CRC ConciserEncyclopedia 01
MATHEMA
CRCConciserEncyclopedia
AICS Eric W, Weisstein
0cpC
Boca Raton
CRC Press London New York
Washington,
D.C,
Library
of Congress
CataloginginPublication
Data
Weisstein, Eric W. The CRC concise encyclopedia of mathematics / Eric W. Weisstein. p. cm. Includes bibliographical references and index. ISBN o849396409 (alk. paper) 1. Mathematics Encyclopedias. I. Title. QA5.W45 1998 5 10â€™.3IX21
9822385 CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. Trademark Notice: Prod uct or corporate without intent to infringe.
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0 1999 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number O8493 96409 Library of Congress Card Number 9822385 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acidfree paper
Introduction
The CRC Concise Encyclopedia of ibfuthemutics 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 unders+ood 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 While a picture may be worth a thousand words, explicit science, and how it is actually implemented. examples are worth at least a few hundred! This work attempts to provide enough details to give the reader While absolute rigor may suffer somewhat, I hope a flavor for a subject without getting lost in minutiae. the improvement in usefulness and readability will more than make up for the deficiencies of this approach. The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs from existing dictionaries of mathematics in a number of important ways. First,, the entire text and all the equations and figures are available in searchable electronic form on CDROM. Second, the entries are extensively crosslinked and crossreferenced, not only to related entries but also to many external sites on the Internet,. This makes locating information very convenient. It also provides a highly efficient way to “navigate” from one related concept to another, a feature that is especially powerful in the electronic version. Standard mathematical references, combined with a few popular ones, are also given at the end of most entries to facilitate additional reading and exploration. In the interests of offering abundant examples, this work also contains a large number of explicit formulas and derivations, providing a ready place to locate a particular formula, as well as including the framework for understanding where it comes from. The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g., Borowski and Borwein’s HarperCollins Dictionary of Mathematics and Jeans and Jeans’ Muthematics Dictionary). 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 t o my mind) aspects of topics are discussed in addition to their technical definitions. For example, in my entry for pi (n), the definition in terms of the diameter and circumference of a circle is supplemented by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. These formulas are comprehensible to readers with only minimal mathematical background, and are interesting to both those with and without formal mathematics training. However, they have not previously been collected in a single convenient location. For this reason, I hope that, in addition to serving as a reference source, this work has some of the same flavor and appeal of Martin Gardner’s delightful Scientific American columns. Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections weave their way through the physical sciences. It frequently transpires that some piece of recently acquired knowledge turns out, to be just what I need to solve some apparently unrelated problem. I have therefore developed the habit of picking up and storing away odd bits of information for future use. This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. I have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I already have some familiarity with the subject. I hope this encyclopedia will provide jumpingoff points for people who are interested in the subjects listed here but who, like me, are not necessarily experts. The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and continuing during graduate school. The initial document was written in Microsoj? 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 OLQX program available for the Mac, I began the task of converting all my documents to 7&X, resulting in a vast improvement in readability. While undertaking the Word to T&X conversion, I also began crossreferencing entries, anticipating that eventually I would be able to convert, the entire document
to hypertext. This hope was realized beginning in 1995, when the Internet explosion was ifi full swing and I learned of Nikos Drakes’s excellent 7QX to HTML converter, UTG2HTML. 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 Inj%ity), “Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its “contour defined; it is as limitless as that space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life.” Several of Sylvester’s points apply particularly to this undertaking. As he points out, mathematics itself cannot be confined to the pages of a book. The results of mathematics, however, are shared and passed on primarily through the printed (and now electronic) medium. While there is no danger of mathematical results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical results simply because they are not aware of them. Not only does collecting many results in one place provide a single starting point for mathematical exploration, but it should also lessen the aggravation of encountering explanations for new concepts which themselves use unfamiliar terminology. In this work, the reader is only a crossreference (or a mouse click) away from the necessary background material. As to Sylvester’s second point, the very fact that the quantity of mathematics is so great means that any attempt to catalog it with any degree of completeness is doomed to failure. This certainly does not mean that it’s not worth trying. Strangely, except for relatively small works usually on particular subjects, there do not appear to have been any substantial attempts to collect and display in a place of prominence the treasure trove of mathematical results that have been discovered (invented?) over the years (one notable exception being Sloane and Plouffe’s Encyclopedia of Integer Sequences). This work, the product of the “gazing” of a single astronomer, attempts to fill that omission. Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, neither table of contents nor index are included. In many cases, a particular entry of interest can be located from a crossreference (indicated in SMALL CAPS TYPEFACE in the text) in a related article. In addition, most articles are followed by a “see also” list of related entries for quick navigation. This can be particularly useful if yolv are looking for a specific entry (say, ‘LZeno’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 &no’s name and thus quickly locate the desired entry. The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning with a numeral are ordered by increasing value and appear before the first entry for “A.” In multipleword entries containing a space or dash, the space or dash is treated as a character which precedes “a,” so entries appear in the following order: Yum,” “Sum P.. . ,” “SumP.. . ,” and “Summary.” One exception is that in a series of entries where a trailing “s” appears in some and not others, the trailing %” is ignored in the alphabetization. Therefore, entries involving Euclid would be alphabetized as follows: “Euclid’s Axioms,” “Euclid Number ,” ” Euclidean Algorithm.” Because of the nonstandard nomenclature that ensues from naming mathematical results after their discoverers, an important result, such as the “Pythagorean Theorem” is written variously as “Pythagoras’s Theorem,” the “Pythagoras Theorem,” etc. In this encyclopedia, I have endeavored to use the most, widely accepted form. I have also tried to consistently give entry titles in the singular (e.g., “Knot” instead of “Knots”). In cases where the same word is applied in different contexts, the context is indicated in parentheses or appended to the end. Examples of the first type are “Crossing Number (Graph)” and “Crossing Number (Link).” Examples of the second type are “Convergent Sequence” and “Convergent Series.” In the case of an entry like “Euler Theorem,” which may describe one of three or four different formulas, I have taken the liberty of adding descriptive words (‘4Euler’s Something Theorem”) to all variations, or kept the standard
name for the examples are SeriesPower possessive ‘s is pronounced. not. Finally,
most commonly used variant and added descriptive words for the others. In cases where specific derived from a general concept, em dashes () are used (for example, “Fourier Series,” “Fourier Series,” “Fourier SeriesSquare Wave,” “ Fourier SeriesTriangle”). The decision to put a at the end of a name or to use a lone trailing apostrophe is based on whether the final “s” ‘LGauss’s Theorem” is therefore written out, whereas “Archimedes’ Recurrence Formula” is 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. lies with me alone. Although the current length makes extensive additions in a printed I plan to continue updating, correcting, and improving the work,
Eric
Weisstein
Charlottesville, August 8, 1998
Virginia
The blame for these version problematic,
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 TEX, 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@, perhaps his shoe box of notes would not have had to await the labors of Erdelyi, Magnus, and Oberhettinger to become a threevolume work on mathematical functions. Andrew TTevorrow’s shareware implementation of QX for the Macintosh, OQjX (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 uTEX2HTML program (wwwdsed.llnl.gov/files/programs/unix/latex2htm~/m~ual/m~ual .html),whichhasallowedmeto easily maintain and update an online version of the encyclopedia long before it existed in book form. I would like to thank Steven Finch of MathSoft, Inc., for his interesting online essays about mathematical constants (www.mathsoft com/asolve/constant/constant .html), and also for his kind permission to reproduce excerpts from some of these essays. I hope that Steven will someday publish his detailed essays in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed and online (www research. att . corn/njas/sequences/) versions of the Encyclopedia of Integer Sequences, an immensely valuable compilation of useful information which represents a truly mindboggling investment of labor. Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript and providing a number of helpful suggestions and additions. Thanks also to algebraic topologist Ryan Budney 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 permission 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 A&uthematica @, but also for permission to include figures from the A&zthematica@ book and MuthSource 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 ccow1” minimal surface, Penrose tiles, polyhedron, and Scherk’s minimal surfaces entries. Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading the entire document for syntax. Thanks also to Bob Stern, my executive editor from CRC Press, for his encouragement, and to Mimi Williams of CRC Press for her careful reading of the manuscript for typographical and formatting errors. As this encyclopedia’s entry on PROOFREADING MISTAKES shows, the number of mistakes that are expected to remain after three independent proofreadings is much lower than the original number, but unfortunately still nonzero. Many thanks to the library staff at the University of Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I would like to thank the hundreds of people who took the time to email me comments and suggestions while this work was in its formative stages. Your continued comments and feedback are very welcome. l
l
0
10
Numerals
3
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 (l! + 2! = 3). It is also the first ODD PRIME. A quantity taken to the POWER 3 is said tobe CUBED. see also 1, 2, 3~ + 1 MAPPING, CUBED, PERIOD THREE THEOREM, SUPER~ NUMBER, TERNARY, THREECOLORABLEJERO 3 is the only
0 see ZERo 1 The number one (1) is the first POSITIVE INTEGER. It is an ODD NUMBER. Although the number I used to be considered a PRIME NUMBER, it requires special treatment 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 RENTS OF UNITY. FRACTIONS having 1 as a NuMERATOR are called UNIT FRACTIONS. Ifonly 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 
lx
1
1
+
x
+
x2
+
x3
+
x4
+
.
.
l
3x + 1 Mapping see COLLATZ PROBLEM 10 The number 10 (ten) is the basis for the DECIMAL system of notation. In this system, each “decimal place” consists of a DIGIT O9 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° = 1s place. For example, the number 1234.56 specifies
l

see also 2, 3, EXACTLY ONE, ROOT OF UNITY, UNIQUE, UNIT FRACTION, ZERO 2
The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 10000, 100000, 10000000, (Sloane’s AO11557), called one, ten, HUNDRED, THOUSAND, ten thousand, hundred thousand, MILLION, 10 million, 100 million, and so on. The names of subsequent decimal places for LARGE NUMBERS differ depending on country. 100000000,
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 FACTORIAL since 2! = 2. A quantity taken to the POWER 2 is said to be SQUARED. The number of times k a given BINARY number b, . . b2 b& is divisible by 2 is given by the position of the first bk = 1, counting from the right, For example, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is divisible by 2 0 times. l
see also 1, BINARY, 3, SQUARED, 2~~0
Xn+l
s 2x,
Then
the number of periodic PRIME) is given by
(mod
1).
ORBITS of period
(1) p (for p
.
l
Any POWER of 10 which can be written as the PRODUCT of two numbers not containing OS must be of the form 2”*5” I, 10n for n an INTEGER such that neither 2” nor 5n contains any ZEROS. The largest known such number
1033 = 233
2x mod 1 Map Let x0 be a REAL NUMBER in the CLOSED INTERVAL [0, 11, and generate a SEQUENCE using the MAP
*
l
533
= 8,589,934,592
A complete
m116,415,321,826,934,814,453,125.
list of known
such numbers
lo1 = 2l
l
is
5l
lo2 = 22  52 103 = 23 53 l
lo4 = 24 ’ 54 lo5 = 25 ’ 55 lo6 = 26  56
Since a typical ORBIT visits each point with equal probability, the NATURAL INVARIANT is given by
lo7 = 27  57
p(x) = 1.
log = 2g 5g 1018 = 21s . 518 l
(3)
see also TENT MAP
1033 = 233 .533
References Ott, E. bridge
Chaos in University
Dynamical
Press,
pp.
Systems. Cambridge: 2631, 1993.
Cam
(Madachy 1979). S ince all POWERS of 2 with exponents n 5 4.6 x lo7 contain at least one ZERO (M. Cook), no
2
1 BPoint Problem
12
other POWER of ten less than 46 million
can be written’ not cant aining OS.
as the PRODU CT of two numbers
also BILLION, DECIMAL, HUNDRED, LARGE NUMBER, MILLIARD, MILLION, THOUSAND, TRILLION, ZERO
see
Madachy, J. S. Mudachy’s Mathematical Recreations. New York: Dover, pp. 127128, 1979. Pickover, C. A, Keys to Infinity. New York: W. H. Freeman, p* 135, 1995. Sloane, N. J. A. Sequence A011557 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
.
ni = 0, so N = 1
References RecmBall, W. W. R. and Coxeter, H. S. M. Mathematical ations and Essays, 13th ed. New York: Dover, pp. 312316, 1987. Bogomolny, A. “Sam Loyd’s Fifteen.” http: //www. cutthe
knot.com/pythagoras/fiftean.html. Bogomolny, A. “Sam Loyd’s Fifteen
12 One DOZEN, or a twelfth
see also
= 1 (2 precedes 1) and all other and the puzzle cannot be solved.
n2
References
of a GROSS.
DOZEN, GROSS
[History].” cuttheknot.com/pythagoras/historyl5.html. Johnson, W. W. “Notes on the ‘15 Puzzle. Math.
2, 397399,
http://www. I.“’
Amer.
J.
1879.
Kasner,
E. and Newman, J. R. Mathematics and the ImagiRedmond, WA: Tempus Books, pp. 177180,1989. Kraitchik, M. “The 15 Puzzle.” 512.2.1 in MathematicaZ Recreations. New York: W. We Norton, pp* 302308, 1942. Story, W. E. “Notes on the ‘15 Puzzle. II.“’ Amer. J. Math. 2, 399404, 1879. nation.
13 A NUMBER traditionally associated with bad luck. A socalled BAKER'S DOZEN is equal to 13. Fear of the number 13 is called TRISKAIDEKAPHOBIA.
see UZSO BAKER'S DOZEN, FRIDAY THE THIRTEENTH, TRISKAIDEKAPHOBIA 15
see 15 PUZZLE, FIFTEEN THEOREM
16Cell A finiteregular4D POLYTOPE with SCHL~FLI SYMBOL (3, 3, 4) and VERTICES which are the PERMUTATIONS of (fl, 0, 0, 0). see also 24CELL, TOPE
120CELL,
600CELL,
CELL, POLY
15 Puzzle
l? 17 is a FERMAT PRIME which means that the 17sided REGULAR POLYGON (the HEPTADECAGON) is CONSTRUCTIBLE using COMPASS and STRAIGHTEDGE (as A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a 4 x 4 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 arrangements, 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(reading the squares in the box ber i appears “before” from left to right and top to bottom) 12 numbers which are less than i, then call it an inversion of order 72, and denote it ~2i. Then define 15
ni =
N$
i=l
lx Iz
ni, 2
where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so n1 must equal 0). If Iv is EVEN, the position is possible, otherwise it is not. This can be formally proved using ALTERNATING GROUPS. For example, in the following arrangement
proved
by Gauss).
CONSTRUCTIBLE POLYGON , FERMAT PRIME, HEPTADECAGON see
aho
References Carr, M. “Snow White and the Seven(teen) Dwarfs.” http:// www + math . harvard . edu / w hmb/ issueZ.l/ SEVENTEEN/seventeen.html. Fischer, R. “Facts About the Number 17.” http: //tsmpo.
harvard.
edu/  rfischer/hcssim/l7facts
/kelly/
kelly. html. Lefevre, V. “Properties of 17.” http : //www . enslyon. f r/ vlef evre/dlXeng . html. Shell Centre for Mathematical Education. “Number 17.” http://acorn.educ.nottingham.ac.uk/ShellCent/ Number/Numl7.html.
18Point
Problem
Place a point somewhere on a LINE SEGMENT. Now place a second point and number it 2 so that each of the points is in a different half of the LINE SEGMENT. Continue, placing every Nth point so that all N points are on different (l/N)th of the LINE SEGMENT. Formally, for a given N, does there exist a sequence of real numbers xl, x2, . . . , ZN such that for every n E {l, . . . , IV} and every k E (1,. . . , n), the inequality kl <Xi<n

k n
24 Cell
196Algorithm it is only (Berlekamp
72 Rule
Steinhaus (1979) gives a 14point solution (0.06, 0.55, 0.77, 0.39, 0.96, 0.28, 0.64, 0.13, 0.88, 0.48, 0.19, 0.71,
120~Cell
holds for some i E { 1, . . . , n}? Surprisingly, possible to place 17 points in this manner and Graham 1970, Warmus 1976).
0.35, 0.82),
and Warmus
(1976)
gives the 17point
see RULE
A finite {5,3,3}
solu
tion
OF 72
regular4D (Coxeter
see also 16 CELL, TOPE FLeferences Coxeter, H. S. M. York:
3
Wiley,
P~LYTOPE
with
SCHL~~FLI
SYMBOL
1969).
24CELL,
Introduction
POLY
CELL,
600~CELL,
to Geometry,
2nd
ed.
New
p. 404, 1969.
144 Warmus (1976) states that there are 768 patterns point solutions (counting reversals as equivalent)
see also
of 17
see
l
1979. Warmus, M. “A Supplementary Note on the Irregularities Distributions.” J. Number Th. 8, 260263, 1976.
A finite regular 4D POLYT~PE with SCHL~FLI SYMBOL {3,4,3}. Coxeter (1969) gives a list of the VERTEX positions. The EVEN coefficients of the Lid lattice are 1, 24, 24, 96, . . . (Sloane’s AOU4011), and the 24 shortest vectors in this lattice form the 24cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995). 16CELL,
120CELL, 600CELL,
CELL, POLY
TOPE
References
INTEGER of two DIGITS or more,reverse 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 number 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. l
.
The first few numbers not known to produce PALINDROMES 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 required to produce a PALINDROMIC NUMBER from n (i.e., = 1 for a PALINDROMIC NUMBER, a(n) = 2 if a 44 PALINDROMIC NUMBER is produced after a single iteration of the 196algorithm, etc.) for n = 1, 2, . are .
Lattices J. H, and Sloane, N. J. A. SpherePackings, and Groups, 2nd ed. New York: SpringerVerlag, 1993. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, Il. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Sloane, N. J. A. Sequences A004011/M5140 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Sloane, N. J. A. and Plouffe, S. Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press,
Conway,
l
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 l
which
require
n = 0, 1, 2, . . . iterations
to reach
drome are 0, 10, 19, 59, 69, 166, 79, 188, A023109).
l
a palin
. . (Sloane’s
see also ADDITIVE PERSISTENCE, DIGITADTTION, MULTIPLICATIVE PERSISTENCE, PALINDROMIC NUMBER, PALINDROMIC NUMBER CONJECTURE, RATS SEQUENCE, RECURRING DIGITAL INVARIANT References
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. Reterences Adams, D. The Hitchhiker’s Ballantine Books, 1997.
196.Algorithm Take any POSITIVE
.
of
24Cell
also
DOZEN
also
DISCREPANCY THEOREM, POINT PICKING
References Berlekamp, E. R. and Graham, R. L. “Irregularities in the Distributions of Finite Sequences.” J. Number Th. 2, 152161, 1970. Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: SpringerVerlag, pp* 3436, 1997. Steinhaus, H. “Distribution on Numbers” and “Generalization.” Problems 6 and 7 in One Hundred Problems in Elementary Mathematics. New York: Dover, pp. 1213,
see
A DOZEN DOZEN, also called a GROSS. 144 is a SQUARE NUMBER and a SUMPRODUCT NUMBER.
Guide
to the Galaxy.
New York:
Gardner, Paradoxes Scientific
M. Mathematical
Circus: More Puzzles, Games, and Other Mathematical Entertainments from American. New York: Knopf, pp. 242245,1979.
Eruenberger, F. “How to Handle Numbers with Thousands of Digits, and Why One Might Want to.” Sci. Amer. 250, 1926, Apr. 1984. Sloane, N. J* A. Sequences A023109, A030547, A033865, and A006960/M5410 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
4
65537gon
239
239
600Cell
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
A finite {3,3,5}.
FORMULA $7r = 4tan(i)
regular 4D POLYTOPE with For VERTICES, see Coxeter
see also TOPE
16CELL,
fteierences Coxeter, H.
tanl(&),
York:
which
is related
S.
Wiley,
M.
24CELL,
Introduction
SCHL;~FLI (1969).
120CELL,
SYMBOL
POLY
CELL,
to Geometry,
2nd
ed.
New
p. 404, 1969.
to the fact that
666
2 ’ 134  1 = 23g2,
A number known as the BEAST NUMBER the Bible and ascribed various numerological
which is why 239/169 is the 7th CONVERGENT of a. Another pair of INVERSE TANGENT FORMULAS involving 239 is
see ah
APOCALYPTIC NUMBER,
VIATHAN
BEAST
appearing in properties. NUMBER,
LE
NUMBER
References tanl(&)
= tan‘($)
Hardy,
tanl(&)
G. H. A Mathematician’s
foreword
= tanl(&)
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).
Cambridge, MA: MIT Artificial Memo AIM239, Feb. 1972.
Intelligence
New
Apology,
reprinted
York: Cambridge
with
a
University
Press, p. 96, 1993.
+ tanl(&).
References Beeler, M.; Gosper, R. W.; and Schroeppel,
by C. P. Snow.
R.
HAKMEM.
Laboratory,
257~gon 257 is a FERMAT PRIME, and the 257gon is therefore a CONSTRUCTIBLE POLYGON using, COMPASS and STRAIGHTEDGE, as proved by Gauss. An illustration of the 257gon is not included here, since its 257 segRichelot and ments so closely resemble a CIRCLE. Schwendenwein found constructions for the 257gon in 1832 (Coxeter 1969). De Temple (1991) gives a construction using 150 CIRCLES (24 of which are CARLYLE CIRCLES) which has GEOMETROGRAPHY symbol 94S1 + 47& + 275C1 + OC2 + 150C3 and SIMPLICITY 566. see dso 65537CON, CONSTRUCTIBLE POLYGON, MAT PRIME, HEPTADECAGON, PENTAGON
FER
References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D, W. “Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions.” Amer. Math. MO mnthly 98, 97108, 1991. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Theory. Rademacher, H. Lectures on Elementary Number New York: Blaisdell, 1964.
2187 The digits NUMBERS:
in the number 2187 form the two VAMPIRE 21 x 87 = 1827 and 2187 = 27 x 81.
References Gardner, 2629,
M. “Lucky Numbers Spring 1997.
and 2187.”
Math.
Intell.
19,
65537gon 65537 is the largest known FERMAT PRIME, and the 65537gonistherefore a CONSTRUCTIBLE POLYGON using COMPASS and STRAIGHTEDGE, as proved by Gauss. The 65537gon has so many sides that it is, for all intents and purposes, indistinguishable from a CIRCLE using any reasonable printing or display methods. Hermes spent 10 years on the construction of the 65537gon at Giittingen around 1900 (Coxeter 1969). De Temple (1991)notesthata GEOMETRIC CONSTRUCTION canbe done using 1332 or fewer CARLYLE CIRCLES. see U~SO 257GON, CONSTRUCTIBLE TADECAGON,~ENTAGON
POLYGON, HEP
Keterences to Geometry, 2nd ed. New H. S. M. Introduction Wiley, 1969. De Temple, D. W. “Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions.” Amer. Math. Monthly 98, 97108, 1991. Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
Coxeter, York:
AAS Theorem
AIntegrable
A AIntegrable A generalization of the LEBESGUE SURABLEFUNCTION f( z ) is called CLOSED INTERVAL [a$] if m{z
: If(z)1
INTEGRAL. Aintegrable
A MEAover the
> n} = O(nl),
where m is the LEBESGUE
(1)
5
Erdiis, P. “Remarks on Number Theory III. Some Problems in Additive Number Theory.” 1Mut. Lupok 13, 2838, 1962. Finch, S. “Favorite Mathematical Constants.” http : //www. mathsoft.com/asolve/constant/erdos/erdos,html* §E28 in Unsolved Problems Guy, R. K. “&Sequences.” irt Number Theory, 2nd ed. New York: SpringerVerlag, pp. 228229, 1994. Levine, E. and O’Sullivan, J. “An Upper Estimate for the Reciprocal Sum of a SumFree Sequence.” Acta Arith. 34, 924, 1977. Zhang, 2. X. “A S urnFree Sequence with Larger Reciprocal Sum.” Unpublished manuscript, 1992.
and
MEASURE,
AAA
Theorem
(2) exists,
whel
fb)ln = References
1
1 . G. “On Conjugate
Titmarsch, Math.
f(z) if If(z)I L n 0 if if(s)1 > 73.
Sot.
29,
4980,
Functions.”
Proc.
SEQUENCE 1 5
London
1928.
ASequence N.B. A detailed online essay by S. Finch ing point for this entry. An INFINITE isfying
(3)
a1
of POSITIVE <
a2
<
u3
<
was the start
INTEGERS .
a; sat
.
=
SUP
all
see also AAS THEOREM, ASA THEOREM, ASS THEOREM, SAS THEOREM, SSS THEOREM, TRIANGLE AAS
/
Specifying determines
(2)
A sequences
Any Asequence satisfies the CHI INEQUALITY (Levine and O’Sullivan 1977)) which gives S(A) < 3.9998. Abbott (1987) and Zhang (1992) have given a bound from below, so the best result to date is < 3.9998.
(3)
Levine and O’Sullivan (1977) conjectured that of RECIPROCALS of an Asequence satisfies
\
two angles A and B and a side a uniquely a TRIANGLE with AREA u2 sin 13 sin@  A  B) a2 sin B sin C 2 sin A 2sinA =
K=
k=l
2.0649 < S(A)
Theorem
(1)
is an Asequence if no ak is the SUM of two or more distinct earlier terms (Guy 1994). Erdk (1962) proved
S(A)
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” (r RADIANS), i.e., C=nAB.
The third
angle
(1)
is given by C=nAB,
(2)
since the sum of angles of a TRIANGLE DIANS). Solving the LAW OF SINES
the sum
l
U
b
sin A
sin B
=
is 180’
(K RA
(3)
for b gives S(A)
OQ 1 2 x = 3.01
,
(4)
where xi are given by the LEVINEO’SULLIVAN ALGORITHM.
GREEDY
k=l
l
l
l
b=Um*
sin B (4)
”
Finally,
see
dso
&SEQUENCE,
MIANCHOWLA
c=bcosA+ucosB=u(sinBcotA+cosB) =usinB(cotA+cotB).
SEQUENCE
References II. L. “On SumFree 9396, 1987.
Abbott,
Sequences.”
Acta
Arith.
48,
see
also
AAA THEOREM, THEOREM,SSS
REM, SAS
(5) (6)
ASA THEOREM, ASS THEOTHEOREM,TRIANGLE
6
Abel’s
Abacus
Abacus A mechanical counting device consisting of a frame holding a series of parallel rods on each of which beads are strung. Each bead represents a counting unit, and each purpose of the abacus rod a place value. The primary 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
Functional
Equation
Abelian
see ABELIAN CATEGORY, ABELIAN DIFFERENTIAL, ABELIAN FUNCTION, ABELIAN GROUP, ABELIAN INTEGRAL, ABELIAN VARIETY, COMMUTATIVE Abelian Category An Abelian category is an abstract mathematical CATEGORY which displays some of the characteristic properties of the CATEGORY of all ABELIAN GROUPS.
see also ABELIAN GROUP, CATEGORY
References Bayer, C. B. and Merzbach, U. C. “The Abacus and Decimal Fractions.” A History of Mathematics, 2nd ed. New York: Wiley, pp. 199201, 1991. Fernandes, L. “The Abacus: The Art of Calculating with Beads ,” http://www.ee.ryerson.ca:8080/elf/abacus. Ch. 18 in Mathematical Circus: Gardner, M. “The Abacus.” More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 232241, 1979. The Joy of Mathematics. San Pappas, T. “The Abacus.” Carlos, CA: Wide World Publ./Tetra, p. 209, 1989. Smith, D. E. “Mechanical Aids to Calculation: The Abacus.” Ch. 3 31 in History of Mathematics, Vol. 2. New York: Dover, pp+ 156196, 1958.
abc Conjecture A CONJECTURE due to J. Oesterlk and D. W. Masser. It states that, for any INFINITESIMAL E > 0, there exists a CONSTANT C, such that for any three RELATIVELY PRIME INTEGERS a,b, c satisfyi %
Abel’s Curve Theorem The sum of the values of an INTEGRAL of the “first” “second” sort x1 3Yl
XNTYN
...+
Pdx+ s
x0 *YU
P(Xl,Yl) Q(xl,yl)
Q
da
J
+
dz
**
xN,yN)
Q(xm
F( z >
Q
x0 1YO
+ p( l
Pdx =
dxN =YN)
or
dF
dz ’
dz
with a from a FIXED POINT to the points of intersection curve depending rationally upon any number of parameters is a RATIONAL FUNCTION of those parameters. References on Algebraic Coolidge, J. L. A Treatise York: Dover, p. 277, 1959.
Plane
Curves.
New
U+b=C,
Abelian Differential An Abelian differential
is an ANALYTIC or MEROMORRIEMANN PHIL DIFFERENTIAL on a COMPACT or SURFACE.
the INEQUALITY max{lal,
1% ICI} 5 CE lI
Pl+”
pbbc
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 z WIEFERICH PRIMES < z for some constant C (Silverman 1988, Vardi 1991).
see UZSO FERMAT'S LAST THEOREM, MASON'S THEOREM,~IEFERICH PRIME Heferences Cox, D. A. “Introduction
to Fermat’s Last Theorem.” Amer. 1994. Goldfeld, De “Beyond the Last Theorem.” The Sciences, 3440, March/April 1996. Math.
Guy,
Monthly
R. K.
101,
Unsolved
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 ~2s~ ABELIAN INTEGRAL, ELLIPTIC FUNCTION References Baker, lied
H. F. Abelian Functions: Abel’s Theorem and the AlTheory, Including the Theory of the Theta Functions.
New York: Cambridge University Press, 1995. Baker, I% F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge University Press, 1907.
314,
Problems
in Number
Theory,
2nd
ed.
New York: SpringerVerlag, pp. 7576, 1994. Silverman, J. “Wieferich’s Criterion and the abc Conjecture.” J. Number Th. 30, 226237, 1988. Vardi, I. Computational Recreations in Mathematics. Reading, MA: AddisonWesley, p+ 66, 1991.
Abel’s Functional Let Liz(x) denote
Equation the DILOGARITHM,
Liz(x)
= 2 n=f
5,
defined
by
A belian Group
Li&c)
+ Lip(y)
Abelian
+ Li2(xy)
+
Li2
$Liz
1  xy
see also DILOGARITHM, ZETA FUNCTION
= 3Li$).
e
( POLYLOGARITHM,
7
a prime cubed p3, then there are three Abelian groups (denoted Zp @ Zp @ Zp, Zp @ Z*Z, and Zp3 ), 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 Zp @ Zp).
x(1Y> ( > p
Group
Another number
>
interesting result is that if a(n) denotes the of nonisomorphic Abelian groups of ORDER r~,
RIEMANN n=l
Abelian Group N.B. A detailed online essay by S. Finch ing point for this entry.
was the start
(i.e., A13 = A GROUP for which the elements COMMUTE 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 SUBGROUPSof 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 of nonisomorphic FINITE GROUPS of a given However, the number of nonisomorphic Abelian GROUPS a(n) of any given ORDER n is given by n as n = where the p; are distinct
PRIME
a(n) = p(w), i
FACTORS,
l
l
l
i:
an0
Srinivasan
= A~lV+A2N1/2+A~N1’3+O[z105/407(ln
x)~],
n=l
(4) where
number
ORDER. FINITE writing
and (is again the RIEMANN ZETA FUNCTION. [Richert (1952) incorrectly gave A3 = 114.1 DeKoninck and Ivic (1980) showed that
(1)
N x
then
1
= BN
+ 0[fi(lnN)1/2],
(6)
44
n=f.
(2)
where P is the PARTITION FUNCTION. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, (Sloane’s AOOOSSS) . 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 0 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). .
where c(s) is the RIEMANN ZETA FUNCTION. (1973) has also shown that
1 F@zj
l
l
The KRONECKER DECOMPOSITION THEOREM states that every FINITE Abelian group can be written as a DIRECT PRODUCT of CYCLIC GROUPS of PRIME POWER ORDERS. Ifthe ORDERS ofa FINITE GROUP isa PRIME p, then there exists a single Abelian group of order p (denoted Zp) and no nonAbelian groups. If the ORDERS is a prime squared p2, then there are two Abelian groups (denoted Zp2 and Zp @IZp. If the ORDERS is
(7) is a product nonisomorphic mann (1969)
over PRIMES. Bounds nonAbelian groups and Pyber (1993).
for the number of are given by Neu
see UZSO FINITE GROUP, GROUP THEORY, KRONECKER DECOMPOSITION
THEOREM,
PARTITION
FUNCTION
P,
RING
References DeKoninck,
J.M.
and Ivic, A. Topics
tions: Asymptotic Formulae Arithmetical Functions and
in Arithmetical Funcfor Sums of Reciprocals of Related Fields. Amsterdam,
Netherlands: NorthHollaGd, 1980. Erdiis, P. and Szekeres, G. “Uber die Anzahl abelscher Gruppen gegebener Ordnung und fiber ein verwandtes zahlentheoretisches Problem.” Acta Sci. Math. (Szeged) 7, 95
102,1935. Finch, S. "Favorite Mathematical Constants.” http: //www. mathsof t . com/asolve/constant/abel/abel . html,. Kendall, D. G. and Rankin, R. A. “On the Number of Abelian Groups of a Given Order.” Quart. J. Oxford 18, 197208,
1947. Kolesnik, Order.”
G. “On the Number J. Reine
Angew.
of Abelian Math.
329,
Groups 164175,
of a Given 1981.
Abel’s Irreducibility
A be1k Identity
8
Neumann, P. M. “An Enumeration Theorem for Finite Groups.” Quart. J. Math. Ser. 2 20, 395401, 1969. Pyber, L. “Enumerating Finite Groups of Given Order.” Ann. Math. 137,,*203220, 1993. Richert, IXE. “Uber die Anzahl abelscher Gruppen gegebener Ordnung I,” Math. Zeitschr. 56, 2132, 1952. Sloane, N. J. A. Sequence AOOO688/M0064 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Srinivasan, B. R. “On the Number of Abelian Groups of a Given Order.” Acta A&h. 23, 195205, 1973.
Abel’s
Given a homogeneous linear SECONDORDER NARY DIFFERENTIAL EQUATION, y" + P(x)y' call the two linearly y&c) Then
+ Q(x)y
independent
ORDI
= 0,
(1)
solutions
y1 (z) and
Y:(X)
+P(x>~:(rc)
+ Q(X)YI
= 0
(2)
Y:(X)
+ P(x>Y;(x>
+ Q(X)YZ = 0.
(3)
Now, take yl x (3)  y2 x (2),
(Y 1; Y


= 0
YIYZ) y5y2)
=
of the WRONSKIAN
y1y;
5
+ Q(X)Yl]
Y;Y~)+Q(YIY~
yzy:')+P(Yly;
Now, use the definition DERIVATIVE,
=
(6)
and take its
y1y;

y;Iy2*
can be rearranged
(8)
(6) gives
Inequality
Let {fn} and {a,} be SEQUENCES with for n = 1, 2, . . . , then
fn 2 fn+l
> 0
> 4. They
are
m Yd n=l
Gafn
< Ah
(9)
P(x)
which can then be directly 1nW
dx
integrated
= Cl
where lna: is the NATURAL tegration then yields Abel’s
p(x)
dx,
LOGARITHM. identity
of integration
see ~1~0 ORDINARY DIFFERENTIAL
and
Boundary
Abel’s
FUNCTION, ELLIPTIC INTEGRAL
Irreducibility
Theorem
F(x) = f(x)Fl(x>,
in
where FI(x)
is also a POLYNOMIAL over K.
see ah ABEL'S LEMMA, KRONECKER'S POLYNOMIAL THEOREM,SCHOENEMANN'S THEOREM (12)
and C2 = ccl.
EQUATIONSEC
R. C. EZementary Value Problems,
ABELIAN
(11)
ONDORDER References Boyce, W. E. and DiPrima,
UZSO
is a POLYNOMIAL of degree HYPERELLIPTIC INTEGRALS.
(10)
A second
P( 5) da: = Cze s 1
where Cl is a constant
where R(t) also called
If one ROOT of the equation f(x) = 0, which is irreducible over a FIELD K, is also a ROOT of the equation F(x) = 0 in K, then all the ROOTS of the irreducible equation f(x) = 0 are ROOTS of F(x) = 0. Equivalently, F(x) can be divided by f(x) without a REMAINDER,
to
s
W(x)
Abelian Integral An INTEGRAL of the form
see
to yield
dW =W
Wiley,
Abel%
(7)
W’+PW=O.
York:
Abel, N. H, “DBmonstration de I’impossibilitG de la &solution alghbraique des kquations g&&ales qui dhpassent le quatrikme degr&” Crelle ‘s J. 1, 1826.
= 0 (5)
0.
y:y2
W and W’ into
Equations
EQUATION, GALOIS'S THEOREM,POLYQUADRATIC EQUATION, QWARTIC EQUATION, QUINTIC EQUATION CUBIC
NOMIAL,
(4)
w' = (YiYh +YlYY)  (YiYb +Y:Y2)
This
see also
+ Q(4~21
(Y~Y; Y~Y:~)+P(Y~Y;
Plugging
Theorem
where
Yz[YY + +>y:
w
Impossibility
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.
References
Identity
YI[Y~’ + P(x>Y;
Abel’s
Theorem
DQferentiul 4th ed. New
pp+ 118, 262, 277, and 355, 1986.
References Abel, N. H. “Mbmoir sur une classe particulihre d’hquations r&solubles alghbraiquement.” Crelle ‘s J. 4, 1829. Dgrrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965.
A be1‘s Lemma Abel’s
A bhyankar ‘s Conjecture
Lemma
The Abel transform is used in calculating the radial mass distribution of galaxies and inverting planetary radio occultation data to obtain atmospheric information.
The pure equation xp = 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.
References A&en, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 875876, 1985. Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987. Bracewell, R. The Fourier Transform and Its Applications. New York: McGrawHill, pp. 262266, 1965.
see also ABEL'S IRREDUCIBILITY THEOREM, GAUSS'S THEOREM, KRONECKER'S POLYNOMIAL THEOREM,SCHOENEMANN'S THEOREM
POLYNOMIAL References Diirrie,
H. 100
Their
Great and
History
of
Problems Solutions
.
Elementary New York:
Mathe
Abel’s Uniform Convergence Test Let {'all} be a SEQUENCE of functions.
,matics:
Dover,
p. 118,
1965.
1. tin(x)
can be written
3.
(i.e., &+1(x)
Abel’s
Theorem a TAYLOR SERIES
4.
00
C,rn
sin(&).
(3)
states
Abelian
that,
if u&8)
and v&0)
Stated
in words,
+ iv(l,
are
Abel’s
0) = lim f(rP),
(4)
Tb1
theorem
guarantees
that,
if a
REAL POWER SERIES CONVERGES for some POSITIVE value of the argument, the DOMAIN of UNIFORM CONVERGENCE extends at least up to and including this point. Furthermore, the continuity extends at least up to and including
of the sum function this point.
References A&en, G. Mathematical lando, FL: Academic
Abel
Methods
Press,
3rd ed. Or
for Physicists,
p. 773, 1985.
The following INTEGRAL TRANSFORM relationship, known as the Abel transform, exists between two functions f(z) and g(;t) for 0 < Q < 1,
f(x)=Jxg@$ g(t)
= 
of Infinite
Series,
T. M. A n Introduc3rd ed. New York:
Variety
see also ALBANESE
VARIETY
References Murty, V. K. Introduction to Abelian R1: Amer. Math. Sot., 1993.
Varieties.
Providence,
Abhyankar’s Conjecture For a FINITE GROUP G, let p(G) be the SUBGROUP generated by all the SYLOW PSUBGROUPS of G. If X is a projective curve in characteristic p > 0, and if 20, . . . , xt are points of X (for t > 0), then a NECESSARY and SUFFICIENT condition that G occur as the GALOIS GROUP of a finite
Transform
0
Theory
An Abelian variety is an algebraic GROUP which is a complete ALGEBRAIC VARIETY. An Abelian variety of DIMENSION 1 is an ELLIPTE CURVE.
CONVERGENT, then U(1,0)
T. J. I'a and MacRobert,
to the
p. 59, 1991. E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cam bridge University Press, p. 17, 1990.
n=O
theorem
CONVERGES
Chelsea, Whittaker,
M
v(r, 0) = x
E [u,b], the SERIES &Jx)
<
CONVERGENCE TESTS
Bromwich, tion
(2)
x
(i.e., 0 < fn(x)
References
PARTS
Abel’s
for all
see also
wherethe COMPLEX NUMBER z has been w .ritten in the polar form z = TeiB, examine the REAL and IMAG INARY cos(n0)
DECREASING SEQUENCE
for all 72, and
UNIFORMLY.
(1)
C/
< fn(x))
If
= a&(x),
fn(x) is BOUNDED in some region M for all II: E [a, b])
then,
00
F(x) = c Cn;sn = x C,rneinO, n=O n=O
u(r, 0) = 2
Us
2. CUE is CONVERGENT, fn(x) is a MONOTONIC
Abel’s Test see ABEL'S UNIFORM CONVERGENCE TEST
Given
9
(1)
covering Y of X, branched only at the points is that the QUOTIENT GROUP G/p(G) has 2g + t generators.
x0,
d
di
sin(rar) 7T
o
(x 
t df
(2)
t)la
dx
o dx (t  x)I~
f(O)
+
pa
l
xt,
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.
dx 7lJt f(x) [J 1(3)
sin(m)
l **)
see also FINITE GROUP, GALOXS GROUP, QUOTIENT GROUP,SYLOW~SUBGROUP
10
AblowitzRamaniSegur
Conjecture
Absolute
References Abhyankar, Math.
S. “Coverings
of Algebraic
Curves,”
Amer.
J.
79, 825856,1957.
American Mathematical Society. “Notices of the AMS, April 1995, 1995 Frank Nelson Cole Prize in Algebra.” http: //
Absolute Error The DIFFERENCE between the measured or inferred value of a quantity ~0 and its actual value x, given by Ax=
WWW.~S.Org/nOtiC8S/~995~~/priZ8CO~8.htm~.
Harbater, D “Abhyankar’s Conjecture on Galois Over Curves.” Invent. Math. 117, l25, 1994. de la droite affine Raynaud, M. “Revetements actkristique p > 0 et conjecture d’Abhyankar.” l
Math.
en carInvent.
116, 425462,1994.
References An Introduction.
Abscissa The CC(horizontal)
Integrability
New York:
in
Wiley,
IVonlinear
Dynamics:
p. 351, 1989.
References Abramowitz,
M.
of Mathematical Mathematical
see also CONDITIONAL
CONVERGENCE, CONVERGENT SERIES, RIEMANN SERIES THEOREM
References
T. J. I’a and MacRobert,
T. M. “Absolute ConCh. 4 in An Introduction to the Theory of InSeries, 3rd ed. New York: Chelsea, pp. 6977, 1991.
Absolute Deviation Let G denote the MEAN of a SET of quantities the absolute deviation is defined by
and
Stegun,
Functions Tables, 9th
C. A.
(Eds.).
with Formulas,. printing. New
Handbook Graphs, and
York:
Dover,
Geometry
GEOMETRY which depends only on the first four of EUCLID'S POSTULATES and notonthe PARALLEL POSTULATE. 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.
axis of a GRAPH.
Absolute Convergence A SERIES c, un is said to CONVERGE absolutely if the SERIES c, /Us/ CONVERGES, where 1~~1 denotes the ABSOLUTE VALUE. Ifa SERIES~~ 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.
finite
see also ERROR PROPAGATION, PERCEN TAGE ERROR, RELATIVE ERRO R
Absolute and
see also AXIS, ORDINATE, REAL LINE, XAXIS, ~AXIS, ZAXIS
Bromwich, vergence.”
(sometimes with the ABSOLUTE VALUE taken) is called The absolute error of the SUM or the absolute error. DIFFERENCE of a number of quantities is less than or equal to the SUM of their absolute errors.
p* 14, 1972.
see also INVERSE SCATTERING METHOD M. Chaos
X0  x
Groups
AblowitzRamaniSegur Conjecture The AblowitzRamaniSegur conjecture states that a nonlinear PARTIAL DIFFERENTIAL EQUATION is solvable by the INVERSE SCATTERING METHOD onlyifevery nonlinear ORDINARY DIFFERENTIAL EQUATION obtained by exact reduction has the PAINLEVI? PROPERTY.
Tabor,
Square
see also AFFINE GEOMETRY, Cements, EUCLID'S PosT~LATES, GEOMETRY, ORDERED GEOMETRY, PARALLEL POSTULATE References Hofstadter, Braid.
D. R. Giidel, Escher, Bach: An Eternal Golden York: Vintage Books, pp. 9091, 1989.
New
Absolute
Pseu
doprime
see CARMICHAEL NUMBER Absolute Square Also known as the squared NORM. The absolute square of a COMPLEX NUMBER x is written ]tl2 and is defined as 2 Ix I = zx*, (1) where z* denotes the COMPLEX CONJUGATE ofx. a REAL NUMBER,(~) simplifies to
ui, then
2
1x I
For
= z2.
(2)
If the COMPLEX NUMBER is written the absolute square can be written
X=
x + iy,
then
12+ iy12 = x2 + y2.
see ah DEVIATION, MEAN DEVIATION, SIGNED DEVIATION, STANDARD DEVIATION An important given by la&be
identity
“I2
involving
the absolute
= (a 41 beib)(a
(3) square
is
zk beis)
= a2 + b2 & ab(ei6 + eBi6) = a2 + b2 k 2abcos6.
(4
Absolute If a = 1, then [1*
Absolutely Continuous Let p be a POSITIVE MEASURE on a SIGMA ALGEBRA M and let X be an arbitrary (real or complex) MEASURE on M. Then X is absolutely continuous with respect to ~1, written X << ~1, if X(E) = 0 for every E E i'W for which p(E) = 0.
(4) becomes
Wi8/2
= 1+b2f2bcosd
= 1 + b2 & 2b[l  2sin2($)] = 1f2b+b2F4bsin2(+6) = (1*
b)2 â€œf: 4bsin2(3).
(5)
11  eib(2 = (1  1)" + 4. 1sin2($)
= 4sin2@).
Analysis.
New
Absorption Law The law appearing in the definition GEBRA which states
+
p212
=
(,ih
+
=
2 +
,i(#2h)
=
2 +
2442
= 4cOS2(&
p2)p'
SINGULAR York:
McGrawHill,
(6)
Finally, Wl
see also CONCENTRATED,MUTUALLY References Rudin, W. Functional pp. 121125, 1991.
If a = 1, and b = 1, then
Ie
11
Abundance
Value
of a BOOLEAN AL
+ei#2) + 
aA(aVb)=aV(aAb)=a
ei(#2411 41)
=
2[1+

cos(q52
 41).
41)]
(7)
for binary operators V and A (which logical OR and logical AND).
most commonly
are
see also BOOLEAN ALGEBRA, LATTICE Absolute
References Birkhoff, G. and Mac Lane, S. 3rd ed. New York: Macmillian,
Value
Abstraction see
Re[Abs
I
1
1
of a number
"o I21
The absolute value of a REAL NUMBER x is denoted and given by X
x sgn(x)
=
1x X
n is the quantity
2
44
1 I=
Algebra,
Operator
21
121
of Modern
p. 317, 1965.
CALCULUS
LAMBDA
Abundance The abundance 2
A Survey
[Z
Ia:
The following table lists special classifications a number n based on the value of A(n).
A(n) < 0 1 0 1 > 0
where SGN is the sign function.
Other NOTATIONS similar to the absolute value are the FLOOR FUNCTION 1x1, NINT function [x], and CEILING FUNCTION [zl* see &SO ABSOLUTE SQUARE, CEILING FUNCTION, FLOOR FUNCTION, MODULUS (COMPLEX NUMBER), NINT, SGN, TRIANGLE FUNCTION, VALUATION
 2n,
where o(n) is the DIVISOR FUNCTION. Kravitz has conjectured that no numbers exist whose abundance is an ODD SQUARE (Guy 1994).
for 2 < 0 for x 5 0,
The same notation is used to denote the MODULUS of a COMPLEX NUMBER x = x+iy, 1~1 E dx2+y2, a pADIC absolute value, or a general VALUATION. The NORM of a VECTOR x is also denoted 1x1, although f 1x1I is more commonly used.
s a(n)
given to
Number deficient number almost perfect number perfect number quasiperfect number abundant number
see also DEFICIENCY References Guy,
Et. K.
New York:
Unsolved
Problems
SpringerVerlag,
in Number
Theory,
pp. 4546, 1994.
2nd
ed.
Abundant
12
Acceleration
Number
Number
Abundant
An abundant
Singh,
is an INTEGER 12 which is not a
number
PERFECT NUMBER and for which
(1)
s(n) E u(n)  n > 72,
o(n) is the DIVISOR FUNCTION. The quantity called the ABUNDANCE. The +4  2n is sometimes first few abundant numbers are 12, 18, 20, 24, 30, 36, . . . (Sloane’s A005 101) Abundant numbers are sometimes called EXCESSIVE NUMBERS.
where
l
There are only 21 abundant numbers less than 100, and they are all EVEN. The first ODD abundant number is
the
S. Fermat’s World’s
Greatest
The Enigma: Mathematical
Epic Quest to Problem. New
Solve
York:
Walker, pp. 11 and 13, 1997. Sloane, N. J. A. Sequence A005101/M4825 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Wall, C. R. “Density Bounds for the Sum of Divisors F’unction.” In The Theory of Arithmetic Functions (Ed. A. A. Gioia and D. L. Goldsmith). New York: SpringerVerlag, pp. 283287, 1971. Wall, C. R.; Crews, P. L.; and Johnson, D. B. “Density Bounds for the Sum of Divisors Function.” Math. Comput. 26, 773777, 1972. Wall, C. R.; Crews, P. L.; and Johnson, D. B. “Density Bounds for the Sum of Divisors Function.” Math. Comput. 31, 616, 1977.
Acceleration That
945 is abundant
(2)
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 VECTORVELOCITY) isthengiven
(3)
bY
can be seen by computing
s(945)
= 975 > 945.
Any multiple of a PERFECT NUMBER or an abundant number is also abundant Every number greater than num20161 can be expressed as a sum of two abundant bers.
&//.
(1)
l
Define
the density
The acceleration the VELOCITY,
SO
function a=
for a POSITIVE REAL NUMBER x, then Davenport (1933) proved that A(x) exists and is continuous for all x, and ErdCs (1934) gave a simplified proof (Finch). Wall (1971) and Wall et al. (1977) showed that
is defined as the time DERIVATIVE of the SCALAR acceleration is given by
dv 2T  d2 S dt2
(2) (3)

(4)
dz dx d2x dY d2Y  d2z ds dt2 + ds dt2 + ds dt2 dr d2r ds’dtz’ 
0.2441
and DelGglise
<
A(2) < 0.2909,
(5)
showed that
0.2474 < A(2) < 0.2480.
(6)
see also ALEQUOT SEQUENCE, DEFICIENT NUMBER, HIGHLY ABUNDANT NUMBER, MULTIAMICABLE NUMBERS,PERFECT NUMBER,~RACTICAL NUMBER, PRIMITIVE ABUNDANT NUMBER,~EIRD NUMBER References Delkglise, dams.” Dickson,
M. “Encadrement de la densitk des nombres abonSubmitted. L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 333, 1952. Erdiis, P. “On the Density of the Abundant Numbers.” J. London
Math.
Sot.
9, 278282,
1934.
http: //uww ,
Finch, S. “Favorite Mathematical Constants.” mathsoft.com/asolve/constant/abund/abund,html. Guy,
R. K.
New York:
Unsolved
Problems
SpringerVerlag,
in
Number
pp. 4546,
Theory,
1994.
2nd
ed,
The VECTOR acceleration
is given
(5) (6)
by
where ?k is the UNIT TANGENT VECTOR, K the CURVATURE, s the ARC LENGTH, and 3 the UNIT NORMAL
VECTOR. Let a particle move along a straight LINE so that the positions at times tl, t2, and t3 are ~1, ~2, and ~3, respectively. Then the particle is uniformly accelerated with acceleration a IFF (s2
a2 [ is a constant

S3)h
(t 1 (Klamkin
+ 
(s3 t2)(t2

s1)tz 
t3)(t3
1995, 1996).
+(a

t1)
s2)t3
1 (8)
Accidental Consider reference
Cancellation
Ackermann
the measurement of acceleration in a rotating frame. Apply the ROTATION OP ERATOR
(9)
40X body
twice to the RADIUS VECTOR r and suppress notation,
the body
see
CHAOS,
UZSO
RIOD

d2r d &2 +z(wXr)+wx
 d2r ~+wx~+rX
dr gwx(wxr) dw dt+WXdt
dr
dr
+wx(wxr).
(10)
and using the definitions of the VELOCand ANGULAR VELOCITY a E dwldt give the expression Grouping
terms
ITY v E dr/dt
and
the
as consisting
the
=
=
aCori0Iis
z
a “body” acceleration, Coriolis acceleration. gives abody
+
1
2w x v,
centrifugal Using these
acoriolis
+
acentrifugal
+
r
X
a,
A(2, y) = 2y + 3
(13)
= 2y+3  3
.2 A(4, y) = G3.
and finally
a uniformly roThe centrifugal b4Jrounds, and for the motions large trajectory missiles.
see also
ARC
ACCELEREION,
= Y +2
A(3,y)
(15)
(2) (3) (4) (5)
Y) = Y + 1
(12)
where the fourth term will vanish in tating frame of reference (Le., .a = 0). acceleration is familiar to riders of merrythe Coriolis acceleration is responsible of hurricanes on Earth and ilecessitates corrections for intercontin~~** KJ Mlistic ANGULAR
(1)
values for INTEGER x include
(14) acceleration, definitions
 1))
ifx=O if y =o otherwise
of A(l,Y)
=wx(wxr),
hzntrifugal
=
&2
PE
Paradox
A(z  1,l) A(z  l,A(x,y
d2r
&pace
Tortoise
4%
abody
LOCKING,
PARADOXES
(11)
expression
MODE
Y+l
Special Now, we can identify three terms
MAP,
Ackermann Function The Ackermann function is the simplest example of a welldefined TOTAL FUNCTION which is COMPUTABLE but not PRIMITIVE RECURSIVE, providing a counterexample to the belief in the early 1900s that every COMPUTABLE FUNCTION was also PRIMITIVE RECURSIVE (D&e1 1991). It grows faster than an exponential function, or even a multiple exponential function. The Ackermann function A(z,y) is defined by
A(X,Y)
d2 &pace = &IZWXv+wx(wxr)+rxa.
LOGISTIC
DOUBLING
see 2~~0%
=($+wx) ($+wxr)
13
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.
Achilles
=R2r= ($+wx)2r &pace
Function
(6)
Y+3 Expressions of the latter form are sometimes called POWER TOWERS. A(O,y) follows trivially from the definition. A(1, y) can be derived as follows,
A(17 Y>
= A(O,A(l,y
 1)) = A&y
= A(O,A(l,Y
 2)) + 1 = A&Y
= .*. = A(1, 0) + y = $0,
 1) + 1  2, + 2
1) + Y = Y + 2. (7)
A( 2, Y) has a similar
derivation,
LENGTH,
JERK,~ELOCITY
42, y) = A(1,42,
y  1)) = 42,
y  1) + 2
References
= A(1, A(2, y  2)) + 2 = A(2, y  2) + 4 = . . .
Klamkin, M. S. "Problem 1481." 1Math. Msg. 68, 307, 1995. Klamkin, M. S. “A Characteristic of Constant Acceleration.” Solution to Problem 1481. IMath. Msg. 69, 308, 1996.
= A(2,O)
Accidental
Cancellation
see ANOMALOUS CANCELLAT
+ 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
 1, F(x,
Y  l)),
(9)
14
Ackermann
but with
the slightly
Acute lliangle
Number different
boundary
values
see
~SO
POWER
Buck’s
ARROW
= Y+ 1
References
F(l,O)
= 2
F(2,0)
= 0
F(x,O)
= 1
Ackermann, W, ‘Zum hilbertschen Zahlen.” Math. Ann. 99, 118133, Conway, J. H. and Guy, R. K. The York: SpringerVerlag, pp. 6061, Crandall, R. E. “The Challenge of Amer. 276, 7479, Feb. 1997. Vardi, I. Computational Recreations wood City, CA: AddisonWesley, 1991.
recurrence
for x=3,4,....
gives
q&Y)
= 2Y
F(3,y)
= 2’
(4 (15) (16) .2
F(4,y)
2 65536
FUNCTION,
F(O, Y)
F(L Y>=2ty
Taking
ACKERMANN TOWER
(17)
=Ga
F(4,n) gives the sequence 1, 2, 4, 16, 65536, ) ..*. Defining e(x) = F(x, x) for IL: = 0, 1, . . .
.2
then gives 1, 3, 4, 8, 65536, 2” v’
(Sloane’s
’’’
A001695),
: a truly
where m. = 22 v’
huge number!
Acnode Another
name
see
CRUNODE,
also
for an ISOLATED SPINODE,
NOTATION,
Aufbau der reellen 1928. Book of Numbers. New 1996. Large Numbers.” Sci. in Muthemutica. Redpp, 11, 227, and 232,
POINT. TACNODE
Acoptic Polyhedron A term invented by I3. Griinbaum in an attempt to promote 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, makingthem ZMANIFOLDS. see
also
HONEYCOMB,
NOLID,
POLYHEDRON,
SPONGE
65536 see
also
TION,
ACKERM GOODSTEIN
TIVE RECURSIVE FUNCTION
ANN NUMBER, SEQUENCE,
FUNCTION,
FUNCP RIMITOTAL
COMPUTABLE POWER TOWER,
TAK
FUNCTION,
Action Let M(X) denote the GROUP of all invertible MAPS X +X and let G be any GROUP. A HOMOMORPHISM B : G + M(X) is called an action of G on X. Therefore, 0 satisfies
References Buck, R. C. “Mathematical Induction and Recursive Definitions.” Amer. Muth, Monthly 70, 128135, 1963+ Dijtael, G. LLA Function to End All Functions.” Algorithm: Recreational Programming 2.4, 1617, 1991. Kleene, S. C. Introduction to Metamathematics. New York: Elsevier, 1971. Peter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 1951. Reingold, E. H. and Shen, X. “More Nearly Optimal Algorithms for Unbounded Searching, Part I: The Finite Case.” SIAM J. Cumput. 20, 156183, 1991, Rose, H, E. Subrecursion, FzLnctions, and Hierarchies. New York: Clarendon Press, 1988. Sloane, N. J. A. Sequence A001695/M2352 in “An OnLine
Version of the Encyclopedia Smith,
H. J. “Ackermann’s
of Integer Function.”
1. For each g E G, 0(g) is a MAP 2. O(gh)x
X + X : x t+ $(g)x,
= wP(h)4~
3. B(e)x = x, where e is the group 4. 8(gl)x
in G,
= qg>‘x.
see also CASCADE,
Flow,
Acute Angle An ANGLE of less than acute angle. also ANGLE, STRAXGH T ANGLE see
Sequences.” http: //www.netcom.
identity
SEMXFLOW
r/2
RADIANS
OBTUSE
ANGLE,
(90”)
is called
RIGHT
an
ANGLE,
corn/hjsmith/Ackerman.html. Spencer, Amer.
3. “Large Math.
Numbers
and Unprovable 1983.
MonthEy 90, 669675, Data Structures and
Tarjan, R. E. Philadelphia PA: SIAM, 1983. Vardi, I. Computational Recreations wood City, CA: AddisonWesley, 1991.
Ackermann A number
Number of the form n
Network
Theorems.”
Triangle
Algorithms.
Red
in Muthemuticu.
pp. 11, 227, and 232,
where ARROW
TION has been used. The fi?st few Ackermann are1~1=1,2~~2=4,and3~~~3=
Acute
A TRIANGLE in which allthree ANGLES are ACUTE ANGLES. A TRIANGLE which is neither acute nor a RIGHT TRIANGLE (i.e., it has an OBTUSE ANGLE) is called an OBTUSE TRIANGLE. A SQUARE canbe dissectedinto as few as 8 acute triangles.
NOTAnymbers
& 7,625,597,484,987
see .
also
OBTUSE
TRIANGLE,
RIGHT
TRIANGLE
AdamsBashforthMoulton AdamsBashforthMoulton see ADAMS’
Addition
Method Method
GILL’S see also TORCORRECTOR
METHOD
Chain
METHOD, MILNE’S METHOD, METHODS, RUNGEKUTTA
I5 PREDICMETHOD
References
Adams’ Adams’ linear
Method method
Abramowita,
is a numerical
FIRSTORDER
METHOD for DIFFERENTIAL
ORDINARY
M.
of
Mathematical Mathematical
solving EQUA
Handbook and Stegun, C. A. (Eds.). Functions with Fo~&las,~ Graphs, and Tables, 9th printing. New York: Dover,
p. 896, 1972. W. H. CRC Standard Mathematical Tables, 28th Boca Raton, FL: CRC Press, p. 455, 1987. K&r&n, T. von and Bid, M. A. Mathematical Methods
TIONS of the form
Beyer,
2
= f(X,Y>.
(1)
h= be the step interval, RIES of y about xn,
X:n+lXn
pp. 1420,194O. Press, W. H.; Flannery, ling, W. T. Numerical
(2)
and consider
the MACLA~RIN
in to the Mathematical TreatNew York: McGrawHill,
Engineering: An Introduction ment of Engineering Problems.
Let
Scientific
SE
B. P.; Teukolsky,
University
S. A.; and Vetter
Recipes in FORTRAN: The Art of 2nd ed. Cambridge, England: Cam
Computing,
bridge
ed.
Press,
p. 741, 1992.
Addend n
(X
Xrb)
(x  xrJ2 + “0
(3)
A quantity to be ADDED to another, also called a SUMMAND. For example, in the expression a+b+c, a, b, 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,
Here, the DERIVATIVES DIFFERENCES
AUGEND,
RADICAND
PLUS,
Addition
of y are given by the BACKWARD
1 1 15
+carries 84addend
1
+ 2 4 geaddend2
4u7+sum
=
etc. Note
that
by (l),
qn

Qn1
qn. is just
For firstorder interpolation, iterating the expression Yn+l
The combining of two or more quantities operator. The individual numbers being called ADDENDS, and the total is called first of several ADDENDS, or “the one to ers are added,” is sometimes called the opposite of addition is SUBTRACTION.
the value of f(xn,yn).
the method
=
$/n
+
proceeds
by
(8)
qnh
where qn G f(xn,yn)* can then be exTh e method tended to arbitrarv order using the finite difference integration formula kom Beyer fi987) 1
While the usual form of adding two ndigit INTEGERS (which consists of summing over the columns right to left and “CARRYING” a 1 to the next column if the sum exceeds 9) requires n. operations (plus carries), two ndigit INTEGERS can be added in about 2 lgn steps by n processors using carrylookahead addition (McGeoch 1993). Here, lg x is the LG function, the LOGARITHM to the base 2. see
also
= (I+
$V + &,V2 + iv3 v4 + g&v5
+
ggy
+
l
l
n)fp (9)
to obtain
SUBTRACTION,
McGeoch, C. C. “Parallel 100, 867871, 1993.
Addition
 Yn
=h(qn
+
$Vqn1
+
&V2qn2
+
gQ3qn3
+
NUMBER,
AUGEND,
MULTIPLICATION,
SUM
gv5qn5
t'..).
(10)
Note that von K&m&n and Biot (1940) confusingly use the symbol normally used for FORWARD DIFFERENCES A to denote BACKWARD DIFFERENCES V.
Addition.”
Amer.
Math,
Monthly
for a number n is a SEQUENCE 1 = = n, such that each member after a0 is the SUM of the two earlier (not necessarily distinct) ones. The number T is called the length of the addition chain. For example, a0
+zv4q,4
DIVISION,
Chain
An addition
Yn+l
AMENABLE
DIFFERENCE,
References
s0
+E
ADDEND,
CARRY, PLUS,
fPdp
using the PLUS combined are the SUM. The which the othAUGEND. The
<
1,1+1
a1
<
chain
l
l
l
<
a,
=2,2+2=4,4+2=6,6+2=8,8+6=14
AdditionMultiplication
16
is an addition
chain
for 14 of length
CHAIN,
see also BRAUER JECTURE
HANSEN
T =
CHAIN,
5 (Guy SCHOLZ
References Guy, R. K. “Addition
Chains. Brauer Chains. Chains.” SC6 in Unsolved Problems in Number 2nd ed. New York: SpringerVerlag, pp. 111113,
AdditionMultiplication
Ad&
Magic Square
Magic
1994).
CON
Hansen Theory,
1994.
Group
(Sloane’s A031286). The smallest numbers of additive persistence n for n = 0, 1, .** are 0, 10, 19, 199,19999999999999999999999,. . . (Sloane’s AOOSOSO). There is no number < 105’ with additive persistence greater than 11. It is conjectured that the maximum 1 with persistence 11 is
number
lacking
the
DIGIT
Square
77777733332222222222222222222 There is a stronger conjecture that there is a maximum number lacking the DIGIT 1 for each persistence 2 2. The maximum additive persistence in base 2 is 1. It is conjectured that all powers of 2 > 215 contain a 0 in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy, 1994). see also DIGITADITION, DIGITAL TIVE PERSISTENCE, NARCISSISTIC RING DIGITAL INVARIANT References Guy, R. K. “The Persistence Problems
A square which is simultaneously
a MAGIC SQUARE and The three squares shown above (the top square has order eight and the bottom two have order nine) have addition MAGIC CONSTANTS (840, 848, 1200) and multiplicative magic constants (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). MULTIPLICATION
see also MAGIC
MAGIC
SQUARE.
SQUARE
References Hunter, J. A. H. and Madachy, in Mathematical Diversions. 1975.
Madachy, York:
J. S. Mudachy’s
Dover,
pp. 8991,
J. S. “Mystic Arrays.” Ch. 3 York: Dover, pp. 3031,
New
Mathematical
Recreations.
New
1979.
Additive Persistence Consider the process of taking a number, adding its DIGITS, 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 persistence of n, and the DIGIT obtained is called the DIGITAL ROOT of 7~. 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 ROUT of 3. The additive 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, 17 2, 1,
in Number
ROOT, MULTIPLICANUMBER, RECUR
of a Number.”
Theory,
SF25 in Unsolved Springer
2nd ed. New York:
Verlag, pp. 262263, 1994. Hinden, H. J. “The Additive Persistence of a Number.” J. Recr. Math. 7, 134135, 1974. Sloane, N. 5. A. Sequences A031286 and A006050/M4683 in “An OnLine Version of the Encyclopedia of Integer Sequences +” Sloane, N. J. A. “The Persistence of a Number.” J. Recr. Math. 6, 9798, 1973.
Addle An element
of an ADELE GROUP, sometimes called a in older literature. Addles arise in both NUMBER FIELDS and FUNCTION FIELDS. The addles of a NUMBER FIELD are the additive SUBGROUPS of all elements in fl k,, where v is the PLACE, whose ABSOLUTE VALUE is < 1 at all but finitely many VS. REPARTITION
Let F be a FUNCTION FIELD of algebraic functions of one variable. Then a MAP T 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 U&T(P)) < 0. see also IDELE
References Chevalley,
C. C. Introduction
of
to
the
Theory
Math.
sot., p* 25, 1951. Knapp, A. W. “Group Representations and Harmonic ysis, Part II.” Not. Amer. Math. Sot. 43, 537549,
Anal1996.
One
Variable.
Providence,
of Algebraic
RI: Amer.
Functions
Addle Group The restricted topological DIRECT PRODUCT of the GROUP Gk, with distinct invariant open subgroups Go,. References Weil, A. Ade’les and Princeton University
Algebraic
Press,
Groups.
1961.
Princeton,
NJ:
Adem
Relations
Adem
Relations
Adjoint Adjacent
Relations in the definition of a STEENR~D ALGEBRA which state that, for i < Zj,
Value
The value nearest
to but still
inside
J. W. Explanatory Data AddisonWesley, p. 667, 1977.
k=O
COMPOSITION and LiJ is
where f o g denotes function the FLOOR FUNCTION.
see also
Reading,
Analysis.
Adjacent Vertices In a GRAPH G, two VERTICES are adjacent joined
MA:
if they are
by an EDGE.
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#&
which
@I#&) This
FENCE.
an inner
References
Tukey,
sqi0Q(x) = p, (j ;_k2;l)sqi+‘’ 0Q”(x),
17
Operator
relationship
KNOTS. see also
is an adequate
= c(K) + c(&).
is postulated
ALTERNATING
knot,
to hold
KNOT,
true
for
all
Adjoint
Curve
A curve which point where a lar points and adjoint to the order 72  3, it References Coolidge, York:
J. L. A Treatise
Adjoint The
Invariant
Matrix
The adjacency matrix of a simple GRAPH is a MATRIX with rows and columns labelled by VERTICES, with a 1 or 0 in position (vi+)) according to whether zli and w~j are ADJACENT or not.
matrix,
sometimes by
also called
New
the ADJU
A+ E (AT)*,
(1)
where the ADJOINT OPERATOR is denoted t and T denotes the TRANSPOSE. If a MATRIX is SELFADJOINT, it is said to be HERMITIAN. The adjoint matrix of a MATRIX product is given by ( a b) +;j G Using
the property
[(OLb>'],'j
m
of transpose
products
(2)
that
INCIDENCE MATRIX [(ab)T]Tj
References Chartrand, G. Introductory Dover, p. 218, 1985.
Adjacency
Gmph
Theory.
New
York:
it follows
Relation
EQUATION
Adjacent
Adjoint
F’raction
see also FAREY SEQUENCE, FORD CIRCLE, FRACTION, NUMERATOR C. A. Keys to Infinity.
p. 119, 1995.
New York:
1
(3)
(4
Operator a SECONDORDER
C,(x) where pi E pi(x) is defined by
pop
ORDINARY
du2
DIFFERENTIAL
du +pl&
and u E U(X),
(1)
+p2%
the adjoint
operator
Lt
L+ uz d2u = PO@ + (@of PI)&
References
= (b’)Ik(aT>;j
(AB)+ = B+A+.
see also
Two FRACTIONS are said to be adjacent if their difference has a unit NUMERATOR. For example, I/3 and 1/4 are adjacent since l/3  l/4 = l/12, but l/2 and l/5 fractions can are not since l/2  l/5 = 3/10. Adjacent be adjacent in a FAREY SEQUENCE.
= (bzaEj)*
that
Given
IRREFLEXIVE, RELATION, SYMMETRIC
= (bTaT)rj
= btik atkj = (btat)ij
The SET E of EDGES of a GRAPH (V, E), being a set of unordered pairs of elements of V, constitutes a RELATION on V. Formally, San adjacency relation is any RELATION which is IRREFLEXIVE and SYMMETRIC.
Pickover,
Curves.
GATE MATRIX, is defined
see also ALGEBRAIC INVARIANT, LYAPUNOV CHARACTERISTIC NUMBER
see also
Plane
Matrix
adjoint
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.
Adjacency
012 Algebraic
Dover, p. 30, 1959.
CROSSING NUMBER
(LINK)
Adiabatic
has at least multiplicity pi  1 at each given curve (having only ordinary singucusps) has a multiplicity Q is called the given curve. When the adjoint curve is of is called a special adjoint curve.
du
+ (PO!!  P1’ fP217.4.
W. H. F’reeman, (2)
18
Adjugate
Affine Hull
Matrix
Write
Affine
y&)
the two LINEARLY INDEPENDENT solutions as operator can also and YZ(X). Then the adjoint be written
The set A2 of all ordered
2+u zs(Y&I y&)dx= ;(YltY2  YlY2’ 1 l
(3)
see also SELFADJOINT THEORY
Adjugate see
ADJOINT
OPERATOR,
STURMLIOUVI
pairs of COMPLEX NUMBERS.
see also AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE GROUP, AFFINE HULL, AFFINE PLANE, AFFINE SPACE, AFFINE TRANSFORMATION, AFFINITY, COMPLEX PLANE,COMPLEXPROJECTIVE PLANE Affine Connection see CONNECTION COEFFICIENT
Matrix Affine
MATRIX
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 adjunction of a.
AdlemanPomeranceRumely
Primality
Equation
A nonhomogeneous nonhomogeneous affine.
LINEAR EQUATION or system of LINEAR
EQUATIONS
is said
to
be
see UZSO AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE GEOMETRY, AFFINE GROUP, AFFINE HULL,AFFINE PLANE,AFFINE SPACE,AFFINE TRANSFORMATION, AFFINITY
Test
A modified MILLER’S PRIMALITY TEST which gives a guarantee of PRIMALITY or COMPOSITENESS. The ALGORITHM'S running time for a number N has been proved to be as O((InN)c’nln’nN) 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 Runlely, R. S. “On Distinguishing Prime Numbers from Corn] %osite Number.” Math.
Plane
LLE
Adjunction
Ann.
Complex
117, 173206,1983.
Bosma, ing.”
W. and van der Hulst, M.P. “Faster Primality Test,In Advances in Cryptology, Proc. Eurocrypt ‘89, Houthalen, April 1013, 1989 (Ed. J.J. Quisquater). New York: SpringerVerlag, 652656, 1990. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b” zk 1, b = 2, 3,5,6,7,10,11,12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Sot., pp* lxxxivlxxxv, 1988. Cohen, K and Lenstra, A. K. “Primality Testing and Jacobi Sums? Math. Comput. 42, 297330, 1984. Cohen, H. and Len&a, A. K. “Implementation of a New Primality Test .” Math. Comput. 48, 103121, 1987. Mihailescu, P. “A Primality Test Using Cyclotomic Extensions.” In Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes (Proc. AAECC6, Rome, July 1988). New York: SpringerVerlag, pp. 310323, 1989.
AdlemanRumely Primality Test see ADLEMANPOMERANCERUMELY PRIMALITY TEST
Affine
Geometry
A GEOMETRY in which properties are preserved by PARALLEL PROJECTION from one PLANE to another. In an affine geometry, the third and fourth of EUCLID’S POSTULATES 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, ORDERED
GEOMETRY
References G. and Mac Lane, S. %Fme
Birkhoff, Survey
of Modern
pp. 268275,
Affine
Algebra,
3rd
ed.
Geometry.” New York:
59.13 in A Macmillan,
1965.
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 SUBGROUPS.
see also AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE
HULL,AFF~NE PLANE, AFFINE SPACE,AFFINE TRANSFORMATION,
AFFINITY
References
Admissible A string or word is said to be admissible if that word appears in a given SEQUENCE. For example, in the SEQUENCE aabaabaabaabaab . . ., a, aa, baab are all admissible, but bb is inadmissible.
see also BLOCK GROWTH
Birkhoff, G. and Mac Lane, S. A Survey of Modern 3rd ed. New York: Macmillan, p. 237, 1965.
Affine Hull The IDEAL generated see
Algebra,
by a SET in a VECTOR SPACE.
AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE GROUP, AFFINE PLANE, AFFINE SPACE, AFFINE TRANSFORMATION, AFFINITY, CONVEX HULL,HULL also
Affine ltansformation
Affine Plane Affine A 2D
Plane GEOMETRY constructed over 8 FINITE For a FIELD F of size n, the affine plane consists
AFFINE
FIELD.
of the F and Adding allows affine
set of points which are ordered pairs of elements in a set of lines which are themselves a set of points. a POINT AT INFINITY and LINE AT INFINITY a PROJECTIVE PLANE to be constructed from an plane. An affine plane of order n is a BLOCK DESIGN of the form (n2, 12, 1). An affine plane of order ’ n exists IFF a PROJECTIVE PLANE of order n exists. see also AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE GROUP, AFFINE HULL, AFFINE SPACE, AFFINE TRANSFORMATION, AFFINITY, PROJECTIVE PLANE
References Lindner, Raton,
C. C. and Rodger, C. A. Design FL: CRC Press, 199%
Theory.
Boca
Affine Transformat ion 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” is a MAP F : R” + Ik” of the form F(p)=Ap+q
DILATION (CONTRACTION, REFLECTION, ROTATION,
of
is ORTENTATIONis ORIENTATION
HOMOTHECY), EXPANSION, and TRANSLATION are all
affine transformations, as are their combinations. A particular example combining ROTATION and EXPANSION is the rotationenlargement transformation
a I[  1 [I [ cos 1 (2) X1
=S
Yl
=S
Separating
Affine Space Let V be a VECTOR SPACE over a FIELD K, and let A be a nonempty SET. Now define addition p + a f A for any VECTOR a E V and element p E A subject to the conditions
transformation
REVERSING.
see also SCHEME Y. (Eds.). “Schemes.” 518E in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 69, 1980.
(1)
for all p f IV, where A is a linear Rn. If det(A) = 1, the transformation PRESERVING; if det(A) = 1, it
Affine Scheme A technical mathematical object defined as the SPECTRUM o(A) of a set of PRIME IDEALS of a commutative RING A regarded as a local ringed space with a structure sheaf.
References Iyanaga, S. and Kawada,
19
 sin Q
sina! cos a
x  x0 Y
Yo
cos a(z  x0) + sin a(y  yo) [ sinClr(x  20) + cosa(y  yo)
’
the equations,
2’ f= (scosa)x Yt = ( ssina)x
 s(x0 cos a + y0 sin a)
+ (ssina)y + (scosa)y
+ ~(50 sina
(3)
 yo cow).
(4) This
can be also written
as
xJ = ax+by+c yt z bx + uy + d,
(5) (6)
1, p + 0 = p, where
2. (p + a) + b = p + (a + b), 3. For any q E A, there EXISTS a unique such that Q = p + a. Here, Then CIENT
VECTOR
a E V
a, b E V. Note that (1) is implied by (2) and (3). A is an affine space and K is called the COEFFI
The scale factor
a=
scosa
(7)
b=
s sin a.
($1
s is then
defined
by
FIELD.
In an affine space, it is possible to fix a point and coordinate axis such that every point in the SPACE can be represented as an ntuple of its coordinates. Every ordered pair of points A and B in an affine space is then associated with a VECTOR AB. see also AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE GROUP, AFFINE HULL, AFFINE PLANE, AFFINE SPACE, AFFINE TRANSFORMATION, AFFINITY
s E 2/a2 + b2 1 and the rotation
ANGLE
a
(9)
by
(10)
see also AFFINE COMPLEX PLANE, AFFINE CONNECTION, 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 Surfaces. Boca Raton, FL: CRC Press, p. 105, 1993.
Affinity
20
Airy Differential
Affinity see AFFINE
Equation
Specializing to the “conventional” Airy differential tion occurs by taking the MINUS SIGN and k2 = 1. Then plug (4) into
TRANSFORMATION
Affix
equasetting
y”  xy = 0
In the archaic terminology of Whittaker and Watson (1990), the COMPLEX NUMBER z representing ~1:+ iy.
(5)
to obtain References Whittaker, E. T. and Watson, Analysis, 4th ed. Cambridge, versity Press, 1990.
G. N. A Course in Modern England: Cambridge Uni
~(~+2)(n+l)a,+21”x~a,ln=0
(6)
nd
n=O
Aggregate An archaic word for infinite ered by Georg Cantor. see
CLASS
also
(SET),
SETS such as those
?(?2+2)(n+
consid
l)Un+2Xn
n=O
SET
00
2a2 + F(n
AGM
+ 2)(n +
 x
l)an+2Xn
n=l
see ARITHMETICGEOMETRIC
+)[(n+z)(n+ /
Witch
see WITCH
and
= 0 (8)
n=l
MEAN 2a2
Agnesi’s
(7)
 eUnz"+l=O
1)%+2

Unl]Xn
0.
=
4
(9)
n=l
In order for this equality to hold must separately be 0. Therefore,
OF AGNESI
for all 3, each term
A&sienne (n + 2)(n +
Agonic
AhlforsBers
Starting
Gl
(11)
with
the n = 3 term and using we obtain
the above RE
5 .4as = 20~5 = a2 = 0.
(12)
CURRENCE RELATION, Theorem
RIEMANN’S MODULI SPACE gives the solution to RXEMANN’S MODULI PROBLEM, which requires an ANALYTIC parameterization of the compact R~EMANN SUR
The
Continuing,
FACES in a fixed HOMEOMORPHISM. Differential
Some authors as
Equation
define a general
This equation expansions
Airy differential
equation
for n = 1, 2,
.
nl

IfI
=
(nt
I)&
U3n1
=
0
terms
of the form
a0
6*5
(6~5)(3*2) a0
94
=
(9a8)(64)(3m2)’

by INDUCTION, a0
 1)][(37113)(3n
 4)]
. v [6 5][3.2]
l
l
(17)
co
x
for n = 1, 2,
.
(n+
a4 =
l)nUn+~Xnl
n=l
l
Finally,
.
+ 2>(n +
1)&a+
n 2x
l
a7 = (4)
look
at terms
of the form
a3n+lt
00
l)?lUn+lXn'
a3n.
3.2
al
a10
=
*
4*3  a4
7 6 a7 %% l
n=O
.
a0

a3n = [(3n)(3n
+d
n=O
= F(7i
l
n=l
00
y” = x(n+
.
(13)
nl
?lanX
00 x n0
=
 a6
n=O 
all
@3== a3
using the
Again 72&X
=
Now examine
.
u3
(1)
can be solved by series solution
l
as
1
by INDUCTION that
it follows
u2 = a5 = (28
y” 31 k2zy = 0.
yt =
(10)
=
l)an+2
Lines
see SKEW LINES
Airy
=0
a2
OF AGNESI
see WITCH

(18)
al
(7.6)(4
l
(19)
3) a1
= (10 . 9)(7
l
6)(4
l
3) 
(20)
AiryFock
Functions
Airy
By INDUCTION, a3n+l
al
=
[(3n + 1)(3n)][(3n
 2)(3n
 3)]  4  [7. 6][4  31 (21) solution is therefore
for n = 1, 2, . . . The general l
Y=ao
‘+I?
[
(3n)(3n
n=l
.2
1)(3n3)(3n4)d~~3
I
+a1
x+
E n=lO"
[
For a general
(3n+1)(3n)(3n22)(3n3)mmm4a3
k2 with
a MINUS Ytt  k2xy
and the solution Y( x>
i&
FIRST KIND. AIRY
= 0,
This
y(x)
($kx3j2)
 BIljJ
(zkx3i”)]
is usually expressed Ai and Bi(x)
= A’ Ai(k2’3x)
and the solutions
,
in terms
+ B’ Bi(Fi2/“2). instead,
of the
(25)
($kx3’2)
where satisfy
FUNCTION
(;kx”‘“)]
AIRY
KIND.
(3)
FUNCTION. 
These
a(+;
(7)
functions
w2(4
2i w2(z7,
(4) (5)
where Z* is the COMPLEX CONJUGATE ofz.
see also AIRY FUNCTIONS References Hazewinkel,
M. (Managing Ed.). Encyclopaedia of MathUpdated and Annotated Translation of the Soviet ‘IMathematical Encyclopaedia. ” Dordrecht, NetherAn
lands: Reidel,
is a MODIFIED BESSEL FUNCTION OF THE the second case can be reexpressed
KIND,
are
(2)
Twlb41*=
p. 65, 1988.
(5)
(8)
wz(z) = 2ei”l”v(w1z),
w(z)
OF THE
KIND
(1)
=
FUNCTION
 ‘dx), sin(rm)
FUNCTIONS,
w1 (z) = 2ei"%(wa)
is an AIRY
1
,
v(z) = $fiAi(z)
Y(Z)
ematics:
= ;‘dx)
OF THE FIRST KIND, MODIFIED
Functions AiryFock functions
Ai
OF THE FIRST
OF THE FIRST
($)I
(26)
+ 13Jli3
CK FUNCTIONS,
FUNCTION
AiryFock The three
h/3
and I(Z) is a MODIFIED BESSEL FIRST KIND. Using the identity K,(x)
(27) AIRYF•
($)
where J(Z) is a BESSEL FUNCTION OF THE FIRST KIND
SECOND
where J(z) is a BESSEL
[I,/,
(4
where K(z)
[AJ1,3
$fi
then
are
FUNCTION
a(+;x> =
(24)
ytt + k2xy = 0
see also
+ J1/3 ($)I
BESSEL FUNCTION OF THE
If the PLUS SIGN is present
BESSEL BESSEL
(g)
[J1,3
(3)
[AI1,3
FUNCTIONS
y(x) = +fi
*($x> = $6
(23)
is
I is a MODIFIED
where
(1) is
(2)
cos(t3 +mxt) dt
’
equation
SIGN,
Airy Functions Watson’s (1966, pp. 188190) definition of an Airy function is the solution to the AIRY DIFFERENTIAL EQUATION at’ dz k2+x = 0 (1)
1
x3n+l
21
which is FINITE at the ORIGIN, where a’ denotes the DERIVATIVE d@/dx, k2 = l/3, and either SIGN is permitted. Call these solutions (l/n)@(*k2, x), then
n
X3
Amctions
A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp, 446447) and illustrated above. This definition identifies the Ai and Bi(x) functions as the two LINEARLY INDEPENDENT solutions to (1) with k2 = 1 and a MINUS SIGN,
y”  yz = 0.
(9)
22
Airy
The solutions
Ai tken ‘s d2 Process
Functions Airy
are then written
Projection
A MAP y(z) = A Ai
+ B Bi(z),
(10)
The inverse equations for 4 are Let the ANGLE of the projection
PROJECTION.
computed by iteration. plane be &. Define
where Ai
E I+(1,
z)
= ;J;;[I&$r3/2)
 r&~3’2)]
(11) Bi(z)
E &
[I&z~‘~)
.
+ I&Z3’2)]
xi = cos ‘{exp[(dm+atanxi)tanzi]}
where r(z) is the GAMMA FUNCTION. This Watson’s expression becomes cos(at3
means
until the change in zi between evaluations than the acceptable tolerance. The (inverse) are then given by
Gi(z)
E 1 7T 1;0
sin( $t3 + zt) dt
(15)
Hi(z)
= 1
exp(+t3
(16)
+ zt) 6%.
7r s 0
=
tanl
Aitken’s
see dso
Tables,
9th printing.
New
York: Dover, pp. 446452, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Bessel Functions of Fractional Order, Airy Functions, Spherical I3essel Functions.” $6.7 in Numerical Art
of Scientific
Computing,
Cambridge University Press, pp. 234245, 1992. Spanier, J. and Oldham, K. B. “The Airy Functions Ai and B(x).” Ch. 56 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 555562, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.
(5)
l
Y
>
S2 Process
(&a+1  hJ2 &+1s&+3.
Abramowitz, M. and Stegun, C. A. (Eds.). “Airy Functions.” $10.4 in Handbook of Mathematical Functions ulith Formu
Recipes in FORTRAN: The 2nd ed. Cambridge, England:
Z
An ALGORITHM which extrapolates the partial sums So of a SERIES c, a72 whose CONVERGENCE is approximately geometric and accelerates its rate of CONVERGENCE. The extrapolated partial sum is given by
CK FUNCTIONS
and Mathematical
(4
(
I%t =
References Graphs,
2xi
by Hardy. X
las,
is smaller equations
dt. (14)
fxt)
 zn 4 1
has been constructed
(3)
that
The ASYMPTOTIC SERIES of Ai has a different form QUADRANTS ofthe COMPLEX PLANE, a fact indifferent known as the STOKES PHENOMENON. Functions related to the Airy futictions have been defined as
AIRYF•
(2)
(13)
r(13
see ah
n
long as zi > 1, take xi+1 = x&2 and iterate again. The first value for which xi < 1 is then the starting point. Then compute
3213
A generalization
xi = lexp[(@TjF+atanxi)tanxi]~
the
AS
= 2,
=
let xi = 7r/6 and compute
is
l
(3 a 11/3.1r Ai(*(3a)l/“2)
For proper convergence, initial point by checking
(12)
curve and Bi(z)
In the above plot, Ai is the solid dashed For zero argument Ai
(1)
otherwise.
EULER’S
SERIES

2s,
+
sn1
l
TRANSFORMATION
References Abramowitz,
M.
of Mathematical Mathematical
and
p. 18, 1972. Press, W. H.; Flannery, ling, W. T. Numerical Scientific
Stegun,
Functions Tables, 9th
A.
B. P.; Teukolsky,
Computing,
bridge University
C.
Press,
(Eds.).
vlith Formulas, printing. New
Handbook Graphs, and
York:
Dover,
S. A.; and Vetter
Recipes in FORTRAN: The Art of 2nd ed. Cambridge, England: Cam
p. 160, 1992.
23
Aitken Interpolation
Albers EqualArea
Aitken Interpolation An algorithm similar to NEVILLE’S ALGORITHM for constructing the LAGRANGE INTERPOLATING POLYNOMIAL. Let f(zlzo, ~1,. . . , zk) be the unique POLYNOMIAL of kth ORDER coinciding with f(z) at ~0, . . . , zk. Then
Albanese Variety An ABELIAN VARIETY which is canonically attached to an ALGEBRAIC VARIETY which is the solution to a certain universal problem. The Albanese variety is dual to the PICARD VARIETY.
Conic Projection
References Hazewinkel,
f 0 20  x
1 f(4~0,22)
=
x2

f2
x0
f(Z120,Xl,X2,X3)
see ~SCI
=
Ed.). Encyclopaedia and Annotated Translation Eric yclopaedia. ” Dordrecht,
lands:
Reidel,
pp. 6768,
Conic
Projection
of Muthof the
Nether
1988.
x2 x
f(zlzo,x1)
1 f(41C0,51,x2)=
M. (Managing
ematics: An Updated Soviet i‘MathematicaZ
Xl
52
Xl
1
f(+o,
x3
22
f (2~113o,X1,~3)
f(x[xo,x2)

x2 Xl?
Albers
x

see ALBERS
x
x2  x
4
x3

x
Albers
EQUALAREA
EqualArea
CONIC Conic
PROJECTION
Projection
*
LAGRANGE INTERPOLATING POLYNOMIAL
References M.
Abramowitx, of Mathematical Mathematical
and
Stegun,
Functions Tables, 9th
C, A.
(Eds.).
urith Formulas, printing. New
Handbook Graphs,
York:
and
Dover,
p. 879, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 9394, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam
bridge University A.jimaMalfatti
Let, &, be the LATITUDE for the origin of the CARTESIAN COORDINATES and X0 its LONGITUDE. Let $1 and $2 be the standard parallels. Then
Press, p* 102, 1992. Points
c
x = pin0
(1)
Y = PO  pose,
(2)
where C  2nsin#
(3) (4)
P=J 8 = n(A  TO)
C  2nsin$o n C = cos2 41 + 272 sin 41
po= A
72 = +n@l
The lines connecting the vertices and corresponding circlecircle intersections in MALFATTI'S TANGENT TRIANGLE PROBLEM coincide in a point Y called the first AjimaMalfatti point (Kimberling and MacDonald 1990, Kimberling 1994). Similarly, letting A”, B”, and C” be the excenters of ABC, then the lines A’A”, BIB”, and C’C” are coincident in another point called the second AjimaMalfatti point. The points are sometimes simply called the MALFATTI POINTY (Kimberling 1994).
The inverse
(5) (6) (7)
d
FORMULAS
+ sin&$
are
$=sinl A=&+,
(C2fnZ)
(8)
8
(9)
n
where
References Points and Central Lines in the Kimberling, C, “Central Math. Mug. 67, 163187, 1994. Plane of a Triangle.” Points.” Kimberling, C. “1st and 2nd AjimaMalfatti http:// www . evansville . edu/ck6/tcenters/recent/
p = dx2 + (PO(J = tanl
z ( POY
Y)”
(10) (11)
> *
ajmalf.html. Kimberling, Solution.
C. and MacDonald, ” Amer.
Math.
I. G. “Problem
Monthly
97,
612613,
E 3251 and 1990*
References J. P. Map ProjectionsA Geological Survey Professional DC: U. S. Government Printing
Snyder,
Working
Manual.
U. S.
Paper 1395. Washington, Office, pp. 98103, 1987.
Alcuin’s
24 Alcuin’s
AlexanderConway
Sequence
Polynomial
Aleph
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, . . m (Sloane’s A005044) given by the COEFFICIENTS of the MACLAURIN SERIES for l/(l~~)(l~~)(l~~). The number of different TRIANGLES which have INTEGRAL sides and PERIMETER n is given by
The SET THEORY symbol an INFINITE SET.
(N) for the CARDINALITY of
see UZSO ALEPH0 (No), ALEPH1 (N1), COUNTABLE SET, COUNTABLY INFINITE SET, FINITE, INFINITE, TRANSFINITE NUMBER, UNCOUNTABLY INFINITE SET Aleph0 (Ho) SET THEORY symbol for a SET having the same CARDINAL NUMBER as the “small” INFINITE SET ofINTEGERS. The ALGEBRAIC NUMBERS also belong to No* The
(2) 
n2 48
I
where Pz(n) and P3(n) are PARTITION FUNCTIONS, with Pk (n) giving the number of ways of writing n as a sum of and 1x1 is the FLOOR k terms, [z] is the NINT function, FUNCTION (Jordan et al. 1979, Andrews 1979, Honsberger 1985). Strangely enough, T(n) for n = 3, 4, . . . is precisely Alcuin’s sequence.
see U~SO PARTITION FUNCTION P, TRIANGLE References Andrews, G. “A Note on Partitions and Triangles with Integer Sides.” Amer. Math. Monthly 86, 477, 1979. Honsberger, FL Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 3947, 1985. with Jordan, J. H.; Walch, R.; and Wisner, R. J1 “Triangles Integer Sides.” Amer. Math. Monthly 88, 686689, 1979. Sloane, N. J. A, Sequence AOO5044/MO146 in “An On Line Version of the Encyclopedia of Integer Sequences.”
AleksandrovTech
Cohomology
which satisfies all the EILENBERGSTEENROD AXIOMS with the possible exception of the LONG EXACT SEQUENCE OF A PAIR AXIOM, as well as a certain additional continuity CONDITION. A theory
References Hazewinkel,
M. (Managing
Ed.). Encyclopaedia of Mathand Annotated Translation of the Encyclopaedia. ” Dordrecht, Netherp. 68, 1988.
ematics: An Updated Soviet “‘Mathematical
lands:
Reidel,
Aleksandrov’s
Uniqueness
properties
satisfied
Theorem
A convex body in EUCLIDEAN nspace that is centrally symmetric with center at the ORIGIN is determined among all such bodies by its brightness function (the VOLUME of each projection).
see also TOMOGRAPHY
(1)
TN0 = No
(2)
No+f =No,
Not.
Amer.
Math.
(3)
where f is any FINITE SET. However,
Nofro  c >
(4)
where C is the CONTINUUM. see also ALEPH1, CARDINAL NUMBER, CONTINUUM, CONTINUUM HYPOTHESIS, COUNTABLY INFINITE SET, FINITE, INFINITE, TRANSFINITE NUMBER, UNCOUNTABLY INFINITE SET Aleph1
(Ni)
The SET THEORY symbol
for the smallest INFINITE SET larger than ALPHAO (NO). The CONTINUUM HYPOTHESIS asserts that & = c, where c is the CARDINALITY of the “large” INFINITE SET of REAL NUMBERS (called the CONTINUUM in SET THEORY). However, the truth of the CONTINUUM HYPOTHESIS depends on the version of SET THEORY you are using and so is UNDECIDABLE. Curiously enough, nD SPACE has the same number of points (c) as 1D SPACE, or any FINITE INTERVAL oflD SPACE (a LINE SEGMENT), as was first recognized by Georg Cantor. see also ALEPH0 (No), CONTINUUM, CONTINUUM HYPOTHESIS, CWNTABLY INFINITE SET, FINITE, INFINITE, TRANSFINITE NUMBER, UNCOUNTABLY INFINITE SET
Alet hit in LOGIC meaning pertaining FALSEHOOD. see also FALSE, PREDICATE, TRUE A term
References Gardner, R. 3. “Geometric Tomography.” Sot. 42, 422429, 1995.
by No include
NOT= No
(3)
for n odd,
{
surprising
for n even
(n+3j2 [48 1
[
Rather
AlexanderConway Polynomial see CONWAY POLYNOMIAL
to TRUTH and
Alexander’s
Horned
Sphere
Alexander’s
Horned
Sphere
Alexander Schroeder, un Infinite 1991.
M.
Fractals, Paradise.
Matrix
25
Chaos, Power Law: Minutes New York: W. H. Freeman,
from p. 58,
Alexander
Ideal IDEAL in A, the RING of integral LAURENT POLYNOMIALS, associatedwithan 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 s3 has a SQUARE presentation MATRIX, its Alexander ideal is PRINCIPAL and it has an ALEXANDER POLYNOMIAL The
order
w see UZS~ALEXANDERINVARIANT,ALEXANDERMATRIX, ALEXANDER POLYNOMIAL l
composed of a countable UNION of COMPACT SETS, is called Alexander’s horned sphere. It is HOMEOMORPHIC with the BALL B3, and its boundary is therefore a SPHERE. It is therefore an example of of the a wild embedding in E3. The outer complement 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
above
solid,
The complement in Iw3 of the bad points for Alexander’s horned sphere is SIMPLY CONNECTED, making it inequivalent to ANTOINE'S HORNED SPHERE. Alexander’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 contains a horned complex. A humorous drawing by Simon Frazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander’s horned sphere growing from his head,
References Rolfsen, Perish
D.
Knots
Press,
Alexander
pp.
and Links. 206207,
Wilmington, 1976.
DE:
Publish
or
Invariant
The Alexander invariant H, (2) MOLOGY of the INFINITE cyclic of K, considered as a MODULE gral LAURENT POLYNOMIALS. for a classical TAME KNOT is only HI is significant.
of a KNOT K is the HOcover of the complement over A, the RING of inteThe Alexander invariant finitely presentable, and
For any KNOT Kn in s”+’ whose complement has the homotopy type of a FINITE COMPLEX, the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a TAME KNOT in s3 has a SQUARE presentation MATRIX, its ALEXANDER IDEAL is PRINCIPAL and it has an ALEXANDER POLYNOMIAL denoted a(t). see also A LEXANDERIDEAL,ALEXANDERMATRIX, EXANDE
R
AL
POLYNOMIAL
References Rolfsen, Perish
Alexander
D. Knots and Links. Press, pp. 206207,
Wilmington, 1976.
DE:
Publish
or
Matrix
A presentation matrix for the ALEXANDER INVARIANT HI(z) of a KNOT K. If V is a SEIFERT MATRIX for a TAME KNOT K in s3, then VT  tV and V  tVT are Alexander matrices for K, where VT denotes the
MATRIX
TRANSPOSE.
IDEAL, ALEXA .NDER IN VARIANT, see also ALEXANDER ALEXAN 'DER POLYNOMIAL, S EIFERT MATRIX see also ANTOINE’S
HORNED
SPHERE
References J, Illustration accompanying “The Game of Math Horizons, p. 9, Spring 1994. “Conway’s Prime Producing Machine.” Math. Mug. 56, 2633, 1983. Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 8081, 1976. Albers, ‘Life’.” Guy, R.
D.
References Rolfsen, Perish
D. Knots and Links. Press, pp. 206207,
Wilmington, 1976.
DE:
Publish
or
Alexander
26
Polynomial
Alexander
Alexander Polynomial A POLYNOMIAL invariant
q be the MATRIX PRODUCT of BRAID WORDS of a KNOT, then Let
det(I
 Xl?) =
where
t
+
l
l
l
+
A(t)
polynomial and det is the Alexander polynomial of a TAME
= det(VT
 tV),
also satisfies
L+
= *l.
L
Lo
A, then there by J. H. Con
way) AL+
(t) 
AL
(t) +
 t1’2)AL,(t)
(t1'2
= o
(9)
corresponding to the above LINK DIAGRAMS (Adams 1994). A slightly different SKEIN RELATIONSHIP convention used by Doll and Hoste (1991) is
VL+ VL
polynomial remained the onEy 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 tl and satisfying A(unknot) = 1 (4) was formulated by 3. H. Conway and is sometimes denoted VL. The NOTATION [a + b + c + . . . is an abbreviation for the Conwaynormalized Alexander polynomial ofa KNOT
+ c(x” + x“)
+ . . ..
(5)
For a description of the NOTATION for LINKS, see Rolfof the ConwayAlexander sen (1976, p. 389). E xamples polynomials for common KNOTS include v~K=[ll=Zl+lx
(6)
VFEK
= [3  1 = Xl
&SK
= [l  1 + 3. = X2  xl
=
zv+
(10)
These relations allow Alexander polynomials to be constructed for arbitrary knots by building them up as a sequence of over and undercrossings. a
KNOT, A&l)
s
1 (mod
8)
5 (mod
8)
+ 3 X + 1  x +x2
(7) (8)
if Arf(K) if Arf(K)
= 0. = 1,
INVARIANT (Jones
where Arf is the ARF a KNOT and
1985).
(11)
If K is
iAK@>l> 3,
Alexander
a + b(x + xl)
is deter
(3)
The Alexander polynomial of a split table link is always 0. Surprisingly, there are known examples of nontrivial KNOTS with Alexander polynomial 1. An example is PRETZEL KNOT. the (3,5,7) The
SIGN
Let an Alexander polynomial be denoted exists a SKEIN RELATIONSHIP (discovered
Rx A(1)
the final
\/
(2)
V is a SEIFERT MATRIX, det is the DETERMINANT, and VT denotes the MATRIX TRANSPOSE. The polynomial
where
f
(1)
AL,
where
Alexander
Alexander polynomial, mined by convention.
Pl
AL is the Alexander
DETERMINANT. The KNOT in s3 satisfies
TREFOIL KNOT, FIGUREOFEIGHT KNOT, and Multiplying SOLOMON'S SEAL KNOT, respectively. through to clear the NEGATIVE POWERS gives the usual
for the
of a KNOT discovered in 1923 by J. W. Alexander (Alexander 1928). In technical language, the Alexander polynomial arises from the HOMOLOGY of the infinitely cyclic cover of a KNOT'S complement Any generator of a PRINCIPAL ALEXANDER IDEAL is called an Alexander polynomial (Rolfsen 1976)* Because the ALEXANDER INVARIANT of a TAME KNOT in s3 has a SQUARE presentation MATRIX, its ALEXANDER IDIZAL is PRINCIPAL and it has an Alexander polynomial denoted A(t).
1+
Polynomial
then K cannot if
be represented
h(e then K cannot 1985).
245)
be represented
as a closed
(12) SBRAID. Also,
> y , as a closed
(13) 4braid
(Jones
POLYNOMIAL P(a, Z) generalizes the AlThe HOMFLY exander polynomial (as well at the JONES POLYNOMIAL) with V(z) = P(l,Z) (14) (Doll
and Hoste
1991).
Rolfsen (1976) gives a tabulation of Alexander polynomials for KNOTS up to 10 CROSSINGS and LINKS up to 9 CROSSINGS.
see UZSO BRAID GROU P, JONES POLYNOMIAL, KNOT, KN OT D ETERMINANT, LINK, SKEIN RELATIOIIJSHIP References Adams, C. C. The to
the
Knot
Mathematical
Freeman,
pp. 165469,
Book: Theory
1994.
An Elementary of Knots. New
Introduction
York:
W. H.
Alexander, J. W. "Topological Invariants of Knots and Links." Trans. Amer. Math. Sot. 30, 275306, 1928.
AlexanderSpanier
Cohomology
Algebra
Alexander, J. W. “A Lemma on a System of Knotted Curves.” Proc. Nat. Acad. Sci. USA 9, 9395, 1923, Doll, H. and Hoste, J. “A Tabulation of Oriented Links.” Math. Comput. 57, 747761, 1991. Jones, V. “A Polynomial Invariant for Knots via von Neumann Algebras.” Bull. Amer. Math. Sot. 12, 103111, 1985. Rolfsen, D. “Table of Knots and Links.” Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280287, 1976. Stoimenow, A. “Alexander Polynomials .” http://www.
informatik.huberlin.de/stoimeno/ptab/alO.html. Stoimenow,
A,
“Conway
Polynomials
.”
http://www.
informatik,huberlin.de/stoimeno/ptab/clO.html. AlexanderSpanier
Cohomology
fundamental result of DE RHAM COHOMOLOGY is that the kth DE RHAM COHOMOLOGY VECTOR SPACE of a MANIFOLD M is canonically isomorphic to the AlexanderSpanier cohomology VECTOR SPACE H” (M; IQ (also called cohomology with compact supIn the case that A4 is COMPACT, Alexanderport). Spanier cohomology is exactly “singular” COHOMOLA
OGY. Alexander’s Theorem LINK can be represented
Any
by a closed
BRAID.
Algebra with GROUP THEstudies number systems and operations within them. The word “algebra” is a distortion of the Arabic title of a treatise by AlKhwarizmi about algebraic methods. Note that mathematicians refer to the “school algebra” generally taught in middle and high school as “ARITHMETIC," reserving for the more advanced aspects of the the word “algebra” subject.
The
branch
of mathematics
dealing
ORY and CODING THEORY which
Formally,
an algebra is a VECTOR SPACE V, over a a MULTIPLICATION which turns it into defined such that, if f f F and X, y E 7V, then
FIELD F with a RING
f (XY>
= (fX)Y
= X(fY)
In addition to the usual algebra of REAL NUMBERS, there are ==: 1151 additional CONSISTENT algebras which can be formulated by weakening the FIELD AXIOMS, at least 200 of which have been rigorously proven to be selfCONSISTENT (Bell 1945). Algebras which have of interest are usually investigators. This (but unenlightening) use with minimal or
been investigated and found to be named after one or more of their practice leads to exoticsounding names which algebraists frequently nonexistent, explanation.
see also ALTERNATE ALGEBRA, ALTERNATING ALGEBRA,B*ALGEBRA,BANACH ALGEBRA,BOOLEAN ALGEBRA, BOREL SIGMA ALGEBRA, C*ALGEBRA, CAYLEY ALGEBRA, CLIFFORD ALGEBRA, COMMUTATIVE
27
ALGEBRA, EXTERIOR ALGEBRA,FUNDAMENTAL OREM OF ALGEBRA,
GRADED
ALGEBRA,
THEGRASSMANN
ALGEBRA,HECKEALGEBRA,HEYTING ALGEBRA, HoMOLOGICAL ALGEBRA, HOPF ALGEBRA, JORDAN ALGEBRA, LIE ALGEBRA, LINEAR ALGEBRA, MEASURE ALGEBRA, NONASSOCIATIVE ALGEBRA, QUATERNION, ROBBINS ALGEBRA, SCHURALGEBRA,SEMISIMPLE ALGEBRA, SIGMA ALGEBRA, SIMPLE ALGEBRA, STEENROD ALGEBRA,VON NEUMANN ALGEBRA References Artin, M. Algebra. Englewood Cliffs, NJ: PrenticeHall, 1991. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGrawHill, pp. 3536, 1945. Bhattacharya, P. B.; Jain, S. K.; and Nagpu, S. R. (Eds.). Basic Algebra, 2nd ed. New York: Cambridge University Press, 1994, Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996. Brown, K. S. “Algebra.” http://www.seanet.com/ksbrown/ ialgebra .htm. Cardano, G. Ars Magna or The Rules of Algebra. New York: Dover, 1993. Chevalley, Functions
C.
C. Introduction One Variable.
to
the
Theory
Providence,
of
Algebraic
Math. SOL, 1951. Chrystal, G. Textbook of AZgebru, 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. Gulois Theory, corrected 2nd printing. New York: SpringerVerlag, 1993. Euler, L. Elements of AZgebru. New York: SpringerVerlag, 1984. Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994, Grove, Lzl. Algebra. New York: Academic Press, 1983, Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to EZementary Algebra for Schools. London: Macmillan, 1960. Harrison, M. A. “The Number of Isomorphism Types of Finit, e Algebras .” Proc. Amer. Math. Sot. 17, 735737, 1966. Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1996. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975. Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989. Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: University of Chicago Press, 1995. Lang, S. Undergraduate Algebra, 2nd ed. New York: SpringerVerlag, 1990. http : // Pedersen, J. “Catalogue of Algebraic Systems.” of
RI: Amer.
tarski.math.usf.edu/algctlg/. Uspensky, J. V. Theory of Equations. New York: McGrawHill, 1948. van der Waerden, B. L. Algebra, Vol. 2. New York: SpringerVerlag, 1991. van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. New York: SpringerVerlag, 1983. van der Waerden, B. L. A History of Algebra: From AZKhwurizmi to Emmy voether. New York: SpringerVerlag, 1985. v. s Algebra in Ancient and Modern Times. Varadarajan, Providence, RI: Amer. M ath. Sot., 1998. 1
Algebraic
Algebraic
Closure
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 Iw. Algebraic
Coding
Theory
see CODING THEORY Algebraic Curve An algebraic curve
f(X,Y)
over
a FIELD
= 0,where f(X,Y)
K
is an equation
~~~POLYNOMIAL~~ X and
Y with COEFFICIENTS in K. A nonsingular algebraic curve is an algebraic curve over K which has no SINGULAR PRINTS over K. A point on an algebraic curve is simply a solution of the equation of the curve. A KRATIONAL POINT is a point (X, Y) on the curve, where X and Y are in the FIELD K. see
U~SOALGEBRAIC GEOMETRY, ALGEBRAIC VARIETY,
CURVE References
Griffiths, dence,
P.
A. Introductz’on
to
Algebraic
Provi
Curves.
RI: Amer. Math. Sot., 1989.
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 U&W ELEMENTARY FUNCTION, TRANSCENDENTAL FUNCTION Algebraic Function Field A finite extension K = Z(Z)(W) of the FIELD c(z) of FUNCTIONS in the indeterminate z, Le., w is RATIONAL
Lang, S. Introduction Interscience, 1958. Pedoe, D. and Hodge, Vol. 1. Cambridge, 1994. Pedoe, D. and Hodge, Vol. 2. Cambridge, 1994. Pedoe, D. and Hodge, Vol. 3. Cambridge, 1994. Seidenberg, A, (Ed.). ington, DC: Math. Weil, A. Foundations idence, RI: Amer.
see UZSO ALGEBRAIC NUMBER FIELD, RIEMANN SURFACE
Geometry.
New
t
York:
W. V. Methods of Algebraic Geometry, England: Cambridge University Press, W. V. Methods of Algebraic Geometry, England: Cambridge University Press, W. V. Methods of Algebraic Geometry, England: Cambridge University Press, Studies
in Algebraic
Amer.,
Assoc.
of Algebraic
Math.
Wash
Geometry.
1980. Geometry,
ed. Prov
enl.
Sot., 1962,
Algebraic Integer If T is a ROOT of the POLYNOMIAL equation xn
+
anlxnl
+
.
l
l
+
UlX
+
a0
=
0,
where the ais are INTEGERS and T satisfies no similar equation of degree < n, then T is an algebraic INTEGER INTEGER is a special case of of degree rz. An algebraic an ALGEBRAIC NUMBER, for which the leading COEFFICIENT a, need not equall. RADICAL INTEGERS are a subring ofthe ALGEBRAIC INTEGERS. A SUM or PRODUCT of algebraic integers is again an alABEL'S IMPOSSIBILITY THEgebraic integer. However, OREM shows that there are algebraic integers of degree > 5 which are not expressible in terms of ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION, andtheextraction of ROOTS on REAL NUMBERS. The GAUSSIAN INTEGER are are algebraic Q(dT), since a + bi are roots of
2
aR00~ofapOLYNOM1ALa~+~~a+u~a~+...+a,a~, where ai f c(z).
to Algebraic
hvarian

2az
integers
of
+ u2 + b2 = 0.
see also ALGEBRAIC NUMBER, EUCLIDEAN NUMBER, RADICAL INTEGER
References Algebraic Geometry CURVES, ALGEBRAIC VARIThe study of ALGEBRAIC ETIES, and their generalization to nD.
ALGEBRAIC CURVE, ALGEBRAIC VARIETY, COMMUTATIVE ALGEBRA, DIFFERENTIAL GEOMETRY, GEOMETRY,~LANE CURVE,SPACE CURVE see
also
References Abhyankar,
S. S. Algebraic
Geometry
for
Scientists
and Bn
gineers. Providence, RI: Amer. Math. Sot., 1990. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, Algorithms: Commutative
Verlag, Eisenbud,
An Introduction Algebra, 2nd
and to Algebraic Geometry and ed. New York: Springer
1996. D. Commutative Algebra with a View gebraic Geometry. New York: SpringerVerlag, Griffiths, P. and Harris, J. Principles of Algebraic New York: Wiley, 1978* Hartshorne, R. Algebraic Geometry, rev. ed. SpringerVerlag, 1997.
Toward
Al
1995. Geometry. New
York:
Hancock, bers,
H. Foundations Vol.
I:
Introduction
of the Theory of Algebraic . to the General Theory.
York: Macmillan, 1931. Hancock, H. Foundations of the Theory hers,
Vol.
2:
The
General
Theory.
New
of Algebraic
York:
NU77b
New NU772
Macr nillan,
1932. Pohst,
M, and Zassenhaus, H, Algorithmic Algebraic kmCambridge, England: Cambridge University Press, 1989. Wagon, S. “Algebraic Numbers.” §10.5 in Mathematics in Action. New York: W. H. Freeman, pp. 347353, 1991. ber
Theory.
Algebraic Invariant A quantity such as a DISCRIMINANT which remains unchanged under a given class of algebraic transformations. Such invariants were originally called HYPERDETERMINANTS by Cayley.
see also DISCRIMINANT QUADRATIC INVARIANT
(POLYNOMIAL),
INVARIANT,
Algebraic
Knot
Algebraic
References
References
Grace, J. H. and Young, A. The Algebra of Invariants. New York: Chelsea, 1965. Gurevich, G. B. Foundations of the Theory of Algebraic Invariants. Groningen, Netherlands: P. Noordhoff, 1964. Hermann, R. and Ackerman, M. Hilbert’s Invariant Theory Papersrookline, MA: Math Sci Press, 1978. of Algebraic Invariants. Cambridge, EngHilbert, D. Theory land: Cambridge University Press, 1993. Mumford, D.; Fogarty, J.; and Kirwan, F. Geometric Inuariant Theory, 3rd enl. ed. New York: SpringerVerlag, 1994.
Conway,
Algebraic Knot A single component
ALGEBRAIC LINK. see also ALGEBRAIC LINK, KNOT, LINK
and links which arises in ALGEBRAIC 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
knots
also ALGEBRAIC
TANGLE, FIBRATION, TANGLE
References Adams, to
the
C. C. The Mathematical
Knot
Book: Theory
An Elementary of Knots. New
Freeman, pp. 4849, 1994. Rolfsen, D. Knots and Links. Perish Press, p. 335, 1976.
Algebraic Number If T is a ROOT ofthe
Wilmington,
Introduction
York:
W, H.
DE: Publish
+
clxnl
+
0,
l
l
l
+
en12
+
cn
=
0
(2)
a(x
then there called the any other satisfy the
number

cy)(x

of degree
p)(x

y)
n satisfying
’ * a,
the
(3)
are n  1 other algebraic numbers p, y, . . . conjugates of or. Furthermore, if a satisfies algebraic equation, then its conjugates also same equation (Conway and Guy 1996).
which is not algebraic is said to be TRANSCENDENTAL. see also ALGEBRAIC INTEGER, EUCLIDEAN NUMBER, HERMITELINDEMANN THEOREM, RADICAL INTEGER, SEMIALGEBRAIC NUMBER,TRANSCENDENTALNUMBER
Any number
Field
Algebraic Surface The set of ROOTS of a POLYNOMIAL f(z,v,x) = 0. An algebraic surface is said to be of degree n = max(i + j + k), where n is the maximum sum of powers of all terms umx~myjmZkm, The following table lists the names of algebraic surfaces of a given degree. Order
Surface
3 4 5 6 7 8 9 10
(1)
are algebraic numbers, then any ROOT of this equation is also an algebraic number. If a is an algebraic POLYNOMIAL
Number
see NUMBER FIELD
or
where the ais are INTEGERS and T satisfies no similar equation of degree < n, then T is an algebraic number of degree n. If T is an algebraic number and a0 = 1, then it is called an ALGEBRAIC INTEGER. It is also true that if the c;s in coxn
of Numbers.
POLYNOMIAL equation
aoxn + UlXnl +... + an12+ a, =
29
J. H. and Guy, R. K. “Algebraic Numbers.” In The New York: SpringerVerlag, pp. 189190, 1996. Courant, R. and Robbins, H. “Algebraic and Transcendental Numbers.” $2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103107, 1996. Hancock, H, Foundations of the Theory of Algebraic Numbers. Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931. Hancock, H. Foundations of the Theory of Algebraic Numbers. Vol. 2: The General Theory. New York: Macmillan, 1932. Wagon, S. “Algebraic Numbers.” $10.5 in Mathematics in Action. New York: W. H. Freeman, pp. 347353, 1991. Book
Algebraic
Algebraic Link A class of fibered
Tangle
cubic quartic quintic sextic heptic octic nonic decic
surface surface surface surface surface surface surface 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.). of Universities
Vieweg,
Mathematical and Museums.
Models
from
the Collections
Braunschweig,
Germany:
p. 7, 1986.
Algebraic Tangle ADDITIONS and MULTIPLICAAny TANGLE obtainedby TIONS of rational TANGLES (Adams 1994).
see also ALGEBRAIC LINK References Adams, to the
C. C. The Knot Mathematical
Freeman,
pp* 4151,
Book: Theory
1994.
An Elementary of Knots. New
Introduction
York:
W. H.
30
Algebraic
Algorithm
Topology
Algebraic Topology The study of intrinsic qualitative aspects of spatial objects (e.g., SURFACES, SPHERES, TORI, CIRCLES, KNOTS, LINKS, configuration spaces, etc.) that remain invariant under bothdirections continuous ONETOONE (HOMEOMORPHIC) transformations. The discipline of algebraic topology is popularly known as '&RUBBERSHEET GEOMETRY" and can also be viewed as the study of DISCONNECTIVITIES. Algebraic topology has a great deal of mathematical machinery for studying different kinds of HOLE structures, and it gets the prefix “algebraic” since many HOLE structures are represented best by algebraic objects like GROUPS and RINGS. A technical way of saying this is that algebraic topology 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 returned object (usually a GROUP or RING) is then a representation of the HOLE structure of the SPACE, in the sense that this algebraic object is a vestige of what the SPACE was like (i.e., much information is lost, original but some sort of “shadow” of the SPACE is retainedjust enough of a shadow to understand some aspect of its HOL&structure, but no more). The idea is that FUNCTORS give much simpler objects to deal with. Because SPACES by themselves are very complicated, they are unmanageable without looking at particular aspects.
C~MBINAT~RI AL TOPOLOGY is a special type of algeth .at uses COMBINATORIAL methods. see also CATEGORY, COMBINATORIAL TOPOLOGY, DIFFERENTIAL TOPOLOGY, FUNCTOR, HOMOTOPY THEORY braic topology
References Dieudonnk, J. A History of Algebraic and D~fierenfial ogy: 19004960. Boston, MA: Birkhguser, 1989.
Topol
Algebraic Variety A generalization to nD of ALGEBRAIC CURVES. More technically, an algebraic variety is a reduced SCHEME of FINITE type over a FIELD K. An algebraic variety V is defined as the SET of points in the REALS Iw" (or the COMPLEX NUMBERS (Cn) satisfying a system of POLYNOMIAL equations fi(~cr, . ,xn) = 0 for i = 1, 2, + +, , According to the HILBERT BASIS THEOREM, a FINITE number of equations suffices. l
l
see also ABELIAN VARIETY, ALBANESE VARIETY, BRAUERSEVERI VARIETY, CHOW VARIETY, PICARD VARIETY
Algebroidal Function An ANALYTIC FUNCTION f(z) algebraic equation Ao(qfk
Ciliberto, tion
sot.,
of Algebraic
1994.
E.; and Somese, A. J* (Eds.). CZass~~caVarieties, Providence, RI: Amer. Math.
+...
+ A&)
= 0
singlevalued MEROMORPHIC functions COMPLEX DOMAIN G is called a kalgebroidal
Aj(z) in a function
References Iyanaga,
S. and Kawada,
Y. (Eds.).
“Algebroidal
$19 in
Encyclopedic
bridge,
MA: MIT Press, pp* 8688, 1980.
Dictionary
of Mathematics.
Functions.” Cam
Algorithm A specific set of instructions for carrying out a proceusually with the requi .rement dure or solving a problem, that the procedure terminate at some point. Specific algorithms sometimes also go by the name METHOD, PROCEDURE, or TECHNIQUE. The word “algorithm” is a distortion of AlKhwarizmi, an Arab mathematician who wrote an influential treatise about algebraic methods.
see u~so~~~ALGORITHM, ALGORITHMIC COMPLEXITY, ARCHIMEDES ALGORITHM, BHASKARABROUCKNER ALGORITHM, BORCHARDTPFAFF ALGORITHM, BRELAZ'S HEURISTIC ALGORITHM, BUCHBERGER'S ALGORITHM, BULIRSCHSTOER ALGORITHM, BUMPING ALGORITHM, CLEAN ALGORITHM, COMPUTABLE FUNCTION, CONTINUED FRACTION FACTORIZATION ALGORITHM, DECISION PROBLEM, DIJKSTRA'S ALGORITHM, EUCLIDEAN ALGORITHM, FERGUSONFORCADE ALGORITHM, FERMAT'S ALGORITHM, FLOYD'S ALGORITHM, GAUSSIAN APPROXIMATION ALGORITHM, GENETIC ALGORITHM, GOSPER'S ALGORITHM, GREEDY ALGORITHM, HASSE'S ALGORITHM, HJLS ALGORITHM, JACOBI ALGORITHM, KRUSKAL'S ALGORITHM, LEVINEO'SULLIVAN GREEDY ALGORITHM, LLL ALGORITHM, MARKOV ALGORITHM, MILLER'S ALGORITHM, NEVILLE'S ALGORITHM, NEWTON'S METHOD, PRIME FACTORIZATION ALGORITHMS, PRIMITIVE RECURSIVE FUNCTION, PROGRAM, PSLQ ALGORITHM, PSOS ALGORITHM, QUOTIENTDIFFERENCE ALGORITHM, RISCH ALGORITHM, S~HRAGE'S ALGORITHM, SHANKS'ALGORITHM,SPIGOT ALGORTTHM,SYRACUSE ALGORITHM, TOTAL FUNCTION, TURING MACHINE, ZASSENHAUSBERLEKAMP ALGORITHM, ZEILBERGER'S ALGORITHM References Aho,
A. V.;
Hopcroft,
and Analysis
AddisonWesley, C.; Laura,
kl
the irreducible
with
sign
References
+ Al(Z)f
satisfying
Baase, S.
J* E.; and Ullman,
of Computer
Algorithms.
J.D. The DeReading, MA:
1974.
Algorithms. Reading, MA: Addison1988+ Brassard, G. and Bratley, P. Fundamentals of Algorithmics. Englewood Cliffs, NJ: PrenticeHall, 1995. Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduction to Algorithms. Cambridge, MA: MIT Press, 1990.
Wesley,
Computer
Algorithmic
Complexity
Aliquant
Greene, D. H. and Knuth, D. E. Mathematics for the Analysis of Algorithms, 3rd ed. Boston: Birkhauser, 1990. Harel, D. Algorithmicsr The Spirit of Computing, 2nd ed. Reading, MA: AddisonWesley, 1992. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed. Reading, MA: AddisonWesley, 1973. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: AddisonWesley, 1981. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: AddisonWesley, 1973. Kozen, D. C. Design and Analysis and Algorithms. New York: SpringerVerlag, 1991. Shen, A. Algorithms and Programming. Boston: Birkhguser,
1996. Skiena, S. S. The Algorithm Design Manual. SpringerVerlag, 1997. Wilf, H. Algorithms and Complexity. Englewood Prentice Hall, 1986, http://www.cis.upenn.edu/uilf/.
Algorithmic
New Cliffs,
NJ:
COMPLEXITY
The problem is called the billiard problem because it corresponds 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  y”)  2Kxy
+ (x2 + y2)(hy
 kx) = 0.
The problem is equivalent to the determination of the point on a spherical mirror where a ray of light will reflect 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 UZSO BILLIARDS, DUPLICATION
BILLIARD
TABLE
PROBLEM,
References Dijrrie, H. “Alhazen’s Billiard Problem.” $41 in 100 Great Problems of Elementary Mathematics: Their History and pp. 197200, 1965. Solutions. New York: Dover, Hogendijk, J. P. “AlMUtaman’s Simplified Lemmas for Solving ‘Alhazen’s Problem’.” From Bughdad to Barcelona/De Bagdad a Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59101, Anu. Filol. Univ. Bare., XIX B2, Univ. Barcelona, Barcelona, 1996. Lohne, J. A. “Alhazens Spiegelproblem.” Nordisk Mat. Tidskr. 18, 535, 1970. Neumann, P, Submitted to Amer. M&h. Monthly. Riede, H. “Reflexion am Kugelspiegel. Oder: das Problem des Alhazen.” Praxis Math. 31, 6570, 1989. Sabra, A. I. “ibn alHaytham’s Lemmas for Solving ‘Alhaxen’s Problem’.” Arch. Hist. Exact Sci. 26, 299324, 1982.
Alhazen’s
Problem
see ALHAZEN’S
Alhazen’s Billiard Problem In agiven 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.
H(x2
31
York:
Complexity
see BIT COMPLEXITYJOLMOGOROV
Divisor
CUBE
Alias’ Paradox Choose between
BILLIARD
PROBLEM
the following
two alternatives:
1. 90% chance of an unknown chance of $1 million, or
amount
SL:and a 10%
2. 89% chance of the same unknown amount x, 10% chance of $2.5 million, and 1% chance of nothing. which choice has the The PARADOX is to determine larger expectation value, 0.9x + $100,000 or 0.89x + $250,000. However, the best choice depends on the unknown amount, even though it is the same in both cases! This appears to violate the IN DEPENDENCE AXIOM. see also INDEPENDENCE LEM, NEWCOMB’S PARA
Axr OM, MONTY
HALL
PROB
.DOX
Aliasing Given a power spectrum (a plot of power vs. frequency), aliasing is a false translation of power falling in some frequency range (fc, &) outside the range. Aliasing can be caused by discrete sampling below the NYQUIST FREQUENCY. The sidelobes of any INSTRUMENT FUNCTION (including the simple SINC SQUARED function obtained simply from FINITE sampling) are also a form of aliasing. Although sidelobe contribution at large offsets can be minimized with the use of an APODIZATION FUNCTION, 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 number which is not an aliquant divisor (i.e., one that does DIVIDE another exactly) is said to be an ALIQUOT DIVISOR. see UZSO ALIQUOT
DIVISOR,
DIVISOR,
PROPER
DIVISOR
,
Aliquot
32 Aliquot see
Cycle
Allegory
Cycle
SOCIABLE
AlladiGrinstead Constant N.B. A detailed online essay by S. Finch ing point for this entry.
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 DIVISOR. The term “aliquot” is frequently used to specifically mean a PROPER DIVISOR, i.e., a DIVISOR of a number other than the number itself. see also SOR
ALIQUANT
Aliquot
DIVISOR,
DIVISOR,
PROPER
was the start
Let N(n) be the number of ways in which the FACTORIAL n! can be decomposed into n FACTORS of the form arranged in nondecreasing order. Also define pkbk m(n) i.e., m(n) appropriat
= max(pl
bl),
(1)
is the LEAST PRIME FACTO R raised to its e POWER in the factorization. Then define
DIVI
a(n) S equence
where In(z)
In m(n) G Inn
is the NATURAL
(2)
LOGARITHM.
For instance,
s(n) E u(n)  n, where a(n) RESTRICTED of numbers s’(n)
is the DIVISOR FUNCTION and s(n) is the DIVISOR FUNCTION. Then the SEQUENCE
= n, 2(n)
is called an aliquot n is bo unded, periodic. given
= s(n), s2(n) = s(s(n)),
q. .
sequence. If the SEQUEN GE for a i t either end s at s(l) = 0 or becom .es
If the SEQUENCE reaches a constant, knownasa PERFECT NUMBER.
the constant
If the SEQUENCE reaches an alternating called an AMICABLE PAIR.
pair,
is it is
If, after Fz iterations, the SEQUENCE yields a cycle of minimum length t of the form skS1 (n), s’+~ (n), sk+t(n), then th ese numbers form a group of ~&ABLE NUMBERS oforder t. It has not been proven that all aliquot sequences eventually terminate and become period. The smallest number whose fate is not known is 276, which has been computed up to ~~‘~(276) (Guy 1994). see also 196ALGO RITHM, ADDITIVE P ERSISTENCE, AMICABLE NUMBE w MULTIAMICABLE NUMBERS, MULTIPERFECT NUMBER, MULTIPLICATIVE PERSISTENCE, PERFECT N UMBER, SOCIABLE NUMBERS, UNITARY A LIQU~T SEQ UENCE References Guy, R. K. “Aliquot Sequences,” §B6 in Unsolved Problems in Number Theory, 2nd ed, New York: SpringerVerlag, pp. 6062, 1994. Guy, R. K. and Selfridge, J. L. “What Drives Aliquot Sequences.” Math. Comput. 29, 101107, 1975. Sloane, N. J+ A. Sequences A003023/M0062 in ‘&An OnLine Version of the Encyclopedia of Integer Sequences.” Sloane, N. J. A, and Plouffe, S. Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
AllPoles ~~~MAXIMUM
a(g) z fiIn3 n
=  In3 2ln3
 1  2’
(4)
For large n, lim a(n) n+m
= 2l
= 0.809394020534.
.. ,
(5)
where .&ln($).
(6)
References Alladi, K. and Grinstead, C. “On the Decomposition of n! into Prime Powers.” J. Number Th. 9, 452458, 1977. Finch, S. “Favorite Mathematical Constants.” http : //wua. mathsoft.com/asolve/constant/aldgms/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: SpringerVerlag, p. 79, 1994.
Allegory A technical resemblance FUNCTIONS
mathematical object which bears the same to binary relations as CATEGORIES do to and SETS.
see also CATEGORY References
Model ENTROPY
so
METHOD
F’reyd, P. J. and Scedrov, A. Categories, Allegories. dam, Netherlands: NorthHolland, 1990.
Amster
Almost
Allometric Allometric Mathematical growth in which oue population grows at a rate PROPORTIONAL to the POWER of another population.
Almost Integer A number which is very close to an INTEGER. prising example involving both e and PI is elr
References Cofrey,
W. J. Geography Towards Approach. London: Routledge,
a General
Spatial
Systems
Chapman
& Hall,
1981.
Almost All Given a property P, if P(z) N x as z + 00 (so the number of numbers less than x not satisfying the property P is o(z)), 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
FOR ALL,NORMAL
also
ory
G. H. and Wright, of Numbers,
Press,
5th
E. M. An Introduction Oxford, England:
ed.
to the The
Clarendon
p. 8, 1979.
Almost Alternating Knot An ALMOST ALTERNATING LINK nent.
see also ALTERNATING KNOT, LINK References Adams,
C. C. The Knot Book: to the Mathematical Theory
An Elementary of Knots. New
Introduction York: W. H.
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 see
ah
=
19.999099979.
Applying
COSINE
cos(7r cos(7r cos(ln(n
l
, (1)
as
(7T + 20)i = 0.9999999992 cos(ln(7r
l
One sur
 0.0000388927i
E 1
+ 20)) $=: 0.9999999992. a few more
times
(2) (3)
gives
+ 20)))) z 1 + 3.9321609261
x 1035.
(4)
0. MEASURE
References Sansone, G. Orthogonal Functions, York: Dover, p. 1, 1991.
This most J. H. tion ered.
curious nearidentity was apparently noticed alsimultaneously around 1988 by N. J. A. Sloane, Conway, and S. Plouffe, but no satisfying explanaas to “why” it has been true has yet been discov
rev.
nearidentity
is given by
a single compo
All nonalternating gcrossing 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, q), (3,4), and (3,5)TORUS KNOTS are almost alternating.
pp* 139146,
7T
can also be written
An interesting with
Almost Alternating Link Call a projection of a LINK an almost alternating projection 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 Every ALTERNATING KNOT no alternating projection. 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.
Freeman,
which

33
ORDER
References Hardy,
Integer
English
ed.
New
= 1+2.480... (W. Dubuque). given by
Other
remarkable
5(1+ d>[r(~)12 = I+ IT(z
nearidentities
are
4 5422,.
. x IO14 7
FUNCTION
(S. Plouffe),
r5 = 0.000017673..
.
the GAMMA
e6 7T4 
(5)
l
e5T/6 fi where
x lol3
(6)
and (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 Ramanujan (191314). Such approximations were also stud(1863), and Smith ied by Hermite (1859), K ronecker (1965). They can be generated using some amazing (and very deep) properties of the ~FUNCTION. Some of the numbers which are closest approximations to INTEGERS are eTm (sometimesknown as the RAMANUJAN CONSTANT and which corresponds to the field Q(dm) which has CLASS NUMBER 1 and is the <IMAGINARY quadratic field of maximal discriminant), exm, I?~, and I?~, the latter three of which have CLASS NUMBER 2 and are due to Ramanujan (Berndt 1994, WaldSchmidt 1988).
34
Almost
The properties of the spectacular identity 1n(6403203 + 744) n(Le Lionnais
Almost
Integer also give rise to the
~FUNCTION
differs see
1
CLASS
l
l
l
x 1O2g
(8)
1983, p. 152).
The list below gives numbers of the form rz 2 1000 for which 1x1  z 2 0.01.
x = erfi
for
7~”
= 884,736,743.999777466...
see
.,.975,825,573.993898311...
J. Pure H. J. S. Report
also
Equations
Paris:
Hermann,
to 19131914. New York:
and Approximations
45, 350372,
Appl. Math. on the Theory
l
QUASIPERFECT
Almost Prime A number n with
~.~771,804,797,161.992372939... ewm
S. “Modular
Quart.
remarquables.
NUMBER
Guy, R. K. “Almost Perfect, QuasiPerfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers.” §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, pp. 16 and 4553, 1994. Singh, S, Fermat’s Enigma: The Epic Quest to Solve the WorZd’s Greatest Mathematical Problem. New York: Walker, p. 13, 1997. Sloane, N. 3. A. Sequence A000079/M1129 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
.~.627,879,007.999848726...
= 28,994,858,898,043,231,996,779,**.
1863,
340345.
References
  212,174,016.997669832... wJ522 e = 14,871,070,263,238,043,663,567,. v
emm
Berlin,
F. Les nombres
l
= 639,355,180,631,208,421,..~
= 288,099,755,064,053,264,917,867,
PI
Almost Perfect Number A number n for which the DIVISOR FUNCTION satisfies c(n) = 2n  1 is called almost perfect. The only known almost perfect numbers are the POWERS of 2, namely . (Sloane’s AOOOO79). Singh (1997) 1, 2, 4, 8, 16, 32, calls almost perfect numbers SLIGHTLY DEFECTIVE.
= 4,309,793,301,730,386,363,005,719.996011651...
e”m
10m5’.
~FUNCTION,
Smith, of Numbers. Chelsea, 1965. Waldschmidt, M. “Some Transcendental Aspects of Ramanujan’s Work.” In Ramanujun Revisited: Proceedings of the Centenury Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, pp. 5776, 1988.
e rrvm =24,591,257,751.999999822213... Trl.m e = 30,197,683,486.993182260... rrvm e =147,197,952,743.999998662454... TTJ74 e = 54,551,812,208.999917467885... ediz = 45,116,546,012,289,599.991830287... lrdi33 ZT e 262,537,412,640,768,743.999999999999250072... ,die = 1,418,556,986,635,586,485.996179355... exvm = 604,729,957,825,300,084,759.999992171526... 7rvm7 e = 19,683,091,854,079,461,001,445.992737040...
edm
Wiss.
Le Lionnais, 1983. Ramanujan,
= 2,197.990869543... +Tvfn e = 422,150.997675680... 7Tx.43 e = 614,551.992885619... ,d5T = 2,508,951.998257553... rrJ25 e = 6,635,623.999341134... ,.Im = 199,148,647.999978046551...
edm
by a mere
Berndt, B, C, Ramanujan’s Notebooks, Part IV. New York: SpringerVerlag, pp* 9091, 1994. Hermite, C. “Sur la th&orie des kquations modulaires.” C. R. Acad. Sci. (Paris) 48, 10791084 and 10951102, 1859. Hermite, C. “Sur la thhorie des kquations modulaires.” C. R. Acad. Sci. (Paris) 49, 3624, 110118, and 141144, 1859. Kronecker, L. “ober die Klassenzahl der aus Werzeln der EinMonatsber. K. Preuss. heit gebildeten komplexen Zahlen.” Akad.
e?rvG
evm
NUMBER,
References
’ = 163 + 2 32167 l
from an INTEGER
also
Prime
prime
factorization
= 3,842,614,373,539,548,891,490,.~.
n= rI
.~~294,277,805,829,192.999987249... e7f%m = 223,070,667,213,077,889,794,379,
Pi
*i
i=l
.623,183,838,336,437.992055118...
is called kalmost prime when the sum of the POWERS r il ai = Ic. The set of kalmost primes is denoted Pk. c 
e 7tv?a = 249,433,117,287,892,229,255,125,.’. .~~388,685,911,710,805.996097323... e”xm
The PRIMES correspond to the “lalmost prime” numbers 2, 3, 5, 7, 11, . . . (Sloane’s AOO0040). The 2almost prime numbers correspond to SEMIPRIMES 4, 6, 9, 10, 14, 15, 21, 22, (Sloane’s A001358). The first few 3almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44,
= 365,698,321,891,389,219,219,142,m~..531,076,638,716,362,775.998259747...
e
x%/m8
= 6,954,830,200,814,801,770,418,837,. 940,281,460,320,666,108.994649611....
l
l
.
45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, Gosper
noted
that
the expression
1  262537412640768744Crm
 196884e2”m
+103378831900730205293632e3”J163.
(9)
99,
l
.
.
(Sloane’s A014612). The first few 4almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, . . . (Sloane’s A014613). The first few 5almost primes are 32, 48, 72, 80, . . . (Sloane’s A014614).
Alpha
Alternate
see als o CHEN'S THEOREM, PRIME
PRIME NUMBER, SEMI
References Sloane, N. J+ A. Sequences A014612, A000040/M0652, and A001358/M3274 Version of the Encyclopedia of Integer
A014613, in “An Sequences.”
A014614, OnLine
Alpha A financial measure giving the difference between a fund’s actual return and its expected level of performance, given its level of risk (as measured by BETA). A POSITIVE alpha indicates that a fund has performed better than expected based un its BETA, whereas a NEGATIVE alpha indicates poorer performance
see also BETA, SHARPE Alpha
35
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 language being used. In English, for example, 5 28 12
22 15 8
18 2 25
4 11 6
9 7 5
where the MAGIC SQUARE on the right the number
of letters
five twentyeight twelve
RATIO
Function
Algebra
8 3, 10 corresponds
to
in twentytwo fifteen eight
eighteen two twentyfive
’
References Sallows, L. C. F. “Alphamagic Squares.” Abacus 4, 2845, 1986. Sallows, L. C. F. “Alphamagic Squares. 2.” Abacus 4, 2029 and 43, 1987. Sallows, L. C. F. “Alpha Magic Squares.” In The Lighter Side of Math ematics (Ed. R. K, Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.
an(z)
The
=
alpha
r
function
Alphametic A CRYPTARITHM in which the letters used to represent distinct DIGITS are derived from related words or meaningful phrases. The term was coined by Hunter in 1955 (Madachy 1979, p. 178).
n t”e%t
=
n!z(n+lkz
x
zk F;r’ .
satisfies
the RECURRENCE RELA
TION see also
BETA FUNCTION (EXPONENTIAL)
Alpha Value An alpha value is a number zobserved) < a is considered a PVALUE.
see
also
0 < QI < 1 such that P(z >WGNIFICANT,” where P is
CONFIDENCE INTERVAL, PVALUE,
References Brooke, Me One Hundred & Fifty Puzzles in CryptArithmetic. New York: Dover, 1963. Hunter, J. A, H. and Madachy, J. S. “Alphametics and the Like .” Ch. 9 in Mathematical Diversions, New York: Dover, pp. 9095, 1975. Madachy, J. S. “Alphametics.” Ch+ 7 in Madachy’s Mathematical Recreations. New York: Dover, pp+ 178200 1979,
Alternate Algebra Let A denote an RALGEBRA, SPACE over R and
so that
A is a VECTOR
AxA+A
SIGNIFI
(1)
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,
Then
A is said to be alternate
(a:. Y)
if, for all x,y
E A,
’ Y = x . (Y . Y>
co
(x.x)*y=x*(xy).
WORD
(4
VECTOR MULTIPLICATION BILINEAR.
Here,
x
l
y is assumed
to be
References Finch, S. “Zero Structures mathsoft.com/asolve/zerodiv/zerodiv.html. Schafer, R. D. An Introduction New York: Dover, 1995.
in
Real
Algebras.”
to NonAssociative
http://www. Algebras.
36
Alternating
Alternating
Algebra
Alternating
Algebra
Erdener,
~~~EXTERIOR
ALGEBRA
Alternating
Group
nating
EVEN PERMUTATION GROUPS A, which are NORMAL SUBGROUPS of the PERMUTATION GROUP of ORDER They are ple LIE GROUPS. 60. Alternating
analogs of The lowest order groups with n > SIMPLE GROUPS. The number of the alternating groups A, for n = 5, 7, 9, (Sloane’s AUO0702), l
l
Table
of all Alter
f tp : //chs
. cusd.
claremont.edu/pub/knot/Rolfsen_table.final.
Kauffman,
n!/Z,
K. and Flynn, R. “Rolfsen’s Diagrams through 9 Crossings.”
Permutation
FINITE
L. LLNew Invariants
in the Theory
of Knots.”
Amer. Murasugi,
Math. MonthEy 95, 195242, 1988. K. “Jones Polynomials and Classical Conjectures in Knot Theory.” Topology 26, 297307, 1987.
5 are nonABELIAN
Sloane, N, J, A. Sequence A002864/M0847 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Thistlethwaite, M. “A Spanning Tree Expansion for the Jones Topology 26, 297309, 1987. Polynomial.”
conjugacy classes in 2, 3, . . . are 1, 3, 4,
Alternating
the families alternating
of simgroup is
1
see also 15 PUZZLE, FINITE GROUP, GROUP, LIE GROUP,~IMPLE GROUP,~YMMETRIC GROUP
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.
References Sloane, N. J. A. Sequence A000702/M2307 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Wilson, R. A. “ATLAS of Finite Group Representation.” http://for.mat.bham.ac.uk/atlas#alt.
Alternating
Link which has a LINK DIAGRAM underpasses and overpasses.
A LINK
see also ALMOST
Alternating Knot An alternating knot is a KNOT which possesses a knot diagram in which crossings alternate between under and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams. The TREFOIL KNOT and FIGUREOFEIGHT KNOT are alternating knots. One of TAIT'S KNOT CONJECTURES states that the number of crossings is the same for any diagram of a reduced alternating knot. Furthermore, 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 4n. Let c(K) be the CROSSING NUMBER. Then an alternating knot &#I& (a KNOT SUM) satisfies
= @I)
QG#Kz)
+
ALTERNATING LINK
References
As many alternating permutations among n elements begin by rising as by falling. The magnitude of the ens does not matter; only the number of them. Let the number of alternating permutations be given by Zn. = 2A,. This quantity can then be computed from 2rmn where T and s pass through that
A002864).
a0 = a1 = 1, and
References the
C. C. The Mathematical
Freeman, pp. Arnold, B.; Au, R.; Muir, J.; nating Knots
Knot
Book: Theory
An Elementary of Knots, New
Introduction
York:
W. H.
159164, 1994. M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, Wu, D.; and Ho&e, J. “Tabulating Alterthrough 14 Crossings.” ftp://chs.cusd.
claremont.edu/pub/knot/paper.TeX.txtand cusd.claremont.edu/pub/knot/AltKnots/.
r+s
=
Ix
Gas,
all INTEGRAL numbers nl,
(1) such (2)
l
see also ADEQUATE KNOT, ALMOST ALTERNATING LINK, ALTERNATING LINK, FLYPING CONJECTURE
to
of
Alternating Permutation An arrangement of the elements cl, . . . , C~ such that no element ci has a magnitude between ci1 and ci+l is The decalled an alternating (or ZIGZAG) permutation. termination of the number of alternating permutations for the set of the first n INTEGERS {1,2, . q . , n} is known as ANDRI?S PROBLEM. An example of an alternating permutation is (1, 3, 2, 5, 4).
In fact, this is true as well for the larger class of ADEQUATE KNOTS and postulated for all KNOTS. The number of PRIME alternating knots of n crossing for n = 1, 2 9 ..’ are 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, + (Sloane’s
Adams,
alternating
Menasco, W. and Thistlethwaite, M. “The Classification Alternating Links.” Ann. Math. 138, 113171, 1993.
c(K2).
l
with
ftp://chs.
A, = n!an.
(3)
numbers A, are sometimes called the EULER ZIGZAG NUMBERS, and the first few are given by 1, 1, 1, 2, 5, 16, 61, 272, . . (Sloane’s AOOOlll). The ODDnumbered A,s are called EULER NUMBERS, SECANT NUMBERS, or ZIG NUMBERS, and the EVENnumbered ones are sometimes called TANGENT NUMBERS or ZAG NUMBERS. The
l
Alternating Curiously LAURIN
Altitude
Series
enough, the SECANT and TANGENT MACcan be written in terms of the A,s as
SERIES
References Arf’ken, G. “Alternating ods for Physicists,
pp* 293294,
secr=Ao+A&+Aq
g+*** .
. X3
tanx=Alx+Ag~+&gr+...,
X5
(5) l
or combining
(4
l
them,
37
3rd
Series.” $5.3 in Mathematical ed. Orlando, FL: Academic
MethPress,
1985.
Bromwich, T. J. I’a and MacRobert, T. M. “Alternating Se$19 in An Introduction to the Theory of Infinite ries .” Series, 3rd ed. New York: Chelsea, pp. 5557, 1991. Pinsky, M. A. “Averaging an Alternating Series.” Math. Mug. 51, 235237,1978.
Alternating Also known
Series Test as the LEIBNIZ CRITERION. An ALTERNATING SERIES CONVERGES if al > a2 > . . and
secx + tanx
l
=Ao+Alz+A~~+A3~+Aq~+A~~+.,., .
.
(6) l
lim ak = 0. k+m
l
see also ENTRINGER NUMBER, EULER NUMBER, EuLER ZIGZAG NUMBER, SECANT NUMBER, SEIDELENTRINGERARNOLD TRIANGLE,TANGENT NUMBER
see UZSO CONVERGENCE
References And&, II.
Alternative Link A category of LINK encompassing KNOTS and TORUS KNOTS.
“Developments de sect: et tanz.” C. R. Acad. Paris 88, 965967, 1879, D. “Memoire SUP le permutations alternhes.” J. IMath. 7,167184, 1881, Updown Numbers AssociArnold, V. I. “BernoulliEuler ated with Function Singularities, Their Combinatorics and Duke Math. J. 63, 537555, 1991. Arithmetics.” Calculus and Combinatorics of BerArnold, V. I. “Snake noulli, Euler, and Springer Numbers for Coxeter Groups+” Russian Math. Surveys 47, 345, 1992, Bauslaugh, B. and Ruskey, F, “Generating Alternating Permutations Lexicographically.” BIT 30, 1726, 1990. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: SpringerVerlag, pp. 110111, 1996. Dijrrie, H. “Andrh’s Deviation of the Secant and Tangent Series .” §lS in 100 Great Problems of Elementary Mathematics: Th eir History and Solutions. New York: Dover, pp, 6469, 1965, Honsberger, R, Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 6975, 1985, Knuth, D. E. and Buckholtz, T. J. “Computation of Tangent, Euler, and Bernoulli Numbers.” Math. Comput. 21, 663Sci. And&,
688,1967. Millar, J.; Sloane, N. J. A.; and Young, N. E. “A New Operation on Sequences: The Boustrophedon Transform.” J. Combin. Th. Ser. A 76, 4454, 1996. Ruskey, F. “Informat ion of Alternating Permutations .” http:// sue . csc . uvic . ca / Y cos / inf / perm / Alternating . html. Sloane, N. J. A. Sequence A000111/M1492 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
see
UZSO
TESTS
ALTERNATING
KNOT,
both
ALTERNATING
LINK,
TORUS
Knot
Theory.”
References Kauffman, Math.
L. “Combinatorics 20, 181200, 1983.
and
A3
The altitudes of a TRIANGLE are the CEVIANS A&& which are PERPENDICULAR to the LEGS A& opposite Ai. They have lengths hi s A& given by hi
sin ai+
= ai+l
22/ ( s
hl
=
s 
= ai+
sin ai+l


al)(s
a~)(5
and
hlh2h3
= 2s~l
 kf1 IE ( 1) ak Or
00
x(l)“ake
(1)
a3)
>
where s is the SEMIPERIMETER interesting FORMULA is
00
k=l
Contemp.
Altitude
a1
Alternating Series A SERIES of the form
KNOT
= A+&.
ai
(2)
Another
(3)
(Johnson 1929, p. 191), where a is the AREA of the TRIANGLE. The three altitudes of any TRIANGLE are CONCURRENT at the ORTHOCENTER IS. This fundamental fact did not appear anywhere in Euclid’s Elements.
k=l
Other
formulas
satisfied
by the altitude
include
see also SERIES 1
G+h+h=
1
2
1 3
1 T
(4)
38
Amicable
Alysoid 1 =
1 h,+r
Tl
I
1 1 +==,
1
T2
T
r3
1
h3
(5)
1
1
2
(6)
Tl
where T is the INRADIUS and ri are the EXRADII son 19.29, p. 189). In addition,
(J 0 hn
For SMOOTH ambiently
HA1 where
9 HH1 l
HH1
= HA2
l
= +(al”
HH2
= HA3
l
HH3
+ az2 + as2)  4R2,
(7) (8
a MAP
is ISOTOPIC
TFF it is
isotopic.
For KNOTS, the equivalence of MANIFOLDS under continuous deformation is independent of the embedding SPACE. KNOTS of opposite CHIRALITY have ambient isotopy, but not REGULAR ISOTOPY. see
HA1
MANIFOLDS,
Numbers
ISOTOPY,
also
REGULAR
References Hirsch, M. W. Diflerential Verlag, 1988.
ISOTOPY
Topology.
New
York:
Springer
R is the CIRCUMRADIUS. Ambiguous An expression is said to be ambiguous (or poorly defined) if its definition does not assign it a unique interpretation or value. An expression which is not ambiguous is said to be WELLDEFINED. see
WELLDEFINED
also
AmbroseKakutani
The points Al, AS, HI, and & (and their permutations with respect to indices) all lie on a CIRCLE, as do the points AS, H3, H, and HI (and their permutaTRIANGLES aA1A2A3 tions with respect to indices). and AAlHzH3 are inversely similar. The triangle HlHzH3 has the minimum PERIMETER of any TRIANGLE inscribed in a given ACUTE TRIANGLE (Johnson 1929, pp. 161165). The PERIMETER of ~lHlHzH3 is 2A/R (Johnson 1929, p. 191). Additional properties involving the FEET of the altitudes are given by Johnson (1929, pp. 261262).
see also CEVIAN, FOOT, ORTHOCENTER, LAR, PERPENDICULAR FOOT
Amenable
Number
A number n which a2, . ..) Uk by either that
can be built up from INTEGERS al, ADDITION or MULTIPLICATION such
f)i i=l
= fiai
=
72.
i=l
The numbers {al, . . . , a,} in the SUM are simply a TITION of n. The first few amenable numbers are
PAR
PERPENDICU
References S. L. Geometry Revisited. Coxeter, H. S. M. and Greitzer, Washington, DC: Math. Assoc, Amer., pp. 9 and 3640, 1967, Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.
2+2=2x2=4 1+2+3=1x2x3=6 1+1+2+4=1x1x2x4=8
1+1+2+2+2 In fact, see
Alysoid see CATENARY Ambient
Theorem
FLOW on a nonatomic PROBABILITY For every ergodic SPACE, there is a MEASURABLE SET intersecting almost every orbit in a discrete set.
Isotopy
An ambient isotopy from an embedding of a MANIFOLD Min Nto another is a HOMOTOPY ofself DIFFEOMORPHISMS (or ISOMORPHISMS, or piecewiselinear transformations, etc.) of N, starting at the IDENTITY MAP, such that the “last” DIFFEOMORPHISM compounded with the first embedding of M is the second embedding of AL 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 along for the ride.” and the embedding is just “coming
also
all COMPOSITE COMPOSITE
References Tamvakis, H. “Problem 463, 1995.
Amicable
=1x1x2x2x2=8. NUMBERS NUMBER, 10454.”
are amenable.
PARTITION, Amer.
Math.
SUM Monthly
102,
Numbers
see AMICABLE PAIR, AMICABLE QUADRUPLE, BLE TRIPLE, MULTIAMICABLE NUMBERS
AMICA
Amicable
Amicable
Pair
Amicable Pair An amicable pair consists of two INTEGERS m, n for which the sum of PROPER DIVISORS (the DIVISORS excluding 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 related, criterion. Symbolically, amicable pairs satisfy s(m)
= n
where s(n) equivalently,
is the RESTRICTED
a(m)
= a(n)
iW and factors regular There m = 0
+ s(n)
FUNCTION
or,
giving
RESTRICTED
s(220)
DIVISOR
l
938304290/1344480478
= 0.697893577..
22
(5)
FUNCTIONS
= x(1,2,4,71,142} = 220.
(7)
The quantity = s(m)
I s(n),
in this case, 220 + 284 = 504, is called
the PAIR
(8)
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 extended to 60. There were 390 known as of 1946 (Scott 1946). There are a total of 236 amicable pairs below lo8 (Cohen 1970), 1427 below lOlo (te Ri?l + 1 ‘cj), 3340 less than 1011 (Moews and MoewF 1”3j, ,I ’ .ess than 2.01 x loll (Moews and Moe: A , : .d 5ir:cjl ress than =2: 3.06 x 1011 (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: CI?6j, (63020, 76084), ... (Sloane’s A002025 and A002046). AFT exhaustive tabulation is maintained by D. Moe*,,. Let an amicable pair be denoted is called a regular amicable
(12)
= 0.9998582519..
..
(13)
te Riele (1986) also found 37 pairs of amicable pairs having the same PAIR SUM. The first such pair is (609928, 686072) and (643336, 652664), which has the PAIR SUM a(m)
(6)
= o(n)
.
and
= ~{1,2,4,5,10,11,20,22,44,55,110}
44
(11)
(4
= 284 ~(284)
m
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
4000783984/4001351168 11.5.22
284 = 71
(10)
N are SQUAREFREE, then the number of PRIME of A& and N are i and j. Pairs which are not are called irregular or exotic (te Riele 1986). are no regular pairs of type (1, j) for j > 1. If (mod 6) and n = a(m)
(3)
= m / n,
where c(n) is the D~WSOR FUNCTION. The smallest amicable pair is (220, 284) which has factorizations 220=
Dr
(2)
DIVISOR
= s(m)
XI
where g = GCD(m,n) is the GREATEST COMMON VISOR, GCD(g, 111) = GCD(g, N) = 1,
(1)
s(n) = m,
Pair
with m < n. pair of type (i, j) if
h 4
= o(n)
= m + n = 1,296,OOO.
(14)
te Riele (1986) found no amicable ntuples having the Moews and same PAIR SUM for n > 2. However, Moews found a triple in 1993, and te Riele found a quadruple in 1995. In November 1997, a quintuple and sextuple were discovered. The sextuple is (1953433861918, 2216492794082), (1968039941816, (1981957651366, 2187969004634), 2201886714184), (1993501042130, 2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962), all having PAIR SUM 4169926656000. Amazingly, the sextuple 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 Nl=CM[(P+Q)p89 Nz = CQ[(P
 M)P8’
13
(15)
 11,
(16)
where
c = plpg
(17)
M=
287155430510003638403359267
(18)
P=
574451143340278962374313859
(19)
Q = 136272576607912041393307632916794623. (20)
P, Q1 (P + Q)pBg 
1, and (P
M)Pgg
 1 are PRIME.
Amicable
40
Pomerance
(1981)
[amicable
Pair
Amicable
has proved
numbers
that
2 n] < ne[‘nC”)l
for large enough n (Guy bound has been proven.
1994).
No
l/3
(21)
nonfinite
lower
see also AMICABLE QUADRUPLE, AMICABLE TRIPLE, AUGMENTED AMICABLE PAIR,BREEDER,CROWD, EuLER'S RULE, FRIENDLY PAIR, MULTIAMICABLE NUMBERS, PAIR SUM, QUASIAMICABLE PAIR, SOCIABLE NUMBERS,
UNITARY
AMICABLE
PAIR
References Alanen, J.; Ore, 0.; and Stemple, J. “Systematic Computations on Amicable Numbers.” Math. Comput. 21, 242245, 1967. Battiato, S. and Borho, W. “Are there Odd Amicable Numbers not Divisible by Three?” Math. Comput. 50, 633637, 1988. R. Item 62 in Beeler, M.; Gosper, R. W.; and Schroeppel, HA KIMEIM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972. Borho, W. and Hoffmann, H. “Breeding Amicable Numbers in Abundance.” Math. Comput. 46, 281293, 1986. Bratley, P.; Lunnon, F.; and McKay, J. “Amicable Numbers and Their Distribution.” Math. Comput. 24, 431432, 1970. Cohen, H. “On A micable and Sociable Numbers.” Math. Comput.
Costello,
24,
423429,
Math.
P. “Amicable
Comput.
30,
M. “Perfect,
ematical Illusions Scientific
Magic Show: and Other American.
1978. Guy, R. K. “Amicable in
Number
Theory,
Ore, 0. Number Theory and Its History. New York: Dover, pp. 96100, 1988. Pedersen, J. M. “Known Amicable Pairs.” http://uuw. vejlehs.dk/staff/jmp/aliquot/knwnap.htm. of Amicable Numbers.” Pomerance, C. “On the Distribution J. reine angew. Math. 293/294, 217222, 1977. Pomerance, C. “On the Distribution of Amicable Numbers, II.” J. reine angew, Math. 325, 182188, 1981. Scott, E. B. E. “Amicable Numbers.” Scripta Math. ‘12, 6172, 1946. Sloane, N. J. A. Sequences AOO2025/M5414 and AOO2046/ M5435 in “An OnLine Version of the Encyclopedia of Integer Sequences.” te Riele, H. J. J. “On Generating New Amicable Pairs from Given Amicable Pairs.” Math. Comput. 42, 219223, 1984. te Riele, H. J. J. “Computation of All the Amicable Pairs Below lOlo ." Math. Comput. 47, 361368 and S9S35, 1986. te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.; and Lee, E, J. “Table of Amicable Pairs Between lOlO and 105?” Centrum voor Wiskunde en Informatica, Note NMN86O3. Amsterdam: Stichting Math. Centrum, 1986. te Riele, H. J. J. “A New Method for Finding Amicable Pairs.” In Mathematics of Computation 19431993: A Half Century of Computational Mathematics (Vancouver, BC, August 913, 1993) (Ed. W. Gautschi). Providence, RI: Amer. Math. Sot., pp. 577581, 1994. @ Weisstein, E. W. “Sociable and Amicable Numbers.” http://www.astro.virginia.edu/eww6n/math/ notebooks/Sociable.m.
1970.
Pairs of Euler’s First Form.” J. Rec. Math. IO, 183189, 19771978. Costello, P. “Amicable Pairs of the Form (;,I) .” Math. Comput. 56, 859865, 1991. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3850, 1952. Erdiis, P, “On A micable Numbers .” Publ. Math. Debrecen 4, 108111, 19551956. Erd&, P. “On Asymptotic Properties of Aliquot Sequences.” Gardner,
lliple
641645,
Sociable.”
More Puzzles, Mathematical
New Numbers.” 2nd
ed.
York:
Quadruple quadruple
as a QUADRUPLE (a, b, c, d) such
u(a) = a(b) = u(c) = u(d) = a + b + c + d, where a(n)
is the DIVISOR FUNCTION.
References
1976.
Amicable,
Amicable An amicable that
Ch. 12 in Math
Games, Diversions, SleightofMind from
Vintage,
pp. 160171,
§B4 in Unsolved Problems New York: SpringerVerlag,
ppm 5559, 1994. Lee, E. 5. “Amicable Numbers and the Bilinear Diophantine Equation.” Math. Comput. 22, 181197, 1968. Lee, E. J. “On Divisibility of the Sums of Even Amicable Pairs.” Math. Comput. 23, 545548, 1969. Lee, E. J. and Madachy, 5. S. “The History and Discovery of Amicable Numbers, I.” J. Rec. Math. 5, 7793, 1972. Lee, E. J. and Madachy, J. S. “The History and Discovery of Amicable Numbers, II.” J. Rec. Math. 5, 153173, 1972, Lee, E. J. and Madachy, J. S. “The History and Discovery of Amicable Numbers, III.” J. Rec. Math. 5, 231249, 1972. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, pp. 145 and 155156, 1979. Moews, D. and Moews, P. C. “A Search for Aliquot Cycles and Amicable Pairs .” Math. Comput. 61, 935938, 1993. Moews, D. and Moews, P, C. “A List of Amicable Pairs Below 2.01 x loll.” Rev. Jan. 8, 1993. http://xraysgi.ims. ucoun.edu:8080/amicable.txt. Moews, D. and Moews, P. C. “A List of the First 5001 Amicable Pairs .” Rev. Jan. 7, 1996. http://xraysgi.ims. uconn.edu:8080/amicableZ.txt.
GUY, R+ K. Unsolved Problems New York: SpringerVerlag,
in
Number
Theory,
2nd
ed.
p. 59, 1994.
Amicable Dickson
Triple (1913, 1952) defined an amicable triple TRIPLE of three numbers (Z,m,n) such that
to be a
s(l) = m + n s(m)=lfn s(n) = I+
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 )=103340640+124015008
= 227355648
)=
123228768+124015008
= 24724377
)=
123228768+10334064
= 226569408,
Amortization
Amplitude
and (1945330728960, for which
2324196638720,
2615631953920),
Amphichiral An object is amphichiral (also called REFLEXIBLE) if it is superposable with its MIRROR IMAGE (i.e., its image in a plane mirror).
= 2324196638720+2615631953920
s(1945330728960)
see also AMPIIICHIRAL HANDEDNESS, MIRROR
= 4939828592640 s(2324196638720)
= 1945330728960
+ 2615631953920
= 4560962682880
s(2615631953920)
= 1945330728960
(22325.
is the DIVISOR 11, 25327, 223271).
see
AMICABLE
also
an amicable
08012, 10079,
= a(b) = o(c) =a++++,
where o(n)
An example
FUNCTION.
PAIR,
AMICABLE
is
QUADRUPLE
Dickson,
08017,
08018,
10081,
10 088,
L. E. “Amicable Number Triples.” Amer. Math. 20, 8492, 1913. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 50, 1952. Unsolved
Problems
in
Number
Theory,
2nd
ed.
New York: SpringerVerlag, p. 59, 1994. Madachy, J. S. Madachy’s Mathematical Recreations. New York: Dover, p. 156, 1979. Mason, T. E. “On Amicable Numbers and Their Generalizations.” Amer. Math. Monthly 28, 195200, 1921. $& Weisstein, E. W. “Sociable and Amicable Numbers.” http://www.astro.virginia.edu/eww6n/math/ notebooks/Sociable.m.
Amortization The payment lar payments. Ampersand
of a debt plus accrued
INTEREST
10017J0033, 10099,
10037,
10109,
by regu
10045, and
and b the
then the KNOT not amphichiral
corresponding to the closed BRAID (Jones 1985) g
see also AMPHICHIRAL, KNOT, MIRROR IMAGE
BRAID
GROUP,
Burde, G. and Zieschang, pp. 311319, 1985. Jones, V. “A Polynomial mann Algebras.” Bull. 1985. Jones, V. “Hecke Algebra and Link Polynomials ,”
INVERTIBLE
H. Knots.
for Knots
Invariant Amer.
Berlin:
Math.
Sot.
Representations Ann.
de Gruyter,
via von Neu12, 103111,
of Braid
126, 335388,
Math.
2 (Y  x2)(x  1)(2x  3) = 4(x2 + y2  2x)2.
is a JACOBI ELLIPTIC FUNCTION. The term is also used to refer to the maximum offset from its baseline level.
References Abramowitz,
References 3rd
dn IL du, s
see UZSO ARGUMENT (ELLIPTIC INTEGRAL), CHARACTERISTIC (ELLIPTIC INTEGRAL), DELTA AMPLITUDE, ELLIPTIC FUNCTION, ELLIPTIC INTEGRAL, JACOBI ELLIPTIC FUNCTIONS, MODULAR ANGLE, MODULUS (ELLIPTIC INTEGRAL), NOME, PARAMETER
equation
A. Mathematical Models, Tarquin Pub., p. 72, 1989.
Groups 1987.
and EL
4 used in ELLIPTIC FUNCTIONS INTEGRALS, which can be defined by
dn(u) “amplitude” of a function
Cundy, H. and Rollett, Stradbroke, England:
b is
References
where
Cartesian
B,.
 n + 1 > 0,
b+  3b
LIPTIC
with
10123
GROUP
Amplitude The variable
Curve
CURVE
10118,
If
4 =amu=
The PLANE
10043,
10115,
(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.
Monthly
R. K.
DISYMMETRIC,
Let b+ be the SUM of POSITIVE exponents, SUM of NEGATIVE exponents in the BRAID
References
Guy,
CHIRAL,
Knot knot is a KNOT which is capable of bedeformed into its own MIRROR IMAGE. knots having ten or fewer crossings are 04001 (FIGUREOFEIGHT KNOT), 06003, 08003, 08009,
= 4269527367680.
da>
KNOT, IMAGE
Amphichiral An amphichiral ing continuously The amphichiral
+ 2324196638720
A second definition (Guy 1994) defines triple as a TRIPLE (a, b, c) such that
41
ed.
of Mathematical Mathematical
M.
and
p. 590, 1972, Fischer, G. (Ed.). &e/Mathematical
Braunschweig,
Stegun,
Functions Tables, 9th
Plate Models,
Germany:
C.
A.
(Eds.).
with Formulas, printing. New
Handbook Graphs, and
York:
132 in Mathematische Bildband/Photograph
Vieweg,
p. 129, 1986.
Dover, ModVolume.
Anallagmatic
42
Anallagmatic
Anchor
Curve
Curve
A curve which is invariant under’ INVERSION Examples include the CARDIOID, CARTESIAN OVALS, CASSINI OVALS, LIMA~ON, STROPHOID, and MACLAURIN TRISECTRRIX.
Anallagmatic see
Pavement
HADAMARD
MATRIX
Analogy Inference by noting
of the TRUTH of an unknown result obtained its similarity to a result already known to be TRUE. In the hands of a skilled mathematician, analogy can be a very powerful tool for suggesting new and extending old results. However, subtleties can render results obtained by analogy incorrect, so rigorous PROOF is still needed. see
also
INDUCTION
Analysis The
study
of how continuous mathematical structures vary around the NEIGHBORHOOD of a point on a SURFACE. Analysis includes CALCULUS, DIFFERENTIAL EQUATIONS, etc.
(FUNCTIONS)
see UZSO ANALYSIS YSIS, FUNCTIONAL
SITUS,
CALCULUS,
COMPLEX
ANAL
ANALYSIS, NONSTANDARD ANALYSIS, REAL ANALYSIS
Analytic Function A FUNCTION in the COMPLEX NUMBERS C is analytic on a region R if it is COMPLEX DIFFERENTIABLE at every point in R. The terms HOLOMORPHIC FUNCTION and REGULAR FUNCTION are sometimes used interchangeably with “analytic function.” If a FUNCTION is analytic, it is infinitely DIFFERENTIABLE. see UZSOBERGMAN SPACE$OMPLEX DIFFERENTIABLE, DIFFERENTIABLE, PSEUDOANALYTIC FUNCTION, SEMIANALYTIC, SUBANALYTIC References P. M, and Feshbach, H. “Analytic Functions.” of Theoretical Physics, Part I. New McGrawHill, pp. 356374, 1953.
Morse,
in Methods
Analytic
Geometry
The study of the GEOMETRY of figures by algebraic represent ation and manipulation of equations describing their positions, configurations, and separations. Analytic geometry is also called COORDINATE GEOMETRY since the objects are described as ntuples of points (where n = 2 in the PLANE and 3 in SPACE) in some
COORDINATE SYSTEM. see UZSOARGAND DIAGRAM,~ARTESIAN COORDINATES, COMPLEX PLANE, GEOMETRY, PLANE, QUADRANT, SPACE, XAXIS, YAXIS, ZAXIS References Courant,
Keierences Bottazzini, U. The “Higher Calculus”: A History of Real and Complex Analysis from Euler to Weierstrafi. New York: SpringerVerlag, 1986. Bressoud, D. M. A Radical Approach to Real Analysis. Washington, DC: Math. Assoc. Amer., 1994. Ehrlich, P. Real Numbers, Generalization of the Reals, & Theories of Continua. Norwell, MA: Kluwer, 1994. Hairer, E. and Wanner, G. Analysis by Its History. New York: SpringerVerlag, 1996. Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 1988. Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
Analysis
Sit us
An archaic
name
Analytic
Continuation
$4.2 York:
H. “Remarks R. and Robbins, try.” $2.3 in What is Mathematics?: proach to Ideas and Methods, 2nd Oxford University Press, pp+ 7277,
Analytic
on Analytic
GeomeAn Elementary Aped. Oxford, England: 1996+
Set
A DEFINABLE SET, also called
a SOUSLIN SET. see also COANALYTIC SET, SOUSLIN SET
Anarboricity Given a GRAPH G, the anarboricity number
of linedisjoint UNION is G.
nonacyclic
is the maximum
SUBGRAPHS whose
see also ARBORICITY Anchor
for TOPOLOGY.
A process of extending FUNCTION is defined.
the region
Ananchoristhe BUNDLE MAP pfroma VECTOR BUNDLE A to the TANGENT BUNDLE Wsatisfying in which
a COMPLEX
~~~~ESOMONODROMYTHEOREM,PERMANENCEOF ALGEBRAIC FORM, PERMANENCE OF MATHEMATICAL RELATIONS PRINCIPLE References A&en, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 378380, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 389390 and 392398, 1953.
1. W>,P(Y)l = P~w7) and 2. [x7 WI = WL yl + (P(X) ’ W? where X and Y are smooth sections of A, 4 is a smooth function of B, and the bracket is the “Jacobilie bracket” ofa VECTOR FIELD.
see also LIE ALGEBROID References Weinstein, A. Symmetry.”
“Groupoids: Not. Amer.
Unifying Internal and Math. Sot. 43, 744752,
External 1996.
Anchor
AndrewsSchur
Ring
Anchor Ring An archaic name References Eisenhart,
L. and
for the TORUS.
P.
A Treatise on the Difierential Geometry of Surfaces. New York: Dover, p. 314, 1960. Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, Curves
p. 239, 1977. Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles & Rigid Bodies, 4th ed. Cambridge, England: Cambridge University Press, p. 21, 1959.
And A term (PREDICATE) in LOGIC which yields TRUE ifone or more conditions are TRUE, and FALSE if any condition is FALSE. A AND B is denoted A&B, A /\ B, or simply AB. The BINARY AND operator has the following TRUTH TABLE: A
B
And&, D. “Solution directe du probleme M. Bertrand.” Comptes Rendus Acad. Sci.
436437,
is
n
Andrew’s The function
105,
Dordrecht, Numbers,
Nether
Their
Gener
13, 6475, 1991.
Intel.
irt Muthematica. 1991.
Read
Sine
sin (f) 1 0,
occurs in estimation
References Press, W. H.; Flannery, ling, W. T. Numerical Scientific
1~51< CT 121 > CT
theory.
Andrews
OPERATORJNTERSECTION, TABLE, XOR
NOT,
OR,
B. P.; Teukolsky, Recipes 2nd ed.
S. A.; and Vetter
in FORTRAN:
Cambridge, Press, p. 697, 1992.
Computing,
bridge University
Ak*
Two binary numbers can have the operation AND performed bitwise with 1 representing TRUE and 0 FALSE. Some computer languages denote this operation on A, B, and c as A&&B&&C orlogand(A,B,C). see also BINARY PREDICATEJ'RUTH
par
Paris
1887.
Comtet, L. Advanced Combinatorics. lands: Reidel, p. 22, 1974. Hilton, P. and Pederson, J. “Catalan alization, and Their Uses.” Math. Vardi, I. Computational Recreations ing, MA: AddisonWesley, p. 185,
which
r&olu
see also SINE
A PRODUCT of ANDs (the AND of 72 conditions) called a CONJUNCTION, and is denoted
A
References
*c z>=
F F F T
43
Andr@s Reflection Method A technique used by Andre (1887) to provide an elegant solution to the BALLOT PROBLEM (Hilton and Pederson 1991).
AAB
FF FT TF TT
Identity
The
England:
Art
of
Cam
Cube
see SEMIPERFECT MAGIC CUBE Andrew+Curtis Link The LINK of 2spheres in Iw4 obtained by SPINNING intertwined arcs. The link consists of a knotted 2sphere and a SPUN TREFOIL KNOT.
see UZSO SPUN KNOT, TREFOIL KNOT AndersonDarling Statistic A statistic defined to improve the KOLMOGOROVSMIRNOV TEST in the TAIL of a distribution. see
KOLMOGOROVSMIRNOV
&O
TEST,
References Rolfsen, D. Knots and Links. Perish Press, p. 94, 1976.
Wilmington,
DE: Publish
KUIPER
STATISTIC
Andrew+Schur
Identity
References Press, W. H.; Flannery, ling, W. T. Numerical Scientific
bridge
B. P+; Teukolsky,
Computing,
University
Press,
S. A.; and Vetter
n
Recipes in FORTRAN: The Art of 2nd ed. Cambridge, England: Cam
p. 621, 1992.
k2+ak
8.x
Q E
[
see
also
ALTERNATING
PERMUTATION
PER
k
k 03
Andr& Problem The determination of the number of ALTERNATING MUTATIONS having elements (1, 2, , , . , n}

10k2+(4al)k
c k=m
Q
+
a
I 2n + 2a + 2 n  5k
I
[IOk + 2a + 21 [zn + 2a + 21
or
Andrica’s
44
Conjecture
Anger Function
where
[x] is a GAUSSIAN POLYNOMIAL. It is a POLYidentity for a = 0, 1 which implies the ROGERSRAMANUJAN IDENTITIES by taking n + 00 and applying the JACOBI TRIPLE PRODUCT identity. A variant of this equation is NOMIAL
n
nkk+a
k2f2ak
holds, where the discrete function A, is plotted above. The largest value among the first 1000 PRIMES is for n = 4, giving J11  fi z 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.
c
1451 Ix

I
30 15k2+(6a+l)k 4
25
 l(n+za+z)/sJ
[lOk+2a+2] x [2n+2a+2] where
the symbol FUNCTION (Paule tity is
'
15
(2)
10
1x1 in the SUM limits is the FLOOR 1994). The RECIPROCAL of the iden
5
100
200
300
400
500
An bears a strong
resemblance to the PRIME DIFFERENCE FUNCTION, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, . . . (Sloane’s A001223).
00 1 
%+')(I
rI
j=O
('


q 20j+4a+4)(
1 
q20j4a+16)
(3)
'
for a = 0, 1 (Paule
1994).
For q = 1, (1) and (2) become
References
n+k+a
nk
>

(4)  l(n+2a+2)/5J
References Andrews, G. E. “A Polynomial Identity which Implies the Scripta Math. 28, 297RogersRamanujan Identities.” 305, 1970. Paule, P. “Short and Easy Computer Proofs of the RogersRamanujan Identities and of Identities of Similar Type.” Electronic J. Combinatorics 1, RlO, 19, 1994. http: // uww.combinatorics.org/Volumel/volurnel.html#RlO.
Andrica’s
see also BROCARD’S CONJECTURE, GOOD PRIME, FORTUNATE PRIME, P~LYA CONJECTURE, PRIME DIFFERENCE FUNCTION, TWIN PEAKS Golomb, S. W+ “Problem E2506: Limits of Differences of Amer. Math. Monthly 83, 6061, 1976. Square Roots.” Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p, 21, 1994. Rivera, C. “Problems & Puzzles (Conjectures): Andrica’s Conjecture.” http://www.sci.net.mx/crivera/ ppp/conj ,008. htm. Sloane, N. J. A. Sequence A001223/M0296 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Anger Function A generalization of the FIRST KIND defined by
BESSEL
FUNCTION
OF
THE
7r cos(v0  z sin 0) d& s0
Conjecture
If v is an INTEGER n, then Jn(z) = Jn(Z), where Jn(Z) is a BESSEL FUNCTION OF THE FIRST KIND. Anger’s original function had an upper limit of 275 but the current NOTATION was standardized by Watson (1966).
0.6 0.5 0.4
see
also BESSEL FUNCTION, MODIFIED STRUVE FUNCTION, PARABOLIC CYLINDER FUNCTION, STRUVE FUNCTION, WEBER FUNCTIONS
0.3 0.2
100
Andrica’s NUMBER,
References
P
0.1
200
conjecture states that, the INEQUALITY
300
400
for pn the nth
500
PRIME
Abramowitz, M. and Stegun, C. A. (Eds.). “Anger and Weber Functioris.” $12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp* 498499, 1972.
Watson,
G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press,
1966.
Angle Bracket
Angle Angle
Angle
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 8 between them. Angles are usually measured in DEGREES (denoted “), RADIANS (denoted rad, or without a unit), or sometimes GRADIANS (denoted grad).
45
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.
One full rotation in these three measures corresponds to 360”, 2~ rad, or 400 grad. IIalf a full ROTATION is called of a full rotation a STRAIGHT ANGLE, and a QUARTER 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
d
da: sina: = cost does not require the insertion of multiplicative constants like r/180. GRADIANS are sometimes used in surveying (they have the nice property that a RIGHT ANGLE is exactly 100 GRADIANS), but are encountered infrequently, if at all, in mathematics. The
concept
can be generalized from the of a SPHERE subtended 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 caXled a PROTRACTOR. A COMPASS can also be used to draw circular ARCS of some angular extent.
see dso
ACUTE ANGLE, ARC MINUTE, ARC SECOND, CENTRAL ANGLE, COMPLEMENTARY ANGLE,DEGREE, DIHEDRAL ANGLE,DIRECTED ANGLE,EULERANGLES, GRADIAN, HORN ANGLE, INSCRIBED ANGLE, OBLIQUE ANGLE, OBTUSE ANGLE, PERIGON, PROTRACTOR, RADIAN, RIGHT ANGLE, SOLID ANGLE, STERADIAN, STRAIGHTANGLE,SUBTEND,SUPPLEMENTARYANGLE, VERTEX ANGLE n
References Dixon, R. Mathographics.
1991.
TRIANGLE
A2
T3
of the bisector of ANGLE Al nA1A2A3 is given by
a1
2 t1
=
a2a3
[
l
(a2
in the above
2
+a3)2
I
'
where ti E AiTi and ai s Aj Ak. The angle bisectors meet at the INCENTER 1, which has TRILINEAR COORDINATES
l&l.
BISECTOR THEOREM, CYCLIC QUADRANGLE, EXTERIOR ANGLE BISECTOR, ISODYNAMIC POINTS, ORTHOCENTRIC SYSTEM, STEINERLEHMUS THEOREM,TRISECTION see
ANGLE
also
References Coxeter, H. S. M. and Greitzer,
Washington, Dixon, Mackay,
DC:
Math.
R. Mathogruphics. J. S. “Properties
sectors
Proc.
of a Triangle.”
S. L. Geometry
Revisited.
Assoc. Amer., pp. 910, 1967. New York: Dover, p. 19, 1991. Concerned with the Angular Bi
Edinburgh
Math.
Sot.
13,
37102,1895,
of an angle
CIRCLE to the SPHERE. The fraction

Al length
The
New
York:
Dover,
pp. 99100,
Angle
Bisector
Theorem
The ANGLE BISECTOR ofan ANGLE ina TRIANGLE divides the opposite side in the same RATIO as the sides adjacent to the ANGLE.
Angle
Bracket
The combination of a bracket) which represents functions or vectors,
(VI4
BRA and the INNER
KET (bra+ket = PRODUCT of two
=v*w.
By itself, the BRA is a COVARIANT ~VECTOR, and the KET isa COVARIANT ONEFORM. Thesetermsarecommonly
used in quantum
see &OBRA,DIFFERENTIAL
mechanics.
~FORM, KET,ONEFORM
Annulus
Angle of Parallelism
46 Angle
Angular
of Parallelism P
A
The
Velocity
angular
velocity
D
WEZ=.
B
Given a point P and a LINE AB, draw the PERPENDICULAR 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 parallelism for perpendicular distance z, and is given by
see also
ANGULAR
This
ACCELERATION,
ANGULAR
DIS
permits
no n
Tiling tiling
is a tiling
which
ISOHEDRAL TILING with n < k.
LOBACHEVSKY'S FORMWLA. GEOMETRY, LOBACHEVSKY'S
is knownas
see also
v r
Anharmonic Ratio see CROSSRATIO
A Lanisohedral
= 2 tan‘(P).
d0, dt
DERIVATIVE of the & PERPENDICU
TANCE
Anisohedral II(x)
w is the time
ANGULAR DISTANCE 8 with direction LAR to the plane of angular motion,
C
Conjecture
HYPERBOLIC
FORMULA References Manning, H. P. Introductory NonEuclidean York: Dover, pp. 3132 and 58, 1963.
Geometry.
New
References J. “Is There a kAnisohedral Amer. Math. Monthly 100, 585588, Klee, V. and Wagon, S. Old and New
Tile
BerElund,
Plane
Geometry
Math.
Assoc.
and
Number
Amer.,
for k 2 5?”
1993. Unsolved
Problems
Washington,
Theory.
in
DC:
1991.
Annihilator Angle FIXsection see TRISECTION Angular
Acceleration
The
angular acceleration CII is defined as the time RIVATIVE of the ANGULAR VELOCITY w, dw =dt
a= see also ACCELERATION,
d2B, @z=;.
DE
region
in common
DISTANCE,
A annulus
ANGUAn interesting
identity
S=27v
x
ANGLES Ai
C2
Ai.
@ the AREA of the shaded
also DESCARTES TOTAL ANGULAR
A
region
A is given by
DEFECT, JUMP
ANGLE
Angular
In the figure,
Cl
The DIFFERENCE between the SUM offace at a VERTEX ofa POLYHEDRON and 2n,
see
is
= n(b2  a”). is as follows.
Defect
CIRCLES of
to two concentric
RADII a and b. The AREA of an annulus
LAR VELOCITY Angular
Annulus The
a
ANGULAR
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.
A=C1+C2.
Distance
The angular distance traveled around a CIRCLE is the number of RADIANS the path subtends,
tk
c G2R=
e . T
see also ANGULAR ACCELERATION, ANGULAR VELOCITY
see also CHORD, CIRCLE, CONCENTRIC CIRCLES,LUNE (PLANE),~PHERICAL SHELL References Pappas, T. “The Amazing Trick.” The Joy San Carlos, CA: Wide World Publ./Tetra,
Annulus ~~~ANNULUS
Conjecture THEOREM
of Mathematics.
p. 69, 1989.
Annulus
Theorem
Annulus
Theorem
Anosov bicollared knots in IF1 or open region between them. closed annulus s” x [O, 11. the theorem was proved by
Kirby, R. C. “Stable Homeomorphisms and jecture.” Ann. Math. 89, 575582, 1969. Rolfsen, D. Knots and Links. Wilmington, Perish Press, p. 38, 1976.
the
Annulus DE:
Con
Publish
or
Cancellation
The simplification of a FRACTION a/b which gives a correct answer by “canceling” DIGITS of a and b. There are only four such cases for NUMERATOR and DENOMINATORS of two DIGITS in base 10: 64/16 = 4/l = 4, 98149 = 814 = 2, 95119 = 511 = 5, and 65126 = 512 (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 DWISOR 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 3241134 = 24. Boas gives a table of solutions for b < 39. The number of solutions is EVEN unless b is an EVEN SQUARE.
3r
b
4
1
6 8 9
2 2 2
10
4 4 2 6
12 14 15 16
7
18
4
20
4
21
10
22
6
24
6
A term in SOCIAL CHOICE of a result under permutation
see also DUAL Anosov
References
26 27 28 30 32 34 35 36 38 39
N 4 6 10 6 4 6 6 21 2 6
MONOTONIC
VOTING
Automorphism
Anosov
Diffeomorphism
An Anosov diffeomorphism is a C1 DIFFEOMORPHISM 4 suchthatthe MANIFOLD ik! is HYPERBOLIC withrespect to 4. Very few classes of Anosov diffeomorphisms are known. The best known is ARNOLD'S CAT MAP.
A HYPERBOLIC linear
map Iw” + Ik” with INTEGER in the transformation MATRIX and DETERMINANT *l is an Anosov diffeomorphism of the ~TORUS. Not every MANIFOLD admits an Anosov diffeomorphism. Anosov diffeomorphisms are EXPANSIVE, and there are no Anosov diffeomorphisms on the CIRCLE.
entries
It is conjectured that if 4 : M + M is an Anosov diffeomorphism on a COMPACT RIEMANNIAN MANIFOLD and the NONWANDERING SET O(4) of 6 is AI, then $ is TOPOLOGICALLY CONJUGATE to a FINITETOONE FACTOR of an ANOSOV AUTOMORPHISM of a NILMANIFOLD. It has been proved that any Anosov diffeomorphismonthe ~TORUS is TOPOLOGICALLY CONJUGATE to an ANOSOV AUTOMORPHISM, and also that Anosov diffeomorphisms are C1 STRUCTURALLY STABLE.
“Geodesic ifolds with Negative A. M. S. 1969. Smale, S. “Different iable Math. Sm. 73, 747817,
in
Number
Anosov A FLOW
Flow on Curvature.” Dynamical
Closed
Riemannian Proc. Steklov
Systems.”
Bull.
ManInst., Amer.
1967.
Flow
defined analogously to the ANOSOV DIFFEOMORPHISM, except that instead of splitting the TANGENT BUNDLE into two invariant subBUNDLES, they are split into three (one exponentially contracting, one expanding, and one which is ldimensional and tangential to the flow direction).
see also Number
invariance
of voters.
A HYPERBOLIC linear map Iw" 3 IIB" with INTEGER entries in the transformation MATRIX and DETERMINANT *l is an ANOSOV DIFFEOMORPHISM of the ~TORUS, called an Anosov automorphism (or HYPERBOLIC AUTOMORPHISM). Here, the term automorphism is used in the GROUP THEORY sense.
References Anosov, D, V.
MathematDC: Math.
see BENFORD'S LAW
VOTING,
THEORY meaning
see UZSO ANOSOV AUTOMORPHISM, AXIOM A DIFFEOMORPHISM,DYNAMICAL SYSTEM
REDUCED
Anomalous
47
Anonymous
Let K,” and K,” be disjoint s n+l and let U denote the Then the closure of U is a Except for the case n = 3, Kirby (1969).
Anomalous
Flow
DYNAMICAL SYSTEM
48
Anosov
Anosov
Map
An ticlas tic
Map Anthropomorphic Polygon A SIMPLE POLYGON with precisely
AN~S~V DIFFEOMORPHISM.
Animportantexampleofa
two EARS and one
MOUTH. [z:]
= [f
:]
References
[:::
, G. “Anthropomorphic
Toussaint
where zn+l,yn+l
are computed
see U~SO ARNOLD'S
mod
Anthyphairetic
ANOVA
References
“Analysis of Variance .” A STATISTICAL TEST for heterogeneity of MEANS by analysis of group VARIANCES. To apply the test, assume random sampling of a variate y with equal VARIANCES, independent errors, and a NORMAL DISTRIBUTION. Let nbethenumberof REPLICATES (sets of identical observations) within each of K FACTOR LEVELS (treatment groups), and yij 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.
Fowler,
the sum of square terms k
SST E )\
n
)‘(y;j
 5)”
2
(ce,
yij i=l

Amer.
Math.
1.
CAT MAP
An archaic
Now define
Polygons.”
122, 3135, 1991.
Monthly
(1)
Yij)a
c;=1
(2 >
Kn
Ratio
word for a CONTINUED FRACTION.
D. H. The Mathematics New York:
of Plato’s
Oxford
Reconstruction.
Antiautomorphism If a MAP f:G+
G’from satisfies f(ab) = f(cz)f(b) to be an antiautomorphism.
Academy:
University
A New
Press, 1987.
a GROUP Gto a GROUP G’ for all a, b e G, then f is said
see UZSO AUTOMORPHISM Anticevian
Triangle
triangle is Given a center c~ : p : y, the anticevian defined as the TRIANGLE with VERTICES a : p : y, QI : 0 : y, and CY: p : 7. If A’B’C’ is the CEVIAN TRIANGLE of X and A”B”C” is an anticevian triangle,thenXand A” are HARMONIC CONJUGATE POINTS with respect to A and A’.
see also CEVIAN TRIANGLE References
j=l
Points and Central Lines in the Kimberling, C. “Central Plane of a Triangle.” Math. Mug. 67, 163487, 1994.
> Antichain k
n
SSE E x y,(yij i=l j=l
 gi)’
(4
= SST  SSA, which are the squares. Here,
total, & is FACTOR LEVEL i, and of means). Compute obtaining the PVALUE FRATIO of the mean
(5) treatment, and error sums of the mean of observations within 5 is the ‘Lgroup” mean (i.e., mean the entries in the following table, corresponding to the calculated squared values
Let P be a finite PARTIALLY ORDERED SET. An antichain 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 themaximum CARDINALITY ofan ANTICHAIN in P. For a PARTIAL ORDER, the size of the longest ANTICHAIN is called the WIDTH.
see also CHAIN, DILWORTH'S LEMMA, PARTIALLY ORDERED SET,WIDTH (PARTIAL ORDER) References Sloane, N. J. A. Sequence A006826/M2469 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Anticlastic Category
SS
OFreedom
Mean
treatment
SSA
Kl
MSA=s
error
SSE
K(n
total
SST
Kn  1
 1)
Squared
FRatio E
MSE=
&
MST
&
If the PVALUE is small, reject the NULL HYPOTHESIS that all MEANS are the same for the different groups. see also
FACTOR
LEVEL,
REPLICATE, VARIANCE
GAUSSIAN CURVATURE K is everywhere NEGATIVE, a SURFACE is called anticlastic and is saddleshaped. A SURFACE on which K is everywhere POSITIVE is called SYNCLASTIC. A point at which the GAGSSIAN CURVATURE is NEGATIVE is called a HYPERBOLIC POINT. see also ELLIPTIC POINT, GAUSSIAN QUADRATURE, HYPERBOLIC POINT, PARABOLIC POINT, PLANAR POINT, SYNCLASTIC Whenthe
Anticommu Anticommutative An OPERATOR anticommutative. see
also
Antimagic
tative
* for which
a * b = b * a is said to be
COMMUTATIVE
Anticommutator For OPERATORS A and fi, the anticommutator
Ant
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.
2. The PRODUCT of distances CENTER to two antihomologous
icomplementary
49
1. Two pairs of antihomologous points form inversely similar triangles with the HOMOTHETIC CENTER.
is defined
bY
see also COMMUTATORJORDAN
Graph
from a HOMOTHETIC points is a constant.
3. Any two pairs of points which are antihomologous with respect to a SIMILITUDE CENTER lie on a CIRCLE.
ALGEBRA
Triangle
4. The tangents to two CIRCLES at antihomologous points make equal ANGLES with the LINE through the points. see also H0~0~0G0u
SIMILIT
s POINTS,
CENTER,
HUMOTHETIC
UDE CENTER
References R. A. Modern
Johnson, on
the
Geometry
MA: Houghton
A TRIANGLE AA’B’C’ which has a given TRIANGLE AABC as its MEDIAL TRIANGLE. The TRILINEAR CoORDINATES of the anticomplementary triangle are
also
MEDIAL
Operator OPERATOR
A[fl(X)
Antiderivative
where
see INTEGRATION
see
Ant
Antilogarithm The INVERSE such that
= Afl(X>
Points B
is the COMPLEX
C*
UZSO
LINEAR
+
following
two
Afd4
The antilogarithm see
Given LAXB + LAYB = 71”RADIANS in the above figure, then X and Y are said to be antigonal points with respect to A and B. Antihomography A CIRCLEpreserving TRANSFORMATION an ODD number of INVERSIONS. HOMOGRAPHY
composed
of
also
FUNCTION
of c.
of the
LOGARITHM,
z) = x = antilog&og,
defined
2).
in base b of z is therefore
COLOGARITHM,
Antimagic A GRAPH
CONJUGATE
OPERATOR
logb(antilogb
also
the
Acf (2) = c*Af (x)7
Antidifferentiation
see
1929.
satisfies
+ fi(X>]
see INTEGRAL
igonal
pp. 1941,
Treatise
Boston,
LAPLACIAN
also
Antilinear An antilinear properties:
TRIANGLE
Mifflin,
An Elementary and the Circle.
Antilaplacian The antilaplacian of u with respect to 2 is a function whose LAPLACIAN with respect to x equals u. The antilaplacian is never unique. see
see
Geometry: of the Triangle
LOGARITHM,
b’.
POWER
(1 2 VJL&X
Graph with e EDGES labeled with distinct elements e} so that the SUM of the EDGE labels at each hiffer .
see
MAGIC
also
GRAPH
References N. and Ringel,
Hartsfield, Comprehensive
Press,
1990.
Introduction.
G.
Pearls
in
Graph
San Diego,
Theory:
CA: Academic
A
Antimagic
50 Ant
imagic
Antipedal
Square Antinomy A PARADOX
Square
Triangle
or contradiction.
Antiparallel
A2
An antimagic square is an n x n ARRAY 1 to n2 such that each row, column, nal produces a different sum such that a SEQUENCE of consecutive integers. special case of a HETEROSQUARE. Antimagic ble, and squares timagic ders 49
of integers from and main diagothese sums form It is therefore a
squares of orders one and two are impossiit is believed that there are also no antimagic of order three. There are 18 families of ansquares of order four. Antimagic squares of orare illustrated above (Madachy 1979).
see also HETEROSQUARE,
SQUARE,
MAGIC
4
A pair of LINES B1, B2 which make the same ANGLES but in opposite order with two other given LINES A1 and Aa, as in the above diagram, are said to be antiparallel to A1 and AZ.
see UZWHYPERPARALLEL, PARALLEL References Phillips, York:
A. W. and Fisher, I. Elements American Book Co., 1896.
of
Geometry.
New
TALISMAN
SQUARE References Abe,
G. “Unsolved
Problems
on Magic
Squares.”
Disc.
Math. 127, 313, 1994. Madachy,
J* S. “Magic
Madachy’s
s
Squares."
and Antimagic
Mathematical
York:
Ch. 4 in Dover,
http : //www
, astro .
Recreations.
pp, 103~113,1979. Weisstein, E. W. “Magic
New
Squares.”
virginia.edu/eww6n/math/notebooks/MagicSqu~es.m. Antimorph A number which can be represented both in the form X02  Dyo2 and in the form Dxr2  y12. This is only possible when the PELL EQUATION X2
is solvable.

Dy2
= 1
The antipedal
triangle
A of a given
 (~+acosC)(y+a~~~B)
Then
TRIANGLE
T is the
TRIANGLE of which T is the PEDAL TRIANGLE. For a TRIANGLE with TRILINEAR COORDINATES QI : p : y and ANGLES A, B, and C, the antipedal triangle has VERTICES with TRILINEAR COORDINATES : (y+acosB)(a+~cosC)
:
(~+acosC)(a+yosB) X2  Dy2 = (x0
 Dyo2)(xn2
= qxoyn see also IDONEAL
References Beiler, A. H. Queen
of
NUMBER,
Recreations Mathematical
1964.
Antimorphic
see ANTIMORPH
 YoGJ2
Number
 Dyn2)
(y+~cosA)(~+acosC)
 (x0&L  Qoy7J2*
: (y+pcosA)(cu+pcosC) (a+~cosC)(/3+~cosA)
POLYMORPH
in the Theory Entertains.
of
Numbers:
The
New
York:
Dover,
The ISOGONAL CONJUGATE ofthe ANTIPEDAL TRIANGLE ofagiven TRIANGLE is HOMOTHETIC withtheoriginal 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
Gallatly, W. The Modern Geometry of the London: Hodgson, pp* 5658, 1913.
Antipersistent Process A FRACTAL PROCESS for which UZSO
PERSISTENT
Triangle,
2nd
ed.
see GEOMETRIC PROBLEMS OF ANTIQUITY Antisnowflake
H < l/2,
SO
see KOCH ANTISNOWFLAKE
T < 0.
PROCESS
Antipodal Map The MAP which takes points on the surface of a SPHERE s2 to their ANTIPODAL POINTS.
Antisquare Number A number of the form p” . 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
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 6 and LONGITUDE X, the antipodal point has LATITUDE 6 and LONGITUDE X & 180” (where the sign is taken so that the result is between 180” and +1800).
see also ANTIPODE, SPHERE
DIAMETER,
GREAT
CIRCLE,
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
51
Antiquity
References
see
Matrix
ANTIPODAL POINTS
NUMBER
Antisymmetric A quantity which changes SIGN when indices are reversed. For example, Aij E ai  aj is antisymmetric since Aij = Aji+
see also ANTISYMMETRIC TENSOR,~YMMETRIC
MATRIX,
Matrix
Antisymmetric An antisymmetric the identity
is a MATRIX
matrix
where AT is the MATRIX notation, this becomes t&j
Letting
so an antisymmetric onal. The general form
TRANSPOSE. In component
= Oiji*
=
= +a2
[ Applying A’ tion gives
[cot (E)
+fi]
(3)
a12
a3
0
a13
a23
a23
to both
0
1 9
sides of the antisymmetry AlAT
+2n(~dL2)
becomes
akk,
a12
+ 2nAn (z)]
(2)
matrix must have zeros on its diag3 x 3 antisymmetric matrix is of the 0
= 2 [$a2cot
= 1.
.
I d
c;.
II
a11
H. S. M. “Polyhedra,” and Essays,
13th
Ch. 5 in York:
ed. New
Dover, p. 130, 1987. Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp+ 8586, 1997. @ Weisstein, E. W. “Prisms and Ant iprisms.” http : / /www. astro.virginia.edu/eww6n/math/n~tebooks/Prism*m.
(6)
c
References Recreations
condi(5)
A= +(A + AT) + +(A  AT).
Mathematical
(4)
Any SQUARE MATRIX can be expressed as the sum of symmetric and antisymmetric parts. Write
see also OCTAHEDRON,PRISM,PRISMOID,TRAPEZOHEDRoN Ball, W. W. R. and Coxeter,
satisfies (1)
k = i = j, the requirement akk
S = 2Angon
which
ACAT
Antiprism
A SEMIREGULAR POLYHEDRON constructed with 2 nis simply the gons and 2n TRIANGLES. The 3antiprism OCTAHEDRON. The DUALS are the TRAPEZOHEDRA. The SURFACE AREA of a ngonal antiprism is
ANTISYMMETRIC
A=
a12
'*'
ah
(7)
Antisymmetric
52
a11 AT
=
“‘”
a21
*
l
;’
an2
l
*
ain
,
l
a2n
”
Antoine’s A topological
ad
. . . ““”
.
1
Apeirogon
Relation
,
. 1 l
arm
’
1
so al2
2Ull WJ
A+AT=
+
+
a21
l
2a22
a21
l
“’
l
aln
+
&xl
a2n
+
an2
j
1
1 /fi\
which
is symmetric,
and
AAT= a12  a21
0 (al2
.
[
(al,
which
. . .
.
l
l
.
.

Gl)
(a2n
..
l
0
~21)
ain  ani a27b
.
an2 l
l
Gt2)
.
l
.
0
*”
1 1
Horned Sphere 2sphere in 3space whose exterior is not SIMPLY CONNECTED. The outer complement of Antoine’s horned sphere is not SIMPLY CONNECTED. Furthermore, the group of the outer complement is not even finitely generated. Antoine’s horned sphere is inequivalent to ALEXANDER'S HORNED SPHERE since the complement in Iw3 of the bad points for ALEXANDER'S HORNED SPHERE is SIMPLY CONNECTED.
see ~SO
ALEXANDER'S HORNED
SPHERE
m I
nererences Alexander, J. W. “An Example of a SimplyConnected Surface Bounding a Region which is not SimplyConnected.” PTOC. Nat. Acad. Sci. 10, 810, 1924. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 7679, 1976.
Antoine’s
Necklace
is antisymmetric.
see also SKEW
MATRIX, SYMMETRIC MA
SYMMETRIC
TRIX
Construct a chain C of 2n components in a solid TORUS V. Now form a chain Cl of 2n solid tori in V, where
Antisymmetric Relation A RELATION R on a SET S is antisymmetric that distinct elements are never both related other. In other words zRy and yRz together z = y. Antisymmetric An antisymmetric which
Tensor tensor
A rnn
as a TENSOR for
is defined =
provided to one animply that
A”“.
m(V

Cl)
%1(VC)
via inclusion. In each component of Cl, construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori Cs. Continue this process a countable number of times, then the intersection A00
A=()Gi
(1)
i=l
Any TENSOR can be written and antisymmetric parts as 1 A 77x72 _ &4””
The antisymmetric special notation
as a sum of SYMMETRIC which is a nonempty Antoine’s necklace.
+ An,)
part is sometimes
denoted
Abbl _ 1 (A”b _ Aba) 2 For a general
 An,)*
+ $(A””
(2)
using the
.
l
acal
a,
x
A a1 *a,
I
(4)
E~~...~,
is the LEVICIVITA
PERMUTATION SYMBOL. see also SYMMETRIC TENSOR
PHIC with the CANTOR SET. see also ALEXANDER'S HORNED SPHEREJECKLACE References Rolfsen, D. Knots Perish Press, pp.
and Links. Wilmington, 7374, 1976.
DE:
Publish
or
Apeirogon The REGULAR POLYGON essentially equivalent to the CIRCLE having an infinite number of sides and denoted with SCHL;~FLI SYMBOL (00).
see also
permutations
where
SUBSET of Iw3 is called is HOMEOMOR
necklace
(3)
TENSOR,
I=A[ al **CL,
compact Antoine’s
SYMBOL, a.k.a.
the
CIRCLE, REGULAR POLYGON
References Coxeter,
H.
S.
M,
Regular
Polytopes,
3rd
ed.
New
York:
Dover, 1973. Schwartzman, S. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994.
Apkry’s Constant Apkry’s
Ap&y ‘s Constant
Constant
N.B. A detailed online ing point for this entry.
Ap&y’s
constant
by S. Finch
essay
is defined
was
the start
whereS,,,isa STIRLING NUMBER OFTHEFIRST This can be rewritten as
53 KIND.
x 5 =2C(3), (10)
by
n=l
c(3) = 1.2020569..
.,
(1)
A002117) where c(z) is the RIEMANN ZETA FUNCTION. Ap&y (1979) proved that c(3) is IRRATIONAL, although it is not known if it is TRANSCENDENTAL. The CONTINUED FRACTION for c(3)& [1,4, 1, 18, 1, 1, 1, 4, 1, . . . ] (Sloane’s A013631). The positions at which the numbers 1, 2, +++ occur in the continued fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, . (Sloane’s
l
Sums related
l
l
C(3)(Castellanos
loo 2x
1988)
1 1 l+s+...+7 (
1 n>
(11)
l
INTEGRALS for c(3) include
to c(3) are
n=
where Hn is the nth HARMONIC NUMBER. Yet another expression for c(3) is
(2)
1
1
C(3) 
O” t2 etl
2 so
(12)
dt
= p [f*lln2+2~~/IDln(sinz)dz]. (used by Aphy),
Gosper =$ l + 1)”
X(3)=2 k=o w
C(3)
(1990)
gave
(3) C(3) 
00
1
2n3
k=O (3k + 1)s
= m
>:
00
1
@k+
00
+
+ SC(3)
7T3 1)3
=
1 (6k
(5)
64 + hc(3)
7r3 1)”
=
a
+
(4)
(6)
%c(3)?
where X(Z) is the DIRICHLET LAMBDA FUNCTION. The above equations are special cases of a general result due to Ramanujan (Berndt 1985). Apkry’s proof relied on showing that the sum
44=z=(;)2(:“>: where (i) ~~~BINOMIAL COEFFICIENT, satisfiesthe CURRENCE RELATION (n + l)3a(n
(13)
and
 1) = 0
A CONTINUED FRACTION involving 6
=5.?..* l6 117
c9
26 535
(14
’
Ap&y’s
constant
n6 34n3 + 51n2 + 27n + 5
(Apery 1979, Le Lionnais 1983). Amdeberhan used WILFZEILBERGER PAIRS (F, G) with F(n
S
1
k) 
(l)kk!2(sn
RE
is
” (15) (1996) l
 k  l)!
(sn + k t l)!(k
+ 1) ’
(16)
= 1 to obtain
(17)
(7) For s = 2,
C(3)
+ 1)  (34n3 + 51n2 + 27n + 5)a(n) +n3a(n
1 O” 30k  11 4 x (2k  l)k3 (ik)” k=l
=
;
&)“’ n=
5”;2;:3;)T 1
5
(3”) n
(inJn3
OS)
n
(8) and for s = 3,
(van der Poorten
1979, Zeilberger
Apery’s
is also given by
constant
C(3) = 2 n=l
$7
UN), C(3) = F; (4” n=O 72(4nn) (“n”)
(9)
’
6120n+ 5265n4 + 13761n2 + 13878n3 (4n I 1)(4n + 3)(n + 1)(3n + 3)2(3n
+ 1040 + 2)2
(l’)
54
Constant
Apkry’s
(Amdeberhan I and i are
1996).
Apoapsis
The corresponding
2(l)kk!2(n
G(n, 1) =
G(n, k) for s =
see
 k)!
(n + k + l)!(n
(20)
+ 1)2
G(n, k) =
Gosper
(1996)
expressed
+ 6n + k + 3) .
c(3) as the MATRIX
(21)
PRODUCT
N
lim Nm
M,
rI
o0
=
PAIR Amdeberhan, T. “Faster and Faster Convergent Series for c(3).” Electronic J. Combinatorics 3, R13, 12, 1996. http://www.combinatorics.org/Volume3/ volume3, html#R13. Amdeberhan, T. and Zeilberger, D. “Hypergeometric Series Acceleration via the WZ Method.” Electronic J. Combinatorics 4, No. 2, R3, l3, 1997. http: //www. combinatorics.org/Volume4/wilftoc.html#R03. Also available at http : //wua *math. temple, edu/zeilberg/ mamarim/mamarimhtml/accel. html. Ap&y, R. “Irrationalitk de c(2) et C(3).” Aste’risque 61, ll
13, 1979.
C(3) i
[
n=l
,
(22)
I
Berndt, B. C. Ramanujan% Notebooks: SpringerVerlag, 1985+ Beukers, F. “A Note on the Irrationality don Math.
where
Analytic
40PB(n+p(n+p
1 (23)
61,
31104(nf~)(n+~)(n+~)
1
0
Number
6798,
I. New
of 5( 3) .” Bull.
1979. P. B. Pi &
Theory
and
New York: Wiley, 1987. Castellanos, D. “The Ubiquitous
24570n4+64161n3+62152n2+28427n+4154
Part
York: Lon
11, 268272,
Sot.
J. M. and Borwein,
Borwein,
(n+lj4
as of
ZETA FUNCTION, WILFZEILBERGER
RIEMANN
also
record
References
and
(l)kk!2(2n  k)!(3 + 4n)(4n2 2(2n+k+2)!(n+Q2(2n+1)2
(Amdeberhan and Zeilberger 1997). The Aug. 1998 was 64 million digits (Plouffe).
the AGM: Computational
Pi.
Part
I.”
A Study in Complexity. Math.
Mug.
1988.
Conway,
J. H. and Guy, R. K. “The Great Enigma.” In The of Numbers. New York: SpringerVerlag, pp. 261262, 1996. Ewell, J. A. “A New Series Representation for c( 3)*” Amer. Book
which
gives 12 bits per term.
which
gives
C(3)
E
423203577229 352066176000
The first few terms
= 1.20205690315732..
..
are
(27)
Given three INTEGERS chosen at random, the probability that no common factor will divide them all is
K(3)11= 1.202l
= 0.832.u.
(28)
B. Haible and T. Papanikolaou computed c(3) to l,OOO,OOO DIGITS using a WILFZEILBERGER PAIR identity with  k  l)!k!3 ‘J(n + k + 1)!2(2n)!3’
Math.
S=
‘) = (‘)
1, and t = 1, giving
the rapidly
converging
97,
219220,
1990.
1979. Zeilberger,
D.
Symb.
Comput.
“The
Method
11, 195204,
of Creative 1991.
Telescoping.”
J.
Apoapsis
G (z>
k n!6(2n
F(n’
Monthly
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 Evaluation of Series of Rational Numbers.” Technical Report ‘11977. 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(S).” http: //lacim. uqam. ca/piDATA/ZetaS, txt. Sloane, N. J. A. Sequences A013631 and A002117/M0020 in “An OnLine Version of the Encyclopedia of Integer Sequences .” van der Poorten, A. “A Proof that Euler Missed.. . Apkry’s Proof of the Irrationality of c(3).” Math. Intel. 1, 196203,
(29)
l
r
F
The greatest radial distance of an ELLIPSE as measured from a FOCUS. Taking II = T in the equation of an
ELLIPSE r=
a(1  e2) 1+ ecosu
Apocalypse
Apodization
Number
gives the apoapsis
Apodization
distance
Apoapsis for an orbit around the Earth is called apogee, and apoapsis for an orbit around the Sun is called aphelion.
Apocalypse
ELLIPSE,
Instrument
Instrument
Function
55 Function
Sidelobes
Bartlett
T+ = a(1 + e).
see also ECCENTRICITY,
Function
Function
Focus,
1.25 0.7' 05 0 5
Blackma n 3 A
20125
1
2
3
0.5
PERIAPSIS Comes
Number
A number having 666 DIGITS (where 666 is the BEAST NUMBER) is called an apocalypse number. The FIBONACCI NUMBER F3184 is an apocalypse number.
see UZSO BEAST NUMBER,LEVIATHAN
1.251
Cosine
NUMBER
References Pickover, C A. Keys 102, 1995.
pp. 97
Gaussian
A number of the form 2” which contains the digits 666 an APOCALYPTIC NUM(the BEAST NUMBER) iscalled BER. 2157 is an apocalyptic number. The first few such powers are 157, 192, 218, 220, . . . (Sloane’s A007356).
Hamming
Apocalyptic
to Infinity.
New York: ,
Wiley,
Number
see UZSO APOCALYPSE NUMBER, LEVIATHAN
Hanning
NUMBER
References
2
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97102, 1995. Sloane, N. J+ A. Sequences A007356/M5405 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Sloane, N. J. A, and Ploufle, S. Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
Apodization
Apodization
0.5
Welch
Apodization
TYPE
of an APODIZATION FUWTION.
The application
1.5
Uniform
Bartlett
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 resolution.
Instrument
a
BI W
dx)
Connes
(l
$)’ 4acaa(2rrd)
Cosine
YqziaF)
Gaussian
2s,”
Hamming
cos(2nk++‘2)
da:
Hm(k)
Hanning
Hw(k) 2a sine
Uniform
The following are apodization functions for symmetrical (2sided) interferograms, together with the INSTRUMENT FUNCTIONS (or APPARATUS FUNCTIONS) they produce and a blowup of the INSTRUMENT FUNCTION sidelobes. The INSTRUMENT FUNCTION I(K) corresponding to a given apodization function A(x) can be computed by taking the finite FOURIER COSINE TRANSFORM,
Function
a sinc’(7rlca)
!A 1
Blackman
Function
Function
Welch
(2nka)
WI (Ic)
where &(x)
= 0.42 + 0.5 cos
BI(k) =
a(0.84  0.36a2k2  2.17 x 10~~su4~4) (1  a2k2)(1  4a2k2)
(2) sinc(2nak)
(3) I(k)
=
I’
cos(2nkx)A(x)
dx.
(1)
HmA(x)
= 0.54 + 0.46~0s
(4)
HmI(k)
a( 1.08  0.64a2k2) sinc(2Tak) = 1  4a2k2
(5)
56
Apodiza
Apollonius
tion Function a1 =
(6)
i
5
=
(7) HnI(k)
+ + sinc(2nlca
+ + sinc(2rka = 626
J3p(2rJEa) (2rka)3/2 cos( 2rak)
.
2a3 k3n3
IFFWHM
Bartlett Blackman Connes Cosine Gaussian Hamming
Hanning Uniform Welch
IF
Peak
Peak
a0 = E ==: 0.5435
(20)
a1 = g$ = 0.2283.
(21)
(1
S.L.
Peak
S.L.
(+)
Peak
Peak
0.00000000
0.84 16 15
0.00106724 0.0411049
0.0471904 0.00124325 0.0128926
1.63941 
r4 1
0.0708048 
0.0292720 
1.81522 2.00000 1.20671 1.59044
1.08 1 2 43
0.00689132 0.0267076 0.217234 0.0861713
0.00734934 0.00843441 0.128375 0.356044
1
apodization SERIES
the COEFFICIENTS
A(x)
function
can be
;
(12)
satisfy
00 ao+2 E The corresponding
apparatus
(13)
function
is
s
4xF2xikx
b
00
To obtain 3/4, use
+ 7m) + sinc(2nkb
a0 sinc( $7r) + ai[sinc( Plugging
 n7r)]
1
an APODIZATION
uo
FUNCTION
with
>
.
(14)
(22) (23) (24)
see ah BARTLETT FUNCTION, BLACKMAN FUNCTION, CONNES FUNCTION, COSINE AP~DIZATION FUNCTION, FULL WIDTH AT HALF MAXIMUM, GAUSSIAN FUNCTION, HAMMING FUNCTION, IIANN FUNCTION, HANNING FUNCTION, MERTZ APODIZATION FUNCTION, PARZEN APODIZATION FUNCTION, UNIFORM APODIZATION FUNCTION, WELCH APODIZATION FUNCTION References Ball, J* A. “The Spectral Resolution in a Correlator System” §4+3.5 in Methods of Experimental Physics 12C (Ed. M. L. Meeks). New York: Academic Press, pp. 5557, 1976. Blackman, R. B. and Tukey, 3. W. “Particular Pairs of Windows.” In The Measurement of Power Spectra, From York: Brault,
of View
of Communications
Engineering.
New
Dover, pp. 95101, 1959. J. W. “Fourier Transform
Spectrometry.” In High in Astronomy: 15th Advanced Course of Society of Astronomy and Astrophysics (Ed.
A. Benz, M. Huber, and M. Mayor), Geneva Observatory, Sauverny, Switzerland, pp. 3132, 1985. Harris, F. 3. “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform.” Proc. IEEE 66, 5183, 1978. Norton, R. He and Beer, R. “New Apodizing Functions for Fourier Spectroscopy.” J. Opt. Sot. Amer. 66, 259264,
(15)
Press, W1 H.; Flannery, ling, W. T. Numerical Scientific
B. P.; Teukolsky,
Computing,
S. A.; and Vetter
Recipes in FORTRAN: The 2nd ed. Cambridge, England:
Art
of
Cam
bridge University Press, pp. 547548, 1992, Schnopper, H. W. and Thompson, R. I. “Fourier Spectrometers.” In Methods of Experimental Physics 12A (Ed. M. L. Meeks). New York: Academic Press, pp. 491529, 1974.
in (13),
Apollonius (l
$=: 0.4266
1976.
zero at ka =
47r) + sinc( +7r) = 0.
3969 9304
=
al = E ==: 0.2483 715 u2 = 18608 =2: 0.0384.
Resolution the Swiss
b
sinc(2xkb
The BLACKMAN FUNCTION is chosen so that the APPARATUS FUNCTION goes to 0 at ku = 5/4 and 9/4, giving
the Point
u,=l.
?I=1
n=
(19)
(10)
n=l
+>:[
g
(11)
A(z) = a0+2fya,cos (y)

la _ =28E’
(9)
1.77179 2.29880 1.90416
A general symmetric written as a FOURIER
I@)
2.5
28
The HAMMING FUNCTION is close to the requirement that the APPARATUS FUNCTION goes to 0 at ka== 5/4, giving
 n)
+ 7r)]
sin( 27rlca)  2rak
=a
where
28
(8)
1  4a2k2
= a[sinc(27da)
Type
a0 = 1  2Ul =
(18)
a sine (274
=
WI(k)
5  28
6.3+2*5
;+g
Circles
There called
 2Q) 32,+al(&+g = +(l
 2~~) +a~(+
Ul(f + 3> = ;
+ 1) = 0
(16)
(17)
Circles
are two completely Apollonius circles:
different
definitions
1. The set of all points whose distances from points are in a constant ratio 1 : p (Ogilvy
of the sotwo fixed 1990).
Apollonius 2. The ate)
Point
eight which
Apolhh’
CIRCLES solve
(two of which are nondegenerAPOLLONIUS’ PROBLEM for three
the has
Problem
LINES AA’, BB’, and CC’ CONCUR in this point. TRIANGLE CENTER FUNCTION
57 It
CIRCLES. Given kngths
one of VERTEX is CENTER is
TRIANGLE,
side of a TRIANGLE and the ratio of the the other two sides, the LOCUS of the third the Apollonius circle (of the first type) whose on the extension of the given side. For a given there are three circles of Apollonius.
Denote the three Apollonius circles (of the first type) of a TRIANGLE by kl, k2, and k3, and their centers L1, Lz, and L3. The center L1 is the intersection of the side AaA3 with the tangent to the CIRCUMCIRCLE at Al. L1 is also the pole of the SYMMEDIAN POINT K with respect to CIRCUMCIRCLE. The centers L1, L2, and L3 are COLLINEAR on the POLAR of K with regard to its CIRCUMCIRCLE, called the LEMOINE LINE. The circle of Apollonius k1 is also the locus of a point whose PEDAL TRIANGLE is ISOSCELES such that PI Pz = PIPa.
a = sin2 Acos2[$(B
 C)].
References Kimberling, C. “Apollonius Point ,” http:~/www. evansville.edu/ck6/tcenters/recent/apollon.htm1. Kimberling, C. “Central Points and Central Lines in the Plane of a Triangle.” Math. Mug. 67, 163187, 1994. F. “Problem 1091 Kimberling, C., . Iwata, S.; and Hidetosi, and Solution.” Crux Math. 13, 128129 and 217218, 1987.
Apollonius’
Problem
U and V be points on the side line BC of a TRIANGLE AABC met by the interior and exterior ANGLE BISECTORS of ANGLES A. The CIRCLE with DIAMETER UV is called the AApollonian circle. Similarly,
Let
construct the B and CApollonian circles. lonian circles pass through the VERTICES A, and through the two ISODYNAMIC PRINTS The VERTICES of the DTRIANGLE lie on the Apollonius circles.
The
Apol
B, and C, S and S’. respective
Geometry: An Elementary on the Geometry of the Triangle and the Circle. MA: Houghton Mifflin, pp. 40 and 294299, 1929. Ogilvy, C. S. Excursions in Geometry. New York:
pp. 1423,
Apollonius
three
objects,
LINE, or CIRCLE, draw
see UZSOAPOLLONIUS' PROBLEM,APOLLONIUS PURSUIT PROBLEM, CASEY'S THEOREM,HART'S THEOREM, IsoDYNAMIC POINTS, SODDY CIRCLES References Johnson, R. A. Modern
each of which may be a POINT, a CIRCLE that is TANGENT to each. There are a total of ten cases. The two easiest 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 Tangencies of Apollonius which was, however, lost. The general problem is, in principle, solvable by STRAIGHTEDGE and COMPASS alone. Given
Treatise Boston, Dover,
1990.
Point
Consider the EXCIRCLES I?A, l?~, and rc of a TRIANGLE, and the CIRCLE r internally TANGENT to all three. Denote the contact point of r and rA by A’, etc. Then
58
Apollonius’
Apollonius
Problem which
Pursuit
Problem
can then be plugged back into the QUADRATIC (1) and solved using the QUADRATIC FOR
EQUATION
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 (illustrated 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 intersections are the points of tangency of two of the eight circles (Johnson 1929, Dijrrie 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. The threeCIRCLE problem was solved by Vikte (Boyer 1968), and the solutions are called AP~LLONIUS CIRCLES. There are eight total solutions. The simplest solution is obtained by solving the three simultaneous quadratic equations (x  Xl)”
+ (Y  y1)2  (T Ik r1)2 = 0
(1)
(x 
+ (y 
y2)2  (r & r2)2 = 0
(2)
If the three CIRCLES are mutually tangent, then the eight solutions collapse to two, known as the SODDY CIRCLES. UZS~APOLLONIUS PURSUIT PROBLEM,BEND (CURVATURE), CASEY'S THEOREM, DESCARTES CIRCLE THEOREM,FOUR COINS PROBLEM,HART'S THEOREM, SODDY CIRCLES
see
References x2)2
(x  x3)2 + (y  y3)2  (r zt T3)2= 0 in the three unknowns x, y, T for the eight signs (Courant and Robbins 1996). Expanding tions gives
(x2+y2 r2)2X&2
yyi&Zrr;
+(xi2
+yi2
(3)
triplets of the equa
ri”)
= 0
(4) for i = 1, 2, 3. Since the first term is the same for each equation, taking (2)  (1) and (3)  (1) gives ax + by + CT = d
(5)
a’x + b’y + C’T = d’,
(6)
where a = 2(x1  x2) b = 2(yl c = F2(?3

(7)
(9)
 73’)
 (xl2
+ y12  r12>
(10)
and similarly for a’, b’, c’ and d’ (where the 2 subscripts are replaced by 3s). Solving these two simultaneous linear equations gives x=
b’d  bd’  b’cr + bc’r aId
Y
New
York:
Problem.”
?: An Elementary What is Mathematics and Methods, 2nd ed. Oxford, England:
Approach
Wiley, $3.3 in to Ideas
Oxford University Press, pp. 117 and 125127, 1996. Dijrrie, H. “The Tangency Problem of Apollonius.” 332 in 100 Great Problems History and Solutions.
1965. Gauss, C. F. We&e,
Vol.
1981. Johnson,
of Elementary New York:
4. New York:
R+ A. Modern Geometry: on the Geometry of the Triangle
MA: Houghton Miffiin, Ogilvy, C. S. Excursions
Mathematics:
Dover,
Their
pp. 154160,
George Olms, p. 399,
An Elementary and the Circle.
Treatise
Boston,
ppm 118121,1929.
in Geometry. New York: Dover, pp. 4851, 1990. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 151, 1989. Simon, M. i!%er die Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin, pp. 97105, 1906. @ Weisstein, E. W. “Plane Geometry.” http: //www +astro . virginia.edu/eww6n/math/notebooks/PlaneGeometry,m.
(8)
~2)
 73)
d = (z22 +y22
Boyer, C. B. A History of Mathematics. p* 159, 1968. Courant, R. and Robbins, H. “Apollonius’
ab’  ba’ + ad’ + ah ab’  a/b
(11)  a& 7
(12)
Apollonius Pursuit Problem Given a ship with a known constant direction and speed ‘u, what course should be taken by a chase ship in pursuit (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 p = w/V, If the CIRCLE cuts the path of the pursued ship, the intersection is the point towards which
Apolhius
Theorem
Appell
ship sho uld steer. If the CIRCLE the pursuit cut the path, then it cannot b caught. see also APOLLONIUS LEM, PURSUIT CURVE
CIRCLES,
APOLLONIUS’
PROB
Appell defined the functions in 1880, and Picard showed in 1881 that they may all be expressed by INTEGRALS of the form 1
Ogilvy, C. S. Solved by M. S. Klamkin. “A Slow Ship Intercepting a Fast Ship.” Problem EWl. Amer. kfath. Monthly
58,
408,
Apollonius
u”(l
#(l
New
York:
3rd
American
ed.
pp. 126138,
1983
Al a3
A2
maz2 + ntQ2 =
Appell Polynomial A type of POLYNOMIAL
P
(m+n)A1P2
References Appell, P. “Sur les fonctions hypergkom&iques de plusieurs variables.” In Me’moir. Sci. Math. Paris: GauthierVillars,
a2
n
m
which
+mPA3
ALS {A&z)}~=~
2 +nFZ2.
r as c?
is defined
= c
A(t)
is a formal
=
?,a&”
series with
k = 0, 1, . . and a0 # 0. l
References Hazewinkel,
M. (Managing
ematics: An Updated Soviet “Mathematical
lands:
RADIUS, SAGITTA, SECTOR,
POWER
Reidel,
Ed.). Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht,
pp. 209210,
of Mathof the
Nether
1988.
SEGMENT Appell Transformation A HOMOGRAPHIC transfurmation
see INSTRUMENT FUNCTION x1 = Appell Hypergeometric Aformalextensionofthe
F’unction
ux + by + c d'x + bfl y I c
HYPERGEOMETRIC FUNCTION
resulting
in four kinds
of functions
a’x + b’y + cl y1
(Apwith
= y;
;fl;
m=O n=o 03 =
(Q)m+n(P)m(P’)n
xmyn
m!n!(r),(r’>
xmyn
dt
= +
vt
m x
n ’
*
M. (Managing
ematics: An Updated Soviet “4Mathematical
lands:
(ff)m+n(P)m+n TYL!n!(~),(~‘)n
to
y
+
c”
)2
l
References
m!n!(y),+n
m=O n=o m z”c m=O no
for t according
xmyn n
Hazewinkel,
= fy
P; Y, A 5, Y) =
kdtl
=
(a”x
(a>m+n(P)m(P’)n PC m=O n=O
tl substituted
m!n!(&+,
(a>m(Q’)n(P)m(P’)n
a’; PJcx”rY)
A,(z)tn,
k=O
Function
F+;P,PhY’;x>Y)
by
where
a =?S.
Fd~;P,P’;r;x,Y)
the BERNOULLI
n=O
Given a CIRCLE, the PERPENDICULAR distance a from the MIDPOINT of a CHORD to the CIRCLE'S clsnter is called the apothem, It is also equal to the RADIUS T minus the SAGITTA s,
to two variables, pell 1925),
includes
POLYNOMIAL, HERMITE POLYNOMIAL, and LAGUERRE POLYNOMIAL as special cases. The series of POLYNOMI
A3
A(t)?
Apothem
Apparatus
du.
1925. Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 73, 1935. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary Cambridge, MA: MIT Press, p. 1461, of Mathematics. 1980.
Theorem
CHORD,
yu)’
Dover,
Al
F&;
XU)~ (I
s 0
1952.
ogilvy, C. S. Excursions in Geometry. p. 17, 1990. Snapshots, Steinhaus, H Mathematical New York: ‘Oxford University Press,
&(a,
59
does not
References
see also
Transformation
Reidel,
Ed.). Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht,
pp. X0211,
1988,
of Mathof the
Nether
60
Arbelos
Apple
Arakelov Theory A formal mathematical theory which introduces “components at infinity” by defining a new type of divisor class group of INTEGERS of a NUMBER FIELD. The divisor class group is called an “arithmetic surface.”
Apple
see also ARITHMETIC
GEOMETRY
Arbelos
A SURFACE OF REVOLUTIO~J defined by Kepler. Itconsists 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 XX PLANE are zh = z&JR2  (x  T)~ for R > T and it: E [(T + R),r surface of a SPINDLE TORUS. see also BUBBLE, LEMON, TION, SPINDLE T ORUS
+ R].
It is the outside
SPHERESPHERE
Approximately Equal If two quantities A and B are approximately is written A z B. see
also
DEFINED,
INTERSEC
&O
LAGRANGE
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 arbitrafy and can be located anywhere along the DIAMETER. The arbelos satisfies (Gardner 1979).
a number
of unexpected
identities
1. Call
equal,
this
EQUAL
the radii of the left and right SEMICIRCLES a and b, respectively, with a + b z R. Then the arc length along the bottom of the arbelos is L = 27~ + 2nb = 2/r(a I b) = 2zR,
Approximation Theory The mathematical study of how given quantities can be approximated by other (usually simpler) ones under appropriate conditions. Approximation theory also studies 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
d
^
so the arc lengths
along
the top and bottom
of the
REMAINDER
References Achieser,
N. I. and Hyman, C+ J. Theory York: Dover, 1993. Akheizer, N. I. Theory of Approximation.
of Approximation.
New
1992. Cheney, E. W, Introduction to Approximation York: McGrawHill, 1966.
New
York: Theory.
Dover, New
Golomb, M. Lectures on Theory of Approximation. Argonne, IL: Argonne National Laboratory, 1962. Jackson, D. The Theory of Approximation. New York: Amer. Math. Sot., 1930. Natanson, I. P. Constructive Function Theory, Vol. 1: Uniform 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 Functions. New York: Dover, 1981. Timan, A. F. Theory of Approximation of Functions of a Real Variable. New York: Dover, 1994.
2. Draw the PERPENDICULAR BD from the tangent of the two SEMICIRCLES to the edge of the large CIRCLE, Then the AREA of the arbelos is the same as the AREA ofthe CIRCLE with DIAMETER BD. 3. The CIRCLES Cl and Cz inscribed on each half of BD on the arbelos (called ARCHIMEDES’ CIRCLES) each have DIAMETER (AB)(BC)/(AC). Furthermore, the smallest CIRCUMCIRCLE of these two circles has an area equal to that of the arbelos. 4. The line tangent to the semicircles AB and BC contains the point E and F which lie on the lines AD and CD, respectively. Furthermore, BD and EF bisect each other, and the points B, D, E, and F are CONCYCLIC.
Arc Length
Arbelos CIRCLES, STEINER
GOLDEN CHAIN
RATIO,
INVERSION,
PAPPUS
61 CHAIN,
References
5. In addition to the ARCHIMEDES' CIRCLES C1 and Cz in the arbelos figure, there is a third circle C3 called to these the B ANKoFF CIRCLE which 1s congruent 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 CIRCLE Cn is (l/n)th PERPENDICULAR distance to the This result is most easbase of the SEMICIRCLE. ily proven using INVERSION, but was known to Pappus, who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). If T = AB/AC, then the radius of the nth circle in the PAwus CHAIN is Tn =
(1  r)r
2[?22(1  r)2 + r].
Bankoff, L. “The Fibonacci Arbelos.” Scripta Math. 20, 218, 1954. Bankoff, L. “The Golden Arbelos.” Scripta Math. 21, 7076, 1955. Circles of Archimedes Really Bankoff, 1;. “Are the Twin Twins?” Math. Mag. 47, X4218, 1974. Bankoff, L. “How Did Pappus Do It?” In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112118, 1981. Bankoff, L. “The Marvelous Arbelos.” In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow), Washington, DC: Math. Assoc. Amer., 1994. Cadwell, J. H. Topics in Recreational Mathematics, Cambridge, England: Cambridge University Press, 1966. Gaba, M. G. “On a Generalization of the Arbelos.” Amer. Math. Monthly 47, 1924, 1940. Gardner, M, “Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.” Sci. Amer, 240, 1828, Jan. 1979. Heath, T. L. The Works of Archimedes with the Method of Archimedes. New York: Dover, 1953. Hood, R. T. “A Chain of Circles.” Math. Teacher 54, 134137, 1961. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp. 116117, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 5455, 1990.
Arborescence A DIGRAPH is called an arborescence if, from a given node x known as the ROOT, there is exactly one elementary path from 2 to every other node y. see also ARBORICITY
Arboricity Given a GRAPH G, the arboricity is the MINIMUM number of linedisjoint acyclic SUBGRAPHS whose UNION is G. see also ANARBORICITY
This general result simplifies to TV = l/(6 + n2) for T = 2/3 (Gardner 1979). Further special cases when AC = I+ AB are considered by Gaba (1940). 7 If B divides AC in the GOLDEN RATIO 4, then the circles in the chain satisfy a number of other special properties (Bankoff 1955).
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, CIRCLECIRCLE INTERSECTION, FIVE DISKS PROBLEM, FLOWER OF LIFE, LEMON, LENS, PIECEWISE CIRCULAR CURVE, REULEAUX POLYGON, REULEAUX TRIANGLE, SALINON, SEED OF LIFE, TRIANGLE ARCS, VENN DIAGRAM, YINYANG Arc Length Arc length is defined
as the length
along
a curve,
b SE
see UZSO ARCHIMEDES’ LOXODROM cox .ETER’S
RCLES, BAN KOF SF CIRCLE, SEQUENCE OF TANGENT
14
l (1)
s a
Defining the line element ds2 E j&.12, parameterizing the curve in terms of a parameter t, and noting that
Arc Minute
62 ds/dt
which
Archimedes
is simply the magnitude of the VELOCITY with the end of the RADIUS VECTOR r moves gives
(2)
s=~bds=~b~dt=~b,~lo(dt
Algorithm
Arc Second A unit of ANGULAR measure equal to l/60 of an ARC MINUTE, or l/3600 of a DEGREE. The arc second is denoted ” (not to be confused with the symbol for inches). Arccosecant
In POLAR COORDINATES,
see INVERSE C~~ECANT
(3)
Arccosine see INVERSE
COSINE
so
Arccotangent
(4)
see INVERSE COTANGENT Arch
(5) In CARTESIAN COORDINATES,
de =xji:+yg
Therefore,
(6)
A 4POLYHEX.
(7)
References Gardner, M.
if the curve is written
Vintage,
r(x) = x2 + f (x)9,
(8)
then s=
(9)
rbdmdx.
If the curve is instead
written
r(t)
= x(t)ji:
+ y(t)y,
x’“(t)
+ y’“(t)
PO)
then S=
Mathematical Magic Show: Games, Diversions, Illusions and Other SleightofMind from Scientific American.
p. 147, 1978.
Archimedes Successive
Algorithm application of ARCHIMEDES' RECURRENCE FORMULA gives the Archimedes algorithm, which can be used to provide successive approximations to K (PI). The algorithm is also called the BORCHARDTPFAFF ALGORITHM. Archimedes obtained the first rigorous approximation of r by CIRCUMSCRIBING and INSCRIBING n = 6 2”gons on a CIRCLE. From ARCHIMEDES' RECURRENCE FORMULA, the CIRCUMFERENCES aandbof the circumscribed and inscribed POLYGONS are l
(11)
dt.
= x(t)ji:
+ y(t)9
(12)
+ z(t)2,
x’“(t)
+ y’“(t)
+ z’“(t)
dt.
(13)
< C = 27~ =
2n
Arc Minute A unit of ANGULAR measure equal to 60 ARC SECONDS, or l/60 of a DEGREE. The arc minute is denoted ’ (not to be confused with the symbol for feet).
(2)
l
1 = 27r < u(n).
(3)
For a HEXAGON, n = 6 and a0 G a(6)
see also CURVATURE, GEODESIC, NORMAL VECTOR, RADIUS OF CURVATURE,RADIUS OF TORSION,~PEED, SURFACE AREA,TANGENTIAL ANGLE,TANGENT VECTOR, TORSION (DIFFERENTIAL GEOMETRY), VELOCITY
E , (n >
where b(n)
so S=
= 2nsin
b(n)
Or, in three dimensions, r(t)
More Puzzles, Mathematical New York:
= 4J3
(4
b. = b(6) = 6,
where arc = ~(62”).
(5)
The first iteration
of ARCHIMEDES'
RECURRENCE FORMULA then gives 2.6+41/3 a’=
24d3
6+4&
=3+2fi
4(2 = 6(&

h)
 h).
l
= 24(2  6)
(6)
6 = 12j/a (7)
Archimedes’
Axiom
Archimedes’
Additional iterations do not have simple closed forms, approximations for k = 0, 1, 2, 3, 4 but the numerical are (corresponding to 6, 12, 24, 48, and 96gons) 3.00000
< TT < 3.46410
(8)
3.10583
< T < 3.21539
(9)
63
consists
of solving
TINE EQUATIONS in INTEGERS
w=
$x+2
(1)
X=&Y+2
(2)
3.~3263
< TT < 3.15966
(10)
Y=
SWt2
(3)
3.13935
< TT < 3.14609
(11)
w=
&(X+x)
(4)
3.14103
< 7r < 3.14271.
(12)
x=&(Y+y)
The = 3.14084.n
< T < $? = 3.14285....
smallest
Axiom
An AXIOM actually which states that
attributed
to Eudoxus
Y=
g<z
Z=
g<w
solution
in
(5) + z)
(6)
+ w).
(7)
INTEGERS is
(13)
References Miel, G. “Of Calculations Past and Present: The Archimedean Algorithm.” Amer. Math. Monthly 90, 1735, 1983. Phillips, G. M. “Archimedes in the Complex Plane.” Amer. Math. Monthly 91, 108114, 1984.
Archimedes’
Problem
the simultaneous DIOPHAN‘w, X, Y, 2 (the number of white, black, spotted, and brown bulls) and w, 5, y, z (the number of white, black, spotted, and brown cows), Solution
By taking k = 4 (a 96gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result 223 71
Cattle
(Boyer
1968)
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)
2=
4,893,246
(13)
y =
3,515,820
(14)
z =
5,439,213.
(15)
a/b = c/d IFF the appropria te one of fol lowing fied for INTEGERS m and n: 1, If ma < nb, then
mc < md.
2. If ma = nd, then
mc = nd.
3. If ma > nd, then
mc
ARCHIMEDES' imedes’
conditions
is satis
> nd.
LEMM A is sometimes
also known
as Arch
axiom
References Boyer, C. B, A History p. 99, 1968.
Archimedes’
Cattle
of Mathematics.
New York:
Wiley,
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 number 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?”
A more complicated version of the problem W+X bea SQUARENUMBER andY+Za NUMBER. The solution to this PROBLEM with 206544 or 206545 digits.
requires
that
TRIANGULAR are numbers
References Amthor, A. and Krumbiegel B. “Das Problema bovinum des Archimedes.” 2. Math. Phys. 25, 1X171, 1880. Archibald, R. C. “Cattle Problem of Archimedes.” Amer. Math. MonthEy 25, 411414, 1918. Beiler, A. H. Recreations in the Theory of Numbers: The New York: Dover, Queen of Mathematics Entertains. pp* 249252, 1966. Bell, A. H. “Solution to the Celebrated Indeterminate Equation x2  ny2 = 1.” Amer. Math. Monthly 1, 240, 1894. Bell, A. H. “‘Cattle Problem.’ By Archimedes 251 BC.” Amer, Math. MonthZy 2, 140, 1895, Bell, A* H. “Cattle Problem of Archimedes.” Math. Mag. 1, 163, 18821884. Calkins, K. G. “Archimedes’ Problema Bovinum.” http: //
uww.andrews.edu/calkins/cattle.html. Dijrrie, Great
H.
“Archimedes Problema Bovinum.” 51 in 100 Problems of Elementary Mathematics: Their Hisand Solutions. New York: Dover, pp* 37, 1965.
tory Grosjean, C. C. and de Meyer, H. E. “A New Contribution to the Mathematical Study of the CattleProblem of Archimedes.” In Constantin Carathe’odory: An International Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ: World Scientific, pp. 404453, 1991. Merriman, M. “Cattle Problem of Archimedes,” Pop. Sci. Monthly 67, 660, 1905. Rorres, C. ‘&The Cattle Problem.” http : //www .mcs. drsxel .
edu/crorres/Archimedes/Cattle/Statement.htmL Vardi,
1. “Archimedes’ Cattle 105, 305319, 1998.
Monthly
Problem.”
Amer.
Math.
64
Archimedes
Archimedes’
’ Circles
Archimedes’
Circles
Archimedes’
Recurrence
Formula
Postulate
see ARCHIMEDES' LEMMA Cl
^
Archimedes’ Problem Cut a SPHERE by a PLANE in such a way that the VOLUMES ofthe SPHERICAL SEGMENTS haveagiven RATIO.
c2
~
I
I
see UZSO SPHERICAL SEGMENT
4
Draw the PERPENDICULAR LINE from the intersection of the two small SEMICIRCLES in the ARBELOS. The two CIRCLES Cl and Cz TANGENT to this line, the large SEMICIRCLE, and each ofthetwo SEMICIRCLES are then congruent and known as Archimedes’ circles.
Archimedes’
Recurrence
Formula
see UZSO ARBELOS, BANK~FF CIRCLE, SEMICIRCLE Archimedes’
Constant
see PI Archimedes’ HatBox Theorem Enclose a SPHERE in a CYLINDER and slice PERPENDICULARLY~~ the CYLINDER'S axis. Then the SURFACE AREA of the of SPHERE slice is equal to the SURFACE AREA of the CYLINDER slice.
a, and b, be the PERIMETERS of the CIRCUMSCRIBED and INSCRIBED ngon and ~2~ and b2n the PERIMETERS ofthe CIRCUMSCRIBED and INSCRIBED 2n
Let
gon. Then
Archimedes’ Lemma Also known as the continuity axiom, this LEMMA SWvives 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
References Boyer, C. B. A p. 100, 1968. Archimedes’
The first
follows
u2n = 
h& an + bn
(1)
b2n
&ix*
(2)
=
from
the fact that
side lengths
AXIOMS ST
History
of the
POLYGONS on a CIRCLE of RADIUS T = 1 are
of
Mathematics.
New York:
(4)
Wiley, so
Midpoint
Theorem
a fl=2ntan
If ( n)
(5)
bn=2nsin
IT . (n >
(6)
M
But  h&n a, + 6, Let A4 be the MIDPOINT of the ARC AMB. Pick at random and pick D such that MD 1 AC (where denotes PERPENDICULAR). Then
202ntan 2ntan
(z) 2nsin (c)
tan (t)
C 1
= 4ntan Using
(z)
+ 2nsin
(E) (z)
sin (f$ + sin (E) *
(7)
the identity
AD=DC+BC. tan(+)
see also
MIDPOINT
References Honsberger, R. More Mathematical Morsels. DC: Math. Assoc. Amer., pp. 3132, 1991.
=
tan II: sin 2 tanz + sinz
(8)
then gives Washington,
* hh
an + b,
= azn.
(9)
Archimedean
Solid
The second follows
Archimedean
the identity sin x = 2 sin( ix) cos( $c)
(11)
gives
=&y/sin’
(&)
=4nsin
(&)
=b2n.
(12)
Successive application gives the ARCHIMEDES ALGORITHM, which can be used to provide successive approximations to PI (K).
see also
ARCHIMEDES ALGORITHM, PI H, 100 History
Their
Great and
Problems Solutions.
of Elementary New York:
Mathematics:
Dover,
p. 186,
1965.
Archimedean Solid The Archimedean solids are convex POLYHEDRA which have a similar arrangement of nonintersecting regular plane CONVEX POLYGONS of two or more different 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 Archimedean solids have a spherical symmetry, while the others have “dihedral” symmetry. The Archimedean solids are sometimes also referred to as the SEMIREGULAR
POLYHEDRA.
Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular TETRAHEDRON so that four of their faces lie on the faces the of that TETRAHEDRON. A method of constructing Archimedean solids using a method known as “expansion” has been enumerated by Stott (Stott 1910; Ball and Coxeter 1987, ppm 139140). Let the cyclic sequence S = (~1, ~2, . . . , pp) 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 2 3 and every member 2. cyzl plane
$
2 iq  1, with
TESSELLATION, and
of S is at least 3, equality
in the case of a
a subse
Condition (1) simply says that the figure consists of two or more polygons, each having at least three sides. Condition (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 polyhedra is to eliminate solutions of conditions (1) and (2) using several classes of arguments and then prove that the solutions left are, in fact, semiregular (Kepler 1864, pp. 116126; Catalan 1865, pp* 2532; Coxeter 1940, p. 394; Coxeter et al. 1954; Lines 1965, pp. 202203; Walsh 1972). The following table gives all possible regular 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. S
References Dijrrie,
65
3. for every ODD NUMBER p E S, S contains quence (b, p, b).
from (10)
Using
Solid
(3, (3, (3, (3, (3, (3, (4, (4, (4, (4, (4, (4, (4, (5, (5, (6, (3, (3,
3, 3) 4 4) 6, 6) 8, 8) 10, 10) 12, 12) 4, 4 4 4) 6, 6) 6, 8) 6, 10) 6, 12) 8, 8) 5, 5) 6, 6) 6, 6) 3, 3, n) 3, 3, 3)
Fg.
Solid
P M
tetrahedron triangular prism truncated tetrahedron truncated cube truncated dodecahedron (plane tessellation) ngonal Prism cube truncated octahedron great, rhombicuboct m
A A A T M P A A A T T P
SchlUi
great, rhombicosidodec.
T M P A
(plane tessellation) (plane tessellation) dodecahedron truncated icosahedron (plane tessellation) ngonal antiprism octahedron cuboctahedron
(3, 5, 3, 5) (3, 6, 3, 6)
A
icosidodecahedron
T
(plane tessellation)
(3, 4, 4, 4)
A
small rhombicuboct.
(3, 4, 5, 4)
A T T P
small rhombicosidodec. (plane tessellation) (plane tessellation) icosahedron
snub cube
(3, 3, 3, 3, 5)
A A
snub dodecahedron
(3, (3, (3, (3,
T T T T
(plane (plane (plane (plane
A
(3, 4, 3, 4)
(3, (4, (3, (3,
4 4, 3, 3, 3, 3, 3, 3,
6, 4, 3, 3, 3, 3, 4, 3,
4) 4) 3, 3) 3, 4) 3, 4 3, 3,
6) 4) 4) 3)
tessellation) tessellation) tessellation) tessellation)
{3,5)
4
s{ 4 > C3?6>
As shown in the above table, there are exactly chimedean solids (Walsh 1972, Ball and Coxeter
13 Ar1987).
66
Archimedean
Solid
Archimedean
Theyarecalledthe CUBOCTAHEDRON, GREAT RHOMBICOSIDODECAHEDRON, GREAT RHOMBICUBOCTAHEDRON, ICOSIDODECAHEDRON, SMALL RHOMBICOSIDODECAHEDRONSMALL RHOMBICUBOCTAHEDRONJNUB CUBE, SNUB DODECAHEDRON, TRUNCATED CUBE, TRUNCATED DODECAHEDRON, TRUNCATED ICOSAHEDRON (soccer ball), TRUNCATED OCTAHEDRON, and TRUNCATED TETRAHEDRON. The Archimedean solids satisfy (2n

a)V
= 4n,
where o is the sum of faceangles 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),
Solid
SchEfli
great
rhombicosidodecahedron
great
rhombicuboctahedron
t lJ
icosidodecahedron small
rhombicosidodecahedron
small
rhombicuboctahedron
snub
cube
snub
dodecahedron
truncated truncated
Wythoff 3 4
cuboctahedron
r
s st
cube dodecahedron
C&R
2 ] 3 4
i
2351
:
2341
3
3 5 12
: 4 3 4 3 5 I
3 4 12
3)
2 3 14
w,
34.4
[ 2 3 5
34.5
5x2 4.62
2 3 ] 3
3.62
t{3,5> t@, 4) t{3,3)
3.102
Solid
V
e
cuboctahedron
12
24
120
180
30
2G
48 30
72 60
12
8
20
60 24
120 48
20 8
30 18 6
icosidodecahedron rhombicos. small
smallrhombicub. snub
cube
3.82
2 5 13 2 4 13
icosahedron octahedron tetrahedron
rhombicub.
3.43
12 3 4
2315
truncated truncated
rhombicos.
(3.5)2 3.4.5.4
t(5,31
truncated
great great
(3.4)2
2 13 5
:
t
Solid
f3
f4
8
24
60
32
snub dodecahedron trunc. cube
60 24
150 36
80 8
trunc. trunc.
dodec. icosahedron
60 60
90 90
trunc. trunc.
octahedron
24
tetrahedron
12
36 18
f5
f6
f8
fl0
6 12 6
12 12
12 6
20
12 12 6
20 8 4
4
Let T be the INRADIUS, p the MIDRADIUS, and R the CIRCUMRADIUS. The following tables give the analytic and numerical values of T, p, and R for the Archimedean solids with EDGES of unit length. Solid
r
cuboctahedron
2 4
great
rhombicosidodecahedron
great
rhombicuboctahedron
&(105
+ 6fi)da
&(14+J@Gz
icosidodecahedron
The following of each type 1989, p. 9).
table lists the symbol for the Archimedean
and number of faces solids (Wenninger
i(5+3vq
small
rhombicosidodecahedron
small
rhombicuboctahedron
$J15+2~)~Fz
&(6+4)d=
snub cube
*
snub
*
dodecahedron
truncated
cube
truncated
dodecahedron
truncated
icosahedron
truncated
octahedron
x m 20
truncated
tetrahedron
&rn
&(5
+
2Jz)&zz
&(17Jz+3JiG)JS
~(21+~)&Gz
Archimedean
Solid
Archimedean
Solid
67
Solid cuboctahedron great
rhombicosidodecahedron
great
rhombicuboctahedron
icosidodecahedron small
rhombicosidodecahedron
small
rhombicuboctahedron
snub
cube
snub
dodecahedron
truncated
cube
truncated
dodecahedron
truncated
icosahedron
truncated
octahedron
truncated
tetrahedron
*The
Here
RADII
complicated analytic expressions for the CIRCUMof these solids are given in the entries for the SNUB CUBE and SNUB DODECAHEDRON.
ALS.
Solid
r
0.86603
great
rhombicosidodecahedron
3.73665
3.76938
3.80239
rhombicuboctahedron
2.20974
2.26303
2.31761
icosidodecahedron
1.46353
1.53884
1.61803
small
2.12099
2.17625
2.23295
small
rhombicuboctahedron
snub
cube
snub
dodecahedron
1.22026
1.30656
1.39897
1.15766
1.24722
1.34371
2.15583
2.03987
2.09705
truncated
cube
1.63828
1.70711
1.77882
truncated
dodecahedron
truncated
icosahedron
truncated
octahedron
truncated
tetrahedron
2.88526 2.37713 1.42302 0.95940
2.92705 2.42705 1.5 1.06066
2.96945 2.47802 1.58114 1.17260
The DWAL~ of the Archimedean solids, sometimes the CATALAN SOLIDS, are given in the following
Archimedean
and
paired
with
their
DU
1
great
rhombicosidodecahedron
solids
R
P
0.75
cuboctahedron
are the Archimedean
called table.
The
CANONICAL
solids
their
DUALS
are
all
POLYHEDRA.
rhombicosidodecahedron
deltoidal
hexecontahedron
see also ARCHIMEDEAN SOLID STELLATION, CATALAN SOLID, DELTAHEDRON, JOHNSON SOLID, KEPLERP~INSOT SOLID, PLATONIC SOLID, SEMXREGULAR
small
rhombicuboctahedron
deltoidal
icositetrahedron
POLYHEDRON,
great
rhombicuboctahedron
disdyakis
dodecahedron
great
rhombicosidodecahedron
Archimedean
Solid
truncated snub
Dual
disdyakis
icosahedron dodecahedron
pentakis (laevo)
pentagonal
triacontahedron dodecahedron hexecontahedron
(dextro) snub
cube
(laevo)
pentagonal
icositetrahedron
(dextro) cuboctahedron
rhombic
dodecahedron
icosidodecahedron
rhombic
triacontahedron
truncated
octahedron
tetrakis
truncated
dodecahedron
triakis
icosahedron
hexahedron
truncated
cube
triakis
octahedron
truncated
tetrahedron
triakis
tetrahedron
Here are the Archimedean DUALS (Holden 1971, Pearce 1978) displayed in alphabetical order (left to right, then continuing to the next row),
UNIFORM
POLYHEDRON
References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 136, 1987. Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2. Cambridge, MA: MIT Press, pp. 269286, 1974, Catalan, E. “M&moire sur la Thkorie des Polyedres.” J. Z’&ole Polytechnique (Paris) 41, l71, 1865. Coxeter, H. S. M. “The Pure Archimedean Polytopes in Six and Seven Dimensions.” Proc. Cambridge Phil. Sot. 24, 19, 1928. Coxeter, H. S. M. “Regular and SemiRegular Polytopes I.” Math. 2. 46, 380407, 1940. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Coxeter, H. S. M.; LonguetHiggins, M. S.; and Miller, J. C. P. “Uniform Polyhedra.” Phil. Trans. Roy. Sot. London Ser. A 246, 401450, 1954. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970.
68
Archimedean
Solid Stellation
Archimedes
Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 7986, 1997. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Holden, A. Shupes, Space, and Symmetry. New York: Dover, p. 54, 1991. Kepler, J. “Harmonice Mundi," Opera Omniu, Vol. 5. Frankfurt, pp. 75334, 1864. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 199207, 1942. Le, Ha. “Archimedean Solids.” http://daisy.uwaterloo. ca/hqle/archimedean .html. Pearce, P. Structure in Nature is a Strategy for Design. Cambridge, MA: MIT Press, pp+ 3435, 1978. Pugh, A. Polyhedra: A Visual Approach. Berkeley: University of California Press, p. 25, 1976. Rawles, B, A. "Platonic and Archimedean SolidsFaces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios.”
http://www.intent.com/sg/polyhedra.html. C. “Archimedean Solids: Pappus.” http : //www .mcs . drexel.edu/crorres/Archimedes/Solids/Pappus.html~
Rorres,
Steinitz, orie
Stott,
E. and Rademacher, Polyheder. Berlin, A. B. Verhundelingen der
schuppen,
H. Vorlesungen p. 11, 1934.
iiber
der
Akad.
Konniklijke
die
The
Weten
11, 1910.
Amsterdam
Walsh, T. R. S. “Characterizing the Vertex Neighbourhoods of SemiRegular Polyhedra.” Geometriue Dedicutu 1, 117123, 1972. Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, 1989.
Archimedean Solid Stellation A large class of POLYHEDRA which includes the DoDECADODECAHEDRON and GREAT ICOSIDODECAHEDRON. No complete enumeration (even with restrictive uniqueness conditions) has been worked out.
’ Spiral
see also ARCHIMEDES' SPIRAL, DAISY, FERMAT'S SPIRAL, HYPERBOLIC SPIRAL, LITWUS, SPIRAL References Gray,
A. Modern Differential Geometry of Curves and SurBoca Raton, FL: CRC Press, pp, 6970, 1993. Lauweirer, H. Fructuls: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, ppm 5960, 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. corn/xah/ Special Plane Curves _ dir / ArchimedeanSpiraldir / faces.
l
archimedeanSpiraLhtm1. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge U niversity Press, p, 175, 1967. MacTutor History of Mathematics Archive. “Spiral of Archimedes .” http: //wwwgroups . dcs.stand.ac.uk/ history/Curves/Spiral .html, Pappas, T. “The Spiral of Archimedes,” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 1989.
Archimedean The INVERSE
Spiral Inverse Curve CURVE ofthe ARCHIMEDEAN r
=
SPIRAL
a$‘/”
with INVERSION CENTER at the origin and inversion SPIRAL DIUS k is the ARCHIMEDEAN
RA
T = kaO1’“.
Archimedes’
Spiral
References Coxeter, H. S. M.; LonguetHiggins, J. C. P. “Uniform Polyhedra.” Phil. don Ser. A 246, 401450, 1954. Wenninger, M. 3. Polyhedron ModeZs. University Press, pp. 6672, 1989.
Archimedean A SPIRAL with
M. S.; and Miller, Trans. New
Roy.
York:
Sot.
Lon
Cambridge
Spiral equation
POLAR
ra0
l/m ,
An ARCHIMEDEAN SPIRAL with
where T is the radial distance, 0 is the polar angle, and ~rz is a constant which determines how tightly the spiral is “wrapped.” The CURVATURE of an Archimedean spiral is given by
kc= Various
special
nP+(l
+ n + n202) a(1 + n202j3i2
hyperbolic
spiral
= a@.
This spiral was studied by Conon, and later by Archimedes in On SpiraEs about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral. spiral can be used for COMPASS and STRAIGHTEDGE division of an ANGLE into n parts (including ANGLE TRISECTION) and can also be used for CIRCLE SQUARING. In addition, the curve can be used Archimedes’
’
cases are given in the followine:
T
POLAR equation
table.
as a cam to convert uniform circular motion into uniform linear motion. The cam consists of one arch of the spiral above the XAXIS together with its reflection in the XAXIS. Rotating this with uniform angular velocity about its center will result in uniform linear motion of the point where it crosses the ~AXIS.
Archimedes
’ Spiral
see also ARCHIMEDEAN
Inverse
AreaPreserving
Map
69
CALCULUS and, in particular,
the INTEGRAL, are powerful tools for computing the AREA between a curve f(z) and the XAXIS over an INTERVAL [a, b], giving
SPIRAL
b A= The AREA
of a POLAR
.Ia
(6)
f (4 dx
curve with
A”$ s
equation
T = ~(0) is
r2 d0.
Archimedes’ Spiral Inverse Taking the ORIGIN as the INVERSION CENTER, ARCHIMEDES' SPIRAL T = aeinvertstothe HYPERBOLIC SPI
Written
in CARTESIAN COORDINATES, this becomes
A=;
RAL T= a/& Archimedean Valuation A VALUATION for which 1x1<  1 IMPLIES 1lt21 < C for the constant C = 1 (independent of 2). Such a VALUATION does not satisfy the strong TRIANGLE INEQUALITY
Ix + YI 5 max(Id Ivl>*
(7)
; /b
dy dx xdt Yz > dt
(8)
dY  Y w
(9)
For the AREA of special surfaces or regions, see the entry for that region. The generalization of AREA to 3D is called VOLUME, and to higher DIMENSIONS is called
CONTENT. see also ARC LENGTH, AREA ELEMENT, CONTENT, SURFACE AREA,VOLUME
Arcsecant
see INVERSE SECANT
References Gray, A. “The Intuitive Idea of Area on a Surface.” 513.2 in Modern Differential Geometry of Curves and Surfaces. Coca Raton, FL: CRC Press, pp+ 259260, 1993.
Arcsine
see INVERSE SINE
Area Element The area element
Ar ctangent see INVERSE TA NGENT Area The AREA of a SURFACE is the amount needed to “cover” it completely. The AREA GLE is given by AA = $lh, where
I is the base length
of material of a TRIAN
where
the
side
= &(s

lengths
and h is the height,
a)(s  b)(s  c>, are
a, b, and
or by
c and
(2) s the
bY
A rectangle
=
ab,
(3)
where the sides are length a and b. This gives the special case of 2 A square = a (4)
for the SQUARE. The AREA ofaregular sides and side length
POLYGON with
s is given by
ALgon= +ns2cot( ; >.
ds2 = Edu2
+ 2Fdudv
RIEMANN~AN
+ Gdv2
is dA = JEGFZdu
SEMIPERIMETER. The AREA of a RECTA NGLE is given
n
a SURFACE with
A dw,
(1)
HERON'S FORMULA AA
for
METRIC
(5)
du A dv is the WEDGE PRODUCT. see also AREA, LINE ELEMENT, RIEMANNIAN METRIC, VOLUME ELEMENT
where
References Gray, A. “The Intuitive Idea of Area on a Surface.” 313.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, ppm 259260, 1993.
AreaPreserving A MAP F from Ik”
Map to R”
~vY4)
is AREApreserving
if
= m(A)
for every subregion A of IV, where m(A) is the nD MEASURE of A. A linear transformation is AREApreserving if its corresponding DETERMINANT is equal to 1.
see also C~NFORMAL MAP, SYMPLECTIC MAP
Area Principle
70 Area
Argoh ‘s Conjec t we
Principle
diagram where the two left and right the different LINKS, then a(K+)
P
The
((AREA principle”
+ l(h,
= a(K)
A&l)
1 (mod 5 (mod
=
8) 8)
belong
to
Lz),
where I is the LINKING NUMBER of Arf invariant can be determined from POLYNOMIAL or JONES POLYNOMIAL AK the ALEXANDER POLYNOMIAL of ant is given by
states that
l&PI 
strands
(1)
L1 and La. The the ALEXANDER for a KNOT. For K, the Arf invari
if Arf(K) if Arf(K)
= 0 = 1
(2)
IAlBCl
\AzPl

(1)
lA2BCl’
(Jones
For the JONES POLYNOMIAL TjvK of a
1985).
KNOT K, This
can also be written
in the form
Arf(K)
(2)
Egl =[a’
(Jones
1985),
= W&)
where i is the IMAGINARY
C. C. The Knot Book: to the Mathematical Theory
AB CD
[ 1
(3)
is the ratio of the lengths [A, B] and [C, D] for ABllCD with a PLUS or MINUS SIGN depending on if these segments have the same or opposite directions, and
(4)
Lifitxzl
is the RATIO of signed AREAS of the TRIANGLES. Griinbaum and Shepard show that CEVA'S THEOREM, HOEHN'S THEOREM, and MENELAUS' THEOREM arethe consequences of this result q
NUMBER.
References Adams,
where
(3)
An Elementary of Knots. New
Introduction
York:
W. H. Freeman, pp. 223231, 1994. Invariant for Knots via von NeuJones, V. “A Polynomial mann Algebras .” Bull. Amer, Math. Sot. 12, 103111, 1985. @ Weisstein, E, W. ‘Xnots.” http: //wwv.astro.virginia.
edu/eww6n/math/notebooks/Knots.m. Argand Diagram A plot of COMPLEX NUMBERS as points z=x+iy
see &o CEVA'S THEOREM, HOEHN'S THEOREM, MENELAUS' THEOREM,~ELFTRANSVERSALITY THEOREM
the XAXIS as the REAL axis and ~AXIS as the IMAGINARY axis. This is also called the COMPLEX PLANE or ARGAND PLANE.
References
Argand
Griinbaum, B. and Shepard, G. C. ‘Ceva, Menelaus, Area Principle.” Math, Mug, 68, 254268, 1995.
Area1
Coordinates
TRILINEAR COORDINATES normalized t1 When
and the
+
t2
+
t3
=I
so that
Plane
see ARGAND DIAGRAM Argoh’s Conjecture Let & be the kth BERNOULLI NUMBER. Then nBnl
1.
they become the AREAS of the PAlAa, and PAZAS, where P is whose coordinates have been specified.
so normalized,
TRIANGLES PA1A2, the point
using
Arf Invariant A LINK invariant which always has the value 0 or 1. A KNOT has ARF INVARIANT 0 if the KNOT is “pass equivalent” to the UNKNOT and 1 if it is pass equivalent to the TREFOIL KNOT. If K+, IL, and L are projections which are identical outside the region of the crossing diagram, and K+ and K are KNOTS while L is a 2component LINK with a nonintersecting crossing
z 1
(mod
does
n)
IFF n is PRIME? For example,
for n = 1, 2, , . . , raBnl (mod n) is 0, 1, 1, 0, 1, 0, 1, 0, 3, 0, 1, ... There are no counterexamples less than 12 = 5,600. Any counterexample to Argoh’s conjecture would be a contradiction to GIUGA'S CONJECTURE, and vice versa. l
see ho
BERNOULLI NUMBER, GIUGA'S CONJECTURE
References Borwein, D.; Borwein, J, M,; Borwein, P. B.; and Girgensohn, R. “Giuga’s Conjecture on Primality.” Amer. Math. Monthly 103, 4050, 1996.
Argument
Addition
Relation
Aristotle’s
Argument Addit ion Relation A mathematical relationship relating
f(z
+ y) to f(z)
and f(Y)see also ARGUMENT MULTIPLICATION RECURRENCE RELATION, REFLECTION TRANSLATION RELATION
(Complex
Argument A COMPLEX
NUMBER
RELATION, RELATION,
Wheel Paradox
Argument Principle If f(z) is MEROMORPHIC in a region R enclosed by a curve y, let Iv be the number of COMPLEX ROOTS of f(z) in y, and P be the number of POLES in y, then 1
NP=%
sY
Defining
Number) z may be represented
w s f(z)
see also
the MODULUS
of x, and 8 is called
f ‘(4 dz f( z > ’
and o G f(r)
as
1
Np=
where IzI is called argument
71
VARIATION
gives dw .
2Ki s d w
OF ARGUMENT
the References
arg(rz: + ;y) = tan’
(2)
Duren, P.; Hengartner, W.; and Laugessen, R. S. “The Argument Principle for Harmonic Functions.” Math. Msg.
103,411415,1996. Therefore, Argument arg(zw)
= arg(lzleie”
= arg(eie”eiew)
lwleisw)
= arg[e qz +0w) 1= arg(x) Extending
this procedure
see VARIATION
+ arg(w).
Aristotle’s
= narg(z).
The argument of a COMPLEX called the PHASE.
is sometimes
NUMBER
References M.
of Mathematical Mathematical p. 16, 1972.
Argument Given
and Stegun, C. A. (Eds.). Functions with Formulas, Tables,
(Elliptic
an AMPLITUDE
argument
u is defined
9th
AMPLITUDE,
printing.
Handbook Graphs, and
York:
New
Dover,
Integral) 4 in an ELLIPTIC
INTEGRAL,
the
by the relation
4E see dso
amu.
ELLIPTIC
INTEGRAL
Argument (F’unct ion) An argument of a FUNCTION f(xl, . . . , xn) is one of the n parameters on which the function’s value depends. For example, the SINE sin z is a oneargument function, the BINOMIAL COEFFICIENT (i) is a twoargument function, and the HYPERGEOMETRIC FUNCTION & (a, b; c; z) is a fourargument function. Argument Multiplication A mathematical relationship INTEGER
Wheel
Paradox
(4)
AFFIX, COMPLEX NUMBER, DE MOIVRE’S EULER FORMULA, MODULUS (COMPLEX PHASE, PHASOR
Abramowitz,
OF ARGUMENT
(3)
gives
arg(r”)
see also IDENTITY, NUMBER),
Variation
Relation relating f (nx)
to f(x)
for
72.
see also ARGUMENT ADDITION RENCE RELATION, REFLECTION TION RELATION
RELATION, RELATION,
RECURTRANSLA
A PARADOX mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric 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 smal .I 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:l correspondence of points means that two curves must have the same length. In fact, the CAIEDINALITIES of points in a LINE SEGMENT of any length (or even an INFINITE LINE, a PLANE, a 3D SPACE, or an infinite dimensional EUCLIDEAN SPACE) are all the same: N1 (ALEPHI), so the points of any of these can be put in a ONETOONE correspondence with those of any other. see UZSO ZENO’S
PARADOXES
References Ballew, 507509,
D. “The
Wheel
of Aristotle.”
Math.
Teacher
65,
1972.
Costabel, P. “The Wheel of Aristotle and French Consideration of Galileo’s Arguments.” Math. Teacher 61, 527534,
1968. Drabkin, I. “Aristotle’s Wheel: Notes on the History of the Paradox.” Osiris 9, 162198, 1950. Gardner, M. Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 24, 1983. Pappas, T. “The Wheel of Paradox Aristotle.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 202, 1989. vos Savant, M. The World’s Most Famous Math Problem. New York: St. Martin’s Press, pp. 4850, 1993.
Arithmetic
72
Arithmetic
Arithmetic The branch of mathematics dealing with INTEGERS or, more generally, numerical computation. Arithmetical operations include ADDITION, CONGRUENCE calculation, DIVISION, FACTORIZATION, MULTIPLICATION, POWER computation, ROOT extraction, and SUBTRAC
The AGM
M(a,b)
= M (;(a+
M&b) The Legendre
The L~WENHEIMERSKOLEM THEOREM, whichisafundamental result in MODEL THEORY, establishes the existence of “nonstandard” models of arithmetic.
see ~2s~ ALGEBRA, CALCULUS, FUNDAMENTAL THEGROUP THEORY, HIGHER OREM OF ARITHMETIC, ARITHMETIC, LINEAR ALGEBRA, L~WENHEIMERSKOLEM THEOREM, MODEL THEORY, NUMBER THEORY,TRIGONOMETRY
J. E.
York:
Van Nostrand
Arithmetic
for
Reinhold,
the
Practical
Man.
New
1973.
(1) b n+1= until an = b,. since
da,
(2)
a, and b, converge
= :(a,
an+1  b,+l
6,
towards
each other
+ bn)  da,b,
form
l
2
b),Jab)
(8)
+ x, 1  x)
(9)
(10)
is given
by
= fi +(l n=O
+ kn),
(11)
2dG ~l+knm
to the differential d2Y x3  x) dx2
are given by [A4(1+ generalization
(12)
equation dY
+(3x21)5+z9=0 x,1  2)ll
of
(13)
and [M(l,
x)]?
ARITHMETICGEOMETRIC
the
MEANLY xp’ (2”
which
is related
+
dx
a~)l/~(x~
to solutions
+
(14)
bp)(p1)/p’
of the differential
x(lxp)y”+[l(p+l)xP]Y’(JIl)xPlY
equation = 0. (15)
transformation When p = 2 or p = 3, there is a modular for the solutions of (15) that are bounded as x + 0. Letting J,(x) be one of these solutions, the transformation takes the form JPW
an2&K+bn
&<
bn
(7)
= FM
k n+l=
A ArithmeticGeometric Mean The arithmeticgeometric mean (AGM) M(a, b) of two numbers a and b is defined by starting with a0 E a and bo E b, then iterating
Ab)
where ko E x and
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,
x2 ) = M(l
M(l,x)
Solutions
References
But
b) = M(Xa,
MU, d1 
The FUNDAMENTAL THEOREM OF ARITHMETIC, also called the UNIQUE FACTORIZATION THEOREM, states that any POSITIVE INTEGER can be represented in exactly one way as a PRODUCT of PRIMES.
Mean
has the properties XM(a,
TION.
Geometric
= ~JPW7
(3)
so
lu x = lf2b, < 2\/a,b,.
NOW, add a,  b,  2Janbn
(4
LL=
(p 
(17)
l)u
1 + (p  1)u
(18)
P
to each side and
an t b, 
2&b,
<anb,
xp+up
(5)
= 1.
(19)
case p = 2 gives the ARITHMETICGEOMETRIC MEAN, and 13 = 3 gives a cubic relative discussed by
The
so an+1
 b,+l
< $ ( a,  b n >.
(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, a) is called GAUSS'S CONSTANT.
Borwein and Borwein (1990, 1991) and Borwein in which, for a, b > 0 and I(a, b) defined by I(a, b) =
t dt [(a3 + t3)(b3 + t3)2]1/3
’
(1996)
(20)
Arithmetic
Arithmetic
Geometry
I@, b) = I (F, For iteration
with
[:(a2
+ ab + b’)])
.
(21)
b n+1
(23)
lim
$(un2
Us =
+ anbn + bn”),
I(%
b)
NAPIER’S INEQUALITY
References without
Words:
Mean
Arithmetic
The
Inequality.”
Math.
Msg.
l
Mudular transformations are known when p = 4 and = 6, but they do not give identities for p = 6 (Borwein 1996).
p
ARITHMETICHARMONIC
>a.
68, 305, 1995.
b, = 
n00
see also
Nelson, R. B. “Proof LogarithmicGeometric
I(17 1)
lim
bu > In b  In a
2
(22)
=
Mean
u+b
a, + 2bn an+1 = p 3
n+cx,
see also
ArithmeticLogarithmicGeometric Inequality
a0 = a and bo = b and
73
Mean
MEAN
Arithmetic Mean For a CONTINUOUS DISTRIBUTION function, the arithmetic mean of the population, denoted p, Z, (x), or A(x), is given by
References Abramowitz, M. and Stegun, C. A. (Eds.), “The Process of the ArithmeticGeometric Mean.” $17.6 in Handbook uf Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York: Dover, pp. 571 ad 598599, 1972. Borwein, J. M. Problem 10281. “A Cubic Relative of the 103, 181183, 1996. AGM.” Amer. Math. Monthly Borwein, J. M. and Borwein, P. B. “A Remarkable Cubic Iterat ion.” In Gompututional Method & Function Theory: Proc. Conference Held in Vulpuruiso, Chile, March 1318, 19890387527680 (Ed. A. Dold, B. Eckmann, F. Takens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C., and R. S. Varga). New York: SpringerVerlag, 1990. Borwein, J. M, and Borwein, P, B. “A Cubic Counterpart of Jacobi’s Identity and the AGM.” Trans. Amer. Math. Sot. 323, 691701, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 906907, 1992.
Arithmetic A vaguely
Geometry defined branch of mathematics dealing with VARIETIES, the MORDELL CONJECTURE, ARAKELOV THEORY, and ELLIPTIC CURVES.
CL =
Cornell, G. and Silverman, J. H. (Eds.), try. New York: SpringerVerlag, 1986. Lorenzini, D. An Invitation to Arithmetic dence, RI: Amer. Math. Sot., 1996.
Arithmetic Geometry,
Geome
Provi
=
DISTRIBUTION,
CL= (f(x)) =
P(Xn)f Cxn> c SO z,“=, p( ) Xn
b T&+1=
2&z
(2) The population
mean
satisfies
(f (4 + s(4) = (f (4) + Mx>>
(3)
(cf b>> = c (f (x>>?
(4)
and
(f (WY))
= (f (4)
(5)
MY))
if x and y are INDEPENDENT STATISTICS. The “sample from a statistical mean, ” which is the mean estimated sample, is an UNBIASED ESTIMATOR for the population mean. samples,
the mean
is more efficient than the 7r/2 less (Kenney and Keepexpression which often holds
MEDIAN and approximately ing 1962, p. 211). A general approximately is
=2: 3(mean
(6)
 median).
an
+ bn '
=
lim an = lim b, = Juobo, n+m nF00
the GEOMETRIC MEAN.
Given
a set of samples
Hoehn
and Niven
{xi},
the arithmetic
mean is
(1)
(2)
Then
whichisjust
n=O
Mean
an+1 = +(a + bn)
A(~o,bo)
P(x)f(x)dx,
where (x) is the EXPECTATION VALUE. For a DISCRETE
mean  mode ArithmeticHarmonic Let
Ia oo
For small References
Lfw
(3)
A(al+v2fc
(1985) ,...,
show that
l
a,+c)=~+A(a~,~~
for any POSITIVE constant c. The arithmetic isfies AXDH, 
,...,
u,)
mean
(8) sat(9)
74
Arithmetic
Mean
Arithmetic
Gis the GEOMETRIC MEAN and H is the HARMONIC MEAN (Hardy et al. 1952; Mitrinovic 1970; Seck
where
Progression
LSTATISTICS for a GAUSSIAN DISTRIBUTION, the UNBIASED ESTIMATOR for the VARIANCE is given by
From
enbach and Bellman 1983; Bullen et al. 1988; Mitrinovid. et al. 1993; Alzer 1996). This can be shown as follows. For a, b > 0,
(&$2zo 1 
2 z+b10
a
11 +ca
(10)
N
1 N n
(11)
 Z)“,
Xi
SO
A& 
(13)
H > G
(14)
(22)
i=l n
(12)
a
2
where SI
1 2
;+;
(21)
var@) The
= j&.
(23)
SQUARE ROOT ofthis,
IFF b = a. To show the second part of the
with equality inequality,
is called (fiJb)2=
a2dx+b>O
a+b
(16)
 (z)“,
(25)
SO
(55”) = var(%) + (5)2 = $
A > H, with equality gives (9).
E (z2)
var(Z)
(15)
>a 
2
the STANDARD ERROR.
+ p2.
(26)
07)
IFF a = b. Combining
(14) and (17) then
Given n independent random GAUSSIAN DISTRIBUTED variates xi, each with population mean pi = p and
VARIANCE oi2 = 02,
see also ARITHMETICGEOMETRIC MEAN, ARITHMETICHARMONIC MEAN, CARLEMAN'S INEQUALITY, CUMULANT, GENERALIZED MEAN, GEOMETRIC MEAN, HARMONIC MEAN, HARMONICGEOMETRIC MEAN, KURTOSIS, MEAN, MEAN DEVIATION, MEDIAN (STATISTICS), MODE, MOMENT, QUADRATIC MEAN, ROOTMEANSQUARE,~AMPLE VARIANCE,~KEWNESS, STANDARD DEVIATION, TRIMEAN, VARIANCE Heterences Abramowitz,
M.
of
1 (Ir:>= N
Iv IL (
i=l
Mathematical Mathematical
= $F
Xi
)
Inequality.”
(19) so the sample mean is an UNBIASED ESTIMATOR of population mean. However, the distribution of z depends on the sample size. For large samples, z is approximately NORMAL. For small samples, STUDENT'S tDISTRIBWTXON should be used.
var(Z)
= var
(i&i)
mean
= &var
is independent
(g)
Stegun,
Functions Tables, 9th
C. A.
Handbook
(Eds.).
with Formulas, printing. New
p. 10, 1972. Alter, H. “A Proof of the Arithmetic
( Xi > i=l
The VARIANCE of the population of the distribution.
and
Graphs,
Dover,
MeanGeometric
Mean
Monthly 103, 585, E. F. and Bellman, R. Inequalities. Amer.
Math.
and
York:
1996.
Beckenbach, New York: SpringerVerlag, 1983. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 471, 1987. Bullen, P. S.; Mitrinovie, D. S.; and VasX, P, M. Means & Their Inequalities. Dordrecht, Netherlands: Reidel, 1988. Hardy, G. H.; Littlewood, J. E.; and Pblya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952. Hoehn, L. and Niven, I. “Averages on the Move.” Math. Mag. 58, 151156, 1985. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt+ 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Mitrinovie, D. S.; PeEarZ, J, E.; and Fink, A. M. Classical and New Inequalities in Analysis. Dordrecht, Netherlands: Kluwer, 1993. Vasic, P. M+ and Mitrinovie, D. S. Analytic Inequalities. New York: SpringerVerlag, 1970,
Arithmetic
Progression
see ARITHMETIC
SERIES
Arithmetic
Sequence
Arnold’s
Arithmetic Sequence A SEQUENCE of n numbers {do + Icd}Lzi 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 {ak}, k = 1, 2, in which each term is computed from the previous one by adding (or subtracting) a constant. Therefore, for k > 1, l
ak
“,
=akl+d=ak2+2d=...=al+d(k1).
The sum given by
of the sequence
(1) of the first
n
terms
is then
Mathematical
Approach
ak = ?[a1

 1) = nal
New
York:
Dover,
Mathematical Tables, CRC Press, pm 8, 1987. H. “The Arithmetical ProgresStandard
Mathematics?: An Elementary and Methods, 2nd ed. Oxford, England:
to Ideas
CA: Wide
l
l
NARCISSISTIC NUMBER
References
+ (k  l)d]
Sloane, N. J. A. Sequence A005188/M0488 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
k=l
nal + dk(k
printing.
75
Armstrong Number The ndigit numbers equal to sum of nth powers of their digits (a finite sequence), also called PLUS PERFECT NUMBERS. They first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, (Sloane’s A005188).
see also k=l
9th
Oxford University Press, pp. 1213, 1996. Pappas, T. The Joy of Mathematics. San Carlos, World Publ./Tetra, p. 164, 1989.
.
S, E 2
Tables,
p. 10, 1972. Beyer, W. H. (Ed.). CRC 28th ed. Boca Raton, FL: Courant, R. and Robbins, sion.” 51.2.2 in What is
Cat Map
+ dk(k
 1)
(2)
Arnold’s Cat Map The best known example of an ANOSOV DIFFEOMORPHISM. It is given by the TRANSFORMATION
(1) Using
the SUM identity n
IE
 $t(n
(3)
+ 1)
where xn+l and yn+l are computed mod 1. The Arnold cat mapping is nonHamiltonian, nonanalytic, and mixing. However,itis AREAPRESERVING since the DETERMINANT~~ 1. The LYAPUNOV CHARACTERISTIC EXPONENTS are given by
then gives la Sn = nal Note,
however,
+ id(n
 1) = $,n[2al
1
2g
1
(4)
+ d(n  I)].
that
 30 + 1 = 0,
=c2
o’f = i(3fJ5).
~1 + a, = al + [al + d(n  l)] = 2al + d(n  l),
(5)
The EIGENVECTORS are found by plugging MATRIX EQUATION
so S, = +(a,
(6)
+a,),
or n times the AVERAGE of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the INTEGERS from 1 to 100 given as busywork by his teacher. While his classmates toiled away doing the ADDITION longhand, Gauss wrote a single number, the correct answer
[
lT*
210*]
For ol+, the solution
[;I
=
[;I
of
l
(2) (3) into the
C4)
is
y=
~(1+J5)xqbx,
where q3 is the GOLDEN RATIO, ized) EIGENVECTOR is
(5) so the unstable
(normal
(6) $00)(1+
100) = 50 ’ 101 = 5050
(7) Similarly,
on his slate. When the answers were examined, proved to be the only correct one.
for U,
Gauss’s
see also ARITHMETIC SEQUENCE, GEOMETRIC SERIES, HARMONIC SERIES, PRIME ARITHMETIC PROGRESSION
the solution
y = i(& so the stable
(normalized)
References Abramowitz, of Mathematical
M. and Stegun, Functions
C. A. with
(Eds.).
Formulas,
Handbook Graphs, and
see
also
ANOSOV MAP
is
 1)x E $lx,
EIGENVECTOR is
(7)
Arnold
76
Diffusion
Array
Arnold Diffusion The nonconservation of ADIABATIC INVARIANTS which arises in systems with three or more DEGREES OF FREE
Arrangement
DOM.
Array An array is a “list of lists” with the length of each level of list the same. The size (sometimes called the “shape”) of a ddimensional array is then indicated as m X n X *v X p. The most common type of array en\ d Y
Arnold Tongue Consider the CIRCLE MAP. If K is NONZERO, then the motion is periodic in some FINITE region surrounding each rational 0. This execution of periodic motion in response to an irrational forcing is known as MODE LOCKING. If a plot is made of K versus 0 with the regions of periodic MODELOCKED parameter space plotted around rational St values (the WINDING NUMBERS), then the regions are seen to widen upward from 0 at K = 0 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 CANTOR SET of dimension d $=: 0.8700. In general, an Arnold tongue is defined as a resonance zone emanating out from RATIONAL NUMBERS in a twodimensional parameter space of variables. see
also
CIRCLE
MAP
Aronhold Process The process used to generate an expression for a covariant in the first degree of any one of the equivalent sets of COEFFICIEKTS for a curve. see also C EBSCH ARON HOLD THAL'S EQ ATION
N OTATIO N, JOA CHIMS
References Coolidge, J. L. A Treufise on Algebraic York: Dover, p* 74, 1959.
Plane
Curves.
New
Number
see PERMUTATION
l
d
countered is the 2D m x n rectangular array having m columns and n, rows. If ~2 = n, a square array results. Sometimes, the order of the elements in an array is significant (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 arrays of a given shape with each element being one of Fz 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 krnn*‘** 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 obtained using the POLYA ENUMERATION THEOREM. For the simple case of an m, x n array, even this proves unnecessary since there are only a few possible symmetry types, allowing the possibilities to be counted explicitly. For example, consider the case of VL 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
Aronson’s Sequence The sequence whose definition is: “t is the first, fourth, eleventh, . . letter of this sentence.” The first few values are 1, 4, 11, 16, 24, 29, 33, 35, 39, . . (Sloane’s
9 
bcldef hiljkl 
m
n
0
P
Q
T
S
t
U
V
w
x7
l
+



l
A005224).
References Hofstadter, D. R, Metamagical Themas: Questing of Mind and Puttern. New York: BasicBooks, p 44, 1985. Sloane, N. J. A. Sequence A005224/M3406 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Arrangement In general, an arrangement of objects is simply a grouping of them. The number of “arrangements” of n, items is given either by a COMBINATION (order is ignored) or PERMUTATION (order is significant) l
The division of SPACE into cells by a collection PERPLANES is also called an arrangement. see also COMBINATION,
DERING,
CUTTING,
HYPERPLANE,
of HYOR
where each a, b, . . . , ~1:can take a value from 1 to k. The total number of possible arrangements is kz4 (km, in general). The number of arrangements which are equivalent to their leftright mirror images is P2 (in general, k mn/2 ), as is the number equal to their updown mirror images, or their rotations through 180”. There are also k” arrangements (in general, kmn/4) with full symmetry. In general,
k
it is therefore
mn/4 mn/2
k mn/2 k
_
kmni4
_
kmd4
k mn/2
k
mn/4
true that with with with with
full symmetry only leftright reflection only updown reflection only 180’ rotation,
so there are
PERMUTATION k
mn
 3k
mn/2 + 2km”/4
’
Arrow
Artin
Notation
arrangements with no symmetry. number of images of each type, with m, n EVEN, is N(m,n,
k) = $kmn
Now dividing by the the result, for m # n
+ (i)(3)@““/”
+ +(k= +krnn
+ ;kmni2
+ 2kmni4) + $kmn/4m
The number is therefore of order O(kmn/4), rection” terms of much smaller order. see
also
ANTIMAGIC
SQUARE,
with
EULER
Group
77
see CZZSOACKERMANNNUMBER,CHAINED ARROW NoTATION, DOWN ARROW NOTATION, LARGE NUMBER, POWER TOWER,~TEINHAUSMOSER NOTATION References
 kmnj4)
_ 3kmni2
Braid
“cor
SQUARE,
KIRKMAN'S SCHOOLGIRL PROBLEM, LATIN RECTANGLE, LATIN SQUARE, MAGIC SQUARE, MATRIX, MRS. PERKINS' QUILT, MULTIPLICATION TABLE, ORTHOGONAL ARRAY,PERFECT SQUARE, PRIME ARRAY, QUOTIENTDIFFERENCE TABLE, ROOM SQUARE, STOLARSKY ARRAY,TRUTH TABLE, WYTHOFF ARRAY
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 5962, 1996. Guy, R. K. and Selfridge, J. L. ‘(The Nesting and ‘Roosting Habits of the Laddered Parenthesis.” Amer. Math. Monthly 80, 868876, 1973. Knuth, D. E. “Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations.”
194, 1235~1242,1976.
Science
Vardi, I.
Computational
Recreations
in
wood City, CA: AddisonWesley,
Arrow’s Paradox Perfect democratic voting principle, impossible.
Red
Mathematics.
pp. 11 and 226229,
is, not just
in practice
1991.
but in
References Arrow Notation A NOTATION invented by Knuth (1976) to represent LARGE NUMBERS in which evaluation proceeds from the right (Conway and Guy 1996, p. SO).
n
For example,
(1)
mTn=m” *m m~~n=m~*~m=mm’ 
n
(2) d
3
=mm
=mTmm
m
(3) .m
m
TTf
2
=
m
(4)
TT v
d
(5)
ARROWHEAD
ILLUMINATION
UZSO
CURVE
PROBLEM
References Honsberger, R. “Chv~tal’s in Mathematical Gems
Art IL
Gallery
Washington,
Theorem.” Ch. 11 DC: Math. Assoc.
Amer., pp. 104110, 1976. O’Rourke, J. Art Gallery Theorems and Algorithms. New York: Oxford University Press, 1987. Stewart, I. “How Many Guards in the Gallery?” Sci. Amer. 270,118120, May 1994, Tucker, A. “The Art Gallery Problem.” Math Horizons,
pp. 2426, Spring Wagon,
S. “The
1994. Gallery Theorem.” $10.3 in MathemaNew York: W. H. Freeman, pp. 333345,
Art
in Action.
Articulation Vertex A VERTEX whose removal called a CUTVERTEX.
will disconnect
a GRAPH,
also
(GRAPH)
References
,m
m TT n is sometimes called a POWER values n T  T n are called ACKERMANN
G. “CutVertices
Chartrand,
m
tory
l
Bewilder
pa 56, 1988+
Curve
see also BRIDGE
l
Mathematical
m’
m’
m
Other
1991.
.m
,m
and
Art Gallery Theorem ART GALLERY THEOREM. If Also called CHV~;TAL’S the walls of an art gallery are made up of n straight LINES SEGMENTS, then the entire gallery can always be supervised by Ln/3] watchmen placed in corners, where 1x1 is the FLOOR FUNCTION. This theorem was proved by V. ChvStal in 1973. It is conjectured that an art gallery with n walls and h HOLES requires l(n + Ii)/31 watchmen.
tica
m
2
Travel
New York: W. H. freeman,
Arrowhead
see n
m~~2=m~m=m~m=m”
m~~3=m~m~m=m~(mjm) L
M. Time
ments.
see SIERPI~~SKI
rnarn•mm
m?n.
Gardner,
TOWER. The NUMBERS.
Artin
Gruph
Theory.
Braid
see BRAID
Group
GROUP
New
and Bridges.” York:
Dover,
52.4 in Introduc
pp. 4549, 1985.
78
Artin’s
Artis tic Series
Conjecture
Artin’s Conjecture There are at least two statements which go by the name The first is the RIEMANN HYof Artin’s conjecture. 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 multiples of a constant known as ARTIN'S CONSTANT. There is an analogous theorem for functions instead of numbers which has been proved by Billharz (Shanks 1993, p* 147). see UZSO ARTIN?
CONSTANT,
RIEMANN
D. Solved and Wnsolued Problems in Number Theory, ed. New York: Chelsea, pp. 31, 8083, and 147, 1993.
4th
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 conjecture was solved in 1967 by C. Hooley. If, in addition, n is not an rth POWER for any T > 1, then Artin conjectured that the density of S(n) relative to the PR .IMES (independent of the choice of n), where is CArtin
c Artin
1
1p
1 =
da  1)
0.3739558136
see also LANGLANDS
l l
l
?
and the PRODUCT is over PRIMES. The significance of this constant is more easily seen by describing it as the DECfraction of PRIMES p for which l/p has a maximal IMAL EXPANSION (Conway and Guy 1996).
Artin
Artin LFunction An Artin Lfunction
over the RATIONALS Q encodes in information about how an irreducible manic POLYNOMIAL over Z factors when reduced modulo each PRIME. For the POLYNOMIAL x2 +l, the Artin Lfunction is a
GENERATING FUNCTION
w, W/Q sgn)= P odd
rI
prime
l
1 ($)ps’
Amer.
Math.
and Harmonic
Anal
43, 537549,
1996.
Sot.
Reciprocity RECIPROCITY
THEOREM
Artin’s Reciprocity Theorem A general RECIPROCITY THEOREM for all orders. If R is a NUMBER FIELD and R’ 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 ~ZSOLANGLANDS PROGRAM Artinian Group A GROUP in which any decreasing CHAIN of distinct SUBGROUPS terminates after a FINITE number. Artinian Ring A noncommutative “descending chain
SEMISIMPLE RING
satisfying
the
condition.”
see also GORENSTEIN RING,
SEMISIMPLE RING
References Artin, Artin,
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, p* 169, 1996. Finch, S. “Favorite Mathematical Constants.” http: //uww. mathsoft. com/asolve/constant/artin/artin,html. Hooley, C. “On Artin’s Conjecture.” J. reine angew. IM&h, 225, 209220, 1967. Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: SpringerVerlag, 1990. Ribenboim, P. The Book of Prime Number Records. New York: SpringerVerlag, 1989. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed, New York: Chelsea, pp, 8083, 1993. Wrench, J. W. “Evaluation of Artin’s Constant and the Twin Prime Constant .” Math. Comput. 15, 396398, 1961.
Representations
ysis, Part 11.” Not.
Abh.
References
RECIPROCITY
References Knapp, A. W. “Group
see ARTIN’S
HYPOTHESIS
References Shanks,
where (l/p) is a LEGENDRE SYMBOL, which is equivalent to the EULER 1;FUNCTION. The definition over arbitrary POLYNOMIALS generalizes the above expression.
Abh.
E. “Zur
Theorie
5, 251260,
IL “Zur
der hyperkomplexer
Arithmetik
5, 261289,
Zahlen.”
Hamb.
Zahlen.”
Humb.
1928.
hyperkomplexer
1928.
Artistic Series A SERIES is called artistic if every three terms have a common threeway ratio (ai P[Ui,
%+1,
G+2]
+
ai+
+
consecutive
&+2)&+1
=
l
ai&+2
A SERIES is also artistic IFF its BIAS is a constant. A GEOMETRIC SERIES with RATIO r > 0 is an artistic series with 1 P= +1+r>3. T
see also BIAS (SERIES) , GEOMETRIC
SERIES,
MELO DIG
SERIES References Duffin,
Co.”
R. J. “On Seeing Progressions Amer.
Math.
MO snthly 100,
of Constant 3847,
1993.
Cross
Ra
.
ASA
Theorem
ASA
Theorem
Associative
Magic
Square
79
Associative In simple terms, let zc, y, and z be members of an ALGEBRA. Then the ALGEBRA is said to be associative if x
l
(y
’
x)
=
(x
l
y)
l
2, (1)
where denotes MULTIPLICATION. More formally, let A denote an Ralgebra, so that A is a VECTOR SPACE over Iw and AxA+A (2) l
Specifying two adjacent ANGLES A and B and the side between them c uniquely determines a TRIANGLE with AREA
2
K
2(cot A;
(1)
cot B) ’
The angle C is given in terms
of A and B by
Then
A is said to be massociative S of A such that
(3)
if there exists an mD
by using
the
(y
l
2)
l
z
=
y
’
(x
’
iz) (4)
OF SINES
a sin A 

b sinB
C
(3)
 sinC
to obtain
for all y,z f A and x f S. Here, VECTOR MULTIPLICATION x y is assumed to be BILINEAR. An nD nassociative ALGEBRA is simply said to be “associative.” n
see
sin sin@ sin sin@ 
U= b=
A A  B)’ B A  B) ”
(4)
COMMUTATIVE,
~2~0
Finch, S. “Zero Structures in Real mathsoft,com/asolve/zerodiv/zerodiv.htmL
Associative
Magic
1
Aschbacher’s Component Theorem Suppose that E(G) (the commuting product of all components of G) is SIMPLE and G contains a SEMISIMPm INv0Lu~10N. Then there is some SEMISIMPLE INVOLUTION z such that CG(X) has a NORMAL SUBGROUP K which is either QUASISIMPLE OF ISOMORPHIC to 0+(4,q)’ and such that Q = C&Y) is TIGHTLY EM
BEDDED. see also INVOLUTION (GROUP), ISOMORPHIC GROUPS, NORMAL SUBGROUP, QUASISIMPLE GROUP, SIMPLE GROUP, TIGHTLY EMBEDDED Theorem
/q c
/\ C
c
Specifying two adjacent side lengths a and b of a TRIANGLE (taking a > b) and one ACUTE ANGLE A opposite a does not, in general, uniquely determine a triangle. If sin A < a/c, there are two possible TRIANGLES satisfying the given conditions. If sin A = a/c, there is one possible TRIANGLE. If sin A > u/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 REM, SSS THEOREM, TRIANGLE
Algebras." http: //www.
(5)
also
/q
DISTRIBUTIVE
References
AAA THEOREM, AAS THEOREM, ASS THEOREM, SAS THEOREM,SSS THEOREM, TRIANGLE
ASS
x ’ y*
(2)
and the sides a and b can be determined
see
*
SUBSPACE
C=rAB,
LAW
(XI Y)
THEOREM, SAS THEO
Square 15
23
7
9
18
24
16
2
8
17
5
14
11
25
An 72 x n MAGIC SQUARE for which every pair of numbers symmetrically opposite the center sum to n2 + 1. The LO SHW 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 associative PANMAGIC SQUARES exist, one of which is illustrated above (Gardner 1988). see
MAGIC
also
SQUARE,
PANMAGIC
SQUARE
References Gardner, Travel
York:
M. and
“Magic Squares and Cubes.” Other
Mathematical
W. H. Freeman,
1988.
Ch. 17 in Time
Bewilderments.
New
80
As troid
As troid with
Astroid
n = 4, s4 = 6a.
(11)
The AREA is given by A, = ( n  wn2 with
 2) m2
(12)
n = 4, A4 = $u2.
(13)
EV~LUTE of an ELLIPSE is a stretched HYPOCYCLOID. The gradient of the TANGENT T from the point
The A 4cusped HYPOCYCLOID which is sometimes also called a TETRACUSPID, CUBOCYCLOID, 0r PARACY CLE. The parametric equations of the astroid can be obtained by plugging in n = a/b = 4 or 4/3 into the equations for a general HYPOCYCLOID, giving x = 3bcos++ y = 3bsin$
bcos(3g5) = 4bcos3q5 = acos34  bsin(3+)
= 4bsin3 4 = asin
$.
(1) (2)
with
parameter
w
+
y2/3
=
In PEDAL COORDINATES with center, the equation is T2 +3p2
t
pa
t
t and TANGENTIAL AN
s(t) =gst
1sin( at’) 1dt’ = % sin2 t
Archive).
a constant
L 4
t
The astroid can also be formed as the ENVELOPE produced 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 vertical 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 (x0,0) and (O,dw), respectively. The equation of the LINE made by the ladder with its foot at (x0,0) is therefore
(5)
JL’iEGq
0
kc(t) =  $ csc(2t)
(6)
4(t) = 4.
(7)
As usual, care must be taken in the evaluation of s(t) for the t > T/2. Since (5) comes from an integral involving ABSOLUTE VALUE of a function, it must be monotonic increasing. Each QUADRANT can be treated correctly by defining 2t n= +l, (8) L 7r 1 where
1x1 is the FLOOR FUNCTION, giving s(t) = (1) l+[n
8a(n  1) n
(10)
(15)
x0>
which can be written U(X,Y,Xo)
(x
= Y+
_ x0) .
x0
(16)
The equation of the ENVELOPE is given by the simult aneous solution of
{ (9)
X
x0
U(x,
The overall ARC LENGTH of the astroid can be computed from the general TYPO CYC LOID formula S r&=
yo=
the formula
(mod 2)) 23 sin2 t + 3 L$n] .
(14)
Let T cut the XAXIS and the yThen the length XY is and is equal to a.
(MacTutor
the PEDAL POINT at the
The ARC LENGTH$URVATURE, GLE are
of this TAN
AXIS at X and Y, respectively.
(3)
= u2.
The equation
II: sinp + y cosp = &zsin(2p)
In CARTESIAN COORDINATES, X
p is  tanp.
GENTS is
au ax0
y,
x0)
= x

J L2,02
y +

X0
(x

xo2>51p
=
x0)
=
0 (17)
2L,2 O,
which is *u MA
x=
Y=
3
(18)
L2 W
2
xo2)3'2
L2
l
(19)
Astroid
As hid Noting
81
Involute
that x2/3 Y
w


allows this to be written x2/3
the equation
xo p/3
2
(20)
L2  202 p/3
implicitly + y2/3
of the astroid,
(21) as
= LV,
(22)
as promised. The astroid LIPSES
is also the ENVELOPE of the family $+L
see
U~SO
(30)
l=O,
(l42
illustrated
of EL
above. DELTOID,
ELLIPSE
ENVELOPE,
LAMI?
CURVE,
NEPHROID, RANUNCULUID References
+AL+L+
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 20 and ANGLE 8 is given by ALcosO
X==
y = .JL2=
(23) + ALsinO.
(24)
Using
Lawrence,
J. D. A Catalog of Special Plane Curves. New Dover, pp. 172175, 1972. http://www.best.com/xah/Special Lee, X. “Astroid.” PlaneCurvesdir/Astroiddir/astroid, html. Lockwood, E. H. “The Astroid.” Ch. 6 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 52York:
61, 1967. MacTutor
of
Yates, R. C. “Astroid.” Properties. Ann Arbor,
Astroid x0 = LCOSO
History
Mathematics
http://wwwgroups.dcs.stand. /Astroid.html.
“Astroid.” Archive. ac .uk/*history/Curves
A Handbook on Curves and Their MI: J. W. Edwards, ppm 13, 1952.
Evolute
(25)
then gives AL x = x0 L
(26)
(27) Solving
(26) for ~0, plugging
into (27) and squaring
then
A HYPOCYCLOID EVOLUTE for n = 4 is another AsTROID scaled by a factor n/(n  2) = 4/2 = 2 and rotated l/(2 4) = l/8 of a turn.
(28
Astroid
l
gives
Y2= L2 L2s2 (AT,>2 (l+g2* Rearranging
produces X2
~@JQ2
Involute
the equation Y2
+ (L + AL)2
= ”
(29)
equation of a (QUADRANT of an) ELLIPSE with SEMIMAJOR and SEMIMINOR AXES of lengths AL and the
L+AL. A HYPOCYCLOID INVOLUTE for n = 4 is another TROID scaled by a factor (n  2)/2 = 2/4 = l/2 rotated l/(2 4) = l/8 of a turn. l
Asand
82
Astroid
Astroid
Pedal
Pedal
Curve
Asymptotic
Curve
u
Asymptote
Curve
asymptotes
The PEDAL CURVE ofan ASTROID at the center is a QUADRIFOLIUM. Astroid
Radial
with
PEDAL
A curve approaching a given curve arbitrarily illustrated in the above diagram, see also CURVE
Curve
ASYMPTOSY,
ASYMPTOTIC,
closely,
as
ASYMPTOTIC
References Giblin, P. J. “What 274284, 1972.
 3a cos(3t)
UZSO ASYMPTOSY, CURVE, ASYMPTOTIC RIES, LIMIT
see
y = yo + 3a sin i! + 3a sin(3+
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
Math.
Gaz.
56,
Asympt 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.
The QUADRIFOLIUM x = x0 + 3acmt
is an Asymptote?”
( a cos u cos v)”
ASYMPTOTE, ASYMPTOTIC DIRECTION, ASYMPTOTIC SE
Asymptotic Curve Given a REGULAR SURFACE M, an asymptotic curve is formally defined as a curve x(t) on M such that the NORMAL CURVATURE is 0 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
Y = (b sin u cos V) 3 x=
ed2 + 2 f&i + pi2 = 0,
(csin7.Q3
for u f [7r/27/2] and 21 E [n, ~1. The special case a = b = c = 1 corresponds to the HYPERBOLIC OCTAHEDRON. see
also
ELLIPSOID,
HYPERBOLIC
(1)
FORMS. where e, f, and g are second FUNDAMENTAL The differential equation for asymptotic curves on a MONGE PATCH (u,v, h(u,v)) is
OCTAHEDRON huuutZ + 2hUUu’v’
+ h,,vt2
= 0,
References Nordstrand, people/nf
Asymptosy ASYMPTOTIC found rarely Dictionary.
T. “Astroidal Ellipsoid.” ytn/asttxt . htm.
http : //wwn.
uib .no/
and on a polar
patch h”(r)rf2
behavior. A useful yet endangered word, outside the captivity of the Oxj%rd English
see also ASYMPTOTE,
ASYMPTOTIC
(T cos 0, T sin 8, h(r)) + h’(r)d2
= 0.
(2)
is (3)
The images below show asymptotic curves for the ELLIPTIC HELICOID, FUNNEL, HYPERBOLIC PARABOLOID, and MONKEY SADDLE.
Asymptotic
AtiyahSinger
Direction
see also RULED
a
Curves,” “Examples of Asymptotic Curves,” “Using Mathematics to Find Asymptotic Curves.” $16.1, 16.2, and 16.3 in Modern Diflerential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 320331, 1993.
1. There
Direction direction at a point p of a REGULAR Iw3 is a direction in which the NORMAL M vanishes.
are no asymptotic are exactly
two asymptotic
directions
at a HY
PERBOLIC POINT. 3. There
is exactly
one asymptotic
direction
at a PAR
at a PLANAR
POINT.
ABOLIC POINT. 4. Every
direction
Asymptotic Notation Let n be a integer variable which tends to infinity and let x be a continuous variable tending to some limit. Also, let 4(n) or $(z) b e a p osi t ive function and f(n) or f(z) any function. Then Hardy and Wright (1979) define 1. f = O(4) t o mean that 1f 1 < A# for some constant A and all values of n and 5, 2. f = o($) to mean that f/4 + 0, 3. f  4 to mean that f/4 + 1, 4. f + 4 to mean
the same as f = o(4),
5. f > 4 to mean
f/4
and is stronger
than
f = O(4).
References G. H. and Wright, Introduction
to the
Clarendon
xnRn(x > =0
x”Rn(x)
lim
to have the properties
=m
for fixed n
(4)
for fixed x.
(5)
nkm
Therefore, 00 
f (2) m > ;
(6)
UnXBn
in the limit x + 00. If a function has an asymptotic expansion, the expansion is unique. The symbol  is SIMILAR. also used to mean directly References of Mathematical Mathematical
M.
and
Stegun,
Functions Tables, 9th
C. A.
with printing.
of Semiconvergent
Methods
for
(Eds.).
Formulas New
Physicists,
Handbook Graphs, and
York: Series.”
3rd
ed.
Dover, $5.10 in Orlando,
FL: Academic Press, pp. 339346, 1985. Bleistein, N. and Handelsman, R. A. Asymptotic Expansions of Integrals. New York: Dover, 1986. Copson, E. T. Asymptotic Expansions. Cambridge, England: Cambridge University Press, 1965. de Bruijn, N. G. Asymptotic Methods in Analysis, 2nd ed. New York: Dover, 1982. Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973. Erdelyi, A. Asymptotic Expansions. New York: Dover, 1987. Morse, P. M. and Feshbach, H. “Asymptotic Series; Method of Steepest Descent.” 54.6 in Methods of Theoretical Physics, Part I. New York: McGrawHill, pp* 434443, 1953. Olver, F. W. J. Asymptotics and Special Functions. New York: Academic Press, 1974. Wasow, W. R. Asymptotic Expansions for Ordinary Differential Equations. New York: Dover, 1987.
+ 00, and
6. f x 4 to mean AlqS < f < A2 for some positive constants A1 and AZ.
England:
lim
a:+00
Mathematical
Gray, A. Modern Differential Geometry of Curves and Surfuces.Boca Raton, FL: CRC Press, pp. 270 and 320, 1993.
An
*
rL,
series is defined
p* 15, 1972. A&en, G. “Asymptotic
References
Hardy,
The asymptotic
Abramowitz,
is asymptotic
see also ASYMPTOTIC CURVE
f = o(4) implies
83
at an ELLIPTIC
directions
POINT.
2. There
Theorem
where
SURFACE
References Gray, A+ “Asymptotic
Asymptotic An asymptotic SURFACE M f CURVATURE of
Index
E. M. “Some Theory
Press,
Notation.”
of Numbers,
5th
ed.
$1.6 in Oxford,
pp. ,78, 1979.
AtiyahSinger Index Theorem A theorem which states that the analytic and topological “indices” are equal for any elliptic differential operator on an nD COMPACT DIFFERENTIABLE Cc” boundaryless MANIFOLD. see also COMPACT
MANIFOLD,
DIFFERENTIABLE MAN
IFOLD fleterences
Asymptotic Series An asymptotic series is a SERIES EXPANSION of a FUNCTION in a variable z which may converge or diverge (Erdelyi 1987, p. l), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough 2. To form an asymptotic series R(z) of f(z), written f (4 take
 R(x),
(1)
Atiyah, M. F. and Singer, I. M. “The Index of Elliptic Operators on Compact Manifolds.” Bull. Amer. Math. Sot. 69, 322433, 1963. Atiyah, M. F. and Singer, I. M. “The Index of Elliptic Operators I, II, III.” Ann. Math. 87, 484604, 1968. Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 4, 1996.
A tkinGoldwasserKilianMorain AtkinGoldwasserKilianMorain Certificate A recursive PRIMALITY CERTIFICATE for a PRIME The certificate consists of a list of 1. A point
on an ELLIPTIC
p.
Pair
with distinct BASINS OF ATTRACTION. This restriction is necessary since a DYNAMICAL SYSTEM may have multiple attractors, each with its own BASIN OF ATTRACConservative systems do not have attractors, since the For dissipative DYNAMICAL SYSmotion is periodic. TEMS, however, volumes shrink exponentially so attractors have 0 volume in nD phase space.
g2 and g3.
such that for 2. A PRIME Q with g > (p1i4 + l)“, some other number k and m = kq with k # 1, mC(s,y,gz,g3,p) is the identity on the curve, but kC(z, y, g2, gs,p) is not the identity. This guaranof Goldwasser tees PRIMALITY of p by a theorem 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 Mathematics@ (Wolfram Research, Champaign, IL) task ProvablePrime [n] therefore generates an AtkinGoldwasserKilianMorain certificate only for numbers above a certain limit (lOlo by default), and a PRATT CERTIFICATE for smaller numbers.
see also ELLIPTIC CURVE PRIMALITY PROVING, ELLIPTIC PSEUDOPRIME, PRATT CERTIFICATE, PRIMALITY CERTIFICATE, WITNESS Heferences A. 0. L. and Morain, F. "Elliptic Curves and ity Proving." Math. Gomput. 61, 2968, 1993. Bressoud, II. M. Factorization and Prime Testing.
Atkin,
Primal
New York: SpringerVerlag, 1989. Goldwasser, S. and Kilian, J. “Almost All Primes Can Be Proc. 18th STOC. pp. 316329, 1986. Quickly Certified.” Morain, F, “Implementation of the AtkinGoldwasserKilian Primality Testing Algorithm.” Rapport de Recherche 911, INRIA, Octobre 1988. Schoof, R. “Elliptic Curves over Finite Fields and the Computation of Square Roots mod p.” Math. Comput. 44, 483494, 1985. Wunderlich, M. C. “A Performance Analysis of a Simple PrimeTesting Algorithm.” Math. Comput. 40, 709714,
1983.
A stable FIXED POINT surrounded by a dissipative region is an attractor known as a SINK. Regular attractors (corresponding to 0 LYAPUNOV CHARACTERISTIC ExPONENTS) act as LIMIT CYCLES, in which trajectories circle around a limiting trajectory which they asymptotically approach, but never reach. STRANGE ATTRACTORS are bounded regions of PHASE SPACE (corresponding to POSITIVE LYAPUNOV CHARACTERISTIC EXPONENTS) having zero MEASURE in the embedding PHASE SPACE and a FRACTAL DIMENSION. Trajectories within a STRANGE ATTRACTOR appear to skip around randomly. 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.
VICKERY AUCTION
see also
Augend The first of several ADDENDS, or “the one to which the others are added,” is sometimes called the augend. Therefore, while a, b, and c are ADDENDS in a + b + c, a is the augend.
see also ADDEND, ADDITION
Atomic Statement In LOGIC, a statement into smaller statements.
which
cannot
be broken
down
Augmented Amicable Pair A PAIR of numbers m and n such that u(m)
Attraction
Amicable
TION.
CURVE C
y2 = x3 + g2El:+ g3 (mod p) for some numbers
Augmented
Certificate
= u(n)
= m + n  1,
Basin
see BASIN OF ATTRACTION
where o(m) (1977)
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 evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors
is the DIVISOR FUNCTION. Beckand
found
11 augmented
amicable
Najar
pairs.
see also AMICABLE PAIR, DIVISOR FUNCTION, QUASIAMICABLE
PAIR
References E. and Najar, R. M. “More Reduced Fib. Q uart. 15, 331332, 1977. Guy, R. K. Unsolved Problems in Number Theory, New York: SpringerVerlag, p. 59, 1994.
Beck, W. Pairs.”
Amicable 2nd
ed.
Augmented
Dodecahedron
Au .gmented see JOHNSON
Authalic
Dodecahedron
Latitude
where h E 2k  1 and
SOLID L2h,i&h=2h+1~2k
.gmented JOHNSON
Hexagonal
Augmented A UNIFORM adjoined.
L3hy
Prism
(7)
= 3h + 1 F 3k
(8)
= 52h + 3 5h t 1 F 5k(5h + 1)a
(9)
l
see also GAUSS’S
SOLID
FORMULA
References
Polyhedron POLYHEDRON
M3h
L&t!&
Pentagonal
Augmented see JOHNSON
Prism
SOLID
Augmented see JOHNSON
85
with one or more other
solids
Sphenocoro
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorixations of b” & 1, b = 2, 3,5,6,7,10,11,12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Sot., pp. lxviiilxxii, 1988. Wagstaff, S. S. Jr. ‘ ‘Aurifeullian Fat t orizat ions and the PeMath. Comriod of the Bell Numbers Modulo a Prime.” put. 65, 383391, 1996.
SOLID
Ausdehnungslehre Augmented see JOHNSON
Triangular
Tridiminished
Truncated
Aureum Gauss’s
P=
Truncated
sin 1
Q
( b > ’
(1)
Do decahedron
SOLID
Augmented see JOHNSON
Cube
SOLID
Au gmented See JOHNSON
ALGEBRA
Authalic Latitude An AUXILIARY LATITUDE which gives a SPHERE equal SURFACE AREA relative to an ELLIPSOID. The authalic latitude is defined by
Icosahedron
SOLID
.gmented JOHNSON
~~~EXTERIOR
SOLID
Augmented see JOHNSON
Prism
Truncated
sin #
q E (le”)
Tetrahedron
Ln(;+:;~;)], i 1  e2 sin2 C$  2e
(2)
SOLID
Theorema name for the QUADRATIC
RECIPROCITY
THE
and qp is 4 evaluated at the north pole (4 = 90’). Let R, be the RADIUS of the SPHERE having the same SURFACE AREA as the ELLIPSOID, then
OREM.
Aurifeuillean A factorization
%=ac ik* 2
Factorization of the form
2 4n+2 + 1 = (22n+1  zn+l
(3)
The series for /? is +1)(22n+1
+2”“l+
0 = q5  ( +e2 + &e4
The factorization for n = 14 was discovered rifeuille, and the general form was subsequently ered by Lucas. The large factors are sometimes as L and M as follows 2 4k2 + 1 = (pl
 2” + @“l
36k3 + 1 = (32k1
+ 1)(32”1
1). (1)
+ 2’” + 1)  3” + 1)(32k1
by Audiscovwritten
+ (&e4
 (& The inverse
(2) + 3’” + l),
L&$=
+ &e6
+ $&e6 + . . .) sin(24) + . , .) sin(44)
e6 + . . .) sin(Sg5) + . . . .
FORMULA
(1  e2 sin2 4)” 2cos4
is found
(4
from
sin g5  9 1  e2 sin2 4 [ 1  e2
(3) which can be written 22h + 1 = &hit&
(4)
33h + 1 = (3h + l)L3hi&h
(5)
5
5h
 1 = (5h  l)L&d!~h,
(6)
where
q = q,sinp
(6)
86
Autocorrelation
Autocorrelation
and 40 = sinl as
(q/2).
This
can be written
4 = p + ($5” + &e4 + ($”
+ Ge”
+( se6 see
+ $f&e6
in series form
+ . . .) sin(20)
+ ....
(7)
mv)
I21
LATITUDE
also
References Adams,
S. “Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert EqualArea Meridional Projections.” Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map ProjectionsA Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 16, 1987.
0.
Autocorrelation The autocorrelation
c,(t)
s
given by
+ . . .) sin(4P)
+ . . m)sin(Sp)
FOURIER TRANSFORM known as the WIENERKHINTCHINE THEOREM. Let F[f(x)] = F(lc), and F* denote the COMPLEX CONJUGATE ofF,thenthe FOURIER TRANSFORM of the ABSOLUTE SQUARE ofF(J
the
=
f*
f
=
function
f"(t)
is defined
* f(t)
=
The autocorrelation C,(4) = C,*(t). other words,
(8)
is a HERMITIAN OPERATOR since f is MAXIMUM at the ORIGIN. In
s
O” f(u)f(u+x)du oo
f”(u)
2
du.
(9)
To see this, let E be a REAL NUMBER. Then
SW [f(u)
by
f*(T)f(t
fk
O” f*(r)f(r+x)dr. oo
+ Ef(u
+ x)12 du > 0
(10)
00
+
7)
d7,
SWm
f2(u)du+2c
O” f(u)f(ufx)du
s w
(1)
* denotes CONVOLUTION and 7t denotes CROSSCORRELATION. A finite autocorrelation is given by
where
Cf (4 = ([y(t) =
lim
s
Tbm
iJlEY@
T/2
+ 7)  4)
[y(t)  d[y(t
+ T)  31 dt*
(3)
SW f”(u)
du + 2~
W
> 0.
(12)
O” f (u>f (u + 4 du s w +E2
If f is a REAL FUNCTION,
f* = f,
(4)
a= SW w
= f (4,
f”(u)du
Define
so that
f k7)
(11)
(2)
T/2
and an EVEN FUNCTION
> 0
f2(u+x)du
+e2
(5)
bE2
f2Wdu
(13)
O” f(u)f(u+x)du. s w
(14
then Cf (t) =
SW w
f (df
(t + 7) dr
(6)
Then plugging
into above,
we have ue2 +b~+c
so b2  4ac < 0, i.e., b/2 < a. It follows But let r’ = 7,
so d7’ = dr,
that
f(u)f(u+x)du Ism sO”
then
f 2(u) du,
OO f(r)f(t
Cf Cc> =
ml
 T)(dr)
L
sm00
with f (r)f f(r)f(tr)dT=
> 0. This
QUADRATIC EQUATION does not have any REAL ROOT,
(t  4 dT f * f.
(7)
The autocorrelation discards phase information, returning only the POWER. It is therefore not reversible. There is also a somewhat surprising and extremely portant relationship between the autocorrelation
imand
the equality
(15)
00
at x = 0. This
proves
that
f * f is
MAXIMUM at the ORIGIN. see UZSO CONVOLUTION, CROSSCORRELATION, QUANTIZATION EFFICIENCY, WIENERKHINTCHINE THEOREM References PRSS, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Correlation and Autocorrelation Using the FFT.” §13.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 538539, 1992.
Automorphic
Function
Autoregressive
Automorphic Function An automorphic function f(z) of a COMPLEX variable z is one which is analytic (except for POLES) in a domain D and which is invariant under a DENUMERABLY INFINITE group of LINEAR FRACTIONAL TRANSFORMATIONS (also known as MOBIUS TRANSFORMATIONS) az + b x1 = 
Automorphic METRIC
functions
FUNCTIONS
UZSO MODULAR
see
are generalizations of TRIGONOand ELLIPTIC FUNCTIONS.
FUNCTION,
J. Recr.
Math.
I,
173179,
1968.
Hunter, J. A. H. “Two Very Special Numbers.” Fib. Quart. 2, 230, 1964. Hunter, J. A. H. “Some Polyautomorphic Numbers.” J, Recr. 5, 27,
1972.
M. “Automorphic Numbers.” $3.8 in MathematRecreations. New York: W. W. Norton, pp. 7778,
Kraitchik,
M~~BIUS
TRANSFORMA
ZETA FUCHSIAN
TIONS,
87
Fairbairn, R. A. “More on Automorphic Numbers.” J. Recr. Math. 2, 170174, 1969. Fairbairn, R. A. Erratum to “More on Automorphic Numbers.” J. Recr. 1MaUz. 2, 245, 1969. de Guerre, V. and Fairbairn, R. A. “Automorphic Numbers.”
Math.
cz+d’
Model
Automorphic Number A number IC such that nk2 has its last digits equal to k is called nautomorphic. For example, 1 52 = 25and 1 mS2 = 36 are 1automorphic and 2 . 8 2 = 12s and 2 882 = 15488 are 2automorphic. de Guerre and Fairbairn (1968) g’rve a history of automorphic numbers.
ical
1942. Madachy, J. S. Msdachy’s Mathematical Recreations. New York: Dover, pp. 3454 and 175176, 1979. Sloane, N. J. A. Sequences A016090, AO03226/M3752, and A007185/M3940 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex: Penguin Books, pp. 171, 178, 191192, 1986.
l
l
The first few lautomorphic numbers are 1, 5, 6, 25, 76, 376, 625, 9376, 90625, . . . (Sloane’s A003226, Wells 1986, pa 130). There are two 1automorphic numbers with a given number of digits, one ending in 5 and one in 6 (except that the ldigit automorphic numbers include I), and each of these contains the previous number with a digit prepended. Using this fact, it is possible to construct automorphic numbers having more than 25,000 digits (Madachy 1979). The first few lautomorphic numbers ending with 5 are 5, 25, 625, 0625, 90625, +, . (Sloane’s AO07185), and the first few ending with 6 are 6, 76, 376, 9376, 09376, . . . (Sloane’s A016090). The lautomorphic numbers a(n > endingin5are IDEMPOTENT J (mod 10n) since [a( (Sloane
and Plouffe
E a n) (mod
10n)
1995)
Automorphism An ISOMORPHISM see
UZSO
ANOSOV
table
give:
the
lodigit
n
nAutomorphic
1 2 3 4 5 6 7
0000000001, 8212890625, 1787109376 0893554688 6666666667, 7262369792, 9404296875 0446777344 3642578125 3631184896 7142857143, 4548984375, 1683872768
8 9
0223388672 5754123264, 3134765625, 8888888889
see
~2~0
Numbers
IDEMPOTENT,
BER PYRAMID,TRIMORPHIC
nautomorphic
see
also
NUMBER,
AUTOMORPHISM
GROUP
Autonomous A differential equation or system of ORDINARY DIFFERENTIAL EQUATIONS is said to be autonomous if it does not explicitly contain the independent variable (usually denoted t). A secondorder autonomous differential equation is of the form F(y, &I/‘) = 0, where y’ = dy/dt = v By the CHAIN RULE, y” can be expressed as l
dv dv dy y” = v’ = dt = dy dt 
,A007185, A016090 A030984 , A030985, A030986 A030987 A030988 A030989 A030990, A030991, A030992 A030993 A030994, A030995,
dx n+l dt
NUM
M.; Gosper, R. W.; and Schroeppel, R, Item 59 in Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972. HAKMEM.
dv &v.
For an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means that all particles pass through a given point in phase space. A nonautonomous system of n firstorder ODES can be written as an autonomous system of n + 1 ODES by letting t E x n+l and increasing the dimension of the system by 1 by adding the equation =l.
References Beeler,
onto itself.
AUTOMORPHISM
OUTER
Sloane
NARCH~ISTIC NUMBER
of objects
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 MULTIPLKATION table, the automorphism group of a GRAPH the INCIDENCE MATRICES, andthatofa FIELD the ADDITION and MULTIPLICATION tables.

The following numbers.
of a system
Autoregressive
Model
~~~MAXIMUM ENTROPYMETHOD
Auxiliary
Circle
Axiom
Auxiliary Circle The CIRCUMCIRCLE 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 dso
CIRCLE, ECCENTRIC ANGLE,
Auxiliary
ELLIPSE
Latitude
LATITUDE, CONFORMAL LATITUDE, see AUTHALIC GEOCENTRIC LATITUDE, ISOMETRIC LATITUDE, LATITUDE, PARAMETRIC LATITUDE, RECTIFYING LATITUDE,
REDUCED
Auxiliary
LATITUDE
Triangle
see MEDIAL TRIANGLE Average see MEAN
Average
Absolute
Deviation
1 N N ID
QrE
xi

j.LJ
=
(1%

PI)
l
i=l
see also ABSOLUTE DEVI ATION,DEVIATION JTANDARD DEVIATION, VARIANCE Average Function If f is CONTINUOUS on a CLOSED INTERVAL [a,b],then there is at least one number x* in [a, b] such that
f (x)dx
= f (x*)(b
Axiom A PROPOSITION regarded as selfevidently TRUE without PROOF. The word “axiom” is a slightly archaic synonym for POSTULATE. Compare CONJECTURE or HYPOTHESIS, both of which connote apparently TRUE but not selfevident statements.
see also ARCHIMEDES' AXIOM, AXIOM OF CHOICE, AxIOMATIC SYSTEM, CANTORDEDEKIND AXIOM, CONGRUENCE AXIOMS, CONJECTURE, CONTINUITY AxIOMS, COUNTABLE ADDITIVITY PROBABILITY AXIOM, DEDEKIND'S AXIOM, DIMENSION AXIOM, EILENBERGSTEENROD AXIOMS, EUCLID'S AXIOMS, EXWION AxIOM,FANO'S AXIOM, FIELD AXIOMS, HAUSDORFF AxIOMS, HILE~ERT'S AXIOMS, HOMOTOPY AXIOM, INACCESSIBLE CARDINALS AXIOM, INCIDENCE AXIOMS, INDEPENDENCE AXIOM, INDUCTION AXIOM, LAW, LEMMA, LONG EXACT SEQUENCE OF A PAIR AXIOM, ORDERING AXIOMS, PARALLEL AXIOM, PASCH'S AxIOM, PEANO'S AXIOMS, PLAYFAIR'S AXIOM, PORISM, POSTULATE, PROBABILITY AXIOMS, PROCLUS' AXIOM, RULE, T~SEPARATI~N AXIOM,THEOREM,ZERMELO'S AXIOM OF CHOICE, ZERMELOFRAENKEL AXIOMS Axiom A Diffeomorphism Let 4: M + M be a C1 DIFFEOMORPHISM on acompact RIEMANNIAN MANIFOLD M. Then 4 satisfies AXiom A if the NONWANDERING set a(4) of $ is hyperbolic and the PERIODIC POINTS of q5 are DENSE in O(4). Although it was conjectured that the first of these conditions implies the second, they were shown to be independent in or around 1977. Examples include the ANOSQV DIFFEOMORPHISMS and SMALE HORSESHOE MAP. In some cases, Axiom A can be replaced by the condition that the DIFFEOMORPHISM is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and
fb
J
A Flow
 a).
a
Pollicott
The average value of the FUNCTION (f) on this interval is then given by f(x*).
~~~MEANVALUE THEOREM
1990).
see also ANOSOV DIFFEOMORPHISM, AXIOM A FLOW, DIFFEOMORPHISM, DYNAMICAL SYSTEM, RIEMANNIAN MANIFOLD,~MALE HORSESHOE MAP References
Average
Seek
Bowen,
Time
see POINTPOINT
Anosov
DISTANCEID
AxKochen Isomorphism Theorem Let P be the SET of PRIMES, and let QP and Z&) be the FIELDS of pADIC NUMBERS and formal POWER series over . . ,p  I). Further, suppose that D is a “nonprincipal maximal filter” on P. Then nPEP U&,/D and npEp Z&)/D are ISOMORPHIC. Zp
=
(O,l,
l
see UZSO HYPERREAL NUMBER,NONSTANDARD SIS Axial
Vector
ANALY
R. Equilibrium Diffeomorphisms.
States
and the Ergodic Theory of New York: SpringerVerlag,
1975. Ott, E. Chaos in Dynamicul Systems. New York: Cambridge University Press, p. 143, 1993. Parry, W. and Pollicott, M. “Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics.” A&risque No. 187188, 1990, Smale, S. “Different iable Dynamical Systems." Bull. Amer, Math. Sot. 73, 747817, 1967.
Axiom A FLOW
A Flow defined analogously to the AXIOM A DIFFEOMORPHISM, except that instead of splitting the TANGENT BUNDLE into two invariant subBUNDLES, they are split into three (one exponentially contracting, one expanding, and one which is ldimensional and tangential to the flow direction).
see also DYNAMICAL SYSTEM
Axiom
Azimuthal
of Choice
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.
Hazewinkel,
M. (Managing
ematics: Soviet
An Updated “Mathematical
lands:
Reidel,
Azimuthal
Projection
Ed.).
and Annotated Encyclopaedia.
pp. 322323,
89
Encyclopaedia Translation
of
” Dordrecht
Mathof the
, Nether
1988.
Equidistant
Projection
It is related to HILBERT'S PROBLEM lo, and was proved to be consistent with other AXIOMS in SET THEORY in 1940 by Gijdel. 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). PROBLEMS, SET THEORY, see also H 'ILBERT'S ORDERED SETJER #MELOFRAENKEL AXIOMS, LEMMA
WELL20~~3
References Boyer, C. B. and Merzbacher, U. C. A History of Mathematics, 2nd ed. New York: Wiley, 1991. Cohen, P. J, “The Independence of the Continuum Hypothesis.” Proc. Nat. Acad. Sci. U. S. A. 50, 11431148, 1963. Cohen, P. J. “The Independence of the Continuum Hypothesis. II.” Proc. Nat. Acad. Sci. U. S. A. 51, 105110, 1964. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 274276, 1996. Moore, G. H. Zermelo’s Axiom of Choice: Its Origin, Development, and InfZ1uence. New York: SpringerVerlag, 1982.
Axiomatic Set Theory A version of SET THEORY in which axioms as uninterpreted rather than as formalizations existing truths. see also NAIVE
SET THEORY,
also
2 = k’ cos 4 sin@  X0) y = k’[cos q51sin 4  sin41
are taken of pre
AXIOMATIC THEORY,THEOREM
THEORY,
= sin&
cost
sin4
+ co+
where c is the angular inverse FORMULAS are
4=
sin ’
for A=
XAXIS, YAXIS, ZAXIS with
References Woods,
F. S. Higher
Methods
in
G eometry:
Analytic
An
Geometry.
Introduction
Dover,
qbl
X0 + for X0 + i for
the angular
to Advanced
New York:
(1) (2)
(3) \ I
(C #
tanl 41 = tanl 41 =
distance c
p. 8,
cos~cos(~
distance
(cos csin&
CONSIS
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). ORDINATE,
 X0)].
and
X0 + tanl
see also ABSCISSA,
cos~cos(~
sin c
SET THEORY
COMPLETE
TENCY,MODEL
An AZIMUTHAL PROJECTION which is neither equalAREA nor CONFORMAL. Let @1 and X0 be the LATITUDE and LONGITUDE of the center of the projection, then the transformation equations are given by
Here. I
Axiomatic System A logical system which possesses an explicitly stated SET of AXIOMS from which THEOREMS can be derived. see

from
+
41
the center.
y sin c cos $1 c > x sin
COS
 X0),
COS
Cy
(4) The
(5)
c sin
41
sin
c >
*90”
(i) 90” (5)) go”, from dx2
(6)
the center + y2.
given by (7)
1961.
References Axonometry A METHOD
for mapping
3D figures
onto the PLANE.
see also CROSSSECTION, MAP PROJECTION,POHLKE'S THEOREM, PROJECTION,STEREOLOGY
Azimuthal
References Coxeter,
H, S. M. Regular
Dover, p. 313, 1973.
Polytopes,
3rd
ed.
New
Snyder, J. P, Map ProjectionsA Geological Survey Professional DC: U. S. Government Printing
York:
Working Manual. U. S. Paper 1395. Washington, Office, pp. 191202, 1987.
Projection
see AZIMUTHAL EQUIDISTANT PROJECTION, LAMBERT AZIMUTHAL EQUALAREA PROJECTION, ORTHOGRAPHIC PROJECTION, STEREOGRAPHIC PROJECTION
BSpline
B*Algebra
B
BPTheorem If 0,f (G) = 1 and if 2 is a pelement
II*Algebra A BANACH
with satisfies
an ANTIAUTOMORPHIC
ALGEBRA
* which
VOLUTION
X
**
=
(1)
(yx)*
BSpline
B
pypc6
satisfies
Ilxx*ll = 1/2112~ is a special
see also BANACH
(5)
type of B*algebra.
ALGEBRA,
essay
l
l
p4
A generalization of the BI&XER CURVE. known as the KNOT VECTOR be defined
called a SIDON SEQUENCE. QUENCE of POSITIVE INTEGERS
Let a vector
T= {to,tl,rtm},
by S. Finch
Also
J.p5
p2
C*ALGEBRA
&Sequence NB, A detailed online ing point for this entry.
is the ~LAYER.
(3) (4)
(cx >* = cx*
A C*ALGEBRA
L,I
(2)
x* + y* = (x + y>*
and whose NORM
of G, then
INwhere
=x
x*y*
91
An
was the start
INFINITE
(1)
where T is a nondecreasing SEQUENCE with ti E [O, 11, and define control points PO, . . . , P,. Define the degree as pEmnl.
SEThe
tp+l,
“knots”
(2)
. . . , tmBpl
are called
INTERNAL
KNOTS.
1 L 61 < such that
all pairwise
b2
<
b3
<
. . .
(1)
sums
h + bj
(2)
Define
the basis functions
N,o(t)
=
N+(t)
= ++ i+p
1 0
as
if ti 5 t < ti+l otherwise Ni,pl(t)
and ti < tt+l +
i
‘;+‘+’ ti+p+1
 ’ N+l,,1  ii+1
for i < j are distinct (Guy 1994). An example is I, 2, 4, 8, 13:21, 31, 45, 66, 81, . . . (Sloane’s AO05282). Zhang
(1993,
S(B2)
=
The definition 1994).
O”
SUP all
B2
see also ASEQUENCE,
(3)
to B,sequences
(Guy
the curve defined
c(t)
bk
k=i
can be extended

1
> 2.1597.
x
sequences
MIANCHOWLA
SEQUENCE
(t). (4)
Then
1994) showed that
(3)
by
= 9 RN,,(t) i=o
(5)
is a Bspline. Specific types include the nonperiodic Bspline (first p + 1 knots equal 0 and last p + 1 equal to 1) and uniform Bspline (INTERNAL KNOTS are equally spaced). A BSpline with no INTERNAL KNOTS is a
References
BI&IER
Finch, S. “Favo:ite Mathematical Constants.” http: //www , mathsoft.com/asolve/constant/erdos/erdos,html. Guy, R. K, “Packing Sums of Pairs,” “ThreeSubsets with Distinct Sums,” and “&Sequences,” and &Sequences Formed by the Greedy Algorithm.” §C9, Cll, E28, and E32 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, pp. 115118,121123,228229, and 232233, 1994. Sloane, N. J. A. Sequence A005282/M1094 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Zhang, 2. X. “A B2Sequence with Larger Reciprocal Sum.”
The degree of a Bspline is independent of the number of control points, so a low order can always be maintained for purposes of numerical stability. Also, a curve is p  k times differentiable at a point where ?C duplicate knot values occur. The knot values determine the extent of the control of the control points.
Math.
Comput.
60,
Zhang, 2. X, “Finding urn
l/2

?? Math.
835839,
Finite
Comput.
1993.
B2Sequences 63,
403414,
with 1994.
Larger
m 
CURVE.
A nonperiodic Bspline is a Bspline whose first p + 1 knots are equal to 0 and last p + 1 knots are equal to 1. A uniform Bspline is a Bspline whose INTERNAL KNOTS are equally spaced. see also B~ZIER
CURVE,
NURBS
CURVE
92
Backtracking
BTree
BTree
BACCAB
Btrees were introduced by Bayer (1972) and McCreight. They are a special mary balanced tree used in databases because their structure allows records to be inserted, deleted, and retrieved with guaranteed worstcase performance. An nnode Btree has height O(lg 2), where LG is the LOGARITHM to base 2. The Apple@ Macintosh@ (Apple Computer, Cupertino, CA) HFS filing system uses Btrees to store disk directories (Benedict 1995). A Btree satisfies the following properties:
see BACCAB
1. The Rook is either two CHILDREN, 2, Each tween
a LEAF
or has at least
the ROOT to a LEAF
(TREE)
has the
Every 23 TREE is a Btree of order 3. The number of Btrees of order n = 1, 2, . . , are 0, 1, 1, 1, 2, 2, 3, 4, 5, (Ruskey, Sloane’s A014535). 8, 14, 23, 32, 43, 63, l
F.
.
“Information
on
BTrees.”
http:
//sue.
“An
OnLine
csc
.uvic
ca/cos/inf/tree/BTrees.html,
Monster
Also known
SPORADIC
Bachet’s
A014535
in
Integer
Bachet
l
GROUP.
The
‘23.31.47.
l
EQUATION x2+k=y3,
which is also an ELLIPTIC CURVE. is still the focus of ongoing study.
Backhouse’s
The general
equation
Constant as the POWER series whose equal to the nth PRIME,
Let P(x) be defined has a COEFFICIENT
nth term
P(x)
= 1+2x+3x2+5x3+7x4+11x5+~~.,
E y$x” k=O
and let Q(x)
be defined
Q(x)
=
by
j+
=
Rqiilk+
k=O
Then
N. Backhouse
lim n+m
4n+l Qn
conjectured
that
= 1.456074948582689671399595351116..
..
The constant jolet.
Identity
The VECTOR
TRIPLE
of Finite
Group
A x (B x C) = B(A
Representation.”
. C)  C(A
see also LAGRANGE’S
B).
l
to nD
x mmwx b,1) bl a2 . bl * I
I h1
Constants.”
Mathematical
http:
//www.
.
l
l
IDENTITY
Transformation for solving classes EQUATIONS. SCATTERING
of nonlinear
PARTIAL
DIF
METHOD
References Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos. Cambridge, England: Cambridge University Press, p. 196, 1990. Miura, R. M. (Ed.) Biicklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. New York:
b1
a0 .
SpringerVerlag,
1974.
a2  b1 l
.
. bl
BScklund
. html.
see also INVERSE
identity
can be generalized
n+l  ( 1)
S. “Favorite
A method FERENTIAL
PRODUCT
x (bl
to exist by P. Fla
References
backhous
http://for.mat.bham.ac.uk/atlas/BM.html, BACCAB
was subsequently
mathsoft.com/asolve/constant/backhous/
GROUP
References Wilson, R. A, “ATLAS
x a,1
Equation
DIOPHANTINE
Finch,
see also MONSTER
a2 x 
THEOREM
Group
as FISCHER’S BABY MONSTER GROUP B. It has ORDER
identity
FOURSQUARE
Version
Sequences.”
2 41 * 313 9 56 72 11 113 4 17.19
This
Conjecture
see LAGRANGE’S
.
Sldane, N. 5. A. Sequence of the Encyclopedia of
Baby
FUNCTION
TREE
References Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Structures and Algorithms. Reading, MA: AddisonWesley, pp. 369374, 1987. Benedict, B. Using Norton Utilities for the Macintosh. Indianapolis, IN: Que, pp. B17B33, 1995. Beyer, R. “Symmetric Binary BTrees: Data Structures and Maintenance Algorithms.” Acta Informat. 1, 290306, 1972. Ruskey,
see BROWN
and LEAVES) has bem CHILDREN, where [xl is the
FUNCTION.
see also REDBLACK
Function
the ROOT
(except [m/2] and
3. Each path from same length.
IDENTITY
Bachelier
The
node
CEILING
(TREE)
Rule
.
a,1
. b1
Backtracking A method of drawing FRACTALS by appropriate numbering of the corresponding tree diagram which does not require storage of intermediate results.
BackusGilbert
Baire Category
Method
BackusGilbert A method which
Method can be used to solve some classes of INTEGRAL EQUATIONS and is especially useful in implementing certain types of data inversion. It has been applied to invert seismic data to obtain density profiles in the Earth. References Backus, Earth
G. and Gilbert, Data.” Geophys.
F. “The Resolving J. Roy.
Astron.
Power Sot.
of Growth
16, 169205,
1968. Backus, G. E. and Gilbert, F. “Uniqueness in the Inversion of Inaccurate Gross Earth Data.” Phil. Truns. Roy. Sot. London Ser. A 266, 123192, 1970. Loredo, T. J. and Epstein, R. I. “Analyzing GammaRay Burst Spectral Data.” Astrophys. J. 336, 896919, 1989. Parker, R. L. “Understanding Inverse Theory.” Ann. Rev. Earth
Planet.
Sci.
5, 3564,
1977.
Press, W. H.; Flannery, B. P.; Teukolsky, line;, W. T. “BackusGilbert Method.” Recipes in FORTRAN: 2nd ed. Cambridge,
pp. 806809,
The
England:
Art
of
Cambridge
Bader, G. and Deuflhard, P. “A SemiImplicit MidPoint Rule for Stiff Systems of Ordinary Differential Equations.” Numer. Math. 41, 373398, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge 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 1 n+l (2 s n+l (2
3
Press,
order differences are obtained by repeated of the backward difference operator, so
oper
0; = V(Vp) = V(fp  fp1) = VP  VfPl = (jp  fp1)  (fp1  L2) = fp  2fpI+ fp2
moving the two end rings, BY simultaneously ber of moves can be reduced to ZnB1  1 27xl
The solution to the theory
(3)
is a BINOMIAL COEFFICIENT.
n even n odd.
of the baguenaudier of GRAY CODES.
is intimately
related
Met
hod
LAMBERT'S METHOD
see
NEWTO N’S BAG KWARD DI FFERENCE FORMULA expresses jp as the sum of the nth backward differences jp
the num
Dubrovsky, V. “Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster.” Quantum 8, 6165 (Mar.) and 5859 (Apr.), 1996. Gardner, M. “The Binary Gray Code.” In Knotted Doughnuts and Other Mathematical Entertainments. New York: W* H. Freeman, pp. 1517, 1986. Kraitchik, M. “Chinese Rings.” $3.12.3 in Mathematical Recreations. New York: W. W. Norton, pp. 8991, 1942. Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, p. 268, 1983.
Bailey’s (L)
n even n odd.
 1)
References
(21
In general,
where
 2)
1992.
Backward Difference The backward difference is a FINITE DIFFERENCE defined by vp = vjp = jp  jpla (1) Higher ations
References
Computing,
University
93
BaderDeuflhard Method A generalization of the BULIRSCHSTOER ALGORITHM for solving ORDINARY DIFFERENTIAL EQUATIONS.
S. A.; and Vetter$18.6 in Numerical
Scientific
Theorem
Bailey’s Theorem Let I?(Z) be the GAMMA FUNCTION, then
= fo+PVo+~P(P+1)V~+~P(P+1)(P+2)v~+..., (5)
where
V;jl is the first
difference
nth
difference
computed
from the
table.
also ADAMS’ METHOD, DIFFERENCE EQUATION, DIVIDED DIFFERENCE, FINITE DIFFERENCE, FORWARD DIFFERENCE, NEWTON'S BACKWARD DIFFERENCE FORMULA, RECIPROCAL DIFFERENCE
see
=
[%&q’[;+
(;)2&+ L
(E)“&+...].
Y
/
m
References Beyer, W. H. CRC Standard Mathematical Tables, 28th Boca Raton, FL: CRC Press, pp. 429 and 433, 1987+
ed.
Baire Category Theorem A nonempty complete METRIC SPACE cannot be represented as the UNION of a COUNTABLE family of nowhere
DENSE SUBSETS.
Picking
Ball IYiangle
Baire Space
94
Baire Space A TOPOLOGICAL SPACE X in which each SUBSET of X of the “first category” has an empty interior. A TOPOLOGICAL SPACE which is HOMEOMORPHIC to a complete METRIC SPACE is a I3aire space.
where p E l QI, X, +Xb 5 1, and x and y are computed mod 1. The Q = 1 QDIMENSION is
D1=1+
aln(i)
+pln($) (4)
dn(&)+pl+J’ Bairstow’s A procedure
Method for finding
the quadratic
factors
for the
If X, = Xb, then the general
CONJUGATE ROOTS of a POLYNOMIAL P(X) with REAL COEFFICIENTS.
QDIMENSION
is
COMPLEX
D,=1+
1
In (aq + pg) InX, ’
q1
(5)
[x  (a + ib)][x  (a  a)] =x2+2ax+(a2+b2)~x2+B~+C.
Now write
the original P(x)
= (x2 + Bx
+ C)Q(x) dR
(2
EdC dC
+ 6B, C + SC) &<B,C)+gdB+gdC
Q(x) dP
+ Rx + S
= R(B,C)+dBdB+
= (x2 + Bx + C)g
=O=(x2+Bx+qaB
+ 2
aQ
x&(S)=(X~+BX+C)~+~+~. Now use the 2D NEWTON's METHOD taneous
dS I dC
+ g
dR
dS
(5) (6)
dR dS + dB + dB
+x&(x)
a’
(3) (4)
dR + Q( .I + ac
~=O=(x2+Bx+C)~
dB
as
POLYNOMIAL
R(B+6B,C+dC) S(B
(1
References Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: SpringerVerlag, p. 60, 1983. Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 8182, 1993. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 32, 1990.
Balanced ANOVA An ANOVA in which the number of REPLICATES (sets of identical observations) is restricted to be the same for group). each FACTOR LEVEL (treatment see also ANOVA Balanced
Incomplete
see BLOCK
DESIGN
Block
Design
(7)
Ball The (8)
to find the simul
solutions.
References
nball, denoted B”, is the interior of a SPHERE and sometimes also called the nDISK. (Al though physicists often use the term “SPHERE" to mean the solid ball, mathematicians definitely do not!) Let Vol(B”) denote the volume of an nD ball of RADIUS T. Then
s n ‘,
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 277 and 283284, 1989.
x
Vol(B”)
= C2
[I+
erf(rfi)],
n=O
where erf (x) is the ERF function.
see also ALEXANDER'S HORNED SPHERE, BANACHTARSKI PARADOX, BING'S THEOREM, BISHOP'S INEQUALITY, BOUNDED, DISK, HYPERSPHERE, SPHERE, WILD POINT
Baker’s Dozen The number 13. see ah
13, DOZEN
Baker’s Map The MAP
References Freden, E. Problem 10207. “Summing a Series of Volumes.” Amer. M&h. Monthly 100, 882, 1993.
Xn+l=2PXn7
(1)
where x is computed modulo map can be defined as
Xn+l=
{
Yn+l{
1. A generalized
XaXn
Yn
<
QI
(lh)+AbXn
Yn
>
Q
Yn
yn
9
yn>CY,
<
Baker’s
(2)
af (3)
Ball The
Triangle Picking 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 nD BALL. Buchta (1986) subsequently gave closed form
Banach Measure
Ballantine evaluations tions being
for Hall’s
integrals,
with
the first few solu
P4 ==:0.39 P5 ==: 0.29.
The case Pz corresponds CUBE TRIANGLE
also
to the usual PICKING,
DISK case. OBTUSE
TRIANGLE
References Buchta, C. “A Note on the Volume of a Random Polytope in a Tetrahedron.” Ill. J. Math, 30, 653659, 1986. Hall, G. R. “Acute Triangles in the nBall,” J. Appl. Prob.
19, 712715,1982. Ballantine see B~RROMEAN
RINGS
Ballieu’s Theorem For any set p = (~1, ~2,. . . , pn) of POSITIVE with ~0 = 0 and
Then all the EIGENVALUES X satisfying P(A) P(X) is the CHARACTERISTIC POLYNOMIAL, DISK
= 0, where lie on the
Banach Algebra An ALGEBRA A over a FIELD F with a NORM that makes A into a COMPLETE METRIC SPACE, and therefore, a BANACH SPACE. F is frequently taken to be the COMPLEX NUMBERS in order to assure that the SPECTRUM fully characterizes an OPERATOR (i.e., the spectral theorems for normal or compact normal operators do not, in general, hold in the SPECTRUM over the REAL NUMBERS).
References Gradshteyn, I, S, and Ryzhik, I. M. Tables of Integrals, Series, 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 B? The solution is a CATALAN NUMBER c 72. A related problem also called “the” ballot problem is to let A receive a votes and B b votes with a > b. This version 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 + b), as first shown by M. Bertrand (Bilton and Pedersen 1991). Another elegant solution was provided by And& (1887) using the socalled ANDRI?S REFLECTION METHOD. The problem can also be generalized (Hilton and Pedersen 1991). Furthermore, the TAK FUNCTION is connected with the ballot problem (Vardi 1991). see also
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 49, 1987. Carlitz, L. “Solution of Certain Recurrences.” SIAM J, Appl. Math, 17, 251259, 1969. Comtet, L. Advanced Combinatorics. Dordrecht, Netherlands: Reidel, p+ 22, 1974. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 6797, 1968. Hilton, P. and Pedersen, J. “The Ballot Problem and Catalan Numbers .” Nieuw Archief voor Wiskunde 8, 209216, 1990. Hilton, P. and Pedersen, J. “Catalan Numbers, Their Generalization, and Their Uses.” Math. Intel. 13, 6475, 1991. Kraitchik, M. “The BallotBox Problem.” $6.13 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942. Motzkin, T, “Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for NonAssociative Products.” Bull. Amer. Math. Sot. 54, 352360, 1948. Vardi, I. Computational Recreations in Mathematics. Redwood City, CA: AddisonWesley, pp. 185187, 1991.
numbers
IzI < ik&.
NUMBER,
par
105,
436437,1887.
4
P2 =    =2: 0.72 8 T2 P3 = $g =2: 0.53
see
References Andrk, D. “Solution directe du probkme &olu M. Bertrand.” Comptes Rendus Acad. Sci. Paris
9
95
REFLECTION FUNCTION
ANDRI?S
TAK
METHOD,
CATALAN
see also B*ALGEBRA Banach Fixed Point Theorem Let f be a contraction mapping from a closed SUBSET F of a BANACH SPACE E into F. Then there exists a unique x E F such that f(z) = z. see
also
FIXED
POINT
THEOREM
References Debnath, L. and Mikusiriski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.
BanachHausdorffTarski see BANACHTARSKI
Paradox PARADOX
Banach Measure An "AREA" which can be defined for every seteven those without a true geometric AREAwhich is rigid and finitely additive.
Banach
96
Space
Bar bier’s Theorem
Banach Space A normed linear SPACE which is COMPLETE in the normdetermined METRIC. A HILBERT SPACE is always a Banach space, but the converse need not hold. see &o BESOV SPACE, FIXED POINT THEOREM BanachSteinhaus see UNIFORM
HILBERT
SPACE,
SCHAUDER
Theorem BOUNDEDNESS
PRINCIPLE
BanachTarski 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. IIowever, the pieces are extremely complicated. A generalization of this theorem is that any two bodies in Iw3 which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (they are are EQUIDECOMPOSABLE).
References Bankoff, L, “Are the Twin Circles Twins?” Math. Mug. 47, 214218, Gardner, M. “Mathematical Games: of Circles that Are Tangent to One 240, 1828, Jan. 1979.
of Archimedes Really 1974. The Diverse Pleasures Another.” SC;. Amer.
Banzhaf Power Index The number of ways in which a group of rz with weights n (one with c i=l wi = 1 can change a losing coalition C wi < l/2) to a winning one, or vice versa. It was proposed by the lawyer J. F. Banzhaf in 1965. References Paulos, York:
J. A. A Muthematician BasicBooks, pp. 910,
Bar (Edge) The term in rigidity see Bar
also
theory
CONFIGURATION,
Reads
the Newspaper.
New
1995.
for the EDGES
of a GRAPH.
FRAMEWORK
Polyhex
Reterences K. “The
Stromberg, Monthly
BanachTarski
86, 3, 1979. S. The BanachTarski
Wagon, bridge University
Press,
Paradox.” Paradox.
New
Amer.
York:
Math.
Cam
1993.
A POLYHEX line.
consistGof
HEXAGONS
arranged
along
a
I.
Bang’s Theorem The lines drawn to the VERTICES HEDRON from the point of contact INSPHERE form three ANGLES at which are the same three ANGLES
of a face of a TETRAof the FACE with the the point of contact in each FACE.
Amer.
B. H. “Theorem Math.
Monthly
of Bang. 33, 224226,
Honsberger, R. Mathematical Gems Math. Assoc. Amer., p. 93, 1976.
Bankoff
References Gardner,
M. Mathematical Magic Show: Games, Diversions, Illusions and Other SleightofMind from Scientific American.
Vintage,
References Brown,
see also BAR POLYIAMOND
Isosceles
More Puzzles, Mathematical New York:
p. 147, 1978.
Tetrahedra.”
1926. II. Washington,
DC:
Circle
Bar
Polyiamond
A POLYIAMOND arranged along see
also
consistingof a line.
EQUILATERAL
TRIANGLES
BAR POLYHEX
References S. W. and Packings,
Golomb,
Press,
Polyominoes: Puzzles, Patterns, 2nd ed. Princeton, NJ: Princeton
Problems,
University
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. In addition to the ARCHIMEDES' CIRCLES Cl and Cz in the ARBELO~ figure, there is a third circle C3 congruent to these two as illustrated in the above figure. see
also
ARBELOS
Barbier’s Theorem All CURVES OF CONSTANT same PERIMETER TW.
WIDTH
ofwidthw
have the
Bare The
Angle
Center
TRIANGLE FUNCTION
97
Barth Decic
Bare Angle Center with
CENTER
TRIANGLE
CENTER
The ATTRACTOR ofthe ITERATED given by the set of “fern functions”
FUNCTION
SYSTEM
a = A. References Kimberling,
C. “Major
Centers
104, 431438,
Monthly
Barnes
of Triangles.”
Amer.
Math.
1997.
GFunction
see GFUNCTION Barnes’ Lemma If a CONTOUR in the COMPLEX PLANE is curved such that it separates the increasing and decreasing sequences of POLES, then 1
im
27rTTis ica
r(a
+ spy@

+ s>qy
 s)r(d
r(a + r>qQI+ qw + r)W + 6)? r(a
where r(z)
 s) ds
is the GAMMA
from the LEECH LAT
LATTICE,
1993,
LATTICE
POINT,
References Barnes, E. S. and Wall, G, E, “Some Extreme Forms Defined Math. Sot, 1, in Terms of Abelian Groups.” J. Austrul. 4763, 1959. Conway, J. H. and Sloane, N. J, A. “The 16Dimensional BarnesWall Lattice A&’ $4.10 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: SpringerVerlag, pp. 127129, 1993.
Wagon 1991). These AFFINE contractions. The tip of the (which resembles the black spleenwort variety of is the fixed point of fl, and the tips of the lowest branches are the images of the main tip under fi f3 (Wagon 1991).
fern fern) two and
see also DYNAMICAL FUNCTION SYSTEM
SYSTEM,
FRACTAL,
ITERATED
Euerywhere, 2nd ed. Boston, MA: AcaBarnsley, M. Fractals demic Press, pp* 86, 90, 102 and Plate 2, 1993. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 238, 1988. Wagon, S. “Biasing the Chaos Game: Barnslefs Fern.” $5.3 in Mathematics in Action. New York: We H. Freeman, pp. 156163, 1991.
Barrier A number n is called a barrier of a numbertheoretic function f(m) if, for all VI < n, m + f(m) < n. Neither the TOTIENT FUNCTION qS(n) nor the DI&OR FUNCTION o(n) has barriers. References GUY?
IL
K.
Unsolved
New York:
Barnsley’s
p. 86;
References
FUNCTION.
BarnesWall Lattice A lattice which can be constructed TICE h24. see also COXETERTODD LEECH LATTICE
+ p + 7 + 6)
(Barnsley
TRANSFORMATIONSare
Fern Barth
Decic
Problems
SpringerVerlag,
in
Number
pp. 6465,
Theory,
1994.
2nd
ed.
Barth
98
Sextic
Bartlett
The Barth decic is a DECIC SURFACE in complex threedimensional projective space having the maximum possible number of ORDINARY DOUBLE POINTS (345). It is given by the implicit equation
the surface is the eightfold (EndraQ
8(X2  44y2)(y2  qb4Z2)(Z2 q5”x”)
References
x(x”
Function
cover of the CAYLEY
CUBIC
SURFACE, BARTH DECIC, CAYLEY DOUBLE POINT, SEXTIC SURFACE
see UZSO ALGEBRAIC CUBIC, ORDINARY
Barth, W. “Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron.” J. Alg. Geom. 5, 173186, 1996. mit vielen Doppelpunkten.” ‘DMVEndraB, S. “F&hen
+ y4 + x4  2x2y2  2z2z2  2y2z2)
+(3+5~)(22+y2+z2w2)2[x2+y2+~2(2~)~232~2 = 0, where 4 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
Mitteilungen
4,
1720,
4/1995.
Endrafi, S. “Barth’s Sextic.” http://wwu.mathematik.unimainz.de/AlgebraischeGeometrie/docs/
Ebarthsextic.shtml. Nordstrand, T. “Barth Sextic.” nf ytn/sexttxt . htm.
http:
//wuw
. uib. no/people/
GROUP. see also ALGEBRAIC
SURFACE,
SURFACE,
ORDINARY
BART’H
DOUBLE
SEXTIC,
DECK
Bartlett
Function 1.251
POINT
4
References Barth, W. “Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron.” J. Alg. Geom. 5, 173186, 1996. Endrafi, S. “FLchen Mitteilungen
4,
mit
vielen Doppelpunkten.” 411995. Decic.” http://wwu.mathematik.uni
DMV
3
1
2
3
0.5
The APODIZATION
FUNCTION
1720,
Endraf3, S+ “Barth’s mainz.de/AlgebraischeGeometrie/docs/ Ebarthdecic. shtml. Nordstrand, T. “Batch nfytn/bdectxt.htm.
Barth
0.7 0 5 0. 5 2$25
Decic.”
http:
//www
.uib .no/people/
f(x)
= 1 F
(1)
which is a generalization of the oneargument TRIANGLE FUNCTION. Its FULL WIDTH AT HALF MAXIMUM is a. It has INSTRUMENTFUNCTION
Sextic I(x)=/)~~~~~
(1F)
= fae2,i,,
(I+
+lae2riXE
Letting
dx
;)
dx
(2)
(1  %> dx.
z’ E 2 in the first part
therefore
gives
[ae2Tikx(l+;) dx=~oe2”“*‘i(1;)(dr’) The Barthsextic is a SEXTIC SURFACE in complex threedimensional projective space having the maximum possible number of ORDINARY DOUBLE POINTS (65). It is given by the implicit equation
=lae2mikx
Rewriting
(2) using I(x)
W 2x2  y”)((b”y”
 z2)(#“z2 (1+
2$)(x2
=
a> ds.
(3)
(3) gives
(e2rikx
 x2)
(l
+
e2xikx
) (l
;)
dx
s a
+ y2 + z2  w2)2w2 = 0.
=2
(4)
0
where 4 is the GOLDEN MEAN, and w is a parameter (EndraB, Nordstrand), taken as w = 1 in the above plot. The Barth sextic is invariant under the IC~SAHEDRAL GROUP. Under the map (X>Y,+v4
+ (x2,Y2,22,w2),
Integrating
the first part and using
s
x cos(bx)
dx = $ cos(bx)
the integral + f sin(bx)
(5)
Barycentric
for the second part
I(x)
= 2
Base (Number)
Coordinates gives
sin( 27&x) 2nk a
cos(2rkz)
+ =&
sin(%kx)
I1 0
sin(&rka)
=2
_ o
27&
1
+ a sin(2rka) 2rk
 $&cos(Zrrlcu)
 l] = a’;;:;;) (6)
where sincx is the SINC 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 0 E nka and solving 2
= 2sinpsinPPcosp P
sinP(sin/?

(7)
2
= 0
pcosp)
sir@  pcosp tanp
=O P
Base (Logarithm) The number used to define a LOGARITHM, which is then written log,. The symbol log x is an abbreviation for log,,~, Ins for log, x (the NATURAL LOGARITHM), and lg x for log, x:. see UZSO E, LG, RITHM,NATURAL
00)
Solving this numerically gives p = 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 E rrka and solving sinc2 x = 3 (11)
binary ternary quaternary quinary senary septenary octal nonary decimal undenary duodecimal hexadecimal vigesimal sexafzesimal
10 11
12 16 x1/2 = rkl/za
with
Name
2
9
for x1/2, yielding
Therefore,
Base 3 4 5 6 7 8
(9)
= p.
= 1.39156.
20 60
(12)
= 2kIi2
0.885895 x a
1.77179 = L .
see also APODIZATION FUNCTION, PARZEN TIoN Fu NCTION, TRIANGLE FUNCT ION References Bartlett, M. S. "Periodogram tra.”
Biometrika
37,
I16,
Analysis 1950.
Barycentric Coordinates Also known as HOMOGENEOUS LINEAR COORDINATES. see TRILINEAR
LOGA
L e 2a, Let the base b representation
FWHM
NAPIERIAN
Base (Number) A REAL NUMBER x can be represented using any INTEGER number b as a base (sometimes also called a RADIX or SCALE). The choice of a base yields to a representation of numbers known as a NUMBER SYSTEM. In base b, the DIGITS 0, 1, . . . . b  1 are used (where, by convention, for bases larger than 10, the symbols A, B, C, . . mare generally used as symbols representing the DECIMAL numbers 10, II, 12, . . m).
(8)
= 0
LOGARITHM, LOGARITHM
LN,
Base (Neighborhood System) A base for a neighborhood system of a point x is a collection IV of OPEN SETS such that x belongs to every member of N, and any OPEN SET containing x also contains a member of Iv as a SUBSET.
II
= a sinc2 (rka),
99
(13)
APODIZA
and Continuous
(a, a,1
of a number
. . . ao. a1 . . .)b,
(1)
(e.g., 123.4561& th en the index of the leading needed to represent the number is n E [log, xJ ,
Spec
where 1x1 is the FLOOR FUNCTION. compute the successive DIGITS COORDINATES
x be written
or TRIai =
COORDINATES
ri LG i ’
DIGIT
(2) Now,
recursively
(3)
where rn =1: x and Base
Curve
see DIRECTRIX
(RULED
SURFACE)
ri1
= 7i  a#
(4
100
Basis
Base Space
for i = n, n  1, . . . , 1, 0, . . . . This gives the base b representation of 2. Note that if IZ: is an INTEGER, then i need only run through 0, and that if II: 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
o19999
l
.
.h.
Some number systems use a mixture of bases for counting. 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 l/Z, 1, 3, 6, 12, 30, 240, and 252, respectively). Knuth
has considered using TRANSCENDENTAL to some rather unfamiliar results, equating 7r to 1 in “base n,” n = 1,.
This
bases. such as
leads
see ah
BINARY, DECIMAL, HEREDITARY REPRESENTATION, HEXADECIM SEXAAL, OCTAL, QUATERNARY, GESIMAL , TERNARY, VIG ,ESIMAL
References Abramowitz, M. and Stegun, C. of Mathematical Functions with Mathematical Tables, 9th printing. p. 28, 1972. A. “Base Converter.” Bogomolny,
knot.com/binary.htmf. Lauwerier, IL Fractals: Endlessly ures. Princeton, NJ: Princeton $#
A. (Eds.). Formulas, New
http:
//www
Repeated
E. W. “Bases.” http : //www edu/eww6n/math/notebooks/Bases.m.
. cutthe
Geometric
University
1991. Weisstein,
Handbook Graphs, and York: Dover,
Press, . astro
Fig
pp. 611,
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 I/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 baseball 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 regions on a SPHERE, then cut them out. However, it is difficult to produce two identical regions in this manner. Thompson (1996) gives mathematical expressions giving baseball cover curves both in the plane and in 3D. J. H. Conway has humorously proposed the following “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 THEOREM,YINYANG References Thompson, R. B. “Designing a Baseball Cover. 1860’s: Patience, Trial, and Error, 1990’s: Geometry, Calculus, and Computation.” http://www.mathsoft.com/asolve/ baseball/baseball. html. Rev. March 5, 1996.
. Virginia.
Base Space The SPACE B of a FIBER BUNDLE given by the MAP f:E + B, where E is the TOTAL SPACE ofthe FIBER
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
BUNDLE.
see ah
FIBER BUNDLE, TOTAL SPACE
Baseball The numbers of baseball. three strikes a walk. The 3 (excluding
basis is any SET of n LINEARLY INDEPENof generating an ndimensional of R". Given a HYPERPLANE defined by
DENT VECTORS capable 3 and 4 appear prominently in the game There are 3 3 = 9 innings in a game, and are an out. However, 4 balls are needed for number of bases can either be regarded as HOME PLATE) or 4 (including it).
SUBSPACE
l
see BASEBALL COVER,HOME PLATE Baseball
Basis A (vector)
Cover
x1+x2
+x3
+x4
+x5
=
0,
a basis is found by solving for x1 in terms and x5* Carrying out this procedure, Xl
=
x2
x3
x4
of x2, x3, 24,
x5,
1
1
‘
0
0 +x3
1 0
a0
+x4
+x5
0 0 1
1 1
and the above VECTOR form an (unnormalized) BASIS. Given a MATRIX A with an orthonormal basis, the MATRIX corresponding to a new basis, expressed in terms of the original 21, . . . , k, is A’ = [A&
Bauer’s Ident ical Congruence Let t(m) denote the set of the 4(m) numbers less than and RELATIVELY PRIME to m, where 4(n) is the ToTIENT FUNCTION. Define
fm(x) = rI(x  t)* ef4
. . . A;I,].
see also BILINE AR BASIS, MODULAR ORTH~N ORMAL BA SIS, To POLOGICAL Basis
SYSTEM BASIS
BASIS,
A theorem
of Lagrange fm(x)
Theorem
see HILBERT
101
Bayes ’ Formula
Basis Theorem
BASIS THEOREM
Basler Problem The problem of analytically finding the value where [is the RIEMANN ZETA FUNCTION.
(1)
states that
= x6(m)  1 (mod
m).
(2)
This can be generalized as follows. Let p be an ODD PRIME DIVISOR of m and p” the highest POWER which divides m, then of c(Z), fm(x)
E (x*l
 l)4(“)‘(p1)
(mod
pa)
(3)
References Castellanos, D. “The 61, 6798, 1988.
Pi.
Ubiquitous
Part
I.”
Math.
Msg.
and, in particular,
fpa(x) E (x*’ Basset
see MODIFIED KIND
BESSEL
Batch A set of values of similar ner.
FUNCTION
OF THE
SECOND
fm(x) E (x2 meaning
l (4
obtained
(mod
1)#(m)/2
2a)
(5)
in any manand, in particular,
Tukey, J. W. Explanatory Data AddisonWesley, p. 667, 1977.
Reading,
Analysis.
f2a(z) E (x2  1)2”2
MA:
see also LEUDESDORF
(mod
2”).
(6)
THEOREM
Function U(+,O,2x)
References Hardy, G. H. and Wright, E. M. “Bauer’s Identical to the Theory of ence.” $8.5 in An Introduction 5th
for ,a=> 0,where U is a CONFLUENT FUNCTION OF THE SECOND KIND.
HYPERGEOMETRIC
see also CONFLU ENT HYPERGEOMETRICDIFFERENTIAL FUNCTION EQ UATION, HYP ERG EOMETRIC Batrachion A class of CURVE defined at INTEGER values which hops from one value to another. Their name derives from the word batrachion, which means “froglike.” Many batrachions are FRACTAL. Examples include the BLANCMANGEFUNCTION,HOFSTADTERCONWAY$~O,OWI SEQUENCE,HOFSTADTER'S QSEQUENCE, and MALLOW'S SEQUENCE. References C. A. “The Crying Ch. 25 in Keys to Infinity. pp. 183191, 1995.
Pickover,
pa)
Furthermore, if m > 2 is EVEN and 2= is the highest POWER of 2 that divides m, then
Heferences
Bateman
(mod
l)pul
Function
Oxford,
ed.
Clarendon
England:
Press,
CongruNumbers,
pp+ 98100,
1979.
Bauer’s
Theorem
~~~BAUER’S
IDENTICAL
CONGRUENCE
Bauspiel
A construction
for the RHOMBIC
DODECAHEDRON.
References Coxeter, Dover,
H.
S. M.
Regular
Polytopes,
3rd
ed.
New
York:
pp. 26 and 50, 1973.
Bayes’ Formula Let A and Bj be SETS. requires that
of Fractal Batrachion 1,489.” New York: W. H. Freeman,
P(An where n denotes P(A
n Bj)
Bj)
CONDITIONAL
= P(A)P(Bj/A),
INTERSECTION = P(Bj
PROBABILITY
(“and”),
(1) and also that
n A) = P(Bj)P(AIBj)
(2)
Bayes’
102
Beam Detector
Theorem
and
P(Bj n A) =P(Bj)P Since (2) and (3) must
(3)
be equal,
P(Bj n A).
\ (41
P(A n Bj)= P(Bj)P(AIBj).
(5)
P(AnBj) F’rom
CAIBj)*
=
(2) and (3),
(5) with
Equating
gives
(2)
=
P(A)P(BjlA)
(6)
P(Bj)P(AIBj),
SO
P(BjIA)
P(Bj)P(AIBJ
=
(7
P(A)
l
Now, let N
SE so
Ai
is an event
UAi,
Ai
is S and
(8
n Aj = 121for i # j, then
A=AnS=An
(9)
Bayesian Analysis A statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a (‘PRIOR DISTRIBUTION" which may be based on anything, including an assessment of the relative likelihoods of parameters or the results of nonBayesian observations. In practice, it is common to assume a UNIFORM DISTRIBUTION over the appropriate range of values for the PRIOR DISTRIBUTION.
Given the PRIOR DISTRIBUTION, collect data to obtain Then calculate the LIKELIthe observed distribution. HOOD of the observed distribution as a function of parameter 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 distribution is then the parameter estimate, and “probability intervals” (the Bayesian analog of CONFIDENCE INTERVALS) can be calculated usingthe 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 statistically. see also MAXIMUM LIKELIHOOD, UNIFORM DISTRIBUTION
PRIOR
DISTRIBUTION,
References Hoel,
(l@ From
(5), this becomes
P(A) = 5 P(Ai)P(E,Ai), i=l
(11)
P. G.; Port,
Statistical 42, 1971.
Theory.
to S. C.; and Stone, C. J. Introduction New York: Houghton Mifflin, pp. 36
Iversen, G. R. Bayesian Statistical Inference. Thousand Oaks, CA: Sage Pub., 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp* 799806, 1992. A Bayesian Tutorial. New York: Sivia, D. S. Data Analysis: Oxford University Press, 1996.
SO P(A,IA)
=
NP(Ai)p(AiA,)
Bays’ Shuffle A shuffling algorithm BER generators.
l (12)
C P(Aj)P(A14)
used in a class of RANDOM
NUM
j=l
References see UZSO CONDITIONAL STATISTICS
PROBABILITY,
INDEPENDENT
Knuth, ming,
References Press, W. H.; Flannery, lin& W. T. Numeri& Scientific
Computing,
bridge University
Bayed
B. P.; Teukolsky,
Press,
Theorem
see BAYES’ FORMULA
S. A.; and Vetter
Recipes in FORTRAN: The 2nd ed. Cambridge, England:
Art
of
Cam
D. E. 53.2 and 3.3 in The Art VoZ. 2:
Seminumerical
of Computer Algorithms, 2nd
Program
ed. Read
ing, MA: AddisonWesley, 1981. Press, W+ H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientijic Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 270271, 1992.
p. 810, 1992.
Beam Detector N.B. A detailed online essay by S. Finch ing point for this entry.
was the start
Beast Number
Bean Curve The PLANE
given by the Cartesian
CURVE
103 equation
ST4+ x2y2 + y4 = x(x” + y”). References Cundy, H. and Rollett, 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 larc beam detector, illustrated in the upper left figure, has length Ll = r + 2. The shortest known 2arc beam detector, illustrated in the right figure, has angles
given by solving
81 $=: 1.286 rad
(1)
02 = 1.191 rad,
(2)
the simultaneous
equations
2 cos 01  sin( $2)
= 0
+sin(+&)[sec2($&)
tan(+&)cos(@)
(3) + I] = 2.
Stradbroke,
England:
Models, A. Mathematical Tarquin Pub., 1989.
3rd
Beast Number The occult “number of the beast” associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:13: “Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666.” The beast number has several interesting properties which numerologists may find particularly interesting (Keith 198283). In particular, the beast number is equal to the sum of the squares of the first 7 PRIMES
(4) 22 + 32 + 52 + 72 + 112 + 132 + 172 = 666,
The corresponding L2
length
is satisfies
= 2n  201  02 + 2 tan( $01) + sec( f&)
 cos( i&)+tan(
i&)
sin( @2) = 4.8189264563..
A more complicated expression 3arc length La = 4.799891547.. L
inf L,
detection
GERS’ CONSTANT.
constant,
It is known
or the TRENCH that L > n.
H, T.; Falconer,
solved
Problems
4(666) where 4 is the T~TIENT
6.6, (2)
FUNCTION,
as well as the sum
DIG
K. J.; and Guy, R. K. §A30 in UnNew York: SpringerVerlag,
in Geometry.
i = 666.
x i=l
The number 666 is a sum and difference 6th POWERS, 666 = l6  26 + 36
(3) of the first three (4)
(Keith). Another curious identity is that there are exactly two ways to insert “+” signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321, 666 = I+
2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9
(5) 666 = 9 + 87 + 6 + 543 + 21 (Keith).
666 is a REPDIGIT,
(6)
and is also a TRIANGJJLAR
NUMBER T6.6
Curve
l
6.6
1991. Faber, V.; Mycielski, J.; and Pedersen, P. “On the Shortest Curve which Meets All Lines which Meet a Circle.” Ann. Polon. Math. 44, 249266, 1984. Faber, V. and Mycielski, J. “The Shortest Curve that Meets All Lines that Meet a Convex Body.” Amer. Math. Monthly 93, 796801, 1986. Finch, S. “Favorite Mathematical Constants.” http : //www . mathsoft .com/asolve/constant/beam/beam.html. Makai, E. “On a Dual of Tarski’s Plank Problem.” In Diskrete Geometric. 2 Kolloq., Inst. Math. Univ. Salzburg, 127132, 1980. Stewart, I, “The Great Drain Robbery.” Sci. Amer., 206207, 106, and 125, Sept. 1995, Dec. 1995, and Feb. 1996.
Bean
= 6
(6)
References Croft,
(1)
the identity
(5)
gives the shortest known . . Finch defines
7221
as the beam
..
ed.
= T36 =
666.
(7)
In fact, it is the largest REPDIGIT TRIANGULAR NUMBER (Bellew and Weger 197576). 666 is also a SMITH NUMBER. The first 144 DIGITS of 7r  3, where x is PI, add to 666. In addition 1997).
144 = (6 + 6) x (6 + 6) (Blatner
A number of the form 2” which contains the digits of the beast number “666” is called an APOCALYPTIC NUMBER, and a number having 666 digits is called an APOCALYPSE
NUMBER.
104
Beatty
see also
BeI
Sequence
APOCALYPSE NUMBER, APOCALYPTIC MONSTER GROUP
.
Bee
NUM
BER, BIMONSTER, References
Bellew, D. W. and Weger, R. C. “Repdigit Triangular Numbers .” J. Recr. Math. 8, 9697, 197576. Blatner, D. The Joy of Pi. New York: Walker, back jacket, 1997. Math. Msg. 61, 153Cast ellanos, D . “The Ubiquitous r.” 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 19821983.
Number
666.”
J. Recr.
Math.
15, 8587,
12QJ
>
13@]
1
’
l
l
1
d+‘=ly P then the Beatty sequences la], 124, together contain all the POSITIVE repetition. l
Gardner,
M. Mathematical Magic Show: More Puzzles, Diversions, Illusions and Other Mathematical SleightofMind from Scientific American. New York: Vintage, p. 147, 1978. Games,
BehrensFisher
Test PROBLEM
Behrmann Cylindrical EqualArea Projection A CYLINDRICAL AREAPRESERVING projection uses 30’ N as the nodistortion parallel.
which
j
1
l
References
see FISHERBEHRENS
Beatty Sequence The Beatty sequence is a SPECTRUM SEQUENCE with an IRRATIONAL base. In other words, the Beatty sequence corresponding to an IRRATIONAL NUMBER 8 is given by where 1x1 is the FLOOR FUNCTION. If CY and p are POSITIVE IRRATIONAL NUMBERS such that 1011
A 4POLYHEX.
. . . and [@J, [ZpJ, INTEGERS without
.
References http:
Dana, P. H. “Map Projections.” depts/grg/gcraft/notes/mapproj/mapproj
//www .utexas.
edu/
.&ml.
Bei
References Gardner,
M. Penrose Tiles and Trupdoar Ciph.ers., + and the of Dr. Matrix, reissue ed. New York: W. H. F’reeman, p* 21, 1989. Graham, R. L.; Lin, S.; and Lin, C.S. “Spectra of Numbers.” Math. Mug. 51, 174176, 1978. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p. 227, 1994. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 2930, 1973. Return
To
RelBei
Im[Bei
zl
zl
IBei
zI
Beauzamy and D6got’s Identity For P, Q, R, and S POLYNOMIALS in n variables [PmQ,RS]
=
Ix ill
~ il!.
The IMAGINARY
A . .&J’
.*.,i&O
PART of
J, (xe3=i/4
where
The special
) = her,(x)
OPERATOR,
[X,Y]
is
(1)
t i bei(
(2)
case v = 0 gives J&&x)
Di = B/8xi is the DIFFERENTIAL the B~MBIERI INNER PRODUCT,
+ i b&(x).
E her(x)
where Jo(z) is the zeroth THE FIRST KIND.
order
BESSEL
FUNCTION
and O” (1)“(;)4”
PC’ 21 ,**&a)
= @l
1
OF
. * . &np.
bei
=
x
[(2n)!]2
(3)
l
n=O
see also REZNIK'S
IDENTITY
see also BER, TIONS, KER
BESSEL
FUNCTION,
KEI,
KELVIN
FUNC
Bell Number
Bell Curve References Abramowitz,
M. and Stegun,
C. A. (Eds.).
“Kelvin
Func
of Mathematical Functions with t ions.” $9.9 in Handbook FormzLlas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379381, 1972.
K. B. “The Kelvin Spanier, J. and Oldham, Ch. 55 in An Atlas of Functions. Washington, sphere, pp. 543554, 1987.
where (E) is a BINOMIAL formula of Comtet (1974)
I
where
Bell Curve see GAUSSIAN DISTRIBUTION,
NORMAL DISTRIBUTION
COEFFICIENT,
B, = 1el
Functions .” DC: Hemi
[x ml denotes
105
272
mn
m!
x
m=l
or using
the
1
(4)
’
I
the CEILING FUNCTION.
The Bell number Bn is also equal to & (l), where & (2) is a BEL #L POLYNOMIAL. DOBI~~SKI'S FORMULA gives the nth Bell number
Bell Number The number of ways a SET of n elements can be PARTITIONED into nonempty SUBSETS is called a BELL NUMBER and is denoted B,. For example, there are five ways the numbers (1, 2, 3) can be partitioned: {{l}, Ph Wh w7 219 WEI WP and ((17 2, 3)), so & = 5. Bell numbers for n = 1, 2, 877, 4140, 21147, 115975, . . numbers are closely related to .
.
l
319 Wh W? (2, w, Bo = 1 and the first few are 1, 2, 5, 15, 52, 203, (Sloane’s AOOOllO). Bell
(5) k=O
Lovk (1993) showed that this formula t otic limit
l
CATALAN NUMBERS. where A(n) is defined
The diagram below shows the constructions giving & = elements 5 and Bd = 15, with line segments representing in the same SUBSET and dots representing subsets containing a single element (IX&au).
implicitly
A variation
= 72.
(7)
of DOBI~~SKI'S FORMULA gives
for 1 < k < n (Pitman asymptotic formula
3 ( 1) c  s s=o
nm
mk m!
1997).
de Bruijn
(8) (1958) gave the
lnlnn
In Bn =lnnlnlnnl+K+G can be defined
by the equation
A(n) log[A(n)]
BI, = k
The INTEGERS B,
gives the asymp
1
n
by the sum
+;
(Ey+o
[fi]
l
(9)
TOUCHARD'S CONGRUENCE states is a STIRLING NUMBER OF THE
SECOND KIND, or by the generating
function
g”1=xO”Bnxn n!
’
(2)
B p+k
=
Bk
+
Bk+l
(mod
p>
7
(10)
when p is PRIME. The only PRIME Bell numbers for n 5 1000 are B2, B3, B7, B13, B42, and B55. The Bell numbers also have the curious property that
n=O
The Bell numbers can also be generated using the BELL TRIANGLE, using the RECURRENCE RELATION
IB 0
&
B2
a0
Bl
B2
B3
a
1 Bn
B
r&+1=
.
.
l
m
l
.
.
l
l
Bn+l
Bn+2
Bn &+I .
. l
. .
.*
.
l
I
n rI i=l
n!
(11)
B2n
(3)
(Lenard
1986).
see also BELL POLYNOMIAL, BELL TRIANGLE, DOBI~ SKI'S FORMULA, STIRLING NUMBER OF THE SECOND KIND,TOUCHARD'S CONGRUENCE
Bell Polynomial
106
Beltrami
References Bell,
E.
T.
“Exponential
Numbers.”
Amer.
Math.
Monthly
41, 411419,1934. Comtet, L. Advanced Combinatorics. Dordrecht, Netherlands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: SpringerVerlag, pp. 9194, 1996. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, pp. 102109, 1958. Dickau, R. M. “Bell Number Diagrams.” http : // forum . swarthmore . edu/advanced/robertd/bell . html. Gardner, M. “The Tinkly Temple Bells.” Ch. 2 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992. Gould, H. W. Bell & Catalan Numbers: Research Bibliography of Two Special Number Sequences, 6th ed. Morgantown, WV: Math Monongliae, 1985. Lenard, A. In Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. (M. Gardner). New York: W. H. Freeman, pp. 3536, 1992. Levine, J. and Dalton, R. E. “Minimum Periods, Modulo p, of First Order Bell Exponential Integrals.” Muth. Comput. 16, 416423, 1962. Lov&z, L. Combinatorial Problems and Exercises, 2nd ed. Amsterdam, Netherlands: NorthHolland, 1993. Pitman, J. “Some Probabilistic Aspects of Set Partitions.” Amer. Math. Monthly 104, 201209, 1997. Rota, G.C. “The Number of Partitions of a Set.” Amer. Math. Monthly 71, 498504, 1964, Sloane, N. J. A. Sequence A000110/M1484 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Bell
The Bell polynomials where B, is a BELL nomials are
are defined NUMBER.
Equation
such that & (1) = B,, The first few Bell poly
@o(x)= 1 +1Cx)=X 42
(2)
=x+x2
$3 b>
= x + 3x2 + x3
+4(x)
=
45b)
= x + 15x2 + 25x3 + 10x4 + x5
x + 7x2 + 6x3 + x4
$6 cx> = x + 31x2 + 90x3 + 65x4 + 15x5 + x6. see
BELL
also
NUMBER
References Bell, E. T. “Exponential 258277, 1934.
Bell
Polynomials.”
Ann.
Math.
35,
Triangle 1 2 5
15 52 203 877 ... 10 37 151 674 ‘** 2 7 27 114 523 l . . 5 20 87 409 *me 15 67 322 m. 52 255 ‘m. 203 ‘a.
1 3
...
A triangle of numbers to be computed using
Polynomial
Differential
which allow the BELL NUMBERS the RECURRENCE RELATION
14 
B
12 
n+l
=
10 8
see also BELL NUMBER, CLARK'S TRIANGLE, LEIBNIZ HARMONIC TRIANGLE, NUMBER TRIANGLE, PASCAL'S TRIANGLE, SEIDELENTRINGERARNOLD TRIANGLE
6
Bellows I
0.2
0.4
0.6
TWO different GENERATING FUNCTIONS polynomials for n > 0 are given by 00 &(x)
E e” IE k=l
kn+k
0.8
1
for the
Bell Beltrami Differential For a measurable function equation is given by
(k  l)!
or
Conjecture
see FLEXIBLE POLYHEDRON Equation p, the Beltrami
f z* = A&, where fL: is a PARTIAL DERIVATIVE and X* denotes
COMPLEX CONJUGATE ofz. see also QUASICONFORMAL where
differential
(L) is a BINOMIAL
COEFFICIENT.
the
MAP
References Iyanaga, S. and Kawada, Y. (Eds.). Encyclupedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1087, 1980.
I
Beltrami A VECTOR
Field u satisfying
FIELD
the vector
identity
ux(vxu)=o where CURL
A x B is the CROSS PRODUCT is said to be a Seltrami field.
see ah FIELD,
BIVERGENCELESS SOLENOIDAL
and V x A is the
IRR~TATI~NAL
FIELD,
FIELD
Beltrami Identity An identity in CALCULUS OF VARIATIONS discovered in The EULERLAGRANGE DIFFEREN1868 by Beltrami. TIAL EQUATION is
Now, examine
the DERIVATIVE
df
w
&=dyYx+dy,Yxx+~. Solving
for the Sf /ay
df
af  G’
co
(1) by yat gives af
d af = 0 YxdJ: ( ayx > 
Ya:aY Substituting
gives
df = zdf  dy,Yxx
&Yx
Now, multiplying
term
af
af
(3) into
Bend (Curvature) Given four mutually tangent circles, their bends are deIf the fined as the signed CURVATURES of the CIRCLES. contacts are all external, the signs are all taken as POSITIVE, whereas if one circle surrounds the other three, the sign of this circle is taken as NEGATIVE (Coxeter 1969). see dso CURVATURE, SODDY CIRCLES
(4)
(4) then gives
(5)
Bend (Knot) A KNOT used to join form a longer length.
useful if fz = 0, since in that cas e (7 >
$(fYxg)=O,
Title
immediately
Newspapers Specific Heat
31.0 33.9 41.3 30.0 24.0
Pressure
29.6
HP.
30.0 26.7 27.1 47.2 25.7 26.8 33.4 32.4 27.9 32.7 31.0 28.9 25.3 27.0 30.6 0.8
Rivers, Area Population
Lost
Drainage
gives
Atomic Wgt. nl,
fy,%
=c, X
&i
Design Reader’s
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 allows straightforward solution of the BRACHISTOCHRONE PROBLEM. see U~SO BRACHISTOCHRONE PROBLEM, CALCULUS OF VARIATIONS, EULERLAGRANGE DIFFERENTIAL EQUATION, SURFACE OF REVOLUTION
2nd
ed.
New
to
PA: Courage, p. 49, 1993.
First
Mol. Wgt.
which
Geometry,
THEOREM,
Benford’s Law Also called the FIRST DIGIT LAW, FIRST DIGIT PHENOMENON, or LEADING DIGIT PHENOMENON. In listings, tables of statistics, etc., the DIGIT 1 tends to occur with PROBABILITY N 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 The table below, taken from Benford (1938), pages. 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.
Constants
This form is especially
to
CIRCLE
the ends of two ropes together
Philadelphia,
12
(6)
DESCARTES
References Coxeter, H. S. M. Introduction York: Wiley, pp. 1314, 1969.
References Owen, P. Knots.
of cc
107
Law
Benford’s
Be1trami Field
Dig.
Cost Data XRay Volts Am.
League
Blackbody Addresses nl,
n2.n!
Death Rate Average Prob.
Error
3 16.4 10.7 20.4.14.2 14.4 4.8 18.0 12.0 18.4 16.2 18.3 12.8 18.4 11.9 25.2 15.4 23.9 13.8 18.7 5.5 20.3 9.7 14.8 14.3 18.5 12.4 18.8 10.1 17.5 14.4 17.6 12.6 17.3 14.1 19.2 12.6 16.0 12.0 18.6 15.7 18.5 12.4 0.4 0.4
Digit
4
5
11.3 8.1 8.6 10.0 14.6 9.8 10.8 10.8 12.6 4.4 6.8 7.5 7.5 10.1 9.0 9.8 8.7 8.8 10.0 9.4 9.4 0.3
7.2 7.2 10.6 8.0 10.6 8.3 8.1 6.7 8.2 6.6 6.6 8.3 7.1 9.8 8.1 7.4 6.6 8.5 8.5 6.7 8.0 0.2
#
6 8.6 6.2 5.8 6.0 4.1 6.4 7.0 5.1 5.0 4.4 6.8 8.4 6.5 5.5 7.4 6.4 7.0 6.4 8.8 6.5 6.4 0.2
7
8
9
5.5 4.2 5.1 335 4.1 3.7 2.2 3259 1.0 2.9
6.0 3.2 5.7 5.1 4.1 5.0 3.3 7.2 7.0 5.5 4.7 5.1 4.9 5.2 5.6 6.8 7.2 5.1 0.2
5.0 4.8 4.4 5.1 2.8 2.5 4.4 8.0 7.3 4.9 5.5 5.8 5.6 4.7 5.0 7.1 4.8 4.9 0.2
10.6
5.0 4.1 4.7 3.6 3.2 1.9 5.5 8.9 5.6 4.2 3.1 4.8 3.0 5.4 5.0 5.5 4.1 4.7 0.3
104 100
1389 703 690 1800 159 91 5000 560 308 741 707 1458 1165 342 900 418 1011
108
Benham’s
In fact, the first
LOGARITHMIC
SIGNIFICANT DISTRIBUTION,
P(n)
Benson’s Formula
Wheel
St: log(n
DIGIT
seems to follow
a
with
Bennequin’s Conjecture A BRAID with AcZ strands and R components with positive crossings and N negative crossings satisfies
+ 1)  logn IpNI<2U+MR<P+N, 
uses CENTRAL LIMITfern= 1, . . . . 9. One explanation like theorems for the MANTISSAS of random variables under MULTIPLICATION. As the number of variables increases, the density function approaches that of a LOG
ARITHMIC DISTRIBUTION. References Benford,
F. “The Law of Anomalous Numbers.” Proc. Amer, 78, 551572, 1938. Boyle, J. “An Application of Fourier Series to the Most Significant Digit Problem.” Amer. Math. Monthly 101, 879886, 1994. Hill, T. P. “BaseInvariance Implies Benford’s Law,” Proc. Amer. Math. Sot. 12, 887895, 1995. Amer. Hill, T. P. “The SignificantDigit Phenomenon.” Phil.
Math.
Sot.
102, 322327,
Monthly
1995+
Hill, T. P. “A Statistical Derivation of the SignificantDigit Stat. Sci. 10, 354363, 1996. Law.” Hill, T. P. “The First, Digit Phenomenon.” Amer. SC;. 86,
358363,1998. Ley, E. ‘(On the Peculiar Distribution of the US* Stock Indices Digits .” Amer. Stat. 50, 311313, 1996. Newcomb, S. “Note on the Frequency of the Use of Digits in Natural Numbers.” Amer. J. Math. 4, 3940, 1881. Nigrini, M. “A Taxpayer Compliance Application of Benford’s Law.” J. Amer. Tax. Assoc. 18, 7291, 1996. Plouffe, S. “Inverse Symbolic Calculator.” http : //www . cacm. sfu.ca/projects/ISC/. Raimi, R. A. “The Peculiar Distribution of First, Digits.” Sci. Amer. 221, 109119, Dec. 1969. Raimi, R. A. “The First Digit Phenomenon,” Amer. Math. Monthly 83, 521538, 1976.
Benham’s
P
Wheel

While the where U is the UNKNOTTING NUMBER. 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 MILNOR'S CONJECTURE (and, independently, using MENASCO'S THEOREM).
see also BRAID,MENASCO'S THEOREM,MILNOR'S JECTUREJNKNOTTING NUMBER
CON
R leferences E ennequin,
D. “L’instanton gordien (d’aprks P. B. Kronheimer et T. S. Mrowka).” Aste’risque 218, 233277,1993. B irman, J. S. and Menasco, W. W. “Studying Links via Closed Braids. II. On a Theorem of Bennequin.” Topology Appl. 40, 7182, 1991. B oileau, M. and Weber, C. ‘(Le problkme de 3. Milnor sur le nombre gordien des noeuds alghbriques.” Enseign. Math.
30,173222,1984. B oileau, M. and Weber, C. ‘(Le problkme nombre gordien des neuds algkbriques.” and Singularities
(PlanssurBex,
de J. Milnor sur le In Knots, Braids 1982). Geneva, SwitzerMath. Vol. 31, pp. 4998,
land: Monograph. Enseign. 1983. Cipra, B. What’s Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Sot., pp* 813, 1994. Property of AlKronheimer, P. B. “The GenusMinimizing gebraic Curves .” Bull. Amer. Math. Sot. 29, 6369, 1993. Kronheimer, P. B. and Mrowka, T. S. “Gauge Theory for 1993. Embedded Surfaces. I.” Topology 32, 773826, Kronheimer, P. B. and Mrowka, T. S. “Recurrence Relations BUll. and Asymptotics for FourManifold Invariants.” Amer. Math. Sot. 30, 215221, 1994. Menasco, W. W+ “The BennequinMilnor Unknotting Conjectures.” C. R. Acad. Sci. Paris Sk. I Math. 318, 831
836,1994. Benson’s Formula An equation for a LATTICE SUM with
43(l)
= i, j,
An optical ILLUSION consisting of a spinnable top marked in black with the pattern shown above. When the wheel is spun (especially slowly), the black broken lines appear as green, blue, and red colored bands! References Cohen, J. and Gordon, D. A. “The PrevostFechnerBenham Subjective Colors.” Psycholog. Bull. 46, 97136, 1949. Festinger, L .; Allyn, M, R.; and White, C. W. “The Perception of Color with Achromatic Stimulation,” Vision Res.
11, 591612,1971. Fineman, Dover, Trolland,
M. The Nature of pp. 148151, 1996. T. L. “The Enigma Physiology 2, 2348, 1921.
Visual
of Color
Illusion.
Vision.”
New Amer.
York: J.
n = 3
Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is nu .merically equal to 1 .74756.. ‘1 a valueknown as “the” MADELUNG CONSTANT.
see &O
MADELUNG CONSTANTS
References Borwein, Analytic
J. M. and Borwein, Number
Theory
P. B. Pi & the AGM: and
Computations1
New York: Wiley, p. 301, 1987. Finch, S. “Favorite Mathematical Constants.” mathsoft.com/asolve/constant/mdlung/mdl~g.html.
A Study in Complexity.
http: //www.
Bernoulli
Ber
Differential
Equation
109
BergerKazdan Comparison Theorem Let M be a compact nD MANIFOLD with INJECTIVITY radius inj (M), Then
Ber
Vol(M)
cn inj(M) 7r
>
1
with equality IFF M is ISOMETRIC to the standard round SPHERE S” with RADIUS inj(M), where en(r) is the VOLUME of the standard TZHYPERSPHERE of RADIUS Re[Ber
z1
Imt3er
r.
21
see 0
0
Im[zl
The REAL PART of
c her(s)
where Jo is the zeroth
+ ibei(z),
order BESSEL
(2)
FUNCTION
OF THE
FIRST KIND.
O” (1)n(;)2+4n her(x) = Ix [(zn + l)!]” ’ TIONS,
BEI, KER
BESSEL
(3)
KEI, KELVIN
FUNCTION,
FUNC
Riemannian
Abramowitz, M. and Stegun, C. A. (Eds.). tions .” $9.9 in Handbook of Mathematical Graphs,
“Kelvin
Func
Functions with Tables, 9th printing.
and Mathematical
New York: Dover, pp. 379381, 1972. Spanier, J. and Oldham, K. B. “The Kelvin Ch. 55 in An Atlas of Functions. Washington, sphere, pp. 543554, 1987.
Beraha Constants The nth Beraha constant
Geometry:
Cambridge
A Modern
University
Press,
Introduction.
1994.
Bergman Kernel A Bergman kernel is a function of a COMPLEX VARIABLE with the “reproducing kernel” property defined for any DOMAIN in which there exist NONZERO ANALYTIC FUNCTIONS of class &(D) with respect to the LEBESGUE MEASURE dV. References Hazewinkel,
is given
Functions.” DC: Hemi
by
M, (Managing
ematics: An Upduted Soviet “Mathematical
lands:
References
Formulas,
References Chavel, I.
+ i bei#.
case Y = 0 gives
J&&z)
see also
IsOMETT~Y
New York:
Jv (Xe3Ti/4 ) = ber&) The special
BLASCHKE CONJECTURE, HYPERSPHERE, IN
also
JE~T~E,
Reidel,
Ed.). Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht,
pp. 356357,
of Mathof the
Nether
1988.
Bergman Space Let G be an open subset of the COMPLEX PLANE c, and let L:(G) denote the collection of all ANALYTIC FUNCTIONS f : G + C whose MODULUS is square integrable sometimes with respect to AREA measure. Then L:(G), also denote’d A2(G), is called the Bergman space for G. Thus, the Bergman space consists of all the ANALYTIC FUNCTIONS in L2 (G). The Bergman space can also be generalized to Lz(G), where 0 < p < 00. Bernoulli
Differential
Equation
The first few are 2
Be1 =4 Be2 = 0
Let 21= yl+
+ P(X)Y
(1)
= 4(4Y”*
for n # 1, then
Bea = 1 2
Be4 = 2 Be5 = i(3 + &)
Rewriting
Bee = 3 Be7 = ~+~cos(~T)
(1) gives
z 3.247....
POLYThey appear to be ROOTS of the CHROMATIC NOMIALS of planar triangular GRAPHS. Be4 is 4 + 1, where 4 is the GOLDEN RATIO, and Be7 is the SILVER CONSTANT.
Y Plugging
7tdY z
= q(x)
(3) into
 p(x)yl”
= Q(X)
remarquables.
Paris:
Hermann,
 UP(X)*
(3)
(2))
g = (1  4&>  VW
References Le Lionnais, F, Les nombres p. 143, 1983.
(2)
= (1  n)yn$*
==:2.618
(4)
110
Bernoulli
Distribution
NOW, this is a linear FIRSTORDER ENTIAL EQUATION of the form
Bernoulli
ORDINARY
DIFFER
about
and the MOMENTS
Function
0 are
/A;=/L=iw(O)=p
g’+ VP(X)= Q(x),
(5)
where P(x) = (1n)p(x) and Q(x) E (ln)&). It can therefore be solved analytically using an INTEGRATING
p; = h!P(o)
(8) = p
ptn = &f@)(O) = p The MOMENTS
about
the MEAN
(9) l (10)
are
FACTOR
s
v=
,s fw
l42= p;  (/.L;)2 = p  p2 = p(1  p)
da: Q(x) dx + C
= p$  3/.&L’, + 2(/L:)3
CL3
eJ m4 dx (1  4 J e

(ld
= PG  P)(l
s dxc) d”q(x) dx + C
e(ln)
(6)
=p(l
where C is a constant equation (1) becomes
of integration.
dY dz = Ykl PI
(7)
dY y = (Q  P> dx
The general
solution
is then,
.
(9)
VARIANCE,
Cl and C2 constants,
dP(l
Y=
I
d=
(13) and KURTOSIS
SKEWNESS,
are
(16)
P)
lJ4 3= PC1  2P)(2P2  2P + 1) _ 3 04 P2(1  PI2  6p2  6p + 1 P(l P> . =
To find an estimator
for a population
(17)
mean,
for n = 1. (P) =
Bernoulli Distribution A DISTRIBUTION given by
pb4 = (;
5
P(&)OYl
for7l=O for n = 1
 p)‘”
for n = 0,l.

(1)
N O>: sp
+
(1
 p>‘”
etnpn(l
= eO(l p)

@INl
(>
+ etp,
n=O
 (y)N”
=
0,
(18)
SO (p) is an UNBIASED ESTIMATOR for 8. The probability of Np successes in N trials is then
;p
= 9
ppl(l
Np=l
=
(2)
The distribution of heads and tails in COIN TOSSING is a Bernoulli distribution with p = q = l/2. The GENERATING FUNCTION of the Bernoulli distribution is
= (etn)
p
Np0
Elp
= p”(l
M(t)
1).
PO  PI0  2P) P3 71 = c3 Ml  PII”‘” 1  2p 
72
2e [P&*w1 CJ
 3p+
 P)(3P2
EL= I4 =P f12 = CL2= P(l  P)
(8) dx
with
(12)
If n = 1, then The MEAN, then
y = C2e J[dx)P(X)1
 2Pl
= Pk 4&d + 6P&q2  3(cl:)4 = p  4p2 + 6p3  3p4
P4
1
J P(X) dx
(11)
= p  3P2 + 2p3
BNP(l
 qNq,
(19)
where
p 
(3) so
M(t)
=
(1

p>
+
pet
(4)
M’(t)
= pet
(5)
M”(t)
= pet
(6)
= pet 1
(7)
M@‘(t)
see
also
Bernoulli
[number
BINOMIAL
of successes] n N =N*
DISTRIBUTION
Fhction
see BERNOULLI
POLYNOMIAL
(20)
Bernoulli
Inequality
Bernoulli
Inequality
Bernoulli Bz Bernoulli ”gral
(1 + x)~ > 1t nx,
= l+nx+$n(nl)x2+in(n1)(7v2)x3+.
may be calculated
J 
(1)
where z f R >  1 # 0, n E z > 1. This inequality can be proven by taking a MACLAURIN SERIES of (1 + x)~, (1+x)”
numbers Bi
=4n
0
and analytically
(ly)”
,..
= lny++t(nl)y2+(nl)(n2)y3+.
from the inte
t2nldt

+t
(3)
 1’
from
P
(2)
Since the series terminates after a finite number of terms for INTEGRAL n, the Bernoulli inequality for x > 0 is obtained by truncating after the firstorder term. When 1 < x < 0, slightly more finesse is needed. In this case, let y = 1x1 = 2 > 0 so that 0 < y < 1, and take
111
Number
for n = 1, 2, . . l , where
2n
=
(4)
is the RIEMANN
c(z)
ZETA
FUNCTION,
The first few Bernoulli
numbers
Bz are
... (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 POSITXVE number. Therefore, Cl
(1 + x)”
Y)” > 1  ny,
> 1 + nx,
for
(4)
l<x<O,
completing the proof of the INEQUALITY of parameters. Bernoulli
(5) over all ranges
Lemniscate
see LEMNISCATE Bernoulli Number There are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines the Bernoulli numbers Bz by the equations
Bernoulli numbers defined by the modern definition are denoted B, and also called “EVENindex” Bernoulli numbers. These are the Bernoulli numbers returned by @ (Wolfram Research, Champaign, IL) the Mathematics function BernoulliB [n] . These Bernoulli numbers are a superset of the archaic ones Bc since

B,

B;x2

2!
&x4 4!
B;x6 + 6!
The B, +***
G
(_21)(n/2~1~;,2 0
can be defined
These relationships function 
(5)
by the identity
X = ex  1
B;x4 + 4!
n = 0
rz = 1 n even n odd.
(1)
for 1x1 < 2n, or
B1”x2 2!
for for for for
1
for (ICI < 7r (Whittaker and Watson 1990, pm 125). Gradshteyn and Ryzhik (1979) denote these numbers Bz, while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted B, The
can be derived
F(G)
B3+x6 + 6! +‘’
O” B,xn n!’
= x
(6)
c n=O
using the generating
O” Bn(x)t” 7’ r
(7)
which converges uniformly for ItI < 27r and all x (Castellanos 1988). Taking the partial derivative gives
wx, 4~ all:
O” Bn1
(x)tn
In
II!/
>: n=O
'

 E
crg B,(x)t” n!
= tF(x,
t).
n=O (8)
Bernoulli
112 The solution
Number
to this differential F(x,
so integrating
Bernoulli
equation
is
cothx, orders,
and cschx.
An analytic
exists for EVEN
solution
(9)
t) = T(t)ext,
B2n _ (1)“l2(2n)! (2 7r>2n
gives
1
Number
O” x
P
2n
=
p=l
1
F(x,
t) dx = T(t)
is the RIEMANN ZETA for n = 1, 2, , . . , where [(2n) FUNCTION. Another intimate connection with the RIEMANN ZETA FUNCTION is provided by the identity
ext dx = T(t)+
s 0
s 0

&(x)
dx Bn = (l)‘?a[(l

Bn(x)
dx = 1
(10)
The DENOMINATOR of&
 n).
(19)
VON STAUDT
isgivenbythe
CLAUSEN THEOREM or  text

etl
O” B,(x)tn x
denom(&)
=
$tcoth($t)
P,
rI
n=O
(20)
p prime
(Castellanos 1988). Setting both sides then gives
x = 0 and adding
bl~Pk
to
t/2
that the DENOMINATOR of B:!k is and Wright 1979). Another curious property is that the fraction part of Bn in DECIMAL has a DECIMAL PERIOD which divides n, and there is a single digit before that period (Conway 1996). which
= x
or.
also implies
SQUAREFREE (Hardy
O” B2nt2n (12)
.
?a=0
Letting
Zk+l
(11)
n!
t = 2ix then gives BO
=l
&zi
2 cot 2 =
(13)
Ba = ; B4=$
The Bernoulli for x E [;lr,7F]. calculated from the integral
&J
n!
numbers
may
B6 = & Ba=$)
 z
dz
&o
(14)
ez  1 ~n+l’
2ni .I
also be
= &
B12 = $&
1
B14 = ; B16 = +g
(15) The Bernoulli
numbers
satisfy
the identity
(k;1)8a+(k;1)Bkl+...+(k;1)~l+~o
=o,
(Sloane’s
A000367
Bls
=
B20
=
B22
=
43,867
798
174 611 330 854,513 138
1
and A002445).
In addition,
06) where
(L) is a BINOMIAL
COEFFICIENT.
An asymptotic
B2n+1
= 0
(21)
FORMULA is for n = 1, 2,
l
l
.q
2n
lim n+m
lBzn[ N 46
(> 7L 7re
.
(17)
Bernoulli numbers appear in expressions of the form ;I1 kp, where p = 1, 2, . . . Bernoulli numbers also c appear in the series expansions of functions involving tanx, cotx, cscx, lnl sinxl, lnl cosxl, lnl tanx/, tanhx, l
Bernoulli
first
used the Bernoulli
numbers while comof the FIGURATE
puting CLxl k p. He used the property NUMBER TRIANGLE that n u23. . x i=o
Cn
+
%j (22)
j+l
’
Bernoulli
Nun? ber
Bernoulli
along with a form for a,j which he derived inductively to compute the sums up to n = 10 (Boyer 1968, p. 85). For p E z > 0, the sum is given by (B + n + l)[p+ll
fp=~ k=l
 Bp+l ?
P+l
k=l
(23)
where the NOTATION dk] means the quantity in quesis raised to the appropriate POWER k, and all terms of the form Bm are replaced with the corresponding Bernoulli numbers B,. Written explicitly in terms of a sum of POWERS, tion
kp
gk+1

k’(pl
k=l
k+
l)!
(24)
l
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 curious infinite sum identities involving Bernoulli numbers (Berndt 1994). G. J. Fee and S. Plouffe have computed &oo,~~o, which has  800,000 DIGITS (Plouffe). Plouffe and collaborators have also calculated B, for n up to 72,000. see also ARGOH’S CONJECTURE, BERNOULLI FUNCTION, BERNOULLI POLYNOMIAL, DEEIYE FUNCTIONS, EULERMACLAURIN INTEGRATION FORMULAS, EULER NUMBER, FIGURATE NUMBER TRIANGLE, GENOCCHI NUMBER, PASCAL’S TRIANGLE, RIEMANN ZETA FUNCTION,VON STAUDTCLAUSEN THEOREM
Polynomial
113
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. Sot. 3, 219234, 1911. Sloane, N. J. A. Sequences A000367/M4039 and A002445/ M4189 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Spanier, J+ and Oldham, K. B. “The Bernoulli Numbers, B ” Ch. 4 in An Atlas of Functions. Washington, DC: Hziisphere, pp. 3538, 1987. Wagstaff, S. S. Jr. “Ramanujan’s Paper on Bernoulli Numbers.” J. Indian Math. Sot. 45, 4965, 1981. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
Bernoulli’s Paradox Suppose the HARMONIC
SERIES
converges
to h:
= h. k=l
Then
rearranging
the terms
in the sum gives
hl=h, which
is a contradiction.
Heierences Boas, R, P. “Some Remarkable Sequences of Integers.” Ch. 3 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 3940, 1979.
Bernoulli
Polynomial
0.15
References Abramowitz, M. and Stegun, C. A. (Eds.). and Euler Polynomials and the EulerMaclaurin 523.1 in Handbook of Mathematical Functions las,
Graphs,
and Mathematical
Tables,
“Bernoulli Formula.”
0.1
with Formu9th printing. New
Dover, pp. 804806, 1972. G. “Bernoulli Numbers, EulerMaclaurin Formula,” $5.9 in Muthemuttical Methods for Physicists, 3rd ed. Qrlando, FL: Academic Press, pp. 327338, 1985. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 71, 1987, Berndt, B. C. Ramunujun’s Notebooks, Part IV. New York: SpringerVerlag, pp, 8185, 1994. Boyer, C. B. A History of Mathematics. New York: Wiley, 1968. Castellanos, D. “The Ubiquitous Pi. Part I.” Math. Mag. 61, 6798, 1988. Conway, J. H. and Guy, R+ K. In The Book of Numbers. New York: SpringerVerlag, pp. 107110, 1996. Gradshteyn, I. S+ and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th edSan Diego, CA: Academic Press, 1980. Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 9193, 1979. Ireland, K. and Rosen, M. “Bernoulli Numbers.” Ch. 15 in
0.05
York: A&en,
A Classical Introduction to Modern Number ed. New York: SpringerVerlag, pp. 228248,
Knuth, D. E. and Buckholtz, T. J. “Computation Euler, and Bernoulli Numbers.” Math. Comput. 688, 1967.
Theory,
0.05
There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by polynomial by Bz (x)* B,(x), an d the archaic Bernoulli These definitions correspond to the BERNOULLI NUMBERS evaluated at 0,
They
2nd
= B,(O)
(1)
B;
E B;(O).
(2)
also satisfy
&z(l) = (l)“&(O)
1990. of Tangent,
21, 663
B,
(3)
and B,(l
 2) = (l)“Bn(x)
(4)
Bernoulli
114 (Lehmer are
1988).
Be(x)
Bernstein’s
Polynomial
The
first
few Bernoulli
References
POLYNOMIALS
Abramowitz, M. and Stegun, C. A. (Eds.). and Euler Polynomials and the EulerMaclaurin $23.1 in Handbook of Mathematical Functions
= 1
las,
Bz(x)=x2
Xii 4x”
+
B3(x)
=
x3

B4(2)
=
x4
 2x3 + x2  $
B5(x)
= x5  %2” + gx3 
B6(X)
= x6  3x5 + %x4  +x2 + A.
ix
iX
Comment.
of
ml knl
=
‘[Bn(m)
Euler terms

n
Bn(o)]*
(5)
(1738) gave the Bernoulli POLYNOMIALS of the generating function
E T
B,(x):.
Bn(x)
in
They
satisfy
recurrence
relation
Acad.
(7)
= nB,l(x)
dx
1882),
and obey the identity
Bernoulli’s
where B” is interpreted the FOURIER SERIES
(8)
here as &(x).
&(x) = n!
2
Hurwitz
n
(27Ti)n kxmk
for 0 < x < 1, and Raabe
(1851)
2rikx
e
OF LARGE
(X
+ t>
1738.
’
gave
Curve
Constant
N.B. A detailed online ing point for this entry.
by S. Finch
was the start
4x> = 1x1, then
= mBnBn(mx).
essay
Let En(f) be the error of the best uniform approximation to a REAL function f(x) on the INTERVAL [l,l] by REAL POLYNOMIALS of degree at most n. If
(9)
found
NUMBERS
see B~ZIER CURVE
Bernstein 0.267..
Bn
6, 6897,
Bernoulli Trial An experiment in which s TRIALS are made of an event, with probability p of success in any given TRIAL.
ml
x
Petropol.
Theorem LAW
Bernstein’s
B=(x) = (B + x)~,
A
Sci.
BernsteinBkier
dBn
(Appell
(6)
.
n=O
with Formu9th printing. New
Tables,
D. H. “A New Approach to Bernoulli Polynomials.” Amer. Math. Monthly. 95, 905911, 1988. Lucas, E. Ch. 14 in The’orie des Nombres. Paris, 1891. Raabe, J. L. YZuriickfiihrung einiger Summen und bestimmten Integrale auf die Jakob Bernoullische Function.” J. reine angew. Math. 42, 348376, 1851. Spanier, J. and Oldham, K. B. “The Bernoulli Polynomial B, (5) l ” Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167173, 1987.
see WEAK &
and Mathematical
Lehmer,
Bernoulli (1713) defined the POLYNOMIALS in terms sums of the POWERS of consecutive integers,
c k=O
Graphs,
“Bernoulli Formula.”
York: Dover, pp. 804806, 1972. Appell, P. E. “Sur une classe de polynomes.” Annales d ‘$coZe Normal Superieur, Ser. 2 9, 119144, 1882. A&en, G. Mathematical Methods for Physicis,ts, 3rd ed. Orlando, FL: Academic Press, pa 330, 1985. Bernoulli, J. Ars conjectandi. Base& Switzerland, p. 97, 1713. Published posthumously. Euler, L. “Methodus generalis summandi progressiones.”
B1(x)=x+ ”
Constant
(10)
(1)
showed that .<
lim
nkm
2nE&)
< 0.286.
(2)
He
conjectured that the lower limit (p) was p = However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed l/( 2fi).
A sum identity
involving
the Bernoulli
POLYNOMIALS
is
fl = 0.2801694990
= (ml)Bm(a+p)+m(~+Pl)Bm&+p)
(11)
for an INTEGER m and arbitrary REAL NUMBERS Q: and P. see also BERNOULLI NUMBER, EULERMACLAURIN INTEGRATION
FORMULASJGULER POLYNOMIAL
l
. ..
For rational approximations p(x)/q(x) degree m and n, D. J. Newman (1964)
(3) for p and q of proved (4
Bernstein’s
BernsteinSzegli
Inequality
for Y+Z> 4. Gonchar (1967) and Bulanov the lower bound to e
?r&iqT
5 En,n(a)
5 3edfi.
Vjacheslavo (1975) proved the constants wz and A4 such that
existence
m<e
=~E,,,(cr)
Bernstein
(1975) improved
(5)
con
M. (Managing
ematics: Soviet
An Updated “Mathematical
lands:
Reidel,
Bernstein
References Bulanov, A. P. “Asymptotics for the Best Rational Approximation of the Function Sign 2.” 1Mat. Sbornik 96, 171178,
1975.
S. “Favorite Mathematical Constants.” mathsoft,com/asolve/constant/bmstn/bmstn.html.
http: //uaw .
Gonchar, A. A. “Estimates for the Growth of Rational Functions and their Applications.” Mat. Sbornik 72, 489503, 1967. Newman, D. J. “Rational Approximation to Ix].” Michigan Math.
J.
Petrushev, Real
11, 1114, 1964. P. P, and Popov,
Functions.
New
V. A.
York:
Rational
Approximation
of
Cambridge
University
Press,
1987. Stahl, [A,
I]*”
Uniform
Russian
Acad. Sci. Computations
Varga, R. S. Scientific Zems
and
Rational
Approximation of ]z[ on Sb. Math. 76, 461487, 1993. on
Philadelphia,
Conjectures.
Mathematical
PA: SIAM,
Prub
1990.
Varga, R. S. and Carpenter, A. J. “On a Conjecture of S. Bernstein in Approximation Theory.” Math. USSR Sbomzik 57, 547560, 1987. Varga, R. S.; Ruttan, A.; and Carpenter, A. J. “Numerical Results on Best Uniform Rational Approximations to 1x1 on I1, t11. Math. USSR Sbornik 74, 271290, 1993. Vjacheslavo, N. S. “On the Uniform Approximation of ]lc] by Rat ion al Functions .” Dokl. Akad. Nauk SSSR 220, 512515, 1975.
Bernstein% Inequality Let P be a POLYNOMIAL of degree 12 with derivative
Encyclopaedia Translation
” Dordrecht,
of
Mathof the
Nether
Polynomial defined
by
a,,(t) = ni $(l  yi, 0 where 02 is a BINOMIAL COEFFICIENT. The Bernstein polynomials of degree n form a basis for the POWER POLYNOMIALS of degree n.
see also
B~ZIER CURVE
Bernstein%
Polynomial
Theorem
If g(0) is a trigonometric POLYNOMIAL of degree m satisfying the condition ]g(B)] 5 1 where 0 is arbitrary and real, then g’(0) 5 rrz* References Szegii,
H. “Best
Ed.).
and Annotated Encyclopaedia.
p. 369, 1988.
The POLYNOMIALS
(7)
Finch,
Theorem
If a MINIMAL SURFACE is given by the equation z = f(z, y) and f has CONTINUOUS first and second PARTIAL DERIVATIVES for all REAL x and y, then f is a PLANE. Hazewinkel,
(6)
(Petrushev 1987, pp. 105106). Varga et al. (1993) jectured and Stahl (1993) proved that
Surface
115
References
of POSITIVE
< M
Minimal
Polynomials
G.
Amer.
Orthogonal
Math.
Polynomials,
4th
ed.
Providence,
RI:
Sot., p. 5, 1975.
BernsteinSzeg6 Polynomials The POLYNOMIALS on the interval with the WEIGHT FUNCTIONS
[1,
l] associated
w(x) = (1  z2)lj2
also called
w(x)
= (1  X2)lj2
w(x)
=
J
1X 1+x’
BERNSTEIN POLYNOMIALS.
P’. References
Then
IIP’II~ 5 4Pllm where llPllm
= max IP(x)le lizI=
SzegG,
Amer.
G.
Orthogonal
Math.
Polynomials,
Sot., pp. 3133,
4th
1975.
ed.
Providence,
RI:
116
Berry Osseen hequali
Bertrand’s
ty
BerryOsseen Inequality Gives an estimate of the deviation of a DISTRIBUTION FUNCTION as a SUM of independent RANDOM VARIABLES with a NORMAL DISTRIBUTION. Reierences Hazewinkel,
M. (Managing
ematics: Soviet
An Updated “Mathematical
lands:
Reidel,
Ed.).
and Annotated Encyclopaedia.
Encyclopaedia Translation
of
” Dordrecht,
Mathof the
Nether
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 hundred and twenty one” can be named by the description:
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 72 and 2n. It was proved in 185051 by Chebyshev, and is therefore sometimes known as CHEBYSHEV’S THEOREM. An elegant proof was later given by Erdcs. An extension of this result is that if n > k, then there is a number containing a PRIME divisor > k in the sequence 72, n+l, . . . , n+k1. (The case n = k + 1 then corresponds to Bertrand’s postulate.) This was first proved by Sylvester, independently by Schur, and a simple proof was given by ErdBs. A related problem is to find the least value of 0 so that there exists at least one PRIME between n and n+O(ne) for sufficiently large n (Berndt 1994). The smallest known value is 8 = 6/11+ E (Lou and Yao 1992).
DE POLIGNAC'S CONJEC
THEORY, NUM BER
see als 0 CHOQUET TURE,
is an inconsistency
in naming
G. J. “The
Berry
Paradox.”
Complexity
1, 2630,
Bertelsen’s An erroneous
Number value of r(lO’), where r(z) 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 corn/ksbrown/kmath049.
Numb er.” htm.
http://www.seanet.
Bertini’s Theorem The general curve of a system which is LINEARLY INDEPENDENT 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.
national University,
Notebooks,
Part
Coolidge, J, 1;. A Treatise on Algebraic York: Dover, p. 115, 1959.
Plane
Curves.
New
Bertrand Curves Two curves which, at any point, have a common principal NORMAL VECTOR are called Bertrand curves. The product of the TORSIONS of Bertrand curves is a constant. Paradox
see BERTRAND'S PROBLEM
Ramanujan
Centenary Dec. 21,
Conference
IV*
New
York:
of the Interheld at Anna
Madras, 1987. (Ed. K. Alladi), New York: SpringerVerlag, pp. l20, 1989. Type of Prime Number Lou, S. and Yau, Q. “A Chebyshev’s Theorem in a Short Interval (II).” HardyRamanujan J.
15,1334992. Bertrand’s Problem What is the PROBABZITY that a CHORD drawn at RANDOM on a CIRCLE of RADIUS T has length > r? The answer, it turns out, depends on the interpretation of “two points drawn at RANDOM." In the usual interpretation that ANGLES 81 and 02 are picked at RANDOM on the
CIRCUMFERE NCE, p=.
7T;
_
2
T
However
References
Bertrand’s
C. Ramanujan’s
SpringerVerlag, p. 135, 1994. Erdijs, P. “Ramanujan and I.” In Proceedings
References Chaitin, 1995.
PRIME
References Berndt, B.
it in this manner!
Problem
1
if a point
3
is instead
placed
at RAN DOM ona
RADIUS of the CIRCLE and a CHORD drawn DICULAR to it, P=
d3 TT T

PERPEN
A 2 .
The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated CIRCLE, 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 interpretation of “RANDOM'~ as a continuous UNIFORM DISTRIBUTION over the RADIUS is the only one possessing all these three invariances. References Paradox.” http : //www . cuttheBogomolny, A. “Bertrand’s knot.com/bertrand.html. Jaynes, E, T. Papers on Probability, Statistics, and Statistical Physics. Dordrecht, Netherlands: Reidel, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 4% 45, 1995.
Bertrand’s
Bessel Differential
Test
Bertrand’s Test A CONVERGENCE TEST also called DE MORGAN’S AND BERTRAND'S TEST. If the ratio of terms of a SERIES { a, I Tzl can be written in the form
then the series converges if lim,,,p, > 1 and diverges if lim,+,p, < 1, where limn+oo is the LOWER LIMIT and limn+oo is the UPPER LIMIT.
References Bromwich, tion
TEST
T. J. I’a and MacRobert,
to the
Theory
of Infinite
T. M.
Series,
3rd
An Introduced. New York:
Equation
2d2Y dY x jp+x&+(x2m2)y=u. dividing
nlnn’
&a+1
see also KUMMER’S
Differential
Equivalently,
1 ;+A
an =1+
Bessel
117
Equation
(1)
through
by x2,
d2Y d22+
(2)
The solutions to this equation define the BESSEL FUNCTIONS. The equation has a regular SINGULARITY at 0 and an irregular SINGULARITY at 00. A transformed version of the Bessel differential given by Bowman (1958) is
equation
Chelsea, p. 40, 1991.
x 2d2Y d22 + (2p+ 1,x$ Bertrand’s
+ (u2x2r + P")y
= 0.
(3)
Theorem
see BERTRAND’S
POSTULATE
Besov Space A type of abstract
The solution
occurs in SPLINE and The Besov space BCY p,q is a complete quasinormed space which is a BANACH SPACE when 1 < p, q 2 00 (Petrushev and Popov 1987).
is
SPACE which
RATIONAL FUNCTION approximations.
References Bergh, J. and Lijfstram, J. Interpolation Spaces. New York: SpringerVerlag, 1976. Peetre, J. New Thoughts on Besov Spaces. Durham, NC: Duke University Press, 1976. Petrushev, P. P. and Popov, V. A. “Besov Spaces.” $7.2 in Rational Approximation of Real Functions. New York: Cambridge University Press, pp, 201203, 1987. Triebel, H. Interpolation Theory, Function Spaces, Differential Operators. New York: Elsevier, 1978. Bessel’s Correction between the The factor (N  1)/N in the relationship VARIANCE u and the EXPECTATION VALUES of the SAMPLE VARIANCE,
N12 ( s2 > =
(1)
u7
N
where s2
E
(x2)

(x)2
l (2)
For two samples,
0A2 =
Ns12
+N2s2
2
Nl + N2  2 
(3)
where 4Jppz, J and Y are the BESSEL SECOND KINDS, and Cl form is given by letting c = pxr (Bowman 1958, 2cll 1 dy x dx
d2Y dx2
VARIANCE
OF THE FIRST and FUNCTIONS and c2 are constants. Another y = x”Jn(/?x’), q = yxdQ, and p. 117), then
p2y2x2r2
+
a2  n2y2
22
>
Y = 0. (6)
The solution
is
xa [AJn(@xr) + BYn(@xY)] AJn(px’) + BJn(Px')]
Y
for integral n for nonintegral
72.
FUNCTION,
BEI,
(7)
see
U~SO
AIRY
FUNCTIONS,
ANGER
BER, BESSEL FUNCTION, BOURGET'S HYPOTHESIS, CATALAN INTEGRALS, CYLINDRICAL FUNCTION, DINI EXPANSION,HANKEL FUNCTION, HANKEL’S INTEGRAL, HEMISPHERICAL FUNCTION, KAPTEYN SERIES, LIPSCHITZ'S INTEGRAL, LOMMEL DIFFERENTIAL EQUATION, LOMMEL FUNCTION, LOMMEL'S INTEGRALS, NEUMANN SERIES (BESSEL FUNCTION), PARSEVAL'S INTEGRAL, POISSON INTEGRAL, RAMANUJAN'S INTEGRAL, RICCATI DIFFERENTIAL EQUATION, SONINE'S INTEGRAL, STRUVE FUNCTION, WEBER FUNCTIONS, WEBER'S
see also SAMPLE VARIANCE,
(5)
DISCONTINUOUS
INTEGRALS
References Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, p. 550, 1953.
118
Bessel’s Finite
Difference
Bessel’s Finite Difference An INTERPOLATION formula
Formula
Bessel Function The
Formula also sometimes
known
as
lutions
of the First Kind
Bessel functions are more frequently to the DIFFERENTIAL EQUATION 2d2Y
x dz2 +x&
dY
+ (x2  m2)y
defined
as so
= 0.
are two classes of solution, called the BESSEL FUNCTION OF THE FIRST KIND J and BESSEL FUNCTION OF THE SECOND KIND Y. (A BESSE'L FUNCTION OF THE THIRD KIND is a special combination of the first
There for P E
WI, B
where Sk the CENTRAL DIFFERENCE and 2n
;Gzn
=
E
B2n+l E G2n+1 E2n E G2n F2n
G
$(Ez, 
$G2n
Gzn+l
G2n+l
G
B2n
+
Fzn)
E
$(Fzn
(2) 
G
B2n

+
B2n+l,
&n)
B2n+1
(3) (4)
(5)
Gk are the COEFFICIENTS from GAUSS'S BACKWARDFORMULA and GAUSS'S FORWARD FORMULA and Ek and Fk are the COEFFZC'IENTS from EVERETT'S FORMULA. The &s also satisfy where
B2n (P) = &n(q) B2,+1
(6) (7)
= &n+l (q),
(p)
for q1p.
see also
(8)
EVERETT'S FORMULA
References Handbook Abramowitz, M. and Stegun, C. A. (Eds.). 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., ppm 9091, 1990, Beyer, W. K CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.
Bessel’s
First
Integral
Jn(x)
= i
x
cos(n0  xsin8)
de,
s 0
where
Jn(x)
is a BESSEL
FUNCTION OF THE FIRST
KIND. Bessel’s
Formula
see BESSEL'S FINITE DIFFERENCE FORMULA, BESSEL'S INTERPOLATION FORMULA, BESSEL'S STATISTICAL FORMULA Bessel Function A function Z(x)
defined
by the RECURRENCE RELA
TIONS
CI
zm+1+
zm1=
f!!!z,
and second kinds.) Several fined by slightly modifying
related functions are also dethe defining equations.
also BESSEL FUNCTION OF THE FIRST BESSEL FUNCTION OF THE SECOND KIND, FUNCTION OF THE THIRD KIND, CYLINDER TION, HEMICYLINDRICAL FUNCTION, MODIFIED SEL FUNCTION OF THE FIRST KIND,MODIFIED FUNCTION OF THE SECOND KIND, SPHERICAL FUNCTION OF THE FIRST KIND, SPHERICAL FUNCTION OF THE SECOND KIND
see
KIND, BESSEL FUNCBES
BESSEL BESSEL BESSEL
References Abramowitz, M. and Stegun, C. A. (Eds.). “Bessel Functions of Integer Order, ” “Bessel Functions of Fractional Order,” and “Integrals of Bessel Functions.” Chs. 911 in Hundbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 355389, 435456, and 480491, 1972. Arfken, G. “Bessel Functions.” Ch. 11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573636, 1985. Bickley, W. G. Bessel Functions and Formulae. Cambridge, England: Cambridge University Press, 1957. Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Gray, A. and Matthews, G. B. A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed. New York: Dover, 1966. New York: Luke, Y. L. Integrals of Bessel Functions.
McGrawHill,
1962.
McLachlan, N. W. Bessel Functions for Engineers, 2nd ed. with corrections. Oxford, England: Clarendon Press, 1961. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Bessel Functions of Integral Order” and
“Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions.” 56.5 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 223229 and 234245, 1992. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.
Bessel
0.4 0.2
X
and 0.2
Function
of the First
Kind
Bessel Function
of the First Kind
The Bessel functions the solutionstothe
of the first kind Jn(x) DIFFERENTIAL
are defined as EQUATION
BESSEL
First, look comes
of the First Kind
at the special F[a,n(n
d2Y dY d22 + xz + (x2  m2)y
x2
Bessel Function
= 0
(1)
case m = l/2,
then
 1) + un2]xm+n
(9) be
= 0,
(10)
n=2 so
which are nonsingular at the origin. They are sometimes also called CYLINDER FUNCTIONS ox: CYLINDRICAL HARMONICS. The above plot shows Jn(x) for n = 1, 2, . . . , 5.
119
1 n(n  1)an2*
a,=
(11)
Now let n E 22, where I = 1, 2, q . . l
1
To solve METHOD
the differential equation, apply FROBENIUS using a series solution of the form
a2t
=
n=O
Plugging
into
x2 )(k
(2)
n=O
_
1)
U212
 [2 . 1 ’ l] ao
l
(12)
211!(2Z  l)!!uo7
(1) yields
which,
using the identity
2’1!(2Z  l)!!
= (22)!, gives
+ n  l)anxk+n2
+ n)(k
/
21(21
1 (  1)  [22(21  1)][2(Z  1)(2z  3)]. 1 (  1) 
00
y = xk x u,xn = x u,xn+“*

(13)
4
7x10
+X x(k
+ n)unxk+nl
+
x
x2
n=O
Similarly,
unxk+n
letting
n E 2Z + 1 1
n=O
@l+l =  (2Z+ q(q
00
m2)\ /
4
n+k = 0
UnX
m1
(3)
( 1)

n=O
[21(2Z + l)][Z(E
1
 1)(22  l)] ’
l
[2 ’ 1 . 3][1]u17
l
(14) x(k
+ n)(k
+ n  l)anxk+n
+ x(/h
n=O
+ n)anxktn
which,
using the identity
2’Z!(2Z + l)!!
n=O 00
00 an22
+>:
k+n
 m2
n+k = 0.
&ax
n=2
a2i+l
(4)
EQUATION,
I
(  1)
211!(2Z + l)!!ul
gives
1 (15)
= (22 + l)! u1*
n=O
Plugging The INDICIAL
(  1)
=
= (2Z+ l)!,
obtained
by setting
back into
(2) with
k = m = l/2
gives
n = 0, is 00
uo[k(k
 1) + k  m2] = ao(k2
 m2) = 0.
Since a0 is defined as the first, NONZERO O,sok=&m. Now, if k=m,
term,
(5)
y
=
Xli2
UnXn x n=O
k2 mm2 = unXn n=1,3,5
n=0,2,4
OQ
C[(
m + n)(m
+ n  1) + (m I n) 
Un2X
/
m+n = 0
(6)
a0
4
n2
 m2]unxm+n
+ 5+2xm+n
n=O
= 0
n)UnXmfn
n=O
+>:
+x
1 xO”(1)
L=O
x2’ w
an25
m+n
=0
!
l)+
(8)
n=2
y[a,n@m+n)+
/ .1 n=2
Jl/2(4
Un2]Xm+n
=
0.
+ al x
l=O
L
(2z+
21+1
l)!x

=
(9)
+2(x)
J2
7TX
=
cosx
&sins, $
1
(16)
of order H/2
00
u1(2m+
OQ ( 1)1 i
20 cos x + u1 sin 2).
The BESSEL FUNCTIONS fined as
n=2
n 2m +
21+1 U21+1X
(7)
00
‘T;7(
1
00
U21X
I=0
= xw(
In( m + n)’
1"'
21
m2]UnXmfn x l=O
+)‘
UnXn
+
1"'
are therefore
de
(17)
(18)
Bessel Function
120
so the general
for m = H/Z
solution
Now, consider quires
a general
nz # l/2.
Bessel Function Returning
is
to equation
of the First Kind
(5) and examining
the case k: =
m,
+ a’lJlp(x) .
= adJ&x)
Y
of the _First Kind
(19>
Equation
(9) re
 2m) + e[unn(n
al(l
 2m) + Un2]Znwm
= 0. (29)
7x=2
a1(2m+ [w+m
1) = 0
(20)
+ n) + u~~]x~+~
= 0
(21)
for n = 2, 3, . . . , so =0
a1
(22)
1
a, = 
n(2m
However, the sign of m is arbitrary, so the solutions must be the same for +m and m. We are therefore free to replace m with ImI, so
al(l
+ 2jmI)
+ x[U&(n
(23)
+ n) an2
+ 2lml)
+
Gb2]z’m’+n
=
01
n=2 (30)
for n = 2, 3, . . . . Let n E 21 + 1, where
2 = 1, 2, , . . ,
then a21+1
=


l
(21t 1),2(:IL .

f
(n,
m)al
=
(4
C” I=0
+ 1) + 1]a211
JA4

.
and we obtain the same solutions replaced by Irnl .
=
as before,
x21+lml
22~+lmll!(jml+l)!
but with
m
for 17721# $
Ecosx
for m = 3
d 5 sinx
form=
0, (24)
i where f (n,m) is the function of I and nz obtained by iterating the recursion relationship down to al. Now let n G 22, where I = 1, 2, . . . , so 1
a21 = 
21(2m
back into
1)
1 o 221mZ!(Z =
1 1  1)]
l
l
. [4. (m + I)] ao*
w
1
U,X >:

n=0,2,4,...
21+m+1 U21f12
a21
+IE
I=0
00 a0
(1)’ [4Z(m + 1)][4(1  l)(m + 2  I)]*.*[4
x
[(l)'m(m a0
[41(m + 1)][4(1  l>(m
c
+
m)!
21'+m x
'
(33)
I'=0
* (m + l)lX
But I’! = 00 for 2’ = m,. . . , 1, SO the DENOMINATOR is infinite and the terms on the right are zero. We
 1) * * *1]2""" + 2  l)]~'[m(m
22l'+yV(lf
21'+m
2I+m
00
1
X
+ m)!
2’+?TL (  1)
L I=0
=
22E’+mE~!(It
00
x
x
I’+m
x
21fm +
x
zz'+m
+ m)!l!
(  1)
I’=m
w
W
l’+m
221’+m(lt
l’+m=O n+m
Ix
(32)
’
00 ( 1) x
=
00
r&=1,3,5,.
21m
 m)!x

J&x>
+
=
by
INTEGER)
Now let I s I’ + m. Then
(9))
00 n=O
(when m is an
(  1) ’
Jm(x)=?
(25) Plugging
(31)
Jm and Jm
1
a212 =  41(m + l, a212
+ 21)
( [4Z(m + Z)][4(Z  l)(m+

We can relate writing
$.
therefore
 l)'.'l]
have
I=0
= a0 xO” (1) 4lZ!(m I=0
1m!
 1m! O” (1) Ix + I)! = ao 221Z!(m + I)! ’ l=O
(26)
where
the
= y4
(l)‘+”
l=O
Now define
FUNCTIONS
JBm(x)
factorials can be generalized to GAMMA for nonintegral m. The above equation then
221+mZ!(I
zrn
= a; Jm(x).
(28)
‘lfrn
= (l)“Jm(x).
Note that the BESSEL DIFFERENTIAL EQUATION is secondorder, so there must be two linearly independent solutions. We have found both only for Irn/ = l/2. For a general nonintegral order, the independent solutions are Jm and Jm. When ~rz is an INTEGER, the general (real) solution is of the form
becomes y = ao2”m!J,(x)
+ m)!’
E
CI
Jm(x> + c2Ym(x),
(35)
Bessel Function
of the First Kind
Bessel Function
where (a.k.a.
Jm is a Bessel function of the first kind, Ym N,) is the BESSEL FUNCTION OF THE SECOND KIND (a.k.a. NEUMANN FUNCTION or: WEBER FUNCTION), and Cl and Cz are constants. Complex solutions are given by the HANKEL FUNCTIONS (a.k.a. BESSEL
which
FUNCTIONS
The Bessel function
OF THE THIRD
e
ia cos
6
= JO(Z) + 2 x
KIND).
theorem
=
Jr&)
x
In terms
+ 1)
is the
r(z) WHITTAKER
GAMMA FUNCTION.
TION OF THE FIRST
KIND,
ROOTS of the FUNCTION
(36)
and
Mo,m
zero
is a
HYPERGEOMETRIC
FUNC
the Bessel function
is written
(+z>y
+ 1; $z”).
= IyY + 1)
A derivative identity for expressing functions in terms of JO(X) is
higher
order
Jo (4
Jl(X>
J2 Cd
3.8317
5.1336
KIND.
is a CHEBYSHEV POLYNOMIAL OF THE Asymptotic forms for the Bessel functions Jm(x)E
’ r(m+l)
(,>m
(39)
2
for x << 1 and
&OS
Ix
(
for x >> 1. A derivative
identity
J5H
8.7715
1983)
=
0.38479
.
.
are given in the following J3W
4.2012 8.0152 11.3459 14.5858 17.7887
can be expressed
5.3175 9.2824 12.6819 15.9641 19.1960
in terms
6.4156 10.5199 13.9872 17.3128 20.5755
of Bessel
27r
1
J,&)
l
(48)
3.0542 6.7061 9.9695 13.1704 16.3475
Jo(z) = 27T s 0
is
JO(X),
l
1.8412 5.3314 8.5363 11.7060 14.8636
3.8317 7.0156 10.1735 13.3237 16.4706
22.2178
1
The ROOTS of its DERIVATIVES table. zero J/(x) Jo’(x) J/W 1 2 3 4 5
20.8269

ei” cos t$ dq5
= ; Ix
cos(zsin8
(49)
 no) de,
(50)
0
(41) which identity
J4W
7.5883
of the Bessel function
ROOT
xn JO(G)
n=l
I
(4
114.9309 ] 16.4706 ] 17.9598 119.4094
(40)
xy;)
d
J3
6.3802
5
Various integrals functions
J&x)==:
are given in the following
/ 11.7915 5f201 1101735 86537 13.3237 710156 1116198 14.7960 814172 / 16.2235 130152 9f610 Ill:o((l 143725 112;386 17.6160 15 7002 18.9801
Bessel
are
An integral
(47)
;
Let zn be the nth then 00
(37)
(38)
FIRST
Jn(s)
2.4048
(Le Lionnais
Tn(x)
Jnm(z)
table. 1
&(v
J”(z)
Mo,m(2ir)j
FUNCTION
of a CONFLUENT
where
states
m=m
pw2pwqy773
where
(46)
addition
Jn(y+z)
p/2 =
in J&z) cos(n0).
n=l
INTEGER,
Jm(z)
121
can also be written
The Bessel functions are ORTHOGONAL in [O, l] with respect to the weight factor 2. Except when 2n is a NEGATIVE
Kind
of the First
is BESSEL'S
FIRST
INTEGRAL,
is 7r
J&z)
U
ut J&i) s
= i” 7T
(42)
du’ = uJ1 (u).
0
(51)
= &
~izcos’~in’d~
J
(52)
0
Some sum identities
are
1 = [Jo(
+ 2[J&)]”
for n = 1, 2,
+ 2Jz(x)
and the JACOBIANGER
d0
27r
Jn(x)
1 = Jo(x)
eizcosO cos(n0) 0
s
+
2[J2(~)]~
+ 2J.42)
+. 4 + s . l
l
l
.,
(43) (44
EXPANSION
42 Jn(z)= 3 for n = 1, 2,
(z,“T l
l
l)!! J
sir?
u cos(x cos u) du (53)
0
.,
00 e
iz cos 0
z
x n=m
inJn(Z)eine,
(45)
J&)
= &
e(z/2)(z1/z)zn1 sY
&
(54)
122
Bessel fine
for n > l/2.
Integrals
tion Fourier involving
Bessel Function
Expansion
J1 (5) include
(I3owman
dx =1 (55) r0Jl(X)
1958, p. 108), so
1
xf(x)Jn(xw)dx
= &&&,,Jn+~~(xa,) r=l = +A Jn+~~(w),
s 0
(56) and the COEFFICIENTS
(57) Al = see also BESSEL FUNCTION OF THE SECOND KIND, DEBYE'S ASYMPTOTIC REPRESENTATION, DIXONFERRAR FORMULA, HANSENBESSEL FORMULA, KAPTEYN SERIES, KNESERSOMMERFELD FORMULA, MEHLER’S BESSEL FUNCTION FORMULA, NICHOLSON'S FORMULA, POISSON’S BESSEL FUNCTION FORMULA, SCHL;~FLI'S FORMULA, SERIES, SOMMERFELD’S FORMULA, FORMULA, WATSON'S FORMULA,~ATSONNICHOLSON FORMULA, WEBER'S DISCONTINUOUS INTEGRALS, WEBER'S FORMULA, WEBERSONINE FORMULA, WEYRICH'S FORMULA
of the Second Kind
are given
(4
by
1
2
xf (x) Jn(xQI1) dx.
J n+l %l>
s0
(5)
References Bowman, Dover,
F. Introduction 1958.
Bessel Function
to
Bessel
Functions.
of the Second
New
York:
Kind
References Abramowitz, M. and Stegun, C. A. (Eds.). “Bessel of Mathematical tions J and Y .” $9.1 in Handbook tions with 9th printing.
Formulas, New
Graphs,
and
Mathematical
FuncFuncTables,
York: Dover, pp. 358364, 1972. Arfken, G. “Bessel Functions of the First Kind, J,(s)” and “Orthogonality.” §ll.l and 11,2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573591 and 591596, 1985. Lehmer, D. H. ‘<Arithmetical Periodicities of Bessel Functions.” Ann. Math. 33, 143150, 1932. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 619622, 1953. Spanier, J. and Oldham, K. B. “The Bessel Coefficients Jo(x) and J&C)” and “The Bessel Function J&C)*” Chs. 5253 Washington, DC: Hemisphere, in An Atlas of Functions. pp. 509520 and 521532, 1987.
A Bessel function of the second kind Y,(x) is a solution to the BESSEL DIFFERENTIAL EQUATION which is singular at the origin. Bessel functions of the second kind are also called NEUMANN FUNCTIONS or WEBER FUNCTIONS. The above plot shows Yn(x) for n = 1, 2, . . . , Let U G J&x) be the first solution and u be the other one (since the BESSEL DIFFERENTIAL EQUATION is secondorder, there are two LINEARLY INDEPENDENT solutions) Then l
Bessel Function
Fourier
Expansion
Let n > l/2 and al, ~2, . . . be the P~SITNE ROOTS of Jn(xj = 0. An expansion of a function in the interval (OJ) in terms of BESSEL FUNCTIONS OF THE FIRST KIND
f(x) = F
&Jn(xm),
xv”
found
(1)
x(uttv  uztll) + dv  ud = 0
as follows:
d z [2 ( uh
1
xf (x)Jn(xcrr)
dx =
Jn(xw)Jn(
xal) dx.
FUNCTION
ROOTS
s0
so x(u%
 uvJ)
= L3, where
ud)]
of BESSEL
(3)
= 0,
B is a constant.
(2)
But
(2)
Take w x (1)  u x (2),
I=1
has COEFFICIENTS
+ 21’ + xw = 0.
(4 Divide
by
(5)
gives 1
xJn(xal:) Jn(xw) s 0
dx = +&,,Jn+~~(w)
(3)
u =A+B 21
dx 1 s
XV2
(6)
of the Third
Bessel Function Rearranging
Kind
and using 2t = Jm (5) gives
u = AJm(x)
+ BJm(x)
Bessel’s Inequality If f(x) is piecewise FOURIER SERIES
& m
J
E A’Jm(x)
CONTINUOUS x
X
+ B’Ym(x),
(7)
of the second
kind
has a general
a&(x)
(1)
WEIGHTING
W(X), it must
FUNCTION
be true that
is defined 2
bY f(x) yrncx)
_
Jm(x)
cos(m4


x
Jm(x)
L21n (3
+
27

bmfk

bk I
m

l ’ xm+21c(m  k  I)! 2m+2kk! T x
J
f”(x)w(x)dx
 2
m = 0, 1, 2, . . . , y is the EULERMASCHERONI
+>:
(9)
m(z)
is given by
O”
7T l
no> d0
equations
=
[ent + e“t(l)“]e“sinht
{
+ y]
2 m (X >
m = 0, x 4X 1 m # 0,x <
Y,(X)=/$Sin(Xy;)
di2(x)w(x)dx
J
f (x)4m(x)w(x)
J
 271ai2
f 2(x)w(x)
(11)
1
X2+1,
is a GAMMA
The inequality EQUALITY
References FuncTables,
Function
of the
FUNCTION
Third
Kind
bk2)
5
can also be derived
2 0
(5)
.
(6)

J
7r
(7)
7T 7r
from
SCHWARZ'S
I (f Id I2 5 (f If) kds>
Func
Arfken, G. “Neumann Functions, Bessel Functions of the Second Kind, NV(x).” s11.3 in Mathematical Methods for Physicists, 3rd ecE. Orlando, FL: Academic Press, pp. 596604, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 625627, 1953. Spanier, 5. and Oldham, K. B. “The Neumann Function Yv(x)? Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533542, 1987.
see HANKEL
1 +
k=l
KIND,BOUR
Formulas, Graphs, and Mathematical New York: Dover, pp. 358364, 1972.
SE
(12)
(ak2
Abramowitz, M. and Stegun, C. A. (Eds.). “Bessel tions J and Y.” 59.1 in Handbook of Mathematical tions with 9th printing.
(3)
Equation (6) is an inequality if the functions & 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 ao, al,  1 a, and bl, . . . , b,, then
FUNCTION.
see UZSOBESSEL FUNCTION OF THE FIRST GET'S HYPOTHESIS, HANKEL FUNCTION
(2)
(4)
dx,
dx 2 >,‘ ai2. i
00 where r(z)
2 0.
FOURIER
+ xai” i
i
J
dt. (lo)
are
 %4 R
2 0
s of the generalized
f2(x)w(x)dx
I
0
z [ln(+x) Ym(X)

0
1

am E so
sin(zsin0
7T
Asymptotic
ai 2 i
But the COEFFICIENT RIES is given by
=1Jr
dx
CON
and
The function
w(x)
dx xi aiJf(x>$i(x)w(x)
(8)
k=O
STANT,
1
a&(x) i
sin( mn)
Bessel
and
i with
where the Bessel function
123
Bessel ‘s Inequality
by expanding of f, 9 =2: C;
g in a superposition Then
IN
(8)
of EIGENFUNCTIONS
aifi*
(9)
124
Bessel’s Interpolation
If g is normalized,
then
Beta Distribution
Formula
(gig) = 1 and
(f If) > x
wi*.
(11)
z
see
also
SCHWARZ’S
INEQUALITY,
TRIANGLE
INEQUAL
ITY References
A general type of statistical DISTRIBUTION which is related to the GAMMA DISTRIBUTION. Beta distributions have two free parameters, which are labeled according to one of two notational conventions. The usual defmition calls these QI and p, and the other uses p’ s p  1 and Q’ E a  1 (Beyer 1987, p* 534). The above plots are for (Q) = (1,l) [solid], (1, 2) [dotted], and (2, 3) [dashed]. The probability function P(X) and DISTRIBUTION FUNCTION D(X) are given by
G. Mathematical Methods for Physicists, 3rd ed. OrFL: Academic Press, pp. 526527, 1985. Eradshteyn, I. S. and Ryzhik, I, M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1102, 1980. A&en,
lando,
p(x)
_ (1  xyxl B(% P)
 ,ri”)$\
Bessel’s
Interpolation
see BESSEL’S
Bessel
Formula
FINITE
DIFFERENCE
D(x) = qx; a, b),
(2)
FORMULA where B(a,b) REGULARIZED
Polynomial
see BESSEL
BETA FUNCTION, 1(x; a$) is the FUNCTION, and 0 < x < 1 where
is the BETA
QI, p > 0. The distribution
FUNCTION
s
is normalized
1
Bessel’s see
Second
t=
dx
=
r(a
0
Statistical
/W’
rb + P> B(aJ) rww)

(l)
FUNCTION
= lFl(a,
4(t)
The
where
x”‘(1
')
MOMENTS
are given
w
F
P(1)

P(2)
N E Nl + N2 .
(2)
ALPHA,
SHARPE
RATIO
(3)
is
(4
by
(x  /.L)~ dx = s
= 1.
r(a + p)r(a + T) r(a + p + +(a)
(5)
l
0
(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 T = P = 1). A fund with a beta of p has performed (p  1) x 100% better (or (T( worse if r < 0) than its benchmark index (after deducting the Tbill rate) in up markets and (T( worse (or (T( better if T < 0) in down markets. see &O
MT =
dx
a + b,it).
1 ti E 21  ii?2
 x)
s 0
The CHARACTERISTIC
23W
m=
+
rIa>w
Formula
clw
since
1
p(,)
Integral
PoIssoN INTEGRAL
Bessel’s
(1)
(1  X)P1Xa1
a
The MEAN
l(l _“)@lx dx r(4w) is
%
+ @) B(a
+ 1, p)

r(+yP>
r(a
and the VARIANCE, g2
(6)
+ P + 1)
SKEWNESS,
and KURTOSIS
are
@
z (ct:
+
P)“b
+
P
+
(7)
1)
qfifi)(Ja+Jp)Jl+~+P
41
(8)
= dQ(a+P+2)
Beta
Distribution 72
=
6(cr2 + a3  4cup  2a2p + p2  2arp2 + p”> @(LY+P+2)(QI+P+3)
. (9)
The MODE
of a variate g=
distributed al a+p2’
as p(qfl)
is (10)
Beta Function In “normal”
Beta Function
form,
the distribution
is written
The beta function
r(a + P> a1(1 x)+1 f(x) = r(a)r(p)”
(11)
cos 2m+1 0 sinzn+’
Rewriting
/&=A a+P QP o2 = (a + P>2(1+ QI+ P> y1 = y2
(P  V(q
r(p
=
42 s
The general
 24
a + @(2a2
(14)
trigonometric
form
l
(15)
Equation
(5)
’
is
$B(n+
can be transformed
(6)
POLYNOMIALS by letting GAMMA
l)!
$n+
$)*
(6)
0
=
also
(4)
 I>!
(p+q
sir? 2 cosm xdx=
+ a2p + 2p2 + alP2)
cup(a+P+q(Q+p+3)
see
(m$n+l)!’
r(PN4)
B(pyq)=
(13)
GP(a + P + 2) 3(1+
m!n!
8 de =
the arguments,
(12)
2(&Jp)(&+dP>J1+a+P
by
s0
SKEWNESS, and KURTOSIS
VARIANCE,
defined
B(m+l,n+l)=B(n+l,m+l) r/2 ,2
and the MEAN, are
is then
125
u
G
to an integral
cos2
DISTRIBUTION
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944945, 1972. Beyer, W. II. CRC Standard Mathematical Tables, 26th ed. Boca Raton, FL: CRC Press, pp. 534535, 1987.
1
m!n! (m+n+q!
B(m+l,n+l)s
References
r(m>r(n> rcm+nl
B(m,n)=
=
s
o urnPu)“du
o l IL ml(l
=
over
0,
)nldu
u
(7) ig)
.
s To put
it in a form
which
can be used to derive x E fi,
the
LEGENDRE DUPLICATI:ON FORMULA, let Beta
Function
U
SO
= x2 and du = 22 dx, and
The beta Whittaker
function is the name used by Legendre and and Watson (1990) for the EULERIAN INTEGRAL OF THE SECOND KIND. To derive the integral representation of the beta function, write the product of two FACTORIALS as
1
B(m,n)
=
m1)
X2(
(1

x2)n1
(2x dx)

x2)n1
dxa
s 0 1
2
=
x~~'(I
(9)
J0 m!n!
=
e “~m
du
e vvn
dv.
(1)
Now, let u E x2, v = y2, so m!n!
B(m+l,n+I)=
= 4
Various
=4
~POLAR
T/2
4
COORDINATES withx
=
TCOS~,
%P,
El> = = nv
00
using the GAUSS MUL
can be derived
r(nP)WW) W(P
dl
+
B(p, q)B(p + +, q) 9 B(P + e, l
a)
l
(11)
e r2(wosi?)2m+1(~sin6)2n+1~drd9
= 4 s 0
=
identities
(10)
TIPLICATION FORMULA
Transforming Y = rsin8 m!n!
To put it in a form which can be used to develop integral representations ofthe BESSEL FUNCTIONS and HYPERGEOMETRIC FUNCTION, let u E x/(1+x), SO
B(q,
s 0
Additional
42
er2T2m+2n+3
s42 &
cos
2m+l
Q sin2n+l
l
l
l
B([n

114,
q>
’
include
r(P)% + 1) 4 r(P + 1)W B(Pp Q + ‘1 = r(p + q + 1) = P WP +
s 0
cos 2m’1 8 sin2n+1
identities
4)
,f) d$
0
= 2(m + n + l)!
q)B(Q,
0 d0.
(3)
114)

%P P
+ 1,4)
(12)
B(P7 4) = B(P + 174) + B(P, 4 + 1)
(13)
126
Beta Function
(Exponential)
Betti
B(P, Q + 1) =
If n
is
Another “BETA FUNCTION" defined in terms of an integral is the “exponential” beta function, given by
(14)
a P~~ITWE INTEGER, then
B(p
+ q)B(p
tne“t
1*2*+*n = p(p+l)...( P +n)
B(p++l)
+ w)
= B(w)%
+ ~3 P>*
qt; 2,st
of the beta function
 u)‘l
is the incomplete
1 +
[2
1Y t+...+
du
beta
= (l)nez
The first few integral
satisfies
the
RECUR
1
 esr
I n&1(~).
(3)
values are
2 sinh z % 2(sinh x  z cash z) = iz2 2( 2 + z2) sinh z  42 cash z . = zz3
(4)
(18)
Pl(4
(5)
FUNCTION, DIRICHLET INREGULARIZED BETA
D2(4

Y>
l
l
l
(n

Y>
p
+ l
n!(x
BETA
function
PO(Z)=
(1
x+1
CENTRAL
also
exponential
RENCE RELATION
0
=t”
(1)
(17)
Z&&C)
ux‘(1
y) =
dt
(15)
The A generalization beta function
see
Group
+ n>
*.
.
TEGRALS, GAMMA FUNCTION, FUNCTION
(6)
see UZSO ALPHA FUNCTION
References Abramowitz, M. and Stegun, C. A. (Eds.). “Beta Function” and “Incomplete Beta Function.” 56.2 and 6.6 in Ilandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258 and 263, 1972. Arfken, G. “The Beta Function.” 5 10.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 560565, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, p. 425, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Gamma Function, Beta Function, Factorials, Binomial Coefficients” and “Incomplete Beta Function, Student’s Distribution, FDistribution, Cumulative Binomial Distribution.” $6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206209 and 219223, 1992. Spanier, J. and Oldham, K. B. “The Incomplete Beta Function B(y; p; x).” Ch. 58 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 57358’0, 1987. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
Beta
Function
(Exponential)
Beta Prime A distribution
Distribution with probability
v +x>“p P(x)= Iz: B(%P) ’ where B is a BETA FUNCTION. distributed as B’(Q, p) is kb. If x is a P’(a,P) variate, variate, If x is a p(qp) are P’(P, a> and P’(% P) and ~(~12) variates, then x2/2 and y2/2 are $112) a p’( l/2,1/2) variate. Bethe
The MODE of a variate
1
then l/z is a p’(p, cw) variate. then (1  x)/x and z/(1  x) variates. If x and y are y(crl) x/y is a ,8’(al, cwz) variate. If variates, then z2 G (x/y)” is
Lattice
see CAYLEY TREE Betrothed
Numbers
see QUASIAMICABLE
t
function
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 Mathematics: An Updated and Annotated Translation of the Soviet “IMathematical Encyclopaedia. ” Dordrecht, Netherlands: 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 extend the POLYHEDRAL FORMULA to higher dimensional spaces. The nth Betti number is the rank of the nth HOMOLOGY GROUP. Let p, be the RANK of the HoMOLOGY GROUP HT ofa TOPOLOGICAL SPACEK. For a closed, orientable surface of GENUS g, the Betti numbers are po = 1, pl = 2g, and p2 = 1. For a nonorientable surface with k CROSSCAPS, the Betti numbers are PO = 1, pl = IC  1, and p2 = 0.
the fact that moving a single control point changes the global shape of the curve. The former is sometimes avoided by smoothly patching together loworder B6zier curves. A generalization of the Ezier curve is the B
see U~SOEULER CHARACTERISTIC,
Bezout Integers
Bkier
POINCARI?
DUALITY
SPLINE. see also BSPLINE, NURBS Bkzier
CURVE
Spline
see BI?ZIER CURVE, SPLINE Numbers (X, p) for a and b such that
Curve
Xa + pb = GCD(a,
PI a
a,, the Bezout For INTEGERS al,..., of numbers kl, . . . , kn such that km
b).
numbers
+ kza2 + . . . + &an
are a set
= d,
where d is the GREATEST COMMON DIVISOR of al,..., an.
see also GREATEST COMMON DIVISOR Given a set of n control points, curve (or BERNSTEINB~ZIER
C(t)
where
the corresponding B&ier CURVE) is given by
= UP&,&), i=o
Bi,, (t) is a BERNSTEIN POLYNOMIAL and
t E
P111A “rational”
Bkzier
curve is defined
by
Bi,&)WiPi C(t) ccyxo yxo&,&)Wi’ where p is the order, Bi,, are the BERNSTEIN POLYNOMIALS, Pi are control points, and the weight wi of Pi is the last ordinate of the homogeneous point Pp. These curves are closed under perspective transformations, and can represent CONIC SECTIONS exactly. The B&ier 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,P,1 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 B6zier curves are their numerical inst abili ty for large numbers of control points, and
Bezout’s Theorem In general, two algebraic curves of degrees no and 12 intersect in m.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 POLYNOMIALS with no roots in common, then there exist two other POLYNOMIALS A and B such that AP + BQ = 1. Similarly, given IV POLYNOMIAL equations of degrees n1, n2, . . . nN in N variables, there are in general nln2 . . . nN common solutions.
see also
POLYNOMIAL
References
Coolidge, J. L. York:
Dover,
A
Treatise
on
Algebraic
Plane
Curves.
New
p. 10, 1959.
Bhargava’s Theorem Let the nth composition of a function f(z) be denoted f(“‘(~), such that f’“)(~) = x and f(l)(z) = f(z). Denote f o g(x) = f(g(z)), and define
~F(a,b,c)=F(a,b,c)+F(b,c,a)+F(c,a,b).
(I)
tet u G (a, b, c> ju\a+b+c II u II =a4+b4+c4,
(2) (3) (4)
BhaskaraBrouckner
128
BIBD
Algorithm Bianchi Identities (Contracted) CONTRACTING X with v in the BIANCHI
f (4 ==I(f&)7 f&L
f3W
(5)
= (a(b  c), b(c  a), c(a  b)) g(u)
+ &mTI;v = 0
(1)
= (9d”Lgdu>mw = ( xa2b,xab2,3ah).
Then
Rx~L/E;~+ J&v;n
(6)
IDENTITIES
(7)
if 1~1 = 0, IIf’”
0 9’“‘(u)II
= 2(ab + bc + Cu)2m+13n 
Ilgo Of’“‘(4lL
where m, n E (0, 1, . . .} and composition terms of components. see also DIOPHANTINE
THE~RE
is done in
or
CL
LLV ;gpvR);,
Bias (Estimator) The bias of an ESTIMATOR
References Berndt, B. C+ Ramanujan’s Notebooks, Part IV. New York: SpringerVerlag, pp. 97400, 1994. of Ramanujan’s Formulas for Bhargava, S. “0 n a Family Sums of Fourth Powers.” Ganita 43, 6367, 1992.
BhaskaraBrouckner
It is therefore
BiConnected Component A maximal SUBGRAPH of an undirected graph such that any two edges in the SUBGRAPH lie on a common simple cycle.
defined
d2SPV ~~ dxQxX Permuting V, K, and 7 (Weinberg gives the Bianchi identities
IDENTITIES
+ ((6)
(5)
for which
B = 0 is said to be UNBIASED.
Bias (Series) The bias of a SERIES
is defined
COMPONENT ai+2
1=
A SERIES is GEO METRIC IFFQ TIC IFF the bias is constant.
by
as ai&+2
(CONTRA
CTED),
RIE
Cosmology: Principles Theory of Relativity.
and New
.
= 0. A SERIES
ARTIS
C ross Ra
Biased An ESTIMATOR
which
exhibits
Pentagonal
see JOHNSON
SOLID
Biaugment
ed Triangular
~~~JOHNSON
So LID
BIAS. Prism
P ‘rism
References Weinberg, S. Gravitation and Applications of the General York: Wiley, 1972.
 GS12
aiai+m+2
d2SXK + d2gpn dxMxv dxvdxX > ’ 1972, pp* 146147)
as
 0) = (e  (8)) + B(6).
UNBIASED
Biaugmented
also BIANCHI MAN N TENSOR
= 0.
# is defined
see also ESTIMATOR,
Q[G, ai+l, Bianchi Identities The RIEMANN TENSOR~~
(4)
true that
8  19= (8  (8))
An ESTIMATOR
CONNECTED
= 0,
B(B) E (6)  8.
Algorithm
~~~SQUAREROOT
see
l&R);,
q2
FORD'S
M
see also STRONGLY
CR CR
QUARTIC,
EQUATION
or
(8)
Biaugmented see JOHNSON
Truncated SOLID
BIBD see BLOCK
DESIGN
Cube
Bicentric
Polygon
Bicentric
Polygon
Bicorn Bicentric Quadrilateral A 4sided BICENTRIC POLYGON, INSCRIPTABLE QUADRILATERAL. References Beyer, W. H. (Ed.) 28th ed. Boca Raton,
also called
129 a CYCLIC
CRC Standard Mathematical Tables, 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. 
c A POLYGON which has both a CIRCUMCIRCLE INCIRCLE, both of which touch all VERTICES. ANGLES are bicentric with R2  s2 = 2Rr,
and an All TRI
(1)
where R is the CIRCUMRADIUS, T is the INRADIUS, and s is the separation of centers. In 1798, N. Fuss characterized bicentric POLYGONS of 12 = 4, 5, 6, 7, and 8 sides. For bicentric QUADRILATERALS (FUSS'S PROBLEM), the CIRCLES satisfy 2r2(R2 (Dijrrie
 s”) = (R2  s”)”  4r2s2
R= 1987).
&ii2 1 (ac + bd)(ad + bc)(ab + cd) 4J abed S
Rolfsen, D. Knots and Links. Wilmington, Perish Press, pp. 3435, 1976.
DE: Publish
or
(4)
In addition, 1
1
(5)
and of a bicentric
quadrilateral
x = asini5 a cos2 t(2 + cost) Y= 3 sin2 t *
is
(7)
A = dabcd.
If the circles permit successive tangents around the INCIRCLE which close the POLYGON for one starting point on the CIRCUMCIRCLE, then they do so for all points on the CIRCUMCIRCLE.
PONCELET'S CLOSURE
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 parametric equations
(6)
a+c=b+d.
see also
References
(3)
1 +(R=s)2 (R  s)~
The AREA
Bicollared A SUBSET X c Y is said to be bicollared in Y if there + Y such that exists an embedding b : X x [I,11 b(x, 0) = x when x E X. The MAP b or its image is then said to be the bicollar.
Bicorn
S
(Beyer
(2)
1965) and T==
see UZSO BLUEEMPTY GRAPH, EXTREMAL COLORING, EXTREMAL GRAPH, MONOCHROMATIC FORCED TRIANGLE, RAMSEY NUMBER
THEOREM
References Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. DSrrie, H. “Fuss’ Problem of the ChordTangent Quadrilateral.” $39 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 188193, 1965.
The graph
is similar
to that
of the COCKED
HAT CURVE.
References Lawrence, J+ D. A Catalog of Special Plane Curves. York: Dover, pp. 147149, 1972. Macmtor History of Mathematics Archive. “Bicorn.” dcs . st and. ac . uk/ history/Curves //wwwgroups.
Bicorn.
html.
New
http : /
Bieberbach
Bicu bit Spline
130
Bidiakis
Bicubic Spline A bicubic spline is a special which uses an interpolation 4 Y(Xl,
22)
=
>:
(XI,
X2)
=
Cij
tiBIUjB1
j=l
):
x(j
i=l
j=l

4
4
YXlX2 C x i=l
>(i
The ISVERTEX 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.
l)Cijti.lzLjm2
 l)(j
 l)CijtiB2T&jB2,
see also BISLIT
CUBE,
CUBE,
CUBICAL
GRAPH
j=l
where cij are constants and u and t are parameters ranging from 0 to 1. For a bicubic spline, however, the partial derivatives at the grid points are determined globally by 1D SPLINES. SPLINE
see also BSPLINE,
Cube
case of bicubic interpolation function of the form
4 >)
ix1
?Jsz
Conjecture
Bieberbach Conjecture The nth COEFFICIENT in the P~~~~series ofa UNIVALENT FUNCTION should be no greater than n. In other words, if f(z)
= a0 + a1z + a2z2
+
l
l
.
+
a,zn
+
.
.
.
References Press, W. H.; Flannery, ling, W. T. Numerical Scientific
bridge
B. P.; Teukolsky,
Computing,
University
Bicupola Two adjoined
S. A.; and Vetter
Recipes in FORTRAN: The 2nd ed. Cambridge, England:
Press,
pp. 118122,
Art
Cam
1992.
CUPOLAS.
see also CUPOLA, ELONGATED GYROBICUPOLA,ELONGATED ORTHOBI~UPOLA, GYR~BI~UP~LA, URTH~BICUPOLA Bicuspid
Curve
of
is a conformal transformation of a unit disk on any domain, thenla,l 2 nlall. In more technical terms, “geometric extremality implies metric extremality.” The conjecture had been proven for the first six terms (the Lowner, cases n = 2, 3, and 4 were done by Bieberbach, and Shiffer 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 MILIN CONJECTURE, which established the ROBERTSON CONJECTURE, which in turn established the Bieberbach conjecture (Stewart 1996). References
The PLANE
CURVE given by the Cartesian
(X2

a")(X
Bicylinder see STEINMETZ
SOLID

a>"
+
(y"

a')'
equation =
0.
de Branges, L. “A Proof of the Bieberbach Conjecture.” Acta Math. 154, 137152, 1985. Hayman, W. K, Multivalent Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1994. Hayman, W. K. and Stewart, F. M. ‘(Real Inequalities with Applications to Function Theory.” Proc. Cambridge Phil. Sot. 50, 250260, 1954. Kazarinoff, N. D. “Special Functions and the Bieberbach Conjecture.” Amer. Math. Monthly 95, 689696, 1988. Bieberbach’s Conjecture and its Korevaar, J. “Ludwig Proof.” Amer. Math. Monthly 93, 505513, 1986. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983. Pederson, R. N. “A Proof of the Bieberbach Conjecture for the Sixth Coefficient .” Arch. Rational Mech. Anal. 31, 331351, 1968/1969. Pederson, R. and Schiffer, M. “A Proof of the Bieberbach Conjecture for the Fifth Coefficient.” Arch. Rational Mech, Anal. 45, 161193, 1972. Stewart, I. ‘&The Bieberbach Conjecture.” In From Here to Infinity: A Guide to Today’s Mathematics. Oxford, England: Oxford University Press, ppm 164166, 1996.
BienaymkChebyshev Bienaymkchebyshev see CHE~Y~HEV
Biharmonic
Inequality Inequality
Equation
131
BIFURCATION, LOGISTIC MAP, PERIOD DOUBLING, PITCHFORK BIFURCATION, TANGENT BIFURCATION, TRANSCRITICAL BIFURCATION
INEQUALITY
Bifoliate
References
The PLANE
‘%
CURVE
given
by the Cartesian
Guckenheimer, in Nonlinear cations
equation
Applied
York:
References A. Mathematical Models, Tarquin Pub., p. 72, 1989.
3rd
P. ‘&Local Bifurcations.”
Oscillations, of Vector Fields,
Dynamical 2nd pr.,
Systems, rev. COW
Ch. 3
and BifurNew York:
SpringerVerlag, pp. 117165, 1983. Lichtenberg, A. J. and Lieberman, M. A. “Bifurcation Phenomena and Transition to Chaos in Dissipative Systems.” Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New York: SpringerVerlag, pp. 457569, 1992. Rasband, S. N. “Asymptotic Sets and Bifurcations.” $2.4 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 2531, 1990. Wiggins, S, “Local Bifurcations.” Ch. 3 in Introduction to
x4 + y4 = 2azy2.
Cundy, H. and Rollett, Stradbroke, England:
J. and Holmes,
Nonlinear
Dynamical
SpringerVerlag,
Systems
pp. 253419,
and
Chaos.
New
1990.
ed.
Bifurcation The study TIONS.
Bifolium
see
Theory of the nature
CHAOS,
also
and
DYNAMICAL
properties
of BIFURCA
SYSTEM
Bigraph see BIPARTITE
GRAPH
Bigyrate Diminished Rhombicosidodecahedron see JOHNSON A F~LIUM with b = 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 Cartesian equation is
SOLID
Biharmonic Equation The differential equation obtained HARMONIC OPERATOR and setting
(x2 + y2)2 = 4azy2 and the POLAR
equation
o”tp = 0.
is
In CARTESIAN is
T = 4a sin2 8 cos 9. see
also
FOLIUM,
QUADRIFOLIUM,
TRIFOLIUM
References Lawrence, J. D. A Catalog of Special York: Dover, pp. 152153, 1972. MacTutor History of Mathematics Folium .” http://wwwgroups.dcs.stand.ac.uk/ history/Curves/Double.htmf.
Plane
Archive.
Curves.
New
“Double
Bifurcation A period doubling, quadrupling, etc., that accompanies the onset of CHAOS. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. Bifurcations come in four basic varieties: FLIP BIFURCATION, FOLD BIFURCATION,PITCHFORK BIFURCATION, and TRANSCRITICAL BIFURCATION (Rasband 1990). see also GODIMENSION, FEIGENBAUM FUNCTION,
by applying to zero.
FEIGENBAUM CONSTANT, FLIP BIFURCATION, HOPF
COORDINATES,
v4qb= V2(V2)& d2 d2  =+ay2+s ( a44 a24+

(1)
the biharmonic
a2
a2 a2 =+dy2+s >(
equation
a2 >4
a44 + d4@ + 2 a44 dy4 dz4 dx2dy2
+2&+2$&o. In POLAR
the BI
COORDINATES
 *32 4r80  14 r2
(2) (Kaplan
rr
1984,p.148)
+ +h9
+ gpr
= 0. (3)
Biharmonic
132 For a radial comes
function
Operator
$(T),
Billiards
the biharmonic
equation
be
Bijection A transformation see
~SO
which
ONETOONE,
is ONETOONE ONTO,
and ONTO.
PERMUTATION
Bilaplacian (4) Writing
the inhomogeneous
equation
as
(5)
V4@ = 64@,
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
we have 64@ dr = d
(6)
Bilinear A bilinear
Basis basis is a BASIS, which satisfies the conditions
(7)
(ax + by) ’ 2 = a(x ’ z) + b(y  z)
(8) (9) (Isp,”
+ Clrlnr
(10)
+ Czr) dr
Now use rlnrdr
= ir21nr
 $+2
(11)
s
2
’
(ax
+
by)
=
a(2
l
x)
+
b(z
’
y).
see also BASIS Billiard Table Problem Given a billiard table with only corner pockets and sides of INTEGER lengths m and n, a ball sent at a 45” angle from a corner will be pocketed in a corner after m+ n  2 bounces, see
also ALHAZEN'S BILLIARD
PROBLEM,
BILLIARDS
to obtain
c3
4pr3+Cirlnr+CJr+
4(r)
r >
= pr4 + Ci ($T” lnr
dr = dqb
(13)
 $“)
++Cir2+C31nr+C4 =@~+~+a~~+b+(~r~+d)ln The homogeneous biharmonic and solvedin 2D BIPOLAR Advanced Calculus, Wesley, 199 1.
Biharmonic Also known
4th
ed.
Operator as the BILAPLACIAN. v4 E (v2)2.
In nD space, 3(15  8n + n2) r5 ’ see
also
BIHARMONIC
k
>
.
(14)
equation can be separated COORDINATES.
References Kaplh, W, Addison
(
EQUATION
Reading,
MA:
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 billiards 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., DIAMONDshape) (Wagon 1991). see UZSOALHAZEN'S BILLIARD BLE PROBLEM,REFLECTION
PROBLEM,BILLIARD PROPERTY
TA
Binary
Billion References
133
Bimedian
Davis, D.; Ewing, C.; Billiards Simulation.”
He, Z.; and Shen, http://serendip.brynawr.edu/
T.
“The
chaos/home.html. Dullin, H. R.; Richter, P.H.; and Wittek, A. “A TwoParameter Study of the Extent of Chaos in a Billiard System.” Chaos 6, 4358, 1996. Billiard Balls.” In Madachy’s Madachy, J. S. “Bouncing Mathematical Recreations. New York: Dover, pp* 231241, 1979. Neumann, P+ Submitted to Amer. Math. Monthly. Pappas, T. “Mathematics of the Billiard Table.” The Joy of Muthematics. San Carlos, CA: Wide World Publ./Tetra, p. 43, 1989. Peterson, I. “Billiards in the Round.” http://www. sciencenews .org/sn_arc97/3l97/mathland.htm. Wagon, S. “Billiard Paths on Elliptical Tables.” 510.2 in Mathematics in Action. New York: W, H. Freeman, pp. 330333, 1991.
A LI NE SEGMENT joining the sides of a QUADRIL ATERAL
MIDPOINTS of opposite
see also MEDIAN (TRIANGLE),
VARIGNON'S THEOREM
Billion The word billion denotes different numbers in American and British usage. In the American system, one billion equals 10’. In the British, French, and German systems, one billion equals 1012.
Bimonster The wreathed product of the MONSTER GROUP by 22. The bimonster is a quotient of the COXETER GROUP with the following COXETERDYNKIN DIAGRAM.
see also
LARGE
NUMBER,
MILLIARD,
MILLION,
1 A’
B
MAB
Bimodal Distribution A DISTRIBUTION having
two separated
peaks.
see also UNIMODAL DISTRIBUTION
TRIL
Bilunabirotunda
see JOHNSON SOLID Bimagic
Square This had been conjectured by Conway, but was proven around 1990 by Ivanov and Norton. If the parameters p, ~,r in Coxeter's NOTATION [3PPpYr] are written side by side, the bimonster can be denoted by the BEAST NUMBER 666. Bin An interval not fall.
If replacing each number by its square in a SQUARE produces another MAGIC SQUARE, the is said to be a bimagic square. The first bimagic (shown above) has order 8 with magic constant addition and 11,180 after squaring. &magic are also called DOUBLY MAGIC SQUARES, and
MULTIMAGIC SQUARES. see also MAGIC SQUARE, MULTIMAGIC TRIMAGIC SQUARE
MAGIC square square 260 for squares are Z
SQUARE,
References Ball, W. W. R. and Coxeter, ations
and
Essays,
13th
H. S. M. Mathematical ed. New York: Dover,
Recre
p. 212,
1987. Hunter, J. A. H. and Madachy, 3. S. “Mystic Arrays.” Ch. 3 in Mathematical Diversions. New York: Dover, p, 31, 1975. Kraitchik, M. “Multimagic Sq uares .” 57.10 in Mathema tical Recreations. New York: W. W. Nor ton, pp. 1 76178,l 942.
into
which
a given
data point
does or does
see also HISTOGRAM Binary The BASE 2 method of counting 0 and 1 are used. In this BASE,
in which only the digits the number 1011 equals 1~2°+142+O*22+1~23 = 11. This BASE is used in computers, 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~ would be written as the complement of 10 = 00001010~, or 11110101. This allows addition to be carried out with the usual carrying and the leftmost digit discarded, so 17  11 = 6 gives 00010001
11110101 00000110
17 11 6
134
Binary
Binary
Bracketing
The number of times k a given binary number b, .   b&l bo is divisible by 2 is given by the position from the right. For examof the first bk = I counting ple, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is divisible by 2 0 times. Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers store information in gbit bytes (where a bit is a single binary digit), depending on the “word size” of the machine, numbers requiring more than 8 bits must be stored in multiple bytes. The usual FORTRAN77 integer size is 4 bytes long. However, a number represented as (byte1 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) numbers, which are represented in binary as a MANTISSA and CHARACTERISTIC, and worse still for long (gbyte) 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.
3I x 0 1
see
also
OCTAL,
0 0 0
1 0 1
BASE (NUMBER), DECIMAL, QUATERNARY, TERNARY
HEXADECIMAL,
Recipes in FORTRAN: The 2nd ed. Cambridge, England:
Art
of Scientific
Computing,
Cambridge University pp. 1821, 276, and 881886, 1992, Weisstein, E. W. “Bases.” http://www.astro.virginia. edu/eww6n/math/notebooks/Bases.m.
= = 1 n+l
(2n)! n!2
CATALAN'S
Schrgder,
E. 15, Sloane, N. J. Version of Sloane, N. J. Physik
clopedia
“Vier combinatorische Probleme.” 2. Math. 361376, 1870. A. Sequences AOOOlOS/M1459 in “An OnLine the Encyclopedia of Integer Sequences.” A. and Plouffe, S. Extended entry in The Encyof Integer Sequences. San Diego: Academic Press,
1995. Stanley, R. P. “Hipparchus, Amer.
Math.
Monthly
Plutarch, 104,
344350,
Schrijder, 1997.
and Hough.”
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 operation include the GROUP, GROUPOID, MONOID, QUASIGROUP, and SEMIGROUP. Sets possessing both a binary multiplication and a binary addition operation include the DIVISION ALGEBRA, FIELD, RING, RINGOID, SEMIRING, and UNIT RING. see also AND, BOOLEAN ALGEBRA, CLOSURE, DIVISION ALGEBRA, FIELD, GROUP, GROUPOID, MONOID, OPERATOR, OR, MONOID, NOT, QUASIGROUP, RING, RINGOID, SEMIGROUP, SEMIRING, XOR, UNIT RING
(2 n! > (n + l)!n!
to Algebraic
Binary Quadratic Form A 2variable QUADRATIC FORM
Q(x, Y> =
u11x2
see also QUADRATIC
Structures.
New
of the form
+2a12xy+a22y2.
FORM,
QUADRATIC
INVARIANT
Press,
Binary Remainder An ALGORITHM for (Stewart 1992).
Binary Bracketing A binary bracketing is a BRACKETING built up entirely of binary operations. The number of binary bracketings of n letters (CATALAN’S PROBLEM) are given by the CATALAN NUMBERS &+where 2n 0 n
NUMBER,
References
Rosenfeld, A. An Introduction York: HoldenDay, 1968.
Lauwerier, H. Fkctals: Endlessly Repeated Geometric FigUPS. Princeton, NJ: Princeton University Press, pp. 69, 1991. Pappas, T. “Computers, Counting, & Electricity.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, pp. 2425, 1989. Dress, W. H.; Flannery, B. P.; Teukolsky, S. A,; and Vetterling, W. T. “Error, Accuracy, and Stability” and “Diagnosing Machine Parameters.” 51.2 and $20.1 in Numerical
G 1 n+l
CATALAN
PROBLEM
References
References
c,
see also B RACKETING,
The
’
where ( 2n > denotes a BINOMIAL COEFFICIENT and n! is the us:al 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(bc))d, (ab)(cd), written in shorthand as ((zz)z)z, a((bc)d), and a(W)), (x(xx>>x, (xX)(xX), x((xx>x>, and x(x(x4).
Method computing
a UNIT
FRACTION
References Stewart, I. “The Riddle of the Vanishing 266, 122124, June 1992.
Camel.”
Sci. Amer.
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<2HL 
Bin&
Forms
Binomial
These extremes correspond to a balanced tree (each node except the LEAVES has a left and right CHILD, and all LEAVES are at the same level) and a degenerate tree (each node has only one outgoing BRANCH), respectively. For a search of data organized into a binary tree, the number of search steps S(n) needed to find an item is bounded by lgn < S(n) < n. Partial balancing of an arbitrary tree into a socalled AVL binary search tree can improve search speed. The number of binary trees with n internal nodes is the CATALAN NUMBER Cn (Sloane’s A000108), and the number of binary trees of height b is given by Sloane’s A001699.
see also BTREE, REDBLACK TREE, BINARY TREE
QUADTREE, QUATERNARY STERNBROCOT TREE,
TREE, WEAKLY
CodEcient
135
Binet’s Formula A special case of the Un BINET FORM with corresponding to the nth FIBONACCI NUMBER, F,
(1+
=
Js)”
 (l
m
=
0,
Jsr
2nJs
’
It was derived by Binet in 1843, although the result was known to Euler and Daniel Bernoulli more than a century earlier. see ah
BINET
FORMS,
FIBONACCI
NUMBER
Bing’s Theorem If M3 is a closed oriented connected SMANIFOLD such that every simple closed curve in M lies interior to a BALL in M, then M is HOMEOMORPHXC with the HYPERSPHERE, see
s3.
BALL,
also
HYPERSPHERE
References
References Lucas, J.; Roelants van Baronaigien, D.; ‘Generating Binary Trees by Rotations.”
and Ruskey, F. J. Algorithms
15,343366,1993. Ranum, D. L. “On Some Applications of Fibonacci Numbers.” Amer. Math. Monthly 102, 640645, 1995. Ruskey, F. “Information on Binary Trees.” http: //sue, csc .uvic.ca/cos/inf/tree/BinaryTrees.html. Ruskey, F. and Proskurowski, A. “Generating Binary Trees by Transpositions.” J. Algorithms 11, 6884, 1990. Sloane, N. J. A. Sequences A000108/M1459 and A001699/ M3087 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Binet Forms The two RECURRENCE SEQUENCES un = m&I
+ Un2
(1)
Vnz
(2)
Vn = ml&l+
with Uo = 0, VI = 1 and VO = 2, V1 = m, can be solved for the individual Un and I&. They are given by P
un = Qln
n
A
Bing,
R. H. “Necessary
Manifold
be S3.”
Rolfsen, D. Knots Perish Press, pp. 251257,
Binomial A POLYNOMIAL see
with
Conditions 1958.
that
a 3
DE: Publish
or
1976.
2 terms.
MONOMIAL,
also
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 NC~ and
0 are used,
where
N

N!
n
= (N
 n)!n!
the latter
is sometimes
(1) known as N f&m the
CHOOSE n. The number of LATTICE PATHS ORIGIN (0,O) to a point (a, b) is the BINOMIAL CIENT (“T”) (Hilton and Pedersen 1991).
For POSITIVE
Vn = Qln + p”j
and Sufficient
Ann. Math. 68, 1737, and Links. Wilmington,
integer
n, the BINOMIAL
THEOREM
(z+gn=e ($xka*.
where
COEFFI
gives
(2)
k=O
aP A useful
related
identity
m+A mA 2
for & see
also
FORMULA
corresponding FIBONACCI
(7)
.
The FINITE DIFFERENCE analog of this identity is known as the CHUVANDERMONDE IDENTITY. A similar formula holds for NEGATIVE INTEGRAL n,
(x+ a)+ =
is
U,l+Un+l
BINET’S
(6)
2
=
I&.
is a special case of the Binet to m = 1. QMATRIX
(8)
A general
(3)
is given by
identity
form (a+!+” a

n c(> j=O
n j
(a  jcy‘(b
+ jc)nj
(4)
Binomial
136
Coefficient
Binomial
(Prudnikov et al. 1986), which gives the BINOMIAL OREM as a special case with c = 0. The binomial
coefficients
satisfy
THE
RECURRENCE
of the sums
RELATIONS
n
the identities:
Coefficient
/\w
(15)
s, =
(a)=(z)=l (;>=(rink)=&)k(k;l) (“:‘>=(;>+(k:l)*
(5)
are given by 2sl(n)
(6) 2(2n
(7) S(n
Sums of powers include
al(n+
+ l)s2(n)
+ l)‘sa(n)
l)=
+ (n +
+ (16
0
(16) = 0
@2(n)
 2fn  7n2)ss(n +(n
(17)
+ 1) + 2) = 0
+ 2)2s&2
(18)
= p
+
+ 1)(4n + 3)(4n + 2(2n
~(I)$)
5)s4(n>
+ 9n I 7)sd(n
+ 3)(3n2
=Q
f(n+
+ 1)
2)3s4(n+
2) = 0.
(19)
k=O
This sequence for s3 cannot be expressed number of hypergeometric terms (PetkovSrek p* 160).
n n
(the BINOMIAL
k
= (l+
kr
c(> k=O
r)”
A fascinating series of identities involving efficients times small powers are
and
THEOREM),
zF&(s
+ I), +(s + 2);s
+ 1,4x) x n=l

1 (2n\
= $ (2nfi
+ l)“dzz’
(11)
>: n=l
1 = ;&=0.6045997881... n ( 2nn > 1 n=
nzf
(Comtet
nT
(n + 1  r  j>“”
0
= n!.
n4
gg(;)Z
 j,nk
= $!.
O”
(  n3
1)
(“k’
2n
1 
52 C(3)
(24)
1
where C(Z) is the RIEMANN ZETA FUNCTION (Le Lionnais 1983, pp. 29, 30, 41, 36, and 35; Guy 1994, pm 257). As shown by Kummer dividing (uT’) is equal
in 1852, the exact to
POWER
of p
is
“> [xn+‘(l
 x>” + (1  Z)n+lxk
k=O
(Reeler
(23)
( n >
Eo+El+*..+Q,
2
(22)
17 36 c'(4)
n
(13)
12 sn
1974, p. 89) and
n=l
(:>,r
identity
1 (2nl
(12)
Another
WI
‘” \n/
I
n = 2r  1 gives
(21)
1
>:
Taking
(20)
1

=
T
.
00
[@j(;)w)nk
j=O
+ 9) = 0.7363998587..
co
\nl
00
+>)(l)j
binomial
2”
(di=G
where 2Fl (a, b; c; Z) is a HYPERGEOMETRIC FUNCTION (Abramowitz and Stegun 1972, p. 555; Graham et al. 1994, p. 203). For NONNEGATIVE INTEGERS n and T with T < n + 1,
$G(;)
as a fixed et al. 1996,
et al. 1972, Item
42).
] = 1
(14
1
(25)
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 exponent
number
Binomial
Coefficient
of a PRIME p dividing (L) of integers j > 0 for which
frac(m/pj)
is given
by the
> frac(nlpj),
(26)
(Vardi
137
Coefficient
1991, p* 66).
The binomial coefficient (r) mod 2 can be computed using the XOR operation n XOR nz, making PASCAL’S TRIANGLE mod 2 very easy to construct.
where frac(z) denotes the FRACTIONAL PART of 2. This inequality may be reduced to the study of the exponential sums '& 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
f( >
(1)(“u/2
n=
for all PRIMES,
and conjectured
that
(mod it holds
n)
(27) only
for
PRIMES. This was disproved when Skiena (1990) found it also holds for the COMPOSITE NUMBER n = 3 x 11 x 179. Vardi
(1991, p. 63) subsequently showed that n = whenever p is a WIEFERICH PRIME and that if n = pk with IG > 3 is a solution, then so is n = P Ic? This allowed him to show that the only solutions for COMPOSITE n < 1.3x lo7 are 5907, 10932, and 35112, where 1093 and 3511 are WIEFERICH PRIMES. p2 is a solution
Consider the binomial coefficients ( 2n1 ) , the first few of which are 1, 3, 10, 35, 126, . . . (STbane’s AOOl7OO). The GENERATING FUNCTION is ; [&I]
=~+32~+102~+352~+.~..
(28)
These numbers are SQUAREFREE only for n = 2, 3, 4, with no others 6, 9, 10, 12, 36, . (Sloane’s A046097), less than n = 10,000. Erdkk showed that the binomial coefficient (E) is never a POWER of an INTEGER for n > 3 where k # 0, 1, n 1, and n (Le Lionnais 1983, p. 48). l
l
( Lnnj21) are called CENTRAL BINOMIAL COEFFICIENTS, where 1x1 is the FLOOR FUNCTION, although the subset of coefficients (:) is Erdes and Graham sometimes also given this name. (1980, p. 71) conjectured that the CENTRAL BINOMIAL COEFFICIENT (v) is never SQUAREFREE for n > 4, and this is sometimes known as the ERD~S SQUAREFREE CONJECTURE. %RK&Y’S THEOREM (Sgrkijzy 1985) provides a partial solution which states that the BINOMIAL COEFFICIENT (r) is never SQUAREFREE for all sufficiently large n > no (Vardi 1991). Granville and Ramare (1996) p roved that the only SQUAREFREE values are n = 2 and 4. Sander (1992) subsequently showed that (““n’“) are also never SQUAREFREE for sufficiently large n as long as d is not “too big.” The
binomial
coefficients
For p, Q, and T distinct satisfies f (Pdf
(P)f W(r)
PRIMES, then the above function
= f (Pdf(P~)Pb)
(mod PV) (29)
The binomial
coefficient C(x,
(Fowler
“function” y) =
can be defined
as
x!
y!(z  y)!
1996)) sh own above.
It has a very complicated to render programs.
GRAPH for NEGATIVE x and y which is difficult using standard
plotting
see also BALLOT PROBLEM, BINOMIAL DISTRIBUTION, BINOMIAL THEOREM, CENTRAL BINOMIAL CoEFFICIENT, CHUVANDERMONDE IDENTITY, COMBINATION, DEFICIENCY, ERD~S SQUAREFREE CONJECTURE, GAUSSIAN COEFFICIENT, GAUSSIAN POLYNOMIAL, KINGS PROBLEM, MULTINOMIAL COEFFICIENT, PERMUTATION, ROMAN COEFFICIENT, S~RK&Y’S THEOREM, STREHLIDENTITY,WOLSTENHOLME'S THEOREM References Abramowita, efficients.” tions with 9th printing.
M. and Stegun, C. A. (Eds.). ‘Binomial Co§24.1.1 in Handbook of Mathematical FuncFormulas,
Graphs,
and
Mathematical
Tables,
New York: Dover, pp, 10 and 822823, 1972. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972. Comtet, L. Advanced Combinatorics. Amsterdam, Netherlands: Kluwer, 1974. Conway, 5. H. and Guy, R. K. In The Book of Numbers. New York: SpringerVerlag, pp. 6674, 1996. Erd&, P.; Graham, R. L.; Nathanson, M. B.; and Jia, X. Old and New Problems Theory. New York:
and
Results
in
Combinatorial
Number
SpringerVerlag, 1998. Fowler, D. “The Binomial Coefficient Function.” Amer. Math. Monthly 103, l17, 1996. Graham, R. L.; Knuth, D. E.; and Patashnik, 0. “Binomial Coefficients .” Ch. 5 in Concrete Mathematics: A Foundation for Computer Science. Reading, MA: AddisonWesley, pp. 153242, 1990. Bounds on ExponenGranville, A. and Ramare, 0. “Explicit tial Sums and the Scarcity of Squarefree Binomial Coeficients.” Mathematika 43, 73107, 1996.
Binomial
138
Distribution
Guy, R. K. “Binomial Coefficients,” ‘&Largest Divisor of a Binomial Coefficient ,” and “Series Associated with the CFunction.” §B31, B33, and F17 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, pp. 8485, 8789, and 257258, 1994. Harborth, H. “Number of Odd Binomial Coefficients.” Not. Amer.
Math.
Sot.
23,
Distribution
of n successes in IV BERNOULLI TRIALS
The probability
P(nJN)
=
Distribution
N 0n
p”(’
p)Nn
= nl(N .
N!
 n),pnqNn.
4, 1976.
Hilton, P. and Pedersen, J. “Catalan Numbers, Their Generalization, and Their Uses.” Math. Intel. 13, 6475, 1991. Jutila, M. “On N urn b ers with a Large Prime Factor.” J. Indian Math. Sot. 37, 4353, 1973. Jutila, M. “On Numbers with a Large Prime Factor. II.” J. Indian Math. Sot. 38, 125130, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Coefficients.” Amer. Math. Ogilvy, C. S. “The Binomial Monthly 57, 551552, 1950. Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Gamma Function, Beta Function, Factorials, Binomial Coefficients.” $6.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp+ 206209, 1992, Prudnikov, A. P.; Marichev, 0. 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: SpringerVerlag, pp. 2324, 1989. Riordan, J. “Inverse Relations and Combinatorial Identities.” Amer. Math. Monthly 71, 485498, 1964. Sander, J. W. “On Prime Divisors of Binomial Coefficients.” Bull. London Math. Sot. 24,.140142, 1992. S&rkijzy, A. “On the Divisors of Binomial Coefficients, I.” J. Number Th. 20, 7080, 1985. Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory zuith Mathematics. Reading, MA: AddisonWesley, p. 262, 1990. Sloane, N. J. A. Sequences A046097 and AOOl700/M2848 in “An OnLine Version of the Encyclopedia of Integer Sequences ?’ Spanier, J. and Oldham, K. B. “The Binomial Coefficients u of Functions. Washington, DC: ( m ) ’ ” Ch. 6 in An Atlas Hemisphere, pp. 4352, 1987. Sved, M. “Counting and Recounting.” Math. Intel. 5, 2126, 1983. Vardi, I. “Application to Binomial Coefficients,” “Binomial Coefficients,” “A Class of Solutions,” “Computing Binomial Coefficients,” and “Binomials Modulo and Integer.” $2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematics. Redwood City, CA: AddisonWesley, pp. 2528 and 6371, 1991. Wolfram, S. “Geometry of Binomial Coefficients.” Amer. Math. Monthly 91, 566571, 1984.
Binomial
Binomial
The probability observed is
of obtaining
more
P=
successes than
= I&+
1, N 
N),
(1) the n
(2)
where I&,
b) 
wx; a) v, B(a, b)
(3)
B(a,b) is the BETA FUNCTION, and B(x;a,b) is the The CHARACTERISTIC incomplete BETA FUNCTION. FUNCTION is 4(t) = (q + pe”t)n. (4 The MOMENTGENERATING tribution is
M for the dis
FUNCTION
M(t) = (P) = k etn(;)pnqNn n=O N
N

n
(pe”>“(l
c( n=O
PIN”
>
= [pet+ (1p)lN
(5)
M’(t)
= N[pet
+ (1 p>lNsl(pet)
(6)
M”(t)
= N(N
 l)[pet
+ N[pet The MEAN
+ (1 p)]N2(pet)2
+ (1  p)]ND1(pet).
(7)
is p = M’(O)
= N(p + 1  p)p = Np.
The MOMENTS about
0 are
/.L; = p = Np
00
PL =
NP(l
PL =
Np( 1  3p + 3Np + 2p2  3NP2
 P + NP)
(10)
dl = Np(1  7p + 7Np + 12p2  UNp2  6p3 + llNp3  6N2p3 + N3p3), so the MOMENTS
about
the MEAN
CL3
=
I&
= NP(l P4
 p) = Npq 
3/&p:
 P)(l
+
+ N2p2)
(11)
+ 6N2p2 (12)
are
p2 = g2 = [NW  l)P2 + NP] = N2p2  Np2 + Np  N2p2 = Np(l
(8)
(NPj2
(13)
2(p1)3

2P)
(14)
= Pi  4/&p: t 6p;(cl’l>”  3(/41)4  Np(l  p)[3p2(2  N) + 3p(N  2) + 11. (15)
Binomial The
Binomial
Distribution
SKEWNESS
and KURT~SIS
+y12L
fT3
since p + q = 1. expansion
are
NP(l  P)(l  2P) WP(l  PI”‘” 1  2p
6P2 6p+l NP(l P)
the terms
in the
B2 F 
1  6Pq
=
Npq
(17)
.

An approximation to the Bernoulli distribution for large N can be obtained by expanding about the value C 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 = fi + q, then l@(n)]
We can now find
(16)
=&qq=x
yz=,3=P4
139
Distribution
= ln[P(fi)]+B~q+@~q2+$B3~3+.
B3 G
. . , (18)

1 N2p2

(1
1 N2q2 2p+p”)
N2p2(1
cl2  P2 N2p2q2
p2
p)”
1  2p = N2p2(1 p)”
(31)
where B4 =
(19) But we are expanding nition,
[
also means
Bz = IB&
ln[P(n)]
NOW,
the maximum,
1
wP(n)l
BI =
This
about
dn
so, by defi2(P2 pq+q2)
(20)
= 0.
N3p3q3
n=fi
qP2  P(l
that B2 is negative, so we can write taking the LOGARITHM of (1) gives
2(3P2 3p+1) N3p3(1 p”)
n !ln(Nn)!+nlnp+(Nn)lnq.
= In N!ln
For large n and N  n we can use STIRLING'S IMATION ln(n!) $=:n 1nnn,
(21) APPROX
Now, treating
P2)1
 P) + (1  2P+ N3p3(1  p3)
(32) l
the distribution
as continuous,
(22) dlmkP(n)
=/P(n)dn=
1”;
P(fi+q)dq=
1.
oo
n=O
(33)
yQln+l)linn d[ln(N
 n)!] dn
(23)
==: $[(N

 n)ln(N
Since each term is of order l/N N l/a2 smaller than the previous, we can ignore terms higher than B2, so
 n)  (N  n)] P(n)
= p(fi)eiB21q2/2a
must
be normalized,
= ln(N
 n),
(24)
The probability
and
P(ii)e‘B2’a2’2
W~b)l dn
(34)
ln(Nn)+(Nn)&+l]
=z: lnn+ln(Nn)+lnplnq.
To find ii, set this expression In
Ntip fi (
(25)
so
(35)
dv = P(G)
and
to 0 and solve for n, =0 cl>
NGPzl fi Q
(26)
(27)
(N  ii)p = iiq
(28)
fi(q + p) = ii = Np,
(29)

Defining
Lexp aJE
[(n[lr)2]
.
(36)
u2 = ZNpq, P(n)
= &exp
[WI,
(37)
Binomial
140
Binomial
Expansion
which is a GAUSSIAN DISTRIBUTION. For p << 1, a different approximation procedure shows that the binomial distribution approaches the POISSON DISTRIBUTION. The first CUMULANT is
nm a
_
b””
=
(am
_
bm)[dm(nl)
+
Series
um(n2)bm
+w+bmcn‘)]. In 1770, Euler
proved
(3)
if (a, b) = 1, then every FAC
that
TOR of
a2n + b2n
CUMULANTS are given by the RECURRENCE RELATION
and subsequent
dnr
&+1
= Pq 0dp
= ilx + y = k) = 
P(x

= i,x P(x
= i, y = k  i)
+ P(x
(I)pi(l
=
form
2”$%
+ 1. If p and Q are
upq  l)(a
 1)
( up  l)(aq
 1)
(5)
1
is DIVISIBLE by every PRIME FACTOR ofapr ing aq  1.
see
notdivid
CUNNINGHAM NUMBER, FERMAT NUMBER, NUMBER, RIESEL NUMBER, SIERPI~KI NUMBER OFTHE SECOND KIND also
MERSENNE
+ y = k)
References
y = k)
Guy, R. K. “When
= i)P(y
= k  i)
Unsolved
P(x
na  nb.” §I347 in 2nd ed. New York:
Does 2”  2b Divide
Problems
in Number
Theory,
SpringerVerlag, pm 102, 1994. Qi, S and MingZhi, 2. “Pairs where 2”  ab Divides no  nb for All n.” Proc. Amer. IMath. Sot. 93, 218220, 1985. Schinzel, A. “On Primitive Prime Factors of an  b”.” Proc.
y = k)  p>“” (kri)p”i(l _ p)m(“i) (“i”)pk(l  p)n+mk
P(x+y=k)
(1)
P(x
2 or of the
PRIMES, then
(39)
Let 61:and y be independent binomial RANDOM VARIABLES characterized by parameters n,p and m,p. The CONDITIONAL PROBABILITY ofz given that x + y = k is P(x
is either
(4)
+
Cambridge
Phil.
Sot.
58,
555562,
1962,
(k:i) (40)
Binomial
Series
For 1x1 < 1, that this is a HYPERGEOMETRIC DISTRIBUTION! see UZSO DE MOIVRELAPLACE THEOREM, HYPERGEOMETRIC DISTRIBUTION, NEGATIVE BINOMIAL DISTRIBUTION Note
(1
+x)n
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 Vetterling, W. T. “Incomplete Beta Function, Student’s Distribution, FDistribution, Cumulative Binomial Distribution.” $6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219223, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGrawHill, p. 108409, 1992.
Binomial
(1)
= (:)x0+ (ll>x’+ ($x2+...
(2)
= l+
l!(n
n! a:+  l)!
The binomial representation
Formula see BINOMIAL SERIES, BINOMIAL
a”b”
= (ab)(a”l
+a n2b+. . l +&n2 +P)
(1)
u”+b”
= (u+b)(a”1
u”2b+.
(2)
. .ubn2+bn1)
(3)
l
. (5)
1.(l+n) l2
x
1 m(1  n) 23
2(2
Number
of the form an & b”, where a, b, and 72 are INTEGERS. They can be factored algebraically
.
nx
THEOREM
A number
.
1
1c L I
Binomial
“’
series also has the CONTINUED FRACTION
1+
Binomial
22+
.
l
SERIES
n!
(n  2)!2!
n(n  1) 22 +
=1+7%x+2
(1 + 2)” =
Expansion
see BINOMIAL
=
x +
34
1+
n)
x
2(2  n>
TX 1t
3(3 + n) 5.6 x
l+
1+...
Binomial
Theorem
see also BINOMIAL NEGATIVE BINOMIA
Biotic , MULTINOMIAL
THEOREM L SERIES
SERIES,
References M.
Abramowitz, of Mathematical Mathematical
and
Stegun,
Functions Tables, 9th
C.
with printing.
A.
(Eds,).
Formulas, New
pp. 1415, 1972. Pappas, T. “Pascal’s Triangle, the Fibonacci nomial Formula.” The Joy of Mathematics. CA: Wide World Publ./Tetra, pp* 4041,
Handbook Graphs, and
York:
Dover,
Sequence & BiSan Carlos, 1989.
Binormal A RULED velopable x(u,
141
Developable A4 is said to be a binormal dey if M can be parameterized by where B is the BINORMAL VEC
SURFACE
of a curve
w) = y(z~)+v&(u),
TOR. see
NORMA L DEVELOPABLE, TANGENT
also
DEVEL
OPABLE References Gray,
A. “Developables.”
ometry
Binomial Theorem The theorem that, for INTEGRAL
Potential
of Curves
Press, pp. 352354,
$17.6 in Modern Surfaces. Boca
and
Differential
Raton,
Ge
FL:
CRC
1993.
POSITIVE n, Binormal
Vector
7t
(x + a)n = BCFXR 
BINOMIAL SERIES, where (E) are BINOMIAL COEFFICIENTS. The theorem was known for the
the socalled
case n = 2 by Euclid around 300 BC, and stated in its modern form by Pascal in 1665. Newton (1676) showed that a similar formula (with INFINITE upper limit) holds
(1) rt x d' (2)
Ir' x r"l'
where the unit TANGENT VECTOR T and unit pal” NORMAL VECTOR N are defined by
“princi
for NEGATIVE INTEGRAL n, (x + a)+
=
k
xa
nk
9
the socalled NEGATIVE BINOMIAL SERIES, which verges for 1x1 > /al.
con
see also BINOMIAL COEFFICIENT, BINOMIAL SERIES, CAUCHY BINOMIAL THEOREM, CHUVANDERMONDE IDENTITY, LOGARITHMIC BINOMIAL FORMULA,NEGATIVE BINOMIAL SERIES, QBINOMIAL THEOREM, RANDOM WALK
[i%,B,jij]=is$ (;). see also FRENET FORMULAS,
References Abramowitz,
M.
o,f Mathematical Mathematical
and
Stegun,
C. A.
(Eds.).
Func tions ulith Formulas Tables, 9th printing. New
Handbook Graphs, and
York: Dover, p* 10, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 307308, 1985. Conway, J. II. and Guy, R. K. “Choice Numbers Are Binomial Coefficients.” In The Book of Numbers. New York: SpringerVerlag, pp 7274, 1996. Coolidge, J. L. “The Story of the Binomial Theorem.” Amer. Math. Monthly 56, 147157, 1949. Courant, R. and Robbins, H. “The Binomial Theorem.” $1.6 in What is Mathematics ?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 1618, 1996.
Binomial
Here,risthe RADIUS V~~~~~,sisthe ARC LENGTH,~ is the TORSION, and K is the CURVATURE. The binormal vector satisfies the remarkable identity
Triangle
see PASCAL'S TRIANGLE
NORMAL
(5) VECTOR,
TAN
GENT VECTOR References Kreyszig, E. ‘LBinormal. Moving Trihedron Geometry. New York: in Differential 1991.
of a Curve.” $13 Dover, p. 3637,
Bioche’s Theorem If two complementary PLUCKER CHARACTERISTICS are equal, then each characteristic is equal to its complement except in four cases where the sum of order and class is 9. References Coolidge, J. L. A Treatise on Algebraic York: Dover, p. 101, 1959.
Biotic
Potential
see LOGISTIC EQUATION
Plane
Curves.
New
142
Bipartite
Bipartite
Graph
Graph
Bipolar
Cylindrical
Coordinates
Twocenter bipolar coordinates are two coordinates giving the distances from, two fixed centers ~1 and ~2, somebipolar coorditimes denoted T and T’. For twocenter nates with centers at (I&O),
Tl 2 = (x + c)” + y2 Tz2= (x  c)” + y2. Combining 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 JGPARTITE GRAPH with Ic = 2,
see
COMPLETE BIPARTITE
also
GRAPH,
K~NIGEGEV~RY
GRAPH,
(8) and (9) gives
T12 Tz2= Solving
Conquest.
(11)
Graph
Theory.
New
1 Y = ~q,J16ca,,z
York:
P. C. The FourColor Problem: New York: Dover, p. 12, 1986.
B = tanl
Bipolar coordinates are a 2D system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by
where u f [O, 27r), v E (oo,oo). ties show that curves of constant
J
(13)
2 8c2(r12
+ 7+z2 Q2

Tz2
2c2)
1
l (14)
References Lockwood,
a sinh ‘u cash II  cosu asinu cash w  cod
(12)
?I2+ 722  2c2
Bipolar
Coordinates
 (r12  r22 +4c2).
for POLAR COORDINATES gives
Solving
T=
Y
COORDINATES x and y gives
for CARTESIAN
Biplanar Double Point see ISOLATED SINGULARITY
X=
w
THEOREM
Chartrand, G. Introductory Dover, p. 116, 1985. Saaty, T. L. and Kainen, and
4cx.
~PARTITE
References
Assaults
(8) (9)
E. H. “Bipolar Coordinates.” Ch. 25 in A Book Cambridge, England: Cambridge University pp. 186490, 1967.
of Curves.
Press,
(2
Bipolar
Cylindrical
Coordinates
The following identiu and II are CIRCLES
in xyspace. x2 + (y  a cot u)2 = a2 csc2 u (X
~0th~)~
(3) (4)
+ y2 = a2 csch2 w.
The SCALE FACTORS are a cash ‘u  cosu a h, = cash ZI  cosu’
(5)
h, =
(6)
The LAPLACIAN is
A set of CURVILINEAR COORDINATES defined a sinh v coshv  cosu usinu Y= cash ZI  cos u z = x,
by
x
v2 = (coshys42
( ;;2
LAPLACE'S EQUATION is separable.
+ ;;2)
*
(7)
(2) (3)
where u E [O, 2n), w E (00, oo), and z E (qoo). There are several notational conventions, and whereas (u, V, z) is used in this work, Arfken (1970) prefers
Biprism
Biquadratic show that Th e f o 11owing identities u and 21 are CIRCLES in z:yspace.
(w5 4. constant
The SCALE
curves
of
(4)
(x  a coth v)” + y2 = a2 csch2 21.
(5)
see also CUBEFREE, PRIME NUMBER, RIEMANN ZETA FUNCTION,SQUAREFREE Sloane, N. J. A, Sequences A046100 Line Version of the Encyclopedia
Biquadrat
FACTORS are
ic Equation
see QUARTIC EQUATION (6)
coshw  cosu a h, = cash v  cosu h, = 1.
(7) (8)
is
Biquadratic Number A biquadratic number is a fourth POWER, n4. 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, . .
v2 = (coshw~cosu)2 (& + &) + !r* (9)
(Sloane’s represent numbers
in BIPOLAR ZD BIPOLAR
l
Academic
Methods Press,
Coordinates for
(<, 77, x).”
Physicists,
pp. 97402,
2nd
,
Every POSITIVE integer
References G. “I3ipolar
l
A002377), and the number of distinct ways to the numbers 1, 2, 3, . . in terms of biquadratic are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2,.. A bruteforce algorithm for enumerating the biquadratic permutations of n is repeated application of the GREEDY ALGORITHM. l
LAPLACE'S EQUATION is not separable CYLINDRICAL COORDINATES, but it isin COORDINATES.
ematical
and A046101 in “An Onof Integer Sequences.”
a
h, =
A&en,
143
References
x2 + (y  a cot u)2 = a2 csc2 u
The LAPLACIAN
Number
$2.9 in Ma& Orlando, FL:
ed.
1970.
Biprism Two slant triangular
PRISMS fused together. see also PRISM, SCHMITTCONWAY BIPRISM
is expressible as a SUM of (at = 19 biquadratic numbers (WARING’S PROBmost) g(4) LEM). Davenport (1939) showed that G(4) = 16, meaning that all sufficiently large integers require only 16 biquadratic numbers. The following table gives the first few numbers which require 1, 2, 3, . . l , 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite.
Bipyramid
see DXPYRAMID Biquadratefree 8C
6C
40
#
Sloane
Numbers
1 2 3 4 5 6 7 8 9 10 11 12
000290 003336 003337 003338 003339 003340 003341 003342 003343 003344 003345 003346
1, 2, 3, 4, 5, 6, 7, 8, 9,
16, 81, 256, 625, 1296, 2401, 4096, ..a 17, 32, 82, 97, 162, 257, 272, 18, 33, 48, 83, 98, 113, 163, .m m 19, 34, 49, 64, 84, 99, 114, 129, . . 20, 35, 50, 65, 80, 85, 100, 115, 9.. 21, 36, 51, 66, 86, 96, 101, 116, .a 22, 37, 52, 67, 87, 102, 112, 117, ..a 23, 38, 53, 68, 88, 103, 118, 128, ..m 24, 39, 54, 69, 89, 104, 119, 134, . . 10, 25, 40, 55, 70, 90, 105, 120, 135, . . . l
.
.
l
l
l
11, 26, 41, 56, 71, 91, 106, 121, 136, . . . 12, 27, 42, 57, 72, 92, 107, 122, 137, . . l
The following table gives the numbers which can be represented in n different ways as a sum of k biquadrates.
20
20
40
60
80
100
A number is said to be biquadratefree if its PRIME decomposition contains no tripled factors. All PRIMES are therefore trivially biquadratefree. The biquadratefrce numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, . . (Sloane’s A046100). The biquadrateful numbers (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 l/5(4) = 90/n4 ==: 0.923938, where c(n) is the RIEMANN ZETA FUNCTION. l
k
n
Sloane
Numbers
1 2
1 2
000290
1, 16, 81, 256, 625, 1296, 2401, 4096, 635318657, 3262811042, 8657437697,m
l
l
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 UZSO CUBIC NUMBER, SQUARE NUMBER, WARING’S PROBLEM References
Davenport, H. Ann.
Math.
“On Waring’s 40, 731747,
Problem 1939.
for Fourth
Powers.”
l
144
Biquadratic
Biquadratic
Reciprocity
Reciprocity
was solved
Birthday
Theorem
x4 E q (mod This
Theorem
by Gauss using
p) .
(1)
the GAUSSIAN
INTEGERS
as
At tack
Birkhoff’s Ergodic Theorem Let T be an ergodic ENDOMORPHISM of the PROBABILITY SPACE X and let f : X + Iw be a realvalued MEASURABLE FUNCTION. Then for ALMOST EVERY x f X, we have = ln fdm f oF3(x)+ n IE s
j=l
where
7~ and 0 are distinct N(a
GAUSSIAN
+ bi) = da2
INTEGER
PRIMES,
+ b2
a ( 7T> 4 1 1,
i, or
if z4 = QI (mod otherwise,
 i
r)
is solvable
(4)
where solvable
means
solvable
in terms
f0x=
(3)
and N is the norm.

as n + 00. To illustrate this, take f to be the characteristic function of some SUBSET A of X so that
of GAUSSIAN
IN
1
{0
ifxfA if x $ A.
The lefthand side of (1) just says how often the orbit of x (that is, the points x, TX, 57’2, . . l ) lies in A, and the righthand side is just the MEASURE of A. Thus, for an ergodic ENDOMORPHISM, “spaceaverages = timeaverages almost everywhere.” Moreover, if T is continuous and uniquely ergodic with BOREL PROBAthen we can BILITY MEASURE m and f is continuous, replace the ALMOST EVERYWHERE convergence in (1) to everywhere.
TEGERS. see
RECIPROCITY
also
Birotunda Two adjoined
THEOREM
Biquaternion A QUATERNION with COMPLEX coefficients. The ALGEBRA of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985). see
QUATERNION
also
References Clifford, W. K. “Preliminary
Sketch of Biquaternions.”
Proc.
Sot. 4, 381395, 1873. Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. London
Math.
Dublin: Hodges and Smith, 1853. Study, E. “Van den Bewegung und Umlegungen,” Ann.
39, 441566,
van der Waerden, Khwarizmi
pp. 188189,
1891. B. L. A History
to Emmy
Noether.
New
of Algebra
Math. from
al
York: SpringerVerlag,
1985.
Birational Transformation A transformation in which coordin ates in two SPACES are expressed rationally in t erms of those in an .other. see UZSO REM
Birch
RIEMANN
CURVE
THEOREM,
Conjecture
see SWINNERTONDYER
BirchSwinnertonDyer see SWINNERTONDYER
CONJECTURE Conjecture CONJECTURE
WEBER’S
THEO
ROTUNDAS.
BILUNABIROTUNDA, CUPOLAROTUNDA, ELONsee also GATED GYROCUPOLAROTUNDA, ELONGATED ORTHOCUPOLAROTUNDA, ELONGATED ORTHOBIROTUNDA, GYROCUPOLAROTUNDA, GYROELONGATED ROTUNDA, ORTHOBIROTUNDA,TRIANGULAR HEBESPHENOROTUNDA
Birthday
Attack attacks are a class of bruteforce techniques used in an attempt to solve a class of cryptographic hash function problems. These methods take advantage of functions which, when supplied with a random input, return one of IC equally likely values. By repeatedly evaluating the function for different inputs, the same output is expected to be obtained after about 1.26 evaluations. Birthday
see also BIRTHDAY
PROBLEM
References Laboratories. “Question 95. What is a Birthday Athttp://www.rsa.com/rsalabs/newfaq/q95.html. “Question 96. How Does the Length of a Hash Value Affect Security?" http://www.rsa.com/rsalabs/newfaq/ q96.html. van Oorschot, P. and Wiener, M. “A Known Plaintext Attack on TwoKey Triple Encryption.” In Advances in RSA
tack.”
CryptologyEurocrypt ‘90. New York: SpringerVerlag, pp, 366377, 1991. Yuval, G. “How to Swindle Rabin.” Cryptologia 3, 187189,
Jul. 1979.
Birthday
Problem
Birthday
Problem
Birthday
Consider the probability Ql(n, d) that no two people of a group of n will have matching birthdays out equally possible birthdays. Start with an arbitrary son’s birthday, then note that the probability that second person’s birthday is different is (d  1)/d, the third person’s birthday is different from the first is [(d  Wl[@  WI, and so on, up through the person. Explicitly,
’
Ql(n,d)
out of d perthe that two nth
In general, let Q&z, d) denote birthday is shared by exactly i out of a group of n people. Then birthday is shared by k or more
Pk(n,d)
145
Problem
the probability that a (and no more) people the probability that a people is given by
= 1  FQi(n,d). I 1 z
Q2 can be computed
explicitly
(8)
as
d  (n  1)
= yC$...
d (dl)(d2)*d*[d(nl)]

Sut
.
dn
this can be written
in terms
Ql(ny
of FACTORIALS
d! d, = (d _ n)!dn’
(1)
as
=
(2)
so the probability P2 (n, 365) that two people out of a group of n do have the sume birthday is therefore = 1  Ql(n,d)
P&,d)
= 1
Q2W) = $ ‘z n!
s(f)
1nPJ
dn x
(,“_a)
d! 2%!(n
 2i)!(d
 n + i)!
i=l

(

n 1)
2 “/2r(l
dn
+ n)PiBd’(fJZ)
r(l+

d)
r(l+dn)
I ’
(9)
d! (d  n)!d”’
(3)
If 365day years have been assumed, i.e., the existence of leap days is ignored, then the number of people needed for there to be at least a 50% chance that two share birthdays is the smallest n such that Pz(n, 365) > l/2. This is given by 12 = 23, since
where
(z)
is a BINOMIAL
GAMMA FUNCTION, CAL POLYNOMIAL. B(n, d) as
COEFFICIENT,
is a
r(n)
and $)(x) is an ULTRASPHERIThis gives the explicit formula for
P&d) = 1  Q&d)  Q&d) (l)“+‘r(n
=1+
+ l)PJ7(21’2)
(1o) l
2”/2d”
Pz(23,365) = 380939047022973907852437082910563905188~6454060947061 75091883268515350125426207425223147563269805908203125 $=: 0.507297.
Q&I, d) cannot be computed in entirely but a partially reduced form is
The probability
l
Pz (n, d) can be estimated P&d)
= 1 e El
where the latter
n(n1)/2d
(lJ&
as
Q ( 3n,
d)
=p
I’(d+
1)
(l)“F(:)
 +!)
r(d  72 + 1)
d”
LnPJ +(l)nr(l+n)x r(d i=l
i)r(i + 1)
 i +
(5) (6)
where
has error F = F(n,d,a) ’ ’
(Sayrafiezadeh
form,
(4)
The number of people needed to obtain Pz(n, 365) > l/2 for n = 1, 2, . , are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, . . . (Sloane’s AO3381O). l
closed
1994).
6(d  n + 1)”
G 13F2
(7) and 3F2 (a, b, C; d, e; z) is a GENERALIZED METRIC FUNCTION. In general, Qk(n, RENCE RELATION
d) can be computed
Qk(n,d)
[
= ‘2 i=l
HYPERGEO
using the RECUR
n!d! diki!(k!);(n
 ik)!(d
 i)!
kl
x~Q~(nk,d~)(dd.ii)lri*] j=l
(13) J
146
Birthday
Problem
Bisection Procedure
(Finch). However, the time to compute this recursive function grows exponentially with JGand so rapidly becomes unwieldy. The minimal number of people to give a 50% probability of having at least n coincident birthdays is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, Diaconis and 1385, 1596, 1813, . . (Sloane’s A014088; Mosteller 1989). l
A good approximation to the number of people n such that p = p&z, d) is some given value can given by solving the equation n,n/(dk)
_
d” ‘k! [
In
(&)
(l
&)]“k
(14)
for n and taking [nj, where [nl is the CEILING FUNCTION (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 0 to 4. A much simpler but also poorer approximation for n such that p = 0.5 for k < 20 is given by n = 47(k
 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 approximation for the minimum number of people needed to get a SO50 chance that two have a match within IG days out of d possible is given by
n(k,d)
(Sevast’yanov
= 1.2
1972, Diaconis
see also BIRTHDAY WORLD PROBLEM
ATTACK,

d
(16)
2k+1
and Mosteller COINCIDENCE,
1989). SMALL
Finch, S. “Puzzle #28 [J une 19971: Coincident Birthdays.” http://wwu.mathsoft.com/mathcad/library/puzzle/ soln28/soln28.html. Gehan, E. A. “Note on the ‘Birthday Problem.“’ Amer. Stat. 22, 28, Apr. 1968. Heuer, G. A. “Estimation in a Certain Probability Problem.” Amer. Math. Monthly 66, 704706, 1959. Hocking, R. L. and Schwertman, N. C. “An Extension of the Birthday Problem to Exactly k Matches.” CoEZege Math. J. 17, 315321, 1986. Hunter, J. A. H. and Madachy, J. S. Muthemutical Diversions. New York: Dover, pp* 102103, 1975. Klamkin, M. S. and Newman, D. J. “Extensions of the Birthday Surprise.” J. Combin. Th. 3, 279282, 1967. Levin, B. “A Representation for Multinomial Cumulative Distribution Functions.” Ann. Statistics 9, 11231126, McKinney, M&h.
E. H. “Generalized Monthly
73,
+5387,
Mises, R. von. “Uber Wahrscheinlichkeiten.” ences
de
Birthday Problem.” Amer. 1966. Aufteilungsund Besetzungs
Revue de la d’Istunbu1, N. S. Papers of Richard
Wniversitk in Selected
Fuculte’
des
4, 145163,
Sci
1939.
Reprinted von Mises, Vol. 2 (Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. SzegB, and G. Birkhoff). Providence, RI: Amer. Math. Sot., pp. 313334, 1964. Riesel, H. Prime Numbers and Computer Methods for Fuc.torizution, 2nd ed. Boston, MA: Birkhauser, pp. 179180, 1994. Sayrafiexadeh, M. “The Birthday Problem Revisited.” Muth. i&g. 67, 220223, 1994. Sevast’yanov, B. A. “Poisson Limit Law for a Scheme of Sums of Dependent Random Variables.” Th. Prob. Appl. 17, 695699,
1972.
Sloane, N. J. A. Sequences A014088 and A033810 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Stewart, I. “What a Coincidence!” Sci. Amer. 278, 9596, June 1998. Tesler, L. “Not a Coincidence!” http: //www .nomodes. corn/ coincidence .html.
Bisected
Perimeter
see NAGEL
POINT
Point
Bisection Procedure Given an interval [a, b], let a, and b, be the endpoints at the nth iteration and rn be the nth approximate solution. Then, the number of iterations required to obtain an error smaller than E is found as follows.
References Abramson, M. and Moser, W. 0. J. “More Birthday Surprises .” Amer. Math. Monthly 77, 856858, 1970. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 4546, 1987. Bloom, D. M. “A Birthday Problem.” Amer. Math. Monthly
rn
+(an
1~~  ~1 2 $(bn  a,)
+
bn)
(2)
= 2n(b
 a) < E
In 2 < In E  ln(b  a),
n
80, 11411142,1973. Bogomolny, A. “Coincidence.” http: //www. cuttheknot. corn/doyou_know/coincidence .html. Clevenson, M. L. and Watkins, W. “Majorization and the Birthday Inequality.” Math. Mug. 64, 183188, 1991. Diaconis, I? and Mosteller, F. “Methods of Studying CoinciStatist. Assoc. 84, 853861, 1989. dences.” J. Amer. Feller, W. An Introduction to Probability Theory and Its ApVol. I, 3rd ed. New York: Wiley, pp. 3132, plications, 1968.
G
n>
ln(ba)1nE In2
(3)
(4
(5)
l
see also ROOT lteterences
Arfken, lando,
G. Mathematical
FL: Academic
Methods
for Physicists,
Press, pp. 964965,
1985.
3rd ed. Or
Bisli t Cube
Bisector Press, W. H.; FJannery,
B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Bracketing and Bisection.” 59.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 343347, 1992.
Bisector Bisection is the division two equal parts (halves).
of a given
curve or figure
147
where [n] is the FLOOR FUNCTION, giving the sequence for n = 1, 2, . . . as 1, 1, 2, 3, 6, 10, 20, 36, . . (Sloane’s A005418). l
into
see also ANGLE BISECTOR, BISECTION PROCEDURE, EXTERIOR ANGLE BISECTOR, HALF, HEMISPHERE, LINE BISECTOR, PERPENDICULAR BISECTOR, TRISECTION Bishop’s Inequality Let V(T) be the volume of a BALL of radius T in a complete nD RIEMANNTAN MANIFOLD with RICCI CURVATURE _> (n  1)~. Then V(T) _> V,+&), where V, is the volume of a BALL in a space having constant SECTIONAL CURVATURE. In addition, if equality holds for some BALL, then this BALL is ISOMETRIC to the BALL of radius T in the space of constant SECTIONAL CURVA
TUREK. References Chavel, I. Riemannian Geometry: New York: Cambridge University
Bishops
A
Modern
Press,
Introduction,
1994.
The minimum number of bishops needed to occupy or attack all squares on an n x n CHESSBOARD is n, arranged as illustrated above.
see also CHESS, KINGS PROBLEM, KNIGHTS PROBLEM, QUEENS PROBLEM,ROOKS PROBLEM Heferences Ahrens, Vol.
W. Mathematische 1, 3rd ed. Leipzig,
Unterhaltungen
Germany:
Dudeney, H. E. “BishopsUnguarded” Guarded.” 3297 and 298 in Amusements New York: Dover, pp. 8889, 1970. Guy, R. K. “The n Queens Problem.” Problems
in Number
Theory,
2nd
und
Teubner,
ed.
Spiele,
p. 271, 1921.
and
“Bishops
in Mathematics.
SC18 in Unsolved New York: Springer
Verlag, pp. 133135, 1994. Madachy, J. Madachy’s Mathematical Recreations. New York: Dover, pp. 3646, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 7475, 1995. Sloane, N. J. A. Sequences AU02465/M3616 and A005418/ MO771 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Problem
Bislit
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 2n2 (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, . . (the EVEN NUMBERS) for n = 2, 3, . . . One maximal solution 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
Cube
l
l
.
l
l
l
B(n)
2+41’2[2(n2)/2 2(n3)/2[2(n3)/2
=
(Dudeney 1970, p. 96; Madachy 1995). An equivalent formula is B(n)
=
+ 11 for n even + I] for n odd 1979, p. 45; Pickover
ZnB3 + 21(n1)/211,
The 8VERTEX graph consisting opposite faces have DIAGONALS LAR to each other.
see also
of a CUBE in which two oriented PERPENDICU
B~DIAKIS CUBE, CUBE, CUBICAL GRAPH
Bispherical
148 Bispherical
BlackScholes
Coordinates
Bit Complexity The number of single operations TRACTION, and MULTIPLICATION) an algorithm.
Coordinates i
Theory
(of ADDITION, SUBrequired to complete
see also STRASSEN FORMULAS References Borodin,
A. and Munro,
of Algebraic
Elsevier,
and
I. The
Computational Complexity Problems. New York: American
Numeric
1975.
Bitangent
~~CURVILINEAR COORDINATES defined
A system
z =
Y= z=
asin&zos+ cash 7  cos < a sin < sin 4 cash 7  cos [ a sinh 7 coshq  cost’
by
A LINE which points. (1)
is TANGENT to a curve
at two distinct
(2)
see also KLEIN'S EQUATION, PLUCKER CHARACTERISTICS, SECANT LINE, SOLOMON'S SEALLINES,TANGENT LINE
(3)
Bivariate
Distribution
see GAUSSIAN BIVARIATE DISTRIBUTION The SCALE FACTORS hE =
are a
(4)
COSTjcost
a h, = coshq  cost asinJ h+ = coshq  cost’
(6
, 3 cosh2 v cot u csc u + cosh3 v csc2 2~1 8 cash w  cosu I a24 a 2 a2 +(cos u  cash v) sinh 21dv + (cosh2 u  cos u) dv2
d
 cos u) (cash v cot u  sin u  cos u cot u) du
a2
(7)
v  cos 2~)~ dU2.
In bispherical coordinates, LAPLACE'S EQUATION is separable, but the HELMHOLTZ DIFFERENTIAL EQUATION is not.
see also LAPLACE'S EQUATIONBISPHERICAL DINATES, TOROIDAL COORDINATES
COOR
References G. “Bispherical
Mathematical
Methods
2 = XabWa AL&
A is the WEDGE PRODUCT (or OUTER PROD
where
UCT).
see TUKEY’S
+
Arfken,
2
Biweight
cos u cot2 u + 3 coshv cot2 u cash v  cosu
+(cosh2
TENSOR of second RANK (a.k.a.
(5
The LAPLACIAN is
+(coshv
Bivector An antisymmetric form).
Coordinates for
Physicists,
(e, 7, 4).” 2nd
ed,
FL: Academic Press, pp. 115117, 1970. Morse, P. M. and Feshbach, H. Methods of Theoretical ics, Part I. New York: McGrawHill, pp. 665666,
$2.14 in Orlando,
BlackScholes Theory The theory underlying financial derivatives which involves “stochastic calculus” and assumes an uncorrelated 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 results of the fullblown theory, and allows approximation of options for which analytic solutions are not known (Price 1996).
see also
GARMANKOHLHAGEN
Phys
FORMULA
References Black, F. and Scholes, M. S. “The Pricing of Options and Corporate Liabilities,” J. Political Econ. 81, 637659, 1973. COX, J. C.; Ross, A.; and Rubenstein, M. “Option Pricing: A J. Financial Economics 7, 229263, Simplified Approach.” 1979.
Price,
J. F. “Optional
Amer.
1953.
BIWEIGHT
Sharpe, ments,
Math.
Sot.
is Not Optional.” Not. 1996. G. J.; and Bailey, J. V. InvestCliffs, NJ: PrenticeHall, 1995.
Mathematics 43,
964971,
We F.; Alexander, 5th ed. Englewood
Hack Spleenwort Black
Spleenwort
see BARNSLEY’S
Blackman
BlecksmithBrillhart
Fern
Gent
Theorem
149
points, where N = 2d, and can be obtained b(0) = b(N) = 0, letting
Fern FERN
b(m + 27
Function
and looping
= 2” + $[b(m)
by setting
+ b(m + 2n)],
over n = d to 1 by steps of 1 and m = 0
to N  1 by steps of 2”.
Bl&l
*
An APODIZATXON A(s)
given by
FUNCTION
= 0.42 +Oe5cos
F + 0.08~0s ( a >
Its FULL
2rx s a )
is 0.810957a.
WIDTH AT HALF MAXIMUM APPARATUS FUNCTION is
(1) The Peitgen
Iv4 a(0.84
The
(
 0.36a2k2  2.17 x 101ga4k4) (1  a2k2)(1  4a2k2)
sin(2rak)
are approximations
COEFFICIENTS
which would have produced and k = (9/4)a. see also
APODIZATION
.
(2)
(3) (4) (5)
zeros of I(k)
at k = (7/4)a
FUNCTION
References Blackman, dows .” York:
R. B. and Tukey, In
Point
The of View
Dover,
Blancmange
J. W. “Particular
Measurement of of Communications
pp. 9899,
Power
(1988) refer to this curve as the TAK
CURVE.
$10,000
HOFSTADTERCONWAY see also WEIERSTRA~ FUNCTION
SEQUENCE,
References
to
3969 ao = 9304 1155 a1 = 4652 715 a2 = 18608’
the
and Saupe
AGI FRACTAL
New York: Dover, pp. 175176 Dixon, R. Mathographics. and 210, 1991. Peitgen, H.O. and Saupe, D. (Eds.). “Midpoint Displacement and Systematic Fractals: The Takagi F’ractal Curve, Its Kin, and the Related Systems.” sA.1.2 in The Science New York: SpringerVerlag, pp. 246of Fractal Images. 248, 1988. Takagi, T. “A Simple Example of the Continuous Function without Derivative .” Proc. Phys. Math. Japan 1, 176177, 1903. Tall, D. 0. “The Blancmange Function, Continuous Everywhere but Differentiable Nowhere.” Math. Gax. 66, U22, 1982. Tall, D. “The Gradient of a Graph.” Math. Teaching 111, 4852, 1985.
Pairs of Win
Spectra, Engineering.
From New
1959.
Function
Blaschke Conjecture The only WIEDERSEHEN MANIFOLDS are the standard round spheres. The conjecture has been proven by combining the BERGERKAZDAN COMPARISON THEOREM with A. Weinstein’s results for n EVEN and C. T. Yang’s for n ODD. References Chavel, I. Riemannian Geometry: New York: Cambridge University
A
Modern
Press,
Introduction.
1994.
Blaschke’s Theorem A convex planar domain in which the minimal > 1 always contains a CIRCLE of RADIUS l/3.
length
is
References A CONTINUOUS FUNCTION which is nowhere DIFFERENTIABLE. The iterations towards the continuous function are BATRACHIONS resembling the HOFSTADTERCONWAY $10,000 SEQUENCE. The first six iterations are illustrated
below.
The dth iteration
contains
N + 1
Le Lionnais, F. Les p. 25, 1983.
nombres
Paris:
remarquables.
BlecksmithBrillhartGerst A generalization of SCHR~TER’S
Hermann,
Theorem FORMULA.
References Berndt, B. C. Ramanujan’s Notebooks, SpringerVerlag, p. 73, 1985.
Part
III.
New
York:
Blichfeld Blichfeldt’s
t’s Lemma
Block
Lemma
see BLICHFELDT'S
Bloch Constant N.B. A detailed online essay by S. Finch ing point for this entry.
THEOREM
Blichfeldt’s Theorem Published in 1914 by Hans Blichfeldt. It states that any bounded planar region with PO~ITWE AREA > A placed in any position of the UNIT SQUARE LATTICE can be TRANSLATED so that the number of LATTICE POINTS inside the region will be at least A + 1. The theorem can be generalized to nD. BLM/Ho A lvariable satisfies
Polynomial unoriented
KNOT POLYNOMIAL
Qunknot and the SKEIN QL+
Q(z).
1
=
was the start
functions f deLet F be the set of COMPLEX analytic fined on an open region containing the closure of the f(0) = 0 and unit disk D = {Z : 1~1 < 1) satisfying df /dz(O) = 1. For each f in F, let b(f) be. the SUPREMUM of all numbers r such that there is a disk S in D on which f is ONETOONE and such that f(S) contains a disk of radius T. In 1925, Bloch (Conway 1978) showed that b(f) 2 l/72. Define Bloch’s constant by
It
(1)
Design
B = inf{b(f) Ahlfors
and Grunsky
: f E F}.
(1937)
derived
RELATIONSHIP QL
+
x(Q~o
=
+
QLA
(2)
0.433012701..
. = $&<B 
It also satisfies
QWL~
=QL~QL~
(3)
where # is the KNOT SUM and QL+
They also conjectured the value of B,
= QL,
= ce,J;ycl
O&m)C‘P~(&m), t
= 22 ‘vL(t)VL(tl
where z = t  tl
(Kanenobu
+ 1 
and Sumi
2t7,
(6)
1993).
fies = F(l,x>
1983).
see also LANDAU Reierences Conway, J. B.
CONSTANT
Functions
of One
Complex
Variable,
2nd
ed.
NewYork: SpringerVerlag, 1989. Finch, S. “Favorite Mathematical Constants.” http: //aww. mathsoftxom/asolve/constant/bloch/bloch.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 25, 1983. Minda, C. D. “Bloch 5484, 1982.
Constants.”
J. d’Analyse
Math.
41,
(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
is actually
1
B=
(Le Lionnais
The POLYNOMIAL was subsequently extended to the 2variable KAUFFMAN POLYNOMIAL F(a, z), which satisQ(x)
limit
(5)
where PL is the HOMFLY POLYNOMIAL. Also, the degree of QL is less than the CROSSING NUMBER of L. If L is a ZBRIDGE KNOT, then QL(z)
the upper
(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 properties. 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 2 in QL(x) is 1  c, and lim xc‘Q+) x+0
that
Invariant
for Unoriented Knots and Links.” Invent. IMath. 84, 563573, 1986. Ho, C. F. “A New Polynomial for Knots and LinksPreliminary Report .” Abstracts Amer. Math. Sot. 6, 300, 1985. Kanenobu, T. and Sumi, T. “Polynomial Invariants of 2Bridge Knots through 22Crossings.” Math. Compuf. 60, 771778 and Sl’lS28, 1993. Stoimenow, A. “BrandtLickorishMillettHo Polynomials.” http://uww.informatik.huberlin.de/stoimeno/ ptab/blmhlO .html. http: //www astro . Virginia. @ Weisstein, E. W. “Knots.” edu/eww6n/math/notebooks/Knots.m. l
BlochLandau
Constant
see LANDAU CONSTANT Block see also
BLOCK DESIGN, SQUARE POLYOMINO
Block Design An incidence system (21, k, X, r, b) 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 JZ points in each block, and each point is contained in T different blocks. It is also generally required that k: < *u, which is where the “incomplete” comes from in the formal term most often encountered
Blow Up
Block Design for block designs, BALANCED INCOMPLETE BLOCK DESIGNS (BIBD). The five parameters are not independent, but satisfy the two relations VT = bk
(1)
A( w  1) = T(k  1).
(2)
151
Block Growth Let (2~2122. . ) 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 aabaabaabuabuub. . , the following words are l
l
A BIBD is therefore commonly written as simply (21, k, X) , since b and T are given in terms of V, JG, and A by b = +  QA k(k  1)
A(v 
(3)
1) (4)
T=nA BIBD k).
is called
Writing
X = {zi}yzl
if b = u (or, equivalently,
SYMMETRIC
T =
and A = {Aj},b=,, of the BIBD is given M defined by
CIDENCE MATRIX
MATRIX
mij
This
matrix
satisfies
(5)
=
(T

x)1
+
XJ,
(6)
I is a 21x v IDENTITY MATRIX and J is a 2t x w of 1s (Dinitz
Examples Block
if zi E A otherwise.
the equation
MMT where matrix
1 0
=
then the INby the v x b
of BIBDs
Design
and Stinson
1992).
are given in the following
table.
b, k, w
affine plane Fano plane Hadamard design projective plane Steiner triple system unital
( n 2 t n, 1) (7, 3, 1)) symmetric (4n + 3, 2n + 1, n) symmetric (n2 + n + 1, n + 1, 1) (w, 3, 1) b3 + 1, 4 + 1, 1)
see UZSO AFFINE PLANE, DESIGN, FANO PLANE, HADAMARD DESIGN,~ARALLEL CLASS,PROJECTIVE PLANE, RESOLUTION, RESOLVABLE, STEINER TRIPLE SYSTEM, SYMMETRIC BLOCK DESIGN, UNITAL
References 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. Dinite and D. R. Stinson).
Dinitz,
New York: Wiley, pp. l12, 1992. Ryser, H. J. “The (b, v, r, k, X)Configuration.” binatorial Mathematics. Buffalo, NY; Math.
pp. 96402,
1963.
$8.1 in Corn
Assoc. Amer.,
ADMISSIBLE Length
Admissible
1 2 3 4
4 au, ub, bu ad, ubu, baa uubu, ubau, buub
so B(1) = 2, B(2) = 3, B(3) on. Notice that B(n) 5 B(n + function is always nondecreasing. ADMISSIBLE word of length n wards to produce an ADMISSIBLE Moreover, suppose B(n) = B(n each admissible word of length ADMISSIBLE word of length n+
Words
= 3, B(4) = 3, and so l), so the block growth This is because any can be extended rightword of length n + 1. + 1) for some n. Then n extends to a unique 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 theorems: 1. If the SEQUENCE is eventually period p, then B(n) is strictly reaches p, and B(n) is constant
periodic, with least increasing until it thereafter.
2. If the SEQUENCE is not eventually periodic, then B(n) is strictly increasing and so B(n) 2 n+l for all n. If a SEQUENCE has the property that B(n) = n+l for all n, then it is said to have minimal block growth, andthe SEQUENCE iscalleda 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 elements are SQUARE MATRICES of any size (possibly even 1 x l), and the offdiagonal elements are 0. Block (Set) One of the disjoint SUBSETS making up a SET PARTITION. A block containing n elements is called an nblock. The partitioning of sets into blocks can be denoted using a RESTRICTED GROWTH STRING.
see also BLOCK DESIGN, STRING, SET PARTITION BlowUp A common mechanism which from smooth initial conditions.
RESTRICTED
generates
GROWTH
SINGULARITIES
BlueEmpty
152 BlueEmpty
Coloring
Bohemian References
Coloring
see BLUEEMPTY
Abramowite,
GRAPH
UlSO
COMPLETE GRAPH, EXTREMAL GRAPH, MON 'OCHROMATIC FORCED TRIA NGLE, RED N ET References Lorden, G. “BlueEmpty Chromatic Manthly 69, 114120, 1962. Sauvk, L. “On Chromatic Graphs.”
Graphs.”
Amer.
Amer.
Math.
Math.
Knot HITCH
= IV(du)j’
+ (RicM
Formulas, New
Handbook Gruphs, and
York:
Dover,
Bogdanov Map A 2D MAP which is conjugate to the H~NON MAP in its nondissipative limit. It is given by XI = x + y’ yl = y + ey + JGx(x  1) + pysee also H~NON MAP References 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. Bifurcation Chaos 3, 803842, 1993. Bogdanov, R. “Bifurcations of a Limit Cycle for a Family
Arrowsmith,
Fields
on the Plane.”
see also FERMAT’S
Selecta
Math.
Soviet
1,
LAST THEOREM
References
Bochner Identity For asmooth HARMONIC MAP u:iV a(lOul”)
C. A. (Eds.).
with printing.
BogomolovMiyaokaYau Inequality Relates invariants of a curve defined over the INTEGERS. If this inequality were proven true, then FERMAT’S LAST THEOREM would follow for sufficiently large exponents. Miyaoka claimed to have proven this inequality in 1988, but the proof contained an error.
see also ROOK NUMBER
see CLOVE
Stegun,
Functions Tables, 9th
373388,198l.
of d x d, where d = (1, 2, . . . , d}.
Boatman’s
and
p. 886, 1972.
of Vector
Monthly
68,107111,1961. Board A subset
M.
of Mathematical Mathematical
BlueEmpty Graph An EXTREMAL GRAPH in which the forced TRIANGLES are all the same color. Call R the number of red MONOCHROMATIC FORCED TRIANGLES and B the number of blue MONOCHROMATIC FORCED TRIANGLES, then a blueempty graph is an EXTREMAL GRAPH with B = 0. For EVEN n, a blueempty graph can be achieved by coloring red two COMPLETE SUBGRAPHS of n/2 points (the RED NET method). There is no blueempty coloring for ODD n except for n = 7 (Lorden 1962). see
Dome
Gox, D. A. “Introduction
+N,
Math.
Vu,Vu)
Bohemian
 (RiemN(u)(Vu,
Vu)Vu,
to Fermat’s
Last Theorem.”
Amer.
101, 314, 1994.
Monthly
Dome
VU),
where V is the GRADIENT, Ric is the RICCI TENSOR, and Riemisthe RIEMANN TENSOR. References Eels, Bull.
J. and Lemaire, London
Math.
L. “A Sot.
IO,
Report l68,
on Harmonic
Maps.”
1978.
Bochner’s Theorem Among the continuous functions on R”, the POSITIVE DEFINITE FUNCTIONS are those functions which are the FOURIER TRANSFORMS of finite measures. Bode’s
Rule
A QUARTIC SURFACE which can be constructed as follows. 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
x5
f(x) dx = &h(7fl
X = acosu
+ 32f2 + l2f3 + 32f4 + 7fs)
s Xl
y = bcosv + asinu ?h7f(“)
0 .
see also HARDY'S RULE, NEWTONC• TES FORMULAS, SIMPSON'S 3/8 RULE,STMPSON'S RULE,TRAPEZOIDAL RULE,~EDDLE'S RULE
z = csinv where u, v E [O, 271). In the above plot, and c = 1.
see also QUARTIC
SURFACE
a = 0.5, b = 1.5,
BohrFavard
Born bieri Norm
Inequalities
References ’
Fischer, G. (Ed.). of Universities
Vieweg,
kfath ematical and
pp+ 1920,
Museums.
Models from Braunschweig,
the
Collections Germany:
1986.
Plate 50 in Mathematische Fischer, G. (Ed.). &e/Mathematical Models, Bildband/Photograph Braunschweig, Germany: Vieweg, p. 50, 1986. Dome.” http://www.uib.no/ Nordstrand, T. “Bohemian people/nf ytn/bodtxt . htm.
ModVolume.
are absolutely
24,
f(x),f’(x),
continuous
and s,“”
,“‘, $P(Yl,.*
X&/lo,...,ynr;yll,
l
l
one which
renders
n
p differential
and Q finite
Yn’) = 0
for Q = l,...,p
,Yn)
for p=
= 0
l,...,q
on the endpoints .
for y=
,Ynl) = 0
l,***,T,
U a minimum.
References Goldstine, H. H. A History of the C5lcuZus of Variations from the 17th through the 19th Century. New York: SpringerVerlag, p. 374, 1980.
llfll O” < Wll~ 2A
f(x) = f(x +
&Y(Yl,***,yn;yl~
as well as the r equations
BohrFavard Inequalities If f has no spectrum in [X, A], then
(Bohr 1935). A related inequality the class of functions such that
find in a class of arcs satisfying equations
153
states
that
if Arc is
Bolzano
Theorem
see BOLZANOWEIERSTRAB THEOREM
==. 7f(k1)(4 f(x)
dx = 0, then
(Northcott 1939). Further, for each value of k, there is always a function f(x) belonging to Arc and not identically zero, for which the above inequality becomes an inare discussed equality (Favard 1936) These inequalities in Mitrinovic et ~2. (1991). l
BolzanoWeierstrafl
Theorem
BOUNDED infinite set in R" has an ACCUMULATION POINT. For n = 1, the theorem can be stated as follows: If a SET in a METRIC SPACE, finitedimensional EUCLIDEAN SPACE, or FIRSTCOUNTABLE SPACE has
Every
infinitely many members within a finite interval x f [a, b], then it has at least one LIMIT POINT x such that x E [a, b]. The theorem can be used to prove the INTER
MEDIATE VALUE THEOREM. Bombieri’s Inequality For HOMOGENEOUS POLYNOMIALS m and n, then
P and Q of degree
References Bohr, H, “Ein allgemeiner Satz iiber die Integration eines trigonometrischen Polynoms.” Prace Matem.Fiz. 43, 1935. Favard, J. “Application de la formule sommatoire d’Euler a la dhmonstration de quelques propri&es extr&males des intkgrale des fonctions pkriodiques ou presquep&iodiques,” Mat. Tidsskr. B, 8194, 1936. [Reviewed in Zentralblatt f. Math. 16, 5859, 1939.1 Mitrinovic, D. S.; Pecaric, J, E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, pp. 7172, 1991. Northcott, D, G. “Some Inequalities Between Periodic FuncJ. London Math. Sot. 14, tions and Their Derivatives.” 198202, 1939. Tikhomirov, V. M. “Approximation Theory.” In Analysis II (Ed. R. V, Gamrelidae). New York: SpringerVerlag, ppm 93255, 1990.
BolyaiGerwein
Problem the functional
where [P . Q] 2 is the BOMBIERI NORM. Ifnz becomes
= 72, this
[P  Q]2 2 [P]2[Q]2*
see also BEAUZAMY AND D~GOT'S IDENTITY, REZNIK'S IDENTITY Bombieri Inner Product For HOMOGENEOUS POLYNOMIALS P and Q of degree
+*. Ga!)(%,*.*,i,bi, ,*..,i,). P, Ql = *c ( 21,...,i&O
Theorem
see WALLACEB~LYAIGERWEIN Bolza Given
rpm Ql22 J(m+n)![P]z[Q]z,
THEOREM Bombieri
Norm
For HOMOGENEOUS POLYNOMIALS P of degree m,
see dso
POLYNOMIAL BAR NORM
154
Bum bieri’s
Bombieri’s Define
Bonne Projection
Theorem ALPHA VALUE for each comparison
Theorem qx;
47 a) = *(x;
4, a>  &7
(1)
where
(2) 1980, p. 121), A(n) is the MANGOLDT FUNCTION, and 4(q) is the TOTIENT FUNCTION. NOW define (Davenport
a>I E(x;4)= max a lE(x;CL
(3)
equal to a/n. Explicitly, given n tests Ti for hypotheses Hi (1 < i < n) under the assumption Ho that all hypotheses I& are false, and if the individual test critical values are < a/n, then the experimentwide critical value is 5 a. In equation form, if P(Ti passes (Ho) 5 2 n for 1 < i < n, then P(some
which
follows
from
Ti passes IHO) 5 a, BONFERRONI’S
INEQUALITY.
(a,q)=l
where the sum (a,q) = 1, and
is over
a RELATIVELY PRIME to Q,
E* (5 q) = Bombieri’s
theorem
max E(y, q).
(4)
YlX
then says that
for A > 0 fixed,
x E*(w) a &Q(ln& that
fi(ln~)~
(5)
Bonferroni, C. E. “II calcolo delle assicurazioni su gruppi di teste.” In Studi in Onore de1 Professore S&&ore Urtu Carboni. Rome: Italy, pp. 1360, 1935. Bonferroni, C. E. “Teoria statistica delle classi e calcolo delle probabilith.” Pubblicaxioni de1 R Istituto Superiore di Scienxe
5 Q 5 &.
References 12, ZOlBombieri, E. “On the Large Sieve.” Muthematika 225, 1965. Davenport, H. “Bombieri’s Theorem,” Ch. 28 in MuZtipZicative Number Theory, 2nd ed. New York: SpringerVerlag, pp. 161168, 1980. Bond
see UZSO ALPHA VALUE, HYPOTHESIS TESTING, STATISTICAL TEST References
s5Q
provided
Another correction instead uses 1(1a)‘? While this choice is applicable for twosided hypotheses, multivariate normal statistics, and positive orthant dependent statistics, it is not, in general, correct (Shaffer 1995).
Percolation
Economiche
e Commerciali
Bonferroni’s Inequality Let P(Ei) be the probability P(U yXI Ei) be the probability are all true. Then
bond percolation
A PERCOLATION which considers relevant entities (left figure).
P
site percolation
the lattice
edges as the
see also PERCOLATION THEORY, SITE PERCOLATION
Bonferroni
Test
see BONFERRONI Bonferroni Correction The Bonferroni correction is a multiplecomparison correction used when several independent STATISTICAL TESTS are being performed simultaneously (since while a given ALPHA VALUE QI may be appropriate for each individual comparison, it is not for the set of all comparisons). 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
di Firenze
8, 362,
1936. Dewey, M. “Carlo Emilio Bonferroni: Life and Works.” http://www.nottingham.ac.uk/~mhzmd/life.html. Miller, R. G. Jr. Simultaneous Statistical Inference. New York: SpringerVerlag, 1991. Wrong with Bonferroni AdjustPerneger, T. V. “What’s ments.” Brit. Med. J. 316, 12361238, 1998. ShafFer, J. P. “Multiple Hypothesis Testing.” Ann. Rev. Psych. 46, 561584, 1995.
Bonne
CORRECTION
Projection ‘.
that that
Ei is true, and El, Ez, . . , E, l
Book Stacking Problem
Boolean Algebra
A MAP PROJECTION which heart. Let 41 be the standard meridian. Then
resembles the shape of a parallel and X0 the central
x = psinE
(1)
y=Rcot&
pcosR,
(2)
where p E
=
coQh+qh

(A

4 X0)
cos@
P
The inverse
(4)
'
are
FORMULAS
4=
(3)
cot(b1+&p
(5)
x=x0+&tanl (cot;y) 7’ @) where p = fJx2
+
(cot
$1 
y)“.
(7)
References Snyder, J. P. Map ProjectionsA Geological Survey Professional DC: U. S. Government Printing
Book
Stacking I
I
Working
Manual.
U. S.
Paper 1395. Washington, Office, pp* 138140, 1987.
(Sloane’s
A001008
155
and AOO2805).
In order to find the number of stacked books required to obtain d booklengths of overhang, solve the d, equation for d, and take the CEILING FUNCTION. For n = 1, 2, . . . booklengths of overhang, 4,31, 227, 1674, 12367,91380, 675214, 4989191, 36865412, 272400600, . . . (Sloane’s AOl4537) books are needed. References Dickau, R. M. “The BookStacking
Problem.” http : //www . 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, 0. Concrete Mathematics:
A Foundation
for
Computer
Science,
Read
ing, MA: AddisonWesley, pp. 272274, 1990. Johnson, P. B. “Leaning Tower of Lire.” Amer. J. Phys. 23, 240, 1955. J. 1, 322, 1953. Sharp, R. T. “Problem 52.” Pi Mu Epsilon Sharp, R. T. “Problem 52.” Pi Mu Epsilon J. 2, 411, 1954. Sloane, N. J. A. Sequences A014537, A001008/M2885, and A002805/M1589 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
BOoleTs
Inequality
Problem +++
I
I
1
I
If Ei and Ej are MUTUALLY EXCLUSIVE for all i and j, then the INEQUALITY becomes an equality.
hi+
Boolean Algebra A mathematical object which is similar to a BOOLEAN RING, but which uses the meet and join operators instead of the usual addition and multiplication operators. A Boolean algebra is a set B of elements a, b, . . . with BINARY OPERATORS + and such that l
How far can a stack of ?z books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible CE, for n books (in terms of book lengths) is half the nth partial sum of the HARMONIC SERIES, given explicitly by
dn= i xn i1= +[r+*(l+ Tit)] Q(z)
is the DIGAMMA
EULERMASCHERONI
If a and b are in the set B, then B.
a + b is in the set
lb.
If a and b are in the set B, then 13.
a b is in the set l
2a. There is an element 2 (zero) such that for every element a. 2b. There is an element U (unity) for every element a.
a+ 2 = a
such that
a .U = a
3a. u+b=b+u
k=l
where
la.
FUNCTION and y is the CONSTANT. The first few values
are
3b. ub= 4a. u+b.c=
b.u (u+b)(u+c)
4b. u.(b+c)=a*b+u.c dl = ; = 0.5
5. For every element u+u’=Uandu~u’=Z.
d2 = ; = 0.75
6. There
d3 = ++ =2: 0.91667 dq = g z 1.04167,
are are least two distinct
B.
(Bell
a there is an element
1937, p* 444).
elements
a’ such that in the set
156
Boolean
Boolean
Algebra
In more modern terms, elements a, b, . . with
a Boolean algebra is a SET B of the following properties:
l
and V
1. B has two binary operations, A (WEDGE) (VEE), which satisfy the IDEMP~TENT laws aAa=aVa=a,
The ALGEBRA defined by commutivity, associativity, and the ROBBINS EQUATION is called ROBBINS ALGEBRA. Computer theorem proving demonstrated that every ROBBINS ALGEBRA satisfies the second WINKLER CONDITION, from which it follows immediately that all ROBBINS ALGEBRAS are Boolean. References
the COMMUTATIVE laws
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. Postulates for Huntington, E. V. “New Sets of Independent Trans. Amer. Math. Sot. 35, 274the Algebra of Logic.” 304, 1933a. Huntington, E. V. “Boolean Algebras: A Correction.” Trans.
aOb/\a aVb=bVa,
and the ASSOCIATIVE laws a A (b /\ c) = (a A b) A c
Amer, a V (b V c> =
2. The operations
satisfy
a A (a 3. The operations
4. B contains
the ABSORPTION
Boolean One ofthe
Boolean A Boolean
A c)
557558,
universal
bounds
0,1
which
http : //uww
.
Connective LOGIC operators
AND A, OR V, and NOT 1.
Function function
in n variables
is a function
satisfy
OVa=a IAa=a
Iva=I.
5. B has a unary operation
a + a’ of complementation
obeys the laws
where each xi can be 0 or I and f is 0 or 1. Determining the number of monotone Boolean functions of n variables 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 th .e number Boolean functions of n variables M(n,O)
aVa’=I
M(n,
1) = 2”
Jqn,
2) = yy2n
M(n,3)
Huntington (1933a, Boolean algebra,
References
b) presented
the following
1. Commutivity.
x + y = y + x.
2. Associativity.
(x + y) + z = x + (y + z).
3. HUNTINGTON
basis for
EQUATION. n(n(x) + y) + n(n(x) + n(Y)) = 5. H. Robbins then conjectured that the HUNTINGTON EQUATION could be replaced with the simpler ROBBINS EQUATION,
= i(2n)(2n
 1)3n
=I x
+2”
 l)(2n  2)  6" + 5" +4"  3".
Beeler, M,; Gosper, R. W., * and Schroeppel, R. HAKMEIM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972, Sloane, N, J. A. Sequences A003182/M0729 and A000372/ MO817 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Boolean A RING
Ring with a unit
element
IDEMPOTENT. + y) + n(x + n(Y)))
of dis #tinct monotone with k mint uts. Then
= 1
(Birkhoff and Mac Lane 1965). Under intersection, union, and complement, the subsets of any set I form a Boolean algebra.
n(n(x
1933.
are Boolean.”
(b AC) = (a V b) A (a Vc).
OAa=O
which
35,
see also QUANTIFIER
distributive
(a A b) V (a
Sot.
LAW
b) = a V (a A b) = a.
V c> =
Math.
McCune, W. “Robbins Algebras mcs.anl.gov/mccune/papers/robbins/.
(a V b) V c.
are mutually
a A (b aV
V
Ring
see also BOOLEAN
ALGEBRA
in which
every element
is
BorchardtPfaff
Algorithm
BorchardtPfaff
Algorithm
see ARCHIMEDES
Border
157
Measure
Bore1 Probability BorelCantelli
Lemma be 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,, n = 1, . . . . Then if Let {An}~O
ALGORITHM
Square
then P(A) A MAGIC SQUARE that remains magic when its border 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
= 1.
References M. (Managing
Hazewinkel, ematics: Soviet
An Updated “‘Mathematical
lands:
Reidel,
Ed.).
and Annotated Encyclopaedia.
pp. 435436,
Encyclopaedia Translation
of
” Dordrecht,
Mathof the
Nether
1988.
SQUARE
References Kraitchik, M. “Border Squares.” 57.7 in Mathematical ations. New York: W. W. Norton, ppm 167170,
Bore1 Determinacy Theorem Let T be a tree defined on a metric over a set of Daths such that the distance between paths p and Q is* l/n, where n is the number of nodes shared by p and Q. Let A be a Bore1 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
Recre
1942.
Bordism A relation between COMPACT boundaryless MANIFOLDS (also called closed MANIFOLDS). TWO closed MANIFOLDS are bordant IFF their disjoint union is the boundary of a compact (n+l)MANIFOLD. Roughly, two MANIFOLDS 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.
GAME. see
Cornell.
edu/rybu/bordism/bordism.html.
The machinery of the bordism group winds important for HOMOTOPY THEORY as well.
edu/rybu/bordism/bordism.html.
THEORY,
Borel’s
Expansion
Let
=
4(t)
c;to
An
STRATEGY
tn be any function
for which
the
integral ”
Bordism Group There are bordism groups, also called C~BORDISM GROUPS or COBORDISM RINGS, and there are singugroups give a framelar bordism groups. The bordism work for getting a grip on the question, “When is a compact boundaryless MANIFOLD the boundary of another MANIFOLD. 3” The answer is, p recisely when all of its STIEFELWHITNEY CLASSES are zero. Singular bordismgroupsgiveinsightinto STEENROD'S REALIZATION “When can homology classes be realized as PROBLEM: the image of fundamental classes of manifolds?” That answer is known, too.
References Budney, R. “The Bordism
GAME
also
Project.”
http:
//math.
up being
Cornell
l
I( x > E converges.
Then
I( x > = w
e tx t”$(t)
dt
sa0
the expansion [Ao+(~+l)~
Al A2
+(P+l)(P+2)~+*.*
where r(z) ASYMPTOTIC
is the GAMMA FUNCTION, SERIES for 1(x).
1 7
is usually
an
Bore1 Measure If F isthe BOREL SIGMA ALGEBRA onsome TOPOLOGICAL SPACE, then a MEASURE m : F + R is said to be a Bore1 measure (or BOREL PROBABILITY MEASURE). For a Bore1 measure, all continuous functions are MEASURABLE. Bore1
Probability
see BOREL
MEASURE
Measure
158
Bore1 Set
Borwein
Bore1 Set A DEFINABLE SET derived
from the REAL LINE by reof intervals. Bore1 sets are a special type of SIGMA AL
moving a FINITE number measurable and constitute GEBRA called a BOREL SIGMA
ALGEBRA.
see also STANDARD SPACE Bore1 Sigma Algebra A SIGMA ALGEBRA which is related
to the TOPOLOGY of a SET. The Bore1 sigmaalgebra is defined to be the SIGMA ALGEBRA generated by the OPEN SETS (or equivalently, by the CLOSED SETS). see also BOREL
MEASURE
Bore1 Space A SET equipped
with
a SIGMA ALGEBRA of SUBSETS.
Conjectures
Borsuk’s Conjecture Borsuk conjectured that it is possible to cut an nD shape of DIAMETER 1 into n + I pieces each with diameter 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 N 1.1 ? Since l.lfi > n + 1 at n = 9162, the conjecture becomes false at high dimensions. In fact, the limit has been pushed back to N 2000.
see also DIAMETER (GENERAL), KELLER’S TURE, LEBESGUE MINIMAL PROBLEM
CONJEC
References Borsuk, K. “uber die Zerlegung einer Euklidischen ndimensionalen Vollkugel in n Mengen.” Verh. Internat. Math.Kongr. Ziirich 2, 192, 1932. Borsuk, K. “Drei Satze fiber die ndimensionale euklidische Sphare.” Fund. Math. 20, 177490, 1933. Cipra, B. “If You Can’t See It, Don’t Believe It. . .” Science l
Borromean
259, 2627,1993.
Rings
r3LC’j)
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 06& and are also called the BALLANTINE. The Borromean rings have BRAID WORD o~%~o~%~~~%~ and are also the simplest BRUNNIAN LINK.
Cipra, B. What’s Happening in the Mathematical Sciences, VoZ. I. Providence, RI: Amer. Math. Sot., pp* 2125, 1993. Griinbaum, 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 1315, 1961. Providence, RI: Amer. Math. Sot., pp. 271284, 1963. Kalai, J. K. G. “A Counterexample to Borsuk’s Conjecture.” Bu22. Amer. Math. Sot. 329, 6062, 1993. Listernik, L. and Schnirelmann, L. Topological Methods in Variational Problems. Moscow, 1930.
Borwein Conjectures Use the definition of the ~SERIES
References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 5859, 1989. Gardner, M. The Unexpected Hanging and Other Muthematical Diversions. Chicago, IL: University of Chicago Press, 1991. Jablan, S. “Borromean Triangles.” http: //members. tripod. corn/modularity/links,htm. Pappas, T, “Trinity of RingsA Topological Model.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p, 31, 1989.
nl
(a;q)n= rI (1  an9 and define (qNMS1;
q)M (2)
(a
Then
P. Borwein
dm
has conjectured
l
that
MIALS An(q), Bn(q), and Cn (q) defined Borrow ~ ~borrows

1234 789 445
12 118’4 789 445
(!?;q3)n(q2; q3)n= An(q3)
 qBn(q3)
(1) the PULYNOby (3)
 q2Cn(q3)
lLI have NONNEGATIVE ALS K(a),
CL(q)7
COEFFICIENTS,
and Cz (q) defined
(2) the POLYNOMIby
(q;q3)Fk12; q”>H= fG(q3) @7*,(q3)  q2G(q3) (4
procedure used in SUBTRACTION to “borrow” 10 the next higher DIGIT column in order to obtain a POSITIVE DIFFERENCE in the column in question.
have NONNEGATIVE
see also CARRY
ALS A:,(Q),
The
from
(1)
j=O
mz),
(3) the POLYNOMI
COEFFICIENTS, G(q),
X(q)1
and E:(q)
defined
bY
(q;Q5)n(Q2; Q5)n(q3; q5)n(q4; q5)n= A~(q5)qB~(q5)q2Cn*(q5)q3D:,(q5)q4E~(q5)
(5)
Bouligand
Dimension
have NONNEGATIVE ALS &(vwz), fined by
Boundary
COEFFICIENTS, (4)the P~LYNOMI@t(m,n,t,q), and Gl(m,n,t,q) de
(q;q3)m(q2; q3)m(ZQ; Q3)n(zq2; C13>n = Fz’[At(m,n,t,q3)
 qBt(m,n,t,q3)
t=o q2ct
(m, n, 6 q”>l
(6)
Bound Variable An occurrence of a variable FREE.
Point
in a LOGIC
159
which
is not
Boundary The set of points, known as BOUNDARY POINTS, which are members of the CLOSURE of a given set S and the CLOSURE of its complement set. The boundary is sometimes called the FRONTIER. see also SURGERY
have NONNEGATIVE COEFFICIENTS, 1 5 a 5 lc/2, consider the expansion (Cl”; Qk)m(4ka;
5) for !C ODD and Boundary Conditions There are several types of boundary conditions commonly encountered in the solution of PARTIAL DIFFERENTIAL EQUATIONS.
qk)n (k1)/2 “2a”F,(qk)
(7)
u=(lk)/2
BOUNDARY CONDITIONS specify value of the function on a surface T = f(r, t).
1. DIRICHLET
2. NEUMANN BOUNDARY CONDITIONS specify mal derivative of the function on a surface,
with
dT d7E = ii  VT /1‘ j=m
(8)
‘
thenif ais RELATIVELY PRIME tokand m = n, the COEFFICIENTS of F,(q) are NONNEGATIVE, and (6) given Q + p < 2’K and K + p 5 n  m < K  QI, consider
x
(9)
the GENERATING FUNCTION for partitions inside an mx n rectangle with hook difference conditions specified by QI, p, and K. Let Q: and p be POSITIVE RATIONAL
NUMBERS and K > 1 an INTEGER such that aK and PK are integers. Then if 1 5 a+P 5 2K 1 (with strict inequalities for K = 2) and K + /3 5 n  m 5 K  QI, then G(cY, /?,K;q) has NONNEGATIVE COEFFICIENTS. see also qSERIES References Andrews, G. E. et al. “Partitions with Prescribed Hook Differences.” Europ. J. Combin. 8, 341350, 1987. Bressoud, D. M. “The Borwein Conjecture and Partitions with Prescribed Hook Differences.” Electronic J. Combinatorics 3, No. 2, R4, l14, 1996. http://vww. combinatorics. org/Volume3/volume32. html#R4.
Bouligand
Dimension
see MINKOWSKIBOULXAND
DIMENSION
see GREATEST
LOWER PER BOUND,~UPREMUM
BOUND,
INFIMUM,
LEAST
UP
the nor
= f (r, y).
3. CAUCHY BOUNDARYCONDITIONS average of first and second kinds.
specify a weighted
4. ROBIN BOUNDARY CONDITIONS. For anelliptic partial differential equation in a region R, Robin boundary conditions specify the sum of QIU and the normal derivative of u = f at all points of the boundary of 0, with QI and f being prescribed. see also BOUNDARY VALUE PROBLEM, DIRICHLET BOUNDARY CONDITIONS, INITIAL VALUE PROBLEM, NEUMANN BOUNDARY CONDITIONS, PARTIAL DIFFERENTIAL EQUATION, ROBIN BOUNDARY CONDITIONS References A&en, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp, 502504, 1985. Morse, P. M. and Feshbach, H. “Boundary Conditions and Eigenfunctions.” Ch. 6 in Methods of Theoretical Physics, Part I, New York: McGrawHill, pp. 495498 and 676790,
Boundary Map The MAP Hn(X,A) EXACT SEQUENCE see also LONG
+ H,l(A) appearing OF A PAIR AXIOM.
EXACT
SEQUENCE
in the LONG
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 Iw”, then a point x E IP 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
Bound
the
160 Boundary
Boundary
Boustrophedon
set Bourget
Set
A (symmetrical) boundary set of RADIUS x0 is the set of all points x such that
Function
T and center Jn,&)=
Ixx01
$/tmnB1 
Boundary
DISK,
Value
OPEN
SET, SPHERE
Problem
d2u 3F I
 V2u = f u(0, t) = IL1 8th (0 ’ t) = u2 8t the
see also BOUNDARY PROBLEM
[$(t
dt
i)]
in n on 80 on &I,
boundary
of a,
CONDITIONS,
1
=
(2 cos 0)” cos(n0  z sine) d0.
7r s 0 see also BESSEL References Hazewinkel, ematics: Soviet lands:
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,
where dCJ denotes problem.
(t+i)kexp
=r.
Let x0 be the ORIGIN. In @, the boundary set is then the pair of points II: = T and =1: = T. In Iw2, the boundary set is a CIRCLE. In R3, the boundary set is a SPHERE. see also CIRCLE,
llansform
FUNCTION
OF THE FIRST
KIND
M. (Managing
Ed.). Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht, p. 465, 1988.
of
An Updated “Mathematical
Reidel,
Bourget’s
Mathof the Nether
Hypothesis
When n is an INTEGER and Jn+m(z) > 0, then J&z) have no common zeros other than at z = 0 for nz an INTEGER J&T) is a BESSEL FUNCTION OF > 1, where THE FIRST KIND. The theorem has been proved true for m=l 2, 3, and 4. References Watson, 2nd
is a boundary INITIAL
VALUE
ed.
G. N. A Treatise on the Theory Cambridge, England: Cambridge
of Bessel Functions, University Press,
1966.
Boustrophedon
Transform
The boustrophedon quence a is given
(“oxplowing”)
transform
b of a se
by
References Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Computationai Differential Equations. Lund: Studentlitteratur, 1996. Press, W. H.; Flannery, B. I?; Teukolsky, S. A.; and Vetterling, We T. “Two Point Boundary Value Problems.” Ch. 17 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 745778, 1992.
ak&k
Un
=
for n > 0, where
E,
(2)
&En.I,
is a SECANT
NUMBER
or
TANGENT
bY
Bounded
see also BOUND,
(;)
k=O
NUMBER defined A SET in a METRIC SPACE (X,d) is bounded if it has a FINITE diameter, i.e., there is an R < 00 such that d(x, y) < R for all 2, y E X. A SET in Ik” is bounded if it is contained inside some BALL xl2 + . . . + xn2 < R2 of FINITE RADIUS R (Adams 1994).
f(l)“”
(1)
00
En%= >:
n=O
The exponential related by
.
generating
(3)
set 61:+ tan 2.
functions
of a and
b are
FINITE B(x)
= (sect:
+ tanx)d(x),
(4
References R. A. Calculus: A Complete AddisonWesley, p. 707, 1994.
Adams,
Bounded
Course,
Reading,
MA:
where
the exponential
A(r)=pAn$. n=o
Variation
A FUNCTION f( z over the CLOSED such that
if, ) is said to have bounded variation INTERVAL it: E [a, b], there exists an M
lf(~i>f(~>I+lf(~2)f(21)1+. for all a < xi < x2 < . . . < xnml
. ~+lfwf(~~1)1 < b.
I M
generating
function
is defined
by
(5)
.
see also ALTERNATING PERMUTATION, ENTRINGER NUMBER, SECANT NUMBER, SEIDEL~NTRINGERARNOLD TRIANGLE,TANGENT NUMBER References Millar, J.; Sloane, N. J. A.; and Young, eration on Sequences: The Boustrophedon Combin. Th. Ser. A 76, 4454, 1996.
N. E. “A New Transform+”
OpJ.
Bovinum
Problema
Sovinum
Problema
see
ARCHIMEDES’
Box Fractal
CATTLE
pins knocked down on the next two bowls. are bowled, unless the last frame is a strike which case an additional bowl is awarded.
PROBLEM
Bow
161 Ten frames or spare, in
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 Bowling 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 (Ed. Score Awful. 7” In The Lighter Side of Mathematics R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.
x4 = x2y  y3. References Cundy, H. and Rollett, Stradbroke, England:
A. Mathematical Models, Tarquin Pub., p 72, 1989.
3rd ed.
Box
see CUBOID Bowditch
Curve
see LISSAJO~S CURVE
BoxandWhisker
Plot i
PB = $(PL +pp),
pi is LASPEYRES' INDEX and Pp is PAASCHE'S
where INDEX.
see also INDEX
References Kenney, J. F. and Keeping, Pt. 1, 3rd ed. Princeton,
Bowley
Ah
known
(Q3
Skewness as QUARTILE

Q2)  (Q2 
QrQl)
E. S. Mathematics NJ: Van Nostrand,
COEFFICIENT,
SKEWNESS
QI)
_

of Statistics,
p, 66, 1962.
QI 
2Q2
QsQl
+
.. @!I
Bowley Index The statistical INDEX
A HISTOGRAMlike method of displaying data invented by J. Tukey ,(1977). Draw a box with ends at the QUARTILES Q1 and Q3. 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 J. W. Explanatory Data Analysis. AddisonWesley, pp. 3941, 1977.
Reading,
Tukey,
+ Q3
MA:
’ Box
where the Qs denote the INTERQUARTILE RANGES. see also SKEWNESS
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
Dimension
see CAPACITY DIMENSION Box
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 T4 = 4(4 + 1)/2 = 10.
Counting
Fractal
l xx
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 *If 3 Ii II,>II II line 1: tt*lIr,>ti* line 2: II*),tr,>il
*
II
II
tr,>tt
II
II
tt,>ll
It
I
line 3:
ft*tl,)lt*
*II
,
BoxMuller
162
Let Nn be the number of black boxes, L, the length a side of a white box, and A, the fractional AREA black boxes after the nth iteration.
Nb = 5” L, z (i)”
CAPACITY d cap
DIMENSION
(1) (2)
“(i,?
(3)
In N, lim 72300 lnL,

In 5 = 1.464973521.. In 3
q.
SIERPI~~SKI
(4
This
ln X:1.cos(2~22)
(1)
252= J2
In zl sin(2rs2).
(2)
by solving
for ~1 and ~2,
x1 = e(r12+r22)/2 1 x:2 = Ytan ~
Taking
the JACOBIAN
27T
FL: CRC
Surfaces.
Standard Curves Press, p. 324, 1993.
DOUBLE
roll possible) on a pair of of rolling boxcars is l/36,
SIXES,
SNAKE
EYES
(3) 1
x2
(

Xl
>
.
(4)
yields
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 CROSSCAP 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). B ased on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured that quartic polynomials were also insufficient (Pinkall 1986). Ap&y’s IMMERSION proved this conjecture wrong, giving the equations explicitly in terms of the standard form for a NONORIENTABLE SURFACE, f&y+)
= +[(2x2y2r2)(x2+Y2+z2) + 2yz(y2
 z”)
+ =(x2
 z2)
+ XY(Y2  x2)1 + xz(x2 f3(GYG4
Theorem
References Honsberger, Math.
PUZZLE, CUBOID, DE BRUIJN'S THEOTHEOREM, SLOTHOUBERGRAATSMA
R. Mathematical
Assoc.
Amer.,
Gems p. 74, 1976.
i(x
+
II. Washington,
DC:
+ y2 + z”>
 x2) + xy(y2 y +
x)[(x
+qY+Y)(x~>l.
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 REM, KLARNER'S PUZZLE

(1)
 z2)(x2
f2 (x9 Y? 4 = yd[(y”
BoxPacking
Boca
Boy Surface
d2
can be verified
Raton,
see also DICE,
Transfwmation
=
References von Seggern, D. CRC
A roll of two 6s (the highest 6sided DICE. The probability or 2.777. . . %.
A transformation which transforms from a 2D continuous UNIFORM DISTRIBUTION to a 2D GAUSSIAN BIvmwrE DISTRIBUTION (or COMPLEX GAussIAN DISTRIBUTION). If ~1 and z2 are uniformly and independently distributed between 0 and 1, then zr and z2 as defined below have a GAUSSIAN DISTRIBUTION with MEAN p = 0 and VARIANCE o2 = 1. 21
STEP FUNCTION.
Boxcars
CARPET,
References * Weisstein, E. W. "l?ract als." http://www. astro,virginia. ,m. edu/#eww6n/mat ,h/notebooks/Fractal
BoxMuller
Function
where H is the HEAVISIDE
  yrn  W”) n>m ln(3“)

DUST,
Boxcar
is therefore
=
see also CANTOR SKI SIEVE
of of
n
=
A, = Ln2Nn The
Boy Surface
Transformation
+
y +
 x2)]
(2)
4"
(3)
Boy Surface Plugging
Boy Surface
in
In Iw4, the parametric x = cosusinw
(4)
y = sinusinv
(5)
z=
(6)
cos zt
and letting u E [0, ;TT]and v E [0, 7r] then gives surface, three views of which are shown above. The EC3 parameterization ficos2
can also be written v cos(221)
the Boy
as
(7)
Y==
2  fisin(3u)
(8)
sin(2w)
(Nordstrand)
for u E [~/2,x/2]
64(x0
21E [o,;rr].
(15)
 x3)3~33
(16)
equation
is
 48(x0
 x3)2x32(3x12
 x&[Z~(X~~
+362/22223(~2~ +
+ x22)2
 3x1~)
9xz2

+
1986).
+ 32~~ + 2~3~)
 24xa2(x12
+ ~2~)~  72xa2(x12
In fact, a HOMOTOPY the ROMAN SURFACE equations
x(21, w) =
(smooth and Boy
&kos(2u)
z(u,v)
=
JZsin(2u)
this
parame
deformation) between surface is given by the
cos2 v + cos u sin(2v)
2  a&in(3u) Y(% v) =
using
sin(2w)
(10)
cos2 v  sinusin(2v)
2  alfisin(3u)
sin(2v)
(11)
3 cos2 v 2  aJz
sin(3u)
sin( 2w)
as a varies from 0 to 1, where QI = 0 corresponds ROMAN SURFACE and or = 1 to the Boy surface shown below.
gives
another
(17)
Letting
version
Xl
=x
x2
=y
x3
=
(18) (19) (20)
z
of the surface
(21)
in Ik3.
see also CROSSCAP, IMMERSION, M~~BIUS NONORIENTABLE SURFACE, REAL PROJECTIVE ROMAN SURFACE, SEXTIC SURFACE
STRIP, PLANE,
References Ap&y, F. “The Boy Surface.” Adv. Math. 61, 185266, 1986. Boy, W. “Uber die Curvatura integra und die Topologie geschlossener F&hen.” Math. Ann 57, 151184, 1903. Brehm, U. “How to Build Minimal Polyhedral Models of the Boy Surface.” Math. Intell. 12, 5156, 1990. Carter, J. S. “On Generalizing Boy SurfaceConstructing a Generator of the 3rd Stable Stem.” Trans. Amer. Math. Sot. 298, 103122, 1986, Fischer, G. (Ed.). Plates 115120 in Muthematische ModeZle/Muthematical
to the (Wang),
+ x2”)
 32~~) + 4x34] = 0
x0 = 1
Three views of the surface obtained terization are shown above.
+ x22)
~3~1
2~3~)
x [81(x12
(Apkry
(14)
 Jzvw)
+lO8dS~1~3(~1~
and
 v2)]
 t12 + duw)
+ v2)(2uv
(9)
sin(2v)
+ II”)  J2vw(3u2
= 3(u2 + v2)2,
+(9x12
3 cos2 v ’ = 2  fisin(3u)
x2 = J2(u2
+12(x0
Jz cos2 21sin( 2u) + cos u sin( 2v)
is
(13)
and the algebraic
+ cosusin(2v) sin(2v)
+ v2 + w2)(u2
Xl = J2 (u” + u2)(u2
x3
X=
2  JZsin(3u)
x0 = 3[(u2
representation
163
Models,
Bildband/Photograph
Volume.
Braunschweig, Germany: Vieweg, pp. 110115, 1986. Geometry Center. “Boy’s Surface.” http : //www . geom .umn. edu/zoo/toptype/pplanne/boy/. Hilbert, D. and CohnVossen, S. $4647 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 reprksentation analytique de la surface de Boy.” C. R. Acad. Sci. Paris S&r. I Math 293, 269272, 1981. Pinkall, U. Mathematical IModels from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 6465, 1986. Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991. Wang, P. “Renderings .” http://www.ugcs.caltech.edu/ peterw/portfolio/renderings/.
164
Brachistochrone
Bra
Bra (C~VARIANT) IVECTOR denoted ($1. The bra is DUALLY the CONTRAVARIANTKET, denoted I$), Taken together, the bra and KET form an ANGLE BRACKET
A
(bra+ket = b racket). The bra is commonly in quantum mechanics.
see ah ANGLE BRACKET, VARIANTVECTOR,DIFFERENTIAL FORM
encountered
BRACKET PRODUCT, !+FoRM,KET,ONE
y’(af
Squaring
both
/ay’)
sides
out friction) from one point to another in the least time. was one of the earliest problems posed in the CALThe solution, a segment of a CULUS OF VARIATIONS. CYCLOID, was found by Leibniz, L’Hospital, Newton, Bernoullis.
which pl
to another
point
s
2 ds
t12
=
I’2
(1)
;.
1
The VELOCITY at any point is given by a simple application of energy conservation equating kinetic energy to gravitational potential energy, +v2
= mgy,
and rearranging
simplifying
slightly
results
then
in
where the square of the old constant C has been expressed in terms of a new (POSITIVE) constant K2. This equation is solved by the parametric equations
This
The time to travel from a point is given by the INTEGRAL
and
(10)
Problem the shape of the CURVE down which a bead sliding rest and ACCELERATED by gravity will slip (with
and the two
f,
from
Co
Brachistochrone Find from
subtracting gives
Problem
arelo
2 = $“(8
 sin@)
(11)
y = $k2(1
 cosq,
(12)
and beholdthe
equations
of a CYCLOID.
If kinetic friction is included, the problem can also be solved analytically, although the solution is significantly messier. In that case, terms corresponding to the normal component of weight and the normal component of the ACCELERATION (present because of path CURVATURE) must be included. Including both terms requires a constrained variational technique (Ashby et al. 1975), but including the normal component of weight only gives an elementary solution. The TANGENT and NORMAL VECTORS are
(2) (13)
SO
v= Plugging
this
into
(3)
J2sv*
(1) then
dy, N=z~+clsy,
2 m s1
The function
(4)
dx=12/zdx. d%
to be varied
f =
is thus
(5)
(1 + y’2)1’2(2gy)1’2.
To proceed, one would blown EULERLAGRANGE
(14)
gives gravity
t12 _
dx A
normally
have to apply
DIFFERENTIAL
and friction
are then
F gravity
= mgy
F friction
=
(15) gravityr;J)T
CL(F
and the components
along
=
the curve
the fullF
EQUATION
pmg$,
(16)
are
dY gravity+
=
mg
ds
(17)
dx = 0.
(6) so Newton’s
However, the function since x does not appear 0, and we can immediately fY/g
f(y, y’,zc) is particularly nice explicitly. Therefore, 8f /ax = use the BELTRAMI IDENTITY I = c.
(7)
Second
But
dv
=21x +”
g
= y’(1+
y’2)1’2(2gy)1’2,
(8)
gives
dw dx dY mdt = mg&  wv&
dt Computing
Law
dv
=
1 d = &v2) dY

4
(19)
(20) (21)
Bracket SO
(23) Using
the EULERLAGRANGE DIFFERENTIAL EQUATION
gives
[1+ y’2](l + py’) + qy  px)y’l = 0. This, can be reduced
(24
to
1+(y’12_ c (1+ /q/q2 Y  PX’
gives a KNOT POLYNOMIAL which is invariant under REGULAR ISOTOPY, and normalizing gives the KAUFFMAN POLYNOMIAL X whichisinvariantunder AMBIENT ISOTOPY. The bracket POLYNOMIAL of the UNKNOT is 1. Thebracket POLYNOMIAL ofthe MIRROR IMAGE K* is the same as for K but with A replaced by A? In terms of the onevariable KAUFFMAN POLYNOMIAT, X, the twovariable KAUFFMAN POLYNOMIAL F and the JONES POLYNOMIAL V, X(A)
(25)
the solution
z cot(@),
(26)
is $“[@I
2=
y = fk2[(1

 sin@ + ~(1  COSB)]
(27)
 cod)
(28)
+ ~(0 + sin@)].
(5)
A + Al)
(6)
CYCLOID,TAUTOCHRONE
is the WRITHE of L. see UZSO SQUARE BRACKET POLYNOMIAL
References C. C. The Knot Book: to the Mathematical Theory
Freeman, Kauffman,
PROBLEM
Amer.
a
Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. “Brachistochrone with Coulomb Friction.” Amer. J. Phys. 43, 1975.
Haws, L. and Kiser, T. “Exploring the Brachistochrone Problem.” Amer. M&h. Monthly 102, 328336, 1995. Wagon, S. Mathematics in Action. New York: W. H. Freeman, pp. 6066 and 385389, 1991.
pp. 148155, 1994. L, “New Invariants
Kauffman, entific, pp* 2629, 1991. $$ Weisstein, E. W. “Knots and Links.” virginia.edu/euw6n/math/notebooks/Knots.m.
Bracket Product The INNER PRODUCT see also ANGLE
see ANGLE BRACKET, BRA, BRACKET POLYNOMIAL, BRACKET PRODUCT, IVERSON BRACKET, KET, LAGRANGE BRACKET,~OISSON BRACKET
(LIo)
d”““,
(1)
u where A and B are the “splitting variables,” g runs through all “St at es” of L obtained by SPLITTING the LINK, (Lla) is the product of “splitting labels” corresponding to 0, and
NJ: World
in an L2 SPACE represented
BRACKET,
by an
Bracketing Take x itself to be a bracketing, then recursively define a bracketing as a sequence B = (Bl, . . . , Bk) where C; > 2 and each Bi is a bracketing. A bracketing can be represented as a parenthesized string of xs, with parentheses removed from any single letter x for clarity of notation (Stanley 1997). Bracketings built up of binary operations only are called BINARY BRACKETING% For example, four letters have 11 possible bracketings: xxxx (x2x)x (4
( xx >xx x(xxx)
(xx>
xW)x)
the last five of which The number
x(xx)x ((xx)x)x x(x(x4)
xx(xx) (x(xx))x 9
are binary.
of bracketings
on n letters
is given
by the
GENERATING FUNCTION
and the
+11x4
sn =
3(2n  3)s,4
 (n  3)5,z n
+45x5
RECURRENCE
RELATION (3) (4
Sci
KET, L2 SPACE, ONE
BRA,
>
of loops in O. Letting
of Knots.”
http: //www. astro.
&6x+x2)= 1870 Stanleylg;7;xz +3x3 where NL is the number
W. H.
BRACKET.
FORM
E x
Introduction
York:
in the Theory
Monthly 95, 195242, 1988. L. Knots and Physics. Teaneck,
Bracket
(L) (A, B,d)
An Elementary of Knots. New
Math.
ANGLE
Bracket Polynomial KNOT POLYNOMIAL related to the JONES A onevariable PO'LYNOMIAL. The bracket polynomial, however, is not a topological invariant, since it is changed by type I REIDEMEISTER MOVES. However, the SPAN of the bracket polynomial is a knot invariant. The bracket polynomial is occasionally given the grandiose name REGULAR ISOTOPY INVARIANT. It is defined by
(7)
where w(L)
Adams,
Keferences 902905,
(L),
W) (A) = VW4), y'
also
= (A3)“(L)
(L) (A) = F(A3,
Now letting
see
165
Bracketing
166
Bradley’s
Theorem
Brahmagupta
(Sloane), giving the sequence for sn as 1, 1, 3, 11, 45, 197, 903, . (Sloane’s AOOlO03). The numbers are also given by l
l
sn =
x il+...+i&=n
for n > 2 (Stanley
K = &=ijcs
s(i1)*s(k)

NUMBERS
BRACKETING,~LUTARCH
References Habsieger, Number 1998. Schriider,
L., * Kazarian, of Plutarch.”
M.; and Lando, Amer.
Math.
S. “On the Second Monthly
105,
446,
combinatorische Probleme.” 2. Math. 1870. Sloane, N. J. A. Sequence A001003/M2898 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Stanley, R. P. “Hipparchus, Plutarch, Schrijder, and Hough.” Amer. Math. Monthly 104, 344350, 1997. Physik
E. “Vier
15, 361376,
Bradley’s Let
m
Theorem
l?(m+jz+l)r(a+p+1+j(a+Q
and QI be a NEGATIVE
9
INTEGER.
S(QI,P, m; 4 = where r(z)
+ ad)(ac
is the GAMMA
r(P + 1  m)
(3) 1
4R
(4)
where R is the RADIUS of the CIRCUMCIRCLE. If the QUADRILATERAL is INSCRIBED inone CIRCLE and CIRCUMSCRIBED onanother,thenthe AREA FORMULA simplifies to K=da. (5)
see
also
BRETSCHNEIDER'S FORMULA,
HERON'S
FOR
MULA References Coxeter, H. S. M, and Greiteer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 5660, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 8182, 1929.
Brahmagupta Let
Identity
det[B(zl,
pi)B(xz,
yz)] = det[B(~
 PP 1 2.
then
~111 WBh
ydI
a Polynomials
.” Fib.
References Suryanarayan, E. R. “The Brahmagupt Quart. 34, 3039, 1996.
Then
in
+ bd)(ab + cd)
where B is the BRAHMAGUPTA MATRIX,
00 r(m +j(x+l))r(P +1+24 (4+j c j=O
&IC
inscribed
 b)(s  c)(s  d)
1997).
The first PLUTARCH NUMBER 103,049 is equal to ~10 (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 + srr)/2 = 310,954 (Habsieger et al. 1998). see also BINARY
QUADRILATERAL (i.e., a QUADRILATERAL a CIRCLE), A + B = Z, SO
Matrix
Brahmagupta
Matrix
r(a+p+lm)’ FUNCTION.
References
It satisfies
Berndt, B. C. Ramanujan’s Notebooks, Part IV. New York: SpringerVerlag, pp. 346348, 1994. Bradley, D. “On a Claim by Ramanujan about Certain Hypergeometric Series.” Proc. Amer. Math. Sot. 121, 11451149, 1994.
Brahmagupta’s Formula For a QUADRILATERAL with d, the AREA K is given by
sides of length
B(a,y$3(~2,
~2) = B(
Powers of the matrix
21x2
*
are defined
ty1y2,wyz
*
YlX2).
by
a, b, c, and The ALS.
zn and yn are called BRAHMAGUPTA POLYNOMIThe Brahmagupta matrices can be extended to
NEGATIVE INTEGERS (s  a)(s  b)(s  c)(s  d)  abcdcos2[$(A
+ B)], (1)
where SE
+(a+b+c+d)
(2)
is the SEMIPERIMETER, A is the ANGLE d, and B is the ANGLE
between
between a and b and c. For a CYCLIC
B“= [;
;I‘= [;“,
,“I,] EB,.
see U~SO BRAHMAGUPTA IDENTITY References Suryanarayan, E. R. “The Brahmagupta Quart. 34, 3039, 1996.
Polynomials.”
Fib.
Brahmagupta
Polynomial
Braid Group
Brahmagupta Polynomial One of the POLYNOMIALS obtained by taking POWERS of the BRAHMAGUPTA MATRIX. They satisfy the recurrence relation
A list of many Explicitly,
Xn+l
= XXn
+ tyyn
(1)
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 gravity and horizontal forces.
Yn+l
=
+
(2)
see also BRAID
others
Xyn
is given
YXn.
by Suryanarayan
(1996).
GROUP
References Christy, J. “Braids.” http://www.mathsource.com/cgibin /MathSource/Applications/Mathematics/O202228.
Xn=o”+t(~)Xn~‘TJ2+t2(~)Xn~4~4+~mm
yn=nXn1y+t(~)Xn3y3+t2(~)Xn5y5
The Brahmagupta
satisfy
POLYNOMIAIJ
dXn ax
 @In  dy
dxn
= ,aYn
dY
dY
first few POLYNOMIALS
The
167

(5)
T&Xn1
(6)
= nt?Jnl.
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 by group, denoted Bn. A general nbraid is constructed iteratively applying the oi (; = 1,. . . , n  1) operator, which switches the lower endpoints of the ith and (; + 1) th stringskeeping the upper endpoints fixedwith the (i + 1)th string brought above the ith string. If the (i + 1)th string passes below the ith string, it is denoted
fTi l ’
are
x0 = 0 Xl
=x
x2
= x2 + ty2
x3
= x3 + 3txy2
x4
= x4 + 6tx2y2
1 2
+ t2y4
Topological a BRAID
i+l
i+l
equivalence for different representations of ni pi and Hi 0: is guaranteed by the
WORD
conditions
Yo =0 Yl = Y y2
=
y3
= 3x2y + ty3
2xy
y4
= 4x3y + 4txy3.
2 gives yn equal to the PELL to half the PellLucas numbers. POLYNOMIALS are related to the MORGANVOYCE POLYNOMIALS, but the relationship given by Suryanarayan (1996) is incorrect. .
as first proved by E. Artin. Any nbraid is expressed as is a BRAID WORD a BRAID WORD, e.g., uIuz~~c~~ ‘al for the braid group B3* When the opposite ends of the braids are connected by nonintersecting lines, KNOTS are formed which are identified by their braid group and gives a BRAID WORD. The BURAU REPRESENTATION matrix representation of the braid groups.
References
References
Taking
x =y=landt= and xn equal The Brahmagupta
NUMBERS
Suryanarayan, Quart.
E. R. “The Brahmagupta 34, 3039, 1996.
Brahmagupta’s Solve the PELL
see
also
Fib.
Problem EQUATION X2
in INTEGERS. 120.
Polynomials.”
 92y2 = 1
The smallest
DIOPHANTINE
solution
EQUATION,
is x = 1151, y = PELL
EQUATION
Birman, J. S. “Braids, Links, and the Mapping Class Groups.” NJ: Ann. Math. Studies, No. 82. Princeton, Princeton University Press, 1976. Birman, J. S. “Recent Developments in Braid and Link Theory.” Math. Intell. 13, 5260, 1991. Christy, 3. “Braids.” http://www.mathsource.com/cgibin /MathSource/Applications/Mathematics/O202228. Jones, V. F. R. “Hecke Algebra Representations of Braid Ann. Math. 126, 335Groups and Link Polynomials.”
388,1987. $# Weisstein, E. W. “Knots and Links.“’ virginia.edu/eww6n/math/notebaoks/Knots.m.
http:
//wwe.
astro.
168
Braid
Braid
Branch
Index
Index
BraikenridgeMaclaurin
Point
Construction
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 projection of a KNOT (Yamada 1987). Also, for a nonsplittable LINK with CROSSING NUMBER c(L) and braid index i(L), c(L) 2 2[i(L)  l]
The converse of PASCAL’S THEOREM. Let Al, B2, Cl, AZ, and & be the five points on a CONIC. Then the CONIC is the LOCUS of the point
(Ohyama 1993). Let E be the largest and e the smallest POWER of e in the HOMFLY POLYNOMIAL of an Then the oriented LINK, and i be the braid index. MORTONFRANKSWILLIAMS INEQUALITY holds,
see
i>
i(Ee)+l
09049,
10132,
10150,
and
4 C1A2).
where x is a line through PASCAL'S
also
BI(Z
l
the point
C1B2),
A&
l
&AZ*
THEOREM
Branch The segments tion (FORKS).
of a TREE
see also FORK,
(Franks and Williams 1987). The inequality is sharp for all PRIME KNOTS up to 10 crossings with the exceptions of09042,
C2 = A+
Branch
LEAF
the points
of connec
(TREE)
Cut
Re[Sqrt21
10156.
between
Im[Sqrtzl
lsqrt 21
References Franks, J. and Williams, R. F. “Braids and the Jones Polynomial.” Trans. Amer. ikth, Sot. 303, 97108, 1987. Jones, V. F. R. “Hecke Algebra Representations of Braid Groups and Link Polynomials.” Ann, Math. 126, 335388, 1987. Ohyama, Y. “On the Minimal Crossing Number and the Brad Index of Links.” Cunad. J. Math. 45, 117131, 1993. Yamada, S. “The Minimal Number of Seifert Circles Equals the Braid Index of a Link.” Invent. 1Muth. 89, 347356, 1987.
Braid
Word
Any
nbraid is expressed as a braid word, e.g., %I is a braid word for the BRAID GROUP B3. 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.
m~2~3~2
Let b+ be the sum of POSITIVE exponents, sum of NEGATIVE exponentsinthe BRAID If b+  3b  n + 1 > 0, then the closed 1985). see
also
BRAID
braid
b is not
and b the GROUP B,.
AMPHICHIRAL
(Jones
Alineinthe COMPLEX PLANE is discontinuous. function branch cut(s) cosl cash’ cotl coth’ csc l csch’ In z set l sech’ sinl sinh’ 6z tanl tanh’ zn,n see
x
(00, 1) and (1,~) (007 1) (4, i) [I, 11 (17 1) (4, i) (00, 01 (171) ho] and (174 ( 00, 1) and (1,~) (ioo,;) and (ilk) (oo,o> (400, 4) and (i, iw) ( 00, 11 and [l,~) (co, 0) for !R[n] 5 0; (00,
z z
z z
z g z
also
acrosswhicha
BRANCH
> 0
POINT
References
Morse, P. M. and Feshbach, H. Methods ics, Part I. New York: McGrawHill,
Jones, V. F. R. “A Polynomial Invariant for Knots via von Neumann Algebras.” Bull. Amer. Math. Sot. 12, 103111,
Branch
1985. 388,1987.
O] for R[n]
KeIerences
GROUP
Jones, V. F. R. ‘(Hecke Algebra Groups and Link Polynomials.”
FUNCTION
Representations Ann. Math.
of Braid
of Theoretical
pp. 399401,
Phys
1953.
Line
SUBBRANCH
CUT
126, 335
Branch
Point
An argument at which identical points in the COMPLEX PLANE are mapped to different points. For example, consider f( z > =P.
Brauer
Breeder
Chain
Then f(8i> = f( 1) = 1, but f(e2r’) = eaxia, despite the fact that eio = e2? PINCH POINTS are also called branch points. see dso BRANCH
CUT,
PINCH
Brauer’s Theorem If, in the GER~ORIN 1ajj
POINT
Hekrences A&en, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 397399, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp, 391392 and 399401, 1953.
see also ADDITION CHAIN
CHAIN,
BRAUER
NUMBER,
HANSEN

Grim
for a given KU,
THEOREM
1 >Aj+Arn
for all j # nz, then exactly the DISK I&.
of A lies in
one EIGENVALUE
References Gradshteyn, ries,
Press,
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.
CIRCLE
169
and
I. S. and Ryzhik, Products,
5th
I. M.
Tables
San Diego,
ed.
of Integrals,
CA:
Se
Academic
p. 1121, 1979.
Braun’s Let B
Conjecture = {b&z,...} b e an INFINITE Abelian SEMIGROUP with linear order bl < b2 < . . such that 61 is the unit element and a < b IMPLIES UC < bc for a, b, c E B. Define a MOBIUS FUNCTION 11on B by &) = 1 and l
References Guy, R. K. “Addition Chains. Brauer Chains. Chains.” SC6 in Unsolved Problems in Number 2nd ed. New York: SpringerVerlag, pp. 111113,
Hansen Theory,
1994,
for 72 = 2, 3, . . . . Further suppose (the true MOBIUS FUNCTION) for
Brauer Group The GROUP of classes of finite dimensional central simple ALGEBRAS over k with respect to a certain equivalence.
Braun’s
References
for all m, n > 1. see UZSO MOBIUS PROBLEM
Hazewinkel, ematics:
M. (Managing
Soviet
An Updated “Muthematical
lands:
Reidel,
Ed.),
Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht,
of
Mathof the
Nether
p. 479, 1988.
Brauer Number A number n for which a shortest chain exists which is a BRAUER CHAIN is called a Brauer number. There are infinitely many nonBrauer numbers. see UZSO BRAUER
CHAIN,
HANSEN
NUMBER
conjecture
Flath, A. and Zulauf, A. “Does the Mijbius Function Determine Multiplicative Arithmetic?” Amer. Math. Monthly 102, 354256, 1995.
Breeder A pair of POSITIVE equations
INTEGERS
(al, a2) such that
K that
M. (Managing
Reidel,
Ed.). Encyclopaedia and Annotated Translation Encyclopaedia. ” Dordrecht,
pp. 480481,
1988.
the
+ 1)
Hansen Theory,
1994.
Variety VARIETY over a FIELD to a PROJECTIVE SPACE.
ematics: An Updated Soviet “Mathematical
lands:
b,b,
References
becomes
have a POSITIVE
INTEGER solution Al:, where a(n) is the DIVISOR FUNCTION. If x is PRIME, then (al, azz) is an AMICABLE PAIR (te Riele 1986). (al, a~) is a “special”
breeder
if
References Hazewinkel,
=
a1 + a2x = cr(a1) = &2)(x
BrauerSeveri An ALGEBRAIC ISOMORPHIC
states that b mn
References Guy, R. K. “Addit ion Chains. Brauer Chains. Chains.” SC6 in Unsolved Problems in Number 2nd ed. New York: SpringerVerlag, pp. 111113,
that p(bn) = p(n) all n > 1. Then
a2 = a,
of Mathof the
Nether
where a and u are RELATIVELY PRIME, (a,~) = 1. If regular amicable pairs of type (i, 1) with i > 2 are of the form (au, up) with p PRIME, then (au, a) are special breeders (te Riele 1986). Heferences te Riele, H. J. J+ “Computation Below 101’? Math. Cornput. 1986.
of All the Amicable Pairs 47, 361368 and S9S35,
Brelaz’s
170 Brelaz’s
Heuristic
An ALGORITHM not necessarily a GRAPH. see
UZSO
Heuristic
Method POLLARD
p FACTORIZATION
uses
xi+1 = xi2  c (mod
20, 176184,
Brent’s
Algo
&x+1=
Method root
[Y  fb)lIY [f(x3)
 f(x3)h  f(xdl[f(xd  f(x3)l lY  fb3)l[Y  f(xd1x2
Subsequent giving
root estimates

+

fb>][f(x2>
are obtained

b)
d, = an2  bn2,
(4
and define the initial conditions to be a0 = 1, bo = l/A. Then iterating a, and b, gives the ARITHMETICGEOMETRIC MEAN, and 7r is given by 4[M(l,
7T=
21’2)]2
4[M(l,

(5)
2jf’dj
1  cE1
21/2)]z
1  cEl
(6)
2'+lCj2.
and the LEGEN
formula
and that
see also ARITHMETICGEOMETRIC
either
MEAN,
may be
PI
References Borwein,
f(xl>l
l
by setting
(’
y = 0
P
J. M. and Borwein,
Analytic
Number
Theory
P. B. Pi 0 the AGIM: and
Computational
New York: Wiley, pp. 4851, 1987. Castellanos, D. “The Ubiquitous Pi. Part II.” 61, 148163, 1988. King, L. V. On the Direct Numerical Calculation Functions
Cambridge,
and Integrals.
England:
A Study in Complexity. Math.
Mag.
of Elliptic
Cambridge
University Press, 1924. Lord, N. J. “Recent Calculations of ;r~: The GaussSalamin Algorithm.” Math. Gax. 76, 231242, 1992.
where P = S[R(R
(3)
=
(1924) showed that this DRE RELATION are equivalent derived from the other.
 fW1
+ [fb>
(2)
G&+1
King
[Y  f(d][!J
+ [fW
(1)
bnfl = d Gabn IN
 f(XdlX3
 f(~dlV(~3)
+(%a+&)
VAN
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 containing a ROOT. Given three points 21, 22, and x3, Brent’s method fits x as a quadratic function of y, then uses the interpolation formula
=
Formula
A formula which uses the ARITHMETICGEOMETRIC MEAN to compute PI. It has quadratic convergence and is also called the GAUSSSALAMIN FORMULA and SALAMIN FORMULA. Let
1980.
A ROOTfinding ALGORITHM which combines bracketing, bisection, and INVERSE QUADRATIC TERPOLATION. It is sometimes known as the WIJNGAARDENDEKERBRENT METHOD.
x
Brent, FL P. Ch. 34 in Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: PrenticeHall, 1973. Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. $7.2 in Computer Methods fur Mathematical Computations. Englewood Cliffs, NJ: PrenticeHall, 1977. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Van WijngaardenDekkerBrent Method.” $9.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 352355, 1992.
BrentSalamin
n).
References Brent, R. “An Improved Monte Carlo Factorization rithm.” Nurdisk Tidskrifi for Informationsbehandlung (BIT)
but for
NUMBER
of the
Formula
References
can be used to find a good, EDGE or VERTEX coloring
Factorization
A modification METHOD which
Bretschneider’s
Algorithm
which minimal,
CHROMATIC
Brent’s
Algorithm
 T)(23
Q = (T  l)(R
 l)(S
 x2)  (1  R)(xz  1)
 x1)]
(3) (4)
Bretschneider’s
of T Using ArithmeticGeometric
30, 565570, 1976.
Formula
Given a general QUADRILATERAL with sides of lengths a, b, c, and d (Beyer 1987), the AREA is given by
with
R = f cx2> f (4
s G f (4 f (Xl) T E f(xl) fk3)
(Press et al. 1992),
Salamin, E. “Computation Mean.” Math. Comput.
(5) (6) (7)
A quadrilateral where
p
=
3’
\/4p2q2
 (b2 + d2  a2  c~)~,
and q are the diagonal
see &o MULA
BRAHMAGUPTA'S
lengths.
FORMULA,
HERON'S
FOR
References Beyer, 28th
W.
H. (Ed.).
ed.
Boca Raton,
CRC
Standard
Mathematical
FL: CRC Press, p. 123, 1987.
Tables,
Brianchon
Point
Brianchon Point The point of CONCURRENCE of the joins of the VERTICES of a TRIANGLE and the points of contact of a CONIC SECTION INSCRIBED inthe TRIANGLE. A CONIC INSCRIBED in a TRIANGLE has an equation of the form f+g+h0u zt
211j
point has TRILINEAR COORDINATES For KIEPERT'S PARABOLA, the Branhas TRIANGLE CENTER FUNCTION
so its Brianchon
(l/f,
l/h).
l/g,
chion point
1
QI=
which
up2
 c2) ’
is the STEINER POINT.
Brianchon’s Theorem The DUAL of PASCAL'S THEOREM. It states that,given a 6sided POLYGON CIRCUMSCRIBED on a CONIC SECTION, thelinesjoining opposite VERTICES (DIAGONALS) meet in a single point.
the chance that one of four players of a single suit is
The probabilities of a given type with probability (l/P)  1 : 1.
Hand 13 top honors
Exact
13card
suit
4 N
12card
suit,
see EULER BRICK, HARMONIC BRICK, PARALLELEPIPED
RECTANGULAR
Bride’s Chair One name for the figure used by Euclid PYTHAGOREAN THEOREM. see also PEACOCK'S Bridge Bridge cards.
TAIL,
Iv ace high
158,753,389,900 1 15&,753,389,900 4 1,469,938,795
v 32
( 13 > N
5,394 9,860,459
48 ( 9 > 20
CN)C 9 N
11 4,165 32 4 >
Hand
Probability
13 top honors 13card suit 12card suit, ace high Yarborough four aces nine honors
6.30 x lol2 6.30 x lol2 2.72 x lo’
5.47 x lo” 2.64 x 1O3 9.51 X 1o3
888,212 93,384,347
Odds 158,753,389,899:1 158,753,389,899:1
367,484,697.8:1 1,827.O:l
377.6:1 104.1:1
see also CARDS, POKER References the
Ball, W. W. R. and Coxeter, H. S. M. IMathematical Recreations and Essays, i3th ed. New York: Dover, pp* 4849, 1987, Kraitchik, M, “Bridge Hands.” 56.3 in Mathematical Recreations. New York: W. W. Norton, pp. 119121, 1942.
with a normal deck of 52 distinct 13card hands is
Bridge (Graph) The bridges of a GRAPH are the EDGES whose removal disconnects the GRAPH.
to prove
WINDMILL
Card Game is a CARD game played The number of possible
Probability 1
4
nine honors
Brick

of being dealt 13card bridge hands are given below. As usual, for a hand P, the ODDS against being dealt it are
DUALITY PRINCIPLE, PASCAL'S THEOREM
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math, Assoc. Amer., pp. 7779, 1967. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p* 110, 1990.
a hand
There are special names for specific types of hands. A ten, jack, queen, king, or ace is called an “honor.” Getting the three top cards (ace, king, and queen) of three suits and the ace, king, and queen, and jack of the remaining suit is called 13 top honors. Getting all cards of the same suit is called a 13card suit. Getting 12 cards of same suit with ace high and the 13th card not an ace is called 2card suit, ace high. Getting no honors is called a Yarborough.
four aces
References
will receive
1
39,688,347,497
Yarborough
see also
171
Bridge (Graph)
see also
= 635,013,559,600.
ARTICULATION VERTEX
References where (;) is a BINOMIAL COEFFICIENT. chances of being dealt a hand of 13 CARDS of the same suit are  4 52 ( 13

>
1 158,753,389,900’
While the (out of 52)
Chartrand, G. “CutVertices and Bridges.” 52.4 in Introductory Graph Theory. New York: Dover, pp* 4549, 1985.
172
Bridge
Index
Bridge Index A numerical KNOT bridge index is the representations of UNKNOT is defined see
&O
BRIDGE
Bring
invariant. For a TAME KNOT K, the least BRIDGE NUMBER of all planar the KNOT. The bridge index of the as 1.
NUMBER,
CROOKEDNESS
References Rolfsen, D. Knots and Links. Wilmington, DE: Publish Perish Press, pJ 114, 1976. Schubert, IL “Uber eine numerische Knotteninvariante.” Math. 2. 61, 245288, 1954.
or
see
see K~NIGSBERG
BRIDGE
= 22 ‘VL(t)vL(tl+
I
22l),
where z = t  tl (Kanenobu and Sumi 1993). Kanenobu and Sumi also give a table containing the number of distinct 2bridge knots of 72 crossings for 72 = 10 to 22, both not counting and counting MIRROR IMAGES as distinct. n K,+K; K, 3 0 0
4 5 6 7 8 9 '
0
45
11
91 176
14 15 16 17 18
INDEX
References Adams, C. C. the
The Mathematical
Knot
pp. 6467,
Book: Theory
0
352 693 1387 2752 5504
19
10965 21931
20 21 22
43776 87552 174933
85 182 341 704 1365 2774 5461 11008 21845
43862 87381 175104 349525
Introduction
York:
W. H.
Wilmington,
DE: Publish
or
BrillNoether Theorem If the total group of the canonical series is divided into two parts, the difference between the number of points in each part and the double of the dimension of the complete series to which it belongs is the same. References Coolidge, J. L. A Treatise on Algebraic York: Dover, p. 263, 1959.
Plane
Curves.
New
BringJerrard Quintic Form A TSCHIRNHAUSEN TRANSFORMATION can be used to algebraically transform a general QUINTIC EQUATION to the form 2 + ClZ + co = 0. (1) In practice, PRINCIPAL
the general quintic QUINTIC FORM
is first, reduced
to the
+ bly + bo = 0
(2)
before the transformation is done. Then, we require that the sum of the third POWERS of the ROOTS vanishes, SO S3(yj) = 0. We assume that the ROOTS zi of the BringJerrard quintic are related to the ROOTS yi of the PRINCIPAL QUINTIC FORM by & = ayi4 + pyi3
+ yy?
+ 6yi + E*
(3)
In a similar manner to the PRINCIPAL QUINTIC transformation, we can express the COEFFICIENTS terms of the bj. see also BRING FORM, QUINTIC
QUINTIC EQUATIO
FORM, N
PRINCIPAL
FORM cj in
QUINTIC
Bring Quintic Form A TSCHIRNHAUSEN TRANSFORMATION can be used to take a general QUINTIC EQUATION to the form x5 Lca==,
References Kanenobu, T. and Sumi, T. “Polynomial Invariants of 2Bridge Knots through 22Crossings.” Math. Comput. 60, 771778 and Sl7S28, 1993. Schubert, H. “Knotten mit zwei Briicken." Math. 2. 65, 133170, 1956.
An Elementary of Knots. New
1994.
y5 + by2
10 12 13
BRIDGE
Rolfsen, D. Knots and Links. Perish Press, p. 115, 1976.
PROBLEM
Bridge Knot An nbridge knot is a knot with BRIDGE NUMBER n. The set of Zbridge knots is identical to the set of rational knots. If L is a ZBRIDGE KNOT, then the BLM/Ho POLYNOMIAL Q and JONES POLYNOMIAL V satisfy QL(z)
UZSO
Freeman,
of Kkigsberg
Form
Bridge Number The least number of unknotted arcs lying above the plane in any projection. The knot 0505 has bridge number 2. Such knots are called ZBRIDGE KNOTS. There is a onetoone correspondence beetween ZBRIDGE KNOTS and rational knots. The knot 080lo is a 3bridge knot. A knotwithbridgenumber bis annEMBEDDABLE KNOT where n < b.
to
Bridge
Quintic
where a may be COMPLEX. see UZSO BRINGJ EQ UATION
RD
QUTNTIC
FORM,
QUINTIC
References Ruppert, W. M, “On the Bring Normal Form of a Quintic Characteristic 5.” Arch. Math. 58, 4446, 1992.
in
Brioschi
Brocard
Formula Brocard
Brioschi Formula For a curve with METRIC ds2 = Edu2
+ Fdudv+Gdv2,
E, F, and G is the first the GAUSSIAN CURVATURE is where
FUNDAMENTAL FORM,
 32
(2)
)
A
where
$E, E F
0 LE, $Y,
ikf2
which
Angle
(1)
M1+M2
K = (EG
173
Angle
$GU F G
(4
c
Define the first BROCARD POINT as the interior point 0 of a TRIANGLE for which the ANGLES LOAB, LOBC, and LfICA are equal. Similarly, define the second BROCARD POINT as the interior point s2’ for which the ANGLES LWAC, LfI’CB, and LfYBA are equal. Then the ANGLES in both cases are equal, and this angle is called the Brocard angle, denoted w. The Brocard the formulas
can also be written
angle w of a TRIANGLE AABC
is given by
cotw = cotA+cotB+cotC
(1)
= (a2+;;+2)
=&[E(&)+i&g)]* see also FUNDAMENTAL FORMS, GAUSSIAN
CURVATURE
References Gray, A. Modern faces.
Boca
Differential Geometry of Curves Raton, FL: CRC Press, pp. 392393,
1+ cos a1 cos a2 cosa3 sin a1 sin a2 sin a3
(3)
 sin2 Qrl + sin2 a2 + sin2 a3
(4)
=
@)
(2)
2 sin QI~ sin ~r2 sin a3 al sin QC~+ a2 sin a2 +

and Sur1993.
sin
a3
EQUATION of the form
at x =
If an ANGLE a of a TRIANGLE is given, possible Brocard angle is given by cotw = $ tan( +)
(Managing Ed.). Encyclopaedia of MathUpdated and Annotated Translation of the Soviet “Mathematical Encyclopaedia. ” Dordrecht, Netherlands: Reidel, pp. 481482, 1988. M.
(7)
where A is the TRIANGLE AREA, A, B, and C are ANGLES, and a, b, and c are side lengths. the maximum
+ $ cos(+).
References An
(6)
a12uz2 + az2as2 + a32a12 ’
where m is a POSITIVE INTEGER, f is ANALYTIC y = 0, f(O,O) = 0, and fb(O, 0) # 0. Hazewinkel, ematics:
(5)
2A
sinw = BriotBouquet Equation An ORDINARY DIFFERENTIAL
u3
a1coscyl + u2 cosa2 + u3 cosa3 csc2w = csc2a1 + csc2a2 + csc2a3
(8)
Let a TRIANGLE have ANGLES A, J3, and C. Then sin A sin B sin C < kABC,
(9)
where k= ( (Le Lionnais
1983).
This
3J3 3 2n >
can be used to prove that
8w3 < ABC (AbiKhuzam
1974).
w
(11)
Brocard
174 see also
Brocard Line
Axis
BROCARD
CIRCLE,
BROCARD
LINE,
EQUI
BROC ARD CENTER, FERMAT POINT, ISOG~NIC CENTERS
see UZSOBROCARD ANGLE, BROCAR'D CARD POINTS
DIAMETER,BRO
References Johnson, R* A. Modern Geometry: An Elementary of the Triangle and the Circle. on the Geometry MA: Houghton Mifflin, p. 272, 1929.
Brocard’s
Brocard Axis The LINE KO passing through the LEMOINE POINT K and CIRCUMCENTER 0 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(B
 C)cy + sin(C
 A)P + sin(A  B)y
= 0
LEMOINE
POINT,
CIRCUMCENTER,
FUNCTION.
Brocard Diameter The LINE SEGMENT KOjoiningthe LEMOINE POINTK and CIRCUMCENTER 0 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
OK=
R& on = cos w
 4sin2 w 3 cos w
ISODYNAMIC
POINTS, and BROCARD MIDPOINT all lie along the Brocard axis. Note that the Brocard to the BROCARD LINE.
.
for n >  2 where 7r is the PRIME COUNTING see also ANDREA'S CONJECTURE
bc(b2 c2)a + ca(c”  a”)@+ &(a2  b2)y = 0. The
Conjecture
Treatise Boston,
axis is not equivalent
where s2 is the first BROCARD POINT, R is the CIRCUMRADIUS, and w is the BROCARD ANGLE.
see also BROCARD AXIS, BROCARD CIRCLE, BROCARD LINE,BROCARD POINTS
CARD LINE Brocard Brocard
Line
Circle
\
/
. .
/ .
, N
,
0
m_d
CIRCLE passing through the first and second BROCARD POINTS fl and s2', the LEMOINE POINT K, and the CIRCUMCENTER 0 ofa given TRIANGLE. The BRO
A
2
The
POINTS 0 and s1’ are symmetrical about the LINE which is called the BROCARD LINE. The LINE SEGMENT KO is called the BROCARD DIAMETER, and it has length
CARD
zo,
()K=OR
cosw 
RJl
A LINE from any of the VERTICES Ai to the first 0 or second n' BROCARD ANGLE at a VERTEX Ai also be denoted the intersections of Ala and AlSl’ with Wz* Then the ANGLES involving these
 4sin2 w 1 cos w
R is the CIRCUMRADIUS and w is the BROCARD ANGLE. The distance between either of the BROCARD POINTS and the LEMOINE POINT is
where
Distances
involving
= 00 tanw.
Ai, and denote AzA3 as WI and points are
LA1S1W3
= Al
(1)
LW3flA2
= A3
(2)
LA2S2W1
= A2.
(3)
the points
A2n= OK = fYK
of a TRIANGLE POINT. Let the
Wi and Wi are given by
zsin A2
sin#
(4
Brocard
Brocard Midpoint AdI A3R W3A1 ~
as2


al
w3A2
where w is the pp* 267268).

ma2
sin(Ag  w) sin w
a2 sin w sin(As  w) 
(6)
’
1929,
The Brocard line, MEDIAN M, and LEMOINE POINT are concurrent, with Alf21, A& and A&f meeting a point P. Similarly, Al R’, A&f, and A& meet a point which is the ISOGONAL CONJUGATE point of (Johnson 1929, pp. 268269).
K at at P
see also BROCARD AXIS, BROCARD DIAMETER, BROCARD POINTS, ISOGONAL CONJUGATE, LEMOINE POINT, MEDIAN (TRIANGLE) References Johnson, on the
R. A. Modern
Geometry: of the Triangle
Geometry
MA: Houghton
Mifflin,
An Elementary and the Circle.
pp. 263286,
175
(5)
ANGLE (Johnson
BROCARD
Points
Treatise
Boston,
1929.
Let CBC be the CIRCLE which passes through the vertices 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 0. Similarly, let C& be the CIRCLE which passes through the vertices B and C and is TANGENT to the line AB at B, and similarly for CkB and C& Then the CIRCLES C’ AB? CL,, and C& intersect in the second Brocard points 0’ (Johnson 1929, pp. 264265).
Brocard Midpoint The MIDPOINT of the BROCARD POINTS. It has TRI
ANGLE CENTER FUNCTION QI =
a(b2
+ c”) = sin(A + w),
where w is the BROCARD CARD AXIS.
ANGLE.
It lies on the BRO
References
PEDAL TRIANGLES of s2 and 0’ are congruent, and SIMILAR to the TRIANGLE AABC (Johnson 1929, p. 269). Lengths involving the Brocard points include
The
Kimberling, C. “Central Points and Central Lines in the Math. Mug. 67, 163187, 1994. Plane of a Triangle.”
Brocard
On=OW=R&4sin2w
Points
(1)
s2fl’ = 2Rsinwdl
B
 4sin2 w.
(2)
Brocard’s third point is related to a given TRIANGLE by the TRIANGLE CENTER FUNCTION a=a
I
A’
0 0% 12 I
2
first Brocard point is the interior point s2 (or ~1 or 2,) of a TRIANGLE for which the ANGLES LflAB, LOBC, and LfXA are equal. The second Brocard point is the interior point 0’ (or 72 or 22) for which the ANGLES LR’AC, LO’CB, and LO’BA are equal. The ANGLES in both cases are equal to the BROCARD ANGLE w,
TRIANGLE'S INCENTER. see &~BROCARD ANGLE,BROCARD MIDPOINT,EQUIBROCARD 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. en2. Dublin: Hodges, Figgis, & Co., p. 66,
1893. Johnson,
R. A. Modern Geometry
MA: Houghton
first
two
Brocard points 1929, p. 266).
are ISOGONAL CONJU
Geometry: of the Triangle
An Elementary and the Circle.
Treatise
Boston, Mifflin, pp. 263286, 1929. Kimberling, C. “Central Points and Central Lines in the Plane of a Triangle.” Muth. Mug. 67, 163187, 1994. Stroeker, R. J. “Brocard Points, Circulant Matrices, and Descartes’ Folium.” Math. Mug. 61, 172187, 1988. on the
GATES (Johnson
(3)
(Casey 1893, Kimberling 1994). The third Brocard point fl" (or 73 or 23) is COLLINEAR with the SPIEKER CENTER and the ISOTOMIC CONJUGATE POINT of its
The
The
3
Brocard
176
Brown
‘s Problem
Find
Brothers A PAIR of consecutive numbers. see also PAIR, SMITH BROTHERS, TWINS
see also BR~WNNUMBERS,FACTORIAL,SQUARE
Any continuous POINT, where
Brocard’s
Problem
the values of n for which n! + 1 is a SQUARE NUM(Brocard 1876, BER ?'& where n! is the FACTORIAL 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.
NUM
Brouwer
Fixed
Function
Point Theorem FUNCTION G : Dn + Dn has a FIXED
BER
Heterences
D”
Brocard, 1876. Brocard,
H. Question
166.
Nouv.
Corres.
1Math.
H. Question 1532, Nouv. Ann. Math, 4, 391, 1885. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p. 193, 1994.
Brocard
and c3 and ci, ck, and CL be the CIRCLES intersecting in the BROCARD POINTS s2 and fl’, respectively. Let the two circles cl and ci tangent at Al to Al A2 and AlA3, and passing respectively through AS and AZ, meet again at Cl. The triangle C&&3 is the second Brocard triangle. Each VERTEX of the second Brocard triangle lies on the second BROCARD CIRCLE. cl,
c2,
The two Brocard
triangles
are in perspective
R. A, Modern
Geometry: of the Triangle
Geometry
MA: Houghton
is t he
The inverse
Mifflin,
An Elementary and the Circle.
277281,
pp.
Treatise
Boston,
1929.
Browkin’s
+
l
.
l
+
Gb2
I
1)

see also
est f (s) ds,
2ni s y&m
CONTOURED the COMPLEX PLANE all singularities
of f(s)
CUBE, SCHINZEL'S THEOREM, SQUARE
References Honsberger, R. Mathematical Gems I. Washington, Math. Assoc. Amer., pp. 121125, 1973.
Brown’s Criterion A SEQUENCE {yi) ofnondecreasing is COMPLETE IFF 1. Ul = 1.
DC:
POSITIVEINTEGERS
2. For all k = 2, 3, . . . , Sk1
=
ul
+
v2
+
. . . +
vk1
2
vk

1.
states that a SEQUENCE for which ~1 = 1 (Honsberger 1985). < 2Vk is COMPLETE
see ah
COMPLETE SEQUENCE
References J.
L. Jr. “Notes on Complete Math.
Sequences
Monthly, 557560, 1961. Mathematical Gems III
Honsberger, R. Math. Assoc. Amer.,
pp, 123130,
of Integers.”
Washington,
DC:
1985.
Brown Function For a FRACTAL PROCESS with values ~(t
7+im
1
Theorem
For every POSITIVE INTEGER n, there exists a SQUARE in the plane with exactly n LATTICE POINTS in its interior. This was extended by Schinzel and Kulikowski to all plane figures of a given shape. The generalization of the SQUARE in 2D to the CUBE in 3D was also proved by Browkin.
Brown,
Integral
whereyisavertical chosen so that
Xl2
nB ALL.
unit
and vk+l
of the LAPLA GE TRANSFORM, given by
w
:
References
Amer.
Bromwich
R”
also FIXED POINT THEOREM
A corollary
References on the
f
at AZ.
see also STEINER POINTS, TARRY POINT Johnson,
{x
Milnor, J. W. Topology from the Difierentiable Viewpoint. Princeton, NJ: Princeton University Press, p, 14, 1965.
Triangles
Let the point of intersection of A&l and A&’ be B1, where s1 and fl' are the BROCARD POINTS, and similarly define Bz and B3. BIB2 B3 is the first Brocard triangle, and is inversely similar to A1 Aa As, It is inscribed in the BROCARD CIRCLE drawn with OK as the DIAMETER. The triangles BlAzAs, B2A3Al, and BSAlAz are IS~SCELES TRIANGLES with base angles w, where w is the BROCARD ANGLE. The sum of the areas of the I~OSCELES TRIANGLES is A, the AREA of TRIANGLE A1 AZ As, The first Brocard triangle is in perspective with the given TRIANGLE, with AIBI, AzBz, and A& CONCURRENT. The MEDIAN POINT ofthefirst Brocard triangle is the MEDIAN POINT M of the original triangle. The Brocard triangles are in perspective at M. Let
=
2, 287,
At), the correlation the Brown function
are to the left of
between
y =
At) and y(t+ these two values is given by
22H1  1,
1tJ.
References Arfken,
G. “Inverse
Laplace
Mathematical
Methods
FL: Academic
Press,
for pp.
Transformation.” Physicists,
853861,
1985.
3rd
515.12 in ed. Orlando,
as the BACHELIER FUNCTION, LEVY FUNCTION, or WIENER FUNCTION.
also known
Numbers
Brown
Brown
Brun’s
Numbers numbers
Brown
the condition
(m, n) of INTEGERS
are PAIRS
of BROCARD'S
PROBLEM,
satisfying
i.e., such that
Constant
177
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. l12, 1992. Gordon, D. M. “The Prime Power Conjecture is True Electronic J. Combinatorics 1, for 7z < Z,OOO,OOO.” erg/Volumel/ R6, l7,1994. http: //www combinatorics. volumei. html#RG. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963. l
n! + 1 = rn2 where n! is the FACTORIAL and m2 is a SQUARE NUMBER. Only three such PAIRS of numbers are known: and Erdes conjectured that these (V), (11,5), (To, are the only three such PAIRS. Le Lionnais (1983) points out that there are 3 numbers less than 200,000 for which (n
l)! + 1 = 0 (mod

BruckRyser
THEOREM
Brun’s Constant The number obtained
n”) ,
TWIN
by adding
the reciprocals
of the
PRIMES,
5, 13, and 563.
namely see
Theorem
~~~BRucKRYSERCHOWLA
BROCARD'S
also
PROBLEM,
FACTORIAL,
B(;+$)+($++)+(A+&)+($+&)+,
SQUARE
(1)
NUMBER Heferences Guy,
R.
K.
Unsolved
Problems
in
Number
Theory,
2nd
ed.
p. 193, 1994.
New York: SpringerVerlag, Le Lionnais, F. Les nombres
Paris: Hermann,
remarquables.
p. 56, 1983. Pickover, C. A. Keys p. 170, 1995.
New York:
to Infinity.
B z 1.90216054,
Broyden’s Method An extension of the secant higher dimensions.
method
of root
References Broyden, C. G. “A Class of Methods Simultaneous
W. H. Freeman,
By BRUN'S THEOREM, the constant converges to a definite number as p + 00. Any finite sum underestimates 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 obtained (Ribenboim 1989, p. 146)
Equations.”
Math.
finding
to
assuming the truth of the first HARDYLITTLEWOOD CONJECTURE. Using TWIN PRIMES up to 1014, Nicely (1996) obtained
for Solving Nonlinear 19, 577593,
Comput.
B sz: 1.9021605778 I
1965. Press, W. H.; Flannery, ling, W. T. Numerical Scientific
B. P.; Teukolsky, Recipes 2nd ed.
Computing,
bridge University
in
FORTRAN:
The
Art
England:
of
Cam
1992.
BruckRyserChowla Theorem If n = 1,2 (mod 4), and the SQUAREFREE part of n is divisible 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 lowing form. If a symmetric (v&X)BLOCK exists, then 1. Ifvis
then Ic
EVEN,
X is a SQUARE
2. If 21 is ODD, the the DIOPHANTINE x2
has a solution see UZSO BLOCK EQUALITY
z
(k

X)y”
+
in integers, DESIGN,
in the folDESIGN
NUMBER,
EQUATION
(3)
not all of which BLOCK
are 0.
DESIGN
(Cipra 1995, 1996), in the process Intel’s@ Pent iumTM microprocessor. Le Lionnais (1983) is incorrect. see
TWIN PRIMES, PRIMES CONSTANT
IN
PRIME
CONJECTURE,
References Ball, W. W. R. and Coxeter,
H. S. M. Mathematical RecreNew York: Dover, p. 64, 1987. Brent, R. P. “Tables Concerning Irregularities in the Distribution of Primes and Twin Primes Up to lo?” Math. ations
Comput.
and
Essays,
30,
379,
13th
ed.
1976.
Cipra, B. “How Number Theory Got the Best of the Pentium Chip.” Science 287, 175, 1995. Cipra, B. “Divide and Conquer.” What’s Happening in the Mathematical Sciences, 1995l 996, Vol. 3. Providence, RI: Amer. Math. Sot,, pp. 3847, 1996. Finch, S. “Favorite Mathematical Constants.” http: //www mathsoft.com/asolve/constant/brun/brun.html~ Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983. Nicely, T. “Enumeration to 1014 of the Twin Primes and Brun’s Constant .” Virginia J. Sci. 48, 195204, 1996. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: SpringerVerlag, 1989. Shanks, D. and Wrench, J. W. “‘Brun’s Constant.” Math. Comput.
References Dinitz, J. H. and Stinson, D. R. “A Brief Introduction Design Theory." Ch. 1 in Contemporary Design Theory:
TWIN
UZSO
TWIN
discovering a bug in The value given by
l
(l)(“1)/2xz2
FISHER'S
zt 2.1 x lo’
S. A.; and Vetter
Cambridge,
pp. 382385,
Press,
(2)
to A
28,
293299,
Wolf, M. “Generalized uni. wroc .pl/mwoff
1974.
Brun’s /.
Constants.”
http : //www
, if t .
BrunnMinkowski
178
Inequality
Buffon’s
BrunnMinkowski Inequality The nth root of the CONTENT of the set sum of two sets in Euclidean nspace is greater than or equal to the sum of the nth roots of the CONTENTS of the individual sets. see also TOMOGRAPHY
Buchberger’s Algorithm The algorithm for the construction from an arbitrary ideal basis. see also GR~BNER
of a GR~BNER
BASIS
BASIS
References Becker,
References Cover, T. M. “The Entropy Power Inequality and the BrunnMinkowski Inequality” $5.10 in In Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: SpringerVerlag, p. 172, 1987. Schneider, R. Convex Bodies: The BrunnMinkowski Theory. Cambridge, England: Cambridge University Press, 1993.
T. and Weispfenning, V. Gr6bner Bases: A ComApproach to Commutative Algebra. New York: SpringerVerlag, pp. 213214, 1993. Buchberger, B. “Theoretical Basis for the Reduction of Polynomials to Canonical Forms.” SIGSAM Bull. 39, 1924, Aug. 1976. Cox, D.; Little, J.; and O’Shea, D. Ideals, Varieties, and putational
Algorithms: Commutative
Verlag,
Brun’s
Needle Problem
An
Introduction Algebra, 2nd
to ed.
Algebraic
Geometry
New
York:
and
Springer
1996.
Sum
see BRuN'S
CONSTANT
Buckminster
Fuller
see GEODESIC 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.
Dome
DOME
BuffonLaplace
Needle
Problem
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 aho BORROMEAN
RINGS
References Rolfsen, D. Knots and Perish Press, 1976.
Brute
Force
see DIRECT
Links.
Wilmington,
DE: Publish
or
Factorization SEARCH
FACTORIZATION
Find the probability P(!, a, b) that a will land on a line, given a floor with spaced PARALLEL LINES distances a l > a, b. 2C(a + b) P(t, a, b) = nab
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 multiple bubbles are joined together. Two outstanding problems involving bubbles are to find the arrangements with the smallest PERIMETER (planar problem) or SURFACE AREA (AREA problem) which enclose and separate r~ given unit areas or volumes in the plane or in space. BUBFor n = 2, the problems are called the DOUBLE BLE CONJECTURE and the solution to both problems is knownto be the DOUBLE BUBBLE. see also PLATEAU
‘S
DOUBLE LAWS,P
BUBBLE
LIATEAU'S
MINIMA PROBLEM
f
L
SURFACE,
References Morgan,
Look
F. “Mathematicians, at Soap Bubbles.” Amer.
Including Math.
Undergraduates, Monthly
101, 343
351,1994. Pappas,
T. “Mathematics & Soap Bubbles.” San Carlos, CA: Wide World p. 219, 1989. Mathematics.
The
Joy
Publ./Tetra,
of
see UZSO BUFFON’S
Buffon’s
Needle
NEEDLE
Problem
PROBLEM
needle of length G a grid of equally and b apart, with t2 l
I3ulirschStoer
Burau Representation
Algorithm
Find the probability P(t, d) that a needle of length e will land on a line, given a floor with equally spaced PARALLEL LINES a distance d apart.
P(t,
d) =
s
2T tl cos81 d0 27T 0 =a4d
42
G
The
cos 8 d0
CURVATURE
s0
see
BUFFONLAPLACE NEEDLE PROBLEM
UZSO
References Badger, L. “Lazzarini’s Lucky Approximation of TV” AI&h. Mug. 67, 8391, 1994. Dijrrie, H. "Buffon's Needle Problem.” $18 in 100 Great Problems Solutions.
of Elementary New York:
Mbthematics:
Their
History
and
Dover, pp. 7377, 1965. Kraitchik, M. “The Needle Problem.” 56.14 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942. Wegert, E. and Trefethen, L. N. “FTom the Buffon Needle Problem to the Kreiss Matrix Theorem.” Amer. Math. Monthly 101, 132139, 1994.
BulirschStow Algorithm An algorithm which finds RATIONAL FUNCTION extrapolations of the form
&(i+l)...(i+m)
=
pcL(x) pi/ (4
ENTIAL
+.
l
l
+q,xv
EQ UATIONS
Bulirsch,
R. and Stoer, J. 52.2 in Introduction to Numerical New York: SpringerVerlag, 1991. Press, W. H.; Flannery, B. P,; Teukolsky, S. A.; and Vetterling, We T. “Richardson Extrapolation and the BulirschStoer Method.” 516.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 718725, 1992. Analysis.
(5) References Lawrence, 3. D. A Catalog of Special York: Dover, pp. 127129, 1972.
Skiena, torics
S. Implementing and
Discrete Theory with
Graph
Burau Gives a in terms appears
0
0
0
0
In parametric
form,
equation bi =
b2 =
la
(1)
Reading,
0 0 0 0 0
l
. l
.
.
0
**
0
0
1
* 1 .
0
1
l
0
.
1
y2
Combina
Representation MATRIX representation bi of a BRAID GROUP of (n  1) x (n  1) MATRICES. A t always in the (i, i) position.
.
x2
Mathematics: Mathematics.
see FIBER BUNDLE
b 1=
a2 
New
Bundle
1
implicit
Curves.
1990.
Nose
curve with
Plane
Bumping Algorithm Given a PERMUTATION {pl,p2,...,p,} 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 bumping algorithm, start with {{PI}}, which is a YOUNG TABLEAU. If p1 through pk have already been inserted, then in order to insert pk+l, 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+l. If there is no such element, append pk+l to the first line and stop. If there is such an element (say, pP), exchange p, for pk+l, search the second line using p,, and so on.
t
A plane
is
ANGLE
MA: AddisonWesley,
References
Bullet
(4)
References
of ORDINARY DIFFER
solution
and can be used in
+q1rr:
(3)
see also YOUNG TABLEAU
po+pla:+...+p,xP = qo
(2)
y = bcott.
3ab cot t csc t (b2 csc4 t + a2 sin2 t)3/2
and the TANGENTIAL
l
x = acost is
K=
Several attempts have been made to experimentally determine 7r by needletossing. For a discussion of the relevant statistics and a critical analysis of one of the more accurate (and least believable) needletossings, see Badger (1994)
179
+*
. l
. .
1
l
a.. ‘a
0
0
l
t
0 0
.
1
1
0 0
0 0
.
.
l
l
.
.
l
l
l
l
(1)
0
.
.
1
l
I
l
l
(2)
180
Burkhardt
b
n1
=
Burnside
Quartic
.
.
;
;
l ‘.
l
m
;
:
l (3)
I Let Q be the MATRIX
0
0
a
0
t
0
0
*a
0
t
of BRAID
PRODTJCT
det(I  q) 1 + t + * * * + t1 where AL is the ALEXANDER the DETERMINANT.
=
WORDS,
(4)
AL,
POLYNOMIAL
then
and det is
References Burau, W. ‘%ber Zopfgruppen und gleichsinnig verdrilte Verkettungen.” Abh. Math. Sem. Hunischen Univ. 11, 171178, 1936e Jones, V. “Hecke Algebra Representation of Braid Groups and Link Polynomials.” Ann. Math. 126, 335388, 1987.
Burkhardt Quartic The VARIETY which is an invariant is given by the equation
of degree
four and
References Burkhardt, H. “Untersuchungen liptischen Modulfunctionen.
aus dem Gebiet der hyperelII.” Math. Ann. 38, 161224,
1890,
Burkhardt, H. “Untersuchungen aus dem Gebiet der hyperelliptischen Modulfunct ionen. III.” Math. Ann. 40, 313343, 1892. Hunt,, B. “The Burkhardt Quartic.” Ch. 5 in The Geometry of Some Special Arithmetic Quotients. New York: SpringerVerlag, pp. 168221, 1996.
Burnside’s Conjecture Every nonABELIAN SIMPLE see
also
ABELIAN
GROUP,
GROUP SIMPLE
has EVEN
ORDER.
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 of R( J) as the equivalence classes under II:  y, which is true if there is some permutation p in R(J) such that p(z) = y. Define the fixed points of p as the elements 5 of X for which p(z) = z. 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 It was subsequently extended and (1900) rediscovery. refined by Pblya (1937) for applications in COMBINATORIAL counting problems. In this form, it is known as P~LYA
ENUMERATION
Burnside Problem A problem originating with W. Burnside (1902), who wrote, CtA still undecided point in the theory of discontinuous groups is whether the ORDER of a GROUP may be not finite, while the order of every operation This question would now be it contains is finite.” phrased as “Can a finitely generated group be infinite while every element in the group has finite order?” (VaughanLee 1990). This question was answered by Golod (1964) when he constructed finitely generated infinite ~GROUPS. These GROUPS, however, do not have a finite exponent. Let FT be the FREE GROUP of RANK T and let N be SUBGROUP generated by the set of nth POWERS subgroup of FT. We ($19 E FT}. Th en Iv is a normal define B(T, n) = FT/N to be the QUOTIENT GROUP. We call B(r, n) the rgenerator Burnside group of exponent n. It is the largest rgenerator group of exponent n, in the sense that every other such group is a HOMEOMORPHIC image of B(T, n)* The Burnside problem is usually stated as: “For which values of T and n is B(r, n) a the
FINITE
d  YO(d+ 3; + y; + y!) + 3yly2y3y4 = 0.
Problem
GROUP?”
An answer is known for the following values. For T = 1, B&n) is a CYCLIC GROUP of ORDER n. For n = 2, B(r, 2) is an elementary ABELTAN 2group of ORDER 2Y For n = 3, B(T, 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 3a where
(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, IB(2,4)1
= 212
(2)
IB(3,4)1
= 26g
(3)
IB(4,4)1
= 2422
(4)
IB(5,4)1
= 22728,
(5)
while for larger values of T the exact value is not yet known. For n = 6, B(T, 6) was proved to be finite by Hall (1958) with ORDER 2”3’, where
THEOREM.
(8)
References P6lya, G. “Kombinatorische Anzahlbestimmungen pen, Graphen, und chemische Verbindungen.” 68, 145254, 1937.
fiir GrupActa
Math.
No other the other
Burnside groups are known to be finite. On hand, for T > 2 and n 2 665, with n ODD,
BusemannPetty
Butterfly
Problem
B(T, n) is infinite (Novikov and Adjan 1968). There similar fact for T > 2 and n a large POWER of 2. E. Zelmanov his solution
is a
Butterfly
181
fiactal
Catastrophe
was awarded a FIELDS MEDAL in 1994 for of the “restricted” Burnside problem.
see also FREE GROUP References Burnside, W. “On an Unsettled Question in the Theory of Discontinuous Groups.” Quart. J. Pure Appl. Math. 33, 230238, 1902. Golod, E. S. “On NilAlgebras and Residually Finite p Isu. Akad. Nauk SSSR Ser. Mat. 28, 273276, Groups.” 1964. Hall, M. “Sofution of the Burnside Problem for Exponent Six.” Ill. J. Math. 2, 764786, 1958. Levi, F. and van der Waerden, B. L. “uber eine besondere Klasse von Gruppen.” Abh. Math. Sem. Univ. Hamburg 9, 154158, 1933, Novikov, P, S. and Adjan, S. I. “Infinite Periodic Groups I, II, III.” Izv. Akad. Nauk SSSR Ser. Mat. 32, 212244, 251524, and 709731, 1968. Sanov, I. N. “Solution of Burnside’s problem for exponent State Univ. Ann. Math. Ser. 10, 166four .” Leningrad 170, 1940. VaughanLee, M. The Restricted Burnside Problem, 2nd ed. New York: Clarendon Press, 1993.
A CATASTROPHE which can occur for four control tors and one behavior axis. The equations
fac
x = c(8at3 + 24t5) Y
display
= c(6at2
such a catastrophe
 15t4)
(von Seggern
1993).
References von Seggern, D. CRC Standard Curves Raton, FL: CRC Press, p. 94, 1993.
Butterfly
and
Boca
Surfaces.
Curve
BusemannPetty Problem If the section function of a centered convex body in Euclidean nspace (n 2 3) is smaller than that of another such body, is its volume also smaller? A PLANE
References Gardner, R. J. ‘&Geometric Tomography.” Sm. 42, 422429, 1995.
Not.
Amer.
CURVE given by the implicit
Math.
equation
y6 = (x2  x6)*
see also DUMBBELL CURVE, EIGHT CURVE, PIRIFORM Busy Beaver A busy beaver is an nstate, 2symbol, 5tuple TURING MACHINE which writes the maximum possible number BB(n) of Is on an initially blank tape before halting. For n = 0, 1, 2, . . , , BB(n) is given by 0, 1, 4, 6, 13, 2 4098, 2 136612, . . . The busy beaver sequence is also known as RADO’S SIGMA FUNCTION. l
see also
HALTING
PROBLEM,
TURING
MACHINE
References Chaitin, G. 3. “Computing the Busy Beaver Function.” 54.4 Problems in Communication and Computation in Open (Ed. T. M. Cover and B. Gopinath). New York: SpringerVerlag, pp. 108112, 1987. Dewdney, A. K. “A Computer Trap for the Busy Beaver, the HardestWorking Turing Machine.” Sci. Amer. 251, 1923, Aug. 1984. Marxen, H. and Buntrock, J. “Attacking the Busy Beaver 5.” Bull. EATCS 40, 247251, Feb. 1990. Sloane, N. J. A. Sequence A028444 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
References Cundy, H. and Rollett, Stradbroke, England:
A. Mathematical Models, Tarquin Pub., p. 72, 1989.
3rd
ed.
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
Butterfly
CHAOS,
LORENZ
SYSTEM
Fkactal
The FRACTALlike f&Y)
curve generated =
by the 2D function
( x2  y”)sin (+) x2 + y2
*
182
Butterfly
Butterfly
Polyiamond
Butterfly
Polyiamond
A 6POLYIAMOND. kkkrences Golomb, and
Puzzles, Patterns, S. W. Polyominoes: Zni ed. Princeton, NJ: Princeton p. 92, 1994.
Packings,
Press,
Butterfly
Problems,
University
Theorem
Given
a CHORD PQ of a CIRCLE, draw any other two AB and CD passing through its MIDPOINT. Call the points where AD and BC meet PQ X and Y. Then iW is the MIDPOINT ofXY. CHORDS
see also
CHORD,
CIRCLE,
MIDPOINT
References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited, Washington, DC: Math. Assoc. Amer., pp. 4546, 1967.
Theorem
c
Cake Cutting
C
183
CTable see CDETERMINANT
of COMPLEX NUMBERS, denoted
The FIELD see also c*,
Cable
c.
NUMBER, I, N, Q, II& Z
COMPLEX
Knot
Let K1 be a TORUS KNOT. Then the SATELLITE KNOT with COMPANION KNOT Kz is a cable knot on Kz*
c*
Reterences
The RIEMANN SPHERE c
see also c, SPHERE,~
Adams,
U(W).
COMPLEX NUMBER, (& R, RIEMANN
C. C. The Knot Book: to the Mathematical Theory
An Elementary of Knots. New
Freeman, p. 118, 1994, Rolfsen, D. Knots and Links. Wilmington, Perish Press, pp. 112 and 283, 1976.
Introduction
York:
W. H.
DE: Publish
or
C*Algebra type of B*ALGEBRA in which the INVOLUTION is the ADJ~INT OPERATOR in a HILBERT SPACE. see also B*ALGEBRA, ~THEORY
A special
Cactus
Fractal
References Davidson, K. R C* Algebras Amer. Math. ‘sot ., 1996.
Providence,
bY Example.
RI:
A MANDELBROT SETlike FRACTAL obtained ing the map
CCurve see LEVY FRACTAL
&x+1
CDeterminant A DETERMINANT
see also FRACTALJULIA appearing
in
Cake ars+1 r/s
Gs+2 l
.
l
.
.
l
l
a,
l
. .
=
l .
a,+1
a,
see also
’
l
mm
.
.
G+s1
PADS APPROXIMANT
CMatrix T
Any SYMMETRIC MATRIX (A = A)or SKEW SYMMETRIC MATRIX (AT = A) C, with diagonal elements 0 and others &l satisfying
cc
T
= (n  l)l,
where I is the IDENTITY matrix
(Ball
and Coxeter
MATRIX,
is known as a CExamples include
1987).
(zo

I)&

zo.
SET,MANDELBROT
SET
PADS APPROXIMANT
identities:
c
Gb 3 +
=
byiterat
Cutting
divide a cake among n It is always possible to “fairly” people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes that everyone has received l/n of the cake according 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 participant believes that he has obtained more than l/n of to his own measure. the cake according Ignoring the height of the cake, the cakecutting problem is really a question of fairly dividing a CIRCLE into n equal AREA pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the FROBENIUSK~NIG THEOREM.
see UZSO CIRCLE CUTTING, CYLINDER CUTTING, ENVYFREE, FROBENIUSK~NIG THEOREM, HAM SANDWICH THEOREM, PANCAKE THEOREM, PIZZA THEOREM, SQUARE CUTTING,TORUS CUTTING References Brams,
S. J. and Taylor,
A. D. “An EnvyFree Cake Division Monthly 102, 919, 1995. Brams, S. J. and Taylor, A. D. Fair Division: From CakeCutting to Dispute Resolution. New York: Cambridge University Press, 1996. Dubbins, L. and Spanier, E. “How to Cut a Cake Fairly.” Amer. Math. Monthly 68, l17, 1961. Gale, D. “Dividing a Cake.” Math. Intel. 15, 50, 1993. Jones, M. L+ “A Note on a Cake Cutting Algorithm of Banach and Knaster.” Amer. Math. Monthly 104, 353355, 1997. Rebman, K. “How to Get (At Least) a Fair Share of the Cake.” In Mathematical Plums (Ed, R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 2237, 1979.
Protocol.”
++o++
C6= +
+

+
0
+

+


+
0
+
+


+
0
References Ball, W. W. R. and Coxeter, ations
and
309, 1987.
Essays,
23th
H. S. M. Mathematical ed.
New
York:
Dover,
Recre
pp. 30%
Amer.
Math.
184
Calculus
Cal “Sur la division 315319, 1949. , W. “How to Cut 87, 640644, 1980.
Steinhaus, (Supp.) Stromquist Monthly
progmatique.”
H.
of Variations
and INTEGRALS
Ekonometrika
17,
a Cake
Fairly.”
Amer.
Math.
respectively. Cal see WALSH Calabi’s
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 WeierstraQ.
FUNCTION Triangle
AREA, CAL CULUS OF VARIsee also ARC LENGTH, THEOREM, DEATIONS, CHANGE OF VARIABLES RIVATIVE, DIFFERENTIAL CALCULUS, ELLIPSOIDAL CALCULUS, EXTENSIONS CALCULUS, FLUENT, FLUXION,FRACTIONAL CALCULUS,FUNCTIONAL CALCULUS, FUNDAMENTAL THEOREMS OF CALCULUS, HEAWSIDE Calculus, INTEGRAL, INTEGRAL CfmcuLus, JAcoBIAN, LAMBDA CALCULUS, KIRBY CALCULUS, MALLIAWN C ALCULUS, P REDICATE CALC ULUS, PROP0 SISLOPE, TENSOR CALCU LUS, u MTION AL CALCULUS BRAL CALCULUSJOLUME
Calabi’s Triangle
Equilateral Triangle
The one TRIANGLE in addition to the EQUILATERAL TRIANGLE for which the largest inscribed SQUARE The racan be inscribed in three different ways. tio of the sides to that of the base is given by =1: = 1.55138752455.. . (Sloane’s A046095), where
1 X=
(23 + 3idm)1'3 3 p/3
3+
+
is the largest
POSITIVE
ROOT
2x3 2x2 which
has CONTINUED
11
3[2(23
l
also
GRAHAM'S
3x+2
FRACTION BIGGEST
Anton, H. Calculus with Analytic Geometry, 5th ed. New York: Wiley, 1995. Apostol, T. M. Calculus, 2nd ed., Vol. 1: One Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, 1967. Apostol, T. M. Calculus, 2nd ed., Vol. 2: MultiVariable Calculus and Linear Algebra, with Applications to Differential Equations and Probability. Waltham, MA: Blaisdell, 1969. Apostol, T. M. A Century of Calculus, 2 ~01s. Pt. 1: 18941968. Pt. 2: 19691991. Washington, DC: Math. Assoc. Amer., 1992. Ayres, F. Jr. and Mendelson, E. Schaum’s Outline of Theory and Problems of Differential and Integral Calculus, 3rd ed. New York: McGrawHill, 1990. Borden, R. S. A Course in Advanced Calculus. New York: Dover, 1998. Boyer, C. B, A History of the Calculus and Its Conceptual Development. New York: Dover, 1989. Brown, K. S. “Calculus and Differential Equations.” http: //
of
5, 2, 1, 3, 1, 1, 390, . . . ] (Sloane’s see
References
+ 3id%7)]1/3
= 0, [l, 1, 1, 4, 2, A046096).
LITTLE
1,
2, 1,
HEXAGON
References Conway, J. H. and Guy, R. K. “Calabi’s Triangle.” In The Book of Numbers. New York: SpringerVerlag, p. 206, 1996. Sloane, N. J. A. Sequences A046095 and A046096 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
CalabiYau Space A structure into which string theory curl up. Calculus In general, in a purely
www.seanet.com/ksbrown/icalculu.htm.
the 6 extra
DIMENSIONS
“a” calculus is an abstract formal way.
theory
of 10D
developed
“The” calculus, more properly called ANALYSIS (or REAL ANALYSIS or, in older literature, INFINITESIMAL ANALYSIS) is the branch of mathematics studying the rate df 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 d
dz
f( x >
’
Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 1. New York: SpringerVerlag, 1990. Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 2. New York: SpringerVerlag, 1990. Hahn, A. Basic Calculus: From Archimedes to Newton to Its Role in Science. New York: SpringerVerlag, 1998. Reading, MA: Kaplan, W. Advanced Calculus, 4 th ed. AddisonWesley, 1992. Marsden, J. E. and Tromba, A. J. Vector Calculus, 4th ed. New York: W. H. Freeman, 1996. Strang, G. Calculus. Wellesley, MA : WellesleyCambridge Press, 1991.
Calculus of Variations A branch of mathematics which is a sort of generalization of CALCULUS. Calculus of variations seeks to find the path, curve, surface, etc., for which a given FUNCTION hasa STATIONARYVALUE (which,inphysical
Cancellation
Calcus problems, matically, integrals
is usually a MINIMUM or MAXIMUM). Mathethis involves finding STATIONARY VALUES of of the form a I
185
Formula
f(x) f(Y, !A 4 dx
sb
Calderh’s
Law
= c1,
(1)
O” (f, Qapb) T,!J~‘~(x)u~ O” s m s m
da db,
where I has an extremum only if the EULERLAGRANGE FERENTIAL EQUATION is satisfied, Le., if
DIF
(2) The FUNDAMENTAL TIONS states that,
LEMMA
OF CALCULUS
OF VARIA
if
This result was originally ANALYSIS, but also follows Caliban
Apuzzle inferred
M(x)h(x)
dx = 0
(3)
derived using HARMONIC from a WAVELETS viewpoint.
Puzzle in LOGIC in which one or more facts must from a set of given facts.
Calvary
be
Cross
J a
for all h(x) with TIVES, then
CONTINUOUS M(x)
second PARTIAL
DERIVA
(4)
= 0
on (a, b).
see also BELTRAMI IDENTITY, BOLZA PROBLEM, PROBLEM, CATENARY, ENVEBRACHISTOCHRONE LOPE THEOREM, EULERLAGRANGE DIFFERENTIAL EQUATION, ISOPERIMETRIC PROBLEM, ISOVOLUME PROBLEM, LINDELOF'S THEOREM, PLATEAU'S PROBLEM, POINTPOINT DISTANCE~D, POINTPOINT DISTANCE~D, ROULETTE, SKEW QUADRILATERAL, SPHERE WITH TUNNEL, UNDULOID, WEIERSTRA% ERDMAN CORNER CONDITION
References A&en, G. “‘Cdculus Methods
for
Cameron’s SumFree Set Constant A set of POSITIVE INTEGERS S is sumfree if the equation x + y = x has no solutions x, y, z f S. The probability that a random sumfree set S consists entirely of ODD INTEGERS satisfies 0.21759
of Variations.”
Physicists,
3rd
ed,
< c < 0.21862.
Ch. 17 in 2MathematicaZ Orlando,
Press, pp. 925962, 1985. Bliss, G. A. Culculus of Variations.
court,
see also CROSS
FL: Academic
References Chicago,
IL:
Open
1925.
Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Fox, C. An Introduction to the CalcuZus of Variations. New York: Dover, 1988. lsenberg, 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. 886890, 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.
Cameron, Graph
P. J. “Cyclic Automorphisms of a Countable and Random SumFree Sets.” Gruphs and Combinatorics 1, 129135, 1985. Cameron, P. J. “Portrait of a Typical SumF’ree Set.” In Surveys in Combinatorics 1987 (Ed. C. Whitehead). New York: Cambridge University Press, 1342, 1987. Finch, S. “Favorite Mathematical Constants.” http: //www. mathsoft.com/asolve/constant/cameron/cameron*html.
Cancellation
see ANOMALOUS CANCELLATION Cancellation Law If bc E bd ( mod a) and (6, a) = 1 (i.e., a and b are RELATIVELY PRIME), then CE d (mod a). see UZSO CONGRUENCE
Calcus
References Courant, ementary
1 calcus S A. see also FRACTION
HALF,
QUARTER,
SCRUPLE,
UNCIA,
UNIT
R. and Robbins, Approach
H, What
to Ideas
is Mathematics?: An Eland Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 36, 1996. Shanks, D. Solved and Unsolved Problems in Number 4th ed. New York: Chelsea, p. 56, 1993.
Theory,
186
Cannonball
Cantor
Problem
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 corresponds to solving the DIOPHANTINE EQUATION k ,2 2 
>: i=l
;k(l+
Ic)(l+
2K) = N2
for some pyramid height k. The only solution N = 70, corresponding to 4900 cannonballs Coxeter 1987, Dickson 1952), as conjectured (1875, 1876) and proved by Watson (1918).
is k = 24, (Ball and by Lucas
SPHERE PACKING, SQUARE NUMB ER,SQUARE NUMBER PY RAMI D,SQUA RE PYRAMIDAL see UZSU
References Ball, W. W. R. and Coxeter, ations
and Essays,
Dickson,
L. E. History
Diophantine
Lucas,
H. S. M. Mathematical Recreed. New York: Dover, p. 59, 1987. of the Theory of Numbers, Vol. 2:
13th
New York: 1180. Nouvelles
Analysis.
I& Question
Chelsea, Ann.
p. 25, 1952.
Math,
Ser.
Ser.
de Question
1180. Nouvelles
Ann.
Math.
2 15, 429432,1876.
J. T. Excursions in Number C. S. and Anderson, New York: Dover, pp. 77 and 152, 1988. The Joy of MathPappas, T. “Cannon Balls & Pyramids.” ematics. San Carlos, CA: Wide World Publ./Tetra, p. 93, 1989. Watson, G. N. “The Problem of the Square Pyramid.” 1Mes
Ogilvy,
Theory.
48, l22,
Math.
Cantor Diagonal Slash A clever and rather abstract technique used by Georg Cantor to show that the INTECER~ and REALS cannot be put into a ONETOONE correspondence (i.e., the INFINITY 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 arranging 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 , CONTIN UUM HYPOTHESIS , DENUMERABLE SET Courant,
R. and Robbins, Approach
every object
in a class in
NORMAL FORM, ONETOONE
References Petkovgek, M.; Wilf, H. S.; and Zeilberger, ley, MA: A. K. Peters, p. 7, 1996.
D. A=B.
Welles
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 combinatorial type of (GENUS zero) polyhedron contains just one canonical version. The ARCHIMEDEAN SOLIDS and their DUALS are all canonical. References polyhedra database.” Conway, J. H. “Re: geometry. f Orurn newsgroup, Aug. 31, 1995.
Cantor
Dust
Posting
t0
A FRACTAL which can be constructed using STRING REWRITING by creating a matrix three times the size of the current matrix using the rules line 1: II*ll,>II* *II ,I1 it,>II II II, II lf,>ll II line 2: II*ll,>II rl*Il>ll* *II, tl fl,>ll II line 3: Let Nn be the number of black boxes, L, the length of a side of a white box, and A, the fractional AREA of black boxes after the nth iteration.
Nn c 5”
Transformat
(2)
= (g)“m
(3)
The CAPACITY DIMENSION is therefore hNn in

Box
SIEVE
ln( sn)
= n
see also
'
= 3”
An = Ln2Nn
ion
Comb
~~~CANTOR SET
(1)
L, c (i)”
see SYMPLECTIC DIFFEOMORPHISM Cantor
is Mathematics?: An Eland Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 8183, 1996. Penrose, R. The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp* 8485, 1989.
d cap =  Jnm Canonical
H. What
to Ideas
1918,
Canonical Form A clearcut way of describing a ONETOONE manner.
see also
see also CARDINAL NUMBER, CONTINUUM HYPOTHESIS, DEDEKIND CUT
ementary
I% Solution
senger.
CantorDedekind Axiom The points on a line can be put into a ONETOONE correspondence with the REAL NUMBERS.
References
2 14,
336,1875. Lucas,
Dust
In 5 = 1.464973521.. In 3
7z5zx3
ln(3“)
..
(4
FRACTAL, SIERPI~SKI CARPET, SIERPI~~
Cantor Square Fractal
Cantor’s Equatim References Dickau, R. M. “Cantor Dust.” http://forum. swarthmore . edu/advanced/robertd/cantor.html. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103104, 1993. @ Weisstein, E. W. “F’ractals.” http: //www, astro .virginia. edu/euw6n/math/notebooks/Fractal .m.
Cantor’s
Equation
Cantor Set The Cantor set (T,) is g iven by taking the interval [OJ] (set To), removing the middle third (Tl), removing the middle third of each of the two remaining pieces (Tz), and continuing this procedure ad infinitum. It is therefore the set of points in the INTERVAL [OJ] whose ternary expansions do not contain I, illustrated below. 
WE = tz,
I
me
where w is an ORDINAL NUMBER and E is an INACCESSIBLE CARDINAL. ORDINAL
NUMBER
References Conway, York:
J. H. and Guy, R. K. The Book SpringerVerlag, p. 274, 1996,
New

X=
(2) such that
:+...+g+...,
(1)
where C~ may equal 0 or 2 for each n. This is an infinite, PERFECT SET. Thetotallengthofthe LINE SEGMENTS in the nth iteration is
c,= 0g y
Cantor Function The function whose values
(2)
are
1 i...+~+z" Cl 5 (2 for any number
of Numbers.

WI
the SET of REAL NUMBERS
This produces
see also INACCESSIBLE CARDINAL,
187
and the number of LINE SEGMENTS is IVn = 2”, so the length of each element is
2 >
e en = = N
between
In 03
(3)
and the CAPACITY DIMENSION is d cap
and b&L+
3
w
.
l
+c”‘+L
Chalice (1991) shows that any realvalues function F(s) on [0, l] which is MONOTONE INCREASING and satisfies
see
UZSO
STAIRCASE
SET, DEVIL’S
References Chalice, Amer.
Math.
Monthly
98,
of the Cantor
255258,
Function.”
1991.
and “Complex in Action.
New
Cantor York:
143149, 1991.
Cantor’s Paradox The SET of all SETS is its own POWER SET. Therefore, the CARDINALITY of the SET of all SETS must be bigger than itself. also
he
Cantor
set is nowhere
..
DENSE,
so
(4
it has LEBESGUE
0.
l
D. R. “A Characterization
Wagon, S. ‘(The Cantor Function” Sets.” $4.2 and 5+1 in Mafhematica W. H. Freeman, pp. 102108 and
see
HO+
nIn2 n
lim
nm
A general Cantor set is a CLOSED SET consisting entirely of BOUNDARY POINTS. Such sets are UNCOUNTABLE and may have 0 or POSITIVE LEBESGUE MEASURE. The Cantor set is the only totally disconnected, perfect, COMPACT METRIC SPACE up to a HOMEOMORPHISM (Willard 19’70)
function.
CANTOR
The
MEASURE
1. F(0) = 0, 2. F(x/3) = F(x)/2, 3. F(l2) = lF(z) is the Cantor
In N = 
lim
In 2 = 0.630929.. In 3
3"'
3m1
= 
CANTOR'S
THEOREM,
POWER
see
ALEXANDER'S HORNED SPHERE, NECKLACE, CANTOR FUNCTION also
References Boas, R. P. Jr.
A Primer
of Real
Functions.
DC: Amer, Math. Sm., 1996. Lauwerier, H. Fractals: Endlessly Repeated ures. Princetqn, NJ: Princeton University 20, 1991. Willard, S. $30.4 in General Topology. AddisonWesley, 1970.
SET Cantor
Square
Fkactal
ANTOINE'S
Washington, Geometric
Press, Reading,
Fig
pp. 15MA:
188
Cantor’s
Theorem
Cardano
A FRACTAL which can be constructed using STRING REWRITING by creating a matrix three times the size of the current matrix using the rules line
1:
tt*tl,>il***ll,
line
2;
tl*fl,>ll*
II *tt
line 3: tl*lt,>tt***tt The first few steps are illustrated The size of the unit
element L,=
and
the number
Il,>lt
II
,tI
1i,>l
II
, If
1t,>rt
If
above.
after the nth iteration
0
is
1 n 3
of elements
is given
Capacity Dimension A DIMENSION also
called the FRACTAL DIMENSION, HAWSDORFF DIMENSION, and HAUSDORFFBESICOVITCH DIMENSION in which nonintegral values
are permitted. Objects whose capacity dimension is different from their TOPOLOGICAL DIMENSION are called FRACTALS. The capacity dimension of a compact METRIC SPACE X is a REAL NUMBER &apacity such that if n(e) denotes the minimum number of open sets of diameter less than or equal to E, then n(c) is proportional to eD as c + 0. Explicitly,
by the RECUR
d capacity
=

lim
In N
HO+
RENCE RELATION Nn = 4N,1
+ 5(9”)
where Nl = 5, and the first few numbers of elements 5, 65, 665, 6305, . . Expanding out gives l
are
l
Nn = 5 2
4nk9k1
z 9”  4”.
k Formula
lnc
(if the limit exists), where N is the number of elements forming a finite COVER of the relevant METRIC SPACE and c 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 element of a FRACTAL is equally likely to be visited, then d capacity is the INFOR= dinformation, w here dinformation MATION DIMENSION. The capacity dimension satisfies
k=O
The CAPACITY D=
DIMENSION
lim
nkm

Since
In Nn xhl L,
ln(gn) ,Jym ln(3“)
the DIMENSION SQUARE is completely not a true FRACTAL.
d correlation
is therefore
where d correlation is conj ectured
lim
ln(9”  4”) lIl(3“) 2 In 3 In 9 = E = In3 = 2* n+m
H. Fructals: Endlessly Princeton, NJ: Princeton
DUST
ures. 83, 199L
Geometric
University
Press,
$@ Weisstein, E. W. “Fractals.” http : //www . astro edu/eww6n/math/notebooks/Fractal.m.
Fig
ppm 82
CANTOR'S PARADOX
Cap
see CROSSCAP, SPHERICAL CAP Capacity
see TRANSFINITE DIAMETER
Dynamics: Methods.
Analytical, New York:
H.O. Images
Wiley, and Richter,
of Complex
to Real
Analysis.
and
Nonlinear Experimental
pp, 538541, 1995. D, H. The Beauty
Dynamical
SpringerVerlag, 1986. Wheeden, R. L, and Zygmund, Introduction
B. Applied
Computational,
Systems.
of FracNew York:
A. Measure and Integral: An New York: M. Dekker,
1977.
. Virginia.
Cantor’s Theorem The CARDINAL NUMBER of any set is lower than CARDINAL NUMBER of the set of all its subsets. COROLLARY is that there is no highest N (ALEPH).
see also
dcapacity
is the CORRELATION DIM ENSION, and to be equal to the LYAPUNOV DIMENSION.
References A. H. and Balachandran,
tals:
Repeated
<
Nayfeh,
Peitgen,
References Lauwerier,
dinformation
see also CORRELATION EXPONENT, DIMENSION, H AUSDORFF DIMENSION,KAPLANYORKE DIMENSION
of the filled part is 2 (i.e., the filled), Cantor’s square fractal is
see also Box FRACTAL, CANTOR
5
Carathkodory Derivative A function f is Carathkodory differentiable at a if there exists a function d which is CONTINUOUS at a such that the A
f(x)  f(a) = 4(x)(x  4 Every function also FR&HET
see also
which
is Carathkodory
differentiable
is
DIFFERENTIABLE. DERIVATIVE, FRI?CHET DERIVATIVE
Carath&odory’s Fundamental Theorem Each point in the CONVEX HULL of a set S in R” is in the convex combination of n + 1 or fewer points of S.
see also CONVEX HULL, HELLY'S THEOREM Cardano’s
Formula
see Cumc
EQUATION
Cardinal
Number
Cardioid
Cardinal In informal in counting
Number usage, a cardinal number is a number used (a COUNTING NUMBER), such as 1, 2, 3, , , ,
and the parametric
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 obtainable by counting a given set. A set has No (ALEPH0) members if it can be put into a ONETOONE correspondence 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 ONETOONE. An INACCESSIBLE CARDINAL cannot be expressed in terms of a smaller number of smaller cardinals.
see
also
ALEPH,
ALEPH0
(No),
ALEPH1
(HI),
CAN
AXIOM, CANTOR DIAGONAL SLASH, CONTINUUM, CONTINUUM HYPOTHESIS, EQUIPOLLENT, INACCESSIBLE CARDINALS AXIOM, INFINITY, ORDINAL NUMBER, POWER SET, SURREAL NUMBER, TORDEDEKIND
UNCOUNTABLE
equations X
l
189
= a cos t(1+
y = asint(l+
cos t)
(4)
cost).
(5)
The cardioid is a degenerate case of the LIMA~ON. It is also a lCUSPED EPICYCLOID (with T = 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 Bxznsactions 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 2~.
SET
References Cantor, G. Uber unendliche, lineare Punktmannigfaltigkeiten, Arbeiten zur MengenEehre aus dem Jahren 187Z1884. Leipzig, Germany: Teubner, 1884. Conway, J. H. and Guy, R. K. “Cardinal Numbers.” In The Book of Numbers. New York: SpringerVerlag, pp. 277282, 1996. Courant, R. and Robbins, H. “Cantor’s ‘Cardinal Numbers.“’ 52.4.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 8386, 1996.
Cardinality
see CARDINAL NUMBER
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,O). Then the RADIUS of a CIRCLE centered at an ANGLE 0 from (1, 0) is
Cardioid r2 = (0  cos8)2 + (1  sin0)2 = cos2 e + 1  2 sin 8 + sin2 8  2(1  sin0).
(6)
The curve given by the POLAR equation T = a(1 + cod), sometimes
The ARC (1 >
CURVATURE, and TANGENTIAL AN
LENGTH,
GLE are t
also written
21 cos( $!)I dt = 4a sin( $0)
S=
(7)
s
T = 2b(1+
cod),
(2 >
3Psec($)l 4a 4  $e. K=
where b = a/2,
the CARTESIAN
equation
(x2 + y2  ax)2 = a2(x2 + y”),
(3)
(8) (9)
As usual, care must be taken in the evaluation of s(t) for t > 7r. Since f7) comes from an integral involving: the
190
Cardioid
ABSOLUTE increasing. by defining
VALUE
of a function,
Each
Cards
Caustic
QUADRANT
it must be monotonic can be treated correctly
Evolute //
/N
l\ \
/ n=
where
Cardioid
(10)
1x1 is the FLIER
s(t) = (1) lf+
The PERIMETER
FUNCTION,
hod
2)14sin($)
giving
\
\
\
the formula
+ 8 Lin]
.
(11)
of the curve is
27r
L=
x = gu+
x
~2acos(~~)~
dB = 4a
cos( +O) d0
Y
s 0
s 0
42 = 4a
n/2 cos 4(2 dg5) = 8a
s0
This
cos 4 d4
~ucose(1
$sinO(l
is a mirrorimage
cod)
c0se).
CARDIOID
with
a’ = a/3*
s0
= 8a[sin#’
= 8~.
(12)
The AREA is 2x
2x
s 0
Cardioid Inverse Curve If the CUSP of the cardioid is taken as the INVERSION CENTER, the cardioid inverts to a PARABOLA. Cardioid
(1+2cos8+cos20)d0
r2 d0 = +a2
A=;
:
\
Involute
s 0 27T

{1+
$U2
2cos8+
g1+
cos(28)]}de
s0 27T 12 
ZU
= &“@I see
aho
SPIRAL,
[; + 2 cos 8 + + cos(20)] d0 s0 + 2sine
+ $ sin(20)]iT
= $a2.
(13)
CIRCLE, CISSOID, CONCHOID, EQUIANGULAR LEMNISCATE, LIMA~ON, MANDELBROT SET
X = zu + 3a cos e(i  cos e) case). Y = 3asin@(l
References Gray, A. “Cardioids.” 53.3 in Modern Difierential Geometry Boca Raton, FL: CRC Press, of Curves and Surfaces. pp. 4142, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 118121, 1972. http://www.best.com/xah/Special Lee, X. “Cardioid.” PlaneCurvesdir/Cardioid,dir/cardioid.html, Lee, X. “Cardioid.” http://www,best.com/xah/Special PlaneCurvesdir/Cardioiddir/cardioidGG.html. Lockwood, E. H. “The Cardioid.” Ch. 4 in A Book of Curves. Cambridge, England: Cambridge University Press, pp+ 34
This
is a mirrorimage
Cardioid
Pedal
CARDIOID
with
a’ = 3a.
Curve
43, 1967. “Cardioid.” MacTntor History of Mathematics Archive. http://wwwgroups.dcs,stand.ac.uk/history/Curves /Cardioid. html. Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. xxvixxvii, 1995. Yates, R. C. “The Cardioid.” Math. Teacher 52, lo14,1959. Yates, R. C. “Cardioid.” A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 47, 1952.
Cardioid Caustic The CATACAUSTIC of a CARDIOID for a RADIANT POINT at the CUSP is a NEPHROID. The CATACAUSTIC for PARALLEL rays crossing a CIRCLE is a CARDIOID.
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 different “suits.” The suits are called clubs (&), diamonds (o), hearts (O), and spades (4). Spades and clubs are
Carleman ‘s hequali
ty
Carlyle
colored black, while hearts and diamonds are colored red. The cards of each suit are numbered I through 13, where the special terms ace (l), jack (ll), queen (12), and king (13) are used instead of numbers 1 and 1113.
INEQUALITY which general
uses a smaller
Circle constant).
191 For the
case
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 modern PROBABILITY theory. see COIN,
also
BRIDGE CARD GAME, CLOCK SOLITAIRE, COIN TOSSING, DICE, POKER, SHUFFLE
c=
l (PS>” w
Carleman’s Let {ai}yzl
Inequality be a SET of POSITIVE numbers. GEOMETRIC MEAN and ARITHMETIC MEAN n
In
Then the satisfy
Sl
n
l/i<
alm***ai)
i=l
e _ ;
IE i=l
P
where
a;.
y = p/J + qx
(6)
t=X
(7)
P/J + Clx
cX=lst Here, the constant that counterexamples INEQUALITY which
e is the best possible, in the sense can be constructed for any stricter uses a smaller constant.
see also ARITHMETIC
MEAN,
MEAN
e, GEOMETRIC
References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1094, 1979. Hardy, G. H.; Littlewood, J. E.; and P6lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 249250, 1988.
CarlsonLevin
Constant
N.B. A detailed online ing point for this entry.
essay
by S. Finch
was the start
Assume that f is a NONNEGATIVE REAL function [0, co) and that the two integrals
on
(8)
and I?(Z) is the GAMMA FUNCTION. References Beckenbach, E. F.; and Bellman, R. Inequalities. New York: SpringerVerlag, 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/cr~slvn.html~ Levin, V. I. “Exact Constants in Inequalities of the Carlson Type.” Doklady Akad. Nauk. SSSR (N. S.) 59, 635638, 1948. English review in Boas (1948), Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, 1991.
Carlsods Theorem If f(z) is regular and of the form 0(W) where k: < n, for S[r] 2 0, and if f(x) = 0 for x = 0, 1, . . , then f(z) is identically zero.
dx (1) r x”‘“[f(x)]”
l
see also GENERALIZED HYPERGEOMETRIC FUNCTION
0
References
x”‘+“[f(x)]”
dx
(2)
exist and are FINITE. If p = Q = 2 and A = ~1 = 1, Carlson (1934) determined
r
f (4 dx 2 fi
(pw
Bailey, W. N. “Carlson’s Theorem.” Hypergeometric Series. Cambridge, University Press, pp. 3640, 1935.
Carlyle
s5.3 in Generalised England: Cambridge
Circle
dx) Ii4
0
l/4
x2[ml2 dz
X
(Iand showed that fi that counterexamples
0
>
(3)
is the best constant (in the sense can be constructed for any stricter
Consider a QUADRATIC EQUATIONX~sx+p= s and p denote signed lengths. The CIRCLE
0 where which has
Carmichael
192
Carmichael
Condition
the points A = (0,l) and B = (s,p) as a DIAMETER is then called the Carlyle circle C,,, of the equation. The CENTER of C,,, is then at the MIDPOINT of AB, IW = (s/2, (1 + p)/2), which is also the MIDPOINT of S = (s,O) and Y = (0,l + p). Call the points at which C,,, crosses the XAXIS HI = (~1~0) and I& = (~2~0) (with ~1 2 ~2)~ Then
Carmichael Function X(n) is the LEAST COMMON MULTIPLE (LCM) of all the FACTORS of the TOTIENT FUNCTION 4(n), except that if 81n, then 2”2 is a FACTOR instead of 2”?
@( 1 n
forn=p”,p=2anda<2, 1 5 4( n > for n = 2a and a > 3 LCM[X(P~“~)]~ for n = nipiai.
A( n=>
s = x1 + x2 p=mx2
(x  x1)(x  x2) = x2  sx +p,
Some special
so x1 and x2 are the ROOTS of the quadratic see also 257~GON,
values
65537CON, HEPTADECAGON,
orp>3 
are W) =1 X(2) = 1
PEN
X(4) = 2 X(27 = 2'2
References and the Lemoine
Simplic
Amer. Math. MonthZy 88, 97408, 1991. Eves, K An Introduction to the History of Mathematics, 6th PA: Saunders, 1990. ed. Philadelphia, Leslie, J. Elements of Geometry and Plane Trigonometry with an Appendix and Very Copious Notes and Illustrations, 4th ed., improved and exp. Edinburgh:
W. & G. Tait,

equation.
TAGON De Temple, D. W. “Carlyle Circles ity of Polygonal Constructions.”
Number
1820.
for T > 3, and X(PT)
= 4(P’)
an ODD PRIME and T > 1. The ORDER of a (mod (R i b en b aim 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 AOll773).
for
p
n) is at most A(n)
l
Carmichael Condition A number n satisfies the Carmichael condition IFF (p l)l(n/p  1) for all PRIME DIVISORS p of n. This is equivalent to the condition (p  1) 1(n  1) for all PRIME DIVISORS
p of n.
NUMBER
see ~2~0 CARMICHAEL References Borwein, D.; Borwein,
J, M.; Conjecture 103, 4050, 1996.
sohn, R. “Giuga’s Monthly
Borwein, P. B.; and Girgenon Primality.” Amer. Math.
Carmichael’s Conjecture Carmichael’s conjecture asserts FINITE number of CARMICHAEL proven by Alford et al. (1994). see U~SO CARMICHAEL
NUMBER,
that
there
NUMBERS.
are an INThis was
CARMICHAEL’S
see also MODULO MULTIPLICATION
GROUP
References Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: SpringerVerlag, p. 27, 1989. Riesel, H, “Carmichael’s Function.” Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhguser, pp. 273275, 1994. Sloane, N. J. A. Sequence A011773 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Vardi, I. Computational Recreations in Mathematics. Redwood City, CA: AddisonWesley, p. 226, 1991.
Carmichael Number A Carmichaelnumberis an ODD COMPOSITE NUMBER n which satisfies FERMAT'S LITTLE THEOREM a n1  1 s 0 (mod
To
n)
TIENT FUNCTION CONJECTURE for every choice
References Alford, W. R.; Granville, A.; and Pomerance, C. “There Are Infinitely Many Carmichael Numbers.” Ann, Math, 139,
703722,1994. Cipra,
B. What’s Happening in the Mathematical Sciences, I. Providence, RI: Amer. Math, Sot., 1993. Conjecture.” §B39 in Unsolved Guy, R. K. “Carmichael’s Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p. 94, 1994. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. “The Pseudoprimes to 25 s IO9 .” Math. Comput. 35, 10031026, Vol.
1980.
Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: SpringerVerlag, pp. 2931, 1989. Schlafly, A. and Wagon, S. “Carmichael’s Conjecture on the Euler Function is Valid Below 101o~ooo~ooo.” Math, Comput.
63,
415419,
1994.
n are
of a satisfying
RELATIVELY PRIME) with
(upt) = 1 (i.e., a and 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 LITTLE THEOREM is sometimes NONZERO, thus identifying a Carmichael number n as COMPOSITES. numbers are sometimes called ABSOLUTE PSEUDOPRIMES and also satisfy KORSELT~ CRITERION.
Carmichael
R. D. Carmichael first noted the existence of such numbers in 1910, computed 15 examples, and conjectured that there were infinitely many (a fact finally proved by Alford et al. 1994).
Carmichael The first
Totient
Carmichael’s
Num her
few Carmichael
numbers are 561, 1105, 1729, 10585, 15841, 29341, .. . (Sloane’s A002997). C armichael numbers have at least For Carmichael numbers with three PRIME FACTORS. exactly three PRIME FACTORS, once one of the PRIMES has been specified, there are only a finite number of Carmichael numbers which can be constructed. Numbers of the form (6Tc + l)( 12k + 1) (18k + 1) are Carmichael numbers if each of the factors is PRIME (Korselt 1899, Ore 1988, Guy 1994). This can be seen since for
2465, 2821, 6601, 8911,
N = (6k+1)(12k+1)(18k+l)
Function
Pinch, R. G. E, ftp: // emu . pmms . cam . ac . uk / pub / Carmichael/. Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. “The Comput. 35, 10034026, Pseudoprimes to 25 lo’+” Math. l
1980.
Riesel,
H. Prime
torization,
2nd
95, 1994. Shanks, D. Solved
Numbers and Computer ed. Basel: Birkhauser,
ppm 8990 and 94
and
in Number
The smallest Carmichael numbers having 3, 4, . . . factors 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 < 106, 2163 < 2.5 x lOlo, 105,212 < 1015, and 246,683 < 1016 (Pinch EN)* Let C(n) d enote the number of Carmichael numbers less than n. Then, for sufficiently large n (n  lo7 from numerical evidence),
C(n)
ed.
P (mod
P(P
numbers
have the following

the
properties:
Carmichael
p  1) implies
that
number n G
I>>*
Carmichael
number
is SQUAREFREE.
3. An ODD COMPOSITE SQUAREFREE number n is a Carmichael number IFF n divides the DENOMINATOR of the BERNOULLI NUMBER &I.
see also
CARMICHAEL
CONDITION, PSEUDOPRIME
References Alford, W. R.; Granville, A.; and Pomerance, C. “There are Infinitely Many Carmichael Numbers.” Ann. Math. 139, 703722,
Theory,
New
Carmichael Sequence A FINITE, INCREASING    ? a,} such that
SEQUENCE of INTEGERS {al,
(Ui  l)l(Ul
*.a&l)
fori= l,..., VYL,where mln indicates that m DIVIDES n. A Carmichael sequence has exclusive EVEN or ODD elements. There are infinitely many Carmichael sequences for every order.
see also
GIUGA SEQUENCE
References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgenon Primality.” Amer. Math. sohn, R. “Giuga’s Conjecture Monthly
103, 4050, 1996.
Carmichael’s
Theorem
are RELATIVELY PRIME so that the GREATEST COMMON DENOMINATOR GCD(a,n) =l,then
If a and n
 n217
1. If a PRIME p divides n, then n = 1 (mod ’
Fac
= 1 (mod
n),
1994).
The Carmichael
2. Every
,Problems
for
York: Chelsea, p. 116, 1993. Sloane, N. J. A. Sequences A002997/M5462 and A006931/ M5463 in “An OnLine Version of the Encyclopedia of Integer Sequences.” 4th
uw et al.
Unsolved
Methods
= 1296k3+396k2+36k+l,
N  1 is a multiple of 36J~ and the LEAST COMMON MULTIPLE of 6k, 12k, and 18/z is 36k, so aNwl E 1 modulo each of the PRIMES 6k + 1, 12k + 1, and 18k + their product. The first 1, hence uRT’ = 1 modulo few such Carmichael numbers correspond to k = 1, 6, 35, 45, 51, 55, 56, . . . and are 1729, 294409, 56052361, The largest known 118901521, . . . (Sloane’s A046025). Carmichael number of this form was found by II. Dubner in 1996 and has 1025 digits.
(Alford
193
Conjecture
1994.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 87, 1987. Numbers.” §A13 in Unsolved ProbGuy, R. K. “Carmichael lems in Number Theory, 2nd ed. New York: SpringerVerlag, pp. 3032, 1994. Korselt , A. “Probkme chinois .” L ‘interme’diaire math. 6, 143143, 1899. Ore, 0. Number Theory and Its History. New York: Dover, 1988. Numbers up to 1015.” Pinch, R. G. E. “The Carmichael Math. Comput. 55, 381391, 1993.
where X is the CARMICHAEL FUNCTION. Carmichael’s
Totient Function Conjecture that the TOTIENT VALENCE FUNCTION N&(m) > 2 (i.e., the TOTIENT VALENCE FUNCTION never takes the value 1). This assertion is called Carmichael’s totient function conjecture and is equivalent to the statement that there exists an m, # n such that #(n) = 4(m) (Ribenboim 1996, pp. 3940). Any counterexample to the conjecture must have more than 10,000 DIGITS (Conway and Guy 1996). Recently, the conjecture was reportedly proven by F. Saidak in November, 1997 with a proof short enough to fit on a postcard. It is thought
see also TOTIENT FUNCTION
FUNCTION,
TOTIENT
VALENCE
References Conway, J. H. and Guy, R. K. The Book York: SpringerVerlag, p. 155, 1996. Ribenboim, P. The New Book of Prime New York: SpringerVerlag, 1996.
of Numbers. Number
New Records.
194
Carnot’s
Polygon
Cartan
Theorem cos(Zm/q)
Carnot’s Polygon Theorem If PI, P2, .. , are the VERTICES of a finite POLYGON with no “minimal sides” and the side Pi Pj meets a curve in the POINTS PQ~ and Pij2, then
Curves.
where
Lq/ZJ,
Pickover, C. A, “Are Infinite CarotidKundalini F’ractal?” Ch. 24 in Keys to Infinity. New Freeman, pp. 179481, 1995.
Carot idKundalini The FUNCTION given
References Plane
.. ,
[zl
is the
References
the DISTANCE from POINT A to B.
Coolidge, J. L. A Treatise on Algebraic York: Dover, p. 190, 1959.
l
Coefficient
CEILING FUNCTION and 1x1 is the FLOOR FUNCTION.
1, where AB denotes
for r = 0, 1,
Torsion
finct by
ion
CK, (x) E cos(nx
New
Functions York: W. H.
cos 1x>,
where n is an INTEGER and 1 < GL:< 1. Carnot’s Theorem Given any TRIANGLE AlAzA3, the signed sum of PERPENDICULAR distances from the CIRCUMCENTER 0 to
see also
CAROTIDKUNDALINI
FRACTAL
Carry
the sides is
1 1
carries
1 5 geaddend + 2 4 g+addend2
4 0 7+sum
where T is the INRADIUS and R is the CIRCUMRADIUS. The sign of the distance is chosen to be POSITIVE IFF the entire segment 00i lies outside the TRIANGLE.
see also JAPANESE TRIANGULATION
1
of shifting the leading DIGITS of an ADDITION into the next column to the left when the SUM of that column exceeds a single DIGIT (i.e., 9 in base 10). The operating
THEOREM
see also
ADDEND, ADDITION, BORROW
References
MA:
Eves, H. W. A Survey of Geometry, Few ed. Boston, Allyn and Bacon, pp, 256 and 262, 1972. Honsberger, R. Mathematical Gems III. Washington, Math. Assoc. Amer., p. 25, 1985.
CarotidKundalini ,r
Fractal 1.
Carrying
Cartan Matrix A MATRIX used in the presentation
Fkactal *,
Valley
1
"
Gaussian "
1
Mtn. "
"
Oscillation I>
"'
Land
Capacity
see LOGISTIC GROWTH CURVE
DC:
1
of a LIE ALGEBRA.
References Jacobson,
N. Lie Algebras.
New
York:
Dover,
p. 121,
1979.
Cartan Relation The relationship Sqi (X  9) = Cj+k=iSqj (x)  Sq” (y) in the definition of the STEENROD ALGEencountered
Cartan Subgroup A type of maximal
Ab elian
SUBGROUP.
References Knapp, A. W. “Group Representations and Harmonic ysis, Part II.” Not. Amer. Math. Sot. 43, 537549,
A fractallike structure is produced for 2 < 0 by superposing plots of CAROTIDKUNDALINI FUNCTIONS CK, of different orders 72. The region 1 < 61: < 0 is called FRACTAL LAND by Pickover (1995), the central region the GAUSSIAN MOUNTAIN RANGE, and the region x > 0 OSCILLATION LAND. The plot above shows n = 1 to 25. Gaps in FRACTAL LAND occur whenever 2 cos I x =
27pP Q
for p and q RELATIVELY PRIM E INTEGERS. At such points 2, the functions assume the [(q + Q/21 values
Anal1996.
Cartan Torsion Coefficient The ANTISYMMETRIC parts of the CONNECTION COEF
FICIENT rxpV.
Cartesian
Coordinates
Cartesian
Coordinates
Cartesian The
GRADIENT
DIVERGENCE
ofthe
195
Ovals is
zaxis A
xaxisu
yaxis
Cartesian coordinates are rectilinear ZD or 3D coordinates (and therefore a special case of CURVILINEAR COORDINATES) which are also called RECTANGULAR CoORDINATES. The three axes of 3D Cartesian coordinates, conventionally denoted the X, y, and ZAXES (a NOTATION due to Descartes) are chosen to be linear and mutually PERPENDICULAR. In 3D, the coordinates X, y, and x may lie anywhere in the INTERVAL (co, 00). The SCALE FACTORS of Cartesian unity, hi = 1. The LINE ELEMENT
coordinates is given by
ds=dxji:+dy?+dzii, and the VOLUME
ELEMENT
(1)
GRADIENT
LAPLACE'S
L 9 BY I3 d
EQUATIQN
au, au, dz+dy+dz
is separable
au, >
.
in Cartesian
(8)
cooral
1.
nates. see
also
COORDINATES,
EQUATIONCARTESIAN
HELMHOLTZ COORDINATES
DIFFERENTIAL
References A&en, G. “Special Coordinate SystemsRectangular Cartesian Coordinates.” $2.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 9495, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys‘Its, Part I. New York: McGrawHill, p. 656, 1953.
by
(2)
dV = dxdydz. The
are all

has a particularly
simple
Cartesian
Ovals
form,
(3) as does the LAPLACIAN
a2 v2=@+T+'22. The
LAPLACIAN V2F
d2
a2
(4)
dY
is
E V
= (VF)
=
d2F
a2F
d2F
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 Fl and F2 in twocenter BIPOLAR COORDINATES satisfy
=+dy2+= mr * nr’
= k,
(1)
where m, 12 are POSITIVE INTEGERS, k is a POSITIVE real, and T and T’ are the distances from Fl and F2. If m = n, the oval becomes an an ELLIPSE. In CARTESIAN COORDINATES, the Cartesian ovals can be written
(5) The DIVERGENCE
(2)
is
V . F = aFx dX
and the CURL
(x + a)2 + y2 = k2
m Jc
+ aFy dy
+ dFz ax ’
(6)
(x2 + y2 + a”)(m”
 n2)  2ax(m2
+ n2)  k2 + a)2 + y2,
= 2n&x
(3)
is
Km2 
 n2)(x2
+ y2 + a”)
2(m2 + n2)(n2
Now
 2ax(m2
+ n2)12
+ y2 + a”)  4ax(m2
 n2)  k2.
(4)
define b E m2  n2
(5)
cIm2
(6)
+n2,
196
Product
Cartesian
Cassini Casimir Operator An OPERATOR
and set a = 1. Then [b(22+~2)2cz+b]2+4b2+k22C=2C(x2+y2)~ If c’ is the distance
between
(7)
Fl and &,
r = ii, on a representation
an alternate
(8)
References Jacobson, N.
Lie
R of a LIE ALGEBRA. Algebras.
New
York:
Dover,
p. 78, 1979.
form is Cassini
[(1m2)(x2+y2)+2m2c’x+at2m2c’2]2 The oval is a one.
R iR ei u
il
and the equation
T+mr’=a is used instead,
0 vah
= 4at2(x2+y2).
(9) curves possess three FOCI. If m = 1, one Cartesian is a central CONIC, while if m = a/c, then the curve LIMACON and the inside oval touches the outside Cartesian ovals are ANALLAGMATIC CWRVES.
Ellipses
see CASSINI
OVALS
Cassini’s Identity For F, the nth FIBONACCI F n1Fn+lFn2
NUMBER,
=(I)“.
References Cundy, H, and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 155157, 1972, Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967. MacTutor History of Mathematics Archive. “Cartesian Oval.” http://wwwgroups.dcs.stand.ac.uk/history/ Curves/Cartesian.htm1.
Cartesian
Product
seel31R~~~
PRODUCT
Cartesian
Trident
see TRIDENT
(SET)
Cartography The study of MAP PROJECTIONS ographical maps.
and the making
of ge
PROJECTION
Cascade A ZACTION or IVACTION. 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 straight LINE IFF t12t34
k
where tij is a common see also PURSER’S
t13t42
to a fifth
zt
TANGENT
t14t23
=
CIRCLE
or a
on
the
Petkovgek, M.; Wilf, H. S.; and Zeilberger, ley, MA: A. K, Peters, pm 12, 1996.
Cassini
Welles
to CIRCLES
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 b2. The shape of the curve depends on b/a. If a < b, the curve is a single loop with an OVAL (left figure above) or dog bone (second figure) shape. The case a = b produces a LEMNISCATE (third figure). If a > b, then the curve consists of two loops (right figure). The curve was first investigated by Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one FOCUS of the oval. Cassini ovals are ANALLAGMATIC CURVES. The Cassini ovals are defined in twocenter BIPOLAR COORDINATES by the equation TIT2 = b2, (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 constant.
i and j*
ovals have the CARTESIAN [(x  a)" + y”][(x
equation
+ a)’ + y”] = b4
(2)
THEOREM
R. A. Modern
MA: Houghton
D. A=B.
Ovals
or the equivalent Geometry
NUMBER
References
The Cassini
0,
References Johnson,
FIBONACCI
<qcxL:‘:I:::x:::I:(:,
OF DESCARTES
see also MAP
see also
Geometry; of the Triangle
Mifflin,
An Elementary and the Circle.
pp* 121l 27, 1929.
form
Treatise
Boston,
(x2 + y2 + u2)2  4u2x2 = b4
(3)
Cassini
Cassini Surface
Ovals
and the polar
Cassini
equation r4 + u4  2a2r2 cos(20)
Solving
= b4.
for r2 using the QUADRATIC
r2 =
2a2 cos(28) + d4a4
Projection
(4)
EQUATION
cos2(20) 2
197
gives
 4(a4  b”)
 u2 cos(28) + db4  u4 sin2(28) =a2
.
[cos(2t9+/pjxG
(5)
A MAP
PROJECTION . x = sinl
If a < b, the curve has AREA f0 A=
has been done over half the curve by two and E(x) is the complete ELLIPTIC INTEGRAL OF THE SECOND KIND. If u = b, the curve becomes
1
r2 = a2 cos(20) + J1sin28 [ is a LEMNISCATE
= 2u2 cos(28),
(8)
,
r = *+os(2S)
where 8 E [&,&I
The inverse
* Jci,”
 sin2(28),
(9)
(2)
 X0).
(3)
FORMULAS are 4 = sin l(sinDcosx)
(7)
(two loops of a curve fi the linear scale of the usual lemniscate r2 = u2 cos( 20), which has area A = u2 /2 for each loop). If a > 6, the curve becomes two disjoint ovals with equations 1
B = cos+in(X
X=X()
A = 2u2
1
where
AREA
having
(1)
tan 4 cos(X  A,) [
Y = tanl
+r2 d0 = 2(i)
where the integral and then multiplied
which
B
+tanl
(4)
(
tan x a
>
,
(5)
where D =y+&.
(6)
References Snyder, J. P. MalJ ProjectionsA Geological Survey Professional DC: U. S. Government Printing
Cassini
Working 1Manual. U. S. Paper 1395. Washington, Office, pp. 9295, 1987.
Surface
and 80 E + sinl
K)I b 
2
l
U
see UZSOCASSINI SURFACE, LEMNISCATE, MANDELBROT SET, OVAL References Gray, A. “Cassinian
Ovals .” $4.2 in Modern Differentiul Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 6365, 1993. Lawrence, 5. D. A Catalog of Special Plane Curves. New York: Dover, pp. 153155, 1972. Lee, X. “Cassinian Oval.” http: // www . best . corn / # xah / SpecialPlaneCurvesdir/CassinianOvaldir/cassinian Oval. html. Lockwood, E. H. A Boolc of Curves. Cambridge, England: Cambridge University Press, pp. 187188, 1967. “Cassinian MacTutor History of Mathematics Archive. ovals.” http: //wwwgroups .dcs. stand.ac .uk/history /Curves/Cassinian.html. Curves.” A Handbook on Curves Yates, R. C. Tassinian Ann Arbor, MI: J. W. Edwards, and Their Properties. pp. 811, 1952,
The QUARTIC SURFACE obtained by replacing the constant c in the equation of the CASSINI OVALS [(x  u)” + y”][(z
+ u)” + Y”] = c2
(1)
by c = z2, obtaining Kx 
u)”
+
y2][(x + a)” + Y”] = z4m
As can be seen by letting
(2)
y = 0 to obtain
(x2  a2)2 = x4
(3)
x2+x2 =u2,
(4)
Castillon’s
198
Catalan’s
Problem
the intersection of the surface with a CIRCLE of RADIUS a.
the y = 0 PLANE
is
Fischer,
G. (Ed.). of Universities
Mathematical and Museums.
Vieweg, p. 20, 1986. G. (Ed.). Plate Fischer, elle/Mathematical
Models,
Braunschweig,
Castillon’s
Curve
Source
Catacaustic
cardioid
cusp not on circumf.
nephroid limaqon
on circumf. point at 00
cardioid
circle circle
References
Germany:
Models
from
the Collections
Braunschweig,
Germany:
51 in Mathematische Bildband/Photograph
Vieweg,
ModVolume.
circle cissoid
of Diocles
1 arch of a cycloid
rays 1 axis point at 00 rays 11 axis
spiral
parabola quadrifolium
Problem
Tschirnhausen
see
UZSO
cubic
CAUSTIC,
nephroid cardioid
focus
deltoid In 2 logarithmic
p. 51, 1986.
Conjecture
2 arches astroid
of a cycloid
catenary
origin
equal
rays 1 axis center
Tschirnhausen astroid
focus
semicubical
logarithmic
spiral cubic
parabola
DIACAUSTIC
References Lawrence, 5, D. A Catalog of Special York: Dover, pp. 60 and 207, 1972.
Inscribe a TRIANGLE in a CIRCLE such that the sides of A, B, the TRIANGLE pass through three given POINTS and C. References Dijrrie,
H. “Castillon’s
of Elementary
New York:
Problem.”
Mathematics:
Dover,
$29 in 100 Their History
pp. 144147,
x
bE x
X3  Y2 = 1.
ai = u*
bi = b*
CIc G=c*, so ab E a*b* must be = c* (mod 9) Casting is sometimes also called “the HINDU CHECK."
out nines
This CONJECTURE has not yet been proved or refuted, although it has been shown to be decidable in a FxNITE (but more than astronomical) number of steps. In particular, if n and TI + 1 are POWERS, then n < expexpexpexp 730 (Guy 1994, pa 155), which follows from R. Tijdeman’s proof that there can be only a FrNITE number of exceptions should the CONJECTURE not hold. Hyyrb and M3kowski 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
of Numbers.
New
also
Map
see ARNOLD’S
CAT MAP
Catacaustic The curve which
is the ENVELOPE
of reflected
rays.
CATALAN'S
References Guy, R. K. “Difference Problems
Cat
New
x2  y" = 1
1965.
References Conway, J. H. and Guy, R. K. The Book York: SpringerVerlag, pp. 2829, 1996.
Curves.
Catalan’s Conjecture 8 and 9 (23 and 32) are the only consecutive POWERS to CATA(excluding 0 and l), i.e., the only solution LAN'S DIOPHANTINE PROBLEM. Solutions to this problem (CATALAN'S DIOPHANTINE PROBLEM) are equivalent to solving the simultaneous DIOPHANTINE EQUATIONS
Great Problems and Solutions.
Casting Out Nines An elementary check of a MULTIPLICATION which makes use of the CONGRUENCE 10" = 1 (mod 9) for n > 2. From this CONGRUENCE, a MULTIPLICATION ab = c must give a
Plane
in Number
DI~PHANTINE of Two Theory,
PROBLEM Power.”
2nd
ed. New
SD9 in Unsolved York: Springer
Verlag, pp. 155157, 1994. Ribenboim, P. Catalan’s Conjecture. Boston, MA: Academic Press, 1994. Amer. Math. Ribenboim, P. “Catalan’s Conjecture.” Monthly 103, 529538, 1996, Ribenboim, P. “Consecutive Powers.” Expositiones Mathematicae 2, 193221, 1984.
Catalan’s
Constant
Catalan’s
Catalan’s Constant A constant which appears in estimates of combinatorial functions. It is usually denoted K, P(2), or G. It is not known if K is IRRATIONAL. Numerically, K = 0.915965594177..
.
K = p(2)
(2)
 k cO” (‘Jk(1)+ 112 k=O‘
.
1
n=l
’ tanl

1
(472 + 1)”  ii
’ + x
O” x n=l
fi where
FOR
[q&]21’2g [&  l]1’(2k+1)p (14) q(m)
=
m*mi(i) 7Fm(2m  1)4”lBm
’
(15)
where Bn is a BERNOULLI NUMBER and $(x) is a POLYGAMMA FUNCTION (Finch). The Catalan constant may also be defined by
K=i2
1
’ K(k)
(4
(4n + 3)2
dk,
where K(k) (not to be confused itself, denoted K) is a complete
xdx
(16)
s 0
with
Catalan’s
constant
ELLIPTIC INTEGRAL OF
THE FIRST KIND.
’ lnxda: s o 1+x2’

used the related
(3)
X
s0
K= 1
’
00

1 1 1 = 12  32 + 52 + ” ’
1991, p. 159). W. Gosper
199
MULA
(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,

(Vardi
Constant
00
K
where p(z) is the DIRICHLET BETA FUNCTION. Interms ofthe P~LYGAMMA FUNCTION XP1(x),
ai ?
+
x i=l
2L(i+1)/2]i2
’
(17)
where {Ui}
= {l,l,
l,O, 1, 1, 1,o)
(18)
is given by the periodic sequence obtained by appending copies of (1, 1, 1, 0, 1, 1, 1, 0) (in other words, for i > 8) and [xJ is the FLOOR ai E a[(i1) (mod 8)]+1 FUNCTION (Nielsen 1909).
see also DIRICHLET BETA FUNCTION CONVERGENCE IMPROVEMENT to (3) gives
Applying
3m  1
) am 4m
K=$(m+l m=l
where C(Z) identity
+ 21,
(10)
ZETA FUNCTION and the
is the RIEMANN
1 1 ~ = T(rn+ (1  3X)2  (1  X)2 m
l)vzm
(11)
1
has been used (Flajolet and Vardi and Vardi algorithm also gives
1 96).
The Flajolet
References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807808, 1972. Adam&k, V. “32 Representations for Catalan’s Constant .” http://www.wolfram.com/victor/articles/ catalan/catalan.html. 3rd ed. OrArfken, G. Mathematical Methods for Physicists, lando, FL: Academic Press, pp. 551552, 1985. Fee, G. J. “Computation of Catalan’s Constant using Ramanujan’s Formula.” ISAAC ‘90. Proc. Internat. Symp. Symbolic Algebraic Comp., Aug. 1990. Reading, MA: AddisonWesley, 1990. Finch, S. “Favorite Mathematical Constants.” http: //www. mathsoft. com/asolve/constant/catalan/catalan.html. Flajolet, P. and Vardi, I. “Zeta Function Expan
sions K=$[(l$)g k=l
where p(z) is the DIRICHLET BETA FUNCTION. Glaisher (1913) gave
K=lx
O”n@n + 1) l@~
n=l
(13)
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  l/3” + l/T  1/7n+1/9n&c for n = 2, 4, 6.” Messenger Math. 42, 3558, 1913. Gosper, R. W. “A Calculus of Series Rearrangements.” In Algorithms and Complexity: New Directions and Recent Results (Ed. J. F. Traub). New York: Academic Press, 1976. Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany: Halle, pp+ 105 and 151, 1909.
Catalan’s
200
Diophantine
Catalan Number
Problem
Plouffe, S. “Plouffe’s Inverter: Table of Current Records for the Computation of Constants.” http: //lacim.uqam. ca/ pi/records. html. Sloane, N. J. A. Sequences A014538 and A006752/M4593 in “An OnLine Version of the Encyclopedia of Integer Sequences .” 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. IMath. Educ.
Sci. Technol. I. Computational
21,
Vardi, ing, MA: AddisonWesley, Yang, S. “Some Properties Math.
Educ.
Sci.
375377, Recreations
1990.
in Mathematics. p. 159, 1991. of Catalan’s Constant
Technol.
23,
Catalan’s Diophantine Find consecutive POWERS,
549556,
ReadG.”
Int.
J.
1992.
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 c2kBl, and the last DIGIT is five for k = 9 to 15. The only PRIME Catalan numbers for n < 215  1 are C2 = 2 and c3
= 5.
The Catalan numbers turn up in many other related types of problems. For instance, the Catalan number of BINARY BRACKETINGS of n c 72l gives the number letters (CATALAN'S PROBLEM). The Catalan numbers also give the solution to the BALLOT PROBLEM, the number of trivalent PLANTED PLANAR TREES (Dickau),
Problem i.e., solutions
to
2  Cd = 1,
excluding 0 and 1. CATALAN'S CONJECTURE isthatthe only solution is 3’  23 = 1, so 8 and 9 (2” and 32) are the only consecutive POWERS (again excluding 0 and 1). see d5o CATALAN’S
CONJECTURE
References Cassels,
J. W. S. “On the Equation
Cambridge
Phil.
Sot.
56,
Inkeri, K. “On Catalan’s 1964.
Catalan Special
97103,
Problem.”
Integrals cases of general
FORMULAS
a”  by = 1. II.” Proc. 1960. Acta Arith. 9, 285290,
due to Bessel.
m
J&m)
= f
eYcosBcos(~sinB)d~, s 0
where Jo is a BESSEL FUNCTION OF THE FIRST Now, let z E 1  z’ and y E 1+ z’. Then Jo(2i&
= 1 7r s 0
r e(l+")cOs~
KIND.
cos[( 1  z) sin 01 de.
Catalan Number The Catalan numbers are an INTEGER SEQUENCE {Cn} which appears in TREE enumeration problems of the type, “In how many ways can a regular ngon be divided into n  2 TRIANGLES if different orientations (EULER'S POLYGON DIVIare counted separately?” SION PROBLEM). The solution is the Catalan number C+z (Dijrrie 1965, Honsberger 1973), as graphically illustrated below (Dickau).
the number of states possible in an nFLEXAGON, the number of different diagonals possible in a FRIEZE PATTERN with 72 + 1 rows, the number of ways of forming an nfold exponential, the number of rooted planar binodes, the number of rooted nary trees with n internal plane bushes with TX 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 people (Conway and Guy 1996), and the number of SEQUENCES with NONNEGATIVE PARTIAL SUMS whichcan be formed from n 1s and n 1s (Bailey 1996, Buraldi 1992)! An explicit
formula
for Cn is given 1 n+
by (2 n! >
(2n)! 1 n!2
= (n+
l)!n!’
(‘I
COEFFICIENT and n! is where (2) d enotes a BINOMIAL the usual FACTORIAL. A RECURRENCE RELATION for Cn is obtained from C n+l cn

(2n + 2)! (n + 2)[(n

(272 + 2)(2n

2(2n + l)(n
(n +
(n + 1)(n!)2
+ 1)!12 + l)(n
(2 72> ! + 1)
2)(n + 1)” + I>"
(n+1)2(n+2)
2(2n + 1) =
M2
’
(2)
Catalan
Catalan
Number
so Other
and the explicit
2(2n + 1) c
c n+1=
n+2
p

2  6  10. . . (4n  2)
(4)
(n + I)! 2”(2n  l)!! (n + l)!
(6)
l)!’
SEGNER'S RECURRENCE FORMULA, given by Segner in 1758, gives the solution to EULER'S POLYGON DIVISION PROBLEM &tEn1+
E3En2
3m
A RECURRENCE
l
l
+
Enl&*
(7)
With El = Ea = 1, the above RECURRENCE RELATION gives the Catalan number Cn2 = E,. The GENERATING FUNCTION for the Catalan is given by
numbers
(12)
d qk=
is given by
RELATION P d qk
(5)
(2 n! > z n!(n+
E, =
formula
(3)
n*
forms include CT&=
201
Number
=
P d Pr,i
x
P
d q+r,j
(13)
wherei,j,r>l,k>l,q<pr,andi+j=k+l (Hilton and Pederson 1991). see &O BALLOT PROBLEM, BINARY BRACKETING, PROBLEM, CATALAN'S BINARY TREE, CATALAN'S TRIANGLE, DELANNOY NUMBER, EULER'S POLYGON DIVISION PROBLEM, FLEXAGON, FRIEZE PATTERN, MOTZKIN NUMBER,~GOOD PATH, PLANTED PLANAR TREE,SCHR~DERNUMBER, SUPERCATALAN NUMBER References Alter,
R. “Some Remarks
Proc. 2nd Louisiana put., 109132, 1971.
and Results Conf.
Comb.,
on Catalan Graph
Th.,
Numbers.” and
Com
Alter, R. and Kubota, K. K. “Prime and Prime Power Divisibility of Catalan Numbers.” J. Combin. Th, 15, 243256, 1973.
= l+a:+2z2+5z3+.
l
l
l
(8)
Bailey, Math.
The asymptotic
form for the Catalan
numbers
is
4” ck
(Vardi
1991, Graham
A generalization
(9)

et al. 1994).
of the Catalan
numbers
is defined
by
(10) for k > 1 (Klarner 1970, Hilton and Pederson 1991). The usual Catalan numbers ck = a& are a special case with p = 2. pdk gives the number of pary TREES with k sourcenodes, the number of ways of associating k applications of a given pary OPERATOR, the number of ways of dividing a convex POLYGON into k disjoint (p + l)gons with nonintersecting DIAGONALS, and the number ofpGOOD PATHS from (0, 1) to (k, (pl)k1) (Hilton and Pederson 1991). A further generalization is obtained as follows. Let p be an INTEGER > 1, let pk = (k, (p  1)k  1) with k > 0, and q < p  1. Then define pdqO = 1 and let pd,k bethe number of ~GOOD PATHS from (1, q  1) to Pk (Hilton and Pederson 1991). Formulas for pdqi include the generalized JONAH FORMULA
(11)
D. F.
“Counting
69, 128131, R. A. Introductory
Mug.
Arrangements 1996.
of l’s
and 1’s.”
Brualdi, Combinatorics, 3rd ed. New York: Elsevier, 1997. Campbell, D. “The Computation of Catalan Numbers.” Math, Mug. 57, 195208, 1984, Chorneyko, I. 2. and Mohanty, Se G* “On the Enumeration of Certain Sets of Planted Trees.” J. combin. Th. Ser, B 18, 209221, 1975. Chu, W. “A New Combinatorial Interpretation for Generalized Catalan Numbers.” Disc. Math. 65, 9194, 1987. Conway, 5. H. and Guy, R. K. In The Book of Numbers. New York: SpringerVerlag, pp. 96106, 1996. Dershowitz, N+ and Zaks, S. “Enumeration of Ordered Trees.” Disc. Math. 31, 928, 1980. Dickau, R. M. “Cat alan Numbers .” http://forum. swarthmore. edu/advanced/robertd/catalan. ht,ml. Dijrrie, H. “Euler’s Problem of Polygon Division.” $7 in IOU Great Problems of Elementary tory and Solutions. New York:
Mathematics:
Their
His
Dover, pp. 2127, 1965. Eggleton, R. B. and Guy, R. K. “Catalan Strikes Again! How Likely is a Function to be Convex?” Math. Mug. 61, 211219, 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, 120125, June 1976. Gould, H. W. Bell & Catalan Numbers: Research Bibliography of Two Special Number Sequences, 6th ed. Morgantown, 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: AddisonWesley, 1994. Guy, R. K. “Dissecting a Polygon Into Triangles.” Bull. Malayan Math. Sot. 5, 5760, 1958. Hilton, P. and Pederson, J. “Catalan Numbers, Their Generalization, and Their Uses.” Math. Int. 13, 6475, 1991. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 130134, 1973.
Catalan’s
202
Problem
Catalan’s
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp* 146150, 1985. Klarner, D. A. “Correspondences Between Plane Trees and Binary Sequences.” J. Comb. Th. 9, 401411, 1970. Rogers, D. G. “Pascal Triangles, Catalan Numbers and Renewal Arrays.” Disc. Math. 22, 301310, 1978. Catalan Numbers.” Disc. Sands, A. D. “On G eneralized 21, 218221, 1978. Math. Singmaster, D. “An Elementary Evaluation of the Catalan Numbers.” Amer. Math. Monthly 85, 366368, 1978. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 1820, 1973. Sloane, N. J. A. Sequences A000108/M1459 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Sloane, N. J. A. and Plouffe, S. Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematics. Redwood City, CA: AddisonWesley, pp. 187188 and 198199, 1991. Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, pp. 121122, 1986.
Catalan’s Problem The problem of finding the number of different ways in of n different ordered FACTORS can be which a PRODUCT calculated by pairs (i.e., the number of BINARY BRACKETINGS of n letters). For example, for the four FACTORS a, b, c, and H, there are five possibilities: ((ab)c)d, (a(bc))d, (ab)(cd), u((bc)d), and u(b(cd)). The solution was given by Catalan in 1838 as CA = andis see
equaltothe
l
NUMBER
H. 100
Their
History
Great and
Here are the Archimedean sponding Cat alan solids.
with
solids
paired
see also ARCHIMEDEAN SOLID, SEMIREGULAR POLYHEDRON
DUAL
CATALAN'S
POLYGON
“.’ k @
DIOPHAN
DIVISION
PROBLEM
Problems Solutions.
of Elementary New York:
Mathematics:
Dover,
p. 23,
1965.
&cole
Catalan Solid The DUAL POLYHEDRA of the ARCHIMEDEAN given in the following table. Solid
SOLIDS,
Dual
rhombicosidodecahedron
deltoidal
hexecontahedron
small
rhombicuboctahedron
deltoidal
icositetrahedron
great great
rhombicuboctahedron rhombicosidodecahedron
disdyakis disdyakis
dodecahedron triacontahedron
truncated
icosahedron
pentakis pentagonal (dextro)
hexecontahedron
snub
pentagonal
icositetrahedron
cube
Holden, 1991.
E. “M&moire Polytechnique A. Shapes,
sur la ThGorie des Polyhdres.” J. (Paris) 41, 171, 1865. Space, and Symmetry. New York: Dover,
Catalan’s
Surface
A MINIMAL
SURFACE
dodecahedron
snub dodecahedron (laevo) (laevo) cuboctahedron
POLYHEDRON,
References Catalan,
Archimedean
the corre
= CL.
Cnl
References D&rie,
(Holden 1971, order (left to
(4n  6)
BRACKETING, EULER’S
Here are the ARCHIMEDEAN DUALS Pearce 1978) displayed in alphabetical right, then continuing to the next row).
?I!
CATALAN
BINARY
also
TINE PROBLEM,
2 m6 9 10
Surface
(dextro) rhombic dodecahedron
icosidodecahedron
rhombic triacontahedron
truncated
octahedron
tetrakis
truncated truncated
dodecahedron cube
triakis triakis
icosahedron octahedron
truncated
tetrahedron
triakis
tetrahedron
given by the parametric
equations
hexahedron
X(U,II)
= u  sinucoshw
Y b7 4 = 1  cosucoshv z(u, v) = 4 sin( $u) sinh( iv)
(1) (2) (3)
Catalan’s (Gray
Categorical
YlIiangle
19X3), or  24
Y(? 99 = a cos(Z+) Z(T, qb) L=: 2av sin #,
+ $a~’ cos(2$)
 +v2
cos(2qb)
(4 (5) (6)
where
1 r
vT+
1986).
References Catalan, E. “Mkmoir sur les surfaces dont les rayons de courburem en chaque point, sont kgaux et des signes contraires.” C. 22. Acad. Sci. Paris 41, 10191023, 1855. do Carmo, M. P. “Catalan’s Surface” §3.5D in Mathematical Models
frum
the
Collections
of
Universities
urns (Ed. G. Fischer). Braunschweig, Germany: pp. 4546, 1986. Fischer, G. (Ed.). Plates 9495 in Mathematische elle/Mathematical
Models,
Bildband/Photogruph
and
Muse
Vieweg, ModVolume.
Braunschweig, Germany: Vieweg, pp. 9091, 1986. Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, pp. 448449, 1993.
Catalan’s A triangle
203
Catastrophe
x(f, 4) = a sin(2$)
(da Carmo
Variable
Triangle of numbers
with
entries
given by
Cnm (n+m)!(nAl) m!(n
see BUTTERFLY CATASTROPHE, CATASTROPHE THEORY, CUSP CATASTROPHE, ELLIPTIC UMBILIC CATASTROPHE, 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 bifurcation theory, nonequilibrium thermodynamics, singularity theory, synergetics, and topological dynamics. For any system that seeks to minimize a function, only seven different local forms of catastrophe “typically” occur for four or fewer variables: (1) FOLD CATASTROPHE, (2) CUSP CATASTROPHE, (3) SWALLOWTAIL CATASTROPHE, (4) BUTTERFLY CATASTROPHE, (5) ELLIPTIC UMBILIC CATASTROPHE, (6) HYPERBOLIC UMBILIC CATASTROPHE, (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
+ l)!
for 0 < 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 Cn.
1 2 3 4
1 Behavior
2 Behavior Axes
Axis
fold

cusp

swallowtail
hyperbolic parabolic
butterfly
umbilic, umbilic
elliptic
umbilic
References 1 1 1 12
2
135 1 1 1
(Sloane’s
5 4 5 6
9 14 20
14 28 48
14 42 90
42 132
132
A009766).
see also BELL TRIANGLE, CLARK'S TRIANGLE, EuLER'S TRIANGLE,LEIBNIZ HARMONIC TRIANGLE,NUMBER TRIANGLE, PASCAL'S TRIANGLE, PRIME TRIANGLE, SEIDELENTRINGERARNOLD TRIANGLE References Sloane, N. J. A. Sequence A009766 in “An OnLine of the Encyclopedia of Integer Sequences.”
Catalan’s
Trisectrix
Version
Arnold, V. 1. Catastrophe Theory, 3rd ed. Berlin: SpringerVerlag, 1992. Gilmore, R, Catastrophe Theory for Scientists and Engineers. New York: Dover, 1993. Goetz, P. “Phil’s Good Enough Complexity Dictionary.” http://www.cs.buffalo.edu/gaetz/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, pm 211, 1987, Thorn, R. Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Reading, MA: Reading, MA: AddisonWesley, 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 TheorySelected Papers 197% 1977. Reading, MA: AddisonWesley, 1977.
Categorical Game A GAME in which no draw is possible.
see TSCHIRNHAUSEN CUBIC Categorical Variable A variable which belongs ber of CATEGORIES.
to exactly
one of a finite
num
204
Category
Catenary
Category A category
consists of two things: an OBJECT and a MORPHISM (sometimes called an “arrow”) An OBJECT is some mathematical structure (e.g., a GROUP, VECTOR SPACE, or DIFFERENTIABLEMANIFOLD) anda MORPHISM is a MAP betweentwo OBJECTS. The MORPHISMS 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 l
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
equation
CARTESIAN
MORPHISM.
y = ia(exia
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, homomorphisms are usually continuous maps between topological spaces. However, there are also other category structures having TOPOLOGICAL SPACESas objects, but they are not nearly as important as the “standard” category of TOPOLOGICAL SPACES and continuous maps.
and the CES~RO
for the catenary
+ Cxia)
is given by
= acosh is
EQUATION
(s2 + a2)bc = a.
The catenary gives the shape of the road over which a regular polygonal ‘cwheel” can travel smoothly. For a regular ngon, the corresponding catenary is (3) where
(4)
see also ABELIAN CATEGORY, ALLEGORY, EILENBERGSTEENROD AXIOMS, GROUPOID, HOLONOMY, LOGOS, MONODROMY, TOPOS References Freyd, P. J. and Scedrov, A. Categories, Allegories. dam, Netherlands: NorthHolland, 1990.
Amster
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 compositions of mappings. The objects studied in category theory are called CATEGORIES.
see also CATEGORY
The ARC LENGTH,~UR~AT;RE, GLE are S=asinh
t 0 
a
and TANGENTIAL AN
,
(5)
(6) # = 2 tar?
[tanh Ml
l
The slope is proportional to the ARC LENGTH sured from the center of symmetry.
(7) as mea
see also CALCULUS OF VARIATIONS, CATENOID, LINDELOF’S THEOREM,~URFACE OF REVOLUTION References
Catenary
The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational 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 unArchive). der gravity would be a PARABOLA (MacTutor 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.
Geometry Center. “The Catenary.” http: //www. geom.umn, edu/zoo/diffgeom/surfspace/catenoid/catenary.html. is a Catenary.” $5.3 Gray, A. “The Evolute of a Tractrix in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 8081, 1993. Lawrence, J. D. A Catalog of Special Plane Cumes. New York: Dover, pp. 195 and 199200, 1972. Lockwood, E. H. “The Tractrix and Catenary.” Ch. 13 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 118124, 1967. “Catenary.” MacTutor History of Mathematics Archive. http://wwwgroups.dcs.stand.ac.uk/lhistory/Curves /Catenary.html. Pappas, T. “The Catenary & the Parabolic Curves.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 34, 1989. Yates, R. C. “Catenary.” A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1214, 1952.
Catenary
Evolute
Catenary
Evolute
Catenoid Catenary
Radial
The KAMPYLE
205
Curve
OF EUDOXUS.
Catenoid
X = a[x

i sinh( 2t)]
Y = 2acosh
Catenary
t.
Involute \ \
\
A CATENARY are the only
\
l_
of REVOLUTION. The catenoid and PLANE SURFACES OF REVOLUTION which are also SURFACES. The catenoid can be given by the equations
MINIMAL
parametric
= ccosh
X
The parametric
equation
for a CATENARY
is
t 1) dt> =u[cosht
Y = ccosh
v

cosu
(1)
z 0c
sinu
(2)
0c
x = 21,
(1)
(3)
where u E [0, 27~). The differentials
are
so dr
dt 121
dx = sinh
[ 1 1 sinh t
=a
(2)
and
ds2 = Jdr2) = a2(l
+ sinh2 t) di2 = a2 cosh2 dt2 ds dt
the LINE
ELEMENT
cash t dt = a sinh t s
(z)
= cosh2 The PRINCIPAL
s=a
This
curve is called
of the INVOLUTE
+ l]
dv2 + cosh2 (z)
dv2 + cosh2 CURVATURES
z
0C
dzL2.
1
C
y = asecht.
(9)
du2 (7)
(8)
is
(8)
(5)
are
(7)
x = a(t  tanht)
a TRACTRTX.
cos u du
is
~2 =  sech2 and the equation
V

0C
(6)
= [sinh2
(5)
Therefore,
sin u dv + cash
(4)
ds2 = dx2 + dy2 + dz2
(6)
= acosht.

0C
cos u dv  cash
dz = du, SO
(4)
V
dY = sinh
(3)
=aJX=acosht
E 0C
The MEAN
CURVATURE
21 . 0c
of the catenoid HO
(9) is
(10)
Caterpillar
206
Cauchy
Graph
CURVATURE is
and the GAUSSIAN
Cauchy
(11)
Boundary
Distribution
Conditions
BOUNDARY CONDITIONS of a PARTIAL DIFFERENTIAL EQUATION which are a weighted AVERAGE of DIRICHLET 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, CAWHY PROBLEM, DIRICHLET BOUNDARY CONDITIONS, NEUMANN BOUNDARY CONDITIONS References Morse,
P. M. and Feshbach,
ics,
Cauchy’s deformed The HELICOID can be continuously catenoid with c = 1 by the transformation
into
x(u, v) = cos cy sinh zt sin u + sin CI:cash v cos u y(u,v)
= cosasinhvcosu+sin~~coshvsin~(13)
z(u,v)
= ucosa
+ vsinar,
where QI = 0 corresponds to a catenoid.
M. P. “The from
the
(Ed. G. Fischer). 1986. Fischer, G. (Ed,). Mathematical
Catenoid.”
of
§3.5A in Wxthematical Universities
Braunschweig, Plate Models,
and
Germany:
Museums
Vieweg,
90 in Mathematische Bildband/Photograph
p* 43,
Modelle/ Volume.
Braunschweig, Germany: Vieweg, p+ 86, 1986, Geometry Center. “The Catenoid.” http: //www. geom.umn, edu/zoo/diffgeom/surfspace/catenoid/. Gray, A. “The Catenoid.” $18.4 Modern Difierential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 367369, 1993. Meusnier, J. B. “M&moire sur la courbure des surfaces.” Me’m. des saztans &angers 10 (lu 1776), 477510, 1785.
Caterpillar Graph A TREE with every NODE on a central EDGE away from the stalk.
stalk
or only one
References Gardner, ments.
M. Wheels,
Life,
and
other
New York: W. H. Freeman,
Amuse
Mathematical
where r(z) is the GAMMA FUNCTION. Cauchy Criterion A NECESSARY and SUFFICIENT condition for a SEQUENCE & to CONVERGE. The Cauchy criterion is satisfied when, for all E > 0, there is a fixed number IV such that ISj  SJ < E for all i, j > IV. Cauchy
Distribution
8 b
\
Problem
Cauchy
p. 160, 1983.
Binomial
= 5
(1)
of Archimedes
see ARCHIMEDES' CATTLE PROBLEM Cauchy
x
distribution, also called the LORENTZIAN DISTRIBUTION, describes resonance behavior. It also describes the distribution of horizontal distances at which a LINE SEGMENT tilted at a random ANGLE cuts the XAXIS. Let 8 represent the ANGLE that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then The
tan0 Cattle
Formula
(14)
to a HELICOID and a = 7r/2
Collections
Integral
Phys
1953.
(12)
References Models
Cosine
Theoretical
pp. 678679,
a
see also CATENARY, COSTA MINIMAL SURFACE, HELISURFACE, SURFACE OF REVOLUTION COID, MINIMAL do Carmo,
of
H. Methods
I. New York: McGrawHill,
Part
0 = tanr
x 0b
d&cL
Theorem
(2)
=
dx
l+$b
bdx b2 + x2 ’
(3)
n
rI(
where
1+
(z) q is a GAUSSIAN COEFFICIENT.
see also qBINOMIAL
YQk)7
so the distribution
of ANGLE
8 is given
by
k=l
m=O
THEOREM
dB
 7r
1 n
bdx b2
+
(4) x2
l
Cauchy Distribution This
Cauchy Inequality
d2 de = 1 7r s
is normalized
over all angles,
since
see ah TION
(5)
x/2
and
s
O” $$$ m
= i I 
[tan’
;[+

GAUSSIAN
M. R,
Spiegel,
(6)
York:
and
Problems
pp. 114115,
is smaller
$r
P(x)=
1 7T(5  p)2 + ($)" 1 1 =  +  tanl 2 7T
D(x)
(7)
XP ( 7’
(8)
>
where r is the FULL WIDTH AT HALF MAXIMUM (r = 2b in the above example) and p is the MEAN (cl = 0 in the above example). The CHARACTERISTIC FUNCTION
Cauchy Functional The fifth of HILBERT’S this equation.
1
1+ x2
eipt s
dx
O” cos(rtx/2)
7T
rn 1+
iptqq/2
=e
uo
dx
is a generalization
+
UlZ
+
ofthe
a2z2
+
l
of
SERIES
TAYLOR ‘0
1 lim (la,l>+’ n+m
OF CONVERGENCE,
Inequality case of the HOLDER
TAYLOR
SUM
SERIES
with
INEQUALITY
p=q=2, (9)
by
p4=QQ,
(10) (11)
where equality
(12)
DEVIATION,
SKEWNESS,
and KUR
TOSIS by u2 = 00 71’
RADIUS
dso
Cauchy A special
(w2)2
p2 = 2 = 00 0 for p = 0 p3 = { 00 for p # 0 and the STANDARD
’
Equation
is
see
.
are given
ARITH
eit(rz/2p)
7r s m 
The MOMENTS
O”
N
PROBLEMS
’ = 
the
N * l ni 2
c
CauchyHadamard Theorem The RADIUS OF CONVERGENCE
iS
4(t)
than
MEAN,
<
dis
and
1992.
EQUATION
l/N
and its cumulative
of Probability
McGrawHill,
Cauchy’s Formula The GEOMETRIC MEAN METIC
The general Cauchy distribution tribution can be written as
DISTRIBU
Equation
see EULER = 1.
Theory
New
Cauchy
($)I
NORMAL
References Statistics.
(!)I,
DISTRIBUTION,
207
O 1 00 72 = 00.
It can be proven
f(aix
by writing
+ bi)2 = f)i2
i=l
(14)
If X and Y are variates with a NORMAL DISTRIBUTION, then 2 = X/Y has a Cauchy distribution with MEAN = 0 and full width IL
OX
In 2D, it becomes
(x + %>2 i
= 0.
(3)
I
(15)
r=20,.
for ak = &.
(a2 + b2)(c2 + a2) > (UC+ bd)2.
(13) for p = 0 for p # 0
holds
If b&i is a constant c, then x = cc. 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 2xaibi
(16)
i=l
X=
& 44
(‘&bi)’ 2Eai2
4’&2xbi2
(4) l
Cauchy
208 In order
Integral
Formula
for this to be COMPLEX,
it must
Cauchy be true that
Integral
Now, let z E x0 + reiB, so dz = irei
de. Then
s dz =lim sf(z) s
f (zo + reie)i
with equality when bi/ai is a constant. derivation is much simpler, (a.
The
VECTOR
(6)
Y
where
de.
(5)
Yr
But we are free to allow the radius
b)2 = a2b2 cos28 < a2b2,
Formula
2  20
T to shrink
f(zo + reie)i
Tb0
d0 =
to 0, so f(xo)i
de
s Yr
Yr
de = 27rif(zo),
= if(zo>
(6)
s Yr and similarly
for b.
and
see also CHEBYSHEV EQUALITY
INEQUALITY,
HOLDER
SUM
IN
1Mathematical
and Stegun, C. Functions with Tables, 9th printing.
A. (Eds.). Formulas, New
Handbook Graphs, and
York:
If multiple loops are made tion (7) becomes
Formula
where n(y,zo)
Q=Q+@ Y
Given
A similar
a CONTOUR INTEGRAL, ofthe
formula
f’(z0) =
'/r
'/b
is the WINDING
lim
holds
fh
f (4 dz s Y z  20 ’
for the derivatives
f(z),
h
form x0)  (z  z.  h)] dz
define a path 70 as an infinitesimal CIRCLE around the point zo (the dot in the above illustration). Define the path “(7 as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around ~0.
= lim

hf (4 dz s y (‘z  20  h)(z
1 2ni
f (4 dz s
(9) (z

z())2
l
Y
f (4 dz
2
f"(xo> = 2ni Continuing BER n,
CAUCHY~NTEGRAL THEOREM, ~~~CONTOUR INTEGRAL along any path not enclosing a POLE is 0.
n(y,zo)f(r'(zo)
(10)
s Y (z  zo)3’
the WINDING NUM
the process and adding
Fromthe
see also
 x0)
again,
(2)
Therefore, the first term in the above equation is 0 since “)‘o does not enclose the POLE, and we are left with
1

h+0 2rih
Iterating
is then Y = “(0 + Yn
of
+h)  f(h)
he0
f (4 dz
path
(8)
NUMBER.
1 / Y z  x0
The total
(7)
the POLE, then equa
1 = 27Ti 
n(wo)f(zo)
Integral
f (4 dz ~ s Y z  zo ’
around
Dover,
p* 11, 1972.
Cauchy
1
2Ti
References Abramowitz, M. of Mathematical
f(z0) = 
= r!27Ti
f(z)dz s
(11) y
(z

z(#+1
l
MORERA'S THEOREM
References
(4
A&en,
G. “Cauchy’s Integral Formula.” $6.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 371376, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, ppm 367372, 1953.
Cauchy Integral Cauchy
Cauchy
Test
Integral
Cauchy Integral Theorem If f is continuous and finite on a simply connected region 2? and has only finitely many points of nondifferentiability in R, then f(z)
dx = 0
(1)
f Y contained
and f(z)
Identity
(aI2 + lQ2 + . * * + (albz
dn”)(bl”
(2)
as
f (2) E u + iv
From lows.
this
identity,
the nD
s sf(x7 sf(x7
udx
anbn1)2.
INEQUALITY
fol
THEOREM
+ i dy)
 vdy+i
v dx + udy.
(4)
.I Y
Y
F'rom GREEN'S
CAUCHY

Theorem
see MACLAURINCAUCHY
(u + iv)@
+ .. .
+(a,&
CauchyMaclaurin = l
 ah)2
(3)
then gives ~fW
+ bz2 + =. . + b,‘)
 a2b1)2 + (albs
in R.
iEx+iy
and uniqueness
PROBLEM
CauchyLagrange

y completely
CONTOUR
see also CAUCHY
209
Test
CauchyKovalevskaya Theorem The theorem which proves the existence of solutions to the CAUCHY PROBLEM.
Test
see INTEGRAL TEST
for any closed Writing x as
Ratio
Cauchy Mean Theorem For numbers > 0, the GEOMETRIC METIC MEAN.
MEAN
< the ARITH
THEOREM, Cauchy
Y) dx  9(x, Y) dY = //(g+g)
Principal
Value
dXdY,
Y
(5)
Y> dY z//($$)
Y) dx+dx7
dxdy
R
PV
(6)
r
f(x) dx = lim f (4 dx R+oo R oo b f(x) dx = lim s
CE
Y
PV
so (4) becomes
E+0
sa
where E > 0 and a < c < b. References +i//
(a
But the CAUCHYRIEMANN
dY
dxdy.
(7)
EQUATIONS require that
au dv z = ay dU
 $)
dV Z dX’
(8) (9)
so .f Y
f( z) dz = 0,
A&en, G. Mathematical lando, FL: Academic
Sansone,
G.
York:
Dover,
Orthogonal
CAUCHY
English
ed.
New
Cauchy Problem If f(x,y) is an ANALYTIC FUNCTION in a NEIGHBORHOOD of the point (x~,yo) (i.e., it can be expanded in a series of NONNEGATIVE INTEGER POWERS of(xso) and (y  yo)), find a solution y(x) of the DIFFERENTIAL
dY
CONNECTED region,
THEOREM,RESIDUE
rev.
3rd ed. Or
1985.
EQUATION
(11) also
Functions,
dx
For a MULTIPLY
for Physicists,
pp. 401403,
p. 158, 1991.
Qa E. D.
see
Methods
Press,
INTEGRAL THEOREM, MORERA'S THEOREM (COMPLEX ANALYSIS)
References Arf’ken, G. “Cauchy’s Integral Theorem.” $6.3 in Muthematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 365371, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 363367, 1953.
f(
x’
)
with initial conditions y = yo and x = x0. The existence and uniqueness of the solution were proven by Cauchy and Kovalevskaya in the CAUCHYKOVALEVSKAYA THEOREM. 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
see RATIO TEST
Test
210
Cauchy Remainder
Cauchy Root
Form
Cauchy Remainder Form The remainder of n terms of a TAYLOR
SERIES
is given
These are known as the They lead to the condition
CauchyRiemann
Test
equations.
bY
R,
=
(x  cy (n 
a2u
cx  4 fbyc),
dxdy
l)!
where a < c < X. CauchyRiemann Let
The CauchyRiemann ten as
Equations f (x, Y> = u(x,
+ iv(x,
Y)


d2V dxdy
equations
(14)
'
may be concisely
writ
(1)
Y),
where x E x + iy,
(2)
so dz=dx+idy. The total computed
(3)
derivative of f with as follows.
respect
COORDINATES,
to z may then be
f (reie)
(4) (5)
iy,
so dY z=i=i da: dz
and 4f dz In terms
af ax
1
=
(6) (7)
1,
af
af
aY
3f
dXdL+dy&=da:2dy
the real, or XAXIS,
df the imaginary,
aflay
dx’
(9)
(10)
r

(18)
a?
l
If u and v satisfy the CauchyRiemann equations, also satisfy LAPLACE'S EQUATION in ZD, since
 du dy’
they
(20)
By picking an arbitrary f(z), solutions can be found which automatically satisfy the CauchyRiemann equations and LAPLACE'S EQUATION. This fact is used to find socalled CONFORMAL SOLUTIONS to physical problems involving scalar potentials such as fluid flow and electrostatics. see also CAUCHY INTEGRAL THEOREM, CONFORMAL SOLUTION,MONOGENIC FUNCTION,POLYGENIC FUNCTION Abramowits,
M.
of
(11)
If f is COMPLEXDIFFERENTIABLE, thenthevalueofthe derivative must be the same for a given dz, regardless of its orientation. Therefore, (10) must equal (ll), which requires that du dv (12) dx  dy dv dz=
(17)
References
af /da: = 0, so
du dv df i+ . dz dy dy
and
 RdO  r do  RdO 
(8)
= 0, so
&+i* or yaxis,
become
dR dr 16R at9
(16)
(g)+gg,=o.
dzdx
Along
equations
of u and zt, (8) becomes
df g+ig,i(E+i$) dz= .du dv  au .dV “ay+ay ( z+“z + ) ( > Along
E R(r, t9)eiecrge),
so the CauchyRiemann
ZX y=i 2 = z 
In POLAR
(13)
Mathematical Mathematical
and
Stegun,
Functions Tables, 9th
p. 17, 1972. Arfken, G. “CauchyRiemann matical
demic
C.
with printing.
Conditions.”
Methods for Physicists, 3rd Press, pp. 3560365, 1985.
Cauchy’s
Rigidity
see RIGIDITY
THEOREM
Cauchy
Root
see ROOT
TEST
Test
A.
Theorem
(Eds.).
Formulas, New
ed.
Handbook Graphs, and
York:
Dover,
56.2 in MatheFL: Aca
Orlando,
CauchySchwarz
Integral Integral
CauchySchwarz
Cayley
Inequality Inequality
Cavalieri’s
Let f(z) and g(z) by any two R ,EAL integrable of [a, b], then
functions
Principle
1. If the lengths of every onedimensional slice are equal for two regions, then the regions have equal AREAS.
AREAS of every twodimensional slice (CROSSSECTION) areequalfortwo SOLIDS, thenthe SOLIDS have equal VOLUMES.
2. If the
b f wl(4
with
211
Cubic
dx
equality
IFF f(x)
= kg(x)
with
k real.
see also CROSSSECTION, PAPPUS'S CENTROID THEOREM
References Gradshteyn, ries, and
I. S. and Products,
Ryzhik,
I. M. Tables San
Diego,
of
SeIntegrals, CA: Academic
Press, p. 1099, 1993
References Beyer, 28th
W. El. (Ed.) CRC Standard ed. Boca Raton, FL: CRC
Mathematical
Press,
Tables,
p. 126 and 132,
1987.
CauchySchwarz
Sum Inequality Cayley
Algebra NONASSOCIATIVE DIVISION ALGEBRA with REAL SCALARS. There is an &square identity corre
lablI lallbl. (gg2s
(g2)
(g2)
Equality holds IFF the sequences are proportional. l
l
The
sponding to this algebra. The elements of a Cayley algebra are called CAYLEY NUMBERS or OCTONIONS.
al, ~2, . . . and bl, 62,
28,
FIBONACCI IDENTITY
References Gradshteyn, ries, and
Press, p.
S. and Ryzhik, I. M. Tables Products, 5th ed. San Diego, 1,092, 1979.
of Integrals, SeCA: Academic
METRIC~(U,,U,)
suchthatthe
satisfies d(a,,
a,)
= 0.
min(m,n)+m
Cauchy
sequences in the rat ionals CONVERGE, but they do CONVERGE
REAL NUMBERS can be defined CUTS or Cauchy sequences. see also DEDEKIND CUT
do not necessarily in the REALS.
using either
DEDEKIND
curve
York:
Chelsea,
pp. 226
Theorem
Eisenbud, D.; Green, M.; and Theorems and Conjectures.” 295324, 1996.
Cayley
which
is the
curve for a light
ENVELOPE of reflected (CAT(DIACAUSTIC) rays of a given
source at a given point (known as the The caustic is the EVOEUTE of the
RADIANT POINT). ORTHOTOMIC. References
J. D. A Catalog of Special Plane Curves. New York: Dover, p. 60, 1972. Lee, X. “Caustics.” http://www.best.com/xah/Special PlaneCurvesdir/Caustics,djr/caustics.html. Lockwood, E. H. “Caustic Curves.” Ch. 24 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 182185, 1967. A Handbook on Curves and Their Yates, R. C. “Caustics.” Properties. Ann Arbor, MI: J. W. Edwards, pp. 1520, 1952.
Harris, Bull.
J. “CayleyBacharach Amer. Math. Sot.
33,
Cubic
a
a_, b ‘I , 0
TEST
ACAUSTIC) or refracted
Lawrence,
New
Let X1, X2 c p2 be CUBIC plane curves meeting in nine points pl, . . . , pg. If X C p2 is any CUBIC containing p8, then X contains pg as well. It is related to PI, “‘I GORENSTEIN RINGS, andis ageneralizationof PAPPUS'S HEXAGON THEOREM and PASCAL'S THEOREM.
Test
Caustic The
Algebra.
References lim
see RATIO
A. G. General 1963+
CayleyBacharach
I.
Cauchy Sequence A SEQUENCEU~, u2,...
Cauchy
References Kurosh,
.
see also
only
A CUBIC RULED SURFACE (Fischer 1986) in which the director line meets the director CONXC SECTION. Cayley’s surface is the unique cubic surface having four ORDINARY DOUBLE POINTS (Hunt), the maximum possible for CUBIC SURFACE (EndraB). The Cayley cubic is invariant under the TETRAHEDRAL GROUT and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endrafi). Ifthe ORDINARY DOUBLE POINTS in projective 3space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, l), then the equation of the surface in projective coordinates is 1
g+
1 1 ;+&+&=0
1
212
Cayley
Cubic
Cayley Hamil ton Theorem
(Hunt). Defining “affine” coordinates finity w = 20 + ~1 + 22 + 2x3 and
with
plane
at in
Endrafi, S. “The Cayley Cubic.” http : //wnw .mathemat ik. unimainz.de/AlgebraischeGeometrie/docs/ Ecayley.shtml. Models from the Collections Fischer, G. (Ed.). Mathematical of Universities and Museums. Braunschweig, Germany: Vieweg, p. 14, 1986. Fischer, G. (Ed.). Plate 33 in Mathematische Modelle/Mathematical
zx
X2 w
then gives the equation 5(22y+x2~+y22ty2Zt~2y+z2x)+2(xy+x~+y~)
plotted in the left different form
figure
= 0
above
(Hunt).
The
slightly
4(x3 + y3 + z3 + w”>  (x + y + x + w)” = 0 is given by EndraB DRAL
COORDINATES,
which, when rewritten becomes
in TETRAHE
Models,
Bildband/Photograph
Volume.
Braunschweig, Germany: Vieweg, p. 33, 1986. Hunt, B. “Algebraic Surfaces.” http : //uww ,mathematik unikl.de/wwwagag/Galerie.html. Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: SpringerVerlag, pp. 115422, 1996. http: //www . uib. no/ Nordstrand, T. “The Cayley Cubic.” people/nfytn/cleytxt.htm,
l
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 New York: Random House, p. 45, 1964.
x2 + y2  x22 + y22 + x2  1 = 0,
plotted
in the right
figure
Their
Graphs.
above. Cayley’s Group Theorem Every FINITE GROUP of order n can be represented a PERMUTATION GROUP on n letters, as first proved Cayley in 1878 (Rotman 1995). see also FINITE
GROUP,
PERMUTATION
as by
GROUP
References Rotman, J. J. An Introduction to the Theory of Groups, ed. New York: SpringerVerlag, p. 52, 1995.
CayleyHamilton
4th
Theorem
Given a11
a12
x
.
The Hessian
of the Cayley
cubic is given by
I
.
.
am2
xo2(XlX2
+
+&x0x1
+
x1X3 X0X3
+ +
x2x3) X1X3)
+ +
Xf(XoX2 X~(XOXl
+ +
X0x3 x0x2
+ +
P2m .
.
.
1 l
l
=
~2x3)
alrn
.. .
l
1
Gd
0 =
“’
a22  x
a21
*
l
arnrn2
l
Xm
+
Cm~Xml
+

l

+
CO,
(l)
x122).
then in homogeneous coordinates x0, 51, x2, and ~3. Taking the plane at infinity as w = 5(x0 + xl + 22 + 2x3)/2 and setting x, y, and z as above gives the equation
A”
+ CmlArnol
+a ma+
~01
= 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
25[~3(y+~)+y3(x+z)+z3(x+y)]+50(x2y2+x2z2+y2z2)
125(x2yz+y2xz+z2xy)+60xyz4(xy+xz+yz)
= 0,
i?X
b C
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 EndraB,
S. “FlZchen mit vielen Doppelpunkten.” 4, 1720, Apr, 1995,
Mitteilungen
DMV
dx
=(ax)(dx)bc
= x2  (a + d)x + (ad  bc) = x2 + clx + c2
(3)
Cayley’s Hypergeometric
Function
Theorem
(4)
CayleyKlein
a2 + bc UC + cd
(a+d)A=
ab + bd bc + d2 I
7a,“cI;f
(adbc)l
=
(5)
adibc
adrbc
=
#bw
cos(
$?)
sin( $0)
y = ie i(+4)12
sin( ;S,
(++w2
Se
(Goldstein
so (a + d)A + (ad  bc)I =

ew+w2
cos( $0)
, I
A2

P
1 (6)
I;;;
213
where Z* denotes the COMPLEX CONJUGATE. In terms of the EULER ANGLES 8, 4, and $, the CayleyKlein parameters are given by a
_ 
Parameters
(8)
The CayleyHamilton theorem states that a n, x n MATRIX A is annihilated by its CHARACTERISTIC POLYNOMIAL det(xI  A), which is manic of degree n.
1960, p. 155).
The transformation matrix is given CayleyKlein parameters by A= f (a” y2+s2p2) $i(a2 + y2  p”  h2) PJ  a7
References
in terms
 a2 + d2  p”) +P2+S2) ib7 + PJ)
fi(7”
+(a”
+r2
7s
of the
 ap
qap+yq ad+@7
.ik, I. M. Tables of Integrals, Seed. San Diego, CA: Academic (Goldstein
the Cay IcyHar nilt on Equation ations .” Amer. Math. Monthly
Hypergeometric
Function
1960, p. 153).
The CayleyKlein parameters may be viewed as parameters of a matrix (denoted Q for its close relationship with QUATERNIONS)
99, 4244,1992.
Cayley’s
Theorem
If (1  q+bC
2Fl(2a,
P Q = Q: y [ sI
2b; 2c; z) = 2 a,?, TL=O which
A(a,b;c+
+; z)

x
see
UZSO
is a HYPERGEOMETRIC
HYPERGEOMETRIC
The
parameters
EULER
ANGLES,
the orientation the identities
(Cn> (c
+
of a linear isfies
FUNCTION.
+
(14)
6v.
(15)
space having
complex
axes. This matrix
I is the IDENTITY as well as
sat(16)
and At the MATRIX
MATRIX
TRANSPOSE,
a, /3, y, and S which, like the three provide a way to uniquely characterize of a solid body. These parameters satisfy
IQl*lQI = 1.
cya*+yy*=1
(1) (2)
= 1
(3)
a*@ + y*6 = 0
(4)
= 1
(17)
In terms ofthe EULER PAR AMETERS ei and the PAULI M ATRICES ui, the Qmatrix can be writ ten as
m*+pp*=1
a6  pr
yu
QtQ  QQt = I 7
FUNCTION
+ ss*
=
p”
Parameters
pp*
the transformations
V’
where
CayleyKlein
(13)
u1 = au+pv
n=O
where 2 Fl (a,b;c;z)
characterizes
2F1 (c  a, c  b; c;; z) 00
1 (12)
Q= e0l+i(elal + (Goldstein
(5 >
and
>
e2u2
+
e3u3)
(18)
1980, p. 156).
see ~SO EULER ANGLES, MATRICES, QUATERNION
EULER
PARAMETERS,
PAULI
References Goldstein, H. “The CayleyKlein Parameters and Related Quantities.” 545 in Classical Mechanics, 2nd ed. Reading, MA: AddisonWesley, pp. 148158, 1980.
CayleyKleinHilbert
214
CayleyKleinHilbert The METRIC of Felix
Metric
Metric Klein’s model
Cayley’s Sextic Evolute
for HYPERBOLIC
GEOMETRY,
a2(1  X2”) 911
= 
(1
Xl2
x,y

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 Archive). Cayley’s sextic is given in POLAR COORDINATES bY
a2x1x2 912

(1
922
see
also
T = acos3($),
(1)
T = 4bcos3(50),
(2)
=
=
‘HYPERBOLIC
Xl2

or
x,2)2
a2(1  21~)
x2y
(1  Xl2 
where b E a/4. tion is
In the latter
case, the CARTESIAN
GEOMETRY
4(x2 +y2 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 + cil + di2 + ei3 +
f id + gi5 +
The parametric
 i!~x)~ = 27a2(x2
equations
+y2)2.
(3)
 1)
(4)
are
x(t)
= 4acos4($)(2cost
y(t)
= 4acos3(+t)
sin( $).
The
ARC 'LENGTH,
CURVATURE, and TANGENTIAL
also COMPLEX NUMBER, QUATERNION, REAL NUMBER
see
DEL
is a quantity PEZZO
which
SURFACE,
References Conway, J. H. and Guy, FL K. “Cayley Numbers.” In The Book of Numbers. New York: SpringerVerlag, pp. 234235, 1996. Okubo, S. Introduction to U&onion and Other NonAssociative Algebras in Physics. New York: Cambridge
Cayley’s
(7)
qs(t) = 2t.
(8)
References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 and 180, 1972. MacTutor History of Mathematics Archive. “Cayley’s Sextic.” http: // wwwgroups , dcs . Stand. ac . ~&/history/
Curves/Cayleys.html. Cayley’s
Sextic
Evolute
Surface
see CAYLEY CUBIC Cayley’s
(6)
kc(t) = + sec2(+t),
Press, 1995. Ruled
AN
GLE are
s(t) = 3(t + sint),
The second type of Cayley number describes a DEL PEZZO SWRFACE.
(5)
his,
where each of the triples (io, ii, is), (ii, i2, id), (& i3, is), (i3, id, is), (id, is, io), (is, is, il), (&, io, i2) behaves like the QUATERNIONS (i, j, k). Cayley numbers are not AsSOCIATIVE. They have been used in the study of 7 and 8D space, and a general rotation in 843 space can be written
University
equa
\\ \ //// / 
Sextic
The EVOLUTE of Cayley’s x=ia+ Ywhich
&a[3 &@sin($t)
is a NEPHROID.
sextic is cos( it)
 cos(2t)l
 sin(2t)],
Cayley
Cellular
Llkee
Cayley Tree A TREE in which each NODE branches. The PERCOLATION tree having z branches is
has a constant number of THRESHOLD for a Cayley 1
pc = z1’ see
CAYLEY
also
GRAPH
Cayleyian Curve The ENVELOPE of the lines connecting corresponding points on the JACOBIAN CURVE and STEINERIAN CURVE. The Cayleyian curve of a net of curves of order 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 York: Dover, pm 150, 1959.
Plane
Curves.
bch Cohomology The direct limit of the COHOMOLOGY groups EFFICIENTS in an ABELIAN GROUP of certain of a TOPOLOGICAL SPACE. Ceiling
New
with COcoverings
Function
[xl Ceiling [xJ Nint (Round)    Lx] Floor fl _.“1 x
4 2
The function [zl which gives the smallest INTEGER 2 x:, 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 cx), where int(x> is the INTEGER PART of zc.
see U~SO FLOOR FUNCTION, INTEGER PART, NINT References Graham, R. L.; Knuth, D. E.; and Patashnik, 0, “Tnt eger Functions.” Ch. 3 in Concrete 1Mathematics: A Found&ion for Computer Science. Reading, MA: AddisonWesley, pp+ 67101, 1990. Iverson, K. E. A Programming Language. New Yor k: Wiley, p. 12, 1962. Schroeder, M. Fructals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 57, 1991.
Cell A finite
regular
see also
16CELL,
215
Automaton
POLYTOPE. 24CELL,
120~CELL,
600CELL
Cellular Automaton A grid (possibly 1D) of cells which evolves according to a set of rules based on the states of surrounding cells. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his %niversal 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, pa 227). 1D automata are called “elementary” and are represented by a row of pixels with states either 0 or 1. These can be represented with an 8bit binary number, as shown by Stephen Wolfram. Wolfram further restricted the number from 28 = 256 to 32 by requiring certain symmetry conditions. The most wellknown cellular automaton is Conway’s game of LIFE, popularized in Martin Gardner’s ScienAlthough the computation of tific American columns. 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 Life. Adami, C. Artificial Buchi, J. R. and Siefkes, Algebras pressions.
and
Wheels,
Life,
Grammars:
Cambridge, MA: MIT Press, 1998. D. (Eds.). Finite Automata, Their Towards
a Theory
of Formal
Ex
New York: SpringerVerlag, 1989. Burks, A. W. (Ed.), Essays on Cellular Automata. UrbanaChampaign, IL: University of Illinois Press, 1970. Cipra, B. “Cellular Automata Offer New Outlook on Life, the In What’s Happening in the Universe, and Everything.” Mathematical Sciences, 1995l 996, Vol. 3. Providence, RI: Amer. Math. Sot., pp. 7081, 1996. Dewdney, A. K. The Armchair Universe: An Explorution of Computer Worlds. New York: W. H. Freeman, 1988. Gardner, M. “The Game of Life, Parts IIII.” Chs. 2022 in and
Other
Mathematical
Amusements.
New
York: W. H. Freeman, pp. 219 and 222, 1983. Gutowitz, H. (Ed.). Cellular Automata: Theory and Experiment. Cambridge, MA: MIT Press, 1991. Levy, S. Artificial Life: A Report from the Frontier Where Computers Meet Biology. New York: Vintage, 1993. Martin, 0.; Odlyzko, A.; and Wolfram, S. “Algebraic Aspects of Cellular Automata.” Communications in Mathematical Physics 93, 219258, 1984. McIntosh, H. V. “Cellular Automat a.” http://www.cs. cinvestav.mx/mcintosh/cellular.html. Preston, K. Jr. and Duff, M. 5. B. Modern Cellular Automata: Theory and Applications. New York: Plenum, 1985. Sigmund, K. Games cf Lifer Explorations in Ecology, Evolution and Behaviour. New York: Penguin, 1995. Sloane, N. 3. A. Sequences A006977/M2497 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Sloane, N. J. A. and Plouffe, S. Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
,
Cellular
216 Toffoli,
T. and Margolus,
A New
Press, Wolfram,
Centered Pentagonal
Space N.
Cellular
Automata
for Modeling.
Environment
1987. S. “Statistical
Mechanics
Centered
Machines:
Cambridge,
Cube
Number
Number
MA: MIT
of Cellular
Automata.”
Rev. Mod. Phys. 55, 601644, 1983. Wolfram, tomata.
Wolfram, Papers.
Wuensche,
S. (Ed.). Reading, S.
and
Theory
Application
MA: AddisonWesley,
Cellular
Automata
and
An
Addison
1992.
Wesley,
Atlas of Basin of Cellular Automata.
Cellular Space A HAUSD~RFF SPACE which called CWCOMPLEX.
Cellular
Complexity:
Reading, MA: AddisonWesley, A. and Lesser, M. The Global
lular Automata: of OneDimensional
of
Au
1986. Collected
1994. Dynamics Attraction
of CelFields
Reading,
has the structure
MA:
of a so
CC&
Center A special placement
POINT which usually has some symmetric with respect to points on a curve or in a SOLID. The center of a CIRCLE is equidistant from all puints 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, CIRCUMCENTER, CURVATURE CENTER, ELLIPSE, EQUIBROCARD CENTER, EXCENTER, HOMOTHETIC CENTER, INCENTER, INVERSION CENTER, Is0G0~1c CENTERS, MAJOR TRIANGLE CENTER, NINEPOINT CENTER, ORTHOCENTER, PERSPECTIVE CENTER, POINT, RADICAL CENTER, SIMILITUDE CENTER, SPHERE, SPIEKER CENTER, TAYLOR CENTER, TRIANGLE CENTER, TRIANGLE CENTER FUNCTION, YFF CENTER OF CONGRUENCE Center
A FIGURATE
NUMBER
ofthe
form,
= n3 + (n  1)” = (2n  l)(n2
 n + 1).
The first few are 1, 9, 35, 91, 189, 341, A005898). The GENERATING FUNCTION tered cube numbers is x(x”
see
+5x2 +5x (x  1)4 CUBIC
also
+ 1)
= 2 + 9x2 + 35x3
. . . (Sloane’s for the cen
+
91x4
+
.
l
.
NUMBER
References Conway, J. H. and
Guy, R. K. The Book of Numbers. New York:. SpringerVerl@, p. 51, 1996. Sloane, N. J. A. Sequence A005898/M4616 in ‘(An OnLine Version of the Encyclopedia of Integer Sequences.”
Centered
Hexagonal
Number
see HEX NUMBER Centered
Pentagonal
Number
Function
~~~TR~ANGLE 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 UZSO IS~CLINIC GROUPS, Center
of Mass
see CENTROID
(GEOMETRIC)
NILPOTENT GROUP
A CENTERED POLYGONAL NUMBER consistingofacentral dot with five dots around it, and then additional dots in the gaps between adjacent dots. The general term is (5n2  5n + 2)/2, and the first few such numbers are 1, 6, 16, 31, 51, 76, . . , (Sloane’s A005891). The GENERATING FUNCTION of the centered pentagonal numbers is x(x2 + 3x + 1) (x  1)”
= x + 6x2 + 16x3 + 31~~ + . . . .
~~~UZSOCENTERED SQUARENUMBER,CENTEREDTRIANGULAR
NUMBER
References Sloane, N. J. A. Sequence A005891/M4112 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
.
Centered
Polygonal
Centered
Polygonal
Central
Number
Beta Function
217
dots in the gaps between adjacent dots. The general term is (3n2  3n + 2112, 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
Number
0 @ l
A FIGURATE NUMBER in which layers of POLYGONS are drawn centered about a point instead of with the point at a VERTEX.
~~~UZSOCENTEREDPENTAGONALNUMBER,CENTERED SQUARE NUMBER,~ENTERED TRIANGULAR NUMBER
x(x2 + x + 1) = x + 4x2 + 10x3 + 19x4 + * ’ (1  x)3
~~~&~CENTERED SQUARE NUMBER
l
l
PENTAGONALNUMBER,CENTERED
References Rekrences Sloane, N. J. A. and Plouffe, S. Extended M3826 in The Encyclopedia of Integer Diego, CA: Academic Press, 1995.
Centered
Square
entry
for sequence Sequences. San
Sloane, N. J. A. Sequence A005448/M3378 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
cent illion In the American
Number R
see also
system,
Central
Angle
4 @3
V v
A CENTERED POLYGONAL NUMBER consistingofacentral dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is n2 + (n  l)“, and the first few such numbers Centered are 1, 5, 13, 25, 41, . . . (Sloane’s A001844). square numbers are the sum of two consecutive SQUARE NUMBERS and are congruent to 1 (mod 4). The GENERATING FUNCTION giving the centered square numbers is
x(x + 1)” (1
x)3
= x + 5x2 + 13x3 + 25x4
10303.
LARGE NUMBER
+
l
.
l
*c
An ANGLE having its VERTEX at a CIRCLE'S center which is formed by two points on the CIRCLE'S CIRCUMFERENCE. For angles with the same endpoints, 8, = 2&, where 0i is the INSCRIBED ANGLE. References Pedoe, D. Circles: ton, DC: Math.
Assoc.
Central
Fhction
View, rev. ed. Washingpp+ xxixxii, 1995.
A Mathematical
Amer.,
l
NUMBER$ENTERED POLYGONAL NUMBER,~ENTERED TRIANGULAR NUMBER,SQUARE NUMBER ~~~~Z~OCENTEREDPENTAGONAL
Beta
References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, p. 41, 1996. Sloane, N. J. A. Sequence A001844/M3826 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
2.5: 5:
Centered
Triangular
Number 7.5:
13Re[b
Im[b
z]
z]
lb
2 2
40 20
121
The central A CENTERED POLYGONAL NUMBER consistingofacentral dot with three dots around it, and then additional
4
121
beta function P(P)
is defined = WP,
P>,
by (1)
218
Central
where B(p, identities
q) is the BETA FUNCTION.
= 2 12pB(p,
P(P)
= 2l”
s 1

 p,p>
(4)
q2p
J(a, b),
dt (7)
JrF
beta function
satisfies
+x>= d(x)
(2+4x)P(1
1 PC 
x1
2
 x)P(x) =2 4x4
(8)
= 27WOt(Xx)
(9)
tan(rx)P(x)
(10)
P(x)P(x + 3> = 24x+17rP(2x)P(2x + $>*
(11)
For p an ODD POSITIVE INTEGER, the central beta func!
tion
satisfies
the identity 1
P(P4
= 3
(P1)/2
n
2x
have GENERATING
+
Y
zn
p1
p
(x+
;)
’
(12)
FE=1
see UZSO BETA FUNCTION, REGULARIZED
BETA FUNC
TION References Borwein, J. M. and Zucker, I. J. “Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominators.” IMA J. Numerical Analysis 12, 519526, 1992.
Central Binomial Coefficient The nth central binomial coefficient
m
(5)
gives the WALLIS FORMULA
J(a, b) =s (1  zx)p(l
which
1
d ’ tl
The central
0 2n n
(3)
tP dt
= 2 ‘Wb
bp(a/b)
The above coefficients are a superset “central” binomial coefficients
(2)
cos(~p)B(;
the latter
where
the
n(n + 2P) (n+p>(n+p)’
nl
With p = l/2, When p = a/b,
It satisfies
+)
o (1+
203 P rI
Central
Coefficient
Binomial
Conic
of the alternative
n!  (2 > n! ( >2’ FUNCTION
= 1 + 2x + 6x2 + 20x3 + 70x4
+
l
.
.
.
The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, . . . (Sloane’s A000984). Erdes and Graham (1980, p. 71) conjectured that the central binomial coefficient (2) is wver SQUAREFREE for n > 4, and this is sometimes known as the ERD~S SQUAREFREE CONJECTURE. S~RK~ZY'S THEOREM (S&k&y 1985) provides a partial solution which states that the BINOMIAL COEFFICIENT (F) 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 (2”n’“) are also never SQUAREFREE for sufficiently large n as long as d is not “too big.” BINOMIAL COEFFICIENT, CENTRAL TRINOsee also MIAL COEFFICIENT, ERD~S SQUAREFREE CONJECTURE,S~LRK~ZY'S THEOREM,QUOTA SYSTEM References Bounds on ExponenGranville, A. and Ramare, 0. “Explicit tial Sums and the Scarcity of Squarefree Binomial Coefficients .” Mathematika 43, 73107, 1996. Sander, J. W. “On Prime Divisors of Binomial Coefficients.” Bull. London Math. Sot. 24, 140142, 1992. S&rkiizy, A. “On Divisors of Binomial Coefficients. I.” J. Number Th. 20, 7080, 1985. Sloane, N. J. A. Sequences A046098, A000984/M1645, and A001405/M0769 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Vardi, I. “Application to Binomial Coefficients,” “Binomial Coefficients, ” “A Class of Solutions,” “Computing Binomial Coefficients,” and “Binomials Module and Integer.” 52.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematics. Redwood City, CA: AddisonWesley, pp. 2528 and 6371, 1991.
Central Conic An ELLIPSE or HYPERBOLA. is defined
as ( ,a2,),
(L) is a BINOMIAL COEFFICIENT and Ln] is the FLOOR FUNCTION. The first, few values are 1, 2,3, 6, 10,
see
also
CONIC
SECTION
where 20,35,
70,126,252,...
binomial l4x2
coefficients  &=22
2(2x3
 x2)
(Sloane’s AOOl405). The central have GENERATING FUNCTION = 1 +2x+3x2
+6x3
+10x4
+....
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.
References Coxeter, H. S. M. and Greitzer,
S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 146150, 1967. Ogilvy, C. S. Excursions in Geometry. New York: ‘Dover, p. 77, 1990.
Central DifKerence Central Difference The central difference intervals fi is defined
Central Limit
219
Now write for a function by
tabulated
at equal
(Xn)
=
fn+1

fw
+ x2 + . . . + x~)~)
= (N“(xl

=&x+1/2 = 6:+1,2
Wn+l/a)
Thoren
Nn(xl+..
.
+
xN)np(xl)
l
l
*I
dxl    dxN,
r m
(1)
(4) Higher order differences ODD powers,
2k
6 n+l/Z
=
may be computed
E(l)’
(y)
for EVEN and
(2)
fn+kj
=
j=O
6 2k+l n+1/2
=
y(1)’
(‘“;
‘>
1I I + xN)n
xP(Q) ** ’ p(xN)
(3)
fn+k+lj.
j=O
2nif(xl

see also BACKWARD DIFFERENCE, ENCE, FORWARD DIFFERENCE
DIVIDED
DIFFER
References Abramowitz, M. and Stegun, C. A. (Eds.). “Differences.” $25.1 in Handbook of Mathematical Functions with Formulas, Graphs,
so we have
and Mathematical
York: Dover, pp. 877478,
Tables, 9th printing.
+ . . . + XN) N
n 1
1n!
rwe2rif (~I++x:N)/N l **p(xN)dxl***dxN
x p(x1)
p(x+.p(xN)dx1
. ..dxN
JW
e2rifxl/N
New
p(a)
1
dxl
1972.
e2xifxN/N
Central Limit Theorem random Let 21,52,. . , XN be a set of Iv INDEPENDENT variates and each xi have an arbitrary probability distribution P(xl, . , XN) with MEAN pi and a finite VARIANCE gi2. Then the normal form variate l
l
dxl . . . dxN

p(xN)
1
dxN
N
1
l
= (1,
[l+
(y)x+;
(y)2x2+...]p(x)dx}N
N
has a limiting distribution which is NORMAL (GAUSSIAN) with MEAN p = 0 and VARIANCE o2 = 1. If conversion to normal form is not performed, then the variate
x’p(x) =
1+ y
(2)  g
(x2)
dx + O(N3)
+ O(iv“)]
1
N
N
XL 
N IL
xi
(2)
= exp
@$
(x2) + O(N3)]
} .
(5) is NORMALLY DISTRIBUTED with PX = pLs and OX = ~,/a. To prove this, consider the INVERSE FOURIER TRANSFORM of P&f).
Now expand ln(1 +x)
= x  +x2 + ix3
+. . . ,
(6)
so Fl
[PX (f)]
E [m
e2Tifxp(X)
dX
00
(2rifx)np(,)
x n=O

dX
n!
x
n=O
n!
+;w
/”
Xnp(X)
= exp
O” Cznif >” lxjn
IE
n!
’
I>
(x)2 + O(N3)
dx
2Tif (2)  (2nf)2((x2)  (‘>“> + o(N2) 2N
(3) $=: exp
2Tifpx

(2rf)2a,2 2N
1 1
I
(7)
220
Central
Limit
Theorem
Centroid
since
(8) (9)
px = (x) 5x2
Taking
E
(x2)

(x)2
l
TRANSFORM,
the FOURIER Px E
e2”ifxF1[Px(f)]
df
(10) This
Central Trinomial Coefficient The nth central binomial coefficient is defined as the coefficient of xn in the expansion of (1 +z + x2)? 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 (PetkovBek et al. 1996, p. 160). The GENERATING FUNCTION is given by
f( x=>
r oo
(Geometric)
1 J(1+
= I+
z)(l
x + 3x2 + 7x3 + , . . .
 3x)
see also CENTRAL BINOMIAL COEFFICIENT References
is of the form
eiafbf2 df,
(11)
where a E Z;rr(p,  x) and b E (27~~)~/2N. But, from Abramowita and Stegun (1972, pa 302, equation 7.4.6), eiaf
df = p2/4b
bf2
Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. A=& Wellesley, MA: A. K. Peters, 1996. Sloane, N. 3. A. Sequence A002426/M2673 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Centralizer The centralizer of a FINITE nonABELIAN GROUP G is an element z of order 2 such that
(12)
Q(z) see also CENTER
Therefore,
SIMPLE
= {g f G : gx = zg}.
(GROUP),
NORMALIZER
Centrode
where 7 is the TORSION, K is the CURVATURE, T is the TANGENT VECTOR, and B is the BINORMAL VECTOR.
(13) But
OX = (r,/fi
and px = px, so
Px = 5x
1 6
e(Px42/2flx2
see
LIMIT
THEOREM,
LINDEBERGFELLER
LYAPUNOV
CONDITION
References M.
Abramowitz, of Mathematical Mathematical
1972. Spiegel,
M.
R.
and
Stegun,
Functions Tables, 9th Theory
Amer.
and
C. A.
(Eds.).
uith Formulas, printing. New Problems
Hand book Graphs, and
York:
of Probability
Bracewell, R. The Fourier New York: McGrawHill,
Monthly
102,
483494,
1995.
Transform
and
pp. 139140
Its Applications.
and 156, 1965.
Centroid (Geometric) The CENTER OF MASS of a 2D planar LAMINA or a 3D solid. The mass of a LAMINA with surface density function a(x,y) is
M= ss4x7
Dover,
Y) dA.
and
New York: McGrawHill, pp. 112113, 1992. S. L. “Alan Turing and the Central Limit Theorem.” Math.
the cenas
References
Statistics.
Zabell,
is defined
r
LINDEBERG CONDITION,
UZSO
CENTROID,
JYm xf (4 dx
(5= >
(14)
l
The ‘Lfuzzy” central limit theorem says that data which are influenced by many small and unrelated random effects are approximately NORMALLY DISTRIBUTED. CENTRAL
Centroid (Function) By analogy with the GEOMETRIC troid of an arbitrary function f(x)
The coordinates
of the centroid
(1)
(also called the CENTER
OF GRAVITY) are a:
=
JJ
Y> dA
X5(X?
M
(2)
Centroid
(Orthocentric
System)
Cen trod (3)
The centroids of several common symmetrical axis are summarized
laminas along the nonin the following table.
Figure parabolic
segment
gh 4r 37r
semicircle
(Wangle)
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 1 1 1 
l


(1)

a’b’c’
In
3D,
P(X,Y, 4
the
mass
of a solid
with
density
function
P(X, 4dv, JJJ
is
M=
(4
Y,
and the coordinates
of the center
Y? 4 dV
(6)
M
(7) Figure
z
cone conical
frust urn
h(R12+2R&+3R22) 4~R12+Rl&+Rz2)
(2)
csc A : csc B : csc C. If the sides of a TRIANGLE
of mass are
(5)
y = JJ YPh
or
are divided
A2P1 
&%
Ad’2 
PA3
P2A1
P3A2
so that P
(3)
d
API P2 P3 is M (Johnson
the centroid of the TRIANGLE 1929, p. 250).
Pick an interior point X. The TRIANGLES SXC, CXA, and AXE? 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,
hemisphere
d = ;(dA
+ dB +dc),
(4
paraboloid pyramid see
also
PAPPUS’S
CENTROID
THEOREM
References Beyer, W. H. CRC Standard Mathematical Tubles, 26th ed. Boca Raton, FL: CRC Press, p. 132, 1987. McLean, W. G. and Nelson, E. W. “First Moments and CenOutline of Theory Mechanics: Statics and
troids .” Ch. 9 in Schaum’s lems of Engineering 4th ed. New York:
McGrawHill,
pp. 134162,
where d, dA, d B, and dc are the distances from the centroid and VERTICES to the line. A TRIANGLE will balance 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 si‘,
and ProbDynamics,
alp2
=
a2p2
=
= ;A,
asp3
1988.
Centroid (Orthocentric System) The centroid of the four points constituting an ORTHOCENTRIC SYSTEM is the center of the common NINEPOINT CIRCLE (Johnson 1929, pa 249). This fact automatically guarantees that the centroid of the INCENTER and EXCENTERS of a TRIANGLE is located at the CIRCUMCENTER.
where A is the AREA of the TRIANGLE. Let P be an arbitrary point, the VERTICES be Al, AZ, and As, and the centroid M. Then m2+PA22+PA32
= m2+m2+MA32+3PM2.
If 0 is the CIRCUMCENTER of the triangle’s then m2 = R2  ;(a” + b2 + c”).
(6) centroid, (7)
References Johnson, on the
R. A. Modern Geometry
MA: Houghton
Geometry: of the Triangle
Mifflin,
1929+
An Elementary and the Circle.
Treatise
Boston,
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 CIRCUMCENTER, POINT,INCENTER,URTHOCENTER References Carr, G. S. 2nd
Formulas ed. New York:
and
EULER
LINE,
Theorems in Pure 622, 1970.
Chelsea, p.
EXMEDIAN
Mathematics,
222
Certificate
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pm 7, 1967. Dixon, R. Mathographics. New York: Dover, pp. 5557, 1991. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mif%n, pp. 173176 and 249, 1929. Kimberling, C. “Central Points and Central Lines in the Plane of a Triangle.” Math. Mag. 67, 163187, 1994. Kimberling, C. ‘Centroid.” http://www.evansville.edu/ ck6/tcenters/class/centroid.html.
Certificate
Certificate
Mean
see FEJES T~TH'S INTEGRAL Ceva’s
Theorem
CERTIFICATE
of Primality
B
see PRIMALITY CERTIFICATE Cesko Equation An INTRINSIC EQUATION which expresses a curve in terms of its ARC LENGTH s and RADIUS OF CURVATURE R (or equivalently, the CURVATURE K).
see &~ARc LENGTH,~NTRINSIC EQUATION, NATURAL EQUATION, RADIUS OF CURVATURE,~HEWELL EQUATION References and
CesAro
of Compositeness
see COMPOSITENESS
Yates,
Ceva k Theorem
of Compositeness
R. C. ‘(Intrinsic Their
Proverties.
pp, 123126,
Cesko
a TRIANGLE with VERTICES A, B, and C along the sides D, E, and F, a NECESSARY SUFFICIENT condition for the CEVIANS AD, BE, CF to be CONCURRENT (intersect in a single point) that BDXEaAF=DC.EA*FB. Given points
and and and is (1)
Vn] be an arbitrary ngon, C a given Let P = [V,..., point, and k a POSITIVE INTEGER such that 1 5 Iz 2 n/2, For i = 1, . . . , n, let Wi be the intersection of the lines CVi and Vi&+k, then
A Handbook on Curves Equations.” Ann Arbor, MI: J. W. Edwards,
f&+1. i=l
li52.
(2)
Fkactal
2x3 A FRACTAL also known as the TORN SQUARE FRACTAL. The base curves and motifs for the two fractals illustrated above are show below.
Here,
ABlICD
and AB CD
[ 1
(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 (Griinbaum and Shepard 1995). Another form of the theorem is that three CONCURRENT lines from the VERTICES of a TRIANGLE divide the opposite 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,
see also
FRACTAL,KOCH
SNOWFLAKE
References Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. 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.
W.
H. (Ed.)
Boca Raton,
CRC
Standard
Mathematical
Tables,
FL: CRC Press, p* 122, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 45, 1967. Griinbaum, B. and Shepard, G. C. “Ceva, Menelaus, and the Area Principle.” Math. Mag. 68, 254268, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 145151, 1929. Pedoe, D. Circles: A MathematicaZ View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xx, 1995. 28th
ed.
Chain Rule
Cevian
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.
Cevian
see also ADDITION CHAIN, ANTICHAINJ~RAUERCHAIN, CHAIN (GRAPH),DILWORTH'S LEMMA,HANSEN CHAIN
b
A line segment which joins a VERTEX of a TRIANGLE with a point on the opposite side (or its extension). In the above figure, b sin a’ ’ = sin(y + a’)
Chain
Fraction
see CONTINUED FRACTION Chain (Graph) Achain ofa GRAPH isa SEQUENCE{~~,X~,...,Z~) such that (q, 4, (~2, Q), . . , (xn1,x,) are EDGES of the
l
l
References
GRAPH.
Thhbault, Monthly
V. “On the Cevians of a Triangle.” 60, 167173, 1953.
Cevian
Conjugate
Amer.
Math.
Chain If g(x)
Rule
is DIFFERENTIABLE at the point x and f(x) is DIFFERENTIABLE at the point g(x), then f o g is DIFFERENTIABLE at x. Furthermore, let y = f(g(z)) and
Point
see ISOTOMIC CONJUGATE POINT
u = g(x), Cevian Transform Vandeghen’s (1965) name for the transformation points to their ISOTOMIC CONJUGATE POINTS.
then dY dz=
taking
see also ISOTOMIC CONJUGATE POINT References
(1)
dz dt= The “general”
Triangle
du dx’
There are a number of related results which also go under the name of “chain rules.” For example, if z = f(x, y), x = g(t), and y = h(f), then
Vandeghen, A. %ome Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.” Amer. Math. Monthly 72, 10911094, 1965.
Cevian
dy adu
(2)
chain rule applies
to two sets of functions
and Ul
=
g&1,
’
l
 7 2,)
l .
4
A2
Given a center a : p : y, the cevian triangle is defined as that with VERTICES 0 : p : y, CY : 0 : y, and Q : p : 0. If A’B’C’ is the CEVIAN TRIANGLE of X and A”B”C” is the ANTICEV~AN TRIANGLE, then X and A” are HARMONIC CONJUGATE POINTS withrespectto A and A’.
l (4)
up
Defining
=gp(x1,**,xn)*
the m x n JACOBI MATRIX
ayl &Ii  3x1 . ( ) [
dY1
by
. ..
8x2
(5)
l
see also
ANTICEVIAN TRIANGLE
3Xj

’ ay7l.L ax1
and similarly
for (ayi/auj)
and (dui /axj)
(g)=(g)($
then gives
(6)
224
Chained
In differential
form,
Arrow this
Champernowne
Notation
Chaitin’s Constant An IRRATIONAL NUMBER s2 which gives the probability that for any set of instructions, a UNIVERSAL TURING MACHINE will halt. The digits in 0 are random and cannot be computed ahead of time.
becomes
l
.
. (7)
(Kaplan see
&SO
1984).
see also VERSAL
HALTING PROBLEM, TURING MACHINE
JACOBIAN,
POWER
RULE,
PROD
the Mysteries
Anton, H. Calculus with Analytic Geometry, 2nd ed, New York: Wiley, p. 165, 1984. and Differentials of Composite Kaplan, W. “Derivatives Functions” and “The General Chain Rule.” $2.8 and 2.9 in Advanced CaZcuZus, 3rd ed. Reading, MA: AddisonWesley, pp. 101105 and 106110, 1984.
Nov.
Chained Arrow Notation A NOTATION which generalizes is defined as
ARROW
NOTATION
Gardner,
York:
UNI
SpringerVerlag,
R. K. The Book p. 61, 1996.
Chait
//www.
Fair
to Hold
241,
2034,
1979,
M. “Chaitin’s
in’s
Omega.”
Ch. 21 in Fractal Recreations
York:
Music, from Sci
W. H. F!reeman,
Properties Information
of ProProc.
Number
CHAITIN’S
CONSTANT
in’s
Omega
see CHAITIN?
of Numbers.
Amer.
http:
and
NOTATION Guy,
Sci.
1992. Kobayashi, K. “Sigma(N)@Complete grams and LartinLof Randomness.” Let. 46, 3742, 1993.
C hait
References Conway, J. H. and
of the Universe.”
Hypercards, and More Mathematical entific American Magazine. New
see
ARROW
MACHINE,
Finch, S. “Favorite Mathematical Constants.” mathsoft.com/asolve/constant/chaitin/chaitin.htm~. Gardner, M. “The Random Number n Bids
References
also
TURING
References
DERIVATIVE,
UCT RULE
see
Constant
New
Chainette
CONSTANT
Champernowne Constant Champernowne’s number 0.1234567891011.. . (Sloane’s AO33307) is the decimal obtained by concatenating the POSITIVE INTEGERS. It is NORMAL in base 10. In 1961, Mahler showed it to also be TRANSCENDENTAL.
see CATENARY The
CONTINUED
FRACTION
stant is [O, 8, 9, 1, 149083, 1, 15,
Chair
of the Champernowne con1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1,
457540111391031076483646628242956118599603939~~~ 710457555000662004393090262659256314937953207~~~ 747128656313864120937550355209460718308998457~~~ 5801469863148833592141783010987, 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, J, 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 lo5 occur at positions 5, 19,41, 102, 163, 247,358,460, . . . and have 6, 166, 2504, 140, 33102, 109, 2468, 136, . . . digits (Plouffe). Interestingly, the COPELANDERD~S CONSTANT, which is the decimal obtained by concatenating the PRIMES, has awellbehaved CONTINUED FRACTION which does not show the “large term” phenomenon. l
A SURFACE with tetrahedral symmetry which, according to Nordstrand, looks like an inflatable chair from the 1970s. It is given by the implicit equation (x2+y2+z2
ak2)2
see also BRIDE’S
A[(
zk)22x2][(Z+k)22y2]
CHAIR
chairtxt
T. “Chair.” . htm.
l
see also COPELANDERD~S SEQUENCES
References Nordstrand,
= 0.
l
http://www.uib.no/people/nfytn/
CONSTANT,~MARANDACHE
L
Change of Variables References
Construction of Decimals NorChampernowne, D. G. “The J. London Math. Sot. 8, 1933. mal in the Scale of Ten.” Finch, S. “Favorite Mathematical Constants.” http: //www. mathsoft.com/asolve/constant/cntfrc/cntfrc.html, Sloane, N. J. A. Sequences A030167 and A033307 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Change of Variables Theorem A theorem which effectively describes how lengths, areas, volumes, and generalized ndimensional volumes (CONTENTS) are distorted by DIFFERENTIABLE FUNCTIONS. In particular, the change of variables theorem reduces the whole problem of figuring out the distortion of the content to understanding the infinitesimal distortion, i.e., the distortion of the DERIVATIVE (a linear MAP), which is given by the linear MAP'S DETERMINANT. So f : R" + R" is an AREAPRESERVING linear MAP IFF Idet(f)l = 1, and in more generality, if S is any subset of Iw”, the CONTENT of its image is given by 1det( f) 1 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 generality of DIFFERENTIAL FORMS on MANIFOLDS, giving the formula
under the conditions that M and W are compact connected oriented MANIFOLDS with nonempty boundaries, f : M + W is a smooth map which is an orientationpreserving DIFFEOMORPHISM of the boundaries. In 2D, the explicit
sf(x,
statement
of the theorem
is
Y> dXdY
R
and in 3D, it is
The change of variables theorem is a simple consequence of the CURL THEOREM and a little DE RHAM COHOMOLOGY. The generalization to nD requires no additional assumptions other than the regularity conditions on the boundary. see
also
is the JACOBIAN, DIFFEOMORPHISM of P).
is th e image
of the original
region
R*,
and f is a global orientationpreserving of R and R* (which are open subsets
IMPLICIT
FUNCTION
THEOREM,
JACOBIAN
Heterences Kaplan,
W.
“Change
of Variables
pp. 238245,
1984.
Chaos A DYNAMICAL 1.
in Integrals.” $4.6 in AdMA: AddisonWesley,
3rd ed. Reading,
vanced Calculus,
SYSTEM
Has a DENSE bits,
2. Is sensitive that initially very different
is chaotic
collection
if it
of points
with
periodic
or
to the initial condition of the system (so nearby points can evolve quickly into states), and
3. IS TOPOLOGICALLY
TRANSITIVE.
Chaotic systems exhibit irregular, unpredictable behavior (the BUTTERFLY EFFECT). The boundary between linear and chaotic behavior is characterized by PERIOD DOUBLING, following by quadrupling, etc. An example of a simple chaotic behavior is the over a plane containing The magnet over which to rest (due to frictional on the starting position (Dickau). Another such pendulum with another
physical system which displays motion of a magnetic pendulum two or more attractive magnets. the pendulum ultimately comes damping) is highly dependent and velocity of the pendulum system is a double pendulum (a pendulum attached to its end).
ACCUMULATION POINT, ATTRACTOR, BASIN see also OF ATTRACTION, BUTTERFLY EFFECT, CHAOS GAME, FEIGENBAUM CONSTANT, FRACTAL DIMENSION, GINGERBREADMAN MAP, HI?NONHEILES EQUATION, HI?NON MAP,LIMIT CYCLE,LOGISTIC EQUATION,LYAPUNOV CHARACTERISTIC EXPONENT, PERIOD THREE THEOREM, PHASE SPACE, QUANTUM CHAOS, RESONANCE OVERLAP METHOD, SARKOVSKII'S THEOREM, SHADOWING THEOREM, SINK (MAP), STRANGE ATTRACTOR 1 References H. Chaos. Singapore: World Scientific, 1984. G. L. and Gollub, J. B. Chaotic Dynamics: An Introduction, 2nd ed. Cambridge: Cambridge University Press, 1996. Cvitanovic, P. Universality in Chaos: A Reprint Selection, 2nd ed. Bristol: Adam Hilger, 1989. http: // f orurn , Dickau, R, M. “Magnetic Pendulum.” / robertd / magnetic swarthmore . edu / advanced pendulum.html. Drazin, P. G. Nonlinear Systems. Cambridge, England: Cambridge University Press, 1992. Field, M. and Golubitsky, M. Symmetry in Chaos: A Search Oxford, for Pattern in Mathematics, Art and Nature. England: Oxford University Press, 1992. Gleick, J. Chaos: Making a New Science. New York: Penguin, 1988. BaiLin, Baker,
where R = f (R*)
225
Chaos
Theorem
Character
Chaos Game
226 Guckenheimer,
J. and Holmes,
Dynamical ed. New
Systems,
and
P. Nonlinear
Bifurcations
of Vector
Oscillations, Fields,
References 3rd
York: SpringerVerlag, 1997. Lichtenberg, A. and Lieberman, M, Regular and Stochastic Motion, 2nd ed. New York: SpringerVerlag, 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 Systems.
of
chaotic
Data
and
the
Exploitation
of
Chaotic
New York: Wiley, 1994. Peitgen, H.O.; Jiirgens, H.; and Saupe, D. Chaos an,d Fructals: New Frontiers of Science. New York: SpringerVerlag, 1992. Poon, L. “Chaos at Maryland.” http: //nwwchaos .umd, edu. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990. Strogatz, S. H. Nonlinear Dynamics and Chaos, with Applications
to Physics,
Biology,
Chemistry,
and
Engineering.
Reading, MA: AddisonWesley, 1994. Tabor, M. Chaos and Integrubility in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989. Tufillaro, N.; Abbott, T. R*; and Reilly, J. An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, CA: AddisonWesley, 1992. Wiggins, S. Global Bifurcations and Chaos: Analytical Methods. New York: SpringerVerlag, 1988. Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: SpringerVerlag, 1990.
Chaos
Table
Game
Pick a point at random inside a regular ngon. Then draw the next point a fraction T of the distance between it and a VERTEX picked at random. Continue the process (after throwing out the first few points). The result of this “chaos game” is sometimes, but not always, a FRACTAL. The case (n, T) = (4,1/2) gives the interior of a SQUARE with all points visited with equal probability. cr **&+ i,A> 05 03 .,%i
Barnsley, M. F. and Rising, H. Fractals Boston, MA: Academic Press, 1993. Dickau, R+ M. “The Chaos Game.”
swarthmore Wagon,
l
Everywhere,
http:// edu/advanced/robertd/chaos*arne
S. Muthematica
in Action.
New
York:
2nd
ed.
forum . html.
.
W. H. F’ree
man, pp. 149163,
1991. @f Weisstein, E. W. ‘Tractals.” http: edu/eww6n/math/notebooks/Fractal
Character
//www. astro
*Virginia.
.m.
(Group)
The GROUP THEORY term for what is known to physicists as the TRACE. All members of the same CONJUGACY 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 SCH~NFLIES SYMBOLS denote different sets of symmetry operations but correspond to the same abstract GROUP and so have the same CHARACTER TABLES.
Character
(Multiplicative) HOMEOMORPHISM of a GROUP into the NONZERO COMPLEX NUMBERS. A multiplicative character w gives a REPRESENTATION on the 1D SPACE c of COMPLEX NUMBERS, wherethe REPRESENTATION acA continuous
tion by g E G is multiplication character is UNITARY ifit erywhere.
by w(g). A multiplicative has ABSOLUTE VALUE 1 ev
References Knapp, A. W+ “Group Representations and Harmonic ysis, Part II.” Not. Amer. Math. Sot. 43, 537549,
Character
(Number
Anal1996.
Theory)
A number theoretic function xk(n) n is a character module k if
for POSITIVE
integral
for all m, n, and Xl;(n)
=
0
if (k, n) # 1. xk can only assume values which are 4(k) ROOTS OF UNITY, where q5is the TOTIENT FUNCTION.
see also Character
see also BARNSLEY’S
FERN
DIRICHLET
Table
LSERIES
Character
Character
Table Dg E AIll 1 A2 1 Bl B2 1 El 2 E2 2
C3 A
E 1
C3
Cs2
1
1
c 2v
E = exp(27ri/3)
>
2
2
z,RZ
x
(XIY)(%RY)
(x2 
IY
2
,z
E
B
C3
1
E
C2
1
1
1
1
Cd3
1
z,RZ
1
Bl
1
2
22
c3v
x+y,z X2  Y2,XY
1
i 1 i
C2
2 2
G&Z)
0
0
0
0
x2 + y2,
x2
2,& (X,Y)(Rx,&) (X6 Y4 ( x2  Y2,XY)
ad(v)
1
1
1 1 1 1 1 1
B2
C4 A
1 1
1
1
A2
Y2,XY)(yz,X4
2c3 cz 3c; 3c; 1 1 1 1 11I 1 1 1 1 I 1 1 1 1
1
E
A1
,XY
2cfJ 1 1 1 1
227
Table
1
z
X2,Y2,ZJ
1 1 1
E
X.1
E
2c4
I
3aw 1
Al A2
E C5 A
E
C5
C5’
C5=
El { ;
;*
;z*
E = exp(2Fi/5) x2 + y2,r2
cdv
Oh
Al
1
(X~YHRaq/~
(YGX4
A2
1
B1
1
1
B2 E
1
1
2
C 5v AI
E 1
Cs4
11111 ;:*
:*
Cx2  Y2, XY)
A
1
B
ll
1
1
1
I
1 1
1
I x2 +y2,x2
I x, Rz
1 1
El t: z+ 1:’ ; :a ;* E2
D2
{
1 E 1 E*
E
E E*
C2(4
C2(y)
1 1
AI
1
Bl
1
B2
1
1
B3
1
1
D3
A1
1 E* 1 E
B2
1
1
1
x 2
E
2
A1
1
A2
1
B1 B2 E
1 1
2
+Y 22 9
1 1
1
1
1 1
1 1 1 1
1 1
(x,
RY)
y)(Rx,
1
2
E2
2
1
x2 
1
1
o2
0
2C5 1
2cs2 1
1
1
22 +Y 7x
z,RZ
1 1
1
x
2
~2)
0
y2
XY
(x7 Y)&
RY)
(x4
Y4
A,sX
2&j
1 1
2c,
x2
229
x2  y2 ZY
0
(2, YmL
2cs2 1 1 2 cos 144” 2 cos 72” C2 30,
+Y
R, 1
0
RY)
(X? Y4
50, 1
x2 + y2, x2
x
3ad
x2 +y2,z2
1
1 1 1 1 1 I 1 1 1 0 1 2 0 1 2
E
C,’
1
lz 1 1
ll
1 1 1 0 0
R,
b, Y)(%,
R&f) b% Y4 cx2  Y21XY)
. . . mu,
1
XY (x2  y2, XY)(XG
1
2Cg
El
x 2 +YJ 22
z,R, o
E
B2
4
1 1
O2
111111% 1 1 1 1 1 1
A2
1
1
B3
Gtl Al
G(x)
E 2c, 3c2 1 1 1
A2
cx2  Y21XY)
2c, 1
2C5 1 1 1 2 2~0s 72” 2 2~0s 144’
Bl
& .?T*
c2 11 1
. *.
El cc II E2 s A
1 1 2 2cos a
... rn..
2 2cos2G
*..
E3 G XD . . .
2 2cos3+ . . .
. .. * . .
..
.
1 1
%
x2 +y2,z2
R, 0 0
(x7 Y>; (REY RY)
(x5 Y4 (x”  Y2, XY)
0 . . .
References Bishop, D. M. “Character Theory and Chemistry.
Tables.” Appendix New York: Dover,
1 in Group pp. 279288,
1993, D5
AI &
E
1 1
B2
2 2 cos
B3
2 2~0s 144”
72”
2cos 144” 2~0s 72O
SC, 1 1 0 0
x2 +y2,z2
z,R, (X7Y)(RmR,)
(XZ,YZ)
(2”  Y2, XY)
Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. of Finite Iyanaga, S. and Kawada, Y. (Eds.). “Characters Groups.” Appendix B, Table 5 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp+ 14961503, 1980.
228
Characteristic
Characteristic
Class
Characteristic Characteristic
Class classes are COHOMOLOGY classes in the BASE SPACE of a VECTOR BUNDLE, defined through OBSTRUCTION theory, which are (perhaps partial) obstructions 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 STIEFELWHITNEY
CLASSES. Characteristic A parameter
(Elliptic
Integral) an ELLIPTIC INTEGRAL
n used to specify
the characteristic
equation
is
a12
a11 t a22
a21
t
l
.
.
l
l
l
C. A.
(Eds.).
with Formulas, printing. New
Handbook Graphs, and
York:
Dover,
p. 590, 1972. Characteristic Equation The equation which is solved to find a MATRIX'S EIGENVALUES, also called the CHARACTERISTIC POLYNOMIAL. Given a 2 x 2 system of equations with MATRIX
(1) the MATRIX
EQUATION is b
d which
&f I[ 1 t 11 x
y=
(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 characteri stic equation
(5) the DETERMINANT of A. For a general
k x k MATRIX all a21 . l .
akl
ries,
.II
al2
alk
a22
a2k .
akk
and
Press,
I. S. and Products,
Ryzhik, I. M. 5th
pp. 11171119,
Characteristic

t
ed.
San
Tables
of Integrals,
Se
Diego, CA: Academic
1979.
(Euler)
Characteristic Factor A characteristic factor is a factor in a particular factorization of the TOTIENT FUNCTION 4(n) such that the product of characteristic factors gives the representation of a corresponding abstract GROUP as a DIRECT PRODUCT. By computing the characteristic factors, any ABELIAN GROUP can be expressed as a DIRECT PRODUCT of CYCLIC SUBGROUPS, for example, 22 8 24 or &@Zz @Z2. 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 FUNCTION
4th
M can have no MATRIX INVERSE, since otherwise
where [AI denotes
akk
References
Shanks,
[;I =M$]= [;I,
(7)
l
References
can be rewritten
[,,, d:t][;I=$I*
0.
.
see also BALLIEU'S THEOREM, CAYLEYHAMILTON THEOREM, PARODI'S THEOREM, ROUTHHURWITZ THEOREM
see EULER CHARACTERISTIC
Functions Tables, 9th
=
l
d..
References
M. and Stegun,
a2k
l
Gradshteyn,
of Muthemutical Muthematicul
alk
. . . .
OF THE THIRD KIND. see UZSO AMPLITUDE, ELLIPTIC INTEGRAL, MODULAR ANGLE, MODULUS (ELLIPTIC INTEGRAL), NOME, PARAMETER Abramowitz,
.** l
ak2
akl
(Field)
Theory,
Characteristic (Field) identity 1, consider For a FIELD K with multiplicative 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 t + 1 $,. . . + l, = 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 II. The FIELDS Q, R, c, and the pADIC NUMBERS Qp have characteristic 0. For p a PRIME, the GALOIS FIELD GF(pâ€?) has characteristic p. If 1y is a SUBFIELD of K, then H and K have the same characteristic.
see also
(6)
D. Solved and Unsolved Problems in Number ed. New York: Chelsea, p. 94, 1993.
FIELD, SUBFIELD
Characteristic
Function
Characteristic The characteristic IER TRANSFORM TION,
Chasles’s Polars Theorem
it follows that dt/ds = 1, dx/ds = 621, and du/ds = 0. Integrating gives t(s) = s, x(s) = 6suo(x), and of integration are 0 u(s) = uo(x), where the constants and uo(x) = u(x,O).
Function function 4(t) is defined as the FOURofthe PROBABILITY DENSITY FUNC
(b(t) =F[P(x)] =sm O” eit”P(x) dx (1) r P(x)
dx + it
xP(x)
+ i(d)”
dx
x2P(x)
dx +.
..
(2)
smm 
i k O” (t> cLlk Ix k=O
=
1+
Characteristic Polynomial Theexpandedformofthe CHARACTERISTIC det(xl
r m
oo
(3)
229
 A),
where A is an n x n MATRIX MATRIX. see
CAYLEYHAMILTON
UZSO
EQUATION.
and
I is the IDENTITY
THEOREM
l
it/L;

+t”p;

&it”&
+
.
&t4pl
+
l
l
l
, (4)
where pk (sometimes also denoted y”) is the nth MOfunction MENT about 0 and & E 1. The characteristic can therefore be used to generate MOMENTS about 0,
(5) or the CUMULANTS
see
MANTISSA,
UZSO
SCIENTIFIC
the char
NOTATION
C harlier’s Check A check which can be used to verify of MOMENTS.
correct
computation
K~,
(6) A DISTRIBUTION is not uniquely MENTS, but is uniquely specified function. see also CUMULANT, FUNCTION,PROBAB
Characteristic (Real Number) For a REAL NUMBER x, 1x1 = int(x) is called acteristic. Here, 1x1 is the FLOOR FUNCTION.
specified by its Moby its characteristic
MOMENT, M 0 MENT ,GENE RATING LITY DENSIT Y FUNC TION
ChaslesCayleyBrill Formula The number of coincidences of a (v, y’> correspondence of value y on a curve of GENUS p is given by
v+vt+2py. see
ZEUTHEN'S
also
THEOREM
References Coolidge, York:
J. L, A Treatise on Dover, p. 129, 1959.
Algebraic
Plane
Curves.
New
References Handbook Abramowits, M. and Stegun, C, A. (Eds.). of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kenney, 3. F. and Keeping, E. S. “MomentGenerating and “Some Examples of MomentCharacteristic Functions,” Generating Functions,” and “Uniqueness Theorem for Characteristic Functions.” $4.64.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 7277, 1951.
Chasles’s Contact Theorem If a oneparameter family of curves has index Iv and class IV, the number tangent to a curve of order 721 and class ml in general position is mlN+nlM.
References Characteristic (Partial Differential Equation) Paths in a 2D plane used to transform PARTIAL FERENTIAL EQUATIONS into systems of ORDINARY FERENTIAL EQUATIONS. They were invented by mann. For an example of the use of characteristics, sider the equation ut  6211~~ = 0. Now let u(s) = u(x(s>, t(s)). du ds 
dx ds”x
Since dt + &%
Coolidge, York:
DIFDIFRiecon
J. L. Dover,
A Treatise on p. 436, 1959.
Algebraic
Plane
Curves.
New
Chasles’s Polars Theorem If the TRILINEAR POLARS of the VERTICES of a TRIANGLE are distinct from the respectively opposite sides, they meet the sides in three COLLINEAR points. see
also
COLLINEAR,
TRIANGLE,
TRILINEAR
POLAR
230
Chasles’s Theorem
Chasles’s
Theorem
Chebyshev see also ONENINTH
If two projective PENCILS of curves of orders 12 and n’ have no common curve, the LOCUS of the intersections of corresponding curves of the two is a curve of order n+n’ through all the centers of either PENCIL. Conversely, if a curve of order n + n’ 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
References Coolidge, J. L. York:
Dover,
Chebyshev
FUNDAMENTAL
A Treatise
THEOREM,
on Algebraic
Plane
PENCIL
Curves.
Using a CHE~YSHEV T, define
RATIONAL
FUNCTION
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 Problems and Conjectures. Philadelphia, PA: SIAM, 1990. Philadelphia, PA: SIAM, 1990.
Chebyshev
Deviation apyf(4  
New
 P(4lW~.
References
Formula
POLYNOMIAL
CONSTANT,
Equation
References Finch, S. “Favorite
p. 33, 1959.
Approximation
Differential
Szeg6,
OF THE FIRST
KIND
G.
Orthogonal
Amer. Math.
Polynomials,
4th
ed.
Providence,
RI:
Sot., p. 41, 1975.
Chebyshev
Differential
Equation
2 d2Y dY ~)~xz+rn~y=O
(1 
(1)
for 1x1 < 1. The Chebyshev differential equation has regular SINGULARITIES at 1, 1, and 00. It can be solved by series solution using the expansions Then Nl
f(x)
= 7;
d%(X)
 fCo*
the Iv zeros of TN(X). This type of apimportant because, when truncated, the smoothly over [1, l]. The Chebyshev formula is very close to the MINIMAX
W
= )‘
7tUnXnl
 n00n+lan+d n=O
(3)
n=O 00
References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Chebyshev Approximation,” “Derivatives or Integrals of a ChebyshevApproximated Function,” and “Polynomial Approximation from Chebyshev Coefficient s.” 55.8, 5.9, and 5.10 in Numerical Recipes in FURTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184188, 189190, and 191492, 1992.
Chebyshev
(2)
n=O w
k=O
It is exact for proximation is error is spread approximation POLYNOMIAL.
anxn
Y=F
Constants
IV.5 A detailed online essay by S. Finch ing point for this entry.
was the start
ytt
=
x(n
+
1
n&+111:
n1
=
F(n
+
l)n&+lXnl
n=l
n=O W
nl2

(72
+
l)an+2Xn
(4
E(
n=O
Now, plug
(24) into the original
(1 x2)
F(n+z)(n+
equation
(1) to obtain
l)Gb+2Xn
n=O
W
The constants
x
x m,n
=
inf TT,
sup le” 2>0 
e(n+
1)?Ln+1Xn+TT12
 r(x)l,
r(x) =
PC 2, > q(x)
n
is the set all RATIONAL cients.
FUNCTIONS
00
n + 2)(n + 1)an+2xn
n=O
p and 4 are nzth and nth order POLYNOMIALS, with
and Rm,n REAL coeffi
(5)
n=O
00
where
UnXn = 0
x
n=O

00 X( n=O

Iu n=O
n
+ 2>(n 00
nt
l)an+lXn+l+
m2 x
n=O
an
Xn
=0
(6)
Chebyshev f)n
Differential
+ 2)(n
+ l)an+2xn
ChebyshevGauss
Equation n(n  l)UnXn+2
 F n=2
n=O
cxl
00 
nUnXn
+
7Yb2
= 0
UnXn
x
c
n=l
n=O
(7)
2 la2 + 3 2a3x  Z 9 ax + m2ao + m2alx l
l
+ ji;[(n
+ 2)(n
+ Q&+2
 n(n
If n is EVEN, then yl terminates and is a POLYNOMIAL solution, whereas if n is ODD, then y2 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 + 1)”  m2 I I an+2 = (n + 2)(n + 3) an
 I)%
n=2
a,
+ m2an]xn
+ F,[(n
+ [(m2  1)~
+ 2)(n
+
e(x) = 0,
PlX
p, so
where the sum is over PRIMES
SO
lim
2a2 + m2a0 = 0
 l)al
(11)
forn=2,3,.... The first two are special recurrence relation is 2
From
this,
cases of the third,
so the general
for n = O,l,.
2) an
obtain
we
(12)
2
an+2 = (n 9 ,xl”,
for the EVEN
...
(13)
ChebyshevGauss Quadrature Also called CHEBYSHEV QUADRATURE. A GAUSSIAN QUADRATURE over the interval 11, l] with WEIGHTING FUNCTION W(X) = l/d. The ABSCISSAS for quadrature order n are given by the roots of the CHEBYSHEV POLYNOMIAL OF THE FIRST KIND 7&(x), which occur symmetrically about 0. The WEIGHTS are A n+lYn
COEFFICIENTS wi
a2 = +rn’ao
A&
AnTA(xi)Tn+l(xi)
=
 2)’  m2] (2 n!>
(15)
ao l
l
l
[m2]
MITE
T!l(xi)T!(xi)
’
of xn in T,(x).
For HER
POLYNOMIALS,
An = 2n1,
a07 (16)
so
A n+l A,
and for the ODD COEFFICIENTS
= 2.
Addit ionally,
1  m2 a3 = a0
(17)
6
32  m2 (3 2  m2)(12  m”) = al (18) 45 5! [(Zn  1)”  m2][(2n  3)2  m”] mm[12  m2] al = (2n + l)!
a5 =
Yn1
A,_1
(1)
where An is the COEFFICIENT
2  m2)(m2) = (2 1.2*3*4
K2 n >2  m2][(2n
=
=
(14
22  m2 a4 = a2 3*4 a272
g=l. O( X >
Xho0
+ 6a3 = 0
(21)
= Tilnp,
(9)
n=2
(m2
O,l,....
Function
+ 6a3]x + (m2  n2)an]xn
l)%+2
for n=
(8)
= 0
Chebyshev (2~2 + m2ao)
231
Quadrature
so
a3
w;
= Tn+&:)Z(xi)
’
(5)
l
a2n1
Since Tn(x)
( 1 9)
the ABSCISSAS So the general
solution O”
y=ao
1+ [
a1
xt
[k”

m2][(k

2)”

k!
m2]
* * * [m2]
xi = cos
k
x
+
k=2,4,...
=
k=3,5,...
cos(n cos’
are given explicitly
is
x
x
=
x by
(2 [ I i  1)n 272
Since [(k  zq2  m2J[(k
 2)"  m2] a* *[12  my
1 (20)
Xk
k!
T:,(xi)
= (  1)
i+1 n
(8)
Qri
Tn+l(xi)
= (l)i
Sinai,
(9)
232
Chebyshev
Inequality
Qri =
(2i  1)X 2n ’
Chebyshev
where
Chebyshev
Integral
Polynomial
Inequality
(10)
all the WEIGHTS are 7r wi = * n
The explicit
b
(11)
<
J
(b  u)~’
is then
FORMULA
f (Xl)f (22) ” fnCx>dX~ l
a
where fl, f2, . . ) fn
are NONNEGATIVE integrable functions on [a, b] which are monotonic increasing or decreasing. l
References Gradshteyn, 1, S. and Ryzhik, ries, and Products, 5th ed. Press, p* 1092, 1979.
k=l
I. M.
n
Xi
Wi
Chebyshev
2 3
*0.707107 0 1tO.866025 1t0.382683 *O92388 0 1t0.587785 zto.951057
1.5708 1.0472 I .0472 0.785398 0.785398 0.628319 0.628319 0.628319
see PRIME QUADRATIC EFFECT
4 5
Tables
of Integrals,
San Diego,
CA:
Se
Academic
Phenomenon
Chebyshev
Polynomial
of the
First
Kind
References Hildebrand, F. B. Introduction to Numerical York: McGrawHill, pp. 330331, 1956.
Chebyshev Inequality Apply MARKOV’S INEQUALITY with
((x Pb

>
I1)”
k21
<

p
Analysis.
New
a E k2 to obtain
d2)
u2 z(1) k2
’
Therefore, if a RANDOM VARIABLE x has a finite p and finite VARIANCE c2, then V IG 2 0,
MEAN
A set of ORTHOGONAL POLYNOMIALS defined as the solutions to the CHEBYSHEV DIFFERENTIAL EQUATION and denoted T,(x). They are used as an approximation to a LEAST SQUARES FIT, and are a special case ofthe ULTRASPHERICAL POLYNOMIAL witha=O. The Chebyshev polynomials of the first kind T.(x) are illustrated above for x E [0, 11 and n= 1, 2, . . , 5. l
p(lx  PI > k>L
(2)
The Chebyshev polynomials tained from the generating
g1(t,x) = 1  l2xt t2 + t2
of the first kind functions = To(x) + 25,(z)t”
of Mathematical Mathematical
p* II,
M.
and
Stegun,
Functions Tables, 9th
C. A. (Eds.).
with Formulas, printing. New
1972.
Chebyshev
(1)
nl
References Abramowita,
can be ob
Integral ~‘(1 J
 2)’ dx.
Handbook Graphs, and
York:
Dover,
and g2(t,x)
=
l  xt = 1  2xt + t2
2Tn(x)t"
(2)
n=O
for 1x1 < 1 and ItI < 1 (Beeler et al. 1972, Item 15). (A closZy related GENERATING FUNCTION is the basis for the definition of CHEBYSHEV POLYNOMIAL OF THE SECOND KIND.) They are normalized such that Tn( 1) = 1. They can also be written
Chebyshev or in terms
Polynomial
Chebyshev
of a DETERMINANT 2 1 0 0 .
Tn =
1 2X 1 0 .
l
0 1 2x 1 .
l
Using a FAST FIBONACCI tion law 0 0 1 2X .
.
.*’ ... ... ..*
0 0 0 0
0 0 0 0
.
.
l
l
l
l
l
(A,B)(C,
Polynomial
233
with
TRANSFORM
D) = (AD I BC + 2xAC,BD
multiplica
 AC)
(14)
l (4)
gives
.
0
(j
i
0
0
l
**
22:
(Tn+l(x),
T,(x))
= (TV,
To(~))(l,0)~.
(15)
In closed form, 1nPJ %x(x)
cos(ncosl
=
x)
n
=
2m
cc m=O
n2m
x
>
( x2  l>“, (5)
where FLOOR
is a BINOMIAL COEFFICIENT and [xJ is the FUNCTION. Therefore, zeros occur when
Using GRAMSCHMIDT URTHONORMALIZATION in the range (1,l) with WEIGHTING FUNCTION (1 x2)(1/2) gives
PO(X)= 1
(i)
(16)
1 =x c(1 =x[x2y2111 1 1 jT1 x(1  x~)“~
p&j=
x
I;,(1
[
[
r(k 
2 = cos
 3>
n
for k = 1, 2, . . . , n. Extrema
1
(6) p2(5)=
(7)
(8)
where 6,, is the KRONECKER DELTA. Chebyshev polynomials of the first kind satisfy the additional discrete identity
m ck=l
for i # 0, j # 0 for i = j = 0,
+TKdij Ti(Xk)Tj(Xk)
=
m
(9)
where xk for k = 1, . . . , m are the m zeros of T,(x). They also satisfy the RECURRENCE RELATIONS Xx+1(x) T,+l(x)
=
= 2xTn(x)
XX(X)
for n > 1. They tion Tn(x)

d(1

=
represent
(1)“fi(1

(10) [C(X)]~}
integral nl

(1
&
dx
X
 x2)l/2
To(x)
such that
Tn(l)
.1
da
(18)
= 1 gives
= 1
Tl cx> =X T2(x) = 2x2  1 T3(x)
= 4x3  3x
T4(x)=8x48x2+1 Wx) T6(x)
= 16x5  20x3 + 5x = 32x”  48x4 + 18x2  1.
The Chebyshev polynomial of the first kind is related to the BESSEL FUNCTION OF THE FIRST KIND Jn(x) and MODIFIED BESSEL FUNCTION OF THE FIRST KIND 1n (x) by the relations
(19)
In(x)
=
Tn.
(20)
(12) Letting x z cos 8 allows the Chebyshev the first kind to be written as
d” [(lX2)7H/2] dxn
dx
= [x  ()1x I 2 = 22  3, 7r
dz
ation
 x2Y2  k)!
JT1 x2(1  x2)li2
representa
(13) l
2n(n
dx
(11)
1  2x2 + z2
s 7
Tn(x)
x2)(1
have a COMPLEX =
and a Rodrigues
 Tnq(x)
j’Tl x3(1  x2)1’2
J:,(l
etc. Normalizing for m # 0, n # 0 for m = n = 0,
(17)
xl’,
JT, x2(1  x2)1’2

Tn(x) = 1, and where k = 0, 1, . . . ,W At maximum, at minimum, Tn(x) = 1. The Chebyshev POLYNOMIALS are ORTHONORMAL with respect to the WEIGHTING FUNCTION (1  x2)li2
dx
x
[
’s
 x2)l/2
sinl
occur for
l
dx
Tn(x)
= cos(n0)
= cos(nco8
polynomials
x).
of
(21)
Chebyshev
234
Polynomial
The second linearly dependent formed differential equation d2Tn
d82
is then given
Chebyshev
solution
+ n2Tn
to the trans
= 0
(22)
by
Polynomial
A modified set of Chebyshev POLYNOMIALS defined by a slightly different GENERATING FUNCTION. Used to develop fourdimensional SPHERICAL HARMONICS in angular momentum theory. They are also a special case ofthe ULTRASPHERICAL POLYNOMIAL with a = 1. The Chebyshev polynomials of the second kind & (2) are illustrated above for x E [0, l] and n = 1, 2, . , 5. l
vn Cx> = sin(n0) which
= sin(ncos1
x),
(23)
can also be written Vn(X)
= \/l

X2
GENERATING of the second
(24)
Unl(X),
& is a CHEBYSHEV POLYNOMIAL OND KIND. Note that V.,Jx) is therefore NOMIAL. where
Q2(t, 2) =
THE SECnot a POLY
OF
 21BnTn(x)
(25)
for k = 0, 1, . . . , n (Beeler see also CHEBYSHEV CHEBYSHEV POLYNOMIAL
L = x 1  2xt + t2
= F
k7r , ( n >
(26)
et al. 1972, Item
15).
APPROXIMATION OF THE SECOND
KIND
of the Chebyshev
is
K+>tn
(1)
To see the relationship OF THE FIRST KIND
to (T),
89 = (l  2xt + t2)2(2x + 2t) 8t = 2(t  x)(1  2xt + t2)2
(of degree n  2) is the POLYNOMIAL of degree < n which stays closest to xn in the interval (1, l)* The maximum at the n + 1 points where deviation is 2l” x = cos
FUNCTION kind
for 1x1 < 1 and It] < 1. a CHEBYSHEV POLYNOMIAL take dgldt,
The POLYNOMIAL Xn
The defining polynomials
l
nU,(x)t”l.
(2)
n=O
(2) by t,
Multiply
FORMULA,
Pt 2 
2Xt)(l
 2Xt + t2)12
=
(3)
xnUn(X)tn
References Abramowitz,
M. and Stegun, C. A. (Eds.). “Orthogonal Polynomials .” Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771802, 1972. A&en, G+ “Chebyshev (Tschebyscheff) Polynomials” and “C hebyshev PolynomialsNumerical Applications .” 5 13 3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731748, 1985. Beeler, M,; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972. Iyanaga, S. and Kawada, Y. (Eds.). “Cebygev (Tschebyscheff) Polynomials.” Appendix A, Table 20.11 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 14781479, 1980. Rivlin, T. J. Chebyshev Polynomials. New York: Wiley,
and take
Pf 2 
(3)(2),
2tx)
t2  1
 (1  2xt + t”)
(l2xt+ty
=
(12xt+t)2
l
(4)
r&=0 The Rodrigues un(X)
=
representation (V(n+
is
l)fi
P+l(n+
+)!(l
dn
 x2)li2
[(I
dX”
x2)n+1/2]

l
(5) The polynomials
1990*
 l)&Jx)t”.
= F(n
Spanier, J. and Oldham, K. B. “The Chebyshev Polynomials T,(x) and Un(x).” Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193207, 1987.
Un(X)
can also be written b421 x (1)’
=
T=o
Chebyshev
Polynomial
of the
Second
Kind = 'z
(Znm++ll>~~~~(x~
 l)",
(6)
1x1 is the FLOOR FUNCTION and [xl is the CEILING FUNCTION, or in terms of a DETERMINANT
where
un =
2x 0 0 .
1 2x 1 .
.
.
0 1 2x .
0 0 1
0
0 0 0
0 0 0
1’
2x
l (7)
. l
.
0
l
l
.
0
** .‘. '*
l
0
1
l
. .
m
.
l
.
Chebyshev
Quadrature
Chebyshev
The first few POLYNOMIALS are Uo(x)
= 1
Ul(X)
= 2x
uz(x)
=4x2
U3(x)
= 8x3 4x
U4(x)
The ABSCISSAS are found by taking the MACLAURIN SERIES of s,(y)
in
+ ln(1  y)
2
terms
= 64x" 80x4
+24x2
up to yn in
( > ( >I1 Y
+1 +6x
235
1 1 
+ln(l+y)
32x5 32x3
Us(x)
{[
= exp
1
= 16x4 12x2
Us(x)=
Quadrature
l+
1
Y
,
and then defining
1.
G,(x) Gxnsn( i ) . X
Letting x =: cos 0 allows the Chebyshev the second kind to be written as
GL(x) =
polynomials
of
The ROOTS of G,(x) few values are
sin[(n + l)Q] sin 0 *
(8)
The second linearly dependent solution to the transformed differential equation is then given by
wk(x) which
=
cos[(n + l)O] I sin tJ
(9)
Go(x)
then give the ABSCISSAS. The first
= 1
Gl(X)
= x
G2(x)
= +(3x2  1)
G3(x)
=
$(2x3
G4(2)
=
&(45x4


G5 (x) = $(72x5
can also be written
G6(2)
Wn(x) = (l
x2)'/"Tn+l(x),
10)
= &(105x6
 105x4 + 21x2  1)
where T, is a CHEBYSHEV POLYNOMIAL OF THE FIRST KIND. Note that WR.(x) is therefore not a POLYNOMIAL.
G(x)
= &(42525x8
see also CHEBYSHEV APPROXIMATION FORMULA, CHEBYSHEV POLYNOMIAL OF THE FIRST KIND, ULTRASPHERICAL POLYNOMIAL
G(x)
Polynomials tions with 9th printing.
C. A. (Eds,).
“Orthogonal
.” Ch, 22 in Handbook of Mathematical FuncFormulas, Graphs, and Mathematical Tables, New York: Dover, pp. 771802, 1972.
2220x2
Quadrature QUADRATURElike FORMULA for numeriof integrals. It uses WEIGHTING FWNCTION w(z) = 1 in the interval [  1, 11 and forces all the weights to be equal. The general FORMULA is
s 1
f(x)dx
1
= ; k i=l
f(xi).
149x)
+20790x4
43) + 15120x5
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
En = n even,
1990.
Chebyshev A GAUSSIAN cal estimation
56700x6
 33600x7 = &(22400x9  2280x3 + 53x).
A&en, G. “Chebyshev (Tschebyscheff) Polynomials” and “Chebyshev PolynomialsNumerical Applications.” 5 13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731748, 1985. Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, Spanier, J. and Oldham, K. B3. “The Chebyshev Polynomials T,(x) and &(x).” Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193207, 1987.
 7560x5 +2142x3
&,(6480x7
=
M. and Stegun,
30x2 + 1)
 60x3 + 7x)
G7(x)
References Abramowitz,
2)
Cn
=
JT1x&(x) dx
n odd
1 s 1 x2Gn(x)
n even.
dx
The first few values of cn are 2/3, 8/45, l/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.
ChebyshevRadau
236
n 2 3 4 5
6
7
9
Chebyshev’s n
Xi *0.57735 0 %0.707107 50.187592 &0.794654 0 zkO.374541 ho.832497 *0.266635 zto.422519 &0.866247 0 *o323912 *0.529657 zk0.883862 0 *O. 167906 zkO.528762 ztO.601019 ho.911589
The ABSCISSAS and weights cally for small n. n
Quadrature Xi
1
0.7745967
2
0.5002990 0.8922365 0.4429861 0.7121545 0.9293066 0.3549416 0.6433097 0.7783202 0.9481574
3
4
wi 0.4303315 0.2393715 0.2393715 0.1599145 0.1599145 0.1599145 0.1223363 0.1223363 0.1223363 0.1223363
References Beyer, W. H. CRC Standard Boca Raton,
Chebyshev If
FL: CRC
Sum
Theorem
Press,
Mathematical Tables, 28th ed. p. 466, 1987.
Inequality al > a2 > ..* > a,
can be computed
analyti
Xi This SE ITY
is true for any distribution. UZSO
CAUCHY
INEQUALITY,H~LDER
SUM INEQUAL
References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979. Hardy, G. H.; Littlewood, J. E.; and P6lya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 4344, 1988.
see
also
CHEBYSHEV
QUADRATURE,
L~BATTO
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 York: McGrawHill, pp. 345351, 1956.
Analysis.
New
ChebyshevRadau Quadrature A GAUSSIAN QUADRATURElike FORMULA over the interval[l,l] whichhas WEIGHTING FUNCTION W(X) = X. The general FORMULA is
s 1
1
z.f(x) dx = &[f(Xi) i=l
 f(%>I*
ChebyshevSylvester Constant In 1891, Chebyshev and Sylvester showed that for sufficiently large x, there exists at least one prime number p satisfying 27 < p < (1+ QI)x, where cy = 0.092.. . . Since the PRIME NUMBER THEOREM shows the above inequality is true for all Q > 0 for sufficiently large x, this constant is only of historical interest. References Le Lionnais,
F. Les nombres p. 22, 1983.
Chebyshev’s see BERTRAND'S
remarquables.
Theorem POSTULATE
Paris:
Hermann,
Checker Jumping
Problem
CheckerJumping
Problem
237
Chern Num her
Seeks the minimum number of checkers placed on a board required to allow pieces to move by a sequence of horizontal or vertical jumps (removing the piece jumped over) n rows beyond the forwardmost initial checker. The first few cases are 2, 4, 8, 20. It is, however, impossible to reach level 5.
see also CY CLIC REDUNDANCY CORRECTING CODE
CHECK,
ERROR
References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Cyclic Redundancy and Other Checksums.” Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of Scientijic Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 888895, 1992.
Kererences Gems II. R. Mathematical Honsberger, Math. Assoc. Amer., pp, 2328, 1976.
Washington,
DC:
see CHESSBOARD
Checkers Beeler et al. (1972, Item 93) estimated that there are about 101’ possible positions. However, this disagrees with the estimate of Jon Schaeffer of 5 x 10zO plausible under the rules of the positions, with 1018 reachable game. Because “solving” checkers may require only the SQUARE ROOT of the number of positions in the search space (i.e., lo’), 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 EULERIAN 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). l
CHECKERBOARD,
CHECKERJUMPING
Finiteness
Theorem
Consider the set of compact nRIEMANNIAN MANIFOLDS M with diameter(M) 5 d, Volume(M) > V, and 1x1 < K where K is the SECTIONAL CURVATURE. Then there is a bound on the number of DIFFEOMORPHISMS classes of this set in terms of the constants n, d, V, and K.
Checkerboard
see also LEM
Cheeger’s
PROB
References Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, Feb. 1972. Dubuque, W. “Re: number of legal chess positions.” mathfun@cs.arieona.edu posting, Aug 15, 1996. Kraitchik, M. “Chess and Checkers” and “Checkers Recre(Draughts).” $12.1.1 and 12.1.10 in Mathematical ations. New York: W. W. Norton, pp. 267276 and 284287, 1942. Challenging Human Schaeffer, J. One Jump Ahead: Supremacy in Checkers. New York: SpringerVerlag, 1997. Sloane, N. J. A. Sequences A006239/M4909 and A006240/ M5271 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Checksum A sum of the digits in a given transmission modulo some number. The simplest form of checksum is a parity bit appended on to 7bit numbers (e.g., ASCII characters) such that the total number of 1s 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 polynomials over the integers (mod 2). It is substantially more reliable in detecting transmission errors, and is one common errorchecking protocol used in modems.
References Chavel, I. Riemannian Geometry: New York: Cambridge University
Chefalo
A Modern
Press,
Introduction.
1994.
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 Every
Theorem “large”
EVEN
p + no where SEMIPRIMES
INTEGER
(i.e., ZALMOST
see UZSO ALMOST
PRIME,
may
be written
as 2n =
and m e PZ is the SET of
p is a PRIME
PRIMES).
PRIME
NUMBER,
SEMIPRIME
References Rivera, C. “Problems & Puzzles (Conjectures): Conjecture.” http://uww.sci,net.mx/crivera/ppp/
Chen’s
conj002.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 ith Chern class is in the (2i)th cohomology group of the base SPACE. see
OBSTRUCTION,
UZSO
WH ITNEY
Chern
PONTRYAGIN
CLASS, STIEFEL
CLASS
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% FUNDAMENTAL CLASS. The resulting number is called the Chern number for that combination of Chern classes. The most important aspect of Chern numbers is that they are C~B~RDI~M invariant. see also NUMBER
PONTRYAGIN
NUMBER,
STIEFELWHITNEY
238
Chernoff
Chess
Face
1903. The history of the determination of the chess sequences is discussed in Schwarzkopf (1994).
Chernoff Face A way to display n variables on a 2D surface. For instance, let II: be eyebrow slant, y be eye size, x be nose length, etc.
Two problems
Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993.
2. What is the smallest number cupy or attack every square.
Chess Chess is a game played on an 8x 8 board, called a CHESSBOARD, 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 (l), knight (Z), pawn (8), queen (l), and rook (2). The object of the game is to king. capt we the opponent’s 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 10120 according to Peterson (1996) However, this value does not agree with the 104’ possible positions given by Beeler et al. (1972, Item 95). This value was obtained by estimating the number of pawn positions (in the nocaptures situation, this is 158), times all pieces in all positions, dividing by 2 for each of the (rook, knight) which are interchangeable, dividing by 2 for each pair of bishops (since half the positions 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 l
64! = 32!(8!)2(2!)6
= 1043m
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 AOO7545). Rex Stout’s fictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: “Wolfe grunted. One hundred and sixtynine million, five hundred and eighteen thousand, eight hundred and twentynine followed by twentyone ciphers. The number of ways the first ten moves, both sides, may be played” (Stout 1983). The number of chess positions after n moves for n = 1, 2, , . are 20, 400, 5362, 71852, 809896?, 9132484?, . . . (Schwarzkopf 1994, Sloane’s AO19319). l
Cunningham (1889) incorrectly found 197,299 games C. Flye and 71,782 positions after the fourth move. St. Marie was the first to find the correct number of positions 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
mathematics
ask
1. How many pieces of a given type can be placed CHESSBOARD without any two attacking.
References
p(4o)
in recreational
The answers are given 1979)
of pieces needed
in the following
table
on a to oc
(Madachy
l
Piece bishops kings knights queens rooks
Max.
Min.
14 16 32
8 9 12
8 8
5 8
see also BISHOPS PROBLEM, CHECKERBOARD,CHECKERS, 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 Recreations and Essays, 13th ed. New York: Dover, pp. 124127, 1987. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239 Feb. 1972. Dawson, T. R. “A Surprise Correction.” The Fairy Chess Review 6, 44, 1946. Dickins, A. “A Guide to Fairy Chess.” p. 28, 1967/1969/ 1971. Dudeney, H. E. “Chessboard Problems.” Amusements in Mathematics. New York: Dover, pp. 84109, 1970. Fabel, K. ‘LNiisse.” Die Schwalbe 84, 196, 1934. Fabel, K. “Weihnachtsniisse.” Die Schwalbe 190, 97, 1947. Fabel, K. “Weihnachtsniisse.” Die Schwalbe 195, 14, 1948. Fabel, K. “Eriiffnungen.” Am Rande des Schachbretts, 3435, 1947. Fabel, K. “Die ersten Schritte.” Rund urn das Schachbrett, 107109, 1955. Fabel, K. “Erijffnungen.” Schach und Zahl8, 1966/1971, Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 8689, 1975, Kraitchik, M. “Chess and Checkers.” §12.1.1 in Mathematical Recreations. New York: W. W. Norton, pp. 267276, 1942. Madachy, J. S. “Chessboard Placement Problems.” Ch. 2 in Madachy’s Mathematical Recreations. New York: Dover, pp. 3454, 1979. Peterson, I. “The Soul of a Chess Machine: Lessons Learned from a Contest Pitting Man Against Computer.” Sci. News 149, 200201, Mar. 30, 1996. PetkovZ, M. Mathematics and Chess. New York: Dover, 1997. Schroeppel, R. “Reprise: Number of legal chess positions.” technews@cs.arizona.edu posting, Aug. 18, 1996. Schwarzkopf, B. “Die ersten Ziige.” Problemkiste, 142143, No. 92, Apr. 1994. Shannon, C. “Programming a Computer for Playing Chess.” Phil. Mag. 41, 256275, 1950. Sloane, N. J. A. Sequences A019319 and A007545/M5100 in “An OnLine Version of the Encyclopedia of Integer Sequences .”
239
Chi Distribution
Chessboard Stout, R. “‘Gambit.” New York: Avenic
In Seven Complete Nero Books, p. 475, 1983.
Wolfe
Novels.
Chevron
Chessboard A 6POLYIAMOND. References Golomb, S. W+
Polyominoes: Puzzles, 2nd ed. Princeton, NJ:
and Packings,
Patterns,
Problems,
Princeton
University
Press, p. 92, 1994. Chi 20: 15: 10 y Jr;
A board containing 8 x 8 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
CHECKERS, CHESS, DOMINO, GOMORY'S AND CHESSBOARD PROBLEM
also
OREM,
5
4
t
15 1 Re[CoshIntegral
ICoshIntegral
z]
z]
THE
WHEAT
References Pappas, T. “The Checkerboard.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 136 and 232, 1989,
Chevalley Finite
Groups SIMPLE
four
3
2
1
5 I 10
families
GROUPS
of linear
PSU(n,
q), PSp(2n,
see
TWISTED
also
of
LIETYPE.
SIMPLE
q>, or =+,
CHEVALLEY
GROUPS:
They include PSL(n, q),
Chi(z)
= y + lnz +
 I dt 7
t
0
where y is the EULERMASCHERONI CONSTANT. The function is given by the Muthematicu@ (Wolfram Research, Champaign, IL) command CoshIntegral CzJ. see also COSINE
INTEGRAL,
SHI, SINE INTEGRAL
References
4).
Abramowitz, M. and Stegun, C. A. (Eds.). ‘&Sine and Cosine Integrals.” 55.2 in Handbook of Mathematical Func
GROUPS
References Wilson, R. A. “ATLAS of Finite Group http://for *mat .bham.ac .uk/atlas#chev.
tions with 9th printing.
Representation,”
C hevalley’s Let f(z)
Theorem be a member of a FINITE FIELD , zcn] and suppose f(O,O,. . ,O) = 0 and n F[a,22,... is greater than the degree of f , then f has at least two zeros in A” (8’).
Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 231233, 1972.
Chi Distribution The probability bution function
density are
EL(x) = h(x)
References d’une hypothese de M. Artin.” Abhand. Math. Sem. Hamburg 11, 7375, 1936. Ireland, K. and Rosen, M. “Chevalley’s Theorem.” 510.2 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: SpringerVerlag, pp. 143144, 1990+
function
and cumulative
distri
21n/2xnlez2/2
l
Chevalley,
I
’ cosht
(1)
r( $4
= Q(+, +x2),
(2)
C. “Dkmonstration
where
Q is the REGULARIZED
GAMMA
FWNCTION.
(3) 2 _ 2[r( 3431 0 2r3($(n 71
=
+ in)
 r’($(n
+ I))] (4)
F2( in) + 1))  3r(+)r(
[r(+b)r(l
+ in)
i(n  r2(+(n
+
i))r(i + + 1))]3/2
$73)
Chi Inequality
240
+ [r(in>r(l+ 72
=
in)
3r4($(n+
ChiSquared
1)) +6r(in) n
(5)
r2(+(n+1))]3/2 +r2(i(n+
l))r(l+
F)  r2(+
;n>rc
in)
+ I))]z
4r2(+)r(~(n+1))r(~)+r3(+)17(~) +
F)  ry++
P? +w(
i))p
where p is the MEAN, o2 the VARIANCE, 71 NESS, and 72 the KURTOSIS. For n = 1, the tion is a HALFNORMAL DISTRIBUTION with n = 2, it is a RAYLEIGH DISTRIBUTION with see UZSO CHISQUARED DISTRIBUTION, DISTRIBUTION, RAYLEIGH DISTRIBUTION
where P(a, Z) is a REGULARIZED GAMMA FUNCTION. The CONFIDENCE INTERVALS can be found by finding the value of 2 for which D, (x) equals a given value. The MOMENTGENERATING FUNCTION ofthe x2 distribution is
’ @)
M(t) = (1  2t)“2 R(t) E In M(t) =  +7+ln(1  2t)
the SKEWx distribu0 = 1. For 0 = 1.
R’w R”(t)
which
(6)
r l
(7)
2t 2r
(8)
= (1T
so p = R’(O) g2 = R”(0)
is satisfied
(5)
HALFNORMAL
Chi Inequality The inequality (j + l)Uj
Distribution
= T
(9)
= 2r
(10)
+ G 2 (j + l>i,
(11) 12
by all ASEQUENCES.
y2=7
(12)
References Levine, E. and O’Sullivan, J. “An Upper Estimate for the Reciprocal Sum of a SumFree Sequence,” Acta Arith. 34, 924, 1977.
ChiSquared Distribution A x2 distribution is 8 GAMMA DISTRIBUTION with t9 = 2 and a E r/2, where r is the number of DEGREES OF FREEDOM. If Yi have NORMAL INDEPENDENT distributions with MEAN 0 and VARIANCE 1, then
The nth MOMENT about zero for a distribution DEGREES OF FREEDOM is
dL= 2”r(n + $1 =r(r+2)*++2n2), r(;T) and the moments
The nth is distributed as x2 with n DEGREES OF FREEDOM. If xi2 are independently distributed according to a x2 distribution with n1, n2, . . . , nk DEGREES 0~ FREEDOM, then k
IE Xj2
with
CUMULANT
about
the MEAN
n
(13) are
P2 = 2r
(14)
p3 = 8r
(15)
p4 = 12n2 + 48n.
(16)
is
&n = anr(n)($) The MOMENTGENERATING
= 2”‘(n FUNCTION
 l)!r.
(17)
is
(2)
j=l
is distributed according GREES OF FREEDOM.
to x2 with
,r/21,x/2
EL(x) = The cumulative
r/2
1
for x < 0.
t sx2
distribution
=
DE
(3)
0
function
42l
aL(x2)
nj
for 0 < x < 00
r(+T)2+
{
n E &
ct” r($r)2+
0
 r(&
$x2>
r($>
is then
(18)
As r + 00, lim M(t) r+00
dt
= P(+,
.
= et2/‘,
(19)
so for large T, ix”),
(4
(20)
ChiSquared
ChiSquared
Distribution
approximately a GAUSSIAN DISTRIBUTION with and VARIANCE o2 = 1. Fisher showed that
is
MEAN 6
241
Test
CONFLUENT HYPERGEOMETRICLIMITFUNCTION and r is the GAMMA FUNCTION. The MEAN, VARIANCE, SKEWNESS, and KURTOSIS are
OF1 isthe
(21) is an improved estimate Wilferty showed that
for moderate
r.
Wilson
(36)
x2 l/3 (>T
12(4A + n)
(22)
is a nearly GAUSSIAN DISTRIBUTION with 1  2/(9r) and VARIANCE o2 = 2/(9r). In a GAUSSIAN
and
MEAN
p =
DISTRIBUTION,
”
(2X + n)2
see also CHI DISTRIBUTION, TION
(37) l
SNEDECOR’S
FDISTRIBU
References Abramowitz,
P(x)
=
(zC1)2/2a2
dx = ?,a”
dx
1
(23)
of Mathematical Mathematical
pp. 940943,
let
(24)
z = (x  p)2/a2. Then 2(x  4 dz = u2
dx = ii@ u
dx
(25)
so
(26)
dx = zdz. 2G
M. and
Stegun,
Functions Tables, 9th
C.
A.
(Eds.).
with Formulas, printing. New
Hand book Graphs, and
York:
Dover,
1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987. Press, W. fL; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Incomplete Gamma Function, Error Function, ChiSquare Probability Function, Cumulative Poisson Function.” 56.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp, 209214, 1992. and Problems of Probability and Spiegel, M. R. Theory Statistics. New York: McGrawHill, pp. 115116, 1992.
But P(z)
dz = 2P(x)
(27)
dx,
so P(x)
This
1 dx = 2 ~ afie
d2
with
dz
(28)
dz
7T
6
2
=
( >:
i=l
xl/2e1/2
dz =
I?( $)21/Z
k
x9
T = 1, since
g/2lee/2
dz =
ez/2
a+
is a x2 distribution P(z)
1
& =
ChiSquared Test Let the probabilities of various classes in a distribution 1 be PI, PZ, . . . y pk. The expected frequency
(29)
rni  NP~)~ NPi
is a measure of the deviation of a sample from expectation. Karl Pearson proved that the limiting distribution of xS2 is x2 (Kenney and Keeping 1951, pp. 114116).
If Xi are independent variates with a NORMAL DISTRIBUTION having MEANS pi and VARIANCES oi2 for i = 1, n, then ’
l
’
?
n
125x
(Xi = x
i=l
is a GAMMA DISTRIBUTION
 pi)2
(30)
2oi2
variate
with
QI = n/Z, kl
2
The noncentral P(x)
chisquared
= 2ni2e
distribution
(X+z)/2xn/2l
=1I &7 (
is given by
F(fn,
$x),
(32)
where F(a,
x) =
oE(;a;z) r(
a >
'
(33)
>
k3
7
2 i
where 1(x, n) is PEARSON'S FUNCTION. There are some subtleties involved in using the x2 test to fit curves (Kenney and Keeping 1951, pp. 118119). When fitting a oneparameter solution using x2, bestfit parameter value can be found by calculating
the x2
Choose
Child
242
at three points, plotting against the parameter values of these points, then finding the minimum of a PARABOLA fit through the points (Cuzzi 1972, pp. 162168). References Cuzzi, J. Thermal
The Subsurface Microwave
Nature Emission.
of Mercury
Ph.D.
and
Mars
Thesis. 1972.
CA: California Institute of Technology, Kenney, J. F. and Keeping, E. S. Mathematics Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand,
from
b,F = 1 (mod mi
mi).
(6)
from
Pasadena,
of
and the bi are determined
Statistics,
1951.
References Ireland, K. and Rosen, M. “The Chinese rem.” 53.4 in A Classical Introduction Theory,
2nd
ed.
York:
New
Remainder to Modern
SpringerVerlag,
TheoNumber
pp. 3438,
1990.
Uspensky, J. V. and Heaslet,
Child A node which is one EDGE further EDGE ina ROOTED TREE.
away from
a given
see also ROOT (TREE), ROOTED TREE, SIBLING Chinese Hypothesis A PRIME p always satisfies is divisible by p. However,
the condition that 2p  2 this condition is not true exckusively for PRIME (e.g., 2341  2 is divisible by 341 = 1131). COMPOSITE NUMBERS n (such as 341) for which 2n  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.
M. A. Elementary Number Theory. New York: McGrawHill, pp. 189191, 1939. Wagon, S. “The Chinese Remainder Theorem.” $8.4 in Muthematica in. Action. New York: We H. Freeman, pp. 260263, 1991.
Chinese Rings see BAGUENAUDIER Chiral Having forms mirrorsymmetric.
HANDEDNESS which are not
of different
see also DISYMMETRIC, ENA .NTIOMER, HANDEDNESS, MIRROR IMAGE,REFLEXIBLE
Choice Axiom see AXIOM OF CHOICE Choice Number see COMBINATION
References Shanks, 4th
D. Solved and ed. New York:
Chinese
Unsolved
Problems
in Number
Theory,
Chelsea, pp, 1920, 1993.
Remainder
Cholesky
Theorem
Let T and s be POSITIVE INTEGERS which are RELATIVELY PRIME and let a and b be any two INTEGERS. Then there is an INTEGER IV such that N s a (mod
r)
(1)
and N G b (mod
s) .
(2)
Moreover, N is uniquely determined module TS. An equivalent statement is that if (r, s) = 1, then every pair of RESIDUE CLASSES modulo T and s corresponds to a simple RESIDUE CLASS modulo TS. The theorem can also be generalized a set of simul taneous CONGRUENCES x E ai (mod fori=
as follows.
Given
(3)
mi)
1, . . . . T and for which the rni are pairwise RELAof the set of CONGRUENCES
TIVELY PRIME, the solution is
where
ikl x=a&+...+a,&ml
M m,
(mod
M),
(4)
Decomposition
Given a symmetric POSITIVE DEFINITE MATRIX A, the Cholesky decomposition is an upper TRIANGULAR MATRIX U such that
A = UTU. see
LU
ah
References Nash, J. C. “The Choleski pact and
Numerical Function
QR DECOMPOSITION
DECOMPOSITION,
Methods Minimisation,
Decomposition.”
Ch. 7 in Com
for
Linear
Computers: 2nd ed.
Adam Hilger, pp. 8493, 1990. Press, Wm H.; Flannery, B. P.; Teukolsky, ling, W. T. “Cholesky Decomposition.” Recipes in FORTRAN: The 2nd ed. Cambridge, England:
pp. 8991,
Art
Bristol,
S. A.; and Vetter52.9 in Numerical
of Scientific
Cambridge
Algebra
England:
Computing,
University
Press,
1992.
Choose An alternative term for a BINOMIAL COEFFICIENT, in which (z) is read as “n choose 3c.” R. K. Guy suggested this pronunciation around 1950, when the notations “CT and & were commonly used. Leo Moser liked the pronunciation 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
Choquet Theory Erd6s proved that there exist at least one PRIME of the form 4k + 1 and at least one PRIME of the form 4k + 3 between n and 2n for all n > 6.
see also
243
Chow Coordinates
Theory
dR24
A=2
(JR2_yz
@Y
s 0
EQUINUMEROUS, PRIME NUMBER 
ydm
+ R2 tan’
Chord dR2
 2ry
1 1
r2
0
I) 1
=rJR2_rZ+R2tan1 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, T is the RADIUS of the CIRCLE, a is called the APOTHEM, and s the
f
R K) r
=R2ta8
Checking
2
1
21
~T&FZ
rdm.
(4
I
the limits,
when
r = R, A = 0 and when
T + 0, A = $R2,
The shaded region in the left figure is called a SECTOR, and the shaded region in the right figure is called a SEG
(5)
see also ANNULUS, APOTHEM, BERTRAND'S PROBLEM, CONCENTRIC CIRCLES, RADIUS, SAGITTA, SECTOR, SEGMENT Chordal
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 T. The AREA enclosed by the CHORD, shown as the shaded region in the above figure, is then
s dR2
A=2
X(Y) dY
(1)
0
Theorem
The LOCUS of the point at possess the same POWER is ULAR to the linejoiningthe and is known as the chordal two
which two given CIRCLES a straight line PERPENDIC
MIDPOINTS of the CIRCLE (or RADICAL AXIS) of the
CIRCLES.
Dijrrie, H. 100 Great Problems Their History and Solutions.
of Elementary New York:
Mathematics:
Dover,
p. 153,
1965.
But = R2,
(2)
so x(y) = Jm
Chordal
References
TV
y2 I (r + x)”
see RADICAL AXIS
 T
(3)
Chow
Coordinates
A generalization
of varieties of degree To jective space. the intersection of
GRASSMANN COORDINATES to mD in P”, where Pn is an nD pro
d
define the Chow coordinates, take a mD VARIETY 2 of degree d by an (n  m)I3 SUBSPACE U of Pn. Then the coordinates of the d points of intersection are algebraic functions of the GRASSMANN COORDINATES of U, and by taking a symmetric function of the algebraic functions,
a hHOMOGENEous form of 2 is obtained.
POLYNOMIAL known as the Chow The
Chow
coordinates
are then
244
Chow Ring
Christoffel
the COEFFICIENTS of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor.
ChristoffelDarboux
Number
Identity
References Chow, W.L. and van der Waerden., B. L. “Zur algebraische Geometrie IX.” Math. Ann. 113, 692704,1937. Wilson, W. S.; Chern, S. S.; Abhyankar, S, S.; Lang, S.; and Igusa, J.I. “WeiLiang Chow.” Not. Amer, Math. Sot. 43, 11171124, 1996.
O” &dx)b(Y) IE Yk
@m+dX)#&)

 bdx>b+dY>

am%&

Y)7
(1) where
are
#k(x) WEIGHTING
ORTHOGONAL FUNCTION W(x),
Chow Ring The intersection product for classes of rational equivalence between cycles on an ALGEBRAIC VARIETY.
Trn
E
POLYNOMIALS
(4
[4m(X)]2W
with
dx,
(2)
s
References Chow, W.L. “On Equivalence Classes of Cycles in an Algebraic Variety.” Ann. Math. 64, 450479, 1956. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.I. “WeiLiang Chow.” Not. Amer. Math. Sot. 43, 11171124, 1996.
Ak+l ak = Al, where Ak is the COEFFICIENT
(3)
of
in &(x).
X'
SCeterences Chow Variety The set (&,@ of all mD varieties projective space P” into an MD
of degree d in an nD projective space PM.
Hildebrand, F. B. Introduction to Numerical . York: McGrawHill, p+ 322, 1956.
S. S.; Abhyankar, Not. Igusa, J.I. “WeiLiang Chow.” 43, 11171124, 1996.
S. S.; Lang, S.; and Amer.
Math.
Sot.
Let
with let
=
Formula ORTHOGONAL
(Anx
+ &&nlx
Cnpnz(x)
(1)
for 7x = 2, 3, . . . , where A, > 0, B,, and C, > 0 are constants. Denoting the highest COEFFICIENT of pn(x) bY
c(x

p(x) da(x)
x1)(x

x2)
l
l
l
(x

x1)
as Pn(X)
Pn+&>
pn(Xl)
P(X>Gdx>
A, = 5
of order 2 which is Then the orthogonal with the distribution in terms of the POLYNO
can be represented
MIALS J&(X)
kn7
* ’ l
pn+l(Xl)
=
’
.
.
.
.
’
l
P,+W
1
.
PTX+;(~~)
Pn[Xl)
n 1
h&&2
(3)
k ?I 12’
Then
l
l
l
.
. .
A,
(Xl)
Pn+l
.
(2) n
c,=._=
on the interval
(for c # 0) be a POLYNOMIAL NONNEGATIVE in this interval. POLYNOMIALS {q(x)} associated
POLYNOMIALS 
=
associated [a, b]. Also
POLYNOMIALS
{P&)1
p
ChristoffelDarboux For three consecutive pn(x)
New
Christoffel
References Wilson, W. S.; Chern,
Formula b e orthogonal the distribution da(x)
Analysis.
*’
Pn+l
(xl)
In the case of a zero xk of multiplicity nz > 1, we replace the corresponding rows by the derivatives of order 0, 1, 2, . . . . m l of the POLYNOMIALS pn(xi), . . . , p,+l(xi) atx=xk.
P&)PO(Y)
+ Pdx>PdY)
+ ..a
kn

In the special [po(x>12
+
l
l

Pn+l(x>P&>
c4)
P&)P~+~Y) l
k
n+l
References Szeg6,
G.
Amer,
XY
Orthogonal
Polynomials,
4th
Sot., pp. 2930,
Math.
ed.
Providence,
RI:
1975.
case of x = y, (4) gives ’
+
Christoffel Number One of the quantities Xi appearing in the GAUSSJACOBI MECHANICAL QUADRATURE. They satisfy
LPn(x>12
+Ph+l(z)P?%(z)
nfl
 PLd4Pn+l(41.
(5)
s b
A1
+
x2
+
.
l
.
+
A,
=
da(x)
a
References Handbook (Eds.). nmulas, Graphs, and New York: Dover, ed.
Providence,
RI:
= a(b)  a(a)
(1)
Christ&e1
Symbol of the First Kind
and are given
by
Christoffel
Symbol of the Second Kind
The Christoffel FUNDAMENTAL d44
symbols are given in terms FORM E,F,and Gby
of
G. Orthogonal
Amer. Math.
(4)
(4
2GFv  GG,  FG, 2(EG  F2)
(5)
pn(x)*
4th
ed.
Providence,
Symbol denoted
RI:
of the First
Kind
and r& symbols
[ij, k], [i Ic j] , rabc, or {ab, c}*
 EE,  FE, 2(EG  F2)
(6)
EG,  FE, 2(EG  F2) EG,  2FF, + FG, 2(EG  F2)
Sot., pp. 4748, 1975.
Christoffel Variously
Polynomials,
(3)
GEv  FG, 2(EG  F2)
References Szeg6,
+ FE,  F2)
(3)
2EF, COEFFICIENT
of the first
GE,  2FF,
(2)
2(EG
where k, is the higher
245
(7) ’
(8)
= IT:, and r& = rT2. If F = 0, the Christoffel of the second kind simplify to
where gmk is the METRIC TENSOR and
w
(2)
(12) But
= [ik, j] + [jk, i], (Gray so [&
c]
=
f (gac,b
References A&en, G. Mathematical lando, FL: Academic
Christoffel Variously

gab&
(4)
The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first
Methods
for Physicists,
3rd ed. Or
&E+&F=
Press, pp. 160167, 1985.
of the Second
Kind
{ i m j } or rg.
= r;
=
gbc,a
FUNDAMENTAL FORM,
Symbol denoted
+
1 p
1993).
= Z”
d&  7 &P
= gkm[ij,
k]
;E,
rt2E
+ I’T2F
= +Ev
ri2E
+ ri2F
= Fv  ;G,
rilF
+ r&G
= Fu  +Ev
rt2F
+ rt2G
= iG,
lYi2F + l?;,G
= ;G,
km
r,‘, + rf,
= (In JEGF2),
ri,
= (In JEGFZ),
(15) (16) (17) (18) (19) (20) (21) (22)
(1)
where r; is a CONNECTION COEFFICIENT is a CHRISTOFFEL SYMBOL OF THE FIRST
and {bc,d} KIND.
(Gray
+ ri2
1993).
For a surface given in MONGE'S = gcd{bc, d}*
(2)
zijxk
FORM
z = F(x, y),
(23)
see UZSO CHRISTOFFEL SYMBOL OF THE FIRST KIND, CONNECTION COEFFICIENT, GAUSS EQUATIONS
Chromatic
246
Number
References 3rd ed. OrA&en, G. Mathematical Methods for Phys icists, lando, FL: Academic Press, pp. 160167 , 1985. Gray, A. “Christoffel Symbols.” 520.3 in M ‘odern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, pp. 397400, 1993. PhysMorse, P. M. and Feshbach, H. Methods of Theoretical its, Part I. New York: McGrawHill, pp . 4748,1953.
Chromatic Number The fewest number of colors r(G) necessary to color a GRAPH or surface. The chromatic number of a surface of GENUS g is given by the HEAWOOD CONJECTURE,
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 Spaces .” http://boole.stanford.edu/chuguide.btml,
ChuVandermonde
Identity 00
(x
on Chu
+
a)n
=
/\
x
(;)
(a)k(x)nk
k=O
where Lxj is the FLOOR FUNCTION. y(g) is sometimes also denoted x(g). For g = 0, 1, . . . , the first few values of x(g) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, . . . (Sloane’s AOOO934). 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, CHROMATIC POLYNOMIAL, EDGECOLORING, EULER CHARACTERISTIC, HEAWOOD CONJECTURE, MAP COLORING,TORUS COLORING
where (L) is a BINOMIAL COEFFICIENT and (a), n+ 1) isthe POCHHAMMER SYMBOL. a@1)+2special case gives the identity
E A
“YJ($ (;>=(:“)* see also BINOMIAL THEOREM,~MBRAL
CALCULUS
References Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 130 and 181182, 1996.
Church’s Theorem No decision procedure
exists for ARITHMETIC.
References Chartrand, G. “A Scheduling Problem: An Introduction to Chromatic Numbers.” $9.2 in IntrodzLctory Graph Theory. New York: Dover, pp. 202209, 1985. Eppstein, D. ‘&The Chromatic Number of the Plane.” http://uww S its , uci . edu / c~ eppstein / junkyard/ planecolor/. in “An OnLine Sloane, N. J. A. Sequence A000934/M3292 Version of the Encyclopedia of Integer Sequences.”
Chromatic Polynomial A POLYNOMIAL P(z) of a graph g which counts the number of ways to color g with exactly z colors. Tutte (1970) showed that the chromatic POLYNOMIALS of planar triangular graphs possess a ROOT close to +2 = 2.618033.. ‘, where # is the GOLDEN MEAN. More precisely, if n is the number of VERTICES of G, then
PG($“) 5 #5n (Le Lionnais
Thesis
see CHURCHTURING
Paris *: He rmann, Le Lionnais, F. Les nombres remarquables. p. 46, 1983. and the Tutte, W. T. “On Chromati .c Polynomials Ratio.” J. Combin. Th. 9, 289296, 1970,
ChurchTuring Thesis The TURING MACHINE concept defines what is meant mathematically by an algorithmic procedure. Stated another way, a function f is effectively COMPUTABLE IFF it can be computed by a TURING MACHINE. U~SO ALGORITHM, COMPUTABLE FUN CTION , TURING MACHINE
see
Heterences Penrose, R. ers,
Minds,
Oxford
C hv6t
The Emperor’s New Mind: Concerning and the Laws of Physics. Oxford,
University
al’s
Art
Press,
Gallery
pp. 4749,
Comput
England:
1989.
Theorem
Chv6tal’s Theorem Let the GRAPH G have VERTICES with VALENCES dl 5 < d,. If for every i < n/2 we have either di 2 i + 1 or d,i 2 n  i, then the GRAPH is HAMILTONIAN. l
l
l
Identity
seeCH~VANDERMONDE
THESIS
see ART GALLERY THEOREM
1983),
References
Chu
Church’s
IDENTITY
see COSINE INTEGRAL
Circle
Ci Cl
.
When normalized, the former gives the equation for the unit TANGENT VECTOR of the circle, ( sin t, cost). The circle can also be parameterized by the rational functions
see COSINE INTEGRAL Cigarettes It is possible each touches p. 115).
to place 7 cigarettes in such a way that the other if Z/d > 7fi/2 (Gardner 1959,
1  t2 x = t(1 + t)
(8)
y= 2t
References
(9)
1+ t2 ’
Gardner, M. The Scientific Puzzles & Diversions. 1959.
American
New
York:
Book
Simon
of Mathematical
and Schuster,
Cin see COSINE
247
but an ELLIPTIC CURVE cannot. The following plots show a sequence of NORMAL and TANGENT VECTORS for the circle.
INTEGRAL
Circle
A circle is the set of points equidistant from a given point 0. The distance T from the CENTER is called the RADIUS, and the point 0 is called the CENTER. Twice the RADIUS is known as the DIAMETER d = 2~. The PERIMETER C of a circle is called the CIRCUMFERENCE, and is given by C = rd = 27~. (1)
u m :_________
The circle is a CONIC SECTION obtained by the intersection of a CONE with a PLANE PERPENDICULAR to the CONE’S symmetry axis. A circle is the degenerate case and semiminor axes of an ELLIPSE with equal semimajor (i.e., with ECCENTRICITY 0). The interior of a circle is called a DISK. The generalization of a circle to 3D is called a SPHERE, and to nD for n 2 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
x = acost y = asin t. For a body
moving
uniformly
around
a are
t
ARC LENGTH S, CURVATURE K, and TANGENTIAL ANGLE 4 of the circle are
The
s(t)=
ds=
dmdt=.t
s 
xlyff @)
4(t)
=
@t2

+
1
yy yt2)3/2
ix(t)dt s
The
=
(11)
;
= 4. a
(12)
1 . a
(13)
CES~RO EQUATION is K=
In POLAR COORDINATES, the equation a particularly simple form.
of the circle has
(2) (3)
r=a is a circle of RADIUS a centered
the circle,
x1 = asint
(4)
y’ = acost,
(5)
at ORIGIN,
T = 2acosO is circle of RADIUS a centered
at (a, 0), and
and
xl1 = acost y" = a sin t.
(10)
s
r = 2asin8
(6) (7)
Circle
Circle
248
is a circle of RADIUS a centered SIAN COORDINATES, the equation a centered on (~0, ~0) is
on (0, a). In CARTEof a circle of RADIUS
(x  xo)2 + (y  ya)2 = u2* In PEDAL COORDINATES center, the equation is
withthe
(17)
PEDAL
POINT
pa = T2. The circle having
(18)
PI P2 as a diameter
(xx1)(x
at the
y2)
=
0.
C = rd = 27~. (19)
The equation of a circle passing through the three points (xi,yi) for i = 1, 2, 3 (the CIRCUMCIRCLE of the TRIANGLE determined by the points) is x
Xl2 +
Y
The CIRCUMFERENCEtoDIAMETER ratio C/d for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor s increases its PERIMETER by s), and d also scales by s. This ratio is denoted 7r (PI), and has been proved TRANSCENDENTAL. With d the DIAMETER and T the RADIUS,
is given by
x2)+(yy1)(y
I x2 + y2
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.
Xl
y1
1
x22 + Y22
x2
y2
1
xs2 + Y32
x3
Y3
1
Knowing C/d, we can then compute the AREA circle either geometrically or using CALCULUS. CALCULUS, 27r
A=
of the From
r
d6 s 0
11
Y12
(30)
rdr
= (27r)(;r2)
= m2.
derivations.
Using
(31)
s 0
=o
’
(20)
Now for a few geometrical strips, we have
concentric
The CENTER and RADIUS of this circle can be identified by assigning coefficients of a QUADRATIC CURVE
f =
ax2 + cy2 + dx + ey +
0,
(21)
where a = c and b  0 (since there is no x$~ cross term). C~MPLETIN G THE SQUARE gives
a(x+$)2+a(y+$)2+f~=~m The
CENTER
(22)
can then be identified
as
d
(23)
x”=tG yo=2a
and the RADIUS
e
As the number of strips increases to infinity, with a TRIANG LE on the right, so A=
i(2m)r
we are left
= m2.
(32)
This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC). If we cut the circle instead into wedges,
(24)
as
(25) As the number of wedges increases left with a RECTANGLE, so
(26) Xl2 d=
+y32
x32
e=
f
+Y12
~2~+y2~


Xl2
+y12
x22
+y22
x32
+
Yl
l
y2
1
y32
1
y3 Xl
(27)
1
x2
1
x3
1
(28)
Xl2
+
Y12
Xl
yr
x22
+
Y22
x2
Y2
x32
+Y32
x3
Y3
.
(29)
A = (m)r
= m2.
to infinity,
we are
(3%
~~~&OARC,BLASCHKE'S THEOREM,BRAHMAGUPTA'S FORMULA, BROCARD CIRCLE, CASEY'S THEOREM, CHORD, CIRCUMCIRCLE, CIRCUMFERENCE, CmFORD'SCIRCLETHEOREM,CLOSEDDISK,CONCENTRIC CIRCLES, COSINE CIRCLE, COTES CIRCLE PROPERTY, DIAMETER, DISK, DROZFARNY CIRCLES, EULER TRIANGLEFORMULA,EXCIRCLE,FEUERBACH'S THEOREM,
CirclesandSquares
fiactal
CircleCircle
Intersection
249
DISKS PROBLEM, FLOWER OF LIFE, FORD CIRCLE, FUHRMANN CIRCLE, GER~GORIN CIRCLE THEOREM, HOPF CIRCLE, INCIRCLE, INVERSIVE DISTANCE, JOHNSON CIRCLE, KINNEY'S SET, LEMOINE CIRCLE, LENS, MAGIC CIRCLES, MALFATTI CIRCLES, MCCAY CIRCLE, MIDCIRCLE, MONGE'S THEOREM, MOSER'S CIRCLE PROBLEM, NEUBERG CIRCLES, NINEPOINT CIRCLE, OPEN DISK, PCIRCLE, PARRY CIRCLE, PI, POLAR CIRCLE, POWER (CIRCLE), PRIME CIRCLE, PTOLEMY'S THEOREM, PURSER'S THEOREM, RADICAL AXIS, RADIUS, REULEAUX TRIANGLE, SEED OF LIFE, SEIFERT CIRCLE, SEMICIRCLE, SODDY CIRCLES, SPHERE, TAYLOR CIRCLE, TRIANGLE INSCRIBING IN A CIRCLE, TRIPLICATERATIO CIRCLE, TUCKER CIRCLES, UNIT CIRCLE, VENN DIAGRAM, VILLARCEAU CIRCLES, YINYANG
Circle Caustic Consider a point light source located at a point (p, 0). The CATACAUSTIC of a unit CIRCLE for the light at p = 00 is the NEPHROID
References
andforthe light 011the CIRCUMFERENCE ofthe P = 1 is the CARDXOID
FIVE
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 Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., reu. enl. Dublin: Hodges, F&is, & Co., pp. 96150, 1893. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 7475, 1996. Dunham, W. “Archimedes’ Determination of Circular Area.” Ch. 4 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 84112, 1990. Eppstein, D. “Circles and Spheres.” http://www . its . uci . edu/eppstein/junkyard/sphere.html. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 6566, 1972. MacTutor History of Mathematics Archive. “Circle.” http:
//wwwgroups. Circle.html.
x=
i[3 cos t  cos(3t)]
y = +[3sint The CATACAUSTIC for the light is the curve II:=
Y=
(1)
 sin(3t)l. at a finite
p(1  3p cos t + 2/L cos3 (1+ 2/G) +3pcost
(2) distance
t)
2~~ sin3 t 1+2/.?
xx;
 3pcost’
cost(1
y = Tj sint(l+ If the point is inside continuous twopart trated below.
+ cost)  + cost).
p > 1
(3) (4) CIRCLE
(5) (6)
the circle, the catacaustic is a discurve. These four cases are illus
dcs.stand.ac.uk/history/Curves/
Pappas, T.
“Infinity & the Circle” and “Japanese Calculus.” of Mathematics. San Carlos, CA: Wide World pp. 68 and 139, 1989. Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995. Yates, R. C. “The Circle,” A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 2125, 1952. The Joy Publ./Tetra,
CirclesandSquares
The CATACAUSTIC for PARALLEL rayscrossinga is a CARDIOID. see also
Fractal
CATACAUSTIC, CAUSTIC
CircleCircle
A FRACTAL produced &2+1=
which results see also
by iteration &I
in a MBIRilike
2 (mod
CIRCLE
Intersection
of the equation m)
pattern.
FRACTAL, MQ)IR~ PATTERN
Let two CIRCLES of RADII R and T and centered at (0,O) and (d, 0) intersect in a LENSshaped region. The equations of the two circles are
x2 + y2 = R2 (x  d)2 + y2 = r2.
(1) (2)
250
Circle Circle Intersection
Combining
(1) and (2) gives
Circle The limiting cases of this expression give 0 when d = R + T and
(x  d)2 + (R2  x2) = r2.
through
and rearranging
Solving
for 2 results
=T2
R2.
(4
d2  r2+R2 2d
(&)
 +dJ4Rzdz
(12)
(5)
’
The line connecting the cusps of the LENS therefore halflength given by plugging z back in to obtain
R)
(13)
when r = R, as expected. In order for half the area of two UNIT DISKS (R = 1) to overlap, set A = 7rR2/2 = r/2 in the above equation
in
x=
to
gives = 2A(;d,
x2  2dx + d2 x2
can be checked
(3)
A = 2R2 cos’ Multiplying
Cutting
;TT = 2cos‘(id) and solve numerically,
has
see
ah
LENS,
 ;dJ4dZ
yielding
SEGMENT,
(14
d ==:0.807946.
SPHERESPHERE
INTERSEC
TION 2d  (d2  r2 + R2)2
 4d2R2 giving
a length
2
d2  r2 + R2
y2 = R2  22 = R2 
4d2
Circle
>
0
(6)
1
Cutting
of
4
2 ;&d2R2
a=
= ;[(d+
T  R)(d
x [(d
’
 (d2  ~~ + R2)2  r + R)
+ T + R)(d + T + R)]‘/“.
This same formulation SPHERE INTERSECTION
applies directly problem.
(7)
to the SPHERE
= Rt2 cosl
(8)
twice, one for each half of the “LENS." Noting heights of the two segment triangles are
2d d2 + r2  R2 2d .
d2=dx=
the
f(l)
= 2
f(2)
= 2t
f(l)
f( n 1 = n+
f(n
f( n >=
n + [(n  1) + f (n  2)]
=n+(nl)+...
I
x k=l
(10)
k  1+ f(1)
J(d
= +(n+
1)  1+ 2 (4)
Evaluating (Sloane’s
for n = 1, 2, . . . gives 2, 4, 7, 11, 16, 22, . . . AOO0124)
A related PROBLEM, a CIRCLE
4 8 1 2 problem, sometimes called MOSER'S CIRCLE is to find the number of pieces into which is divided if n points on its CIRCUMFERENCE
l
d2 + r2  R2
sl
2dr

>
d2 + R2  r2
+ R2 cos1 1 2
= &(l)
n
(9)
+ A(R2,dz) (

2 + f(l)
is
A = A(Rd) = T2 cos
 1)
Therefore,
= i(n2+n+2).
The result
(1) (2) (3)
k=2
d2  r2+R2
dl=x=
that
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
To find the AREA of the asymmetric “LENS” in which the CIRCLES intersect, simply use the formula for the circular SEGMENT of radius R’and triangular height d’ A(R’,d’)
7
2dR T 
R)(d
>
+ r  R)(d
 T + R)(d
+ T + R). (11)
Circle Evolute
Circle Involute
are joined by CHORDS answer is
with no three CONCURRENT.
The
and the EVOLUTE degenerates GIN.
see also
(5)  &(n4
 6n3 + 23n2  18n + 24),
(6)
(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where (z) is a BINOMIAL COEFFICIENT. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, . . . (Sloane’s AO00127). This sequence and problem are an example of the danger in making assumptions based on limited trials. While the series starts off like 2n1, it begins differing from this GEOMETRIC SERIES at n = 6. see also CAKE CUTTING, CYLINDER CUTTING, SANDWICH THEOREM, PANCAKE THEOREM, THEOREM,~QUARE CUTTING,TORUS CUTTING
HAM PIZZA
References Conway, J. H. and Guy, R. K. “How
Many Regions.” In The Book of Numbers. New York: SpringerVerlag, pp* 7679, 1996. Amer. Guy, R. K. “The Strong Law of Small Numbers.” Math. Monthly 95, 697712, 1988. Noy, M. “A Short Solution of a Problem in Combinatorial Geometry.” Math. Mug. 69, 5253, 1996. Sloane, N. J. A. Sequences AO00124/M1041 and A000127/ Ml119 in “An OnLine Version of the Encyclopedia of Integer Sequences.” Yaglom, A. M. and Yaglom, I. M, Problem 47. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, 1987.
Circle
251
to a POINT at the ORI
CIRCLE INVOLUTE
References Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 77, 1993, Lauwerier, H. Fractals: Endlessly ures. Princeton, NJ: Princeton
Repeated
University
Geometric
Fig
Press, pp. 55
59, 1991, Circle Inscribing If T is the RADIUS of a CIRCLE inscribed in a RIGHT TRIANGLE with sides a and b and HYPOTENUSE c, then
T = +(a+bc). 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 circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a CYCLOID.
Evolute x = cost
x’ = sini!
y=sint
y’=cost
so the RADIUS OF CURVATURE R
=
(xl2 ylxl

+
xl1 = cost
(1)
y” =  sin t,
(2)
equations a 1 , the parametric deriva tives are given by
x = cost
x’ = sint
xtl = cost
Y = sin t
yt = cost
y” =  sin t.
The TANGENTVECTOR
xl'y'
(sin2 t + ~0s’ t)3/2 (sint)(sint)  (cost)cost
 1, 
(3) and the ARC LENGTH
along
the circle
(4) Therefore,
sin7G.f
t(t)
= x  Rsin7
= cost  1 xost
is given by
(5) (6)
=cos&
q(t)=y+Rcos~=sint+l*(sint)=O
is
(4 so the involute
sint
(2)
is
(3)
and the TANGENT VECTOR is
cos~=~~ji:=
of
is
y'2)3/2 
For a CIRCLE with the circle and their
= 0
(7) (8)
x = a(cost
+ tsint)
(6)
y = a(sint
 tcost).
(7)
Circle Involute
252
Pedal Curve
Circle Lattice
Points
THEOREM shows that for every POSITIVE INTEGER n, there exists a CIRCLE in the PLANE having exactly n LATTICE POINTS on its CIRCUMFERENCE. The theorem also explicitly identifies such ~CHINZEL SCHINZEL’S
The ARC GLE are
C~R&JRE,
LENGTH,
and TANGENTIAL
AN
CIRCLES”
as (x  $>” + y2 =
dmdt
= +t2
(9)
at (b= t. The CES~RO
(10)
EQUATION
see also CIRCLE, INVOLUTE
CIRCLE
is
EVOLUTE,
ELLIPSE
INVOLUTE,
References Differential Geometry of Curves and Gray, A. Modern faces. Boca Raton, FL: CRC Press, p. 83, 1993, Lawrence, J. D. A Catalog of Special Plane Curves. York: Dover, pp. 190191, 1972. MacTutor History of Mathematics Archive. “Involute Circle*” http://wwwgroups.dcs.stand.ac.uk/history /Curves/Involute*html.
Circle
Involute
Pedal
( X
(8)
Sur
New
of a
for n = 2k forn.=2k+l.
(I)
Note, however, that these solutions do not necessarily have the smallest possible RADIUS. For example, while the SCHINZEL CIRCLE centered at (l/3, 0) and with RADIUS 625/3 has nine lattice points on its CIRCUMFERENCE, so does the CIRCLE centered at (l/3, 0) with RADIUS 65/3. Let T be the smallest INTEGER RADIUS of a CIRCLE centered at the ORIGIN (0, 0) with L(r) LATTICE POINTS. In order to find the number of lattice points of the CIRCLE, it is only necessary to find the number in the first octant, i.e., those with 0 5 y 5 Lr/fi], where [zj is the FLOOR FUNCTION. Calling this N(T), then for T 2 I, L(T) = IN  4, so L(r) E 4 (mod 8). The multiphcation by eight counts all octants, and the subtraction by four eliminates points on the axes which the multi(Since 1/2 is IRRATIONAL, the plication counts twice. MIDPOINT of a are is never a LATTICE POINT.) GAUSS'S CIRCLE tice points within
Curve
$“’
;)‘+IJ~=;~~’
N(T)
=
PROBLEM a CIRCLE
4
1 +
LT]
asks for the number of RADIUS T
+
4E
Id=]
l
oflat
(2)
i=l
Gauss showed that N(r) The PEDAL
CURVE
of CIRCLE
INVOLUTE
g = sint  tcost with the center DES' SPIRAL
as the PEDAL
2
POINT
is the ARCHIME
= tsint
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 exactly n lattice points in its interior.
(3)
where IE(r)l
f = cost + tsint
= m2 + E(T),
< 2&m
............ . .... .......... .......... ............ . ........ . ........ ............ .......... .......... ..... ............ @
....
(4
. . .
.....
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 T < 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
L( r >
T
1 4 12 20 28 36 44 52 60 68 76 84 92 100 108 132 140 180 252 300 324
0 1 5 25 125 65 3,125 15,625 325 _< 390,625 < 1,953,125 1,625 c 48,828,125 4,225 1,105 40,625 21,125 5,525 27,625 71,825 32,045
and given by 1, 5, 25, 125, 65, 3125, (Sloane’s A046112).
(x  u)” + (y  b)2 + (z  h)” where a and b are the coordinates socalled SCHINZEL CIRCLE and berger 1973). see
UZSO
PROBLEM, SCHINZEL
c
= c2 + 2,
of the center of the is its RADIUS (Hons
CIRCLE, CIRCUMFERENCE, GAUSS’S KULIKOWSKI’S THEOREM, LATTICE CIRCLE, SCHINZEL’S THEOREM
CIRCLE POINT,
References Honsberger, R. “Circles, Squares, and Lattice Points.” Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117127, 1973. Kulikowski, T. “Sur l’existence d’une sphere passant par un nombre don& aux coordon&es enti&res.” L ‘Enseignement Math. Ser. 2 5, 8990, 1959. Schinael, A. ‘(Sur l’existence d’un cercle passant par un nombre don& de points aux coordonnkes entikres.” L’Enseignement Math. Ser. 2 4, 7172, 1958. Sierpiriski, W. “Sur quelques probl&mes concernant les points aux coordonnees entikres.” L’Enseignement Math. Ser. 2 4, 2531, 1958. Sierpiliski, W. “Sur un probEme de H. Steinhaus concernant les ensembles de points sur le plan.” Fund. Math. 46, 191194, 1959. Sierpiriski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. $$ Weisstein, E. W. “Circle Lattice Points.” http : // w w w . astro . Virginia. edu /  eww6n/math / notebooks / Circle LatticePoints .m.
Circle If the CIRCLE is instead centered at (l/2, 0), then the CIRCLES of RADII l/2, 312, 512, . . . 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 (l/3, 0), th en the number of lattice points on the CIRCUMFERENCE of the CIRCLES of RADIUS l/3, 2/3, 413, 513, 713, 813, . . . are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 5, 3, . . . (Sloane’s A0461 11). Let 1. a, be the RADIUS of the CIRCLE centered at (0, 0) having 8n+4 lattice points on its CIRCUMFERENCE, 2. b,/2 be the RADIUS of the CIRCLE centered at (l/2, 0) having 4n + 2 lattice points on its CIRCUMFER
Lattice
see GAUSS’S
Circle
Theorem
CIRCLE
PROBLEM
Map
A 1D MAP
which
maps
0)
Then the sequences {a,}, {bn}, and {cn} are equal, with the exception that b, = 0 if 21n and cn = 0 if 31n. However, the sequences of smallest radii having the above numbers of lattice points are equal in the three cases
a CIRCLE
onto itself
K
0 n+l=en+f2
G
SiK$2dn),
(1)
where 8n+l is computed mod 1. Note that the circle map has two parameters: n and K. 0 can be interpreted as an externally applied frequency, and K as a strength of nonlinearity. The 1D JACOBIAN is do n+l ~ aa
ENCE,
of CIRCLE centered at (l/3, points on its CIRCUMFERENCE.
325, . . .
KULIKOWSKI’S THEOREM states that for every POXTIVE INTEGER n, there exists a 3D SPHERE which has exactly n LATTICE PRINTS on its surface. The SPHERE is given by the equation
..e......
3. cn/3 be the RADIUS having 2n + 1 lattice
15625,
253
= 1  Kcos(2nOn),
so the circle map is not AREAPRESERVING. to the STAN ‘DARD MAP K
I n+1
=
0 n+l
= 0, + L+I,
172
+
G
sin( 27&)
(2) It is related
(3) (4)
Circle Method
254
for I and 8 computed
en+l
mod
Circle Packing
1. Writing
= 6, + In + g
&+I
as
sin(27&)
(5)
gives the circle map with Ifl = s1 and K = K. unperturbed circle map has the form
8n+l If s1 is RATIONAL, NUMBER, defined
The
=&+S2.
(6)
then it is known by
If a plot is made of K vs. n with the regions of periodic MODELOCKED parameter space plotted around RATIONAL 0 values (WINDING NUMBERS), then the regions are seen to widen upward from 0 at K = 0 to some finite width at K = 1. The region surrounding each RATIONAL NUMBER is knownasan ARNOLD TONGUE. At 0, the ARNOLD TONGUES are an isolated set of K= MEASURE zero. At K = 1, they form a CANTOR SET of DIMENSION d =2: 0.08’700. For K > 1, the tongues overlap, and the circle map becomes noninvertible. The circle map has a FEIGENBAUM CONSTANT
n+m
en  en’ 8 n+l 8,
see also ARNOLD TONGUE, LOCKING, Circle
WINDING
NUMBER
= 2 833 
FUNCTION
DEVIL'S STAIRCASE, (MAP)
References Steinhaus, New York:
H. Mathematical Oxford
University
POSETS,
PARTIALLY
ORDERED
Orthotomic
The
of the CIRCLE
ORTHOTOMIC
X =
represented
cost
(1)
y = sin t with
(2)
a source at (x, y) is x = xcos(2t)
 ysin(2i)
y = x sin(2t)
Circle
by
+ 2sint
 y cos(2t)
(3)
+ 2 cost.
(4)
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
P
Circle Notation A NOTATION for LARGE NUMBERS due to Steinhaus (1983) in which @ is defined in terms of STEINHAUSMOSER NOTATION as 72 in n SQUARES. The particular number known as the MEGA is then defined as follows.
MEGISTRON,
Circle
ISOMORPHIC
MODE
Circle Negative Pedal Curve The NEGATIVE PEDAL CURVE of a circle is an ELLIPSE if the PEDAL POINT is inside the CIRCLE, and a HYPERBOLA if the PEDAL POINT is outside the CIRCLE.
see U~SO MEGA, TATION
&SO
(8)
Method
see PARTITION
see SET
to a SET
as the map WINDING
and implies a periodic trajectory, since 8, will return to the same point (at most) every 4 ORBITS. If 0 is IRRATIONAL, then the motion is quasiperiodic. If K is NONZERO, then the motion may be periodic in some finite region surrounding each RATIONAL 0. This execution of periodic motion in response to an IRRATIONAL forcing is known as MODE LOCKING.
S G lim
Circle Order A POSET P is a circle order if it is ISOMORPHIC of DISKS ordered by containment.
STEINHAUSMOSER
Snapshots,
3rd American
Press, pp. 2829, 1983.
No
ed.
qZ2w3
= 0.9068996821..
..
Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes T6th 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 optimal for n = 1 through 10 (Kravitz 1967). The best known results are summarized in the following table.
CirclePoint
Circle Packing d approx.
d exact
1.00000
1
2.00000 2.15470.. 2.41421.. 2.70130.. 3.00000 3.00000 3.30476.
.
3.61312.. 3.82.. . 4.02..
Packings McCaughan, F. “Circle ac.uk/gjml1/cpacking/info.html. Molland, M. and Payan, Charles.
.”
.
m
Schaer,
n
a SQUARE,
d exact 1
1 2 3 4 5 6 7 8 9 10
1 5
of Ten
3
J. “The
Densest
Packing
Equal
Circles
in a
Circles
in
a
l
Pedal
Curve
proofs are known
d approx. 1.000 0.58.. . 0.500. . * 0.500 0.41.. . 0.37. . . 0.348.. . 0.341. 0.333.. . 0.148204. . . l
I
http://www.pmms.cam.
Square.” Math. Mag. 44, 139140, 1971. Valette, G. “A Better Packing of Ten Equal Square.” Discrete Math. 76, 5759, 1989.
Circle For CIRCLE packing inside only for n = 1 to 9.
255
Theorem
“A Better Packing of Ten Discrete Math. 84, 303305, Equal Circles in a Square.” 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. Mug. 48, 3337, 1975. Schaer, J. “The Densest Packing of Nine Circles in a Square.” Can. Math, Bul. 8, 273277, 1965.
. .
l
Midpoint
The PEDAL CURVE of a CIRCLE is a CARDIOID PEDAL POINT is taken on the CIRCUMFERENCE,
l
if the
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). and otherwise
see also HYPERSPHERE PACKING, MALFATTI’S RIGHT TRIANGLE PROBLEM, MERGELYANWESLER THEOREM, SPHERE PACKING
CirclePoint
a LIMA~ON. Midpoint
Theorem
References Conway, J. H. and Sloane, N. J. A. Sphere Packings, and Groups, 2nd ed. New York: SpringerVerlag, Eppstein, D. “Covering and Packing.” http: //www
Lattices, 1992.
. its . uci
.edu/eppstein/junkyard/cover.html.
Folkman, J. H. and Graham, R. “A Packing Compact Convex Subsets of the Plane.”
Bull. Gardner,
12, 745752,
Inequality
Canad.
for
Math.
1969.
M. “Mathematical
Games: The Diverse Pleasures of Circles that Are Tangent to One Another.” Sci. Amer. 240, 1828, Jan. 1979. Gardner, M. “Tangent Circles ,” Ch. 10 in Fractul Music, Hypercurds, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992. Goldberg,
M. “Problem E1924.” Amer. Math. Monthly 75, 195, 1968. Goldberg, M. “The Packing of Equal Circles in a Square.” Math. Mag. 43, 2430, 1970.
Goldberg, Circle.”
M. “Packing Math. Mag.
of 14,
16,
17,
and
20 Circles
in
a
44, 134139, 1971. Graham, R. L. and Luboachevsky, B, D. “Repeated Patterns of Dense Packings of Equal Disks in a Square.” Electronic J. Combinatorics 3, R16, l17, 1996. http://www. combinatorics.org/Volume3/volume3.html#Rl6.
Kravitz,
Math.
S. “Packing Cylinders into Mug. 40, 6570, 1967.
Cylindrical
Containers
.”
Taking the locus of MIDPOINTS from a fixed point to a circle of radius r results in a circle If radius r/2. This follows trivially from
r(O) =[a] +f ([z~  +ose ix 1 [ + sin8
 EI> X
0
’
References Johnson, R. A. Modern Geometry: An Elementary on the Geometry of the Triangle and the Circle. MA: Houghton Mifflin, p. 17, 1929.
Treatise Boston,
Circle Radial Curve
256 Circle
Radial
Circle Tangents
Curve
Circle Strophoid The STROPHOID of a CIRCLE with pole at the center and fixed point on the CIRCUMFERENCE is a FREETH’S NEPHROID.
The RADIAL POINT (sJ)
CURVE
is another
of a unit CIRCLE from a RADIAL CIRCLE with parametric equa
Circle Tangents There are four CIRCLES that touch all the sides of a given TRIANGLE. These are all touched by the CIRCLE through the intersection of the ANGLE BISECTORS of the TRIANGLE, known as the NINEPOINT CIRCLE.
x(t) =xcost y(t) =  sin&
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 finally 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 K = 3.1415926.. .. Ramanujan (191314) and Olds (1963) give geomet. Gardric constructions for 355/113 = 3.1415929.. ner (1966, pp. 9293) gives a geometric construction for 3 + 16/113 = 3.1415929. . Dixon (1991) gives constructions for 6/5(1 + 4) = 3.141640 . . and 4 + [3  tan(30°)] = 3.141533. . l
l
Given
the above figure,
GE = FH,
since
AB=AG+GB=GE+GF=GEf(GE+EF) =2G+EF CD=CH+HD=EH+FH=FH+(FH+EF) = EF + 2FH. Because
AB = CD,
it follows
that
GE = FH.
l
l
l
While the circle cannot SPACE, it can in GAUSSB• (Gray 1989). see also GEOMETRIC SQUARING
be
l
squared
in
LYAILOBACHEVSKY
CONSTRUCTION,
EUCLIDEAN SPACE
QUADRAT
URE,
References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 190191, 1996. New York: Dover, pp. 4449 and Dixon, R. Muthogruphics. 5253, 1991. Dunham, W. “Hippocrates’ Quadrature of the Lune.” Ch, 1 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 2026, 1990. Gardner, M. “The Transcendental Number Pi.” Ch. 8 in hlurtin entific
Gardner’s New Mathematical Diversions from SciAmerican. New York: Simon and Schuster, 1966. 3. Ideas of Space. Oxford, England: Oxford University
Gray, Press, 1989. Meyers, L. F. “Update on William Wernick’s ‘Triangle Constructions with Three Located Points.“’ Math. Mag. 69, 4649,
The
line tangent
to a CIRCLE
of RADIUS
a centered
(X? Y>
2’ = x+acost y' = y+ through
(0,O) can be found
asint by solving
the equation
1996.
Olds, C. D. Continued Fractions. New York: Random House, pp. 5960, 1963. Ramanujan, S. “Modular Equations and Approximations to K” Quart. J. Pure. Appl. Math. 45, 350372, 19131914.
giving
t= zkcosl (
ax zt &Idx2 + y2
. 1
at
Circular
Circuit Two of these four solutions trated above.
give tangent
lines,
as illus
also KISSING CIRCLES PROBLEM, MIQUEL POINT, MONGE'S PROBLEM, PEDAL CIRCLE, TANGENT LINE, TRIANGLE  #. fleferences
see
Dixon, R. Mathogruphics. New York: Dover, p. 21, 1991. Washington, R. More Mathematical Morsels. Honsberger, DC: Math. ,A ssoc. Amer., pp, 45, 1991.
257
Circulant Graph A GRAPH of n VERTICES in which the ith VERTEX is adjacent to the (i + j)th and (i  j)th VERTICES for each j in a list 2. Circulant An n x
Matrix n
MATRIX C defined
C
Circuit
I

(
(GRAPH)
see CYCLE
Functions
l
as follows,
(7)
(Y)
l
o7
. mm
l
*
(nnl)
n
n
1
’
>
.
l
.
(n22) l
.
1 Circuit Rank Also known as the CYCL~MATIC NUMBER. The circuit rank is the smallest number of EDGES y which must be removed from a GRAPH of IV EDGES and n nodes such that no CIRCUIT remains.
nl
C
n[(l
+Wj)n
where
wo E
1, ~1, Circulant
Determinant and Ryzhik
Xl
x2
x3
“’
Xn
Xl
x2
..
Xn
Xl
“’
Xn1 l
.
l
.
.
.
.
.
l
x2
x3
 rI(
(1970)
DETERMINANT
Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994. Stroeker, R. J. “Brocard Points, Circulant Matrices, and Descartes’ Folium.” Math, Mag. 61, 172187, 1988. Vardi, I. Computational Recreations in Mathematics. Reading, MA: AddisonWesley, p. 114, 1991.
l . .
“’ +
are the nth ROOTS . . . , tin1 matrices are examples of LATIN
References
by
X722
.
Xl
circulants
Xn1
l
x4
define
xn
l
113
SQUARES.
see also CIRCULANT Circulant Gradshteyn

j=O
OF UNITY.
Y =Nn+l.
=
x1
X2Wj
Circular
+
X3Wj2
+
l
.
+
XnWj
n1L
(1)
Cylindrical
Coordinates
see CYLINDRICAL COORDINATES
j=l
where wj is the nth ROOT OF UNITY. The secondorder circulant determinant is
I I Xl
x2
x2
Xl
=
and the third Xl
x2
x3
X3
Xl
52
152
X3
x11
=
order
(Xl
+x2)(x1
(2)
x2),
Circular Functions The functions describing the horizontal and vertical positions of a point on a CIRCLE as a function of ANGLE (COSINE and SINE) and those functions derived from them:
is
1 E =tan x 1 csc x = sin x
(x1+x2
+x3)(x1
+
wx2
+
w2x3)(x1
+
w2x2
I WXQ),
set x = (3)
where
cos 2 sin x
cotx
w and
w2 are the COMPLEX CUBE ROOTS of
(1) (2)
1 cos 2
tanz
sin 2 = cosx’
of circular
functions
(3)
(4)
UNITY. The EIGENVALUES X of the corresponding matrix are Xj
see
=
Xl
+
X2Wj
CIRCULANT
also
+
X3Ldj2
+
m  +
n
XnWj
x n circulant n1

(4)
MATRIX
References Gradshteyn, ries,
and
I. S. and Ryzhik, Products,
5th
ed.
I. M. Tables of Integrals, SeSan Diego, CA: Academic
Press, pp. 11111112, 1979. Vardi, I. Computational Recreations in Mathematics. ing, MA: AddisonWesley, p. 114, 1991.
Read
The
study
is called
TRIGONOME
see &SO COSECANT, COSINE, COTANGENT, ELLIPTIC FUNCTION, GENERALIZED HYPERBOLIC FUNCTIONS, HYPERBOLIC FUNCTIONS, SECANT, SINE, TANGENT, TRIGONOMETRY Keierences Abramowitz, M. and Stegun, C. A. (Eds.). tions.” $4.3 in Handbook of Mathematical Formulas, New York:
Graphs,
Dover,
and Mathematical
pp. 7179,
1972.
“Circular
Func
Functions with Tables, 9th printing.
Circular
258
Circumcircle
Permutation
Circular Permutation The number of ways to arrange n distinct a CIRCLE is Pn = (n  l)!.
objects
along
The number is (n  l)! instead of the usual FACTORIAL n! since all CYCLIC PERMUTATIONS of objects are equivalent because the CIRCLE can be rotated.
see also
PERMUTATION, PRIME CIRCLE
Circumcenter
The circumcenter
The center 0 of a TRIANGLE’S CIRCUMCIRCLE. It can be found as the intersection of the PERPENDICULAR BISECTORS. If the TRIANGLE is ACUTE, the circumcenter is in the interior of the TRIANGLE. In a RIGHT TRIANGLE, the circumcenter is the MIDPOINT of the HYPOTENUSE.


001+002+003=R+r,
(1)
where Oi are the MIDPOINTS of sides Ai, R is the CIRCUMRADIUS, and T is the INRADIUS (Johnson 1929, pa 190). The TRILINEAR COORDINATES of the circumcenter are cosA: and the exact trilinears RcosA
COSB : cosC,
(2)
are therefore : RcosB
: RcosC.
(3)
A, +bcot B, +ccot C).
also
H of the PEDAL TRIANGLE the CIRCUMCENTER 0 concurs 0 itself, as illustrated above. The on the EULER LINE.
BROCARD DIAMETER, CARNOT'S THEOREM,
CENTROID (TRIANGLE), TER, ORTHOCENTER
(4)
The distance between the INCENTER and circumcenter is @(R  2~). Given an interior point, the distances to the VERTICES are equal IFF this point is the circumcenter. It lies on the BROCARD AXIS.
CIRCLE,
EULER
LINE,
INCEN
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. of Triangles.” http: //www . its Eppstein, D. “Circumcenters .uci.edu/eppstein/junkyard/circumcenter.html.
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. Mug. 6?, 163187, 1994. Kimberling, C. “Circumcenter.” edu/ck6/tcenters/class/ccenter.html.
The AREAL COORDINATES are (+ot
ORTHOCENTER The A010203 formed by with the circumcenter circumcenter also lies see

0 and ORTHOCENTER H are ISOGO
Circumcircle
http:
//www.
evansville,
Circumradius
Circumcircle A TRIANGLE'S circumscribed circle. Its center 0 is called the CIRCUMCENTER, and its RADIUS R the CIRThe circumcircle can be specified using CUMRADIUS. TRILINEAR COORDINATES as &a.+ The STEINER circumcircle.
yab + apt
POINT
= 0.
S and TARRY
(1)
POINT
T lie on the
A GEOMETRIC CONSTRUCTION for the circumcircle is given by Pedoe (1995, pp. xiixiii). The equation for the circumcircle ofthe TRIANGLE with VERTICES (x,,Yi) for i= 1, 2, 3 is
Circumference The PERIMETER ETER d = 2r,
of a CIRCLE.
Expanding
x
y
1
Xl
Yl
1 =o
x22
+y22
x2
y2
1
x32
+
x3
y3
1
Y32
(2)
.
the DETERMINANT, a(x2 + y”)
+ Ma: + 2fy
+ g = 0,
(3)
a=
Xl
Yl
1
x2
Y2 Y3
1 1
x3
d=$
;
f
see
also
CIRCLE,
DIAMETER,
Circuminscribed Given two closed curves, simultaneously INSCRIBED CUMSCRIBED on the inner
Xl2
+y12
y1
1
+y22
y2
1
x32
+y32
y3
1
Xl
x22+y22
22
1
+y32
x3
1
Xl2 + Y12
Xl
y1
x22+y22
x2
y2
x3
y3
x32
COMPLETING
(4)
a2+y12
g=
+y32
THE SQUARE
is a CIRCLE
with
CLOSURE
Circumradius The radius of a TRIANGLE'S POLYHEDRON'S CIRCUMSPHERE, ANGLE,
.
+ b + c)(b + c  a)(c + a  b)(a + b  c)’
where the side lengths
(8)
2
 Yo)
=
r2,
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 Ol,Oz,and 03 and the radii ~1, 73, and 73, and using a = 71 + 72
00) (11)
and CIRCUMRADIUS
+d2 a2
Circles:
A Mathematical
Assoc.
Amer.,
View,
1995.
(4)
=
7l
+
73

(rl
+
T2)(Tl
+
r3)(T2
+
T3) (5)
+T2
+r3)
l
(12)
CIRCUMCENTER, CIRCUMRADIUS, ExCIRCLE,INCIRCLE,PARRY POINT,PURSER'S THEOREM, STEINER POINTS, TARRY POINT
ton, DC: Math.
(3)
c
lTP?Q(Tl
9 a’
(2)
b = r2 + r3
then gives R
see also CIRCLE,
References Pedoe, D.
are a, b, and c.
(9)
a
J
of the TRIANGLE
(7)
x0 =  d a f yoz
f2 
CIRCUMCIRCLE or of a denoted R. For a TRI
abc
R=
CIRCUMCENTER
T=
THEOREM
(6)
gives
‘I2+ (Y
the circuminscribed curve is in the outer one and CIRone.
1
of the form
( x  x0
PI, RADIUS
(5)
a(x+~)2+a(y+~)2~~+~=0 which
PERIMETER,
(1)
222
x32
T or DIAM
where K is PI.
J(a
where
For RADIUS
C = 27~ = nd,
see also PONCELET'S x2 + y2 Xl2 + Y12
259
rev.
ed.
Washing
If 0 is the CIRCUMCENTER TROID, then
m2zR2
and M is the triangle
;(a” + b2 + c”). ala2a3
Rr = 
4s
CEN
(6)
260
Circumscribed COSQil
+COSaz
Cissoid of Diodes = 1+
+COSa3
i
(8)
r = 2R cos a1 cos a2 cos a3
U12+
+
az2
a32
= 4TR
(Johnson 1929, pp. 189191). Let between INRADIUS T and circumradius
(9)
+ 8R2
(10)
d be the distance R, d = rR. Then
R2  d2 = 2Rr 1
1
1
SOLID,
for an ARCHIMEDEAN see
are
expressing the circumraT and MIDRADIUS p gives
SOLID.
CARNOT'S
also
THEOREM,
1
Curve
2
Pole
Cissoid
any point
line
center
conchoid of Nicomedes
on C on C opp.
oblique cissoid
parallel
line
circle
circle circle
tangent tangent
circle circle
radial line concentric circle
on C center
circle
same circle
bfi,
line
line line
(12)
(Mackay 188687). These and many other identities given in Johnson (1929, pp. 186190).
INRADIUS
Curve
line
(11)
Rd+=;
For an ARCHIMEDEAN dius in terms of the
Cissoid Given two curves Cl and C2 and a fixed point 0, let a line from 0 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.”
CIRCUMCIRCLE,
CIR
CUMSPHERE
cissoid of Diocles
tangent
see also CISSOID
strophoid circle 0)
lemniscate
OF DIoCLES
References Lawrence, J. D. A Catalog of SpeciaE Plane Curves. New York: Dover, pp. 5356 and 205, 1972. Lee, X. “Cissoid.” http://awv.best.com/xah/Special PlaneCurves,dir/Cissoiddir/cissoid.html, Lockwood, E. H. “Cissoids.” Ch. 15 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130433, 1967. Yates R . C “Cissoid ” A Handbook on Curves and Their Prdpertiei. Ann Arbor, MI: J. W. Edwards, pp* 2630, 1952.
References Johnson, on the
R. A. Modern Geometry
Geometry: of the Triangle
MA: Houghton Mifflin, Mackay, J. S. “Historical and its Developments Math.
Sot.
5, 6278,
An Elementary and the Circle.
1929, Notes on a Geometrical [18th Century].” Proc.
of Diocles
Theorem Edinburgh
18864887.
Circumscribed A geometric figure which touches only the VERTICES other extremities) of another figure. see also SCRIBED,
Cissoid
Treatise Boston,
CIRCUMCENTER,
CIRCUMRADIUS,
Circumsphere A SPHERE circumscribed called the CIRCUMRADIUS.
CIRCUMCIRCLE, INSCRIBED
in a given
solid.
see also INSPHERE Cis Cis x S eix = cost
+ isinx.
(or
CIRCUMIN
Its radius
is
A curve invented by Diocles in ibout 180 BC in connection with his attempt to duplicate the cube by geometrical methods. The name Vissoid” 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 0 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.
Clark’s
Cissoid of Diodes The cissoid of Diocles 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 t ions
of Diocles
is given by the parametric
y=
0
261
Cissoid of Diocles Caustic The CAUSTIC ofthe 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 D~OCLES is taken as the INVERSION CENTER, then the cissoid inverts to a
PARABOLA.
equaCissoid
x = 2asin2
Wangle
of Diocles
Pedal
Curve
(1)
2a sin3 8
case
(2)
’
these to POLAR COORDINATES gives
Converting
r2 = x2 + y2 = 4a2 = 4a2 sin4
ep
(
sin6 8 sin4 0 + 
~0~2e >
+ tan2 0) = 4a2 sin4 Osec2 0,
(3)
PEDAL CURVE of the cissoid, when the PEDAL POINT is on the axis beyond the ASYMPTOTE at a dis
The
so
T = 2asin2 6&d?
(4
= 2asinOtanO.
In CARTESIAN COORDINATES, ~ X3 2ax
8a3 sin6 0

2a  2asin2
= 4a An equivalent
0
= 4a2
sin6 e 1  sin2 e
2=y2. sin6 e ~0~2
parametric
y(t)
Equation
(9)
+ 4)3/Z ’
Cissoid of Diocles.” s3.4 in Modern DifferGeometry of Curves and Surfaces.Boca Raton, FL: Press, pp. 4346, 1993. Lawrence, J. D. A Catulog of Special Plane Curves. New York: Dover, pp. 98100, 1972. Lee, X. “Cissoid of Diocles.” http://www.best.com/xah/ SpecialPlaneCurves_dir/CissoidClfDioclesAir/cissoid
Of Diocles.html. E. H. A Book
of Curves. Cambridge, England: University Press, pp. 130133, 1967. History of Mathematics Archive. “Cissoid of Dio
cles .” http://wwwgroups.dcs.stand+ac.uk/history/ Curves/Cissoid.html. Yates R . C . “Cissoid ” A Handbook on Curves Prdperties. Ann Aibor, MI: J. W. Edwards,
Y = PX +
and
fCP>7
see &O D'ALEMBERT'S EQUATION Boyer, C. B. A History p. 494, 1968.
A. “The
1952.
Differential
References
ential CRC
MacTutor
of the
(8)
References
Cambridge
that
(7)
g ives the CURVATURE as 3
Lockwood,
is four times
where f is a FUNCTION of one variable and p E dy/dx. The general solution is y = cx + f(c). The singular ENVELOPES are x = f’(c) and y = f(c) solution cf’(4
form
2at3 = jq
IEM = altl(f2
Gray,
cusp which
(6)
2at2
1993),
the
or
x(t) = ~ 1 + t2
(Gray
Clairaut’s
form is
the alternative
from
(5)
e
x(x2 + y2) = 2ay2. Using
tance
ASYMPTOTE is a CARDIOID.
Their
pp+ 2630,
Clarity The RATIO of a measure of a “residual.”
of Mathematics.
New York:
of the size of a “fit”
Wiley,
to the size
References Tukey, J. W. Explanatory Data AddisonWesley, p. 667, 1977.
Clark’s
Analysis.
Triangle (m1)3 n2 6 1 d 12 7 1 J 18 19 8 1 24 37 27 9 1 30 61 64 36 10 1 36 91 125100 46 11 1 0
Reading,
MA:
Clark’s
262
Triangle
Class Number
A NUMBER TRIANGLE created by equal to 0, filling one diagonal with onal with multiples of an INTEGER remaining entries by summing the side from one row above. Call the and the last column vz = n so that
setting
the VERTEX Is, the other diag
f, and filling in the elements on either first column n = 0
= 1,
(2)
n) = c(m  1, n  1) + c(m  1,
to compute
the rest of the entries.
 1)(6m2
c(m, 3) = $ (m  1)2(m
(1)
then use the RECURRENCE RELATION c(m,
c(m, 2) = i(m
 12m + 6)
= (m  1)3
c(m, 0) = fm c(m,m)
So far, this has just been relatively boring ALGEBRA. But the amazing part is that if f = 6 is chosen as the INTEGER, then c(m, 2) and c(m, 3) simplify to
n)
(3)
For n = 1, we have
(13)  2)2,
(14)
which are consecutive CUBES (m  1)3 and nonconsecutive SQUARES n2 = [(m  l)(m  2)/212.
see UZSO BELL TRIANGLE, CATALAN'S TRIANGLE, EULER'S TRIANGLE, LEIBNIZ HARMONIC TRIANGLE, NUMBER TRIANGLE, PASCAL'S TRIANGLE, SEIDELENTRINGERARNOLD TRIANGLE, SUM References
c(m,
1) = c(m  1,O) + c(m  1, 1)
c(m, 1)  c(m  1,1) = c(m  1,O) = f(m For arbitrary m, the value MING this RECURRENCE,
c(m,
(4)
 1).
can be computed
(5)
by SUM
1) =
Clark,
J. E. “Clark’s
Triangle.”
Math.
Student
26, No. 2,
p* 4, Nov. 1978. Class
see CHARACTERISTIC CLAss, Cuss (MULTIPLY PERFECT NUMBER), CLASS (SET)$ONJUGACY CLASS Class
INTERVAL, CLASS CLASS NUMBER,
(Group)
see CONJUGACY Cuss Now, for n = 2 we have c(m,
2) = c(m
 1,l)
+ c(m
 1,2)
(7)
= $f(ml)m+l,
(8)
Class Interval The constant bin size in a HISTOGRAM.
see also c(m, 2)c(m1,2) SO
= c(ml,l)
Class (Map) A MAP u : Iw” + R” from a DOMAIN G is called of class CT if each component of
SUMMING the RECURRENCE gives
c(m,Z)
= x[+fk(k
 1) + 11 = fJ+fk’
k=l + l)(Zm


Taking
 &fk + 1)
+ I)]

$f[$2(m
+ l)]
+ m
(9
i(ml)(fm22fm+6).
for n = 3 we have
c(m, 3)  c(m
 1,3)
+l)m
(f +2).
is of class C’ (0 5 T 2 00 or r = w) in G, where denotes a continuous function which is differentiable times.
(10
the SUM,
= kn for a MULTIPLY PERFECT NUMBER is called its class.
Class Number For any IDEAL I, there
m
i fk3  fk2 +
(+f +
1)k  (f + 2). (11)
where z is a PRINCIPAL
the SUM gives
c(m, 3) = & (m  l)(m
 2)(fm2
 3fm+
12).
(12)
is an IDEAL Ii such that I&
k=2
Evaluating
Cd d
Class (Multiply Perfect Number) The number IG in the expression s(n)
= c(m  1,2)
 fm” +(+f
ifm3
c(m, 3) = x
a map
k=l
= +f[&(m
Similarly,
SHEPPARD'S CORRECTION
= x,
(1)
IDEAL, (i.e., an IDEAL of rank 1). Moreover, there is a finite list of ideals Ii 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 number is a measure of the failure in the original number ring.
of unique
factorization
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 cyclotomic integers, as well as for any subring of the cyclotomic integers. Finding the class number is a computationally difficult problem. Let h(d) denote correspondingtothe
with
the class number
of a quadratic
ring,
BINARY QUADRATIC FORM ax2 + bxy + cyz,
(2)
d c b2  4~.
(3)
DISCRIMINANT
Then the class number h(d) for DISCRIMINANT d gives the number of possible factoriaations of ax2 + bxy + cy2 in the QUADRATIC FIELD Q(d). Here, the factors are of the form x + yJd, with z and y half INTEGERS. Some fairly sophisticated mathematics shows that the class number for discriminant d can be given by the
CLASS NUMBER FORMULA
h(d) =
 &
x,“r:(djr)
 #
crA;‘(dlr)r
In sin (7)
for d > 0
(4)
for d < 0,
(dir) is the KRONECKER SYMBOL, r](d) is the FUNDAMENTAL UNIT, w(d) is the number of substitutions which leave the BINARY QUADRATIC FORM un
where
changed for d = 3 for d = 4 otherwise,
6 4
w(d) =
2
(5)
over all terms where the KRON(Cohn 1980). The class number for d > 0 can also be written and the sums are taken
ECKER SYMBOL is defined
2hW
=
(6)
rI T=1
for d > 0, where the PRODUCT is taken over terms which the KRONECKER SYMBOL is defined. The class number
to the DIRICHLET
is related
for
The Mathematics@ (Wolfram Research, Champaign, IL) function NumberTheory ’ NumberTheoryFunct ions ’ ClassNumber [n] gives the class number h(d) for d a NEGATIVE SQUAREFREE number ofthe form 4k+l.
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 nz. This problem has been solved for n < 7 and ODD 72 < 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 (Cohen 1993, p. 229), Sloane’s A046002, Sloane’s A046003, The complete set of negative discriminants havf&lass numbers l5 and ODD 723 are known. Buell (1977) gives the smallest and largest fundamental class numbers for d < 4,000,000, partitioned into EVEN discriminants, discriminants 1 (mod 8)) and discriminants 5 (mod 8). Arno et al. (1993) give complete lists of values of d with h(d) = k for ODD k = 5, 7, 9, . . . , 23. Wagner gives complete lists of values for Tc = 5, 6, and l
+
+
7.
Lists of NEGATIVE discriminants corresponding to IMAGINARY QUADRATIC FIELDS Q(Jdo) having small class numbers h( d) are given in the table below. In the table, Iv is the number of “fundamental” values of d with a given class number h(d), where “fundamental” means that d is not divisible by any SQUARE NUMBER s2 such that h(d/s2) < h(d). For example, although h( 63) = 2, 63 is not a fundamental dis= h(7) = criminant since 63 = 32 7 and h(63/32) 1 < h(63). EVEN values 8 5 h(d) 5 18 have been The number of negative discomputed by Weisstein. 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). l
LSERIES
bY
 Ld(l> h(d) 44 ' where n(d) is the DIRICHLET Wagner (1996) sh ows that the INEQUALITY
for d < 0, where 1x1 is the FLOOR FUNCTION, the product is orer PRIMES dividing d, and the * indicates that the GREATEST PRIME FACTOR of d isomittedfrom the product.
l
dl rl
263
(7)
STRUCTURE CONSTANT.
class number
h(d)
In d,
satisfies
(8)
The following table lists the numbers with small class numbers 5 11. Lists including larger class numbers are given by Weisstein. h(d)
N 1
9
2
18
3
16
d 3, 4, 7, 8, 11, 15, 20, 24, 35, 123,148,187, 23, 31, 59, 83, 331, 379,499,
19, 43, 67, 163 40, 51, 52, 88, 91, 115, 232, 235,267,403,427 107, 139, 211, 283, 307, 547,643,883, 907
Class Number
264 h&d)
N
4
5
d
Class Num her h(d)
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 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,
7
8
9
10
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 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 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 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 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,
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
2683
6
N
51
The table below gives lists of POSITIVE fundamental discriminants d having small class numbers h(d), corresponding to REAL quadratic fields. All POSITIVE SQUAREFREE values of d < 97 (for which the KRONECKER SYMBOL is defined) are included.
h(d) d 1 2
5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77 65 The POSITIVE d for which h(d) = 1 is given by Sloane’s A014539. see also
CLASS NUMBER FORMULA, DIRICHLET LSERIES, DHCRIMINANT (BINARY QUADRATIC FORM), GAUSS’S CLASS NUMBER CONJECTURE, GAUSS’S CLASS NUMBER PROBLEM, HEEGNER NUMBER, IDEAL,
References Arno,
4."
S. “The Imaginary Acta
Arith.
Quadratic
40, 321334,
Fields
of Class Number
1992.
Arno, S.; Robinson, M. L.; and Wheeler, F. S. “Imaginary Quadratic F ie Id s with Small Odd Class Number.” http: // www.math.uiuc.edu/AlgebraicNumberTheory/OOO9/. Buell, D. A. “Small Class Numbers and Extreme Values of LFunctions of Quadratic Fields.” Math. Comput. 139, 786796, 1977. Cohen, H. A Course in Computational Algebraic Number Theory. New York: SpringerVerlag, 1993. Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 and 234, 1980. Cox, D. A. Primes of the Form x2 fny2: 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: SpringerVerlag, pp. 4353, 1980. Iyanaga, S. and Kawada, Y. (Eds.). “Class Numbers of Algebraic Number Fields.” Appendix B, Table 4 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp, 14941496, 1980. Montgomery, H. and Weinberger, P. “Notes on Small Class Numbers.” Acta. Arith. 24, 529542, 1974. Sloane, N. J. A. Sequences A014539, A038552, A046125, and A003657/M2332 in “An OnLine Version of the Encyclopedia of Integer Sequences.”
Class Number
Clausen
Formula
Stark, II. M. “A Complete Determination Quadratic Fields of Class Number One."
J. 14, l27, Stark,
ber Two." Wagner,
Math.
1967.
H. M. “On Complex Math.
Compwt.
C, “Class
Number
Quadratic
Fields with
29, 289302, 5, 6, and 7.”
Class Num
1975.
Math.
Comput.
65,
785800,1996. Weisstein, E. W. “Class Numbers.” http virginia.edu/eww6n/math/note~ooks/ClassN~bers.m.
: //www
. astro
.
1. CYCLIC
GROUPS
2. ALTERNATING 3. LIETYPE
Class Number Formula A class number formula is a finite series giving exactly the CLASS NUMBER ofa RING. For a RING ofquadratic integers, the class number is denoted h(d), where d is the discriminant. A class number formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. This formula includes the quadratic case as well as many cubic and higherorder rings. see also CLASS NUMBER Class Representative A set of class representatives contains exactly one element
is a SUBSET of X which from each EQUIVALENCE
CLASS.
TITS 2F4(2n)t,
Classical Groups The four following
FINITE
see also GROUP
GROUPS
E&)r
pwn, a) 7 4)) GROUPS or the
h(q),
Es(q),
F.(q),
2B(2”),
GROWP,
GROUP,
~FUNCTION,
SIMPLE
M. “Ten Thousand
Pages to Prove
Scientist
109, 2630, 1985.
B. “Are
Group
Theorists
Simpleminded?”
Happening in the Mathematical Sciences, Vol. 3. Providence, RI: Amer. Math. Sot., pp.
Cipra,
B. “Sl imming
794795,
Simplicity.”
an Outsized
Theorem."
What’s 19951996,
8299, 1996. Science
267,
Sci. Amer.
253,
1995.
Gorenstein, D. “The Enormous 104~115,Dec. 1985.
Theorem.”
Solomon, R. “On Finite Simple Groups and Their Classification.” Not. Amer. Math. Sot. 42, 231239, 1995.
Clausen Clausen’s
Formula 4 F3 identity b
c
e f
9
d
”
1 =
(24 IdI b + b) 14 (Zb) I4
(2a + 2b)ldpldlbldl ’
holds for a + b + c  d = l/2, e = a + b + l/2, a + f = d a nonpositive integer, and (a), is the P~CHHAMMER SYMBOL (PetkovSrek et al. 1996). d+l=b+g,
which were studied before more exotic types of groups (such as the SPORADIC GROUPS) were discovered. LINEAR G ROUP, GROUP J NITARY
of degree at least five,
References
4F3
1. LINEAR GROUPS, 2. ORTHOGONAL GROUPS, 3. SYMPLECTIC GROUPS, and 4. UNITARY GROUPS,
see UZSO GROUP, GRO UP, SYMPLECTIC
GROUP) 3D4(q), G&), “G2(sn),
a
types of GROUPS,
A,
CHEVALLEY
The “PROOF,, of this theorem is spread throughout the mathematical literature and is estimated to be approximately 15,000 pages in length.
Cipra,
see also AGGREGATE,RUSSELL'S PARADOX, SET
GROUPS
ORDER,
5. SPORADIC GROUPS k&, Mlz, M22, Mz3, M2*, Jz = HJ, SW, HS, McL, Cog, Coz, Co1, He, Fia2, Fi29, Fik4, HN, Th, B, M, JI, U’N, J3, Ly, Ru, J4.
New
kind of SET invented to get around RUSSELL'S PARADOX while retaining the arbitrary criteria 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).
Z& of PRIME
psw 4)' pswn, 4)' and Pf+, 4. LIETYPE (TWISTED CHEVALLEY
Cartwright,
Class (Set) A class is a special
265
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 completely into
of the Complex
Michigan
Formula
ORTHOGONAL GROUP
Classifxat ion The classification of a collection of objects generally means that a list has been constructed with exactly one member from each IS~M~RPHISM type among the objects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of mathematical objects which have been classified include the finite SIMPLE GROUPS and ZMANIFOLDS but not, for example, KNOTS.
Another identity ascribed to Clausen which involves the HYPERGEOMMTRIC FUNCTION &(u,~;c;z) and the GENERALIZED HYPERGEOMETRIC FUNCTION 3F2 (CL, b, c; d, e; z) is given by
=
3F2
2a, a + b, 2b a+b++,2a+2bix
see also GENERALIZED HYPERG EOMETRIC HYPERG .EOMETRIC FUN 'CTION
> ’
FUNCTION,
References Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. AB. ley, MA: A. K. Peters, pp. 43 and 127, 1996.
Welles
266
Clausen
Clausen
CLEAN
Function
Algorithm
References
Function
Abramowitz, M. and Stegun, C. A. (Eds.). “Clausen’s Integral and Related Summations” $27.8 in Handbook of Mathematical matical
Functions with Formulas, Tables, 9th printing. New
Graphs,
York:
and
Dover,
Mathe
pp. 1005
1006, 1972. A&en, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 783, 1985. Clausen, R. “Uber die Zerlegung reeller gebrochener Funktionen.” J. reine angew. Math. 8, 298300, 1832. Concerning the Computation of Grosjean, C. C. “Formulae the Clausen Integral Cl,(cy)J’ J. Comput. Appl. Math. 11,
331342,1984. Jolley, L. EL W. Summation of Series. London: Chapman, 1925. Wheelon, A. II. A Short Table of Summable Series. Report No. SM14642. Santa Monica, CA: Douglas Aircraft Co., 1953.
Define
(1) Clausen’s
k=l
Integral
(2) k=l
and write Cl,(x)
Then ically
=
S,(X)
= CTxl
+$J
n even
&(x)
= Crxl
w
n odd.
(3)
the Clausen function Cl,(x) can be given symbolin terms of the POLYLOGARITHM as The Cl,(x)
li[Li,(e““) $Li,(eeiz)
=
 Li,(ei”)] + Lin(ei2)]
n even n odd.
CLAUSEN FUNCTION 0
Cl&?) For n = 1, the function Cl,(x)
takes on the special
= C,(x)
= In
= 
ln[2 sin( it)]
dt.
s0
form
]2sin($x)]
(4)
see also CLAUSEN FUNCTION References
and for n = 2, it becomes
Abramowitz,
INTEGRAL
CLAUSEN’S X
Cl,(x)
= &(x)
ln[2 sin( it)]
= s
M.
of Mathematical Mathematical
dt.
(5)
0
sin(ne) ” Math.
65,nrd56.n2 sums of opposite parity are summable and the first few are given by +x2
Cz(x)
=
Cd(X)
=
&
Sl(X)
=
+ CT
S3(x)
=
in22
S5(4
=
&7r4x

$x
+
&x2x2

+x2 +
+x3

&x4
(7) ($1
2) 
$x2 
$T2x3
for 0 5 x < 27r (Abramowitz
+
&x3 +
(9) &x4
and Stegun

&x5
(10)
1972).
see do CLAUSEN'S INTEGRAL, POLYGAMMA FUNCTION, POLYLOGARITHM'
C.
A.
Tables
Aids
Handbook
(Eds.).
with Formulas, printing. New
Graphs,
York:
and
Dover,
of the Function $(e) = Comp. IO, 54 and 57
l
Clausen, R. ‘%ber t ionen.” J. reine
(6)
Stegun,
Functions Tables, 9th
pp* 10051006, 1972. Ashour, A. and Sabri, A. “Tabulation C”
The symbolic symbolically,
and
die Zerlegung
reeller
angew.
8, 298300,
Math.
gebrochener
Funk
1832.
CLEAN Algorithm An iterative algorithm which DECONVOLVES a sampling function (the “DIRTY BEAM") from an observed brightness ("D