's Conjecture
Anger Function
where [x] is a GAUSSIAN POLYNOMIAL. It is a POLY-
NOMIAL identity for a = 0, 1 which implies the Ro.GERS-
Ramanujan Identities by taking n -t oo and apply-
ing the Jacobi Triple Product identity. A variant of
this equation is
£ -
fc=-|_a/2j
k 2 +2ak
n 4- k + a
n — k
|n/5j
-L(n+2a+2)/5j
15fc 2 +(6a+l)fc
2n + 2a + 2
5-5/z
[10A; + 2a + 2]
[2n 4- 2a + 2] '
(2)
where the symbol [xj in the Sum limits is the Floor
Function (Paule 1994). The Reciprocal of the iden-
tity is
00 k 2 +2ak
Z^ in-
(kq)
2fc+a
11(1 -q-
3 =
1
2j + l)(1 _ g20j+4a+4)(l _ g20j-4a+16)
(3)
for a = 0, 1 (Paule 1994). For g = 1, (1) and (2) become
£
-La/2j
n + A; -J- a
n — k
[n/5j
£
-|_(n+2a+2)/5j
2n + 2a + 2\ 5fc + a + 1
n — 5A;
n + a + 1
(4)
References
Andrews, G. E. "A Polynomial Identity which Implies the
Rogers-Ramanujan Identities." Scripta Math. 28, 297—
305, 1970.
Paule, P. "Short and Easy Computer Proofs of the Rogers-
Ramanujan Identities and of Identities of Similar Type."
Electronic J. Combinatorics 1, RIO, 1-9, 1994. http://
www. combinatorics . org/Volume JYvolumel .html#R10.
Andrica's Conjecture
100 200 300 400 500
Andrica's conjecture states that, for p n the nth PRIME
Number, the Inequality
A n = ^/Pn+l — \/Pn < 1
holds, where the discrete function A n is plotted above.
The largest value among the first 1000 PRIMES is for
n = 4, giving y/u. - \ft « 0.670873. Since the Andrica
function falls asymptotically as n increases so a PRIME
Gap of increasing size is needed at large n, it seems
likely the CONJECTURE is true. However, it has not yet
been proven.
100 200 300 400 500
An bears a strong resemblance to the PRIME DIFFER-
ENCE Function, plotted above, the first few values of
which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, . . . (Sloane's
A001223).
see also Brocard's Conjecture, Good Prime, For-
tunate Prime, Polya Conjecture, Prime Differ-
ence Function, Twin Peaks
References
Golomb, S. W. "Problem E2506: Limits of Differences of
Square Roots." Amer. Math. Monthly 83, 60-61, 1976.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 21, 1994.
Rivera, C. "Problems & Puzzles (Conjectures): An-
drica's Conjecture." http://www.sci.net.mx/-crivera/
ppp/conj _008 . htm.
Sloane, N. J. A. Sequence A001223/M0296 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Anger Function
A generalization of the Bessel Function OF the
First Kind defined by
Mz)
-tf
cos(v9 — zsinO) dQ.
If v is an INTEGER n, then J n (z) = J n (z), where J n (z)
is a Bessel Function of the First Kind. Anger's
original function had an upper limit of 27T, but the cur-
rent Notation was standardized by Watson (1966).
see also BESSEL FUNCTION, MODIFIED STRUVE FUNC-
TION, Parabolic Cylinder Function, Struve
Function, Weber Functions
References
Abramowitz, M. and Stegun, C A. (Eds.). "Anger and We-
ber Functions." §12.3 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 498-499, 1972.
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Angle
Angle
Angle Bracket 45
Given two intersecting Lines or Line Segments, the
amount of ROTATION about the point of intersection
(the Vertex) required to bring one into correspondence
with the other is called the angle 6 between them. An-
gles are usually measured in Degrees (denoted °), Ra-
dians (denoted rad, or without a unit), or sometimes
Gradians (denoted grad).
One full rotation in these three measures corresponds to
360°, 27r rad, or 400 grad. Half a full ROTATION is called
a Straight Angle, and a Quarter of a full rotation
is called a RIGHT ANGLE. An angle less than a RIGHT
Angle is called an Acute Angle, and an angle greater
than a Right Angle is called an Obtuse Angle.
The use of Degrees to measure angles harks back to
the Babylonians, whose SEXAGESIMAL number system
was based on the number 60. 360° likely arises from the
Babylonian year, which was composed of 360 days (12
months of 30 days each). The DEGREE is further divided
into 60 Arc Minutes, and an Arc Minute into 60
Arc Seconds. A more natural measure of an angle is
the Radian. It has the property that the Arc Length
around a CIRCLE is simply given by the radian angle
measure times the Circle Radius. The Radian is also
the most useful angle measure in CALCULUS because the
Derivative of Trigonometric functions such as
dx
does not require the insertion of multiplicative constants
like 7r/180. GRADIANS are sometimes used in surveying
(they have the nice property that a Right Angle is ex-
actly 100 Gradians), but are encountered infrequently,
if at all, in mathematics.
The concept of an angle can be generalized from the
Circle to the Sphere. The fraction of a Sphere sub-
tended by an object is measured in StERADIANS, with
the entire Sphere corresponding to 4n Steradians.
A ruled Semicircle used for measuring and drawing
angles is called a Protractor. A Compass can also
be used to draw circular ARCS of some angular extent.
see also Acute Angle, Arc Minute, Arc Second,
Central Angle, Complementary Angle, Degree,
Dihedral Angle, Directed Angle, Euler Angles,
Gradian, Horn Angle, Inscribed Angle, Oblique
Angle, Obtuse Angle, Perigon, Protractor,
Radian, Right Angle, Solid Angle, Steradian,
Straight Angle, Subtend, Supplementary Angle,
Vertex Angle
References
Dixon, R. Mathographics.
1991.
Angle Bisector
interior angle
bisector
exterior angle
^ bisection
The (interior) bisector of an Angle is the LINE or Line
Segment which cuts it into two equal Angles on the
same "side" as the Angle.
Ai h A 2
The length of the bisector of Angle A± in the above
Triangle AA!A 2 A 3 is given by
,, 2
ti
a 2 a%
ax
(a 2 +a 3 ) 2
where U = A& and a\ = AjA^. The angle bisectors
meet at the Incenter J, which has Trilinear Coor-
dinates 1:1:1.
see also Angle Bisector Theorem, Cyclic Quad-
rangle, Exterior Angle Bisector, Isodynamic
Points, Orthocentric System, Steiner-Lehmus
Theorem, Trisection
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 9-10, 1967.
Dixon, R. Mathographics. New York: Dover, p. 19, 1991.
Mackay, J. S. "Properties Concerned with the Angular Bi-
sectors of a Triangle." Proc. Edinburgh Math. Soc. 13,
37-102, 1895.
Angle Bisector Theorem
The Angle Bisector of an Angle in a Triangle di-
vides the opposite side in the same RATIO as the sides
adjacent to the ANGLE.
Angle Bracket
The combination of a Bra and Ket (bra+ket =
bracket) which represents the INNER PRODUCT of two
functions or vectors,
(f\9)
(V|W) :
f(x)g(x)dx
New York: Dover, pp. 99-100,
By itself, the Bra is a Covariant 1- Vector, and the
Ket is a Covariant One-Form. These terms are com-
monly used in quantum mechanics.
see also Bra, Differential &-Form, Ket, One-Form
46 Angle of Parallelism
Annulus Conjecture
Angle of Parallelism
P
Yl(x)
A C D B
Given a point P and a Line AB, draw the PERPENDIC-
ULAR through P and call it PC, Let PD be any other
line from P which meets CB in D. In a Hyperbolic
Geometry, as D moves off to infinity along CB, then
the line PD approaches the limiting line PE, which is
said to be parallel to CB at P. The angle LCPE which
PE makes with PC is then called the angle of paral-
lelism for perpendicular distance x, and is given by
n(x)-2tan- 1 (e- x ).
This is known as Lobachevsky's FORMULA.
see also Hyperbolic Geometry, Lobachevsky's
Formula
References
Manning, H. P. Introductory Non-Euclidean Geometry. New
York: Dover, pp. 31-32 and 58, 1963.
Angle Trisection
see Trisection
Angular Acceleration
The angular acceleration ct is defined as the time DE-
RIVATIVE of the Angular Velocity u>,
a ~
~dt
d 2 6 „ _ a
di 2 *' r'
see also Acceleration, Angular Distance, Angu-
lar Velocity
Angular Defect
The Difference between the Sum of face Angles Ai
at a Vertex of a Polyhedron and 27r,
5 = 2ir-^2Ai.
see also Descartes Total Angular Defect, Jump
Angle
Angular Velocity
The angular velocity U) is the time DERIVATIVE of the
Angular Distance with direction z Perpendicu-
lar to the plane of angular motion,
d0„ v
io = — z = — .
dt r
see also ANGULAR ACCELERATION, ANGULAR DIS-
TANCE
Anharmonic Ratio
see Cross-Ratio
Anisohedral Tiling
A fc-anisohedral tiling is a tiling which permits no n-
ISOHEDRAL TILING with n < k.
References
Berglund, J. "Is There a A;-Anisohedral Tile for k > 5?"
Amer. Math. Monthly 100, 585-588, 1993.
Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory. Washington, DC:
Math. Assoc. Amer., 1991.
Annihilator
The term annihilator is used in several different ways in
various aspects of mathematics. It is most commonly
used to mean the SET of all functions satisfying a given
set of conditions which is zero on every member of a
given SET.
Annulus
The region in common to two concentric CIRCLES of
RADII a and b. The AREA of an annulus is
Aannulus = ?t(& — CL ).
An interesting identity is as follows. In the figure,
the AREA of the shaded region A is given by
A = d + C 2 .
Angular Distance
The angular distance traveled around a CIRCLE is the
number of RADIANS the path subtends,
0= 7^2tt= -.
27TT r
see also CHORD, CIRCLE, CONCENTRIC CIRCLES, LUNE
(Plane), Spherical Shell
References
Pappas, T, "The Amazing Trick," The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 69, 1989.
see also ANGULAR ACCELERATION, ANGULAR VELOC-
ITY
Annulus Conjecture
see Annulus Theorem
Annulus Theorem
Anosov Flow 47
Annulus Theorem
Let Ki and K^ be disjoint bicollared knots in W n+ or
S and let U denote the open region between them.
Then the closure of U is a closed annulus S n x [0,1].
Except for the case n = 3, the theorem was proved by
Kirby (1969).
References
Kirby, R. C. "Stable Homeomorphisms and the Annulus Con-
jecture." Ann. Math. 89, 575-582, 1969.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 38, 1976.
Anomalous Cancellation
The simplification of a FRACTION a/b which gives a cor-
rect answer by "canceling" DIGITS of a and b. There
are only four such cases for NUMERATOR and DENOM-
INATORS of two Digits in base 10: 64/16 = 4/1 = 4,
98/49 ^ 8/4 = 2, 95/19 = 5/1 = 5, and 65/26 = 5/2
(Boas 1979).
The concept of anomalous cancellation can be extended
to arbitrary bases. PRIME bases have no solutions, but
there is a solution corresponding to each PROPER DIVI-
SOR of a Composite b. When b - 1 is Prime, this type
of solution is the only one. For base 4, for example,
the only solution is 324/ 134 = 24. Boas gives a table of
solutions for b < 39. The number of solutions is EVEN
unless b is an EVEN SQUARE.
6
N
b
N
4
1
26
4
6
2
27
6
8
2
28
10
9
2
30
6
10
4
32
4
12
4
34
6
14
2
35
6
15
6
36
21
16
7
38
2
18
4
39
6
20
4
21
10
22
6
24
6
see also Fraction, Printer's Errors, Reduced
Fraction
References
Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathemat-
ical Plums (Ed. R. Honsberger). Washington, DC: Math.
Assoc. Amer., pp. 113-129, 1979.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 86-87, 1988.
Anomalous Number
see Benford's Law
Anonymous
A term in SOCIAL CHOICE Theory meaning invariance
of a result under permutation of voters.
see also Dual Voting, Monotonic Voting
Anosov Automorphism
A Hyperbolic linear map R n -» R n with Integer en-
tries in the transformation Matrix and Determinant
±1 is an Anosov Diffeomorphism of the n-ToRUS,
called an Anosov automorphism (or HYPERBOLIC AU-
TOMORPHISM). Here, the term automorphism is used in
the Group Theory sense.
Anosov Diffeomorphism
An Anosov diffeomorphism is a C x DIFFEOMORPHISM <f>
such that the Manifold M is Hyperbolic with respect
to (j>. Very few classes of Anosov diffeomorphisms are
known. The best known is ARNOLD'S Cat Map.
A Hyperbolic linear map W 1 — > W 1 with Integer
entries in the transformation Matrix and Determi-
nant ±1 is an Anosov diffeomorphism of the n-TORUS.
Not every MANIFOLD admits an Anosov diffeomorphism.
Anosov diffeomorphisms are EXPANSIVE, and there are
no Anosov diffeomorphisms on the CIRCLE.
It is conjectured that if <f> : M —> M is an Anosov dif-
feomorphism on a Compact Riemannian Manifold
and the Nonwandering Set Q(<f>) of <f> is M, then <f>
is TOPOLOGICALLY CONJUGATE to a FlNITE-TO-ONE
Factor of an Anosov Automorphism of a Nilman-
ifold. It has been proved that any Anosov diffeomor-
phism on the n-TORUS is TOPOLOGICALLY CONJUGATE
to an ANOSOV AUTOMORPHISM, and also that Anosov
diffeomorphisms are C 1 STRUCTURALLY STABLE.
see also ANOSOV AUTOMORPHISM, AXIOM A DIFFEO-
MORPHISM, Dynamical System
References
Anosov, D. V. "Geodesic Flow on Closed Riemannian Man-
ifolds with Negative Curvature." Proc. Steklov Inst,
A. M. S. 1969.
Smale, S. "Differentiable Dynamical Systems." Bull. Amer.
Math. Soc. 73, 747-817, 1967.
Anosov Flow
A Flow defined analogously to the Anosov Diffeo-
morphism, except that instead of splitting the TAN-
GENT BUNDLE into two invariant sub-BUNDLES, they
are split into three (one exponentially contracting, one
expanding, and one which is 1-dimensional and tangen-
tial to the flow direction).
see also DYNAMICAL SYSTEM
48
Anosov Map
Anticlastic
Anosov Map
An important example of a ANOSOV DlFFEOMORPHISM.
Xn+l
=
2 l"
1 1
where x n +i,y n +i are computed mod 1.
see also ARNOLD'S CAT MAP
ANOVA
"Analysis of Variance." A Statistical Test for het-
erogeneity of Means by analysis of group VARIANCES.
To apply the test, assume random sampling of a vari-
ate y with equal VARIANCES, independent errors, and a
Normal Distribution. Let n be the number of Repli-
cates (sets of identical observations) within each of K
FACTOR LEVELS (treatment groups), and y^ be the jth
observation within FACTOR LEVEL i. Also assume that
the ANOVA is "balanced" by restricting n to be the
same for each Factor Level.
Now define the sum of square terms
k n
P\2
SST = £) £(j/ - J)
(1)
\ 2 / u „ v 2
i=l j = l
k
k n
"*-:E E« -eIE* (3)
k n
v j=rl j=l
.-\2
i=l j = l
= SST - SSA,
(4)
(5)
which are the total, treatment, and error sums of
squares. Here, yi is the mean of observations within
FACTOR Level i, and y is the "group" mean (i.e., mean
of means). Compute the entries in the following table,
obtaining the P- Value corresponding to the calculated
F- Ratio of the mean squared values
F =
MSA
MSE*
(6)
Category SS ° Freedom Mean Squared F- Ratio
treatment SSA K-l MSA = |P^ §g
error SSE K(n - 1) MSE = ^^
total
SST Kn - 1
MST=J^r_
If the P- VALUE is small, reject the NULL HYPOTHESIS
that all Means are the same for the different groups.
see also Factor Level, Replicate, Variance
Anthropomorphic Polygon
A Simple Polygon with precisely two Ears and one
Mouth.
References
Toussaint, G. "Anthropomorphic Polygons." Amer. Math.
Monthly 122, 31-35, 1991.
Anthyphairetic Ratio
An archaic word for a Continued Fraction.
References
Fowler, D. H. The Mathematics of Plato's Academy: A New
Reconstruction. New York: Oxford University Press, 1987.
Antiautomorphism
If a Map / : G -> G* from a Group G to a Group G'
satisfies f(ab) = f(a)f(b) for all a, 6 £ G, then / is said
to be an antiautomorphism.
see also AUTOMORPHISM
Anticevian Triangle
Given a center a : /3 : 7, the anticevian triangle is
defined as the TRIANGLE with VERTICES -a : /3 : 7,
a : -0 : 7, and a : f3 : -7. If A'B'C is the CEVIAN
TRIANGLE of X and A"B"G" is an anticevian trian-
gle, then X and A" are HARMONIC CONJUGATE POINTS
with respect to A and A 1 .
see also Cevian Triangle
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Antichain
Let P be a finite PARTIALLY ORDERED Set. An an-
tichain in P is a set of pairwise incomparable elements
(a family of SUBSETS such that, for any two members,
one is not the Subset of another). The WIDTH of P is
the maximum CARDINALITY of an ANTICHAIN in P. For
a Partial Order, the size of the longest Antichain
is called the Width.
see also Chain, Dilworth's Lemma, Partially Or-
dered Set, Width (Partial Order)
References
Sloane, N. J. A. Sequence A006826/M2469 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Anticlastic
When the Gaussian Curvature K is everywhere Neg-
ative, a SURFACE is called anticlastic and is saddle-
shaped. A Surface on which K is everywhere Posi-
tive is called Synclastic. A point at which the Gaus-
sian Curvature is Negative is called a Hyperbolic
Point.
see also Elliptic Point, Gaussian Quadrature,
Hyperbolic Point, Parabolic Point, Planar
Point, Synclastic
Anticommutative
Antimagic Graph 49
Anticommutative
An Operator * for which a * b = —6 * a is said to be
anticommutative.
see also Commutative
Anticommutator
For Operators A and B, the anticommutator is defined
by
{i,B} = AB + Si.
see a/50 Commutator, Jordan Algebra
Anticomplementary Triangle
A Triangle AA'B'C* which has a given Triangle
AABC as its Medial Triangle. The Trilinear Co-
ordinates of the anticomplementary triangle are
-1 L-i ^-1
-a : : c
A'
B = a : —0 : c
s^r -1 7-1 -1
C = a :b : — c .
see ateo MEDIAL TRIANGLE
Antiderivative
see Integral
Antihomologous Points
Two points which are COLLINEAR with respect to
a Similitude Center but are not Homologous
Points. Four interesting theorems from Johnson (1929)
follow.
1. Two pairs of antihomologous points form inversely
similar triangles with the HoMOTHETIC CENTER.
2. The Product of distances from a HOMOTHETIC
Center to two antihomologous points is a constant.
3. Any two pairs of points which are antihomologous
with respect to a Similitude Center lie on a Cir-
cle.
4. The tangents to two CIRCLES at antihomologous
points make equal ANGLES with the LINE through
the points.
see also HOMOLOGOUS POINTS, HOMOTHETIC CENTER,
Similitude Center
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 19-21, 1929.
Antilaplacian
The antilaplacian of u with respect to a? is a function
whose LAPLACIAN with respect to x equals u. The an-
tilaplacian is never unique.
see also LAPLACIAN
Antilinear Operator
An antilinear OPERATOR satisfies the following two
properties:
A[h(x) + f 2 (x)] = Ah{x) + Af 2 (x)
Acf(x) = c*Af(x),
Antidifferentiation
see INTEGRATION
where c* is the Complex Conjugate of c.
see also LINEAR OPERATOR
Antigonal Points
B
Given LAXB + IAYB — n RADIANS in the above fig-
ure, then X and Y are said to be antigonal points with
respect to A and B.
Antihomography
A ClRCLE-preserving TRANSFORMATION composed of
an Odd number of Inversions.
see also HOMOGRAPHY
Antilogarithm
The Inverse Function of the Logarithm, defined
such that
log 6 (antilog 6 z) = z = antilogy (log b z).
The antilogarithm in base b of z is therefore b z .
see also Cologarithm, Logarithm, Power
Antimagic Graph
A GRAPH with e EDGES labeled with distinct elements
{l,2,...,e}so that the Sum of the EDGE labels at each
VERTEX differ.
see also MAGIC GRAPH
References
Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A
Comprehensive Introduction. San Diego, CA: Academic
Press, 1990.
50 Antimagic Square
Antipedal Triangle
Antimagic Square
15
2
12
4
1
14
10
5
8
9
3
16
11
13
6
7
21
18
6
17
4
7
3
13
16
24
5
20
23
11
1
15
8
19
2
25
14
12
9
22
10
10
25
32
13
16
9
22
7
3
24
21
30
20
27
18
26
11
6
1
31
23
33
17
8
19
5
36
12
15
29
34
14
2
4
35
28
14
3
34
21
47
29
22
43
16
13
25
6
26
44
30
48
24
8
12
9
45
10
5
11
38
49
46
19
4
41
37
36
33
27
1
39
17
40
20
7
35
23
31
42
18
32
28
2
15
49
16
50
10
19
28
24
56
42
43
11
15
44
38
55
5
25
21
48
46
9
37
6
63
29
47
8
40
51
30
52
1
45
22
54
23
20
34
2
62
14
59
18
33
41
26
61
13
36
12
58
32
27
64
3
35
17
39
7
57
53
4
60
31
52
19
81
22
29
15
42
31
76
61
10
67
23
54
79
25
33
16
57
9
71
24
38
1
51
47
75
26
78
7
69
66
77
13
27
12
39
21
74
20
37
17
49
55
64
8
65
4
62
50
34
73
41
40
56
68
2
63
14
72
35
44
6
53
30
60
32
36
3
46
43
58
11
70
5
59
48
80
28
45
18
An antimagic square is an n x n ARRAY of integers from
1 to n 2 such that each row, column, and main diago-
nal produces a different sum such that these sums form
a Sequence of consecutive integers. It is therefore a
special case of a HETEROSQUARE.
Antimagic squares of orders one and two are impossi-
ble, and it is believed that there are also no antimagic
squares of order three. There are 18 families of an-
timagic squares of order four. Antimagic squares of or-
ders 4-9 are illustrated above (Madachy 1979).
see also HETEROSQUARE, MAGIC SQUARE, TALISMAN
Square
References
Disc.
Abe, G. "Unsolved Problems on Magic Squares."
Math. 127, 3-13, 1994.
Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 103-113, 1979.
# Weisstein, E. W. "Magic Squares." http: //www. astro.
Virginia, edu/~eww6n/math/notebooks/MagicSquares .m.
Antimorph
A number which can be represented both in the form
xo 2 — Dyo 2 and in the form Dx\ 2 — y\ 2 . This is only
possible when the PELL EQUATION
2 n 2
x — Dy
Antinomy
A Paradox or contradiction.
Antiparallel
A pair of LINES B\ , B2 which make the same ANGLES
but in opposite order with two other given LINES A\ and
A2, as in the above diagram, are said to be antiparallel
to A\ and A2.
see also HYPERPARALLEL, PARALLEL
References
Phillips, A. W. and Fisher, I. Elements of Geometry. New
York: American Book Co., 1896.
Antipedal Triangle
The antipedal triangle A of a given TRIANGLE T is the
Triangle of which T is the Pedal Triangle. For
a Triangle with Trilinear Coordinates a : j3 : 7
and Angles A, B, and C, the antipedal triangle has
Vertices with Trilinear Coordinates
is solvable. Then
x 2 - Dy 2 = ~(x - Dy 2 )(x n 2 - Dy n 2 )
= D(x y n - y x n ) 2 - {x x n - Dy y n ) 2 .
see also Idoneal Number, Polymorph
References
Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematical Entertains. New York: Dover,
1964.
Antimorphic Number
see Antimorph
— (/? + a cos C) (7 + a cos i?) : (7 + aicosI?)(a + /?cosC) :
(0 + a cos C) (a + 7 cos B)
(7 + cos A)(/3 + a cos C) : -(7 + ^cos A)(a + 0cosC) :
(a + cos C) (0 + 7 cos A)
(0 + 7 cos A) (7 + acosi?) : (a + 7 cos B) (7 + ficosA) :
— (a + jcosB)(0 + 7 cos A).
The Isogonal Conjugate of the Antipedal Trian-
gle of a given TRIANGLE is HOMOTHETIC with the orig-
inal Triangle. Furthermore, the Product of their
Areas equals the Square of the Area of the original
Triangle (Gallatly 1913).
see also Pedal Triangle
Antipersistent Process
Antisymmetric Matrix 51
References
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, pp. 56-58, 1913.
Antipersistent Process
A Fractal Process for which H < 1/2, so r < 0.
see also Persistent Process
Antipodal Map
The Map which takes points on the surface of a Sphere
S 2 to their ANTIPODAL POINTS.
Antipodal Points
Two points are antipodal (i.e., each is the ANTIPODE of
the other) if they are diametrically opposite. Examples
include endpoints of a Line SEGMENT, or poles of a
Sphere. Given a point on a Sphere with Latitude S
and Longitude A, the antipodal point has Latitude
~6 and LONGITUDE A ± 180° (where the sign is taken
so that the result is between —180° and +180°).
see also Antipode, Diameter, Great Circle,
Sphere
Antipode
Given a point A, the point B which is the ANTIPODAL
Point of A is said to be the antipode of A.
see also ANTIPODAL POINTS
Antiprism
Antiquity
see Geometric Problems of Antiquity
Antisnowflake
see Koch Antisnowflake
Antisquare Number
A number of the form p a • A is said to be an antisquare
if it fails to be a Square Number for the two reasons
that a is ODD and A is a nonsquare modulo p.
see also Square Number
Antisymmetric
A quantity which changes Sign when indices are re-
versed. For example, Aij = a, — aj is antisymmetric
since Aij = —Aji.
see also ANTISYMMETRIC MATRIX, ANTISYMMETRIC
Tensor, Symmetric
Antisymmetric Matrix
An antisymmetric matrix is a MATRIX which satisfies
the identity
A=-A*
(i)
where A T is the MATRIX TRANSPOSE. In component
notation, this becomes
an = —a-*
Letting k = i = j, the requirement becomes
cikk — —a-kkj
(2)
(3)
A Semiregular Polyhedron constructed with 2 n-
gons and 2n TRIANGLES. The 3-antiprism is simply the
Octahedron. The Duals are the Trapezohedra.
The Surface Area of a n-gonal antiprism is
-2[|na 2 cot(^)]+2n(|v / 3a 2 )
cot(£)+V3\
= \ na2
see also Octahedron, Prism, Prismoid, Trapezohe-
DRON
References
Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in
Mathematical Recreations and Essays, 13ili ed. New York;
Dover, p, 130, 1987.
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, pp. 85-86, 1997.
Weisstein, E. W. "Prisms and Antiprisms." http://www.
astro .virginia.edu/-eww6n/math/notebooks/Pr ism. m.
so an antisymmetric matrix must have zeros on its diag-
onal. The general 3x3 antisymmetric matrix is of the
form
ai2 ai3~
-aw a 2 3 • (4)
. — ai3 — G&23
Applying A" 1 to both sides of the antisymmetry condi-
tion gives
-A^A 1 = I. (5)
Any SQUARE MATRIX can be expressed as the sum of
symmetric and antisymmetric parts. Write
A=i(A + A T ) + f(A-A T ).
an
a>2i
ai2
«22
0,2n
a n i a n 2
(6)
(7)
52 Antisymmetric Relation
Apeirogon
an
a2i
a n i
A T =
ai2
a22 " *
0>n2
)
_ain
a2n • • ■
Q>nn _
2an
ai2 + C121
flln + «nl
0,12 + &21
2a22
fl2n + «n2
_ain +
a-Tii
&2n + a„2
Z(l nn
(8)
A + A T =
which is symmetric, and
A-A T =
ai2 - fltei
-(ai2 - a2i)
-(flln — flnl) — (tl2n — ^n2)
(9)
Oln - Q>nl
din — &n2
(10)
which is antisymmetric.
see ateo Skew Symmetric Matrix, Symmetric Ma-
trix
Antisymmetric Relation
A RELATION R on a SET S is antisymmetric provided
that distinct elements are never both related to one an-
other. In other words xRy and yRx together imply that
x~y.
Antisymmetric Tensor
An antisymmetric tensor is denned as a TENSOR for
which
A mn = _ A r,m t ^
Any Tensor can be written as a sum of Symmetric
and antisymmetric parts as
The antisymmetric part is sometimes denoted using the
special notation
A [ab] = U A ab _ A bay
For a general TENSOR,
(3)
(4)
permutations
where e ai -a. n is the Levi-Civita Symbol, a.k.a. the
Permutation Symbol.
see also Symmetric Tensor
Antoine's Horned Sphere
A topological 2-sphere in 3-space whose exterior is not
Simply Connected. The outer complement of An-
toine's horned sphere is not Simply Connected. Fur-
thermore, the group of the outer complement is not
even finitely generated. Antoine's horned sphere is in-
equivalent to Alexander's Horned Sphere since the
complement in E 3 of the bad points for Alexander's
Horned Sphere is Simply Connected.
see also Alexander's Horned Sphere
References
Alexander, J. W. "An Example of a Simply-Connected Sur-
face Bounding a Region which is not Simply-Connected."
Proc. Nat Acad. Sci. 10, 8-10, 1924.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 76-79, 1976.
Antoine's Necklace
Construct a chain C of 2n components in a solid TORUS
V. Now form a chain C± of 2n solid tori in V, where
ir x (V - Ci) <* iri(V - C)
via inclusion. In each component of Ci, construct a
smaller chain of solid tori embedded in that component.
Denote the union of these smaller solid tori C^. Con-
tinue this process a countable number of times, then the
intersection
A=f|C
which is a nonempty compact SUBSET of IR. is called
Antoine's necklace. Antoine's necklace is HOMEOMOR-
PHIC with the CANTOR SET.
see also ALEXANDER'S HORNED SPHERE, NECKLACE
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 73-74, 1976,
Apeirogon
The Regular Polygon essentially equivalent to the
CIRCLE having an infinite number of sides and denoted
with Schlafli Symbol {oo}.
see also CIRCLE, REGULAR POLYGON
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Schwartzman, S. The Words of Mathematics: An Etymolog-
ical Dictionary of Mathematical Terms Used in English.
Washington, DC: Math. Assoc. Amer., 1994.
Apery 's Constant
Apery's Constant 53
Apery's Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Apery's constant is defined by
C(3) = 1.2020569...
(1)
(Sloane's A002117) where f(z) is the RlEMANN Zeta
Function. Apery (1979) proved that £(3) is Irra-
tional, although it is not known if it is TRANSCEN-
DENTAL. The Continued Fraction for £(3) is [1, 4, 1,
18, 1, 1, 1, 4, 1, ...] (Sloane's A013631). The positions
at which the numbers 1, 2, . . . occur in the continued
fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, ... .
Sums related to £(3) are
c(3) _ 5 f^ (-1)- 1 ,.5f (-ir'(M)»
(used by Apery), and
oo
(2fc + 1)3
2tt 3
(3* + l)» " 81^/3 ' 27
OO
^ (3& + 1)
oo 3
2-> (4/k + l) 3 = 64 + " ^ 3 )
+ H C(3) (4)
(5)
OO
^ (6Jfe + 1
+ iC(3), (6)
(6fc+l)3 36^3 2
where X(z) is the Dirichlet Lambda Function. The
above equations are special cases of a general result due
to Ramanujan (Berndt 1985). Apery's proof relied on
showing that the sum
<»>-£©"Cr)". o
where (£) is a Binomial Coefficient, satisfies the Re-
currence Relation
(n + l) 3 a(n + 1) - (34n 3 + bin 2 + 27n + 5)a(n)
+n 3 a(n-l) = (8)
(van der Poorten 1979, Zeilberger 1991).
Apery's constant is also given by
Sn,\
«3)=x; 2 T i '
*-*t nln
(9)
where 5 n , m is a Stirling Number of the First Kind.
This can be rewritten as
E§= 2 « 3 )'
(10)
where H n is the nth HARMONIC NUMBER, Yet another
expression for £(3) is
««-*£;?(•♦*+••■ + ;)
(11)
(Castellanos 1988).
Integrals for C(3) include
CO)
i r e
2io c'-l
cK
\W
r ir/4
= ^ | j7r J ln2 + 2 / a: In (sin a;) da;
(12)
(13)
Gosper (1990) gave
30& - 11
4 £? (2* -!)*»(?)'
(14)
A Continued Fraction involving Apery's constant is
JL = 5 _J^ t_ rf
C(3) 117- 535- ' ' * 34n 3 + 51n 2 + 27n + 5- * * '
(15)
(Apery 1979, Le Lionnais 1983). Amdeberhan (1996)
used Wilf- Zeilberger Pairs (F,G) with
F{n,k)
_ (-l) k k\ 2 (sn-k-l)\
(sn + & + l)!(fc + l)
(16)
s — 1 to obtain
c(3) = §f;(-ir- 1 «i ? . (it)
For 5 = 2,
oo „
ffl)- 1 ^ nn-i 56n 2 -32 + 5 1
and for s = 3,
(-i) n
((3) = V { ~ X)
^72( 4n )( 3n )
6120n + 5265n 4 + 13761n 2 + 13878n 3 + 1040
(4n-fl)(4n + 3)(n+l)(3n+l) 2 (3n + 2) 2 ^ >
54 Apery's Constant
Apoapsis
(Amdeberhan 1996). The corresponding G(n,k) for s =
1 and 2 are
G(n ' fc) -(n + fc + l)!(n+l)' (20)
and
<3(n,fc) =
(-l) fc fci 2 (2n - fe)!(3 + 4n)(4n 2 + 6n + k + 3)
2(2n-hA; + 2)!(n + l) 2 (2n-hl) 2
Gosper (1996) expressed C(3) as the MATRIX PRODUCT
N
(21)
lim TTM n =
C(3)
1
(22)
where
M n =
" (n + l) 4 24570Tt 4 + 64161n 3 +62152n 2 +26427n.+4154
4096(n+f)2(n+J)2 31104(n+|)(n+±)(n+§)
1
(23)
which gives 12 bits per term. The first few terms are
(24)
(25)
(26)
which gives
C( 3 ) * IllZlVwl = 1-20205690315732 .... (27)
Given three INTEGERS chosen at random, the probabil-
ity that no common factor will divide them all is
r i
2077 "
1728
1
Mi =
19600
M 2 =
1
9801
7561 "
4320
1
r °
50501
20160
1
-1
M 3 =
67600
[CO)]'
1.202 -1 =0.832..
(28)
B. Haible and T. Papanikolaou computed £(3) to
1,000,000 Digits using a Wilf-Zeilberger Pair iden-
tity with
_ fc n! 6 (2n-fc-l)!fc! 3
*(n,k)-( 1) 2(n + A . + 1)!2(2rl )!3> W
5 = 1, and t = 1, giving the rapidly converging
,vn-V\ 1 ^ rc! 1 °(205n 2 + 250n + 77)
QW-Z^l- 1 ) 64(2n+l)!«
(Amdeberhan and Zeilberger 1997). The record as of
Aug. 1998 was 64 million digits (Plouffe).
see also Riemann Zeta Function, Wilf-Zeilberger
Pair
References
Amdeberhan, T. "Faster and Faster Convergent Se-
ries for C(3)-" Electronic J. Combinatorics 3, R13,
1—2, 1996. http: //www. combinatorics. org/Volume^/
volume3 ,html#R13.
Amdeberhan, T. and Zeilberger, D. "Hypergeometric Se-
ries Acceleration via the WZ Method." Electronic J.
Combinatorics 4, No. 2, R3, 1-3, 1997. http: //www.
combinatorics . org/Volume_4/wilf toe .html#R03. Also
available at http : //www . math . temple . edu/~zeilberg/
mamarim/mamarimhtml/accel . html.
Apery, R. "Irrationalite de £(2) et C(3)." Asterisque 61, 11-
13, 1979.
Berndt, B. C. Ramanujan's Notebooks: Part J. New York:
Springer- Verlag, 1985.
Beukers, F. "A Note on the Irrationality of C(3)." Bull. Lon-
don Math. Soc. 11, 268-272, 1979.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Conway, J. H. and Guy, R. K. "The Great Enigma." In The
Book of Numbers. New York: Springer- Verlag, pp. 261—
262, 1996.
Ewell, J. A. "A New Series Representation for C(3)." Amer.
Math. Monthly 97, 219-220, 1990.
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsoft.com/asolve/constant/apery/apery.html.
Gosper, R. W. "Strip Mining in the Abandoned Orefields
of Nineteenth Century Mathematics." In Computers in
Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks).
New York: Marcel Dekker, 1990.
Haible, B. and Papanikolaou, T. "Fast Multiprecision Eval-
uation of Series of Rational Numbers." Technical Report
TI-97-7. Darmstadt, Germany: Darmstadt University of
Technology, Apr. 1997.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 36, 1983.
Plouffe, S. "Plouffe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records. html.
Plouffe, S. "32,000,279 Digits of Zeta(3)." http://lacim.
uqam.ca/piDATA/Zet a3.txt.
Sloane, N. J. A. Sequences A013631 and A002117/M0020 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
van der Poorten, A. "A Proof that Euler Missed. . . Apery's
Proof of the Irrationality of £(3)." Math. Intel. 1,196-203,
1979.
Zeilberger, D. "The Method of Creative Telescoping." J.
Symb. Comput. 11, 195-204, 1991.
Apoapsis
(30)
The greatest radial distance of an Ellipse as measured
from a FOCUS. Taking v = n in the equation of an
Ellipse
a(l-e 2 )
r =
1 + e cos v
Apocalypse Number
Apodization Function 55
gives the apoapsis distance
r+ =a(l + e).
Apoapsis for an orbit around the Earth is called apogee,
and apoapsis for an orbit around the Sun is called aphe-
lion.
see also Eccentricity, Ellipse, Focus, Periapsis
Apocalypse Number
A number having 666 Digits (where 666 is the Beast
Number) is called an apocalypse number. The FI-
BONACCI NUMBER F3184 is an apocalypse number.
see also Beast Number, Leviathan Number
References
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97-
102, 1995.
Apocalyptic Number
A number of the form 2 n which contains the digits 666
(the Beast Number) is called an Apocalyptic Num-
ber. 2 157 is an apocalyptic number. The first few such
powers are 157, 192, 218, 220, . . . (Sloane's A007356).
see also Apocalypse Number, Leviathan Number
References
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97-
102, 1995.
Sloane, N. J. A. Sequences A007356/M5405 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
Apodization
The application of an APODIZATION FUNCTION.
Apodization Function
A function (also called a Tapering Function) used to
bring an interferogram smoothly down to zero at the
edges of the sampled region. This suppresses sidelobes
which would otherwise be produced, but at the expense
of widening the lines and therefore decreasing the reso-
lution.
The following are apodization functions for symmetrical
(2-sided) interferograms, together with the Instrument
Functions (or Apparatus Functions) they produce
and a blowup of the Instrument Function sidelobes.
The Instrument Function I(k) corresponding to a
given apodization function A(x) can be computed by
taking the finite FOURIER COSINE TRANSFORM,
Apodization Function
Instrument Function
1.25
Instrument Function Sidelobes
I(k)
/a
■a
cos(27r kx)A(x) dx.
(1)
Bartlett
Connes
Harming
Uniform
Welch
-1 -0.5 0,5 1
Type
Apodization Function
Instrument Function
Bartlett
1 _ i£i
a sinc 2 (7r/:a)
Blackman
B A (x)
Bi(fc)
Connes
o-sr
8aV27r (2ir*a)«/»
Cosine
c °s(ff)
4aca B (2jrafc)
TT(l-160 2 fc 2 )
Gaussian
e -*V(2» a )
2j a cos(27rfca:)e- l3/(2 ' 2) dx
Hamming
Hm.A{x)
Hmj{k)
Hanning
Hn A (x)
Hnt(k)
Uniform
1
2a sine (27r/ea)
Welch
1-S
W:(k)
where
B A (x) =
Bj(k) =
Hm A (x) =
Hrm(k) =
(1ZX \ / 2-7TX \
— J +0.08 cos f J
a(0.84 - 0.36a 2 k 2 -2.17 x 1Q- X9 a 4 fc 4 ) sinc(27rafc)
(2)
(l-a 2 A: 2 )(l-4a 2 fc 2 )
0.54 + 0.46 cos (—)
a(1.08 - 0.64a 2 fc 2 ) sinc(27rafe)
" l-4a 2 fc 2 ""
(3)
(4)
(5)
56 Apodization Function
Apollonius Circles
Hn A (x) = cos 2 I — )
1 + cos
(?)
Hrnik)
1
" 2
a sine (2irak)
' l-4a 2 fc 2
= a[sinc(27rfca) + ^ sinc(27rA;a — 7r)
+ ^ sinc(27rA;a + 7r)]
W}(fc) =a2V2?r
J3/2
(27rA;a)
(27rfca) 3 / 2
sin(27rfca) — 2nak cos(2-7rafc)
2a 3 fe 3 7T 3 '
(6)
(7)
(8)
(9)
(10)
(11)
Type
IF FWHM
IF Peak
Peak (-) S.L.
Peak
Peak (+) S.L.
Peak
Bartlett
1.77179
1
0.00000000
0.0471904
Blackmail
2.29880
0.84
-0.00106724
0.00124325
Cormes
1.90416
16
15
-0.0411049
0.0128926
Cosine
1.63941
4.
-0.0708048
0.0292720
Gaussian
—
1
—
—
Hamming
1.81522
1.08
-0.00689132
0.00734934
Hanning
2.00000
1
-0.0267076
0.00843441
Uniform
1.20671
2
-0.217234
0.128375
Welch
1.59044
4
3
-0.0861713
0.356044
A general symmetric apodization function A(x) can be
written as a FOURIER SERIES
oo
a n cosl— -J, (12)
n=l
where the COEFFICIENTS satisfy
oo
a + 2^a„ = 1. (13)
n = l
The corresponding apparatus function is
I(t) = J A{x)e~ 2 ' Ktkx dx = 26Ja sinc(27r£;&)
oo
+ y^[sinc(27rA:& + mr) + sinc(27rA;6 - nir)] |. (14)
n=l
To obtain an APODIZATION FUNCTION with zero at ka =
3/4, use
ao sinc(|7r) + ai[sinc(|7r) + sinc(^7r) = 0. (15)
Plugging in (13),
-d-^ +*(£ + £)
= -|(l-2o 1 ) + oi(i + l) = (16)
ai =
ao
- 5
3 ° _ 5
| + | 6-3 + 2-5 28
1 n„ 28 — 2 • 5 18
1 lai = 9* ~" 28 -
9
14'
(18)
(19)
The Hamming Function is close to the requirement
that the Apparatus Function goes to at ka — 5/4,
giving
a = § « 0.5435
ai
21
92
0.2283.
(20)
(21)
The Blackman Function is chosen so that the Appa-
ratus Function goes to at ka — 5/4 and 9/4, giving
ao
ai =
a 2 =
3969 ,.
9304 n
1155 „
4652 "
715
18608
0.4266
(22)
0.2483
(23)
i 0.0384.
(24)
ai(! + !) =
^5 ' 3^ 3
(IT)
see also Bartlett Function, Blackman Function,
Connes Function, Cosine Apodization Function,
Full Width at Half Maximum, Gaussian Func-
tion, Hamming Function, Hann Function, Han-
ning Function, Mertz Apodization Function,
Parzen Apodization Function, Uniform Apodiza-
tion Function, Welch Apodization Function
References
Ball, J. A. "The Spectral Resolution in a Correlator Sys-
tem" §4,3.5 in Methods of Experimental Physics 12C (Ed.
M. L. Meeks). New York: Academic Press, pp. 55-57,
1976.
Blackman, R. B. and Tukey, J. W. "Particular Pairs of Win-
dows." In The Measurement of Power Spectra, From
the Point of View of Communications Engineering. New
York: Dover, pp. 95-101, 1959.
Brault, J. W. "Fourier Transform Spectrometry." In High
Resolution in Astronomy: 15th Advanced Course of
the Swiss Society of Astronomy and Astrophysics (Ed.
A. Benz, M. Huber, and M. Mayor), Geneva Observatory,
Sauverny, Switzerland, pp. 31-32, 1985.
Harris, F. J. "On the Use of Windows for Harmonic Analysis
with the Discrete Fourier Transform." Proc. IEEE 66, 51-
83, 1978.
Norton, R. H. and Beer, R. "New Apodizing Functions for
Fourier Spectroscopy." J. Opt. Soc. Amer. 66, 259-264,
1976.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 547-548, 1992.
Schnopper, H. W. and Thompson, R. I. "Fourier Spectrom-
eters." In Methods of Experimental Physics 12 A (Ed.
M. L. Meeks). New York: Academic Press, pp. 491-529,
1974.
Apollonius Circles
There are two completely different definitions of the so-
called Apollonius circles:
1 . The set of all points whose distances from two fixed
points are in a constant ratio 1 : \i (Ogilvy 1990).
Apollonius Point
Apollonius 3 Problem 57
2. The eight CIRCLES (two of which are nondegener-
ate) which solve APOLLONIUS ' PROBLEM for three
Circles.
Given one side of a Triangle and the ratio of the
lengths of the other two sides, the LOCUS of the third
VERTEX is the Apollonius circle (of the first type) whose
Center is on the extension of the given side. For a given
Triangle, there are three circles of Apollonius.
Denote the three Apollonius circles (of the first type)
of a Triangle by &i, fo, and £3, and their centers Li,
L 2) and L 3 . The center L\ is the intersection of the side
A2A3 with the tangent to the ClRCUMCIRCLE at A\.
L\ is also the pole of the SYMMEDIAN POINT K with
respect to ClRCUMCIRCLE. The centers Li, Z/ 2 , and Lz
are COLLINEAR on the POLAR of K with regard to its
ClRCUMCIRCLE, called the Lemoine Line. The circle of
Apollonius ki is also the locus of a point whose Pedal
Triangle is Isosceles such that P1P2 = P1P3.
Let U and V be points on the side line BC of a TRI-
ANGLE AABC met by the interior and exterior ANGLE
Bisectors of Angles A. The Circle with Diame-
ter UV is called the A-Apollonian circle. Similarly,
construct the B- and C-Apollonian circles. The Apol-
lonian circles pass through the VERTICES A, £?, and C,
and through the two ISODYNAMIC POINTS S and S' .
The Vertices of the D-Triangle lie on the respective
Apollonius circles.
see also Apollonius' Problem, Apollonius Pursuit
Problem, Casey's Theorem, Hart's Theorem, Iso-
dynamic Points, Soddy Circles
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 40 and 294-299, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 14-23, 1990.
Apollonius Point
Consider the Excircles Fa, T b , and Tc of a Trian-
gle, and the CIRCLE T internally TANGENT to all three.
Denote the contact point of T and Fa by A f , etc. Then
the Lines AA\ BB f , and CC' CONCUR in this point. It
has Triangle Center Function
a = sin 2 ,4 cos 2 [§(£-<?)].
References
Kiinherling, C. "Apollonius Point." http://vvv.
evansville . edu/~ck6/t centers/re cent /apollon. html.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C; Iwata, S.; and Hidetosi, F. "Problem 1091
and Solution." Crux Math. 13, 128-129 and 217-218,
1987.
Apollonius' Problem
•3 ^€J •£)
© #;(•}
^
Given three objects, each of which may be a Point,
Line, or Circle, draw a Circle that is Tangent to
each. There are a total of ten cases. The two easi-
est involve three points or three LINES, and the hardest
involves three CIRCLES. Euclid solved the two easiest
cases in his Elements, and the others (with the exception
of the three CIRCLE problem), appeared in the Tangen-
cies of Apollonius which was, however, lost. The general
problem is, in principle, solvable by STRAIGHTEDGE and
Compass alone.
58
Apollonius 7 Problem
Apollonius Pursuit Problem
The three-ClRCLE problem was solved by Viete (Boyer
1968), and the solutions are called Apollonius Cir-
cles. There are eight total solutions. The simplest
solution is obtained by solving the three simultaneous
quadratic equations
(x - x x f + (y - Vl ) 2 - (r ± n) 2 = (1)
(x - x 2 f + (y - y 2 f - (r ± r 2 f = (2)
(x - x z f + (y - y 3 ) 2 - (r ± r 3 ) 2 - (3)
in the three unknowns x, y y r for the eight triplets of
signs (Courant and Robbins 1996). Expanding the equa-
tions gives
OOO O ^
(x + y -r )-2xXi-2yyi±2rri+(xi +yi -n ) =
(4)
for i — 1, 2, 3. Since the first term is the same for each
equation, taking (2) — (1) and (3) — (1) gives
where
ax 4- by + cr = d
(5)
ax + by + cr=-d,
(6)
a = 2(a?i — x 2 )
(7)
b= 2(yi -y 2 )
(8)
c = q=2(ri - r 2 )
(9)
1/2. 2 2\/2, 2
a = (x 2 +2/2 - r-2 ) - (xi + yi -
-n 2 )
(10)
and similarly for a , 6' , c and d' (where the 2 subscripts
are replaced by 3s). Solving these two simultaneous lin-
ear equations gives
b'd - bd! - b'cr + bc'r
ab 1 - ba !
—ad + ad' + o! cr — ac'r
ab' -a'b '
(11)
(12)
which can then be plugged back into the QUADRATIC
Equation (1) and solved using the Quadratic For-
mula.
Perhaps the most elegant solution is due to Gergonne.
It proceeds by locating the six HOMOTHETIC CENTERS
(three internal and three external) of the three given
CIRCLES. These lie three by three on four lines (illus-
trated above). Determine the Poles of one of these
with respect to each of the three CIRCLES and connect
the Poles with the Radical Center of the Circles.
If the connectors meet, then the three pairs of intersec-
tions are the points of tangency of two of the eight circles
(Johnson 1929, Dorrie 1965). To determine which two
of the eight Apollonius circles are produced by the three
pairs, simply take the two which intersect the original
three CIRCLES only in a single point of tangency. The
procedure, when repeated, gives the other three pairs of
Circles.
If the three CIRCLES are mutually tangent, then the
eight solutions collapse to two, known as the Soddy
Circles.
see also Apollonius Pursuit Problem, Bend (Cur-
vature), Casey's Theorem, Descartes Circle
Theorem, Four Coins Problem, Hart's Theorem,
Soddy Circles
References
Boyer, C. B. A History of Mathematics. New York: Wiley,
p. 159, 1968.
Courant, R. and Robbins, H. "Apollonius' Problem." §3.3 in
What is Mathematics? : An Elementary Approach to Ideas
and Methods , 2nd ed. Oxford, England: Oxford University
Press, pp. 117 and 125-127, 1996.
Dorrie, H. "The Tangency Problem of Apollonius." §32 in
100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 154-160,
1965.
Gauss, C. F. Werke, Vol. 4. New York: George Olms, p. 399,
1981.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 118-121, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 48-51, 1990.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 151, 1989.
Simon, M. Uber die Entwicklung der Element argeometrie im
XIX Jahrhundert. Berlin, pp. 97-105, 1906.
^ Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia . edu/-eww6n/math/notebooks/PlaneGeometry . m.
Apollonius Pursuit Problem
Given a ship with a known constant direction and speed
v 1 what course should be taken by a chase ship in pur-
suit (traveling at speed V) in order to intersect the other
ship in as short a time as possible? The problem can be
solved by finding all points which can be simultaneously
reached by both ships, which is an APOLLONIUS CIRCLE
with fi = v/V. If the CIRCLE cuts the path of the pur-
sued ship, the intersection is the point towards which
Apollonius Theorem
Appell Transformation 59
the pursuit ship should steer. If the CIRCLE does not
cut the path, then it cannot be caught.
see also Apollonius Circles, Apollonius' Prob-
lem, Pursuit Curve
References
Ogilvy, C. S. Solved by M. S. Klamkin. "A Slow Ship In-
tercepting a Fast Ship." Problem E991. Amer. Math.
Monthly 59, 408, 1952.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
p. 17, 1990.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 126-138, 1983.
Apollonius Theorem
ma 2 2 + na 3 2 = (m + n)AiP 2 + mPA 3 2 + nPA 2 2 .
Apothem
Given a CIRCLE, the PERPENDICULAR distance a from
the Midpoint of a Chord to the Circle's center is
called the apothem. It is also equal to the RADIUS r
minus the SAGITTA s,
a — r — s.
see also Chord, Radius, Sagitta, Sector, Segment
Apparatus Function
see Instrument Function
Appell Hypergeometric Function
A formal extension of the Hypergeometric Function
to two variables, resulting in four kinds of functions (Ap-
pell 1925),
oo oo
F 1 (a;/3,/3'; 7 ;x,y) = ^^
(a)™+«GS)m(/3')«
m = n =
oo oo
-x y
x y
m = n =
F 3 (a,a ;/3,/3 i7i *,y) = ^ JL m!n !( 7 ) m+ „
m = Q ti-0
oo oo
77 / a < \ V^ V^ ( Q )m + n(^)m + r 1 ^ mj ,
Appell defined the functions in 1880, and Picard showed
in 1881 that they may all be expressed by INTEGRALS
of the form
/'
Jo
u a (l - uf{l - xuy(l - yu) S du.
References
Appell, P. "Sur les fonctions hypergeometriques de plusieurs
variables." In Memoir. Sci. Math. Paris: Gauthier-Villars,
1925.
Bailey, W. N. Generalised Hypergeometric Series. Cam-
bridge, England: Cambridge University Press, p. 73, 1935.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1461,
1980.
Appell Polynomial
A type of Polynomial which includes the Bernoulli
Polynomial, Hermite Polynomial, and Laguerre
POLYNOMIAL as special cases. The series of POLYNOMI-
ALS {A n (z)}™ =0 is defined by
where
A(t)e** = ^TA n (z)t n ,
A(t) = ^2a k t k
is a formal POWER series with k = 0, 1, . . . and ao ^ 0.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, pp. 209-210, 1988.
Appell Transformation
A HOMOGRAPHIC transformation
ax + by -\- c
a"x + b"y + c
ax + b'y + c'
a n x + b"y + c"
with t\ substituted for t according to
X! —
yi
kdti
dt
{a"x + b"y + c") 2 '
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, pp. 210-211, 1988.
m = ti=:0
m!n!(7) m (7') 71
A Surface of Revolution defined by Kepler. It con-
sists of more than half of a circular ARC rotated about
an axis passing through the endpoints of the Arc. The
equations of the upper and lower boundaries in the x-z
Plane are
z± = ± V / R 2 -(x-r) 2
for R > r and x E [— (r + R), r + R]. It is the outside
surface of a Spindle TORUS.
see also Bubble, Lemon, Sphere-Sphere Intersec-
tion, Spindle Torus
Approximately Equal
If two quantities A and B are approximately equal, this
is written A « B.
see also Defined, Equal
Approximation Theory
The mathematical study of how given quantities can be
approximated by other (usually simpler) ones under ap-
propriate conditions. Approximation theory also stud-
ies the size and properties of the ERROR introduced by
approximation. Approximations are often obtained by
POWER SERIES expansions in which the higher order
terms are dropped.
see also LAGRANGE REMAINDER
References
Achieser, N. I. and Hyman, C. J. Theory of Approximation.
New York: Dover, 1993.
Akheizer, N. I. Theory of Approximation. New York: Dover,
1992.
Cheney, E. W. Introduction to Approximation Theory. New
York: McGraw-Hill, 1966.
Golomb, M. Lectures on Theory of Approximation. Argonne,
IL: Argonne National Laboratory, 1962.
Jackson, D, The Theory of Approximation. New York:
Amer. Math. Soc, 1930.
Natanson, I. P. Constructive Function Theory, Vol. 1: Uni-
form Approximation. New York: Ungar, 1964.
Petrushev, P. P. and Popov, V. A. Rational Approximation of
Real Functions. New York: Cambridge University Press,
1987.
Rivlin, T. J. An Introduction to the Approximation of Func-
tions. New York: Dover, 1981.
Timan, A. F. Theory of Approximation of Functions of a
Real Variable. New York: Dover, 1994.
Arbelos
Arakelov Theory
A formal mathematical theory which introduces "com-
ponents at infinity" by defining a new type of divisor
class group of Integers of a Number Field. The di-
visor class group is called an "arithmetic surface."
see also ARITHMETIC GEOMETRY
Arbelos
The term "arbelos" means SHOEMAKER'S KNIFE in
Greek, and this term is applied to the shaded AREA
in the above figure which resembles the blade of a knife
used by ancient cobblers (Gardner 1979). Archimedes
himself is believed to have been the first mathematician
to study the mathematical properties of this figure. The
position of the central notch is arbitrary and can be lo-
cated anywhere along the DIAMETER.
The arbelos satisfies a number of unexpected identities
(Gardner 1979).
1. Call the radii of the left and right SEMICIRCLES a
and 6, respectively, with a + b = R. Then the arc
length along the bottom of the arbelos is
L = 27va + 2tt6 = 2?r(a + b) = 2tvR,
so the arc lengths along the top and bottom of the
arbelos are the same.
2. Draw the PERPENDICULAR BD from the tangent of
the two Semicircles to the edge of the large Cir-
cle. Then the Area of the arbelos is the same as
the Area of the Circle with Diameter BD.
3. The CIRCLES C\ and C2 inscribed on each half of
BD on the arbelos (called ARCHIMEDES' CIRCLES)
each have DIAMETER (AB)(BC)/(AC). Further-
more, the smallest ClRCUMCIRCLE of these two cir-
cles has an area equal to that of the arbelos.
4. The line tangent to the semicircles AB and BC con-
tains the point E and F which lie on the lines AD
and CD, respectively. Furthermore, BD and EF bi-
sect each other, and the points B, D, E, and F are
CONCYCLIC.
Arbelos
Arc Length 61
5. In addition to the ARCHIMEDES' CIRCLES C± and C 2
in the arbelos figure, there is a third circle Cz called
the Bankoff Circle which is congruent to these
two.
6. Construct a chain of TANGENT CIRCLES starting
with the Circle Tangent to the two small ones
and large one. The centers of the CIRCLES lie on
an Ellipse, and the Diameter of the nth Cir-
cle C n is (l/n)th Perpendicular distance to the
base of the Semicircle. This result is most eas-
ily proven using INVERSION, but was known to Pap-
pus, who referred to it as an ancient theorem (Hood
1961, Cadwell 1966, Gardner 1979, Bankoff 1981). If
r = AB/AC, then the radius of the nth circle in the
Pappus Chain is
_ (1 — r)r
n 2[n 2 (l-r) 2 +r]"
This general result simplifies to r n = 1/(6 -f n 2 ) for
r = 2/3 (Gardner 1979). Further special cases when
AC = 1 + AB are considered by Gaba (1940).
If B divides AC in the GOLDEN RATIO 0, then the
circles in the chain satisfy a number of other special
properties (Bankoff 1955).
see also Archimedes' Circles, Bankoff Circle,
Coxeter's Loxodromic Sequence of Tangent
Circles, Golden Ratio, Inversion, Pappus Chain,
Steiner Chain
References
Bankoff, L. "The Fibonacci Arbelos." Scripta Math. 20,
218, 1954.
Bankoff, L. "The Golden Arbelos." Scripta Math. 21, 70-76,
1955.
Bankoff, L. "Are the Twin Circles of Archimedes Really
Twins?" Math. Mag. 47, 214-218, 1974,
Bankoff, L. "How Did Pappus Do It?" In The Mathematical
Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber,
and Schmidt, pp. 112-118, 1981.
Bankoff, L. "The Marvelous Arbelos." In The Lighter Side of
Mathematics (Ed. R. K. Guy and R. E. Woodrow). Wash-
ington, DC: Math. Assoc. Amer., 1994.
Cadwell, J. H. Topics in Recreational Mathematics. Cam-
bridge, England: Cambridge University Press, 1966.
Gaba, M. G. "On a Generalization of the Arbelos." Amer.
Math. Monthly 47, 19-24, 1940.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979.
Heath, T. L. The Works of Archimedes with the Method of
Archimedes. New York: Dover, 1953.
Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134-
137, 1961.
Johnson, R. A. Modern Geometry; An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 116-117, 1929.
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 54-55, 1990.
Arborescence
A Digraph is called an arborescence if, from a given
node x known as the ROOT, there is exactly one ele-
mentary path from x to every other node y.
see also Arboricity
Arboricity
Given a GRAPH G, the arboricity is the MINIMUM num-
ber of line-disjoint acyclic SUBGRAPHS whose UNION is
G.
see also ANARBORICITY
Arc
In general, any smooth curve joining two points. In
particular, any portion (other than the entire curve) of
a Circle or Ellipse.
see also APPLE, ClRCLE-ClRCLE INTERSECTION, FlVE
Disks Problem, Flower of Life, Lemon, Lens,
Piecewise Circular Curve, Reuleaux Polygon,
Reuleaux Triangle, Salinon, Seed of Life, Tri-
angle Arcs, Venn Diagram, Yin- Yang
Arc Length
Arc length is defined as the length along a curve,
J a
\d£\.
(1)
Defining the line element ds 2 = \d£\ 2 , parameterizing
the curve in terms of a parameter t, and noting that
62 Arc Minute
Archimedes Algorithm
ds/dt is simply the magnitude of the VELOCITY with
which the end of the Radius Vector r moves gives
= / ds = I ft dt= I |r ' ( * )|dt - (2)
In Polar Coordinates,
d£ = rdr + r§d6= (^-r + rd\ dd, (3)
so
ds=\de\ = X /r*+(j£) d0
In Cartesian Coordinates,
di = x± + yy
Therefore, if the curve is written
r(x) = xx-\- f(x)y,
then
J a
* = / x/l + f' 2 {x)dx.
If the curve is instead written
r(t) = x(t)x + y(t)y t
then
J a
(4)
=J m =C^ 2+ (%) 2de - (5)
(6)
ds= ^dx 2 + dy 2 = A/(£) +ldx. (7)
(8)
(9)
(10)
'= I ^x"(t) + y*(t)dt. (11)
J a
Or, in three dimensions,
r(t) = x(t)x + y(t)y + z(t)z, (12)
(t)+y' 2 {t) + z' 2 (t)dt. (13)
see also Curvature, Geodesic, Normal Vector,
Radius of Curvature, Radius of Torsion, Speed,
Surface Area, Tangential Angle, Tangent Vec-
tor, Torsion (Differential Geometry), Veloc-
ity
Arc Minute
A unit of Angular measure equal to 60 Arc Seconds,
or 1/60 of a DEGREE. The arc minute is denoted ' (not
to be confused with the symbol for feet).
Arc Second
A unit of Angular measure equal to 1/60 of an Arc
MINUTE, or 1/3600 of a DEGREE. The arc second is de-
noted " (not to be confused with the symbol for inches).
Arccosecant
see Inverse Cosecant
Arccosine
see Inverse Cosine
Arccotangent
see Inverse Cotangent
Arch
A 4-POLYHEX.
References
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.
Archimedes Algorithm
Successive application of ARCHIMEDES' RECURRENCE
FORMULA gives the Archimedes algorithm, which can
be used to provide successive approximations to it (Pi).
The algorithm is also called the Borchardt-Pfaff Al-
gorithm. Archimedes obtained the first rigorous ap-
proximation of TV by Circumscribing and Inscribing
n = 6 • 2 fe -gons on a CIRCLE. Prom ARCHIMEDES' RE-
CURRENCE Formula, the Circumferences a and b of
the circumscribed and inscribed POLYGONS are
a(n) = 2ntan ( — )
b(n) = 2nsin ( — ) ,
(i)
(2)
where
b(n) < C = 27rr = 2tt • 1 = 2tt < a(n). (3)
For a Hexagon, n = 6 and
a = a(6) = 4\/3 (4)
feo = 6(6) = 6,
(5)
where a^ = a(6-2 k ). The first iteration of ARCHIMEDES'
Recurrence Formula then gives
2-6 -4^3 24^ nA , n /-,
ffll = 7Tvr = ^vi = 24(2 -^ ) (6)
h = yj 24(2 - V3) • 6 = 12\/2- \/3
-6(v / 6-v / 2). (7)
Archimedes 7 Axiom
Archimedes' Cattle Problem
63
Additional iterations do not have simple closed forms,
but the numerical approximations for k = 0, 1, 2, 3, 4
(corresponding to 6-, 12-, 24-, 48-, and 96-gons) are
3.00000 < TV < 3.46410
3.10583 <tt< 3.21539
3.13263 < 7T < 3.15966
3.13935 < TV < 3.14609
3.14103 < 7T < 3.14271.
(8)
(9)
(10)
(11)
(12)
By taking k = 4 (a 96-gon) and using strict inequalities
to convert irrational bounds to rational bounds at each
step, Archimedes obtained the slightly looser result
^ =3.14084... <tt < f
: 3.14285.
(13)
References
Miel, G. "Of Calculations Past and Present: The Archimed-
ean Algorithm." Amer. Math. Monthly 90, 17-35, 1983.
Phillips, G. M. "Archimedes in the Complex Plane." Amer.
Math. Monthly 91, 108-114, 1984.
Archimedes' Axiom
An Axiom actually attributed to Eudoxus (Boyer 1968)
which states that
a/6 = c/d
IFF the appropriate one of following conditions is satis-
fied for Integers m and n:
1. If ma < nb, then mc < md.
2. If ma — rid, then mc = nd.
3. If ma > nd, then mc > nd.
Archimedes' Lemma is sometimes also known as Arch-
imedes' axiom.
References
Boyer, C. B. A History of Mathematics. New York: Wiley,
p. 99, 1968.
Archimedes' Cattle Problem
Also called the Bovinum PROBLEMA. It is stated as
follows: "The sun god had a herd of cattle consisting of
bulls and cows, one part of which was white, a second
black, a third spotted, and a fourth brown. Among the
bulls, the number of white ones was one half plus one
third the number of the black greater than the brown;
the number of the black, one quarter plus one fifth the
number of the spotted greater than the brown; the num-
ber of the spotted, one sixth and one seventh the number
of the white greater than the brown. Among the cows,
the number of white ones was one third plus one quarter
of the total black cattle; the number of the black, one
quarter plus one fifth the total of the spotted cattle; the
number of spotted, one fifth plus one sixth the total of
the brown cattle; the number of the brown, one sixth
plus one seventh the total of the white cattle. What
was the composition of the herd?"
Solution consists of solving the simultaneous DlOPHAN-
tine Equations in Integers W, X, Y, Z (the number
of white, black, spotted, and brown bulls) and w y x y y, z
(the number of white, black, spotted, and brown cows),
w ^ \x + z
_9_
20 J
42 '
_7_
12 v
Y + Z
W + Z
(X + x)
x =±(Y + y)
(W + w).
_ 13
(i)
(2)
(3)
(4)
(5)
(6)
(7)
The smallest solution in INTEGERS is
W = 10,366,482
(8)
X = 7,460,514
(9)
Y = 7,358,060
(10)
Z = 4,149,387
(11)
w= 7,206,360
(12)
x = 4,893,246
(13)
y= 3,515,820
(14)
z = 5,439,213.
(15)
A more complicated version of the problem requires that
W+X be a Square Number and Y+Z a Triangular
Number. The solution to this Problem are numbers
with 206544 or 206545 digits.
References
Amthor, A. and Krumbiegel B. "Das Problema bovinum des
Archimedes." Z. Math. Phys. 25, 121-171, 1880.
Archibald, R. C. "Cattle Problem of Archimedes." Amer.
Math. Monthly 25, 411-414, 1918.
Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains. New York: Dover,
pp. 249-252, 1966.
Bell, A. H. "Solution to the Celebrated Indeterminate Equa-
tion x 2 - ny 2 = 1." Amer. Math. Monthly 1, 240, 1894.
Bell, A. H. "'Cattle Problem.' By Archimedes 251 BC."
Amer. Math. Monthly 2, 140, 1895.
Bell, A. H. "Cattle Problem of Archimedes." Math. Mag. 1,
163, 1882-1884.
Calkins, K. G. "Archimedes' Problema Bovinum." http://
www. andrews . edu/-calkins/cattle .html.
Dorrie, H. "Archimedes' Problema Bovinum" §1 in 100
Great Problems of Elementary Mathematics: Their His-
tory and Solutions. New York: Dover, pp. 3-7, 1965.
Grosjean, C. C. and de Meyer, H. E. "A New Contribution
to the Mathematical Study of the Cattle-Problem of Arch-
imedes." In Constantin Caratheodory: An International
Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ:
World Scientific, pp. 404-453, 1991.
Merriman, M. "Cattle Problem of Archimedes." Pop. Sci.
Monthly 67, 660, 1905.
Rorres, C. "The Cattle Problem." http: //www. mcs.drexel.
edu/-crorres/Archimedes/Cattle/Statement .html.
Vardi, I. "Archimedes' Cattle Problem." Amer. Math.
Monthly 105, 305-319, 1998.
64 Archimedes' Circles
Archimedes' Circles
Draw the PERPENDICULAR LINE from the intersection
of the two small SEMICIRCLES in the ARBELOS. The
two Circles Ci and Ci Tangent to this line, the large
SEMICIRCLE, and each of the two SEMICIRCLES are then
congruent and known as Archimedes' circles.
see also ARBELOS, BANKOFF CIRCLE, SEMICIRCLE
Archimedes' Constant
see Pi
Archimedes' Hat-Box Theorem
Enclose a Sphere in a Cylinder and slice Perpen-
dicularly to the Cylinder's axis. Then the Surface
Area of the of Sphere slice is equal to the Surface
Area of the Cylinder slice.
Archimedes' Lemma
Also known as the continuity axiom, this Lemma sur-
vives in the writings of Eudoxus (Boyer 1968). It states
that, given two magnitudes having a ratio, one can find
a multiple of either which will exceed the other. This
principle was the basis for the EXHAUSTION METHOD
which Archimedes invented to solve problems of Area
and Volume.
see also Continuity Axioms
References
Boyer, C. B. A History of Mathematics. New York: Wiley,
p. 100, 1968.
Archimedes' Midpoint Theorem
Let M be the Midpoint of the Arc AMB. Pick C
at random and pick D such that MD _L AC (where J_
denotes PERPENDICULAR). Then
AD = DC + BC.
see also MIDPOINT
References
Honsberger, R. More Mathematical Morsels.
DC: Math. Assoc. Amer., pp. 31-32, 1991.
Washington,
Archimedes 7 Recurrence Formula
Archimedes' Postulate
see Archimedes' Lemma
Archimedes' Problem
Cut a Sphere by a Plane in such a way that the VOL-
UMES of the Spherical Segments have a given Ratio.
see also SPHERICAL SEGMENT
Archimedes' Recurrence Formula
Let a n and b n be the Perimeters of the Circum-
scribed and Inscribed n-gon and a2 n and fen the
Perimeters of the Circumscribed and Inscribed 2n-
gon. Then
di
2a n b n
a n + b n
&2n = V d2nK •
(1)
(2)
The first follows from the fact that side lengths of the
Polygons on a Circle of Radius r = 1 are
SR
2 tan
2 sin
CD
But
a n = 2ntan ( — )
b n — 2nsin f — 1 .
2an b n 2-2ntan(i)-2nsin(^)
(3)
(4)
(5)
(6)
a n +b n 2ntan(^) +2nsin(^)
tan(S) sin (?)
An
tan(^)+sin(^)-
Using the identity
tan(|x) =
tan x sin x
tan x + sin x
then gives
2a n frn
a n + b n
— 4ntan
(-
V2n
2n)
&2n
(7)
(8)
(9)
Archimedean Solid
The second follows from
Archimedean Solid
65
\Zd2nbn = W4ntan ( — j • 2nsin (-) (10)
Using the identity
since = 2sin(|x) cos(|x)
gives
(ii)
y^X = 2n v /2tan (£) - 2 sin (£) cos ( j)
= 4n v sin2 (£) = 4nsin (£) = b2n - (12)
Successive application gives the Archimedes Algo-
rithm, which can be used to provide successive approx-
imations to Pi (it).
see also ARCHIMEDES ALGORITHM, Pi
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 186,
1965.
Archimedean Solid
The Archimedean solids are convex Polyhedra which
have a similar arrangement of nonintersecting regu-
lar plane Convex Polygons of two or more differ-
ent types about each VERTEX with all sides the same
length. The Archimedean solids are distinguished from
the Prisms, Antiprisms, and Elongated Square
GYROBICUPOLA by their symmetry group: the Arch-
imedean solids have a spherical symmetry, while the
others have "dihedral" symmetry. The Archimedean
solids are sometimes also referred to as the SEMIREG-
ular Polyhedra.
Pugh (1976, p. 25) points out the Archimedean solids
are all capable of being circumscribed by a regular Tet-
rahedron so that four of their faces lie on the faces
of that Tetrahedron. A method of constructing the
Archimedean solids using a method known as "expan-
sion" has been enumerated by Stott (Stott 1910; Ball
and Coxeter 1987, pp. 139-140).
Let the cyclic sequence S = (pi,P2, . . . ,p q ) represent the
degrees of the faces surrounding a vertex (i.e., S is a list
of the number of sides of all polygons surrounding any
vertex). Then the definition of an Archimedean solid
requires that the sequence must be the same for each
vertex to within ROTATION and REFLECTION. Walsh
(1972) demonstrates that S represents the degrees of the
faces surrounding each vertex of a semiregular convex
polyhedron or TESSELLATION of the plane IFF
1. q > 3 and every member of S is at least 3,
2. ^2? =1 ~ > \q — 1, with equality in the case of a
plane TESSELLATION, and
3. for every ODD NUMBER p £ 5, S contains a subse-
quence (6, p, 6).
Condition (1) simply says that the figure consists of two
or more polygons, each having at least three sides. Con-
dition (2) requires that the sum of interior angles at a
vertex must be equal to a full rotation for the figure to
lie in the plane, and less than a full rotation for a solid
figure to be convex.
The usual way of enumerating the semiregular polyhe-
dra is to eliminate solutions of conditions (1) and (2)
using several classes of arguments and then prove that
the solutions left are, in fact, semiregular (Kepler 1864,
pp. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940,
p. 394; Coxeter et al. 1954; Lines 1965, pp. 202-203;
Walsh 1972). The following table gives all possible reg-
ular and semiregular polyhedra and tessellations. In
the table, 'P } denotes PLATONIC SOLID, 'M' denotes a
PRISM or ANTIPRISM, 'A' denotes an Archimedean solid,
and 'T' a plane tessellation.
Fg. Solid
Schlafli
3,3)
4,4)
6,6)
8,8)
10, 10)
12, 12)
4,n)
4, 4)
6,6)
6,8)
6,10)
6,12)
8, 8)
5,5)
6,6)
6,6)
3, 3, n)
3, 3, 3)
4, 3, 4)
5, 3, 5)
6, 3, 6)
4, 4, 4)
4, 5, 4)
4, 6, 4)
4, 4, 4)
o, o, o,
3, 3, 3,
o, o, o,
o, o, o,
3, 3, 4,
3, 4, 3,
3, 3, 3,
P tetrahedron {3j3}
M triangular prism t{2,3}
A truncated tetrahedron t{3, 3}
A truncated cube t{4, 3}
A truncated dodecahedron t{5,3}
T (plane tessellation) t{6,3}
M n-gonal Prism t{2,n}
P cube {4, 3}
A truncated octahedron t{3,4}
A great rhombicuboct.
A great rhombicosidodec.
T (plane tessellation)
T (plane tessellation)
P dodecahedron
A truncated icosahedron
T (plane tessellation)
M n-gonal antiprism
P octahedron
A cuboctahedron
A icosidodecahedron
T (plane tessellation)
A small rhombicuboct.
A small rhombicosidodec.
T (plane tessellation)
T (plane tessellation)
P icosahedron
A snub cube
A snub dodecahedron
T (plane tessellation)
T (plane tessellation) —
T (plane tessellation) s 1 4 J
T (plane tessellation) {3,6}
As shown in the above table, there are exactly 13 Ar-
chimedean solids (Walsh 1972, Ball and Coxeter 1987).
66
Archimedean Solid
Archimedean Solid
They are called the CUBOCTAHEDRON, GREAT RHOMB-
ICOSIDODECAHEDRON, GREAT RHOMBICUBOCTAHE-
DRON, ICOSIDODECAHEDRON, SMALL RHOMBICOSIDO-
DECAHEDRON, SMALL RHOMBICUBOCTAHEDRON, SNUB
Cube, Snub Dodecahedron, Truncated Cube,
Truncated Dodecahedron, Truncated Icosahe-
dron (soccer ball), Truncated Octahedron, and
Truncated Tetrahedron. The Archimedean solids
satisfy
(27T- <t)V — 4tt,
where a is the sum of face- angles at a vertex and V is
the number of vertices (Steinitz and Rademacher 1934,
Ball and Coxeter 1987).
Here are the Archimedean solids shown in alphabetical
order (left to right, then continuing to the next row).
ry
\ /
^m
n A ,
Li
/^
aM
LV
The following table lists the symbol and number of faces
of each type for the Archimedean solids (Wenninger
1989, p. 9).
Solid
Schlafli
Wythoff
C&R
cuboctahedron
i 3 \
X 4 1
2 | 34
(3.4) 2
great rhombicosidodecahedron
*{*}
2 3 5 |
great rhombicuboctahedron
*{:}
234 |
icosidodecahedron
/ 3 \
1 5 J
2 | 3 5
(3-5) 2
small rhombicosidodecahedron
'it)
3 5 | 2
3.4.5.4
small rhombicuboctahedron
r l:l
3 4)2
3.4 3
snub cube
s i:i
| 2 3 4
3 4 .4
snub dodecahedron
*{*}
| 2 3 5
3 4 .5
truncated cube
t{4,3}
2 3 | 4
3.8 2
truncated dodecahedron
t{5,3}
23[5
3.10 2
truncated icosahedron
t{3,5}
2 5 | 3
5.6 2
truncated octahedron
t{3,4}
2 4 | 3
4.6 2
truncated tetrahedron
t{3,3}
23 | 3
3.6 2
Solid
V
e
h
h
A
h
h
/io
cuboctahedron
12
24
8
6
great rhombicos.
120
180
30
2G
12
great rhombicub.
48
72
12
8
6
icosidodecahedron
30
60
20
12
small rhombicos.
60
120
20
30
12
small rhombicub.
24
48
8
18
snub cube
24
60
32
6
snub dodecahedron
60
150
80
12
trunc. cube
24
36
8
6
trunc. dodec.
60
90
20
12
trunc. icosahedron
60
90
12
20
trunc. octahedron
24
36
6
8
trunc. tetrahedron
12
18
4
4
Let r be the INRADIUS, p the MIDRADIUS, and R the
ClRCUMRADIUS. The following tables give the analytic
and numerical values of r, p, and R for the Archimedean
solids with EDGES of unit length.
Solid
r
cuboctahedron
great rhombicosidodecahedron
great rhombicuboctahedron
icosidodecahedron
small rhombicosidodecahedron
small rhombicuboctahedron
snub cube
snub dodecahedron
truncated cube
truncated dodecahedron
truncated icosahedron
truncated octahedron
truncated tetrahedron
3
4
aii (105 + 6^5 )\/31 4- 12 VE
£(14 + >/2)\/l3 + 6^
±(5 + 3^5)
^(15 + 2^)^11 + 4^5
^r(6 + v / 2)V /s + 2 v / 2
*
*
£(5 + 2^)^7 + 4^
4§s (17V2 + 3</l0 ) ^37 + ISn/5
? f^(21 + Vo")V /58 + 18 v / 5
£v^2
Archimedean Solid
Archimedean Solid 67
Solid
P
"i2
cuboctahedron
great rhombicosidodecahedron
ivs.
1
1^/30 + 12^
1^31 + 12 V5
great rhombicuboctahedron
IA/12 + 6X/2
I ^13 + 6x72
icosidodecahedron
small rhombicosidodecahedron
§\/ 5 + 2 >/5
|- V / 11 + 4 ^
\y/io + ±y/z
small rhombicuboctahedron
J-/4 + 2V2
| ^5 + 2y/2
snub cube
*
*
snub dodecahedron
truncated cube
*
i(2 + v/2)
*
i ^/V + 4V5
truncated dodecahedron
^(5 + 3^)
^V /t4 + 3 °v / 5
truncated icosahedron
ia + V5)
JV/58+18X/5
truncated octahedron
3
2
Iv'lO
truncated tetrahedron
f\/2
IV22
*The complicated analytic expressions for the ClRCUM-
RADII of these solids are given in the entries for the SNUB
Cube and Snub Dodecahedron.
Solid
r
P
R
cuboctahedron
0.75
0.86603
1
great rhombicosidodecahedron
3.73665
3.76938
3.80239
great rhombicuboctahedron
2.20974
2.26303
2.31761
icosidodecahedron
1.46353
1.53884
1.61803
small rhombicosidodecahedron
2.12099
2.17625
2.23295
small rhombicuboctahedron
1.22026
1.30656
1.39897
snub cube
1.15766
1.24722
1.34371
snub dodecahedron
2.03987
2.09705
2.15583
truncated cube
1.63828
1.70711
1.77882
truncated dodecahedron
2.88526
2.92705
2.96945
truncated icosahedron
2.37713
2.42705
2.47802
truncated octahedron
1.42302
1.5
1.58114
truncated tetrahedron
0.95940
1.06066
1.17260
The Duals of the Archimedean solids, sometimes called
the Catalan Solids, are given in the following table.
Archimedean Solid
Dual
rhombicosidodecahedron
small rhombicuboctahedron
great rhombicuboctahedron
great rhombicosidodecahedron
truncated icosahedron
snub dodecahedron (laevo)
snub cube (laevo)
cuboctahedron
icosidodecahedron
truncated octahedron
truncated dodecahedron
truncated cube
truncated tetrahedron
deltoidal hexecontahedron
deltoidal icositetrahedron
disdyakis dodecahedron
disdyakis triacontahedron
pentakis dodecahedron
pentagonal hexecontahedron
(dextro)
pentagonal icositetrahedron
(dextro)
rhombic dodecahedron
rhombic triacontahedron
tctrakis hexahedron
triakis icosahedron
triakis octahedron
triakis tetrahedron
Here are the Archimedean DUALS (Holden 1971, Pearce
1978) displayed in alphabetical order (left to right, then
continuing to the next row).
Here are the Archimedean solids paired with their DU-
ALS.
The Archimedean solids and their DUALS are all
Canonical Polyiiedra.
see also Archimedean Solid Stellation, Cata-
lan Solid, Deltahedron, Johnson Solid, Kepler-
Poinsot Solid, Platonic Solid, Semiregular
Polyhedron, Uniform Polyhedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 136,
1987.
Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.).
Fundamentals of Mathematics, Vol. 2. Cambridge, MA:
MIT Press, pp. 269-286, 1974.
Catalan, E. "Memoire sur la Theorie des Polyedres." J.
I'Ecole Polytechnique (Paris) 41, 1-71, 1865.
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six
and Seven Dimensions." Proc. Cambridge Phil Soc. 24,
1-9, 1928.
Coxeter, H. S. M. "Regular and Semi- Regular Polytopes I."
Math. Z. 46, 380-407, 1940.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller,
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon-
don Ser. A 246, 401-450, 1954.
Critchlow, K. Order in Space: A Design Source Book. New
York: Viking Press, 1970.
68
Archimedean Solid Stellation
Archimedes' Spiral
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, pp. 79-86, 1997.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
p. 54, 1991.
Kepler, J. "Harmonice Mundi." Opera Omnia, Vol. 5.
Frankfurt, pp. 75-334, 1864.
Kraitchik, M. Mathematical Recreations. New York:
W. W. Norton, pp. 199-207, 1942.
Le, Ha. "Archimedean Solids." http : //daisy, uwaterloo.
ca/~hqle/archimedean.htnil.
Pearce, P. Structure in Nature is a Strategy for Design. Cam-
bridge, MA: MIT Press, pp. 34-35, 1978.
Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer-
sity of California Press, p. 25, 1976.
Rawles, B. A. "Platonic and Archimedean Solids — Faces,
Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios."
http://www.intent.com/sg/polyhedra.html.
Rorres, C. "Archimedean Solids: Pappus." http://www.mcs.
drexel.edu/-crorres/Archimedes/Solids/Pappus.html.
Steinitz, E. and Rademacher, H. Vorlesungen uber die The-
orie der Polyheder. Berlin, p. 11, 1934.
Stott, A. B. Verhandelingen der Konniklijke Akad. Weten-
schappen, Amsterdam 11, 1910.
Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods
of Semi- Regular Polyhedra." Geometriae Dedicata 1, 117-
123, 1972.
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, 1989.
Archimedean Solid Stellation
A large class of Polyhedra which includes the Do-
DECADODECAHEDRON and GREAT ICOSIDODECAHE-
DRON. No complete enumeration (even with restrictive
uniqueness conditions) has been worked out.
References
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller,
J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon-
don Ser. A 246, 401-450, 1954.
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, pp. 66-72, 1989.
Archimedean Spiral
A Spiral with Polar equation
r = a0 1/7n ,
see also Archimedes' Spiral, Daisy, Fermat's Spi-
ral, Hyperbolic Spiral, Lituus, Spiral
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993.
Lauweirer, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 59-
60, 1991.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 186 and 189, 1972.
Lee, X. "Archimedean Spiral." http://www.best.com/-xah/
Special Plane Curves _ dir / Archimedean Spiral _ dir /
archimedeanSpiral .html.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 175, 1967.
MacTutor History of Mathematics Archive. "Spiral of Arch-
imedes." http: // www - groups . dcs . st - and .ac.uk/
-history/Curves/Spiral.html.
Pappas, T. "The Spiral of Archimedes." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 149,
1989.
Archimedean Spiral Inverse Curve
The Inverse Curve of the Archimedean Spiral
1/77
aO
with Inversion Center at the origin and inversion Ra-
dius k is the Archimedean Spiral
r = ka6 l/m .
Archimedes' Spiral
An Archimedean Spiral with Polar equation
where r is the radial distance, 6 is the polar angle, and m
is a constant which determines how tightly the spiral is
"wrapped." The Curvature of an Archimedean spiral
is given by
_ n(9 1 - 1 / n (l + n + n 2 l9 2 )
K ~ a(l + n 2 2 ) 3 / 2
Various special cases are given in the following table.
Name
lituus
hyperbolic spiral
Archimedes' spiral
Fermat's spiral
m
-2
-1
1
2
This spiral was studied by Conon, and later by Archi-
medes in On Spirals about 225 BC. Archimedes was able
to work out the lengths of various tangents to the spiral.
Archimedes' spiral can be used for COMPASS and
Straightedge division of an Angle into n parts (in-
cluding Angle Trisection) and can also be used for
Circle Squaring. In addition, the curve can be used
as a cam to convert uniform circular motion into uni-
form linear motion. The cam consists of one arch of the
spiral above the cc-AxiS together with its reflection in
the z-AxiS. Rotating this with uniform angular veloc-
ity about its center will result in uniform linear motion
of the point where it crosses the y-AxiS.
Archimedes' Spiral Inverse
Area-Preserving Map 69
see also ARCHIMEDEAN SPIRAL
References
Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: Chicago University Press,
pp. 106-107, 1991.
Gray, A- Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 186-187, 1972.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, pp. 173 164, 1967.
Archimedes' Spiral Inverse
Taking the ORIGIN as the INVERSION CENTER, ARCHI-
MEDES' Spiral r = aO inverts to the Hyperbolic Spi-
ral r = a/6.
Archimedean Valuation
A Valuation for which |a;| < 1 Implies |1 + z| < C for
the constant C — 1 (independent of x). Such a VALUA-
TION does not satisfy the strong TRIANGLE INEQUALITY
\x + y\< maxO|,|y|).
Arcsecant
see Inverse Secant
Arcsine
see Inverse Sine
Calculus and, in particular, the Integral, are power-
ful tools for computing the AREA between a curve f(x)
and the x-Axis over an INTERVAL [a, 6], giving
A = f f(x)
J a
c)dx. (6)
The Area of a Polar curve with equation r = r(8) is
A= | fr 2 dO. (7)
Written in CARTESIAN COORDINATES, this becomes
*-j/(-2-'i)* (8 >
-\!
(xdy — ydx).
(9)
For the AREA of special surfaces or regions, see the en-
try for that region. The generalization of AREA to 3-D
is called Volume, and to higher Dimensions is called
Content.
see also ARC LENGTH, AREA ELEMENT, CONTENT,
Surface Area, Volume
References
Gray, A. "The Intuitive Idea of Area on a Surface." §13.2
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 259-260, 1993.
Arctangent
see Inverse Tangent
Area
The Area of a Surface is the amount of material
needed to "cover" it completely. The AREA of a Trian-
gle is given by
A A = \lh, (1)
where I is the base length and h is the height, or by
Heron's Formula
Aa = a/ s(s — a)(s — b)($ — c),
(2)
where the side lengths are a, b, and c and s the
Semiperimeter. The Area of a Rectangle is given
by
^rectangle
a6,
(3)
where the sides are length a and b. This gives the special
case of
^square = & yQ)
for the Square. The Area of a regular Polygon with
n sides and side length s is given by
-^Ti-gon — 4^^ COt I 1
(5)
Area Element
The area element for a Surface with Riemannian
Metric
ds 2 = Edu 2 + 2Fdudv + Gdv 2
dA = y^EG - F 2 du A dv,
where du A dv is the WEDGE PRODUCT.
see also Area, Line Element, Riemannian Metric,
Volume Element
References
Gray, A. "The Intuitive Idea of Area on a Surface." §13.2
in Modern Differential Geometry of Curves and Surfaces,
Boca Raton, FL; CRC Press, pp. 259-260, 1993.
Area-Preserving Map
A Map F from R n to W 1 is AREA-preserving if
m{F(A)) = m(A)
for every subregion A of M n , where m(A) is the n-
D Measure of A. A linear transformation is AREA-
preserving if its corresponding DETERMINANT is equal
to 1.
see also Conformal Map, Symplectic Map
70 Area Principle
Area Principle
The "AREA principle" states that
|i4iP| _ \A,BC\
\A 2 P\ \A 2 BC\'
This can also be written in the form
[ AiP -1 = \AiBCl
IA 2 P\ [A2BCI '
where
AB
CD
(1)
(2)
(3)
is the ratio of the lengths [A, B] and [C, D] for AB\\CD
with a PLUS or MINUS SIGN depending on if these seg-
ments have the same or opposite directions, and
ABC 1
DEFGl
(4)
is the Ratio of signed Areas of the Triangles.
Griinbaum and Shepard show that Ceva'S THEOREM,
Hoehn's Theorem, and Menelaus' Theorem are the
consequences of this result.
see also Ceva's Theorem, Hoehn's Theorem, Men-
elaus' Theorem, Self-Transversality Theorem
References
Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254-268, 1995.
Areal Coordinates
Trilinear Coordinates normalized so that
tl +*2+*3 = 1.
When so normalized, they become the AREAS of the
Triangles PAiA 2 , PAiA$, and PA 2 A 3 , where P is
the point whose coordinates have been specified.
Arf Invariant
A LINK invariant which always has the value or 1.
A Knot has Arf Invariant if the Knot is "pass
equivalent" to the UNKNOT and 1 if it is pass equiv-
alent to the Trefoil Knot. If iC+, if_, and L are
projections which are identical outside the region of the
crossing diagram, and K+ and K- are Knots while L
is a 2-component LINK with a nonintersecting crossing
Argoh's Conjecture
diagram where the two left and right strands belong to
the different LINKS, then
a{K+)=a(K-) + l{L u L 2 ),
(1)
where I is the Linking Number of L\ and L 2 - The
Arf invariant can be determined from the ALEXANDER
Polynomial or Jones Polynomial for a Knot. For
A K the Alexander Polynomial of K, the Arf invari-
ant is given by
a*(-i;
■{i
(mod 8)
5 (mod 8)
if Arf(K) =
if Arf(J0 = 1
(2)
(Jones 1985). For the Jones Polynomial W K of a
Knot K ,
Arf(K) = W K (i) (3)
(Jones 1985), where i is the Imaginary Number.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 223-231, 1994.
Jones, V. "A Polynomial Invariant for Knots via von Neu-
mann Algebras." Bull. Amer. Math. Soc. 12, 103-111,
1985.
# Weisstein, E. W. "Knots." http://www. astro. Virginia.
edu/-eww6n/math/notebooks/Knots.m.
Argand Diagram
A plot of Complex Numbers as points
z = x + iy
using the x-AxiS as the Real axis and y-AxiS as the
Imaginary axis. This is also called the Complex
Plane or Argand Plane.
Argand Plane
see Argand Diagram
Argon's Conjecture
Let B k be the fcth BERNOULLI NUMBER. Then does
nBn-i = —1 (mod n)
Iff n is Prime? For example, for n = 1, 2, . . . , nB n -i
(mod n) is 0, -1, -1, 0, -1, 0, -1, 0, -3, 0, -1, ....
There are no counterexamples less than n = 5, 600. Any
counterexample to Argon's conjecture would be a con-
tradiction to Giuga's Conjecture, and vice versa.
see also BERNOULLI NUMBER, GlUGA'S CONJECTURE
References
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-
sohn, R. "Giuga's Conjecture on Primality." Amer. Math.
Monthly 103, 40-50, 1996.
Argument Addition Relation
Aristotle's Wheel Paradox
71
Argument Addition Relation
A mathematical relationship relating f(x + y) to f(x)
and f(y).
see also ARGUMENT MULTIPLICATION RELATION,
Recurrence Relation, Reflection Relation,
Translation Relation
Argument (Complex Number)
A Complex Number z may be represented as
z = x + iy = \z\e ld ,
(i)
where \z\ is called the Modulus of z, and is called the
argument
wg(x + iy) = tern' 1 (^j. (2)
Therefore,
arg(^) = argGzlc^Me"") = oxg(e ie 'e i&v> )
= arg[e i( ^ + ^ } ] = arg(z) + arg(u/). (3)
Extending this procedure gives
arg(z n ) = narg(z).
(4)
The argument of a COMPLEX NUMBER is sometimes
called the PHASE.
see also Affix, Complex Number, de Moivre's
Identity, Euler Formula, Modulus (Complex
Number), Phase, Phasor
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 16, 1972.
Argument (Elliptic Integral)
Given an Amplitude <f> in an Elliptic Integral, the
argument u is defined by the relation
4> = am u.
see also Amplitude, Elliptic Integral
Argument (Function)
An argument of a FUNCTION /(xi,...,x n ) is one of
the n parameters on which the function's value de-
pends. For example, the SINE since is a one-argument
function, the BINOMIAL Coefficient (™) is a two-
argument function, and the Hypergeometric Func-
tion 2 Fi (a, b; c; z) is a four-argument function.
Argument Multiplication Relation
A mathematical relationship relating f(nx) to f(x) for
Integer n.
see also Argument Addition Relation, Recur-
rence Relation, Reflection Relation, Transla-
tion Relation
Argument Principle
If f(z) is MEROMORPHIC in a region R enclosed by a
curve 7, let N be the number of COMPLEX ROOTS of
f(z) in 7, and P be the number of POLES in 7, then
J(z)dz
2ttz
N
J_ [ f'(z)d.
2iriL f(z)
Defining w = f(z) and a = / (7) gives
1 f dw
2-xi I w
N ■
see also VARIATION OF ARGUMENT
References
Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Ar-
gument Principle for Harmonic Functions." Math. Mag.
103, 411-415, 1996.
Argument Variation
see Variation of Argument
Aristotle's Wheel Paradox
<a=®
A PARADOX mentioned in the Greek work Mechanica,
dubiously attributed to Aristotle. Consider the above
diagram depicting a wheel consisting of two concen-
tric Circles of different Diameters (a wheel within
a wheel). There is a 1:1 correspondence of points on
the large CIRCLE with points on the small CIRCLE, so
the wheel should travel the same distance regardless of
whether it is rolled from left to right on the top straight
line or on the bottom one. This seems to imply that
the two Circumferences of different sized Circles
are equal, which is impossible.
The fallacy lies in the assumption that a 1:1 correspon-
dence of points means that two curves must have the
same length. In fact, the CARDINALITIES of points in
a Line Segment of any length (or even an Infinite
Line, a Plane, a 3-D Space, or an infinite dimensional
Euclidean Space) are all the same: Hi (Aleph-1), so
the points of any of these can be put in a One-TO-One
correspondence with those of any other.
see also ZENO'S PARADOXES
References
Ballew, D. "The Wheel of Aristotle." Math. Teacher 65,
507-509, 1972.
Costabel, P. "The Wheel of Aristotle and French Considera-
tion of Galileo's Arguments." Math. Teacher 61, 527-534,
1968.
Drabkin, I. "Aristotle's Wheel: Notes on the History of the
Paradox." Osiris 9, 162-198, 1950.
Gardner, M. Wheels, Life, and other Mathematical Amuse-
ments. New York: W. H. Freeman, pp. 2-4, 1983.
Pappas, T. "The Wheel of Paradox Aristotle." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
p. 202, 1989.
vos Savant, M. The World's Most Famous Math Problem.
New York: St. Martin's Press, pp. 48-50, 1993.
72
Arithmetic
Arithmetic
The branch of mathematics dealing with Integers
or, more generally, numerical computation. Arithmeti-
cal operations include Addition, Congruence cal-
culation, Division, Factorization, Multiplication,
Power computation, Root extraction, and SUBTRAC-
TION.
The Fundamental Theorem of Arithmetic, also
called the Unique Factorization Theorem, states
that any Positive Integer can be represented in ex-
actly one way as a PRODUCT of PRIMES.
The Lowenheimer-Skolem Theorem, which is a fun-
damental result in Model Theory, establishes the ex-
istence of "nonstandard" models of arithmetic.
see also Algebra, Calculus, Fundamental The-
orem of Arithmetic, Group Theory, Higher
Arithmetic, Linear Algebra, Lowenheimer-
Skolem Theorem, Model Theory, Number The-
ory, Trigonometry
References
Karpinski, L. C. The History of Arithmetic. Chicago, IL:
Rand, McNally, & Co., 1925.
Maxfield, J. E. and Maxfield, M. W. Abstract Algebra and
Solution by Radicals. Philadelphia, PA: Saunders, 1992.
Thompson, J. E. Arithmetic for the Practical Man. New
York: Van Nostrand Reinhold, 1973.
Arithmetic-Geometric Mean
The arithmetic-geometric mean (AGM) M(a, b) of two
numbers a and b is defined by starting with clq = a and
bo = &, then iterating
CLn + l = 2 ( a ™ + kn)
b n + l = yCLnbn
(1)
(2)
until a n = b n . a n and b n converge towards each other
since
a n +i - b n +i = \{a n -\- b n ) - ydnb n
a n — 2\/a n b n + b n
(3)
But "s/Sn < V^"» SO
2b n < 2^a n b n . (4)
Now, add a n — b n — 2y/a n b n to each side
a n + b n — 2\/a n b n < a n — b n > (5)
CLn + l - b n + l < 2^ an ~ bn)-
(6)
The AGM is very useful in computing the values of
complete Elliptic Integrals and can also be used
for finding the INVERSE TANGENT. The special value
l/M(l,\/2) is called Gauss's Constant.
Arithmetic- Geometric Mean
The AGM has the properties
AM(a,6) = M(Aa,A6) (7)
M(a,6) = M(£(a + 6),>/S) (8)
M(l, V 1 - x 2 ) = M (! + s, 1 ~ x) (9)
The Legendre form is given by
Af(l,x) = JJi(l + fc„), (11)
where ko = x and
Solutions to the differential equation
(x 3 -z)^| + (3x 2 - l)^-+xy = (13)
ax* ax
are given by [M(l + x, 1 - x)] 1 and[M(l,x)] 1 .
A generalization of the Arithmetic-Geometric
Mean is
f°° x p ~ 2 dx
(14)
which is related to solutions of the differential equation
x(l-x p )Y" + [l-fr+l)x p ]Y'-(p-l)x p - 1 Y = 0. (15)
When p = 2 or p = 3, there is a modular transformation
for the solutions of (15) that are bounded as x — > 0. Let-
ting J p (x) be one of these solutions, the transformation
takes the form
J p (\) = M J p (z), (16)
where
A
1-u
and
1 + (p - l)u
1 + (p - l)u
x p + u p = 1.
(17)
(18)
(19)
The case p = 2 gives the Arithmetic- Geometric
Mean, and p = 3 gives a cubic relative discussed by
Borwein and Borwein (1990, 1991) and Borwein (1996)
in which, for a, b > and I(a, b) defined by
/(a, 6)
Jo V& +
tdt
*3)(6 3 +f 3 ) 2 ] 1 / 3 '
(20)
Arithmetic Geometry
Iia , b) = l(^,[^+ab + b 2 )]). (21)
For iteration with ao = a and bo = 6 and
a n +i
a n + 26 n
fcn + l = — (dn + fln&Ti + b n )
lim a n = lim b n
Hhi)
I(a,b)-
(22)
(23)
(24)
Modular transformations are known when p = 4 and
p = 6, but they do not give identities for p = 6 (Borwein
1996).
see also Arithmetic-Harmonic Mean
References
Abramowitz, M. and Stegun, C. A. (Eds.), "The Process
of the Arithmetic-Geometric Mean." §17.6 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 571 ad 598-599, 1972.
Borwein, J. M. Problem 10281. "A Cubic Relative of the
AGM." Amer, Math. Monthly 103, 181-183, 1996.
Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic It-
eration." In Computational Method & Function Theory:
Proc. Conference Held in Valparaiso, Chile, March 13-
18, i9SP0387527680 (Ed. A. Dold, B. Eckmann, F. Tak-
ens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C, and
R, S. Varga). New York: Springer- Vcrlag, 1990.
Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of
Jacobi's Identity and the AGM." Trans. Amer. Math. Soc.
323, 691-701, 1991.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 906-907, 1992.
Arithmetic Geometry
A vaguely defined branch of mathematics dealing with
Varieties, the Mordell Conjecture, Arakelov
Theory, and Elliptic Curves.
References
Cornell, G. and Silverman, J. H. (Eds.). Arithmetic Geome-
try. New York: Springer- Verlag, 1986.
Lorenzini, D. An Invitation to Arithmetic Geometry. Provi-
dence, RI: Amer. Math. Soc, 1996.
Arithmetic-Harmonic Mean
Let
a n+ x = \{a n + b n )
b n
+i
a n + b n
Then
A(ao,bo) = lim a n = lim b n
n — ► oo n— »-oo
which is just the GEOMETRIC MEAN.
(1)
(2)
'aobo, (3)
Arithmetic Mean 73
Arithmetic-Logarithmic-Geometric Mean
Inequality
a + b b — a i—r
—— > — — : — > Vab.
2 In o — In a
see also Napier's Inequality
References
Nelson, R. B. "Proof without Words: The Arithmetic-
Logarithmic-Geometric Mean Inequality." Math. Mag.
68, 305, 1995.
Arithmetic Mean
For a Continuous Distribution function, the arith-
metic mean of the population, denoted /*, x, {x) t or
A(x) t is given by
-/.
H=(f(x))= / P(x)f(x)dx,
(1)
where (x) is the EXPECTATION VALUE. For a DISCRETE
Distribution,
„ = </(*)> ss E ^°/ (a " )/( ; n) = 5><*.)/<*.).
l^n = Q F V Xn ) n=0
(2)
The population mean satisfies
{f(x)+g(x)) = {f(x)) + (g(x)} (3)
<c/(x))=c </(*)>, (4)
and
{f(x)g(y)) = </(*)> (g(y))
(5)
if x and y are INDEPENDENT STATISTICS. The "sample
mean," which is the mean estimated from a statistical
sample, is an UNBIASED ESTIMATOR for the population
mean.
For small samples, the mean is more efficient than the
Median and approximately tt/2 less (Kenney and Keep-
ing 1962, p. 211). A general expression which often holds
approximately is
mean — mode « 3(mean — median). (6)
Given a set of samples {a;*}, the arithmetic mean is
N
A(x) =x = hee{x) = ^^2 Xi ' W
Hoehn and Niven (1985) show that *
A(a± +c,a 2 +c, . ..,a n +c) = c + j4(ai,a2,...,a n ) (8)
for any POSITIVE constant c. The arithmetic mean sat-
isfies
(9)
74 Arithmetic Mean
Arithmetic Progression
where G is the Geometric Mean and H is the Har-
monic Mean (Hardy et al. 1952; Mitrinovic 1970; Beck-
enbach and Bellman 1983; Bullen et ah 1988; Mitrinovic
et al. 1993; Alzer 1996). This can be shown as follows.
For a, b > 0,
P--^Y>
1 2 1 rt
1 1^2
- + r > -7=
a ~ b
H>G,
(10)
(11)
(12)
(13)
(14)
with equality Iff b = a. To show the second part of the
inequality,
{yfa-Vbf = a-2\/a6 + &> (15)
<> + b
> Vab
2
A> H<
(16)
(17)
with equality Iff a = b. Combining (14) and (17) then
gives (9).
Given n independent random GAUSSIAN DISTRIBUTED
variates #», each with population mean fii = \i and
Variance <n 2 = a 2 ,
= ^E^=^)=/i, (19)
Z = l
so the sample mean is an Unbiased Estimator of
population mean. However, the distribution of x de-
pends on the sample size. For large samples, x is ap-
proximately Normal. For small samples, Student's
^-Distribution should be used.
The Variance of the population mean is independent
of the distribution.
var(z) = var I - > Xi\ = —— var > x {
n N 2
From /c-Statistics for a GAUSSIAN DISTRIBUTION, the
Unbiased Estimator for the Variance is given by
N
N - 1
where
var (a:) =
JV-1
The Square Root of this,
s
is called the Standard Error.
var(x) = (% 2 ) ~~ (^) 2 >
(21)
(22)
(23)
(24)
(25)
(x 2 )=var(x) + (a-) 2 = ^+/A (26)
see also Arithmetic-Geometric Mean, Arith-
metic-Harmonic Mean, Carleman's Inequal-
ity, Cumulant, Generalized Mean, Geomet-
ric Mean, Harmonic Mean, Harmonic-Geometric
Mean, Kurtosis, Mean, Mean Deviation, Median
(Statistics), Mode, Moment, Quadratic Mean,
Root-Mean-Square, Sample Variance, Skewness,
Standard Deviation, Trimean, Variance
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 10, 1972.
Alzer, H. "A Proof of the Arithmetic Mean-Geometric Mean
Inequality." Amer. Math. Monthly 103, 585, 1996.
Beckenbach, E. F. and Bellman, R. Inequalities. New York:
Springer- Verlag, 1983.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 471, 1987.
Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means &
Their Inequalities. Dordrecht, Netherlands: Reidel, 1988.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities.
Cambridge, England: Cambridge University Press, 1952.
Hoehn, L. and Niven, I. "Averages on the Move." Math.
Mag. 58, 151-156, 1985.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical
and New Inequalities in Analysis. Dordrecht, Netherlands:
Kluwer, 1993.
Vasic, P. M. and Mitrinovic, D. S. Analytic Inequalities. New
York: Springer- Verlag, 1970.
Arithmetic Progression
see Arithmetic Series
(20)
Arithmetic Sequence
Arnold's Cat Map 75
Arithmetic Sequence
A Sequence of n numbers {do 4- kd} 7 ^ such that the
differences between successive terms is a constant d.
see also ARITHMETIC SERIES, SEQUENCE
Arithmetic Series
An arithmetic series is the Sum of a SEQUENCE {a^},
k = 1, 2, ..., in which each term is computed from
the previous one by adding (or subtracting) a constant.
Therefore, for k > 1,
a>k = a,k-i + d = afc-2 + 2d •
:ai+d(fc-l). (1)
The sum of the sequence of the first n terms is then
given by
n n
S n = ]Ta fc =J^[ai + (* - l)d]
k=l k=l
n n
= nai + d^ik — 1) = noi + d /(& - 1)
fc = l k = 2
n-l
= nai -\- dj k (2)
Using the SUM identity
]T = §n(n+l)
(3)
then gives
S n = nai + \d(n - 1) = \ n[2ai + d(n - 1)]. (4)
Note, however, that
ai + a n = ai + [a\ + d(n — 1)] — 2ai + d(n — 1), (5)
5 n = \n{a\ +a n ),
(6)
or n times the AVERAGE of the first and last terms!
This is the trick Gauss used as a schoolboy to solve
the problem of summing the INTEGERS from 1 to 100
given as busy-work by his teacher. While his classmates
toiled away doing the ADDITION longhand, Gauss wrote
a single number, the correct answer
|(100)(1 + 100) = 50 ■ 101 = 5050
(7)
on his slate. When the answers were examined, Gauss's
proved to be the only correct one.
see also Arithmetic Sequence, Geometric Series,
Harmonic Series, Prime Arithmetic Progression
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 10, 1972.
Beyer, W. H. (Ed.), CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.
Courant, R. and Robbins, H. "The Arithmetical Progres-
sion." §1.2.2 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 12-13, 1996.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 164, 1989.
Armstrong Number
The n-digit numbers equal to sum of nth powers of their
digits (a finite sequence), also called PLUS PERFECT
NUMBERS. They first few are given by 1, 2, 3, 4, 5,
6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748,
... (Sloane's A005188).
see also NARCISSISTIC NUMBER
References
Sloane, N. J. A. Sequence A005188/M0488 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Arnold's Cat Map
The best known example of an ANOSOV DlFFEOMOR-
PHISM. It is given by the TRANSFORMATION
(i)
where x n +i and y n +i are computed mod 1. The Arnold
cat mapping is non-Hamiltonian, nonanalytic, and mix-
ing. However, it is Area- Preserving since the Deter-
minant is 1. The Lyapunov Characteristic Expo-
nents are given by
Xn+l
=
"l l"
1 2
x n
y n _
l-a 1
1 2-<T
3(7 + 1 = 0,
(2)
ct± = |(3±v / 5). (3)
The Eigenvectors are found by plugging <r± into the
Matrix Equation
1
1
2 - cr±
(4)
For <r+, the solution is
y=\{l + ^)x = 4>x, (5)
where <j> is the GOLDEN RATIO, so the unstable (normal-
ized) Eigenvector is
^+ = ^\ / 50-10v / 5
1(1 + V5)
Similarly, for <j- , the solution is
y = -±(V5-l)x~(/>- 1 x y
so the stable (normalized) Eigenvector is
£_ = ^\/50 + 10v / 5
see also Anosov Map
1(1 -v/5)
(6)
(7)
(8)
76
Arnold Diffusion
Array
Arnold Diffusion
The nonconservation of ADIABATIC INVARIANTS which
arises in systems with three or more DEGREES OF FREE-
DOM.
Arnold Tongue
Consider the Circle Map. If K is Nonzero, then
the motion is periodic in some FINITE region surround-
ing each rational Q. This execution of periodic motion
in response to an irrational forcing is known as MODE
LOCKING. If a plot is made of K versus Q with the re-
gions of periodic MODE-LOCKED parameter space plot-
ted around rational ft values (the WINDING Numbers),
then the regions are seen to widen upward from at
K = to some FINITE width at K = 1. The region
surrounding each RATIONAL NUMBER is known as an
Arnold Tongue.
At K — 0, the Arnold tongues are an isolated set of
MEASURE zero. At K = 1, they form a general CAN-
TOR Set of dimension d w 0.8700. In general, an Arnold
tongue is defined as a resonance zone emanating out
from RATIONAL NUMBERS in a two-dimensional param-
eter space of variables.
see also Circle Map
Aronhold Process
The process used to generate an expression for a covari-
ant in the first degree of any one of the equivalent sets
of Coefficients for a curve.
see also Clebsch-Aronhold Notation, Joachims-
thal's Equation
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 74, 1959.
Aronson's Sequence
The sequence whose definition is: "t is the first, fourth,
eleventh, . . . letter of this sentence." The first few val-
ues are 1, 4, 11, 16, 24, 29, 33, 35, 39, ... (Sloane's
A005224).
References
Hofstadter, D. R. Metamagical Themas: Questing of Mind
and Pattern. New York: BasicBooks, p. 44, 1985.
Sloane, N. J. A. Sequence A005224/M3406 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Arrangement
In general, an arrangement of objects is simply a group-
ing of them. The number of "arrangements" of n items
is given either by a COMBINATION (order is ignored) or
Permutation (order is significant).
The division of Space into cells by a collection of Hy-
PERPLANES is also called an arrangement.
see also COMBINATION, CUTTING, HYPERPLANE, OR-
DERING, Permutation
Arrangement Number
see Permutation
Array
An array is a "list of lists" with the length of each
level of list the same. The size (sometimes called the
"shape") of a d-dimensional array is then indicated as
m x n x • • • x p. The most common type of array en-
d
countered is the 2-D m x n rectangular array having m
columns and n rows. If m = n, a square array results.
Sometimes, the order of the elements in an array is sig-
nificant (as in a MATRIX), whereas at other times, arrays
which are equivalent modulo reflections (and rotations,
in the case of a square array) are considered identical
(as in a MAGIC SQUARE or PRIME ARRAY).
In order to exhaustively list the number of distinct ar-
rays of a given shape with each element being one of k
possible choices, the naive algorithm of running through
each case and checking to see whether it's equivalent to
an earlier one is already just about as efficient as can
be. The running time must be at least the number of
answers, and this is so close to k mn '" p that the difference
isn't significant.
However, finding the number of possible arrays of a given
shape is much easier, and an exact formula can be ob-
tained using the POLYA ENUMERATION THEOREM. For
the simple case of an m x n array, even this proves un-
necessary since there are only a few possible symmetry
types, allowing the possibilities to be counted explicitly.
For example, consider the case of m and n EVEN and
distinct, so only reflections need be included. To take a
specific case, let m = 6 and n = 4 so the array looks like
a
b
c
1
d
e
f
9
h
i
1
+
1
3
k
I
m
n
V
Q
r
s
t
u
1
V
w
X,
where each a, 6, . . . , x can take a value from 1 to k. The
total number of possible arrangements is k 24 (k mn in
general). The number of arrangements which are equiv-
alent to their left-right mirror images is k 1 (in general,
k mn/2 ), as is the number equal to their up-down mirror
images, or their rotations through 180°. There are also
k Q arrangements (in general, fc mn/4 ) with full symmetry.
In general, it is therefore true that
jL7TiTl/4
j^mn/2 _ fcmn/4
femn/2 _ pn/4
femn/2 _ femn/4
with full symmetry
with only left-right reflection
with only up-down reflection
with only 180° rotation,
so there are
3k
ran/2 , rw mn./4
Arrow Notation
Artin Braid Group 77
arrangements with no symmetry. Now dividing by the
number of images of each type, the result, for m -£ n
with m, n EVEN, is
N(m,n,k) = |A; mn + (|)(3)(fc mn/2 - A; mn/4 )
mn i
+ 1*"
4. \(k mn -3k mn/2 + 2k mn/4 )
4\
lfcrnn _,_ 3 j.mn/2 _,_ lj.mn/4
+ ifc"
The number is therefore of order C>(fc mn /4), with "cor-
rection" terms of much smaller order.
see also Antimagic Square, Euler Square,
Kirkman's Schoolgirl Problem, Latin Rect-
angle, Latin Square, Magic Square, Matrix,
Mrs. Perkins' Quilt, Multiplication Table, Or-
thogonal Array, Perfect Square, Prime Array,
Quotient-Difference Table, Room Square, Sto-
larsky Array, Truth Table, Wythoff Array
Arrow Notation
A Notation invented by Knuth (1976) to represent
Large Numbers in which evaluation proceeds from the
right (Conway and Guy 1996, p. 60).
m t n
m * m- - -m
n
m ttt n m tt m tt ■ ' ' tt m
n
For example,
m t n ~ m n
m"[ J \-2 = m J [m~m^m = rn 71
2
m tt 3 = m t rn t m = m t ( m t rn)
v v /
3
= m|m m = m mm
m ttt 2 = mtt^ = ^tt^ = mm
m ttt 3 = m tt rn tt ™ = m tt ™ m
v v ' ' ^~
3 m
= m t • • * t rn — m m
(1)
(2)
(3)
(4)
(5)
m tt n is sometimes called a Power Tower. The
values nt • * • t n are called ACKERMANN NUMBERS.
see also Ackermann Number, Chained Arrow No-
tation, Down Arrow Notation, Large Number,
Power Tower, Steinhaus-Moser Notation
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New-
York: Springer- Verlag, pp. 59-62, 1996.
Guy, R. K. and Selfridge, J. L. "The Nesting and Roost-
ing Habits of the Laddered Parenthesis." Amer. Math.
Monthly 80, 868-876, 1973.
Knuth, D. E. "Mathematics and Computer Science: Coping
with Finiteness. Advances in Our Ability to Compute are
Bringing Us Substantially Closer to Ultimate Limitations."
Science 194, 1235-1242, 1976.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 11 and 226-229, 1991.
Arrow's Paradox
Perfect democratic voting is, not just in practice but in
principle, impossible.
References
Gardner, M. Time Travel and Other Mathematical Bewilder-
ments. New York: W. H. Freeman, p. 56, 1988.
Arrowhead Curve
see Sierpinski Arrowhead Curve
Art Gallery Theorem
Also called Chvatal's Art Gallery Theorem. If
the walls of an art gallery are made up of n straight
Lines Segments, then the entire gallery can always be
supervised by [n/3\ watchmen placed in corners, where
[x\ is the Floor Function. This theorem was proved
by V. Chvatal in 1973. It is conjectured that an art
gallery with n walls and h HOLES requires [(n + h)/3j
watchmen.
see also Illumination Problem
References
Honsberger, R. "Chvatal's Art Gallery Theorem." Ch. 11
in Mathematical Gems II. Washington, DC: Math. Assoc.
Amer., pp. 104-110, 1976.
O'Ronrke, J. Art Gallery Theorems and Algorithms. New-
York: Oxford University Press, 1987.
Stewart, I. "How Many Guards in the Gallery?" Sci. Amer.
270, 118-120, May 1994.
Tucker, A. "The Art Gallery Problem." Math Horizons,
pp. 24-26, Spring 1994.
Wagon, S. "The Art Gallery Theorem." §10.3 in Mathema-
tica in Action. New York: W. H. Freeman, pp. 333-345,
1991.
Articulation Vertex
A VERTEX whose removal will disconnect a GRAPH, also
called a Cut- Vertex.
see also Bridge (Graph)
References
Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introduc-
tory Graph Theory. New York: Dover, pp. 45—49, 1985.
Artin Braid Group
see Braid Group
78 Artin's Conjecture
Artistic Series
Artin's Conjecture
There are at least two statements which go by the name
of Artin's conjecture. The first is the RlEMANN HY-
POTHESIS. The second states that every INTEGER not
equal to —1 or a SQUARE NUMBER is a primitive root
modulo p for infinitely many p and proposes a density
for the set of such p which are always rational multi-
ples of a constant known as ARTIN'S CONSTANT. There
is an analogous theorem for functions instead of num-
bers which has been proved by Billharz (Shanks 1993,
p. 147).
see also ARTIN'S CONSTANT, RlEMANN HYPOTHESIS
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 31, 80-83, and 147, 1993.
Artin's Constant
If n / -1 and n is not a PERFECT SQUARE, then Artin
conjectured that the SET S(n) of all PRIMES for which n
is a PRIMITIVE ROOT is infinite. Under the assumption
of the Extended Riemann Hypothesis, Artin's con-
jecture was solved in 1967 by C. Hooley. If, in addition,
n is not an rth POWER for any r > 1, then Artin con-
jectured that the density of S(n) relative to the Primes
is CArtin (independent of the choice of n) , where
CAn
n
1-
1
<?(<?-!)
= 0.3739558136..
and the PRODUCT is over Primes. The significance of
this constant is more easily seen by describing it as the
fraction of PRIMES p for which 1/p has a maximal DEC-
IMAL Expansion (Conway and Guy 1996).
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer-Verlag, p. 169, 1996.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/artin/artin.html.
Hooley, C. "On Artin's Conjecture." J. reine angew. Math.
225, 209-220, 1967.
Ireland, K. and Rosen, M. A Classical Introduction to Mod-
ern Number Theory, 2nd ed. New York: Springer-Verlag,
1990.
Ribenboim, P. The Book of Prime Number Records. New
York: Springer-Verlag, 1989.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 80-83, 1993.
Wrench, J. W. "Evaluation of Artin's Constant and the Twin
Prime Constant." Math. Comput. 15, 396-398, 1961.
Artin L- Function
An Artin L-function over the Rationals Q encodes in
a Generating Function information about how an
irreducible monic POLYNOMIAL over Z factors when re-
duced modulo each PRIME. For the POLYNOMIAL x 2 + l,
the Artin L-function is
L(s,Q(i)/Q,sgn):
n itt^
where ( — 1/p) is a Legendre Symbol, which is equiv-
alent to the Euler L-FUNCTION. The definition over
arbitrary POLYNOMIALS generalizes the above expres-
sion.
see also Langlands Reciprocity
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996.
Artin Reciprocity
see Artin's Reciprocity Theorem
Artin's Reciprocity Theorem
A general RECIPROCITY Theorem for all orders. If R
is a NUMBER FIELD and R f a finite integral extension,
then there is a SURJECTION from the group of fractional
IDEALS prime to the discriminant, given by the Artin
symbol. For some cycle c, the kernel of this SURJECTION
contains each Principal fractional Ideal generated by
an element congruent to 1 mod c.
see also LANGLANDS PROGRAM
Artinian Group
A GROUP in which any decreasing CHAIN of distinct
Subgroups terminates after a Finite number.
Artinian Ring
A noncommutative Semisimple RING satisfying the
"descending chain condition."
see also GORENSTEIN RING, SEMISIMPLE RING
References
Artin, E. "Zur Theorie der hyperkomplexer Zahlen." Hamb.
Abh. 5, 251-260, 1928.
Artin, E. "Zur Arithmetik hyperkomplexer Zahlen." Hamb.
Abh. 5, 261-289, 1928.
Artistic Series
A Series is called artistic if every three consecutive
terms have a common three-way ratio
-P[ai,ai+i,a; + 2]
(ai + aj+i + ai+2)ai+i
aiOi+2
A Series is also artistic Iff its BIAS is a constant. A
Geometric Series with Ratio r > is an artistic
series with
P=i + l + r>3.
r
see also BIAS (SERIES), GEOMETRIC SERIES, MELODIC
Series
References
Duffin, R. J. "On Seeing Progressions of Constant Cross Ra-
tio." Amer. Math. Monthly 100, 38-47, 1993.
p odd prime
(?)*"
ASA Theorem
ASA Theorem
Specifying two adjacent ANGLES A and B and the side
between them c uniquely determines a Triangle with
Area
K =
2(coti4 + cot£)"
The angle C is given in terms of A and B by
C = n-A-B,
(1)
(2)
and the sides a and b can be determined by using the
Law of Sines
to obtain
sin A sin B sin C
sin A
sin(7r — A — B)
sinB
sin(7r — A — B)
(3)
(4)
(5)
see also AAA Theorem, AAS Theorem, ASS Theo-
rem, SAS Theorem, SSS Theorem, Triangle
Aschbacher's Component Theorem
Suppose that E(G) (the commuting product of all com-
ponents of G) is SIMPLE and G contains a SEMISIM-
ple Involution. Then there is some Semisimple
Involution x such that C G (x) has a Normal Sub-
group K which is either QUASISIMPLE or ISOMORPHIC
to + (4,q)' and such that Q — C G {K) is Tightly Em-
bedded.
see also Involution (Group), Isomorphic Groups,
Normal Subgroup, Quasisimple Group, Simple
Group, Tightly Embedded
ASS Theorem
c c c
Specifying two adjacent side lengths a and b of a TRIAN-
GLE (taking a > b) and one ACUTE ANGLE A opposite
a does not, in general, uniquely determine a triangle.
If sin A < a/cy there are two possible TRIANGLES satis-
fying the given conditions. If sin A = a/c, there is one
possible Triangle. If sin A > a/c, there are no possible
TRIANGLES. Remember: don't try to prove congruence
with the ASS theorem or you will make make an ASS
out of yourself.
see also AAA Theorem, AAS Theorem, SAS Theo-
rem, SSS Theorem, Triangle
Associative Magic Square 79
Associative
In simple terms, let x, y, and z be members of an Al-
gebra. Then the Algebra is said to be associative
if
x - (y - z) = (x - y) • z, (1)
where • denotes MULTIPLICATION. More formally, let A
denote an IR-algebra, so that A is a VECTOR SPACE over
Rand
Ax A-+ A (2)
(x,y) \->x-y.
(3)
Then A is said to be m-associative if there exists an m-D
Subspace S of A such that
(y-x)-z = y-(x-z)
(4)
for all y,z € A and x € S. Here, VECTOR MULTIPLI-
CATION x • y is assumed to be Bilinear. An n-D n-
associative ALGEBRA is simply said to be "associative."
see also COMMUTATIVE, DISTRIBUTIVE
References
Finch, S. "Zero Structures in Real Algebras." http://www.
mathsoft.com/asolve/zerodiv/zerodiv.html.
Associative Magic Square
1
15
24
8
17
23
7
16
5
14
20
4
13
22
6
12
21
10
19
3
9
18
2
11
25
An n x n Magic Square for which every pair of num-
bers symmetrically opposite the center sum to n 2 + 1.
The Lo Shu is associative but not PANMAGIC. Order
four squares can be PANMAGIC or associative, but not
both. Order five squares are the smallest which can be
both associative and PANMAGIC, and 16 distinct asso-
ciative PANMAGIC Squares exist, one of which is illus-
trated above (Gardner 1988).
see also Magic Square, Panmagic Square
References
Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H, Freeman, 1988.
80 Astroid
Astroid
A 4-cusped HYPOCYCLOID which is sometimes also
called a Tetracuspid, Cubocycloid, or Paracycle.
The parametric equations of the astroid can be obtained
by plugging in n = a/b = 4 or 4/3 into the equations for
a general HYPOCYCLOID, giving
x = 3bcos(j> + 6cos(30) = 46 cos 3 <j> — a cos 3 <j) (1)
y — 36 sin ^ — bs'm(3<f>) = 46 sin <j) = asin <j>. (2)
In Cartesian Coordinates,
2/3 . 2/3 2/3
(3)
In Pedal Coordinates with the Pedal Point at the
center, the equation is
2 , o 2 2
r + op — a .
(4)
v J
I J
r ^
n
The Arc Length, Curvature, and Tangential An-
gle are
s(t)
/ |sin(2*')|d*'
Jo
f sin 2 t (5)
K(t) = -|csc(2t)
4>(t) = -t.
(6)
(7)
As usual, care must be taken in the evaluation of s(t) for
t > it/ 2. Since (5) comes from an integral involving the
Absolute Value of a function, it must be monotonic
increasing. Each QUADRANT can be treated correctly
by defining
'»=[fj+l. (8)
where [x] is the FLOOR FUNCTION, giving the formula
S (t) = (-l) 1+ l" < mod 2 » | sin 2 t + 3 [|nj . (9)
The overall Arc Length of the astroid can be com-
puted from the general HYPOCYCLOID formula
&a(n - 1)
(10)
with n = 4,
54 = 6a.
Astroid
(ll)
The Area is given by
An = ("-D("-2) ro .
with n = 4,
I- 2
(12)
(13)
The Evolute of an Ellipse is a stretched Hypocy-
CLOID. The gradient of the TANGENT T from the point
with parameter p is — tan p. The equation of this TAN-
GENT T is
xsinp + ycosp = |asin(2p)
(14)
(MacTutor Archive). Let T cut the z-Axis and the y-
Axis at X and Y, respectively. Then the length XY is
a constant and is equal to a.
t
L
The astroid can also be formed as the ENVELOPE pro-
duced when a Line Segment is moved with each end
on one of a pair of PERPENDICULAR axes (e.g., it is the
curve enveloped by a ladder sliding against a wall or a
garage door with the top corner moving along a verti-
cal track; left figure above). The astroid is therefore
a GLISSETTE. To see this, note that for a ladder of
length L, the points of contact with the wall and floor
are (xo,0) and (0, y/L 2 — xq 1 ), respectively. The equa-
tion of the Line made by the ladder with its foot at
(xojO) is therefore
VL 2 - xq 2 , .
y — = (x - xo)
-xq
(15)
which can be written
U{x,y,xo) = y + — (x-x ). (16)
Xo
The equation of the Envelope is given by the simulta-
neous solution of
U(x,y,x ) = y+ V L xn X ° (x-xo) =
ax o *oV^ 2 -*o 2
-0,
which is
Xq
I?
(L 2 - xq 2 ) 3 / 2
L 2
(17)
(18)
(19)
Astroid
Noting that
~ 2
2/3 _ ^0
~ L 4 /3
(20)
r 2 2
2/3 _ -k — x
y ~~ TAlZ
(21)
Astroid Involute
81
allows this to be written implicitly as
x 2/3 +y 2/3 = L 2/3 ,
the equation of the astroid, as promised.
A
/~\
slotted
track
^
(22)
The related problem obtained by having the "garage
door" of length L with an "extension" of length AL
move up and down a slotted track also gives a surprising
answer. In this case, the position of the "extended" end
for the foot of the door at horizontal position xo and
Angle 6 is given by
x = — ALcosO
y = \/L 2 - xo 2 + ALsin0.
(23)
(24)
The astroid is also the Envelope of the family of El-
lipses
y
(1 - c)'<
-1 = 0,
(30)
illustrated above.
see also Deltoid, Ellipse Envelope, Lame Curve,
Nephroid, Ranunculoid
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 172-175, 1972.
Lee, X. "Astroid." http://www.best.com/-xah/Special
PlaneCurves_dir/Astroid_dir/astroid.html.
Lockwood, E. H. "The Astroid." Ch. 6 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 52-
61, 1967.
MacTutor History of Mathematics Archive. "Astroid."
http: //www-groups .dcs .st-and.ac.uk/-history/Curves
/Astroid. html.
Yates, R. C. "Astroid." A Handbook on Curves and Their
Properties. Ann Arbor, Ml: J. W. Edwards, pp. 1-3, 1952.
Using
then gives
xq = L cos
AL
x = —xo
= yz^^( 1 + ^)
(25)
(26)
(27)
Astroid Evolute
Solving (26) for xo, plugging into (27) and squaring then
gives
(Ai) 2
Rearranging produces the equation
2 f2 Z/V / ALV
y
(AL) 2 (L + AL) 2
(28)
(29)
the equation of a (QUADRANT of an) ELLIPSE with
Semimajor and Semiminor Axes of lengths AL and
L + AL.
A Hypocycloid Evolute for n = 4 is another As-
troid scaled by a factor n/(n — 2) = 4/2 = 2 and
rotated 1/(2 • 4) = 1/8 of a turn.
Astroid Involute
V
/
\
/
\
/
\
/V-
--7V
y \
/ ^
--
^,
'^i-
4.""
\
\
\
/
/
/
/
/
A Hypocycloid Involute for n = 4 is another As-
troid scaled by a factor (n — 2)/2 = 2/4 = 1/2 and
rotated 1/(2 • 4) = 1/8 of a turn.
82 Astroid Pedal Curve
Astroid Pedal Curve
Asymptotic Curve
The Pedal Curve of an Astroid with Pedal Point
at the center is a QUADRIFOLIUM.
Astroid Radial Curve
The QUADRIFOLIUM
x = xo + 3a cos t — 3a cos(3£)
y = y -\- 3a sin t + 3a sin(3t) .
Astroidal Ellipsoid
The surface which is the inverse of the ELLIPSOID in the
sense that it "goes in" where the ELLIPSOID "goes out."
It. is given by the parametric equations
x = (acosticosv)
y = (b sin u cost;) 3
z — (csinv) 3
for u € [— 7r/2,7r/2] and v G [— 7r,7r]. The special case
a = b = c = 1 corresponds to the HYPERBOLIC OCTA-
HEDRON.
see also Ellipsoid, Hyperbolic Octahedron
References
Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/
people/nfytn/ asttxt.htm.
Asymptosy
Asymptotic behavior. A useful yet endangered word,
found rarely outside the captivity of the Oxford English
Dictionary.
see also ASYMPTOTE, ASYMPTOTIC
Asymptote
asymptotes
A curve approaching a given curve arbitrarily closely, as
illustrated in the above diagram.
see also ASYMPTOSY, ASYMPTOTIC, ASYMPTOTIC
Curve
References
Giblin, P. J. "What is an Asymptote?" Math. Gaz. 56,
274-284, 1972.
Asymptotic
Approaching a value or curve arbitrarily closely (i.e.,
as some sort of Limit is taken). A Curve A which is
asymptotic to given CURVE C is called the ASYMPTOTE
of C.
see also ASYMPTOSY, ASYMPTOTE, ASYMPTOTIC
Curve, Asymptotic Direction, Asymptotic Se-
ries, Limit
Asymptotic Curve
Given a Regular Surface M, an asymptotic curve
is formally defined as a curve x(i) on M such that the
Normal Curvature is in the direction x'(t) for all
t in the domain of x. The differential equation for the
parametric representation of an asymptotic curve is
eu -\-2fuv + gv = 0,
(i)
where e, /, and g are second FUNDAMENTAL FORMS.
The differential equation for asymptotic curves on a
Monge Patch (u,v,h(u t v)) is
h uu u + 2h U uU v + h vv v = 0,
and on a polar patch (r cos0,rsin#, h(r)) is
ti'(r)r ,2 +ti{r)rd' 2 =0.
(2)
(3)
The images below show asymptotic curves for the EL-
LIPTIC Helicoid, Funnel, Hyperbolic Paraboloid,
and Monkey Saddle.
Asymptotic Direction
Atiyah-Singer Index Theorem 83
see also RULED SURFACE
References
Gray, A. "Asymptotic Curves," "Examples of Asymp-
totic Curves," "Using Mathematica to Find Asymptotic
Curves." §16.1, 16.2, and 16.3 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp, 320-331, 1993.
Asymptotic Direction
An asymptotic direction at a point p of a REGULAR
Surface M e M 3 is a direction in which the NORMAL
Curvature of M vanishes.
1. There are no asymptotic directions at an Elliptic
Point.
2. There are exactly two asymptotic directions at a HY-
PERBOLIC Point.
3. There is exactly one asymptotic direction at a PAR-
ABOLIC Point.
4. Every direction is asymptotic at a Planar Point.
see also ASYMPTOTIC CURVE
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces.Boca, Raton, FL: CRC Press, pp. 270 and 320, 1993.
Asymptotic Notation
Let n be a integer variable which tends to infinity and let
xbea continuous variable tending to some limit. Also,
let 4>(n) or (j){x) be a positive function and f(n) or f{x)
any function. Then Hardy and Wright (1979) define
1. / = 0{(j>) to mean that |/| < A<f> for some constant
A and all values of n and x y
2. f = o(<j>) to mean that f/<j> — y 0,
3. / ~ <j> to mean that f /<j> — > 1,
4. / -< <j> to mean the same as / = o((f>) 7
5. f y </> to mean f/<j> — > oo, and
6. / x <fc to mean Ai<j> < / < A 2 for some positive
constants A± and A 2 .
f = o(<j>) implies and is stronger than / = 0(<}>).
References
Hardy, G. H. and Wright, E. M. "Some Notation." §1.6 in
An Introduction to the Theory of Numbers, 5th ed. Oxford,
England: Clarendon Press, pp. 7-8, 1979.
Asymptotic Series
An asymptotic series is a SERIES EXPANSION of a FUNC-
TION in a variable x which may converge or diverge
(Erdelyi 1987, p. 1), but whose partial sums can be made
an arbitrarily good approximation to a given function
for large enough x. To form an asymptotic series R(x)
of /(#), written
/(*) ~ R(x), (1)
where
c / \ — i ai i a2 i i an
S n (x) = a H h -=■ + ... + — -■
(3)
The asymptotic series is defined to have the properties
lim x n R n (x) = for fixed n
(4)
lim x n R n (x) = oo for fixed x. (5)
Therefore,
f(x) « 22 anX
(6)
in the limit x — > oo. If a function has an asymptotic
expansion, the expansion is unique. The symbol ~ is
also used to mean directly Similar.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 15, 1972.
Arfken, G. "Asymptotic of Semiconvergent Series." §5.10 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 339-346, 1985.
Bleistein, N. and Handelsman, R. A. Asymptotic Expansions
of Integrals. New York: Dover, 1986.
Copson, E. T. Asymptotic Expansions. Cambridge, England:
Cambridge University Press, 1965.
de Bruijn, N. G. Asymptotic Methods in Analysis, 2nd ed.
New York: Dover, 1982.
Dingle, R. B. Asymptotic Expansions: Their Derivation and
Interpretation. London: Academic Press, 1973.
Erdelyi, A. Asymptotic Expansions. New York: Dover, 1987.
Morse, P. M. and Feshbach, H. "Asymptotic Series; Method
of Steepest Descent." §4.6 in Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 434-443, 1953.
Olver, F. W. J. Asymptotics and Special Functions. New
York: Academic Press, 1974.
Wasow, W. R. Asymptotic Expansions for Ordinary Differ-
ential Equations. New York: Dover, 1987.
Atiyah-Singer Index Theorem
A theorem which states that the analytic and topological
"indices" are equal for any elliptic differential operator
on an n-D Compact Differentiable C°° boundary-
less Manifold.
see also Compact Manifold, Differentiable Man-
ifold
References
Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Op-
erators on Compact Manifolds." Bull. Amer. Math. Soc.
69, 322-433, 1963.
Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Oper-
ators I, II, III." Ann. Math. 87, 484-604, 1968.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 4, 1996.
take
x n R n (x)=x n [f(x)-S n (x)],
(2)
84
Atkin-Goldwasser-Kilian-Morain Certificate
Augmented Amicable Pair
Atkin-Goldwasser-Kilian-Morain Certificate
A recursive PRIMALITY CERTIFICATE for a PRIME p.
The certificate consists of a list of
1. A point on an ELLIPTIC CURVE C
y 2 - x 3 + 92X + p 3 (mod p)
for some numbers £2 and #3-
2. A Prime g with q > (p 1 ^ 4 + l) 2 , such that for
some other number k and m = kq with k ^ 1,
mC{X)y,g2 ) g$ ) p) is the identity on the curve, but
kC(x,y,g2 ) gz,p) is not the identity. This guaran-
tees PRIMALITY of p by a theorem of Goldwasser
and Kilian (1986).
3. Each q has its recursive certificate following it. So if
the smallest q is known to be PRIME, all the numbers
are certified PRIME up the chain.
A Pratt Certificate is quicker to generate for
small numbers. The Mathematica® (Wolfram Re-
search, Champaign, IL) task ProvablePrime [n] there-
fore generates an Atkin-Goldwasser-Kilian-Morain cer-
tificate only for numbers above a certain limit (10 10 by
default), and a Pratt CERTIFICATE for smaller num-
bers.
see also Elliptic Curve Primality Proving, Ellip-
tic Pseudoprime, Pratt Certificate, Primality
Certificate, Witness
References
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math. Comput. 61, 29-68, 1993.
Bressoud, D, M. Factorization and Prime Testing. New
York: Springer- Verlag, 1989.
Goldwasser, S. and Kilian, J. "Almost All Primes Can Be
Quickly Certified." Proc. 18th STOC. pp. 316-329, 1986.
Morain, F. "Implementation of the Atkin-Goldwasser-Kilian
Primality Testing Algorithm." Rapport de Recherche 911,
INRIA, Octobre 1988.
Schoof, R. "Elliptic Curves over Finite Fields and the Com-
putation of Square Roots mod p." Math. Comput. 44,
483-494, 1985.
Wunderlich, M. C "A Performance Analysis of a Simple
Prime-Testing Algorithm." Math. Comput. 40, 709-714,
1983.
Atomic Statement
In LOGIC, a statement which cannot be broken down
into smaller statements.
Attraction Basin
see Basin of Attraction
Attractor
An attractor is a Set of states (points in the Phase
Space), invariant under the dynamics, towards which
neighboring states in a given Basin of Attraction
asymptotically approach in the course of dynamic evo-
lution. An attractor is denned as the smallest unit which
cannot be itself decomposed into two or more attractors
with distinct BASINS OF ATTRACTION. This restriction
is necessary since a Dynamical System may have mul-
tiple attractors, each with its own Basin OF Attrac-
tion.
Conservative systems do not have attractors, since the
motion is periodic. For dissipative Dynamical Sys-
tems, however, volumes shrink exponentially so attrac-
tors have volume in n-D phase space.
A stable FIXED Point surrounded by a dissipative re-
gion is an attractor known as a SINK. Regular attractors
(corresponding to Lyapunov Characteristic Ex-
ponents) act as Limit Cycles, in which trajectories
circle around a limiting trajectory which they asymp-
totically approach, but never reach. STRANGE ATTRAC-
TORS are bounded regions of PHASE SPACE (correspond-
ing to Positive Lyapunov Characteristic Expo-
nents) having zero MEASURE in the embedding PHASE
Space and a Fractal Dimension. Trajectories within
a Strange Attractor appear to skip around ran-
domly.
see also Barnsley's Fern, Basin of Attraction,
Chaos Game, Fractal Dimension, Limit Cycle,
Lyapunov Characteristic Exponent, Measure,
Sink (Map), Strange Attractor
Auction
A type of sale in which members of a group of buyers
offer ever increasing amounts. The bidder making the
last bid (for which no higher bid is subsequently made
within a specified time limit: "going once, going twice,
sold") must then purchase the item in question at this
price. Variants of simple bidding are also possible, as in
a Vickery Auction.
see also Vickery Auction
Augend
The first of several Addends, or "the one to which
the others are added," is sometimes called the augend.
Therefore, while a, 6, and c are ADDENDS in a + 6 -J- c,
a is the augend.
see also ADDEND, ADDITION
Augmented Amicable Pair
A Pair of numbers m and n such that
a(m) — cr(n) = m + n — 1,
where a{m) is the DIVISOR FUNCTION. Beck and Najar
(1977) found 11 augmented amicable pairs.
see also Amicable Pair, Divisor Function, Quasi-
amicable Pair
References
Beck, W. E. and Najar, R. M. "More Reduced Amicable
Pairs." Fib. Quart. 15, 331-332, 1977.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 59, 1994.
Augmented Dodecahedron
Authalic Latitude
85
Augmented Dodecahedron
see Johnson Solid
Augmented Hexagonal Prism
see Johnson Solid
where h = 2k — 1 and
L 2hi M 2h = 2 h + lT2 (7)
L 3hi M 3h = 3 h + lT3 k (8)
L 6h , M 5h = 5 2/l + 3 • S' 1 + 1 =F 5* (5* + 1). (9)
Augmented Pentagonal Prism
see Johnson Solid
Augmented Polyhedron
A Uniform Polyhedron with one or more other solids
adjoined.
Augmented Sphenocorona
see Johnson Solid
Augmented Triangular Prism
see Johnson Solid
Augmented Tridiminished Icosahedron
see Johnson Solid
Augmented Truncated Cube
see Johnson Solid
Augmented Truncated Dodecahedron
see Johnson Solid
Augmented Truncated Tetrahedron
see Johnson Solid
Aureum Theorema
Gauss's name for the QUADRATIC RECIPROCITY THE-
OREM.
Aurifeuillean Factorization
A factorization of the form
2 4n + 2 + x = ^2n + l _ ^ + 1 + 1 )( 2 2n+1 + 2 n + 1 + 1). (1)
The factorization for n — 14 was discovered by Au-
rifeuille, and the general form was subsequently discov-
ered by Lucas. The large factors are sometimes written
as L and M as follows
2 4fc-2 + : = ^ 2 fe-i _ 2 fc + 1 )( 2 2 *- 1 + 2 k + 1) (2)
3 6fc-3 + x = ^2*-i + 1 j( 3 2fc-i _ 3 fc + i)^ 2 *" 1 + 3 fc + 1),
(3)
see also GAUSS'S FORMULA
References
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; WagstafT, S. S. Jr.;
and Tuckerman, B. Factorizations of b n ± 1, b = 2,
3, 5j 6, 7j 10, 11, 12 Up to High Powers, rev. ed. Providence,
RI: Amer. Math. Soc, pp. lxviii-lxxii, 1988.
WagstafT, S. S. Jr. "Aurifeullian Factorizations and the Pe-
riod of the Bell Numbers Modulo a Prime." Math. Corn-
put. 65, 383-391, 1996.
Ausdehnungslehre
see Exterior Algebra
Authalic Latitude
An Auxiliary Latitude which gives a Sphere equal
Surface Area relative to an Ellipsoid. The authalic
latitude is defined by
(i)
/^sin- 1 -*-
where
Q=(l-e 2 )
-
sin0 1 ^
1-
- e 2 sin 2 4> 2e
Li n f 1 ~ esin( A
le \ l-\-es'm<j)J
. (2)
and q p is q evaluated at the north pole (0 = 90°). Let R q
be the Radius of the Sphere having the same Surface
Area as the Ellipsoid, then
Rq
V 2
(3)
The series for j3 is
/3 = ^-Ge 2 + ^e 4 + a i e 6 + ...)sin(2<A)
+ (^ e4 + lio e6 + ---)sin(4^)
-(4lfoe 6 + ...)sin(60) + ....
(4)
The inverse FORMULA is found from
A<f> =
(l-e 2 sin 2 0) 2
2cos0
sin<j)
1-e 2
+
1 — e 2 sin <j>
which can be written
1^/l-esin.A
le VI -f esin<j> J
(5)
2 in + 1 = L 2h M 2h
3 3h + 1 = (3 h + l)L 3h M 3h
(4)
(5)
(6)
where
q = q p sm/3
(6)
86 Autocorrelation
and (f>o — s'm~ 1 (q/2). This can be written in series form
+ (^e* + ^ e 6 + ...)sin(4/3)
(7)
see a/so LATITUDE
References
Adams, O. S. "Latitude Developments Connected with
Geodesy and Cartography with Tables, Including a Table
for Lambert Equal-Area Meridional Projections." Spec.
Pub. No. 67. U. S. Coast and Geodetic Survey, 1921.
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, p. 16, 1987.
Autocorrelation
The autocorrelation function is denned by
F
C f {t) = f*f = f(-t)*nt)= I r(T)f(t + r)dr,
(1)
where * denotes CONVOLUTION and • denotes CROSS-
CORRELATION. A finite autocorrelation is given by
Cf(r) = ([y(t)-y][y(t + r)-y]) (2)
pT/2
= lim / [y(t)-y][y(t + r)-y]dt. (3)
If / is a Real Function,
/* = /,
and an Even Function so that
f(-r) = / (r),
then
(4)
(5)
(6)
Cf(t)= I f(r)f(t + r)dr.
J — OO
But let t' = — r, so dr ~ —dr, then
/» — OO
Cf{t)= f(-r)f(t-r)(-dr)
J OO
OO
f(-r)f(t-r)dr
f(r)f(t-r)dr = f*f. (7)
-F
-F
The autocorrelation discards phase information, return-
ing only the POWER. It is therefore not reversible.
There is also a somewhat surprising and extremely im-
portant relationship between the autocorrelation and
Autocorrelation
the Fourier Transform known as the Wiener-
Khintchine Theorem. Let FF[f{x)] — F(fc), and F*
denote the COMPLEX CONJUGATE of F, then the FOUR-
IER Transform of the Absolute Square of F(k) is
given by
n\F(k)\'}= r r(r)f(r + x)dr. (8)
t/-oo
The autocorrelation is a Hermitian Operator since
Cf(-t) = C f *(t). /*/ is Maximum at the Origin. In
other words,
/oo /»oo
f(u)f(u + x)du< / f 2 (u)du. (9)
•oo J — oo
To see this, let e be a Real Number. Then
/oo
[f{u) + ef(u + x)] 2 du>Q (10)
■oo
/oo /»oo
f(u)du + 2e l f{u)f(u + x)du
-oo J —oo
/oo
f 2 (u + x)du> (11)
■oo
+e
/ f 2 (u)du + 2e /
J — oo J —
) du + 2e / f{u)f(u + x) du
+e
/oo
-oo
) du > 0. (12)
Define
/oo
f(u)du (13)
■oo
/oo
f(u)f(u + x)du. (14)
■oo
Then plugging into above, we have ae 2 +be-\-c > 0. This
Quadratic Equation does not have any Real Root,
so b 2 — 4ac < 0, i.e., 6/2 < a. It follows that
F
f(u)f(u + x) du <
/oo
f{u)du,
-OO
(15)
with the equality at x — 0. This proves that / * / is
Maximum at the Origin.
see also CONVOLUTION, CROSS-CORRELATION, QUAN-
TIZATION Efficiency, Wiener-Khintchine Theo-
rem
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Correlation and Autocorrelation Using the
FFT." §13.2 in Numerical Recipes in FORTRAN: The Art
of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 538-539, 1992.
Automorphic Function
Automorphic Function
An automorphic function f(z) of a COMPLEX variable
z is one which is analytic (except for POLES) in a do-
main D and which is invariant under a DENUMERABLY
Infinite group of Linear Fractional Transforma-
tions (also known as MOBIUS TRANSFORMATIONS)
, az + 6
z = -.
cz + a
Automorphic functions are generalizations of TRIGONO-
METRIC Functions and Elliptic Functions.
see also Modular Function, Mobius Transforma-
tions, Zeta Fuchsian
Automorphic Number
A number k such that nk 2 has its last digits equal to
k is called n-automorphic. For example, 1 • 5 2 = 25
and 1 ■ 6 2 = 36 are 1-automorphic and 2 ■ 8 2 — 128
and 2 • 88 2 = 15488 are 2-automorphic. de Guerre and
Fairbairn (1968) give a history of automorphic numbers.
The first few 1-automorphic numbers are 1, 5, 6, 25,
76, 376, 625, 9376, 90625, . . . (Sloane's A003226, Wells
1986, p. 130). There are two 1-automorphic numbers
with a given number of digits, one ending in 5 and one in
6 (except that the 1-digit automorphic numbers include
1), and each of these contains the previous number with
a digit prepended. Using this fact, it is possible to con-
struct automorphic numbers having more than 25,000
digits (Madachy 1979). The first few 1-automorphic
numbers ending with 5 are 5, 25, 625, 0625, 90625, . . .
(Sloane's A007185), and the first few ending with 6 are
6, 76, 376, 9376, 09376, . . . (Sloane's A016090). The 1-
automorphic numbers a(n) ending in 5 are IDEMPOTENT
(mod 10") since
[a(n)] 2 = a(n) (mod 10 n )
(Sloane and Plouffe 1995).
The following table gives the 10-digit n-automorphic
numbers.
n n-Automorphic Numbers Sloane
1 0000000001, 8212890625, 1787109376
2 0893554688
3 6666666667, 7262369792, 9404296875
4 0446777344
5 3642578125
6 3631184896
7 7142857143, 4548984375, 1683872768
8 0223388672
9 5754123264, 3134765625, 8888888889
— , A007185, A016090
A030984
— , A030985, A030986
A030987
A030988
A030989
A030990, A030991,
A030992
A030993
A030994, A030995, —
see also IDEMPOTENT, NARCISSISTIC NUMBER, NUM-
BER Pyramid, Trimorphic Number
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 59 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, Feb. 1972.
Autoregressive Model 87
Fairbairn, R. A. "More on Automorphic Numbers." J. Recr.
Math. 2, 170-174, 1969.
Fairbairn, R. A. Erratum to "More on Automorphic Num-
bers." J. Recr. Math. 2, 245, 1969.
de Guerre, V. and Fairbairn, R. A. "Automorphic Numbers."
J. Recr. Math. 1, 173-179, 1968.
Hunter, J. A. H. "Two Very Special Numbers." Fib. Quart
2, 230, 1964.
Hunter, J. A. H. "Some Polyautomorphic Numbers." J. Recr.
Math. 5, 27, 1972.
Kraitchik, M. "Automorphic Numbers." §3.8 in Mathemat-
ical Recreations. New York: W. W. Norton, pp. 77-78,
1942.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 34-54 and 175-176, 1979.
Sloane, N. J. A. Sequences A016090, A003226/M3752, and
A007185/M3940 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Wells, D. The Penguin Dictionary of Curious and Interesting
Numbers. Middlesex: Penguin Books, pp. 171, 178, 191-
192, 1986.
Automorphism
An Isomorphism of a system of objects onto itself.
see also ANOSOV AUTOMORPHISM
Automorphism Group
The GROUP of functions from an object G to itself which
preserve the structure of the object, denoted Aut(G).
The automorphism group of a GROUP preserves the
Multiplication table, the automorphism group of a
Graph the Incidence Matrices, and that of a Field
the Addition and Multiplication tables,
see also Outer Automorphism Group
Autonomous
A differential equation or system of ORDINARY DIFFER-
ENTIAL EQUATIONS is said to be autonomous if it does
not explicitly contain the independent variable (usu-
ally denoted i). A second-order autonomous differen-
tial equation is of the form F{y,y \y") — 0, where
y = dy/dt = v. By the CHAIN RULE, y" can be ex-
pressed as
y
dv
It
dv dy __ dv
dy dt dy
For an autonomous ODE, the solution is independent of
the time at which the initial conditions are applied. This
means that all particles pass through a given point in
phase space. A nonautonomous system of n first-order
ODEs can be written as an autonomous system of n + 1
ODEs by letting t = x n +i and increasing the dimension
of the system by 1 by adding the equation
dx
Tl + l
dt
1.
Autoregressive Model
see Maximum Entropy Method
88 Auxiliary Circle
Axiom A Flow
Auxiliary Circle
The ClRCUMCIRCLE of an ELLIPSE, i.e., the CIRCLE
whose center corresponds with that of the ELLIPSE and
whose Radius is equal to the Ellipse's Semimajor
Axis.
see also CIRCLE, ECCENTRIC ANGLE, ELLIPSE
Auxiliary Latitude
see Authalic Latitude, Conformal Latitude,
Geocentric Latitude, Isometric Latitude, Lat-
itude, Parametric Latitude, Rectifying Lati-
tude, Reduced Latitude
Auxiliary Triangle
see Medial Triangle
Average
see Mean
Average Absolute Deviation
N
a= — ^\xi- fi\ = (\xi-n\).
i=l
see also ABSOLUTE DEVIATION, DEVIATION, STANDARD
Deviation, Variance
Average Function
If / is Continuous on a Closed Interval [a, 6], then
there is at least one number x* in [a, 6] such that
/
J a
f(x)dx = f(x*)(b- a).
The average value of the FUNCTION (/) on this interval
is then given by f(x*).
see Mean- Value Theorem
Average Seek Time
see POINT-POINT DlSTANCE-
-1-D
Ax-Kochen Isomorphism Theorem
Let P be the Set of PRIMES, and let Q p and Z p (t) be the
Fields of p-ADic Numbers and formal Power series
over Z p = (0, 1, ... ,p — 1). Further, suppose that D is a
"nonprincipal maximal filter" on P. Then Y[ GP Q p /D
and Y[ ep Z p (t)/D are ISOMORPHIC.
see also Hyperreal Number, Nonstandard Analy-
sis
Axial Vector
see PSEUDOVECTOR
Axiom
A Proposition regarded as self-evidently True with-
out Proof. The word "axiom" is a slightly archaic syn-
onym for Postulate. Compare Conjecture or Hy-
pothesis, both of which connote apparently TRUE but
not self- evident statements.
see also ARCHIMEDES' AXIOM, AXIOM OF CHOICE, AX-
IOMATIC System, Cantor-Dedekind Axiom, Con-
gruence Axioms, Conjecture, Continuity Ax-
ioms, Countable Additivity Probability Axiom,
Dedekind's Axiom, Dimension Axiom, Eilenberg-
Steenrod Axioms, Euclid's Axioms, Excision Ax-
iom, Fano's Axiom, Field Axioms, Hausdorff Ax-
ioms, Hilbert's Axioms, Homotopy Axiom, In-
accessible Cardinals Axiom, Incidence Axioms,
Independence Axiom, Induction Axiom, Law,
Lemma, Long Exact Sequence of a Pair Axiom,
Ordering Axioms, Parallel Axiom, Pasch's Ax-
iom, Peano's Axioms, Playfair's Axiom, Porism,
Postulate, Probability Axioms, Proclus' Axiom,
Rule, T2-Separation Axiom, Theorem, Zermelo's
Axiom of Choice, Zermelo-Fraenkel Axioms
Axiom A Diffeomorphism
Let 4> : M -¥ M be a C 1 Diffeomorphism on a com-
pact Riemannian Manifold M. Then <f> satisfies Ax-
iom A if the Nonwandering set Q(4>) of is hyperbolic
and the Periodic Points of <j> are Dense in Q(<f>). Al-
though it was conjectured that the first of these condi-
tions implies the second, they were shown to be indepen-
dent in or around 1977. Examples include the AN0S0V
Diffeomorphisms and Smale Horseshoe Map.
In some cases, Axiom A can be replaced by the condi-
tion that the DIFFEOMORPHISM is a hyperbolic diffeo-
morphism on a hyperbolic set (Bowen 1975, Parry and
Pollicott 1990).
see also Anosov Diffeomorphism, Axiom A Flow,
Diffeomorphism, Dynamical System, Riemannian
Manifold, Smale Horseshoe Map
References
Bowen, R. Equilibrium States and the Ergodic Theory of
Anosov Diffeomorphisms. New York: Springer- Verlag,
1975.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 143, 1993.
Parry, W. and Pollicott, M. "Zeta Functions and the Peri-
odic Orbit Structure of Hyperbolic Dynamics," Asterisque
No. 187-188, 1990.
Smale, S. "Different iable Dynamical Systems." Bull Amer.
Math. Soc. 73, 747-817, 1967.
Axiom A Flow
A Flow defined analogously to the Axiom A Diffeo-
morphism, except that instead of splitting the Tan-
gent Bundle into two invariant sub-BUNDLES, they
are split into three (one exponentially contracting, one
expanding, and one which is 1-dimensional and tangen-
tial to the flow direction).
see also DYNAMICAL SYSTEM
Axiom of Choice
Azimuthal Projection 89
Axiom of Choice
An important and fundamental result in Set Theory
sometimes called Zermelo'S Axiom of Choice. It was
formulated by Zermelo in 1904 and states that, given any
Set of mutually exclusive nonempty SETS, there exists
at least one Set that contains exactly one element in
common with each of the nonempty SETS.
It is related to HlLBERT'S PROBLEM IB, and was proved
to be consistent with other Axioms in Set Theory in
1940 by GodeL In 1963, Cohen demonstrated that the
axiom of choice is independent of the other Axioms in
Cantorian Set Theory, so the Axiom cannot be proved
within the system (Boyer and Merzbacher 1991, p. 610).
see also Hilbert's Problems, Set Theory, Well-
Ordered Set, Zermelo-Fraenkel Axioms, Zorn's
Lemma
References
Boyer, C. B. and Merzbacher, U. C, A History of Mathemat-
ics, 2nd ed. New York: Wiley, 1991.
Cohen, P. J, "The Independence of the Continuum Hypoth-
esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963.
Cohen, P. J. "The Independence of the Continuum Hypothe-
sis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 274-276, 1996.
Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Devel-
opment, and Influence. New York: Springer- Verlag, 1982.
Axiomatic Set Theory
A version of Set Theory in which axioms are taken
as uninterpreted rather than as formalizations of pre-
existing truths.
see also Naive Set Theory, Set Theory
Axiomatic System
A logical system which possesses an explicitly stated
Set of Axioms from which Theorems can be derived.
see also Complete Axiomatic Theory, Consis-
tency, Model Theory, Theorem
Axis
A LINE with respect to which a curve or figure is drawn,
measured, rotated, etc. The term is also used to refer
to a Line Segment through a Range (Woods 1961).
see also Abscissa, Ordinate, cc-AxiS, y-AxiS, z-Axis
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 8,
1961.
Axonometry
A Method for mapping 3-D figures onto the Plane.
see also CROSS-SECTION, Map Projection, Pohlke's
Theorem, Projection, Stereology
References
Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York:
Dover, p. 313, 1973.
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, pp. 322-323, 1988.
Azimuthal Equidistant Projection
'^rk
An Azimuthal Projection which is neither equal-
Area nor CONFORMAL. Let <p± and Ao be the LATI-
TUDE and LONGITUDE of the center of the projection,
then the transformation equations are given by
x - fc'cos0sin(A - Ao)
y = fc'jcos^i sin0 — sin<^i cos<£cos(A — Ao)].
Here,
and
k'
c
sine
(i)
(2)
(3)
cose — sin 0i sin0 + cos^i cos0cos(A — Ao), (4)
where c is the angular distance from the center. The
inverse FORMULAS are
-(
= sin I cose sin 0i +
y sin c cos <f>.
l ) (5)
and
( A + tan" 1 ( -r xsinc . . .
u V ccos <pi cos c — y sin q>\ sin c
for 0! ^ ±90°
Ao+tan-^-l)
for 0! = 90°
Ao+tan- 1 ^),
for 0i = -90°,
)
(6)
with the angular distance from the center given by
c = V^ + y 2 . (7)
References
Snyder, J. P. Map Projections — A Working Manual U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 191-202, 1987.
Azimuthal Projection
see Azimuthal Equidistant Projection, Lam-
bert Azimuthal Equal-Area Projection, Ortho-
graphic Projection, Stereographic Projection
B* -Algebra
B
B-Spline 91
E*-Algebra
A Banach Algebra with an Antiautomorphic In-
volution * which satisfies
(5)
A C*-Algebra is a special type of i?*-algebra.
see also Banach Algebra, C*-Algebra
i?2- Sequence
N. B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Also called a Sidon Sequence. An Infinite Se-
quence of Positive Integers
1 < h < 6 2 < h <
such that all pairwise sums
bi + bj
(i)
(2)
for i < j are distinct (Guy 1994). An example is 1, 2, 4,
8, 13, 21, 31, 45, 66, 81, . . . (Sloane's A005282).
Zhang (1993, 1994) showed that
S(B2) = sup V — > 2.1597.
all B2 sequences ~~j ®k
(3)
The definition can be extended to B n -sequences (Guy
1994).
see also ^-Sequence, Mian-Chowla Sequence
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/erdos/erdos .html.
Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with
Distinct Sums," and "^-Sequences," and B 2 -Sequences
Formed by the Greedy Algorithm." §C9, Cll, E28, and
E32 in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 115-118, 121-123, 228-229, and
232-233, 1994.
Sloane, N. J. A. Sequence A005282/M1094 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum."
Math. Comput. 60, 835-839, 1993.
Zhang, Z. X. "Finding Finite B2-Sequences with Larger m —
a™ 1 ' 2 ." Math. Comput. 63, 403-414, 1994.
B p - Theorem
If Op' (G) = 1 and if a? is a p-element of G, then
L p ,(C g (x)<E(Cg(x)),
where L p > is the p-LAYER.
X = X
(i)
(2)
B
-Spline
Po®t
x* +y* = {x + y)*
(3)
(ex)* = ex*
satisfies
(4)
•
Pi
A generalization of the Bezier Curve. Let a vector
known as the KNOT VECTOR be defined
T = {£o,£ij • ■ , tm}i
(i)
where T is a nondecreasing SEQUENCE with U 6 [0, 1],
and define control points Po, . . . , Pn- Define the degree
as
p = m — n — 1. (2)
The "knots" £ p +i, ..., tm- P -i are called Internal
Knots.
Define the basis functions as
at / ,\ _ f 1 if ti < t < ti+i and U < tt+i
1 otherwise
ti + v+l — t
N ilP (t)
t-U
ti-\-p Ci
(3)
ii+p+1 — *i+l
Then the curve defined by
C(t) = £p<M,p(t)
(4)
(5)
is a B-spline. Specific types include the nonperiodic B-
spline (first p + 1 knots equal and last p + 1 equal to
1) and uniform B-spline (INTERNAL KNOTS are equally
spaced). A B-Spline with no INTERNAL KNOTS is a
Bezier Curve.
The degree of a B-spline is independent of the number of
control points, so a low order can always be maintained
for purposes of numerical stability. Also, a curve is p — k
times differentiate at a point where k duplicate knot
values occur. The knot values determine the extent of
the control of the control points.
A nonperiodic B-spline is a B-spline whose first p + 1
knots are equal to and last p -f 1 knots are equal to
1. A uniform B-spline is a B-spline whose INTERNAL
Knots are equally spaced.
see also Bezier Curve, NURBS Curve
92
B-Tree
Backtracking
B-Tree
B-trees were introduced by Bayer (1972) and Mc-
Creight. They are a special m-ary balanced tree used in
databases because their structure allows records to be
inserted, deleted, and retrieved with guaranteed worst-
case performance. An n-node £?-tree has height C(lg2),
where Lg is the LOGARITHM to base 2. The Apple®
Macintosh® (Apple Computer, Cupertino, CA) HFS fil-
ing system uses B-trees to store disk directories (Bene-
dict 1995). A B-tree satisfies the following properties:
1. The Root is either a Leaf (Tree) or has at least
two Children,
2. Each node (except the ROOT and LEAVES) has be-
tween \m/2\ and m Children, where \x\ is the
Ceiling Function.
3. Each path from the Root to a Leaf (Tree) has the
same length.
Every 2-3 Tree is a B-tree of order 3. The number of
B-trees of order n = 1, 2, . . . are 0, 1, 1, 1, 2, 2, 3, 4, 5,
8, 14, 23, 32, 43, 63, . . . (Ruskey, Sloane's A014535).
see also Red-Black Tree
References
Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc-
tures and Algorithms. Reading, MA: Addison-Wesley,
pp. 369-374, 1987.
Benedict, B. Using Norton Utilities for the Macintosh. Indi-
anapolis, IN: Que, pp. B-17-B-33, 1995.
Beyer, R. "Symmetric Binary jB-Trees: Data Structures and
Maintenance Algorithms." Acta Informat. 1, 290-306,
1972.
Ruskey, F. "Information on B-Trees." http://sue.csc.uvic
. ca/~cos/inf /tree/BTrees .html.
Sloane, N. J. A. Sequence A014535 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Baby Monster Group
Also known as FISCHER'S BABY MONSTER GROUP. The
Sporadic Group B. It has Order
2 4i . 3 i3 . 5 6 . 7 2 . ii . 13 . 17 . 19 . 23 • 31 ■ 47.
see also MONSTER GROUP
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac .uk/atlas/BM . html.
BAC-CAB Identity
The Vector Triple Product identity
A x (B x C) = B(A ■ C) - C(A • B).
This identity can be generalized to n-D
a 2 x ■ • • x a n _i x (bi x ■ • • x b n _i)
h x
a 2 ■ In
= (-*)
a n _i ■ bi
See also LAGRANGE'S IDENTITY
b n -i
a 2 • b n _i
&n-i " b n _i
BAC-CAB Rule
see BAC-CAB IDENTITY
Bachelier Function
see Brown Function
Bachet's Conjecture
see Lagrange's Four-Square Theorem
Bachet Equation
The Diophantine Equation
x 2 + k = y 3 ,
which is also an Elliptic Curve. The general equation
is still the focus of ongoing study.
Backhouse's Constant
Let P(x) be defined as the POWER series whose nth term
has a Coefficient equal to the nth Prime,
oo
P(x) = Y^PhX k = l + 2z + 3z 2 + 5z 3 + 7z 4 -hllz 5 + ...,
and let Q(x) be defined by
on
1
Q(*) =
P(x)
y^qkX h .
k=o
Then N. Backhouse conjectured that
lim
n—t-oc
<7n+l
q n
1.456074948582689671399595351116. . . .
The constant was subsequently shown to exist by P. Fla-
jolet.
References
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsof t . com/asolve/constant/backhous/
backhous .html.
Backlund Transformation
A method for solving classes of nonlinear Partial Dif-
ferential Equations.
see also INVERSE SCATTERING METHOD
References
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and
Chaos. Cambridge, England: Cambridge University Press,
p. 196, 1990.
Miura, R. M. (Ed.) Backlund Transformations, the Inverse
Scattering Method, Solitons, and Their Applications. New
York: Springer- Verlag, 1974.
Backtracking
A method of drawing FRACTALS by appropriate num-
bering of the corresponding tree diagram which does not
require storage of intermediate results.
Backus-Gilbert Method
Baire Category Theorem 93
Backus-Gilbert Method
A method which can be used to solve some classes of
INTEGRAL EQUATIONS and is especially useful in im-
plementing certain types of data inversion. It has been
applied to invert seismic data to obtain density profiles
in the Earth.
References
Backus, G. and Gilbert, F. "The Resolving Power of Growth
Earth Data." Geophys. J. Roy. Astron. Soc. 16, 169-205,
1968.
Backus, G. E. and Gilbert, F. "Uniqueness in the Inversion
of Inaccurate Gross Earth Data." Phil Trans. Roy. Soc.
London Ser. A 266, 123-192, 1970.
Loredo, T. J. and Epstein, R. I. "Analyzing Gamma-Ray
Burst Spectral Data." Astrophys. J. 336, 896-919, 1989.
Parker, R. L. "Understanding Inverse Theory." Ann. Rev.
Earth Planet Sci. 5, 35-64, 1977.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Backus-Gilbert Method." §18.6 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 806-809, 1992.
Backward Difference
The backward difference is a Finite Difference de-
fined by
Vp = V/ p s/p-/p_i. (1)
Higher order differences are obtained by repeated oper-
ations of the backward difference operator, so
Vp = V(Vp) = V(/ p - /„_!) = V/ p - V/,_i (2)
= {fp ~ fp-i) ~ (fp-i ~ /p-z)
= fp~ 2 /p-l + fp-2
(3)
In general,
v5 = vv, = £(-ir(*W* +m >
(4)
where (^) is a BINOMIAL COEFFICIENT.
Newton's Backward Difference Formula ex-
presses f p as the sum of the nth backward differences
/ P = /o+pVo + ^p(p + l)V? + J T p(p + l)(p + 2)Vg + ...,
(5)
where Vq is the first nth difference computed from the
difference table.
see also Adams' Method, Difference Equation,
Divided Difference, Finite Difference, For-
ward Difference, Newton's Backward Differ-
ence Formula, Reciprocal Difference
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 429 and 433, 1987.
Bader-Deuflhard Method
A generalization of the Bulirsch-Stoer Algorithm
for solving Ordinary Differential Equations.
References
Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point
Rule for Stiff Systems of Ordinary Differential Equations."
Numer. Math. 41, 373-398, 1983.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 730, 1992.
Baguenaudier
A Puzzle involving disentangling a set of rings from a
looped double rod (also called CHINESE Rings). The
minimum number of moves needed for n rings is
§(2 n+1 -2)
|(2 n+1 -l)
n even
n odd.
By simultaneously moving the two end rings, the num-
ber of moves can be reduced to
f 2 n_1 -In even
I 2 71 " 1 n odd.
The solution of the baguenaudier is intimately related
to the theory of GRAY CODES.
References
Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Pro-
duce a Chinese Monster." Quantum 6, 61-65 (Mar.) and
58-59 (Apr.), 1996.
Gardner, M. "The Binary Gray Code." In Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, pp. 15-17, 1986.
Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical
Recreations. New York: W. W. Norton, pp. 89-91, 1942.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, p. 268, 1983.
Bailey's Method
see Lambert's Method
Bailey's Theorem
Let T(z) be the GAMMA FUNCTION, then
r(m+|)
V(m)
[i (-Y — — (-Y—!—
I>+§)
V(n)
1 /iy_j_ /i-3\ 2 1
n + \2J n-M + \2.4y n + 2 + '
Baire Category Theorem
A nonempty complete Metric Space cannot be repre-
sented as the Union of a Countable family of nowhere
Dense Subsets.
94 Baire Space
Ball Triangle Picking
Baire Space
A Topological Space X in which each Subset of X
of the "first category" has an empty interior. A TOPO-
LOGICAL Space which is Homeomorphic to a complete
Metric Space is a Baire space.
Bairstow's Method
A procedure for rinding the quadratic factors for the
Complex Conjugate Roots of a Polynomial P(x)
with Real Coefficients.
[x — (a + ib)][x - (a — ib)]
= x 2 + 2ax + (a 2 + b 2 ) = x 2 + Bx + C. (1)
Now write the original POLYNOMIAL as
P(x) = (x 2 +Bx + C)Q{x) + Rx + S (2)
R(B + SB,C + 6C)KR(B,C) + ^dB+^dC (3)
dB
8C
S(B + 5B,C + 5C)*d(B,C) + ^dB+^dC (4)
£ — <.- + «.♦* ,« + «, + g5 + » ( .>
. QW = ( I . + B , + C )g + g + f (6)
"*«M = <** + «* + C >I + 1 + !' (8)
Now use the 2-D Newton's Method to find the simul-
taneous solutions.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in C: The Art of Scientific
Computing. Cambridge, England: Cambridge University
Press, pp. 277 and 283-284, 1989.
Baker's Dozen
The number 13.
see also 13, DOZEN
Baker's Map
The Map
X n +1 = 2^£ n ,
(1)
where x is computed modulo 1. A generalized Baker's
map can be defined as
Vn < a
(2)
Xn+1 -\(1-X b ) + X b x n y n >a
where (3 = 1 — a, A + A 6 < 1, and x and y are computed
mod 1. The q = 1 g-DlMENSION is
aln(±)+/31n(|)
D 1 = 1 + Va) )*' ■ (4)
' ta (£)+*»»(*)'
If A a = A&, then the general g-DlMENSION is
1 In (a q +f3 q )
D q = l +
q — 1 In A
(5)
References
Lichtenberg, A. and Lieberman, M. Regular and Stochastic
Motion. New York: Springer- Verlag, p. 60, 1983.
Ott, E. Chaos in Dynamical Systems. Cambridge, England:
Cambridge University Press, pp. 81-82, 1993.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, p. 32, 1990.
Balanced ANOVA
An ANOVA in which the number of REPLICATES (sets
of identical observations) is restricted to be the same for
each Factor Level (treatment group).
see also ANOVA
Balanced Incomplete Block Design
see Block Design
Ball
The n-ball, denoted B n , is the interior of a SPHERE
S™" 1 , and sometimes also called the n-DlSK. (Al-
though physicists often use the term "SPHERE" to mean
the solid ball, mathematicians definitely do not!) Let
Vol(B n ) denote the volume of an n-D ball of RADIUS r.
Then
oo
Y^ Vol(S n ) = e^ 2 [1 + erf (r^ )],
where erf(x) is the ERF function.
see also Alexander's Horned Sphere, Banach-
Tarski Paradox, Bing's Theorem, Bishop's In-
equality, Bounded, Disk, Hypersphere, Sphere,
Wild Point
References
Preden, E. Problem 10207. "Summing a Series of Volumes."
Amer. Math. Monthly 100, 882, 1993.
Ball Triangle Picking
The determination of the probability for obtaining an
Obtuse Triangle by picking 3 points at random in
the unit Disk was generalized by Hall (1982) to the n-
D Ball. Buchta (1986) subsequently gave closed form
Ballantine
Banach Measure
95
evaluations for Hall's integrals, with the first few solu-
tions being
9 4
P 2 = - - — « 0.72
8 7V d
P 4 « 0.39
P 5 « 0.29.
The case P^ corresponds to the usual DISK case.
see also Cube Triangle Picking, Obtuse Triangle
References
Buchta, C. "A Note on the Volume of a Random Polytope in
a Tetrahedron." III. J. Math. 30, 653-659, 1986.
Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob.
19, 712-715, 1982.
Ballantine
see Borromean Rings
Ballieu's Theorem
For any set fi = (^1,^2, ■ ■ ■ ,fi n ) of POSITIVE numbers
with ^o = and
M M = max
flk + {ln\b n -k\
0<k<n-l /ifc + 1
Then all the EIGENVALUES A satisfying P(X) = 0, where
P{\) is the Characteristic Polynomial, lie on the
Disk \z\ < M M .
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1119, 1979.
Ballot Problem
Suppose A and B are candidates for office and there are
2n voters, n voting for A and n for B, In how many ways
can the ballots be counted so that A is always ahead of
or tied with B1 The solution is a CATALAN NUMBER
A related problem also called "the" ballot problem is to
let A receive a votes and B b votes with a > b. This ver-
sion of the ballot problem then asks for the probability
that A stays ahead of B as the votes are counted (Vardi
1991). The solution is (a — b)/(a + 6), as first shown
by M. Bertrand (Hilton and Pedersen 1991). Another
elegant solution was provided by Andre (1887) using the
so-called Andre's Reflection Method.
The problem can also be generalized (Hilton and Ped-
ersen 1991). Furthermore, the TAK FUNCTION is con-
nected with the ballot problem (Vardi 1991).
see also Andre's Reflection Method, Catalan
Number, TAK Function
References
Andre, D. "Solution directe du probleme resolu par
M. Bertrand." Comptes Rendus Acad. Sci. Paris 105,
436-437, 1887.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 49, 1987.
Carlitz, L. "Solution of Certain Recurrences." SIAM J. Appl.
Math. 17, 251-259, 1969.
Comtet, L. Advanced Combinatorics. Dordrecht, Nether-
lands: Reidel, p. 22, 1974.
Feller, W. An Introduction to Probability Theory and Its Ap-
plications, Vol. 1, 3rd ed. New York: Wiley, pp. 67-97,
1968.
Hilton, P. and Pedersen, J. "The Ballot Problem and Cata-
lan Numbers." Nieuw Archief voor Wiskunde 8, 209-216,
1990.
Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel. 13, 64-75, 1991.
Kraitchik, M. "The Ballot-Box Problem." §6.13 in Mathe-
matical Recreations. New York: W. W. Norton, p. 132,
1942.
Motzkin, T. "Relations Between Hypersurface Cross Ratios,
and a Combinatorial Formula for Partitions of a Polygon,
for Permanent Preponderance, and for Non-Associative
Products." Bull Amer. Math. Soc. 54, 352-360, 1948.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 185-187, 1991.
Banach Algebra
An Algebra A over a Field F with a Norm that
makes A into a COMPLETE METRIC SPACE, and there-
fore, a Banach Space. F is frequently taken to be the
Complex Numbers in order to assure that the Spec-
trum fully characterizes an Operator (i.e., the spec-
tral theorems for normal or compact normal operators
do not, in general, hold in the Spectrum over the Real
Numbers).
see also £?*-Algebra
Banach Fixed Point Theorem
Let / be a contraction mapping from a closed SUBSET
F of a Banach Space E into F. Then there exists a
unique z £ F such that f(z) = z.
see also FIXED POINT THEOREM
References
Debnath, L. and Mikusiriski, P. Introduction to Hilbert
Spaces with Applications. San Diego, CA: Academic Press,
1990.
Banach-Hausdorff-Tarski Paradox
see Banach- Tarski Paradox
Banach Measure
An "Area" which can be defined for every set — even
those without a true geometric AREA — which is rigid
and finitely additive.
96
Banach Space
Baibiefs Theorem
Banach Space
A normed linear Space which is Complete in the norm-
determined Metric. A Hilbert Space is always a Ba-
nach space, but the converse need not hold.
see also Besov Space, Hilbert Space, Schauder
Fixed Point Theorem
Banach-Steinhaus Theorem
see Uniform Boundedness Principle
Banach- Tarski Paradox
First stated in 1924, this theorem demonstrates that it
is possible to dissect a Ball into six pieces which can
be reassembled by rigid motions to form two balls of
the same size as the original. The number of pieces was
subsequently reduced to five. However, the pieces are
extremely complicated. A generalization of this theo-
rem is that any two bodies in R which do not extend
to infinity and each containing a ball of arbitrary size
can be dissected into each other (they are are EQUIDE-
composable).
References
Stromberg, K. "The Banach- Tarski Paradox." Amer. Math.
Monthly 86, 3, 1979.
Wagon, S. The Banach-Tarski Paradox. New York: Cam-
bridge University Press, 1993.
Bang's Theorem
The lines drawn to the Vertices of a face of a Tetra-
hedron from the point of contact of the FACE with the
INSPHERE form three ANGLES at the point of contact
which are the same three ANGLES in each FACE.
References
Brown, B. H. "Theorem of Bang. Isosceles Tetrahedra."
Amer. Math. Monthly 33, 224-226, 1926.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 93, 1976.
Bankoff Circle
References
Bankoff, L. "Are the Twin Circles of Archimedes Really
Twins?" Math. Mag. 47, 214-218, 1974.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979.
Banzhaf Power Index
The number of ways in which a group of n with weights
X^r=i Wi = 1 can cnan g e a losing coalition (one with
^2 w i < 1/2) to a winning one, or vice versa. It was
proposed by the lawyer J. F. Banzhaf in 1965.
References
Paulos, J. A. A Mathematician Reads the Newspaper. New
York: BasicBooks, pp. 9-10, 1995.
Bar (Edge)
The term in rigidity theory for the EDGES of a GRAPH.
see also Configuration, Framework
Bar Polyhex
A Polyhex consisting of Hexagons arranged along a
line.
see also Bar Polyiamond
References
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.
Bar Polyiamond
In addition to the ARCHIMEDES' CIRCLES d and C 2 in
the Arbelos figure, there is a third circle C3 congruent
to these two as illustrated in the above figure.
see also ARBELOS
A Polyiamond consisting of Equilateral Triangles
arranged along a line.
see also Bar Polyhex
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Barber Paradox
A man of Seville is shaved by the Barber of Seville IFF
the man does not shave himself. Does the barber shave
himself? Proposed by Bertrand Russell.
Barbier's Theorem
All Curves of Constant Width of width w have the
same Perimeter ttw.
Bare Angle Center
Barth Decic
97
Bare Angle Center
The Triangle Center with Triangle Center
Function
a = A.
References
Kimberling, C. "Major Centers of Triangles." Amer. Math.
Monthly 104, 431-438, 1997.
Barnes G- Function
see G-Function
Barnes' Lemma
If a Contour in the Complex Plane is curved such
that it separates the increasing and decreasing sequences
of Poles, then
2-m . v
</ —too
+ s)r(0 + s)T('y-s)r(5-s)ds
= T(a + 7)r(a + 6)r{(3 + j)T{p + 6)
r(a + /3 + 7 + <5)
where T(z) is the Gamma Function.
Barnes- Wall Lattice
A lattice which can be constructed from the LEECH LAT-
TICE A 2 4-
see also Coxeter-Todd Lattice, Lattice Point,
Leech Lattice
References
Barnes, E. S. and Wall, G. E. "Some Extreme Forms Denned
in Terms of Abelian Groups." J. Austral Math. Soc. 1,
47-63, 1959.
Conway, J. H. and Sloane, N. J, A, "The 16- Dimensional
Barnes- Wall Lattice Ai 6 ." §4.10 in Sphere Packings, Lat-
tices, and Groups, 2nd ed. New York: Springer- Verlag,
pp. 127-129, 1993,
Barnsley's Fern
^■,;f^'
*^7&g$^~~
0.85 0.04"
X
+
"o.oo"
(1)
-0.04 0.85
_y
1.60
-0.15 0.28"
X
+
"o.oo"
(2)
0.26 0.24
y '.
0.44
0.20 -0.26'
X
+
"o.oo"
(3)
0.23 0.22
y .
1.60
0.00 0.00 "
X
(4)
0.00 0.16
y
The Attractor of the Iterated Function System
given by the set of "fern functions"
h(x,y) =
fs(x,y) =
U(x,y) =
(Barnsley 1993, p. 86; Wagon 1991). These Affine
Transformations are contractions. The tip of the
fern (which resembles the black spleehwort variety of
fern) is the fixed point of /i , and the tips of the lowest
two branches are the images of the main tip under J2
and f z (Wagon 1991).
see also Dynamical System, Fractal, Iterated
Function System
References
Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Aca-
demic Press, pp. 86, 90, 102 and Plate 2, 1993.
Gleick, J. Chaos: Making a New Science. New York: Pen-
guin Books, p. 238, 1988.
Wagon, S. "Biasing the Chaos Game: Barnslej^s Fern." §5.3
in Mathematica in Action. New York: W. H. Freeman,
pp. 156-163, 1991.
Barrier
A number n is called a barrier of a number-theoretic
function f(m) if, for all m < n, m + f(m) < n. Neither
the Totient Function <p(n) nor the Divisor Func-
tion o-(n) has barriers.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 64-65, 1994.
Barth Decic
\ 1 7 / .
98
Barth Sextic
Bartlett Function
The Barth decic is a Decic Surface in complex three-
dimensional projective space having the maximum pos-
sible number of ORDINARY DOUBLE POINTS (345). It is
given by the implicit equation
■A 2 )
■2yV)
x( :c 4 + y 4 + z 4 -2 2; V-
+(3 + 50)(;r 2 +y 2 +z 2 -w 2 ) 2 [x 2 4-y 2 \z 2 -(2-0)u? 2 ]V
= 0,
where is the Golden Mean and w is a parameter
(EndraB, Nordstrand), taken as w = 1 in the above plot.
The Barth decic is invariant under the ICOSAHEDRAL
Group.
see also ALGEBRAIC SURFACE, BARTH SEXTIC, DECIC
Surface, Ordinary Double Point
References
Barth, W. "Two Projective Surfaces with Many Nodes Ad-
mitting the Symmetries of the Icosahedron." J. Alg. Geom.
5, 173-186, 1996.
Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV-
Mitteilungen 4, 17-20, 4/1995.
Endrafi, S. "Barth's Decic." http://www.mathematik.uni-
mainz . de/AlgebraischeGeometrie/docs/
Ebarthdecic . shtml.
Nordstrand, T. "Batch Decic." http://www.uib.no/people/
nf ytn/bdectxt .htm.
Barth Sextic
The Barth-sextic is a SEXTIC SURFACE in complex
three-dimensional projective space having the maximum
possible number of ORDINARY DOUBLE POINTS (65). It
is given by the implicit equation
A{4> 2 x 2 -y 2 ){4> 2 y 2 ~z'){<t>-z- -x
2 2 2w ,2 2 2x
-(1 + 2<P)(x 2 + y 2 + z 2 - w 2 ) 2 w 2
0.
where 4> is the GOLDEN Mean, and w is a parameter
(Endrafi, Nordstrand), taken as w — 1 in the above plot.
The Barth sextic is invariant under the ICOSAHEDRAL
Group. Under the map
/ \ v / 2 2 2 2\
(x,y,z,w) -+ (x ,y ,z 9 w ),
the surface is the eightfold cover of the Cayley Cubic
(Endrafi).
see also ALGEBRAIC SURFACE, BARTH DECIC, CAYLEY
Cubic, Ordinary Double Point, Sextic Surface
References
Barth, W. "Two Projective Surfaces with Many Nodes Ad-
mitting the Symmetries of the Icosahedron." J. Alg. Geom.
5, 173-186, 1996.
Endrafl, S. "Flachen mit vielen Doppelpunkten." DMV-
Mitteilungen 4, 17-20, 4/1995.
Endrafl, S. "Barth's Sextic." http://www.mathematik.uni-
mainz.de/AlgebraischeGeometrie/docs/
Ebarthsextic . shtml.
Nordstrand, T. "Barth Sextic." http://www.uib.no/people/
nf ytn/sexttxt .htm.
Bartlett Function
o.is
o.c
o.c
oflc
J '-0725
-1 -0.5 ' 075 1 -0.5
The Apodization Function
L
f{x) = 1
(1)
which is a generalization of the one-argument TRIANGLE
Function. Its Full Width at Half Maximum is a.
It has Instrument Function
I(x) = ^ e~ 2 * ikx (l - M) dx
v —a
+ fe- J,iJ, (l-j)<b. (2)
Letting x' = —x in the first part therefore gives
f° e- 2 " ikx (l + |) dx = I e Mk *' (l - ^\ (-dx')
Rewriting (2) using (3) gives
(3)
7-/ \ / 2irikx . — 2-rrikx\ ( -, % \
I(x) = (e +e H aj
dx
= 2 / cos(27rfcz) (l - -J dx.
(4)
Integrating the first part and using the integral
/
x cos(bx) dx — — cos(6;c) + — sin(for) (5)
b 1 b
Barycentric Coordinates
for the second part gives
sin(27rA;a;)
I(x) = 2
2irk
[s\n(2Trk
2™fe~
= 2 { l" sin ( 27rfc a ) __
cos(27rfca) — 1 asm.{2nka)
47T 2 fc 2
27r 2 a/c 2
: a sine (7rka),
[cos(27r&a) — 1] = a
2ttA;
sin 2 (7rfca)
7r 2 k 2 a 2
(6)
where sine x is the SlNC FUNCTION. The peak (in units
of a) is 1. The function I(x) is always positive, so there
are no Negative sidelobes. The extrema are given by
letting j3 = nka and solving
d ( sin j3
2 sin/9sin/3-/3cos/9 .
P
P
sin/3(sin/?-/?cos/3) =
sin/3-/3cos/3 =
tan/3 = /3.
(8)
(9)
(10)
Solving this numerically gives j3 = 4.49341 for the first
maximum, and the peak POSITIVE sidelobe is 0.047190.
The full width at half maximum is given by setting x =
nka and solving
sine x = | (11)
for #1/2, yielding
Ei/2 = 7rfci /2 a = 1.39156. (12)
Therefore, with L = 2a,
FWHM = 2fei /2 =
0.885895 1.77179
a
(13)
see a/so APODIZATION FUNCTION, PARZEN ApODIZA-
tion Function, Triangle Function
References
Bartlett, M. S. "Periodogram Analysis and Continuous Spec-
tra." Biometrika 37, 1-16, 1950.
Barycentric Coordinates
Also known as HOMOGENEOUS COORDINATES or TRI-
linear Coordinates.
see Trilinear Coordinates
Base Curve
see Directrix (Ruled Surface)
Base (Number) 99
Base (Logarithm)
The number used to define a LOGARITHM, which is then
written log 6 . The symbol logo; is an abbreviation for
log 10 x, In as for log e x (the Natural Logarithm), and
lga: for log 2 x.
see also e, Lg, Ln, Logarithm, Napierian Loga-
rithm, Natural Logarithm
Base (Neighborhood System)
A base for a neighborhood system of a point x is a col-
lection N of Open Sets such that x belongs to every
member of iV, and any Open Set containing x also con-
tains a member of N as a Subset.
Base (Number)
A Real Number x can be represented using any Inte-
ger number b as a base (sometimes also called a RADIX
or SCALE). The choice of a base yields to a representa-
tion of numbers known as a Number System. In base
6, the DIGITS 0, 1, . . . , b - 1 are used (where, by con-
vention, for bases larger than 10, the symbols A, B, C,
. . . are generally used as symbols representing the DEC-
IMAL numbers 10, 11, 12, . . . ).
Base
Name
2
binary
3
ternary
4
quaternary
5
quinary
6
senary
7
septenary
8
octal
9
nonary
10
decimal
11
undenary
12
duodecimal
16
hexadecimal
20
vigesimal
60
sexagesimal
Let the base b representation of a number x be written
(a n Cin-i ... ao- a_i . . .)*,,
(1)
(e.g., 123.456io), then the index of the leading DIGIT
needed to represent the number is
n = |k>g 6 x\ ,
(2)
where \_x\ is the FLOOR FUNCTION. Now, recursively
compute the successive Digits
ai = L?J •
where r n = x and
n-! = n
(lib 1
(3)
(4)
100
Base Space
Basis
for i = n, n — 1, . . . , 1,0, This gives the base b
representation of x. Note that if x is an Integer, then
i need only run through 0, and that if x has a fractional
part, then the expansion may or may not terminate.
For example, the HEXADECIMAL representation of 0.1
(which terminates in DECIMAL notation) is the infinite
expression 0.19999. . .h-
Some number systems use a mixture of bases for count-
ing. Examples include the Mayan calendar and the old
British monetary system (in which ha'pennies, pennies,
threepence, sixpence, shillings, half crowns, pounds, and
guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240,
and 252, respectively).
Knuth has considered using TRANSCENDENTAL bases.
This leads to some rather unfamiliar results, such as
equating -k to 1 in "base 7r," 7r = I*..
see also Binary, Decimal, Hereditary Represen-
tation, Hexadecimal, Octal, Quaternary, Sexa-
gesimal, Ternary, Vigesimal
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 28, 1972.
Bogomolny, A. "Base Converter." http : //www . cut-the-
knot . com/binary .html.
Lauwerier, II. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 6-11,
1991.
i$ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Bases.m.
Base Space
The Space B of a Fiber Bundle given by the Map
/ : E -> B, where E is the Total Space of the Fiber
Bundle.
see also FIBER BUNDLE, TOTAL SPACE
Baseball
The numbers 3 and 4 appear prominently in the game
of baseball. There are 3*3 = 9 innings in a game, and
three strikes are an out. However, 4 balls are needed for
a walk. The number of bases can either be regarded as
3 (excluding HOME Plate) or 4 (including it).
see Baseball Cover, Home Plate
A pair of identical plane regions (mirror symmetric
about two perpendicular lines through the center) which
can be stitched together to form a baseball (or tennis
ball). A baseball has a CIRCUMFERENCE of 9 1/8 inches.
The practical consideration of separating the regions far
enough to allow the pitcher a good grip requires that
the "neck" distance be about 1 3/16 inches. The base-
ball cover was invented by Elias Drake as a boy in the
1840s. (Thompson's attribution of the current design
to trial and error development by C. H. Jackson in the
1860s is apparently unsubstantiated, as discovered by
George Bart.)
One way to produce a baseball cover is to draw the re-
gions on a Sphere, then cut them out. However, it is
difficult to produce two identical regions in this man-
ner. Thompson (1996) gives mathematical expressions
giving baseball cover curves both in the plane and in
3-D. J. H. Conway has humorously proposed the follow-
ing "baseball curve conjecture:" no two definitions of
"the" baseball curve will give the same answer unless
their equivalence was obvious from the start.
see also Baseball, Home Plate, Tennis Ball The-
orem, Yin- Yang
References
Thompson, R. B. "Designing a Baseball Cover. 1860's: Pa-
tience, Trial, and Error. 1990's: Geometry, Calculus,
and Computation," http://www.mathsoft.com/asolve/
baseball/baseball. html. Rev. March 5, 1996.
Basin of Attraction
The set of points in the space of system variables such
that initial conditions chosen in this set dynamically
evolve to a particular Attractor.
see also Wada Basin
Basis
A (vector) basis is any Set of n LINEARLY INDEPEN-
DENT Vectors capable of generating an n-dimensional
SUBSPACE of R n . Given a IlYPERPLANE defined by
xi + x 2 + X3 4- x 4 + x$ = 0,
a basis is found by solving for Xi in terms of #2, #3, 2:4,
and £5. Carrying out this procedure,
Baseball Cover
Xi
-X2 — X3 — X4 — £5,
~Xi~
--1-
--1-
--1-
--1-
X2
1
X3
= x 2
+£3
1
~\-X4
-\-x 5
X4
1
-335-
. .
- -
. .
. 1 -
Basis Theorem
B ayes' Formula 101
and the above VECTOR form an (unnormalized) BASIS.
Given a MATRIX A with an orthonormal basis, the MA-
TRIX corresponding to a new basis, expressed in terms
of the original xi , . . . , x n is
A' = [Axi
Ax n ].
see also Bilinear Basis, Modular System Basis,
Orthonormal Basis, Topological Basis
Basis Theorem
see Hilbert Basis Theorem
Basler Problem
The problem of analytically finding the value of C(2),
where £ is the Riemann Zeta Function.
References
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Basset Function
see Modified Bessel Function of the Second
Kind
Batch
A set of values of similar meaning obtained in any man-
ner.
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison-Wesley, p. 667, 1977.
Bateman Function
Mx) - r(i < +V ( "^'°' fa)
for x > 0, where U is a Confluent Hypergeometric
Function of the Second Kind.
see also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL
Equation, Hypergeometric Function
Batrachion
A class of CURVE defined at Integer values which hops
from one value to another. Their name derives from the
word batrachion, which means "frog- like." Many ba-
trachions are FRACTAL. Examples include the BLANC-
MANGE Function, Hofstadter-Conway $10,000 Se-
quence, Hofstadter's Q-Sequence, and Mallow's
Sequence.
References
Pickover, C. A. "The Crying of Fractal Batrachion 1,489."
Ch. 25 in Keys to Infinity. New York: W. H. Freeman,
pp. 183-191, 1995.
Bauer's Identical Congruence
Let t(m) denote the set of the </>(m) numbers less than
and Relatively Prime to m, where <f>(n) is the To-
tient Function. Define
f m {x)= n (*-*)•
(i)
t(m)
A theorem of Lagrange states that
f m {x) = x Hm) -1 (mod to). (2)
This can be generalized as follows. Let p be an ODD
Prime Divisor of m and p a the highest Power which
divides to, then
f m (x) = (x*- 1 - l)*^)/^- 1 ) (mod p») (3)
and, in particular,
/„.(*) = (a*" 1 -l)*" -1 (mod/). (4)
Furthermore, if to > 2 is EVEN and 2 a is the highest
POWER of 2 that divides m, then
/ m (a:) = (a: 2 -l)* (m)/2 (mod 2 a ) (5)
and, in particular,
f 2a ( x ) = ( x 2 -l) 2a ~ 2 (mod2 a ). (6)
see also Leudesdorf Theorem
References
Hardy, G. H. and Wright, E. M. "Bauer's Identical Congru-
ence." §8.5 in An Introduction to the Theory of Numbers,
5th ed. Oxford, England: Clarendon Press, pp. 98-100,
1979.
Bauer's Theorem
see Bauer's Identical Congruence
Bauspiel
A construction for the RHOMBIC DODECAHEDRON.
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 26 and 50, 1973.
Bayes' Formula
Let A and Bj be Sets. Conditional Probability
requires that
P(AC\B j )=P(A)P(B j \Al (1)
where n denotes INTERSECTION ("and"), and also that
P(A n Bj) = P(Bj n A) - P{Bj)P{A\Bj) (2)
102 Bayes' Theorem
and
P{B j nA)=P{B j )P{A\B j ). (3)
Since (2) and (3) must be equal,
P(AnB j ) = P(B j nA). (4)
Prom (2) and (3),
P(AnB j ) = P(B j )P(A\B j ). (5)
Equating (5) with (2) gives
P(A)P(B j \A) = P(B i )P(A\B j ), (6)
so
P(Bj\A)
PjB^PjAlBj)
P(A) ■
(7)
Now, let
S=U^> (8)
i=l
so Ai is an event is S and A» O Aj = for i ^ j, then
/ N \ JV
A = A n 5 - A n ( (J ^ J = (J (A n Ai) (9)
\ N
P(A) = Pl\J(AnA i )\=Y i P(AnA i ). (10)
Prom (5), this becomes
N
P(A) = Y,P(Ai)P(E\Ai), (11)
i=l
SO
P{Ai)P(A\Ai)
P(Ai\A) N
£ P(Ai)P(A\Ai)
3 = 1
(12)
5ee also CONDITIONAL PROBABILITY, INDEPENDENT
Statistics
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 810, 1992.
Bayes' Theorem
see Bayes' Formula
Beam Detector
Bayesian Analysis
A statistical procedure which endeavors to estimate pa-
rameters of an underlying distribution based on the ob-
served distribution. Begin with a "PRIOR Distribu-
tion" which may be based on anything, including an
assessment of the relative likelihoods of parameters or
the results of non-Bayesian observations. In practice, it
is common to assume a UNIFORM DISTRIBUTION over
the appropriate range of values for the PRIOR Distri-
bution.
Given the Prior Distribution, collect data to obtain
the observed distribution. Then calculate the LIKELI-
HOOD of the observed distribution as a function of pa-
rameter values, multiply this likelihood function by the
PRIOR Distribution, and normalize to obtain a unit
probability over all possible values. This is called the
Posterior Distribution. The Mode of the distribu-
tion is then the parameter estimate, and "probability
intervals" (the Bayesian analog of Confidence Inter-
vals) can be calculated using the standard procedure.
Bayesian analysis is somewhat controversial because the
validity of the result depends on how valid the PRIOR
DISTRIBUTION is, and this cannot be assessed statisti-
cally.
see also Maximum Likelihood, Prior Distribution,
Uniform Distribution
References
Hoel, P. G.; Port, S. C; and Stone, C. J. Introduction to
Statistical Theory. New York: Houghton Mifflin, pp. 36-
42, 1971.
Iversen, G. R. Bayesian Statistical Inference. Thousand
Oaks, CA: Sage Pub., 1984.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 799-806, 1992.
Sivia, D. S. Data Analysis: A Bayesian Tutorial. New York:
Oxford University Press, 1996.
Bays' Shuffle
A shuffling algorithm used in a class of RANDOM NUM-
BER generators.
References
Knuth, D. E. §3.2 and 3.3 in The Art of Computer Program-
ming, Vol. 2: Seminumerical Algorithms, 2nd ed. Read-
ing, MA: Addison-Wesley, 1981.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 270-271, 1992.
Beam Detector
N. B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Bean Curve
A "beam detector" for a given curve C is defined as
a curve (or set of curves) through which every Line
tangent to or intersecting C passes. The shortest 1-
arc beam detector, illustrated in the upper left figure,
has length L\ — n + 2. The shortest known 2-arc beam
detector, illustrated in the right figure, has angles
Ox « 1.286 rad
6 2 « 1.191 rad,
(1)
(2)
given by solving the simultaneous equations
2 cos <9i -sin(§0 2 ) = (3)
tan(§0i)cos(f 2 ) + sm{±0 2 )[sec 2 {±6 2 ) + 1] = 2. (4)
The corresponding length is
L 2 =2tt-26>i -0 2 + 2tan(§0i)+sec(|0 2 )
- cos(§<9 2 )+tan(§6>i) sin(±<9 2 ) = 4.8189264563. . . . (5)
A more complicated expression gives the shortest known
3-arc length L 3 = 4.799891547. . .. Finch defines
L = inf L n
n>l
(6)
as the beam detection constant, or the Trench Dig-
gers' Constant. It is known that L>n.
References
Croft, H, T.; Falconer, K, J.; and Guy, R. K. §A30 in Un-
solved Problems in Geometry. New York: Springer- Verlag,
1991.
Faber, V.; Mycielski, J.; and Pedersen, P. "On the Shortest
Curve which Meets All Lines which Meet a Circle." Ann.
Polon. Math. 44, 249-266, 1984.
Faber, V. and Mycielski, J. "The Shortest Curve that Meets
All Lines that Meet a Convex Body." Amer. Math.
Monthly 93, 796-801, 1986.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/beam/beam.html.
Makai, E. "On a Dual of Tarski's Plank Problem." In
Diskrete Geometric 2 Kolloq., Inst. Math. Univ. Salzburg,
127-132, 1980.
Stewart, L "The Great Drain Robbery." Sci. Amer., 206-
207, 106, and 125, Sept. 1995, Dec. 1995, and Feb. 1996.
Bean Curve
Beast Number 103
The Plane Curve given by the Cartesian equation
x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 ).
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.
Beast Number
The occult "number of the beast" associated in the Bible
with the Antichrist. It has figured in many numerolog-
ical studies. It is mentioned in Revelation 13:13: "Here
is wisdom. Let him that hath understanding count the
number of the beast: for it is the number of a man; and
his number is 666."
The beast number has several interesting properties
which numerologists may find particularly interesting
(Keith 1982-83). In particular, the beast number is
equal to the sum of the squares of the first 7 PRIMES
2 2 + 3 2 -h 5 2 + 7 2 + ll 2 + 13 2 + 17 2 = 666, (1)
satisfies the identity
0(666) = 6-6-6, (2)
where 4> is the Totient Function, as well as the sum
^2 = 666.
(3)
The number 666 is a sum and difference of the first three
6th Powers,
666 = l 6 - 2 6 + 3 6 (4)
(Keith). Another curious identity is that there are ex-
actly two ways to insert "+" signs into the sequence
123456789 to make the sum 666, and exactly one way
for the sequence 987654321,
666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9
(5)
666 = 9 + 87 + 6 + 543 + 21 (6)
(Keith). 666 is a Repdigit, and is also a Triangular
Number
T 6 . 6 = T 36 = 666. (7)
In fact, it is the largest Repdigit Triangular Num-
ber (Bellew and Weger 1975-76). 666 is also a Smith
Number. The first 144 Digits of n - 3, where n is Pi,
add to 666. In addition 144 = (6 + 6) x (6 + 6) (Blatner
1997).
A number of the form 2 1 which contains the digits of the
beast number "666" is called an Apocalyptic Num-
ber, and a number having 666 digits is called an APOC-
ALYPSE Number.
104 Beatty Sequence
Bei
see also Apocalypse Number, Apocalyptic Num-
ber, Bimonster, Monster Group
References
Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num-
bers." J. Recr. Math. 8, 96-97, 1975-76.
Blatner, D. The Joy of Pi. New York: Walker, back jacket,
1997.
Castellanos, D. "The Ubiquitous tt." Math. Mag. 61, 153-
154, 1988.
Hardy, G. H. A Mathematician's Apology, reprinted with a
foreword by C. P. Snow. New York: Cambridge University
Press, p, 96, 1993.
Keith, M. "The Number of the Beast." http://users.aol.
com/s6sj7gt/mike666.htm.
Keith, M. "The Number 666." J. Recr. Math. 15, 85-87,
1982-1983.
Bee
A 4-P0LYHEX.
References
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of -Mind from Scientific American. New York:
Vintage, p. 147, 1978.
Behrens-Fisher Test
see Fisher-Behrens Problem
Beatty Sequence
The Beatty sequence is a Spectrum Sequence with an
Irrational base. In other words, the Beatty sequence
corresponding to an Irrational Number 6 is given by
[0J, [20 \, [30J, . . . , where \_x\ is the Floor Function.
If a and f3 are Positive Irrational Numbers such
that
1 1 ,
a p
then the Beatty sequences [a J , [2aJ , . . . and [f3\ , \_W\ >
. . . together contain all the POSITIVE INTEGERS without
repetition.
References
Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the
Return of Dr. Matrix, reissue ed. New York: W. H. Free-
man, p. 21, 1989.
Graham, R. L.; Lin, S.; and Lin, C.-S. "Spectra of Numbers."
Math. Mag. 51, 174-176, 1978.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 227, 1994.
Sloane, N. J. A. A Handbook of Integer Sequences. Boston,
MA: Academic Press, pp. 29-30, 1973.
Beauzamy and Degot's Identity
For P, Q, R, and S POLYNOMIALS in n variables
[PQ,RS]= J^
A
ii,...,t n >0
iil---i n ] -
vhere
A=[Rl i i>-"M(D li ...,D n )Q(x u ... i x n )
XP (il, - ,iB) (ft 2?n)5(Xl,.. M In)]
Di = d/dxi is the Differential Operator, [X,Y] is
the Bombieri Inner Product, and
p(ti,...,i™) =D i 1 1 -.-D i r TP.
Behrraann Cylindrical Equal- Area
Projection
A Cylindrical Area-Preserving projection which
uses 30° N as the no-distortion parallel.
References
Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraft/notes/mapproj/mapproj ,html,
Bei
I Bei z|
10000
.10 5000
-1000UE^^^^^/5 -500
Re[z] ^i^-lO Re[z]" 5 ^10
The Imaginary Part of
J„(xe 3vi/4 ) = ber„(a;) +ibei„(x). (1)
The special case v = gives
Jo(iVix) = ber(rc) + ibei(sc), (2)
where J Q (z) is the zeroth order BESSEL FUNCTION OF
the First Kind.
bei (x) = ^ [(2n) , ]2
(3)
see also Reznik's Identity
see also Ber, Bessel Function, Kei, Kelvin Func-
tions, Ker
Bell Curve
Bell Number 105
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func-
tions." §9.9 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 379-381, 1972.
Spanier, J. and Oldham, K. B. "The Kelvin Functions."
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 543-554, 1987.
Bell Curve
see Gaussian Distribution, Normal Distribution
Bell Number
The number of ways a Set of n elements can be PARTI-
TIONED into nonempty Subsets is called a Bell Num-
ber and is denoted B n . For example, there are five
ways the numbers {1, 2, 3} can be partitioned: {{1},
{2}, {3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{1}, {2, 3}},
and {{1, 2, 3}}, so B 3 = 5. B = 1 and the first few
Bell numbers for n = 1, 2, ... are 1, 2, 5, 15, 52, 203,
877, 4140, 21147, 115975, ... (Sloane's A000110). Bell
numbers are closely related to CATALAN NUMBERS.
The diagram below shows the constructions giving B 3 =
5 and B± = 15, with line segments representing elements
in the same Subset and dots representing subsets con-
taining a single element (Dickau).
B,
The Integers B n can be defined by the sum
fc=i ^ J
= {l} isa
(i)
where s£ fc) = i '," *> is a STIRLING NUMBER OF THE
SECOND Kind, or by the generating function
B„
6 = 2^
(2)
The Bell numbers can also be generated using the BELL
Triangle, using the Recurrence Relation
Jn+l
(3)
where (£) is a Binomial Coefficient, or using the
formula of Comtet (1974)
B n
-E
m
(4)
where \x] denotes the Ceiling Function.
The Bell number B n is also equal to n (l), where <t> n (x)
is a Bell Polynomial. Dobinski's Formula gives
the nth Bell number
oo
(5)
Lovasz (1993) showed that this formula gives the asymp-
totic limit
-1/2
[A(n)]
n+l/2 A(n)-n-l
where A(n) is defined implicitly by the equation
A(n)log[A(n)] = n.
A variation of DOBINSKI'S FORMULA gives
- -«■ ( _ 1)S
B * = E 5- E
(6)
(?)
(8)
for 1 < k < n (Pitman 1997). de Bruijn (1958) gave the
asymptotic formula
InBn , , , In Inn 1
= lnn — Inlnn — 1 + — h - —
n Inn Inn
WlnlnnX 2
^2 V Inn /
In Inn
(Inn) 2
Touchard's Congruence states
B p+k = B k + B k+1 (mod p) ,
(9)
(10)
when p is Prime. The only PRIME Bell numbers for
n < 1000 are B 2 , B 3i B 7 , B 13 , B 42 , and £55. The Bell
numbers also have the curious property that
Bq B\
B\ £2
B n
?n + l
B 2
B 3
B n ^
B n
B n +i
B 2n
J[n\ (11)
(Lenard 1986).
see also Bell Polynomial, Bell Triangle, Dobin-
ski's Formula, Stirling Number of the Second
Kind, Touchard's Congruence
106 Bell Polynomial
Beltrami Differential Equation
References
Bell, E. T. "Exponential Numbers." Amer. Math. Monthly
41, 411-419, 1934.
Comtet, L. Advanced Combinatorics. Dordrecht, Nether-
lands: Reidel, 1974.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 91-94, 1996.
de Bruijn, N. G. Asymptotic Methods in Analysis. New York:
Dover, pp. 102-109, 1958.
Dickau, R. M. "Bell Number Diagrams." http:// forum .
swarthmore.edu/advanced/robertd/bell.html.
Gardner, M. "The Tinkly Temple Bells." Ch. 2 in Fractal
Music, HyperCards, and More Mathematical Recreations
from Scientific American Magazine. New York: W. H.
Freeman, 1992.
Gould, H. W. Bell & Catalan Numbers: Research Bibliogra-
phy of Two Special Number Sequences, 6th ed. Morgan-
town, WV: Math Monongliae, 1985.
Lenard, A. In Fractal Music, HyperCards, and More Math-
ematical Recreations from Scientific American Magazine.
(M. Gardner). New York: W. H. Freeman, pp. 35-36,
1992.
Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p,
of First Order Bell Exponential Integrals." Math. Comput.
16, 416-423, 1962.
Lovasz, L. Combinatorial Problems and Exercises, 2nd ed.
Amsterdam, Netherlands: North-Holland, 1993.
Pitman, J. "Some Probabilistic Aspects of Set Partitions."
Amer. Math. Monthly 104, 201-209, 1997.
Rota, G.-C. "The Number of Partitions of a Set." Amer.
Math. Monthly 71, 498-504, 1964.
Sloane, N. J. A. Sequence A000110/M1484 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Bell Polynomial
0.2 0.4 0.6 0.8 1
Two different GENERATING FUNCTIONS for the Bell
polynomials for n > are given by
<t> n {x) =e x ^
k n ~ 1 x k
The Bell polynomials are denned such that <f> n (l) = B nj
where B n is a Bell NUMBER. The first few Bell poly-
nomials are
4>o(x
<pi(x
4>2(x
fo{x
(J>a{x
<p 6 (x
= 1
= X
= x + x 2
- x + 3z 2 + x 3
= x + 7x 2 + 6x 3 + x 4
= x 4- 15x 2 + 25a; 3 + 10z 4 + x 5
= x + Six 2 + 90x 3 + 65z 4 + 15a; 5 + x 6 .
see also Bell Number
References
Bell, E. T. "Exponential Polynomials."
258-277, 1934.
Ann. Math. 35,
Bell Triangle
12 5 15 52 203 877 ...
1 3 10 37 151 674 \
2 7 27 114 523 \
5 20 87 409 \
15 67 322 \
52 255 •■.
203 ■-.
A triangle of numbers which allow the Bell Numbers
to be computed using the Recurrence Relation
= Va
B n+1 = 2^B k { n k
k-o
see also Bell Number, Clark's Triangle, Leibniz
Harmonic Triangle, Number Triangle, Pascal's
Triangle, Seidel-Entringer-Arnold Triangle
Bellows Conjecture
see Flexible Polyhedron
Beltrami Differential Equation
For a measurable function /z, the Beltrami differential ,
equation is given by
n~ 1 s v
4> n (x) = x^2 [ k-1 j^" 1 ^)'
where (£) is a Binomial Coefficient.
where f z is a PARTIAL DERIVATIVE and z* denotes the
Complex Conjugate of z.
see also QUASICONFORMAL MAP
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1087,
1980.
Beltrami Field
Ben ford's Law 107
Beltrami Field
A Vector Field u satisfying the vector identity
u x (V x u) =
where A x B is the CROSS Product and V x A is the
CURL is said to be a Beltrami field.
see also DlVERGENCELESS FIELD, IRROTATIONAL
Field, Solenoidal Field
Beltrami Identity
An identity in CALCULUS OF VARIATIONS discovered in
1868 by Beltrami. The Euler-Lagrange Differen-
tial Equation is
d£__d_
dy dx
(&)-*
Now, examine the DERIVATIVE of x
— ~ l/x T" n t/xx ~r •
ax oy oy x ox
Solving for the df /dy term gives
dy 1
dx dy x
0/ b
dx'
Now, multiplying (1) by y x gives
(i)
(2)
(3)
(4)
(5)
(6)
This form is especially useful if f x = 0, since in that case
0/ _ d_
oy ax
dy* J
Substituting (3) into (4) then gives
dx
dy x Vxz dx
x dx \dy x
dx dx \
- y *dyZ) =
dx
which immediately gives
/
dy x
= 0,
dy x
(7)
(8)
where C is a constant of integration.
The Beltrami identity greatly simplifies the solution for
the minimal AREA SURFACE OF REVOLUTION about
a given axis between two specified points. It also al-
lows straightforward solution of the BRACHISTOCHRONE
Problem.
see also Brachistochrone Problem, Calculus of
Variations, Euler-Lagrange Differential Equa-
tion, Surface of Revolution
Bend (Curvature)
Given four mutually tangent circles, their bends are de-
fined as the signed CURVATURES of the CIRCLES. If the
contacts are all external, the signs are all taken as Pos-
itive, whereas if one circle surrounds the other three,
the sign of this circle is taken as NEGATIVE (Coxeter
1969).
see also Curvature, Descartes Circle Theorem,
Soddy Circles
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, pp. 13-14, 1969.
Bend (Knot)
A Knot used to join the ends of two ropes together to
form a longer length.
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 49, 1993.
Benford's Law
Also called the FIRST DIGIT LAW, FIRST DIGIT PHE-
NOMENON, or Leading Digit Phenomenon. In list-
ings, tables of statistics, etc., the DIGIT 1 tends to oc-
cur with Probability ~ 30%, much greater than the
expected 10%. This can be observed, for instance, by
examining tables of LOGARITHMS and noting that the
first pages are much more worn and smudged than later
pages. The table below, taken from Benford (1938),
shows the distribution of first digits taken from several
disparate sources. Of the 54 million real constants in
Plouffe's "Inverse Symbolic Calculator" database, 30%
begin with the Digit 1.
Title
First Digit
#
12 3
4
5 6 7 8
9
Rivers, Area
31.0 16.4 10.7 11.3
7.2 8.6 5.5 4.2
5.1
335
Population
33.9 20.4 14.2
8.1
7.2 6.2 4.1 3.7
2.2 3259
Constants
41.3 14.4 4.8
8.6 10.6 5.8 1.0 2.9 10.6
104
Newspapers
30.0 18.0 12.0
10.0
8.0 6.0 6.0 5.0
5.0
100
Specific Heat
24.0 18.4 16.2
14.6 10.6 4.1 3.2 4.8
4.1
1389
Pressure
29.6 18.3 12.8
9.8
8.3 6.4 5.7 4.4
4.7
703
H.P. Lost
30.0 18.4 11.9
10.8
8.1 7.0 5.1 5.1
3.6
690
Mol. Wgt.
26.7 25.2 15.4
10.8
6.7 5.1 4.1 2.8
3.2 1800
Drainage
27.1 23.9 13.8
12.6
8.2 5.0 5.0 2.5
1.9
159
Atomic Wgt.
47.2 18.7 5.5
4.4
6.6 4.4 3.3 4.4
5.5
91
n" 1 , sfn
25.7 20.3 9.7
6.8
6.6 6.8 7.2 8.0
8.9 5000
Design
26.8 14.8 14.3
7.5
8.3 8.4 7.0 7.3
5.6
560
Reader's Dig.
33.4 18.5 12.4
7.5
7.1 6.5 5.5 4.9
4.2
308
Cost Data
32.4 18.8 10.1
10.1
9.8 5.5 4.7 5.5
3.1
741
X-Ray Volts
27.9 17.5 14.4
9.0
8.1 7.4 5.1 5.8
4.8
707
Am. League
32.7 17.6 12.6
9.8
7.4 6.4 4.9 5.6
3.0
1458
Blackbody
31.0 17.3 14.1
8.7
6.6 7.0 5.2 4.7
5.4 1165
Addresses
28.9 19.2 12.6
8.8
8.5 6.4 5.6 5.0
5.0
342
n 1 , n 2 - • - n\
25.3 16.0 12.0 10.0
8.5 8.8 6.8 7.1
5.5
900
Death Rate
27.0 18.6 15.7
9.4
6.7 6.5 7.2 4.8
4.1
418
Average
30.6 18.5 12.4
9.4
8.0 6.4 5.1 4.9
4.7 1011
Prob. Error
0.8 0.4 0.4
0.3
0.2 0.2 0.2 0.2
0.3
108
Benham's Wheel
Benson's Formula
In fact, the first SIGNIFICANT DIGIT seems to follow a
Logarithmic Distribution, with
P(n) « log(n + 1) - logn
for n — 1, . . . , 9. One explanation uses Central Limit-
like theorems for the MANTISSAS of random variables
under Multiplication. As the number of variables in-
creases, the density function approaches that of a LOG-
ARITHMIC DISTRIBUTION.
References
Benford, F. "The Law of Anomalous Numbers." Proc. Amer.
Phil Soc. 78, 551-572, 1938.
Boyle, J. "An Application of Fourier Series to the Most Sig-
nificant Digit Problem." Amer. Math. Monthly 101, 879™
886, 1994.
Hill, T. P. "Base-Invariance Implies Benford 's Law." Proc.
Amer. Math. Soc. 12, 887-895, 1995.
Hill, T. P. "The Significant-Digit Phenomenon." Amer.
Math. Monthly 102, 322-327, 1995.
Hill, T. P. "A Statistical Derivation of the Significant-Digit
Law." Stat Sci. 10, 354-363, 1996.
Hill, T. P. "The First Digit Phenomenon." Amer. Sci. 86,
358-363, 1998.
Ley, E. "On the Peculiar Distribution of the U.S. Stock In-
dices Digits." Amer. Stat. 50, 311-313, 1996.
Newcomb, S. "Note on the Frequency of the Use of Digits in
Natural Numbers." Amer. J. Math. 4, 39-40, 1881.
Nigrini, M. "A Taxpayer Compliance Application of Ben-
ford's Law." J. Amer. Tax. Assoc. 18, 72-91, 1996.
Plouffe, S. "Inverse Symbolic Calculator." http://www.cecm.
sfu.ca/projects/ISC/.
Raimi, R. A. "The Peculiar Distribution of First Digits." Sci.
Amer. 221, 109-119, Dec. 1969.
Raimi, R. A. "The First Digit Phenomenon." Amer. Math,
Monthly 83, 521-538, 1976.
Benham's Wheel
An optical ILLUSION consisting of a spinnable top
marked in black with the pattern shown above. When
the wheel is spun (especially slowly), the black broken
lines appear as green, blue, and red colored bands!
References
Cohen, J. and Gordon, D. A. "The Prevost-Fechner-Benham
Subjective Colors." Psycholog. Bull. 46, 97-136, 1949.
Festinger, L.; Allyn, M. R.; and White, C. W. "The Percep-
tion of Color with Achromatic Stimulation." Vision Res.
11, 591-612, 1971.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. 148-151, 1996.
Trolland, T. L. "The Enigma of Color Vision." Amer. J.
Physiology 2, 23-48, 1921.
Bennequin's Conjecture
A BRAID with M strands and R components with P
positive crossings and N negative crossings satisfies
\P - N\ < 2U + M - R < P + iV,
where U is the UNKNOTTING NUMBER. While the
second part of the Inequality was already known to
be true (Boileau and Weber, 1983, 1984) at the time
the conjecture was proposed, the proof of the entire
conjecture was completed using results of Kronheimer
and Mrowka on MlLNOR'S CONJECTURE (and, indepen-
dently, using Menasco's Theorem).
see also Braid, Menasco's Theorem, Milnor's Con-
jecture, Unknotting Number
References
Bennequin, D. "L'instanton gordien (d'apres P. B. Kron-
heimer et T. S. Mrowka)." Asterisque 216, 233-277, 1993.
Birman, J. S. and Menasco, W. W. "Studying Links via
Closed Braids. II. On a Theorem of Bennequin." Topology
Appl. 40, 71-82, 1991.
Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le
nombre gordien des nceuds algebriques." Enseign. Math.
30, 173-222, 1984.
Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le
nombre gordien des nceuds algebriques." In Knots, Braids
and Singularities (Plans- sur- Bex, 1982). Geneva, Switzer-
land: Monograph. Enseign. Math. Vol. 31, pp. 49-98,
1983.
Cipra, B. What's Happening in the Mathematical Sciences,
Vol. 2. Providence, RI: Amer. Math. Soc, pp. 8-13, 1994.
Kronheimer, P. B. "The Genus-Minimizing Property of Al-
gebraic Curves." Bull. Amer. Math. Soc. 29, 63-69, 1993.
Kronheimer, P. B. and Mrowka, T. S, "Gauge Theory for
Embedded Surfaces. I." Topology 32, 773-826, 1993.
Kronheimer, P. B. and Mrowka, T. S. "Recurrence Relations
and Asymptotics for Four-Manifold Invariants." Bull.
Amer. Math. Soc. 30, 215-221, 1994.
Menasco, W. W. "The Bennequin-Milnor Unknotting Con-
jectures." C. R. Acad. Sci. Paris Ser. I Math. 318, 831-
836, 1994,
Benson's Formula
An equation for a LATTICE SUM with n = 3
i+i+fe+l
i, j,k= — oo V J
= 12?r ^ sech 2 (!7iVm 2 +n 2 ).
m, n=l, 3, ...
Here, the prime denotes that summation over (0, 0, 0) is
excluded. The sum is numerically equal to —1.74756 . . .,
a value known as "the" MADELUNG CONSTANT.
see also MADELUNG CONSTANTS
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, p. 301, 1987.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/mdlung/mdlTing.html.
Ber
Ber
| Ber z |
The Real Part of
J„(xe 3ni/4 ) = beT v (x)+ibei v (x).
The special case v = gives
Jo(iV^x) = ber(:r) + zbei(z),
(1)
(2)
where J is the zeroth order BESSEL FUNCTION OF THE
First Kind.
i 2+4n
ber( !B ) = ^ [(2n + 1)!] 2 ■
(3)
see a/so Bei, Bessel Function, Kei, Kelvin Func-
tions, Ker
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func-
tions." §9.9 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 379-381, 1972.
Spanier, J. and Oldham, K. B. "The Kelvin Functions."
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 543-554, 1987.
Beraha Constants
The nth Beraha constant is given by
'2tt\
Be„ = 2 + 2 cos
(!)-
The first few are
Bei =4
Be 2 =
Be 3 = 1
Be 4 = 2
Be 5 = |(3 + \/5)« 2.618
Be 6 = 3
Be 7 = 2 + 2cos(|7r) « 3.247....
They appear to be ROOTS of the CHROMATIC POLY-
NOMIALS of planar triangular GRAPHS. Be 4 is 0+1,
where <p is the Golden Ratio, and Be 7 is the Silver
Constant.
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 143, 1983.
Bernoulli Differential Equation 109
Berger-Kazdan Comparison Theorem
Let M be a compact n-D Manifold with Injectivity
radius inj(M). Then
Vol(M) >
qnj(M)
with equality IFF M is ISOMETRIC to the standard round
Sphere S n with Radius inj(M), where c n {r) is the
Volume of the standard u-Hypersphere of Radius
r.
see also Blaschke Conjecture, Hypersphere, In-
jective, Isometry
References
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Bergman Kernel
A Bergman kernel is a function of a COMPLEX VARI-
ABLE with the "reproducing kernel" property defined
for any Domain in which there exist NONZERO Ana-
lytic Functions of class L 2 (D) with respect to the
Lebesgue Measure dV.
References
Hazewinkel, M. (Managing Ed,). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, pp. 356-357, 1988.
Bergman Space
Let G be an open subset of the COMPLEX PLANE C, and
let Ll(G) denote the collection of all Analytic Func-
tions / : G — y C whose Modulus is square integrable
with respect to Area measure. Then L 2 a {G), sometimes
also denoted A 2 (G), is called the Bergman space for G.
Thus, the Bergman space consists of all the ANALYTIC
FUNCTIONS in L 2 (G). The Bergman space can also be
generalized to L?(G), where < p < oo.
Bernoulli Differential Equation
-£ + p(x)y = q(x)y n .
Let v ~ y 1 " 71 for n ^ 1, then
dv , . - n dy
— — (l - n)y — .
dx v }y dx
Rewriting (1) gives
y~ n f; = q(x) - p{x)y'~ n = q(x) - vp(x).
Plugging (3) into (2),
dv
— = (I - n)[q(x) - vp(x)].
(1)
(2)
(3)
(4)
110
Bernoulli Distribution
Bernoulli Function
Now, this is a linear First-Order Ordinary Differ-
ential Equation of the form
^+vP(x) = Q{x),
(5)
where P(x) = (l-n)p(x) and Q(x) = (l-n)q(x). It can
therefore be solved analytically using an Integrating
Factor
/
Jri'U
c Q(x)dx + C
J P(x)dx
_ (1 - n) / e (1 - n) / pW dx g(x) dx + C
(1-n) J p(x) dx
(6)
where C is a constant of integration. If n = 1, then
equation (1) becomes
dy
dx
= y(q-p)
— = {q-p)dx
(?)
(8)
y = C 2 ef [q{x) - p{x)]dx . (9)
The general solution is then, with C\ and C2 constants,
l/(l-n)
y= <
'(l-nlp 1 - 10 /*
a;) da;
4(3;) das+Ci
(1
r^j
p(ar) da:
for n ^ 1
C e/ te(s)-p(x)]dx
for n = 1.
(10)
Bernoulli Distribution
A Distribution given by
p M = {l
q = 1 — p for n =
for n — 1
— p n (l— p) n for n = 0,1.
(i)
(2)
The distribution of heads and tails in COIN TOSSING is
a Bernoulli distribution with p = q — 1/2. The GENER-
ATING FUNCTION of the Bernoulli distribution is
1
M « = (*'"> = E ^w - p) 1_n = e °( 1 - p) + e ^
(3)
SO
M(t) = (l-p)+pe t
(4)
M'{t) = pe
(5)
M"(t)=pe t
(6)
M (n) (t)=pe\
(7)
and the Moments about are
Ml=M = M'(0)=p (8)
& = M"(0) = p (9)
M ; = M ( " ) (0)=p. (10)
The Moments about the Mean are
P2 = p' 2 -(pi) 2 =P-P 2 =p(l-p) (11)
p.3=p 3 - 3^2Pi + 2(p' 1 ) 3 = p - 3p 2 + 2p 3
= p(l-p)(l-2p) (12)
P4 = A»4 - 4/i3pi + 6^2 (m!) 2 - 3(pi) 4
= p - 4p 2 + 6p 3 - 3p 4
= p(l-p)(3p 2 -3p+l).
(13)
The Mean, Variance, Skewness, and Kurtosis are
then
P = Pi = P
cr 2 - p.2 = p(l - p)
_fi 3 _ p(l-p)(l-2p)
71 <T 3 [p(l - p)]3/ 2
_ l-2p
H4 p(l-2p)(2p 2 -2p+l)
72 = —t - 3 =
P 2 (l-P) 2
6p 2 - 6p + 1
p(l-p)
To find an estimator for a population mean,
V^ ( N
(14)
(15)
(16)
(17)
■0)
JVp=0 v /
Np=l
= e[e + (i-8)] N - 1 = e,
(18)
so (p) is an Unbiased Estimator for 9, The probabil-
ity of Np successes in N trials is then
N
Np
e Np (i-o) Nq ,
(19)
where
__ [number of successes] _ n
p- x =77- _ (20)
see also BINOMIAL DISTRIBUTION
Bernoulli Function
see Bernoulli Polynomial
Bernoulli Inequality
Bernoulli Inequality
(l + x) n > 1 + nx,
(1)
where x£l> — 1^0, n€Z> 1. This inequality can
be proven by taking a MACLAURIN SERIES of (1 + x) n ,
Bernoulli Number 111
B n Bernoulli numbers may be calculated from the inte-
gral
(3)
Bn=4n L **=r
and analytically from
(l+x) n = l+n^+|n(n-l)x +|n(n-l)(n-2)a; +....
(2)
Since the series terminates after a finite number of terms
for INTEGRAL n, the Bernoulli inequality for x > is
obtained by truncating after the first-order term. When
— 1 < x < 0, slightly more finesse is needed. In this case,
let y = \x\ = — cc > so that < y < 1, and take
(l-y) n = l-ny+in(n-l)y 2 - in(n-l)(n-2)y 3 + . . . .
(3)
Since each Power of y multiplies by a number < 1 and
since the ABSOLUTE VALUE of the COEFFICIENT of each
subsequent term is smaller than the last, it follows that
the sum of the third order and subsequent terms is a
Positive number. Therefore,
(i - vT > i
ny,
(4)
(1 -f x) n > 1 + nx, for - 1 < x < 0, (5)
completing the proof of the INEQUALITY over all ranges
of parameters.
Bernoulli Lemniscate
see Lemniscate
Bernoulli Number
There are two definitions for the Bernoulli numbers. The
older one, no longer in widespread use, defines the Ber-
noulli numbers B* by the equations
-12 *-*>
n — 1 r>* ™2n
i-rr^B^x
(2n)!
B{x 2 B$x A Btx [
2!
+
4! 6!
-f ... (1)
for \x\ < 27r, or
2(2n)! v .
p=i
2(2n)!
(2tt) 2 "
C(2r
(4)
where ((z) is the RlEMANN Zeta
Function.
The first few Bernoulli numbers B* are
b;
=
i
6
b;
=
1
30
b;
=
1
42
bx
=
1
30
b;
=
5
66
bi
=
691
2,730
b;
=
7
6
B' 8
=
3,617
510
b;
=
43,867
798
^10
=
174,611
330
*n
=
854,513
138
Bernoulli numbers defined by the modern definition are
denoted B n and also called "EVEN-index" Bernoulli
numbers. These are the Bernoulli numbers returned by
the Mathematical (Wolfram Research, Champaign, IL)
function BernoulliB[n] . These Bernoulli numbers are
a superset of the archaic ones B n since
r 1
B n
for n =
for n = 1
(-l)^/ 2 )- 1 ^;^ for n even
< for n odd.
The B n can be defined by the identity
B n x n
(5)
(6)
, x (x\ ^ B n x 2r
'- 2 COt (2J S T,~§M
2!
+
B* 2 x A
4!
+
D* ™6
-P3^
6!
+ ... (2)
for \x\ < 7T (Whittaker and Watson 1990, p. 125). Grad-
shteyn and Ryzhik (1979) denote these numbers B n ,
while Bernoulli numbers defined by the newer (National
Bureau of Standards) definition are denoted B, The
These relationships can be derived using the generating
function
F(*,t) = £*££,
(7)
which converges uniformly for \t\ < 2tt and all x (Castel-
lanos 1988). Taking the partial derivative gives
dF(x,t) _ A B n ^(x)t n _ + ^ B n {x)t n
dx
Z— < ( n - i)! Z-, n \
(8)
112 Bernoulli Number
The solution to this differential equation is
F(x,t) = T(t)e xt ,
so integrating gives
/ F(x,t)dx = T(t) / e xt dx = T{t)^—- -
./o Jo l
00 *«- r 1
n = l * / °
(9)
(a;) da?
1 +
te
_ 1 ~ 2^ n :
(a;)da; = 1 (10)
(11)
(Castellanos 1988). Setting x = and adding t/2 to
both sides then gives
B2nt
itcoth(It) = ^
n—O
Letting t = 2ix then gives
00 . 2
xcotx = ^(-i)"^*^
n=0
(12)
2a 2
(2n)!
(13)
for x 6 [— 7r,7r], The Bernoulli numbers may also be
calculated from the integral
n! f z dz
n=r 2^7 ^TT^+T'
(14)
(15)
or from
Bn= \ dn x '
[dx n e x — 1_
The Bernoulli numbers satisfy the identity
*t>H*r)*- + - + (*i> +fl —-
(16)
where (£) is a BINOMIAL COEFFICIENT. An asymptotic
Formula is
lim \B 2n \ ~4,^{ — \ U . (17)
n-voo \7re/
Bernoulli numbers appear in expressions of the form
X^fe = i k P y wnere V — I? 2, Bernoulli numbers also
appear in the series expansions of functions involving
tanx, cotx, csccc, ln|sinx|, ln|cosa?|, ln|tanx|, tanhx,
Bernoulli Number
cothx, and cschx. An analytic solution exists for EVEN
orders,
B 2
(-l)- 1 2(2n)! ^ -2n _ (-l)- 1 2(2n)!
(2»)
ir) 2 n ^—~f
P
p=i
(2w) 2n
: C(2n)
(18)
for n = 1, 2, ..., where ((2n) is the RlEMANN ZETA
FUNCTION. Another intimate connection with the RlE-
MANN Zeta Function is provided by the identity
£ n = (-l) n+1 nC(l-n).
(19)
The Denominator of B 2k is given by the von Staudt-
Clausen Theorem
2fc + l
denom(B 2 fc) = fj P>
(20)
p prime
(p-l)|2fc
which also implies that the DENOMINATOR of B 2 k is
Squarefree (Hardy and Wright 1979). Another curi-
ous property is that the fraction part of B n in DECIMAL
has a Decimal Period which divides n, and there is a
single digit before that period (Conway 1996).
B =
1
B 1 =
1
2
B 2 =
1
6
£4 =
1
30
B<> =
1
42
B 8 =
1
30
3io =
5
66
B12 = —
B14 = 6
Big = —
691
2,730
798
174,611
518
#20
D 854,513
^22 - i3 8
(Sloane's A000367 and A002445). In addition,
B2n+1 —
(21)
for n = 1, 2,
Bernoulli first used the Bernoulli numbers while com-
puting X)fc=i ^ P - l* e used the property of the FlGURATE
Number Triangle that
£<
(n + l)a n
i + i
(22)
Bernoulli Number
Bernoulli Polynomial 113
along with a form for a n j which he derived inductively
to compute the sums up to n = 10 (Boyer 1968, p. 85).
For p € Z > 0, the sum is given by
where the NOTATION B^ means the quantity in ques-
tion is raised to the appropriate POWER fc, and all terms
of the form B™ are replaced with the corresponding Ber-
noulli numbers B m . Written explicitly in terms of a sum
of Powers,
I> =
B kP l
fc!(p-fc + l)!
j-Hl
(24)
Plouffe, S. "Plouffe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records .html.
Ramanujan, S. "Some Properties of Bernoulli's Numbers."
J. Indian Math. Soc. 3, 219-234, 1911.
Sloane, N. J. A. Sequences A000367/M4039 and A002445/
M4189 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Spanier, J. and Oldham, K. B. "The Bernoulli Numbers,
B n ." Ch. 4 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 35-38, 1987,
Wagstaff, S. S. Jr. "Ramanujan's Paper on Bernoulli Num-
bers." J. Indian Math. Soc. 45, 49-65, 1981.
Whit taker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Bernoulli's Paradox
Suppose the Harmonic Series converges to h:
It is also true that the COEFFICIENTS of the terms in
such an expansion sum to 1 (which Bernoulli stated
without proof). Ramanujan gave a number of curi-
ous infinite sum identities involving Bernoulli numbers
(Berndt 1994).
G. J. Fee and S. Plouffe have computed #200,000? which
has ~ 800,000 Digits (Plouffe). Plouffe and collabora-
tors have also calculated B n for n up to 72,000.
see also Argoh's Conjecture, Bernoulli Func-
tion, Bernoulli Polynomial, Debye Functions,
Euler-Maclaurin Integration Formulas, Euler
Number, Figurate Number Triangle, Genocchi
Number, Pascal's Triangle, Riemann Zeta Func-
tion, von Staudt-Clausen Theorem
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli
and Euler Polynomials and the Euler-Maclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804-806, 1972.
Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula."
§5.9 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 327-338, 1985.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 71, 1987.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 81-85, 1994.
Boyer, C. B. A History of Mathematics. New York: Wiley,
1968.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 107-110, 1996.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1980.
Hardy, G. H. and Wright, W. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 91-93, 1979.
Ireland, K. and Rosen, M. "Bernoulli Numbers." Ch. 15 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 228-248, 1990.
Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent,
Euler, and Bernoulli Numbers." Math. Comput. 21, 663-
688, 1967.
00
Then rearranging the terms in the sum gives
h-l = h,
which is a contradiction.
References
Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3
in Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., pp. 39-40, 1979.
Bernoulli Polynomial
There are two definitions of Bernoulli polynomials in
use. The nth Bernoulli polynomial is denoted here by
B n (x)i and the archaic Bernoulli polynomial by -B*(x).
These definitions correspond to the BERNOULLI NUM-
BERS evaluated at 0,
B n = B n (0)
b: = s;(o).
They also satisfy
and
B„(l) = (-l) n B n (0)
B n (l-x) = (-l) n B n (x)
(1)
(2)
(3)
(4)
114 Bernoulli Polynomial
Bernstein's Constant
(Lehmer 1988). The first few Bernoulli POLYNOMIALS
are
B (x) = l
B!(x) = x- \
' B 2 (x) = x 2 -i+ |
B 3 (x) = x 3 - §z 2 + \x
B A {x) = x 4 -2x z + x 2 - ^
B 5 (x) = x 5 -%x 4 + lx 3 -±x
B 6 (x) = x 6 - 3x 5 + f x 4 ~ \x 2 + ^.
Bernoulli (1713) defined the POLYNOMIALS in terms of
sums of the Powers of consecutive integers,
fc=0
&"- 1 = -[B n {m) - B„(0)].
(5)
Euler (1738) gave the Bernoulli POLYNOMIALS B n (x) in
terms of the generating function
e 4 - 1 ^-^ n\
They satisfy recurrence relation
dB n
T = nB - l(l)
(Appell 1882), and obey the identity
B n (x) = (B + x) n ,
(6)
(7)
(8)
where B k is interpreted here as Bk(x). Hurwitz gave
the Fourier Series
B n {x)
(2«)
- ^ A-V"^ (9)
for < x < 1, and Raabe (1851) found
m-l
~ 1Z B " ( X + ) = m " n5 "( mX )' ( 10 )
fc=0
A sum identity involving the Bernoulli POLYNOMIALS is
f2(™)B k (a)B m - k (0)
= _( m -l)B m (a + /3)+m(a + /3-l)B m _i(a + /3) (11)
for an INTEGER m and arbitrary REAL NUMBERS a and
P.
see also Bernoulli Number, Euler-Maclaurin In-
tegration Formulas, Euler Polynomial
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli
and Euler Polynomials and the Euler-Maclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804-806, 1972.
Appell, P. E. "Sur une classe de polynomes." Annales d'Ecole
Normal Superieur, Ser. 2 9, 119-144, 1882.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 330, 1985.
Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713.
Published posthumously.
Euler, L. "Methodus generalis summandi progressiones."
Comment. Acad. Set. Petropol. 6, 68-97, 1738.
Lehmer, D. H. "A New Approach to Bernoulli Polynomials."
Amer. Math. Monthly. 95, 905-911, 1988.
Lucas, E. Ch. 14 in Theorie des Nombres. Paris, 1891.
Raabe, J. L. "Zuruckfiihrung einiger Summen und bes-
timmten Integrale auf die Jakob Bernoullische Function."
J. reine angew. Math. 42, 348-376, 1851.
Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial
B n (x)" Ch. 19 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 167-173, 1987.
Bernoulli's Theorem
see Weak Law of Large Numbers
Bernoulli Trial
An experiment in which s TRIALS are made of an event,
with probability p of success in any given TRIAL.
Bernstein-Bezier Curve
see Bezier Curve
Bernstein's Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let E n (f) be the error of the best uniform approxima-
tion to a Real function f(x) on the Interval [—1,1]
by Real Polynomials of degree at most n. If
«(*) = M> (i)
then Bernstein showed that
0.267... < lim 2nE 2n {a) < 0.286. (2)
n— >oo
He conjectured that the lower limit {(5) was f3 —
1/(2^/7?). However, this was disproven by Varga and
Carpenter (1987) and Varga (1990), who computed
/? = 0.2801694990....
(3)
For rational approximations p(x)/q(x) for p and q of
degree m and n, D. J. Newman (1964) proved
i e _ 9v ^ < Enn ( a) < 3e -^
(4)
Bernstein's Inequality
Bernstein-Szego Polynomials 115
for n > 4. Gonchar (1967) and Bulanov (1975) improved
the lower bound to
-7rVn+T
< K,„(a) < 3e~^\
(5)
Vjacheslavo (1975) proved the existence of POSITIVE
constants m and M such that
m<e Vy/K E^ n [pL) <M
(6)
(Petrushev 1987, pp. 105-106). Varga et al (1993) con-
jectured and Stahl (1993) proved that
lim e 2n i?2Ti,2n,(a) = 8.
n—too
(7)
Bernstein Minimal Surface Theorem
If a Minimal Surface is given by the equation z =
f(x, y) and / has CONTINUOUS first and second PARTIAL
Derivatives for all Real x and y, then / is a Plane.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, p. 369, 1988.
Bernstein Polynomial
The Polynomials defined by
B itn (t)= ('.') **(!-*)*
References
Bulanov, A. P. "Asymptotics for the Best Rational Approxi-
mation of the Function Sign a." Mat. Sbornik 96, 171-178,
1975.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/brnstn/brnstn.html.
Gonchar, A. A. "Estimates for the Growth of Rational Func-
tions and their Applications." Mat. Sbornik 72, 489-503,
1967.
Newman, D. J. "Rational Approximation to |x|." Michigan
Math. J. 11, 11-14, 1964.
Petrushev, P. P. and Popov, V. A. Rational Approximation of
Real Functions. New York: Cambridge University Press,
1987.
Stahl, H. "Best Uniform Rational Approximation of \x\ on
[-1,1]." Russian Acad. Sci. Sb. Math. 76, 461-487, 1993.
Varga, R. S. Scientific Computations on Mathematical Prob-
lems and Conjectures. Philadelphia, PA: SIAM, 1990.
Varga, R. S. and Carpenter, A. J. "On a Conjecture of
S. Bernstein in Approximation Theory." Math. USSR
Sbornik 57, 547-560, 1987.
Varga, R. S.; Rut tan, A.; and Carpenter, A. J. "Numerical
Results on Best Uniform Rational Approximations to |x|
on [-1,+1]. Math. USSR Sbornik 74, 271-290, 1993.
Vjacheslavo, N. S. "On the Uniform Approximation of \x\ by
Rational Functions." Dokl Akad. Nauk SSSR 220, 512-
515, 1975.
Bernstein's Inequality
Let P be a POLYNOMIAL of degree n with derivative P' .
Then
HP'lloo <n||P||oo,
where (™) is a BINOMIAL COEFFICIENT. The Bernstein
polynomials of degree n form a basis for the POWER
Polynomials of degree n.
see also Bezier Curve
Bernstein's Polynomial Theorem
If g(9) is a trigonometric POLYNOMIAL of degree m sat-
isfying the condition \g(0) \ < 1 where 6 is arbitrary and
real, then g'{9) < m.
References
Szego, G. Orthogonal Polynomials, ^.th ed. Providence, RI:
Amer. Math. Soc, p. 5, 1975.
Bernstein-Szego Polynomials
The POLYNOMIALS on the interval [-1,1] associated
with the Weight Functions
w{x) — (1 - z 2 ) _1/
w(x) = (1 - x 2 ) 1/2
w(x) -
1 + x*
also called BERNSTEIN POLYNOMIALS.
References
Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI:
Amer. Math. Soc, pp. 31-33, 1975.
where
|F||oo = mK|PW|.
116 Berry-Osseen Inequality
Bertrand's Problem
Berry-Osseen Inequality
Gives an estimate of the deviation of a DISTRIBUTION
Function as a Sum of independent Random Vari-
ables with a Normal Distribution.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, p. 369, 1988.
Berry Paradox
There are several versions of the Berry paradox, the
original version of which was published by Bertrand
Russell and attributed to Oxford University librarian
Mr. G. Berry. In one form, the paradox notes that the
number "one million, one hundred thousand, one hun-
dred and twenty one" can be named by the description:
"the first number not nameable in under ten words."
However, this latter expression has only nine words, so
the number can be named in under ten words, so there
is an inconsistency in naming it in this manner!
References
Chaitin, G. J. "The Berry Paradox."
1995.
Complexity 1, 26-30,
Bertelsen's Number
An erroneous value of 7r(10 9 ), where tt(x) is the PRIME
Counting Function. Bertelsen's value of 50,847,478
is 56 lower than the correct value of 50,847,534.
References
Brown, K. S. "Bertelsen's Number." http://www.seanet .
com/-ksbrown/kmath049.htm.
Bertini's Theorem
The general curve of a system which is LINEARLY IN-
DEPENDENT on a certain number of given irreducible
curves will not have a singular point which is not fixed
for all the curves of the system.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves
York: Dover, p. 115, 1959.
New
Bertrand Curves
Two curves which, at any point, have a common princi-
pal Normal Vector are called Bertrand curves. The
product of the TORSIONS of Bertrand curves is a con-
stant.
Bertrand's Paradox
see Bertrand's Problem
Bertrand's Postulate
If n > 3, there is always at least one PRIME between n
and 2n — 2. Equivalently, if n > 1, then there is always
at least one PRIME between n and 2n, It was proved
in 1850-51 by Chebyshev, and is therefore sometimes
known as Chebyshev's Theorem. An elegant proof
was later given by Erdos. An extension of this result is
that if n > k, then there is a number containing a Prime
divisor > k in the sequence n, n + 1, . . . , n + k — 1. (The
case n = k + 1 then corresponds to Bertrand's postu-
late.) This was first proved by Sylvester, independently
by Schur, and a simple proof was given by Erdos.
A related problem is to find the least value of 8 so that
there exists at least one PRIME between n and n + O(n )
for sufficiently large n (Berndt 1994). The smallest
known value is 9 = 6/11 -f e (Lou and Yao 1992).
see also Choquet Theory, de Polignac's Conjec-
ture, Prime Number
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, p. 135, 1994.
Erdos, P. "Ramanujan and I." In Proceedings of the Inter-
national Ramanujan Centenary Conference held at Anna
University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New
York: Springer- Verlag, pp. 1-20, 1989.
Lou, S. and Yau, Q. "A Chebyshev's Type of Prime Number
Theorem in a Short Interval (II)." Hardy- Ramanujan J.
15, 1-33, 1992.
Bertrand's Problem
What is the Probability that a Chord drawn at Ran-
dom on a Circle of Radius r has length > r? The an-
swer, it turns out, depends on the interpretation of "two
points drawn at RANDOM." In the usual interpretation
that Angles #i and 6i are picked at Random on the
Circumference,
t, 7r " f 2
P= *-=-•
7T 3
However, if a point is instead placed at RANDOM on a
Radius of the Circle and a Chord drawn Perpen-
dicular to it,
r 2
The latter interpretation is more satisfactory in the
sense that the result remains the same for a rotated CIR-
CLE, a slightly smaller CIRCLE INSCRIBED in the first,
or for a CIRCLE of the same size but with its center
slightly offset. Jaynes (1983) shows that the interpre-
tation of "Random" as a continuous Uniform Distri-
bution over the RADIUS is the only one possessing all
these three invariances.
References
Bogomolny, A. "Bertrand's Paradox." http: //www. cut-the-
knot . com/bertrand.html.
Jaynes, E. T. Papers on Probability, Statistics, and Statisti-
cal Physics. Dordrecht, Netherlands: Reidel, 1983.
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42-
45, 1995.
Bertrand's Test
Bertrand's Test
A Convergence Test also called de Morgan's and
Bertrand's Test. If the ratio of terms of a Series
{flnj^Li can be written in the form
an
1
= 1 + - +
Pn
n n In n '
then the series converges if lim n ->oo pn > 1 and diverges
if lim n _^oo/0n < 1, where lim w ->oo is the Lower Limit
and lim n _>.oo is the Upper Limit.
see also Rummer's Test
References
Bromwich, T. J. Pa and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 40, 1991.
Bertrand's Theorem
see Bertrand's Postulate
Bessel Differential Equation
Bessel Differential Equation
m?)y = 0.
Equivalently, dividing through by z 2 ,
2d 2 y dy 2 „^
117
(i)
The solutions to this equation define the BESSEL FUNC-
TIONS. The equation has a regular SINGULARITY at
and an irregular SINGULARITY at oo.
A transformed version of the Bessel differential equation
given by Bowman (1958) is
* 2 § + (2p+l)sg + (aV r + /? 2 )y = 0. (3)
The solution is
Besov Space
A type of abstract Space which occurs in Spline and
Rational Function approximations. The Besov space
Bp yQ is a complete quasinormed space which is a Ba-
NACH Space when 1 < p, q < oo (Petrushev and Popov
1987).
References
Bergh, J. and Lofstrom, J. Interpolation Spaces. New York:
Springer- Verlag, 1976.
Peetre, J. New Thoughts on Besov Spaces. Durham, NC:
Duke University Press, 1976.
Petrushev, P. P. and Popov, V. A. "Besov Spaces." §7.2
in Rational Approximation of Real Functions. New York:
Cambridge University Press, pp. 201-203, 1987.
Triebel, H. Interpolation Theory, Function Spaces, Differen-
tial Operators. New York: Elsevier, 1978.
Bessel's Correction
The factor (N — 1)/N in the relationship between the
Variance a and the Expectation Values of the Sam-
ple Variance,
y = x p
I 2\ N-l 2
s 2 = (x 1 ) - (x) 2 .
N lSl 2 +N 2 s 2 2
Ni+N 2 -2 '
see also Sample Variance, Variance
where
For two samples,
(i)
(2)
(3)
c 1 J q/r (^-)+c 2 r g/r (^)
where
q = vV - P\
(4)
(5)
J and Y are the Bessel Functions of the First and
SECOND KINDS, and C\ and Ci are constants. Another
form is given by letting y = x a J n (/3x' y ) i tj — yx~ a , and
£ = 0x 7 (Bowman 1958, p. 117), then
(6)
The solution is
= f x a [AJ n {(3x' r ) + BYniPx 1 )] for integral n
V \ AJniffx 7 ) + BJ-niPx 1 )] for nonintegral u.
(?)
see also AlRY FUNCTIONS, ANGER FUNCTION, Bei,
Ber, Bessel Function, Bourget's Hypothesis,
Catalan Integrals, Cylindrical Function, Dini
Expansion, Hankel Function, Hankel's Integral,
Hemispherical Function, Kapteyn Series, Lip-
schitz's Integral, Lommel Differential Equa-
tion, Lommel Function, Lommel's Integrals,
Neumann Series (Bessel Function), Parseval's
Integral, Poisson Integral, Ramanujan's Inte-
gral, Riccati Differential Equation, Sonine's
Integral, Struve Function, Weber Functions,
Weber's Discontinuous Integrals
References
Bowman, F. Introduction to Bessel Functions. New York:
Dover, 1958.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 550, 1953.
118 BesseVs Finite Difference Formula
Bessel Function of the First Kind
Bessel's Finite Difference Formula
An Interpolation formula also sometimes known as
The Bessel functions are more frequently defined as so-
lutions to the Differential Equation
f P = fo+ pSi/2 + B 2 {Sl + <$i ) + B 3 8l /2
+ B 4 05$ + tf) + B 5 *? /a + ... ) (1)
for p e [0, 1], where 6 is the Central Difference and
Bin = ^Gln = g ("^2n + i*2n) (2)
B2n + 1 = G2n + 1 ~ 2^ 2n ~ 2 (^ 2ri ~ ^2n) (**)
£?2n = ^2n — G 2n +1 = Bin — #2n + l (4)
F 2 n = t?2n+l = B 2n + #2n+l> (5)
where Gk are the COEFFICIENTS from GAUSS'S BACK-
WARD Formula and Gauss's Forward Formula and
E k and Fk are the Coefficients from Everett's FOR-
MULA. The i?fcS also satisfy
B 2n {p) = B 2n (q)
B 2n+X {p) = -B 2n +i(q),
for
(6)
(7)
(8)
q = l-p.
see also Everett's Formula
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 880, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing,
Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Bessel's First Integral
i r
J n (x) = — / cos(n# - xsinO) d8,
77 Jo
where J n (x) is a BESSEL FUNCTION OF THE FIRST
Kind.
Bessel's Formula
see Bessel's Finite Difference Formula, Bes-
sel's Interpolation Formula, Bessel's Statisti-
cal Formula
Bessel Function
A function Z(x) defined by the RECURRENCE RELA-
TIONS
Zm + l + Z m — 1 — Zm
and
&m+l — ^m-1
Zm~l — —2
dx
2d 2 y dy 2
X dx^ +X dx- + {x
m )y — 0.
There are two classes of solution, called the BESSEL
Function of the First Kind J and Bessel Func-
tion of the Second Kind Y. (A Bessel Function
OF THE THIRD Kind is a special combination of the first
and second kinds.) Several related functions are also de-
fined by slightly modifying the defining equations.
see also Bessel Function of the First Kind,
Bessel Function of the Second Kind, Bessel
Function of the Third Kind, Cylinder Func-
tion, Hemicylindrical Function, Modified Bes-
sel Function of the First Kind, Modified Bessel
Function of the Second Kind, Spherical Bessel
Function of the First Kind, Spherical Bessel
Function of the Second Kind
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions
of Integer Order," "Bessel Functions of Fractional Order,"
and "Integrals of Bessel Functions." Chs. 9-11 in Hand-
book of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover,
pp. 355-389, 435-456, and 480-491, 1972.
Arfken, G. "Bessel Functions." Ch. 11 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 573-636, 1985.
Bickley, W. G. Bessel Functions and Formulae. Cambridge,
England: Cambridge University Press, 1957.
Bowman, F. Introduction to Bessel Functions. New York:
Dover, 1958.
Gray, A. and Matthews, G. B. A Treatise on Bessel Func-
tions and Their Applications to Physics, 2nd ed. New
York: Dover, 1966.
Luke, Y. L. Integrals of Bessel Functions. New York:
McGraw-Hill, 1962.
McLachlan, N. W. Bessel Functions for Engineers, 2nd ed.
with corrections. Oxford, England: Clarendon Press, 1961.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Bessel Functions of Integral Order" and
"Bessel Functions of Fractional Order, Airy Functions,
Spherical Bessel Functions." §6.5 and 6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 223-229 and 234-245, 1992.
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Bessel Function of the First Kind
-0.2
Bessel Function of the First Kind
Bessel Function of the First Kind 119
The Bessel functions of the first kind J n {x) are defined as
the solutions to the BESSEL DIFFERENTIAL EQUATION
which are nonsingular at the origin. They are some-
times also called Cylinder Functions or Cylindri-
cal Harmonics. The above plot shows J n (x) for n = 1,
2,..., 5.
To solve the differential equation, apply FROBENIUS
METHOD using a series solution of the form
First, look at the special case m = —1/2, then (9) be-
comes
oo
^[a n n(n-l) + a n _ 2 ]x m +" = 0, (10)
n(n — 1)
Now let n = 2/, where / = 1, 2,
fln-2.
(11)
&21
1
2/(2/ - 1)
0*21-2
(-i)'
y = x k ^ a n x n = JT a n x n+k .
(2)
n=0 n=0
Plugging into (1) yields
oo
x 2 ^{k + n)(k + n~ l)a n x k+n - 2
n—0
oo oo
+X Y^(k + n)OnX h+n - 1 +X 2 J2 a nX k+n
n~ n —
oo
-m 2 ^2a n x n + k = (3)
[2/(2/ - 1)][2(Z - 1)(2Z - 3)] ■ * - [2 - 1 • 1]
7«0,
do
2 l l\(2l~l)\\
which, using the identity 2 l l\(2l - 1)!! = (2/)!, gives
a 2/ = /rtlXI Q0>
(12)
(20! °'
(13)
Similarly, letting n = 21 + 1
fl2i+i = —
1
(2/ + l)(2/)
«2i-l
(-1)'
[2/(2/ + 1)][2(I - 1)(2/ - 1)] ... [2 • 1 • 3][1] au
(14)
^(fc + n)(fc + n - l)a n x fc+n + ^(fc + n)a n x fe+n which, using the identity 2 l l\(2l + 1)!! = (2/ + 1)!, gives
]Ta n - 2 x k+n -m 2 J2a n x n+k = 0. (4)
n=2 n=0
The INDICIAL EQUATION, obtained by setting n = 0, is
a [fc(£; - 1) + k - m 2 ] = a (k 2 - m 2 ) = 0. (5)
Since ao is defined as the first NONZERO term, k 2 —m 2 =
0, so k = ±ra. Now, if k ~ m,
oo
^[(m + n)(m + n - 1) + (m + n) - m 2 }a n x 7n+n
n~0
oo
+ Y. an ~ 2^ m+n = (6)
n^2
oo oo
£][(m + n) 2 - m 2 ]a n x m+n + ^ a„_ 2 x m+n = (7)
n=0 n = 2
oo oo
^ n(2m + n)a n i m+ " + J] a„. 2 2 ra+n = (8)
n — n=2
OO
ai(2m + 1) 4- ^[a n n(2m + n) + a„_ 2 ]a; m+n = 0. (9)
(-1) 1 (-l) z
a2/+1 " 2^/!(2/ + l)!! ai = (27TI)! ai ' (15)
Plugging back into (2) with k = m = —1/2 gives
2/ = x 1/2 N a n a; n
t=0
oo oo
2. a n x n + N^ a n x n
_n~l,3,5,... n-0,2,4
oo oo
E2J . V~^
CL21X + J> ^21 + lX
-1/2
-1/2
n = 0,2,4,...
2J + 1
-1/2
J = 1=0
(-1)
2^ ( 2 n! X +ai 2^(2Z + l)!
;=o ■ ' z=o
(20!
-1/2/ , • \
= x ' (ao cos a; + a\ since).
(2/ + 1)
(16)
The BESSEL FUNCTIONS of order ±1/2 are therefore de-
fined as
(17)
(18)
J-i/ 2 (x) =4/ — cosa;
' U 7TX
Ji/ 2 (x) = 4/ — sinx,
17 7TZ
120 Bessel Function of the First Kind
so the general solution for m = ±1/2 is
y = a' J- 1/2 {x) + a 1 J 1/2 (x). (19)
Now, consider a general m ^ —1/2. Equation (9) re-
quires
ai(2m+l)=0 (20)
[a n n(2m + n) + a n _ 2 ]z m+n = (21)
for n = 2, 3, . . . , so
ai =0
n(2m + n
■fln-2
(22)
(23)
for n = 2, 3, Let n = 2Z + 1, where Z = 1, 2, . . . ,
then
«2/ + l
1
Tfl2Z-l
(2Z + l)[2(m + l) + l]
= ... = /(n,m)ai =0, (24)
where f(n,m) is the function of Z and m obtained by
iterating the recursion relationship down to a\ . Now let
n = 2Z, where Z = 1, 2, . . . , so
1 1
a 2* = ~~ 77777; r~^ a 2/-2 = —777 — 77^-2
2l{2m + 2l) " 4Z(m + Z)
tn
[4Z(m + Z)][4(Z - l)(m + Z - 1)] • • ■ [4 • (m + 1)]
ao-
(25)
Plugging back into (9),
a n x = > a n x + y a n x
n = n = l,3,5,... n = 0,2,4,,,,
E2I + m + l . \~^ 2I + m
G 2 i-M^ + > a 2lX
1=0 (=0
„ v^ (_z}Y « + ™
= tin 7 X
Z^ [4i(m + l)][4{l - l)(m + I - 1)] • • . [4 • (m + 1)]
1=0
[(-l) f m(m- l)---l]x 2t+m
[4/(m + i)][4(i - l)(m + i - 1)] • ■ ■ [m(m - 1) ■ • • 1]
Bessel Function of the First Kind
Returning to equation (5) and examining the case k —
— m,
00
ai(l-2m) + ^[a n ra(ra-2m) + a n _ 2 ]a; n ~ m = 0. (29)
However, the sign of m is arbitrary, so the solutions must
be the same for +ra and — m. We are therefore free to
replace — m with — |m|, so
00
oi(l + 2|m|) + ^[o n n(n + 2|m|) + a n _ 2 ]x |m|+n = 0,
n = 2
. (30)
and we obtain the same solutions as before, but with m
replaced by \m\.
*J<m\X) '■
v^oo (-1)' 2Z+|m| f or | rn |^_I
for m = — |
for m = |.
(31)
We can relate J m and J_ m (when m is an Integer) by
writing
( — lV
1=0 v '
(32)
Now let 1 = 1' + m. Then
J-m(x) = ^
(-1)
Z' + m
Z' + m=0
-1
2 2 <'+™(Z' + m)!Z!
(-1)''+™
2l'+m
V I- 1 ) 2Z'+m
Z^ 2 2 <'+™Z'!(Z'+m)!
l' = — m
+ 2-1 2«'+"Z'!(Z'+m)! a;2 ' +m ' (33)
i'=o v
But Z'! = oo for Z' = -m, ...,-1, so the Denomina-
tor is infinite and the terms on the right are zero. We
therefore have
--Ej^--£?&- w '-<*>- Es.J^fe'" 4 "-'- 1 '"^
\(m + l)\ ~ u ^ 2 2 <Z!(m + Z)
Z=0 ' 1=0 /
Now define
OO ;
Jm(x) = Jj 2 2 <+™Z!(m + Z)! x2 ' +m ' (27)
where the factorials can be generalized to Gamma
FUNCTIONS for nonintegral m. The above equation then
becomes
(34)
Note that the Bessel Differential Equation is
second-order, so there must be two linearly independent
solutions. We have found both only for \m\ = 1/2. For
a general nonintegral order, the independent solutions
are J m and J~ m . When m is an INTEGER, the general
(real) solution is of the form
Z m = C 1 J m (x) + C 2 Y rn (x),
(35)
y = a 2 m m\J m (x) — a' J m (x).
(28)
Bessel Function of the First Kind
Bessel Function of the First Kind 121
where J m is a Bessel function of the first kind, F m
(a.k.a. iV m ) is the BESSEL FUNCTION OF THE SECOND
Kind (a.k.a. Neumann Function or Weber Func-
tion), and C\ and C 2 are constants. Complex solutions
are given by the Hankel Functions (a.k.a. Bessel
Functions of the Third Kind).
The Bessel functions are ORTHOGONAL in [0, 1] with re-
spect to the weight factor x. Except when 2n is a NEG-
ATIVE Integer,
Jrn(z)
-1/2
2 2m+l/2 i m + l/2 r ( m+1 ^
Mo im (2iz) ) (36)
where T(x) is the Gamma Function and M , m is a
Whittaker Function.
In terms of a Confluent Hypergeometric Func-
tion of the First Kind, the Bessel function is written
Mz)
^fryo^^ + i;-^ 2 )- (37)
A derivative identity for expressing higher order Bessel
functions in terms of Jo(x) is
Jn(x) — i n T n li-j-) Jo(
(38)
where T n (x) is a Chebyshev Polynomial of the
First Kind. Asymptotic forms for the Bessel functions
are
J - {x) * fd+T) (!)
for x <^ 1 and
J m (x) :
/ ran tt\
x
V 2 4/
for x ^> 1. A derivative identity is
d
dx
[x^Jmix)] = X^Jm-lix).
An integral identity is
uJo(u)du —uJ\{u).
F
Jo
Some sum identities are
1 = [Jo(x)] 2 + 2[J 1 {x)f + 2[J 2 (x)] 2 + ,
1 = J (x) + 2J 2 {x) + 2J A {x) + . .
and the Jacobi-Anger Expansion
% J n (z)e
(39)
(40)
(41)
(42)
(43)
(44)
(45)
which can also be written
00
e tzcose = J (z) + 2^2i n J n (z)cos(n8). (46)
n=l
The Bessel function addition theorem states
00
My + z) = ^ J™{y) J n-m{z). (47)
m=-oo
ROOTS of the FUNCTION J n (x) are given in the following
table.
zero
J Q (x)
Ji(x)
J 2 {x)
Mx)
Mx)
J*(x)
1
2.4048
3.8317
5.1336
6.3802
7.5883
8.7715
2
5.5201
7.0156
8.4172
9.7610
11.0647
12.3386
3
8.6537
10.1735
11.6198
13.0152
14.3725
15.7002
4
11.7915
13.3237
14.7960
16.2235
17.6160
18.9801
5
14.9309
16.4706
17.9598
19.4094
20,8269
22.2178
Let x n be the nth ROOT of the Bessel function Jo(#),
then
Y —
71 = 1
(Le Lionnais 1983).
2"n*J§y£n}
= 0.38479...
(48)
The Roots of its Derivatives are given in the following
table.
zero
Jo'(x)
•V(z)
•V(s)
J 3 '(x)
J 4 '(x)
J 5 '(x)
1
3.8317
1.8412
3.0542
4.2012
5,3175
6.4156
2
7.0156
5.3314
6.7061
8.0152
9.2824
10.5199
3
10.1735
8.5363
9.9695
11.3459
12.6819
13.9872
4
13.3237
11.7060
13.1704
14.5858
15.9641
17.3128
5
16.4706
14.8636
16.3475
17.7887
19.1960
20.5755
Various integrals can be expressed in terms of Bessel
functions
1 f 2 "
* w - s y «••
' cos <j> d(f>
i
i r
J n (z) = — / cos(z sin — n6) d8 ,
n Jo
which is BESSEL'S FIRST INTEGRAL,
(49)
(50)
.-71 f*
./«(*) = —/ e izcose cos(n9)d0 (51)
w Jo
Jn{z)
JL_ [ 2 \i
V"* d<t>
z cos <p in.1
(52)
J, . . . ,
for n = 1, 2
2 x
J»W
7r (2m
for n = I, 2, . . . ,
sin n u cos(x cos u) du (53)
71— — OO
T f~\ 1 I (x/2)(z-l/z) -71-1 ,
Jtl(x) = - — ; / e K ' A ' } z dz
2tvi J
(54)
122 Bessel Function Fourier Expansion
Bessel Function of the Second Kind
for n > —1/2. Integrals involving J\(x) include
(Bowman 1958, p. 108), so
/ J\ (x) dx = 1
Jo
(55)
r[¥
dx = h
(56)
cm
xdx = — .
2
(57)
see also BESSEL FUNCTION OF THE SECOND KIND, DE-
bye's Asymptotic Representation, Dixon-Ferrar
Formula, Hansen-Bessel Formula, Kapteyn Se-
ries, Kneser-Sommerfeld Formula, Mehler's
Bessel Function Formula, Nicholson's Formula,
Poisson's Bessel Function Formula, Schlafli's
Formula, Schlomilch's Series, Sommerfeld's
Formula, Sonine-Schafheitlin Formula, Wat-
son's Formula, Watson-Nicholson Formula, We-
ber's Discontinuous Integrals, Weber's For-
mula, Weber-Sonine Formula, Weyrich's For-
mula
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func-
tions J and V." §9.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 358-364, 1972.
Arfken, G. "Bessel Functions of the First Kind, J„(;r)" and
"Orthogonality." §11.1 and 11,2 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 573-591 and 591-596, 1985.
Lehmer, D. H. "Arithmetical Periodicities of Bessel Func-
tions." Ann. Math. 33, 143-150, 1932.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 25, 1983.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 619-622, 1953.
Spanier, J. and Oldham, K. B. "The Bessel Coefficients Jq(x)
and Ji(x)" and "The Bessel Function J u (x)." Chs. 52-53
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 509-520 and 521-532, 1987.
Bessel Function Fourier Expansion
Let n > 1/2 and e*i, a 2 , ...be the POSITIVE ROOTS
of J n (x) — 0. An expansion of a function in the inter-
val (0,1) in terms of BESSEL FUNCTIONS OF THE FIRST
Kind
f( x ) = /]A r J n (xa r ),
(i)
has Coefficients found as follows:
/ xf(x)J n (xai)dx = y^A r / xJ n (xa r )Jn(xai)dx.
Jo r=1 Jo
(2)
But Orthogonality of Bessel Function Roots
gives
/'
Jo
xJ n (xai)J n (xa r )dx = ^Sl^Jn + 1 (&r) (3)
ol °°
/ xf(x)J n (xai)dx - \ }^ Ar5i, r J n +i 2 (xa r )
J° r=l
I A. T . . 2 (^,.\ U)
= ^AiJ n+1 (on),
and the COEFFICIENTS are given by
2
A t =
J n+ i 2 (ai)
f
Jo
xf(x)Jn(xai)dx. (5)
References
Bowman, F. Introduction to Bessel Functions. New York:
Dover, 1958.
Bessel Function of the Second Kind
A Bessel function of the second kind Y n (x) is a solution
to the Bessel Differential Equation which is sin-
gular at the origin. Bessel functions of the second kind
are also called Neumann Functions or Weber Func-
tions. The above plot shows Y n {x) for n = 1, 2, . . . ,
5.
Let v = Jm{x) be the first solution and u be the
Other one (since the BESSEL DIFFERENTIAL EQUATION
is second-order, there are two Linearly Independent
solutions). Then
xu + u + xu =
XV + V + XV = 0.
Take v x (1) - u x (2),
x{u v — uv ) -\- u v — uv =0
— \x(uv — uv')] = 0,
ax
(i)
(2)
(3)
(4)
so x(uv — uv) = B, where B is a constant. Divide by
xv 2 ,
uv — uv _ d /u\ _ B ( .
v 2 dx \v ) xv 2
V
f-
J &
,2*
(6)
Bessel Function of the Third Kind
Rearranging and using v = J m (x) gives
u = AJm(x) + BJ m (x)
I.
dx
XJrn \X~)
= A , J m (x)-{-B'Y rn (x), (7)
where the Bessel function of the second kind is denned
by
Y m (x)
J m (x) cos(mir) — J_ m (x)
sin(m7r)
* Z. 2«+»*!(m + *)! [ 2 ln 1 2 j + 27 " bm+k ~ bk
1 v^ x~ m+2k (m -k-l)\
--J2
2-m4-2fcfc|
(8)
m = 0, 1, 2, . . . , 7 is the Euler-Mascheroni Con-
stant, and
Jo k = 0,
(9)
The function is given by
Y n (z) = - / sin(z sin d-n0)d0
* Jo
I!?-"
— nt ( ,\ni —z sinh t
+ e~ nt (-l) n ]e
dt, (10)
Asymptotic equations are
m() ~l-^(f) m m^0,x«l (U)
rmW = V^ sm r T"4J * >>x ' (12)
where r(z) is a Gamma Function.
see also Bessel Function of the First Kind, Bour-
get's Hypothesis, Hankel Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func-
tions J and Y. n §9.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 358-364, 1972.
Arfken, G. "Neumann Functions, Bessel Functions of the Sec-
ond Kind, N v (x). n §11.3 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596-
604, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 625-627, 1953.
Spanier, J. and Oldham, K. B. "The Neumann Function
Y u (x)" Ch. 54 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 533-542, 1987.
Bessel Function of the Third Kind
see Hankel Function
Bessel's Inequality 123
Bessel's Inequality
If f(x) is piecewise CONTINUOUS and has a general
Fourier Series
5^Mi(x) (1)
i
with Weighting Function w(x), it must be true that
/
/O) - y^ai<fc(aQ
w(x)dx > (2)
+ 5^a< 2 <t>i 2 (x)w{x)dx>0. (3)
i J
But the Coefficient of the generalized Fourier Se-
ries is given by
a m = / f(x)<f> m (x)w(x)dx, (4)
so
/ f 2 (x)w(x)dx-2^2ai 2 -h^di 2 > (5)
i i
f{x)w(x)dx>Y^ai 2 - (6)
i
Equation (6) is an inequality if the functions <j>i are not
Complete. If they are Complete, then the inequality
(2) becomes an equality, so (6) becomes an equality and
is known as PARSEVAL's THEOREM. If f(x) has a simple
Fourier Series expansion with Coefficients a , ai,
. . . , a n and &i, . . . , b ni then
ia 2 + ^(a fc 2 +6 fc 2 )<- / [f(x)] 2 dx. (7)
fc = l n J—*
The inequality can also be derived from SCHWARZ'S IN-
EQUALITY
I (f\g) I 2 < {/I/} (g\g) (8)
by expanding g in a superposition of ElGENFUNCTlONS
0f/,S= Yji a ifc- Then
(/|5) = X)°* </!/*> ^Z) fli -
(9)
(f\g) r <
Y< ai
= 5> 4 a«' < </|/> <s| S ) . (10)
124 BesseVs Interpolation Formula
If g is normalized, then (g\g) = 1 and
</!/>> 5> t a t *. (11)
see also Schwarz's Inequality, Triangle Inequal-
ity
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 526-527, 1985.
Gradshteyn, I. S. and Ryzhik, L M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1102, 1980.
BessePs Interpolation Formula
see Bessel's Finite Difference Formula
Bessel Polynomial
see Bessel Function
Bessel's Second Integral
see POISSON INTEGRAL
Bessel's Statistical Formula
W — UJ W — U)
•'** t lzL
(i)
(wi-w) 2
N(N-l)
where
w = X\ — X2
u = M(i) - M(2)
N = N 1 + N 2 .
(2)
(3)
(4)
Beta
A financial measure of a fund's sensitivity to market
movements which measures the relationship between a
fund's excess return over Treasury Bills and the excess
return of a benchmark index (which, by definition, has
j3 = 1). A fund with a beta of (3 has performed r —
(j3 - 1) x 100% better (or \r\ worse if r < 0) than its
benchmark index (after deducting the T-bill rate) in up
markets and \r\ worse (or \r\ better if r < 0) in down
markets.
see also Alpha, Sharpe Ratio
Beta Distribution
/^ "\ (a.6) = (l. 1)
Q
/ //ia.G) = {l, 1)
<2.3)-\
/(J. 2} ^
Beta Distribution
A general type of statistical DISTRIBUTION which is re-
lated to the Gamma Distribution. Beta distributions
have two free parameters, which are labeled according
to one of two notational conventions. The usual defini-
tion calls these a and /?, and the other uses /?' = j3 — 1
and d = a - 1 (Beyer 1987, p. 534). The above plots
are for (a,/3) = (1,1) [solid], (1, 2) [dotted], and (2, 3)
[dashed]. The probability function P(x) and DISTRIBU-
TION Function D(x) are given by
P{x)
0-l„a-l
(l-xf- l x'
B(a,0)
T(a)T((3)
D{x) = I(x; a, 6),
(l-xf^x
/3-1 a-1
(1)
(2)
where B(a,b) is the BETA FUNCTION, J(x;a,6) is the
Regularized Beta Function, and < x < 1 where
a, f3 > 0. The distribution is normalized since
Jo
P(x) dx :
r(a)r(/3)
r(a + /3)
Jo
(l-xf^dx
B(a,0) = l. (3)
T(a)T(/3)
The Characteristic Function is
</>(*) = ^faa + bjit)
The Moments are given by
: + /3)r(a + r
P 1 T(rv
M r = (a- fi) r dx= ~^—
Jo r ( a
+ /3 + r)r(a)
(4)
(5)
The Mean is
M -r(a)r ( ^y (1 x)
T(a + P)
B(a + l,f3)
r(a + y3)r(a + l)r(/3) _ a
r(a)r(/?)r(a + /? + i) a + /?'
and the Variance, SKEWNESS, and KURTOSIS are
2__ a/3
a {a + f3) 2 {a + (3 + l)
_ 2(yff- y^)( % /S+V^)Vl + q + /?
71 ~ V^p(a + /3 + 2)
_ 6(a 2 + a 3 - Aaj3 - 2a 2 (3 + (3 2 - 2af3 2 + /3 3 )
72 ~ a/3(a + /3 + 2)(a + /3 + 3)
(6)
(7)
(8)
(9)
The Mode of a variate distributed as /3(a,/3) is
- - Q ~ 1
(10)
Beta Function
In "normal" form, the distribution is written
and the MEAN, VARIANCE, SKEWNESS, and KURTOSIS
are
A* =
a + /3
2 OL0
a =
7i =
72
(a + /?) 2 (l + a + /?)
_ 2(Vo:-^)( v /a + v / g)Vl + tt + /3
V^(a + /? + 2)
3(1 + a + /?)(2a 2 - 2a/? + a 2 /3 + 2/3 2 + a/3 2 )
(12)
(13)
(14)
a/?(a + /3 + 2)(a + /? + 3)
(15)
see a/so GAMMA DISTRIBUTION
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 944-945, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 534-535, 1987.
Beta Function
The beta function is the name used by Legendre and
Whittaker and Watson (1990) for the Eulerian Inte-
gral of the Second Kind. To derive the integral
representation of the beta function, write the product
of two Factorials as
POO no
i\n\= I e- u u m du I
Jo Jo
Now, let u = a? 2 , v = y 2 , so
e v v n dv. (1)
dy
/oo />oo
e- x2 x 2rn+1 dx e~ y2 y 2n+ \
/oo poo
/ e-^ + ^x 2m+1 y 2m+1 dxdy. (2)
-oo «/ — oo
Transforming to POLAR COORDINATES with x ~ rcos9,
y — r sin 6
pTT/2 pO
n! = 4/ / e~ r '(r cos dY m+1 (r sin 6) 2n+1 rdrc
Jo Jo
poo /' 7r /2
A I -r 2 2m+2n+3 i / 2ro+l n ■ 2n+l n jq
■ 4 / e r dr cos v sin ^ v dv
Jo Jo
tt/2
2(m +
n + 1)! /
Jo
cos" m+1 sin n+1 (9 d0. (3)
Beta Function 125
The beta function is then defined by
B(m + l,n + l) = 5(71 + 1,771+ 1)
/.tt/2
= 2 / cos 2 ™ +1 flsin 2 " +1 ^= / m]n ' „ .
Jo (m + n + 1)!
(4)
Rewriting the arguments,
B( P a) - r &™ - (P-D'(g-l)! (5)
The general trigonometric form is
o
/•tt/2
/ sin n a;cos m ;rdx = \B(n+ |,m+ |). (6)
Equation (6) can be transformed to an integral over
Polynomials by letting u = cos 2 0,
— = u (1-u) du.
' n ) Jo
B(m,n)
T(m)T(r,
r(m + i
du
(7)
(8)
To put it in a form which can be used to derive the
Legendre Duplication Formula, let x = y/u, so
u = x and du — 2x dx, and
B(m y n)= / x 2irn ~ 1) (l-x 2 ) n - 1 {2xdx)
Jo
-'f-
Jo
2m — 1/-. 2\n~l
(l-x^^dx. (9)
To put it in a form which can be used to develop integral
representations of the Bessel Functions and Hyper-
geometric Function, let u = x/(l + x), so
£(m + l,n + l)= H , "">
(10)
Various identities can be derived using the GAUSS MUL-
TIPLICATION Formula
B(np, nq)
T(np)T(nq)
T[n(p + q)}
_ - nq B(p,q)B(p+ l, t )- B(p+ 2=1, q)
B(q,q)B(2q,q)---B([n-l]q,q) ' ( >
Additional identities include
B(va ^) = r(p)F(9 + 1) = g T(p + l)r(q)
(P ' q+ ' T(p + q + l) p r(\p+l]q)
= |s(p+l, ff ) (12)
B(p,q) = B(p+l,q) + B(p,q+l) (13)
126 Beta Function (Exponential)
B{p,q+1)
P + Q
If n is a Positive Integer, then
B(p,q).
(14)
1 * 2 • • • 71 ,„ ^ x
B(p, n + 1 = , . x (15)
p(p + 1) • • • (p + n)
S(P,p)5(P+iP+5) =
(16)
2'^ ' 2> 2 4 P" X p
5(p + <?)#(p + 9, r) = £(<?, r)B(q + r,p). (17)
A generalization of the beta function is the incomplete
beta function
B(t;x,y)= r«"- 1 (l-u)*- 1
Jo
+P |1 , i-w.
\ X X + 1
(l-y)---(n-y), w
n!(rr + n)
r + . . •
(18)
see aZso Central Beta Function, Dirichlet In-
tegrals, Gamma Function, Regularized Beta
Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Beta Function"
and "Incomplete Beta Function." §6.2 and 6.6 in Hand-
book of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover,
pp. 258 and 263, 1972.
Arfken, G. "The Beta Function." §10.4 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 560-565, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 425, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Gamma Function, Beta Function, Facto-
rials, Binomial Coefficients" and "Incomplete Beta Func-
tion, Student's Distribution, F-Distribution, Cumulative
Binomial Distribution." §6.1 and 6.2 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 206-209 and 219-223, 1992.
Spanier, J. and Oldham, K. B. "The Incomplete Beta Func-
tion B(v\mx)" Ch, 58 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 573-580, 1987.
Whittaker, E. T. and Watson, G. N. A Course of Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Beta Function (Exponential)
Betti Group
Another "Beta Function" defined in terms of an in-
tegral is the "exponential" beta function, given by
/?»(*)
5 />-"
dt
i!*-< n+1 >
'£
(-i)*
L fc=o
fc!
2-r fc!
fc=0
(1)
(2)
The exponential beta function satisfies the Recur-
rence Relation
z(3 n (z) = (-l) n e z - e- z +n(3 n ^(z).
The first few integral values are
2 sinh z
2 (sinh z — z cosh z)
_ , , 2(2 + z 2 ) sinh z - 4z cosh z
02(a) = ^ .
see also ALPHA FUNCTION
Beta Prime Distribution
A distribution with probability function
(3)
(4)
(5)
(6)
P{x) =
x a - 1 (l + xy
B(a,l3)
-OL-P
where B is a Beta Function. The Mode of a variate
distributed as (3 f (a,(3) is
. a-1
+ 1'
If x is a f (a,0) variate, then 1/x is a j9'(/3,a) variate.
If x is a j3(a,/3) variate, then (1 - x)/x and x/(l — x)
are 0\0 ) ct) and 0'{a,0) variates. If x and y are 7( a i)
and 7(0:2) variates, then x/y is a /?' (0:1,0:2) variate. If
x 2 /2 and y 2 /2 are 7(1/2) variates, then z 2 = (x/y) 2 is
a £'(1/2, 1/2) variate.
Bethe Lattice
see Cayley Tree
Betrothed Numbers
see QUASIAMICABLE PAIR
Betti Group
The free part of the Homology Group with a domain
of Coefficients in the Group of Integers (if this
Homology Group is finitely generated).
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, p. 380, 1988.
Betti Number
Bhargava's Theorem 127
Betti Number
Betti numbers are topological objects which were proved
to be invariants by Poincare, and used by him to ex-
tend the Polyhedral Formula to higher dimensional
spaces. The nth Betti number is the rank of the nth
Homology Group. Let p r be the Rank of the Ho-
mology Group H r of a Topological Space K. For
a closed, orientable surface of GENUS g, the Betti num-
bers are po = 1, Pi = 2#, and p 2 = I. For a nonori-
entable surface with k CROSS-CAPS, the Betti numbers
are po = 1, Pi = fc - 1, and p<z = 0.
see also Euler Characteristic, Poincare Duality
Bezier Curve
the fact that moving a single control point changes the
global shape of the curve. The former is sometimes
avoided by smoothly patching together low-order Bezier
curves. A generalization of the Bezier curve is the B-
Spline.
see also B-Spline, NURBS Curve
Bezier Spline
see Bezier Curve, Spline
Bezout Numbers
Integers (A,//) for a and b such that
Aa + fib = GCD(a,6).
Given a set of n control points, the corresponding Bezier
curve (or BernSTEIN-Bezier Curve) is given by
C(t) = 5^P.B i>n (t),
where Bi n (t) is a Bernstein Polynomial and t €
[0,1]-
A "rational" Bezier curve is defined by
C(*) =
jy; =0 B itP (t)wii>i
where p is the order, B itP are the BERNSTEIN POLYNO-
MIALS, Pi are control points, and the weight Wi of Pi is
the last ordinate of the homogeneous point P™. These
curves are closed under perspective transformations, and
can represent CONIC SECTIONS exactly.
The Bezier curve always passes through the first and
last control points and lies within the CONVEX Hull of
the control points. The curve is tangent to Pi — Po and
P n -P n _i at the endpoints. The "variation diminishing
property" of these curves is that no line can have more
intersections with a Bezier curve than with the curve
obtained by joining consecutive points with straight line
segments. A desirable property of these curves is that
the curve can be translated and rotated by performing
these operations on the control points.
Undesirable properties of Bezier curves are their numer-
ical instability for large numbers of control points, and
For Integers ai, . . . , a n , the Bezout numbers are a set
of numbers k\ , . . . , k n such that
k\a\ + k-2<i2 + . . . + k n a n = d,
where d is the Greatest Common Divisor of ai, . . . ,
a n .
see also GREATEST COMMON DIVISOR
Bezout's Theorem
In general, two algebraic curves of degrees m and n in-
tersect inm-n points and cannot meet in more than m-n
points unless they have a component in common (i.e.,
the equations defining them have a common factor).
This can also be stated: if P and Q are two POLYNOMI-
ALS with no roots in common, then there exist two other
Polynomials A and B such that AP + BQ = 1. Simi-
larly, given N Polynomial equations of degrees m, ri2,
. . . tin in N variables, there are in general niti2 • • • tin
common solutions.
see also POLYNOMIAL
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 10, 1959.
Bhargava's Theorem
Let the nth composition of a function f(x) be denoted
/ (n) (x), such that / (0) (z) = x and / Cl) (z) = f(x). De-
note / o g(x) = f(g(x)), and define
Let
u = (a, 6, c)
\u\ = a-h b + c
u\\ = a 4 + 6 4 + c 4 ,
(2)
(3)
(4)
128 Bhaskara-Brouckner Algorithm
and
/(«) = (/i(«),/ a («),/3(t*)) (5)
= (a(b - c), b(c - a),c(a - &)) (6)
S(w) = (5i( u )»P2H,53(«))
= (^a 2 6,^a& 2 ,3a&c) . (7)
Then if |u| = 0,
||/ (m) o 5 (n) (tx)|| = 2(a6 + 6c + ca) 2m+l3 "
= llff (n) o/ (m) (u)|| )
(8)
where 771, n E {0, 1, ...} and composition is done in
terms of components.
see also DlOPHANTINE EQUATION — QUARTIC, FORD'S
Theorem
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 97-100, 1994.
Bhargava, S. "On a Family of Ramanujan's Formulas for
Sums of Fourth Powers." Ganita 43, 63-67, 1992.
Bhaskara-Brouckner Algorithm
see Square Root
Bi-Connected Component
A maximal SUBGRAPH of an undirected graph such that
any two edges in the SUBGRAPH lie on a common simple
cycle.
see also Strongly Connected Component
Bianchi Identities
The Riemann Tensor is defined by
-IJLf
dx K dxv-
2 q2 q2
9^u a g\ K a g^ K
dx K dx x dx^dx u dx u dx x
Permuting 1/, «, and 77 (Weinberg 1972, pp. 146-147)
gives the Bianchi identities
see also BlANCHI IDENTITIES (CONTRACTED), RlE-
mann Tensor
References
Weinberg, S. Gravitation and Cosmology: Principles and
Applications of the General Theory of Relativity. New
York: Wiley, 1972.
BIBD
Bianchi Identities (Contracted)
Contracting A with v in the Bianchi Identities
gives
(2)
Contracting again,
R-n — R n\ii ~ R n-,v — 0, (3)
or
{R% - i<J%fi) ;M = 0,
(fl"" - \!TR);* = 0.
(4)
(5)
Bias (Estimator)
The bias of an ESTIMATOR 9 is defined as
b0) = (e) - e.
It is therefore true that
6 -6 = (8- (§)) + ((6) -$) = (0 - (§)) + B(0).
An Estimator for which B = is said to be Unbiased.
see also ESTIMATOR, UNBIASED
Bias (Series)
The bias of a Series is defined as
Q[ai, at+i,a»+2] '■■
A Series is Geometric Iff Q = 0. A Series is Artis-
tic Iff the bias is constant.
see also Artistic Series, Geometric Series
References
Duffin, R. J. "On Seeing Progressions of Constant Cross Ra-
tio." Amer. Math. Monthly 100, 38-47, 1993.
Biased
An Estimator which exhibits Bias.
Biaugmented Pentagonal Prism
see Johnson Solid
Biaugmented Triangular Prism
see Johnson Solid
Biaugmented Truncated Cube
see Johnson Solid
BIBD
see Block Design
Bicentric Polygon
Dicentric Polygon
Bicorn
129
A Polygon which has both a Circumcircle and an
INCIRCLE, both of which touch all VERTICES. All TRI-
ANGLES are bicentric with
R 2 -s 2 = 2Rr,
(1)
where R is the ClRCUMRADlUS, r is the Inradius, and s
is the separation of centers. In 1798, N. Puss character-
ized bicentric POLYGONS of n = 4, 5, 6, 7, and 8 sides.
For bicentric QUADRILATERALS (FUSS'S PROBLEM), the
Circles satisfy
2r 2 (R 2 ~s 2 )
(Dorrie 1965) and
Vabcd
(R 2 -s 2 ) 2 -4r 2 s 2
1 {ac + bd)(ad + bc)(ab + cd)
4 V abed
(Beyer 1987). In addition,
1 1
+
{R-s) 2 {R + s
and
a + c = b + d.
The Area of a bicentric quadrilateral is
A = vabed.
(2)
(3)
(4)
(5)
(6)
(7)
If the circles permit successive tangents around the In-
CIRCLE which close the POLYGON for one starting point
on the CIRCUMCIRCLE, then they do so for all points on
the Circumcircle.
see also PONCELET'S CLOSURE THEOREM
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 124, 1987.
Dorrie, H. "Fuss' Problem of the Chord- Tangent Quadrilat-
eral," §39 in 100 Great Problems of Elementary Mathe-
matics: Their History and Solutions. New York: Dover,
pp. 188-193, 1965.
Bicentric Quadrilateral
A 4-sided Bicentric Polygon, also called a Cyclic-
Inscriptable Quadrilateral.
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 124, 1987.
Bichromatic Graph
A Graph with Edges of two possible "colors," usually
identified as red and blue. For a bichromatic graph with
R red EDGES and B blue Edges,
R + B>2.
see also Blue-Empty Graph, Extremal Coloring,
Extremal Graph, Monochromatic Forced Tri-
angle, Ramsey Number
Bicollared
A SUBSET X C Y is said to be bicollared in Y if there
exists an embedding 6 : X x [-1, 1] -> Y such that
b(x, 0) = x when x £ X. The MAP 6 or its image is then
said to be the bicollar.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 34-35, 1976.
Bicorn
The bicorn is the name of a collection of QUARTIC
CURVES studied by Sylvester in 1864 and Cayley in 1867
(MacTutor Archive). The bicorn is given by the para-
metric equations
V
asint
a cos 2 t(2 + cost)
3 sin 2 t
The graph is similar to that of the COCKED HAT CURVE.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New-
York: Dover, pp. 147-149, 1972.
MacTutor History of Mathematics Archive. "Bicorn." http:
// www - groups . des . st - and .ac.uk/ -history / Curves /
Bicorn.html.
130 Bicubic Spline
Bieberbach Conjecture
Bicubic Spline
A bicubic spline is a special case of bicubic interpolation
which uses an interpolation function of the form
4 4
t=l j = l
4 4
Bidiakis Cube
J- 2
4 4
y X2 (xi,x 2 ) = 5^ 5^0" - l)cijt"~V
4 4
t=l J=l
where Cij are constants and u and £ are parameters rang-
ing from to 1. For a bicubic spline, however, the partial
derivatives at the grid points are determined globally by
1-D Splines.
see also B-Spline, Spline
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 118-122, 1992.
Bicupola
Two adjoined CUPOLAS.
see also Cupola, Elongated Gyrobicupola, Elon-
gated Orthobicupola, Gyrobicupola, Orthobi-
CUPOLA
Bicuspid Curve
The Plane Curve given by the Cartesian equation
(x 2 - a 2 )(x - a) 2 + (y 2 - a 2 ) 2 = 0.
Bicylinder
see Steinmetz Solid
f^
The 12- Vertex graph consisting of a Cube in which two
opposite faces (say, top and bottom) have edges drawn
across them which connect the centers of opposite sides
of the faces in such a way that the orientation of the
edges added on top and bottom are PERPENDICULAR to
each other.
see also Bislit Cube, Cube, Cubical Graph
Bieberbach Conjecture
The nth. Coefficient in the Power series of a Univa-
lent Function should be no greater than n. In other
words, if
f(z) = a + aiz 4- a 2 z 2 + . . . + a n z n + ...
is a conformal transformation of a unit disk on any do-
main, then|a n | < n|ai|. In more technical terms, "ge-
ometric extremality implies metric extremality." The
conjecture had been proven for the first six terms (the
cases n = 2, 3, and 4 were done by Bieberbach, Lowner,
and Sniffer and Garbedjian, respectively), was known
to be false for only a finite number of indices (Hayman
1954), and true for a convex or symmetric domain (Le
Lionnais 1983). The general case was proved by Louis
de Branges (1985). De Branges proved the MlLlN CON-
JECTURE, which established the ROBERTSON CONJEC-
TURE, which in turn established the Bieberbach conjec-
ture (Stewart 1996).
References
de Branges, L. "A Proof of the Bieberbach Conjecture." Acta
Math. 154, 137-152, 1985.
Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge,
England: Cambridge University Press, 1994.
Hayman, W. K. and Stewart, F. M. "Real Inequalities with
Applications to Function Theory." Proc. Cambridge Phil.
Soc. 50, 250-260, 1954.
Kazarinoff, N. D. "Special Functions and the Bieberbach
Conjecture." Amer. Math. Monthly 95, 689-696, 1988.
Korevaar, J. "Ludwig Bieberbach's Conjecture and its
Proof." Amer. Math. Monthly 93, 505-513, 1986.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 53, 1983.
Pederson, R. N. "A Proof of the Bieberbach Conjecture for
the Sixth Coefficient." Arch. Rational Mech. Anal. 31,
331-351, 1968/1969.
Pederson, R. and SchifFer, M. "A Proof of the Bieberbach
Conjecture for the Fifth Coefficient." Arch. Rational
Mech. Anal. 45, 161-193, 1972.
Stewart, I. "The Bieberbach Conjecture." In From Here to
Infinity: A Guide to Today's Mathematics. Oxford, Eng-
land: Oxford University Press, pp. 164-166, 1996.
Bienayme-Chebyshev Inequality
Biharmonic Equation 131
Bienayme-Chebyshev Inequality
see Chebyshev Inequality
Bifoliate
The Plane Curve given by the Cartesian equation
x A + y = 2axy .
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Bifolium
A Folium with 6 = 0. The bifolium is the Pedal
Curve of the Deltoid, where the Pedal Point is the
Midpoint of one of the three curved sides. The Carte-
sian equation is
(x 2 +y 2 ) 2 =4axy 2
and the POLAR equation is
r = 4a sin 2 OcosO.
see also FOLIUM, QuADRIFOLIUM, TRIFOLIUM
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 152-153, 1972.
MacTutor History of Mathematics Archive. "Double
Folium." http : // www - groups . dcs . st - and .ac.uk/
-history/Curves/Double .html.
Bifurcation
A period doubling, quadrupling, etc., that accompanies
the onset of Chaos. It represents the sudden appear-
ance of a qualitatively different solution for a nonlin-
ear system as some parameter is varied. Bifurcations
come in four basic varieties: FLIP BIFURCATION, FOLD
Bifurcation, Pitchfork Bifurcation, and Trans-
critical Bifurcation (Rasband 1990).
see also CODIMENSION, FEIGENBAUM CONSTANT,
Feigenbaum Function, Flip Bifurcation, Hopf
Bifurcation, Logistic Map, Period Doubling,
Pitchfork Bifurcation, Tangent Bifurcation,
Transcritical Bifurcation
References
Guckenheimer, J. and Holmes, P. "Local Bifurcations." Ch. 3
in Nonlinear Oscillations, Dynamical Systems, and Bifur-
cations of Vector Fields, 2nd pr., rev. corr. New York:
Springer- Verlag, pp. 117-165, 1983.
Lichtenberg, A. J. and Lieberman, M. A. "Bifurcation Phe-
nomena and Transition to Chaos in Dissipative Systems."
Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New-
York: Springer- Verlag, pp. 457-569, 1992.
Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4
in Chaotic Dynamics of Nonlinear Systems. New York:
Wiley, pp. 25-31, 1990.
Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to
Applied Nonlinear Dynamical Systems and Chaos. New
York: Springer- Verlag, pp. 253-419, 1990.
Bifurcation Theory
The study of the nature and properties of BIFURCA-
TIONS.
see also CHAOS, DYNAMICAL SYSTEM
Digraph
see Bipartite Graph
Bigyrate Diminished
Rhombicosidodecahedron
see Johnson Solid
Biharmonic Equation
The differential equation obtained by applying the Bi-
harmonic Operator and setting to zero.
vV = o. (i)
In Cartesian Coordinates, the biharmonic equation
V 2 (V 2 )0
dx 2 + dy 2 + dz 2 ) \dx 2 + dy 2 + dz 2 J *
-4 + ^-t + -Tnr +^--
dx 4 dy 4 dz 4 dx 2 dy 2
0.
d A (j> n d 4 <f>
(2)
dy 2 dz 2 dx 2 dz 2
In Polar Coordinates (Kaplan 1984, p. 148)
2 12
V (p = (prrrr H 2^ rr9$ ~* 4^0090 H 4>rrr
2 14 1
~<t>rdd ~4>rr + ~7<l>e0 + ~^4>r = 0. (3)
132 Biharmonic Operator
Billiards
For a radial function </>(r), the biharmonic equation be-
comes
Id f d [1 d
r dr \ dr [ r dr V dr J J J
2 11
Vr-rr + ~<firrr ~ ^<t>rr + -3 0r = 0. (4)
Writing the inhomogeneous equation as
V 4 = 64/3,
we have
M rdr = d{r±\ 1 -±(r^)]}
I dr lr dr \ dr / J J
2 Vlnr- |r 2
to obtain
#
(5)
dr L r dr V dr / J
r dr \ dr J
(16j3r 3 + Cir Inr + C 2 r) dr = d {r*j-\ . (10)
Now use
/ r In r dr = \
(6)
(7)
(8)
(9)
(11)
4/3r 4 + d(±r 2 lnr - \r 2 ) + §C 2 r 2 + ^ 3 = r^ (12)
(4/3r 3 + C> In r + C 2 r+— \ dr = d<f> (13)
</>(r)=/?r 4 -f C[ (|r 2 lnr- \r 2 )
+ §C 2 r 2 + C 3 lnr + C 4
= /?r 4 + or 2 + 6 4- (cr 2 + d) In (?-) . (14)
The homogeneous biharmonic equation can be separated
and solved in 2-D Bipolar Coordinates.
References
Kaplan, W. Advanced Calculus, ^th ed. Reading, MA:
Addison-Wesley, 1991.
Biharmonic Operator
Also known as the BlLAPLAClAN.
In n-D space,
V 4 = (V 2 ) 2 .
, 4 /'1\ _ 3(15 -8n + n 2 )
(;)-
Bijection
A transformation which is One-TO-One and ONTO.
see also One-to-One, Onto, Permutation
Bilaplacian
see Biharmonic Operator
Bilinear
A function of two variables is bilinear if it is linear with
respect to each of its variables. The simplest example is
f(x,y) =xy.
Bilinear Basis
A bilinear basis is a BASIS, which satisfies the conditions
(ax + by) • z = a(x * z) + 6(y • z)
z • (ax 4- by) = a(z • x) + 6(z • y).
see also Basis
Billiard Table Problem
Given a billiard table with only corner pockets and sides
of Integer lengths m and n, a ball sent at a 45° angle
from a corner will be pocketed in a corner after m+n-2
bounces.
see also Alhazen's Billiard Problem, Billiards
Billiards
The game of billiards is played on a RECTANGULAR table
(known as a billiard table) upon which balls are placed.
One ball (the "cue ball") is then struck with the end
of a "cue" stick, causing it to bounce into other balls
and Reflect off the sides of the table. Real billiards
can involve spinning the ball so that it does not travel
in a straight LINE, but the mathematical study of bil-
liards generally consists of REFLECTIONS in which the
reflection and incidence angles are the same. However,
strange table shapes such as CIRCLES and Ellipses are
often considered. Many interesting problems can arise.
For example, Alhazen's BILLIARD PROBLEM seeks to
find the point at the edge of a circular "billiards" table
at which a cue ball at a given point must be aimed in
order to carom once off the edge of the table and strike
another ball at a second given point. It was not until
1997 that Neumann proved that the problem is insoluble
using a COMPASS and RULER construction.
On an ELLIPTICAL billiard table, the ENVELOPE of a
trajectory is a smaller ELLIPSE, a HYPERBOLA, a LINE
through the FOCI of the ELLIPSE, or periodic curve (e.g.,
DlAMOND-shape) (Wagon 1991).
see also Alhazen's Billiard Problem, Billiard Ta-
ble Problem, Reflection Property
see also Biharmonic Equation
Billion
Binary 133
References
Davis, D.; Ewing, C; He, Z.; and Shen, T. "The
Billiards Simulation." http : //serendip .brynmawr . edu/
chao s /home . html .
Dullin, H. R.; Richter, RH.; and Wittek, A. "A Two-
Parameter Study of the Extent of Chaos in a Billiard Sys-
tem." Chaos 6, 43-58, 1996.
Madachy, J. S. "Bouncing Billiard Balls." In Madachy's
Mathematical Recreations. New York: Dover, pp. 231—
241, 1979.
Neumann, P. Submitted to Amer. Math. Monthly.
Pappas, T. "Mathematics of the Billiard Table." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
p. 43, 1989.
Peterson, I. "Billiards in the Round." http : //www .
sciencenews.org/sn_arc97/3-l_97/mathland.htm.
Wagon, S. "Billiard Paths on Elliptical Tables." §10.2 in
Mathematica in Action. New York: W. H. Freeman,
pp. 330-333, 1991.
Billion
The word billion denotes different numbers in American
and British usage. In the American system, one billion
equals 10 9 . In the British, French, and German systems,
one billion equals 10 12 .
see also LARGE NUMBER, MILLIARD, MILLION, TRIL-
LION
Bilunabirotunda
see Johnson Solid
Bimagic Square
16
41
36
5
27
62
55
18
26
63
54
19
13
44
33
8
1
40
45
12
22
51
58
31
23
50
59
30
4
37
48
9
38
3
10
47
49
24
29
60
52
21
32
57
39
2
11
46
43
14
7
34
64
25
20
53
61
28
17
56
42
15
6
35
If replacing each number by its square in a MAGIC
Square produces another Magic Square, the square
is said to be a bimagic square. The first bimagic square
(shown above) has order 8 with magic constant 260 for
addition and 11,180 after squaring. Bimagic squares
are also called Doubly Magic Squares, and are 2-
Multimagic Squares.
see also MAGIC SQUARE, MULTIMAGIC SQUARE,
Trimagic Square
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 212,
1987.
Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3
in Mathematical Diversions. New York: Dover, p. 31,
1975.
Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical
Recreations. New York: W. W. Norton, pp. 176-178, 1942.
M BC
A' M AB
A Line Segment joining the Midpoints of opposite
sides of a QUADRILATERAL.
see also Median (Triangle), Varignon's Theorem
Bimodal Distribution
A Distribution having two separated peaks.
see also Unimodal Distribution
Bimonster
The wreathed product of the Monster Group by Z 2 .
The bimonster is a quotient of the Coxeter Group
with the following Coxeter-Dynkin Diagram.
This had been conjectured by Conway, but was proven
around 1990 by Ivanov and Norton. If the parameters
p,<?, r in Coxeter's NOTATION [3 F,q>r ] are written side
by side, the bimonster can be denoted by the BEAST
Number 666.
Bin
An interval into which a given data point does or does
not fall.
see also HISTOGRAM
Binary
The BASE 2 method of counting in which only the digits
and 1 are used. In this Base, the number 1011 equals
l-2° + l-2 + 0-2 2 + l-2 3 = 11. This Base is used in com-
puters, since all numbers can be simply represented as
a string of electrically pulsed ons and offs. A NEGATIVE
— n is most commonly represented as the complement of
the Positive number n - 1, so -11 = 00001011 2 would
be written as the complement of 10 — OOOOIOIO2, or
11110101. This allows addition to be carried out with
the usual carrying and the left-most digit discarded, so
17 — 11 = 6 gives
00010001 17
11110101 -11
00000110 6
134 Binary Bracketing
Binary Tree
The number of times k a given binary number
b n ■ • -&2&1&0 is divisible by 2 is given by the position
of the first bk = 1 counting from the right. For exam-
ple, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is
divisible by 2 times.
Unfortunately, the storage of binary numbers in com-
puters is not entirely standardized. Because computers
store information in 8-bit bytes (where a bit is a sin-
gle binary digit), depending on the "word size" of the
machine, numbers requiring more than 8 bits must be
stored in multiple bytes. The usual F0RTRAN77 integer
size is 4 bytes long. However, a number represented as
(bytel byte2 byte3 byte4) in a VAX would be read and
interpreted as (byte4 byte3 byte2 bytel) on a Sun. The
situation is even worse for floating point (real) num-
bers, which are represented in binary as a MANTISSA
and Characteristic, and worse still for long (8-byte)
reals!
Binary multiplication of single bit numbers (0 or 1) is
equivalent to the AND operation, as can be seen in the
following Multiplication Table.
X
1
1
1
see also Base (Number), Decimal, Hexadecimal,
Octal, Quaternary, Ternary
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 6-9,
1991.
Pappas, T. "Computers, Counting, & Electricity." The Joy
of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 24-25, 1989.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Error, Accuracy, and Stability" and "Diag-
nosing Machine Parameters." §1.2 and §20.1 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 18-21, 276, and 881-886, 1992.
^ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Bases.m.
Binary Bracketing
A binary bracketing is a BRACKETING built up entirely
of binary operations. The number of binary bracket ings
of n letters (Catalan's Problem) are given by the
Catalan Numbers C n _i, where
C n =
n + 1
2n\ _ 1 (2ra)! _
n ) n+ 1 n! 2
(2n)!
(n+l)!n!
where ( 2 ™) denotes a Binomial Coefficient and n\
is the usual FACTORIAL, as first shown by Catalan in
1838, For example, for the four letters a, 6, c, and d
there are five possibilities: ({ab)c)d, (a(6c))d, (a&)(cd),
a((bc)d), and a(6(cd)), written in shorthand as {(xx)x)x }
(x(xx))x, (xx)(xx), x((xx)x), and x(x(xx)).
see also BRACKETING, CATALAN NUMBER, CATALAN'S
Problem
References
Schroder, E. "Vier combinatorische Probleme." Z. Math.
Physik 15, 361-376, 1870.
Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough."
Amer. Math. Monthly 104, 344-350, 1997.
Binary Operator
An Operator which takes two mathematical objects
as input and returns a value is called a binary operator.
Binary operators are called compositions by Rosenfeld
(1968). Sets possessing a binary multiplication opera-
tion include the Group, Groupoid, Monoid, Quasi-
group, and Semigroup. Sets possessing both a bi-
nary multiplication and a binary addition operation in-
clude the Division Algebra, Field, Ring, Ringoid,
Semiring, and Unit Ring.
see also AND, BOOLEAN ALGEBRA, CLOSURE, DIVI-
SION Algebra, Field, Group, Groupoid, Monoid,
Operator, Or, Monoid, Not, Quasigroup, Ring,
Ringoid, Semigroup, Semiring, XOR, Unit Ring
References
Rosenfeld, A. An Introduction to Algebraic Structures. New
York: Holden-Day, 1968.
Binary Quadratic Form
A 2-variable QUADRATIC FORM of the form
Q(x, y) = aux 2 + 2a\ixy + a 2 2V .
see also QUADRATIC FORM, QUADRATIC INVARIANT
Binary Remainder Method
An Algorithm for computing a Unit Fraction
(Stewart 1992).
References
Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer.
266, 122-124, June 1992.
Binary Tree
A Tree with two Branches at each Fork and with
one or two Leaves at the end of each Branch. (This
definition corresponds to what is sometimes known as
an "extended" binary tree.) The height of a binary tree
is the number of levels within the TREE. For a binary
tree of height H with n nodes,
H < n < 2 H - 1,
Binet Forms
Binomial Coefficient 135
These extremes correspond to a balanced tree (each
node except the Leaves has a left and right Child,
arid all LEAVES are at the same level) and a degenerate
tree (each node has only one outgoing BRANCH), respec-
tively. For a search of data organized into a binary tree,
the number of search steps S(n) needed to find an item
is bounded by
lgn < S(n) < n.
Partial balancing of an arbitrary tree into a so-called
AVL binary search tree can improve search speed.
The number of binary trees with n internal nodes is
the Catalan Number C n (Sloane's A000108), and the
number of binary trees of height b is given by Sloane's
A001699.
see also S-Tree, Quadtree, Quaternary Tree,
Red-Black Tree, Stern-Brocot Tree, Weakly
Binary Tree
References
Lucas, J.; Roelants van Baronaigien, D.; and Ruskey, F.
"Generating Binary Trees by Rotations." J. Algorithms
15, 343-366, 1993.
Ranum, D. L. "On Some Applications of Fibonacci Num-
bers." Amer. Math. Monthly 102, 640-645, 1995.
Ruskey, F. "Information on Binary Trees." http://sue.csc
,uvic.ca/~cos/inf/tree/BinaryTrees.html.
Ruskey, F. and Proskurowski, A. "Generating Binary Trees
by Transpositions." J. Algorithms 11, 68-84, 1990.
Sloane, N. J. A. Sequences A000108/M1459 and A001699/
M3087 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Binet Forms
The two Recurrence Sequences
U n — mUn-l + U n ~2
V n =mV n - 1 + V n - 2
(1)
(2)
with Uo = 0, Ui = 1 and Vo = 2, V\ — m, can be solved
for the individual U n and V n . They are given by
" ~ P (3)
(4)
(5)
(6)
(7)
(8)
Binet' S Formula is a special case of the Binet form
for U n corresponding to m = 1.
see also Fibonacci Q-Matrix
yJ-n —
A
V n =
= a n + ^,
where
A =
\/m 2 + 4
a =
771+ A
2
=
m - A
2
A useful related
identity
is
Un-l+Un+l = Vn-
Binet 's Formula
A special case of the U n Binet Form with m
corresponding to the nth FIBONACCI NUMBER,
_ _ (1 + V5)"-(1-a/5) b
0,
2 n VE
It was derived by Binet in 1843, although the result
was known to Euler and Daniel Bernoulli more than a
century earlier.
see also Binet Forms, Fibonacci Number
Bing's Theorem
If M 3 is a closed oriented connected 3-MANIFOLD such
that every simple closed curve in M lies interior to a
BALL in Af , then M is HOMEOMORPHIC with the Hy-
persphere, S 3 .
see also Ball, Hypersphere
References
Bing, R. H. "Necessary and Sufficient Conditions that a 3-
Manifold be S 3 ." Ann. Math. 68, 17-37, 1958.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 251-257, 1976.
Binomial
A Polynomial with 2 terms.
see also Monomial, Polynomial, Trinomial
Binomial Coefficient
The number of ways of picking n unordered outcomes
from N possibilities. Also known as a COMBINATION.
The binomial coefficients form the rows of PASCAL'S
Triangle. The symbols N C n and
n) = (N - n)\n\
(1)
are used, where the latter is sometimes known as N
CHOOSE n. The number of LATTICE PATHS from the
Origin (0, 0) to a point (a, b) is the Binomial Coeffi-
cient (°+ 6 ) (Hilton and Pedersen 1991).
For Positive integer n, the Binomial Theorem gives
i.— n V /
(2)
The Finite Difference analog of this identity is
known as the Chu-Vandermonde Identity. A sim-
ilar formula holds for Negative Integral n,
~ n \ k-Tl-k
,x a
k
(3)
A general identity is given by
(a + b) n
= Eu)( a -* : ) i " 1 ( 6 +j' c ) B "' w
136 Binomial Coefficient
(Prudnikov et ol. 1986), which gives the BINOMIAL THE-
OREM as a special case with c = 0.
The binomial coefficients satisfy the identities:
n
n
k
n + 1
A:
„-»-<-<":"'
n \ I n
(5)
(6)
(7)
Sums of powers include
fc=0 x y
E<-')"(I)=»
k=0 x 7
t(;V -(•+')•
(8)
(9)
(10)
(the Binomial Theorem), and
^2n + s
£
a; n = 2^1(1(5 4- 1), §(* + 2); s + 1, 4x)
2 s
(\/l - 4x + l)Vl - 4a;
, (11)
where 2F1 (a, 6; c;z) is a Hypergeometric Function
(Abramowitz and Stegun 1972, p. 555; Graham et al.
1994, p. 203). For NONNEGATIVE INTEGERS n and r
with r < n + 1,
££?(;) [D-*G")"- fl '
fc^o v 7 L j=o v y
+Et- 1 ) i (")( n+l - r -j) n - fc
Taking n = 2r — 1 gives
r-l
n!. (12)
-fc _ i
in!. (13)
Another identity is
E ( n £ k ) [xn+1(1 - x)fc + (1 - x)n+1 ^ = 1 w
Binomial Coefficient
Recurrence Relations of the sums
EI
— . <15)
are given by
2si(n)-si(n + l) = (16)
-2(2n + l)s 2 (n) + (n + l)s 2 (n) = (17)
-8(n + l) 2 s 3 (n) + (-16 - 21n - 7n 2 )s 3 (n + 1)
+(n + 2) 2 53 (n + 2) = (18)
-4(n + l)(4n + 3)(4n + 5)s 4 (n)
-2(2n + 3)(3n 2 + 9n + 7)s 4 (n + 1)
+(n + 2) 3 s 4 (n + 2) = 0. (19)
This sequence for S3 cannot be expressed as a fixed
number of hypergeometric terms (Petkovsek et a/. 1996,
p. 160).
A fascinating series of identities involving binomial co-
efficients times small powers are
00
£ 72^Y = 27 ( 27r ^ + 9 ) = 0.7363998587 . . .
n=l V n )
00
E— 1_ = Ittv^ = 0.6045997881 . . .
n( 2n ) 9
n=l \n J
n=l V n 7
°° 1
/ ^ 4(2n\ 36 ^W 3240 /l
71=1 V n /
(Comtet 1974, p. 89) and
- ( _ 1} n-i
£
8 a
= I C(3),
(20)
(21)
(22)
(23)
(24)
where ((z) is the Riemann Zeta Function (Le Lion-
nais 1983, pp. 29, 30, 41, 36, and 35; Guy 1994, p. 257).
As shown by Kummer in 1852, the exact Power of p
dividing ( a ^ b ) is equal to
eo + ei + . . . + e £ ,
(25)
(Beeler et al 1972, Item 42).
where this is the number of carries in performing the
addition of a and b written in base b (Graham et al.
1989, Exercise 5.36; Ribenboim 1989; Vardi 1991, p. 68).
Kummer's result can also be stated in the form that the
Binomial Coefficient
Binomial Coefficient
137
exponent of a Prime p dividing (j^j is given by the
number of integers j > for which
frac(ra/p J ) > frac (n/p 3 ).
(26)
where frac(cc) denotes the FRACTIONAL PART of x. This
inequality may be reduced to the study of the exponen-
tial sums ^2 n A(n)e(x/n), where A(n) is the MANGOLDT
FUNCTION. Estimates of these sums are given by Jutila
(1974, 1975), but recent improvements have been made
by Granville and Ramare (1996).
R. W. Gosper showed that
/( n ) = (l(n~-l)) ~ (- 1 ) < "" 1)/2 ( mod ") (27)
for all Primes, and conjectured that it holds only for
Primes. This was disproved when Skiena (1990) found
it also holds for the Composite Number n = 3xllx
179. Vardi (1991, p. 63) subsequently showed that n =
p 2 is a solution whenever p is a Wieferich Prime and
that if n = p k with k > 3 is a solution, then so is n =
p k ~ 1 . This allowed him to show that the only solutions
for Composite n < 1.3xl0 7 are 5907, 1093 2 , and 3511 2 ,
where 1093 and 3511 are Wieferich PRIMES.
Consider the binomial coefficients ( n ~ )•> the first few
of which are 1, 3, 10, 35, 126, ... (Sloane's A001700).
The Generating Function is
Vl-4o;
: x + 3x 2 + 10x 3 + 35x 4 + .
(28)
These numbers are SQUAREFREE only for n = 2, 3, 4,
6, 9, 10, 12, 36, . . . (Sloane's A046097), with no others
less than n = 10, 000. Erdos showed that the binomial
coefficient (™) is never a Power of an Integer for n >
3 where A; ^ 0, 1, n— 1, and n (Le Lionnais 1983, p. 48).
The binomial coefficients (| n / 2 |) are called CENTRAL
Binomial Coefficients, where |xj is the Floor
Function, although the subset of coefficients ( 2 ™) is
sometimes also given this name. Erdos and Graham
(1980, p. 71) conjectured that the Central Binomial
Coefficient ( 2 ^) is never Squarefree for n > 4, and
this is sometimes known as the Erdos SQUAREFREE
Conjecture. Sarkozy's Theorem (Sarkozy 1985)
provides a partial solution which states that the BINO-
MIAL Coefficient ( 2 ^) is never Squarefree for all
sufficiently large n > no (Vardi 1991). Granville and
Ramare (1996) proved that the only SQUAREFREE val-
ues are n = 2 and 4. Sander (1992) subsequently showed
that ( 2n ^ d ) are also never SQUAREFREE for sufficiently
large n as long as d is not "too big."
For p, qr, and r distinct PRIMES, then the above function
satisfies
f(pqr)f(p)f(q)f(r) = f {pq) f (pr)p(qr) (mod pqr)
(29)
(Vardi 1991, p. 66).
The binomial coefficient (™) mod 2 can be computed
using the XOR operation n XOR m, making Pascal's
Triangle mod 2 very easy to construct.
The binomial coefficient "function" can be defined as
C{z,y)
y\(x - y)\
(30)
(Fowler 1996), shown above. It has a very complicated
Graph for Negative x and y which is difficult to render
using standard plotting programs.
see also BALLOT PROBLEM, BINOMIAL DISTRIBU-
TION, Binomial Theorem, Central Binomial Co-
efficient, Chu-Vandermonde Identity, Combi-
nation, Deficiency, Erdos Squarefree Conjec-
ture, Gaussian Coefficient, Gaussian Polynom-
ial, Kings Problem, Multinomial Coefficient,
Permutation, Roman Coefficient, Sarkozy's
Theorem, Strehl Identity, Wolstenholme's The-
orem
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Binomial Co-
efficients. " §24.1.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 10 and 822-823, 1972.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Comtet, L. Advanced Combinatorics. Amsterdam, Nether-
lands: Kluwer, 1974.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 66-74, 1996.
Erdos, P.; Graham, R. L.; Nathanson, M. B.; and Jia, X. Old
and New Problems and Results in Combinatorial Number
Theory. New York: Springer- Verlag, 1998,
Fowler, D. "The Binomial Coefficient Function." Amer.
Math. Monthly 103, 1-17, 1996.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial
Coefficients." Ch. 5 in Concrete Mathematics: A Foun-
dation for Computer Science. Reading, MA: Addison-
Wesley, pp. 153-242, 1990.
Granville, A. and Ramare, O. "Explicit Bounds on Exponen-
tial Sums and the Scarcity of Squarefree Binomial Coeffi-
cients." Mathematika 43, 73-107, 1996.
138
Binomial Distribution
Binomial Distribution
Guy, R. K. "Binomial Coefficients," "Largest Divisor of a
Binomial Coefficient," and "Series Associated with the £-
Function." §B31, B33, and F17 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 84-85, 87-89, and 257-258, 1994.
Harborth, H. "Number of Odd Binomial Coefficients." Not.
Amer. Math. Soc. 23, 4, 1976.
Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel 13, 64-75, 1991.
Jutila, M. "On Numbers with a Large Prime Factor." J.
Indian Math. Soc. 37, 43-53, 1973.
Jutila, M. "On Numbers with a Large Prime Factor. II," J.
Indian Math. Soc. 38, 125-130, 1974.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Ogilvy, C. S. "The Binomial Coefficients." Amer. Math.
Monthly 57, 551-552, 1950.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Gamma Function, Beta Function, Factorials,
Binomial Coefficients." §6.1 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, Qnd ed. Cam-
bridge, England: Cambridge University Press, pp. 206—
209, 1992.
Prudnikov, A. P.; Marichev, O. I.; and Brychkow, Yu. A.
Formula 41 in Integrals and Series, Vol. 1: Elementary
Functions. Newark, NJ: Gordon & Breach, p. 611, 1986.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 23-24, 1989.
Riordan, J. "Inverse Relations and Combinatorial Identities."
Amer. Math. Monthly 71, 485-498, 1964.
Sander, J. W. "On Prime Divisors of Binomial Coefficients."
Bull. London Math. Soc. 24,140-142, 1992.
Sarkozy, A. "On the Divisors of Binomial Coefficients, I." J.
Number Th. 20, 70-80, 1985.
Skiena, S, Implementing Discrete Mathematics: Combina-
torics and Graph Theory with Mathematica. Reading,
MA: Addison- Wesley, p. 262, 1990.
Sloane, N. J. A. Sequences A046097 and A001700/M2848 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Spanier, J. and Oldham, K. B. "The Binomial Coefficients
(^)." Ch. 6 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 43-52, 1987.
Sved, M. "Counting and Recounting." Math. Intel 5, 21-26,
1983.
Vardi, I. "Application to Binomial Coefficients," "Binomial
Coefficients," "A Class of Solutions," "Computing Bino-
mial Coefficients," and "Binomials Modulo and Integer."
§2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations
in Mathematica. Redwood City, CA: Addison- Wesley,
pp. 25-28 and 63-71, 1991.
Wolfram, S. "Geometry of Binomial Coefficients." Amer.
Math. Monthly 91, 566-571, 1984.
Binomial Distribution
The probability of n successes in N BERNOULLI TRIALS
is
n N — n
P Q
(1)
The probability of obtaining more successes than the n
observed is
*=E
k = n + l
N ^p k (l-p) N - k =I p (n + l,N-N), (2)
where
Ix{a,b)
B{x\ a y b)
B(a,b) '
(3)
B(a,b) is the Beta Function, and B(x\a,b) is the
incomplete BETA FUNCTION. The CHARACTERISTIC
Function is
<f>(t) = ( q + pe it )\ (4)
The Moment-Generating Function M for the dis-
tribution is
N y v
M(t) = <e tn > = XX n ( ^ V<? N " n
n=0 ^ '
= E(?V)"(i-p) n -"
= \pe t + (l-p)] N
At (t) = Nfre* + (1 - p)] JV - 1 (pe')
M"it) = NiN - l)[pc* + (1 - p)]"- V) 2
+ N\pe t + il-p)} N - 1 ipe t ).
The Mean is
H = M'(0) = Nip + 1 - p)p = Np.
The Moments about are
(5)
(6)
(7)
(8)
/*!=/* = Np (9)
l& = Np(l-p + Np) (10)
^ = Np(l - 3p + 3Np + 2p 2 - 3NP 2 + N 2 p 2 ) (11)
/4 = Np(l - 7p + 7Np + 12p 2 - ISNp 2 + 6iVV
- 6p 3 + HATp 3 - 6iVV + A^p 3 ), (12)
so the Moments about the Mean are
M2 = a 2 = [N(N - l)p 2 + Np] - {Np) 2
= N 2 p 2 - Np 2 +Np- N 2 p 2
= Np(l -p) = Npq
(13)
(14)
Pz = P3 ~ 3p f 2Pi + 2{pif
= Np(l-p)(l-2p)
li4 = fJ,4- 4/4/ii + 6/i2(/ii) 2 - 3(/ii) 4
= Np(l - p)[3p 2 (2 - N) + 3p(N - 2) + 1]. (15)
Binomial Distribution
The SKEWNESS and KURTOSIS are
/is = Np(l-p){l-2p)
<7 3 [iV>(l-p)] 3 /2
7i
l-2p
q-p
y/Np(l-p) y/Npq
7 2 = ^-3:
cr 4
6p — 6p + 1 1 — 6pg
iVp(l-p)
A^pg
(16)
(17)
An approximation to the Bernoulli distribution for large
N can be obtained by expanding about the value n
where P(n) is a maximum, i.e., where dP/dn = 0. Since
the Logarithm function is Monotonic, we can instead
choose to expand the LOGARITHM. Let n = h + to, then
ln[F(n)]-ln[P(n)] + B 1 7?+|B2T? 2 + |jS 3 7 ? 3 + ..., (18)
where
(19)
B k =
d k \n[P(n)]
dn k
But we are expanding about the maximum, so, by defi
nition,
~dln[P(n)]
Bi
dn
= 0.
(20)
This also means that B2 is negative, so we can write
B 2 = — 1B 2 1 . Now, taking the LOGARITHM of (1) gives
ln[P(n)] = lnNl-\nn\-ln(N-n)\ + nlnp+(N-n)\nq.
(21)
For large n and N — n we can use STIRLING'S APPROX-
IMATION
ln(n!) « toIxito-to, (22)
so
d[ln(w!)]
dn
d[\n(N-n)\]
dn
« (Inn -I- 1) - 1 = lnn
d
(23)
dn
[{N - n) \vv{N -n)-(N -
= -ln(7V-n),
(24)
and
dln{P(n)} ^ _ lnn + ln(JV _ w) + lnp _ lnq / 25 v
dn
To find n, set this expression to and solve for ra,
(26)
N — hp
h (^)='
1
n q
(N — n)p = hq
n(q + p) = h = Np,
(27)
(28)
(29)
Binomial Distribution 139
since p + q — 1. We can now find the terms in the
expansion
B 2
d 2 \n[P(n)]
dn 2
1 1
h N ~ h
1 1 _ _ J_ (\ l\
Np N(l-p)~ N\p + q)
i (p + q
B 3 =
N \ pq
d*\n[P(n)]
dn 3
1 1
1
1
Npq N(l-p)
1 1
(30)
h 2 (N - h) 2
„2 2
N 2 p 2 N 2 q 2 N 2 p 2 q 2
(l-2p + p 2 )-p 2 _
l-2p
B 4 =
N 2 p 2 (l-p) 2 N 2 p 2 {l-p) 2
d 4 \n[P(n)]
(31)
dn 4
h 3 (n — h) 3
-2
3
_J_ _J_\ = 2(P 3 +Q
N 3 p 3 N 3 q 3 J N 3 p 3 q<
= 2(p 2 -pq + q 2 )
N z p 3 q 3
= 2\p 2 -p{l-p) + {l-2p + p 2 )]
N 3 p 3 (l-p 3 )
= 2(3p 2 -3p+l)
N 3 p 3 (l-p 3 ) '
Now, treating the distribution as continuous,
(32)
■W p /»oo
lim y^P(n)^ P(n)dn= / P(h + to) dn - 1.
(33)
Since each term is of order 1/iV ~ 1/<t 2 smaller than the
previous, we can ignore terms higher than B 2 , so
P(n) = P(n)e- |B2| " 2/2 .
The probability must be normalized, so
(34)
J~ P(fi)e-W 2 '*dT, = P(n)^=l, (35)
and
P(n)
\Bl\ -\B 2 \(n-n) 2 /2
2tt
yj2nNpq
exp
Defining a 2 = 2Npq,
P(n)
(7V27T
: exp
(n - iVp) 2
2iVpg
(to - n) 2
2a 2
(36)
(37)
140 Binomial Expansion
Binomial Series
which is a GAUSSIAN DISTRIBUTION. For p < 1, a
different approximation procedure shows that the bi-
nomial distribution approaches the PoiSSON DISTRIBU-
TION. The first Cumulant is
m = np,
(38)
and subsequent Cumulants are given by the RECUR-
RENCE Relation
kv+i = pq
dp '
(39)
Let x and y be independent binomial Random Vari-
ables characterized by parameters n,p and m,p. The
Conditional Probability of x given that x + y = k
is
P(a; = i|a; + y — k)
P(x = i y x + y = k)
P{x + y = k)
P(x = i,y = k-i) _ P(x = i)P(y = k-i)
P(x-\-y = k) ~ P{x + y = k)
( n t m )p fc (i-p) n+m - fc
(40)
Note that this is a Hypergeometric Distribution!
see also de Moivre-Laplace Theorem, Hypergeo-
metric Distribution, Negative Binomial Distri-
bution
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 531, 1987,
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Incomplete Beta Function, Student's Distribu-
tion, F-Distribution, Cumulative Binomial Distribution."
§6.2 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 219-223, 1992.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 108-109, 1992.
Binomial Expansion
see Binomial Series
Binomial Formula
see Binomial Series, Binomial Theorem
Binomial Number
A number of the form a n ± b n , where a, 6, and n are
Integers. They can be factored algebraically
a n -& n = (a-6)(a ri ~ 1 +a Tl - 2 b + ... + a6 n " 2 +b n - 1 ) (1)
a n + b n -(a + 6)(a n - 1 -a n - 2 6 + ...-ab n - 2 +6 n - 1 ) (2)
a nm - b nTn = (a m - 6 m )[a m(n ~ 1) + a m(n ' 2) 6 m
+ ... + 6 m(n_1) ]. (3)
In 1770, Euler proved that if (a, b) = 1, then every FAC-
TOR of
o a "+6 jn (4)
is either 2 or of the form 2 n+1 K + 1. If p and q are
Primes, then
a pq -l)(a-l)
(aP-l)(a«-l]
- 1
(5)
is Divisible by every Prime Factor of a p 1 not divid-
ing a q — 1.
see also CUNNINGHAM NUMBER, FERMAT NUMBER,
Mersenne Number, Riesel Number, Sierpinski
Number of the Second Kind
References
Guy, R. K. "When Does 2 a - 2 b Divide n a - n 6 ." §B47 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, p. 102, 1994.
Qi, S and Ming-Zhi, Z. "Pairs where 2 a - a b Divides n a - n h
for All n." Proc. Amer. Math. Soc. 93, 218-220, 1985.
Schinzel, A. "On Primitive Prime Factors of a n — 6 n ." Proc.
Cambridge Phil Soc. 58, 555-562, 1962.
Binomial Series
For Id < 1,
(i + x y
- £(:)■*
fc=o v 7
(i)
= i +
;x +
l!(n-l)! (n-2)!2!
" ! x= + ...(3)
n(n — 1) o
l + nz+ -^ — -x 2 + .
(4)
The binomial series also has the CONTINUED FRACTION
representation
(1 + *)" =
-. (5)
1 +
l-(l + n)
1-2
1 +
1 ■(!-") ,
2-3
1 +
2(2 + n)
3-4
1+-
2(2 -n) ,
4-5
3(3 + n)
5-6
1 + ...
1 +
Binomial Theorem
Biotic Potential
141
see also Binomial Theorem, Multinomial Series,
Negative Binomial Series
References
Abramowitz, M. and Stegun, C, A, (Eds,). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 14-15, 1972.
Pappas, T. "Pascal's Triangle, the Fibonacci Sequence &; Bi-
nomial Formula." The Joy of Mathematics. San Carlos,
CA: Wide World Publ./Tetra, pp. 40-41, 1989.
Binomial Theorem
The theorem that, for INTEGRAL POSITIVE n,
Z_/ kun -
A;=0
(n-k)\
^r
k=0
the so-called Binomial Series, where (™) are Bino-
mial Coefficients. The theorem was known for the
case n = 2 by Euclid around 300 BC, and stated in its
modern form by Pascal in 1665. Newton (1676) showed
that a similar formula (with Infinite upper limit) holds
for Negative Integral n,
(* + a)-» = £; ( fc n y a -
the so-called Negative Binomial Series, which con-
verges for |x| > \a\.
see also BINOMIAL COEFFICIENT, BINOMIAL SERIES,
Cauchy Binomial Theorem, Chu-Vandermonde
Identity, Logarithmic Binomial Formula, Nega-
tive Binomial Series, <?-Binomial Theorem, Ran-
dom Walk
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 10, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 307-308, 1985.
Conway, J. H. and Guy, R. K. "Choice Numbers Are Bino-
mial Coefficients." In The Book of Numbers. New York:
Springer- Verlag, pp. 72-74, 1996.
Coolidge, J. L. "The Story of the Binomial Theorem," Amer.
Math. Monthly 56, 147-157, 1949.
Courant, R. and Robbins, H. "The Binomial Theorem." §1.6
in What is Mathematics?: An Elementary Approach to
Ideas and Methods, 2nd ed. Oxford, England: Oxford Uni-
versity Press, pp. 16-18, 1996.
Binomial Triangle
see Pascal's Triangle
Binormal Developable
A Ruled Surface M is said to be a binormal de-
velopable of a curve y if M can be parameterized by
x(«,v) = y(u)+t;B(u), where B is the BINORMAL VEC-
TOR.
see also NORMAL DEVELOPABLE, TANGENT DEVEL-
OPABLE
References
Gray, A. "Developables." §17.6 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 352-354, 1993.
Binormal Vector
:TxN
r' x r"
' |r' xr'T
(1)
(2)
where the unit TANGENT VECTOR T and unit "princi-
pal" NORMAL VECTOR N are defined by
t - r'(s)
N:
|r'( S )|
\t"(s)\
(3)
(4)
Here, r is the Radius Vector, s is the Arc Length, r
is the TORSION, and « is the Curvature. The binormal
vector satisfies the remarkable identity
[B,B,B1
ds
(")
(5)
see also Frenet Formulas, Normal Vector, Tan-
gent Vector
References
Kreyszig, E. "Binormal. Moving Trihedron of a Curve." §13
in Differential Geometry. New York: Dover, p. 36—37,
1991.
Bioche's Theorem
If two complementary PLUCKER CHARACTERISTICS are
equal, then each characteristic is equal to its comple-
ment except in four cases where the sum of order and
class is 9.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 101, 1959.
Biotic Potential
see Logistic Equation
142 Bipartite Graph
Bipartite Graph
A set of VERTICES decomposed into two disjoint sets
such that no two VERTICES within the same set are
adjacent. A bigraph is a special case of a &- Partite
Graph with k = 2.
see also Complete Bipartite Graph, /c-Partite
Graph, Konig-Egevary Theorem
References
Chartrand, G. Introductory Graph Theory. New York:
Dover, p. 116, 1985.
Saaty, T. L. and Kainen, P. C. The Four- Color Problem:
Assaults and Conquest, New York: Dover, p. 12, 1986.
Biplanar Double Point
see Isolated Singularity
Bipolar Coordinates
Bipolar coordinates are a 2-D system of coordinates.
There are two commonly defined types of bipolar co-
ordinates, the first of which is defined by
a sinh v
y =
cosh v — cos u
as'mu
cosh v — cos u '
(i)
(2)
where u € [0,27r), v G (—00,00). The following identi-
ties show that curves of constant u and v are CIRCLES
in ay-space.
x 2 + {y — a cot u) 2 — a 2 esc 2 u
(x — a coth v) 2 + y = a 2 csch 2 v.
The Scale Factors
are
h u -
a
coshi; —
cosu
h v -
a
coshv —
cosu
The Laplacian is
^2 _ (coshi;
\2
— cos u)
a 2
( d 2
\du 2
+
d 2
dv 2
(3)
(4)
(5)
(6)
(7)
Laplace's Equation is separable.
Bipolar Cylindrical Coordinates
Two-center bipolar coordinates are two coordinates giv-
ing the distances from two fixed centers r\ and V2 , some-
times denoted r and r'. For two-center bipolar coordi-
nates with centers at (±c, 0),
ri 2 = ( x + c) 2 +y 2
2 / \2 , 2
r 2 ■ = (x - c) + y .
Combining (8) and (9) gives
2 2 A
ri — ri = 4cx.
(8)
(9)
(10)
Solving for CARTESIAN COORDINATES x and y gives
* 2 „ 2
Ti — 7*2
4c
(11)
y = ±^y/l6c 2 n 2 - (n 2 - r 2 2 + 4c 2 ). (12)
Solving for POLAR COORDINATES gives
ri 2 + r 2 2 -2c 2
8 — tan
8c 2 (n 2 +r 2 2 -2c 2 )
(13)
(14)
References
Lockwood, E. H. "Bipolar Coordinates." Ch. 25 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 186-190, 1967.
Bipolar Cylindrical Coordinates
A set of Curvilinear Coordinates defined by
a sinh v
cosh v — cos u
asinu
cosh v — cos u
z = z,
2/ =
(1)
(2)
(3)
where u 6 [0,27r), v £ (-00,00), and z e (—00,00).
There are several notational conventions, and whereas
(u,v,z) is used in this work, Arfken (1970) prefers
Biprism
Biquadratic Number 143
(77, £, z). The following identities show that curves of
constant u and v are CIRCLES in xy- space.
2 , / x \2 2 2
x -\- (y — a cot u) = a esc it
(x — acothv) + y =a csch v.
The Scale Factors are
a
h u =
h v =
cosh v — cos u
a
cosh v — cos u
1.
The Laplacian is
2 (cosh v — cos u) 2 ( d 2 d 2
(4)
(5)
(6)
(7)
(8)
d 2
Laplace's Equation is not separable in Bipolar
Cylindrical Coordinates, but it is in 2-D Bipolar
Coordinates.
References
Arfken, G. "Bipolar Coordinates (£, 77, z)." §2.9 in Math-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 97-102, 1970.
Biprism
Two slant triangular PRISMS fused together.
see also Prism, Schmitt-Conway Biprism
Bipyramid
see Dipyramid
Biquadratefree
60
40
20 40 60 80 100
A number is said to be biquadratefree if its Prime de-
composition contains no tripled factors. All PRIMES are
therefore trivially biquadratefree. The biquadratefree
numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 17, . . . (Sloane's A046100). The biquadrateful num-
bers (i.e., those that contain at least one biquadrate)
are 16, 32, 48, 64, 80, 81, 96, ... (Sloane's A046101).
The number of biquadratefree numbers less than 10, 100,
1000, ... are 10, 93, 925, 9240, 92395, 923939, . . . , and
their asymptotic density is 1/C(4) = 90/tt 4 « 0.923938,
where C(n) is the Riemann Zeta Function.
see also Cubefree, Prime Number, Riemann Zeta
Function, Squarefree
References
Sloane, N. J. A. Sequences A046100 and A046101 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Biquadratic Equation
see Quartic Equation
Biquadratic Number
A biquadratic number is a fourth POWER, n 4 . The first
few biquadratic numbers are 1, 16, 81, 256, 625, ...
(Sloane's A000583). The minimum number of squares
needed to represent the numbers 1, 2, 3, . . . are 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, . . .
(Sloane's A002377), and the number of distinct ways to
represent the numbers 1, 2, 3, . . . in terms of biquadratic
numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2,
2, 2, 2, A brute-force algorithm for enumerating the
biquadratic permutations of n is repeated application of
the Greedy Algorithm.
Every POSITIVE integer is expressible as a SUM of (at
most) 5(4) = 19 biquadratic numbers (WARING'S PROB-
LEM). Davenport (1939) showed that G(4) = 16, mean-
ing that all sufficiently large integers require only 16
biquadratic numbers. The following table gives the first
few numbers which require 1, 2, 3, . . . , 19 biquadratic
numbers to represent them as a sum, with the sequences
for 17, 18, and 19 being finite.
# Sloane Numbers
1, 16, 81, 256, 625, 1296, 2401, 4096, ...
2, 17, 32, 82, 97, 162, 257, 272, . . .
3, 18, 33, 48, 83, 98, 113, 163, ...
4, 19, 34, 49, 64, 84, 99, 114, 129, . .
5, 20, 35, 50, 65, 80, 85, 100, 115, ..
6, 21, 36, 51, 66, 86, 96, 101, 116, ..
7, 22, 37, 52, 67, 87, 102, 112, 117, .
8, 23, 38, 53, 68, 88, 103, 118, 128, .
9, 24, 39, 54, 69, 89, 104, 119, 134, .
10, 25, 40, 55, 70, 90, 105, 120, 135,
11, 26, 41, 56, 71, 91, 106, 121, 136,
12, 27, 42, 57, 72, 92, 107, 122, 137,
1
000290
2
003336
3
003337
4
003338
5
003339
6
003340
7
003341
8
003342
9
003343
10
003344
11
003345
12
003346
The following table gives the numbers which can be rep-
resented in n different ways as a sum of k biquadrates.
k n Sloane Numbers
1 1 000290 1, 16, 81, 256, 625, 1296, 2401, 4096, . . .
2 2 635318657, 3262811042, 8657437697, ...
The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 18, 19, 20, 21, ... (Sloane's A046039) cannot be
represented using distinct biquadrates.
see also CUBIC NUMBER, SQUARE NUMBER, WARING'S
Problem
References
Davenport, H. "On Waring's Problem for Fourth Powers."
Ann. Math. 40, 731-747, 1939.
144 Biquadratic Reciprocity Theorem
Biquadratic Reciprocity Theorem
x = q (mod p) .
(i)
This was solved by Gauss using the GAUSSIAN INTEGERS
as
(J).®.-'-"""*'-
)/4][(JV(<r)-l)/4]
(2)
'4 \TV / 4
where n and a are distinct GAUSSIAN INTEGER PRIMES,
N(a + hi) = yja? + b 2
and N is the norm.
(3)
■{
1 if x 4 = a (mod 7r) is solvable
— l,i, or — i otherwise,
(4)
where solvable means solvable in terms of Gaussian In-
tegers.
see also RECIPROCITY THEOREM
Biquaternion
A Quaternion with Complex coefficients. The Alge-
bra of biquaternions is isomorphic to a full matrix ring
over the complex number field (van der Waerden 1985).
see also Quaternion
References
Clifford, W. K. "Preliminary Sketch of Biquaternions." Proc.
London Math. Soc. 4, 381-395, 1873.
Hamilton, W. R. Lectures on Quaternions: Containing a
Systematic Statement of a New Mathematical Method.
Dublin: Hodges and Smith, 1853.
Study, E. "Von den Bewegung und Umlegungen." Math.
Ann. 39, 441-566, 1891.
van der Waerden, B. L. A History of Algebra from al-
Khwarizmi to Emmy Noether. New York: Springer- Verlag,
pp. 188-189, 1985.
Birational Transformation
A transformation in which coordinates in two SPACES
are expressed rationally in terms of those in another.
see also Riemann Curve Theorem, Weber's Theo-
rem
Birch Conjecture
see Swinnerton-Dyer Conjecture
Birch-Swinnerton-Dyer Conjecture
see Swinnerton-Dyer Conjecture
Birthday Attack
Birkhoff 's Ergodic Theorem
Let T be an ergodic ENDOMORPHISM of the PROBABIL-
ITY SPACE X and let / : X -t R be a real-valued MEA-
SURABLE Function. Then for Almost Every x € X,
we have
-^TfoF j (x)^ If dm
as n — v oo. To illustrate this, take / to be the charac-
teristic function of some Subset A of X so that
/(*)={;
if xe A
if x £ A.
The left-hand side of (-1) just says how often the or-
bit of x (that is, the points x, Tx, T 2 x, . . . ) lies in
A, and the right-hand side is just the MEASURE of A.
Thus, for an ergodic ENDOMORPHISM, "space-averages
= time- averages almost everywhere." Moreover, if T is
continuous and uniquely ergodic with BOREL PROBA-
BILITY MEASURE m and / is continuous, then we can
replace the Almost Everywhere convergence in (-1)
to everywhere.
Birotunda
Two adjoined ROTUNDAS.
see also BlLUNABIROTUNDA, CUPOLAROTUNDA, ELON-
GATED Gyrocupolarotunda, Elongated Ortho-
CUPOLAROTUNDA, ELONGATED ORTHOBIROTUNDA,
Gyrocupolarotunda, Gyroelongated Rotunda,
ORTHOBIROTUNDA, TRIANGULAR HEBESPHENOROTUN-
DA
Birthday Attack
Birthday attacks are a class of brute-force techniques
used in an attempt to solve a class of cryptographic
hash function problems. These methods take advantage
of functions which, when supplied with a random in-
put, return one of k equally likely values. By repeatedly
evaluating the function for different inputs, the same
output is expected to be obtained after about 1.2\/fc
evaluations.
see also Birthday Problem
References
RSA Laboratories. "Question 95. What is a Birthday At-
tack." http : //www . rsa . com/rsalabs/newf aq/q95 . html.
"Question 96. How Does the Length of a Hash Value
Affect Security?" http : //www . rsa . com/r salabs/newf aq/
q96.html.
van Oorschot, P. and Wiener, M. "A Known Plaintext At-
tack on Two-Key Triple Encryption." In Advances in
Cryptology — Eurocrypt '90. New York: Springer- Verlag,
pp. 366-377, 1991.
Yuval, G. "How to Swindle Rabin." Cryptologia 3, 187-189,
Jul. 1979.
Birthday Problem
Birthday Problem 145
Birthday Problem
Consider the probability Qi(n, d) that no two people out
of a group of n will have matching birthdays out of d
equally possible birthdays. Start with an arbitrary per-
son's birthday, then note that the probability that the
second person's birthday is different is (d — l)/d, that
the third person's birthday is different from the first two
is [(d — l)/d][(d — 2)/d], and so on, up through the nth
person. Explicitly,
Qi(n,d) =
Id- 2 d-(n-l)
d d d
_ (d-l)(d-2)---[d-(n-l)]
d n
But this can be written in terms of FACTORIALS as
dl
Qi(n,d)
(d-n)\d>
71 '
(1)
(2)
so the probability P 2 (n, 365) that two people out of a
group of n do have the same birthday is therefore
P 2 (n,d) = 1-Qi(n,d) = 1
d\
(d-n)\d n '
(3)
If 365-day years have been assumed, i.e., the existence of
leap days is ignored, then the number of people needed
for there to be at least a 50% chance that two share
birthdays is the smallest n such that p2(^, 365) > 1/2.
This is given by n — 23, since
P 2 (23,365) =
38093904702297390785243708291056390518886454060947061
75091883268515350125426207425223147563269805908203125
« 0.507297. (4)
The number of people needed to obtain Pzin, 365) > 1/2
for n = 1, 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, . . .
(Sloane's A033810).
The probability P2(n,d) can be estimated as
P 2 (n,d)^l-e- n(n " 1)/2d (5)
— ('-sr- <■>
where the latter has error
C < — 7~7TK (7)
6(d-n + l) 2
(Sayrafiezadeh 1994).
In general, let Qi(n,d) denote the probability that a
birthday is shared by exactly i (and no more) people
out of a group of n people. Then the probability that a
birthday is shared by k or more people is given by
P k {n,d) = l-^Qi(n,d).
Qi can be computed explicitly as
(8)
L«/2J
~ ~cF 2^i 2H\(n
dl
(-l) n
(n~2i)l(d- n + i)\
(9)
where (™) is a BINOMIAL COEFFICIENT, T(n) is a
Gamma Function, and Pj[ x \x) is an Ultraspheri-
cal Polynomial. This gives the explicit formula for
P^n^d) as
Pz(n 7 d) = 1 - Qi(n,d) - Q 2 (n,d)
(-1)^(71+ l)Pi~ d) (2-^)
" ^ 2 n / 2 d n * K J
Qz{n,d) cannot be computed in entirely closed form,
but a partially reduced form is
Qz{n,d) =
r(d+i)
d n
(-irF(f)-F(-f)
T(d-n + l)
+(-i) r(i + n) ^ r(d-i + i)r(i + i)
(ii)
where
F = F(n,d, a) = 1-3^2
■ i(l_„),l(2-n),-I
i(d-n+l) i(d-n + 2)''
(12)
and 3^2 (a, 6, c; d, e; z) is a GENERALIZED HYPERGEO-
metric Function.
In general, Qk(n,d) can be computed using the RECUR-
RENCE Relation
[n/kj
Qk(n,d) — y^
n!rf!
d ik i\(k\y(n-ik)\(d-i)\
x^2Qj{n-k,d-i)
; Jd-iy
j=i
(Jn — ik
(13)
146 Birthday Problem
Bisection Procedure
(Finch). However, the time to compute this recursive
function grows exponentially with k and so rapidly be-
comes unwieldy. The minimal number of people to give
a 50% probability of having at least n coincident birth-
days is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181,
1385, 1596, 1813, . . . (Sloane's A014088; Diaconis and
Mosteller 1989).
A good approximation to the number of people n such
that p = Pk(n,d) is some given value can given by solv-
ing the equation
ne
-n/(dk)
d* _1 fc!ln
1
1-
d(fc + l)
i/fc
(14)
for n and taking [n], where [n] is the CEILING Func-
tion (Diaconis and Mosteller 1989). For p = 0.5 and
k — 1, 2, 3, ... , this formula gives n = 1, 23, 88, 187,
313, 459, 722, 797, 983, 1179, 1382, 1592, 1809, ...,
which differ from the true values by from to 4. A
much simpler but also poorer approximation for n such
that p — 0.5 for k < 20 is given by
n = 47(fe-1.5)'
3/2
(15)
(Diaconis and Mosteller 1989), which gives 86, 185, 307,
448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for k = 3,
4,....
The "almost" birthday problem, which asks the number
of people needed such that two have a birthday within
a day of each other, was considered by Abramson and
Moser (1970), who showed that 14 people suffice. An ap-
proximation for the minimum number of people needed
to get a 50-50 chance that two have a match within k
days out of d possible is given by
n(k y d) = 1.2
d
2k + 1
(16)
(Sevast'yanov 1972, Diaconis and Mosteller 1989).
see also Birthday Attack, Coincidence, Small
World Problem
References
Abramson, M. and Moser, W. O. J. "More Birthday Sur-
prises." Amer. Math. Monthly 77, 856-858, 1970.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 45-46,
1987.
Bloom, D. M. "A Birthday Problem." Amer. Math. Monthly
80, 1141-1142, 1973.
Bogomolny, A. "Coincidence." http://www.cut-* the-knot .
com/do_you_know/coincidence.html.
Clevenson, M. L. and Watkins, W. "Majorization and the
Birthday Inequality." Math. Mag. 64, 183-188, 1991.
Diaconis, P. and Mosteller, F. "Methods of Studying Coinci-
dences." J. Amer. Statist. Assoc. 84, 853-861, 1989.
Feller, W. An Introduction to Probability Theory and Its Ap-
plications, Vol. 1, 3rd ed. New York: Wiley, pp. 31-32,
1968.
Finch, S. "Puzzle #28 [June 1997]: Coincident Birthdays."
http: //www. maths oft . com/mathcad/library /puzzle/
soln28/soln28.html.
Gehan, E. A. "Note on the 'Birthday Problem.'" Amer. Stat.
22, 28, Apr. 1968.
Heuer, G. A. "Estimation in a Certain Probability Problem."
Amer. Math. Monthly 66, 704-706, 1959.
Hocking, R. L. and Schwertman, N. C. "An Extension of the
Birthday Problem to Exactly k Matches." College Math.
J. 17, 315-321, 1986.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 102-103, 1975.
Klamkin, M. S. and Newman, D. J. "Extensions of the Birth-
day Surprise." J. Combin. Th. 3, 279-282, 1967.
Levin, B. "A Representation for Multinomial Cumulative
Distribution Functions." Ann. Statistics 9, 1123-1126,
1981.
McKinney, E. H. "Generalized Birthday Problem." Amer.
Math. Monthly 73, 385-387, 1966.
Mises, R. von. "Uber Aufteilungs — und Besetzungs-
Wahrscheinlichkeiten." Revue de la Faculte des Sci-
ences de VUniversite d'Istanbul, N. S. 4, 145—163, 1939.
Reprinted in Selected Papers of Richard von Mises, Vol. 2
(Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. Szego,
and G. BirkhofF), Providence, RI: Amer. Math. Soc,
pp. 313-334, 1964.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 179-180,
1994.
Sayrafiezadeh, M. "The Birthday Problem Revisited." Math.
Mag. 67, 220-223, 1994.
Sevast'yanov, B. A. "Poisson Limit Law for a Scheme of Sums
of Dependent Random Variables." Th. Prob. Appl. 17,
695-699, 1972.
Sloane, N. J. A. Sequences A014088 and A033810 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96,
June 1998.
Tesler, L. "Not a Coincidence!" http://www.nomodes.com/
coincidence .html.
Bisected Perimeter Point
see Nagel Point
Bisection Procedure
Given an interval [a, &], let a n and b n be the endpoints
at the nth iteration and r n be the nth approximate solu-
tion. Then, the number of iterations required to obtain
an error smaller than e is found as follows.
1
(1)
(2)
\r n -r\<±{b n - a n ) = 2~ n (b - a) < e (3)
— n In 2 < In e — ln(6 — a),
ln(6 — a) — In e
n >
In 2
(4)
(5)
so
see also ROOT
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 964-965, 1985.
Bisector
Bislit Cube
147
Press, W. H.; Fiannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Bracketing and Bisection." §9.1 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 343-347, 1992.
Bisector
Bisection is the division of a given curve or figure into
two equal parts (halves).
see also Angle Bisector, Bisection Procedure,
Exterior Angle Bisector, Half, Hemisphere,
Line Bisector, Perpendicular Bisector, Trisec-
TION
Bishop's Inequality
Let V{r) be the volume of a BALL of radius r in a com-
plete 7l-D RlEMANNIAN MANIFOLD with RlCCI CURVA-
TURE > (n - 1)k. Then V(r) > V K (r), where V K is
the volume of a Ball in a space having constant Sec-
tional Curvature. In addition, if equality holds for
some Ball, then this Ball is Isometric to the Ball
of radius r in the space of constant SECTIONAL CURVA-
TURE K.
References
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Bishops Problem
B
B
B
B
B
B
B
B
B
B
B
B
B
B
Find the maximum number of bishops B(n) which can
be placed on an n x n Chessboard such that no two
attack each other. The answer is 2n — 2 (Dudeney 1970,
Madachy 1979), giving the sequence 2, 4, 6, 8, . . . (the
Even Numbers) for n = 2, 3, One maximal so-
lution for n = 8 is illustrated above. The number of
distinct maximal arrangements of bishops for n — 1, 2,
... are 1, 4, 26, 260, 3368, . . . (Sloane's A002465). The
number of rotationally and reflectively distinct solutions
on an n x n board for n > 2 is
B(n)
/ 2 (n-4)/2 [2 (n-2)/2 + y ^ n ey( . Q
| 2 (n-3)/2 [2 („-3)/2 + 1 ] fornodd
where |nj is the FLOOR FUNCTION, giving the sequence
for n = 1, 2, . . . as 1, 1, 2, 3, 6, 10, 20, 36, . . . (Sloane's
A005418).
B
B
B
B
B
B
B
B
The minimum number of bishops needed to occupy or
attack all squares on an n x n Chessboard is n, ar-
ranged as illustrated above.
see also Chess, Kings Problem, Knights Problem,
Queens Problem, Rooks Problem
References
Ahrens, W. Mathematische Unterhaltungen und Spiele,
Vol 1, 3rd ed. Leipzig, Germany: Teubner, p. 271, 1921.
Dudeney, H. E. "Bishops — Unguarded" and "Bishops —
Guarded." §297 and 298 in Amusements in Mathematics.
New York: Dover, pp. 88-89, 1970.
Guy, R. K. "The n Queens Problem." §C18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 133-135, 1994.
Madachy, J. Madachy's Mathematical Recreations. New
York: Dover, pp. 36-46, 1979.
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 74-
75, 1995.
Sloane, N. J. A. Sequences A002465/M3616 and A005418/
M0771 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Bislit Cube
The 8- Vertex graph consisting of a Cube in which two
opposite faces have DIAGONALS oriented PERPENDICU-
LAR to each other.
see also Bidiakis Cube, Cube, Cubical Graph
(Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover
1995). An equivalent formula is
J B(n) = 2 n - 3 + 2 L( ' l - 1)/2J - 1 ,
148 Bispherical Coordinates
Bispherical Coordinates
A system of CURVILINEAR COORDINATES defined by
a sin £ cos <fi
y-
cosh 77 — cos £
a sin £ sin <\>
cosh 77 — cos £
a sinh 77
cosh T] — cos £
The Scale Factors are
h a
h v
cos 77 — cos £
a
The Laplacian is
2 _ / — cos u co ^ 2 u '
\ cosh t>
cosh 77 — cos £
asin£
cosh 77 — cos £
+ 3 cosh v cot u
a)
(2)
(3)
(4)
(5)
(6)
-3 cosh 2 v cot u esc u + cosh vcsc u
cosh v — cos ti
d(j> 2
+ (cosu — cosh v) sinh v~ — h (cosh v - cosu) ■^-^
<%
0v 2
a
+ (cosh v — cos ti) (cosh v cot w — sin u — cos u cot u) —
+(cosh 2 i; — cos u) -^—7 .
ou 2
(7)
In bispherical coordinates, LAPLACE'S EQUATION is sep-
arable, but the Helmholtz Differential Equation
is not.
see also Laplace's Equation— Bispherical Coor-
dinates, Toroidal Coordinates
References
Arfken, G. "Bispherical Coordinates (£, 77, <£)." §2.14 in
Mathematical Methods for Physicists, 2nd ed. Orlando,
FL: Academic Press, pp. 115-117, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.
Black-Scholes Theory
Bit Complexity
The number of single operations (of ADDITION, SUB-
TRACTION, and Multiplication) required to complete
an algorithm.
see also STRASSEN FORMULAS
References
Borodin, A. and Munro, I. The Computational Complexity
of Algebraic and Numeric Problems. New York: American
Elsevier, 1975.
Bitangent
bitangent
A Line which is Tangent to a curve at two distinct
points.
see also Klein's Equation, Plucker Characteris-
tics, Secant Line, Solomon's Seal Lines, Tangent
Line
Bivariate Distribution
see Gaussian Bivariate Distribution
Bivector
An antisymmetric TENSOR of second Rank (a.k.a. 2-
form) .
X = X ah u) a A u) ,
where A is the Wedge Product (or Outer Prod-
uct).
Biweight
see TUKEY'S BIWEIGHT
Black-Scholes Theory
The theory underlying financial derivatives which in-
volves "stochastic calculus" and assumes an uncor-
rected Log Normal Distribution of continuously
varying prices. A simplified "binomial" version of the
theory was subsequently developed by Sharpe et al.
(1995) and Cox et al (1979). It reproduces many re-
sults of the full-blown theory, and allows approximation
of options for which analytic solutions are not known
(Price 1996).
see also Garman-Kohlhagen Formula
References
Black, F. and Scholes, M. S. "The Pricing of Options and
Corporate Liabilities." J. Political Econ. 81, 637-659,
1973.
Cox, J. C; Ross, A.; and Rubenstein, M. "Option Pricing: A
Simplified Approach." J. Financial Economics 7, 229-263,
1979.
Price, J. F. "Optional Mathematics is Not Optional." Not.
Amer. Math. Soc. 43, 964-971, 1996.
Sharpe, W. F.; Alexander, G. J,; and Bailey, J. V. Invest-
ments, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Black Spleenwort Fern
Black Spleenwort Fern
see BARNSLEY'S FERN
Blackman Function
Blecksmith-Brillhart- Gerst Theorem
149
-1 -0.5 0.5 1 -0.5
An Apodization Function given by
A(x) = 0.42 + 0.5 cos
(?)
+ 0.08 cos
(^)
a)
Its Full Width at Half Maximum is 0.810957a. The
Apparatus Function is
I(k) =
a(0.84 - 0.36a 2 fc 2 - 2.17 x 10~ x Vfe 4 ) sin(27raA:)
(l-a 2 fc 2 )(l-4a 2 A; 2 )
The Coefficients are approximations to
ao
ai
a 2 =
3969
9304
1155
4652
715
18608'
(2)
(3)
(4)
(5)
which would have produced zeros of I(k) at k = (7/4)a
and k = (9/4)a.
see also APODIZATION FUNCTION
References
Blackman, R. B. and Tukey, J, W. "Particular Pairs of Win-
dows." In The Measurement of Power Spectra, From
the Point of View of Communications Engineering. New
York: Dover, pp. 98-99, 1959.
Blancmange Function
A Continuous Function which is nowhere Differ-
ENTIABLE. The iterations towards the continuous func-
tion are Batrachions resembling the Hofstadter-
Conway $10,000 Sequence. The first six iterations
are illustrated below. The dth iteration contains TV + 1
points, where TV = 2 d , and can be obtained by setting
6(0) = b(N) = 0, letting
b{m + 2 71 " 1 ) = 2 n + \[b{m) + b{m + 2 n )],
and looping over n = d to 1 by steps of —1 and m =
to TV- 1 by steps of 2 n .
Peitgen and Saupe (1988) refer to this curve as the Tak-
agi Fractal Curve.
see also HOFSTADTER-CONWAY $10,000 SEQUENCE,
Weierstrak Function
References
Dixon, R. Mathographics. New York: Dover, pp. 175-176
and 210, 1991.
Peitgen, H.-O. and Saupe, D. (Eds.). "Midpoint Displace-
ment and Systematic Fractals: The Takagi Fractal Curve,
Its Kin, and the Related Systems." §A.1.2 in The Science
of Fractal Images. New York: Springer- Verlag, pp. 246-
248, 1988.
Takagi, T. "A Simple Example of the Continuous Function
without Derivative." Proc. Phys. Math. Japanl y 176-177,
1903.
Tall, D. O. "The Blancmange Function, Continuous Every-
where but DifTerentiable Nowhere." Math. Gaz. 66,11-22,
1982.
Tall, D. "The Gradient of a Graph." Math. Teaching 111,
48-52, 1985.
Blaschke Conjecture
The only WlEDERSEHEN MANIFOLDS are the standard
round spheres. The conjecture has been proven by com-
bining the Berger-Kazdan Comparison Theorem
with A. Weinstein's results for n Even and C. T. Yang's
for n Odd.
References
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Blaschke's Theorem
A convex planar domain in which the minimal length is
> 1 always contains a Circle of Radius 1/3.
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 25, 1983.
Blecksmith-Brillhart-Gerst Theorem
A generalization of Schroter'S FORMULA.
References
Berndt, B. C Ramanujan's Notebooks, Part III. New York:
Springer- Verlag, p. 73, 1985.
BlichfeldVs Lemma
Blichfeldt's Lemma
see Blichfeldt's Theorem
Blichfeldt's Theorem
Published in 1914 by Hans Blichfeldt. It states that any
bounded planar region with POSITIVE AREA > A placed
in any position of the UNIT SQUARE LATTICE can be
Translated so that the number of Lattice Points
inside the region will be at least A + 1. The theorem
can be generalized to n-D.
BLM/Ho Polynomial
A 1-variable unoriented Knot Polynomial Q(x). It
satisfies
Qunknot = 1 (l)
and the SKEIN RELATIONSHIP
Ql^+Ql^ =x(Q Lq + Q Lqo ). (2)
It also satisfies
Qlx#l 2 = Ql y Ql 2 , (3)
where # is the KNOT Sum and
Ql*=Ql> (4)
where L* is the Mirror Image of L. The BLM/Ho
polynomials of Mutant KNOTS are also identical.
Brandt et al. (1986) give a number of interesting prop-
erties. For any Link L with > 2 components, Ql — 1 is
divisible by 2 (x — 1). If L has c components, then the
lowest POWER of x in Ql(x) is 1 — c, and
lim x c
lim (-m) c - 1 P L (£,m) J (5)
(^m)-4(l,0) n V '
where P L is the HOMFLY Polynomial. Also, the de-
gree of Ql is less than the Crossing Number of L. If
L is a 2-Bridge Knot, then
Q L (z) = 2z~' 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ), (6)
where z = -t - r -1 (Kanenobu and Sumi 1993).
The Polynomial was subsequently extended to the 2-
variable Kauffman Polynomial F(a i z) y which satis-
fies
Q(x) = F{l,x). (7)
Brandt et al. (1986) give a listing of Q POLYNOMIALS
for KNOTS up to 8 crossings and links up to 6 crossings.
References
Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. "A
Polynomial Invariant for Unoriented Knots and Links." In-
vent Math. 84, 563-573, 1986.
Ho, C. F. "A New Polynomial for Knots and Links —
Preliminary Report." Abstracts Amer. Math. Soc. 6, 300,
1985.
Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-
Bridge Knots through 22-Crossings." Math. Comput. 60,
771-778 and S17-S28, 1993.
Stoimenow, A. "Brandt-Lickorish-Millett-Ho Polynomi-
als." http: //www, informatik.hu-berlin.de/-stoimeno/
ptab/blmhlO . html.
^ Weisstein, E. W. "Knots." http: //www. astro. Virginia,
edu/ - eww6n/math/not ebooks/Knot s . m.
Block Design
Bloch Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let F be the set of Complex analytic functions / de-
fined on an open region containing the closure of the
unit disk D = {z : \z\ < 1} satisfying /(0) = and
df/dz(Q) = 1. For each / in F, let b(f) be the SUPRE-
MUM of all numbers r such that there is a disk S in D on
which / is ONE-TO-ONE and such that f(S) contains a
disk of radius r. In 1925, Bloch (Conway 1978) showed
that b(f) > 1/72. Define Bloch's constant by
B = mi{btf):f£F}.
Ahlfors and Grunsky (1937) derived
0.433012701...= \VZ<B
i r(i)r(i§)
4
<
v / nm r (?)
f^- < 0.4718617.
They also conjectured that the upper limit is actually
the value of B,
1
r(j)r(M)
v/TTv! r (i)
iV
= 0.4718617X
/
V
^
°4?
(Le Lionnais 1983).
see also Landau Constant
References
Conway, J. B. Functions of One Complex Variable, 2nd ed.
New York: Springer- Verlag, 1989.
Finch, S, "Favorite Mathematical Constants." http: //www.
mathsof t . com/asolve/constant/bloch/bloch.html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 25, 1983.
Minda, CD. "Bloch Constants." J. d Analyse Math. 41,
54-84, 1982.
BIoch-Landau Constant
see Landau Constant
Block
see also Block Design, Square Polyomino
Block Design
An incidence system (v, fc, A, r, 6) in which a set X
of v points is partitioned into a family A of b subsets
(blocks) in such a way that any two points determine A
blocks, there are k points in each block, and each point
is contained in r different blocks. It is also generally
required that k < v , which is where the "incomplete"
comes from in the formal term most often encountered
Block Design
Blow-Up 151
for block designs, Balanced Incomplete Block De-
signs (BIBD). The five parameters are not independent,
but satisfy the two relations
bk
X(v~ 1) = r(fc-l).
(1)
(2)
A BIBD is therefore commonly written as simply (v, &,
A), since b and r are given in terms of u, k, and A by
v(v - 1)A
k(k - 1)
(3)
(4)
A BIBD is called SYMMETRIC if b = v (or, equivalently,
r = k).
Writing X = {^}Li and A — {Aj} b j=1 , then the IN-
CIDENCE Matrix of the BIBD is given by the v x b
Matrix M defined by
1J I othe
GA
otherwise.
This matrix satisfies the equation
MM T = (r-A)l + AJ,
(5)
(6)
where I is a v x v IDENTITY MATRIX and J is a v x v
matrix of Is (Dinitz and Stinson 1992).
Examples of BIBDs are given in the following table.
Block Design
(v, K A)
affine plane (n , n, 1)
Fano plane (7, 3, 1))
Hadamard design symmetric (An + 3, 2n -f- 1, n)
projective plane symmetric (n 2 + n -j- 1, n + 1, 1)
Steiner triple system (v, 3, 1)
unital (g 3 + 1, q+ 1, 1)
see also Affine Plane, Design, Fano Plane, Hada-
mard Design, Parallel Class, Projective Plane,
Resolution, Resolvable, Steiner Triple System,
Symmetric Block Design, Unital
References
Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to
Design Theory." Ch. 1 in Contemporary Design Theory: A
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson).
New York: Wiley, pp. 1-12, 1992.
Ryser, H. J. "The {b,v,r, k, A)-Configuration." §8.1 in Com-
binatorial Mathematics. Buffalo, NY: Math. Assoc. Amer.,
pp. 96-102, 1963.
Block Growth
Let (rco£i£2 • • •) be a sequence over a finite ALPHABET
A (all the entries are elements of A). Define the block
growth function B(n) of a sequence to be the number
of Admissible words of length n. For example, in the
sequence aabaabaabaabaab. . . , the following words are
Admissible
Length Admissible Words
a, b
aa, a&, ba
aab, aba, baa
aaba, abaa, baab
so B(l) = 2, B(2) = 3, 5(3) = 3, B(4) = 3, and so
on. Notice that B(n) < B(n + 1), so the block growth
function is always nondecreasing. This is because any
Admissible word of length n can be extended right-
wards to produce an Admissible word of length n + 1.
Moreover, suppose B(n) = B(n + 1) for some n. Then
each admissible word of length n extends to a unique
Admissible word of length n + 1.
For a SEQUENCE in which each substring of length n
uniquely determines the next symbol in the SEQUENCE,
there are only finitely many strings of length n, so the
process must eventually cycle and the SEQUENCE must
be eventually periodic. This gives us the following the-
orems:
1. If the Sequence is eventually periodic, with least
period p, then B(n) is strictly increasing until it
reaches p, and B(n) is constant thereafter.
2. If the Sequence is not eventually periodic, then
B(n) is strictly increasing and so B(n) > n + 1 for all
n. If a Sequence has the property that B(n) = n+1
for all n, then it is said to have minimal block growth,
and the Sequence is called a Sturmian Sequence.
The block growth is also called the GROWTH FUNCTION
or the Complexity of a Sequence.
Block Matrix
A square Diagonal Matrix in which the diagonal ele-
ments are Square Matrices of any size (possibly even
lxl), and the off-diagonal elements are 0.
Block (Set)
One of the disjoint Subsets making up a Set Parti-
tion. A block containing n elements is called an n-
block. The partitioning of sets into blocks can be de-
noted using a RESTRICTED GROWTH STRING.
see also Block Design, Restricted Growth
String, Set Partition
Blow-Up
A common mechanism which generates SINGULARITIES
from smooth initial conditions.
152 Blue-Empty Coloring
Bohemian Dome
Blue-Empty Coloring
see Blue-Empty Graph
Blue-Empty Graph
An Extremal Graph in which the forced Trian-
gles are all the same color. Call R the number of
red Monochromatic Forced Triangles and B the
number of blue Monochromatic Forced Triangles,
then a blue-empty graph is an Extremal Graph with
B = 0. For Even n, a blue-empty graph can be
achieved by coloring red two Complete SUBGRAPHS
of n/2 points (the RED Net method). There is no blue-
empty coloring for Odd n except for n = 7 (Lorden
1962).
see also Complete Graph, Extremal Graph,
Monochromatic Forced Triangle, Red Net
References
Lorden, G. "Blue-Empty Chromatic Graphs." Amer. Math.
Monthly 69, 114-120, 1962.
Sauve, L. "On Chromatic Graphs." Amer. Math. Monthly
68, 107-111, 1961.
Board
A subset of d x d, where d = {1, 2, . . . , d}.
see also Rook Number
Boatman's Knot
see Clove Hitch
Bochner Identity
For a smooth Harmonic Map u : M -► TV,
A(|Vu| 2 ) = \V(du)\ 2 + {RicM Vu,Vu)
- (Riem N (u)(Vu, Vu)Vu, Vu> ,
where V is the GRADIENT, Ric is the RlCCl TENSOR,
and Riem is the Riemann Tensor.
References
Eels, J. and Lemaire, L. "A Report on Harmonic Maps."
Bull. London Math. Soc. 10, 1-68, 1978.
Bochner's Theorem
Among the continuous functions on R n , the POSITIVE
Definite Functions are those functions which are the
Fourier Transforms of finite measures.
Bode's Rule
J XI
f{x) dx = ^/i(7/i + 32/ 2 + 12/ 3 + 32/ 4 + 7/5)
-sfeW'K).
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 886, 1972.
Bogdanov Map
A 2-D MAP which is conjugate to the Henon Map in
its nondissipative limit. It is given by
x = x + y
y' =y + ey + kx(x - l) + [ixy.
see also Henon Map
References
Arrowsmith, D. K.; Cartwright, J. H. E.; Lansbury, A. N.;
and Place, C. M. "The Bogdanov Map: Bifurcations, Mode
Locking, and Chaos in a Dissipative System." Int. J. Bi-
furcation Chaos 3, 803-842, 1993.
Bogdanov, R. "Bifurcations of a Limit Cycle for a Family
of Vector Fields on the Plane." Selecta Math. Soviet 1,
373-388, 1981.
Bogomolov-Miyaoka-Yau Inequality
Relates invariants of a curve defined over the INTEGERS.
If this inequality were proven true, then FERMAT'S Last
THEOREM would follow for sufficiently large exponents.
Miyaoka claimed to have proven this inequality in 1988,
but the proof contained an error.
see also FERMAT'S LAST THEOREM
References
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
Bohemian Dome
see also Hardy's Rule, Newton-Cotes Formulas,
Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal
Rule, Weddle's Rule
A Quartic Surface which can be constructed as fol-
lows. Given a CIRCLE C and PLANE E PERPENDICULAR
to the Plane of C, move a second Circle K of the
same Radius as C through space so that its Center
always lies on C and it remains PARALLEL to E. Then
K sweeps out the Bohemian dome. It can be given by
the parametric equations
x = a cos u
y = b cos v + a sin u
z — csinv
where u, v 6 [0, 27r). In the above plot, a = 0.5, b = 1.5,
and c = 1.
see also Quartic Surface
Bohr-Favard Inequalities
Bombieri Norm
153
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 19-20, 1986.
Fischer, G. (Ed.). Plate 50 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 50, 1986.
Nordstrand, T. "Bohemian Dome." http://www.uib.no/
people/nf ytn/bodtxt .htm.
Bohr-Favard Inequalities
If / has no spectrum in [—A, A], then
saii'i
(Bohr 1935). A related inequality states that if Ak is
the class of functions such that
/(*) = /(* + 2*), /(*),/'(*),... ./^(a:)
are absolutely continuous and f w f(x) dx = 0, then
4
1 _1)^
S ^2^ ( 2 ^+ l)M-i N/ wi
(Northcott 1939). Further, for each value of k, there is
always a function f(x) belonging to Ak and not identi-
cally zero, for which the above inequality becomes an in-
equality (Favard 1936). These inequalities are discussed
in Mitrinovic et al. (1991).
References
Bohr, H. "Ein allgemeiner Satz iiber die Integration eines
trigonometrischen Polynoms." Prace Matem.-Fiz. 43,
1935.
Favard, J. "Application de la formule soiiimaloire d'Euler
a la demonstration de quelques proprietes extremales des
integrale des fonctions periodiques ou presqueperiodiqu.es."
Mat Tidsskr. B, 81-94, 1936. [Reviewed in Zentralblatt f.
Math. 16, 58-59, 1939.]
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives.
Dordrecht, Netherlands: Kluwer, pp. 71-72, 1991.
Northcott, D. G. "Some Inequalities Between Periodic Func-
tions and Their Derivatives." J. London Math. Soc. 14,
198-202, 1939.
Tikhomirov, V. M. "Approximation Theory." In Analysis
II (Ed. R. V. Gamrelidze). New York: Springer- Verlag,
pp." 93-255, 1990.
Bolyai-Gerwein Theorem
see Wallace-Bolyai-Gerwein Theorem
Bolza Problem
Given the functional
U= /(2/l,---,S/n;3/l\...,2/n') d *
Jt
+G(yi , . . . , 2Mr; 2/11, • ■ ■ , 2/m),
find in a class of arcs satisfying p differential and q finite
equations
<M3/i,-..,2/n;3/i',...,3/n') = ° for a = l,...,p
VV3(yi»--->yn) = for = l,...,g
as well as the r equations on the endpoints
X7(yio,---)2/nr;3/ii,...,2/ni) = for 7 = 1, . . . , r,
one which renders U a minimum.
References
Goldstine, H. H, A History of the Calculus of Variations from
the 17th through the 19th Century. New York: Springer-
Verlag, p. 374, 1980.
Bolzano Theorem
see Bolzano- WeierstraB Theorem
Bolzano- Weierstrafl Theorem
Every Bounded infinite set in W 1, has an ACCUMULA-
TION Point. For n = 1, the theorem can be stated as
follows: If a Set in a METRIC SPACE, finite-dimensional
Euclidean Space, or First-Countable Space has
infinitely many members within a finite interval x 6
[a, 6], then it has at least one Limit Point x such that
x e [a, &]. The theorem can be used to prove the Inter-
mediate Value Theorem.
Bombieri's Inequality
For Homogeneous Polynomials P and Q of degree
m and n, then
[P ■ Qh >
tM.
(m + n)\
-jiPhlQb,
where [P • Q] 2 is the BOMBIERI Norm. If m = n, this
becomes
[P'Qh>[P]2[Q]2.
see also Beauzamy and Degot's Identity, Reznik's
Identity
Bombieri Inner Product
For Homogeneous Polynomials P and Q of degree
n,
[P,Q]= J2 C*! 1 "-^ 1 )^,..^^!,..^)-
ii,...,i„>0
Bombieri Norm
For Homogeneous Polynomials P of degree m,
mV
[P], = y/frF]=\ J2 S |a '
y |a|=m
see also POLYNOMIAL BAR NORM
154 BombievVs Theorem
Bonne Projection
Bombieri's Theorem
Define
E(x;q,a) = ip(x\q,a) -
<KqV
where
■tP(x;q,a)= ^ A(n)
(1)
(2)
n<x
n = a (mod g)
(Davenport 1980, p. 121), A(n) is the MANGOLDT
Function, and <j>(q) is the Totient Function. Now
define
E(x;q)= max \E(x\q y a)\ (3)
(a,q°) = l
where the sum is over a RELATIVELY PRIME to q,
(a,g) = 1, and
E*(x,q) = ma,xE{y,q). (4)
y<x
Bombieri's theorem then says that for A > fixed,
^E*(x,q) « ^Q{\nx)\ (5)
q<Q
provided that ^(lnx)" 4 < Q < \fx.
References
Bombieri, E. "On the Large Sieve." Mathematika 12, 201-
225, 1965.
Davenport, H. "Bombieri's Theorem." Ch. 28 in Multiplica-
tive Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 161-168, 1980.
Bond Percolation
bond percolation site percolation
A Percolation which considers the lattice edges as the
relevant entities (left figure).
see also Percolation Theory, Site Percolation
Bonferroni Correction
The Bonferroni correction is a multiple-comparison cor-
rection used when several independent STATISTICAL
TESTS are being performed simultaneously (since while
a given Alpha Value a may be appropriate for each
individual comparison, it is not for the set of all com-
parisons). In order to avoid a lot of spurious positives,
the Alpha Value needs to be lowered to account for
the number of comparisons being performed.
The simplest and most conservative approach is the
Bonferroni correction, which sets the ALPHA VALUE for
the entire set of n comparisons equal to a by taking the
Alpha Value for each comparison equal to cx/n. Ex-
plicitly, given n tests Ti for hypotheses Hi (1 < i < n)
under the assumption Ho that all hypotheses Hi are
false, and if the individual test critical values are < a/n,
then the experiment-wide critical value is < a. In equa-
tion form, if
P(Ti passes \H Q ) < -
n
for 1 < i < ra, then
P(some Ti passes \H ) < a,
which follows from BONFERRONl'S INEQUALITY.
Another correction instead uses 1 — (1— a) 1 / 71 . While this
choice is applicable for two-sided hypotheses, multivari-
ate normal statistics, and positive orthant dependent
statistics, it is not, in general, correct (Shaffer 1995).
see also ALPHA VALUE, HYPOTHESIS TESTING, STATIS-
TICAL Test
References
Bonferroni, C. E. "II calcolo delle assicurazioni su gruppi di
teste." In Studi in Onore del Professore Salvatore Ortu
Carboni. Rome: Italy, pp. 13-60, 1935.
Bonferroni, C. E. "Teoria statistica delle classi e calcolo delle
probabilita." Pubblicazioni del R Istituto Superiore di
Scienze Economiche e Commerciali di Firenze 8, 3-62,
1936.
Dewey, M. "Carlo Emilio Bonferroni: Life and Works."
http://www.nottingham.ac.uk/-mh2md/life.html.
Miller, R. G. Jr. Simultaneous Statistical Inference. New
York: Springer- Verlag, 1991.
Perneger, T. V. "What's Wrong with Bonferroni Adjust-
ments." Brit Med. J. 316, 1236-1238, 1998.
Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev.
Psych. 46, 561-584, 1995.
Bonferroni's Inequality
Let P(Ei) be the probability that £?» is true, and
P(U" =1 ^i) be the probability that E u E 2j ..., E n
are all true. Then
fU 1 *") *!><*>•
Bonferroni Test
see Bonferroni Correction
Bonne Projection
Book Stacking Problem
Boolean Algebra 155
A Map Projection which resembles the shape of a
heart. Let (pi be the standard parallel and Ao the central
meridian. Then
where
x = p sin E
y — R cot 0i — p cos R )
p = cot (pi + 0i — <(>
(A- Aq)cos0
The inverse FORMULAS are
<p = cot 01 + (f>i — p
A = A +
COS0
■ tan
-l ( x
\ cot (pi -y
where
p = ±\/x 2 + (cot 0i -y) 2 .
(1)
(2)
(3)
(4)
(5)
(6)
(7)
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 138-140, 1987.
(Sloane's A001008 and A002805).
In order to find the number of stacked books required to
obtain d book-lengths of overhang, solve the d n equation
for d, and take the Ceiling Function. For n = 1, 2, . . .
book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380,
675214, 4989191, 36865412, 272400600, ... (Sloane's
A014537) books are needed.
References
Dickau, R. M. "The Book-Stacking Problem." http://wwv.
prairienet.org/-pops/BookStacking.html.
Eisner, L. "Leaning Tower of the Physical Review." Amer.
J. Phys. 27, 121, 1959.
Gardner, M. Martin Gardner's Sixth Book of Mathematical
Games from Scientific American. New York: Scribner's,
p. 167, 1971.
Graham, R. L.; Knuth, D, E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science. Read-
ing, MA: Addison- Wesley, pp. 272-274, 1990.
Johnson, P. B. "Leaning Tower of Lire." Amer. J. Phys. 23,
240, 1955.
Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 1, 322, 1953.
Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 2, 411, 1954.
Sloane, N. J. A. Sequences A014537, A001008/M2885, and
A002805/M1589 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Boole's Inequality
Book Stacking Problem
How far can a stack of n books protrude over the edge
of a table without the stack falling over? It turns out
that the maximum overhang possible d n for n books (in
terms of book lengths) is half the nth partial sum of the
Harmonic Series, given explicitly by
d n
n
where <&(z) is the DiGAMMA FUNCTION and 7 is the
Euler-Mascheroni Constant. The first few values
are
di = -= 0.5
3
4
— 11
0.75
d 3 = i| « 0.91667
A — 25
rf 4 - 24
p U £ 0^E p ^)-
1.04167,
If Ei and Ej are Mutually Exclusive for all i and j,
then the INEQUALITY becomes an equality.
Boolean Algebra
A mathematical object which is similar to a BOOLEAN
RING, but which uses the meet and join operators in-
stead of the usual addition and multiplication operators.
A Boolean algebra is a set B of elements a, 6, ... with
Binary Operators + and * such that
la. If a and b are in the set S, then a + b is in the set
B.
lb. If a and b are in the set B, then a • b is in the set
B.
2a. There is an element Z (zero) such that a + Z = a
for every element a.
2b. There is an element U (unity) such that a • U = a
for every element a.
3a. a + 6 = b + a
3b. a - b = b ■ a
4a. a + 6 ■ c = (a + b) (a + c)
4b. a ■ (b-\- c) — a - b-\- a ■ c
5. For every element a there is an element a such that
a + a' — U and a ■ a' = Z.
6. There are are least two distinct elements in the set
B.
(Bell 1937, p. 444).
156 Boolean Algebra
Boolean Ring
In more modern terms, a Boolean algebra is a Set B of
elements a, 6, ... with the following properties:
1. B has two binary operations, A (Wedge) and V
(Vee), which satisfy the IDEMPOTENT laws
aAa = a\/a = a,
the Commutative laws
a A b — b A a
aVb^bV a,
and the Associative laws
a A (b A c) = (a A b) A c
aV(6Vc) = (aVb) V c.
2. The operations satisfy the ABSORPTION LAW
a A (a V b) = a V (a A 6) = a.
3. The operations are mutually distributive
a A (6Vc) = (a A 6) V (a Ac)
a V (6 A c) = (a V 6) A (a V c).
4. I? contains universal bounds 0,/ which satisfy
OAa =
O Va = a
/ A a = a
/Vfl = J.
5. B has a unary operation a —± a' of complementation
which obeys the laws
a A a = O
aV a = I
(Birkhoff and Mac Lane 1965). Under intersection,
union, and complement, the subsets of any set I form a
Boolean algebra.
Huntington (1933a, b) presented the following basis for
Boolean algebra,
1. Commutivity. x + y = y + x.
2. Associativity, (x + y) + z = x + (y + z).
3. Huntington Equation. n(n(x) + y) + n(n(a;) +
n(y)) = x.
H. Robbins then conjectured that the Huntington
Equation could be replaced with the simpler Robbins
Equation,
n(n(x + y) + n(x + n(j/))) = x.
The Algebra defined by commutivity, associativity,
and the Robbins EQUATION is called ROBBINS ALGE-
BRA. Computer theorem proving demonstrated that ev-
ery Robbins Algebra satisfies the second Winkler
Condition, from which it follows immediately that all
Robbins Algebras are Boolean.
References
Bell, E. T. Men of Mathematics. New York: Simon and
Schuster, 1986.
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillian, p. 317, 1965.
Halmos, P. Lectures on Boolean Algebras. Princeton, NJ:
Van Nostrand, 1963.
Huntington, E. V. "New Sets of Independent Postulates for
the Algebra of Logic." Trans. Amer. Math. Soc. 35, 274-
304, 1933a.
Huntington, E. V. "Boolean Algebras: A Correction." Trans.
Amer. Math. Soc. 35, 557-558, 1933.
McCune, W. "Robbins Algebras are Boolean." http://www.
mcs.anl.gov/-mccune/papers/robbins/.
Boolean Connective
One of the Logic operators And A, Or V, and Not ->.
see also QUANTIFIER
Boolean Function
A Boolean function in n variables is a function
J\Xi , . . . , x n J,
where each Xi can be or 1 and / is or 1. Determining
the number of monotone Boolean functions of n vari-
ables is known as Dedekind'S Problem. The number
of monotonic increasing Boolean functions of n variables
is given by 2, 3, 6, 20, 168, 7581, 7828354, . . . (Sloane's
A000372, Beeler et al. 1972, Item 17). The number of
inequivalent monotone Boolean functions of n variables
is given by 2, 3, 5, 10, 30, . . . (Sloane's A003182).
Let M(n, k) denote the number of distinct monotone
Boolean functions of n variables with k mincuts. Then
M(n,0) = 1
M(n,l)-2 n
M(n, 2) = 2 n " 1 (2 n - 1) - 3 n + 2 n
M(n,3) = |(2 n )(2 n - l)(2 n - 2) - 6 n + 5" + 4 n - 3 n .
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Sloane, N, J. A. Sequences A003182/M0729 and A000372/
M0817 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Boolean Ring
A Ring with a unit element in which every element is
IDEMPOTENT.
see also BOOLEAN ALGEBRA
Borchardt-Pfaff Algorithm
Borel Probability Measure 157
Borchardt-Pfaff Algorithm
see Archimedes Algorithm
Border Square
40
1
2
3
42
41
46
38
31
13
14
32
35
12
39
30
26
21
28
20
11
43
33
27
25
23
17
7
6
16
22
29
24
34
44
5
15
37
36
18
19
45
4
49
48
47
8
9
10
31
13
14
32
35
30
26
21
28
20
33
27
25
23
17
16
22
29
24
34
15
37
36
18
19
26
21
28
27
25
23
22
29
24
A MAGIC SQUARE that remains magic when its bor-
der is removed. A nested magic square remains magic
after the border is successively removed one ring at a
time. An example of a nested magic square is the order
7 square illustrated above (i.e., the order 7, 5, and 3
squares obtained from it are all magic).
see also MAGIC SQUARE
References
Kraitchik, M. "Border Squares." §7.7 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 167-170, 1942.
Bordism
A relation between Compact boundaryless Manifolds
(also called closed Manifolds). Two closed Mani-
folds are bordant IFF their disjoint union is the bound-
ary of a compact (n+l)-MANlFOLD. Roughly, two Man-
ifolds are bordant if together they form the boundary
of a Manifold. The word bordism is now used in place
of the original term COBORDISM.
References
Budney, R. "The Bordism Project." http: //math. Cornell.
eduArybu/bordism/bordism.html.
Bordism Group
There are bordism groups, also called Cobordism
Groups or Cobordism Rings, and there are singu-
lar bordism groups. The bordism groups give a frame-
work for getting a grip on the question, "When is a
compact boundaryless MANIFOLD the boundary of an-
other Manifold?" The answer is, precisely when all of
its Stiefel- Whitney Classes are zero. Singular bor-
dism groups give insight into STEENROD's REALIZATION
PROBLEM: "When can homology classes be realized as
the image of fundamental classes of manifolds?" That
answer is known, too.
The machinery of the bordism group winds up being
important for HOMOTOPY THEORY as well.
References
Budney, R. "The Bordism Project." http: //math. Cornell.
edu/-rybu/bordism/bordism.html.
Borel-Cantelli Lemma
Let {^4.n}£Lo De a Sequence of events occurring with a
certain probability distribution, and let A be the event
consisting of the occurrence of a finite number of events
A ni n = 1, Then if
then
^2p(A n ) < oo,
P(A) = 1.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet u Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, pp. 435-436, 1988.
Borel Determinacy Theorem
Let T be a tree defined on a metric over a set of paths
such that the distance between paths p and q is 1/n,
where n is the number of nodes shared by p and q. Let
A be a Borel set of paths in the topology induced by this
metric. Suppose two players play a game by choosing a
path down the tree, so that they alternate and each time
choose an immediate successor of the previously chosen
point. The first player wins if the chosen path is in A,
Then one of the players has a winning STRATEGY in this
Game.
see also Game Theory, Strategy
BorePs Expansion
Let <p(t) = Xl^lo ^nt 71 ^ e any function for which the
integral
/>oo
I(x) = / e- tx t v 4>{t) dt
Jo
converges. Then the expansion
I(x)
XP+
■^l[Ao + ( P + iy-
+ (p+l)(p + 2)^ + ...
where F(z) is the Gamma Function, is usually an
Asymptotic Series for I(x).
Borel Measure
If F is the Borel Sigma Algebra on some Topolog-
ical Space, then a Measure m : F -+ R is said to be
a Borel measure (or BOREL PROBABILITY MEASURE).
For a Borel measure, all continuous functions are MEA-
SURABLE.
Borel Probability Measure
see BOREL MEASURE
158
Borel Set
Borwein Conjectures
Borel Set
A Definable Set derived from the Real Line by re-
moving a Finite number of intervals. Borel sets are
measurable and constitute a special type of Sigma Al-
gebra called a BOREL SIGMA ALGEBRA.
see also Standard Space
Borel Sigma Algebra
A Sigma Algebra which is related to the Topology
of a Set, The Borel sigma-algebxa is defined to be
the Sigma Algebra generated by the Open Sets (or
equivalently, by the CLOSED Sets).
see also Borel MEASURE
Borel Space
A Set equipped with a Sigma Algebra of Subsets.
Borromean Rings
Three mutually interlocked rings named after the Italian
Renaissance family who used them on their coat of arms.
No two rings are linked, so if one of the rings is cut, all
three rings fall apart. They are given the Link symbol
O603, and are also called the Ballantine. The Bor-
romean rings have BRAID WORD c^ -1 o- 2 o'i~ 1 <J 2 ai _1 &2
and are also the simplest Brunnian Link.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 58-59, 1989.
Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: University of Chicago Press,
1991.
Jablan, S. "Borromean Triangles." http:/ /members. tripod,
com/ -modularity/links .htm.
Pappas, T. "Trinity of Rings— A Topological Model." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 31, 1989.
Borrow
1 2 3
-78
borrows
4 4 5
The procedure used in SUBTRACTION to "borrow" 10
from the next higher Digit column in order to obtain a
Positive Difference in the column in question.
see also Carry
Borsuk's Conjecture
Borsuk conjectured that it is possible to cut an n-D
shape of DIAMETER 1 into n + 1 pieces each with di-
ameter smaller than the original. It is true for n = 2,
3 and when the boundary is "smooth." However, the
minimum number of pieces required has been shown to
increase as ~ 1.1^. Since 1.1^ > n + 1 at n = 9162,
the conjecture becomes false at high dimensions. In fact,
the limit has been pushed back to ~ 2000.
see also DIAMETER (GENERAL), KELLER'S CONJEC-
TURE, Lebesgue Minimal Problem
References
Borsuk, K. "Uber die Zerlegung einer Euklidischen n-
dimensionalen Vollkugel in n Mengen." Verh. Internat.
Math.-Kongr. Zurich 2, 192, 1932.
Borsuk, K. "Drei Satze iiber die n-dimensionale euklidische
Sphare." Fund. Math. 20, 177-190, 1933.
Cipra, B. "If You Can't See It, Don't Believe It. . . ." Science
259, 26-27, 1993.
Cipra, B. What's Happening in the Mathematical Sciences,
Vol. 1. Providence, RI: Amer. Math, Soc, pp. 21-25, 1993,
Grunbaum, B. "Borsuk's Problem and Related Questions."
In Convexity, Proceedings of the Seventh Symposium in
Pure Mathematics of the American Mathematical Society,
Held at the University of Washington, Seattle, June 13-
15, 1961. Providence, RI: Amer. Math. Soc, pp. 271-284,
1963.
Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture."
Bull. Amer. Math. Soc. 329, 60-62, 1993. Listernik, L.
and Schnirelmann, L. Topological Methods in Variational
Problems. Moscow, 1930.
Borwein Conjectures
Use the definition of the q- Series
{a\q)n = JJ(l-ag')
j=o
and define
N
M
[q \q)m
(Q\Q)m
(1)
(2)
Then P. Borwein has conjectured that (1) the Polyno-
mials A n (q), B n (q), and C n (q) defined by
(q\ </ 3 W<Z 2 ; qX = A n (q 3 ) - qB n (q 3 ) - q 2 C n (q 3 ) (3)
have NONNEGATIVE COEFFICIENTS, (2) the POLYNOMI-
ALS A* n {q), B*{q), and C*(q) defined by
(q;qX(q 2 ;q 3 )l = A:(q S )- q B:(q S )~ q 2 C:(q 3 ) (4)
have Nonnegative Coefficients, (3) the Polynomi-
als A* n {q), B*{q), C*(q), D*(q), and E* n (q) defined by
(9; 5 )n(q ;q )n(q ;q)n(q;q)n-
AUq 5 )-qB* n {f , )-q 2 C* n {qS)-q 3 Dl{q 5 )-q 4 EUq 5 ) (5)
Bouligand Dimension
Boundary Point 159
have NONNEGATIVE COEFFICIENTS, (4) the POLYNOMI-
ALS Al l (m i n,t,q) 1 £* (m,n,£, g), and C^m^n^t^q) de-
fined by-
Bound Variable
An occurrence of a variable in a LOGIC which is not
Free.
(?; q 3 )m(q 2 ; q Z )m{zq\ q 3 ) n {zq 2 ; q 3 ) n
2m
= > z [A* (m,n, £, q ) — qB* (m, n,t,q )
t=Q
-q 2 C\m,n,t,q 3 )} (6)
have Nonnegative Coefficients, (5) for k Odd and
1 < a < k/2, consider the expansion
(q a ;q k U(q k - a ;q k )n
(fc-D/2
E
t/=(l-fc)/2
(_ 1 )^M- 2 +-)/2-a,^ ( ^ ) (7)
with
oo
_ V^ f-lY 3(k 2 j + 2ku + k-2a)/2
m 4- n
m + v + kj
(8)
then if a is Relatively Prime to k and m = n, the CO-
EFFICIENTS of F^qr) are NONNEGATIVE, and (6) given
a J rf3< 2'K and — K + /? < n — m < K — a, consider
G(a,0,K;q) = ^(_i)V 1JC(a+w+lf(a+/9)1/a
ra + n
171+ Kj
, (9)
the Generating Function for partitions inside an mx
n rectangle with hook difference conditions specified by
a, /?, and if. Let a and /? be POSITIVE RATIONAL
Numbers and K > 1 an Integer such that aK and
/3Jf are integers. Then if 1 < a + < 2K-1 (with strict
inequalities for K = 2) and —if + /3<n — m < K — a,
then G(a,j3,K;q) has NONNEGATIVE COEFFICIENTS,
see ateo ^-SERIES
References
Andrews, G. E. ei al. "Partitions with Prescribed Hook Dif-
ferences." Europ. J. Combin. 8, 341-350, 1987.
Bressoud, D. M. "The Borwein Conjecture and Partitions
with Prescribed Hook Differences. " Electronic J. Com-
binatorics 3, No. 2, R4, 1-14, 1996. http://www.
combinatorics. org/Volume^3/volume3_2.html#R4.
Bouligand Dimension
see MlNKOWSKI-BOULIGAND DIMENSION
Bound
see Greatest Lower Bound, Infimum, Least Up-
per Bound, Supremum
Boundary
The set of points, known as Boundary Points, which
are members of the CLOSURE of a given set 5 and the
CLOSURE of its complement set. The boundary is some-
times called the FRONTIER.
see also SURGERY
Boundary Conditions
There are several types of boundary conditions com-
monly encountered in the solution of PARTIAL DIFFER-
ENTIAL Equations.
1. Dirichlet Boundary Conditions specify the
value of the function on a surface T = /(r,£).
2. Neumann Boundary Conditions specify the nor-
mal derivative of the function on a surface,
dT
dn
_=fi-vr = /(r, y ).
3. Cauchy Boundary Conditions specify a weighted
average of first and second kinds.
4. Robin Boundary Conditions. For an elliptic par-
tial differential equation in a region Q, Robin bound-
ary conditions specify the sum of au and the normal
derivative of u = / at all points of the boundary of
Q } with a and / being prescribed.
see also BOUNDARY VALUE PROBLEM, DlRICHLET
Boundary Conditions, Initial Value Problem,
Neumann Boundary Conditions, Partial Differ-
ential Equation, Robin Boundary Conditions
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 502-504, 1985.
Morse, P. M. and Feshbach, H. "Boundary Conditions and
Eigcnfunctions." Ch. 6 in Methods of Theoretical Physics,
Part L New York: McGraw-Hill, pp. 495-498 and 676-790,
1953.
Boundary Map
The Map H n {X, A) -► H n - 1 (A) appearing in the Long
Exact Sequence of a Pair Axiom.
see also Long Exact Sequence of a Pair Axiom
Boundary Point
A point which is a member of the Closure of a given
set S and the CLOSURE of its complement set. If A is a
subset of M n , then a point x € M. n is a boundary point
of A if every NEIGHBORHOOD of x contains at least one
point in A and at least one point not in A.
see also BOUNDARY
160
Boundary Set
Boustrophedon Transform
Boundary Set
A (symmetrical) boundary set of RADIUS r and center
xq is the set of all points x such that
Bourget Function
x- x = r.
Let xo be the ORIGIN. In IR , the boundary set is then
\ the
-r. In
the pair of points x — r and x
boundary set is a CIRCLE. In R
is a Sphere.
see also Circle, Disk, Open Set, Sphere
the boundary set
Boundary Value Problem
A boundary value problem is a problem, typically an
Ordinary Differential Equation or a Partial
Differential Equation, which has values assigned
on the physical boundary of the Domain in which the
problem is specified. For example,
u(O t t)
V 2 u = f
m
f*(0,t)=u 2
in Q
on dQ
on dQ,
where dCl denotes the boundary of O, is a boundary
problem.
see also Boundary Conditions, Initial Value
Problem
References
Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Compu-
tational Differential Equations. Lund: Studentlitteratur,
1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Two Point Boundary Value Problems." Ch. 17
in Numerical Recipes in FORTRAN: The Art of Scientific
Computing, 2nd ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 745-778, 1992.
Bounded
A Set in a Metric Space (X,d) is bounded if it has
a FINITE diameter, i.e., there is an R < oo such that
d(#, y) < R for all x, y € X. A Set in W 1 is bounded if
it is contained inside some Ball x\ 2 + . . . + x n 2 < R 2
of Finite Radius R (Adams 1994).
see also Bound, Finite
References
Adams, R. A. Calculus: A Complete Course, Reading, MA:
Addison- Wesley, p. 707, 1994.
Bounded Variation
A Function f(x) is said to have bounded variation if,
over the Closed Interval x e [a, b], there exists an M
such that
\f(xi)-f(a)\ + \f(x2)-f(x 1 )\ + . . .+ |/(6)-/(x„_i)| < M
for all a < xi < X2 < ■ . . < x n -i < b.
J -*u-hf t ""( ,+ \Y"*[i'('-\)]*
* Jo
(2 cos d) k cos(n(9 - z sin 6) d0.
see also Bessel Function of the First Kind
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia." Dordrecht, Nether-
lands: Reidel, p. 465, 1988.
Bourget's Hypothesis
When n is an INTEGER > 0, then J n (z) and J n +m(z)
have no common zeros other than at z = for m an
Integer > 1, where J n (z) is a Bessel Function of
THE First Kind. The theorem has been proved true
for m=l 2, 3, and 4.
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Boustrophedon Transform
The boustrophedon ( "ox-plowing" ) transform b of a se-
quence a is given by
bn = 7 7 \dkEn-k
k=o v /
— ±(-')-(0
fc=0 x '
bkEn~k
(1)
(2)
for n > 0, where E n is a Secant Number or Tangent
Number defined by
Ex n
E n — 7 = sec X +
tanz.
(3)
The exponential generating functions of a and b are
related by
B(x) = (sec a? + tanz)^4(#), (4)
where the exponential generating function is defined by
A(x) = Y,An
x
(5)
see also ALTERNATING PERMUTATION, ENTRINGER
Number, Secant Number, Seidel-Entringer-
Arnold Triangle, Tangent Number
References
Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op-
eration on Sequences: The Boustrophedon Transform." J.
Combin. Th. Ser. A 76, 44-54, 1996.
Bovinum Problema
Box Fractal 161
Bovinum Problema
see Archimedes' Cattle Problem
Bow
4 2 3
x = x y — y .
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Bowditch Curve
see Lissajous Curve
Bowley Index
The statistical Index
where P L is Laspeyres' Index and P P is Paasche's
Index.
see also INDEX
References
Kenney, J. F, and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962.
Bowley Skewness
Also known as Quartile Skewness Coefficient,
(Q 3 - Q 2 ) - (<?2 - <2i) _ Qi - 2Q 2 + Q 3
Qz-Qi)
Qz-Qi
where the Qs denote the Interquartile Ranges.
see also SKEWNESS
Bowling
Bowling is a game played by rolling a heavy ball down
a long narrow track and attempting to knock down ten
pins arranged in the form of a TRIANGLE with its vertex
oriented towards the bowler. The number 10 is, in fact,
the Triangular Number T 4 = 4(4 4- l)/2 = 10.
Two "bowls" are allowed per "frame." If all the pins are
knocked down in the two bowls, the score for that frame
is the number of pins knocked down. If some or none of
the pins are knocked down on the first bowl, then all the
pins knocked down on the second, it is called a "spare,"
and the number of points tallied is 10 plus the number
of pins knocked down on the bowl of the next frame.
If all of the pins are knocked down on the first bowl,
the number of points tallied is 10 plus the number of
pins knocked down on the next two bowls. Ten frames
are bowled, unless the last frame is a strike or spare, in
which case an additional bowl is awarded.
The maximum number of points possible, corresponding
to knocking down all 10 pins on every bowl, is 300.
References
Cooper, C N. and Kennedy, R. E. "A Generating Function
for the Distribution of the Scores of All Possible Bowl-
ing Games." In The Lighter Side of Mathematics (Ed.
R. K. Guy and R. E. Woodrow). Washington, DC: Math.
Assoc. Amer., 1994.
Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling
Score Awful?" In The Lighter Side of Mathematics (Ed.
R. K. Guy and R. E. Woodrow). Washington, DC: Math.
Assoc. Amer., 1994.
Box
see Cuboid
Box-and- Whisker Plot
X
T
i
A HlSTOGRAM-like method of displaying data invented
by J. Tukey (1977). Draw a box with ends at the QUAR-
TILES Qi and Q 3 . Draw the MEDIAN as a horizontal
line in the box. Extend the "whiskers" to the farthest
points. For every point that is more than 3/2 times the
Interquartile Range from the end of a box, draw a
dot on the corresponding top or bottom of the whisker.
If two dots have the same value, draw them side by side.
References
Tukey, J. W. Explanatory Data Analysis.
Addison- Wesley, pp. 39-41, 1977.
Box Counting Dimension
see Capacity Dimension
Box Fractal
Reading, MA:
A Fractal which can be constructed using String
Rewriting by creating a matrix with 3 times as
many entries as the current matrix using the rules
line 1
line 2
line 3
11 jkii — S 11 sk " "
'■_>11 "
162
Box-Muller Transformation
Boy Surface
Let N n be the number of black boxes, L n the length of
a side of a white box, and A n the fractional AREA of
black boxes after the nth iteration.
N„=5 n
Ln = (!)"= 3""
The Capacity Dimension is therefore
(i)
(2)
(3)
lniV n
d cap = - lim lnL
lim
ln(5")
n— >oo
In 5
m~3
n-voo ln(3- n )
1.464973521....
(4)
Boxcar Function
y = c[H(x-o)-H{x-b)],
where H is the Heaviside Step Function.
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 324, 1993.
see also Cantor Dust, Sierpinski Carpet, Sierpin-
ski Sieve
References
$ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/~evw6n/math/notebooks/Fractal.m.
Box-Muller Transformation
A transformation which transforms from a 2-D contin-
uous Uniform Distribution to a 2-D Gaussian Bi-
variate Distribution (or Complex Gaussian Dis-
tribution). If xi and X2 are uniformly and indepen-
dently distributed between and 1, then z\ and z 2 as de-
fined below have a Gaussian Distribution with Mean
li = and Variance <t 2 = 1.
z\ — y — 21na;i cos(27ra;2)
Z2 = v — 21n#i sin(27ra;2)-
This can be verified by solving for x\ and x 2 ,
-( Zl 2 + Z2 2 )/2
x x
X 2
2tt
■ tan
■■(!)•
(1)
(2)
(3)
(4)
Taking the Jacobian yields
d(xi,x 2 )
d(z u z 2 )
d%i dxi
dz± dz 2
dx 2 dx 2
£zi dz 2
1 -Z! 2 /2
— e * '
2tt
\/2^
^ 2 2 /2
(5)
Box-Packing Theorem
The number of "prime" boxes is always finite, where a
set of boxes is prime if it cannot be built up from one
or more given configurations of boxes.
see also Conway Puzzle, Cuboid, de Bruijn's Theo-
rem, Klarner's Theorem, Slothouber-Graatsma
Puzzle
References
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 74, 1976.
Boxcars
A roll of two 6s (the highest roll possible) on a pair of
6-sided DICE, The probability of rolling boxcars is 1/36,
or 2.777...%.
see also Dice, Double Sixes, Snake Eyes
Boy Surface
A Nonorientable Surface which is one of the three
possible Surfaces obtained by sewing a Mobius Strip
to the edge of a DISK. The other two are the CROSS-
Cap and ROMAN SURFACE. The Boy surface is a model
of the Projective Plane without singularities and is
a Sextic Surface.
The Boy surface can be described using the general
method for NONORIENTABLE SURFACES, but this was
not known until the analytic equations were found by
Apery (1986). Based on the fact that it had been proven
impossible to describe the surface using quadratic poly-
nomials, Hopf had conjectured that quartic polynomials
were also insufficient (Pinkall 1986). Apery's IMMER-
SION proved this conjecture wrong, giving the equations
explicitly in terms of the standard form for a NONORI-
ENTABLE Surface,
/i(*,y,s) = |[(2x 2 - y 2 - z 2 )(x 2 + y 2 + z 2 )
+ 2yz(y — z ) + zx(x — z )
+ xy(y 2 -x 2 )] (1)
Mx,y,z) = \s/Z[{y 2 - z 2 )(x 2 + y 2 + z 2 )
+ zx{z 2 - x 2 ) + xy(y 2 - x 2 )} (2)
f 3 (x, y,z) = i(x + y + z)[(x + y + z) 3
+ A(y-x)(z-y)(x-z)]. (3)
Boy Surface
Plugging in
x = cos u sin v
y = sin u sin v
Z = COS V
(4)
(5)
(6)
and letting u G [0, tv] and v € [0, 7r] then gives the Boy
surface, three views of which are shown above.
The K. parameterization can also be written as
V =
V2cos 2 vcos(2it) + cosusin(2t;)
2- v / 2sin(3u)sin(2v)
__ \/2cos 2 vsin(2u) + cos^sin(2i;)
2- v / 2sin(3u)sin(2i;)
3 cos 2 v
2- V2sin(3u)sin(2t;)
(Nordstrand) for u 6 [-7r/2,7r/2] and i; G [0,7r].
(7)
(8)
(9)
Three views of the surface obtained using this parame-
terization are shown above.
In fact, a HOMOTOPY (smooth deformation) between
the Roman Surface and Boy surface is given by the
equations
x(u,v) =
y{u,v) =
Z(U y V) =
\[2 cos(2n) cos 2 v + cos u sin(2v)
2 — a\/2 sin(3ti) sin(2t;)
\/2sin(2u) cos 2 v — sinusin(2t>)
2-aA/2sin(3tx)sin(2i;)
3 cos 2 v
2 — a\/2 sin(3u) sin(2v)
(10)
(11)
(12)
as a varies from to 1, where a — corresponds to the
Roman Surface and a = 1 to the Boy surface (Wang),
shown below.
Boy Surface 163
In K. , the parametric representation is
xq = 3[(u + v +w )(u + v ) — V2vw(3u — v )]
(13)
X! = V2(u 2 + v 2 )(u 2 - v 2 + v^uty) (14)
a?2 = V2(u 2 + v 2 )(2wu - V2vw) (15)
X3 = 3(u 2 + v 2 ) 2 , (16)
and the algebraic equation is
64(x — £3) 3 #3 3 — 48(x — ^3) 2 ^3 2 (32;i 2 + Sx2 2 + 2x 3 2 )
+12(z - x 3 )x 3 [27(x 1 2 + z 2 2 ) 2 - 24z 3 2 (zi 2 + x 2 2 )
+36^3:2^3 (x2 2 — 3cci 2 ) + X3 4 ]
+(9zi 2 +92 2 2 - 2x 3 2 )
x[-81(^i 2 + x 2 2 ) 2 - 72x 3 2 (xi 2 + x 2 2 )
+10%V2x 1 x 3 {x 1 2 - 3z 2 2 ) + 4z 3 4 ] = (17)
(Apery 1986). Letting
Xq — 1
Xi = X
x 2 =y
X3 = z
(18)
(19)
(20)
(21)
gives another version of the surface in M .
see also Cross-Cap, Immersion, Mobius Strip,
nonorientable surface, real projective plane,
Roman Surface, Sextic Surface
References
Apery, F. "The Boy Surface." Adv. Math. 61, 185-266, 1986.
Boy, W. "Uber die Curvatura Integra und die Topologie
geschlossener Flachen." Math. Ann 57, 151-184, 1903.
Brehm, U. "How to Build Minimal Polyhedral Models of the
Boy Surface." Math. Intell. 12, 51-56, 1990.
Carter, J. S. "On Generalizing Boy Surface — Constructing a
Generator of the 3rd Stable Stem." Trans. Amer. Math.
Soc. 298, 103-122, 1986.
Fischer, G. (Ed.). Plates 115-120 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 110-115, 1986.
Geometry Center. "Boy's Surface." http://www.geom.umn.
edu/zoo/toptype/pplane/boy/.
Hilbert, D. and Cohn-Vossen, S. §46—47 in Geometry and the
Imagination. New York: Chelsea, 1952.
Nordstrand, T. "Boy's Surface." http : //www . uib . no/
people/nf ytn/boytxt . htm.
Petit, J .-P. and Souriau, J. "Une representation analytique
de la surface de Boy." C. R. Acad. Sci. Paris Sir. 1 Math
293, 269-272, 1981.
Pinkall, U. Mathematical Models from the Collections of Uni-
versities and Museums (Ed. G. Fischer). Braunschweig,
Germany: Vieweg, pp. 64-65, 1986.
Stewart, I. Game, Set and Math. New York: Viking Penguin,
1991.
Wang, P. "Renderings." http: //www.ugcs . caltech.edu/
-pet erw/portf olio/renderings/.
164
Bra
Brachistochrone Problem
Bra
A (COVARIANT) 1-VECTOR denoted (V>|- The bra is
Dual to the Contravariant Ket, denoted \ip). Taken
together, the bra and KET form an ANGLE BRACKET
(bra+ket = bracket). The bra is commonly encountered
in quantum mechanics.
see also Angle Bracket, Bracket Product, Co-
variant Vector, Differential /.-Form, Ket, One-
Form
Brachistochrone Problem
Find the shape of the CURVE down which a bead sliding
from rest and Accelerated by gravity will slip (with-
out friction) from one point to another in the least time.
This was one of the earliest problems posed in the CAL-
CULUS of Variations. The solution, a segment of a
Cycloid, was found by Leibniz, L'Hospital, Newton,
and the two Bernoullis.
The time to travel from a point Pi to another point Pi
is given by the INTEGRAL
= C -
(i)
The VELOCITY at any point is given by a simple appli-
cation of energy conservation equating kinetic energy to
gravitational potential energy,
1 2
2 mv
mgy,
v = y/2gy.
Plugging this into (1) then gives
tl2
i:
a/i + y' 2
s/5gy
dx ■
i:
l + y' 2
tgy
dx.
The function to be varied is thus
f = (l + y ,2 ) 1/2 (2gy)-^.
(2)
(3)
(4)
(5)
subtracting y'{df/dy') from /, and simplifying then
gives
C. (9)
V^gy^i + y' 2
Squaring both sides and rearranging slightly results in
1 +
[dx)
2gC*
(10)
where the square of the old constant C has been ex-
pressed in terms of a new (POSITIVE) constant k 2 . This
equation is solved by the parametric equations
x -
y
\k 2 {e-s\n9)
§fc 2 (l-cos6>),
(11)
(12)
which are — lo and behold — the equations of a CYCLOID.
If kinetic friction is included, the problem can also be
solved analytically, although the solution is significantly
messier. In that case, terms corresponding to the normal
component of weight and the normal component of the
Acceleration (present because of path Curvature)
must be included. Including both terms requires a con-
strained variational technique (Ashby et al. 1975), but
including the normal component of weight only gives an
elementary solution. The Tangent and Normal Vec-
tors are
(13)
(14)
T =
dx „
ds
dy-
ds
N =
dy ~
ds
dx ^
gravity and friction are then
• gravity
: mgy
dx r
Ff r i c tion = ~M( F gravityN)T = - flTTig — T,
and the components along the curve are
(15)
(16)
To proceed, one would normally have to apply the full-
blown Euler-Lagrange Differential Equation
21
dy
dx \dy'J
0.
(6)
However, the function f{y,y' } x) is particularly nice
since x does not appear explicitly. Therefore, df /dx =
0, and we can immediately use the Beltrami Identity
>-<%-<>■
Computing
8y'
y {l + y
/2\-l/2
(2gy)
-1/2
(7)
(8)
-T gravity J- :
■T friction -L
dy
m9 dS
-fj,mg
dx
ds '
so Newton's Second Law gives
dv
dy
m— — mg- 1
dt ds
limg
dx
ds
But
dv dv
— = v —
at as
1 d 2 ^
2dS^ V)
\v 2 = g{y - fix)
v = y/2g{y - fix),
(17)
(18)
(19)
(20)
(21)
(22)
Bracket
Bracketing 165
-Jyi
+ (y') 2
dx.
(23)
2< ? (y - fix) ~' v
Using the Euler-Lagrange Differential Equation
gives
[i + y 2 ](i + aV) + 2(2/ - ^)y" - 0. (24)
This can be reduced to
i + (y') 2 _ c
Now letting
the solution is
(1 + /X2/') 2 y- iix'
y'=cot(±0),
(25)
(26)
a; = ffc 2 [(0-sm0)+Ai(l-cos6O] (27)
y = |A; 2 [(1 - cos<9) + ^(<9 + sin0)]. (28)
see also Cycloid, Tautochrone Problem
References
Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. "Bra-
chistochrone with Coulomb Friction." Amer. J. Phys. 43,
902-905, 1975.
Haws, L. and Kiser, T. "Exploring the Brachistochrone Prob-
lem." Amer. Math. Monthly 102, 328-336, 1995.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 60-66 and 385-389, 1991.
Bracket
see Angle Bracket, Bra, Bracket Polynomial,
Bracket Product, Iverson Bracket, Ket, La-
grange Bracket, Poisson Bracket
Bracket Polynomial
A one- variable KNOT POLYNOMIAL related to the JONES
Polynomial. The bracket polynomial, however, is not
a topological invariant, since it is changed by type I REI-
demeister Moves. However, the Span of the bracket
polynomial is a knot invariant. The bracket polynom-
ial is occasionally given the grandiose name REGULAR
Isotopy Invariant. It is defined by
<L)(A,*,d) = ^<2W
Ikll
(1)
where A and B are the "splitting variables," a runs
through all "states" of L obtained by Splitting the
LINK, (L\a) is the product of "splitting labels" corre-
sponding to cr, and
\W\\ = N L -1, (2)
where JV& is the number of loops in er. Letting
-l
B = A
d^-A 2 -A' 2
(3)
(4)
gives a Knot Polynomial which is invariant under
Regular Isotopy, and normalizing gives the Kauff-
man Polynomial X which is invariant under Ambient
Isotopy. The bracket Polynomial of the Unknot is
1. The bracket Polynomial of the Mirror Image K*
is the same as for K but with A replaced by A -1 . In
terms of the one-variable KAUFFMAN POLYNOMIAL X,
the two-variable KAUFFMAN POLYNOMIAL F and the
Jones Polynomial V\
X(A)
(L) (A)
(
-A*y
-«.(£)
(L),
F(-A 3 ,A + A- 1 )
(5)
(6)
(7)
<L> (A) = V{A~%
where w(L) is the WRITHE of L.
see also SQUARE BRACKET POLYNOMIAL
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 148-155, 1994.
Kauffman, L. "New Invariants in the Theory of Knots."
Amer. Math. Monthly 95, 195-242, 1988.
Kauffman, L. Knots and Physics. Teaneck, NJ: World Sci-
entific, pp. 26-29, 1991.
i$ Weisstein, E. W. "Knots and Links." http: //www. astro.
virginia.edu/~eww6n/math/notebooks/Knots .m.
Bracket Product
The Inner Product in an Li Space represented by an
Angle Bracket.
see also Angle Bracket, Bra, Ket, L 2 Space, One-
Form
Bracketing
Take x itself to be a bracketing, then recursively de-
fine a bracketing as a sequence B = (jBi, . . . , Bk) where
k > 2 and each Bi is a bracketing. A bracketing can be
represented as a parenthesized string of xs, with paren-
theses removed from any single letter x for clarity of
notation (Stanley 1997). Bracketings built up of binary
operations only are called BINARY BRACKETINGS. For
example, four letters have 11 possible bracketings:
xxxx (xx)xx x(xx)x xx(xx)
(xxx)x x(xxx) ((xx)x)x (x(xx))x
{xx)(xx) x((xx)x) x(x(xx)),
the last five of which are binary.
The number of bracketings on n letters is given by the
Generating Function
\(l + x- y/l ~6x + x 2 ) = x + x 2 + 3x 3 + llx 4 + 45x 5
(Schroder 1870, Stanley 1997) and the RECURRENCE
Relation
_ 3(2n — 3)s n -i — (n — 3)s n -2
166
Bradley's Theorem
Brahmagupta Matrix
(Sloane), giving the sequence for s n as 1, 1, 3, 11, 45,
197, 903, . . . (Sloane's A001003). The numbers are also
given by
s n = ^ s(ii) • • - s(i k )
for n > 2 (Stanley 1997).
The first PLUTARCH NUMBER 103,049 is equal to $io
(Stanley 1997), suggesting that Plutarch's problem of
ten compound propositions is equivalent to the number
of bracketings. In addition, Plutarch's second number
310,954 is given by (sio + sn)/2 = 310,954 (Habsieger
et al. 1998).
see also Binary Bracketing, Plutarch Numbers
References
Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second
Number of Plutarch." Amer. Math. Monthly 105, 446,
1998.
Schroder, E. "Vier combinatorische Probleme." Z. Math.
Physik 15, 361-376, 1870.
Sloane, N. J. A. Sequence A001003/M2898 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough."
Amer. Math. Monthly 104, 344-350, 1997.
Bradley's Theorem
Let
S(a,(3,m;z) =
y> T(m + j(z + l))rpg + 1 + jz) (a) + j
m 2^ T{m + jz + l)r(a + + 1 + j(z + 1)) j!
j =
and a be a Negative Integer. Then
T(/3 + 1 - m)
S{a,(3,m\z) =
r(a + /3 + l-m)'
where T(z) is the GAMMA FUNCTION.
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 346-348, 1994.
Bradley, D. "On a Claim by Ramanujan about Certain Hy-
pergeometric Series." Proc. Amer. Math. Soc. 121, 1145-
1149, 1994.
Brahmagupta's Formula
For a Quadrilateral with sides of length a, 6, c, and
d, the Area K is given by
K :
J(s - a)(s - b)(s - c)(s - d) - abcdcos 2 [\{A + B)],
(1)
where
s= |(a + 6 + c + d)
(2)
is the Semiperimeter, A is the Angle between a and
d, and B is the Angle between b and c. For a Cyclic
Quadrilateral (i.e., a Quadrilateral inscribed in
a Circle), A + B — 7r, so
K = ^/(s-a)(s-b){s-c){s-d) (3)
y/(bc + ad)(ac + bd)(ab -f- cd)
4R
(4)
where R is the RADIUS of the CiRCUMClRCLE. If the
Quadrilateral is Inscribed in one Circle and Cir-
cumscribed on another, then the Area Formula sim-
plifies to
K = \fabc~d. (5)
see also Bretschneider's Formula, Heron's For-
mula
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 81-82, 1929.
Brahmagupta Identity
Let
0=\B\^x 2 -ty\
where B is the Brahmagupta Matrix, then
det[B(x u yi)B(x 2 ,y2)] = det[B(x u yi )] det[B(x 2 , y 2 )]
= Pifo.
References
Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib.
Quart. 34, 30-39, 1996.
Brahmagupta Matrix
5(z,y) =
x y
±ty ±x
It satisfies
B(x!,yi)B(x 27 y2) = B{xxx 2 ±tyiy2,x 1 y 2 ±2/1X2).
Powers of the matrix are defined by
B n =
X
y
n
X<n
Vn
ty
X
ty n
Xn
= B n .
The x n and y n are called BRAHMAGUPTA POLYNOMI-
ALS. The Brahmagupta matrices can be extended to
Negative Integers
n-n _ x y _ *-n y-n _ d
X
y
— n
X — n
y-
ty
X
ty-n
X-
see also Brahmagupta Identity
References
Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib.
Quart. 34, 30-39, 1996.
Brahmagupta Polynomial
Braid Group 167
Brahmagupta Polynomial
One of the POLYNOMIALS obtained by taking POWERS
of the Brahmagupta Matrix. They satisfy the recur-
rence relation
x n +! = xx n + tyy n
y n+1 = xy n + yx n .
(1)
(2)
A list of many others is given by Suryanarayan (1996).
Explicitly,
X +t
(;)«-v+« a (j)«-- 4 » 4 +... (3)
n-l . .i n \ n-3 3 . .2l n \ n-5 5 .
rix y + t[\x y +t i ]x y +
The Brahmagupta POLYNOMIALS satisfy
dx
dx n
dy
nx n -i
dyn
dy
,dy n
t—- = ntyn-L
dy
(4)
(5)
(6)
The first few POLYNOMIALS are
xo =
xi = x
x 2 — x 2 + ty 2
xz = x 3 + 3txy 2
4 . «, 2 2 , ,2 4
X4 = x + otx y +t y
and
yo-o
2/1=2/
y 2 = 2xy
2/3 = 3x 2 y + ty 3
2/4 = ^x z y -\- Atxy z .
Taking x = i/ = 1 and £ = 2 gives j/„ equal to the PELL
Numbers and x n equal to half the Pell-Lucas num-
bers. The Brahmagupta POLYNOMIALS are related to
the Morgan- Voyce Polynomials, but the relation-
ship given by Suryanarayan (1996) is incorrect.
References
Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib.
Quart. 34, 30-39, 1996.
Brahmagupta's Problem
Solve the PELL EQUATION
x 2 - 92y 2 = 1
in Integers. The smallest solution is x = 1151, y =
120.
see also Diophantine Equation, Pell Equation
Braid
An intertwining of strings attached to top and bottom
"bars" such that each string never "turns back up." In
other words, the path of a braid in something that a
falling object could trace out if acted upon only by grav-
ity and horizontal forces.
see also Braid GROUP
References
Christy, J. "Braids." http://www.mathsource.com/cgi-bin
/MathSource/Applications/Mathematics/0202-228.
Braid Group
Also called Artin Braid Groups. Consider n strings,
each oriented vertically from a lower to an upper "bar."
If this is the least number of strings needed to make a
closed braid representation of a LINK, n is called the
Braid Index. Now enumerate the possible braids in a
group, denoted B n . A general n-braid is constructed by
iteratively applying the <Tj (i = 1, . . . ,n — 1) operator,
which switches the lower endpoints of the ith and (i +
l)th strings — keeping the upper endpoints fixed — with
the (i + l)th string brought above the ith string. If the
(i + l)th string passes below the zth string, it is denoted
1 2
Topological equivalence for different representations of
a BRAID Word JJ o~i and J^ a^ is guaranteed by the
conditions
CTiCTj — <Tj<Ti
/ it
O'iO'i + iO'i — 0'i-\-\(T i <Ti + i
for \i-j\ >2
for all i
as first proved by E. Artin. Any n-braid is expressed as
a Braid Word, e.g., G^aicr^a^ a\ is a Braid Word
for the braid group #3 . When the opposite ends of the
braids are connected by nonintersecting lines, KNOTS
are formed which are identified by their braid group and
Braid Word. The Burau Representation gives a
matrix representation of the braid groups.
References
Birman, J. S. "Braids, Links, and the Mapping Class
Groups." Ann. Math. Studies, No. 82. Princeton, NJ:
Princeton University Press, 1976.
Birman, J. S. "Recent Developments in Braid and Link The-
ory." Math. Intell. 13, 52-60, 1991.
Christy, J. "Braids." http://www.mathsource.com/cgi-bin
/MathSource/Applications/Mathematics/0202-228.
Jones, V. F. R. "Hecke Algebra Representations of Braid
Groups and Link Polynomials." Ann. Math. 126, 335-
388, 1987.
^ Weisstein, E. W. "Knots and Links." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/Knots .m.
168
Braid Index
Branch Point
Braid Index
The least number of strings needed to make a closed
braid representation of a LINK. The braid index is equal
to the least number of Seifert Circles in any projec-
tion of a Knot (Yamada 1987). Also, for a nonsplit-
table Link with Crossing Number c(L) and braid in-
dex i{L) y
c(L) > 2[i(L) - 1]
(Ohyama 1993). Let E be the largest and e the small-
est Power of £ in the HOMFLY Polynomial of an
oriented LINK, and i be the braid index. Then the
Morton-Franks- Williams Inequality holds,
i>\{E-e) + l
(Franks and Williams 1987). The inequality is sharp for
all Prime Knots up to 10 crossings with the exceptions
of 09 42, 09 49, IO132, IO150, and 10i 5 6-
References
Franks, J. and Williams, R. F. "Braids and the Jones Poly-
nomial." Trans. Amer. Math. Soc. 303, 97-108, 1987.
Jones, V. F. R. "Hecke Algebra Representations of Braid
Groups and Link Polynomials." Ann. Math. 126, 335-
388, 1987.
Ohyama, Y. "On the Minimal Crossing Number and the Brad
Index of Links," Canad. J. Math. 45, 117-131, 1993.
Yamada, S. "The Minimal Number of Seifert Circles Equals
the Braid Index of a Link." Invent. Math. 89, 347-356,
1987.
Braid Word
Any n-braid is expressed as a braid word, e.g.,
o-i^osa^ <y\ is a braid word for the Braid Group S3.
By Alexander's Theorem, any LINK is representable
by a closed braid, but there is no general procedure for
reducing a braid word to its simplest form. However,
Markov's Theorem gives a procedure for identifying
different braid words which represent the same LINK.
Let 6+ be the sum of Positive exponents, and 6_ the
sum of Negative exponents in the Braid Group B n .
If
b + - 36_ - n+ 1 > 0,
then the closed braid b is not AMPHICHIRAL (Jones
1985).
see also Braid GROUP
References
Jones, V. F. R. "A Polynomial Invariant for Knots via von
Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111,
1985.
Jones, V. F. R. "Hecke Algebra Representations of Braid
Groups and Link Polynomials." Ann. Math. 126, 335-
388, 1987.
Braikenridge-Maclaurin Construction
The converse of PASCAL'S THEOREM. Let Ai, B 2i Ci,
A 2 , and Br be the five points on a Conic. Then the
Conic is the Locus of the point
C 2 =Ax{z> dA 2 ) ■ B x (z • C1B2),
where z is a line through the point AiB 2 • B\A 2 .
see also PASCAL'S THEOREM
Branch
The segments of a TREE between the points of connec-
tion (Forks).
see also FORK, LEAF (TREE)
Branch Cut
|Sqrt z|
A line in the COMPLEX PLANE across which a FUNCTION
is discontinuous.
function
branch cut(s)
cos -1 z
(— 00, — 1) and (l,oo)
cosh -1
(-oo,l)
cot -1 z
(-i,i)
coth" 1
[-1,1]
esc -1 z
(-1,1)
csch -1
(-m)
In z
(-oo,0]
sec" 1 z
(-1,1)
sech -1
(oo,0] and (1, 00)
sin - z
(— 00,— 1) and (l,oo)
sinh -1
(—200, —i) and (2,200)
v/i
(-oo,0)
tan x z
(-ioo, -i) and (2,200)
tanh -1
( — 00, —1] and [1, 00)
z n ,n<£Z
(-oo,0) for R[n] < 0; (-
-oo,0] for R[n] >
see also Branch Point
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 399-401, 1953.
Branch Line
see Branch Cut
Branch Point
An argument at which identical points in the COMPLEX
PLANE are mapped to different points. For example,
consider
Brauer Chain
Breeder
169
Then f(e oi ) = /(l) = 1, but f(e 27ri ) = e 2 ™, despite
the fact that e i0 = e 2ni . Pinch Points are also called
branch points.
see also BRANCH CUT, PlNCH POINT
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 397-399, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 391-392 and 399-
401, 1953.
Brauer Chain
A Brauer chain is an ADDITION CHAIN in which each
member uses the previous member as a summand. A
number n for which a shortest chain exists which is a
Brauer chain is called a BRAUER NUMBER.
see also Addition Chain, Brauer Number, Hansen
Chain
References
Guy, R. K. "Addition Chains, Brauer Chains. Hansen
Chains." §C6 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994.
Brauer Group
The GROUP of classes of finite dimensional central sim-
ple Algebras over k with respect to a certain equiva-
lence.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, p. 479, 1988.
Brauer Number
A number n for which a shortest chain exists which is
a BRAUER Chain is called a Brauer number. There are
infinitely many non-Brauer numbers.
see also Brauer Chain, Hansen Number
References
Guy, R. K. "Addition Chains. Brauer Chains. Hansen
Chains." §C6 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994.
Brauer- Severi Variety
An Algebraic Variety over a Field K that becomes
Isomorphic to a Projective Space.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, pp. 480-481, 1988.
Brauer's Theorem
If, in the Gersgorin Circle Theorem for a given m,
for all j f^ m, then exactly one EIGENVALUE of A lies in
the Disk F m .
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1121, 1979.
Braun's Conjecture
Let B = {&i,& 2 ,...} be an Infinite Abelian Semi-
group with linear order &i < & 2 < . . . such that &i is the
unit element and a < b Implies ac < be for a,b,c 6 B.
Define a Mobius Function jj, on B by /x(6i) = 1 and
Yl ^ = °
b d \b n
for n = 2, 3, Further suppose that /x(6 n ) = M n )
(the true MOBIUS FUNCTION) for all n > 1. Then
Braun's conjecture states that
for all m,n> 1.
see also MOBIUS PROBLEM
References
Flath, A. and Zulauf, A. "Does the Mobius Function Deter-
mine Multiplicative Arithmetic?" Amer. Math. Monthly
102, 354-256, 1995.
Breeder
A pair of POSITIVE INTEGERS (ai,a 2 ) such that the
equations
a\ 4- a 2 x = cr(a\) — a(a 2 )(x 4- 1)
have a POSITIVE INTEGER solution x, where a(n) is the
DIVISOR FUNCTION. If x is Prime, then (ai,a 2 x) is an
Amicable Pair (te Riele 1986). (ai,a 2 ) is a "special"
breeder if
a± = au
a 2 = a,
where a and u are Relatively Prime, (a, u) — 1. If
regular amicable pairs of type (i,l) with i > 2 are of
the form (au,ap) with p PRIME, then (au,a) are special
breeders (te Riele 1986).
References
te Riele, H. J. J. "Computation of All the Amicable Pairs
Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35,
1986.
170 Brelaz's Heuristic Algorithm
Bretschneider's Formula
Brelaz's Heuristic Algorithm
An Algorithm which can be used to find a good, but
not necessarily minimal, EDGE or VERTEX coloring for
a Graph.
see also Chromatic Number
Brent's Factorization Method
A modification of the POLLARD p FACTORIZATION
Method which uses
Xi+i = Xi — c (mod n).
References
Brent, R. "An Improved Monte Carlo Factorization Algo-
rithm." Nordisk Tidskrift for Informationsbehandlung
(BIT) 20, 176-184, 1980.
Brent's Method
A RoOT-finding ALGORITHM which combines root
bracketing, bisection, and Inverse Quadratic In-
terpolation. It is sometimes known as the VAN
Wijngaarden-Deker-Brent Method.
Brent's method uses a LAGRANGE INTERPOLATING
Polynomial of degree 2. Brent (1973) claims that this
method will always converge as long as the values of the
function are computable within a given region contain-
ing a ROOT. Given three points asi, x 2 , and £3, Brent's
method fits x as a quadratic function of y, then uses the
interpolation formula
[y-f(*i)][y-f{ x 2)] x 3
[/(**)
+
f(x 1 )][f(x 3 )-f(x 2 )}
[y- /Qg2)][y- f(x s )]xi
[f(x 1 )-f(x 2 )][f(x 1 )-f(x 3 )]
[y- f(x3)][y- f(xi)]x 2
+
[f( X 2)-f(x 3 )][f(x 2 )-f(x 1 )Y
(1)
Subsequent root estimates are obtained by setting y = 0,
giving
, P
(2)
where
P = S[R(R - T)(x 3 - x 2 ) - (1 - R)(x 2 - zi)] (3)
Q = (T-1)(R-1)(S-1) (4)
with
R =
f(X2)
/(*s)
a - /(*»)
" f(xi)
T _/(*l)
(5)
(6)
(7)
References
Brent, R. P. Ch. 3-4 in Algorithms for Minimization Without
Derivatives. Englewood Cliffs, NJ: Prentice- Hall, 1973.
Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. §7.2 in
Computer Methods for Mathematical Computations. En-
glewood Cliffs, NJ: Prentice-Hall, 1977.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Van Wijngaarden-Dekker-Brent Method."
§9.3 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 352—355, 1992.
Brent- Salamin Formula
A formula which uses the Arithmetic-Geometric
MEAN to compute Pi. It has quadratic convergence
and is also called the Gauss-Salamin Formula and
Salamin Formula. Let
CLn + l = 2 {,Q>n + O n )
(i)
&n+l = ydnbn
(2)
C n +i = 2 ( a n — b n )
(3)
A — 2 h 2
0, n = €L n On y
(4)
and define the initial conditions to be ao = 1, &o =
l/\/2- Then iterating a„ and 6„ gives the ARITHMETIC-
GEOMETRIC MEAN, and it is given by
4[M(1,2- 1 / 2 )] 2
4[M(l,2- 1 / 2 )] 2
l-£~i2 i+ V
(5)
(6)
King (1924) showed that this formula and the LEGEN-
DRE RELATION are equivalent and that either may be
derived from the other.
see also Arithmetic-Geometric Mean, Pi
References
Borwein, J. M. and Borwein, P. B, Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 48-51, 1987.
Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag.
61, 148-163, 1988.
King, L. V. On the Direct Numerical Calculation of Elliptic
Functions and Integrals. Cambridge, England: Cambridge
University Press, 1924.
Lord, N. J. "Recent Calculations of n: The Gauss-Salamin
Algorithm." Math. Gaz. 76, 231-242, 1992.
Salamin, E. "Computation of n Using Arithmetic-Geometric
Mean." Math. Comput. 30, 565-570, 1976.
Bretschneider's Formula
Given a general QUADRILATERAL with sides of lengths
a, 6, c, and d (Beyer 1987), the Area is given by
(Press et al. 1992).
^quadrilateral = \ ^4p 2 q 2 - (b 2 + d 2 - d 2 - C 2 ) 2 ,
where p and q are the diagonal lengths.
see also BRAHMAGUPTA'S FORMULA, HERON'S FOR-
MULA
References
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.
Brianchon Point
Bridge (Graph) 171
Brianchon Point
The point of CONCURRENCE of the joins of the VER-
TICES of a Triangle and the points of contact of a
Conic Section Inscribed in the Triangle. A Conic
Inscribed in a Triangle has an equation of the form
the chance that one of four players will receive a hand
of a single suit is
39,688,347,497
/ 9 h
- + - + -
U V w
o,
an it.s HrianrTinn -nrnnt Viae Trttttmrar nnrmnTM atpq
There are special names for specific types of hands. A
ten, jack, queen, king, or ace is called an "honor." Get-
suits and the ace, king, and queen, and jack of the re-
maining suit is called 13 top honors. Getting all cards of
the same suit is called a 13-card suit. Getting 12 cards
of same suit with ace high and the 13th card not an
ace is called 2-card suit, ace high. Getting no honors is
called a Yarborough.
The probabilities of being dealt 13-card bridge hands
of a given type are given below. As usual, for a hand
with probability P, the Odds against being dealt it are
(1/P) -1:1.
(1//, l/g,l/h). For Kiepert's Parabola, the Bran-
chion point has TRIANGLE CENTER FUNCTION
a(b 2 -
which is the Steiner Point.
2).
Brianchon's Theorem
The Dual of Pascal's Theorem. It states that, given
a 6-sided Polygon Circumscribed on a Conic SEC-
TION, the lines joining opposite VERTICES (DIAGONALS)
meet in a single point.
see also DUALITY PRINCIPLE, PASCAL'S THEOREM
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
p. 110, 1990.
Brick
see Euler Brick, Harmonic Brick, Rectangular
Parallelepiped
Bride's Chair
One name for the figure used by Euclid to prove the
Pythagorean Theorem.
see also Peacock's Tail, Windmill
Bridge Card Game
Bridge is a CARD game played with a normal deck of 52
cards. The number of possible distinct 13-card hands is
N =
635,013,559,600.
where (£) is a Binomial Coefficient. While the
chances of being dealt a hand of 13 CARDS (out of 52)
of the same suit are
4 1
Hand
Exact
Probability
13 top honors
high
4
N
4
N
4-12-36
N
(S)
N
ill
mm
AT
i
158,753,389,900
1
12-card suit, ace
Yarborough
four aces
nine honors
158,753,389,900
4
1,469,938,795
5,394
9,860,459
11
4,165
888,212
93,384,347
Hand
Probability
Odds
13 top honors
13-card suit
12-card suit, ace high
Yarborough
four aces
nine honors
6.30
6,30
2.72
5.47
2.64
9.51
x 10~ 12
x 10" 12
x 10" 9
x 10" 4
x 10~ 3
x 10" 3
158,753,389,899:1
158,753,389,899:1
367,484,697.8:1
1,827.0:1
377.6:1
104.1:1
see also CARDS, POKER
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 48-49,
1987.
Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 119-121, 1942.
Bridge (Graph)
The bridges of a Graph are the Edges whose removal
disconnects the Graph.
see also Articulation Vertex
References
Chartrand, G. "Cut- Vertices and Bridges." §2.4 in Introduc-
tory Graph Theory. New York: Dover, pp. 45-49, 1985.
(«) 158,753,389,900'
172 Bridge Index
Bring Quintic Form
Bridge Index
A numerical KNOT invariant. For a TAME KNOT K, the
bridge index is the least BRIDGE NUMBER of all planar
representations of the Knot. The bridge index of the
Unknot is defined as 1.
see also Bridge Number, Crookedness
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 114, 1976.
Schubert, H. "Uber eine numerische Knotteninvariante."
Math. Z. 61, 245-288, 1954.
Bridge of Konigsberg
see Konigsberg Bridge Problem
Bridge Number
The least number of unknotted arcs lying above the
plane in any projection. The knot 05os has bridge num-
ber 2. Such knots are called 2-BRIDGE KNOTS. There is
a one-to-one correspondence between 2-Bridge KNOTS
and rational knots. The knot O8010 is a 3-bridge knot. A
knot with bridge number b is an n-EMBEDDABLE KNOT
where n < b.
see also BRIDGE INDEX
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 64-67, 1994.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 115, 1976.
Bridge Knot
An n-bridge knot is a knot with BRIDGE Number n.
The set of 2-bridge knots is identical to the set of rational
knots. If L is a 2-Bridge Knot, then the BLM/Ho
Polynomial Q and Jones Polynomial V satisfy
Q L (z) = 2z- 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ),
where z = — t — i" 1 (Kanenobu and Sumi 1993). Ka-
nenobu and Sumi also give a table containing the num-
ber of distinct 2-bridge knots of n crossings for n — 10
to 22, both not counting and counting MIRROR IMAGES
as distinct.
n
K n
K n + K n
3
4
5
6
7
8
9
10
45
85
11
91
182
12
176
341
13
352
704
14
693
1365
15
1387
2774
16
2752
5461
17
5504
11008
18
10965
21845
19
21931
43862
20
43776
87381
21
87552
175104
22
174933
349525
References
Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-
Bridge Knots through 22-Crossings." Math. Comput. 60,
771-778 and S17-S28, 1993.
Schubert, H. "Knotten mit zwei Briicken." Math. Z. 65,
133-170, 1956.
Brill-Noether Theorem
If the total group of the canonical series is divided into
two parts, the difference between the number of points
in each part and the double of the dimension of the
complete series to which it belongs is the same.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 263, 1959.
Bring-Jerrard Quintic Form
A TSCHIRNHAUSEN TRANSFORMATION can be used to
algebraically transform a general Quintic Equation
to the form
z + C\Z + Co == 0.
(1)
y + b 2 y 2 + hy + b -0
In practice, the general quintic is first reduced to the
Principal Quintic Form
(2)
before the transformation is done. Then, we require that
the sum of the third POWERS of the ROOTS vanishes,
so ss(yj) = 0. We assume that the ROOTS Zi of the
Bring-Jerrard quintic are related to the ROOTS yi of the
Principal Quintic Form by
Zi = ayi 4 + j3yi 3 + jyi 2 + 6yi + e. (3)
In a similar manner to the Principal Quintic Form
transformation, we can express the COEFFICIENTS Cj in
terms of the bj .
see also Bring Quintic Form, Principal Quintic
Form, Quintic Equation
Bring Quintic Form
A TSCHIRNHAUSEN Transformation can be used to
take a general Quintic Equation to the form
x — x — a :
0,
where a may be Complex.
see also Bring-Jerrard Quintic Form, Quintic
Equation
References
Ruppert, W. M. "On the Bring Normal Form of a Quintic in
Characteristic 5." Arch. Math. 58, 44-46, 1992.
Brioschi Formula
Brocard Angle 173
Brioschi Formula
For a curve with METRIC
Brocard Angle
ds 2 = E du + F dudv + G dv 2 ,
(1)
where E, F, and G is the first FUNDAMENTAL FORM,
the Gaussian Curvature is
Mi + M 2 /0 v
where
Mi =
M 2
-F 2
r
2 U1; ~t" -^tit; 2 uu
2 Eu
F u
- ^E v
i*V — 2^1*
E
F
2^«
F
G
(3)
2 ^v 2 ""
§£ v £ F
i
(4)
\G U F G
which can also be written
K =
d_ (j_dVG\ d_ ( i d^E\
r EG [du \^E du J dv \^Q dv J _
d ( G u \ 3 ( E v \
du \y/EGj dv \<JEGJ
2VEG
(5)
(6)
see also Fundamental Forms, Gaussian Curvature
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 392-393, 1993.
Briot-Bouquet Equation
An Ordinary Differential Equation of the form
where m is a Positive Integer, / is Analytic at x ~
y = 0, /(0,0) = 0, and /i(0, 0)^0.
References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math-
ematics: An Updated and Annotated Translation of the
Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether-
lands: Reidel, pp. 481-482, 1988.
A C
Define the first Brocard Point as the interior point Q
of a Triangle for which the Angles ICIAB, IQBC,
and iVtCA are equal. Similarly, define the second BRO-
CARD Point as the interior point Cl' for which the AN-
GLES IQ'AC, /-0,'CB, and IQ'BA are equal. Then the
Angles in both cases are equal, and this angle is called
the Brocard angle, denoted u).
The Brocard angle u> of a Triangle AABC is given by
the formulas
cot u) = cot A 4- cot B + cot C
1 + cos ai cos ct2 cos az
sin ct\ sin 0:2 sin otz
_ sin 2 ai + sin 2 c*2 + sin 2 0:3
2sinai sina2 sin 0:3
_ ai sin ai + 02 sin 0:2 + az sin a<3
a± cos a± + a2 cos 0:2 + &z cos a3
2
; a2
2A
2 2,2,2
csc w = csc a± + csc a.2 + esc otz
-s/ai 2 a2 2 + a2 2 a 3 2 + a 3 2 ai 2
(i)
(2)
(3)
(4)
(5)
(6)
(7)
where A is the Triangle Area, A, B, and C are An-
gles, and a, b, and c are side lengths.
If an Angle a of a Triangle is given, the maximum
possible Brocard angle is given by
coto; = § tan(ia) + 5COs(|a). (8)
Let a Triangle have Angles A, B, and C. Then
sin A sin B sin C < kABC,
(9)
where
k=[^) (10)
(Le Lionnais 1983). This can be used to prove that
8a; 3 < ABC (11)
(Abi-Khuzam 1974).
174
Brocard Axis
Brocard Line
see also BROCARD CIRCLE, BROCARD LINE, EQUI-
Brocard Center, Fermat Point, Isogonic Cen-
ters
References
Abi-Khuzam, F. "Proof of YfFs Conjecture on the Brocard
Angle of a Triangle." Elem. Math. 29, 141-142, 1974.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 263-286 and 289-294, 1929.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 28, 1983.
Brocard Axis
The Line KO passing through the Lemoine Point K
and Circumcenter O of a Triangle. The distance
OK is called the Brocard Diameter. The Brocard
axis is Perpendicular to the Lemoine Axis and is
the Isogonal Conjugate of Kiepert's Hyperbola.
It has equations
sin(£ - C)a + sin(C - A)f3 + sin(A - B)j =
bc(b 2 - c 2 )a + ca(c 2 - a 2 )p + ab(a 2 - 6 2 ) 7 = 0.
The Lemoine Point, Circumcenter, Isodynamic
Points, and BROCARD Midpoint all lie along the Bro-
card axis. Note that the Brocard axis is not equivalent
to the Brocard Line.
see also Brocard Circle, Brocard Diameter, Bro-
card Line
Brocard Circle
The CIRCLE passing through the first and second Bro-
card Points ft and ft', the Lemoine Point K, and
the Circumcenter O of a given Triangle. The Bro-
card Points ft and ft' are symmetrical about the Line
KO'
which is called the Brocard Line. The Line
Segment KO is called the Brocard Diameter, and
it has length
OK :
on
COS UJ
R^Jl -4sin 2 cj
cos a;
where R is the ClRCUMRADlUS and u> is the BROCARD
Angle. The distance between either of the Brocard
Points and the Lemoine Point is
OK = TVK = Tld tan a;.
see also Brocard Angle, Brocard Diameter, Bro-
card Points
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 272, 1929.
Brocard's Conjecture
7r(pn+i 2 ) -7r(Pn 2 ) > 4
for n > 2 where tt is the Prime Counting Function.
see also ANDRICA'S CONJECTURE
Brocard Diameter
The Line Segment KO joining the Lemoine Point K
and Circumcenter O of a given Triangle. It is the
Diameter of the Triangle's Brocard Circle, and
lies along the BROCARD Axis. The Brocard diameter
has length
— — On R\/l - 4 sin 2 w
OK = — ,
COS UJ COS U)
where ft is the first Brocard Point, R is the Circum-
RADIUS, and w is the Brocard Angle.
see also Brocard Axis, Brocard Circle, Brocard
Line, Brocard Points
Brocard Line
^3 "3
A Line from any of the Vertices Ai of a Triangle
to the first ft or second ft' BROCARD POINT, Let the
Angle at a Vertex A» also be denoted A i} and denote
the intersections of A±Q and Aifl' with A2A3 as Wi and
W2. Then the ANGLES involving these points are
LA&Wz^Ax
(1)
IW Z QA 2 = A 3 (2)
LA 2 £IW 1 =A 2 . (3)
Distances involving the points Wi and W[ are given by
a 3
,4 2 ft
sin A2
(4)
Brocard Midpoint
Brocard Points 175
A 2 Q
A 3 n
_ a 3 2 _ sin(^4 3 - ll>)
aia2 sin a;
W 3 Ai _
W3A2
a2 sin u) (0,2
ai sin(A3 — uj) \a%
(5)
(6)
where uj is the Brocard Angle (Johnson 1929,
pp. 267-268).
The Brocard line, MEDIAN M, and LEMOINE POINT K
are concurrent, with A1Q1, A2K ', and A3M meeting at
a point P. Similarly, AiQ' , A2M, and A3K meet at
a point which is the ISOGONAL CONJUGATE point of P
(Johnson 1929, pp. 268-269).
see also Brocard Axis, Brocard Diameter, Bro-
card Points, Isogonal Conjugate, Lemoine
Point, Median (Triangle)
References
Johnson, It. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 263-286, 1929.
Brocard Midpoint
The Midpoint of the Brocard Points. It has Tri-
angle Center Function
a = a(b + c ) — sin(^4 -f- a;),
where uj is the Brocard Angle. It lies on the Bro-
card Axis.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Brocard Points
A C
The first Brocard point is the interior point H (or n
or Zx) of a Triangle for which the Angles IQAB,
ZfiBC, and IQCA are equal. The second Brocard point
is the interior point fi' (or T2 or Z2) for which the An-
gles IQ'AC, IQ'CB, and Itt'BA are equal. The AN-
GLES in both cases are equal to the BROCARD ANGLE
uj = IttAB = IttBC = mCA
= in' ac = m'CB - iq'ba.
The first two Brocard points are ISOGONAL Conju-
gates (Johnson 1929, p. 266).
Let Cbc be the CIRCLE which passes through the ver-
tices B and C and is TANGENT to the line AC at C, and
similarly for Cab and Cbc- Then the CIRCLES Cab,
Cbc, and Cac intersect in the first Brocard point Q.
Similarly, let C' BC be the CIRCLE which passes through
the vertices B and C and is TANGENT to the line AB at
B, and similarly for C' AB and C' AC . Then the CIRCLES
C A b j C'bC) anc * Cac intersect in the second Brocard
points £V (Johnson 1929, pp. 264-265).
a c a c
The Pedal Triangles of Q and 0! are congruent,
and Similar to the Triangle AABC (Johnson 1929,
p. 269). Lengths involving the Brocard points include
OQ = OW = R\/l-4sm 2 uj
nO' = 2Rs\xiu\/l -4sin 2 u>.
(i)
(2)
Brocard's third point is related to a given TRIANGLE by
the Triangle Center Function
(3)
(Casey 1893, Kimberling 1994). The third Brocard
point Q" (or r 3 or Z z ) is COLLINEAR with the SPIEKER
Center and the Isotomic Conjugate Point of its
Triangle's Incenter.
see also Brocard Angle, Brocard Midpoint, Equi-
Brocard Center, Yff Points
References
Casey, J. A Treatise on the Analytical Geometry of the Point,
Line, Circle, and Conic Sections, Containing an Account
of Its Most Recent Extensions, with Numerous Examples,
2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66,
1893.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 263-286, 1929.
Kimberling, C "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Stroeker, R. J. "Brocard Points, Circulant Matrices, and
Descartes' Folium." Math. Mag. 61, 172-187, 1988.
176
Brocard's Problem
Brown Function
Brocard's Problem
Find the values of n for which n! + 1 is a SQUARE NUM-
BER m 2 , where n! is the FACTORIAL (Brocard 1876,
1885). The only known solutions are n = 4, 5, and
7, and there are no other solutions < 1027. The pairs of
numbers (m,n) are called Brown NUMBERS.
see also BROWN NUMBERS, FACTORIAL, SQUARE NUM-
BER
References
Brocard, H. Question 166. Nouv. Corres. Math. 2, 287,
1876.
Brocard, H. Question 1532. Nouv. Ann. Math. 4, 391, 1885.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 193, 1994.
Brocard Triangles
Let the point of intersection of A 2 ^l and Azfl' be Bi,
where Q and fl f are the Brocard Points, and similarly
define B2 and £3. B1B2BZ is the first Brocard trian-
gle, and is inversely similar to A1A2A3. It is inscribed
in the BROCARD CIRCLE drawn with OK as the DIAM-
ETER. The triangles B1A2A3, £ 2 A 3 Ai, and B3A1A2
are ISOSCELES TRIANGLES with base angles lj, where u;
is the Brocard Angle. The sum of the areas of the
Isosceles Triangles is A, the Area of Triangle
A1A2A3. The first Brocard triangle is in perspective
with the given TRIANGLE, with AtB^ A 2 B 2 , and A3B3
Concurrent. The Median Point of the first Brocard
triangle is the MEDIAN POINT M of the original triangle.
The Brocard triangles are in perspective at M.
Let ci, c 2 , and c 3 and ci, c 2 , and c 3 be the CIRCLES
intersecting in the Brocard Points Q and Q' , respec-
tively. Let the two circles c\ and c[ tangent at A\ to
A1A2 and A\A$, and passing respectively through As
and A 2 , meet again at C\. The triangle C1C2C3 is the
second Brocard triangle. Each Vertex of the second
Brocard triangle lies on the second Brocard Circle.
The two Brocard triangles arc in perspective at M.
see also Steiner Points, Tarry Point
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 277-281, 1929.
Bromwich Integral
The inverse of the Laplace Transform, given by
2iri I
J -y — to
'7—100
where 7 is a vertical Contour in the Complex Plane
chosen so that all singularities of f(s) are to the left of
it.
References
Arfken, G. "Inverse Laplace Transformation." §15.12 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 853-861, 1985.
Brothers
A Pair of consecutive numbers.
see also Pair, Smith Brothers, Twins
Brouwer Fixed Point Theorem
Any continuous FUNCTION G : D n -> D n has a FIXED
Point, where
£> n = {x€M n :xi 2 + ... + a;„ 2 <1}
is the unit n-BALL.
see also FIXED POINT THEOREM
References
Milnor, J. W. Topology from the Differentiate Viewpoint.
Princeton, NJ: Princeton University Press, p. 14, 1965.
Browkin's Theorem
For every Positive Integer n, there exists a Square
in the plane with exactly n Lattice Points in its inte-
rior. This was extended by Schinzel and Kulikowski to
all plane figures of a given shape. The generalization of
the Square in 2-D to the Cube in 3-D was also proved
by Browkin.
see also Cube, Schinzel's Theorem, Square
References
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 121-125, 1973.
Brown's Criterion
A Sequence {^} of nondecreasing Positive Integers
is Complete Iff
1. 1/1 = 1.
2. For all k = 2, 3, . . . ,
S k -1 = v\ + ^2 + . . • + ffc-l > Vk - 1.
A corollary states that a Sequence for which v\ = 1
and v>k+i < 2vk is COMPLETE (Honsberger 1985).
see also COMPLETE SEQUENCE
References
Brown, J. L. Jr. "Notes on Complete Sequences of Integers."
Amer. Math. Monthly, 557-560, 1961.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 123-130, 1985.
Brown Function
For a Fractal Process with values y(t — At) and y(t+
At) j the correlation between these two values is given by
the Brown function
1,
also known as the Bachelier Function, Levy Func-
tion, or Wiener Function.
Brown Numbers
Brun's Constant
177
Brown Numbers
Brown numbers are Pairs (m, n) of Integers satisfying
the condition of Brocard's Problem, i.e., such that
n! + 1 = m
where n! is the FACTORIAL and m 2 is a SQUARE Num-
ber. Only three such Pairs of numbers are known:
(5,4), (11,5), (71,7), and Erdos conjectured that these
are the only three such Pairs. Le Lionnais (1983) points
out that there are 3 numbers less than 200,000 for which
(n-l)! + l = (mod n 2 ) ,
namely 5, 13, and 563.
see also Brocard's Problem, Factorial, Square
Number
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 193, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 56, 1983.
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 170, 1995.
Broyden's Method
An extension of the secant method of root finding to
higher dimensions.
References
Broyden, C. G. "A Class of Methods for Solving Nonlinear
Simultaneous Equations." Math. Comput. 19, 577-593,
1965.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 382-385, 1992.
Bruck-Ryser-Chowla Theorem
If n = 1, 2 (mod 4), and the SQUAREFREE part of n is di-
visible by a Prime p = 3 (mod 4), then no Difference
Set of ORDER n exists. Equivalently, if a PROJECTIVE
PLANE of order n exists, and n — 1 or 2 (mod 4), then
n is the sum of two SQUARES.
Dinitz and Stinson (1992) give the theorem in the fol-
lowing form. If a symmetric (v, k, A)-BLOCK DESIGN
exists, then
1. If v is Even, then k - A is a Square Number,
2. If v is Odd, the the Diophantine Equation
x 2 ^(k-\)y 2 + (-l)
(f-l)/2
\z z
has a solution in integers, not all of which are 0.
see also Block Design, Fisher's Block Design In-
equality
References
Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to
Design Theory." Ch. 1 in Contemporary Design Theory: A
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson).
New York: Wiley, pp. 1-12, 1992.
Gordon, D. M. "The Prime Power Conjecture is True
for n < 2,000,000." Electronic J. Combinatorics 1,
R6, 1-7, 1994. http://www.combinatorics.org/Volume_l/
volume 1 ,html#R6.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., 1963.
Bruck-Ryser Theorem
see BRUCK-RYSER-CHOWLA Theorem
Brun's Constant
The number obtained by adding the reciprocals of the
Twin Primes,
(1)
By Brun's Theorem, the constant converges to a def-
inite number as p — > oo. Any finite sum underesti-
mates B. Shanks and Wrench (1974) used all the Twin
PRIMES among the first 2 million numbers. Brent (1976)
calculated all Twin Primes up to 100 billion and ob-
tained (Ribenboim 1989, p. 146)
B « 1.90216054,
(2)
assuming the truth of the first HARDY-LlTTLEWOOD
Conjecture. Using Twin Primes up to 10 14 , Nicely
(1996) obtained
B^ 1.9021605778 ±2.1 x 10
-9
(3)
(Cipra 1995, 1996), in the process discovering a bug in
Intel's® Pentium™ microprocessor. The value given by
Le Lionnais (1983) is incorrect.
see also Twin Primes, Twin Prime Conjecture,
Twin Primes Constant
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 64, 1987.
Brent, R. P. "Tables Concerning Irregularities in the Distri-
bution of Primes and Twin Primes Up to 10 11 ." Math.
Comput 30, 379, 1976.
Cipra, B. "How Number Theory Got the Best of the Pentium
Chip." Science 267, 175, 1995.
Cipra, B. "Divide and Conquer." What's Happening in the
Mathematical Sciences, 1995-1996, Vol 3. Providence,
RI: Amer. Math. Soc, pp. 38-47, 1996.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/brun/brun.html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 41, 1983.
Nicely, T. "Enumeration to 10 14 of the Twin Primes and
Brun's Constant." Virginia J. Sci. 46, 195-204, 1996.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, 1989.
Shanks, D. and Wrench, J. W. "Brun's Constant." Math.
Comput 28, 293-299, 1974.
Wolf, M. "Generalized Brun's Constants." http://www.ift.
uni.wroc.pl/-mwolf/.
178 Brunn-Minkowski Inequality
Buffon's Needle Problem
Brunn-Minkowski Inequality
The nth root of the Content of the set sum of two sets
in Euclidean n-space is greater than or equal to the sum
of the nth roots of the Contents of the individual sets.
see also TOMOGRAPHY
References
Cover, T. M. "The Entropy Power Inequality and the Brunn-
Minkowski Inequality" §5.10 in In Open Problems in Com-
munications and Computation. (Ed. T. M. Cover and
B. Gopinath). New York: Springer- Verlag, p. 172, 1987.
Schneider, R. Convex Bodies: The Brunn-Minkowski The-
ory. Cambridge, England: Cambridge University Press,
1993.
Brun's Sum
see Brun's Constant
Brun's Theorem
The series producing Brun's Constant Converges
even if there are an infinite number of TWIN PRIMES.
Proved in 1919 by V. Brun.
Brunnian Link
A Brunnian link is a set of n linked loops such that
each proper sublink is trivial, so that the removal of any
component leaves a set of trivial unlinked Unknots.
The Borromean Rings are the simplest example and
have n = 3.
see also Borromean Rings
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, 1976.
Brute Force Factorization
see Direct Search Factorization
Bubble
A bubble is a MINIMAL SURFACE of the type that is
formed by soap film. The simplest bubble is a single
Sphere. More complicated forms occur when multi-
ple bubbles are joined together. Two outstanding prob-
lems involving bubbles are to find the arrangements with
the smallest Perimeter (planar problem) or Surface
Area (Area problem) which enclose and separate n
given unit areas or volumes in the plane or in space.
For n — 2, the problems are called the DOUBLE BUB-
BLE CONJECTURE and the solution to both problems is
known to be the DOUBLE Bubble.
see also Double Bubble, Minimal
Plateau's Laws, Plateau's Problem
Surface,
References
Morgan, F. "Mathematicians, Including Undergraduates,
Look at Soap Bubbles." Amer. Math. Monthly 101, 343-
351, 1994.
Pappas, T. "Mathematics & Soap Bubbles." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
p. 219, 1989.
Buchberger's Algorithm
The algorithm for the construction of a GROBNER BASIS
from an arbitrary ideal basis.
see also GROBNER BASIS
References
Becker, T. and Weispfenning, V. Grobner Bases: A Com-
putational Approach to Commutative Algebra. New York:
Springer- Verlag, pp. 213-214, 1993.
Buchberger, B. "Theoretical Basis for the Reduction of Poly-
nomials to Canonical Forms." SIGSAM Bull 39, 19-24,
Aug. 1976.
Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and
Algorithms: An Introduction to Algebraic Geometry and
Commutative Algebra, 2nd ed. New York: Springer-
Verlag, 1996.
Buckminster Fuller Dome
see Geodesic Dome
Buffon-Laplace Needle Problem
-4 1 X
t v
/ v
£~
/
, ^
r ,%
^ i
^ +
b
h
Find the probability P(£, a, b) that a needle of length £
will land on a line, given a floor with a grid of equally
spaced Parallel Lines distances a and b apart, with
£ > a,b.
2£(a + b)-P
P(*,a,6) =
-nab
see also BUFFON'S NEEDLE PROBLEM
BufFon's Needle Problem
/
^
/
/
Bulirsch-Stoer Algorithm
Burau Representation 179
Find the probability P(£>d) that a needle of length £
will land on a line, given a floor with equally spaced
Parallel Lines a distance d apart.
P&d)
-f
Jo
£\cosO\ dd _ t
= -[8in*] '
27r 2nd
- *L
ird
/.tt/2
7 '
Jo
cos 8 dO
Several attempts have been made to experimentally de-
termine 7r by needle- tossing. For a discussion of the
relevant statistics and a critical analysis of one of the
more accurate (and least believable) needle-tossings, see
Badger (1994).
see also Buffon-Laplace Needle Problem
References
Badger, L. "Lazzarini's Lucky Approximation of 7r." Math.
Mag. 67, 83-91, 1994.
Dorrie, H. "Buffon's Needle Problem." §18 in 100 Great
Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, pp. 73-77, 1965.
Kraitchik, M. "The Needle Problem." §6.14 in Mathematical
Recreations. New York: W. W. Norton, p. 132, 1942.
Wegert, E. and Trefethen, L, N. "Prom the Buffon Needle
Problem to the Kreiss Matrix Theorem." Amer. Math.
Monthly 101, 132-139, 1994.
Bulirsch-Stoer Algorithm
An algorithm which finds RATIONAL FUNCTION extrap-
olations of the form
Ri(i + l)---(i+m)
Py(x) __ po + p\x + . . . +p^x M
P„(x) qo + qix + . . . + q u x v
and can be used in the solution of Ordinary Differ-
ential Equations.
References
Bulirsch, R. and Stoer, J. §2.2 in Introduction to Numerical
Analysis. New York: Springer- Verlag, 1991.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Richardson Extrapolation and the Bulirsch-
Stoer Method." §16.4 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 718-725, 1992.
Bullet Nose
A plane curve with implicit equation
x 1 y 2
(1)
The Curvature is
x = a cost
y = b cot t.
Sab cot t esc t
(6 2 csc 4 i + a 2 sin 2 i) 3 / 2
and the TANGENTIAL ANGLE is
■ = tan
_i /bcsc 3 A
(2)
(3)
(4)
(5)
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 127-129, 1972.
Bumping Algorithm
Given a Permutation {pi,f>2, ■ ■ • ,Vn) of {1, . . . , n},
the bumping algorithm constructs a standard YOUNG
Tableau by inserting the pi one by one into an already
constructed YOUNG TABLEAU. To apply the bump-
ing algorithm, start with {{pi}}, which is a YOUNG
TABLEAU. If p\ through pk have already been inserted,
then in order to insert pfc+i, start with the first line of
the already constructed YOUNG TABLEAU and search
for the first element of this line which is greater than
Pk+i- If there is no such element, append Pk+\ to the
first line and stop. If there is such an element (say, p p ),
exchange p p for pjt+i, search the second line using p p ,
and so on.
see also YOUNG TABLEAU
References
Skiena, S. Implementing Discrete Mathematics: Combina-
torics and Graph Theory with Mathematica. Reading,
MA: Addison- Wesley, 1990.
Bundle
see Fiber Bundle
Burau Representation
Gives a Matrix representation b* of a Braid Group
in terms of (n - 1) x (n - 1) Matrices. A -t always
appears in the (i,i) position.
bi =
-too
-1 1
1
(1)
bi =
In parametric form,
1
••
•
•
■ ■ -t
•
•
•• -t
o .
•
• ■ -1
1 ■
•
'.
••
•
• 1
(2)
180 Burkhardt Quartic
Burnside Problem
b n _
rl
•
•
1 ■
•
1 —
■
•
—
Lo
•
•
—
(3)
Let * be the Matrix Product of Braid Words, then
det(l - 9) _ , ,
l + t + . .. + t»-i " AL ' (4)
where A L is the ALEXANDER POLYNOMIAL and det is
the Determinant.
References
Burau, W. "Uber Zopfgruppen und gleichsinnig verdrilte Ver-
kettungen." Abh. Math. Sem. Hanischen Univ. 11, 171-
178, 1936.
Jones, V. "Hecke Algebra Representation of Braid Groups
and Link Polynomials." Ann. Math. 126, 335-388, 1987.
Burkhardt Quartic
The Variety which is an invariant of degree four and
is given by the equation
yt
2/0(2/?
3 , 3
■ 2/2 + yz ■
■2/1) + 32/12/22/32/4 = 0.
References
Burkhardt, H. "Untersuchungen aus dem Gebiet der hyperel-
liptischen Modulfunctionen. II." Math. Ann. 38, 161-224,
1890.
Burkhardt, H. "Untersuchungen aus dem Gebiet der hyper-
elliptischen Modulfunctionen. III." Math. Ann. 40, 313-
343, 1892.
Hunt, B. "The Burkhardt Quartic." Ch. 5 in The Geom-
etry of Some Special Arithmetic Quotients. New York:
Springer- Verlag, pp. 168-221, 1996.
Burnside's Conjecture
Every non-ABELIAN SIMPLE GROUP has EVEN ORDER.
see also Abelian Group, Simple Group
Burnside's Lemma
Let J be a Finite Group and the image R(J) be a
representation which is a HOMEOMORPHISM of J into a
Permutation Group S(X), where S(X) is the Group
of all permutations of a Set X. Define the orbits o£R(J)
as the equivalence classes under x ~ y, which is true if
there is some permutation p in R( J) such that p(x) = y.
Define the fixed points of p as the elements x of X for
which p(x) = x. Then the AVERAGE number of FIXED
POINTS of permutations in R(J) is equal to the number
of orbits of R(J).
The LEMMA was apparently known by Cauchy (1845) in
obscure form and Frobenius (1887) prior to Burnside's
(1900) rediscovery. It was subsequently extended and
refined by Polya (1937) for applications in COMBINATO-
RIAL counting problems. In this form, it is known as
Polya Enumeration Theorem.
References
Polya, G. "Kombinatorische Anzahlbestimmungen fur Grup-
pen, Graphen, und chemische Verbindungen." Acta Math.
68, 145-254, 1937.
Burnside Problem
A problem originating with W. Burnside (1902), who
wrote, "A still undecided point in the theory of dis-
continuous groups is whether the Order of a Group
may be not finite, while the order of every operation
it contains is finite." This question would now be
phrased as "Can a finitely generated group be infinite
while every element in the group has finite order?"
(Vaughan-Lee 1990). This question was answered by
Golod (1964) when he constructed finitely generated in-
finite p-GROUPS. These GROUPS, however, do not have
a finite exponent.
Let F r be the Free Group of Rank r and let N be
the Subgroup generated by the set of nth POWERS
{g n \g e F r }. Then TV is a normal subgroup of F r . We
define B(r, n) = F r /N to be the QUOTIENT GROUP. We
call B(r,n) the r-generator Burnside group of exponent
n. It is the largest r-generator group of exponent n, in
the sense that every other such group is a HOMEOMOR-
PHIC image of B(r, n). The Burnside problem is usually
stated as: "For which values of r and n is £(r,n) a
Finite Group?"
An answer is known for the following values. For r = 1,
5(1,77) is a Cyclic Group of Order n. For n = 2,
B(r, 2) is an elementary Abelian 2-group of Order 2 n ,
For n = 3, B(r, 3) was proved to be finite by Burnside.
The ORDER of the B(r,3) groups was established by
Levi and van der Waerden (1933), namely 3 a where
:r +
(1)
where (™) is a Binomial COEFFICIENT. For n = 4,
B(r> 4) was proved to be finite by Sanov (1940). Groups
of exponent four turn out to be the most complicated
for which a POSITIVE solution is known. The precise
nilpotency class and derived length are known, as are
bounds for the ORDER. For example,
|S(2,4)| = 2 12
|B(3,4)| = 2 69
|S(4,4)| = 2 422
|B(5,4)|=2 2728
(2)
(3)
(4)
(5)
while for larger values of r the exact value is not yet
known. For n = 6, B(r,6) was proved to be finite by
Hall (1958) with ORDER 2 a 3 6 , where
a = 1 + (r - 1)3 C
6 = l + (r-l)2 r
c = r +
+
(6)
(7)
(8)
No other Burnside groups are known to be finite. On
the other hand, for r > 2 and n > 665, with n ODD,
Busemann-Petty Problem
B(r,n) is infinite (Novikov and Adjan 1968). There is a
similar fact for r > 2 and n a large Power of 2.
E. Zelmanov was awarded a Fields Medal in 1994 for
his solution of the "restricted" Burnside problem.
see also FREE GROUP
References
Burnside, W. "On an Unsettled Question in the Theory of
Discontinuous Groups." Quart. J. Pure Appl. Math. 33,
230-238, 1902.
Golod, E. S. "On Nil-Algebras and Residually Finite p-
Groups." Isv. Akad. Nauk SSSR Ser. Mat. 28, 273-276,
1964.
Hall, M. "Solution of the Burnside Problem for Exponent
Six." Ill J. Math. 2, 764-786, 1958. „
Levi, F. and van der Waerden, B. L. "Uber eine besondere
Klasse von Gruppen." Abh. Math. Sem. Univ. Hamburg
9, 154-158, 1933.
Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I,
II, III." Izv. Akad. Nauk SSSR Ser. Mat 32, 212-244,
251-524, and 709-731, 1968.
Sanov, I. N. "Solution of Burnside's problem for exponent
four." Leningrad State Univ. Ann. Math. Ser. 10, 166—
170, 1940.
Vaughan-Lee, M. The Restricted Burnside Problem, 2nd ed.
New York: Clarendon Press, 1993.
Busemann-Petty Problem
If the section function of a centered convex body in Eu-
clidean n-space (n > 3) is smaller than that of another
such body, is its volume also smaller?
References
Gardner, R. J. "Geometric Tomography." Not. Amer. Math.
Soc. 42, 422-429, 1995.
Busy Beaver
A busy beaver is an n-state, 2-symbol, 5-tuple Turing
MACHINE which writes the maximum possible number
BB(n) of Is on an initially blank tape before halting.
For n = 0, 1, 2, ... , BB(n) is given by 0, 1, 4, 6, 13,
> 4098, > 136612, The busy beaver sequence is
also known as Rado's Sigma Function.
see also HALTING PROBLEM, TURING MACHINE
References
Chaitin, G. J. "Computing the Busy Beaver Function." §4.4
in Open Problems in Communication and Computation
(Ed. T. M. Cover and B. Gopinath). New York: Springer-
Verlag, pp. 108-112, 1987.
Dewdney, A. K. "A Computer Trap for the Busy Beaver,
the Hardest- Working Turing Machine." Sci. Amer. 251,
19-23, Aug. 1984.
Marxen, H. and Buntrock, J. "Attacking the Busy Beaver 5."
Bull. EATCS40, 247-251, Feb. 1990.
Sloane, N. J. A. Sequence A028444 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Butterfly Fractal 181
Butterfly Catastrophe
A Catastrophe which can occur for four control fac-
tors and one behavior axis. The equations
x = c(Sat 3 + 24t 5 )
y = c(-6ai 2 - 15t 4 )
display such a catastrophe (von Seggern 1993).
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 94, 1993.
Butterfly Curve
A Plane Curve given by the implicit equation
y =(x -x ).
see also DUMBBELL CURVE, EIGHT CURVE, PIRIFORM
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Butterfly Effect
Due to nonlinearities in weather processes, a butterfly
flapping its wings in Tahiti can, in theory, produce a
tornado in Kansas. This strong dependence of outcomes
on very slightly differing initial conditions is a hallmark
of the mathematical behavior known as CHAOS.
see also Chaos, Lorenz System
Butterfly Fractal
The FRACTAL-like curve generated by the 2-D function
(z 2 -y 2 )sin(^)
ffay) =
x 2 +y 2
182 Butterfly Polyiamond Butterfly Theorem
Butterfly Polyiamond
A 6-POLYIAMOND.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Butterfly Theorem
A
Given a Chord PQ of a Circle, draw any other two
CHORDS AB and CD passing through its MIDPOINT.
Call the points where AD and BC meet PQ X and Y.
Then M is the Midpoint of XY.
see also CHORD, CIRCLE, MIDPOINT
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited,
Washington, DC: Math. Assoc. Amer., pp. 45-46, 1967.
Cake Cutting 183
C
C-Table
see C-Determinant
The Field of Complex Numbers, denoted C.
see also C\ Complex Number, I, N, Q, R, Z
C*
The Riemann Sphere C U {oo},
see also C, Complex Number, Q, R, Riemann
Sphere, Z
C*-Algebra
A special type of B* -Algebra in which the Involu-
tion is the Adjoint Operator in a Hilbert Space.
see also £*-ALGEBRA, fc-THEORY
References
Davidson, K. R. C* -Algebras by Example. Providence, RI:
Amer. Math. Soc, 1996.
C- Curve
see Levy Fractal
Cable Knot
Let Ki be a Torus Knot. Then the Satellite Knot
with Companion Knot K 2 is a cable knot on K 2 .
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 118, 1994.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 112 and 283, 1976.
Cactus Fractal
•m
A Mandelbrot SET-like Fractal obtained by iterat-
ing the map
Zn+l = Z n + (ZQ — l)z n — Zq -
C-Determinant
A Determinant appearing in Pade Approximant
identities:
a
s + l <Xr-s+2
a r +\
Gr+s-1
see also Pade APPROXIMANT
C-Matrix
Any Symmetric Matrix (A t = A) or Skew Symmet-
ric Matrix (A t = -A) C™ with diagonal elements
and others ±1 satisfying
CC T = (n-l)l,
where I is the IDENTITY MATRIX, is known as a C-
matrix (Ball and Coxeter 1987), Examples include
c 4 =
+
+
+
-
-
+
-
+
-
-
-
+
0_
+
+
+
+
+
+
+
-
-
+
+
+
+
+
__
+
-
+
+
-
+
-
-
+
+
+
+
-
-
+
c 6 =
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 308-
309, 1987.
see also FRACTAL, JULIA SET, MANDELBROT SET
Cake Cutting
It is always possible to "fairly" divide a cake among n
people using only vertical cuts. Furthermore, it is pos-
sible to cut and divide a cake such that each person
believes that everyone has received 1/n of the cake ac-
cording to his own measure. Finally, if there is some
piece on which two people disagree, then there is a way
of partitioning and dividing a cake such that each par-
ticipant believes that he has obtained more than 1/n of
the cake according to his own measure.
Ignoring the height of the cake, the cake-cutting problem
is really a question of fairly dividing a CIRCLE into n
equal Area pieces using cuts in its plane. One method
of proving fair cake cutting to always be possible relies
on the Frobenius-Konig Theorem.
see also CIRCLE CUTTING, CYLINDER CUTTING, EN-
VYFREE, FROBENIUS-KONIG THEOREM, HAM SAND-
WICH Theorem, Pancake Theorem, Pizza Theo-
rem, Square Cutting, Torus Cutting
References
Brams, S. J. and Taylor, A. D. "An Envy-Free Cake Division
Protocol." Amer. Math. Monthly 102, 9-19, 1995.
Brams, S. J. and Taylor, A. D. Fair Division: From Cake-
Cutting to Dispute Resolution. New York: Cambridge Uni-
versity Press, 1996.
Dubbins, L. and Spanier, E. "How to Cut a Cake Fairly."
Amer. Math. Monthly 68, 1-17, 1961.
Gale, D. "Dividing a Cake." Math. Intel. 15, 50, 1993.
Jones, M. L. "A Note on a Cake Cutting Algorithm of Banach
and Knaster." Amer. Math. Monthly 104, 353-355, 1997.
Rebman, K. "How to Get (At Least) a Fair Share of the
Cake." In Mathematical Plums (Ed. R. Honsberger).
Washington, DC: Math. Assoc. Amer., pp. 22-37, 1979.
184
Cal
Calculus of Variations
rsi
Steinhaus, H. "Sur la division progmatique." Ekonometrika
(Supp.) 17, 315-319, 1949.
Stromquist, W. "How to Cut a Cake Fairly." Amer. Math.
Monthly 87, 640-644, 1980.
Cal
see Walsh Function
Calabi's Triangle
and Integrals
/
f(x) dx,
Equilateral Triangle Calabi's Triangle
The one TRIANGLE in addition to the EQUILATERAL
Triangle for which the largest inscribed Square
can be inscribed in three different ways. The ra-
tio of the sides to that of the base is given by x =
1.55138752455. . . (Sloane's A046095), where
11
_ 1 (-23 + 3zy / 237) 1/3
X ~ 3 + 3-2 2 /3 + 3[ 2 (-23 + 3iv / 237)] 1 / 3
is the largest POSITIVE ROOT of
2x 3 - 2x 2 - 3z + 2 = 0,
which has CONTINUED FRACTION [1, 1, 1, 4, 2, 1, 2, 1,
5, 2, 1, 3, 1, 1, 390, . . .] (Sloane's A046096).
see also GRAHAM'S BIGGEST LITTLE HEXAGON
References
Conway, J. H. and Guy, R. K. "Calabi's Triangle." In The
Book of Numbers. New York: Springer- Verlag, p. 206,
1996,
Sloane, N. J. A. Sequences A046095 and A046096 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Calabi-Yau Space
A structure into which the 6 extra Dimensions of 10-D
string theory curl up.
Calculus
In general, "a" calculus is an abstract theory developed
in a purely formal way.
"The" calculus, more properly called ANALYSIS (or
Real Analysis or, in older literature, Infinitesimal
Analysis) is the branch of mathematics studying the
rate of change of quantities (which can be interpreted as
Slopes of curves) and the length, Area, and Volume
of objects. The CALCULUS is sometimes divided into
Differential and Integral Calculus, concerned
with Derivatives
respectively.
While ideas related to calculus had been known for some
time (Archimedes' Exhaustion Method was a form
of calculus), it was not until the independent work of
Newton and Leibniz that the modern elegant tools and
ideas of calculus were developed. Even so, many years
elapsed until the subject was put on a mathematically
rigorous footing by mathematicians such as Weierstraft.
see also Arc Length, Area, Calculus of Vari-
ations, Change of Variables Theorem, De-
rivative, Differential Calculus, Ellipsoidal
Calculus, Extensions Calculus, Fluent, Flux-
ion, Fractional Calculus, Functional Calculus,
Fundamental Theorems of Calculus, Heaviside
Calculus, Integral, Integral Calculus, Jaco-
bian, Lambda Calculus, Kirby Calculus, Malli-
avin Calculus, Predicate Calculus, Proposi-
tional Calculus, Slope, Tensor Calculus, Um-
bral Calculus, Volume
References
Anton, H. Calculus with Analytic Geometry, 5th ed. New
York: Wiley, 1995.
Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Cal-
culus, with an Introduction to Linear Algebra. Waltham,
MA: Blaisdell, 1967.
Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Cal-
culus and Linear Algebra, with Applications to Differential
Equations and Probability. Waltham, MA: Blaisdell, 1969.
Apostol, T. M. A Century of Calculus, 2 vols. Pt. 1: 1894~
1968. Pt. 2: 1969-1991. Washington, DC: Math. Assoc.
Amer., 1992.
Ayres, F. Jr. and Mendelson, E. Schaum's Outline of Theory
and Problems of Differential and Integral Calculus, 3rd ed.
New York: McGraw-Hill, 1990.
Borden, R. S, A Course in Advanced Calculus. New York:
Dover, 1998.
Boyer, C B. A History of the Calculus and Its Conceptual
Development. New York: Dover, 1989.
Brown, K. S. "Calculus and Differential Equations." http://
www. seanet . com/-ksbrown/icalculu.htm.
Courant, R. and John, F. Introduction to Calculus and Anal-
ysis, Vol. 1. New York: Springer- Verlag, 1990.
Courant, R. and John, F. Introduction to Calculus and Anal-
ysis, Vol. 2. New York: Springer- Verlag, 1990.
Hahn, A. Basic Calculus: From Archimedes to Newton to Its
Role in Science. New York: Springer- Verlag, 1998.
Kaplan, W. Advanced Calculus, 4th ed. Reading, MA:
Addison- Wesley, 1992.
Marsden, J. E. and Tromba, A. J. Vector Calculus, ^i/i ed.
New York: W. H. Freeman, 1996.
Strang, G. Calculus. Wellesley, MA: Wellesley-Cambridge
Press, 1991.
Calculus of Variations
A branch of mathematics which is a sort of general-
ization of CALCULUS. Calculus of variations seeks to
find the path, curve, surface, etc., for which a given
Function has a Stationary Value (which, in physical
Calcus
problems, is usually a Minimum or Maximum). Mathe-
matically, this involves finding STATIONARY VALUES of
integrals of the form
'= / /(y.y,
x) dx.
(i)
J has an extremum only if the Euler-Lagrange Dif-
ferential Equation is satisfied, i.e., if
dy
dx \dyj
(2)
The Fundamental Lemma of Calculus of Varia-
tions states that, if
t/ a
M(x)h(x)dx =
(3)
for all h(x) with CONTINUOUS second PARTIAL DERIVA-
TIVES, then
M(x) = (4)
on (a, 6).
see also BELTRAMI IDENTITY, BOLZA PROBLEM,
Brachistochrone Problem, Catenary, Enve-
lope Theorem, Euler-Lagrange Differential
Equation, Isoperimetric Problem, Isovolume
Problem, Lindelof's Theorem, Plateau's Prob-
lem, Point-Point Distance — 2-D, Point-Point
Distance— 3-D, Roulette, Skew Quadrilateral,
Sphere with Tunnel, Unduloid, WeierstraB-
Erdman Corner Condition
References
Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 925-962, 1985.
Bliss, G. A. Calculus of Variations. Chicago, IL: Open
Court, 1925.
Forsyth, A. R. Calculus of Variations. New York: Dover,
1960.
Fox, C An Introduction to the Calculus of Variations. New-
York: Dover, 1988.
Isenberg, C The Science of Soap Films and Soap Bubbles.
New York: Dover, 1992.
Menger, K. "What is the Calculus of Variations and What
are Its Applications?" In The World of Mathematics (Ed.
K. Newman). Redmond, WA: Microsoft Press, pp. 886-
890, 1988.
Sagan, H. Introduction to the Calculus of Variations. New
York: Dover, 1992.
Todhunter, I. History of the Calculus of Variations During
the Nineteenth Century. New York: Chelsea, 1962.
Weinstock, R. Calculus of Variations, with Applications to
Physics and Engineering. New York: Dover, 1974.
Calcus
1 calcus =
see also Half, Quarter, Scruple, Uncia, Unit
Fraction
Cancellation Law 185
Calderon's Formula
/oo /*oo
/ (f,tp a ' b )i> a - b (x)a.- 2 dadb,
-oo J — CO
where
r' b (x) = \a\-^(^.).
This result was originally derived using HARMONIC
Analysis, but also follows from a Wavelets viewpoint.
Caliban Puzzle
A puzzle in LOGIC in which one or more facts must be
inferred from a set of given facts.
Calvary Cross
see also CROSS
Cameron's Sum-Free Set Constant
A set of POSITIVE INTEGERS S is sum-free if the equa-
tion x 4- y = z has no solutions x, y, z 6 S. The proba-
bility that a random sum-free set S consists entirely of
Odd Integers satisfies
0.21759 < c < 0.21862.
References
Cameron, P. J. "Cyclic Automorphisms of a Countable
Graph and Random Sum-Free Sets." Graphs and Com-
binatorics 1, 129-135, 1985.
Cameron, P. J. "Portrait of a Typical Sum- Free Set." In
Surveys in Combinatorics 1987 (Ed. C. Whitehead). New
York: Cambridge University Press, 13-42, 1987.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/cameron/cameron.html.
Cancellation
see Anomalous Cancellation
Cancellation Law
If be = bd (mod a) and (6, a) — 1 (i.e., a and b are
Relatively Prime), then c~ d (mod a).
see also CONGRUENCE
References
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 36, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 56, 1993.
186
Cannonball Problem
Cantor Dust
Cannonball Problem
Find a way to stack a SQUARE of cannonballs laid out on
the ground into a Square Pyramid (i.e., find a Square
Number which is also Square Pyramidal). This cor-
responds to solving the DlOPHANTINE EQUATION
Cantor-Dedekind Axiom
The points on a line can be put into a One-to-One
correspondence with the REAL NUMBERS.
see also Cardinal Number, Continuum Hypothe-
sis, Dedekind Cut
£V = I*(1 + *)(! + 2*)
N 2
for some pyramid height k. The only solution is k = 24,
N = 70, corresponding to 4900 cannonballs (Ball and
Coxeter 1987, Dickson 1952), as conjectured by Lucas
(1875, 1876) and proved by Watson (1918).
see also Sphere Packing, Square Number, Square
Pyramid, Square Pyramidal Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, p. 25, 1952.
Lucas, E. Question 1180. Nouvelles Ann. Math. Ser. 2 14,
336, 1875.
Lucas, E. Solution de Question 1180. Nouvelles Ann. Math.
Ser. 2 15, 429-432, 1876.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 77 and 152, 1988.
Pappas, T. "Cannon Balls & Pyramids." The Joy of Math-
ematics. San Carlos, CA: Wide World Publ./Tetra, p. 93,
1989.
Watson, G. N. "The Problem of the Square Pyramid." Mes-
senger. Math. 48, 1-22, 1918.
Canonical Form
A clear-cut way of describing every object in a class in
a One-to-One manner.
see also Normal Form, One-to-One
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 7, 1996.
Canonical Polyhedron
A Polyhedron is said to be canonical if all its Edges
touch a Sphere and the center of gravity of their contact
points is the center of that Sphere. Each combinato-
rial type of (GENUS zero) polyhedron contains just one
canonical version. The ARCHIMEDEAN SOLIDS and their
DUALS are all canonical.
References
Conway, J. H. "Re: polyhedra database."
geometry. forum newsgroup, Aug. 31, 1995.
Canonical Transformation
see Symplectic Diffeomorphism
Cantor Comb
see Cantor Set
Posting to
Cantor Diagonal Slash
A clever and rather abstract technique used by Georg
Cantor to show that the Integers and Reals cannot be
put into a One-to-One correspondence (i.e., the INFIN-
ITY of Real Numbers is "larger" than the Infinity of
INTEGERS), It proceeds by constructing a new member
S' of a Set from already known members S by arrang-
ing its nth term to differ from the nth term of the nth
member of S. The tricky part is that this is done in
such a way that the Set including the new member has
a larger CARDINALITY than the original SET S.
see also Cardinality, Continuum Hypothesis, De-
NUMERABLE SET
References
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 81-83, 1996.
Penrose, R. The Emperor's New Mind: Concerning Comput-
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, pp. 84-85, 1989.
Cantor Dust
A Fractal which can be constructed using String Re-
writing by creating a matrix three times the size of the
current matrix using the rules
line 1: "*"->"* *",'' "->" "
line 2: "*"->" "," *'->"
line 3: "*»->"* *",» »->"
Let N n be the number of black boxes, L n the length of
a side of a white box, and A n the fractional Area of
black boxes after the nth iteration.
iVn-5 71
A n = L n 2 N n = ($) n .
The Capacity Dimension is therefore
(1)
(2)
(3)
ln(5 n )
r lniV n
= - hm - — — = - hm /0 _ .
n-J-oo III L n n->-oo Ul(cJ n )
In 5
ln3
1.464973521.
(4)
see also Box FRACTAL, SlERPINSKI CARPET, SlERPIN-
ski Sieve
Cantor's Equation
Cantor Square Fractal 187
References
Dickau, R. M. "Cantor Dust." http://f orum . swarthmore .
edu/advanced/robertd/cantor .html.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, pp. 103-104, 1993.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Fractal.m.
Cantor's Equation
Cantor Set
The Cantor set (Too) is given by taking the interval [0,1]
(set To), removing the middle third (Ti), removing the
middle third of each of the two remaining pieces (T2),
and continuing this procedure ad infinitum. It is there-
fore the set of points in the INTERVAL [0,1] whose ternary
expansions do not contain 1, illustrated below.
where uj is an Ordinal Number and e is an Inacces-
sible Cardinal,
see also INACCESSIBLE CARDINAL, ORDINAL NUMBER
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 274, 1996.
Cantor Function
The function whose values are
2 V 2
Cm-l . 2
Orn — 1 Orn
for any number between
This produces the Set of Real Numbers {x} such that
(i)
Cl C n
X= 3+--- + F + ---'
where c n may equal or 2 for each n. This is an infinite,
Perfect Set. The total length of the Line Segments
in the nth iteration is
*•-(!)"■
(2)
and the number of LINE SEGMENTS is N n = 2 n , so the
length of each element is
tn - N~ (3)
(3)
Cl Cm-l
"3 * ' ' 3™" 1
and
Cl
+ ■
Cm-l _2_
Chalice (1991) shows that any real- values function F(x)
on [0, 1] which is MONOTONE INCREASING and satisfies
1. F(0) = 0,
2. F(x/S) = F{x)/2,
3. F(l-x) = 1-F(x)
is the Cantor function.
see also CANTOR SET, DEVIL'S STAIRCASE
References
Chalice, D. R. "A Characterization of the Cantor Function."
Amer. Math. Monthly 98, 255-258, 1991.
Wagon, S. "The Cantor Function" and "Complex Cantor
Sets." §4.2 and 5.1 in Mathematica in Action. New York:
W. H. Freeman, pp. 102-108 and 143-149, 1991.
Cantor's Paradox
The Set of all Sets is its own Power Set. Therefore,
the Cardinality of the Set of all Sets must be bigger
than itself.
see also CANTOR'S THEOREM, POWER SET
and the Capacity DIMENSION is
In AT
lim _
€-►0+ me
lim
nln2
00 — nln3
In 2
In 3
0.630929...
(4)
The Cantor set is nowhere Dense, so it has LEBESGUE
MEASURE 0.
A general Cantor set is a CLOSED SET consisting en-
tirely of BOUNDARY POINTS. Such sets are UNCOUNT-
ABLE and may have or POSITIVE LEBESGUE MEA-
SURE. The Cantor set is the only totally disconnected,
perfect, Compact Metric Space up to a Homeomor-
PHISM (Willard 1970).
see also Alexander's Horned Sphere, Antoine's
Necklace, Cantor Function
References
Boas, R. P. Jr. A Primer of Real Functions. Washington,
DC: Amer. Math. Soc, 1996.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princetqn, NJ: Princeton University Press, pp. 15-
20, 1991.
Willard, S. §30.4 in General Topology. Reading, MA:
Addison- Wesley, 1970.
Cantor Square Fractal
188
Cantor's Theorem
Cardano's Formula
A Fractal which can be constructed using String Re-
writing by creating a matrix three times the size of the
current matrix using the rules
line 1: "*"->"***"," "->" "
line 2: "*"->"* *"," "->" "
line 3: "*"->"***",» "->" "
The first few steps are illustrated above.
The size of the unit element after the nth iteration is
L n
G)"
and the number of elements is given by the RECUR-
RENCE Relation
N n = 4JV n _i + 5(9 n )
where Ni = 5, and the first few numbers of elements are
5, 65, 665, 6305, Expanding out gives
N n
5 \p 4 n-fc g fc-l =9 n_ 4 n_
fc=0
The Capacity Dimension is therefore
liml^-lim^ 9 "- 4 ")
th-oo In L n
ln(9 n
n-+oo ln(3- n )
n-^oo ln(3" n )
ln9 _ 21n3 _
ln3 "" In 3 ~
2.
Since the DIMENSION of the filled part is 2 (i.e., the
SQUARE is completely filled), Cantor's square fractal is
not a true FRACTAL.
see also Box Fractal, Cantor Dust
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 82-
83, 1991.
^ Weisstein, E. W. "Fractals." http://www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Cantor's Theorem
The Cardinal Number of any set is lower than the
Cardinal Number of the set of all its subsets. A
Corollary is that there is no highest N (Aleph).
see also Cantor's Paradox
Cap
see Cross-Cap, Spherical Cap
Capacity
see Transfinite Diameter
Capacity Dimension
A Dimension also called the Fractal Dimen-
sion, Hausdorff Dimension, and Hausdorff-
Besicovitch Dimension in which nonintegral values
are permitted. Objects whose capacity dimension is dif-
ferent from their TOPOLOGICAL Dimension are called
Fractals. The capacity dimension of a compact Met-
ric Space X is a Real Number capacity such that if
n(e) denotes the minimum number of open sets of diam-
eter less than or equal to e, then n(e) is proportional to
e~ D as e — > 0. Explicitly,
-^capacity
,. miV
hm
€-►0+ hie
(if the limit exists), where N is the number of elements
forming a finite Cover of the relevant Metric SPACE
and e is a bound on the diameter of the sets involved
(informally, e is the size of each element used to cover
the set, which is taken to to approach 0). If each ele-
ment of a Fractal is equally likely to be visited, then
^capacity = ^information, where ^information is the INFOR-
MATION Dimension. The capacity dimension satisfies
^correlation S: ^information S: ^capacity
where correlation is the Correlation Dimension, and
is conjectured to be equal to the LYAPUNOV DIMENSION.
see also CORRELATION EXPONENT, DIMENSION, HAUS-
DORFF Dimension, Kaplan- Yorke Dimension
References
Nayfeh, A. H. and Balachandran, B. Applied Nonlinear
Dynamics: Analytical, Computational, and Experimental
Methods. New York: Wiley, pp. 538-541, 1995.
Peitgen, H.-O. and Richter, D. H. The Beauty of Frac-
tals: Images of Complex Dynamical Systems. New York:
Springer- Verlag, 1986.
Wheeden, R. L. and Zygmund, A. Measure and Integral: An
Introduction to Real Analysis. New York: M. Dekker,
1977.
Caratheodory Derivative
A function / is Caratheodory differentiate at a if there
exists a function which is CONTINUOUS at a such that
f(x) -/(a) = <t>(x)(x-a).
Every function which is Caratheodory differentiable is
also FRECHET DIFFERENTIABLE.
see also Derivative, Frechet Derivative
Caratheodory's Fundamental Theorem
Each point in the CONVEX Hull of a set S in R n is in
the convex combination of n + 1 or fewer points of 5.
see also Convex Hull, Helly's Theorem
Cardano's Formula
see Cubic Equation
Cardinal Number
Cardioid 189
Cardinal Number
In informal usage, a cardinal number is a number used
in counting (a Counting Number), such as 1, 2, 3,
Formally, a cardinal number is a type of number defined
in such a way that any method of counting SETS using it
gives the same result. (This is not true for the ORDINAL
Numbers.) In fact, the cardinal numbers are obtained
by collecting all ORDINAL NUMBERS which are obtain-
able by counting a given set. A set has No (ALEPH-0)
members if it can be put into a One-TO-One correspon-
dence with the finite ORDINAL NUMBERS.
Two sets are said to have the same cardinal number if
all the elements in the sets can be paired off One-to-
One. An Inaccessible Cardinal cannot be expressed
in terms of a smaller number of smaller cardinals.
see also Aleph, Aleph-0 (Ho), Aleph-1 (Hi), Can-
tor-Dedekind Axiom, Cantor Diagonal Slash,
Conttnuum, Continuum Hypothesis, Equipol-
lent, Inaccessible Cardinals Axiom, Infinity,
Ordinal Number, Power Set, Surreal Number,
Uncountable Set
References
Cantor, G. Uber unendliche, lineare Punktmannigfaltig-
keiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-
1884. Leipzig, Germany: Teubner, 1884.
Conway, J. H. and Guy, R. K. "Cardinal Numbers." In The
Book of Numbers. New York: Springer- Verlag, pp. 277-
282, 1996.
Courant, R. and Robbins, H. "Cantor's 'Cardinal Numbers.'"
§2.4.3 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England: Oxford
University Press, pp. 83-86, 1996.
Cardinality
see Cardinal Number
Cardioid
and the parametric equations
The curve given by the POLAR equation
r = a(l + cos#),
sometimes also written
r = 26(1 + cos 0),
where b = a/2, the Cartestan equation
/ 2 . 2 n2 2/ 2 . 2\
[x + y -ax) — a (x +y ),
(1)
(2)
(3)
x = acost(l + cost)
y = asini(l + cost).
(4)
(5)
The cardioid is a degenerate case of the LlMA<JON. It is
also a 1-CuSPED EPICYCLOID (with r = R) and is the
CAUSTIC formed by rays originating at a point on the
circumference of a CIRCLE and reflected by the Circle.
The name cardioid was first used by de Castillon in
Philosophical Transactions of the Royal Society in 1741.
Its Arc Length was found by La Hire in 1708. There
are exactly three PARALLEL TANGENTS to the cardioid
with any given gradient. Also, the TANGENTS at the
ends of any Chord through the Cusp point are at
Right Angles. The length of any Chord through the
Cusp point is 2a.
The cardioid may also be generated as follows. Draw
a CIRCLE C and fix a point A on it. Now draw a set
of Circles centered on the Circumference of C and
passing through A. The ENVELOPE of these Circles
is then a cardioid (Pedoe 1995). Let the CIRCLE C be
centered at the origin and have RADIUS 1, and let the
fixed point be A — (1, 0). Then the RADIUS of a CIRCLE
centered at an ANGLE 9 from (1, 0) is
r 2 = (0-cos(9) 2 + (l-sin(9) 2
= cos 2 0+l-2sin0 + sin 2
= 2(1- sin 0).
(6)
J
^
The Arc Length, Curvature, and Tangential An-
gle are
/'
Jo
2|cos(!i)|dt = 4asin(i0)
3|sec(i0)|
4o
(7)
(8)
(9)
As usual, care must be taken in the evaluation of s(t)
for t > n. Since (7) comes from an integral involving the
190
Cardioid Caustic
Cards
ABSOLUTE Value of a function, it must be monotonic
increasing- Each Quadrant can be treated correctly
by defining
+ 1, (10)
Cardioid Evolute
l_7T
where [a; J is the FLOOR FUNCTION, giving the formula
s(t) = (~l) 1+[n (mod 2)] 4sin(|i) + 8 Ll n J ' ( U >
The Perimeter of the curve is
/»2tt
/
Jo
|2acos(|i9)|d0 = 4a
/ cos (|
Jo
9)dB
/•7r/2 / 1 t/2
= 4a / cos <j>(2 d<$>) — 8a / cos (j)d<fi
Jo Jo
-8a[sin0]o /2 = 8a. (12)
The Area is
/•27T
A= \ I r 2 d6=\a I (1 + 2cos<9 + cos 2 6) dO
Jo Jo
= 2 a /
Jo
{1 + 2 cos + | [1 + cos(26>)]} d0
/»27T
= |a 2 / [§ + 2cos(9+|cos(26>)]dl9
Jo
= \A¥ + 2sin # + \ sin^lo" =
2tt _ 3 2
(13)
see also Circle, Cissoid, Conchoid, Equiangular
Spiral, Lemniscate, LiMAgoN, Mandelbrot Set
References
Gray, A. "Cardioids." §3.3 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 41-42, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 118-121, 1972.
Lee, X. "Cardioid." http : //www .best . com/ ~xah/ Special
PlaneCurves_dir/Cardioid_dir/cardioid.html.
Lee, X. "Cardioid." http://www.best.com/-xah/Special
PlaneCurves_dir/Cardioid_dir/cardioidGG.html.
Lockwood, E. H. "The Cardioid." Ch. 4 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 34-
43, 1967.
MacTutor History of Mathematics Archive. "Cardioid."
http : //www-groups . dcs . st-and. ac .uk/ -history/Curves
/Cardioid. html.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995.
Yates, R. C. "The Cardioid." Math. Teacher 52, 10-14, 1959.
Yates, R. C. "Cardioid." A Handbook on Curves and Their
Properties. Ann Arbor, Ml: J. W. Edwards, pp. 4-7, 1952.
Cardioid Caustic
The Catacaustic of a Cardioid for a Radiant Point
at the Cusp is a Nephroid. The Catacaustic for
Parallel rays crossing a Circle is a Cardioid.
y^
~-^
/
V
/
\
/
\
/
\
\
/^~^\
\
\
\
\
f \
\
V
\ I
1
/
\
u
;
/
/
/
\
/
\
/
/
N.
/
*"■'
**"
x = -a + |a cos 0(1 — cos#)
y = |asin#(l — cos#).
This is a mirror-image Cardioid with a = a/3.
Cardioid Inverse Curve
If the Cusp of the cardioid is taken as the Inversion
Center, the cardioid inverts to a Parabola.
Cardioid Involute
x — 2a + 3a cos 9(1 — cos 0)
y = 3a sin 0(1 — cos#).
This is a mirror-image CARDIOID with a 1 = 3a.
Cardioid Pedal Curve
/ /
/ /
y
-
-
NX
V \
/ /
\
\ \
\ \
//
-
The Pedal Curve of the Cardioid where the Pedal
Point is the Cusp is Cayley's Sextic.
Cards
Cards are a set of n rectangular pieces of cardboard
with markings on one side and a uniform pattern on the
other. The collection of all cards is called a "deck," and
a normal deck of cards consists of 52 cards of four dif-
ferent "suits." The suits are called clubs (Jt), diamonds
(<0>), hearts (\?), and spades (♦). Spades and clubs are
Carleman's Inequality
colored black, while hearts and diamonds are colored
red. The cards of each suit are numbered 1 through 13,
where the special terms ace (1), jack (11), queen (12),
and king (13) are used instead of numbers 1 and 11-13.
The randomization of the order of cards in a deck is
called Shuffling. Cards are used in many gambling
games (such as POKER), and the investigation of the
probabilities of various outcomes in card games was one
of the original motivations for the development of mod-
ern Probability theory.
see also Bridge Card Game, Clock Solitaire,
Coin, Coin Tossing, Dice, Poker, Shuffle
Carleman's Inequality-
Let {a,i}™ =1 be a Set of Positive numbers. Then the
Geometric Mean and Arithmetic Mean satisfy
n n
^J(aia 2 • • • a;) 1/j < - ^J a».
Here, the constant e is the best possible, in the sense
that counterexamples can be constructed for any stricter
Inequality which uses a smaller constant.
see also Arithmetic Mean, e, Geometric Mean
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1094, 1979.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 249-250, 1988.
Carlson-Levin Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Assume that / is a Nonnegative Real function on
[0, oo) and that the two integrals
Carlyle Circle 191
/
Jo
x P ~ [f(x)] p dx
*'- 1+M [/(aO]' dx
(1)
(2)
exist and are FINITE. If p = q — 2 and A = /x = 1,
Carlson (1934) determined
/ f(x)dx< \M /
(I
[f(x)] 2 dx
1/4
x / x*[f{x)Ydx\ (3)
1/4
and showed that ^pK is the best constant (in the sense
that counterexamples can be constructed for any stricter
INEQUALITY which uses a smaller constant). For the
general case
/ f(x)dx<cl x p - 1 - x [f(x)] p dx\
C 9 - 1+ "[/(x)]* dx
and Levin (1948) showed that the best constant
r(;)r(i)
(4)
(pa)*(qty
(A + / «)r(4±i)
where
t =
ppL + qX
A
pfi + q\
a = 1 — s — t
and T(z) is the GAMMA FUNCTION.
(5)
(6)
(7)
(8)
References
Beckenbach, E. F.; and Bellman, R. Inequalities. New York:
Springer- Verlag, 1983.
Boas, R. P. Jr. Review of Levin, V. I. "Exact Constants
in Inequalities of the Carlson Type." Math. Rev. 9, 415,
1948.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/crlslvn/crlslvn.htnil.
Levin, V. L "Exact Constants in Inequalities of the Carlson
Type." Doklady Akad. Nauk. SSSR (N. S.) 59, 635-638,
1948. English review in Boas (1948).
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives.
Kluwer, 1991.
Carlson's Theorem
If f(z) is regular and of the form <9(e fc '*') where k < tt,
for K[z] > 0, and if f(z) = for z = 0, 1, . . . , then f(z)
is identically zero.
see also Generalized Hypergeometric Function
References
Bailey, W. N. "Carlson's Theorem." §5.3 in Generalised
Hypergeometric Series. Cambridge, England: Cambridge
University Press, pp. 36—40, 1935.
Carlyle Circle
n
A = (1,0)
B = {s,p)
Y=(0,p+l)
C=(0,p)
H 2 S = (j, 0)
Consider a Quadratic Equation x 2 -sx+p = where
s and p denote signed lengths. The CIRCLE which has
192
Carmichael Condition
Carmichael Number
the points A = (0,1) and B — (s,p) as a DIAMETER
is then called the Carlyle circle C S>P of the equation.
The Center of C SjP is then at the Midpoint of AB,
M = (s/2,(l +p)/2), which is also the Midpoint of
S = (s, 0) and Y = (0, 1 + p). Call the points at which
C SiP crosses the x-AxiS Hi = (2:1,0) and #2 = (#2,0)
(with x\ > X2)> Then
s = Xi -\- X2
p = X1X2
(# — x\)(x — X2) = x 2 — sx + p,
so xi and X2 are the ROOTS of the quadratic equation.
see also 257-gon, 65537-gon, Heptadecagon, Pen-
tagon
References
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.
Eves, H. An Introduction to the History of Mathematics, 6th
ed. Philadelphia, PA: Saunders, 1990.
Leslie, J. Elements of Geometry and Plane Trigonome-
try with an Appendix and Very Copious Notes and Il-
lustrations, J^th ed., improved and exp. Edinburgh:
W. & G. Tait, 1820.
Carmichael Condition
A number n satisfies the Carmichael condition IFF (p —
l)\(n/p - 1) for all PRIME DIVISORS p of n. This is
equivalent to the condition (p - l)\(n - 1) for all Prime
Divisors pofn.
see also Carmichael Number
References
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-
sohn, R. "Giuga's Conjecture on Primality." Amer. Math.
Monthly 103, 40-50, 1996.
Carmichael's Conjecture
CarmichaeFs conjecture asserts that there are an In-
finite number of Carmichael Numbers. This was
proven by Alford et al. (1994).
see also CARMICHAEL NUMBER, CARMICHAEL'S TO-
tient Function Conjecture
References
Alford, W. R.; Granville, A.; and Pomerance, C. "There Are
Infinitely Many Carmichael Numbers." Ann. Math. 139,
703-722, 1994.
Cipra, B. What's Happening in the Mathematical Sciences,
Vol 1. Providence, RI: Amer. Math. Soc, 1993.
Guy, R. K. "Carmichael's Conjecture." §B39 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, p. 94, 1994.
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The
Pseudoprimesto25-10 9 ." Math. Comput. 35,1003-1026,
1980.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 29-31, 1989.
Schlafly, A. and Wagon, S. "Carmichael's Conjecture on the
Euler Function is Valid Below lO 10 - 000 - 000 ." Math. Com-
put. 63, 415-419, 1994.
Carmichael Function
A(n) is the LEAST COMMON MULTIPLE (LCM) of all the
Factors of the Totient Function <j>(n), except that
if 8|n, then 2 a ~ 2 is a FACTOR instead of 2 a ~ 1 .
\{n) = <
0(n)
for n = p a ,p = 2 and a < 2, or p > 3
\<t>{n)
for n = 2 a and a > 3
LCM[X(jH ai )]i
for n = YiiPi ai
Some special values are
for r > 3, and
A(l) = 1
A(2) = 1
A(4) = 2
A(2 r ) - 2 r ~ 2
X(p r ) = 4>tf)
for p an ODD PRIME and r > 1. The ORDER of a (mod
n) is at most A(n) (Ribenboim 1989). The values of A(n)
for the first few n are 1, 1, 2, 2, 4, 2, 6, 4, 10, 2, 12, . . .
(Sloane's A011773).
see also MODULO MULTIPLICATION GROUP
References
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 27, 1989.
Riesel, H. "Carmichael's Function." Prime Numbers and
Computer Methods for Factorization, 2nd ed. Boston,
MA: Birkhauser, pp. 273-275, 1994.
Sloane, N. J. A. Sequence A011773 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, p. 226, 1991.
Carmichael Number
A Carmichael number is an Odd Composite Number
n which satisfies Fermat's Little Theorem
a n_1 -1 = (mod n)
for every choice of a satisfying (a,n) = 1 (i.e., a and
n are Relatively Prime) with 1 < a < n. A Car-
michael number is therefore a PSEUDOPRIMES to any
base. Carmichael numbers therefore cannot be found
to be Composite using Fermat's Little Theorem.
However, if (a,n) ^ 1, the congruence of Fermat's Lit-
tle Theorem is sometimes Nonzero, thus identifying
a Carmichael number n as COMPOSITES,
Carmichael numbers are sometimes called ABSOLUTE
PSEUDOPRIMES and also satisfy KORSELT'S CRITERION.
R. D. Carmichael first noted the existence of such num-
bers in 1910, computed 15 examples, and conjectured
that there were infinitely many (a fact finally proved by
Alford et al. 1994).
Carmichael Number
CarmichaeFs Totient Function Conjecture 193
The first few Carmichael numbers are 561, 1105, 1729,
2465, 2821, 6601, 8911, 10585, 15841, 29341, ...
(Sloane's A002997). Carmichael numbers have at least
three PRIME FACTORS. For Carmichael numbers with
exactly three PRIME FACTORS, once one of the PRIMES
has been specified, there are only a finite number of Car-
michael numbers which can be constructed. Numbers of
the form (6fc + l)(12fc + l)(18fc + l) are Carmichael num-
bers if each of the factors is Prime (Korselt 1899, Ore
1988, Guy 1994). This can be seen since for
N = (6fc+l)(12fc+l)(18fc+l)
1296fc 3 +396/c 2 +36£;+l,
N - 1 is a multiple of 36k and the LEAST COMMON
Multiple of 6fc, 12fc, and 18k is 36fc, so a^" 1 = 1
modulo each of the PRIMES 6A; + 1, 12k + 1, and lSk +
1, hence a N ~ x = 1 modulo their product. The first
few such Carmichael numbers correspond to k = 1, 6,
35, 45, 51, 55, 56, ... and are 1729, 294409, 56052361,
118901521, ... (Sloane's A046025). The largest known
Carmichael number of this form was found by H. Dubner
in 1996 and has 1025 digits.
The smallest Carmichael numbers having 3, 4, ... fac-
tors are 561 = 3 x 11 x 17, 41041 = 7 x 11 x 13 x 41,
825265, 321197185, ... (Sloane's A006931). In total,
there are only 43 Carmichael numbers < 10 6 , 2163
< 2.5 x 10 10 , 105,212 < 10 15 , and 246,683 < 10 16 (Pinch
1993). Let C(n) denote the number of Carmichael num-
bers less than n. Then, for sufficiently large n (n ~ 10 7
from numerical evidence),
C(n)
2/7
(Alford et al. 1994).
The Carmichael numbers have the following properties:
1. If a PRIME p divides the Carmichael number
n, then n = 1 (mod p — 1) implies that n =
p (mod p(p — 1)).
2. Every Carmichael number is SQUAREFREE.
3. An Odd Composite Squarefree number n is a
Carmichael number Iff n divides the DENOMINATOR
of the Bernoulli Number B n -\.
see also CARMICHAEL CONDITION, PSEUDOPRIME
References
Alford, W. R.; Granville, A.; and Pomerance, C. "There are
Infinitely Many Carmichael Numbers." Ann. Math. 139,
703-722, 1994.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 87, 1987.
Guy, R. K. "Carmichael Numbers." §A13 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 30-32, 1994.
Korselt, A. "Probleme chinois." L 'intermediate math. 6,
143-143, 1899.
Ore, 0. Number Theory and Its History. New York: Dover,
1988.
Pinch, R. G. E. "The Carmichael Numbers up to 10 15 ."
Math. Comput. 55, 381-391, 1993.
Pinch, R. G. E. ftp:// emu . pmms . cam .ac.uk/ pub /
Carmichael/.
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The
Pseudoprimesto25'10 9 ." Math. Corn-put 35, 1003-1026,
1980.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Basel: Birkhauser, pp. 89-90 and 94-
95, 1994.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p'. 116, 1993.
Sloane, N. J. A. Sequences A002997/M5462 and A006931/
M5463 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Carmichael Sequence
A Finite, Increasing Sequence of Integers {a ly
. . . , a m } such that
(en - l)|(ai •■ -ai-i)
for i = 1, . . . , ?n, where m\n indicates that m DIVIDES n.
A Carmichael sequence has exclusive EVEN or Odd ele-
ments. There are infinitely many Carmichael sequences
for every order.
see also GlUGA SEQUENCE
References
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-
sohn, R. "Giuga's Conjecture on Primality." ^4mer. Math.
Monthly 103, 40-50, 1996.
CarmichaePs Theorem
If a and n are RELATIVELY PRIME so that the GREATEST
Common Denominator GCD(a,n) = 1, then
a ^ = 1 (mod n) ,
where A is the Carmichael Function.
CarmichaePs Totient Function Conjecture
It is thought that the Totient Valence Function
AT (m) > 2 (i.e., the TOTIENT VALENCE FUNCTION
never takes the value 1). This assertion is called Car-
michael's totient function conjecture and is equivalent
to the statement that there exists an m ^ n such
that <t>{n) = <p(m) (Ribenboim 1996, pp. 39-40). Any
counterexample to the conjecture must have more than
10,000 DIGITS (Conway and Guy 1996). Recently,
the conjecture was reportedly proven by F. Saidak in
November, 1997 with a proof short enough to fit on a
postcard.
see also Totient Function, Totient Valence
Function
References
Conway, J. H, and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 155, 1996.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, 1996.
194 Carnot's Polygon Theorem
Cartan Torsion Coefficient
Carnot's Polygon Theorem
If Pi, P2, . • ■ , are the VERTICES of a finite POLYGON
with no "minimal sides" and the side PiPj meets a curve
in the POINTS Piji and Pj-,2, then
Ui^ P ^Ui P 2P23i--Ui P ^ P ^
= 1,
where AB denotes the DISTANCE from POINT A to B.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 190, 1959.
Carnot's Theorem
Given any TRIANGLE A 1 A2A$ i the signed sum of PER-
PENDICULAR distances from the C IRC UM CENTER O to
the sides is
OOi + OO2 + OO3 = R + r,
where r is the INRADIUS and R is the ClRCUMRADIUS.
The sign of the distance is chosen to be POSITIVE IFF
the entire segment OOi lies outside the TRIANGLE.
see also JAPANESE TRIANGULATION THEOREM
References
Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA:
Allyn and Bacon, pp. 256 and 262, 1972.
Honsbergcr, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., p. 25, 1985.
Carotid-Kundalini Fractal
Fractal Valley Gaussian Mtn. Oscillation Land
0.5
-1
, ,11:.,
111
//>;'); V
. Mmmm
I
'''■-jffi/i
x m
\ i
m
0.5
A fractal-like structure is produced for x < by super-
posing plots of Carotid-Kundalini Functions CK n
of different orders n. The region — 1 < x < is called
FRACTAL LAND by Pickover (1995), the central region
the Gaussian Mountain Range, and the region x >
Oscillation Land. The plot above shows n — 1 to 25.
Gaps in FRACTAL LAND occur whenever
cos(27rr/<?) for r = 0, 1, ..., [q/2\, where \z\ is the
Ceiling Function and L^J is the Floor Function.
References
Pickover, C. A. "Are Infinite Carotid-Kundalini Functions
Fractal?" Ch. 24 in Keys to Infinity. New York: W. H.
Freeman, pp. 179-181, 1995.
Carotid-Kundalini Function
The Function given by
CK n (x) = cos(nxcos _1 x),
where n is an Integer and — 1 < x < 1.
see also Carotid-Kundalini Fractal
Carry
l 1
1 5 8-
H 249 -
407-
-carries
- addend 1
- addend 2
-sum
The operating of shifting the leading DIGITS of an AD-
DITION into the next column to the left when the Sum of
that column exceeds a single DIGIT (i.e., 9 in base 10).
see also ADDEND, ADDITION, BORROW
Carrying Capacity
see Logistic Growth Curve
Cartan Matrix
A Matrix used in the presentation of a Lie Algebra.
References
Jacobson, N. Lie Algebras. New York: Dover, p. 121, 1979.
Cartan Relation
The relationship Sq*(x ^ y) = Z j+k =iSq j (x) -- Sq k {y)
encountered in the definition of the Steenrod Alge-
bra.
Cartan Subgroup
A type of maximal Abelian SUBGROUP.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Cartan Torsion Coefficient
The Antisymmetric parts of the Connection Coef-
ficient r A u „.
-i p
xcos X = 2it-
Q
for p and q RELATIVELY PRIME INTEGERS. At such
points #, the functions assume the \(q + l)/2] values
Cartesian Coordinates
Cartesian Ovals 195
Cartesian Coordinates
2-axis
A
The Gradient of the Divergence is
y-axis
Cartesian coordinates are rectilinear 2-D or 3-D coordi-
nates (and therefore a special case of CURVILINEAR CO-
ORDINATES) which are also called Rectangular Co-
ordinates. The three axes of 3-D Cartesian coordi-
nates, conventionally denoted the a>, y-, and z-Axes (a
Notation due to Descartes) are chosen to be linear and
mutually PERPENDICULAR. In 3-D, the coordinates x,
y, and z may lie anywhere in the INTERVAL ( — 00,00).
The Scale Factors of Cartesian coordinates are all
unity, hi = l. The Line Element is given by
ds — dx x + dy y + dz z,
and the Volume Element by
dV = dx dy dz.
The Gradient has a particularly simple form,
„J?_ ,d_ ^d_
dx dy dz '
as does the Laplacian
dx 2 dy 2 dz 2 *
(i)
(2)
(3)
(4)
The Laplacian is
V 2 F = V-(VF)
d 2 F d 2 F
dx 2 dy 2
d 2 F
dz 2
+ y
+ z
The Divergence is
V-F -
and the CURL is
d 2 F x d 2 F 2
dx 2 +
d 2 F v
dx 2
d 2 F z
dx 2
dy 2
d 2 F y
dy 2
d 2 F x
dz 2
d 2 F,
+
+
d^F z
dy 2
+
dz 2
d 2 F z
dz 2
dF x
dx
dF v . 8F X
dy
+
dz
(5)
(6)
V x F :
x
_d_
dx
F x
y
a
+
(dF z
\ dy
y dx dy
z
d_
dz
F z
dFy
dz
dF Q
x +
( dF x
V dz
V(V-u)
a ( du_x_ 1 du y 1 <t
x "I" dy ~r c
du x 1 9uy_ du z \
dx ~T~ dy ^~ dz J
dy
o 1 du x 1 dv-y 1
*~ l dx ~*~ dy ^
r A.
%
dy
_d_
dz
du x du v du z
___ _j * _j
dx dy dz
(8)
Laplace's Equation is separable in Cartesian coordi-
nates.
see also COORDINATES, HELMHOLTZ DIFFERENTIAL
Equation— Cartesian Coordinates
References
Arfken, G. "Special Coordinate Systems— Rectangular
Cartesian Coordinates." §2.3 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94-
95, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 656, 1953.
Cartesian Ovals
A curve consisting of two ovals which was first studied
by Descartes in 1637. It is the locus of a point P whose
distances from two FOCI F\ and F2 in two-center BIPO-
LAR Coordinates satisfy
mr ± nr = k,
(i)
where m,n are Positive Integers, A; is a Positive
real, and r and r are the distances from F\ and F2. If
m = n, the oval becomes an an ELLIPSE. In CARTESIAN
Coordinates, the Cartesian ovals can be written
iy/(x - a) 2 + y 2 + ny/(x + a) 2 + 1
(2)
/ 2 , 2 . 2w 2 2\ / 2 . 2\ 7 2
(x -\- y + a ){m — n ) — 2ax{m + n ) — k
= -2n^{x + a) 2 + y 2 , (3)
[(m 2 - n 2 )(x 2 + y 2 + a 2 ) - 2ax(m 2 + n 2 )] 2
= 2(m 2 + n 2 )(n 2 + y 2 + a 2 ) - 4ax(m 2 - n 2 ) - A; 2 . (4)
(5)
(6)
dF z \ „
Now define
(7)
,22
— 771 — n
_ 2 . 2
c = m +n ,
196
Cartesian Product
Cassini Ovals
and set a = 1. Then
[b(x 2 +y 2 )-2cx + bf +Abx + k 2 -2c = 2c(x 2 +y 2 ). (7)
If c is the distance between Fi and F2, and the equation
r 4- mr = a (8)
is used instead, an alternate form is
[(l-m 2 )(x 2 +y 2 )+2m 2 c'x+a' 2 -m 2 c 12 } 2 = 4a' 2 (x 2 +y 2 ).
(9)
The curves possess three Foci. If m — 1, one Cartesian
oval is a central CONIC, while if m = a/c % then the curve
is a LlMAgON and the inside oval touches the outside
one. Cartesian ovals are ANALLAGMATIC CURVES.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 35, 1989.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 155-157, 1972.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 188, 1967.
MacTutor History of Mathematics Archive. "Cartesian
Oval." http : //www-groups . dcs . st -and . ac . uk/ -history/
Curves /Cart esian.html.
Cartesian Product
see Direct Product (Set)
Cartesian Trident
see Trident of Descartes
Cartography
The study of Map Projections and the making of ge-
ographical maps.
see also Map Projection
Cascade
A Z-Action or N- Action. A cascade and a single Map
X — ¥ X are essentially the same, but the term "cascade"
is preferred by many Russian authors.
see also Action, Flow
Casey's Theorem
Four Circles are Tangent to a fifth Circle or a
straight Line Iff
£12^34 i £13^42 db £14^23 = 0,
where Uj is a common TANGENT to CIRCLES i and j.
see also PURSER'S THEOREM
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 121-127, 1929.
Casimir Operator
An Operator
on a representation R of a LIE ALGEBRA.
References
Jacobson, N. Lie Algebras. New York: Dover, p. 78, 1979.
Cassini Ellipses
see Cassini Ovals
Cassini's Identity
For F n the nth FIBONACCI NUMBER,
Fn~iF n +i — F n — (— l) n .
see also Fibonacci Number
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 12, 1996.
Cassini Ovals
The curves, also called CASSINI ELLIPSES, described by
a point such that the product of its distances from two
fixed points a distance 2a apart is a constant b . The
shape of the curve depends on b/a. If a < 6, the curve
is a single loop with an Oval (left figure above) or dog
bone (second figure) shape. The case a = b produces
a Lemniscate (third figure). If a > b, then the curve
consists of two loops (right figure). The curve was first
investigated by Cassini in 1680 when he was studying
the relative motions of the Earth and the Sun. Cassini
believed that the Sun traveled around the Earth on one
of these ovals, with the Earth at one FOCUS of the oval.
Cassini ovals are Anallagmatic Curves. The Cassini
ovals are defined in two-center Bipolar Coordinates
by the equation
T\T2 = b ,
(1)
with the origin at a FOCUS. Even more incredible curves
are produced by the locus of a point the product of
whose distances from 3 or more fixed points is a con-
stant.
The Cassini ovals have the CARTESIAN equation
[(x-a) 2 +y 2 ][(x + a) 2 +2/ 2 ] = 6 4 (2)
or the equivalent form
(x 4- y + a ) — 4a x = b (3)
Cassini Ovals
Cassini Surface 197
and the polar equation
Cassini Projection
4 . 4
r 4- a
2aVcos(2(9) = & 4 .
(4)
Solving for r 2 using the QUADRATIC Equation gives
2
r =
2a 2 cos(2(9) + ^a 4 cos 2 (20) - 4(a 4 - b 4 )
= a 2 003(20) + V / a 4 cos 2 (2(9) + 6 4 -a 4
= a 2 cos(20) v/a 4 [cos 2 (20) - 1] + fe 4
= a 2 cos(20) + ^b 4 - a 4 sin 2 (20)
cos(20) + J(-} -sin 2 (20)
(5)
If a < 6, the curve has Area
A= L r i de = 2 (l) f r 2 c
J-tv/4
a J +6^(- ), (6)
where the integral has been done over half the curve
and then multiplied by two and E(x) is the complete
Elliptic Integral of the Second Kind. If a = 6,
the curve becomes
r 2 = a 2 |cos(20) + >/l-sin 2 0l = 2a 2 cos(2<9), (7)
which is a Lemniscate having Area
A = 2a 2 (8)
(two loops of a curve y/2 the linear scale of the usual
lemniscate r 2 — a 2 cos(2#), which has area A = a 2 /2
for each loop). If a > 6, the curve becomes two disjoint
ovals with equations
r = ±aJ cos(20) ± J (~) -sin 2 (20), (9)
where £ [— 0o,9q] and
A — 1 * "I
t/o = f sin
&'
(10)
see a/so Cassini Surface, Lemniscate, Mandelbrot
Set, Oval
References
Gray, A. "Cassinian Ovals." §4.2 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 63-65, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 153-155, 1972.
Lee, X. "Cassinian Oval," http : // www . best . com / - xah /
SpecialPlane Curves _dir/CassinianOval_dir/ cassinian
Oval.html.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, pp. 187-188, 1967.
MacTutor History of Mathematics Archive. "Cassinian
Ovals." http: //www-groups .dcs .st-and.ac .uk/ -history
/Curves/Cassinian.html.
Yates, R. C. "Cassinian Curves." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 8-11, 1952.
iCTION.
x — sin - B
(i)
y = tan -
tan<£
(2)
cos(A — Ao)
where
B = cos</>sin(A - Ao).
The inverse FORMULAS are
<t> = sin -1 (sin D cos x)
-l ( tan x \
A = Ao + tan ( — 1 ,
V cos D J
where
D = y + <f> .
(3)
(4)
(5)
(6)
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 92-95, 1987.
Cassini Surface
The QUARTIC SURFACE obtained by replacing the con-
stant c in the equation of the CASSINI OVALS
{(x-a) 2 +y 2 ][(x + af + y 2 ] = c 2
by c = z 2 , obtaining
[(x-a) 2 +y 2 }[(x + a) 2 +y 2 ] = z 4 .
As can be seen by letting y = to obtain
/ 2 2\2 4
(x — a ) — z
2.2 2
x + z = a ,
(i)
(2)
(3)
(4)
198
Castillon's Problem
Catalan's Conjecture
the intersection of the surface with the y — PLANE is
a Circle of Radius a.
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 20, 1986.
Fischer, G. (Ed.). Plate 51 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 51, 1986.
Castillon's Problem
Inscribe a TRIANGLE in a CIRCLE such that the sides of
the Triangle pass through three given Points A, B,
and C>
References
Dorrie, H. "Castillon's Problem." §29 in 100 Great Problems
of Elementary Mathematics: Their History and Solutions.
New York: Dover, pp. 144-147, 1965.
Casting Out Nines
An elementary check of a Multiplication which makes
use of the CONGRUENCE 10 n = 1 (mod 9) for n > 2.
Prom this CONGRUENCE, a MULTIPLICATION ab — c
must give
a = > a,i = a*
bi = b*
C = 2~J Ci — c* ,
so ab = a*b* must be = c* (mod 9). Casting out nines
is sometimes also called "the Hindu Check."
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 28-29, 1996.
Cat Map
see Arnold's Cat Map
Catacaustic
The curve which is the ENVELOPE of reflected rays.
Curve
Source
Catacaustic
cardioid
cusp
nephroid
circle
not on circumf.
lima^on
circle
on circumf.
cardioid
circle
point at oo
nephroid
cissoid of Diocles
focus
cardioid
1 arch of a cycloid
rays _L axis
2 arches of a cycloid
deltoid
point at oo
astroid
In x
rays || axis
catenary
logarithmic spiral
origin
equal logarithmic spiral
parabola
rays _L axis
Tschirnhausen cubic
quadrifolium
center
astroid
Tschirnhausen cubic
focus
semicubical parabola
see also CAUSTIC,
DlACAUSTIC
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 60 and 207, 1972.
Catalan's Conjecture
8 and 9 (2 3 and 3 2 ) are the only consecutive POWERS
(excluding and 1), i.e., the only solution to Cata-
lan's Diophantine PROBLEM. Solutions to this prob-
lem (Catalan's Diophantine Problem) are equiva-
lent to solving the simultaneous Diophantine Equa-
tions
X 2 - Y s = 1
X 3 -Y 2 = 1.
This Conjecture has not yet been proved or refuted,
although it has been shown to be decidable in a Fi-
nite (but more than astronomical) number of steps.
In particular, if n and n H- 1 are POWERS, then n <
exp exp exp exp 730 (Guy 1994, p. 155), which follows
from R. Tijdeman's proof that there can be only a FI-
NITE number of exceptions should the CONJECTURE not
hold.
Hyyro and Makowski proved that there do not exist
three consecutive POWERS (Ribenboim 1996), and it is
also known that 8 and 9 are the only consecutive CUBIC
and Square Numbers (in either order).
see also Catalan's Diophantine Problem
References
Guy, R. K. "Difference of Two Power." §D9 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 155-157, 1994.
Ribenboim, P. Catalan's Conjecture. Boston, MA: Academic
Press, 1994.
Ribenboim, P. "Catalan's Conjecture." Amer. Math.
Monthly 103, 529-538, 1996.
Ribenboim, P. "Consecutive Powers." Expositiones Mathe-
maticae 2, 193-221, 1984.
Catalan's Constant
Catalan's Constant
199
Catalan's Constant
A constant which appears in estimates of combinatorial
functions. It is usually denoted K, /3(2), or G. It is not
known if K is IRRATIONAL. Numerically,
K = 0.915965594177...
(1)
(Sloane's A006752). The CONTINUED FRACTION for K
is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (Sloane's A014538).
K can be given analytically by the following expressions,
K = /3(2)
(-l) fe _ J__jL 1
(2fc-fl) 2 ~ l 2 3 2 + 5 2 + "
(2)
(3)
= 1 +
71 = 1
oo oo
^ (4n + l) 2 ~ 9 ~ ^ (4n + 3) 2 ^
I
Jo
(4
tan -1 xdx
l
In xdx
(5)
(6)
where (3(z) is the Dirichlet Beta Function. In terms
of the POLYGAMMA FUNCTION *i(as),
*=£*iU)-£Mi) (7)
= ^*i(A) + ^*i(A)-> 2 (8)
= i* 1 (l)-i* 1 (|)-i^. (9)
Applying CONVERGENCE IMPROVEMENT to (3) gives
^=^E( TO + 1 )^C(m + 2), (10)
where ((z) is the Riemann Zeta Function and the
identity
1 1__ _ ^ 3 m -l
(l-3^) 2 (I-*) 2 ~ 2^ TO + 1 > 4 „
has been used (Flajolet and Vardi 1996). The Flajolet
and Vardi algorithm also gives
K - - 1 - n (i - ±-\ W-l
V2 11 \ 2»V/3(2*)
k^i^/i 2 ^ 1 )
(12)
where f3(z) is the Dirichlet Beta Function. Glaisher
(1913) gave
*-i-E
nC(2n + l)
16 n
(13)
(Vardi 1991, p. 159). W. Gosper used the related FOR-
MULA
K =
where
V2
*(2) - 1
n
-il/(2 fe+1 )
*(m)
-*(2 fc ) -1
K-m^rn _ l)4 m - 1 S m '
(14)
(15)
where B n is a Bernoulli Number and ip(x) is a Poly-
gamma Function (Finch). The Catalan constant may
also be defined by
Jo
K{k) dk,
(16)
where K(k) (not to be confused with Catalan's constant
itself, denoted K) is a complete Elliptic Integral of
the First Kind.
K =
7rln2
8
+£
at
2L(i+l)/2Ji2>
where
{o<} = {1,1,1,0,-1,-1,-1,0}
(17)
(18)
is given by the periodic sequence obtained by appending
copies of {1, 1, 1, 0, — 1, — 1, — 1, 0} (in other words,
en = a[(t-i) (mod 8)]+i for i > 8) and [x\ is the FLOOR
Function (Nielsen 1909).
see also Dirichlet Beta Function
References
Abramowitz, M, and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 807-808, 1972.
Adamchik, V. "32 Representations for Catalan's Con-
stant." http://www.wolfram.com/-victor/articles/
catalan/catalan.html.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 551—552, 1985.
Fee, G. J. "Computation of Catalan's Constant using Ra-
in anuj an' s Formula." ISAAC '90. Proc. Internal. Symp.
Symbolic Algebraic Cornp., Aug. 1990. Reading, MA:
Addison-Wesley, 1990.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/catalan/catalan. html.
Flajolet, P. and Vardi, I. "Zeta Function Expan-
sions of Classical Constants." Unpublished manu-
script. 1996. http://pauillac.inria.fr/algo/flajolet/
Publications/landau. ps.
Glaisher, J. W. L. "Numerical Values of the Series 1 - 1/3" +
1/5" - 1/7" + 1/9" - &c for n = 2, 4, 6." Messenger Math.
42, 35-58, 1913.
Gosper, R. W. "A Calculus of Series Rearrangements." In
Algorithms and Complexity: New Directions and Recent
Results (Ed. J. F. Traub). New York: Academic Press,
1976.
Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany:
Halle, pp. 105 and 151, 1909.
200 Catalan's Diophantine Problem
Catalan Number
Plouffe, S. "PloufiVs Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records .html.
Sloane, N. J. A. Sequences A014538 and A006752/M4593 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Srivastava, H. M. and Miller, E. A. "A Simple Reducible
Case of Double Hypergeometric Series involving Catalan's
Constant and Riemann's Zeta Function." Int. J. Math.
Educ. Sci. Technol. 21, 375-377, 1990.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. 159, 1991.
Yang, S. "Some Properties of Catalan's Constant G." Int. J.
Math. Educ. Sci. Technol 23, 549-556, 1992.
Catalan's Diophantine Problem
Find consecutive POWERS, i.e., solutions to
b d -,
a — c = 1,
excluding and 1. CATALAN'S CONJECTURE is that the
only solution is 3 2 - 2 3 = 1, so 8 and 9 (2 3 and 3 2 ) are
the only consecutive POWERS (again excluding and 1).
see also CATALAN'S CONJECTURE
References
Cassels, J. W. S. "On the Equation a x - 6^ = 1. II." Proc.
Cambridge Phil Soc. 56, 97-103, 1960.
Inkeri, K. "On Catalan's Problem." Acta Arith. 9, 285-290,
1964.
Catalan Integrals
Special cases of general FORMULAS due to Bessel.
Jo(\A 2 -2/ 2 ) = - / e ycosd cos(z sin 0)d6,
77 Jo
where J is a BESSEL FUNCTION OF THE FIRST KIND.
Now, let z = 1 — z' and y = 1 + z' . Then
Jo(2iv^) = - / e (1+z)cos6 cos[(l-z)sm0]d6.
n Jo
Catalan Number
The Catalan numbers are an INTEGER SEQUENCE {C n }
which appears in TREE enumeration problems of the
type, "In how many ways can a regular n-gon be di-
vided into n — 2 TRIANGLES if different orientations
are counted separately?" (EULER'S POLYGON DIVI-
SION Problem). The solution is the Catalan number
Cn-2 (Dorrie 1965, Honsberger 1973), as graphically il-
lustrated below (Dickau).
The first few Catalan numbers are 1, 2, 5, 14, 42, 132,
429, 1430, 4862, 16796, ... (Sloane's A000108). The
only Odd Catalan numbers are those of the form c 2 fc_i,
and the last DIGIT is five for k = 9 to 15. The only
PRIME Catalan numbers for n < 2 15 - 1 are C 2 = 2 and
C 3 = 5.
The Catalan numbers turn up in many other related
types of problems. For instance, the Catalan number
C n -i gives the number of BINARY BRACKETINGS of n
letters (CATALAN'S Problem). The Catalan numbers
also give the solution to the Ballot PROBLEM, the
number of trivalent Planted Planar Trees (Dickau),
^J^O^^
the number of states possible in an n-FLEXAGON, the
number of different diagonals possible in a FRIEZE PAT-
TERN with n+1 rows, the number of ways of forming
an n-fold exponential, the number of rooted planar bi-
nary trees with n internal nodes, the number of rooted
plane bushes with n EDGES, the number of extended
Binary Trees with n internal nodes, the number of
mountains which can be drawn with n upstrokes and
n downstrokes, the number of noncrossing handshakes
possible across a round table between n pairs of peo-
ple (Conway and Guy 1996), and the number of SE-
QUENCES with NONNEGATIVE PARTIAL SUMS which can
be formed from n Is and n -Is (Bailey 1996, Buraldi
1992)!
An explicit formula for C n is given by
'2n\ _ _^_ (2n)! _ (2n)!
n
C n —
1
n+1
n + 1 n! 2 (n + l)!n!'
(1)
■&mQm<s><^
where ( 2 ™) denotes a BINOMIAL COEFFICIENT and n\ is
the usual Factorial. A Recurrence Relation for
C n is obtained from
Cn+i (2n + 2)! (n+l)(n!) 2
C n (n + 2)[(n+l)!] 2 (2n)!
__ (2n + 2)(2n + l)(n + l)
(n + 2)(n-f-l) 2
_ 2(2n + l)(n + l) 2 _ 2(2n+l)
(n+l) 2 (n + 2) ~ n + 2 '
(2)
Catalan Number
Catalan Number 201
_ 2(2n + l)
t-'n+l — T~^ ^n*
n + 2
Other forms include
C n —
2-6-10---(4n-2)
(n + 1)!
2 n (2n~l)!!
(n + 1)!
(2n)!
n!(n+l)f
(3)
(4)
(5)
(6)
Segner's Recurrence Formula, given by Segner in
1758, gives the solution to Euler's POLYGON DIVISION
Problem
E n = E^En-x + EsE n -2 + . . . + E n -iE2. (7)
With Ei = E 2 = 1, the above RECURRENCE RELATION
gives the Catalan number C n _2 = Z2 n .
The Generating Function for the Catalan numbers
is given by
1 VI 4x = Y CnX " = i + x + 2x 2 + bx s + .... (8)
n=0
The asymptotic form for the Catalan numbers is
C k
v^FP/2
(9)
(Vardi 1991, Graham et al. 1994).
A generalization of the Catalan numbers is defined by
if pk \_ 1 (pk
(10)
for k > 1 (Klarner 1970, Hilton and Pederson 1991).
The usual Catalan numbers Ck = 2<ih are a special case
with j) —2. p dk gives the number of p-ary TREES with k
source-nodes, the number of ways of associating k appli-
cations of a given p-ary OPERATOR, the number of ways
of dividing a convex POLYGON into k disjoint (p + 1)-
gons with nonintersecting DIAGONALS, and the number
of p-GoOD PATHS from (0, -1) to (]fe, (p-l)k-l) (Hilton
and Pederson 1991).
A further generalization is obtained as follows. Let p
be an INTEGER > 1, let P k = (k,(p - l)k - 1) with
k > 0, and q < p - 1. Then define p d q o = 1 and let p d q k
be the number of p-GoOD PATHS from (1, q — 1) to Pk
(Hilton and Pederson 1991). Formulas for p d q i include
the generalized JONAH FORMULA
k
z = l
- pi
(11)
and the explicit formula
p^qk
p-q (pk - q\
ok — qyk ~ 1 J
A Recurrence Relation is given by
pd q k - 2_^
p&p — T,i P^Q-^Tyj
(12)
(13)
k + 1
where i,j, r > 1, k > 1, q < p — r, and i 4- j
(Hilton and Pederson 1991).
see also BALLOT PROBLEM, BINARY BRACKETING,
Binary Tree, Catalan's Problem, Catalan's
Triangle, Delannoy Number, Euler's Polygon
Division Problem, Flexagon, Frieze Pattern,
Motzkin Number, p-Good Path, Planted Planar
Tree, Schroder Number, Super Catalan Number
References
Alter, R. "Some Remarks and Results on Catalan Numbers."
Proc. 2nd Louisiana Conf. Comb., Graph Th., and Corn-
put, 109-132, 1971.
Alter, R. and Kubota, K. K. "Prime and Prime Power Divis-
ibility of Catalan Numbers." J. Combin. Th. 15,243-256,
1973.
Bailey, D. F. "Counting Arrangements of l's and — l's."
Math. Mag. 69, 128-131, 1996.
Brualdi, R. A. Introductory Combinatorics, 3rd ed. New
York: Elsevier, 1997.
Campbell, D. "The Computation of Catalan Numbers."
Math. Mag. 57, 195-208, 1984,
Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration
of Certain Sets of Planted Trees." J. Combin. Th. Ser. B
18, 209-221, 1975.
Chu, W. "A New Combinatorial Interpretation for General-
ized Catalan Numbers." Disc. Math. 65, 91-94, 1987.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New-
York: Springer- Verlag, pp. 96-106, 1996.
Dershowitz, N. and Zaks, S. "Enumeration of Ordered Trees."
Disc, Math. 31, 9-28, 1980.
Dickau, R. M. "Catalan Numbers." http: //forum.
swarthmore.edu/advanced/robertd/catalan.html.
Dorrie, H. "Euler's Problem of Polygon Division." §7 in 100
Great Problems of Elementary Mathematics: Their His-
tory and Solutions. New York: Dover, pp. 21-27, 1965.
Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How
Likely is a Function to be Convex?" Math. Mag. 61, 211—
219, 1988.
Gardner, M. "Catalan Numbers." Ch. 20 in Time Travel and
Other Mathematical Bewilderments. New York: W. H.
Freeman, 1988.
Gardner, M. "Catalan Numbers: An Integer Sequence that
Materializes in Unexpected Places." Sci. Amer. 234, 120-
125, June 1976.
Gould, H. W. Bell & Catalan Numbers: Research Bibliogra-
phy of Two Special Number Sequences, 6th ed. Morgan-
town, WV: Math Monongliae, 1985.
Graham, R. L.; Knuth, D. E.; and Patashnik, 0. Exercise
9.8 in Concrete Mathematics: A Foundation for Computer
Science, 2nd ed. Reading, MA: Addison- Wesley, 1994.
Guy, R. K. "Dissecting a Polygon Into Triangles." Bull.
Malayan Math. Soc. 5, 57-60, 1958.
Hilton, P. and Pederson, J. "Catalan Numbers, Their Gen-
eralization, and Their Uses." Math. Int. 13, 64-75, 1991.
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 130-134, 1973.
202
Catalan's Problem
Catalan's Surface
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 146-150, 1985,
Klarner, D. A. "Correspondences Between Plane Trees and
Binary Sequences." J. Comb. Th. 9, 401-411, 1970.
Rogers, D. G. "Pascal Triangles, Catalan Numbers and Re-
newal Arrays." Disc. Math. 22, 301-310, 1978.
Sands, A. D. "On Generalized Catalan Numbers." Disc.
Math. 21, 218-221, 1978.
Singmaster, D. "An Elementary Evaluation of the Catalan
Numbers." Amer. Math. Monthly 85, 366-368, 1978.
Sloane, N. J. A. A Handbook of Integer Sequences. Boston,
MA: Academic Press, pp. 18-20, 1973.
Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addis on- Wesley, pp. 187-188 and 198-199,
1991.
Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Numbers. London: Penguin, pp. 121-122, 1986.
Catalan's Problem
The problem of finding the number of different ways in
which a PRODUCT of n different ordered FACTORS can be
calculated by pairs (i.e., the number of BINARY Brack-
ETINGS of n letters). For example, for the four FAC-
TORS a, 6, c, and d } there are five possibilities: ((ab)c)d,
(a(bc))d, (ab)(cd), a((bc)d) y and a(b(cd)). The solution
was given by Catalan in 1838 as
c: =
2 ■ 6 • 10 • (4n - 6)
r\
C'
and is equal to the CATALAN NUMBER C n -i
see also Binary Bracketing, Catalan's Diophan-
tine Problem, Euler's Polygon Division Problem
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 23,
1965.
Catalan Solid
The Dual Polyhedra of the Archimedean Solids,
given in the following table.
Archimedean Solid
Dual
rhombicosidodecahedron
small rhombicuboctahedron
great rhombicuboctahedron
great rhombicosidodecahedron
truncated icosahedron
snub dodecahedron
(laevo)
snub cube
(laevo)
cuboctahedron
icosidodecahedron
truncated octahedron
truncated dodecahedron
truncated cube
truncated tetrahedron
deltoidal hexecontahedron
deltoidal icositetrahedron
disdyakis dodecahedron
disdyakis triacontahedron
pentakis dodecahedron
pentagonal hexecontahedron
(dextro)
pentagonal icositetrahedron
(dextro)
rhombic dodecahedron
rhombic triacontahedron
tetrakis hexahedron
triakis icosahedron
triakis octahedron
triakis tetrahedron
Here are the Archimedean DUALS (Holden 1971,
Pearce 1978) displayed in alphabetical order (left to
right, then continuing to the next row).
Here are the Archimedean solids paired with the corre-
sponding Catalan solids.
O
© Q
© €
© w
see also Archimedean Solid, Dual Polyhedron,
Semiregular Polyhedron
References
Catalan, E. "Memoire sur la Theorie des Polyedres." J.
I'Ecole Polytechnique (Paris) 41, 1-71, 1865.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Catalan's Surface
A Minimal Surface given by the parametric equations
x(u, v) = u — sin u cosh v
y(u, v) = 1 — cos u cosh v
z(u,v) = 4sin(|w)sinh(|u)
(i)
(2)
(3)
Catalan's Triangle
Categorical Variable 203
(Gray 1993), or
x(r, <j>) = asin(2</>) — 2a<fi + \o>v 2 cos(2<fi)
y(r, <j>) = — acos(2<p) — ~av 2 cos(2(p)
z(r,(fi) = 2avsin0,
where
-r +
(4)
(5)
(6)
(?)
(do Carmo 1986).
References
Catalan, E. "Memoir sur les surfaces dont les rayons de
courburem en chaque point, sont egaux et des signes con-
traires." C. R. Acad. Sci. Paris 41, 1019-1023, 1855.
do Carmo, M. P. "Catalan's Surface" §3.5D in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 45-46, 1986.
Fischer, G. (Ed.). Plates 94-95 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 90-91, 1986.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp, 448-449, 1993.
Catalan's Triangle
A triangle of numbers with entries given by
(n + m)\{n — m + 1)
Cnrn= m!(n+l)!
for < m < n, where each element is equal to the one
above plus the one to the left. Furthermore, the sum
of each row is equal to the last element of the next row
and also equal to the CATALAN NUMBER C n .
5
14 14
14 28 42 42
20 48 90 132 132
(Sloane's A009766).
see also Bell Triangle, Clark's Triangle, Eu-
ler's Triangle, Leibniz Harmonic Triangle, Num-
ber Triangle, Pascal's Triangle, Prime Trian-
gle, Seidel-Entringer-Arnold Triangle
References
Sloane, N. J. A. Sequence A009766 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Catalan's Trisectrix
see TSCHIRNHAUSEN CUBIC
Catastrophe
see Butterfly Catastrophe, Catastrophe The-
ory, Cusp Catastrophe, Elliptic Umbilic Catas-
trophe, Fold Catastrophe, Hyperbolic Umbilic
Catastrophe, Parabolic Umbilic Catastrophe,
Swallowtail Catastrophe
Catastrophe Theory
Catastrophe theory studies how the qualitative nature
of equation solutions depends on the parameters that
appear in the equations. Subspecializations include bi-
furcation theory, nonequilibrium thermodynamics, sin-
gularity theory, synergetics, and topological dynamics.
For any system that seeks to minimize a function, only
seven different local forms of catastrophe "typically" oc-
cur for four or fewer variables: (1) FOLD CATASTROPHE,
(2) Cusp Catastrophe, (3) Swallowtail Catastro-
phe, (4) Butterfly Catastrophe, (5) Elliptic Um-
bilic Catastrophe, (6) Hyperbolic Umbilic Catas-
trophe, (7) Parabolic Umbilic Catastrophe.
More specifically, for any system with fewer than five
control factors and fewer than three behavior axes, these
are the only seven catastrophes possible. The following
tables gives the possible catastrophes as a function of
control factors and behavior axes (Goetz).
Control
Factors
1 Behavior
Axis
2 Behavior
Axes
fold
cusp
swallowtail
butterfly-
hyperbolic umbilic, elliptic umbilic
parabolic umbilic
References
Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin: Springer-
Verlag, 1992.
Gilmore, R. Catastrophe Theory for Scientists and Engi-
neers. New York: Dover, 1993.
Goetz, P. "Phil's Good Enough Complexity Dictionary."
http ; //www . cs .but f alo . edu/~goetz/dict .html.
Saunders, P. T. An Introduction to Catastrophe Theory.
Cambridge, England: Cambridge University Press, 1980.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, p. 211, 1987.
Thorn, R. Structural Stability and Morphogenesis: An Out-
line of a General Theory of Models. Reading, MA: Read-
ing, MA: Addison- Wesley, 1993.
Thompson, J. M. T. Instabilities and Catastrophes in Science
and Engineering. New York: Wiley, 1982.
Woodcock, A. E. R. and Davis, M. Catastrophe Theory. New
York: E. P. Dutton, 1978.
Zeeman, E. C. Catastrophe Theory — Selected Papers 1972-
1977. Reading, MA: Addis on- Wesley, 1977.
Categorical Game
A Game in which no draw is possible.
Categorical Variable
A variable which belongs to exactly one of a finite num-
ber of Categories.
204 Category
Catenary
Category
A category consists of two things: an OBJECT and a
MORPHISM (sometimes called an "arrow"). An OB-
JECT is some mathematical structure (e.g., a GROUP,
Vector Space, or Differentiable Manifold) and a
Morphism is a Map between two Objects. The Mor-
PHISMS are then required to satisfy some fairly natural
conditions; for instance, the IDENTITY MAP between
any object and itself is always a Morphism, and the
composition of two MORPHISMS (if defined) is always a
Morphism.
One usually requires the MORPHISMS to preserve the
mathematical structure of the objects. So if the objects
are all groups, a good choice for a MORPHISM would be
a group HOMOMORPHISM. Similarly, for vector spaces,
one would choose linear maps, and for differentiable
manifolds, one would choose differentiable maps.
In the category of TOPOLOGICAL SPACES, homomor-
phisms are usually continuous maps between topologi-
cal spaces. However, there are also other category struc-
tures having TOPOLOGICAL SPACES as objects, but they
are not nearly as important as the "standard" category
of Topological Spaces and continuous maps.
see also Abelian Category, Allegory, Eilenberg-
Steenrod Axioms, Groupoid, Holonomy, Logos,
monodromy, topos
References
Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster-
dam, Netherlands: North-Holland, 1990.
Category Theory
The branch of mathematics which formalizes a number
of algebraic properties of collections of transformations
between mathematical objects (such as binary relations,
groups, sets, topological spaces, etc.) of the same type,
subject to the constraint that the collections contain the
identity mapping and are closed with respect to compo-
sitions of mappings. The objects studied in category
theory are called CATEGORIES.
see also CATEGORY
Catenary
The curve a hanging flexible wire or chain assumes when
supported at its ends and acted upon by a uniform grav-
itational force. The word catenary is derived from the
Latin word for "chain." In 1669, Jungius disproved
Galileo's claim that the curve of a chain hanging un-
der gravity would be a PARABOLA (MacTutor Archive).
The curve is also called the ALYSOID and CHAINETTE.
The equation was obtained by Leibniz, Huygens, and
Johann Bernoulli in 1691 in response to a challenge by
Jakob Bernoulli.
Huygens was the first to use the term catenary in a letter
to Leibniz in 1690, and David Gregory wrote a treatise
on the catenary in 1690 (MacTutor Archive). If you roll
a PARABOLA along a straight line, its FOCUS traces out
a catenary. As proved by Euler in 1744, the catenary is
also the curve which, when rotated, gives the surface of
minimum SURFACE Area (the Catenoid) for the given
bounding CIRCLE.
The Cartesian equation for the catenary is given by
y =l a (e x/a + e- K/a ) = acoshg), (1)
and the Cesaro Equation is
{s 2 +a 2 )K=-a. (2)
The catenary gives the shape of the road over which a
regular polygonal "wheel" can travel smoothly. For a
regular n-gon, the corresponding catenary is
where
y = -Acosh I — j ,
A = R cos
(3)
(4)
The Arc Length, Curvature, and Tangential An-
gle are
s = asinh ( — ) , (5)
n=--sedi 2 (-) y (6)
a \a/
<f>= -2 tan" 1 [tanh (^-)1 * (?)
The slope is proportional to the Arc Length as mea-
sured from the center of symmetry.
see also Calculus of Variations, Catenoid, Linde-
lof's Theorem, Surface of Revolution
References
Geometry Center. "The Catenary." http://www.geom.umn.
edu/zoo/diffgeom/surf space/catenoid/catenary.html.
Gray, A. "The E volute of a Tract rix is a Catenary." §5.3
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 80-81, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 195 and 199-200, 1972.
Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A
Book of Curves. Cambridge, England: Cambridge Univer-
sity Press, pp. 118-124, 1967,
MacTutor History of Mathematics Archive. "Catenary."
http : //www-groups . dcs . st-and . ac . uk/ -history/Curves
/Catenary .html.
Pappas, T. "The Catenary & the Parabolic Curves." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 34, 1989.
Yates, R. C. "Catenary." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 12-14,
1952.
Catenary Evolute
Catenary Evolute
x = a[x — \ sinh(2t)]
y = 2a cosh t.
Catenary Involute
\
\
\
/
y
/
The parametric equation for a Catenary is
dx
dt
dr
dt
1
sinh 2
ayl + sinh 2 t = acoshi
and
dr
rpi dt
i dt |
secht
tanhi
(1)
(2)
(3)
(4)
ds 2 = \dr 2 \ = a 2 (I + sinh 2 t) dt 2 = a 2 cosh 2 <ft 2 (5)
dt
a cosh i.
Therefore,
-•/
cosh tdt = a sinh £
and the equation of the INVOLUTE is
x = a(t — tanht)
y — asechi.
This curve is called a TRACTRIX.
(6)
(7)
(8)
(9)
Catenoid 205
Catenary Radial Curve
\
/
\
/
/ ^
"^^v^
S^^***'^
^^^_
^^^^
The Kampyle of Eudoxus.
Catenoid
A Catenary of Revolution. The catenoid and Plane
are the only SURFACES OF Revolution which are also
Minimal Surfaces. The catenoid can be given by the
parametric equations
x = ccosh
cosu
y = c cosh ( - J sin u
(i)
(2)
(3)
where u G [0, 2w). The differentials are
dx — sinh ( - j cos u dv - cosh ( - J sin u du (4)
dy = sinh I - J sin u dv -f cosh [ - j cos u du (5)
dz = du, (6)
so the Line Element is
ds 2 = dx 2 + dy 2 + dz 2
= [sinh 2 Q) + l] dv 2 + cosh 2 Q) du *
= cosh 2 f^\ dv 2 + cosh 2 (-) du 2 . (7)
The Principal Curvatures are
Kl — — sech 2 f - j
K2 — - sech 2 ( - ) •
The Mean Curvature of the catenoid is
(8)
(9)
(10)
206
Caterpillar Graph
Cauchy Distribution
and the GAUSSIAN CURVATURE is
(i)
(ii)
The HELICOID can be continuously deformed into a
catenoid with c = 1 by the transformation
x(u, v) = cos a sinh v sin u + sin a cosh v cos u (12)
y(u, v) = — cos a sinh v cos u -f sin a cosh f sin u (13)
z(?z, u) = u cos a + v sin a, (14)
where a = corresponds to a HELICOID and a = n/2
to a catenoid.
see also CATENARY, COSTA MINIMAL SURFACE, HELI-
COID, Minimal Surface, Surface of Revolution
References
do Carmo, M. P. "The Catenoid." §3.5A in Mathematical
Models from the Collections of Universities and Museums
(Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43,
1986.
Fischer, G. (Ed.). Plate 90 in Mathematische Modelle/
Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 86, 1986.
Geometry Center. "The Catenoid." http://www.geom.umn,
edu/zoo/diffgeom/surf space/catenoid/.
Gray, A. "The Catenoid." §18.4 Modern Differential Geom-
etry of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 367-369, 1993.
Meusnier, J. B. "Memoire sur la courbure des surfaces."
Mem. des savans etrangers 10 (lu 1776), 477-510, 1785.
Caterpillar Graph
A TREE with every NODE on a central stalk or only one
EDGE away from the stalk.
References
Gardner, M. Wheels, Life, and other Mathematical Amuse-
ments. New York: W. H. Freeman, p. 160, 1983.
Cattle Problem of Archimedes
see Archimedes' Cattle Problem
Cauchy Binomial Theorem
V^ y m q m(m+l)/2 ( tl
m=0 ^
J[(l + yq k ),
where ( n ) is a Gaussian Coefficient.
\m/ q
see also g-BlNOMIAL THEOREM
Cauchy Boundary Conditions
Boundary Conditions of a Partial Differential
Equation which are a weighted Average of Dirich-
let Boundary Conditions (which specify the value
of the function on a surface) and Neumann Boundary
CONDITIONS (which specify the normal derivative of the
function on a surface).
see also Boundary Conditions, Cauchy Prob-
lem, Dirichlet Boundary Conditions, Neumann
Boundary Conditions
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 678-679, 1953.
Cauchy's Cosine Integral Formula
/.tt/2
/ '
J-rr/2
a + u-2 ni0(v.-v+2£)
dO
7rV(fl + V ~ 1)
2*+"- 2 r( M + 0rV-0'
where F(z) is the GAMMA Function.
Cauchy Criterion
A Necessary and Sufficient condition for a Se-
quence Si to CONVERGE. The Cauchy criterion is sat-
isfied when, for all e > 0, there is a fixed number N such
that \Sj - Si\ < e for all i,j > N.
Cauchy Distribution
The Cauchy distribution, also called the Lorentzian
Distribution, describes resonance behavior. It also de-
scribes the distribution of horizontal distances at which
a Line Segment tilted at a random Angle cuts the
x-AxiS. Let 6 represent the ANGLE that a line, with
fixed point of rotation, makes with the vertical axis, as
shown above. Then
tan# :
b
6 = tan~
■(?)
dx
bdx
1 + fJ 6 b 2 -rx 2 '
so the distribution of ANGLE is given by
<W_ _ 1 bdx
7T 7T b 2 + X 2 '
(i)
(2)
(3)
(4)
Cauchy Distribution
This is normalized over all angles, since
/7T/2
■tt/2
d9
= 1
(5)
and
f
J — c
i feds _ i [-,/nr
■K b 2 +X 2 7T L VX/J _oo
= i[i w -(-i ff )] = l.
(6)
The general Cauchy distribution and its cumulative dis-
tribution can be written as
P(x)
2 X
7r(x- M ) 2 + (|r)2
.(*)=I + i tan -l(^)
(7)
(8)
where T is the FULL WIDTH AT HALF MAXIMUM (r =
26 in the above example) and /x is the MEAN (/x — in
the above example). The Characteristic Function
is
<m
dx
~ 7T / 1
t/ — oo
_ e -i M t-r|ti/2^
1 + x 2
cos(Tta/2)
+ (r^/2) 2
dz
The Moments are given by
2
\i2 = cr = oo
for ji =
M3
.oo for fi /
/44 = oo,
(9)
(10)
(11)
(12)
and the STANDARD DEVIATION, SKEWNESS, and KUR-
TOSIS by
_ f for fj, =
71 ~ I oo for /x #
72 = oo.
(13)
(14)
(15)
If X and Y are variates with a NORMAL DISTRIBUTION,
then Z = X/Y has a Cauchy distribution with MEAN
fi — and full width
(16)
Cauchy Inequality 207
see a/so Gaussian Distribution, Normal Distribu-
tion
References
Spiegel, M. R, Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, pp. 114-115, 1992.
Cauchy Equation
see Euler Equation
Cauchy's Formula
The Geometric Mean is smaller than the Arith-
metic Mean,
1/JV
n~) <%=■
Cauchy Functional Equation
The fifth of HlLBERT'S PROBLEMS is a generalization of
this equation.
Cauchy-Hadamard Theorem
The Radius of Convergence of the Taylor Series
ao + cl\z + aiz + . . .
is
1
r =
lim (Kl) 1 /"
n— too
see also RADIUS OF CONVERGENCE, TAYLOR SERIES
Cauchy Inequality
A special case of the HOLDER SUM INEQUALITY with
y ^flfc&fc
E- 2 E»* 2 • w
Ok I < ^^
. k=l / \ k-1 / \ k=l
where equality holds for ak = cbk- In 2-D, it becomes
(2)
It can be proven by writing
(a 2 +6 2 )(c 2 + a 2 ) > {ac + bdf.
Y^iatx + bi) 2 = f> 2 (x+ ^-) 2 = 0. (3)
i=l i=l
If bi/di is a constant c, then x = — c. If it is not a
constant, then all terms cannot simultaneously vanish
for REAL x, so the solution is COMPLEX and can be
found using the QUADRATIC EQUATION
2j2a i b i ±^4&a i b i ) -4^a, 2 ^6 i 2
2J>^
• (4)
208 Cauchy Integral Formula
In order for this to be COMPLEX, it must be true that
$>* <£«.'£«
(5)
with equality when hi /at is a constant. The VECTOR
derivation is much simpler,
(a-b) 2 = aV cos 2 6 < ab 2 ,
yhere
= E^ 2
2 _ V^ 2
a = a • a — x
(6)
(7)
and similarly for b.
see also Chebyshev Inequality, Holder Sum In-
equality
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.
Cauchy Integral Formula
r Yo Y r
Given a Contour Integral of the form
/
/(*) dz
Z — Zo
(1)
define a path 70 as an infinitesimal CIRCLE around the
point zo (the dot in the above illustration). Define the
path 7 r as an arbitrary loop with a cut line (on which
the forward and reverse contributions cancel each other
out) so as to go around zq.
The total path is then
7 = 7o + It
(2)
tm±=tm*+tixr=L. (3 )
L z ~ z ° L a z ~ z o L r ~ -
Z - Zq
Prom the Cauchy Integral Theorem, the Contour
Integral along any path not enclosing a Pole is 0.
Therefore, the first term in the above equation is since
70 does not enclose the Pole, and we are left with
r Hz)dz = r f_(z)dz
Cauchy Integral Formula
Now, let z = z + re iB , so dz = ire w d9. Then
f fWdz = f
A z ~ Zo A,
-I
f{Zo + r / ) ire ig d0
re™
f{zo + re ie )id9.
(5)
But we are free to allow the radius r to shrink to 0, so
f Hz)dz = lim f f f ZQ + re ™\ id0 = f f( ZQ )idO
/ Z - ZQ r->Q / /
= if(zo) [ dd = 2<Kif(z ), (6)
J It
and
/(*>)
2 ™L
f(z) dz
z — Zq'
(7)
If multiple loops are made around the POLE, then equa-
tion (7) becomes
t/ 7
)dz
(8)
where 71(7, z ) is the WINDING NUMBER.
A similar formula holds for the derivatives of f(z),
f(zo) = i im n«+h)-m
h^t-0 h
= ]im J-([ f^ dz - f M*z\
/i^o 2izih \Jz — zo-~h J z — zo I
_ y 1 f f(z)[(z - z ) - (z - zo - h)} dz
h^o 2nih j
lim
h
im — — - /
■-+0 27Vih I
2 ™ 7 7 ( z - >
(z - zo - h)(z - zo)
hf(z) dz
(z — zo — h)(z — zq)
Iterating again,
™-ht£
z) dz
zo) 3
(9)
(10)
Continuing the process and adding the WINDING Num-
ber n,
see also Morera's Theorem
References
Arfken, G. "Cauchy's Integral Formula." §6.4 in Mathemati-
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 371-376, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 367-372, 1953.
Cauchy Integral Test
Cauchy Ratio Test 209
Cauchy Integral Test
see Integral Test
Cauchy Integral Theorem
If / is continuous and finite on a simply connected region
R and has only finitely many points of nondifferentia-
bility in i£, then
£
f(z)dz =
(1)
for any closed CONTOUR 7 completely contained in R.
Writing z as
z = x + iy (2)
and f(z) as
f(z)=u + iv (3)
then gives
(p f(z) dz — \ (u + iv)(dx + idy)
= / udx -vdy + i / vdx + udy. (4)
Prom Green's Theorem,
J f(x J y)dx-g(x J y)dy=- fj (f| + fj) <**<fo
/ f(x,y)dx+g{x,y)dy^ // (
so (4) becomes
<9x %
(5)
dxdy (6)
-h//(£-£')«M* (7)
But the Cauchy-Riemann Equations require that
du _ <9v
dx dy
du dv
dy dx '
(8)
(9)
£
f(z)dz = 0,
Q. E. D.
For a Multiply Connected region,
f f(z)dz= f f(z)dz.
(10)
(11)
see also Cauchy Integral Theorem, Morera's
Theorem, Residue Theorem (Complex Analysis)
References
Arfken, G. "Cauchy's Integral Theorem." §6.3 in Mathemati-
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 365-371, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 363-367, 1953.
Cauchy-Kovalevskaya Theorem
The theorem which proves the existence and uniqueness
of solutions to the Cauchy Problem.
see also Cauchy Problem
Cauchy-Lagrange Identity
(ax 2 + a 2 2 + ■ • • + an 2 )(&i 2 + b 2 2 + . . . + b n 2 )
= (aib 2 - a 2 h) 2 + (ai& 3 - a3&i) 2 + • • •
+(a n -i&n - a n &n-i) •
From this identity, the n-D Cauchy Inequality fol-
lows.
Cauchy-Maclaurin Theorem
see Maclaurin-Cauchy Theorem
Cauchy Mean Theorem
For numbers > 0, the Geometric Mean < the Arith-
metic Mean.
Cauchy Principal Value
fix) dx = lim
-00
/ f(x)dx
J-R
I
J a
PV I f{x)dx = lim
I f{x)dx+ / f(x)dx
J a J c+e
where e > and a < c <b.
References
Arfken, G. Mathematical Methods for Physicists f 3rd ed. Or-
lando, FL: Academic Press, pp. 401-403, 1985.
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 158, 1991.
Cauchy Problem
Tf f(x,y) is an ANALYTIC FUNCTION in a NEIGHBOR-
HOOD of the point (xo,yo) (i.e., it can be expanded in
a series of Nonnegative Integer Powers of (x - x )
and (y — yo)), find a solution y(x) of the DIFFERENTIAL
Equation
dy
dx
/(*),
with initial conditions y = yo and x = xq. The existence
and uniqueness of the solution were proven by Cauchy
and Kovalevskaya in the Cauchy-Kovalevskaya The-
orem. The Cauchy problem amounts to determining
the shape of the boundary and type of equation which
yield unique and reasonable solutions for the CAUCHY
Boundary Conditions.
see also Cauchy Boundary Conditions
Cauchy Ratio Test
see Ratio Test
210 Cauchy Remainder Form
Cauchy Root Test
Cauchy Remainder Form
The remainder of n terms of a TAYLOR Series is given
by
(x-c) n_1 (a;-a)
Rn —
where a < c < x.
(n-l)!
r'(c),
Cauchy- Riemann Equations
Let
f(x,y) = u(x,y) + iv(x,y) y
where
z = x + iy,
(1)
(2)
These are known as the Cauchy- Riemann equations.
They lead to the condition
d 2 u
d 2 v
dxdy dxdy
(14)
The Cauchy-Riemann equations may be concisely writ-
ten as
(du .dv\ . ( du .dv\
\dx dx) \dy dy J
df _ df df _ (du ( . dv \ t . ( du t . dv
)x dy
du dv
dx) \dy
. . du dv\
+ * -^- + -^- =0.
dz* dx dy \dx dx) \dy dy
du dv
dx dy J ' " \dy dx
(15)
dz = dx -\- i dy.
(3)
The total derivative of / with respect to z may then be
computed as follows.
(4)
(5)
x = z - ty,
dy __ 1
dz
dx
dz
and
In terms of u and t>, (8) becomes
df __ / du .dv\ . I du .dv
\dx dx) \dy dy
dz \dx dx j
(6)
(7)
V = dldx + dldy = dl_ i dl
dz dx dz dy dz dx dy'
(du ,dv\ ( .du dv\ ,„,
= U + ^) + (-^ + ^J- (9)
Along the real, or as- Axis, df /dy = 0, so
df _ du .dv . .
dz dx dx
Along the imaginary, or y-axis, df /dx = 0, so
df _ .du dv . .
dz dy dy *
If / is Complex Differentiable, then the value of the
derivative must be the same for a given dz, regardless of
its orientation. Therefore, (10) must equal (11), which
requires that
and
dv du
dx dy'
(13)
In Polar Coordinates,
f(re ie ) = R(r,0)e i@(r ' e \
so the Cauchy-Riemann equations become
dR
dr
IdR
RdQ
r d6
— = -*$©.
r dd dr
(16)
(17)
(18)
If u and v satisfy the Cauchy-Riemann equations, they
also satisfy Laplace's Equation in 2-D, since
d^u d?u
dx 2 dy 2
d_ (dv
dx \dy
)+ £(-£)- " 9 >
d 2 v d 2 v _ d ( du\ d (du\_ { .
dx 2 dy 2 dx \ dy J dy \dx)
By picking an arbitrary f(z), solutions can be found
which automatically satisfy the Cauchy-Riemann equa-
tions and Laplace's Equation. This fact is used to
find so-called Conformal Solutions to physical prob-
lems involving scalar potentials such as fluid flow and
electrostatics.
see also Cauchy Integral Theorem, Conformal
Solution, Monogenic Function, Polygenic Func-
tion
References
Abramowitz, M . and Stegun, C . A . (Eds . ) . Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 17, 1972.
Arfken, G. "Cauchy-Riemann Conditions." §6.2 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 3560-365, 1985.
Cauchy's Rigidity Theorem
see Rigidity Theorem
Cauchy Root Test
see Root Test
Cauchy-Schwarz Integral Inequality
Cayley Cubic 211
Cauchy-Schwarz Integral Inequality
Let f(x) and g(x) by any two Real integrable functions
of [a, 6], then
'/"■
x)g(x) dx
<
nb "I r pb
I f 2 (x)dx / g 2 (x)dx
yd J Lv a
with equality IFF f(x) = kg(x) with k real.
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1099, 1993.
Cauchy-Schwarz Sum Inequality
|a-b|<|ailb|.
E
, fe = l
akbk
Equality holds IFF the sequences ai, a2, ... and &i, 62,
. . . are proportional.
see also Fibonacci Identity
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1092, 1979.
Cauchy Sequence
A Sequence ai, 02, . . . such that the Metric d(a m , a n )
satisfies
lim d(a m , a n ) = 0.
min(m,n)— ^00
Cauchy sequences in the rationals do not necessarily
Converge, but they do Converge in the Reals.
Real Numbers can be defined using either Dedekind
Cuts or Cauchy sequences.
see also Dedekind Cut
Cauchy Test
see Ratio Test
Caustic
The curve which is the ENVELOPE of reflected (CAT-
ACAUSTIC) or refracted (DIACAUSTIC) rays of a given
curve for a light source at a given point (known as the
Radiant Point). The caustic is the Evolute of the
Orthotomic.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York; Dover, p. 60, 1972.
Lee, X. "Caustics." http://www.best.com/-xah/Special
PlaneCurves_dir/Caustics-dir/caustics.html.
Lockwood, E. H. "Caustic Curves." Ch. 24 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 182-185, 1967.
Yates, R. C. "Caustics." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 15-20,
1952.
Cavalieri's Principle
1. If the lengths of every one-dimensional slice are equal
for two regions, then the regions have equal Areas.
2. If the AREAS of every two-dimensional slice (CROSS-
Section) are equal for two SOLIDS, then the SOLIDS
have equal Volumes.
see also Cross-Section, Pappus's Centroid Theo-
rem
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 126 and 132,
1987.
Cayley Algebra
The only Nonassociative Division Algebra with
REAL SCALARS. There is an 8-square identity corre-
sponding to this algebra. The elements of a Cayley al-
gebra are called CAYLEY NUMBERS or OCTONIONS.
References
Kurosh, A. G. General Algebra. New York: Chelsea, pp. 226-
28, 1963.
Cayley-Bacharach Theorem
Let Xi, X 2 C P 2 be CUBIC plane curves meeting in nine
points pi, . . . , pq. If X C P 2 is any CUBIC containing
Pi, ■ - ■ , Ps, then X contains pg as well. It is related to
GORENSTEIN RINGS, and is a generalization of PAPPUS'S
Hexagon Theorem and Pascal's Theorem,
References
Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach
Theorems and Conjectures." Bull. Amer. Math. Soc. 33,
295-324, 1996.
Cayley Cubic
*
**4
A Cubic Ruled Surface (Fischer 1986) in which the
director line meets the director CONIC SECTION. Cay-
ley's surface is the unique cubic surface having four OR-
DINARY Double Points (Hunt), the maximum possible
for Cubic Surface (EndraB). The Cayley cubic is in-
variant under the TETRAHEDRAL GROUP and contains
exactly nine lines, six of which connect the four nodes
pairwise and the other three of which are coplanar (En-
draB).
If the Ordinary Double Points in projective 3-space
are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0,
0, 0, 1), then the equation of the surface in projective
coordinates is
1 1 1 1
— + — + — + — =0
Xq X\ X2 X3
212
Cay ley Cubic
Cayley -Hamilton Theorem
(Hunt). Denning "affine" coordinates with plane at in-
finity v — Xq 4- x\ + X2 + 2^3 and
Xq
v
v
X 2
then gives the equation
-b(x 2 y+x 2 z+y 2 x+y 2 z+z 2 y+z 2 x)+2(xy+xz+yz) =
plotted in the left figure above (Hunt). The slightly
different form
4(x 3 + y + z 3 + w ) - (x + y'+ z + •
is given by Endrafi which, when rewritten in Tetrahe-
dral Coordinates, becomes
x + y — xz + yz-\-z — 1 = 0,
plotted in the right figure above.
The Hessian of the Cayley cubic is given by
= Z 2 (xia:2 + X 1 X 3 + X2Xz) + X l (x X2 + X0X3 + Z2Z3)
+xl(x Xi + XqX 3 + X1X3) + xI(xqX! + X X 2 + X1X2).
in homogeneous coordinates xq, #1, x 2 , and X3. Taking
the plane at infinity as v = 5(:ro + x\ + #2 + 2xz)j2 and
setting a;, y, and 2 as above gives the equation
25[x 3 (y+z)+y 3 (x+z)+z 3 {x+y)]+b0(x 2 y 2 +x 2 z 2 +y 2 z 2 )
— 125(x 2 yz + y xz-\-z xy)-{-60xyz — 4(xy-{-xz-\-yz) = 0,
plotted above (Hunt). The Hessian of the Cayley cubic
has 14 ORDINARY Double Points, four more than a
the general Hessian of a smooth CUBIC SURFACE (Hunt).
References
Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV-
Mitteilungen 4, 17-20, Apr. 1995.
Endrafi, S. "The Cayley Cubic." http://www.mathematik.
uni-mainz . de/AlgebraischeGeometrie/docs/
Ecayley.shtml.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 14, 1986.
Fischer, G. (Ed.). Plate 33 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 33, 1986.
Hunt, B. "Algebraic Surfaces." http://www.mathematik.
uni-kl . de/-wwwagag/Galerie . html.
Hunt, B. The Geometry of Some Special Arithmetic Quo-
tients. New York: Springer- Verlag, pp. 115-122, 1996.
Nordstrand, T. "The Cayley Cubic." http://www.uib.no/
people/nfytn/cleytxt.htm.
Cayley Graph
The representation of a GROUP as a network of directed
segments, where the vertices correspond to elements and
the segments to multiplication by group generators and
their inverses.
see also Cayley Tree
References
Grossman, I. and Magnus, W. Groups and Their Graphs.
New York: Random House, p. 45, 1964.
Cayley's Group Theorem
Every Ftntte GROUP of order n can be represented as
a Permutation Group on n letters, as first proved by
Cayley in 1878 (Rotman 1995).
see also Finite Group, Permutation Group
References
Rotman, J, J. An Introduction to the Theory of Groups, J^th
ed. New York: Springer- Verlag, p. 52, 1995.
Cayley-Hamilton Theorem
Given
a>\\ ~
X
ai2
aim
0,21
&22 —
X
ft2m
dml
dm2
a>mrn X
— X ~T~ Cjn — \X
771—1 ,
• + c ,
(1)
then
A m + c m - 1 A m - 1 + ... + c l = 0,
(2)
where I is the Identity Matrix. Cayley verified this
identity for m = 2 and 3 and postulated that it was true
for all m. For m = 2, direct verification gives
a — x b
c d — x
= (a — x)(d — x) — be
— x 2 — (a + d)x + {ad — be) = x 2 + c\x + C2 (3)
Cayley's Hypergeometric Function Theorem
Cayley-Klein Parameters 213
A =
A 2 =
-{a + d)A =
(ad — be) I =
a b
c d
a b
a b
c d
c d
a 2 + be ab + bd
ac + cd
be + d 2
—a — ad —ab —
bd'
—ac — dc —ad — d 2
ad — be
ad — be
)
\-(ad-bc)\ =
"o o"
(4)
(5)
(6)
(7)
(8)
The Cayley-Hamilton theorem states that a n x n MA-
TRIX A is annihilated by its Characteristic Poly-
nomial det(xl — A), which is monic of degree n.
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed, San Diego, CA: Academic
Press, p. 1117, 1979.
Segercrantz, J. "Improving the Cayley-Hamilton Equation
for Low-Rank Transformations." Amer. Math. Monthly
99, 42-44, 1992.
Cayley's Hypergeometric Function Theorem
If
oo
(1 - z) a+h ~ c 2 Fi (2a, 26; 2c; z) = VJ a n z n ,
n =
then
2 Fi(a,6;c+ \- 1 z) 2 F 1 (c- a,c- b;e\\z)
oo
= E
( c )" a ,»
where 2 Fi (a, b; c; z) is a HYPERGEOMETRIC FUNCTION.
see also Hypergeometric Function
Cayley-Klein Parameters
The parameters a, f3, 7, and S which, like the three
Euler Angles, provide a way to uniquely characterize
the orientation of a solid body. These parameters satisfy
the identities
and
aa* + 77* = 1
aa* + 00* = 1
00* + SS* = 1
a*/? + 7*5 =
a5 — /?7 = 1
/3 = -7*
5 = a*,
(i)
(2)
(3)
(4)
(5)
(6)
(7)
where z* denotes the COMPLEX CONJUGATE. In terms
of the EULER ANGLES 8, </>, and tj>, the Cayley-Klein
parameters are given by
a = e *(*+*)/ a OOB(i#)
s i(V>-«)/2,
d = c -(*+*)/ a cos(itf)
/3 = te* l *- w/:, 8in(ie)
■y = le " v ' r ' r " J sin(^)
(8)
(9)
(10)
(11)
(Goldstein 1960, p. 155).
The transformation matrix is given in terms of the
Cayley-Klein parameters by
A =
I (a 2 - 7 2 + S 2 - (3 2 ) |i( 7 2 - a 2 + S 2 - /3 2 ) 7* - ct{3
\i{a 2 + 7 2 - P 2 - 6 2 ) i (a 2 + 7 2 + ^ 2 + <* 2 ) -i(a/3 + 7 tf)
/3£ — a7
(Goldstein 1960, p. 153).
i(ay + p8)
a<5 + /37
(12)
The Cayley-Klein parameters may be viewed as param-
eters of a matrix (denoted Q for its close relationship
with Quaternions)
Q =
a
7 6
which characterizes the transformations
u = au + 0v
(13)
(14)
(15)
of a linear space having complex axes. This matrix sat-
isfies
Q f Q = 0(^ = 1, (16)
where I is the IDENTITY MATRIX and A f the MATRIX
Transpose, as well as
iQriQi = i.
(17)
In terms of the Euler Parameters a and the Pauli
MATRICES cr iy the Q-matrix can be written as
Q = e l + z(ei<n + e 2 a 2 + e 3 cr 3 )
(18)
(Goldstein 1980, p. 156).
see also EULER ANGLES, EULER PARAMETERS, PAULI
Matrices, Quaternion
References
Goldstein, H. "The Cayley-Klein Parameters and Related
Quantities." §4-5 in Classical Mechanics, 2nd ed. Read-
ing, MA: Addison- Wesley, pp. 148-158, 1980.
214 Cayley-Klein-Hilbert Metric
Cayley's Sextic Evolute
Cayley-Klein-Hilbert Metric
The METRIC of Felix Klein's model for HYPERBOLIC
Geometry,
9ii
912
922
a 2 (l-x 2 2 )
(1-Z! 2 -Z 2 2 ) 2
a X\X2
(1-Zl 2 ~X 2 2 ) 2
a 2 (l-X! 2 )
(1-xi 2 -X2 2 ) 2 '
see also HYPERBOLIC GEOMETRY
Cayley Number
There are two completely different definitions of Cayley
numbers. The first type Cayley numbers is one of the
eight elements in a Cayley Algebra, also known as
an OCTONION. A typical Cayley number is of the form
a + bio + ci\ + dii + ei 3 + fU + gh + hi Qi
where each of the triples (10,11,13), (n,^,^), (22,^3,25),
(z3,i4)*6)) (i4,*5,*o)» (*5»*6j*i), (*e,*o,«2) behaves like
the QUATERNIONS (i,j,k). Cayley numbers are not AS-
SOCIATIVE. They have been used in the study of 7- and
8-D space, and a general rotation in 8-D space can be
written
x ' -> {{{{{( xc i)c2)c3)c 4 )c 5 )cq)c 7 .
The second type of Cayley number is a quantity which
describes a Del Pezzo Surface.
see also Complex Number, Del Pezzo Surface,
Quaternion, Real Number
References
Conway, J. H. and Guy, R. K. "Cayley Numbers." In The
Book of Numbers. New York: Springer- Ver lag, pp. 234-
235, 1996.
Okubo, S. Introduction to Octonion and Other Non-
Associative Algebras in Physics. New York: Cambridge
University Press, 1995.
Cayley's Ruled Surface
see Cayley Cubic
Cayley's Sextic
A plane curve discovered by Maclaurin but first studied
in detail by Cayley. The name Cayley's sextic is due
to R. C. Archibald, who attempted to classify curves in
a paper published in Strasbourg in 1900 (MacTutor Ar-
chive). Cayley's sextic is given in POLAR COORDINATES
by
r = acos 3 (|0), (1)
or
r = 4&cos 3 (§0), (2)
where b = a/4. In the latter case, the CARTESIAN equa-
tion is
4(x 2 + y 2 - bxf - 27a 2 (x 2 + y 2 ) 2 .
The parametric equations are
x(t) = 4a cos 4 (I t) (2 cost - 1)
y(t) =4acos 3 (|t)sin(|t).
(3)
(4)
(5)
JV_
The Arc Length, Curvature, and Tangential An-
gle are
s(t) = 3(i + sini),
K(i) = !sec 2 (£t),
<f>(t) = 2t.
(6)
(7)
(8)
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 178 and 180, 1972.
MacTutor History of Mathematics Archive. "Cayley's Sex-
tic." http: //www-groups . dcs . st-and. ac . uk/ -history/
Curves/Cayleys.html.
Cayley's Sextic Evolute
/
\
/
\
1
\
\
\
\
'"""N ^
\ I
A \
\ A
) 1
S^T
s <
/
) j
/ \
^J i
1
^-^ 1
\
/
\
/
\
/
\
/
y
•^ y
The Evolute of Cayley's sextic is
x=\a + ^a[3cos(|t) - cos(2<)]
y=^a[3sin(|t)-sin(2t)] )
which is a Nephroid.
Cayley Tree
Cellular Automaton
215
Cayley Tree
A Tree in which each NODE has a constant number of
branches. The PERCOLATION THRESHOLD for a Cayley
tree having z branches is
1
Pc
see also CAYLEY GRAPH
1"
Cayleyian Curve
The Envelope of the lines connecting correspond-
ing points on the JACOBIAN CURVE and STEINERIAN
CURVE. The Cayleyian curve of a net of curves of or-
der n has the same Genus (Curve) as the JACOBIAN
Curve and Steinerian Curve and, in general, the
class 3n(n— 1).
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 150, 1959.
Cech Cohomology
The direct limit of the COHOMOLOGY groups with CO-
EFFICIENTS in an ABELIAN GROUP of certain coverings
of a Topological Space.
Ceiling Function
1**1 Ceiling
[x] Nint (Round)
|jc| Floor
-4
-2
Jj_:
i
JT
i
i
JT
i
L
i
_ j
u
' -2
The function \x] which gives the smallest INTEGER > as,
shown as the thick curve in the above plot. Schroeder
(1991) calls the ceiling function symbols the "Gallows"
because of the similarity in appearance to the structure
used for hangings. The name and symbol for the ceiling
function were coined by K. E. Iverson (Graham et al.
1990). It can be implemented as ceil(x)=-int (-x),
where int(x) is the INTEGER PART of x.
set also Floor Function, Integer Part, Nint
References
Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer
Functions." Ch. 3 in Concrete Mathematics: A Foun-
dation for Computer Science. Reading, MA: Addison-
Wesley, pp. 67-101, 1990.
Iverson, K. E. A Programming Language. New York: Wiley,
p. 12, 1962.
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, p. 57,
1991.
Cell
A finite regular POLYTOPE.
see also 16-Cell, 24-Cell, 120-Cell, 600-Cell
Cellular Automaton
A grid (possibly 1-D) of cells which evolves according to
a set of rules based on the states of surrounding cells,
von Neumann was one of the first people to consider
such a model, and incorporated a cellular model into
his "universal constructor." von Neumann proved that
an automaton consisting of cells with four orthogonal
neighbors and 29 possible states would be capable of
simulating a TURING MACHINE for some configuration
of about 200,000 cells (Gardner 1983, p. 227).
l-D automata are called "elementary" and are repre-
sented by a row of pixels with states either or 1.
These can be represented with an 8-bit binary num-
ber, as shown by Stephen Wolfram. Wolfram further
restricted the number from 2 8 = 256 to 32 by requiring
certain symmetry conditions.
The most well-known cellular automaton is Conway's
game of Life, popularized in Martin Gardner's Scien-
tific American columns. Although the computation of
successive Life generations was originally done by hand,
the computer revolution soon arrived and allowed more
extensive patterns to be studied and propagated.
see Life, Langton's Ant
References
Adami, C. Artificial Life. Cambridge, MA: MIT Press, 1998.
Buchi, J. R. and Siefkes, D. (Eds.). Finite Automata, Their
Algebras and Grammars: Towards a Theory of Formal Ex-
pressions. New York: Springer- Verlag, 1989.
Burks, A. W. (Ed.). Essays on Cellular Automata. Urbana-
Champaign, IL: University of Illinois Press, 1970.
Cipra, B. "Cellular Automata Offer New Outlook on Life, the
Universe, and Everything." In What's Happening in the
Mathematical Sciences, 1995-1996, Vol 3. Providence,
RI: Amer. Math. Soc, pp. 70-81, 1996.
Dewdney, A. K. The Armchair Universe: An Exploration of
Computer Worlds. New York: W. H. Freeman, 1988.
Gardner, M. "The Game of Life, Parts I— III." Chs. 20-22 in
Wheels, Life, and Other Mathematical Amusements. New
York: W. H. Freeman, pp. 219 and 222, 1983.
Gutowitz, H. (Ed.). Cellular Automata: Theory and Exper-
iment. Cambridge, MA: MIT Press, 1991.
Levy, S. Artificial Life: A Report from the Frontier Where
Computers Meet Biology. New York: Vintage, 1993.
Martin, O.; Odlyzko, A.; and Wolfram, S. "Algebraic Aspects
of Cellular Automata." Communications in Mathematical
Physics 93, 219-258, 1984.
Mcintosh, H. V. "Cellular Automata." http://www.es.
cinvestav.mx/mcintosh/cellular.html.
Preston, K. Jr. and Duff, M. J. B. Modern Cellular Au-
tomata: Theory and Applications. New York: Plenum,
1985.
Sigmund, K. Games of Life: Explorations in Ecology, Evo-
lution and Behaviour. New York: Penguin, 1995.
Sloane, N. J. A. Sequences A006977/M2497 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.
216 Cellular Space
Toffoli, T. and Margolus, N. Cellular Automata Machines:
A New Environment for Modeling. Cambridge, MA: MIT
Press, 1987.
Wolfram, S. "Statistical Mechanics of Cellular Automata."
Rev. Mod. Phys. 55, 601-644, 1983.
Wolfram, S. (Ed.). Theory and Application of Cellular Au-
tomata. Reading, MA: Addis on- Wesley, 1986.
Wolfram, S. Cellular Automata and Complexity: Collected
Papers. Reading, MA: Addison- Wesley, 1994.
Wuensche, A. and Lesser, M. The Global Dynamics of Cel-
lular Automata: An Atlas of Basin of Attraction Fields
of One- Dimensional Cellular Automata. Reading, MA:
Addison- Wesley, 1992.
Cellular Space
A Hausdorff Space which has the structure of a so-
called CW-COMPLEX.
Center
A special POINT which usually has some symmetric
placement with respect to points on a curve or in a
SOLID. The center of a CIRCLE is equidistant from all
points on the CIRCLE and is the intersection of any two
distinct DIAMETERS. The same holds true for the center
of a Sphere.
see also Center (Group), Center of Mass, Cir-
CUMCENTER, CURVATURE CENTER, ELLIPSEj EQUI-
Brocard Center, Excenter, Homothetic Cen-
ter, Incenter, Inversion Center, Isogonic Cen-
ters, Major Triangle Center, Nine-Point Cen-
ter, Orthocenter, Perspective Center, Point,
Radical Center, Similitude Center, Sphere,
Spieker Center, Taylor Center, Triangle Cen-
ter, Triangle Center Function, Yff Center of
Congruence
Center Function
see Triangle Center Function
Center of Gravity
see Center of Mass
Center (Group)
The center of a GROUP is the set of elements which
commute with every member of the GROUP. It is equal
to the intersection of the Centralizers of the Group
elements.
see also ISOCLINIC GROUPS, NlLPOTENT GROUP
Center of Mass
see Centroid (Geometric)
Centered Pentagonal Number
Centered Cube Number
A Figurate Number of the form,
CCub n = n +(n- l) 3 = (2n - l)(n 2 - n + 1).
The first few are 1, 9, 35, 91, 189, 341, ... (Sloane's
A005898). The Generating Function for the cen-
tered cube numbers is
x(x 3 + 5z 2 + 5x + 1) n 2 * ^ 4
-^ -, -^j [ — - = x + 9x 2 + 35z + 91a? 4 + . . . .
(x- l) 4
see also Cubic Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 51, 1996.
Sloane, N. J. A. Sequence A005898/M4616 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Centered Hexagonal Number
see Hex Number
Centered Pentagonal Number
A Centered Polygonal Number consisting of a cen-
tral dot with five dots around it, and then additional
dots in the gaps between adjacent dots. The general
term is (5n 2 - 5n + 2)/2, and the first few such num-
bers are 1, 6, 16, 31, 51, 76, ... (Sloane's A005891).
The Generating Function of the centered pentago-
nal numbers is
x(x 2 + Sx + 1)
(z-1) 3
x + 6x 2 + 16z 3 + 31z 4 + . . . .
see also CENTERED SQUARE NUMBER, CENTERED TRI-
ANGULAR Number
References
Sloane, N. J. A. Sequence A005891/M4112 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Centered Polygonal Number
Centered Polygonal Number
N, / ^"*--~- / ^ — • — • — d
A Figurate Number in which layers of Polygons are
drawn centered about a point instead of with the point
at a Vertex.
see also Centered Pentagonal Number, Centered
Square Number, Centered Triangular Number
References
Sloane, N. J. A. and Plouffe, S. Extended entry for sequence
M3826 in The Encyclopedia of Integer Sequences. San
Diego, CA: Academic Press, 1995.
Centered Square Number
A Centered Polygonal Number consisting of a cen-
tral dot with four dots around it, and then additional
dots in the gaps between adjacent dots. The general
term is n 2 + (n — l) 2 , and the first few such numbers
are 1, 5, 13, 25, 41, ... (Sloane's A001844). Centered
square numbers are the sum of two consecutive SQUARE
Numbers and are congruent to 1 (mod 4). The Gen-
erating Function giving the centered square numbers
is
(1 — x) 6
see also Centered Pentagonal Number, Centered
Polygonal Number, Centered Triangular Num-
ber, Square Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 41, 1996.
Sloane, N. J. A. Sequence A001844/M3826 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Centered Triangular Number
A Centered Polygonal Number consisting of a cen-
tral dot with three dots around it, and then additional
Central Beta Function 217
dots in the gaps between adjacent dots. The general
term is (3n — 3n + 2)/2, and the first few such numbers
are 1, 4, 10, 19, 31, 46, 64, . . . (Sloane's A005448). The
Generating Function giving the centered triangular
numbers is
x{x' + x + l) =x + 4x * + 10x > + 19x * + ....
(1 — X) 6
see also CENTERED PENTAGONAL NUMBER, CENTERED
Square Number
References
Sloane, N. J. A. Sequence A005448/M3378 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Centillion
In the American system, 10 303 .
see also Large Number
Central Angle
An Angle having its Vertex at a Circle's center
which is formed by two points on the CIRCLE'S Cir-
cumference. For angles with the same endpoints,
C = 29 i,
where 0; is the INSCRIBED ANGLE.
References
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., pp. xxi— xxii, 1995.
Central Beta Function
10r
;im[zj
;im[z]
MzH*?
The central beta function is defined by
f3(p) = B(p,p),
(1)
218 Central Binomial Coefficient
Central Conic
where B(p,q) is the BETA FUNCTION. It satisfies the
identities
^(p) = 2 1 - ap B(p > i)
(2)
= 2 1 - 2p cos(7rp)B(f-p,p) (3)
1 t p dt
_ 2 T-r n(n + 2p)
V *1 (n + p)(n + p)"
(4)
(5)
With p = 1/2, the latter gives the WALLIS FORMULA.
When p = a/b,
b/3(a/b) = 2 1 - 2a/b J(a,b),
where
a
a,b)= f
Jo
1 1*- 1 dt
The central beta function satisfies
(2 -\- 4x)0(l -\- x) = x0(x)
(1 - 2x)j8(l - x)f3(x) = 27rcot(7nr)
P(\ - x) = 2 4x_1 t<m(7rx)/3(x)
(6)
(7)
(8)
(9)
(10)
P(x)0(x + |) = 2 4 * +1 7r/?(2z)/3(2; C + §). (11)
For p an Odd Positive Integer, the central beta func-
tion satisfies the identity
^ )= vP n -^
V fc=l fc=0
n>(" + i;J- < 12 >
see a/so BETA FUNCTION, REGULARIZED BETA FUNC-
TION
References
Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua-
tion of the Gamma Function at Rational Values of Small
Denominators." IMA J. Numerical Analysis 12, 519—526,
1992.
Central Binomial Coefficient
The nth central binomial coefficient is defined as ( i n / 2 i ) >
where (™) is a BINOMIAL COEFFICIENT and [n\ is the
Floor Function. The first few values are 1, 2, 3, 6, 10,
20, 35, 70, 126, 252, . . . (Sloane's A001405). The central
binomial coefficients have GENERATING FUNCTION
2(2# 3 - x 2 )
The central binomial coefficients are SQUAREFREE only
for n = 1, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, . . . (Sloane's
A046098), with no others less than 1500.
The above coefficients are a superset of the alternative
"central" binomial coefficients
CD-
(2n)!
(n!) 2 '
which have GENERATING FUNCTION
v 7 ! - 4z
: 1 + 2x + 6x 2 + 20z 3 + 70x 4 + . . . .
The first few values are 2, 6, 20, 70, 252, 924, 3432,
12870, 48620, 184756, ... (Sloane's A000984).
Erdos and Graham (1980, p. 71) conjectured that
the central binomial coefficient ( 2 ^) is never SQUARE-
FREE for n > 4, and this is sometimes known as the
Erdos Squarefree Conjecture. Sarkozy's The-
orem (Sarkozy 1985) provides a partial solution which
states that the BINOMIAL COEFFICIENT ( 2 ") is never
Squarefree for all sufficiently large n > no (Vardi
1991). Granville and Ramare (1996) proved that the
only Squarefree values are n — 2 and 4. Sander
(1992) subsequently showed that ( 2n T f d ) are also never
SQUAREFREE for sufficiently large n as long as d is not
"too big."
see also BINOMIAL COEFFICIENT, CENTRAL TRINO-
MIAL Coefficient, Erdos Squarefree Conjec-
ture, Sarkozy's Theorem, Quota System
References
Granville, A. and Ramare, O. "Explicit Bounds on Exponen-
tial Sums and the Scarcity of Squarefree Binomial Coeffi-
cients." Mathematika 43, 73-107, 1996.
Sander, J. W. "On Prime Divisors of Binomial Coefficients."
Bull London Math. Soc. 24, 140-142, 1992.
Sarkozy, A. "On Divisors of Binomial Coefficients. I." J.
Number Th. 20, 70-80, 1985.
Sloane, N. J. A. Sequences A046098, A000984/M1645, and
A001405/M0769 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Vardi, I. "Application to Binomial Coefficients," "Binomial
Coefficients," "A Class of Solutions," "Computing Bino-
mial Coefficients," and "Binomials Modulo and Integer."
§2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations
in Mathematica. Redwood City, CA: Addison- Wesley,
pp. 25-28 and 63-71, 1991.
Central Conic
An Ellipse or Hyperbola.
see also CONIC Section
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited,
Washington, DC: Math. Assoc. Amer., pp. 146-150, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
p. 77, 1990.
Central Difference
Central Limit Theorem 219
Central Difference
The central difference for a function tabulated at equal
intervals fi is defined by
^(/n+l/2) = <Wi/ 2 = $n + l/2 = /n+1 - fn- (1)
Higher order differences may be computed for Even and
Odd powers,
2fc / \
C +1 /2 =£(-1)' 2 f/n+ fc
2fc+l / \
(2)
+fc+i-j- (3)
see a/so Backward Difference, Divided Differ-
ence, Forward Difference
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Differences."
§25.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 877-878, 1972.
Central Limit Theorem
Let x\ , X2 , . . . , xn be a set of AT INDEPENDENT random
variates and each Xi have an arbitrary probability distri-
bution P(a?i, . . . , xn) with MEAN fii and a finite VARI-
ANCE cr^ 2 . Then the normal form variate
A norra —
v^
(1)
Vi*
has a limiting distribution which is NORMAL (GAUS-
SIAN) with Mean \l = and Variance a 2 ~ 1. If
conversion to normal form is not performed, then the
variate
X
^5>
(2)
is Normally Distributed with fi x = \x x and a x =
o~ x /y/N. To prove this, consider the Inverse Fourier
Transform of Px{}).
/oo
e 2 * ifX p(X)dX
-OO
J —c
sr ( 27ri
n=Q
•J —oo
p{X) dx
(2^/) n /„Xn
E^<*>
(3)
Now write
(X n ) = (AT n (xi +X2 + ... + x N ) n )
/OO
N~ n (xi + . .. + xn) u p(xi) - • -p(xN)dxi ---cIxn,
•oo
(4)
so we have
(2«/)»
*■ — ' n
n = «/-oo
/*°° y^ r 27rz/(x 1 + ... + x JV ) l" 1
<J — oo _ rt
+ .., + x^) n
x p(xi) • • -p(x N ) dxi ■ • ■ dxj\
x p(#i) • * -p(xjyf) dx\ • • 'dxj\
/oo
pix^dx!
F
V — c
w p(xn) dxjsr
p(x) dx
}'
= / 6 a-</-/JVp(a.) da .
= / p(x)dx-\ / xp(x) dx
L 1 ' — oo <J — oo
Now expand
ln(l + x) = x-\x 2 + \x 3 + ...
(5)
(6)
w exp < AT
N {X) 2N* \ X I
+ ^<*> 2 + <^- 3 )
: exp
J exp
(2nf) 2 ((x 2 ) - (x) 2 )
2iviffi x
(27T/)V, 2
2N
(7)
220
Central Limit Theorem
Centroid (Geometric)
Hx = (x)
a 2 = (x 2 ) - (x) 2
Taking the FOURIER TRANSFORM,
(8)
(9)
/OO
e -wr-i[P x (f)]df
-oo
= f°° e 2^if(^ x -x)-(2^f) 2 a x 2 /2N d , ^
J — oo
This is of the form
/CO
iaf -
e
-CO
bf
df,
(11)
where a = 2iz(ti x — x) and 6 = (27ro~ x ) 2 /2N. But, from
Abramowitz and Stegun (1972, p. 302, equation 7.4.6),
/CO
e iaf-
-oo
bf 2
df = e
-a 2 /ib /W
(12)
Therefore,
7T J -[27T(fl x -X)} 2
exp ' — —
2AT
27TJV
47T 2 <7 X 2
exp
4 (2™*) 2 f
^ 2AT J
4tt 2 (^ -x) 2 2iV
4 • 47T 2 cr x 2
ViV
But ax = <7 x /VN and //x = Man so
P x = \ c -(mx-^) 2 ^x 2
(rx\/27r
(13)
(14)
Central Trinomial Coefficient
The nth central binomial coefficient is denned as the co-
efficient of x n in the expansion of (l-\-x-\-x 2 ) n . The first
few are 1, 3, 7, 19, 51, 141, 393, . . . (Sloane's A002426).
This sequence cannot be expressed as a fixed number
of hypergeometric terms (Petkovsek et al. 1996, p. 160).
The Generating Function is given by
/(*) =
1
^(l + z)(l-3x)
= 1 + x + 3z 2 + 7x 3 + . . . .
see also Central Binomial Coefficient
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Sloane, N. J. A. Sequence A002426/M2673 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Centralizer
The centralizer of a Finite non-ABELiAN Simple
Group G is an element z of order 2 such that
C G (z) = {geG:gz = zg}.
see also Center (Group), Normauzer
Centrode
C = rT + kB,
where r is the TORSION, k is the CURVATURE, T is the
Tangent Vector, and B is the Binormal Vector.
Centroid (Function)
By analogy with the GEOMETRIC CENTROID, the cen-
troid of an arbitrary function f(x) is defined as
{x} =
IZo f( x ) dx
The "fuzzy" central limit theorem says that data which
are influenced by many small and unrelated random ef-
fects are approximately NORMALLY DISTRIBUTED.
see also LlNDEBERG Condition, Lindeberg-Feller
Central Limit Theorem, Lyapunov Condition
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972,
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, pp. 112-113, 1992.
Zabell, S. L. "Alan Turing and the Central Limit Theorem."
Amer. Math. Monthly 102, 483-494, 1995.
References
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 139-140 and 156, 1965.
Centroid (Geometric)
The Center of Mass of a 2-D planar Lamina or a
3-D solid. The mass of a LAMINA with surface density
function o~(x,y) is
M--
: //' (x '
y)dA.
(1)
The coordinates of the centroid (also called the CENTER
of Gravity) are
ff xo~(x,y) dA
M
(2)
Centroid (Orthocentric System)
Jfya(x,y)dA
y
M
(3)
The centroids of several common laminas along the non-
symmetrical axis are summarized in the following table.
Figure
y
parabolic segment |/t
3tt
semicircle
In 3-D , the mass of a solid with density function
p(x,y,z) is
Iff**'*
M= I I I p(x,y,z)dV, (4)
and the coordinates of the center of mass are
_ _ fffxp(x,y,z)dV
M
JJfyp(x,y,z)dV
M
JfJzp(x,y,z)dV
M
(5)
(6)
(7)
Figure
cone ^ h
conical frustum ^Y^t 3 ^
hemisphere
paraboloid
pyramid
4(R 1 2 +R 1 R 2 +R2 2 )
\h
see also Pappus's Centroid Theorem
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 132, 1987.
McLean, W. G. and Nelson, E. W. "First Moments and Cen-
troids." Ch. 9 in Schaum's Outline of Theory and Prob-
lems of Engineering Mechanics: Statics and Dynamics,
4th ed. New York: McGraw-Hill, pp. 134-162, 1988.
Centroid (Orthocentric System)
The centroid of the four points constituting an ORTHO-
CENTRIC System is the center of the common Nine-
Point Circle (Johnson 1929, p. 249). This fact auto-
matically guarantees that the centroid of the Incenter
and Excenters of a Triangle is located at the Cir-
cumcenter.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Centroid (Triangle) 221
Centroid (Triangle)
The centroid (Center of Mass) of the Vertices of
a Triangle is the point M (or G) of intersection of
the Triangle's three Medians, also called the Median
Point (Johnson 1929, p. 249). The centroid is always
in the interior of the TRIANGLE, and has TRILINEAR
Coordinates
csc A : esc B : esc C.
If the sides of a TRIANGLE are divided so that
A 2 Pi A3P2 A ± P 2
PiA 3 P 2 A X P 3 A 2
P
9
(2)
(3)
the centroid of the TRIANGLE AP1P2P3 is M (Johnson
1929, p. 250).
Pick an interior point X. The TRIANGLES BXC, CXA,
and AXB have equal areas IFF X corresponds to the
centroid. The centroid is located one third of the way
from each Vertex to the Midpoint of the opposite side.
Each median divides the triangle into two equal areas;
all the medians together divide it into six equal parts,
and the lines from the Median Point to the Vertices
divide the whole into three equivalent TRIANGLES. In
general, for any line in the plane of a Triangle ABC,
d= l{d A + d B + d c ),
(4)
where d } d A , ds, and dc are the distances from the cen-
troid and Vertices to the line. A Triangle will bal-
ance at the centroid, and along any line passing through
the centroid. The Trilinear Polar of the centroid is
called the Lemoine Axis. The Perpendiculars from
the centroid are proportional to s^ -1 ,
CL1P2 = CL2P2 = dtps - § A, (5)
where A is the Area of the Triangle. Let P be an
arbitrary point, the Vertices be Ai, A 2) and A 3) and
the centroid M. Then
PA X +PA 2 +PA 3 = MA! +Mi 2 +MA Z +3PM .
(6)
If O is the ClRCUMCENTER of the triangle's centroid,
then
OM 2 =,R 2 -|(a 2 + 6 2 +c 2 ). (7)
The centroid lies on the EULER LINE.
The centroid of the PERIMETER of a TRIANGLE is the
triangle's Spieker Center (Johnson 1929, p. 249).
see also ClRCUMCENTER, EULER LlNE, EXMEDIAN
Point, Incenter, Orthocenter
References
Carr, G. S. Formulas and Theorems in Pure Mathematics,
2nd ed. New York: Chelsea, p. 622, 1970.
222
Certificate of Compositeness
Ceva's Theorem
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., p. 7, 1967.
Dixon, R. Mathographics. New York: Dover, pp. 55-57, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 173-176 and 249, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Centroid." http : //www . evansville . edu/
-ck6/tcenters/class/centroid.html.
Certificate of Compositeness
see Compositeness Certificate
Certificate of Primality
see Primality Certificate
Cesaro Equation
An Intrinsic Equation which expresses a curve in
terms of its ARC LENGTH s and RADIUS OF CURVA-
TURE R (or equivalently, the CURVATURE k).
see also Arc Length, Intrinsic Equation, Natural
Equation, Radius of Curvature, Whewell Equa-
tion
References
Yates, R. C. "Intrinsic Equations." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 123-126, 1952.
Cesaro Fractal
A Fractal also known as the Torn Square Frac-
tal. The base curves and motifs for the two fractals
illustrated above are show below.
see also Fractal, Koch Snowflake
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, p. 43,
1991.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 79, 1989.
^ Weisstein, E. W. "Fractals." http://www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Cesaro Mean
see FEJES TOTH'S INTEGRAL
Ceva's Theorem
Given a Triangle with Vertices A, £?, and C and
points along the sides D, E, and F, a NECESSARY and
Sufficient condition for the Cevians AD, BE, and
CF to be Concurrent (intersect in a single point) is
that
BDCE-AF^DCEA- FB. (1)
Let P = [Vi, . . . , V^] be an arbitrary n-gon, C a given
point, and k a Positive Integer such that 1 < k <
n/2. For i = 1, . . . , n, let Wi be the intersection of the
lines CVi and Vi-kV i+ k, then
n
Vi-kWi
WtVi
i+k
= 1.
Here, AB\\CD and
AB
VCD\
(2)
(3)
is the Ratio of the lengths [A, B] and [C, D] with a plus
or minus sign depending on whether these segments have
the same or opposite directions (Grunbaum and Shepard
1995).
Another form of the theorem is that three Concurrent
lines from the Vertices of a Triangle divide the op-
posite sides in such fashion that the product of three
nonadjacent segments equals the product of the other
three (Johnson 1929, p. 147).
see also Hoehn's Theorem, Menelaus' Theorem
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 122, 1987.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967.
Grunbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254-268, 1995.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 145-151, 1929.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., p. xx, 1995.
Cevian
Cevian
A line segment which joins a Vertex of a Triangle
with a point on the opposite side (or its extension). In
the above figure,
6 sin a
sin(7 + a')
References
Thebault, V. "On the Cevians of a Triangle." Amer. Math.
Monthly 60, 167-173, 1953.
Cevian Conjugate Point
see ISOTOMIC CONJUGATE POINT
Cevian Transform
Vandeghen's (1965) name for the transformation taking
points to their ISOTOMIC CONJUGATE POINTS.
see also Isotomic Conjugate Point
References
Vandeghen, A. "Some Remarks on the Isogonal and Cevian
Transforms. Alignments of Remarkable Points of a Trian-
gle." Amer. Math. Monthly 72, 1091-1094, 1965.
Cevian Triangle
Given a center a : (3 : 7, the cevian triangle is defined
as that with VERTICES : : 7, a : : 7, and a :
P : 0. If A'B'C is the CEVIAN TRIANGLE of X and
A"B"C" is the Anticevian Triangle, then X and
A" are Harmonic Conjugate Points with respect to
A and A',
see also Anticevian Triangle
Chain Rule 223
Chain
Let P be a finite Partially Ordered Set. A chain
in P is a set of pairwise comparable elements (i.e., a
Totally Ordered subset). The Width of P is the
maximum CARDINALITY of an Antichain in P. For a
Partial Order, the size of the longest Chain is called
the Width.
see also Addition Chain, Antichain, Brauer Chain,
Chain (Graph), Dilworth's Lemma, Hansen Chain
Chain Fraction
see Continued Fraction
Chain (Graph)
A chain of a GRAPH is a SEQUENCE {x u z 2 , . . ■ , x n } such
that {x u x 2 ), (052,2:3), .--, (z„_i,a:n) are EDGES of the
Graph.
Chain Rule
If g(x) is DlFFERENTlABLE at the point x and f(x) is
DlFFERENTIABLE at the point g(x), then / o g is DlF-
FERENTlABLE at x. Furthermore, let y = f(g(x)) and
u = g(x), then
dy _ dy du
dx du dx
(i)
There are a number of related results which also go un-
der the name of "chain rules." For example, if z —
f(x,y), x = g{t), and y - h(t), then
dz _ dz dx
dt dx dt
dz dy
dy dt '
(2)
The "general" chain rule applies to two sets of functions
yi
/1 (ui,..., «p)
and
:(3)
y m - fm{ui i ... J Up)
U± = £l(25i, . .. ,X n )
:(4)
U P = 0p(#l»- • • j^n)-
Defining the m X n JACOBI MATRIX by
dyi
dx.
dvi .
dx 2
dyi
dx n
dxi
dy m
8x2
dXn
(5)
and similarly for (dyi/duj) and (diii/dxj) then gives
dyi
dx.
-(£)(£)■ m
224
Chained Arrow Notation
Champernowne Constant
In differential form, this becomes
dpi du p
du p dxi
d _ | dy^diH
* dui dxi
+ ^^L ]dxi
I dmdu± ^dup\ ^
du\ &X2 ' ' ' du p 8x2 J
(Kaplan 1984).
see also Derivative, Jacobian, Power Rule, Prod-
uct Rule
References
Anton, H. Calculus with Analytic Geometry, 2nd ed. New
York: Wiley, p. 165, 1984.
Kaplan, W. "Derivatives and Differentials of Composite
Functions" and "The General Chain Rule." §2.8 and 2.9
in Advanced Calculus, 3rd ed. Reading, MA: Addison-
Wesley, pp. 101-105 and 106-110, 1984.
Chained Arrow Notation
A Notation which generalizes Arrow Notation and
is defined as
a\-<*"\h = a^b^>c.
see also Arrow Notation
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 61, 1996.
Chainette
see Catenary
Chair
Chaitin's Constant
An Irrational Number Q which gives the probability
that for any set of instructions, a Universal Turing
MACHINE will halt. The digits in are random and
cannot be computed ahead of time.
see also Halting Problem, Turing Machine, Uni-
versal Turing Machine
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/chaitin/chaitin.html.
Gardner, M. "The Random Number Bids Fair to Hold
the Mysteries of the Universe." Set. Amer. 241, 20-34,
Nov. 1979.
Gardner, M. "Chaitin's Omega." Ch. 21 in Fractal Music,
HyperCards, and More Mathematical Recreations from Sci-
entific American Magazine. New York: W. H. Freeman,
1992.
Kobayashi, K. "Sigma(N)0-Complete Properties of Pro-
grams and Lartin-Lof Randomness." Information Proc.
Let 46, 37-42, 1993.
Chaitin's Number
see Chaitin's Constant
Chaitin's Omega
see Chaitin's Constant
Champernowne Constant
Champernowne's number 0.1234567891011. . . (Sloane's
A033307) is the decimal obtained by concatenating the
Positive Integers. It is Normal in base 10. In 1961,
Mahler showed it to also be TRANSCENDENTAL.
The Continued Fraction of the Champernowne con-
stant is [0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1,
1, 15,
A Surface with tetrahedral symmetry which, according
to Nordstrand, looks like an inflatable chair from the
1970s. It is given by the implicit equation
(x 2 +y 2 + z 2 -ak 2 ) 2 -b[(z-k) 2 -2x 2 ][{z + k) 2 ~2y 2 ] = 0.
see also Bride's Chair
References
Nordstrand, T. "Chair." http://www.uib.no/people/nfytn/
chairtxt.htm.
457540111391031076483646628242956118599603939- • •
710457555000662004393090262659256314937953207- - •
747128656313864120937550355209460718308998457* • *
5801469863148833592141783010987,
6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4,
58, 8, 54, ...] (Sloane's A030167). The next term of
the Continued Fraction is huge, having 2504 digits.
In fact, the coefficients eventually become unbounded,
making the continued fraction difficult to calculate for
too many more terms. Large terms greater than 10 5 oc-
cur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and
have 6, 166, 2504, 140, 33102, 109, 2468, 136, . . . digits
(Plouffe). Interestingly, the Copeland-Erdos Con-
stant, which is the decimal obtained by concatenating
the Primes, has a well-behaved Continued Fraction
which does not show the "large term" phenomenon.
see also COPELAND-ERDOS CONSTANT, SMARANDACHE
Sequences
Change of Variables Theorem
Chaos 225
References
Champernowne, D. G. "The Construction of Decimals Nor-
mal in the Scale of Ten." J. London Math. Soc. 8, 1933.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/cntfrc/cntfrc.html.
Sloane, N. J. A. Sequences A030167 and A033307 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Change of Variables Theorem
A theorem which effectively describes how lengths, ar-
eas, volumes, and generalized n-dimensional volumes
(Contents) are distorted by Differentiable Func-
tions. In particular, the change of variables theorem
reduces the whole problem of figuring out the distortion
of the content to understanding the infinitesimal dis-
tortion, i.e., the distortion of the DERIVATIVE (a linear
Map), which is given by the linear Map's Determi-
nant. So / : R n -► W 1 is an Area-Preserving linear
MAP Iff |det(/)| = 1, and in more generality, if S is
any subset of MJ 1 , the CONTENT of its image is given by
I det(/)| times the CONTENT of the original. The change
of variables theorem takes this infinitesimal knowledge,
and applies CALCULUS by breaking up the DOMAIN into
small pieces and adds up the change in AREA, bit by
bit.
The change of variable formula persists to the general-
ity of Differential Forms on Manifolds, giving the
formula
/ (/*w) = f (u
Jm Jw
under the conditions that M and W are compact con-
nected oriented MANIFOLDS with nonempty boundaries,
/ : M — > W is a smooth map which is an orientation-
preserving DlFFEOMORPHISM of the boundaries.
In 2-D, the explicit statement of the theorem is
/.
f(x,y)dxdy
-L
f[x(u,v),y(u,v)]
d(x,y)
d(u,v)
dudv
and in 3-D, it is
/
/(a;, y, z) dx dy dz
■ I f[x(u, v,w), y(u, v, w) } z(u, u,
J R*
W)]
d(x,y,z)
du dv dw ,
d(u, v, w)
where R = f(R*) is the image of the original region R* ,
d(u,v,w)
is the JACOBIAN, and / is a global orientation-preserving
DlFFEOMORPHISM of R and R* (which are open subsets
ofM n ).
The change of variables theorem is a simple consequence
of the Curl Theorem and a little de Rham Cohomol-
OGY. The generalization to n-D requires no additional
assumptions other than the regularity conditions on the
boundary.
see also Implicit Function Theorem, Jacobian
References
Kaplan, W. "Change of Variables in Integrals." §4.6 in Ad-
vanced Calculus, 3rd ed. Reading, MA: Addison- Wesley,
pp. 238-245, 1984.
Chaos
A Dynamical System is chaotic if it
1. Has a Dense collection of points with periodic or-
bits,
2. Is sensitive to the initial condition of the system (so
that initially nearby points can evolve quickly into
very different states), and
3. Is TOPOLOGICALLY TRANSITIVE.
Chaotic systems exhibit irregular, unpredictable behav-
ior (the Butterfly Effect). The boundary between
linear and chaotic behavior is characterized by PERIOD
DOUBLING, following by quadrupling, etc.
An example of a simple physical system which displays
chaotic behavior is the motion of a magnetic pendulum
over a plane containing two or more attractive magnets.
The magnet over which the pendulum ultimately comes
to rest (due to frictional damping) is highly dependent
on the starting position and velocity of the pendulum
(Dickau). Another such system is a double pendulum (a
pendulum with another pendulum attached to its end).
see also Accumulation Point, Attractor, Basin
of Attraction, Butterfly Effect, Chaos Game,
Feigenbaum Constant, Fractal Dimension, Gin-
gerbreadman Map, Henon-Heiles Equation,
Henon Map, Limit Cycle, Logistic Equation, Lya-
punov Characteristic Exponent, Period Three
Theorem, Phase Space, Quantum Chaos, Reso-
nance Overlap Method, Sarkovskii's Theorem,
Shadowing Theorem, Sink (Map), Strange At-
tractor
References
Bai-Lin, H. Chaos. Singapore: World Scientific, 1984.
Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Intro-
duction, 2nd ed. Cambridge: Cambridge University Press,
1996.
Cvitanovic, P. Universality in Chaos: A Reprint Selection,
2nd ed. Bristol: Adam Hilger, 1989.
Dickau, R. M. "Magnetic Pendulum." http:// forum .
swarthmore . edu / advanced / robertd / magnetic
pendulum . html .
Drazin, P. G. Nonlinear Systems. Cambridge, England:
Cambridge University Press, 1992.
Field, M. and Golubitsky, M. Symmetry in Chaos: A Search
for Pattern in Mathematics, Art and Nature. Oxford,
England: Oxford University Press, 1992.
Gleick, J. Chaos: Making a New Science. New York: Pen-
guin, 1988.
226
Chaos Game
Character Table
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields, 3rd
ed. New York: Springer- Verlag, 1997.
Lichtenberg, A. and Lieberman, M. Regular and Stochastic
Motion, 2nd ed. New York: Springer- Verlag, 1994.
Lorenz, E. N. The Essence of Chaos. Seattle, WA: University
of Washington Press, 1996.
Ott, E, Chaos in Dynamical Systems. New York: Cambridge
University Press, 1993.
Ott, E.; Sauer, T.; and Yorke, J. A. Coping with Chaos:
Analysis of Chaotic Data and the Exploitation of Chaotic
Systems. New York: Wiley, 1994.
Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Sprhiger-
Verlag, 1992.
Poon, L. "Chaos at Maryland." http://www-chaos.umd.edu.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, 1990.
Strogatz, S. H. Nonlinear Dynamics and Chaos, with Appli-
cations to Physics, Biology, Chemistry, and Engineering.
Reading, MA: Addis on- Wesley, 1994.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989,
Tufillaro, N.; Abbott, T. R.; and Reilly, J. An Experimental
Approach to Nonlinear Dynamics and Chaos. Redwood
City, CA: Addison-Wesley, 1992.
Wiggins, S. Global Bifurcations and Chaos: Analytical Meth-
ods. New York: Springer- Verlag, 1988.
Wiggins, S. Introduction to Applied Nonlinear Dynamical
Systems and Chaos. New York: Springer- Verlag, 1990.
Chaos Game
Pick a point at random inside a regular n-gon. Then
draw the next point a fraction r of the distance between
it and a Vertex picked at random. Continue the pro-
cess (after throwing out the first few points). The result
of this "chaos game" is sometimes, but not always, a
Fractal. The case (n,r) = (4,1/2) gives the interior
of a SQUARE with all points visited with equal probabil-
ity.
<Tfc
********
A A
4%
Ah;
f\ A.
A
********
(3,1/2)
&& pig
(5,1/3)
&
hi
(5,3/8)
see a/so Barnsley's Fern
(6,1/3)
References
Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed.
Boston, MA: Academic Press, 1993.
Dickau, R. M. "The Chaos Game." http:// forum .
swarthmore.edu/advanced/robertd/chaos_game.html.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 149-163, 1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Character (Group)
The Group Theory term for what is known to physi-
cists as the Trace. All members of the same Conju-
GACY Class in the same representation have the same
character. Members of other Conjugacy Classes may
also have the same character, however. An (abstract)
Group can be uniquely identified by a listing of the
characters of its various representations, known as a
Character Table. Some of the Schonflies Sym-
bols denote different sets of symmetry operations but
correspond to the same abstract GROUP and so have the
same Character Tables.
Character (Multiplicative)
A continuous HOMEOMORPHISM of a GROUP into the
Nonzero Complex Numbers. A multiplicative char-
acter w gives a REPRESENTATION on the 1-D SPACE C
of Complex Numbers, where the Representation ac-
tion by g 6 G is multiplication by uj(g). A multiplicative
character is UNITARY if it has ABSOLUTE VALUE 1 ev-
erywhere.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Character (Number Theory)
A number theoretic function Xk(n) for POSITIVE integral
n is a character modulo k if
X*(l) = l
Xk{n) = Xk(n + k)
Xk(m)xk(n) = Xk(mn)
for all m^n, and
X*(») = 0-
if (fc,n) ^ 1. Xk can only assume values which are (j>{k)
Roots of Unity, where <j> is the Totient Function.
see also DlRlCHLET L-SERIES
Character Table
C x
E
A
1
C 8
E
CTh
A
B
1
1
1
-1
-3 j JXx , •t* j y
2 2 2
x ,y ,z
yz,xz
xy
Character Table
Character Table 227
a
E
i
A 9
A u
1
1
1
-1
x,y,z
x 2 ,y 2 ,z 2 ,xy,xz
yz
c 2
E
c 2
A
B
1
1
1
-1
z,R z
x,y,R x ,R y
x'\y\z'\xy
yz,xz
C 3
E C3 Cz
e = exp(27rz/3)
A
E
111
{! I- f }
z,R z
{x,y)(R x ,R y )
222
x ,y ,z ,xy
(x 2 -y 2 ,xy){yz,xz)
c 4
E O3 C 2 C4
A
B
E
1111
1-1 1-1
ri i -1 -n
ll-i 1 i)
z,R z
(x,y)(R x ,R y )
2,22
x +y ,z
x 2 -y 2 ,xy
(yz,xz)
D 6
E 2C 6
2O3 O2 3Gj 3G 2
A,
1 1
1111
x 2 +y\z 2
A 2
1 1
1 1-1-1
z, R z
B 1
1 -1
1-1 1-1
B 2
1 -1
1-1-1 1
(x^yXR^Ry)
E 1
2 1
-1-2
(xz,yz)
E 2
2 -1
-12
(x 2 -y 2 ,xy)
C2v
E C 2
cr v (xz)
°'v{yz)
A 1
1 1
1
1
z
2 2
x ,y
z 2
A 2
1 1
-1
-1
Rz
xy
3i
1 -1
1
-1
X, ity
xz
B 2
1 -1
-1
1
y,Rx
yz
c$ v
Ai
A 2
E
E 2 C3 3<x v
1 1 1
1 1 -1
2-10
z
Rz
(x,y)(R x ,R y )
~^2~, 2 2~~
x +y ,z
(x 2 -y 2 ,xy)(xz,yz)
c &
E C 5 C 5 2
c 5 3
c 5 4
e = exp(27ri/5)
A
11 1
1
1
2,H,
2,22
a; 4- y ,z
E,
fie e 2
tl e * e 2 '
e 2 *
e 2
r}
(x,*/)^,^)
(yz, xz)
E 2
(1 £ 2 e*
ll e 2 * e
e
£*
?}
(x 2 — y 2 ,xy)
c.
E
c 6
c 3
C 2 Gz
<V
£ = exp(27rt/6)
A
1
1
1
1 1
1
z,R x
2 1 2 2
x + y ,z
B
1
_i
1
-1 1
-1
Ei
(I
£
— £*
— e
-1 -£
~1 ~<T
I'}
(s,y)
(R x , Ry)
(f,^)
E 2
(i
— £
— £*
— £
— £*
1 -£*
1 -£
-:■}
(x 2 - y 2 , xy)
Z>2
E C 2 (z) C 2 {y) C 2 (x)
A 1
1111
2,22
x +y ,z
B 1
1 1-1-1
z,R z
xy
B 2
1-1 1-1
y,Ry
xz
B 3
1-1-1 1
z,R z
yz
D 3
A 1
A 2
E
E 2C 3 3C 2
111
1 1 -1
2 -1
z,R z
(x,y)(R x ,R y )
ar -\-y,z z
xy
(x 2 -y 2 ,xy){xz,yz)
D 4
E 2C4 C 2 2C 2 2C2
Ai
11111
2,22
x +y ,z
A 2
1 11-1-1
z,R z
Bi
1-11 1-1
2 2
x -y
B 2
1-11-1 1
xy
E
2 0-20
(x,y)(R x ,R y )
(xz,yz)
D 5
E 2C 5
2C 5 2
5C 2
A x
Bi
B 2
£3
1 1
1 1
2 2 cos 72°
2 2 cos 144°
1
1
2 cos 144°
2 cos 72°
1
-1
z,R z
(x,y)(R x ,R y )
x 2 ^y 2 ,z 2
{xz,yz)
{x 2 -y 2 ,xy)
Cav
E
2C4 O2 2(T V 2(Td
A 2
B 1
B 2
E
1
1
1
1
2
1111
1 1-1-1
-11 1-1
-11-1 1
0-200
z
Rz
(x,y)(R x ,R y )
2,22
x z +y,z z
2 2
x -y
xy
(xz,yz)
c 5v
E 2C 5
2C 5 2
5<7v
A x
Bi
B 2
B3
1 1
1 1
2 2 cos 72°
2 2 cos 144°
1
1
2 cos 144°
2 cos 72°
1
-1
z
R z
(x,y)(R x ,R y )
x 2 +y 2 ,z 2
(xz.yz)
(x 2 -y 2 ,xy)
c, v
E
2C 6
2C 3
C 2 3cr v
3cr d
A!
1
1 1
1
z
* 2 \y\z 2
A 2
1
1 -1
-1
Rz
B l
1
-1
-1 1
-1
B 2
1
-1
-1 _i
1
E l
2
-1
-2
(x,y)(R x ,R y )
(xz,yz)
E 2
2
-1
-1
2
(x 2 - y 2 ,xy)
c».
£7
Coo* •
oocr„
A x = S +
1
1
1
z
x 2 +y 2 ,z 2
A 2 = E"
1
1
.. -1
Rz
E x = n
2
2 cos <£
(x,y);(R x ,R y )
(xz,yz)
£? 2 = A
2
2 cos 2*
(x 2 - y 2 ,xy)
S 3 =*
2
2 cos 3*
References
Bishop, D. M. "Character Tables." Appendix 1 in Group
Theory and Chemistry. New York: Dover, pp. 279—288,
1993.
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, 1990.
Iyanaga, S. and Kawada, Y. (Eds.). "Characters of Finite
Groups." Appendix B, Table 5 in Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, pp. 1496-
1503, 1980.
228
Characteristic Class
Characteristic (Field)
Characteristic Class
Characteristic classes are Cohomology classes in the
Base Space of a Vector Bundle, defined through
Obstruction theory, which are (perhaps partial) ob-
structions to the existence of k everywhere linearly
independent vector Fields on the Vector Bundle.
The most common examples of characteristic classes
are the Chern, Pontryagin, and Stiefel- Whitney
Classes.
Characteristic (Elliptic Integral)
A parameter n used to specify an ELLIPTIC INTEGRAL
of the Third Kind.
see also AMPLITUDE, ELLIPTIC INTEGRAL, MODULAR
Angle, Modulus (Elliptic Integral), Nome, Pa-
rameter
References
Abramowitz, M. and Stegun, C. A, (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 590, 1972.
Characteristic Equation
The equation which is solved to find a Matrix's Eigen-
values, also called the CHARACTERISTIC POLYNOMIAL.
Given a 2 x 2 system of equations with MATRIX
M
the Matrix Equation is
a b
c d
a b
c d
X
= t
X
y_
which can be rewritten
(i)
(2)
(3)
(4)
which contradicts our ability to pick arbitrary x and y.
Therefore, M has no inverse, so its Determinant is 0.
This gives the characteristic equation
a — t b
c d — t
= t
M can have no Matrix Inverse, since otherwise
X
= M" 1
"o"
=
V
a — t b
c d — t
= 0,
(5)
where | A| denotes the Determinant of A. For a general
k x k Matrix
(6)
an
ai2 ■
- • aifc
021
^22 •
. . Q>2k
afci
&k2 .
•• a>kh
the characteristic equation is
an — t a 12
0,21 CL22 — t
CLkl
ak2
aifc
a2fc
&kk — t
(7)
see also Ballieu's Theorem, Cayley-Hamilton
Theorem, Parodi's Theorem, Routh-Hurwitz
Theorem
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1117-1119, 1979.
Characteristic (Euler)
see Euler Characteristic
Characteristic Factor
A characteristic factor is a factor in a particular fac-
torization of the Totient Function <j>(n) such that
the product of characteristic factors gives the represen-
tation of a corresponding abstract Group as a Direct
PRODUCT. By computing the characteristic factors, any
Abelian Group can be expressed as a Direct Prod-
uct of Cyclic Subgroups, for example, Z 2 ® Z 4 or
Z2® Z2® Z 2 . There is a simple algorithm for determining
the characteristic factors of Modulo Multiplication
Groups.
see also Cyclic Group, Direct Product (Group),
Modulo Multiplication Group, Totient Func-
tion
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 94, 1993.
Characteristic (Field)
For a FIELD K with multiplicative identity 1, consider
the numbers 2 = 1 + 1, 3 = 1 + 1 + 1,4 = 1 + 1 + 1 + 1,
etc. Either these numbers are all different, in which
case we say that K has characteristic 0, or two of them
will be equal. In this case, it is straightforward to show
that, for some number p, we have 1 + 1 + .. . + 1 = 0.
p times
If p is chosen to be as small as possible, then p will
be a Prime, and we say that K has characteristic p.
The Fields Q, E, C, and the /?-adic Numbers Q p
have characteristic 0. For p a Prime, the Galois Field
GF(p n ) has characteristic p.
If H is a Subfield of K, then H and K have the same
characteristic.
see also Field, Subfield
Characteristic Function
Chasles's Polars Theorem 229
Characteristic Function
The characteristic function <j>(t) is defined as the Four-
ier Transform of the Probability Density Func-
tion,
/CO
e iix P{x)dx (1)
■oo
/OO /"OO
P(x)dx + it / xP(x)dx
■oo J — OO
/OO
x 2 P(z)dx + ... (2)
OO
= 1 + ii/i'i - ^2 - ^f« 3 /*3 + ^Vi + . . . , (4)
where fi f n (sometimes also denoted i/ n ) is the nth MO-
MENT about and {j! = 1. The characteristic function
can therefore be used to generate MOMENTS about 0,
or the Cumulants « n ,
OO
z — ' n!
(5)
(6)
A Distribution is not uniquely specified by its Mo-
ments, but is uniquely specified by its characteristic
function.
see also Cumulant, Moment, Moment-Generating
Function, Probability Density Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 928, 1972.
Kenney, J. F. and Keeping, E. S. "Moment-Generating and
Characteristic Functions," "Some Examples of Moment-
Generating Functions," and "Uniqueness Theorem for
Characteristic Functions." §4.6—4.8 in Mathematics of
Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand,
pp. 72-77, 1951.
Characteristic (Partial Differential
Equation)
Paths in a 2-D plane used to transform Partial Dif-
ferential Equations into systems of Ordinary Dif-
ferential EQUATIONS. They were invented by Rie-
mann. For an example of the use of characteristics, con-
sider the equation
Ut - 6uu x = 0.
Now let u(s) = u(x(s))t(s)). Since
it follows that dt/ds = 1, dx/ds = — 6u, and du/ds =
0. Integrating gives t(s) = s, x(s) — -6su (x) J and
u(s) = uo(x) 7 where the constants of integration are
and Uq(x) = u(x, 0).
Characteristic Polynomial
The expanded form of the CHARACTERISTIC EQUATION.
det(al - A),
where A is an n x n MATRIX and I is the IDENTITY
Matrix.
see also Cayley-Hamilton Theorem
Characteristic (Real Number)
For a Real Number x, [^J = int(x) is called the char-
acteristic. Here, [x\ is the FLOOR FUNCTION.
see also MANTISSA, SCIENTIFIC NOTATION
Charlier's Check
A check which can be used to verify correct computation
of Moments.
Chasles-Cayley-Brill Formula
The number of coincidences of a (i/, i/') correspondence
of value 7 on a curve of Genus p is given by
v + v + 2^7.
see also Zeuthen's Theorem
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 129, 1959.
Chasles's Contact Theorem
If a one-parameter family of curves has index N and
class M, the number tangent to a curve of order m and
class mi in general position is
mi TV -hm M.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 436, 1959.
Chasles's Polars Theorem
If the Trilinear Polars of the Vertices of a Tri-
angle are distinct from the respectively opposite sides,
they meet the sides in three Collinear points.
see also COLLINEAR, TRIANGLE, TRILINEAR POLAR
du
ds
dx dt
~ru x + -j-u t ,
ds ds
230
Chasles's Theorem
Chasles's Theorem
If two projective PENCILS of curves of orders n and n'
have no common curve, the LOCUS of the intersections of
corresponding curves of the two is a curve of order n + n f
through all the centers of either PENCIL. Conversely, if
a curve of order n + n 1 contains all centers of a PENCIL
of order n to the multiplicity demanded by Noether'S
Fundamental Theorem, then it is the Locus of the
intersections of corresponding curves of this PENCIL and
one of order n projective therewith.
see also Noether's Fundamental Theorem, Pencil
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 33, 1959.
Chebyshev Approximation Formula
Using a Chebyshev Polynomial of the First Kind
T\ define
Cj = ^^f{x k )Tj{x k )
k=i
N
= NZ^ f cos {^v— / cos {
k=l L y J J ^
"*i(*-§)
jv
Then
f{x)K^c k T k (x)-\c .
It is exact for the TV zeros of T N (x). This type of ap-
proximation is important because, when truncated, the
error is spread smoothly over [—1,1]. The Chebyshev
approximation formula is very close to the MlNIMAX
Polynomial.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and
Vetterling, W. T. "Chebyshev Approximation," "Deriva-
tives or Integrals of a Chebyshev- Approximated Function,"
and "Polynomial Approximation from Chebyshev Coeffi-
cients." §5.8, 5.9, and 5.10 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 184-
188, 189-190, and 191-192, 1992.
Chebyshev Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
The constants
where
inf sup \e x ~ r(x)\,
reRm, n x >o
r(x) =
P(s)
q{xY
p and q are mth and nth order POLYNOMIALS, and R mt n
is the set all RATIONAL FUNCTIONS with REAL coeffi-
cients.
Chebyshev Differential Equation
see also One-Ninth Constant, Rational Function
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/onenin/onenin.html.
Petrushev, P. P. and Popov, V. A. Rational Approximation of
Real Functions. New York: Cambridge University Press,
1987.
Varga, R. S. Scientific Computations on Mathematical Prob-
lems and Conjectures. Philadelphia, PA: SIAM, 1990.
Philadelphia, PA: SIAM, 1990.
Chebyshev Deviation
max {|/(x) - p(x)\w(x)}.
a<x<b
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 41, 1975.
Chebyshev Differential Equation
( 1 -^)S-S+-'» = o W
for | x | < 1. The Chebyshev differential equation has reg-
ular Singularities at -1, 1, and oo. It can be solved
by series solution using the expansions
y = ^2a n x n (2)
OO oo
y = \ na n x n ~ = y na n x n ~
n=0 n=l
oo
= J^(™ + l)a n +ix n (3)
71 =
oo oo
y" = ^(n + l)na ri+ ix n " 1 = ^(n + ^na^ix 71 ' 1
n=0 n=l
oo
= ^(n + 2)(n + l)a n+2 x n . (4)
71 =
Now, plug (2-4) into the original equation (1) to obtain
oo
(1 - x 2 ) ^(n + 2)(n + l)a n+2 x n
n-0
oo oo
-x "^(n + l)n n+1 x n + rn ^ a n x n = (5)
n=0 n=0
oo oo
^(n + 2)(n + l)a n+2 x n - ^(n + 2)(n + l)a n + 2 x n+2
n=0 n=0
oo oo
- J](n+l)a n+1 x n+1 +m 2 ^a„x n = (6)
Chebyshev Differential Equation
OO CO
VVn + 2)(n + l)a n+ 2X n - V^ n(n - l)a n x n+2
n=0 n=2
OO OO
— > na n x n -\- m /, a nX n = (?)
2 2
2 * la2 + 3 • 2a%x — 1 • ax + m ao + m aiz
+ y^[(" + 2)(n + l)a n+2 - n(n - l)a„
— na n + m an]/ = (8)
n=2
(2a 2 4- m 2 a ) + [(m 2 - l)ai + 6a 3 ]a
+ ^[(n + 2)(n + l)a n+2 + (m 2 - nVl^ = °> ( 9 )
2a 2 +771 ao =
(m 2 — l)ai + 6a3 =
a n +2
2 2
n — m
for n = 2, 3, .
(10)
(11)
(12)
(n + l)(n + 2)
The first two are special cases of the third, so the general
recurrence relation is
n 2 — m
for n = 0, 1,
(n+l)(n + 2)
Prom this, we obtain for the EVEN COEFFICIENTS
a 2 = -|m 2 ao
a4
a2n
-a 2 =
(2 2 - m 2 )(-m 2 )
ao
3*4 ~* 1-2*3*4
[(2n) 2 - m 2 ][(2n - 2) 2 - m 2 } • • • [-m 2 ]
(2n)!
ao,
(13)
(14)
(15)
(16)
and for the Odd Coefficients
So the general solution is
[A,* _ m 2 ][(k - 2) 2 -m 2 ]---[-m 2 ] r
V = a
1 +
E
z +
E
[{k - 2) 2 - m 2 ][(Jfe - 2) 2 - m 2 ] ■ • • [I 2 - m 2 ]
3
fc!
Chebyshev- Gauss Quadrature 231
If n is Even, then y\ terminates and is a Polynomial
solution, whereas if n is ODD, then y 2 terminates and
is a Polynomial solution. The Polynomial solutions
defined here are known as CHEBYSHEV POLYNOMIALS
of the First Kind. The definition of the Chebyshev
Polynomial of the Second Kind gives a similar, but
distinct, recurrence relation
, (n+ l) 2 - m 2 , , .
fln+2 = ; , w .^ n for n = 0, 1, . . . . (21)
(n + 2)(n + 3)
Chebyshev Function
0(z) = ^lnp,
p<a:
where the sum is over PRIMES p, so
hm -^-r = 1.
Chebyshev-Gauss Quadrature
Also called Chebyshev Quadrature. A Gaussian
Quadrature over the interval [—1,1] with Weight-
ing Function W(x) = l/\/i - z 2 - The Abscissas for
quadrature order n are given by the roots of the CHEBY-
SHEV Polynomial of the First Kind T n (x), which
occur symmetrically about 0. The WEIGHTS are
Wi ■
A n +l7n
A n
7n-l
' A n Tk(xi)T n +i(xi) A n -! T n - l (x i )T n (x t )'
(1)
where A n is the COEFFICIENT of x n in T n (x). For HER-
mite Polynomials,
1-m 2
o
(17)
Additionally,
3 2 -m 2 (3 2 -m 2 )(l 2 -m 2 )
a 5 = 4 5 a 3 = 5 ,
(18)
so
[(2n - l) 2 - m 2 ][{2n - 3) 2 - m 2 ] ■ ■ ■ [l 2 -
-m 2 ]
Since
a ' n - L ~ (2n + l)!
ai-
(19)
A n = 2
A n+1
A n
In =
= 2.
|tt,
WJi =
T n+1 (xi)T n (xi)'
T n {x) = cos(ncos x),
the ABSCISSAS are given explicitly by
(2i- 1)tt"
Since
Xi = cos
T' n {Xi) =
In
(~1)' +1 »
(20)
T„ + i(o;i) = (-l)'sinai,
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
232 Chebyshev Inequality
where
on =
(2i - 1)tt
2n '
all the Weights are
Wi
(10)
(11)
The explicit Formula is then
f(x)dx
i:
vr
Zt'hF^)]*^'™®- < 12 >
11^
2 ±0.707107 1.5708
3 1.0472
±0.866025 1.0472
4 ±0.382683 0.785398
±0.92388 0.785398
5 0.628319
±0.587785 0.628319
±0.951057 0.628319
References
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 330-331, 1956.
Chebyshev Inequality
Apply Markov's Inequality with a = k 2 to obtain
P[{x-fxf >k 2 } <
((x-nf) _a 2
k 2
= h- (^
Therefore, if a RANDOM Variable x has a finite Mean
H and finite VARIANCE <r 2 , then V ft > 0,
P(\x - fi\ > ft) < -^
P(\x - fi\> ka) <
(2)
(3)
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.
Chebyshev Integral
x p (l-x) q dx.
/■
Chebyshev Polynomial
Chebyshev Integral Inequality
/ fi(x)dx I f 2 (x)dx--- I f n (x)dx
«/ a J a J a
<{b-
J a
f(xi)f(x 2 )"-f n (x)dx t
where /i, / 2 , . . . , f n are NONNEGATIVE integrable func-
tions on [a, 6] which are monotonic increasing or decreas-
ing.
References
Gradshteyn, IS. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1092, 1979.
Chebyshev Phenomenon
see Prime Quadratic Effect
Chebyshev Polynomial of the First Kind
0.5
-0.5
A set of Orthogonal Polynomials defined as the so-
lutions to the Chebyshev Differential Equation
and denoted T n (x). They are used as an approxima-
tion to a Least Squares Fit, and are a special case
of the Ultraspherical Polynomial with a = 0. The
Chebyshev polynomials of the first kind T n (x) are illus-
trated above for x £ [0, 1] and n = 1, 2, . . . , 5.
The Chebyshev polynomials of the first kind can be ob-
tained from the generating functions
9i(t>n)
\-t z
1 - 2xt + t 2
= T {x) + 2j2T n (x)t n (1)
and
9*(t,z)= , \j\^ =Y. T ^ tn ( 2 )
l-2xt + t 2
n=0
for \x\ < 1 and \t\ < 1 (Beeler et al 1972, Item 15).
(A closely related Generating Function is the basis
for the definition of Chebyshev Polynomial of the
Second Kind.) They are normalized such that T„(l) =
1. They can also be written
(3)
Chebyshev Polynomial
Chebyshev Polynomial 233
or in terms of a DETERMINANT
X
1
■
••
1
2x
1
■
■■
1
2x
1 ■
■■
1
2x ■
■•
*
•• 1
2x
(4)
In closed form,
L«/2J / v
T n (x) = cosmos" 1 z) = ^ I £)* n ~ 2m (* 2 " 1)™
m=0 ^ '
(5)
where (™) is a BINOMIAL COEFFICIENT and \_x\ is the
Floor Function. Therefore, zeros occur when
*(*-§)
for k — 1, 2, . . . , n. Extrema occur for
X — cos I — J ,
(6)
(7)
where k = 0, 1, . . . , n. At maximum, T n (x) = 1, and
at minimum, T n (x) = -1. The Chebyshev POLYNOMI-
ALS are Orthonormal with respect to the Weighting
Function (1 - x 2 )~ 1/2
/',
T m (x)T n {x)dx
Vl-x 2
{I
ir8 n m for m ^ 0, n ^
for m = n = 0,
(8)
where £ m n is the KRONECKER DELTA. Chebyshev poly-
nomials of the first kind satisfy the additional discrete
identity
m s
-£-' m for % = 7 = 0,
where Xk for fc = 1, . . . , m are the m zeros of T m (x).
They also satisfy the Recurrence Relations
T n+1 (x) = 2xT n (x) - T n _i(x) (10)
T n+ i(a:) - xT„(x) - ^/(l- x 2){l-[T n (x)}2} (11)
for n > 1. They have a Complex integral representa-
tion
Tn{x) = 4ri I l-2 X z + z> (12)
and a Rodrigues representation
Using a FAST FIBONACCI TRANSFORM with multiplica-
tion law
(A, B)(C, D) = (AD + BC + 2xAC, BD - AC) (14)
gives
(T n+ i(aO,-T n (aO) = (Ti(aO,-T (aO)(l,0) n . (15)
Using Gram-Schmidt Orthonormalization in the
range (-1,1) with Weighting Function (1-x 2 ) c ~ 1/2)
gives
Po(x) =
pi(x) =
p 2 {x) =
/^^(1-x 2 )- 1 / 2 ^
/^(l-a: 2 )- 1 ^^
[-(l-* a ) 1/3 ]li =g '
[sin 1 a:]l: 1
/^(l-x 2 )- 1 / 2 ^
/^^(l-o: 2 )- 1 / 2 ^
f\(l - x 2 )- 1 / 2 dx
(16)
(17)
X -
• 1
= [x — 0]x — - = x — h,
etc. Normalizing such that T n (l) = 1 gives
T (x) = 1
Tx(x) = x
T 2 (x) = 2x 2 -1
T 3 (x) = 4x 3 -Sx
T 4 (x) = 8x 4 -8x 2 + l
Ts(x) = 16z 5 -20z 3 + 5z
T 6 (x) = 32z 6 - 48a; 4 + 18x 2 - 1.
(18)
The Chebyshev polynomial of the first kind is related
to the Bessel Function of the First Kind J„(x)
and Modified Bessel Function of the First Kind
I n {x) by the relations
J n (x) = i n T n (i-j^j Jo(x) (19)
I n {x)=T n (J^)lo(x). (20)
Letting x = cos 8 allows the Chebyshev polynomials of
the first kind to be written as
T n (x) = cos(rz0) = cos(ncos~ x). (21)
234 Chebyshev Polynomial
Chebyshev Polynomial
The second linearly dependent solution to the trans-
formed differential equation
d T n t 2
d9 2
+ ri T n =
(22)
is then given by
V n (x) = sin(n#) = sin(ncos~ a;), (23)
which can also be written
V n (x) = Vl-X 2 C/„-i(x),
(24)
where U n is a Chebyshev Polynomial of the Sec-
ond Kind. Note that V n (x) is therefore not a Poly-
nomial.
The Polynomial
x n - 2 L - n T n (x)
(25)
(of degree n — 2) is the POLYNOMIAL of degree < n which
stays closest to x n in the interval (—1,1). The maximum
deviation is 2 1 ~ n at the n -+- 1 points where
(26)
for k = 0, 1, . . . , n (Beeler et al. 1972, Item 15).
see also Chebyshev Approximation Formula,
Chebyshev Polynomial of the Second Kind
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal
Polynomials." Ch. 22 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 771-802, 1972.
Arfken, G. "Chebyshev (TschebyschefF) Polynomials" and
"Chebyshev Polynomials — Numerical Applications." §13.3
and 13.4 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 731-748, 1985.
Beeler, M.; Gosper, R. W.; and Schroeppel, R HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM 239, Feb. 1972.
Iyanaga, S. and Kawada, Y. (Eds.). "Cebysev (Tscheby-
schefF) Polynomials." Appendix A, Table 20.11 in Encyclo-
pedic Dictionary of Mathematics. Cambridge, MA: MIT
Press, pp. 1478-1479, 1980.
Rivlin, T. J. Chebyshev Polynomials. New York: Wiley,
1990.
Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi-
als T n (x) and U n (x)" Ch. 22 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 193-207, 1987.
Chebyshev Polynomial of the Second Kind
A modified set of Chebyshev Polynomials defined by a
slightly different GENERATING FUNCTION. Used to de-
velop four- dimensional SPHERICAL HARMONICS in an-
gular momentum theory. They are also a special case
of the Ultraspherical Polynomial with a = 1. The
Chebyshev polynomials of the second kind U n (x) are
illustrated above for x 6 [0, 1] and n— 1, 2, ..., 5.
The defining GENERATING FUNCTION of the Chebyshev
polynomials of the second kind is
g<2(t,x) =
1
1 - 2xt + t 2
Y,Un(x)t n (1)
for \x\ < 1 and \t\ < 1. To see the relationship to
a Chebyshev Polynomial of the First Kind (T),
take dg/Ot,
^ = -(1 - 2xt + t 2 )~\~2x + 2t)
- 2(t - x){l - 2xt -\- 1 2 )~ 2
oo
= \ nC/n(x)£ n-1 .
n—
Multiply (2) by t,
oo
{2t 2 -2xt){l-2xt-rt 2 )~ 2 = ^nU n {x)t n
n=0
and take (3) -(2),
{2t 2 - 2tx) - (1 - 2xt + t 2 ) _ t 2 - 1
(2)
(3)
(l-2xt + t 2 ) 2
{l-2xt + t) 2
oo
= 5> -!)£/„(*)*"• (4)
The Rodrigues representation is
Un{x) =
(-i)> + iysF
2\n+l/2i
[(1 _ X *)W}
2»+ 1 (n+ |)!(1 -x 2 y/*dx n
The polynomials can also be written
u n {x)= X)(-ir( n /)(2xr- a -
rv 2 i / x
^ \2m + l/ v }
(5)
(6)
where [a; J is the Floor Function and \x] is the Ceil-
ing Function, or in terms of a Determinant
U n
2x 1
2x 1
1 2x 1
1 2x
(7)
Chebyshev Quadrature
Chebyshev Quadrature 235
The first few POLYNOMIALS are
U (x)
= 1
Ui(x)
= 2x
U 2 {x)
= 4x 2 - 1
U 3 (x)
= 8x 3 - 4x
Ut{x)
= 16z 4 - 12z 2 + 1
U 5 (x)
= 32a; 5 - 32a; 3 + 6a;
U 6 (x)
= 64a; 6 - 80a; 4 + 24a; 2 - 1
Letting x = cos 6 allows the Chebyshev polynomials of
the second kind to be written as
U n (x) =
sin[(ra+l)fl]
sin#
(8)
The second linearly dependent solution to the trans-
formed differential equation is then given by
W n (x)
cos[(n+l)fl]
sin#
which can also be written
W n (x) = {l-x 2 )- 1/2 T n + 1 (x),
(9)
(10)
where T n is a CHEBYSHEV POLYNOMIAL OF THE FIRST
Kind. Note that W n (x) is therefore not a Polynomial.
see also Chebyshev Approximation Formula,
Chebyshev Polynomial of the First Kind, Ultra-
spherical Polynomial
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal
Polynomials." Ch. 22 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 771-802, 1972.
Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and
"Chebyshev Polynomials — Numerical Applications." §13.3
and 13.4 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 731-748, 1985.
Rivlin, T. J. Chebyshev Polynomials, New York: Wiley,
1990.
Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi-
als T n (x) and U n [x). n Ch. 22 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 193-207, 1987.
Chebyshev Quadrature
A Gaussian QuADRATURE-like Formula for numeri-
cal estimation of integrals. It uses Weighting Func-
tion W(x) = 1 in the interval [-1, 1] and forces all the
weights to be equal. The general FORMULA is
/;
f(x)dx = - \ }{xi).
n *■ — ^
The ABSCISSAS are found by taking terms up to y n in
the MACLAURIN SERIES of
Sn(y) = exp <
| in -2 + ln(l-y)(l-i)
+ ln(l + y)
H)]}
and then defining
G n (x) = x n s n (-)
The ROOTS o£G n (x) then give the ABSCISSAS. The first
few values are
G (x) = 1
G\{x) = x
G 2 (x) = l(3x 2 ~l)
G s {x) = l(2x 3 -x)
G*(x) = ^(45z 4 -30:£ 2 + i)
G s (x) = ^(72a; 5 - 60x 3 + 7x)
G G {x) = ^(105x 6 - 105x 4 + 21z 2 - 1)
Gr{x)
G 8 (x)
G 9 (x) =
j^ (6480a; 7 - 7560a; 5 + 2142a; 3 - 149a;)
56700x 6 + 20790a; 4
6480
42k (42525a; 8
- 2220a; 2 - 43)
22^ (22400a; 9 - 33600x 7 + 15120a; 5
2280a; 3 + 53a;).
Because the ROOTS are all REAL for n < 7 and n = 9
only (Hildebrand 1956), these are the only permissible
orders for Chebyshev quadrature. The error term is
_ I c n (n+1)! n
n ~) c f {n+2) U) _
I ° Tl (n+2)! U
odd
even.
where
{J_ xG n (x)dx n odd
I-i x 2 Gn{x)dx n even.
The first few values of c n are 2/3, 8/45, 1/15, 32/945,
13/756, and 16/1575 (Hildebrand 1956). Beyer (1987)
gives abscissas up to n = 7 and Hildebrand (1956) up
to n = 9.
236 Chebyshev-Radau Quadrature
Chebyshev's Theorem
cally for small n.
n
x»
2
±0.57735
3
±0.707107
4
±0.187592
±0.794654
5
±0.374541
±0.832497
6
±0.266635
±0.422519
±0.866247
7
±0.323912
±0.529657
±0.883862
9
±0.167906
±0.528762
±0.601019
±0.911589
d w
eights can be
n
Xi
2
±|V3
3
±|V2
4
5
i ■ 1 y/h-2
± V sVs
±\^-^F
.1 /s+x/TT
=C 2 V 3
see a/so Chebyshev Quadrature, Lobatto Quad-
rature
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 466, 1987.
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 345-351, 1956.
Chebyshev-Radau Quadrature
A Gaussian QuADRATURE-like Formula over the in-
terval [-1, 1] which has Weighting Function W(x) =
x. The general FORMULA is
/l "
xf(x)dx = ^Wilfixt) - f(-Xi)].
1 i=i
n
Xi
Wi
1
0.7745967
0.4303315
2
0.5002990
0.2393715
0.8922365
0.2393715
3
0.4429861
0.1599145
0.7121545
0.1599145
0.9293066
0.1599145
4
0.3549416
0.1223363
0.6433097
0.1223363
0.7783202
0.1223363
0.9481574
0.1223363
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 466, 1987.
Chebyshev Sum Inequality
If
Cb\ > 0,2 > . • • > 0,-n
h >b 2 >...>6n,
then
n z2 akbk - ( Z-s ak } [ z2^ k J '
k^i \ fc=i / \ k=i /
This is true for any distribution.
see also CAUCHY INEQUALITY, HOLDER SUM INEQUAL-
ITY
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1092, 1979.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 43-44, 1988.
Chebyshev-Sylvester Constant
In 1891, Chebyshev and Sylvester showed that for suf-
ficiently large x, there exists at least one prime number
p satisfying
x < p < (1 + a)x,
where a = 0.092.... Since the PRIME NUMBER THE-
OREM shows the above inequality is true for all a >
for sufficiently large x t this constant is only of historical
interest.
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 22, 1983.
Chebyshev's Theorem
see Bertrand's Postulate
Checker-Jumping Problem
Chern Number 237
Checker-Jumping Problem
Seeks the minimum number of checkers placed on a
board required to allow pieces to move by a sequence of
horizontal or vertical jumps (removing the piece jumped
over) n rows beyond the forward-most initial checker.
The first few cases are 2, 4, 8, 20. It is, however, impos-
sible to reach level 5.
References
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 23-28, 1976.
Checkerboard
see Chessboard
Checkers
Beeler et al. (1972, Item 93) estimated that there are
about 10 12 possible positions. However, this disagrees
with the estimate of Jon Schaeffer of 5 x 10 20 plausible
positions, with 10 18 reachable under the rules of the
game. Because "solving" checkers may require only the
Square Root of the number of positions in the search
space (i.e., 10 9 ), so there is hope that some day checkers
may be solved (i.e., it may be possible to guarantee a
win for the first player to move before the game is even
started; Dubuque 1996).
Depending on how they are counted, the number of Eu-
LERIAN CIRCUITS on an n x n checkerboard are either
1, 40, 793, 12800, 193721, ... (Sloane's A006240) or 1,
13, 108, 793, 5611, 39312, . . . (Sloane's A006239).
see also Checkerboard, Checker-Jumping Prob-
lem
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Dubuque, W. "Re: number of legal chess positions." math-
fun@cs.arizona.edu posting, Aug 15, 1996.
Kraitchik, M. "Chess and Checkers" and "Checkers
(Draughts)." §12.1.1 and 12.1.10 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 267-276 and 284-
287, 1942.
Schaeffer, J. One Jump Ahead: Challenging Human
Supremacy in Checkers. New York: Springer- Verlag, 1997.
Sloane, N. J. A. Sequences A006239/M4909 and A006240/
M5271 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Checksum
A sum of the digits in a given transmission modulo some
number. The simplest form of checksum is a parity bit
appended on to 7-bit numbers (e.g., ASCII characters)
such that the total number of Is is always EVEN ("even
parity") or Odd ("odd parity"). A significantly more
sophisticated checksum is the CYCLIC REDUNDANCY
Check (or CRC), which is based on the algebra of poly-
nomials over the integers (mod 2). It is substantially
more reliable in detecting transmission errors, and is
one common error- checking protocol used in modems.
see also Cyclic Redundancy Check, Error-
Correcting Code
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Cyclic Redundancy and Other Checksums."
Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 888-895, 1992.
Cheeger's Finiteness Theorem
Consider the set of compact n-RlEMANNlAN MANIFOLDS
M with diameter(M) < d, Volume(M) > V, and \K\ <
k where k is the Sectional Curvature. Then there
is a bound on the number of DlFFEOMORPHlSMS classes
of this set in terms of the constants n, d, V, and k.
References
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Chefalo Knot
A fake KNOT created by tying a SQUARE Knot, then
looping one end twice through the KNOT such that when
both ends are pulled, the KNOT vanishes.
Chen's Theorem
Every "large" EVEN INTEGER may be written as 2n =
p -J- m where p is a Prime and m 6 P2 is the Set of
Semiprimes (i.e., 2-Almost Primes).
see also ALMOST PRIME, PRIME NUMBER, SEMIPRIME
References
Rivera, C "Problems & Puzzles (Conjectures): Chen's
Conjecture." http://www.sci.net .mx/-crivera/ppp/
conj_002.htm.
Chern Class
A Gadget defined for Complex Vector Bundles.
The Chern classes of a Complex Manifold are the
Chern classes of its Tangent Bundle. The ith Chern
class is an OBSTRUCTION to the existence of (n — i +
1) everywhere COMPLEX linearly independent VECTOR
Fields on that Vector Bundle. The zth Chern class
is in the (2z)th cohomology group of the base SPACE.
see also OBSTRUCTION, PONTRYAGIN CLASS, STIEFEL-
Whitney Class
Chern Number
The Chern number is defined in terms of the Chern
Class of a Manifold as follows. For any collection
Chern Classes such that their cup product has the
same Dimension as the Manifold, this cup product
can be evaluated on the Manifold's Fundamental
CLASS. The resulting number is called the Chern num-
ber for that combination of Chern classes. The most
important aspect of Chern numbers is that they are
COBORDISM invariant.
see also Pontryagin Number, Stiefel-Whitney
Number
238
Chemoff Face
Chess
Chernoff Face
A way to display n variables on a 2-D surface. For in-
stance, let x be eyebrow slant, y be eye size, z be nose
length, etc.
References
Gonick, L. and Smith, W. The Cartoon Guide to Statistics.
New York: Harper Perennial, p. 212, 1993.
Chess
Chess is a game played on an 8x8 board, called a CHESS-
BOARD, of alternating black and white squares. Pieces
with different types of allowed moves are placed on the
board, a set of black pieces in the first two rows and
a set of white pieces in the last two rows. The pieces
are called the bishop (2), king (1), knight (2), pawn (8),
queen (1), and rook (2). The object of the game is to
capture the opponent's king. It is believed that chess
was played in India as early as the sixth century AD.
In a game of 40 moves, the number of possible board
positions is at least 10 120 according to Peterson (1996).
However, this value does not agree with the 10 pos-
sible positions given by Beeler et al. (1972, Item 95).
This value was obtained by estimating the number of
pawn positions (in the no-captures situation, this is 15 ),
times all pieces in all positions, dividing by 2 for each
of the (rook, knight) which are interchangeable, divid-
ing by 2 for each pair of bishops (since half the posi-
tions will have the bishops on the same color squares).
There are more positions with one or two captures, since
the pawns can then switch columns (Schroeppel 1996).
Shannon (1950) gave the value
P(40) :
64!
32!(8!) 2 (2!) 6
10 4
The number of chess games which end in exactly n plies
(including games that mate in fewer than n plies) for
n = 1, 2, 3, . . . are 20, 400, 8902, 197742, 4897256,
119060679, 3195913043, ... (K. Thompson, Sloane's
A007545). Rex Stout's fictional detective Nero Wolfe
quotes the number of possible games after ten moves as
follows: "Wolfe grunted. One hundred and sixty-nine
million, five hundred and eighteen thousand, eight hun-
dred and twenty-nine followed by twenty-one ciphers.
The number of ways the first ten moves, both sides,
may be played" (Stout 1983). The number of chess
positions after n moves for n — 1, 2, . , . are 20, 400,
5362, 71852, 809896?, 9132484?, . . . (Schwarzkopf 1994,
Sloane's A019319).
Cunningham (1889) incorrectly found 197,299 games
and 71,782 positions after the fourth move. C. Flye
St. Marie was the first to find the correct number of po-
sitions after four moves: 71,852. Dawson (1946) gives
the source as Intermediare des Mathematiques (1895),
but K. Fabel writes that Flye St. Marie corrected the
number 71,870 (which he found in 1895) to 71,852 in
1903. The history of the determination of the chess se-
quences is discussed in Schwarzkopf (1994).
Two problems in recreational mathematics ask
1. How many pieces of a given type can be placed on a
Chessboard without any two attacking.
2. What is the smallest number of pieces needed to oc-
cupy or attack every square.
The answers are given in the following table (Madachy
1979).
Piece
Max.
Min.
bishops
14
8
kings
16
9
knights
32
12
queens
8
5
rooks
8
8
see also BISHOPS PROBLEM, CHECKERBOARD, CHECK-
ERS, Fairy Chess, Go, Gomory's Theorem, Hard
Hexagon Entropy Constant, Kings Problem,
Knight's Tour, Magic Tour, Queens Problem,
Rooks Problem, Tour
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 124-
127, 1987.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Dawson, T. R. "A Surprise Correction." The Fairy Chess
Review 6, 44, 1946.
Dickins, A. "A Guide to Fairy Chess." p. 28, 1967/1969/
1971.
Dudeney, H. E. "Chessboard Problems," Amusements in
Mathematics. New York: Dover, pp. 84-109, 1970.
Fabel, K. "Nusse." Die Schwalbe 84, 196, 1934.
Fabel, K. "Weihnachtsniisse." Die Schwalbe 190, 97, 1947.
Fabel, K. "Weihnachtsniisse." Die Schwalbe 195, 14, 1948.
Fabel, K. "Eroffnungen." Am Rande des Schachbretts, 34—
35, 1947.
Fabel, K. "Die ersten Schritte." Rund um das Schachbrett,
107-109, 1955.
Fabel, K. "Eroffnungen." Schach und Zahl 8, 1966/1971.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 86-89, 1975.
Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathemati-
cal Recreations. New York: W. W. Norton, pp. 267-276,
1942.
Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 34-54, 1979.
Peterson, I. "The Soul of a Chess Machine: Lessons Learned
from a Contest Pitting Man Against Computer." Sci.
News 149, 200-201, Mar. 30, 1996.
Petkovic, M. Mathematics and Chess. New York: Dover,
1997.
Schroeppel, R. "Reprise: Number of legal chess positions."
tech-news@cs.arizona.edu posting, Aug. 18, 1996.
Schwarzkopf, B. "Die ersten Ziige." Problemkiste, 142—143,
No. 92, Apr. 1994.
Shannon, C. "Programming a Computer for Playing Chess."
Phil. Mag. 41, 256-275, 1950.
Sloane, N. J. A. Sequences A019319 and A007545/M5100 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Chessboard
Chi Distribution 239
Stout, R. "Gambit." In Seven Complete Nero Wolfe Novels.
New York: Avenic Books, p. 475, 1983.
Chessboard
A board containing 8x8 squares alternating in color
between black and white on which the game of Chess is
played. The checkerboard is identical to the chessboard
except that chess's black and white squares are colored
red and white in CHECKERS. It is impossible to cover a
chessboard from which two opposite corners have been
removed with DOMINOES.
see also Checkers, Chess, Domino, Gomory's The-
orem, Wheat and Chessboard Problem
References
Pappas, T. "The Checkerboard." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 136 and 232,
1989.
Chevalley Groups
Finite Simple Groups of Lie-Type. They include
four families of linear SIMPLE GROUPS: PSL(n,q),
PSU(n,q), PSp(2n,q), or PQ € (n,q).
see also Twisted Chevalley Groups
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //f or . mat . bham . ac . uk/atlas#chev.
Chevalley's Theorem
Let f{x) be a member of a Finite Field
F[xx, #2, . . • jX n ] and suppose /(0,0,...,0) = and n
is greater than the degree of /, then / has at least two
zeros in A n {F).
References
Chevalley, C "Demonstration d'une hypothese de M. Artin."
Abhand. Math. Sem. Hamburg 11, 73-75, 1936.
Ireland, K. and Rosen, M. "Chevalley's Theorem." §10.2 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 143-144, 1990.
Chevron
A 6-Polyiamond.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Chi
^, ./ s , f Z cosht — 1 . ■
Chi(jz) = 7 + In z + / dt }
Jo ^
where 7 is the Euler-Mascheroni Constant. The
function is given by the Mathematica® (Wolfram Re-
search, Champaign, IL) command CoshlntegralEz] .
see also Cosine Integral, Shi, Sine Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co-
sine Integrals." §5.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 231-233, 1972.
Chi Distribution
The probability density function and cumulative distri-
bution function are
Pn{x)
2 l-n/2 x n~l e -x 2 /2
D n {x) = Q{\n,\x 2 ),
where Q is the Regularized Gamma Function.
v^r(i(n + i))
M= r(in)
2 ^ 2[r(in)r(l + ln)-r 2 (f(n+l))]
= 2T*{\{n + 1)) - 3r(|n)r(§(n + l))r(l + \n)
71 [r(In)r(l + in)-r»(i(n + l))] 8 / a
(1)
(2)
(3)
(4)
240
72 =
Chi Inequality
[r(in)r(l + in)-H(I(„ + l))]3/2
-3r*(i(n + 1)) + er(§n) + r 2 (|(n + i))r(i + i n )
(5)
[r(|n)r(^)-r»(i(n + i))]»
-AT*C-n)T(\(n + i))r(*±=) + r»(±n)r(±±*)
[r(^)r(^)-r 2 (i(n + i))] 2
(6)
where m is the MEAN, <r 2 the VARIANCE, 71 the Skew-
ness, and 72 the Kurtosis. For n = 1, the x distribu-
tion is a Half-Normal Distribution with = 1. For
n = 2, it is a Rayleigh Distribution with a = 1.
see a/50 Chi-Squared Distribution, Half-Normal
Distribution, Rayleigh Distribution
Chi Inequality
The inequality
(j + l)aj -V ca> (j + l)i,
which is satisfied by all ^-SEQUENCES.
References
Levine, E. and O'Sullivan, J. "An Upper Estimate for the
Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34,
9-24, 1977.
Chi-Squared Distribution
A x 2 distribution is a Gamma Distribution with = 2
and a = r/2, where r is the number of DEGREES OF
Freedom. If Y» have Normal Independent distribu-
tions with MEAN and VARIANCE 1, then
-£* 2
(i)
is distributed as x* witn n DEGREES OF FREEDOM. If
Xi 2 are independently distributed according to a x 2 dis-
tribution with m, 712, . . . , n*. DEGREES OF FREEDOM,
then
Xj
(2)
is distributed according to x with n = X] n =i n J DE-
GREES of Freedom.
P n (x) = \ r(|r)2-/2 - (3)
for x < 0.
The cumulative distribution function is then
_ , a , f x t^e-^dt
Chi-Squared Distribution
where P(a, z) is a REGULARIZED GAMMA FUNCTION.
The Confidence Intervals can be found by finding
the value of x for which D n (x) equals a given value.
The Moment-Generating Function of the x 2 distri-
bution is
M(t)--
= (1
- 2t)~ T/2
(5)
R(t) =
Eblj
M(t) = -
§rln(l-
-2t)
(6)
R'(t) --
1-
r
-2t
(7)
R"(t) =
2r
(8)
(1
-2ty
M
= R'(0) =
= r
(9)
2
= R"(0)--
= 2r
(10)
71
12
(11)
72
(12)
The nth Moment about zero for a distribution with n
Degrees of Freedom is
m' n = 2- r( ' 1 1 ^ = r(r + 2) ■ ■ ■ (r + 2n - 2), (13)
and the moments about the MEAN are
fJL2 = 2r (14)
A*3 = 8r (15)
p 4 = 12n 2 + 48n. (16)
The nth CUMULANT is
« n = 2 n r(n)(|r) = 2 n - x (n - l)!r, (17)
The Moment-Generating Function is
-r/2
9 *\/2A
-r/2
As r* — ► 00,
so for large r,
lim M(t) = e* 2/2 ,
r/2
^i/E
(x< - /J,) 2
<Ti'
(18)
(19)
(20)
Chi-Squared Distribution
Chi-Squared Test 241
is approximately a Gaussian Distribution with
MEAN y/2r and VARIANCE <t 2 = 1. Fisher showed that
X 2 ~r
V27--1
(21)
is an improved estimate for moderate r. Wilson and
Hilferty showed that
1/3
(22)
is a nearly GAUSSIAN DISTRIBUTION with MEAN \i =
1 - 2/(9r) and VARIANCE a 2 = 2/(9r).
In a Gaussian Distribution,
P(x) dx = ~^=e~ (x ~ » )2/2(r2 dx, (23)
let
Then
so
But
z = (x — fi) I a .
dx = — -=dz.
2v^
P(z)dz = 2P(x)dx,
r(f)2V2
\/27r
(24)
dz ^2(x-^ dx= 2^z dx
(T z (7
(26)
(27)
P(x) dx = 2 —^—e-^ 2 dz = -^=e~ z/2 dz. (28)
This is a \ 2 distribution with r = 1, since
1/2-1 -z/2 1/2-1/2
P(z) ^ = e d* = -L dz. (29)
oFi is the Confluent Hypergeometric Limit Func-
tion and T is the GAMMA FUNCTION. The Mean,
Variance, Skewness, and Kurtosis are
\i = A + n
2
7i
72
2(2A + n)
2y / 2(3A + n)
(2A + n)3/2
12(4A + n)
(2A + n) 2 *
(34)
(35)
(36)
(37)
see also Chi Distribution, Snedecor's F-Distribu-
tion
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 940-943, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 535, 1987.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Incomplete Gamma Function, Error Func-
tion, Chi-Square Probability Function, Cumulative Poisson
Function." §6.2 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 209-214, 1992.
Spiegel, M . R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, pp. 115-116, 1992.
Chi-Squared Test
Let the probabilities of various classes in a distribution
be pi , p2 , . . . , Pk • The expected frequency
£
(mi - Npj) 2
N Pi
is a measure of the deviation of a sample from expecta-
tion. Karl Pearson proved that the limiting distribution
of \s 2 is x 2 (Kenney and Keeping 1951, pp. 114-116).
If Xi are independent variates with a NORMAL DISTRI-
BUTION having MEANS \i{ and VARIANCES a 2 for i = 1,
. . . , n, then
i 2 _ v^ (Xi -in)
= £
2 A ~ £^ 2<7i 2
1=1
(30)
is a Gamma Distribution variate with a = n/2,
r( ? n)
(31)
The noncentral chi-squared distribution is given by
P(x) = 2-" /2 e - (A+l)/2 x n/2 - 1 F(in, f Ax), (32)
where
F(a,z) =
oFi(;a;z)
T(a) '
(33)
Pr(* 2 >X* 2 )= f^ f(x 2 )d( X 2 )
Jxs 2
2\M)/2
= 1
~ 2
= 1
f
(*)
,V/3
r(ft=i)
d(x 2 )
= 1-1
Xs
k-3
V^^T)' 2
where I(x i n) is PEARSON'S FUNCTION. There are some
subtleties involved in using the x 2 test to fit curves (Ken-
ney and Keeping 1951, pp. 118-119).
When fitting a one-parameter solution using x 2 > the
best-fit parameter value can be found by calculating % 2
242
Child
Choose
at three points, plotting against the parameter values of
these points, then finding the minimum of a PARABOLA
fit through the points (Cuzzi 1972, pp. 162-168).
References
Cuzzi, J. The Subsurface Nature of Mercury and Mars from
Thermal Microwave Emission. Ph.D. Thesis. Pasadena,
CA: California Institute of Technology, 1972.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Child
A node which is one EDGE further away from a given
Edge in a Rooted Tree.
see also Root (Tree), Rooted Tree, Sibling
Chinese Hypothesis
A Prime p always satisfies the condition that 2 P — 2
is divisible by p. However, this condition is not true
exclusively for PRIME (e.g., 2 341 — 2 is divisible by 341 =
11*31). Composite Numbers n (such as 341) for which
2 n - 2 is divisible by n are called Poulet Numbers,
and are a special class of Fermat Pseudoprimes. The
Chinese hypothesis is a special case of FERMAT's LITTLE
Theorem.
see also Carmichael Number, Euler's Theorem,
Fermat's Little Theorem, Fermat Pseudoprime,
Poulet Number, Pseudoprime
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 19-20, 1993.
Chinese Remainder Theorem
Let r and s be Positive Integers which are Rela-
tively Prime and let a and b be any two Integers.
Then there is an INTEGER N such that
and the 6, are determined from
M
and
N = a (mod r)
N = b (mod 5) .
(i)
(2)
Moreover, iV is uniquely determined modulo rs. An
equivalent statement is that if (r,s) = 1, then every
pair of Residue Classes modulo r and s corresponds
to a simple RESIDUE CLASS modulo rs.
The theorem can also be generalized as follows. Given
a set of simultaneous CONGRUENCES
x = a,i (mod rrii) (3)
for i — 1, . . . , r and for which the rrti are pairwise Rela-
tively Prime, the solution of the set of Congruences
is
x = aibi (- . . . -h a r b r (mod M), (4)
mi m r
bi — = 1 (mod rrii).
TTli
(6)
where
M = m\m2 - - *rn r
(5)
References
Ireland, K. and Rosen, M. "The Chinese Remainder Theo-
rem." §3.4 in A Classical Introduction to Modern Number
Theory, 2nd ed. New York: Springer- Verlag, pp. 34-38,
1990.
Uspensky, J. V. and Heaslet, M. A. Elementary Number The-
ory. New York: McGraw-Hill, pp. 189-191, 1939.
Wagon, S. "The Chinese Remainder Theorem." §8.4 in Math-
ematica in Action. New York: W. H. Freeman, pp. 260-
263, 1991.
Chinese Rings
see Baguenaudier
Chiral
Having forms of different HANDEDNESS which are not
mirror-symmetric.
see also Disymmetric, Enantiomer, Handedness,
Mirror Image, Reflexible
Choice Axiom
see Axiom of Choice
Choice Number
see Combination
Cholesky Decomposition
Given a symmetric POSITIVE DEFINITE MATRIX A, the
Cholesky decomposition is an upper TRIANGULAR MA-
TRIX U such that
A-U T U.
see also LU Decomposition, QR Decomposition
References
Nash, J. C. "The Choleski Decomposition." Ch. 7 in Com-
pact Numerical Methods for Computers: Linear Algebra
and Function Minimisation, 2nd ed. Bristol, England:
Adam Hilger, pp. 84-93, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Cholesky Decomposition." §2.9 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 89-91, 1992.
Choose
An alternative term for a BINOMIAL COEFFICIENT, in
which C?} is read as "n choose k" R. K. Guy suggested
this pronunciation around 1950, when the notations n C r
and n C r were commonly used. Leo Moser liked the pro-
nunciation and he and others spread it around. It got
the final seal of approval from Donald Knuth when he
incorporated it into the TeX mathematical typesetting
language as {n\choose k}.
Choquet Theory
Chow Coordinates
243
Choquet Theory
Erdos proved that there exist at least one Prime of the
form Ak + 1 and at least one Prime of the form 4k -f 3
between n and 2n for all n > 6.
see also Equinumerous, Prime Number
Chord
chord^
The Line Segment joining two points on a curve. The
term is often used to describe a LINE Segment whose
ends lie on a CIRCLE. In the above figure, r is the RA-
DIUS of the CIRCLE, a is called the Apothem, and s the
Sagitta.
s s_
The shaded region in the left figure is called a Sector,
and the shaded region in the right figure is called a SEG-
MENT.
All ANGLES inscribed in a Circle and subtended by
the same chord are equal. The converse is also true:
The LOCUS of all points from which a given segment
subtends equal ANGLES is a CIRCLE.
Let a Circle of Radius R have a Chord at distance r.
The Area enclosed by the Chord, shown as the shaded
region in the above figure, is then
f , v / J? 2„ 7 .2
A = 2 / x(y) dy.
Jo
But
y 2 + (r + x) 2 = R 2 ,
x(y) = \/R 2 - y 2 - r
(1)
(2)
(3)
and
A = 2
/ (y/R 2 -y 2
Jo
r)dy
y^R 2 -y 2 +R 2 tan" 1
■i^
2ry
■.ry/B? ™r 2 + J^ 2 tan" 1
sfR?
:i)'-
= i^tan" 1
(f) :
- r^R 2 - \
2r^R? - r 2
(4)
Checking the limits, when r = R, A = and when
r->0,
A=\kR\ (5)
see also Annulus, Apothem, Bertrand's Problem,
Concentric Circles, Radius, Sagitta, Sector,
Segment
Chordal
see Radical Axis
Chordal Theorem
The LOCUS of the point at which two given CIRCLES
possess the same POWER is a straight line PERPENDIC-
ULAR to the line joining the MIDPOINTS of the CIRCLE
and is known as the chordal (or RADICAL Axis) of the
two Circles.
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 153,
1965.
Chow Coordinates
A generalization of GRASSMANN COORDINATES to m-D
varieties of degree d in P n , where P n is an n-D pro-
jective space. To define the Chow coordinates, take
the intersection of a m-D VARIETY Z of degree d by
an (n - m)-D SUBSPACE U of P n . Then the coordi-
nates of the d points of intersection are algebraic func-
tions of the Grassmann Coordinates of U, and by
taking a symmetric function of the algebraic functions,
a hHOMOGENEOUS POLYNOMIAL known as the Chow
form of Z is obtained. The Chow coordinates are then
244 Chow Ring
the Coefficients of the Chow form. Chow coordinates
can generate the smallest field of definition of a divisor.
References
Chow, W.-L. and van der Waerden., B. L. "Zur algebraische
Geometrie IX." Math. Ann. 113, 692-704, 1937.
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc.
43, 1117-1124, 1996.
Chow Ring
The intersection product for classes of rational equiva-
lence between cycles on an Algebraic Variety.
References
Chow, W.-L. "On Equivalence Classes of Cycles in an Alge-
braic Variety." Ann. Math. 64, 450-479, 1956.
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.-L "Wei-Liang Chow." Not. Amer. Math. Soc.
43, 1117-1124, 1996.
Chow Variety
The set C n ,m,d of all rn-D varieties of degree d in an n-D
projective space P n into an M-D projective space P M .
References
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc.
43, 1117-1124, 1996.
Christoffel-Darboux Formula
For three consecutive ORTHOGONAL POLYNOMIALS
Pn(x) = (A n X + B n )p n -lX ~ C n p n -2(x) (l)
for n = 2, 3, . . . , where A n > 0, B n , and C n > are
constants. Denoting the highest Coefficient of p n (x)
by fc n ,
A n =
kn-l
•A-n rCn^n — 2
A n -i kn-i 2
(2)
(3)
Then
Po(x)po{y) 4- . . -+p n (x)p n {y)
= k n Pn + l(x)p n (y) - Pn(x)p n + l(y)
kn+x x-y
In the special case of x = y, (4) gives
(4)
\P0(X)} 2 + . . . + \p n (x)]
k
kn+l
\Pn+l{x)Pn{x) ~ P n ( X )Pn+l(x)}. (5)
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 785, 1972.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, pp. 42-44, 1975.
Christoffel Number
Christoffel-Darboux Identity
<f> k (x)(f) k (y) _ <p m +l(x)(f>m(y) - (t>m{x)<f>m+l{y)
a m ^ m {x -y),
k=0 "" ' x
(1)
where 4>k{x) are ORTHOGONAL POLYNOMIALS with
Weighting Function W(x) y
7m= J[cj>r n {x)fW{x)dx, (2)
and
Q>k —
Z±±i
A k
(3)
where A k is the COEFFICIENT of x k in <f>k(x).
References
Hildebrand, F. B. Introduction to Numerical Analysis. New
. York: McGraw-Hill, p. 322, 1956.
Christoffel Formula
Let {p n {x)} be orthogonal Polynomials associated
with the distribution da(x) on the interval [a, 6]. Also
let
p = c(x — Xi)(x - X2) ' ' • (x — Xi)
(for c ^ 0) be a Polynomial of order I which is
NONNEGATIVE in this interval. Then the orthogonal
Polynomials {q(x)} associated with the distribution
p(x) da(x) can be represented in terms of the POLYNO-
MIALS p n {x) as
p{x)q n {x) =
Pn(x) p n + l(x)
Pn(xi) Pn + l(xi)
Pn(Xl) Pn+l{xi)
Pn+l{x)
Pn+l(xi)
Pn+l{xi)
In the case of a zero x k of multiplicity m > 1, we replace
the corresponding rows by the derivatives of order 0, 1,
2, . . . , m - 1 of the POLYNOMIALS p n (xi), . . . , p n +l{xi)
at x — — x k .
References
Szego, G. Orthogonal Polynomials, J^.th ed. Providence, RI:
Amer. Math. Soc, pp. 29-30, 1975.
Christoffel Number
One of the quantities Xi appearing in the GAUSS-JACOBI
Mechanical Quadrature. They satisfy
Ai + A 2 +
. . . + A„ = /
J a
da(x) = a{b) - a(a) (1)
Christoffel Symbol of the First Kind
Christoffel Symbol of the Second Kind 245
and are given by
J a [Pn(x v )(X - X,
A„ =
&n
+ 1
1
k n Pn+l(Xv)Pn(Xv)
k n 1
da(x) (2)
(3)
(4)
(5)
k n -\ p n -r{xu)Pk{x u )
where A; n is the higher COEFFICIENT of p n (x).
References
Szego, G. Orthogonal Polynomials, ^th ed. Providence, RI:
Amer. Math. Soc, pp. 47-48, 1975.
Christoffel Symbol of the First Kind
Variously denoted [ij,k], [\ J ], r obc , or {ab,c}.
[ij, k] :
(i)
where p mfc is the METRIC TENSOR and
But
df
dq k ~ dq* [€i ' 6j) " a 9 * ' ej ei ■ a<? fc
= [»M + b"M, (3)
so
[ab,c]= \{9ac,b+ 9bc,a- 9ab,c)' (4)
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 160-167, 1985.
Christoffel Symbol of the Second Kind
Variously denoted { . m . } or rg.
-*m C76i kmr • • j i
_ 1 fcm ( <9ffi
2 5 \ dgJ
rJ d^ dq k
(1)
where rjj is a CONNECTION COEFFICIENT and {6c, d}
is a Christoffel Symbol of the First Kind.
\ b a c j =9ad{bc,d}.
(2)
The Christoffel symbols are given in terms of the first
Fundamental Form E, F, and G by
r 12
1 _ GE U - 2FF U + FE V
2(EG-F 2 )
GE V — FG U
2{EG - F 2 )
2GF V — GG U — FG V
2(EG - F 2 )
2£F U - EE V - FE U
2(EG-F 2 )
EG U — FE V
r*22
r 2 -
1 11 —
r 2 -
1 12 —
2(£G-F 2 )
■p2 SGd — 2FF V + FG U
1 22 —
2(£G - F 2 )
(3)
(4)
(5)
(6)
(7)
(8)
and T^ = T\ 2 and T^ = r? 3 . If F = 0, the Christoffel
symbols of the second kind simplify to
(9)
(10)
(11)
(12)
(13)
(14)
(Gray 1993).
The following relationships hold between the Christoffel
symbols of the second kind and coefficients of the first
Fundamental Form,
r 1
1 ii
=
E u
2E
r 1
1 12
=
E v
2E
r 2 2
=
G u
2E
r 2
-L 11
=
E v
2G
r 2
1 12
=
G u
2G
r 2
1 22
=
G v
2G
T\ 1 E + T\ 1 F=\E U
T 12 E + T 12 F — ^E v
^22^ + 1^22^ ~ -Pw — 2^* u
^nF + T 1X G = F u — -E v
r"l2-^ + ^12^? = ^G u
1^22^ + T22G = oG v
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
Fii + T? 2 = (In y/EG - F=> )„
ria + Im = (In yjEG - F* ).
(Gray 1993).
For a surface given in Monge'S Form 2 = F(x,y),
r k - = ZijZk C2S^
see also Christoffel Symbol of the First Kind,
Connection Coefficient, Gauss Equations
246
Chromatic Number
ci
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 160-167, 1985.
Gray, A. "Christoffel Symbols." §20.3 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 397-400, 1993.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 47-48, 1953.
Chromatic Number
The fewest number of colors j(G) necessary to color a
Graph or surface. The chromatic number of a surface
of GENUS g is given by the HEAWOOD CONJECTURE,
l(9)= §(7+7485 + 1)
where [x\ is the Floor Function. j(g) is sometimes
also denoted x(p)- For g = 0, 1, ... , the first few values
of x(9) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15,
16, ... (Sloane's A000934).
The fewest number of colors necessary to color each
Edge of a Graph so that no two Edges incident on the
same Vertex have the same color is called the "Edge
chromatic number."
see also Brelaz's Heuristic Algorithm, Chro-
matic Polynomial, Edge-Coloring, Euler Char-
acteristic, Heawood Conjecture, Map Color-
ing, Torus Coloring
References
Chartrand, G. "A Scheduling Problem: An Introduction to
Chromatic Numbers." §9.2 in Introductory Graph Theory.
New York: Dover, pp. 202-209, 1985.
Eppstein, D. "The Chromatic Number of the Plane."
http:// www . ics . uci . edu / - eppstein / junkyard /
plane-color/.
Sloane, N. J. A. Sequence A000934/M3292 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Chromatic Polynomial
A Polynomial P(z) of a graph g which counts the
number of ways to color g with exactly z colors. Tutte
(1970) showed that the chromatic POLYNOMIALS of pla-
nar triangular graphs possess a ROOT close to <j> 2 =
2.618033 . . ., where <j> is the GOLDEN Mean. More pre-
cisely, if n is the number of VERTICES of G, then
(Le Lionnais 1983).
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 46, 1983.
Tutte, W. T. "On Chromatic Polynomials and the Golden
Ratio." J. Corabin. Th. 9, 289-296, 1970.
Chu Space
A Chu space is a binary relation from a Set A to an
antiset X which is defined as a Set which transforms
via converse functions.
References
Stanford Concurrency Group. "Guide to Papers on Chu
Spaces." http : //boole . Stanford . edu/ chuguide .html.
Chu-Vandermonde Identity
(x + a) n = Y^ Uj(a)fcO*On-fc
where (™) is a Binomial Coefficient and (a) n =
a(a - 1) • • • (a - n + 1) is the Pochhammer Symbol. A
special case gives the identity
max(fe,n)
£
( =
m
k-l
i)-\ k )■
see also BINOMIAL THEOREM, UMBRAL CALCULUS
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 130 and 181-182, 1996.
Church's Theorem
No decision procedure exists for Arithmetic.
Church's Thesis
see Church-Turing Thesis
Church- Turing Thesis
The Turing Machine concept defines what is meant
mathematically by an algorithmic procedure. Stated
another way, a function / is effectively COMPUTABLE
Iff it can be computed by a Turing Machine.
see also ALGORITHM, COMPUTABLE FUNCTION, TUR-
ING Machine
References
Penrose, R. The Emperor's New Mind: Concerning Comput-
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, pp. 47-49, 1989.
Chvatal's Art Gallery Theorem
see Art Gallery Theorem
Chvatal's Theorem
Let the Graph G have Vertices with Valences di <
. . . < d m . If for every i < n/2 we have either efc > i + 1
or d n -i > n - 2, then the Graph is Hamiltonian.
Chu Identity
see Chu-Vandermonde Identity
ci
see Cosine Integral
a
Circle 247
Ci
see Cosine Integral
Cigarettes
It is possible to place 7 cigarettes in such a way that
each touches the other if l/d > 7\/3/2 (Gardner 1959,
p. 115).
References
Gardner, M. The Scientific American Book of Mathematical
Puzzles & Diversions. New York: Simon and Schuster,
1959.
Cin
see Cosine Integral
Circle
A circle is the set of points equidistant from a given
point O. The distance r from the Center is called the
Radius, and the point O is called the Center. Twice
the Radius is known as the Diameter d = 2r. The
Perimeter C of a circle is called the Circumference,
and is given by
C = ird = 2tt7\ (1)
The circle is a Conic SECTION obtained by the intersec-
tion of a Cone with a Plane Perpendicular to the
Cone's symmetry axis. A circle is the degenerate case
of an Ellipse with equal semimajor and semiminor axes
(i.e., with ECCENTRICITY 0). The interior of a circle is
called a Disk. The generalization of a circle to 3-D is
called a SPHERE, and to n-D for n > 4 a HYPERSPHERE.
The region of intersection of two circles is called a LENS.
The region of intersection of three symmetrically placed
circles (as in a VENN DIAGRAM), in the special case of
the center of each being located at the intersection of
the other two, is called a Reuleaux Triangle.
The parametric equations for a circle of RADIUS a are
x — a cos t
y = a sin t.
For a body moving uniformly around the circle,
X
t
y
-asint
a cost,
and
x = —a cost
y" = —asint.
(2)
(3)
(4)
(5)
(6)
(7)
When normalized, the former gives the equation for the
unit Tangent Vector of the circle, (-sint,cost). The
circle can also be parameterized by the rational func-
tions
x =
y-
2t
1 + t 2 '
(8)
(9)
but an Elliptic Curve cannot. The following plots
show a sequence of NORMAL and TANGENT VECTORS
for the circle.
The Arc Length s, Curvature k, and Tangential
ANGLE <j> of the circle are
s(t) = ds= \/x f2 + y' 2 dt = at (10)
(j>(t) = I K(t)dt= -. (12)
The Cesaro Equation is
K=~. (13)
a
In POLAR COORDINATES, the equation of the circle has
a particularly simple form.
r = a (14)
is a circle of RADIUS a centered at Origin,
r = 2acos9 (15)
is circle of RADIUS a centered at (a, 0), and
r = 2asm6 (16)
248
Circle
Circle
is a circle of RADIUS a centered on (0, a). In CARTE-
SIAN Coordinates, the equation of a circle of Radius
a centered on (xo,2/o) is
(x - x ) 2 + (y-yo) 2
(17)
In Pedal Coordinates with the Pedal Point at the
center, the equation is
pa = r 2 . (18)
The circle having P1P2 as a diameter is given by
(x - xi)(x - x 2 ) + (2/ - yi){y - 2/2) = 0. (19)
The equation of a circle passing through the three points
(xi,yi) for i = 1, 2, 3 (the Circumcircle of the Tri-
angle determined by the points) is
(20)
The Center and Radius of this circle can be identified
by assigning coefficients of a Quadratic Curve
2 , 2
x +y
X
y
1
2 1 2
xi +2/1
Xi
2/1
1
2 1 2
x 2 +t/2
X 2
2/2
1
2 , 2
XZ +J/3
xz
2/3
1
ax 2 + cy 2 + dx + ey + / = 0,
(21)
where a — c and 6 = (since there is no xy cross term) .
Completing the Square gives
The Center can then be identified as
Xq
2/o
2a
e
2a
and the Radius as
where
d 2 + e 2 /
a
4a 2
(23)
(24)
(25)
e =
xi 2/1 I
#2 2/2 1
(26)
xz 2/3 1
#i 2 +2/i 2 2/i 1
Z2 2 +2/2 2 2/2 1
(27)
£3 2 +2/3 2 2/3 1
zi 2 +2/i 2 X! 1
Z2 2 +2/2 2 £ 2 1
(28)
Xz 2 + 2/3 2 #3 1
#i 2 +2/1 2 asi 2/i
Z2 2 +2/2 2 Z 2 2/2
(29)
#3 2 +2/3^
! xz
2/3
Four or more points which lie on a circle are said to be
Concyclic. Three points are trivially concyclic since
three noncollinear points determine a circle.
The ClRCUMFERENCE-to-DlAMETER ratio C/d for a cir-
cle is constant as the size of the circle is changed (as
it must be since scaling a plane figure by a factor s in-
creases its Perimeter by s), and d also scales by s. This
ratio is denoted -k (Pi), and has been proved Transcen-
dental. With d the Diameter and r the Radius,
C == 7rd = 27r?\
(30)
Knowing C/d, we can then compute the Area of the
circle either geometrically or using CALCULUS. From
Calculus,
A =
p1t\ nr
Jo Jo
rdr = (27r)(^r ) = irr
(31)
Now for a few geometrical derivations. Using concentric
strips, we have
As the number of strips increases to infinity, we are left
with a Triangle on the right, so
A = \{2nr)r = nr .
(32)
This derivation was first recorded by Archimedes in
Measurement of a Circle (ca. 225 BC). If we cut the
circle instead into wedges,
^ *+ nr ►
As the number of wedges increases to infinity, we are
left with a RECTANGLE, so
(-Kr)r = nr .
(33)
see also Arc, Blaschke's Theorem, Brahmagupta's
Formula, Brocard Circle, Casey's Theorem,
Chord, Circumcircle, Circumference, Clif-
ford's Circle Theorem, Closed Disk, Concentric
Circles, Cosine Circle, Cotes Circle Property,
Diameter, Disk, Droz-Farny Circles, Euler Tri-
angle Formula, Excircle, Feuerbach's Theorem,
Circles-and-Squares Fractal
Circle-Circle Intersection
249
Five Disks Problem, Flower of Life, Ford Cir-
cle, Fuhrmann Circle, Gersgorin Circle Theo-
rem, Hopf Circle, Incircle, Inversive Distance,
Johnson Circle, Kinney's Set, Lemoine Circle,
Lens, Magic Circles, Malfatti Circles, McCay
Circle, Midcircle, Monge's Theorem, Moser's
Circle Problem, Neuberg Circles, Nine-Point
Circle, Open Disk, P-Circle, Parry Circle, Pi,
Polar Circle, Power (Circle), Prime Circle,
Ptolemy's Theorem, Purser's Theorem, Radi-
cal Axis, Radius, Reuleaux Triangle, Seed of
Life, Seifert Circle, Semicircle, Soddy Circles,
Sphere, Taylor Circle, Triangle Inscribing in
a Circle, Triplicate-Ratio Circle, Tucker Cir-
cles, Unit Circle, Venn Diagram, Villarceau
Circles, Yin- Yang
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 125 and 197, 1987.
Casey, J. "The Circle." Ch. 3 in A Treatise on the Analyt-
ical Geometry of the Point, Line, Circle, and Conic Sec-
tions, Containing an Account of Its Most Recent Exten-
sions, with Numerous Examples, 2nd ed., rev. enl. Dublin:
Hodges, Figgis, & Co., pp. 96-150, 1893.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 74-75, 1996.
Dunham, W. "Archimedes' Determination of Circular Area."
Ch. 4 in Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 84-112, 1990.
Eppstein, D. "Circles and Spheres." http://www. ics . uci .
edu/*eppstein/ junkyard/sphere. html.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 65-66, 1972.
MacTutor History of Mathematics Archive. "Circle." http:
//www -groups . dcs . st -and .ac.uk/ -history /Curves/
Circle.html.
Pappas, T. "Infinity & the Circle" and "Japanese Calculus."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, pp. 68 and 139, 1989.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.
Yates, R. C "The Circle." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 21-25,
1952.
Circles-and-Squares Fractal
m
A FRACTAL produced by iteration of the equation
Zn+i = z n (mod m)
which results in a M0IRE-Iike pattern.
see also FRACTAL, M0IRE PATTERN
Circle Caustic
Consider a point light source located at a point (//,0).
The CATACAUSTIC of a unit CIRCLE for the light at fi =
oo is the Nephroid
x = ~ [3 cost - cos(3£)]
y = \ [3 sin t — sin(3i)].
a)
(2)
The CATACAUSTIC for the light at a finite distance fx > 1
is the curve
V :
fi(l — 3/j, cos t + 2fi cos 3 t)
-(l-\-2fi 2 )-r3ficost
2fi 2 sin 3 t
1 + 2// 2 — 3/xcost '
(3)
(4)
and for the light on the CIRCUMFERENCE of the CIRCLE
{i — 1 is the CARDIOID
x = | cos t(l + cos t) - |
y — | sini(l + cost).
(5)
(6)
If the point is inside the circle, the catacaustic is a dis-
continuous two-part curve. These four cases are illus-
trated below.
The CATACAUSTIC for PARALLEL rays crossing a CIRCLE
is a Cardioid.
see also CATACAUSTIC, CAUSTIC
Circle-Circle Intersection
Let two Circles of Radii R and r and centered at (0, 0)
and (d, 0) intersect in a LENS-shaped region. The equa-
tions of the two circles are
2,2 D 2
x +y — R
(x - df +y 2 = r 2
(1)
(2)
250 Circle-Circle Intersection
Combining (1) and (2) gives
(x-d) 2 + (R 2 -x 2 ) = r 2 .
Multiplying through and rearranging gives
x 2 - 2dx + d 2 - x 2 = r 2 - R 2 .
Solving for x results in
d 2 - r 2 + R 2
2d
(3)
(4)
(5)
The line connecting the cusps of the LENS therefore has
half-length given by plugging x back in to obtain
2 D 2 2 D 2 / d - r + R
y = R — x = R
2d
Ad 2 R 2 -{d 2 -r 2 +R 2 ) 2
Ad?
(6)
giving a length of
a= ^V 4 ^ 1 * 2 ~ ( d2 ~ r2 + R2 ) 2
= h(-d + r-R)(-d-r + R)
a
x [(-d + r + R){d + r + R)] 1/2 . (7)
This same formulation applies directly to the SPHERE-
Sphere Intersection problem.
To find the AREA of the asymmetric "Lens" in which
the Circles intersect, simply use the formula for the
circular SEGMENT of radius i^'and triangular height d'
A{R!,d') = i^cos" 1 f^\ -d'^R' 2 -d<* (8)
twice, one for each half of the "Lens." Noting that the
heights of the two segment triangles are
di = x ■
d 2 -r 2 + R 2
dz = d — x ■■
2d
d 2 +r 2 - R 2
2d
(9)
(10)
The result is
A = A(Ri,d 1 )+A(R 2 ,d 2 )
_i (d 2 + r 2 -R 2
2
r cos
2dr
+ R* cos
/ d 2 +E 2 -r 2 \
^ 2dR )
- \^{d - r - R)(d + r - R){d - r + R)(d + r + R).
(11)
Circle Cutting
The limiting cases of this expression can be checked to
give when d — R + r and
A = 2R 2 cos" 1 (^) - \d\/AR? - d? (12)
= 2A{\d,R)
(13)
when r = i2, as expected. In order for half the area of
two Unit Disks (R = 1) to overlap, set A = irR 2 /2 =
7r/2 in the above equation
|tt = 2cos~ l (±d) - \d^J\ - d? (14)
and solve numerically, yielding d w 0.807946.
see also Lens, Segment, Sphere-Sphere Intersec-
tion
Circle Cutting
2 4 7 11
Determining the maximum number of pieces in which
it is possible to divide a CIRCLE for a given number of
cuts is called the circle cutting, or sometimes PANCAKE
Cutting, problem. The minimum number is always
n + 1, where n is the number of cuts, and it is always
possible to obtain any number of pieces between the
minimum and maximum. The first cut creates 2 regions,
and the nth cut creates n new regions, so
/(l) = 2
(1)
/(2) = 2 + /(l)
(2)
/(n) = n+/(n-l).
(3)
Therefore,
f(n) = n+[(n-l) + f(n-2)}
n
= n + (n-l) + ... + 2 + /(l) = J^ k fW
fc-2
n
= ^fc-l + /(l)-in(n+l)-l + 2
k = l
= §(n 2 +n + 2). (4)
Evaluating for n = 1, 2, . . . gives 2, 4, 7, 11, 16, 22, . . .
(Sloane's A000124).
OO
12 4 8
A related problem, sometimes called Moser's CIRCLE
PROBLEM, is to find the number of pieces into which
a Circle is divided if n points on its Circumference
Circle Evolute
Circle Involute 251
are joined by Chords with no three Concurrent. The
answer is
»<»>=(:)+©+>
= 5j(n 4 - 6n 3 + 23n 2 - 18n + 24),
(5)
(6)
(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy
1996, Noy 1996), where (£) is a Binomial Coeffi-
cient. The first few values are 1, 2, 4, 8, 16, 31, 57,
99, 163, 256, ... (Sloane's A000127). This sequence
and problem are an example of the danger in making
assumptions based on limited trials. While the series
starts off like 2 n ~ 1 , it begins differing from this GEO-
METRIC Series at n = 6.
see also Cake Cutting, Cylinder Cutting, Ham
Sandwich Theorem, Pancake Theorem, Pizza
Theorem, Square Cutting, Torus Cutting
References
Conway, J. H. and Guy, R. K. "How Many Regions." In The
Book of Numbers. New York: Springer- Verlag, pp. 76-79,
1996.
Guy, R. K. "The Strong Law of Small Numbers." Amer.
Math. Monthly 95, 697-712, 1988.
Noy, M. "A Short Solution of a Problem in Combinatorial
Geometry." Math. Mag. 69, 52-53, 1996.
Sloane, N. J. A. Sequences A000124/M1041 and A000127/
M1119 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Yaglom, A. M. and Yaglom, I. M. Problem 47. Challenging
Mathematical Problems with Elementary Solutions, Vol. 1.
New York: Dover, 1987.
Circle Evolute
x = cos t x = — sin t x ~
— cost
(i)
y = sin t y = cos t y = -
- sin t,
(2)
so the Radius of Curvature is
^_(x' 2 +y' 2 ) 3/2
y" x' — x"y'
(sin 2 t + cos 2 t) 3/2
— i
i"*t
(— sint)(— sint) — (— cost) cost
and the TANGENT VECTOR is
— sint
cost
Therefore,
cos r —T • x = — sin t
sin r ~T • y = cos t,
(4)
(5)
(6)
and the EVOLUTE degenerates to a POINT at the ORI-
GIN.
see also CIRCLE INVOLUTE
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 77, 1993.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 55-
59, 1991.
Circle Inscribing
If r is the Radius of a Circle inscribed in a Right
Triangle with sides a and b and Hypotenuse c, then
r = i( a + 6-c).
see Inscribed, Polygon
Circle Involute
First studied by Huygens when he was considering clocks
without pendula for use on ships at sea. He used the cir-
cle involute in his first pendulum clock in an attempt to
force the pendulum to swing in the path of a CYCLOID.
For a Circle with a = 1, the parametric equations of
the circle and their derivatives are given by
x = cost x =— sint x =— cost (1)
y — sin t y = cos t
The Tangent Vector is
- sin t.
T =
— sint
cost
and the Arc LENGTH along the circle is
so the involute is given by
(2)
(3)
(4)
n = r - sT =
cost
sint
j
-t
— sint
cost
=
cos t + t sin t
sin t — t cos t
(5)
£(t) = x — R sin r — cos t — 1 • cos t = (7)
>q(t) = y + Rcosr = sint + 1 * (-sint) = 0, (8)
x = a(cost -f tsint)
y = a(sint — tcost).
(6)
(7)
252
Circle Involute Pedal Curve
Circle Lattice Points
The Arc Length, Curvature, and Tangential An-
gle are
J ds= / ^x' 2 + y' 2 dt = \
1
K =
= i.
The Cesaro Equation is
Vas'
at 2 (8)
(9)
(10)
(11)
see also Circle, Circle Evolute, Ellipse Involute,
Involute
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 83, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 190-191, 1972.
MacTutor History of Mathematics Archive. "Involute of a
Circle." http://www-groups.dcs.st-and.ac.uk/-history
/Curves/Involute. html.
Circle Involute Pedal Curve
The Pedal Curve of Circle Involute
/ = cos t + t sin t
g = sin t — t cos t
with the center as the PEDAL Point is the ARCHIME-
DES' Spiral
x ~ t sin t
y = — tcost.
Circle Lattice Points
For every POSITIVE INTEGER n, there exists a CIRCLE
which contains exactly n lattice points in its interior.
H. Steinhaus proved that for every POSITIVE INTEGER
n, there exists a Circle of Area n which contains ex-
actly n lattice points in its interior.
Schinzel's Theorem shows that for every Positive
Integer n, there exists a Circle in the Plane hav-
ing exactly n LATTICE POINTS on its CIRCUMFERENCE.
The theorem also explicitly identifies such "Schinzel
Circles" as
{x
(x
l) 2 + y 2
1 cfc-]
4 5
1 r2fc
9 5
for n = 2k
for n = 2fc + 1.
(1)
Note, however, that these solutions do not necessarily
have the smallest possible RADIUS, For example, while
the Schinzel Circle centered at (1/3, 0) and with
RADIUS 625/3 has nine lattice points on its CIRCUM-
FERENCE, so does the CIRCLE centered at (1/3, 0) with
Radius 65/3.
Let r be the smallest INTEGER RADIUS of a CIRCLE cen-
tered at the Origin (0, 0) with L(r) Lattice Points.
In order to find the number of lattice points of the Cir-
cle, it is only necessary to find the number in the first
octant, i.e., those with < y < [r/v^J , where [z\ is the
Floor Function. Calling this N(r% then for r > 1,
L(r) = 8N(r) - 4, so L(r) = 4 (mod 8). The multipli-
cation by eight counts all octants, and the subtraction
by four eliminates points on the axes which the multi-
plication counts twice. (Since ^/2 is IRRATIONAL, the
MIDPOINT of a are is never a LATTICE POINT.)
Gauss's Circle Problem asks for the number of lat-
tice points within a CIRCLE of RADIUS r
N(r) = 1 + 4 [rj + 4 ^ ^r 2 - i 2 .
Gauss showed that
where
N(r) = nr 2 + E(r),
\E(r)\ < 2V2nr.
(2)
(3)
(4)
i
The number of lattice points on the CIRCUMFERENCE of
circles centered at (0, 0) with radii 0, 1, 2, . . . are 1, 4, 4,
4, 4, 12, 4, 4, 4, 4, 12, 4, 4, . . . (Sloane's A046109). The
following table gives the smallest RADIUS r < 111,000
for a circle centered at (0, 0) having a given number of
LATTICE POINTS L(r). Note that the high water mark
radii are always multiples of five.
Circle Lattice Points
Circle Map 253
L(r)
r
1
4
1
12
5
20
25
28
125
36
65
44
3,125
52
15,625
60
325
68
< 390,625
76
< 1,953,125
84
1,625
92
< 48,828,125
100
4,225
108
1,105
132
40,625
140
21,125
180
5,525
252
27,625
300
71,825
324
32,045
*
If the CIRCLE is instead centered at (1/2, 0), then the
Circles of Radii 1/2, 3/2, 5/2, . . . have 2, 2, 6, 2, 2,
2, 6, 6, 6, 2, 2, 2, 10, 2, . . . (Sloane's A046110) on their
Circumferences. If the Circle is instead centered
at (1/3, 0), then the number of lattice points on the
Circumference of the Circles of Radius 1/3, 2/3,
4/3, 5/3, 7/3, 8/3, ... are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1,
3, 1, 3, 1, 1, 5, 3, . . . (Sloane's A046111).
Let
1. a n be the RADIUS of the CIRCLE centered at (0, 0)
having 8n + 4 lattice points on its CIRCUMFERENCE,
2. b n /2 be the RADIUS of the Circle centered at (1/2,
0) having 4n + 2 lattice points on its CIRCUMFER-
ENCE,
3. c n /3 be the Radius of Circle centered at (1/3, 0)
having 2n + 1 lattice points on its CIRCUMFERENCE.
Then the sequences {a n }, {&n}, and {c n } are equal, with
the exception that b n — if 2|n and c n = if 3|n. How-
ever, the sequences of smallest radii having the above
numbers of lattice points are equal in the three cases
and given by 1, 5, 25, 125, 65, 3125, 15625, 325, ...
(Sloane's A046112).
Kulikowski's Theorem states that for every Posi-
tive Integer n, there exists a 3-D Sphere which has
exactly n Lattice Points on its surface. The Sphere
is given by the equation
(x-a) 2 + {y-b) 2 + (z-^) 2
: C + 2,
where a and b are the coordinates of the center of the
so-called Schinzel Circle and c is its Radius (Hons-
berger 1973).
see also CIRCLE, CIRCUMFERENCE, GAUSS'S CIRCLE
Problem, Kulikowski's Theorem, Lattice Point,
Schinzel Circle, Sciiinzel's Theorem
References
Honsberger, R. "Circles, Squares, and Lattice Points."
Ch. 11 in Mathematical Gems I. Washington, DC: Math.
Assoc. Amer., pp. 117-127, 1973.
Kulikowski, T. "Sur l'existence d'une sphere passant par un
nombre donne aux coordonnees entieres." L'Enseignement
Math. Ser. 2 5, 89-90, 1959.
Schinzel, A. "Sur l'existence d'un cercle passant par un
nombre donne de points aux coordonnees entieres."
L'Enseignement Math. Ser. 2 4, 71-72, 1958.
Sierpiiiski, W. "Sur quelques problemes concernant les points
aux coordonnees entieres." L'Enseignement Math. Ser. 2
4, 25-31, 1958.
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant
les ensembles de points sur le plan." Fund. Math. 46,
191-194, 1959.
Sierpinski, W. A Selection of Problems in the Theory of
Numbers. New York: Pergamon Press, 1964.
# Weisstein, E. W. "Circle Lattice Points." http:// www .
astro . Virginia . edu/ -eww6n/ math /notebooks /Circle
LatticePoints .m.
Circle Lattice Theorem
see Gauss's Circle Problem
Circle Map
A 1-D Map which maps a CIRCLE onto itself
0n+i = n + Q-^- sin(27r0„), (1)
where # n +i is computed mod 1. Note that the circle map
has two parameters: Q and K. Q can be interpreted as
an externally applied frequency, and K as a strength of
nonlinearity. The 1-D JACOBIAN is
d9,
n+l
d0 n
l-ii:cos(27r(9n),
(2)
so the circle map is not Area-Preserving. It is related
to the Standard Map
/n+l = Jn + — sin(27r0 n )
@n + l — n + /n + l,
(3)
(4)
254
Circle Method
for / and computed mod 1. Writing 8 n +i as
n+ i = n + /„ + ^- sin(27rl9 n ) (5)
gives the circle map with I n = Q, and K = —K. The
unperturbed circle map has the form
0n + l=0n+fi. (6)
If fi is RATIONAL, then it is known as the map WINDING
Number, defined by
(7)
and implies a periodic trajectory, since n will return
to the same point (at most) every q ORBITS. If Q is
Irrational, then the motion is quasiperiodic. If K is
NONZERO, then the motion may be periodic in some
finite region surrounding each RATIONAL Q. This exe-
cution of periodic motion in response to an IRRATIONAL
forcing is known as Mode Locking.
If a plot is made of K vs. Q with the regions of pe-
riodic MODE-LOCKED parameter space plotted around
Rational Q values (Winding Numbers), then the re-
gions are seen to widen upward from at K = to some
finite width at K = 1. The region surrounding each Ra-
tional Number is known as an Arnold Tongue. At
K = 0, the Arnold Tongues are an isolated set of
Measure zero. At K = 1, they form a Cantor Set
of Dimension d « 0.08700. For K > 1, the tongues
overlap, and the circle map becomes noninvertible. The
circle map has a Feigenbaum Constant
6= lim
n—¥oo U n + 1
On — On-1
n
2.833.
(8)
see also Arnold Tongue, Devil's Staircase, Mode
Locking, Winding Number (Map)
Circle Method
see Partition Function P
Circle Negative Pedal Curve
The Negative Pedal Curve of a circle is an Ellipse
if the Pedal Point is inside the Circle, and a Hy-
perbola if the Pedal Point is outside the Circle.
Circle Notation
A Notation for Large Numbers due to Steinhaus
(1983) in which is defined in terms of STEINHAUS-
Moser Notation as n in n SQUARES. The particular
number known as the MEGA is then defined as follows.
©-E
A-\A
4 4
256
see also Mega, Megistron, Steinhaus-Moser No-
tation
References
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 28-29, 1983.
Circle Packing
Circle Order
A Poset P is a circle order if it is Isomorphic to a Set
of Disks ordered by containment.
see also ISOMORPHIC POSETS, PARTIALLY ORDERED
Set
Circle Orthotomic
The Orthotomic of the Circle represented by
X = cos t
y = sin t
with a source at (x, y) is
(1)
(2)
x = x cos(2£) - y sin(2t) + 2 sin t (3)
y = ~x sin(2i) - y cos(2t) + 2 cos t. (4)
Circle Packing
The densest packing of spheres in the PLANE is the
hexagonal lattice of the bee's honeycomb (illustrated
above), which has a Packing Density of
2\/3
= 0.9068996821..
Gauss proved that the hexagonal lattice is the densest
plane lattice packing, and in 1940, L. Fejes Toth proved
that the hexagonal lattice is indeed the densest of all
possible plane packings.
Solutions for the smallest diameter CIRCLES into which
n Unit Circles can be packed have been proved op-
timal for n = 1 through 10 (Kravitz 1967). The best
known results are summarized in the following table.
Circle Packing
Circle-Point Midpoint Theorem 255
n
d exact
d approx.
1
2
3
4
5
6
7
8
9
10
11
12
1
2
l+fx/3
1 + V2
1.00000
2.00000
2.15470...
2.41421...
2.70130...
3.00000
3.00000
3.30476...
3.61312...
3.82...
4.02...
1 + \/2(l + l/\/5)
3
3
1 + csc(tt/7)
1 + ^/2(2 + ^/2)
For Circle packing inside a Square, proofs are known
only for n = 1 to 9.
n
d exact
d approx.
1
1
1.000
2
0.58...
3
0.500...
4
i
2
0.500
5
0.41...
6
0.37. . .
7
0.348...
8
0.341...
9
1
3
0.333. . .
10
0.148204...
The smallest Square into which two Unit Circles,
one of which is split into two pieces by a chord, can be
packed is not known (Goldberg 1968, Ogilvy 1990).
see also Hypersphere Packing, Malfatti's Right
Triangle Problem, Mergelyan-Wesler Theorem,
Sphere Packing
References
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices,
and Groups, 2nd ed. New York: Springer- Verlag, 1992.
Eppstein, D. "Covering and Packing." http://www.ics.uci
. edu/-eppstein/ junkyard/cover, html.
Folkman, J. H. and Graham, R. "A Packing Inequality for
Compact Convex Subsets of the Plane." Canad. Math,
Bull. 12, 745-752, 1969.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." ScL Amer.
240, 18-28, Jan. 1979.
Gardner, M. "Tangent Circles." Ch. 10 in Fractal Music,
HyperCards, and More Mathematical Recreations from Sci-
entific American Magazine. New York: W. H. Freeman,
1992.
Goldberg, M. "Problem E1924." Amer. Math. Monthly 75,
195, 1968.
Goldberg, M. "The Packing of Equal Circles in a Square."
Math. Mag. 43, 24-30, 1970.
Goldberg, M. "Packing of 14, 16, 17, and 20 Circles in a
Circle." Math. Mag. 44, 134-139, 1971.
Graham, R. L. and Luboachevsky, B, D, "Repeated Patterns
of Dense Packings of Equal Disks in a Square." Elec-
tronic J. Combinatorics 3, R16, 1-17, 1996. http://www.
combinatorics. org/Volume^3/volume3.html#R16.
Kravitz, S. "Packing Cylinders into Cylindrical Containers."
Math. Mag. 40, 65-70, 1967.
McCaughan, F, "Circle Packings." http://www.pmms.cam.
ac, uk/ -gj ml i/cpacking/ info. html.
Molland, M. and Payan, Charles. "A Better Packing of Ten
Equal Circles in a Square." Discrete Math. 84, 303-305,
1990.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
p. 145, 1990.
Reis, G. E. "Dense Packing of Equal Circle within a Circle."
Math. Mag. 48, 33-37, 1975.
Schaer, J. "The Densest Packing of Nine Circles in a Square."
Can. Math. Bui. 8, 273-277, 1965.
Schaer, J. "The Densest Packing of Ten Equal Circles in a
Square." Math. Mag. 44, 139-140, 1971.
Valette, G. "A Better Packing of Ten Equal Circles in a
Square." Discrete Math. 76, 57-59, 1989.
Circle Pedal Curve
/ s*
\ 1
/ /
/ /
\^\
/ /
\ \
/
1/
1
y
1 /
/ /
/ /
/ /
^ = ^^;
— " ^s^
The Pedal Curve of a Circle is a Cardioid if the
Pedal Point is taken on the Circumference,
and otherwise a LlMAQON.
Circle-Point Midpoint Theorem
Taking the locus of MIDPOINTS from a fixed point to a
circle of radius r results in a circle of radius r/2. This
follows trivially from
r(0)
—x
+K
rcosS
rsinO
-
—x
~r cos9 — \x
-
\ sin
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 17, 1929.
256 Circle Radial Curve
Circle Radial Curve
The Radial Curve of a unit Circle from a Radial
Point (x, 0) is another Circle with parametric equa-
tions
x(t) = x — cost
y(i) = — sint.
Circle Squaring
Construct a SQUARE equal in Area to a CIRCLE using
only a STRAIGHTEDGE and COMPASS. This was one of
the three Geometric Problems of Antiquity, and
was perhaps first attempted by Anaxagoras. It was fi-
nally proved to be an impossible problem when Pi was
proven to be TRANSCENDENTAL by Lindemann in 1882.
However, approximations to circle squaring are given
by constructing lengths close to tt = 3.1415926....
Ramanujan (1913-14) and Olds (1963) give geomet-
ric constructions for 355/113 = 3.1415929.... Gard-
ner (1966, pp. 92-93) gives a geometric construc-
tion for 3+ 16/113 = 3.1415929.... Dixon (1991)
gives constructions for 6/5(1 + <fi) = 3.141640... and
y / 4+[3-tan(30°)] = 3.141533 . . ..
While the circle cannot be squared in EUCLIDEAN
Space, it can in Gauss-Bolyai-Lobachevsky Space
(Gray 1989).
see also GEOMETRIC CONSTRUCTION, QUADRATURE,
Squaring
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 190-191, 1996.
Dixon, R. M athographics. New York: Dover, pp. 44-49 and
52-53, 1991.
Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 20-26, 1990.
Gardner, M. "The Transcendental Number Pi." Ch. 8 in
Martin Gardner's New Mathematical Diversions from Sci-
entific American. New York: Simon and Schuster, 1966.
Gray, J. Ideas of Space. Oxford, England: Oxford University
Press, 1989.
Meyers, L. F. "Update on William Wernick's 'Triangle Con-
structions with Three Located Points,"' Math. Mag. 69,
46-49, 1996.
Olds, C. D. Continued Fractions. New York: Random House,
pp. 59-60, 1963.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914,
Circle Tangents
Circle Strophoid
The Strophoid of a Circle with pole at the center
and fixed point on the CIRCUMFERENCE is a FREETH'S
Nephroid.
Circle Tangents
There are four CIRCLES that touch all the sides of a
given TRIANGLE. These are all touched by the CIRCLE
through the intersection of the ANGLE BISECTORS of
the Triangle, known as the Nine-Point Circle.
Given the above figure, GE — FH, since
AB = AG 4- GB = GE + GF = GE + {GE + EF)
= 2G + EF
CD = CH + HD = EH + FH = FH + (FH + EF)
= EF + 2FH.
Because AB = CD, it follows that GE = FH.
The line tangent to a CIRCLE of RADIUS a centered at
(a,y)
x — x + a cos t
V — V + o, sin t
through (0,0) can be found by solving the equation
x + a cos t
y 4- a sin t
a cost
a sint
giving
t — db cos
—ax db y\/x 2 -\- y 2 — a 2
x 2 + y 2
Circuit
Two of these four solutions give tangent lines, as illus-
trated above.
see also KISSING CIRCLES PROBLEM, MlQUEL POINT,
Monge's Problem, Pedal Circle, Tangent Line,
Triangle
References
Dixon, R. Mathographics. New York: Dover, p. 21, 1991.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 4-5, 1991.
Circuit
see Cycle (Graph)
Circuit Rank
Also known as the Cyclomatic Number. The circuit
rank is the smallest number of EDGES 7 which must be
removed from a GRAPH of N EDGES and n nodes such
that no Circuit remains.
7 = N - n + 1.
Circulant Determinant
Gradshteyn and Ryzhik (1970) define circulants by
Xn
X n -1
Xn-2
Circular Functions
257
Xl
X 2
X3
Xn
Xl
x 2
m-1
X n
Xl
X2 Xz X4
Xl
= Y\( Xl + X 2ti>j +X3Wj 2 + .- ■ +Xn(Jj n ), (1)
i=i
where u>j is the nth ROOT OF Unity. The second-order
circulant determinant is
Xl X2
X2 Xi
and the third order is
Xl X2 Xz
Xz Xi X2
X2 Xz Xi
= (xi -\-x 2 )(xi - x 2 ),
(2)
= (xi + x 2 + X3)(asi + ujx 2 + oj xz){xi + OJ X2 + UJXz),
(3)
where u) and u 2 are the COMPLEX CUBE ROOTS of
Unity.
The Eigenvalues A of the corresponding n x n circulant
matrix are
\j = xi -f- X20JJ 4- Xz^j + . . . + x n ujj n
see also CIRCULANT MATRIX
(4)
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1111-1112, 1979.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. 114, 1991.
Circulant Graph
A Graph of n Vertices in which the zth Vertex is
adjacent to the (i + j)th and (i - j)th Vertices for
each j in a list I.
Circulant Matrix
An n x n MATRIX C defined as follows,
1 (?) G) - UO
L (?) (?) (?)■•• i
c = n[(i+u,,r-i],
3 =
where u;o = 1, cji, ..., u) n -i are the nth ROOTS
OF UNITY. Circulant matrices are examples of LATIN
Squares.
see also CIRCULANT DETERMINANT
References
Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea,
1994.
Stroeker, R. J. "Brocard Points, Circulant Matrices, and
Descartes' Folium." Math. Mag. 61, 172-187, 1988.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. 114, 1991.
Circular Cylindrical Coordinates
see Cylindrical Coordinates
Circular Functions
The functions describing the horizontal and vertical po-
sitions of a point on a Circle as a function of Angle
(COSINE and Sine) and those functions derived from
them:
cot a; =
tana; =
tana;
1
sinx
1
cos a;
sinx
(i)
(2)
(3)
(4)
The study of circular functions is called TRIGONOME-
TRY.
see also COSECANT, COSINE, COTANGENT, ELLIPTIC
Function, Generalized Hyperbolic Functions,
Hyperbolic Functions, Secant, Sine, Tangent,
Trigonometry
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
258
Circular Permutation
Circumcircle
Circular Permutation
The number of ways to arrange n distinct objects along
a Circle is
P n = (n- 1)1
The number is (n - 1)! instead of the usual FACTORIAL
n! since all Cyclic Permutations of objects are equiv-
alent because the CIRCLE can be rotated.
see also Permutation, Prime Circle
Circumcenter
The center O of a TRIANGLE'S CIRCUMCIRCLE. It can
be found as the intersection of the PERPENDICULAR BI-
SECTORS. If the Triangle is Acute, the circumcenter
is in the interior of the TRIANGLE. In a RIGHT TRI-
ANGLE, the circumcenter is the Midpoint of the Hy-
potenuse.
OOi + OQ 2 + OOz =R + r,
(1)
where Oi are the MIDPOINTS of sides Ai, R is the
Circumradius, and r is the INRADIUS (Johnson 1929,
p. 190), The Trilinear Coordinates of the circum-
center are
cos A : cos B : cos C, (2)
and the exact trilinears are therefore
R cos A : R cos B : R cos C.
The Areal Coordinates are
(^acotA, \bcotB, |ccotC).
(3)
(4)
The distance b etween the Incenter and circumcenter
is ^R(R — 2r). Given an interior point, the distances
to the Vertices are equal Iff this point is the circum-
center. It lies on the BROCARD AXIS.
The circumcenter O and ORTHOCENTER H are ISOGO-
nal Conjugates.
The Orthocenter H of the Pedal Triangle
AO1O2O3 formed by the CIRCUMCENTER O concurs
with the circumcenter O itself, as illustrated above. The
circumcenter also lies on the EULER LINE.
see also Brocard Diameter, Carnot's Theorem,
Centroid (Triangle), Circle, Euler Line, Incen-
ter, Orthocenter
References
Carr, G. S. Formulas and Theorems in Pure Mathematics,
2nd ed. New York: Chelsea, p, 623, 1970.
Dixon, R. Mathographics. New York: Dover, p. 55, 1991.
Eppstein, D. "Circumcenters of Triangles." http://www.ics
.uci.edu/-eppstein/junkyard/circumcenter.htnil.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Circumcenter." http://vvv.evansville,
edu/-ck6/tcenters/class/ccenter.html.
Circumcircle
Circumcircle
Circumradius 259
A Triangle's circumscribed circle. Its center O is
called the Circumcenter, and its Radius R the Cir-
cumradius. The circumcircle can be specified using
Trilinear Coordinates as
Pya. + yab + a/3c = 0.
(i)
The Steiner Point S and Tarry Point T lie on the
circumcircle.
A Geometric Construction for the circumcircle is
given by Pedoe (1995, pp. xii-xiii). The equation for the
circumcircle of the Triangle with Vertices (zu, yi) for
i = 1, 2, 3 is
2 , 2
x + y
X
y
1
2 i 2
xi +2/i
X\
yi
1
2 , 2
X 2 +V2
X2
2/2
1
2 , 2
xz +2/3
xz
2/3
1
= 0.
Expanding the DETERMINANT,
a(x 2 + y 2 ) + 2dx + 2/y + 5 = 0,
where
(2)
(3)
Xi
yi l
a —
X 2 2/2 1
X3 2/3 1
d=-
1
2
xi 2 +2/
# 2 2 +2/
Z3 2 +2/
2
1
2
2
2
3
yi
2/2
2/3
Xi 2 +2/i 2
Xi
J 2
x 2 2 + 2/2 2
x 2
2 i 2
Xz +2/3
xz
2 , 2
Xl +2/1
Xi
9 = ~
2 , 2
#2 +2/2
x 2
2 , 2
Z3 +2/3
xz
COMPLETING THE SQUARE gives
a { x+ lY +a (" + z?-
a
which is a CIRCLE o
: the form
1
1
1
2/1
2/2
2/3
(x - zo) 2 + (y- yo) 2 = r 2 ,
with ClRCUMCENTER
Xq
yo
a
./
a
and Circumradius
P±&_9
a 2 a
(4)
(5)
(6)
(?)
+ 5 = (8)
(9)
(10)
(11)
(12)
see also CIRCLE, ClRCUMCENTER, CIRCUMRADIUS, EX-
CIRCLE, INCIRCLE, PARRY POINT, PURSER'S THEOREM,
Steiner Points, Tarry Point
References
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.
Circumference
The Perimeter of a Circle. For Radius r or Diam-
eter d = 2r,
C = 27vr = ltd,
where tv is Pi.
see also Circle, Diameter, Perimeter, Pi, Radius
Circuminscribed
Given two closed curves, the circuminscribed curve is
simultaneously INSCRIBED in the outer one and CIR-
CUMSCRIBED on the inner one.
see also Poncelet's Closure Theorem
Circumradius
The radius of a TRIANGLE'S CIRCUMCIRCLE or of a
Polyhedron's Circumsphere, denoted R. For a Tri-
angle,
R =
abc
y/(a + b + c)(b + c - a)(c + a - b)(a + b - c)
(1)
where the side lengths of the TRIANGLE are a, 6, and c.
This equation can also be expressed in terms of the
Radii of the three mutually tangent Circles centered
at the Triangle's Vertices. Relabeling the diagram
for the SODDY CIRCLES with VERTICES Oi, O2, and 3
and the radii 7*1, r 2 , and rz, and using
a = T\ + V2
b = V2 + 7"3
c — r\-\-rz
(2)
(3)
(4)
then gives
R = (n +r 2 )(n + r 3 )(r 2 +r 3 )
4^/Vir 2 r3(ri + r 2 + rz)
If O is the ClRCUMCENTER and M is the triangle Cen-
TROID, then
OM 2 =R 2 - §(a 2 + 6 2 + c 2 ).
Rr =
Q1Q2Q3
As
(6)
(?)
260
Circumscribed
Cissoid of Diodes
COS CKi + COS Ct2 + cos 0:3 — 1 +
R
v = 2R cos ai cos 0:2 cos a$
ai 2 + a 2 2 + a 3 2 = 4r# + 8iZ 2
(8)
(9)
(10)
(Johnson 1929, pp. 189-191). Let d be the_distance
between INRADIUS r and circumradius R, d = rR. Then
= 2Rr
1 1
R- d R+d
(11)
(12)
(Mackay 1886-87). These and many other identities are
given in Johnson (1929, pp. 186-190).
For an ARCHIMEDEAN SOLID, expressing the circumra-
dius in terms of the INRADIUS r and MlDRADIUS p gives
tf =±(r + xA 2 +a 2 )
s>
(13)
(14)
for an Archimedean Solid.
see also Carnot's Theorem, Circumcircle, Cir-
CUMSPHERE
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Mackay, J. S. "Historical Notes on a Geometrical Theorem
and its Developments [18th Century]." Proc. Edinburgh
Math. Soc. 5, 62-78, 1886-1887.
Circumscribed
A geometric figure which touches only the VERTICES (or
other extremities) of another figure.
see also ClRCUMCENTER, CIRCUMCIRCLE, ClRCUMIN-
scribed, Circumradius, Inscribed
Circumsphere
A Sphere circumscribed in a given solid. Its radius is
called the CIRCUMRADIUS.
see also Insphere
Cis
Cis x = e 1 '
■ cosx 4- i since.
Cissoid
Given two curves C\ and C2 and a fixed point O, let a
line from O cut C at Q and C at R. Then the LOCUS of
a point P such that OP = QR is the cissoid. The word
cissoid means "ivy shaped."
Curve 1 Curve 2
Pole
Cissoid
line
line
circle
circle
circle
circle
circle
parallel line
circle
tangent line
tangent line
radial line
concentric circle
same circle
any point
center
on C
on C opp.
tangent
on C
center
(0A0)
line
conchoid of
Nicomedes
oblique cissoid
cissoid of Diocles
strophoid
circle
lemniscate
see also Cissoid of Diocles
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 53-56 and 205, 1972.
Lee, X. "Cissoid." http : //www . best . com/~xah/Special
PlaneCurves^dir/Cissoid_dir/c issoid.html.
Lockwood, E. H. "Cissoids." Ch. 15 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 130-133, 1967.
Yates, R. C. "Cissoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30,
1952.
Cissoid of Diocles
A curve invented by Diocles in about 180 BC in con-
nection with his attempt to duplicate the cube by geo-
metrical methods. The name "cissoid" first appears in
the work of Geminus about 100 years later. Fermat and
Roberval constructed the tangent in 1634. Huygens and
Wallis found, in 1658, that the Area between the curve
and its asymptote was 3a (MacTutor Archive). From a
given point there are either one or three TANGENTS to
the cissoid.
Given an origin O and a point P on the curve, let S be
the point where the extension of the line OP intersects
the line x — 2a and R be the intersection of the CIRCLE
of RADIUS a and center (a, 0) with the extension of OP.
Then the cissoid of Diocles is the curve which satisfies
OP = RS.
Cissoid of Diodes
Clark's Triangle 261
The cissoid of Diodes is the Roulette of the Vertex
of a Parabola rolling on an equal Parabola. Newton
gave a method of drawing the cissoid of Diocles using
two line segments of equal length at RIGHT ANGLES. If
they are moved so that one line always passes through a
fixed point and the end of the other line segment slides
along a straight line, then the MIDPOINT of the sliding
line segment traces out a cissoid of Diocles.
The cissoid of Diocles is given by the parametric equa-
tions
x = 2a sin 6
_ 2a sin 3
V ~ cos 6
Converting these to POLAR COORDINATES gives
sin 6 '
(1)
(2)
2 2.2
r = x + y
: 4a 2 [ sin 4 +
cos 2 (9,
: 4a 2 sin 4 0(1 + tan 2 6) = 4a 2 sin 4 6 sec 2 0, (3)
so
r = 2a sin 2 sec = 2a sin 6 tan 0.
In Cartesian Coordinates,
,3 Qrt 3 • 6/1
(4)
X
2a -x 2a — 2a sin 2
. 2 sin 6 2
= 4a — = y .
= 4a*
sin
1 - sin 2 8
s 2 9
An equivalent form is
x(x 2 -\-y 2 ) = 2ay .
Using the alternative parametric form
*(*) =
y(t)
2at 2
1 + i 2
2at 3
1 + t 2
(Gray 1993), gives the Curvature as
*<*)= a \t\{t* + 4)3/2-
(5)
(6)
(7)
(8)
(9)
References
Gray, A. "The Cissoid of Diocles." §3.4 in Modern Differ-
ential Geometry of Curves and Surf 'aces. Roca Raton, FL:
CRC Press, pp. 43-46, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 98-100, 1972.
Lee, X. "Cissoid of Diocles." http://www.best.com/-xah/
SpecialPlaneCurvesjdir/CissoidOf Diocles jdir/cissoid
OfDiocles.html.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, pp. 130-133, 1967.
MacTutor History of Mathematics Archive. "Cissoid of Dio-
cles." http: //www-groups . dcs . st-and.ac.uk/-history/
Curves/Cissoid.html.
Yates, R. C. "Cissoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30,
1952.
Cissoid of Diocles Caustic
The Caustic of the cissoid where the Radiant Point
is taken as (8a, 0) is a CARDIOID.
Cissoid of Diocles Inverse Curve
If the cusp of the CISSOID OF DIOCLES is taken as
the Inversion Center, then the cissoid inverts to a
PARABOLA.
Cissoid of Diocles Pedal Curve
\
\
\
The Pedal Curve of the cissoid, when the Pedal
Point is on the axis beyond the Asymptote at a dis-
tance from the cusp which is four times that of the
Asymptote is a Cardioid.
Clairaut's Differential Equation
- x ^y + f f^M.\
dx V dx )
or
y = px + f(p),
where / is a Function of one variable and p = dy/dx.
The general solution is y — ex + /(c). The singular
solution ENVELOPES are x — ~f f (c) and y = f(c) -
cf'(c).
see also d'Alembert's Equation
References
Boyer, C B. A History of Mathematics. New York: Wiley,
p. 494, 1968.
Clarity
The Ratio of a measure of the size of a "fit" to the size
of a "residual."
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 667, 1977.
Clark's Triangle
(m-l)3
12 7 1 t>
18 19 8 1
24 37 27 9 1
30 61 64 36 10 1
36 91 125 100 46 11 1
//
262 Clark's Triangle
A Number Triangle created by setting the Vertex
equal to 0, filling one diagonal with Is, the other diag-
onal with multiples of an INTEGER /, and rilling in the
remaining entries by summing the elements on either
side from one row above. Call the first column n =
and the last column m = nso that
c(m, 0) = fm
c(rri) m) = 1,
(1)
(2)
then use the Recurrence Relation
c(m, n) = c(m — 1, n — 1) + c(m — 1, n) (3)
to compute the rest of the entries. For n = 1, we have
c(m, 1) = c(m -1,0) + c(m - 1, 1) (4)
c(m, 1) - c(m - 1, 1) = c(m -1,0) = f(m - 1). (5)
For arbitrary m, the value can be computed by Sum-
ming this Recurrence,
c(m, 1) = / j J2 k I + X = l/ m ( m - 1) + 1. (6)
Now, for n = 2 we have
c(m, 2) = c(m - 1, 1) + c(m - 1, 2) (7)
c(m,2)-c(m-l,2) = c(m-l,l) = |/(m-l)m+l, (8)
so Summing the Recurrence gives
c(m, 2) = 5}±/*(* - 1) + 1] = ]T(§/fc 2 " 3** + X )
fc=i
fc=i
= \f[\m{m + l)(2m + 1)] - \f[\m{m + 1)] + m
= ±(m-l)(/m 2 -2/m + 6). (9)
Similarly, for n = 3 we have
c(m, 3) - c(m -1,3) = c(m - 1, 2)
= |/m 3 -/m 2 + (^/ + l)m-(/ + 2). (10)
Taking the Sum,
m
c(m,3) = ^ i/fc 3 - /fc 2 + (ff + l)k - (/ + 2). (11)
fc = 2
Evaluating the Sum gives
c(m,3) = ^(m- l)(m-2)(/m 2 -3/m+12). (12)
Ciass Number
So far, this has just been relatively boring Algebra.
But the amazing part is that if / = 6 is chosen as the
Integer, then c(m, 2) and c(tm, 3) simplify to
c(m, 2) = \{m - l)(6m 2 - 12m + 6)
-(m-1) 3 (13)
c(m,3)=|(m-l) 2 (m-2) 2 , (14)
which are consecutive Cubes (m — l) 3 and nonconsecu-
tive Squares n 2 = [(m - l)(m - 2)/2] 2 .
see a/so Bell Triangle, Catalan's Triangle,
Euler's Triangle, Leibniz Harmonic Triangle,
Number Triangle, Pascal's Triangle, Seidel-
Entringer-Arnold Triangle, Sum
References
Clark, J. E. "Clark's Triangle." Math. Student 26, No. 2,
p. 4, Nov. 1978.
Class
see Characteristic Class, Class Interval, Class
(Multiply Perfect Number), Class Number,
Class (Set), Conjugacy Class
Class (Group)
see Conjugacy Class
Class Interval
The constant bin size in a HISTOGRAM,
see also Sheppard's Correction
Class (Map)
A Map u : R n -► R n from a Domain G is called a map
of class C r if each component of
u(x) - (ui(zi,...,Xn),...,u m (a;i J ...,x„))
is of class C r (0 < r < 00 or r — w) in G, where C d
denotes a continuous function which is differentiable d
times.
Class (Multiply Perfect Number)
The number k in the expression s(n) — kn for a Mul-
tiply Perfect Number is called its class.
Class Number
For any IDEAL 7, there is an IDEAL 7* such that
Hi = z,
(1)
where z is a Principal IDEAL, (i.e., an IDEAL of rank
1). Moreover, there is a finite list of ideals h such that
this equation may be satisfied for every I. The size
of this list is known as the class number. When the
class number is 1, the Ring corresponding to a given
IDEAL has unique factorization and, in a sense, the class
Class Number
Class Number
263
number is a measure of the failure of unique factorization
in the original number ring.
A finite series giving exactly the class number of a Ring
is known as a CLASS NUMBER FORMULA. A CLASS
Number Formula is known for the full ring of cyclo-
tomic integers, as well as for any subring of the cyclo-
tomic integers. Finding the class number is a computa-
tionally difficult problem.
Let h(d) denote the class number of a quadratic ring,
corresponding to the Binary Quadratic Form
ax + bxy + cy ,
with Discriminant
d = b — 4ac.
(2)
(3)
Then the class number h(d) for DISCRIMINANT d gives
the number of possible factorizations of ax 2 + bxy + cy 2
in the QUADRATIC Field Q(y/d). Here, the factors are
of the form x 4- yVd, with x and y half INTEGERS.
Some fairly sophisticated mathematics shows that the
class number for discriminant d can be given by the
Class Number Formula
,, f-^E^VWlnsin(^) ford>0 /x
mElt\d\r)r
for d < 0,
where (d\r) is the Kronecker Symbol, 77(d) is the
Fundamental Unit, w(d) is the number of substitu-
tions which leave the Binary Quadratic Form un-
changed
( 6 for d = -3
w(d) ^<4 for d = -4 (5)
[ 2 otherwise,
and the sums are taken over all terms where the Kron-
ecker SYMBOL is defined (Cohn 1980). The class num-
ber for d > can also be written
^M-) = TJ Bin -(-|r)^^
(6)
for d > 0, where the PRODUCT is taken over terms for
which the Kronecker Symbol is defined.
The class number is related to the DlRlCHLET L-Series
by
L„(l)
h(d) =
K{d)
(7)
where /c(d) is the DlRlCHLET STRUCTURE CONSTANT.
Wagner (1996) shows that class number h(—d) satisfies
the Inequality
-»^(>-M)
lnd,
(8)
for -d < 0, where [x] is the Floor Function, the
product is over PRIMES dividing d, and the * indicates
that the Greatest Prime Factor of d is omitted from
the product.
The Mathematica® (Wolfram Research, Champaign,
IL) function NumberTheory'NumberTheoryFunct ions'
ClassNumber [n] gives the class number h{d) for d a
Negative Squarefree number of the form 4k -f 1,
Gauss's Class Number Problem asks to determine
a complete list of fundamental DISCRIMINANTS — d such
that the CLASS Number is given by h(—d) = m for
a given m. This problem has been solved for n < 7
and Odd n < 23. Gauss conjectured that the class
number h(—d) of an IMAGINARY quadratic field with
Discriminant —d tends to infinity with d, an assertion
now known as Gauss's Class Number Conjecture.
The discriminants d having h(~d) = 1, 2, 3, 4, 5, ...
are Sloane's A014602 (Cohen 1993, p. 229; Cox 1997,
p. 271), Sloane's A014603 (Cohen 1993, p. 229), Sloane's
A006203 (Cohen 1993, p. 504), Sloane's A013658 (Co-
hen 1993, p. 229), Sloane's A046002, Sloane's A046003,
The complete set of negative discriminants hav-
ing class numbers 1-5 and Odd 7-23 are known. Buell
(1977) gives the smallest and largest fundamental class
numbers for d < 4, 000, 000, partitioned into EVEN dis-
criminants, discriminants 1 (mod 8), and discriminants
5 (mod 8). Arno et al. (1993) give complete lists of val-
ues of d with h{-d) = k for ODD k = 5, 7, 9, . . . , 23.
Wagner gives complete lists of values for k = 5, 6, and
7.
Lists of NEGATIVE discriminants co rrespon ding to
Imaginary Quadratic Fields Q(y/—d(n) ) having
small class numbers h{—d) are given in the table below.
In the table, N is the number of "fundamental" values
of — d with a given class number h{—d)^ where "funda-
mental" means that — d is not divisible by any SQUARE
Number s 2 such that h(—d/s 2 ) < h(—d). For example,
although h(— 63) = 2, —63 is not a fundamental dis-
criminant since 63 = 3 2 • 7 and h(-63/3 2 ) = h(-7) =
1 < h(-63). Even values 8 < h(-d) < 18 have been
computed by Weisstein. The number of negative dis-
criminants having class number 1, 2, 3, . . . are 9, 18,
16, 54, 25, 51, 31, ... (Sloane's A046125). The largest
negative discriminants having class numbers 1, 2, 3, . . .
are 163, 427, 907, 1555, 2683, . . . (Sloane's A038552).
The following table lists the numbers with small class
numbers < 11. Lists including larger class numbers are
given by Weisstein.
h(-d) N d
1 9 3, 4, 7, 8, 11, 19, 43, 67, 163
2 18 15, 20, 24, 35, 40, 51, 52, 88, 91, 115,
123, 148, 187, 232, 235, 267, 403, 427
3 16 23, 31, 59, 83, 107, 139, 211, 283, 307,
331, 379, 499, 547, 643, 883, 907
264
Class Number
Class Number
h(-d) N d
4 54 39, 55, 56, 68, 84, 120, 132, 136, 155,
168, 184, 195, 203, 219, 228, 259, 280,
291, 292, 312, 323, 328, 340, 355, 372,
388, 408, 435, 483, 520, 532, 555, 568,
595, 627, 667, 708, 715, 723, 760, 763,
772, 795, 955, 1003, 1012, 1027, 1227,
1243, 1387, 1411, 1435, 1507, 1555
5 25 47, 79, 103, 127, 131, 179, 227, 347, 443,
523, 571, 619, 683, 691, 739, 787, 947,
1051, 1123, 1723, 1747, 1867, 2203, 2347,
2683
6 51 87, 104, 116, 152, 212, 244, 247, 339,
411, 424, 436, 451, 472, 515, 628, 707,
771, 808, 835, 843, 856, 1048, 1059, 1099,
1108, 1147, 1192, 1203, 1219, 1267, 1315,
1347, 1363, 1432, 1563, 1588, 1603, 1843,
1915, 1963, 2227, 2283, 2443, 2515, 2563,
2787, 2923, 3235, 3427, 3523, 3763
7 31 71, 151, 223, 251, 463, 467, 487, 587,
811, 827, 859, 1163, 1171, 1483, 1523,
1627, 1787, 1987, 2011, 2083, 2179, 2251,
2467, 2707, 3019, 3067, 3187, 3907, 4603,
5107, 5923
8 131 95, 111, 164, 183, 248, 260, 264, 276,
295, 299, 308, 371, 376, 395, 420, 452,
456, 548, 552, 564, 579, 580, 583, 616,
632, 651, 660, 712, 820, 840, 852, 868,
904, 915, 939, 952, 979, 987, 995, 1032,
1043, 1060, 1092, 1128, 1131, 1155,
1195, 1204, 1240, 1252, 1288, 1299, 1320,
1339, 1348, 1380, 1428, 1443, 1528, 1540,
1635, 1651, 1659, 1672, 1731, 1752, 1768,
1771, 1780, 1795, 1803, 1828, 1848, 1864,
1912, 1939, 1947, 1992, 1995, 2020, 2035,
2059, 2067, 2139, 2163, 2212, 2248, 2307,
2308, 2323, 2392, 2395, 2419, 2451, 2587,
2611, 2632, 2667, 2715, 2755, 2788, 2827,
2947, 2968, 2995, 3003, 3172, 3243, 3315,
3355, 3403, 3448, 3507, 3595, 3787, 3883,
3963, 4123, 4195, 4267, 4323, 4387, 4747,
4843, 4867, 5083, 5467, 5587, 5707, 5947,
6307
9 34 199, 367, 419, 491, 563, 823, 1087, 1187,
1291, 1423, 1579, 2003, 2803, 3163, 3259,
3307, 3547, 3643, 4027, 4243, 4363, 4483,
4723, 4987, 5443, 6043, 6427, 6763, 6883,
7723, 8563, 8803, 9067, 10627
10 87 119, 143, 159, 296, 303, 319, 344, 415,
488, 611, 635, 664, 699, 724, 779, 788,
803, 851, 872, 916, 923, 1115, 1268,
1384, 1492, 1576, 1643, 1684, 1688, 1707,
1779, 1819, 1835, 1891, 1923, 2152, 2164,
h(~d) N d
2363, 2452, 2643, 2776, 2836, 2899, 3028,
3091, 3139, 3147, 3291, 3412, 3508, 3635,
3667, 3683, 3811, 3859, 3928, 4083, 4227,
4372, 4435, 4579, 4627, 4852, 4915, 5131,
5163, 5272, 5515, 5611, 5667, 5803, 6115,
6259, 6403, 6667, 7123, 7363, 7387, 7435,
7483, 7627, 8227, 8947, 9307, 10147,
10483, 13843
11 41 167, 271, 659, 967, 1283, 1303, 1307,
1459, 1531, 1699, 2027, 2267, 2539, 2731,
2851, 2971, 3203, 3347, 3499, 3739, 3931,
4051, 5179, 5683, 6163, 6547, 7027, 7507,
7603, 7867, 8443, 9283, 9403, 9643, 9787,
10987, 13003, 13267, 14107, 14683, 15667
The table below gives lists of Positive fundamental
discriminants d having small class numbers h(d), cor-
responding to Real quadratic fields. All Positive
SQUAREFREE values of d < 97 (for which the KRON-
ECKER SYMBOL is defined) are included.
h(d) d
1 5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77
2 65
The POSITIVE d for which h(d) = 1 is given by Sloane's
A014539.
see also Class Number Formula, Dirichlet L-
Series, Discriminant (Binary Quadratic Form),
Gauss's Class Number Conjecture, Gauss's
Class Number Problem, Heegner Number, Ideal,
j-FUNCTION
References
Arno, S. "The Imaginary Quadratic Fields of Class Number
4." Acta Arith. 40, 321-334, 1992.
Arno, S.; Robinson, M. L«; and Wheeler, F. S. "Imaginary
Quadratic Fields with Small Odd Class Number." http://
www.math.uiuc . edu/Algebraic -Number-Theory/ 0009/.
Buell, D. A. "Small Class Numbers and Extreme Values of
//-Functions of Quadratic Fields." Math. Comput. 139,
786-796, 1977.
Cohen, H. A Course in Computational Algebraic Number
Theory. New York: Springer- Verlag, 1993.
Cohn, H. Advanced Number Theory. New York: Dover,
pp. 163 and 234, 1980.
Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field
Theory and Complex Multiplication. New York: Wiley,
1997.
Davenport, H. "Dirichlet's Class Number Formula." Ch. 6
in Multiplicative Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 43-53, 1980.
Iyanaga, S. and Kawada, Y. (Eds.). "Class Numbers of Al-
gebraic Number Fields." Appendix B, Table 4 in Encyclo-
pedic Dictionary of Mathematics. Cambridge, MA: MIT
Press, pp. 1494-1496, 1980.
Montgomery, H. and Weinberger, P. "Notes on Small Class
Numbers." Acta. Arith. 24, 529-542, 1974.
Sloane, N. J. A. Sequences A014539, A038552, A046125, and
A003657/M2332 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Class Number Formula
Clausen Formula 265
Stark, H. M. "A Complete Determination of the Complex
Quadratic Fields of Class Number One." Michigan Math.
J. 14, 1-27, 1967.
Stark, H, M. "On Complex Quadratic Fields with Class Num-
ber Two." Math. Comput. 29, 289-302, 1975.
Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65,
785-800, 1996.
# Weisstein, E. W. "Class Numbers." http: //www. astro .
Virginia. edu/~eww6n/math/notebooks/ClassNumbers .m.
Class Number Formula
A class number formula is a finite series giving exactly
the Class Number of a Ring. For a Ring of quadratic
integers, the class number is denoted h(d) y where d is the
discriminant. A class number formula is known for the
full ring of cyclotomic integers, as well as for any subring
of the cyclotomic integers. This formula includes the
quadratic case as well as many cubic and higher-order
rings.
see also Class Number
Class Representative
A set of class representatives is a SUBSET of X which
contains exactly one element from each Equivalence
Class.
Class (Set)
A class is a special kind of Set invented to get around
RUSSELL'S PARADOX while retaining the arbitrary cri-
teria for membership which leads to difficulty for Sets.
The members of classes are Sets, but it is possible to
have the class C of "all Sets which are not members of
themselves" without producing a paradox (since C is a
proper class (and not a Set), it is not a candidate for
membership in C).
see also Aggregate, Russell's Paradox, Set
Classical Groups
The four following types of GROUPS,
1. Linear Groups,
2. Orthogonal Groups,
3. Symplectic Groups, and
4. Unitary Groups,
which were studied before more exotic types of groups
(such as the SPORADIC GROUPS) were discovered.
see also GROUP, LINEAR GROUP, ORTHOGONAL
Group, Symplectic Group, Unitary Group
Classification
The classification of a collection of objects generally
means that a list has been constructed with exactly one
member from each ISOMORPHISM type among the ob-
jects, and that tools and techniques can effectively be
used to identify any combinatorially given object with
its unique representative in the list. Examples of math-
ematical objects which have been classified include the
finite Simple Groups and 2-Manifolds but not, for
example, Knots.
Classification Theorem
The classification theorem of FINITE Simple GROUPS,
also known as the ENORMOUS THEOREM, which states
that the Finite Simple Groups can be classified com-
pletely into
1. Cyclic Groups Z p of Prime Order,
2. Alternating Groups A n of degree at least five,
3. Lie-Type Chevalley Groups PSL(n,q),
PSU(n,q), PsP(2n,g), and Pft € (n,g),
4. Lie-Type (Twisted Chevalley Groups or the
Tits Group) s D 4 (q) y E Q (q) y E 7 (q), E s (q), F 4 (g),
2 F 4 (2*% G 2 (q), 2 G 2 (3 n ), 2 B(2 n ),
5. Sporadic Groups Mu, M i2 , M 22 , M23, M 24 , Ji =
HJ, Suz, HS, McL, Co 3 , Co 2 , C01, He, Fi 22} ^'23,
Fi' 24 , HN, Th, B, M, J u OW, J 3 , Ly, Ru, J 4 .
The "Proof" of this theorem is spread throughout the
mathematical literature and is estimated to be approx-
imately 15,000 pages in length.
see also FINITE GROUP, GROUP, j-FUNCTION, SIMPLE
Group
References
Cartwright, M. "Ten Thousand Pages to Prove Simplicity."
New Scientist 109, 26-30, 1985.
Cipra, B. "Are Group Theorists Simpleminded?" What's
Happening in the Mathematical Sciences, 1995-1996,
Vol 3. Providence, RJ: Amer. Math. Soc, pp. 82-99, 1996.
Cipra, B. "Slimming an Outsized Theorem." Science 267,
794-795, 1995.
Gorenstein, D. "The Enormous Theorem," Set Amer, 253,
104-115, Dec. 1985.
Solomon, R. "On Finite Simple Groups and Their Classifica-
tion." Not Amer. Math. Soc. 42, 231-239, 1995.
Clausen Formula
Clausen's 4^3 identity
/ 9
(2a)| d |(a + %|(26)| d |
(2a + 2b)\d\a\ d \b\d\
holds for a + b + c- d= 1/2, e = a + 6 + 1/2, a + / =
d+l = 6 + p, da nonpositive integer, and (a) n is the
POCHHAMMER Symbol (Petkovsek tt al. 1996).
Another identity ascribed to Clausen which in-
volves the Hypergeometric Function 2 i*i(a, b\c\z)
and the GENERALIZED HYPERGEOMETRIC FUNCTION
3F2 (a, 6, c; d, e; z) is given by
a, 6
a + b+k'' X
= 3-^2
(•
2a, a + b, 2b
+ 6+|,2a + 26 ;:C
see also GENERALIZED HYPERGEOMETRIC FUNCTION,
HYPERGEOMETRIC FUNCTION
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K, Peters, pp. 43 and 127, 1996.
266 Clausen Function
Clausen Function
sin(kx)
*.(*>- £n£
C n {x) = J2
cos(kx)
(i)
(2)
and write
cl " (x) = \c„(x)=Er=i ££ ^ 1 "° dd -
Then the Clausen function Cl n (x) can be given symbol-
ically in terms of the Polylogarithm as
/ii[Li n (e-")-Li n (e-)] r.
Ol nW - | i [Lin(e -i*) + Li n (e-)] n
even
odd.
For n = l, the function takes on the special form
Cli(x) = Ci(x) = -ln|2sin(|x)|
and for n = 2, it becomes Clausen's Integral
Cl 2 (a:) - S 2 (x) = - / ln[2sin(ft)]dt.
(4)
(5)
The symbolic sums of opposite parity are summable
symbolically, and the first few are given by
i~ 2
1_ 4
48^
C 2 (ac) = ±tt - ±ttx+±x
C 4 (z) = ^ - T^ 2 ^ 2 + T2 7 ™ 3 ~ -h*
5i(x)=§(7T-x)
(6)
(7)
(8)
(9)
5 5 (x) = i7r 4 x-^7rV + ^7rx 4 -^x 5 (10)
for < x < 27r (Abramowitz and Stegun 1972).
see also CLAUSEN'S INTEGRAL, POLYGAMMA FUNC-
TION, Polylogarithm
CLEAN Algorithm
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Clausen's Inte-
gral and Related Summations" §27.8 in Handbook of Math-
ematical Functions with Formulas, Graphs, and Mathe-
matical Tables, 9th printing. New York: Dover, pp. 1005-
1006, 1972.
Arfken, G. Mathematical Methods {or Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 783, 1985.
Clausen, R. "Uber die Zerlegung reeller gebrochener . Funk-
tionen." J. reine angew. Math. 8, 298-300, 1832.
Grosjean, C. C. "Formulae Concerning the Computation of
the Clausen Integral Cl 2 (a)." J. Comput. Appl. Math. 11,
331-342, 1984.
Jolley, L. B. W. Summation of Series. London: Chapman,
1925.
Wheelon, A. D. A Short Table of Summable Series. Report
No. SM-14642. Santa Monica, CA: Douglas Aircraft Co.,
1953.
Clausen's Integral
0.5
-1-
The Clausen Function
C1 2 (0) = - / \n[2sm(lt)]dt
t/0
see also CLAUSEN FUNCTION
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 1005-1006, 1972.
Ashour, A. and Sabri, A. "Tabulation of the Function ip(0) =
V°° -i£l»£i.» Math. Tables Aids Comp. 10, 54 and 57-
65, 1956.
Clausen, R. "Uber die Zerlegung reeller gebrochener Funk-
tionen." J. reine angew. Math. 8, 298-300, 1832.
CLEAN Algorithm
An iterative algorithm which DECONVOLVES a sampling
function (the "Dirty Beam") from an observed bright-
ness ("DIRTY Map") of a radio source. This algorithm
is of fundamental importance in radio astronomy, where
it is used to create images of astronomical sources which
are observed using arrays of radio telescopes ( "synthesis
imaging"). As a result of the algorithm's importance to
synthesis imaging, a great deal of effort has gone into
optimizing and adjusting the ALGORITHM. CLEAN is a
nonlinear algorithm, since linear DECONVOLUTION algo-
rithms such as Wiener Filtering and inverse filtering
CLEAN Algorithm
CLEAN Algorithm 267
are inapplicable to applications with invisible distribu-
tions (i.e., incomplete sampling of the spatial frequency
plane) such as map obtained in synthesis imaging.
The basic CLEAN method was developed by Hogbom
(1974). It was originally designed for point sources, but
it has been found to work well for extended sources
as well when given a reasonable starting model. The
Hogbom CLEAN constructs discrete approximations I n
to the CLEAN Map in the (£,77) plane from the CON-
VOLUTION equation
b' *I = /',
(1)
where b' is the Dirty Beam, I' is the Dirty Map (both
in the (£>r?) Plane), and f*g denotes a Convolution.
The CLEAN algorithm starts with an initial approxi-
mation Jo = 0. At the nth iteration, it then searches for
the largest value in the residual map
I n ^ I' - b' * I n -1.
(2)
A Delta Function is then centered at the location of
the largest residual flux and given an amplitude /x (the
so-called "Loop Gain") times this value. An antenna's
response to the Delta FUNCTION, the DlRTY Beam, is
then subtracted from I n -i to yield I n . Iteration con-
tinues until a specified iteration limit N is reached, or
until the peak residual or Root-Mean-Square resid-
ual decreases to some level. The resulting final map is
denoted In, and the position of each Delta Function
is saved in a "CLEAN component" table in the CLEAN
Map file. At the point where component subtraction is
stopped, it is assumed that the residual brightness dis-
tribution consists mainly of NOISE.
To diminish high spatial frequency features which may
be spuriously extrapolated from the measured data,
each CLEAN component is convolved with the so-called
CLEAN Beam 6, which is simply a suitably smoothed
version of the sampling function ("Dirty Beam"). Usu-
ally, a Gaussian is used. A good CLEAN Beam should:
1. Have a unity FOURIER TRANSFORM inside the sam-
pled region of (u, v) space,
2. Have a FOURIER TRANSFORM which tends to out-
side the sampled (u, v) region as quickly as possible,
and
3. Not have any effects produced by NEGATIVE side-
lobes larger than the NOISE level.
A CLEAN Map is produced when the final residual
map is added to the the approximate solution,
[clean map] = In * b -\- [I — b * In]
in order to include the NOISE.
(3)
CLEAN will always converge to one (of possibly many)
solutions if the following three conditions are satisfied
(Schwarz 1978):
1. The beam must be symmetric.
2. The Fourier Transform of the Dirty Beam is
NONNEGATIVE (positive definite or positive semidef-
inite).
3. There must be no spatial frequencies present in the
dirty image which are not also present in the Dirty
Beam.
These conditions are almost always satisfied in practice.
If the number of CLEAN components does not exceed
the number of independent (u,v) points, CLEAN con-
verges to a solution which is the least squares fit of the
Fourier Transforms of the Delta Function com-
ponents to the measured visibility (Thompson et al.
1986, p. 347). Schwarz claims that the CLEAN algo-
rithm is equivalent to a least squares fitting of cosine
and sine parts in the (u, v) plane of the visibility data.
Schwab has produced a NOISE analysis of the CLEAN
algorithm in the case of least squares minimization of
a noiseless image which involves am N x M MATRIX.
However, no NOISE analysis has been performed for a
Real image.
Poor modulation of short spacings results in an under-
estimation of the flux, which is manifested in a bowl of
negative surface brightness surrounding an object. Pro-
viding an estimate of the "zero spacing" flux (the to-
tal flux of the source, which cannot be directly mea-
sured by an interferometer) can considerably reduce
this effect. Modulations or stripes can occur at spa-
tial frequencies corresponding to undersampled parts
of the (u,v) plane. This can result in a golf ball-like
mottling for disk sources such as planets, or a corru-
gated pattern of parallel lines of peaks and troughs
("stripes"). A more accurate model can be used to sup-
press the "golf ball" modulations, but may not elimi-
nate the corrugations. A tapering function which de-
emphasizes data near (u, v) = (0,0) can also be used.
Stripes can sometimes be eliminated using the Cornwell
smoothness-stabilized CLEAN (a.k.a. Prussian helmet
algorithm; Thompson et al 1986). CLEANing part way,
then restarting the CLEAN also seems to eliminate the
stripes, although this fact is more disturbing than reas-
suring. Stability the the CLEAN algorithm is discussed
by Tan (1986).
In order to CLEAN a map of a given dimension, it is nec-
essary to have a beam pattern twice as large so a point
source can be subtracted from any point in the map.
Because the CLEAN algorithm uses a Fast FOURIER
Transform, the size must also be a Power of 2.
There are many variants of the basic Hogbom CLEAN
which extend the method to achieve greater speed and
produce more realistic maps. Alternate nonlinear De-
convolution methods, such as the Maximum En-
tropy Method, may also be used, but are gener-
ally slower than the CLEAN technique. The Astro-
nomical Image Processing Software (AIPS) of the Na-
tional Radio Astronomical Observatory includes 2-D
268 CLEAN Algorithm
CLEAN Algorithm
DECONVOLUTION algorithms in the tasks DCONV and
UVMAP. Among the variants of the basic Hogbom CLEAN
are Clark, Cornwell smoothness stabilized (Prussian
helmet), Cotton-Schwab, Gerchberg-Saxton (Fienup),
Steer, Steer-Dewdney-Ito, and van Cittert iteration.
In the Clark (1980) modification, CLEAN picks out only
the largest residual points, and subtracts approximate
point source responses in the (£,77) plane during minor
(Hogbom CLEAN) cycles. It only occasionally (dur-
ing major cycles) computes the full /„, residual map by
subtracting the identified point source responses in the
(ujv) plane using a Fast Fourier Transform for the
Convolution. The Algorithm then returns to a mi-
nor cycle. This algorithm modifies the Hogbom method
to take advantage of the array processor (although it also
works without one). It is therefore a factor of 2-10 faster
than the simple Hogbom routine. It is implemented as
the AIPS task APCLN.
The Cornwell smoothness stabilized variant was devel-
oped because, when dealing with two-dimensional ex-
tended structures, CLEAN can produce artifacts in the
form of low-level high frequency stripes running through
the brighter structure. These stripes derive from poor
interpolations into unsampled or poorly sampled re-
gions of the (u, v) plane. When dealing with quasi-one-
dimensional sources (i.e., jets), the artifacts resemble
knots (which may not be so readily recognized as spuri-
ous). APCLN can invoke a modification of CLEAN that
is intended to bias it toward generating smoother solu-
tions to the deconvolution problem while preserving the
requirement that the transform of the CLEAN compo-
nents list fits the data. The mechanism for introducing
this bias is the addition to the Dirty Beam of a Delta
FUNCTION (or "spike") of small amplitude (PHAT) while
searching for the CLEAN components. The beam used
for the deconvolution resembles the helmet worn by Ger-
man military officers in World War I, hence the name
"Prussian helmet" CLEAN.
The theory underlying the Cornwell smoothness stabi-
lized algorithm is given by Cornwell (1982, 1983), where
it is described as the smoothness stabilized CLEAN. It
is implemented in the AIPS tasks APCLN and MX. The
spike performs a NEGATIVE feedback into the dirty im-
age, thus suppressing features not required by the data.
Spike heights of a few percent and lower than usual loop
gains are usually needed. Also according to the MX doc-
umentation,
PHAT ;
(noise) 1
2(signal) 2 ~ 2(SNR) 2
Unfortunately, the addition of a Prussian helmet gen-
erally has "limited success," so resorting to another de-
convolution method such as the MAXIMUM ENTROPY
METHOD is sometimes required.
The Cotton-Schwab uses the Clark method, but the
major cycle subtractions of CLEAN components are
performed on ungridded visibility data. The Cotton-
Schwab technique is often faster than the Clark variant.
It is also capable of including the w baseline term, thus
removing distortions from noncoplanar baselines. It is
often faster than the Clark method. The Cotton-Schwab
technique is implemented as the AIPS task MX.
The Gerchberg-Saxton variant, also called the Fienup
variant, is a technique originally introduced for solv-
ing the phase problem in electron microscopy. It was
subsequently adapted for visibility amplitude measure-
ments only. A Gerchberg-Saxton map is constrained to
be Nonzero, and positive. Data and image plane con-
straints are imposed alternately while transforming to
and from the image plane. If the boxes to CLEAN are
chosen to surround the source snugly, then the algorithm
will converge faster and will have more chance of finding
a unique image. The algorithm is slow, but should be
comparable to the Clark technique (APCLN) if the map
contains many picture elements. However, the resolu-
tion is data dependent and varies across the map. It is
implemented as the AIPS task APGS (Pearson 1984).
The Steer variant is a modification of the Clark variant
(Cornwell 1982). It is slow, but should be comparable
to the Clark algorithm if the map contains many pic-
ture elements. The algorithm used in the program is
due to David Steer. The principle is similar to Barry
Clark's CLEAN except that in the minor cycle only
points above the (trim level) x (peak in the residual map)
are selected. In the major cycle these are removed us-
ing a Fast Fourier Transform. If boxes are chosen
to surround the source snugly, then the algorithm will
converge faster and will have more chance of finding a
unique image. It is implemented in AIPS as the exper-
imental program STEER and as the Steer-Dewdney-Ito
variant combined with the Clark algorithm as SDCLN.
The Steer-Dewdney-Ito variant is similar to the Clark
variant, but the components are taken as all pixels
having residual flux greater than a cutoff value times
the current peak residual. This method should avoid
the "ripples" produced by the standard CLEAN on ex-
tended emission. The AIPS task SDCLN does an AP-
based CLEAN of the the Clark type, but differs from
APCLN in that it offers the option to switch to the Steer-
Dewdney-Ito method.
Finally, van Cittert iteration consists of two steps:
1. Estimate a correction to add to the current map es-
timate by multiplying the residuals by some weight.
In the classical van Cittert algorithm, this weight is
a constant, where as in CLEAN the weight is zero
everywhere except at the peak of the residuals.
2. Add the step to the current estimate, and subtract
the estimate, convolved with the DIRTY BEAM, from
the residuals.
CLEAN Beam
Clebsch Diagonal Cubic 269
Though it is a simple algorithm, it works well (if slowly)
for cases where the DlRTY BEAM is positive semidefmite
(as it is in astronomical observations). The basic idea is
that the DlRTY MAP is a reasonably good estimate of
the deconvolved map. The different iterations vary only
in the weight to apply to each residual in determining
the correction step, van Cittert iteration is implemented
as the AIPS task APVC, which is a rather experimental
and ad hoc procedure. In some limiting cases, it reduces
to the standard CLEAN algorithm (though it would be
unpractically slow).
see also CLEAN Beam, CLEAN Map, Dirty Beam,
Dirty Map
References
Christiansen, W. N. and Hogbom, J. A. Radiotelescopes, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 214-216, 1985,
Clark, B, G. "An Efficient Implementation of the Algorithm
'CLEAN'." Astron. Astrophys, 89, 377-378, 1980.
Cornwell, T. J. "Can CLEAN be Improved?" VLA Scientific
Memorandum No. 141, 1982.
Cornwell, T\ J. "Image Restoration (and the CLEAN Tech-
nique)." Lecture 9. NRAO VLA Workshop on Synthesis
Mapping, p. 113, 1982,
Cornwell, T. J. "A Method of Stabilizing the CLEAN Algo-
rithm." Astron. Astrophys. 121, 281-285, 1983.
Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Syn-
thesis Imaging in Radio Astronomy: Third NRAO Sum-
mer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and
A. H. Bridle). San Francisco, CA: Astronomical Society of
the Pacific, pp. 178-179, 1989.
Hogbom, J. A. "Aperture Synthesis with a Non-Regular Dis-
tribution of Interferometric Baselines." Astron. Astrophys.
Supp. 15, 417-426, 1974.
National Radio Astronomical Observatory. Astronomical Im-
age Processing Software (AIPS) software package. APCLN,
MX, and UVMAP tasks.
Pearson, T. J. and Readhead, A. C. S. "Image Formation by
Self-Calibration in Radio Astronomy." Ann. Rev. Astron.
Astrophys. 22, 97-130, 1984.
Schwarz, U. J. "Mathematical-Statistical Description of the
Iterative Beam Removing Technique (Method CLEAN)."
Astron. Astrophys. 65, 345-356, 1978.
Tan, S. M. "An Analysis of the Properties of CLEAN and
Smoothness Stabilized CLEAN — Some Warnings." Mon.
Not. Royal Astron. Soc. 220, 971-1001, 1986.
Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
Inter jerometry and Synthesis in Radio Astronomy. New
York: Wiley, p. 348, 1986.
CLEAN Beam
An Elliptical Gaussian fit to the Dirty Beam in
order to remove sidelobes. The CLEAN beam is con-
volved with the final CLEAN iteration to diminish spu-
rious high spatial frequencies.
see also CLEAN Algorithm, CLEAN Map, Decon-
volution, Dirty Beam, Dirty Map
CLEAN Map
The deconvolved map extracted from a finitely sampled
Dirty Map by the CLEAN Algorithm, Maximum
Entropy Method, or any other Deconvolution pro-
cedure.
see also CLEAN Algorithm, CLEAN Beam, Decon-
volution, Dirty Beam, Dirty Map
Clebsch- Aronhold Notation
A notation used to describe curves. The fundamen-
tal principle of Clebsch-Aronhold notation states that
if each of a number of forms be replaced by a POWER of
a linear form in the same number of variables equal to
the order of the given form, and if a sufficient number
of equivalent symbols are introduced by the ARONHOLD
Process so that no actual Coefficient appears except
to the first degree, then every identical relation holding
for the new specialized forms holds for the general ones.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 79, 1959.
Clebsch Diagonal Cubic
A Cubic Algebraic Surface given by the equation
xo 3 + xi S + x 2 3 + x 3 3 + X4 3 = 0, (1)
with the added constraint
xo + Xi + X2 + £3 + X4
0.
(2)
The implicit equation obtained by taking the plane at
infinity as xq + x\ + x 2 + x$/2 is
81(x -hy -\-z ) — 189(x y-\-x z + y x-\-y z + z x + z y)
+54xyz + 126(xy + xz + yz) - 9(x 2 + y 2 + z 2 )
-9(x + y + z) + 1 = (3)
(Hunt, Nordstrand). On Clebsch's diagonal surface,
all 27 of the complex lines (Solomon's Seal Lines)
present on a general smooth CUBIC SURFACE are real.
In addition, there are 10 points on the surface where 3
of the 27 lines meet. These points are called ECKARDT
POINTS (Fischer 1986, Hunt), and the Clebsch diago-
nal surface is the unique CUBIC SURFACE containing 10
such points (Hunt).
If one of the variables describing Clebsch's diagonal sur-
face is dropped, leaving the equations
xq 3 + xi 3 + x 2 3 + #3 3 = 0,
(4)
270 Clebsch-Gordon Coefficient
x + xi + x 2 + xz = 0, (5)
the equations degenerate into two intersecting Planes
given by the equation
{x + y)(x + z){y + z) = Q. (6)
see also Cubic Surface, Eckardt Point
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 9-11, 1986.
Fischer, G. (Ed.). Plates 10-12 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 13-15, 1986.
Hunt, B. The Geometry of Some Special Arithmetic Quo-
tients. New York: Springer- Verlag, pp. 122-128, 1996.
Nordstrand, T. "Clebsch Diagonal Surface." http://www.
uib , no/people/nf ytn/clebtxt . htm.
Clebsch-Gordon Coefficient
A mathematical symbol used to integrate products of
three SPHERICAL HARMONICS. Clebsch-Gordon coeffi-
cients commonly arise in applications involving the ad-
dition of angular momentum in quantum mechanics. If
products of more than three SPHERICAL HARMONICS
are desired, then a generalization known as WlGNER
6J-SYMBOLS or WlGNER 9?'-Symb0LS is used. The
Clebsch-Gordon coefficients are written
C J mim2 = UiJ2mim 2 \jiJ2Jm) (1)
and are denned by
^jm = 2_^ Cm 1 m 2 ^m 1 m 2 , ( 2 )
M=Mi+M 2
where J = Ji 4- J 2 - The Clebsch-Gordon coefficients
are sometimes expressed using the related RACAH V-
COEFFICIENTS
V(jiJ2J;m 1 7n 2 7n)
(3)
or Wigner 3 j- Symbols. Connections among the three
are
(jiJ2mim2\jiJ2m)
(jiJ2m 1 m 2 \jiJ2Jm)
3i
mi
32
7712
(4)
V(ji32J;rn 1 m 2 m) = (-1)"
-h+32+3 I 3i 32 3i
m 2 mi m 2
Clenshaw Recurrence Formula
They have the symmetry
(jiJ2mim 2 \jij 2 jm) = (-iyi+w (j 2 j 1 Tn 2 m 1 \j 2 jijm),
(7)
and obey the orthogonality relationships
"y y j (jiJ2Tn 1 m2\jiJ2Jm)(jiJ2Jm\j 1 J2Tn' 1 tn' 2 )
= S,
>6„
Tn, l TTl i Tri 2 Tn {
(8)
^ (ji J2mim 2 \jiJ2Jm)(jiJ2J'm'\jiJ2mim2)
see also Racah ^-Coefficient, Racah ^-Coef-
ficient, Wigner 3j-Symbol, Wigner 6j-Symbol,
WlGNER 9J-SYMBOL
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition
Coefficients." §27.9 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 1006-1010, 1972.
Cohen- Tannoudji, C; Diu, B.; and Laloe, F. "Clebsch-
Gordon Coefficients." Complement B x in Quantum Me-
chanics, Vol 2. New York: Wiley, pp. 1035-1047, 1977.
Condon, E. U. and Shortley, G. §3.6-3.14 in The Theory of
Atomic Spectra. Cambridge, England: Cambridge Univer-
sity Press, pp. 56-78, 1951.
Fano, U. and Fano, L. Basic Physics of Atoms and Molecules.
New York: Wiley, p. 240, 1959.
Messiah, A. "Clebsch-Gordon (C.-G.) Coefficients and 'Sf
Symbols." Appendix C.I in Quantum Mechanics, Vol. 2.
Amsterdam, Netherlands: North-Holland, pp. 1054-1060,
1962.
Shore, B. W. and Menzel, D. H. "Coupling and Clebsch-
Gordon Coefficients." §6.2 in Principles of Atomic Spectra.
New York: Wiley, pp. 268-276, 1968.
Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec-
tra and Radiative Transitions, 2nd ed. Berlin: Springer-
Verlag, 1992.
Clement Matrix
see Kac Matrix
Clenshaw Recurrence Formula
The downward Clenshaw recurrence formula evaluates a
sum of products of indexed COEFFICIENTS by functions
which obey a recurrence relation. If
f( X ) = Y, ckFk ^
fc-0
(-l) i+ "V2j + lV{jij 2 j;mim 2 - m) (5) and
F n +i(x) = a(n,x)F n (x) + f3(n,x)F n -i(x),
(6)
Cliff Random Number Generator
where the CfcS are known, then define
VN+2 = Vn+i =
y k = a(/c, x)y k +i + 0{k + 1, x)y k+2 + c k
for k ~ N, N - 1, . . . and solve backwards to obtain y 2
and yi.
Cfc = J/* - a(fe, ^)y fc+ i - /?(fc + 1, x)y fc +2
N
f(x) = ^2c k F k (x)
fc=0
- coFo(x) + [t/i - a(l,x)y 2 - /3(2,x)y 3 ]F 1 (x)
+ [y 2 - a(2,x)y<i - (3(3,x)y4]F 2 (x)
+ [ys - a(3,x)y 4 - j3(4,x)y5]F s {x)
+ [y 4 - a(4, a) 3/5 - /3(5, x)y 6 ]i ? 4(x) + . . .
= c Fo(x) + yi Fi (a:) + y 2 [F 2 (x) - a(l, z)Pi(z)]
+ ys[F 3 (x) - a(2, z)P 2 (:r) - 0(2, a)]
+ 2/ 4 [F 4 (x) - a(3,z)F 3 (x) - 0(3, x)] + . . .
= c Fo(x) + y2[{a(l,x)F 1 ( : r)+/?(l,x)Fo( : r)}
~a{l t x)F 1 (x)]+yiF 1 (x)
= c F Q {x) + yiFi(a) + 0(l,x)F o (x)y 2 .
The upward Clenshaw recurrence formula is
y-2 = y-i =
_ 1
y *~/?(fc+l,x)
for fe = 0, 1,..., N - 1.
[y fc _ 2 - a(k 1 x)y k -i - c k ]
f(x) = c N F N {x) - P(N ) x)F N -i(x)y N -i - F N (x)y N - 2 .
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Recurrence Relations and Clenshaw's Recur-
rence Formula." §5.5 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 172-178, 1992.
Cliff Random Number Generator
A Random Number generator produced by iterating
X n+ i = 1 100 In X„ (mod 1)|
for a Seed X = 0.1. This simple generator passes
the NOISE SPHERE test for randomness by showing no
structure.
see also RANDOM NUMBER, SEED
References
Pickover, C. A. "Computers, Randomness, Mind, and In-
finity." Ch. 31 in Keys to Infinity. New York: W. H.
Freeman, pp. 233-247, 1995.
Clique Number 271
Clifford Algebra
Let V be an n-D linear Space over a Field K, and let Q
be a Quadratic Form on V. A Clifford algebra is then
defined over the T{V)/I(Q), where T(V) is the tensor
algebra over V and I is a particular Ideal of T(V).
References
Iyanaga, S. and Kawada, Y. (Eds.). "Clifford Algebras." §64
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 220-222, 1980.
Lounesto, P. "Counterexamples to Theorems Published and
Proved in Recent Literature on Clifford Algebras, Spinors,
Spin Groups, and the Exterior Algebra." http://www.hit.
f i/~lounesto/counterexamples .htm.
Clifford's Circle Theorem
Let Ci, <7 2 , C 3 , and C 4 be four CIRCLES of GENERAL
POSITION through a point P. Let Pij be the second
intersection of the CIRCLES C» and Cj. Let dj k be
the Circle PijP ik Pjk- Then the four Circles P234,
Pi34, P124, and P123 all pass through the point P1234.
Similarly, let C 5 be a fifth CIRCLE through P. Then the
five points P2345, P1345, P1245, A235 and P1234 all lie on
one Circle C12345. And so on.
see also CIRCLE, Cox's THEOREM
Clifford's Curve Theorem
The dimension of a special series can never exceed half
its order.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 263, 1959.
Clique
In a Graph of N Vertices, a subset of pairwise ad-
jacent Vertices is known as a clique. A clique is a
fully connected subgraph of a given graph. The prob-
lem of finding the size of a clique for a given GRAPH is
an NP-Complete Problem. The number of graphs on
n nodes having 3 cliques are 0, 0, 1, 4, 12, 31, 67, ...
(Sloane's A005289).
see also Clique Number, Maximum Clique Prob-
lem, Ramsey Number, Turan's Theorem
References
Sloane, N. J. A. Sequence A005289/M3440 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Clique Number
The number of VERTICES in the largest CLIQUE of G,
denoted u)(G). For an arbitrary GRAPH,
^— ' n-di
where di is the DEGREE of VERTEX i.
References
Aigner, M. "Turan's Graph Theorem." Amer. Math.
Monthly 102, 808-816, 1995.
272
Clock Solitaire
Closure
Clock Solitaire
A solitaire game played with Cards. The chance of
winning is 1/13, and the AVERAGE number of CARDS
turned up is 42.4.
References
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, pp. 244-247, 1978.
Close Packing
see Sphere Packing
Closed Curve
closed curves open curves
A CURVE with no endpoints which completely encloses
an AREA. A closed curve is formally denned as the con-
tinuous Image of a Closed Set.
see also SIMPLE CURVE
Closed Curve Problem
Find Necessary and Sufficient conditions that de-
termine when the integral curve of two periodic func-
tions k(s) and t(s) with the same period L is a CLOSED
Curve.
Closed Disk
An n-D closed disk of Radius r is the collection of points
of distance < r from a fixed point in EUCLIDEAN n-
space.
see also Disk, Open Disk
Closed Form
A discrete FUNCTION A(n,k) is called closed form (or
sometimes "hypergeometric" ) in two variables if the ra-
tios A(n-rl,k)/A(n, k) and A(n,k-\-l)/A(n i k) are both
Rational Functions. A pair of closed form functions
(F, G) is said to be a Wilf-Zeilberger Pair if
F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k).
see also Rational Function, Wilf-Zeilberger Pair
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 141, 1996.
Zeilberger, D. "Closed Form (Pun Intended!)." Contempo-
rary Math. 143, 579-607, 1993.
Closed Graph Theorem
A linear Operator between two Banach Spaces is
continuous IFF it has a "closed" GRAPH.
see also Banach SPACE
References
Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Closed Interval
An Interval which includes its Limit Points. If the
endpoints of the interval are Finite numbers a and b,
then the Interval is denoted [a, 6]. If one of the end-
points is ±oo, then the interval still contains all of its
Limit Points, so [a, oo) and ( — 00,6] are also closed
intervals.
see also Half-Closed Interval, Open Interval
Closed Set
There are several equivalent definitions of a closed Set.
A Set S is closed if
1. The Complement of S is an Open Set,
2. S is its own CLOSURE,
3. Sequences/nets/filters in S which converge do so
within 5,
4. Every point outside S has a NEIGHBORHOOD disjoint
from 5.
The Point-Set Topological definition of a closed set
is a set which contains all of its Limit POINTS. There-
fore, a closed set C is one for which, whatever point x
is picked outside of C, x can always be isolated in some
Open Set which doesn't touch C.
see also CLOSED INTERVAL
Closure
A Set S and a Binary Operator * are said to ex-
hibit closure if applying the Binary Operator to two
elements S returns a value which is itself a member of
S.
The term "closure" is also used to refer to a "closed"
version of a given set. The closure of a Set can be
denned in several equivalent ways, including
1. The Set plus its Limit Points, also called "bound-
ary" points, the union of which is also called the
"frontier,"
2. The unique smallest CLOSED Set containing the
given Set,
3. The Complement of the interior of the Comple-
ment of the set,
4. The collection of all points such that every NEIGH-
BORHOOD of them intersects the original Set in a
nonempty SET.
In topologies where the T2-Separation Axiom is as-
sumed, the closure of a finite Set S is S itself.
Clothoid
Cobordism
273
see also Binary Operator, Existential Closure,
Reflexive Closure, Tight Closure, Transitive
Closure
Clothoid
see also CORNU SPIRAL
Clove Hitch
A Hitch also called the Boatman's Knot or Peg
Knot.
References
Owen, P. Knots. Philadelphia, PA: Courage, pp. 24-27, 1993.
Clump
see Run
Cluster
Given a lattice, a cluster is a group of filled cells which
are all connected to their neighbors vertically or hori-
zontally.
see also Cluster Perimeter, Percolation Theory,
s-Cluster, s-Run
References
StaufFer, D. and Aharony, A. Introduction to Percolation
Theory, 2nd ed. London: Taylor & Francis, 1992.
Cluster Perimeter
The number of empty neighbors of a CLUSTER.
see also PERIMETER POLYNOMIAL
Coanalytic Set
A Definable Set which is the complement of an An-
alytic Set.
see also Analytic Set
Coastline Paradox
Determining the length of a country's coastline is not
as simple as it first appears, as first considered by
L. F. Richardson (1881-1953). In fact, the answer de-
pends on the length of the RULER you use for the mea-
surements. A shorter RULER measures more of the sin-
uosity of bays and inlets than a larger one, so the esti-
mated length continues to increase as the Ruler length
decreases.
In fact, a coastline is an example of a Fractal, and
plotting the length of the Ruler versus the measured
length of the coastline on a log-log plot gives a straight
line, the slope of which is the FRACTAL DIMENSION of
the coastline (and will be a number between 1 and 2).
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 29-
31, 1991.
Coates- Wiles Theorem
In 1976, Coates and Wiles showed that Elliptic
Curves with Complex Multiplication having an in-
finite number of solutions have //-functions which are
zero at the relevant fixed point. This is a special case of
the Swinnerton-Dyer Conjecture.
References
Cipra, B. "Fermat Prover Points to Next Challenges." Sci-
ence 271, 1668-1669, 1996.
Coaxal Circles
Circles which share a Radical Line with a given cir-
cle are said to be coaxal. The centers of coaxal circles
are COLLINEAR. It is possible to combine the two types
of coaxal systems illustrated above such that the sets
are orthogonal.
see also Circle, Coaxaloid System, Gauss-
Bodenmiller Theorem, Radical Line
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 35-36 and 122,
1967.
Dixon, R. Mathographics. New York: Dover, pp, 68-72, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 34-37, 199, and 279, 1929.
Coaxal System
A system of COAXAL CIRCLES.
Coaxaloid System
A system of circles obtained by multiplying each Radius
in a Coaxal System by a constant.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 276-277, 1929.
Cobordant Manifold
Two open MANIFOLDS M and M' are cobordant if there
exists a MANIFOLD with boundary W n+1 such that an
acceptable restrictive relationship holds.
see also COBORDISM, /i-COBORDISM THEOREM, MORSE
Theory
Cobordism
see Bordism, /i-Cobordism
274 Cobordism Group
Code
Cobordism Group
see Bordism Group
Cobordism Ring
see Bordism Group
with Inversion Center at the Origin and inversion
radius k is the QuADRATRIX OF HlPPIAS.
x = kt cot
y = kt.
(2)
(3)
Cochleoid
The cochleoid, whose name means "snail-form" in Latin,
was first discussed by J. Peck in 1700 (MacTutor Ar-
chive). The points of contact of PARALLEL TANGENTS
to the cochleoid lie on a Strophoid.
In Polar Coordinates,
asin#
In Cartesian Coordinates,
(x 2 + 2/ 2 )tan- 1 (|)
The Curvature is
_ 2y / 2l9 3 [2l9-sin(2fl)]
ay.
[1 + 20 2 - cos(2(9) - 2(9 sin(2<9)] 3 / 2 '
(1)
(2)
(3)
see also QUADRATRIX OF HlPPIAS
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 192 and 196, 1972.
MacTutor History of Mathematics Archive. "Cochleoid."
http: //www-groups . dcs . st-and.ac . uk/ -history /Curves
/Cochleoid. html.
Cochleoid Inverse Curve
The Inverse Curve of the Cochleoid
Cochloid
see Conchoid of Nicomedes
Cochran's Theorem
The converse of FISHER'S THEOREM.
Cocked Hat Curve
The Plane Curve
(x 2 + 2ay - a 2 ) 2 = y 2 (a 2 - x 2 ),
which is similar to the BlCORN.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Cocktail Party Graph
(i)
A Graph consisting of two rows of paired nodes in which
all nodes but the paired ones are connected with an
Edge. It is the complement of the Ladder Graph.
Coconut
see Monkey and Coconut Problem
Codazzi Equations
see MAINARDI-CODAZZI EQUATIONS
Code
A code is a set of n-tuples of elements ("WORDS") taken
from an ALPHABET.
see also Alphabet, Coding Theory, Encoding,
Error-Correcting Code, Gray Code, Huffman
Coding, ISBN, Linear Code, Word
Codimension
Coin
275
Codimension
The minimum number of parameters needed to fully de-
scribe all possible behaviors near a nonstructurally sta-
ble element.
see also BIFURCATION
Coding Theory
Coding theory, sometimes called ALGEBRAIC CODING
THEORY, deals with the design of ERROR-CORRECTING
CODES for the reliable transmission of information
across noisy channels. It makes use of classical and
modern algebraic techniques involving Finite Fields,
Group Theory, and polynomial algebra. It has con-
nections with other areas of DISCRETE MATHEMATICS,
especially Number Theory and the theory of experi-
mental designs.
see also Encoding, Error-Correcting Code, Ga-
lois Field, Hadamard Matrix
References
Alexander, B. "At the Dawn of the Theory of Codes." Math.
Intel 15, 20-26, 1993.
Golomb, S. W.; Peile, R. E.; and Scholtz, R. A. Basic Con-
cepts in Information Theory and Coding: The Adventures
of Secret Agent 00111. New York: Plenum, 1994.
Humphreys, O. F. and Prest, M. Y. Numbers, Groups, and
Codes. New York: Cambridge University Press, 1990.
MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-
Correcting Codes. New York: Elsevier, 1978.
Roman, S. Coding and Information Theory. New York:
Springer- Verlag, 1992.
Coefficient
A multiplicative factor (usually indexed) such as one of
the constants ai in the Polynomial a n x n + a n -i£ n_1 4-
. . . + aix 2 -f- a\x + a .
see also Binomial Coefficient, Cartan Tor-
sion Coefficient, Central Binomial Coeffi-
cient, Clebsch-Gordon Coefficient, Coeffi-
cient Field, Commutation Coefficient, Con-
nection Coefficient, Correlation Coefficient,
Cross-Correlation Coefficient, Excess Coef-
ficient, Gaussian Coefficient, Lagrangian Co-
efficient, Multinomial Coefficient, Pearson's
Skewness Coefficients, Product-Moment Co-
efficient of Correlation, Quartile Skewness
Coefficient, Quartile Variation Coefficient,
Racah V-Coefficient, Racah ^-Coefficient, Re-
gression Coefficient, Roman Coefficient, Tri-
angle Coefficient, Undetermined Coefficients
Method, Variation Coefficient
Coercive Functional
A bilinear FUNCTIONAL <j> on a normed SPACE E is called
coercive (or sometimes Elliptic) if there exists a POS-
ITIVE constant K such that
<i>(x,x)>K\\x\\ 2
for all x £ E.
see also Lax-Milgram Theorem
References
Debnath, L. and Mikusinski, P. Introduction to Hilbert
Spaces with Applications. San Diego, CA: Academic Press,
1990.
Cofactor
The Minor of a Determinant is another Determi-
nant |C| formed by omitting the ith row and jth col-
umn of the original DETERMINANT |M|.
dj = (-l) i+J 'oiM y .
see also Determinant Expansion by Minors, Minor
Cohen-Kung Theorem
Guarantees that the trajectory of Langton's Ant is
unbounded.
Cohomology
Cohomology is an invariant of a TOPOLOGICAL SPACE,
formally "dual" to HOMOLOGY, and so it detects "holes'*
in a SPACE. Cohomology has more algebraic structure
than Homology, making it into a graded ring (multi-
plication given by "cup product"), whereas HOMOLOGY
is just a graded Abelian Group invariant of a Space.
A generalized homology or cohomology theory must sat-
isfy all of the Eilenberg-Steenrod Axioms with the
exception of the dimension axiom.
see also Aleksandrov-Cech Cohomology, Alexan-
der-Spanier Cohomology, Cech Cohomology, de
Rham Cohomology, Homology (Topology)
Cohomotopy Group
Cohomotopy groups are similar to HOMOTOPY GROUPS.
A cohomotopy group is a Group related to the Homo-
topy classes of Maps from a Space X into a Sphere
see also HOMOTOPY GROUP
Coefficient Field
Let V be a Vector Space over a Field K, and let A be
a nonempty Set. For an appropriately defined Affine
Space A, K is called the Coefficient field.
Coin
A flat disk which acts as a two-sided Die.
see Bernoulli Trial, Cards, Coin Paradox, Coin
Tossing, Dice, Feller's Coin-Tossing Constants,
Four Coins Problem, Gambler's Ruin
References
Brooke, M. Fun for the Money. New York: Scribner's, 1963.
276 Coin Flipping
Coin Tossing
Coin Flipping
see Coin Tossing
Coin Paradox
After a half rotation of the coin on the left around the
central coin (of the same RADIUS), the coin undergoes
a complete rotation.
References
Pappas, T. "The Coin Paradox." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 220, 1989.
Coin Problem
Let there be n > 2 INTEGERS < a\ < . . . < a n with
(ai,a 2 ,...,a n ) = 1 (all Relatively Prime). For large
enough N = X^-i a i x ii there is a solution in NoNNEG-
ATIVE INTEGERS xi. The greatest N — g(ai,a 2 , ...a n )
for which there is no solution is called the coin problem.
Sylvester showed
g(ai,a 2 ) - {a\ - l)(o 3 - 1) - 1,
and an explicit solution is known for n — 3, but no
closed form solution is known for larger N.
References
Guy, R. K. "The Money- Changing Problem." §C7 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 113-114, 1994.
Coin Tossing
An idealized coin consists of a circular disk of zero thick-
ness which, when thrown in the air and allowed to fall,
will rest with either side face up ("heads" H or "tails" T)
with equal probability. A coin is therefore a two-sided
Die. A coin toss corresponds to a Bernoulli Distri-
bution with p = 1/2. Despite slight differences between
the sides and NONZERO thickness of actual coins, the
distribution of their tosses makes a good approximation
to a p = 1/2 Bernoulli Distribution.
There are, however, some rather counterintuitive prop-
erties of coin tossing. For example, it is twice as likely
that the triple TTH will be encountered before THT
than after it, and three times as likely that THH will
precede HTT. Furthermore, it is six times as likely that
HTT will be the first of HTT, TTH, and TTT to oc-
cur (Honsberger 1979). More amazingly still, spinning
a penny instead of tossing it results in heads only about
30% of the time (Paulos 1995).
Let w(n) be the probability that no RUN of three consec-
utive heads appears in n independent tosses of a Coin.
The following table gives the first few values of w{n).
n
w(n)
1
1
1
2
1
3
7
8
4
13
16
5
3
4
Feller (1968, pp. 278-279) proved that
lim w(n)a
n + l
■0,
(1)
vhere
a = f [(136 + 24v / 33) 1/3 - 8(136 + 24v / 33)~ 1/3 - 2]
- 1.087378025.
and
= ^ — — = 1.236839845 . .
4 — 3a
(2)
(3)
The corresponding constants for a RUN of k > 1 heads
are a*, the smallest Positive Root of
and
i -* + (!*)
k =
k + 1
o,
k + 1 — kak
(4)
(5)
These arc modified for unfair coins with P(H) = p and
P(T) = q = 1 - p to a' k , the smallest Positive Root
of
l-z + <2pV +1 -0, (6)
and
& =
P&k
(7)
(k + 1 -ka' k )p
(Feller 1968, pp. 322-325).
see also BERNOULLI DISTRIBUTION, CARDS, COIN,
Dice, Gambler's Ruin, Martingale, Run, Saint
Petersburg Paradox
References
Feller, W. An Introduction to Probability Theory and Its Ap-
plication, Vol. 1, 3rd ed. New York: Wiley, 1968.
Finch, S. u Favorite Mathematical Constants.' 1 http://www.
mathsoft.com/asolve/constant/feller/feller.htnil.
Ford, J. "How Random is a Coin Toss?" Physics Today 36,
40-47, 1983.
Honsberger, R. "Some Surprises in Probability." Ch. 5 in
Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., pp. 100-103, 1979.
Keller, J. B. "The Probability of Heads." Amer. Math.
Monthly 93, 191-197, 1986.
Paulos, J. A. A Mathematician Reads the Newspaper. New
York: BasicBooks, p. 75, 1995.
Peterson, I. Islands of Truth: A Mathematical Mystery
Cruise. New York: W. H. Freeman, pp. 238-239, 1990.
Spencer, J. "Combinatorics by Coin Flipping." Coll. Math.
J., 17, 407-412, 1986.
Coincidence
Collatz Problem
277
Coincidence
A coincidence is a surprising concurrence of events, per-
ceived as meaningfully related, with no apparent causal
connection (Diaconis and Mosteller 1989).
see also Birthday Problem, Law of Truly Large
Numbers, Odds, Probability, Random Number
References
Bogomolny, A. "Coincidence." http://www.cut— the-knot .
com/ do_you_know/coincidence. html.
Falk, R. "On Coincidences." Skeptical Inquirer 6, 18—31,
1981-82.
Falk, R. "The Judgment of Coincidences: Mine Versus
Yours." Amer. J. Psych. 102, 477-493, 1989.
Falk, R. and MacGregor, D. "The Surprisingness of Coinci-
dences." In Analysing and Aiding Decision Processes (Ed.
P. Humphreys, O. Svenson, and A. Vari). New York: El-
sevier, pp. 489-502, 1984.
Diaconis, P. and Mosteller, F. "Methods of Studying Coinci-
dences." J. Amer. Statist. Assoc. 84, 853-861, 1989.
Jung, C. G. Synchronicity: An Acausal Connecting Princi-
ple. Princeton, NJ: Princeton University Press, 1973.
Kammerer, P. Das Gesetz der Serie: Eine Lehre von
den Wiederholungen im Lebens — und im Weltgeschehen.
Stuttgart, Germany: Deutsche Verlags-Anstahlt, 1919.
Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96,
June 1998.
Colatitude
The polar angle on a SPHERE measured from the North
Pole instead of the equator. The angle <j> in SPHERICAL
Coordinates is the Colatitude. It is related to the
Latitude 5 by <p = 90° - S.
see also LATITUDE, LONGITUDE, SPHERICAL COORDI-
NATES
Colinear
see COLLINEAR
Collatz Problem
A problem posed by L. Collatz in 1937, also called the
3x + 1 Mapping, Hasse's Algorithm, Kakutani's
Problem, Syracuse Algorithm, Syracuse Prob-
lem, Thwaites Conjecture, and Ulam's Problem
(Lagarias 1985). Thwaites (1996) has offered a £1000
reward for resolving the Conjecture. Let n be an In-
teger. Then the Collatz problem asks if iterating
fin)
i 1
I 3
3n+l
for n even
for n odd
(i)
always returns to 1 for POSITIVE n. This question
has been tested and found to be true for all numbers
< 5.6 x 10 13 (Leavens and Vermeulen 1992), and more
recently, 10 15 (Vardi 1991, p. 129). The members of
the SEQUENCE produced by the Collatz are sometimes
known as Hailstone NUMBERS. Because of the dif-
ficulty in solving this problem, Erdos commented that
"mathematics is not yet ready for such problems" (La-
garias 1985). If NEGATIVE numbers are included, there
are four known cycles (excluding the trivial cycle): (4,
2, 1), (-2, -1), (-5, -7, -10), and (-17, -25, -37,
-55, -82, -41, -61, -91, -136, -68, -34). The num-
ber of tripling steps needed to reach 1 for n = 1, 2, ...
are 0, 0, 2, 0, 1, 2, 5, 0, 6, . . . (Sloane's A006667).
The Collatz problem was modified by Terras (1976,
1979), who asked if iterating
T(x)
-{I
X
(Sx + 1)
for x even
for x odd
(2)
always returns to 1. If NEGATIVE numbers are included,
there are 4 known cycles: (1, 2), (-1), (-5, -7, -10),
and (-17, -25, -37, -55, -82, -41, -61, -91, -136,
—68, —34). It is a special case of the "generalized Collatz
problem" with d = 2, mo = 1, mi = 3, ro — 0, and
ri = -1. Terras (1976, 1979) also proved that the set
of Integers Sk = {n : n has stopping time < k} has a
limiting asymptotic density F(h), so the limit
F(k)= lim -,
a:-»oo X
(3)
for {n : n < x and cr(n) < k} exists. Furthermore,
F(k) — >- 1 as k -4 oo, so almost all INTEGERS have a
finite stopping time. Finally, for all k > 1,
1 - F(k)
lim - < 2
£->00 X
-T]k
where
7] = 1-H(0) = 0.05004...
H (x) = —x lg x — (1 — x) lg(l — x)
"Si
(4)
(5)
(6)
(7)
(Lagarias 1985).
Conway proved that the original Collatz problem has
no nontrivial cycles of length < 400. Lagarias (1985)
showed that there are no nontrivial cycles with length
< 275,000. Conway (1972) also proved that Collatz-
type problems can be formally Undecidable.
A generalization of the COLLATZ PROBLEM lets d > 2 be
a Positive Integer and mo, . . . , md-i be Nonzero
Integers. Also let r»eZ satisfy
n = irfii (mod d) .
Then
T(x) =
mix — Ti
(8)
(9)
for x = i (mod d) defines a generalized Collatz mapping.
An equivalent form is
w-L'r
+ x t
(10)
278
Collatz Problem
Collineation
for x = i (mod d) where Xo, . . . , Xd-\ are INTEGERS
and [r\ is the FLOOR FUNCTION. The problem is con-
nected with Ergodic Theory and Markov Chains
(Matthews 1995). Matthews (1995) obtained the fol-
lowing table for the mapping
Tk(x)
\i(3x
for x = (mod 2)
+ k) for x = 1 (mod 2),
(11)
where k = T*\
k
# Cycles
Max.
Cycle Length
5
27
1
10
34
2
13
118
3
17
118
4
19
118
5
21
165
6
23
433
Matthews and Watts (1984) proposed the following con-
jectures.
1. If | mo ■ ■ -rrid-il < d d , then all trajectories {T K (n)}
for n € Z eventually cycle.
2. If |mo---md-i| > <2 d , then almost all trajectories
{T K (n)} for n € Z are divergent, except for an ex-
ceptional set of Integers n satisfying
#{n £S\-X<n<X} = o(X).
3. The number of cycles is finite.
4. If the trajectory {T K (n)} for n 6 Z is not eventually
cyclic, then the iterates are uniformly distribution
mod d a for each a > 1, with
1
lim
iv^oo AT+ 1
card{if < N\T K (n) = j (mod d a )}
(12)
for < j < d a - 1.
Matthews believes that the map
T(x)
"{i
v 3
7a: + 3
(7a: + 2)
3^-2)
for x = (mod 3)
for x = 1 (mod 3)
for x = 2 (mod 3)
(13)
will either reach (mod 3) or will enter one of the cycles
( — 1) or (-2,-4), and offers a $100 (Australian?) prize
for a proof.
see also HAILSTONE Number
References
Applegate, D. and Lagarias, J. C. "Density Bounds for the
3z + 1 Problem 1. Tree-Search Method." Math. Comput
64, 411-426, 1995.
Applegate, D. and Lagarias, J. C. "Density Bounds for the
Sx + 1 Problem 2. Krasikov Inequalities." Math. Comput.
64, 427-438, 1995.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Burckel, S. "Functional Equations Associated with Congru-
ential Functions." Theor. Comp. Set. 123, 397-406, 1994.
Conway, J. H. "Unpredictable Iterations." Proc. 1972 Num-
ber Th. Conf., University of Colorado, Boulder, Colorado,
pp. 49-52, 1972.
Crandall, R. "On the ( 3z + 1' Problem." Math. Comput 32,
1281-1292, 1978.
Everett, C. "Iteration of the Number Theoretic Function
f(2n) = n, f(2n + 1) = f(3n + 2)." Adv. Math. 25,
42-45, 1977.
Guy, R. K. "Collatz's Sequence." §E16 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 215-218, 1994.
Lagarias, J. C. "The 3x + l Problem and Its Generalizations."
Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm,
sfu. ca/organics/papers/lagarias/.
Leavens, G. T. and Vermeulen, M. "3x + l Search Programs."
Comput. Math. Appl. 24, 79-99, 1992.
Matthews, K. R. "The Generalized 3x+l Mapping." http://
www.maths.uq.oz.au/-krm/survey.dvi. Rev. Sept. 10,
1995.
Matthews, K. R. "A Generalized 3z + 1 Conjecture." [$100
Reward for a Proof.] ftp://www.maths.uq.edu.au/pub/
krm/gnubc/challenge.
Matthews, K. R. and Watts, A. M. "A Generalization of
Hasses's Generalization of the Syracuse Algorithm." Acta
Arith. 43, 167-175, 1984.
Sloane, N. J. A. Sequence A006667/M0019 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Terras, R. "A Stopping Time Problem on the Positive Inte-
gers." Acta Arith. 30, 241-252, 1976.
Terras, R. "On the Existence of a Density." Acta Arith. 35,
101-102, 1979.
Thwaites, B. "Two Conjectures, or How to win £1100."
Math.Gaz. 80, 35-36, 1996.
Vardi, I. "The 3# + 1 Problem." Ch. 7 in Computational
Recreations in Mathematica. Redwood City, CA: Addison-
Wesley, pp. 129-137, 1991.
Collinear
Three or more points Pi, P2, P3, . .., are said to be
collinear if they lie on a single straight LINE L. (Two
points are always collinear.) This will be true IFF the
ratios of distances satisfy
X2 - xi : y 2 - yi : Z2 - zi = x 3 - xi : y 3 — yi : zs - zi.
Two points are trivially collinear since two points deter-
mine a Line.
see also Concyclic, Directed Angle, N-Cluster,
Sylvester's Line Problem
Collineation
A transformation of the plane which transforms COL-
LINEAR points into COLLINEAR points. A projective
collineation transforms every 1-D form projectively, and
a perspective collineation is a collineation which leaves
all lines through a point and points through a line invari-
ant. In an ELATION, the center and axis are incident; in
Cologarithm
Combination 279
a HOMOLOGY they are not. For further discussion, see
Coxeter (1969, p. 248).
see also Affinity, Correlation, Elation, Equi-
affinity, Homology (Geometry), Perspective
Collineation, Projective Collineation
References
Coxeter, H. S. M. "Collineations and Correlations." §14.6
in Introduction to Geometry, 2nd ed. New York: Wiley,
pp. 247-251, 1969.
Cologarithm
The Logarithm of the Reciprocal of a number, equal
to the Negative of the Logarithm of the number it-
self,
colog x = log ( — J — — log x.
see also Antilogarithm, Logarithm
Colon Product
Let AB and CD be Dyads. Their colon product is
defined by
AB : CD = C AB D = (A C)(B D).
Colorable
Color each segment of a KNOT DIAGRAM using one of
three colors. If
1. at any crossing, either the colors are all different or
all the same, and
2. at least two colors are used,
then a KNOT is said to be colorable (or more specif-
ically, Three- Colorable). Color ability is invariant
under REIDEMEISTER Moves, and can be generalized.
For instance, for five colors 0, 1, 2, 3, and 4, a KNOT is
five-colorable if
1. at any crossing, three segments meet. If the overpass
is numbered a and the two underpasses B and C,
then 2a = b -f c (mod 5), and
2. at least two colors are used.
Colorability cannot alway distinguish HANDEDNESS.
For instance, three-colorability can distinguish the mir-
ror images of the TREFOIL KNOT but not the FlGURE-
OF-ElGHT KNOT. Five-colorability, on the other hand,
distinguishes the MIRROR Images of the FlGURE-OF-
Eight Knot but not the Trefoil Knot.
see also Coloring, Three-Colorable
Coloring
A coloring of plane regions, Link segments, etc., is an
assignment of a distinct labelling (which could be a
number, letter, color, etc.) to each component. Col-
oring problems generally involve TOPOLOGICAL consid-
erations (i.e., they depend on the abstract study of the
arrangement of objects), and theorems about colorings,
such as the famous Four-Color THEOREM, can be ex-
tremely difficult to prove.
see also COLORABLE, EDGE-COLORING, FOUR-COLOR
Theorem, ^-Coloring, Polyhedron Coloring,
Six-Color Theorem, Three-Colorable, Vertex
Coloring
References
Eppstein, D. "Coloring," http://vvv . ics . uci . edu / -
eppstein/ junkyard/color. html.
Saaty, T. L. and Kainen, P. C The Four-Color Problem:
Assaults and Conquest. New York: Dover, 1986.
Columbian Number
see Self Number
Colunar Triangle
Given a SCHWARZ TRIANGLE (p q r), replacing each
Vertex with its antipodes gives the three colunar
Spherical Triangles
(p q r'),(p q r f ),(p q r),
where
P P
q q'
r r
see also SCHWARZ TRIANGLE, SPHERICAL TRIANGLE
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, p. 112, 1973.
Comb Function
see Shah Function
Combination
The number of ways of picking r unordered outcomes
from n possibilities. Also known as the Binomial Co-
efficient or Choice Number and read "n choose r."
t.Ct. =
rl(n -
where n\ is a FACTORIAL.
see also Binomial Coefficient, Derangement, Fac-
torial, Permutation, Subfactorial
References
Conway, J. H. and Guy, R. K. "Choice Numbers." In The
Book of Numbers. New York: Springer- Verlag, pp. 67-68,
1996.
Ruskey, F. "Information on Combinations of a Set."
http://sue . esc . uvic . ca/~cos/inf /comb/Combinations
Info.html.
280
Combination Lock
Combinatorics
Combination Lock
Let a combination of n buttons be a SEQUENCE of dis-
joint nonempty Subsets of the Set {1, 2, . . . , n}. If
the number of possible combinations is denoted a n , then
a n satisfies the RECURRENCE RELATION
i— n \ /
with ao = 1. This can also be written
2 / , 2 k '
k=0
(1)
(2)
where the definition 0° = 1 has been used. Furthermore,
a n = 2^i n ,fe2 n = ^^^4n,fc2 ,
(3)
fc = l
where A n ,k are EULERIAN NUMBERS. In terms of the
Stirling Numbers of the Second Kind s{n,k),
a n = \, k\s(n,k).
k = l
a n can also be given in closed form as
a n — 2 Ll -n(2)>
(4)
(5)
where Li n (z) is the POLYLOGARITHM. The first few
values of a n for n = 1, 2, ... are 1, 3, 13, 75, 541,
4683, 47293, 545835, 7087261, 102247563, ... (Sloane's
A000670).
The quantity
b n =
satisfies the inequality
1
2(ln2) n
<b n <
(ln2) n *
(6)
(7)
References
Sloane, N. J. A. Sequence A000670/M2952 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Velleman, D. J. and Call, G. S. "Permutations and Combi-
nation Locks." Math. Mag. 68, 243-253, 1995.
Combinatorial Species
see Species
Combinatorial Topology
Combinatorial topology is a special type of Algebraic
Topology that uses Combinatorial methods. For
example, Simplicial Homology is a combinatorial
construction in ALGEBRAIC TOPOLOGY, so it belongs
to combinatorial topology.
see also ALGEBRAIC TOPOLOGY, SlMPLICIAL HOMO-
LOGY, Topology
Combinatorics
The branch of mathematics studying the enumeration,
combination, and permutation of sets of elements and
the mathematical relations which characterize these
properties.
see also Antichain, Chain, Dilworth's Lemma,
Diversity Condition, Erdos-Szekeres Theo-
rem, Inclusion-Exclusion Principle, Kirkman's
Schoolgirl Problem, Kirkman Triple System,
Length (Partial Order), Partial Order, Pigeon-
hole Principle, Ramsey's Theorem, Schroder-
Bernstein Theorem, Schur's Lemma, Sperner's
Theorem, Total Order, van der Waerden's The-
orem, Width (Partial Order)
References
Abramowitz, M. and Stegun, C A. (Eds.). "Combinatorial
Analysis." Ch. 24 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 821-8827, 1972.
Aigner, M. Combinatorial Theory. New York: Springer-
Verlag, 1997.
Bellman, R. and Hall, M. Combinatorial Analysis. Amer.
Math. Soc, 1979.
Biggs, N. L. "The Roots of Combinatorics." Historia Math-
ematica 6, 109-136, 1979.
Bose, R. C. and Manvel, B. Introduction to Combinatorial
Theory. New York: Wiley, 1984.
Brown, K. S. "Combinatorics." http://www.seanet.com/
-ksbrown/icombina.htm.
Cameron, P. J. Combinatorics: Topics, Techniques, Algo-
rithms. New York: Cambridge University Press, 1994.
Cohen, D. Basic Techniques of Combinatorial Theory. New
York: Wiley, 1978.
Cohen, D. E. Combinatorial Group Theory: A Topological
Approach. New York: Cambridge University Press, 1989.
Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi-
natorial Designs. Boca Raton, FL: CRC Press, 1996.
Comtet, L. Advanced Combinatorics. Dordrecht, Nether-
lands: Reidel, 1974.
Coolsaet, K. "Index of Combinatorial Objects." http://www.
hogent.be/~kc/ico/.
Dinitz, J. H. and Stinson, D. R. (Eds.). Contemporary De-
sign Theory: A Collection of Surveys. New York: Wiley,
1992.
Electronic Journal of Combinatorics. http : //www .
combinatorics.org/previousjvolumes.html.
Eppstein, D. "Combinatorial Geometry." http://www.ics.
uci.edu/-eppstein/junkyard/combinatorial.html.
Erickson, M. J. Introduction to Combinatorics. New York:
Wiley, 1996.
Fields, J. "On-Line Dictionary of Combinatorics." http://
math.uic.edu/-fields/dic/.
Godsil, C. D. "Problems in Algebraic Combinatorics." Elec-
tronic J. Combinatorics 2, Fl, 1-20, 1995. http: //www.
combinatorics. org/Volume_2/volume2.html#Fl.
Graham, R. L.; Grotschel, M.; and Lovasz, L. (Eds.). Hand-
book of Combinatorics, 2 vols. Cambridge, MA: MIT
Press, 1996.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd
ed. Reading, MA: Add is on- Wesley, 1994.
Hall, M. Jr. Combinatorial Theory, 2nd ed. New York: Wi-
ley, 1986.
Knuth, D. E. (Ed.). Stable Marriage and Its Relation to
Other Combinatorial Problems. Providence, RI: Amer.
Math. Soc, 1997.
Comma Derivative
Commutation Coefficient 281
Kucera, L. Combinatorial Algorithms. Bristol, England:
Adam Hilger, 1989.
Liu, C. L. Introduction to Combinatorial Mathematics. New-
York: McGraw-Hill, 1968.
MacMahon, P. A. Combinatory Analysis. New York:
Chelsea, 1960.
Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for
Computers and Calculators, 2nd ed. New York: Academic
Press, 1978.
Riordan, J. Combinatorial Identities, reprint ed. with correc-
tions. Huntington, NY: Krieger, 1979.
Riordan, J. An Introduction to Combinatorial Analysis. New
York: Wiley, 1980.
Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ:
Prentice-Hall, 1984.
Rota, G.-C. (Ed.). Studies in Combinatorics. Providence,
RI: Math. Assoc. Amer., 1978.
Ruskey, F. "The (Combinatorial) Object Server." http://
sue.csc.uvic.ca/-cos.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., 1963.
Skiena, S. S. Implementing Discrete Mathematics: Combi-
natorics and Graph Theory with Mathematica. Reading,
MA: Addison- Wesley, 1990.
Sloane, N. J. A. "An On-Line Version of the Encyclopedia
of Integer Sequences." http://www.research.att.com/
-njas/sequences/eisonline.html.
Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer
Sequences. San Diego, CA: Academic Press, 1995.
Street, A. P. and Wallis, W. D. Combinatorial Theory: An
Introduction. Winnipeg, Manitoba: Charles Babbage Re-
search Center, 1977.
Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley,
1995.
van Lint, J. H. and Wilson, R. M. A Course in Combina-
torics. New York: Cambridge University Press, 1992.
Wilf, H. S. Combinatorial Algorithms: An Update. Philadel-
phia, PA: SIAM, 1989.
Comma Derivative
9k dx k
see also COVARIANT DERIVATIVE, SEMICOLON DERIV-
ATIVE
Comma of Didymus
The musical interval by which four fifths exceed a sev-
enteenth (i.e., two octaves and a major third),
(I)
2 2(|) 2^.5
81
80
1.0125,
also called a Syntonic Comma.
see also COMMA OF PYTHAGORAS, DlESIS, SCHISMA
Comma of Pythagoras
The musical interval by which twelve fifths exceed seven
octaves,
ill
2 7
3^
2 19
531441
524288
1.013643265.
Successive CONTINUED FRACTION CONVERGENTS to
log 2/ log (3/2) give increasingly close approximations
m/n of m fifths by n octaves as 1, 2, 5/3, 12/7, 41/24,
53/31, 306/179, 665/389, ... (Sloane's A005664 and
A046102; Jeans 1968, p. 188), shown in bold in the ta-
ble below. All near-equalities of m fifths and n octaves
having
R.
(§r
2^ Om+n
with \R — 1| < 0.02 are given in the following table.
m
n
Ratio
m
n
Ratio
12
7
1.013643265
265
155
1.010495356
41
24
0.9886025477
294
172
0.9855324037
53
31
1.002090314
306
179
0.9989782832
65
38
1.015762098
318
186
1.012607608
94
55
0.9906690375
347
203
0.9875924759
106
62
1.004184997
359
210
1.001066462
118
69
1.017885359
371
217
1.014724276
147
86
0.9927398469
400
234
0.9896568543
159
93
1.006284059
412
241
1.003159005
188
110
0.9814251419
424
248
1.016845369
200
117
0.994814985
453
265
0.9917255479
212
124
1.008387509
465
272
1.005255922
241
141
0.9834766286
477
279
1.018970895
253
148
0.9968944607
494
289
0.9804224033
see also COMMA OF DlDYMUS, DlESIS, SCHISMA
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 257, 1995.
Guy, R. K. "Small Differences Between Powers of 2 and 3."
§F23 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 261, 1994.
Sloane, N. J. A. Sequences A005664 and A046102 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Common Cycloid
see Cycloid
Common Residue
The value of fr, where a = b (mod m), taken to be NON-
NEGATIVE and smaller than m.
see also Minimal Residue, Residue (Congruence)
Commutation Coefficient
A coefficient which gives the difference between partial
derivatives of two coordinates with respect to the other
coordinate,
c ap^ — [^cn^a] = V^e/3 - V^e a .
see also CONNECTION COEFFICIENT
282
Commutative
Compactness Theorem
Commutative
Let A denote an M- algebra, so that A is a VECTOR
Space over R and
A x A ->■ A
(x,y) M- x-y.
Now define
Z = {x e a ; x • y foi some y 6 A / 0},
where € Z. An ASSOCIATIVE R-algebra is commuta-
tive if x • y = y * x for all x, y € A. Similarly, a Ring is
commutative if the MULTIPLICATION operation is com-
mutative, and a LIE ALGEBRA is commutative if the
Commutator [A, B] is for every A and B in the LIE
Algebra.
see also Abelian, Associative, Transitive
References
Finch, S. "Zero Structures in Real Algebras." http://www.
mathsoft.com/asolve/zerodiv/zerodiv.html.
MacDonald, I. G. and Atiyah, M. F. Introduction to Com-
mutative Algebra. Reading, MA: Addison- Wesley, 1969.
Commutative Algebra
An Algebra in which the + operators and x are Com-
mutative.
see also Algebraic Geometry, Grobner Basis
References
MacDonald, I. G. and Atiyah, M. F. Introduction to Com-
mutative Algebra. Reading, MA: Addison-Wesley, 1969.
Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and
Algorithms: An Introduction to Algebraic Geometry and
Commutative Algebra, 2nd ed. New York: Springer-
Verlag, 1996.
Samuel, P. and Zariski, O, Commutative Algebra, Vol. 2.
New York: Springer- Verlag, 1997.
Commutator
Let A, £, . . .be Operators. Then the commutator of
A and B is defined as
[A,B] = AB-BA. (1)
Let a, 6, ... be constants. Identities include
[/(*),*] = (2)
[A,A]=0 (3)
[A,B] = -[B,A] (4)
[A,BC] = [A,B]C + B[A,C] (5)
[AB, C] = [A, C]B + A[B, C] (6)
[a + A,b + B] = [A,B] (7)
[A + B,C + D} = [A,C] + [A,D] + [B,C] + [B,D].
(8)
The commutator can be interpreted as the "infinitesi-
mal" of the commutator of a Lie Group.
Let A and B be Tensors. Then
[A,B]=X? a B-VbA. (9)
see also Anticommutator, Jacobi Identities
Compact Group
If the parameters of a LIE GROUP vary over a CLOSED
Interval, the GROUP is compact. Every representation
of a compact group is equivalent to a UNITARY repre-
sentation.
Compact Manifold
A Manifold which can be "charted" with finitely many
Euclidean Space charts. The Circle is the only com-
pact l-D Manifold. The Sphere and n-ToRUS are
the only compact 2-D MANIFOLDS. It is an open ques-
tion if the known compact MANIFOLDS in 3-D are com-
plete, and it is not even known what a complete list in
4-D should look like. The following terse table there-
fore summarizes current knowledge about the number
of compact manifolds N(D) of D dimensions.
D N(D)
see also Tychonof Compactness Theorem
Compact Set
The Set S is compact if, from any Sequence of ele-
ments Xi, X 2y ...of S, a subsequence can always be
extracted which tends to some limit element X of S.
Compact sets are therefore closed and bounded.
Compact Space
A Topological Space is compact if every open cover
of X has a finite subcover. In other words, if X is the
union of a family of open sets, there is a finite subfamily
whose union is X. A subset A of a Topological Space
X is compact if it is compact as a TOPOLOGICAL Space
with the relative topology (i.e., every family of open
sets of X whose union contains A has a finite subfamily
whose union contains A).
Compact Surface
A surface with a finite number of TRIANGLES in its TRI-
angulation. The Sphere and TORUS are compact,
but the PLANE and TORUS minus a Disk are not.
Compactness Theorem
Inside a Ball B in R 3 ,
{rectifiable currents 5 in BL Area S < c,
length dS < c}
is compact under the Flat Norm.
References
Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Companion Knot
Complete Axiomatic Theory 283
Companion Knot
Let Ki be a knot inside a TORUS. Now knot the TORUS
in the shape of a second knot (called the companion
knot) K2. Then the new knot resulting from K\ is called
the Satellite Knot K 3 .
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots, New York: W. H.
Freeman, pp. 115-118, 1994.
Comparability Graph
The comparability graph of a POSET P = (X, <) is the
Graph with vertex set X for which vertices x and y are
adjacent IFF either x < y or y < x in P.
see also INTERVAL GRAPH, PARTIALLY ORDERED SET
Comparison Test
Let J2 ak and J2^ k be a Series with Positive terms
and suppose a\ < &i, 02 < ta,
1. If the bigger series CONVERGES, then the smaller
series also Converges.
2. If the smaller series DIVERGES, then the bigger series
also Diverges.
see also Convergence Tests
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 280-281, 1985.
Compass
A tool with two arms joined at their ends which can
be used to draw Circles. In Geometric Construc-
tions, the classical Greek rules stipulate that the com-
pass cannot be used to mark off distances, so it must
"collapse" whenever one of its arms is removed from
the page. This results in significant complication in the
complexity of GEOMETRIC CONSTRUCTIONS,
see also Constructible Polygon, Geometric Con-
struction, Geometrography, Mascheroni Con-
struction, Plane Geometry, Polygon, Poncelet-
Steiner Theorem, Ruler, Simplicity, Steiner
Construction, Straightedge
References
Dixon, R. "Compass Drawings." Ch. 1 in Mathographics.
New York: Dover, pp. 1-78, 1991.
Compatible
Let 1 1 A 1 1 be the MATRIX NORM associated with the MA-
TRIX A and ||x|| be the Vector Norm associated with
a Vector x. Let the product Ax be defined, then ||A||
and ||x|| are said to be compatible if
l|Ax||<||A||||x||.
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1115, 1980.
Complement Graph
The complement Graph G of G has the same Vertices
as G but contains precisely those two-element SUBSETS
which are not in G.
Complement Knot
see Knot Complement
Complement Set
Given a set S with a subset F, the complement of E is
defined as
E' = {F:FeS,F^E}. (1)
If E = 5, then
E' = S' = 0,
(2)
where is the EMPTY SET. Given a single Set, the
second Probability Axiom gives
1 = P(S) = P(EUE').
Using the fact that E n E f = 0,
1 = P(E) + P(E')
P(E') = 1-P(E).
(3)
(4)
(5)
This demonstrates that
P(S') = P{0) = 1 - P(S) = 1-1-0. (6)
Given two Sets,
P(E O F') = P(E) - P(E F) (7)
P(E' r\F f ) = l- P(E) - P{F) + P(E O F). (8)
Complementary Angle
Two ANGLES a and 7r/2 - a are said to be complemen-
tary.
see also ANGLE, SUPPLEMENTARY ANGLE
Complete
see Complete Axiomatic Theory, Complete Bi-
graph, Complete Functions, Complete Graph,
Complete Quadrangle, Complete Quadrilat-
eral, Complete Sequence, Complete Space,
Completeness Property, Weakly Complete Se-
quence
Complete Axiomatic Theory
An axiomatic theory (such as a Geometry) is said to be
complete if each valid statement in the theory is capable
of being proven true or false.
see also CONSISTENCY
284 Complete Bigraph
Complete Graph
Complete Bigraph
see Complete Bipartite Graph
Complete Bipartite Graph
Complete Graph
A Bipartite Graph (i.e., a set of Vertices decom-
posed into two disjoint sets such that there are no two
VERTICES within the same set are adjacent) such that
every pair of VERTICES in the two sets are adjacent. If
there are p and q VERTICES in the two sets, the complete
bipartite graph (sometimes also called a COMPLETE Bl-
GRAPH) is denoted K p , q . The above figures show K^^
and i^2,5*
see also Bipartite Graph, Complete Graph,
Complete ^-Partite Graph, ^-Partite Graph,
Thomassen Graph, Utility Graph
References
Saaty, T. L. and Kainen, P. C. The Four-Color Problem;
Assaults and Conquest. New York: Dover, p. 12, 1986.
Complete Functions
A set of Orthonormal Functions </> n (x) is termed
complete in the CLOSED INTERVAL x € [a, b] if, for every
piecewise CONTINUOUS Function f(x) in the interval,
the minimum square error
E n = ||/-(ci0i + ... + c n n )|| 2
(where || denotes the Norm) converges to zero as n be-
comes infinite. Symbolically, a set of functions is com-
plete if
lim
771— »-00
f
f(x) - y^an4>n(x)
n=Q
w(x) dx — 0,
where w(x) is a Weighting Function and the above
is a Lebesgue Integral.
see also BESSEL'S INEQUALITY, HlLBERT SPACE
References
Arfken, G. "Completeness of Eigenfunctions." §9.4 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 523-538, 1985.
A Graph in which each pair of VERTICES is connected
by an EDGE. The complete graph with n VERTICES is
denoted K n . In older literature, complete GRAPHS are
called UNIVERSAL GRAPHS.
K 4 is the Tetrahedral Graph and is therefore a PLA-
NAR GRAPH. K$ is nonplanar. Conway and Gordon
(1983) proved that every embedding of K G is INTRINSI-
CALLY Linked with at least one pair of linked triangles.
They also showed that any embedding of Kj contains a
knotted Hamiltonian Cycle.
The number of Edges in K v is v(v — l)/2, and the
Genus is (v — 3)(v — 4)/12 for v > 3. The number of dis-
tinct variations for K n (GRAPHS which cannot be trans-
formed into each other without passing nodes through
an EDGE or another node) for n — 1, 2, . . . are 1, 1, 1,
1, 1, 1, 6, 3, 411, 37, The Adjacency Matrix of
the complete graph takes the particularly simple form
of all Is with Os on the diagonal.
It is not known in general if a set of Trees with 1,2,...,
n — 1 Edges can always be packed into K n . However,
if the choice of TREES is restricted to either the path or
star from each family, then the packing can always be
done (Zaks and Liu 1977, Honsberger 1985).
References
Chartrand, G. Introductory Graph Theory. New York:
Dover, pp. 29-30, 1985.
Conway, J. H. and Gordon, C. M. "Knots and Links in Spatial
Graphs." J. Graph Th. 7, 445-453, 1983.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 60-63, 1985.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem,:
Assaults and Conquest. New York: Dover, p. 12, 1986.
Zaks, S. and Liu, C. L. "Decomposition of Graphs into
Trees." Proc. Eighth Southeastern Conference on Com-
binatorics, Graph Theory, and Computing, pp. 643-654,
1977.
Complete k-Partite Graph
Complete fc-Partite Graph
A A;-Partite Graph (i.e., a set of Vertices decom-
posed into k disjoint sets such that no two VERTICES
within the same set are adjacent) such that every pair
of Vertices in the k sets are adjacent. If there are
p, q, . . . , r Vertices in the k sets, the complete bi-
partite graph is denoted i^ P) ^,...,r- The above figure
Shows 1^2,3,5-
see also COMPLETE GRAPH, COMPLETE fc-PARTITE
Graph, ^-Partite Graph
References
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest, New York: Dover, p. 12, 1986.
Complete Metric Space
A complete metric space is a METRIC SPACE in which
every CAUCHY SEQUENCE is CONVERGENT. Examples
include the Real Numbers with the usual metric and
the p-ADic Numbers.
Complete Permutation
see Derangement
Complete Quadrangle
If the four points making up a Quadrilateral are
joined pairwise by six distinct lines, a figure known as
a complete quadrangle results. Note that a complete
quadrilateral is defined differently from a COMPLETE
Quadrangle.
The midpoints of the sides of any complete quadrangle
and the three diagonal points all lie on a CONIC known
as the Nine-Point Conic If it is an Orthocentric
Quadrilateral, the Conic reduces to a Circle. The
Orthocenters of the four Triangles of a complete
quadrangle are COLLINEAR on the RADICAL Line of the
Circles on the diameters of a Quadrilateral.
see also Complete Quadrangle, Ptolemy's Theo-
rem
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 230-231, 1969.
Demir, H. "The Compleat [sic] Cyclic Quadrilateral." Amer.
Math. Monthly 79, 777-778, 1972.
Complete Sequence 285
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 61-62, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 101-104, 1990.
Complete Quadrilateral
The figure determined by four lines and their six points
of intersection (Johnson 1929, pp. 61-62). Note that
this is different from a COMPLETE QUADRANGLE. The
midpoints of the diagonals of a complete quadrilateral
are COLLINEAR (Johnson 1929, pp. 152-153).
A theorem due to Steiner (Mention 1862, Johnson 1929,
Steiner 1971) states that in a complete quadrilateral, the
bisectors of angles are CONCURRENT at 16 points which
are the incenters and EXCENTERS of the four TRIAN-
GLES. Furthermore, these points are the intersections of
two sets of four CIRCLES each of which is a member of
a conjugate coaxal system. The axes of these systems
intersect at the point common to the ClRCUMCIRCLES
of the quadrilateral.
see also COMPLETE QUADRANGLE, GAUSS-BODENMIL-
ler Theorem, Polar Circle
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 230-231, 1969.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 61-62, 149, 152-153, and 255-
256, 1929.
Mention, M. J. "Demonstration d'un Theoreme de
M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 16-20, 1862.
Mention, M. J. "Demonstration d'un Theoreme de
M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 65-67, 1862.
Steiner, J. Gesammelte Werke, 2nd ed, Vol. 1. New York:
Chelsea, p. 223, 1971.
Complete Residue System
A set of numbers clq, cli, ..., a m -i (mod m) form a
complete set of residues if they satisfy
ai = i (mod m)
for i = 0, 1, . . . , m — 1. In other words, a complete
system of residues is formed by a base and a modulus if
the residues r; in b l = Vi (mod m) for i = 1, . . . , m - 1
run through the values 1, 2, ..., m — 1.
see also Haupt-Exponent
Complete Sequence
A Sequence of numbers V — {u n } is complete if every
Positive Integer n is the sum of some subsequence of
V, i.e., there exist a; = or 1 such that
/ v aM
(Honsberger 1985, pp. 123-126). The Fibonacci Num-
bers are complete. In fact, dropping one number still
286
Complete Space
Complex Analysis
leaves a complete sequence, although dropping two num-
bers does not (Honsberger 1985, pp. 123 and 126). The
Sequence of Primes with the element {1} prepended,
{1,2,3,5,7,11,13,17,19,23,...}
is complete, even if any number of Primes each > 7 are
dropped, as long as the dropped terms do not include
two consecutive PRIMES (Honsberger 1985, pp. 127—
128). This is a consequence of BERTRAND'S POSTU-
LATE.
see also Bertrand's Postulate, Brown's Cri-
terion, Fibonacci Dual Theorem, Greedy Al-
gorithm, Weakly Complete Sequence, Zeck-
endorf's Theorem
References
Brown, J. L. Jr. "Unique Representations of Integers as Sums
of Distinct Lucas Numbers." Fib. Quart. 7,243-252,1969.
Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for
Fibonacci Numbers. XIL" Fib. Quart. 11, 317-331, 1973.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., 1985.
Complete Space
A Space of Complete Functions.
see also COMPLETE METRIC SPACE
Completely Regular Graph
A POLYHEDRAL Graph is completely regular if the
Dual Graph is also Regular. There are only five
types. Let p be the number of EDGES at each node, p*
the number of EDGES at each node of the DUAL GRAPH,
V the number of VERTICES, E the number of EDGES,
and F the number of faces in the Platonic Solid cor-
responding to the given graph. The following table sum-
marizes the completely regular graphs.
Type
9
P*
V
E
F
Tetrahedral
3
3
4
6
4
Cubical
3
4
8
12
6
Dodecahedral
3
5
20
39
12
Octahedral
4
3
6
12
8
Icosahedral
5
3
12
30
20
Completeness Property
All lengths can be expressed as Real Numbers.
Completing the Square
The conversion of an equation of the form ax 2 + bx + c
to the form
a { x + ^)
+ ic -4-al'
which, defining B = b/2a and C = c — b 2 /4a, simplifies
to
a(x + B) 2 + C.
Complex
A finite Set of SlMPLEXES such that no two have a
common point. A 1-D complex is called a GRAPH.
see also CW-Complex, Simplicial Complex
Complex Analysis
The study of Complex NUMBERS, their DERIVATIVES,
manipulation, and other properties. Complex analysis is
an extremely powerful tool with an unexpectedly large
number of practical applications to the solution of phys-
ical problems. CONTOUR INTEGRATION, for example,
provides a method of computing difficult INTEGRALS by
investigating the singularities of the function in regions
of the Complex Plane near and between the limits of
integration.
The most fundamental result of complex analysis is the
Cauchy-Riemann Equations, which give the condi-
tions a Function must satisfy in order for a com-
plex generalization of the Derivative, the so-called
Complex Derivative, to exist. When the Complex
Derivative is defined "everywhere," the function is
said to be ANALYTIC. A single example of the unex-
pected power of complex analysis is PlCARD'S Theo-
rem, which states that an Analytic Function as-
sumes every Complex Number, with possibly one ex-
ception, infinitely often in any NEIGHBORHOOD of an
Essential Singularity!
see also ANALYTIC CONTINUATION, BRANCH CUT,
Branch Point, Cauchy Integral Formula, Cau-
chy Integral Theorem, Cauchy Principal Value,
Cauchy-Riemann Equations, Complex Number,
Conformal Map, Contour Integration, de
Moivre's Identity, Euler Formula, Inside-
Outside Theorem, Jordan's Lemma, Laurent Se-
ries, Liouville's Conformality Theorem, Mono-
genic Function, Morera's Theorem, Permanence
of Algebraic Form, Picard's Theorem, Pole,
Polygenic Function, Residue (Complex Analy-
sis)
References
Arfken, G. "Functions of a Complex Variable I: Analytic
Properties, Mapping" and "Functions of a Complex Vari-
able II: Calculus of Residues." Chs. 6—7 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 352-395 and 396-436, 1985.
Boas, R. P. Invitation to Complex Analysis. New York: Ran-
dom House, 1987.
Churchill, R. V. and Brown, J. W. Complex Variables and
Applications, 6th ed. New York: McGraw-Hill, 1995.
Conway, J. B. Functions of One Complex Variable, 2nd ed.
New York: Springer- Verlag, 1995.
Forsyth, A. R. Theory of Functions of a Complex Variable,
3rd ed, Cambridge, England: Cambridge University Press,
1918. .
Lang, S. Complex Analysis, 3rd ed. New York: Springer-
Verlag, 1993.
Morse, P. M. and Feshbach, H. "Functions of a Complex Vari-
able" and "Tabulation of Properties of Functions of Com-
plex Variables." Ch. 4 in Methods of Theoretical Physics,
Part I. New York: McGraw-Hill, pp. 348-491 and 480-485,
1953.
Complex Conjugate
Complex Number 287
Complex Conjugate
The complex conjugate of a Complex Number z =
a+bi is defined to be z* = a— hi. The complex conjugate
is Associative, (zi + z 2 )* = zi* + z 2 *, since
(ai H- M)* + (a 2 + M)* — ai - ibi + a 2 - i&2
= (ai - ibi) + (a 2 - ib 2 )
= (ai+6i)* + (a 2 + b 2 )*,
and Distributive, (ziz 2 ) m = zi*z 2 *, since
[(ai + bii)(a 2 + 62*)]* = [( a i a 2 - 6162) + i(ai& 2 + 0261)]*
= (ai(X2 — &ifr 2 ) — i(ai6 2 + a 2 6i)
= (ai - z6i)(a 2 - i6 2 )
— (ai + i6i)*(a2 + 162)*.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 16, 1972.
Complex Derivative
A Derivative of a Complex function, which must sat-
isfy the Cauchy-Riemann Equations in order to be
Complex Differentiable.
see also Cauchy-Riemann Equations,
Differentiable, Derivative
Complex
Complex Differentiable
If the Cauchy-Riemann Equations are satisfied for a
function f(x) = u(x) + iv(x) and the PARTIAL DERIVA-
TIVES of u(x) and v(x) are Continuous, then the Com-
plex Derivative df/dz exists.
see also Analytic Function, Cauchy-Riemann
Equations, Complex Derivative, Pseudoanalytic
Function
Complex Function
A Function whose Range is in the Complex Num-
bers is said to be a complex function.
see also Real Function, Scalar Function, Vector
Function
Complex Matrix
A Matrix whose elements may contain Complex Num-
bers. The Matrix Product of two 2x2 complex
matrices is given by
xu + 2/i 1* Z12 + y 12 i
£21 + V2ii £22 + 2/22*
uu -\-Vni
U12
+ V121
U21 -\-v21i
1*22 + ^22^
R11 R12
-H
111 1 12
R21 R22
hi 1
22
where
R11 — u\\x\\ + u 2 ixi 2 — viij/11 — v 2 iyi 2
Rl2 — Wl2Xll + ^22^12 - V122/11 - U222/12
R 2 1 = U11X21 + U21X22 - Ul 12/21 - V21J/22
R 22 = Ui 2 X 2 ± + u 22 x 22 — vi 2 y 2 i — V 222/22
In = vnxii + ^21X12 + wnyii + U21IJ12
111 = V12X11 + ^22^12 + U122/11 + ^222/12
^21 = ^113521 + ^21^22 + U112/21 + ^212/22
i~22 = V\ 2 X 2 1 + V 22 #22 + ^122/21 + ^222/22-
see a/so Real Matrix
Complex Multiplication
Two Complex Numbers x = a + ib and y = c + id are
multiplied as follows:
xy — (a + i&)(c + zd) = ac + ibc + zad — 6d
= (ac - bd) + i(ad -f 6c).
However, the multiplication can be carried out using
only three REAL multiplications, ac, bd, and (a+b)(c-\-d)
as
R[(a + ib)(c + id)] = ac - bd
9f[(a + ifc)(c + id)] = (a + 6)(c + d) - ac - bd.
Complex multiplication has a special meaning for EL-
LIPTIC Curves.
see also Complex Number, Elliptic Curve, Imagi-
nary Part, Multiplication, Real Part
References
Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field
Theory and Complex Multiplication. New York: Wiley,
1997.
Complex Number
The complex numbers are the Field C of numbers of the
form x + iy, where x and y are REAL NUMBERS and i is
the Imaginary Number equal to >/-!• When a single
letter z - x + iy is used to denote a complex number, it
is sometimes called an "AFFIX." The FIELD of complex
numbers includes the Field of Real Numbers as a
Subfield.
Through the Euler FORMULA, a complex number
z = x -f iy (1)
may be written in "PHASOR" form
z = \z\ (cos + i sin 6) = \z\e ie . (2)
Here, \z\ is known as the Modulus and 9 is known as
the Argument or Phase. The Absolute Square of
288 Complex Number
Complex Structure
z is defined by \z\ 2 — zz* , and the argument may be
computed from
Complex Plane
arg(z) — = tan I — J
(3)
de Moivre's Identity relates Powers of complex
numbers
z n = |z| n [cos(n#) + zsin(n#)].
(4)
Finally, the Real R(z) and Imaginary Parts $s(z) are
given by
»w = i(^+o (5)
*(*> = ^^ = ~W - O = 5*(** " *)■ ( 6 )
2z
The Powers of complex numbers can be written in
closed form as follows:
-0
ri-2 2 . I n \ n-4 4
x y + 1 4 p y
+ i
>~v
3 F y +..
(7)
The first few are explicitly
z 2 = (x 2 - y 2 ) -{- i(2xy)
z = (x — 3xy ) + i(3x y — y )
z 4 = (x 4 - 6x 2 y 2 + y 4 ) 4- i(4z 3 y - 4xy 3 )
z 5 = ( x 5 - I0x 3 y 2 + 5zy 4 ) + i{$x A y - 10xV + y 5 )
(8)
(9)
(10)
(11)
(Abramowitz and Stegun 1972).
see also Absolute Square, Argument (Complex
Number), Complex Plane, i, Imaginary Number,
Modulus, Phase, Phasor, Real Number, Surreal
Number
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 16-17, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 353-357, 1985.
Courant, R. and Robbins, H. "Complex Numbers." §2.5 in
What is Mathematics? : An Elementary Approach to Ideas
and Methods, 2nd ed. Oxford, England: Oxford University-
Press, pp. 88-103, 1996.
Morse, P. M. and Feshbach, H. "Complex Numbers and Vari-
ables." §4.1 in Methods of Theoretical Physics, Part I. New
York: McGraw-Hill, pp. 349-356, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Complex Arithmetic." §5.4 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 171-172, 1992.
Imaginary
The plane of COMPLEX Numbers spanned by the vec-
tors 1 and i, where i is the IMAGINARY NUMBER. Every
Complex Number corresponds to a unique Point in
the complex plane. The LINE in the plane with i = is
the Real Line. The complex plane is sometimes called
the Argand Plane or Gauss Plane, and a plot of
Complex Numbers in the plane is sometimes called
an Argand Diagram.
see also AFFINE COMPLEX PLANE, ARGAND DIAGRAM,
Argand Plane, Bergman Space, Complex Projec-
tive Plane
References
Courant, R. and Robbins, H. "The Geometric Interpretation
of Complex Numbers." §5.2 in What is Mathematics?: An
Elementary Approach to Ideas and Methods, 2nd ed. Ox-
ford, England: Oxford University Press, pp. 92-97, 1996.
Complex Projective Plane
The set P 2 is the set of all Equivalence Classes
[a, 6,c] of ordered triples (a, 6, c) E C 3 \(0,0,0) under
the equivalence relation (a, 6, c) ~ (a', &', c') if (a, 6, c) =
(Aa', A6',Ac') for some Nonzero Complex Number A.
Complex Representation
see Phasor
Complex Structure
The complex structure of a point x =
PLANE is defined by the linear MAP J : '.
J{Xi,X 2 ) - (-Z2,Zl),
X\ , X2 in the
and corresponds to a clockwise rotation by rr/2. This
map satisfies
J 2 = -I
(Jx).(Jy) = x.y
( Jx) • x = 0,
where / is the IDENTITY MAP.
More generally, if V is a 2-D Vector SPACE, a linear
map J : V — > V such that J 2 = — I is called a complex
structure on7. If V = M. , this collapses to the previous
definition.
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 3 and 229, 1993.
Complexity (Number)
Complexity (Number)
The number of Is needed to represent an INTEGER us-
ing only additions, multiplications, and parentheses are
called the integer's complexity. For example,
1 = 1
2 = 1 + 1
3=1+1+1
4=(1 + 1)(1 + 1) = 1 + 1 + 1 + 1
5 = (1 + 1)(1 + 1) + 1 = 1 + 1 + 1 + 1 + 1
6 = (1 + 1)(1 + 1 + 1)
7 = (1 + 1)(1 + 1 + 1) + 1
8 = (1 + 1)(1 + 1)(1 + 1)
9=(1 + 1 + 1)(1 + 1 + 1)
10 = (1 + 1 + 1)(1 + 1 + 1) + 1
= (1 + 1)(1 + 1 + 1 + 1 + 1)
So, for the first few n, the complexity is 1, 2, 3, 4, 5, 5,
6, 6, 6, 7, 8, 7, 8, . . . (Sloane's A005245).
References
Guy, R. K. "Expressing Numbers Using Just Ones." §F26 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, p. 263, 1994.
Guy, R. K. "Some Suspiciously Simple Sequences." Amer.
Math. Monthly 93, 186-190, 1986.
Guy, R. K. "Monthly Unsolved Problems, 1969-1987."
Amer. Math. Monthly 94, 961-970, 1987.
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.
Rawsthorne, D. A. "How Many l's are Needed?" Fib. Quart.
27, 14-17, 1989.
Sloane, N. J. A. Sequence A005245/M0457 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Complexity (Sequence)
see Block Growth
Complexity Theory
Divides problems into "easy" and "hard" categories.
A problem is easy and assigned to the P-Problem
(Polynomial time) class if the number of steps needed
to solve it is bounded by some Power of the prob-
lem's size. A problem is hard and assigned to the NP-
PROBLEM (nondeterministic POLYNOMIAL time) class if
the number of steps is not bounded and may grow ex-
ponentially.
However, if a solution is known to an NP-Problem, it
can be reduced to a single period verification. A prob-
lem is NP-Complete if an Algorithm for solving it
can be translated into one for solving any other NP-
Problem. Examples of NP-Complete Problems in-
clude the Hamiltonian Cycle and Traveling Sales-
man Problems. Linear Programming, thought to
be an NP-PROBLEM, was shown to actually be a P-
PROBLEM by L. Khachian in 1979. It is not known if all
apparently NP-PROBLEMS are actually P-PROBLEMS.
Composite Number 289
see also Bit Complexity, NP-Complete Problem,
NP-Problem, P-Problem
References
Bridges, D. S. Computability. New York: Springer- Verlag,
1994.
Brookshear, J. G. Theory of Computation: Formal Lan-
guages, Automata, and Complexity. Redwood City, CA:
Benjamin/Cummings, 1989.
Cooper, S. B.; Slaman, T. A.; and Wainer, S. S. (Eds.). Com-
putability, Enumerability, Unsolvability: Directions in Re-
cursion Theory. New York: Cambridge University Press,
1996.
Garey, M. R. and Johnson, D. S. Computers and Intractabil-
ity: A Guide to the Theory of NP- Completeness. New
York: W. H. Freeman, 1983.
Goetz, P. "Phil Goetz's Complexity Dictionary." http://
www . cs .buf f alo . edu/~goetz/dict .html.
Hopcroft, J. E. and Ullman, J. D. Introduction to Auto-
mated Theory, Languages, and Computation. Reading,
MA: Addison-Wesley, 1979.
Lewis, H. R. and Papadimitriou, C. H. Elements of the
Theory of Computation, 2nd ed. Englewood Cliffs, NJ:
Prentice-Hall, 1997.
Sudkamp, T. A. Language and Machines: An Introduction
to the Theory of Computer Science, 2nd ed. Reading, MA:
Addison-Wesley, 1996.
Welsh, D. J. A. Complexity: Knots, Colourings and Count-
ing. New York: Cambridge University Press, 1993.
Component
A Group L is a component of H if L is a Quasisimple
Group which is a Subnormal Subgroup of H,
see also GROUP, QUASISIMPLE GROUP, SUBGROUP,
Subnormal
Composite Knot
A Knot which is not a Prime Knot. Composite knots
are special cases of Satellite Knots.
see also Knot, Prime Knot, Satellite Knot
Composite Number
A Positive Integer which is not Prime (i.e., which
has FACTORS other than 1 and itself).
A composite number C can always be written as a
Product in at least two ways (since 1 ■ C is always
possible). Call these two products
C = ab = cd.
(i)
then it is obviously the case that C\ab (C divides ab).
Set
c = mn 1 (2)
where m is the part of C which divides a, and n the part
of C which divides n. Then there are p and q such that
a = mp
b = nq.
(3)
(4)
290 Composite Runs
Composition Theorem
Solving ab = cd for d gives
ab __ (mp)(nqr) _
£j — — — pq t
c mn
(5)
It then follows that
S = a 2 + b 2 + c 2 + d 2
2 2. 22. 22. 22
= m p + n q +m n -\- p q
= (m 2 + q 2 )(n 2 +p 2 ).
(6)
It therefore follows that a 2 + b 2 + c 2 + d 2 is never Prime!
In fact, the more general result that
S = a k + 6 fc + c fc -f d k
(7)
is never Prime for k an Integer > also holds (Hons-
berger 1991).
There are infinitely many integers of the form |_(3/2) Tt J
and L(4/3) n J which are composite, where [^J is the
Floor Function (Forman and Shapiro, 1967; Guy
1994, p. 220). The first few composite |_(3/2) n J occur
for n = 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
23, ..., and the the few composite |_(4/3) n J occur for
n = 5, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... .
see also Amenable Number, Grimm's Conjecture,
Highly Composite Number, Prime Factorization
Prime Gaps, Prime Number
References
Forman, W. and Shapiro, H. N. "An Arithmetic Property of
Certain Rational Powers." Comm. Pure AppL Math. 20,
561-573, 1967.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, 1994.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 19-20, 1991.
Sloane, N. J. A. Sequence A002808/M3272 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Composite Runs
see Prime Gaps
Compositeness Certificate
A compositeness certificate is a piece of information
which guarantees that a given number p is COMPOSITE.
Possible certificates consist of a Factor of a number
(which, in general, is much quicker to check by direct
division than to determine initially), or of the determi-
nation that either
a p_1 ^ 1 (modp),
(i.e., p violates Fermat's Little Theorem), or
a ^ —1, 1 and a = 1 (mod p) .
A quantity a satisfying either property is said to be a
Witness to p's compositeness.
see also Adleman-Pomerance-Rumely Primality
Test, Fermat's Little Theorem, Miller's Pri-
mality Test, Primality Certificate, Witness
Compositeness Test
A test which always identifies Prime numbers correctly,
but may incorrectly identify a Composite Number as
a Prime.
see also Primality Test
Composition
The combination of two FUNCTIONS to form a single new
Operator. The composition of two functions / and g
is denoted fog and is defined by
/°S = /(#(#))
when / and g are both functions of x.
An operation called composition is also defined on Bi-
nary Quadratic Forms. For two numbers repre-
sented by two forms, the product can then be repre-
sented by the composition. For example, the composi-
tion of the forms 2x 2 + 15y 2 and 3x 2 + 10y 2 is given by
6x 2 + 5y 2 , and in this case, the product of 17 and 13
would be represented as (6 * 36 + 5 - 1 = 221). There
are several algorithms for computing binary quadratic
form composition, which is the basis for some factoring
methods.
see also ADEM RELATIONS, BINARY OPERATOR, BI-
NARY Quadratic Form
Composition Series
Every Finite GROUP G of order greater than one pos-
sesses a finite series of SUBGROUPS, called a composition
series, such that
/ C H s C . . . C H 2 C H! C G,
where if;+i is a maximal subgroup of Hi. The Quo-
tient Groups G/Hi, H 1 /H 2 , ■•-, H 3 -i/H 3 , H s are
called composition quotient groups.
see also Finite Group, Jordan-Holder Theorem,
Quotient Group, Subgroup
References
Lomont, J. S. Applications of Finite Groups. New York:
Dover, p. 26, 1993.
Composition Theorem
Let
Q( X ,y)= X 2 +y 2 .
Then
Q(x,y)Q(x,y) = Q{xx - yy \xy + xy),
{x 2 + y 2 ){x 2 + y 2 ) = (xx - yy 1 ) 2 + (xy + xy) 2
2 /2 , 2 12 , /2 2 , 2/2
= x x +y y +x y +x y .
see also Genus Theorem
Compound Interest
Concatenation 291
Compound Interest
Let P be the Principal (initial investment), r be the
annual compounded rate, v n > the "nominal rate," rt be
the number of times INTEREST is compounded per year
(i.e., the year is divided into n CONVERSION PERIODS),
and t be the number of years (the "term"). The INTER-
EST rate per CONVERSION PERIOD is then
-■(«)
(1)
If interest is compounded n times at an annual rate of r
(where, for example, 10% corresponds to r = 0.10), then
the effective rate over 1/n the time (what an investor
would earn if he did not redeposit his interest after each
compounding) is
(i + '
Nl/n
(2)
The total amount of holdings A after a time t when
interest is re-invested is then
A = P[l + —) =P(l + r) nt . (3)
Note that even if interest is compounded continuously,
the return is still finite since
lim (l + -V
n— >oo \ 71 J
(4)
where e is the base of the NATURAL LOGARITHM.
The time required for a given PRINCIPAL to double (as-
suming n = l Conversion Period) is given by solving
2P = P(l + r) t ,
In 2
ln(l + r)'
(5)
(6)
where Ln is the NATURAL LOGARITHM. This function
can be approximated by the so-called RULE OF 72:
0.72
r
(?)
see also e, Interest, Ln, Natural Logarithm, Prin-
cipal, Rule of 72, Simple Interest
References
Kellison, S. G. The Theory of Interest, 2nd ed. Burr Ridge,
IL: Richard D. Irwin, pp. 14-16, 1991.
Milanfar, P. "A Persian Folk Method of Figuring Interest."
Math. Mag. 69, 376, 1996.
Compound Polyhedron
see Polyhedron Compound
Comput ability
see Complexity Theory
Computable Function
Any computable function can be incorporated into a
Program using while-loops (i.e., "while something is
true, do something else"). For-loops (which have a fixed
iteration limit) are a special case of while-loops, so com-
putable functions could also be coded using a combina-
tion of for- and while-loops. The ACKERMANN FUNC-
TION is the simplest example of a well-defined TOTAL
Function which is computable but not Primitive Re-
cursive, providing a counterexample to the belief in
the early 1900s that every computable function was also
primitive recursive (Dotzel 1991).
see also Ackermann Function, Church's Thesis,
Computable Number, Primitive Recursive Func-
tion, Turing Machine
References
Dotzel, G. "A Function to End All Functions." Algorithm:
Recreational Programming 2, 16—17, 1991.
Computable Number
A number which can be computed to any number of
Digits desired by a Turing Machine. Surprisingly,
most Irrationals are not computable numbers!
References
Penrose, R. The Emperor's New Mind: Concerning Comput-
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, 1989.
Computational Complexity
see Complexity Theory
Concatenated Number Sequences
see Consecutive Number Sequences
Concatenation
The concatenation of two strings a and b is the string ab
formed by joining a and b. Thus the concatenation of
the strings "book" and "case" is the string "bookcase".
The concatenation of two strings a and 6 is often de-
noted ab, a\\b, or (in Mathematica® (Wolfram Research,
Champaign, IL) a <> b. Concatenation is an asso-
ciative operation, so that the concatenation of three or
more strings, for example abc, abed, etc., is well-defined.
The concatenation of two or more numbers is the num-
ber formed by concatenating their numerals. For exam-
ple, the concatenation of 1, 234, and 5678 is 12345678.
The value of the result depends on the numeric base,
which is typically understood from context.
The formula for the concatenation of numbers p and q
in base b is
p\\q=pb lM +q,
where
i(«) = |>g fc «J + i
is the Length of q in base b and [x\ is the Floor
Function.
292
Concave
Conchoid
see also CONSECUTIVE NUMBER SEQUENCES, LENGTH
(Number), Smarandache Sequences
Concave
A Set in R is concave if it does not contain all the
Line Segments connecting any pair of its points. If
the Set does contain all the Line Segments, it is called
Convex.
see also CONNECTED SET, CONVEX FUNCTION, CON-
VEX Hull, Convex Optimization Theory, Convex
Polygon, Delaunay Triangulation, Simply Con-
nected
Concave Function
A function f(x) is said to be concave on an interval [a, b]
if, for any points x\ and X2 in [a, 6], the function —f(x)
is Convex on that interval. If the second Derivative
of/
/"(*) > 0,
on an open interval (a, fo) (where f"(x) is the second
Derivative), then / is concave up on the interval. If
/"(*) < o
on the interval, then / is concave down on it.
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1100, 1980.
Concentrated
Let fi be a POSITIVE MEASURE on a SlGMA ALGEBRA
M, and let A be an arbitrary (real or complex) MEASURE
on M. If there is a Set A € M such that X(E) =
A(Afl^) for every E G M, then lambda is said to be
concentrated on A. This is equivalent to requiring that
X(E) = whenever E n A = 0.
see also Absolutely Continuous, Mutually Singu-
lar
References
Rudin, W. Functional Analysis. New York: McGraw-Hill,
p. 121, 1991.
Concentric
Two geometric figures are said to be concentric if their
Centers coincide. The region between two concentric
Circles is called an Annulus.
see also Annulus, Concentric Circles, Concyclic,
Eccentric
Concentric Circles
The region between two CONCENTRIC circles of different
Radii is called an Annulus.
Given two concentric circles with RADII R and 2R, what
is the probability that a chord chosen at random from
the outer circle will cut across the inner circle? Depend-
ing on how the "random" CHORD is chosen, 1/2, 1/3, or
1/4 could all be correct answers.
1. Picking any two points on the outer circle and con-
necting them gives 1/3.
2. Picking any random point on a diagonal and then
picking the Chord that perpendicularly bisects it
gives 1/2.
3. Picking any point on the large circle, drawing a line
to the center, and then drawing the perpendicularly
bisected CHORD gives 1/4.
So some care is obviously needed in specifying what is
meant by "random" in this problem.
Given an arbitrary Chord BB' to the larger of two
concentric CIRCLES centered on O, the distance be-
tween inner and outer intersections is equal on both
sides (AB = A'B'). To prove this, take the PERPEN-
DICULAR to BB' passing through O and crossing at P.
By symmetry, it must be true that PA and PA' are
equal. Similarly, PB and PB' must be equal. There-
fore, PB - PA = AB equals PB' - PA' = A'B'. Inci-
dentally, this is also true for HOMEOIDS, but the proof
is nontrivial.
see also Annulus
Concho-Spiral
The Space Curve with parametric equations
u
r = fj, a
6 = u
u
Z = fJ, C.
see also CONICAL SPIRAL, SPIRAL
Conchoid
A curve whose name means "shell form." Let C be a
curve and O a fixed point. Let P and P f be points
on a line from O to C meeting it at Q, where P'Q =
QP — k, with k a given constant. For example, if C is a
CIRCLE and O is on C, then the conchoid is a LlMAQON,
while in the special case that k is the DIAMETER of C,
Conchoid of de Sluze
Concordant Form 293
then the conchoid is a CARDIOID. The equation for a
parametrically represented curve (f(t),g(t)) with O =
(x Qi yo) is
x = f±
y = g±
k(f - xp)
y/(f-x )* + {g-yo) 2
fe(g ~ 2/Q )
A/(/-*o) a + (5-yd) 2 '
see a/so CONCHO-SPIRAL, CONCHOID OF DE SLUZE,
Conchoid of Nicomedes, Conical Spiral, Durer's
Conchoid
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 49-51, 1972.
Lee, X. "Conchoid." http : //www. best . com/ -xah/ Special
PlaneCurves-dir/Conchoid_dir/conchoid.html.
Lockwood, E. H. "Conchoids." Ch. 14 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 126-129, 1967.
Yates, R. C. "Conchoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33,
1952.
Conchoid of de Sluze
A curve first constructed by Rene de Sluze in 1662. In
Cartesian Coordinates,
a(x — a){x + y ) = k x ,
and in POLAR COORDINATES,
k 2 cos 8 „
r = h a sec v.
a
The above curve has k 2 fa — 1, a = —0.5.
Conchoid of Nicomedes
A curve studied by the Greek mathematician Nicomedes
in about 200 BC, also called the Cochloid. It is the
LOCUS of points a fixed distance away from a line as
measured along a line from the FOCUS point (MacTutor
Archive). Nicomedes recognized the three distinct forms
seen in this family. This curve was a favorite with 17th
century mathematicians and could be used to solve the
problems of CUBE DUPLICATION and ANGLE TRISEC-
TION.
In Polar Coordinates,
r = b + asecO.
In Cartesian Coordinates,
/ \2/ 2 . 2\ ,2 2
{x — a) [x + y ) — b x .
(i)
(2)
The conchoid has x — a as an asymptote and the Area
between either branch and the ASYMPTOTE is infinite.
The Area of the loop is
+ 6 2 cos- 1 (JJ. (3)
see also CONCHOID
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 135-139, 1972.
Lee, X. "Conchoid of Nicomedes." http://www.best.com/
-xah/SpecialPlaneCurvesjdir/ConchoidOfNicomedesjdir
/conchoidOf Nicomedes .html.
MacTutor History of Mathematics Archive. "Conchoid."
http : //www-groups . dcs . st-and . ac . uk/ -history/Curves
/Conchoid. html.
Pappas, T. "Conchoid of Nicomedes." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, pp. 94-95,
1989.
Yates, R. C. "Conchoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33,
1952.
Concordant Form
A concordant form is an integer TRIPLE (a, 6, N) where
fa 2 +6 2 =c 2
\a 2 +Nb 2 = d 2 ,
with c and d integers. Examples include
/ 14663 2 + 111384 2 = 112345 2
14663 2 + 47 • 111384 2 = 763751 2
1141 2 + 13260 2 = 13309 2
1141 2 + 53-13260 2 =96541 2
2873161 2 + 2401080 2 = 3744361 2
2873161 2 + 83 • 2401080 2 = 22062761 2
Dickson (1962) states that C. H. Brooks and S. Watson
found in The Ladies' and Gentlemen's Diary (1857) that
x 2 + y 2 and x 2 + Ny 2 can be simultaneously squares for
N < 100 only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27,
30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61,
68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97,
294
Concur
Conditional Convergence
99, and 100 (which evidently omits 47, 53, and 83 from
above). The list of concordant primes less than 1000
is now complete with the possible exception of the 16
primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593,
607, 641, 743, 821, 929, and 971 (Brown).
see also Congruum
References
Brown, K. S. "Concordant Forms." http : //www . seanet .
com/~ksbrown/kmath286 .htm.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, p. 475,
1952.
Concur
Two or more lines which intersect in a POINT are said
to concur.
see also CONCURRENT
Concurrent
Two or more LINES are said to be concurrent if they
intersect in a single point. Two LINES concur if their
Trilinear Coordinates satisfy
(1)
Three Lines concur if their Trilinear Coordinates
satisfy
Concyclic
h
mi
m
h
7U2
n 2
h
ms
ri3
ha + m\f3 -\- nij =
hoc + 777,2/? + ri27 =
/3a + m3/3 + ri3j = 0,
(2)
(3)
(4)
in which case the point is
?7i2n3 — ri27nz : ri2h ~ hns : hrns — 7712/3- (5)
Three lines
Aiz-b£iy + Ci =0
A 2 x + B 2 y + C 2 =
A 3 x + B 3 y + C 3 = 0.
are concurrent if their COEFFICIENTS satisfy
A t
Si
Cx
A 2
B 2
c 2
A 3
B 3
c 3
(6)
(7)
(8)
(9)
Four or more points Pi, P 2 , Ps, P4, • • - which lie on a
Circle C are said to be concyclic. Three points are
trivially concyclic since three noncollinear points deter-
mine a CIRCLE. The number of the n 2 LATTICE POINTS
x>y £ [l,n] which can be picked with no four concyclic
is 0(n
2/3
(Guy 1994).
A theorem states that if any four consecutive points of
a POLYGON are not concyclic, then its Area can be
increased by making them concyclic. This fact arises in
some PROOFS that the solution to the ISOPERIMETRIC
Problem is the Circle.
see also Circle, Collinear, Concentric, Cyclic
Hexagon, Cyclic Pentagon, Cyclic Quadrilat-
eral, Eccentric, N-Cluster
References
Guy, R. K. "Lattice Points, No Four on a Circle." §F3 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, p. 241, 1994.
Condition
A requirement NECESSARY for a given statement or the-
orem to hold. Also called a Criterion.
see also Boundary Conditions, Carmichael Con-
dition, Cauchy Boundary Conditions, Condition
Number, Dirichlet Boundary Conditions, Diver-
sity Condition, Feller-Levy Condition, Holder
Condition, Lichnerowicz Conditions, Lindeberg
Condition, Lipschitz Condition, Lyapunov Con-
dition, Neumann Boundary Conditions, Robert-
son Condition, Robin Boundary Conditions, Tay-
lor's Condition, Triangle Condition, Weier-
straB-Erdman Corner Condition, Winkler Con-
ditions
Condition Number
The ratio of the largest to smallest Singular Value of
a system. A system is said to be singular if the condition
number is Infinite, and ill-conditioned if it is too large.
Conditional Convergence
If the Series
see also CONCYCLIC, POINT
XX
Converges, but
I>»i
Conditional Probability
Cone
295
does not, where \x\ is the ABSOLUTE VALUE, then the
Series is said to be conditionally Convergent.
see also Absolute Convergence, Convergence
Tests, Riemann Series Theorem, Series
Conditional Probability
The conditional probability of A given that B has oc-
curred, denoted P(A\B), equals
P{A\B)
P{A n b)
P(B) '
(1)
which can be proven directly using a Venn Diagram.
Multiplying through, this becomes
P(A\B)P(B) = P(A H £), (2)
which can be generalized to
P(A l)BUC) = P(A)P{B\A)P(C\A U B). (3)
Rearranging (1) gives
P(B n A)
P(B\A) =
P(A)
(4)
Solving (4) for P(B n A) = P(A n B) and plugging in
to (1) gives
P(A\B) =
P(A)P{B\A)
P(B) ■
(5)
see also BAYES' FORMULA
Condom Problem
see Glove Problem
Condon-Shortley Phase
The ( — l) m phase factor in some definitions of the
Spherical Harmonics and associated Legendre
POLYNOMIALS. Using the Condon-Shortley convention
gives
Y?{6,4>) = {-1Y
2n+ 1 (n-m)\
47r (n + to)!
Pn(cos9)e
im<f>
see also Legendre Polynomial, Spherical Har-
monic
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 682 and 692, 1985.
Condon, E. U. and Shortley, G, The Theory of Atomic Spec-
tra. Cambridge, England: Cambridge University Press,
1951.
Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra.
New York: Wiley, p. 158, 1968.
Conductor
see j-CONDUCTOR
Cone
A cone is a Pyramid with a circular Cross-Section.
A right cone is a cone with its vertex above the center
of its base. A right cone of height h can be described by
the parametric equations
x = r(h — z) cos 9
(i)
y = r (h — z)sin#
(2)
z = z
(3)
for z e [0, h] and 9 e [0, 2tt). The VOLUME of a cone is
therefore
V = \A h h, (4)
where At is the base Area and h is the height. If the
base is circular, then
V
1 2,
7j7rr a.
(5)
This amazing fact was first discovered by Eudoxus, and
other proofs were subsequently found by Archimedes in
On the Sphere and Cylinder (ca. 225 BC) and Euclid in
Proposition XII. 10 of his Elements (Dunham 1990).
The CENTROID can be obtained by setting R2 = in the
equation for the centroid of the CONICAL Frustum,
{Z)_ _ /l(Hi 2 +i?1^2+^2 2 )
v ~ 4(i^l 2 + 2i^li^2 + 3.R 2 2 ) ,
(Beyer 1987, p. 133) yielding
For a right circular cone, the Slant Height s is
(6)
(7)
s = VV 2 + h 2 (8)
and the surface Area (not including the base) is
S = irrs = irr\/r 2 + h 2 . (9)
In discussions of Conic Sections, the word cone is of-
ten used to refer to two similar cones placed apex to
apex. This allows the HYPERBOLA to be defined as the
296 Cone Graph
intersection of a PLANE with both NAPPES (pieces) of
the cone.
The LOCUS of the apex of a variable cone containing
an Ellipse fixed in 3-space is a Hyperbola through
the Foci of the Ellipse. In addition, the Locus of
the apex of a cone containing that Hyperbola is the
original Ellipse. Furthermore, the Eccentricities of
the ELLIPSE and HYPERBOLA are reciprocals.
see also Conic Section, Conical Frustum, Cylin-
der, Nappe, Pyramid, Sphere
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 129 and 133,
1987.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 76-77, 1990.
Yates, R. C. "Cones." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 34-35,
1952.
Cone Graph
A Graph C n + K m , where C n is a Cyclic Graph and
Km is a Complete Graph.
Cone Net
The mapping of a grid of regularly ruled squares onto a
CONE with no overlap or misalignment. Cone nets are
possible for vertex angles of 90°, 180°, and 270°, and
are beautifully illustrated by Steinhaus (1983).
References
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 224-228, 1983.
Confidence Interval
(l + ~^j - 2x x + y 2 (l + ^ - 2y y
2z
+(x 2 + yo 2 + zo 2 -r 2 )-^- ^ x 2 + y 2 = 0. (4)
Therefore, x and y are connected by a complicated
Quartic Equation, and x, y, and z by a Quadra-
tic Equation.
If the CONE- SPHERE intersection is on-axis so that a
Cone of opening parameter c and vertex at (0, 0, zq) is
oriented with its Axis along a radial of the Sphere of
radius r centered at (0,0,0), then the equations of the
curve of intersection are
- zo)
2 . 2
2 vr + y
x 2 +y 2 + z 2 =r 2 .
(5)
(6)
(7)
(8)
z 2 {c 2 + 1) - 2c 2 z z + (z V - r 2 ) = 0. (9)
Using the QUADRATIC EQUATION gives
2c 2 zo ± ^4c 4 z 2 - 4(c 2 + l)(zo 2 c 2 - r 2 )
2(c 2 + 1)
Combining (5) and (6) gives
c (z — zo) + z2 = r
c 2 (z 2 -2z z + zo 2 ) + z 2 =r 2
c 2 z D ± x /c 2 (r 2 -z 2 ) + r 2
c 2 + l
(10)
Cone (Space)
The Join of a Topological Space X and a point P,
C(X) = X*P.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 6, 1976.
So the curve of intersection is planar. Plugging (10)
into (5) shows that the curve is actually a Circle, with
Radius given by
= y/r 2 - ;
(11)
Cone-Sphere Intersection
Let a CONE of opening parameter c and vertex at (0, 0, 0)
intersect a SPHERE of RADIUS r centered at (xo, yo, ^o),
with the Cone oriented such that its axis does not pass
through the center of the SPHERE. Then the equations
of the curve of intersection are
2 , 2
(x - xq) 2 + (y - y ) 2 + (z - z ) 2
(i)
(2)
Confidence Interval
The probability that a measurement will fall within a
given Closed Interval [a, b]. For a continuous distri-
bution,
CI(
a,b) = J
P(x) dx.
(i)
where P(x) is the Probability Distribution Func-
tion. Usually, the confidence interval of interest is sym-
metrically placed around the mean, so
Combining (1) and (2) gives
(3)
CI(x) = CI(iJL-x,fjL + x) = / P(x)dx, (2)
J fJ, — X
Configuration
Confluent Hypergeometric Function.
297
where fi is the Mean. For a Gaussian Distribution,
the probability that a measurement falls within ncr of
the mean pi is
CI(twt)
fj,-\-ncr
aV2n
2
e^*^ /2tr dx
fj, — ncr
-(x-fi) 2 /2<T 2
dx.
(3)
Now let u = (x — \i)j\[2cr, so du = dx/^/2a. Then
VV5 „
Cl(rwr)
-4=^2* r
CTV27T ./o
/
du
2
e" u dw = erf | -^= 1 , (4)
where erf(x) is the so-called Erf function. The variate
value producing a confidence interval CI is often denoted
#CIj so
xci = v^err 1 (CI). (5)
range
CI
<T
0.6826895
2(7
0.9544997
3(7
0.9973002
4(7
0.9999366
5(7
0.9999994
To find the standard deviation range corresponding to
a given confidence interval, solve (4) for n.
n = V^err^CI)
(6)
CI
range
0.800
±1.28155<r
0.900
±1.64485*
0.950
±1.95996*
0.990
±2.57583*
0.995
±2.80703*
0.999
±3.29053o-
Configuration
A finite collection of points p = (p± } . . . ,p n ), Pi € M. ,
where R d is a EUCLIDEAN SPACE.
see also Bar (Edge), Euclidean Space, Frame-
work. Rigid
Confluent Hypergeometric Differential
Equation
xy" + (6 - x)y -ay = 0, (1)
where y' = dy/dx and with boundary conditions
iFi(o;6;0) = l (2)
[dx
iFi(a\b]x)
(3)
The equation has a Regular Singular Point at
and an irregular singularity at oo. The solutions are
called Confluent Hypergeometric Function of
the First or Second Kinds. Solutions of the first
kind are denoted ii*i(a; 6; x) or M(a 7 b,x).
see also HYPERGEOMETRIC DIFFERENTIAL EQUATION,
Whittaker Differential Equation
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 504, 1972.
Arfken, G. "Confluent Hypergeometric Functions." §13.6 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 753-758, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 551-555, 1953.
Confluent Hypergeometric Function
see Confluent Hypergeometric Function of the
First Kind, Confluent Hypergeometric Func-
tion of the Second Kind
Confluent Hypergeometric Function of the
First Kind
The confluent hypergeometric function a degenerate
form the HYPERGEOMETRIC FUNCTION 2 F 1 (a,b]C]z)
which arises as a solution the the Confluent Hyper-
geometric Differential Equation. It is commonly
denoted iFi(a\b;z), M(a,6,z), or $(a;6;z), and is also
known as Kummer's Function of the first kind. An
alternate form of the solution to the Confluent Hyper-
geometric Differential Equation is known as the Whit-
taker Function.
The confluent hypergeometric function has a HYPERGE-
OMETRIC Series given by
iF!(a;b\z)
i i a - ■ a(a+l) z 2
+ & + 6(6+1) 2!
+ ...
fc=0
v^ (a)fc z k
^ (b) h k\ '
(1)
where (a)k and (b)k are Pochhammer Symbols. If a
and 6 are INTEGERS, a < 0, and either b > or b < a,
then the series yields a POLYNOMIAL with a finite num-
ber of terms. If b is an Integer < 0, then iFi(a; 6; z) is
undefined. The confluent hypergeometric function also
has an integral representation
^^^J'fT^w/ 1 ^^ 1 -') 6 ""' 1
(Abramowitz and Stegun 1972, p. 505).
dt
(2)
Bessel Functions, the Error Function, the incom-
plete Gamma Function, Hermite Polynomial', La-
GUERRE POLYNOMIAL, as well as other are all special
298 Confluent Hypergeometric Function.
cases of this function (Abramowitz and Stegun 1972,
p. 509).
Kummer's Second Formula gives
m+l/2
iFi(|+m;2m + l;jz) = M , m {z) = z
z 2p
+ Z^ 2 4 Pp\(m + 1) (m + 2) • ■ • (m + p)
, (3)
where uFi is the Confluent HYPERGEOMETRIC FUNC-
TION and m # -1/2, -1, -3/2, ....
5ee also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL
Equation, Confluent Hypergeometric Function
of the Second Kind, Confluent Hypergeomet-
ric Limit Function, Generalized Hypergeomet-
ric Function, Heine Hypergeometric Series,
hypergeometric function, hypergeometric se-
RIES, Kummer's Formulas, Weber-Sonine For-
mula, Whittaker Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Confluent Hy-
pergeometric Functions." Ch. 13 in Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathemat-
ical Tables, 9th printing. New York: Dover, pp. 503-515,
1972.
Arfken, G. "Confluent Hypergeometric Functions." §13.6 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 753-758, 1985.
Iyanaga, S. and Kawada, Y. (Eds.). "Hypergeometric Func-
tion of Confluent Type." Appendix A, Table 19.1 in En-
cyclopedic Dictionary of Mathematics. Cambridge, MA:
MIT Press, p. 1469, 1980.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 551-554 and 604-
605, 1953.
Slater, L. J, Confluent Hypergeometric Functions. Cam-
bridge, England: Cambridge University Press, 1960.
Spanier, J. and Oldham, K. B. "The Kummer Function
M(a; c; a;)." Ch. 47 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 459-469, 1987.
Confluent Hypergeometric Function of the
Second Kind
Gives the second linearly independent solution to the
Confluent Hypergeometric Differential Equa-
tion. It is also known as the Kummer's Function of
the second kind, the TRICOMI FUNCTION, or the GOR-
DON Function. It is denoted U{a ) b ) z) and has an in-
tegral representation
U(a,b.
' 2) = fW)l
— zt.a-
e t
\i + t)
6-a-l
dt
(Abramowitz and Stegun 1972, p. 505). The WHIT-
TAKER FUNCTIONS give an alternative form of the solu-
tion. For small z, the function behaves asz 1 " .
see also Bateman Function, Confluent Hyperge-
ometric Function of the First Kind, Conflu-
ent Hypergeometric Limit Function, Coulomb
Wave Function, Cunningham Function, Gordon
Confocal Conies
Function, Hypergeometric Function, Poisson-
Charlier Polynomial, Toronto Function, We-
ber Functions, Whittaker Function
References
Abramowitz, M. and Stegun, C A. (Eds.). "Confluent Hy-
pergeometric Functions." Ch. 13 in Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathemat-
ical Tables, 9th printing. New York: Dover, pp. 503-515,
1972.
Arfken, G. "Confluent Hypergeometric Functions." §13.6 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 753-758, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 671-672, 1953.
Spanier, J. and Oldham, K. B. "The Tricomi Function
U(a;c\x). n Ch. 48 in An Atlas of Functions. Washing-
ton, DC: Hemisphere, pp. 471-477, 1987.
Confluent Hypergeometric Limit Function
Fi(;a;js) = lim 1F1 f q;a; - ) . (1)
g->oo y qj
It has a series expansion
oo
oFi(;a;z) = Y] 7-^—7
^— ' (ajnnl
(2)
and satisfies
dz 2 dz
^+ fl x-» = «-
(3)
A Bessel Function of the First Kind can be ex-
pressed in terms of this function by
J n (x) = ^-y-oFi(;n + 1; ~\x z ) (4)
(Petkovsek et al 1996).
see also Confluent Hypergeometric Function,
Generalized Hypergeometric Function, Hyper-
geometric Function
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 38, 1996.
Confocal Conies
Confocal conies are Conic SECTIONS sharing a common
Focus. Any two confocal Central Conics are orthog-
onal (Ogilvy 1990, p. 77).
see also Conic Section, Focus
References
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 77-78, 1990.
Confocal Ellipsoidal Coordinates
Confocal Ellipsoidal Coordinates
The confocal ellipsoidal coordinates (called simply el-
lipsoidal coordinates by Morse and Feshbach 1953) are
given by the equations
+
y
+
a 2 +£ &+£ c 2 +£
a 2 + 7] b 2 + 7] c 2 + 7]
= 1
+
+
a 2 +C & 2 + C c 2 + C
(1)
(2)
(3)
where -c 2 < £ < oo, -b 2 < r) < -c 2 , and -a 2 <
£ < — b 2 . Surfaces of constant £ are confocal Ellip-
soids, surfaces of constant rj are one-sheeted HYPER-
BOLOIDS, and surfaces of constant C are two- sheeted
HYPERBOLOIDS, For every (x y y, z), there is a unique
set of ellipsoidal coordinates. However, (£, r/, C) specifies
eight points symmetrically located in octants. Solving
for cc, y, and z gives
2 (a2 + fl( g ' + l? )(a' + C)
(6 2 -a 2 )(c 2 -a 2 )
2 (6 2 +Q(b 2 +^)(b 2 + C)
y (a 2 - 6 2 )(c 2 - ft 2 )
2 (c 2 +g)(c 2 +> ? )(c 2 + C)
(a 2 - c 2 )(6 2 - c 2 )
(4)
(5)
(6)
The Laplacian is
V 2 * = (»7-C)/(0
0£
/(€)
+(C-«/fa)
where
cfy
+ K-*)/(0
ac
/(C)
/(x) = v /(a: + a 2 )(x + 6 2 )(a ; + c 2 ).
Another definition is
x
■ +
a 2 _ x b 2 - A c 2 - A
„2
X
+
2/
a 2 — /a b 2 — pi c 2 ~ fi
= 1
= 1
+
6 2
+
z^ c^ - f
1,
where
A < c 2 < // < b 2 < v < a 2
(7)
(8)
(9)
(10)
(11)
(12)
(Arfken 1970, pp. 117-118). Byerly (1959, p. 251) uses a
slightly different definition in which the Greek variables
are replaced by their squares, and a = 0. Equation (9)
represents an ELLIPSOID, (10) represents a one-sheeted
Confocal Ellipsoidal Coordinates 299
HYPERBOLOID, and (11) represents a two-sheeted Hy-
PERBOLOID. In terms of CARTESIAN COORDINATES,
2 (a 2 -A)(q 2 -A*)(a 2 -^)
X (a 2 -6 2 )(a 2 -c 2 )
2 (& 2 ~A)(fe 2 ^)(b 2 -^)
V ~ (b 2 - a 2 )(b 2 - c 2 )
2 _ (c 2 -A)(c 2 -^)(c 2 -^)
Z (c 2 -a 2 )(c 2 -6 2 ) *
The Scale Factors are
hx =
h,, =
h u =
(/j-A)(i/-A)
4(a 2 -A)(6 2 -A)(c 2 -A)
(1/ - a>) (A - li)
4(a 2 -M)(6 2 -/x)(c 2 - M )
(A-tQQi-i/)
4(a 2 -^)(6 2 -i/)(c 2 -i/)'
(13)
(14)
(15)
(16)
(17)
(18)
The Laplacian is
a 2 b 2 + a 2 c 2 + b 2 c 2 - 2z/(a 2 + fe 2 -f c 2 ) + 3^ 2 d
V 2 =-2
+
(^-i/)(i/- A)
4(a 2 -i/)(6 2 -i/)(c 3 -i/) 9 2
0i/
4-2
+
+ 2
+
(/i-i/)(i/-A) 5i/ 2
a 2 6 2 + a 2 c 2 + fr 2 c 2 - 2^(a 2 + b 2 + c 2 ) + 3^ 2 j9_
(i/-/*)(/i- A) 0a*
(/i-A)(i/-M) a M 2
(a 2 6 2 + a 2 c 2 + b 2 c 2 ) + 2A(a 2 + b 2 + c 2 ) - 3A 2 a
(a*-A)(z/-A) d\
4(a 2 -A)(b 2 -A)(c 2 -A) a 2
(j*-A)(z/-A) dA 2 *
(19)
Using the NOTATION of Byerly (1959, pp. 252-253), this
can be reduced to
V 2 = (M 2 - 2 )|^ + (A 2 -, 2 )& + (A 2 -/, 2 )|^, (20)
} da 2
y <9 7 2
where
a — c
t
d\
v / (A 2_ 6 2 )(A 2_ c2)
(21)
/? = c
/
„ v/(c 2 - a* 2 )(m 2 - *> 2 )
= F
1-^sin" 1
(22)
V'(6 2 -^)(c 2 ^P)
(23)
300
Confocal Parabolic Coordinates
Conformal Latitude
Here, F is an Elliptic Integral of the First Kind.
In terms of a, (3, and 7,
, = ede (a, - J
v = 6sn
(24)
(25)
(26)
where dc, nd and sn are Jacobi Elliptic Functions.
The Helmholtz Differential Equation is separable
in confocal ellipsoidal coordinates.
see also Helmholtz Differential Equation —
Confocal Ellipsoidal Coordinates
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Definition of
Elliptical Coordinates." §21.1 in Handbook of Mathemat-
ical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 752, 1972.
Arfken, G, "Confocal Ellipsoidal Coordinates (^1,^2,^3)-"
§2.15 in Mathematical Methods for Physicists, 2nd ed.
New York: Academic Press, pp. 117-118, 1970.
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, 1959.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 663, 1953.
Confocal Parabolic Coordinates
see Confocal Paraboloidal Coordinates
Confocal Paraboloidal Coordinates
= z-\ (1)
= z-n (2)
= z-u, (3)
x 2
+
y 2
6 2 -
o 2 -
X
X
x 2
a? -
M
+
y 2
b 2 -
v>
x 2
+
y 2
a* — v b 2 — v
i2\ .. r- (1? „2\
where A e (-00, b ), ^ e (6 ,a ), and v € (a 2 , 00).
2 _ (a 2 -A)(a 2 -^)(a 2 -^)
(b 2 - a 2 )
{b 2 „ X )(b 2 -y)(b 2 -v)
* (a 2 - 6 2 )
z = X + fi-h v — a 2 — 6 2 .
The Scale Factors are
( M -A)(i/-A)
4(a 2 -A)(6 2 -A)
_ , (u-fi)(X-fi)
M A/ 4(a 2 -^)(6 2 -/x)
h v =
(A - v)(n - u)
16(a 2 - i/)(6 2 - v) '
(4)
(5)
(6)
(7)
(8)
(9)
The Laplacian is
v2 _ 2(a 2 + b 2 -2v) d 4(a 2 - v)(y - b 2 ) d 2
+
{ji - v)(v - X) dv (/1 - u){v - A) v 2
2(a 2 + b 2 - 2fi) d 4(a 2 - /*)(/* - b 2 ) d 2
+ ■
(/a — A)(i> — /x) ^ (// — \)(v — fi) dy?
2(2A-a 2 -6 2 ) a 4(A-q 2 )(A-6 2 ) <9 2
(//-A)(i/-A) dA + (/a-A)(i/-A) dA 2 '
(10)
The Helmholtz Differential Equation is Separa-
ble.
see also Helmholtz Differential Equation —
Confocal Paraboloidal Coordinates
References
Arfken, G. "Confocal Parabolic Coordinates (£1, £ 2) £3)-"
§2.17 in Mathematical Methods for Physicists, 2nd ed. Or-
lando, FL: Academic Press, pp. 119-120, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 664, 1953.
Conformal Latitude
An Auxiliary Latitude defined by
X = 2tan 1 <han(±7r+§<£)
1 — e sin <j>
e/2"
= 2 tan
1 + sin <j>
1 + e sin <
e-i 1/2
(1 — e sin <p \
1 + e sin 4> J
1 — sin <f>
*-(& + h** + |e 6 + ^e 8 + . . .) sin(20)
^48
U80 C ~ 13440
13 6 , 461
480 e "•" :
1237 rt 8
ii52o- e +---)sin(4^)
■f . . .) sin(60)
+ (lMio e +---)sin(8<^) + ....
The inverse is obtained by iterating the equation
e/2l
» = 2 tan
/i 1 \ ( 1 + e sin <£ \
tan ±tt+±x - r-^
* y 1 — e sin y
using = x as the first trial. A series form is
^ = X + {\e 2 + £e 4 + £e 6 + ^e 8 + . . .) sin(2 X )
+ (^- 4 + ^ 6 + ^i5 e 9 + ...)sin(4 X )
+ (no e6 +if5o e8 + ---)sin(6 X )
+ (Tflio e8 + ---)sin(8x) + ...
The conformal latitude was called the ISOMETRIC LAT-
ITUDE by Adams (1921), but this term is now used to
refer to a different quantity.
see also Auxiliary Latitude, Latitude
References
Adams, O. S. "Latitude Developments Connected with
Geodesy and Cartography with Tables, Including a Table
for Lambert Equal-Area Meridianal Projections." Spec.
Pub. No. 67. U. S. Coast and Geodetic Survey, pp. 18 and
84-85, 1921.
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 15-16, 1987.
Conforms,! Map
Conformal Transformation
301
Conformal Map
A Transformation which preserves Angles is known
as conformal. For a transformation to be conformal, it
must be an Analytic Function and have a Nonzero
Derivative. Let 9 and <j> be the tangents to the curves
7 and /(7) at zq and wo y
wo = f(z) - f{z )
/(*)-/(*>)
Z — Zq
zo)
(1)
arg(w — wo) = arg
/(*) - /(*>)
z — Zq
Then as w — > Wq and z — v zq,
+ &rg(z-z ). (2)
<f> = arg/'(z ) + 6
M = l/'(*o)||4
(3)
(4)
see also Analytic Function, Harmonic Function,
Mobius Transformation, Quasiconformal Map,
Similar
References
Arfken, G. "Conformal Mapping." §6.7 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 392-394, 1985.
Bergman, S. The Kernel Function and Conformal Mapping.
New York: Amer. Math. Soc, 1950.
Katznelson, Y. An Introduction to Harmonic Analysis. New
York: Dover, 1976.
Morse, P. M. and Feshbach, H. "Conformal Mapping." §4.7
in Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 358-362 and 443-453, 1953.
Nehari, Z. Conformal Map. New York: Dover, 1982.
Conformal Solution
By letting w = f(z), the REAL and IMAGINARY PARTS of
w must satisfy the CAUCHY-RlEMANN EQUATIONS and
Laplace's Equation, so they automatically provide a
scalar POTENTIAL and a so-called stream function. If a
physical problem can be found for which the solution is
valid, we obtain a solution — which may have been very
difficult to obtain directly — by working backwards. Let
Az n = ArV
the Real and Imaginary Parts then give
For n
$ = Ar n cos(n9)
ip = Ar n sin(n0).
<f> = — cos(2(9)
^:
-sin(2(9),
(1)
(2)
(3)
(4)
(5)
which is a double system of LEMNISCATES (Lamb 1945,
p. 69). For n = -1,
: cos#
ip — sin0.
(6)
(T)
This solution consists of two systems of CIRCLES, and
<f> is the Potential Function for two Parallel op-
posite charged line charges (Feynman et al. 1989, §7-5;
Lamb 1945, p. 69). For n = 1/2,
A 1/2
At l cos
(f)=V
y/x 2 + y 2 + x
i>
= av> 8m y =xy
(8)
(9)
(p gives the field near the edge of a thin plate (Feynman
et al. 1989, §7-5). For n = 1,
<j> = At cos 9 = Ax
tp = At sin 6 = Ay.
(10)
(ii)
This is two straight lines (Lamb 1945, p. 68). For n
3/2,
w = At
3/2 3i8/2
(12)
</> gives the field near the outside of a rectangular corner
(Feynman et al 1989, §7-5). For n = 2,
w = A(x + iyf = A[(x 2 - y 2 ) + 2ixy] (13)
(j> = A(x 2 - y 2 ) = At 2 cos(20) (14)
i/> = 2Axy = Ar 2 sin(2^). (15)
These are two PERPENDICULAR HYPERBOLAS, and <j> is
the Potential Function near the middle of two point
charges or the field on the opening side of a charged
Right Angle conductor (Feynman 1989, §7-3).
see also Cauchy-Riemann Equations, Conformal
Map, Laplace's Equation
References
Feynman, R. P.; Leighton, R. B.; and Sands, M. The Feyn-
man Lectures on Physics, Vol. 1. Redwood City, CA:
Addison-Wesley, 1989.
Lamb, H. Hydrodynamics, 6th ed. New York: Dover, 1945.
Conformal Tensor
see Weyl Tensor
Conformal Transformation
see Conformal Map
302 Congruence
Congruence
If b — c is integrally divisible by a, then b and c are said
to be congruent with MODULUS a. This is written math-
ematically as b = c (mod a). If 6 — c is not divisible by a,
then we say b ^ c (mod a). The (mod a) is sometimes
omitted when the MODULUS a is understood for a given
computation, so care must be taken not to confuse the
symbol = with that for an EQUIVALENCE. The quantity
b is called the RESIDUE or REMAINDER. The COMMON
RESIDUE is taken to be NONNEGATIVE and smaller than
m, and the MINIMAL RESIDUE is b or b - m, whichever
is smaller in Absolute Value. In many computer lan-
guages (such as FORTRAN or Mathematic®), the COMMON
Residue of c (mod a) is written mod(c,a).
Congruence arithmetic is perhaps most familiar as a
generalization of the arithmetic of the clock: 40 min-
utes past the hour plus 35 minutes gives 40 + 35 =
15 (mod 60), or 15 minutes past the hour, and 10 o'clock
a.m. plus five hours gives 10 + 5 = 3 (mod 12), or 3
o'clock p.m. Congruences satisfy a number of impor-
tant properties, and are extremely useful in many areas
of Number Theory. Using congruences, simple DI-
VISIBILITY TESTS to check whether a given number is
divisible by another number can sometimes be derived.
For example, if the sum of a number's digits is divisible
by 3 (9), then the original number is divisible by 3 (9).
Congruences also have their limitations. For example, if
a = b and c = d (mod n), then it follows that a x = 6 X ,
but usually not that x c = x d or a c = b d . In addition,
by "rolling over," congruences discard absolute informa-
tion. For example, knowing the number of minutes past
the hour is useful, but knowing the hour the minutes are
past is often more useful still.
Let a = a (mod m) and b = b' (mod m), then im-
portant properties of congruences include the following,
where => means "Implies":
1. Equivalence: a = b (mod 0) => a = 6.
2. Determination: either a = b (mod m) or a ^
b (mom m).
3. Reflexivity: a = a (mod m).
4. Symmetry: a = b (mod ra) => 6 = a (mod m).
5. Transitivity: a = b (mod m) and b =
c (mod ra) ^ a = c (mod ra).
6. a-\-b = a' + 6' (mod m).
7. a — b = a' — b' (mod m).
8. ab = a'b' (mod ra).
9. a = b (mod ra) => ka = kb (mod ra).
10. a = b (mod m) => a n = b n (mod m).
11. a = b (mod mi) and a = b (mod 1712) =>■ a =
b (mod [mi, 7712]), where [7711,7712] is the LEAST
Common Multiple.
12. ak = bk (mod ra) =^ a = b f mod t^t j , where
(fc,ra) is the Greatest Common Divisor.
Congruence
13. If a = b (mod ra), then P(a) = P(6) (mod ra), for
P(x) a Polynomial.
Properties (6-8) can be proved simply by denning
a = a + rd
b = b' + sd,
where r and s are INTEGERS. Then
(1)
(2)
a + & = a' + &' + (r + s)d (3)
a _ 5 = a' _ &' + ( r _ s)d (4)
a& = a' 6' + (a's + b'r + rsd)d, (5)
so the properties are true.
Congruences also apply to FRACTIONS. For example,
note that (mod 7)
2x4:
3x3 = 2 6x6 = 1 (mod 7), (6)
; 6 (mod 7). (7)
To find p/q mod ra, use an ALGORITHM similar to the
Greedy Algorithm. Let q = q and find
Po
(8)
where \x] is the CEILING FUNCTION, then compute
q x = q po (mod m). (9)
Iterate until q n = 1, then
p \ I pi (mod ra).
(10)
This method always works for m PRIME, and sometimes
even for ra COMPOSITE. However, for a COMPOSITE m,
the method can fail by reaching (Conway and Guy
1996).
A Linear Congruence
ax = b (mod m)
is solvable Iff the congruence
b = (mod (a, ra))
(11)
(12)
is solvable, where d = (a, ra) is the GREATEST COMMON
Divisor, in which case the solutions are #o, xo + m/d,
xo + 2m/ d, . . . , xq + (d — l)m/d, where xo < m/d. If
d = 1, then there is only one solution.
Congruence Axioms
Congruent Numbers 303
A general Quadratic Congruence
a-ix + a\x + ao ^ (mod n)
can be reduced to the congruence
x = q (mod p)
(13)
(14)
and can be solved using EXCLUDENTS. Solution of the
general polynomial congruence
a m x m + . . . + a 2 x 2 -f a±x + a = (mod n) (15)
is intractable. Any polynomial congruence will give con-
gruent results when congruent values are substituted.
Two simultaneous congruences
x = a (mod m)
x = b (mod n)
(16)
(17)
are solvable only when x = b (mod (m,n)), and the
single solution is
x = Xo (mod [m,n]) ,
(18)
where xo < m/d.
see also Cancellation Law, Chinese Remainder
Theorem, Common Residue, Congruence Axioms,
Divisibility Tests, Greatest Common Divisor,
Least Common Multiple, Minimal Residue, Mod-
ulus (Congruence), Quadratic Reciprocity Law,
Residue (Congruence)
References
Conway, J. H. and Guy, R. K. "Arithmetic Modulo p." In The
Book of Numbers. New York: Springer- Verlag, pp. 130-
132, 1996.
Courant, R. and Robbins, H. "Congruences." §2 in Supple-
ment to Ch. 1 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 31-40, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 55, 1993.
# Weisstein, E. W. "Fractional Congruences." http://wvw .
astro . Virginia . edu / - eww6n / math / notebooks / Mod
Fraction. m.
Congruence Axioms
The five of Hilbert's Axioms which concern geometric
equivalence.
see also CONGRUENCE AXIOMS, CONTINUITY AXIOMS,
Hilbert's Axioms, Incidence Axioms, Ordering
Axioms, Parallel Postulate
References
Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago,
IL: Open Court, 1980,
Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert's System of Ax-
ioms." §163B in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 544-545, 1980.
Congruence (Geometric)
Two geometric figures are said to be congruent if they
are equivalent to within a ROTATION. This relationship
is written A = B. (Unfortunately, this symbol is also
used to denote ISOMORPHIC GROUPS.)
see also SIMILAR
Congruence Transformation
A transformation of the form g — D 77 D, where det(D)
^ and det(D) is the DETERMINANT.
see also SYLVESTER'S INERTIA LAW
Congruent
A number a is said to be congruent to b modulo m if
m\a - b (m Divides a - b).
Congruent Incircles Point
The point Y for which TRIANGLES BYC, CYA, and
AYB have congruent INCIRCLES. It is a special case of
an Elkies Point.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Congruent Isoscelizers Point
A B
In 1989, P. Yff proved there is a unique configuration of
Isoscelizers for a given Triangle such that all three
have the same length. Furthermore, these ISOSCELIZERS
meet in a point called the congruent isoscelizers point,
which has Triangle Center Function
a = cos(|B) + cos(|C) - cos(^A).
see also CONGRUENT ISOSCELIZERS POINT, ISOSCE-
LIZER
References
Kimberling, C. "Congruent Isoscelizers Point." http://wvw.
evansville.edu/~ck6/tcenters/recent/conisos.htial.
Congruent Numbers
A set of numbers (a } x,y, t) such that
x 2 4- ay 2 — z 2
x 2 -ay 2 = t 2 .
304 Congruum
Conic Equidistant Projection
They are a generalization of the CONGRUUM PROBLEM,
which is the case y = 1. For a = 101, the smallest
solution is
x = 2015242462949760001961
y = 118171431852779451900
z = 2339148435306225006961
t = 1628124370727269996961.
see also CONGRUUM
References
Guy, R. K. "Congruent Number." §D76 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 195-197, 1994.
Congruum
A number h which satisfies the conditions of the CON-
GRUUM Problem:
and
2 i L 2
x + h = a
x 2 -h = b 2 .
see also CONCORDANT FORM, CONGRUUM PROBLEM
Congruum Problem
Find a Square Number x 2 such that, when a given
number h is added or subtracted, new SQUARE Num-
bers are obtained so that
x + h ■
and
■h = F
(1)
(2)
This problem was posed by the mathematicians
Theodore and Jean de Palerma in a mathematical tour-
nament organized by Frederick II in Pisa in 1225. The
solution (Ore 1988, pp. 188-191) is
2 , 2
x = m -\- n
h — 4mn(m 2 — n 2
(3)
(4)
where m and n are INTEGERS. Fibonacci proved that
all numbers h (the CONGRUA) are divisible by 24. Fer-
mat's Right Triangle Theorem is equivalent to the
result that a congruum cannot be a Square Number.
A table for small m and n is given in Ore (1988, p. 191),
and a larger one (for h < 1000) by Lagrange (1977).
2 1 24 5
3 1 96 10
3 2 120 13
4 1 240 17
4 3 336 25
see also CONCORDANT FORM, CONGRUENT NUMBERS,
Square Number
References
Alter, R. and Curtz, T. B. "A Note on Congruent Numbers."
Math. Comput. 28, 303-305, 1974.
Alter, R.; Curtz, T. B.; and Kubota, K. K. "Remarks and
Results on Congruent Numbers." In Proc. Third South-
eastern Conference on Combinatorics, Graph Theory, and
Computing, 1972, Boca Raton, FL. Boca Raton, FL:
Florida Atlantic University, pp. 27-35, 1972.
Bastien, L. "Nombres congruents." Intermed. des Math. 22,
231-232, 1915.
Gerardin, A. "Nombres congruents." Intermed. des Math.
22, 52-53, 1915.
Lagrange, J. "Construction d'une table de nombres congru-
ents." Calculateurs en Math., Bull Soc. math. France.,
Memoire 49-50, 125-130, 1977.
Ore, 0. Number Theory and Its History. New York: Dover,
1988.
Conic
see Conic Section
Conic Constant
K = -e 2 ,
where e is the ECCENTRICITY of a CONIC SECTION.
see also CONIC SECTION, ECCENTRICITY
Conic Double Point
see Isolated Singularity
Conic Equidistant Projection
A Map Projection with transformation equations
x = psin#
y = po - pcosO,
where
9 = n(\- A )
Po = (G- 0o)
COS 01
(1)
(2)
G
n
cos 01
COS 02
02 - (f>l
(3)
(4)
(5)
(6)
(7)
Conic Projection
The inverse FORMULAS are given by
A = A + -,
n
where
p = sgn(n) yz 2 + (p - t/) 2
fl=tan -if_E_y
(8)
(9)
(10)
(11)
Conic Projection
see Albers Equal- Area Conic Projection, Conic
Equidistant Projection, Lambert Azimuthal
Equal- Area Projection, Polyconic Projection
Conic Section
Parabola
Hyperbola
Ellipse
The conic sections are the nondegenerate curves gener-
ated by the intersections of a Plane with one or two
Nappes of a Cone. For a Plane parallel to a Cross-
Section, a Circle is produced. The closed curve pro-
duced by the intersection of a single Nappe with an
inclined PLANE is an ELLIPSE or PARABOLA. The curve
produced by a PLANE intersecting both NAPPES is a
HYPERBOLA. The Ellipse and HYPERBOLA are known
as Central Conics.
Because of this simple geometric interpretation, the
conic sections were studied by the Greeks long before
their application to inverse square law orbits was known,
Apollonius wrote the classic ancient work on the subject
entitled On Conics. Kepler was the first to notice that
planetary orbits were Ellipses, and Newton was then
able to derive the shape of orbits mathematically us-
ing CALCULUS, under the assumption that gravitational
force goes as the inverse square of distance. Depending
on the energy of the orbiting body, orbit shapes which
are any of the four types of conic sections are possible.
A conic section may more formally be defined as the
locus of a point P that moves in the Plane of a fixed
point F called the FOCUS and a fixed line d called the
Conic Section Tangent 305
Directrix (with F not on d) such that the ratio of the
distance of P from F to its distance from d is a constant
e called the ECCENTRICITY. For a FOCUS (0,0) and
Directrix x — -a, the equation is
y
2 (x + a) 2
If e = 1, the conic is a PARABOLA, if e < 1, the conic is
an Ellipse, and if e > 1, it is a Hyperbola.
In standard form, a conic section is written
y = 2Rx-{l-e 2 )x 2 ,
where R is the RADIUS OF CURVATURE and e is the
ECCENTRICITY. Five points in a plane determine a conic
(Le Lionnais 1983, p. 56).
see also Brianchon's Theorem, Central Conic,
Circle, Cone, Eccentricity, Ellipse, Fermat
Conic, Hyperbola, Nappe, Parabola, Pascal's
Theorem, Quadratic Curve, Seydewitz's Theo-
rem, Skew Conic, Steiner's Theorem
References
Besant, W. H. Conic Sections, Treated Geometrically, 8th
ed. rev. Cambridge, England: Deighton, Bell, 1890.
Casey, J. "Special Relations of Conic Sections" and "Invari-
ant Theory of Conics." Chs. 9 and 15 in A Treatise on
the Analytical Geometry of the Point, Line, Circle, and
Conic Sections, Containing an Account of Its Most Re-
cent Extensions, with Numerous Examples, 2nd ed., rev.
enl. Dublin: Hodges, Figgis, & Co., pp. 307-332 and 462-
545, 1893.
Coolidge, J. L. A History of the Conic Sections and Quadric
Surfaces. New York: Dover, 1968.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 138-141, 1967.
Downs, J. W. Conic Sections. Dale Seymour Pub., 1993.
Iyanaga, S. and Kawada, Y. (Eds.). "Conic SecHons." §80
in Encyclopedic Dictionary of Mathematics, Cambridge,
MA: MIT Press, pp. 271-276, 1980.
Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
p. 56, 1983.
Lee, X. "Conic Sections." http://www . best . com/ - xah /
Special Plane Curves _ dir / Conic Sections _ dir / conic
Sections.html.
Ogilvy, C. S. "The Conic Sections." Ch. 6 in Excursions in
Geometry. New York: Dover, pp. 73-85, 1990.
Pappas, T. "Conic Sections." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 196-197, 1989.
Salmon, G. Conic Sections, 6th ed. New York: Chelsea,
1954.
Smith, C. Geometric Conics. London: MacMillan, 1894.
Sommerville, D. M. Y. Analytical Conics, 3rd ed. London:
G. Bell and Sons, 1961.
Yates, R. C. "Conics." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 36-56,
1952.
Conic Section Tangent
Given a Conic Section
x 2 + y + 2gx + 2fy + c = 0,
the tangent at (x\,yi) is given by the equation
xxi + 2/2/1 + g(x 4- xi) 4- f(y + yi) + c = 0.
306
Conical Coordinates
Conical Frustum
Conical Coordinates
Arfken (1970) and Morse and Feshbach (1953) use
slightly different definitions of these coordinates. The
system used in Mathematical (Wolfram Research, Inc.,
Champaign, Illinois) is
Conical Frustum
X\xv
ab
z =
A
/(m 2 -
-a?)(v 2 -
-a 2 )
a \
/
a? -b 2
A
k» 2 '
-b 2 ){u 2 -
-b 2 )
b 2
(1)
(2)
(3)
where b 2 > \j? > c 2 > v 2 . The Notation of Byerly
replaces A with r, and a and b with b and c. The above
equations give
x 2 +y 2 + z 2 = X 2 (4)
2 2
n ~V n
+
a 2 \x 2 -b 2
=
x 2
^7 +
y
+
v 2 -b 2
= o.
The Scale Factors are
\ 2 {li 2 -v 2 )
'" A/ (fi 2 - a 2 )(b 2 ~ fi 2 )
h v
\2{n 2 -v 2 )
(i/ 2 - a 2 )(u 2 - b 2 )
(5)
(6)
(7)
(8)
(9)
The Laplacian is
2 _ i/(2z/ 2 - a 2 - b 2
+
+
+
{jjb -i/)(/x + i^)A 2 <9i/
(a-i/)(a + i/)(i/-6)(i/ + 6) d 2
(i/- M )(^ + m)A 2 dv 2
fi(2^i 2 -a 2 -b 2 ) &
{v-li){v + li)\ 2 diL
{li-b){ti + b)(fjL-a){fi + a) d 2
(v-riiv + ^X 2 OfX 2
1JL _?!_
+ AdA + dX 2 '
(10)
The Helmholtz Differential Equation is separable
in conical coordinates.
see also Helmholtz Differential Equation —
Conical Coordinates
References
Arfken, G. "Conical Coordinates (&, f 2 , &)." §2.16 in Math-
ematical Methods for Physicists, 2nd ed, Orlando, FL:
Academic Press, pp. 118-119, 1970.
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics,
New York: Dover, p. 263, 1959.
Morse, P. M. and Feshbach, FL Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 659, 1953.
Spence, R. D. "Angular Momentum in Sphero-Conal Coor-
dinates." Amer. J, Phys. 27, 329-335, 1959.
A conical frustum is a FRUSTUM created by slicing the
top off a Cone (with the cut made parallel to the base).
For a right circular CONE, let s be the slant height and
R\ and Ri the top and bottom RADII. Then
8= yJ(Ri-R2) 2 +h 2 .
(1)
The SURFACE AREA, not including the top and bottom
Circles, is
A = tt(R 1 +R 2 )s = tt(R 1 +R 2 )^(Ri - R2) 2 + h 2 . (2)
The VOLUME of the frustum is given by
V = tt / [r{z)fdz. (3)
Jo
But
r(z) = R 1 + {R*-Ri)^
(4)
V = n I [Ri + (R2-Ri)j^ dz
= |tt^i 2 +R1R2 + R2 2 ). (5)
This formula can be generalized to any PYRAMID by
letting Ai be the base AREAS of the top and bottom of
the frustum. Then the VOLUME can be written as
V = \h(A x + A 2 + VA1A2 )• (6)
The weighted mean of z over the frustum is
(z)
Jo
z[r(z)] 2 dz = ±h 2 (Ri + 2R 1 R 2 + 3i? 2 2 ).
The CENTROID is then given by
. _ (z) _ h(R! 2 + R1R2 + R2 2 )
V 4(i?i 2 + 2i2i#2 + 3^2 2 )
(7)
(8)
(Beyer 1987, p. 133). The special case of the Cone is
given by taking R 2 = 0, yielding z = /i/4.
see also Cone, Frustum, Pyramidal Frustum,
Spherical Segment
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 129-130 and
133, 1987.
Conical Function
Conjecture 307
Conical Function
Functions which can be expressed in terms of LEGENDRE
Functions of the First and Second Kinds. See
Abramowitz and Stegun (1972, p. 337).
* Jo
cosh(pt) dt
-y/2(cos t — cos 9)
, > , , I cos(pt) dt
^-i/2 T » P v w /y Q ^ 2 (coshi + cos0)
/* cosh(jtf) dt
y / 2(cos £ — cos 0)
see a/so Toroidal Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Conical Func-
tions." §8.12 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 337, 1972.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA; MIT Press, p. 1464,
1980.
Conical Spiral
A surface modeled after the shape of a Seashell. One
parameterization (left figure) is given by
x = 2[l - e u/K ^ } }cosucos\\v)
y = 2[-l + e u/(6lv) ] cos 2 (|t;)sinu
z — \ — e /y } — sin v + e
i/(6ir)
smi),
(1)
(2)
(3)
where v e [0,2tt), and u G [0,6tt) (Wolfram). Nord-
strand gives the parameterization
x = ( 1 ) (1 + costi) + c\ cos(nv) (4)
x = f 1 — -— J (1 + cos w) + c sin(nv) (5)
bv
2^
+ asinu(l-£)
(6)
for u, v e [0,27r] (right figure with a = 0.2, b = 1, c
0.1, and n = 2).
References
Gray, A. "Sea Shells." §11.6 in Modern Differential Geome-
try of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 223-223, 1993.
Nordstrand, T. "Conic Spiral or Seashell." http://www.uib.
no/people/nf ytn/shelltxt .htm.
Wolfram Research "Mathematica Version 2.0 Graphics
Gallery." http : // www . maths our ce . com/cgi-bin/ Math
Source/Applications/Graphics/3D/0207-155.
Conical Wedge
The Surface also called the Conocuneus of Wallis
and given by the parametric equation
X = u cos V
y = usinv
z = c(l — 2 cos 2 v).
see also Cylindrical Wedge, Wedge
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 302, 1993.
Conjecture
A proposition which is consistent with known data, but
has neither been verified nor shown to be false. It is
synonymous with HYPOTHESIS.
see also abc Conjecture, Abhyankar's Conjec-
ture, Ablowitz-Ramani-Segur Conjecture, An-
drica's Conjecture, Annulus Conjecture, Ar-
goh's Conjecture, Artin's Conjecture, Ax-
iom, Bachet's Conjecture, Bennequin's Conjec-
ture, Bieberbach Conjecture, Birch Conjec-
ture, Blaschke Conjecture, Borsuk's Conjec-
ture, Borwein Conjectures, Braun's Conjec-
ture, Brocard's Conjecture, Burnside's Con-
jecture, Carmichael's Conjecture, Catalan's
Conjecture, Cramer Conjecture, de Polig-
nac's Conjecture, Diesis, Dodecahedral Con-
jecture, Double Bubble Conjecture, Eber-
hart's Conjecture, Euler's Conjecture, Euler
Power Conjecture, Euler Quartic Conjecture,
Feit-Thompson Conjecture, Fermat's Conjec-
ture, Flyping Conjecture, Gilbreath's Conjec-
ture, Giuga's Conjecture, Goldbach Conjec-
ture, Grimm's Conjecture, Guy's Conjecture,
Hardy-Littlewood Conjectures, Hasse's Con-
jecture, Heawood Conjecture, Hypothesis, Ja-
cobian Conjecture, Kaplan- Yorke Conjecture,
Keller's Conjecture, Kelvin's Conjecture, Ke-
pler Conjecture, Kreisel Conjecture, Rum-
mer's Conjecture, Lemma, Local Density Con-
jecture, Mertens Conjecture, Milin Conjec-
ture, Milnor's Conjecture, Mordell Conjec-
ture, Netto's Conjecture, Nirenberg's Con-
jecture, Ore's Conjecture, Pade Conjecture,
308
Conjugacy Class
Conjunction
Palindromic Number Conjecture, Pillai's Con-
jecture, Poincare Conjecture, Polya Con-
jecture, Porism, Prime /c-Tuples Conjecture,
Prime Patterns Conjecture, Prime Power Con-
jecture, Proof, Quillen-Lichtenbaum Conjec-
ture, Ramanujan-Petersson Conjecture, Ro-
bertson Conjecture, Safarevich Conjecture,
Sausage Conjecture, Schanuel's Conjecture,
schisma, scholz conjecture, seifert conjec-
TURE, Selfridge's Conjecture, Shanks' Con-
jecture, Smith Conjecture, Swinnerton-Dyer
Conjecture, Szpiro's Conjecture, Tait's Ham-
iltonian Graph Conjecture, Tait's Knot Con-
jectures, Taniyama-Shimura Conjecture, Tau
Conjecture, Theorem, Thurston's Geometriza-
tion Conjecture, Thwaites Conjecture, Vo-
jta's Conjecture, Wang's Conjecture, Waring's
Prime Conjecture, Waring's Sum Conjecture,
Zarankiewicz's Conjecture
References
Rivera, C. "Problems &; Puzzles (Conjectures)." http://
www.sci.net.mx/-crivera/ppp/conjectures.htm.
Conjugacy Class
A complete set of mutually conjugate GROUP elements.
Each element in a GROUP belongs to exactly one class,
and the identity (I = 1) element is always in its own
class. The Orders of all classes must be integral Fac-
tors of the Order of the Group. Prom the last two
statements, a Group of Prime order has one class for
each element. More generally, in an Abelian GROUP,
each element is in a conjugacy class by itself. Two opera-
tions belong to the same class when one may be replaced
by the other in a new COORDINATE SYSTEM which is ac-
cessible by a symmetry operation (Cotton 1990, p. 52).
These sets correspond directly to the sets of equivalent
operation.
Let G be a Finite Group of Order |G|. If |G| is Odd,
then
|G| = s (mod 16)
(Burnside 1955, p. 295). Furthermore, if every Prime
Pi Dividing |G| satisfies pi = 1 (mod 4), then
|G| = s (mod 32)
(Burnside 1955, p. 320). Poonen (1995) showed that if
every Prime pi Dividing |G| satisfies pi = 1 (mod m)
for m > 2, then
|G| = s (mod 2m 2 ) .
References
Burnside, W. Theory of Groups of Finite Order, 2nd ed. New
York: Dover, 1955.
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, 1990.
Poonen, B. "Congruences Relating the Order of a Group to
the Number of Conjugacy Classes." Amer. Math. Monthly
102, 440-442, 1995.
Conjugate Element
Given a GROUP with elements A and X, there must
be an element B which is a SIMILARITY TRANSFORMA-
TION of A, B = X~ 1 AX so A and B are conjugate with
respect to X. Conjugate elements have the following
properties:
1. Every element is conjugate with itself.
2. If A is conjugate with B with respect to X ) then B
is conjugate to A with respect to X.
3. If .A is conjugate with B and C, then B and C are
conjugate with each other.
see also CONJUGACY CLASS, CONJUGATE SUBGROUP
Conjugate Gradient Method
An Algorithm for calculating the Gradient V/(P)
of a function at an n-D point P. It is more robust than
the simpler Steepest Descent Method.
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. 413-417, 1992.
Conjugate Points
see Harmonic Conjugate Points, Isogonal Con-
jugate, Isotomic Conjugate Point
Conjugate Subgroup
A SUBGROUP H of an original GROUP G has elements ft*.
Let x be a fixed element of the original GROUP G which
is not a member of H. Then the transformation xhiX~ l ,
(i = 1, 2, ... ) generates a conjugate SUBGROUP xHx~ x .
If, for all x, xHx' 1 = H, then H is a SELF-CONJUGATE
(also called Invariant or Normal) Subgroup. All
Subgroups of an Abelian Group are invariant.
Conjugation
1 2 n-1
I
I n
-1
B
A
A type I Markov Move.
see also Markov Moves, Stabilization
Conjunction
A product of Ands, denoted
A*.
fc+i
see also And, Disjunction
Connected Graph
Connection Coefficient
309
Connected Graph
1 •
2 • •
A
ummh
A GRAPH which is connected (as a TOPOLOGICAL
SPACE), i.e., there is a path from any point to any other
point in the Graph. The number of n- Vertex (unla-
beled) connected graphs for n = 1, 2, ... are 1, 1, 2, 6,
21, 112, 853, 11117, . . . (Sloane's A001349).
References
Chartrand, G. "Connected Graphs." §2.3 in Introductory
Graph Theory. New York: Dover, pp. 41-45, 1985.
Sloane, N. J. A. Sequence A001349/M1657 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Connected Set
A connected set is a Set which cannot be partitioned
into two nonempty SUBSETS which are open in the rel-
ative topology induced on the Set. Equivalently, it is
a Set which cannot be partitioned into two nonempty
Subsets such that each Subset has no points in com-
mon with the closure of the other.
The Real Numbers are a connected set.
see also Closed Set, Empty Set, Open Set, Set,
Subset
Connected Space
A SPACE D is connected if any two points in D can be
connected by a curve lying wholly within D. A SPACE
is O-connected (a.k.a. Pathwise-Connected) if every
MAP from a O-Sphere to the SPACE extends contin-
uously to the 1-DlSK. Since the 0-Sphere is the two
endpoints of an interval (I-Disk), every two points have
a path between them. A space is 1-connected (a.k.a.
Simply Connected) if it is O-connected and if every
Map from the 1-Sphere to it extends continuously to
a Map from the 2-DlSK. In other words, every loop
in the SPACE is contractible. A SPACE is n-MULTlPLY
Connected if it is (ra — l)-connected and if every Map
from the n-SPHERE into it extends continuously over the
(n + 1)-Disk.
A theorem of Whitehead says that a SPACE is infinitely
connected Iff it is contractible.
see also CONNECTIVITY, LOCALLY PATHWISE-CON-
nected Space, Multiply Connected, Pathwise-
CONNECTED, SIMPLY CONNECTED
Connected Sum
The connected sum Mi#M 2 of n-manifolds Mi and M 2
is formed by deleting the interiors of n-BALLS B™ in
M™ and attaching the resulting punctured MANIFOLDS
Mi-Bi to each other by a HOMEOMORPHISM h : dB 2 ->
dBi, so
Mi#M 2 - (Ma - Si) (J(M 2 - B 2 ).
Bi is required to be interior to Mi and dBi bicollared in
Mi to ensure that the connected sum is a MANIFOLD.
The connected sum of two Knots is called a KNOT Sum.
see also KNOT SUM
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 39, 1976.
Connected Sum Decomposition
Every COMPACT 3-MANIFOLD is the CONNECTED SUM
of a unique collection of Prime 3-Manifolds.
see also JACO-SHALEN-JOHANNSON TORUS DECOMPO-
SITION
Connection
see Connection Coefficient, Gauss-Manin Con-
nection
Connection Coefficient
A quantity also known as a CHRISTOFFEL Symbol OF
the Second Kind. Connection Coefficients are de-
fined by
'r^ = e a ■ (V^) (1)
(long form) or
T£ 7 EEe" -(V 7 e>),
(abbreviated form), and satisfy
(long form) and
W* = r|^e«
V 7 e/3 = r^ 7 e Q
(2)
(3)
(4)
(abbreviated form).
Connection COEFFICIENTS are not TENSORS, but have
TENSOR-like Contravariant and Covariant indices.
A fully Covariant connection Coefficient is given
by
r Q/ a 7 = 2 G/a0,7 + 9<*lS + C <*/37 + C a70 — C Piol), (5)
where the gs are the Metric Tensors, the cs are Com-
mutation Coefficients, and the commas indicate the
310
Connection Coefficient
Consecutive Number Sequences
Comma Derivative. In an Orthonormal Basis,
Pa/3,7 = and # M7 = £ M7 , so
F a {3y
~~ * a/3#M7 _1 ^ — 2 1 Cq: /37 "•" c »7/3 —
C/3 7 a) (6)
and
r ijfc = for z ^ j ^ k
(7)
Tiik = -\iB i0Ti * k
(8)
r r 1 d °"
m - m - 2 Qxj
(9)
Tij =0 fOTi^j^k
(10)
r? i= J **' forz/fc
2fffcfc era*
(11)
r i p i 1 Qgu Idlngu
y - j. - 25 .. ^ - 2 9xJ
(12)
For Tensors of Rank 3, the connection Coefficients
may be concisely summarized in Matrix form:
r" =
l 9r
1 4>r
1 vB
1 99
V 9 1
L 9
T 9
(13)
Connection COEFFICIENTS arise in the computation of
Geodesics. The Geodesic Equation of free motion
is
dr 2 = - Va0 de df , (14)
(15)
t2/-q:
Expanding,
d (dZ a dx*\ _ di a d 2 x» d 2 C dx» dx u
dr \dx^ dr ) ~ dx^ dr 2 dx^dx u dr dr
8C d 2 x" 0x x d 2 C dx» dx u dx x _
dxv dr 2 d£ a ' dx^dx" dr dr d£ a ~
But
dr dx x x
so
,A^ ,
r d 2 c dx x ~
dx" dx u
** dr 2 +
dx"dx v d£ a
dr dr
_ d 2 x x A dx» dx u
dr 2 " v dr dr
where
TpA
d 2 c dx x
dx^dx lJ d£ a '
=
(16)
(17)
(18)
(19)
(20)
see also CARTAN TORSION COEFFICIENT, CHRISTOF-
fel Symbol of the First Kind, Christoffel Sym-
bol of the Second Kind, Comma Derivative, Com-
mutation Coefficient, Curvilinear Coordinates,
Semicolon Derivative, Tensor
Connectivity
see Connected Space, Edge Connectivity, Ver-
tex Connectivity
Connes Function
0^5
r~ '3
-1 ^OTS 075 1
The Apodization Function
A(x)
(-5)"-
Its Full Width at Half Maximum is a/ 4 - 2y/2a,
and its Instrument Function is
J 5 / 2 (27rka)
I(x) = 8aV27r
(27rfca)
5/2 '
where J n (z) is a BESSEL FUNCTION OF THE FIRST
Kind.
see also APODIZATION FUNCTION
Conocuneus of Wallis
see Conical Wedge
Conoid
see Plucker's Conoid, Right Conoid
Consecutive Number Sequences
Consecutive number sequences are sequences con-
structed by concatenating numbers of a given type.
Many of these sequences were considered by Smaran-
dache, so they are sometimes known as SMARANDACHE
Sequences.
The nth term of the consecutive integer sequence con-
sists of the concatenation of the first n Positive inte-
gers: 1, 12, 123, 1234, ... (Sloane's A007908; Smaran-
dache 1993, Dumitrescu and Seleacu 1994, sequence 1;
Mudge 1995; Stephen 1998). This sequence gives the
digits of the Champernowne Constant and contains
no Primes in the first 4,470 terms (Weisstein). This
is roughly consistent with simple arguments based on
the distribution of prime which suggest that only a sin-
gle prime is expected in the first 15,000 or so terms.
The number of digits of the n term can be computed
by noticing the pattern in the following table, where
d — [log 10 nj + 1 is the number of digits in n.
d n Range Digits
_ __ _
2 10-99 9 + 2(n - 9)
3 100-999 9 + 90 • 2 + 3(n - 99)
4 1000-9999 9 + 90 • 2 + 900 ■ 3 + 4(n - 999)
Consecutive Number Sequences
Consistency 311
Therefore, the number of digits D(n) in the nth term
can be written
D(n) = d(n+l- 10 d_1 ) + ^ 9k ■ 10 fc
= (n + l)d-
10 d -l
where the second term is the Repunit Rd.
The nth term of the reverse integer sequence consists
of the concatenation of the first n POSITIVE integers
written backwards: 1, 21, 321, 4321, ... (Sloane's
A000422; Smarandache 1993, Dumitrescu and Seleacu
1994, Stephen 1998). The only Prime in the first
3,576 terms (Weisstein) of this sequence is the 82nd
term 828180... 321 (Stephen 1998), which has 155 dig-
its. This is roughly consistent with simple arguments
based on the distribution of prime which suggest that a
single prime is expected in the first 15,000 or so terms.
The terms of the reverse integer sequence have the same
number of digits as do the consecutive integer sequence.
The concatenation of the first n PRIMES gives 2, 23,
235, 2357, 235711, ... (Sloane's A019518; Smith 1996,
Mudge 1997). This sequence converges to the digits
of the Copeland-Erdos Constant and is Prime for
terms 1, 2, 4, 128, 174, 342, 435, 1429, ... (Sloane's
A046035; Ibstedt 1998, pp. 78-79), with no others less
than 2,305 (Weisstein).
The concatenation of the first n ODD NUMBERS gives
1, 13, 135, 1357, 13579, ... (Sloane's A019519; Smith
1996, Marimutha 1997, Mudge 1997). This sequence is
PRIME for terms 2, 10, 16, 34, 49, 2570, . . . (Sloane's
A046036; Weisstein, Ibstedt 1998, pp. 75-76), with no
others less than 2,650 (Weisstein). The 2570th term,
given by 1 3 5 7. . . 5137 5139, has 9725 digits and was
discovered by Weisstein in Aug. 1998.
The concatenation of the first n EVEN NUMBERS gives
2, 24, 246, 2468, 246810, ... (Sloane's A019520; Smith
1996; Marimutha 1997; Mudge 1997; Ibstedt 1998,
pp. 77-78).
The concatenation of the first n SQUARE NUMBERS gives
1, 14, 149, 14916, ... (Sloane's A019521; Marimutha
1997). The only PRIME in the first 2,090 terms is the
third term, 149, (Weisstein).
The concatenation of the first n CUBIC NUMBERS gives
1, 18, 1827, 182764, ... (Sloane's A019522; Marimutha
1997). There are no Primes in the first 1,830 terms
(Weisstein) .
see also CHAMPERNOWNE CONSTANT, CONCATENA-
TION, Copeland-Erdos Constant, Cubic Num-
ber, Demlo Number, Even Number, Odd Number,
Smarandache Sequences, Square Number
References
Dumitrescu, C. and Seleacu, V. (Ed.). Some Notions and
Questions in Number Theory. Glendale, AZ: Erhus Uni-
versity Press, 1994.
Ibstedt, H. "Smarandache Concatenated Sequences." Ch. 5
in Computer Analysis of Number Sequences. Lupton, AZ:
American Research Press, pp. 75-79, 1998.
Marimutha, H. "Smarandache Concatenate Type Se-
quences." Bull. Pure Appl. Set. 16E, 225-226, 1997.
Mudge, M. "Top of the Class." Personal Computer World,
674-675, June 1995.
Mudge, M. "Not Numerology but Numeralogy!" Personal
Computer World, 279-280, 1997.
Smarandache, F. Only Problems, Not Solutions!, J^th ed.
Phoenix, AZ: Xiquan, 1993.
Smith, S. "A Set of Conjectures on Smarandache Sequences."
Bull. Pure Appl. Sci. 15E, 101-107, 1996.
Stephen, R. W. "Factors and Primes in Two Smarandache Se-
quences." Smarandache Notions J. 9, 4—10, 1998. http://
www.tmt.de/-stephen/sm.ps.gz.
Conservation of Number Principle
A generalization of Poncelet's PERMANENCE OF MATH-
EMATICAL Relations Principle made by H. Schubert
in 1874-79. The conservation of number principle as-
serts that the number of solutions of any determinate
algebraic problem in any number of parameters under
variation of the parameters is invariant in such a man-
ner that no solutions become Infinite. Schubert called
the application of this technique the CALCULUS of Enu-
merative Geometry.
see also DUALITY PRINCIPLE, HlLBERT'S PROBLEMS,
Permanence of Mathematical Relations Princi-
ple
References
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, p. 340, 1945.
Conservative Field
The following conditions are equivalent for a conserva-
tive Vector Field:
1. For any oriented simple closed curve C, the Line
Integral § c F * ds = 0.
2. For any two oriented simple curves C\ and Ci with
the same endpoints, J F * ds — J F * ds.
3. There exists a SCALAR POTENTIAL FUNCTION /
such that F = V/, where V is the GRADIENT.
4. The Curl V x F = 0.
see also CURL, GRADIENT, LINE INTEGRAL, POTENTIAL
Function, Vector Field
Consistency
The absence of contradiction (i.e., the ability to prove
that a statement and its Negative are both true) in an
Axiomatic Theory is known as consistency,
see also Complete Axiomatic Theory, Consis-
tency Strength
312 Consistency Strength
Constant
Consistency Strength
If the CONSISTENCY of one of two propositions implies
the Consistency of the other, the first is said to have
greater consistency strength.
Constant
Any Real Number which is "significant" (or interest-
ing) in some way. In this work, the term "constant" is
generally reserved for REAL nonintegral numbers of in-
terest, while "NUMBER" is reserved for interesting INTE-
GERS (e.g., Brun's Constant, but Beast Number).
Certain constants are known to many Decimal Digits
and recur throughout many diverse areas of mathemat-
ics, often in unexpected and surprising places (e.g., Pi,
e, and to some extent, the Euler-Mascheroni Con-
stant 7). Other constants are more specialized and
may be known to only a few DIGITS. S. Plouffe main-
tains a site about the computation and identification of
numerical constants. Plouffe's site also contains a page
giving the largest number of DIGITS computed for the
most common constants. S. Finch maintains a delight-
ful, more expository site containing detailed essays and
references on constants both common and obscure.
see also Abundant Number, Alladi-Grinstead
Constant, Apery's Constant, Archimedes' Con-
stant, Artin's Constant, Backhouse's Constant,
Beraha Constants, Bernstein's Constant, Bloch
Constant, Brun's Constant, Cameron's Sum-
Free Set Constant, Carlson-Levin Constant,
Catalan's Constant, Chaitin's Constant, Cham-
pernowne Constant, Chebyshev Constants,
Chebyshev-Sylvester Constant, Comma of Didy-
mus, Comma of Pythagoras, Conic Constant,
Constant Function, Constant Problem, Con-
tinued Fraction Constant, Conway's Constant,
Copeland-Erdos Constant, Copson-de Bruijn
Constant, de Bruijn-Newman Constant, Delian
Constant, Diesis, Du Bois Raymond Constants, e,
Ellison-Mendes-France Constant, Erdos Recip-
rocal Sum Constants, Euler-Mascheroni Con-
stant, Extreme Value Distribution, Favard
Constants, Feller's Coin-Tossing Constants,
Fransen-Robinson Constant, Freiman's Con-
stant, Gauss's Circle Problem, Gauss's Con-
stant, Gauss-Kuzmin-Wirsing Constant, Gel-
fond-Schneider Constant, Geometric Proba-
bility Constants, Gibbs Constant, Glaisher-
Kinkelin Constant, Golden Mean, Golomb
Constant, Golomb-Dickman Constant, Gom-
pertz Constant, Grossman's Constant, Gro-
thendieck's Majorant, Hadamard-Vallee Pous-
sin Constants, Hafner-Sarnak-McCurley Con-
stant, Halphen Constant, Hard Square En-
tropy Constant, Hardy-Littlewood Constants,
Hermite Constants, Hilbert's Constants, Infi-
nite Product, Iterated Exponential Constants,
Khintchine's Constant, Khintchine-Levy Con-
stant, Koebe's Constant, Kolmogorov Con-
stant, Lal's Constant, Landau Constant, Lan-
dau-Kolmogorov Constants, Landau-Ramanujan
Constant, Lebesgue Constants (Fourier Se-
ries), Lebesgue Constants (Lagrange Interpo-
lation), Legendre's Constant, Lehmer's Con-
stant, Lengyel's Constant, Levy Constant, Lin-
nik's Constant, Liouville's Constant, Liouville-
Roth Constant, Ludolph's Constant, Madelung
Constants, Magic Constant, Magic Geometric
Constants, Masser-Gramain Constant, Mertens
Constant, Mills' Constant, Moving Sofa Con-
stant, Napier's Constant, Nielsen-Ramanujan
Constants, Niven's Constant, Omega Constant,
One-Ninth Constant, Otter's Tree Enumera-
tion Constants, Parity Constant, Pi, Pisot-
Vijayaraghavan Constants, Plastic Constant,
Plouffe's Constant, Polygon Circumscribing
Constant, Polygon Inscribing Constant, Por-
ter's Constant, Pythagoras's Constant, Quad-
ratic Recurrence, Quadtree, Rabbit Constant,
Ramanujan Constant, Random Walk, Renyi's
Parking Constants, Robbin Constant, Salem
Constants, Self-Avoiding Walk, Shah-Wilson
Constant, Shallit Constant, Shapiro's Cyclic
Sum Constant, Sierpinski Constant, Silver Con-
stant, Silverman Constant, Smarandache Con-
stants, Soldner's Constant, Sphere Packing,
Stieltjes Constants, Stolarsky-Harborth Con-
stant, Sylvester's Sequence, Thue Constant,
Thue-Morse Constant, Totient Function Con-
stants, Traveling Salesman Constants, Tree
Searching, Twin Primes Constant, Varga's
Constant, W2-Constant, WeierstraJ3 Constant,
Whitney-Mikhlin Extension Constants, Wil-
braham-Gibbs Constant, Wirtinger-Sobolev Iso-
perimetric Constants
References
Borwein, J. and Borwein, P. A Dictionary of Real Numbers.
London: Chapman & Hall, 1990.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/constant .html.
Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
1983.
Plouffe, S. "Inverse Symbolic Calculator Table of Constants."
http://www.cecm.sfu.ca/projects/ISC/Ijd.html.
Plouffe, S. "Plouffe's Inverter." http://www.lacim.uqam.ca/
Pi/-
Plouffe, S. "Plouffe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records . html.
Wells, D. W. The Penguin Dictionary of Curious and In-
teresting Numbers. Harmondsworth, England: Penguin
Books, 1986.
Constant Function
Constant Function
-1 -0.5 0.5 1
A Function f(x) = c which does not change as its
parameters vary. The Graph of a 1-D constant Func-
tion is a straight LINE. The DERIVATIVE of a constant
Function c is
d o, (1)
dx
and the INTEGRAL is
/
cdx ■
(2)
The Fourier Transform of the constant function
f(x) = 1 is given by
m
f
J — c
e- 2 " ikx dx = 6(k),
(3)
where 5(k) is the DELTA FUNCTION.
see also Fourier Transform — 1
References
Spanier, J. and Oldham, K. B. "The Constant Function c."
Ch. 1 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 11-14, 1987.
Constant Precession Curve
see Curve of Constant Precession
Constant Problem
Given an expression involving known constants, integra-
tion in finite terms, computation of limits, etc., deter-
mine if the expression is equal to Zero. The constant
problem is a very difficult unsolved problem in Trans-
cendental Number theory. However, it is known
that the problem is UNDECIDABLE if the expression in-
volves oscillatory functions such as Sine. However, the
Ferguson-Forcade Algorithm is a practical algo-
rithm for determining if there exist integers ai for given
real numbers Xi such that
Constructible Number 313
References
Bailey, D. H. "Numerical Results on the Transcendence of
Constants Involving 7r, e, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.
Sackell, J. "Zero-Equivalence in Function Fields Defined by
Algebraic Differential Equations." Trans. Amer. Math.
Soc. 336, 151-171, 1993.
Constant Width Curve
see Curve of Constant Width
Constructible Number
A number which can be represented by a Finite num-
ber of Additions, Subtractions, Multiplications,
Divisions, and Finite Square Root extractions of in-
tegers. Such numbers correspond to LINE SEGMENTS
which can be constructed using only STRAIGHTEDGE
and Compass.
All RATIONAL NUMBERS are constructible, and all con-
structible numbers are ALGEBRAIC NUMBERS (Courant
and Robbins 1996, p. 133). If a CUBIC EQUATION with
rational coefficients has no rational root, then none of
its roots is constructible (Courant and Robbins, p. 136).
In particular, let F be the Field of RATIONAL NUM-
BERS. Now construct an extension field Fi of con-
structible numbers by the adjunction of y/ko, where ko
is in Fo, but y/ko is not, consisting of all numbers of the
form ao 4- &o Vko, where ao, bo € Fo- Next, construct an
extension field F^ of F± by the adjunction of \/ki, de-
fined as the numbers a\ +b\\fk\, where ai, b\ E i*i, and
fci is a number in F\ for which y/kl does not lie in F\.
Continue the process n times. Then constructible num-
bers are precisely those which can be reached by such
a sequence of extension fields F n , where n is a measure
of the "complexity" of the construction (Courant and
Robbins 1996).
see also Algebraic Number, Compass, Con-
structible Polygon, Euclidean Number, Ratio-
nal Number, Straightedge
References
Courant, R. and Robbins, H. "Constructible Numbers and
Number Fields." §3.2 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 127-134, 1996.
a±xi + CL2X2 + ...-(- a n x n = 0,
or else establish bounds within which no relation can
exist (Bailey 1988).
see also Ferguson-Forcade Algorithm, Integer
Relation, Schanuel's Conjecture
314 Constructible Polygon
Contact Triangle
Constructible Polygon
B
Pentagon 17-gon
Compass and Straightedge constructions dating
back to Euclid were capable of inscribing regular poly-
gons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48,
64, . . . , sides. However, this listing is not a complete
enumeration of "constructible" polygons. A regular n-
gon (n > 3) can be constructed by STRAIGHTEDGE and
Compass Iff
n = 2 fc pip 2 ■ • -Pa,
where k is in INTEGER > and the pt are distinct Fer-
mat Primes. Fermat Numbers are of the form
F m = 2 2m + 1,
where m is an INTEGER > 0. The only known PRIMES of
this form are 3, 5, 17, 257, and 65537. The fact that this
condition was SUFFICIENT was first proved by Gauss in
1796 when he was 19 years old. That this condition was
also Necessary was not explicitly proven by Gauss, and
the first proof of this fact is credited to Wantzel (1836).
see also Compass, Constructible Number, Ge-
ometric Construction, Geometrography, Hep-
tadecagon, Hexagon, Octagon, Pentagon, Poly-
gon, Square, Straightedge, Triangle
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 94-96,
1987.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 119, 1996.
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.
Dixon, R. "Compass Drawings." Ch. 1 in Mathographics.
New York: Dover, pp. 1-78, 1991.
Gauss, C F. §365 and 366 in Disquisitiones Arithmeticae.
Leipzig, Germany, 1801. Translated by A. A. Clarke. New
Haven, CT: Yale University Press, 1965.
Kazarinoff, N. D. "On Who First Proved the Impossibil-
ity of Constructing Certain Regular Polygons with Ruler
and Compass Alone." Amer. Math. Monthly 75, 647-648,
1968.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 137-138, 1990.
Wantzel, P. L. "Recherches sur les moyens de reconnaitre si
un Probleme de Geometrie peut se resoudre avec la regie
et le compas." J. Math, pures appliq. 1, 366-372, 1836.
Construction
see Geometric Construction
Constructive Dilemma
A formal argument in LOGIC in which it is stated that
(1) P => Q and R => S (where => means "IMPLIES"),
and (2) either P or R is true, from which two statements
it follows that either Q or S is true.
see also Destructive Dilemma, Dilemma
Contact Angle
contact
angle
The Angle a between the normal vector of a Sphere
(or other geometric object) at a point where a PLANE is
tangent to it and the normal vector of the plane. In the
above figure,
-(I)
-fir)-
see also Spherical Cap
Contact Number
see Kissing Number
Contact Triangle
The TRIANGLE formed by the points of intersection of
a Triangle T's Incircle with T. This is the Pedal
Triangle of T with the Incenter as the Pedal Point
(c.f., Tangential Triangle). The contact triangle
Content
Continued Fraction
315
and Tangential Triangle are perspective from the
Gergonne Point.
see also GERGONNE POINT, PEDAL TRIANGLE, TAN-
GENTIAL Triangle
References
Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Content
The generalized Volume for an n-D object (the "Hy-
pervolume").
see also VOLUME
Contiguous Function
A HYPERGEOMETRIC FUNCTION in which one parame-
ter changes by +1 or —1 is said to be contiguous. There
are 26 functions contiguous to 2-Fi(a, 0, c; x) taking one
pair at a time. There are 325 taking two or more pairs
at a time. See Abramowitz and Stegun (1972, pp. 557-
558).
see also Hypergeometric Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.
Continued Fraction
A "general" continued fraction representation of a Real
Number x is of the form
x = 6 +
ai
6i +
a 2
(1)
&2 +
a3
& 3 + . . .
which can be written
x = & +
a\ 0,2
h+b~2~+
(2)
The Simple Continued Fraction representation of x
(which is usually what is meant when the term "contin-
ued fraction" is used without qualification) of a number
is given by
is the integral part of a (where [x\ is the FLOOR FUNC-
TION),
= 1 — 1 (6 >
L x - ao J
ai
is the integral part of the RECIPROCAL of x — ao, ai is the
integral part of the reciprocal of the remainder, etc. The
quantities a» are called PARTIAL QUOTIENTS. An ar-
chaic word for a continued fraction is ANTHYPHAIRETIC
Ratio.
Continued fractions provide, in some sense, a series of
"best" estimates for an IRRATIONAL NUMBER. Func-
tions can also be written as continued fractions, pro-
viding a series of better and better rational approxima-
tions. Continued fractions have also proved useful in
the proof of certain properties of numbers such as e and
TV (Pi). Because irrationals which are square roots of
Rational Numbers have periodic continued fractions,
an exact representation for a tabulated numerical value
(i.e., 1.414... for PYTHAGORAS'S CONSTANT, \/2) can
sometimes be found.
Continued fractions are also useful for finding near com-
mensurabilities between events with different periods.
For example, the Metonic cycle used for calendrical pur-
poses by the Greeks consists of 235 lunar months which
very nearly equal 19 solar years, and 235/19 is the sixth
Convergent of the ratio of the lunar phase (synodic)
period and solar period (365.2425/29.53059). Continued
fractions can also be used to calculate gear ratios, and
were used for this purpose by the ancient Greeks (Guy
1990).
If only the first few terms of a continued fraction are
kept, the result is called a CONVERGENT. Let P n /Q n
be convergent s of a nonsimple continued fraction. Then
P_i = l Q-i=0
Po = b Qo = l
(7)
(8)
and subsequent terms are calculated from the Recur-
rence Relations
P^bjPj-i+ajPj-2 (9)
Qj ^bjQj-i+ajQj-2 (10)
for j = 1, 2, . . . , n. It is also true that
x = ao +
ai +
a 2 +
(3)
0,3 + ...
which can be written in a compact abbreviated NOTA-
TION as
x = [a ,ai,a 2 ,a 3) . . .]. (4)
Here,
ao = [x]
(5)
± nUcn
P n -iQ n = (-l) n - 1 l[[a k . (11)
The error in approximating a number by a given CON-
VERGENT is roughly the MULTIPLICATIVE INVERSE of
the square of the DENOMINATOR of the first neglected
term.
A finite simple continued fraction representation termi-
nates after a finite number of terms. To "round" a con-
tinued fraction, truncate the last term unless it is ±1,
316 Continued Fraction
Continued Fraction
in which case it should be added to the previous term
(Beeler et al. 1972, Item 101A). To take one over a con-
tinued fraction, add (or possibly delete) an initial term.
To negate, take the NEGATIVE of all terms, optionally-
using the identity
[-a, -6, -c, -d, . - .] = [-a - 1, 1, b - 1, c, d, . . .]. (12)
A particularly beautiful identity involving the terms of
the continued fraction is
[ao,ai, . . . ,a n ] _ [a n ,a n _i, . . . , oi, ao]
[ao, ai, . . . , a n -i] [a n ,a n _i, . . . , ai]
(13)
Finite simple fractions represent rational numbers and
all rational numbers are represented by finite continued
fractions. There are two possible representations for a
finite simple fraction:
r rt i _ J [ai,- ■ ■ ,a n -i,a n - 1,1
[ill j • • • j U-nJ — S r -,
^ [ai, . . . ,a n _2,a n _i -+• 1
] for a n > 1
] for a n = 1.
(14)
On the other hand, an infinite simple fraction represents
a unique IRRATIONAL Number, and each IRRATIONAL
NUMBER has a unique infinite continued fraction.
Consider the CONVERGENTS p n /q n of a simple continued
fraction, and define
p_i =0 g-i = 1
po = 1 q =
pi = ai gi = 1.
(15)
(16)
(17)
Then subsequent terms can be calculated from the RE-
CURRENCE Relations
Pi = CLiPi-i -\-pi~2
qi = a%qi-i + qi-2-
(18)
(19)
The Continued Fraction Fundamental Recur-
rence Relation for simple continued fractions is
(20)
p n qn-i -
Pn-iq-n
—
(-
-i)"
It
is also true that if a\
^0,
Pn _
Pn-l
[On j O n ~
i»-
,ai]
q n
qn-i
[a n ,...
a-2
■
Furthermore,
Pn _ Pn+1 ~ Pn-1
qn qn+i — q n ~i
(21)
(22)
(23)
p n = (n- l)p n -i + (n - l)p n -2 + (n - 2)p n _ 3
+ ... + 3p 2 + 2pi +pi + 1. (24)
Also, if p/g > 1 and
then
Similarly, if p/q < 1 so
then
The convergents also satisfy
v r
- = [0,1,0,2, •
q
■ * ) On\,
(25)
- = [0,oi,.
P
. . ,a n ].
(26)
1 so
- = [0,ai,.
■ • ) &nj,
(27)
Max,..
P
,a n ].
(28)
Cn C n _ i
C n -2 =
(-1)"
5n^n-l
an(-l)"- 1
<Zn<Zn-2
(29)
(30)
The Odd convergents C2n+i of an infinite simple contin-
ued fraction form an INCREASING SEQUENCE, and the
Even convergents c 2n form a Decreasing Sequence
(so any Odd convergent is less than any EVEN conver-
gent). Summarizing,
Ci < C 3 < C 5 < • • • < C 2n +1 < • • "
< c 2n < • - < c 6 < c 4 < c 2 . (31)
Furthermore, each convergent for n > 3 lies between
the two preceding ones. Each convergent is nearer to the
value of the infinite continued fraction than the previous
one. Let p n /q n be the nth continued fraction represen-
tation. Then
1
(a n +i + 2)q n 2
Pn
q n
a n +iq n
(32)
The Square Root of a Squarefree Integer has a
periodic continued fraction of the form
\/n = [ai , a 2 , . . . , a„ , 2ai ]
(33)
(Rose 1994, p. 130). Furthermore, if D is not a Square
Number, then the terms of the continued fraction of
VD satisfy
< a n < 2VT>.
In particular,
[1,3]
|ac, a\
a + y/a 2 + 4
2
-1 + Vl + 4a
2
\/a 2 + l
b+yjb 2 + 4c
(34)
(35)
(36)
(37)
(38)
Continued Fraction
Continued Fraction 317
[oi, . . . ,a n ]
-(gn-i - Prx) + y/(q n -i - Pn) 2 + 4g n p n _i
2^
[ai,6i, . . . , b n ] = ai +
1
[bi, . . . ,6 n ]
[6i,. . . ,6 n ] =
[bi,.. -,b n ]pn +Pn-1
[bl, . . . , bn]^ + qfn_i
(39)
(40)
(41)
The first follows from
a. — n -\ —
n+-
n +
n +
(42)
n +
n +
Therefore,
(43)
n +
n +
n + .
so plugging (43) into (42) gives
1
a — n +
i / | = n + -
n + (a — n) a
Expanding
a 2 — na — 1 = 0,
and solving using the Quadratic Formula gives
n+ y/n 2 + 4
(44)
(45)
(46)
The analog of this treatment in the general case gives
_ ap n +Pn~l
ocq n + qn-i
(47)
The following table gives the repeating simple continued
fractions for the square roots of the first few integers
(excluding the trivial Square Numbers).
N a VN
N a VN
2 [1,2]
3 [1,1,2]
5 [2,4]
6 [2,2,4]
22 [4,1,2,4,2,1,8]
23 [4,1,3,1,8]
24 [4,1,8]
26 [5,10]
7 [2,1,1,1,4]
8 [2,1,4]
10 [3,6]
11 [3,376]
12 [3,276]
13 [3,1,1,1,1,6]
14 [3,1,2,1,6]
15 [3,1,6]
17 [4,8]
18 [4,4,8]
27 [5,5,10]
28 [5,3,2,3,10]
29 [5,2,1,1,2,10]
30 [5,2,10]
31 [5,1,1,3,5,3,1,1,10]
32 [5,1,1,1,10]
33 [5,1,2,1,10]
34 [5,1,4,1,10]
35 [5,1,10]
37 [6,12]
19 [4,2,1,3,1,2,8]
20 [4,2,8]
38 [6,6,12]
39 [6,4,12]
21 [4,1,1,2,1,1,8]
40 [6,3,12]
The periods of the continued fractions of the square
roots of the first few nonsquare integers 2, 3, 5, 6, 7,
8, 10, 11, 12, 13, ... (Sloane's A000037) are 1, 2, 1, 2,
4, 2, 1, 2, 2, 5, ... (Sloane's A013943; Williams 1981,
Jacobson et al. 1995). An upper bound for the length is
roughly 0{\nDy/D).
An even stronger result is that a continued fraction is
periodic Iff it is a Root of a quadratic Polynomial.
Calling the portion of a number x remaining after a
given convergent the "tail," it must be true that the
relationship between the number x and terms in its tail
is of the form
x = — -, (48)
cd + d y J
which can only lead to a Quadratic EQUATION.
LOGARITHMS log 6o b\ can be computed by defining b2,
. . . and the POSITIVE INTEGER m, ... such that
bi ni < bo < bi ni+1 b 2 -
b 2 n2 <6i <b 2 n2+1
b 3 =
bo
bi" 1
bi
b 2 n2
and so on. Then
lo g6 fe l = [*li, 712,713,...].
(49)
(50)
(51)
318
Continued Fraction
Continued Fraction
A geometric interpretation for a reduced FRACTION y/x
consists of a string through a Lattice of points with
ends at (1,0) and (x,y) (Klein 1907, 1932; Steinhaus
1983; Ball and Coxeter 1987, pp. 86-87; Davenport
1992). This interpretation is closely related to a simi-
lar one for the GREATEST Common DIVISOR. The pegs
it presses against (xi,yi) give alternate CONVERGENTS
y%/xi, while the other CONVERGENTS are obtained from
the pegs it presses against with the initial end at (0, 1).
The above plot is for e — 2, which has convergents 0, 1,
2/3,3/4,5/7, ....
Let the continued fraction for x be written [ai , a<i , . . . ,
a n ]. Then the limiting value is almost always KHINT-
chine's Constant
K = lim ( ai a 2 • * * a n ) 1/n = 2.68545 . .
(52)
Continued fractions can be used to express the Posi-
tive Roots of any Polynomial equation. Continued
fractions can also be used to solve linear DlOPHANTINE
Equations and the Pell Equation. Euler showed
that if a Convergent Series can be written in the
form
c\ + cic 2 + cic 2 c 3 + . . . , (53)
then it is equal to the continued fraction
Ci
1-
c 2
(54)
l + c 2
cz
l + c 3
Gosper has invented an ALGORITHM for performing ana-
lytic Addition, Subtraction, Multiplication, and
Division using continued fractions. It requires keep-
ing track of eight INTEGERS which are conceptually ar-
ranged at the Vertices of a Cube. The Algorithm
has not, however, appeared in print (Gosper 1996).
An algorithm for computing the continued fraction for
{ax + b)/(cx + d) from the continued fraction for x is
given by Beeler et al (1972, Item 101), Knuth (1981,
Exercise 4.5.3.15, pp. 360 and 601), and Fowler (1991).
(In line 9 of Knuth's solution, Xk <r- \_AjC\ should be
replaced by X k <- min([^/Cj , [{A + B)/(C + D)\).)
Beeler et al. (1972) and Knuth (1981) also mention the
bivariate case (axy -\-bx + cy + d) / '(Axy + Bx + Cy + D) .
see also GAUSSIAN BRACKETS, HURWITZ'S IRRA-
TIONAL Number Theorem, Khintchine's Con-
stant, Lagrange's Continued Fraction Theo-
rem, Lame's Theorem, Levy Constant, Pade Ap-
proximant, Partial Quotient, Pi, Quadratic Ir-
rational Number, Quotient-Difference Algo-
rithm, Segre's Theorem
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 19, 1972.
Acton, F. S. "Power Series, Continued Fractions, and Ra-
tional Approximations." Ch. 11 in Numerical Methods
That Work, 2nd printing. Washington, DC: Math. As-
soc. Amer., 1990.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 54-57
and 86-87, 1987.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, pp. 36-44, Feb. 1972.
Beskin, N. M. Fascinating Fractions. Moscow: Mir Publish-
ers, 1980.
Brezinski, C. History of Continued Fractions and Pade Ap-
proximants. New York: Springer- Verlag, 1980.
Conway, J. H. and Guy, R. K. "Continued Fractions." In The
Book of Numbers. New York: Springer- Verlag, pp. 176-
179, 1996.
Courant, R. and Robbins, H. "Continued Fractions. Dio-
phantine Equations." §2.4 in Supplement to Ch. 1 in What
is Mathematics?: An Elementary Approach to Ideas and
Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 49-51, 1996.
Davenport, H. §IV.12 in The Higher Arithmetic: An Intro-
duction to the Theory of Numbers, 6th ed. New York:
Cambridge University Press, 1992.
Euler, L. Introduction to Analysis of the Infinite, Book I.
New York: Springer- Verlag, 1980.
Fowler, D. H. The Mathematics of Plato's Academy. Oxford,
England: Oxford University Press, 1991.
Guy, R. K. "Continued Fractions" §F20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 259, 1994.
Jacobson, M. J. Jr.; Lukes, R. F.; and Williams, H. C. "An
Investigation of Bounds for the Regulator of Quadratic
Fields." Experiment. Math. 4, 211-225, 1995.
Khinchin, A. Ya. Continued Fractions. New York: Dover,
1997.
Kimberling, C. "Continued Fractions." http://www.
evansville.edu/-ck6/integer/contfr.html.
Klein, F. Ausgewahlte Kapitel der Zahlentheorie. Germany:
Teubner, 1907.
Klein, F. Elementary Number Theory. New York, p. 44,
1932.
Kline, M. Mathematical Thought from Ancient to Modern
Times. New York: Oxford University Press, 1972.
Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison- Wesley, p. 316, 1981.
Moore, C. D. An Introduction to Continued Fractions.
Washington, DC: National Council of Teachers of Math-
ematics, 1964.
Olds, C. D. Continued Fractions. New York: Random House,
1963.
Pettofrezzo, A. J. and Bykrit, D. R. Elements of Number
Theory. Englewood Cliffs, NJ: Prentice-Hall, 1970.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Evaluation of Continued Fractions." §5.2
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 163-167, 1992.
Rose, H. E. A Course in Number Theory, 2nd ed. Oxford,
England: Oxford University Press, 1994.
Rosen, K. H. Elementary Number Theory and Its Applica-
tions. New York: Addis on- Wesley, 1980.
Sloane, N. J. A. Sequences A013943 and A000037/M0613 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 39-42, 1983.
Continued Fraction Constant
Continued Fraction Unit Fraction Algorithm 319
VanTuyl, A. L. "Continued Fractions." http://www.calvin.
edu/academic/math/confrac/.
Wagon, S. "Continued Fractions." §8.5 in Mathematica in
Action. New York: W. H. Freeman, pp. 263-271, 1991.
Wall, H. S. Analytic Theory of Continued Fractions. New
York: Chelsea, 1948.
Williams, H. C. "A Numerical Investigation into the Length
of the Period of the Continued Fraction Expansion of y/D."
Math. Comp. 36, 593-601, 1981.
Continued Fraction Constant
A continued fraction with partial quotients which in-
crease in Arithmetic Progression is
[A + D, ,4 + 2D, ,4 + 3D,...]
*A/D
(*)
ll + A/
D (£),
where I n (x) is a Modified Bessel Function of the
First Kind (Beeler et ah 1972, Item 99). A special case
is
C = +
1 +
1
2 +
1
3 +
4 +
5 + ..
which has the value
C = -4lv = 0.697774658 . . .
Jo (2)
(Lehmer 1973, Rabinowitz 1990).
see also e, Golden Mean, Modified Bessel Func-
tion of the First Kind, Pi, Rabbit Constant,
Thue-Morse Constant
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/cntfrc/cntfrc.html.
Guy, R. K. "Review: The Mathematics of Plato's Academy."
Amer. Math. Monthly 97, 440-443, 1990.
Lehmer, D. H. "Continued Fractions Containing Arithmetic
Progressions." Scripta Math. 29, 17-24, 1973.
Rabinowitz, S. Problem E3264. "Asymptotic Estimates
from Convergents of a Continued Fraction." Amer. Math.
Monthly 97, 157-159, 1990.
Continued Fraction Factorization Algorithm
A Prime Factorization Algorithm which uses
R esid ues produced in the Continued Fraction of
y/rnN for some suitably chosen m to obtain a SQUARE
Number. The Algorithm solves
x = y (mod n)
by finding an m for which m? (mod n) has the small-
est upper bound. The method requires (by conjecture)
about exp ( a/2 log n log log n ) steps, and was the fastest
Prime Factorization Algorithm in use before the
Quadratic Sieve Factorization Method, which
eliminates the 2 under the SQUARE ROOT (Pomerance
1996), was developed.
see also EXPONENT VECTOR, PRIME FACTORIZATION
Algorithms
References
Morrison, M. A. and Brillhart, J. "A Method of Factoring
and the Factorization of i*V." Math. Comput. 29, 183-
205, 1975.
Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.
Continued Fraction Fundamental
Recurrence Relation
For a Simple Continued Fraction a = [a ,ai,...]
with Convergents Pn/q n , the fundamental Recur-
rence Relation is given by
Pnqn-l -pn-lQn = (-!)"■
References
Olds, C, D. Continued Fractions. New York: Random House,
p. 27, 1963.
Continued Fraction Map
0.8
0.4
'M-i-iy
for x e [0,1], where [x\ is the Floor Function. The
Invariant Density of the map is
p(y)
(l + y) In 2
References
Beck, C. and Schlogl, F. Thermodynamics of Chaotic Sys-
tems. Cambridge, England: Cambridge University Press,
pp. 194-195, 1995.
Continued Fraction Unit Fraction Algorithm
An algorithm for computing a UNIT FRACTION, called
the FAREY SEQUENCE method by Bleicher (1972).
References
Bleicher, M. N. "A New Algorithm for the Expansion of Con-
tinued Fractions." J. Number Th. 4, 342-382, 1972.
320
Continued Square Root
Continuous Distribution
Continued Square Root
Expressions of the form
lim xo + y xi + V x 2 + V- ■ • +
Xfc.
Herschfeld (1935) proved that a continued square root
of REAL NONNEGATIVE terms converges IFF (x n ) 2 is
bounded. He extended this result to arbitrary POWERS
(which include continued square roots and CONTINUED
Fractions as well), which is known as Herschfeld's
Convergence Theorem.
see also Continued Fraction, Herschfeld's Con-
vergence Theorem, Square Root
References
Herschfeld, A. "On Infinite Radicals." Amer. Math. Monthly
42, 419-429, 1935.
Polya, G. and Szego, G. Problems and Theorems in Analysis,
Vol 1. New York: Springer- Verlag, 1997.
Sizer, W. S. "Continued Roots." Math. Mag. 59, 23-27,
1986.
Continuity Correction
A correction to a discrete Binomial Distribution to
approximate a continuous distribution.
[a < X <
b)
-1
np
<i
-P) ~
b-\-
r. < 2
np
y/np(l
-p)
lere
z = —
-m)
is a continuous variate with a NORMAL DISTRIBUTION
and X is a variate of a Binomial Distribution.
see also BINOMIAL DISTRIBUTION, NORMAL DISTRIBU-
TION
Continuity Principle
see PERMANENCE OF MATHEMATICAL RELATIONS
Principle
Continued Vector Product
see Vector Triple Product
Continuity
The property of being Continuous.
see also CONTINUITY AXIOMS, CONTINUITY CORREC-
TION, Continuity Principle, Continuous Distri-
bution, Continuous Function, Continuous Space,
Fundamental Continuity Theorem
Continuity Axioms
"The" continuity axiom is an additional Axiom which
must be added to those of Euclid's Elements in order to
guarantee that two equal CIRCLES of RADIUS r intersect
each other if the separation of their centers is less than
2r (Dunham 1990). The continuity axioms are the three
of Hilbert's Axioms which concern geometric equiva-
lence.
Archimedes' Lemma is sometimes also known as "the
continuity axiom."
see also Congruence Axioms, Hilbert's Axioms, In-
cidence Axioms, Ordering Axioms, Parallel Pos-
tulate
References
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 38, 1990.
Hilbert, D. The Foundations of Geometry. Chicago, IL:
Open Court, 1980.
Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert's System of Ax-
ioms." §163B in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 544-545, 1980.
Continuous
A general mathematical property obeyed by mathemat-
ical objects in which all elements are within a NEIGH-
BORHOOD of nearby points.
see also ABSOLUTELY CONTINUOUS, CONTINUOUS DIS-
TRIBUTION, Continuous Function, Continuous
Space, Jump
Continuous Distribution
A Distribution for which the variables may take on
a continuous range of values. Abramowitz and Stegun
(1972, p. 930) give a table of the parameters of most
common discrete distributions.
see also Beta Distribution, Bivariate Distribu-
tion, Cauchy Distribution, Chi Distribution,
Chi-Squared Distribution, Correlation Coef-
ficient, Discrete Distribution, Double Ex-
ponential Distribution, Equally Likely Out-
comes Distribution, Exponential Distribution,
Extreme Value Distribution, F-Distribution,
Fermi-Dirac Distribution, Fisher's ^-Distribu-
tion, Fisher-Tippett Distribution, Gamma Dis-
tribution, Gaussian Distribution, Half-Normal
Distribution, Laplace Distribution, Lattice Dis-
tribution, Levy Distribution, Logarithmic Dis-
tribution, Log-Series Distribution, Logistic Dis-
tribution, Lorentzian Distribution, Maxwell
Distribution, Normal Distribution, Pareto Dis-
tribution, Pascal Distribution, Pearson Type
III Distribution, Poisson Distribution, Polya
Distribution, Ratio Distribution, Rayleigh Dis-
tribution, Rice Distribution, Snedecor's F-
Distribution, Student's ^-Distribution, Stu-
dent's ^-Distribution, Uniform Distribution,
Weibull Distribution
Continuous Function
Contour Integration 321
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 927 and 930, 1972.
Continuous Function
A continuous function is a FUNCTION / : X — »■ Y where
the pre-image of every Open Set in Y is Open in X.
A function f{x) in a single variable x is said to be con-
tinuous at point Xq if
1. f{xo) is defined, so .To is the DOMAIN of /.
2. lim x _> xo f(x) exists.
3. lim^^ f(x) = /(x ),
where lim denotes a Limit. If / is Differentiable at
point xq, then it is also continuous at xq. If / and g are
continuous at aso, then
1. / + g is continuous at xq.
2. / — g is continuous at xq.
3. / x g is continuous at xq.
4. / -r g is continuous at x*o if g(xo) 7^ anu is discon-
tinuous at xo if p(xo) = 0.
5. / o g is continuous, where o denotes using g as the
argument to /.
see also CRITICAL POINT, DIFFERENTIABLE, LIMIT,
Neighborhood, Stationary Point
Continuous Space
A Topological Space.
see also NET
Continuum
The nondenumerable set of Real Numbers, denoted
C. It satisfies
Ko + C-C (1)
and
C r = C, (2)
where K is ^0 (Aleph-0). It is also true that
Ko*° = C. (3)
However,
C c = F (4)
is a Set larger than the continuum. Paradoxically, there
are exactly as many points C on a Line (or Line Seg-
ment) as in a PLANE, a 3-D SPACE, or finite Hyper-
SPACE, since all these Sets can be put into a One-TO-
One correspondence with each other.
The Continuum Hypothesis, first proposed by Georg
Cantor, holds that the Cardinal Number of the con-
tinuum is the same as that of Ni. The surprising truth
is that this proposition is Undecidable, since neither it
nor its converse contradicts the tenets of Set Theory.
see also Aleph-0 (N ), Aleph-1 (Ni), Continuum Hy-
pothesis, Denumerable Set
Continuum Hypothesis
The proposal originally made by Georg Cantor that
there is no infinite Set with a Cardinal Number be-
tween that of the "small" infinite Set of INTEGERS K
and the "large" infinite set of REAL NUMBERS C (the
"Continuum"). Symbolically, the continuum hypoth-
esis is that Ki = C. Godel showed that no contra-
diction would arise if the continuum hypothesis were
added to conventional Zermelo-Fraenkel Set The-
ory. However, using a technique called FORCING, Paul
Cohen (1963, 1964) proved that no contradiction would
arise if the negation of the continuum hypothesis was
added to Set Theory. Together, Godel's and Cohen's
results established that the validity of the continuum
hypothesis depends on the version of Set Theory be-
ing used, and is therefore Undecidable (assuming the
Zermelo-Fraenkel Axioms together with the Axiom
of Choice).
Conway and Guy (1996) give a generalized version of
the Continuum Hypothesis which is also UNDECIDABLE:
is 2** = K a+ i for every a?
see also ALEPH-0 (No), ALEPH-1 (Ni), AXIOM OF
Choice, Cardinal Number, Continuum, Denumer-
able Set, Forcing, Hilbert's Problems, Lebesgue
Measurability Problem, Undecidable, Zermelo-
Fraenkel Axioms, Zermelo-Fraenkel Set The-
ory
References
Cohen, P. J. "The Independence of the Continuum Hypoth-
esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963.
Cohen, P. J. "The Independence of the Continuum Hypothe-
sis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964.
Cohen, P. J. Set Theory and the Continuum Hypothesis. New
York: W. A. Benjamin, 1966.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 282, 1996,
Godel, K. The Consistency of the Continuum- Hypothesis.
Princeton, NJ: Princeton University Press, 1940.
McGough, N. "The Continuum Hypothesis." http://www.
jazzie, com/ii/math/ch and http: //www. best . com/-ii/
math/ch/.
Contour
A path in the Complex Plane over which Contour
Integration is performed.
see also Contour Integration
Contour Integral
see Contour Integration
Contour Integration
Let P(x) and Q(x) be POLYNOMIALS of DEGREES n and
m with Coefficients b ni . . . , 60 and c m , . . . , co. Take
the contour in the upper half-plane, replace x by 2, and
write z = Re i9 . Then
f
J — c
P{z) dz
= lim
/;
P(z) dz
(1)
322 Contour Integration
Contraction (Tensor)
Define a path ^r which is straight along the REAL axis
from — R to R and makes a circular arc to connect the
two ends in the upper half of the COMPLEX PLANE. The
Residue Theorem then gives
P{z) dz
Q(Re*°)
P(z)
Q(z)
(2)
lim /
v f R p(z)dz ,. r
= hm / ^/ + hm /
= 2-ki y Res
3[z]>0
where Res denotes the Residues. Solving,
= 2?ri V Resg4- lim f ' ?V^liRf*de. (3)
l^ Q(z) r^™J Q{Re e )
3[z]>0
Define
— ^ r^e 10 <20
Q(ife")
i im / ^^ + K-^r-^... + i>o iRde
n *'-... + Co
I r = iim /
= lim / —{Re i6 ) n - m iRdB
^n + 1-m-^i^n-m^
and set
e = — (n + 1 — m),
then equation (4) becomes
/rS lim j_^l r e *
H-kx> R e Cm J
(n — m)6
dO.
Now,
lim R~ e =
R^oo
(4)
(5)
(6)
(7)
for e > 0. That means that for — n — 1 + m > 1, or
m > n + 2, i^ = 0, so
/ Q(z) ^
3f[z]>0
£(f)
Q(*)
(8)
for m > n + 2. Apply Jordan's Lemma with /(x) =
P(x)/Q(x). We must have
lim /(x) = 0,
(9)
so we require m > n -f 1. Then
Q{z) ^
/
£(£) iaz
Q(zf
(10)
for m > n + 1.
Since this must hold separately for REAL and IMAGI-
NARY Parts, this result can be extended to
f
f
J — o
— t-4 cos(az) dx = 27rJJ < > Res
Q(x) \ ^
P(x)
It is also true that
sin(ax) dx ~ 2x9 ^ YJ Res
sw>o
Q(*)
P(2) i,
Q(z)
Q(z)
ln(a^) dz = 0.
(13)
see also CAUCHY INTEGRAL FORMULA, CAUCHY IN-
TEGRAL Theorem, Inside-Outside Theorem, Jor-
dan's Lemma, Residue (Complex Analysis), Sine
Integral
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 353-356, 1953.
Contracted Cycloid
see Curtate Cycloid
Contraction
see Dilation
Contraction (Graph)
The merging of nodes in a GRAPH by eliminating seg-
ments between two nodes.
Contraction (Tensor)
The contraction of a TENSOR is obtained by setting un-
like indices equal and summing according to the EIN-
STEIN Summation convention. Contraction reduces the
Rank of a Tensor by 2. For a second Rank Tensor,
contr(^) = B':
fi dx'i dxi k dxt fc i k k
* = dx~:dx 71 = dx~: 1 =Sh i *•
Therefore, the contraction is invariant, and must be a
Scalar. In fact, this Scalar is known as the Trace
of a Matrix in Matrix theory.
References
Arfken, G. "Contraction, Direct Product." §3.2 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 124-126, 1985.
Contradiction Law
Convective Operator 323
Contradiction Law
No A is not- A.
see also NOT
Contravariant Vector
A Contravariant Tensor of Rank 1.
see also Contravariant Tensor, Vector
Contravariant Tensor
A contravariant tensor is a TENSOR having specific
transformation properties (c.f., a COVARIANT TENSOR).
To examine the transformation properties of a contra-
variant tensor, first consider a Tensor of Rank 1 (a
Vector)
dr.
for which
dxilti + dx2X 2 4- dx3X 3 ,
dx'i
dx'i
dx^
(1)
(2)
Now let Ai = dxi, then any set of quantities Aj which
transform according to
A
or, defining
according to
A\
Br'-
dxj J '
dxj '
: Q>ij -f*.j
(3)
(4)
(5)
is a contravariant tensor. Contravariant tensors are in-
dicated with raised indices, i.e., a M .
Covariant Tensors are a type of Tensor with differ-
ing transformation properties, denoted a u . However, in
3-D Cartesian Coordinates,
dxj_
dx'
dxj
(6)
for i,j = 1, 2, 3, meaning that contravariant and co-
variant tensors are equivalent . The two types of tensors
do differ in higher dimensions, however. Contravariant
Four- Vectors satisfy
a M = K%a v ,
where A is a LORENTZ TENSOR.
(7)
To turn a COVARIANT TENSOR into a contravariant ten-
sor, use the Metric Tensor g^ to write
U, LLV
a = g a v
(8)
Covariant and contravariant indices can be used simul-
taneously in a Mixed Tensor.
see also COVARIANT TENSOR, FOUR- VECTOR, LOR-
entz Tensor, Metric Tensor, Mixed Tensor,
Tensor
References
Arfken, G. "Noncartesian Tensors, Covariant Differentia-
tion." §3.8 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 158-164, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 44-46, 1953.
Control Theory
The mathematical study of how to manipulate the pa-
rameters affecting the behavior of a system to produce
the desired or optimal outcome.
see also Kalman Filter, Linear Algebra, Pon-
tryagin Maximum Principle
References
Zabczyk, J. Mathematical Control Theory: An Introduction.
Boston, MA: Birkhauser, 1993.
Convective Acceleration
The acceleration of an element of fluid, given by the
Convective Derivative of the Velocity v,
Dv dv „
= + v . Vv,
Dt dt
where V is the GRADIENT operator.
see also ACCELERATION, CONVECTIVE DERIVATIVE,
Convective Operator
References
Batchelor, G K. An Introduction to Fluid Dynamics. Cam-
bridge, England: Cambridge University Press, p. 73, 1977.
Convective Derivative
A Derivative taken with respect to a moving coordi-
nate system, also called a LAGRANGIAN DERIVATIVE. It
is given by
D d
= h v • V,
Dt dt '
where V is the GRADIENT operator and v is the VE-
LOCITY of the fluid. This type of derivative is especially
useful in the study of fluid mechanics. When applied to
v,
Dt dt
+ (Vx v) x v + V(fv 2 ).
see also CONVECTIVE OPERATOR, DERIVATIVE, VE-
LOCITY
References
Batchelor, G K. An Introduction to Fluid Dynamics. Cam-
bridge, England: Cambridge University Press, p. 73, 1977.
Convective Operator
Defined for a Vector Field A by (A • V), where V is
the Gradient operator.
Applied in arbitrary orthogonal 3-D coordinates to a
Vector Field B, the convective operator becomes
[(A • V)B],
3 r
fc=i
A k dBj
+
hk dqk hkhj
Aj—^- -A k
dqk
dh k
(1)
324 Convergence Acceleration
where the his are related to the Metric TENSORS by
hi = yfgTi. In Cartesian Coordinates,
+ \ Ax dx + Ay d y +Az dz I y
In Cylindrical Coordinates,
(A . v)B= ^fn + ^ + ^_^ lr
<£
I r dr r d<fi dz r
In Spherical Coordinates,
(A ■ V)B
~~ V T dr r d$ rsinO d<f> r )*
/ dB e A e 0B 9 A^ 8B e A e B T A^B* cot 6 \ A
+ A - ~^~~ + ^~ + — ^~H "^X" + W
\ 9r r o9 r sin a<p r r )
dB e Ae dB 9 | A dB,, ; i(,B r A^B,
dr r
V dr r
r sin d<f>
OB^ A^ dBj, A^B,. A (b B l
89 r sin 6 d(f> r
(4)
see also Convective Acceleration, Convective
Derivative, Curvilinear Coordinates, Gradient
Convergence Acceleration
see Convergence Improvement
Convergence Improvement
The improvement of the convergence properties of a Se-
ries, also called CONVERGENCE ACCELERATION, such
that a Series reaches its limit to within some accuracy
with fewer terms than required before. Convergence im-
provement can be effected by forming a linear combina-
tion with a Series whose sum is known. Useful sums
include
oo
^ n(n+ 1)
n = l
oo
y I . = !
^ n(n + l)(n + 2) 4
y = JL
^ n(n + l)(ra + 2)(n + 3) 18
n = l
y I = _U
^— •* n(n -f- 1) • • ■ (n 4- p) p • p\
(i)
(2)
(3)
(4)
Convergence Improvement
Rummer's transformation takes a convergent series
S = VJo/fc
k=0
(5)
and another convergent series
c=J2 ck
*=0
with known c such that
lim 5* = X / 0.
fc— voo Cfc
(6)
(7)
Then a series with more rapid convergence to the same
value is given by
Ac+Vj(l-A^)a fc
(8)
(Abramowitz and Stegun 1972).
Euler'S Transform takes a convergent alternating se-
ries
2j(-l) fc Ofe = a - ai +
a 2
(9)
into a series with more rapid convergence to the same
value to
(-l)*A fc a
E
2*+i
(10)
where
A fc a = ^= (-l) m (Ma fc _ m (11)
m=0 ^ '
(Abramowitz and Stegun 1972; Beeler et al. 1972, Item
120).
Given a series of the form
S
£>G)
(12)
where f(z) is an Analytic at and on the closed unit
Disk, and
f{z)\ z ^ = G{z% (13)
then the series can be rearranged to
oo oo
*-£S>(i)"
n=\ m = 2
oo oo oo
= EE'-G) m = E /««"), (14)
Convergence Tests
Convergent Series 325
where
the convergents are
/CO = E f "
(15)
is the MACLAURIN SERIES of / and ((z) is the RlEMANN
ZETA FUNCTION (Flajolet and Vardi 1996). The trans-
formed series exhibits geometric convergence. Similarly,
if f(z) is Analytic in \z\ < l/n for some Positive
Integer n , then
n — i ■
n=l
+ E/»KM-^-----(^hp • w
m=:2 L J
which converges geometrically (Flajolet and Vardi
1996). (16) can also be used to further accelerate the
convergence of series (14).
see also Euler's TRANSFORM, Wilf-Zeilberger Pair
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.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 288-289, 1985.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Flajolet, P. and Vardi, I. "Zeta Function Expan-
sions of Classical Constants." Unpublished manu-
script. 1996. http://pauillac.inria.fr/algo/flajolet/
Publicat ions/landau. ps.
Convergence Tests
A test to determine if a given Series Converges or
Diverges.
see also ABEL'S UNIFORM CONVERGENCE TEST,
Bertrand's Test, d'Alembert Ratio Test, Diver-
gence Tests, Ermakoff's Test, Gauss's Test, In-
tegral Test, Rummer's Test, Raabe's Test, Ra-
tio Test, Riemann Series Theorem, Root Test
References
Arfken, G. "Convergence Tests." §5.2 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 280-293, 1985.
Bromwich, T. J. Fa and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, pp. 55-57, 1991.
Convergent
The Rational Number obtained by keeping only a
limited number of terms in a CONTINUED FRACTION is
called a convergent. For example, in the Simple Con-
tinued Fraction for the Golden Ratio,
</>=! + l — ,
> i+ l 2' l + ± 3'
The word convergent is also used to describe a Conver-
gent Sequence or Convergent Series.
see also CONTINUED FRACTION, CONVERGENT SE-
QUENCE, Convergent Series, Partial Quotient,
Simple Continued Fraction
Convergent Sequence
A Sequence S n converges to the limit S
lim S n = S
n— J-oo
if, for any e > 0, there exists an N such that \S n —
S\ < e for n > N. If S n does not converge, it is said
to Diverge. Every bounded Monotonic Sequence
converges. Every unbounded SEQUENCE diverges. This
condition can also be written as
lim S n — lim S n = S.
see also CONDITIONAL CONVERGENCE, STRONG CON-
VERGENCE, Weak Convergence
Convergent Series
The infinite Series ^2 n ° =1 a n is convergent if the Se-
quence of partial sums
n
S n = / CLk
fc=l
is convergent. Conversely, a SERIES is divergent if the
SEQUENCE of partial sums is divergent. If ^ttfc and
^2, v k are convergent SERIES, then ^2(uk + Vk) and
^2(v>k — Vk) are convergent. If c ^ 0, then ^Uk and
c Yl Uk both converge or both diverge. Convergence
and divergence are unaffected by deleting a finite num-
ber of terms from the beginning of a series. Constant
terms in the denominator of a sequence can usually
be deleted without affecting convergence. All but the
highest Power terms in Polynomials can usually be
deleted in both NUMERATOR and DENOMINATOR of a
Series without affecting convergence. If a Series con-
verges absolutely, then it converges.
see also CONVERGENCE TESTS, RADIUS OF CONVER-
GENCE
References
Bromwich, T. J. Pa. and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, 1991.
! + ■
1 + ...
326
Conversion Period
Convex Polyhedron
Conversion Period
The period of time between INTEREST payments.
see also Compound Interest, Interest, Simple In-
terest
Convex
A Set in Euclidean Space M. is convex if it contains
all the Line Segments connecting any pair of its points.
If the Set does not contain all the Line Segments, it
is called Concave.
see also CONNECTED SET, CONVEX FUNCTION, CON-
VEX Hull, Convex Optimization Theory, Convex
Polygon, Delaunay Triangulation, Minkowski
Convex Body Theorem, Simply Connected
References
Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Convexity."
Ch. A in Unsolved Problems in Geometry. New York:
Springer- Verlag, pp. 6-47, 1994.
Convex Function
t
concavedown
concaveup
A function whose value at the Midpoint of every In-
terval in its Domain does not exceed the Average of
its values at the ends of the Interval. In other words,
a function f(x) is convex on an INTERVAL [a, b] if for any
two points x\ and X2 in [a, 6],
f[±(xi+X 2 )]<±[f(x 1 ) + f(x 2 )].
If f(x) has a second Derivative in [a, 6], then a Nec-
essary and Sufficient condition for it to be convex on
that Interval is that the second Derivative f"(x) >
for all x in [a, b].
see also Concave Function, Logarithmically Con-
vex Function
References
Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How
Likely is a Function to be Convex?" Math. Mag. 61, 211-
219, 1988.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1100, 1980.
Convex Hull
The convex hull of a set of points S is the INTERSECTION
of all convex sets containing 5. For TV points pi, . . . ,
Pn, the convex hull C is then given by the expression
V 3 = 1
^jPj '■ ^j > f° r au 3 an d 2, ^ = 1
j = l
see also CARATHEODORY'S FUNDAMENTAL THEO-
REM, Cross Polytope, Groemer Packing, Groe-
mer Theorem, Sausage Conjecture, Sylvester's
Four-Point Problem
References
Santalo, L. A. Integral Geometry and Geometric Probability.
Reading, MA: Addison-Wesley, 1976.
Convex Optimization Theory
The problem of maximizing a linear function over a con-
vex polyhedron, also known as OPERATIONS RESEARCH
or Optimization Theory. The general problem of con-
vex optimization is to find the minimum of a convex (or
quasiconvex) function / on a FlNlTE-dimensional con-
vex body A. Methods of solution include Levin's al-
gorithm and the method of circumscribed ELLIPSOIDS,
also called the Nemirovsky-Yudin-Shor method.
References
Tokhomirov, V. M. "The Evolution of Methods of Convex
Optimization." Amer. Math. Monthly 103, 65-71, 1996.
Convex Polygon
A Polygon is Convex if it contains all the Line Seg-
ments connecting any pair of its points. Let f(n) be
the smallest number such that when W is a set of more
than f(n) points in General POSITION (with no three
points Collinear) in the plane, all of the VERTICES of
some convex n-gon are contained in W. The answers for
n ~ 2, 3, and 4 are 2, 4, and 8. It is conjectured that
/(n) — 2™ -2 , but only proven that
(2n - 4\
where (£) is a Binomial Coefficient.
Convex Polyhedron
A Polyhedron for which a line connecting any two
(noncoplanar) points on the surface always lies in the
interior of the polyhedron. The 92 convex polyhedra
having only REGULAR POLYGONS as faces are called the
Johnson Solids, which include the Platonic Solids
and Archimedean Solids. No method is known for
computing the VOLUME of a general convex polyhedron
(Ogilvy 1990, p. 173).
see also Archimedean Solid, Deltahedron, John-
son Solid, Kepler-Poinsot Solid, Platonic Solid,
Regular Polygon
References
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
1990.
2"~ 2 < f(n) <
Convolution
Convolution
A convolution is an integral which expresses the amount
of overlap of one function g(t) as it is shifted over an-
other function f(t). It therefore "blends" one function
with another. For example, in synthesis imaging, the
measured DlRTY MAP is a convolution of the "true"
CLEAN Map with the Dirty Beam (the Fourier
Transform of the sampling distribution). The con-
volution is sometimes also known by its German name,
FALTUNG ("folding"). A convolution over a finite range
[0,i] is given by
f(t)*g{t)
f
Jo
f(r)g(t-T)dr,
(1)
where the symbol f*g (occasionally also written as f®g)
denotes convolution of / and g. Convolution is more
often taken over an infinite range,
f(t)*9(t)
f
J — C
f(r)g(t-T)dT= I g(r)f(t-r)dr.
J — OO
(2)
Let /, g, and h be arbitrary functions and a a constant.
Convolution has the following properties:
f*g=g*f
f*(g*h) = (f*g)*h
f*(9 + h) = (f*g) + (f*h)
a(f*g) = («/)* 5 = /*(a^)-
The Integral identity
px px px
/ / f(t)dtdx= / (x-t)f(t)dt
*J a " a, J a
(3)
(4)
(5)
(6)
(7)
also gives a convolution. Taking the Derivative of a
convolution gives
d (J: v df dg
dx KJ y} dx y J dx
(8)
The AREA under a convolution is the product of areas
under the factors,
/oo poo |~ poo
(f*g)dx = \ f(u)g(x - u) du
-oo t/-oo U-co
/oo |~ /»oo
f(u) / g(x-u)dx
-oo L<J — oo
= / /(u)chJ / g(x)dx
dx
du
(9)
J L*J — oo
The horizontal CENTROIDS add
{x(f * g)) dx = (xf) + (xg) , (10)
J — c
Convolution Theorem
as do the VARIANCES
/CO
(x 2 (f*g))dx=(x 2 f) + (x 2 g) )
-CO
where
{* n f) -
J^ oo x n f(x)dx
327
(11)
(12)
see also Autocorrelation, Convolution Theorem,
Cross-Correlation, Wiener-Khintchine Theo-
rem
References
Bracewell, R. "Convolution." Ch. 3 in The Fourier Trans-
form and Its Applications. New York: McGraw-Hill,
pp. 25-50, 1965.
Hirschman, I. 1. and Widder, D. V. The Convolution Trans-
form. Princeton, NJ: Princeton University Press, 1955.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 464-465, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Convolution and Deconvolution Using the
FFT." §13.1 in Numerical Recipes in FORTRAN: The Art
of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 531-537, 1992.
Convolution Theorem
Let f(t) and f(t) be arbitrary functions of time t with
Fourier Transforms. Take
/(*)
git)
/oo
F{u)e 2irivt du
-oo
/oo
G(u)e 2wivt du,
-oo
(1)
(2)
where T' 1 denotes the inverse FOURIER TRANSFORM
(where the transform pair is defined to have constants
A = 1 and B = -2ir). Then the Convolution is
/oo
g(t')f(t - t')dt'
-OO
/oo r /*oo
ff (t') / F(i/)e 2 " i " (t - t ' ) du dt'.
■OO L" — oo
(3)
Interchange the order of integration,
/oo r poo
F(u) / g(t')e- 2 ™
-oo Y.J — oo
dt'
e du
f
F(u)G{u)e'
= T-\F{u)G{u)].
du
(4)
So, applying a FOURIER TRANSFORM to each side, we
have
T[f*9]=F\f\T\3\' (5)
328 Conway- Alexander Polynomial
Conway Notation
The convolution theorem also takes the alternate forms
(6)
(7)
(8)
see also AUTOCORRELATION, CONVOLUTION, FOURIER
Transform, Wiener-Khintchine Theorem
References
Arfken, G. "Convolution Theorem." §15.5 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 810-814, 1985.
Bracewell, R. "Convolution Theorem." The Fourier Trans-
form and Its Applications. New York: McGraw-Hill,
pp. 108-112, 1965.
Conway- Alexander Polynomial
see Alexander Polynomial
Conway's Constant
The constant
A- 1.303577269034296...
(Sloane's A014715) giving the asymptotic rate of growth
C\ k of the number of Digits in the &th term of the
Look and Say Sequence. A is given by the largest
Root of the Polynomial
= x
69
67 ,
X + 2x
2x
x vv — x 4- 2x -f bx + 3x
53 o 52
60
+ 2x 65
57 , o..56
64 63 62
- X — X — X
~64
55
2z - 10x
54
- 3x 53 - 2x 52 + 6x 51 + 6x 50 + x 49 + 9x 48 - 3x 47
_ 7x 46 _ 8a ,45 _ ^44 + 1()x 43 + ^42 + ^41 _ ^C
- 12z 39 + 7x 38 - 7x 37 + 7x 36 + x 35 - 3x 34 + 10x 33
+ x ** - 6x 31 - 2x S0 - I0x 29 - 3x 2S + 2x 27 + 9x 26
- 3x 25 + Ux 24 - 8x 23 - 7x 21 + 9x 20 - 3x 19 - 4x 18
- 10x 17 - 7x 16 + 12x 15 + 7x 14 + 2z 13 -
— 4x — 2x — 5x 4 x — 7x
4- 7x 5 - 4x 4 + 12x 3 - 6x 2 + 3x - 6.
12a 1
The POLYNOMIAL given in Conway (1987, p. 188) con-
tains a misprint. The CONTINUED FRACTION for A is 1,
3, 3, 2, 2, 54, 5, 2, 1, 16, 1, 30, 1, 1, 1, 2, 2, 1, 14, 1, . . .
(Sloane's A014967).
see also Conway Sequence, Cosmological Theo-
rem, Look and Say Sequence
References
Conway, J. H. "The Weird and Wonderful Chemistry of
Audioactive Decay." §5.11 in Open Problems in Com-
munications and Computation (Ed. T. M. Cover and
B. Gopinath). New York: Springer-Verlag, pp. 173-188,
1987.
Conway, J. H. and Guy, R. K. "The Look and Say Sequence."
In The Book of Numbers. New York: Springer-Verlag,
pp. 208-209, 1996.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/cnvy/cnwy .html.
Sloane, N. J. A. Sequence A014967 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, pp. 13-14, 1991.
Conway's Game of Life
see Life
Conway Groups
The Automorphism Group Coi of the Leech Lat-
tice modulo a center of order two is called "the"
Conway group. There are 15 exceptional Conjugacy
Classes of the Conway group. This group, combined
with the GROUPS C02 and C03 obtained similarly from
the Leech Lattice by stabilization of the 1-D and 2-D
sublattices, are collectively called Conway groups. The
Conway groups are Sporadic Groups.
see also LEECH LATTICE, SPORADIC GROUP
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/Col.html, Co2.html,
Co3.html.
Conway's Knot
The Knot with Braid Word
3 _i _2 -1 -1
(J2 0~\0~Z 0~2 O'\0~2 0~\&$
61 The Jones Polynomial of Conway's knot is
t" 4 (-l + 2t - 2t 2 + 2i 3 + t G - 2t 7 + 2t 8 - 2t g + 1 10 ),
the same as for the Kinoshita-Terasaka Knot.
Conway's Knot Notation
A concise Notation based on the concept of the TAN-
GLE used by Conway (1967) to enumerate KNOTS up
to 11 crossings. An ALGEBRAIC KNOT containing no
NEGATIVE signs in its Conway knot NOTATION is an
Alternating Knot.
References
Conway, J. H. "An Enumeration of Knots and Links, and
Some of Their Algebraic Properties." In Computation
Problems in Abstract Algebra (Ed. J. Leech). Oxford, Eng-
land: Pergamon Press, pp. 329-358, 1967.
Conway's Life
see Life
Conway Notation
see Conway's Knot Notation, Conway Polyhe-
dron Notation
Conway Polyhedron Notation
Copeland-Erdos Constant 329
Conway Polyhedron Notation
A Notation for Polyhedra which begins by speci-
fying a "seed" polyhedron using a capital letter. The
Platonic Solids are denoted T (Tetrahedron), O
(Octahedron), C (Cube), I (Icosahedron), and D
(DODECAHEDRON), according to their first letter. Other
polyhedra include the PRISMS, Pn, ANTIPRISMS, An,
and Pyramids, Yn, where n > 3 specifies the number
of sides of the polyhedron's base.
Operations to be performed on the polyhedron are then
specified with lower-case letters preceding the capital
letter.
see also POLYHEDRON, SCHLAFLI SYMBOL, WYTHOFF
Symbol
References
Hart, G. "Conway Notation for Polyhedra." http://www.li.
net /-george/virtual-polyhedra/conway .notation, html.
Conway Polynomial
see Alexander Polynomial
Conway Puzzle
Construct a 5 x 5 x 5 cube from 13 1 x 2 x 4 blocks, 1
2x2x2 block, 11x2x2 and 31x1x3 blocks.
see also Box-Packing Theorem, Cube Dissection,
de Bruijn's Theorem, Klarner's Theorem, Poly-
cube, Slothouber-Graatsma Puzzle
References
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 77-80, 1976.
Conway Sequence
The LOOK AND SAY SEQUENCE generated from a start-
ing DIGIT of 3, as given by Vardi (1991).
see also Conway's Constant, Look and Say Se-
quence
References
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 13-14, 1991.
Conway Sphere
A sphere with four punctures occurring where a KNOT
passes through the surface.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 94, 1994.
Coordinate Geometry
see Analytic Geometry
Coordinate System
A system of COORDINATES.
Coordinates
A set of n variables which fix a geometric object. If the
coordinates are distances measured along PERPENDICU-
LAR axes, they are known as CARTESIAN COORDINATES.
The study of GEOMETRY using one or more coordinate
systems is known as ANALYTIC GEOMETRY.
see also Areal Coordinates, Barycentric Coor-
dinates, Bipolar Coordinates, Bipolar Cylin-
drical Coordinates, Bispherical Coordinates,
Cartesian Coordinates, Chow Coordinates, Cir-
cular Cylindrical Coordinates, Confocal El-
lipsoidal Coordinates, Confocal Paraboloidal
Coordinates, Conical Coordinates, Curvilinear
Coordinates, Cyclidic Coordinates, Cylindrical
Coordinates, Ellipsoidal Coordinates, Elliptic
Cylindrical Coordinates, Gaussian Coordinate
System, Grassmann Coordinates, Harmonic Co-
ordinates, Homogeneous Coordinates, Oblate
Spheroidal Coordinates, Orthocentric Coordi-
nates, Parabolic Coordinates, Parabolic Cylin-
drical Coordinates, Paraboloidal Coordinates,
Pedal Coordinates, Polar Coordinates, Pro-
late Spheroidal Coordinates, Quadriplanar Co-
ordinates, Rectangular Coordinates, Spherical
Coordinates, Toroidal Coordinates, Trilinear
Coordinates
References
Arfken, G. "Coordinate Systems." Ch. 2 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 85-117, 1985.
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 1,
1961.
Coordination Number
see Kissing Number
Copeland-Erdos Constant
The decimal 0.23571113171923... (Sloane's A033308)
obtained by concatenating the PRIMES: 2, 23, 235, 2357,
235711, ... (Sloane's A033308; one of the Smaran-
dache Sequences). In 1945, Copeland and Erdos
showed that it is a NORMAL NUMBER, The first few
digits of the Continued Fraction of the Copeland-
Erdos are 0, 4, 4, 8, 16, 18, 5, 1, . . . (Sloane's A030168).
The positions of the first occurrence of n in the CON-
TINUED Fraction are 8, 16, 20, 2, 7, 15, 12, 4, 17,
254, . . . (Sloane's A033309). The incrementally largest
terms are 1, 27, 154, 1601, 2135, . . . (Sloane's A033310),
which occur at positions 2, 5, 11, 19, 1801, . . . (Sloane's
A033311).
330 Coplanar
Cornish-Fisher Asymptotic Expansion
see also Champernowne Constant, Prime Number
References
Sloane, N. J. A. Sequences A030168, A033308, A033309,
A033310, and A033311 in "An On-Line Version of the En-
cyclopedia of Integer Sequences."
Coplanar
Three noncollinear points determine a plane and so are
trivially coplanar. Four points are coplanar Iff the vol-
ume of the Tetrahedron defined by them is 0,
xi yi z\
x 2 2/2 Z 2
Xs T/3 z 3
CC4 7/4 Z4
Coprime
see Relatively Prime
Copson-de Bruijn Constant
see de Bruijn Constant
Copson's Inequality
Let {a n } be a Nonnegative Sequence and f(x) a
NONNEGATIVE integrable function. Define
k=l
OO
dk
dk
and
F(x)
-f
Jo
x)= r
J x
f(t) dt
f(t)dt,
and take < p < 1. For integrals,
p / \ p
dx >
f[¥
)P poo
(i)
(2)
(3)
(4)
x)] p dx (5)
(unless / is identically 0). For sums,
K-M*' + f:(£)'>(^T
p OO
n = l
(6)
(unless all a n = 0).
References
Beesack, P. R. "On Some Integral Inequalities of E. T. Cop-
son." In General Inequalities 2 (Ed. E. F. Beckenbach).
Basel: Birkhauser, 1980.
Copson, E. T. "Some Integral Inequalities." Proc. Royal Soc.
Edinburgh 75A, 157-164, 1975-1976.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Theorems
326-327, 337-338, and 345 in Inequalities. Cambridge,
England: Cambridge University Press, 1934.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M, Inequalities
Involving Functions and Their Integrals and Derivatives.
Dordrecht, Netherlands: Kluwer, 1991.
Copula
A function that joins univariate distribution functions to
form multivariate distribution functions. A 2-D copula
is a function C : I 2 -» I such that
and
C(0 ) t) = C(*,0) =
C(M) = C(i,l) = t
for all t € I, and
C(u 2 ,v 2 ) -C(ui,v 2 ) -C(u 2 ,vi) + C'(ui,vi) >
for all ui t U2 J vi i v — 2 € J such that u\ < Ui and v\ <
v-2.
see also Sklar's Theorem
Cork Plug
A 3-D Solid which can stopper a Square, Triangu-
lar, or Circular Hole. There is an infinite family of
such shapes. The one with smallest VOLUME has TRI-
ANGULAR Cross-Sections and V = 7rr 3 ; that with the
largest VOLUME is made using two cuts from the top
diameter to the EDGE and has VOLUME V = 47rr 3 /3.
see also Stereology, Trip-Let
Corkscrew Surface
A surface also called the TWISTED SPHERE.
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces.Boca Raton, FL: CRC Press, pp. 493-494, 1993.
Cornish-Fisher Asymptotic Expansion
y « 771 + civ.
where
w-x-\- [71 Ma)] + [72M2O +7i 2 ^n(^)]
+ [73^3(2) +7172^12(2) +71 hm(x)]
+ [74M2O + 72^22 (») +7173^13(2:)
+ 7i 2 72/iii2(z) +Ji 4 h 111 i(x)] + ...,
Cornu Spiral
Cornu Spiral 331
where
hi(x) = |He2(x)
h 2 (x) = ~He 3 (a;)
hu{x) = -±[2Re3(x) + He 1 (x)]
M&) = T20 He 4(z)
/ii 2 (z) = -~[He 4 (z) + He 2 (x)]
^mW = sk[ 12H e4(x) + 19He 2 (a:)]
h *( x ) = 72o He s(aO
h 2 2 (s) = - 3k P He 5 (a;) + 6 He 3 (x) + 2 Hd (a;)]
Ms) = -iJoPHes +3He 3 (x)]
h ll2 (x) = ^g[14He 5 (x) + 37He 3 (a:)+8He 1 (x)]
^mi(ar) = -^[252He 5 (x) + 832He 3 (x) + 227Hei(x)].
see also Edgeworth Series, Gram-Charlier Series
References
Abramowitz, M. and Stegun, C. A, (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 935, 1972.
Cornu Spiral
is plotted above.
A plot in the Complex Plane of the points
B(z) - C{t) + iS(t) = [ e™ x2/2 dx, (1)
Jo
where C(z) and S(z) are the Fresnel INTEGRALS. The
Cornu spiral is also known as the CLOTHOID or EULER'S
Spiral. A Cornu spiral describes diffraction from the
edge of a half-plane.
12 3
The Slope of the Cornu spiral
m(t)
C(t)
(2)
The Slope of the curve's Tangent Vector (above
right figure) is
m T (t) = §|| = tan(±irt 9 ),
(3)
plotted below.
10
The Cesaro Equation for a Cornu spiral is p = c 2 /s,
where p is the Radius OF Curvature and 5 the Arc
Length. The Torsion is r = 0.
Gray (1993) defines a generalization of the Cornu spiral
given by parametric equations
du
xlt) = a I sin
J Q v n + 1
r ( u n+i \
y(t) = a / cos — — - du
Jo \ n + 1 J
(4)
(5)
332
Cornucopia
Correlation Coefficient
The Arc Length, Curvature, and Tangential An-
gle of this curve are
s(t) = at
t n
K(t) = --
a
<p(t)
The Cesaro Equation is
t
n+l
n+1
(3)
(4)
(5)
(6)
Dillen (1990) describes a class of "polynomial spirals"
for which the CURVATURE is a polynomial function of the
Arc Length. These spirals are a further generalization
of the Cornu spiral.
see also Fresnel Integrals, Nielsen's Spiral
References
Dillen, F. "The Classification of Hypersurfaces of a Euclidean
Space with Parallel Higher Fundamental Form." Math. Z.
203, 635-643, 1990.
Gray, A. "Clothoids." §3.6 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 50-52, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 190-191, 1972.
Cornucopia
The SURFACE given by the parametric equations
x = e cos v H- e cos u cos v
bv • , av
y — e sin v + e cos u sin v
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 304, 1993.
Corona (Polyhedron)
see Augmented Sphenocorona, Hebesphenomega-
corona, Sphenocorona, Sphenomegacorona
Corona (Tiling)
The first corona of a TILE is the set of all tiles that have
a common boundary point with that tile (including the
original tile itself) . The second corona is the set of tiles
that share a point with something in the first corona,
and so on.
References
Eppstein, D. "Heesch's Problem." http://www.ics.uci.edu
/-eppstein/junkyard/heesch.
Correlation
see Autocorrelation, Correlation Coefficient,
Correlation (Geometric), Correlation (Statis-
tical), Cross-Correlation
Correlation Coefficient
The correlation coefficient is a quantity which gives the
quality of a LEAST SQUARES FITTING to the original
data. To define the correlation coefficient, first consider
the sum of squared values ss xx , ss xy , and ss yy of a set
of n data points (xi,yi) about their respective means,
ss xa; = T,(xi — x) = TiX — 2xY>x + ££
= Sec — 2nx -f- nx — Ttx — nx (1)
ssyy = T>(yi - y) 2 — Sy 2 - 2y£y + Sy 2
= Sy 2 - 2ny 2 + ny 2 = Sy 2 - ny 2 (2)
ss xy = S(a;i - x)(yi - y) = %{xiVi - xyi - xiy + xy)
= T,xy - nxy — nxy + nxy — T,xy — nxy. (3)
For linear Least Squares Fitting, the Coefficient
b in
y = a + bx (4)
is given by
SSxx
nJ2 x2 ~ (X»
and the Coefficient b' in
is given by
x = a + b y
y _ n Yj x v-Y, x Y^y
n Y,y 2 - (T,y)
(6)
(7)
Corollary
An immediate consequence of a result already proved.
Corollaries usually state more complicated THEOREMS
in a language simpler to use and apply.
see also LEMMA, PORISM, THEOREM
Correlation Coefficient
3^=0.820841
The correlation coefficient r 2 (sometimes also denoted
R 2 ) is then defined by
r = Vbb' =
^E^y-E^Ey
™£> 2 - (X» ^E^ 2 - (Ez>)
which can be written more simply as
(8)
SSxajSSyy
(9)
The correlation coefficient is also known as the
Product-Moment Coefficient of Correlation or
Pearson's Correlation. The correlation coefficients
for linear fits to increasingly noise data are shown above.
The correlation coefficient has an important physical in-
terpretation. To see this, define
A = (Ex — nx )~
(10)
and denote the "expected" value for yi as yi. Sums of
yi are then
yi = a + bxi = y ~ bx + bxi = x + b(xi — x)
= A{yTiX — xTixy + XiUxy — nxyxi)
= A[yEx + (xi — x)Exy — nxyxi]
Hyi — A(nyYiX — nxy)
Ey 2 — yl 2 [ny 2 (E:r 2 ) 2 - n 2 x 2 y 2 (Ex 2 )
(11)
(12)
- 2nxy{Y,xy){Y>x 2 ) + 2n 2 x 3 y(Exy)
+ (Ez 2 )(Ezy) 2 - nx 2 (Exy)} (13)
Eyiyi = AT,[yiyEx 2 + yi(xi - x)Exy - nxyxiyi]
= A[ny Ex + (Exy) — nxyTixy — nxy(T,xy)]
= A[ny 2 Ex 2 + (Ezy) 2 - 2nxyZxy]. (14)
The sum of squared residuals is then
SSR = E(y* - y) 2 = E(y 2 - 2fffc + y 2 )
A (E#y — nxy) (T>x — nx )
(Exy — nxy) 2
Ex 2 — nx 2
SSx y
SSx
— SSyyT 1 — SSxxj v-'-^'j
Correlation Coefficient 333
and the sum of squared errors is
SSE = E(y< - xjif = E(y; -y + bx- b Xi ) 2
= E[y* - y - b(xi - x)] 2
= E(y, - y) 2 + 6 2 E(x, - xf - 2bY i {x i - x){ Vi - y)
■ SSyy ~\~ O SSjpa: AOoox
But
b
=
SSa;x
2
ss^ 2
SSuSSyy
(16)
(17)
(18)
SS;
ss ;
00£j — SSyy ~\~ ty SSxa; ■" SS^y
SSxx SSxx
2
SS.
SSx jj
yy
ss.
xy \
xx J
= SSyy ( 1 ~ ^^2 ] = SSyy(l - V )
„ _ 2 2
Sy Sy ,
(19)
(20)
and
SSE + SSR = ssy y (l - r 2 ) + ss yy r 2 = ss yy , (21)
The square of the correlation coefficient r 2 is therefore
given by
2 _ SSR _ ssxy 2 _ (Exy - nxy) 2
ssyy ss xx ssyy (Ecc 2 — na? 2 )(Ey 2 — ny 2 ) '
(22)
In other words, r 2 is the proportion of ss yy which is
accounted for by the regression,
If there is complete correlation, then the lines obtained
by solving for best-fit (a, b) and (a', 6') coincide (since
all data points lie on them), so solving (6) for y and
equating to (4) gives
a x
y = -» + v= a+bx -
Therefore, a = —a'/b' and 6 = 1/6', giving
r 2 = 66' = 1.
(23)
(24)
The correlation coefficient is independent of both origin
and scale, so
r(u,v) = r(x,y), (25)
where
x — Xq
h~
y-yo
h '
(26)
(27)
334
Correlation Coefficient — Gaussian.
Correlation Coefficient — Gaussian.
see also Correlation Index, Correlation Coeffi-
cient — Gaussian Bivariate Distribution, Corre-
lation Ratio, Least Squares Fitting, Regression
Coefficient
References
Acton, F. S. Analysis of Straight- Line Data. New York:
Dover, 1966.
Kenney, J. F. and Keeping, E. S. "Linear Regression and
Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1,
3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285, 1962.
Gonick, L. and Smith, W. The Cartoon Guide to Statistics.
New York: Harper Perennial, 1993.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Linear Correlation." §14.5 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 630-633, 1992.
Correlation Coefficient — Gaussian Bivariate
Distribution
For a Gaussian Bivariate Distribution, the distri-
bution of correlation COEFFICIENTS is given by
P( r) = i C ;V-2)(l-r 2 r 4)/2 U-p 2
dp
F
Jo
(cosh/3 — pr) N ~ l
= -(JV-2)(l-r 3 ) 1 '
2 (1-P 2 ) (
W V{N - 1)
2 V{N - I)
x(l - P r ) 2^1(2' 2' "2"' ~T~ )
(N - 2)T(N - 1)(1 - p a )t w -^ 2 (l - r 2 )<*- 4 > /2
y/2^V{N - i)(l-p P )w-3/a
9 (pr+1) 2
[ 4 2N - 1
16 (2JV - 1)(2AT + 1)
(i)
where p is the population correlation COEFFICIENT,
2 Fi(a,6;c;ic) is a Hypergeometric Function, and
T(z) is the Gamma Function (Kenney and Keeping
1951, pp. 217-221). The MOMENTS are
(r) =
" 2n
var(r) =
(1-P 2 ) 2
n
('♦
HP 2
2n
7i =
S( ,+
77 p 2 - 30
12n
72 =
-(12p 2 -
!) + •
* * ?
(4)
where n = N — 1. If the variates are uncorr elated, then
p = and
2^H2' 2' ~~ 2~> 2 J _ 2^U2> 2' ~~ 2~~ ' 2
") = 2^1(3) 2'
r(jv
l\ 3/2-iV /—
^ )2 ^, (5)
m)?
_ (iv-2)r(iv-i) _ 2)(JV - 4 )/ 2
r(iv-^)2 3 / 2 - Jtf ^
[r(f)p
2 1 - JV (JV-2)r(Ar-i) ,^-4/2) f61
[r(* )P l J ' U
But from the Legendre Duplication Formula,
Ar(iv-i) = 2 N - 2 r(f)r(V). ( 7 )
pl , _ (2 1 - jv )(2^- 2 )(tv - 2 )r(f )r(^) a ( „. 4)/a
P(r) " ^[r(f)F { ]
(iv-2)r(^) a(w - 4)/a
~~ 2^r(f) ^ J
_ 1 flX 2 )/, _ 2^-2)/2
= 4= E Pr( 1 -^" 2)/2 - (8)
The uncorrelated case can be derived more simply by
letting be the true slope, so that 77 = a + 0x. Then
t-fr-w^m-^i/fc! <•>
1-r 2
is distributed as STUDENT'S t with u = N -2 Degrees
of Freedom. Let the population regression Coeffi-
cient p be 0, then = 0, so
t = r
1-r 2 '
and the distribution is
(3)
1 rf £i±i ')
P{t) dt=-j= Vf.M-H/2
(10)
eft. (11)
Plugging in for t and using
'v / T^7 2 "-r(|)(-2r)(l-r 2 )- 1/2
d£ = v^
1-r 2
<ir
1-r 2
(l-r)3
1 - r 2 + r 2
1-r 2
dr
dr
(12)
Correlation Coefficient — Gaussian. . .
gives
P(t) dt ■■
r(^)
^r( f )
1 +
2 V i^+^va-o
r 2 u ]
dr
(l-r 2 )
2^-3/2 T(^ ±1 )
Correlation Coefficient — Gaussian. . . 335
But v is Odd, so v — 1 ~ 2n is Even. Therefore
2 T(^) _ 2 r(n + l) = 2 n!
^ r(f) - v^r(n + §)" v^ ^- 1 ?^
2 2 n n! __ 2 (2n)!!
^ r(f)( T ^)
1 \(<H-l)/2
dr
= ^ r rS( 1 -^ 2 )" 3/2 ( 1 - r2 ) ( ^ 1)/2 ^
1 M 2 / /-i 2\(t/-2)/2 j
vtt r^)
w v^ r(f)
2)/2
as before. See Bevington (1969, pp. 122-123) or Pugh
and Winslow (1966, §12-8). If we are interested instead
in the probability that a correlation COEFFICIENT would
be obtained > |r|, where r is the observed COEFFICIENT,
then
pl *\r\
P c (r,N) = 2 / P(r',N)dr =1-2 I P{r,N)dr
J\r\ Jo
v±l\ p\r\
2 r(^) f
V* r(|) J
(1 _ r 2)(-2)/2 dr
(15)
Let I = \{v - 2). For Even v, the exponent / is an
Integer so, by the Binomial Theorem,
c-')'-i:fO<
and
P c {r) = 1
= 1
2 r, *> ( -:,'
/!
0F r(|) v ' (i-k)\k\
r \r\ I
Jo k=0
(16)
2fc , /
dr
2 r(*±i)
A rm
(-i)*
J2fc-|-l
(/-&)!*;! 2k + 1
(17)
For ODD i/, the integral is
/>|r|
P c (r) = l-2 / P(r')dr'
_2_r(^i) f|r|
V^ r(§)
Let r = since so dr = cos x dx, then
Jo
1 -r 2 )"- 1 dr. (18)
Pc(r) = 1
2 r[(*±i)] f sin ~ lM
I
Jo
^ r(|)
2 r(^±i) r in 1|r|
cos 1 " a? cos x dx
cos" * x t£c. (19)
7r(2n-l)!! tt (2n - 1)!!
. (20)
Combining with the result from the COSINE INTEGRAL
gives
(13) P c (r)
(14)
_2(2n)M(2n-l)ll
tv (2n - l)!!(2n)!!
n-l
/—/ Oh 4- 1V!
( 2fc)!! 2 fc+ i .
cos X + X
(2fc + l)!
sin \r\
(21)
Use
2fc-l /-, 2\(2fc-l)/2 /-, 2x(fc-l/2) /r>r>\
cos x = (1 — r y = (1 — r y , (/<5j
and define J = n-l = (i/- 3)/2, then
Pc(r)
7T
\fe+l/2
(23)
(In Bevington 1969, this is given incorrectly.) Combin-
ing the correct solutions
Pc(r) = <
1_ _2_EKi
^ r(
for i^even
1 -
+d/2] v IV n fe J! iH 2fc+1 1
i//2) Z^ V A ; (7-fc)!fc! 2fc + l
fc = L J
-- x M + IH£^(i-r 2 ) fc+1
/2
for i/ odd
(24)
If p z£ 0, a skew distribution is obtained, but the variable
z defined by
(25)
z = tanh 1 r
is approximately normal with
ix z = tanh p
2_ 1
(26)
(27)
(Kenney and Keeping 1962, p. 266).
Let bj be the slope of a best-fit line, then the multiple
correlation COEFFICIENT is
^£( 6 'TH=£(^r r '0' (28)
3=1 X ' 3=1
where Sj y is the sample VARIANCE,
336
Correlation Dimension
Correlation Exponent
On the surface of a Sphere,
JfgdQ
j fdnfgdtl'
(29)
where dQ is a differential Solid Angle. This definition
guarantees that — l<r<l. If/ and g are expanded in
Real Spherical Harmonics,
/(*, <f>) = Y, J^ [CrYr(0, <f>) sin(m0)
1 = m =
oo I
1 = 7Tl =
Then
n
+ B, m y, m '(tf,^)].
EL =0 ( g r^r + stbd
(30)
(31)
VELo( c r 2 + a,™ VeLoW 12 + s « m2 )
The confidence levels are then given by
Gi(r) = r
(32)
G 2 (r) = r(l + | 5 2 )- |r(3-r 2 )
I,
8'
G 4 (r) = r{l+| S 2 [l+| S 2 (l+| S 2 )]}
= ^r(35 - 35r 2 + 21r 4 - 5r 6 ),
G 3 (r) = r[l + |s 2 (l + f s 2 )] = |r(15 - 10r 2 + 3r 4 )
where
VT"
(33)
(Eckhardt 1984).
see also FISHER'S ^'-TRANSFORMATION, SPEARMAN
Rank Correlation, Spherical Harmonic
References
Bevington, P. R. Data Reduction and Error Analysis for the
Physical Sciences. New York: McGraw-Hill, 1969.
Eckhardt, D. H. "Correlations Between Global Features of
Terrestrial Fields." Math. Geology 16, 155-171, 1984.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Pugh, E. M. and Winslow, G. H. The Analysis of Physical
Measurements. Reading, MA: Addis on- Wesley, 1966.
Correlation Dimension
Define the correlation integral as
oo
C(e) = lim -L J2 H(e - \\x t - xj\\), (1)
i*3
where H is the Heaviside Step Function. When the
below limit exists, the correlation dimension is then de-
fined as
D 2 = d c ,
m C(e f
lim
e,e'->0+
If v is the Correlation Exponent, then
lim v
c-+0
D 2 .
It satisfies
< d in f < dcap = di^y
(2)
(3)
(4)
To estimate the correlation dimension of an M-
dimensional system with accuracy (1 — Q) requires iV m i n
data points, where
JV=
g(2 - Q)
2(1 -Q)
•\ M
(5)
where R > 1 is the length of the "plateau region." If
an Attractor exists, then an estimate of D 2 saturates
above some M given by
M> 2D + 1,
(6)
which is sometimes known as the fractal Whitney em-
bedding prevalence theorem.
see also CORRELATION EXPONENT, <?-DlMENSION
References
Nayfeh, A. H. and Balachandran, B. Applied Nonlinear
Dynamics: Analytical, Computational, and Experimental
Methods. New York: Wiley, pp. 547-548, 1995.
Correlation Exponent
A measure v of a STRANGE ATTRACTOR which allows
the presence of CHAOS to be distinguished from random
noise. It is related to the Capacity Dimension D and
INFORMATION DIMENSION cr, satisfying
It satisfies
v< a< D.
v < D K y,
(1)
(2)
where Dry is the Kaplan- Yorke Dimension. As the
cell size goes to zero,
lim v
e-+0
D 2 ,
(3)
where D 2 is the CORRELATION DIMENSION.
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Correlation (Geometric)
Correlation (Statistical) 337
Correlation (Geometric)
A point-to-line and line-to-point TRANSFORMATION
which transforms points A into lines a' and lines 6 into
points B' such that a passes through B' Iff A! lies on
6.
see also Polarity
Correlation Index
__ a yy
SySy
r c 2 = % = 1
SSE
see also Correlation Coefficient
where
Nrj 2
2(1 -n 2 )
L _ n2
o — — -,
(6)
(7)
(8)
and 1 F 1 (a,b;z) is the CONFLUENT HYPERGEOMETRIC
Limit Function. If A = 0, then
f(E 2 )=(3(a,b)
(9)
(Kenney and Keeping 1951, pp. 323-324).
see also Correlation Coefficient, Regression Co-
efficient
Correlation Integral
Consider a set of points X; on an ATTRACTOR, then the
correlation integral is
where / is the number of pairs (2,7) whose distance |X; —
Xj| < I. For small /,
where v is the CORRELATION EXPONENT.
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Correlation Ratio
Let there be N% observations of the zth phenomenon,
where i = 1, . . . , p and
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Correlation (Statistical)
For two variables x and y }
cor(x,y) =
cov(x,y)
(1)
where er x denotes STANDARD DEVIATION and cov(x,y)
is the COVARIANCE of these two variables. For the gen-
eral case of variables z» and Xj, where i, j = 1, 2, . . . ,
71,
COv(Xi,Xj)
cor(#i, Xj) —
y/W~»
(2)
where Vu are elements of the Covariance Matrix. In
general, a correlation gives the strength of the relation-
ship between variables. The variance of any quantity is
alway NONNEGATIVE by definition, so
Then
J^yx —
nY.^,**
EjNijyi-y) 2
EiEc.(2/--J/) 2 '
(i)
(2)
(3)
(4)
Let 7] yx be the population correlation ratio. If Ni = Nj
for i y£ j, then
f(E')
t2 . _ e-'HE 2 )- 1 ^ - E 2 ) t '- 1 1 F 1 (a,fc;AE 2 )
B(a, b)
(5)
var [ A + JL ] > 0.
(3)
From a property of VARIANCES, the sum can be ex-
panded
\o~ x )
var I — J + var ( — ) + 2 cov f — , — ) > (4)
\<? X &y
112
— var(x) H var(y) H cov(z, y) > (5)
2 2
1 + H cov(x, y) = 2 H cov(x, y) > 0. (6)
Therefore,
cor(x, ;/ )= COv(a: ' y) >-l.
(7)
338 Cosecant
Similarly,
var
£)-(£)*• <8)
Vcr x /
+ var ( -— ) +2cov ( — ,--^- ) > (9)
<Jy J \a X CTy
112
— - var(z) H var(y) cov(z, y) > (10)
fli <T y J a x a y
2 2
1 _l_ 1 cov(x,y) = 2 cov(x,y) > 0. (11)
<T X (Jy <T X <7y
Therefore,
, . cov(:r,?/) ^ „
cor(x,y) = ^^ <1,
(12)
so —1 < cox{x,y) < 1. For a linear combination of two
variables,
var(y — bx) = var(y) + var(— bx) + 2 cov(y, — bx)
= var(y) + 6 var(x) — 26cov(a;,y)
— <y y -\- (Tx -2bcov(x,y). (13)
Examine the cases where cor(x,y) = ±1,
(14)
_ cov(z,;y)
cor(x,y) = = ±1
var(y — bx) — b a x -\-cr y ^f2ba x cr y — (bcr x ^a y ) . (15)
The VARIANCE will be zero if b = ±a y /a x , which re-
quires that the argument of the VARIANCE is a constant.
Therefore, y — bx = a, so y — a + foe. If cor(#, y) = zbl,
y is either perfectly correlated (b > 0) or perfectly anti-
correlated (b < 0) with x.
see also Covariance, Covariance Matrix, Vari-
ance
Cosecant
10
7 .5
5
2.5
Cosine
The function defined by esc a: = 1/sinz, where since
is the Sine. The Maclaurin Series of the cosecant
function is
1 1 7 3
CSCX = x ~^~ Q X ^ 360^ + 15120
X° + . . .
+
l) n+1 2(2 2
(2n)!
l)^2n 2n-l ,
a? + ,
where £2™ is a Bernoulli Number.
see also Inverse Cosecant, Secant, Sine
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Spanier, J. and Oldham, K. B. "The Secant sec(a;) and Cose-
cant csc(x) Functions." Ch. 33 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 311-318, 1987.
Coset
Consider a countable SUBGROUP H with ELEMENTS hi
and an element x not in H, then
xhi
hiX
(i)
(2)
for i = 1, 2, ... are left and right cosets of the Sub-
group H with respect to x. The coset of a SUBGROUP
has the same number of ELEMENTS as the SUBGROUP.
The Order of any Subgroup is a divisor of the Order
of the GROUP. The original GROUP can be represented
by
G = H + xiH + X2H + .... (3)
For G a not necessarily FINITE GROUP with H a Sub-
group of G y define an EQUIVALENCE RELATION x ~ y
if x = hy for some h in H. Then the EQUIVALENCE
Classes are the left (or right, depending on conven-
tion) cosets of H in G, namely the sets
{x G G : x = ha for some h in H},
where a is an element of G.
see also EQUIVALENCE CLASS, GROUP, SUBGROUP
Cosh
see Hyperbolic Cosine
Cosine
sin 6
(4)
Cosine
Cosine 339
Let 6 be an Angle measured counterclockwise from the
£-axis along the arc of the unit CIRCLE. Then cos0
is the horizontal coordinate of the arc endpoint. As a
result of this definition, the cosine function is periodic
with period 2ir.
| Cos z|
The cosine function has a Fixed Point at 0.739085.
The cosine function can be defined algebraically using
the infinite sum
cos a? ;
-2^ (2n)! 2! + 4! 6! + ' ' ' ' l '
7X3=0
or the Infinite Product
cos a; :
n
4x 2
7r 2 (2n-l) 2 _
A close approximation to cos(x) for x € [0,7r/2] is
(2)
(§■)
1-
x + (i-*)ya=-
(3)
(Hardy 1959). The difference between cos x and Hardy's
approximation is plotted below.
The Fourier Transform of cos(27rfc £) is given by
/oo
e~ 27rikx cos(27vk x) dx
-GO
= ±[8(k-k )+8(k + ko)], (4)
where S(k) is the Delta Function.
The cosine sum rule gives an expansion of the COSINE
function of a multiple ANGLE in terms of a sum of POW-
ERS of sines and cosines,
cos(n(9) = 2cos<9cos[(n - 1)0] - cos[(n - 2)0}
= cos n •
n \ n-2 a • 2 /,
I cos t/sm v
+ I" I cos n - 4 0sin 4 9
(5)
Summing the COSINE of a multiple angle from n = to
AT - 1 can be done in closed form using
\^ cos(nx) = 5R
£■
(6)
The Exponential Sum Formulas give
JV-1
^^«^ — so '.
_ sin(|a;)
i(|iV^)
> cos(nx) = 5ft
n=0
sm(±Nx) i(N . 1)x/2
cos[±ic(iV-l)]
sin(|x)
Similarly,
/P cos(nz) — £ft
En in
p e
(7)
(8)
where \p\ < 1. The EXPONENTIAL Sum FORMULA gives
1 — pe
> p n cos(nx) — Sft
1 — 2p cos re + p 2
1 — p cos a;
1 — 2p cos a; + p 2
(9)
Cvijovic and Klinowski (1995) note that the following
series
^ cos(2fc + l)a
fc=o
(10)
has closed form for ^ = 2n,
C2 - (Q) = 4(2.-1)! " ^-'WJ'
(11)
where £7„(a:) is an EULER POLYNOMIAL,
see also EULER POLYNOMIAL, EXPONENTIAL SUM FOR-
MULAS, Fourier Transform — Cosine, Hyperbolic
Cosine, Sine, Tangent, Trigonometric Functions
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
340
Cosine Apodization Function
Cosine Integral
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 68, 1959.
Cvijovic, D. and Klinowski, J. "Closed- Form Summation of
Some Trigonometric Series." Math. Comput. 64, 205-210,
1995.
Hansen, E. R. A Table of Series and Products. Englewood
Cliffs, NJ: Prentice-Hall, 1975.
Project Mathematics! Sines and Cosines, Parts I-IIL Video-
tapes (28, 30, and 30 minutes). California Institute of
Technology. Available from the Math. Assoc. Amer.
Spanier, J. and Oldham, K. B. "The Sine sin(a:) and Co-
sine cos(x) Functions." Ch. 32 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 295-310, 1987.
Cosine Apodization Function
The Apodization Function
A(z)=cosg).
Its Full Width at Half Maximum is 4a/3. Its In-
strument Function is
/(*) =
4acos(27rafc)
7r(l-16a 2 fc 2 )'
see also APODIZATION FUNCTION
Cosine Circle
Also called the second Lemoine Circle. Draw lines
through the Lemoine Point K and Parallel to the
sides of the Triangles. The points where the antiparal-
lel lines intersect the sides then lie on a Circle known as
the cosine circle with center at K. The CHORDS P2Q3,
F3Q1, and P1Q2 are proportional to the COSINES of the
Angles of AAi^4 2 ^4.3, giving the circle its name.
Triangles P1P2P2. and AA1A2A3 are directly similar,
and Triangles AQ1Q2Q3 and A1A2A3 are similar.
The Miquel Point of AP1P2P3 is at the Brocard
Point n of AP1P2P3.
see also Brocard Points, Lemoine Circle, Miquel
Point, Tucker Circles
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 271-273, 1929.
Cosine Integral
0.5r
There are (at least) three types of "cosine integrals,"
denoted ci(x), Ci(cc), and Cin(a;):
c\{x) = -
J X
cos t dt
t
= l[ei(ix) + ei(-ix)]
= -±[E 1 (ix) + E l (-ix)] J
Ci(,)^ 7 + lnz+ f Z ^l
Jo t
„. , , f z (1 — cost) dt
Cm(a Wo i
= — Ci(a;) + In x + 7.
dt
(1)
(2)
(3)
(4)
(5)
(6)
Here, ei(x) is the EXPONENTIAL INTEGRAL, E n (x) is
the E n -FUNCTiON, and 7 is the Euler-Mascheroni
CONSTANT. ci(x) is the function returned by the
Mathematical (Wolfram Research, Champaign, IL)
command Coslntegral [x] and displayed above.
To compute the integral of an Even power times a co-
sine,
I,J X - COs{mx)dx , (7)
use Integration by Parts. Let
dv = cos(mx) dx
du = 2nx 2n ~ l dx
v = — sin(mx).
m
(8)
(9)
Cosine Integral
T 1 2n . / x 2n
I = — x smfraxj
m 77i
/2n-l
X
sin(mx) dx. (10)
Using Integration by Parts again,
u = x 2 ™ -1 dv = sin(Tnx) dx (11)
du = (2n - l)x 2n 2 dx v = cos(rox), (12)
and
1 2n • / \ 2n [ 1 2n-l / x
= — x smfraxj x cosimx)
m m I m
2n-
2n • / \ - ^ 2n-l / \
-x sin(mx) -\ -x cos(mx)
(2n)(2n
ii/,-
1 cos(mx) dx
2n
= —x sin(mx) + — ^x cos(mx)
+ ...+
(2
pT J x cos
(mx) dx
1 2n • / \ , 277 2n — 1 / \
= —x sin(mx) H -x cos(mx)
(2n)! . , ,
+ ••■ + \„li sm(mx)
m 2n+l
• / ^/ ^fc + 1 ( 2yi ) ! „,2n-2fc
— Sin(?7lX) > (—1) 7T TTT1 . , n ^
v ; Z_^ V ' (2n-2fc)!77i 2fc+1
+ cos(mx) /(-I)
(2n-2fc)l
*+i (2n)!
2n-2fc + l
(2k - 2n - l)lm 2k
(13)
Letting k' ~ n — k,
n
= sin(mx) > (— l) n ~ fc+1 7 rtT N1 «. . ,
(2n)! ^
k=0
n-1
+C0S ( mx )Y / (-D n ' k+1 w ^
(2")l ^fc+i
= (-l) n+1 (2n)!
sin(m:r) N
(2fc- l)!m 2 "- 2fc "
(-1)" .»
fc~0
(2ifc)!m :
2n-2fc + l
/ \ V^ \~ l ) 2fc-l
+ cos(m,)^ (2jfc _ 3)!m2n _ 2fc+2 ,
. (14)
Cosine Integral 341
To find a closed form for an integral power of a cosine
function,
1=1 cos m xdx, (15)
/■
perform an INTEGRATION BY PARTS so that
u = cos m_ x dv — cosxdx (16)
du = — (m — 1) cos m ~ 2 xsinxdx v — sinx. (17)
Therefore
/ = sinx cos m x + (m — 1) / cos m x sin x dx
/■
: sin x cos m x
+ (m - 1)
= sin x cos m ~ x + (m — 1)
/m — 2 j / m j
cos xdx — I cos a: ax
/ cos m ~ 2 xdx — J , (18)
/■
J[l + (ra — 1)] = sin x cos m x x + (771 — 1) j cos m 2 x dx
(19)
/
I — I cos m x dx
sinxcos m x x m-l f m _ 2 , /on x
j 1 cos xdx. (20)
ra m J
Now, if 7n is Even som = 2n, then
;dx
/2n
cos x t
sin x cos 271 * x 2n-
2n
2n
V
cos x dx
sinxcos 2n *x 2n — 1
2n n
sin x cos 2n 3 x
+
2n- 2
2n - 3 / 2n-4
2n-2
/
in — <± j
cos x ax
1 2n-l , 277 - 1 2n-3
-— cos X + /n s/n -7 cos X
2n (2n)(2n~2)
+
(2n-l)(2;
(2n)(2n
L2n
— / cos n xdx
-2) J
271-1 2n-3 ,
+ /„ x/rt ^7 COS X + . . .
(2n)(2n-2)
(2n-
:n-l)(2n-3)---l / T ,
-; r-; T / COS X OtX
(2n)(2n-2)---2 J
, ^ (2n-2fe)H (2n-l)!! *«-»+! „
'2^ (2n)!! (2n-2fc + l)!!
fc=i
(277-1)!!
+L ^)ir a: - (21)
342
Cosine Integral
Cosmological Theorem
Now let k r = n — k -f 1, so n — k = k' — 1,
/-*
x da;
x (2fe-2)!! (2 n-l)!! M _ t , (2n-l)ll
sin a; > ,„ ,.. -^r; 7-^7 cos x + ,„ ,,. a
E
_ (2n-l)!!
(2n)!!
(2n)!! (2fc-l)!!
(2n)!!
sinx >
(2fc)!
(2A + 1)!!
COS X + X
(22)
Now if m is Odd so m = 2n 4- 1, then
/
cos xdx
sinxcos 2n x 2n
2n+ 1
+
2n +
i/
cos 2n * x dx
sinxcos^x 2n sinxcos^^x
2n + 1
2n
■ +
2n + 1
+
2n
3J
2n-3 j
cos x dx
1 2n
: sinxl cos x +
2n- 1
2n
2n + 1
(2n+l)(2n- 1)
cos 2n_2 x
+ (2
= sin a;
(2n)(2w-2) f
ra + l)(2n-l) 7
2n-3 j
cos o? ax
1
L2n+1
2n
2n
cos X
(2n + l)(2n- 1)
cos 2 "" 2 x + . . .
(2n)(2n-2)---2 f
+ y-^ — ^Vt ^ / cos xdx
(2n+l)(2n- 1)---3J
E (2n-2fc-l)!! (2n)
(2n+l)H (2n
fc=0
(2n+l)H (2n-2fe)!
(23)
Now let k l = n — k,
— __! — ill— s i n # \
(2n+l)!! ^
The general result is then
/
2„ , (2n)!
cos x ax —
(2n+l)!! ^ (2k)l\
(2fe-l)H 2fe
cos X.
(24)
The infinite integral of a cosine times a Gaussian can
also be done in closed form,
F
e~ ax2 cos(kx)dx = yf e" fc2/4 °.
(26)
see also Chi, Damped Exponential Cosine Inte-
gral, Nielsen's Spiral, Shi, Sici Spiral, Sine In-
tegral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co-
sine Integrals." §5.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 231-233, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 342-343, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Fresnel Integrals, Cosine and Sine Integrals."
§6.79 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 248-252, 1992.
Spanier, J. and Oldham, K. B. "The Cosine and Sine Inte-
grals." Ch. 38 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 361—372, 1987.
Cosines Law
see Law of Cosines
Cosmic Figure
see Platonic Solid
Cosmological Theorem
There exists an INTEGER N such that every string in
the Look and Say Sequence "decays" in at most N
days to a compound of "common" and "transuranic el-
ements."
The table below gives the periodic table of atoms asso-
ciated with the Look and Say Sequence as named
by Conway (1987). The "abundance" is the average
number of occurrences for long strings out of every mil-
lion atoms. The asymptotic abundances are zero for
transuranic elements, and 27.246. . . for arsenic (As), the
next rarest element. The most common element is hy-
drogen (H), having an abundance of 91,970.383 The
starting element is U, represented by the string "3," and
subsequent terms are those giving a description of the
current term: one three (13); one one, one three (1113);
three ones, one three (3113), etc.
/
cos m x dx = <
-"*£S cos2l+1 * + *
fe=0
for 77i = In
(2n)!! - V^ (2fc-l)!! 2fc
for 7n = 2n + 1.
(25)
Cosmological Theorem
Costa-Hoffman-Meeks Minimal Surface 343
Abundance n E n E n is the derivate of E n+1
3
12
1113
3113
132113
1113122113
311311222113
Ho.1322113
1113222113
3113322113
Pm.123222113
111213322113
31121123222113
132112211213322113
111312212221121123222113
3113112211322112211213322113
1321132122211322212221121123222113
1131221131211322113321132211221121
3322113
Ge.Ca. 312211322212221121123222113
13112221133211322112211213322113
11132. Pa.H.Ca.W
311312
1321131112
11131221133112
311311222. Ca.Co
1321132.Pm
111312211312
3113112221131112
Ho. 13221133112
1113222. Ca.Co
311332
132.Ca.Zn
111312
31131112
1321133112
11131. H. Ca.Co
311311
13211321
11131221131211
311311222113111221
Ho. 1322113312211
Eu.Ca. 3112221
Pm. 13211
11131221
3113112211
132113212221
111312211312113211
311311222113111221131221
Ho. 132211331222113112211
Eu.Ca.311322113212221
13211322211312113211
1113122113322113111221131221
Er. 12322211331222113112211
1112133. H.Ca.Tc
3112112.U
1321122112
11131221222112
3113112211322112
102.56285249
92
U
9883.5986392
91
Pa
7581,9047125
90
Th
6926.9352045
89
Ac
5313.7894999
88
Ra
4076.3134078
87
Fr
3127.0209328
86
Rn
2398.7998311
85
At
1840.1669683
84
Po
1411.6286100
83
Bi
1082.8883285
82
Pb
830.70513293
81
Tl
637.25039755
80
Hg
488.84742982
79
Au
375.00456738
78
Pt
287.67344775
77
Ir
220.68001229
76
Os
169.28801808
75
Re
315.56655252
74
W
242.07736666
73
Ta
2669.0970363
72
Hf
2047.5173200
71
Lu
1570.6911808
70
Yb
1204.9083841
69
Tm
1098.5955997
68
Er
47987.529438
67
Ho
36812.186418
66
Dy
28239.358949
65
Tb
21662.972821
64
Gd
20085.668709
63
Eu
15408.115182
62
Sm
29820.456167
61
Pm
22875.863883
60
Nd
17548.529287
59
Pr
13461.825166
58
Ce
10326.833312
57
La
7921.9188284
56
Ba
6077.0611889
55
Cs
4661.8342720
54
Xe
3576.1856107
53
I
2743.3629718
52
Te
2104.4881933
51
Sb
1614.3946687
50
Sn
1238.4341972
49
In
950.02745646
48
Cd
728.78492056
47
Ag
559.06537946
46
Pd
428.87015041
45
Rh
328.99480576
44
Ru
386.07704943
43
Tc
296.16736852
42
Mo
227.19586752
41
Nb
174.28645997
40
Zr
133.69860315
39
Y
102.56285249
38
Sr
78.678000089
37
Rb
60.355455682
36
Kr
46.299868152
35
Br
Abundance
n
E*
E n is the derivate of E n+1
35.517547944
34
Se
13211321222113222112
27.246216076
33
As
11131221131211322113322112
1887.4372276
32
Ge
31131122211311122113222.Na
1447.8905642
31
Ga
Ho.13221133122211332
23571.391336
30
Zn
Eu.Ca.Ac.H.Ca.312
18082.082203
29
Cu
131112
13871.123200
28
Ni
11133112
45645.877256
27
Co
Zn.32112
35015.858546
26
Fe
13122112
26861.360180
25
Mn
111311222112
20605.882611
24
Cr
31132.Si
15807.181592
23
V
13211312
12126.002783
22
Ti
11131221131112
9302.0974443
21
Sc
3113112221133112
56072.543129
20
Ca
Ho. Pa. H. 12. Co
43014.360913
19
K
1112
32997.170122
18
Ar
3112
25312.784218
17
CI
132112
19417.939250
16
s
1113122112
14895.886658
15
p
311311222112
32032.812960
14
Si
Ho. 1322112
24573.006696
13
Al
1113222112
18850.441228
12
Mg
3113322112
14481.448773
11
Na
Pm.123222112
11109.006696
10
Ne
111213322112
8521.9396539
9
F
31121123222112
6537.3490750
8
o
132112211213322112
5014.9302464
7
N
111312212221121123222112
3847.0525419
6
C
3113112211322112211213322112
2951.1503716
5
B
1321132122211322212221121123222112
2263.8860325
4
Be
11131221131211322113321132211221121
3322112
4220.0665982
3
Li
Ge.Ca.312211322212221121123222122
3237.2968588
2
He
13112221133211322112211213322112
91790.383216
1
H
Hf.Pa.22.Ca.Li
see also Conway's Constant, Look and Say Se-
quence
References
Conway, J. H. "The Weird and Wonderful Chemistry of Au-
dioactive Decay." §5.11 in Open Problems in Communica-
tion and Computation (Ed. T. M. Cover and B. Gopinath).
New York: Springer- Verlag, pp. 173-188, 1987.
Conway, J. H. "The Weird and Wonderful Chemistry of Au-
dioactive Decay." Eureka, 5-18, 1985.
Ekhad, S. B. and Zeilberger, D. "Proof of Conway's
Lost Cosmological Theorem." Electronic Research An-
nouncement of the Amer. Math. Soc. 3, 78-82,
1997. http : //www .mathtemple . edu/~zeilberg/mamarim/
mamarimhtml/horton.html.
Costa-Hoffman-Meeks Minimal Surface
see Costa Minimal Surface
344 Costa Minimal Surface
Costa Minimal Surface
A complete embedded MINIMAL SURFACE of finite to-
pology. It has no BOUNDARY and does not intersect
itself. It can be represented parametrically by
Cotangent Bundle
see also Brocard Angle, Brocard Circle, Bro-
card Points, Brocard Triangles, Circumcircle,
Lemoine Point, Symmedian Line
Cotangent
iv) + ITU +
+ iv) + 7TV +
iv^rln
p(u-\-iv) - ei
p(u + if) + ei
where C(z) is the WeierstraB Zeta Function,
p(P2,5a;z) is the WEIERSTRAfi ELLIPTIC FUNCTION,
c = 189.07272, ei = 6.87519, and the invariants are
given by 52 — c and £3 = 0.
References
Costa, A. "Examples of a Complete Minimal Immersion in
R 3 of Genus One and Three Embedded Ends." Bil. Soc.
Bras. Mat. 15, 47-54, 1984.
do Carmo, M. P. Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, p. 43, 1986.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces.Boca Raton, FL: CRC Press, 1993.
Gray, A. Images of the Costa surface, ftp://bianchi.iund.
edu/pub/COSTAPS/.
Nordstrand, T. "Costa-Hoffman-Meeks Minimal Surface."
http : //www . uib . no/people/nf ytn/costatxt . htm.
Peterson, I. "The Song in the Stone: Developing the Art of
Telecarving a Minimal Surface." Sci. News 149, 110-111,
Feb. 17, 1996.
Cosymmedian Triangles
Extend the Symmedian Lines of a Triangle
AA1A2A3 to meet the Circumcircle at Pi, P 2 , iV
Then the Lemoine Point K of AA1A2A3 is also
the Lemoine Point of AP1P2P3. The Triangles
AAiA 2 ^3 and AP1P2P3 are cosymmedian triangles,
and have the same BROCARD CIRCLE, second BROCARD
Triangle, Brocard Angle, Brocard Points, and
Circumcircle.
The function defined by cot a; = 1/tana?, where tana; is
the Tangent. The Maclaurin Series for cot x is
cot a;
l T _ ± x * _ JL X 5 i- X 7 -
3 X 45 ^ 945 ^ 4725*^
n+lrt2n 1
(-l) n+1 2 2n B;
(2n)!
where B n is a BERNOULLI NUMBER.
00
7v cottnx) — — h 2x \ — ■=
v ' x ^— ' x 2 - n 2
It is known that, for n > 3, cot(7r/n) is rational only for
n = 4,
see also Hyperbolic Cotangent, Inverse Cotan-
gent, Lehmer's Constant, Tangent
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Spanier, J. and Oldham, K. B. "The Tangent tan(z) and
Cotangent cot (as) Functions." Ch. 34 in An Atlas of Func-
tions. Washington, DC: Hemisphere, pp. 319-330, 1987.
Cotangent Bundle
The cotangent bundle of a MANIFOLD is similar to the
Tangent Bundle, except that it is the set (x, /) where
x e M and / is a dual vector in the TANGENT SPACE
toiG M. The cotangent bundle is denoted by T*M.
see also Tangent Bundle
Cotes Circle Property
Cotes Circle Property
2n , 1
X + 1 =
X — 2x COS
x — 2x cos
(2n - 1)tt
a; — 2x cos
2n
+ 1
Cotes Number
The numbers X vn in the Gaussian Quadrature for-
mula
Q n {f) = ^TK n f(x„ n ).
see also GAUSSIAN QUADRATURE
References
Cajori, F. A History of Mathematical Notations, Vols. 1-2,
New York: Dover, p. 42, 1993.
Cotes' Spiral
The planar orbit of a particle under a r -3 force field. It
is an EPISPIRAL.
Coth
see Hyperbolic Cotangent.
Coulomb Wave Function
A special case of the Confluent Hypergeometric
Function of the First Kind. It gives the solution to
the radial Schrodinger equation in the Coulomb poten-
tial (1/r) of a point nucleus
d 2 W
dp 2
+
2r) L(L + 1)
W = Q.
(1)
P P-
The complete solution is
W = CiF L (ri,p) + C 2 G L (v,p)- (2)
The Coulomb function of the first kind is
F L (r ) ,p) = C L {r))p L+1 e- i \F l {L + l--i m 2L + 2-2ip),
where
(3)
_ 2*e-^ 2 |r(L + l + ir,)\
Cl{v) = r(2L + 2) ' (4)
iFi(a\b\z) is the Confluent Hypergeometric
Function, T(z) is the Gamma Function, and the
Coulomb function of the second kind is
2/7
GL(r},p)= c 2/ x ^fap)
ln(2p) +
+
{2L + l)C L {v)
■p-i J2 <£wp k+l > (*>
Counting Number 345
where qh, pz,, and a£ are defined in Abramowitz and
Stegun (1972, p. 538).
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Coulomb Wave
Functions." Ch. 14 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 537-544, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 631-633, 1953.
Count
The largest n such that \z n \ < 4 in a MANDELBROT Set.
Points of different count are often assigned different col-
ors.
Countable Additivity Probability Axiom
For a Countable Set of n disjoint events Eu #2, • • • ,
E n
p U* =E p (*>-
\i=i / i=i
see also COUNTABLE Set
Countable Set
A Set which is either Finite or Countably Infinite.
see also Aleph-0, Aleph-1, Countably Infinite Set,
Finite, Infinite, Uncountably Infinite Set
Countable Space
see First-Countable Space
Countably Infinite Set
Any Set which can be put in a One-to-One correspon-
dence with the Natural Numbers (or Integers), and
so has Cardinal Number Ko. Examples of countable
sets include the Integers and Algebraic Numbers.
Georg Cantor showed that the number of Real NUM-
BERS is rigorously larger than a countably infinite set,
and the postulate that this number, the "Continuum,"
is equal to Ni is called the CONTINUUM HYPOTHESIS.
see also Aleph-0, Aleph-1, Cantor Diagonal
Slash, Cardinal Number, Continuum Hypothesis,
Countable Set,
Counting Generalized Principle
If r experiments are performed with n; possible out-
comes for each experiment i=l,2,...,r, then there are
a total of ni=i Ui P oss ibl e outcomes.
Counting Number
A Positive Integer: 1, 2, 3, 4, . . . (Sloane's A000027),
also called a NATURAL Number. However, is some-
times also included in the list of counting numbers. Due
to lack of standard terminology, the following terms
are recommended in preference to "counting number,"
"Natural Number," and "Whole Number."
346 Coupon Collector's Problem
Set
Name
Symbol
. . , , -2, -1, 0, 1, 2, . . . integers Z
1, 2, 3, 4, ... positive integers Z
0, 1, 2, 3, 4 . . . nonnegative integers Z*
— 1, —2, —3, —4, . . . negative integers Z~
see a/so Natural Number, Whole Number, Z, Z",
Z + ,Z*
References
Sloane, N. J. A. Sequence A000027/M0472 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Coupon Collector's Problem
Let n objects be picked repeatedly with probability pi
that object i is picked on a given try, with
E* = L
Find the earliest time at which all n objects have been
picked at least once.
References
Hildebrand, M. V. "The Birthday Problem." Amer, Math.
Monthly 100, 643, 1993.
Covariance
Given n sets of variates denoted {xi}, ..., {x n } , a
quantity called the COVARIANCE MATRIX is denned by
Vij — cov(x i7 Xj)
= {(Xi ~ fii)(Xj - fly))
— \%iXj) ~ \ x i) \%j) j
(i)
(2)
(3)
where pbi = (xi) and jjlj = (xj) are the MEANS of xi
and Xj, respectively. An individual element Vij of the
Covariance Matrix is called the covariance of the
two variates Xi and Xj, arid provides a measure of how
strongly correlated these variables are. In fact, the de-
rived quantity
cor(xi i Xj) =
COv(Xj,Xj)
<J%<J
(4)
iv 3
where a, crj are the STANDARD DEVIATIONS, is called
the Correlation of Xi and xj. Note that if x% and Xj
are taken from the same set of variates (say, x), then
cov(x,x) = (x 2 ) — (x) 2 = var(x), (5)
giving the usual VARIANCE var(x). The covariance is
also symmetric since
cov(x,y) — cov(y, x).
(6)
For two variables, the covariance is related to the VARI-
ANCE by
Covariance Matrix
For two independent variates x = xi and y = Xj,
cov(x, y) — (xy) - \i^\i y = (x) (y) - ^xVy = 0, (8)
so the covariance is zero. However, if the variables are
correlated in some way, then their covariance will be
Nonzero. In fact, if cov(x,y) > 0, then y tends to
increase as x increases. If cov(x, y) < 0, then y tends to
decrease as x increases.
The covariance obeys the identity
cov(x + z, y) = {(x + z)y - (x + z) (y))
= (xy) + (zy) - ((x) + (z)) {y)
= (xy) - (x) {y) + (zy) - (z) (y)
= cov(x,y) + cov(z,y). (9)
By induction, it therefore follows that
(n \ n
i=i / »=i
Cn m \ n / m \
X] Xi ']C^ ) = X] cov ( Xi ^^yi J ( n )
i = l j = l / i=l \ j = l /
n / m \
= ^cov I ^yj,Xi J (12)
»=i \j=i /
n m
= ^^cov(y J -,x i ) (13)
i=i j=i
n m
= ^2^2cov(xi, yj ). (14)
i=i j=i
see also Correlation (Statistical), Covariance
Matrix, Variance
Covariance Matrix
Given n sets of variates denoted {xi}, . . . , {x n } , the
first-order covariance matrix is defined by
Vij = COv(x ii Xj) = ((Xi - lli){Xj - flj)) ,
where (Jbi is the MEAN. Higher order matrices are given
by
V™ n = ((xi - fj.i) m (xj - {ij) n ) .
An individual matrix element Vij = cov(x;,Xj) is called
the Covariance of Xi and xj.
see also CORRELATION (STATISTICAL), COVARiANCE,
Variance
var(x + y) — var(x) + var(y) + 2 cov(x, y). (7)
Covariant Derivative
Cover 347
Covariant Derivative
The covariant derivative of a TENSOR A a (also called the
Semicolon Derivative since its symbol is a semicolon)
A" ]a = V.A = A k k + T k jk A j ,
and of Aj is
* fc ~ 5 fc * dxk X ;* A *>
(1)
(2)
where T is a CONNECTION COEFFICIENT.
see also Connection Coefficient, Covariant Ten-
sor, Divergence
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 48-50, 1953,
Covariant Tensor
A covariant tensor isaTENSOR having specific transfor-
mation properties (c.f., a Contravariant Tensor).
To examine the transformation properties of a covariant
tensor, first consider the Gradient
for which
„ , 96 A d(b „ 06 „
V(j> = -^-Xi + ^X 2 + -^X 3 ,
OXi OX2 OX3
d(j) d(f> dx,j
dx', Ox, dx'- '
where (p(xi , xi , x% ) = </>' (x[ , x' 2 , x z ) . Now let
d<j>
Ai =
dxi '
(i)
(2)
(3)
then any set of quantities Aj which transform according
to
Finr* •
(4)
or, defining
according to
dij —
dxj
J\% — Q'ij **-j
(5)
(6)
is a covariant tensor. Covariant tensors are indicated
with lowered indices, i.e., a M .
Contravariant Tensors are a type of Tensor with
differing transformation properties, denoted a u . How-
ever, in 3-D Cartesian Coordinates,
dxj
dx',
dx'- dxj
(7)
for i,j = 1, 2, 3, meaning that contravariant and covari-
ant tensors are equivalent. The two types of tensors do
differ in higher dimensions, however. Covariant FOUR-
VECTORS satisfy
a M = A^dt/,
where A is a Lorentz Tensor.
(8)
To turn a Contravariant TENSOR into a covariant
tensor, use the METRIC TENSOR g^ to write
a M = g^a .
(9)
Covariant and contravariant indices can be used simul-
taneously in a Mixed Tensor.
see also Contravariant Tensor, Four-Vector,
Lorentz Tensor, Metric Tensor, Mixed Tensor,
Tensor
References
Arfken, G. "Noncartesian Tensors, Covariant Differentia-
tion." §3.8 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 158-164, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 44-46, 1953.
Covariant Vector
A Covariant Tensor of Rank 1.
Cover
A group C of Subsets of X whose Union contains the
given set X (U{S : S E C} = X) is called a cover (or
a Covering). A Minimal Cover is a cover for which
removal of one member destroys the covering property.
There are various types of specialized covers, includ-
ing proper covers, antichain covers, minimal covers, k~
covers, and fc*-covers. The number of possible covers for
a set of N elements is
the first few of which are 1, 5, 109, 32297, 2147321017,
9223372023970362989, ... (Sloane's A003465). The
number of proper covers for a set of N elements is
lo2"
\C'{N)\ = \C{N)\-\2
N / x N
fc=0 x '
the first few of which are 0, 1, 45, 15913, 1073579193,
... (Sloane's A007537).
see also Minimal Cover
References
Eppstein, D. "Covering and Packing." http://www.ics.uci
. edu/-eppstein/ junkyard/ cover. html.
Macula, A. J. "Covers of a Finite Set." Math. Mag. 67,
141-144, 1994.
Sloane, N. J. A. Sequences A003465/M4024 and A007537/
M5287 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
348
Cover Relation
Coxeter's Loxodromic Sequence of Tangent Circles
Cover Relation
The transitive reflexive reduction of a PARTIAL ORDER.
An element z of a POSET (X, <) covers another element
x provided that there exists no third element y in the
poset for which x < y < z. In this case, z is called an
"upper cover" of x and x a "lower cover" of z.
Covering
see Cover
Covering Dimension
see Lebesgue Covering Dimension
Covering System
A system of congruences cii mod rii with 1 < i < k
is called a covering system if every INTEGER y satisfies
y = di (mod n) for at least one value of i.
see also Exact Covering System
References
Guy, R. K. "Covering Systems of Congruences." §F13 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 251-253, 1994.
Coversine
covers ^4 = 1 — sin A,
where sin A is the Sine.
see also Exsecant, Haversine, Sine, Versine
References
Abramowitz, M. and Stegun, C A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 78, 1972.
Cox's Theorem
Let en, . . . , cr 4 be four Planes in General Position
through a point P and let Pij be a point on the Line
Ci -aj. Let o~ijk denote the PLANE PijPikPjk- Then the
four Planes <X234, cti 34 , 0-124, 0-123 all pass through one
point P1234. Similarly, let <7i, . . . , as be five PLANES
in General Position through P. Then the five points
^2345, ^1345, ^1245, A235, and P1234 all lie in one Plane.
And so on.
see also Clifford's Circle Theorem
Coxeter Diagram
see COXETER-DYNKIN DIAGRAM
Coxeter-Dynkin Diagram
A labeled graph whose nodes are indexed by the gen-
erators of a COXETER GROUP having (Pi,Pj) as an
Edge labeled by Mij whenever Mij > 2, where Mij is
an element of the Coxeter Matrix. Coxeter-Dynkin
diagrams are used to visualize Coxeter Groups, A
Coxeter-Dynkin diagram is associated with each RATIO-
NAL Double Point (Fischer 1986).
see also COXETER GROUP, DYNKIN DIAGRAM, RATIO-
NAL Double Point
References
Arnold, V. I. "Critical Points of Smooth Functions." Proc.
Int. Congr. Math. 1, 19-39, 1974.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 12-13, 1986.
Coxeter Graph
see Coxeter-Dynkin Diagram
Coxeter Group
A group generated by the elements Pi for i ■
subject to
1, ..
(PiPjV
= 1,
where Mij are the elements of a COXETER MATRIX.
Coxeter used the NOTATION [3 p,g,r ] for the Coxeter
group generated by the nodes of a Y-shaped COXETER-
DYNKIN DIAGRAM whose three arms have p, q, and r
Edges. A Coxeter group of this form is finite Iff
■ +
+
p + q q+1 r + 1
>1.
see also BlMONSTER
References
Arnold, V. I. "Snake Calculus and Combinatorics of Ber-
noulli, Euler, and Springer Numbers for Coxeter Groups."
Russian Math. Surveys 47, 3-45, 1992.
Coxeter's Loxodromic Sequence of Tangent
Circles
An infinite sequence of CIRCLES such that every four
consecutive Circles are mutually tangent, and the Cir-
cles' Radii ..., R- n , ..., R- ly R Q , R lt R 2 , #3, R4,
. . . , R n , R n + 1, . . . , are in GEOMETRIC PROGRESSION
with ratio
where <f) is the GOLDEN Ratio (Gardner 1979ab). Cox-
eter (1968) generalized the sequence to SPHERES.
see also Arbelos, Golden Ratio, Hexlet, Pappus
Chain, Steiner Chain
References
Coxeter, D. "Coxeter on 'Firmament.'" http://wvw.bangor.
ac . uk/SculMath/image/donald . htm.
Coxeter, H. S. M. "Loxodromic Sequences of Tangent
Spheres." Aequationes Math. 1, 112-117, 1968.
Coxeter Matrix
Cramer's Rule 349
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Set. Amer.
240, 18-28, Jan. 1979a.
Gardner, M. "Mathematical Games: How to be a Psychic,
Even if You are a Horse or Some Other AnimaL" Set.
Amer. 240, 18-25, May 1979b.
Coxeter Matrix
Annxn SQUARE Matrix M with
M» = 1
Mij = Mji > 1
for all i,j = 1, . . . , n.
see also COXETER GROUP
Coxeter- Todd Lattice
The complex LATTICE Ag corresponding to real lattice
K12 having the densest Hypersphere Packing (KISS-
ING Number) in 12-D. The associated Automorphism
GROUP Go was discovered by Mitchell (1914). The order
of Go is given by
| Aut(A£)| = 2 9 • 3 7 • 5 • 7 = 39, 191, 040.
The order of the AUTOMORPHISM GROUP of K\ 2 is given
by
|Aut(JTi 2 )| = 2 10 .3 7 -5-7
(Conway and Sloane 1983).
see also Barnes- Wall Lattice, Leech Lattice
References
Conway, J. H. and Sloane, N. J. A. "The Coxeter- Todd Lat-
tice, the Mitchell Group and Related Sphere Packings."
Math. Proc. Camb. Phil. Soc. 93, 421-440, 1983.
Conway, J. H. and Sloane, N. J. A. "The 12-Dimensional
Coxeter-Todd Lattice K 12 " §4.9 in Sphere Packings, Lat-
tices, and Groups, 2nd ed. New York: Springer- Verlag,
pp. 127-129, 1993.
Coxeter, H. S. M. and Todd, J. A. "As Extreme Duodenary
Form." Canad. J. Math. 5, 384-392, 1953.
Mitchell, H. H. "Determination of All Primitive Collineation
Groups in More than Four Variables." Amer. J. Math. 36,
1-12, 1914.
Todd, J. A. "The Characters of a Collineation Group in Five
Dimensions." Proc. Roy. Soc. London Ser. A 200, 320-
336, 1950.
Cramer Conjecture
An unproven CONJECTURE that
lim
Pn+l ~ Pn
= 1,
, b -rw (lnp n ) 2
where p n is the nth Prime.
References
Cramer, H. "On the Order of Magnitude of the Difference
Between Consecutive Prime Numbers." Acta Arith. 2,
23-46, 1936.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 7, 1994.
Riesel, H. "The Cramer Conjecture." Prime Numbers and
Computer Methods for Factorization, 2nd ed. Boston,
MA: Birkhauser, pp. 79-82, 1994.
Rivera, C. "Problems & Puzzles (Conjectures): Cramer's
Conjecture." http : //www . sci . net .mx/~crivera/ppp/
conj_007.htm.
Cramer-Euler Paradox
A curve of order n is generally determined by n(n 4-
3)/2 points. So a CONIC SECTION is determined by five
points and a CUBIC CURVE should require nine. But the
Maclaurin-Bezout Theorem says that two curves of
degree n intersect in n 2 points, so two CUBICS intersect
in nine points. This means that n(n + 3)/2 points do
not always uniquely determine a single curve of order n.
The paradox was publicized by Stirling, and explained
by P Kicker.
see also Cubic Curve, Maclaurin-Bezout Theorem
Cramer's Rule
Given a set of linear equations
a 2 x + b 2 y + c 2 z = d 2
, azx + bzy + csz — cfe,
(1)
consider the DETERMINANT
D =
ai
6i
C\
a 2
b 2
c 2
a 3
63
C3
(2)
Now multiply D by x, and use the property of Determi-
nants that Multiplication by a constant is equivalent
to MULTIPLICATION of each entry in a given row by that
constant
a\
61
Ci
a 2
b 2
c 2
=
a 3
h
c 3
a\x 61 c\
a 2 x b 2 c 2
0,3 x 63 Ci
(3)
Another property of DETERMINANTS enables us to add
a constant times any column to any column and obtain
the same Determinant, so add y times column 2 and
z times column 3 to column 1,
xD
aix 4- hy + c\z b\ c\
a 2 x -h b 2 y + c 2 z b 2 c 2
C13X + bsy + c 3 z 63 c 3
dx
61
Ci
d 2
62
C2
d 3
63
C3
(4)
If d = 0, then (4) reduces to xD — 0, so the system
has nondegenerate solutions (i.e., solutions other than
(0, 0, 0)) only if D — (in which case there is a family
of solutions). If d 7^ and D = 0, the system has no
unique solution. If instead d / and D ^ 0, then
solutions are given by
(5)
di
61
ci
d 2
62
C2
d 3
h
C3
D
350
Cramer's Theorem
Criss-Cross Method
and similarly for
y =
CLi
dx
Ci
a 2
d 2
C2
<*3
dz
C3
D
ai
61
di
a 2
62
d 2
a3
63
d 3
Z)
(6)
(7)
This procedure can be generalized to a set of n equations
so, given a system of n linear equations
as 1 1 r di
an ai2
ttlnl «n2
let
L> =
ain
an ai2
fllnl 1n2
d n
ain
(8)
(9)
If d = 0, then nondegenerate solutions exist only if D =
0. If d ^ and £> = 0, the system has no unique
solution. Otherwise, compute
^i(fe-i) d± ai(fc+i) ■■• air,
D k =
an
a-ni
a n (k-i) d n a n (fc_|_i)
(10)
Then Xk = D^/D for 1 < k < n. In the 3-D case, the
VECTOR analog of Cramer's rule is
(AxB)x(CxD) = (A-BxD)C-(A-BxC)D. (II)
see also Determinant, Linear Algebra, Matrix,
System of Equations, Vector
Cramer's Theorem
If X and Y are Independent variates and X + Y is
a Gaussian Distribution, then both X and Y must
have Gaussian Distributions, This was proved by
Cramer in 1936.
Craps
A game played with two Dice. If the total is 7 or 11
(a "natural"), the thrower wins and retains the Dice
for another throw. If the total is 2, 3, or 12 ("craps"),
the thrower loses but retains the Dice. If the total is
any other number (called the thrower's "point"), the
thrower must continue throwing and roll the "point"
value again before throwing a 7. If he succeeds, he wins
and retains the Dice, but if a 7 appears first, the player
loses and passes the dice. The probability of winning is
244/495 w 0.493 (Kraitchik 1942).
References
Kenney, J, F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2 } 2nd ed. Princeton, NJ: Van Nostrand, pp. 12-13,
1951.
Kraitchik, M. "Craps." §6.5 in Mathematical Recreations.
New York: W. W. Norton, pp. 123-126, 1942.
CRC
see Cyclic Redundancy Check
Creative Telescoping
see Telescoping Sum, Zeilberger's Algorithm
Cremona Transformation
An entire Cremona transformation is a BlRATIONAL
Transformation of the Plane. Cremona transfor-
mations are MAPS of the form
Xi+i = f{xi,yi)
Vi+i = g(xiiVi),
in which / and g are Polynomials. A quadratic Cre-
mona transformation is always factorable.
see also NOETHER'S TRANSFORMATION THEOREM
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 203-204, 1959.
Cribbage
Cribbage is a game in which each of two players is dealt a
hand of six Cards. Each player then discards two of his
six cards to a four-card "crib" which alternates between
players. After the discard, the top card in the remaining
deck is turned up. Cards are then alternating played out
by the two players, with points being scored for pairs,
runs, cumulative total of 15 and 31, and playing the
last possible card ("go") not giving a total over 31. All
face cards are counted as 10 for the purpose of playing
out, but the normal values of Jack =11, Queen — 12,
King = 13 are used to determine runs. Aces are always
low (ace = 1). After all cards have been played, each
player counts the four cards in his hand taken in con-
junction with the single top card. Points are awarded
for pairs, flushes, runs, and combinations of cards giv-
ing 15. A Jack having the same suit as a top card is
awarded an additional point for "nobbs." The crib is
then also counted and scored. The winner is the first
person to "peg" a certain score, as recorded on a "crib-
bage board."
The best possible score in a hand is 29, corresponding
to three 5s and a Jack with a top 5 the same suit as
the Jack. Hands with scores of 25, 26, and 27 are not
possible.
see also Bridge Card Game, Cards, Poker
Criss-Cross Method
A standard form of the LINEAR PROGRAMMING problem
of maximizing a linear function over a CONVEX POLY-
HEDRON is to maximize c ■ x subject to mx < b and
x > 0, where m is a given s x d matrix, c and b are
given d- vector and s- vectors, respectively. The Criss-
cross method always finds a Vertex solution if an op-
timal solution exists.
see also CONVEX POLYHEDRON, LINEAR PROGRAM-
MING, Vertex (Polyhedron)
Criterion
Cross-Cap 351
Criterion
A requirement NECESSARY for a given statement or the-
orem to hold. Also called a Condition.
see also BROWN'S CRITERION, CAUCHY CRITERION,
Euler's Criterion, Gauss's Criterion, Korselt's
Criterion, Leibniz Criterion, Pocklington's Cri-
terion, Vandiver's Criteria, Weyl's Criterion
Critical Line
The Line R(s) = 1/2 in the Complex Plane on which
the RlEMANN HYPOTHESIS asserts that all nontrivial
(Complex) Roots of the Riemann Zeta Function
£(s) lie. Although it is known that an INFINITE number
of zeros lie on the critical line and that these comprise
at least 40% of all zeros, the RlEMANN HYPOTHESIS is
still unproven.
see also RlEMANN HYPOTHESIS, RlEMANN ZETA FUNC-
TION
References
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 142, 1991.
Critical Point
A Function y = f(x) has critical points at all points
xq where f'(x ) = or f(x) is not DlFFERENTlABLE.
A Function z = f(x,y) has critical points where the
Gradient V/ = or df/dx or the Partial Deriva-
tive df/dy is not defined.
see also FIXED POINT, INFLECTION POINT, ONLY CRIT-
ICAL Point in Town Test, Stationary Point
Critical Strip
see Critical Line
Crook
A 6-Polyiamond.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Crookedness
Let a Knot K be parameterized by a Vector Func-
tion v(i) with teS 1 , and let w be a fixed Unit VEC-
TOR in R 3 . Count the number of RELATIVE MINIMA of
the projection function w-v(r-). Then the Minimum such
number over all directions w and all K of the given type
is called the crookedness fJ>(K). Milnor (1950) showed
that 27Tfj,(K) is the INFIMUM of the total curvature of
K. For any TAME KNOT K in .
b(K) is the Bridge Index.
see also BRIDGE INDEX
% fi(K) = b(K) where
References
Milnor, J. W. "On the Total Curvature of Knots." Ann.
Math. 52, 248-257, 1950.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 115, 1976.
Cross
In general, a cross is a figure formed by two intersect-
ing Line Segments. In Linear Algebra, a cross is
defined as a set of n mutually PERPENDICULAR pairs
of VECTORS of equal magnitude from a fixed origin in
Euclidean u-Space.
The word "cross" is also used to denote the operation
of the CROSS Product, so a x b would be pronounced
"a cross b."
see also CROSS PRODUCT, DOT, EUTACTIC STAR,
Gaullist Cross, Greek Cross, Latin Cross, Mal-
tese Cross, Papal Cross, Saint Andrew's Cross,
Saint Anthony's Cross, Star
Cross- Cap
The self-intersection of a one-sided Surface. It can be
described as a circular HOLE which, when entered, exits
from its opposite point (from a topological viewpoint,
both singular points on the cross-cap are equivalent).
The cross-cap has a segment of double points which ter-
minates at two "Pinch Points" known as Whitney
Singularities.
The cross-cap can be generated using the general
method for Nonorientable Surfaces using the poly-
nomial function
f(x,y 7 z) = (xz,yz,±(z - x ))
(1)
(Pinkall 1986). Transforming to Spherical Coordi-
nates gives
x(u, v) = \ cosxtsin(2f ) (2)
y(u,v) = ! siniisin(2v) (3)
z(ii, v) = | (cos 2 v — cos usin v) (4)
for u G [0, 2tt) and v € [0,7r/2]. To make the equa-
tions slightly simpler, all three equations are normally
multiplied by a factor of 2 to clear the arbitrary scaling
constant. Three views of the cross-cap generated using
this equation are shown above. Note that the middle one
looks suspiciously like MAEDER'S Owl MINIMAL SUR-
FACE.
352 Cross-Cap
Cross-Correlation Theorem
Another representation is
f (x, y, z) = (yz y 2xy, x 2 - y 2 ), (5)
(Gray 1993), giving parametric equations
x = \ sinusin(2t>) (6)
y — sin(2it) sin v (7)
z = cos(2u) sin t>, (8)
(Geometry Center) where, for aesthetic reasons, the y-
and ^-coordinates have been multiplied by 2 to produce
a squashed, but topologically equivalent, surface. Nord-
strand gives the implicit equation
4x V + y + z 2 + z) + y 2 {y 2 + z 2 - 1) = (9)
which can be solved for z to yield
-2a: 2 ± v /(y 2 + 2x 2 )(l-4x 2 - y 2 )
Ax 2 + y 2
(10)
Taking the inversion of a cross-cap such that (0, 0, —1/2)
is sent to oo gives a Cylindroid, shown above (Pinkall
1986).
The cross-cap is one of the three possible SURFACES ob-
tained by sewing a MOBIUS Strip to the edge of a Disk.
The other two are the Boy Surface and Roman Sur-
face.
see also BOY SURFACE, MOBIUS STRIP, NONORI-
ENTABLE SURFACE, PROJECTIVE PLANE, ROMAN SUR-
FACE
References
Fischer, G. (Ed.). Plate 107 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 108, 1986.
Geometry Center. "The Crosscap." http://www.geom.ximn.
edu/zoo/toptype/pplane/cap/,
Pinkall, U. Mathematical Models from the Collections of Uni-
versities and Museums (Ed. G. Fischer). Braunschweig,
Germany: Vieweg, p. 64, 1986.
Cross-Correlation
Let • denote cross-correlation. Then the cross-
correlation of two functions f(t) and g(t) of a real vari-
able t is defined by
f*g = r(-t)*g(t),
(1)
where * denotes CONVOLUTION and f*(t) is the COM-
PLEX Conjugate of f{t). The Convolution is defined
by
f(t)*g(t)
-F
J — c
f(r)g(t - r)dr,
therefore
/oo
r{- T )g{t-r)dT.
■OO
Let r = — r, so dr = — dr and
f*9= I °° f m (r')g(t + r')(-dT')
J OO
(2)
(3)
/OO
■oo
(r)g(t + T)dT.
The cross-correlation satisfies the identity
(g*h)*{g*h) = (g*g)*(h*h).
If / or g is EVEN, then
f*9 = f*9,
(4)
(5)
(6)
where * denotes Convolution.
see also AUTOCORRELATION, CONVOLUTION, CROSS-
CORRELATION THEOREM
Cross- Correlation Coefficient
The Coefficient p in a Gaussian Bivariate Distri-
bution.
Cross-Correlation Theorem
Let f + g denote the CROSS-CORRELATION of functions
/(*) and g(t). Then
/CO
r(r)g(t + T)dT
oo
/co r poo poo
/ F*{v)e 2lTiUT dv / G(i/)e- 2wi "'<* +T >di/ dr
■oo L" — oo v — oo
/oo /»eo poo
/ / F*(i/)G(i/>- 2w4rC *'-^ awiv '*drdi/di/
CO w -oo v -oo
/oo poo r poo
/ F"(u)G{^)e~ 2niv,t / e- 27riT( "'- u) dr dv dv'
■ co J — oo L.J — oo
/oo poo
I F m {v)G{v l )e- 2iriv ' t 8{y t -v)dv f dv
■oo J — CO
/CO
F m (v)G(v)e- 2nil/t di>
oo
= F[F'{u)G{y)\, (1)
Cross Curve
where T denotes the FOURIER TRANSFORM and
/(*) = nn
g(t) = T[G{
/CO
-oo
/CO
■CO
' dt
)e at.
(2)
(3)
Applying a FOURIER TRANSFORM on each side gives the
cross-correlation theorem,
f*g = T\F'{y)G[y)\.
(4)
If F = Gj then the cross-correlation theorem reduces to
the Wiener-Khintchine Theorem.
see also Fourier Transform, Wiener-Khintchine
Theorem
Cross Curve
see Cruciform
Cross Fractal
see Cantor Square Fractal
Cross of Lorraine
see GAULLIST CROSS
Cross Polytope
A regular POLYTOPE in n-D (generally assumed to sat-
isfy n > 5) corresponding to the CONVEX HULL of the
points formed by permuting the coordinates (± 1, 0, 0,
..., 0). It is denoted j3 n and has Schlafli Symbol
{3 n_2 ,4}. In 3-D, the cross polytope is the OCTAHE-
DRON.
see also MEASURE POLYTOPE, SIMPLEX
Cross Product
For Vectors u and v,
uxv — x(u y v z -u z v y )-y(u x v z -u z v x )+z(u x Vy-UyV x ).
(1)
This can be written in a shorthand NOTATION which
takes the form of a Determinant
(2)
X
y
z
u x
Uy
u z
v x
Vy
V z
It is also true that
u x v = u v sine
|u||v|Vl-(u-v)2,
(3)
(4)
where is the angle between u and v, given by the DOT
Product
cos# = u • v. (5)
Cross-Ratio 353
Identities involving the cross product include
|[n(t) x r 2 (t)] = n(i) x ^ + ^ x r 2 (t) (6)
A x B = -B x A (7)
A x (B + C) = A x B + A x C (8)
(tA) xB = t(AxB). (9)
For a proof that A x B is a Pseudovector, see Arfken
(1985, pp. 22-23). In Tensor notation,
AxB = e ijk A j B k ,
(10)
where ei jk is the Levi-Civita Tensor.
see also Dot Product, Scalar Triple Product
References
Arfken, G. "Vector or Cross Product." §1.4 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 18-26, 1985.
Cross- Ratio
[a, 6, c, d]
{a~b)(c-d)
(a-d)(c-fc)'
For a MOBIUS TRANSFORMATION /,
[a,b,c,d} = [f(a)J(b)J(c)J(d)].
(1)
(2)
There are six different values which the cross-ratio may
take, depending on the order in which the points are
chosen. Let A = [a, 6, c, d\. Possible values of the cross-
ratio are then A, 1 — A, 1/A, (A — 1)/A, 1/(1 — A), and
A/(A-1).
Given lines a, 6, c, and d which intersect in a point O,
let the lines be cut by a line /, and denote the points of
intersection of I with each line by A, B, (7, and D. Let
the distance between points A and B be denoted AB>
etc. Then the cross-ratio
[AB y CD]
(AB)(CD)
(BC)(AD)
(3)
is the same for any position of the / (Coxeter
and Greitzer 1967), Note that the definitions
(AB/AD)/(BC/CD) and (CA/CB)/(DA/DB) are
used instead by Kline (1990) and Courant and Robbins
(1966), respectively. The identity
[AD,BC] + [AB,DC] = 1 (4)
holds IFF AC//BD, where // denotes SEPARATION.
The cross-ratio of four points on a radial line of an IN-
VERSION Circle is preserved under Inversion (Ogilvy
1990, p. 40).
see also Mobius Transformation, Separation
354
Cross-Section
Crossing Number (Graph)
References
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, 1996.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 107-108, 1967.
Kline, M. Mathematical Thought from Ancient to Modern
Times, Vol. 1. Oxford, England: Oxford University Press,
1990.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 39-41, 1990.
Cross-Section
The cross-section of a SOLID is a LAMINA obtained by
its intersection with a Plane. The cross-section of an
object therefore represents an infinitesimal "slice" of a
solid, and may be different depending on the orientation
of the slicing plane. While the cross-section of a Sphere
is always a DISK, the cross-section of a CUBE may be a
Square, Hexagon, or other shape.
see also AXONOMETRY, CAVALIERl'S PRINCIPLE, LAM-
INA, Plane, Projection, Radon Transform,
Stereology
Crossed Ladders Problem
Given two crossed LADDERS resting against two build-
ings, what is the distance between the buildings? Let
the height at which they cross be c and the lengths of
the LADDERS a and b. The height at which b touches
the building k is then obtained by solving
k 4 -2ck 3 + k 2 (a 2
■b 2 )-
■2ck(a 2 -b 2 ) + c 2 {a 2 - b 2 ) = 0.
Call the horizontal distance from the top of a to the
crossing it, and the distance from the top of 6, v. Call
the height at which a touches the building h. There are
solutions in which a, 6, h, k y u y and v are all INTEGERS.
One is a — 119, 6 = 70, c = 30, and u + v = 56.
see also Ladder
References
Gardner, M. Mathematical Circus: More Puzzles, Games,
Paradoxes and Other Mathematical Entertainments from
Scientific American. New York: Knopf, pp. 62-64, 1979.
Crossed Trough
The Surface
2 2
ex y .
see also Monkey Saddle
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 286, 1993.
Crossing Number (Graph)
Given a "good" Graph (i.e., one for which all intersect-
ing Edges intersect in a single point and arise from
four distinct Vertices), the crossing number is the
minimum possible number of crossings with which the
GRAPH can be drawn. A GRAPH with crossing num-
ber is a Planar Graph. Garey and Johnson (1983)
showed that determining the crossing number is an NP-
Complete Problem. Guy's Conjecture suggests
that the crossing number for the COMPLETE GRAPH K n
which can be rewritten
in(n-2) 2 (n-4)
£(n-l) 2 (n-3) 2
for n even
for n odd.
(2)
The first few predicted and known values are given in
the following table (Sloane's A000241).
Order
Predicted
Known
1
2
3
4
5
1
1
6
3
3
7
9
9
8
18
18
9
36
36
10
60
60
11
100
12
150
13
225
14
315
15
441
16
588
Zarankiewicz's Conjecture asserts that the crossing
number for a COMPLETE BlGRAPH is
n-1
m — 1
(3)
It has been checked up to m, n = 7, and Zarankiewicz
has shown that, in general, the FORMULA provides an
upper bound to the actual number. The table below
gives known results. When the number is not known ex-
actly, the prediction of ZARANKIEWICZ'S CONJECTURE
is given in parentheses.
1
2
3
4
5
6
7
1
2
3
1
2
4
6
9
4
4
8
12
18
5
16
24
36
6
36
54
7
77,
79, or (81)
Crossing Number (Graph)
Crucial Point 355
Consider the crossing number for a rectilinear GRAPH
G which may have only straight EDGES, denoted v(G).
For a Complete Graph of order n > 10, the rectilinear
crossing number is always larger than the general graph
crossing number. The first few values for COMPLETE
Graphs are 0, 0, 0, 0, 1, 3, 9, 19, 36, 61 or 62, ...
(Sloane's A014540). The n = 10 lower limit is from
Singer (1986), who proved that
H K n) < 3T2 (5n 4 - 39n 3 + 91n 2 - 57n).
(4)
Jensen (1971) has shown that
Crossing Number (Link)
The least number of crossings that occur in any pro-
jection of a LINK. In general, it is difficult to find the
crossing number of a given Link.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 67-69, 1994.
Crout's Method
A ROOT finding technique used in LU DECOMPOSITION.
It solves the N 2 equations
u(K n )< ^n+G{n z ).
(5)
Consider the crossing number for a toroidal GRAPH. For
a Complete Graph, the first few are 0, 0, 0, 0, 0, 0,
0, 4, 9, 23, 42, 70, 105, 154, 226, 326, ... (Sloane's
A014543). The toroidal crossing numbers for a COM-
PLETE BlGRAPH are given in the following table.
1
2
3
4
5
6
7
1
2
3
4
2
5
5
8
6
12
7
see also Guy's Conjecture, Zarankiewicz's Con-
jecture
References
Gardner, M. "Crossing Numbers." Ch. 11 in Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, pp. 133-144, 1986.
Garey, M. R. and Johnson, D. S. "Crossing Number is NP-
Complete." SIAM J. Alg. Discr. Meth. 4, 312-316, 1983.
Guy, R. K. "Latest Results on Crossing Numbers." In Re-
cent Trends in Graph Theory, Proc. New York City Graph
Theory Conference, 1st, 1970. (Ed. New York City Graph
Theory Conference Staff). New York: Springer- Verlag,
1971.
Guy, R. K. and Jenkyns, T. "The Toroidal Crossing Number
of Km.n." J. Comb. Th. 6, 235-250, 1969.
Guy, R. K.; Jenkyns, T.; and Schaer, J. "Toroidal Crossing
Number of the Complete Graph." J. Comb. Th. 4, 376-
390, 1968.
Jensen, H. F. "An Upper Bound for the Rectilinear Crossing
Number of the Complete Graph." J. Comb. Th. Ser. B
10, 212-216, 1971.
Kleitman, D. J. "The Crossing Number of /C 5in ." J. Comb.
Th. 9, 315-323, 1970.
Singer, D. Unpublished manuscript "The Rectilinear Cross-
ing Number of Certain Graphs," 1971. Quoted in Gard-
ner, M. Knotted Doughnuts and Other Mathematical En-
tertainments. New York: W. H. Freeman, 1986.
Sloane, N. J. A. Sequences A014540, A014543, and A000241/
M2772 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Tutte, W. T. "Toward a Theory of Crossing Numbers." J.
Comb. Th. 8, 45-53, 1970.
i < j C*il01j 4" Oi i2 p2j + . . . + CtiiPij — CLij
i = j anPij + ai202j + . . . + ocuPjj = aij
i > j oluPij + ai2p2j + . . . + ocijPjj = aij
for the N 2 + N unknowns otij and 0ij .
see also LU DECOMPOSITION
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. 36-38, 1992.
Crowd
A group of Sociable Numbers of order 3.
Crown
A 6-POLYIAMOND.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Crucial Point
The HOMOTHETIC CENTER of the ORTHIC TRIANGLE
and the triangular hull of the three EXCIRCLES. It has
Triangle Center Function
a = tan A = sin(2£) + sin(2C) - sin(2i4).
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Lyness, R. and Veldkamp, G. R. Problem 682 and Solution.
Crux Math. 9, 23-24, 1983.
356 Cruciform
Cruciform
A plane curve also called the CROSS Curve and Po-
liceman on Point Duty Curve (Cundy and Rollett
1989). It is given by the equation
2 2 2 2 2 2 n
x y -ax -ay =0,
which is equivalent to
or, rewriting,
!- — - — = °
a 2 b 2 ,
~2 + T = l '
x z y z
2 2
2 ax
In parametric form,
The Curvature is
x = a sec t
y = bcsci.
3a6csc £sec t
(b 2 cos 2 * esc 2 t + a 2 sec 2 t tan 2 t) 3 / 2 *
(1)
(2)
(3)
(4)
(5)
(6)
(7)
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 71, 1989.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 127 and 130-131, 1972.
Crunode
A point where a curve intersects itself so that two
branches of the curve have distinct tangent lines. The
MACLAURIN Trisectrix, shown above, has a crunode
at the origin.
see also ACNODE, Spinode, Tacnode
Crystallography Restriction
Cryptarithm
see Cryptaritiimetic
Cryptarithmetic
A number Puzzle in which a group of arithmetical oper-
ations has some or all of its DIGITS replaced by letters or
symbols, and where the original Digits must be found.
In such a puzzle, each letter represents a unique digit.
see also Alphametic, Digimetic, Skeleton Division
References
Bogomolny, A. "Cryptarithms." http://www.cut— the-knot .
com/st_crypto.html.
Brooke, M. One Hundred & Fifty Puzzles in Crypt-
Arithmetic. New York: Dover, 1963.
Kraitchik, M. "Cryptarithmetic." §3.11 in Mathematical
Recreations. New York: W. W. Norton, pp. 79-83, 1942.
Marks, R. W. The New Mathematics Dictionary and Hand-
book. New York: Bantam Books, 1964.
Cryptography
The science and mathematics of encoding and decoding
information.
see also CRYPTARITHM, KNAPSACK PROBLEM, PUBLIC-
Key Cryptography
References
Davies, D. W. The Security of Data in Networks, Los Ange-
les, CA: IEEE Computer Soc, 1981.
Dime, W. and Hellman, M. "New Directions in Cryptogra-
phy." IEEE Trans. Info. Th. 22, 644-654, 1976.
Honsberger, R. "Four Clever Schemes in Cryptography."
Ch. 10 in Mathematical Gems III. Washington, DC: Math.
Assoc. Amer., pp. 151-173, 1985.
Simmons, G. J. "Cryptology, The Mathematics of Secure
Communications." Math. Intel. 1, 233-246, 1979.
Crystallography Restriction
If a discrete GROUP of displacements in the plane has
more than one center of rotation, then the only rotations
that can occur are by 2, 3, 4, and 6. This can be shown
as follows. It must be true that the sum of the interior
angles divided by the number of sides is a divisor of 360°.
180°(n-2) _ 360°
where m is an INTEGER. Therefore, symmetry will be
possible only for
2n
: m,
n-2
where m is an INTEGER. This will hold for 1-, 2-, 3-, 4-,
and 6-fold symmetry. That it does not hold for n > 6 is
seen by noting that n = 6 corresponds to m = 3. The
m = 2 case requires that n = n — 2 (impossible), and
the m = 1 case requires that n = —2 (also impossible).
see also POINT GROUPS, SYMMETRY
Csaszar Polyhedron
Cube 357
Csaszar Polyhedron
A Polyhedron topologically equivalent to a Torus
discovered in the late 1940s. It has 7 VERTICES, 14
faces, and 21 EDGES, and is the DUAL POLYHEDRON of
the Szilassi Polyhedron. Its Skeleton is Isomor-
phic to the Complete Graph K 7 .
see also Szilassi Polyhedron, Toroidal Polyhe-
dron
References
Csaszar, A. "A Polyhedron without Diagonals." Acta Sci.
Math. 13, 140-142, 1949-1950.
Gardner, M. "The Csaszar Polyhedron." Ch. 11 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H. Freeman, 1988.
Gardner, M. Fractal Music, HyperCards, and More: Math-
ematical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 118-120, 1992.
Hart, G. "Toroidal Polyhedra." http://www.li.net/
-george/virtual-polyhedr a/toroidal. html.
Csch
see Hyperbolic Cosecant
Cube
The three-dimensional Platonic Solid (P 3 ) which is
also called the HEXAHEDRON. The cube is composed of
six Square faces 6{4} which meet each other at Right
Angles, and has 8 Vertices and 12 Edges. It is de-
scribed by the Schlafli Symbol {4,3}. It is a Zono-
hedron. It is also the Uniform Polyhedron Uq with
Wythoff Symbol 3|24. It has the Oh Octahedral
Group of symmetries. The DUAL Polyhedron of the
cube is the Octahedron.
Because the Volume of a cube of side length n is given
by n 3 , a number of the form n 3 is called a CUBIC NUM-
BER (or sometimes simply "a cube"). Similarly, the op-
eration of taking a number to the third Power is called
Cubing.
The cube cannot be Stellated. A Plane passing
through the MIDPOINTS of opposite sides (perpendic-
ular to a C% axis) cuts the cube in a regular HEXAG-
ONAL CROSS-SECTION (Gardner 1960; Steinhaus 1983,
p. 170; Cundy and Rollett 1989, p. 157; Holden 1991,
pp. 22-23). Since there are four such axes, there are four
possibly hexagonal cross-sections. If the vertices of the
cube are (d=l,=bl±l), then the vertices of the inscribed
HEXAGON are (0,-1,-1), (1,0,-1), (1,1,0), (0,1,1),
(-1,0,1), and (-1,-1,0). The largest Square which
will fit inside a cube of side a has each corner a distance
1/4 from a corner of a cube. The resulting SQUARE has
side length 3\/2a/4, and the cube containing that side
is called Prince Rupert's Cube.
The solid formed by the faces having the sides of the
Stella Octangula (left figure) as Diagonals is a
cube (right figure; Ball and Coxeter 1987).
The Vertices of a cube of side length 2 with face-
centered axes are given by (±1,±1,±1). If the cube is
oriented with a space diagonal alon g the z-axis, the coor-
dinates are (0, 0, V3), (0, 2^/2/3, l/\/3), (v^, y/2/3,
-i/V3), (A -v^Ts. VV3), (o, -2J2JI, -1/V3),
(->/2, -\/2A 1A/3 ), ("A y/2/3* -1/V3), and the
negatives of these vectors. A Faceted version is the
Great Cubicuboctahedron.
A cube of side length 1 has INRADIUS, MIDRADIUS, and
ClRCUMRADIUS of
r=i = 0.5
(1)
p= i\/2« 0.70710
(2)
R= j\/3« 0.86602.
(3)
The cube has a DIHEDRAL ANGLE of
a = ±tt. (4)
The Surface Area and Volume of the cube are
2
S = 6a
a
(5)
(6)
see also AUGMENTED TRUNCATED CUBE, BlAUG-
mented Truncated Cube, Bidiakis Cube, Bis-
lit Cube, Browkin's Theorem, Cube Dissection,
Cube Dovetailing Problem, Cube Duplication,
Cubic Number, Cubical Graph, Hadwiger Prob-
lem, Hypercube, Keller's Conjecture, Prince
358
Cube 2-Compound
Cube 5-Compound
Rupert's Cube, Rubik's Cube, Soma Cube, Stella
octangula, tesseract
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.
Cundy, H. and Rollett, A. "Hexagonal Section of a Cube."
§3.15.1 in Mathematical Models, 3rd ed. Stradbroke, Eng-
land: Tarquin Pub., p. 157, 1989.
Davie, T. "The Cube (Hexahedron)." http://www.dcs.
st-and.ac.uk/~d/mathrecs/polyhedra/cube.html.
Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.
edu/~eppstein/junkyard/rect .html.
Gardner, M. "Mathematical Games: More About the Shapes
that Can Be Made with Complex Dominoes." Sci. Amer,
203, 186-198, Nov. 1960.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, 1983.
Cube 2-Compound
Cube 4-Compound
A Polyhedron Compound obtained by allowing two
Cubes to share opposite Vertices, then rotating one a
sixth of a turn (Holden 1971, p. 34).
see also CUBE, CUBE 3-COMPOUND, CUBE 4-
Compound, Cube 5-Compound, Polyhedron Com-
pound
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Cube 3-Compound
A compound with the symmetry of the Cube which
arises by joining three Cubes such that each shares two
C 2 axes (Holden 1971, p. 35).
see also Cube, Cube 2-Compound, Cube 4-
Compound, Cube 5-Compound, Polyhedron Com-
pound
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
A compound with the symmetry of the CUBE which
arises by joining four CUBES such that each Cz axis falls
along the C$ axis of one of the other CUBES (Holden
1971, p. 35).
see also Cube, Cube 2-Compound, Cube 3-
Compound, Cube 5-Compound, Polyhedron Com-
pound
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Cube 5-Compound
60 x
30 x
A Polyhedron Compound consisting of the arrange-
ment of five Cubes in the Vertices of a Dodecahe-
dron. In the above figure, let a be the length of a Cube
Edge. Then
x= ±a(3- V5)
^ = tan- 1 f^^ N ) ^20°54'
.1 (y/l-l
(f> = tan' 1 ( ^— - — 1 « 31 u 43'
V> = 90°-<£^58°17'
a = 90° - « 69°6\
The compound is most easily constructed using pieces
like the ones in the above line diagram. The cube 5-
compound has the 30 facial planes of the Rhombic Tri-
ACONTAHEDRON (Ball and Coxeter 1987).
see also Cube, Cube 2-Compound, Cube 3-
Compound, Cube 4-Compound, Dodecahedron,
Cube Dissection
Cube Duplication 359
Polyhedron Compound, Rhombic Triacontahe-
dron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 135 and
137, 1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 135-136, 1989.
Cube Dissection
A CUBE can be divided into n subcubes for only n = 1,
8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and
n > 48 (Sloane's A014544).
#^«$
The seven pieces used to construct the 3x3x3 cube dis-
section known as the Soma Cube are one 3-Polycube
and six 4-POLYCUBES (1-3 + 6*4 = 27), illustrated
above.
Another 3x3x3 cube dissection due to Steinhaus uses
three 5-POLYCUBES and three 4-POLYCUBES (3-5+3-4 =
27), illustrated above.
It is possible to cut a 1 x 3 Rectangle into two identical
pieces which will form a Cube (without overlapping)
when folded and joined. In fact, an INFINITE number of
solutions to this problem were discovered by C. L. Baker
(Hunter and Madachy 1975).
see also Conway Puzzle, Dissection, Hadwiger
Problem, Polycube, Slothouber-Graatsma Puz-
zle, Soma Cube
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 112-
113, 1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 203-205, 1989.
Gardner, M. "Block Packing." Ch. 18 in Time Travel and
Other Mathematical Bewilderments. New York: W. H.
Freeman, pp. 227-239, 1988.
Gardner, M. Fractal Music, HyperCards, and More: Math-
ematical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 297-298, 1992.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 75-80, 1976.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 69-70, 1975.
Sloane, N. J. A. Sequence A014544 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Cube Dovetailing Problem
Given the figure on the left (without looking at the so-
lution on the right), determine how to disengage the
two slotted CUBE halves without cutting, breaking, or
distorting.
References
Dudeney, H. E. Amusements in Mathematics. New York:
Dover, pp. 145 and 249, 1958.
Ogilvy, C. S. Excursions in Mathematics. New York: Dover,
pp. 57, 59, and 143, 1994.
Cube Duplication
Also called the Delian Problem or Duplication of
the Cube. A classical problem of antiquity which, given
the Edge of a Cube, requires a second Cube to be
constructed having double the VOLUME of the first using
only a STRAIGHTEDGE and COMPASS.
Under these restrictions, the problem cannot be solved
because the DELIAN CONSTANT 2 1/3 (the required RA-
TIO of sides of the original CUBE and that to be con-
structed) is not a EUCLIDEAN NUMBER. The problem
can be solved, however, using a NEUSIS CONSTRUCTION.
see also ALHAZEN'S BILLIARD PROBLEM, COMPASS,
Cube, Delian Constant, Geometric Problems of
Antiquity, Neusis Construction, Straightedge
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 93-94,
1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 190-191, 1996.
Courant, R. and Robbins, H. "Doubling the Cube" and "A
Classical Construction for Doubling the Cube." §3.3.1 and
3.5.1 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England: Oxford
University Press, pp. 134-135 and 146, 1996.
Dorrie, H. "The Delian Cube-Doubling Problem." §35 in
100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 170-172,
1965.
360 Cube-Octahedron Compound
Cube-Octahedron Compound
Cube Point Picking
A Polyhedron Compound composed of a Cube and
its Dual Polyhedron, the Octahedron. The 14 ver-
tices are given by (±1, ±1, ±1), (±2, 0, 0), (0, ±2, 0),
(0, 0, ±2).
The solid common to both the CUBE and OCTAHEDRON
(left figure) in a cube-octahedron compound is a Cub-
octahedron (middle figure). The edges intersecting
in the points plotted above are the diagonals of RHOM-
BUSES, and the 12 RHOMBUSES form a RHOMBIC DO-
DECAHEDRON (right figure; Ball and Coxeter 1987).
see also CUBE, CUBOCTAHEDRON, OCTAHEDRON,
Polyhedron Compound
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 158, 1969.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 130, 1989.
Cube Point Picking
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let two points be picked randomly from a unit n-D Hy-
PERCUBE. The expected distance between the points
A(iV) is then
A(l) = |
A(2) = ^[\/2 + 2 + 51n(l + \/2)] = 0.521405433...
A(3) = Ik [4 + 17V"2 -6^3 + 21 ln(l + y/2)
+ 42 ln(2 + V^ ) - 7tt] = 0.661707182 . . .
A (4) -0.77766...
A(5) =0.87852...
A(6) = 0.96895...
A(7) = 1.05159...
A(8) = 1.12817....
The function A(n) satisfies
\n^<Hn)<{\n?'* y
(Anderssen et al. 1976).
1 + 2(1-
(-£)1
Pick N points pi, , , , , pN randomly in a unit n-cube.
Let C be the CONVEX HULL, so
{N N \
y^ XjPj : Aj > for all j and ^Aj = l|,
Let V(n,iV) be the expected n-D Volume (the Con-
tent) of C, S(n t N) be the expected (n-l)-D Surface
Area of C, and P(n, N) the expected number of Ver-
tices on the Polygonal boundary of C. Then
N[1-V{2,N)] _ 8
lnAT ~ 3
lim VN[4-S(2,N)]
lim
N^oo
V2^ 2
/v
Jo
i + 1 2 - i)r 3/2 dt
= 4.2472965...,
and
lim P(2,N)-
\]nN= |( 7 -In 2)
-0.309150708..
(Renyi and Sulanke 1963, 1964). The average DISTANCE
between two points chosen at random inside a unit cube
is
^(4 + 17^-6^+21 ln(l + v / 2) + 421n(2 + v / 3)-77r)
(Robbins 1978, Le Lionnais 1983).
Pick n points on a CUBE, and space them as far apart
as possible. The best value known for the minimum
straight LINE distance between any two points is given
in the following table.
n
d(n)
5
1.1180339887498
6
1.0606601482100
7
1
8
1
9
0.86602540378463
10
0.74999998333331
11
0.70961617562351
12 0.70710678118660
13 0.70710678118660
14 0.70710678118660
15 0.625
Cube Power
Cubefree 361
see also CUBE TRIANGLE PICKING, DISCREPANCY THE-
OREM, Point Picking
References
Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P.
"Concerning J * - • f y/xi 2 + . . . + Xk 2 dx\ • • • dxk and a
Taylor Series Method." SIAM J. Appl. Math. 30, 22-30,
1976.
Bolis, T. S. Solution to Problem E2629. "Average Distance
Between Two Points in a Box." Amer. Math. Monthly 85,
277-278, 1978.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/geom/geom.html.
Ghosh, B. "Random Distances within a Rectangle and Be-
tween Two Rectangles." Bull. Calcutta Math. Soc. 43,
17-24, 1951.
Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution
to Problem E3217, "Minimum Average Distance Between
Points in a Rectangle." Amer. Math. Monthly 96, 64-65,
1989.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 30, 1983.
Renyi, A. and Sulanke, R. "Uber die konvexe Hiille von n
zufallig gewahlten Punkten, I." Z. Wahrscheinlichkeits 2,
75-84, 1963.
Renyi, A. and Sulanke, R. "Uber die konvexe Hiille von n
zufallig gewahlten Punkten, II." Z. Wahrscheinlichkeits 3,
138-147, 1964.
Robbins, D. "Average Distance Between Two Points in a
Box." Amer. Math. Monthly 85, 278, 1978.
Santalo, L. A. Integral Geometry and Geometric Probability.
Reading, MA: Addison- Wesley, 1976.
Cube Power
A number raised to the third POWER, a; 3 is read as "x
cubed."
see also Cubic Number
Cube Root
0.5
Given a number z, the cube root of z, denoted %fz or
z 1/3 (z to the 1/3 POWER), is a number a such that
a 3 = z. There are three (not necessarily distinct) cube
roots for any number.
~2 -1
For real arguments, the cube root is an INCREASING
Function, although the usual derivative test cannot
be used to establish this fact at the ORIGIN since the
the derivative approaches infinity there (as illustrated
above).
see also Cube Duplication, Cubed, Delian Con-
stant, Geometric Problems of Antiquity, k-
Matrix, Square Root
Cube Triangle Picking
Pick 3 points at random in the unit n-HYPERCUBE. De-
note the probability that the three points form an Ob-
tuse Triangle LT(n). Langford (1969) proved
n(2) = ^ + ~7r = 0.725206483 ....
see also Ball Triangle Picking, Cube Point Pick-
ing
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/geom/geoin.htnil.
Langford, E. "The Probability that a Random Triangle is
Obtuse." Biometrika 56, 689-690, 1969.
Santalo, L. A. Integral Geometry and Geometric Probability.
Reading, MA: Addison- Wesley, 1976.
Cubed
A number to the POWER 3 is said to be cubed, so that
x 3 is called "x cubed."
see also Cube Root, Squared
Cubefree
60
20 40 60 80 100
A number is said to be cubefree if its Prime decom-
position contains no tripled factors. All PRIMES are
therefore trivially cubefree. The cubefree numbers are
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...
(Sloane's A004709). The cubeful numbers (i.e., those
that contain at least one cube) are 8, 16, 24, 27, 32, 40,
48, 54, ... (Sloane's A046099). The number of cube-
free numbers less than 10, 100, 1000, ... are 9, 85, 833,
362
Cubic Curve
Cubic Equation
8319, 83190, 831910, . . . , and their asymptotic density
is 1/C(3) « 0.831907, where ((n) is the Riemann Zeta
Function.
see also BlQUADRATEFREE, PRIME NUMBER, RlEMANN
Zeta Function, Squarefree
References
Sloane, N. J. A. Sequences A004709 and A046099 in "An On-
line Version of the Encyclopedia of Integer Sequences."
Cubic Curve
A cubic curve is an Algebraic Curve of degree 3.
An algebraic curve over a FIELD K is an equation
f(X, Y) = 0, where /(X, Y) is a POLYNOMIAL in X and
Y with Coefficients in K , and the degree of / is the
Maximum degree of each of its terms (Monomials).
Newton showed that all cubics can be generated by the
projection of the five divergent cubic parabolas. New-
ton's classification of cubic curves appeared in the chap-
ter "Curves" in Lexicon Technicum by John Harris pub-
lished in London in 1710. Newton also classified all cu-
bics into 72 types, missing six of them. In addition, he
showed that any cubic can be obtained by a suitable
projection of the Elliptic Curve
2 3 , t 2 .
y = ax + ox -\- ex ■
-<*,
(1)
where the projection is a Birational Transforma-
tion, and the general cubic can also be written as
V
x + ax + 6.
(2)
Newton's first class is equations of the form
xy 2 + ey — ax 3 + bx 2 -f ex + d. (3)
This is the hardest case and includes the Serpentine
Curve as one of the subcases. The third class was
ay 2 = x(x — 2bx + c),
(4)
which is called Newton's Diverging Parabolas.
Newton's 66th curve was the Trident of Newton.
Newton's classification of cubics was criticized by Euler
because it lacked generality. Pliicker later gave a more
detailed classification with 219 types.
Pick a point P, and draw the tangent to the curve at P.
Call the point where this tangent intersects the curve Q.
Draw another tangent and call the point of intersection
with the curve R. Every curve of third degree has the
property that, with the areas in the above labeled figure,
B = 16A
(5)
(Honsberger 1991).
see also Cayley-Bacharach Theorem, Cubic Equa-
tion
References
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 114-118, 1991.
Newton, I. Mathematical Works, Vol 2. New York: Johnson
Reprint Corp., pp. 135-161, 1967.
Wall, C. T. C. "Affine Cubic Functions III." Math. Proc.
Cambridge Phil Soc. 87, 1-14, 1980.
Westfall, R. S. Never at Rest: A Biography of Isaac Newton.
New York: Cambridge University Press, 1988.
Yates, R. C "Cubic Parabola." A Handbook on Curves and
Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 56-
59, 1952.
Cubic Equation
A cubic equation is a POLYNOMIAL equation of degree
three. Given a general cubic equation
z 3 + a2Z 2 + <i\z + ao =
(i)
(the Coefficient a 3 of z 3 may be taken as 1 without
loss of generality by dividing the entire equation through
by 03), first attempt to eliminate the a 2 term by making
a substitution of the form
z = x — A.
(2)
Then
(x - A) 3 + a 2 (x - A) 2 + ai(x - A) + a = (3)
(x 3 - 3Xx 2 + 3X 2 x - A 3 ) + a 2 (x 2 - 2 Ax + A 2 )
+ai(x - A) + a = (4)
x 3 + x 2 (a 2 - 3A) + x(ai - 2a 2 A + 3A 2 )
+(a - aiA + a 2 A 2 - A 3 ) = 0. (5)
The x 2 is eliminated by letting A = aa/3, so
z = x •
1&2-
(6)
Then
z 3 = (x - \a 2 ) 3 — x 3 ~ a 2 x 2 + \a 2 x - ~^a 2 (7)
2 / 1\2 2 2 2^.13 / Q \
a 2 z = a 2 (x - 2<i2) — a 2 x - ^a 2 x + $a 2 {&)
aiz — ai{x — \a 2 ) = a\x — \a\a 2 , (9)
so equation (1) becomes
x 3 + (—02 + a 2 )x 2 + (\a 2 — f a 2 + a\)x
-(^a 2 3 - \a 2 3 + iaioa - a ) = (10)
Cubic Equation
+ (ai - \a 2 2 )x - (|aia 2 - ^a 2 - a ) = (11)
3 , 3ai — a 2 rt 9aia 2 — 27a - 2a 2
z +3-
Defining
-x-2-
54
3ai — a 2
9aia 2 - 27a Q - 2a 2 J
27
:0. (12)
(13)
(14)
then allows (12) to be written in the standard form
x + px — q.
(15)
The simplest way to proceed is to make VlETA'S SUB-
STITUTION
(16)
P
x — w — — ,
3u/
which reduces the cubic to the equation
w 3 - ^ - q = 0,
27w 3
(17)
which is easily turned into a QUADRATIC EQUATION in
w 3 by multiplying through by w 2 to obtain
(w>)*-q{w a )-±p a =
(18)
(Birkhoff and Mac Lane 1965, p. 106). The result from
the Quadratic Equation is
= i (q ± y/F+&?} = h ± y[\* + h.
= R±y/lt? + Q 3 ,
(19)
where Q and R are are sometimes more useful to deal
with than are p and q. There are therefore six solutions
for w (two corresponding to each sign for each ROOT
of w 3 ). Plugging w back in to (17) gives three pairs
of solutions, but each pair is equal, so there are three
solutions to the cubic equation.
Equation (12) may also be explicitly factored by at-
tempting to pull out a term of the form (a; — B) from
the cubic equation, leaving behind a quadratic equa-
tion which can then be factored using the QUADRATIC
Formula. This process is equivalent to making Vieta's
substitution, but does a slightly better job of motivat-
ing Vieta's "magic" substitution, and also at producing
the explicit formulas for the solutions. First, define the
intermediate variables
Q
R:
3ai — a 2 2
9a 2 ai - 27ao - 2a2
54
(20)
(21)
Cubic Equation 363
(which are identical to p and q up to a constant factor).
The general cubic equation (12) then becomes
x 3 + SQx - 2R = 0.
(22)
Let B and C be, for the moment, arbitrary constants.
An identity satisfied by PERFECT CUBIC equations is
that
x 3 -B 3
(x-B^x'+Bx + B*).
The general cubic would therefore be directly factorable
if it did not have an x term (i.e., if Q = 0). However,
since in general Q^0, add a multiple of (x — B) — say
C(x-B) — to both sides of (23) to give the slightly messy
identity
(x 3 -B 3 ) + C(x- B) = (x- B)(x 2 + Bx + B 2 + C) = 0,
(24)
which, after regrouping terms, is
x 3 + Cx-(B 3 + BC) = (x-B)[x 2 + Bx + {B 2 +C)] = 0.
(25)
We would now like to match the COEFFICIENTS C and
-{B 3 + BC) with those of equation (22), so we must
have
C = 3Q (26)
B 3 + BC = 2R. (27)
Plugging the former into the latter then gives
B 3 + 3QB = 2R. (28)
Therefore, if we can find a value of B satisfying the above
identity, we have factored a linear term from the cubic,
thus reducing it to a QUADRATIC EQUATION. The trial
solution accomplishing this miracle turns out to be the
symmetrical expression
B = [R + VQ 3 + R 2 ] 1/3 + [R- \/Q 3 + R2 ] 1/3 - ( 29 )
Taking the second and third POWERS of B gives
B 2 = [R + y/Q*+R 2 ] a/ * + 2[R 2 - (Q 3 + R 2 )} 1/3
+ [R - y/Q 3 + R 2 ] 2/3
= [a + y/Q* + R 2 ] 2 " + [R - \A? 3 + ^ 2 1 2/3 - 2Q (30)
B 3 = -2QB + j[# + y/q 3 + R 2 ] 1/3 + [R - a/q 3 + H 2 ] 1/3 }
x j [R + ^/Q 3 + i? 2 ] 2/3 + [R- yV + R 2 } 2/3 }
= [R + yV + R 2 ] + [R- a/<2 3 + fl2 ]
+ [H - t/q' + H 8 ] 1 ^!* - y/Q* + R*] 2/ *
+ [R - ^/q 3 + il 2 ] 2/3 [H - V /Q 3 + R 2 ] 1/3 - 2QB
= -2QB + 21* + [il 2 - (Q 3 + i* 2 )] 1/3
(* + v/QH^j +(r- y/Q 3 - * 2 )
-2QB + 2R ~ QB = -3QB + 2R.
(31)
364 Cubic Equation
Plugging B 3 and B into the left side of (28) gives
{-3QB + 2R) + 3QB = 2R, (32)
so we have indeed found the factor (x — B) of (22), and
we need now only factor the quadratic part. Plugging
C = 3Q into the quadratic part of (25) and solving the
resulting
x 2 +Bx + (B 2 +3Q) = (33)
then gives the solutions
x = \[-B± V^ 2 -4(B 2 + 3Q)]
= -±5±fv / 3i v / B 2 + 4Q.
These can be simplified by denning
(34)
A^[R^y/^+^) lfz -[R-y/^T^f ,z (35)
A 2 = [J* -I- y/Q 3 +B?] 2/ * - 2[R 2 - (Q 3 + tf 2 )] 1/3
+ [B - v/Q 3 +i? 2 ] 2/3
= [r + VQ 3 + ^ 2 1 2/3 + [* - v / Q 3 + ^ 2 ] 2/3 + 2 Q
= B 2 + 4Q, (36)
so that the solutions to the quadratic part can be written
x = -\B±\yf?>iA. (37)
Denning
D = Q 3 + R 2 (38)
5= V^+v^D
(39)
(40)
where L> is the Discriminant (which is defined slightly
differently, including the opposite Sign, by Birkhoff and
Mac Lane 1965) then gives very simple expressions for
A and B, namely
B = S + T
A = S-T.
(41)
(42)
Therefore, at last, the ROOTS of the original equation
in z are then given by
z 1 = -\a 2 + {S + T) (43)
z 2 = -\a 2 - \(S + T) + \%y/l{S - T) (44)
z 3 = -\a 2 - \{S + T) - I*V3 (S - T), (45)
with a 2 the COEFFICIENT of z 2 in the original equation,
and S and T as defined above. These three equations
Cubic Equation
giving the three ROOTS of the cubic equation are some-
times known as Cardano's Formula. Note that if the
equation is in the standard form of Viet a
x -\-px = q,
(46)
in the variable x, then a 2 = 0, a\ ~ p y and ao = —q,
and the intermediate variables have the simple form (c.f.
Beyer 1987)
Q
R:
D = Q 3 + R 2
(f)'+(i)'
(47)
(48)
(49)
The equation for z\ in CARDANO'S FORMULA does not
have an % appearing in it explicitly while z 2 and z% do,
but this does not say anything about the number of
Real and Complex Roots (since S and T are them-
selves, in general, COMPLEX). However, determining
which ROOTS are REAL and which are COMPLEX can
be accomplished by noting that if the DISCRIMINANT
D > 0, one Root is Real and two are Complex Con-
jugates; if D = 0, all ROOTS are REAL and at least
two are equal; and if D < 0, all ROOTS are REAL and
unequal. If D < 0, define
= cos
R
y^o 3
Then the Real solutions are of the form
zi = 2 v / ^Qcos (-) - \a 2
2^-Qcos{ — — J - |
^ r~^ /0 + 4?r\ i
2^-Qws{— — J -|
z 2
a 2
zz
a 2 .
(50)
(51)
(52)
(53)
This procedure can be generalized to find the Real
ROOTS for any equation in the standard form (46) by
using the identity
sin 3 6 - f sin# + \ sin(3<9) =
(Dickson 1914) and setting
_ [Mp\
x = \ - L - L v
V 3 y
(Birkhoff and Mac Lane 1965, pp. 90-91), then
-f) y'+W-fy = «
y° +
*\p\ V \*\p\)
3/2
(54)
(55)
(56)
(57)
Cubic Equation
Cubic Equation 365
4y +3sgn(p)y = ^q
*•<$
3/2
c.
If p > 0, then use
sinh(30) = 4 sinh 3 6 + 3 sinh
to obtain
2/ = sinh(| sinh C).
If p< and |C| > 1, use
cosh(30) = 4 cosh 3 (9-3 cosh (9,
and if p < and \C\ < 1, use
cos(30) = 4 cos 3 0- 3 cos 0,
to obtain
(58)
(59)
(60)
(61)
(62)
{cosh(i cosh -1 C) for C > 1
- cosh(| cosh" 1 |C|) for C < -1
cos(|cos _1 C) [three solutions] for |C| < 1.
(63)
The solutions to the original equation are then
Xi = 2\l^rVi- i<*2.
(64)
An alternate approach to solving the cubic equation is
to use Lagrange Resolvents. Let w = e 27r * /3 , define
(l,xi) = #1 + x 2 + # 3 (65)
(a;, xi) — x\ + UX2 -\- uj xs (66)
(a; 2 , xi) = xi + u; 2 :c2 + a;x3, (67)
where ^^ are the ROOTS of
x z +px + q = 0, (68)
and consider the equation
[a - (ui +it2)][a; - (u^i H-uj 2 ^)^ - (u> 2 ui + u>ti2)] = 0,
(69)
where u\ and u 2 are COMPLEX NUMBERS. The ROOTS
are then
Xj = u; J ^i + u> 23 u 2 (70)
for j = 0, 1, 2. Multiplying through gives
x 3 - 3u lU2 x - (u L 3 + u 2 S ) = 0, (71)
or
x + j>x + gr — 0,
where
Wi 3 + ti2 3 = -<7
3 3
U\ U2
G)'
(72)
(73)
(74)
The solutions satisfy Newton's Identities
zi + z 2 + 23 = -a 2 (75)
^iz 2 +^2^3 + ^i^3 = ai (76)
Z1Z223 = — ao. (77)
In standard form, 02 = 0, ai = p, and ao = —q, so we
have the identities
2
P = 21-22 — Zz
(78)
{ Zl - z 2 f = -(4p - 3z 3 2 )
(79)
ZX +Z2 2 +Z 3 2 =~2p.
(80)
Some curious identities involving the roots of a cubic
equation due to Ramanujan are given by Berndt (1994).
see also Quadratic Equation, Quartic Equation,
Quintic Equation, Sextic Equation
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.
Berger, M. §16.4.1-16.4.11.1 in Geometry L New York:
Springer- Verlag, 1994.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 22-23, 1994.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 9-11, 1987.
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, pp. 90-91, 106-107, and
414-417, 1965.
Dickson, L. E. "A New Solution of the Cubic Equation."
Amer. Math. Monthly 5, 38-39, 1898.
Dickson, L. E. Elementary Theory of Equations. New York:
Wiley, pp. 36-37, 1914.
Dunham, W. "Cardano and the Solution of the Cubic."
Ch. 6 in Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 133-154, 1990.
Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Alge-
bra. Boston, MA: PWS-Kent, 1991.
Jones, J. "Omar Khayyam and a Geometric Solution of the
Cubic." http : // j wilson . coe . uga.edu/emt669/Student .
Folders/Jones. June/omar/omarpaper. html.
Kennedy, E. C. "A Note on the Roots of a Cubic." Amer.
Math. Monthly 40, 411-412, 1933.
King, R. B. Beyond the Quartic Equation. Boston, MA:
Birkhauser, 1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Quadratic and Cubic Equations." §5.6
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 178-180, 1992.
Spanier, J. and Oldham, K. B. "The Cubic Function x z +
ox 2 + bx + c and Higher Polynomials." Ch. 17 in An Atlas
of Functions. Washington, DC: Hemisphere, pp. 131—147,
1987.
van der Waerden, B. L. §64 in Algebra. New York: Frederick
Ungar, 1970.
366 Cubic Number
Cubic Number
A FlGURATE NUMBER of the form n 3 , for n a POSITIVE
Integer. The first few are 1, 8, 27, 64, . ., (Sloane's
A000578). The Generating Function giving the cu-
bic numbers is
x(x 2 +4x + 1)
(*-l) 4
= x + 8x 2 + 27a; 3 +
(1)
The HEX Pyramidal Numbers are equivalent to the
cubic numbers (Conway and Guy 1996).
The number of positive cubes needed to represent the
numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4,
5, 6, 7, 8, 2, . . . (Sloane's A02376), and the number of
distinct ways to represent the numbers 1, 2, 3, ... in
terms of positive cubes are 1, 1, 1, 1, 1, 1, 1, 2, 2, 2,
2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5,
. . . (Sloane's A003108). In the early twentieth century,
Dickson, Pillai, and Niven proved that every POSITIVE
INTEGER is the sum of not more than nine CUBES (so
g(3) = 9 in Waring's Problem).
In 1939, Dickson proved that the only INTEGERS requir-
ing nine CUBES are 23 and 239. Wieferich proved that
only 15 Integers require eight Cubes: 15, 22, 50, 114,
167, 175, 186, 212, 213, 238, 303, 364, 420, 428, and 454
(Sloane's A018889). The quantity G(3) in Waring's
PROBLEM therefore satisfies G(3) < 7, and the largest
number known requiring seven cubes is 8042. The fol-
lowing table gives the first few numbers which require
at least iV = 1, 2, 3, . . . , 9 (positive) cubes to represent
them as a sum.
N Sloane Numbers
1 000578 1, 8, 27, 64, 125, 216, 343, 512, .
2 003325 2, 9, 16, 28, 35, 54, 65, 72, 91, . .
3 003072 3, 10, 17, 24, 29, 36, 43, 55, 62, .
4 003327 4, 11, 18, 25, 30, 32, 37, 44, 51, .
5 003328 5, 12, 19, 26, 31, 33, 38, 40, 45, .
6 6, 13, 20, 34, 39, 41, 46, 48, 53, .
7 018890 7, 14, 21, 42, 47, 49, 61, 77, . . .
8 018889 15, 22, 50, 114, 167, 175, 186, . . . ,
9 — 23, 239
There is a finite set of numbers which cannot be ex-
pressed as the sum of distinct cubes: 2, 3, 4, 5, 6, 7, 10,
Cubic Number
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, . . . (Sloane's A001476). The following table gives the
numbers which can be represented in W different ways
as a sum of N positive cubes. For example,
157 = 4 3 +4 3 + 3 3 + l 3 + l 3
5 3 +2 3 + 2 3 + 2 3 + 2 3 (2)
can be represented in W = 2 ways by N = 5 cubes. The
smallest number representable in W = 2 ways as a sum
of N = 2 cubes,
1729 = l 3 + 12 3 = 9 3
10 d
(3)
is called the Hardy- Ramanujan Number and has spe-
cial significance in the history of mathematics as a result
of a story told by Hardy about Ramanujan. Sloane's
A001235 is defined as the sequence of numbers which
are the sum of cubes in two or more ways, and so ap-
pears identical in the first few terms.
N W Sloane Numbers
1, 8, 27, 64, 125, 216, 343, 512, . . .
2, 9, 16, 28, 35, 54, 65, 72, 91, ...
1729, 4104, 13832, 20683, 32832, . . .
87539319, 119824488, 143604279, ...
6963472309248, 12625136269928, . . .
48988659276962496, . . .
8230545258248091551205888, ...
3, 10, 17, 24, 29, 36, 43, 55, 62, . . .
It is believed to be possible to express any number as a
Sum of four (positive or negative) cubes, although this
has not been proved for numbers of the form 9n ± 4. In
fact, all numbers not of the form 9n ± 4 are known to
be expressible as the Sum of three (positive or negative)
cubes except 30, 33, 42, 52, 74, 110, 114, 156, 165, 195,
290, 318, 366, 390, 420, 435, 444, 452, 462, 478, 501,
530, 534, 564, 579, 588, 600, 606, 609, 618, 627, 633,
732, 735, 758, 767, 786, 789, 795, 830, 834, 861, 894,
903, 906, 912, 921, 933, 948, 964, 969, and 975 (Guy
1994, p. 151).
The following table gives the possible residues (mod n)
for cubic numbers for n = 1 to 20, as well as the number
of distinct residues s(n).
1
1
000578
2
1
025403
2
2
2
3
003825
2
4
003826
2
5
2
6
3
1
025395
Cubic Number
Cubic Spline 367
n s(n) x 3 (mod n)
0,1
0,1,2
0,1,3
0, 1, 2, 3, 4
0, 1, 2, 3, 4, 5
0,1,6
0, 1, 3, 5, 7
0,1,8
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
0, 1, 3, 4, 5, 7, 8, 9, 11
0, 1, 5, 8, 12
0, 1, 6, 7, 8, 13
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
0, 1, 3, 5, 7, 8, 9, 11, 13, 15
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
0, 1, 8, 9, 10, 17
0, 1, 7, 8, 11, 12, 18
0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19
Dudeney found two RATIONAL NUMBERS other than 1
and 2 whose cubes sum to 9,
2
2
3
3
4
3
5
5
6
6
7
3
8
5
9
3
10
10
11
11
12
9
13
5
14
6
15
15
16
10
17
17
18
6
19
7
20
15
415280564497 „ 676702467503
and
348671682660
348671682660
(4)
The problem of finding two RATIONAL NUMBERS whose
cubes sum to six was "proved" impossible by Legendre.
However, Dudeney found the simple solutions 17/21 and
37/21.
The only three consecutive INTEGERS whose cubes sum
to a cube are given by the Diophanttne Equation
3 3 + 4 3 + 5 3
(5)
Catalan's Conjecture states that 8 and 9 (2 3 and
3 2 ) are the only consecutive POWERS (excluding and
1), i.e., the only solution to Catalan's Diophantine
Problem. This Conjecture has not yet been proved
or refuted, although R. Tijdeman has proved that there
can be only a finite number of exceptions should the
Conjecture not hold. It is also known that 8 and 9
are the only consecutive cubic and Square Numbers
(in either order).
There are six POSITIVE INTEGERS equal to the sum of
the DIGITS of their cubes: 1, 8, 17, 18, 26, and 27 (Moret
Blanc 1879), There are four POSITIVE INTEGERS equal
to the sums of the cubes of their digits:
153 = l 3 4- 5 3 + 3 3
370 =
= 3 3 + 7 3 + 3
371 =
= 3 3 + 7 3 + l 3
407 =
= 4 3 + 3 + 7 3
(6)
(7)
(8)
(9)
(Ball and Coxeter 1987). There are two Square Num-
bers of the form n 3 -4: 4 = 2 3 -4 and 121 = 5 3 -4(Le
Lionnais 1983). A cube cannot be the concatenation of
two cubes, since if c 3 is the concatenation of a 3 and 6 3 ,
then c 3 = 10 fc a 3 + 6 3 , where k is the number of digits
in b 3 . After shifting any powers of 1000 in 10 into a ,
the original problem is equivalent to finding a solution
to one of the DIOPHANTINE EQUATIONS
c — b = a
c 3 -b 3 = 10a 3
c 3 -6 3 = 100a 3 .
(10)
(11)
(12)
None of these have solutions in integers, as proved in-
dependently by Sylvester, Lucas, and Pepin (Dickson
1966, pp. 572-578).
see also BIQUADRATIC NUMBER, CENTERED CUBE
Number, Clark's Triangle, Diophantine Equa-
tion — Cubic, Hardy-Ramanujan Number, Parti-
tion, Square Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 14, 1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 42-44, 1996.
Davenport, H. "On Waring's Problem for Cubes." Acta
Math. 71, 123-143, 1939.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1966.
Guy, R. K. "Sum of Four Cubes." §D5 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 151-152, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 53, 1983.
Sloane, N. J. A. Sequences A000578/M4499, A02376/M0466,
and A003108/M0209 in "An On-Line Version of the Ency-
clopedia of Integer Sequences."
Cubic Reciprocity Theorem
A Reciprocity Theorem for the case n = 3 solved by
Gauss using "Integers" of the form a + bp, when p is
a root if x 2 + x + 1 = and a, b are Integers.
see also RECIPROCITY THEOREM
References
Ireland, K. and Rosen, M. "Cubic and Biquadratic Reci-
procity." Ch, 9 in A Classical Introduction to Modem
Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 108-137, 1990.
Cubic Spline
A cubic spline is a Spline constructed of piecewise third-
order POLYNOMIALS which pass through a set of control
points. The second DERIVATIVE of each POLYNOMIAL
is zero at the endpoints.
References
Burden, R. L.; Faires, J. D.; and Reynolds, A. C. Numerical
Analysis, 6th ed. Boston, MA: Brooks/Cole, pp. 120-121,
1997.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Cubic Spline Interpolation." §3.3 in Numeri-
cal Recipes in FORTRAN: The Art of Scientific Comput-
ing, 2nd ed. Cambridge, England: Cambridge University
Press, pp. 107-110, 1992.
368
Cubic Surface
Cubical Hyperbola
Cubic Surface
An Algebraic Surface of Order 3. Schlafli and
Cayley classified the singular cubic surfaces. On the
general cubic, there exists a curious geometrical struc-
ture called Double Sixes, and also a particular ar-
rangement of 27 (possibly complex) lines, as discovered
by Schlafli (Salmon 1965, Fischer 1986) and sometimes
called Solomon's Seal Lines. A nonregular cubic sur-
face can contain 3, 7, 15, or 27 real lines (Segre 1942,
Le Lionnais 1983). The Clebsch Diagonal Cubic
contains all possible 27. The maximum number of Or-
dinary Double Points on a cubic surface is four, and
the unique cubic surface having four ORDINARY DOU-
BLE Points is the Cayley Cubic.
Schoutte (1910) showed that the 27 lines can be put
into a One-TO-One correspondence with the vertices of
a particular POLYTOPE in 6-D space in such a manner
that all incidence relations between the lines are mir-
rored in the connectivity of the POLYTOPE and con-
versely (Du Val 1931). A similar correspondence can
be made between the 28 bitangents of the general plane
Quartic Curve and a 7-D Polytope (Coxeter 1928)
and between the tritangent planes of the canonical curve
of genus 4 and an 8-D POLYTOPE (Du Val 1933).
A smooth cubic surface contains 45 TRITANGENTS
(Hunt). The Hessian of smooth cubic surface contains
at least 10 Ordinary Double Points, although the
Hessian of the CAYLEY Cubic contains 14 (Hunt).
see also Cayley Cubic, Clebsch Diagonal Cubic,
Double Sixes, Eckardt Point, Isolated Singu-
larity, Nordstrand's Weird Surface, Solomon's
Seal Lines, Tritangent
References
Bruce, J. and Wall, C. T. C. "On the Classification of Cubic
Surfaces." J. London Math. Soc. 19, 245-256, 1979.
Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy.
Soc. 159, 231-326, 1869.
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six
and Seven Dimensions." Proc. Cambridge Phil. Soc. 24,
7-9, 1928.
Du Val, P. "On the Directrices of a Set of Points in a Plane."
Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 9-14, 1986.
Fladt, K. and Baur, A. Analytische Geometrie spezieler
Fldchen und Raumkurven. Braunschweig, Germany:
Vieweg, pp. 248-255, 1975.
Hunt, B. "Algebraic Surfaces." http: //www.mathematik.
uni-kl . de/-wwwagag/Galerie . html.
Hunt, B. "The 27 Lines on a Cubic Surface" and "Cubic
Surfaces." Ch. 4 and Appendix B.4 in The Geometry of
Some Special Arithmetic Quotients. New York: Springer-
Verlag, pp. 108-167 and 302-310, 1996.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 49, 1983.
Rodenberg, C. "Zur Classification der Flachen dritter Ord-
nung." Math. Ann. 14, 46-110, 1878.
Salmon, G. Analytic Geometry of Three Dimensions. New
York: Chelsea, 1965.
Schlafli, L. "On the Distribution of Surface of Third Order
into Species." Phil. Trans. Roy. Soc. 153, 193-247, 1864.
Schoutte, P. H. "On the Relation Between the Vertices of a
Definite Sixdimensional Polytope and the Lines of a Cubic
Surface." Proc. Roy. Acad. Amsterdam 13, 375-383, 1910.
Segre, B. The Nonsingular Cubic Surface. Oxford, England:
Clarendon Press, 1942.
Cubical Conic Section
see Cubical Ellipse, Cubical Hyperbola, Cubical
Parabola, Skew Conic
Cubical Ellipse
An equation of the form
y = ax 3 + bx 2 + ex + d
where only one Root is real.
see also CUBICAL CONIC SECTION, CUBICAL HYPER-
BOLA, Cubical Parabola, Cubical Parabolic Hy-
perbola, Ellipse, Skew Conic
Cubical Graph
An 8-vertex POLYHEDRAL GRAPH.
see also Bidiakis Cube, Bislit Cube, Dodecahedral
Graph, Icosahedral Graph, Octahedral Graph,
Tetrahedral Graph
Cubical Hyperbola
Cubical Parabola
An equation of the form
y — ax 3 + bx 2 + ex + d,
where the three Roots are Real and distinct, i.e.,
y = a(x — ri)(x — V2){x — rz)
= a[x 3 - (n + r2 + 7"3)z 2 + (^1^*2 + rirz + nr^x
— nr2T3].
Cuboctahedron 369
where two of the ROOTS of the equation coincide (and
all three are therefore real), i.e.,
y = a (x -r\f{x - r 2 )
= a[x 3 — (2r*i + r 2 )# 2 + ri(n + 2r2)x — ri 2 r 2 ].
see a/50 CUBICAL CONIC SECTION, CUBICAL ELLIPSE,
Cubical Hyperbola, Cubical Parabola, Hyper-
bola
see also Cubical Conic Section, Cubical Ellipse,
Cubical Hyperbola, Cubical Parabola, Hyper-
bola
Cubicuboctahedron
see Great Cubicuboctahedron, Small Cubicuboc-
tahedron
Cubical Parabola
An equation of the form
y = ax -f bx + ex + d,
where the three ROOTS of the equation coincide (and
are therefore real), i.e.,
y = a(x — r) 3 = a(x 3 — Srx 2 — 3r 2 x - r 3 ).
Cubique d'Agnesi
see Witch of Agnesi
Cubitruncated Cuboctahedron
The Uniform Polyhedron Ui 6 whose Dual is the
Tetradyakis Hexahedron. It has Wythoff Sym-
bol 3 | 4|. Its faces are 8{6} + 6{8} + 6{|}. It is a
Faceted Octahedron. The Circumradius for a cu-
bitruncated cuboctahedron of unit edge length is
R=\yfi.
see also Cubical Conic Section, Cubical Ellipse,
Cubical Hyperbola, Cubical Parabolic Hyper-
bola, Parabola, Semicubical Parabola
Cubical Parabolic Hyperbola
An equation of the form
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 113-114, 1971.
Cuboctahedron
y — ax -\-bx + ex + d,
An Archimedean Solid (also called the Dymaxion or
Heptaparallelohedron) whose Dual is the Rhom-
bic Dodecahedron. It is one of the two convex
QUASIREGULAR POLYHEDRA and has SCHLAFLI SYM-
BOL { I }. It is also Uniform Polyhedron t/ 7 and has
Wythoff Symbol 2 | 34. Its faces are {3} + 6{4}. It
has the Oh OCTAHEDRAL GROUP of symmetries.
370
Cuboctatruncated Cuboctahedron
Cumulant
The Vertices of a cuboctahedron with Edge length
of 72 are (0,±1,±1), (±1,0, ±1), and (±1,±1,0). The
Inradius, Midradius, and Circumradius for a = 1
are
3 -i
4
0.75
p= \y/i^ 0.86602
i*=l.
Faceted versions include the Cubohemioctahedron
and OCTAHEMIOCTAHEDRON.
The solid common to both the Cube and Octahedron
(left figure) in a CUBE-OCTAHEDRON COMPOUND is a
Cuboctahedron (right figure; Ball and Coxeter 1987).
see also Archimedean Solid, Cube, Cube-Octahe-
dron Compound, Cubohemioctahedron, Octahe-
dron, OCTAHEMIOCTAHEDRON, QUASIREGULAR POLY-
HEDRON, Rhombic Dodecahedron, Rhombus
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Ghyka, M. The Geometry of Art and Life. New York: Dover,
p. 54, 1977.
Cuboctatruncated Cuboctahedron
see Cubitruncated Cuboctahedron
Cubocycloid
see ASTROID
Cubohemioctahedron
The Uniform Polyhedron C/" 15 whose Dual is the
Hexahemioctahedron. It has Wythoff Symbol
| 4 | 3. Its faces are 4{6} + 6{4}. It is a Faceted ver-
sion of the Cuboctahedron. Its Circumradius for
unit edge length is
R=l.
References
Wenninger, M, J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 121-122, 1971.
Cuboid
A rectangular Parallelepiped.
see also Euler Brick, Parallelepiped, Spider and
Fly Problem
Cullen Number
A number of the form
C n = 2 n n+1.
The first few are 3, 9, 25, 65, 161, 385, ... (Sloane's
A002064). The only Cullen numbers C n for n < 300, 000
which are Prime are for n = 1, 141, 4713, 5795, 6611,
18496, 32292, 32469, 59656, 90825, 262419, . . . (Sloane's
A005849; Ballinger). Cullen numbers are DIVISIBLE by
p = 2n - 1 if p is a Prime of the form 8A; ± 3.
see also Cunningham Number, Fermat Number,
Sierpinski Number of the First Kind, Woodall
Number
References
Ballinger, R. "Cullen Primes: Definition and Status."
http://ballingerr.xray.ufl.edu/proths/cullen.html.
Guy, R. K. "Cullen Numbers." §B20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 77, 1994.
Keller, W. "New Cullen Primes." Math. Comput. 64, 1733-
1741, 1995.
Leyland, P. ftp : //sable . ox . ac . uk/pub/math/f actors/
cullen.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 360-361, 1996.
Sloane, N. J. A. Sequences A002064/M2795 and A005849/
M5401 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Cumulant
Let (f>(t) be the Characteristic Function, defined as
the Fourier Transform of the Probability Den-
sity Function,
0(i) = F[P{x)\ = / e itx P(x) dx. (1)
J — OO
Then the cumulants « n are denned by
ln^) S 5> n ^-. (2)
Taking the Maclaurin Series gives
ln«£(i) = (;<)Mi + §W 2 (M2-//i 2 )
+ 5i(it) 3 (2/4 - 3/4/4 +/4)
+ ^(it) 4 (-6M'i 4 + 12 M i 2 M2 - 3M2 2 - 4/4/4 + /4)
+ | f (it) 5 [-2V 1 5 + 6O//1V2 + 20/4 2 /4 + 10/4/4
+5/4(6/4 -/4) + j4] + •••> (3)
Cumulant-Generating Function
where Mn are MOMENTS about 0, so
Kl = Mi
k 2 = M 2 - Mi
k z = 2/ii - 3miM2 + Ms
«4 = — 6/xi + 12^1
M2
/3 /
3^ 2
(4)
(5)
(6)
- ViZ-4 + /4 (7)
K5 = -24/i'i 5 + 60/ii 3 p2 + 20//i P3 + 10/^2^3
+ 5^i (6p' 2 - pi) + Ms •
In terms of the MOMENTS ju„ about the MEAN,
(8)
Ki = /l
w
2
«2 = M2 = O"
(10)
«3 = M3
(11)
K4 = M4 — 3^2
(12)
K5 = M5 - 10/X2M3,
(13)
where \i is the Mean and a 2 = \i2 is the Variance.
The ^-Statistics are Unbiased Estimators of the
cumulants.
see also Characteristic Function, Cumulant-
Generating Function, /^-Statistic, Kurtosis,
Mean, Moment, Sheppard's Correction, Skew-
ness, 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. 928, 1972.
Kenney, J. F. and Keeping, E. S. "Cumulants and the
Cumulant-Generating Function," "Additive Property of
Cumulants," and "Sheppard's Correction." §4.10-4.12 in
Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ:
Van Nostrand, pp. 77-82, 1951.
Cumulant-Generating Function
Let M(h) be the Moment-Generating Function.
Then
K(h) = laM(h) = mh + ±h 2 K 2 + ^/i 3 «3 +
If
M
l = y ^ cjxj
is a function of N independent variables, the cumulant
generating function for L is then
N
K{h) = Y, K ^ h y
J"=l
see also Cumulant, Moment-Generating Function
References
Abramowitz, M. and Stegun, C. A. (Eds,). Handbook
of Mathematical Functions with Formulas, Graphs, and
Cunningham Chain 371
Mathematical Tables, 9th printing. New York: Dover,
p. 928, 1972.
Kenney, J. F. and Keeping, E. S. "Cumulants and the
Cumulant-Generating Function" and "Additive Property
of Cumulants." §4.10-4.11 in Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77-80,
1951.
Cumulative Distribution Function
see Distribution Function
Cundy and Rollett's Egg
An Oval dissected into pieces which are to used to cre-
ate pictures. The resulting figures resemble those con-
structed out of Tangrams.
see also DISSECTION, EGG, OVAL, TANGRAM
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 19-21, 1989.
Dixon, R. Mathographics. New York: Dover, p. 11, 1991.
Cunningham Chain
A Sequence of Primes gi < <?2 < • • . < qk is a Cun-
ningham chain of the first kind (second kind) of length
h if q i+1 = 2qi + 1 (g i+ i = 2q { - 1) for i = 1, . . . ,
k-l. Cunningham PRIMES of the first kind are SOPHIE
Germain Primes.
The two largest known Cunningham chains (of the
first kind) of length three are (384205437 ■ 2 4000 -
1, 384205437 • 2 4001 - 1, 384205437 • 2 4002 - 1) and
(651358155 • 2 3291 - 1, 651358155 ■ 2 3292 - 1, 651358155 •
2 3293 _ -^ both discoverec i by W. Roonguthai in 1998.
see also Prime Arithmetic Progression, Prime
Cluster
References
Guy, R. K. "Cunningham Chains." §A7 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 18-19, 1994.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, p. 333, 1996,
Roonguthai, W. "Yves Gallot's Proth and Cunningham
Chains." http : //ksc9 . th . com/warut/ Cunningham .html.
372
Cunningham Function
Cupola
Cunningham Function
Sometimes also called the Pearson-Cunningham
Function. It can be expressed using WHITTAKER
FUNCTIONS (Whittaker and Watson 1990, p. 353).
.(*) =
7ri(m/2 — n) + x
I\l+n- \m)
U(\m — n, 1 + m,x)
where U is a Confluent Hypergeometric Function
of the Second Kind (Abramowitz and Stegun 1972,
p. 510).
see also CONFLUENT HYPERGEOMETRIC FUNCTION OF
the Second Kind, Whittaker Function
References
Abramowitz, M. and Stegun, C, A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Cunningham Number
A Binomial Number of the form C^fc, n) = b n ± 1.
Bases b k which are themselves powers need not be con-
sidered since they correspond to (b k ) n ± 1 = b kn ± 1.
Prime Numbers of the form (7^(6,™) are very rare.
A Necessary (but not Sufficient) condition for
C + (2,n) = 2 n + 1 to be Prime is that n be of the
form n = 2 m . Numbers of the form F m = C + (2,2 m ) =
2 2 ™ + 1 are called FERMAT NUMBERS, and the only
known PRIMES occur for C + (2,l) = 3, C + (2,2) = 5,
C + (2,4) = 17, C + (2,8) = 257, and C + (2,16) = 65537
(i.e., n = 0, 1, 2, 3, 4). The only other PRIMES
C + (b,n) for nontrivial b < 11 and 2 < n < 1000 are
C + (6,2) = 37, C + (6,4) = 1297, andC + (10,2) = 101.
Primes of the form C~ (6, n) are also very rare. The
Mersenne Numbers M n = C~(2,n) = 2 n - 1 are
known to be prime only for 37 values, the first few
of which are n = 2, 3, 5, 7, 13, 17, 19, . , . (Sloane's
A000043). There are no other PRIMES C~(b,n) for non-
trivial b < 20 and 2 < n < 1000.
In 1925, Cunningham and Woodall (1925) gathered to-
gether all that was known about the Primality and
factorization of the numbers C (6, n) and published a
small book of tables. These tables collected from scat-
tered sources the known prime factors for the bases 2 and
10 and also presented the authors' results of 30 years'
work with these and other bases.
Since 1925, many people have worked on filling in these
tables. D. H. Lehmer, a well-known mathematician who
died in 1991, was for many years a leader of these efforts.
Lehmer was a mathematician who was at the forefront
of computing as modern electronic computers became
a reality. He was also known as the inventor of some
ingenious pre-electronic computing devices specifically
designed for factoring numbers.
Updated factorizations were published in Brillhart et al.
(1988). The current archive of Cunningham number fac-
torizations for 6 = 1, ... , ±12 is kept on ftp: //sable,
ox.ac.uk/pub/math/cuiniinghain. The tables have been
extended by Brent and te Riele (1992) to b = 13, ... ,
100 with m < 255 for b < 30 and m < 100 for b > 30.
All numbers with exponent 58 and smaller, and all com-
posites with < 90 digits have now been factored.
see also BINOMIAL NUMBER, CULLEN NUMBER, FER-
MAT NUMBER, MERSENNE NUMBER, REPUNIT, RlESEL
Number, Sierpinski Number of the First Kind,
Woodall Number
References
Brent, R. P. and te Riele, H. J. J. "Factorizations of
a n ± 1, 13 < a < 100." Report NM-R9212, Centrum
voor Wiskunde en Informatica. Amsterdam, June 1992.
The text is available electronically at ftp://sable.ox.
ac.uk/pub/math/factors/BMtR_13-99.dvi, and the files at
BMtR_13-99. Updates are given in BMtR_13-99_updatel (94-
09-01) and BMtR_i3-99_update2 (95-06-01).
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. Provi-
dence, Rl: Amer. Math. Soc, 1988. Updates are avail-
able electronically from ftp://sable.ox.ac.uk/pub/math/
Cunningham/.
Cunningham, A. J. C. and Woodall, H. J. Factorisation of
y ™ ip i, y = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers
(n). London: Hodgson, 1925.
Mudge, M. "Not Numerology but Numeralogy!" Personal
Computer World, 279-280, 1997.
Ribenboim, P. "Numbers k x 2 n ± 1." §5.7 in The New Book
of Prime Number Records. New York: Springer- Verlag,
pp. 355-360, 1996.
Sloane, N. J. A. Sequence A000043/M0672 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Cunningham Project
see Cunningham Number
Cupola
An n-gonal cupola Q n (possible for only n = 3, 4, 5) is
a Polyhedron having n Triangular and n Square
faces separating an {n} and a {2n} Regular Polygon.
The coordinates of the base VERTICES are
.Rcos
7r(2fc+l)
2n
, .Rsin
7r(2fc + l)
2n
,oj,
and the coordinates of the top VERTICES are
[ 2k7v
I n
,rsin
2fc7T]
L n J
-)■
(i)
(2)
Cupolarotunda
where R and r are the ClRCUMRADll of the base and top
r= 2 aCSC (l)> ( 4 )
and z is the height, obtained by letting k = in the
equations (1) and (2) to obtain the coordinates of neigh-
boring bottom and top VERTICES,
b =
t =
tfcos(^)
Since all side lengths are a,
|b-t| 2 =a 2 .
Solving for z then gives
[,„(i)- r ]\*W(i) + .
z 2 + R 2 + r 2 - 2rRcos (^-\ = <?
(5)
(6)
(7)
(8)
(9)
z = \ a 2 — 2rRcos
\2n)
.2 _ R 2
= a V 1_|CSC2 (n)'
(10)
see also BlCUPOLA, ELONGATED Cupola, Gyro-
elongated cupola, pentagonal cupola, square
Cupola, Triangular Cupola
References
Johnson, N. W. "Convex Polyhedra with Regular Faces."
Canad. J. Math. 18, 169-200, 1966.
Cupolarotunda
A CUPOLA adjoined to a ROTUNDA.
see also Gyrocupolarotunda, Orthocupolaro-
TUNDA
Curl
The curl of a TENSOR field is given by
(V x A) a = e a ^A u; ^
(1)
where €i jk is the LEVI-ClVlTA TENSOR and ";" is the
Covariant Derivative. For a Vector Field, the
curl is denoted
Curl Theorem 373
and V x F is normal to the PLANE in which the "circula-
tion" is Maximum. Its magnitude is the limiting value
of circulation per unit AREA,
(V x F) • n = lim
A-+0
/c F - rfs
Let
and
then
F = FiUx + F 2 u 2 + F 3 \is
hi =
V x F =
1
h\h 2 hz
1
dui
/11U1 /12U2 hsuz
d d _d_
du\ du 2 du§
/11F1 h 2 F 2 h 3 F 3
(3)
(4)
(5)
h 2 hs
1
d
8u 2
d
+
hihs
1
h\h2
(h 3 F 3 )-—(h 2 F 2 )^u 1
-(h 2 F 2 )-'£-(h l F 1 )}u 3 . (6)
du
Special cases of the curl formulas above can be given for
Curvilinear Coordinates,
see also Curl Theorem, Divergence, Gradient,
Vector Derivative
References
Arfken, G. "Curl, Vx." §1.8 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 42-
47, 1985.
Curl Theorem
A special case of STOKES ' THEOREM in which F is a
Vector Field and M is an oriented, compact embed-
ded 2-Manifold with boundary in M 3 , given by
/ (V x F) • da = / F • ds.
Js Jes
IS JdS
There are also alternate forms. If
F = cF,
then
and if
then
/ da. x VF = / Fds.
Js Jc
c x P,
f(dax V) xP= /
Js Jc
ds x P.
(1)
(2)
(3)
(4)
(5)
curl(F) eVxF,
(2)
see also Change of Variables Theorem, Curl,
Stores' Theorem
References
Arfken, G. "Stokes's Theorem." §1.12 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 61-64, 1985.
374 Curlicue Fractal
Curlicue Fractal
Curtate Cycloid Evolute
The curlicue fractal is a figure obtained by the following
procedure. Let s be an IRRATIONAL NUMBER. Begin
with a line segment of unit length, which makes an An-
gle 4>q = to the horizontal. Then define n iteratively
by
n +i = (6> n + 27rs) (mod 2tt),
with #o = 0. To the end of the previous line segment,
draw a line segment of unit length which makes an angle
07i+l = 0n~\- <pn (mod 271"),
to the horizontal (Pickover 1995). The result is a FRAC-
TAL, and the above figures correspond to the curlicue
fractals with 10,000 points for the GOLDEN RATIO <j>,
In 2, e, \/2, the Euler-Mascheroni Constant 7, 7r,
and Feigenbaum Constant S.
The Temperature of these curves is given in the fol-
lowing table.
Constant
Temperature
golden ratio 4>
46
In 2
51
e
58
s/2
58
Euler-Mascheroni constant 7
63
7T
90
Feigenbaum constant $
92
References
Berry, M. and Goldberg, J. "Renormalization of Curlicues."
Nonlinearity 1, 1-26, 1988.
Moore, R. and van der Poorten, A. "On the Thermodynamics
of Curves and Other Curlicues." McQuarie Univ. Math.
Rep. 89-0031, April 1989.
Pickover, C. A. "The Fractal Golden Curlicue is Cool."
Ch. 21 in Keys to Infinity. New York: W. H. Freeman,
pp. 163-167, 1995.
Pickover, C. A. Mazes for the Mind: Computers and the
Unexpected. New York: St. Martin's Press, 1993.
Sedgewick, R. Algorithms. Reading, MA: Addison- Wesley,
1988.
Stewart, I. Another Fine Math You've Got Me Into. . . . New
York: W. H, Freeman, 1992.
Current
A linear FUNCTIONAL on a smooth differential form.
see also Flat Norm, Integral Current, Rectifi-
able Current
Curtate Cycloid
The path traced out by a fixed point at a Radius b < a,
where a is the RADIUS of a rolling CIRCLE, sometimes
also called a Contracted Cycloid.
a<f> — b sin (f>
a ~ b cos 6.
The Arc Length from <f> = is
s = 2{a + b)E{u),
where
sin(|^>) = snu
k 2 =
4ab
(1)
(2)
(3)
(4)
(5)
(a + c) 2 '
and E(u) is a complete Elliptic Integral of the
Second Kind and snu is a Jacobi Elliptic Func-
tion.
see also CYCLOID, PROLATE CYCLOID
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 46-50, 1991.
Curtate Cycloid Evolute
The Evolute of the Curtate Cycloid
x —
a<j> —
6sin</>
(1)
y =
a — b
COS0.
(2)
is
given
by
x -
a[-
-2b(f> +
2acf>
COS0 -
- 2a sin
0+6
sin(2#]
(3)
2(acos0 — b)
y =
a(a — bcos<j)) 2
b(a cos 4> — b)
(4)
Curvature
Curvature 375
Curvature
In general, there are two important types of curva-
ture: Extrinsic Curvature and Intrinsic Curva-
ture. The Extrinsic Curvature of curves in 2- and
3-space was the first type of curvature to be studied his-
torically, culminating in the Frenet Formulas, which
describe a Space Curve entirely in terms of its "cur-
vature," TORSION, and the initial starting point and
direction.
and
dt
x'y" - y'x"
1 + tan 2 <p x' 2
1 x'y" - y'x" _ x'y" - y'x"
x' 2 + y' 2
Combining (2) and (4) gives
(4)
After the curvature of 2- and 3-D curves was studied,
attention turned to the curvature of surfaces in 3-space.
The main curvatures which emerged from this scrutiny
are the MEAN CURVATURE, GAUSSIAN CURVATURE, and
the Weingarten Map. Mean Curvature was the
most important for applications at the time and was
the most studied, but Gauss was the first to recognize
the importance of the GAUSSIAN CURVATURE.
Because Gaussian Curvature is "intrinsic," it is de-
tectable to 2-dimensional "inhabitants" of the surface,
whereas MEAN CURVATURE and the WEINGARTEN MAP
are not detectable to someone who can't study the 3-
dimensional space surrounding the surface on which he
resides. The importance of GAUSSIAN CURVATURE to
an inhabitant is that it controls the surface Area of
SPHERES around the inhabitant.
Riemann and many others generalized the concept of
curvature to SECTIONAL CURVATURE, SCALAR CURVA-
TURE, the Riemann Tensor, Ricci Curvature, and
a host of other Intrinsic and Extrinsic Curvatures.
General curvatures no longer need to be numbers, and
can take the form of a Map, GROUP, GROUPOID, tensor
field, etc.
The simplest form of curvature and that usually first
encountered in Calculus is an Extrinsic Curvature.
In 2-D, let a Plane Curve be given by Cartesian
parametric equations x — x(t) and y = y(t). Then the
curvature k is defined by
ds 4*
dt
64
dt
d±
dt
m 2 +m 2 >/**+<*'
a)
where <f> is the Polar ANGLE and s is the Arc Length.
As can readily be seen from the definition, curvature
therefore has units of inverse distance. The d<f>/dt de-
rivative in the above equation can be eliminated by using
the identity
dy dy/dt y'
tan (p — -— — — — 77 — —
dx dx dt x'
so
d 2 ,d4> x'y"
-(tan0) = sec </,- =
y x
(2)
(3)
x'y" - y'x"
(x' 2 +2/' 2 ) 3 / 2 '
(5)
For a 2-D curve written in the form y — f(x), the equa-
tion of curvature becomes
1^
[i + (£) 2 ]
3/2 *
(6)
If the 2-D curve is instead parameterized in POLAR CO-
ORDINATES, then
r + 2r$ — rree
( r 2 +rfl 2)3/2 '
(7)
where r e = dr/dO (Gray 1993). In PEDAL COORDI-
NATES, the curvature is given by
ldp
r dr'
(8)
The curvature for a 2-D curve given implicitly by
g(x, y) — is given by
(9)
(Gray 1993).
Now consider a parameterized Space Curve r(t) in 3-D
for which the TANGENT VECTOR T is denned as
—
rp _ dt
~ \dr\
I dt I
dr
dt
ds '
dt
Therefore,
dr _ ds *
di ~ di
d*r d*s~ ds dT ds^~ . (ds\ 2
de dt 1 ^ dt dt dt 2 ^ \dtj '
where N is the NORMAL VECTOR. But
'ds** 3
(10)
(11)
(12)
dr d r _ ds d s
~dt X ~dt? ~ diW
ds^
It)
(Txf) + «g) (TxN)
«(£) 3 (TxN)
(13)
376
Curvature
fds\ 3
= K \ t l — «
dr 3
dT
ds
dr dTr
dt A dt*
(14)
(15)
dt |
The curvature of a 2-D curve is related to the RADIUS OF
Curvature of the curve's Osculating Circle. Con-
sider a CIRCLE specified parametrically by
x — a cos t
y — a sin t
(16)
(17)
which is tangent to the curve at a given point. The
curvature is then
xy -yx
1
(a;/2 +2/2)3/2 a 3
(18)
or one over the Radius of Curvature. The curvature
of a Circle can also be repeated in vector notation. For
the Circle with < t < 27r, the Arc Length is
*>- [J ®'+®'
dt
l \/a 2 cos 2 t + a 2 sin 2 t dt = at, (19)
Jo
so t = s/a and the equations of the Circle can be
rewritten as
x = a cos I — )
y = asin {0-
The Position Vector is then given by
(20)
(21)
r(s) = a cos ( - J x + a sin ( - J y,
and the Tangent Vector is
dr
= — = - sin - x + cos - y,
ds V a / V a /
(22)
(23)
so the curvature is related to the RADIUS OF CURVA-
TURE a by
dT
ds
- - cos I — x sin I — y
a \a/ a \a/
co^(f)+sin 2 (f) i
as expected.
r= kN + k(tB - kT)
Curvature
Four very important derivative relations in differential
geometry related to the FRENET FORMULAS are
r = T (25)
r = kN (26)
(27)
(28)
where T is the TANGENT VECTOR, N is the NORMAL
Vector, B is the Binormal Vector, and r is the
Torsion (Coxeter 1969, p. 322).
The curvature at a point on a surface takes on a variety
of values as the PLANE through the normal varies. As
tz varies, it achieves a minimum and a maximum (which
are in perpendicular directions) known as the PRINCIPAL
CURVATURES. As shown in Coxeter (1969, pp. 352-353),
[r,r,rj
2
k r,
k -^6JK + det(6j) =
(29)
k 2 - 2Hk + K = 0, (30)
where K is the Gaussian Curvature, H is the Mean
Curvature, and det denotes the Determinant.
The curvature k is sometimes called the First Curva-
ture and the TORSION r the SECOND CURVATURE. In
addition, a THIRD CURVATURE (sometimes called TO-
TAL Curvature)
\/ds T 2 +ds B 2 (31)
is also defined. A signed version of the curvature of a
CIRCLE appearing in the DESCARTES CIRCLE THEOREM
for the radius of the fourth of four mutually tangent
circles is called the Bend.
see also Bend (Curvature), Curvature Center,
Curvature Scalar, Extrinsic Curvature, First
Curvature, Four- Vertex Theorem, Gaussian
Curvature, Intrinsic Curvature, Lancret Equa-
tion, Line of Curvature, Mean Curvature, Nor-
mal Curvature, Principal Curvatures, Radius of
Curvature, Ricci Curvature, Riemann Tensor,
Second Curvature, Sectional Curvature, Soddy
Circles, Third Curvature, Torsion (Differen-
tial Geometry), Weingarten Map
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, 1969.
Fischer, G. (Ed.). Plates 79-85 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 74—81, 1986.
Gray, A. "Curvature of Curves in the Plane," "Drawing Plane
Curves with Assigned Curvature," and "Drawing Space
Curves with Assigned Curvature." §1.5, 6.4, and 7.8 in
Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 11-13, 68-69, 113-118,
and 145-147, 1993.
(24) Kreyszig, E. "Principal Normal, Curvature, Osculating Cir-
cle." §12 in Differential Geometry. New York: Dover,
pp. 34-36, 1991.
Yates, R. C. "Curvature." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 60-64,
1952.
Curvature Center
Curve 377
Curvature Center
The point on the POSITIVE RAY of the NORMAL VEC-
TOR at a distance p(s), where p is the RADIUS OF CUR-
VATURE. It is given by
Other simple curves can be simply defined only implic-
itly, i.e., in the form
/(xi,z 2 ,...) = 0.
z = x + pN = x + p
ds 1
(1)
where N is the Normal Vector and T is the Tangent
Vector. It can be written in terms of x explicitly as
= x +
x"(x'-x') a -x'(x'-x')(x'-x")
(2)
(x'-x')(x" -x") -(x' -x") 2
For a CURVE represented parametrically by {f{t),g(t)),
a = f-
= 9+^
f'9" ~ f"9'
(f' 2 -9' 2 )f
f'9" ~ f"9' '
(3)
(4)
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, 1993.
Curvature Scalar
The curvature scalar is given by
R = g Rfj,K ,
where g^ is the Metric Tensor and R^ is the Ricci
Tensor.
see also Curvature, Gaussian Curvature, Mean
Curvature, Metric Tensor, Radius of Curva-
ture, Ricci Tensor, Riemann-Christoffel Ten-
sor
Curvature Vector
K
dT
ds '
where T is the TANGENT VECTOR defined by
ds
1 5*5 I
I ds |
Curve
A Continuous Map from a 1-D Space to an n-D
SPACE. Loosely speaking, the word "curve" is often used
to mean the Graph of a 2- or 3-D curve. The simplest
curves can be represented parametrically in n-D SPACE
as
xi = fi(t)
x 2 = h(t)
X n = / n (t).
see also Archimedean Spiral, Astroid, Asymp-
totic Curve, Baseball Cover, Batrachion, Bi-
corn, Bifolium, Bow, Bullet Nose, Butterfly
Curve, Cardioid, Cassini Ovals, Catalan's Tri-
sectrix, Catenary, Caustic, Cayley's Sextic,
Cesaro Equation, Circle, Circle Involute, Cis-
soid, clssoid of dlocles, cochleoid, conchoid,
Conchoid of Nicomedes, Cross Curve, Cruci-
form, Cubical Parabola, Curve of Constant
Precession, Curve of Constant Width, Cur-
tate Cycloid, Cycloid, Delta Curve, Deltoid,
Devil's Curve, Devil on Two Sticks, Dumbbell
Curve, Durer's Conchoid, Eight Curve, Electric
Motor Curve, Ellipse, Ellipse Involute, Ellip-
tic Curve, Envelope, Epicycloid, Equipotential
Curve, Eudoxus's Kampyle, Evolute, Exponen-
tial Ramp, Fermat Conic, Folium of Descartes,
Freeth's Nephroid, Frey Curve, Gaussian Func-
tion, Gerono Lemniscate, Glissette, Guder-
mannian Function, Gutschoven's Curve, Hip-
popede, Horse Fetter, Hyperbola, Hyperel-
lipse, Hypocycloid, Hypoellipse, Involute, Isop-
tic Curve, Kappa Curve, Keratoid Cusp, Knot
Curve, Lame Curve, Lemniscate, L'Hospital's
Cubic, Limaqon, Links Curve, Lissajous Curve,
Lituus, Logarithmic Spiral, Maclaurin Trisec-
trix, Maltese Cross, Mill, Natural Equation,
Negative Pedal Curve, Nephroid, Nielsen's Spi-
ral, Orthoptic Curve, Parabola, Pear Curve,
Pear-Shaped Curve, Pearls of Sluze, Pedal
Curve, Peg Top, Piriform, Plateau Curves, Po-
liceman on Point Duty Curve, Prolate Cycloid,
Pursuit Curve, Quadratrix of Hippias, Radial
Curve, Rhodonea, Rose, Roulette, Semicubical
Parabola, Serpentine Curve, Sici Spiral, Sig-
moid Curve, Sinusoidal Spiral, Space Curve,
Strophoid, Superellipse, Swastika, Sweep Sig-
nal, Talbot's Curve, Teardrop Curve, Tractrix,
Trident, Trident of Descartes, Trident of New-
ton, Trochoid, Tschirnhausen Cubic, Versiera,
Watt's Curve, Whewell Equation, Witch of Ag-
nesi
References
Cundy, H, and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 71-75, 1989.
"Geometry." The New Encyclopaedia Britannica, 15th ed.
19, pp. 946-951, 1990.
Gray, A. "Famous Plane Curves." Ch. 3 in Modern Differen-
tial Geometry of Curves and Surfaces. Boca Raton, FL:
CRC Press, pp. 37-55, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, 1972.
Lee, X. "A Catalog of Special Plane Curves." http://vvw.
best, com/ -xah/ Special Plane Curves jdir/specialPlane
Curves.html.
378
Curve of Constant Breadth
Curvilinear Coordinates
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, 1961.
MacTutor History of Mathematics Archive, http:// www -
groups . dcs . st - and .ac.uk/- history / Curves /
Curves.html.
Oakley, C. O. Analytic Geometry, New York: Barnes and
Noble, 1957.
Shikin, E. V. Handbook and Atlas of Curves. Boca Raton,
FL: CRC Press, 1995.
Smith, P. F.; Gale, A. S.; and Neelley, J. H. New Analy-
tic Geometry, Alternate Edition. Boston, MA: Ginn and
Company, 1938.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, 1993.
Walker, R. J. Algebraic Curves. New York: Springer- Verlag,
1978.
# Weisstein, E. W. "Plane Curves." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/Curves.ni.
Yates, R. C. A Handbook on Curves and Their Properties.
Ann Arbor, MI: J. W. Edwards, 1947.
Yates, R. C. The Trisection Problem. Reston, VA: National
Council of Teachers of Mathematics, 1971.
Zwillinger, D. (Ed.). "Algebraic Curves." §8.1 in CRC Stan-
dard Mathematical Tables and Formulae, 3rd ed. Boca
Raton, FL: CRC Press, 1996. http: //www. geom.umn.edu/
docs/reference/CRC-f ormulas/node33.html.
Curve of Constant Breadth
see Curve of Constant Width
Curve of Constant Precession
A curve whose Centrode revolves about a fixed axis
with constant Angle and Speed when the curve is tra-
versed with unit SPEED. The TANGENT INDICATRIX of a
curve of constant precession is a Spherical Helix. An
Arc Length parameterization of a curve of constant
precession with NATURAL EQUATIONS
k(s) = — u)sin(fis)
t(s) = ujcos(fxs)
(i)
(2)
is
( x _ a + \i sin[(q - fj,)s] a - p sin[(q + fx)s] ,.
x ^ s ) — ~^7. Z 7 WI. 7TTT. W
2a a — \x
2a
a + ji
. , _ a + fx cos[(a — /x)s] a — /x cos[(a + fx)s]
V ^ ~ ~~2a a- ii + ~2a a + M
z(s) — — sin(^s),
fJLOL
where
a = y a; 2 + fi 2
(4)
(5)
(6)
and u) y and \i are constant. This curve lies on a circular
one-sheeted Hyperboloid
2,2 M 2 2 4/i 2
X +2/ ~^ Z = ^" (7)
The curve is closed Iff fx/a is Rational.
References
Scofield, P. D. "Curves of Constant Precession." Amer.
Math. Monthly 102, 531-537, 1995.
Curve of Constant Slope
see Generalized Helix
Curve of Constant Width
Curves which, when rotated in a square, make contact
with all four sides. The "width" of a closed convex
curve is defined as the distance between parallel lines
bounding it ("supporting lines"). Every curve of con-
stant width is convex. Curves of constant width have
the same "width" regardless of their orientation between
the parallel lines. In fact, they also share the same Per-
imeter (Barbier's Theorem). Examples include the
Circle (with largest Area), and Reuleaux Triangle
(with smallest Area) but there are an infinite number.
A curve of constant width can be used in a special drill
chuck to cut square "HOLES."
A generalization gives solids of constant width. These
do not have the same surface AREA for a given width,
but their shadows are curves of constant width with the
same width!
see also Delta Curve, Kakeya Needle Problem,
Reuleaux Triangle
References
Bogomolny, A. "Shapes of Constant Width." http: //www.
cut-the-knot.com/do_you_know/cwidth.html.
Bohm, J. "Convex Bodies of Constant Width." Ch. 4 in
Mathematical Models from the Collections of Universities
and Museums (Ed. G. Fischer). Braunschweig, Germany:
Vieweg, pp. 96-100, 1986.
Fischer, G. (Ed.). Plates 98-102 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986.
Gardner, M. Ch. 18 in The Unexpected Hanging and Other
Mathematical Diversions. Chicago, IL: Chicago University
Press, 1991.
Goldberg, M. "Circular-Arc Rotors in Regular Polygons."
Amer. Math. Monthly 55, 393-402, 1948.
Kelly, P. Convex Figures. New York: Harcourt Brace, 1995.
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, 1957.
Yaglom, I. M. and Boltyanski, V. G. Convex Figures. New
York: Holt, Rinehart, and Winston, 1961.
Curvilinear Coordinates
A general METRIC g^ has a LINE ELEMENT
ds = g^ u du^du u ',
(i)
where EINSTEIN SUMMATION is being used. Curvilinear
coordinates are defined as those with a diagonal METRIC
so that
Qixv — U tin
(2)
where 5% is the KRONECKER Delta. Curvilinear coor-
dinates therefore have a simple Line Element
ds 2 = 5^h fl 2 du^du u = hffdu** 2 ,
(3)
Curvilinear Coordinates
Cushion 379
which is just the Pythagorean Theorem, so the dif-
ferential Vector is
or
dr = hftdUtxUf^
dr , dr . dr _
dr = - — dux H- « — «^2 + « — au 3 ,
aiti au2 a^3
where the SCALE FACTORS are
dr
hi
Bui
and
dui
Ui = -
1 dr
I C*Ui I
Equation (5) may therefore be re-expressed as
dr = hiduiiix + h 2 du 2 vi2 + hzdu$u&-
The Gradient is
(4)
(5)
(6)
(7)
(8)
,,,, „, 1 8<j) „ 1 5<A „ 1 9^.
grad(0) S V0 = -^- Ul + -^-u a + ^^u 3 ,
the Divergence is
^^•^^b^ 3 ^
+^-(h 3 h 1 F 2 ) + ^-{h 1 h 2 F 3 )
du 2 ou 3
(9)
(10)
and the Curl is
V x F :
1
hxh,2hz
1
JllUl feU2 /I3U3
a a _a_
dui 9u2 5ti3
/il^l A2F2 ^3^3
W 3 L^ (W) -i (M)
Ui
+
U3.(ll)
hxh
dux du 2
Orthogonal curvilinear coordinates satisfy the addi-
tional constraint that
Ui * \ij = Sij.
(12)
Therefore, the Line Element is
ds = dr • dr = hx dux + h 2 du 2 + h$ du% (13)
and the Volume Element is
dV = |(hiuidui) * {h 2 u. 2 du 2 ) x (/i3U3dit3)|
= h\h 2 hz dux du 2 du$
dr dr dr
dux du 2 du3
dx dx dx
du\ du^ du$
dy dy dy
dui dw2, duz
dz dz dz
dui dii2 du%
du\ du 2 duz
du\ du2 du3
d{x,y 1 z)
d(ux,u 2 ,u 3 )
dux du 2 dus,
(14)
where the latter is the JACOBIAN.
Orthogonal curvilinear coordinate systems include
Bipolar Cylindrical Coordinates, Bispherical
Coordinates, Cartesian Coordinates, Confo-
cal Ellipsoidal Coordinates, Confocal Parabo-
loidal Coordinates, Conical Coordinates, Cy-
clidic Coordinates, Cylindrical Coordinates,
Ellipsoidal Coordinates, Elliptic Cylindrical
Coordinates, Oblate Spheroidal Coordinates,
Parabolic Coordinates, Parabolic Cylindrical
Coordinates, Paraboloidal Coordinates, Polar
Coordinates, Prolate Spheroidal Coordinates,
Spherical Coordinates, and Toroidal Coordi-
nates. These are degenerate cases of the CONFOCAL
Ellipsoidal Coordinates.
see also Change of Variables Theorem, Curl, Di-
vergence, Gradient, Jacobian, Laplacian
References
Arfken, G. "Curvilinear Coordinates" and "Differential Vec-
tor Operators." §2.1 and 2.2 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 86-
90 and 90-94, 1985.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1084-1088, 1980.
Morse, P. M. and Feshbach, H. "Curvilinear Coordinates"
and "Table of Properties of Curvilinear Coordinates." §1.3
in Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 21-31 and 115-117, 1953.
Cushion
The QUARTIC SURFACE resembling a squashed round
cushion on a barroom stool and given by the equation
2zx 2 + 2z z + x 2
/ 2 \2 4 22 2 2 , 2 . 2 n
-{x - z) -y -2xy ~yz -\-2yz-\-y = 0.
380
Cusp
CW-Complex
see also Quartic Surface
References
Nordstrand, T. "Surfaces.'
nfytn/surf aces. htm.
Cusp
Cusp Map
http : //www . uib . no/people/
A function f(x) has a cusp (also called a Spinode) at a
point x if f(x) is CONTINUOUS at x and
lim f'{x) = oo
from one side while
lim f'(x) = — oo
x— >xo
from the other side, so the curve is Continuous but the
Derivative is not. A cusp is a type of Double Point.
The above plot shows the curve x 3 — y 2 = 0, which has
a cusp at the ORIGIN.
see also Double Cusp, Double Point, Ordinary
Double Point, Ramphoid Cusp, Salient Point
References
Walker, R. J. Algebraic Curves. New York: Springer- Verlag,
pp. 57-58, 1978.
Cusp Catastrophe
A Catastrophe which can occur for two control factors
and one behavior axis. The equation y — x 2 ' 3 has a cusp
catastrophe.
see also CATASTROPHE
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 28, 1993.
Cusp Form
A cusp form on T (N), the group of INTEGER matri-
ces with determinant 1 which are upper triangular mod
iV, is an ANALYTIC FUNCTION on the upper half-plane
consisting of the COMPLEX NUMBERS with POSITIVE
Imaginary Part. Weight n cusp forms satisfy
f az + b
'(Sts) -<-+«■«■>
for all matrices
a b
c d
eTo(N).
The function
see also MODULAR FORM
/(*) = 1 - 2M 1 ' 2
for x e [-1, 1]. The Invariant Density is
p(y)=Ul-y).
References
Beck, C. and Schlogl, F. Thermodynamics of Chaotic Sys-
tems. Cambridge, England: Cambridge University Press,
p. 195, 1995.
Cusp Point
see Cusp
Cut- Vert ex
see Articulation Vertex
Cutting
see Arrangement, Cake Cutting, Circle Cut-
ting, Cylinder Cutting, Pancake Cutting, Pie
Cutting, Square Cutting, Torus Cutting
CW- Approximation Theorem
If X is any SPACE, then there is a CW-Complex Y
and a MAP / : Y -> X inducing ISOMORPHISMS on all
Homotopy, Homology, and Cohomology groups.
CW- Complex
A CW- complex is a homotopy- theoretic generalization
of the notion of a Simplicial Complex. A CW-
complex is any SPACE X which can be built by starting
off with a discrete collection of points called X°, then
attaching 1-D DISKS D 1 to X° along their boundaries
5°, writing X 1 for the object obtained by attaching the
D 1 s to X°, then attaching 2-D DISKS D 2 to X 1 along
their boundaries S 1 , writing X 2 for the new SPACE, and
so on, giving spaces X n for every n. A CW-complex
is any SPACE that has this sort of decomposition into
SUBSPACES X n built up in such a hierarchical fashion
(so the X n s must exhaust all of X). In particular, X n
may be built from X" -1 by attaching infinitely many
n-DlSKS, and the attaching Maps 5 n ~ 1 -+ X 71 ' 1 may
be any continuous MAPS.
Cycle (Circle)
Cyclic Graph 381
The main importance of CW-complexes is that, for
the sake of HOMOTOPY, HOMOLOGY, and COHOMOL-
OGY groups, every SPACE is a CW-complex. This is
called the CW-Approximation Theorem. Another
is Whitehead's Theorem, which says that Maps be-
tween CW-complexes that induce ISOMORPHISMS on all
Homotopy Groups are actually Homotopy equiva-
lences.
see also Cohomology, CW-Approximation Theo-
rem, Homology Group, Homotopy Group, Sim-
plicial Complex, Space, Subspace, Whitehead's
Theorem
Cycle (Circle)
A CIRCLE with an arrow indicating a direction.
Cycle (Graph)
A subset of the EDGE-set of a graph that forms a CHAIN
(Graph), the first node of which is also the last (also
called a CIRCUIT).
see also Cyclic Graph, Hamiltonian Cycle, Walk
Cycle Graph
z 2 z 3 z 4
A cycle graph is a Graph which shows cycles of a
GROUP as well as the connectivity between the cycles.
Several examples are shown above. For Z4, the group
elements At satisfy Af = 1, where 1 is the Identity
Element, and two elements satisfy A\ 2
A3 2
For a Cyclic Group of Composite Order n (e.g.,
Z4, Zq, Zs), the degenerate subcycles corresponding to
factors dividing n are often not shown explicitly since
their presence is implied.
see also Characteristic Factor, Cyclic Group
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 83-98, 1993.
Cycle (Map)
An n-cycle is a finite sequence of points Y 0i . . . , Y n -i
such that, under a MAP G,
Yi = G(Y )
Y 2 = G{Y!)
Y n -i = G(Y n -2)
Y = G(Y n - 1 ).
In other words, it is a periodic trajectory which comes
back to the same point after n iterations of the cycle.
Every point Yj of the cycle satisfies Yj = G n {Yj) and is
therefore a Fixed Point of the mapping G n . A fixed
point of G is simply a CYCLE of period 1.
Cycle (Permutation)
A Subset of a Permutation whose elements trade
places with one another. A cycle decomposition of a
Permutation can therefore be viewed as a Class of
a Permutation Group. For example, in the Per-
mutation GROUP {4, 2, 1, 3}, {1, 3, 4} is a 3-cycle
(1 -> 3, 3 ->> 4, and 4 -> 1) and {2} is a 1-cycle
(2 -> 2). Every PERMUTATION GROUP on n symbols
can be uniquely expressed as a product of disjoint cycles.
The cyclic decomposition of a PERMUTATION can be
computed in Mathematical (Wolfram Research, Cham-
paign, IL) with the function ToCycles and the PERMU-
TATION corresponding to a cyclic decomposition can be
computed with FromCycles. According to Vardi (1991),
the Mathematica code for ToCycles is one of the most
obscure ever written.
To find the number JV(m, n) of m cycles in a Permu-
tation Group of order n, take
N(n,m) = (-l) n " m 5i(n,m),
where Si is the Stirling Number of the First Kind.
see also Golomb-Dickman Constant, Permuta-
tion, Permutation Group, Subset
References
Skiena, S. Implementing Discrete Mathematics: Combina-
torics and Graph Theory with Mathematica. Reading,
MA: Addison- Wesley, p. 20, 1990.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addis on- Wesley, p. 223, 1991.
Cyclic Graph
c 3 ^ 4 v. 5
A GRAPH of n nodes and n edges such that node i is
connected to the two adjacent nodes z + 1 and i — 1 (mod
n), where the nodes are numbered 0, 1, . . . , n — 1.
see also Cycle (Graph), Cycle Graph, Star
Graph, Wheel Graph
382 Cyclic Group
Cyclic Number
Cyclic Group
A cyclic group Z n of Order n is a Group defined by
the element X (the GENERATOR) and its n POWERS up
to
X n = 1,
where / is the IDENTITY ELEMENT, Cyclic groups are
both ABELIAN and SIMPLE. There exists a unique cyclic
group of every order n > 2, so cyclic groups of the same
order are always isomorphic (Shanks 1993, p. 74), and
all Groups of Prime Order are cyclic.
Examples of cyclic groups include Z2, Z3, Z4, and
the Modulo Multiplication Groups M m such that
m = 2, 4, p n , or 2p n , for p an Odd Prime and n > 1
(Shanks 1993, p. 92). By computing the CHARACTERIS-
TIC FACTORS, any ABELIAN GROUP can be expressed as
a Direct Product of cyclic Subgroups, for example,
Z 2 ® Z 4 or Z 2 ® Z 2 ® Z 2 .
see also ABELIAN GROUP, CHARACTERISTIC FAC-
TOR, Finite Group — Z 2 , Finite Group — Z 3 , Finite
Group — Z 2 , Finite Group — Z&, Finite Group — Z 6 ,
Modulo Multiplication Group, Simple Group
References
Lomont, J. S. "Cyclic Groups." §3. 10. A in Applications of
Finite Groups. New York: Dover, p. 78, 1987.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, 1993.
Cyclic Hexagon
A hexagon (not necessarily regular) on whose VERTICES
a Circle may be Circumscribed. Let
Then define
Ti = y en a,j
(i)
i,j,...,n-l
where the sum runs over all distinct permutations of the
Squares of the six side lengths, so
2 . 2 . 2 , 2 , 2 . 2 /n\
cri = a\ + ai +0,3 + 04 + a 5 + &§ (2)
22. 22. 22. 22, 22
<t 2 =ai 02 + ai as + a\ a$ + a\ as + a± a&
+ 0,2 0,3 + a2 04 -\- 0,2 a$ + 0*2 clq
22 22 22
+ 03 04 + a$ as + a% a$
+ 0,40,5 + 04 ae + as a& (3)
222. 222, 222, 222
o~3 = ai a2 a3 + ai a2 a4 + ai a2 as + a\ 02 ae
, 222, 222, 222
+ a2 az 04 + a2 a>z as 4- a2 03 oq
222 222 222
4- a3 a4 as + 03 a4 a6 4- a4 as ae
2222 2222 2222
CF4 = ai a 2 a3 a4 4- &\ 02 0,3 as 4- ai 0,2 0,3 ae
(4)
2222 2222
4- ai a3 a4 as 4- o>\ 03 04 ae
2222 2222
+ a± 03 as ae + a\ 04 as ae
_, 2„ 2„ 2^ 2 _. 2„ 2„ 2^ 2 , ^ 2„ 2^ 2^ 2
4--fl2 ^3 a4 as -+- a 2 03 a4 a6 + a2 a3 as a6
1 n 2 n 2„ 2„ 2 , n 1 n 2„ 2„ 2
+ 02 0,4 as ae + 03 04 as oq
22222 22222
0"s = ai a2 a3 a4 as 4- 0,1 a 2 a3 a4 ae
22222 22222
4- tii 0,2 0,3 as ae 4- ai a 2 Gk as ae
_L ~ 2 ~ 2 ~ 2 „ 2 r, 2 _L ^ 2 „ 2 ~ 2 ~ 2 ^ 2
+ ai 03 a4 as a6 + a2 03 04 as a6
222222
erg = a\ a 2 az a4 a§ a§ .
(5)
(6)
(7)
t2 ~ u — 4cr 2 4- 0"i
£3 = 80*3 + <7it2 — lO-y/o^
£4 — t 2 - 64cr4 + 64(7i yfae~
t 5 = 128a 5 + 32i 2 V^6
(8)
(9)
(10)
(11)
(12)
u=161T\
The Area of the hexagon then satisfies
ut 4 3 + t 3 2 t 4 2 - 16tz 3 t 5 - 18u< 3 *4*5 - 27u 2 t 5 2 = 0, (13)
or this equation with y/b~e replaced by — -^/o^, a seventh
order POLYNOMIAL in u. This is l/(4^ 2 ) times the DIS-
CRIMINANT of the Cubic Equation
z 3 + 2t 3 z 2 -ut4Z + 2y 2 t 5 .
(14)
see also Concyclic, Cyclic Pentagon, Cyclic
Polygon, Fuhrmann's Theorem
References
Robbins, D. P. "Areas of Polygons Inscribed in a Circle."
Discr. Comput. Geom. 12, 223-236, 1994.
Robbins, D. P. "Areas of Polygons Inscribed in a Circle."
Amer. Math. Monthly 102, 523-530, 1995.
Cyclic-Inscriptable Quadrilateral
see BlCENTRIC QUADRILATERAL
Cyclic Number
A number having n— 1 Digits which, when Multiplied
by 1, 2, 3, . . . , n — 1, produces the same digits in a dif-
ferent order. Cyclic numbers are generated by the Unit
Fractions 1/n which have maximal period Decimal
Expansions (which means n must be Prime). The first
few numbers which generate cyclic numbers are 7, 17,
19, 23, 29, 47, 59, 61, 97, ... (Sloane's A001913). A
much larger generator is 17389.
It has been conjectured, but not yet proven, that an
Infinite number of cyclic numbers exist. In fact, the
FRACTION of PRIMES which generate cyclic numbers
seems to be approximately 3/8. See Yates (1973) for a
table of Prime period lengths for Primes < 1,370,471.
When a cyclic number is multiplied by its generator, the
result is a string of 9s. This is a special case of Midy's
Theorem.
07 = 0.142857
17 = 0.0588235294117647
19 = 0.052631578947368421
23 = 0.0434782608695652173913
29 = 0.0344827586206896551724137931
47 = 0.021276595744680851063829787234042553191- ■ •
■ ■ ■ 4893617
59 = 0.016949152542372881355932203389830508474- ■ ■
■•■5762711864406779661
61 = 0.016393442622950819672131147540983606557- • ■
Cyclic Pentagon
Cyclic Polygon 383
■ ■ * 377049180327868852459
97 = 0.010309278350515463917525773195876288659* • •
• . • 79381443298969072164948453608247422680412 • • .
... 3711340206185567
see also DECIMAL EXPANSION, Midy's THEOREM
References
Gardner, M. Ch. 10 in Mathematical Circus: More Puz-
zles, Games, Paradoxes and Other Mathematical Enter-
tainments from Scientific American, New York: Knopf,
1979.
Guttman, S. "On Cyclic Numbers." Amer. Math. Monthly
44, 159-166, 1934.
Kraitchik, M. "Cyclic Numbers." §3.7 in Mathematical
Recreations. New York: W. W. Norton, pp. 75-76, 1942.
Rao, K. S. "A Note on the Recurring Period of the Reciprocal
of an Odd Number." Amer. Math. Monthly 62, 484-487,
1955.
Sloane, N. J. A. Sequence A001913/M4353 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Yates, S. Primes with Given Period Length. Trondheim, Nor-
way: Universitetsforlaget, 1973.
Cyclic Pentagon
A cyclic pentagon is a not necessarily regular Pentagon
on whose Vertices a Circle may be Circumscribed.
Let
CTi= 2_^ di 2 dj 2 • • • (in 2 , (1)
i,j,...,n = l
where the Sum runs over all distinct PERMUTATIONS of
the Squares of the 5 side lengths, so
2 2 2 2 2
a\ = a\ + a2 + a3 H~ a4 + a>h
22 22 22 22 22
o~2 =ai a2 + a\ az +ai 04 -J- a\ a$ + a-i az
22 22 22 22
+ a2 CI4 + d2 as +«3 A4 4" CI3 GL5
2 2
+ a$ as
222 222 222
(T3 = a± a2 as + cl\ Q>2 &4 + ai a2 as
(2)
(3)
, 222, 222. 222 / A \
-\- a2 a3 a± + a2 a3 as + az a± as (4)
2222 2222 2222
<74 = ai a2 03 a4 + a± a^ az as + a\ az a$ as
2222
+ a2 a3 a4 as
22222
<75 = a\ a<i az a± as .
Then define
(5)
(6)
t 2 =U- 4(72 + 01
tz = 8*73 + 0~\t2
£4 = — 64CT4 + t%
t 6 = 1280-5
(7)
(8)
(9)
(10)
(11)
The Area of the pentagon then satisfies
uU z + tz 2 U 2 - 16t 3 3 i 5 - l&utzUh - 27u 2 t 5 2 = 0, (12)
a seventh order POLYNOMIAL in u. This is l/(4u 2 ) times
the Discriminant of the Cubic Equation
see also Concyclic, Cyclic Hexagon, Cyclic Poly-
gon
References
Robbins, D. P. "Areas of Polygons Inscribed in a Circle."
Discr. Comput. Geom. 12, 223-236, 1994.
Robbins, D. P. "Areas of Polygons Inscribed in a Circle."
Amer. Math. Monthly 102, 523-530, 1995.
Cyclic Permutation
A Permutation which shifts all elements of a Set by a
fixed offset, with the elements shifted off the end inserted
back at the beginning. For a Set with elements ao, ai,
. . . , a n _i, this can be written at —» a i+k (mod n) f° r a
shift of k.
see also Permutation
Cyclic Polygon
A cyclic polygon is a POLYGON with VERTICES upon
which a CIRCLE can be CIRCUMSCRIBED. Since every
Triangle has a Circumcircle, every Triangle is
cyclic. It is conjectured that for a cyclic polygon of
2m 4- 1 sides, 16K 2 (where K is the Area) satisfies a
Monic Polynomial of degree A m , where
m — 1 / \
fc=0
1
2
(2m +1)
V 7
-2 d
(1)
(2)
(Robbins 1995). It is also conjectured that a cyclic poly-
gon with 2m + 2 sides satisfies one of two Polynomials
of degree A m . The first few values of A m are 1, 7, 38,
187, 874, . . . (Sloane's A000531).
For Triangles (n = 3 = 2 • 1 + 1), the Polynomial is
Heron's Formula, which may be written
16/r
2a 2 b 2 + 2a 2 c 2 + 26 V - a 4 - b 4 - c 4 , (3)
and which is of order Ai = 1 in 16K 2 . For a CYCLIC
Quadrilateral, the Polynomial is Brahmagupta's
Formula, which may be written
16HT
+ 2a 2 b 2 -b 4 + 2a 2 c 2 +2b 2 c 2 -c 4
+ Sabcd + 2a 2 d 2 + 2b 2 d 2 + 2c 2 d 2
d\ (4)
z 3 -\-2tzz 2 -ut 4 z + 2y 2 t 5 .
(13)
which is of order Ai - 1 in 16K 2 . Robbins (1995)
gives the corresponding FORMULAS for the CYCLIC PEN-
TAGON and Cyclic Hexagon.
see also CONCYCLIC, CYCLIC HEXAGON, CYCLIC PEN-
TAGON, Cyclic Quadrangle, Cyclic Quadrilat-
eral
References
Robbins, D. P. "Areas of Polygons Inscribed in a Circle."
Discr. Comput. Geom. 12, 223-236, 1994.
Robbins, D. P. "Areas of Polygons Inscribed in a Circle,"
Amer. Math. Monthly 102, 523-530, 1995.
Sloane, N. J. A. Sequence A000531 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
384
Cyclic Quadrangle
Cyclic Quadrilateral
Cyclic Quadrangle
Let Ai, A 2 , A3, and A 4 be four Points on a Circle,
and H U H 2 ,H Z , H 4 the Orthocenters of Triangles
AA2A3A4, etc. If, from the eight POINTS, four with
different subscripts are chosen such that three are from
one set and the fourth from the other, these POINTS
form an ORTHOCENTRIC SYSTEM. There are eight such
systems, which are analogous to the six sets of ORTHO-
CENTRIC SYSTEMS obtained using the feet of the ANGLE
Bisectors, Orthocenter, and Vertices of a generic
Triangle.
On the other hand, if all the POINTS are chosen from one
set, or two from each set, with all different subscripts,
the four POINTS lie on a CIRCLE. There are four pairs
of such Circles, and eight Points lie by fours on eight
equal CIRCLES.
The Simson Line of A 4 with regard to Triangle
A'AiA 2 A$ is the same as that of H4 with regard to the
Triangle AifiA 2 A 3 .
see also ANGLE BISECTOR, CONCYCLIC, CYCLIC POLY-
GON, Cyclic Quadrilateral, Orthocentric Sys-
tem
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 251-253, 1929.
Cyclic Quadrilateral
Solving for the ClRCUMRADlUS gives
A Quadrilateral for which a Circle can be circum-
scribed so that it touches each Vertex. The Area is
then given by a special case of BRETSCHNEIDER'S FOR-
MULA. Let the sides have lengths a, 6, c, and d, let s be
the Semiperimeter
3= f(a + 6 + c + d),
and let R be the ClRCUMRADlUS. Then
A = ^J(s — a){s — b)(s — c)(s — d)
y{ac + bd)(ad + bc)(ab -j- cd)
(1)
(2)
(3)
o _ 1 I (ac + bd)(ad + bc)(ab + cd) ,,
R= *y (s - a)(s - b){s - c){s - d) ' ()
The Diagonals of a cyclic quadrilateral have lengths
(5)
(ab + cd)(ac + bd)
ad + be
q =
(ac + bd)(ad + be)
ab + cd
(6)
so that pq — ac 4- bd. In general, there are three essen-
tially distinct cyclic quadrilaterals (modulo Rotation
and REFLECTION) whose edges are permutations of the
lengths a, 6, c, and d. Of the six corresponding DIAG-
ONAL lengths, three are distinct. In addition to p and
q, there is therefore a "third" DIAGONAL which can be
denoted r. It is given by the equation
(ad + bc)(ab + cd)
ac-\-bd
(7)
This allows the Area formula to be written in the par-
ticularly beautiful and simple form
A =
pqr
4R*
(8)
The DIAGONALS are sometimes also denoted p, q, and
r.
The Area of a cyclic quadrilateral is the Maximum
possible for any QUADRILATERAL with the given side
lengths. Also, the opposite ANGLES of a cyclic quadri-
lateral sum to 7r Radians (Dunham 1990).
A cyclic quadrilateral with RATIONAL sides a, 6, c, and
d, Diagonals p and g, Circumradius R, and Area
A is given by a = 25, b = 33, c = 39, d = 65, p = 60,
q = 52, R = 65/2, and A = 1344.
Let AHBO be a Quadrilateral such that the angles
IHAB and LHOB are Right Angles, then AHBO is
a cyclic quadrilateral (Dunham 1990). This is a COROL-
LARY of the theorem that, in a RIGHT TRIANGLE, the
Midpoint of the Hypotenuse is equidistant from the
Cyclic Redundancy Check
Cyclide 385
three VERTICES. Since M is the MIDPOINT of both
Right Triangles AAHB and ABOH, it is equidis-
tant from all four VERTICES, so a CIRCLE centered at
M may be drawn through them. This theorem is one
of the building blocks of Heron's derivation of Heron's
Formula.
Cyclide
Place four equal CIRCLES so that they intersect in a
point. The quadrilateral ABCD is then a cyclic quadri-
lateral (Honsberger 1991). For a CONVEX cyclic quad-
rilateral Q, consider the set of Convex cyclic quadri-
laterals Q\\ whose sides are Parallel to Q. Then the
Qll of maximal AREA is the one whose DIAGONALS are
Perpendicular (Gurel 1996).
see also BRETSCHNEIDER'S FORMULA, CONCYCLIC,
Cyclic Polygon, Cyclic Quadrangle, Euler
Brick, Heron's Formula, Ptolemy's Theorem,
Quadrilateral
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 123, 1987.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 121, 1990.
Gurel, E. Solution to Problem 1472. "Maximal Area of
Quadrilaterals." Math. Mag. 69, 149, 1996.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 36-37, 1991.
Cyclic Redundancy Check
A sophisticated Checksum (often abbreviated CRC),
which is based on the algebra of polynomials 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 CHECKSUM, ERROR-CORRECTING CODE
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Cyclic Redundancy and Other Checksums."
Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 888-895, 1992.
Cyclid
see Cyclide
A pair of focal conies which are the envelopes of two
one-parameter families of spheres, sometimes also called
a Cyclid. The cyclide is a Quartic Surface, and the
lines of curvature on a cyclide are all straight lines or
circular arcs (Pinkall 1986). The STANDARD TORI and
their inversions in a SPHERE S centered at a point Xo
and of RADIUS r, given by
J(xo,r) = x +
x - x r
|x-x | 2
are both cyclides (Pinkall 1986). Illustrated above are
Ring Cyclides, Horn Cyclides, and Spindle Cy-
clides. The figures on the right correspond to xo lying
on the torus itself, and are called the PARABOLIC Ring
Cyclide, Parabolic Horn Cyclide, and Parabolic
Spindle Cyclide, respectively.
see also Cyclidic Coordinates, Horn Cyclide,
Parabolic Horn Cyclide, Parabolic Ring Cy-
clide, Ring Cyclide, Spindle Cyclide, Standard
Tori
References
Bierschneider-Jakobs, A. "Cyclides." http://www.mi.uni-
erlangen.de/-biersch/cyclides.html.
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, p. 273, 1959.
Eisenhart, L. P, "Cyclides of Dupin." §133 in A Treatise on
the Differential Geometry of Curves and Surfaces. New
York: Dover, pp. 312-314, 1960.
386 Cyclidic Coordinates
Cycloid
Fischer, G. (Ed.). Plates 71-77 in Mathematiscke Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 66-72, 1986.
Nordstrand, T. "Dupin Cyclide." http://www.uib.no/
people/nf ytn/dupintxt .htm.
Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models
from the Collections of Universities and Museums (Ed.
G. Fischer). Braunschweig, Germany: Vieweg, pp. 28—30,
1986.
Salmon, G. Analytic Geometry of Three Dimensions. New
York: Chelsea, p. 527, 1979,
Cyclidic Coordinates
A general system of CURVILINEAR COORDINATES based
on the Cyclide in which Laplace's Equation is Sep-
arable.
References
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, p. 273, 1959.
Cycloid
in the 17th century, the cycloid became known as the
"Helen of Geometers" (Boyer 1968, p. 389).
The cycloid is the Catacaustic of a Circle for a Ra-
diant Point on the circumference, as shown by Jakob
and Johann Bernoulli in 1692. The CAUSTIC of the cy-
cloid when the rays are parallel to the y-axis is a cycloid
with twice as many arches. The RADIAL CURVE of a
Cycloid is a Circle. The Evolute and Involute of
a cycloid are identical cycloids.
If the cycloid has a Cusp at the Origin, its equation in
Cartesian Coordinates is
x — a cos
In parametric form, this becomes
x — a(t — sint)
y = a(l — cost).
(i)
(2)
(3)
If the cycloid is upside-down with a cusp at (0,a), (2)
and (3) become
= 2asin 1 (|-) + y/2ay - y*
(4)
x — a(t + sint)
y = a(l — cost)
(5)
(6)
(sign of sint flipped for x).
The Derivatives of the parametric representation (2)
and (3) are
x = a(l — cost) (7)
y = a sint (8)
The cycloid is the locus of a point on the rim of a CIRCLE
of Radius a rolling along a straight LINE, It was studied
and named by Galileo in 1599. Galileo attempted to
find the Area by weighing pieces of metal cut into the
shape of the cycloid. Torricelli, Fermat, and Descartes
all found the Area. The cycloid was also studied by
Roberval in 1634, Wren in 1658, Huygens in 1673, and
Johann Bernoulli in 1696. Roberval and Wren found the
Arc Length (MacTutor Archive). Gear teeth were also
made out of cycloids, as first proposed by Desargues in
the 1630s (Cundy and Rollett 1989).
In 1696, Johann Bernoulli challenged other mathemati-
cians to find the curve which solves the Brachisto-
chrone Problem, knowing the solution to be a cy-
cloid. Leibniz, Newton, Jakob Bernoulli and L'Hospital
all solved Bernoulli's challenge. The cycloid also solves
the Tautochrone Problem. Because of the frequency
with which it provoked quarrels among mathematicians
dy _ y' _ a sint sint
dx x f a(l — cost) 1 — cost
_ 2sin(|t)cos(|t) _
2sin 2 (it)
The squares of the derivatives are
3t(|t).
x n = a 2 (l-2cost + cos 2 t)
11 2-2,
y = a sin t,
so the Arc Length of a single cycle is
//*27T
ds= VV 2 + y' 2 dt
/>2tt
= a I y (1 — 2 cos t + cos 2 t) + sin 2 1 1
Jo
/2tt p2n nr
Vl - cos tdt = 2a / J-
= 2a |sin(|t)| dt.
Jo
(9)
(10)
(11)
— cost
dt
(12)
Cycloid
Now let u = t/2 so du = dt/2. Then
L — 4a / sin udu = 4a[— cos u]J
= -4a[(-l) - 1] = 8a.
(13)
The Arc Length, Curvature, and Tangential An-
gle are
\ = Sasm(lt)
; = — |acsc
> = -fat
'(!*)
(14)
(15)
(16)
The Area under a single cycle is
,2. .27T
A= ydx~a I (1 — cos </>)(! — cos (f>)d</>
Jo Jo
(1 — COS0) d</>
=•7
= * 2 / [§-
= a 2 [§0-2 B in^+ Jsin(2^)]3*
(1 — 2 cos + cos <j)) d<f>
{1 - 2 cos + \ [1 + cos(2</>)] } a>
2cos0+ f cos(20)]d<£
a 2 |27r = 37ra 2 .
The Normal is
T =
y/2 - 2 cos t
1 — cos £
sint
(17)
(18)
see also Curtate Cycloid, Cyclide, Cycloid Evo-
lute, Cycloid Involute, Epicycloid, Hypocy-
cloid, Prolate Cycloid, Trochoid
References
Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/
pythagoras/cycloids .html.
Boyer, C. B. A History of Mathematics. New York: Wiley,
1968.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.
Gray, A. "Cycloids," §3.1 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 37-39, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves, New
York: Dover, pp. 192 and 197, 1972.
Lee, X. "Cycloid." http://www.best.com/-xah/Special
PlaneCurves_dir/Cycloid_dir/cycloid.html.
Cycloid Involute 387
Lockwood, E. H. "The Cycloid." Ch. 9 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 80-
89, 1967.
MacTutor History of Mathematics Archive. "Cycloid."
http: //www-groups . dcs . st-and.ac ,uk/*history/Curves
/Cycloid. html.
Muterspaugh, J.; Driver, T.; and Dick, J. E. "The Cycloid
and Tautochronism." http://ezinfo.ucs.indiana.edu/
-jedick/project/ intro.html.
Pappas, T. "The Cycloid— The Helen of Geometry." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 6-8, 1989.
Wagon, S. "Rolling Circles." Ch, 2 in Mathematica in Ac-
tion. New York: W. H. Freeman, pp. 39-66, 1991.
Yates, R. C. "Cycloid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 65-70,
1952.
Cycloid Evolute
/
\
/
X
/
\
The Evolute of the Cycloid
x(t) — a(t — sint)
y(t) = a(l — cost)
is given by
x(t) = a(t + sint)
y(t) = a(cost - 1).
As can be seen in the above figure, the Evolute is
simply a shifted copy of the original CYCLOID, so the
Cycloid is its own Evolute.
Cycloid Involute
The Involute of the Cycloid
x(t) = a(t — sint)
y(t) = a(l — cost)
is given by
x(t) = a(t + sint)
y(t) = a(3 + cost).
As can be seen in the above figure, the INVOLUTE is
simply a shifted copy of the original CYCLOID, so the
Cycloid is its own Involute!
388 Cycloid Radial Curve
Cycloid Radial Curve
The Radial Curve of the Cycloid is the Circle
x = Xq + 2asin<£
y = —2a + yo + 2a cos 0.
Cyclomatic Number
see Circuit Rank
Cyclotomic Equation
The equation
1,
where solutions Cfc = e 27rifc/p are the ROOTS OF UNITY
sometimes called DE Moivre NUMBERS. Gauss showed
that the cyclotomic equation can be reduced to solving a
series of QUADRATIC EQUATIONS whenever p is a Fer-
mat Prime. Wantzel (1836) subsequently showed that
this condition is not only SUFFICIENT, but also NECES-
SARY. An "irreducible" cyclotomic equation is an ex-
pression of the form
x- 1
- x +x p " 2 + .
. + 1 = 0,
where p is PRIME. Its ROOTS Z{ satisfy \zi\ = 1.
see also CYCLOTOMIC POLYNOMIAL, DE MOIVRE NUM-
BER, Polygon, Primitive Root of Unity
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. 99-100, 1996.
Scott, C. A. "The Binomial Equation x p - 1 = 0." Amer. J.
Math. 8, 261-264, 1886.
Wantzel, P. L. "Recherches sur les moyens de reconnaitre si
un Probleme de Geometrie peut se resoudre avec la regie
et le compas." J. Math, pures appliq. 1, 366-372, 1836.
Cyclotomic Factorization
z p -y p = (z-y)(z-ty)...(z-( p - 1 y) J
where C, = e 27ri/p (a de Moivre Number) and p is a
Prime.
Cyclotomic Polynomial
Cyclotomic Field
The smallest field containing m G Z > 1 with ( a PRIME
Root of Unity is denoted M m (0-
Specific cases are
K 3 =Q(V^3)
R4=Q(v^I)
R 6 =Q( X /Z3),
where Q denotes a Quadratic Field.
Cyclotomic Integer
A number of the form
a + ai( + ... + a p _ 1 C P ,
where
C = e
2-Ki/p
is a de Moivre Number and p is a Prime number.
Unique factorizations of cyclotomic INTEGERS fail for
p > 23.
Cyclotomic Invariant
Let p be an Odd PRIME and F n the CYCLOTOMIC FIELD
of p n+1 th ROOTS of unity over the rational FIELD. Now
let p e(n) be the POWER of p which divides the CLASS
Number h n of F n . Then there exist Integers ^ p , X p >
and u p such that
e(n) = fipp 71 + \ p n + u p
for all sufficiently large n. For Regular Primes, p p =
X p — Up = 0.
References
Johnson, W. "Irregular Primes and Cyclotomic Invariants."
Math. Comput. 29, 113-120, 1975.
Cyclotomic Number
see de Moivre Number, Sylvester Cyclotomic
Number
Cyclotomic Polynomial
A polynomial given by
<s> d (x) = f[(x-<; k ),
(i)
where & are the primitive dth. ROOTS OF UNITY in C
given by Cfc = e 27rifc / d . The numbers £fc are sometimes
called de Moivre Numbers. $d{x) is an irreducible
Cyclotomic Polynomial
Polynomial in Z[as] with degree <t>(d), where 4> is the
Totient Function. For d Prime,
**
p-i
(2)
i.e., the coefficients are all 1. $105 has coefficients of — 2
for x 7 and x 41 , making it the first cyclotomic polynomial
to have a coefficient other than ±1 and 0. This is true
because 105 is the first number to have three distinct
Odd Prime factors, i.e., 105 = 3 • 5 • 7 (McClellan and
Rader 1979, Schroeder 1997). Migotti (1883) showed
that Coefficients of $ pq for p and q distinct Primes
can be only 0, ±1. Lam and Leung (1996) considered
pq-l
®pq - 2_^
a k x
(3)
for p, q PRIME. Write the TOTIENT FUNCTION as
<j>(pq) = (p - l)(q - 1) = rp + sq (4)
and let
0<*<(p-l)(9-l), (5)
ip + jq for some i G [0, r] and j G
then
1. a k = 1 Iff k
[0,5],
2. a k = —1 IFF k + pq = ip + jq for i E [r + 1, q — 1]
and j G [s + l,p - 1],
3. otherwise a k = 0.
The number of terms having a k = 1 is (r + l)(s-f 1), and
the number of terms having a k = — 1 is (p — s — l)(q —
r — 1). Furthermore, assume q > p, then the middle
Coefficient of $ pq is (-l) r .
The LOGARITHM of the cyclotomic polynomial
$ n ( X ) = jj(i - x n/d r (d) (6)
is the Mobius Inversion Formula (Vardi 1991,
p. 225).
The first few cyclotomic POLYNOMIALS are
$i(z
$2(2
$3(#
$4(2?'
$s(£
$e(z
$7(2
$8(z
$g(x
$io(z
= z-l
= x + 1
= x 2 + a; + 1
= :r 2 + l
= x 4 + x 3 + x 2 + a: + 1
x
■ x + 1
= z 6 + z 5 + x 4 + x 3 -f z 2 + x + 1
= o: 4 + l
= X + X +1
4 3.2
= x — x -j- X
Cyclotomic Polynomial 389
The smallest values of n for which 3> n has one or more
coefficients ±1, ±2, ±3, ... are 0, 105, 385, 1365, 1785,
2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465,
10465, 11305, ... (Sloane's A013594).
The Polynomial x n - 1 can be factored as
x n -l = Y[$ d {x),
d\n
(7)
where *d(x) is a Cyclotomic Polynomial. Further-
more,
„ , , X 2 "" 1 IL| 3 „ *.«(*) -i-r. , . , fi .
d\m
The Coefficients of the inverse of the cyclotomic
Polynomial
1 -, t 3 4,6 7.9 10,
= 1 — x + x — x + x — x H- x — x +...
1 + X + X 1
— z2 cna
(9)
n=0
can also be computed from
■ x + 1.
Cn = l-2 [l(n + 2)J + [|(n + 1)J + |>J , (10)
where [x\ is the FLOOR FUNCTION.
see also Aurifeuillean Factorization, Mobius In-
version Formula
References
Beiter, M. "The Midterm Coefficient of the Cyclotomic Poly-
nomial F pq (x)." Amer. Math. Monthly 71, 769-770, 1964.
Beiter, M. "Magnitude of the Coefficients of the Cyclotomic
Polynomial F pq ." Amer. Math. Monthly 75, 370-372,
1968.
Bloom, D. M. "On the Coefficients of the Cyclotomic Poly-
nomials." Amer. Math. Monthly 75, 372-377, 1968.
Carlitz, L. "The Number of Terms in the Cyclotomic Poly-
nomial F pq (x)." Amer. Math. Monthly 73, 979-981, 1966.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer-Verlag, 1996.
de Bruijn, N. G. "On the Factorization of Cyclic Groups."
Indag. Math. 15, 370-377, 1953.
Lam, T. Y. and Leung, K. H. "On the Cyclotomic Polynomial
* pg (X)." Amer. Math. Monthly 103, 562-564, 1996.
Lehmer, E. "On the Magnitude of Coefficients of the Cyclo-
tomic Polynomials." Bull Amer. Math. Soc. 42, 389-392,
1936.
McClellan, J. H. and Rader, C. Number Theory in Digital
Signal Processing. Englewood Cliffs, NJ: Prentice- Hall,
1979.
Migotti, A. "Zur Theorie der Kreisteilungsgleichung."
Sitzber. Math.-Naturwiss. Classe der Kaiser. Akad. der
Wiss., Wien 87, 7-14, 1883.
Schroeder, M. R. Number Theory in Science and Communi-
cation, with Applications in Cryptography, Physics, Dig-
ital Information, Computing, and Self- Similarity, 3rd ed.
New York: Springer-Verlag, p. 245, 1997.
Sloane, N. J. A. Sequence A013594 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 8 and 224-225, 1991.
390 Cylinder
Cylinder
.--— !
L-X.
A cylinder is a solid of circular CROSS-SECTION in which
the centers of the Circles all lie on a single Line. The
cylinder was extensively studied by Archimedes in his
2- volume work On the Sphere and Cylinder in ca. 225
BC.
A cylinder is called a right cylinder if it is "straight"
in the sense that its cross-sections lie directly on top
of each other; otherwise, the cylinder is called oblique.
The surface of a cylinder of height h and RADIUS r can
be described parametrically by
x — r cos 8
y = r sin
z = z,
(i)
(2)
(3)
for z € [0,/i] and € [0, 27r). These are the basis for
Cylindrical Coordinates. The Surface Area (of
the sides) and VOLUME of the cylinder of height h and
Radius r are
S = 2irrh
V — 7rr h.
(4)
(5)
Therefore, if top and bottom caps are added, the
volume- to- surface area ratio for a cylindrical container
is
V nr 2 h
5
1 /l l\~ l
(6)
2nrh + 2irr 2
which is related to the HARMONIC Mean of the radius
r and height h.
see also Cone, Cylinder-Sphere Intersection,
Cylindrical Segment, Elliptic Cylinder, Gen-
eralized Cylinder, Sphere, Steinmetz Solid, Vi-
viani's Curve
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 129, 1987.
Cylinder Cutting
The maximum number of pieces into which a cylinder
can be divided by n oblique cuts is given by
/(") :
Cr)-
+ l = £(n + 2)(n + 3),
Cylinder Function
where (£) is a BINOMIAL COEFFICIENT. This problem is
sometimes also called CAKE CUTTING or PIE CUTTING.
For n = 1, 2, ... cuts, the maximum number of pieces
is 2, 4, 8, 15, 26, 42, , . . (Sloane's A000125).
see also Circle Cutting, Ham Sandwich Theorem,
Pancake Theorem, Torus Cutting
References
Bogomolny, A. "Can You Cut a Cake into 8 Pieces
with Three Movements." http://www.cut-the-knot.com/
do_you_know/cake .html.
Sloane, N. J. A. Sequence A000125/M1100 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Cylinder-Cylinder Intersection
see Steinmetz Solid
Cylinder Function
The cylinder function is denned as
L for ^x 2 -f y 2 > a.
The BESSEL FUNCTIONS are sometimes also called cyl-
inder functions. To find the FOURIER TRANSFORM of
the cylinder function, let
k x = k cos a
ky — k sin a
x = r cos
y = r sin 9,
(2)
(3)
(4)
(5)
Then
F{k,a.)=F(C{x,y))
n2iz />a
/ / i(fccosarcos0+fcsinarsin0) t j/i
Jo Jo
ft e ikrco S (e- a ) rdrde
Jo Jo
(6)
Let b - 6 - a, so db = d9. Then
/It: — at pa
/
■a JO
e ikrcosb rdrd0
p2rr pa
/ / e ikrcosb rdrdd
Jo Jo
2-k \ Jo(kr)rdr,
Jo
(7)
where J is a zeroth order BESSEL FUNCTION OF THE
First Kind. Let u = fer, so du = kdr, then
F(k,a) = Ty / Jo(u)udu= -p-[uJi(u)]o°
2lTa t fu \ o ^ Jijka)
— J 1 (ka) = 2na- j ^ r .
(8)
Cylinder-Sphere Intersection
As defined by Watson (1966), a "cylinder function" is
any function which satisfies the RECURRENCE RELA-
TIONS
Cv- 1 (z)+C„+i(z) = —C v {z) (9)
Cylindrical Coordinates 391
C v - 1 {z)-C v +i{z) = 2&(z).
(10)
This class of functions can be expressed in terms of Bes-
sel Functions.
see also BESSEL FUNCTION OF THE FIRST KIND, CYLIN-
DER Function, Cylindrical Function, Hemispher-
ical Function
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Cylinder-Sphere Intersection
see Viviani's Curve
Cylindrical Coordinates
Cylindrical coordinates are a generalization of 2-D PO-
LAR Coordinates to 3-D by superposing a height (z)
axis. Unfortunately, there are a number of different no-
tations used for the other two coordinates. Either r or
p is used to refer to the radial coordinate and either <f>
or 9 to the azimuthal coordinates. Arfken (1985), for
instance, uses (p,<j>,z), while Beyer (1987) uses (r,0,z).
In this work, the NOTATION (r, 0, z) is used.
r = yfx 2 + y 2
z = z,
(1)
(2)
(3)
where r e [0, oo), 9 € [0, 27r), and z € (—00,00). In
terms of x, y, and z
x = r cos 9
y = r sin 9
(4)
(5)
(6)
Morse and Feshbach (1953) define the cylindrical coor-
dinates by
x = £i&
z = ii ..
(7)
(8)
(9)
where £1 = r and £ 2 = cos#. The METRIC elements of
the cylindrical coordinates are
1
2
9
9ee — r
9zz = 1,
so the Scale Factors are
g r — 1
99 = r
The Line Element is
ds ~ dr r + r d9 4- dz z,
and the VOLUME ELEMENT is
dV = rdr d9 dz.
The Jacobian is
d(x,y,z)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
A Cartesian Vector is given in Cylindrical Coor-
dinates by
" r cos 9
rsin#
z
To find the UNIT VECTORS,
r =
(19)
dr
" cos "
dr __
1 dr 1
sin 9
1 dr 1
L o J
dr
" — sin 9
d9 _
1 dr i
cos 9
1 de 1
L o
dr
[01
ds __
1 dr 1
1 dz 1
1
(20)
0=-^-= cos 9 (21)
(22)
Derivatives of unit VECTORS with respect to the coor-
dinates are
dr
dr
=
dr
d9 ~
dz
dr
ae
89 ~
£--
dr
d9
— sin0
costf
=
cos 6
— sin 6
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
392
Cylindrical Coordinates
Ve
■•£+»
. d
so the Gradient components become
V r r =
V e r =
id
r
V Z T =
V r 0-
V 9 =
1.
— r
r
v z e =
V r z =
V^z-
V z z =
0.
The Gradient of a Vector Field in cylindrical coor-
dinates is given by
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
Now, since the Connection Coefficients are defined
by
T) k = ±i • (V fc x,0, (42)
"0 0"
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
"0
01
r =
~~r
.0
o r
T e =
'0
.0
0.
"0
0"
T* =
.0
0.
The COVARIANT Derivatives, given by
A r ;z —
A e -r =
dA r
dr
IdA,
r d6
1 A,
r 86
dA r
dz
8Ae_
dr
Id A,
1 dAj _
-T) k Ai,
. r* A
dA r
dr
Id A e
rd^-e
— T r0 Ai = — T rd A>
r or
_Ab_
/a -^
lrz l ~ dz
TlrAi
dA 9
dr
i8;z
r» a . - x dA$ r r a
r 86 r d6
ldA e A r
r d6 r
dA & r* A dA$
~dz~~ l&zAl -^z~
A -^-
or
idA z
r dd
dA z
dz
A 2 ; Z =
■ r* a —
- rUAi
- r* A- —
dA z
dr
- l dAz
~ r d0
dA z
dz '
Cylindrical Coordinates
Cross Products of the coordinate axes are
f x z = -6 (56)
x z = r (57)
fx^ = z. (58)
The Commutation Coefficients are given by
C a/3^M = [£*, ep] = V a e> - V^ea, (59)
But
[r,r] = [0,0] = [0,0] = O, (60)
so Crr = Cg B — c^ = 0, where a = T, 0, <£. Also
[r,0] = -[0,r] = VrO-Ver = - -9 = -±0, (61)
so cj fl = -(&. = -J, c r r9 = cf 8 = 0. Finally,
[r,01 = [M] = o.
Summarizing,
"0
0"
c r =
.0
6
C —
'0
1
.0
0.
"0
0'
c* =
_0
0.
Time Derivatives of the Vector are
- fr -\-r6 9 -\- zz
cos Br ~ r sin
sin Or + r cos 6
(62)
(63)
(64)
(65)
(66)
— sin 6 r$ 4- cos r — sin 6r6 — r cos 2 — r sin 6 9
cos 6 r6 + sin f + cos 6 r9 — r sin $ 2 + r cos £
-2 sin 6 f6 + cos $ r — r cos 2 - r sin
2 cos 6 r6 4- sin f - r sin 2 + r cos
= (f - r0 2 )r + (2r0 4- r0)0 + z z.
Speed is given by
v = |r| = \/V 2 +r 2 2 +i 2 .
Time derivatives of the unit VECTORS are
■■69
9 =
-_
sin00"
cos
"-COS00"
-sin0
"0"
-0.
.0.
= -9r
(67)
(68)
(69)
(70)
(71)
Cylindrical Coordinates
Cylindrical Coordinates 393
Cross Products of the axes are
f x z = — 6
X z = f
r x = z.
The Convective Derivative is
Dr ( d . _\ . dr . __.
To rewrite this, use the identity
(72)
(73)
(74)
(75)
V(A-B) = Ax(VxB)+Bx(VxA)+(A-V)B+(B-V)A
(76)
and set A = B, to obtain
V(A • A) = 2A x (V x A) -f 2(A • V)A, (77)
so
Then
(A ■ V)A = V(± A d ) - A x (V x A). (78)
Dr
— =r+V(|r 3 )-rx(Vxr) = r+(Vxr)xr-hV(fr ).
(79)
The Curl in the above expression gives
V xf= -i-(r9)z = 2Qz,
r or
(80)
-r x (V x r) = -28(rr X z + rOd x z)
^ -26{-r0 + rOv) = 2r00 - 2r0 2 v. (81)
We have already computed r, so combining all three
pieces gives
— = {f- rO 2 - 2r$ 2 )r + (2f$ + 2f$ + r6)6 + zz
(87)
= (r - 3r0 2 )r + (4r0 + rd)0 + zz.
The Divergence is
v * a = A* r = a;; + (r; r a* + r; P A° + r zr A z ) + a? 6
+ Af, + (r;, A r + t z 9z a 9 + rLA*)
= A% + Af, + Af, + (0 + + 0) + (i + + o)
+ (0 + + 0)
g T dr g e d0 g z dz r
= [fr + rJ^ + rdO^ + te*''
or, in Vector notation
V rdr [ r) ^ r d9 + dz
(88)
(89)
V x F= (-
The Cross Product is
^r dO dz
1
r Vdr
and the LAPLACIAN is
~ r dr \ dr J
)•+(£-£)»
+
(90)
i a 2 / a 2 /
+ r 2 d0 2 + dz 2
5V 10/ i a 2 / a 2 /
dr 2 r 9r r 2 06> 2 0z 2 '
(91)
The vector LAPLACIAN is
We expect the gradient term to vanish since Speed does
not depend on position. Check this using the identity
V(/ 2 ) = 2/V/,
V(ir 2 ) = i V(r 2 + r 2 e 2 + z 2 ) = rVr + r6V{r$) + zVz.
(82)
Examining this term by term,
._ . . d _ . d ^ .x .a?*
rwr — r—vr = r — r = rr = rt/tf
ai dt
(83)
r(9V(r<9) = r<9 \r-~VB + 0Vrl = r0 \r^- (-0\ + 0r
= -0f + r0(-0r) + r6 2 v = ~6r9 (84)
(85)
zvz = z—vz = z— Z - zz = 0,
dr. d£
so, as expected,
V(|r 2 ) = 0.
(86)
V 2 v =
8*v r
Or 2
d 2
Or 2
+
a*v r
a 2 v r
O0 2
9^
Or
+ ■
d 2 v
+
e^ 2
2_ Q^0
r 2 a<^>
' t- 2 a^
• i a^»
1£
(92)
The Helmholtz Differential Equation is separable
in cylindrical coordinates and has StACKEL DETERMI-
NANT S = 1 (for r, 6, z) or S = 1/(1 - £ 2 2 ) (for Morse
and Feshbach's £i, £2, &)•
see also ELLIPTIC CYLINDRICAL COORDINATES, HELM-
holtz Differential Equation — Circular Cylin-
drical Coordinates, Polar Coordinates, Spher-
ical Coordinates
References
Arfken, G. "Circular Cylindrical Coordinates." §2.4 in Math-
ematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 95—101, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 212, 1987.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 657, 1953.
394 Cylindrical Equal- Area Projection
Cylindrical Equal- Area Projection
The Map Projection having transformation equa-
tions,
x — (A — Ao)cos0 5
sin0
y
COS0 S
(1)
(2)
for the normal aspect, and inverse transformation equa-
tions
<j> = sin (ycos0 s )
+ A .
COS (pi
(3)
(4)
An oblique form of the cylindrical equal-area projection
is given by the equations
cos 0i sin <f>2 cos Ai — sin 0i cos 02 cos A2 \
, sin 0i cos 02 sin A2 — cos 0i sin 02 sin Ai J
X p = tan
-1
» = tan
-1
cos(A p — Ai)
tan 0i
! Formulas are
(5)
(6)
and the inverse
= sin -1 (2/ sin P + y 1 — y 2 cos P sin x) (7)
-1 / v 1 ~~ 2/ 2 sm 0p sin a: — y cos P \
A = A + tan I — ^=^= J .
Y yj 1 - y 2 cos X J
(8)
Cylindrical Function
A transverse form of the cylindrical equal-area projec-
tion is given by the equations
x — cos0sin(A — Ao)
y — tan
tan0
cos (A — Ao)
and the inverse FORMULAS are
<£o,
= sin 1 [yl -x 2 sin(y + 0o)]
A = Ao + tan
s/T
■. cos(y + 4>o)
(9)
(10)
(11)
(12)
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 76-85, 1987.
Cylindrical Equidistant Projection
The Map Projection having transformation equations
x = (A — Ao) cos 0i
y = <f>,
and the inverse FORMULAS are
= y
A = Ao +
COS 01
(1)
(2)
(3)
(4)
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 90-91, 1987.
Cylindrical Function
R m (x,y) =
Sm(x,y) =
Jm(x)Y^(y) ~ Jin(y)Y m (x)
jU(x)Y™(y) ~ Jm(y)Y^(x) ^
Jm{x)Ym{y) ~ J m (y)Ym{x) '
see also Cylinder Function, Hemispherical Func-
tion
Cylindrical Harmonics
Cylindroid 395
Cylindrical Harmonics
see BESSEL FUNCTION OF THE FIRST KIND
Cylindrical Hoof
see Cylindrical Wedge
Cylindrical Projection
see Behrmann Cylindrical Equal-Area Projec-
tion, Cylindrical Equal-Area Projection, Cyl-
indrical Equidistant Projection, Gall's Stereo-
graphic Projection, Mercator Projection, Mil-
ler Cylindrical Projection, Peters Projection,
Pseudocylindrical Projection
Cylindrical Segment
The solid portion of a CYLINDER below a cutting Plane
which is oriented Parallel to the Cylinder's axis of
symmetry. For a CYLINDER of RADIUS r and length
L, the Volume of the cylindrical segment is given by
multiplying the Area of a circular SEGMENT of height
hby L,
V = Lr 2 cos™ 1 ( ! ^) - (" " h)L^2rh-h 2 .
see also Cylindrical Wedge, Sector, Segment,
Spherical Segment
Cylindrical Wedge
(r, 0, h)
(0,0,0) J
(0, -r, 0) i
The solid cut from a Cylinder by a tilted Plane pass-
ing through a Diameter of the base. It is also called a
Cylindrical Hoof. Let the height of the wedge be h
and the radius of the Cylinder from which it is cut r.
Then plugging the points (0,-r, 0), (0,r, 0), and (r, 0,/i)
into the 3-point equation for a PLANE gives the equation
for the plane as
hx-rz = 0. (1)
Combining with the equation of the CIRCLE which de-
scribes the curved part remaining of the cylinder (and
writing t = x) then gives the parametric equations of
the "tongue" of the wedge as
x — t
±y/r*
ht
r
y
t 2
(2)
(3)
(4)
for t £ [0,r]. To examine the form of the tongue, it
needs to be rotated into a convenient plane. This can
be accomplished by first rotating the plane of the curve
by 90° about the a;- Axis using the Rotation Matrix
R :e (90 o ) and then by the ANGLE
— tan
■■(*)
(5)
above the z-AxiS. The transformed plane now rests in
the xz-pl&ne and has parametric equations
Wh 2 + r 2
= ±\fr
t 2
(6)
(7)
and is shown below.
The length of the tongue (measured down its middle) is
obtained by plugging t = r into the above equation for
x, which becomes
L= yjh? + r 2
(8)
(and which follows immediately from the PYTHAGO-
REAN Theorem). The Volume of the wedge is given
by
V = lr 2 h. (9)
see also CONICAL WEDGE, CYLINDRICAL SEGMENT
Cylindroid
see Plucker's Conoid
d'Alembert's Equation
D
d'Alembert's Equation
The Ordinary Differential Equation
y = xf(y')+g(y ),
where y' = dy/dx and / and g are given functions.
d'Alembert Ratio Test
see Ratio Test
d'Alembert's Solution
A method of solving the 1-D Wave Equation.
see also Wave Equation
d'Alembert's Theorem
If three CIRCLES A, B, and C are taken in pairs, the ex-
ternal similarity points of the three pairs lie on a straight
line. Similarly, the external similarity point of one pair
and the two internal similarity points of the other two
pairs lie upon a straight line, forming a similarity axis
of the three CIRCLES.
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 155,
1965.
d'Alembertian Operator
Written in the Notation of Partial Derivatives,
c 2 dt 2 '
where c is the speed of light. Writing in Tensor nota-
tion
Daisy 397
□V =(/"*;*),..=$■
A* d 2 d>
dx x dx li
see also Harmonic Coordinates
dx x '
d- Analog
The ^-analog of Infinity Factorial is given by
[0Ol]d :
n('-S)
This Infinite Product can be evaluated in closed form
for small Positive integral d > 2.
see also q- ANALOG
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/infprd/infprd.htinl.
D-Number
A Natural Number n > 3 such that
n|(a n " 2 -a)
whenever (a, n) - 1 (a and n are Relatively Prime)
and a < n. There are an infinite number of such
numbers, the first few being 9, 15, 21, 33, 39, 51, ...
(Sloane's A033553).
see also Knodel Numbers
References
Makowski, A. "Generalization of Morrow's D-Numbers." Si-
mon Stevin 36, 71, 1962/1963.
Sloane, N. J. A. Sequence A033553 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
D-Statistic
see KOLMOGOROV-SMIRNOV TEST
D- Triangle
Let the circles c 2 and c' 3 used in the construction of the
BROCARD POINTS which are tangent to A2A3 at A 2 and
A 3 , respectively, meet again at D\. The points D1D2D3
then define the D-triangle. The VERTICES of the D-
triangle lie on the respective APOLLONIUS CIRCLES.
see also Apollonius Circles, Brocard Points
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, pp. 284-285, 296 and 307, 1929.
Daisy
Hi
•aw
••
A figure resembling a daisy or sunflower in which copies
of a geometric figure of increasing size are placed at regu-
lar intervals along a spiral. The resulting figure appears
to have multiple spirals spreading out from the center.
see also Phyllotaxis, Spiral, Swirl, Whirl
References
Dixon, R. "On Drawing a Daisy." §5.1 in Mathographics.
New York: Dover, pp. 122-143, 1991.
398 Damped Exponential Cosine Integral
Damped Exponential Cosine Integral
Dandelin Spheres
f
Jo
e u cos(o;£) duj.
Integrate by parts with
u = e dv — cos(o?r.) du>
du = —Te~ w dco v = - sin(wi),
(i)
(2)
(3)
SO
/
e wT cos(u;t) do?
?/<
e " sm(o;£) H / e wT sin(o;i) da>. (4)
Now integrate
by parts. Let
/
e w sin(o;£) do;
dv = sin(atf) da;
(5)
(6)
d^= -Te _u;r da> v- — cos(u;t), (7)
so
I e ut sin(aJi) du> — cos(a;r.) / e cos(u;t) da;
(8)
and
/
e w cos(o;£)da; = -e w sin(a;i)
T/tt2 /*
- — e cos(a>£) ' -
t 2
t 2 J
e'" 1 cos{u)t) du> (9)
1 +
?)/
e w cos(a;i) du;
1 T
- sin(aJi) ^ cos(a>£)
(10)
t -\-T f _uT / ,\ ,
/ e cos(a;i) aa;
-ut
= -^-[tsin(a;T) - Tcos(wt)] (11)
cos(a;i) da; :
t 2 + T 2
Therefore,
/ e _w cos(a;
t) da; = +
[tsin(aJi) -Tcos(uT)].
(12)
• (13)
t 2 +T 2 1 2 + T 2
see also Cosine Integral, Fourier Transform-
LORENTZIAN FUNCTION, LORENTZIAN FUNCTION
Dandelin Spheres
The inner and outer SPHERES TANGENT internally to a
Cone and also to a Plane intersecting the Cone are
called Dandelin spheres.
The SPHERES can be used to show that the intersection
of the Plane with the Cone is an Ellipse. Let 7r be
a Plane intersecting a right circular Cone with vertex
O in the curve E. Call the SPHERES TANGENT to the
Cone and the Plane S x and 5 2 , and the CIRCLES on
which the Circles are Tangent to the Cone Ri and
R 2 . Pick a line along the CONE which intersects Ri at
Q, E at P, and R 2 at T. Call the points on the Plane
where the CIRCLES are Tangent Pi and F 2 . Because
intersecting tangents have the same length,
PiP = QP
F 2 P = TP.
Therefore,
PF X + PF 2 = QP + PT = QT,
which is a constant independent of P, so E is an ELLIPSE
with a = QT/2.
see also CONE, SPHERE
References
Honsberger, R. "Kepler's Conies." Ch. 9 in Mathematical
Plums (Ed. R. Honsberger). Washington, DC: Math. As-
soc. Amer., p. 170, 1979.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 40-44, 1991,
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 80-81, 1990.
Ogilvy, C. S. Excursions in Mathematics. New York: Dover,
pp. 68-69, 1994.
Danielson-Lanczos Lemma
Darling's Products 399
Danielson-Lanczos Lemma
The Discrete Fourier Transform of length N
(where N is Even) can be rewritten as the sum of two
Discrete Fourier Transforms, each of length N/2.
One is formed from the EVEN numbered points; the
other from the Odd numbered points. Denote the kth
point of the DISCRETE FOURIER TRANSFORM by F n .
Then
F n = Y, h*
2-Kink/N
fc=0
N/2-1
N/2-1
= E
-2irikn/(N/2)
hk+w n J2 e
-2irikn/(N/2)
T2k + 1
k=0
= F: + W n F°,
where W = e ~ 27ri/N and n = 0, . . . , N. This procedure
can be applied recursively to break up the N/2 even
and Odd points to their N/4 Even and Odd points.
If N is a Power of 2, this procedure breaks up the
original transform into lg N transforms of length 1. Each
transform of an individual point has F^ eo '" = fk for
some k. By reversing the patterns of evens and odds,
then letting e = and o = 1, the value of k in BINARY
is produced. This is the basis for the Fast Fourier
Transform.
see also Discrete Fourier Transform, Fast Four-
ier Transform, Fourier Transform
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. 407-411, 1989.
Darboux Integral
A variant of the Riemann Integral defined when the
Upper and Lower Integrals, taken as limits of the
Lower Sum
L(/;0;iV) = 2^m(/;<S r )-0(a; r -i)
and Upper Sum
n
T = l
are equal. Here, f(x) is a Real Function, <p(x) is
a monotonic increasing function with respect to which
the sum is taken, m(f;S) denotes the lower bound of
f(x) over the interval 5, and M(/; S) denotes the upper
bound.
see also Lower Integral, Lower Sum, Riemann In-
tegral, Upper Integral, Upper Sum
References
Kestelman, H. Modern Theories of Integration, 2nd rev. ed.
New York: Dover, p. 250, 1960.
Darboux-Stieltjes Integral
see Darboux Integral
Darboux Vector
The rotation VECTOR of the TRIHEDRON of a curve with
CURVATURE k ^ when a point moves along a curve
with unit SPEED. It is given by
D = tT + kB, (1)
where r is the TORSION, T the TANGENT VECTOR, and
B the BlNORMAL VECTOR. The Darboux vector field
satisfies
T = D x T
N = D x N
B = D x B.
(2)
(3)
(4)
see also BlNORMAL VECTOR, CURVATURE, TANGENT
Vector, Torsion (Differential Geometry)
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 151, 1993.
Darling's Products
A generalization of the HYPERGEOMETRIC FUNCTION
identity
2 F 1 (a,0; T , z) 2 F X (1 - a, 1 - (3; 2 - 7; z)
= 2F1 (a + 1-7,0 + 1-7; 2 -7;^) 2^1 (7 -a, 7-/3; 71*0
(1)
to the Generalized Hypergeometric Function
3i ? 2(a, 6, c;d, e;x). Darling's products are
a,/?,7;z
e-8'
i-a,i-/M- 7 ;*
2 - & 2 - e
a + l-$,/? + l-<5,7 + l-<*;2
2-<S,e+l-<S
X3F2
8 — a,£ — j3 y S — 7; z
8, 8 + 1 - e
+ 7 3F2
a + l-e,/? + l-e,7+l-e;z
2-€,tf + l-e
X3F2
e- a,e-/3,e-7;z
e,e + l-<5
(2)
and
(l-2) a+ < 3+7 - <5 - £ 3 F 2
3^2
a,/3,7;2
S,e
8 — a, 8 — (3,8 — 7; z
S, 8 + 1 - e
X3F2
e - a, e - (3, e - 7; z
e-l,e+l-£
+ -= 3F2
e - a, e- P,e -j;z
e,e + l-S
X 3F2
8 — a, 8 — (3,8 — 7; z
£-l,J + l-e
(3)
400
Dart
Dawson's Integral
which reduce to (1) when 7 = e — ► 00.
References
Bailey, W. N. "Darling's Theorems of Products." §10.3 in
Generalised Hypergeometric Series. Cambridge, England:
Cambridge University Press, pp. 88-92, 1935.
Dart
see Penrose Tiles
Darwin-de Sitter Spheroid
A Surface of Revolution of the form
r((f>) = a[l-esin 2 0- (|e 2 + A;) sin 2 (20)],
where fc is a second-order correction to the figure of a
rotating fluid.
see also Oblate Spheroid, Prolate Spheroid,
Spheroid
References
Zharkov, V. N. and Trubitsyn, V. P. Physics of Planetary
Interiors. Tucson, AZ: Pachart Publ. House, 1978.
Darwin's Expansions
Series expansions of the PARABOLIC CYLINDER FUNC-
TION U(a,x) and W(a,x). The formulas can be found
in Abramowitz and Stegun (1972).
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 689-690 and 694-695, 1972.
Data Structure
A formal structure for the organization of information.
Examples of data structures include the LIST, QUEUE,
Stack, and Tree.
Database
A database can be roughly defined as a structure con-
sisting of
1. A collection of information (the data),
2. A collection of queries that can be submitted, and
3. A collection of algorithms by which the structure
responds to queries, searches the data, and returns
the results.
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K, Peters, p. 48, 1996.
Daubechies Wavelet Filter
A Wavelet used for filtering signals. Daubechies (1988,
p. 980) has tabulated the numerical values up to order
p= 10.
References
Daubechies, I. "Orthonormal Bases of Compactly Supported
Wavelets." Comm. Pure Appl. Math. 41, 909-996, 1988.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Interpolation and Extrapolation." Ch. 3
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed, Cambridge, England: Cambridge
University Press, pp. 584-586, 1992.
Davenport- Schinzel Sequence
Form a sequence from an Alphabet of letters [1, n] such
that there are no consecutive letters and no alternating
subsequences of length greater than d. Then the se-
quence is a Davenport-Schinzel sequence if it has max-
imal length Nd(n). The value of Ni(n) is the trivial
sequence of Is: 1,1,1,... (Sloane's A000012). The val-
ues of N 2 (n) are the POSITIVE INTEGERS 1, 2, 3, 4, . . .
(Sloane's A000027). The values of N 3 (n) are the ODD
INTEGERS 1, 3, 5, 7, ... (Sloane's A005408). The first
nontrivial Davenport-Schinzel sequence N^n) is given
by 1, 4, 8, 12, 17, 22, 27, 32, ... (Sloane's A002004).
Additional sequences are given by Guy (1994, p. 221)
and Sloane.
References
Davenport, H. and Schinzel, A. "A Combinatorial Problem
Connected with Differential Equations." Amer. J. Math.
87, 684-690, 1965.
Guy, R. K. "Davenport-Schinzel Sequences." §E20 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 220-222, 1994.
Roselle, D. P. and Stanton, R. G. "Results of Davenport-
Schinzel Sequences." In Proc. Louisiana Conference on
Combinatorics, Graph Theory, and Computing. Louisiana
State University, Baton Rouge, March 1-5, 1970 (Ed.
R. C. Mullin, K. B. Reid, and D. P. Roselle). Winnipeg,
Manitoba: Utilitas Mathematica, pp. 249-267, 1960.
Sharir, M. and Agarwal, P. Davenport-Schinzel Sequences
and Their Geometric Applications. New York: Cambridge
University Press, 1995.
Sloane, N. J. A. Sequences A000012/M0003, A000027/
M0472, A002004/M3328, and A005408/M2400 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Dawson's Integral
0.4
0.2
7.5
5
2.5
An Integral which arises in computation of the Voigt
lineshape:
D(x)
f
Jo
e y dy.
(i)
Day of Week
It is sometimes generalized such that
D±(x) =e^ I e ±y2 dy,
Jo
giving
D + (x) = \\fne x erfi(ai)
D-(x) = ^e"'ett(x),
(2)
(3)
(4)
where erf (z) is the ERF function and ern(z) is the imag-
inary error function Erfi. D+(x) is illustrated in the
left figure above, and D~ (x) in the right figure. D+ has
a maximum at D'+(x) = 0, or
1 - y/ne'^x 2 erfi(a;) = 0, (5)
giving
D+ (0.9241388730) = 0.5410442246, (6)
and an inflection at D+{x) — 0, or
-2z + y/^e~ x2 (2x 2 - l)erfi(z) = 0, (7)
giving
£>+(l. 5019752683) = 0.4276866160. (8)
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 298, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Dawson's Integrals." §6.10 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 252-254, 1992.
Spanier, J. and Oldham, K. B. "Dawson's Integral." Ch. 42
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 405-410, 1987.
Day of Week
see Friday the Thirteenth, Weekday
de Bruijn Constant
Also called the Copson-de Bruijn Constant. It is
defined by
J^a n <cJ2
n=l
where
a n 2 + a n +i 2 + a n +2 2
c= 1.0164957714....
References
Copson, E. T. "Note on Series of Positive Terms." J. London
Math. Soc. 2, 9-12, 1927.
Copson, E. T. "Note on Series of Positive Terms." J. London
Math. Soc. 3, 49-51, 1928.
de Bruijn, N. G. Asymptotic Methods in Analysis. New York:
Dover, 1981.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/copson/copson.html.
de Bruijn-Newman Constant 401
de Bruijn Diagram
see de Bruijn Graph
de Bruijn Graph
A graph whose nodes are sequences of symbols from
some Alphabet and whose edges indicate the sequences
which might overlap.
References
Golomb, S. W. Shift Register Sequences. San Francisco, CA:
Holden-Day, 1967.
Ralston, A. "de Bruijn Sequences — A Model Example of the
Interaction of Discrete Mathematics and Computer Sci-
ence." Math. Mag. 55, 131-143, 1982.
de Bruijn-Newman Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let H be the Xi Function defined by
S(fe) = i(z 2 - i),r-*> 2 -4r(±z + i)C(* + §). (1)
S(z/2)/8 can be viewed as the FOURIER TRANSFORM of
the signal
*(t) = ^(27r 2 n 4 e 9t - STrnV'^-™ 2 * 4 ' (2)
for t € R > 0. Then denote the FOURIER TRANSFORM
of$(i)e At asH(\,z),
^[$(*)e A
H(X,z).
(3)
de Bruijn (1950) proved that H has only Real zeros
for A > 1/2. C. M. Newman (1976) proved that there
exists a constant A such that H has only REAL zeros
Iff A > A. The best current lower bound (Csordas et
al. 1993, 1994) is A > -5.895 x 10" 9 . The Riemann
Hypothesis is equivalent to the conjecture that A < 0.
References
Csordas, G.; Odlyzko, A.; Smith, W.; and Varga, R. S. "A
New Lehmer Pair of Zeros and a New Lower Bound for
the de Bruijn-Newman Constant." Elec. Trans. Numer.
Analysis 1, 104-111, 1993.
Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of
Zeros, the de Bruijn-Newman Constant and the Riemann
Hypothesis." Constr. Approx. 10, 107-129, 1994.
de Bruijn, N. G. "The Roots of Trigonometric Integrals."
Duke Math. J. 17, 197-226, 1950.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/dbnwm/dbnwm.html.
Newman, C. M. "Fourier Transforms with only Real Zeros."
Proc. Amer. Math. Soc. 61, 245-251, 1976.
402 de Bruijn Sequence
de Moivre's Identity
de Bruijn Sequence
The shortest sequence such that every string of length
n on the ALPHABET a occurs as a contiguous subrange
of the sequence described by a. Every de Bruijn se-
quence corresponds to an EULERIAN CYCLE on a "DE
Bruijn Graph." Surprisingly, it turns out that the
lexicographic sequence of LYNDON WORDS of lengths
Divisible by n gives the lexicographically smallest de
Bruijn sequence (Ruskey).
References
Ruskey, F. "Information on Necklaces, Lyndon Words, de
Bruijn Sequences." http://sue.csc.uvic.ca/-cos/inf/
neck/Necklacelnf o .html.
de Bruijn's Theorem
A box can be packed with a Harmonic Brick axabx
abc Iff the box has dimensions ap x abq x abcr for some
natural numbers p, <?, r (i.e., the box is a multiple of the
brick) .
see also Box-Packing Theorem, Conway Puzzle,
de Bruijn's Theorem, Klarner's Theorem
References
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 69-72, 1976.
de Jonquieres Theorem
The total number of groups of a g r N consisting in a point
of multiplicity ki, one of multiplicity &2, . . . , one of mul-
tiplicity k p , where
J2^ ki - !) = r >
(1)
(2)
(3)
and where ai points have one multiplicity, c*2 another
etc., and
II = kik2 • • -k p
Up(p - 1) • • • (p - p)
OLl\(X2 ] -' **
n
P- P
v^ an y^ d 2 u *]
2^/i dki ^ij dkidkj
p— p+l p— p + 2
Referenct
3S
(4)
Coolidge, J. L. A Treatise on Algebraic Plane Curves, New
York: Dover, p. 288, 1959.
de Jonquieres Transformation
A transformation which is of the same type as its inverse.
A de Jonquieres transformation is always factorable.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 203-204, 1959.
de la Loubere's Method
A method for constructing Magic SQUARES of Odd or-
der, also called the Siamese Method.
see also Magic Square
de Longchamps Point
The reflection of the ORTHOCENTER about the ClRCUM-
CENTER. This point is also the Orthocenter of the
Anticomplementary Triangle. It has Triangle
Center Function
a = cos A — cos B cos C.
It lies on the Euler LINE.
References
Altshiller-Court, N. "On the de Longchamps Circle of the
Triangle." Amer. Math. Monthly 33, 368-375, 1926.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Vandeghen, A. "Soddy's Circles and the de Longchamps
Point of a Triangle." Amer. Math. Monthly 71, 176-179,
1964.
de Mere's Problem
The probability of getting at least one "6" in four rolls
of a single 6-sided DIE is
1-(|) 4 = 0.518...,
(1)
which is slightly higher than the probability of at least
one double 6 in 24 throws,
1 -(i) 24 = 0.491.
(2)
de Mere suspected that (1) was higher than (2). He
posed the question to Pascal, who solved the problem
and proved de Mere correct.
see also DICE
References
Kraitchik, M. "A Dice Problem." §6.2 in Mathematical
Recreations. New York: W. W. Norton, pp. 118-119, 1942.
de Moivre's Identity
e i{n6) = (e ie ) n . (1)
Prom the EULER FORMULA it follows that
cos(n0) + zsin(n0) = (costf + 2sin0) n . (2)
A similar identity holds for the HYPERBOLIC FUNC-
TIONS,
(coshz + sinhz)" = cosh(nz) + sinh(nz). (3)
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 356-357, 1985.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 96-100, 1996.
de Moivre Number
de Polignac's Conjecture 403
de Moivre Number
A solution ( k = e 27Vik/d to the Cyclotomic Equation
where
X — 1.
The de Moivre numbers give the coordinates in the
Complex Plane of the Vertices of a regular Poly-
gon with d sides and unit RADIUS.
n de Moivre Numbers
2 ±1
3 1, |(-l±n/3)
4 ±l,±i
5 \,\(-\ + y/S±(\ + y/$)J*^i\,
l+\/5 _j_ y/5-y/E •
4 ^ 2 ^ l
6 ±l,±±(±l + 2\/3)
see a/so Cyclotomic Equation, Cyclotomic Poly-
nomial, Euclidean Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, 1996.
de Moivre-Laplace Theorem
The sum of those terms of the BINOMIAL SERIES of (p +
q) s for which the number of successes x falls between d\
and d2 is approximately
Q:
dt,
where
tl
t 2
=
d 1
1
2
■ sp
d 2
— sp
a
cr = yjspq.
Uspensky (1937) has shown that
«-£jf V " , * +! ^[< 1 -«5
-£ 2 /2
+ n,
where
|fl|< 0.12 + 0.18b - g | +e - te/ ,
(1)
(2)
(3)
(4)
*2
*1
(5)
(6)
for a > 5.
A Corollary states that the probability that x suc-
cesses in s trials will differ from the expected value sp
by more than d is
P 5 « 1 - 2 / 4>(t) dt
I
Jo
(7)
d+i
Uspensky (1937) showed that
Q Sl = P(\x - sp\ < d)
/•5i
7.
<j>(t) dt +
I-61.-62
(8)
*(*i) + ni, (9)
where
61 = (sq + d) - [sq + d\
6 2 = (sp + d)- [sp + d\
(10)
(11)
(12)
and
for a > 5.
N< awH +e . k/2]
(7
References
Uspensky, J. V. Introduction to Mathematical Probability.
New York: McGraw-Hill, 1937.
de Moivre's Quintic
x 5 + ax 3 + lax + b = 0.
see also QuiNTIC EQUATION
de Morgan's and Bertrand's Test
see Bertrand's Test
de Morgan's Duality Law
For every proposition involving logical addition and mul-
tiplication ("or" and "and"), there is a corresponding
proposition in which the words "addition" and "multi-
plication" are interchanged.
de Morgan's Laws
Let U represent "or" , n represent "and" , and ' represent
"not." Then, for two logical units E and F y
(E U F)' = E'nF f
(EHF)' = E'l)F'.
de Polignac's Conjecture
Every EVEN NUMBER is the difference of two consec-
utive PRIMES in infinitely many ways. If true, taking
the difference 2, this conjecture implies that there are
infinitely many TWIN PRIMES (Ball and Coxeter 1987).
The CONJECTURE has never been proven true or refuted.
see also Even Number, Twin Primes
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 64, 1987.
de Polignac, A. "Six propositions arithmologiques deduites
de crible d'Eratosthene." Nouv. Ann. Math. 8, 423-429,
1849.
404 de Rham Cohomology
Decagon
de Rham Cohomology
de Rham cohomology is a formal set-up for the analytic
problem: If you have a DIFFERENTIAL fc-FORM wona
Manifold M, is it the Exterior Derivative of an-
other Differential fc-FORM a/? Formally, if uj = duj'
then do; = 0. This is more commonly stated as dod — 0,
meaning that if a; is to be the EXTERIOR DERIVATIVE of
a DIFFERENTIAL A;-FORM, a NECESSARY condition that
uj must satisfy is that its EXTERIOR DERIVATIVE is zero.
de Rham cohomology gives a formalism that aims to
answer the question, "Are all differential A;-forms on a
Manifold with zero Exterior Derivative the Ex-
terior Derivatives of (k -f l)-forms?" In particular,
the kth de Rham cohomology vector space is defined to
be the space of all fc-forms with Exterior Derivative
0, modulo the space of all boundaries of (k + l)-forms.
This is the trivial Vector Space Iff the answer to our
question is yes.
The fundamental result about de Rham cohomology
is that it is a topological invariant of the MANIFOLD,
namely: the kth de Rham cohomology VECTOR SPACE
of a MANIFOLD M is canonically isomorphic to the
Alexander-Spanier Cohomology Vector Space
H k (M;W) (also called cohomology with compact sup-
port). In the case that M is compact, ALEXANDER-
Spanier Cohomology is exactly singular cohomology.
see also Alexander-Spanier Cohomology, Change
of Variables Theorem, Differential &-Form, Ex-
terior Derivative, Vector Space
de Sluze Conchoid
see Conchoid of de Sluze
de Sluze Pearls
see Pearls of Sluze
Debye's Asymptotic Representation
An asymptotic expansion for a HANKEL FUNCTION OF
the First Kind
H„{x) ~ —= exp{zx[cos a + (a — 7r/2) sin a]}
0r
Atv/4
X
+ (| + ^tan 2 a)
3e
3tH/4
2X 3
where
+ (155 + ^ tana +ll tan4 *)^f^- + •
V
— = sin a,
x
1 V ^ 3 1/2
1 > —v '
and
X = a /— xcos(|a)
see also Hankel Function of the First Kind
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1475,
1980.
Debye Functions
F
Jo
t n dt
e l -1
E
B 2k x 2
n 2(n+l) ^ (2k + n)(2k\)
(1)
where \x\ < 2k and B n are BERNOULLI NUMBERS.
t n dt _ <sr^ _ kx
fc=l
aT nx n ~ l
k + k 2
n(n - l)x n ~ 2 n!
+ — TV + ■■■ +
k 3 ■■■ fcn+1
where x > 0. The sum of these two integrals is
t n dt
f
Jo
e l - 1
n!<(n+l),
(2)
(3)
where ((z) is the Riemann Zeta Function.
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Debye Func-
tions." §27.1 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 998, 1972.
Decagon
The constructible regular 10-sided POLYGON with
Schlafli Symbol {10}. The Inradius r, Circum-
RADIUS i?, and AREA can be computed directly from
the formulas for a general regular POLYGON with side
length s and n = 10 sides,
r = |scot (^-\ = 1^25-10^5 (1)
tf-i S csc(^)= |(l + 75) 5 = 5 (2)
A= ±ns 2 cot(^) = § V / 5 + 2v / 5s 2 . (3)
Here, <p is the GOLDEN MEAN.
see also DECAGRAM, DODECAGON, TRIGONOMETRY
Values — 7r/l0, Undecagon
References
Dixon, R. Mathographics. New York: Dover, p. 18, 1991.
Decagonal Number
Decagonal Number
x + 10x 2 + 27a: 3 + 52z 4 + .
A FiGURATE NUMBER of the form 4n 2 - 3to. The first
few are 1, 10, 27, 52, 85, . . . (Sloane's A001107). The
Generating Function giving the decagonal numbers
is
x(7x + 1)
(1-z) 3
References
Sloane, N. J. A. Sequence A001107/M4690 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Decagram
The Star Polygon {™}-
see also DECAGON, STAR POLYGON
Decic Surface
A Surface which can be represented implicitly by a
Polynomial of degree 10 in x, y, and z. An example
is the Barth Decic.
see also BARTH DECIC, CUBIC SURFACE, QUADRATIC
Surface, Quartic Surface
Decidable
A "theory" in LOGIC is decidable if there is an Algo-
rithm that will decide on input <fi whether or not is a
Sentence true of the Field of Real Numbers R.
see also CHURCH'S THESIS, GODEL'S COMPLETE-
NESS Theorem, Godel's Incompleteness Theorem,
Kreisel Conjecture, Tarski's Theorem, Undecid-
able, Universal Statement
References
Kemeny, J. G. "Undecidable Problems of Elementary Num-
ber Theory." Math. Ann. 135, 160-169, 1958.
Decimal Expansion 405
Decillion
In the American system, 10 33 .
see also Large Number
Decimal
The base 10 notational system for representing Real
Numbers.
see also 10, BASE (NUMBER), BINARY, HEXADECIMAL,
Octal
References
Pappas, T. "The Evolution of Base Ten." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, pp. 2-3,
1989.
$ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Bases.m.
Decimal Expansion
The decimal expansion of a number is its representation
in base 10. For example, the decimal expansion of 25
is 625, of 7T is 3.14159. . . , and of 1/9 is 0.1111. . . .
If r = p/q has a finite decimal expansion, then
Q>1 . Q>2 .
r = h
10 10 2
.+
all0 Tl - 1 +a2l0 T,
10 n
2 + . . . + a n
10"
ailO^+aslO 71 "
' + ... + a n
2 Tl 5 Tl
FACTORING possible common multiples gives
= p
T 2 a 5^'
(1)
(2)
where p ^ (mod 2, 5). Therefore, the numbers with
finite decimal expansions are fractions of this form. The
number of decimals is given by max(a,/3). Numbers
which have a finite decimal expansion are called REGU-
LAR Numbers.
Any NONREGULAR fraction m/n is periodic, and has a
period A(n) independent of m, which is at most to — 1
Digits long. If n is Relatively Prime to 10, then the
period of m/n is a divisor of <j>(n) and has at most <j>(n)
Digits, where <f> is the Totient Function. When a
rational number m/n with (m, to) = 1 is expanded, the
period begins after s terms and has length t, where s
and t are the smallest numbers satisfying
10 2 = 10 a+t
(mod to) .
(3)
When to ^ (mod 2, 5), s = 0, and this becomes a
purely periodic decimal with
(4)
10* = 1 (mod n) .
n example, consider to = 84.
10° = 1 10 1 = 10 10 2 = 16
10 3 =
-8
10 4 =4 10 5 = 40 10 6 = -20
10 7 =
-32,
10 8 = 16
406 Decimal Expansion
Decimal Expansion
so s = 2, t = 6. The decimal representation is 1/84 =
0.01190476. When the DENOMINATOR of a fraction m/n
has the form n = no2 OL 6 (3 with (no, 10) = 1, then the
period begins after max(a,/3) terms and the length of
the period is the exponent to which 10 belongs (mod no),
i.e., the number x such that 10 x = 1 (mod no). If q is
Prime and X(q) is Even, then breaking the repeating
Digits into two equal halves and adding gives all 9s.
For example, 1/7 = 0.142857, and 142 + 857 = 999.
For 1/q with a PRIME DENOMINATOR other than 2 or 5,
all cycles n/q have the same length (Conway and Guy
1996).
If n is a Prime and 10 is a Primitive Root of n, then
the period A(n) of the repeating decimal 1/n is given by
A(n) = <£(n),
(5)
where <j>{n) is the Totient FUNCTION. Furthermore,
the decimal expansions for p/n, with p=l,2,...,n — 1
have periods of length n — 1 and differ only by a cyclic
permutation. Such numbers are called Long Primes
by Conway and Guy (1996). An equivalent definition is
that
10* = 1 (mod n) (6)
for i = n — 1 and no i less than this. In other words, a
Necessary (but not Sufficient) condition is that the
number 9^-1 (where R n is a Repunit) is DIVISIBLE
by n, which means that R n is Divisible by n.
The first few numbers with maximal decimal expansions,
called Full Reptend Primes, are 7, 17, 19, 23, 29,
47, 59, 61, 97, 109, 113, 131, 149, 167, ... (Sloane's
A001913). The decimals corresponding to these are
called Cyclic Numbers. No general method is known
for finding FULL REPTEND Primes. Artin conjectured
that Artin's Constant C = 0.3739558136... is the
fraction of Primes p for with 1/p has decimal maximal
period (Conway and Guy 1996). D. Lehmer has gen-
eralized this conjecture to other bases, obtaining values
which are small rational multiples of C.
To find DENOMINATORS with short periods, note that
10 1
-1
= 3 2
10 2
-1
= 3 2
11
10 3
-1
= 3 3
37
10 4
-1
= 3 2
11-101
10 5
-1
= 3 2
41 • 271
10 6
-1
= 3 3
7 -11 -13 -37
10 7
-1
= 3 2
239 ■ 4649
10 8
-1
= 3 2
11 -73 -101 -137
10 9
- 1
= 3 4
37 • 333667
10 10
-1
= 3 2
11 ■ 41 • 271 ■ 9091
10 11
- 1
= 3 2
21649-513239
10 12
- 1
= 3 3
7 -11 -13 -37 -101 -9901
The period of a fraction with DENOMINATOR equal to a
Prime Factor above is therefore the Power of 10 in
which the factor first appears. For example, 37 appears
in the factorization of 10 — 1 and 10 — 1, so its period
is 3. Multiplication of any FACTOR by a 2 a 5^ still gives
the same period as the FACTOR alone. A DENOMINA-
TOR obtained by a multiplication of two FACTORS has
a period equal to the first POWER of 10 in which both
Factors appear. The following table gives the Primes
having small periods (Sloane's A046106, A046107, and
A046108; Ogilvy and Anderson 1988).
period
primes
1
3
2
11
3
37
4
101
5
41, 271
6
7, 13
7
239, 4649
8
73, 137
9
333667
10
9091
11
21649, 513239
12
9901
13
53, 79, 265371653
14
909091
15
31, 2906161
16
17, 5882353
17
2071723, 5363222357
18
19, 52579
19
1111111111111111111
20
3541, 27961
A table of the periods e of small Primes other than the
special p = 5, for which the decimal expansion is not
periodic, follows (Sloane's A002371).
V
e
V
e
P
e
3
1
31
15
67
33
7
6
37
3
71
35
11
2
41
5
73
8
13
6
43
21
79
13
17
16
47
46
83
41
19
18
53
13
89
44
23
22
59
58
97
96
29
28
61
60
101
4
Shanks (1873ab) computed the periods for all Primes
f£\up to 120,000 and published those up to 29,989.
see also FRACTION, MlDY'S THEOREM, REPEATING
Decimal
References
Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci-
mals." In The Book of Numbers. New York: Springer-
Verlag, pp. 157-163 and 166-171, 1996.
Das, R. C. "On Bose Numbers," Amer. Math. Monthly 56,
87-89, 1949.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 159-
179, 1952.
Decimal Period
Dedekind's Axiom 407
Lehmer, D. H. "A Note on Primitive Roots." Scripta Math.
26, 117-119, 1963.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, p. 60, 1988.
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 147-163,
1957.
Rao, K. S. "A Note on the Recurring Period of the Reciprocal
of an Odd Number." Amer. Math. Monthly 62, 484-487,
1955.
Shanks, W. "On the Number of Figures in the Period of the
Reciprocal of Every Prime Number Below 20,000." Proc.
Roy. Soc. London 22, 200, 1873a.
Shanks, W. "On the Number of Figures in the Period of the
Reciprocal of Every Prime Number Between 20,000 and
30,000." Proc. Roy. Soc. London 22, 384, 1873b.
Shiller, J. K. "A Theorem in the Decimal Representation of
Rationals." Amer. Math. Monthly 66, 797-798, 1959.
Sloane, N. J. A. Sequences A001913/M4353 and A002371/
M4050 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Decimal Period
see Decimal Expansion
Decision Problem
Does there exist an ALGORITHM for deciding whether
or not a specific mathematical assertion does or does
not have a proof? The decision problem is also known
as the Entscheidungsproblem (which, not so coinci-
dentally, is German for "decision problem"). Using the
concept of the Turing Machine, Turing showed the an-
swer to be Negative for elementary Number Theory.
J. Robinson and Tarski showed the decision problem is
undecidable for arbitrary FIELDS.
Decision Theory
A branch of GAME THEORY dealing with strategies to
maximize the outcome of a given process in the face of
uncertain conditions.
see also NEWCOMB'S PARADOX, OPERATIONS RE-
SEARCH, Prisoner's Dilemma
Decomposition
A rewriting of a given quantity (e.g., a Matrix) in terms
of a combination of "simpler" quantities.
see also Cholesky Decomposition, Connected Sum
Decomposition, Jaco-Shalen-Johannson Torus
Decomposition, LU Decomposition, QR Decom-
position, Singular Value Decomposition
Deconvolution
The inversion of a Convolution equation, i.e.
lution for / of an equation of the form
the SO-
given g and h, where e is the Noise and * denotes the
CONVOLUTION. Deconvolution is ill-posed and will usu-
ally not have a unique solution even in the absence of
Noise,
Linear deconvolution Algorithms include Inverse
Filtering and Wiener Filtering. Nonlinear Algo-
rithms include the CLEAN ALGORITHM, MAXIMUM
Entropy Method, and LUCY.
see also CLEAN Algorithm, Convolution, LUCY,
Maximum Entropy Method, Wiener Filter
References
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. 167-183, 1989.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Convolution and Deconvolution Using the
FFT." §13.1 in Numerical Recipes in FORTRAN: The Art
of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 531-537, 1992.
Decreasing Function
A function f(x) decreases on an INTERVAL i" if f(b) <
f(a) for all b > a, where a, b £ I. Conversely, a function
f(x) increases on an Interval I if /(&) > f(a) for all
b > a with a,b E I.
If the Derivative f'(x) of a Continuous Function
f(x) satisfies f(x) < on an Open Interval (a, 6),
then f(x) is decreasing on (a,b). However, a function
may decrease on an interval without having a derivative
defined at all points. For example, the function -x 1/3
is decreasing everywhere, including the origin x = 0,
despite the fact that the DERIVATIVE is not defined at
that point.
see also Derivative, Increasing Function, Nonde-
creasing Function, Nonincreasing Function
Decreasing Sequence
A Sequence {ai, a 2 , . . ■} for which ai > a 2 > . . ..
see also INCREASING SEQUENCE
Decreasing Series
A Series si, s 2 , . . . for which si > s 2 >
Dedekind's Axiom
For every partition of all the points on a line into two
nonempty Sets such that no point of either lies between
two points of the other, there is a point of one Set which
lies between every other point of that Set and every
point of the other Set.
/ *g = h + e,
408
Dedekind Cut
Dedekind Cut
A set partition of the Rational Numbers into two
nonempty subsets Si and S2 such that all members of
Si are less than those of £2 and such that 5i has no
greatest member. Real Numbers can be defined using
either Dedekind cuts or Cauchy Sequences.
see also CANTOR-DEDEKIND AXIOM, CAUCHY SE-
QUENCE
References
Courant, R. and Robbins, H. "Alternative Methods of Defin-
ing Irrational Numbers. Dedekind Cuts." §2.2.6 in What
is Mathematics?: An Elementary Approach to Ideas and
Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 71-72, 1996.
Dedekind Eta Function
Re[DedekindEta z]
Im[DedekindEta z]
IDedekindEta :
Let
2iriz
q = e~~, (1)
then the Dedekind eta function is defined by
V (z) = q 1/2 *l[(l- q n,
(2)
which can be written as
V (z) = q 1/24 1 1 + g(-i)-v (3 "- 1)/2 + q «*»+W] I
(3)
(Weber 1902, pp. 85 and 112; Atkin and Morain 1993).
7? is a Modular Form. Letting £ 2 4 = 2 2irl/24 be a
Root of Unity, rj(z) satisfies
77(2 + 1)^24^)
(4)
(5)
(Weber 1902, p. 113; Atkin and Morain 1993).
see also Dirichlet Eta Function, Theta Function,
Weber Functions
References
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math, Comput. 61, 29-68, 1993.
Weber, H. Lehrbuch der Algebra, Vols, I-II. New York:
Chelsea, 1902.
Dedekind Sum
Dedekind Function
il>(n)=n JJ (1+P _1 ),
distinct prime
factors p of n
where the Product is over the distinct Prime Factors
of n. The first few values are 1, 3, 4, 6, 6, 12, 8, 12, 12,
18, ... (Sloane's A001615).
see also DEDEKIND ETA FUNCTION, EULER PRODUCT,
Totient Function
References
Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field
Theory and Complex Multiplication. New York: Wiley,
p. 228, 1997.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 96, 1994.
Sloane, N. J. A. Sequence A001615/M2315 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Dedekind's Problem
The determination of the number of monotone
Boolean Functions of n variables is called Dedekind's
problem.
Dedekind Ring
A abstract commutative RING in which every NONZERO
Ideal is a unique product of Prime Ideals.
Dedekind Sum
Given RELATIVELY PRIME INTEGERS p and q, the
Dedekind sum is defined by
X — 1
where
Dedekind sums obey 2-term
(2)
-(P, fl ) + .fa,p) = -| + ^(f + | + ^]. (3)
and 3-term
s(bc\a) + s(ca\b) + s(ab\c) = -1 + -L (£ + - + £)
(4)
reciprocity laws, where a, 6, c are pairwise COPRIME and
aa = 1 (mod b)
bb r = 1 (mod c)
cc = 1 (mod a) .
(5)
(6)
(7)
Deducible
Definite Integral 409
Let p, g, u, v G N with (p, q) = (it, v) = 1 (i.e., are
pairwise RELATIVELY Prime), then the Dedekind sums
also satisfy
Deficient Number
Numbers which are not PERFECT and for which
s(p,q) + s(u, v) = s(pu — qv 7 pv -f qu)
■5U + 5 + 5'- (8)
where t = pv + gu, and u' , v' are any INTEGERS such
that uu + vv r = 1 (Pommersheim 1993).
References
Pommersheim, J. "Toric Varieties, Lattice Points, and
Dedekind Sums." Math. Ann. 295, 1-24, 1993.
Deducible
If q is logically deducible from p, this is written p h q.
Deep Theorem
Qualitatively, a deep theorem is a theorem whose proof
is long, complicated, difficult, or appears to involve
branches of mathematics which are not obviously related
to the theorem itself (Shanks 1993). Shanks (1993) cites
the Quadratic Reciprocity Theorem as an example
of a deep theorem.
see also THEOREM
References
Shanks, D. "Is the Quadratic Reciprocity Law a Deep Theo-
rem?" §2,25 in Solved and Unsolved Problems in Number
Theory, 4th ed. New York: Chelsea, pp. 64-66, 1993.
Defective Matrix
A Matrix whose Eigenvectors are not Complete.
s(N) = er(N) - N < N,
or equivalently
<j(n) < 2n,
where a(N) is the Divisor Function. Deficient num-
bers are sometimes called Defective Numbers (Singh
1997). Primes, Powers of Primes, and any divisors
of a Perfect or deficient number are all deficient. The
first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11,
13, 14, 15, 16, 17, 19, 21, 22, 23, . . . (Sloane's A002855).
see also Abundant Number, Least Deficient Num-
ber, Perfect Number
References
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 3—33,
1952.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 45, 1994.
Singh, S. FermaVs Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem. New York:
Walker, p. 11, 1997.
Sloane, N. J. A. Sequence A002855/M0514 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Definable Set
An Analytic, Borel, or Coanalytic Set.
Defined
If A and B are equal by definition (i.e., A is defined
as 5), then this is written symbolically as A = B or
A:=B.
Defective Number
see Deficient Number
Deficiency
The deficiency of a BINOMIAL COEFFICIENT ( n £ fc ) with
k < n as the number of i for which bi = 1, where
a>ibi,
1 < i < &, the Prime factors of bi are > &, and Y\ a i —
k\, where hi is the Factorial.
see also Abundance
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 84-85, 1994.
Definite Integral
An Integral
«/ a
f(x) dx
with upper and lower limits. The first Fundamental
Theorem of Calculus allows definite integrals to be
computed in terms of INDEFINITE INTEGRALS, since if
F is the Indefinite Integral for f(x), then
/'
J a
f(x)dx = F(b)-F(a).
see also Calculus, Fundamental Theorems of
Calculus, Indefinite Integral, Integral
410
Degenerate
Dehn Invariant
Degenerate
A limiting case in which a class of object changes its na-
ture so as to belong to another, usually simpler, class.
For example, the POINT is a degenerate case of the Cir-
cle as the RADIUS approaches 0, and the CIRCLE is
a degenerate form of an Ellipse as the Eccentric-
ity approaches 0. Another example is the two identical
ROOTS of the second-order Polynomial (x-1) 2 . Since
the n ROOTS of an nth degree POLYNOMIAL are usually
distinct, Roots which coincide are said to be degener-
ate. Degenerate cases often require special treatment in
numerical and analytical solutions. For example, a sim-
ple search for both ROOTS of the above equation would
find only a single one: 1
The word degenerate also has several very specific and
technical meanings in different branches of mathematics.
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed, Or-
lando, FL: Academic Press, pp. 513-514, 1985.
Degree
The word "degree" has many meanings in mathematics.
The most common meaning is the unit of Angle mea-
sure defined such that an entire rotation is 360°. This
unit harks back to the Babylonians, who used a base 60
number system. 360° likely arises from the Babylonian
year, which was composed of 360 days (12 months of 30
days each). The degree is subdivided into 60 Minutes
per degree, and 60 SECONDS per MINUTE.
see also Arc Minute, Arc Second, Degree of
Freedom, Degree (Map), Degree (Polynomial),
Degree (Vertex), Indegree, Local Degree, Out-
degree
Degree (Algebraic Surface)
see Order (Algebraic Surface)
Degree of Freedom
The number of degrees of freedom in a problem, distri-
bution, etc., is the number of parameters which may be
independently varied.
see also Likelihood Ratio
Degree (Map)
Let / : M 4 JV be a Map between two compact,
connected, oriented n-D MANIFOLDS without boundary.
Then / induces a HOMEOMORPHISM /* from the HO-
MOLOGY GROUPS H n (M) to H n (N) t both canonically
isomorphic to the INTEGERS, and so /* can be thought
of as a Homeomorphism of the Integers. The Inte-
ger d(f) to which the number 1 gets sent is called the
degree of the MAP /.
There is an easy way to compute d{f) if the MANIFOLDS
involved are smooth. Let x G N, and approximate /
by a smooth map HOMOTOPIC to / such that a; is a
"regular value" of / (which exist and are everywhere by
Sard's Theorem). By the Implicit Function The-
orem, each point in / _1 (x) has a NEIGHBORHOOD such
that / restricted to it is a DlFFEOMORPHlSM. If the
DIFFEOMORPHISM is orientation preserving, assign it the
number +1, and if it is orientation reversing, assign it
the number — 1. Add up all the numbers for all the
points in f~ l (x), and that is the d(f), the degree of
/. One reason why the degree of a map is important is
because it is a HOMOTOPY invariant. A sharper result
states that two self-maps of the n-sphere are homotopic
Iff they have the same degree. This is equivalent to the
result that the nth HOMOTOPY GROUP of the n-SPHERE
is the set Z of INTEGERS. The ISOMORPHISM is given
by taking the degree of any representation.
One important application of the degree concept is that
homotopy classes of maps from n-spheres to n-spheres
are classified by their degree (there is exactly one homo-
topy class of maps for every INTEGER n, and n is the
degree of those maps).
Degree (Polynomial)
see Order (Polynomial)
Degree Sequence
Given an (undirected) Graph, a degree sequence is a
monotonic nonincreasing sequence of the degrees of its
VERTICES. A degree sequence is said to be fc-connected
if there exists some fc-CONNECTED GRAPH correspond-
ing to the degree sequence. For example, while the de-
gree sequence {1, 2, 1} is 1- but not 2-connected, {2, 2,
2} is 2-connected. The number of degree sequences for
n = 1, 2, ... is given by 1, 2, 4, 11, 31, 102, . . . (Sloane's
A004251).
see also GRAPHICAL PARTITION
References
Ruskey, F. "Information on Degree Sequences." http://sue
. esc. uvic,ca/-cos/inf /nump/DegreeSequences. html.
Ruskey, F.; Cohen, R.; Eades, P.; and Scott, A. "Alley CATs
in Search of Good Homes." Congres. Numer. 102, 97-110,
1994.
Sloane, N. J. A. Sequence A004251/M1250 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Degree (Vertex)
see Vertex Degree
Dehn Invariant
An invariant defined using the angles of a 3-D POLYHE-
DRON. It remains constant under solid DISSECTION and
reassembly. However, solids with the same volume can
have different Dehn invariants. Two Polyhedra can
be dissected into each other only if they have the same
volume and the same Dehn invariant.
see also Dissection, Ehrhart Polynomial
Dehn's Lemma
Delian Constant 411
Dehn's Lemma
If you have an embedding of a 1-Sphere in a 3-
MANIFOLD which exists continuously over the 2-DlSK,
then it also extends over the Disk as an embedding.
It was proposed by Dehn in 1910, but a correct proof
was not obtained until the work of Papakyriakopoulos
(1957ab).
References
Hempel, J. 3- Manifolds. Princeton, NJ: Princeton University
Press, 1976.
Papakyriakopoulos, C. D. "On Dehn's Lemma and the As-
phericity of Knots." Proc. Nat. Acad. Sci. USA 43, 169-
172, 1957a.
Papakyriakopoulos, C. D. "On Dehn's Lemma and the As-
phericity of Knots." Ann. Math. 66, 1-26, 1957.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 100-101, 1976.
Dehn Surgery
The operation of drilling a tubular Neighborhood of a
Knot K in S 3 and then gluing in a solid TORUS so that
its meridian curve goes to a (p, g)-curve on the TORUS
boundary of the Knot exterior. Every compact con-
nected 3-MANIF0LD comes from Dehn surgery on a Link
inS 3 .
see also KlRBY CALCULUS
References
Adams, C, C, The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 260, 1994.
Del
see Gradient
Del Pezzo Surface
A Surface which is related to Cayley Numbers.
References
Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York:
Dover, p. 211, 1973.
Hunt, B. "Del Pezzo Surfaces." §4.1.4 in The Geometry of
Some Special Arithmetic Quotients. New York: Springer-
Verlag, pp. 128-129, 1996.
Delannoy Number
The Delannoy numbers are defined by
D(a t b) = D(a - 1,6) + D{a,b- 1) -f D{a - 1,6 - 1),
where jD(0, 0) = 1. They are the number of lattice paths
from (0,0) to (6, a) in which only east (1, 0), north (0,
1), and northeast (1,1) steps are allowed (i.e, — >, 1\ and
233B
Z
Z
z
For n = a = 6, the Delannoy numbers are the number
of "king walks"
£)(n,n) = P„(3),
where P n (x) is a Legendre Polynomial (Moser 1955,
Vardi 1991). Another expression is
k=o \ / \
n \ ( n + h
2 Fi(-n,n + l;l,-l),
where (*) is a BINOMIAL COEFFICIENT and
2 Fi(a,b\c\z) is a Hypergeometric Function. The
values of D(n t n) for n = 1, 2, ... are 3, 13, 63, 321,
1683, 8989, 48639, . . . (Sloane's A001850).
The Schroder Numbers bear the same relation to the
Delannoy numbers as the Catalan Numbers do to the
Binomial Coefficients.
see also BINOMIAL COEFFICIENT, CATALAN NUMBER,
Motzkin Number, Schroder Number
References
Sloane, N. J. A. Sequence A001850/M2942 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Delaunay Triangulation
The Nerve of the cells in a Voronoi Diagram, which
is the triangular of the CONVEX HULL of the points in
the diagram. The Delaunay triangulation of a VORONOI
DIAGRAM in M is the diagram's planar dual.
see also TRIANGULATION
Delian Constant
The number 2 1/3 (the Cube Root of 2) which is to be
constructed in the Cube DUPLICATION problem. This
number is not a Euclidean Number although it is an
Algebraic of third degree.
References
Conway, J. H. and Guy, R. K. "Three Greek Problems."
In The Book of Numbers. New York: Springer- Verlag,
pp. 192-194, 1996.
412 Delian Problem
Delian Problem
see Cube Duplication
Delta Amplitude
Given an AMPLITUDE </> and a MODULUS m in an EL-
LIPTIC Integral,
A(0) = y/l -msin 2 <p.
see also Amplitude, Elliptic Integral, Modulus
(Elliptic Integral)
Delta Curve
A curve which can be turned continuously inside an
Equilateral Triangle. There are an infinite num-
ber of delta curves, but the simplest are the CIRCLE and
lens-shaped A-biangle. All the A curves of height h have
the same PERIMETER 2irh/3. Also, at each position of
a A curve turning in an EQUILATERAL TRIANGLE, the
perpendiculars to the sides at the points of contact are
Concurrent at the instantaneous center of rotation.
see also Reuleaux Triangle
References
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 56-59, 1973.
Delta Function
Defined as the limit of a class of DELTA SEQUENCES.
Sometimes called the Impulse Symbol. The most com-
monly used (equivalent) definitions are
j:/ \ _ r * sin[(7i+ \)x]
S(x)= hm — — yH; — (1)
(the so-called Dirichlet Kernel) and
sin(nx)
S(x) = lim
n— foo TTX
= i r e- ikx dk
«/ — oo
i r -<■
dk
(2)
(3)
(4)
(5)
where T is the FOURIER TRANSFORM. Some identities
include
6(x - a) = (6)
for x ^ a,
pa + c
J '
o a — e
S(x — a)dx = 1, (7)
where e is any POSITIVE number, and
f(x)S(x-a)dx = f(a) (8)
J — c
Delta Function
/oo
f{x)S f (x - a)dx - -/'(a) (9)
-oo
x / f(x)S(x — xo) dx = xo I f(x)5(x — xo) dx (10)
/oo
8'{a-x)f(x)dx = f{x) (11)
-oo
/oo
\8'{x)\dx = oo (12)
-oo
x 2 6'(x) =
S'(-x) = -S'(x)
xS'(x) = —5(x).
(13)
(14)
(15)
(15) can be established using INTEGRATION BY PARTS
as follows:
/f(x)x5'(x)dx = — / S(x)-—[xf(x)]dx
J dx
= - f 5[f(x) + xf(x)]dx
= - / f(x)S(x)dx. (16)
Additional identities are
S(ax) = -5(x)
(17)
_1_
2a l
S(x 2 - a 2 ) = ~[S(x + a) + 5(x - a)] (18)
i
where the XiS are the ROOTS of g. For example, examine
S(x 2 + x - 2) = <5[(z - l)(x + 2)]. (20)
Theng'(x) = 2x + l, so p'(xi) = p'(l) = 3 ands'(z 2 ) =
<?'( — 2) = —3, and we have
S(x 2 + x - 2) = f <5(z - 1) + |tf(x + 2). (21)
A Fourier Series expansion of 5(x — a) gives
i r 1
a n = — I 6(x — a) cos(nx) dx = — cos(no) (22)
7T / 7T
«/ — 7T
«/ — 7T
b n — — I $( x — a ) sin(nx) dx = — sin(na), (23)
7T
SO
oo
S(x — a) = 1 y [cos(na) cos(nx) + sin(na) sin(nx)]
n=l
oo
= ir- + ~ Y^ cos[n(x - a)]. (24)
2tt n *-^
Delta Sequence
Deltahedron 413
The Fourier Transform of the delta function is
-2-nikx^f^ _ \ j m _ — 2-jrik.XQ
F[5{x - Xq)]
f
8(x — xq) dx = e
(25)
Delta functions can also be defined in 2-D, so that in
2-D Cartesian Coordinates
S 2 (x -x ,y - y ) = S(x - x )S(y - y ), (26)
and in 3-D, so that in 3-D CARTESIAN COORDINATES
5 3 (x-x ,y~yo,z-zo) = 5{x - x )8(y - y )S(z - z ),
in Cylindrical Coordinates
S(r)6(z)
(27)
(28)
VI /
and in SPHERICAL COORDINATES,
A series expansion in CYLINDRICAL COORDINATES gives
S 3 (ri - r 2 ) = —6(ri - r 2 )S(<f>i - <t>2)S(zi - z 2 )
(29)
= -^"^) 27r
rn= — oo
1 °° 1 f°°
Z_ V^ e irn(<t>i-4> 2 ) Ji_ / e^* 1- * 2 ^ dk
J — oo
(30)
The delta function also obeys the so-called Sifting
Property
/
/(y)«S(x-y)dy = /( X ).
(31)
see also DELTA SEQUENCE, DOUBLET FUNCTION,
Fourier Transform — Delta Function
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 481-485, 1985.
Spanier, J. and Oldham, K. B. "The Dirac Delta Function
S(x — a)." Ch. 10 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 79-82, 1987.
Delta Sequence
A SEQUENCE of strongly peaked functions for which
lim /
n— >oo /
so that in the limit as n — > oo, the sequences become
Delta Functions. Examples include
(2)
(3)
(4)
(5)
(6)
(7)
*n sin \^xj
where (8) is known as the Dirichlet KERNEL.
Delta Variation
see Variation
Deltahedron
A semiregular Polyhedron whose faces are all Equi-
lateral Triangles. There are an infinite number of
deltahedra, but only eight convex ones (Freudenthal and
van der Waerden 1947). They have 4, 6, 8, 10, 12, 14,
16, and 20 faces. These are summarized in the table
below, and illustrated in the following figures.
$n(x) =
(0 *<~L
U x>±
=
n -n 2 x 2
V7T
=
n . , N sin(ncc)
— smcfaa:) = — -
7T 7VX
i „inx _ — inx
1 e — e
7TX 2i
=
o 1 - W X T
2tnx L J ~ n
=
-!- / e ixt dt
2?r /
J —n
1 sin [(n + £) x]
n
Name
4
tetrahedron
6
triangular dipyramid
8
octahedron
10
pentagonal dipyramid
12
snub disphenoid
14
triaugmented triangular prism
16
gyroelongated square dipyramid
20
icosahedron
S n {x)f(x)dx = f(n)
(i)
The Stella OCTANGULA is a concave deltahedron with
24 sides:
414
Deltoid
Deltoid
Another with 60 faces is a "caved in" DODECAHEDRON
which is ICOSAHEDRON STELLATION J 2 0-
Cundy (1952) identifies 17 concave deltahedra with two
kinds of Vertices.
see also Gyroelongated Square Dipyramid, Icos-
ahedron, Octahedron, Pentagonal Dipyramid,
Snub Disphenoid Tetrahedron, Triangular Di-
pyramid, Triaugmented Triangular Prism
References
Cundy, H. M. "Deltahedra." Math. Gaz. 36, 263-266, 1952.
Preudenthal, H. and van der Waerden, B. L. "On an Assertion
of Euclid." Simon Stevin 25, 115-121, 1947.
Gardner, M. Fractal Music, HyperCards, and More: Math-
ematical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 40, 53, and 58-60, 1992.
Pugh, A. Polyhedra: A Visual Approach. Berkeley, CA: Uni-
versity of California Press, pp. 35-36, 1976.
Deltoid
.*■*
~~
—
-
""""^
/
/
/
/
/
/
\
\
\
\
\^ \
^^-> 1
1
\
^^* \
\
/^ 1
/
\
/
\
/
\
/
\
y
-».
*-■
—
s-
A 3-cusped Hypocycloid, also called a Tricuspoid,
which has n = a/b — 3 or 3/2, where a is the Radius
of the large fixed CIRCLE and b is the RADIUS of the
small rolling CIRCLE. The deltoid was first considered
by Euler in 1745 in connection with an optical prob-
lem. It was also investigated by Steiner in 1856 and
is sometimes called Steiner's Hypocycloid (MacTu-
tor Archive). The equation of the deltoid is obtained
by setting n = 3 in the equation of the HYPOCYCLOID,
yielding the parametric equations
»= [§cos0- |cos(20)]a = 26cos0 + 6cos(20) (1)
y = [|sin0+ I s'm(2(j))]a = 2bsin<t> - 6sin(20). (2)
The Arc Length, Curvature, and Tangential An-
gle are
s(t) = 4 f | sin(|t')| df = f sin 2 (f t) (3)
Jo
«(t) = -|csc(|t) (4)
<f>(t) = -\t. (5)
As usual, care must be taken in the evaluation of s(t)
for t > 27r/3. Since the form given above comes from an
integral involving the ABSOLUTE VALUE of a function,
it must be monotonic increasing. Each branch can be
treated correctly by defining
»=l£J +1 « (6)
where [xj is the Floor Function, giving the formula
a(t) = (-l) 1+ l" < mod 2 "f sin 2 (|t) + f |>J • W
The total Arc Length is computed from the general
Hypocycloid equation
8a(n - 1)
With n = 3, this gives
s 3 = fa.
The Area is given by
with n = 3,
An = (» -D("- 2) TO ,
n- 2
A 3 = l-rra
(8)
(9)
(10)
(11)
The length of the tangent to the tricuspoid, measured
between the two points P, Q in which it cuts the curve
again, is constant and equal to 4a. If you draw Tan-
gents at P and Q, they are at Right Angles.
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 53, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 131-135, 1972.
Deltoid Caustic
Lee, X. "Deltoid." http://www.best.com/~xah/Special
PlaneCurves-dir/Deltoid_dir/deltoid.html.
Lockwood, E. H. "The Deltoid." Ch. 8 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 72-
79, 1967.
Macbeth, A. M. "The Deltoid, I." Eureka 10, 20-23, 1948.
Macbeth, A. M. "The Deltoid, II." Eureka 11, 26-29, 1949.
Macbeth, A. M. "The Deltoid, III." Eureka 12, 5-6, 1950.
MacTutor History of Mathematics Archive. "Tricuspoid."
http: //www-groups .dcs . st-and, ac ,uk/ -history/Curves
/Tricuspoid. html.
Yates, R. C. "Deltoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 71-74,
1952.
Deltoid Caustic
The caustic of the Deltoid when the rays are Parallel
in any direction is an ASTROID.
Deltoid Evolute
A Hypocycloid Evolute for n
TOID scaled by a factor n/(n — 2) =
1/(2 • 3) = 1/6 of a turn.
Deltoid Involute
= 3 is another Del-
3/1 = 3 and rotated
V
f-^
1^
4
' -1
1/
y
A Hypocycloid Involute for n =
TOID scaled by a factor (n — 2)/n
1/(2 - 3) = 1/6 of a turn.
Deltoid Pedal Curve
3 is another DEL-
= 1/3 and rotated
-fe>
The Pedal Curve for a Deltoid with the Pedal
Point at the Cusp is a Folium. For the Pedal Point
at the Cusp (Negative ^-intercept), it is a Bifolium.
At the center, or anywhere on the inscribed Equilat-
eral Triangle, it is a Trifolium.
Deltoidal Icositetrahedron
Deltoid Radial Curve
415
The Trifolium
x = £o + 4a cos — 4a cos(20)
y = 2/o + 4a sin + 4asin(20).
Deltoidal Hexecontahedron
The Dual Polyhedron of the Rhombicosidodeca-
hedron.
Deltoidal Icositetrahedron
416
Demlo Number
Denumerably Infinite
The Dual Polyhedron of the Small Rhombicub-
OCTAHEDRON. It is also called the TRAPEZOIDAL ICOS-
ITETRAHEDRON.
Demlo Number
The initially PALINDROMIC NUMBERS 1, 121, 12321,
1234321, 123454321, ... (Sloane's A002477). For the
first through ninth terms, the sequence is given by the
Generating Function
10x+l
(x- l)(10aj-l)(100x- 1)
1 + 121a + 12321x 2 + 1234321a 3 + ,
(Plouffe 1992, Sloane and Plouffe 1995). The definition
of this sequence is slightly ambiguous from the tenth
term on.
see also Consecutive Number Sequences, Palin-
dromic Number
References
Kaprekar, D. R. "On Wonderful Demlo Numbers." Math.
Student 6, 68-70, 1938.
Plouffe, S. "Approximations de Series Generatrices et
quelques conjectures." Montreal, Canada: Universite du
Quebec a Montreal, Memoire de Maitrise, UQAM, 1992.
Sloane, N. J. A. Sequence A00247T/M5386 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Dendrite Fractal
A Julia Set with c = i.
Denjoy Integral
A type of INTEGRAL which is an extension of both
the Riemann Integral and the Lebesgue Integral.
The original Denjoy integral is now called a Denjoy inte-
gral "in the restricted sense," and a more general type is
now called a Denjoy integral "in the wider sense." The
independently discovered PERON INTEGRAL turns out to
be equivalent to the Denjoy integral "in the restricted
sense."
see also INTEGRAL, LEBESGUE INTEGRAL, PERON IN-
TEGRAL, Riemann Integral
References
Iyanaga, S. and Kawada, Y. (Eds.). "Denjoy Integrals." §103
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 337-340, 1980.
Kestelman, H. "General Denjoy Integral." §9.2 in Modern
Theories of Integration, 2nd rev. ed. New York: Dover,
pp. 217-227, 1960.
Denominator
The number q in a FRACTION p/q.
see also Fraction, Numerator, Ratio, Rational
Number
Dense
A set A in a First-Countable Space is dense in B if
B = AUL, where L is the limit of sequences of elements
of A. For example, the rational numbers are dense in
the reals. In general, a SUBSET A of X is dense if its
Closure c\(A) = X.
see also Closure, Density, Derived Set, Perfect
Set
Density
see Density (Polygon), Density (Sequence), Nat-
ural Density
Density (Polygon)
The number q in a STAR POLYGON {£}.
see also STAR POLYGON
Density (Sequence)
Let a SEQUENCE {ai}~i be strictly increasing and com-
posed of Nonnegative Integers. Call A(x) the num-
ber of terms not exceeding x. Then the density is given
by linix-^oo A(x)/x if the Limit exists.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 199, 1994.
Denumerable Set
A Set is denumerable if a prescription can be given
for identifying its members one at a time. Such a set is
said to have CARDINAL NUMBER No. Examples of denu-
merable sets include Algebraic Numbers, Integers,
and Rational Numbers. Once one denumerable set 5
is given, any other set which can be put into a One-TO-
One correspondence with S is also denumerable. Ex-
amples of nondenumerable sets include the REAL, Com-
plex, Irrational, and Transcendental Numbers.
see also ALEPH-0, ALEPH-1, CANTOR DIAGONAL
Slash, Continuum, Hilbert Hotel
References
Courant, R. and Robbins, H. "The Denumerability of the Ra-
tional Number and the Non- Denumerability of the Contin-
uum." §2.4.2 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 79-83, 1996.
Denumerably Infinite
see Denumerable Set
Depth (Graph)
Derivative 417
Depth (Graph)
The depth E(G) of a GRAPH G is the minimum num-
ber of Planar Graphs Pi needed such that the union
UiPi = G.
see also Planar Graph
Depth (Size)
The depth of a box is the horizontal DISTANCE from
front to back (usually not necessarily defined to be
smaller than the WIDTH, the horizontal DISTANCE from
side to side).
see also Height, Width (Size)
Depth (Statistics)
The smallest RANK (either up or down) of a set of data.
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 30, 1977.
Depth (Tree)
The depth of a RESOLVING TREE is the number of lev-
els of links, not including the top. The depth of the link
is the minimal depth for any RESOLVING TREE of that
link. The only links of length are the trivial links. A
KNOT of length 1 is always a trivial Knot and links
of depth one are always Hopf Links, possibly with a
few additional trivial components (Bleiler and Scharle-
mann). The Links of depth two have also been classified
(Thompson and Scharlemann).
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 169, 1994.
D erangement
A Permutation of n ordered objects in which none of
the objects appears in its natural place. The function
giving this quantity is the SUBFACTORIAL !n, defined by
In = n\y
fc!
"■[t]-
(1)
(2)
where k\ is the usual Factorial and [x] is the NlNT
function. These are also called Rencontres NUMBERS
(named after rencontres solitaire), or COMPLETE PER-
MUTATIONS, or derangements. The number of derange-
ments \n = d(n) of length n satisfy the Recurrence
Relations
d(n) = (n - l)[d(n - 1) + d(n - 2)] (3)
with d(l) = and d(2) = 1- The first few are 0, 1, 2,
9, 44, 265, 1854, ... (Sloane's A000166). This sequence
cannot be expressed as a fixed number of hypergeometric
terms (Petkovsek et al. 1996, pp. 157-160).
see also MARRIED COUPLES PROBLEM, PERMUTATION,
Root, Subfactorial
References
Aitken, A. C. Determinants and Matrices. Westport, CT:
Greenwood Pub., p. 135, 1983.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 46-47,
1987.
Coolidge, J. L. An Introduction to Mathematical Probability.
Oxford, England: Oxford University Press, p. 24, 1925.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 115-116, 1996.
de Montmort, P. R. Essai d'analyse sur les jeux de hasard.
Paris, p. 132, 1713.
Dickau, R. M. "Derangements." http://f orum.swarthmore.
edu/advanced/robertd/derangements . html.
Durell, C. V\ and Robson, A. Advanced Algebra. London,
p. 459, 1937.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ:
Prentice-Hall, 1984.
Ruskey, F. "Information on Derangements." http:// sue .
esc .uvic . ca/-cos/inf /perm/Derangements .html.
Sloane, N. J. A. Sequence A000166/M1937 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Stanley, R. P. Enumerative Combinatorics, Vol. 1. New
York: Cambridge University Press, p. 67, 1986.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 123, 1991.
Derivative
The derivative of a FUNCTION represents an infinites-
imal change in the function with respect to whatever
parameters it may have. The "simple" derivative of a
function / with respect to x is denoted either f'(x) or ^
(and often written in-line as df/dx). When derivatives
are taken with respect to time, they are often denoted
using Newton's FLUXION notation, ^f = x. The deriva-
tive of a function f(x) with respect to the variable x is
defined as
f'(x) = lim
f(x + h)-f(x)
(1)
and
d{n) = nd(n-l) + (-l) n ,
(4)
Note that in order for the limit to exist, both lim^^ +
and lim/^o- must exist and be equal, so the Function
must be continuous. However, continuity is a NECES-
SARY but not SUFFICIENT condition for differentiabil-
ity. Since some DISCONTINUOUS functions can be inte-
grated, in a sense there are "more" functions which can
be integrated than differentiated. In a letter to Stielt-
jes, Hermite wrote, "I recoil with dismay and horror at
this lamentable plague of functions which do not have
derivatives."
A 3-D generalization of the derivative to an arbitrary
direction is known as the DIRECTIONAL DERIVATIVE.
418
Derivative
In general, derivatives are mathematical objects which
exist between smooth functions on manifolds. In this
formalism, derivatives are usually assembled into "TAN-
GENT Maps."
Simple derivatives of some simple functions follow.
dx
— ln|x| = i
dx x
d .
— sin x = cos x
dx
d
cos a; :
- since
dx
d d /since \ cos x cos x — since (— since)
— tancc = — = * '-
dx dx \ cos cc / cos J cc
(2)
(3)
(4)
(5)
)
(6)
GE d . . \~1 /. \ —2
— esc cc = —(sin a;) = —(since)
dx dx
= — esc x cot X
cos x = — -
cosx
— sec cc = — — (cos x)
dx dx
d _ d f cos x\ _
dx dx V since/
1
2
COS 2 £
d x
dx
= e x
d x
dx
_ _^_ In a* _
dx
^ xln
dx
=
(lna)e Ilno =
- (In a) a
d .
-— sin
-x- l
dx '
- sech x ■
(7)
/ n-2/ . \ sin a;
(cosx) ( — since) = —
cos 2 x
sec cc tan x (8)
_ d f cos ai \ _ sin cc ( — sin cc) — cos cc cos x
cos 2 cc
(S)
(10)
(ii)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
dx
y/l-X 2
d
cos 1 X
1
dx
y/l-X 2
d
tan - cc
1
dx
1 + Z 2
d
cot - X
1
dx
1 + CC 2
d
sec - x
1
dx
x\/x 2 — 1
d
esc -1 X
1
dx
x\fx 2 — 1
d
dx
sinh x —
cosh x
d
dx
cosh x —
-■ sinh x
d
dx
tanhec =
- sech x
— coth x = — csch x
dx
A.
dx
d_
dx
d_
dx
d_
dx
csch x — — csch cc coth cc
sn cc ~ en cc dn cc
en x — — sn x dn cc
dn cc = — A; sn cc en x.
Derivative
(23)
(24)
(25)
(26)
Derivatives of sums are equal to the sum of derivatives
so that
[/(cc) + . . . + h(x)]' = f'(x) + . . . + h'(z). (27)
In addition, if c is a constant,
^[cf(x)] = cf'(x).
(28)
Furthermore,
■^[f(x)g(x)} = /(*)*'(*) + /'(*)*(*), (29)
where /' denotes the Derivative of / with respect to x.
This derivative rule can be applied iteratively to yield
derivate rules for products of three or more functions,
for example,
[fgh]' = {fg)ti + {fg)'h = fgti + (fg f + /^)/i
= f'gh + fg'h + fgti. (30)
Other rules involving derivatives include the Chain
Rule, Power Rule, Product Rule, and Quotient
Rule. Miscellaneous other derivative identities include
dy
_ dt
dx is=
dt
dy
dy
dx
dx
dy
If F(x> y) = C, where C is a constant, then
OF OF
dF=?-dy+^-dx = 0,
dy ox
dy
dx
&f
dx
dF '
dy
(31)
(32)
(33)
(34)
A vector derivative of a vector function
X(t)
- sech x tanh x
'xi(ty
x 2 (t)
Xk{t)_
(35)
Derivative Test
Desargues' Theorem 419
can be defined by
dX
dt
dt
dx2
dt
dt '
where w is a parameter (Endraft). The surface can also
be described by the equation
(36)
see also BLANCMANGE FUNCTION, CARATHEODORY
Derivative, Comma Derivative, Convective De-
rivative, Covariant Derivative, Directional De-
rivative, Euler-Lagrange Derivative, Fluxion,
Fractional Calculus, Frechet Derivative, La-
grangian derivative, lie derivative, power
Rule, Schwarzian Derivative, Semicolon Deriva-
tive, WeierstraB 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. 11, 1972.
Anton, H. Calculus with Analytic Geometry, 5th ed. New
York: Wiley, 1987.
Beyer, W. H. "Derivatives." CRC Standard Mathematical
Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 229-
232, 1987.
Derivative Test
see First Derivative Test, Second Derivative
Test
Derived Set
The Limit Points of a Set P, denoted P'.
see also Dense, Limit Point, Perfect Set
Dervish
A QuiNTIC SURFACE having the maximum possible
number of ORDINARY DOUBLE POINTS (31), which was
constructed by W. Barth in 1994 (Endrafi). The implicit
equation of the surface is
4x 3 w — lOx y
a 2 2
■ 4x w
64 (x — w)[x ■
+16xw 3 - 20xy 2 w + 5j/ 4 + 16w 4 - 20y 2 w 2 ]
-5\/5-V5(2z - y/b-VSw)
x [4(z 2 + y 2 + z 2 ) + (1 + 3^5 )w 2 ] 2 ,
where
a F + q = 0,
F = h\h2h^h^hsi
hi = x
and
/2tt\ . /2tt\
hi = cos I — - 1 x - sm ( — - I y - z
h z - cos f — j x - sin ( — J y - z
/6tt\ . /6tt\
h 4 = cos I — I x — sm I — - 1 y - z
/8tt\ . /8tt\
h 5 = cos ( -— J x - sm I — 1 y - z
q=(l- cz)(x 2 + y 2 - 1 + rz 2 ) 2 ,
r=J(l + V5)
.__§(, + £) ,/iTvi
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(Nordstrand).
The dervish is invariant under the GROUP Ds and con-
tains exactly 15 lines. Five of these are the intersection
of the surface with a Ds-invariant cone containing 16
nodes, five are the intersection of the surface with a D5-
invariant plane containing 10 nodes, and the last five
are the intersection of the surface with a second D&-
invariant plane containing no nodes (Endrafi).
References
Endrafi, S. "Togliatti Surfaces." http://www . mathematik .
uni - mainz . de / Algebraische Geometrie / docs /
Etogliatti . shtml.
Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV-
Mitteilungen 4, 17-20, 4/1995.
Endrafi, S. Symmetrische Flache mit vielen gewohnlichen
Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996.
Nordstrand, T. "Dervish." http://www.uib.no/people/
nfytn/dervtxt .htm.
Desargues' Theorem
420
Descartes Circle Theorem
Descartes 7 Sign Rule
If the three straight LINES joining the corresponding
Vertices of two Triangles ABC and A'B'C all meet
in a point (the Perspective Center), then the three
intersections of pairs of corresponding sides lie on a
straight LINE (the PERSPECTIVE Axis). Equivalently, if
two Triangles are Perspective from a Point, they
are Perspective from a Line.
Desargues' theorem is essentially its own dual according
to the Duality Principle of Projective Geometry.
see also Duality Principle, Pappus's Hexagon
Theorem, Pascal Line, Pascal's Theorem, Per-
spective Axis, Perspective Center, Perspective
Triangles
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 70-72, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 89-92, 1990.
Descartes Circle Theorem
A special case of Apollonius' Problem requiring the
determination of a CIRCLE touching three mutually tan-
gent Circles (also called the Kissing Circles Prob-
lem). There are two solutions: a small circle surrounded
by the three original CIRCLES, and a large circle sur-
rounding the original three. Frederick Soddy gave the
Formula for finding the Radius of the so-called inner
and outer SODDY CIRCLES given the RADII of the other
three. The relationship is
2(«i + «2 + «3
2 ~\~ K 4 2 ) = (m + n 2 + «3 4- tt4) 2 ,
where m are the CURVATURES of the CIRCLES. Here,
the Negative solution corresponds to the outer Soddy
Circle and the Positive solution to the inner SODDY
Circle. This formula was known to Descartes and Viete
(Boyer and Merzbach 1991, p. 159), but Soddy extended
it to Spheres. In n-D space, n + 2 mutually touching
n-SPHERES can always be found, and the relationship of
their CURVATURES is
Descartes Folium
see Folium of Descartes
Descartes' Formula
see Descartes Total Angular Defect
Descartes Ovals
see Cartesian Ovals
Descartes' Sign Rule
A method of determining the maximum number of POS-
ITIVE and Negative Real Roots of a Polynomial.
For Positive Roots, start with the Sign of Coeffi-
cient of the lowest (or highest) Power. Count the
number of SIGN changes n as you proceed from the low-
est to the highest POWER (ignoring POWERS which do
not appear). Then n is the maximum number of Pos-
itive ROOTS. Furthermore, the number of allowable
ROOTS is n, n — 2, n — 4, For example, consider the
Polynomial
f(x) = x 7 + x 6 -x 4
•X* +X-1.
Since there are three SIGN changes, there are a maxi-
mum of three possible Positive Roots.
For Negative Roots, starting with a Polynomial
/(as), write a new POLYNOMIAL g{x) with the SIGNS
of all ODD POWERS reversed, while leaving the SIGNS of
the Even Powers unchanged. Then proceed as before
to count the number of SIGN changes n. Then n is the
maximum number of NEGATIVE ROOTS. For example,
consider the POLYNOMIAL
f[x) — x + x
2 ,
■ x + x
i,
and compute the new POLYNOMIAL
/ \ 7,6 4,3 2 -,
g(x) = —x -f x — x -f x — x — x — 1.
see also Apollonius' Problem, Four Coins Prob-
lem, Soddy Circles, Sphere Packing
References
Boyer, C. B. and Merzbach, U. C A History of Mathematics,
2nd ed. New York: Wiley, 1991.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 13-16, 1969.
Wilker, J . B . "Four Proofs of a Generalization of the
Descartes Circle Theorem." Amer. Math. Monthly 76,
278-282, 1969.
There are four SIGN changes, so there are a maximum
of four Negative Roots.
see also BOUND, STURM FUNCTION
References
Anderson, B.; Jackson, J.; and Sitharam, M. "Descartes'
Rule of Signs Revisited." Amer. Math. Monthly 105, 447-
451, 1998.
Hall, H. S. and Knight, S. R. Higher Algebra: A Sequel
to Elementary Algebra for Schools. London: Macmillan,
pp. 459-460, 1950.
Struik, D. J. (Ed.). A Source Book in Mathematics 1200-
1800. Princeton, NJ: Princeton University Press, pp. 89-
93, 1986.
Descartes-Euler Polyhedral Formula
see Polyhedral Formula
Descartes Total Angular Defect
Determinant 421
Descartes Total Angular Defect
The total angular defect is the sum of the Angular
Defects over all Vertices of a Polyhedron, where
the Angular Defect 8 at a given Vertex is the dif-
ference between the sum of face angles and 2tt. For any
convex POLYHEDRON, the Descartes total angular defect
is
A = ]T]& =4tt. (1)
This is equivalent to the POLYHEDRAL FORMULA for a
closed rectilinear surface, which satisfies
A = 2tt(V -E + F).
(2)
References
Assmus, E. F. Jr. and Key, J. D. Designs and Their Codes.
New York: Cambridge University Press, 1993.
Colbourn, C* J. and Dinitz, J. H. CRC Handbook of Combi-
natorial Designs. Boca Raton, FL: CRC Press, 1996.
Dinitz, J. H. and Stinson, D. R. (Eds.). "A Brief Introduction
to Design Theory." Ch. 1 in Contemporary Design Theory:
A Collection of Surveys. New York: Wiley, pp. 1-12, 1992.
Desmic Surface
Let Ai, A2, and A3 be tetrahedra in projective 3-space
F 3 . Then the tetrahedra are said to be desmically re-
lated if there exist constants a, /?, and 7 such that
oAi +/?A 2 +7A 3 =0.
A Polyhedron with N equivalent Vertices is called a
Platonic Solid and can be assigned a Schlafli Sym-
bol {p, q}. It then satisfies
and
Nc
N = T
27r-q(l~
I)
4p
'~2p + 2q-
-pq
(3)
(4)
(5)
see also Angular Defect, Platonic Solid, Poly-
hedral Formula, Polyhedron
Descriptive Set Theory
The study of DEFINABLE SETS and functions in POLISH
Spaces.
References
Becker, H. and Kechris, A. S. The Descriptive Set Theory of
Polish Group Actions. New York: Cambridge University
Press, 1996.
Design
A formal description of the constraints on the possi-
ble configurations of an experiment which is subject to
given conditions. A design is sometimes called an EX-
PERIMENTAL Design.
see also Block Design, Combinatorics, Design
Theory, Hadamard Design, Howell Design,
Spherical Design, Symmetric Block Design,
Transversal Design
A desmic surface is then defined as a QuARTlC SURFACE
which can be written as
aAi 4- 6A 2 + cA 3 =
for desmically related tetrahedra Ai, A2, and A3.
Desmic surfaces have 12 ORDINARY DOUBLE POINTS,
which are the vertices of three tetrahedra in 3-space
(Hunt).
see also QuARTlC SURFACE
References
Hunt, B. "Desmic Surfaces." §B.5.2 in The Geometry of
Some Special Arithmetic Quotients. New York: Springer-
Verlag, pp. 311-315, 1996.
Jessop, C. §13 in Quartic Surfaces with Singular Points.
Cambridge, England: Cambridge University Press, 1916.
Destructive Dilemma
A formal argument in LOGIC in which it is stated that
1. P => Q and R => S (where => means "Implies"),
and
2. Either not-Q or not-5 is true, from which two state-
ments it follows that either not-P or not-R is true.
see also CONSTRUCTIVE DILEMMA, DILEMMA
Determinant
Determinants are mathematical objects which are very-
useful in the analysis and solution of systems of linear
equations. As shown in CRAMER'S RULE, a nonhomo-
geneous system of linear equations has a nontrivial so-
lution IFF the determinant of the system's MATRIX is
Nonzero (so that the Matrix is nonsingular) . A 2 x 2
determinant is defined to be
Design Theory
The study of DESIGNS and, in particular, NECESSARY
and Sufficient conditions for the existence of a Block
Design.
see also Bruck-Ryser-Chowla Theorem, Fisher's
Block Design Inequality
det
a b
c d
a b
c d
= ad ~ be.
(i)
422 Determinant
Determinant
A k x k determinant can be expanded by Minors to
obtain
and
an
012
013
'" Olfc
(221 a22
CL2Z
* * ' »2fc
0>kl &k2
Ctk3
ttfc/s
022
023
■ ' • 02fc
021
023 • • •
02fc
- ail
— ai2
Ofc2
Ofc3
■ ■ ' Ofcfc
Ofcl
Ofc3 * * *
Ofcfc
a 2
1 O22
02(fc-l)
+ ..
• ± Olfc
;
• (2
CLk
1 Ofc2
Ofc(fc-l)
A general determinant for a Matrix A has a value
|A| = £o„a«, (3)
with no implied summation over i and where a 13 is the
Cofactor of a,ij defined by
a t} = (-1) ,+, C«
(4)
Here, C is the (n - 1) x (n - 1) MATRIX formed by
eliminating row i and column j from A, i.e., by DETER-
MINANT Expansion by Minors.
Given an n x n determinant, the additive inverse is
|-A| = (-1)»|A|. (5)
Determinants are also DISTRIBUTIVE, so
|AB| = |A||B|. (6)
This means that the determinant of a Matrix Inverse
can be found as follows:
|l| = |AA- X | = |A| ia- 1 ! = 1,
where I is the IDENTITY MATRIX, so
|A| =
IA" 1 !'
(7)
(8)
Determinants are Multilinear in rows and columns,
since
Oi
02
a 3
Ol
CI4
as
a&
=
04 as a&
a 7
a s
ag
a-j as ag
a 2
+
a4 as ag
+
a-j a$
ag
a-s
a4
as
ae
a 7
a 8
ag
(9)
ai a2 as
a4 05 aQ
aj a% ag
ai a2 a3
as ae
as ag
a2 as
a4 as aQ
+
as ag
a 2
o 3
as
a6
7
a 8
ag
• (io)
The determinant of the SIMILARITY TRANSFORMATION
of a matrix is equal to the determinant of the original
Matrix
(BAB" 1 ! = |B| |A| IB" 1 ] = |B| |A|-^- = |A|. (11)
|B |
The determinant of a similarity transformation minus a
multiple of the unit MATRIX is given by
|B _1 AB - Al| = |B _1 AB - B _1 AIB| = |B _1 (A - Al)B|
= |B- 1 ||A-AI||B| = |A-AI|. (12)
The determinant of a Matrix Transpose equals the
determinant of the original Matrix,
|A| = |A T |,
(13)
and the determinant of a COMPLEX CONJUGATE is equal
to the Complex Conjugate of the determinant
|A'| = |A|*.
Let e be a small number. Then
|l + eA| = l + eTr(A) + 0(e 2 ),
(14)
(15)
where Tr(A) is the trace of A. The determinant takes on
a particularly simple form for a TRIANGULAR MATRIX
an a2i
022
Ofcl
Ofc2
ajtfc
M o nn .
(16)
Important properties of the determinant include the fol-
lowing.
1. Switching two rows or columns changes the sign.
2. Scalars can be factored out from rows and columns.
3. Multiples of rows and columns can be added together
without changing the determinant's value.
4. Scalar multiplication of a row by a constant c multi-
plies the determinant by c.
5. A determinant with a row or column of zeros has
value 0.
6. Any determinant with two rows or columns equal has
value 0.
Determinant
Determinant (Binary Quadratic Form) 423
Property 1 can be established by induction. For a 2 x 2
Matrix, the determinant is
ai 6i
a 2 b 2
: ai&2 — b\a 2 = — (6ia2 — aib 2 )
6i ai
62 CL2
(17)
For a 3 x 3 Matrix, the determinant is
a\ bi c\
o>i 62 C2
as &3 C3
&2 C2
: ai
&3 c 3
■61
0-2 C2
as C3
+ ci
a2 &2
as bs
= -
01
C2
C3
b 2
bs
01
Ci
61
= -
a 2
c 2
&2
a3
C3
&3
+ &1
C2 a2
C3 ^3
Cl
a2 ^2
a3 &3
"-(■
■ai
62 C2
63 C3
+ 61
a 2 C2
a3 c 3
+ Ci
62 a 2
63 as
&i ai ci
62 a2 C2
bs as C3
( -ai
C 2 &2
C3 &3
-61
a 2
a 3
c 2
c 3
+ Ci
&2
&3
a 2
a 3
ci 61 ai
c 2 & 2 a 2
C3 6
3 a 3
(18)
Property 2 follows likewise. For 2x2 and 3x3 matrices,
ka\ b\
ka 2 62
= k(aib2) — k(bia,2) — k
a\ 61
a 2 62
(19)
and
ka\ b\ c\
fca 2 6 2 c 2
/ca3 63 C3
ka\
+ Ci
6 2
63
ka 2
kas
kd2 C 2
kas cs
= k
ai
a 2
a 3
61
b 2
bs
(20)
Property 3 follows from the identity
(ai + kbt)
a\ 4- kbi bi a
a 2 + kb 2 62 C2
as -f &&3 &3 c 3
a + kb 2 c 2
b 2 c 2
bs cs
-h
as + kbs cs
+ ci
a 2 + &&2
a 3 4- &&3
(21)
by the column vectors [o»,i], . . . , [a itn ] in M. n . Here, "ori-
ented" means that, up to a change of 4- or — SIGN, the
number is the n-dimensional CONTENT, but the SIGN
depends on the "orientation" of the column vectors in-
volved. If they agree with the standard orientation,
there is a + Sign; if not, there is a - Sign. The Par-
allelepiped spanned by the n-D vectors Vi through v*
is the collection of points
tlVi + ... +UVi,
(22)
where tj is a REAL NUMBER in the CLOSED INTERVAL
[0,1].
There are an infinite number of 3 x 3 determinants with
no or ±1 entries having unity determinant. One para-
metric family is
-8n 2 - 8n
— 4n 2 — An
-4 n 2 - 4n - 1
2n + l
n+ 1
4n
2n+l
2n-l
(23)
Specific examples having small entries include
2 3 2
2 3 5
2
3
6
4 2 3
)
3 2 3
1
3
2
3
9 6 7
9 5 7
17
11
16
(24)
(Guy 1989, 1994).
see also Circulant Determinant, Cofactor,
Hessian Determinant, Hyperdeterminant, Im-
manant, Jacobian, Knot Determinant, Matrix,
Minor, Permanent, Vandermonde Determinant,
Wronskian
References
Arfken, G. "Determinants." §4.1 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 168-176, 1985.
Guy, R. K. "Unsolved Problems Come of Age." ^4mer. Math.
Monthly 96, 903-909, 1989.
Guy, R. K. "A Determinant of Value One." §F28 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 265-266, 1994.
Determinant (Binary Quadratic Form)
The determinant of a Binary Quadratic Form
An + 2Buv + Cv 2
D = B 2 - AC.
It is equal to 1/4 of the corresponding DISCRIMINANT.
If aij is an n X n MATRIX with a»j REAL NUMBERS,
then det [aij] has the interpretation as the oriented n-
dimensional CONTENT of the PARALLELEPIPED spanned
424 Determinant Expansion by Minors
Devil's Curve
Determinant Expansion by Minors
Also known as Laplacian Determinant Expansion
by Minors. Let | IS/I | denote the Determinant of a
Matrix M, then
Devil's Curve
IMI
J2(-l) i+j ai M ijt
where Mij is called a MINOR,
where Cij is called a COFACTOR.
see also COFACTOR, DETERMINANT
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 169-170, 1985.
Determinant (Knot)
see Knot Determinant
Determinant Theorem
Given a MATRIX m, the following are equivalent:
i. m^o.
2. The columns of m are linearly independent.
3. The rows of m are linearly independent.
4. Range(m) = W 1 .
5. Null(m) = {0}.
6. m has a Matrix Inverse.
see also Determinant, Matrix Inverse, Nullspace,
Range (Image)
Developable Surface
A surface on which the GAUSSIAN CURVATURE K is ev-
erywhere 0.
see also Binormal Developable, Normal Devel-
opable, Synclastic, Tangent Developable
Deviation
The Difference of a quantity from some fixed value,
usually the "correct" or "expected" one.
see Absolute Deviation, Average Absolute Devi-
ation, Difference, Dispersion (Statistics), Mean
Deviation, Signed Deviation, Standard Devia-
tion
Q
The devil's curve was studied by G. Cramer in 1750 and
Lacroix in 1810 (MacTutor Archive). It appeared in
Nouvelles Annales in 1858. The Cartesian equation is
4 2 2 4 1 2 2
y —ay = x — b x ,
equivalent to
2/2 2\ 2/2 i2\
y {y -a ) — x (x - b ),
the polar equation is
r (sin 9 — cos 6) — a sin 9 — b cos (
and the parametric equations are
x = cos t
a 2 sin 2 t — b 2 cos 2 t
sin 2 t — cos 2 t
. t / a 2 sin 2 t — b 2 cos 2 t
y = sin *v — 7 ~2~. tt~~-
V sur t — cos" 1 1
(i)
(2)
(3)
(4)
(5)
A special case of the Devil's curve is the so-called ELEC-
TRIC Motor Curve:
(6)
y 2 (y 2 -96) = x 2 (x 2 - 100)
(Cundy and Rollett 1989).
see also ELECTRIC MOTOR CURVE
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 71, 1989.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 71, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves, New-
York: Dover, pp. 151-152, 1972.
MacTutor History of Mathematics Archive. "Devil's Curve."
http: //www-groups .dcs .st-and.ac .uk/ -history /Curves
/Devils. html.
Devil's Staircase
Diagonal (Polygon) 425
Devil's Staircase
A plot of the Winding Number W resulting from
Mode Locking as a function of fl for the Circle Map
with K = 1. At each value of Q, the WINDING NUM-
BER is some Rational NUMBER. The result is a mono-
tonic increasing "staircase" for which the simplest Ra-
tional Numbers have the largest steps. For K = 1, the
Measure of quasiperiodic states (fi Irrational) on
the O-axis has become zero, and the measure of MODE-
LOCKED state has become 1. The DIMENSION of the
Devil's staircase w 0.8700 ± 3.7 x 10" 4 .
see also Cantor Function
References
Mandelbrot, B. B. The Fractal Geometry of Nature. New
York: W. H. Freeman, 1983,
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, 1993.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, p. 132, 1990.
Devil on Two Sticks
see Devil's Curve
Diabolical Cube
A 6-piece Polycube Dissection of the 3x3 Cube.
see also Cube Dissection, Soma Cube
References
Gardner, M. "Polycubes." Ch. 3 in Knotted Doughnuts and
Other Mathematical Entertainments. New York: W. H.
Freeman, pp. 29-30, 1986.
Diabolical Square
see Panmagic Square
Given a MATRIX equation of the form
an * * * din 1 r Ai ■ * -
_ 0t n i * * * Q>nn J L ^ " * " A n _
Ai ■ * • 1 fan
• • • A„ J |_ a m
multiply through to obtain
anAi **• ainAnl [ aiiAi
air,
. (3)
.flniAi
Q>nnA n
_flnlA n
ainAi
0>nn^n
(4)
Since in general, \i ^ Xj for i ^ j, this can be true only
if off-diagonal components vanish. Therefore, A must
be diagonal.
Given a diagonal matrix T,
yn
tk
h n
t 2 n
t k n
(5)
see also Matrix, Triangular Matrix, Tridiagonal
Matrix
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 181-184 and 217-229,
1985.
Diabolo
A 2-POLYABOLO.
Diacaustic
The Envelope of refracted rays for a given curve.
see also CATACAUSTIC, CAUSTIC
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 60, 1972.
Diagonal Matrix
A diagonal matrix is a Matrix A of the form
&ij — Cidiji
(i)
where S is the Kronecker Delta, a are constants,
and there is no summation over indices. The general
diagonal matrix is therefore SQUARE and of the form
d
c 2
(2)
Diagonal Metric
A Metric gij which is zero for i ^ j.
see also Metric
Diagonal (Polygon)
A Line Segment connecting two nonadjacent VER-
TICES of a POLYGON. The number of ways a fixed con-
vex n-gon can be divided into TRIANGLES by noninter-
secting diagonals is C n -2 (with C n -3 diagonals), where
C n is a Catalan Number. This is Euler's Polygon
Division Problem. Counting the number of regions
determined by drawing the diagonals of a regular n-gon
is a more difficult problem, as is determining the num-
ber of n-tuples of CONCURRENT diagonals (Beller et ai.
1972, Item 2).
The number of regions which the diagonals of a CONVEX
POLYGON divide its center if no three are concurrent in
its interior is
iv=i 4 i +
= ij(n-l)(n-2)(n 2 -3n+12).
426 Diagonal (Polyhedron)
Dice
The first few values are 0, 0, 1, 4, 11, 25, 50, 91, 154,
246, . . . (Sloane's A006522).
see also Catalan Number, Diagonal (Polyhe-
dron), Euler's Polygon Division Problem, Poly-
gon, Vertex (Polygon)
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Sloane, N. J. A. Sequence A006522/M3413 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Diagonal (Polyhedron)
A Line Segment connecting two nonadjacent sides
of a Polyhedron, The only simple Polyhedron
with no diagonals is the Tetrahedron. The only
known TOROIDAL POLYHEDRON with no diagonals is the
CSASZAR POLYHEDRON.
see also Diagonal (Polygon), Euler Brick, POLY-
HEDRON, Space Diagonal
Diagonal Ramsey Number
A Ramsey Number of the form R(k, k\ 2).
see also RAMSEY NUMBER
Diagonal Slash
see Cantor Diagonal Slash
Diagonal (Solidus)
see SOLIDUS
Diagonalization
see Matrix Diagonalization
Diagonals Problem
see Euler Brick
Diagram
A schematic mathematical illustration showing the rela-
tionships between or properties of mathematical objects.
see also Alternating Knot Diagram, Argand Di-
agram, Coxeter-Dynkin Diagram, de Bruijn Dia-
gram, Dynkin Diagram, Ferrers Diagram, Hasse
Diagram, Heegaard Diagram, Knot Diagram,
Link Diagram, Stem-and-Leaf Diagram, Venn Di-
agram, Voronoi Diagram, Young Diagram
Diameter
The diameter of a CIRCLE is the DISTANCE from a point
on the Circle to point 7r Radians away. If r is the
Radius, d = 2r.
see also BROCARD DIAMETER, CIRCUMFERENCE, DI-
AMETER (General), Diameter (Graph), Pi, Ra-
dius, Transfinite Diameter
Diameter (General)
The farthest DISTANCE between two points on the
boundary of a closed figure.
see also BORSUK'S CONJECTURE
References
Eppstein, D. "Width, Diameter, and Geometric
Inequalities." http://www . ics . uci . edu / - eppstein/
junkyard/diam.html.
Diameter (Graph)
The length of the "longest shortest path" between two
VERTICES of a GRAPH. In other words, a graph's di-
ameter is the largest number of vertices which must be
traversed in order to travel from one vertex to another
when paths which backtrack, detour, or loop are ex-
cluded from consideration.
Diamond
A convex Quadrilateral having sides of equal length
and Perpendicular Planes of symmetry passing
through opposite pairs of VERTICES. The LOZENGE is a
special case of a diamond.
see also KITE, LOZENGE, PARALLELOGRAM, QUADRI-
LATERAL, RHOMBUS
Dice
A die (plural "dice") is a Solid with markings on each of
its faces. The faces are usually all the same shape, mak-
ing Platonic Solids and Archimedean Solid Duals
the obvious choices. The die can be "rolled" by throw-
ing it in the air and allowing it to come to rest on one
of its faces. Dice are used in many games of chance as a
way of picking RANDOM NUMBERS on which to bet, and
are used in board or roll-playing games to determine the
number of spaces to move, results of a conflict, etc. A
Coin can be viewed as a degenerate 2-sided case of a
die.
The most common type of die is a six-sided CUBE with
the numbers 1-6 placed on the faces. The value of the
roll is indicated by the number of "spots" showing on the
top. For the six-sided die, opposite faces are arranged to
always sum to seven. This gives two possible MIRROR
Image arrangements in which the numbers 1, 2, and 3
may be arranged in a clockwise or counterclockwise or-
der about a corner. Commercial dice may, in fact, have
either orientation. The illustrations below show 6-sided
dice with counterclockwise and clockwise arrangements,
respectively.
• • •
• • •
_ _ ^ ^ _ I • • • • •
• • • •
• • • • • • • • • • • •
• • •
• • •
Dice
Dice 427
The Cube has the nice property that there is an upward-
pointing face opposite the bottom face from which the
value of the "roll" can easily be read. This would not
be true, for instance, for a TETRAHEDRAL die, which
would have to be picked up and turned over to reveal the
number underneath (although it could be determined
by noting which number 1-4 was not visible on one of
the upper three faces), The arrangement of spots /*
corresponding to a roll of 5 on a six-sided die is called
the QUINCUNX. There are also special names for certain
rolls of two six-sided dice: two Is are called Snake Eyes
and two 6s are called BOXCARS.
Shapes of dice other than the usual 6-sided CUBE are
commercially available from companies such as Dice &
Games, Ltd.®
Diaconis and Keller (1989) show that there exist "fair"
dice other than the usual Platonic Solids and duals
of the Archimedean Solids, where a fair die is one for
which its symmetry group acts transitively on its faces.
However, they did not explicitly provide any examples.
The probability of obtaining p points (a roll of p) on n
s-sided dice can be computed as follows. The number of
ways in which p can be obtained is the COEFFICIENT of
x p in
f(x) = (x + x + ...+£*
(1)
since each possible arrangement contributes one term.
f(x) can be written as a Multinomial Series
/(*)
( l-x 3 \ n
\l-x) '
(2)
so the desired number c is the COEFFICIENT of x p in
x n (l-xT{l-x)~ n -
(3)
Expanding,
n / \ oo
fc=0 ^ ' z=o
^M^Wi+j-i)^ (4)
so in order to get the COEFFICIENT of x p y include all
terms with
p = n + sk + l. (5)
c is therefore
-tri^
n \ ( V — s k — 1
p — sk — n
(6)
But p — sk — n > only when k < (p — n)/s, so the other
terms do not contribute. Furthermore,
p — sk — 1
p — sk — n
p — sk — 1
n-1
(7)
l(p-n)/s}
UP-nj/sj / \ / i. i\
k=0
where [^J is the FLOOR FUNCTION, and
l(p-n)/a\
"*".«>-? t <-'>*(:) (';.."')■ w
Consider now s = 6. For n = 2 six-sided dice,
p-2
&max —
! h:
for 2 < p < 7
for 12 < p < 8,
(10)
and
1 fcmax
=4E<-i>
2!
(p - 6k - 1)
k\(2-ky.
= 36E( 1_2fc ^ fc + 1) ( p - 6fc - 1)
fc =
fc=0
_J L fp-l for2<p<7
~ 36 I 13 - j
- p for 8 < p < 12
6 - |p - 7|
36
for 2 < p < 12.
(11)
The most common roll is therefore seen to be a 7, with
probability 6/36 = 1/6, and the least common rolls are
2 and 12, both with probability 1/36.
for 3 < p < 8
1 for9<p<14 (12)
2 for 15 < p < 18,
P(p,3,6)
-iD-Orr 1 )
4D-"'s
3! (p- 6fe- l)(p-6fc - 2)
(3-fc)!
216
(p-l)(p-2) o (p-T)(p-g)
216 \i(U
for 3 < p < 8
for 9 < p < 14
t"- 1 ^- 2 ) _ 3 (p-7)fp-») + 3 (p-13Mp-14) for 15 < p < 18
i(p- l)(p-2) for 3 <p < 8
+ 21p - 83 for 9 < p < 14 (13)
19 - p)(20 - p) for 15 < p < 18.
For three six-sided dice, the most common rolls are 10
and 11, both with probability 1/8; and the least common
rolls are 3 and 18, both with probability 1/216.
428
Dice
Diesis
For four six-sided dice, the most common roll is 14, with
probability 73/648; and the least common rolls are 4 and
24, both with probability 1/1296.
In general, the likeliest roll pl for n s-sided dice is given
by
PL(n,s)=[±n{8-+l)\, (14)
which can be written explicitly as
PL(n,s)
( \n{s + 1)
{ |[n(5 + l)-
Un( S +l)
for n even
for n odd, s even (15)
for n odd, s odd.
For 6-sided dice, the likeliest rolls are given by
{|n for n even
|(7n - 1) for n odd, s even
|n for n odd, s odd,
(16)
or 7, 10, 14, 17, 21, 24, 28, 31, 35, . . . for n = 2, 3, . . .
(Sloane's A030123) dice. The probabilities correspond-
ing to the most likely rolls can be computed by plugging
p = pl into the general formula together with
in
for n even
k L (n,s) = { l ^'a?" 1 ] for " ° dd - s even
[=^iij for n odd, s odd.
(17)
Unfortunately, P(pL,n y s) does not have a simple closed-
form expression in terms of s and n. However, the proba-
bilities of obtaining the likeliest roll totals can be found
explicitly for a particular s. For n 6-sided dice, the
probabilities are 1/6, 1/8, 73/648, 65/648, 361/3888,
24017/279936, 7553/93312, ... for n = 2, 3, . . . .
References
Diaconis, P. and Keller, J. B. "Fair Dice." Amer. Math.
Monthly 96, 337-339, 1989.
Dice & Games, Ltd. "Dice & Games Hobby Games Acces-
sories." http : //www . dice . co . uk/hob . htm.
Gardner, M. "Dice." Ch. 18 in Mathematical Magic Show:
More Puzzles, Games, Diversions, Illusions and Other
Mathematical Sleight- of- Mind from Scientific American.
New York: Vintage, pp. 251-262, 1978.
Robertson, L. C; Shortt, R. M.; Landry, S. G. "Dice with
Fair Sums." Amer. Math. Monthly 95, 316-328, 1988.
Sloane, N. J. A. Sequence A030123 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Dichroic Polynomial
A POLYNOMIAL Zc(q,v) in two variables for abstract
Graphs. A Graph with one Vertex has Z - q.
Adding a Vertex not attached by any Edges multiplies
the Z by q. Picking a particular EDGE of a GRAPH <3,
the Polynomial for G is defined by adding the POLY-
NOMIAL of the GRAPH with that EDGE deleted to v times
the Polynomial of the graph with that Edge collapsed
to a point. Setting v = -1 gives the number of distinct
Vertex colorings of the Graph. The dichroic Poly-
nomial of a Planar Graph can be expressed as the
Square Bracket Polynomial of the corresponding
Alternating Link by
Z G {q,v)
q ^l(g))
where N is the number of VERTICES in G.
Polynomials for some simple Graphs are
Zk x =q
Zk 2 =q 2 +vq
Zk 3 = q 3 + 3vq 2 + 3v 2 q + v 3 q.
Dichroic
0.150
0.125
0.100
0.075
0.050
0.025
0.12
0.10
0.08
0.06
0.04
0.02
0.06
0.04
0.02
3 4 5 6 7 8 9 10 12 14 16
4 5 6 7 8 10 12 14 16 18 20 22 24
three dice four dice
The probabilities for obtaining a given total using n 6-
sided dice are shown above for n = 1, 2, 3, and 4 dice.
They can be seen to approach a GAUSSIAN DISTRIBU-
TION as the number of dice is increased.
see also Boxcars, Coin Tossing, Craps, de Mere's
Problem, Efron's Dice, Poker, Quincunx, Sicher-
man Dice, Snake Eyes
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 231-235, 1994.
Dido's Problem
Find the figure bounded by a line which has the maxi-
mum AREA for a given PERIMETER. The solution is a
Semicircle.
see also ISOPERIMETRIC PROBLEM, ISOVOLUME PROB-
LEM, Perimeter, Semicircle
Diesis
The musical interval by which an octave exceeds three
major thirds,
(I) 3
128
125
1.024.
Taking Continued Fraction Convergents of
log(5/4)/log(2) gives the increasing accurate approxi-
mations m/n of m octaves and n major thirds: 1/3,
Diffeomorphism
Difference Set 429
9/28, 19/59, 47/146, 207/643, 1289/4004, . . . (Sloane's
A046103 and A046104). Other near equalities of m oc-
taves and n major thirds having
R
(IY
with \R — 1| < 0.02 are given in the following table.
ra
n
Ratio
m
n
Ratio
9
28
0.9903520314
104
323
1.012011267
10
31
1.01412048
113
351
1.002247414
18
56
0.9807971462
122
379
0.9925777621
19
59
1.004336278
123
382
1.016399628
28
87
0.9946464728
131
407
0.983001403
29
90
1.018517988
132
410
1.006593437
37
115
0.9850501549
141
438
0.9968818549
38
118
1.008691359
150
466
0.9872639701
47
146
0.9989595361
151
469
1.010958305
56
174
0.9893216059
160
497
1.001204611
57
177
1.013065324
169
525
0.9915450208
66
205
1.003291302
170
528
1.015342101
75
233
0.9936115791
178
553
0.9819786256
76
236
1.017458257
179
556
1.005546113
84
261
0.9840252458
188
584
0.9958446353
85
264
1.007641852
189
587
1.019744907
94
292
0.9979201548
197
612
0.9862367575
103
320
0.9882922525
198
615
1.00990644
see also Comma of Didymus,
RAS, SCHISMA
Comma of Pythago-
References
Sloane, N. J. A. Sequences A046103 and A046104 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Diffeomorphism
A diffeomorphism is a MAP between MANIFOLDS which
is DlFFERENTlABLE and has a DlFFERENTlABLE inverse.
see also Anosov Diffeomorphism, Axiom A Diffeo-
morphism, Symplectic Diffeomorphism, Tangent
Map
Examples of difference equations often arise in DYNAM-
ICAL SYSTEMS. Examples include the iteration involved
in the MANDELBROT and JULIA Set definitions,
/(n + l) = /(n) 2 + c,
(3)
with c a constant, as well as the LOGISTIC EQUATION
/(n+l) = r/(n)[l-/(n)], (4)
with r a constant.
see also Finite Difference, Recurrence Relation
References
Batchelder, P. M. An Introduction to Linear Difference
Equations. New York: Dover, 1967.
Bellman, R. E. and Cooke, K. L. Differential- Difference
Equations. New York: Academic Press, 1963.
Beyer, W. H. "Finite Differences." CRC Standard Math-
ematical Tables, 28th ed. Boca Raton, FL: CRC Press,
pp. 429-460, 1988.
Brand, L. Differential and Difference Equations. New York:
Wiley, 1966.
Goldberg, S. Introduction to Difference Equations, with Il-
lustrative Examples from Economics, Psychology, and So-
ciology. New York: Dover, 1986.
Levy, H. and Lessman, F. Finite Difference Equations. New
York: Dover, 1992.
Richtmyer, R. D. and Morton, K. W. Difference Methods for
Initial-Value Problems, 2nd ed. New York: Interscience
Publishers, 1967.
Difference Operator
see Backward Difference, Forward Difference
Difference Quotient
A fM _ f(* + h)-f(x)
A h f(x) =
_____
h *
It gives the slope of the Secant Line passing through
f(x) and f(x + h). In the limit n — > 0, the difference
quotient becomes the PARTIAL DERIVATIVE
HmA. (h) /(_,y)=g.
Difference
The difference of two numbers ni and n_ is n\ — ri2,
where the MINUS sign denotes SUBTRACTION.
see also Backward Difference, Finite Difference,
Forward Difference
Difference Equation
A difference equation is the discrete analogue of a DIF-
FERENTIAL Equation. A difference equation involves
a FUNCTION with iNTEGER-valued arguments /(ra) in a
form like
f(n)-f(n-l)=g(n), (1)
where g is some FUNCTION. The above equation is the
discrete analog of the first-order ORDINARY DIFFEREN-
TIAL Equation
f'(x) = g(x). (2)
Difference Set
Let G be a Group of Order h and D be a set of k
elements of G. If the set of differences di — dj contains
every NONZERO element of G exactly A times, then D
is a (ft, fc, A)-difference set in G of Order n — k - X. If
A = 1, the difference set is called planar. The quadratic
residues in the GALOIS FIELD GF(11) form a difference
set. If there is a difference set of size A; in a group G,
then 2(2) must be a multiple of \G\ — 1, where (£) is a
Binomial Coefficient.
see also BRUCK-RYSER-CHOWLA THEOREM, FIRST
Multiplier Theorem, Prime Power Conjecture
References
Gordon, D. M. "The Prime Power Conjecture is True
for n < 2,000,000." Electronic J. Combinatorics 1,
R6, 1-7, 1994. http://www.combinatorics.org/Volume_l/
volume! .html#R6.
430
Difference of Successes
Differential Equation
Difference of Successes
If Xi/ni and X2/TI2 are the observed proportions from
standard NORMALLY DISTRIBUTED samples with pro-
portion of success 0, then the probability that
_ Xi X2
ri\ ri2
will be as great as observed is
Ps = l
7 '
Jo
cp(t) dt,
where
0~w
v^GFI)
e~
Xi -J- X2
Til + Tl2
(1)
(2)
(3)
(4)
(5)
Here, 6 is the Unbiased Estimator. The Skewness
and KURTOSIS of this distribution are
71
72
(rai -n 2 ) 2 1-40(1-0)
run 2 (ni 4- n 2 ) 0(1 - 0)
m 2 -n 1 n 2 +n 2 2 1-60(1-0)
mn 2 (ni +n 2 ) 0(1 - §)
(6)
(7)
Difference Table
A table made by subtracting adjacent entries in a se-
quence, then repeating the process with those numbers.
see also Finite Difference, Quotient-Difference
Table
Different
Two quantities are said to be different (or "unequal") if
they are not EQUAL.
The term "different" also has a technical usage related to
Modules. Let a Module M in an Integral Domain
Di for R(y/D) be expressed using a two-element basis
as
m = Ki,6],
where £1 and £2 are in D\, Then the different of the
Module is defined as
A = A(M)
6 £2
6^-^i6-
The different A ^ Iff £* and £ 2 are linearly indepen-
dent. The Discriminant is denned as the square of the
different.
see also Discriminant (Module), Equal, Module
References
Cohn, H. Advanced Number Theory. New York: Dover,
pp. 72-73, 1980.
Different Prime Factors
see Distinct Prime Factors
Differentiable
A FUNCTION is said to be differentiable at a point if its
Derivative exists at that point. Let z = x + iy and
f(z) ~ u(x, y)-\-iv(x i y) on some region G containing the
point zo. If f(z) satisfies the Cauchy-Riemann Equa-
tions and has continuous first PARTIAL DERIVATIVES
at zo, then f'(zo) exists and is given by
/'(*<>)
lim
Z^ZQ
/(*)-/(*>)
z - Zo
and the function is said to be Complex Differen-
tiable. Amazingly, there exist CONTINUOUS FUNC-
TIONS which are nowhere differentiable. Two exam-
ples are the BLANCMANGE FUNCTION and WeierstraB
Function.
see also BLANCMANGE FUNCTION, CAUCHY-RlEMANN
Equations, Complex Differentiable, Continuous
Function, Derivative, Partial Derivative, Wei-
erstraB Function
Differentiable Manifold
see Smooth Manifold
Differential
A Differential 1-Form.
see also Exact Differential, Inexact Differen-
tial
Differential Calculus
That portion of "the" Calculus dealing with Deriva-
tives.
see also INTEGRAL CALCULUS
Differential Equation
An equation which involves the Derivatives of a func-
tion as well as the function itself. If Partial Deriva-
tives are involved, the equation is called a PARTIAL
Differential Equation; if only ordinary Deriva-
tives are present, the equation is called an Ordinary
Differential Equation. Differential equations play
an extremely important and useful role in applied math,
engineering, and physics, and much mathematical and
numerical machinery has been developed for the solution
of differential equations.
see also INTEGRAL EQUATION, ORDINARY DIFFEREN-
TIAL Equation, Partial Differential Equation
References
Arfken, G. "Differential Equations." Ch. 8 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 437-496, 1985.
Dormand, J. R. Numerical Methods for Differential Equa-
tions: A Computational Approach. Boca Raton, FL: CRC
Press, 1996.
Differential Form
Differentiation 43 1
Differential Form
see Differential &-Form
Differential Geometry
Differential geometry is the study of RlEMANNIAN MAN-
IFOLDS. Differential geometry deals with metrical no-
tions on Manifolds, while Differential Topology
deals with those nonmetrical notions of MANIFOLDS,
see also Differential Topology
References
Eisenhart, L. P. A Treatise on the Differential Geometry of
Curves and Surfaces. New York: Dover, 1960.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press. 1993.
Kreyszig, E. Differential Geometry. New York: Dover, 1991.
Lipschutz, M. M. Theory and Problems of Differential Ge-
ometry. New York: McGraw-Hill, 1969.
Spivak, M. A Comprehensive Introduction to Differential Ge-
ometry, 2nd ed, 5 vols. Berkeley, CA: Publish or Perish
Press, 1979.
Struik, D. J. Lectures on Classical Differential Geometry.
New York: Dover, 1988.
We at herb urn, C. E. Differential Geometry of Three Dimen-
sions, 2 vols. Cambridge, England: Cambridge University
Press, 1961.
Differential ft- Form
A differential fc-form is a TENSOR of RANK k which is
antisymmetric under exchange of any pair of indices.
The number of algebraically independent components in
n-D is ( n ), where this is a BINOMIAL COEFFICIENT. In
particular, a 1-form (often simply called a "differential" )
is a quantity
w = 6i dx\ + 62 dx2, (1)
where b\ = bi{xi,x 2 ) and b 2 — &2(#i,#2) are the com-
ponents of a Covariant Tensor. Changing variables
from x to y gives
where
i=l j = l
*i=i>
dxi
dyj'
(2)
(3)
which is the covariant transformation law. 2-forms can
be constructed from the WEDGE PRODUCT of 1-forms.
Let
0i = 61 dxi 4- 62 dx2 (4)
62 = c\ dx\ 4- C2 dx2,
(5)
then di A 02 is a 2-form denoted a; 2 . Changing variables
211(3/1,2/2) to x 2 (2/1,3/2) gives
dxx = -^dyx + ~±dy 2 (6)
dyi dyi
dx\ A dx2
dxi 8x2 dx
dyi dy 2 dy 2
d{x u x 2 )
1 0x 2 \
2 dyi J
dyi A dy 2
d(yi,y2)
dyi Ady 2 .
(8)
Similarly, a 4-form can be constructed from WEDGE
PRODUCTS of two 2-forms or four 1-forms
uj 4 = u>i 2 A u>2 2 — (uji 1 A co 2 l ) A (cvs 1 A W4 1 ). (9)
see also Angle Bracket, Bra, Exterior Deriva-
tive, Ket, One-Form, Symplectic Form, Wedge
Product
References
Weintraub, S. H, Differential Forms: A Complement to Vec-
tor Calculus. San Diego, CA: Academic Press, 1996.
Differential Operator
The OPERATOR representing the computation of a DE-
RIVATIVE,
d
D
dx
The second derivative is then denoted D 2 , the third I) 3 ,
etc. The INTEGRAL is denoted D~ x .
see also Convective Derivative, Derivative, Frac-
tional Derivative, Gradient
Differential Structure
see Exotic R4, Exotic Sphere
Differential Topology
The motivating force of TOPOLOGY, consisting of the
study of smooth (differentiable) MANIFOLDS. Differen-
tial topology deals with nonmetrical notions of MAN-
IFOLDS, while Differential Geometry deals with
metrical notions of MANIFOLDS.
see also DIFFERENTIAL GEOMETRY
References
Dieudonne, J. A History of Algebraic and Differential Topol-
ogy: 1900-1960. Boston, MA: Birkhauser, 1989.
Munkres, J. R. Elementary Differential Topology. Princeton,
NJ: Princeton University Press, 1963.
Differentiation
The computation of a Derivative.
see also CALCULUS, DERIVATIVE, INTEGRAL, INTEGRA-
TION
0x2 dx2
dx 2 = -r — dy-i + ——dy 2 ,
dyi dy 2
(?)
432 Digamma Function
Digamma Function
Digamma Function
Two notations are used for the digamma function. The
^(z) digamma function is defined by
*M s S tar W = ?$.
(i)
where Y is the Gamma Function, and is the
function returned by the function PolyGammaCz] in
Mathematical (Wolfram Research, Champaign, IL).
The F digamma function is defined by
i^),-lnz!
(2)
and is equal to
F(z) = *(z + 1). (3)
From a series expansion of the FACTORIAL function,
F(z) = — lim [In n! + z In n
(XZ n—>oo
- ]n{z + 1) - ln(z + 2) - . . . - ln(z + n)] (4)
= limflnn-^ L__..._^_)
n->-oo V z + 1 z + 2 z + nJ
oo
^-^ Vz + n nJ
= -7 + £
n(n + z)
oo
— in z + > - — — .
2z ^ 2nz 2
n-l
(5)
(6)
(7)
(8)
where j is the Euler-Mascheroni Constant and B 2n
are Bernoulli Numbers.
The nth DERIVATIVE of *(z) is called the POLYGAMMA
FUNCTION and is denoted ip n (z). Since the digamma
-f(T-lS0*
(9)
function is the zeroth derivative of ^(z) (i.e., the func-
tion itself), it is also denoted ipo(z).
The digamma function satisfies
For integral z = n,
71-1
*(") = -7 + X! I = -7 + tf»-i, (10)
where 7 is the Euler-Mascheroni Constant and H n
is a HARMONIC Number. Other identities include
d^_ y- 1
dz ~ 2^f ( z +
{z + nY
71 = U
*(1 - z) - *(z) = TTCOt(wz)
*(* + 1) = ¥(*) + J
*(2z) = i*(z) + i*(2 + i) + In 2.
Special values are
¥(i) = - 7 -21n2
*(1) - -7.
At integral values,
(11)
(12)
(13)
(14)
(15)
(16)
(17)
V-o(n + l) = -7 + ^p
k = X
and at half-integral values,
n
Vo(|±n) = -ln(4 7 ) + 2^^ T . (18)
fc=i
At rational arguments, ipo(p/q) is given by the explicit
equation
ip (?\ =-7-ln(2g)-l7rcot(^7rj
+2 g cos (^ ln [ sin ^
(19)
for < p < q (Knuth 1973). These give the special
values
M§) = -7 -2 In 2 (20)
Ml) = £(-67-7n/3-91n3) (21)
^o(!) = !(-67 + *V3-91n3) (22)
^0(J) = -7-5ff-3In2 (23)
^)(|) = |(-27 + ir-61n2) (24)
Vo(l) = -7, (25)
Digimetic
Digitadition 433
where 7 is the Euler-Mascheroni Constant. Sums
and differences of ipi(r/s) for small integral r and s can
be expressed in terms of CATALAN'S CONSTANT and it.
see also Gamma Function, Harmonic Number,
Hurwitz Zeta Function, Polygamma Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Psi (Digamma)
Function." §6.3 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 258-259, 1972.
Arfken, G. "Digamma and Polygamma Functions." §10.2 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 549-555, 1985.
Knuth, D. E. The Art of Computer Programming, Vol 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, p. 94, 1973.
Spanier, J. and Oldham, K. B. "The Digamma Function
ip{x)y Ch. 44 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 423-434, 1987.
Digimetic
A CRYPTARITHM in which DIGITS are used to represent
other Digits.
Digit
The number of digits D in an INTEGER n is the number
of numbers in some base (usually 10) required to repre-
sent it. The numbers 1 to 9 are therefore single digits,
while the numbers 10 to 99 are double digits. Terms such
as "double-digit inflation" are occasionally encountered,
although this particular usage has thankfully not been
needed in the U.S. for some time. The number of (base
10) digits in a number n can be calculated as
D= [log 10 n + lj,
where [x\ is the FLOOR FUNCTION.
see also 196-Algorithm, Additive Persistence,
Digitadition, Digital Root, Factorion, Figures,
Length (Number), Multiplicative Persistence,
Narcissistic Number, Scientific Notation, Sig-
nificant Digits, Smith Number
If the process is generalized so that the fcth (instead of
first) powers of the digits of a number are repeatedly
added, a periodic sequence of numbers is eventually ob-
tained for any given starting number n. If the original
number n is equal to the sum of the kth powers of its dig-
its, it is called a NARCISSISTIC NUMBER. If the original
number is the smallest number in the eventually periodic
sequence of numbers in the repeated fc-digitaditions, it
is called a RECURRING DIGITAL Invariant. Both Nar-
cissistic Numbers and Recurring Digital Invari-
ants are relatively rare.
The only possible periods for repeated 2- digit adit ions
are 1 and 8, and the periods of the first few positive
integers are 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, .... The possi-
ble periods p for n-digitaditions are summarized in the
following table, together with digitaditions for the first
few integers and the corresponding sequence numbers.
n Sloane ps
n-Digitaditions
2 031176
3 031178
4 031182
5 031186
6 031195
7 031200
8 031211
9 031212
10 031212
1, 8
1,2,3
1,2,7
1, 2, 4, 6,
10, 12, 22, 28
1, 2, 3, 4,
10, 30
1, 2, 3, 6,
12, 14, 21, 27,
30, 56, 92
1, 25, 154
1, 2, 3, 4, 8,
10, 19, 24, 28,
30, 80, 93
1, 6, 7, 17,
81, 123
1, 8, 8, 8, 8, 8, 1, 8, 8, ...
1,1,1,3,1,1,1,1,1,...
1,7,7,7,7,7,7,7,7,...
1, 12, 22, 4, 10, 22, 28,
10, 22, 1, ...
1, 10, 30, 30, 30, 10, 10,
10, 3, 1, 10, ...
1, 92, 14, 30, 92, 56, 6,
92, 56, 1, 92, 27, ...
1, 25, 154, 154, 154, 154,
25, 154, 154, 1, 25, ...
1, 30, 93, 1, 19, 80, 4, 30,
80, 1, 30,93,4, 10, ...
1, 30, 93, 1, 19, 80, 4, 30,
80, 1, 30, 93, 4, 10, . . .
The numbers having period- 1 2-digitaded sequences are
also called Happy Numbers. The first few numbers
having period p n-digitaditions are summarized in the
following table, together with their sequence numbers.
Digitadition
Start with an Integer n, known as the Generator.
Add the Sum of the Generator's digits to the Gen-
erator to obtain the digitadition ri . A number can
have more than one GENERATOR. If a number has no
Generator, it is called a Self Number. The sum of
all numbers in a digitadition series is given by the last
term minus the first plus the sum of the DIGITS of the
last.
If the digitadition process is performed on n to yield its
digitadition n", on n" to yield n" ', etc., a single-digit
number, known as the DIGITAL ROOT of n, is eventually
obtained. The digital roots of the first few integers are
1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 9, 1, . . .
(Sloane's A010888).
434 Digitadition
Dihedral Angle
p Sloane Members
2
1
007770
2
8
031177
3
1
031179
3
2
031180
3
3
031181
4
1
031183
4
2
031184
4
7
031185
5
1
031187
5
2
031188
5
4
031189
5
6
031190
5
10
031191
5
12
031192
5
22
031193
5
28
031194
6
1
011557
6
2
031357
6
3
031196
6
4
031197
6
10
031198
6
30
031199
7
1
031201
7
2
031202
7
3
031203
7
6
7
12
031204
7
14
031205
7
21
031206
7
27
031207
7
30
031208
7
56
031209
7
92
031210
8
1
8
25
8
154
9
1
9
2
9
3
9
4
9
8
9
10
9
19
9
24
9
28
9
30
9
80
9
93
10
1
011557
10
6
10
7
10
17
10
81
10 123
1, 7, 10, 13, 19, 23, 28, 31, 32, . . .
2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, ..
1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, . . .
49, 94, 136, 163, 199, 244, 316, . . .
4, 13, 16, 22, 25, 28, 31, 40, 46, ..
1, 10, 12, 17, 21, 46, 64, 71, 100, .
66, 127, 172, 217, 228, 271, 282, .,
2, 3,4, 5, 6, 7, 8, 9, 11, 13, 14, . . .
1, 10, 100, 145, 154, 247, 274, ...
133, 139, 193, 199, 226, 262, . . .
4, 37, 40, 55, 73, 124, 142, ...
16, 61, 106, 160, 601, 610, 778, . . .
5, 8, 17, 26,35,44, 47, 50,53, ...
2, 11, 14, 20, 23, 29, 32, 38, 41, ..
3, 6, 9, 12, 15, 18, 21, 24, 27, . . .
7, 13, 19, 22, 25, 28, 31, 34, 43, ..
1, 10, 100, 1000, 10000, 100000, ..
3468, 3486, 3648, 3684, 3846, . . .
9, 13, 31, 37, 39, 49, 57, 73, 75, . .
255, 466, 525, 552, 646, 664, ...
2, 6, 7, 8, 11, 12, 14, 15, 17, 19, ..
3, 4, 5, 16, 18, 22, 29, 30, 33, . . .
1, 10, 100, 1000, 1259, 1295, ...
22, 202, 220, 256, 265, 526, 562, ..
124, 142, 148, 184, 214, 241, 259, ,
7, 70, 700, 7000, 70000, 700000, . .
17, 26, 47, 59, 62, 71, 74, 77, 89, .
3, 30, 111, 156, 165, 249, 294, . . .
19, 34, 43, 91, 109, 127, 172, 190,
12, 18, 21, 24, 39, 42, 45, 54, 78, .
4, 13, 16, 25, 28, 31, 37, 40, 46, ..
6, 9, 15, 27, 33, 36,48, 51, 57, . . .
2, 5, 8, 11, 14, 20, 23, 29, 32, 35, .
1, 10, 14, 17, 29, 37, 41, 71, 73, ..
2, 7, 11, 15, 16, 20, 23, 27, 32, . . .
3, 4, 5, 6, 8, 9, 12, 13, 18, 19, . . .
1, 4, 10, 40, 100, 400, 1000, 1111,
127, 172, 217, 235, 253, 271, 325, .
444, 4044, 4404, 4440, 4558, . . .
7, 13, 31, 67, 70, 76, 103, 130, ...
22, 28, 34, 37, 43, 55, 58, 73, 79, .
14, 38, 41,44,83, 104, 128, 140, ..
5, 26, 50, 62, 89, 98, 155, 206, ...
16,61, 106, 160,337,373,445, ...
19, 25, 46, 49, 52, 64, 91, 94, . . .
2, 8, 11, 17, 20, 23, 29, 32, 35, . . .
6, 9, 15, 18, 24, 33, 42, 48, 51, . . .
3, 12, 21, 27, 30, 36, 39, 45, 54, . . .
1, 10, 100, 1000, 10000, 100000, ..
266, 626, 662, 1159, 1195, 1519, ..
46, 58, 64, 85, 122, 123, 132, ...
2, 4, 5, 11, 13, 20, 31, 38, 40, . . .
17, 18, 37, 71, 73, 81, 107, 108, . . .
3, 6, 7, 8, 9, 12, 14, 15, 16, 19, . . .
Narcissistic Number, Recurring Digital Invari-
ant
Digital Root
Consider the process of taking a number, adding its DIG-
ITS, then adding the DIGITS of numbers 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 Per-
sistence 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 digital
roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1,
2, 3, 4, 5, 6, 7, 9, 1, . . . (Sloane's A010888). The digital
root of an INTEGER n can therefore be computed with-
out actually performing the iteration using the simple
congruence formula
n (mod 9)
9
n ^ (mod 9)
n = (mod 9).
see also ADDITIVE PERSISTENCE, DIGITADITION,
Kaprekar Number, Multiplicative Digital Root,
Multiplicative Persistence, Narcissistic Num-
ber, Recurring Digital Invariant, Self Number
References
Sloane, N. J. A. Sequences A010888 and A007612/M1114 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Digon
The Degenerate Polygon (corresponding to a Line
Segment) with Schlafli Symbol {2}.
see also LINE SEGMENT, POLYGON, TRIGONOMETRY
Values — n/2
Digraph
see Directed Graph
Dihedral Angle
The ANGLE between two PLANES. The dihedral angle
between the planes
see also 196-Algorithm, Additive Persistence,
Digit, Digital Root, Multiplicative Persistence,
cos#
Aix + Biy + Ciz + Di =
A 2 x + B 2 y + C 2 z + D 2 =
A1A2 + B1B2+C1C2
^Ai 2 +tf! 2 + CiVV +B 2 2 + C2 2 '
see also Angle, Plane, Vertex Angle
(i)
(2)
(3)
Dihedral Group
Dimension
435
Dihedral Group
A Group of symmetries for an n-sided Regular Poly-
gon, denoted D n . The Order of D n is 2n.
see also Finite Group — D 3 , Finite Group — £> 4
References
Arfken, G. "Dihedral Groups, D n ." Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
p. 248, 1985.
Lomont, J. S. "Dihedral Groups." §3.10.B in Applications of
Finite Groups. New York: Dover, pp. 78-80, 1987.
Dijkstra's Algorithm
An Algorithm for finding the shortest path between
two Vertices.
see also FLOYD'S ALGORITHM
Dijkstra Tree
The shortest path-spanning Tree from a Vertex of a
Graph.
Dilation
An Affine Transformation in which the scale is re-
duced. A dilation is also known as a CONTRACTION or
HOMOTHECY. Any dilation which is not a simple trans-
lation has a unique FIXED POINT. The opposite of a
dilation is an EXPANSION.
see also Affine Transformation, Expansion, Ho-
MOTHECY
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 94-95, 1967.
Dilemma
Informally, a situation in which a decision must be made
from several alternatives, none of which is obviously the
optimal one. In formal LOGIC, a dilemma is a spe-
cific type of argument using two conditional statements
which may take the form of a Constructive Dilemma
or a Destructive Dilemma.
see also CONSTRUCTIVE DILEMMA, DESTRUCTIVE
Dilemma, Monty Hall Problem, Paradox, Pris-
oner's Dilemma
Dilogarithm
A special case of the Polylogarithm Li n (z) for n = 2.
It is denoted Li2(^), or sometimes £2(2), and is defined
by the sum
Li 2 M=x;
k 2
or the integral
Li 2 (z) = J
ln(l - 1) dt
t
There are several remarkable identities involving the
Polylogarithm function.
see also Abel's Functional Equation, Polyloga-
rithm, Spence's Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Dilogarithm,"
§27.7 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 1004-1005, 1972.
Dilworth's Lemma
The Width of a set P is equal to the minimum num-
ber of Chains needed to Cover P. Equivalent ly, if a
set P of ab + 1 elements is PARTIALLY ORDERED, then
P contains a CHAIN of size a + 1 or an ANTICHAIN of
size 6 + 1. Letting JV be the Cardinality of P, W
the Width, and L the Length, this last statement
says N < LW. Dil worth's lemma is a generalization
of the Erdos-Szekeres Theorem. Ramsey's Theo-
rem generalizes Dilworth's Lemma.
see also Combinatorics, Erdos-Szekeres Theorem,
Ramsey's Theorem
Dilworth's Theorem
see Dilworth's Lemma
Dimension
The notion of dimension is important in mathematics
because it gives a precise parameterization of the con-
ceptual or visual complexity of any geometric object. In
fact, the concept can even be applied to abstract ob-
jects which cannot be directly visualized. For example,
the notion of time can be considered as one-dimensional,
since it can be thought of as consisting of only "now,"
"before" and "after." Since "before" and "after," re-
gardless of how far back or how far into the future they
are, are extensions, time is like a line, a 1-dimensional
object.
To see how lower and higher dimensions relate to each
other, take any geometric object (like a Point, Line,
CIRCLE, PLANE, etc.), and "drag" it in an opposing di-
rection (drag a Point to trace out a Line, a Line to
trace out a box, a Circle to trace out a Cylinder, a
Disk to a solid Cylinder, etc.). The result is an object
which is qualitatively "larger" than the previous object,
"qualitative" in the sense that, regardless of how you
drag the original object, you always trace out an ob-
ject of the same "qualitative size." The Point could be
made into a straight Line, a Circle, a Helix, or some
other Curve, but all of these objects are qualitatively
of the same dimension. The notion of dimension was
invented for the purpose of measuring this "qualitative"
topological property.
Making things a bit more formal, finite collections of ob-
jects (e.g., points in space) are considered 0-dimensionaL
Objects that are "dragged" versions of 0-dimensional
objects are then called 1-dimensional. Similarly, ob-
jects which are dragged 1-dimensional objects are 2-
dimensional, and so on. Dimension is formalized in
436
Dimension
Dini 's Surface
mathematics as the intrinsic dimension of a TOPO-
LOGICAL Space. This dimension is called the Lebes-
gue Covering Dimension (also known simply as the
Topological Dimension). The archetypal example
is EUCLIDEAN n-space R n , which has topological di-
mension n. The basic ideas leading up to this result
(including the DIMENSION INVARIANCE THEOREM, DO-
MAIN Invariance Theorem, and Lebesgue Cover-
ing Dimension) were developed by Poincare, Brouwer,
Lebesgue, Urysohn, and Menger.
There are several branchings and extensions of the no-
tion of topological dimension. Implicit in the notion
of the Lebesgue Covering Dimension is that dimen-
sion, in a sense, is a measure of how an object fills space.
If it takes up a lot of room, it is higher dimensional, and
if it takes up less room, it is lower dimensional. HAUS-
dorff Dimension (also called Fractal Dimension) is
a fine tuning of this definition that allows notions of ob-
jects with dimensions other than Integers. Fractals
are objects whose HAUSDORFF DIMENSION is different
from their TOPOLOGICAL DIMENSION.
The concept of dimension is also used in ALGEBRA, pri-
marily as the dimension of a VECTOR SPACE over a
Field. This usage stems from the fact that Vector
Spaces over the reals were the first Vector Spaces
to be studied, and for them, their topological dimension
can be calculated by purely algebraic means as the CAR-
DINALITY of a maximal linearly independent subset. In
particular, the dimension of a SUBSPACE of W 1 is equal
to the number of LINEARLY INDEPENDENT VECTORS
needed to generate it (i.e., the number of VECTORS in
its BASIS). Given a transformation A of R n ,
dim[Range(^)] + dim[Null(A)] = dim(R n ).
see also Capacity Dimension, Codimension, Corre-
lation Dimension, Exterior Dimension, Fractal
Dimension, Hausdorff Dimension, Hausdorff-
Besicovitch Dimension, Kaplan- Yorke Dimen-
sion, Krull Dimension, Lebesgue Covering Di-
mension, Lebesgue Dimension, Lyapunov Dimen-
sion, Poset Dimension, ^-Dimension, Similarity
Dimension, Topological Dimension
References
Abbott, E. A. Flatland: A Romance of Many Dimensions.
New York: Dover, 1992.
Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health
Research, 1993.
Manning, H. The Fourth Dimension Simply Explained, Mag-
nolia, MA: Peter Smith, 1990.
Manning, H. Geometry of Four Dimensions. New York:
Dover, 1956.
Neville, E. H. The Fourth Dimension. Cambridge, England:
Cambridge University Press, 1921.
Rucker, R. von Bitter. The Fourth Dimension: A Guided
Tour of the Higher Universes. Boston, MA: Houghton
Mifflin, 1984.
Sommerville, D. M. Y. An Introduction to the Geometry of
n Dimensions. New York: Dover, 1958.
Dimension Axiom
One of the Eilenberg-Steenrod Axioms. Let X be
a single point space. H n (X) = unless n = 0, in which
case H (X) = G where G are some Groups. The H are
called the COEFFICIENTS of the HOMOLOGY THEORY
H(-).
see also Eilenberg-Steenrod Axioms, Homology
(Topology)
Dimension Invariance Theorem
M. n is Homeomorphic to M m Iff n = m. This theorem
was first proved by Brouwer.
see also Domain Invariance Theorem
Dimensionality Theorem
For a finite GROUP of h elements with an ruth dimen-
sional ith irreducible representation,
2_, n * = ^*
Diminished Polyhedron
A Uniform Polyhedron with pieces removed.
Diminished Rhombicosidodecahedron
see Johnson Solid
Dini Expansion
An expansion based on the ROOTS of
x - n [xJUx) + HJ n (x)]=Q,
where J n (x) is a BESSEL FUNCTION OF THE FIRST
KIND, is called a Dini expansion.
see also BESSEL FUNCTION FOURIER EXPANSION
References
Bowman, F. Introduction to Bessel Functions. New York:
Dover, p. 109, 1958.
Dini's Surface
A surface of constant NEGATIVE CURVATURE obtained
by twisting a PSEUDOSPHERE and given by the paramet-
ric equations
x = a cos u sin v
y — a sin u sin v
z — a{cosf + ln[tan(|t;)]} + bu.
(i)
(2)
(3)
Dini's Test
Diophantine Equation 437
The above figure corresponds to a = 1, 6 = 0.2, u €E
[0,471-], and v€ (0,2].
see also PSEUDOSPHERE
References
Geometry Center. "Dini's Surface." http://www.geom.umn.
edu/zoo/diffgeom/surf space/dini/.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 494-495, 1993.
Nordstrand, T. "Dini's Surface." http://www.uib.no/
people/nf ytn/dintxt .htm.
Dini's Test
A test for the convergence of FOURIER SERIES. Let
<t> x {t) = f(x + t) + f{x -t)- 2/(x),
then if
/"
t/0
\4>xjt)\dt
t
is Finite, the Fourier Series converges to f{x) at x.
see also Fourier Series
References
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, pp. 65-68, 1991.
Dinitz Problem
Given any assignment of n-element sets to the n 2 loca-
tions of a square n x n array, is it always possible to
find a Partial Latin Square? The fact that such a
Partial Latin Square can always be found for a 2 x 2
array can be proven analytically, and techniques were
developed which also proved the existence for 4 x 4 and
6x6 arrays. However, the general problem eluded solu-
tion until it was answered in the affirmative by Galvin in
1993 using results of Janssen (1993ab) and F. Maffray.
see also Partial Latin Square
References
Chetwynd, A. and Haggkvist, R. "A Note on List-Colorings."
J. Graph Th. 13, 87-95, 1989.
Cipra, B. "Quite Easily Done." In What's Happening in the
Mathematical Sciences 2, pp. 41—46, 1994.
Erdos, P.; Rubin, A.; and Taylor, H. "Choosability in
Graphs." Congr. Numer. 26, 125-157, 1979.
Haggkvist, R. "Towards a Solution of the Dinitz Problem?"
Disc. Math. 75, 247-251, 1989.
Janssen, J. C. M. "The Dinitz Problem Solved for Rectan-
gles." Bull. Amer. Math. Soc. 29, 243-249, 1993a.
Janssen, J. C. M. Even and Odd Latin Squares. Ph.D. thesis.
Lehigh University, 1993b.
Kahn, J. "Recent Results on Some Not-So-Recent Hyper-
graph Matching and Covering Problems." Proceedings
of the Conference on Extremal Problems for Finite Sets.
Visegrad, Hungary, 1991.
Kahn, J. "Coloring Nearly-Disjoint Hypergraphs with n 4
o(n) Colors." J. Combin. Th. Ser. A 59, 31-39, 1992.
Diophantine Equation
An equation in which only INTEGER solutions are al-
lowed. Hilbert's 10th Problem asked if a technique
for solving a general Diophantine existed. A general
method exists for the solution of first degree Diophan-
tine equations. However, the impossibility of obtaining a
general solution was proven by Julia Robinson and Mar-
tin Davis in 1970, following proof of the result that the
equation n — F 2m (where F 2m is a FIBONACCI Num-
ber) is Diophantine by Yuri Matijasevic (Matijasevic
1970, Davis 1973, Davis and Hersh 1973, Matijasevic
1993).
No general method is known for quadratic or higher
Diophantine equations. Jones and Matijasevic (1982)
proved that no ALGORITHMS can exist to determine if
an arbitrary Diophantine equation in nine variables has
solutions. Ogilvy and Anderson (1988) give a number
of Diophantine equations with known and unknown so-
lutions.
D. Wilson has compiled a list of the smallest nth Pow-
ers which are the sums of n distinct smaller nth POW-
ERS. The first few are 3, 5, 6, 15, 12, 25, 40, . . . (Sloane's
A030052):
3 1 =:l 1 +2 1
5 2 = 3 2 + 4 2
6 3 = 3 3 + 4 3 + 5 3
15 4 = 4 4 + 6 4 + 8 4 + 9 4 + 14 4
12 5 = 4 5 + 5 5 4 6 5 4- 7 5 4- 9 5 4 ll 5
25 6 = l 6 + 2 6 4 3 6 + 5 6 4- 6 6 4 7 6 4 8 6 + 9 6 4- 10 6
4 12 6 4- 13 6 + 15 6 + 16 6 4 17 6 + 18 6 + 23 6
40 7 = l 7 4 3 7 + 5 7 4 9 7 4- 12 7 4 14 7 + 16 7 + 17 7
4 18 7 + 20 7 + 21 7 + 22 7 + 25 7 4- 28 7 + 39 7
84 8 = l 8 + 2 8 + 3 8 + 5 8 + 7 8 4 9 8 + 10 8 4 ll 8
4 12 8 + 13 8 4 14 8 4- 15 8 + 16 8 + 17 8 4 18 8
4 19 8 + 21 8 + 23 8 4 24 8 4 25 8 + 26 8 4 27 8
4 29 8 + 32 8 + 33 8 + 35 8 4 37 8 + 38 8 + 39 8
+ 41 8 4 42 8 + 43 8 + 45 8 + 46 8 + 47 8 + 48 8
+ 49 8 4- 51 8 4 52 8 4- 53 8 + 57 8 + 58 8 4 59 8
4 61 8 + 63 8 + 69 8 + 73 8
47 9 = X 9 + 2 9 + 4 9 + 7 9 + U 9 + U 9 + ^9 + lg 9
+ 26 9 + 27 9 + 30 9 + 31 9 + 32 9 + 33 9
+ 36 9 4 38 9 + 39 9 4 43 9
63 10 = l 10 4- 2 10 + 4 10 + 5 10 + 6 10 + 8 10 + 12 10
+ is 10 4 16 10 + 17 10 + 20 10 4 21 10 + 25 10
4 26 10 4 27 10 + 28 10 + 30 10 4 36 10 4 37 10
4 38 10 + 40 10 4 51 10 4 62 10 .
Diocles's Cissoid
see Cissoid of Diocles
see also abc Conjecture, Archimedes' Cat-
tle Problem, Bachet Equation, Brahmagupta's
438 Diophantine Equation
Diophantine Equation — 5th Powers
Problem, Cannonball Problem, Catalan's Prob-
lem, Diophantine Equation — Linear, Diophan-
tine Equation — Quadratic, Diophantine Equa-
tion — Cubic, Diophantine Equation — Quartic,
Diophantine Equation — 5th Powers, Diophan-
tine Equation — 6th Powers, Diophantine Equa-
tion — 7th Powers, Diophantine Equation — 8th
Powers, Diophantine Equation — 9th Powers,
Diophantine Equation — 10th Powers, Diophan-
tine Equation — nra Powers, Diophantus Prop-
erty, Euler Brick, Euler Quartic Conjecture,
Fermat's Last Theorem, Fermat Sum Theo-
rem, Genus Theorem, Hurwitz Equation, Markov
Number, Monkey and Coconut Problem, Multi-
grade Equation, p-adic Number, Pell Equation,
Pythagorean Quadruple, Pythagorean Triple
Diophantine Equation — 5th Powers
The 2-1 fifth-order Diophantine equation
A 5 + B 5 = C 5
(1)
is a special case of Fermat's Last Theorem with
n = 5, and so has no solution. No solutions to the
2-2 equation
A 5 + B 5 = C 5 + D 5 (2)
are known, despite the fact that sums up to 1.02 x 10 26
have been checked (Guy 1994, p. 140), improving on
the results on Lander et al. (1967), who checked up to
2.8 x 10 14 . (In fact, no solutions are known for POWERS
of 6 or 7 either.)
No solutions to the 3-1 equation
References
Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains. New York: Dover,
1966.
Carmichael, R. D. The Theory of Numbers, and Diophantine
Analysis. New York: Dover, 1959.
Chen, S. "Equal Sums of Like Powers: On the Integer Solu-
tion of the Diophantine System." http://www.nease.net/
-chin/eslp/.
Chen, S. "References." http://www.nease.net/-chin/eslp/
referenc.htm
Davis, M. "Hilbert's Tenth Problem is Unsolvable." Amer.
Math. Monthly 80, 233-269, 1973.
Davis, M. and Hersh, R. "Hilbert's 10th Problem." Sci.
Amer., pp. 84-91, Nov. 1973.
Dorrie, H. "The Fermat-Gauss Impossibility Theorem." §21
in 100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 96-104,
1965.
Guy, R. K. "Diophantine Equations." Ch. D in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 139-198, 1994.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.
Hunter, J. A. H. and Madachy, J. S. "Diophantos and All
That." Ch. 6 in Mathematical Diversions. New York:
Dover, pp. 52-64, 1975.
Ireland, K. and Rosen, M. "Diophantine Equations." Ch. 17
in A Classical Introduction to Modern Number Theory,
2nd ed. New York: Springer- Verlag, pp. 269-296, 1990.
Jones, J. P. and Matijasevic, Yu. V. "Exponential Diophan-
tine Representation of Recursively Enumerable Sets." Pro-
ceedings of the Herbrand Symposium, Marseilles, 1981.
Amsterdam, Netherlands: North-Holland, pp. 159—177,
1982.
Lang, S. Introduction to Diophantine Approximations, 2nd
ed. New York: Springer- Verlag, 1995.
Matijasevic, Yu. V. "Solution to of the Tenth Problem of
Hilbert." Mat. Lapok 21, 83-87, 1970.
Matijasevic, Yu. V. Hilbert's Tenth Problem. Cambridge,
MA: MIT Press, 1993.
Mordell, L. J. Diophantine Equations. New York: Academic
Press, 1969.
Nagel, T. Introduction to Number Theory. New York: Wiley,
1951.
Ogilvy, C. S. and Anderson, J. T. "Diophantine Equations."
Ch. 6 in Excursions in Number Theory. New York: Dover,
pp. 65-83, 1988.
Sloane, N. J. A. Sequence A030052 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
/ + B 5 + c 5
D>
(3)
are known (Lander et al. 1967), nor are any 3-2 solutions
up to 8 x 10 12 (Lander et al. 1967).
Parametric solutions are known for the 3-3 (Guy 1994,
pp. 140 and 142). Swinnerton-Dyer (1952) gave two
parametric solutions to the 3-3 equation but, forty years
later, W. Gosper discovered that the second scheme has
an unfixable bug. The smallest primitive 3-3 solutions
are
24 5 + 28 5 + 67 5 = 3 5 + 54 5 + 62 5
18 5 + 44 5 + 66 5
21 5 +43 5 + 76 5
13 5 + 51 5 + 64 5
8 5 + 62 5 + 68 5
56 5 + 67 5 + 83 5 = 53 5 + 72 5 + 81 5
49 5 + 75 5 + 107 5
39 5 + 92 5 + 100 5
(4)
(5)
(6)
(7)
(8)
(Moessner 1939, Moessner 1948, Lander et al. 1967).
For 4 fifth POWERS, we have the 4-1 equation
27 5 + 84 5 + 110 5 + 133 5
144°
(9)
(Lander and Parkin 1967, Lander et al, 1967), but it is
not known if there is a parametric solution (Guy 1994,
p. 140). Sastry's (1934) 5-1 solution gives some 4-2 so-
lutions. The smallest primitive 4-2 solutions are
4 5 + 10 5 + 20 5 + 28 5
5 5 + 13 5 + 25 5 + 37 5
26 5 + 29 5 + 35 5 + 50 5
5 5 + 25 5 + 62 5 + 63 5
6 5 + 50 5 + 53 5 + 82 5
56 5 + 63 5 + 72 5 + 86 5
44 5 + 58 5 + 67 5 + 94 5
ll 5 + 13 5 + 37 5 +99 5
48 5 + 57 5 + 76 5 + 100 5
58 5 + 76 5 + 79 5 4- 102 5
3 5 + 29 5
(10)
12 5 + 38 5
(11)
28 5 + 52 5
(12)
61 5 + 64 5
(13)
16 5 + 85 5
(14)
31 5 + 96 5
(15)
14 5 + 99 5
(16)
63 5 + 97 5
(17)
25 5 + 106 5
(18)
54* + 111 5
(19)
Diophantine Equation — 5th Powers
Diophantine Equation — 6th Powers 439
(Rao 1934, Moessner 1948, Lander et al 1967).
A two-parameter solution to the 4-3 equation was given
by Xeroudakes and Moessner (1958). Gloden (1949) also
gave a parametric solution. The smallest solution is
l 5 + 8 5 + 14 5 + 27 5 = 3 5 + 22 5 + 25 5 (20)
(Rao 1934, Lander et al. 1967). Several parametric so-
lutions to the 4-4 equation were found by Xeroudakes
and Moessner (1958). The smallest 4-4 solution is
5 5 + 6 5 + 6 5 + 8 5 = 4 5 + 7 5 + 7 5 + 7 5
(21)
(Rao 1934, Lander et al. 1967). The first 4-4-4 equation
3 5 + 48 5 + 52 5 + 61 5 = 13 5 + 36 5 + 51 5 + 64 5
(Rao 1934, Lander et al. 1967).
The 6-1 equation has solutions
4 5 +5 5 +6 5 + 7 5 + 9 5 -hll 5
5 5 + 10 5 + ll 5 + 16 5 + 19 5 + 29 5
15 5 + 16 5 + 17 5 + 22 5 + 24 5 + 28 5
13 5 + 18 5 + 23 5 + 31 5 + 36 5 4- 66 5
7 5 + 20 5 + 29 5 + 31 5 + 34 5 + 66 5
22 5 + 35 5 + 48 5 + 58 5 + 61 5 + 64 5
4 5 + 13 5 4- 19 5 + 20 5 + 67 5 + 96 5
6 5 + 17 5 + 60 5 + 64 5 + 73 5 + 89 5
12 D
= 30 5
32 5
= 67 5
= 67 5
:78 5
99 5
99 5
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(Martin 1887, 1888, Lander and Parkin 1967, Lander et
al 1967).
18 5 + 36 5 + 44 5 + 66 5 (22) The smallest 7-1 solution is
(Lander et al 1967).
I 5 + 7 5 + 8 5 + 14 5 + 15 5 + 18 5 + 20 5
23 a
(50)
Sastry (1934) found a 2-parameter solution for 5-1 equa-
tions
*5\5
25v 5 ) 5
(75v* - uy + (u & + 25v D ) D + (u
+(10uV) 5 + (50m; 4 ) 5 = (u 5 + 75u 5 ) 5 (23)
(quoted in Lander and Parkin 1967), and Lander and
Parkin (1967) found the smallest numerical solutions.
Lander et al. (1967) give a list of the smallest solutions,
the first few being
19 5 + 43 5 + 46 5 + 47 5 + 67 5 = 72 5
21 5 + 23 5 + 37 5 + 79 5 + 84 5
94"
t + 43 & + 57 & + 80 5 + 100 5 = 107 &
8 5 + 120 5 + 191 5 + 259 5 + 347 5
365 5
79" + 202" + 258 + 261 + 395 = 415
427 D
4 5 + 26 5 + 139 5 + 296 5 + 412 5
31 5 + 105 5 + 139 5 4- 314 5 + 416 5
54 5 + 91 5 + 101 5 + 404 5 4- 430 5 :
19 5 4- 201 5 4- 347 5 4- 388 5 + 448 5 :
159 5 4- 172 5 4- 200 5 + 356 5 + 513 5 =
218 5 + 276 5 + 385 5 + 409 5 4- 495 5 =
:435°
:480 5
:503 5
530 5
553 5
2 5 + 298 5 + 351 5 + 474 5 + 500 5 = 575 5
(Lander and Parkin 1967, Lander et al 1967).
The smallest primitive 5-2 solutions are
4 5 4- 5 5 4- 7 5 4- 16 5 + 21 5 = l 5 + 22 5
5 4- ll 5 4- 14 5 + 18 5 + 30 5 = 23 5 + 29 5
16 5 + 38 5
24 5 + 42 5
30 5 + 44 5
36 5 + 42 5
10 5 + 14 5 + 26 5 + 31 5 4- 33 5
4 5 + 22 5 + 29 5 + 35 5 4- 36 5
8 5 + 15 5 + 17 5 + 19 5 ■
•45 a
5 5 + 6 5 + 26 5 + 27 5 + 44 5
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(Lander et al 1967).
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, p. 95, 1994.
Gloden, A. "Uber mehrgeradige Gleichungen." Arch. Math.
1, 482-483, 1949.
Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 139-144, 1994.
Lander, L. J. and Parkin, T. R. "A Counterexample to Eu-
ler's Sum of Powers Conjecture." Math. Comput. 21,101-
103, 1967.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Martin, A. "Methods of Finding nth-Power Numbers Whose
Sum is an nth Power; With Examples." Bull. Philos. Soc.
Washington 10, 107-110, 1887.
Martin, A. Smithsonian Misc. Coll. 33, 1888.
Martin, A. "About Fifth-Power Numbers whose Sum is a
Fifth Power." Math. Mag. 2, 201-208, 1896.
Moessner, A. "Einige numerische Identitaten," Proc. Indian
Acad. Sci. Sect. A 10, 296-306, 1939.
Moessner, A. "Alcune richerche di teoria dei numeri e prob-
lemi diofantei." Bol. Soc. Mat. Mexicana 2, 36-39, 1948.
Rao, K. S. "On Sums of Fifth Powers." J. London Math.
Soc. 9, 170-171, 1934.
Sastry, S. "On Sums of Powers." J. London Math. Soc. 9,
242-246, 1934.
Swinnerton-Dyer, H. P. F. "A Solution of A 5 + B 5 + C 5 =
£>5 + £5 + F s» p roc Cambridge Phil. Soc. 48, 516-518,
1952.
Xeroudakes, G. and Moessner, A. "On Equal Sums of Like
Powers." Proc. Indian Acad. Sci. Sect A 48, 245-255,
1958.
Diophantine Equation-
The 2-1 equation
-6th Powers
A 6 +B 6
(1)
440 Diophantine Equation — 6th Powers
is a special case of Fermat's Last Theorem with n =
6, and so has no solution. Ekl (1996) has searched and
found no solutions to the 2-2
A 6 +B 6
C b 4 D b
(2)
with sums less than 7.25 x 10 26 .
No solutions are known to the 3-1 or 3-2 equations. How-
ever, parametric solutions are known for the 3-3 equa-
tion
A 6 + B 6 + C 6 = D 6 + E Q + F 6 (3)
(Guy 1994, pp. 140 and 142). Known solutions are
10 D 4 15° + 23 d
15 6 4 52 6 + 65 6
3 6 + 19 6 + 22 6
36 6 + 37 6 + 67 6
33 6 4 47 6 + 74 6 = 23 6 4 54 6 4 73 6
32 6 + 43 6 + 81 6
37 6 + 50 6 + 81 6
25 6 + 62 6 + 138 6
51 6 4 113 6 + 136 6 = 40 6 + 125 6 + 129
71 6 + 92 6 + 147 6
111 6 + 121 6 -h 230 6
75 6 + 142 6 4 245 6
: 3 d + 55° + 80 D
:ll 6 + 65 6 + 78 6
: §2 6 4 92 6 + 135 6
l 6 + 132 6 + 133 6
26 6 + 169 6 + 225 6
14 6 4 163 6 + 243 6
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(Rao 1934, Lander et al 1967).
No solutions are known to the 4-1 or 4-2 equations. The
smallest primitive 4-3 solutions are
41 6 + 58 6 4 73 6 = 15 6 + 32 6 -f 65 6 4 70 6
61 6 4- 62 6 4 85 6 = 52 6 + 56 6 4 69 6 4 83 6
61 6 4 74 6 + 85 6 = 26 6 + 56 6 4 71 6 4 87 6
ll 6 4 88 6 + 90 6 = 21 6 4- 74 6 4 78 6 4 92 6
26 6 4 83 6 4- 95 6 = 23 6 4 24 6 4 28 6 + 101
(14)
(15)
(16)
(17)
(18)
(Lander et al. 1967). Moessner (1947) gave three para-
metric solutions to the 4-4 equation. The smallest 4-4
solution is
2 6 4 2 6 4 9 6 + 9 6 = 3 6 + 5 6 4 6 6 4 10 6 (19)
(Rao 1934, Lander et al 1967). The smallest 4-4-4 so-
lution is
I 6 + 34 6 + 49 6 + in 6 = 7 6 4 43 6 4 69 6 4 HO 6
= 18 6 + 25 6 4 77 6 4 109 6 (20)
(Lander et al. 1967).
No n-1 solutions are known for n < 6 (Lander et al.
1967). No solution to the 5-1 equation is known (Guy
1994, p. 140) or the 5-2 equation.
No solutions are known to the 6-1 or 6-2 equations.
Diophantine Equation — 6th Powers
The smallest 7-1 solution is
74 6 4 234 6 4 402 6 + 474 6 4 702 6 + 894 6 + 1077 6 = 1141 6
(21)
(Lander et al 1967). The smallest 7-2 solution is
18 6 4 22 6 4 36 6 + 58 6 4 69 6 4 78 6 4 78 6 = 56 6 4 91 6 (22)
(Lander et al 1967).
The smallest primitive 8-1 solutions are
8 6 4 12 6 4 30 6 + 78 6 4 102 6
4138 6 4 165 6 4 246 6 = 251 6 (23)
48 6 4- HI 6 + 156 6 + 186 6 + 188 6
4228 6 4 240 6 4 426 6 = 431 6 (24)
93 6 + 93 6 + 195 6 + 197 6 4 303 6
4303 6 4 303 6 4 411 6 = 440 6 (25)
219 6 4 255 6 4 261 6 + 267 6 4 289 6
4351 6 4 351 6 + 351 6 = 440 6 (26)
12 6 + 66 6 4-138 6 + 174 6 + 212 6
4288 6 + 306 6 + 441 6 = 455 6 (27)
12 6 4- 48 6 4 222 6 4 236 6 + 333 6
4384 6 + 390 6 + 426 6 = 493 6 (28)
66 6 + 78 6 4 144 6 4 228 6 + 256 6
4288 6 + 435 6 + 444 6 = 499 6 (29)
16 6 4 24 6 4 60 6 4- 156 6 + 204 6
4276 6 + 330 6 + 492 6 = 502 6 (30)
61 6 + 96 6 4 156 6 4 228 6 + 276 6
4318 6 + 354 6 4- 534 6 = 547 6 (31)
170 6 4 177 6 4 276 6 4 312 6 + 312 6
4408 6 + 450 6 + 498 6 = 559 6 (32)
60 6 4 102 6 + 126 6 4- 261 6 4 270 6
4338 6 + 354 6 4- 570 6 = 581 6 (33)
57 6 4 146 6 + 150 6 + 360 6 4 390 6
4402 6 + 444 6 + 528 6 = 583 6 (34)
33 6 4 72 6 4 122 6 + 192 6 + 204 6
4390 6 4- 534 6 + 534 6 = 607 6 (35)
12 6 4 90 6 4 114 6 4 H4 6 + 273 6
4306 6 4- 492 5 + 592 6 = 623 6 (36)
(Lander et al. 1967). The smallest 8-2 solution is
8 6 4 10 6 4 12 6 4 15 6 4 24 6 + 30 6 + 33 6 4 36 6 = 35 6 4 37 6
(37)
(Lander et al 1967).
The smallest 9-1 solution is
l 6 4 17 6 4 19 6 4 22 6 4 31 6 4 37 6 4 37 6 + 41 6 4 49 6 = 54 6
(38)
Diophantine Equation — 7th Powers
(Lander et al. 1967). The smallest 9-2 solution is
l 6 + 5 6 + 5 6 + 7 6 4 13 6 4 13 6 4 13 6 4 17 6 4 19 6 = 6 6 4 21 6
(39)
(Lander et al. 1967).
The smallest 10-1 solution is
2 6 +4 6 + 7 6 + 14 6 + 16 6 + 26 6 + 26 6 + 30 6 +32 6 +32 6 = 39 6
(40)
(Lander et al 1967). The smallest 10-2 solution is
l 6 + l 6 + l 6 +4 6 +4 6 +7 6 +9 6 + ll 6 + ll 6 + ll 6 = 12 6 + 12 6
(41)
(Lander et al 1967).
The smallest 11-1 solution is
2 6 + 5 6 +5 6 + 5 6 + 7 6 + 7 6 +9 6 +9 6 + 10 6 + 14 6 + 17 6 = ^
(Lander et al. 1967).
There is also at least one 16-1 identity,
(42)
l 6 + 2 6 4 4 6 4 5 6 + 6 6 4 7 6 4 9 6 4 12 6 + 13 6 4 15 6
4 16 6 + 18 6 4 20 6 + 21 6 + 22 6 4 23 6 = 28 6 (43)
(Martin 1893). Moessner (1959) gave solutions for 16-1,
18-1, 20-1, and 23-1.
References
Ekl, R. L. "Equal Sums of Four Seventh Powers." Math.
Comput 65, 1755-1756, 1996.
Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 139-144, 1994.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Martin, A. "On Powers of Numbers Whose Sum is the Same
Power of Some Number." Quart. J. Math. 26, 225-227,
1893.
Moessner, A. "On Equal Sums of Like Powers." Math. Stu-
dent 15, 83-88, 1947.
Moessner, A. "Einige zahlentheoretische Untersuchungen
und diophantische Probleme." Glasnik Mat.-Fiz. Astron.
Drustvo Mat. Fiz. Hrvatske Ser. 2 14, 177-182, 1959.
Rao, S. K. "On Sums of Sixth Powers." J. London Math.
Soc. 9, 172-173, 1934.
Diophantine Equation — 7th Powers
The 2-1 equation
A 7 4 B r = C 7
(1)
is a special case of Fermat's Last Theorem with
n = 7, and so has no solution. No solutions to the
2-2 equation
A 7 + B 7 = C 7 + D 7 (2)
Diophantine Equation — 7th Powers 441
No solutions to the 3-1 or 3-2 equations are known, nei-
ther are solutions to the 3-3 equation
A 7 + B 7 + C 7 = D 7 4 E 7 4 F 7
(3)
(Ekl 1996).
are known
No 4-1, 4-2, or 4-3 solutions are known. Guy (1994,
p. 140) asked if a 4-4 equation exists for 7th POWERS.
An affirmative answer was provided by (Ekl 1996),
149 7 4 123 7 + 14 7 4 10 7 = 146 7 + 129 7 4 90 7 + 15 7 (4)
194 7 + 150 7 + 105 7 + 23 7 = 192 7 -f 152 7 + 132 7 + 38 7 . (5)
A 4-5 solution is known.
No 5-1, 5-2, or 5-3 solutions are known. Numerical so-
lutions to the 5-4 equation are given by Gloden (1948).
The smallest 5-4 solution is
3 7 + ll 7 + 26 7 + 29 7 + 52 7 = 12 7 + 16 7 + 43 7 + 50 7 (6)
(Lander et al. 1967). Gloden (1949) gives parametric
solutions to the 5-5 equation. The first few 5-5 solutions
are
8 7 + 8 7 + 13 7 + 16 7 + 19 7
= 2 7 + 12 7 + 15 7 + 17 7 + 18 7 (7)
4 7 + 8 7 + 14 7 + 16 7 + 23 7
= 7 7 + 7 7 + 9 7 + 20 7 + 22 7 (8)
ll 7 + 12 7 + 18 7 + 21 7 + 26 7
= 9 7 + 10 7 + 22 7 + 23 7 + 24 7 (9)
6 T + 12 7 + 20 T + 22 7 4 27 7
= 10 7 4 13 7 4 13 7 4 25 7 4 26 7 (10)
3 7 4 13 7 4 17 7 4 24 7 4 38 7
= 14 7 4 26 7 4 32 7 4 32 7 4 33 7 (11)
(Lander et al. 1967).
No 6-1, 6-2, or 6-3 solutions are known. A parametric
solution to the 6-6 equation was given by Sastry and Rai
(1948). The smallest is
2 7 43 7 46 7 46 7 410 7 413 7 = 1 7 41 7 47 7 47 7 412 7 412 7
(12)
(Lander et al. 1967).
There are no known solutions to the 7-1 equation (Guy
1994, p. 140). A 7 2 -2 solution is
2 7 4 26 7
= 4 7 4 8 7 + 13 7 + 14 7 4 14 7 4 16 7 + 18 7 4 22 7 4 23 7 4 23 7
= 7 7 4 7 7 4 9 7 4 13 7 4 14 7 4 18 7 4 20 7 4 22 7 4 22 7 4 23 7
(13)
442 Diophantine Equation — 8th Powers
(Lander et al 1967). The smallest 7-3 solution is
7 7 +7 7 + 12 7 + 16 7 +27 7 +28 7 +31 T = 26 7 +30 7 +30 7 (14)
(Lander et al 1967).
The smallest 8-1 solution is
12 7 + 35 7 + 53 7 + 58 7 + 64 7 + 83 7 + 85 7 + 90 7 = 102 7 (15)
(Lander et al. 1967). The smallest 8-2 solution is
5 7 +6 7 +7 7 + 15 7 + 15 7 +20 7 +28 7 +31 7 = io 7 +33 7 (16)
(Lander et al. 1967).
The smallest 9-1 solution is
6 7 + 14 7 + 20 7 + 22 7 + 27 7 + 33 7 + 41 7 + 50 7 + 59 7 = 62 7
(17)
(Lander et al 1967).
References
Ekl, R. L. "Equal Sums of Four Seventh Powers." Math.
Comput. 65, 1755-1756, 1996.
Gloden, A. "Zwei Parameterlosungen einer mehrgeradigen
Gleichung." Arch. Math. 1, 480-482, 1949.
Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 139-144, 1994.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Sastry, S. and Rai, T. "On Equal Sums of Like Powers."
Math. Student 16, 18-19, 1948.
Diophantine Equation — 8th Powers
The 2-1 equation
A* + B* = C
8 ^8
(1)
is a special case of Fermat'S Last THEOREM with n =
8, and so has no solution. No 2-2 solutions are known.
No 3-1, 3-2, or 3-3 solutions are known.
No 4-1, 4-2, 4-3, or 4-4 solutions are known.
No 5-1, 5-2, 5-3, or 5-4 solutions are known, but Letac
(1942) found a solution to the 5-5 equation. The small-
est 5-5 solution is
l 8 + 10 8 + ll 8 + 20 8 -h43 8 = 5 8 +28 8 +32 8 +35 8 +41 8 (2)
(Lander et al 1967).
No 6-1, 6-2, 6-3, or 6-4 solutions are known. Moessner
and Gloden (1944) found solutions to the 6-6 equation.
The smallest 6-6 solution is
3 8 +6 8 +8 8 + 10 8 + 15 8 +23 8 = 5 8 +9 8 +9 8 + 12 8 +20 8 + 22
Diophantine Equation — 8th Powers
No 7-1, 7-2, or 7-3 solutions are known. The smallest
7-4 solution is
7 8 +9 8 + 16 8 + 22 8 + 22 8 + 28 8 + 34 8 = 6 8 + ll 8 +20 8 + 35 8
(4)
(Lander et al. 1967). Moessner and Gloden (1944) found
solutions to the 7-6 equation. Parametric solutions to
the 7-7 equation were given by Moessner (1947) and
Gloden (1948). The smallest 7-7 solution is
l 8 + 3 8 + 5 8 + 6 8 + 6° + 8° + 13
8 + 8 8 ■
= 4 8 + 7 8 + 9 8 + 9 s + 10 s + 11 s + 12* (5)
(Lander et al 1967).
No 8-1 or 8-2 solutions are known. The smallest 8-3
solution is
6 8 + 12 8 + 16 8 + 16 8 + 38 8 + 38 8 +40 8 +47 8 = 8 8 + 17 8 + 50 8
(6)
(Lander et al. 1967). Sastry (1934) used the smallest
17-1 solution to give a parametric 8-8 solution. The
smallest 8-8 solution is
l 8 + 3 8 + 7 8 + 7 8 + 7 8 + 10 8 + 10 8 + 12 8
= 4 8 + 5 8 + 5 8 + 6 8 + 6 8 + ll 8 + ll 8 + ll 8 (7)
(Lander et al 1967).
No solutions to the 9-1 equation is known. The smallest
9-2 solution is
2 8 + 7 8 + 8 8 + 16 8 + 17 8 + 20 8 +20 8 + 24 8 + 24 8 = ll 8 + 27 8
(8)
(Lander et al 1967). Letac (1942) found solutions to
the 9-9 equation.
No solutions to the 10-1 equation are known.
The smallest 11-1 solution is
14 8 + 18 8 + 2 • 44 8 + 66 8 + 70 8 + 92 8
+93 8 + 96 8 + 106 8 + 112 8 = 125 8 (9)
(Lander et al. 1967).
The smallest 12-1 solution is
2 • 8 8 -f 10 8 + 3 • 24 8 + 26 8 + 30 8
(Lander et al 1967).
(3)
+34 8 + 44 8 + 52 8 + 63 s = 65 8 (10)
(Lander et al 1967).
The general identity
(2 8fc+4 + 1)8 = (2 8fc+4 _ 1)8 + ( 2 7fc+4j8
+(2 fc+1 ) 8 + 7[(2 5fc+3 ) 8 + (2 3fe+2 ) 8 ] (11)
gives a solution to the 17-1 equation (Lander et al 1967).
Diophantine Equation — 9th Powers
Diophantine Equation — 10th Powers 443
References
Gloden, A. "Parametric Solutions of Two Multi-Degreed
Equalities." Amer. Math. Monthly 55, 86-88, 1948.
Lander, L. J.; Parkin, T, R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Corn-put. 21,446-459,
1967.
Letac, A. Gazetta Mathematica 48, 68-69, 1942.
Moessner, A. "On Equal Sums of Like Powers." Math. Stu-
dent 15, 83-88, 1947.
Moessner, A. and Gloden, A. "Einige Zahlentheoretische Un-
tersuchungen und Resultante." Bull. Sci. Ecole Polytech.
de Timisoara 11, 196-219, 1944.
Sastry, S. "On Sums of Powers." J. London Math. Soc. 9,
242-246, 1934.
Diophantine Equation — 9th Powers
The 2-1 equation
A 9 + B 9 - C 9
(1)
is a special case of Fermat's Last Theorem with
n — 9, and so has no solution. There is no known 2-
2 solution.
There are no known 3-1, 3-2, or 3-3 solutions.
There are no known 4-1, 4-2, 4-3, or 4-4 solutions.
There are no known 5-1, 5-2, 5-3, 5-4, or 5-5 solutions.
There are no known 6-1, 6-2, 6-3, 6-4, or 6-5 solutions.
The smallest 6-6 solution is
I 9 + 13 9 + 13 9 + 14 9 + 18 9 + 23 9
= 5 9 + 9 9 + 10 9 + 15 9 + 21 9 + 22 9 (2)
(Lander et al. 1967).
There are no known 7-1, 7-2, 7-3, 7-4, or 7-5 solutions.
There are no known 8-1, 8-2, 8-3, 8-4, or 8-5 solutions.
There are no known 9-1, 9-2, 9-3, 9-4, or 9-5 solutions.
There are no known 10-1, 10-2, or 10-3 solutions. The
smallest 10-4 solution is
(Lander et al 1967). Palama (1953) gave a solution to
the 11-11 equation.
There is no known 12-1 solution. The smallest 12-2 so-
lution is
4 • 2 9 + 2 • 3 9 + 4 9 + 7 9 + 16 9 + 17 9 + 2 • 19 9
= 15 9 +21 9 (6)
(Lander et al. 1967).
There are no known 13-1 or 14-1 solutions. The smallest
15-1 solution is
2 9 + 2 9 + 4 9 + 6 9 + 6 9 + 7 9 + 9 9 + 9 9 + 10 9 + 15 9
+18 9 + 21 9 + 21 9 + 23 9 + 23 9 = 26 9 (7)
(Lander et al 1967).
References
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Moessner, A. "On Equal Sums of Like Powers." Math. Stu-
dent 15, 83-88, 1947.
Palama, G. "Diophantine Systems of the Type J^-i a '* =
ELi bi fe (^ = 1, 2, . . . , n, n + 2, n + 4, . . . , n + 2r).»
Scripta Math. 19, 132-134, 1953.
Diophantine Equation — 10th Powers
The 2-1 equation
A 10 + B 10 = C 10
(1)
is a special case of Fermat's Last Theorem with n =
10, and so has no solution. The smallest values for which
n-1, n-2, etc., have solutions are 23, 19, 24, 23, 16, 27,
and 7, corresponding to
5 . I 10 + 2 10 + 3 10 + 6 10 + 6 • 7 10 + 4 . 9 10
+10 10 + 2 • 12 10 + 13 10 + 14 10 = 15 10 (2)
2 9 + 6 9 + 6 9 + 9 9 + 10 9 + ll 9 4- 14 9 + 18 9 + 2 ■ 19 9
= 5 9 + 12 9 + 16 9 + 21 9 (3)
(Lander et al. 1967). No 10-5 solution is known. Moess-
ner (1947) gives a parametric solution to the 10-10 equa-
tion.
There are no known 11-1 or 11-2 solutions. The smallest
11-3 solution is
5.2 10 + 5 10 + 6 lo + 10 lo + 6-ll 10
,10
+2 • 12 1U + 3 • 15 1U = 9 1U + 17 iu (3)
1*° + 2 10 + 3 10 + 10 • 4 10 + 7 10 + 7 ■ 8 10
,10 . n r>10 . n „10 ,,10 . 1C 10
+10 1U + 12 1U + 16 1U = 11 1U + 2 • 15 1U (4)
2 9 + 3 9 + 6 9 + 7 9 + 9 9 + 9 9 + 19 9 + 19 9 + 21 9 + 25 9 + 29 9
= 13 9 + 16 9 + 30 9 (4)
(Lander et al. 1967). The smallest 11-5 solution is
3 9 + 5 » + 5 9 + 9 9 + 9 » + 12 9 + 15 9 + 15 9 + 16 9 + 21 9 + 21 9
= 7 9 + 8 9 + 14 9 + 20 9 + 22 9 (5)
5.lio + 2 .2 10 + 3-3 10 +4 10 + 4.6 10
+3-7 10 +8 10 +2-10 10 +2-14 10 + 15 10 = 3-ll 10 +16 10 (5)
4 . !"> + 2 10 + 2 ■ 4 10 + 6 10 + 2 • 12 10
+5 • 13 10 + 15 10 = 2 • 3 10 + 8 10 + 14 10 + 16 10 (6)
444 Diophantine Equation— Cubic
Diophantine Equation — Cubic
l 10 + 4.3 10 + 2.4 10 + 2.5 10 + 7.6 10
+9.7 10 + 10 10 + 13 10
2-2 lo + 8 10 + ll 1O + 2.12 10 (7)
(Berndt 1994, p. 107). Another form due to Ramanujan
is
l"> + 28 10 + 31 10 + 32 10 + 55 10 + 61 10
•68 1
= 17 10 + 20 10 + 23 10 + 44 10 + 49 10 + 64 10 + 67 10 (8)
(Lander et al. 1967).
References
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Diophantine Equation — Cubic
The 2-1 equation
I 3 + B 3
(1)
is a case of Fermat'S Last Theorem with n = 3. In
fact, this particular case was known not to have any
solutions long before the general validity of FERMAT'S
Last Theorem was established. The 2-2 equation
A 3 + B 3
C 3 + D 3
(2)
has a known parametric solution (Dickson 1966,
pp. 550-554; Guy 1994, p. 140), and 10 solutions with
sum < 10 5 ,
1729 =
: l 3 + 12 3 =
= 9 3 + 10 3
(3)
4104 =
: 2 3 + 16 3 =
= 9 3 + 15 3
(4)
13832 =
: 2 3 + 24 3 =
= 18 3 + 20 3
(5)
20683 =
: 10 3 + 27 3
= 19 3 + 24 3
(6)
32832 =
: 4 3 + 32 3 =
= 18 3 + 30 3
(7)
39312 =
: 2 3 + 34 3 =
= 15 3 + 33 3
(8)
40033 =
: 9 3 + 34 3 =
= 16 3 + 33 3
(9)
46683 =
: 3 3 + 36 3 =
: 16 3 + 33 3
(10)
64232 =
: 17 3 + 39 3
= 26 3 + 36 3
(11)
65728 =
: 12 3 + 40 3
= 31 3 + 33 3
(12)
(Sloane's A001235; Moreau 1898). The first number
(Madachy 1979, pp. 124 and 141) in this sequence, the
so-called Hardy-Ramanujan Number, is associated
with a story told about Ramanujan by G. H. Hardy,
but was known as early as 1657 (Berndt and Bhargava
1993). The smallest number representable in n ways as
a sum of cubes is called the nth Taxicab NUMBER.
Ramanujan gave a general solution to the 2-2 equation
(a + A 2 7 ) 3 + (A/3 + 7 ) 3 = (Aa + 7 ) 3 + (P + A 2 7 ) 3 (13)
where
{A 2 + 7AB - 9B 2 ) 3 + {2A 2 - 4AB + 12B 2 ) 3
= (2A 2 + 10B 2 ) 3 + {A 2 - SAB - B 2 ) 3 . (15)
Hardy and Wright (1979, Theorem 412) prove that there
are numbers that are expressible as the sum of two cubes
in n ways for any n (Guy 1994, pp. 140-141). The proof
is constructive, providing a method for computing such
numbers: given RATIONALS NUMBERS r and 5, compute
r(r 3 + 2s 3
Then
V
r 3 — s 3
U =
s(2r 3 + s 3 )
r 3 — s 3
V —
t(t 3 - 2u 3 )
t 3 + n 3
w =
u{2t 3 -u 3 )
t 3 + u 3 '
3
r
+
* 3 =
.3 3 3
t — U = V
+ w 3
(16)
(17)
(18)
(19)
(20)
The Denominators can now be cleared to produce an
integer solution. If r/s is picked to be large enough,
the v and w will be POSITIVE. If r/s is still larger, the
v/w will be large enough for v and w to be used as
the inputs to produce a third pair, etc. However, the
resulting integers may be quite large, even for n = 2.
E.g., starting with 3 3 + l 3 = 28, the algorithm finds
Oft — / 28340511 \3 , / 63284705 \3
V 21446828^ ' V 21446828^ J
(21)
giving
28 ■ 21446828 3 = (3 • 21446828) 3 + 21446828 3 (22)
(23)
28340511 3 + 63284705 3 .
The numbers representable in three ways as a sum of
two cubes (a 2-2-2 equation) are
87539319 = 167 3 + 436 3 = 228 3 + 423 3 = 255 3 + 414 3
(24)
i 3
(25)
L23 3
(26)
!5 3
(27)
i80 3
(28)
119824488 = ll 3 + 493 3 = 90 3 + 492 3 = 346 3 + 428 3
143604279 = 111 3 + 522 3 = 359 3 + 460 3 = 408 3 + 423 3
175959000 = 70 3 + 560 3 = 198 3 + 552 3 = 315 3 + 525 3
327763000 = 300 3 + 670 3 = 339 3 -h 661 3 = 510 3 + 580 3
a 2 + a{3 + p 2 = 3A7 2
(14)
Diophantine Equation — Cubic
Diophantine Equation — Cubic 445
(Guy 1994, Sloane's A003825). Wilson (1997) found 32
numbers representable in four ways as the sum of two
cubes (a 2-2-2-2 equation). The first is
6963472309248 = 2421 2 + 19083 2 = 5436 2 + 18948 2
= 102020 3 + 18072 2 = 13322 3 + 15530 3 . (29)
The smallest known numbers so representable
are 6963472309248, 12625136269928, 21131226514944,
26059452841000,... (Sloane's A003826). Wilson also
found six five-way sums,
48988659276962496 = 38787 3 + 365757 3
= 107839 3 + 362753 s
= 205292 3 + 342952 s
= 221424 s + 336588 s
= 231518 3 + 331954 s (30)
490593422681271000 = 48369 s + 788631 3
= 233775 3 + 781785 3
= 285120 3 + 776070 3
= 543145 s + 691295 s
= 579240 s + 666630 3 (31)
6355491080314102272 = 103113 s + 1852215 3
= 580488 3 + 1833120 3
= 788724 3 + 1803372 3
= 1150792 3 + 1690544 3
= 1462050 s + 1478238 3 (32)
27365551142421413376 = 167751 s + 3013305 s
= 265392 s + 3012792 3
= 944376 s + 2982240 s
= 1283148 s + 2933844 s
= 1872184 3 + 2750288 s (33)
1199962860219870469632 = 591543 s + 10625865 3
= 935856 s + 10624056 3
= 3330168 3 + 10516320 3
= 6601912 3 + 9698384 s
= 8387550 3 + 8480418 s (34)
111549833098123426841016 = 1074073 s + 48137999 3
= 8787870 3 + 48040356 3
= 13950972 s + 47744382 s
= 24450192 s + 45936462 s
= 33784478 3 + 41791204 3 , (35)
and a single six-way sum
8230545258248091551205888
= 11239317 3 -h 201891435 s
= 17781264 3 + 201857064 s
= 63273192 3 + 199810080 3
= 85970916 3 + 196567548 s
= 125436328 3 + 184269296 3
= 159363450 3 + 161127942 3 . (36)
The first rational solution to the 3-1 equation
A 3 + B 3 + C 3 = D 3
(37)
was found by Euler and Vieta (Dickson 1966, pp. 550-
554). Hardy and Wright (1979, pp. 199-201) give a so-
lution which can be based on the identities
a\a z + 6 3 ) 3 = b s (a 3 + b 3 ) 3 + a s (a 3 - 2b 3 ) 3
+ 6 3 (2a 3 -6 3 ) 3 (38)
„ 3 /„ 3 i QJ^ 3 3 /« 3 l. 3 ^ i k 3 / 3 k 3 \ 3
a (a + lb ) ~ a (a — b ) + b (a — 6 )
+ 6 3 (2a 3 + fe 3 ) 3 . (39)
This is equivalent to the general 2-2 solution found by
Ramanujan (Berndt 1994, pp. 54 and 107). The smallest
integral solutions are
3 s + 4 s + 5 s = 6 3
l 3 + 6 s + 8 s = 9 3
7 3 + 14 3 + 17 s = 20 s
ll 3 + 15 3 + 27 3 = 29 s
28 s + 53 3 + 75 3 = 84 s
26 3 + 55 3 + 78 3 = 87 3
33 3 -f 70 s + 92 3 = 105 s
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(Beeler et al. 1972; Madachy 1979, pp. 124 and 141).
Other general solutions have been found by Binet (1841)
and Schwering (1902), although Ramanujan's formula-
tion is the simplest. No general solution giving all Posi-
tive integral solutions is known (Dickson 1966, pp. 550-
561).
4-1 equations include
(47)
(48)
(49)
11 s + 12 3 + 13 3 + 14 s = 20 3
5 3 + 7 3 +9 3 + 10 3 = 13 3 .
A solution to the 4-4 equation is
2 3 + 3 3 + 10 3 + ll 3 = l 3 + 5 3 + 8 3 + 12 3
(Madachy 1979, pp. 118 and 133).
5-1 equations
l 3 + 3 3 + 4 3 + 5 3 + 8 s = 9 3
3 3 +4 3 + 5 3 +8 3 + 10 3 = 12 3 ,
and a 6-1 equation is given by
l 3 + 5 3 + 6 3 + 7 3 + 8 3 + 10 3 = 13 3 . (52)
(50)
(51)
A 6-6 equation also exists:
l 3 + 2 3 + 4 3 + 8 3 + 9 3 + 12 3 =3 3 + 5 3 + 6 3 + 7 3 + 10 3 + ll 3
(Madachy 1979, p. 142).
(53)
446 Diophantine Equation — Linear
Euler gave the general solution to
A 3 + B 3 = C 2 (54)
A = 3n + 6n — n
(55)
B = -3n 3 + Qn + n
(56)
C = 6n 2 (3n 2 + 1).
(57)
see also Cannonball Problem, Hardy-Ramanujan
Number, Super-3 Number, Taxicab Number, Tri-
morphic Number
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 58 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, Feb. 1972.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Amer. Math. Monthly 100, 645-656, 1993.
Binet, J. P. M. "Note sur une question relative a la theorie
des nombres." C. R. Acad. Set. (Paris) 12, 248-250, 1841.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1966.
Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 139-144, 1994.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 68, 1959.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, 1979.
Moreau, C. "Plus petit nombre egal a la somme de deux cubes
de deux fac,ons." L Tntermediaire Math. 5, 66, 1898.
Schwering, K. "Vereinfachte Losungen des Eulerschen Auf-
gabe: x 3 + y 3 + z 3 + v s = 0." Arch. Math. Phys. 2,
280-284, 1902.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 157, 1993.
Sloane, N. J. A. Sequences A001235 and A003825 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Wilson, D. Personal communication, Apr. 17, 1997.
Diophantine Equation — Linear
A linear Diophantine equation (in two variables) is an
equation of the general form
ax + by — c,
(i)
where solutions are sought with a, 6, and c INTEGERS.
Such equations can be solved completely, and the first
known solution was constructed by Brahmagupta, Con-
sider the equation
ax + by = 1.
(2)
Diophantine Equation — Linear
Now use a variation of the EUCLIDEAN ALGORITHM,
letting a — n and b = r%
r\ = qiT2 +r 3
T2 ~ Q2' r 3 + ?*4
Tn-Z = q n -3T n -2 + T n -\
r n -2 = q n -2T n -x + 1.
Starting from the bottom gives
1 = r n -2
r n -i = r n -3
■ qn-2r n -i
■ q n -3r n -2,
(3)
(4)
(5)
(6)
(7)
(8)
1 = r n -2 — q n -2(r n -3 — qn~3r n -2)
= -qn~2T n -3 .+ (1 ~ ?n-2<?n-3)r n -2. (9)
Continue this procedure all the way back to the top.
Take as an example the equation
1027x + 712y = 1.
(10)
Proceed as follows.
027=712-1+315 |
1 =
-165- 1027+ 238- 712 t
712=315-2+ 82 |
1 =
73- 712-165-315
315= 82-3+ 69
1 =
-19- 315+ 73* 82
82= 69-1+ 13
1 =
16- 82- 19- 69
69= 13-5+ 4
1 =
-3- 69+ 16- 13
13= 4-3+ 1|
1 =
1- 13- 3- 4
1 =
0- 4+ 1- 1, |
The solution is therefore x = —165, y = 238. The above
procedure can be simplified by noting that the two left-
most columns are offset by one entry and alternate signs,
as they must since
1 = -A i+ in + Ain+i (11)
T*t + 1 = Ti-\ — Tiqi-x (12)
1 = Ain-x - (Aiqt-x + A*+i), (13)
so the Coefficients of n-i and r»+i are the same and
Ai-
-(Aiqi-! + Ai+i).
(14)
Repeating the above example using this information
therefore gives
1027=712-1+315 |
(-)
165-1+73
= 238 t
712=315-2+ 82
(+)
73*2+19
= 165
315= 82-3+ 69
H
19-3+16
= 73 |
82= 69-1+ 13
(+)
16- 1+ 3
= 19
69= 13-5+ 4 |
(")
3-5+ 1
= 16
13= 4-3+ 11
(+)
1-3+
= 3
(-)
01+ 1
= 1
Diophantine Equation — Linear
and we recover the above solution.
Call the solutions to
ax + by = 1 (15)
#o and yo. If the signs in front of ax or by are NEGATIVE,
then solve the above equation and take the signs of the
solutions from the following table:
Diophantine Equation — nth Powers 447
equation
X
y
ax + by = 1
Xo
yo
ax -by = 1
Xq
-yo
—ax + by ~ \
—xo
yo
—ax — by = 1
— Xq
"2/o
In fact, the solution to the equation
ax — by = 1 (16)
is equivalent to finding the CONTINUED FRACTION for
a/6, with a and 6 Relatively Prime (Olds 1963). If
there are n terms in the fraction, take the (n — l)th
convergent p n -i/<Zn-i. But
Pnqn-l -Pn-iqn = ("l)", (17)
so one solution is xq = ( — l) n q n -i, Vo = ( — l) n p n -i,
with a general solution
x = Xo + kb (18)
y = yo + ka (19)
with k an arbitrary INTEGER. The solution in terms
of smallest POSITIVE INTEGERS is given by choosing an
appropriate fc.
Now consider the general first-order equation of the form
ax + by = c. (20)
The Greatest Common Divisor d = GCD(a,6) can
be divided through yielding
ax + b'y = c, (21)
where a f = a/d, b' = b/d, and d = c/d. If d\c, then d is
not an INTEGER and the equation cannot have a solu-
tion in Integers. A necessary and sufficient condition
for the general first-order equation to have solutions in
Integers is therefore that d\c. If this is the case, then
solve
a'x + b'y = l (22)
and multiply the solutions by c', since
a , (c'x)+b'(cy) = c. (23)
References
Courant, R. and Robbins, H. "Continued Fractions. Dio-
phantine Equations." §2.4 in Supplement to Ch. 1 in What
is Mathematics?: An Elementary Approach to Ideas and
Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 49-51, 1996.
Dickson, L. E. "Linear Diophantine Equations and Congru-
ences." Ch. 2 in History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, pp. 41-99,
1952.
Olds, CD. Ch. 2 in Continued Fractions. New York: Ran-
dom House, 1963.
Diophantine Equation — nth Powers
The 2-1 equation
A 71 -f B n = C n
(1)
is a special case of Fermat's Last Theorem and so
has no solutions for n > 3. Lander et ai. (1967) give a
table showing the smallest n for which a solution to
xi ' 4- x 2 + .
+ Xm k = yi k + yi h + ■ ■ ■ + J/n ,
with 1 < m < n is known.
m234567 8 9 10
1 2 3 3 4 7 8 11 15 23
2222478 9 12 19
3 3 3 7 8 11 24
4 4 7 10 23
5 5 5 11 16
6 6 27
_J_ 7_
Take the results from the RAMANUJAN 6-10-8 IDENTITY
that for ad = 6c, with
F 2m (a,6,c,d) = (a + 6 + c) 2m + (6 + c + d) 2m
-(c + d + a) 2
(d + a + 6) 2m + (a - d) 2m - (6 - c) 2m
(2)
and
2m
f2m(x, y) = (1 4- x + y) 2m + (x + y + xy)
-(y+xy + l) 2m -(xy + l+x) 2m + (l-xy) 2m -(x-y) 2
then
Using
F 2m (a,b,c,d) -af 2m (x,y).
h(x,y) =
U(x,y) =
(3)
(4)
(5)
(6)
now gives
(a + b + c) n + (b + c + d) n + (a - d) n
-(c + d + a) n + (^ + a + 6) n + (6-c) n (7)
for n = 2 or 4.
see also RAMANUJAN 6-10-8 IDENTITY
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, p. 101, 1994.
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Amer. Math. Monthly 100, 644-656, 1993.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, pp. 653—657,
1966.
448 Diophantine Equation — Quadratic
Diophantine Equation — Quadratic
Gloden, A. Mehrgradige Gleichungen. Groningen, Nether-
lands: P. Noordhoff, 1944.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, 1994.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21, 446-459,
1967.
Reznick, B. "Sums of Even Powers of Real Linear Forms."
Mem. Amer. Math. Soc. No. 463, 96. Providence, RI:
Amer. Math. Soc, 1992.
Diophantine Equation — Quadratic
An equation of the form
Dy 2 = 1,
(1)
can also be solved for certain values of c and D, but the
procedure is more complicated (Chrystal 1961). How-
ever, if a single solution to the above equation is known,
other solutions can be found. Let p and q be solutions
to (8), and r and s solutions to the "unit" form". Then
(p 2 - Dq 2 )(r 2 - Ds 2 ) = ±c (9)
(pr ± Dqs) 2 - D(ps ± qr) 2 = ±c. (10)
Call a Diophantine equation consisting of finding m
Powers equal to a sum of n equal Powers an "m — n
equation." The 2-1 equation
where D is an INTEGER is called a PELL EQUATION.
Pell equations, as well as the analogous equation with
a minus sign on the right, can be solved by finding the
Continued Fraction for yfD. (The trivial solution
x = 1, y = is ignored in all subsequent discussion.)
Let p n /q n denote the nth CONVERGENT [ai, a2, . . . ,aj,
then we are looking for a convergent which obeys the
identity
p n 2 -Dq n 2 = (-l) n , (2)
which turns out to always be possible since the Contin-
ued Fraction of a Quadratic Surd always becomes
periodic at some term a r +i, where a r +i = 2ai, i.e.,
VD = [ai,a 2) . . . ,a r ,2ai]
(3)
Writing n = rk gives
2 r-\ 2 / i \rk
p rk -Dq rk = (-1) ,
(4)
for k a Positive Integer. If r is Odd, solutions to
x 2 - Dy 2 = ±1
(5)
can be obtained if k is chosen to be EVEN or Odd, but
if r is EVEN, there are no values of k which can make
the exponent Odd.
If r is Even, then (-1) 7 " is Positive and the solution
in terms of smallest Integers is x = p r and y = q ri
where p r /q r is the rth Convergent. If r is Odd, then
(— l) r is Negative, but we can take k — 2 in this case,
to obtain
p 2r - Dq 2r ~ 1, (6)
so the solution in smallest INTEGERS is x = p^r, y = qiv
Summarizing,
(x,y) ■
f (Pr^
\ (P2r,
for r even
Qr)
p2r) for r odd.
(7)
The more complicated equation
B 2 + C 2 ,
(11)
which corresponds to finding a PYTHAGOREAN TRIPLE
(A, B, C) has a well-known general solution (Dickson
1966, pp. 165-170). To solve the equation, note that
every Prime of the form Ax + 1 can be expressed as the
sum of two Relatively Prime squares in exactly one
way. To find in how many ways a general number m
can be expressed as a sum of two squares, factor it as
follows
m = 2 a V 2ai -**Pn 20n < 7 i bl -'V r
(12)
where the ps are primes of the form 4x — 1 and the qs
are primes of the form x + 1. If the as are integral, then
define
B = (2b! + 1)(26 2 + 1) • • • (2b r + 1) - 1.
Then m is a sum of two unequal squares in
(13)
N(m)
for any ai half-integral
|(6i + l)(6a + l)-"(6r + l)
for all ai integral, B odd
|(6i + l)(6 a + l)"-(6r + l)-§
for all ai integral, B even.
(14)
If zero is counted as a square, both Positive and Neg-
ative numbers are included, and the order of the two
squares is distinguished, Jacobi showed that the num-
ber of ways a number can be written as the sum of two
squares is four times the excess of the number of Divi-
sors of the form 4x + 1 over the number of DIVISORS of
the form Ax — 1.
A set of Integers satisfying the 3-1 equation
A 2 + B 2 -f C 2 = D 2
(15)
is called a PYTHAGOREAN QUADRUPLE. Parametric so-
lutions to the 2-2 equation
Dy 2 = ±c
(8)
A 2 + B 2 = C 2 + D 2
(16)
Diophantine Equation — Quadratic
Diophantine Equation — Quartic 449
are known (Dickson 1966; Guy 1994, p. 140).
Solutions to an equation of the form
(A 2 + B 2 )(C 2 + D 2 ) = E 2 + F 2
(17)
are given by the FIBONACCI IDENTITY
(a 2 +6 2 )(c 2 +d 2 ) = {ac±bdf + (bc^adf = e 2 +/ 2 . (18)
Another similar identity is the ElJLER FOUR- SQUARE
Identity
(<h 2 + a 2 2 )(6i 2 + & 2 2 )(ci 2 + c 2 2 )(d 1 2 + d 2 2 )
= ei 2 + e 2 2 +e 3 2 + e 4 2 (19)
(ai 2 + a 2 2 + a 3 2 + a 4 2 )(&i 2 + b 2 2 + &3 2 + 6 4 2 )
= (aibi — a2&2 — ^363 — CI4&4)
+ (ai&2 + a2&i + c&3&4 — 04&3)
+ (0163 — a2&4 + 0361 + CI462)
+ (ai& 4 + a 2 &3 ~~ a3&2 + a 4&i) • ( 20 )
Degen's eight-square identity holds for eight squares, but
no other number, as proved by Cay ley. The two-square
identity underlies much of TRIGONOMETRY, the four-
square identity some of Quaternions, and the eight-
square identity, the CAYLEY Algebra (a noncommuta-
tive nonassociative algebra; Bell 1945).
Ramanujan's Square Equation
T - 7 = x 2
(21)
has been proved to have only solutions n = 3, 4, 5, 7,
and 15 (Beeler et al. 1972, Item 31).
see also Algebra, Cannonball Problem, Contin-
ued Fraction, Fermat Difference Equation, La-
grange Number (Diophantine Equation), Pell
Equation, Pythagorean Quadruple, Pythago-
rean Triple, Quadratic Residue
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Beiler, A. H. "The Pellian." Ch. 22 in Recreations in the The-
ory of Numbers: The Queen of Mathematics Entertains.
New York: Dover, pp. 248-268, 1966.
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, p. 159, 1945.
Chrystal, G. Textbook of Algebra, 2 vols. New York: Chelsea,
1961.
Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817.
Dickson, L. E. "Number of Representations as a Sum of 5,
6, 7, or 8 Squares." Ch. 13 in Studies in the Theory of
Numbers. Chicago, IL: University of Chicago Press, 1930.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1966.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, 1994.
Lam, T. Y. The Algebraic Theory of Quadratic Forms. Read-
ing, MA: W. A. Benjamin, 1973.
Rajwade, A. R. Squares. Cambridge, England: Cambridge
University Press, 1993.
Scharlau, W. Quadratic and Hermitian Forms. Berlin:
Springer- Verlag, 1985.
Shapiro, D. B. "Products of Sums and Squares." Expo. Math.
2, 235-261, 1984.
Smarandache, F. "Un metodo de resolucion de la ecuacion
diofantica." Gaz. Math. 1, 151-157, 1988.
Smarandache, F. " Method to Solve the Diophantine Equa-
tion ax 2 - by 2 + c = 0." In Collected Papers, Vol 1.
Bucharest, Romania: Tempus, 1996.
Taussky, O. "Sums of Squares." Amer. Math. Monthly 77,
805-830, 1970.
Whit ford, E. E. Pell Equation. New York: Columbia Uni-
versity Press, 1912.
Diophantine Equation — Quartic
Call an equation involving quartics m-n if a sum of m
quartics is equal to a sum of n fourth POWERS. The 2-1
equation
A 4 + B 4 = C 4 (1)
is a case of Fermat's Last Theorem with n = 4 and
therefore has no solutions. In fact, the equations
A 4 ± B 4 = C 2
also have no solutions in INTEGERS.
Parametric solutions to the 2-2 equation
A 4 + B 4 = C 4 + D 4
(2)
(3)
are known (Euler 1802; Gerardin 1917; Guy 1994,
pp. 140-141). A few specific solutions are
59 4 4- 158 4 = 133 4 + 134 4 = 635,318,657 (4)
7 4 + 239 4 = 157 4 + 227 4 = 3,262,811,042 (5)
193 4 + 292 4 = 256 4 + 257 4 = 8,657,437,697 (6)
298 4 + 497 4 = 271 4 + 502 4 = 68,899,596,497 (7)
514 4 + 359 4 = 103 4 + 542 4 = 86,409,838,577 (8)
222 4 + 631 4 = 503 4 + 558 4 = 160,961,094,577 (9)
2i 4 + 717 4 = 471 4 + 681 4 = 264,287,694,402 (10)
76 4 + 1203 4 = 653 4 + 1176 4 = 2,094,447,251,857
(11)
997 4 + 1342 4 = 878 4 -I- 1381 4 = 4,231,525,221,377
(12)
27 4 + 2379 4 = 577 4 + 728 4 = 32,031,536,780,322
(13)
(Sloane's A001235; Richmond 1920, Leech 1957), the
smallest of which is due to Euler. Lander et al. (1967)
give a list of 25 primitive 2-2 solutions. General (but
incomplete) solutions are given by
x = a + b
y = c-d
u — a — b
v = c + d,
(14)
(15)
(16)
(17)
450
where
Diophantine Equation — Quartic
a = n{rn 4 n)(-m A + 18m 2 n - n 4 )
b = 2m(rn + 10m 4 n 2 4 run 4 + 4n 6 )
c = 2n(4m 6 + m 4 n 2 + 10m 2 n 4 n 6 )
d = m{rn 4- n 2 )(-m 4 + 18m 2 n 2 - n 4 )
(Hardy and Wright 1979).
In 1772, Euler proposed that the 3-1 equation
A 4 + B 4 + C 4 = D 4
(18)
(19)
(20)
(21)
(22)
had no solutions in Integers (Lander et al. 1967). This
assertion is known as the EULER QUARTIC CONJEC-
TURE. Ward (1948) showed there were no solutions
for D < 10,000, which was subsequently improved to
D < 220,000 by Lander et al. (1967). However, the Eu-
ler Quartic Conjecture was disproved in 1987 by
Noam D. Elkies, who, using a geometric construction,
found
2,682,440 4 4 15,365,639 4 + 18,796,760 4
20,615,673 4
(23)
and showed that infinitely many solutions existed (Guy
1994, p. 140). In 1988, Roger Frye found
95,800 4 + 217,519 4 + 414,560 4 = 422,481 4 (24)
and proved that there are no solutions in smaller INTE-
GERS (Guy 1994, p. 140). Another solution was found
by Allan MacLeod in 1997,
638,523,249 4
=- 630,662,624 4 + 275,156,240 4 4 219,076,465 4 . (25)
It is not known if there is a parametric solution.
In contrast, there are many solutions to the 3-1 equation
A 4 4 B 4 4 C 4 = 2D 4
(see below).
Parametric solutions to the 3-2 equation
A 4 + B A = C A + D 4 + E 4
(26)
(27)
are known (Gerardin 1910, Ferrari 1913). The smallest
3-2 solution is
3 4 + 5 4 +8 4 = 7 4 + 7 4
(Lander et al 1967).
Ramanujan gave the 3-3 equations
2 4 + 4 4 + 7 4
3 4 + 6 4 4- 6 4
3 4 + 7 4 + 8 4 = l 4 + 2 4 + 9 4
6 4 4 9 4 4- 12 4 - 2 4 + 2 4 4 13 4
(28)
(29)
(30)
(31)
Diophantine Equation — Quartic
(Berndt 1994, p. 101). Similar examples can be found
in Martin (1896). Parametric solutions were given by
Gerardin (1911).
Ramanujan also gave the general expression
3 4 + (2x 4 - l) 4 4 (4x 5 4 x) 4
= (4x 4 + l) 4 + (6z 4 - 3) 4 4 (4x 5 - 5x) 4 (32)
(Berndt 1994, p. 106). Dickson (1966, pp. 653 655) cites
several FORMULAS giving solutions to the 3-3 equation,
and Haldeman (1904) gives a general FORMULA.
The 4-1 equation
A 4 4 B 4 + C 4 + D 4 - E 4
(33)
has solutions
30 4 + 120 4 + 272 4 + 315 4 =
:353 4
(34)
240 4 4 340 4 4 430 4 + 599 4 =
:651 4
(35)
435 4 + 710 4 + 1384 4 + 2420 4 =
: 2487 4
(36)
1130 4 4 H90 4 4- 1432 4 + 2365 4 =
: 2501 4
(37)
850 4 4 1010 4 + 1546 4 + 2745 4 =
: 2829 4
(38)
2270 4 4 2345 4 4 2460 4 4 3152 4 =
= 3723 4
(39)
350 4 H- 1652 4 + 3230 4 + 3395 4 =
: 3973 4
(40)
205 4 + 1060 4 + 2650 4 + 4094 4 =
: 4267 4
(41)
1394 4 + 1750 4 + 3545 4 + 3670 4 =
= 4333 4
(42)
699 4 + 700 4 4 2840 4 + 4250 4 -
= 4449 4
(43)
380 4 + 1660 4 4 1880 4 + 4907 4 =
: 4949 4
(44)
1000 4 + 1120 4 + 3233 4 + 5080 4 =
: 5281 4
(45)
410 4 + 1412 4 4 3910 4 4- 5055 4 =
: 5463 4
(46)
955 4 4- 1770 4 4 2634 4 + 5400 4 =
: 5491 4
(47)
30 4 + 1680 4 4 3043 4 + 5400 4 =
: 5543 4
(48)
1354 4 + 1810 4 4 4355 4 + 5150 4 =
: 5729 4
(49)
542 4 + 2770 4 4 4280 4 + 5695 4 =
: 6167 4
(50)
50 4 + 885 4 + 5000 4 + 5984 4 =
: 6609 4
(51)
1490 4 + 3468 4 4 4790 4 + 6185 4 =
: 6801 4
(52)
1390 4 + 2850 4 4 5365 4 + 6368 4 =
: 7101 4
(53)
160 4 4 1345 4 + 2790 4 4 7166 4 -
: 7209 4
(54)
800 4 4 3052 4 4 5440 4 + 6635 4 =
: 7339 4
(55)
2230 4 4 3196 4 + 5620 4 + 6995 4 =
: 7703 4
(56)
(Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964,
Lander et al. 1967), but it is not known if there is a
parametric solution (Guy 1994, p. 139).
Ramanujan gave the 4-2 equation
3 4 +9 4 =:::5 4 + 5 4 +6 4 +6 4 5
(57)
Diophantine Equation — Quartic
Diophantine Equation — Quartic 451
and the 4-3 identities
where
a +6+c=0
2 4 + 2 4 + 7 4 = 4 4 + 4 4 + 5 4 + 6 4 (58)
3 4 + g 4 + 14 4 = 7 4 + 8 4 + 10 4 + 13 4 (59)
7 4 + 10 4 + 13 4 = 5 4 + 5 4 + 6 4 + 14 4 (60)
(Berndt 1994, p. 101). Haldeman (1904) gives general
Formulas for 4-2 and 4-3 equations.
There are an infinite number of solutions to the 5-1 equa-
tion
A 4 + B 4 + C 4 + D 4 + E 4 = F 4 . (61)
Some of the smallest are
2 4 + 2 4 + 3 4 + 4 2 + 4 2 - 5 4 (62)
4 4 + 6 4 + 8 4 + 9 4 + 14 4 = 15 4 (63)
4 4 + 21 4 + 22 4 + 26 4 + 28 4 = 35 4 (64)
l 4 + 2 4 + 12 4 + 24 4 + 44 4 = 45 4 (65)
l 4 + 8 4 + 12 4 + 32 4 + 64 4 = 65 4 {m)
2 4 + 39 4 + 44 4 + 46 4 + 52 4 - 65 4 (67)
22 4 + 52 4 + 57 4 4- 74 4 + 76 4 = 95 4 (68)
22 4 + 28 4 + 63 4 + 72 4 + 94 4 - 105 4 (69)
(Berndt 1994). Berndt and Bhargava (1993) and Berndt
(1994, pp. 94-96) give Ramanujan's solutions for arbi-
trary s, £, m, and n,
(8s 2 + 40si - 24t 2 ) 4 4- (6s 2 - Ust - 18i 2 ) 4
+ (14s 2 - 4st - 42t 2 ) 4 + {9s 2 4- 27t 2 ) 4 + (4s 2 + 12i 2 ) 4
= (15s 2 + 45t 2 ) 4 , (70)
and
(4m 2 - 12n 2 ) 4 + (3m 2 + 9n 2 ) 4 4- (2m 2 - 12mn - 6n 2 ) 4
+ (4m 2 + 12n 2 ) 4 + (2m 2 + 12mn-6n 2 ) 4 = (5m 2 + 15n 2 ) 4 .
(71)
These are also given by Dickson (1966, p. 649), and two
general Formulas are given by Beiler (1966, p. 290).
Other solutions are given by Fauquembergue (1898),
Haldeman (1904), and Martin (1910).
Ramanujan gave
2(ab + ac + be) 2 = a 4 4- b 4 + c 4 (72)
2(a6+ac+6c) 4 = a 4 (6-c) 4 + 6 4 (c-a) 4 + c 4 (a-6) 4 (73)
2(ab + ac + be) 6 = (a 2 b + b 2 c + c 2 a) 4
+(a& 2 4- be 2 + ca 2 ) 4 + (3a6c) 4 (74)
2(ab + ac + be) 8 = (a 3 + 2abc) 4 (b - c) 4
+ (6 3 + 2abe) 4 (c - a) 4 + (c 3 + 2abc) 4 (a - 6) 4 , (75)
(76)
(Berndt 1994, pp. 96-97). FORMULA (73) is equivalent
to Ferrari's Identity
(a 2 + 2ae - 2bc - & 2 ) 4 + (b 2 - 2ab - 2ac - c 2 ) 4
+(c 2 + 2a& + 26c-a 2 ) 4 = 2(a 2 + 6 2 + c 2 -ab + ac + be) 4 .
(77)
Bhargava's Theorem is a general identity which gives
the above equations as a special case, and may have
been the route by which Ramanujan proceeded. An-
other identity due to Ramanujan is
(a + 6 + c) 4 + (Hc + d) 4 4(a-d) 4
= (c + d+a) 4 4(rf+a + 6) 4 4(6- c) 4 , (78)
where a/b = c/d, and 4 may also be replaced by 2 (Ra-
manujan 1957, Hirschhorn 1998).
V. Kyrtatas noticed that a
e — 38, and / = 39 satisfy
a 4- o + c
3, b = 7, c = 20, d = 25,
a + b + c
d 4 + e 4 + / 4 d+e + f
(79)
and asks if there are any other distinct integer solutions.
The first few numbers n which are a sum of two or more
fourth Powers (m — 1 equations) are 353, 651, 2487,
2501, 2829, ... (Sloane's A003294). The only number
of the form
4z 4 4V (80)
which is Prime is 5 (Baudran 1885, Le Lionnais 1983).
see also Bhargava's Theorem, Ford's Theorem
References
Barbette, E. Les sommes de p-iemes puissances distinctes
egales a une p-ieme puissance. Doctoral Dissertation,
Liege, Belgium. Paris: Gauthier-Villars, 1910.
Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains. New York: Dover,
1966.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Am. Math. Monthly 100, 645-656, 1993.
Bhargava, S. "On a Family of Ramanujan's Formulas for
Sums of Fourth Powers." Ganita 43, 63-67, 1992.
Brudno, S. "A Further Example of A 4 +£ 4 + C 4 +£> 4 = E 4 ."
Proc. Cambridge Phil Soc. 60, 1027-1028, 1964.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1966.
Euler, L. Nova Acta Acad. Petrop. as annos 1795-1796 13,
45, 1802.
Fauquembergue, E. L 'intermediaire des Math. 5, 33, 1898.
Ferrari, F. L 'intermediaire des Math. 20, 105-106, 1913.
Guy, R. K. "Sums of Like Powers. Euler's Conjecture" and
"Some Quartic Equations." §D1 and D23 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Veriag, pp. 139-144 and 192-193, 1994.
452 Diophantine Quadruple
Dipyramid
Haldeman, C. B. "On Biquadrate Numbers." Math. Mag. 2,
285-296, 1904.
Hardy, G. H. and Wright, E. M. §13.7 in An Introduction to
the Theory of Numbers, 5th ed. Oxford, England: Claren-
don Press, 1979.
Hirschhorn, M. D. "Two or Three Identities of Ramanujan."
Amer. Math. Monthly 105, 52-55, 1998.
Lander, L. J,; Parkin, T. R.; and Selfridge, J, L, "A Survey of
Equal Sums of Like Powers." Math. Comput. 21,446-459,
1967.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 56, 1983.
Leech, J. "Some Solutions of Diophantine Equations." Proc.
Cambridge Phil. Soc. 53, 778-780, 1957.
Leech, J. "On A 4 + B 4 + C 4 + D 4 = £ 4 ." Proc. Cambridge
Phil. Soc. 54, 554-555, 1958.
Martin, A. "About Biquadrate Numbers whose Sum is a Bi-
quadrate." Math. Mag. 2, 173-184, 1896.
Martin, A. "About Biquadrate Numbers whose Sum is a
Biquadrate— II." Math. Mag. 2, 325-352, 1904.
Norrie, R. University of St. Andrews 500th Anniversary
Memorial Volume. Edinburgh, Scotland: pp. 87-89, 1911.
Patterson, J. O. "A Note on the Diophantine Problem of
Finding Four Biquadrates whose Sum is a Biquadrate."
Bull. Amer. Math. Soc. 48, 736-737, 1942.
Ramanujan, S. Notebooks. New York: Springer- Verlag,
pp. 385-386, 1987.
Richmond, H. W. "On Integers Which Satisfy the Equation
t 3 ± x 3 ± y 3 ± z 3 = 0." Trans. Cambridge Phil. Soc. 22,
389-403, 1920.
Sloane, N. J. A. Sequences A001235 and A003294/M5446 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Ward, M. "Euler's Problem on Sums of Three Fourth Pow-
ers." Duke Math. J. 15, 827-837, 1948.
Diophantine Quadruple
see DIOPHANTINE SET
Diophantine Set
A set 5 of Positive integers is said to be Diophantine
Iff there exists a Polynomial Q with integral coeffi-
cients in 7n > 1 indeterminates such that
S = {Q(xi,...,z m ) > 1 : xi > l,...,x m > 1}.
It has been proved that the set of PRIME numbers is a
Diophantine set.
References
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 189-192, 1995.
Diophantus Property
A set of Positive Integers S = {ai,...,a m } satisfies
the Diophantus property D(n) of order n if, for all i, j =
1, . . . , m with i ^ j,
diCLj + n = bij , (1)
where n and bij are INTEGERS. The set S is called a
Diophantine n-tuple. Fermat found the first D(l) quad-
ruple: {1,3,8,120}. General D(l) quadruples are
{i<2n, i*2n+2, F2n+4» 4F2n+l-f 7 2n+2-f 7 2n+3 , } (2)
where F n are FIBONACCI NUMBERS, and
{n, n + 2, 4n + 4, 4(n + l)(2n + l)(2n + 3)}. (3)
The quadruplet
{2F n -i,2F n +i i 2Fn F n +iF n +2,
2F n + 1 F n + 2 F n+z (2F n+1 2 - F n 2 )} (4)
is D(F n 2 ) (Dujella 1996). Dujella (1993) showed there
exist no Diophantine quadruples D(4k + 2).
References
Aleksandriiskii, D. Arifmetika i kniga o mnogougoVnyh chis-
lakh. Moscow: Nauka, 1974.
Brown, E. "Sets in Which xy + k is Always a Square." Math.
Comput 45, 613-620, 1985.
avenport, H. and Baker, A. "
and 8x 2 - 7 = z 2 ." Quart. J. Math. (Oxford) Ser. 2 20,
129-137, 1969.
Dujella, A. "Generalization of a Problem of Diophantus."
Acta Arithm. 65, 15-27, 1993.
Dujella, A. "Diophantine Quadruples for Squares of Fi-
bonacci and Lucas Numbers." Portugaliae Math. 52, SOS-
SIS, 1995.
Dujella, A. "Generalized Fibonacci Numbers and the Prob-
lem of Diophantus." Fib. Quart. 34, 164-175, 1996.
Hoggatt, V, E. Jr. and Bergum, G. E. "A Problem of Fermat
and the Fibonacci Sequence." Fib. Quart. 15, 323-330,
1977.
Jones, B. W. "A Variation of a Problem of Davenport and
Diophantus." Quart. J. Math. (Oxford) Ser. (2) 27, 349-
353, 1976.
Diophantus' Riddle
"Diophantus' youth lasts 1/6 of his life. He grew a beard
after 1/12 more of his life. After 1/7 more of his life,
Diophantus married. Five years later, he had a son.
The son lived exactly half as long as his father, and
Diophantus died just four years after his son's death.
All of this totals the years Diophantus lived."
Let D be the number of years Diophantus lived, and let
S be the number of years his son lived. Then the above
word problem gives the two equations
^ = (i+i^ + 7)^ + 5 + S + 4
S=±D.
Solving this simultaneously gives S = 42 as the age of
the son and D = 84 as the age of Diophantus.
References
Pappas, T. "Diophantus' Riddle." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 123 and 232,
1989.
Dipyramid
Two PYRAMIDS symmetrically placed base-to-base, also
called a BlPYRAMID. They are the DUALS of the Archi-
medean Prisms.
Dirac Delta Function
see also Elongated Dipyramid, Pentagonal Di-
pyramid, Prism, Pyramid, Trapezohedron, Trian-
gular Dipyramid, Trigonal Dipyramid
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 117, 1989.
Dirac Delta Function
see Delta Function
Dirac Matrices
Define the 4x4 matrices
(Ti = I (g> (Ti, Pauli
Pi = &i, Pauli ® I,
(1)
(2)
where <n, Pau ii are the Pauli Matrices, I is the Iden-
tity Matrix, i = 1, 2, 3, and A (g> B is the matrix
Direct Product. Explicitly,
1 =
CTl =
<T<2
<?3 =
Pi
P2
PS
'1
0"
1
1
„0
1-
'0
1
0"
1
1
_0
1 0-
"0
—i
"
i
-i
_0
i _
"1
-1
1
_0
-1
"0
1 0"
1
1
_0
1
0_
"0
-i "
-i
i
_0
i
_
'1
1
-1
-
-1
(3)
(4)
(5)
(6)
(7)
(8)
(9)
These matrices satisfy the anticommutation identities
(JiCTj H- CTjCTi — 28ij\ (10)
pipj 4- pjpi = 2Sij\, (11)
where Sij is the KRONECKER Delta, the commutation
identity
[<n,Pj] = 0iPj ~ PjVi = 0) ( 12 )
Dirac Matrices 453
and are cyclic under permutations of indices
<Ti<jj = io*k (13)
PiPj = ipk- (14)
A total of 16 Dirac matrices can be defined via
: PiVj
(15)
for i, j = 0, 1, 2, 3 and where <jq = po = I. These matrix
satisfy
1. |Eij| = 1, where |A| is the DETERMINANT,
2 E 2 - I
3. Ejj = Etj., making them Hermitian, and therefore
unitary,
4. tr(E^j) = 0, except tr(Eoo) = 4,
5. Any two E^ multiplied together yield a Dirac matrix
to within a multiplicative factor of —i or ±i,
6. The Eij are linearly independent,
7. The Eij form a complete set, i.e., any 4x4 constant
matrix may be written as
A — y ^ Cij c.ij ,
(16)
i,j—0
where the Cij are real or complex and are given by
itr(AE m „)
(17)
(Arfken 1985).
Dirac's original matrices were written a% and were de-
fined by
oci = Eii = p\&i
a 4 = E 30 = p3,
for i = 1, 2, 3, giving
on = Eii =
C*2 = E2i =
OLZ = E3i =
CK4 = E 30 =
1
10
10
10
-r
i
-i
i _
10
0-1
10
0-100
10
10
0-10
0-1
(18)
(19)
(20)
(21)
(22)
(23)
454
Dirac Matrices
Direct Product (Set)
The additional matrix
a 5 — E 2 o —Pi —
-i
-i
i
i
(24)
is sometimes defined. Other sets of Dirac matrices are
sometimes defined as
(25)
(26)
(27)
Vi
= E«
2/4 = E 30
2/5 = -E10
and
Si — E 3i
for i = 1, 2, 3 (Arfken 1985) and
H =
Gi
-<Ti
74 =
" 1 0"
21 -1
(28)
(29)
(30)
for i = 1, 2, 3 (Goldstein 1980).
Any of the 15 Dirac matrices (excluding the identity
matrix) commute with eight Dirac matrices and anti-
commute with the other eight. Let M = |(1 + E^),
then
M 2 = M.
In addition
"Oti"
' OLX~
a 2
X
OL2
.olz_
_0£ 3 _
: 2icr.
The products of cti and y, satisfy
01020:30405 = 1
2/12/22/32/42/5 = 1.
(31)
(32)
(33)
(34)
The 16 Dirac matrices form six anticommuting sets of
five matrices each:
1. cti, a 2) a 3 , a 4 , a 5 ,
2. yi, 2/2, 2/3, 2/4, 2/5,
3. £1, £2, £3 j pi, P2,
4. ai, 2/1, <5i, cr2, cr3,
5. a 2) 2/2, £2, cri, cr 3)
6. 0:3, 2/3, $3, (Ti, <7 2 .
see a/50 Pauli Matrices
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 211-213, 1985.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison- Wesley, p. 580, 1980.
Dirac's Theorem
A Graph with n > 3 Vertices in which each Vertex
has Valency > n/2 has a Hamiltonian Circuit.
see also Hamiltonian Circuit
Direct Product (Group)
The expression of a GROUP as a product of SUBGROUPS.
The Characters of the representations of a direct
product are equal to the products of the Characters
of the representations based on the individual sets of
functions. For R\ and ife,
The representation of a direct product Tab will con-
tain the totally symmetric representation only if the ir-
reducible Fa equals the irreducible Ts.
Direct Product (Matrix)
Given two n x m Matrices, their direct product C =
A <g> B is an (mn) x (nm) Matrix with elements defined
by
C a [3 = AijBkh (1)
where
a = n(i — 1) + k
= n(j - 1) + J.
For a 2 x 2 Matrix,
A®B =
anB
CI12B
a2iB
&22B
aii&n
G11&12
ai2&n
ai2&i2
an&2i
ail622
ai2&2i
ai2&22
021&11
«2l6i2
a22&n
G22&12
G21&21
(I21&22
«22&21
a22&22
(2)
(3)
(4)
(5)
Direct Product (Set)
The direct product of two sets A and B is defined to
be the set of all points (a, b) where a e A and b e B.
The direct product is denoted A x B or A <g) B and
is also called the Cartesian Product, since it orig-
inated in Descartes' formulation of analytic geometry.
In the Cartesian view, points in the plane are speci-
fied by their vertical and horizontal coordinates, with
points on a line being specified by just one coordinate.
The main examples of direct products are EUCLIDEAN
3-space (M <g> M <g> M, where R are the Real Numbers),
and the plane (M x K).
Direct Product (Tensor)
Directed Graph 455
Direct Product (Tensor)
For a first-RANK Tensor (i.e., a Vector),
alb'*
~ dx\ dxi dx'i dxi
(1)
which is a second- Rank Tensor. The Contraction of
a direct product of first-RANK TENSORS is the SCALAR
contr(ciib 3 ) = a^b % = dkb .
For a second-RANK Tensor,
<ikl
AjBki — Cj
OXi UX n ^fe <J%1 (~,mpq
-iikV
3 dxm dx'- dx p dx q
(2)
(3)
(4)
For a general Tensor, the direct product of two Ten-
sors is a Tensor of Rank equal to the sum of the two
initial RANKS. The direct product is ASSOCIATIVE, but
not Commutative.
References
Arfken, G. "Contraction, Direct Product." §3.2 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 124-126, 1985.
Direct Search Factorization
Direct search factorization is the simplest Prime Fac-
torization Algorithm. It consists of searching for
factors of a number by systematically performing TRIAL
DIVISIONS, usually using a sequence of increasing num-
bers. Multiples of small PRIMES are commonly excluded
to reduce the number of trial DIVISORS, but just includ-
ing them is sometimes faster than the time required to
exclude them. This approach is very inefficient, and can
be used only with fairly small numbers.
When using this method on a number n, only DIVISORS
up to [y/n\ (where [x\ is the FLOOR Function) need
to be tested. This is true since if all INTEGERS less than
this had been tried, then
L%/^J + i
< y/n.
(1)
In other words, all possible FACTORS have had their Co-
factors already tested. It is also true that, when the
smallest PRIME FACTOR p of n is > <J/n, then its COFAC-
TOR m (such that n = pm) must be PRIME. To prove
this, suppose that the smallest p is > %fn. If m = a&,
then the smallest value a and b could assume is p. But
then
n = pm = pab = p > n,
(2)
which cannot be true. Therefore, m must be PRIME, so
n = piP2-
(3)
see also PRIME FACTORIZATION ALGORITHMS, TRIAL
Division
Direct Sum (Module)
The direct sum of two MODULES V and W over the same
Ring R is given by V <g> W with Module operations
defined by
r - (v,w) = (rv, rw)
(v, w) © (y, z) = (v + y, w + z).
The direct sum of an arbitrary family of MODULES over
the same Ring is also defined. If J is the indexing set
for the family of Modules, then the direct sum is repre-
sented by the collection of functions with finite support
from J to the union of all these MODULES such that
the function sends j £ J to an element in the MODULE
indexed by j.
The dimension of a direct sum is the product of the
dimensions of the quantities summed. The significant
property of the direct sum is that it is the coproduct
in the category of MODULES. This general definition
gives as a consequence the definition of the direct sum
of ABELIAN GROUPS (since they are MODULES over the
Integers) and the direct sum of Vector Spaces (since
they are Modules over a Field).
Directed Angle
The symbol LABC denotes the directed angle from AB
to BC, which is the signed angle through which AB
must be rotated about B to coincide with BC. Four
points ABCD lie on a Circle (i.e., are Concyclic)
Iff IABC = IADC. It is also true that
£hl2 + £hh=Q° or 180°.
Three points A, B, and C are COLLINEAR Iff ZABC =
0. For any four points, ^4, S, C, and D,
IABC + LCD A = IB AD + IDCB.
see also ANGLE, COLLINEAR, CONCYCLIC, MlQUEL
Equation
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 11-15, 1929.
Directed Graph
sink
A Graph in which each Edge is replaced by a directed
Edge, also called a Digraph or Reflexive Graph.
A Complete directed graph is called a Tournament.
If G is an undirected connected GRAPH, then one can
456
Direction Cosine
Direction Cosine
always direct the circuit EDGES of G and leave the SEP-
ARATING EDGES undirected so that there is a directed
path from any node to another. Such a Graph is said
to be transitive if the adjacency relation is transitive.
The number of directed graphs of n nodes for n = 1, 2,
... are 1, 1, 3, 16, 218, 9608, . . . (Sloane's A000273).
see also Arborescence, Cayley Graph, Indegree,
Network, Outdegree, Sink (Directed Graph),
Source, Tournament
References
Sloane, N. J. A. Sequence A000273/M3032 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Direction Cosine
Let a be the Angle between v and x, b the Angle
between v and y, and c the ANGLE between v and z.
Then the direction cosines are equivalent to the (x,y, z)
coordinates of a Unit Vector v,
a = cos a ;
|v|
v ■ y
= cos b = -7-7-
m
7 = cos c :
V • z
From these definitions, it follows that
a 2 +/3 2 + 7 2 = 1-
(1)
(2)
(3)
(4)
To find the JACOBIAN when performing integrals over
direction cosines, use
-sin" 1 (va 2 +/? 2 )
■(f)
<p — tan
7 = a/1 - a 2 -
The Jacobian is
d(0,<f>)
d(a,(3)
89 d6_
da 8(3
8± d±
da d[3
(5)
(6)
(7)
(8)
Using
6(0,4)
d(a,0)
d , . _! . _ 1
(9)
dx ( " m X ' v/l-x*
d u -1 ^ 1
(10)
^(tan x)- 1 + a;21
±(a 2 +{3 2 )~ 1 / 2 2a i(a 2 +^ 2 )- 1 / 2 2/3
V / l-a2_ /9 2 y/l-a 2 -(3 2
l+i
(a 2 + (3 2 )- 1 ' 2
^
■0 2
1
1 -1- P
1 +
y/(cfl+ff*){l- a»-0')'
(11)
dU = sm6d</>d6 = y/o? + 2
dad/3 _ da d(i
~ y/l - a 2 - W ~ 7
9(0, <t>)
d(a,0)
dad/3
(12)
Direction cosines can also be defined between two sets
of Cartesian Coordinates,
CKi = X -X
a 2 = x • y
a 3 = x' • z
0i = y • x
P2 = y - y
03 = y * z
71 = z' • x
72 = z' • y
73 = z' • z.
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
Projections of the unprimed coordinates onto the primed
coordinates yield
x = (x' • x)x + (x - y)y 4 (x' • z)z =■ aix 4 a 2 y 4 a 3 z
(22)
y = (y • x)x + (y ■ y)y + (y • z)z = ftx + /3 2 y + ftz
(23)
z = (z • x)x + (z' • x)y 4 (z' • z)z = 71 x + 72y 4 73Z,
(24)
and
x = r • x' = a.\x 4 022/ + 0132 (25)
y'^r-y =0!X + 02y + 03Z (26)
z' = r • z' = 71 x + 72^ + 73-2- (27)
Projections of the primed coordinates onto the unprimed
coordinates yield
and
x = (x ■ x')x' + (x • y')y' + (x • z')z'
= aix 4piy +71Z
y = (y • x)x 4 (y • y )y 4 (y • z')z
= a 2 x +^2y +72Z
z = (z • x )x ; + (z ■ x )y' + (z • z ; )z'
= a 3 x' +^3y' + 73Z 7 ,
x = r ■ x = aix + ^iy + Jiz
y = r ■ y = a 2 x 4 /3 2 y 4 72^
(28)
(29)
(30)
(31)
(32)
Directional Derivative
Dirichlet Beta Function
457
z = r • z = a 3 x + /fey + 73^-
(33)
Using the orthogonality of the coordinate system, it
must be true that
(34)
(35)
x-y = y-z = z*x =
x-x = y-y = z-z = l,
giving the identities
(XiOtm + PtP m + 7/ 7m = (36)
for 1,171 = 1, 2, 3 and / ^ m, and
ai 2 +/?i 2 +7i 2 = l ( 37 )
for / = 1,2,3. These two identities may be combined
into the single identity
OLiam + 010m + 7*7m = ftm, (38)
where £* m is the Kronecker Delta.
Directional Derivative
U „ ,:„ /( X + ku ) ~ /(*)
V u / = V/ • t-t oc lim
U h-+0
ft
(1)
Vu/(a50) S/Oj^o) is the rate at which the function iu =
f{x,y,z) changes at (xo,yo,^o) in the direction u. Let
u be a Unit Vector in Cartesian Coordinates, so
then
|u| = y/u x 2 +U y 2 + U Z 2 = 1,
^ , df df df
V » f= te U *+dy Uy+ d-z U -
(2)
(3)
The directional derivative is often written in the nota-
tion
d ~ _ d d ...
dS = s ^ = s *te + s »di + s *d- z - {4)
Directly Similar
directly similar
Two figures are said to be Similar when all correspond-
ing ANGLES are equal, and are directly similar when all
corresponding ANGLES are equal and described in the
same rotational sense.
see also FUNDAMENTAL THEOREM OF DIRECTLY SIMI-
LAR Figures, Inversely Similar, Similar
Director Curve
The curve d(u) in the Ruled Surface parameteriza-
tion
x(u, v) = h(u) + vd(u).
see also Directrix (Ruled Surface), Ruled Sur-
face, Ruling
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 333, 1993.
Directrix (Conic Section)
directrix
ellipse parabola
The Line which, together with the point known as the
Focus, serves to define a Conic Section.
see also CONIC SECTION, ELLIPSE, FOCUS, HYPER-
BOLA, Parabola
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 115-116, 1969.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 141-144, 1967.
Directrix (Graph)
A Cycle.
Directrix (Ruled Surface)
The curve h(u) in the Ruled Surface parameteriza-
tion
x(ti, v) = b(«) + vd(u)
is called the directrix (or BASE Curve).
see also DIRECTOR CURVE, RULED SURFACE
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 333, 1993.
Dirichlet Beta Function
458 Dirichlet Boundary Conditions
Dirichlet Eta Function
Re[DirichletBeta z]
Im[DirichletBeta z]
Im[z]
|DirichletBeta z|
0(x) = J2(-l) n (2n + iy
) 8( a! ) = 2-*(-l,a!,i) )
(1)
(2)
where $ is the LERCH TRANSCENDENT. The beta func-
tion can be written in terms of the HuRWITZ Zeta
Function ((x,a) by
0(*) = £[C(*.i)-Ct>,f)].
(3)
The beta function can be evaluated directly for POSI-
TIVE Odd x as
P(2k+i) = t^frr+\ ( 4)
where E n is an EULER Number. The beta function
can be defined over the whole Complex Plane using
Analytic Continuation,
i3{l-z)=(^)\inC^z)T{z)t3{z) )
(5)
(6)
(7)
(8)
where T(z) is the Gamma Function.
Particular values for (5 are
W) = \*
0(2) = K
0(3) = ^ 3 ,
where K is CATALAN'S CONSTANT.
see also Catalan's Constant, Dirichlet Eta Func-
tion, Dirichlet Lambda Function, Hurwitz Zeta
Function, Lerch Transcendent, Riemann Zeta
Function, Zeta Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 807-808, 1972.
Spanier, J. and Oldham, K. B. "The Zeta Numbers and Re-
lated Functions." Ch. 3 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 25-33, 1987.
Dirichlet Boundary Conditions
Partial Differential Equation Boundary Condi-
tions which give the value of the function on a surface,
e.g.,T = /(r,t).
see also Boundary Conditions, Cauchy Boundary
Conditions
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 679, 1953.
Dirichlet 's Box Principle
A.k.a. the PIGEONHOLE Principle. Given n boxes and
m > n objects, at least one box must contain more than
one object. This statement has important applications
in number theory and was first stated by Dirichlet in
1834.
see also Fubini PRINCIPLE
References
Chartrand, G. Introductory Graph Theory. New York:
Dover, p. 38, 1985.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 161, 1993.
Dirichlet's Boxing-In Principle
see Dirichlet's Box Principle
Dirichlet Conditions
see Dirichlet Boundary Conditions, Dirichlet
Fourier Series Conditions
Dirichlet Divisor Problem
Let d(n) ~ v(n) = o~o(n) be the number of DIVISORS
of n (including n itself). For a PRIME p, v(p) — 2. In
general,
n
^2 v( k ) = n Inn + (2 7 - l)n 4- 0(n°) y
where 7 is the Euler-Mascheroni Constant. Dirich-
let originally gave 6 « 1/2. As of 1988, this had been
reduced to 6 « 7/22.
see also Divisor Function
Dirichlet Energy
Let h be a real-valued HARMONIC FUNCTION on a
bounded DOMAIN CI, then the Dirichlet energy is de-
fined as J a \Vh\ 2 dx, where V is the GRADIENT.
see also ENERGY
Dirichlet Eta Function
10
7.5
5 ■
2,5 ■
-:.o
-2.5
-5
Re[DirichletEta z] Im[DirichletEta z]
10
|DirichletEta zj
Im[z] -10
lCrlO
Dirichlet's Formula
Dirichlet Integrals 459
r 7 (x)^^(-l)"- 1 n^ = (l-2 1 - I )C(a
(1)
where n = 1, 2, . . . , and ((x) is the RlEMANN Zeta
Function. Particular values are given in Abramowitz
and Stegun (1972, p. 811). The eta function is related to
the Riemann Zeta Function and Dirichlet Lambda
Function by
C(*) = X(u) = V (v)
2 U 2 V - 1 2" - 2
(2)
and
CM + *?H = 2AH (3)
(Spanier and Oldham 1987). The value t?(1) may be
computed by noting that the Maclaurin Series for
ln(l + x) for -1 < x < 1 is
ln(l + x) = x - \x 2 + \x z - \x 4 + . . . . (4)
Therefore,
ln2 = ln(l + l) = l-i + I-i + ...
~ (-1)- 1
£
*?(!)•
(5)
Values for EVEN INTEGERS are related to the analytical
values of the RlEMANN ZETA FUNCTION. 77(0) is defined
to be \.
V(0) = \
77(1) = ln2
t?(3) = 0.90154.
7tt 4
7?(4)
720"
see also Dedekind Eta Function, Dirichlet Beta
Function, Dirichlet Lambda Function, Riemann
Zeta Function, Zeta Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and-
Mathematical Tables, 9th printing. New York: Dover,
pp. 807-808, 1972.
Spanier, J. and Oldham, K. B. "The Zeta Numbers and Re-
lated Functions." Ch. 3 in An Atlas of Functions. Wash-
ington. DC: Hemisphere, pp. 25-33, 1987.
Dirichlet's Formula
If g is continuous and ^t, v > 0, then
[ (t-zy-'dt- [ (t-xy-'g&xjdx
Jo Jo
= f dx f (t-tr- 1 (t-x)"- 1 g(t,x)dt.
Jo J x
Dirichlet Fourier Series Conditions
A piecewise regular function which
1. Has a finite number of finite discontinuities and
2. Has a finite number of extrema
can be expanded in a FOURIER Series which converges
to the function at continuous points and the mean of
the Positive and Negative limits at points of discon-
tinuity.
see also Fourier Series
Dirichlet Function
Let c and d / c be REAL NUMBERS (usually taken as
c = 1 and d = 0). The Dirichlet function is defined by
D(x) = {
c for x rational
d for x irrational.
The function is CONTINUOUS at IRRATIONAL x and dis-
continuous at Rational points. The function can be
written analytically as
D(x) = lim cos[(m!7rx) n ].
m,n—i- 00
^|i^iL
AuJk
Jll^llj^jjjjlll^
Because the Dirichlet function cannot be plotted with-
out producing a solid blend of lines, a modified version
can be defined as
D M (x) = J
for x rational
b for x = a/b with a/b a reduced fraction
(Dixon 1991), illustrated above.
see also CONTINUOUS FUNCTION, IRRATIONAL NUM-
BER, Rational Number
References
Dixon, R, Mathographics. New York: Dover, pp. 177 and
184-186, 1991.
Tall, D. "The Gradient of a Graph." Math. Teaching 111,
48-52, 1985.
Dirichlet Integrals
There are several types of integrals which go under the
name of a "Dirichlet integral." The integral
D[u] = J \Vu\'
dV
(1)
460 Dirichlet Integrals
appears in DlRICHLET'S PRINCIPLE.
The integral
2tt ]_ n sin(| a:)
where the kernel is the DIRICHLET Kernel, gives the
nth partial sum of the FOURIER SERIES.
Another integral is denoted
4
1 r sm* k p k &ipklk dpk= h
for |7 fc | > a k
for J-y-fc | < a k
(3)
for k = 1, . . . , n.
There are two types of Dirichlet integrals which are de-
noted using the letters C, D, /, and J. The type 1
Dirichlet integrals are denoted /, J, and /J, and the
type 2 Dirichlet integrals are denoted C, D, and CD,
The type 1 integrals are given by
//■/
/(*l+*2 + ...+t n )
Ol\— 1, <*2 — 1 J. «n — 1
_ r(ai)r(a a )
t n n dt\ dt 2 dt n
r(E„«»)
where T(^) is the GAMMA FUNCTION. In the case n = 2,
/ - //■ xVdxdv = p!gl = g (P+ 1 »g+ 1 )
yy T y y (p+?+2)! p+ g +2 -
(5)
where the integration is over the TRIANGLE T bounded
by the z-axis, y-axis, and line x + y = 1 and B(x,y) is
the Beta Function.
The type 2 integrals are given for &-D vectors a and r,
and < c < ft,
Ci b) (r,m) =
T(m + R)
rwnLi r (M
x r n n°
JO JO M _|_
(i + £L*«)
Di 6 )(r,m)
T(m + R)
r(m)IlU r (n)
poo />oo rj"
/ '" / ~~^
Jai J a k (1-|-
1 li=l *^* flXj
(i + EU**)
(7)
CDi c ' d - c) (r,m) =
r(m)n! =1 r(f«)
/»a c /»oo /«oo i
JO A c+1 Ja b M
I JL — i *^i CLXi
(i + EU**)
m+H '
(8)
where
* = 5>
a» i
Pi
i-e: =1 p<
Dirichlet Integrals
(9)
(10)
and ^ are the cell probabilities. For equal probabilities,
Oi = 1. The Dirichlet D integral can be expanded as a
Multinomial Series as
1
D™{r,m):
(• + EL)-
y ... y* f m - 1 + ELi a! A
^ ^ \m-l,xi,...,x b l
xi<t- 1 x b <r[ )
S'NS:
(11)
flfc
For small 6, C and D can be expressed analytically either
partially or fully for general arguments and a» = 1.
T(n + r 2 ) 2Fi(r* 2 , ri + r 2 ; 1 + r 2 ; -1)
C{ 1) (r 2 ;r 1 ) =
C[ 2 \r 2 ,r 3 ;ri) =
r 2 r(n)r(r 2 )
r(ri + r 2 + r 3 )
(12)
r 2 r(ri)r(r2)r(r 3 )
x / 2F 1 y r *- 1 (l + y)- {ri+r2+r * ) dy,
Jo
(13)
where
2 F X = 2 F 1 (r 2 , n + r 2 + r 3 ; 1 + r 2 , -(1 + y)" 1 ) (14)
is a Hypergeometric Function.
I?i 1) (r a ;n) =
£>i 2) (r2,r 3 ;ri) =
F(n + t 2 ) 2 Fi(r 1 ,r 1 + r 2 ; 1 + n; -1)
(15)
rir(n)r(r 2 )
F(n +r 2 + r 3 )
(ri+r 3 )r(ri)r(r 2 )r(r 3 )
/oo
2 Fj y^- 1 rfy,
(16)
where
2F1 ~ 2 Fi (ri+r 3) 7*1 +r 2 +r 3 ;l + ri+r 3 ;-l-y). (17)
References
Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Se-
lected Tables in Mathematical Statistics, Vol. 4 : Dirichlet
Distribution — Type 1. Providence, RI: Amer. Math. Soc,
1977.
Sobel, M.; Uppuluri, R. R.; and Frankowski, K. Selected Ta-
bles in Mathematical Statistics, Vol. 9: Dirichlet Integrals
of Type 2 and Their Applications. Providence, RI: Amer.
Math. Soc, 1985.
^ Weisstein, E. W. "Dirichlet Integrals." http : //www . astro
. Virginia . edu / - eww6n / math / notebooks / Dirichlet
Integrals.m.
Dkichlet Kernel
Dirichlet L-Series 461
Dirichlet Kernel
The Dirichlet kernel D„ is obtained by integrating the
Character e i{€,x) over the Ball |£| < M,
D M =
1 d ~m
27rr dr
The Dirichlet kernel of a Delta Sequence is given by
1 sin[(n+ |)x]
S n (x) =
27r sin(|a;)
The integral of this kernal is called the DIRICHLET In-
tegral D[u].
see also Delta Sequence, Dirichlet Integrals,
Dirichlet's Lemma
Dirichlet L-Series
Series of the form
Lk{s,x) = ^2xk{n)n s ,
(1)
where the Character (Number THEORY) Xk(n) is an
Integer function with period m. These series appear
in number theory (they were used, for instance, to prove
Dirichlet's Theorem) and can be written as sums of
Lerch Transcendents with z a Power of e 27ri / m .
The Dirichlet Eta Function
n = l
(for s # l) and Dirichlet Beta Function
(2)
and Riemann Zeta Function
L +1 (s) = C(s)
(3)
(4)
are Dirichlet series (Borwein and Borwein 1987, p. 289).
Xk is called primitive if the Conductor -/(x) = k. Oth-
erwise, Xk is imprimitive. A primitive L-series modulo
k is then defined as one for which Xk{ n ) is primitive.
All imprimitive L-series can be expressed in terms of
primitive L-series.
Let P = 1 or P — Y\ i=1 Pi\ where pi are distinct Odd
PRIMES. Then there are three possible types of prim-
itive L-series with Real Coefficients. The require-
ment of Real Coefficients restricts the Character
to Xk{ri) = ±1 for all k and n. The three type are then
1. If k = P (e.g., k = 1, 3, 5, . . . ) or k = 4P (e.g.,
k = 4, 12, 20, dots), there is exactly one primitive
L-series.
2. If k — SP (e.g., k = 8, 24, . . . ), there are two primi-
tive L-series.
3. If k = 2P, Ppi, or 2 a P where a > 3 (e.g., k = 2, 6,
9, . . . ), there are no primitive L-series
(Zucker and Robertson 1976). All primitive L-series are
algebraically independent and divide into two types ac-
cording to
X*(*-l) = ±l. (5)
Primitive L-series of these types are denoted L± . For
a primitive L-series with Real Character (Number
Theory), if k = P, then
■{
L_ fc
L k
if P = 3 (mod 4)
if P = 1 (mod 4).
If k = 4P, then
L- k
L k
if P:
if Pi
: 1 (mod 4)
3 (mod 4),
(6)
(7)
and if k = 8P, then there is a primitive function of each
type (Zucker and Robertson 1976).
The first few primitive NEGATIVE L-series are L_ 3 , L_ 4 ,
L_7, L_8, L_ii, L_i5, L-19, L-20, L_23, L_24, L_3i,
L_35, L_39, L_40, L_43, L_47, L-51, L_52, L-55, L_56,
L_59, L_67i L_68, L_7i, L_79, L_83, L-84, L_87> L_s8>
L_9i, L-95, ... (Sloane's A003657), corresponding to
the negated discriminants of imaginary quadratic fields.
The first few primitive POSITIVE L-series are L+i, L+5,
L+8, L+12, L+13, L+17, L+21, L+24, L+28) L+29) L+33,
L+37, L+40, L.j-4!, L+44, L + 53, L+56, L+57, L+60, L+61,
L+65 T L+69, L+73, L+76, L+77, L+85, L+88, L+89, L+92,
L+93, L+97, . . . (Sloane's A046113).
The Kronecker Symbol is a Real Character mod-
ulo k, and is in fact essentially the only type of REAL
primitive Character (Ayoub 1963). Therefore,
L +d ( 5 ) = X>|n)n" a
n=l
oo
L- d (s) = J](-d|n)n- 3 ,
(8)
(9)
where (d\n) is the Kronecker Symbol. The functional
equations for L± are
L- k (s) = 2 5 7r a - 1 fc- s+1/2 r(l - s)cos(| S 7r)L_ fc (l - s)
(10)
L +k (s) = 2 s 7r a - 1 A;- s+1/2 r(l - s)sm(±sir)L +k (l - s).
(11)
462 Dirichlet L-Series
For m a Positive Integer
L +k (-2m) =
L- k (l - 2m) =
L +k {2m) = Rk~ 1/2 n 2,n
L_ fc (2m - 1) = .R'AT^V" 1 - 1
(-l) m (2m- 1)!J2
L +fc (l - 2m)
£-fc(-2fc)
(2&) 2 " 1 - 1
(-l) m fl'(2m)!
(2ft) 2m
(12)
(13)
(14)
(15)
(16)
(17)
where R and R' are Rational Numbers. L+k(l) can
be expressed in terms of transcendent als by
L d (l) = ft(d)«(d),
(18)
where /i(d) is the CLASS NUMBER and «(d) is the
Dirichlet Structure Constant. Some specific val-
ues of primitive L-series are
£-16(1
£-n(l
£-8(1
£-7(1
£-4(1
£-3(1
£+5(1
£+8(1
£ + 12(1
£ + 13(1
£ + 17(1
£+2l(l
£+24(1
2tt
VTE
7T
Vn
TV
2\/2
7T
V7
\«
TV
2
In
ln(2 + V3)
2
V13
2
a/17
ln(4 + \/l7)
-£-(
5 + A/21
1ii(5 + 2a/6)
No general forms are known for L-k(2m) and L + k(2m~
1) in terms of known transcendentals. For example,
L_ 4 (2)=/3(2) = K,
where K is defined as CATALAN'S CONSTANT.
(19)
see also Dirichlet Beta Function, Dirichlet Eta
Function
Dirichlet Lambda Function
References
Ayoub, R. G. An Introduction to the Analytic Theory of
Numbers. Providence, Rl: Amer. Math. Soc, 1963.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Buell, D. A. "Small Class Numbers and Extreme Values of
L- Functions of Quadratic Fields." Math. CompuL 139,
786-796, 1977.
Ireland, K. and Rosen, M. "Dirichlet L-Functions." Ch. 16 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 249-268, 1990.
Sloane, N. J. A. Sequences A046113 and A003657/M2332 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
$ Weisstein, E. W. "Class Numbers." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/ClassNumbers.rn.
Zucker, I. J. and Robertson, M. M. "Some Properties of
Dirichlet L-Series." J. Phys. A: Math. Gen. 9, 1207-1214,
1976.
Dirichlet Lambda Function
6 ■
4 -
2 ■
L
^
5
10
-2 ■
-4 ■
Re[DirichletLambda z] Im[DirichletLambda z] | DirichletLambda z|
A(x) = £(2n + l)-* = (l-2-*K(z) (1)
n =
for a: = 2, 3, . . . , where £(x) is the RlEMANN ZETA
Function. The function is undefined at x = 1. It can
be computed in closed form where £(x) can, that is for
Even Positive n. It is related to the Riemann Zeta
Function and Dirichlet Eta Function by
CM
2"
AM _ rt{v)
2»
2" -2
and
C(u) + r,{v) = 2AM
(2)
(3)
(Spanier and Oldham 1987). Special values of A(n) in-
clude
A(2):
A(4):
96'
(4)
(5)
Dirichlet's Lemma
Dirichlet's Theorem
463
see also Dirichlet Beta Function, Dirichlet Eta
Function, Riemann Zeta Function, Zeta 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,
pp. 807-808, 1972.
Spanier, J. and Oldham, K. B. "The Zeta Numbers and Re-
lated Functions." Ch. 3 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 25-33, 1987.
Dirichlet's Lemma
Dirichlet Structure Constant
I
sin[(n + \)x] j _
o 2sm(|x)
where the Kernel is the Dirichlet Kernel.
References
Cohn, H. Advanced Number Theory. New York: Dover, p. 37,
1980.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1101, 1979.
Dirichlet's Principle
Also known as THOMSON'S PRINCIPLE. There exists a
function u that minimizes the functional
D[u]
-I
VuVdV
(called the DIRICHLET INTEGRAL) for Q C IR 2 or M 3
among all the functions u E C (1) (Q)nC {0) (fi) which take
on given values / on the boundary dQ of Q, and that
function u satisfies V 2 = in fi, u\qu ~ f t u E C^ 2 '(f2)n
C^(Q). WeierstraB showed that Dirichlet's argument
contained a subtle fallacy. As a result, it can be claimed
only that there exists a lower bound to which D[u] comes
arbitrarily close without being forced to actually reach
it. Kneser, however, obtained a valid proof of Dirichlet's
principle.
see also Dirichlet's Box Principle, Dirichlet In-
tegrals
Dirichlet Region
see Voronoi Polygon
Dirichlet Series
A sum ^2 a>ne XnZ i where a n and z are COMPLEX and A n
is Real and Monotonic increasing.
see also Dirichlet L-Series
w(d)<
(d)*J\d\
for d >
for d < 0,
where 77(d) is the FUNDAMENTAL UNIT and w(d) is the
number of substitutions which leave the binary quadra-
tic form unchanged
w(d) =
6 for d = -3
4 for d = —4
2 otherwise.
see also CLASS NUMBER, DIRICHLET L-SERIES
References
$ Weisstein, E. W. "Class Numbers." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/ClassNumbers.m.
Dirichlet Tessellation
see Voronoi Diagram
Dirichlet's Test
Let
J2 a »
<K,
where K is independent of p. Then if f n > f n -\-\ >
and
it follows that
lim f n = 0,
n— voo
/ ^ anfn
Converges.
see also CONVERGENCE TESTS
Dirichlet's Theorem
Given an Arithmetic Series of terms an+fe, for n = 1,
2, . . . , the series contains an infinite number of PRIMES if
a and b are RELATIVELY PRIME, i.e., (a, 6) = 1. Dirich-
let proved this theorem using DIRICHLET L-SERIES.
see also PRIME ARITHMETIC PROGRESSION, PRIME
Patterns Conjecture, Relatively Prime, Sier-
pinski's Prime Sequence Theorem
References
Courant, R. and Robbins, H. "Primes in Arithmetical Pro-
gressions." §1.2b in Supplement to Ch. 1 in What is Math-
ematics?: An Elementary Approach to Ideas and Methods,
2nd ed. Oxford, England: Oxford University Press, pp. 26—
27, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 22-23, 1993.
464 Dirty Beam
Discordant Permutation
Dirty Beam
The Fourier Transform of the (u,v) sampling distri-
bution in synthesis imaging
Now note that
6' = ^- 1 (S),
(1)
also called the Synthesized Beam. It is called a
"beam" by way of analogy with the DlRTY MAP
I' =T~ 1 (VS) =T~ 1 [V]^T~ 1 [S]
= I*F- X (S) = /*&', (2)
where * denotes Convolution. Here, J' is the intensity
which would be observed for an extended source by an
antenna with response pattern b\ ,
/' ^b 1 (9 ,, )*I{9 tt ).
(3)
The dirty beam is often a complicated function. In order
to avoid introducing any high spatial frequency features
when CLEANing, an elliptical Gaussian is usually fit
to the dirty beam, producing a CLEAN BEAM which is
Convolved with the final iteration.
see also CLEAN Algorithm, CLEAN Map, Dirty
Map
Dirty Map
From the van Cittert-Zernicke theorem, the relationship
between observed visibility function V(u, v) and source
brightness I(£, 77) in synthesis imaging is given by
/oo /»oo
/ V(u,v)
■00 J — oo
= T-\V{u,v)}.
2niteu+r]v)
dudv
(i)
But the visibility function is sampled only at discrete
points S(u,v) (finite sampling), so only an approxima-
tion to I, called the "dirty map" and denoted /', is mea-
sured. It is given by
/oo />oo
/ S{u,v)V(u,v)e 27Ti(iu+T,v) dudv
■001/-00
= F- 1 [VS], (2)
where S(u, v) is the sampling function and V(u,v) is
the observed visibility function. Let * denote CONVO-
LUTION and rearrange the CONVOLUTION THEOREM,
Hf*9\=F\f]H9\ (3)
into the form
T[F- 1 [f]*F- 1 \g]] = f9, (4)
from which it follows that
r- 1 [f]*r- 1 \ g ] = r- 1 [fg]. (5)
I = F-\V]
(6)
is the CLEAN Map, and define the "Dirty Beam"
as the inverse FOURIER TRANSFORM of the sampling
function,
b'^T^iS]. (7)
The dirty map is then given by
I' =f- 1 [VS] =f- 1 [V]*F- 1 [S] = I*b'. (8)
In order to deconvolve the desired CLEAN Map I from
the measured dirty map /' and the known DIRTY BEAM
6\ the CLEAN Algorithm is often used.
see also CLEAN Algorithm, CLEAN Map, Dirty
Beam
Disc
see DISK
Disconnected Form
A FORM which is the sum of two FORMS involving sep-
arate sets of variables.
Disconnectivity
Disconnectivities are mathematical entities which stand
in the way of a Space being contractible (i.e., shrunk to
a point, where the shrinking takes place inside the Space
itself). When dealing with TOPOLOGICAL SPACES, a
disconnectivity is interpreted as a "Hole" in the space.
Disconnectivities in SPACE are studied through the EX-
TENSION Problem or the Lifting Problem.
see also Extension Problem, Hole, Lifting Prob-
lem
Discontinuity
discontinuity
A point at which a mathematical object is Discontin-
uous.
Discontinuous
Not Continuous. A point at which a function is dis-
continuous is called a DISCONTINUITY, or sometimes a
Jump.
References
Yates, R. C. "Functions with Discontinuous Properties." A
Handbook on Curves and Their Properties. Ann Arbor,
MI: J. W. Edwards, pp. 100-107, 1952.
Discordant Permutation
see Married Couples Problem
Discrepancy Theorem
Discrete Fourier Transform 465
Discrepancy Theorem
Let si, S2, - • • be an infinite series of real numbers lying
between and 1. Then corresponding to any arbitrar-
ily large K, there exists a positive integer n and two
subintervals of equal length such that the number of s v
with v = 1, 2, . . . , n which lie in one of the subintervals
differs from the number of such s u that lie in the other
subinterval by more than K (van der Corput 1935ab,
van Aardenne-Ehrenfest 1945, 1949, Roth 1954).
This statement can be refined as follows. Let N be a
large integer and S\, 52, . . . , sjv be a sequence of N real
numbers lying between and 1. Then for any integer
1 < rc < N an d any real number a satisfying < a < 1,
let Dn(ct) denote the number of s u with v = 1, 2, . . . , n
that satisfy < s v < a. Then there exist n and a such
that
\D n {a) — na\ > a
where c\ is a positive constant.
In In AT
In In In AT
This result can be further strengthened, which is most
easily done by reformulating the problem. Let N > 1
be an integer and Pi, P2, - • - , Pn be N (not necessarily
distinct) points in the square 0<£<1,0<2/<1.
Then
/ / [S{z,
Jo Jo
y) — Nxy] 2 dx dy > c^ In N,
where C2 is a positive constant and S(u, v) is the number
of points in the rectangle < x < it, Q < y < v (Roth
1954). Therefore,
\S(x,y) - Nxy\ > csVlnN,
and the original result can be stated as the fact that
there exist n and a such that
\D n (a) - na\ > C4VI11JV.
The randomly distributed points shown in the above
squares have \S(x,y) — Nxy\ 2 = 6.40 and 9.11, respec-
tively.
Similarly, the discrepancy of a set of N points in a unit
d-HYPERCUBE satisfies
(Roth 1954, 1976, 1979, 1980).
see also 18-Point Problem, Cube Point Picking
References
Berlekamp, E. R. and Graham, R. L. "Irregularities in the
Distributions of Finite Sequences." J. Number Th. 2, 152-
161, 1970.
Roth, K. F. "On Irregularities of Distribution." Mathematika
1, 73-79, 1954.
Roth, K. F. "On Irregularities of Distribution. II." Comm.
Pure Appl. Math. 29, 739-744, 1976.
Roth, K. F. "On Irregularities of Distribution. III." Acta
Arith. 35, 373-384, 1979.
Roth, K. F. "On Irregularities of Distribution. IV." Acta
Arith. 37, 67-75, 1980
van Aardenne-Ehrenfest, T. "Proof of the Impossibility of a
Just Distribution of an Infinite Sequence Over an Interval."
Proc. Kon. Ned. Akad. Wetensch. 48, 3-8, 1945.
van Aardenne-Ehrenfest, T. Proc. Kon. Ned. Akad. Weten-
sch. 52, 734-739, 1949.
van der Corput, J. G. Proc. Kon. Ned. Akad. Wetensch. 38,
813-821, 1935a.
van der Corput, J. G. Proc. Kon. Ned. Akad. Wetensch. 38,
1058-1066, 1935b.
Discrete Distribution
A Distribution whose variables can take on only dis-
crete values. Abramowitz and Stegun (1972, p. 929)
give a table of the parameters of most common discrete
distributions.
see also Bernoulli Distribution, Binomial Distri-
bution, Continuous Distribution, Distribution,
Geometric Distribution, Hypergeometric Dis-
tribution, Negative Binomial Distribution, Pois-
son Distribution, Probability, Statistics, Uni-
form 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. 927 and 929, 1972.
Discrete Fourier Transform
The Fourier Transform is defined as
/
\S(x,y)-Nxy\ > c(lnJV)
(d-l)/2
/M=^[/(*)]= / f(t)e' Mvt dt (1)
J —00
Now consider generalization to the case of a discrete
function, f(t) -> f(t k ) by letting f k = f(t k ), where
tk ^ &A, with k = 0, . . . , N — 1. Choose the frequency
step such that
^ = ]VA> (2)
with n = -N/2, . . . , 0, . . . , N/2. There are iV+1 values
of n, so there is one relationship between the frequency
components. Writing this out as per Press et al (1989)
JV-l N-l
T[f{t)] - Y, f k e- 2 " i(n/NA)kA A = A ^ f k e~ 27rink/N ,
k =0 k=0
(3)
466
Discrete Mathematics
Discriminant (Metric)
N-l
and
The inverse transform is
-~2irink/N
N-l
h
-y
N
F n e
2-xink/N
(4)
(5)
Note that F- n = F N -
formulation is
n = 1, 2, ..., so an alternate
"» = m' (6)
where the NEGATIVE frequencies — v c < v < have
N/2 + l<n<JV-l, Positive frequencies < v < i/ c
have 1 < n < N/2 — 1, with zero frequency n =■ 0.
n = N/2 corresponds to both u = v c and v = — v c .
The discrete Fourier transform can be computed using
a Fast Fourier Transform.
The discrete Fourier transform is a special case of the
z-Transform.
see also Fast Fourier Transform, Fourier Trans-
form, Hartley Transform, Winograd Trans-
form, z-Transform
References
Arfken, G. "Discrete Orthogonality — Discrete Fourier Trans-
form." §14.6 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 787-792, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Fourier Transform of Discretely Sampled
Data." §12.1 in Numerical Recipes in C: The Art of Sci-
entific Computing. Cambridge, England: Cambridge Uni-
versity Press, pp. 494-498, 1989.
Discrete Mathematics
The branch of mathematics dealing with objects which
can assume only certain "discrete" values. Discrete ob-
jects can be characterized by INTEGERS (or RATIONAL
Numbers), whereas continuous objects require Real
Numbers. The study of how discrete objects combine
with one another and the probabilities of various out-
comes is known as COMBINATORICS.
see also COMBINATORICS
References
Balakrishnan, V. K. Introductory Discrete Mathematics.
New York: Dover, 1997.
Bobrow, L. S. and Arbib, M. A. Discrete Mathematics:
Applied Algebra for Computer and Information Science.
Philadelphia, PA: Saunders, 1974,
Dossey, J. A.; Otto, A. D.; Spence, L.; and Eynden, C. V.
Discrete Mathematics, 3rd ed. Reading, MA: Addison-
Wesley, 1997.
Skiena, S. S. Implementing Discrete Mathematics. Reading,
MA: Addison-Wesley, 1990.
Discrete Set
A finite Set or an infinitely Countable Set of ele-
ments.
Discrete Uniform Distribution
see Equally Likely Outcomes Distribution
Discriminant
A discriminant is a quantity (usually invariant under
certain classes of transformations) which characterizes
certain properties of a quantity's ROOTS. The con-
cept of the discriminant is used for BINARY QUADRATIC
Forms, Elliptic Curves, Metrics, Modules, Poly-
nomials, Quadratic Curves, Quadratic Fields,
Quadratic Forms, and in the Second Derivative
Test.
Discriminant (Binary Quadratic Form)
The discriminant of a Binary Quadratic Form
is defined by
au + buv + cv
d = b — 4ac.
It is equal to four times the corresponding DETERMI-
NANT.
see also CLASS NUMBER
Discriminant (Elliptic Curve)
An Elliptic Curve is of the form
y — x -\- a2X -\-aix-\-<ZQ.
Let the ROOTS of y 2 be n, r2, and r^. The discriminant
is then defined as
A = k(n - r 2 ) 2 (ri - r 3 ) 2 (r2 - ^j) 2 *
see also Frey Curve, Minimal Discriminant
Discriminant (Metric)
Given a Metric g a y the discriminant is defined by
g = det(g a p) =
#11 912
921 <?22
= Pll#22 — (512) ■ (1)
Let g be the discriminant and g the transformed dis-
criminant, then
9 = D 2 g
9 = D 2 g,
where
D
D =
d(u\u 2 )
d{u\v?)
d{u\u 2 )
d(u\u 2 )
du l du 1
du 1 du 2
du 2 du 2
du 1 du 2
du 1 da 1
du 1 du 2
du 2 du 2
du 1 du 2
(2)
(3)
(4)
(5)
Discriminant (Module)
Discriminant (Polynomial) 467
Discriminant (Module)
Let a Module M in an Integral Domain D x for
R(y/D) be expressed using a two-element basis as
where £1 and £ 2 are in Di. Then the DIFFERENT of the
Module is defined as
A = A(M) =
6 6
= 6€i-£6
and the discriminant is defined as the square of the DlF
FERENT (Cohn 1980).
For Imaginary Quadratic Fields
0), the discriminants are given in the
n) (with n <
following table.
-1
-2 2
-33
-2 2 -3- 11
-67
-67
-2
-2 3
-34
-2 3 • 17
-69
-2 2 - 3 - 23
-3
-3
-35
-5-7
-70
-2 3 -5-7
-5
-2 2 -5
-37
-2 2 ■ 37
-71
-71
-6
-2 3 -3
-39
-3- 13
-73
-2 2 ■ 73
-7
-7
-41
-2 2 . 41
-74
-2 3 ■ 37
-10
-2 3 -5
-42
-2 3 -3-7
-77
-2 2 -7-11
-11
-11
-43
-43
-78
-2 3 • 3 • 13
-13
-2 2 • 13
-46
-2 3 • 23
-79
-79
-14
-2 3 -7
-47
-47
-82
-2 3 ■ 41
-15
-3-5
-51
-3- 17
-83
-83
-17
-2 2 • 17
-53
-2 2 • 53
-85
-2 2 ■ 5 ■ 17
-19
-19
-55
-5-11
-86
-2 3 -43
-21
-2 2 • 3 • 7
-57
-2 2 • 3 ■ 19
-87
-3-29
-22
-2 3 • 11
-58
-2 3 ■ 29
-89
-2 2 ■ 89
-23
-23
-59
-59
-91
-7-13
-26
-2 3 • 13
-61
-2 2 -61
-93
-2 2 -3-31
-29
-2 2 -29
-62
-2 3 -31
-94
-2 3 • 47
-30
-2 3 .3-5
-65
-2 2 ■ 5 ■ 13
-95
-5-19
-31
-31
-66
-2 3 -3*11
-97
-2 2 • 97
The discriminants of Real Quadratic Fields Q(V™ )
(n > 0) are given in the following table.
2
2 3
34
2 3 -17
67
67 -2 2
3
3-2 2
35
7 • 2 2 • 5
69
3-23
5
5
37
37
70
7 - 2 3 ■ 5
6
3-2 3
38
19 • 2 3
71
71 -2 2
7
7-2 2
39
3-2 2 -13
73
73
10
2 3 ^5
41
41
74
2 3 -37
11
11- 2 2
42
3 • 2 3 • 7
77
7-11
13
13
43
43- 2 2
78
3 ■ 2 3 ■ 13
14
7-2 3
46
23 *2 3
79
79- 2 2
15
3 • 2 2 • 5
47
47 *2 2
82
2 3 *41
17
17
51
3-2 2 -17
83
83 -2 2
19
19 -2 2
53
53
85
5-17
21
3-7
55
11 -2 2 - 5
86
43 -2 3
22
11 -2 3
57
3-19
87
3 • 2 2 • 13
23
23 -2 2
58
2 3 *29
89
89
26
2 3 -13
59
59* 2 2
91
7 - 2 2 - 13
29
29
61
61
93
3-31
30
3 • 2 3 • 5
62
31 -2 3
94
47- 2 3
31
31 -2 2
65
5*13
95
19 - 2 2 • 5
33
3- 11
66
3-2 3 -11
97
97
see also Different, Fundamental Discriminant,
Module
References
Cohn, H. Advanced Number Theory. New York: Dover,
pp. 72-73 and 261-274, 1980.
Discriminant (Polynomial)
The Product of the Squares of the differences of. the
Polynomial Roots a:*. For a Polynomial of degree
n,
D n = Y±( Xi ~ x j) 2 -
(1)
i<j
The discriminant of the QUADRATIC EQUATION
ax + bx + c =
is usually taken as
D = b 2 - 4ac.
(2)
(3)
However, using the general definition of the POLYNOM-
IAL Discriminant gives
D = Y[(zi - Zjf
(4)
where Zi are the ROOTS.
The discriminant of the CUBIC EQUATION
z 3 + a2Z 2 + a\z + ao =
is commonly defined as
where
Q =
R =
D = Q^ +R Z
3a\ — a?,
9
9a2fli — 27ao — 2a2 3
54
(5)
(6)
(7)
(8)
However, using the general definition of the polynomial
discriminant for the standard form CUBIC EQUATION
z + pz = q
(9)
gives
D = Y[(zi - ztf =P 2 = -V - 27q\ (10)
i<3
where Zi are the ROOTS and
P = {zi- z 2 )(z 2 - z 3 )(zi - z 3 ). (11)
468 Discriminant (Quadratic Curve)
Discriminant (Quadratic Curve)
The discriminant of a Quartic Equation
x + a%x + a-ix + aix + o,q = (12)
— 27ai 4 + 18a 3 a2ai 3 - 4a 3 3 ai 3 - 4a 2 3 ai 2 + a 3 2 a 2 2 ai 2
+ao(144a2ai 2 -6a3 2 ai 2 -80a3a2 2 ai + 18a3 3 a2ai + 16a2 4
-4a 3 2 a 2 3 ) + a 2 (-192a 3 ai - 128a 2 2 + 144a 3 2 a 2 - 27a 3 4 )
-256a 3 (13)
(Beeler et al 1972, Item 4).
see also RESULTANT
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Discriminant (Quadratic Curve)
Given a geaeral Quadratic Curve
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, (1)
the quantity X is known as the discriminant, where
X = B 2 - 4AG, (2)
and is invariant under Rotation. Using the Coeffi-
cients from Quadratic Equations for a rotation by
an angle 0,
A' = \A[\ + cos(20)] + \B sin(20) + §G[1 - cos(2<9)]
= ^T^ + f sin(2 ^ + ^T^ cos{2e) (3)
B' = Gcos ^ + S - |) - Gsin(2l9 + 5) (4)
C' = \A[1 - cos(2<9)] - |J3sin(20) + \)C[l + cos(20)]
-^4 ~\~ C B . /„y.v G — yl /-/i\ /-\
= — 2 ysin(20) + — ^— cos(20). (5)
Now let
G = ^/S 2 + (A - C) 2 (6)
^tan-^^^cot-^), (8)
and use
cot l (x) = |7r — tan 1 (x)
(9)
(10)
to rewrite the primed variables
A + C
A'
+ §Gcos(20 + £)
(11)
2 ' 2
B' = B cos(20) + {C - A) sin(2(9) = G cos(20 + S 2 )
(12)
Prom (11) and (13), it follows that
(13)
4A'C' = (A + Cf - G 2 cos(2(9 + S). (14)
Combining with (12) yields, for an arbitrary 9
X = B' 2 - 4A'C'
= G 2 sin 2 (20 + S) + G 2 cos 2 (20 + S) - (A + G) 2
= g 2 - (A + G) 2 = B 2 + (A - C) 2 -{A + G) 2
- B 2 - 4AG,
(15)
which is therefore invariant under rotation. This invari-
ant therefore provides a useful shortcut to determining
the shape represented by a Quadratic Curve. Choos-
ing 9 to make B' = (see Quadratic Equation), the
curve takes on the form
A'x 2 + C'y 2 + D'x + E'y + F = Q.
(16)
Completing the Square and defining new variables
gives
AV 2 +GV 2 =tf- (17)
Without loss of generality, take the sign of H to be pos-
itive. The discriminant is
X = B' 2 -£A!C = -4j4'C.
(18)
Now, if -4A'G' < 0, then A' and G' both have the
same sign, and the equation has the general form of an
Ellipse (if A' and B' are positive). If -AA'C > 0,
then A' and C' have opposite signs, and the equation
has the general form of a HYPERBOLA. If -4A'G' = 0,
then either A' or C r is zero, and the equation has the
general form of a Parabola (if the Nonzero A' or C'
is positive). Since the discriminant is invariant, these
conclusions will also hold for an arbitrary choice of 9, so
they also hold when — 4A f C f is replaced by the original
B 2 - AAC. The general result is
1. If B 2 — AAC < 0, the equation represents an ELLIPSE,
a Circle (degenerate Ellipse), a Point (degener-
ate Circle), or has no graph.
2. If B 2 - AAC > 0, the equation represents a Hyper-
bola or pair of intersecting lines (degenerate HY-
PERBOLA).
3. If B 2 — 4AC = 0, the equation represents a
Parabola, a Line (degenerate Parabola), a pair
of PARALLEL lines (degenerate Parabola), or has
no graph.
Discriminant (Quadratic Form)
Disk Covering Problem 469
Discriminant (Quadratic Form)
see Discriminant (Binary Quadratic Form)
Discriminant (Second Derivative Test)
D = Jxxjyy — Jxyjyx — Jxxjyy ~ Jxy ,
where fa are Partial Derivatives.
see also SECOND DERIVATIVE TEST
Disjoint
see Mutually Exclusive
Disjunction
A product of Ors, denoted
V*-
fc+i
Disdyakis Dodecahedron
The Dual Polyhedron of the Archimedean Great
Rhombicuboctahedron, also called the Hexakis
Octahedron.
see also GREAT DlSDYAKIS DODECAHEDRON
Disdyakis Triacontahedron
The Dual Polyhedron of the Archimedean Great
Rhombicosidodecahedron. It is also called the Hex-
AKIS ICOSAHEDRON.
see also CONJUNCTION, OR
Disjunctive Game
see Nim-Heap
Disk
An n-D disk (or Disc) of RADIUS r is the collection of
points of distance < r (Closed Disk) or < r (Open
Disk) from a fixed point in EUCLIDEAN n-space. A disk
is the Shadow of a Ball on a Plane Perpendicular
to the Ball-Radiant Point line.
The n-disk for n > 3 is called a Ball, and the boundary
of the n-disk is a (n - 1)-HYPERSPHERE. The standard
n-disk, denoted O n (or B n ), has its center at the ORIGIN
and has Radius r = 1.
see also Ball, Closed Disk, Disk Covering
Problem, Five Disks Problem, Hypersphere,
Mergelyan-Wesler Theorem, Open Disk, Poly-
disk, Sphere, Unit Disk
Disk Covering Problem
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Given a UNIT Disk, find the smallest RADIUS r(n) re-
quired for n equal disks to completely cover the Unit
Disk. For a symmetrical arrangement with n = 5
(the Five Disks Problem), r(5) = <t> - 1 = l/4> —
0.6180340. . ., where 4> is the Golden Ratio. However,
the radius can be reduced in the general disk covering
problem where symmetry is not required. The first few
such values are
r(l) = 1
r(2) = 1
r(3) = *v/3
r(4) = ±V2
r(5) = 0.609382864...
r-(6) = 0.555
r(7) = i
r(8) = 0.437
r(9) = 0.422
r(10) = 0.398.
470 Disk Covering Problem
Dispersion (Sequence)
Here, values for n = 6, 8, 9, 10 were obtained using
computer experimentation by Zahn (1962). The value
r(5) is equal to cos(0 + <£/2), where and <f) are solutions
to
2sin0-sin(0 + \<j> + V) - sin(</> - 9 - \<j>) = (1)
2 sin <j> - sin(0 + \<j> + x) - sin(x - - \<l>) = (2)
2 sin 8 + sin(x + 6) - sin(x - 0) - sin(^ + <t>)
- sin(V> - <f>) ~ 2sin(^ - 20) = (3)
cos(2i/> — x + 0) — cos(2V» + x — 0) — 2 cos x
+ cos(2t/> + x - 29) + cos(2-0 - x - 28) = (4)
(Neville 1915). It is also given by 1/z, where x is the
largest real root of
a{y)x G ~ b(y)x 5 + c(y)x 4 - d(y)x 3
+e(y)x 2 -f(y)x + g(y) = Q (5)
maximized over all y, subject to the constraints
V2 < x < 2y + 1 (6)
-1 < 2/ < 1,
(7)
and with
a{y) = 80y 2 + 64y (8)
6(y) = 416z/ 3 + 384y 2 + 64?/ (9)
c(y) = 848y 4 + 928y 3 + 352y 2 4- 32y (10)
d(y) = 768y 5 + 992y 4 + 736y 3 + 2SSy 2 + 96y
e(y) = 256y 6 + 384/ + 592y 4 + 4S0y 3 + 336y 2
-f 96y + 16 (11)
f(y) = 128y 5 + 192y 4 + 256y 3 + 160y 2 + 96y + 32
(12)
g(y) = 6% 2 + Uy + 16 (13)
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/circle/circle .html.
Kershner, R. "The Number of Circles Covering a Set." Amer.
J. Math. 61, 665-671, 1939.
Neville, E. H. "On the Solution of Numerical Functional
Equations, Illustrated by an Account of a Popular Puz-
zle and of its Solution." Proc. London Math. Soc. 14,
308-326, 1915.
Verblunsky, S. "On the Least Number of Unit Circles which
Can Cover a Square." J. London Math. Soc. 24, 164-170,
1949.
Zahn, C. T. "Black Box Maximization of Circular Coverage."
J. Res. Nat. Bur. Stand. B 66, 181-216, 1962.
Disk Lattice Points
see Gauss's Circle Problem
Dispersion Numbers
see Magic Geometric Constants
Dispersion Relation
Any pair of equations giving the REAL PART of a func-
tion as an integral of its Imaginary Part and the Imag-
inary Part as an integral of its Real Part. Dispersion
relationships imply causality in physics. Let
then
f(x ) = u(x ) + iv(x Q ) y
u( X0 ) = ±pv[ V -<^L
* J -co X ~ X °
v{zo) = --PV
7T
u(x) dx
X — Xq
(i)
(2)
(3)
where PV denotes the Cauchy Principal Value and
u(xo) and v(xo) are Hilbert TRANSFORMS of each
other. If the COMPLEX function is symmetric such that
f(-x) = f*(x), then
(Bezdek 1983, 1984).
Letting N(e) be the smallest number of Disks of Radius
e needed to cover a disk D, the limit of the ratio of the
Area of D to the Area of the disks is given by
lim
0+ e 2 7V(e)
3\/3
2tt
(14)
(Kershner 1939, Verblunsky 1949).
see also FIVE DISKS PROBLEM
References
Ball, W. W. R. and Coxeter, H. S. M. "The Five-Disc Prob-
lem." In Mathematical Recreations and Essays, 13th ed.
New York: Dover, pp. 97-99, 1987.
Bezdek, K. "Uber einige Kreisiiberdeckungen." Beitrage Al-
gebra Geom.14, 7-13, 1983.
Bezdek, K. "Uber einige optimale Konfigurationen von
Kreisen." Ann. Univ. Sci. Budapest Eotvos Sect. Math.
27, 141-151, 1984.
u(xq)
2 py f°° xv(x) dx
* Jo x2 ~ x ° 2
2 f°° xu(x)dx
-M = --pvJ q ^^.
(4)
(5)
Dispersion (Sequence)
An array B = 6^, i, j > 1 of POSITIVE INTEGERS is
called a dispersion if
1. The first column of B is a strictly increasing se-
quence, and there exists a strictly increasing se-
quence {sk} such that
2. b 12 = Si > 2,
3. The complement of the Set {bn : i > 1} is the Set
4. bij = Sbi j _ 1 for all j > 3 for i = 1 and for all g > 2
for all i > 2.
Dispersion (Statistics)
Dissection
471
If an array B = bij is a dispersion, then it is an INTER-
SPERSION.
see also Interspersion
References
Kimberling, C. "Interspersions and Dispersions." Proc.
Amer. Math. Soc. 117, 313-321, 1993.
Dispersion (Statistics)
(Au) a . = (u,-«) 2 .
see also Absolute Deviation, Signed Deviation,
Variance
Disphenocingulum
see Johnson Solid
Disphenoid
A Tetrahedron with identical Isosceles or Scalene
faces.
Dissection
Any two rectilinear figures with equal Area can be dis-
sected into a finite number of pieces to form each other.
This is the WaLLACE-BoLYAI-GeRWEIN THEOREM. For
minimal dissections of a TRIANGLE, PENTAGON, and
Octagon into a Square, see Stewart (1987, pp. 169-
170) and Ball and Coxeter (1987, pp. 89-91). The TRI-
ANGLE to Square dissection (Haberdasher's Prob-
lem) is particularly interesting because it can be built
from hinged pieces which can be folded and unfolded
to yield the two shapes (Gardner 1961; Stewart 1987,
p. 169; Pappas 1989).
xjCk?<7
Laczkovich (1988) proved that the CIRCLE can be
squared in a finite number of dissections (~ 10 50 ). Fur-
thermore, any shape whose boundary is composed of
smoothly curving pieces can be dissected into a Square.
The situation becomes considerably more difficult mov-
ing from 2-D to 3-D. In general, a POLYHEDRON can-
not be dissected into other POLYHEDRA of a specified
type. A Cube can be dissected into n 3 Cubes, where
n is any INTEGER. In 1900, Dehn proved that not ev-
ery Prism cannot be dissected into a Tetrahedron
(Lenhard 1962, Ball and Coxeter 1987) The third of
Hilbert's Problems asks for the determination of two
TETRAHEDRA which cannot be decomposed into con-
gruent TETRAHEDRA directly or by adjoining congru-
ent Tetrahedra. Max Dehn showed this could not be
done in 1902, and W. F. Kagon obtained the same re-
sult independently in 1903. A quantity growing out of
Dehn's work which can be used to analyze the possibil-
ity of performing a given solid dissection is the DEHN
Invariant.
The table below is an updated version of the one given
in Gardner (1991, p. 50). Many of the improvements
are due to G. Theobald (Frederickson 1997). The mini-
mum number of pieces known to dissect a regular n-gon
(where n is a number in the first column) into a fc-gon
(where fc is a number is the bottom row) is read off by
the intersection of the corresponding row and column.
In the table, {n} denotes a regular n-gon, GR a GOLDEN
Rectangle, GC a Greek Cross, LC a Latin Cross,
MC a Maltese Cross, SW a Swastika, {5/2} a five-
point star (solid PENTAGRAM), {6/2} a six-point star
(i.e., Hexagram or solid Star of David), and {8/3}
the solid OCTAGRAM.
{4}
4
{5}
6
6
{6}
5
5
7
{7}
8
7
9
8
{8}
7
5
9
8
11
{9}
8
9
12
11
14
13
{10}
7
7
10
9
11
10
13
{12}
8
6
10
6
11
10
14
12
GR
4
3
6
5
7
6
9
6
7
GC
5
4
7
7
9
9
12
10
6
LC
5
5
8
6
8
8
11
10
7
MC
7
14
SW
6
12
{5/2}
7
7
9
9
11
10
14
6
12
{6/2}
5
5
8
6
9
8
11
9
9
{8/3}
8
8
9
9
12
6
13
12
12
{3}
{4}
{5}
{6}
{7}
{8}
{9}
{10}
{12}
GC
5
LC
5
7
MC
8
SW
8
9
{5/2}
7
12
10
10
{6/2}
5
8
8
11
{8/3}
7
10
11
13
10
GR
GC
LC
MC
SW
{5/2}
{6/2}
The best-known dissections of one regular convex n-gon
into another are shown for n = 3, 4, 5, 6, 7, 8, 9, 10,
and 12 in the following illustrations due to Theobald.
472 Dissection
Dissection
4-3
12-4
12-5
5-3
6-3
7-3
8-3
5-4
6-4
7-4
8-4
6-5
8-5
12-7
12-8
12-9
12-10
The best-known dissections of regular concave poly-
gons are illustrated below for {5/2}, {6/2}, and {8/3}
(Theobald).
V
%-5
9-3
9-4
9-5
%-9
%-3
A
V
%-4
9-8
5 /,-12
%-9
6/ .5/
'2 '2
6 / 9 -12
10-9
W
V'2
Dissection
Distance 473
The best-known dissections of various crosses are illus-
trated below (Theobald).
MC-4
The best-known dissections of the GOLDEN RECTANGLE
are illustrated below (Theobald).
R-10
see also Banach-Tarski Paradox, Cundy and Rol-
lett's Egg, Decagon, Dehn Invariant, Diaboli-
cal Cube, Dissection Puzzles, Dodecagon, Ehr-
hart Polynomial, Equidecomposable, Equilat-
eral Triangle, Golden Rectangle, Heptagon
Hexagon, Hexagram, Hilbert's Problems, Latin
Cross, Maltese Cross, Nonagon, Octagon, Oc-
tagram, Pentagon, Pentagram, Polyhedron Dis-
section, Pythagorean Square Puzzle, Pythag-
orean Theorem, Rep-Tile, Soma Cube, Square,
Star of Lakshmi, Swastika, T-Puzzle, Tangram,
Wallace-Bolyai-Gerwein Theorem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 87-94,
1987.
Coffin, S. T. The Puzzling World of Polyhedral Dissections.
New York: Oxford University Press, 1990.
Cundy, H. and Rollett, A. Ch. 2 in Mathematical Models, 3rd
ed. Stradbroke, England: Tarquin Pub., 1989.
Eppstein, D. "Dissection." http://www . ics . uci . edu / -
eppstein/ junkyard/dissect. html.
Eppstein, D. "Dissection Tiling." http://www.ics.uci.edu
/ -eppstein/ junkyard/distile.
Eriksson, K. "Splitting a Polygon into Two Congruent
Pieces." Amer. Math. Monthly 103, 393-400, 1996.
Frederickson, G. Dissections: Plane and Fancy. New York:
Cambridge University Press, 1997.
Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New
York: Simon and Schuster, 1961,
Gardner, M. "Paper Cutting." Ch. 5 in Martin Gardner's
New Mathematical Diversions from Scientific American.
New York: Simon and Schuster, 1966.
Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: Chicago University Press,
1991.
Hunter, J, A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 65-67, 1975.
Kraitchik, M. "Dissection of Plane Figures." §8.1 in Mathe-
matical Recreations. New York: W. W. Norton, pp. 193-
198, 1942.
Laczkovich, M. "Von Neumann's Paradox with Translation."
Fund. Math. 131, 1-12, 1988.
Lenhard, H.-C. "Uber fiinf neue Tetraeder, die einem Wurfel
aquivalent sind." Elemente Math. 17, 108-109, 1962.
Lindgren, H. "Geometric Dissections." Austral. Math.
Teacher 7, 7-10, 1951.
Lindgren, H. "Geometric Dissections." Austral. Math.
Teacher 9, 17-21, 1953.
Lindgren, H. "Going One Better in Geometric Dissections."
Math. Gaz. 45, 94-97, 1961.
Lindgren, H. Recreational Problems in Geometric Dissection
and How to Solve Them. New York: Dover, 1972.
Madachy, J. S. "Geometric Dissection." Ch. 1 in Madachy's
Mathematical Recreations. New York: Dover, pp. 15-33,
1979.
Pappas, T. "A Triangle to a Square." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, pp. 9 and
230, 1989.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, 1987.
Dissection Puzzles
A puzzle in which one object is to be converted to an-
other by making a finite number of cuts and reassem-
bling it. The cuts are often, but not always, restricted to
straight lines. Sometimes, a given puzzle is precut and
is to be re-assembled into two or more given shapes.
see also Cundy and Rollett's Egg, Pythagorean
Square Puzzle, T-Puzzle, Tangram
Dissipative System
A system in which the phase space volume contracts
along a trajectory. This means that the generalized Di-
vergence is less than zero,
dxt
where EINSTEIN SUMMATION has been used.
Distance
Let 7(t) be a smooth curve in a MANIFOLD M from x to
y with 7(0) = x and 7(1) = y. Then Y(t) e T 7 ( t ), where
474 Distance
Distinct Prime Factors
T x is the Tangent Space of M at x. The Length of
7 with respect to the Riemannian structure is given by
/'
Jo
HVWII-rW*.
(1)
and the distance d(x, y) between x and y is the shortest
distance between x and y given by
d{x,y)= inf /|| 7 '(t)|| 7 (t)dt.
-y:x to y J
(2)
In order to specify the relative distances of n > 1 points
in the plane, l+2(n — 2) = 2n— 3 coordinates are needed,
since the first can always be taken as (0, 0) and the sec-
ond as (z,0), which defines the x-AxiS. The remaining
n — 2 points need two coordinates each. However, the
total number of distances is
2!(n-2)!
\n{n- 1),
(3)
where (™) is a BINOMIAL COEFFICIENT. The distances
between n > 1 points are therefore subject to m rela-
tionships, where
m = \n(n - 1) - (2n - 3) = \{n - 2)(n - 3). (4)
For n = 1, 2, . . . , this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28,
... (Sloane's A000217) relationships, and the number
of relationships between n points is the TRIANGULAR
Number T„_ 3 .
Although there are no relationships for n = 2 and n =
3 points, for n = 4 (a QUADRILATERAL), there is one
(Weinberg 1972):
= di2 ds4 + <ii3 d,24 + di4 Gfo3 + <^23 ^14
-f" ^24^13 + ^34^12
-h d 12 d 2 3dsi + ^12^24^41 + ^13^34^41
+ ^23^34^42 ~~ <^12^23^34 ~" ^13^32^24
— C?i2^24^43 "~ ^14^42^23 — ^13^34*^42
— ^14^43^32 — ^23^31^14 " ^21^13^34
^24^41 ^13 ~~ ^21^14^43 — ^31^12^24
"32"21"14*
(5)
This equation can be derived by writing
dij = yJixi-XjY + iyi-Vj) 2 ( 6 )
and eliminating Xi and yj from the equations for di2,
di3, di4, ^23? ^24 1 and d^.
see also Arc Length, Cube Point Picking, Ex-
pansive, Length (Curve), Metric, Planar Dis-
tance, Point-Line Distance — 2-D, Point-Line
Distance — 3-D, Point-Plane Distance, Point-
Point Distance — 1-D, Point-Point Distance — 2-
D, Point-Point Distance — 3-D, Space Distance,
Sphere
References
Gray, A. "The Intuitive Idea of Distance on a Surface." §13.1
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 251-255, 1993.
Sloane, N. J. A. Sequence A000217/M2535 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Weinberg, S. Gravitation and Cosmology: Principles and
Applications of the General Theory of Relativity. New
York: Wiley, p. 7, 1972.
Distinct Prime Factors
3
2.5
ii j 1
2
1.5
1
III
w
hrn
11
)
0.5
1
20
40 60 80
100
200 400 600 800 1000
The number of distinct prime factors of a number n is
denoted w(n). The first few values for n = 1, 2, ...
are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1,
2, . . . (Sloane's A001221). The first few values of the
SUMMATORY FUNCTION
n
k-2
are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21,
. . . (Sloane's A013939), and the asymptotic value is
n
y" uj(k) = nlnlnn + Bin + o(n),
where B\ is Mertens Constant. In addition,
n
Y^["(k)] 2 = n(lnlnn) 2 + O(nlnlnn).
see also Divisor Function, Greatest Prime Fac-
tor, Hardy- Ramanujan Theorem, Heteroge-
neous Numbers, Least Prime Factor, Mertens
Constant, Prime Factors
References
Hardy, G. H. and Wright, E. M. "The Number of Prime Fac-
tors of n" and "The Normal Order of w(n) and fi(ra)."
§22.10 and 22.11 in An Introduction to the Theory of Num-
bers, 5th ed. Oxford, England: Clarendon Press, pp. 354-
358, 1979.
Sloane, N. J. A. Sequences A013939 and A001221/M0056 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Distribution
Distribution Function 475
Distribution
The distribution of a variable is a description of the rel-
ative numbers of times each possible outcome will occur
in a number of trials. The function describing the distri-
bution is called the Probability Function, and the
function describing the probability that a given value or
any value smaller than it will occur is called the DIS-
TRIBUTION Function.
Formally, a distribution can be defined as a normalized
Measure, and the distribution of a Random Variable
x is the MEASURE P x on §' defined by setting
F x (A / ) = P{5G5:a;(6)G^},
where (S,8,P) is a PROBABILITY SPACE, (5, S) is a
Measurable Space, and P a Measure on § with
P(S) = 1.
see also CONTINUOUS DISTRIBUTION, DISCRETE DIS-
TRIBUTION, Distribution Function, Measurable
Space, Measure, Probability, Probability Den-
sity Function, Random Variable, Statistics
References
Doob, J. L. "The Development of Rigor in Mathematical
Probability (1900-1950)." Amer. Math. Monthly 103,
586-595, 1996.
Distribution Function
The distribution function D(x), sometimes also called
the Probability Distribution Function, describes
the probability that a trial X takes on a value less than
or equal to a number x. The distribution function is
therefore related to a continuous Probability Density
Function P(x) by
/x
P(x f )dx,
-oo
(1)
so P(x) (when it exists), is simply the derivative of the
distribution function
Similarly, a multiple distribution function can be defined
if outcomes depend on n parameters:
D(ai,...,a„) = P(xi <ai,... 9 x n < a n )- (7)
Given a continuous P(x), assume you wish to generate
numbers distributed as P(x) using a random number
generator. If the random number generator yields a uni-
formly distributed value y% in [0,1] for each trial i, then
compute
D(x)
-r
P(x) dx.
(8)
The FORMULA connecting yi with a variable distributed
as P(x) is then
(9)
Xi = D 1 (y i ) i
where D 1 (x) is the inverse function of D(x). For ex-
ample, if P(x) were a Gaussian Distribution so that
D(x) = -
1 + erf
\ try/2 J,
then
Xi = <rV2eri 1 (2y i -!) + /*•
(10)
(11)
If P(x) = Cx n for x £ (z m in, x max ), then normalization
gives
/"
n+lixn
P{x)dx — C-
n+ 1
= 1,
n+ 1
(12)
(13)
Let y be a uniformly distributed variate on [0, 1], Then
1 dx
D(x)= J P{x)dx = C I x n
= -^-( X ^- Xmin ^) = y, (14)
n+1
P(x) = D'(x) = [P(x')]-oo = P{x) - P(-oo). (2) and the variate given by
Similarly, the distribution function is related to a dis-
crete probability P(x) by
l/(n+l)
D(x) = P{X <x)=^2 p ( x )'
(3)
In general, there exist distributions which are neither
continuous nor discrete.
A Joint Distribution Function can be defined if
outcomes are dependent on two parameters:
D(x,y) = P(X <x,Y <y)
D x (x) = D(x y oo)
Dy(y) = D(oo,y).
(4)
(5)
(6)
^ [(iC m ax ^min )y ~T~ 3?min J \*-^)
is distributed as P(x).
A distribution with constant VARIANCE of y for all val-
ues of x is known as a HOMOSCEDASTIC distribution.
The method of finding the value at which the distribu-
tion is a maximum is known as the MAXIMUM LIKELI-
HOOD method,
see also Bernoulli Distribution, Beta Distri-
bution, Binomial Distribution, Bivariate Dis-
tribution, Cauchy Distribution, Chi Distribu-
tion, Chi-Squared Distribution, Cornish-Fisher
476 Distribution (Functional)
Distributive
Asymptotic Expansion, Correlation Coeffi-
cient, Distribution, Double Exponential Distri-
bution, Equally Likely Outcomes Distribution,
Exponential Distribution, Extreme Value Dis-
tribution, F-Distribution, Fermi-Dirac Distri-
bution, Fisher's z-Distribution, Fisher-Tippett
Distribution, Gamma Distribution, Gaussian
Distribution, Geometric Distribution, Half-
Normal Distribution, Hypergeometric Distri-
bution, Joint Distribution Function, Laplace
Distribution, Lattice Distribution, Levy Dis-
tribution, Logarithmic Distribution, Log-Series
Distribution, Logistic Distribution, Lorentzian
Distribution, Maxwell Distribution, Negative
Binomial Distribution, Normal Distribution,
Pareto Distribution, Pascal Distribution, Pear-
son Type III Distribution, Poisson Distri-
bution, Polya Distribution, Ratio Distribu-
tion, Rayleigh Distribution, Rice Distribu-
tion, Snedecor's F-Distribution, Student's t-
Distribution, Student's z-Distribution, Uniform
Distribution, Weibull Distribution
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Probability
Functions." Ch. 26 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 925-964, 1972.
Iyanaga, S. and Kawada, Y. (Eds.). "Distribution of Typical
Random Variables." Appendix A, Table 22 in Encyclopedic
Dictionary of Mathematics. Cambridge, MA: MIT Press,
pp. 1483-1486, 1980.
Distribution (Functional)
A functional distribution, also called a GENERALIZED
FUNCTION, is a generalization of the concept of a func-
tion. Functional distributions are defined as continuous
linear FUNCTIONALS over a SPACE of infinitely differen-
tiable functions such that all continuous functions have
SCHWARZIAN DERIVATIVES which are themselves distri-
butions. The most commonly encountered functional
distribution is the DELTA FUNCTION.
see also DELTA FUNCTION, GENERALIZED FUNCTION,
Schwarzian Derivative
References
Friedlander, F. G. Introduction to the Theory of Distribu-
tions. Cambridge, England: Cambridge University Press,
1982.
Gel'fand, I. M. and Shilov, G. E. Generalized Functions,
Vol. 1: Properties and Operations. New York: Harcourt
Brace, 1977.
Gel'fand, I. M. and Shilov, G. E. Generalized Functions,
Vol. 2: Spaces of Fundamental and Generalized Functions.
New York: Harcourt Brace, 1977.
Gel'fand, I. M. and Shilov, G. E. Generalized Functions,
Vol. 3: Theory of Differential Equations. New York: Har-
court Brace, 1977.
Gel'fand, I. M. and Vilenkin, N. Ya. Generalized Functions,
Vol. 4 : Applications of Harmonic Analysis. New York:
Harcourt Brace, 1977.
Gel'fand, I. M.; Graev, M. I.; and Vilenkin, N. Ya. General-
ized Functions, Vol. 5: Integral Geometry and Represen-
tation Theory. New York: Harcourt Brace, 1977.
Griffel, D. H. Applied Functional Analysis. Englewood Cliffs,
NJ: Prentice-Hall, 1984.
Halperin, I. and Schwartz, L. Introduction to the Theory
of Distributions, Based on the Lectures Given by Laurent
Schwarz. Toronto, Canada: University of Toronto Press,
1952.
Lighthill, M. J. Introduction to Fourier Analysis and Gen-
eralised Functions. Cambridge, England: Cambridge Uni-
versity Press, 1958.
Richards, I. and Young, H. The Theory of Distributions: A
Nontechnical Introduction. New York: Cambridge Univer-
sity Press, 1995.
Rudin, W. Functional Analysis, 2nd ed. New York:
McGraw-Hill, 1991.
Strichartz, R. Fourier Transforms and Distribution Theory.
Boca Raton, FL: CRC Press, 1993.
Zemanian, A. H. Distribution Theory and Transform Anal-
ysis: An Introduction to Generalized Functions, with Ap-
plications. New York: Dover, 1987.
Distribution Parameter
The distribution parameter of a NONCYLINDRICAL
Ruled Surface parameterized by
c(ii, v) = <t(u) + vS(u),
(1)
where a is the Striction Curve and 8 the Director
CURVE, is the function p defined by
det(<r'65')
6' ■ *' "
(2)
The Gaussian Curvature of a Ruled Surface is
given in terms of its distribution parameter by
K = -
\p(u)} 2
{\p{u)Y+v*y
(3)
see also NONCYLINDRICAL RULED SURFACE, STRICTION
Curve
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 347-348, 1993.
Distribution (Statistical)
The set of probabilities for each possible event.
see Distribution Function
Distributive
Elements of an Algebra which obey the identity
A(B 4- C) = AB + AC
are said to be distributive over the operation +.
see also ASSOCIATIVE, COMMUTATIVE, TRANSITIVE
Distributive Lattice
Divergence Theorem 477
Distributive Lattice
A Lattice which satisfies the identities
(x A y) V (a; A z) — x A (y V z)
(x V y) A (x V z) = x V (y A z)
is said to be distributive.
see also Lattice, Modular Lattice
References
Gratzer, G. Lattice Theory: First Concepts and Distributive
Lattices. San Francisco, CA: W. H. Freeman, pp. 35—36,
1971.
Disymmetric
An object which is not superimposable on its MIRROR
Image is said to be disymmetric. All asymmetric ob-
jects are disymmetric, and an object with no IMPROPER
Rotation (rotoinversion) axis must also be disymmet-
Ditrigonal Dodecadodecahedron
The Uniform Polyhedron L/41 , also called the
Ditrigonal Dodecahedron, whose Dual Polyhe-
dron is the Medial Triambic Icosahedron. It has
Wythoff Symbol 3 | f 5. Its faces are 12{§} + 12{5}.
It is a Faceted version of the Small Ditrigonal
Icosidodecahedron. The Circumradius for unit
edge length is
R=±y/3.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 123-124, 1989.
Ditrigonal Dodecahedron
see Ditrigonal Dodecadodecahedron
Divergence
The divergence of a Vector Field F is given by
$ c F • da
div(F) ~ V-F= lim Js
Define
v->o V
F = Fiiii + F 2 U2 + F3U3.
(1)
(2)
Then in arbitrary orthogonal CURVILINEAR COORDI-
NATES,
div(F) = V • F =
h\li2hz
d_
{h 2 h z F 1 )
d
+ J-(h 3 h l F 2 ) + J~(h 1 h 2 F 3 )] . (3)
If V • F = 0, then the field is said to be a DlVERGENCE-
LESS FIELD. For divergence in individual coordinate sys-
tems, see Curvilinear Coordinates.
Ax __ Tr(A) x T (Ax)
ix| |x| |xr
The divergence of a Tens OR A is
V-A = A% = A%+r%A',
(4)
(5)
where ; is the COVARIANT DERIVATIVE. Expanding the
terms gives
A% = A% + (T^A* + T% a A* + T^A*)
+ A^ + (Tl y A a + T} y A* + r^>r) . (6)
see also Curl, Curl Theorem, Gradient, Green's
Theorem, Divergence Theorem, Vector Deriva-
tive
References
Arfken, G. "Divergence, V-." §1.7 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 37-42, 1985.
Divergence Tests
If
lim uh ^ 0,
k— J- 00
then the series {u n } diverges.
see also Convergence Tests, Convergent Series,
Dini's Test, Series
Divergence Theorem
A.k.a. Gauss's Theorem. Let V be a region in space
with boundary 8V. Then
J (V-F)dV= J F
Jv J dV
■ da, (1)
Let S be a region in the plane with boundary dS.
(2)
/ V -FdA= / F-nds.
Js Jas
If the Vector Field F satisfies certain constraints,
simplified forms can be used. If F(cc,y, z) — v(x i y,z)c
where c is a constant vector / 0, then
/ F • da = c • / v da.
Js Js
(3)
478
But
so
Divergenceless Field
V-(/v) = (V/)-v + /(V-v),
(4)
/ V - (cv) dV = c * / ( Vv + uV • c) dF = c - / Vv dV
(5)
(6)
[ vda- J S/vdVj = 0.
But c^O, and c ■ f(v) must vary with v so that c ■ f (v)
cannot always equal zero. Therefore,
/ vda.— / Vv
Js Jv
dV.
(7)
If F(x, y y z) = c x P(x, y, z), where c is a constant vector
^ 0, then
/daXP=: f
Js Jv
V x P dV.
(8)
see also Curl Theorem, Gradient, Green's Theo-
rem
References
Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 57-61, 1985.
Divergenceless Field
A divergenceless field, also called a SOLENOIDAL FIELD,
is a FIELD for which V • F = 0. Therefore, there exists a
G such that F = V x G. Furthermore, F can be written
as
?2/
F - V x (Tr) + V z (Sr) = T + S,
where
T = V x (Tr) = -
S = V 2 (Sr) = V
r x (VT)
■ d
dr
(rS)
rV 2 5.
Following Lamb, T and S are called TOROIDAL Field
and Poloidal Field.
see also BELTRAMI FIELD, IRROTATIONAL FIELD,
Poloidal Field, Solenoidal Field, Toroidal
Field
Divergent Sequence
A divergent sequence is a Sequence for which the Limit
exists but is not Convergent.
see also CONVERGENT SEQUENCE, DIVERGENT SERIES
Divergent Series
A Series which is not Convergent. Series may di-
verge by marching off to infinity or by oscillating.
see also CONVERGENT SERIES, DIVERGENT SEQUENCE
References
Bromwich, T. J. I'a and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, 1991.
Divided Difference
Diversity Condition
For any group of k men out of N, there must be at least
k jobs for which they are collectively qualified.
Divide
To divide is to perform the operation of DIVISION, i.e.,
to see how many time a DIVISOR d goes into another
number n. n divided by d is written n/d or n ~ d. The
result need not be an INTEGER, but if it is, some addi-
tional terminology is used. d\n is read "d divides n n and
means that d is a Proper Divisor of n. In this case, n
is said to be Divisible by d. Clearly, l|n and n\n. By
convention, n|0 for every n except (Hardy and Wright
1979). The "divided" operation satisfies
b\a and c\b => c\a
b\a => bc\ac
c\a and c\b => c\(ma + nb).
d!\n is read "d' does not divide n" and means that d' is
not a Proper Divisor of n. a k \\b means a k divides b
exactly.
see also Congruence, Divisible, Division, Divisor
References
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 1, 1979.
Divided Difference
The divided difference /[xi, X2, • ■ • , x n ] on n points Xi,
X2, . ■ • , x n of a function f(x) is denned by f[xi] = f(x\)
and
, r i /[xi,...,a n ] -/[x 2 ,...,x n ] m
/[Xi,X 2) ... ,X n j = UJ
X\ x n
for n > 2. The first few differences are
/o-/i
Xo — Xi
[Xq,Xi] — [Xi,X2J
Xo — X2
[xo,xi] =
[X0,#1,X2] =
(2)
(3)
[ao,ai,.^n]= [a ° ^ [si,---,*^ (4)
Xo — x n
Defining
7T n (x) = (X - X )(X - Xi) ■ • ■ (X - X n ) (5)
and taking the DERIVATIVE
^(Xfc) = (Xfc-Xo) * ■ ■ (x fc -Xfe-l)(Xfc-Xfc4.l) ' * * (Xfc-Xn)
(6)
gives the identity
n fk
[x ,Xi, . . . ,x n ] = \ — r — r.
^— ' TTn(Xfc)
(7)
Divine Proportion
Division Algebra 479
Consider the following question: does the property
f[x lj x 2l ... i x n ] = h(xi +£ 2 + ... + x n ) (8)
for n > 2 and h(x) a given function guarantee that
f(x) is a Polynomial of degree < n? Aczel (1985)
showed that the answer is "yes" for n = 2, and Bailey
(1992) showed it to be true for n = 3 with differen-
tiate f(x). Schwaiger (1994) and Andersen (1996) sub-
sequently showed the answer to be "yes" for all n > 3
with restrictions on f(x) or h(x).
see also Newton's Divided Difference Interpola-
tion Formula, Reciprocal Difference
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 877-878, 1972.
Aczel, J. "A Mean Value Property of the Derivative of Quad-
ratic Polynomials — Without Mean Values and Deriva-
tives." Math. Mag. 58, 42-45, 1985.
Andersen, K. M. "A Characterization of Polynomials."
Math. Mag. 69, 137-142, 1996.
Bailey, D. F. "A Mean- Value Property of Cubic Polynomi-
als—Without Mean Values." Math. Mag. 65, 123-124,
1992,
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 439-440, 1987.
Schwaiger, J. "On a Characterization of Polynomials by Di-
vided Differences." Aequationes Math. 48, 317-323, 1994.
Divine Proportion
see Golden Ratio
Divisibility Tests
Write a decimal number a out digit by digit in the form
a n . . . a3(i2a\aQ. It is always true that 10° = 1 = 1 for
any base.
2 10 1 = 0, so 10" ~ for n > 1. Therefore, if the last
digit ao is divisible by 2 (i.e., is Even), then so is
a.
10 1 == 2, 10 2 = 4, 10 3
0,
10" = 0. There-
3 10 1 = 1, 10 2 = 1,
10" = 1. Therefore, if
SILi ai ls divisible by 3, so is a.
4 10 1 = 2, 10 2 = 0, . . . 10" = 0. So if the last two
digits are divisible by 4, more specifically if r =
ao + 2ai is, then so is a.
5 10 1 = 0, so 10 n = for n > 1. Therefore, if the last
digit ao is divisible by 5 (i.e., is 5 or 0), then so is
ao.
6 10 1 == -2, 10 2 = -2, so 10" = -2. Therefore, if
r = ao — 2 ^y? =1 d{ is divisible by 6, so is a. If a is
divisible by 3 and is EVEN, it is also divisible by 6.
7 10 1 ~ 3, 10 2 = 2, 10 3 = -1, 10 4 = -3, 10 5 = -2,
10 6 = 1, and the sequence then repeats. Therefore,
if r = (ao + 3ai + 2a 2 — a$ — 3a4 — 2a$ ) + (a& + 3a7 +
...) + ... is divisible by 7, so is a.
fore, if the last three digits are divisible by 8, more
specifically if r = ao + 2ai -f 4a 2 is, then so is a.
9 10 1 = 1, 10 2 = 1, ..., 10 3 = 1. Therefore, if
S"^i ai IS divisible by 9, so is a.
10 10 1 = 0, so if the last digit is 0, then a is divisible
by 10.
11 10 1 = -1, 10 2 = 1, 10 3 = -1, 10 4 = 1, . . . . There-
fore, if r = ao — ai + a 2 — a3 + . . . is divisible by 11,
then so is a.
12 10 1 = -2, 10 2 = 4, 10 3 = 4, . . . . Therefore, if
r = ao - 2ai + 4(a 2 + az + . . .) is divisible by 12,
then so is a. Divisibility by 12 can also be checked
by seeing if a is divisible by 3 and 4.
13 10 1 = -3, 10 2 = -4, 10 3 = -1, 10 4 = 3, 10 5 = 4,
10 6 = 1, and the pattern repeats. Therefore, if r =
(ao-3ai-4a 2 -a3+3a4+4a 5 ) + (a6-3a7 + . . .) + . . •
is divisible by 13, so is a.
For additional tests for 13, see Gardner (1991).
References
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 337-
346, 1952.
Gardner, M. Ch. 14 in The Unexpected Hanging and Other
Mathematical Diversions. Chicago, IL: Chicago University
Press, 1991,
Divisible
A number n is said to be divisible by d if d is a PROPER
Divisor of n. The sum of any n consecutive Integers
is divisible by n!, where n! is the FACTORIAL.
see also DIVIDE, DIVISOR, DIVISOR FUNCTION
References
Guy, R. K. "Divisibility." Ch. B in Unsolved Problems in
Number Theory, 2nd ed. New York: Springer-Verlag,
pp. 44-104, 1994.
Division
Taking the RATIO x/y of two numbers x and y, also writ-
ten x+y. Here, y is called the Divisor. The symbol "/"
is called a SOLIDUS (or DIAGONAL), and the symbol "-=-"
is called the OBELUS. Division in which the fractional
(remainder) is discarded is called Integer Division,
and is sometimes denoted using a backslash, \.
see also Addition, Divide, Integer Division, Long
Division, Multiplication, Obelus, Odds, Ratio,
Skeleton Division, Solidus, Subtraction, Trial
Division
Division Algebra
A division algebra, also called a Division Ring or Skew
Field, is a Ring in which every NONZERO element has a
multiplicative inverse, but multiplication is not Commu-
tative. Explicitly, a division algebra is a set together
with two Binary Operators 5(+, *) satisfying the fol-
lowing conditions:
480
Division Lemma
Divisor
1. Additive associativity: For all a,b,c 6 S, (a+6) + c =
a+(b + c),
2. Additive commutativity: For all a, b £ 5, a + b =
6 + a,
3. Additive identity: There exists an element 6 5
such that for all a G 5, + a = a + — a,
4. Additive inverse: For every a 6 S there exists a —a E
S such that a + (—a) = (—a) + a = 0,
5. Multiplicative associativity: For all a, 6, c G 5, (a*
6) *c— a* (6* c),
6. Multiplicative identity: There exists an element 1 G
S not equal to such that for all a £ S, 1 * a =
a * 1 = a,
7. Multiplicative inverse: For every a G S not equal to
0, there exists a"" 1 G 5, a* a -1 = a -1 * a = 1,
8. Left and right distributivity: For all a,b,c £ S, a *
(6+c) = (a*b) + (a*c) and (6 + c)*a = (6*a) + (c*a).
Thus a division algebra (S } +, *) is a UNIT RING for
which (S — {0}, *) is a GROUP. A division algebra must
contain at least two elements. A COMMUTATIVE division
algebra is called a FIELD.
In 1878 and 1880, Frobenius and Peirce proved that the
only associative REAL division algebras are real num-
bers, Complex Numbers, and Quaternions. The
Cayley Algebra is the only Nonassociative Di-
vision Algebra. Hurwitz (1898) proved that the
Algebras of Real Numbers, Complex Numbers,
Quaternions, and Cayley Numbers are the only
ones where multiplication by unit "vectors" is distance-
preserving. Adams (1956) proved that n-D vectors form
an Algebra in which division (except by 0) is always
possible only for n = 1, 2, 4, and 8.
see also Cayley Number, Field, Group, Nonassoc-
iative Algebra, Quaternion, Unit Ring
References
Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL:
University of Chicago Press, 1923.
Dixon, G. M. Division Algebras: Octonions, Quaternions,
Complex Numbers and the Algebraic Design of Physics.
Dordrecht, Netherlands: Kluwer, 1994.
Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley,
pp. 326-329, 1975.
Hurwitz, A. "Ueber die Composition der quadratischen For-
men von beliebig vielen Variabeln." Nachr. Gesell. Wiss.
Gottingen, Math.-Phys. Klasse, 309-316, 1898.
Kurosh, A. G. General Algebra. New York: Chelsea, pp. 221-
243, 1963.
Petro, J. "Real Division Algebras of Dimension > 1 contain
C." Amer. Math. Monthly 94, 445-449, 1987.
Division Lemma
When ac is Divisible by a number b that is Relatively
PRIME to a, then c must be DIVISIBLE by b.
Division Ring
see Division Algebra
Divisor
A divisor of a number N is a number d which Divides
N, also called a FACTOR. The total number of divisors
for a given number N can be found as follows. Write a
number in terms of its Prime Factorization
iNr = pi ai p a aa ---p r ° r . (i)
For any divisor d of N, N = dd f where
d = PiSa' a "V r , (2)
SO
d! = Pl ai - $1 p 2 a2 - 62 ..." a — S r
•••Pr
(3)
Now, Si = 0, 1, . . . , ai, so there are ai + 1 possible val-
ues. Similarly, for J n , there are a n + 1 possible values,
so the total number of divisors v(N) of N is given by
i/(JV) = J|(a„ + l).
(4)
The function v(N) is also sometimes denoted d(N) or
ao(N). The product of divisors can be found by writing
the number N in terms of all possible products
d^d'W
N =
(5)
N^ N) = [d w ---d^}[d' w d' M ]
n*n*'=(n*)'.
i=l i~l
and
Y[d = N"W /2 .
The Geometric Mean of divisors is
WiV)/2
(6)
(7)
(IH
[iv ,(n)/2 ]1 /,(N) = ^ (g)
The sum of the divisors can be found as follows. Let
N = ab with a ^ b and (a, b) = 1. For any divisor d
of AT, d — dibi, where a, is a divisor of a and bi is a
divisor of b. The divisors of a are 1, a±, a^, . . . , and a.
The divisors of b are 1, bi, fe 2 , • • • , b. The sums of the
divisors are then
<r(a) = 1 + ai + a 2 + . . . + a (9)
a(b) = l + 6i+6 2 -K-. + 6. (10)
For a given a;,
ai(l + 6i + b 2 + . . . + b) = <n<T(b). (11)
Divisor
Summing over all at,
(1 + a x + a 2 + . . . + a)a(b) = <r(a)<r(b), (12)
so cr(N) = cr(afc) = <r(a)cr(&). Splitting a and 6 into
prime factors,
a(N) = *(pi ai )<j( P 2 a >)---v(p T a "). (13)
For a prime Power pi ai , the divisors are 1, p;, p* 2 , . . . ,
Pz a S so
„( Pi <*t) = 1 + Pi + Pi 2 + . . . +Pi ai = P ^ 1+1 ~ 1 . (14)
Pi — 1
For JV, therefore.
Divisor Function
481
(15)
For the special case of N a Prime, (15) simplifies to
(16)
o 2 — 1
<t(p) = - — t=p + 1 -
2 a+l _ x>
p-1
For AT a Power of two, (15) simplifies to
The Arithmetic Mean is
A(N) =
The Harmonic Mean is
But AT = dd', so I ^j = ^ and
Ed = ^E d ' = ^I> =
o(A0
and we have
1 <t(N) _ A(N)
H(N) N(v) N N
N = A(N)H(N).
(17)
(18)
(19)
(20)
(21)
(22)
Given three INTEGERS chosen at random, the probabil-
ity that no common factor will divide them all is
[<(3)] _1 w 1.202" 1 = 0.832 . . . , (23)
where £(3) is Apery's Constant.
Let f(n) be the number of elements in the greatest sub-
set of [l,n] such that none of its elements are divisible
by two others. For n sufficiently large,
0.6725... < ^^ < 0.673.
(24)
(Le Lionnais 1983, Lebensold 1976/1977).
see also Aliquant Divisor, Aliquot Divisor,
Aliquot Sequence, Dirichlet Divisor Problem,
Divisor Function, e-DivisoR, Exponential Divi-
sor, Greatest Common Divisor, Infinary Divisor,
fc-ARY Divisor, Perfect Number, Proper Divisor,
Unitary Divisor
References
Guy, R. K. "Solutions of d(n) = d(n + 1)." §B18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 73-75, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 43, 1983.
Lebensold, K. "A Divisibility Problem." Studies Appl. Math.
56, 291-294, 1976/1977.
Divisor Function
12
20 40 60 80 100
20 40 60 80 100
1,000,000
800,000
600,000
400,000
200,000
<t,(/i)
20 40 60 80 100
20 40 60 80 100
o~k (n) is defined as the sum of the fcth POWERS of the
DIVISORS of n. The function <To(n) gives the total num-
ber of Divisors of n and is often denoted d(n) y v(n),
r(n), or fi(n) (Hardy and Wright 1979, pp. 354-355).
The first few values of (T (n) are 1, 2, 2, 3, 2, 4, 2, 4, 3,
4, 2, 6, ... (Sloane's A000005). The function (n(n) is
equal to the sum of DIVISORS of n and is often denoted
a(n). The first few values of cr(n) are 1, 3, 4, 7, 6, 12, 8,
15, 13, 18, . . . (Sloane's A000203). The first few values
of cr 2 (n) are 1, 5, 10, 21, 26, 50, 50, 85, 91, 130, ...
(Sloane's A001157). The first few values of tr 3 (n) are 1,
9, 28, 73, 126, 252, 344, 585, 757, 1134, ... (Sloane's
A001158).
The sum of the DIVISORS of n excluding n itself (i.e.,
the Proper Divisors of n) is called the Restricted
Divisor Function and is denoted s(n). The first few
values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, . . . (Sloane's
A001065).
482
Divisor Function
As an illustrative example, consider the number 140,
which has Divisors dk = 1, 2, 4, 5, 7, 10, 14, 20, 28, 35,
70, and 140 (for a total of N = 12 of them). Therefore,
d(140) = N = 12
N
<r(140) = ]T<£ = 336
i
N
o-2(140) = ]P* 2 = 27,300
i
N
a 3 (140) = ]P d; 3 = 3, 164, 112.
The cr(n) function has the series expansion
(i)
(2)
(3)
(4)
cr(n) = g7r n
1 +
(-l) n 2cos(|n7r)
2 2
3 2
+
2cos(|n7r) 2[cos(|n7r) + cos(|n7r)]
■ +
+
42 ■ 52
(Hardy 1959). It also satisfies the INEQUALITY
*W < e 7 + 2(l-y / 2)+7-ln(47r)
nlnlnn ~ vlnnlnlnn
(5)
+0
t ■
\ Vlnn (In Inn)"
(6)
where 7 is the EULER-MASCHERONI CONSTANT (Robin
1984, Erdos 1989).
Let a number n have Prime factorization
r
3=1
then
pj - 1
(7)
o-(«) = li-
es)
j=i
(Berndt 1985). Gronwall's Theorem states that
hm V = e 7 ,
twoo nlnlnn
where 7 is the Euler-Mascheroni Constant.
(9)
0000 100000
Divisor Function
In general,
<Tk(n) = y^v.
d|n
(10)
In 1838, Dirichlet showed that the average number of
Divisors of all numbers from 1 to n is asymptotic to
£7=1*0 (i)
~ In n -f 27 — 1
(11)
(Conway and Guy 1996), as illustrated above, where the
thin solid curve plots the actual values and the thick
dashed curve plots the asymptotic function.
A curious identity derived using MODULAR FORM the-
ory is given by
<r 7 (n) = a 3 {n) + 120^2 (T 3 (k)a 3 (n ~ k).
(12)
fc=i
The asymptotic SUMMATORY FUNCTION of a (n) =
Q(n) is given by
^n(Jfe) = nlnlnn + £ 2 +0(1%), (13)
k=2
where
B 2 = 1 + Yj
ln ( 1 -^ + ^T
1.034653
(14)
(Hardy and Wright 1979, p. 355). This is related to
the Dirichlet Divisor Problem. The Summatory
Functions for a a with a > 1 are
C(a+1) a+1
^— ' a+1
(15)
For a = 1,
^<ri(fc) = ^- n 2 + 0(nlnn).
(16)
The divisor function is Odd Iff n is a Square Num-
ber or twice a Square Number. The divisor function
satisfies the CONGRUENCE
ncr(n) = 2 (mod <t>(n)) ,
(17)
for all Primes and no Composite Numbers with the
exception of 4, 6, and 22 (Subbarao 1974). r(n) is
PRIME whenever cr(n) is (Honsberger 1991). Factoriza-
tions of cr(p a ) for PRIME p are given by Sorli.
see also Dirichlet Divisor Problem, Divisor, Fac-
tor, Greatest Prime Factor, Gronwall's The-
orem, Least Prime Factor, Multiply Perfect
Divisor Theory
Number, Ore's Conjecture, Perfect Number,
r(n), Restricted Divisor Function, Silverman
Constant, Tau Function, Totient Function, To-
tient Valence Function, Twin Peaks
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Divisor Func-
tions." §24.3.3 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, p. 827, 1972.
Berndt, B. C. Ramanujan's Notebooks: Part I. New York:
Springer- Verlag, p. 94, 1985.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 260-261, 1996.
Dickson, L, E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 279-
325, 1952.
Dirichlet, G. L. "Sur l'usage des series infinies dans la theorie
des nombres." J. reine angew. Math. 18, 259-274, 1838.
Erdos, P. "Ramanujan and I." In Proceedings of the Inter-
national Ramanujan Centenary Conference held at Anna
University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New
York: Springer- Verlag, pp. 1-20, 1989.
Guy, R. K. "Solutions of ma(m) = ncr(n)," "Analogs with
d(n), <r k {n), n "Solutions of <r(n) = a(n + 1)," and "Solu-
tions of a{q) + cr{r) = a(q + r)." §B11, B12, B13 and B15
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 67-70, 1994.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 141, 1959.
Hardy, G. H. and Weight, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 354-355, 1979.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 250-251, 1991.
Robin, G. "Grandes valeurs de la fonction somme des di-
viseurs et hypothese de Riemann." J. Math. Pures Appl.
63, 187-213, 1984.
Sloane, N. J. A. Sequences A000005/M0246, A000203/
M2329, A001065/M2226, A001157/M3799, A001158/
M4605 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Sorli, R. "Factorization Tables." http://www.maths.uts.
edu.au/staff/ron/fact/fact.html.
Subbarao, M. V. "On Two Congruences for Primality." Pa-
cific J. Math. 52, 261-268, 1974.
Divisor Theory
A generalization by Kronecker of Rummer's theory of
Prime Ideal factors. A divisor on a full subcategory C
of mod(A) is an additive mapping \ on C witn values
in a Semigroup of Ideals on A.
see also IDEAL, IDEAL NUMBER, PRIME IDEAL, SEMI-
GROUP
References
Edwards, H. M, Divisor Theory. Boston, MA: Birkhauser,
1989.
Vasconcelos, W. V. Divisor Theory in Module Categories.
Amsterdam, Netherlands: North-Holland, pp. 63-64, 1974.
Dixon's Factorization Method
In order to find INTEGERS x and y such that
Dixon's Factorization Method 483
(a modified form of Fermat's Factorization
Method), in which case there is a 50% chance that
GCD(n,x - y) is a Factor of n, choose a Random
Integer n, compute
gin) = ri 2 (mod n) ,
(2)
and try to factor g{ri). If g(ri) is not easily factorable
(up to some small trial divisor d), try another n. In
practice, the trial rs are usually taken to be [v^J + &>
with k = 1, 2, . . . , which allows the QUADRATIC SIEVE
Factorization Method to be used. Continue finding
and factoring g(n)s until N = nd are found, where tt is
the Prime Counting Function. Now for each gin),
write
9(r i )=pu ait P2i aat ...pm a * ri , (3)
and form the EXPONENT VECTOR
v(r<) =
an
Q>2i
CiNi
(4)
Now, if aki are even for any k, then g(ri) is a SQUARE
Number and we have found a solution to (1). If not,
look for a linear combination ^2n CiW ^ ri ) sucn that the
elements are all even, i.e.,
Cl
an
ai2
auv
G21
+ C 2
a22
-r . ■ • + c N
CL2N
_Q>ni _
_CIN2 _
_&nn _
"0"
=
_0_
(mod 2) (5)
an
a i2
diN
" Ci "
"0"
(121
a22
CL2N
c 2
=
(mod 2).
_o,n\
Q>N2
■ • ai
VN _
CN m
_0_
(6)
Since this must be solved only mod 2, the problem can
be simplified by replacing the a,ijS with
for dij even
bij - \l foray odd. { }
Gaussian Elimination can then be used to solve
be = z (8)
for c, where z is a VECTOR equal to (mod 2). Once c
is known, then we have
x = y (mod n)
(i)
JJfl(r fc ) = rjr fc 2 (modn),
(9)
484
Dixon-Ferrar Formula
Dobinski's Formula
where the products are taken over all k for which Ck = 1.
Both sides are Perfect Squares, so we have a 50%
chance that this yields a nontrivial factor of n. If it
does not, then we proceed to a different z and repeat the
procedure. There is no guarantee that this method will
yield a factor, but in practice it produces factors faster
than any method using trial divisors. It is especially
amenable to parallel processing, since each processor can
work on a different value of r.
References
Bressoud, D. M. Factorization and Prime Testing. New
York: Springer- Verlag, pp. 102-104, 1989.
Dixon, J. D. "Asymptotically Fast Factorization of Integers."
Math. Comput 36, 255-260, 1981.
Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Num-
ber Theory." In Handbook of Theoretical Computer Sci-
ence, Volume A: Algorithms and Complexity (Ed. J. van
Leeuwen). New York: Elsevier, pp. 673-715, 1990.
Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.
Dixon-Ferrar Formula
Let J v (z) be a Bessel Function of the First Kind,
Y v (z) a Bessel Function of the Second Kind, and
K v (z) a Modified Bessel Function of the First
Kind. Also let 9ft [z] > and |5R[z]| < 1/2. Then
Jl(z) + Y?{z) =
8cos(i/7r)
Jo
i^2^(2zsinhi) dt.
see also Nicholson's Formula, Watson's Formula
References
Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.518 in Tables of
Integrals, Series, and Products, 5th ed. San Diego, CA:
Academic Press, p. 671, 1979.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1476,
1980.
where 1 + a/2 — b — c has a positive Real Part, d —
a - b + 1, and e = a - c + 1. The identity can also be
written as the beautiful symmetric sum
»-"i°:9 (:::)(:;
{a + b-rc)\
alblcl
(Petkovsek 1996).
see also Dougall-Ramanujan Identity, General-
ized Hypergeometric Function
References
Bailey, W. N. Generalised Hypergeometric Series. Cam-
bridge, England: Cambridge University Press, 1935.
Cartier, P. and Foata, D. Problemes combinatoires de com-
mutation et rearrangements. New York: Springer- Verlag,
1969.
Knuth, D. E. The Art of Computer Programming, Vol. 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973,
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 43, 1996.
Zeilberger, D. and Bressoud, D. "A Proof of Andrew's q-
Dyson Conjecture." Disc. Math. 54, 201-224, 1985.
Dobinski's Formula
Gives the nth Bell Number,
B n
(i)
It can be derived by dividing the formula for a Stirling
Number of the Second Kind by m!, yielding
-{*}(«*-
k
*)!'
(2)
Then
Dixon's Random Squares Factorization
Method
see Dixon's Factorization Method
Dixon's Theorem
F n,-x,-y
a: + n + l,y + n + 1
= T(x + n + l)T{y + n + l)r(f n + l)r(x + y + \ n -f 1)
xT(n + l)r(z + y + n + l)V(x + \n + \)T{y + \n + 1),
where zF 2 {a,b,c\d,e\z) is a Generalized Hypergeo-
metric Function and V(z) is the Gamma Function.
It can be derived from the DOUGALL-RAMANUJAN
Identity. It can be written more symmetrically as
3 F 2 (a,6,c;d,e;l) =
E
and
£
£■
(3)
(4)
a\{\a ~ b)\{\a - c)\{a - b - c)\ '
Now setting A = 1 gives the identity (Dobinski 1877;
Rota 1964; Berge 1971, p. 44; Comtet 1974, p. 211; Ro-
man 1984, p. 66; Lupas 1988; Wilf 1990, p. 106; Chen
and Yeh 1994; Pitman 1997).
References
Berge, C. Principles of Combinatorics. New York: Academic
Press, 1971.
Chen, B. and Yeh, Y.-N. "Some Explanations of Dobinski's
Formula." Studies Appl. Math. 92, 191-199, 1994.
Comtet, L. Advanced Combinatorics. Boston, MA: Reidel,
1974.
Dobinski, G. "Summierung der Reihe ^n m /n! fiir m = 1,
2, 3, 4, 5, " Grunert Archiv (Arch. Math. Phys.) 61,
333-336, 1877.
Dodecadodecahedron
Dodecagram 485
Foata, D. La serie generatrice exponentielle dans les
problemes d 'enumeration. Vol. 54 of Seminaire de
Mathematiques superieures. Montreal, Canada: Presses
de l'Universite de Montreal, 1974.
Lupas, A. "Dobinski-Type Formula for Binomial Polynomi-
als." Stud. Univ. Babes-Bolyai Math. 33, 30-44, 1988.
Pitman, J. "Some Probabilistic Aspects of Set Partitions."
Amer. Math. Monthly 104, 201-209, 1997.
Roman, S. The Umbral Calculus. New York: Academic
Press, 1984.
Rota, G.-C. "The Number of Partitions of a Set." Amer.
Math. Monthly 71, 498-504, 1964.
Wilf, H. Generatingfunctionology, 2nd ed. San Diego, CA:
Academic Press, 1990.
Dodecadodecahedron
The Uniform Polyhedron U Z q whose Dual Poly-
hedron is the Medial Rhombic Triacontahedron.
The solid is also called the Great Dodecadodec-
ahedron, and its Dual Polyhedron is also called
the Small Stellated Triacontahedron. It can be
obtained by TRUNCATING a GREAT DODECAHEDRON
or Faceting a Icosidodecahedron with Pentagons
and covering remaining open spaces with PENTAGRAMS
(Holden 1991, p. 103). A Faceted version is the
GREAT DODECAHEMICOSAHEDRON. The dodecadodec-
ahedron is an Archimedean Solid Stellation. The
dodecadodecahedron has Schlafli Symbol {§,5} and
Wythoff Symbol 2 | § 5. Its faces are 12{f } + 12{5},
and its ClRCUMRADIUS for unit edge length is
iE= 1.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 123, 1989.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 112, 1989.
Dodecagon
The constructive regular 12-sided POLYGON with
Schlafli Symbol {12}. The Inradius r, Circum-
radius i£, and Area A can be computed directly from
the formulas for a general regular POLYGON with side
length s and n = 12 sides,
r=i S cot(^)=!(2 + VS) S (1)
i?=i S cot(^)= i(y/2 + y/6)s (2)
' cot (^)= 3(2 + a/3), 3 . (3)
1 2
4^ '
A Plane Perpendicular to a C 5 axis of a Dodec-
ahedron or ICOSAHEDRON cuts the solid in a regular
Decagonal Cross-Section (Holden 1991, pp. 24-25).
The Greek, Latin, and Maltese Crosses are all ir-
regular dodecagons.
oft
see also DECAGON, DODECAGRAM, DODECAHEDRON,
Greek Cross, Latin Cross, Maltese Cross,
Trigonometry Values — 7r/l2, Undecagon
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Dodecagram
The Star Polygon { x 5 2 }.
see also Star Polygon, Trigonometry Values-
tt/12
486 Dodecahedral Conjecture
Dodecahedron
Dodecahedral Conjecture
In any unit SPHERE PACKING, the volume of any
VORONOI CELL around any sphere is at least as large as
a regular DODECAHEDRON of INRADIUS 1. If true, this
would provide a bound on the densest possible sphere
packing greater than any currently known. It would not,
however, be sufficient to establish the KEPLER CONJEC-
TURE.
Dodecahedral Graph
A Polyhedral Graph.
see also CUBICAL GRAPH, ICOSAHEDRAL GRAPH, OCT-
AHEDRAL Graph, Tetrahedral Graph
Dodecahedral Space
see Poincare Manifold
Dodecahedron
The regular dodecahedron is the PLATONIC SOLID (P 4 )
composed of 20 VERTICES, 30 EDGES, and 12 PENTAG-
ONAL Faces. It is given by the symbol 12{5}, the
SchlafliSymbol {5,3}. It is also Uniform Poly-
hedron £/23 and has WYTHOFF SYMBOL 3 | 2 5. The
dodecahedron has the Icosahedral Group Ih of sym-
metries.
A Plane Perpendicular to a C 3 axis of a dodeca-
hedron cuts the solid in a regular HEXAGONAL CROSS-
Section (Holden 1991, p. 27). A Plane Perpendic-
ular to a C5 axis of a dodecahedron cuts the solid in
a regular DECAGONAL CROSS-SECTION (Holden 1991,
p. 24).
The Dual Polyhedron of the dodecahedron is the
Icosahedron.
When the dodecahedron with edge length y 10 — 2\f%
is oriented with two opposite faces parallel to the xy-
Plane, the vertices of the top and bottom faces lie at
z = ±(0+1) and the other VERTICES lie at z = ±(0-1),
where is the GOLDEN RATIO. The explicit coordinates
are
± (2cos(f7ri),2sin(§7ri),0+ l) (1)
± (20cos(§7ri),20sin(f7n),0- l)
(2)
with i = 0, 1, . . . , 4, where is the GOLDEN RATIO.
Explicitly, these coordinates are
xf = ±(2,0,f(3+v / 5))
(3)
x± = ±(i(>/5 - l),|\/l + 2x/5,f (3 + V5)) (4)
x± = ±(-i(l + v^), f \/l0 - 2^5, |(3 + y/E)) (5)
- ±(-|(l + a/5),-§ViO-2a/5, H 3 + ^5))
x 13
x 14
(6)
=(§(V5-l),-i>/lO + 2V5,i(3+>/5)) (7)
x£ = ±(l + A0,£(V5-l))
x± = ±(1, y/h + 2>/5, \{y/l- 1))
(8)
(9)
x£ = ±(-|(3 + >/5),i\/l0 + 2>/5,i(>/5-l))(10)
x£ = ±(-i(3 + V5),-iVlO + 2>/5,i(V5-l))
l£ = ±(1, -^5 + 2^5, 1(>/5 - 1)),
(11)
(12)
where xj^ are the top vertices, x^ are the vertices above
the mid-plane, x^ are the vertices below the mid-plane,
and x^ are the bottom vertices. The VERTICES of a
dodecahedron can be given in a simple form for a do-
decahedron of side length a = v5 — 1 by (0, ±0 _1 , ±0),
(±<t>, 0, ±<t>- 1 ), (±</r\ ±4>, 0), and (±1, ±1, ±1).
Dodecahedron
Dodecahedron 487
For a dodecahedron of unit edge length a — 1, the ClR-
cumradius R' and Inradius t of a Pentagonal Face
Now,
R' = ±y/bO + 10y/E
^\/25 + 10V5.
(13)
(14)
The SAGITTA a; is then given by
x = R' -r =± \/l25 - 10\/5. (15)
Now consider the following figure.
Using the PYTHAGOREAN THEOREM on the figure then
gives
^i + m = (it + r)
z 2 2 J r{m-xf = 1
(16)
(17)
(^±^) 2 + ^^(^) 2 + (m + /) 2 . (18)
Equation (18) can be written
Zl z 2 +r 2 = {m + rf. (19)
Solving (16), (17), and (19) simultaneously gives
m = r = ± V25 + 10a/5 (20)
Zl = 2r = I a/25 + 10>/5
Z2 = R f = ^V5uTl0V5.
(21)
(22)
The INRADIUS of the dodecahedron is then given by
r= ±(z 1 +z 2 ) J (23)
so
r 2 = \ ^-^50+10^5+1^25 + 10^)
= ^(25 + 11^), (24)
and
y 25 + ^ = ^^250+110^5 = 1.11351....
(25)
R 2 = R' 2 + r 2 = [^(50 + l(h/5) + ^(250+110^5)]
= |(3 + v/5), (26)
and the ClRCUMRADIUS is
iJ = o^/|(3 + V / 5) = |(v / 15 + v / 3) = 1-40125...:
(27)
The INTERRADIUS is given by
p 2 = r ' 2 + r 2 = [^(25 + l(h/5 ) + ^(250 + lloVS)]
= 1(7 + 3^), (28)
so
p= |(3 + V5) = 1.30901.... (29)
The Area of a single Face is the Area of a Pentagon,
A = |V25 + 10Vs . (30)
The Volume of the dodecahedron can be computed by
summing the volume of the 12 constituent PENTAGONAL
Pyramids,
V = 12{\At)
12(|)(|\/25 + W5)(^\/250 + 110^5)
^(75 + 35\/5 ) - |(15 +7^/5 ). (31)
Apollonius showed that the VOLUME V and SURFACE
Area A of the dodecahedron and its Dual the ICOSA-
HEDRON are related by
Vicosahedro
-^Mcosahedron
Vdo decahedron ^dodecahedron
(32)
The HEXAGONAL SCALENOHEDRON is an irregular do-
decahedron.
see also AUGMENTED DODECAHEDRON, AUGMENTED
Truncated Dodecahedron, Dodecagon, Dodeca-
hedron-icosahedron compound, elongated do-
DECAHEDRON, Great Dodecahedron, Great Stel-
lated Dodecahedron, Hyperbolic; Dodecahe-
dron, ICOSAHEDRON, METABIAUGMENTED DODECA-
HEDRON, Metabiaugmented Truncated Dodeca-
hedron, Parabiaugmented Dodecahedron, Para-
biaugmented Truncated Dodecahedron, Pyrito-
hedron, Rhombic Dodecahedron, Small Stel-
lated Dodecahedron, Triaugmented Dodeca-
hedron, Triaugmented Truncated Dodecahe-
dron, Trigonal Dodecahedron, Trigonometry
Values — 7r/5 Truncated Dodecahedron
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.
Davie, T. "The Dodecahedron." http://vvw.dcs.st-and.
ac.uk/-ad/mathrecs/polyhedra/dodecahedron.html.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
488 Dodecahedron-Icosahedron Compound
Domino
Dodecahedron- Icosahedron Compound
A Polyhedron Compound of a Dodecahedron and
Icosahedron which is most easily constructed by
adding 20 triangular Pyramids, constructed as above,
to an Icosahedron. In the compound, the Dodecahe-
dron and ICOSAHEDRON are rotated 7r/5 radians with
respect to each other, and the ratio of the ICOSAHEDRON
to Dodecahedron edges lengths are the Golden Ra-
tio <j>.
References
Bulatov, V.v "270 Stellations of Deformed Dodecahedron."
http:// www . physics . orst . edu/ - bulatov /polyhedra/
dodeca270/.
Dodecahedron 2-Compound
A compound of two dodecahedra with the symmetry
of the Cube arises by combining the two dodecahedra
rotated 90° with respect to each other about a common
C 2 axis (Holden 1991, p. 37).
see also Polyhedron Compound
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Domain
A connected OPEN SET. The term domain is also used
to describe the set of values D for which a Function
is defined. The set of values to which D is sent by the
function (Map) is then called the Range.
see also Map, One-to-One, Onto, Range (Image),
Reinhardt Domain
The above figure shows compounds composed of a DO-
DECAHEDRON of unit edge length and ICOSAHEDRA hav-
ing edge lengths varying from \/5/2 (inscribed in the
dodecahedron) to 2 (circumscribed about the dodecahe-
dron) .
The intersecting edges of the compound form the DIAG-
ONALS of 30 Rhombuses comprising the Triaconta-
hedron, which is the the Dual Polyhedron of the
ICOSIDODECAHEDRON (Ball and Coxeter 1987). The
dodecahedron-icosahedron is the first Stellation of
the ICOSIDODECAHEDRON.
see also DODECAHEDRON, ICOSAHEDRON, ICOSIDODEC-
AHEDRON, Polyhedron Compound
References
Cundy, H. and Rollett, A. Mathematical Models, 2nd ed.
Stradbroke, England: Tarquin Pub., p. 131, 1989.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 76, 1989.
Dodecahedron Stellations
The dodecahedron has three STELLATIONS: the
Great Dodecahedron, Great Stellated Dodec-
ahedron, and Small Stellated Dodecahedron.
The only STELLATIONS of PLATONIC SOLIDS which are
Uniform Polyhedra are these three and one Icosa-
hedron Stellation. Bulatov has produced 270 stel-
lations of a deformed dodecahedron.
see also ICOSAHEDRON STELLATIONS,
Polyhedron, Stellation
Stellated
Domain Invariance Theorem
The Invariance of Domain Theorem is that if / : A — >
W 1, is a ONE-TO-ONE continuous MAP from A, a com-
pact subset of R n , then the interior of A is mapped to
the interior of f(A).
see also Dimension Invariance Theorem
Dome
see Bohemian Dome, Geodesic Dome, Hemisphere,
Spherical Cap, Torispherical Dome, Vault
Dominance
The dominance Relation on a Set of points in EUCLID-
EAN n-space is the INTERSECTION of the n coordinate-
wise orderings. A point p dominates a point q provided
that every coordinate of p is at least as large as the
corresponding coordinate of q.
The dominance orders in M 71 are precisely the POSETS
of Dimension at most n.
see also Partially Ordered Set, Realizer
Domino
The unique 2-POLYOMINO consisting of two equal
squares connected along a complete EDGE.
The Fibonacci Number F n +i gives the number of ways
for 2 x 1 dominoes to cover a 2 x n CHECKERBOARD, as
illustrated in the following diagrams (Dickau).