CRC Concise Encyclopedia MAmEMAfJCS CRC Concise Encyclopedia MAfflEMAffG Eric W. Weisstein CRC Press Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Weisstein, Eric W. The CRC concise encyclopedia of mathematics / Eric W. Weisstein. p. cm. Includes bibliographical references and index. ISBN 0-8493-9640-9 (alk. paper) 1. Mathematics- -Encyclopedias. I. Title. QA5.W45 1998 510'.3— DC21 98-22385 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are only used for identification and explanation, without intent to infringe. © 1999 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-9640-9 Library of Congress Card Number 98-22385 Printed in the United States of America 1234567890 Printed on acid-free paper Introduction The CRC Concise Encyclopedia of Mathematics is a compendium of mathematical definitions, formulas, figures, tabulations, and references. It is written in an informal style intended to make it accessible to a broad spectrum of readers with a wide range of mathematical backgrounds and interests. Although mathematics is a fascinating subject, it all too frequently is clothed in specialized jargon and dry formal exposition that make many interesting and useful mathematical results inaccessible to laypeople. This problem is often further compounded by the difficulty in locating concrete and easily understood examples. To give perspective to a subject, I find it helpful to learn why it is useful, how it is connected to other areas of mathematics and science, and how it is actually implemented. While a picture may be worth a thousand words, explicit examples are worth at least a few hundred! This work attempts to provide enough details to give the reader a flavor for a subject without getting lost in minutiae. While absolute rigor may suffer somewhat, I hope the improvement in usefulness and readability will more than make up for the deficiencies of this approach. The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs from existing dictionaries of mathematics in a number of important ways. First, the entire text and all the equations and figures are available in searchable electronic form on CD-ROM. Second, the entries are extensively cross-linked and cross-referenced, not only to related entries but also to many external sites on the Internet. This makes locating information very convenient. It also provides a highly efficient way to "navigate" from one related concept to another, a feature that is especially powerful in the electronic version. Standard mathematical references, combined with a few popular ones, are also given at the end of most entries to facilitate additional reading and exploration. In the interests of offering abundant examples, this work also contains a large number of explicit formulas and derivations, providing a ready place to locate a particular formula, as well as including the framework for understanding where it comes from. The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g., Borowski and Borwein's HarperCollins Dictionary of Mathematics and Jeans and Jeans' Mathematics Dictio- nary). At the same time, the descriptions are more accessible than in "technical" mathematical encyclopedias (e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of Mathematics), While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate, research scientist, or recreational mathematician. In this work, the most useful, interesting, and entertaining (at least to my mind) aspects of topics are discussed in addition to their technical definitions. For example, in my entry for pi (71-), the definition in terms of the diameter and circumference of a circle is supplemented by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. These formulas are comprehensible to readers with only minimal mathematical background, and are interesting to both those with and without formal mathematics training. However, they have not previously been collected in a single convenient location. For this reason, I hope that, in addition to serving as a reference source, this work has some of the same flavor and appeal of Martin Gardner's delightful Scientific American columns. Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections weave their way through the physical sciences. It frequently transpires that some piece of recently acquired knowledge turns out to be just what I need to solve some apparently unrelated problem. I have therefore developed the habit of picking up and storing away odd bits of information for future use. This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. I have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for people who are interested in the subjects listed here but who, like me, are not necessarily experts. The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and continuing during graduate school. The initial document was written in Microsoft Word® on a Mac Plus® computer, and had reached about 200 pages by the time I started graduate school in 1990. When Andrew Treverrow made his OzTgX program available for the Mac, I began the task of converting all my documents to T^X, resulting in a vast improvement in readability. While undertaking the Word to T^}K conversion, I also began cross-referencing entries, anticipating that eventually I would be able to convert the entire document to hypertext. This hope was realized beginning in 1995, when the Internet explosion was in full swing and I learned of Nikos Drakos's excellent I^X to HTML converter, I£TgX2HTML. After some additional effort, I was able to post an HTML version of my encyclopedia to the World Wide Web, currently located at www . astro . Virginia . edu/ - eww6n/math/. The selection of topics included in this compendium is not based on any fixed set of criteria, but rather reflects my own random walk through mathematics. In truth, there is no good way of selecting topics in such a work. The mathematician James Sylvester may have summed up the situation most aptly. According to Sylvester (as quoted in the introduction to Ian Stewart's book From Here to Infinity), "Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its "contour defined; it is as limitless as that space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life." Several of Sylvester's points apply particularly to this undertaking. As he points out, mathematics itself cannot be confined to the pages of a book. The results of mathematics, however, are shared and passed on primarily through the printed (and now electronic) medium. While there is no danger of mathematical results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical results simply because they are not aware of them. Not only does collecting many results in one place provide a single starting point for mathematical exploration, but it should also lessen the aggravation of encountering explanations for new concepts which themselves use unfamiliar terminology. In this work, the reader is only a cross-reference (or a mouse click) away from the necessary background material. As to Sylvester's second point, the very fact that the quantity of mathematics is so great means that any attempt to catalog it with any degree of completeness is doomed to failure. This certainly does not mean that it's not worth trying. Strangely, except for relatively small works usually on particular subjects, there do not appear to have been any substantial attempts to collect and display in a place of prominence the treasure trove of mathematical results that have been discovered (invented?) over the years (one notable exception being Sloane and Plouffe's Encyclopedia of Integer Sequences), This work, the product of the "gazing" of a single astronomer, attempts to fill that omission. Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, neither table of contents nor index are included. In many cases, a particular entry of interest can be located from a cross-reference (indicated in SMALL CAPS TYPEFACE in the text) in a related article. In addition, most articles are followed by a "see also" list of related entries for quick navigation. This can be particularly useful if you are looking for a specific entry (say, "Zeno's Paradoxes"), but have forgotten the exact name. By examining the "see also" list at bottom of the entry for "Paradox," you will likely recognize Zeno's name and thus quickly locate the desired entry. The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning with a numeral are ordered by increasing value and appear before the first entry for "A." In multiple-word entries containing a space or dash, the space or dash is treated as a character which precedes "a," so entries appear in the following order: "Sum," "Sum P. . . ," "Sum-P. . . ," and "Summary." One exception is that in a series of entries where a trailing "s" appears in some and not others, the trailing "s" is ignored in the alphabetization. Therefore, entries involving Euclid would be alphabetized as follows: "Euclid's Axioms," "Euclid Number," "Euclidean Algorithm." Because of the non-standard nomenclature that ensues from naming mathematical results after their discoverers, an important result such as the "Pythagorean Theorem" is written variously as "Pythagoras 's Theorem," the "Pythagoras Theorem," etc. In this encyclopedia, I have endeavored to use the most widely accepted form. I have also tried to consistently give entry titles in the singular (e.g., "Knot" instead of "Knots"). In cases where the same word is applied in different contexts, the context is indicated in parentheses or appended to the end. Examples of the first type are "Crossing Number (Graph)" and "Crossing Number (Link)." Examples of the second type are "Convergent Sequence" and "Convergent Series." In the case of an entry like "Euler Theorem," which may describe one of three or four different formulas, I have taken the liberty of adding descriptive words ("Euler's Something Theorem") to all variations, or kept the standard name for the most commonly used variant and added descriptive words for the others. In cases where specific examples are derived from a general concept, em dashes ( — ) are used (for example, "Fourier Series," "Fourier Series — Power Series," "Fourier Series — Square Wave," "Fourier Series — Triangle"). The decision to put a possessive 's at the end of a name or to use a lone trailing apostrophe is based on whether the final "s" is pronounced. "Gauss's Theorem" is therefore written out, whereas "Archimedes' Recurrence Formula" is not. Finally, given the absence of a definitive stylistic convention, plurals of numerals are written without an apostrophe (e.g., 1990s instead of 1990's). In an endeavor of this magnitude, errors and typographical mistakes are inevitable. The blame for these lies with me alone. Although the current length makes extensive additions in a printed version problematic, I plan to continue updating, correcting, and improving the work. Eric Weisstein Charlottesville, Virginia August 8, 1998 Acknowledgments Although I alone have compiled and typeset this work, many people have contributed indirectly and directly to its creation. I have not yet had the good fortune to meet Donald Knuth of Stanford University, but he is unquestionably the person most directly responsible for making this work possible. Before his mathematical typesetting program TfeX, it would have been impossible for a single individual to compile such a work as this. Had Prof. Bateman owned a personal computer equipped with T£jX, perhaps his shoe box of notes would not have had to await the labors of Erdelyi, Magnus, and Oberhettinger to become a three- volume work on mathematical functions. Andrew Trevorrow's shareware implementation of I^X for the Macintosh, OzI]eX (www.kagi.com/authors/akt/oztex.html), was also of fundamental importance. Nikos Drakos and Ross Moore have provided another building block for this work by developing the IM]gX2HTML program (www-dsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html), which has allowed me to easily maintain and update an on-line version of the encyclopedia long before it existed in book form. I would like to thank Steven Finch of MathSoft, Inc., for his interesting on-line essays about mathemat- ical constants (www.mathsoft.com/asolve/constant/constant.html), and also for his kind permission to reproduce excerpts from some of these essays. I hope that Steven will someday publish his detailed essays in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed and on-line (www.research.att.com/-njas/sequences/) versions of the Encyclopedia of Integer Sequences, an immensely valuable compilation of useful information which represents a truly mind-boggling investment of labor. Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript and providing a number of helpful suggestions and additions. Thanks also to algebraic topologist Ryan Bud- ney for sharing some of his expertise, to Charles Walkden for his helpful comments about dynamical systems theory, and to Lambros Lambrou for his contributions. Thanks to David W. Wilson for a number of helpful comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permis- sion to reproduce their beautiful knot and link diagrams. Thanks to Gavin Theobald for providing diagrams of his masterful polygonal dissections. Thanks to Wolfram Research, not only for creating an indispensable mathematical tool in Mathematica® , but also for permission to include figures from the Mathematical book and MathSource repository for the braid, conical spiral, double helix, Enneper's surfaces, Hadamard matrix, helicoid, helix, Henneberg's minimal surface, hyperbolic polyhedra, Klein bottle, Maeder's "owl" minimal surface, Penrose tiles, polyhedron, and Scherk's minimal surfaces entries. Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading the entire document for syntax. Thanks also to Bob Stern, my executive editor from CRC Press, for his encouragement, and to Mimi Williams of CRC Press for her careful reading of the manuscript for typographical and formatting errors. As this encyclopedia's entry on Proofreading Mistakes shows, the number of mistakes that are expected to remain after three independent proofreadings is much lower than the original number, but unfortunately still nonzero. Many thanks to the library staff at the University of Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I would like to thank the hundreds of people who took the time to e-mail me comments and suggestions while this work was in its formative stages. Your continued comments and feedback are very welcome. 10 Numerals see Zero The number one (1) is the first Positive Integer. It is an Odd Number. Although the number 1 used to be considered a PRIME Number, it requires special treat- ment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. The number 1 is sometimes also called "unity," so the nth roots of 1 are often called the nth Roots of Unity. Fractions having 1 as a Nu- merator are called Unit Fractions. If only one root, solution, etc., exists to a given problem, the solution is called Unique. The Generating Function have all Coefficients 1 is given by 1 ii ,2.3.4. 1 + x + x -\- x + x + l~x see also 2, 3, Exactly One, Root of Unity, Unique, Unit Fraction, Zero The number two (2) is the second POSITIVE INTEGER and the first PRIME NUMBER. It is Even, and is the only Even Prime (the Primes other than 2 are called the Odd Primes). The number 2 is also equal to its Fac- torial since 2! = 2. A quantity taken to the Power 2 is said to be SQUARED. The number of times k a given BINARY number & n --*&2&i&o is divisible by 2 is given by the position of the first 6^ = 1, counting from the right. For example, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is divisible by 2 times. see also 1, BINARY, 3, SQUARED, ZERO 2x mod 1 Map Let xo be a Real Number in the Closed Interval [0, 1], and generate a SEQUENCE using the MAP Xn+i = 2x n (mod 1). (i) Then the number of periodic Orbits of period p (for p Prime) is given by N„ 2 p -2 V (2) Since a typical Orbit visits each point with equal prob- ability, the Natural Invariant is given by P {x) = 1. (3). see also Tent Map References Ott, E. Chaos in Dynamical Systems. Cambridge: Cam- bridge University Press, pp. 26-31, 1993. 3 is the only INTEGER which is the sum of the preceding Positive Integers (1 + 2 = 3) and the only number which is the sum of the FACTORIALS of the preceding Positive Integers (1! + 2! = 3). It is also the first Odd Prime. A quantity taken to the Power 3 is said to be Cubed. see also 1, 2, 3^ + 1 Mapping, Cubed, Period Three Theorem, Super-3 Number, Ternary, Three- Colorable, Zero 3x + 1 Mapping see Collatz Problem 10 The number 10 (ten) is the basis for the DECIMAL sys- tem of notation. In this system, each "decimal place" consists of a DIGIT 0-9 arranged such that each Digit is multiplied by a POWER of 10, decreasing from left to right, and with a decimal place indicating the 10° = Is place. For example, the number 1234.56 specifies Ixl0 3 +2xl0 2 +3xl0 1 +4xl0° + 5xl0~ 1 +6xl0~ 2 . The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 10000, 100000, 10000000, 100000000, ... (Sloane's A011557), called one, ten, HUNDRED, THOUSAND, ten thousand, hundred thou- sand, Million, 10 million, 100 million, and so on. The names of subsequent decimal places for Large Num- bers differ depending on country. Any Power of 10 which can be written as the PRODUCT of two numbers not containing 0s must be of the form 2 n • 5 n — 10 n for n an INTEGER such that neither 2 n nor 5 n contains any ZEROS. The largest known such number 10 33 - 2 33 * 5 33 = 8, 589, 934, 592 ■ 116, 415, 321, 826, 934, 814, 453, 125. A complete list of known such numbers is 10 1 = 2 1 10 2 = 2 2 10 4 10' 10 9 10 18 10 33 : 2 9 * 5 9 : 2 18 ■ 5 1 2 33 • 5 3 (Madachy 1979). Since all POWERS of 2 with exponents n < 4.6 X 10 7 contain at least one ZERO (M. Cook), no 12 18-Point Problem other POWER of ten less than 46 million can be written as the PRODUCT of two numbers not containing Os. see also Billion, Decimal, Hundred, Large Num- ber, Milliard, Million, Thousand, Trillion, Zero References Madachy, J. S. Madachy J s Mathematical Recreations. New York: Dover, pp. 127-128, 1979. Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, p. 135, 1995. Sloane, N. J. A. Sequence A011557 in "An On-Line Version of the Encyclopedia of Integer Sequences." 12 One Dozen, or a twelfth of a Gross. see also DOZEN, GROSS 13 A Number traditionally associated with bad luck. A so-called Baker's Dozen is equal to 13. Fear of the number 13 is called Triskaidekaphobia. see also Baker's Dozen, Friday the Thirteenth, Triskaidekaphobia 15 see 15 Puzzle, Fifteen Theorem 15 Puzzle 2 1 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a 4x4 box leaving one position out of the 16 empty. The goal is to rearrange the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrange- ments, this rearrangement is possible, but for others, it is not. To address the solubility of a given initial arrangement, proceed as follows. If the SQUARE containing the num- ber i appears "before" (reading the squares in the box from left to right and top to bottom) n numbers which are less than £, then call it an inversion of order n, and denote it rii. Then define N — X^ n * = 5Z n *' where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so n\ must equal 0). If AT is EVEN, the position is possible, otherwise it is not. This can be formally proved using Alternating Groups. For example, in the following arrangement ri2 = 1 (2 precedes 1) and all other rii = 0, so N — 1 and the puzzle cannot be solved. References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, pp. 312- 316, 1987. Bogomolny, A. "Sam Loyd's Fifteen." http://www.cut— the- knot.com/pythagoras/fifteen.html. Bogomolny, A. "Sam Loyd's Fifteen [History]." http://www. cut-the-knot .com/pythagoras/historyl5.html. Johnson, W. W. "Notes on the '15 Puzzle. I.'" Amer. J. Math. 2, 397-399, 1879. Kasner, E. and Newman, J. R. Mathematics and the Imagi- nation. Redmond, WA: Tempus Books, pp. 177-180, 1989. Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical Recreations. New York: W. W. Norton, pp. 302-308, 1942. Story, W. E. "Notes on the '15 Puzzle. II.*" Amer. J. Math. 2, 399-404, 1879. 16-Cell A finite regular 4-D POLYTOPE with SCHLAFLI SYMBOL {3, 3, 4} and Vertices which are the PERMUTATIONS of (±1, 0, 0, 0). see also 24-Cell, 120-Cell, 600-Cell, Cell, Poly- tope 17 17 is a FERMAT PRIME which means that the 17-sided Regular Polygon (the Heptadecagon) is Con- STRUCTIBLE using COMPASS and STRAIGHTEDGE (as proved by Gauss). see also CONSTRUCTIBLE POLYGON , FERMAT PRIME, HEPTADECAGON References Carr, M. "Snow White and the Seven(teen) Dwarfs." http:// www . math . harvard . edu / - hmb / issue2.1 / SEVENTEEN/seventeen.html. Fischer, R. "Facts About the Number 17." http: //tempo, harvard . edu / - rf ischer / hcssim / 17_f acts / kelly / kelly.html. Lefevre, V. "Properties of 17." http://www.ens-lyon.fr/ -vlefevre/dl7_eng.html. Shell Centre for Mathematical Education. "Number 17." http : //acorn . educ . nott ingham . ac . uk/ShellCent/ Number /Num 17 .html. 18-Point Problem Place a point somewhere on a Line Segment. Now place a second point and number it 2 so that each of the points is in a different half of the Line SEGMENT. Con- tinue, placing every ATth point so that all N points are on different (l/iV)th of the Line Segment. Formally, for a given N y does there exist a sequence of real num- bers xi t X2, • • • , #jv such that for every n £ {1, . - . , N} and every k £ {1, . . . , n}, the inequality fc- 1 ^ k — < Xi < - n n 24-Cell 196-Algorithm holds for some i € {l,...,n}? Surprisingly, it is only possible to place 17 points in this manner (Berlekamp and Graham 1970, Warmus 1976). Steinhaus (1979) gives a 14-point solution (0.06, 0.55, 0.77, 0.39, 0.96, 0.28, 0.64", 0.13, 0.88, 0.48, 0.19, 0.71, 0.35, 0.82), and Warmus (1976) gives the 17-point solu- tion | < a* < ■&> f < X2 < £, jf < x 3 < 1, £ < x 4 < ^, IT < ** < IS- H < ** < h 1 < ^ < £, if < ** < h I <x 9 < ±,$ <x 10 < *,± <zu < £, 17 < ^12 < 12 > 2 — Xl2 < 17' U — Xl4 < 17' 13 ^ ^ ^ 4 5 ^ _ ^ 6 10 ^ ^ ^ 11 Warmus (1976) states that there are 768 patterns of 17- point solutions (counting reversals as equivalent). see also Discrepancy Theorem, Point Picking References Berlekamp, E. R. and Graham, R. L. "Irregularities in the Distributions of Finite Sequences." J. Number Th. 2, 152- 161, 1970. Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer- Verlag, pp. 34-36, 1997. Steinhaus, H. "Distribution on Numbers" and "Generaliza- tion." Problems 6 and 7 in One Hundred Problems in Elementary Mathematics. New York: Dover, pp. 12-13, 1979. Warmus, M. "A Supplementary Note on the Irregularities of Distributions." J. Number Th. 8, 260-263, 1976. 24-Cell A finite regular 4-D Polytope with SCHLAFLI Symbol {3,4,3}. Coxeter (1969) gives a list of the VERTEX po- sitions. The Even coefficients of the D 4 lattice are 1, 24, 24, 96, ... (Sloane's A004011), and the 24 shortest vectors in this lattice form the 24-cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995). see also 16-Cell, 120-Cell, 600-Cell, Cell, Poly- TOPE References Conway, J. H. and Sloane, N. J. A. Sphere- Packings, Lattices and Groups, 2nd ed. New York: Springer- Verlag, 1993. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Sloane, N. J. A. Sequences A004011/M5140 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. 42 According to Adams, 42 is the ultimate answer to life, the universe, and everything, although it is left as an exercise to the reader to determine the actual question leading to this result. References Adams, D. The Hitchhiker's Guide to the Galaxy. New York: Ballantine Books, 1997. 72 Rule see Rule of 72 120-Cell A finite regular 4-D Polytope with Schlafli Symbol {5,3,3} (Coxeter 1969). see also 16-Cell, 24-Cell, 600-Cell, Cell, Poly- tope Preferences Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. 144 A Dozen Dozen, also called a Gross. 144 is a Square Number and a Sum-Product Number. see also Dozen 196-Algorithm Take any POSITIVE INTEGER of two DIGITS or more, re- verse the DIGITS, and add to the original number. Now repeat the procedure with the SUM so obtained. This procedure quickly produces PALINDROMIC NUMBERS for most INTEGERS. For example, starting with the num- ber 5280 produces (5280, 6105, 11121, 23232). The end results of applying the algorithm to 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, ... (Sloane's A033865). The value for 89 is especially large, being 8813200023188. The first few numbers not known to produce PALIN- DROMES are 196, 887, 1675, 7436, 13783, . . . (Sloane's A006960), which are simply the numbers obtained by iteratively applying the algorithm to the number 196. This number therefore lends itself to the name of the Algorithm. The number of terms a(n) in the iteration sequence re- quired to produce a Palindromic Number from n (i.e., a(n) = 1 for a PALINDROMIC NUMBER, a(n) = 2 if a Palindromic Number is produced after a single iter- ation of the 196-algorithm, etc.) for n = 1, 2, . . . are 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, ... (Sloane's A030547). The smallest numbers which require n = 0, 1, 2, . . . iterations to reach a palin- drome are 0, 10, 19, 59, 69, 166, 79, 188, . . . (Sloane's A023109). see also Additive Persistence, Digitadition, Mul- tiplicative Persistence, Palindromic Number, Palindromic Number Conjecture, RATS Se- quence, Recurring Digital Invariant References Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 242-245, 1979. Gruenberger, F. "How to Handle Numbers with Thousands of Digits, and Why One Might Want to." Sci. Amer. 250, 19-26, Apr. 1984. Sloane, N. J. A. Sequences A023109, A030547, A033865, and A006960/M5410 in "An On-Line Version of the Encyclo- pedia of Integer Sequences." 239 65537-gon 239 Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Beeler et al. (1972, Item 63). 239 appears in Machin's Formula | 7 r = 4tan(|)-tan- 1 (^), which is related to the fact that 2 * 13 - 1 239 2 , which is why 239/169 is the 7th CONVERGENT of y/2 . Another pair of INVERSE TANGENT FORMULAS involv- ing 239 is tan" 1 ^) = tan" 1 ^) - tan" 1 ^) = tan x (^)+tan l (^). 239 needs 4 SQUARES (the maximum) to express it, 9 Cubes (the maximum, shared only with 23) to express it, and 19 fourth POWERS (the maximum) to express it (see Waring'S Problem). However, 239 doesn't need the maximum number of fifth POWERS (Beeler et al 1972, Item 63). References Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. 257-gon 257 is a FERMAT PRIME, and the 257-gon is there- fore a Constructible Polygon using Compass and Straightedge, as proved by Gauss. An illustration of the 257-gon is not included here, since its 257 seg- ments so closely resemble a Circle. Richelot and Schwendenwein found constructions for the 257-gon in 1832 (Coxeter 1969). De Temple (1991) gives a con- struction using 150 Circles (24 of which are Car- lyle Circles) which has Geometrography symbol 945i + 475 2 + 275Ci + 0C 2 + 150C 3 and Simplicity 566. see also 65537-GON, CONSTRUCTIBLE POLYGON, Fer- mat Prime, Heptadecagon, Pentagon References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- ity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964. 600-Cell A finite regular 4-D Polytope with Schlafli Symbol {3,3,5}. For Vertices, see Coxeter (1969). see also 16-Cell, 24-Cell, 120-Cell, Cell, Poly- tope References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. 666 A number known as the Beast Number appearing in the Bible and ascribed various numerological properties. see also Apocalyptic Number, Beast Number, Le- viathan Number References Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 96, 1993. 2187 The digits in the number 2187 form the two VAMPIRE NUMBERS: 21 x 87 = 1827 and 2187 = 27 x 81. References Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19, 26-29, Spring 1997. 65537-gon 65537 is the largest known Fermat Prime, and the 65537-gon is therefore a CONSTRUCTIBLE POLYGON us- ing Compass and Straightedge, as proved by Gauss. The 65537-gon has so many sides that it is, for all in- tents and purposes, indistinguishable from a CIRCLE us- ing any reasonable printing or display methods. Her- mes spent 10 years on the construction of the 65537-gon at Gottingen around 1900 (Coxeter 1969). De Temple (1991) notes that a Geometric Construction can be done using 1332 or fewer Carlyle Circles. see also 257-GON, CONSTRUCTIBLE POLYGON, HEP- TADECAGON, Pentagon References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- ity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. A-Integrable A A-Integrable A generalization of the Lebesgue INTEGRAL. A MEA- SURABLE Function f(x) is called A-integrable over the Closed Interval [a, b] if m{x:\f(x)\>n} = 0(n- 1 ), (1) where m is the LEBESGUE MEASURE, and lim / [f(x)] n dx (2) exists, where tf(xW -IfW if 1/0*01 <" |« l/(*)J»-| if|/( x )|>„. W References Titmarsch, E. G. "On Conjugate Functions." Proc. London Math. Soc. 29, 49-80, 1928. A- Sequence N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. An Infinite Sequence of Positive Integers ai sat- isfying 1 < ai < a-2. < az < ■ • . (1) is an A-sequence if no a^ is the SUM of two or more distinct earlier terms (Guy 1994). Erdos (1962) proved oo S{A) = sup Y^ ~ < 103 - ( 2 ) all A sequences , a k Any A-sequence satisfies the Chi Inequality (Levine and O'Sullivan 1977), which gives 5(A) < 3.9998. Ab- bott (1987) and Zhang (1992) have given a bound from below, so the best result to date is AAS Theorem Erdos, P. "Remarks on Number Theory III. Some Problems in Additive Number Theory." Mat. Lapok 13, 28-38, 1962. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/erdos/erdos.html. Guy, R. K. "B 2 -Sequences." §E28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 228-229, 1994. Levine, E. and O'Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34, 9-24, 1977. Zhang, Z. X. "A Sum-Free Sequence with Larger Reciprocal Sum." Unpublished manuscript, 1992. AAA Theorem Specifying three ANGLES A, B, and C does not uniquely define a Triangle, but any two TRIANGLES with the same Angles are SIMILAR. Specifying two ANGLES of a TRIANGLE automatically gives the third since the sum of Angles in a Triangle sums to 180° (it Radians), i.e., C = tt-A-B. see also AAS Theorem, ASA Theorem, ASS Theo- rem, SAS Theorem, SSS Theorem, Triangle AAS Theorem Specifying two angles A and B and a side a uniquely determines a TRIANGLE with AREA K ■ a 2 sin B sin C a 2 sin B sin(7r — A — B) 2 sin A 2 sin A The third angle is given by C = ir - A- B, (1) (2) 2.0649 < 5(A) < 3.9998. (3) since the sum of angles of a Triangle is 180° (n Ra- dians). Solving the Law of Sines Levine and O'Sullivan (1977) conjectured that the sum of Reciprocals of an A-sequence satisfies oo (4) where %% are given by the Levine-O'Sullivan Greedy Algorithm. see also B 2 -Sequence, Mian-Chowla Sequence References Abbott, H. L. "On Sum-Free Sequences." Acta Arith. 48, 93-96, 1987. for b gives Finally, sin A sin B sinB b = a—r sin A (3) (4) c = b cos A + a cos B = a(sin B cot A -f cos B) (5) = a sin B(cot A -f cot B) . (6) see also AAA Theorem, ASA Theorem, ASS Theo- rem, SAS Theorem, SSS Theorem, Triangle 6 Abacus AbeVs Functional Equation Abacus A mechanical counting device consisting of a frame hold- ing a series of parallel rods on each of which beads are strung. Each bead represents a counting unit, and each rod a place value. The primary purpose of the abacus is not to perform actual computations, but to provide a quick means of storing numbers during a calculation. Abaci were used by the Japanese and Chinese, as well as the Romans. see also Roman Numeral, Slide Rule References Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal Fractions." A History of Mathematics, 2nd ed. New York: Wiley, pp. 199-201, 1991. Fernandes, L. "The Abacus: The Art of Calculating with Beads." http : //www . ee . ryerson . ca : 8080/-elf /abacus. Gardner, M. "The Abacus." Ch. 18 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathemati- cal Entertainments from Scientific American. New York: Knopf, pp. 232-241, 1979. Pappas, T. "The Abacus." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 209, 1989. Smith, D. E. "Mechanical Aids to Calculation: The Abacus." Ch. 3 §1 in History of Mathematics, Vol. 2. New York: Dover, pp. 156-196, 1958. abc Conjecture A Conjecture due to J. Oesterle and D. W. Masser. It states that, for any INFINITESIMAL e > 0, there exists a Constant C e such that for any three Relatively Prime Integers a, 6, c satisfying a 4- b = c, the Inequality max{|a|,|6|,|c|}<a JJ p 1+e p\abc holds, where p\abc indicates that the PRODUCT is over Primes p which Divide the Product abc. If this Conjecture were true, it would imply Fermat's Last Theorem for sufficiently large Powers (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are at least C In x WlEFERlCH PRIMES < x for some constant C (Silverman 1988, Vardi 1991). see also Fermat's Last Theorem, Mason's Theo- rem, Wieferich Prime References Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. Math. Monthly 101, 3-14, 1994. Goldfeld, D. "Beyond the Last Theorem." The Sciences, 34- 40, March/April 1996. Guy, R. K, Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 75-76, 1994. Silverman, J. " Wieferich's Criterion and the abc Conjecture." J. Number Th. 30, 226-237, 1988. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison- Wesley, p. 66, 1991. Abelian see Abelian Category, Abelian Differential, Abelian Function, Abelian Group, Abelian In- tegral, Abelian Variety, Commutative Abelian Category An Abelian category is an abstract mathematical CAT- EGORY which displays some of the characteristic prop- erties of the Category of all Abelian Groups. see also Abelian Group, Category Abel's Curve Theorem The sum of the values of an INTEGRAL of the "first" or "second" sort f XltV1 Pdr [*n,vn pd / ^ + - + J ^ = F ^ and P(xi,2/i) dxx P(xn,Vn) dx N Q(rci,yi) dz Q(xn,Vn) dz dF dz ' from a FIXED Point to the points of intersection with a curve depending rationally upon any number of param- eters is a Rational Function of those parameters. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959. Abelian Differential An Abelian differential is an ANALYTIC or MEROMOR- phic Differential on a Compact or closed Riemann Surface. Abelian Function An Inverse Function of an Abelian Integral. Abelian functions have two variables and four periods. They are a generalization of ELLIPTIC FUNCTIONS, and are also called Hyperelliptic Functions. see also Abelian Integral, Elliptic Function References Baker, H. F. Abelian Functions: Abel's Theorem and the Al- lied Theory, Including the Theory of the Theta Functions. New York: Cambridge University Press, 1995. Baker, H. F. An Introduction to the Theory of Multiply Pe- riodic Functions. London: Cambridge University Press, 1907. Abel's Functional Equation Let Li2(x) denote the DlLOGARITHM, defined by — n Abelian Group then Li 2 (a) + Li 2 (y) + lA 2 {xy) + + (1-2/) see a/50 DlLOGARITHM, POLYLOGARITHM, RlEMANN Zeta Function Abelian Group N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. A Group for which the elements Commute (i.e., AB = BA for all elements A and B) is called an Abelian group. All Cyclic Groups are Abelian, but an Abelian group is not necessarily CYCLIC. All SUBGROUPS of an Abelian group are NORMAL. In an Abelian group, each element is in a CONJUGACY CLASS by itself, and the CHARACTER TABLE involves POWERS of a single element known as a Generator. No general formula is known for giving the number of nonisomorphic Finite GROUPS of a given ORDER. However, the number of nonisomorphic Abelian FINITE Groups a(n) of any given Order n is given by writing n as n = Y[pi"\ (1) i where the pt are distinct PRIME FACTORS, then a(n) =Y[P( ai ), (2) where P is the Partition Function. This gives 1,1, 1, 2, 1, 1, 1, 3, 2, . . . (Sloane's A000688). The smallest orders for which n = 1, 2, 3, ... nonisomorphic Abelian groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 64, 1800, 0, 288, 128, ... (Sloane's A046056), where denotes an impossible number (i.e., not a product of partition numbers) of nonisomorphic Abelian, groups. The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, ... (Sloane's A046064). The incrementally largest numbers of Abelian groups as a function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, . . . (Sloane's A046054), which occur for orders 1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... (Sloane's A046055). The Kronecker Decomposition Theorem states that every Finite Abelian group can be written as a Di- rect Product of Cyclic Groups of Prime Power Orders. If the Orders of a Finite Group is a Prime p, then there exists a single Abelian group of order p (denoted Z p ) and no non-Abelian groups. If the Or- ders is a prime squared p 2 , then there are two Abelian groups (denoted Z p 2 and Z p & Z p . If the Orders is Abelian Group 7 a prime cubed p 3 , then there are three Abelian groups (denoted Z p <g> Z p (g> Z p , Z p % Z p 2, and Z p a), and five groups total. If the order is a PRODUCT of two primes p and q, then there exists exactly one Abelian group of order pq (denoted Z p ® Z q ). Another interesting result is that if a(n) denotes the number of nonisomorphic Abelian groups of ORDER n, then ^a(n)n- s = CWC(2s)C(3 S )- (3) n=l where ((s) is the Riemann Zeta Function. Srinivasan (1973) has also shown that N Y, a (n) = A 1 N+A 2 N 1/2 +A 3 N 1/3 +O[x 105/407 (]nx) 2 ], n=l (4) where ( 2.294856591... for k = 1 Ak = n^(i) = \ - 14 -6475663... for k = 2 (5) j=i V } { 118.6924619 ... for k = 3, and ( is again the Riemann Zeta Function. [Richert (1952) incorrectly gave As = 114.] DeKoninck and Ivic (1980) showed that ^J-^BN + Oi^ilnN)- 1 ' 2 }, i(n) (6) where nKE P(k - 2) P(k) 0.752 . . (7) is a product over Primes. Bounds for the number of nonisomorphic non-Abelian groups are given by Neu- mann (1969) and Pyber (1993). see also Finite Group, Group Theory, Kronecker Decomposition Theorem, Partition Function P, Ring References DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func- tions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North- Holland, 1980. Erdos, P. and Szekeres, G. "Uber die Anzahl abelscher Grup- pen gegebener Ordnung und iiber ein verwandtes zahlen- theoretisches Problem." Acta Sci. Math. (Szeged) 7, 95- 102, 1935. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/abel/abel.html. Kendall, D. G. and Rankin, R. A. "On the Number of Abelian Groups of a Given Order." Quart J. Oxford 18, 197-208, 1947. Kolesnik, G. "On the Number of Abelian Groups of a Given Order." J. Reine Angew. Math. 329, 164-175, 1981. 8 Abel's Identity Abel's Irreducibility Theorem Neumann, P. M. "An Enumeration Theorem for Finite Groups." Quart J. Math. Ser. 2 20, 395-401, 1969. Pyber, L. "Enumerating Finite Groups of Given Order." Ann. Math. 137, 203-220, 1993. Richert, H.-E. "Uber die Anzahl abelscher Gruppen gegebener Ordnung L" Math. Zeitschr. 56, 21-32, 1952. Sloane, N. J. A. Sequence A000688/M0064 in "An On-Line Version of the Encyclopedia of Integer Sequences." Srinivasan, B. R. "On the Number of Abelian Groups of a Given Order." Acta Arith. 23, 195-205, 1973. Abel's Identity Given a homogeneous linear SECOND-ORDER ORDI- NARY Differential Equation, y" + P(x)y' + Q(x)y = 0, (1) call the two linearly independent solutions yi(x) and y 2 (as). Then y'l{x) + P{x)y , l {x) + Q{x)y 1 = ^ (2) y' 2 ' (x) + P(x)y' 2 (x) + Q(x)y 2 = 0. (3) Now, take yi x (3) - y 2 x (2), yilvZ + P(x)y2 + Q(x)y 2 ] -V2[yi+P(x)y' 1 +Q(x)y l ]=Q (4) (yiy% -y2y")+P(yiy2-yiy2)+Q(yiy2-yiy2) = (5) (2/12/2 - 2/22/") + P(2/i2/2 - 2/i2/2) = 0. (6) Now, use the definition of the Wronskian and take its Derivative, W = y t y 2 -2/12/2 (7) W = (y[y 2 + yiyi) - (yiyi + 2/12/2) = 2/12/2-2/1^2. (8) Plugging W and W into (6) gives W' 4- PW = 0. This can be rearranged to yield w = - p ^ dx which can then be directly integrated to lnl^ = -Ci / P(x)dx, (9) (10) (11) where In as is the Natural Logarithm. A second in- tegration then yields AbePs identity W(x)=C 2 e~f P(x)dx , (12) where C\ is a constant of integration and C 2 = e Cl . see alsa Ordinary Differential Equation — Sec- ond-Order References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, J^th ed. New York: Wiley, pp. 118, 262, 277, and 355, 1986. Abel's Impossibility Theorem In general, Polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of Additions, Multiplications, and Root extractions. see also Cubic Equation, Galois's Theorem, Poly- nomial, Quadratic Equation, Quartic Equation, Quintic Equation References Abel, N. H, "Demonstration de l'impossibilite de la resolution algebraique des equations generates qui depassent le qua- trieme degre." Crelle's J. 1, 1826. Abel's Inequality Let {f n } and {a n } be Sequences with f n > fn+i > for n = 1, 2, . . . , then / ^CLnfn <Ah, where A = max{|ai|, |ai + a 2 \ , . . - , |ai + a 2 + . . . 4- a m |}. Abelian Integral An Integral of the form Jo dt where R(t) is a POLYNOMIAL of degree > 4. They are also called Hyperelliptic Integrals. see also Abelian Function, Elliptic Integral Abel's Irreducibility Theorem If one ROOT of the equation f(x) = 0, which is irre- ducible over a Field K, is also a ROOT of the equation F(x) = in K, then all the ROOTS of the irreducible equation f(x) = are ROOTS of F(x) = 0. Equivalently, F(x) can be divided by f(x) without a Remainder, F(x) = f{x)F 1 (x) i where Fi(x) is also a POLYNOMIAL over K. see also ABEL'S LEMMA, KRONECKER'S POLYNOMIAL Theorem, Schoenemann's Theorem References Abel, N. H. "Memoir sur une classe particuliere d'equations resolubles algebraiquement." Crelle's J. 4, 1829. Dorrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965. Abel's Lemma Abhyankar's Conjecture 9 Abel's Lemma The pure equation x p = C of PRIME degree p is irreducible over a FIELD when C is a number of the FIELD but not the pth Power of an element of the Field. see also Abel's Irreducibility Theorem, Gauss's Polynomial Theorem, Kronecker's Polynomial Theorem, Schoenemann's Theorem References Dorrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965. Abel's Test see Abel's Uniform Convergence Test Abel's Theorem Given a Taylor Series F(z) = J2CnZ n = ^Tc n r n e i " (1) 71=0 n=0 where the COMPLEX NUMBER z has been written in the polar form z = re t& , examine the REAL and IMAGINARY Parts u(r,8) = ^Tc n r n cos(n6) n=0 oo v(r,9) = ^2c n r n sin(n0). (2) (3) Abel's theorem states that, if u(l,9) and v(l,0) are Convergent, then u{l,0)+iv{\,9) = lim f(re iB ). (4) Stated in words, Abel's theorem guarantees that, if a Real Power Series Converges for some Positive value of the argument, the Domain of Uniform Con- vergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point. References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, p. 773, 1985. Abel Transform The following INTEGRAL Transform relationship, known as the Abel transform, exists between two func- tions f(x) and g(t) for < a < 1, a(t \ = sin(7TQ) d f l f(x) d yK > tt dtj {x-ty _ sin(7ra) 7V u: dx dx df dx | /(0) dx{t-xY~ a t 1 -" (1) (2) (3) The Abel transform is used in calculating the radial mass distribution of galaxies and inverting planetary ra- dio occultation data to obtain atmospheric information. References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 875-876, 1985. Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987. Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 262-266, 1965. Abel's Uniform Convergence Test Let {u n (x)} be a Sequence of functions. If 1. u n (x) can be written u n (x) — a n f n (x) 1 2. ^a n is Convergent, 3. fn(x) is a Monotonic Decreasing Sequence (i.e., fn+i(x) < f n (x)) for all n, and 4. f n (x) is Bounded in some region (i.e., < f n (x) < M for all x e [a, b]) then, for all x e [a, 6], the Series Yl Un ( x ) Converges Uniformly. see also CONVERGENCE TESTS References Bromwich, T. J. Pa and MacRobert, T. M. An Introduc- tion to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 59, 1991. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, p. 17, 1990. Abelian Variety An Abelian variety is an algebraic GROUP which is a complete Algebraic Variety. An Abelian variety of Dimension 1 is an Elliptic Curve. see also Albanese Variety References Murty, V. K. Introduction to Abelian Varieties. Providence, RI: Amer. Math, Soc, 1993. Abhyankar's Conjecture For a Finite Group G, let p(G) be the Subgroup gen- erated by all the Sylow p-SuBGROUPS of G. If X is a projective curve in characteristic p > 0, and if xq, ...,xt are points of X (for t > 0), then a NECESSARY and SUF- FICIENT condition that G occur as the GALOIS GROUP of a finite covering Y of X, branched only at the points a;o, . .., x ti is that the Quotient GROUP G/p{G) has 2g + 1 generators. Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution. see also FINITE GROUP, GALOIS GROUP, QUOTIENT Group, Sylow p-Subgroup 10 Ablowitz-Ramani-Segur Conjecture Absolute Square References Abhyankar, S. "Coverings of Algebraic Curves." Airier. J. Math. 79, 825-856, 1957. American Mathematical Society. "Notices of the AMS, April 1995, 1995 Prank Nelson Cole Prize in Algebra." http:// www. ams . org/notices/199504/prize-cole .html. Harbater, D. "Abhyankar's Conjecture on Galois Groups Over Curves." Invent. Math. 117, 1-25, 1994. Raynaud, M. "Revetements de la droite affine en car- acteristique p > et conjecture d' Abhyankar." Invent. Math. 116, 425-462, 1994. Ablowitz-Ramani-Segur Conjecture The Ablowitz-Ramani-Segur conjecture states that a nonlinear Partial Differential Equation is solv- able by the Inverse Scattering Method only if ev- ery nonlinear Ordinary Differential Equation ob- tained by exact reduction has the Painleve Property. see also Inverse Scattering Method References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 351, 1989. Abscissa The x- (horizontal) axis of a Graph. see also Axis, Ordinate, Real Line, a;- Axis, y-Axis, z-Axis Absolute Convergence A Series J^ n u n is said to Converge absolutely if the Series J^ |u n | Converges, where |u n | denotes the Absolute Value. If a Series is absolutely convergent, then the sum is independent of the order in which terms are summed. Furthermore, if the SERIES is multiplied by another absolutely convergent series, the product series will also converge absolutely. see also Conditional Convergence, Convergent Series, Riemann Series Theorem References Bromwich, T. J. Pa and MacRobert, T. M. "Absolute Con- vergence." Ch. 4 in An Introduction to the Theory of In- finite Series, 3rd ed. New York: Chelsea, pp. 69-77, 1991. Absolute Deviation Let u denote the Mean of a Set of quantities m, then the absolute deviation is denned by Aui = \m — u\. Absolute Error The Difference between the measured or inferred value of a quantity xq and its actual value x, given by Ax = Xq — x (sometimes with the ABSOLUTE VALUE taken) is called the absolute error. The absolute error of the Sum or Difference of a number of quantities is less than or equal to the SUM of their absolute errors. see also Error Propagation, Percentage Error, Relative Error References Abramowitz, M. and Stegun, C A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972. Absolute Geometry Geometry which depends only on the first four of Eu- clid's Postulates and not on the Parallel Postu- late. Euclid himself used only the first four postulates for the first 28 propositions of the Elements, but was forced to invoke the PARALLEL POSTULATE on the 29th. see also Affine Geometry, Elements, Euclid's Pos- tulates, Geometry, Ordered Geometry, Paral- lel Postulate References Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 90-91, 1989. Absolute Pseudoprime see Carmichael Number Absolute Square Also known as the squared NORM. The absolute square of a Complex Number z is written \z\ 2 and is defined zz , (1) where z* denotes the COMPLEX CONJUGATE of z. For a Real Number, (1) simplifies to I i2 2 \z\ = Z . (2) If the Complex Number is written z — x + iy, then the absolute square can be written see also Deviation, Mean Deviation, Signed Devi- ation, Standard Deviation k + w\ 2 = x +y 2 > (3) An important identity involving the absolute square is given by a ± be' ld | 2 = (a ± be' ld ){a ± be ld ) - a 2 -h b 2 ± ab(e i5 + e~ i5 ) — a + b 2 ± 2ab cos S. (4) Absolute Value If a = 1, then (4) becomes Abundance 11 |l±&e~ ilS | 2 = l + b 2 ±2bcos8 = l + & 2 ±26[l-2sin 2 (f£)] = l±26 + & 2 =F46sin 2 (^) - (l±&) 2 q= 4&sin 2 (^). (5) If a = 1, and 6=1, then |1 - e~ iS \ 2 = (1 - l) 2 + 4 ■ lsin 2 (!<5) = 4sin 2 (±<5). (6) Finally, u^i+e** 3 ! 2 : l) I e -i(<t>2-4>i) - 2 + e n < =:2 + 2cos(02-<^i) = 2[l + cos(^ 2 -0i)] = 4 COS (02 - 0l). (7) Absolute Value The absolute value of a REAL Number x is denoted \x\ and given by , , f x f -x for x < |x|=x 8 gn(*) = | a . forx ^ 0j where SGN is the sign function. The same notation is used to denote the M ODULUS of a Complex Number z — x + iy, \z\ = y/x 2 + t/ 2 , a p-ADlC absolute value, or a general Valuation. The Norm of a Vector x is also denoted |x|, although ||x|| is more commonly used. Other Notations similar to the absolute value are the Floor Function [zj, Nint function [x], and Ceiling Function [af|. see also Absolute Square, Ceiling Function, Floor Function, Modulus (Complex Number), Nint, Sgn, Triangle Function, Valuation Absolutely Continuous Let // be a Positive Measure on a Sigma Algebra M and let A be an arbitrary (real or complex) MEASURE on M. Then A is absolutely continuous with respect to //, written A < /z, if X(E) = for every E e M for which fj.(E) = 0. see also Concentrated, Mutually Singular References Rudin, W. Functional Analysis. New York: McGraw-Hill, pp. 121-125, 1991. Absorption Law The law appearing in the definition of a Boolean Al- gebra which states a A (a V b) = a V (a A b) = a for binary operators V and A (which most commonly are logical OR and logical And). see also BOOLEAN ALGEBRA, LATTICE References BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillian, p. 317, 1965. Abstraction Operator see Lambda Calculus Abundance The abundance of a number n is the quantity A(n) = o~(n) — 2n, where <x(n) is the DIVISOR FUNCTION. Kravitz has con- jectured that no numbers exist whose abundance is an Odd Square (Guy 1994). The following table lists special classifications given to a number n based on the value of A(n). A(n) Number < deficient number — 1 almost perfect number perfect number 1 quasiperfect number > abundant number see also DEFICIENCY References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 45-46, 1994. 12 Abundant Number Acceleration Abundant Number An abundant number is an INTEGER n which is not a Perfect Number and for which s(n) = <r(n) ~ n > n, (1) where <r(n) is the DIVISOR FUNCTION. The quantity cr(n) — 2n is sometimes called the ABUNDANCE. The first few abundant numbers are 12, 18, 20, 24, 30, 36, . . . (Sloane's A005101). Abundant numbers are sometimes called Excessive Numbers. There are only 21 abundant numbers less than 100, and they are all Even. The first Odd abundant number is 945 = 3 3 -7-5. (2) That 945 is abundant can be seen by computing s(945) = 975 > 945. (3) Any multiple of a PERFECT NUMBER or an abundant number is also abundant. Every number greater than 20161 can be expressed as a sum of two abundant num- bers. Define the density function \{n : <x(n) > xn}\ A(x) = lim (4) for a POSITIVE Real Number x, then Davenport (1933) proved that A(x) exists and is continuous for all x, and Erdos (1934) gave a simplified proof (Finch). Wall (1971) and Wall et at. (1977) showed that 0.2441 < A(2) < 0.2909, and Deleglise showed that 0.2474 < A(2) < 0.2480. (5) (6) A number which is abundant but for which all its Proper Divisors are Deficient is called a Primitive Abundant Number (Guy 1994, p. 46). see also Aliquot Sequence, Deficient Number, Highly Abundant Number, Multiamicable Num- bers, Perfect Number, Practical Number, Prim- itive Abundant Number, Weird Number References Deleglise, M. "Encadrement de la densite des nombres abon- dants." Submitted. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3—33, 1952. Erdos, P. "On the Density of the Abundant Numbers." J. London Math. Soc. 9, 278-282, 1934. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/abund/abund* html. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 45-46, 1994. Singh, S. FermaVs Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, pp. 11 and 13, 1997. Sloane, N. J. A. Sequence A005101/M4825 in "An On-Line Version of the Encyclopedia of Integer Sequences." Wall, C. R. "Density Bounds for the Sum of Divisors Func- tion." In The Theory of Arithmetic Functions (Ed. A. A. Gioia and D. L. Goldsmith). New York: Springer- Verlag, pp. 283-287, 1971. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 26, 773-777, 1972. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 31, 616, 1977. Acceleration Let a particle travel a distance s(t) as a function of time t (here, s can be thought of as the ARC LENGTH of the curve traced out by the particle). The SPEED (the Scalar Norm of the Vector Velocity) is then given §=V(§r + (§)' + (s' <" The acceleration is defined as the time DERIVATIVE of the Velocity, so the SCALAR acceleration is given by dv di (2) d 2 s dt 2 (3) dx d 2 x _i_ dji d 2 y , dz d 2 z dt dt 7 " ~*~ dt df2 "T" dt di? (4) jm 2 +m 2 +m* dx d 2 x dy d 2 y dz d 2 z ds~dF + dsdi? + ds~d¥ (5) dr d 2 r ds ' dt 2 ' (6) The Vector acceleration is given by dv dt dfr d's~ fdsY <T d* = dt*- T+K {di) N - (7) where T is the UNIT TANGENT VECTOR, k the CURVA- TURE, s the Arc Length, and N the Unit Normal Vector. Let a particle move along a straight LINE so that the positions at times £i, £2, and £3 are si, 52, and S3, re- spectively. Then the particle is uniformly accelerated with acceleration a Iff a = 2 ($2 - S 3 )h + (33 - Si)t2 + (Si - 52)^3 (tl - t 2 )(t2 ~ t 3 )(t 3 - ti) (8) is a constant (Klamkin 1995, 1996). Accidental Cancellation Ackermann Function 13 Consider the measurement of acceleration in a rotating reference frame. Apply the ROTATION OPERATOR - f d \ ■■-( \. Ctt / body + u;x (9) twice to the RADIUS VECTOR r and suppress the body notation, R 2 r (^ +WX )(S +WXr ) d 2 r d ( . dr d 2 r dr du: dv — — + u> x — +r x — - +u> x — dt 2 dt dt dt + u?x (u; x r). (10) Grouping terms and using the definitions of the VELOC- ITY v = dr/dt and Angular Velocity a = du/dt give the expression 9-space dt 2 + 2u? x v + u; x (u> x r) 4- r x ex. (11) Now, we can identify the expression as consisting of three terms = d*r a b ody - df2 , aCoriolis = 2u? X V, a ce ntrifugal = <*> X (u> X I*) , (12) (13) (14) a "body" acceleration, centrifugal acceleration, and Coriolis acceleration. Using these definitions finally gives &space = <*body "r ^Coriolis ~~r ^centrifugal + T X Of, (15) where the fourth term will vanish in a uniformly ro- tating frame of reference (i.e., ex = 0). The centrifugal acceleration is familiar to riders of merry ^j-rounds, and the Coriolis acceleration is responsible for the motions of hurricanes on Earth and necessitates large trajectory corrections for intercontinfv: L al ballistic missiles. see also Angular Acceleration, Arc Length, Jerk, Velocity References Klamkin, M. S. "Problem 1481." Math. Mag. 68, 307, 1995. Klamkin, M. S. "A Characteristic of Constant Acceleration." Solution to Problem 1481. Math. Mag. 69, 308, 1996. Accidental Cancellation see Anomalous Cancellation Accumulation Point An accumulation point is a Point which is the limit of a Sequence, also called a Limit Point. For some Maps, periodic orbits give way to Chaotic ones beyond a point known as the accumulation point. see also Chaos, Logistic Map, Mode Locking, Pe- riod Doubling Achilles and the Tortoise Paradox see Zeno's Paradoxes Ackermann Function The Ackermann function is the simplest example of a well-defined TOTAL FUNCTION which is COMPUTABLE but not Primitive Recursive, providing a counterex- ample to the belief in the early 1900s that every COM- PUTABLE Function was also Primitive Recursive (Dotzel 1991). It grows faster than an exponential func- tion, or even a multiple exponential function. The Ack- ermann function A(x } y) is defined by (y+l if x = A(x,y)= I A(x-l,l) if 2/ — [ A{x — 1, A(x, y — 1)) otherwise. Special values for Integer x include (i) A(0,y) = y + 1 (2) A(l,y) = y + 2 (3) A(2,y) = 2y + 3 (4) A(3,y) = 2"+ 3 - 3 (5) .4(4,2/) = 2^-3. (6) V+3 Expressions of the latter form are sometimes called Power Towers. A(0,y) follows trivially from the def- inition. A(l,y) can be derived as follows, A(l,y) = A(0,A(l,y- 1)) = A(l,y- 1) + 1 = A(0,A(l,y- 2)) + 1 = A(l,y- 2) + 2 = . . . = .4(1, 0) + y = A(0, l) + y = y + 2. (7) A(2,y) has a similar derivation, A(2,y) = A(l,A(2,y-l)) = A(2,y-.l) + 2 = i4(l ) A(2,y-2))+2 = i4(2 I y-2) + 4 = ... = A(2, 0) + 2y = A(l, 1) + 2y = 2y + 3. (8) Buck (1963) defines a related function using the same fundamental Recurrence Relation (with arguments flipped from Buck's convention) F(x,y) = F(x-l,F(x t y-l)), (9) 14 Ackermann Number Acute Triangle but with the slightly different boundary values ^(0, y) = V + 1 **(1,0) = 2 F(2,0) = F(x,0) = 1 for x = 3,4, Buck's recurrence gives F(l,») = 2 + i/ F(2,y) = 2y f(3,y) = 2» .2 F(4,j,) = 2 2 . (10) (11) (12) (13) (14) (15) (16) (17) Taking F(4,n) gives the sequence 1, 2, 4, 16, 65536, 2 65536 , .... Defining ip(x) = F(x, x) for x = 0, 1, ... .2 then gives 1, 3, 4, 8, 65536, 2 2 ' , . . . (Sloane's A001695), where m = 2 2 , a truly huge number! 65536 see a/50 Ackermann Number, Computable Func- tion, Goodstein Sequence, Power Tower, Primi- tive Recursive Function, TAK Function, Total Function References Buck, R. C. "Mathematical Induction and Recursive Defini- tions." Amer. Math. Monthly 70, 128-135, 1963. Dotzel, G. "A Function to End All Functions." Algorithm: Recreational Programming 2.4, 16-17, 1991. Kleene, S. C. Introduction to Metamathematics. New York: Elsevier, 1971. Peter, R. Rekursive Funktionen. Budapest: Akad. Kiado, 1951. Reingold, E. H. and Shen, X. "More Nearly Optimal Algo- rithms for Unbounded Searching, Part I: The Finite Case." SIAM J. Corn-put. 20, 156-183, 1991. Rose, H. E. Subrecursion, Functions, and Hierarchies. New York: Clarendon Press, 1988. Sloane, N. J. A. Sequence A001695/M2352 in "An On-Line Version of the Encyclopedia of Integer Sequences." Smith, H. J. "Ackermann's Function." http://www.netcom. com/-hj smith/Ackerman . html. Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669-675, 1983. Tarjan, R. E. Data Structures and Network Algorithms. Philadelphia PA: SIAM, 1983. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison- Wesley, pp. 11, 227, and 232, 1991. Ackermann Number A number of the form n t • • • T™> where Arrow Nota- n TION has been used. The first few Ackermann numbers .3 are 1 t 1 = 1, 2 tt 2 = 4, and 3 ttt 3 = 3 3 7,625,597,484,987 see also Ackermann Function, Arrow Notation, Power Tower References Ackermann, W. "Zum hilbertschen Aufbau der reellen Zahlen." Math. Ann. 99, 118-133, 1928. Conway, J. H. and Guy, R. K, The Book of Numbers. New York: Springer-Verlag, pp. 60-61, 1996. Crandall, R. E. "The Challenge of Large Numbers." Sci. Amer. 276, 74-79, Feb. 1997. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison- Wesley, pp. 11, 227, and 232, 1991. Acnode Another name for an ISOLATED POINT. see also Crunode, Spinode, Tacnode Acoptic Polyhedron A term invented by B. Griinbaum in an attempt to pro- mote concrete and precise POLYHEDRON terminology. The word "coptic" derives from the Greek for "to cut," and acoptic polyhedra are defined as POLYHEDRA for which the FACES do not intersect (cut) themselves, mak- ing them 2-Manifolds. see also Honeycomb, Nolid, Polyhedron, Sponge Action Let M(X) denote the GROUP of all invertible MAPS X -> X and let G be any GROUP. A HOMOMORPHISM 6 :G -> M(X) is called an action of G on X. Therefore, 6 satisfies 1. For each g € G, 6(g) is a Map X -> X : x \-> 0(g)x, 2. 0(gh)x = 6{g)(O(h)x), 3. 0(e) a; = x, where e is the group identity in G, 4. 0(g- 1 )x = 6(g)- 1 x. see also CASCADE, FLOW, SEMIFLOW Acute Angle An Angle of less than 7r/2 Radians (90°) is called an acute angle. see also ANGLE, OBTUSE ANGLE, RIGHT ANGLE, Straight Angle Acute Triangle A Triangle in which all three Angles are Acute An- gles. A Triangle which is neither acute nor a RIGHT Triangle (i.e., it has an Obtuse Angle) is called an Obtuse Triangle. A Square can be dissected into as few as 8 acute triangles. see also Obtuse Triangle, Right Triangle Adams-Bashforth-Moulton Method Addition Chain 15 Adams-Bashforth-Moulton Method see Adams' Method Adams' Method Adams' method is a numerical METHOD for solving linear First-Order Ordinary Differential Equa- tions of the form dy dx f{x>y)- Let : 3?n + l X n (i) (2) be the step interval, and consider the Maclaurin Se- ries of y about x n , y n +i = y n + ( -T-) ( x ~ x n) (x - x n ) 2 + . V dx J n + 1 \dxj n \ dx 2 J (3) (4) Here, the Derivatives of y are given by the Backward Differences \dx/ n Xn+i ~ X 3/n+i - y n h (5) (6) (7) etc. Note that by (1), q n is just the value of f{x ni y n ). For first-order interpolation, the method proceeds by iterating the expression 2/n+i = yn + q n h (8) where q n = /(x n ,2/n). The method can then be ex- tended to arbitrary order using the finite difference in- tegration formula from Beyer (1987) /* Jo / p ^=(l+IV+£,V 2 + fV 3 ,251 V 4 + J95_V 5 ~720 v ~ 288 v 19087 V7 6 V° + ...)/p (9) to obtain 2/n+i -y n = h(q n + \ Vq n -i + ^ V 2 q n -2 + f V 3 g n - 12 95 288 + ffivV-4 + ^V 5 g n _5 + ...)■ (10) Note that von Karman and Biot (1940) confusingly use the symbol normally used for FORWARD DIFFERENCES A to denote BACKWARD DIFFERENCES V. see also Gill's Method, Milne's Method, Predic- TOR-CORRECTOR METHODS, RUNGE-KUTTA METHOD References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 455, 1987. Karman, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat- ment of Engineering Problems. New York: McGraw-Hill, pp. 14-20, 1940. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 741, 1992. Addend A quantity to be Added to another, also called a Sum- MAND. For example, in the expression a + 6 + c, a, 6, and c are all addends. The first of several addends, or "the one to which the others are added" (a in the previous example), is sometimes called the AUGEND. see also Addition, Augend, Plus, Radicand Addition i i - 15 8- J- 249 * 407- Y carries (-addend 1 Y addend 2 hsum The combining of two or more quantities using the PLUS operator. The individual numbers being combined are called ADDENDS, and the total is called the Sum. The first of several ADDENDS, or "the one to which the oth- ers are added," is sometimes called the AUGEND. The opposite of addition is SUBTRACTION. While the usual form of adding two n-digit INTEGERS (which consists of summing over the columns right to left and "Carrying" a 1 to the next column if the sum exceeds 9) requires n operations (plus carries), two n- digit INTEGERS can be added in about 21gn steps by n processors using carry-lookahead addition (McGeoch 1993). Here, lgx is the Lg function, the LOGARITHM to the base 2. see also Addend, Amenable Number, Augend, Carry, Difference, Division, Multiplication, Plus, Subtraction, Sum References McGeoch, C. C. "Parallel Addition." Amer. Math. Monthly 100, 867-871, 1993. Addition Chain An addition chain for a number n is a SEQUENCE 1 = ao < ai < . . . < a T = n, such that each member after ao is the SUM of the two earlier (not necessarily distinct) ones. The number r is called the length of the addition chain. For example, 1,1 + 1 = 2,2 + 2 = 4,4 + 2 = 6,6 + 2 = 8,8 + 6 = 14 16 Addition-Multiplication Magic Square Adele Group is an addition chain for 14 of length r = 5 (Guy 1994). see also BRAUER CHAIN, HANSEN CHAIN, SCHOLZ CON- JECTURE References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. Addition-Multiplication Magic Square 46 81 117 102 15 76 200 203 19 60 232 175 54 69 153 78 216 161 17 52 171 90 58 75 135 114 50 87 184 189 13 68 150 261 45 38 91 136 92 27 119 104 108 23 174 225 57 30 116 25 133 120 51 26 162 207 39 34 138 243 100 29 105 152 102207290 38 115216171 102207290 3 115216171 A square which is simultaneously a MAGIC SQUARE and Multiplication Magic Square. The three squares shown above (the top square has order eight and the bottom two have order nine) have addition MAGIC CON- STANTS (840, 848, 1200) and multiplicative magic con- stants (2,058,068,231,856,000; 5,804,807,833,440,000; 1,619,541,385,529,760,000), respectively (Hunter and Madachy 1975, Madachy 1979). see also MAGIC SQUARE References Hunter, J, A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp, 30-31, 1975. Madachy, J. S. Madachy 's Mathematical Recreations. New York: Dover, pp. 89-91, 1979. Additive Persistence Consider the process of taking a number, adding its DIG- ITS, then adding the DIGITS of number derived from it, etc., until the remaining number has only one DIGIT. The number of additions required to obtain a single DIGIT from a number n is called the additive persis- tence of n, and the DIGIT obtained is called the DIGITAL Root of n. For example, the sequence obtained from the starting number 9876 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a DIGITAL ROOT of 3. The ad- ditive persistences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, . . . (Sloane's A031286). The smallest numbers of ad- ditive persistence n for n = 0, 1, . . . are 0, 10, 19, 199, 19999999999999999999999, . . . (Sloane's A006050). There is no number < 10 5 ° with additive persistence greater than 11. It is conjectured that the maximum number lacking the DIGIT 1 with persistence 11 is 77777733332222222222222222222 There is a stronger conjecture that there is a maximum number lacking the DIGIT 1 for each persistence > 2. The maximum additive persistence in base 2 is 1. It is conjectured that all powers of 2 > 2 15 contain a in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy, 1994). see also Digitadition, Digital Root, Multiplica- tive Persistence, Narcissistic Number, Recur- ring Digital Invariant References Guy, R. K. "The Persistence of a Number." §F25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 262-263, 1994. Hinden, H. J. "The Additive Persistence of a Number." J. Recr. Math. 7, 134-135, 1974. Sloane, N. J. A. Sequences A031286 and A006050/M4683 in "An On-Line Version of the Encyclopedia of Integer Se- quences." Sloane, N. J. A. "The Persistence of a Number." J. Recr. Math. 6, 97-98, 1973. Adele An element of an Adele GROUP, sometimes called a Repartition in older literature. Adeles arise in both Number Fields and Function Fields. The adeles of a Number Field are the additive Subgroups of all ele- ments in Yl kvi where v is the PLACE, whose ABSOLUTE Value is < 1 at all but finitely many i/s. Let F be a Function Field of algebraic functions of one variable. Then a MAP r which assigns to every PLACE P of F an element r(P) of F such that there are only a finite number of PLACES P for which v P (r(P)) < 0. see also Idele References Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc, p. 25, 1951. Knapp, A. W. "Group Representations and Harmonic Anal- ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. Adele Group The restricted topological Direct Product of the GROUP Gk v with distinct invariant open subgroups Go v , References Weil, A. Adeles and Algebraic Groups. Princeton University Press, 1961. Princeton, NJ: Adem Relations Adjoint Operator 17 Adem Relations Relations in the definition of a Steenrod Algebra which state that, for i < 2j, L*J j - k - l\ i+j-k Sq* o Sq*(x) = Y.[ 3 i- 2k ' W +J ~" ° S <^' where fog denotes function COMPOSITION and |_*J is the Floor Function. see also STEENROD ALGEBRA Adequate Knot A class of Knots containing the class of Alternating Knots. Let c(K) be the CROSSING Number. Then for KNOT Sum Ki#K 2 which is an adequate knot, c(K 1 #K 2 )^c(Ki) + c(K2). This relationship is postulated to hold true for all Knots. see also Alternating Knot, Crossing Number (Link) Adiabatic Invariant A property of motion which is conserved to exponential accuracy in the small parameter representing the typical rate of change of the gross properties of the body. see also ALGEBRAIC INVARIANT, LYAPUNOV CHARAC- TERISTIC Number Adjacency Matrix The adjacency matrix of a simple Graph is a Matrix with rows and columns labelled by VERTICES, with a 1 or in position (vi,Vj) according to whether Vi and Vj are ADJACENT or not. see also INCIDENCE MATRIX References Chartrand, G. Introductory Graph Theory. Dover, p. 218, 1985. New York: Adjacency Relation The Set E of Edges of a Graph (V,E), being a set of unordered pairs of elements of V, constitutes a RE- LATION on V. Formally, an adjacency relation is any Relation which is Irreflexive and Symmetric. see also Irreflexive, Relation, Symmetric Adjacent Fraction Two FRACTIONS are said to be adjacent if their differ- ence has a unit NUMERATOR. For example, 1/3 and 1/4 are adjacent since 1/3 - 1/4 = 1/12, but 1/2 and 1/5 are not since 1/2 — 1/5 = 3/10. Adjacent fractions can be adjacent in a Farey SEQUENCE. see also FAREY SEQUENCE, FORD CIRCLE, FRACTION, Numerator References Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, p. 119, 1995. Adjacent Value The value nearest to but still inside an inner FENCE. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison- Wesley, p. 667, 1977. Adjacent Vertices In a GRAPH G, two VERTICES are adjacent if they are joined by an EDGE. Adjoint Curve A curve which has at least multiplicity Vi — 1 at each point where a given curve (having only ordinary singu- lar points and cusps) has a multiplicity vi is called the adjoint to the given curve. When the adjoint curve is of order n — 3, it is called a special adjoint curve. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 30, 1959. Adjoint Matrix The adjoint matrix, sometimes also called the Ad JU- GATE Matrix, is defined by a* = (A T r, (i) where the ADJOINT OPERATOR is denoted * and T de- notes the Transpose. If a Matrix is Self- Adjoint, it is said to be HERMITIAN. The adjoint matrix of a Matrix product is given by (oft)^. = [(a6) T ]*, . (2) Using the property of transpose products that [(a&) T ];, = (6 T a-% = (&<&■)• = (b T ): k (a T y kj = b lAj = ( fot «% > ( 3 ) it follows that (AB) f = BW. (4) Adjoint Operator Given a Second-Order Ordinary Differential Equation - , . du du t v Cu(x) - p — + Pl — + P2 u, (1) where pi = Pi(x) and u = u(x), the adjoint operator & is defined by d " ^ (PoU) " di^ PlU) +PaU d 2 u f t ,du ( „ , , -P°ZT^ + ( 2 Po -pi)^~ + (po -pi +P2)U. 'dx 2 dx (2) 18 Adjugate Matrix Affine Hull Write the two Linearly Independent solutions as t/i (x) and 2/2 (#)■ Then the adjoint operator can also be written ?../ (y 2 Cyi ~yi£y 2 )dx = — {yi 2/2 - 2/13/2 ) Po (3) see a/50 Self-Adjoint Operator, Sturm-Liouville Theory Adjugate Matrix see Adjoint Matrix Adjunction If a is an element of a Field F over the PRIME Field P, then the set of all RATIONAL FUNCTIONS of a with Coefficients in P is a Field derived from P by ad- junction of a. Adleman-Pomerance-Rumely Primality Test A modified Miller's Primality Test which gives a guarantee of Primality or COMPOSITENESS. The Al- gorithm's running time for a number N has been provedtobeasO((lniV) clnlnlnJV ) for some c> 0. It was simplified by Cohen and Lenstra (1984), implemented by Cohen and Lenstra (1987), and subsequently optimized by Bosma and van der Hulst (1990). References Adleman, L. M.; Pomerance, C; and Rumely, R. S. "On Distinguishing Prime Numbers from Composite Number." Ann. Math. 117, 173-206, 1983. Bosma, W. and van der Hulst, M.-P. "Faster Primality Test- ing." In Advances in Cryptology, Proc. Eurocrypt '89, Houthalen, April 10-13, 1989 (Ed. J.-J. Quisquater). New- York: Springer- Verlag, 652-656, 1990. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b n ± 1, b — 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc, pp. lxxxiv-lxxxv, 1988. Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi Sums." Math. Comput. 42, 297-330, 1984. Cohen, H. and Lenstra, A. K. "Implementation of a New Primality Test." Math. Comput 48, 103-121, 1987. Mihailescu, P. "A Primality Test Using Cyclotomic Exten- sions." In Applied Algebra, Algebraic Algorithms and Error- Correcting Codes (Proc. AAECC-6, Rome, July 1988). New York: Springer- Verlag, pp. 310-323, 1989. Adleman- Rumely Primality Test see Adleman-Pomerance-Rumely Primality Test Admissible A string or word is said to be admissible if that word appears in a given SEQUENCE. For example, in the SE- QUENCE aabaabaabaabaab . . ., a, aa, baab are all admis- sible, but bb is inadmissible. see also BLOCK GROWTH Affine Complex Plane The set A 2 of all ordered pairs of COMPLEX NUMBERS. see also Affine Connection, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transforma- tion, Affinity, Complex Plane, Complex Projec- tive Plane Affine Connection see Connection Coefficient Affine Equation A nonhomogeneous Linear Equation or system of nonhomogeneous LINEAR EQUATIONS is said to be affine. see also AFFINE COMPLEX PLANE, AFFINE CONNEC- TION, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Trans- formation, Affinity Affine Geometry A GEOMETRY in which properties are preserved by PAR- ALLEL Projection from one Plane to another. In an affine geometry, the third and fourth of Euclid's Pos- tulates become meaningless. This type of GEOMETRY was first studied by Euler. see also ABSOLUTE GEOMETRY, AFFINE COMPLEX Plane, Affine Connection, Affine Equation, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity, Or- dered Geometry References Birkhoff, G. and Mac Lane, S. "Affine Geometry." §9.13 in A Survey of Modern Algebra, 3rd ed. New York: Macmillan, pp. 268-275, 1965. Affine Group The set of all nonsingular Affine TRANSFORMATIONS of a Translation in Space constitutes a Group known as the affine group. The affine group contains the full linear group and the group of TRANSLATIONS as SUB- GROUPS. see also AFFINE COMPLEX PLANE, AFFINE CONNEC- TION, Affine Equation, Affine Geometry, Affine Hull, Affine Plane, Affine Space, Affine Trans- formation, Affinity References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillan, p. 237, 1965. Affine Hull The IDEAL generated by a SET in a VECTOR SPACE. see also Affine Complex Plane, Affine Connec- tion, Affine Equation, Affine Geometry, Affine Group, Affine Plane, Affine Space, Affine Transformation, Affinity, Convex Hull, Hull AfRne Plane Affine Transformation 19 Affine Plane A 2-D Affine Geometry constructed over a Finite Field. For a Field F of size n, the affine plane consists of the set of points which are ordered pairs of elements in F and a set of lines which are themselves a set of points. Adding a Point at Infinity and Line at Infinity allows a Projective Plane to be constructed from an affine plane. An affine plane of order n is a BLOCK DESIGN of the form (n 2 , n, 1). An affine plane of order n exists Iff a PROJECTIVE PLANE of order n exists. see also Affine Complex Plane, Affine Connec- tion, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Space, Affine Trans- formation, Affinity, Projective Plane References Lindner, C. C. and Rodger, C. A. Design Theory. Raton, FL: CRC Press, 1997. Boca Affine Scheme A technical mathematical object defined as the SPEC- TRUM ct(A) of a set of Prime Ideals of a commutative RING A regarded as a local ringed space with a structure sheaf. see also SCHEME References Iyanaga, S. and Kawada, Y. (Eds.). "Schemes." §18E in En- cyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 69, 1980. Affine Space Let V be a VECTOR Space over a FIELD K, and let A be a nonempty SET. Now define addition p -f a € A for any VECTOR a E V and element p e A subject to the conditions 1. p + 0=p, 2. (p + a)+b = p+(a + b), 3. For any q G A, there EXISTS a unique VECTOR a 6 V such that q = p + a. Here, a, b £ V. Note that (1) is implied by (2) and (3). Then A is an affine space and K is called the COEFFI- CIENT Field. In an affine space, it is possible to fix a point and co- ordinate axis such that every point in the SPACE can be represented as an n-tuple of its coordinates. Every ordered pair of points A and B in an affine space is then associated with a VECTOR AB. see also Affine Complex Plane, Affine Connec- tion, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity Affine Transformation Any Transformation preserving Collinearity (i.e., all points lying on a Line initially still lie on a Line after TRANSFORMATION). An affine transformation is also called an AFFINITY. An affine transformation of R n is a Map F : R n -> W 1 of the form F(p) = Ap + q (1) for all p € M n , where A is a linear transformation of W 1 . If det(A) = 1, the transformation is Orientation- Preserving; if det(A) = -1, it is Orientation- Reversing. Dilation (Contraction, Homothecy), Expansion, Reflection, Rotation, and Translation are all affine transformations, as are their combinations. A par- ticular example combining ROTATION and EXPANSION is the rotation-enlargement transformation V = s = s cos a — sin a sin a cos a x — Xo y-yo cos a(x — Xo) + sin a(y — yo) — sina(x — Xo) + cos a(y — yo) (2) Separating the equations, x — (s cos a)x + (s sin a)y — s(xo cos a + yo sin a) (3) y = (— s sin a)x + (5 cos a)y + s(xq sin a — yo cos a). (4) This can be also written as where x = ax + by + c y = bx + ay + d, a = s cos a b = —3 sin a. The scale factor 5 is then defined by 8= \/a 2 +6 2 , and the rotation Angle by ■'(-!)■ a = tan (5) (6) (7) (8) (9) (10) see also Affine Complex Plane, Affine Connec- tion, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity, Equiaffinity, Euclidean Motion References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, p. 105, 1993. 20 Affinity Affinity see AFFINE TRANSFORMATION Affix In the archaic terminology of Whittaker and Watson (1990), the Complex Number z representing x + iy. References Whittaker, E. T. and Watson, G. N. A Course in Modem Analysis, ^th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Aggregate An archaic word for infinite SETS such as those consid- ered by Georg Cantor. see also Class (Set), Set AGM see Arithmetic-Geometric Mean Agnesi's Witch see Witch of Agnesi Agnesienne see Witch of Agnesi Agonic Lines see Skew Lines Ahlfors-Bers Theorem The Riemann's Moduli Space gives the solution to Riemann's Moduli Problem, which requires an An- alytic parameterization of the compact RlEMANN SUR- FACES in a fixed HOMEOMORPHISM. Airy Differential Equation Some authors define a general Airy differential equation as y" ± k xy — 0. (1) This equation can be solved by series solution using the expansions y = ^a n z n (2) 71 = OO CO / V^ n-1 V^ "-1 y = > na n x = y ^na n x n=0 n=l OO = ^^(n + l)a n +ix n (3) TX-0 OO OO y" — /.( n + l)na n +ix n ~~ = 2_^^ n ~*~ l) na n+i# n ~ n=0 n=l oo = J^(n + 2)(n + l)a n+2 x n . (4) Airy Differential Equation Specializing to the "conventional" Airy differential equa- tion occurs by taking the Minus Sign and setting k 2 = 1. Then plug (4) into y" -xy = (5) to obtain OO oo ^(n + 2)(n + l)a n+2 x n - x ^ a ^ = ° ( 6 ) n=0 n—0 OO oo ^(n + 2)(n + l)a n+2 z n -^a n :r n+1 =0 (7) Tl = Tl = OO oo 2a 2 + ^(n + 2)(n + l)a n+2 z n - ^T ^-ix n = (8) n=l n— 1 OO 2a 2 + J^[(n + 2)(n + l)a n+2 - a n _i]a; n = 0. (9) n = l In order for this equality to hold for all #, each term must separately be 0. Therefore, a 2 = (10) (n + 2)(n + l)a n+2 = a n _i. (11) Starting with the n = 3 term and using the above RE- CURRENCE Relation, we obtain 5-4a 5 = 20a 5 = a 2 = 0. (12) Continuing, it follows by INDUCTION that a 2 = a$ = ag = an = . . . a3n-i = (13) for n = 1, 2, Now examine terms of the form £3^. (14) a 3 = ae = ao 3^2 ^3 = 6-5 ~ (6-5)(3-2) a& ao ao (15) (16) 9-8 (9-8)(6-5)(3-2)' Again by INDUCTION, _ _ao 0,371 " f(3n)(3n - l)][(3n - 3)(3n - 4)] • ■ • [6 * 5] [3 ■ 2] (17) for n = 1, 2, Finally, look at terms of the form a3n+l, a^ a 7 ai 4-3 a4 7^6 = (7-6)(4-3) ar 01 ai 10-9 (10-9)(7-6)(4-3)" (18) (19) (20) Airy-Fock Functions By Induction, d3n+l = 0,1 [(3n + l)(3n)][(3n - 2)(3n - 3)] • * - [7 ■ 6] [4 ■ 3] (21) for n = 1, 2, The general solution is therefore y = a>o + ai n=l oo (3n)(3n - l)(3n - 3)(3n - 4) • • • 3 ■ 2 (3n + l)(3n)(3n - 2)(3n - 3) ■ ■ ■ 4 • 3 (22) For a general k 2 with a MINUS SIGN, equation (1) is y" - k 2 xy = 0, (23) and the solution is y(x) = fvS [A/_ 1/3 (§W /2 ) - S/ 1/3 (f fcx 3 / 2 )] , (24) where I is a Modified Bessel Function of the First Kind. This is usually expressed in terms of the Airy Functions Ai(#) and Bi(#) y(x) = A' Ai{k 2/3 x) + B' Bi(fc 2/3 x). (25) If the Plus Sign is present instead, then y +k xy = (26) and the solutions are y(x) = \& [AJ. 1/3 (\kx z ? 2 ) + BJ 1/Z (f kx^ 2 )] , (27) where J(z) is a Bessel Function of the First Kind. see also Airy-Fock Functions, Airy Functions, Bessel Function of the First Kind, Modified Bessel Function of the First Kind Airy-Fock Functions The three Airy-Fock functions are v{z) = ~y/irAi(z) wi(z) = 2e l7T/6 u(ujz) W2(z) = 2e~ t7r/ v(uj~ z) (i) (2) (3) where Ai(z) is an Airy Function. These functions satisfy v{z) = ^W-^W ( 4) [w 1 {z)]*=w 2 {z*), (5) where z* is the COMPLEX CONJUGATE of z. see also AlRY FUNCTIONS References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 65, 1988. Airy Functions 21 Airy Functions Watson's (1966, pp. 188-190) definition of an Airy func- tion is the solution to the Airy Differential EQUA- TION $" ±k 2 $x = (1) which is Finite at the Origin, where <£' denotes the Derivative d$/dx, k 2 — 1/3, and either Sign is per- mitted. Call these solutions (l/7r)$(±fc 2 ,:c), then £•<*»'*> =jf cos(£ 3 ± xi) dt (2) *(§;*) = Wf *(-§;*) = Wf (2x 3/2 \ (2x 3/2 \ (3) r . 2x 3 »\ T fix*'* 3 3 / 2 3 3 / 2 (4) where J(z) is a Bessel Function of the First Kind and I(z) is a MODIFIED BESSEL FUNCTION OF THE First Kind. Using the identity K n (x) TV I-n(x) - I n (x) 2 sin(n7r) (5) where K{z) is a MODIFIED BESSEL FUNCTION OF THE Second Kind, the second case can be re-expressed (8) 1 /Fir f 2 * 3/2 ^ A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp. 446-447) and illustrated above. This definition identifies the Ai(x) and Bi(a?) functions as the two LINEARLY INDE- PENDENT solutions to (1) with k 2 = 1 and a MINUS Sign, y -yz^o. (9) 22 Airy Functions The solutions are then written y(z) = AAi(z) + BBi(z) 7 where (10) Ai(*) = -*(-l,z) = |^[/_ 1/3 (I^ /2 )-/ 1/ 3(Iz 3/2 )] = ^^/3(I^ /2 ) ("J Bi(z) = y|[7_ a/ 3(fz 3/2 ) + / 1 /3(!/ /2 )]. (12) In the above plot, Ai(z) is the solid curve and Bi(z) is dashed. For zero argument, Ai(0) 3 -2/3 (13) where T(z) is the GAMMA FUNCTION. This means that Watson's expression becomes /»oo (3a)- 1/3 7rAi(±(3a)- 1/3 z)= / cos(at 3 ±xt)dt. (14) Jo A generalization has been constructed by Hardy. The Asymptotic Series of Ai(z) has a different form in different QUADRANTS of the COMPLEX PLANE, a fact known as the STOKES PHENOMENON, Functions related to the Airy functions have been defined as Gi(z) HiW * Jo t + zt) dt (15) exp(-f* 3 +2t)<ft. (16) see also AlRY-FoCK FUNCTIONS References Abramowitz, M. and Stegun, C. A. (Eds.). "Airy Functions." §10.4 in Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446-452, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234-245, 1992. Spanier, J. and Oldham, K. B. "The Airy Functions Ai(x) and Bi(x)." Ch. 56 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 555-562, 1987. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nded. Cambridge, England: Cambridge University Press, 1966. Aitken's 5 2 Process Airy Projection A Map Projection. The inverse equations for <j> are computed by iteration. Let the ANGLE of the projection plane be 0&. Define for 9 b a— < ln[2 cos( -^ it -e b )] I* t—y otherwise. tan[f(|ir-0 b )] (1) For proper convergence, let Xi = 7r/6 and compute the initial point by checking Xi = exp[-(^fx 2 + y 2 +atanxi)tan#i] . (2) As long as x» > 1, take x i+ \ = Xi/2 and iterate again. The first value for which Xi < 1 is then the starting point. Then compute Xi = cos' 1 {exp[-(^/x 2 ~+y 2 -{- atanxi) ta,nxi]} (3) until the change in xi between evaluations is smaller than the acceptable tolerance. The (inverse) equations are then given by ^7T - 2Xi - tan -(-;) (4) (5) Aitken's 5 2 Process An Algorithm which extrapolates the partial sums s n of a Series J^ a n whose Convergence is approxi- mately geometric and accelerates its rate of CONVER- GENCE. The extrapolated partial sum is given by Sn = S n +1 (S n +1 — S n ) S n +1 — 2s n + Sn-1 see also EULER'S SERIES TRANSFORMATION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 18, 1972. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 160, 1992. Aitken Interpolation Albers Equal-Area Conic Projection 23 Aitken Interpolation An algorithm similar to Neville's Algorithm for con- structing the Lagrange Interpolating Polynom- ial. Let f(x\xo, x\, . . • , Xk) be the unique POLYNOMIAL of kth ORDER coinciding with f(x) at xq, . . . , Xfc. Then f(x\xo,Xi) = f(x\x Qy x 2 ) = f(x\xo>x ly x 2 ) = f(x\x 0i x 1 ,x 2 ,X3) = 1 Xl - Xo 1 X2 — Xo 1 X 2 - x± 1 /o Xo — X A Xl — X /o Xo — X A X 2 — X X 3 — X2 /(x|x ,xi) Xi - X /(x|x 0) x 2 ) x 2 - x /(x|x ,Xi,X 2 ) X 2 - X /(x|x ,Xi,X 3 ) X 3 - X see a/so LAGRANGE INTERPOLATING POLYNOMIAL References Abramowitz, M. and Stegun, C. A. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 879, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 93-94, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 102, 1992. Ajima-Malfatti Points The lines connecting the vertices and corresponding circle-circle intersections in Malfatti's Tangent Tri- angle Problem coincide in a point Y called the first Ajima-Malfatti point (Kimberling and MacDonald 1990, Kimberling 1994). Similarly, letting A", £", and C" be the excenters of ABC, then the lines A 1 A", B'B", and C'C" are coincident in another point called the second Ajima-Malfatti point. The points are sometimes simply called the Malfatti Points (Kimberling 1994). References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Kimberling, C. "1st and 2nd Ajima-Malfatti Points." http://vvw . evansville . edu/ -ck6/ tcenters/ recent / ajmalf.html. Kimberling, C. and MacDonald, I. G. "Problem E 3251 and Solution. " Amer. Math. Monthly 97, 612-613, 1990. Albanese Variety An Abelian Variety which is canonically attached to an Algebraic Variety which is the solution to a cer- tain universal problem. The Albanese variety is dual to the Picard Variety. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, pp. 67-68, 1988. Albers Conic Projection see Albers Equal- Area Conic Projection Albers Equal- Area Conic Projection Let <fro be the Latitude for the origin of the Cartesian Coordinates and Ao its Longitude. Let 0i and <j>2 be the standard parallels. Then x = p sin v 11) y = po - pcosO, (2) where \JC — In sin (3) e = n(X- Ao) (4) yJC — 2nsin<^o po = n (5) C = cos 2 0i + 2n sin 0i (6) n = ~ (sin 0i + sin 02 ) . (7) The inverse FORMULAS are (8) A = A + -, (9) where P= \A 2 + (po - y) 2 = tan x po-y (10) (ii) References Snyder, J. P, Map Projections — A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 98-103, 1987. 24 Alcuin's Sequence Alexander- Conway Polynomial Alcuin's Sequence The Integer Sequence 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, . . . (Sloane's A005044) given by the COEFFICIENTS of the Maclaurin Series for 1/(1 - x 2 )(l - x 3 )(l - x 4 ). The number of different TRIANGLES which have INTEGRAL sides and Perimeter n is given by T(n) = P 3 (n) - J2 P2 W l<j<ln/2\ [si - lij m 48 j for n even for n odd, (1) (2) (3) where P2(n) and Ps{n) are PARTITION FUNCTIONS, with Pk{n) giving the number of ways of writing n as a sum of k terms, [x] is the NiNT function, and |_^J is the FLOOR Function (Jordan et al 1979, Andrews 1979, Hons- berger 1985). Strangely enough, T(n) for n = 3, 4, . . . is precisely Alcuin's sequence. see also PARTITION FUNCTION P, TRIANGLE References Andrews, G. "A Note on Partitions and Triangles with Inte- ger Sides." Amer. Math. Monthly 86, 477, 1979. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 39-47, 1985. Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with Integer Sides." Amer. Math. Monthly 86, 686-689, 1979. Sloane, N. J. A. Sequence A005044/M0146 in 'An On-Line Version of the Encyclopedia of Integer Sequences." Aleksandrov-Cech Cohomology A theory which satisfies all the ElLENBERG-STEENROD Axioms with the possible exception of the LONG EX- ACT Sequence of a Pair Axiom, as well as a certain additional continuity CONDITION. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 68, 1988. Aleksandrov's Uniqueness Theorem A convex body in Euclidean n-space that is centrally symmetric with center at the ORIGIN is determined among all such bodies by its brightness function (the Volume of each projection). see also TOMOGRAPHY References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422-429, 1995. Aleph The Set Theory symbol (N) for the Cardinality of an Infinite Set. see also Aleph-0 (N ), Aleph-1 (Ni), Countable Set, Countably Infinite Set, Finite, Infinite, Transfinite Number, Uncountably Infinite Set Aleph-0 (N ) The Set Theory symbol for a Set having the same Cardinal Number as the "small" Infinite Set of In- tegers. The Algebraic Numbers also belong to N . Rather surprising properties satisfied by N include N r = No rN = N N + / = N , where / is any FINITE SET. However, No* = C, (1) (2) (3) (4) where C is the CONTINUUM. see also ALEPH-1, CARDINAL NUMBER, CONTINUUM, Continuum Hypothesis, Countably Infinite Set, Finite, Infinite, Transfinite Number, Uncount- ably Infinite Set Aleph-1 (Ni) The Set Theory symbol for the smallest Infinite Set larger than Alpha-0 (N ). The CONTINUUM HYPOTH- ESIS asserts that Ni = c, where c is the CARDINALITY of the "large" Infinite Set of Real Numbers (called the CONTINUUM in Set Theory). However, the truth of the Continuum Hypothesis depends on the version of Set Theory you are using and so is Undecidable. Curiously enough, n-D SPACE has the same number of points (c) as 1-D Space, or any Finite Interval of 1- D Space (a Line Segment), as was first recognized by Georg Cantor. see also Aleph-0 (N ), Continuum, Continuum Hy- pothesis, Countably Infinite Set, Finite, Infi- nite, Transfinite Number, Uncountably Infinite Set Alethic A term in LOGIC meaning pertaining to TRUTH and Falsehood. see also False, Predicate, True Alexander- Conway Polynomial see Conway Polynomial Alexander's Horned Sphere Alexander's Horned Sphere Alexander Matrix 25 The above solid, composed of a countable UNION of Compact Sets, is called Alexander's horned sphere. It is Homeomorphic with the BALL B 3 , and its bound- ary is therefore a SPHERE. It is therefore an example of a wild embedding in E 3 . The outer complement of the solid is not SIMPLY CONNECTED, and its fundamental GROUP is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor Set. The complement in K of the bad points for Alexan- der's horned sphere is SIMPLY CONNECTED, making it inequivalent to Antoine'S Horned Sphere. Alexan- der's horned sphere has an uncountable infinity of Wild POINTS, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any NEIGHBORHOOD of a limit con- tains a horned complex. A humorous drawing by Simon Prazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander's horned sphere grow- ing from his head. see also Antoine's Horned Sphere References Albers, D. J. Illustration accompanying "The Game of 'Life'." Math Horizons, p. 9, Spring 1994. Guy, R. "Conway's Prime Producing Machine." Math. Mag. 56, 26-33, 1983. Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 80-81, 1976. Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 58, 1991. Alexander Ideal The order IDEAL in A, the RING of integral LAURENT Polynomials, associated with an Alexander Matrix for a Knot K. Any generator of a principal Alexander ideal is called an Alexander Polynomial. Because the Alexander Invariant of a Tame Knot in S 3 has a Square presentation Matrix, its Alexander ideal is Principal and it has an Alexander Polynomial A(t). see also Alexander Invariant, Alexander Matrix, Alexander Polynomial References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206-207, 1976. Alexander Invariant The Alexander invariant i7* (X) of a Knot K is the HO- MOLOGY of the Infinite cyclic cover of the complement of K, considered as a MODULE over A, the RING of inte- gral Laurent Polynomials. The Alexander invariant for a classical Tame Knot is finitely presentable, and only Hi is significant. For any KNOT K n in § n+ whose complement has the homotopy type of a FINITE COMPLEX, the Alexander invariant is finitely generated and therefore finitely pre- sentable. Because the Alexander invariant of a Tame Knot in S 3 has a Square presentation Matrix, its Alexander Ideal is Principal and it has an Alex- ander Polynomial denoted A(t). see also Alexander Ideal, Alexander Matrix, Al- exander Polynomial References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206-207, 1976. Alexander Matrix A presentation matrix for the Alexander Invariant Hi(X) of a Knot K. If V is a Seifert Matrix for a Tame Knot K in S 3 , then V T - tV and V - tV T are Alexander matrices for K, Matrix Transpose. where V denotes the see also Alexander Ideal, Alexander Invariant, Alexander Polynomial, Seifert Matrix References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206-207, 1976. 26 Alexander Polynomial Alexander Polynomial Alexander Polynomial A Polynomial invariant of a Knot discovered in 1923 by J. W. Alexander (Alexander 1928). In technical lan- guage, the Alexander polynomial arises from the HO- MOLOGY of the infinitely cyclic cover of a Knot's com- plement. Any generator of a PRINCIPAL ALEXANDER Ideal is called an Alexander polynomial (Rolfsen 1976). Because the Alexander Invariant of a Tame Knot in S 3 has a Square presentation Matrix, its Alex- ander Ideal is Principal and it has an Alexander polynomial denoted A(i). Let * be the MATRIX PRODUCT of BRAID WORDS of a Knot, then det(l - V) 1 + *+...+ t*- = Az (1) where Az, is the Alexander polynomial and det is the Determinant. The Alexander polynomial of a Tame Knot in S 3 satisfies A(t) = det(V T -tV), (2) where V is a Seifert Matrix, det is the Determi- nant, and V T denotes the Matrix TRANSPOSE. The Alexander polynomial also satisfies A(l) = ±l. (3) The Alexander polynomial of a splittable link is always 0. Surprisingly, there are known examples of nontrivial Knots with Alexander polynomial 1. An example is the (-3,5,7) Pretzel Knot. The Alexander polynomial remained the only known Knot Polynomial until the Jones Polynomial was discovered in 1984. Unlike the Alexander polynomial, the more powerful JONES POLYNOMIAL does, in most cases, distinguish HANDEDNESS. A normalized form of the Alexander polynomial symmetric in t and £ _1 and satisfying A(unknot) = 1 (4) was formulated by J. H. Conway and is sometimes de- noted Vl • The Notation [a 4- b + c + . . . is an abbrevi- ation for the Conway-normalized Alexander polynomial of a Knot a + b(x + x ) + c(x + x ) + . (5) For a description of the NOTATION for Links, see Rolf- sen (1976, p. 389). Examples of the Conway-Alexander polynomials for common KNOTS include Vtk Vfek VsSK [1-1 = [3-1 = [l - i + : -x" 1 + 1 _1 +3-x (6) (7) _1 + l-a: + x 2 (8) for the Trefoil Knot, Figure-of-Eight Knot, and Solomon's Seal Knot, respectively. Multiplying through to clear the NEGATIVE POWERS gives the usual Alexander polynomial, where the final SIGN is deter- mined by convention. \, \ )( s s u L J + M) Let an Alexander polynomial be denoted A, then there exists a Skein Relationship (discovered by J. H. Con- way) A L+ (t)-A L _(t) + (t- 1/2 -t 1/2 )A Lo (t) = (9) corresponding to the above Link Diagrams (Adams 1994). A slightly different Skein RELATIONSHIP con- vention used by Doll and Hoste (1991) is V i+ -V £ _ =zV Lo . (10) These relations allow Alexander polynomials to be con- structed for arbitrary knots by building them up as a sequence of over- and undercrossings. For a Knot, * , n _fl(mod8) ifArf(tf) = 0. (n) Ak(-1)= j 5(modg) ifArf(K) = 1) (11) where Arf is the Arf Invariant (Jones 1985). If K is a Knot and |A*(i)|>3, (12) then K cannot be represented as a closed 3-BRAID. Also, if A K (e 27ri/5 )> f, (13) then K cannot be represented as a closed 4-braid (Jones 1985). The HOMFLY POLYNOMIAL P{a, z) generalizes the Al- exander polynomial (as well at the JONES POLYNOMIAL) with V(z) = P{l t z) (14) (Doll and Hoste 1991). Rolfsen (1976) gives a tabulation of Alexander polyno- mials for Knots up to 10 Crossings and Links up to 9 Crossings. see also Braid Group, Jones Polynomial, Knot, Knot Determinant, Link, Skein Relationship References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 165-169, 1994. Alexander, J. W. "Topological Invariants of Knots and Links." Trans. Amer. Math. Soc. 30, 275-306, 1928. Alexander-Spanier Cohomology Algebra 27 Alexander, J. W. "A Lemma on a System of Knotted Curves." Proc. Nat. Acad. Set. USA 9, 93-95, 1923, Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747-761, 1991. Jones, V. "A Polynomial Invariant for Knots via von Neu- mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985. Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 280-287, 1976. Stoimenow, A. "Alexander Polynomials." http://www. informatik.hu-berlin.de/-stoimeno/ptab/alO.html. Stoimenow, A. "Conway Polynomials." http://www. informatik.hu-berlin.de/-stoimeno/ptab/clO.html. Alexander-Spanier Cohomology A fundamental result of DE RHAM COHOMOLOGY is that the fcth de Rham Cohomology Vector Space of a Manifold M is canonically isomorphic to the Alexander-Spanier cohomology VECTOR SPACE H k (M;M) (also called cohomology with compact sup- port). In the case that M is Compact, Alexander- Spanier cohomology is exactly "singular" COHOMOL- OGY. Alexander's Theorem Any Link can be represented by a closed Braid. Algebra The branch of mathematics dealing with GROUP The- ory and Coding Theory which studies number sys- tems and operations within them. The word "algebra" is a distortion of the Arabic title of a treatise by Al- Khwarizmi about algebraic methods. Note that mathe- maticians refer to the "school algebra" generally taught in middle and high school as "Arithmetic," reserving the word "algebra" for the more advanced aspects of the subject. Formally, an algebra is a Vector Space V, over a Field F with a Multiplication which turns it into a RING defined such that, if / 6 F and x, y G V, then f{*y) = (fx)y = x(fy)- In addition to the usual algebra of Real Numbers, there are as 1151 additional Consistent algebras which can be formulated by weakening the FIELD AXIOMS, at least 200 of which have been rigorously proven to be self-CONSlSTENT (Bell 1945). Algebras which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice leads to exotic-sounding (but unenlightening) names which algebraists frequently use with minimal or nonexistent explanation. see also ALTERNATE ALGEBRA, ALTERNATING ALGE- BRA, i?*-ALGEBRA, BANACH ALGEBRA, BOOLEAN AL- GEBRA, Borel Sigma Algebra, C*-Algebra, Cay- ley Algebra, Clifford Algebra, Commutative Algebra, Exterior Algebra, Fundamental The- orem of Algebra, Graded Algebra, Grassmann Algebra, Hecke Algebra, Heyting Algebra, Ho- mological Algebra, Hopf Algebra, Jordan Al- gebra, Lie Algebra, Linear Algebra, Measure Algebra, Nonassociative Algebra, Quaternion, Robbins Algebra, Schur Algebra, Semisimple Al- gebra, Sigma Algebra, Simple Algebra, Steen- rod Algebra, von Neumann Algebra References Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, pp. 35-36, 1945. Bhattacharya, P. B,; Jain, S. K.; and Nagpu, S. R. (Eds.). Basic Algebra, 2nd ed. New York: Cambridge University Press, 1994. BirkhofF, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996. Brown, K. S. "Algebra." http://www.seanet.com/-ksbrown/ ialgebra.htm. Cardano, G. Ars Magna or The Rules of Algebra. New York: Dover, 1993. Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc, 1951. Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover, 1961. Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923. Dickson, L. E. Modern Algebraic Theories. Chicago, IL: H. Sanborn, 1926. Edwards, H. M. Galois Theory, corrected 2nd printing. New York: Springer- Verlag, 1993. Euler, L. Elements of Algebra. New York: Springer- Verlag, 1984. Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lex- ington, MA: D. C. Heath, 1994. Grove, L. Algebra. New York: Academic Press, 1983. Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to El- ementary Algebra for Schools. London: Macmillan, 1960. Harrison, M. A. "The Number of Isomorphism Types of Fi- nite Algebras." Proc. Amer. Math. Soc. 17, 735-737, 1966. Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1996. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975. Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989. Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: Uni- versity of Chicago Press, 1995. Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer- Verlag, 1990. Pedersen, J. "Catalogue of Algebraic Systems." http:// tarski.math.usf .edu/algctlg/. Uspensky, J. V. Theory of Equations. New York: McGraw- Hill, 1948. van der Waerden, B. L. Algebra, Vol. 2. New York: Springer- Verlag, 1991. van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. New York: Springer- Verlag, 1983. van der Waerden, B. L. A History of Algebra: From Al- Khwarizmi to Emmy Noether. New York: Springer- Verlag, 1985. Varadarajan, V. S. Algebra in Ancient and Modern Times. Providence, RI: Amer. Math. Soc, 1998. 28 Algebraic Closure Algebraic Invariant Algebraic Closure The algebraic closure of a Field K is the "smallest" Field containing K which is algebraically closed. For example, the FIELD of COMPLEX NUMBERS C is the algebraic closure of the Field of Reals R. Algebraic Coding Theory see Coding Theory Algebraic Curve An algebraic curve over a Field K is an equation f(X,Y) = 0, where f{X,Y) is a POLYNOMIAL in X and Y with Coefficients in K. A nonsingular algebraic curve is an algebraic curve over K which has no SIN- GULAR Points over K. A point on an algebraic curve is simply a solution of the equation of the curve. A K- Rational Point is a point (X, Y) on the curve, where X and Y are in the FIELD K. see also Algebraic Geometry, Algebraic Variety, Curve References Griffiths, P. A. Introduction to Algebraic Curves. dence, RI: Amer. Math. Soc, 1989. Provi- Algebraic Function A function which can be constructed using only a finite number of ELEMENTARY FUNCTIONS together with the Inverses of functions capable of being so constructed. see also Elementary Function, Transcendental Function Algebraic Function Field A finite extension K = Z(z)(w) of the Field C(z) of Rational Functions in the indeterminate z, i.e., w is a Root of a Polynomial a +aia + a 2 a 2 + . . . + a n a: n , where a; € C(z). see also Algebraic Number Field, Riemann Sur- face Algebraic Geometry The Study of ALGEBRAIC CURVES, ALGEBRAIC VARI- ETIES, and their generalization to n-D. see also Algebraic Curve, Algebraic Variety, Commutative Algebra, Differential Geometry, Geometry, Plane Curve, Space Curve References Abhyankar, S. S. Algebraic Geometry for Scientists and En- gineers. Providence, RI: Amer. Math. Soc, 1990. Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Eisenbud, D. Commutative Algebra with a View Toward Al- gebraic Geometry. New York: Springer- Verlag, 1995. Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1978. Hartshorne, R. Algebraic Geometry, rev. ed. New York: Springer- Verlag, 1997. Lang, S. Introduction to Algebraic Geometry. New York: Interscience, 1958. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 1. Cambridge, England: Cambridge University Press, 1994. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 2. Cambridge, England: Cambridge University Press, 1994. Pedoe, D. and Hodge, W. V. Methods of Algebraic Geometry, Vol. 3. Cambridge, England: Cambridge University Press, 1994. Seidenberg, A. (Ed.). Studies in Algebraic Geometry. Wash- ington, DC: Math. Assoc. Amer., 1980. Weil, A. Foundations of Algebraic Geometry, enl. ed. Prov- idence, RI: Amer. Math. Soc, 1962. Algebraic Integer If r is a Root of the Polynomial equation x n + a n -ix n ~ + . . . + aiz + ao = 0, where the a^s are INTEGERS and r satisfies no similar equation of degree < n, then r is an algebraic INTEGER of degree n. An algebraic INTEGER is a special case of an Algebraic Number, for which the leading Coef- ficient a n need not equal 1. RADICAL INTEGERS are a subring of the ALGEBRAIC INTEGERS. A Sum or Product of algebraic integers is again an al- gebraic integer. However, Abel's IMPOSSIBILITY THE- OREM shows that there are algebraic integers of degree > 5 which are not expressible in terms of ADDITION, Subtraction, Multiplication, Division, and the ex- traction of Roots on Real Numbers. The Gaussian Integer are are algebraic integers of -1 ), since a + bi are roots of z 2 - 2az + a 2 + b 2 = 0. see also Algebraic Number, Euclidean Number, Radical Integer References Hancock, H. Foundations of the Theory of Algebraic Num- bers, Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931. Hancock, H. Foundations of the Theory of Algebraic Num- bers, Vol. 2: The General Theory. New York: Macmillan, 1932. Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num- ber Theory. Cambridge, England: Cambridge University- Press, 1989. Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347-353, 1991. Algebraic Invariant A quantity such as a Discriminant which remains un- changed under a given class of algebraic transforma- tions. Such invariants were originally called HYPERDE- TERMINANTS by Cayley. see also DISCRIMINANT (POLYNOMIAL), INVARIANT, Quadratic Invariant Algebraic Knot Algebraic Tangle 29 References Grace, J. H. and Young, A. The Algebra of Invariants. New York: Chelsea, 1965. Gurevich, G. B. Foundations of the Theory of Algebraic In- variants. Groningen, Netherlands: P. NoordhofF, 1964. Hermann, R. and Ackerman, M. Hilbert's Invariant Theory Papers.rookline, MA: Math Sci Press, 1978. Hilbert, D. Theory of Algebraic Invariants. Cambridge, Eng- land: Cambridge University Press, 1993. Mumford, D.; Fogarty, J.; and Kirwan, F. Geometric Invari- ant Theory, 3rd enl. ed. New York: Springer- Verlag, 1994. Algebraic Knot A single component ALGEBRAIC LINK. see also Algebraic Link, Knot, Link Algebraic Link A class of fibered knots and links which arises in Al- gebraic Geometry. An algebraic link is formed by connecting the NW and NE strings and the SW and SE strings of an ALGEBRAIC Tangle (Adams 1994). see also Algebraic Tangle, Fibration, Tangle References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 48-49, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 335, 1976. Algebraic Number If r is a ROOT of the POLYNOMIAL equation a$x -\- a±x . . + a n - 1 x -\- a n = 0, (i) where the a^s are Integers and r satisfies no similar equation of degree < n, then r is an algebraic number of degree n. If r is an algebraic number and ao = 1, then it is called an ALGEBRAIC INTEGER. It is also true that if the c;s in CQX + ClX n + . . . + Cn-lX + C n - (2) are algebraic numbers, then any ROOT of this equation is also an algebraic number. If a is an algebraic number of degree n satisfying the Polynomial a(x — a)(x — j3)(x — 7) ■ (3) then there are n — 1 other algebraic numbers (3, 7, ... called the conjugates of ex. Furthermore, if a satisfies any other algebraic equation, then its conjugates also satisfy the same equation (Conway and Guy 1996). Any number which is not algebraic is said to be TRANS- CENDENTAL. see also ALGEBRAIC INTEGER, EUCLIDEAN NUMBER, Hermite-Lindemann Theorem, Radical Integer, Semialgebraic Number, Transcendental Number References Conway, J. H. and Guy, R. K. "Algebraic Numbers." In The Book of Numbers. New York: Springer- Verlag, pp. 189— 190, 1996. Courant, R. and Robbing, H. "Algebraic and Transcendental Numbers." §2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103-107, 1996. Hancock, H. Foundations of the Theory of Algebraic Num- bers. Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931. Hancock, H. Foundations of the Theory of Algebraic Num- bers. Vol. 2: The General Theory. New York: Macmillan, 1932. Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347-353, 1991. Algebraic Number Field see Number Field Algebraic Surface The set of ROOTS of a POLYNOMIAL f(x,y,z) = 0. An algebraic surface is said to be of degree n = max(i + J + fc), where n is the maximum sum of powers of all terms amX lrn y jrn z krn . The following table lists the names of algebraic surfaces of a given degree. Order Surface 3 cubic surface 4 quartic surface 5 quintic surface 6 sextic surface 7 heptic surface 8 octic surface 9 nonic surface 10 decic surface see also Barth Decic, Barth Sextic, Boy Surface, Cayley Cubic, Chair, Clebsch Diagonal Cubic, Cushion, Dervish, Endrass Octic, Heart Surface, Kummer Surface, Order (Algebraic Surface), Roman Surface, Surface, Togliatti Surface References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 7, 1986. Algebraic Tangle Any Tangle obtained by Additions and Multiplica- tions of rational TANGLES (Adams 1994). see also Algebraic Link References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 41-51, 1994. 30 Algebraic Topology Algorithm Algebraic Topology The study of intrinsic qualitative aspects of spatial objects (e.g., SURFACES, SPHERES, TORI, CIRCLES, Knots, Links, configuration spaces, etc.) that re- main invariant under both-directions continuous ONE- TO-One (HOMEOMORPHIC) transformations. The dis- cipline of algebraic topology is popularly known as "Rubber-Sheet Geometry" and can also be viewed as the study of Disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of HOLE structures, and it gets the prefix "algebraic" since many Hole structures are represented best by algebraic objects like GROUPS and RINGS. A technical way of saying this is that algebraic topol- ogy is concerned with FUNCTORS from the topological Category of Groups and Homomorphisms. Here, the FUNCTORS are a kind of filter, and given an "input" SPACE, they spit out something else in return. The re- turned object (usually a Group or Ring) is then a rep- resentation of the HOLE structure of the SPACE, in the sense that this algebraic object is a vestige of what the original SPACE was like (i.e., much information is lost, but some sort of "shadow" of the SPACE is retained — just enough of a shadow to understand some aspect of its HOLE-structure, but no more). The idea is that FUNC- TORS give much simpler objects to deal with. Because SPACES by themselves are very complicated, they are unmanageable without looking at particular aspects. COMBINATORIAL TOPOLOGY is a special type of alge- braic topology that uses COMBINATORIAL methods. see also CATEGORY, COMBINATORIAL TOPOLOGY, DIF- FERENTIAL TOPOLOGY, FUNCTOR, HOMOTOPY THE- ORY References Dieudonne, J. A History of Algebraic and Differential Topol- ogy: 1900-1960. Boston, MA: Birkhauser, 1989. Algebraic Variety A generalization to n-D of ALGEBRAIC CURVES. More technically, an algebraic variety is a reduced SCHEME of Finite type over a Field K. An algebraic variety V is defined as the Set of points in the Reals W 1 (or the Complex Numbers C n ) satisfying a system of Poly- nomial equations fi(xi, . . . , x n ) = for i = 1, 2, According to the Hilbert Basis Theorem, a Finite number of equations suffices. see also Abelian Variety, Albanese Variety, Brauer-Severi Variety, Chow Variety, Picard Variety References Ciliberto, C; Laura, E.; and Somese, A. J. (Eds.). Classifica- tion of Algebraic Varieties. Providence, RI: Amer. Math. Soc, 1994. Algebroidal Function An Analytic Function f(z) satisfying the irreducible algebraic equation A (z)f k + Ai(z)/*- 1 + . . . + A k (z) = with single- valued MEROMORPHIC functions Aj(z) in a Complex Domain G is called a fc-algebroidal function in G. References Iyanaga, S. and Kawada, Y. (Eds.). "Algebroidal Functions." §19 in Encyclopedic Dictionary of Mathematics. Cam- bridge, MA: MIT Press, pp. 86-88, 1980. Algorithm A specific set of instructions for carrying out a proce- dure or solving a problem, usually with the requirement that the procedure terminate at some point. Specific algorithms sometimes also go by the name Method, Procedure, or Technique. The word "algorithm" is a distortion of Al-Khwarizmi, an Arab mathematician who wrote an influential treatise about algebraic meth- ods. see also 196- ALGORITHM, ALGORITHMIC COMPLEXITY, Archimedes Algorithm, Bhaskara-Brouckner Algorithm, Borchardt-Pfaff Algorithm, Bre- laz's Heuristic Algorithm, Buchberger's Algo- rithm, Bulirsch-Stoer Algorithm, Bumping Al- gorithm, CLEAN Algorithm, Computable Func- tion, Continued Fraction Factorization Algo- rithm, Decision Problem, Dijkstra's Algorithm, Euclidean Algorithm, Ferguson-Forcade Al- gorithm, Fermat's Algorithm, Floyd's Algo- rithm, Gaussian Approximation Algorithm, Ge- netic Algorithm, Gosper's Algorithm, Greedy Algorithm, Hasse's Algorithm, HJLS Algo- rithm, Jacobi Algorithm, Kruskal's Algorithm, Levine-O 'Sullivan Greedy Algorithm, LLL Al- gorithm, Markov Algorithm, Miller's Algo- rithm, Neville's Algorithm, Newton's Method, Prime Factorization Algorithms, Primitive Re- cursive Function, Program, PSLQ Algorithm, PSOS Algorithm, Quotient-Difference Algo- rithm, Risch Algorithm, Schrage's Algorithm, Shanks' Algorithm, Spigot Algorithm, Syracuse Algorithm, Total Function, Turing Machine, Zassenhaus-Berlekamp Algorithm, Zeilberger's Algorithm References Aho, A. V.; Hopcroft, J. E.; and Ullman, J.D. The De- sign and Analysis of Computer Algorithms. Reading, MA: Addison- Wesley, 1974. Baase, S. Computer Algorithms. Reading, MA: Addison- Wesley, 1988. Brassard, G. and Bratley, P. Fundamentals of Algorithmics. Englewood Cliffs, NJ: Prentice-Hall, 1995. Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduc- tion to Algorithms. Cambridge, MA: MIT Press, 1990. Algorithmic Complexity Aliquant Divisor 31 Greene, D. H. and Knuth, D. E. Mathematics for the Analysis of Algorithms, 3rd ed. Boston: Birkhauser, 1990. Harel, D. Algorithmics: The Spirit of Computing, 2nd ed. Reading, MA: Addison- Wesley, 1992. Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed. Reading, MA: Addison- Wesley, 1973. Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison- Wesley, 1981. Knuth, D. E. The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison- Wesley, 1973. Kozen, D. C. Design and Analysis and Algorithms. New York: Springer- Verlag, 1991. Shen, A. Algorithms and Programming. Boston: Birkhauser, 1996. Skiena, S. S. The Algorithm Design Manual. New York: Springer- Verlag, 1997. Wilf, H. Algorithms and Complexity. Englewood Cliffs, NJ: Prentice Hall, 1986. http://www.cis.upenn.edu/-wilf/. References Dorrie, H. "Alhazen's Billiard Problem." §41 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 197-200, 1965. Hogendijk, J. P. "Al-Mutaman's Simplified Lemmas for Solv- ing 'Alhazen's Problem'." From Baghdad to Barcelona/De Bagdad a Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59- 101, Anu. Filol. Univ. Bare, XIX B-2, Univ. Barcelona, Barcelona, 1996. Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tid~ skr. 18, 5-35, 1970. Neumann, P. Submitted to Amer. Math. Monthly. Riede, H. "Reflexion am Kugelspiegel. Oder: das Problem des Alhazen." Praxis Math. 31, 65-70, 1989. Sabra, A. I. "ibn al-Haytham's Lemmas for Solving 'Al- hazen's Problem'." Arch. Hist Exact Sci. 26, 299-324, 1982. Alhazen's Problem see Alhazen's Billiard Problem Algorithmic Complexity see Bit Complexity, Kolmogorov Complexity Alhazen's Billiard Problem In a given Circle, find an Isosceles Triangle whose Legs pass through two given Points inside the Circle. This can be restated as: from two POINTS in the Plane of a Circle, draw Lines meeting at the Point of the Circumference and making equal Angles with the Normal at that Point. The problem is called the billiard problem because it cor- responds to finding the POINT on the edge of a circular "BILLIARD" table at which a cue ball at a given POINT must be aimed in order to carom once off the edge of the table and strike another ball at a second given Point. The solution leads to a BIQUADRATIC EQUATION of the form H{x 2 V ) 2Kxy + {x 2 -r y 2 ){hy - kx) = 0. The problem is equivalent to the determination of the point on a spherical mirror where a ray of light will re- flect in order to pass from a given source to an observer. It is also equivalent to the problem of finding, given two points and a Circle such that the points are both inside or outside the Circle, the Ellipse whose Foci are the two points and which is tangent to the given CIRCLE. The problem was first formulated by Ptolemy in 150 AD, and was named after the Arab scholar Alhazen, who discussed it in his work on optics. It was not until 1997 that Neumann proved the problem to be insoluble using a COMPASS and RULER construction because the solution requires extraction of a CUBE ROOT, This is the same reason that the CUBE DUPLICATION problem is insoluble. see also Billiards, Billiard Table Problem, Cube Duplication Alias' Paradox Choose between the following two alternatives: 1. 90% chance of an unknown amount x and a 10% chance of $1 million, or 2. 89% chance of the same unknown amount x, 10% chance of $2.5 million, and 1% chance of nothing. The Paradox is to determine which choice has the larger expectation value, 0.9x + $100,000 or 0.89:r -f $250,000. However, the best choice depends on the un- known amount, even though it is the same in both cases! This appears to violate the INDEPENDENCE Axiom. see also Independence Axiom, Monty Hall Prob- lem, Newcomb's Paradox Aliasing Given a power spectrum (a plot of power vs. frequency), aliasing is a false translation of power falling in some fre- quency range ( — / c ,/ c ) outside the range. Aliasing can be caused by discrete sampling below the NYQUIST FRE- QUENCY. The sidelobcs of any INSTRUMENT FUNCTION (including the simple SlNC SQUARED function obtained simply from FINITE sampling) are also a form of alias- ing. Although sidelobe contribution at large offsets can be minimized with the use of an APODIZATION FUNC- TION, the tradeoff is a widening of the response (i.e., a lowering of the resolution). see also Apodization Function, Nyquist Fre- quency Aliquant Divisor A number which does not DIVIDE another exactly. For instance, 4 and 5 are aliquant divisors of 6. A num- ber which is not an aliquant divisor (i.e., one that does Divide another exactly) is said to be an Aliquot Di- visor. see also ALIQUOT DIVISOR, DIVISOR, PROPER DIVISOR 32 Aliquot Cycle Allegory Aliquot Cycle see Sociable Numbers Aliquot Divisor A number which DIVIDES another exactly. For instance, 1, 2, 3, and 6 are aliquot divisors of 6, A number which is not an aliquot divisor is said to be an ALIQUANT DI- VISOR. The term "aliquot" is frequently used to specif- ically mean a PROPER DIVISOR, i.e., a DIVISOR of a number other than the number itself. see also ALIQUANT DIVISOR, DIVISOR, PROPER DIVI- SOR Aliquot Sequence Let s(n) = cr(n) — n, where a(n) is the DIVISOR FUNCTION and s(n) is the Restricted Divisor Function. Then the Sequence of numbers s°(n) = n, s 1 (n) = s(n), s (n) — s(s(n)), . . . is called an aliquot sequence. If the SEQUENCE for a given n is bounded, it either ends at s(l) = or becomes periodic. 1. If the Sequence reaches a constant, the constant is known as a PERFECT NUMBER. 2. If the SEQUENCE reaches an alternating pair, it is called an AMICABLE PAIR. 3. If, after k iterations, the SEQUENCE yields a cycle of minimum length t of the form s fc+1 (n), s fc+2 (n), ..., s k+t (n), then these numbers form a group of Sociable Numbers of order t. It has not been proven that all aliquot sequences eventu- ally terminate and become period. The smallest number whose fate is not known is 276, which has been computed up to s 487 (276) (Guy 1994). see also 196-Algorithm, Additive Persistence, Amicable Numbers, Multiamicable Numbers, Multiperfect Number, Multiplicative Persis- tence, Perfect Number, Sociable Numbers, Uni- tary Aliquot Sequence References Guy, R. K. "Aliquot Sequences." §B6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 60-62, 1994. Guy, R. K. and Selfridge, J. L. "What Drives Aliquot Se- quences." Math. Corn-put. 29, 101-107, 1975. Sloane, N. J. A. Sequences A003023/M0062 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. All-Poles Model see Maximum Entropy Method Alladi-Grinstead Constant N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Let N(n) be the number of ways in which the Facto- rial n! can be decomposed into n Factors of the form Pk bk arranged in nondecreasing order. Also define m(n) = max(pi 1 ), (1) i.e., m(n) is the Least Prime Factor raised to its appropriate POWER in the factorization. Then define a(n) = lnm(n) Inn (2) where ln(x) is the NATURAL LOGARITHM. For instance, 2 • 2 ■ 2 2 ■ 5 • 7 • 3 4 23-5-7-2 3 -3 3 2 • 5 - 7 ■ 2 3 • 3 2 • 3 2 9! = 2 2 2 = 2 2 2 = 2 2 2 = 2 2 2 = 2 2 2- = 2 2 2- = 2 2 3- = 2 2 3- = .2 3 3- = 2 3 3 = 2 3 3- = 3 3 3- 2 2 • 2 2 5 * 7 • 3 2 • 3 2 3 • 3 • 5 • 7 ■ 3 2 • 2 4 3 ■ 2 2 • 5 • 7 • 2 3 ■ 3 2 3 • 3 ■ 3 • 5 • 7 • 2 5 2 2 • 2 2 • 2 2 ■ 5 • 7 ■ 3 2 3-3-2 2 .5-7-2 4 3-3-5 7 • 2 3 - 2 3 a(9) = 3-2 2 -2 2 .5-7-2 3 , In 3 In 3 1 In 9 21n3 2 For large n, lim a(n) = e c_1 = 0.809394020534 . . . , n— kx> where -£WA)- (3) (4) (5) (6) References Alladi, K. and Grinstead, C. "On the Decomposition of n! into Prime Powers." J. Number Th, 9, 452-458, 1977. Finch, S. "Favorite Mathematical Constants." http://www. mathsof t . c om/ as olve/ const ant /aldgrns/aldgrns .html. Guy, R. K. "Factorial n as the Product of n Large Factors." §B22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 79, 1994. Allegory A technical mathematical object which bears the same resemblance to binary relations as CATEGORIES do to Functions and Sets. see also CATEGORY References Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- dam, Netherlands: North-Holland, 1990. Allometric Almost Integer 33 Allometric Mathematical growth in which one population grows at a rate PROPORTIONAL to the POWER of another popu- lation. References Cofrey, W. J. Geography Towards a General Spatial Systems Approach. London: Routledge, Chapman & Hall, 1981, Almost All Given a property P, if P{x) ~ x as x — > oo (so the num- ber of numbers less than x not satisfying the property P is o(x)), then P is said to hold true for almost all numbers. For example, almost all positive integers are Composite Numbers (which is not in conflict with the second of Euclid's Theorems that there are an infinite number of PRIMES). see also For All, Normal Order References Hardy, G. H. and Wright, E. M. An Introduction to the The- ory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 8, 1979. Almost Alternating Knot An Almost Alternating Link with a single compo- nent. Almost Alternating Link Call a projection of a LINK an almost alternating pro- jection if one crossing change in the projection makes it an alternating projection. Then an almost alternating link is a Link with an almost alternating projection, but no alternating projection. Every ALTERNATING KNOT has an almost alternating projection. A PRIME KNOT which is almost alternating is either a Torus Knot or a Hyperbolic Knot. Therefore, no Satellite Knot is an almost alternating knot. All nonalternating 9-crossing PRIME KNOTS are almost alternating. Of the 393 nonalternating with 11 or fewer crossings, all but five are known to be nonalternating (3 of these have 11 crossings). The fate of the remaining five is not known. The (2,qr), (3,4), and (3,5)-TORUS KNOTS are almost alternating. see also Alternating Knot, Link References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 139-146, 1994. Almost Everywhere A property of X is said to hold almost everywhere if the SET of points in X where this property fails has Measure 0. see also MEASURE References Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 1, 1991. Almost Integer A number which is very close to an INTEGER. One sur- prising example involving both e and Pi is 7T = 19.999099979. (1) which can also be written as (tt + 20)* = -0.9999999992 - 0.0000388927i & -1 (2) cos(ln(7r + 20)) « -0.9999999992. (3) Applying Cosine a few more times gives COs(7T COS(7T COs(ln(7T + 20)))) « -1 + 3.9321609261 x 10" 35 . (4) This curious near-identity was apparently noticed al- most simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explana- tion as to "why" it has been true has yet been discov- ered. An interesting near-identity is given by i[cos(^) + cosh(^) + 2cos(^^)cosh(^V / 2)] = 1 + 2.480... x 10" 13 (5) (W. Dubuque). Other remarkable near-identities are given by 5(1 + we )[ g !)]2 =l + 4.5422 -x 10- (6) where T(z) is the Gamma FUNCTION (S. Plouffe), and e 6 - 7v 4 - tt 5 = 0.000017673 ... (7) (D. Wilson). A whole class of IRRATIONAL "almost integers" can be found using the theory of MODULAR FUNCTIONS, and a few rather spectacular examples are given by Ramanu- jan (1913-14). Such approximations were also stud- ied by Hermite (1859), Kronecker (1863), and Smith (1965). They can be generated using some amazing (and very deep) properties of the j-FUNCTlON. Some of the numbers which are closest approximations to INTEGERS are e*^ 1 ^ (sometimes known as the R A MANU J AN Con- stant and which corresponds to the field Q(V"163) which has Class Number 1 and is the Imaginary quadratic field of maximal discriminant), e 22 , e 71 " 37 , and e"^, the latter three of which have Class Num- ber 2 and are due to Ramanujan (Berndt 1994, Wald- schmidt 1988). 34 Almost Integer Almost Prime The properties of the j-FUNCTlON also give rise to the spectacular identity ln(640320 3 + 744) 163 + 2.32167... x 10" (8) (Le Lionnais 1983, p. 152). The list below gives numbers of the form x = e 71 "^ for n < 1000 for which \x] - x < 0.01. e^: e - e = jt-v/25 e = nVTf e = e = e : e = e = e = tvvT49 e ttvT63 2,197.990 869 543... = 422, 150.997 675 680. . . = 614,551.992 885619... = 2,508,951.998 257 553. . . = 6,635,623.999 341134... = 199, 148, 647.999 978 046 551 .. . = 884, 736, 743.999 777 466 .. . = 24, 591, 257, 751.999 999 822 213 .. . = 30, 197, 683, 486.993 182 260 .. . = 147, 197, 952, 743.999 998 662 454 .. . = 54,551,812,208.999917467 885... = 45, 116, 546, 012, 289, 599.991 830 287 . . . = 262, 537, 412, 640, 768, 743.999 999 999 999 250 072 . = 1, 418, 556, 986, 635, 586, 485.996 179 355 .. . = 604, 729, 957, 825, 300, 084, 759.999 992 171 526 .. . = 19, 683, 091, 854, 079, 461, 001, 445.992 737 040 .. . = 4, 309, 793, 301, 730, 386, 363, 005, 719.996 011 651 . = 639, 355, 180, 631, 208, 421, • • • ■ ■ - 212, 174, 016.997 669 832 . = 14, 871, 070, 263, 238, 043, 663, 567, • - • • • • 627, 879, 007.999 848 726 . = 288, 099, 755, 064, 053, 264, 917, 867, • - • •■• 975, 825, 573. 993 898 311. = 28, 994, 858, 898, 043, 231, 996, 779, - • - ■ • ■ 771, 804, 797, 161.992 372 939 . = 3, 842, 614, 373, 539, 548, 891, 490, • • - ' • ■ • 294, 277, 805, 829, 192.999 987 249 . = 223, 070, 667, 213, 077, 889, 794, 379, - - - - - ■ 623, 183, 838, 336, 437.992 055 118 . = 249, 433, 117, 287, 892, 229, 255, 125, • ■ • • • • 388, 685, 911, 710, 805.996 097 323 . = 365, 698, 321, 891, 389, 219, 219, 142, ■ ■ - ■ • - 531, 076, 638, 716, 362, 775.998 259 747 . = 6, 954, 830, 200, 814, 801, 770, 418, 837, - - ■ 940, 281, 460, 320, 666, 108.994 649 611 . . Gosper noted that the expression differs from an Integer by a mere 10 see also Class Number, j-Function, Pi References Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer- Verlag, pp. 90-91, 1994. Hermite, C. "Sur la theorie des equations modulaires." C. R. Acad. Sci. (Paris) 48, 1079-1084 and 1095-1102, 1859. Hermite, C. "Sur la theorie des equations modulaires." C. R. Acad. Sci. (Paris) 49, 16-24, 110-118, and 141-144, 1859. Kronecker, L. "Uber die Klassenzahl der aus Werzeln der Ein- heit gebildeten komplexen Zahlen." Monatsber. K. Preuss. Akad. Wiss. Berlin, 340-345. 1863. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Ramanujan, S. "Modular Equations and Approximations to 7T." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914. Smith, H. J, S. Report on the Theory of Numbers. New York: Chelsea, 1965. Waldschmidt, M. "Some Transcendental Aspects of Ramanu- jan's Work." In Ramanujan Revisited: Proceedings of the Centenary Conference (Ed. G. E« Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, pp. 57-76, 1988. Almost Perfect Number A number n for which the DIVISOR FUNCTION satisfies cr(n) = 2n — 1 is called almost perfect. The only known almost perfect numbers are the POWERS of 2, namely 1, 2, 4, 8, 16, 32, ... (Sloane's A000079). Singh (1997) calls almost perfect numbers SLIGHTLY DEFECTIVE. see also QuASIPERFECT NUMBER References Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Num- bers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 16 and 45—53, 1994. Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997. Sloane, N. J. A. Sequence A000079/M1129 in "An On-Line Version of the Encyclopedia of Integer Sequences." Almost Prime A number n with prime factorization =n»- ■ 2625374126407G8744e -7TV163 196884e -27TN/163 +103378831900730205293632e~ 37rv/I ^. (9) is called ^-almost prime when the sum of the POWERS J^^ l di = k. The set of fc-almost primes is denoted Ph. The Primes correspond to the "1-almost prime" num- bers 2, 3, 5, 7, 11, . . . (Sloane's A000040). The 2-almost prime numbers correspond to SEMIPRIMES 4, 6, 9, 10, 14, 15, 21, 22, ... (Sloane's A001358). The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (Sloane's A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, . . . (Sloane's A014613). The first few 5-almost primes are 32, 48, 72, 80, ... (Sloane's A014614). Alpha Alternate Algebra 35 see also Chen's Theorem, Prime Number, Semi- prime References Sloane, N. J. A. Sequences A014612, A014613, A014614, A000040/M0652, and A001358/M3274 in "An On-Line Version of the Encyclopedia of Integer Sequences." Alpha A financial measure giving the difference between a fund's actual return and its expected level of perfor- mance, given its level of risk (as measured by Beta). A POSITIVE alpha indicates that a fund has performed better than expected based on its Beta, whereas a Neg- ative alpha indicates poorer performance see also Beta, Sharpe Ratio Alphamagic Square A Magic Square for which the number of letters in the word for each number generates another MAGIC Square. This definition depends, of course, on the lan- guage being used. In English, for example, 5 22 18 4 9 8 28 15 2 11 7 3 12 8 25 6 5 10 where the MAGIC SQUARE on the right corresponds to the number of letters in five twenty-eight twelve twenty-two eighteen fifteen two eight twenty-five Alpha Function a n (z) = / t n e~ zt dt = n\z- (n+1) e- z ^ k\ The alpha function satisfies the Recurrence Rela- tion za n (z) = e~ z + na n -i(z). see also BETA FUNCTION (Exponential) Alpha Value An alpha value is a number < a < 1 such that P(z > ^observed) < « is considered "Significant," where P is a P- Value. see also Confidence Interval, P- Value, Signifi- cance Alphabet A Set (usually of letters) from which a Subset is drawn. A sequence of letters is called a WORD, and a set of Words is called a Code. see also CODE, WORD References Sallows, L. C. F. "Alphamagic Squares." Abacus 4, 28-45, 1986. Sallows, L. C. F. "Alphamagic Squares. 2." Abacus 4, 20-29 and 43, 1987. Sallows, L. C. F. "Alpha Magic Squares." In The Lighter Side of Mathematics (Ed. R. K. Guy and R, E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Alphametic A CRYPTARITHM in which the letters used to represent distinct DIGITS are derived from related words or mean- ingful phrases. The term was coined by Hunter in 1955 (Madachy 1979, p. 178). References Brooke, M. One Hundred & Fifty Puzzles in Crypt- Arithmetic. New York: Dover, 1963. Hunter, J. A. H. and Madachy, J. S. "Alphametics and the Like." Ch. 9 in Mathematical Diversions, New York: Dover, pp. 90-95, 1975. Madachy, J. S. "Alphametics." Ch. 7 in Madachy p s Mathe- matical Recreations. New York: Dover, pp. 178-200 1979. Alternate Algebra Let A denote an R-Algebra, so that A is a Vector Space over R and AxA^A (1) (x,y) \->x-y. (2) Then A is said to be alternate if, for all x,y £ A, (x-y)-y-x-(yy) (3) (x-x)-y = x-(x-y). (4) Here, VECTOR MULTIPLICATION x • y is assumed to be Bilinear. References Finch, S. "Zero Structures in Real Algebras." http://www. raathsof t . com/asolve/zerodiv/zerodiv .html. Schafer, R. D. An Introduction to Non- Associative Algebras. New York: Dover, 1995. 36 Alternating Algebra Alternating Permutation Alternating Algebra see Exterior Algebra Alternating Group Even Permutation Groups A n which are Normal Subgroups of the Permutation Group of Order n!/2. They are Finite analogs of the families of sim- ple Lie GROUPS. The lowest order alternating group is 60. Alternating groups with n > 5 are non-ABELIAN Simple Groups. The number of conjugacy classes in the alternating groups A n for n = 2, 3, . . . are 1, 3, 4, 5, 7, 9, ... (Sloane's A000702). see also 15 Puzzle, Finite Group, Group, Lie Group, Simple Group, Symmetric Group References Sloane, N. J, A. Sequence A000702/M2307 in "An On-Line Version of the Encyclopedia of Integer Sequences." Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.nk/atlas#alt. Alternating Knot An alternating knot is a KNOT which possesses a knot diagram in which crossings alternate between under- and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams. The Trefoil Knot and Figure-of-Eight Knot are alternating knots. One of Tait's Knot Conjectures states that the number of crossings is the same for any diagram of a reduced alternating knot. Further- more, a reduced alternating projection of a knot has the least number of crossings for any projection of that knot. Both of these facts were proved true by Kauffman (1988), Thistlethwaite (1987), and Murasugi (1987). If K has a reduced alternating projection of n crossings, then the Span of K is An. Let c(K) be the Crossing Number. Then an alternating knot K±#K 2 (a Knot Sum) satisfies Erdener, K. and Flynn, R. "Rolfsen's Table of all Alter- nating Diagrams through 9 Crossings." ftp://chs.cusd. claremont . e du/pub/knot /Rolf sen_t able .final. Kauffman, L. "New Invariants in the Theory of Knots." Amer. Math. Monthly 95, 195-242, 1988. Murasugi, K. "Jones Polynomials and Classical Conjectures in Knot Theory." Topology 26, 297-307, 1987. Sloane, N. J. A. Sequence A002864/M0847 in "An On-Line Version of the Encyclopedia of Integer Sequences," Thistlethwaite, M. "A Spanning Tree Expansion for the Jones Polynomial." Topology 26, 297-309, 1987. Alternating Knot Diagram A Knot Diagram which has alternating under- and overcrossings as the KNOT projection is traversed. The first KNOT which does not have an alternating diagram has 8 crossings. Alternating Link A Link which has a Link Diagram with alternating underpasses and overpasses. see also Almost Alternating Link References Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113-171, 1993. Alternating Permutation An arrangement of the elements ci, ..., c n such that no element a has a magnitude between a-\ and Ci + i is called an alternating (or Zigzag) permutation. The de- termination of the number of alternating permutations for the set of the first n INTEGERS {1, 2, ... , n} is known as Andre's Problem. An example of an alternating permutation is (1, 3, 2, 5, 4). As many alternating permutations among n elements begin by rising as by falling. The magnitude of the c n s does not matter; only the number of them. Let the number of alternating permutations be given by Z n = 2A n . This quantity can then be computed from In fact, this is true as well for the larger class of Ade- quate KNOTS and postulated for all KNOTS. The num- ber of Prime alternating knots of n crossing for n = 1, 2, . . . are 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, . . . (Sloane's A002864). see also ADEQUATE KNOT, ALMOST ALTERNATING Link, Alternating Link, Flyping Conjecture References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 159-164, 1994. Arnold, B.; Au, M.; Candy, C; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. "Tabulating Alter- nating Knots through 14 Crossings." ftp://chs.cusd. claremont.edu/pub/knot/paper.TeX.txt and ftp://chs. cusd. claremont ,edu/pub/knot/AltKnots/. 2na n J2 ar (1) where r and s pass through all INTEGRAL numbers such that r + 5==n _l ) (2) ao = a\ = 1, and A n = n\a n . (3) The numbers A n are sometimes called the EULER Zigzag Numbers, and the first few are given by 1, 1, 1, 2, 5, 16, 61, 272, ... (Sloane's A000111). The Odd- numbered A n s are called Euler Numbers, Secant Numbers, or Zig Numbers, and the EvEN-numbered ones are sometimes called TANGENT NUMBERS or ZAG Numbers. Alternating Series Altitude 37 Curiously enough, the SECANT and TANGENT MAC- LAURIN SERIES can be written in terms of the A n s as X X sec x = A + A 2 — - + A 4 — + . . 2! 4! X X tan x = AiX + A 3 — - + A 5 — - + . o! 5! (4) (5) or combining them, sec x + tan x t 2 r 3 r 4 A x -A 5 - + .. (6) see also Entringer Number, Euler Number, Eu- ler Zigzag Number, Secant Number, Seidel- Entringer-Arnold Triangle, Tangent Number References Andre, D. "Developments de seccc et tan a?." C. R. Acad. Sci. Paris 88, 965-967, 1879. Andre, D. "Memoire sur le permutations alternees." J. Math. 7, 167-184, 1881. Arnold, V. I. "Bernoulli-Euler Updown Numbers Associ- ated with Function Singularities, Their Combinatorics and Arithmetics." Duke Math. J. 63, 537-555, 1991. Arnold, V. I. "Snake Calculus and Combinatorics of Ber- noulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47, 3-45, 1992. Bauslaugh, B. and Ruskey, F. "Generating Alternating Per- mutations Lexicographically." BIT 30, 17-26, 1990. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer- Verlag, pp. 110-111, 1996. Dorrie, H. "Andre's Deviation of the Secant and Tangent Series." §16 in 100 Great Problems of Elementary Math- ematics: Their History and Solutions. New York: Dover, pp. 64-69, 1965. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 69-75, 1985. Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663- 688, 1967. Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op- eration on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76, 44-54, 1996. Ruskey, F. "Information of Alternating Permutations." http:// sue . esc . uvic . ca / - cos / inf / perm / Alternat ing . html. Sloane, N. J. A. Sequence A000111/M1492 in "An On-Line Version of the Encyclopedia of Integer Sequences." Alternating Series A Series of the form k=l 00 D-d a k ajt. see also SERIES References Arfken, G. "Alternating Series." §5.3 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 293-294, 1985. Bromwich, T. J. Pa and MacRobert, T. M. "Alternating Se- ries." §19 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 55-57, 1991. Pinsky, M. A. "Averaging an Alternating Series." Math. Mag. 51, 235-237, 1978. Alternating Series Test Also known as the Leibniz Criterion. An Alternat- ing Series Converges if a± > a 2 > . . . and lim ak = 0. see also CONVERGENCE TESTS Alternative Link A category of Link encompassing both ALTERNATING Knots and Torus Knots. see also Alternating Knot, Link, Torus Knot References Kauffman, L. "Combinatorics and Knot Theory." Contemp. Math. 20, 181-200, 1983. Altitude A r H 3 A 2 The altitudes of a TRIANGLE are the Cevians AiHi which are Perpendicular to the Legs AjAk opposite Ai. They have lengths hi = AiHi given by hi = at+i sinai+2 = ^+2 sinaii+i hi = 2^/s(s — ai)(s — 0,2) {s — as) where s is the Semiperimeter and a% interesting FORMULA is hihzhz = 2sA AiA k (1) (2) Another (3) (Johnson 1929, p. 191), where A is the Area of the Tri- angle. The three altitudes of any TRIANGLE are CON- CURRENT at the ORTHOCENTER H. This fundamental fact did not appear anywhere in Euclid's Elements. Other formulas satisfied by the altitude include _1_ 1_ l_ _ 1 h\ h? /13 v (4) 38 Alysoid Amicable Numbers 1 = 1 h~ 2 + 1 1 hx~ 1 r 2 + 1 = 1 r 1 2 " hx (5) (6) where r is the INRADIUS and n are the Exradii (John- son 1929, p. 189). In addition, HA 1 • HHi = HA 2 • HH 2 = HA Z . HH 3 (7) Jf Ai • HHi = |(ai 2 + a 2 2 + a 3 2 ) - 4# 2 , (8) where R is the ClRCUMRADlUS. The points Ai, A 3 , #i, and H 3 (and their permuta- tions with respect to indices) all lie on a Circle, as do the points A3, Hz, H, and Hi (and their permuta- tions with respect to indices). TRIANGLES AA1A2A3 and AA\H 2 H 3 are inversely similar. The triangle H±H 2 H 3 has the minimum PERIMETER of any TRIANGLE inscribed in a given Acute TRIAN- GLE (Johnson 1929, pp. 161-165). The PERIMETER of AHxH 2 H 3 is 2A/R (Johnson 1929, p. 191). Additional properties involving the Feet of the altitudes are given by Johnson (1929, pp. 261-262). see also Cevian, Foot, Orthocenter, Perpendicu- lar, Perpendicular Foot References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Alysoid see Catenary Ambient Isotopy An ambient isotopy from an embedding of a MANIFOLD M in N to another is a Homotopy of self Diffeomor- phisms (or Isomorphisms, or piecewise-linear transfor- mations, etc.) of JV, starting at the IDENTITY Map, such that the "last" DlFFEOMORPHISM compounded with the first embedding of M is the second embedding of M. In other words, an ambient isotopy is like an Isotopy except that instead of distorting the embedding, the whole ambient SPACE is being stretched and distorted and the embedding is just "coming along for the ride." For Smooth Manifolds, a Map is Isotopic Iff it is ambiently isotopic. For KNOTS, the equivalence of MANIFOLDS under con- tinuous deformation is independent of the embedding Space. Knots of opposite Chirality have ambient isotopy, but not REGULAR ISOTOPY. see also ISOTOPY, REGULAR ISOTOPY References Hirsch, M. W. Differential Topology. New York: Springer- Verlag, 1988. Ambiguous An expression is said to be ambiguous (or poorly de- fined) if its definition does not assign it a unique inter- pretation or value. An expression which is not ambigu- ous is said to be Well-Defined. see also Well-Defined Ambrose-Kakutani Theorem For every ergodic Flow on a nonatomic PROBABILITY Space, there is a Measurable Set intersecting almost every orbit in a discrete set. Amenable Number A number n which can be built up from INTEGERS ax, a 2 , . . . , afc by either ADDITION or MULTIPLICATION such that k k / a i — \\ a i — n - i=x i=X The numbers {ai, . . . , a n } in the Sum are simply a Par- tition of n. The first few amenable numbers are 2+2=2x2=4 1+2+3= 1x2x3=6 1+1+2+4=1x1x2x4=8 1 + 1 + 2 + 2 + 2 = 1x1x2x2x2 = 8. In fact, all COMPOSITE NUMBERS are amenable. See also COMPOSITE NUMBER, PARTITION, SUM References Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102, 463, 1995. Amicable Numbers see Amicable Pair, Amicable Quadruple, Amica- ble Triple, Multiamicable Numbers Amicable Pair Amicable Pair 39 Amicable Pair An amicable pair consists of two Integers m,n for which the sum of PROPER DIVISORS (the DIVISORS ex- cluding the number itself) of one number equals the other. Amicable pairs are occasionally called FRIENDLY Pairs, although this nomenclature is to be discouraged since FRIENDLY PAIRS are defined by a different, if re- lated, criterion. Symbolically, amicable pairs satisfy s(m) — n s(n) = m, (i) (2) where s(n) is the RESTRICTED Divisor FUNCTION or, equivalently, cr(m) = cr(n) = s(m) + s(n) = m -f n, (3) where <x(n) is the DIVISOR FUNCTION. The smallest amicable pair is (220, 284) which has factorizations 220= 11-5-2^ 284 = 71 • 2 2 giving RESTRICTED DIVISOR FUNCTIONS s(220) = ^{1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} = 284 S (284) = ^{1,2,4,71,142} = 220. (4) (5) The quantity <r{m) = cr(n) — s(m) + s(n). (6) (7) (8) in this case, 220 + 284 = 504, is called the Pair Sum. In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056). By 1747, Euler had found 30 pairs, a number which he later ex- tended to 60. There were 390 known as of 1946 (Scott 1946). There are a total of 236 amicable pairs below 10 8 (Cohen 1970), 1427 below 10 10 (te RhI • 1 ^6), 3340 less than 10 11 (Moews and Moew? 1 r "3), J' ,ess than 2.01 x 10 11 (Moews and Moe^ _., < .d 5001 .ess than ft* 3.06 x 10 11 (Moews and Moews). The first few amicable pairs are (2, 0, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296 : u -116), (63020, 76084), ... (Sloane's A002025 and AQ02046). An exhaustive tab- ulation is maintained by D. Moevvo. Let an amicable pair be denoted (m, n) with m < n. (m,n) is called a regular amicable pair of type (i, j) if (m,n) = (gM,gN), (9) where g = GCD(m,n) is the Greatest Common DI- VISOR, GCD( 5 ,M) = GCD{g,N) = 1, (10) M and N are SQUAREFREE, then the number of Prime factors of M and N are i and j. Pairs which are not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type (l,j) for j > 1. If m = (mod 6) and n = cr(m) — m (ii) is Even, then (m,n) cannot be an amicable pair (Lee 1969). The minimal and maximal values of m/n found by te Riele (1986) were 938304290/1344480478 = 0.697893577. . . (12) and 4000783984/4001351168 = 0.9998582519 .... (13) te Riele (1986) also found 37 pairs of amicable pairs hav- ing the same Pair Sum. The first such pair is (609928, 686072) and (643336, 652664), which has the Pair Sum a(m) = cr(n) = m + n = 1,296,000. (14) te Riele (1986) found no amicable n-tuples having the same Pair Sum for n > 2. However, Moews and Moews found a triple in 1993, and te Riele found a quadruple in 1995. In November 1997, a quin- tuple and sextuple were discovered. The sextuple is (1953433861918, 2216492794082), (1968039941816, 2201886714184), (1981957651366, 2187969004634), (1993501042130, 2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962), all having PAIR SUM 4169926656000. Amazingly, the sex- tuple is smaller than any known quadruple or quintuple, and is likely smaller than any quintuple. On October 4, 1997, Mariano Garcia found the largest known amicable pair, each of whose members has 4829 Digits. The new pair is N x = CM[(P + Q)P 89 - 1] (15) N 2 - CQ[(P ~ M)P S9 - 1], (16) where C = 2 1X P 89 (17) M = 287155430510003638403359267 (18) P = 574451143340278962374313859 (19) Q = 136272576607912041393307632916794623. (20) P, Q, (P + Q)P 89 - 1, and (P - M)P 89 - 1 are Prime. 40 Amicable Pair Amicable Triple Pomerance (1981) has proved that [amicable numbers < n] < ne~^ n ^ J (21) for large enough n (Guy 1994). No nonfinite lower bound has been proven. see also Amicable Quadruple, Amicable Triple, Augmented Amicable Pair, Breeder, Crowd, Eu- ler's Rule, Friendly Pair, Multiamicable Num- bers, Pair Sum, Quasiamicable Pair, Sociable Numbers, Unitary Amicable Pair References Alanen, J.; Ore, 0.; and Stemple, J. "Systematic Computa- tions on Amicable Numbers." Math. Comput. 21, 242— 245, 1967. Battiato, S. and Borho, W. "Are there Odd Amicable Num- bers not Divisible by Three?" Math. Comput. 50, 633- 637, 1988. Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 62 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Borho, W. and Hoffmann, H. "Breeding Amicable Numbers in Abundance." Math. Comput 46, 281-293, 1986. Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers and Their Distribution." Math. Comput. 24, 431-432, 1970. Cohen, H. "On Amicable and Sociable Numbers." Math. Comput. 24, 423-429, 1970. Costello, P. "Amicable Pairs of Euler's First Form." J. Rec. Math. 10, 183-189, 1977-1978. Costello, P. "Amicable Pairs of the Form (i,l)." Math. Com- put. 56, 859-865, 1991. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 38-50, 1952. Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4, 108-111, 1955-1956. Erdos, P. "On Asymptotic Properties of Aliquot Sequences." Math. Comput. 30, 641-645, 1976. Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math- ematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of~Mind from Scientific American. New York: Vintage, pp. 160-171, 1978. Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 55-59, 1994. Lee, E. J. "Amicable Numbers and the Bilinear Diophantine Equation." Math. Comput. 22, 181-197, 1968. Lee, E. J. "On Divisibility of the Sums of Even Amicable Pairs." Math. Comput. 23, 545-548, 1969. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, 1." J. Rec. Math. 5, 77-93, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, II." J. Rec. Math. 5, 153-173, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, HI." J. Rec. Math. 5, 231-249, 1972. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 145 and 155-156, 1979. Moews, D. and Moews, P. C. "A Search for Aliquot Cycles and Amicable Pairs." Math. Comput. 61, 935-938, 1993. Moews, D. and Moews, P. C. "A List of Amicable Pairs Below 2.01 x 10 u ." Rev. Jan. 8, 1993. http://xraysgi.ims. uconn . edu : 8080/amicable . txt . Moews, D. and Moews, P. C. "A List of the First 5001 Am- icable Pairs." Rev. Jan. 7, 1996. http://xraysgi.ims. uconn.edu: 8080/amicable2. txt. Ore, 0. Number Theory and Its History. New York: Dover, pp. 96-100, 1988. Pedersen, J. M. "Known Amicable Pairs." http://www. vejlehs.dk/staff/jmp/aliquot/knwnap.htm. Pomerance, C. "On the Distribution of Amicable Numbers." J. reine angew. Math. 293/294, 217-222, 1977. Pomerance, C. "On the Distribution of Amicable Numbers, II." J. reine angew. Math. 325, 182-188, 1981. Scott, E. B. E. "Amicable Numbers." Scripta Math. 12, 61-72, 1946. Sloane, N. J. A. Sequences A002025/M5414 and A002046/ M5435 in "An On-Line Version of the Encyclopedia of In- teger Sequences." te Riele, H. J. J. "On Generating New Amicable Pairs from Given Amicable Pairs." Math. Comput. 42, 219-223, 1984. te Riele, H. J. J. "Computation of All the Amicable Pairs Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35, 1986. te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.; and Lee, E. J. "Table of Amicable Pairs Between 10 x and 10 52 ." Centrum voor Wiskunde en Informatica, Note NM- N8603. Amsterdam: Stichting Math. Centrum, 1986. te Riele, H. J. J. "A New Method for Finding Amicable Pairs." In Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (Vancouver, BC, August 9-13, 1993) (Ed. W. Gautschi). Providence, Rl: Amer. Math. Soc, pp. 577-581, 1994. $$ Weisstein, E. W. "Sociable and Amicable Num- bers." http : //www . astro . Virginia, edu/ -eww6n/math/ notebooks/Sociable .m. Amicable Quadruple An amicable quadruple as a QUADRUPLE (a, b, c, d) such that a(a) = a(b) — a(c) — cr(d) — a + b + c + d, where cr(n) is the DIVISOR FUNCTION. References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 59, 1994. Amicable Triple Dickson (1913, 1952) defined an amicable triple to be a TRIPLE of three numbers (Z,m, n) such that s(/) = m + n ${m) = I + n s(n) = / + m, where s(n) is the Restricted Divisor Function (Madachy 1979). Dickson (1913, 1952) found eight sets of amicable triples with two equal numbers, and two sets with distinct numbers. The latter are (123228768, 103340640, 124015008), for which s(12322876) = 103340640 + 124015008 = 227355648 s(103340640) = 123228768 + 124015008 = 24724377 5(124015008) = 123228768 + 10334064 = 226569408, Amortization Amplitude 41 and (1945330728960, 2324196638720, 2615631953920), for which s(1945330728960) = 2324196638720+2615631953920 = 4939828592640 s(2324196638720) = 1945330728960 + 2615631953920 = 4560962682880 5(2615631953920) = 1945330728960 + 2324196638720 = 4269527367680. A second definition (Guy 1994) defines an amicable triple as a TRIPLE (a, &, c) such that a (a) = a(b) — o~(c) = a + b + c, where a(n) is the DIVISOR FUNCTION. An example is (2 2 3 2 5- 11, 2 5 3 2 7, 2 2 3 2 71). see also Amicable Pair, Amicable Quadruple References Dickson, L. E. "Amicable Number Triples." Amer. Math. Monthly 20, 84-92, 1913. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 50, 1952. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 59, 1994. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 156, 1979. Mason, T. E. "On Amicable Numbers and Their Generaliza- tions." Amer, Math. Monthly 28, 195-200, 1921. $$ Weisstein, E. W. "Sociable and Amicable Num- bers." http : //www . astro . Virginia . edu/~eww6n/math/ notebooks/Sociable .m. Amortization The payment of a debt plus accrued INTEREST by regu- lar payments. Ampersand Curve The Plane CURVE with Cartesian equation (y 2 - x 2 ){x - l)(2a> - 3) = 4(z 2 + y 2 - 2x) 2 . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989, Amphichiral An object is amphichiral (also called Reflexible) if it is superposable with its Mirror Image (i.e., its image in a plane mirror). see also Amphichiral Knot, Chiral, Disymmetric, Handedness, Mirror Image Amphichiral Knot An amphichiral knot is a Knot which is capable of be- ing continuously deformed into its own MIRROR IMAGE. The amphichiral knots having ten or fewer crossings are 04 O oi (Figure-of-Eight Knot), O6003, O8003, O8009, 08oi2j 08oi7j O8018) 10oi7,10o33, IO037, IO043, 10o45, 10o79, IO081, IO088, IO099, IO109, IO115, IO118, and IO123 (Jones 1985). The HOMFLY Polynomial is good at identifying amphichiral knots, but sometimes fails to identify knots which are not. No complete invariant (an invariant which always definitively determines if a Knot is Amphichiral) is known. Let 6+ be the Sum of Positive exponents, and 6_ the Sum of Negative exponents in the Braid Group B n . If b + - 3b- - n + 1 > 0, then the Knot corresponding to the closed BRAID b is not amphichiral (Jones 1985), see also Amphichiral, Braid Group, Invertible Knot, Mirror Image References Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, pp. 311-319, 1985. Jones, V. "A Polynomial Invariant for Knots via von Neu- mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985. Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388, 1987. Amplitude The variable <j> used in ELLIPTIC FUNCTIONS and EL- LIPTIC Integrals, which can be defined by = / dnudu, where dn(u) is a JACOBI ELLIPTIC FUNCTION. The term "amplitude" is also used to refer to the maximum offset of a function from its baseline level. see also Argument (Elliptic Integral), Charac- teristic (Elliptic Integral), Delta Amplitude, Elliptic Function, Elliptic Integral, Jacobi El- liptic Functions, Modular Angle, Modulus (El- liptic Integral), Nome, Parameter References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Fischer, G. (Ed.). Plate 132 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, p. 129, 1986. 42 Anallagmatic Curve Anchor Anallagmatic Curve A curve which is invariant under" Inversion. Exam- ples include the Cardioid, Cartesian Ovals, Cassini Ovals, Limaqon, Strophoid, and Maclaurin Tri- SECTRIX. Anallagmatic Pavement see Hadamard Matrix Analogy Inference of the Truth of an unknown result obtained by noting its similarity to a result already known to be TRUE. In the hands of a skilled mathematician, anal- ogy can be a very powerful tool for suggesting new and extending old results. However, subtleties can render re- sults obtained by analogy incorrect, so rigorous PROOF is still needed. see also INDUCTION Analysis The study of how continuous mathematical structures (Functions) vary around the Neighborhood of a point on a Surface. Analysis includes Calculus, Dif- ferential Equations, etc. see also Analysis Situs, Calculus, Complex Anal- ysis, Functional Analysis, Nonstandard Analy- sis, Real Analysis References Bottazzini, U. The "Higher Calculus": A History of Real and Complex Analysis from Euler to Weierstraft. New York: Springer-Verlag, 1986. Bressoud, D. M. A Radical Approach to Real Analysis. Washington, DC: Math. Assoc. Amer., 1994. Ehrlich, P. Real Numbers, Generalization of the Reals, & Theories of Continua. Norwell, MA: Kluwer, 1994. Hairer, E. and Wanner, G. Analysis by Its History. New York: Springer-Verlag, 1996. Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 1988. Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, J^th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Analytic Function A Function in the Complex Numbers C is analy- tic on a region R if it is COMPLEX DlFFERENTIABLE at every point in R. The terms HOLOMORPHIC FUNC- TION and Regular Function are sometimes used in- terchangeably with "analytic function." If a Function is analytic, it is infinitely DlFFERENTIABLE. see also BERGMAN SPACE, COMPLEX DlFFERENTIABLE, DlFFERENTIABLE, PSEUDOANALYTIC FUNCTION, SEMI- ANALYTIC, SUBANALYTIC References Morse, P. M. and Feshbach, H. "Analytic Functions." §4.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 356-374, 1953. Analytic Geometry The study of the GEOMETRY of figures by algebraic rep- resentation and manipulation of equations describing their positions, configurations, and separations. Ana- lytic geometry is also called Coordinate Geometry since the objects are described as n-tuples of points (where n = 2 in the PLANE and 3 in Space) in some Coordinate System. see also Argand Diagram, Cartesian Coordinates, Complex Plane, Geometry, Plane, Quadrant, Space, x-Axis, y-Axis, z-Axis References Courant, R. and Robbins, H. "Remarks on Analytic Geome- try." §2.3 in What is Mathematics?: An Elementary Ap- proach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 72-77, 1996. Analytic Set A Definable Set, also called a Souslin Set. see also COANALYTIC SET, SOUSLIN Set Anarboricity Given a Graph G, the anarboricity is the maximum number of line- disjoint nonacyclic SUBGRAPHS whose UNION is G. see also ARBORICITY Analysis Situs An archaic name for TOPOLOGY. Analytic Continuation A process of extending the region in which a COMPLEX FUNCTION is defined. see also Monodromy Theorem, Permanence of Al- gebraic Form, Permanence of Mathematical Re- lations Principle References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 378-380, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 389-390 and 392- 398, 1953. Anchor An anchor is the Bundle Map p from a Vector Bun- dle A to the Tangent Bundle TB satisfying 1. [p(X),p(Y)] = p([X,r])and 2. [x,0y] = 0[x,y] + ( P (x).0)y, where X and Y are smooth sections of A, <j> is a smooth function of B, and the bracket is the "Jacobi-Lie bracket" of a VECTOR FIELD. see also Lie Algebroid References Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996. Anchor Ring Andrews-Schur Identity 43 Anchor Ring An archaic name for the TORUS. References Eisenhart, h. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, p. 314, 1960. Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley, p. 239, 1977. Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles & Rigid Bodies, J^th ed. Cambridge, England: Cambridge University Press, p. 21, 1959. And A term (PREDICATE) in LOGIC which yields TRUE if one or more conditions are TRUE, and FALSE if any condi- tion is False. A AND B is denoted Ak,B, A A B, or simply AB. The Binary AND operator has the follow- ing Truth Table: A B AAB F F F F T F T F F T T T A PRODUCT of ANDs (the AND of n conditions) is called a CONJUNCTION, and is denoted A*- Andre's Reflection Method A technique used by Andre (1887) to provide an elegant solution to the BALLOT PROBLEM (Hilton and Pederson 1991). References Andre, D. "Solution directe du probleme resohi par M, Bertrand." Comptes Rendus Acad. Sci. Paris 105, 436-437, 1887. Comtet, L. Advanced Combinatorics. Dordrecht, Nether- lands: Reidel, p. 22, 1974. Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener- alization, and Their Uses." Math. Intel. 13, 64-75, 1991. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison- Wesley, p. 185, 1991. Andrew's Sine The function *(*)■■ {sin o, (f) < C7T > C7T which occurs in estimation theory. see also SlNE References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 697, 1992. Two binary numbers can have the operation AND per- formed bitwise with 1 representing TRUE and FALSE. Some computer languages denote this operation on A, B, and C as A&&B&&C or logand(A,B,C). see also BINARY OPERATOR, INTERSECTION, NOT, OR, Predicate, Truth Table, XOR Anderson-Darling Statistic A statistic defined to improve the Kolmogorov- SMIRNOV TEST in the TAIL of a distribution. see also Kolmogorov-Smirnov Test, Kuiper Statistic References Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 621, 1992. Andre's Problem The determination of the number of ALTERNATING PER- MUTATIONS having elements {1, 2, . . . , n} see also ALTERNATING PERMUTATION Andrews Cube see Semiperfect Magic Cube Andrews- Curtis Link The Link of 2-spheres in M 4 obtained by Spinning in- tertwined arcs. The link consists of a knotted 2-sphere and a Spun Trefoil Knot. see also Spun Knot, Trefoil Knot References Rolfsen, D. Knots and Links. Perish Press, p. 94, 1976. Wilmington, DE: Publish or Andrews-Schur Identity £« fc2+a fc=0 2n — k + a k _ V~^ 10fc 2 + (4a-l)fc 2n + 2a + 2 n — 5k [lOfc + 2a + 2] [2n -r 2a + 2] ' (1) 44 Andrica's Conjecture Anger Function where [x] is a GAUSSIAN POLYNOMIAL. It is a POLY- NOMIAL identity for a = 0, 1 which implies the Ro.GERS- Ramanujan Identities by taking n -t oo and apply- ing the Jacobi Triple Product identity. A variant of this equation is £ - fc=-|_a/2j k 2 +2ak n 4- k + a n — k |n/5j -L(n+2a+2)/5j 15fc 2 +(6a+l)fc 2n + 2a + 2 5-5/z [10A; + 2a + 2] [2n 4- 2a + 2] ' (2) where the symbol [xj in the Sum limits is the Floor Function (Paule 1994). The Reciprocal of the iden- tity is 00 k 2 +2ak Z^ in- (kq) 2fc+a 11(1 -q- 3 = 1 2j + l)(1 _ g20j+4a+4)(l _ g20j-4a+16) (3) for a = 0, 1 (Paule 1994). For g = 1, (1) and (2) become £ -La/2j n + A; -J- a n — k [n/5j £ -|_(n+2a+2)/5j 2n + 2a + 2\ 5fc + a + 1 n — 5A; n + a + 1 (4) References Andrews, G. E. "A Polynomial Identity which Implies the Rogers-Ramanujan Identities." Scripta Math. 28, 297— 305, 1970. Paule, P. "Short and Easy Computer Proofs of the Rogers- Ramanujan Identities and of Identities of Similar Type." Electronic J. Combinatorics 1, RIO, 1-9, 1994. http:// www. combinatorics . org/Volume JYvolumel .html#R10. Andrica's Conjecture 100 200 300 400 500 Andrica's conjecture states that, for p n the nth PRIME Number, the Inequality A n = ^/Pn+l — \/Pn < 1 holds, where the discrete function A n is plotted above. The largest value among the first 1000 PRIMES is for n = 4, giving y/u. - \ft « 0.670873. Since the Andrica function falls asymptotically as n increases so a PRIME Gap of increasing size is needed at large n, it seems likely the CONJECTURE is true. However, it has not yet been proven. 100 200 300 400 500 An bears a strong resemblance to the PRIME DIFFER- ENCE Function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, . . . (Sloane's A001223). see also Brocard's Conjecture, Good Prime, For- tunate Prime, Polya Conjecture, Prime Differ- ence Function, Twin Peaks References Golomb, S. W. "Problem E2506: Limits of Differences of Square Roots." Amer. Math. Monthly 83, 60-61, 1976. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 21, 1994. Rivera, C. "Problems & Puzzles (Conjectures): An- drica's Conjecture." http://www.sci.net.mx/-crivera/ ppp/conj _008 . htm. Sloane, N. J. A. Sequence A001223/M0296 in "An On-Line Version of the Encyclopedia of Integer Sequences." Anger Function A generalization of the Bessel Function OF the First Kind defined by Mz) -tf cos(v9 — zsinO) dQ. If v is an INTEGER n, then J n (z) = J n (z), where J n (z) is a Bessel Function of the First Kind. Anger's original function had an upper limit of 27T, but the cur- rent Notation was standardized by Watson (1966). see also BESSEL FUNCTION, MODIFIED STRUVE FUNC- TION, Parabolic Cylinder Function, Struve Function, Weber Functions References Abramowitz, M. and Stegun, C A. (Eds.). "Anger and We- ber Functions." §12.3 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 498-499, 1972. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Angle Angle Angle Bracket 45 Given two intersecting Lines or Line Segments, the amount of ROTATION about the point of intersection (the Vertex) required to bring one into correspondence with the other is called the angle 6 between them. An- gles are usually measured in Degrees (denoted °), Ra- dians (denoted rad, or without a unit), or sometimes Gradians (denoted grad). One full rotation in these three measures corresponds to 360°, 27r rad, or 400 grad. Half a full ROTATION is called a Straight Angle, and a Quarter of a full rotation is called a RIGHT ANGLE. An angle less than a RIGHT Angle is called an Acute Angle, and an angle greater than a Right Angle is called an Obtuse Angle. The use of Degrees to measure angles harks back to the Babylonians, whose SEXAGESIMAL number system was based on the number 60. 360° likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The DEGREE is further divided into 60 Arc Minutes, and an Arc Minute into 60 Arc Seconds. A more natural measure of an angle is the Radian. It has the property that the Arc Length around a CIRCLE is simply given by the radian angle measure times the Circle Radius. The Radian is also the most useful angle measure in CALCULUS because the Derivative of Trigonometric functions such as dx does not require the insertion of multiplicative constants like 7r/180. GRADIANS are sometimes used in surveying (they have the nice property that a Right Angle is ex- actly 100 Gradians), but are encountered infrequently, if at all, in mathematics. The concept of an angle can be generalized from the Circle to the Sphere. The fraction of a Sphere sub- tended by an object is measured in StERADIANS, with the entire Sphere corresponding to 4n Steradians. A ruled Semicircle used for measuring and drawing angles is called a Protractor. A Compass can also be used to draw circular ARCS of some angular extent. see also Acute Angle, Arc Minute, Arc Second, Central Angle, Complementary Angle, Degree, Dihedral Angle, Directed Angle, Euler Angles, Gradian, Horn Angle, Inscribed Angle, Oblique Angle, Obtuse Angle, Perigon, Protractor, Radian, Right Angle, Solid Angle, Steradian, Straight Angle, Subtend, Supplementary Angle, Vertex Angle References Dixon, R. Mathographics. 1991. Angle Bisector interior angle bisector exterior angle ^ bisection The (interior) bisector of an Angle is the LINE or Line Segment which cuts it into two equal Angles on the same "side" as the Angle. Ai h A 2 The length of the bisector of Angle A± in the above Triangle AA!A 2 A 3 is given by ,, 2 ti a 2 a% ax (a 2 +a 3 ) 2 where U = A& and a\ = AjA^. The angle bisectors meet at the Incenter J, which has Trilinear Coor- dinates 1:1:1. see also Angle Bisector Theorem, Cyclic Quad- rangle, Exterior Angle Bisector, Isodynamic Points, Orthocentric System, Steiner-Lehmus Theorem, Trisection References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9-10, 1967. Dixon, R. Mathographics. New York: Dover, p. 19, 1991. Mackay, J. S. "Properties Concerned with the Angular Bi- sectors of a Triangle." Proc. Edinburgh Math. Soc. 13, 37-102, 1895. Angle Bisector Theorem The Angle Bisector of an Angle in a Triangle di- vides the opposite side in the same RATIO as the sides adjacent to the ANGLE. Angle Bracket The combination of a Bra and Ket (bra+ket = bracket) which represents the INNER PRODUCT of two functions or vectors, (f\9) (V|W) : f(x)g(x)dx New York: Dover, pp. 99-100, By itself, the Bra is a Covariant 1- Vector, and the Ket is a Covariant One-Form. These terms are com- monly used in quantum mechanics. see also Bra, Differential &-Form, Ket, One-Form 46 Angle of Parallelism Annulus Conjecture Angle of Parallelism P Yl(x) A C D B Given a point P and a Line AB, draw the PERPENDIC- ULAR through P and call it PC, Let PD be any other line from P which meets CB in D. In a Hyperbolic Geometry, as D moves off to infinity along CB, then the line PD approaches the limiting line PE, which is said to be parallel to CB at P. The angle LCPE which PE makes with PC is then called the angle of paral- lelism for perpendicular distance x, and is given by n(x)-2tan- 1 (e- x ). This is known as Lobachevsky's FORMULA. see also Hyperbolic Geometry, Lobachevsky's Formula References Manning, H. P. Introductory Non-Euclidean Geometry. New York: Dover, pp. 31-32 and 58, 1963. Angle Trisection see Trisection Angular Acceleration The angular acceleration ct is defined as the time DE- RIVATIVE of the Angular Velocity u>, a ~ ~dt d 2 6 „ _ a di 2 *' r' see also Acceleration, Angular Distance, Angu- lar Velocity Angular Defect The Difference between the Sum of face Angles Ai at a Vertex of a Polyhedron and 27r, 5 = 2ir-^2Ai. see also Descartes Total Angular Defect, Jump Angle Angular Velocity The angular velocity U) is the time DERIVATIVE of the Angular Distance with direction z Perpendicu- lar to the plane of angular motion, d0„ v io = — z = — . dt r see also ANGULAR ACCELERATION, ANGULAR DIS- TANCE Anharmonic Ratio see Cross-Ratio Anisohedral Tiling A fc-anisohedral tiling is a tiling which permits no n- ISOHEDRAL TILING with n < k. References Berglund, J. "Is There a A;-Anisohedral Tile for k > 5?" Amer. Math. Monthly 100, 585-588, 1993. Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991. Annihilator The term annihilator is used in several different ways in various aspects of mathematics. It is most commonly used to mean the SET of all functions satisfying a given set of conditions which is zero on every member of a given SET. Annulus The region in common to two concentric CIRCLES of RADII a and b. The AREA of an annulus is Aannulus = ?t(& — CL ). An interesting identity is as follows. In the figure, the AREA of the shaded region A is given by A = d + C 2 . Angular Distance The angular distance traveled around a CIRCLE is the number of RADIANS the path subtends, 0= 7^2tt= -. 27TT r see also CHORD, CIRCLE, CONCENTRIC CIRCLES, LUNE (Plane), Spherical Shell References Pappas, T, "The Amazing Trick," The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 69, 1989. see also ANGULAR ACCELERATION, ANGULAR VELOC- ITY Annulus Conjecture see Annulus Theorem Annulus Theorem Anosov Flow 47 Annulus Theorem Let Ki and K^ be disjoint bicollared knots in W n+ or S and let U denote the open region between them. Then the closure of U is a closed annulus S n x [0,1]. Except for the case n = 3, the theorem was proved by Kirby (1969). References Kirby, R. C. "Stable Homeomorphisms and the Annulus Con- jecture." Ann. Math. 89, 575-582, 1969. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 38, 1976. Anomalous Cancellation The simplification of a FRACTION a/b which gives a cor- rect answer by "canceling" DIGITS of a and b. There are only four such cases for NUMERATOR and DENOM- INATORS of two Digits in base 10: 64/16 = 4/1 = 4, 98/49 ^ 8/4 = 2, 95/19 = 5/1 = 5, and 65/26 = 5/2 (Boas 1979). The concept of anomalous cancellation can be extended to arbitrary bases. PRIME bases have no solutions, but there is a solution corresponding to each PROPER DIVI- SOR of a Composite b. When b - 1 is Prime, this type of solution is the only one. For base 4, for example, the only solution is 324/ 134 = 24. Boas gives a table of solutions for b < 39. The number of solutions is EVEN unless b is an EVEN SQUARE. 6 N b N 4 1 26 4 6 2 27 6 8 2 28 10 9 2 30 6 10 4 32 4 12 4 34 6 14 2 35 6 15 6 36 21 16 7 38 2 18 4 39 6 20 4 21 10 22 6 24 6 see also Fraction, Printer's Errors, Reduced Fraction References Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathemat- ical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86-87, 1988. Anomalous Number see Benford's Law Anonymous A term in SOCIAL CHOICE Theory meaning invariance of a result under permutation of voters. see also Dual Voting, Monotonic Voting Anosov Automorphism A Hyperbolic linear map R n -» R n with Integer en- tries in the transformation Matrix and Determinant ±1 is an Anosov Diffeomorphism of the n-ToRUS, called an Anosov automorphism (or HYPERBOLIC AU- TOMORPHISM). Here, the term automorphism is used in the Group Theory sense. Anosov Diffeomorphism An Anosov diffeomorphism is a C x DIFFEOMORPHISM <f> such that the Manifold M is Hyperbolic with respect to (j>. Very few classes of Anosov diffeomorphisms are known. The best known is ARNOLD'S Cat Map. A Hyperbolic linear map W 1 — > W 1 with Integer entries in the transformation Matrix and Determi- nant ±1 is an Anosov diffeomorphism of the n-TORUS. Not every MANIFOLD admits an Anosov diffeomorphism. Anosov diffeomorphisms are EXPANSIVE, and there are no Anosov diffeomorphisms on the CIRCLE. It is conjectured that if <f> : M —> M is an Anosov dif- feomorphism on a Compact Riemannian Manifold and the Nonwandering Set Q(<f>) of <f> is M, then <f> is TOPOLOGICALLY CONJUGATE to a FlNITE-TO-ONE Factor of an Anosov Automorphism of a Nilman- ifold. It has been proved that any Anosov diffeomor- phism on the n-TORUS is TOPOLOGICALLY CONJUGATE to an ANOSOV AUTOMORPHISM, and also that Anosov diffeomorphisms are C 1 STRUCTURALLY STABLE. see also ANOSOV AUTOMORPHISM, AXIOM A DIFFEO- MORPHISM, Dynamical System References Anosov, D. V. "Geodesic Flow on Closed Riemannian Man- ifolds with Negative Curvature." Proc. Steklov Inst, A. M. S. 1969. Smale, S. "Differentiable Dynamical Systems." Bull. Amer. Math. Soc. 73, 747-817, 1967. Anosov Flow A Flow defined analogously to the Anosov Diffeo- morphism, except that instead of splitting the TAN- GENT BUNDLE into two invariant sub-BUNDLES, they are split into three (one exponentially contracting, one expanding, and one which is 1-dimensional and tangen- tial to the flow direction). see also DYNAMICAL SYSTEM 48 Anosov Map Anticlastic Anosov Map An important example of a ANOSOV DlFFEOMORPHISM. Xn+l = 2 l" 1 1 where x n +i,y n +i are computed mod 1. see also ARNOLD'S CAT MAP ANOVA "Analysis of Variance." A Statistical Test for het- erogeneity of Means by analysis of group VARIANCES. To apply the test, assume random sampling of a vari- ate y with equal VARIANCES, independent errors, and a Normal Distribution. Let n be the number of Repli- cates (sets of identical observations) within each of K FACTOR LEVELS (treatment groups), and y^ be the jth observation within FACTOR LEVEL i. Also assume that the ANOVA is "balanced" by restricting n to be the same for each Factor Level. Now define the sum of square terms k n P\2 SST = £) £(j/ - J) (1) \ 2 / u „ v 2 i=l j = l k k n "*-:E E« -eIE* (3) k n v j=rl j=l .-\2 i=l j = l = SST - SSA, (4) (5) which are the total, treatment, and error sums of squares. Here, yi is the mean of observations within FACTOR Level i, and y is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtaining the P- Value corresponding to the calculated F- Ratio of the mean squared values F = MSA MSE* (6) Category SS ° Freedom Mean Squared F- Ratio treatment SSA K-l MSA = |P^ §g error SSE K(n - 1) MSE = ^^ total SST Kn - 1 MST=J^r_ If the P- VALUE is small, reject the NULL HYPOTHESIS that all Means are the same for the different groups. see also Factor Level, Replicate, Variance Anthropomorphic Polygon A Simple Polygon with precisely two Ears and one Mouth. References Toussaint, G. "Anthropomorphic Polygons." Amer. Math. Monthly 122, 31-35, 1991. Anthyphairetic Ratio An archaic word for a Continued Fraction. References Fowler, D. H. The Mathematics of Plato's Academy: A New Reconstruction. New York: Oxford University Press, 1987. Antiautomorphism If a Map / : G -> G* from a Group G to a Group G' satisfies f(ab) = f(a)f(b) for all a, 6 £ G, then / is said to be an antiautomorphism. see also AUTOMORPHISM Anticevian Triangle Given a center a : /3 : 7, the anticevian triangle is defined as the TRIANGLE with VERTICES -a : /3 : 7, a : -0 : 7, and a : f3 : -7. If A'B'C is the CEVIAN TRIANGLE of X and A"B"G" is an anticevian trian- gle, then X and A" are HARMONIC CONJUGATE POINTS with respect to A and A 1 . see also Cevian Triangle References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Antichain Let P be a finite PARTIALLY ORDERED Set. An an- tichain in P is a set of pairwise incomparable elements (a family of SUBSETS such that, for any two members, one is not the Subset of another). The WIDTH of P is the maximum CARDINALITY of an ANTICHAIN in P. For a Partial Order, the size of the longest Antichain is called the Width. see also Chain, Dilworth's Lemma, Partially Or- dered Set, Width (Partial Order) References Sloane, N. J. A. Sequence A006826/M2469 in "An On-Line Version of the Encyclopedia of Integer Sequences." Anticlastic When the Gaussian Curvature K is everywhere Neg- ative, a SURFACE is called anticlastic and is saddle- shaped. A Surface on which K is everywhere Posi- tive is called Synclastic. A point at which the Gaus- sian Curvature is Negative is called a Hyperbolic Point. see also Elliptic Point, Gaussian Quadrature, Hyperbolic Point, Parabolic Point, Planar Point, Synclastic Anticommutative Antimagic Graph 49 Anticommutative An Operator * for which a * b = —6 * a is said to be anticommutative. see also Commutative Anticommutator For Operators A and B, the anticommutator is defined by {i,B} = AB + Si. see a/50 Commutator, Jordan Algebra Anticomplementary Triangle A Triangle AA'B'C* which has a given Triangle AABC as its Medial Triangle. The Trilinear Co- ordinates of the anticomplementary triangle are -1 L-i ^-1 -a : : c A' B = a : —0 : c s^r -1 7-1 -1 C = a :b : — c . see ateo MEDIAL TRIANGLE Antiderivative see Integral Antihomologous Points Two points which are COLLINEAR with respect to a Similitude Center but are not Homologous Points. Four interesting theorems from Johnson (1929) follow. 1. Two pairs of antihomologous points form inversely similar triangles with the HoMOTHETIC CENTER. 2. The Product of distances from a HOMOTHETIC Center to two antihomologous points is a constant. 3. Any two pairs of points which are antihomologous with respect to a Similitude Center lie on a Cir- cle. 4. The tangents to two CIRCLES at antihomologous points make equal ANGLES with the LINE through the points. see also HOMOLOGOUS POINTS, HOMOTHETIC CENTER, Similitude Center References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 19-21, 1929. Antilaplacian The antilaplacian of u with respect to a? is a function whose LAPLACIAN with respect to x equals u. The an- tilaplacian is never unique. see also LAPLACIAN Antilinear Operator An antilinear OPERATOR satisfies the following two properties: A[h(x) + f 2 (x)] = Ah{x) + Af 2 (x) Acf(x) = c*Af(x), Antidifferentiation see INTEGRATION where c* is the Complex Conjugate of c. see also LINEAR OPERATOR Antigonal Points B Given LAXB + IAYB — n RADIANS in the above fig- ure, then X and Y are said to be antigonal points with respect to A and B. Antihomography A ClRCLE-preserving TRANSFORMATION composed of an Odd number of Inversions. see also HOMOGRAPHY Antilogarithm The Inverse Function of the Logarithm, defined such that log 6 (antilog 6 z) = z = antilogy (log b z). The antilogarithm in base b of z is therefore b z . see also Cologarithm, Logarithm, Power Antimagic Graph A GRAPH with e EDGES labeled with distinct elements {l,2,...,e}so that the Sum of the EDGE labels at each VERTEX differ. see also MAGIC GRAPH References Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press, 1990. 50 Antimagic Square Antipedal Triangle Antimagic Square 15 2 12 4 1 14 10 5 8 9 3 16 11 13 6 7 21 18 6 17 4 7 3 13 16 24 5 20 23 11 1 15 8 19 2 25 14 12 9 22 10 10 25 32 13 16 9 22 7 3 24 21 30 20 27 18 26 11 6 1 31 23 33 17 8 19 5 36 12 15 29 34 14 2 4 35 28 14 3 34 21 47 29 22 43 16 13 25 6 26 44 30 48 24 8 12 9 45 10 5 11 38 49 46 19 4 41 37 36 33 27 1 39 17 40 20 7 35 23 31 42 18 32 28 2 15 49 16 50 10 19 28 24 56 42 43 11 15 44 38 55 5 25 21 48 46 9 37 6 63 29 47 8 40 51 30 52 1 45 22 54 23 20 34 2 62 14 59 18 33 41 26 61 13 36 12 58 32 27 64 3 35 17 39 7 57 53 4 60 31 52 19 81 22 29 15 42 31 76 61 10 67 23 54 79 25 33 16 57 9 71 24 38 1 51 47 75 26 78 7 69 66 77 13 27 12 39 21 74 20 37 17 49 55 64 8 65 4 62 50 34 73 41 40 56 68 2 63 14 72 35 44 6 53 30 60 32 36 3 46 43 58 11 70 5 59 48 80 28 45 18 An antimagic square is an n x n ARRAY of integers from 1 to n 2 such that each row, column, and main diago- nal produces a different sum such that these sums form a Sequence of consecutive integers. It is therefore a special case of a HETEROSQUARE. Antimagic squares of orders one and two are impossi- ble, and it is believed that there are also no antimagic squares of order three. There are 18 families of an- timagic squares of order four. Antimagic squares of or- ders 4-9 are illustrated above (Madachy 1979). see also HETEROSQUARE, MAGIC SQUARE, TALISMAN Square References Disc. Abe, G. "Unsolved Problems on Magic Squares." Math. 127, 3-13, 1994. Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy 's Mathematical Recreations. New York: Dover, pp. 103-113, 1979. # Weisstein, E. W. "Magic Squares." http: //www. astro. Virginia, edu/~eww6n/math/notebooks/MagicSquares .m. Antimorph A number which can be represented both in the form xo 2 — Dyo 2 and in the form Dx\ 2 — y\ 2 . This is only possible when the PELL EQUATION 2 n 2 x — Dy Antinomy A Paradox or contradiction. Antiparallel A pair of LINES B\ , B2 which make the same ANGLES but in opposite order with two other given LINES A\ and A2, as in the above diagram, are said to be antiparallel to A\ and A2. see also HYPERPARALLEL, PARALLEL References Phillips, A. W. and Fisher, I. Elements of Geometry. New York: American Book Co., 1896. Antipedal Triangle The antipedal triangle A of a given TRIANGLE T is the Triangle of which T is the Pedal Triangle. For a Triangle with Trilinear Coordinates a : j3 : 7 and Angles A, B, and C, the antipedal triangle has Vertices with Trilinear Coordinates is solvable. Then x 2 - Dy 2 = ~(x - Dy 2 )(x n 2 - Dy n 2 ) = D(x y n - y x n ) 2 - {x x n - Dy y n ) 2 . see also Idoneal Number, Polymorph References Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964. Antimorphic Number see Antimorph — (/? + a cos C) (7 + a cos i?) : (7 + aicosI?)(a + /?cosC) : (0 + a cos C) (a + 7 cos B) (7 + cos A)(/3 + a cos C) : -(7 + ^cos A)(a + 0cosC) : (a + cos C) (0 + 7 cos A) (0 + 7 cos A) (7 + acosi?) : (a + 7 cos B) (7 + ficosA) : — (a + jcosB)(0 + 7 cos A). The Isogonal Conjugate of the Antipedal Trian- gle of a given TRIANGLE is HOMOTHETIC with the orig- inal Triangle. Furthermore, the Product of their Areas equals the Square of the Area of the original Triangle (Gallatly 1913). see also Pedal Triangle Antipersistent Process Antisymmetric Matrix 51 References Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 56-58, 1913. Antipersistent Process A Fractal Process for which H < 1/2, so r < 0. see also Persistent Process Antipodal Map The Map which takes points on the surface of a Sphere S 2 to their ANTIPODAL POINTS. Antipodal Points Two points are antipodal (i.e., each is the ANTIPODE of the other) if they are diametrically opposite. Examples include endpoints of a Line SEGMENT, or poles of a Sphere. Given a point on a Sphere with Latitude S and Longitude A, the antipodal point has Latitude ~6 and LONGITUDE A ± 180° (where the sign is taken so that the result is between —180° and +180°). see also Antipode, Diameter, Great Circle, Sphere Antipode Given a point A, the point B which is the ANTIPODAL Point of A is said to be the antipode of A. see also ANTIPODAL POINTS Antiprism Antiquity see Geometric Problems of Antiquity Antisnowflake see Koch Antisnowflake Antisquare Number A number of the form p a • A is said to be an antisquare if it fails to be a Square Number for the two reasons that a is ODD and A is a nonsquare modulo p. see also Square Number Antisymmetric A quantity which changes Sign when indices are re- versed. For example, Aij = a, — aj is antisymmetric since Aij = —Aji. see also ANTISYMMETRIC MATRIX, ANTISYMMETRIC Tensor, Symmetric Antisymmetric Matrix An antisymmetric matrix is a MATRIX which satisfies the identity A=-A* (i) where A T is the MATRIX TRANSPOSE. In component notation, this becomes an = —a-* Letting k = i = j, the requirement becomes cikk — —a-kkj (2) (3) A Semiregular Polyhedron constructed with 2 n- gons and 2n TRIANGLES. The 3-antiprism is simply the Octahedron. The Duals are the Trapezohedra. The Surface Area of a n-gonal antiprism is -2[|na 2 cot(^)]+2n(|v / 3a 2 ) cot(£)+V3\ = \ na2 see also Octahedron, Prism, Prismoid, Trapezohe- DRON References Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in Mathematical Recreations and Essays, 13ili ed. New York; Dover, p, 130, 1987. Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 85-86, 1997. Weisstein, E. W. "Prisms and Antiprisms." http://www. astro .virginia.edu/-eww6n/math/notebooks/Pr ism. m. so an antisymmetric matrix must have zeros on its diag- onal. The general 3x3 antisymmetric matrix is of the form ai2 ai3~ -aw a 2 3 • (4) . — ai3 — G&23 Applying A" 1 to both sides of the antisymmetry condi- tion gives -A^A 1 = I. (5) Any SQUARE MATRIX can be expressed as the sum of symmetric and antisymmetric parts. Write A=i(A + A T ) + f(A-A T ). an a>2i ai2 «22 0,2n a n i a n 2 (6) (7) 52 Antisymmetric Relation Apeirogon an a2i a n i A T = ai2 a22 " * 0>n2 ) _ain a2n • • ■ Q>nn _ 2an ai2 + C121 flln + «nl 0,12 + &21 2a22 fl2n + «n2 _ain + a-Tii &2n + a„2 Z(l nn (8) A + A T = which is symmetric, and A-A T = ai2 - fltei -(ai2 - a2i) -(flln — flnl) — (tl2n — ^n2) (9) Oln - Q>nl din — &n2 (10) which is antisymmetric. see ateo Skew Symmetric Matrix, Symmetric Ma- trix Antisymmetric Relation A RELATION R on a SET S is antisymmetric provided that distinct elements are never both related to one an- other. In other words xRy and yRx together imply that x~y. Antisymmetric Tensor An antisymmetric tensor is denned as a TENSOR for which A mn = _ A r,m t ^ Any Tensor can be written as a sum of Symmetric and antisymmetric parts as The antisymmetric part is sometimes denoted using the special notation A [ab] = U A ab _ A bay For a general TENSOR, (3) (4) permutations where e ai -a. n is the Levi-Civita Symbol, a.k.a. the Permutation Symbol. see also Symmetric Tensor Antoine's Horned Sphere A topological 2-sphere in 3-space whose exterior is not Simply Connected. The outer complement of An- toine's horned sphere is not Simply Connected. Fur- thermore, the group of the outer complement is not even finitely generated. Antoine's horned sphere is in- equivalent to Alexander's Horned Sphere since the complement in E 3 of the bad points for Alexander's Horned Sphere is Simply Connected. see also Alexander's Horned Sphere References Alexander, J. W. "An Example of a Simply-Connected Sur- face Bounding a Region which is not Simply-Connected." Proc. Nat Acad. Sci. 10, 8-10, 1924. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 76-79, 1976. Antoine's Necklace Construct a chain C of 2n components in a solid TORUS V. Now form a chain C± of 2n solid tori in V, where ir x (V - Ci) <* iri(V - C) via inclusion. In each component of Ci, construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori C^. Con- tinue this process a countable number of times, then the intersection A=f|C which is a nonempty compact SUBSET of IR. is called Antoine's necklace. Antoine's necklace is HOMEOMOR- PHIC with the CANTOR SET. see also ALEXANDER'S HORNED SPHERE, NECKLACE References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 73-74, 1976, Apeirogon The Regular Polygon essentially equivalent to the CIRCLE having an infinite number of sides and denoted with Schlafli Symbol {oo}. see also CIRCLE, REGULAR POLYGON References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Schwartzman, S. The Words of Mathematics: An Etymolog- ical Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994. Apery 's Constant Apery's Constant 53 Apery's Constant N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Apery's constant is defined by C(3) = 1.2020569... (1) (Sloane's A002117) where f(z) is the RlEMANN Zeta Function. Apery (1979) proved that £(3) is Irra- tional, although it is not known if it is TRANSCEN- DENTAL. The Continued Fraction for £(3) is [1, 4, 1, 18, 1, 1, 1, 4, 1, ...] (Sloane's A013631). The positions at which the numbers 1, 2, . . . occur in the continued fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, ... . Sums related to £(3) are c(3) _ 5 f^ (-1)- 1 ,.5f (-ir'(M)» (used by Apery), and oo (2fc + 1)3 2tt 3 (3* + l)» " 81^/3 ' 27 OO ^ (3& + 1) oo 3 2-> (4/k + l) 3 = 64 + " ^ 3 ) + H C(3) (4) (5) OO ^ (6Jfe + 1 + iC(3), (6) (6fc+l)3 36^3 2 where X(z) is the Dirichlet Lambda Function. The above equations are special cases of a general result due to Ramanujan (Berndt 1985). Apery's proof relied on showing that the sum <»>-£©"Cr)". o where (£) is a Binomial Coefficient, satisfies the Re- currence Relation (n + l) 3 a(n + 1) - (34n 3 + bin 2 + 27n + 5)a(n) +n 3 a(n-l) = (8) (van der Poorten 1979, Zeilberger 1991). Apery's constant is also given by Sn,\ «3)=x; 2 T i ' *-*t nln (9) where 5 n , m is a Stirling Number of the First Kind. This can be rewritten as E§= 2 « 3 )' (10) where H n is the nth HARMONIC NUMBER, Yet another expression for £(3) is ««-*£;?(•♦*+••■ + ;) (11) (Castellanos 1988). Integrals for C(3) include CO) i r e 2io c'-l cK \W r ir/4 = ^ | j7r J ln2 + 2 / a: In (sin a;) da; (12) (13) Gosper (1990) gave 30& - 11 4 £? (2* -!)*»(?)' (14) A Continued Fraction involving Apery's constant is JL = 5 _J^ t_ rf C(3) 117- 535- ' ' * 34n 3 + 51n 2 + 27n + 5- * * ' (15) (Apery 1979, Le Lionnais 1983). Amdeberhan (1996) used Wilf- Zeilberger Pairs (F,G) with F{n,k) _ (-l) k k\ 2 (sn-k-l)\ (sn + & + l)!(fc + l) (16) s — 1 to obtain c(3) = §f;(-ir- 1 «i ? . (it) For 5 = 2, oo „ ffl)- 1 ^ nn-i 56n 2 -32 + 5 1 and for s = 3, (-i) n ((3) = V { ~ X) ^72( 4n )( 3n ) 6120n + 5265n 4 + 13761n 2 + 13878n 3 + 1040 (4n-fl)(4n + 3)(n+l)(3n+l) 2 (3n + 2) 2 ^ > 54 Apery's Constant Apoapsis (Amdeberhan 1996). The corresponding G(n,k) for s = 1 and 2 are G(n ' fc) -(n + fc + l)!(n+l)' (20) and <3(n,fc) = (-l) fc fci 2 (2n - fe)!(3 + 4n)(4n 2 + 6n + k + 3) 2(2n-hA; + 2)!(n + l) 2 (2n-hl) 2 Gosper (1996) expressed C(3) as the MATRIX PRODUCT N (21) lim TTM n = C(3) 1 (22) where M n = " (n + l) 4 24570Tt 4 + 64161n 3 +62152n 2 +26427n.+4154 4096(n+f)2(n+J)2 31104(n+|)(n+±)(n+§) 1 (23) which gives 12 bits per term. The first few terms are (24) (25) (26) which gives C( 3 ) * IllZlVwl = 1-20205690315732 .... (27) Given three INTEGERS chosen at random, the probabil- ity that no common factor will divide them all is r i 2077 " 1728 1 Mi = 19600 M 2 = 1 9801 7561 " 4320 1 r ° 50501 20160 1 -1 M 3 = 67600 [CO)]' 1.202 -1 =0.832.. (28) B. Haible and T. Papanikolaou computed £(3) to 1,000,000 Digits using a Wilf-Zeilberger Pair iden- tity with _ fc n! 6 (2n-fc-l)!fc! 3 *(n,k)-( 1) 2(n + A . + 1)!2(2rl )!3> W 5 = 1, and t = 1, giving the rapidly converging ,vn-V\ 1 ^ rc! 1 °(205n 2 + 250n + 77) QW-Z^l- 1 ) 64(2n+l)!« (Amdeberhan and Zeilberger 1997). The record as of Aug. 1998 was 64 million digits (Plouffe). see also Riemann Zeta Function, Wilf-Zeilberger Pair References Amdeberhan, T. "Faster and Faster Convergent Se- ries for C(3)-" Electronic J. Combinatorics 3, R13, 1—2, 1996. http: //www. combinatorics. org/Volume^/ volume3 ,html#R13. Amdeberhan, T. and Zeilberger, D. "Hypergeometric Se- ries Acceleration via the WZ Method." Electronic J. Combinatorics 4, No. 2, R3, 1-3, 1997. http: //www. combinatorics . org/Volume_4/wilf toe .html#R03. Also available at http : //www . math . temple . edu/~zeilberg/ mamarim/mamarimhtml/accel . html. Apery, R. "Irrationalite de £(2) et C(3)." Asterisque 61, 11- 13, 1979. Berndt, B. C. Ramanujan's Notebooks: Part J. New York: Springer- Verlag, 1985. Beukers, F. "A Note on the Irrationality of C(3)." Bull. Lon- don Math. Soc. 11, 268-272, 1979. Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988. Conway, J. H. and Guy, R. K. "The Great Enigma." In The Book of Numbers. New York: Springer- Verlag, pp. 261— 262, 1996. Ewell, J. A. "A New Series Representation for C(3)." Amer. Math. Monthly 97, 219-220, 1990. Finch, S. "Favorite Mathematical Constants." http: //www. mathsoft.com/asolve/constant/apery/apery.html. Gosper, R. W. "Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics." In Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Marcel Dekker, 1990. Haible, B. and Papanikolaou, T. "Fast Multiprecision Eval- uation of Series of Rational Numbers." Technical Report TI-97-7. Darmstadt, Germany: Darmstadt University of Technology, Apr. 1997. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 36, 1983. Plouffe, S. "Plouffe's Inverter: Table of Current Records for the Computation of Constants." http://lacim.uqam.ca/ pi/records. html. Plouffe, S. "32,000,279 Digits of Zeta(3)." http://lacim. uqam.ca/piDATA/Zet a3.txt. Sloane, N. J. A. Sequences A013631 and A002117/M0020 in "An On-Line Version of the Encyclopedia of Integer Se- quences." van der Poorten, A. "A Proof that Euler Missed. . . Apery's Proof of the Irrationality of £(3)." Math. Intel. 1,196-203, 1979. Zeilberger, D. "The Method of Creative Telescoping." J. Symb. Comput. 11, 195-204, 1991. Apoapsis (30) The greatest radial distance of an Ellipse as measured from a FOCUS. Taking v = n in the equation of an Ellipse a(l-e 2 ) r = 1 + e cos v Apocalypse Number Apodization Function 55 gives the apoapsis distance r+ =a(l + e). Apoapsis for an orbit around the Earth is called apogee, and apoapsis for an orbit around the Sun is called aphe- lion. see also Eccentricity, Ellipse, Focus, Periapsis Apocalypse Number A number having 666 Digits (where 666 is the Beast Number) is called an apocalypse number. The FI- BONACCI NUMBER F3184 is an apocalypse number. see also Beast Number, Leviathan Number References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97- 102, 1995. Apocalyptic Number A number of the form 2 n which contains the digits 666 (the Beast Number) is called an Apocalyptic Num- ber. 2 157 is an apocalyptic number. The first few such powers are 157, 192, 218, 220, . . . (Sloane's A007356). see also Apocalypse Number, Leviathan Number References Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97- 102, 1995. Sloane, N. J. A. Sequences A007356/M5405 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. Apodization The application of an APODIZATION FUNCTION. Apodization Function A function (also called a Tapering Function) used to bring an interferogram smoothly down to zero at the edges of the sampled region. This suppresses sidelobes which would otherwise be produced, but at the expense of widening the lines and therefore decreasing the reso- lution. The following are apodization functions for symmetrical (2-sided) interferograms, together with the Instrument Functions (or Apparatus Functions) they produce and a blowup of the Instrument Function sidelobes. The Instrument Function I(k) corresponding to a given apodization function A(x) can be computed by taking the finite FOURIER COSINE TRANSFORM, Apodization Function Instrument Function 1.25 Instrument Function Sidelobes I(k) /a ■a cos(27r kx)A(x) dx. (1) Bartlett Connes Harming Uniform Welch -1 -0.5 0,5 1 Type Apodization Function Instrument Function Bartlett 1 _ i£i a sinc 2 (7r/:a) Blackman B A (x) Bi(fc) Connes o-sr 8aV27r (2ir*a)«/» Cosine c °s(ff) 4aca B (2jrafc) TT(l-160 2 fc 2 ) Gaussian e -*V(2» a ) 2j a cos(27rfca:)e- l3/(2 ' 2) dx Hamming Hm.A{x) Hmj{k) Hanning Hn A (x) Hnt(k) Uniform 1 2a sine (27r/ea) Welch 1-S W:(k) where B A (x) = Bj(k) = Hm A (x) = Hrm(k) = (1ZX \ / 2-7TX \ — J +0.08 cos f J a(0.84 - 0.36a 2 k 2 -2.17 x 1Q- X9 a 4 fc 4 ) sinc(27rafc) (2) (l-a 2 A: 2 )(l-4a 2 fc 2 ) 0.54 + 0.46 cos (—) a(1.08 - 0.64a 2 fc 2 ) sinc(27rafe) " l-4a 2 fc 2 "" (3) (4) (5) 56 Apodization Function Apollonius Circles Hn A (x) = cos 2 I — ) 1 + cos (?) Hrnik) 1 " 2 a sine (2irak) ' l-4a 2 fc 2 = a[sinc(27rfca) + ^ sinc(27rA;a — 7r) + ^ sinc(27rA;a + 7r)] W}(fc) =a2V2?r J3/2 (27rA;a) (27rfca) 3 / 2 sin(27rfca) — 2nak cos(2-7rafc) 2a 3 fe 3 7T 3 ' (6) (7) (8) (9) (10) (11) Type IF FWHM IF Peak Peak (-) S.L. Peak Peak (+) S.L. Peak Bartlett 1.77179 1 0.00000000 0.0471904 Blackmail 2.29880 0.84 -0.00106724 0.00124325 Cormes 1.90416 16 15 -0.0411049 0.0128926 Cosine 1.63941 4. -0.0708048 0.0292720 Gaussian — 1 — — Hamming 1.81522 1.08 -0.00689132 0.00734934 Hanning 2.00000 1 -0.0267076 0.00843441 Uniform 1.20671 2 -0.217234 0.128375 Welch 1.59044 4 3 -0.0861713 0.356044 A general symmetric apodization function A(x) can be written as a FOURIER SERIES oo a n cosl— -J, (12) n=l where the COEFFICIENTS satisfy oo a + 2^a„ = 1. (13) n = l The corresponding apparatus function is I(t) = J A{x)e~ 2 ' Ktkx dx = 26Ja sinc(27r£;&) oo + y^[sinc(27rA:& + mr) + sinc(27rA;6 - nir)] |. (14) n=l To obtain an APODIZATION FUNCTION with zero at ka = 3/4, use ao sinc(|7r) + ai[sinc(|7r) + sinc(^7r) = 0. (15) Plugging in (13), -d-^ +*(£ + £) = -|(l-2o 1 ) + oi(i + l) = (16) ai = ao - 5 3 ° _ 5 | + | 6-3 + 2-5 28 1 n„ 28 — 2 • 5 18 1 lai = 9* ~" 28 - 9 14' (18) (19) The Hamming Function is close to the requirement that the Apparatus Function goes to at ka — 5/4, giving a = § « 0.5435 ai 21 92 0.2283. (20) (21) The Blackman Function is chosen so that the Appa- ratus Function goes to at ka — 5/4 and 9/4, giving ao ai = a 2 = 3969 ,. 9304 n 1155 „ 4652 " 715 18608 0.4266 (22) 0.2483 (23) i 0.0384. (24) ai(! + !) = ^5 ' 3^ 3 (IT) see also Bartlett Function, Blackman Function, Connes Function, Cosine Apodization Function, Full Width at Half Maximum, Gaussian Func- tion, Hamming Function, Hann Function, Han- ning Function, Mertz Apodization Function, Parzen Apodization Function, Uniform Apodiza- tion Function, Welch Apodization Function References Ball, J. A. "The Spectral Resolution in a Correlator Sys- tem" §4,3.5 in Methods of Experimental Physics 12C (Ed. M. L. Meeks). New York: Academic Press, pp. 55-57, 1976. Blackman, R. B. and Tukey, J. W. "Particular Pairs of Win- dows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 95-101, 1959. Brault, J. W. "Fourier Transform Spectrometry." In High Resolution in Astronomy: 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics (Ed. A. Benz, M. Huber, and M. Mayor), Geneva Observatory, Sauverny, Switzerland, pp. 31-32, 1985. Harris, F. J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proc. IEEE 66, 51- 83, 1978. Norton, R. H. and Beer, R. "New Apodizing Functions for Fourier Spectroscopy." J. Opt. Soc. Amer. 66, 259-264, 1976. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 547-548, 1992. Schnopper, H. W. and Thompson, R. I. "Fourier Spectrom- eters." In Methods of Experimental Physics 12 A (Ed. M. L. Meeks). New York: Academic Press, pp. 491-529, 1974. Apollonius Circles There are two completely different definitions of the so- called Apollonius circles: 1 . The set of all points whose distances from two fixed points are in a constant ratio 1 : \i (Ogilvy 1990). Apollonius Point Apollonius 3 Problem 57 2. The eight CIRCLES (two of which are nondegener- ate) which solve APOLLONIUS ' PROBLEM for three Circles. Given one side of a Triangle and the ratio of the lengths of the other two sides, the LOCUS of the third VERTEX is the Apollonius circle (of the first type) whose Center is on the extension of the given side. For a given Triangle, there are three circles of Apollonius. Denote the three Apollonius circles (of the first type) of a Triangle by &i, fo, and £3, and their centers Li, L 2) and L 3 . The center L\ is the intersection of the side A2A3 with the tangent to the ClRCUMCIRCLE at A\. L\ is also the pole of the SYMMEDIAN POINT K with respect to ClRCUMCIRCLE. The centers Li, Z/ 2 , and Lz are COLLINEAR on the POLAR of K with regard to its ClRCUMCIRCLE, called the Lemoine Line. The circle of Apollonius ki is also the locus of a point whose Pedal Triangle is Isosceles such that P1P2 = P1P3. Let U and V be points on the side line BC of a TRI- ANGLE AABC met by the interior and exterior ANGLE Bisectors of Angles A. The Circle with Diame- ter UV is called the A-Apollonian circle. Similarly, construct the B- and C-Apollonian circles. The Apol- lonian circles pass through the VERTICES A, £?, and C, and through the two ISODYNAMIC POINTS S and S' . The Vertices of the D-Triangle lie on the respective Apollonius circles. see also Apollonius' Problem, Apollonius Pursuit Problem, Casey's Theorem, Hart's Theorem, Iso- dynamic Points, Soddy Circles References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 40 and 294-299, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 14-23, 1990. Apollonius Point Consider the Excircles Fa, T b , and Tc of a Trian- gle, and the CIRCLE T internally TANGENT to all three. Denote the contact point of T and Fa by A f , etc. Then the Lines AA\ BB f , and CC' CONCUR in this point. It has Triangle Center Function a = sin 2 ,4 cos 2 [§(£-<?)]. References Kiinherling, C. "Apollonius Point." http://vvv. evansville . edu/~ck6/t centers/re cent /apollon. html. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Kimberling, C; Iwata, S.; and Hidetosi, F. "Problem 1091 and Solution." Crux Math. 13, 128-129 and 217-218, 1987. Apollonius' Problem •3 ^€J •£) © #;(•} ^ Given three objects, each of which may be a Point, Line, or Circle, draw a Circle that is Tangent to each. There are a total of ten cases. The two easi- est involve three points or three LINES, and the hardest involves three CIRCLES. Euclid solved the two easiest cases in his Elements, and the others (with the exception of the three CIRCLE problem), appeared in the Tangen- cies of Apollonius which was, however, lost. The general problem is, in principle, solvable by STRAIGHTEDGE and Compass alone. 58 Apollonius 7 Problem Apollonius Pursuit Problem The three-ClRCLE problem was solved by Viete (Boyer 1968), and the solutions are called Apollonius Cir- cles. There are eight total solutions. The simplest solution is obtained by solving the three simultaneous quadratic equations (x - x x f + (y - Vl ) 2 - (r ± n) 2 = (1) (x - x 2 f + (y - y 2 f - (r ± r 2 f = (2) (x - x z f + (y - y 3 ) 2 - (r ± r 3 ) 2 - (3) in the three unknowns x, y y r for the eight triplets of signs (Courant and Robbins 1996). Expanding the equa- tions gives OOO O ^ (x + y -r )-2xXi-2yyi±2rri+(xi +yi -n ) = (4) for i — 1, 2, 3. Since the first term is the same for each equation, taking (2) — (1) and (3) — (1) gives where ax 4- by + cr = d (5) ax + by + cr=-d, (6) a = 2(a?i — x 2 ) (7) b= 2(yi -y 2 ) (8) c = q=2(ri - r 2 ) (9) 1/2. 2 2\/2, 2 a = (x 2 +2/2 - r-2 ) - (xi + yi - -n 2 ) (10) and similarly for a , 6' , c and d' (where the 2 subscripts are replaced by 3s). Solving these two simultaneous lin- ear equations gives b'd - bd! - b'cr + bc'r ab 1 - ba ! —ad + ad' + o! cr — ac'r ab' -a'b ' (11) (12) which can then be plugged back into the QUADRATIC Equation (1) and solved using the Quadratic For- mula. Perhaps the most elegant solution is due to Gergonne. It proceeds by locating the six HOMOTHETIC CENTERS (three internal and three external) of the three given CIRCLES. These lie three by three on four lines (illus- trated above). Determine the Poles of one of these with respect to each of the three CIRCLES and connect the Poles with the Radical Center of the Circles. If the connectors meet, then the three pairs of intersec- tions are the points of tangency of two of the eight circles (Johnson 1929, Dorrie 1965). To determine which two of the eight Apollonius circles are produced by the three pairs, simply take the two which intersect the original three CIRCLES only in a single point of tangency. The procedure, when repeated, gives the other three pairs of Circles. If the three CIRCLES are mutually tangent, then the eight solutions collapse to two, known as the Soddy Circles. see also Apollonius Pursuit Problem, Bend (Cur- vature), Casey's Theorem, Descartes Circle Theorem, Four Coins Problem, Hart's Theorem, Soddy Circles References Boyer, C. B. A History of Mathematics. New York: Wiley, p. 159, 1968. Courant, R. and Robbins, H. "Apollonius' Problem." §3.3 in What is Mathematics? : An Elementary Approach to Ideas and Methods , 2nd ed. Oxford, England: Oxford University Press, pp. 117 and 125-127, 1996. Dorrie, H. "The Tangency Problem of Apollonius." §32 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 154-160, 1965. Gauss, C. F. Werke, Vol. 4. New York: George Olms, p. 399, 1981. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 118-121, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 48-51, 1990. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 151, 1989. Simon, M. Uber die Entwicklung der Element argeometrie im XIX Jahrhundert. Berlin, pp. 97-105, 1906. ^ Weisstein, E. W. "Plane Geometry." http: //www. astro. Virginia . edu/-eww6n/math/notebooks/PlaneGeometry . m. Apollonius Pursuit Problem Given a ship with a known constant direction and speed v 1 what course should be taken by a chase ship in pur- suit (traveling at speed V) in order to intersect the other ship in as short a time as possible? The problem can be solved by finding all points which can be simultaneously reached by both ships, which is an APOLLONIUS CIRCLE with fi = v/V. If the CIRCLE cuts the path of the pur- sued ship, the intersection is the point towards which Apollonius Theorem Appell Transformation 59 the pursuit ship should steer. If the CIRCLE does not cut the path, then it cannot be caught. see also Apollonius Circles, Apollonius' Prob- lem, Pursuit Curve References Ogilvy, C. S. Solved by M. S. Klamkin. "A Slow Ship In- tercepting a Fast Ship." Problem E991. Amer. Math. Monthly 59, 408, 1952. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 17, 1990. Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, pp. 126-138, 1983. Apollonius Theorem ma 2 2 + na 3 2 = (m + n)AiP 2 + mPA 3 2 + nPA 2 2 . Apothem Given a CIRCLE, the PERPENDICULAR distance a from the Midpoint of a Chord to the Circle's center is called the apothem. It is also equal to the RADIUS r minus the SAGITTA s, a — r — s. see also Chord, Radius, Sagitta, Sector, Segment Apparatus Function see Instrument Function Appell Hypergeometric Function A formal extension of the Hypergeometric Function to two variables, resulting in four kinds of functions (Ap- pell 1925), oo oo F 1 (a;/3,/3'; 7 ;x,y) = ^^ (a)™+«GS)m(/3')« m = n = oo oo -x y x y m = n = F 3 (a,a ;/3,/3 i7i *,y) = ^ JL m!n !( 7 ) m+ „ m = Q ti-0 oo oo 77 / a < \ V^ V^ ( Q )m + n(^)m + r 1 ^ mj , Appell defined the functions in 1880, and Picard showed in 1881 that they may all be expressed by INTEGRALS of the form /' Jo u a (l - uf{l - xuy(l - yu) S du. References Appell, P. "Sur les fonctions hypergeometriques de plusieurs variables." In Memoir. Sci. Math. Paris: Gauthier-Villars, 1925. Bailey, W. N. Generalised Hypergeometric Series. Cam- bridge, England: Cambridge University Press, p. 73, 1935. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1461, 1980. Appell Polynomial A type of Polynomial which includes the Bernoulli Polynomial, Hermite Polynomial, and Laguerre POLYNOMIAL as special cases. The series of POLYNOMI- ALS {A n (z)}™ =0 is defined by where A(t)e** = ^TA n (z)t n , A(t) = ^2a k t k is a formal POWER series with k = 0, 1, . . . and ao ^ 0. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 209-210, 1988. Appell Transformation A HOMOGRAPHIC transformation ax + by -\- c a"x + b"y + c ax + b'y + c' a n x + b"y + c" with t\ substituted for t according to X! — yi kdti dt {a"x + b"y + c") 2 ' References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 210-211, 1988. m = ti=:0 m!n!(7) m (7') 71 A Surface of Revolution defined by Kepler. It con- sists of more than half of a circular ARC rotated about an axis passing through the endpoints of the Arc. The equations of the upper and lower boundaries in the x-z Plane are z± = ± V / R 2 -(x-r) 2 for R > r and x E [— (r + R), r + R]. It is the outside surface of a Spindle TORUS. see also Bubble, Lemon, Sphere-Sphere Intersec- tion, Spindle Torus Approximately Equal If two quantities A and B are approximately equal, this is written A « B. see also Defined, Equal Approximation Theory The mathematical study of how given quantities can be approximated by other (usually simpler) ones under ap- propriate conditions. Approximation theory also stud- ies the size and properties of the ERROR introduced by approximation. Approximations are often obtained by POWER SERIES expansions in which the higher order terms are dropped. see also LAGRANGE REMAINDER References Achieser, N. I. and Hyman, C. J. Theory of Approximation. New York: Dover, 1993. Akheizer, N. I. Theory of Approximation. New York: Dover, 1992. Cheney, E. W. Introduction to Approximation Theory. New York: McGraw-Hill, 1966. Golomb, M. Lectures on Theory of Approximation. Argonne, IL: Argonne National Laboratory, 1962. Jackson, D, The Theory of Approximation. New York: Amer. Math. Soc, 1930. Natanson, I. P. Constructive Function Theory, Vol. 1: Uni- form Approximation. New York: Ungar, 1964. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Rivlin, T. J. An Introduction to the Approximation of Func- tions. New York: Dover, 1981. Timan, A. F. Theory of Approximation of Functions of a Real Variable. New York: Dover, 1994. Arbelos Arakelov Theory A formal mathematical theory which introduces "com- ponents at infinity" by defining a new type of divisor class group of Integers of a Number Field. The di- visor class group is called an "arithmetic surface." see also ARITHMETIC GEOMETRY Arbelos The term "arbelos" means SHOEMAKER'S KNIFE in Greek, and this term is applied to the shaded AREA in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be lo- cated anywhere along the DIAMETER. The arbelos satisfies a number of unexpected identities (Gardner 1979). 1. Call the radii of the left and right SEMICIRCLES a and 6, respectively, with a + b = R. Then the arc length along the bottom of the arbelos is L = 27va + 2tt6 = 2?r(a + b) = 2tvR, so the arc lengths along the top and bottom of the arbelos are the same. 2. Draw the PERPENDICULAR BD from the tangent of the two Semicircles to the edge of the large Cir- cle. Then the Area of the arbelos is the same as the Area of the Circle with Diameter BD. 3. The CIRCLES C\ and C2 inscribed on each half of BD on the arbelos (called ARCHIMEDES' CIRCLES) each have DIAMETER (AB)(BC)/(AC). Further- more, the smallest ClRCUMCIRCLE of these two cir- cles has an area equal to that of the arbelos. 4. The line tangent to the semicircles AB and BC con- tains the point E and F which lie on the lines AD and CD, respectively. Furthermore, BD and EF bi- sect each other, and the points B, D, E, and F are CONCYCLIC. Arbelos Arc Length 61 5. In addition to the ARCHIMEDES' CIRCLES C± and C 2 in the arbelos figure, there is a third circle Cz called the Bankoff Circle which is congruent to these two. 6. Construct a chain of TANGENT CIRCLES starting with the Circle Tangent to the two small ones and large one. The centers of the CIRCLES lie on an Ellipse, and the Diameter of the nth Cir- cle C n is (l/n)th Perpendicular distance to the base of the Semicircle. This result is most eas- ily proven using INVERSION, but was known to Pap- pus, who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). If r = AB/AC, then the radius of the nth circle in the Pappus Chain is _ (1 — r)r n 2[n 2 (l-r) 2 +r]" This general result simplifies to r n = 1/(6 -f n 2 ) for r = 2/3 (Gardner 1979). Further special cases when AC = 1 + AB are considered by Gaba (1940). If B divides AC in the GOLDEN RATIO 0, then the circles in the chain satisfy a number of other special properties (Bankoff 1955). see also Archimedes' Circles, Bankoff Circle, Coxeter's Loxodromic Sequence of Tangent Circles, Golden Ratio, Inversion, Pappus Chain, Steiner Chain References Bankoff, L. "The Fibonacci Arbelos." Scripta Math. 20, 218, 1954. Bankoff, L. "The Golden Arbelos." Scripta Math. 21, 70-76, 1955. Bankoff, L. "Are the Twin Circles of Archimedes Really Twins?" Math. Mag. 47, 214-218, 1974, Bankoff, L. "How Did Pappus Do It?" In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112-118, 1981. Bankoff, L. "The Marvelous Arbelos." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Wash- ington, DC: Math. Assoc. Amer., 1994. Cadwell, J. H. Topics in Recreational Mathematics. Cam- bridge, England: Cambridge University Press, 1966. Gaba, M. G. "On a Generalization of the Arbelos." Amer. Math. Monthly 47, 19-24, 1940. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240, 18-28, Jan. 1979. Heath, T. L. The Works of Archimedes with the Method of Archimedes. New York: Dover, 1953. Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134- 137, 1961. Johnson, R. A. Modern Geometry; An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 116-117, 1929. Ogilvy, C S. Excursions in Geometry. New York: Dover, pp. 54-55, 1990. Arborescence A Digraph is called an arborescence if, from a given node x known as the ROOT, there is exactly one ele- mentary path from x to every other node y. see also Arboricity Arboricity Given a GRAPH G, the arboricity is the MINIMUM num- ber of line-disjoint acyclic SUBGRAPHS whose UNION is G. see also ANARBORICITY Arc In general, any smooth curve joining two points. In particular, any portion (other than the entire curve) of a Circle or Ellipse. see also APPLE, ClRCLE-ClRCLE INTERSECTION, FlVE Disks Problem, Flower of Life, Lemon, Lens, Piecewise Circular Curve, Reuleaux Polygon, Reuleaux Triangle, Salinon, Seed of Life, Tri- angle Arcs, Venn Diagram, Yin- Yang Arc Length Arc length is defined as the length along a curve, J a \d£\. (1) Defining the line element ds 2 = \d£\ 2 , parameterizing the curve in terms of a parameter t, and noting that 62 Arc Minute Archimedes Algorithm ds/dt is simply the magnitude of the VELOCITY with which the end of the Radius Vector r moves gives = / ds = I ft dt= I |r ' ( * )|dt - (2) In Polar Coordinates, d£ = rdr + r§d6= (^-r + rd\ dd, (3) so ds=\de\ = X /r*+(j£) d0 In Cartesian Coordinates, di = x± + yy Therefore, if the curve is written r(x) = xx-\- f(x)y, then J a * = / x/l + f' 2 {x)dx. If the curve is instead written r(t) = x(t)x + y(t)y t then J a (4) =J m =C^ 2+ (%) 2de - (5) (6) ds= ^dx 2 + dy 2 = A/(£) +ldx. (7) (8) (9) (10) '= I ^x"(t) + y*(t)dt. (11) J a Or, in three dimensions, r(t) = x(t)x + y(t)y + z(t)z, (12) (t)+y' 2 {t) + z' 2 (t)dt. (13) see also Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion, Speed, Surface Area, Tangential Angle, Tangent Vec- tor, Torsion (Differential Geometry), Veloc- ity Arc Minute A unit of Angular measure equal to 60 Arc Seconds, or 1/60 of a DEGREE. The arc minute is denoted ' (not to be confused with the symbol for feet). Arc Second A unit of Angular measure equal to 1/60 of an Arc MINUTE, or 1/3600 of a DEGREE. The arc second is de- noted " (not to be confused with the symbol for inches). Arccosecant see Inverse Cosecant Arccosine see Inverse Cosine Arccotangent see Inverse Cotangent Arch A 4-POLYHEX. References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight- of- Mind from Scientific American. New York: Vintage, p. 147, 1978. Archimedes Algorithm Successive application of ARCHIMEDES' RECURRENCE FORMULA gives the Archimedes algorithm, which can be used to provide successive approximations to it (Pi). The algorithm is also called the Borchardt-Pfaff Al- gorithm. Archimedes obtained the first rigorous ap- proximation of TV by Circumscribing and Inscribing n = 6 • 2 fe -gons on a CIRCLE. Prom ARCHIMEDES' RE- CURRENCE Formula, the Circumferences a and b of the circumscribed and inscribed POLYGONS are a(n) = 2ntan ( — ) b(n) = 2nsin ( — ) , (i) (2) where b(n) < C = 27rr = 2tt • 1 = 2tt < a(n). (3) For a Hexagon, n = 6 and a = a(6) = 4\/3 (4) feo = 6(6) = 6, (5) where a^ = a(6-2 k ). The first iteration of ARCHIMEDES' Recurrence Formula then gives 2-6 -4^3 24^ nA , n /-, ffll = 7Tvr = ^vi = 24(2 -^ ) (6) h = yj 24(2 - V3) • 6 = 12\/2- \/3 -6(v / 6-v / 2). (7) Archimedes 7 Axiom Archimedes' Cattle Problem 63 Additional iterations do not have simple closed forms, but the numerical approximations for k = 0, 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are 3.00000 < TV < 3.46410 3.10583 <tt< 3.21539 3.13263 < 7T < 3.15966 3.13935 < TV < 3.14609 3.14103 < 7T < 3.14271. (8) (9) (10) (11) (12) By taking k = 4 (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result ^ =3.14084... <tt < f : 3.14285. (13) References Miel, G. "Of Calculations Past and Present: The Archimed- ean Algorithm." Amer. Math. Monthly 90, 17-35, 1983. Phillips, G. M. "Archimedes in the Complex Plane." Amer. Math. Monthly 91, 108-114, 1984. Archimedes' Axiom An Axiom actually attributed to Eudoxus (Boyer 1968) which states that a/6 = c/d IFF the appropriate one of following conditions is satis- fied for Integers m and n: 1. If ma < nb, then mc < md. 2. If ma — rid, then mc = nd. 3. If ma > nd, then mc > nd. Archimedes' Lemma is sometimes also known as Arch- imedes' axiom. References Boyer, C. B. A History of Mathematics. New York: Wiley, p. 99, 1968. Archimedes' Cattle Problem Also called the Bovinum PROBLEMA. It is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the num- ber of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?" Solution consists of solving the simultaneous DlOPHAN- tine Equations in Integers W, X, Y, Z (the number of white, black, spotted, and brown bulls) and w y x y y, z (the number of white, black, spotted, and brown cows), w ^ \x + z _9_ 20 J 42 ' _7_ 12 v Y + Z W + Z (X + x) x =±(Y + y) (W + w). _ 13 (i) (2) (3) (4) (5) (6) (7) The smallest solution in INTEGERS is W = 10,366,482 (8) X = 7,460,514 (9) Y = 7,358,060 (10) Z = 4,149,387 (11) w= 7,206,360 (12) x = 4,893,246 (13) y= 3,515,820 (14) z = 5,439,213. (15) A more complicated version of the problem requires that W+X be a Square Number and Y+Z a Triangular Number. The solution to this Problem are numbers with 206544 or 206545 digits. References Amthor, A. and Krumbiegel B. "Das Problema bovinum des Archimedes." Z. Math. Phys. 25, 121-171, 1880. Archibald, R. C. "Cattle Problem of Archimedes." Amer. Math. Monthly 25, 411-414, 1918. Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 249-252, 1966. Bell, A. H. "Solution to the Celebrated Indeterminate Equa- tion x 2 - ny 2 = 1." Amer. Math. Monthly 1, 240, 1894. Bell, A. H. "'Cattle Problem.' By Archimedes 251 BC." Amer. Math. Monthly 2, 140, 1895. Bell, A. H. "Cattle Problem of Archimedes." Math. Mag. 1, 163, 1882-1884. Calkins, K. G. "Archimedes' Problema Bovinum." http:// www. andrews . edu/-calkins/cattle .html. Dorrie, H. "Archimedes' Problema Bovinum" §1 in 100 Great Problems of Elementary Mathematics: Their His- tory and Solutions. New York: Dover, pp. 3-7, 1965. Grosjean, C. C. and de Meyer, H. E. "A New Contribution to the Mathematical Study of the Cattle-Problem of Arch- imedes." In Constantin Caratheodory: An International Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ: World Scientific, pp. 404-453, 1991. Merriman, M. "Cattle Problem of Archimedes." Pop. Sci. Monthly 67, 660, 1905. Rorres, C. "The Cattle Problem." http: //www. mcs.drexel. edu/-crorres/Archimedes/Cattle/Statement .html. Vardi, I. "Archimedes' Cattle Problem." Amer. Math. Monthly 105, 305-319, 1998. 64 Archimedes' Circles Archimedes' Circles Draw the PERPENDICULAR LINE from the intersection of the two small SEMICIRCLES in the ARBELOS. The two Circles Ci and Ci Tangent to this line, the large SEMICIRCLE, and each of the two SEMICIRCLES are then congruent and known as Archimedes' circles. see also ARBELOS, BANKOFF CIRCLE, SEMICIRCLE Archimedes' Constant see Pi Archimedes' Hat-Box Theorem Enclose a Sphere in a Cylinder and slice Perpen- dicularly to the Cylinder's axis. Then the Surface Area of the of Sphere slice is equal to the Surface Area of the Cylinder slice. Archimedes' Lemma Also known as the continuity axiom, this Lemma sur- vives in the writings of Eudoxus (Boyer 1968). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the EXHAUSTION METHOD which Archimedes invented to solve problems of Area and Volume. see also Continuity Axioms References Boyer, C. B. A History of Mathematics. New York: Wiley, p. 100, 1968. Archimedes' Midpoint Theorem Let M be the Midpoint of the Arc AMB. Pick C at random and pick D such that MD _L AC (where J_ denotes PERPENDICULAR). Then AD = DC + BC. see also MIDPOINT References Honsberger, R. More Mathematical Morsels. DC: Math. Assoc. Amer., pp. 31-32, 1991. Washington, Archimedes 7 Recurrence Formula Archimedes' Postulate see Archimedes' Lemma Archimedes' Problem Cut a Sphere by a Plane in such a way that the VOL- UMES of the Spherical Segments have a given Ratio. see also SPHERICAL SEGMENT Archimedes' Recurrence Formula Let a n and b n be the Perimeters of the Circum- scribed and Inscribed n-gon and a2 n and fen the Perimeters of the Circumscribed and Inscribed 2n- gon. Then di 2a n b n a n + b n &2n = V d2nK • (1) (2) The first follows from the fact that side lengths of the Polygons on a Circle of Radius r = 1 are SR 2 tan 2 sin CD But a n = 2ntan ( — ) b n — 2nsin f — 1 . 2an b n 2-2ntan(i)-2nsin(^) (3) (4) (5) (6) a n +b n 2ntan(^) +2nsin(^) tan(S) sin (?) An tan(^)+sin(^)- Using the identity tan(|x) = tan x sin x tan x + sin x then gives 2a n frn a n + b n — 4ntan (- V2n 2n) &2n (7) (8) (9) Archimedean Solid The second follows from Archimedean Solid 65 \Zd2nbn = W4ntan ( — j • 2nsin (-) (10) Using the identity since = 2sin(|x) cos(|x) gives (ii) y^X = 2n v /2tan (£) - 2 sin (£) cos ( j) = 4n v sin2 (£) = 4nsin (£) = b2n - (12) Successive application gives the Archimedes Algo- rithm, which can be used to provide successive approx- imations to Pi (it). see also ARCHIMEDES ALGORITHM, Pi References Dorrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965. Archimedean Solid The Archimedean solids are convex Polyhedra which have a similar arrangement of nonintersecting regu- lar plane Convex Polygons of two or more differ- ent types about each VERTEX with all sides the same length. The Archimedean solids are distinguished from the Prisms, Antiprisms, and Elongated Square GYROBICUPOLA by their symmetry group: the Arch- imedean solids have a spherical symmetry, while the others have "dihedral" symmetry. The Archimedean solids are sometimes also referred to as the SEMIREG- ular Polyhedra. Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular Tet- rahedron so that four of their faces lie on the faces of that Tetrahedron. A method of constructing the Archimedean solids using a method known as "expan- sion" has been enumerated by Stott (Stott 1910; Ball and Coxeter 1987, pp. 139-140). Let the cyclic sequence S = (pi,P2, . . . ,p q ) represent the degrees of the faces surrounding a vertex (i.e., S is a list of the number of sides of all polygons surrounding any vertex). Then the definition of an Archimedean solid requires that the sequence must be the same for each vertex to within ROTATION and REFLECTION. Walsh (1972) demonstrates that S represents the degrees of the faces surrounding each vertex of a semiregular convex polyhedron or TESSELLATION of the plane IFF 1. q > 3 and every member of S is at least 3, 2. ^2? =1 ~ > \q — 1, with equality in the case of a plane TESSELLATION, and 3. for every ODD NUMBER p £ 5, S contains a subse- quence (6, p, 6). Condition (1) simply says that the figure consists of two or more polygons, each having at least three sides. Con- dition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for the figure to lie in the plane, and less than a full rotation for a solid figure to be convex. The usual way of enumerating the semiregular polyhe- dra is to eliminate solutions of conditions (1) and (2) using several classes of arguments and then prove that the solutions left are, in fact, semiregular (Kepler 1864, pp. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940, p. 394; Coxeter et al. 1954; Lines 1965, pp. 202-203; Walsh 1972). The following table gives all possible reg- ular and semiregular polyhedra and tessellations. In the table, 'P } denotes PLATONIC SOLID, 'M' denotes a PRISM or ANTIPRISM, 'A' denotes an Archimedean solid, and 'T' a plane tessellation. Fg. Solid Schlafli 3,3) 4,4) 6,6) 8,8) 10, 10) 12, 12) 4,n) 4, 4) 6,6) 6,8) 6,10) 6,12) 8, 8) 5,5) 6,6) 6,6) 3, 3, n) 3, 3, 3) 4, 3, 4) 5, 3, 5) 6, 3, 6) 4, 4, 4) 4, 5, 4) 4, 6, 4) 4, 4, 4) o, o, o, 3, 3, 3, o, o, o, o, o, o, 3, 3, 4, 3, 4, 3, 3, 3, 3, P tetrahedron {3j3} M triangular prism t{2,3} A truncated tetrahedron t{3, 3} A truncated cube t{4, 3} A truncated dodecahedron t{5,3} T (plane tessellation) t{6,3} M n-gonal Prism t{2,n} P cube {4, 3} A truncated octahedron t{3,4} A great rhombicuboct. A great rhombicosidodec. T (plane tessellation) T (plane tessellation) P dodecahedron A truncated icosahedron T (plane tessellation) M n-gonal antiprism P octahedron A cuboctahedron A icosidodecahedron T (plane tessellation) A small rhombicuboct. A small rhombicosidodec. T (plane tessellation) T (plane tessellation) P icosahedron A snub cube A snub dodecahedron T (plane tessellation) T (plane tessellation) — T (plane tessellation) s 1 4 J T (plane tessellation) {3,6} As shown in the above table, there are exactly 13 Ar- chimedean solids (Walsh 1972, Ball and Coxeter 1987). 66 Archimedean Solid Archimedean Solid They are called the CUBOCTAHEDRON, GREAT RHOMB- ICOSIDODECAHEDRON, GREAT RHOMBICUBOCTAHE- DRON, ICOSIDODECAHEDRON, SMALL RHOMBICOSIDO- DECAHEDRON, SMALL RHOMBICUBOCTAHEDRON, SNUB Cube, Snub Dodecahedron, Truncated Cube, Truncated Dodecahedron, Truncated Icosahe- dron (soccer ball), Truncated Octahedron, and Truncated Tetrahedron. The Archimedean solids satisfy (27T- <t)V — 4tt, where a is the sum of face- angles at a vertex and V is the number of vertices (Steinitz and Rademacher 1934, Ball and Coxeter 1987). Here are the Archimedean solids shown in alphabetical order (left to right, then continuing to the next row). ry \ / ^m n A , Li /^ aM LV The following table lists the symbol and number of faces of each type for the Archimedean solids (Wenninger 1989, p. 9). Solid Schlafli Wythoff C&R cuboctahedron i 3 \ X 4 1 2 | 34 (3.4) 2 great rhombicosidodecahedron *{*} 2 3 5 | great rhombicuboctahedron *{:} 234 | icosidodecahedron / 3 \ 1 5 J 2 | 3 5 (3-5) 2 small rhombicosidodecahedron 'it) 3 5 | 2 3.4.5.4 small rhombicuboctahedron r l:l 3 4)2 3.4 3 snub cube s i:i | 2 3 4 3 4 .4 snub dodecahedron *{*} | 2 3 5 3 4 .5 truncated cube t{4,3} 2 3 | 4 3.8 2 truncated dodecahedron t{5,3} 23[5 3.10 2 truncated icosahedron t{3,5} 2 5 | 3 5.6 2 truncated octahedron t{3,4} 2 4 | 3 4.6 2 truncated tetrahedron t{3,3} 23 | 3 3.6 2 Solid V e h h A h h /io cuboctahedron 12 24 8 6 great rhombicos. 120 180 30 2G 12 great rhombicub. 48 72 12 8 6 icosidodecahedron 30 60 20 12 small rhombicos. 60 120 20 30 12 small rhombicub. 24 48 8 18 snub cube 24 60 32 6 snub dodecahedron 60 150 80 12 trunc. cube 24 36 8 6 trunc. dodec. 60 90 20 12 trunc. icosahedron 60 90 12 20 trunc. octahedron 24 36 6 8 trunc. tetrahedron 12 18 4 4 Let r be the INRADIUS, p the MIDRADIUS, and R the ClRCUMRADIUS. The following tables give the analytic and numerical values of r, p, and R for the Archimedean solids with EDGES of unit length. Solid r cuboctahedron great rhombicosidodecahedron great rhombicuboctahedron icosidodecahedron small rhombicosidodecahedron small rhombicuboctahedron snub cube snub dodecahedron truncated cube truncated dodecahedron truncated icosahedron truncated octahedron truncated tetrahedron 3 4 aii (105 + 6^5 )\/31 4- 12 VE £(14 + >/2)\/l3 + 6^ ±(5 + 3^5) ^(15 + 2^)^11 + 4^5 ^r(6 + v / 2)V /s + 2 v / 2 * * £(5 + 2^)^7 + 4^ 4§s (17V2 + 3</l0 ) ^37 + ISn/5 ? f^(21 + Vo")V /58 + 18 v / 5 £v^2 Archimedean Solid Archimedean Solid 67 Solid P "i2 cuboctahedron great rhombicosidodecahedron ivs. 1 1^/30 + 12^ 1^31 + 12 V5 great rhombicuboctahedron IA/12 + 6X/2 I ^13 + 6x72 icosidodecahedron small rhombicosidodecahedron §\/ 5 + 2 >/5 |- V / 11 + 4 ^ \y/io + ±y/z small rhombicuboctahedron J-/4 + 2V2 | ^5 + 2y/2 snub cube * * snub dodecahedron truncated cube * i(2 + v/2) * i ^/V + 4V5 truncated dodecahedron ^(5 + 3^) ^V /t4 + 3 °v / 5 truncated icosahedron ia + V5) JV/58+18X/5 truncated octahedron 3 2 Iv'lO truncated tetrahedron f\/2 IV22 *The complicated analytic expressions for the ClRCUM- RADII of these solids are given in the entries for the SNUB Cube and Snub Dodecahedron. Solid r P R cuboctahedron 0.75 0.86603 1 great rhombicosidodecahedron 3.73665 3.76938 3.80239 great rhombicuboctahedron 2.20974 2.26303 2.31761 icosidodecahedron 1.46353 1.53884 1.61803 small rhombicosidodecahedron 2.12099 2.17625 2.23295 small rhombicuboctahedron 1.22026 1.30656 1.39897 snub cube 1.15766 1.24722 1.34371 snub dodecahedron 2.03987 2.09705 2.15583 truncated cube 1.63828 1.70711 1.77882 truncated dodecahedron 2.88526 2.92705 2.96945 truncated icosahedron 2.37713 2.42705 2.47802 truncated octahedron 1.42302 1.5 1.58114 truncated tetrahedron 0.95940 1.06066 1.17260 The Duals of the Archimedean solids, sometimes called the Catalan Solids, are given in the following table. Archimedean Solid Dual rhombicosidodecahedron small rhombicuboctahedron great rhombicuboctahedron great rhombicosidodecahedron truncated icosahedron snub dodecahedron (laevo) snub cube (laevo) cuboctahedron icosidodecahedron truncated octahedron truncated dodecahedron truncated cube truncated tetrahedron deltoidal hexecontahedron deltoidal icositetrahedron disdyakis dodecahedron disdyakis triacontahedron pentakis dodecahedron pentagonal hexecontahedron (dextro) pentagonal icositetrahedron (dextro) rhombic dodecahedron rhombic triacontahedron tctrakis hexahedron triakis icosahedron triakis octahedron triakis tetrahedron Here are the Archimedean DUALS (Holden 1971, Pearce 1978) displayed in alphabetical order (left to right, then continuing to the next row). Here are the Archimedean solids paired with their DU- ALS. The Archimedean solids and their DUALS are all Canonical Polyiiedra. see also Archimedean Solid Stellation, Cata- lan Solid, Deltahedron, Johnson Solid, Kepler- Poinsot Solid, Platonic Solid, Semiregular Polyhedron, Uniform Polyhedron References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 136, 1987. Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2. Cambridge, MA: MIT Press, pp. 269-286, 1974. Catalan, E. "Memoire sur la Theorie des Polyedres." J. I'Ecole Polytechnique (Paris) 41, 1-71, 1865. Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil Soc. 24, 1-9, 1928. Coxeter, H. S. M. "Regular and Semi- Regular Polytopes I." Math. Z. 46, 380-407, 1940. Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon- don Ser. A 246, 401-450, 1954. Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970. 68 Archimedean Solid Stellation Archimedes' Spiral Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 79-86, 1997. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 54, 1991. Kepler, J. "Harmonice Mundi." Opera Omnia, Vol. 5. Frankfurt, pp. 75-334, 1864. Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 199-207, 1942. Le, Ha. "Archimedean Solids." http : //daisy, uwaterloo. ca/~hqle/archimedean.htnil. Pearce, P. Structure in Nature is a Strategy for Design. Cam- bridge, MA: MIT Press, pp. 34-35, 1978. Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer- sity of California Press, p. 25, 1976. Rawles, B. A. "Platonic and Archimedean Solids — Faces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios." http://www.intent.com/sg/polyhedra.html. Rorres, C. "Archimedean Solids: Pappus." http://www.mcs. drexel.edu/-crorres/Archimedes/Solids/Pappus.html. Steinitz, E. and Rademacher, H. Vorlesungen uber die The- orie der Polyheder. Berlin, p. 11, 1934. Stott, A. B. Verhandelingen der Konniklijke Akad. Weten- schappen, Amsterdam 11, 1910. Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi- Regular Polyhedra." Geometriae Dedicata 1, 117- 123, 1972. Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, 1989. Archimedean Solid Stellation A large class of Polyhedra which includes the Do- DECADODECAHEDRON and GREAT ICOSIDODECAHE- DRON. No complete enumeration (even with restrictive uniqueness conditions) has been worked out. References Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. Lon- don Ser. A 246, 401-450, 1954. Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 66-72, 1989. Archimedean Spiral A Spiral with Polar equation r = a0 1/7n , see also Archimedes' Spiral, Daisy, Fermat's Spi- ral, Hyperbolic Spiral, Lituus, Spiral References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993. Lauweirer, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 59- 60, 1991. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 189, 1972. Lee, X. "Archimedean Spiral." http://www.best.com/-xah/ Special Plane Curves _ dir / Archimedean Spiral _ dir / archimedeanSpiral .html. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967. MacTutor History of Mathematics Archive. "Spiral of Arch- imedes." http: // www - groups . dcs . st - and .ac.uk/ -history/Curves/Spiral.html. Pappas, T. "The Spiral of Archimedes." The Joy of Mathe- matics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 1989. Archimedean Spiral Inverse Curve The Inverse Curve of the Archimedean Spiral 1/77 aO with Inversion Center at the origin and inversion Ra- dius k is the Archimedean Spiral r = ka6 l/m . Archimedes' Spiral An Archimedean Spiral with Polar equation where r is the radial distance, 6 is the polar angle, and m is a constant which determines how tightly the spiral is "wrapped." The Curvature of an Archimedean spiral is given by _ n(9 1 - 1 / n (l + n + n 2 l9 2 ) K ~ a(l + n 2 2 ) 3 / 2 Various special cases are given in the following table. Name lituus hyperbolic spiral Archimedes' spiral Fermat's spiral m -2 -1 1 2 This spiral was studied by Conon, and later by Archi- medes in On Spirals about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral. Archimedes' spiral can be used for COMPASS and Straightedge division of an Angle into n parts (in- cluding Angle Trisection) and can also be used for Circle Squaring. In addition, the curve can be used as a cam to convert uniform circular motion into uni- form linear motion. The cam consists of one arch of the spiral above the cc-AxiS together with its reflection in the z-AxiS. Rotating this with uniform angular veloc- ity about its center will result in uniform linear motion of the point where it crosses the y-AxiS. Archimedes' Spiral Inverse Area-Preserving Map 69 see also ARCHIMEDEAN SPIRAL References Gardner, M. The Unexpected Hanging and Other Mathemat- ical Diversions. Chicago, IL: Chicago University Press, pp. 106-107, 1991. Gray, A- Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186-187, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 173 164, 1967. Archimedes' Spiral Inverse Taking the ORIGIN as the INVERSION CENTER, ARCHI- MEDES' Spiral r = aO inverts to the Hyperbolic Spi- ral r = a/6. Archimedean Valuation A Valuation for which |a;| < 1 Implies |1 + z| < C for the constant C — 1 (independent of x). Such a VALUA- TION does not satisfy the strong TRIANGLE INEQUALITY \x + y\< maxO|,|y|). Arcsecant see Inverse Secant Arcsine see Inverse Sine Calculus and, in particular, the Integral, are power- ful tools for computing the AREA between a curve f(x) and the x-Axis over an INTERVAL [a, 6], giving A = f f(x) J a c)dx. (6) The Area of a Polar curve with equation r = r(8) is A= | fr 2 dO. (7) Written in CARTESIAN COORDINATES, this becomes *-j/(-2-'i)* (8 > -\! (xdy — ydx). (9) For the AREA of special surfaces or regions, see the en- try for that region. The generalization of AREA to 3-D is called Volume, and to higher Dimensions is called Content. see also ARC LENGTH, AREA ELEMENT, CONTENT, Surface Area, Volume References Gray, A. "The Intuitive Idea of Area on a Surface." §13.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 259-260, 1993. Arctangent see Inverse Tangent Area The Area of a Surface is the amount of material needed to "cover" it completely. The AREA of a Trian- gle is given by A A = \lh, (1) where I is the base length and h is the height, or by Heron's Formula Aa = a/ s(s — a)(s — b)($ — c), (2) where the side lengths are a, b, and c and s the Semiperimeter. The Area of a Rectangle is given by ^rectangle a6, (3) where the sides are length a and b. This gives the special case of ^square = & yQ) for the Square. The Area of a regular Polygon with n sides and side length s is given by -^Ti-gon — 4^^ COt I 1 (5) Area Element The area element for a Surface with Riemannian Metric ds 2 = Edu 2 + 2Fdudv + Gdv 2 dA = y^EG - F 2 du A dv, where du A dv is the WEDGE PRODUCT. see also Area, Line Element, Riemannian Metric, Volume Element References Gray, A. "The Intuitive Idea of Area on a Surface." §13.2 in Modern Differential Geometry of Curves and Surfaces, Boca Raton, FL; CRC Press, pp. 259-260, 1993. Area-Preserving Map A Map F from R n to W 1 is AREA-preserving if m{F(A)) = m(A) for every subregion A of M n , where m(A) is the n- D Measure of A. A linear transformation is AREA- preserving if its corresponding DETERMINANT is equal to 1. see also Conformal Map, Symplectic Map 70 Area Principle Area Principle The "AREA principle" states that |i4iP| _ \A,BC\ \A 2 P\ \A 2 BC\' This can also be written in the form [ AiP -1 = \AiBCl IA 2 P\ [A2BCI ' where AB CD (1) (2) (3) is the ratio of the lengths [A, B] and [C, D] for AB\\CD with a PLUS or MINUS SIGN depending on if these seg- ments have the same or opposite directions, and ABC 1 DEFGl (4) is the Ratio of signed Areas of the Triangles. Griinbaum and Shepard show that Ceva'S THEOREM, Hoehn's Theorem, and Menelaus' Theorem are the consequences of this result. see also Ceva's Theorem, Hoehn's Theorem, Men- elaus' Theorem, Self-Transversality Theorem References Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254-268, 1995. Areal Coordinates Trilinear Coordinates normalized so that tl +*2+*3 = 1. When so normalized, they become the AREAS of the Triangles PAiA 2 , PAiA$, and PA 2 A 3 , where P is the point whose coordinates have been specified. Arf Invariant A LINK invariant which always has the value or 1. A Knot has Arf Invariant if the Knot is "pass equivalent" to the UNKNOT and 1 if it is pass equiv- alent to the Trefoil Knot. If iC+, if_, and L are projections which are identical outside the region of the crossing diagram, and K+ and K- are Knots while L is a 2-component LINK with a nonintersecting crossing Argoh's Conjecture diagram where the two left and right strands belong to the different LINKS, then a{K+)=a(K-) + l{L u L 2 ), (1) where I is the Linking Number of L\ and L 2 - The Arf invariant can be determined from the ALEXANDER Polynomial or Jones Polynomial for a Knot. For A K the Alexander Polynomial of K, the Arf invari- ant is given by a*(-i; ■{i (mod 8) 5 (mod 8) if Arf(K) = if Arf(J0 = 1 (2) (Jones 1985). For the Jones Polynomial W K of a Knot K , Arf(K) = W K (i) (3) (Jones 1985), where i is the Imaginary Number. References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223-231, 1994. Jones, V. "A Polynomial Invariant for Knots via von Neu- mann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985. # Weisstein, E. W. "Knots." http://www. astro. Virginia. edu/-eww6n/math/notebooks/Knots.m. Argand Diagram A plot of Complex Numbers as points z = x + iy using the x-AxiS as the Real axis and y-AxiS as the Imaginary axis. This is also called the Complex Plane or Argand Plane. Argand Plane see Argand Diagram Argon's Conjecture Let B k be the fcth BERNOULLI NUMBER. Then does nBn-i = —1 (mod n) Iff n is Prime? For example, for n = 1, 2, . . . , nB n -i (mod n) is 0, -1, -1, 0, -1, 0, -1, 0, -3, 0, -1, .... There are no counterexamples less than n = 5, 600. Any counterexample to Argon's conjecture would be a con- tradiction to Giuga's Conjecture, and vice versa. see also BERNOULLI NUMBER, GlUGA'S CONJECTURE References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996. Argument Addition Relation Aristotle's Wheel Paradox 71 Argument Addition Relation A mathematical relationship relating f(x + y) to f(x) and f(y). see also ARGUMENT MULTIPLICATION RELATION, Recurrence Relation, Reflection Relation, Translation Relation Argument (Complex Number) A Complex Number z may be represented as z = x + iy = \z\e ld , (i) where \z\ is called the Modulus of z, and is called the argument wg(x + iy) = tern' 1 (^j. (2) Therefore, arg(^) = argGzlc^Me"") = oxg(e ie 'e i&v> ) = arg[e i( ^ + ^ } ] = arg(z) + arg(u/). (3) Extending this procedure gives arg(z n ) = narg(z). (4) The argument of a COMPLEX NUMBER is sometimes called the PHASE. see also Affix, Complex Number, de Moivre's Identity, Euler Formula, Modulus (Complex Number), Phase, Phasor References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Argument (Elliptic Integral) Given an Amplitude <f> in an Elliptic Integral, the argument u is defined by the relation 4> = am u. see also Amplitude, Elliptic Integral Argument (Function) An argument of a FUNCTION /(xi,...,x n ) is one of the n parameters on which the function's value de- pends. For example, the SINE since is a one-argument function, the BINOMIAL Coefficient (™) is a two- argument function, and the Hypergeometric Func- tion 2 Fi (a, b; c; z) is a four-argument function. Argument Multiplication Relation A mathematical relationship relating f(nx) to f(x) for Integer n. see also Argument Addition Relation, Recur- rence Relation, Reflection Relation, Transla- tion Relation Argument Principle If f(z) is MEROMORPHIC in a region R enclosed by a curve 7, let N be the number of COMPLEX ROOTS of f(z) in 7, and P be the number of POLES in 7, then J(z)dz 2ttz N J_ [ f'(z)d. 2iriL f(z) Defining w = f(z) and a = / (7) gives 1 f dw 2-xi I w N ■ see also VARIATION OF ARGUMENT References Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Ar- gument Principle for Harmonic Functions." Math. Mag. 103, 411-415, 1996. Argument Variation see Variation of Argument Aristotle's Wheel Paradox <a=® A PARADOX mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concen- tric Circles of different Diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large CIRCLE with points on the small CIRCLE, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. This seems to imply that the two Circumferences of different sized Circles are equal, which is impossible. The fallacy lies in the assumption that a 1:1 correspon- dence of points means that two curves must have the same length. In fact, the CARDINALITIES of points in a Line Segment of any length (or even an Infinite Line, a Plane, a 3-D Space, or an infinite dimensional Euclidean Space) are all the same: Hi (Aleph-1), so the points of any of these can be put in a One-TO-One correspondence with those of any other. see also ZENO'S PARADOXES References Ballew, D. "The Wheel of Aristotle." Math. Teacher 65, 507-509, 1972. Costabel, P. "The Wheel of Aristotle and French Considera- tion of Galileo's Arguments." Math. Teacher 61, 527-534, 1968. Drabkin, I. "Aristotle's Wheel: Notes on the History of the Paradox." Osiris 9, 162-198, 1950. Gardner, M. Wheels, Life, and other Mathematical Amuse- ments. New York: W. H. Freeman, pp. 2-4, 1983. Pappas, T. "The Wheel of Paradox Aristotle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 202, 1989. vos Savant, M. The World's Most Famous Math Problem. New York: St. Martin's Press, pp. 48-50, 1993. 72 Arithmetic Arithmetic The branch of mathematics dealing with Integers or, more generally, numerical computation. Arithmeti- cal operations include Addition, Congruence cal- culation, Division, Factorization, Multiplication, Power computation, Root extraction, and SUBTRAC- TION. The Fundamental Theorem of Arithmetic, also called the Unique Factorization Theorem, states that any Positive Integer can be represented in ex- actly one way as a PRODUCT of PRIMES. The Lowenheimer-Skolem Theorem, which is a fun- damental result in Model Theory, establishes the ex- istence of "nonstandard" models of arithmetic. see also Algebra, Calculus, Fundamental The- orem of Arithmetic, Group Theory, Higher Arithmetic, Linear Algebra, Lowenheimer- Skolem Theorem, Model Theory, Number The- ory, Trigonometry References Karpinski, L. C. The History of Arithmetic. Chicago, IL: Rand, McNally, & Co., 1925. Maxfield, J. E. and Maxfield, M. W. Abstract Algebra and Solution by Radicals. Philadelphia, PA: Saunders, 1992. Thompson, J. E. Arithmetic for the Practical Man. New York: Van Nostrand Reinhold, 1973. Arithmetic-Geometric Mean The arithmetic-geometric mean (AGM) M(a, b) of two numbers a and b is defined by starting with clq = a and bo = &, then iterating CLn + l = 2 ( a ™ + kn) b n + l = yCLnbn (1) (2) until a n = b n . a n and b n converge towards each other since a n +i - b n +i = \{a n -\- b n ) - ydnb n a n — 2\/a n b n + b n (3) But "s/Sn < V^"» SO 2b n < 2^a n b n . (4) Now, add a n — b n — 2y/a n b n to each side a n + b n — 2\/a n b n < a n — b n > (5) CLn + l - b n + l < 2^ an ~ bn)- (6) The AGM is very useful in computing the values of complete Elliptic Integrals and can also be used for finding the INVERSE TANGENT. The special value l/M(l,\/2) is called Gauss's Constant. Arithmetic- Geometric Mean The AGM has the properties AM(a,6) = M(Aa,A6) (7) M(a,6) = M(£(a + 6),>/S) (8) M(l, V 1 - x 2 ) = M (! + s, 1 ~ x) (9) The Legendre form is given by Af(l,x) = JJi(l + fc„), (11) where ko = x and Solutions to the differential equation (x 3 -z)^| + (3x 2 - l)^-+xy = (13) ax* ax are given by [M(l + x, 1 - x)] 1 and[M(l,x)] 1 . A generalization of the Arithmetic-Geometric Mean is f°° x p ~ 2 dx (14) which is related to solutions of the differential equation x(l-x p )Y" + [l-fr+l)x p ]Y'-(p-l)x p - 1 Y = 0. (15) When p = 2 or p = 3, there is a modular transformation for the solutions of (15) that are bounded as x — > 0. Let- ting J p (x) be one of these solutions, the transformation takes the form J p (\) = M J p (z), (16) where A 1-u and 1 + (p - l)u 1 + (p - l)u x p + u p = 1. (17) (18) (19) The case p = 2 gives the Arithmetic- Geometric Mean, and p = 3 gives a cubic relative discussed by Borwein and Borwein (1990, 1991) and Borwein (1996) in which, for a, b > and I(a, b) defined by /(a, 6) Jo V& + tdt *3)(6 3 +f 3 ) 2 ] 1 / 3 ' (20) Arithmetic Geometry Iia , b) = l(^,[^+ab + b 2 )]). (21) For iteration with ao = a and bo = 6 and a n +i a n + 26 n fcn + l = — (dn + fln&Ti + b n ) lim a n = lim b n Hhi) I(a,b)- (22) (23) (24) Modular transformations are known when p = 4 and p = 6, but they do not give identities for p = 6 (Borwein 1996). see also Arithmetic-Harmonic Mean References Abramowitz, M. and Stegun, C. A. (Eds.), "The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 571 ad 598-599, 1972. Borwein, J. M. Problem 10281. "A Cubic Relative of the AGM." Amer, Math. Monthly 103, 181-183, 1996. Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic It- eration." In Computational Method & Function Theory: Proc. Conference Held in Valparaiso, Chile, March 13- 18, i9SP0387527680 (Ed. A. Dold, B. Eckmann, F. Tak- ens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C, and R, S. Varga). New York: Springer- Vcrlag, 1990. Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of Jacobi's Identity and the AGM." Trans. Amer. Math. Soc. 323, 691-701, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 906-907, 1992. Arithmetic Geometry A vaguely defined branch of mathematics dealing with Varieties, the Mordell Conjecture, Arakelov Theory, and Elliptic Curves. References Cornell, G. and Silverman, J. H. (Eds.). Arithmetic Geome- try. New York: Springer- Verlag, 1986. Lorenzini, D. An Invitation to Arithmetic Geometry. Provi- dence, RI: Amer. Math. Soc, 1996. Arithmetic-Harmonic Mean Let a n+ x = \{a n + b n ) b n +i a n + b n Then A(ao,bo) = lim a n = lim b n n — ► oo n— »-oo which is just the GEOMETRIC MEAN. (1) (2) 'aobo, (3) Arithmetic Mean 73 Arithmetic-Logarithmic-Geometric Mean Inequality a + b b — a i—r —— > — — : — > Vab. 2 In o — In a see also Napier's Inequality References Nelson, R. B. "Proof without Words: The Arithmetic- Logarithmic-Geometric Mean Inequality." Math. Mag. 68, 305, 1995. Arithmetic Mean For a Continuous Distribution function, the arith- metic mean of the population, denoted /*, x, {x) t or A(x) t is given by -/. H=(f(x))= / P(x)f(x)dx, (1) where (x) is the EXPECTATION VALUE. For a DISCRETE Distribution, „ = </(*)> ss E ^°/ (a " )/( ; n) = 5><*.)/<*.). l^n = Q F V Xn ) n=0 (2) The population mean satisfies {f(x)+g(x)) = {f(x)) + (g(x)} (3) <c/(x))=c </(*)>, (4) and {f(x)g(y)) = </(*)> (g(y)) (5) if x and y are INDEPENDENT STATISTICS. The "sample mean," which is the mean estimated from a statistical sample, is an UNBIASED ESTIMATOR for the population mean. For small samples, the mean is more efficient than the Median and approximately tt/2 less (Kenney and Keep- ing 1962, p. 211). A general expression which often holds approximately is mean — mode « 3(mean — median). (6) Given a set of samples {a;*}, the arithmetic mean is N A(x) =x = hee{x) = ^^2 Xi ' W Hoehn and Niven (1985) show that * A(a± +c,a 2 +c, . ..,a n +c) = c + j4(ai,a2,...,a n ) (8) for any POSITIVE constant c. The arithmetic mean sat- isfies (9) 74 Arithmetic Mean Arithmetic Progression where G is the Geometric Mean and H is the Har- monic Mean (Hardy et al. 1952; Mitrinovic 1970; Beck- enbach and Bellman 1983; Bullen et ah 1988; Mitrinovic et al. 1993; Alzer 1996). This can be shown as follows. For a, b > 0, P--^Y> 1 2 1 rt 1 1^2 - + r > -7= a ~ b H>G, (10) (11) (12) (13) (14) with equality Iff b = a. To show the second part of the inequality, {yfa-Vbf = a-2\/a6 + &> (15) <> + b > Vab 2 A> H< (16) (17) with equality Iff a = b. Combining (14) and (17) then gives (9). Given n independent random GAUSSIAN DISTRIBUTED variates #», each with population mean fii = \i and Variance <n 2 = a 2 , = ^E^=^)=/i, (19) Z = l so the sample mean is an Unbiased Estimator of population mean. However, the distribution of x de- pends on the sample size. For large samples, x is ap- proximately Normal. For small samples, Student's ^-Distribution should be used. The Variance of the population mean is independent of the distribution. var(z) = var I - > Xi\ = —— var > x { n N 2 From /c-Statistics for a GAUSSIAN DISTRIBUTION, the Unbiased Estimator for the Variance is given by N N - 1 where var (a:) = JV-1 The Square Root of this, s is called the Standard Error. var(x) = (% 2 ) ~~ (^) 2 > (21) (22) (23) (24) (25) (x 2 )=var(x) + (a-) 2 = ^+/A (26) see also Arithmetic-Geometric Mean, Arith- metic-Harmonic Mean, Carleman's Inequal- ity, Cumulant, Generalized Mean, Geomet- ric Mean, Harmonic Mean, Harmonic-Geometric Mean, Kurtosis, Mean, Mean Deviation, Median (Statistics), Mode, Moment, Quadratic Mean, Root-Mean-Square, Sample Variance, Skewness, Standard Deviation, Trimean, Variance References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Alzer, H. "A Proof of the Arithmetic Mean-Geometric Mean Inequality." Amer. Math. Monthly 103, 585, 1996. Beckenbach, E. F. and Bellman, R. Inequalities. New York: Springer- Verlag, 1983. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 471, 1987. Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means & Their Inequalities. Dordrecht, Netherlands: Reidel, 1988. Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952. Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962. Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. Dordrecht, Netherlands: Kluwer, 1993. Vasic, P. M. and Mitrinovic, D. S. Analytic Inequalities. New York: Springer- Verlag, 1970. Arithmetic Progression see Arithmetic Series (20) Arithmetic Sequence Arnold's Cat Map 75 Arithmetic Sequence A Sequence of n numbers {do 4- kd} 7 ^ such that the differences between successive terms is a constant d. see also ARITHMETIC SERIES, SEQUENCE Arithmetic Series An arithmetic series is the Sum of a SEQUENCE {a^}, k = 1, 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant. Therefore, for k > 1, a>k = a,k-i + d = afc-2 + 2d • :ai+d(fc-l). (1) The sum of the sequence of the first n terms is then given by n n S n = ]Ta fc =J^[ai + (* - l)d] k=l k=l n n = nai + d^ik — 1) = noi + d /(& - 1) fc = l k = 2 n-l = nai -\- dj k (2) Using the SUM identity ]T = §n(n+l) (3) then gives S n = nai + \d(n - 1) = \ n[2ai + d(n - 1)]. (4) Note, however, that ai + a n = ai + [a\ + d(n — 1)] — 2ai + d(n — 1), (5) 5 n = \n{a\ +a n ), (6) or n times the AVERAGE of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the INTEGERS from 1 to 100 given as busy-work by his teacher. While his classmates toiled away doing the ADDITION longhand, Gauss wrote a single number, the correct answer |(100)(1 + 100) = 50 ■ 101 = 5050 (7) on his slate. When the answers were examined, Gauss's proved to be the only correct one. see also Arithmetic Sequence, Geometric Series, Harmonic Series, Prime Arithmetic Progression References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Beyer, W. H. (Ed.), CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. Courant, R. and Robbins, H. "The Arithmetical Progres- sion." §1.2.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 12-13, 1996. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 164, 1989. Armstrong Number The n-digit numbers equal to sum of nth powers of their digits (a finite sequence), also called PLUS PERFECT NUMBERS. They first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane's A005188). see also NARCISSISTIC NUMBER References Sloane, N. J. A. Sequence A005188/M0488 in "An On-Line Version of the Encyclopedia of Integer Sequences." Arnold's Cat Map The best known example of an ANOSOV DlFFEOMOR- PHISM. It is given by the TRANSFORMATION (i) where x n +i and y n +i are computed mod 1. The Arnold cat mapping is non-Hamiltonian, nonanalytic, and mix- ing. However, it is Area- Preserving since the Deter- minant is 1. The Lyapunov Characteristic Expo- nents are given by Xn+l = "l l" 1 2 x n y n _ l-a 1 1 2-<T 3(7 + 1 = 0, (2) ct± = |(3±v / 5). (3) The Eigenvectors are found by plugging <r± into the Matrix Equation 1 1 2 - cr± (4) For <r+, the solution is y=\{l + ^)x = 4>x, (5) where <j> is the GOLDEN RATIO, so the unstable (normal- ized) Eigenvector is ^+ = ^\ / 50-10v / 5 1(1 + V5) Similarly, for <j- , the solution is y = -±(V5-l)x~(/>- 1 x y so the stable (normalized) Eigenvector is £_ = ^\/50 + 10v / 5 see also Anosov Map 1(1 -v/5) (6) (7) (8) 76 Arnold Diffusion Array Arnold Diffusion The nonconservation of ADIABATIC INVARIANTS which arises in systems with three or more DEGREES OF FREE- DOM. Arnold Tongue Consider the Circle Map. If K is Nonzero, then the motion is periodic in some FINITE region surround- ing each rational Q. This execution of periodic motion in response to an irrational forcing is known as MODE LOCKING. If a plot is made of K versus Q with the re- gions of periodic MODE-LOCKED parameter space plot- ted around rational ft values (the WINDING Numbers), then the regions are seen to widen upward from at K = to some FINITE width at K = 1. The region surrounding each RATIONAL NUMBER is known as an Arnold Tongue. At K — 0, the Arnold tongues are an isolated set of MEASURE zero. At K = 1, they form a general CAN- TOR Set of dimension d w 0.8700. In general, an Arnold tongue is defined as a resonance zone emanating out from RATIONAL NUMBERS in a two-dimensional param- eter space of variables. see also Circle Map Aronhold Process The process used to generate an expression for a covari- ant in the first degree of any one of the equivalent sets of Coefficients for a curve. see also Clebsch-Aronhold Notation, Joachims- thal's Equation References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- York: Dover, p. 74, 1959. Aronson's Sequence The sequence whose definition is: "t is the first, fourth, eleventh, . . . letter of this sentence." The first few val- ues are 1, 4, 11, 16, 24, 29, 33, 35, 39, ... (Sloane's A005224). References Hofstadter, D. R. Metamagical Themas: Questing of Mind and Pattern. New York: BasicBooks, p. 44, 1985. Sloane, N. J. A. Sequence A005224/M3406 in "An On-Line Version of the Encyclopedia of Integer Sequences." Arrangement In general, an arrangement of objects is simply a group- ing of them. The number of "arrangements" of n items is given either by a COMBINATION (order is ignored) or Permutation (order is significant). The division of Space into cells by a collection of Hy- PERPLANES is also called an arrangement. see also COMBINATION, CUTTING, HYPERPLANE, OR- DERING, Permutation Arrangement Number see Permutation Array An array is a "list of lists" with the length of each level of list the same. The size (sometimes called the "shape") of a d-dimensional array is then indicated as m x n x • • • x p. The most common type of array en- d countered is the 2-D m x n rectangular array having m columns and n rows. If m = n, a square array results. Sometimes, the order of the elements in an array is sig- nificant (as in a MATRIX), whereas at other times, arrays which are equivalent modulo reflections (and rotations, in the case of a square array) are considered identical (as in a MAGIC SQUARE or PRIME ARRAY). In order to exhaustively list the number of distinct ar- rays of a given shape with each element being one of k possible choices, the naive algorithm of running through each case and checking to see whether it's equivalent to an earlier one is already just about as efficient as can be. The running time must be at least the number of answers, and this is so close to k mn '" p that the difference isn't significant. However, finding the number of possible arrays of a given shape is much easier, and an exact formula can be ob- tained using the POLYA ENUMERATION THEOREM. For the simple case of an m x n array, even this proves un- necessary since there are only a few possible symmetry types, allowing the possibilities to be counted explicitly. For example, consider the case of m and n EVEN and distinct, so only reflections need be included. To take a specific case, let m = 6 and n = 4 so the array looks like a b c 1 d e f 9 h i 1 + 1 3 k I m n V Q r s t u 1 V w X, where each a, 6, . . . , x can take a value from 1 to k. The total number of possible arrangements is k 24 (k mn in general). The number of arrangements which are equiv- alent to their left-right mirror images is k 1 (in general, k mn/2 ), as is the number equal to their up-down mirror images, or their rotations through 180°. There are also k Q arrangements (in general, fc mn/4 ) with full symmetry. In general, it is therefore true that jL7TiTl/4 j^mn/2 _ fcmn/4 femn/2 _ pn/4 femn/2 _ femn/4 with full symmetry with only left-right reflection with only up-down reflection with only 180° rotation, so there are 3k ran/2 , rw mn./4 Arrow Notation Artin Braid Group 77 arrangements with no symmetry. Now dividing by the number of images of each type, the result, for m -£ n with m, n EVEN, is N(m,n,k) = |A; mn + (|)(3)(fc mn/2 - A; mn/4 ) mn i + 1*" 4. \(k mn -3k mn/2 + 2k mn/4 ) 4\ lfcrnn _,_ 3 j.mn/2 _,_ lj.mn/4 + ifc" The number is therefore of order C>(fc mn /4), with "cor- rection" terms of much smaller order. see also Antimagic Square, Euler Square, Kirkman's Schoolgirl Problem, Latin Rect- angle, Latin Square, Magic Square, Matrix, Mrs. Perkins' Quilt, Multiplication Table, Or- thogonal Array, Perfect Square, Prime Array, Quotient-Difference Table, Room Square, Sto- larsky Array, Truth Table, Wythoff Array Arrow Notation A Notation invented by Knuth (1976) to represent Large Numbers in which evaluation proceeds from the right (Conway and Guy 1996, p. 60). m t n m * m- - -m n m ttt n m tt m tt ■ ' ' tt m n For example, m t n ~ m n m"[ J \-2 = m J [m~m^m = rn 71 2 m tt 3 = m t rn t m = m t ( m t rn) v v / 3 = m|m m = m mm m ttt 2 = mtt^ = ^tt^ = mm m ttt 3 = m tt rn tt ™ = m tt ™ m v v ' ' ^~ 3 m = m t • • * t rn — m m (1) (2) (3) (4) (5) m tt n is sometimes called a Power Tower. The values nt • * • t n are called ACKERMANN NUMBERS. see also Ackermann Number, Chained Arrow No- tation, Down Arrow Notation, Large Number, Power Tower, Steinhaus-Moser Notation References Conway, J. H. and Guy, R. K. The Book of Numbers. New- York: Springer- Verlag, pp. 59-62, 1996. Guy, R. K. and Selfridge, J. L. "The Nesting and Roost- ing Habits of the Laddered Parenthesis." Amer. Math. Monthly 80, 868-876, 1973. Knuth, D. E. "Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations." Science 194, 1235-1242, 1976. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison- Wesley, pp. 11 and 226-229, 1991. Arrow's Paradox Perfect democratic voting is, not just in practice but in principle, impossible. References Gardner, M. Time Travel and Other Mathematical Bewilder- ments. New York: W. H. Freeman, p. 56, 1988. Arrowhead Curve see Sierpinski Arrowhead Curve Art Gallery Theorem Also called Chvatal's Art Gallery Theorem. If the walls of an art gallery are made up of n straight Lines Segments, then the entire gallery can always be supervised by [n/3\ watchmen placed in corners, where [x\ is the Floor Function. This theorem was proved by V. Chvatal in 1973. It is conjectured that an art gallery with n walls and h HOLES requires [(n + h)/3j watchmen. see also Illumination Problem References Honsberger, R. "Chvatal's Art Gallery Theorem." Ch. 11 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 104-110, 1976. O'Ronrke, J. Art Gallery Theorems and Algorithms. New- York: Oxford University Press, 1987. Stewart, I. "How Many Guards in the Gallery?" Sci. Amer. 270, 118-120, May 1994. Tucker, A. "The Art Gallery Problem." Math Horizons, pp. 24-26, Spring 1994. Wagon, S. "The Art Gallery Theorem." §10.3 in Mathema- tica in Action. New York: W. H. Freeman, pp. 333-345, 1991. Articulation Vertex A VERTEX whose removal will disconnect a GRAPH, also called a Cut- Vertex. see also Bridge (Graph) References Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introduc- tory Graph Theory. New York: Dover, pp. 45—49, 1985. Artin Braid Group see Braid Group 78 Artin's Conjecture Artistic Series Artin's Conjecture There are at least two statements which go by the name of Artin's conjecture. The first is the RlEMANN HY- POTHESIS. The second states that every INTEGER not equal to —1 or a SQUARE NUMBER is a primitive root modulo p for infinitely many p and proposes a density for the set of such p which are always rational multi- ples of a constant known as ARTIN'S CONSTANT. There is an analogous theorem for functions instead of num- bers which has been proved by Billharz (Shanks 1993, p. 147). see also ARTIN'S CONSTANT, RlEMANN HYPOTHESIS References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 31, 80-83, and 147, 1993. Artin's Constant If n / -1 and n is not a PERFECT SQUARE, then Artin conjectured that the SET S(n) of all PRIMES for which n is a PRIMITIVE ROOT is infinite. Under the assumption of the Extended Riemann Hypothesis, Artin's con- jecture was solved in 1967 by C. Hooley. If, in addition, n is not an rth POWER for any r > 1, then Artin con- jectured that the density of S(n) relative to the Primes is CArtin (independent of the choice of n) , where CAn n 1- 1 <?(<?-!) = 0.3739558136.. and the PRODUCT is over Primes. The significance of this constant is more easily seen by describing it as the fraction of PRIMES p for which 1/p has a maximal DEC- IMAL Expansion (Conway and Guy 1996). References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 169, 1996. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/artin/artin.html. Hooley, C. "On Artin's Conjecture." J. reine angew. Math. 225, 209-220, 1967. Ireland, K. and Rosen, M. A Classical Introduction to Mod- ern Number Theory, 2nd ed. New York: Springer-Verlag, 1990. Ribenboim, P. The Book of Prime Number Records. New York: Springer-Verlag, 1989. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 80-83, 1993. Wrench, J. W. "Evaluation of Artin's Constant and the Twin Prime Constant." Math. Comput. 15, 396-398, 1961. Artin L- Function An Artin L-function over the Rationals Q encodes in a Generating Function information about how an irreducible monic POLYNOMIAL over Z factors when re- duced modulo each PRIME. For the POLYNOMIAL x 2 + l, the Artin L-function is L(s,Q(i)/Q,sgn): n itt^ where ( — 1/p) is a Legendre Symbol, which is equiv- alent to the Euler L-FUNCTION. The definition over arbitrary POLYNOMIALS generalizes the above expres- sion. see also Langlands Reciprocity References Knapp, A. W. "Group Representations and Harmonic Anal- ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996. Artin Reciprocity see Artin's Reciprocity Theorem Artin's Reciprocity Theorem A general RECIPROCITY Theorem for all orders. If R is a NUMBER FIELD and R f a finite integral extension, then there is a SURJECTION from the group of fractional IDEALS prime to the discriminant, given by the Artin symbol. For some cycle c, the kernel of this SURJECTION contains each Principal fractional Ideal generated by an element congruent to 1 mod c. see also LANGLANDS PROGRAM Artinian Group A GROUP in which any decreasing CHAIN of distinct Subgroups terminates after a Finite number. Artinian Ring A noncommutative Semisimple RING satisfying the "descending chain condition." see also GORENSTEIN RING, SEMISIMPLE RING References Artin, E. "Zur Theorie der hyperkomplexer Zahlen." Hamb. Abh. 5, 251-260, 1928. Artin, E. "Zur Arithmetik hyperkomplexer Zahlen." Hamb. Abh. 5, 261-289, 1928. Artistic Series A Series is called artistic if every three consecutive terms have a common three-way ratio -P[ai,ai+i,a; + 2] (ai + aj+i + ai+2)ai+i aiOi+2 A Series is also artistic Iff its BIAS is a constant. A Geometric Series with Ratio r > is an artistic series with P=i + l + r>3. r see also BIAS (SERIES), GEOMETRIC SERIES, MELODIC Series References Duffin, R. J. "On Seeing Progressions of Constant Cross Ra- tio." Amer. Math. Monthly 100, 38-47, 1993. p odd prime (?)*" ASA Theorem ASA Theorem Specifying two adjacent ANGLES A and B and the side between them c uniquely determines a Triangle with Area K = 2(coti4 + cot£)" The angle C is given in terms of A and B by C = n-A-B, (1) (2) and the sides a and b can be determined by using the Law of Sines to obtain sin A sin B sin C sin A sin(7r — A — B) sinB sin(7r — A — B) (3) (4) (5) see also AAA Theorem, AAS Theorem, ASS Theo- rem, SAS Theorem, SSS Theorem, Triangle Aschbacher's Component Theorem Suppose that E(G) (the commuting product of all com- ponents of G) is SIMPLE and G contains a SEMISIM- ple Involution. Then there is some Semisimple Involution x such that C G (x) has a Normal Sub- group K which is either QUASISIMPLE or ISOMORPHIC to + (4,q)' and such that Q — C G {K) is Tightly Em- bedded. see also Involution (Group), Isomorphic Groups, Normal Subgroup, Quasisimple Group, Simple Group, Tightly Embedded ASS Theorem c c c Specifying two adjacent side lengths a and b of a TRIAN- GLE (taking a > b) and one ACUTE ANGLE A opposite a does not, in general, uniquely determine a triangle. If sin A < a/cy there are two possible TRIANGLES satis- fying the given conditions. If sin A = a/c, there is one possible Triangle. If sin A > a/c, there are no possible TRIANGLES. Remember: don't try to prove congruence with the ASS theorem or you will make make an ASS out of yourself. see also AAA Theorem, AAS Theorem, SAS Theo- rem, SSS Theorem, Triangle Associative Magic Square 79 Associative In simple terms, let x, y, and z be members of an Al- gebra. Then the Algebra is said to be associative if x - (y - z) = (x - y) • z, (1) where • denotes MULTIPLICATION. More formally, let A denote an IR-algebra, so that A is a VECTOR SPACE over Rand Ax A-+ A (2) (x,y) \->x-y. (3) Then A is said to be m-associative if there exists an m-D Subspace S of A such that (y-x)-z = y-(x-z) (4) for all y,z € A and x € S. Here, VECTOR MULTIPLI- CATION x • y is assumed to be Bilinear. An n-D n- associative ALGEBRA is simply said to be "associative." see also COMMUTATIVE, DISTRIBUTIVE References Finch, S. "Zero Structures in Real Algebras." http://www. mathsoft.com/asolve/zerodiv/zerodiv.html. Associative Magic Square 1 15 24 8 17 23 7 16 5 14 20 4 13 22 6 12 21 10 19 3 9 18 2 11 25 An n x n Magic Square for which every pair of num- bers symmetrically opposite the center sum to n 2 + 1. The Lo Shu is associative but not PANMAGIC. Order four squares can be PANMAGIC or associative, but not both. Order five squares are the smallest which can be both associative and PANMAGIC, and 16 distinct asso- ciative PANMAGIC Squares exist, one of which is illus- trated above (Gardner 1988). see also Magic Square, Panmagic Square References Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H, Freeman, 1988. 80 Astroid Astroid A 4-cusped HYPOCYCLOID which is sometimes also called a Tetracuspid, Cubocycloid, or Paracycle. The parametric equations of the astroid can be obtained by plugging in n = a/b = 4 or 4/3 into the equations for a general HYPOCYCLOID, giving x = 3bcos(j> + 6cos(30) = 46 cos 3 <j> — a cos 3 <j) (1) y — 36 sin ^ — bs'm(3<f>) = 46 sin <j) = asin <j>. (2) In Cartesian Coordinates, 2/3 . 2/3 2/3 (3) In Pedal Coordinates with the Pedal Point at the center, the equation is 2 , o 2 2 r + op — a . (4) v J I J r ^ n The Arc Length, Curvature, and Tangential An- gle are s(t) / |sin(2*')|d*' Jo f sin 2 t (5) K(t) = -|csc(2t) 4>(t) = -t. (6) (7) As usual, care must be taken in the evaluation of s(t) for t > it/ 2. Since (5) comes from an integral involving the Absolute Value of a function, it must be monotonic increasing. Each QUADRANT can be treated correctly by defining '»=[fj+l. (8) where [x] is the FLOOR FUNCTION, giving the formula S (t) = (-l) 1+ l" < mod 2 » | sin 2 t + 3 [|nj . (9) The overall Arc Length of the astroid can be com- puted from the general HYPOCYCLOID formula &a(n - 1) (10) with n = 4, 54 = 6a. Astroid (ll) The Area is given by An = ("-D("-2) ro . with n = 4, I- 2 (12) (13) The Evolute of an Ellipse is a stretched Hypocy- CLOID. The gradient of the TANGENT T from the point with parameter p is — tan p. The equation of this TAN- GENT T is xsinp + ycosp = |asin(2p) (14) (MacTutor Archive). Let T cut the z-Axis and the y- Axis at X and Y, respectively. Then the length XY is a constant and is equal to a. t L The astroid can also be formed as the ENVELOPE pro- duced when a Line Segment is moved with each end on one of a pair of PERPENDICULAR axes (e.g., it is the curve enveloped by a ladder sliding against a wall or a garage door with the top corner moving along a verti- cal track; left figure above). The astroid is therefore a GLISSETTE. To see this, note that for a ladder of length L, the points of contact with the wall and floor are (xo,0) and (0, y/L 2 — xq 1 ), respectively. The equa- tion of the Line made by the ladder with its foot at (xojO) is therefore VL 2 - xq 2 , . y — = (x - xo) -xq (15) which can be written U{x,y,xo) = y + — (x-x ). (16) Xo The equation of the Envelope is given by the simulta- neous solution of U(x,y,x ) = y+ V L xn X ° (x-xo) = ax o *oV^ 2 -*o 2 -0, which is Xq I? (L 2 - xq 2 ) 3 / 2 L 2 (17) (18) (19) Astroid Noting that ~ 2 2/3 _ ^0 ~ L 4 /3 (20) r 2 2 2/3 _ -k — x y ~~ TAlZ (21) Astroid Involute 81 allows this to be written implicitly as x 2/3 +y 2/3 = L 2/3 , the equation of the astroid, as promised. A /~\ slotted track ^ (22) The related problem obtained by having the "garage door" of length L with an "extension" of length AL move up and down a slotted track also gives a surprising answer. In this case, the position of the "extended" end for the foot of the door at horizontal position xo and Angle 6 is given by x = — ALcosO y = \/L 2 - xo 2 + ALsin0. (23) (24) The astroid is also the Envelope of the family of El- lipses y (1 - c)'< -1 = 0, (30) illustrated above. see also Deltoid, Ellipse Envelope, Lame Curve, Nephroid, Ranunculoid References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172-175, 1972. Lee, X. "Astroid." http://www.best.com/-xah/Special PlaneCurves_dir/Astroid_dir/astroid.html. Lockwood, E. H. "The Astroid." Ch. 6 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 52- 61, 1967. MacTutor History of Mathematics Archive. "Astroid." http: //www-groups .dcs .st-and.ac.uk/-history/Curves /Astroid. html. Yates, R. C. "Astroid." A Handbook on Curves and Their Properties. Ann Arbor, Ml: J. W. Edwards, pp. 1-3, 1952. Using then gives xq = L cos AL x = —xo = yz^^( 1 + ^) (25) (26) (27) Astroid Evolute Solving (26) for xo, plugging into (27) and squaring then gives (Ai) 2 Rearranging produces the equation 2 f2 Z/V / ALV y (AL) 2 (L + AL) 2 (28) (29) the equation of a (QUADRANT of an) ELLIPSE with Semimajor and Semiminor Axes of lengths AL and L + AL. A Hypocycloid Evolute for n = 4 is another As- troid scaled by a factor n/(n — 2) = 4/2 = 2 and rotated 1/(2 • 4) = 1/8 of a turn. Astroid Involute V / \ / \ / \ /V- --7V y \ / ^ -- ^, '^i- 4."" \ \ \ / / / / / A Hypocycloid Involute for n = 4 is another As- troid scaled by a factor (n — 2)/2 = 2/4 = 1/2 and rotated 1/(2 • 4) = 1/8 of a turn. 82 Astroid Pedal Curve Astroid Pedal Curve Asymptotic Curve The Pedal Curve of an Astroid with Pedal Point at the center is a QUADRIFOLIUM. Astroid Radial Curve The QUADRIFOLIUM x = xo + 3a cos t — 3a cos(3£) y = y -\- 3a sin t + 3a sin(3t) . Astroidal Ellipsoid The surface which is the inverse of the ELLIPSOID in the sense that it "goes in" where the ELLIPSOID "goes out." It. is given by the parametric equations x = (acosticosv) y = (b sin u cost;) 3 z — (csinv) 3 for u € [— 7r/2,7r/2] and v G [— 7r,7r]. The special case a = b = c = 1 corresponds to the HYPERBOLIC OCTA- HEDRON. see also Ellipsoid, Hyperbolic Octahedron References Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/ people/nfytn/ asttxt.htm. Asymptosy Asymptotic behavior. A useful yet endangered word, found rarely outside the captivity of the Oxford English Dictionary. see also ASYMPTOTE, ASYMPTOTIC Asymptote asymptotes A curve approaching a given curve arbitrarily closely, as illustrated in the above diagram. see also ASYMPTOSY, ASYMPTOTIC, ASYMPTOTIC Curve References Giblin, P. J. "What is an Asymptote?" Math. Gaz. 56, 274-284, 1972. Asymptotic Approaching a value or curve arbitrarily closely (i.e., as some sort of Limit is taken). A Curve A which is asymptotic to given CURVE C is called the ASYMPTOTE of C. see also ASYMPTOSY, ASYMPTOTE, ASYMPTOTIC Curve, Asymptotic Direction, Asymptotic Se- ries, Limit Asymptotic Curve Given a Regular Surface M, an asymptotic curve is formally defined as a curve x(i) on M such that the Normal Curvature is in the direction x'(t) for all t in the domain of x. The differential equation for the parametric representation of an asymptotic curve is eu -\-2fuv + gv = 0, (i) where e, /, and g are second FUNDAMENTAL FORMS. The differential equation for asymptotic curves on a Monge Patch (u,v,h(u t v)) is h uu u + 2h U uU v + h vv v = 0, and on a polar patch (r cos0,rsin#, h(r)) is ti'(r)r ,2 +ti{r)rd' 2 =0. (2) (3) The images below show asymptotic curves for the EL- LIPTIC Helicoid, Funnel, Hyperbolic Paraboloid, and Monkey Saddle. Asymptotic Direction Atiyah-Singer Index Theorem 83 see also RULED SURFACE References Gray, A. "Asymptotic Curves," "Examples of Asymp- totic Curves," "Using Mathematica to Find Asymptotic Curves." §16.1, 16.2, and 16.3 in Modern Differential Ge- ometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp, 320-331, 1993. Asymptotic Direction An asymptotic direction at a point p of a REGULAR Surface M e M 3 is a direction in which the NORMAL Curvature of M vanishes. 1. There are no asymptotic directions at an Elliptic Point. 2. There are exactly two asymptotic directions at a HY- PERBOLIC Point. 3. There is exactly one asymptotic direction at a PAR- ABOLIC Point. 4. Every direction is asymptotic at a Planar Point. see also ASYMPTOTIC CURVE References Gray, A. Modern Differential Geometry of Curves and Sur- faces.Boca, Raton, FL: CRC Press, pp. 270 and 320, 1993. Asymptotic Notation Let n be a integer variable which tends to infinity and let xbea continuous variable tending to some limit. Also, let 4>(n) or (j){x) be a positive function and f(n) or f{x) any function. Then Hardy and Wright (1979) define 1. / = 0{(j>) to mean that |/| < A<f> for some constant A and all values of n and x y 2. f = o(<j>) to mean that f/<j> — y 0, 3. / ~ <j> to mean that f /<j> — > 1, 4. / -< <j> to mean the same as / = o((f>) 7 5. f y </> to mean f/<j> — > oo, and 6. / x <fc to mean Ai<j> < / < A 2 for some positive constants A± and A 2 . f = o(<j>) implies and is stronger than / = 0(<}>). References Hardy, G. H. and Wright, E. M. "Some Notation." §1.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7-8, 1979. Asymptotic Series An asymptotic series is a SERIES EXPANSION of a FUNC- TION in a variable x which may converge or diverge (Erdelyi 1987, p. 1), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough x. To form an asymptotic series R(x) of /(#), written /(*) ~ R(x), (1) where c / \ — i ai i a2 i i an S n (x) = a H h -=■ + ... + — -■ (3) The asymptotic series is defined to have the properties lim x n R n (x) = for fixed n (4) lim x n R n (x) = oo for fixed x. (5) Therefore, f(x) « 22 anX (6) in the limit x — > oo. If a function has an asymptotic expansion, the expansion is unique. The symbol ~ is also used to mean directly Similar. References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 15, 1972. Arfken, G. "Asymptotic of Semiconvergent Series." §5.10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339-346, 1985. Bleistein, N. and Handelsman, R. A. Asymptotic Expansions of Integrals. New York: Dover, 1986. Copson, E. T. Asymptotic Expansions. Cambridge, England: Cambridge University Press, 1965. de Bruijn, N. G. Asymptotic Methods in Analysis, 2nd ed. New York: Dover, 1982. Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation. London: Academic Press, 1973. Erdelyi, A. Asymptotic Expansions. New York: Dover, 1987. Morse, P. M. and Feshbach, H. "Asymptotic Series; Method of Steepest Descent." §4.6 in Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 434-443, 1953. Olver, F. W. J. Asymptotics and Special Functions. New York: Academic Press, 1974. Wasow, W. R. Asymptotic Expansions for Ordinary Differ- ential Equations. New York: Dover, 1987. Atiyah-Singer Index Theorem A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-D Compact Differentiable C°° boundary- less Manifold. see also Compact Manifold, Differentiable Man- ifold References Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Op- erators on Compact Manifolds." Bull. Amer. Math. Soc. 69, 322-433, 1963. Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Oper- ators I, II, III." Ann. Math. 87, 484-604, 1968. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, p. 4, 1996. take x n R n (x)=x n [f(x)-S n (x)], (2) 84 Atkin-Goldwasser-Kilian-Morain Certificate Augmented Amicable Pair Atkin-Goldwasser-Kilian-Morain Certificate A recursive PRIMALITY CERTIFICATE for a PRIME p. The certificate consists of a list of 1. A point on an ELLIPTIC CURVE C y 2 - x 3 + 92X + p 3 (mod p) for some numbers £2 and #3- 2. A Prime g with q > (p 1 ^ 4 + l) 2 , such that for some other number k and m = kq with k ^ 1, mC{X)y,g2 ) g$ ) p) is the identity on the curve, but kC(x,y,g2 ) gz,p) is not the identity. This guaran- tees PRIMALITY of p by a theorem of Goldwasser and Kilian (1986). 3. Each q has its recursive certificate following it. So if the smallest q is known to be PRIME, all the numbers are certified PRIME up the chain. A Pratt Certificate is quicker to generate for small numbers. The Mathematica® (Wolfram Re- search, Champaign, IL) task ProvablePrime [n] there- fore generates an Atkin-Goldwasser-Kilian-Morain cer- tificate only for numbers above a certain limit (10 10 by default), and a Pratt CERTIFICATE for smaller num- bers. see also Elliptic Curve Primality Proving, Ellip- tic Pseudoprime, Pratt Certificate, Primality Certificate, Witness References Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal- ity Proving." Math. Comput. 61, 29-68, 1993. Bressoud, D, M. Factorization and Prime Testing. New York: Springer- Verlag, 1989. Goldwasser, S. and Kilian, J. "Almost All Primes Can Be Quickly Certified." Proc. 18th STOC. pp. 316-329, 1986. Morain, F. "Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm." Rapport de Recherche 911, INRIA, Octobre 1988. Schoof, R. "Elliptic Curves over Finite Fields and the Com- putation of Square Roots mod p." Math. Comput. 44, 483-494, 1985. Wunderlich, M. C "A Performance Analysis of a Simple Prime-Testing Algorithm." Math. Comput. 40, 709-714, 1983. Atomic Statement In LOGIC, a statement which cannot be broken down into smaller statements. Attraction Basin see Basin of Attraction Attractor An attractor is a Set of states (points in the Phase Space), invariant under the dynamics, towards which neighboring states in a given Basin of Attraction asymptotically approach in the course of dynamic evo- lution. An attractor is denned as the smallest unit which cannot be itself decomposed into two or more attractors with distinct BASINS OF ATTRACTION. This restriction is necessary since a Dynamical System may have mul- tiple attractors, each with its own Basin OF Attrac- tion. Conservative systems do not have attractors, since the motion is periodic. For dissipative Dynamical Sys- tems, however, volumes shrink exponentially so attrac- tors have volume in n-D phase space. A stable FIXED Point surrounded by a dissipative re- gion is an attractor known as a SINK. Regular attractors (corresponding to Lyapunov Characteristic Ex- ponents) act as Limit Cycles, in which trajectories circle around a limiting trajectory which they asymp- totically approach, but never reach. STRANGE ATTRAC- TORS are bounded regions of PHASE SPACE (correspond- ing to Positive Lyapunov Characteristic Expo- nents) having zero MEASURE in the embedding PHASE Space and a Fractal Dimension. Trajectories within a Strange Attractor appear to skip around ran- domly. see also Barnsley's Fern, Basin of Attraction, Chaos Game, Fractal Dimension, Limit Cycle, Lyapunov Characteristic Exponent, Measure, Sink (Map), Strange Attractor Auction A type of sale in which members of a group of buyers offer ever increasing amounts. The bidder making the last bid (for which no higher bid is subsequently made within a specified time limit: "going once, going twice, sold") must then purchase the item in question at this price. Variants of simple bidding are also possible, as in a Vickery Auction. see also Vickery Auction Augend The first of several Addends, or "the one to which the others are added," is sometimes called the augend. Therefore, while a, 6, and c are ADDENDS in a + 6 -J- c, a is the augend. see also ADDEND, ADDITION Augmented Amicable Pair A Pair of numbers m and n such that a(m) — cr(n) = m + n — 1, where a{m) is the DIVISOR FUNCTION. Beck and Najar (1977) found 11 augmented amicable pairs. see also Amicable Pair, Divisor Function, Quasi- amicable Pair References Beck, W. E. and Najar, R. M. "More Reduced Amicable Pairs." Fib. Quart. 15, 331-332, 1977. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 59, 1994. Augmented Dodecahedron Authalic Latitude 85 Augmented Dodecahedron see Johnson Solid Augmented Hexagonal Prism see Johnson Solid where h = 2k — 1 and L 2hi M 2h = 2 h + lT2 (7) L 3hi M 3h = 3 h + lT3 k (8) L 6h , M 5h = 5 2/l + 3 • S' 1 + 1 =F 5* (5* + 1). (9) Augmented Pentagonal Prism see Johnson Solid Augmented Polyhedron A Uniform Polyhedron with one or more other solids adjoined. Augmented Sphenocorona see Johnson Solid Augmented Triangular Prism see Johnson Solid Augmented Tridiminished Icosahedron see Johnson Solid Augmented Truncated Cube see Johnson Solid Augmented Truncated Dodecahedron see Johnson Solid Augmented Truncated Tetrahedron see Johnson Solid Aureum Theorema Gauss's name for the QUADRATIC RECIPROCITY THE- OREM. Aurifeuillean Factorization A factorization of the form 2 4n + 2 + x = ^2n + l _ ^ + 1 + 1 )( 2 2n+1 + 2 n + 1 + 1). (1) The factorization for n — 14 was discovered by Au- rifeuille, and the general form was subsequently discov- ered by Lucas. The large factors are sometimes written as L and M as follows 2 4fc-2 + : = ^ 2 fe-i _ 2 fc + 1 )( 2 2 *- 1 + 2 k + 1) (2) 3 6fc-3 + x = ^2*-i + 1 j( 3 2fc-i _ 3 fc + i)^ 2 *" 1 + 3 fc + 1), (3) see also GAUSS'S FORMULA References Brillhart, J.; Lehmer, D. H.; Selfridge, J.; WagstafT, S. S. Jr.; and Tuckerman, B. Factorizations of b n ± 1, b = 2, 3, 5j 6, 7j 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc, pp. lxviii-lxxii, 1988. WagstafT, S. S. Jr. "Aurifeullian Factorizations and the Pe- riod of the Bell Numbers Modulo a Prime." Math. Corn- put. 65, 383-391, 1996. Ausdehnungslehre see Exterior Algebra Authalic Latitude An Auxiliary Latitude which gives a Sphere equal Surface Area relative to an Ellipsoid. The authalic latitude is defined by (i) /^sin- 1 -*- where Q=(l-e 2 ) - sin0 1 ^ 1- - e 2 sin 2 4> 2e Li n f 1 ~ esin( A le \ l-\-es'm<j)J . (2) and q p is q evaluated at the north pole (0 = 90°). Let R q be the Radius of the Sphere having the same Surface Area as the Ellipsoid, then Rq V 2 (3) The series for j3 is /3 = ^-Ge 2 + ^e 4 + a i e 6 + ...)sin(2<A) + (^ e4 + lio e6 + ---)sin(4^) -(4lfoe 6 + ...)sin(60) + .... (4) The inverse FORMULA is found from A<f> = (l-e 2 sin 2 0) 2 2cos0 sin<j) 1-e 2 + 1 — e 2 sin <j> which can be written 1^/l-esin.A le VI -f esin<j> J (5) 2 in + 1 = L 2h M 2h 3 3h + 1 = (3 h + l)L 3h M 3h (4) (5) (6) where q = q p sm/3 (6) 86 Autocorrelation and (f>o — s'm~ 1 (q/2). This can be written in series form + (^e* + ^ e 6 + ...)sin(4/3) (7) see a/so LATITUDE References Adams, O. S. "Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921. Snyder, J. P. Map Projections — A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 16, 1987. Autocorrelation The autocorrelation function is denned by F C f {t) = f*f = f(-t)*nt)= I r(T)f(t + r)dr, (1) where * denotes CONVOLUTION and • denotes CROSS- CORRELATION. A finite autocorrelation is given by Cf(r) = ([y(t)-y][y(t + r)-y]) (2) pT/2 = lim / [y(t)-y][y(t + r)-y]dt. (3) If / is a Real Function, /* = /, and an Even Function so that f(-r) = / (r), then (4) (5) (6) Cf(t)= I f(r)f(t + r)dr. J — OO But let t' = — r, so dr ~ —dr, then /» — OO Cf{t)= f(-r)f(t-r)(-dr) J OO OO f(-r)f(t-r)dr f(r)f(t-r)dr = f*f. (7) -F -F The autocorrelation discards phase information, return- ing only the POWER. It is therefore not reversible. There is also a somewhat surprising and extremely im- portant relationship between the autocorrelation and Autocorrelation the Fourier Transform known as the Wiener- Khintchine Theorem. Let FF[f{x)] — F(fc), and F* denote the COMPLEX CONJUGATE of F, then the FOUR- IER Transform of the Absolute Square of F(k) is given by n\F(k)\'}= r r(r)f(r + x)dr. (8) t/-oo The autocorrelation is a Hermitian Operator since Cf(-t) = C f *(t). /*/ is Maximum at the Origin. In other words, /oo /»oo f(u)f(u + x)du< / f 2 (u)du. (9) •oo J — oo To see this, let e be a Real Number. Then /oo [f{u) + ef(u + x)] 2 du>Q (10) ■oo /oo /»oo f(u)du + 2e l f{u)f(u + x)du -oo J —oo /oo f 2 (u + x)du> (11) ■oo +e / f 2 (u)du + 2e / J — oo J — ) du + 2e / f{u)f(u + x) du +e /oo -oo ) du > 0. (12) Define /oo f(u)du (13) ■oo /oo f(u)f(u + x)du. (14) ■oo Then plugging into above, we have ae 2 +be-\-c > 0. This Quadratic Equation does not have any Real Root, so b 2 — 4ac < 0, i.e., 6/2 < a. It follows that F f(u)f(u + x) du < /oo f{u)du, -OO (15) with the equality at x — 0. This proves that / * / is Maximum at the Origin. see also CONVOLUTION, CROSS-CORRELATION, QUAN- TIZATION Efficiency, Wiener-Khintchine Theo- rem References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Correlation and Autocorrelation Using the FFT." §13.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 538-539, 1992. Automorphic Function Automorphic Function An automorphic function f(z) of a COMPLEX variable z is one which is analytic (except for POLES) in a do- main D and which is invariant under a DENUMERABLY Infinite group of Linear Fractional Transforma- tions (also known as MOBIUS TRANSFORMATIONS) , az + 6 z = -. cz + a Automorphic functions are generalizations of TRIGONO- METRIC Functions and Elliptic Functions. see also Modular Function, Mobius Transforma- tions, Zeta Fuchsian Automorphic Number A number k such that nk 2 has its last digits equal to k is called n-automorphic. For example, 1 • 5 2 = 25 and 1 ■ 6 2 = 36 are 1-automorphic and 2 ■ 8 2 — 128 and 2 • 88 2 = 15488 are 2-automorphic. de Guerre and Fairbairn (1968) give a history of automorphic numbers. The first few 1-automorphic numbers are 1, 5, 6, 25, 76, 376, 625, 9376, 90625, . . . (Sloane's A003226, Wells 1986, p. 130). There are two 1-automorphic numbers with a given number of digits, one ending in 5 and one in 6 (except that the 1-digit automorphic numbers include 1), and each of these contains the previous number with a digit prepended. Using this fact, it is possible to con- struct automorphic numbers having more than 25,000 digits (Madachy 1979). The first few 1-automorphic numbers ending with 5 are 5, 25, 625, 0625, 90625, . . . (Sloane's A007185), and the first few ending with 6 are 6, 76, 376, 9376, 09376, . . . (Sloane's A016090). The 1- automorphic numbers a(n) ending in 5 are IDEMPOTENT (mod 10") since [a(n)] 2 = a(n) (mod 10 n ) (Sloane and Plouffe 1995). The following table gives the 10-digit n-automorphic numbers. n n-Automorphic Numbers Sloane 1 0000000001, 8212890625, 1787109376 2 0893554688 3 6666666667, 7262369792, 9404296875 4 0446777344 5 3642578125 6 3631184896 7 7142857143, 4548984375, 1683872768 8 0223388672 9 5754123264, 3134765625, 8888888889 — , A007185, A016090 A030984 — , A030985, A030986 A030987 A030988 A030989 A030990, A030991, A030992 A030993 A030994, A030995, — see also IDEMPOTENT, NARCISSISTIC NUMBER, NUM- BER Pyramid, Trimorphic Number References Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 59 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Autoregressive Model 87 Fairbairn, R. A. "More on Automorphic Numbers." J. Recr. Math. 2, 170-174, 1969. Fairbairn, R. A. Erratum to "More on Automorphic Num- bers." J. Recr. Math. 2, 245, 1969. de Guerre, V. and Fairbairn, R. A. "Automorphic Numbers." J. Recr. Math. 1, 173-179, 1968. Hunter, J. A. H. "Two Very Special Numbers." Fib. Quart 2, 230, 1964. Hunter, J. A. H. "Some Polyautomorphic Numbers." J. Recr. Math. 5, 27, 1972. Kraitchik, M. "Automorphic Numbers." §3.8 in Mathemat- ical Recreations. New York: W. W. Norton, pp. 77-78, 1942. Madachy, J. S. Madachy 's Mathematical Recreations. New York: Dover, pp. 34-54 and 175-176, 1979. Sloane, N. J. A. Sequences A016090, A003226/M3752, and A007185/M3940 in "An On-Line Version of the Encyclo- pedia of Integer Sequences." Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex: Penguin Books, pp. 171, 178, 191- 192, 1986. Automorphism An Isomorphism of a system of objects onto itself. see also ANOSOV AUTOMORPHISM Automorphism Group The GROUP of functions from an object G to itself which preserve the structure of the object, denoted Aut(G). The automorphism group of a GROUP preserves the Multiplication table, the automorphism group of a Graph the Incidence Matrices, and that of a Field the Addition and Multiplication tables, see also Outer Automorphism Group Autonomous A differential equation or system of ORDINARY DIFFER- ENTIAL EQUATIONS is said to be autonomous if it does not explicitly contain the independent variable (usu- ally denoted i). A second-order autonomous differen- tial equation is of the form F{y,y \y") — 0, where y = dy/dt = v. By the CHAIN RULE, y" can be ex- pressed as y dv It dv dy __ dv dy dt dy For an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means that all particles pass through a given point in phase space. A nonautonomous system of n first-order ODEs can be written as an autonomous system of n + 1 ODEs by letting t = x n +i and increasing the dimension of the system by 1 by adding the equation dx Tl + l dt 1. Autoregressive Model see Maximum Entropy Method 88 Auxiliary Circle Axiom A Flow Auxiliary Circle The ClRCUMCIRCLE of an ELLIPSE, i.e., the CIRCLE whose center corresponds with that of the ELLIPSE and whose Radius is equal to the Ellipse's Semimajor Axis. see also CIRCLE, ECCENTRIC ANGLE, ELLIPSE Auxiliary Latitude see Authalic Latitude, Conformal Latitude, Geocentric Latitude, Isometric Latitude, Lat- itude, Parametric Latitude, Rectifying Lati- tude, Reduced Latitude Auxiliary Triangle see Medial Triangle Average see Mean Average Absolute Deviation N a= — ^\xi- fi\ = (\xi-n\). i=l see also ABSOLUTE DEVIATION, DEVIATION, STANDARD Deviation, Variance Average Function If / is Continuous on a Closed Interval [a, 6], then there is at least one number x* in [a, 6] such that / J a f(x)dx = f(x*)(b- a). The average value of the FUNCTION (/) on this interval is then given by f(x*). see Mean- Value Theorem Average Seek Time see POINT-POINT DlSTANCE- -1-D Ax-Kochen Isomorphism Theorem Let P be the Set of PRIMES, and let Q p and Z p (t) be the Fields of p-ADic Numbers and formal Power series over Z p = (0, 1, ... ,p — 1). Further, suppose that D is a "nonprincipal maximal filter" on P. Then Y[ GP Q p /D and Y[ ep Z p (t)/D are ISOMORPHIC. see also Hyperreal Number, Nonstandard Analy- sis Axial Vector see PSEUDOVECTOR Axiom A Proposition regarded as self-evidently True with- out Proof. The word "axiom" is a slightly archaic syn- onym for Postulate. Compare Conjecture or Hy- pothesis, both of which connote apparently TRUE but not self- evident statements. see also ARCHIMEDES' AXIOM, AXIOM OF CHOICE, AX- IOMATIC System, Cantor-Dedekind Axiom, Con- gruence Axioms, Conjecture, Continuity Ax- ioms, Countable Additivity Probability Axiom, Dedekind's Axiom, Dimension Axiom, Eilenberg- Steenrod Axioms, Euclid's Axioms, Excision Ax- iom, Fano's Axiom, Field Axioms, Hausdorff Ax- ioms, Hilbert's Axioms, Homotopy Axiom, In- accessible Cardinals Axiom, Incidence Axioms, Independence Axiom, Induction Axiom, Law, Lemma, Long Exact Sequence of a Pair Axiom, Ordering Axioms, Parallel Axiom, Pasch's Ax- iom, Peano's Axioms, Playfair's Axiom, Porism, Postulate, Probability Axioms, Proclus' Axiom, Rule, T2-Separation Axiom, Theorem, Zermelo's Axiom of Choice, Zermelo-Fraenkel Axioms Axiom A Diffeomorphism Let 4> : M -¥ M be a C 1 Diffeomorphism on a com- pact Riemannian Manifold M. Then <f> satisfies Ax- iom A if the Nonwandering set Q(4>) of is hyperbolic and the Periodic Points of <j> are Dense in Q(<f>). Al- though it was conjectured that the first of these condi- tions implies the second, they were shown to be indepen- dent in or around 1977. Examples include the AN0S0V Diffeomorphisms and Smale Horseshoe Map. In some cases, Axiom A can be replaced by the condi- tion that the DIFFEOMORPHISM is a hyperbolic diffeo- morphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990). see also Anosov Diffeomorphism, Axiom A Flow, Diffeomorphism, Dynamical System, Riemannian Manifold, Smale Horseshoe Map References Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. New York: Springer- Verlag, 1975. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 143, 1993. Parry, W. and Pollicott, M. "Zeta Functions and the Peri- odic Orbit Structure of Hyperbolic Dynamics," Asterisque No. 187-188, 1990. Smale, S. "Different iable Dynamical Systems." Bull Amer. Math. Soc. 73, 747-817, 1967. Axiom A Flow A Flow defined analogously to the Axiom A Diffeo- morphism, except that instead of splitting the Tan- gent Bundle into two invariant sub-BUNDLES, they are split into three (one exponentially contracting, one expanding, and one which is 1-dimensional and tangen- tial to the flow direction). see also DYNAMICAL SYSTEM Axiom of Choice Azimuthal Projection 89 Axiom of Choice An important and fundamental result in Set Theory sometimes called Zermelo'S Axiom of Choice. It was formulated by Zermelo in 1904 and states that, given any Set of mutually exclusive nonempty SETS, there exists at least one Set that contains exactly one element in common with each of the nonempty SETS. It is related to HlLBERT'S PROBLEM IB, and was proved to be consistent with other Axioms in Set Theory in 1940 by GodeL In 1963, Cohen demonstrated that the axiom of choice is independent of the other Axioms in Cantorian Set Theory, so the Axiom cannot be proved within the system (Boyer and Merzbacher 1991, p. 610). see also Hilbert's Problems, Set Theory, Well- Ordered Set, Zermelo-Fraenkel Axioms, Zorn's Lemma References Boyer, C. B. and Merzbacher, U. C, A History of Mathemat- ics, 2nd ed. New York: Wiley, 1991. Cohen, P. J, "The Independence of the Continuum Hypoth- esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963. Cohen, P. J. "The Independence of the Continuum Hypothe- sis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, pp. 274-276, 1996. Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Devel- opment, and Influence. New York: Springer- Verlag, 1982. Axiomatic Set Theory A version of Set Theory in which axioms are taken as uninterpreted rather than as formalizations of pre- existing truths. see also Naive Set Theory, Set Theory Axiomatic System A logical system which possesses an explicitly stated Set of Axioms from which Theorems can be derived. see also Complete Axiomatic Theory, Consis- tency, Model Theory, Theorem Axis A LINE with respect to which a curve or figure is drawn, measured, rotated, etc. The term is also used to refer to a Line Segment through a Range (Woods 1961). see also Abscissa, Ordinate, cc-AxiS, y-AxiS, z-Axis References Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 8, 1961. Axonometry A Method for mapping 3-D figures onto the Plane. see also CROSS-SECTION, Map Projection, Pohlke's Theorem, Projection, Stereology References Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York: Dover, p. 313, 1973. Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 322-323, 1988. Azimuthal Equidistant Projection '^rk An Azimuthal Projection which is neither equal- Area nor CONFORMAL. Let <p± and Ao be the LATI- TUDE and LONGITUDE of the center of the projection, then the transformation equations are given by x - fc'cos0sin(A - Ao) y = fc'jcos^i sin0 — sin<^i cos<£cos(A — Ao)]. Here, and k' c sine (i) (2) (3) cose — sin 0i sin0 + cos^i cos0cos(A — Ao), (4) where c is the angular distance from the center. The inverse FORMULAS are -( = sin I cose sin 0i + y sin c cos <f>. l ) (5) and ( A + tan" 1 ( -r xsinc . . . u V ccos <pi cos c — y sin q>\ sin c for 0! ^ ±90° Ao+tan-^-l) for 0! = 90° Ao+tan- 1 ^), for 0i = -90°, ) (6) with the angular distance from the center given by c = V^ + y 2 . (7) References Snyder, J. P. Map Projections — A Working Manual U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 191-202, 1987. Azimuthal Projection see Azimuthal Equidistant Projection, Lam- bert Azimuthal Equal-Area Projection, Ortho- graphic Projection, Stereographic Projection B* -Algebra B B-Spline 91 E*-Algebra A Banach Algebra with an Antiautomorphic In- volution * which satisfies (5) A C*-Algebra is a special type of i?*-algebra. see also Banach Algebra, C*-Algebra i?2- Sequence N. B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Also called a Sidon Sequence. An Infinite Se- quence of Positive Integers 1 < h < 6 2 < h < such that all pairwise sums bi + bj (i) (2) for i < j are distinct (Guy 1994). An example is 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, . . . (Sloane's A005282). Zhang (1993, 1994) showed that S(B2) = sup V — > 2.1597. all B2 sequences ~~j ®k (3) The definition can be extended to B n -sequences (Guy 1994). see also ^-Sequence, Mian-Chowla Sequence References Finch, S. "Favorite Mathematical Constants." http://www. mathsof t . com/asolve/constant/erdos/erdos .html. Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with Distinct Sums," and "^-Sequences," and B 2 -Sequences Formed by the Greedy Algorithm." §C9, Cll, E28, and E32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 115-118, 121-123, 228-229, and 232-233, 1994. Sloane, N. J. A. Sequence A005282/M1094 in "An On-Line Version of the Encyclopedia of Integer Sequences." Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum." Math. Comput. 60, 835-839, 1993. Zhang, Z. X. "Finding Finite B2-Sequences with Larger m — a™ 1 ' 2 ." Math. Comput. 63, 403-414, 1994. B p - Theorem If Op' (G) = 1 and if a? is a p-element of G, then L p ,(C g (x)<E(Cg(x)), where L p > is the p-LAYER. X = X (i) (2) B -Spline Po®t x* +y* = {x + y)* (3) (ex)* = ex* satisfies (4) • Pi A generalization of the Bezier Curve. Let a vector known as the KNOT VECTOR be defined T = {£o,£ij • ■ , tm}i (i) where T is a nondecreasing SEQUENCE with U 6 [0, 1], and define control points Po, . . . , Pn- Define the degree as p = m — n — 1. (2) The "knots" £ p +i, ..., tm- P -i are called Internal Knots. Define the basis functions as at / ,\ _ f 1 if ti < t < ti+i and U < tt+i 1 otherwise ti + v+l — t N ilP (t) t-U ti-\-p Ci (3) ii+p+1 — *i+l Then the curve defined by C(t) = £p<M,p(t) (4) (5) is a B-spline. Specific types include the nonperiodic B- spline (first p + 1 knots equal and last p + 1 equal to 1) and uniform B-spline (INTERNAL KNOTS are equally spaced). A B-Spline with no INTERNAL KNOTS is a Bezier Curve. The degree of a B-spline is independent of the number of control points, so a low order can always be maintained for purposes of numerical stability. Also, a curve is p — k times differentiate at a point where k duplicate knot values occur. The knot values determine the extent of the control of the control points. A nonperiodic B-spline is a B-spline whose first p + 1 knots are equal to and last p -f 1 knots are equal to 1. A uniform B-spline is a B-spline whose INTERNAL Knots are equally spaced. see also Bezier Curve, NURBS Curve 92 B-Tree Backtracking B-Tree B-trees were introduced by Bayer (1972) and Mc- Creight. They are a special m-ary balanced tree used in databases because their structure allows records to be inserted, deleted, and retrieved with guaranteed worst- case performance. An n-node £?-tree has height C(lg2), where Lg is the LOGARITHM to base 2. The Apple® Macintosh® (Apple Computer, Cupertino, CA) HFS fil- ing system uses B-trees to store disk directories (Bene- dict 1995). A B-tree satisfies the following properties: 1. The Root is either a Leaf (Tree) or has at least two Children, 2. Each node (except the ROOT and LEAVES) has be- tween \m/2\ and m Children, where \x\ is the Ceiling Function. 3. Each path from the Root to a Leaf (Tree) has the same length. Every 2-3 Tree is a B-tree of order 3. The number of B-trees of order n = 1, 2, . . . are 0, 1, 1, 1, 2, 2, 3, 4, 5, 8, 14, 23, 32, 43, 63, . . . (Ruskey, Sloane's A014535). see also Red-Black Tree References Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc- tures and Algorithms. Reading, MA: Addison-Wesley, pp. 369-374, 1987. Benedict, B. Using Norton Utilities for the Macintosh. Indi- anapolis, IN: Que, pp. B-17-B-33, 1995. Beyer, R. "Symmetric Binary jB-Trees: Data Structures and Maintenance Algorithms." Acta Informat. 1, 290-306, 1972. Ruskey, F. "Information on B-Trees." http://sue.csc.uvic . ca/~cos/inf /tree/BTrees .html. Sloane, N. J. A. Sequence A014535 in "An On-Line Version of the Encyclopedia of Integer Sequences." Baby Monster Group Also known as FISCHER'S BABY MONSTER GROUP. The Sporadic Group B. It has Order 2 4i . 3 i3 . 5 6 . 7 2 . ii . 13 . 17 . 19 . 23 • 31 ■ 47. see also MONSTER GROUP References Wilson, R. A. "ATLAS of Finite Group Representation." http : //for . mat . bham . ac .uk/atlas/BM . html. BAC-CAB Identity The Vector Triple Product identity A x (B x C) = B(A ■ C) - C(A • B). This identity can be generalized to n-D a 2 x ■ • • x a n _i x (bi x ■ • • x b n _i) h x a 2 ■ In = (-*) a n _i ■ bi See also LAGRANGE'S IDENTITY b n -i a 2 • b n _i &n-i " b n _i BAC-CAB Rule see BAC-CAB IDENTITY Bachelier Function see Brown Function Bachet's Conjecture see Lagrange's Four-Square Theorem Bachet Equation The Diophantine Equation x 2 + k = y 3 , which is also an Elliptic Curve. The general equation is still the focus of ongoing study. Backhouse's Constant Let P(x) be defined as the POWER series whose nth term has a Coefficient equal to the nth Prime, oo P(x) = Y^PhX k = l + 2z + 3z 2 + 5z 3 + 7z 4 -hllz 5 + ..., and let Q(x) be defined by on 1 Q(*) = P(x) y^qkX h . k=o Then N. Backhouse conjectured that lim n—t-oc <7n+l q n 1.456074948582689671399595351116. . . . The constant was subsequently shown to exist by P. Fla- jolet. References Finch, S. "Favorite Mathematical Constants." http: //www. mathsof t . com/asolve/constant/backhous/ backhous .html. Backlund Transformation A method for solving classes of nonlinear Partial Dif- ferential Equations. see also INVERSE SCATTERING METHOD References Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos. Cambridge, England: Cambridge University Press, p. 196, 1990. Miura, R. M. (Ed.) Backlund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. New York: Springer- Verlag, 1974. Backtracking A method of drawing FRACTALS by appropriate num- bering of the corresponding tree diagram which does not require storage of intermediate results. Backus-Gilbert Method Baire Category Theorem 93 Backus-Gilbert Method A method which can be used to solve some classes of INTEGRAL EQUATIONS and is especially useful in im- plementing certain types of data inversion. It has been applied to invert seismic data to obtain density profiles in the Earth. References Backus, G. and Gilbert, F. "The Resolving Power of Growth Earth Data." Geophys. J. Roy. Astron. Soc. 16, 169-205, 1968. Backus, G. E. and Gilbert, F. "Uniqueness in the Inversion of Inaccurate Gross Earth Data." Phil Trans. Roy. Soc. London Ser. A 266, 123-192, 1970. Loredo, T. J. and Epstein, R. I. "Analyzing Gamma-Ray Burst Spectral Data." Astrophys. J. 336, 896-919, 1989. Parker, R. L. "Understanding Inverse Theory." Ann. Rev. Earth Planet Sci. 5, 35-64, 1977. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Backus-Gilbert Method." §18.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 806-809, 1992. Backward Difference The backward difference is a Finite Difference de- fined by Vp = V/ p s/p-/p_i. (1) Higher order differences are obtained by repeated oper- ations of the backward difference operator, so Vp = V(Vp) = V(/ p - /„_!) = V/ p - V/,_i (2) = {fp ~ fp-i) ~ (fp-i ~ /p-z) = fp~ 2 /p-l + fp-2 (3) In general, v5 = vv, = £(-ir(*W* +m > (4) where (^) is a BINOMIAL COEFFICIENT. Newton's Backward Difference Formula ex- presses f p as the sum of the nth backward differences / P = /o+pVo + ^p(p + l)V? + J T p(p + l)(p + 2)Vg + ..., (5) where Vq is the first nth difference computed from the difference table. see also Adams' Method, Difference Equation, Divided Difference, Finite Difference, For- ward Difference, Newton's Backward Differ- ence Formula, Reciprocal Difference References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429 and 433, 1987. Bader-Deuflhard Method A generalization of the Bulirsch-Stoer Algorithm for solving Ordinary Differential Equations. References Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 730, 1992. Baguenaudier A Puzzle involving disentangling a set of rings from a looped double rod (also called CHINESE Rings). The minimum number of moves needed for n rings is §(2 n+1 -2) |(2 n+1 -l) n even n odd. By simultaneously moving the two end rings, the num- ber of moves can be reduced to f 2 n_1 -In even I 2 71 " 1 n odd. The solution of the baguenaudier is intimately related to the theory of GRAY CODES. References Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Pro- duce a Chinese Monster." Quantum 6, 61-65 (Mar.) and 58-59 (Apr.), 1996. Gardner, M. "The Binary Gray Code." In Knotted Dough- nuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 15-17, 1986. Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical Recreations. New York: W. W. Norton, pp. 89-91, 1942. Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, p. 268, 1983. Bailey's Method see Lambert's Method Bailey's Theorem Let T(z) be the GAMMA FUNCTION, then r(m+|) V(m) [i (-Y — — (-Y—!— I>+§) V(n) 1 /iy_j_ /i-3\ 2 1 n + \2J n-M + \2.4y n + 2 + ' Baire Category Theorem A nonempty complete Metric Space cannot be repre- sented as the Union of a Countable family of nowhere Dense Subsets. 94 Baire Space Ball Triangle Picking Baire Space A Topological Space X in which each Subset of X of the "first category" has an empty interior. A TOPO- LOGICAL Space which is Homeomorphic to a complete Metric Space is a Baire space. Bairstow's Method A procedure for rinding the quadratic factors for the Complex Conjugate Roots of a Polynomial P(x) with Real Coefficients. [x — (a + ib)][x - (a — ib)] = x 2 + 2ax + (a 2 + b 2 ) = x 2 + Bx + C. (1) Now write the original POLYNOMIAL as P(x) = (x 2 +Bx + C)Q{x) + Rx + S (2) R(B + SB,C + 6C)KR(B,C) + ^dB+^dC (3) dB 8C S(B + 5B,C + 5C)*d(B,C) + ^dB+^dC (4) £ — <.- + «.♦* ,« + «, + g5 + » ( .> . QW = ( I . + B , + C )g + g + f (6) "*«M = <** + «* + C >I + 1 + !' (8) Now use the 2-D Newton's Method to find the simul- taneous solutions. References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 277 and 283-284, 1989. Baker's Dozen The number 13. see also 13, DOZEN Baker's Map The Map X n +1 = 2^£ n , (1) where x is computed modulo 1. A generalized Baker's map can be defined as Vn < a (2) Xn+1 -\(1-X b ) + X b x n y n >a where (3 = 1 — a, A + A 6 < 1, and x and y are computed mod 1. The q = 1 g-DlMENSION is aln(±)+/31n(|) D 1 = 1 + Va) )*' ■ (4) ' ta (£)+*»»(*)' If A a = A&, then the general g-DlMENSION is 1 In (a q +f3 q ) D q = l + q — 1 In A (5) References Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer- Verlag, p. 60, 1983. Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 81-82, 1993. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 32, 1990. Balanced ANOVA An ANOVA in which the number of REPLICATES (sets of identical observations) is restricted to be the same for each Factor Level (treatment group). see also ANOVA Balanced Incomplete Block Design see Block Design Ball The n-ball, denoted B n , is the interior of a SPHERE S™" 1 , and sometimes also called the n-DlSK. (Al- though physicists often use the term "SPHERE" to mean the solid ball, mathematicians definitely do not!) Let Vol(B n ) denote the volume of an n-D ball of RADIUS r. Then oo Y^ Vol(S n ) = e^ 2 [1 + erf (r^ )], where erf(x) is the ERF function. see also Alexander's Horned Sphere, Banach- Tarski Paradox, Bing's Theorem, Bishop's In- equality, Bounded, Disk, Hypersphere, Sphere, Wild Point References Preden, E. Problem 10207. "Summing a Series of Volumes." Amer. Math. Monthly 100, 882, 1993. Ball Triangle Picking The determination of the probability for obtaining an Obtuse Triangle by picking 3 points at random in the unit Disk was generalized by Hall (1982) to the n- D Ball. Buchta (1986) subsequently gave closed form Ballantine Banach Measure 95 evaluations for Hall's integrals, with the first few solu- tions being 9 4 P 2 = - - — « 0.72 8 7V d P 4 « 0.39 P 5 « 0.29. The case P^ corresponds to the usual DISK case. see also Cube Triangle Picking, Obtuse Triangle References Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." III. J. Math. 30, 653-659, 1986. Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 19, 712-715, 1982. Ballantine see Borromean Rings Ballieu's Theorem For any set fi = (^1,^2, ■ ■ ■ ,fi n ) of POSITIVE numbers with ^o = and M M = max flk + {ln\b n -k\ 0<k<n-l /ifc + 1 Then all the EIGENVALUES A satisfying P(X) = 0, where P{\) is the Characteristic Polynomial, lie on the Disk \z\ < M M . References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1119, 1979. Ballot Problem Suppose A and B are candidates for office and there are 2n voters, n voting for A and n for B, In how many ways can the ballots be counted so that A is always ahead of or tied with B1 The solution is a CATALAN NUMBER A related problem also called "the" ballot problem is to let A receive a votes and B b votes with a > b. This ver- sion of the ballot problem then asks for the probability that A stays ahead of B as the votes are counted (Vardi 1991). The solution is (a — b)/(a + 6), as first shown by M. Bertrand (Hilton and Pedersen 1991). Another elegant solution was provided by Andre (1887) using the so-called Andre's Reflection Method. The problem can also be generalized (Hilton and Ped- ersen 1991). Furthermore, the TAK FUNCTION is con- nected with the ballot problem (Vardi 1991). see also Andre's Reflection Method, Catalan Number, TAK Function References Andre, D. "Solution directe du probleme resolu par M. Bertrand." Comptes Rendus Acad. Sci. Paris 105, 436-437, 1887. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 49, 1987. Carlitz, L. "Solution of Certain Recurrences." SIAM J. Appl. Math. 17, 251-259, 1969. Comtet, L. Advanced Combinatorics. Dordrecht, Nether- lands: Reidel, p. 22, 1974. Feller, W. An Introduction to Probability Theory and Its Ap- plications, Vol. 1, 3rd ed. New York: Wiley, pp. 67-97, 1968. Hilton, P. and Pedersen, J. "The Ballot Problem and Cata- lan Numbers." Nieuw Archief voor Wiskunde 8, 209-216, 1990. Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener- alization, and Their Uses." Math. Intel. 13, 64-75, 1991. Kraitchik, M. "The Ballot-Box Problem." §6.13 in Mathe- matical Recreations. New York: W. W. Norton, p. 132, 1942. Motzkin, T. "Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for Non-Associative Products." Bull Amer. Math. Soc. 54, 352-360, 1948. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison- Wesley, pp. 185-187, 1991. Banach Algebra An Algebra A over a Field F with a Norm that makes A into a COMPLETE METRIC SPACE, and there- fore, a Banach Space. F is frequently taken to be the Complex Numbers in order to assure that the Spec- trum fully characterizes an Operator (i.e., the spec- tral theorems for normal or compact normal operators do not, in general, hold in the Spectrum over the Real Numbers). see also £?*-Algebra Banach Fixed Point Theorem Let / be a contraction mapping from a closed SUBSET F of a Banach Space E into F. Then there exists a unique z £ F such that f(z) = z. see also FIXED POINT THEOREM References Debnath, L. and Mikusiriski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990. Banach-Hausdorff-Tarski Paradox see Banach- Tarski Paradox Banach Measure An "Area" which can be defined for every set — even those without a true geometric AREA — which is rigid and finitely additive. 96 Banach Space Baibiefs Theorem Banach Space A normed linear Space which is Complete in the norm- determined Metric. A Hilbert Space is always a Ba- nach space, but the converse need not hold. see also Besov Space, Hilbert Space, Schauder Fixed Point Theorem Banach-Steinhaus Theorem see Uniform Boundedness Principle Banach- Tarski Paradox First stated in 1924, this theorem demonstrates that it is possible to dissect a Ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five. However, the pieces are extremely complicated. A generalization of this theo- rem is that any two bodies in R which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (they are are EQUIDE- composable). References Stromberg, K. "The Banach- Tarski Paradox." Amer. Math. Monthly 86, 3, 1979. Wagon, S. The Banach-Tarski Paradox. New York: Cam- bridge University Press, 1993. Bang's Theorem The lines drawn to the Vertices of a face of a Tetra- hedron from the point of contact of the FACE with the INSPHERE form three ANGLES at the point of contact which are the same three ANGLES in each FACE. References Brown, B. H. "Theorem of Bang. Isosceles Tetrahedra." Amer. Math. Monthly 33, 224-226, 1926. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 93, 1976. Bankoff Circle References Bankoff, L. "Are the Twin Circles of Archimedes Really Twins?" Math. Mag. 47, 214-218, 1974. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." Sci. Amer. 240, 18-28, Jan. 1979. Banzhaf Power Index The number of ways in which a group of n with weights X^r=i Wi = 1 can cnan g e a losing coalition (one with ^2 w i < 1/2) to a winning one, or vice versa. It was proposed by the lawyer J. F. Banzhaf in 1965. References Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, pp. 9-10, 1995. Bar (Edge) The term in rigidity theory for the EDGES of a GRAPH. see also Configuration, Framework Bar Polyhex A Polyhex consisting of Hexagons arranged along a line. see also Bar Polyiamond References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight- of- Mind from Scientific American. New York: Vintage, p. 147, 1978. Bar Polyiamond In addition to the ARCHIMEDES' CIRCLES d and C 2 in the Arbelos figure, there is a third circle C3 congruent to these two as illustrated in the above figure. see also ARBELOS A Polyiamond consisting of Equilateral Triangles arranged along a line. see also Bar Polyhex References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton University Press, p. 92, 1994. Barber Paradox A man of Seville is shaved by the Barber of Seville IFF the man does not shave himself. Does the barber shave himself? Proposed by Bertrand Russell. Barbier's Theorem All Curves of Constant Width of width w have the same Perimeter ttw. Bare Angle Center Barth Decic 97 Bare Angle Center The Triangle Center with Triangle Center Function a = A. References Kimberling, C. "Major Centers of Triangles." Amer. Math. Monthly 104, 431-438, 1997. Barnes G- Function see G-Function Barnes' Lemma If a Contour in the Complex Plane is curved such that it separates the increasing and decreasing sequences of Poles, then 2-m . v </ —too + s)r(0 + s)T('y-s)r(5-s)ds = T(a + 7)r(a + 6)r{(3 + j)T{p + 6) r(a + /3 + 7 + <5) where T(z) is the Gamma Function. Barnes- Wall Lattice A lattice which can be constructed from the LEECH LAT- TICE A 2 4- see also Coxeter-Todd Lattice, Lattice Point, Leech Lattice References Barnes, E. S. and Wall, G. E. "Some Extreme Forms Denned in Terms of Abelian Groups." J. Austral Math. Soc. 1, 47-63, 1959. Conway, J. H. and Sloane, N. J, A, "The 16- Dimensional Barnes- Wall Lattice Ai 6 ." §4.10 in Sphere Packings, Lat- tices, and Groups, 2nd ed. New York: Springer- Verlag, pp. 127-129, 1993, Barnsley's Fern ^■,;f^' *^7&g$^~~ 0.85 0.04" X + "o.oo" (1) -0.04 0.85 _y 1.60 -0.15 0.28" X + "o.oo" (2) 0.26 0.24 y '. 0.44 0.20 -0.26' X + "o.oo" (3) 0.23 0.22 y . 1.60 0.00 0.00 " X (4) 0.00 0.16 y The Attractor of the Iterated Function System given by the set of "fern functions" h(x,y) = fs(x,y) = U(x,y) = (Barnsley 1993, p. 86; Wagon 1991). These Affine Transformations are contractions. The tip of the fern (which resembles the black spleehwort variety of fern) is the fixed point of /i , and the tips of the lowest two branches are the images of the main tip under J2 and f z (Wagon 1991). see also Dynamical System, Fractal, Iterated Function System References Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Aca- demic Press, pp. 86, 90, 102 and Plate 2, 1993. Gleick, J. Chaos: Making a New Science. New York: Pen- guin Books, p. 238, 1988. Wagon, S. "Biasing the Chaos Game: Barnslej^s Fern." §5.3 in Mathematica in Action. New York: W. H. Freeman, pp. 156-163, 1991. Barrier A number n is called a barrier of a number-theoretic function f(m) if, for all m < n, m + f(m) < n. Neither the Totient Function <p(n) nor the Divisor Func- tion o-(n) has barriers. References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 64-65, 1994. Barth Decic \ 1 7 / . 98 Barth Sextic Bartlett Function The Barth decic is a Decic Surface in complex three- dimensional projective space having the maximum pos- sible number of ORDINARY DOUBLE POINTS (345). It is given by the implicit equation ■A 2 ) ■2yV) x( :c 4 + y 4 + z 4 -2 2; V- +(3 + 50)(;r 2 +y 2 +z 2 -w 2 ) 2 [x 2 4-y 2 \z 2 -(2-0)u? 2 ]V = 0, where is the Golden Mean and w is a parameter (EndraB, Nordstrand), taken as w = 1 in the above plot. The Barth decic is invariant under the ICOSAHEDRAL Group. see also ALGEBRAIC SURFACE, BARTH SEXTIC, DECIC Surface, Ordinary Double Point References Barth, W. "Two Projective Surfaces with Many Nodes Ad- mitting the Symmetries of the Icosahedron." J. Alg. Geom. 5, 173-186, 1996. Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV- Mitteilungen 4, 17-20, 4/1995. Endrafi, S. "Barth's Decic." http://www.mathematik.uni- mainz . de/AlgebraischeGeometrie/docs/ Ebarthdecic . shtml. Nordstrand, T. "Batch Decic." http://www.uib.no/people/ nf ytn/bdectxt .htm. Barth Sextic The Barth-sextic is a SEXTIC SURFACE in complex three-dimensional projective space having the maximum possible number of ORDINARY DOUBLE POINTS (65). It is given by the implicit equation A{4> 2 x 2 -y 2 ){4> 2 y 2 ~z'){<t>-z- -x 2 2 2w ,2 2 2x -(1 + 2<P)(x 2 + y 2 + z 2 - w 2 ) 2 w 2 0. where 4> is the GOLDEN Mean, and w is a parameter (Endrafi, Nordstrand), taken as w — 1 in the above plot. The Barth sextic is invariant under the ICOSAHEDRAL Group. Under the map / \ v / 2 2 2 2\ (x,y,z,w) -+ (x ,y ,z 9 w ), the surface is the eightfold cover of the Cayley Cubic (Endrafi). see also ALGEBRAIC SURFACE, BARTH DECIC, CAYLEY Cubic, Ordinary Double Point, Sextic Surface References Barth, W. "Two Projective Surfaces with Many Nodes Ad- mitting the Symmetries of the Icosahedron." J. Alg. Geom. 5, 173-186, 1996. Endrafl, S. "Flachen mit vielen Doppelpunkten." DMV- Mitteilungen 4, 17-20, 4/1995. Endrafl, S. "Barth's Sextic." http://www.mathematik.uni- mainz.de/AlgebraischeGeometrie/docs/ Ebarthsextic . shtml. Nordstrand, T. "Barth Sextic." http://www.uib.no/people/ nf ytn/sexttxt .htm. Bartlett Function o.is o.c o.c oflc J '-0725 -1 -0.5 ' 075 1 -0.5 The Apodization Function L f{x) = 1 (1) which is a generalization of the one-argument TRIANGLE Function. Its Full Width at Half Maximum is a. It has Instrument Function I(x) = ^ e~ 2 * ikx (l - M) dx v —a + fe- J,iJ, (l-j)<b. (2) Letting x' = —x in the first part therefore gives f° e- 2 " ikx (l + |) dx = I e Mk *' (l - ^\ (-dx') Rewriting (2) using (3) gives (3) 7-/ \ / 2irikx . — 2-rrikx\ ( -, % \ I(x) = (e +e H aj dx = 2 / cos(27rfcz) (l - -J dx. (4) Integrating the first part and using the integral / x cos(bx) dx — — cos(6;c) + — sin(for) (5) b 1 b Barycentric Coordinates for the second part gives sin(27rA;a;) I(x) = 2 2irk [s\n(2Trk 2™fe~ = 2 { l" sin ( 27rfc a ) __ cos(27rfca) — 1 asm.{2nka) 47T 2 fc 2 27r 2 a/c 2 : a sine (7rka), [cos(27r&a) — 1] = a 2ttA; sin 2 (7rfca) 7r 2 k 2 a 2 (6) where sine x is the SlNC FUNCTION. The peak (in units of a) is 1. The function I(x) is always positive, so there are no Negative sidelobes. The extrema are given by letting j3 = nka and solving d ( sin j3 2 sin/9sin/3-/3cos/9 . P P sin/3(sin/?-/?cos/3) = sin/3-/3cos/3 = tan/3 = /3. (8) (9) (10) Solving this numerically gives j3 = 4.49341 for the first maximum, and the peak POSITIVE sidelobe is 0.047190. The full width at half maximum is given by setting x = nka and solving sine x = | (11) for #1/2, yielding Ei/2 = 7rfci /2 a = 1.39156. (12) Therefore, with L = 2a, FWHM = 2fei /2 = 0.885895 1.77179 a (13) see a/so APODIZATION FUNCTION, PARZEN ApODIZA- tion Function, Triangle Function References Bartlett, M. S. "Periodogram Analysis and Continuous Spec- tra." Biometrika 37, 1-16, 1950. Barycentric Coordinates Also known as HOMOGENEOUS COORDINATES or TRI- linear Coordinates. see Trilinear Coordinates Base Curve see Directrix (Ruled Surface) Base (Number) 99 Base (Logarithm) The number used to define a LOGARITHM, which is then written log 6 . The symbol logo; is an abbreviation for log 10 x, In as for log e x (the Natural Logarithm), and lga: for log 2 x. see also e, Lg, Ln, Logarithm, Napierian Loga- rithm, Natural Logarithm Base (Neighborhood System) A base for a neighborhood system of a point x is a col- lection N of Open Sets such that x belongs to every member of iV, and any Open Set containing x also con- tains a member of N as a Subset. Base (Number) A Real Number x can be represented using any Inte- ger number b as a base (sometimes also called a RADIX or SCALE). The choice of a base yields to a representa- tion of numbers known as a Number System. In base 6, the DIGITS 0, 1, . . . , b - 1 are used (where, by con- vention, for bases larger than 10, the symbols A, B, C, . . . are generally used as symbols representing the DEC- IMAL numbers 10, 11, 12, . . . ). Base Name 2 binary 3 ternary 4 quaternary 5 quinary 6 senary 7 septenary 8 octal 9 nonary 10 decimal 11 undenary 12 duodecimal 16 hexadecimal 20 vigesimal 60 sexagesimal Let the base b representation of a number x be written (a n Cin-i ... ao- a_i . . .)*,, (1) (e.g., 123.456io), then the index of the leading DIGIT needed to represent the number is n = |k>g 6 x\ , (2) where \_x\ is the FLOOR FUNCTION. Now, recursively compute the successive Digits ai = L?J • where r n = x and n-! = n (lib 1 (3) (4) 100 Base Space Basis for i = n, n — 1, . . . , 1,0, This gives the base b representation of x. Note that if x is an Integer, then i need only run through 0, and that if x has a fractional part, then the expansion may or may not terminate. For example, the HEXADECIMAL representation of 0.1 (which terminates in DECIMAL notation) is the infinite expression 0.19999. . .h- Some number systems use a mixture of bases for count- ing. Examples include the Mayan calendar and the old British monetary system (in which ha'pennies, pennies, threepence, sixpence, shillings, half crowns, pounds, and guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240, and 252, respectively). Knuth has considered using TRANSCENDENTAL bases. This leads to some rather unfamiliar results, such as equating -k to 1 in "base 7r," 7r = I*.. see also Binary, Decimal, Hereditary Represen- tation, Hexadecimal, Octal, Quaternary, Sexa- gesimal, Ternary, Vigesimal References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 28, 1972. Bogomolny, A. "Base Converter." http : //www . cut-the- knot . com/binary .html. Lauwerier, II. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 6-11, 1991. i$ Weisstein, E. W. "Bases." http: //www. astro. Virginia. edu/~eww6n/math/notebooks/Bases.m. Base Space The Space B of a Fiber Bundle given by the Map / : E -> B, where E is the Total Space of the Fiber Bundle. see also FIBER BUNDLE, TOTAL SPACE Baseball The numbers 3 and 4 appear prominently in the game of baseball. There are 3*3 = 9 innings in a game, and three strikes are an out. However, 4 balls are needed for a walk. The number of bases can either be regarded as 3 (excluding HOME Plate) or 4 (including it). see Baseball Cover, Home Plate A pair of identical plane regions (mirror symmetric about two perpendicular lines through the center) which can be stitched together to form a baseball (or tennis ball). A baseball has a CIRCUMFERENCE of 9 1/8 inches. The practical consideration of separating the regions far enough to allow the pitcher a good grip requires that the "neck" distance be about 1 3/16 inches. The base- ball cover was invented by Elias Drake as a boy in the 1840s. (Thompson's attribution of the current design to trial and error development by C. H. Jackson in the 1860s is apparently unsubstantiated, as discovered by George Bart.) One way to produce a baseball cover is to draw the re- gions on a Sphere, then cut them out. However, it is difficult to produce two identical regions in this man- ner. Thompson (1996) gives mathematical expressions giving baseball cover curves both in the plane and in 3-D. J. H. Conway has humorously proposed the follow- ing "baseball curve conjecture:" no two definitions of "the" baseball curve will give the same answer unless their equivalence was obvious from the start. see also Baseball, Home Plate, Tennis Ball The- orem, Yin- Yang References Thompson, R. B. "Designing a Baseball Cover. 1860's: Pa- tience, Trial, and Error. 1990's: Geometry, Calculus, and Computation," http://www.mathsoft.com/asolve/ baseball/baseball. html. Rev. March 5, 1996. Basin of Attraction The set of points in the space of system variables such that initial conditions chosen in this set dynamically evolve to a particular Attractor. see also Wada Basin Basis A (vector) basis is any Set of n LINEARLY INDEPEN- DENT Vectors capable of generating an n-dimensional SUBSPACE of R n . Given a IlYPERPLANE defined by xi + x 2 + X3 4- x 4 + x$ = 0, a basis is found by solving for Xi in terms of #2, #3, 2:4, and £5. Carrying out this procedure, Baseball Cover Xi -X2 — X3 — X4 — £5, ~Xi~ --1- --1- --1- --1- X2 1 X3 = x 2 +£3 1 ~\-X4 -\-x 5 X4 1 -335- . . - - . . . 1 - Basis Theorem B ayes' Formula 101 and the above VECTOR form an (unnormalized) BASIS. Given a MATRIX A with an orthonormal basis, the MA- TRIX corresponding to a new basis, expressed in terms of the original xi , . . . , x n is A' = [Axi Ax n ]. see also Bilinear Basis, Modular System Basis, Orthonormal Basis, Topological Basis Basis Theorem see Hilbert Basis Theorem Basler Problem The problem of analytically finding the value of C(2), where £ is the Riemann Zeta Function. References Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988. Basset Function see Modified Bessel Function of the Second Kind Batch A set of values of similar meaning obtained in any man- ner. References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 667, 1977. Bateman Function Mx) - r(i < +V ( "^'°' fa) for x > 0, where U is a Confluent Hypergeometric Function of the Second Kind. see also CONFLUENT HYPERGEOMETRIC DIFFERENTIAL Equation, Hypergeometric Function Batrachion A class of CURVE defined at Integer values which hops from one value to another. Their name derives from the word batrachion, which means "frog- like." Many ba- trachions are FRACTAL. Examples include the BLANC- MANGE Function, Hofstadter-Conway $10,000 Se- quence, Hofstadter's Q-Sequence, and Mallow's Sequence. References Pickover, C. A. "The Crying of Fractal Batrachion 1,489." Ch. 25 in Keys to Infinity. New York: W. H. Freeman, pp. 183-191, 1995. Bauer's Identical Congruence Let t(m) denote the set of the </>(m) numbers less than and Relatively Prime to m, where <f>(n) is the To- tient Function. Define f m {x)= n (*-*)• (i) t(m) A theorem of Lagrange states that f m {x) = x Hm) -1 (mod to). (2) This can be generalized as follows. Let p be an ODD Prime Divisor of m and p a the highest Power which divides to, then f m (x) = (x*- 1 - l)*^)/^- 1 ) (mod p») (3) and, in particular, /„.(*) = (a*" 1 -l)*" -1 (mod/). (4) Furthermore, if to > 2 is EVEN and 2 a is the highest POWER of 2 that divides m, then / m (a:) = (a: 2 -l)* (m)/2 (mod 2 a ) (5) and, in particular, f 2a ( x ) = ( x 2 -l) 2a ~ 2 (mod2 a ). (6) see also Leudesdorf Theorem References Hardy, G. H. and Wright, E. M. "Bauer's Identical Congru- ence." §8.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 98-100, 1979. Bauer's Theorem see Bauer's Identical Congruence Bauspiel A construction for the RHOMBIC DODECAHEDRON. References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 26 and 50, 1973. Bayes' Formula Let A and Bj be Sets. Conditional Probability requires that P(AC\B j )=P(A)P(B j \Al (1) where n denotes INTERSECTION ("and"), and also that P(A n Bj) = P(Bj n A) - P{Bj)P{A\Bj) (2) 102 Bayes' Theorem and P{B j nA)=P{B j )P{A\B j ). (3) Since (2) and (3) must be equal, P(AnB j ) = P(B j nA). (4) Prom (2) and (3), P(AnB j ) = P(B j )P(A\B j ). (5) Equating (5) with (2) gives P(A)P(B j \A) = P(B i )P(A\B j ), (6) so P(Bj\A) PjB^PjAlBj) P(A) ■ (7) Now, let S=U^> (8) i=l so Ai is an event is S and A» O Aj = for i ^ j, then / N \ JV A = A n 5 - A n ( (J ^ J = (J (A n Ai) (9) \ N P(A) = Pl\J(AnA i )\=Y i P(AnA i ). (10) Prom (5), this becomes N P(A) = Y,P(Ai)P(E\Ai), (11) i=l SO P{Ai)P(A\Ai) P(Ai\A) N £ P(Ai)P(A\Ai) 3 = 1 (12) 5ee also CONDITIONAL PROBABILITY, INDEPENDENT Statistics References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, p. 810, 1992. Bayes' Theorem see Bayes' Formula Beam Detector Bayesian Analysis A statistical procedure which endeavors to estimate pa- rameters of an underlying distribution based on the ob- served distribution. Begin with a "PRIOR Distribu- tion" which may be based on anything, including an assessment of the relative likelihoods of parameters or the results of non-Bayesian observations. In practice, it is common to assume a UNIFORM DISTRIBUTION over the appropriate range of values for the PRIOR Distri- bution. Given the Prior Distribution, collect data to obtain the observed distribution. Then calculate the LIKELI- HOOD of the observed distribution as a function of pa- rameter values, multiply this likelihood function by the PRIOR Distribution, and normalize to obtain a unit probability over all possible values. This is called the Posterior Distribution. The Mode of the distribu- tion is then the parameter estimate, and "probability intervals" (the Bayesian analog of Confidence Inter- vals) can be calculated using the standard procedure. Bayesian analysis is somewhat controversial because the validity of the result depends on how valid the PRIOR DISTRIBUTION is, and this cannot be assessed statisti- cally. see also Maximum Likelihood, Prior Distribution, Uniform Distribution References Hoel, P. G.; Port, S. C; and Stone, C. J. Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 36- 42, 1971. Iversen, G. R. Bayesian Statistical Inference. Thousand Oaks, CA: Sage Pub., 1984. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 799-806, 1992. Sivia, D. S. Data Analysis: A Bayesian Tutorial. New York: Oxford University Press, 1996. Bays' Shuffle A shuffling algorithm used in a class of RANDOM NUM- BER generators. References Knuth, D. E. §3.2 and 3.3 in The Art of Computer Program- ming, Vol. 2: Seminumerical Algorithms, 2nd ed. Read- ing, MA: Addison-Wesley, 1981. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 270-271, 1992. Beam Detector N. B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Bean Curve A "beam detector" for a given curve C is defined as a curve (or set of curves) through which every Line tangent to or intersecting C passes. The shortest 1- arc beam detector, illustrated in the upper left figure, has length L\ — n + 2. The shortest known 2-arc beam detector, illustrated in the right figure, has angles Ox « 1.286 rad 6 2 « 1.191 rad, (1) (2) given by solving the simultaneous equations 2 cos <9i -sin(§0 2 ) = (3) tan(§0i)cos(f 2 ) + sm{±0 2 )[sec 2 {±6 2 ) + 1] = 2. (4) The corresponding length is L 2 =2tt-26>i -0 2 + 2tan(§0i)+sec(|0 2 ) - cos(§<9 2 )+tan(§6>i) sin(±<9 2 ) = 4.8189264563. . . . (5) A more complicated expression gives the shortest known 3-arc length L 3 = 4.799891547. . .. Finch defines L = inf L n n>l (6) as the beam detection constant, or the Trench Dig- gers' Constant. It is known that L>n. References Croft, H, T.; Falconer, K, J.; and Guy, R. K. §A30 in Un- solved Problems in Geometry. New York: Springer- Verlag, 1991. Faber, V.; Mycielski, J.; and Pedersen, P. "On the Shortest Curve which Meets All Lines which Meet a Circle." Ann. Polon. Math. 44, 249-266, 1984. Faber, V. and Mycielski, J. "The Shortest Curve that Meets All Lines that Meet a Convex Body." Amer. Math. Monthly 93, 796-801, 1986. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/beam/beam.html. Makai, E. "On a Dual of Tarski's Plank Problem." In Diskrete Geometric 2 Kolloq., Inst. Math. Univ. Salzburg, 127-132, 1980. Stewart, L "The Great Drain Robbery." Sci. Amer., 206- 207, 106, and 125, Sept. 1995, Dec. 1995, and Feb. 1996. Bean Curve Beast Number 103 The Plane Curve given by the Cartesian equation x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 ). References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989. Beast Number The occult "number of the beast" associated in the Bible with the Antichrist. It has figured in many numerolog- ical studies. It is mentioned in Revelation 13:13: "Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666." The beast number has several interesting properties which numerologists may find particularly interesting (Keith 1982-83). In particular, the beast number is equal to the sum of the squares of the first 7 PRIMES 2 2 + 3 2 -h 5 2 + 7 2 + ll 2 + 13 2 + 17 2 = 666, (1) satisfies the identity 0(666) = 6-6-6, (2) where 4> is the Totient Function, as well as the sum ^2 = 666. (3) The number 666 is a sum and difference of the first three 6th Powers, 666 = l 6 - 2 6 + 3 6 (4) (Keith). Another curious identity is that there are ex- actly two ways to insert "+" signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321, 666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 (5) 666 = 9 + 87 + 6 + 543 + 21 (6) (Keith). 666 is a Repdigit, and is also a Triangular Number T 6 . 6 = T 36 = 666. (7) In fact, it is the largest Repdigit Triangular Num- ber (Bellew and Weger 1975-76). 666 is also a Smith Number. The first 144 Digits of n - 3, where n is Pi, add to 666. In addition 144 = (6 + 6) x (6 + 6) (Blatner 1997). A number of the form 2 1 which contains the digits of the beast number "666" is called an Apocalyptic Num- ber, and a number having 666 digits is called an APOC- ALYPSE Number. 104 Beatty Sequence Bei see also Apocalypse Number, Apocalyptic Num- ber, Bimonster, Monster Group References Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num- bers." J. Recr. Math. 8, 96-97, 1975-76. Blatner, D. The Joy of Pi. New York: Walker, back jacket, 1997. Castellanos, D. "The Ubiquitous tt." Math. Mag. 61, 153- 154, 1988. Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p, 96, 1993. Keith, M. "The Number of the Beast." http://users.aol. com/s6sj7gt/mike666.htm. Keith, M. "The Number 666." J. Recr. Math. 15, 85-87, 1982-1983. Bee A 4-P0LYHEX. References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight- of -Mind from Scientific American. New York: Vintage, p. 147, 1978. Behrens-Fisher Test see Fisher-Behrens Problem Beatty Sequence The Beatty sequence is a Spectrum Sequence with an Irrational base. In other words, the Beatty sequence corresponding to an Irrational Number 6 is given by [0J, [20 \, [30J, . . . , where \_x\ is the Floor Function. If a and f3 are Positive Irrational Numbers such that 1 1 , a p then the Beatty sequences [a J , [2aJ , . . . and [f3\ , \_W\ > . . . together contain all the POSITIVE INTEGERS without repetition. References Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the Return of Dr. Matrix, reissue ed. New York: W. H. Free- man, p. 21, 1989. Graham, R. L.; Lin, S.; and Lin, C.-S. "Spectra of Numbers." Math. Mag. 51, 174-176, 1978. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 227, 1994. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 29-30, 1973. Beauzamy and Degot's Identity For P, Q, R, and S POLYNOMIALS in n variables [PQ,RS]= J^ A ii,...,t n >0 iil---i n ] - vhere A=[Rl i i>-"M(D li ...,D n )Q(x u ... i x n ) XP (il, - ,iB) (ft 2?n)5(Xl,.. M In)] Di = d/dxi is the Differential Operator, [X,Y] is the Bombieri Inner Product, and p(ti,...,i™) =D i 1 1 -.-D i r TP. Behrraann Cylindrical Equal- Area Projection A Cylindrical Area-Preserving projection which uses 30° N as the no-distortion parallel. References Dana, P. H. "Map Projections." http://www.utexas.edu/ depts/grg/gcraft/notes/mapproj/mapproj ,html, Bei I Bei z| 10000 .10 5000 -1000UE^^^^^/5 -500 Re[z] ^i^-lO Re[z]" 5 ^10 The Imaginary Part of J„(xe 3vi/4 ) = ber„(a;) +ibei„(x). (1) The special case v = gives Jo(iVix) = ber(rc) + ibei(sc), (2) where J Q (z) is the zeroth order BESSEL FUNCTION OF the First Kind. bei (x) = ^ [(2n) , ]2 (3) see also Reznik's Identity see also Ber, Bessel Function, Kei, Kelvin Func- tions, Ker Bell Curve Bell Number 105 References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972. Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 543-554, 1987. Bell Curve see Gaussian Distribution, Normal Distribution Bell Number The number of ways a Set of n elements can be PARTI- TIONED into nonempty Subsets is called a Bell Num- ber and is denoted B n . For example, there are five ways the numbers {1, 2, 3} can be partitioned: {{1}, {2}, {3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{1}, {2, 3}}, and {{1, 2, 3}}, so B 3 = 5. B = 1 and the first few Bell numbers for n = 1, 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (Sloane's A000110). Bell numbers are closely related to CATALAN NUMBERS. The diagram below shows the constructions giving B 3 = 5 and B± = 15, with line segments representing elements in the same Subset and dots representing subsets con- taining a single element (Dickau). B, The Integers B n can be defined by the sum fc=i ^ J = {l} isa (i) where s£ fc) = i '," *> is a STIRLING NUMBER OF THE SECOND Kind, or by the generating function B„ 6 = 2^ (2) The Bell numbers can also be generated using the BELL Triangle, using the Recurrence Relation Jn+l (3) where (£) is a Binomial Coefficient, or using the formula of Comtet (1974) B n -E m (4) where \x] denotes the Ceiling Function. The Bell number B n is also equal to n (l), where <t> n (x) is a Bell Polynomial. Dobinski's Formula gives the nth Bell number oo (5) Lovasz (1993) showed that this formula gives the asymp- totic limit -1/2 [A(n)] n+l/2 A(n)-n-l where A(n) is defined implicitly by the equation A(n)log[A(n)] = n. A variation of DOBINSKI'S FORMULA gives - -«■ ( _ 1)S B * = E 5- E (6) (?) (8) for 1 < k < n (Pitman 1997). de Bruijn (1958) gave the asymptotic formula InBn , , , In Inn 1 = lnn — Inlnn — 1 + — h - — n Inn Inn WlnlnnX 2 ^2 V Inn / In Inn (Inn) 2 Touchard's Congruence states B p+k = B k + B k+1 (mod p) , (9) (10) when p is Prime. The only PRIME Bell numbers for n < 1000 are B 2 , B 3i B 7 , B 13 , B 42 , and £55. The Bell numbers also have the curious property that Bq B\ B\ £2 B n ?n + l B 2 B 3 B n ^ B n B n +i B 2n J[n\ (11) (Lenard 1986). see also Bell Polynomial, Bell Triangle, Dobin- ski's Formula, Stirling Number of the Second Kind, Touchard's Congruence 106 Bell Polynomial Beltrami Differential Equation References Bell, E. T. "Exponential Numbers." Amer. Math. Monthly 41, 411-419, 1934. Comtet, L. Advanced Combinatorics. Dordrecht, Nether- lands: Reidel, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer- Verlag, pp. 91-94, 1996. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, pp. 102-109, 1958. Dickau, R. M. "Bell Number Diagrams." http:// forum . swarthmore.edu/advanced/robertd/bell.html. Gardner, M. "The Tinkly Temple Bells." Ch. 2 in Fractal Music, HyperCards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992. Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- phy of Two Special Number Sequences, 6th ed. Morgan- town, WV: Math Monongliae, 1985. Lenard, A. In Fractal Music, HyperCards, and More Math- ematical Recreations from Scientific American Magazine. (M. Gardner). New York: W. H. Freeman, pp. 35-36, 1992. Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p, of First Order Bell Exponential Integrals." Math. Comput. 16, 416-423, 1962. Lovasz, L. Combinatorial Problems and Exercises, 2nd ed. Amsterdam, Netherlands: North-Holland, 1993. Pitman, J. "Some Probabilistic Aspects of Set Partitions." Amer. Math. Monthly 104, 201-209, 1997. Rota, G.-C. "The Number of Partitions of a Set." Amer. Math. Monthly 71, 498-504, 1964. Sloane, N. J. A. Sequence A000110/M1484 in "An On-Line Version of the Encyclopedia of Integer Sequences." Bell Polynomial 0.2 0.4 0.6 0.8 1 Two different GENERATING FUNCTIONS for the Bell polynomials for n > are given by <t> n {x) =e x ^ k n ~ 1 x k The Bell polynomials are denned such that <f> n (l) = B nj where B n is a Bell NUMBER. The first few Bell poly- nomials are 4>o(x <pi(x 4>2(x fo{x (J>a{x <p 6 (x = 1 = X = x + x 2 - x + 3z 2 + x 3 = x + 7x 2 + 6x 3 + x 4 = x 4- 15x 2 + 25a; 3 + 10z 4 + x 5 = x + Six 2 + 90x 3 + 65z 4 + 15a; 5 + x 6 . see also Bell Number References Bell, E. T. "Exponential Polynomials." 258-277, 1934. Ann. Math. 35, Bell Triangle 12 5 15 52 203 877 ... 1 3 10 37 151 674 \ 2 7 27 114 523 \ 5 20 87 409 \ 15 67 322 \ 52 255 •■. 203 ■-. A triangle of numbers which allow the Bell Numbers to be computed using the Recurrence Relation = Va B n+1 = 2^B k { n k k-o see also Bell Number, Clark's Triangle, Leibniz Harmonic Triangle, Number Triangle, Pascal's Triangle, Seidel-Entringer-Arnold Triangle Bellows Conjecture see Flexible Polyhedron Beltrami Differential Equation For a measurable function /z, the Beltrami differential , equation is given by n~ 1 s v 4> n (x) = x^2 [ k-1 j^" 1 ^)' where (£) is a Binomial Coefficient. where f z is a PARTIAL DERIVATIVE and z* denotes the Complex Conjugate of z. see also QUASICONFORMAL MAP References Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1087, 1980. Beltrami Field Ben ford's Law 107 Beltrami Field A Vector Field u satisfying the vector identity u x (V x u) = where A x B is the CROSS Product and V x A is the CURL is said to be a Beltrami field. see also DlVERGENCELESS FIELD, IRROTATIONAL Field, Solenoidal Field Beltrami Identity An identity in CALCULUS OF VARIATIONS discovered in 1868 by Beltrami. The Euler-Lagrange Differen- tial Equation is d£__d_ dy dx (&)-* Now, examine the DERIVATIVE of x — ~ l/x T" n t/xx ~r • ax oy oy x ox Solving for the df /dy term gives dy 1 dx dy x 0/ b dx' Now, multiplying (1) by y x gives (i) (2) (3) (4) (5) (6) This form is especially useful if f x = 0, since in that case 0/ _ d_ oy ax dy* J Substituting (3) into (4) then gives dx dy x Vxz dx x dx \dy x dx dx \ - y *dyZ) = dx which immediately gives / dy x = 0, dy x (7) (8) where C is a constant of integration. The Beltrami identity greatly simplifies the solution for the minimal AREA SURFACE OF REVOLUTION about a given axis between two specified points. It also al- lows straightforward solution of the BRACHISTOCHRONE Problem. see also Brachistochrone Problem, Calculus of Variations, Euler-Lagrange Differential Equa- tion, Surface of Revolution Bend (Curvature) Given four mutually tangent circles, their bends are de- fined as the signed CURVATURES of the CIRCLES. If the contacts are all external, the signs are all taken as Pos- itive, whereas if one circle surrounds the other three, the sign of this circle is taken as NEGATIVE (Coxeter 1969). see also Curvature, Descartes Circle Theorem, Soddy Circles References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New- York: Wiley, pp. 13-14, 1969. Bend (Knot) A Knot used to join the ends of two ropes together to form a longer length. References Owen, P. Knots. Philadelphia, PA: Courage, p. 49, 1993. Benford's Law Also called the FIRST DIGIT LAW, FIRST DIGIT PHE- NOMENON, or Leading Digit Phenomenon. In list- ings, tables of statistics, etc., the DIGIT 1 tends to oc- cur with Probability ~ 30%, much greater than the expected 10%. This can be observed, for instance, by examining tables of LOGARITHMS and noting that the first pages are much more worn and smudged than later pages. The table below, taken from Benford (1938), shows the distribution of first digits taken from several disparate sources. Of the 54 million real constants in Plouffe's "Inverse Symbolic Calculator" database, 30% begin with the Digit 1. Title First Digit # 12 3 4 5 6 7 8 9 Rivers, Area 31.0 16.4 10.7 11.3 7.2 8.6 5.5 4.2 5.1 335 Population 33.9 20.4 14.2 8.1 7.2 6.2 4.1 3.7 2.2 3259 Constants 41.3 14.4 4.8 8.6 10.6 5.8 1.0 2.9 10.6 104 Newspapers 30.0 18.0 12.0 10.0 8.0 6.0 6.0 5.0 5.0 100 Specific Heat 24.0 18.4 16.2 14.6 10.6 4.1 3.2 4.8 4.1 1389 Pressure 29.6 18.3 12.8 9.8 8.3 6.4 5.7 4.4 4.7 703 H.P. Lost 30.0 18.4 11.9 10.8 8.1 7.0 5.1 5.1 3.6 690 Mol. Wgt. 26.7 25.2 15.4 10.8 6.7 5.1 4.1 2.8 3.2 1800 Drainage 27.1 23.9 13.8 12.6 8.2 5.0 5.0 2.5 1.9 159 Atomic Wgt. 47.2 18.7 5.5 4.4 6.6 4.4 3.3 4.4 5.5 91 n" 1 , sfn 25.7 20.3 9.7 6.8 6.6 6.8 7.2 8.0 8.9 5000 Design 26.8 14.8 14.3 7.5 8.3 8.4 7.0 7.3 5.6 560 Reader's Dig. 33.4 18.5 12.4 7.5 7.1 6.5 5.5 4.9 4.2 308 Cost Data 32.4 18.8 10.1 10.1 9.8 5.5 4.7 5.5 3.1 741 X-Ray Volts 27.9 17.5 14.4 9.0 8.1 7.4 5.1 5.8 4.8 707 Am. League 32.7 17.6 12.6 9.8 7.4 6.4 4.9 5.6 3.0 1458 Blackbody 31.0 17.3 14.1 8.7 6.6 7.0 5.2 4.7 5.4 1165 Addresses 28.9 19.2 12.6 8.8 8.5 6.4 5.6 5.0 5.0 342 n 1 , n 2 - • - n\ 25.3 16.0 12.0 10.0 8.5 8.8 6.8 7.1 5.5 900 Death Rate 27.0 18.6 15.7 9.4 6.7 6.5 7.2 4.8 4.1 418 Average 30.6 18.5 12.4 9.4 8.0 6.4 5.1 4.9 4.7 1011 Prob. Error 0.8 0.4 0.4 0.3 0.2 0.2 0.2 0.2 0.3 108 Benham's Wheel Benson's Formula In fact, the first SIGNIFICANT DIGIT seems to follow a Logarithmic Distribution, with P(n) « log(n + 1) - logn for n — 1, . . . , 9. One explanation uses Central Limit- like theorems for the MANTISSAS of random variables under Multiplication. As the number of variables in- creases, the density function approaches that of a LOG- ARITHMIC DISTRIBUTION. References Benford, F. "The Law of Anomalous Numbers." Proc. Amer. Phil Soc. 78, 551-572, 1938. Boyle, J. "An Application of Fourier Series to the Most Sig- nificant Digit Problem." Amer. Math. Monthly 101, 879™ 886, 1994. Hill, T. P. "Base-Invariance Implies Benford 's Law." Proc. Amer. Math. Soc. 12, 887-895, 1995. Hill, T. P. "The Significant-Digit Phenomenon." Amer. Math. Monthly 102, 322-327, 1995. Hill, T. P. "A Statistical Derivation of the Significant-Digit Law." Stat Sci. 10, 354-363, 1996. Hill, T. P. "The First Digit Phenomenon." Amer. Sci. 86, 358-363, 1998. Ley, E. "On the Peculiar Distribution of the U.S. Stock In- dices Digits." Amer. Stat. 50, 311-313, 1996. Newcomb, S. "Note on the Frequency of the Use of Digits in Natural Numbers." Amer. J. Math. 4, 39-40, 1881. Nigrini, M. "A Taxpayer Compliance Application of Ben- ford's Law." J. Amer. Tax. Assoc. 18, 72-91, 1996. Plouffe, S. "Inverse Symbolic Calculator." http://www.cecm. sfu.ca/projects/ISC/. Raimi, R. A. "The Peculiar Distribution of First Digits." Sci. Amer. 221, 109-119, Dec. 1969. Raimi, R. A. "The First Digit Phenomenon." Amer. Math, Monthly 83, 521-538, 1976. Benham's Wheel An optical ILLUSION consisting of a spinnable top marked in black with the pattern shown above. When the wheel is spun (especially slowly), the black broken lines appear as green, blue, and red colored bands! References Cohen, J. and Gordon, D. A. "The Prevost-Fechner-Benham Subjective Colors." Psycholog. Bull. 46, 97-136, 1949. Festinger, L.; Allyn, M. R.; and White, C. W. "The Percep- tion of Color with Achromatic Stimulation." Vision Res. 11, 591-612, 1971. Fineman, M. The Nature of Visual Illusion. New York: Dover, pp. 148-151, 1996. Trolland, T. L. "The Enigma of Color Vision." Amer. J. Physiology 2, 23-48, 1921. Bennequin's Conjecture A BRAID with M strands and R components with P positive crossings and N negative crossings satisfies \P - N\ < 2U + M - R < P + iV, where U is the UNKNOTTING NUMBER. While the second part of the Inequality was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronheimer and Mrowka on MlLNOR'S CONJECTURE (and, indepen- dently, using Menasco's Theorem). see also Braid, Menasco's Theorem, Milnor's Con- jecture, Unknotting Number References Bennequin, D. "L'instanton gordien (d'apres P. B. Kron- heimer et T. S. Mrowka)." Asterisque 216, 233-277, 1993. Birman, J. S. and Menasco, W. W. "Studying Links via Closed Braids. II. On a Theorem of Bennequin." Topology Appl. 40, 71-82, 1991. Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le nombre gordien des nceuds algebriques." Enseign. Math. 30, 173-222, 1984. Boileau, M. and Weber, C. "Le probleme de J. Milnor sur le nombre gordien des nceuds algebriques." In Knots, Braids and Singularities (Plans- sur- Bex, 1982). Geneva, Switzer- land: Monograph. Enseign. Math. Vol. 31, pp. 49-98, 1983. Cipra, B. What's Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc, pp. 8-13, 1994. Kronheimer, P. B. "The Genus-Minimizing Property of Al- gebraic Curves." Bull. Amer. Math. Soc. 29, 63-69, 1993. Kronheimer, P. B. and Mrowka, T. S, "Gauge Theory for Embedded Surfaces. I." Topology 32, 773-826, 1993. Kronheimer, P. B. and Mrowka, T. S. "Recurrence Relations and Asymptotics for Four-Manifold Invariants." Bull. Amer. Math. Soc. 30, 215-221, 1994. Menasco, W. W. "The Bennequin-Milnor Unknotting Con- jectures." C. R. Acad. Sci. Paris Ser. I Math. 318, 831- 836, 1994, Benson's Formula An equation for a LATTICE SUM with n = 3 i+i+fe+l i, j,k= — oo V J = 12?r ^ sech 2 (!7iVm 2 +n 2 ). m, n=l, 3, ... Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to —1.74756 . . ., a value known as "the" MADELUNG CONSTANT. see also MADELUNG CONSTANTS References Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 301, 1987. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/mdlung/mdlTing.html. Ber Ber | Ber z | The Real Part of J„(xe 3ni/4 ) = beT v (x)+ibei v (x). The special case v = gives Jo(iV^x) = ber(:r) + zbei(z), (1) (2) where J is the zeroth order BESSEL FUNCTION OF THE First Kind. i 2+4n ber( !B ) = ^ [(2n + 1)!] 2 ■ (3) see a/so Bei, Bessel Function, Kei, Kelvin Func- tions, Ker References Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func- tions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972. Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemi- sphere, pp. 543-554, 1987. Beraha Constants The nth Beraha constant is given by '2tt\ Be„ = 2 + 2 cos (!)- The first few are Bei =4 Be 2 = Be 3 = 1 Be 4 = 2 Be 5 = |(3 + \/5)« 2.618 Be 6 = 3 Be 7 = 2 + 2cos(|7r) « 3.247.... They appear to be ROOTS of the CHROMATIC POLY- NOMIALS of planar triangular GRAPHS. Be 4 is 0+1, where <p is the Golden Ratio, and Be 7 is the Silver Constant. References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983. Bernoulli Differential Equation 109 Berger-Kazdan Comparison Theorem Let M be a compact n-D Manifold with Injectivity radius inj(M). Then Vol(M) > qnj(M) with equality IFF M is ISOMETRIC to the standard round Sphere S n with Radius inj(M), where c n {r) is the Volume of the standard u-Hypersphere of Radius r. see also Blaschke Conjecture, Hypersphere, In- jective, Isometry References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Bergman Kernel A Bergman kernel is a function of a COMPLEX VARI- ABLE with the "reproducing kernel" property defined for any Domain in which there exist NONZERO Ana- lytic Functions of class L 2 (D) with respect to the Lebesgue Measure dV. References Hazewinkel, M. (Managing Ed,). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, pp. 356-357, 1988. Bergman Space Let G be an open subset of the COMPLEX PLANE C, and let Ll(G) denote the collection of all Analytic Func- tions / : G — y C whose Modulus is square integrable with respect to Area measure. Then L 2 a {G), sometimes also denoted A 2 (G), is called the Bergman space for G. Thus, the Bergman space consists of all the ANALYTIC FUNCTIONS in L 2 (G). The Bergman space can also be generalized to L?(G), where < p < oo. Bernoulli Differential Equation -£ + p(x)y = q(x)y n . Let v ~ y 1 " 71 for n ^ 1, then dv , . - n dy — — (l - n)y — . dx v }y dx Rewriting (1) gives y~ n f; = q(x) - p{x)y'~ n = q(x) - vp(x). Plugging (3) into (2), dv — = (I - n)[q(x) - vp(x)]. (1) (2) (3) (4) 110 Bernoulli Distribution Bernoulli Function Now, this is a linear First-Order Ordinary Differ- ential Equation of the form ^+vP(x) = Q{x), (5) where P(x) = (l-n)p(x) and Q(x) = (l-n)q(x). It can therefore be solved analytically using an Integrating Factor / Jri'U c Q(x)dx + C J P(x)dx _ (1 - n) / e (1 - n) / pW dx g(x) dx + C (1-n) J p(x) dx (6) where C is a constant of integration. If n = 1, then equation (1) becomes dy dx = y(q-p) — = {q-p)dx (?) (8) y = C 2 ef [q{x) - p{x)]dx . (9) The general solution is then, with C\ and C2 constants, l/(l-n) y= < '(l-nlp 1 - 10 /* a;) da; 4(3;) das+Ci (1 r^j p(ar) da: for n ^ 1 C e/ te(s)-p(x)]dx for n = 1. (10) Bernoulli Distribution A Distribution given by p M = {l q = 1 — p for n = for n — 1 — p n (l— p) n for n = 0,1. (i) (2) The distribution of heads and tails in COIN TOSSING is a Bernoulli distribution with p = q — 1/2. The GENER- ATING FUNCTION of the Bernoulli distribution is 1 M « = (*'"> = E ^w - p) 1_n = e °( 1 - p) + e ^ (3) SO M(t) = (l-p)+pe t (4) M'{t) = pe (5) M"(t)=pe t (6) M (n) (t)=pe\ (7) and the Moments about are Ml=M = M'(0)=p (8) & = M"(0) = p (9) M ; = M ( " ) (0)=p. (10) The Moments about the Mean are P2 = p' 2 -(pi) 2 =P-P 2 =p(l-p) (11) p.3=p 3 - 3^2Pi + 2(p' 1 ) 3 = p - 3p 2 + 2p 3 = p(l-p)(l-2p) (12) P4 = A»4 - 4/i3pi + 6^2 (m!) 2 - 3(pi) 4 = p - 4p 2 + 6p 3 - 3p 4 = p(l-p)(3p 2 -3p+l). (13) The Mean, Variance, Skewness, and Kurtosis are then P = Pi = P cr 2 - p.2 = p(l - p) _fi 3 _ p(l-p)(l-2p) 71 <T 3 [p(l - p)]3/ 2 _ l-2p H4 p(l-2p)(2p 2 -2p+l) 72 = —t - 3 = P 2 (l-P) 2 6p 2 - 6p + 1 p(l-p) To find an estimator for a population mean, V^ ( N (14) (15) (16) (17) ■0) JVp=0 v / Np=l = e[e + (i-8)] N - 1 = e, (18) so (p) is an Unbiased Estimator for 9, The probabil- ity of Np successes in N trials is then N Np e Np (i-o) Nq , (19) where __ [number of successes] _ n p- x =77- _ (20) see also BINOMIAL DISTRIBUTION Bernoulli Function see Bernoulli Polynomial Bernoulli Inequality Bernoulli Inequality (l + x) n > 1 + nx, (1) where x£l> — 1^0, n€Z> 1. This inequality can be proven by taking a MACLAURIN SERIES of (1 + x) n , Bernoulli Number 111 B n Bernoulli numbers may be calculated from the inte- gral (3) Bn=4n L **=r and analytically from (l+x) n = l+n^+|n(n-l)x +|n(n-l)(n-2)a; +.... (2) Since the series terminates after a finite number of terms for INTEGRAL n, the Bernoulli inequality for x > is obtained by truncating after the first-order term. When — 1 < x < 0, slightly more finesse is needed. In this case, let y = \x\ = — cc > so that < y < 1, and take (l-y) n = l-ny+in(n-l)y 2 - in(n-l)(n-2)y 3 + . . . . (3) Since each Power of y multiplies by a number < 1 and since the ABSOLUTE VALUE of the COEFFICIENT of each subsequent term is smaller than the last, it follows that the sum of the third order and subsequent terms is a Positive number. Therefore, (i - vT > i ny, (4) (1 -f x) n > 1 + nx, for - 1 < x < 0, (5) completing the proof of the INEQUALITY over all ranges of parameters. Bernoulli Lemniscate see Lemniscate Bernoulli Number There are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines the Ber- noulli numbers B* by the equations -12 *-*> n — 1 r>* ™2n i-rr^B^x (2n)! B{x 2 B$x A Btx [ 2! + 4! 6! -f ... (1) for \x\ < 27r, or 2(2n)! v . p=i 2(2n)! (2tt) 2 " C(2r (4) where ((z) is the RlEMANN Zeta Function. The first few Bernoulli numbers B* are b; = i 6 b; = 1 30 b; = 1 42 bx = 1 30 b; = 5 66 bi = 691 2,730 b; = 7 6 B' 8 = 3,617 510 b; = 43,867 798 ^10 = 174,611 330 *n = 854,513 138 Bernoulli numbers defined by the modern definition are denoted B n and also called "EVEN-index" Bernoulli numbers. These are the Bernoulli numbers returned by the Mathematical (Wolfram Research, Champaign, IL) function BernoulliB[n] . These Bernoulli numbers are a superset of the archaic ones B n since r 1 B n for n = for n = 1 (-l)^/ 2 )- 1 ^;^ for n even < for n odd. The B n can be defined by the identity B n x n (5) (6) , x (x\ ^ B n x 2r '- 2 COt (2J S T,~§M 2! + B* 2 x A 4! + D* ™6 -P3^ 6! + ... (2) for \x\ < 7T (Whittaker and Watson 1990, p. 125). Grad- shteyn and Ryzhik (1979) denote these numbers B n , while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted B, The These relationships can be derived using the generating function F(*,t) = £*££, (7) which converges uniformly for \t\ < 2tt and all x (Castel- lanos 1988). Taking the partial derivative gives dF(x,t) _ A B n ^(x)t n _ + ^ B n {x)t n dx Z— < ( n - i)! Z-, n \ (8) 112 Bernoulli Number The solution to this differential equation is F(x,t) = T(t)e xt , so integrating gives / F(x,t)dx = T(t) / e xt dx = T{t)^—- - ./o Jo l 00 *«- r 1 n = l * / ° (9) (a;) da? 1 + te _ 1 ~ 2^ n : (a;)da; = 1 (10) (11) (Castellanos 1988). Setting x = and adding t/2 to both sides then gives B2nt itcoth(It) = ^ n—O Letting t = 2ix then gives 00 . 2 xcotx = ^(-i)"^*^ n=0 (12) 2a 2 (2n)! (13) for x 6 [— 7r,7r], The Bernoulli numbers may also be calculated from the integral n! f z dz n=r 2^7 ^TT^+T' (14) (15) or from Bn= \ dn x ' [dx n e x — 1_ The Bernoulli numbers satisfy the identity *t>H*r)*- + - + (*i> +fl —- (16) where (£) is a BINOMIAL COEFFICIENT. An asymptotic Formula is lim \B 2n \ ~4,^{ — \ U . (17) n-voo \7re/ Bernoulli numbers appear in expressions of the form X^fe = i k P y wnere V — I? 2, Bernoulli numbers also appear in the series expansions of functions involving tanx, cotx, csccc, ln|sinx|, ln|cosa?|, ln|tanx|, tanhx, Bernoulli Number cothx, and cschx. An analytic solution exists for EVEN orders, B 2 (-l)- 1 2(2n)! ^ -2n _ (-l)- 1 2(2n)! (2») ir) 2 n ^—~f P p=i (2w) 2n : C(2n) (18) for n = 1, 2, ..., where ((2n) is the RlEMANN ZETA FUNCTION. Another intimate connection with the RlE- MANN Zeta Function is provided by the identity £ n = (-l) n+1 nC(l-n). (19) The Denominator of B 2k is given by the von Staudt- Clausen Theorem 2fc + l denom(B 2 fc) = fj P> (20) p prime (p-l)|2fc which also implies that the DENOMINATOR of B 2 k is Squarefree (Hardy and Wright 1979). Another curi- ous property is that the fraction part of B n in DECIMAL has a Decimal Period which divides n, and there is a single digit before that period (Conway 1996). B = 1 B 1 = 1 2 B 2 = 1 6 £4 = 1 30 B<> = 1 42 B 8 = 1 30 3io = 5 66 B12 = — B14 = 6 Big = — 691 2,730 798 174,611 518 #20 D 854,513 ^22 - i3 8 (Sloane's A000367 and A002445). In addition, B2n+1 — (21) for n = 1, 2, Bernoulli first used the Bernoulli numbers while com- puting X)fc=i ^ P - l* e used the property of the FlGURATE Number Triangle that £< (n + l)a n i + i (22) Bernoulli Number Bernoulli Polynomial 113 along with a form for a n j which he derived inductively to compute the sums up to n = 10 (Boyer 1968, p. 85). For p € Z > 0, the sum is given by where the NOTATION B^ means the quantity in ques- tion is raised to the appropriate POWER fc, and all terms of the form B™ are replaced with the corresponding Ber- noulli numbers B m . Written explicitly in terms of a sum of Powers, I> = B kP l fc!(p-fc + l)! j-Hl (24) Plouffe, S. "Plouffe's Inverter: Table of Current Records for the Computation of Constants." http://lacim.uqam.ca/ pi/records .html. Ramanujan, S. "Some Properties of Bernoulli's Numbers." J. Indian Math. Soc. 3, 219-234, 1911. Sloane, N. J. A. Sequences A000367/M4039 and A002445/ M4189 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Spanier, J. and Oldham, K. B. "The Bernoulli Numbers, B n ." Ch. 4 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 35-38, 1987, Wagstaff, S. S. Jr. "Ramanujan's Paper on Bernoulli Num- bers." J. Indian Math. Soc. 45, 49-65, 1981. Whit taker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Bernoulli's Paradox Suppose the Harmonic Series converges to h: It is also true that the COEFFICIENTS of the terms in such an expansion sum to 1 (which Bernoulli stated without proof). Ramanujan gave a number of curi- ous infinite sum identities involving Bernoulli numbers (Berndt 1994). G. J. Fee and S. Plouffe have computed #200,000? which has ~ 800,000 Digits (Plouffe). Plouffe and collabora- tors have also calculated B n for n up to 72,000. see also Argoh's Conjecture, Bernoulli Func- tion, Bernoulli Polynomial, Debye Functions, Euler-Maclaurin Integration Formulas, Euler Number, Figurate Number Triangle, Genocchi Number, Pascal's Triangle, Riemann Zeta Func- tion, von Staudt-Clausen Theorem References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972. Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula." §5.9 in Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 327-338, 1985. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 71, 1987. Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer- Verlag, pp. 81-85, 1994. Boyer, C. B. A History of Mathematics. New York: Wiley, 1968. Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer- Verlag, pp. 107-110, 1996. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, 1980. Hardy, G. H. and Wright, W. M. An Introduction to the The- ory of Numbers, 5th ed. Oxford, England: Oxford Univer- sity Press, pp. 91-93, 1979. Ireland, K. and Rosen, M. "Bernoulli Numbers." Ch. 15 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, pp. 228-248, 1990. Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663- 688, 1967. 00 Then rearranging the terms in the sum gives h-l = h, which is a contradiction. References Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 39-40, 1979. Bernoulli Polynomial There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by B n (x)i and the archaic Bernoulli polynomial by -B*(x). These definitions correspond to the BERNOULLI NUM- BERS evaluated at 0, B n = B n (0) b: = s;(o). They also satisfy and B„(l) = (-l) n B n (0) B n (l-x) = (-l) n B n (x) (1) (2) (3) (4) 114 Bernoulli Polynomial Bernstein's Constant (Lehmer 1988). The first few Bernoulli POLYNOMIALS are B (x) = l B!(x) = x- \ ' B 2 (x) = x 2 -i+ | B 3 (x) = x 3 - §z 2 + \x B A {x) = x 4 -2x z + x 2 - ^ B 5 (x) = x 5 -%x 4 + lx 3 -±x B 6 (x) = x 6 - 3x 5 + f x 4 ~ \x 2 + ^. Bernoulli (1713) defined the POLYNOMIALS in terms of sums of the Powers of consecutive integers, fc=0 &"- 1 = -[B n {m) - B„(0)]. (5) Euler (1738) gave the Bernoulli POLYNOMIALS B n (x) in terms of the generating function e 4 - 1 ^-^ n\ They satisfy recurrence relation dB n T = nB - l(l) (Appell 1882), and obey the identity B n (x) = (B + x) n , (6) (7) (8) where B k is interpreted here as Bk(x). Hurwitz gave the Fourier Series B n {x) (2«) - ^ A-V"^ (9) for < x < 1, and Raabe (1851) found m-l ~ 1Z B " ( X + ) = m " n5 "( mX )' ( 10 ) fc=0 A sum identity involving the Bernoulli POLYNOMIALS is f2(™)B k (a)B m - k (0) = _( m -l)B m (a + /3)+m(a + /3-l)B m _i(a + /3) (11) for an INTEGER m and arbitrary REAL NUMBERS a and P. see also Bernoulli Number, Euler-Maclaurin In- tegration Formulas, Euler Polynomial References Abramowitz, M. and Stegun, C. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972. Appell, P. E. "Sur une classe de polynomes." Annales d'Ecole Normal Superieur, Ser. 2 9, 119-144, 1882. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, p. 330, 1985. Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. Published posthumously. Euler, L. "Methodus generalis summandi progressiones." Comment. Acad. Set. Petropol. 6, 68-97, 1738. Lehmer, D. H. "A New Approach to Bernoulli Polynomials." Amer. Math. Monthly. 95, 905-911, 1988. Lucas, E. Ch. 14 in Theorie des Nombres. Paris, 1891. Raabe, J. L. "Zuruckfiihrung einiger Summen und bes- timmten Integrale auf die Jakob Bernoullische Function." J. reine angew. Math. 42, 348-376, 1851. Spanier, J. and Oldham, K. B. "The Bernoulli Polynomial B n (x)" Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167-173, 1987. Bernoulli's Theorem see Weak Law of Large Numbers Bernoulli Trial An experiment in which s TRIALS are made of an event, with probability p of success in any given TRIAL. Bernstein-Bezier Curve see Bezier Curve Bernstein's Constant N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Let E n (f) be the error of the best uniform approxima- tion to a Real function f(x) on the Interval [—1,1] by Real Polynomials of degree at most n. If «(*) = M> (i) then Bernstein showed that 0.267... < lim 2nE 2n {a) < 0.286. (2) n— >oo He conjectured that the lower limit {(5) was f3 — 1/(2^/7?). However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed /? = 0.2801694990.... (3) For rational approximations p(x)/q(x) for p and q of degree m and n, D. J. Newman (1964) proved i e _ 9v ^ < Enn ( a) < 3e -^ (4) Bernstein's Inequality Bernstein-Szego Polynomials 115 for n > 4. Gonchar (1967) and Bulanov (1975) improved the lower bound to -7rVn+T < K,„(a) < 3e~^\ (5) Vjacheslavo (1975) proved the existence of POSITIVE constants m and M such that m<e Vy/K E^ n [pL) <M (6) (Petrushev 1987, pp. 105-106). Varga et al (1993) con- jectured and Stahl (1993) proved that lim e 2n i?2Ti,2n,(a) = 8. n—too (7) Bernstein Minimal Surface Theorem If a Minimal Surface is given by the equation z = f(x, y) and / has CONTINUOUS first and second PARTIAL Derivatives for all Real x and y, then / is a Plane. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 369, 1988. Bernstein Polynomial The Polynomials defined by B itn (t)= ('.') **(!-*)* References Bulanov, A. P. "Asymptotics for the Best Rational Approxi- mation of the Function Sign a." Mat. Sbornik 96, 171-178, 1975. Finch, S. "Favorite Mathematical Constants." http://www. mathsof t . com/asolve/constant/brnstn/brnstn.html. Gonchar, A. A. "Estimates for the Growth of Rational Func- tions and their Applications." Mat. Sbornik 72, 489-503, 1967. Newman, D. J. "Rational Approximation to |x|." Michigan Math. J. 11, 11-14, 1964. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Stahl, H. "Best Uniform Rational Approximation of \x\ on [-1,1]." Russian Acad. Sci. Sb. Math. 76, 461-487, 1993. Varga, R. S. Scientific Computations on Mathematical Prob- lems and Conjectures. Philadelphia, PA: SIAM, 1990. Varga, R. S. and Carpenter, A. J. "On a Conjecture of S. Bernstein in Approximation Theory." Math. USSR Sbornik 57, 547-560, 1987. Varga, R. S.; Rut tan, A.; and Carpenter, A. J. "Numerical Results on Best Uniform Rational Approximations to |x| on [-1,+1]. Math. USSR Sbornik 74, 271-290, 1993. Vjacheslavo, N. S. "On the Uniform Approximation of \x\ by Rational Functions." Dokl Akad. Nauk SSSR 220, 512- 515, 1975. Bernstein's Inequality Let P be a POLYNOMIAL of degree n with derivative P' . Then HP'lloo <n||P||oo, where (™) is a BINOMIAL COEFFICIENT. The Bernstein polynomials of degree n form a basis for the POWER Polynomials of degree n. see also Bezier Curve Bernstein's Polynomial Theorem If g(9) is a trigonometric POLYNOMIAL of degree m sat- isfying the condition \g(0) \ < 1 where 6 is arbitrary and real, then g'{9) < m. References Szego, G. Orthogonal Polynomials, ^.th ed. Providence, RI: Amer. Math. Soc, p. 5, 1975. Bernstein-Szego Polynomials The POLYNOMIALS on the interval [-1,1] associated with the Weight Functions w{x) — (1 - z 2 ) _1/ w(x) = (1 - x 2 ) 1/2 w(x) - 1 + x* also called BERNSTEIN POLYNOMIALS. References Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc, pp. 31-33, 1975. where |F||oo = mK|PW|. 116 Berry-Osseen Inequality Bertrand's Problem Berry-Osseen Inequality Gives an estimate of the deviation of a DISTRIBUTION Function as a Sum of independent Random Vari- ables with a Normal Distribution. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 369, 1988. Berry Paradox There are several versions of the Berry paradox, the original version of which was published by Bertrand Russell and attributed to Oxford University librarian Mr. G. Berry. In one form, the paradox notes that the number "one million, one hundred thousand, one hun- dred and twenty one" can be named by the description: "the first number not nameable in under ten words." However, this latter expression has only nine words, so the number can be named in under ten words, so there is an inconsistency in naming it in this manner! References Chaitin, G. J. "The Berry Paradox." 1995. Complexity 1, 26-30, Bertelsen's Number An erroneous value of 7r(10 9 ), where tt(x) is the PRIME Counting Function. Bertelsen's value of 50,847,478 is 56 lower than the correct value of 50,847,534. References Brown, K. S. "Bertelsen's Number." http://www.seanet . com/-ksbrown/kmath049.htm. Bertini's Theorem The general curve of a system which is LINEARLY IN- DEPENDENT on a certain number of given irreducible curves will not have a singular point which is not fixed for all the curves of the system. References Coolidge, J. L. A Treatise on Algebraic Plane Curves York: Dover, p. 115, 1959. New Bertrand Curves Two curves which, at any point, have a common princi- pal Normal Vector are called Bertrand curves. The product of the TORSIONS of Bertrand curves is a con- stant. Bertrand's Paradox see Bertrand's Problem Bertrand's Postulate If n > 3, there is always at least one PRIME between n and 2n — 2. Equivalently, if n > 1, then there is always at least one PRIME between n and 2n, It was proved in 1850-51 by Chebyshev, and is therefore sometimes known as Chebyshev's Theorem. An elegant proof was later given by Erdos. An extension of this result is that if n > k, then there is a number containing a Prime divisor > k in the sequence n, n + 1, . . . , n + k — 1. (The case n = k + 1 then corresponds to Bertrand's postu- late.) This was first proved by Sylvester, independently by Schur, and a simple proof was given by Erdos. A related problem is to find the least value of 8 so that there exists at least one PRIME between n and n + O(n ) for sufficiently large n (Berndt 1994). The smallest known value is 9 = 6/11 -f e (Lou and Yao 1992). see also Choquet Theory, de Polignac's Conjec- ture, Prime Number References Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer- Verlag, p. 135, 1994. Erdos, P. "Ramanujan and I." In Proceedings of the Inter- national Ramanujan Centenary Conference held at Anna University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer- Verlag, pp. 1-20, 1989. Lou, S. and Yau, Q. "A Chebyshev's Type of Prime Number Theorem in a Short Interval (II)." Hardy- Ramanujan J. 15, 1-33, 1992. Bertrand's Problem What is the Probability that a Chord drawn at Ran- dom on a Circle of Radius r has length > r? The an- swer, it turns out, depends on the interpretation of "two points drawn at RANDOM." In the usual interpretation that Angles #i and 6i are picked at Random on the Circumference, t, 7r " f 2 P= *-=-• 7T 3 However, if a point is instead placed at RANDOM on a Radius of the Circle and a Chord drawn Perpen- dicular to it, r 2 The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated CIR- CLE, a slightly smaller CIRCLE INSCRIBED in the first, or for a CIRCLE of the same size but with its center slightly offset. Jaynes (1983) shows that the interpre- tation of "Random" as a continuous Uniform Distri- bution over the RADIUS is the only one possessing all these three invariances. References Bogomolny, A. "Bertrand's Paradox." http: //www. cut-the- knot . com/bertrand.html. Jaynes, E. T. Papers on Probability, Statistics, and Statisti- cal Physics. Dordrecht, Netherlands: Reidel, 1983. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42- 45, 1995. Bertrand's Test Bertrand's Test A Convergence Test also called de Morgan's and Bertrand's Test. If the ratio of terms of a Series {flnj^Li can be written in the form an 1 = 1 + - + Pn n n In n ' then the series converges if lim n ->oo pn > 1 and diverges if lim n _^oo/0n < 1, where lim w ->oo is the Lower Limit and lim n _>.oo is the Upper Limit. see also Rummer's Test References Bromwich, T. J. Pa and MacRobert, T. M. An Introduc- tion to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40, 1991. Bertrand's Theorem see Bertrand's Postulate Bessel Differential Equation Bessel Differential Equation m?)y = 0. Equivalently, dividing through by z 2 , 2d 2 y dy 2 „^ 117 (i) The solutions to this equation define the BESSEL FUNC- TIONS. The equation has a regular SINGULARITY at and an irregular SINGULARITY at oo. A transformed version of the Bessel differential equation given by Bowman (1958) is * 2 § + (2p+l)sg + (aV r + /? 2 )y = 0. (3) The solution is Besov Space A type of abstract Space which occurs in Spline and Rational Function approximations. The Besov space Bp yQ is a complete quasinormed space which is a Ba- NACH Space when 1 < p, q < oo (Petrushev and Popov 1987). References Bergh, J. and Lofstrom, J. Interpolation Spaces. New York: Springer- Verlag, 1976. Peetre, J. New Thoughts on Besov Spaces. Durham, NC: Duke University Press, 1976. Petrushev, P. P. and Popov, V. A. "Besov Spaces." §7.2 in Rational Approximation of Real Functions. New York: Cambridge University Press, pp. 201-203, 1987. Triebel, H. Interpolation Theory, Function Spaces, Differen- tial Operators. New York: Elsevier, 1978. Bessel's Correction The factor (N — 1)/N in the relationship between the Variance a and the Expectation Values of the Sam- ple Variance, y = x p I 2\ N-l 2 s 2 = (x 1 ) - (x) 2 . N lSl 2 +N 2 s 2 2 Ni+N 2 -2 ' see also Sample Variance, Variance where For two samples, (i) (2) (3) c 1 J q/r (^-)+c 2 r g/r (^) where q = vV - P\ (4) (5) J and Y are the Bessel Functions of the First and SECOND KINDS, and C\ and Ci are constants. Another form is given by letting y = x a J n (/3x' y ) i tj — yx~ a , and £ = 0x 7 (Bowman 1958, p. 117), then (6) The solution is = f x a [AJ n {(3x' r ) + BYniPx 1 )] for integral n V \ AJniffx 7 ) + BJ-niPx 1 )] for nonintegral u. (?) see also AlRY FUNCTIONS, ANGER FUNCTION, Bei, Ber, Bessel Function, Bourget's Hypothesis, Catalan Integrals, Cylindrical Function, Dini Expansion, Hankel Function, Hankel's Integral, Hemispherical Function, Kapteyn Series, Lip- schitz's Integral, Lommel Differential Equa- tion, Lommel Function, Lommel's Integrals, Neumann Series (Bessel Function), Parseval's Integral, Poisson Integral, Ramanujan's Inte- gral, Riccati Differential Equation, Sonine's Integral, Struve Function, Weber Functions, Weber's Discontinuous Integrals References Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, p. 550, 1953. 118 BesseVs Finite Difference Formula Bessel Function of the First Kind Bessel's Finite Difference Formula An Interpolation formula also sometimes known as The Bessel functions are more frequently defined as so- lutions to the Differential Equation f P = fo+ pSi/2 + B 2 {Sl + <$i ) + B 3 8l /2 + B 4 05$ + tf) + B 5 *? /a + ... ) (1) for p e [0, 1], where 6 is the Central Difference and Bin = ^Gln = g ("^2n + i*2n) (2) B2n + 1 = G2n + 1 ~ 2^ 2n ~ 2 (^ 2ri ~ ^2n) (**) £?2n = ^2n — G 2n +1 = Bin — #2n + l (4) F 2 n = t?2n+l = B 2n + #2n+l> (5) where Gk are the COEFFICIENTS from GAUSS'S BACK- WARD Formula and Gauss's Forward Formula and E k and Fk are the Coefficients from Everett's FOR- MULA. The i?fcS also satisfy B 2n {p) = B 2n (q) B 2n+X {p) = -B 2n +i(q), for (6) (7) (8) q = l-p. see also Everett's Formula References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing, Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Bessel's First Integral i r J n (x) = — / cos(n# - xsinO) d8, 77 Jo where J n (x) is a BESSEL FUNCTION OF THE FIRST Kind. Bessel's Formula see Bessel's Finite Difference Formula, Bes- sel's Interpolation Formula, Bessel's Statisti- cal Formula Bessel Function A function Z(x) defined by the RECURRENCE RELA- TIONS Zm + l + Z m — 1 — Zm and &m+l — ^m-1 Zm~l — —2 dx 2d 2 y dy 2 X dx^ +X dx- + {x m )y — 0. There are two classes of solution, called the BESSEL Function of the First Kind J and Bessel Func- tion of the Second Kind Y. (A Bessel Function OF THE THIRD Kind is a special combination of the first and second kinds.) Several related functions are also de- fined by slightly modifying the defining equations. see also Bessel Function of the First Kind, Bessel Function of the Second Kind, Bessel Function of the Third Kind, Cylinder Func- tion, Hemicylindrical Function, Modified Bes- sel Function of the First Kind, Modified Bessel Function of the Second Kind, Spherical Bessel Function of the First Kind, Spherical Bessel Function of the Second Kind References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Functions of Integer Order," "Bessel Functions of Fractional Order," and "Integrals of Bessel Functions." Chs. 9-11 in Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 355-389, 435-456, and 480-491, 1972. Arfken, G. "Bessel Functions." Ch. 11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-636, 1985. Bickley, W. G. Bessel Functions and Formulae. Cambridge, England: Cambridge University Press, 1957. Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Gray, A. and Matthews, G. B. A Treatise on Bessel Func- tions and Their Applications to Physics, 2nd ed. New York: Dover, 1966. Luke, Y. L. Integrals of Bessel Functions. New York: McGraw-Hill, 1962. McLachlan, N. W. Bessel Functions for Engineers, 2nd ed. with corrections. Oxford, England: Clarendon Press, 1961. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Bessel Functions of Integral Order" and "Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions." §6.5 and 6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 223-229 and 234-245, 1992. Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Bessel Function of the First Kind -0.2 Bessel Function of the First Kind Bessel Function of the First Kind 119 The Bessel functions of the first kind J n {x) are defined as the solutions to the BESSEL DIFFERENTIAL EQUATION which are nonsingular at the origin. They are some- times also called Cylinder Functions or Cylindri- cal Harmonics. The above plot shows J n (x) for n = 1, 2,..., 5. To solve the differential equation, apply FROBENIUS METHOD using a series solution of the form First, look at the special case m = —1/2, then (9) be- comes oo ^[a n n(n-l) + a n _ 2 ]x m +" = 0, (10) n(n — 1) Now let n = 2/, where / = 1, 2, fln-2. (11) &21 1 2/(2/ - 1) 0*21-2 (-i)' y = x k ^ a n x n = JT a n x n+k . (2) n=0 n=0 Plugging into (1) yields oo x 2 ^{k + n)(k + n~ l)a n x k+n - 2 n—0 oo oo +X Y^(k + n)OnX h+n - 1 +X 2 J2 a nX k+n n~ n — oo -m 2 ^2a n x n + k = (3) [2/(2/ - 1)][2(Z - 1)(2Z - 3)] ■ * - [2 - 1 • 1] 7«0, do 2 l l\(2l~l)\\ which, using the identity 2 l l\(2l - 1)!! = (2/)!, gives a 2/ = /rtlXI Q0> (12) (20! °' (13) Similarly, letting n = 21 + 1 fl2i+i = — 1 (2/ + l)(2/) «2i-l (-1)' [2/(2/ + 1)][2(I - 1)(2/ - 1)] ... [2 • 1 • 3][1] au (14) ^(fc + n)(fc + n - l)a n x fc+n + ^(fc + n)a n x fe+n which, using the identity 2 l l\(2l + 1)!! = (2/ + 1)!, gives ]Ta n - 2 x k+n -m 2 J2a n x n+k = 0. (4) n=2 n=0 The INDICIAL EQUATION, obtained by setting n = 0, is a [fc(£; - 1) + k - m 2 ] = a (k 2 - m 2 ) = 0. (5) Since ao is defined as the first NONZERO term, k 2 —m 2 = 0, so k = ±ra. Now, if k ~ m, oo ^[(m + n)(m + n - 1) + (m + n) - m 2 }a n x 7n+n n~0 oo + Y. an ~ 2^ m+n = (6) n^2 oo oo £][(m + n) 2 - m 2 ]a n x m+n + ^ a„_ 2 x m+n = (7) n=0 n = 2 oo oo ^ n(2m + n)a n i m+ " + J] a„. 2 2 ra+n = (8) n — n=2 OO ai(2m + 1) 4- ^[a n n(2m + n) + a„_ 2 ]a; m+n = 0. (9) (-1) 1 (-l) z a2/+1 " 2^/!(2/ + l)!! ai = (27TI)! ai ' (15) Plugging back into (2) with k = m = —1/2 gives 2/ = x 1/2 N a n a; n t=0 oo oo 2. a n x n + N^ a n x n _n~l,3,5,... n-0,2,4 oo oo E2J . V~^ CL21X + J> ^21 + lX -1/2 -1/2 n = 0,2,4,... 2J + 1 -1/2 J = 1=0 (-1) 2^ ( 2 n! X +ai 2^(2Z + l)! ;=o ■ ' z=o (20! -1/2/ , • \ = x ' (ao cos a; + a\ since). (2/ + 1) (16) The BESSEL FUNCTIONS of order ±1/2 are therefore de- fined as (17) (18) J-i/ 2 (x) =4/ — cosa; ' U 7TX Ji/ 2 (x) = 4/ — sinx, 17 7TZ 120 Bessel Function of the First Kind so the general solution for m = ±1/2 is y = a' J- 1/2 {x) + a 1 J 1/2 (x). (19) Now, consider a general m ^ —1/2. Equation (9) re- quires ai(2m+l)=0 (20) [a n n(2m + n) + a n _ 2 ]z m+n = (21) for n = 2, 3, . . . , so ai =0 n(2m + n ■fln-2 (22) (23) for n = 2, 3, Let n = 2Z + 1, where Z = 1, 2, . . . , then «2/ + l 1 Tfl2Z-l (2Z + l)[2(m + l) + l] = ... = /(n,m)ai =0, (24) where f(n,m) is the function of Z and m obtained by iterating the recursion relationship down to a\ . Now let n = 2Z, where Z = 1, 2, . . . , so 1 1 a 2* = ~~ 77777; r~^ a 2/-2 = —777 — 77^-2 2l{2m + 2l) " 4Z(m + Z) tn [4Z(m + Z)][4(Z - l)(m + Z - 1)] • • ■ [4 • (m + 1)] ao- (25) Plugging back into (9), a n x = > a n x + y a n x n = n = l,3,5,... n = 0,2,4,,,, E2I + m + l . \~^ 2I + m G 2 i-M^ + > a 2lX 1=0 (=0 „ v^ (_z}Y « + ™ = tin 7 X Z^ [4i(m + l)][4{l - l)(m + I - 1)] • • . [4 • (m + 1)] 1=0 [(-l) f m(m- l)---l]x 2t+m [4/(m + i)][4(i - l)(m + i - 1)] • ■ ■ [m(m - 1) ■ • • 1] Bessel Function of the First Kind Returning to equation (5) and examining the case k — — m, 00 ai(l-2m) + ^[a n ra(ra-2m) + a n _ 2 ]a; n ~ m = 0. (29) However, the sign of m is arbitrary, so the solutions must be the same for +ra and — m. We are therefore free to replace — m with — |m|, so 00 oi(l + 2|m|) + ^[o n n(n + 2|m|) + a n _ 2 ]x |m|+n = 0, n = 2 . (30) and we obtain the same solutions as before, but with m replaced by \m\. *J<m\X) '■ v^oo (-1)' 2Z+|m| f or | rn |^_I for m = — | for m = |. (31) We can relate J m and J_ m (when m is an Integer) by writing ( — lV 1=0 v ' (32) Now let 1 = 1' + m. Then J-m(x) = ^ (-1) Z' + m Z' + m=0 -1 2 2 <'+™(Z' + m)!Z! (-1)''+™ 2l'+m V I- 1 ) 2Z'+m Z^ 2 2 <'+™Z'!(Z'+m)! l' = — m + 2-1 2«'+"Z'!(Z'+m)! a;2 ' +m ' (33) i'=o v But Z'! = oo for Z' = -m, ...,-1, so the Denomina- tor is infinite and the terms on the right are zero. We therefore have --Ej^--£?&- w '-<*>- Es.J^fe'" 4 "-'- 1 '"^ \(m + l)\ ~ u ^ 2 2 <Z!(m + Z) Z=0 ' 1=0 / Now define OO ; Jm(x) = Jj 2 2 <+™Z!(m + Z)! x2 ' +m ' (27) where the factorials can be generalized to Gamma FUNCTIONS for nonintegral m. The above equation then becomes (34) Note that the Bessel Differential Equation is second-order, so there must be two linearly independent solutions. We have found both only for \m\ = 1/2. For a general nonintegral order, the independent solutions are J m and J~ m . When m is an INTEGER, the general (real) solution is of the form Z m = C 1 J m (x) + C 2 Y rn (x), (35) y = a 2 m m\J m (x) — a' J m (x). (28) Bessel Function of the First Kind Bessel Function of the First Kind 121 where J m is a Bessel function of the first kind, F m (a.k.a. iV m ) is the BESSEL FUNCTION OF THE SECOND Kind (a.k.a. Neumann Function or Weber Func- tion), and C\ and C 2 are constants. Complex solutions are given by the Hankel Functions (a.k.a. Bessel Functions of the Third Kind). The Bessel functions are ORTHOGONAL in [0, 1] with re- spect to the weight factor x. Except when 2n is a NEG- ATIVE Integer, Jrn(z) -1/2 2 2m+l/2 i m + l/2 r ( m+1 ^ Mo im (2iz) ) (36) where T(x) is the Gamma Function and M , m is a Whittaker Function. In terms of a Confluent Hypergeometric Func- tion of the First Kind, the Bessel function is written Mz) ^fryo^^ + i;-^ 2 )- (37) A derivative identity for expressing higher order Bessel functions in terms of Jo(x) is Jn(x) — i n T n li-j-) Jo( (38) where T n (x) is a Chebyshev Polynomial of the First Kind. Asymptotic forms for the Bessel functions are J - {x) * fd+T) (!) for x <^ 1 and J m (x) : / ran tt\ x V 2 4/ for x ^> 1. A derivative identity is d dx [x^Jmix)] = X^Jm-lix). An integral identity is uJo(u)du —uJ\{u). F Jo Some sum identities are 1 = [Jo(x)] 2 + 2[J 1 {x)f + 2[J 2 (x)] 2 + , 1 = J (x) + 2J 2 {x) + 2J A {x) + . . and the Jacobi-Anger Expansion % J n (z)e (39) (40) (41) (42) (43) (44) (45) which can also be written 00 e tzcose = J (z) + 2^2i n J n (z)cos(n8). (46) n=l The Bessel function addition theorem states 00 My + z) = ^ J™{y) J n-m{z). (47) m=-oo ROOTS of the FUNCTION J n (x) are given in the following table. zero J Q (x) Ji(x) J 2 {x) Mx) Mx) J*(x) 1 2.4048 3.8317 5.1336 6.3802 7.5883 8.7715 2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386 3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002 4 11.7915 13.3237 14.7960 16.2235 17.6160 18.9801 5 14.9309 16.4706 17.9598 19.4094 20,8269 22.2178 Let x n be the nth ROOT of the Bessel function Jo(#), then Y — 71 = 1 (Le Lionnais 1983). 2"n*J§y£n} = 0.38479... (48) The Roots of its Derivatives are given in the following table. zero Jo'(x) •V(z) •V(s) J 3 '(x) J 4 '(x) J 5 '(x) 1 3.8317 1.8412 3.0542 4.2012 5,3175 6.4156 2 7.0156 5.3314 6.7061 8.0152 9.2824 10.5199 3 10.1735 8.5363 9.9695 11.3459 12.6819 13.9872 4 13.3237 11.7060 13.1704 14.5858 15.9641 17.3128 5 16.4706 14.8636 16.3475 17.7887 19.1960 20.5755 Various integrals can be expressed in terms of Bessel functions 1 f 2 " * w - s y «•• ' cos <j> d(f> i i r J n (z) = — / cos(z sin — n6) d8 , n Jo which is BESSEL'S FIRST INTEGRAL, (49) (50) .-71 f* ./«(*) = —/ e izcose cos(n9)d0 (51) w Jo Jn{z) JL_ [ 2 \i V"* d<t> z cos <p in.1 (52) J, . . . , for n = 1, 2 2 x J»W 7r (2m for n = I, 2, . . . , sin n u cos(x cos u) du (53) 71— — OO T f~\ 1 I (x/2)(z-l/z) -71-1 , Jtl(x) = - — ; / e K ' A ' } z dz 2tvi J (54) 122 Bessel Function Fourier Expansion Bessel Function of the Second Kind for n > —1/2. Integrals involving J\(x) include (Bowman 1958, p. 108), so / J\ (x) dx = 1 Jo (55) r[¥ dx = h (56) cm xdx = — . 2 (57) see also BESSEL FUNCTION OF THE SECOND KIND, DE- bye's Asymptotic Representation, Dixon-Ferrar Formula, Hansen-Bessel Formula, Kapteyn Se- ries, Kneser-Sommerfeld Formula, Mehler's Bessel Function Formula, Nicholson's Formula, Poisson's Bessel Function Formula, Schlafli's Formula, Schlomilch's Series, Sommerfeld's Formula, Sonine-Schafheitlin Formula, Wat- son's Formula, Watson-Nicholson Formula, We- ber's Discontinuous Integrals, Weber's For- mula, Weber-Sonine Formula, Weyrich's For- mula References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func- tions J and V." §9.1 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972. Arfken, G. "Bessel Functions of the First Kind, J„(;r)" and "Orthogonality." §11.1 and 11,2 in Mathematical Meth- ods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-591 and 591-596, 1985. Lehmer, D. H. "Arithmetical Periodicities of Bessel Func- tions." Ann. Math. 33, 143-150, 1932. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 619-622, 1953. Spanier, J. and Oldham, K. B. "The Bessel Coefficients Jq(x) and Ji(x)" and "The Bessel Function J u (x)." Chs. 52-53 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987. Bessel Function Fourier Expansion Let n > 1/2 and e*i, a 2 , ...be the POSITIVE ROOTS of J n (x) — 0. An expansion of a function in the inter- val (0,1) in terms of BESSEL FUNCTIONS OF THE FIRST Kind f( x ) = /]A r J n (xa r ), (i) has Coefficients found as follows: / xf(x)J n (xai)dx = y^A r / xJ n (xa r )Jn(xai)dx. Jo r=1 Jo (2) But Orthogonality of Bessel Function Roots gives /' Jo xJ n (xai)J n (xa r )dx = ^Sl^Jn + 1 (&r) (3) ol °° / xf(x)J n (xai)dx - \ }^ Ar5i, r J n +i 2 (xa r ) J° r=l I A. T . . 2 (^,.\ U) = ^AiJ n+1 (on), and the COEFFICIENTS are given by 2 A t = J n+ i 2 (ai) f Jo xf(x)Jn(xai)dx. (5) References Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958. Bessel Function of the Second Kind A Bessel function of the second kind Y n (x) is a solution to the Bessel Differential Equation which is sin- gular at the origin. Bessel functions of the second kind are also called Neumann Functions or Weber Func- tions. The above plot shows Y n {x) for n = 1, 2, . . . , 5. Let v = Jm{x) be the first solution and u be the Other one (since the BESSEL DIFFERENTIAL EQUATION is second-order, there are two Linearly Independent solutions). Then xu + u + xu = XV + V + XV = 0. Take v x (1) - u x (2), x{u v — uv ) -\- u v — uv =0 — \x(uv — uv')] = 0, ax (i) (2) (3) (4) so x(uv — uv) = B, where B is a constant. Divide by xv 2 , uv — uv _ d /u\ _ B ( . v 2 dx \v ) xv 2 V f- J & ,2* (6) Bessel Function of the Third Kind Rearranging and using v = J m (x) gives u = AJm(x) + BJ m (x) I. dx XJrn \X~) = A , J m (x)-{-B'Y rn (x), (7) where the Bessel function of the second kind is denned by Y m (x) J m (x) cos(mir) — J_ m (x) sin(m7r) * Z. 2«+»*!(m + *)! [ 2 ln 1 2 j + 27 " bm+k ~ bk 1 v^ x~ m+2k (m -k-l)\ --J2 2-m4-2fcfc| (8) m = 0, 1, 2, . . . , 7 is the Euler-Mascheroni Con- stant, and Jo k = 0, (9) The function is given by Y n (z) = - / sin(z sin d-n0)d0 * Jo I!?-" — nt ( ,\ni —z sinh t + e~ nt (-l) n ]e dt, (10) Asymptotic equations are m() ~l-^(f) m m^0,x«l (U) rmW = V^ sm r T"4J * >>x ' (12) where r(z) is a Gamma Function. see also Bessel Function of the First Kind, Bour- get's Hypothesis, Hankel Function References Abramowitz, M. and Stegun, C. A. (Eds.). "Bessel Func- tions J and Y. n §9.1 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972. Arfken, G. "Neumann Functions, Bessel Functions of the Sec- ond Kind, N v (x). n §11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596- 604, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 625-627, 1953. Spanier, J. and Oldham, K. B. "The Neumann Function Y u (x)" Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533-542, 1987. Bessel Function of the Third Kind see Hankel Function Bessel's Inequality 123 Bessel's Inequality If f(x) is piecewise CONTINUOUS and has a general Fourier Series 5^Mi(x) (1) i with Weighting Function w(x), it must be true that / /O) - y^ai<fc(aQ w(x)dx > (2) + 5^a< 2 <t>i 2 (x)w{x)dx>0. (3) i J But the Coefficient of the generalized Fourier Se- ries is given by a m = / f(x)<f> m (x)w(x)dx, (4) so / f 2 (x)w(x)dx-2^2ai 2 -h^di 2 > (5) i i f{x)w(x)dx>Y^ai 2 - (6) i Equation (6) is an inequality if the functions <j>i are not Complete. If they are Complete, then the inequality (2) becomes an equality, so (6) becomes an equality and is known as PARSEVAL's THEOREM. If f(x) has a simple Fourier Series expansion with Coefficients a , ai, . . . , a n and &i, . . . , b ni then ia 2 + ^(a fc 2 +6 fc 2 )<- / [f(x)] 2 dx. (7) fc = l n J—* The inequality can also be derived from SCHWARZ'S IN- EQUALITY I (f\g) I 2 < {/I/} (g\g) (8) by expanding g in a superposition of ElGENFUNCTlONS 0f/,S= Yji a ifc- Then (/|5) = X)°* </!/*> ^Z) fli - (9) (f\g) r < Y< ai = 5> 4 a«' < </|/> <s| S ) . (10) 124 BesseVs Interpolation Formula If g is normalized, then (g\g) = 1 and </!/>> 5> t a t *. (11) see also Schwarz's Inequality, Triangle Inequal- ity References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 526-527, 1985. Gradshteyn, I. S. and Ryzhik, L M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1102, 1980. BessePs Interpolation Formula see Bessel's Finite Difference Formula Bessel Polynomial see Bessel Function Bessel's Second Integral see POISSON INTEGRAL Bessel's Statistical Formula W — UJ W — U) •'** t lzL (i) (wi-w) 2 N(N-l) where w = X\ — X2 u = M(i) - M(2) N = N 1 + N 2 . (2) (3) (4) Beta A financial measure of a fund's sensitivity to market movements which measures the relationship between a fund's excess return over Treasury Bills and the excess return of a benchmark index (which, by definition, has j3 = 1). A fund with a beta of (3 has performed r — (j3 - 1) x 100% better (or \r\ worse if r < 0) than its benchmark index (after deducting the T-bill rate) in up markets and \r\ worse (or \r\ better if r < 0) in down markets. see also Alpha, Sharpe Ratio Beta Distribution /^ "\ (a.6) = (l. 1) Q / //ia.G) = {l, 1) <2.3)-\ /(J. 2} ^ Beta Distribution A general type of statistical DISTRIBUTION which is re- lated to the Gamma Distribution. Beta distributions have two free parameters, which are labeled according to one of two notational conventions. The usual defini- tion calls these a and /?, and the other uses /?' = j3 — 1 and d = a - 1 (Beyer 1987, p. 534). The above plots are for (a,/3) = (1,1) [solid], (1, 2) [dotted], and (2, 3) [dashed]. The probability function P(x) and DISTRIBU- TION Function D(x) are given by P{x) 0-l„a-l (l-xf- l x' B(a,0) T(a)T((3) D{x) = I(x; a, 6), (l-xf^x /3-1 a-1 (1) (2) where B(a,b) is the BETA FUNCTION, J(x;a,6) is the Regularized Beta Function, and < x < 1 where a, f3 > 0. The distribution is normalized since Jo P(x) dx : r(a)r(/3) r(a + /3) Jo (l-xf^dx B(a,0) = l. (3) T(a)T(/3) The Characteristic Function is </>(*) = ^faa + bjit) The Moments are given by : + /3)r(a + r P 1 T(rv M r = (a- fi) r dx= ~^— Jo r ( a + /3 + r)r(a) (4) (5) The Mean is M -r(a)r ( ^y (1 x) T(a + P) B(a + l,f3) r(a + y3)r(a + l)r(/3) _ a r(a)r(/?)r(a + /? + i) a + /?' and the Variance, SKEWNESS, and KURTOSIS are 2__ a/3 a {a + f3) 2 {a + (3 + l) _ 2(yff- y^)( % /S+V^)Vl + q + /? 71 ~ V^p(a + /3 + 2) _ 6(a 2 + a 3 - Aaj3 - 2a 2 (3 + (3 2 - 2af3 2 + /3 3 ) 72 ~ a/3(a + /3 + 2)(a + /3 + 3) (6) (7) (8) (9) The Mode of a variate distributed as /3(a,/3) is - - Q ~ 1 (10) Beta Function In "normal" form, the distribution is written and the MEAN, VARIANCE, SKEWNESS, and KURTOSIS are A* = a + /3 2 OL0 a = 7i = 72 (a + /?) 2 (l + a + /?) _ 2(Vo:-^)( v /a + v / g)Vl + tt + /3 V^(a + /? + 2) 3(1 + a + /?)(2a 2 - 2a/? + a 2 /3 + 2/3 2 + a/3 2 ) (12) (13) (14) a/?(a + /3 + 2)(a + /? + 3) (15) see a/so GAMMA DISTRIBUTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944-945, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534-535, 1987. Beta Function The beta function is the name used by Legendre and Whittaker and Watson (1990) for the Eulerian Inte- gral of the Second Kind. To derive the integral representation of the beta function, write the product of two Factorials as POO no i\n\= I e- u u m du I Jo Jo Now, let u = a? 2 , v = y 2 , so e v v n dv. (1) dy /oo />oo e- x2 x 2rn+1 dx e~ y2 y 2n+ \ /oo poo / e-^ + ^x 2m+1 y 2m+1 dxdy. (2) -oo «/ — oo Transforming to POLAR COORDINATES with x ~ rcos9, y — r sin 6 pTT/2 pO n! = 4/ / e~ r '(r cos dY m+1 (r sin 6) 2n+1 rdrc Jo Jo poo /' 7r /2 A I -r 2 2m+2n+3 i / 2ro+l n ■ 2n+l n jq ■ 4 / e r dr cos v sin ^ v dv Jo Jo tt/2 2(m + n + 1)! / Jo cos" m+1 sin n+1 (9 d0. (3) Beta Function 125 The beta function is then defined by B(m + l,n + l) = 5(71 + 1,771+ 1) /.tt/2 = 2 / cos 2 ™ +1 flsin 2 " +1 ^= / m]n ' „ . Jo (m + n + 1)! (4) Rewriting the arguments, B( P a) - r &™ - (P-D'(g-l)! (5) The general trigonometric form is o /•tt/2 / sin n a;cos m ;rdx = \B(n+ |,m+ |). (6) Equation (6) can be transformed to an integral over Polynomials by letting u = cos 2 0, — = u (1-u) du. ' n ) Jo B(m,n) T(m)T(r, r(m + i du (7) (8) To put it in a form which can be used to derive the Legendre Duplication Formula, let x = y/u, so u = x and du — 2x dx, and B(m y n)= / x 2irn ~ 1) (l-x 2 ) n - 1 {2xdx) Jo -'f- Jo 2m — 1/-. 2\n~l (l-x^^dx. (9) To put it in a form which can be used to develop integral representations of the Bessel Functions and Hyper- geometric Function, let u = x/(l + x), so £(m + l,n + l)= H , ""> (10) Various identities can be derived using the GAUSS MUL- TIPLICATION Formula B(np, nq) T(np)T(nq) T[n(p + q)} _ - nq B(p,q)B(p+ l, t )- B(p+ 2=1, q) B(q,q)B(2q,q)---B([n-l]q,q) ' ( > Additional identities include B(va ^) = r(p)F(9 + 1) = g T(p + l)r(q) (P ' q+ ' T(p + q + l) p r(\p+l]q) = |s(p+l, ff ) (12) B(p,q) = B(p+l,q) + B(p,q+l) (13) 126 Beta Function (Exponential) B{p,q+1) P + Q If n is a Positive Integer, then B(p,q). (14) 1 * 2 • • • 71 ,„ ^ x B(p, n + 1 = , . x (15) p(p + 1) • • • (p + n) S(P,p)5(P+iP+5) = (16) 2'^ ' 2> 2 4 P" X p 5(p + <?)#(p + 9, r) = £(<?, r)B(q + r,p). (17) A generalization of the beta function is the incomplete beta function B(t;x,y)= r«"- 1 (l-u)*- 1 Jo +P |1 , i-w. \ X X + 1 (l-y)---(n-y), w n!(rr + n) r + . . • (18) see aZso Central Beta Function, Dirichlet In- tegrals, Gamma Function, Regularized Beta Function References Abramowitz, M. and Stegun, C. A. (Eds.). "Beta Function" and "Incomplete Beta Function." §6.2 and 6.6 in Hand- book of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258 and 263, 1972. Arfken, G. "The Beta Function." §10.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 560-565, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part L New York: McGraw-Hill, p. 425, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Gamma Function, Beta Function, Facto- rials, Binomial Coefficients" and "Incomplete Beta Func- tion, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209 and 219-223, 1992. Spanier, J. and Oldham, K. B. "The Incomplete Beta Func- tion B(v\mx)" Ch, 58 in An Atlas of Functions. Wash- ington, DC: Hemisphere, pp. 573-580, 1987. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis, 4th ed. Cambridge, England: Cambridge Uni- versity Press, 1990. Beta Function (Exponential) Betti Group Another "Beta Function" defined in terms of an in- tegral is the "exponential" beta function, given by /?»(*) 5 />-" dt i!*-< n+1 > '£ (-i)* L fc=o fc! 2-r fc! fc=0 (1) (2) The exponential beta function satisfies the Recur- rence Relation z(3 n (z) = (-l) n e z - e- z +n(3 n ^(z). The first few integral values are 2 sinh z 2 (sinh z — z cosh z) _ , , 2(2 + z 2 ) sinh z - 4z cosh z 02(a) = ^ . see also ALPHA FUNCTION Beta Prime Distribution A distribution with probability function (3) (4) (5) (6) P{x) = x a - 1 (l + xy B(a,l3) -OL-P where B is a Beta Function. The Mode of a variate distributed as (3 f (a,(3) is . a-1 + 1' If x is a f (a,0) variate, then 1/x is a j9'(/3,a) variate. If x is a j3(a,/3) variate, then (1 - x)/x and x/(l — x) are 0\0 ) ct) and 0'{a,0) variates. If x and y are 7( a i) and 7(0:2) variates, then x/y is a /?' (0:1,0:2) variate. If x 2 /2 and y 2 /2 are 7(1/2) variates, then z 2 = (x/y) 2 is a £'(1/2, 1/2) variate. Bethe Lattice see Cayley Tree Betrothed Numbers see QUASIAMICABLE PAIR Betti Group The free part of the Homology Group with a domain of Coefficients in the Group of Integers (if this Homology Group is finitely generated). References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, p. 380, 1988. Betti Number Bhargava's Theorem 127 Betti Number Betti numbers are topological objects which were proved to be invariants by Poincare, and used by him to ex- tend the Polyhedral Formula to higher dimensional spaces. The nth Betti number is the rank of the nth Homology Group. Let p r be the Rank of the Ho- mology Group H r of a Topological Space K. For a closed, orientable surface of GENUS g, the Betti num- bers are po = 1, Pi = 2#, and p 2 = I. For a nonori- entable surface with k CROSS-CAPS, the Betti numbers are po = 1, Pi = fc - 1, and p<z = 0. see also Euler Characteristic, Poincare Duality Bezier Curve the fact that moving a single control point changes the global shape of the curve. The former is sometimes avoided by smoothly patching together low-order Bezier curves. A generalization of the Bezier curve is the B- Spline. see also B-Spline, NURBS Curve Bezier Spline see Bezier Curve, Spline Bezout Numbers Integers (A,//) for a and b such that Aa + fib = GCD(a,6). Given a set of n control points, the corresponding Bezier curve (or BernSTEIN-Bezier Curve) is given by C(t) = 5^P.B i>n (t), where Bi n (t) is a Bernstein Polynomial and t € [0,1]- A "rational" Bezier curve is defined by C(*) = jy; =0 B itP (t)wii>i where p is the order, B itP are the BERNSTEIN POLYNO- MIALS, Pi are control points, and the weight Wi of Pi is the last ordinate of the homogeneous point P™. These curves are closed under perspective transformations, and can represent CONIC SECTIONS exactly. The Bezier curve always passes through the first and last control points and lies within the CONVEX Hull of the control points. The curve is tangent to Pi — Po and P n -P n _i at the endpoints. The "variation diminishing property" of these curves is that no line can have more intersections with a Bezier curve than with the curve obtained by joining consecutive points with straight line segments. A desirable property of these curves is that the curve can be translated and rotated by performing these operations on the control points. Undesirable properties of Bezier curves are their numer- ical instability for large numbers of control points, and For Integers ai, . . . , a n , the Bezout numbers are a set of numbers k\ , . . . , k n such that k\a\ + k-2<i2 + . . . + k n a n = d, where d is the Greatest Common Divisor of ai, . . . , a n . see also GREATEST COMMON DIVISOR Bezout's Theorem In general, two algebraic curves of degrees m and n in- tersect inm-n points and cannot meet in more than m-n points unless they have a component in common (i.e., the equations defining them have a common factor). This can also be stated: if P and Q are two POLYNOMI- ALS with no roots in common, then there exist two other Polynomials A and B such that AP + BQ = 1. Simi- larly, given N Polynomial equations of degrees m, ri2, . . . tin in N variables, there are in general niti2 • • • tin common solutions. see also POLYNOMIAL References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 10, 1959. Bhargava's Theorem Let the nth composition of a function f(x) be denoted / (n) (x), such that / (0) (z) = x and / Cl) (z) = f(x). De- note / o g(x) = f(g(x)), and define Let u = (a, 6, c) \u\ = a-h b + c u\\ = a 4 + 6 4 + c 4 , (2) (3) (4) 128 Bhaskara-Brouckner Algorithm and /(«) = (/i(«),/ a («),/3(t*)) (5) = (a(b - c), b(c - a),c(a - &)) (6) S(w) = (5i( u )»P2H,53(«)) = (^a 2 6,^a& 2 ,3a&c) . (7) Then if |u| = 0, ||/ (m) o 5 (n) (tx)|| = 2(a6 + 6c + ca) 2m+l3 " = llff (n) o/ (m) (u)|| ) (8) where 771, n E {0, 1, ...} and composition is done in terms of components. see also DlOPHANTINE EQUATION — QUARTIC, FORD'S Theorem References Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer- Verlag, pp. 97-100, 1994. Bhargava, S. "On a Family of Ramanujan's Formulas for Sums of Fourth Powers." Ganita 43, 63-67, 1992. Bhaskara-Brouckner Algorithm see Square Root Bi-Connected Component A maximal SUBGRAPH of an undirected graph such that any two edges in the SUBGRAPH lie on a common simple cycle. see also Strongly Connected Component Bianchi Identities The Riemann Tensor is defined by -IJLf dx K dxv- 2 q2 q2 9^u a g\ K a g^ K dx K dx x dx^dx u dx u dx x Permuting 1/, «, and 77 (Weinberg 1972, pp. 146-147) gives the Bianchi identities see also BlANCHI IDENTITIES (CONTRACTED), RlE- mann Tensor References Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972. BIBD Bianchi Identities (Contracted) Contracting A with v in the Bianchi Identities gives (2) Contracting again, R-n — R n\ii ~ R n-,v — 0, (3) or {R% - i<J%fi) ;M = 0, (fl"" - \!TR);* = 0. (4) (5) Bias (Estimator) The bias of an ESTIMATOR 9 is defined as b0) = (e) - e. It is therefore true that 6 -6 = (8- (§)) + ((6) -$) = (0 - (§)) + B(0). An Estimator for which B = is said to be Unbiased. see also ESTIMATOR, UNBIASED Bias (Series) The bias of a Series is defined as Q[ai, at+i,a»+2] '■■ A Series is Geometric Iff Q = 0. A Series is Artis- tic Iff the bias is constant. see also Artistic Series, Geometric Series References Duffin, R. J. "On Seeing Progressions of Constant Cross Ra- tio." Amer. Math. Monthly 100, 38-47, 1993. Biased An Estimator which exhibits Bias. Biaugmented Pentagonal Prism see Johnson Solid Biaugmented Triangular Prism see Johnson Solid Biaugmented Truncated Cube see Johnson Solid BIBD see Block Design Bicentric Polygon Dicentric Polygon Bicorn 129 A Polygon which has both a Circumcircle and an INCIRCLE, both of which touch all VERTICES. All TRI- ANGLES are bicentric with R 2 -s 2 = 2Rr, (1) where R is the ClRCUMRADlUS, r is the Inradius, and s is the separation of centers. In 1798, N. Puss character- ized bicentric POLYGONS of n = 4, 5, 6, 7, and 8 sides. For bicentric QUADRILATERALS (FUSS'S PROBLEM), the Circles satisfy 2r 2 (R 2 ~s 2 ) (Dorrie 1965) and Vabcd (R 2 -s 2 ) 2 -4r 2 s 2 1 {ac + bd)(ad + bc)(ab + cd) 4 V abed (Beyer 1987). In addition, 1 1 + {R-s) 2 {R + s and a + c = b + d. The Area of a bicentric quadrilateral is A = vabed. (2) (3) (4) (5) (6) (7) If the circles permit successive tangents around the In- CIRCLE which close the POLYGON for one starting point on the CIRCUMCIRCLE, then they do so for all points on the Circumcircle. see also PONCELET'S CLOSURE THEOREM References Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. Dorrie, H. "Fuss' Problem of the Chord- Tangent Quadrilat- eral," §39 in 100 Great Problems of Elementary Mathe- matics: Their History and Solutions. New York: Dover, pp. 188-193, 1965. Bicentric Quadrilateral A 4-sided Bicentric Polygon, also called a Cyclic- Inscriptable Quadrilateral. References Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 124, 1987. Bichromatic Graph A Graph with Edges of two possible "colors," usually identified as red and blue. For a bichromatic graph with R red EDGES and B blue Edges, R + B>2. see also Blue-Empty Graph, Extremal Coloring, Extremal Graph, Monochromatic Forced Tri- angle, Ramsey Number Bicollared A SUBSET X C Y is said to be bicollared in Y if there exists an embedding 6 : X x [-1, 1] -> Y such that b(x, 0) = x when x £ X. The MAP 6 or its image is then said to be the bicollar. References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 34-35, 1976. Bicorn The bicorn is the name of a collection of QUARTIC CURVES studied by Sylvester in 1864 and Cayley in 1867 (MacTutor Archive). The bicorn is given by the para- metric equations V asint a cos 2 t(2 + cost) 3 sin 2 t The graph is similar to that of the COCKED HAT CURVE. References Lawrence, J. D. A Catalog of Special Plane Curves. New- York: Dover, pp. 147-149, 1972. MacTutor History of Mathematics Archive. "Bicorn." http: // www - groups . des . st - and .ac.uk/ -history / Curves / Bicorn.html. 130 Bicubic Spline Bieberbach Conjecture Bicubic Spline A bicubic spline is a special case of bicubic interpolation which uses an interpolation function of the form 4 4 t=l j = l 4 4 Bidiakis Cube J- 2 4 4 y X2 (xi,x 2 ) = 5^ 5^0" - l)cijt"~V 4 4 t=l J=l where Cij are constants and u and £ are parameters rang- ing from to 1. For a bicubic spline, however, the partial derivatives at the grid points are determined globally by 1-D Splines. see also B-Spline, Spline References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 118-122, 1992. Bicupola Two adjoined CUPOLAS. see also Cupola, Elongated Gyrobicupola, Elon- gated Orthobicupola, Gyrobicupola, Orthobi- CUPOLA Bicuspid Curve The Plane Curve given by the Cartesian equation (x 2 - a 2 )(x - a) 2 + (y 2 - a 2 ) 2 = 0. Bicylinder see Steinmetz Solid f^ The 12- Vertex graph consisting of a Cube in which two opposite faces (say, top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are PERPENDICULAR to each other. see also Bislit Cube, Cube, Cubical Graph Bieberbach Conjecture The nth. Coefficient in the Power series of a Univa- lent Function should be no greater than n. In other words, if f(z) = a + aiz 4- a 2 z 2 + . . . + a n z n + ... is a conformal transformation of a unit disk on any do- main, then|a n | < n|ai|. In more technical terms, "ge- ometric extremality implies metric extremality." The conjecture had been proven for the first six terms (the cases n = 2, 3, and 4 were done by Bieberbach, Lowner, and Sniffer and Garbedjian, respectively), was known to be false for only a finite number of indices (Hayman 1954), and true for a convex or symmetric domain (Le Lionnais 1983). The general case was proved by Louis de Branges (1985). De Branges proved the MlLlN CON- JECTURE, which established the ROBERTSON CONJEC- TURE, which in turn established the Bieberbach conjec- ture (Stewart 1996). References de Branges, L. "A Proof of the Bieberbach Conjecture." Acta Math. 154, 137-152, 1985. Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1994. Hayman, W. K. and Stewart, F. M. "Real Inequalities with Applications to Function Theory." Proc. Cambridge Phil. Soc. 50, 250-260, 1954. Kazarinoff, N. D. "Special Functions and the Bieberbach Conjecture." Amer. Math. Monthly 95, 689-696, 1988. Korevaar, J. "Ludwig Bieberbach's Conjecture and its Proof." Amer. Math. Monthly 93, 505-513, 1986. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983. Pederson, R. N. "A Proof of the Bieberbach Conjecture for the Sixth Coefficient." Arch. Rational Mech. Anal. 31, 331-351, 1968/1969. Pederson, R. and SchifFer, M. "A Proof of the Bieberbach Conjecture for the Fifth Coefficient." Arch. Rational Mech. Anal. 45, 161-193, 1972. Stewart, I. "The Bieberbach Conjecture." In From Here to Infinity: A Guide to Today's Mathematics. Oxford, Eng- land: Oxford University Press, pp. 164-166, 1996. Bienayme-Chebyshev Inequality Biharmonic Equation 131 Bienayme-Chebyshev Inequality see Chebyshev Inequality Bifoliate The Plane Curve given by the Cartesian equation x A + y = 2axy . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Bifolium A Folium with 6 = 0. The bifolium is the Pedal Curve of the Deltoid, where the Pedal Point is the Midpoint of one of the three curved sides. The Carte- sian equation is (x 2 +y 2 ) 2 =4axy 2 and the POLAR equation is r = 4a sin 2 OcosO. see also FOLIUM, QuADRIFOLIUM, TRIFOLIUM References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972. MacTutor History of Mathematics Archive. "Double Folium." http : // www - groups . dcs . st - and .ac.uk/ -history/Curves/Double .html. Bifurcation A period doubling, quadrupling, etc., that accompanies the onset of Chaos. It represents the sudden appear- ance of a qualitatively different solution for a nonlin- ear system as some parameter is varied. Bifurcations come in four basic varieties: FLIP BIFURCATION, FOLD Bifurcation, Pitchfork Bifurcation, and Trans- critical Bifurcation (Rasband 1990). see also CODIMENSION, FEIGENBAUM CONSTANT, Feigenbaum Function, Flip Bifurcation, Hopf Bifurcation, Logistic Map, Period Doubling, Pitchfork Bifurcation, Tangent Bifurcation, Transcritical Bifurcation References Guckenheimer, J. and Holmes, P. "Local Bifurcations." Ch. 3 in Nonlinear Oscillations, Dynamical Systems, and Bifur- cations of Vector Fields, 2nd pr., rev. corr. New York: Springer- Verlag, pp. 117-165, 1983. Lichtenberg, A. J. and Lieberman, M. A. "Bifurcation Phe- nomena and Transition to Chaos in Dissipative Systems." Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New- York: Springer- Verlag, pp. 457-569, 1992. Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 25-31, 1990. Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer- Verlag, pp. 253-419, 1990. Bifurcation Theory The study of the nature and properties of BIFURCA- TIONS. see also CHAOS, DYNAMICAL SYSTEM Digraph see Bipartite Graph Bigyrate Diminished Rhombicosidodecahedron see Johnson Solid Biharmonic Equation The differential equation obtained by applying the Bi- harmonic Operator and setting to zero. vV = o. (i) In Cartesian Coordinates, the biharmonic equation V 2 (V 2 )0 dx 2 + dy 2 + dz 2 ) \dx 2 + dy 2 + dz 2 J * -4 + ^-t + -Tnr +^-- dx 4 dy 4 dz 4 dx 2 dy 2 0. d A (j> n d 4 <f> (2) dy 2 dz 2 dx 2 dz 2 In Polar Coordinates (Kaplan 1984, p. 148) 2 12 V (p = (prrrr H 2^ rr9$ ~* 4^0090 H 4>rrr 2 14 1 ~<t>rdd ~4>rr + ~7<l>e0 + ~^4>r = 0. (3) 132 Biharmonic Operator Billiards For a radial function </>(r), the biharmonic equation be- comes Id f d [1 d r dr \ dr [ r dr V dr J J J 2 11 Vr-rr + ~<firrr ~ ^<t>rr + -3 0r = 0. (4) Writing the inhomogeneous equation as V 4 = 64/3, we have M rdr = d{r±\ 1 -±(r^)]} I dr lr dr \ dr / J J 2 Vlnr- |r 2 to obtain # (5) dr L r dr V dr / J r dr \ dr J (16j3r 3 + Cir Inr + C 2 r) dr = d {r*j-\ . (10) Now use / r In r dr = \ (6) (7) (8) (9) (11) 4/3r 4 + d(±r 2 lnr - \r 2 ) + §C 2 r 2 + ^ 3 = r^ (12) (4/3r 3 + C> In r + C 2 r+— \ dr = d<f> (13) </>(r)=/?r 4 -f C[ (|r 2 lnr- \r 2 ) + §C 2 r 2 + C 3 lnr + C 4 = /?r 4 + or 2 + 6 4- (cr 2 + d) In (?-) . (14) The homogeneous biharmonic equation can be separated and solved in 2-D Bipolar Coordinates. References Kaplan, W. Advanced Calculus, ^th ed. Reading, MA: Addison-Wesley, 1991. Biharmonic Operator Also known as the BlLAPLAClAN. In n-D space, V 4 = (V 2 ) 2 . , 4 /'1\ _ 3(15 -8n + n 2 ) (;)- Bijection A transformation which is One-TO-One and ONTO. see also One-to-One, Onto, Permutation Bilaplacian see Biharmonic Operator Bilinear A function of two variables is bilinear if it is linear with respect to each of its variables. The simplest example is f(x,y) =xy. Bilinear Basis A bilinear basis is a BASIS, which satisfies the conditions (ax + by) • z = a(x * z) + 6(y • z) z • (ax 4- by) = a(z • x) + 6(z • y). see also Basis Billiard Table Problem Given a billiard table with only corner pockets and sides of Integer lengths m and n, a ball sent at a 45° angle from a corner will be pocketed in a corner after m+n-2 bounces. see also Alhazen's Billiard Problem, Billiards Billiards The game of billiards is played on a RECTANGULAR table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and Reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight LINE, but the mathematical study of bil- liards generally consists of REFLECTIONS in which the reflection and incidence angles are the same. However, strange table shapes such as CIRCLES and Ellipses are often considered. Many interesting problems can arise. For example, Alhazen's BILLIARD PROBLEM seeks to find the point at the edge of a circular "billiards" table at which a cue ball at a given point must be aimed in order to carom once off the edge of the table and strike another ball at a second given point. It was not until 1997 that Neumann proved that the problem is insoluble using a COMPASS and RULER construction. On an ELLIPTICAL billiard table, the ENVELOPE of a trajectory is a smaller ELLIPSE, a HYPERBOLA, a LINE through the FOCI of the ELLIPSE, or periodic curve (e.g., DlAMOND-shape) (Wagon 1991). see also Alhazen's Billiard Problem, Billiard Ta- ble Problem, Reflection Property see also Biharmonic Equation Billion Binary 133 References Davis, D.; Ewing, C; He, Z.; and Shen, T. "The Billiards Simulation." http : //serendip .brynmawr . edu/ chao s /home . html . Dullin, H. R.; Richter, RH.; and Wittek, A. "A Two- Parameter Study of the Extent of Chaos in a Billiard Sys- tem." Chaos 6, 43-58, 1996. Madachy, J. S. "Bouncing Billiard Balls." In Madachy's Mathematical Recreations. New York: Dover, pp. 231— 241, 1979. Neumann, P. Submitted to Amer. Math. Monthly. Pappas, T. "Mathematics of the Billiard Table." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 43, 1989. Peterson, I. "Billiards in the Round." http : //www . sciencenews.org/sn_arc97/3-l_97/mathland.htm. Wagon, S. "Billiard Paths on Elliptical Tables." §10.2 in Mathematica in Action. New York: W. H. Freeman, pp. 330-333, 1991. Billion The word billion denotes different numbers in American and British usage. In the American system, one billion equals 10 9 . In the British, French, and German systems, one billion equals 10 12 . see also LARGE NUMBER, MILLIARD, MILLION, TRIL- LION Bilunabirotunda see Johnson Solid Bimagic Square 16 41 36 5 27 62 55 18 26 63 54 19 13 44 33 8 1 40 45 12 22 51 58 31 23 50 59 30 4 37 48 9 38 3 10 47 49 24 29 60 52 21 32 57 39 2 11 46 43 14 7 34 64 25 20 53 61 28 17 56 42 15 6 35 If replacing each number by its square in a MAGIC Square produces another Magic Square, the square is said to be a bimagic square. The first bimagic square (shown above) has order 8 with magic constant 260 for addition and 11,180 after squaring. Bimagic squares are also called Doubly Magic Squares, and are 2- Multimagic Squares. see also MAGIC SQUARE, MULTIMAGIC SQUARE, Trimagic Square References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 212, 1987. Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, p. 31, 1975. Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 176-178, 1942. M BC A' M AB A Line Segment joining the Midpoints of opposite sides of a QUADRILATERAL. see also Median (Triangle), Varignon's Theorem Bimodal Distribution A Distribution having two separated peaks. see also Unimodal Distribution Bimonster The wreathed product of the Monster Group by Z 2 . The bimonster is a quotient of the Coxeter Group with the following Coxeter-Dynkin Diagram. This had been conjectured by Conway, but was proven around 1990 by Ivanov and Norton. If the parameters p,<?, r in Coxeter's NOTATION [3 F,q>r ] are written side by side, the bimonster can be denoted by the BEAST Number 666. Bin An interval into which a given data point does or does not fall. see also HISTOGRAM Binary The BASE 2 method of counting in which only the digits and 1 are used. In this Base, the number 1011 equals l-2° + l-2 + 0-2 2 + l-2 3 = 11. This Base is used in com- puters, since all numbers can be simply represented as a string of electrically pulsed ons and offs. A NEGATIVE — n is most commonly represented as the complement of the Positive number n - 1, so -11 = 00001011 2 would be written as the complement of 10 — OOOOIOIO2, or 11110101. This allows addition to be carried out with the usual carrying and the left-most digit discarded, so 17 — 11 = 6 gives 00010001 17 11110101 -11 00000110 6 134 Binary Bracketing Binary Tree The number of times k a given binary number b n ■ • -&2&1&0 is divisible by 2 is given by the position of the first bk = 1 counting from the right. For exam- ple, 12 = 1100 is divisible by 2 twice, and 13 = 1101 is divisible by 2 times. Unfortunately, the storage of binary numbers in com- puters is not entirely standardized. Because computers store information in 8-bit bytes (where a bit is a sin- gle binary digit), depending on the "word size" of the machine, numbers requiring more than 8 bits must be stored in multiple bytes. The usual F0RTRAN77 integer size is 4 bytes long. However, a number represented as (bytel byte2 byte3 byte4) in a VAX would be read and interpreted as (byte4 byte3 byte2 bytel) on a Sun. The situation is even worse for floating point (real) num- bers, which are represented in binary as a MANTISSA and Characteristic, and worse still for long (8-byte) reals! Binary multiplication of single bit numbers (0 or 1) is equivalent to the AND operation, as can be seen in the following Multiplication Table. X 1 1 1 see also Base (Number), Decimal, Hexadecimal, Octal, Quaternary, Ternary References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 6-9, 1991. Pappas, T. "Computers, Counting, & Electricity." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, pp. 24-25, 1989. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Error, Accuracy, and Stability" and "Diag- nosing Machine Parameters." §1.2 and §20.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 18-21, 276, and 881-886, 1992. ^ Weisstein, E. W. "Bases." http: //www. astro. Virginia. edu/~eww6n/math/notebooks/Bases.m. Binary Bracketing A binary bracketing is a BRACKETING built up entirely of binary operations. The number of binary bracket ings of n letters (Catalan's Problem) are given by the Catalan Numbers C n _i, where C n = n + 1 2n\ _ 1 (2ra)! _ n ) n+ 1 n! 2 (2n)! (n+l)!n! where ( 2 ™) denotes a Binomial Coefficient and n\ is the usual FACTORIAL, as first shown by Catalan in 1838, For example, for the four letters a, 6, c, and d there are five possibilities: ({ab)c)d, (a(6c))d, (a&)(cd), a((bc)d), and a(6(cd)), written in shorthand as {(xx)x)x } (x(xx))x, (xx)(xx), x((xx)x), and x(x(xx)). see also BRACKETING, CATALAN NUMBER, CATALAN'S Problem References Schroder, E. "Vier combinatorische Probleme." Z. Math. Physik 15, 361-376, 1870. Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough." Amer. Math. Monthly 104, 344-350, 1997. Binary Operator An Operator which takes two mathematical objects as input and returns a value is called a binary operator. Binary operators are called compositions by Rosenfeld (1968). Sets possessing a binary multiplication opera- tion include the Group, Groupoid, Monoid, Quasi- group, and Semigroup. Sets possessing both a bi- nary multiplication and a binary addition operation in- clude the Division Algebra, Field, Ring, Ringoid, Semiring, and Unit Ring. see also AND, BOOLEAN ALGEBRA, CLOSURE, DIVI- SION Algebra, Field, Group, Groupoid, Monoid, Operator, Or, Monoid, Not, Quasigroup, Ring, Ringoid, Semigroup, Semiring, XOR, Unit Ring References Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968. Binary Quadratic Form A 2-variable QUADRATIC FORM of the form Q(x, y) = aux 2 + 2a\ixy + a 2 2V . see also QUADRATIC FORM, QUADRATIC INVARIANT Binary Remainder Method An Algorithm for computing a Unit Fraction (Stewart 1992). References Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer. 266, 122-124, June 1992. Binary Tree A Tree with two Branches at each Fork and with one or two Leaves at the end of each Branch. (This definition corresponds to what is sometimes known as an "extended" binary tree.) The height of a binary tree is the number of levels within the TREE. For a binary tree of height H with n nodes, H < n < 2 H - 1, Binet Forms Binomial Coefficient 135 These extremes correspond to a balanced tree (each node except the Leaves has a left and right Child, arid all LEAVES are at the same level) and a degenerate tree (each node has only one outgoing BRANCH), respec- tively. For a search of data organized into a binary tree, the number of search steps S(n) needed to find an item is bounded by lgn < S(n) < n. Partial balancing of an arbitrary tree into a so-called AVL binary search tree can improve search speed. The number of binary trees with n internal nodes is the Catalan Number C n (Sloane's A000108), and the number of binary trees of height b is given by Sloane's A001699. see also S-Tree, Quadtree, Quaternary Tree, Red-Black Tree, Stern-Brocot Tree, Weakly Binary Tree References Lucas, J.; Roelants van Baronaigien, D.; and Ruskey, F. "Generating Binary Trees by Rotations." J. Algorithms 15, 343-366, 1993. Ranum, D. L. "On Some Applications of Fibonacci Num- bers." Amer. Math. Monthly 102, 640-645, 1995. Ruskey, F. "Information on Binary Trees." http://sue.csc ,uvic.ca/~cos/inf/tree/BinaryTrees.html. Ruskey, F. and Proskurowski, A. "Generating Binary Trees by Transpositions." J. Algorithms 11, 68-84, 1990. Sloane, N. J. A. Sequences A000108/M1459 and A001699/ M3087 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Binet Forms The two Recurrence Sequences U n — mUn-l + U n ~2 V n =mV n - 1 + V n - 2 (1) (2) with Uo = 0, Ui = 1 and Vo = 2, V\ — m, can be solved for the individual U n and V n . They are given by " ~ P (3) (4) (5) (6) (7) (8) Binet' S Formula is a special case of the Binet form for U n corresponding to m = 1. see also Fibonacci Q-Matrix yJ-n — A V n = = a n + ^, where A = \/m 2 + 4 a = 771+ A 2 = m - A 2 A useful related identity is Un-l+Un+l = Vn- Binet 's Formula A special case of the U n Binet Form with m corresponding to the nth FIBONACCI NUMBER, _ _ (1 + V5)"-(1-a/5) b 0, 2 n VE It was derived by Binet in 1843, although the result was known to Euler and Daniel Bernoulli more than a century earlier. see also Binet Forms, Fibonacci Number Bing's Theorem If M 3 is a closed oriented connected 3-MANIFOLD such that every simple closed curve in M lies interior to a BALL in Af , then M is HOMEOMORPHIC with the Hy- persphere, S 3 . see also Ball, Hypersphere References Bing, R. H. "Necessary and Sufficient Conditions that a 3- Manifold be S 3 ." Ann. Math. 68, 17-37, 1958. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 251-257, 1976. Binomial A Polynomial with 2 terms. see also Monomial, Polynomial, Trinomial Binomial Coefficient The number of ways of picking n unordered outcomes from N possibilities. Also known as a COMBINATION. The binomial coefficients form the rows of PASCAL'S Triangle. The symbols N C n and n) = (N - n)\n\ (1) are used, where the latter is sometimes known as N CHOOSE n. The number of LATTICE PATHS from the Origin (0, 0) to a point (a, b) is the Binomial Coeffi- cient (°+ 6 ) (Hilton and Pedersen 1991). For Positive integer n, the Binomial Theorem gives i.— n V / (2) The Finite Difference analog of this identity is known as the Chu-Vandermonde Identity. A sim- ilar formula holds for Negative Integral n, ~ n \ k-Tl-k ,x a k (3) A general identity is given by (a + b) n = Eu)( a -* : ) i " 1 ( 6 +j' c ) B "' w 136 Binomial Coefficient (Prudnikov et ol. 1986), which gives the BINOMIAL THE- OREM as a special case with c = 0. The binomial coefficients satisfy the identities: n n k n + 1 A: „-»-<-<":"' n \ I n (5) (6) (7) Sums of powers include fc=0 x y E<-')"(I)=» k=0 x 7 t(;V -(•+')• (8) (9) (10) (the Binomial Theorem), and ^2n + s £ a; n = 2^1(1(5 4- 1), §(* + 2); s + 1, 4x) 2 s (\/l - 4x + l)Vl - 4a; , (11) where 2F1 (a, 6; c;z) is a Hypergeometric Function (Abramowitz and Stegun 1972, p. 555; Graham et al. 1994, p. 203). For NONNEGATIVE INTEGERS n and r with r < n + 1, ££?(;) [D-*G")"- fl ' fc^o v 7 L j=o v y +Et- 1 ) i (")( n+l - r -j) n - fc Taking n = 2r — 1 gives r-l n!. (12) -fc _ i in!. (13) Another identity is E ( n £ k ) [xn+1(1 - x)fc + (1 - x)n+1 ^ = 1 w Binomial Coefficient Recurrence Relations of the sums EI — . <15) are given by 2si(n)-si(n + l) = (16) -2(2n + l)s 2 (n) + (n + l)s 2 (n) = (17) -8(n + l) 2 s 3 (n) + (-16 - 21n - 7n 2 )s 3 (n + 1) +(n + 2) 2 53 (n + 2) = (18) -4(n + l)(4n + 3)(4n + 5)s 4 (n) -2(2n + 3)(3n 2 + 9n + 7)s 4 (n + 1) +(n + 2) 3 s 4 (n + 2) = 0. (19) This sequence for S3 cannot be expressed as a fixed number of hypergeometric terms (Petkovsek et a/. 1996, p. 160). A fascinating series of identities involving binomial co- efficients times small powers are 00 £ 72^Y = 27 ( 27r ^ + 9 ) = 0.7363998587 . . . n=l V n ) 00 E— 1_ = Ittv^ = 0.6045997881 . . . n( 2n ) 9 n=l \n J n=l V n 7 °° 1 / ^ 4(2n\ 36 ^W 3240 /l 71=1 V n / (Comtet 1974, p. 89) and - ( _ 1} n-i £ 8 a = I C(3), (20) (21) (22) (23) (24) where ((z) is the Riemann Zeta Function (Le Lion- nais 1983, pp. 29, 30, 41, 36, and 35; Guy 1994, p. 257). As shown by Kummer in 1852, the exact Power of p dividing ( a ^ b ) is equal to eo + ei + . . . + e £ , (25) (Beeler et al 1972, Item 42). where this is the number of carries in performing the addition of a and b written in base b (Graham et al. 1989, Exercise 5.36; Ribenboim 1989; Vardi 1991, p. 68). Kummer's result can also be stated in the form that the Binomial Coefficient Binomial Coefficient 137 exponent of a Prime p dividing (j^j is given by the number of integers j > for which frac(ra/p J ) > frac (n/p 3 ). (26) where frac(cc) denotes the FRACTIONAL PART of x. This inequality may be reduced to the study of the exponen- tial sums ^2 n A(n)e(x/n), where A(n) is the MANGOLDT FUNCTION. Estimates of these sums are given by Jutila (1974, 1975), but recent improvements have been made by Granville and Ramare (1996). R. W. Gosper showed that /( n ) = (l(n~-l)) ~ (- 1 ) < "" 1)/2 ( mod ") (27) for all Primes, and conjectured that it holds only for Primes. This was disproved when Skiena (1990) found it also holds for the Composite Number n = 3xllx 179. Vardi (1991, p. 63) subsequently showed that n = p 2 is a solution whenever p is a Wieferich Prime and that if n = p k with k > 3 is a solution, then so is n = p k ~ 1 . This allowed him to show that the only solutions for Composite n < 1.3xl0 7 are 5907, 1093 2 , and 3511 2 , where 1093 and 3511 are Wieferich PRIMES. Consider the binomial coefficients ( n ~ )•> the first few of which are 1, 3, 10, 35, 126, ... (Sloane's A001700). The Generating Function is Vl-4o; : x + 3x 2 + 10x 3 + 35x 4 + . (28) These numbers are SQUAREFREE only for n = 2, 3, 4, 6, 9, 10, 12, 36, . . . (Sloane's A046097), with no others less than n = 10, 000. Erdos showed that the binomial coefficient (™) is never a Power of an Integer for n > 3 where A; ^ 0, 1, n— 1, and n (Le Lionnais 1983, p. 48). The binomial coefficients (| n / 2 |) are called CENTRAL Binomial Coefficients, where |xj is the Floor Function, although the subset of coefficients ( 2 ™) is sometimes also given this name. Erdos and Graham (1980, p. 71) conjectured that the Central Binomial Coefficient ( 2 ^) is never Squarefree for n > 4, and this is sometimes known as the Erdos SQUAREFREE Conjecture. Sarkozy's Theorem (Sarkozy 1985) provides a partial solution which states that the BINO- MIAL Coefficient ( 2 ^) is never Squarefree for all sufficiently large n > no (Vardi 1991). Granville and Ramare (1996) proved that the only SQUAREFREE val- ues are n = 2 and 4. Sander (1992) subsequently showed that ( 2n ^ d ) are also never SQUAREFREE for sufficiently large n as long as d is not "too big." For p, qr, and r distinct PRIMES, then the above function satisfies f(pqr)f(p)f(q)f(r) = f {pq) f (pr)p(qr) (mod pqr) (29) (Vardi 1991, p. 66). The binomial coefficient (™) mod 2 can be computed using the XOR operation n XOR m, making Pascal's Triangle mod 2 very easy to construct. The binomial coefficient "function" can be defined as C{z,y) y\(x - y)\ (30) (Fowler 1996), shown above. It has a very complicated Graph for Negative x and y which is difficult to render using standard plotting programs. see also BALLOT PROBLEM, BINOMIAL DISTRIBU- TION, Binomial Theorem, Central Binomial Co- efficient, Chu-Vandermonde Identity, Combi- nation, Deficiency, Erdos Squarefree Conjec- ture, Gaussian Coefficient, Gaussian Polynom- ial, Kings Problem, Multinomial Coefficient, Permutation, Roman Coefficient, Sarkozy's Theorem, Strehl Identity, Wolstenholme's The- orem References Abramowitz, M. and Stegun, C. A. (Eds.). "Binomial Co- efficients. " §24.1.1 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 10 and 822-823, 1972. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Comtet, L. Advanced Combinatorics. Amsterdam, Nether- lands: Kluwer, 1974. Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer- Verlag, pp. 66-74, 1996. Erdos, P.; Graham, R. L.; Nathanson, M. B.; and Jia, X. Old and New Problems and Results in Combinatorial Number Theory. New York: Springer- Verlag, 1998, Fowler, D. "The Binomial Coefficient Function." Amer. Math. Monthly 103, 1-17, 1996. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Foun- dation for Computer Science. Reading, MA: Addison- Wesley, pp. 153-242, 1990. Granville, A. and Ramare, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coeffi- cients." Mathematika 43, 73-107, 1996. 138 Binomial Distribution Binomial Distribution Guy, R. K. "Binomial Coefficients," "Largest Divisor of a Binomial Coefficient," and "Series Associated with the £- Function." §B31, B33, and F17 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 84-85, 87-89, and 257-258, 1994. Harborth, H. "Number of Odd Binomial Coefficients." Not. Amer. Math. Soc. 23, 4, 1976. Hilton, P. and Pedersen, J. "Catalan Numbers, Their Gener- alization, and Their Uses." Math. Intel 13, 64-75, 1991. Jutila, M. "On Numbers with a Large Prime Factor." J. Indian Math. Soc. 37, 43-53, 1973. Jutila, M. "On Numbers with a Large Prime Factor. II," J. Indian Math. Soc. 38, 125-130, 1974. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983. Ogilvy, C. S. "The Binomial Coefficients." Amer. Math. Monthly 57, 551-552, 1950. Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients." §6.1 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, Qnd ed. Cam- bridge, England: Cambridge University Press, pp. 206— 209, 1992. Prudnikov, A. P.; Marichev, O. I.; and Brychkow, Yu. A. Formula 41 in Integrals and Series, Vol. 1: Elementary Functions. Newark, NJ: Gordon & Breach, p. 611, 1986. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer- Verlag, pp. 23-24, 1989. Riordan, J. "Inverse Relations and Combinatorial Identities." Amer. Math. Monthly 71, 485-498, 1964. Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24,140-142, 1992. Sarkozy, A. "On the Divisors of Binomial Coefficients, I." J. Number Th. 20, 70-80, 1985. Skiena, S, Implementing Discrete Mathematics: Combina- torics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, p. 262, 1990. Sloane, N. J. A. Sequences A046097 and A001700/M2848 in "An On-Line Version of the Encyclopedia of Integer Se- quences." Spanier, J. and Oldham, K. B. "The Binomial Coefficients (^)." Ch. 6 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 43-52, 1987. Sved, M. "Counting and Recounting." Math. Intel 5, 21-26, 1983. Vardi, I. "Application to Binomial Coefficients," "Binomial Coefficients," "A Class of Solutions," "Computing Bino- mial Coefficients," and "Binomials Modulo and Integer." §2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematica. Redwood City, CA: Addison- Wesley, pp. 25-28 and 63-71, 1991. Wolfram, S. "Geometry of Binomial Coefficients." Amer. Math. Monthly 91, 566-571, 1984. Binomial Distribution The probability of n successes in N BERNOULLI TRIALS is n N — n P Q (1) The probability of obtaining more successes than the n observed is *=E k = n + l N ^p k (l-p) N - k =I p (n + l,N-N), (2) where Ix{a,b) B{x\ a y b) B(a,b) ' (3) B(a,b) is the Beta Function, and B(x\a,b) is the incomplete BETA FUNCTION. The CHARACTERISTIC Function is <f>(t) = ( q + pe it )\ (4) The Moment-Generating Function M for the dis- tribution is N y v M(t) = <e tn > = XX n ( ^ V<? N " n n=0 ^ ' = E(?V)"(i-p) n -" = \pe t + (l-p)] N At (t) = Nfre* + (1 - p)] JV - 1 (pe') M"it) = NiN - l)[pc* + (1 - p)]"- V) 2 + N\pe t + il-p)} N - 1 ipe t ). The Mean is H = M'(0) = Nip + 1 - p)p = Np. The Moments about are (5) (6) (7) (8) /*!=/* = Np (9) l& = Np(l-p + Np) (10) ^ = Np(l - 3p + 3Np + 2p 2 - 3NP 2 + N 2 p 2 ) (11) /4 = Np(l - 7p + 7Np + 12p 2 - ISNp 2 + 6iVV - 6p 3 + HATp 3 - 6iVV + A^p 3 ), (12) so the Moments about the Mean are M2 = a 2 = [N(N - l)p 2 + Np] - {Np) 2 = N 2 p 2 - Np 2 +Np- N 2 p 2 = Np(l -p) = Npq (13) (14) Pz = P3 ~ 3p f 2Pi + 2{pif = Np(l-p)(l-2p) li4 = fJ,4- 4/4/ii + 6/i2(/ii) 2 - 3(/ii) 4 = Np(l - p)[3p 2 (2 - N) + 3p(N - 2) + 1]. (15) Binomial Distribution The SKEWNESS and KURTOSIS are /is = Np(l-p){l-2p) <7 3 [iV>(l-p)] 3 /2 7i l-2p q-p y/Np(l-p) y/Npq 7 2 = ^-3: cr 4 6p — 6p + 1 1 — 6pg iVp(l-p) A^pg (16) (17) An approximation to the Bernoulli distribution for large N can be obtained by expanding about the value n where P(n) is a maximum, i.e., where dP/dn = 0. Since the Logarithm function is Monotonic, we can instead choose to expand the LOGARITHM. Let n = h + to, then ln[F(n)]-ln[P(n)] + B 1 7?+|B2T? 2 + |jS 3 7 ? 3 + ..., (18) where (19) B k = d k \n[P(n)] dn k But we are expanding about the maximum, so, by defi nition, ~dln[P(n)] Bi dn = 0. (20) This also means that B2 is negative, so we can write B 2 = — 1B 2 1 . Now, taking the LOGARITHM of (1) gives ln[P(n)] = lnNl-\nn\-ln(N-n)\ + nlnp+(N-n)\nq. (21) For large n and N — n we can use STIRLING'S APPROX- IMATION ln(n!) « toIxito-to, (22) so d[ln(w!)] dn d[\n(N-n)\] dn « (Inn -I- 1) - 1 = lnn d (23) dn [{N - n) \vv{N -n)-(N - = -ln(7V-n), (24) and dln{P(n)} ^ _ lnn + ln(JV _ w) + lnp _ lnq / 25 v dn To find n, set this expression to and solve for ra, (26) N — hp h (^)=' 1 n q (N — n)p = hq n(q + p) = h = Np, (27) (28) (29) Binomial Distribution 139 since p + q — 1. We can now find the terms in the expansion B 2 d 2 \n[P(n)] dn 2 1 1 h N ~ h 1 1 _ _ J_ (\ l\ Np N(l-p)~ N\p + q) i (p + q B 3 = N \ pq d*\n[P(n)] dn 3 1 1 1 1 Npq N(l-p) 1 1 (30) h 2 (N - h) 2 „2 2 N 2 p 2 N 2 q 2 N 2 p 2 q 2 (l-2p + p 2 )-p 2 _ l-2p B 4 = N 2 p 2 (l-p) 2 N 2 p 2 {l-p) 2 d 4 \n[P(n)] (31) dn 4 h 3 (n — h) 3 -2 3 _J_ _J_\ = 2(P 3 +Q N 3 p 3 N 3 q 3 J N 3 p 3 q< = 2(p 2 -pq + q 2 ) N z p 3 q 3 = 2\p 2 -p{l-p) + {l-2p + p 2 )] N 3 p 3 (l-p 3 ) = 2(3p 2 -3p+l) N 3 p 3 (l-p 3 ) ' Now, treating the distribution as continuous, (32) ■W p /»oo lim y^P(n)^ P(n)dn= / P(h + to) dn - 1. (33) Since each term is of order 1/iV ~ 1/<t 2 smaller than the previous, we can ignore terms higher than B 2 , so P(n) = P(n)e- |B2| " 2/2 . The probability must be normalized, so (34) J~ P(fi)e-W 2 '*dT, = P(n)^=l, (35) and P(n) \Bl\ -\B 2 \(n-n) 2 /2 2tt yj2nNpq exp Defining a 2 = 2Npq, P(n) (7V27T : exp (n - iVp) 2 2iVpg (to - n) 2 2a 2 (36) (37) 140 Binomial Expansion Binomial Series which is a GAUSSIAN DISTRIBUTION. For p < 1, a different approximation procedure shows that the bi- nomial distribution approaches the PoiSSON DISTRIBU- TION. The first Cumulant is m = np, (38) and subsequent Cumulants are given by the RECUR- RENCE Relation kv+i = pq dp ' (39) Let x and y be independent binomial Random Vari- ables characterized by parameters n,p and m,p. The Conditional Probability of x given that x + y = k is P(a; = i|a; + y — k) P(x = i y x + y = k) P{x + y = k) P(x = i,y = k-i) _ P(x = i)P(y = k-i) P(x-\-y = k) ~ P{x + y = k) ( n t m )p fc (i-p) n+m - fc (40) Note that this is a Hypergeometric Distribution! see also de Moivre-Laplace Theorem, Hypergeo- metric Distribution, Negative Binomial Distri- bution References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987, Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Incomplete Beta Function, Student's Distribu- tion, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Sci- entific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 219-223, 1992. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 108-109, 1992. Binomial Expansion see Binomial Series Binomial Formula see Binomial Series, Binomial Theorem Binomial Number A number of the form a n ± b n , where a, 6, and n are Integers. They can be factored algebraically a n -& n = (a-6)(a ri ~ 1 +a Tl - 2 b + ... + a6 n " 2 +b n - 1 ) (1) a n + b n -(a + 6)(a n - 1 -a n - 2 6 + ...-ab n - 2 +6 n - 1 ) (2) a nm - b nTn = (a m - 6 m )[a m(n ~ 1) + a m(n ' 2) 6 m + ... + 6 m(n_1) ]. (3) In 1770, Euler proved that if (a, b) = 1, then every FAC- TOR of o a "+6 jn (4) is either 2 or of the form 2 n+1 K + 1. If p and q are Primes, then a pq -l)(a-l) (aP-l)(a«-l] - 1 (5) is Divisible by every Prime Factor of a p 1 not divid- ing a q — 1. see also CUNNINGHAM NUMBER, FERMAT NUMBER, Mersenne Number, Riesel Number, Sierpinski Number of the Second Kind References Guy, R. K. "When Does 2 a - 2 b Divide n a - n 6 ." §B47 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 102, 1994. Qi, S and Ming-Zhi, Z. "Pairs where 2 a - a b Divides n a - n h for All n." Proc. Amer. Math. Soc. 93, 218-220, 1985. Schinzel, A. "On Primitive Prime Factors of a n — 6 n ." Proc. Cambridge Phil Soc. 58, 555-562, 1962. Binomial Series For Id < 1, (i + x y - £(:)■* fc=o v 7 (i) = i + ;x + l!(n-l)! (n-2)!2! " ! x= + ...(3) n(n — 1) o l + nz+ -^ — -x 2 + . (4) The binomial series also has the CONTINUED FRACTION representation (1 + *)" = -. (5) 1 + l-(l + n) 1-2 1 + 1 ■(!-") , 2-3 1 + 2(2 + n) 3-4 1+- 2(2 -n) , 4-5 3(3 + n) 5-6 1 + ... 1 + Binomial Theorem Biotic Potential 141 see also Binomial Theorem, Multinomial Series, Negative Binomial Series References Abramowitz, M. and Stegun, C, A, (Eds,). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 14-15, 1972. Pappas, T. "Pascal's Triangle, the Fibonacci Sequence &; Bi- nomial Formula." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 40-41, 1989. Binomial Theorem The theorem that, for INTEGRAL POSITIVE n, Z_/ kun - A;=0 (n-k)\ ^r k=0 the so-called Binomial Series, where (™) are Bino- mial Coefficients. The theorem was known for the case n = 2 by Euclid around 300 BC, and stated in its modern form by Pascal in 1665. Newton (1676) showed that a similar formula (with Infinite upper limit) holds for Negative Integral n, (* + a)-» = £; ( fc n y a - the so-called Negative Binomial Series, which con- verges for |x| > \a\. see also BINOMIAL COEFFICIENT, BINOMIAL SERIES, Cauchy Binomial Theorem, Chu-Vandermonde Identity, Logarithmic Binomial Formula, Nega- tive Binomial Series, <?-Binomial Theorem, Ran- dom Walk References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 307-308, 1985. Conway, J. H. and Guy, R. K. "Choice Numbers Are Bino- mial Coefficients." In The Book of Numbers. New York: Springer- Verlag, pp. 72-74, 1996. Coolidge, J. L. "The Story of the Binomial Theorem," Amer. Math. Monthly 56, 147-157, 1949. Courant, R. and Robbins, H. "The Binomial Theorem." §1.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford Uni- versity Press, pp. 16-18, 1996. Binomial Triangle see Pascal's Triangle Binormal Developable A Ruled Surface M is said to be a binormal de- velopable of a curve y if M can be parameterized by x(«,v) = y(u)+t;B(u), where B is the BINORMAL VEC- TOR. see also NORMAL DEVELOPABLE, TANGENT DEVEL- OPABLE References Gray, A. "Developables." §17.6 in Modern Differential Ge- ometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 352-354, 1993. Binormal Vector :TxN r' x r" ' |r' xr'T (1) (2) where the unit TANGENT VECTOR T and unit "princi- pal" NORMAL VECTOR N are defined by t - r'(s) N: |r'( S )| \t"(s)\ (3) (4) Here, r is the Radius Vector, s is the Arc Length, r is the TORSION, and « is the Curvature. The binormal vector satisfies the remarkable identity [B,B,B1 ds (") (5) see also Frenet Formulas, Normal Vector, Tan- gent Vector References Kreyszig, E. "Binormal. Moving Trihedron of a Curve." §13 in Differential Geometry. New York: Dover, p. 36—37, 1991. Bioche's Theorem If two complementary PLUCKER CHARACTERISTICS are equal, then each characteristic is equal to its comple- ment except in four cases where the sum of order and class is 9. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- York: Dover, p. 101, 1959. Biotic Potential see Logistic Equation 142 Bipartite Graph Bipartite Graph A set of VERTICES decomposed into two disjoint sets such that no two VERTICES within the same set are adjacent. A bigraph is a special case of a &- Partite Graph with k = 2. see also Complete Bipartite Graph, /c-Partite Graph, Konig-Egevary Theorem References Chartrand, G. Introductory Graph Theory. New York: Dover, p. 116, 1985. Saaty, T. L. and Kainen, P. C. The Four- Color Problem: Assaults and Conquest, New York: Dover, p. 12, 1986. Biplanar Double Point see Isolated Singularity Bipolar Coordinates Bipolar coordinates are a 2-D system of coordinates. There are two commonly defined types of bipolar co- ordinates, the first of which is defined by a sinh v y = cosh v — cos u as'mu cosh v — cos u ' (i) (2) where u € [0,27r), v G (—00,00). The following identi- ties show that curves of constant u and v are CIRCLES in ay-space. x 2 + {y — a cot u) 2 — a 2 esc 2 u (x — a coth v) 2 + y = a 2 csch 2 v. The Scale Factors are h u - a coshi; — cosu h v - a coshv — cosu The Laplacian is ^2 _ (coshi; \2 — cos u) a 2 ( d 2 \du 2 + d 2 dv 2 (3) (4) (5) (6) (7) Laplace's Equation is separable. Bipolar Cylindrical Coordinates Two-center bipolar coordinates are two coordinates giv- ing the distances from two fixed centers r\ and V2 , some- times denoted r and r'. For two-center bipolar coordi- nates with centers at (±c, 0), ri 2 = ( x + c) 2 +y 2 2 / \2 , 2 r 2 ■ = (x - c) + y . Combining (8) and (9) gives 2 2 A ri — ri = 4cx. (8) (9) (10) Solving for CARTESIAN COORDINATES x and y gives * 2 „ 2 Ti — 7*2 4c (11) y = ±^y/l6c 2 n 2 - (n 2 - r 2 2 + 4c 2 ). (12) Solving for POLAR COORDINATES gives ri 2 + r 2 2 -2c 2 8 — tan 8c 2 (n 2 +r 2 2 -2c 2 ) (13) (14) References Lockwood, E. H. "Bipolar Coordinates." Ch. 25 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 186-190, 1967. Bipolar Cylindrical Coordinates A set of Curvilinear Coordinates defined by a sinh v cosh v — cos u asinu cosh v — cos u z = z, 2/ = (1) (2) (3) where u 6 [0,27r), v £ (-00,00), and z e (—00,00). There are several notational conventions, and whereas (u,v,z) is used in this work, Arfken (1970) prefers Biprism Biquadratic Number 143 (77, £, z). The following identities show that curves of constant u and v are CIRCLES in xy- space. 2 , / x \2 2 2 x -\- (y — a cot u) = a esc it (x — acothv) + y =a csch v. The Scale Factors are a h u = h v = cosh v — cos u a cosh v — cos u 1. The Laplacian is 2 (cosh v — cos u) 2 ( d 2 d 2 (4) (5) (6) (7) (8) d 2 Laplace's Equation is not separable in Bipolar Cylindrical Coordinates, but it is in 2-D Bipolar Coordinates. References Arfken, G. "Bipolar Coordinates (£, 77, z)." §2.9 in Math- ematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 97-102, 1970. Biprism Two slant triangular PRISMS fused together. see also Prism, Schmitt-Conway Biprism Bipyramid see Dipyramid Biquadratefree 60 40 20 40 60 80 100 A number is said to be biquadratefree if its Prime de- composition contains no tripled factors. All PRIMES are therefore trivially biquadratefree. The biquadratefree numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, . . . (Sloane's A046100). The biquadrateful num- bers (i.e., those that contain at least one biquadrate) are 16, 32, 48, 64, 80, 81, 96, ... (Sloane's A046101). The number of biquadratefree numbers less than 10, 100, 1000, ... are 10, 93, 925, 9240, 92395, 923939, . . . , and their asymptotic density is 1/C(4) = 90/tt 4 « 0.923938, where C(n) is the Riemann Zeta Function. see also Cubefree, Prime Number, Riemann Zeta Function, Squarefree References Sloane, N. J. A. Sequences A046100 and A046101 in "An On- Line Version of the Encyclopedia of Integer Sequences." Biquadratic Equation see Quartic Equation Biquadratic Number A biquadratic number is a fourth POWER, n 4 . The first few biquadratic numbers are 1, 16, 81, 256, 625, ... (Sloane's A000583). The minimum number of squares needed to represent the numbers 1, 2, 3, . . . are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, . . . (Sloane's A002377), and the number of distinct ways to represent the numbers 1, 2, 3, . . . in terms of biquadratic numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, A brute-force algorithm for enumerating the biquadratic permutations of n is repeated application of the Greedy Algorithm. Every POSITIVE integer is expressible as a SUM of (at most) 5(4) = 19 biquadratic numbers (WARING'S PROB- LEM). Davenport (1939) showed that G(4) = 16, mean- ing that all sufficiently large integers require only 16 biquadratic numbers. The following table gives the first few numbers which require 1, 2, 3, . . . , 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite. # Sloane Numbers 1, 16, 81, 256, 625, 1296, 2401, 4096, ... 2, 17, 32, 82, 97, 162, 257, 272, . . . 3, 18, 33, 48, 83, 98, 113, 163, ... 4, 19, 34, 49, 64, 84, 99, 114, 129, . . 5, 20, 35, 50, 65, 80, 85, 100, 115, .. 6, 21, 36, 51, 66, 86, 96, 101, 116, .. 7, 22, 37, 52, 67, 87, 102, 112, 117, . 8, 23, 38, 53, 68, 88, 103, 118, 128, . 9, 24, 39, 54, 69, 89, 104, 119, 134, . 10, 25, 40, 55, 70, 90, 105, 120, 135, 11, 26, 41, 56, 71, 91, 106, 121, 136, 12, 27, 42, 57, 72, 92, 107, 122, 137, 1 000290 2 003336 3 003337 4 003338 5 003339 6 003340 7 003341 8 003342 9 003343 10 003344 11 003345 12 003346 The following table gives the numbers which can be rep- resented in n different ways as a sum of k biquadrates. k n Sloane Numbers 1 1 000290 1, 16, 81, 256, 625, 1296, 2401, 4096, . . . 2 2 635318657, 3262811042, 8657437697, ... The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, ... (Sloane's A046039) cannot be represented using distinct biquadrates. see also CUBIC NUMBER, SQUARE NUMBER, WARING'S Problem References Davenport, H. "On Waring's Problem for Fourth Powers." Ann. Math. 40, 731-747, 1939. 144 Biquadratic Reciprocity Theorem Biquadratic Reciprocity Theorem x = q (mod p) . (i) This was solved by Gauss using the GAUSSIAN INTEGERS as (J).®.-'-"""*'- )/4][(JV(<r)-l)/4] (2) '4 \TV / 4 where n and a are distinct GAUSSIAN INTEGER PRIMES, N(a + hi) = yja? + b 2 and N is the norm. (3) ■{ 1 if x 4 = a (mod 7r) is solvable — l,i, or — i otherwise, (4) where solvable means solvable in terms of Gaussian In- tegers. see also RECIPROCITY THEOREM Biquaternion A Quaternion with Complex coefficients. The Alge- bra of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985). see also Quaternion References Clifford, W. K. "Preliminary Sketch of Biquaternions." Proc. London Math. Soc. 4, 381-395, 1873. Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. Dublin: Hodges and Smith, 1853. Study, E. "Von den Bewegung und Umlegungen." Math. Ann. 39, 441-566, 1891. van der Waerden, B. L. A History of Algebra from al- Khwarizmi to Emmy Noether. New York: Springer- Verlag, pp. 188-189, 1985. Birational Transformation A transformation in which coordinates in two SPACES are expressed rationally in terms of those in another. see also Riemann Curve Theorem, Weber's Theo- rem Birch Conjecture see Swinnerton-Dyer Conjecture Birch-Swinnerton-Dyer Conjecture see Swinnerton-Dyer Conjecture Birthday Attack Birkhoff 's Ergodic Theorem Let T be an ergodic ENDOMORPHISM of the PROBABIL- ITY SPACE X and let / : X -t R be a real-valued MEA- SURABLE Function. Then for Almost Every x € X, we have -^TfoF j (x)^ If dm as n — v oo. To illustrate this, take / to be the charac- teristic function of some Subset A of X so that /(*)={; if xe A if x £ A. The left-hand side of (-1) just says how often the or- bit of x (that is, the points x, Tx, T 2 x, . . . ) lies in A, and the right-hand side is just the MEASURE of A. Thus, for an ergodic ENDOMORPHISM, "space-averages = time- averages almost everywhere." Moreover, if T is continuous and uniquely ergodic with BOREL PROBA- BILITY MEASURE m and / is continuous, then we can replace the Almost Everywhere convergence in (-1) to everywhere. Birotunda Two adjoined ROTUNDAS. see also BlLUNABIROTUNDA, CUPOLAROTUNDA, ELON- GATED Gyrocupolarotunda, Elongated Ortho- CUPOLAROTUNDA, ELONGATED ORTHOBIROTUNDA, Gyrocupolarotunda, Gyroelongated Rotunda, ORTHOBIROTUNDA, TRIANGULAR HEBESPHENOROTUN- DA Birthday Attack Birthday attacks are a class of brute-force techniques used in an attempt to solve a class of cryptographic hash function problems. These methods take advantage of functions which, when supplied with a random in- put, return one of k equally likely values. By repeatedly evaluating the function for different inputs, the same output is expected to be obtained after about 1.2\/fc evaluations. see also Birthday Problem References RSA Laboratories. "Question 95. What is a Birthday At- tack." http : //www . rsa . com/rsalabs/newf aq/q95 . html. "Question 96. How Does the Length of a Hash Value Affect Security?" http : //www . rsa . com/r salabs/newf aq/ q96.html. van Oorschot, P. and Wiener, M. "A Known Plaintext At- tack on Two-Key Triple Encryption." In Advances in Cryptology — Eurocrypt '90. New York: Springer- Verlag, pp. 366-377, 1991. Yuval, G. "How to Swindle Rabin." Cryptologia 3, 187-189, Jul. 1979. Birthday Problem Birthday Problem 145 Birthday Problem Consider the probability Qi(n, d) that no two people out of a group of n will have matching birthdays out of d equally possible birthdays. Start with an arbitrary per- son's birthday, then note that the probability that the second person's birthday is different is (d — l)/d, that the third person's birthday is different from the first two is [(d — l)/d][(d — 2)/d], and so on, up through the nth person. Explicitly, Qi(n,d) = Id- 2 d-(n-l) d d d _ (d-l)(d-2)---[d-(n-l)] d n But this can be written in terms of FACTORIALS as dl Qi(n,d) (d-n)\d> 71 ' (1) (2) so the probability P 2 (n, 365) that two people out of a group of n do have the same birthday is therefore P 2 (n,d) = 1-Qi(n,d) = 1 d\ (d-n)\d n ' (3) If 365-day years have been assumed, i.e., the existence of leap days is ignored, then the number of people needed for there to be at least a 50% chance that two share birthdays is the smallest n such that p2(^, 365) > 1/2. This is given by n — 23, since P 2 (23,365) = 38093904702297390785243708291056390518886454060947061 75091883268515350125426207425223147563269805908203125 « 0.507297. (4) The number of people needed to obtain Pzin, 365) > 1/2 for n = 1, 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, . . . (Sloane's A033810). The probability P2(n,d) can be estimated as P 2 (n,d)^l-e- n(n " 1)/2d (5) — ('-sr- <■> where the latter has error C < — 7~7TK (7) 6(d-n + l) 2 (Sayrafiezadeh 1994). In general, let Qi(n,d) denote the probability that a birthday is shared by exactly i (and no more) people out of a group of n people. Then the probability that a birthday is shared by k or more people is given by P k {n,d) = l-^Qi(n,d). Qi can be computed explicitly as (8) L«/2J ~ ~cF 2^i 2H\(n dl (-l) n (n~2i)l(d- n + i)\ (9) where (™) is a BINOMIAL COEFFICIENT, T(n) is a Gamma Function, and Pj[ x \x) is an Ultraspheri- cal Polynomial. This gives the explicit formula for P^n^d) as Pz(n 7 d) = 1 - Qi(n,d) - Q 2 (n,d) (-1)^(71+ l)Pi~ d) (2-^) " ^ 2 n / 2 d n * K J Qz{n,d) cannot be computed in entirely closed form, but a partially reduced form is Qz{n,d) = r(d+i) d n (-irF(f)-F(-f) T(d-n + l) +(-i) r(i + n) ^ r(d-i + i)r(i + i) (ii) where F = F(n,d, a) = 1-3^2 ■ i(l_„),l(2-n),-I i(d-n+l) i(d-n + 2)'' (12) and 3^2 (a, 6, c; d, e; z) is a GENERALIZED HYPERGEO- metric Function. In general, Qk(n,d) can be computed using the RECUR- RENCE Relation [n/kj Qk(n,d) — y^ n!rf! d ik i\(k\y(n-ik)\(d-i)\ x^2Qj{n-k,d-i) ; Jd-iy j=i (Jn — ik (13) 146 Birthday Problem Bisection Procedure (Finch). However, the time to compute this recursive function grows exponentially with k and so rapidly be- comes unwieldy. The minimal number of people to give a 50% probability of having at least n coincident birth- days is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, . . . (Sloane's A014088; Diaconis and Mosteller 1989). A good approximation to the number of people n such that p = Pk(n,d) is some given value can given by solv- ing the equation ne -n/(dk) d* _1 fc!ln 1 1- d(fc + l) i/fc (14) for n and taking [n], where [n] is the CEILING Func- tion (Diaconis and Mosteller 1989). For p = 0.5 and k — 1, 2, 3, ... , this formula gives n = 1, 23, 88, 187, 313, 459, 722, 797, 983, 1179, 1382, 1592, 1809, ..., which differ from the true values by from to 4. A much simpler but also poorer approximation for n such that p — 0.5 for k < 20 is given by n = 47(fe-1.5)' 3/2 (15) (Diaconis and Mosteller 1989), which gives 86, 185, 307, 448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for k = 3, 4,.... The "almost" birthday problem, which asks the number of people needed such that two have a birthday within a day of each other, was considered by Abramson and Moser (1970), who showed that 14 people suffice. An ap- proximation for the minimum number of people needed to get a 50-50 chance that two have a match within k days out of d possible is given by n(k y d) = 1.2 d 2k + 1 (16) (Sevast'yanov 1972, Diaconis and Mosteller 1989). see also Birthday Attack, Coincidence, Small World Problem References Abramson, M. and Moser, W. O. J. "More Birthday Sur- prises." Amer. Math. Monthly 77, 856-858, 1970. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, pp. 45-46, 1987. Bloom, D. M. "A Birthday Problem." Amer. Math. Monthly 80, 1141-1142, 1973. Bogomolny, A. "Coincidence." http://www.cut-* the-knot . com/do_you_know/coincidence.html. Clevenson, M. L. and Watkins, W. "Majorization and the Birthday Inequality." Math. Mag. 64, 183-188, 1991. Diaconis, P. and Mosteller, F. "Methods of Studying Coinci- dences." J. Amer. Statist. Assoc. 84, 853-861, 1989. Feller, W. An Introduction to Probability Theory and Its Ap- plications, Vol. 1, 3rd ed. New York: Wiley, pp. 31-32, 1968. Finch, S. "Puzzle #28 [June 1997]: Coincident Birthdays." http: //www. maths oft . com/mathcad/library /puzzle/ soln28/soln28.html. Gehan, E. A. "Note on the 'Birthday Problem.'" Amer. Stat. 22, 28, Apr. 1968. Heuer, G. A. "Estimation in a Certain Probability Problem." Amer. Math. Monthly 66, 704-706, 1959. Hocking, R. L. and Schwertman, N. C. "An Extension of the Birthday Problem to Exactly k Matches." College Math. J. 17, 315-321, 1986. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 102-103, 1975. Klamkin, M. S. and Newman, D. J. "Extensions of the Birth- day Surprise." J. Combin. Th. 3, 279-282, 1967. Levin, B. "A Representation for Multinomial Cumulative Distribution Functions." Ann. Statistics 9, 1123-1126, 1981. McKinney, E. H. "Generalized Birthday Problem." Amer. Math. Monthly 73, 385-387, 1966. Mises, R. von. "Uber Aufteilungs — und Besetzungs- Wahrscheinlichkeiten." Revue de la Faculte des Sci- ences de VUniversite d'Istanbul, N. S. 4, 145—163, 1939. Reprinted in Selected Papers of Richard von Mises, Vol. 2 (Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. Szego, and G. BirkhofF), Providence, RI: Amer. Math. Soc, pp. 313-334, 1964. Riesel, H. Prime Numbers and Computer Methods for Fac- torization, 2nd ed. Boston, MA: Birkhauser, pp. 179-180, 1994. Sayrafiezadeh, M. "The Birthday Problem Revisited." Math. Mag. 67, 220-223, 1994. Sevast'yanov, B. A. "Poisson Limit Law for a Scheme of Sums of Dependent Random Variables." Th. Prob. Appl. 17, 695-699, 1972. Sloane, N. J. A. Sequences A014088 and A033810 in "An On- Line Version of the Encyclopedia of Integer Sequences." Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96, June 1998. Tesler, L. "Not a Coincidence!" http://www.nomodes.com/ coincidence .html. Bisected Perimeter Point see Nagel Point Bisection Procedure Given an interval [a, &], let a n and b n be the endpoints at the nth iteration and r n be the nth approximate solu- tion. Then, the number of iterations required to obtain an error smaller than e is found as follows. 1 (1) (2) \r n -r\<±{b n - a n ) = 2~ n (b - a) < e (3) — n In 2 < In e — ln(6 — a), ln(6 — a) — In e n > In 2 (4) (5) so see also ROOT References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 964-965, 1985. Bisector Bislit Cube 147 Press, W. H.; Fiannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Bracketing and Bisection." §9.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 343-347, 1992. Bisector Bisection is the division of a given curve or figure into two equal parts (halves). see also Angle Bisector, Bisection Procedure, Exterior Angle Bisector, Half, Hemisphere, Line Bisector, Perpendicular Bisector, Trisec- TION Bishop's Inequality Let V{r) be the volume of a BALL of radius r in a com- plete 7l-D RlEMANNIAN MANIFOLD with RlCCI CURVA- TURE > (n - 1)k. Then V(r) > V K (r), where V K is the volume of a Ball in a space having constant Sec- tional Curvature. In addition, if equality holds for some Ball, then this Ball is Isometric to the Ball of radius r in the space of constant SECTIONAL CURVA- TURE K. References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Bishops Problem B B B B B B B B B B B B B B Find the maximum number of bishops B(n) which can be placed on an n x n Chessboard such that no two attack each other. The answer is 2n — 2 (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, . . . (the Even Numbers) for n = 2, 3, One maximal so- lution for n = 8 is illustrated above. The number of distinct maximal arrangements of bishops for n — 1, 2, ... are 1, 4, 26, 260, 3368, . . . (Sloane's A002465). The number of rotationally and reflectively distinct solutions on an n x n board for n > 2 is B(n) / 2 (n-4)/2 [2 (n-2)/2 + y ^ n ey( . Q | 2 (n-3)/2 [2 („-3)/2 + 1 ] fornodd where |nj is the FLOOR FUNCTION, giving the sequence for n = 1, 2, . . . as 1, 1, 2, 3, 6, 10, 20, 36, . . . (Sloane's A005418). B B B B B B B B The minimum number of bishops needed to occupy or attack all squares on an n x n Chessboard is n, ar- ranged as illustrated above. see also Chess, Kings Problem, Knights Problem, Queens Problem, Rooks Problem References Ahrens, W. Mathematische Unterhaltungen und Spiele, Vol 1, 3rd ed. Leipzig, Germany: Teubner, p. 271, 1921. Dudeney, H. E. "Bishops — Unguarded" and "Bishops — Guarded." §297 and 298 in Amusements in Mathematics. New York: Dover, pp. 88-89, 1970. Guy, R. K. "The n Queens Problem." §C18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 133-135, 1994. Madachy, J. Madachy's Mathematical Recreations. New York: Dover, pp. 36-46, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 74- 75, 1995. Sloane, N. J. A. Sequences A002465/M3616 and A005418/ M0771 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Bislit Cube The 8- Vertex graph consisting of a Cube in which two opposite faces have DIAGONALS oriented PERPENDICU- LAR to each other. see also Bidiakis Cube, Cube, Cubical Graph (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula is J B(n) = 2 n - 3 + 2 L( ' l - 1)/2J - 1 , 148 Bispherical Coordinates Bispherical Coordinates A system of CURVILINEAR COORDINATES defined by a sin £ cos <fi y- cosh 77 — cos £ a sin £ sin <\> cosh 77 — cos £ a sinh 77 cosh T] — cos £ The Scale Factors are h a h v cos 77 — cos £ a The Laplacian is 2 _ / — cos u co ^ 2 u ' \ cosh t> cosh 77 — cos £ asin£ cosh 77 — cos £ + 3 cosh v cot u a) (2) (3) (4) (5) (6) -3 cosh 2 v cot u esc u + cosh vcsc u cosh v — cos ti d(j> 2 + (cosu — cosh v) sinh v~ — h (cosh v - cosu) ■^-^ <% 0v 2 a + (cosh v — cos ti) (cosh v cot w — sin u — cos u cot u) — +(cosh 2 i; — cos u) -^—7 . ou 2 (7) In bispherical coordinates, LAPLACE'S EQUATION is sep- arable, but the Helmholtz Differential Equation is not. see also Laplace's Equation— Bispherical Coor- dinates, Toroidal Coordinates References Arfken, G. "Bispherical Coordinates (£, 77, <£)." §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 665-666, 1953. Black-Scholes Theory Bit Complexity The number of single operations (of ADDITION, SUB- TRACTION, and Multiplication) required to complete an algorithm. see also STRASSEN FORMULAS References Borodin, A. and Munro, I. The Computational Complexity of Algebraic and Numeric Problems. New York: American Elsevier, 1975. Bitangent bitangent A Line which is Tangent to a curve at two distinct points. see also Klein's Equation, Plucker Characteris- tics, Secant Line, Solomon's Seal Lines, Tangent Line Bivariate Distribution see Gaussian Bivariate Distribution Bivector An antisymmetric TENSOR of second Rank (a.k.a. 2- form) . X = X ah u) a A u) , where A is the Wedge Product (or Outer Prod- uct). Biweight see TUKEY'S BIWEIGHT Black-Scholes Theory The theory underlying financial derivatives which in- volves "stochastic calculus" and assumes an uncor- rected Log Normal Distribution of continuously varying prices. A simplified "binomial" version of the theory was subsequently developed by Sharpe et al. (1995) and Cox et al (1979). It reproduces many re- sults of the full-blown theory, and allows approximation of options for which analytic solutions are not known (Price 1996). see also Garman-Kohlhagen Formula References Black, F. and Scholes, M. S. "The Pricing of Options and Corporate Liabilities." J. Political Econ. 81, 637-659, 1973. Cox, J. C; Ross, A.; and Rubenstein, M. "Option Pricing: A Simplified Approach." J. Financial Economics 7, 229-263, 1979. Price, J. F. "Optional Mathematics is Not Optional." Not. Amer. Math. Soc. 43, 964-971, 1996. Sharpe, W. F.; Alexander, G. J,; and Bailey, J. V. Invest- ments, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. Black Spleenwort Fern Black Spleenwort Fern see BARNSLEY'S FERN Blackman Function Blecksmith-Brillhart- Gerst Theorem 149 -1 -0.5 0.5 1 -0.5 An Apodization Function given by A(x) = 0.42 + 0.5 cos (?) + 0.08 cos (^) a) Its Full Width at Half Maximum is 0.810957a. The Apparatus Function is I(k) = a(0.84 - 0.36a 2 fc 2 - 2.17 x 10~ x Vfe 4 ) sin(27raA:) (l-a 2 fc 2 )(l-4a 2 A; 2 ) The Coefficients are approximations to ao ai a 2 = 3969 9304 1155 4652 715 18608' (2) (3) (4) (5) which would have produced zeros of I(k) at k = (7/4)a and k = (9/4)a. see also APODIZATION FUNCTION References Blackman, R. B. and Tukey, J, W. "Particular Pairs of Win- dows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98-99, 1959. Blancmange Function A Continuous Function which is nowhere Differ- ENTIABLE. The iterations towards the continuous func- tion are Batrachions resembling the Hofstadter- Conway $10,000 Sequence. The first six iterations are illustrated below. The dth iteration contains TV + 1 points, where TV = 2 d , and can be obtained by setting 6(0) = b(N) = 0, letting b{m + 2 71 " 1 ) = 2 n + \[b{m) + b{m + 2 n )], and looping over n = d to 1 by steps of —1 and m = to TV- 1 by steps of 2 n . Peitgen and Saupe (1988) refer to this curve as the Tak- agi Fractal Curve. see also HOFSTADTER-CONWAY $10,000 SEQUENCE, Weierstrak Function References Dixon, R. Mathographics. New York: Dover, pp. 175-176 and 210, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). "Midpoint Displace- ment and Systematic Fractals: The Takagi Fractal Curve, Its Kin, and the Related Systems." §A.1.2 in The Science of Fractal Images. New York: Springer- Verlag, pp. 246- 248, 1988. Takagi, T. "A Simple Example of the Continuous Function without Derivative." Proc. Phys. Math. Japanl y 176-177, 1903. Tall, D. O. "The Blancmange Function, Continuous Every- where but DifTerentiable Nowhere." Math. Gaz. 66,11-22, 1982. Tall, D. "The Gradient of a Graph." Math. Teaching 111, 48-52, 1985. Blaschke Conjecture The only WlEDERSEHEN MANIFOLDS are the standard round spheres. The conjecture has been proven by com- bining the Berger-Kazdan Comparison Theorem with A. Weinstein's results for n Even and C. T. Yang's for n Odd. References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Blaschke's Theorem A convex planar domain in which the minimal length is > 1 always contains a Circle of Radius 1/3. References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Blecksmith-Brillhart-Gerst Theorem A generalization of Schroter'S FORMULA. References Berndt, B. C Ramanujan's Notebooks, Part III. New York: Springer- Verlag, p. 73, 1985. BlichfeldVs Lemma Blichfeldt's Lemma see Blichfeldt's Theorem Blichfeldt's Theorem Published in 1914 by Hans Blichfeldt. It states that any bounded planar region with POSITIVE AREA > A placed in any position of the UNIT SQUARE LATTICE can be Translated so that the number of Lattice Points inside the region will be at least A + 1. The theorem can be generalized to n-D. BLM/Ho Polynomial A 1-variable unoriented Knot Polynomial Q(x). It satisfies Qunknot = 1 (l) and the SKEIN RELATIONSHIP Ql^+Ql^ =x(Q Lq + Q Lqo ). (2) It also satisfies Qlx#l 2 = Ql y Ql 2 , (3) where # is the KNOT Sum and Ql*=Ql> (4) where L* is the Mirror Image of L. The BLM/Ho polynomials of Mutant KNOTS are also identical. Brandt et al. (1986) give a number of interesting prop- erties. For any Link L with > 2 components, Ql — 1 is divisible by 2 (x — 1). If L has c components, then the lowest POWER of x in Ql(x) is 1 — c, and lim x c lim (-m) c - 1 P L (£,m) J (5) (^m)-4(l,0) n V ' where P L is the HOMFLY Polynomial. Also, the de- gree of Ql is less than the Crossing Number of L. If L is a 2-Bridge Knot, then Q L (z) = 2z~' 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ), (6) where z = -t - r -1 (Kanenobu and Sumi 1993). The Polynomial was subsequently extended to the 2- variable Kauffman Polynomial F(a i z) y which satis- fies Q(x) = F{l,x). (7) Brandt et al. (1986) give a listing of Q POLYNOMIALS for KNOTS up to 8 crossings and links up to 6 crossings. References Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. "A Polynomial Invariant for Unoriented Knots and Links." In- vent Math. 84, 563-573, 1986. Ho, C. F. "A New Polynomial for Knots and Links — Preliminary Report." Abstracts Amer. Math. Soc. 6, 300, 1985. Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Knots through 22-Crossings." Math. Comput. 60, 771-778 and S17-S28, 1993. Stoimenow, A. "Brandt-Lickorish-Millett-Ho Polynomi- als." http: //www, informatik.hu-berlin.de/-stoimeno/ ptab/blmhlO . html. ^ Weisstein, E. W. "Knots." http: //www. astro. Virginia, edu/ - eww6n/math/not ebooks/Knot s . m. Block Design Bloch Constant N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Let F be the set of Complex analytic functions / de- fined on an open region containing the closure of the unit disk D = {z : \z\ < 1} satisfying /(0) = and df/dz(Q) = 1. For each / in F, let b(f) be the SUPRE- MUM of all numbers r such that there is a disk S in D on which / is ONE-TO-ONE and such that f(S) contains a disk of radius r. In 1925, Bloch (Conway 1978) showed that b(f) > 1/72. Define Bloch's constant by B = mi{btf):f£F}. Ahlfors and Grunsky (1937) derived 0.433012701...= \VZ<B i r(i)r(i§) 4 < v / nm r (?) f^- < 0.4718617. They also conjectured that the upper limit is actually the value of B, 1 r(j)r(M) v/TTv! r (i) iV = 0.4718617X / V ^ °4? (Le Lionnais 1983). see also Landau Constant References Conway, J. B. Functions of One Complex Variable, 2nd ed. New York: Springer- Verlag, 1989. Finch, S, "Favorite Mathematical Constants." http: //www. mathsof t . com/asolve/constant/bloch/bloch.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983. Minda, CD. "Bloch Constants." J. d Analyse Math. 41, 54-84, 1982. BIoch-Landau Constant see Landau Constant Block see also Block Design, Square Polyomino Block Design An incidence system (v, fc, A, r, 6) in which a set X of v points is partitioned into a family A of b subsets (blocks) in such a way that any two points determine A blocks, there are k points in each block, and each point is contained in r different blocks. It is also generally required that k < v , which is where the "incomplete" comes from in the formal term most often encountered Block Design Blow-Up 151 for block designs, Balanced Incomplete Block De- signs (BIBD). The five parameters are not independent, but satisfy the two relations bk X(v~ 1) = r(fc-l). (1) (2) A BIBD is therefore commonly written as simply (v, &, A), since b and r are given in terms of u, k, and A by v(v - 1)A k(k - 1) (3) (4) A BIBD is called SYMMETRIC if b = v (or, equivalently, r = k). Writing X = {^}Li and A — {Aj} b j=1 , then the IN- CIDENCE Matrix of the BIBD is given by the v x b Matrix M defined by 1J I othe GA otherwise. This matrix satisfies the equation MM T = (r-A)l + AJ, (5) (6) where I is a v x v IDENTITY MATRIX and J is a v x v matrix of Is (Dinitz and Stinson 1992). Examples of BIBDs are given in the following table. Block Design (v, K A) affine plane (n , n, 1) Fano plane (7, 3, 1)) Hadamard design symmetric (An + 3, 2n -f- 1, n) projective plane symmetric (n 2 + n -j- 1, n + 1, 1) Steiner triple system (v, 3, 1) unital (g 3 + 1, q+ 1, 1) see also Affine Plane, Design, Fano Plane, Hada- mard Design, Parallel Class, Projective Plane, Resolution, Resolvable, Steiner Triple System, Symmetric Block Design, Unital References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1-12, 1992. Ryser, H. J. "The {b,v,r, k, A)-Configuration." §8.1 in Com- binatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 96-102, 1963. Block Growth Let (rco£i£2 • • •) be a sequence over a finite ALPHABET A (all the entries are elements of A). Define the block growth function B(n) of a sequence to be the number of Admissible words of length n. For example, in the sequence aabaabaabaabaab. . . , the following words are Admissible Length Admissible Words a, b aa, a&, ba aab, aba, baa aaba, abaa, baab so B(l) = 2, B(2) = 3, 5(3) = 3, B(4) = 3, and so on. Notice that B(n) < B(n + 1), so the block growth function is always nondecreasing. This is because any Admissible word of length n can be extended right- wards to produce an Admissible word of length n + 1. Moreover, suppose B(n) = B(n + 1) for some n. Then each admissible word of length n extends to a unique Admissible word of length n + 1. For a SEQUENCE in which each substring of length n uniquely determines the next symbol in the SEQUENCE, there are only finitely many strings of length n, so the process must eventually cycle and the SEQUENCE must be eventually periodic. This gives us the following the- orems: 1. If the Sequence is eventually periodic, with least period p, then B(n) is strictly increasing until it reaches p, and B(n) is constant thereafter. 2. If the Sequence is not eventually periodic, then B(n) is strictly increasing and so B(n) > n + 1 for all n. If a Sequence has the property that B(n) = n+1 for all n, then it is said to have minimal block growth, and the Sequence is called a Sturmian Sequence. The block growth is also called the GROWTH FUNCTION or the Complexity of a Sequence. Block Matrix A square Diagonal Matrix in which the diagonal ele- ments are Square Matrices of any size (possibly even lxl), and the off-diagonal elements are 0. Block (Set) One of the disjoint Subsets making up a Set Parti- tion. A block containing n elements is called an n- block. The partitioning of sets into blocks can be de- noted using a RESTRICTED GROWTH STRING. see also Block Design, Restricted Growth String, Set Partition Blow-Up A common mechanism which generates SINGULARITIES from smooth initial conditions. 152 Blue-Empty Coloring Bohemian Dome Blue-Empty Coloring see Blue-Empty Graph Blue-Empty Graph An Extremal Graph in which the forced Trian- gles are all the same color. Call R the number of red Monochromatic Forced Triangles and B the number of blue Monochromatic Forced Triangles, then a blue-empty graph is an Extremal Graph with B = 0. For Even n, a blue-empty graph can be achieved by coloring red two Complete SUBGRAPHS of n/2 points (the RED Net method). There is no blue- empty coloring for Odd n except for n = 7 (Lorden 1962). see also Complete Graph, Extremal Graph, Monochromatic Forced Triangle, Red Net References Lorden, G. "Blue-Empty Chromatic Graphs." Amer. Math. Monthly 69, 114-120, 1962. Sauve, L. "On Chromatic Graphs." Amer. Math. Monthly 68, 107-111, 1961. Board A subset of d x d, where d = {1, 2, . . . , d}. see also Rook Number Boatman's Knot see Clove Hitch Bochner Identity For a smooth Harmonic Map u : M -► TV, A(|Vu| 2 ) = \V(du)\ 2 + {RicM Vu,Vu) - (Riem N (u)(Vu, Vu)Vu, Vu> , where V is the GRADIENT, Ric is the RlCCl TENSOR, and Riem is the Riemann Tensor. References Eels, J. and Lemaire, L. "A Report on Harmonic Maps." Bull. London Math. Soc. 10, 1-68, 1978. Bochner's Theorem Among the continuous functions on R n , the POSITIVE Definite Functions are those functions which are the Fourier Transforms of finite measures. Bode's Rule J XI f{x) dx = ^/i(7/i + 32/ 2 + 12/ 3 + 32/ 4 + 7/5) -sfeW'K). References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972. Bogdanov Map A 2-D MAP which is conjugate to the Henon Map in its nondissipative limit. It is given by x = x + y y' =y + ey + kx(x - l) + [ixy. see also Henon Map References Arrowsmith, D. K.; Cartwright, J. H. E.; Lansbury, A. N.; and Place, C. M. "The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System." Int. J. Bi- furcation Chaos 3, 803-842, 1993. Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373-388, 1981. Bogomolov-Miyaoka-Yau Inequality Relates invariants of a curve defined over the INTEGERS. If this inequality were proven true, then FERMAT'S Last THEOREM would follow for sufficiently large exponents. Miyaoka claimed to have proven this inequality in 1988, but the proof contained an error. see also FERMAT'S LAST THEOREM References Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. Math. Monthly 101, 3-14, 1994. Bohemian Dome see also Hardy's Rule, Newton-Cotes Formulas, Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal Rule, Weddle's Rule A Quartic Surface which can be constructed as fol- lows. Given a CIRCLE C and PLANE E PERPENDICULAR to the Plane of C, move a second Circle K of the same Radius as C through space so that its Center always lies on C and it remains PARALLEL to E. Then K sweeps out the Bohemian dome. It can be given by the parametric equations x = a cos u y = b cos v + a sin u z — csinv where u, v 6 [0, 27r). In the above plot, a = 0.5, b = 1.5, and c = 1. see also Quartic Surface Bohr-Favard Inequalities Bombieri Norm 153 References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 19-20, 1986. Fischer, G. (Ed.). Plate 50 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, p. 50, 1986. Nordstrand, T. "Bohemian Dome." http://www.uib.no/ people/nf ytn/bodtxt .htm. Bohr-Favard Inequalities If / has no spectrum in [—A, A], then saii'i (Bohr 1935). A related inequality states that if Ak is the class of functions such that /(*) = /(* + 2*), /(*),/'(*),... ./^(a:) are absolutely continuous and f w f(x) dx = 0, then 4 1 _1)^ S ^2^ ( 2 ^+ l)M-i N/ wi (Northcott 1939). Further, for each value of k, there is always a function f(x) belonging to Ak and not identi- cally zero, for which the above inequality becomes an in- equality (Favard 1936). These inequalities are discussed in Mitrinovic et al. (1991). References Bohr, H. "Ein allgemeiner Satz iiber die Integration eines trigonometrischen Polynoms." Prace Matem.-Fiz. 43, 1935. Favard, J. "Application de la formule soiiimaloire d'Euler a la demonstration de quelques proprietes extremales des integrale des fonctions periodiques ou presqueperiodiqu.es." Mat Tidsskr. B, 81-94, 1936. [Reviewed in Zentralblatt f. Math. 16, 58-59, 1939.] Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, pp. 71-72, 1991. Northcott, D. G. "Some Inequalities Between Periodic Func- tions and Their Derivatives." J. London Math. Soc. 14, 198-202, 1939. Tikhomirov, V. M. "Approximation Theory." In Analysis II (Ed. R. V. Gamrelidze). New York: Springer- Verlag, pp." 93-255, 1990. Bolyai-Gerwein Theorem see Wallace-Bolyai-Gerwein Theorem Bolza Problem Given the functional U= /(2/l,---,S/n;3/l\...,2/n') d * Jt +G(yi , . . . , 2Mr; 2/11, • ■ ■ , 2/m), find in a class of arcs satisfying p differential and q finite equations <M3/i,-..,2/n;3/i',...,3/n') = ° for a = l,...,p VV3(yi»--->yn) = for = l,...,g as well as the r equations on the endpoints X7(yio,---)2/nr;3/ii,...,2/ni) = for 7 = 1, . . . , r, one which renders U a minimum. References Goldstine, H. H, A History of the Calculus of Variations from the 17th through the 19th Century. New York: Springer- Verlag, p. 374, 1980. Bolzano Theorem see Bolzano- WeierstraB Theorem Bolzano- Weierstrafl Theorem Every Bounded infinite set in W 1, has an ACCUMULA- TION Point. For n = 1, the theorem can be stated as follows: If a Set in a METRIC SPACE, finite-dimensional Euclidean Space, or First-Countable Space has infinitely many members within a finite interval x 6 [a, 6], then it has at least one Limit Point x such that x e [a, &]. The theorem can be used to prove the Inter- mediate Value Theorem. Bombieri's Inequality For Homogeneous Polynomials P and Q of degree m and n, then [P ■ Qh > tM. (m + n)\ -jiPhlQb, where [P • Q] 2 is the BOMBIERI Norm. If m = n, this becomes [P'Qh>[P]2[Q]2. see also Beauzamy and Degot's Identity, Reznik's Identity Bombieri Inner Product For Homogeneous Polynomials P and Q of degree n, [P,Q]= J2 C*! 1 "-^ 1 )^,..^^!,..^)- ii,...,i„>0 Bombieri Norm For Homogeneous Polynomials P of degree m, mV [P], = y/frF]=\ J2 S |a ' y |a|=m see also POLYNOMIAL BAR NORM 154 BombievVs Theorem Bonne Projection Bombieri's Theorem Define E(x;q,a) = ip(x\q,a) - <KqV where ■tP(x;q,a)= ^ A(n) (1) (2) n<x n = a (mod g) (Davenport 1980, p. 121), A(n) is the MANGOLDT Function, and <j>(q) is the Totient Function. Now define E(x;q)= max \E(x\q y a)\ (3) (a,q°) = l where the sum is over a RELATIVELY PRIME to q, (a,g) = 1, and E*(x,q) = ma,xE{y,q). (4) y<x Bombieri's theorem then says that for A > fixed, ^E*(x,q) « ^Q{\nx)\ (5) q<Q provided that ^(lnx)" 4 < Q < \fx. References Bombieri, E. "On the Large Sieve." Mathematika 12, 201- 225, 1965. Davenport, H. "Bombieri's Theorem." Ch. 28 in Multiplica- tive Number Theory, 2nd ed. New York: Springer- Verlag, pp. 161-168, 1980. Bond Percolation bond percolation site percolation A Percolation which considers the lattice edges as the relevant entities (left figure). see also Percolation Theory, Site Percolation Bonferroni Correction The Bonferroni correction is a multiple-comparison cor- rection used when several independent STATISTICAL TESTS are being performed simultaneously (since while a given Alpha Value a may be appropriate for each individual comparison, it is not for the set of all com- parisons). In order to avoid a lot of spurious positives, the Alpha Value needs to be lowered to account for the number of comparisons being performed. The simplest and most conservative approach is the Bonferroni correction, which sets the ALPHA VALUE for the entire set of n comparisons equal to a by taking the Alpha Value for each comparison equal to cx/n. Ex- plicitly, given n tests Ti for hypotheses Hi (1 < i < n) under the assumption Ho that all hypotheses Hi are false, and if the individual test critical values are < a/n, then the experiment-wide critical value is < a. In equa- tion form, if P(Ti passes \H Q ) < - n for 1 < i < ra, then P(some Ti passes \H ) < a, which follows from BONFERRONl'S INEQUALITY. Another correction instead uses 1 — (1— a) 1 / 71 . While this choice is applicable for two-sided hypotheses, multivari- ate normal statistics, and positive orthant dependent statistics, it is not, in general, correct (Shaffer 1995). see also ALPHA VALUE, HYPOTHESIS TESTING, STATIS- TICAL Test References Bonferroni, C. E. "II calcolo delle assicurazioni su gruppi di teste." In Studi in Onore del Professore Salvatore Ortu Carboni. Rome: Italy, pp. 13-60, 1935. Bonferroni, C. E. "Teoria statistica delle classi e calcolo delle probabilita." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 8, 3-62, 1936. Dewey, M. "Carlo Emilio Bonferroni: Life and Works." http://www.nottingham.ac.uk/-mh2md/life.html. Miller, R. G. Jr. Simultaneous Statistical Inference. New York: Springer- Verlag, 1991. Perneger, T. V. "What's Wrong with Bonferroni Adjust- ments." Brit Med. J. 316, 1236-1238, 1998. Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev. Psych. 46, 561-584, 1995. Bonferroni's Inequality Let P(Ei) be the probability that £?» is true, and P(U" =1 ^i) be the probability that E u E 2j ..., E n are all true. Then fU 1 *") *!><*>• Bonferroni Test see Bonferroni Correction Bonne Projection Book Stacking Problem Boolean Algebra 155 A Map Projection which resembles the shape of a heart. Let (pi be the standard parallel and Ao the central meridian. Then where x = p sin E y — R cot 0i — p cos R ) p = cot (pi + 0i — <(> (A- Aq)cos0 The inverse FORMULAS are <p = cot 01 + (f>i — p A = A + COS0 ■ tan -l ( x \ cot (pi -y where p = ±\/x 2 + (cot 0i -y) 2 . (1) (2) (3) (4) (5) (6) (7) References Snyder, J. P. Map Projections — A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 138-140, 1987. (Sloane's A001008 and A002805). In order to find the number of stacked books required to obtain d book-lengths of overhang, solve the d n equation for d, and take the Ceiling Function. For n = 1, 2, . . . book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (Sloane's A014537) books are needed. References Dickau, R. M. "The Book-Stacking Problem." http://wwv. prairienet.org/-pops/BookStacking.html. Eisner, L. "Leaning Tower of the Physical Review." Amer. J. Phys. 27, 121, 1959. Gardner, M. Martin Gardner's Sixth Book of Mathematical Games from Scientific American. New York: Scribner's, p. 167, 1971. Graham, R. L.; Knuth, D, E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Read- ing, MA: Addison- Wesley, pp. 272-274, 1990. Johnson, P. B. "Leaning Tower of Lire." Amer. J. Phys. 23, 240, 1955. Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 1, 322, 1953. Sharp, R. T. "Problem 52." Pi Mu Epsilon J. 2, 411, 1954. Sloane, N. J. A. Sequences A014537, A001008/M2885, and A002805/M1589 in "An On-Line Version of the Encyclo- pedia of Integer Sequences." Boole's Inequality Book Stacking Problem How far can a stack of n books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible d n for n books (in terms of book lengths) is half the nth partial sum of the Harmonic Series, given explicitly by d n n where <&(z) is the DiGAMMA FUNCTION and 7 is the Euler-Mascheroni Constant. The first few values are di = -= 0.5 3 4 — 11 0.75 d 3 = i| « 0.91667 A — 25 rf 4 - 24 p U £ 0^E p ^)- 1.04167, If Ei and Ej are Mutually Exclusive for all i and j, then the INEQUALITY becomes an equality. Boolean Algebra A mathematical object which is similar to a BOOLEAN RING, but which uses the meet and join operators in- stead of the usual addition and multiplication operators. A Boolean algebra is a set B of elements a, 6, ... with Binary Operators + and * such that la. If a and b are in the set S, then a + b is in the set B. lb. If a and b are in the set B, then a • b is in the set B. 2a. There is an element Z (zero) such that a + Z = a for every element a. 2b. There is an element U (unity) such that a • U = a for every element a. 3a. a + 6 = b + a 3b. a - b = b ■ a 4a. a + 6 ■ c = (a + b) (a + c) 4b. a ■ (b-\- c) — a - b-\- a ■ c 5. For every element a there is an element a such that a + a' — U and a ■ a' = Z. 6. There are are least two distinct elements in the set B. (Bell 1937, p. 444). 156 Boolean Algebra Boolean Ring In more modern terms, a Boolean algebra is a Set B of elements a, 6, ... with the following properties: 1. B has two binary operations, A (Wedge) and V (Vee), which satisfy the IDEMPOTENT laws aAa = a\/a = a, the Commutative laws a A b — b A a aVb^bV a, and the Associative laws a A (b A c) = (a A b) A c aV(6Vc) = (aVb) V c. 2. The operations satisfy the ABSORPTION LAW a A (a V b) = a V (a A 6) = a. 3. The operations are mutually distributive a A (6Vc) = (a A 6) V (a Ac) a V (6 A c) = (a V 6) A (a V c). 4. I? contains universal bounds 0,/ which satisfy OAa = O Va = a / A a = a /Vfl = J. 5. B has a unary operation a —± a' of complementation which obeys the laws a A a = O aV a = I (Birkhoff and Mac Lane 1965). Under intersection, union, and complement, the subsets of any set I form a Boolean algebra. Huntington (1933a, b) presented the following basis for Boolean algebra, 1. Commutivity. x + y = y + x. 2. Associativity, (x + y) + z = x + (y + z). 3. Huntington Equation. n(n(x) + y) + n(n(a;) + n(y)) = x. H. Robbins then conjectured that the Huntington Equation could be replaced with the simpler Robbins Equation, n(n(x + y) + n(x + n(j/))) = x. The Algebra defined by commutivity, associativity, and the Robbins EQUATION is called ROBBINS ALGE- BRA. Computer theorem proving demonstrated that ev- ery Robbins Algebra satisfies the second Winkler Condition, from which it follows immediately that all Robbins Algebras are Boolean. References Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillian, p. 317, 1965. Halmos, P. Lectures on Boolean Algebras. Princeton, NJ: Van Nostrand, 1963. Huntington, E. V. "New Sets of Independent Postulates for the Algebra of Logic." Trans. Amer. Math. Soc. 35, 274- 304, 1933a. Huntington, E. V. "Boolean Algebras: A Correction." Trans. Amer. Math. Soc. 35, 557-558, 1933. McCune, W. "Robbins Algebras are Boolean." http://www. mcs.anl.gov/-mccune/papers/robbins/. Boolean Connective One of the Logic operators And A, Or V, and Not ->. see also QUANTIFIER Boolean Function A Boolean function in n variables is a function J\Xi , . . . , x n J, where each Xi can be or 1 and / is or 1. Determining the number of monotone Boolean functions of n vari- ables is known as Dedekind'S Problem. The number of monotonic increasing Boolean functions of n variables is given by 2, 3, 6, 20, 168, 7581, 7828354, . . . (Sloane's A000372, Beeler et al. 1972, Item 17). The number of inequivalent monotone Boolean functions of n variables is given by 2, 3, 5, 10, 30, . . . (Sloane's A003182). Let M(n, k) denote the number of distinct monotone Boolean functions of n variables with k mincuts. Then M(n,0) = 1 M(n,l)-2 n M(n, 2) = 2 n " 1 (2 n - 1) - 3 n + 2 n M(n,3) = |(2 n )(2 n - l)(2 n - 2) - 6 n + 5" + 4 n - 3 n . References Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Sloane, N, J. A. Sequences A003182/M0729 and A000372/ M0817 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Boolean Ring A Ring with a unit element in which every element is IDEMPOTENT. see also BOOLEAN ALGEBRA Borchardt-Pfaff Algorithm Borel Probability Measure 157 Borchardt-Pfaff Algorithm see Archimedes Algorithm Border Square 40 1 2 3 42 41 46 38 31 13 14 32 35 12 39 30 26 21 28 20 11 43 33 27 25 23 17 7 6 16 22 29 24 34 44 5 15 37 36 18 19 45 4 49 48 47 8 9 10 31 13 14 32 35 30 26 21 28 20 33 27 25 23 17 16 22 29 24 34 15 37 36 18 19 26 21 28 27 25 23 22 29 24 A MAGIC SQUARE that remains magic when its bor- der is removed. A nested magic square remains magic after the border is successively removed one ring at a time. An example of a nested magic square is the order 7 square illustrated above (i.e., the order 7, 5, and 3 squares obtained from it are all magic). see also MAGIC SQUARE References Kraitchik, M. "Border Squares." §7.7 in Mathematical Recre- ations. New York: W. W. Norton, pp. 167-170, 1942. Bordism A relation between Compact boundaryless Manifolds (also called closed Manifolds). Two closed Mani- folds are bordant IFF their disjoint union is the bound- ary of a compact (n+l)-MANlFOLD. Roughly, two Man- ifolds are bordant if together they form the boundary of a Manifold. The word bordism is now used in place of the original term COBORDISM. References Budney, R. "The Bordism Project." http: //math. Cornell. eduArybu/bordism/bordism.html. Bordism Group There are bordism groups, also called Cobordism Groups or Cobordism Rings, and there are singu- lar bordism groups. The bordism groups give a frame- work for getting a grip on the question, "When is a compact boundaryless MANIFOLD the boundary of an- other Manifold?" The answer is, precisely when all of its Stiefel- Whitney Classes are zero. Singular bor- dism groups give insight into STEENROD's REALIZATION PROBLEM: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too. The machinery of the bordism group winds up being important for HOMOTOPY THEORY as well. References Budney, R. "The Bordism Project." http: //math. Cornell. edu/-rybu/bordism/bordism.html. Borel-Cantelli Lemma Let {^4.n}£Lo De a Sequence of events occurring with a certain probability distribution, and let A be the event consisting of the occurrence of a finite number of events A ni n = 1, Then if then ^2p(A n ) < oo, P(A) = 1. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet u Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, pp. 435-436, 1988. Borel Determinacy Theorem Let T be a tree defined on a metric over a set of paths such that the distance between paths p and q is 1/n, where n is the number of nodes shared by p and q. Let A be a Borel set of paths in the topology induced by this metric. Suppose two players play a game by choosing a path down the tree, so that they alternate and each time choose an immediate successor of the previously chosen point. The first player wins if the chosen path is in A, Then one of the players has a winning STRATEGY in this Game. see also Game Theory, Strategy BorePs Expansion Let <p(t) = Xl^lo ^nt 71 ^ e any function for which the integral />oo I(x) = / e- tx t v 4>{t) dt Jo converges. Then the expansion I(x) XP+ ■^l[Ao + ( P + iy- + (p+l)(p + 2)^ + ... where F(z) is the Gamma Function, is usually an Asymptotic Series for I(x). Borel Measure If F is the Borel Sigma Algebra on some Topolog- ical Space, then a Measure m : F -+ R is said to be a Borel measure (or BOREL PROBABILITY MEASURE). For a Borel measure, all continuous functions are MEA- SURABLE. Borel Probability Measure see BOREL MEASURE 158 Borel Set Borwein Conjectures Borel Set A Definable Set derived from the Real Line by re- moving a Finite number of intervals. Borel sets are measurable and constitute a special type of Sigma Al- gebra called a BOREL SIGMA ALGEBRA. see also Standard Space Borel Sigma Algebra A Sigma Algebra which is related to the Topology of a Set, The Borel sigma-algebxa is defined to be the Sigma Algebra generated by the Open Sets (or equivalently, by the CLOSED Sets). see also Borel MEASURE Borel Space A Set equipped with a Sigma Algebra of Subsets. Borromean Rings Three mutually interlocked rings named after the Italian Renaissance family who used them on their coat of arms. No two rings are linked, so if one of the rings is cut, all three rings fall apart. They are given the Link symbol O603, and are also called the Ballantine. The Bor- romean rings have BRAID WORD c^ -1 o- 2 o'i~ 1 <J 2 ai _1 &2 and are also the simplest Brunnian Link. References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 58-59, 1989. Gardner, M. The Unexpected Hanging and Other Mathemat- ical Diversions. Chicago, IL: University of Chicago Press, 1991. Jablan, S. "Borromean Triangles." http:/ /members. tripod, com/ -modularity/links .htm. Pappas, T. "Trinity of Rings— A Topological Model." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 31, 1989. Borrow 1 2 3 -78 borrows 4 4 5 The procedure used in SUBTRACTION to "borrow" 10 from the next higher Digit column in order to obtain a Positive Difference in the column in question. see also Carry Borsuk's Conjecture Borsuk conjectured that it is possible to cut an n-D shape of DIAMETER 1 into n + 1 pieces each with di- ameter smaller than the original. It is true for n = 2, 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as ~ 1.1^. Since 1.1^ > n + 1 at n = 9162, the conjecture becomes false at high dimensions. In fact, the limit has been pushed back to ~ 2000. see also DIAMETER (GENERAL), KELLER'S CONJEC- TURE, Lebesgue Minimal Problem References Borsuk, K. "Uber die Zerlegung einer Euklidischen n- dimensionalen Vollkugel in n Mengen." Verh. Internat. Math.-Kongr. Zurich 2, 192, 1932. Borsuk, K. "Drei Satze iiber die n-dimensionale euklidische Sphare." Fund. Math. 20, 177-190, 1933. Cipra, B. "If You Can't See It, Don't Believe It. . . ." Science 259, 26-27, 1993. Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math, Soc, pp. 21-25, 1993, Grunbaum, B. "Borsuk's Problem and Related Questions." In Convexity, Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, Held at the University of Washington, Seattle, June 13- 15, 1961. Providence, RI: Amer. Math. Soc, pp. 271-284, 1963. Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture." Bull. Amer. Math. Soc. 329, 60-62, 1993. Listernik, L. and Schnirelmann, L. Topological Methods in Variational Problems. Moscow, 1930. Borwein Conjectures Use the definition of the q- Series {a\q)n = JJ(l-ag') j=o and define N M [q \q)m (Q\Q)m (1) (2) Then P. Borwein has conjectured that (1) the Polyno- mials A n (q), B n (q), and C n (q) defined by (q\ </ 3 W<Z 2 ; qX = A n (q 3 ) - qB n (q 3 ) - q 2 C n (q 3 ) (3) have NONNEGATIVE COEFFICIENTS, (2) the POLYNOMI- ALS A* n {q), B*{q), and C*(q) defined by (q;qX(q 2 ;q 3 )l = A:(q S )- q B:(q S )~ q 2 C:(q 3 ) (4) have Nonnegative Coefficients, (3) the Polynomi- als A* n {q), B*{q), C*(q), D*(q), and E* n (q) defined by (9; 5 )n(q ;q )n(q ;q)n(q;q)n- AUq 5 )-qB* n {f , )-q 2 C* n {qS)-q 3 Dl{q 5 )-q 4 EUq 5 ) (5) Bouligand Dimension Boundary Point 159 have NONNEGATIVE COEFFICIENTS, (4) the POLYNOMI- ALS Al l (m i n,t,q) 1 £* (m,n,£, g), and C^m^n^t^q) de- fined by- Bound Variable An occurrence of a variable in a LOGIC which is not Free. (?; q 3 )m(q 2 ; q Z )m{zq\ q 3 ) n {zq 2 ; q 3 ) n 2m = > z [A* (m,n, £, q ) — qB* (m, n,t,q ) t=Q -q 2 C\m,n,t,q 3 )} (6) have Nonnegative Coefficients, (5) for k Odd and 1 < a < k/2, consider the expansion (q a ;q k U(q k - a ;q k )n (fc-D/2 E t/=(l-fc)/2 (_ 1 )^M- 2 +-)/2-a,^ ( ^ ) (7) with oo _ V^ f-lY 3(k 2 j + 2ku + k-2a)/2 m 4- n m + v + kj (8) then if a is Relatively Prime to k and m = n, the CO- EFFICIENTS of F^qr) are NONNEGATIVE, and (6) given a J rf3< 2'K and — K + /? < n — m < K — a, consider G(a,0,K;q) = ^(_i)V 1JC(a+w+lf(a+/9)1/a ra + n 171+ Kj , (9) the Generating Function for partitions inside an mx n rectangle with hook difference conditions specified by a, /?, and if. Let a and /? be POSITIVE RATIONAL Numbers and K > 1 an Integer such that aK and /3Jf are integers. Then if 1 < a + < 2K-1 (with strict inequalities for K = 2) and —if + /3<n — m < K — a, then G(a,j3,K;q) has NONNEGATIVE COEFFICIENTS, see ateo ^-SERIES References Andrews, G. E. ei al. "Partitions with Prescribed Hook Dif- ferences." Europ. J. Combin. 8, 341-350, 1987. Bressoud, D. M. "The Borwein Conjecture and Partitions with Prescribed Hook Differences. " Electronic J. Com- binatorics 3, No. 2, R4, 1-14, 1996. http://www. combinatorics. org/Volume^3/volume3_2.html#R4. Bouligand Dimension see MlNKOWSKI-BOULIGAND DIMENSION Bound see Greatest Lower Bound, Infimum, Least Up- per Bound, Supremum Boundary The set of points, known as Boundary Points, which are members of the CLOSURE of a given set 5 and the CLOSURE of its complement set. The boundary is some- times called the FRONTIER. see also SURGERY Boundary Conditions There are several types of boundary conditions com- monly encountered in the solution of PARTIAL DIFFER- ENTIAL Equations. 1. Dirichlet Boundary Conditions specify the value of the function on a surface T = /(r,£). 2. Neumann Boundary Conditions specify the nor- mal derivative of the function on a surface, dT dn _=fi-vr = /(r, y ). 3. Cauchy Boundary Conditions specify a weighted average of first and second kinds. 4. Robin Boundary Conditions. For an elliptic par- tial differential equation in a region Q, Robin bound- ary conditions specify the sum of au and the normal derivative of u = / at all points of the boundary of Q } with a and / being prescribed. see also BOUNDARY VALUE PROBLEM, DlRICHLET Boundary Conditions, Initial Value Problem, Neumann Boundary Conditions, Partial Differ- ential Equation, Robin Boundary Conditions References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 502-504, 1985. Morse, P. M. and Feshbach, H. "Boundary Conditions and Eigcnfunctions." Ch. 6 in Methods of Theoretical Physics, Part L New York: McGraw-Hill, pp. 495-498 and 676-790, 1953. Boundary Map The Map H n {X, A) -► H n - 1 (A) appearing in the Long Exact Sequence of a Pair Axiom. see also Long Exact Sequence of a Pair Axiom Boundary Point A point which is a member of the Closure of a given set S and the CLOSURE of its complement set. If A is a subset of M n , then a point x € M. n is a boundary point of A if every NEIGHBORHOOD of x contains at least one point in A and at least one point not in A. see also BOUNDARY 160 Boundary Set Boustrophedon Transform Boundary Set A (symmetrical) boundary set of RADIUS r and center xq is the set of all points x such that Bourget Function x- x = r. Let xo be the ORIGIN. In IR , the boundary set is then \ the -r. In the pair of points x — r and x boundary set is a CIRCLE. In R is a Sphere. see also Circle, Disk, Open Set, Sphere the boundary set Boundary Value Problem A boundary value problem is a problem, typically an Ordinary Differential Equation or a Partial Differential Equation, which has values assigned on the physical boundary of the Domain in which the problem is specified. For example, u(O t t) V 2 u = f m f*(0,t)=u 2 in Q on dQ on dQ, where dCl denotes the boundary of O, is a boundary problem. see also Boundary Conditions, Initial Value Problem References Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Compu- tational Differential Equations. Lund: Studentlitteratur, 1996. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Two Point Boundary Value Problems." Ch. 17 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge Uni- versity Press, pp. 745-778, 1992. Bounded A Set in a Metric Space (X,d) is bounded if it has a FINITE diameter, i.e., there is an R < oo such that d(#, y) < R for all x, y € X. A Set in W 1 is bounded if it is contained inside some Ball x\ 2 + . . . + x n 2 < R 2 of Finite Radius R (Adams 1994). see also Bound, Finite References Adams, R. A. Calculus: A Complete Course, Reading, MA: Addison- Wesley, p. 707, 1994. Bounded Variation A Function f(x) is said to have bounded variation if, over the Closed Interval x e [a, b], there exists an M such that \f(xi)-f(a)\ + \f(x2)-f(x 1 )\ + . . .+ |/(6)-/(x„_i)| < M for all a < xi < X2 < ■ . . < x n -i < b. J -*u-hf t ""( ,+ \Y"*[i'('-\)]* * Jo (2 cos d) k cos(n(9 - z sin 6) d0. see also Bessel Function of the First Kind References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Nether- lands: Reidel, p. 465, 1988. Bourget's Hypothesis When n is an INTEGER > 0, then J n (z) and J n +m(z) have no common zeros other than at z = for m an Integer > 1, where J n (z) is a Bessel Function of THE First Kind. The theorem has been proved true for m=l 2, 3, and 4. References Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966. Boustrophedon Transform The boustrophedon ( "ox-plowing" ) transform b of a se- quence a is given by bn = 7 7 \dkEn-k k=o v / — ±(-')-(0 fc=0 x ' bkEn~k (1) (2) for n > 0, where E n is a Secant Number or Tangent Number defined by Ex n E n — 7 = sec X + tanz. (3) The exponential generating functions of a and b are related by B(x) = (sec a? + tanz)^4(#), (4) where the exponential generating function is defined by A(x) = Y,An x (5) see also ALTERNATING PERMUTATION, ENTRINGER Number, Secant Number, Seidel-Entringer- Arnold Triangle, Tangent Number References Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Op- eration on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76, 44-54, 1996. Bovinum Problema Box Fractal 161 Bovinum Problema see Archimedes' Cattle Problem Bow 4 2 3 x = x y — y . References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Bowditch Curve see Lissajous Curve Bowley Index The statistical Index where P L is Laspeyres' Index and P P is Paasche's Index. see also INDEX References Kenney, J. F, and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962. Bowley Skewness Also known as Quartile Skewness Coefficient, (Q 3 - Q 2 ) - (<?2 - <2i) _ Qi - 2Q 2 + Q 3 Qz-Qi) Qz-Qi where the Qs denote the Interquartile Ranges. see also SKEWNESS Bowling Bowling is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in the form of a TRIANGLE with its vertex oriented towards the bowler. The number 10 is, in fact, the Triangular Number T 4 = 4(4 4- l)/2 = 10. Two "bowls" are allowed per "frame." If all the pins are knocked down in the two bowls, the score for that frame is the number of pins knocked down. If some or none of the pins are knocked down on the first bowl, then all the pins knocked down on the second, it is called a "spare," and the number of points tallied is 10 plus the number of pins knocked down on the bowl of the next frame. If all of the pins are knocked down on the first bowl, the number of points tallied is 10 plus the number of pins knocked down on the next two bowls. Ten frames are bowled, unless the last frame is a strike or spare, in which case an additional bowl is awarded. The maximum number of points possible, corresponding to knocking down all 10 pins on every bowl, is 300. References Cooper, C N. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowl- ing Games." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling Score Awful?" In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994. Box see Cuboid Box-and- Whisker Plot X T i A HlSTOGRAM-like method of displaying data invented by J. Tukey (1977). Draw a box with ends at the QUAR- TILES Qi and Q 3 . Draw the MEDIAN as a horizontal line in the box. Extend the "whiskers" to the farthest points. For every point that is more than 3/2 times the Interquartile Range from the end of a box, draw a dot on the corresponding top or bottom of the whisker. If two dots have the same value, draw them side by side. References Tukey, J. W. Explanatory Data Analysis. Addison- Wesley, pp. 39-41, 1977. Box Counting Dimension see Capacity Dimension Box Fractal Reading, MA: A Fractal which can be constructed using String Rewriting by creating a matrix with 3 times as many entries as the current matrix using the rules line 1 line 2 line 3 11 jkii — S 11 sk " " '■_>11 " 162 Box-Muller Transformation Boy Surface Let N n be the number of black boxes, L n the length of a side of a white box, and A n the fractional AREA of black boxes after the nth iteration. N„=5 n Ln = (!)"= 3"" The Capacity Dimension is therefore (i) (2) (3) lniV n d cap = - lim lnL lim ln(5") n— >oo In 5 m~3 n-voo ln(3- n ) 1.464973521.... (4) Boxcar Function y = c[H(x-o)-H{x-b)], where H is the Heaviside Step Function. References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 324, 1993. see also Cantor Dust, Sierpinski Carpet, Sierpin- ski Sieve References $ Weisstein, E. W. "Fractals." http: //www. astro. Virginia. edu/~evw6n/math/notebooks/Fractal.m. Box-Muller Transformation A transformation which transforms from a 2-D contin- uous Uniform Distribution to a 2-D Gaussian Bi- variate Distribution (or Complex Gaussian Dis- tribution). If xi and X2 are uniformly and indepen- dently distributed between and 1, then z\ and z 2 as de- fined below have a Gaussian Distribution with Mean li = and Variance <t 2 = 1. z\ — y — 21na;i cos(27ra;2) Z2 = v — 21n#i sin(27ra;2)- This can be verified by solving for x\ and x 2 , -( Zl 2 + Z2 2 )/2 x x X 2 2tt ■ tan ■■(!)• (1) (2) (3) (4) Taking the Jacobian yields d(xi,x 2 ) d(z u z 2 ) d%i dxi dz± dz 2 dx 2 dx 2 £zi dz 2 1 -Z! 2 /2 — e * ' 2tt \/2^ ^ 2 2 /2 (5) Box-Packing Theorem The number of "prime" boxes is always finite, where a set of boxes is prime if it cannot be built up from one or more given configurations of boxes. see also Conway Puzzle, Cuboid, de Bruijn's Theo- rem, Klarner's Theorem, Slothouber-Graatsma Puzzle References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 74, 1976. Boxcars A roll of two 6s (the highest roll possible) on a pair of 6-sided DICE, The probability of rolling boxcars is 1/36, or 2.777...%. see also Dice, Double Sixes, Snake Eyes Boy Surface A Nonorientable Surface which is one of the three possible Surfaces obtained by sewing a Mobius Strip to the edge of a DISK. The other two are the CROSS- Cap and ROMAN SURFACE. The Boy surface is a model of the Projective Plane without singularities and is a Sextic Surface. The Boy surface can be described using the general method for NONORIENTABLE SURFACES, but this was not known until the analytic equations were found by Apery (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic poly- nomials, Hopf had conjectured that quartic polynomials were also insufficient (Pinkall 1986). Apery's IMMER- SION proved this conjecture wrong, giving the equations explicitly in terms of the standard form for a NONORI- ENTABLE Surface, /i(*,y,s) = |[(2x 2 - y 2 - z 2 )(x 2 + y 2 + z 2 ) + 2yz(y — z ) + zx(x — z ) + xy(y 2 -x 2 )] (1) Mx,y,z) = \s/Z[{y 2 - z 2 )(x 2 + y 2 + z 2 ) + zx{z 2 - x 2 ) + xy(y 2 - x 2 )} (2) f 3 (x, y,z) = i(x + y + z)[(x + y + z) 3 + A(y-x)(z-y)(x-z)]. (3) Boy Surface Plugging in x = cos u sin v y = sin u sin v Z = COS V (4) (5) (6) and letting u G [0, tv] and v € [0, 7r] then gives the Boy surface, three views of which are shown above. The K. parameterization can also be written as V = V2cos 2 vcos(2it) + cosusin(2t;) 2- v / 2sin(3u)sin(2v) __ \/2cos 2 vsin(2u) + cos^sin(2i;) 2- v / 2sin(3u)sin(2i;) 3 cos 2 v 2- V2sin(3u)sin(2t;) (Nordstrand) for u 6 [-7r/2,7r/2] and i; G [0,7r]. (7) (8) (9) Three views of the surface obtained using this parame- terization are shown above. In fact, a HOMOTOPY (smooth deformation) between the Roman Surface and Boy surface is given by the equations x(u,v) = y{u,v) = Z(U y V) = \[2 cos(2n) cos 2 v + cos u sin(2v) 2 — a\/2 sin(3ti) sin(2t;) \/2sin(2u) cos 2 v — sinusin(2t>) 2-aA/2sin(3tx)sin(2i;) 3 cos 2 v 2 — a\/2 sin(3u) sin(2v) (10) (11) (12) as a varies from to 1, where a — corresponds to the Roman Surface and a = 1 to the Boy surface (Wang), shown below. Boy Surface 163 In K. , the parametric representation is xq = 3[(u + v +w )(u + v ) — V2vw(3u — v )] (13) X! = V2(u 2 + v 2 )(u 2 - v 2 + v^uty) (14) a?2 = V2(u 2 + v 2 )(2wu - V2vw) (15) X3 = 3(u 2 + v 2 ) 2 , (16) and the algebraic equation is 64(x — £3) 3 #3 3 — 48(x — ^3) 2 ^3 2 (32;i 2 + Sx2 2 + 2x 3 2 ) +12(z - x 3 )x 3 [27(x 1 2 + z 2 2 ) 2 - 24z 3 2 (zi 2 + x 2 2 ) +36^3:2^3 (x2 2 — 3cci 2 ) + X3 4 ] +(9zi 2 +92 2 2 - 2x 3 2 ) x[-81(^i 2 + x 2 2 ) 2 - 72x 3 2 (xi 2 + x 2 2 ) +10%V2x 1 x 3 {x 1 2 - 3z 2 2 ) + 4z 3 4 ] = (17) (Apery 1986). Letting Xq — 1 Xi = X x 2 =y X3 = z (18) (19) (20) (21) gives another version of the surface in M . see also Cross-Cap, Immersion, Mobius Strip, nonorientable surface, real projective plane, Roman Surface, Sextic Surface References Apery, F. "The Boy Surface." Adv. Math. 61, 185-266, 1986. Boy, W. "Uber die Curvatura Integra und die Topologie geschlossener Flachen." Math. Ann 57, 151-184, 1903. Brehm, U. "How to Build Minimal Polyhedral Models of the Boy Surface." Math. Intell. 12, 51-56, 1990. Carter, J. S. "On Generalizing Boy Surface — Constructing a Generator of the 3rd Stable Stem." Trans. Amer. Math. Soc. 298, 103-122, 1986. Fischer, G. (Ed.). Plates 115-120 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, pp. 110-115, 1986. Geometry Center. "Boy's Surface." http://www.geom.umn. edu/zoo/toptype/pplane/boy/. Hilbert, D. and Cohn-Vossen, S. §46—47 in Geometry and the Imagination. New York: Chelsea, 1952. Nordstrand, T. "Boy's Surface." http : //www . uib . no/ people/nf ytn/boytxt . htm. Petit, J .-P. and Souriau, J. "Une representation analytique de la surface de Boy." C. R. Acad. Sci. Paris Sir. 1 Math 293, 269-272, 1981. Pinkall, U. Mathematical Models from the Collections of Uni- versities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 64-65, 1986. Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991. Wang, P. "Renderings." http: //www.ugcs . caltech.edu/ -pet erw/portf olio/renderings/. 164 Bra Brachistochrone Problem Bra A (COVARIANT) 1-VECTOR denoted (V>|- The bra is Dual to the Contravariant Ket, denoted \ip). Taken together, the bra and KET form an ANGLE BRACKET (bra+ket = bracket). The bra is commonly encountered in quantum mechanics. see also Angle Bracket, Bracket Product, Co- variant Vector, Differential /.-Form, Ket, One- Form Brachistochrone Problem Find the shape of the CURVE down which a bead sliding from rest and Accelerated by gravity will slip (with- out friction) from one point to another in the least time. This was one of the earliest problems posed in the CAL- CULUS of Variations. The solution, a segment of a Cycloid, was found by Leibniz, L'Hospital, Newton, and the two Bernoullis. The time to travel from a point Pi to another point Pi is given by the INTEGRAL = C - (i) The VELOCITY at any point is given by a simple appli- cation of energy conservation equating kinetic energy to gravitational potential energy, 1 2 2 mv mgy, v = y/2gy. Plugging this into (1) then gives tl2 i: a/i + y' 2 s/5gy dx ■ i: l + y' 2 tgy dx. The function to be varied is thus f = (l + y ,2 ) 1/2 (2gy)-^. (2) (3) (4) (5) subtracting y'{df/dy') from /, and simplifying then gives C. (9) V^gy^i + y' 2 Squaring both sides and rearranging slightly results in 1 + [dx) 2gC* (10) where the square of the old constant C has been ex- pressed in terms of a new (POSITIVE) constant k 2 . This equation is solved by the parametric equations x - y \k 2 {e-s\n9) §fc 2 (l-cos6>), (11) (12) which are — lo and behold — the equations of a CYCLOID. If kinetic friction is included, the problem can also be solved analytically, although the solution is significantly messier. In that case, terms corresponding to the normal component of weight and the normal component of the Acceleration (present because of path Curvature) must be included. Including both terms requires a con- strained variational technique (Ashby et al. 1975), but including the normal component of weight only gives an elementary solution. The Tangent and Normal Vec- tors are (13) (14) T = dx „ ds dy- ds N = dy ~ ds dx ^ gravity and friction are then • gravity : mgy dx r Ff r i c tion = ~M( F gravityN)T = - flTTig — T, and the components along the curve are (15) (16) To proceed, one would normally have to apply the full- blown Euler-Lagrange Differential Equation 21 dy dx \dy'J 0. (6) However, the function f{y,y' } x) is particularly nice since x does not appear explicitly. Therefore, df /dx = 0, and we can immediately use the Beltrami Identity >-<%-<>■ Computing 8y' y {l + y /2\-l/2 (2gy) -1/2 (7) (8) -T gravity J- : ■T friction -L dy m9 dS -fj,mg dx ds ' so Newton's Second Law gives dv dy m— — mg- 1 dt ds limg dx ds But dv dv — = v — at as 1 d 2 ^ 2dS^ V) \v 2 = g{y - fix) v = y/2g{y - fix), (17) (18) (19) (20) (21) (22) Bracket Bracketing 165 -Jyi + (y') 2 dx. (23) 2< ? (y - fix) ~' v Using the Euler-Lagrange Differential Equation gives [i + y 2 ](i + aV) + 2(2/ - ^)y" - 0. (24) This can be reduced to i + (y') 2 _ c Now letting the solution is (1 + /X2/') 2 y- iix' y'=cot(±0), (25) (26) a; = ffc 2 [(0-sm0)+Ai(l-cos6O] (27) y = |A; 2 [(1 - cos<9) + ^(<9 + sin0)]. (28) see also Cycloid, Tautochrone Problem References Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. "Bra- chistochrone with Coulomb Friction." Amer. J. Phys. 43, 902-905, 1975. Haws, L. and Kiser, T. "Exploring the Brachistochrone Prob- lem." Amer. Math. Monthly 102, 328-336, 1995. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 60-66 and 385-389, 1991. Bracket see Angle Bracket, Bra, Bracket Polynomial, Bracket Product, Iverson Bracket, Ket, La- grange Bracket, Poisson Bracket Bracket Polynomial A one- variable KNOT POLYNOMIAL related to the JONES Polynomial. The bracket polynomial, however, is not a topological invariant, since it is changed by type I REI- demeister Moves. However, the Span of the bracket polynomial is a knot invariant. The bracket polynom- ial is occasionally given the grandiose name REGULAR Isotopy Invariant. It is defined by <L)(A,*,d) = ^<2W Ikll (1) where A and B are the "splitting variables," a runs through all "states" of L obtained by Splitting the LINK, (L\a) is the product of "splitting labels" corre- sponding to cr, and \W\\ = N L -1, (2) where JV& is the number of loops in er. Letting -l B = A d^-A 2 -A' 2 (3) (4) gives a Knot Polynomial which is invariant under Regular Isotopy, and normalizing gives the Kauff- man Polynomial X which is invariant under Ambient Isotopy. The bracket Polynomial of the Unknot is 1. The bracket Polynomial of the Mirror Image K* is the same as for K but with A replaced by A -1 . In terms of the one-variable KAUFFMAN POLYNOMIAL X, the two-variable KAUFFMAN POLYNOMIAL F and the Jones Polynomial V\ X(A) (L) (A) ( -A*y -«.(£) (L), F(-A 3 ,A + A- 1 ) (5) (6) (7) <L> (A) = V{A~% where w(L) is the WRITHE of L. see also SQUARE BRACKET POLYNOMIAL References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 148-155, 1994. Kauffman, L. "New Invariants in the Theory of Knots." Amer. Math. Monthly 95, 195-242, 1988. Kauffman, L. Knots and Physics. Teaneck, NJ: World Sci- entific, pp. 26-29, 1991. i$ Weisstein, E. W. "Knots and Links." http: //www. astro. virginia.edu/~eww6n/math/notebooks/Knots .m. Bracket Product The Inner Product in an Li Space represented by an Angle Bracket. see also Angle Bracket, Bra, Ket, L 2 Space, One- Form Bracketing Take x itself to be a bracketing, then recursively de- fine a bracketing as a sequence B = (jBi, . . . , Bk) where k > 2 and each Bi is a bracketing. A bracketing can be represented as a parenthesized string of xs, with paren- theses removed from any single letter x for clarity of notation (Stanley 1997). Bracketings built up of binary operations only are called BINARY BRACKETINGS. For example, four letters have 11 possible bracketings: xxxx (xx)xx x(xx)x xx(xx) (xxx)x x(xxx) ((xx)x)x (x(xx))x {xx)(xx) x((xx)x) x(x(xx)), the last five of which are binary. The number of bracketings on n letters is given by the Generating Function \(l + x- y/l ~6x + x 2 ) = x + x 2 + 3x 3 + llx 4 + 45x 5 (Schroder 1870, Stanley 1997) and the RECURRENCE Relation _ 3(2n — 3)s n -i — (n — 3)s n -2 166 Bradley's Theorem Brahmagupta Matrix (Sloane), giving the sequence for s n as 1, 1, 3, 11, 45, 197, 903, . . . (Sloane's A001003). The numbers are also given by s n = ^ s(ii) • • - s(i k ) for n > 2 (Stanley 1997). The first PLUTARCH NUMBER 103,049 is equal to $io (Stanley 1997), suggesting that Plutarch's problem of ten compound propositions is equivalent to the number of bracketings. In addition, Plutarch's second number 310,954 is given by (sio + sn)/2 = 310,954 (Habsieger et al. 1998). see also Binary Bracketing, Plutarch Numbers References Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly 105, 446, 1998. Schroder, E. "Vier combinatorische Probleme." Z. Math. Physik 15, 361-376, 1870. Sloane, N. J. A. Sequence A001003/M2898 in "An On-Line Version of the Encyclopedia of Integer Sequences." Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough." Amer. Math. Monthly 104, 344-350, 1997. Bradley's Theorem Let S(a,(3,m;z) = y> T(m + j(z + l))rpg + 1 + jz) (a) + j m 2^ T{m + jz + l)r(a + + 1 + j(z + 1)) j! j = and a be a Negative Integer. Then T(/3 + 1 - m) S{a,(3,m\z) = r(a + /3 + l-m)' where T(z) is the GAMMA FUNCTION. References Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer- Verlag, pp. 346-348, 1994. Bradley, D. "On a Claim by Ramanujan about Certain Hy- pergeometric Series." Proc. Amer. Math. Soc. 121, 1145- 1149, 1994. Brahmagupta's Formula For a Quadrilateral with sides of length a, 6, c, and d, the Area K is given by K : J(s - a)(s - b)(s - c)(s - d) - abcdcos 2 [\{A + B)], (1) where s= |(a + 6 + c + d) (2) is the Semiperimeter, A is the Angle between a and d, and B is the Angle between b and c. For a Cyclic Quadrilateral (i.e., a Quadrilateral inscribed in a Circle), A + B — 7r, so K = ^/(s-a)(s-b){s-c){s-d) (3) y/(bc + ad)(ac + bd)(ab -f- cd) 4R (4) where R is the RADIUS of the CiRCUMClRCLE. If the Quadrilateral is Inscribed in one Circle and Cir- cumscribed on another, then the Area Formula sim- plifies to K = \fabc~d. (5) see also Bretschneider's Formula, Heron's For- mula References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 81-82, 1929. Brahmagupta Identity Let 0=\B\^x 2 -ty\ where B is the Brahmagupta Matrix, then det[B(x u yi)B(x 2 ,y2)] = det[B(x u yi )] det[B(x 2 , y 2 )] = Pifo. References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34, 30-39, 1996. Brahmagupta Matrix 5(z,y) = x y ±ty ±x It satisfies B(x!,yi)B(x 27 y2) = B{xxx 2 ±tyiy2,x 1 y 2 ±2/1X2). Powers of the matrix are defined by B n = X y n X<n Vn ty X ty n Xn = B n . The x n and y n are called BRAHMAGUPTA POLYNOMI- ALS. The Brahmagupta matrices can be extended to Negative Integers n-n _ x y _ *-n y-n _ d X y — n X — n y- ty X ty-n X- see also Brahmagupta Identity References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34, 30-39, 1996. Brahmagupta Polynomial Braid Group 167 Brahmagupta Polynomial One of the POLYNOMIALS obtained by taking POWERS of the Brahmagupta Matrix. They satisfy the recur- rence relation x n +! = xx n + tyy n y n+1 = xy n + yx n . (1) (2) A list of many others is given by Suryanarayan (1996). Explicitly, X +t (;)«-v+« a (j)«-- 4 » 4 +... (3) n-l . .i n \ n-3 3 . .2l n \ n-5 5 . rix y + t[\x y +t i ]x y + The Brahmagupta POLYNOMIALS satisfy dx dx n dy nx n -i dyn dy ,dy n t—- = ntyn-L dy (4) (5) (6) The first few POLYNOMIALS are xo = xi = x x 2 — x 2 + ty 2 xz = x 3 + 3txy 2 4 . «, 2 2 , ,2 4 X4 = x + otx y +t y and yo-o 2/1=2/ y 2 = 2xy 2/3 = 3x 2 y + ty 3 2/4 = ^x z y -\- Atxy z . Taking x = i/ = 1 and £ = 2 gives j/„ equal to the PELL Numbers and x n equal to half the Pell-Lucas num- bers. The Brahmagupta POLYNOMIALS are related to the Morgan- Voyce Polynomials, but the relation- ship given by Suryanarayan (1996) is incorrect. References Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34, 30-39, 1996. Brahmagupta's Problem Solve the PELL EQUATION x 2 - 92y 2 = 1 in Integers. The smallest solution is x = 1151, y = 120. see also Diophantine Equation, Pell Equation Braid An intertwining of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of a braid in something that a falling object could trace out if acted upon only by grav- ity and horizontal forces. see also Braid GROUP References Christy, J. "Braids." http://www.mathsource.com/cgi-bin /MathSource/Applications/Mathematics/0202-228. Braid Group Also called Artin Braid Groups. Consider n strings, each oriented vertically from a lower to an upper "bar." If this is the least number of strings needed to make a closed braid representation of a LINK, n is called the Braid Index. Now enumerate the possible braids in a group, denoted B n . A general n-braid is constructed by iteratively applying the <Tj (i = 1, . . . ,n — 1) operator, which switches the lower endpoints of the ith and (i + l)th strings — keeping the upper endpoints fixed — with the (i + l)th string brought above the ith string. If the (i + l)th string passes below the zth string, it is denoted 1 2 Topological equivalence for different representations of a BRAID Word JJ o~i and J^ a^ is guaranteed by the conditions CTiCTj — <Tj<Ti / it O'iO'i + iO'i — 0'i-\-\(T i <Ti + i for \i-j\ >2 for all i as first proved by E. Artin. Any n-braid is expressed as a Braid Word, e.g., G^aicr^a^ a\ is a Braid Word for the braid group #3 . When the opposite ends of the braids are connected by nonintersecting lines, KNOTS are formed which are identified by their braid group and Braid Word. The Burau Representation gives a matrix representation of the braid groups. References Birman, J. S. "Braids, Links, and the Mapping Class Groups." Ann. Math. Studies, No. 82. Princeton, NJ: Princeton University Press, 1976. Birman, J. S. "Recent Developments in Braid and Link The- ory." Math. Intell. 13, 52-60, 1991. Christy, J. "Braids." http://www.mathsource.com/cgi-bin /MathSource/Applications/Mathematics/0202-228. Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335- 388, 1987. ^ Weisstein, E. W. "Knots and Links." http: //www. astro. Virginia. edu/-eww6n/math/notebooks/Knots .m. 168 Braid Index Branch Point Braid Index The least number of strings needed to make a closed braid representation of a LINK. The braid index is equal to the least number of Seifert Circles in any projec- tion of a Knot (Yamada 1987). Also, for a nonsplit- table Link with Crossing Number c(L) and braid in- dex i{L) y c(L) > 2[i(L) - 1] (Ohyama 1993). Let E be the largest and e the small- est Power of £ in the HOMFLY Polynomial of an oriented LINK, and i be the braid index. Then the Morton-Franks- Williams Inequality holds, i>\{E-e) + l (Franks and Williams 1987). The inequality is sharp for all Prime Knots up to 10 crossings with the exceptions of 09 42, 09 49, IO132, IO150, and 10i 5 6- References Franks, J. and Williams, R. F. "Braids and the Jones Poly- nomial." Trans. Amer. Math. Soc. 303, 97-108, 1987. Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335- 388, 1987. Ohyama, Y. "On the Minimal Crossing Number and the Brad Index of Links," Canad. J. Math. 45, 117-131, 1993. Yamada, S. "The Minimal Number of Seifert Circles Equals the Braid Index of a Link." Invent. Math. 89, 347-356, 1987. Braid Word Any n-braid is expressed as a braid word, e.g., o-i^osa^ <y\ is a braid word for the Braid Group S3. By Alexander's Theorem, any LINK is representable by a closed braid, but there is no general procedure for reducing a braid word to its simplest form. However, Markov's Theorem gives a procedure for identifying different braid words which represent the same LINK. Let 6+ be the sum of Positive exponents, and 6_ the sum of Negative exponents in the Braid Group B n . If b + - 36_ - n+ 1 > 0, then the closed braid b is not AMPHICHIRAL (Jones 1985). see also Braid GROUP References Jones, V. F. R. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985. Jones, V. F. R. "Hecke Algebra Representations of Braid Groups and Link Polynomials." Ann. Math. 126, 335- 388, 1987. Braikenridge-Maclaurin Construction The converse of PASCAL'S THEOREM. Let Ai, B 2i Ci, A 2 , and Br be the five points on a Conic. Then the Conic is the Locus of the point C 2 =Ax{z> dA 2 ) ■ B x (z • C1B2), where z is a line through the point AiB 2 • B\A 2 . see also PASCAL'S THEOREM Branch The segments of a TREE between the points of connec- tion (Forks). see also FORK, LEAF (TREE) Branch Cut |Sqrt z| A line in the COMPLEX PLANE across which a FUNCTION is discontinuous. function branch cut(s) cos -1 z (— 00, — 1) and (l,oo) cosh -1 (-oo,l) cot -1 z (-i,i) coth" 1 [-1,1] esc -1 z (-1,1) csch -1 (-m) In z (-oo,0] sec" 1 z (-1,1) sech -1 (oo,0] and (1, 00) sin - z (— 00,— 1) and (l,oo) sinh -1 (—200, —i) and (2,200) v/i (-oo,0) tan x z (-ioo, -i) and (2,200) tanh -1 ( — 00, —1] and [1, 00) z n ,n<£Z (-oo,0) for R[n] < 0; (- -oo,0] for R[n] > see also Branch Point References Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 399-401, 1953. Branch Line see Branch Cut Branch Point An argument at which identical points in the COMPLEX PLANE are mapped to different points. For example, consider Brauer Chain Breeder 169 Then f(e oi ) = /(l) = 1, but f(e 27ri ) = e 2 ™, despite the fact that e i0 = e 2ni . Pinch Points are also called branch points. see also BRANCH CUT, PlNCH POINT References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 397-399, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 391-392 and 399- 401, 1953. Brauer Chain A Brauer chain is an ADDITION CHAIN in which each member uses the previous member as a summand. A number n for which a shortest chain exists which is a Brauer chain is called a BRAUER NUMBER. see also Addition Chain, Brauer Number, Hansen Chain References Guy, R. K. "Addition Chains, Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. Brauer Group The GROUP of classes of finite dimensional central sim- ple Algebras over k with respect to a certain equiva- lence. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, p. 479, 1988. Brauer Number A number n for which a shortest chain exists which is a BRAUER Chain is called a Brauer number. There are infinitely many non-Brauer numbers. see also Brauer Chain, Hansen Number References Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 111-113, 1994. Brauer- Severi Variety An Algebraic Variety over a Field K that becomes Isomorphic to a Projective Space. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, pp. 480-481, 1988. Brauer's Theorem If, in the Gersgorin Circle Theorem for a given m, for all j f^ m, then exactly one EIGENVALUE of A lies in the Disk F m . References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1121, 1979. Braun's Conjecture Let B = {&i,& 2 ,...} be an Infinite Abelian Semi- group with linear order &i < & 2 < . . . such that &i is the unit element and a < b Implies ac < be for a,b,c 6 B. Define a Mobius Function jj, on B by /x(6i) = 1 and Yl ^ = ° b d \b n for n = 2, 3, Further suppose that /x(6 n ) = M n ) (the true MOBIUS FUNCTION) for all n > 1. Then Braun's conjecture states that for all m,n> 1. see also MOBIUS PROBLEM References Flath, A. and Zulauf, A. "Does the Mobius Function Deter- mine Multiplicative Arithmetic?" Amer. Math. Monthly 102, 354-256, 1995. Breeder A pair of POSITIVE INTEGERS (ai,a 2 ) such that the equations a\ 4- a 2 x = cr(a\) — a(a 2 )(x 4- 1) have a POSITIVE INTEGER solution x, where a(n) is the DIVISOR FUNCTION. If x is Prime, then (ai,a 2 x) is an Amicable Pair (te Riele 1986). (ai,a 2 ) is a "special" breeder if a± = au a 2 = a, where a and u are Relatively Prime, (a, u) — 1. If regular amicable pairs of type (i,l) with i > 2 are of the form (au,ap) with p PRIME, then (au,a) are special breeders (te Riele 1986). References te Riele, H. J. J. "Computation of All the Amicable Pairs Below 10 10 ." Math. Comput. 47, 361-368 and S9-S35, 1986. 170 Brelaz's Heuristic Algorithm Bretschneider's Formula Brelaz's Heuristic Algorithm An Algorithm which can be used to find a good, but not necessarily minimal, EDGE or VERTEX coloring for a Graph. see also Chromatic Number Brent's Factorization Method A modification of the POLLARD p FACTORIZATION Method which uses Xi+i = Xi — c (mod n). References Brent, R. "An Improved Monte Carlo Factorization Algo- rithm." Nordisk Tidskrift for Informationsbehandlung (BIT) 20, 176-184, 1980. Brent's Method A RoOT-finding ALGORITHM which combines root bracketing, bisection, and Inverse Quadratic In- terpolation. It is sometimes known as the VAN Wijngaarden-Deker-Brent Method. Brent's method uses a LAGRANGE INTERPOLATING Polynomial of degree 2. Brent (1973) claims that this method will always converge as long as the values of the function are computable within a given region contain- ing a ROOT. Given three points asi, x 2 , and £3, Brent's method fits x as a quadratic function of y, then uses the interpolation formula [y-f(*i)][y-f{ x 2)] x 3 [/(**) + f(x 1 )][f(x 3 )-f(x 2 )} [y- /Qg2)][y- f(x s )]xi [f(x 1 )-f(x 2 )][f(x 1 )-f(x 3 )] [y- f(x3)][y- f(xi)]x 2 + [f( X 2)-f(x 3 )][f(x 2 )-f(x 1 )Y (1) Subsequent root estimates are obtained by setting y = 0, giving , P (2) where P = S[R(R - T)(x 3 - x 2 ) - (1 - R)(x 2 - zi)] (3) Q = (T-1)(R-1)(S-1) (4) with R = f(X2) /(*s) a - /(*») " f(xi) T _/(*l) (5) (6) (7) References Brent, R. P. Ch. 3-4 in Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice- Hall, 1973. Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. §7.2 in Computer Methods for Mathematical Computations. En- glewood Cliffs, NJ: Prentice-Hall, 1977. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Van Wijngaarden-Dekker-Brent Method." §9.3 in Numerical Recipes in FORTRAN: The Art of Sci- entific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 352—355, 1992. Brent- Salamin Formula A formula which uses the Arithmetic-Geometric MEAN to compute Pi. It has quadratic convergence and is also called the Gauss-Salamin Formula and Salamin Formula. Let CLn + l = 2 {,Q>n + O n ) (i) &n+l = ydnbn (2) C n +i = 2 ( a n — b n ) (3) A — 2 h 2 0, n = €L n On y (4) and define the initial conditions to be ao = 1, &o = l/\/2- Then iterating a„ and 6„ gives the ARITHMETIC- GEOMETRIC MEAN, and it is given by 4[M(1,2- 1 / 2 )] 2 4[M(l,2- 1 / 2 )] 2 l-£~i2 i+ V (5) (6) King (1924) showed that this formula and the LEGEN- DRE RELATION are equivalent and that either may be derived from the other. see also Arithmetic-Geometric Mean, Pi References Borwein, J. M. and Borwein, P. B, Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 48-51, 1987. Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988. King, L. V. On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge, England: Cambridge University Press, 1924. Lord, N. J. "Recent Calculations of n: The Gauss-Salamin Algorithm." Math. Gaz. 76, 231-242, 1992. Salamin, E. "Computation of n Using Arithmetic-Geometric Mean." Math. Comput. 30, 565-570, 1976. Bretschneider's Formula Given a general QUADRILATERAL with sides of lengths a, 6, c, and d (Beyer 1987), the Area is given by (Press et al. 1992). ^quadrilateral = \ ^4p 2 q 2 - (b 2 + d 2 - d 2 - C 2 ) 2 , where p and q are the diagonal lengths. see also BRAHMAGUPTA'S FORMULA, HERON'S FOR- MULA References Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987. Brianchon Point Bridge (Graph) 171 Brianchon Point The point of CONCURRENCE of the joins of the VER- TICES of a Triangle and the points of contact of a Conic Section Inscribed in the Triangle. A Conic Inscribed in a Triangle has an equation of the form the chance that one of four players will receive a hand of a single suit is 39,688,347,497 / 9 h - + - + - U V w o, an it.s HrianrTinn -nrnnt Viae Trttttmrar nnrmnTM atpq There are special names for specific types of hands. A ten, jack, queen, king, or ace is called an "honor." Get- suits and the ace, king, and queen, and jack of the re- maining suit is called 13 top honors. Getting all cards of the same suit is called a 13-card suit. Getting 12 cards of same suit with ace high and the 13th card not an ace is called 2-card suit, ace high. Getting no honors is called a Yarborough. The probabilities of being dealt 13-card bridge hands of a given type are given below. As usual, for a hand with probability P, the Odds against being dealt it are (1/P) -1:1. (1//, l/g,l/h). For Kiepert's Parabola, the Bran- chion point has TRIANGLE CENTER FUNCTION a(b 2 - which is the Steiner Point. 2). Brianchon's Theorem The Dual of Pascal's Theorem. It states that, given a 6-sided Polygon Circumscribed on a Conic SEC- TION, the lines joining opposite VERTICES (DIAGONALS) meet in a single point. see also DUALITY PRINCIPLE, PASCAL'S THEOREM References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 110, 1990. Brick see Euler Brick, Harmonic Brick, Rectangular Parallelepiped Bride's Chair One name for the figure used by Euclid to prove the Pythagorean Theorem. see also Peacock's Tail, Windmill Bridge Card Game Bridge is a CARD game played with a normal deck of 52 cards. The number of possible distinct 13-card hands is N = 635,013,559,600. where (£) is a Binomial Coefficient. While the chances of being dealt a hand of 13 CARDS (out of 52) of the same suit are 4 1 Hand Exact Probability 13 top honors high 4 N 4 N 4-12-36 N (S) N ill mm AT i 158,753,389,900 1 12-card suit, ace Yarborough four aces nine honors 158,753,389,900 4 1,469,938,795 5,394 9,860,459 11 4,165 888,212 93,384,347 Hand Probability Odds 13 top honors 13-card suit 12-card suit, ace high Yarborough four aces nine honors 6.30 6,30 2.72 5.47 2.64 9.51 x 10~ 12 x 10" 12 x 10" 9 x 10" 4 x 10~ 3 x 10" 3 158,753,389,899:1 158,753,389,899:1 367,484,697.8:1 1,827.0:1 377.6:1 104.1:1 see also CARDS, POKER References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, pp. 48-49, 1987. Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recre- ations. New York: W. W. Norton, pp. 119-121, 1942. Bridge (Graph) The bridges of a Graph are the Edges whose removal disconnects the Graph. see also Articulation Vertex References Chartrand, G. "Cut- Vertices and Bridges." §2.4 in Introduc- tory Graph Theory. New York: Dover, pp. 45-49, 1985. («) 158,753,389,900' 172 Bridge Index Bring Quintic Form Bridge Index A numerical KNOT invariant. For a TAME KNOT K, the bridge index is the least BRIDGE NUMBER of all planar representations of the Knot. The bridge index of the Unknot is defined as 1. see also Bridge Number, Crookedness References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 114, 1976. Schubert, H. "Uber eine numerische Knotteninvariante." Math. Z. 61, 245-288, 1954. Bridge of Konigsberg see Konigsberg Bridge Problem Bridge Number The least number of unknotted arcs lying above the plane in any projection. The knot 05os has bridge num- ber 2. Such knots are called 2-BRIDGE KNOTS. There is a one-to-one correspondence between 2-Bridge KNOTS and rational knots. The knot O8010 is a 3-bridge knot. A knot with bridge number b is an n-EMBEDDABLE KNOT where n < b. see also BRIDGE INDEX References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 64-67, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 115, 1976. Bridge Knot An n-bridge knot is a knot with BRIDGE Number n. The set of 2-bridge knots is identical to the set of rational knots. If L is a 2-Bridge Knot, then the BLM/Ho Polynomial Q and Jones Polynomial V satisfy Q L (z) = 2z- 1 V L (t)V L (t- 1 + 1 - 2Z" 1 ), where z = — t — i" 1 (Kanenobu and Sumi 1993). Ka- nenobu and Sumi also give a table containing the num- ber of distinct 2-bridge knots of n crossings for n — 10 to 22, both not counting and counting MIRROR IMAGES as distinct. n K n K n + K n 3 4 5 6 7 8 9 10 45 85 11 91 182 12 176 341 13 352 704 14 693 1365 15 1387 2774 16 2752 5461 17 5504 11008 18 10965 21845 19 21931 43862 20 43776 87381 21 87552 175104 22 174933 349525 References Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2- Bridge Knots through 22-Crossings." Math. Comput. 60, 771-778 and S17-S28, 1993. Schubert, H. "Knotten mit zwei Briicken." Math. Z. 65, 133-170, 1956. Brill-Noether Theorem If the total group of the canonical series is divided into two parts, the difference between the number of points in each part and the double of the dimension of the complete series to which it belongs is the same. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 263, 1959. Bring-Jerrard Quintic Form A TSCHIRNHAUSEN TRANSFORMATION can be used to algebraically transform a general Quintic Equation to the form z + C\Z + Co == 0. (1) y + b 2 y 2 + hy + b -0 In practice, the general quintic is first reduced to the Principal Quintic Form (2) before the transformation is done. Then, we require that the sum of the third POWERS of the ROOTS vanishes, so ss(yj) = 0. We assume that the ROOTS Zi of the Bring-Jerrard quintic are related to the ROOTS yi of the Principal Quintic Form by Zi = ayi 4 + j3yi 3 + jyi 2 + 6yi + e. (3) In a similar manner to the Principal Quintic Form transformation, we can express the COEFFICIENTS Cj in terms of the bj . see also Bring Quintic Form, Principal Quintic Form, Quintic Equation Bring Quintic Form A TSCHIRNHAUSEN Transformation can be used to take a general Quintic Equation to the form x — x — a : 0, where a may be Complex. see also Bring-Jerrard Quintic Form, Quintic Equation References Ruppert, W. M. "On the Bring Normal Form of a Quintic in Characteristic 5." Arch. Math. 58, 44-46, 1992. Brioschi Formula Brocard Angle 173 Brioschi Formula For a curve with METRIC Brocard Angle ds 2 = E du + F dudv + G dv 2 , (1) where E, F, and G is the first FUNDAMENTAL FORM, the Gaussian Curvature is Mi + M 2 /0 v where Mi = M 2 -F 2 r 2 U1; ~t" -^tit; 2 uu 2 Eu F u - ^E v i*V — 2^1* E F 2^« F G (3) 2 ^v 2 "" §£ v £ F i (4) \G U F G which can also be written K = d_ (j_dVG\ d_ ( i d^E\ r EG [du \^E du J dv \^Q dv J _ d ( G u \ 3 ( E v \ du \y/EGj dv \<JEGJ 2VEG (5) (6) see also Fundamental Forms, Gaussian Curvature References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, pp. 392-393, 1993. Briot-Bouquet Equation An Ordinary Differential Equation of the form where m is a Positive Integer, / is Analytic at x ~ y = 0, /(0,0) = 0, and /i(0, 0)^0. References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Math- ematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia. " Dordrecht, Nether- lands: Reidel, pp. 481-482, 1988. A C Define the first Brocard Point as the interior point Q of a Triangle for which the Angles ICIAB, IQBC, and iVtCA are equal. Similarly, define the second BRO- CARD Point as the interior point Cl' for which the AN- GLES IQ'AC, /-0,'CB, and IQ'BA are equal. Then the Angles in both cases are equal, and this angle is called the Brocard angle, denoted u). The Brocard angle u> of a Triangle AABC is given by the formulas cot u) = cot A 4- cot B + cot C 1 + cos ai cos ct2 cos az sin ct\ sin 0:2 sin otz _ sin 2 ai + sin 2 c*2 + sin 2 0:3 2sinai sina2 sin 0:3 _ ai sin ai + 02 sin 0:2 + az sin a<3 a± cos a± + a2 cos 0:2 + &z cos a3 2 ; a2 2A 2 2,2,2 csc w = csc a± + csc a.2 + esc otz -s/ai 2 a2 2 + a2 2 a 3 2 + a 3 2 ai 2 (i) (2) (3) (4) (5) (6) (7) where A is the Triangle Area, A, B, and C are An- gles, and a, b, and c are side lengths. If an Angle a of a Triangle is given, the maximum possible Brocard angle is given by coto; = § tan(ia) + 5COs(|a). (8) Let a Triangle have Angles A, B, and C. Then sin A sin B sin C < kABC, (9) where k=[^) (10) (Le Lionnais 1983). This can be used to prove that 8a; 3 < ABC (11) (Abi-Khuzam 1974). 174 Brocard Axis Brocard Line see also BROCARD CIRCLE, BROCARD LINE, EQUI- Brocard Center, Fermat Point, Isogonic Cen- ters References Abi-Khuzam, F. "Proof of YfFs Conjecture on the Brocard Angle of a Triangle." Elem. Math. 29, 141-142, 1974. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286 and 289-294, 1929. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983. Brocard Axis The Line KO passing through the Lemoine Point K and Circumcenter O of a Triangle. The distance OK is called the Brocard Diameter. The Brocard axis is Perpendicular to the Lemoine Axis and is the Isogonal Conjugate of Kiepert's Hyperbola. It has equations sin(£ - C)a + sin(C - A)f3 + sin(A - B)j = bc(b 2 - c 2 )a + ca(c 2 - a 2 )p + ab(a 2 - 6 2 ) 7 = 0. The Lemoine Point, Circumcenter, Isodynamic Points, and BROCARD Midpoint all lie along the Bro- card axis. Note that the Brocard axis is not equivalent to the Brocard Line. see also Brocard Circle, Brocard Diameter, Bro- card Line Brocard Circle The CIRCLE passing through the first and second Bro- card Points ft and ft', the Lemoine Point K, and the Circumcenter O of a given Triangle. The Bro- card Points ft and ft' are symmetrical about the Line KO' which is called the Brocard Line. The Line Segment KO is called the Brocard Diameter, and it has length OK : on COS UJ R^Jl -4sin 2 cj cos a; where R is the ClRCUMRADlUS and u> is the BROCARD Angle. The distance between either of the Brocard Points and the Lemoine Point is OK = TVK = Tld tan a;. see also Brocard Angle, Brocard Diameter, Bro- card Points References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 272, 1929. Brocard's Conjecture 7r(pn+i 2 ) -7r(Pn 2 ) > 4 for n > 2 where tt is the Prime Counting Function. see also ANDRICA'S CONJECTURE Brocard Diameter The Line Segment KO joining the Lemoine Point K and Circumcenter O of a given Triangle. It is the Diameter of the Triangle's Brocard Circle, and lies along the BROCARD Axis. The Brocard diameter has length — — On R\/l - 4 sin 2 w OK = — , COS UJ COS U) where ft is the first Brocard Point, R is the Circum- RADIUS, and w is the Brocard Angle. see also Brocard Axis, Brocard Circle, Brocard Line, Brocard Points Brocard Line ^3 "3 A Line from any of the Vertices Ai of a Triangle to the first ft or second ft' BROCARD POINT, Let the Angle at a Vertex A» also be denoted A i} and denote the intersections of A±Q and Aifl' with A2A3 as Wi and W2. Then the ANGLES involving these points are LA&Wz^Ax (1) IW Z QA 2 = A 3 (2) LA 2 £IW 1 =A 2 . (3) Distances involving the points Wi and W[ are given by a 3 ,4 2 ft sin A2 (4) Brocard Midpoint Brocard Points 175 A 2 Q A 3 n _ a 3 2 _ sin(^4 3 - ll>) aia2 sin a; W 3 Ai _ W3A2 a2 sin u) (0,2 ai sin(A3 — uj) \a% (5) (6) where uj is the Brocard Angle (Johnson 1929, pp. 267-268). The Brocard line, MEDIAN M, and LEMOINE POINT K are concurrent, with A1Q1, A2K ', and A3M meeting at a point P. Similarly, AiQ' , A2M, and A3K meet at a point which is the ISOGONAL CONJUGATE point of P (Johnson 1929, pp. 268-269). see also Brocard Axis, Brocard Diameter, Bro- card Points, Isogonal Conjugate, Lemoine Point, Median (Triangle) References Johnson, It. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286, 1929. Brocard Midpoint The Midpoint of the Brocard Points. It has Tri- angle Center Function a = a(b + c ) — sin(^4 -f- a;), where uj is the Brocard Angle. It lies on the Bro- card Axis. References Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Brocard Points A C The first Brocard point is the interior point H (or n or Zx) of a Triangle for which the Angles IQAB, ZfiBC, and IQCA are equal. The second Brocard point is the interior point fi' (or T2 or Z2) for which the An- gles IQ'AC, IQ'CB, and Itt'BA are equal. The AN- GLES in both cases are equal to the BROCARD ANGLE uj = IttAB = IttBC = mCA = in' ac = m'CB - iq'ba. The first two Brocard points are ISOGONAL Conju- gates (Johnson 1929, p. 266). Let Cbc be the CIRCLE which passes through the ver- tices B and C and is TANGENT to the line AC at C, and similarly for Cab and Cbc- Then the CIRCLES Cab, Cbc, and Cac intersect in the first Brocard point Q. Similarly, let C' BC be the CIRCLE which passes through the vertices B and C and is TANGENT to the line AB at B, and similarly for C' AB and C' AC . Then the CIRCLES C A b j C'bC) anc * Cac intersect in the second Brocard points £V (Johnson 1929, pp. 264-265). a c a c The Pedal Triangles of Q and 0! are congruent, and Similar to the Triangle AABC (Johnson 1929, p. 269). Lengths involving the Brocard points include OQ = OW = R\/l-4sm 2 uj nO' = 2Rs\xiu\/l -4sin 2 u>. (i) (2) Brocard's third point is related to a given TRIANGLE by the Triangle Center Function (3) (Casey 1893, Kimberling 1994). The third Brocard point Q" (or r 3 or Z z ) is COLLINEAR with the SPIEKER Center and the Isotomic Conjugate Point of its Triangle's Incenter. see also Brocard Angle, Brocard Midpoint, Equi- Brocard Center, Yff Points References Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 1893. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286, 1929. Kimberling, C "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988. 176 Brocard's Problem Brown Function Brocard's Problem Find the values of n for which n! + 1 is a SQUARE NUM- BER m 2 , where n! is the FACTORIAL (Brocard 1876, 1885). The only known solutions are n = 4, 5, and 7, and there are no other solutions < 1027. The pairs of numbers (m,n) are called Brown NUMBERS. see also BROWN NUMBERS, FACTORIAL, SQUARE NUM- BER References Brocard, H. Question 166. Nouv. Corres. Math. 2, 287, 1876. Brocard, H. Question 1532. Nouv. Ann. Math. 4, 391, 1885. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 193, 1994. Brocard Triangles Let the point of intersection of A 2 ^l and Azfl' be Bi, where Q and fl f are the Brocard Points, and similarly define B2 and £3. B1B2BZ is the first Brocard trian- gle, and is inversely similar to A1A2A3. It is inscribed in the BROCARD CIRCLE drawn with OK as the DIAM- ETER. The triangles B1A2A3, £ 2 A 3 Ai, and B3A1A2 are ISOSCELES TRIANGLES with base angles lj, where u; is the Brocard Angle. The sum of the areas of the Isosceles Triangles is A, the Area of Triangle A1A2A3. The first Brocard triangle is in perspective with the given TRIANGLE, with AtB^ A 2 B 2 , and A3B3 Concurrent. The Median Point of the first Brocard triangle is the MEDIAN POINT M of the original triangle. The Brocard triangles are in perspective at M. Let ci, c 2 , and c 3 and ci, c 2 , and c 3 be the CIRCLES intersecting in the Brocard Points Q and Q' , respec- tively. Let the two circles c\ and c[ tangent at A\ to A1A2 and A\A$, and passing respectively through As and A 2 , meet again at C\. The triangle C1C2C3 is the second Brocard triangle. Each Vertex of the second Brocard triangle lies on the second Brocard Circle. The two Brocard triangles arc in perspective at M. see also Steiner Points, Tarry Point References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277-281, 1929. Bromwich Integral The inverse of the Laplace Transform, given by 2iri I J -y — to '7—100 where 7 is a vertical Contour in the Complex Plane chosen so that all singularities of f(s) are to the left of it. References Arfken, G. "Inverse Laplace Transformation." §15.12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 853-861, 1985. Brothers A Pair of consecutive numbers. see also Pair, Smith Brothers, Twins Brouwer Fixed Point Theorem Any continuous FUNCTION G : D n -> D n has a FIXED Point, where £> n = {x€M n :xi 2 + ... + a;„ 2 <1} is the unit n-BALL. see also FIXED POINT THEOREM References Milnor, J. W. Topology from the Differentiate Viewpoint. Princeton, NJ: Princeton University Press, p. 14, 1965. Browkin's Theorem For every Positive Integer n, there exists a Square in the plane with exactly n Lattice Points in its inte- rior. This was extended by Schinzel and Kulikowski to all plane figures of a given shape. The generalization of the Square in 2-D to the Cube in 3-D was also proved by Browkin. see also Cube, Schinzel's Theorem, Square References Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 121-125, 1973. Brown's Criterion A Sequence {^} of nondecreasing Positive Integers is Complete Iff 1. 1/1 = 1. 2. For all k = 2, 3, . . . , S k -1 = v\ + ^2 + . . • + ffc-l > Vk - 1. A corollary states that a Sequence for which v\ = 1 and v>k+i < 2vk is COMPLETE (Honsberger 1985). see also COMPLETE SEQUENCE References Brown, J. L. Jr. "Notes on Complete Sequences of Integers." Amer. Math. Monthly, 557-560, 1961. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 123-130, 1985. Brown Function For a Fractal Process with values y(t — At) and y(t+ At) j the correlation between these two values is given by the Brown function 1, also known as the Bachelier Function, Levy Func- tion, or Wiener Function. Brown Numbers Brun's Constant 177 Brown Numbers Brown numbers are Pairs (m, n) of Integers satisfying the condition of Brocard's Problem, i.e., such that n! + 1 = m where n! is the FACTORIAL and m 2 is a SQUARE Num- ber. Only three such Pairs of numbers are known: (5,4), (11,5), (71,7), and Erdos conjectured that these are the only three such Pairs. Le Lionnais (1983) points out that there are 3 numbers less than 200,000 for which (n-l)! + l = (mod n 2 ) , namely 5, 13, and 563. see also Brocard's Problem, Factorial, Square Number References Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 193, 1994. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Pickover, C. A. Keys to Infinity. New York: W. H. Freeman, p. 170, 1995. Broyden's Method An extension of the secant method of root finding to higher dimensions. References Broyden, C. G. "A Class of Methods for Solving Nonlinear Simultaneous Equations." Math. Comput. 19, 577-593, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 382-385, 1992. Bruck-Ryser-Chowla Theorem If n = 1, 2 (mod 4), and the SQUAREFREE part of n is di- visible by a Prime p = 3 (mod 4), then no Difference Set of ORDER n exists. Equivalently, if a PROJECTIVE PLANE of order n exists, and n — 1 or 2 (mod 4), then n is the sum of two SQUARES. Dinitz and Stinson (1992) give the theorem in the fol- lowing form. If a symmetric (v, k, A)-BLOCK DESIGN exists, then 1. If v is Even, then k - A is a Square Number, 2. If v is Odd, the the Diophantine Equation x 2 ^(k-\)y 2 + (-l) (f-l)/2 \z z has a solution in integers, not all of which are 0. see also Block Design, Fisher's Block Design In- equality References Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to Design Theory." Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1-12, 1992. Gordon, D. M. "The Prime Power Conjecture is True for n < 2,000,000." Electronic J. Combinatorics 1, R6, 1-7, 1994. http://www.combinatorics.org/Volume_l/ volume 1 ,html#R6. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963. Bruck-Ryser Theorem see BRUCK-RYSER-CHOWLA Theorem Brun's Constant The number obtained by adding the reciprocals of the Twin Primes, (1) By Brun's Theorem, the constant converges to a def- inite number as p — > oo. Any finite sum underesti- mates B. Shanks and Wrench (1974) used all the Twin PRIMES among the first 2 million numbers. Brent (1976) calculated all Twin Primes up to 100 billion and ob- tained (Ribenboim 1989, p. 146) B « 1.90216054, (2) assuming the truth of the first HARDY-LlTTLEWOOD Conjecture. Using Twin Primes up to 10 14 , Nicely (1996) obtained B^ 1.9021605778 ±2.1 x 10 -9 (3) (Cipra 1995, 1996), in the process discovering a bug in Intel's® Pentium™ microprocessor. The value given by Le Lionnais (1983) is incorrect. see also Twin Primes, Twin Prime Conjecture, Twin Primes Constant References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 64, 1987. Brent, R. P. "Tables Concerning Irregularities in the Distri- bution of Primes and Twin Primes Up to 10 11 ." Math. Comput 30, 379, 1976. Cipra, B. "How Number Theory Got the Best of the Pentium Chip." Science 267, 175, 1995. Cipra, B. "Divide and Conquer." What's Happening in the Mathematical Sciences, 1995-1996, Vol 3. Providence, RI: Amer. Math. Soc, pp. 38-47, 1996. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/brun/brun.html. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983. Nicely, T. "Enumeration to 10 14 of the Twin Primes and Brun's Constant." Virginia J. Sci. 46, 195-204, 1996. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer- Verlag, 1989. Shanks, D. and Wrench, J. W. "Brun's Constant." Math. Comput 28, 293-299, 1974. Wolf, M. "Generalized Brun's Constants." http://www.ift. uni.wroc.pl/-mwolf/. 178 Brunn-Minkowski Inequality Buffon's Needle Problem Brunn-Minkowski Inequality The nth root of the Content of the set sum of two sets in Euclidean n-space is greater than or equal to the sum of the nth roots of the Contents of the individual sets. see also TOMOGRAPHY References Cover, T. M. "The Entropy Power Inequality and the Brunn- Minkowski Inequality" §5.10 in In Open Problems in Com- munications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer- Verlag, p. 172, 1987. Schneider, R. Convex Bodies: The Brunn-Minkowski The- ory. Cambridge, England: Cambridge University Press, 1993. Brun's Sum see Brun's Constant Brun's Theorem The series producing Brun's Constant Converges even if there are an infinite number of TWIN PRIMES. Proved in 1919 by V. Brun. Brunnian Link A Brunnian link is a set of n linked loops such that each proper sublink is trivial, so that the removal of any component leaves a set of trivial unlinked Unknots. The Borromean Rings are the simplest example and have n = 3. see also Borromean Rings References Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976. Brute Force Factorization see Direct Search Factorization Bubble A bubble is a MINIMAL SURFACE of the type that is formed by soap film. The simplest bubble is a single Sphere. More complicated forms occur when multi- ple bubbles are joined together. Two outstanding prob- lems involving bubbles are to find the arrangements with the smallest Perimeter (planar problem) or Surface Area (Area problem) which enclose and separate n given unit areas or volumes in the plane or in space. For n — 2, the problems are called the DOUBLE BUB- BLE CONJECTURE and the solution to both problems is known to be the DOUBLE Bubble. see also Double Bubble, Minimal Plateau's Laws, Plateau's Problem Surface, References Morgan, F. "Mathematicians, Including Undergraduates, Look at Soap Bubbles." Amer. Math. Monthly 101, 343- 351, 1994. Pappas, T. "Mathematics & Soap Bubbles." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 219, 1989. Buchberger's Algorithm The algorithm for the construction of a GROBNER BASIS from an arbitrary ideal basis. see also GROBNER BASIS References Becker, T. and Weispfenning, V. Grobner Bases: A Com- putational Approach to Commutative Algebra. New York: Springer- Verlag, pp. 213-214, 1993. Buchberger, B. "Theoretical Basis for the Reduction of Poly- nomials to Canonical Forms." SIGSAM Bull 39, 19-24, Aug. 1976. Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Buckminster Fuller Dome see Geodesic Dome Buffon-Laplace Needle Problem -4 1 X t v / v £~ / , ^ r ,% ^ i ^ + b h Find the probability P(£, a, b) that a needle of length £ will land on a line, given a floor with a grid of equally spaced Parallel Lines distances a and b apart, with £ > a,b. 2£(a + b)-P P(*,a,6) = -nab see also BUFFON'S NEEDLE PROBLEM BufFon's Needle Problem / ^ / / Bulirsch-Stoer Algorithm Burau Representation 179 Find the probability P(£>d) that a needle of length £ will land on a line, given a floor with equally spaced Parallel Lines a distance d apart. P&d) -f Jo £\cosO\ dd _ t = -[8in*] ' 27r 2nd - *L ird /.tt/2 7 ' Jo cos 8 dO Several attempts have been made to experimentally de- termine 7r by needle- tossing. For a discussion of the relevant statistics and a critical analysis of one of the more accurate (and least believable) needle-tossings, see Badger (1994). see also Buffon-Laplace Needle Problem References Badger, L. "Lazzarini's Lucky Approximation of 7r." Math. Mag. 67, 83-91, 1994. Dorrie, H. "Buffon's Needle Problem." §18 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 73-77, 1965. Kraitchik, M. "The Needle Problem." §6.14 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942. Wegert, E. and Trefethen, L, N. "Prom the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132-139, 1994. Bulirsch-Stoer Algorithm An algorithm which finds RATIONAL FUNCTION extrap- olations of the form Ri(i + l)---(i+m) Py(x) __ po + p\x + . . . +p^x M P„(x) qo + qix + . . . + q u x v and can be used in the solution of Ordinary Differ- ential Equations. References Bulirsch, R. and Stoer, J. §2.2 in Introduction to Numerical Analysis. New York: Springer- Verlag, 1991. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Richardson Extrapolation and the Bulirsch- Stoer Method." §16.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, Eng- land: Cambridge University Press, pp. 718-725, 1992. Bullet Nose A plane curve with implicit equation x 1 y 2 (1) The Curvature is x = a cost y = b cot t. Sab cot t esc t (6 2 csc 4 i + a 2 sin 2 i) 3 / 2 and the TANGENTIAL ANGLE is ■ = tan _i /bcsc 3 A (2) (3) (4) (5) References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 127-129, 1972. Bumping Algorithm Given a Permutation {pi,f>2, ■ ■ • ,Vn) of {1, . . . , n}, the bumping algorithm constructs a standard YOUNG Tableau by inserting the pi one by one into an already constructed YOUNG TABLEAU. To apply the bump- ing algorithm, start with {{pi}}, which is a YOUNG TABLEAU. If p\ through pk have already been inserted, then in order to insert pfc+i, start with the first line of the already constructed YOUNG TABLEAU and search for the first element of this line which is greater than Pk+i- If there is no such element, append Pk+\ to the first line and stop. If there is such an element (say, p p ), exchange p p for pjt+i, search the second line using p p , and so on. see also YOUNG TABLEAU References Skiena, S. Implementing Discrete Mathematics: Combina- torics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, 1990. Bundle see Fiber Bundle Burau Representation Gives a Matrix representation b* of a Braid Group in terms of (n - 1) x (n - 1) Matrices. A -t always appears in the (i,i) position. bi = -too -1 1 1 (1) bi = In parametric form, 1 •• • • ■ ■ -t • • •• -t o . • • ■ -1 1 ■ • '. •• • • 1 (2) 180 Burkhardt Quartic Burnside Problem b n _ rl • • 1 ■ • 1 — ■ • — Lo • • — (3) Let * be the Matrix Product of Braid Words, then det(l - 9) _ , , l + t + . .. + t»-i " AL ' (4) where A L is the ALEXANDER POLYNOMIAL and det is the Determinant. References Burau, W. "Uber Zopfgruppen und gleichsinnig verdrilte Ver- kettungen." Abh. Math. Sem. Hanischen Univ. 11, 171- 178, 1936. Jones, V. "Hecke Algebra Representation of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388, 1987. Burkhardt Quartic The Variety which is an invariant of degree four and is given by the equation yt 2/0(2/? 3 , 3 ■ 2/2 + yz ■ ■2/1) + 32/12/22/32/4 = 0. References Burkhardt, H. "Untersuchungen aus dem Gebiet der hyperel- liptischen Modulfunctionen. II." Math. Ann. 38, 161-224, 1890. Burkhardt, H. "Untersuchungen aus dem Gebiet der hyper- elliptischen Modulfunctionen. III." Math. Ann. 40, 313- 343, 1892. Hunt, B. "The Burkhardt Quartic." Ch. 5 in The Geom- etry of Some Special Arithmetic Quotients. New York: Springer- Verlag, pp. 168-221, 1996. Burnside's Conjecture Every non-ABELIAN SIMPLE GROUP has EVEN ORDER. see also Abelian Group, Simple Group Burnside's Lemma Let J be a Finite Group and the image R(J) be a representation which is a HOMEOMORPHISM of J into a Permutation Group S(X), where S(X) is the Group of all permutations of a Set X. Define the orbits o£R(J) as the equivalence classes under x ~ y, which is true if there is some permutation p in R( J) such that p(x) = y. Define the fixed points of p as the elements x of X for which p(x) = x. Then the AVERAGE number of FIXED POINTS of permutations in R(J) is equal to the number of orbits of R(J). The LEMMA was apparently known by Cauchy (1845) in obscure form and Frobenius (1887) prior to Burnside's (1900) rediscovery. It was subsequently extended and refined by Polya (1937) for applications in COMBINATO- RIAL counting problems. In this form, it is known as Polya Enumeration Theorem. References Polya, G. "Kombinatorische Anzahlbestimmungen fur Grup- pen, Graphen, und chemische Verbindungen." Acta Math. 68, 145-254, 1937. Burnside Problem A problem originating with W. Burnside (1902), who wrote, "A still undecided point in the theory of dis- continuous groups is whether the Order of a Group may be not finite, while the order of every operation it contains is finite." This question would now be phrased as "Can a finitely generated group be infinite while every element in the group has finite order?" (Vaughan-Lee 1990). This question was answered by Golod (1964) when he constructed finitely generated in- finite p-GROUPS. These GROUPS, however, do not have a finite exponent. Let F r be the Free Group of Rank r and let N be the Subgroup generated by the set of nth POWERS {g n \g e F r }. Then TV is a normal subgroup of F r . We define B(r, n) = F r /N to be the QUOTIENT GROUP. We call B(r,n) the r-generator Burnside group of exponent n. It is the largest r-generator group of exponent n, in the sense that every other such group is a HOMEOMOR- PHIC image of B(r, n). The Burnside problem is usually stated as: "For which values of r and n is £(r,n) a Finite Group?" An answer is known for the following values. For r = 1, 5(1,77) is a Cyclic Group of Order n. For n = 2, B(r, 2) is an elementary Abelian 2-group of Order 2 n , For n = 3, B(r, 3) was proved to be finite by Burnside. The ORDER of the B(r,3) groups was established by Levi and van der Waerden (1933), namely 3 a where :r + (1) where (™) is a Binomial COEFFICIENT. For n = 4, B(r> 4) was proved to be finite by Sanov (1940). Groups of exponent four turn out to be the most complicated for which a POSITIVE solution is known. The precise nilpotency class and derived length are known, as are bounds for the ORDER. For example, |S(2,4)| = 2 12 |B(3,4)| = 2 69 |S(4,4)| = 2 422 |B(5,4)|=2 2728 (2) (3) (4) (5) while for larger values of r the exact value is not yet known. For n = 6, B(r,6) was proved to be finite by Hall (1958) with ORDER 2 a 3 6 , where a = 1 + (r - 1)3 C 6 = l + (r-l)2 r c = r + + (6) (7) (8) No other Burnside groups are known to be finite. On the other hand, for r > 2 and n > 665, with n ODD, Busemann-Petty Problem B(r,n) is infinite (Novikov and Adjan 1968). There is a similar fact for r > 2 and n a large Power of 2. E. Zelmanov was awarded a Fields Medal in 1994 for his solution of the "restricted" Burnside problem. see also FREE GROUP References Burnside, W. "On an Unsettled Question in the Theory of Discontinuous Groups." Quart. J. Pure Appl. Math. 33, 230-238, 1902. Golod, E. S. "On Nil-Algebras and Residually Finite p- Groups." Isv. Akad. Nauk SSSR Ser. Mat. 28, 273-276, 1964. Hall, M. "Solution of the Burnside Problem for Exponent Six." Ill J. Math. 2, 764-786, 1958. „ Levi, F. and van der Waerden, B. L. "Uber eine besondere Klasse von Gruppen." Abh. Math. Sem. Univ. Hamburg 9, 154-158, 1933. Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I, II, III." Izv. Akad. Nauk SSSR Ser. Mat 32, 212-244, 251-524, and 709-731, 1968. Sanov, I. N. "Solution of Burnside's problem for exponent four." Leningrad State Univ. Ann. Math. Ser. 10, 166— 170, 1940. Vaughan-Lee, M. The Restricted Burnside Problem, 2nd ed. New York: Clarendon Press, 1993. Busemann-Petty Problem If the section function of a centered convex body in Eu- clidean n-space (n > 3) is smaller than that of another such body, is its volume also smaller? References Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422-429, 1995. Busy Beaver A busy beaver is an n-state, 2-symbol, 5-tuple Turing MACHINE which writes the maximum possible number BB(n) of Is on an initially blank tape before halting. For n = 0, 1, 2, ... , BB(n) is given by 0, 1, 4, 6, 13, > 4098, > 136612, The busy beaver sequence is also known as Rado's Sigma Function. see also HALTING PROBLEM, TURING MACHINE References Chaitin, G. J. "Computing the Busy Beaver Function." §4.4 in Open Problems in Communication and Computation (Ed. T. M. Cover and B. Gopinath). New York: Springer- Verlag, pp. 108-112, 1987. Dewdney, A. K. "A Computer Trap for the Busy Beaver, the Hardest- Working Turing Machine." Sci. Amer. 251, 19-23, Aug. 1984. Marxen, H. and Buntrock, J. "Attacking the Busy Beaver 5." Bull. EATCS40, 247-251, Feb. 1990. Sloane, N. J. A. Sequence A028444 in "An On-Line Version of the Encyclopedia of Integer Sequences." Butterfly Fractal 181 Butterfly Catastrophe A Catastrophe which can occur for four control fac- tors and one behavior axis. The equations x = c(Sat 3 + 24t 5 ) y = c(-6ai 2 - 15t 4 ) display such a catastrophe (von Seggern 1993). References von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 94, 1993. Butterfly Curve A Plane Curve given by the implicit equation y =(x -x ). see also DUMBBELL CURVE, EIGHT CURVE, PIRIFORM References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Butterfly Effect Due to nonlinearities in weather processes, a butterfly flapping its wings in Tahiti can, in theory, produce a tornado in Kansas. This strong dependence of outcomes on very slightly differing initial conditions is a hallmark of the mathematical behavior known as CHAOS. see also Chaos, Lorenz System Butterfly Fractal The FRACTAL-like curve generated by the 2-D function (z 2 -y 2 )sin(^) ffay) = x 2 +y 2 182 Butterfly Polyiamond Butterfly Theorem Butterfly Polyiamond A 6-POLYIAMOND. References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton University Press, p. 92, 1994. Butterfly Theorem A Given a Chord PQ of a Circle, draw any other two CHORDS AB and CD passing through its MIDPOINT. Call the points where AD and BC meet PQ X and Y. Then M is the Midpoint of XY. see also CHORD, CIRCLE, MIDPOINT References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited, Washington, DC: Math. Assoc. Amer., pp. 45-46, 1967. Cake Cutting 183 C C-Table see C-Determinant The Field of Complex Numbers, denoted C. see also C\ Complex Number, I, N, Q, R, Z C* The Riemann Sphere C U {oo}, see also C, Complex Number, Q, R, Riemann Sphere, Z C*-Algebra A special type of B* -Algebra in which the Involu- tion is the Adjoint Operator in a Hilbert Space. see also £*-ALGEBRA, fc-THEORY References Davidson, K. R. C* -Algebras by Example. Providence, RI: Amer. Math. Soc, 1996. C- Curve see Levy Fractal Cable Knot Let Ki be a Torus Knot. Then the Satellite Knot with Companion Knot K 2 is a cable knot on K 2 . References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 118, 1994. Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 112 and 283, 1976. Cactus Fractal •m A Mandelbrot SET-like Fractal obtained by iterat- ing the map Zn+l = Z n + (ZQ — l)z n — Zq - C-Determinant A Determinant appearing in Pade Approximant identities: a s + l <Xr-s+2 a r +\ Gr+s-1 see also Pade APPROXIMANT C-Matrix Any Symmetric Matrix (A t = A) or Skew Symmet- ric Matrix (A t = -A) C™ with diagonal elements and others ±1 satisfying CC T = (n-l)l, where I is the IDENTITY MATRIX, is known as a C- matrix (Ball and Coxeter 1987), Examples include c 4 = + + + - - + - + - - - + 0_ + + + + + + + - - + + + + + __ + - + + - + - - + + + + - - + c 6 = References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, pp. 308- 309, 1987. see also FRACTAL, JULIA SET, MANDELBROT SET Cake Cutting It is always possible to "fairly" divide a cake among n people using only vertical cuts. Furthermore, it is pos- sible to cut and divide a cake such that each person believes that everyone has received 1/n of the cake ac- cording to his own measure. Finally, if there is some piece on which two people disagree, then there is a way of partitioning and dividing a cake such that each par- ticipant believes that he has obtained more than 1/n of the cake according to his own measure. Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a CIRCLE into n equal Area pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the Frobenius-Konig Theorem. see also CIRCLE CUTTING, CYLINDER CUTTING, EN- VYFREE, FROBENIUS-KONIG THEOREM, HAM SAND- WICH Theorem, Pancake Theorem, Pizza Theo- rem, Square Cutting, Torus Cutting References Brams, S. J. and Taylor, A. D. "An Envy-Free Cake Division Protocol." Amer. Math. Monthly 102, 9-19, 1995. Brams, S. J. and Taylor, A. D. Fair Division: From Cake- Cutting to Dispute Resolution. New York: Cambridge Uni- versity Press, 1996. Dubbins, L. and Spanier, E. "How to Cut a Cake Fairly." Amer. Math. Monthly 68, 1-17, 1961. Gale, D. "Dividing a Cake." Math. Intel. 15, 50, 1993. Jones, M. L. "A Note on a Cake Cutting Algorithm of Banach and Knaster." Amer. Math. Monthly 104, 353-355, 1997. Rebman, K. "How to Get (At Least) a Fair Share of the Cake." In Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 22-37, 1979. 184 Cal Calculus of Variations rsi Steinhaus, H. "Sur la division progmatique." Ekonometrika (Supp.) 17, 315-319, 1949. Stromquist, W. "How to Cut a Cake Fairly." Amer. Math. Monthly 87, 640-644, 1980. Cal see Walsh Function Calabi's Triangle and Integrals / f(x) dx, Equilateral Triangle Calabi's Triangle The one TRIANGLE in addition to the EQUILATERAL Triangle for which the largest inscribed Square can be inscribed in three different ways. The ra- tio of the sides to that of the base is given by x = 1.55138752455. . . (Sloane's A046095), where 11 _ 1 (-23 + 3zy / 237) 1/3 X ~ 3 + 3-2 2 /3 + 3[ 2 (-23 + 3iv / 237)] 1 / 3 is the largest POSITIVE ROOT of 2x 3 - 2x 2 - 3z + 2 = 0, which has CONTINUED FRACTION [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, . . .] (Sloane's A046096). see also GRAHAM'S BIGGEST LITTLE HEXAGON References Conway, J. H. and Guy, R. K. "Calabi's Triangle." In The Book of Numbers. New York: Springer- Verlag, p. 206, 1996, Sloane, N. J. A. Sequences A046095 and A046096 in "An On- Line Version of the Encyclopedia of Integer Sequences." Calabi-Yau Space A structure into which the 6 extra Dimensions of 10-D string theory curl up. Calculus In general, "a" calculus is an abstract theory developed in a purely formal way. "The" calculus, more properly called ANALYSIS (or Real Analysis or, in older literature, Infinitesimal Analysis) is the branch of mathematics studying the rate of change of quantities (which can be interpreted as Slopes of curves) and the length, Area, and Volume of objects. The CALCULUS is sometimes divided into Differential and Integral Calculus, concerned with Derivatives respectively. While ideas related to calculus had been known for some time (Archimedes' Exhaustion Method was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Weierstraft. see also Arc Length, Area, Calculus of Vari- ations, Change of Variables Theorem, De- rivative, Differential Calculus, Ellipsoidal Calculus, Extensions Calculus, Fluent, Flux- ion, Fractional Calculus, Functional Calculus, Fundamental Theorems of Calculus, Heaviside Calculus, Integral, Integral Calculus, Jaco- bian, Lambda Calculus, Kirby Calculus, Malli- avin Calculus, Predicate Calculus, Proposi- tional Calculus, Slope, Tensor Calculus, Um- bral Calculus, Volume References Anton, H. Calculus with Analytic Geometry, 5th ed. New York: Wiley, 1995. Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Cal- culus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, 1967. Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Cal- culus and Linear Algebra, with Applications to Differential Equations and Probability. Waltham, MA: Blaisdell, 1969. Apostol, T. M. A Century of Calculus, 2 vols. Pt. 1: 1894~ 1968. Pt. 2: 1969-1991. Washington, DC: Math. Assoc. Amer., 1992. Ayres, F. Jr. and Mendelson, E. Schaum's Outline of Theory and Problems of Differential and Integral Calculus, 3rd ed. New York: McGraw-Hill, 1990. Borden, R. S, A Course in Advanced Calculus. New York: Dover, 1998. Boyer, C B. A History of the Calculus and Its Conceptual Development. New York: Dover, 1989. Brown, K. S. "Calculus and Differential Equations." http:// www. seanet . com/-ksbrown/icalculu.htm. Courant, R. and John, F. Introduction to Calculus and Anal- ysis, Vol. 1. New York: Springer- Verlag, 1990. Courant, R. and John, F. Introduction to Calculus and Anal- ysis, Vol. 2. New York: Springer- Verlag, 1990. Hahn, A. Basic Calculus: From Archimedes to Newton to Its Role in Science. New York: Springer- Verlag, 1998. Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison- Wesley, 1992. Marsden, J. E. and Tromba, A. J. Vector Calculus, ^i/i ed. New York: W. H. Freeman, 1996. Strang, G. Calculus. Wellesley, MA: Wellesley-Cambridge Press, 1991. Calculus of Variations A branch of mathematics which is a sort of general- ization of CALCULUS. Calculus of variations seeks to find the path, curve, surface, etc., for which a given Function has a Stationary Value (which, in physical Calcus problems, is usually a Minimum or Maximum). Mathe- matically, this involves finding STATIONARY VALUES of integrals of the form '= / /(y.y, x) dx. (i) J has an extremum only if the Euler-Lagrange Dif- ferential Equation is satisfied, i.e., if dy dx \dyj (2) The Fundamental Lemma of Calculus of Varia- tions states that, if t/ a M(x)h(x)dx = (3) for all h(x) with CONTINUOUS second PARTIAL DERIVA- TIVES, then M(x) = (4) on (a, 6). see also BELTRAMI IDENTITY, BOLZA PROBLEM, Brachistochrone Problem, Catenary, Enve- lope Theorem, Euler-Lagrange Differential Equation, Isoperimetric Problem, Isovolume Problem, Lindelof's Theorem, Plateau's Prob- lem, Point-Point Distance — 2-D, Point-Point Distance— 3-D, Roulette, Skew Quadrilateral, Sphere with Tunnel, Unduloid, WeierstraB- Erdman Corner Condition References Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 925-962, 1985. Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, 1925. Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Fox, C An Introduction to the Calculus of Variations. New- York: Dover, 1988. Isenberg, C The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Menger, K. "What is the Calculus of Variations and What are Its Applications?" In The World of Mathematics (Ed. K. Newman). Redmond, WA: Microsoft Press, pp. 886- 890, 1988. Sagan, H. Introduction to the Calculus of Variations. New York: Dover, 1992. Todhunter, I. History of the Calculus of Variations During the Nineteenth Century. New York: Chelsea, 1962. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974. Calcus 1 calcus = see also Half, Quarter, Scruple, Uncia, Unit Fraction Cancellation Law 185 Calderon's Formula /oo /*oo / (f,tp a ' b )i> a - b (x)a.- 2 dadb, -oo J — CO where r' b (x) = \a\-^(^.). This result was originally derived using HARMONIC Analysis, but also follows from a Wavelets viewpoint. Caliban Puzzle A puzzle in LOGIC in which one or more facts must be inferred from a set of given facts. Calvary Cross see also CROSS Cameron's Sum-Free Set Constant A set of POSITIVE INTEGERS S is sum-free if the equa- tion x 4- y = z has no solutions x, y, z 6 S. The proba- bility that a random sum-free set S consists entirely of Odd Integers satisfies 0.21759 < c < 0.21862. References Cameron, P. J. "Cyclic Automorphisms of a Countable Graph and Random Sum-Free Sets." Graphs and Com- binatorics 1, 129-135, 1985. Cameron, P. J. "Portrait of a Typical Sum- Free Set." In Surveys in Combinatorics 1987 (Ed. C. Whitehead). New York: Cambridge University Press, 13-42, 1987. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/cameron/cameron.html. Cancellation see Anomalous Cancellation Cancellation Law If be = bd (mod a) and (6, a) — 1 (i.e., a and b are Relatively Prime), then c~ d (mod a). see also CONGRUENCE References Courant, R. and Robbins, H. What is Mathematics?: An El- ementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 36, 1996. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 56, 1993. 186 Cannonball Problem Cantor Dust Cannonball Problem Find a way to stack a SQUARE of cannonballs laid out on the ground into a Square Pyramid (i.e., find a Square Number which is also Square Pyramidal). This cor- responds to solving the DlOPHANTINE EQUATION Cantor-Dedekind Axiom The points on a line can be put into a One-to-One correspondence with the REAL NUMBERS. see also Cardinal Number, Continuum Hypothe- sis, Dedekind Cut £V = I*(1 + *)(! + 2*) N 2 for some pyramid height k. The only solution is k = 24, N = 70, corresponding to 4900 cannonballs (Ball and Coxeter 1987, Dickson 1952), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). see also Sphere Packing, Square Number, Square Pyramid, Square Pyramidal Number References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, p. 59, 1987. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, p. 25, 1952. Lucas, E. Question 1180. Nouvelles Ann. Math. Ser. 2 14, 336, 1875. Lucas, E. Solution de Question 1180. Nouvelles Ann. Math. Ser. 2 15, 429-432, 1876. Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 77 and 152, 1988. Pappas, T. "Cannon Balls & Pyramids." The Joy of Math- ematics. San Carlos, CA: Wide World Publ./Tetra, p. 93, 1989. Watson, G. N. "The Problem of the Square Pyramid." Mes- senger. Math. 48, 1-22, 1918. Canonical Form A clear-cut way of describing every object in a class in a One-to-One manner. see also Normal Form, One-to-One References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, p. 7, 1996. Canonical Polyhedron A Polyhedron is said to be canonical if all its Edges touch a Sphere and the center of gravity of their contact points is the center of that Sphere. Each combinato- rial type of (GENUS zero) polyhedron contains just one canonical version. The ARCHIMEDEAN SOLIDS and their DUALS are all canonical. References Conway, J. H. "Re: polyhedra database." geometry. forum newsgroup, Aug. 31, 1995. Canonical Transformation see Symplectic Diffeomorphism Cantor Comb see Cantor Set Posting to Cantor Diagonal Slash A clever and rather abstract technique used by Georg Cantor to show that the Integers and Reals cannot be put into a One-to-One correspondence (i.e., the INFIN- ITY of Real Numbers is "larger" than the Infinity of INTEGERS), It proceeds by constructing a new member S' of a Set from already known members S by arrang- ing its nth term to differ from the nth term of the nth member of S. The tricky part is that this is done in such a way that the Set including the new member has a larger CARDINALITY than the original SET S. see also Cardinality, Continuum Hypothesis, De- NUMERABLE SET References Courant, R. and Robbins, H. What is Mathematics?: An El- ementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 81-83, 1996. Penrose, R. The Emperor's New Mind: Concerning Comput- ers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 84-85, 1989. Cantor Dust A Fractal which can be constructed using String Re- writing by creating a matrix three times the size of the current matrix using the rules line 1: "*"->"* *",'' "->" " line 2: "*"->" "," *'->" line 3: "*»->"* *",» »->" Let N n be the number of black boxes, L n the length of a side of a white box, and A n the fractional Area of black boxes after the nth iteration. iVn-5 71 A n = L n 2 N n = ($) n . The Capacity Dimension is therefore (1) (2) (3) ln(5 n ) r lniV n = - hm - — — = - hm /0 _ . n-J-oo III L n n->-oo Ul(cJ n ) In 5 ln3 1.464973521. (4) see also Box FRACTAL, SlERPINSKI CARPET, SlERPIN- ski Sieve Cantor's Equation Cantor Square Fractal 187 References Dickau, R. M. "Cantor Dust." http://f orum . swarthmore . edu/advanced/robertd/cantor .html. Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103-104, 1993. ^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia. edu/~eww6n/math/notebooks/Fractal.m. Cantor's Equation Cantor Set The Cantor set (Too) is given by taking the interval [0,1] (set To), removing the middle third (Ti), removing the middle third of each of the two remaining pieces (T2), and continuing this procedure ad infinitum. It is there- fore the set of points in the INTERVAL [0,1] whose ternary expansions do not contain 1, illustrated below. where uj is an Ordinal Number and e is an Inacces- sible Cardinal, see also INACCESSIBLE CARDINAL, ORDINAL NUMBER References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 274, 1996. Cantor Function The function whose values are 2 V 2 Cm-l . 2 Orn — 1 Orn for any number between This produces the Set of Real Numbers {x} such that (i) Cl C n X= 3+--- + F + ---' where c n may equal or 2 for each n. This is an infinite, Perfect Set. The total length of the Line Segments in the nth iteration is *•-(!)"■ (2) and the number of LINE SEGMENTS is N n = 2 n , so the length of each element is tn - N~ (3) (3) Cl Cm-l "3 * ' ' 3™" 1 and Cl + ■ Cm-l _2_ Chalice (1991) shows that any real- values function F(x) on [0, 1] which is MONOTONE INCREASING and satisfies 1. F(0) = 0, 2. F(x/S) = F{x)/2, 3. F(l-x) = 1-F(x) is the Cantor function. see also CANTOR SET, DEVIL'S STAIRCASE References Chalice, D. R. "A Characterization of the Cantor Function." Amer. Math. Monthly 98, 255-258, 1991. Wagon, S. "The Cantor Function" and "Complex Cantor Sets." §4.2 and 5.1 in Mathematica in Action. New York: W. H. Freeman, pp. 102-108 and 143-149, 1991. Cantor's Paradox The Set of all Sets is its own Power Set. Therefore, the Cardinality of the Set of all Sets must be bigger than itself. see also CANTOR'S THEOREM, POWER SET and the Capacity DIMENSION is In AT lim _ €-►0+ me lim nln2 00 — nln3 In 2 In 3 0.630929... (4) The Cantor set is nowhere Dense, so it has LEBESGUE MEASURE 0. A general Cantor set is a CLOSED SET consisting en- tirely of BOUNDARY POINTS. Such sets are UNCOUNT- ABLE and may have or POSITIVE LEBESGUE MEA- SURE. The Cantor set is the only totally disconnected, perfect, Compact Metric Space up to a Homeomor- PHISM (Willard 1970). see also Alexander's Horned Sphere, Antoine's Necklace, Cantor Function References Boas, R. P. Jr. A Primer of Real Functions. Washington, DC: Amer. Math. Soc, 1996. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princetqn, NJ: Princeton University Press, pp. 15- 20, 1991. Willard, S. §30.4 in General Topology. Reading, MA: Addison- Wesley, 1970. Cantor Square Fractal 188 Cantor's Theorem Cardano's Formula A Fractal which can be constructed using String Re- writing by creating a matrix three times the size of the current matrix using the rules line 1: "*"->"***"," "->" " line 2: "*"->"* *"," "->" " line 3: "*"->"***",» "->" " The first few steps are illustrated above. The size of the unit element after the nth iteration is L n G)" and the number of elements is given by the RECUR- RENCE Relation N n = 4JV n _i + 5(9 n ) where Ni = 5, and the first few numbers of elements are 5, 65, 665, 6305, Expanding out gives N n 5 \p 4 n-fc g fc-l =9 n_ 4 n_ fc=0 The Capacity Dimension is therefore liml^-lim^ 9 "- 4 ") th-oo In L n ln(9 n n-+oo ln(3- n ) n-^oo ln(3" n ) ln9 _ 21n3 _ ln3 "" In 3 ~ 2. Since the DIMENSION of the filled part is 2 (i.e., the SQUARE is completely filled), Cantor's square fractal is not a true FRACTAL. see also Box Fractal, Cantor Dust References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 82- 83, 1991. ^ Weisstein, E. W. "Fractals." http://www. astro. Virginia. edu/-eww6n/math/notebooks/Fractal.m. Cantor's Theorem The Cardinal Number of any set is lower than the Cardinal Number of the set of all its subsets. A Corollary is that there is no highest N (Aleph). see also Cantor's Paradox Cap see Cross-Cap, Spherical Cap Capacity see Transfinite Diameter Capacity Dimension A Dimension also called the Fractal Dimen- sion, Hausdorff Dimension, and Hausdorff- Besicovitch Dimension in which nonintegral values are permitted. Objects whose capacity dimension is dif- ferent from their TOPOLOGICAL Dimension are called Fractals. The capacity dimension of a compact Met- ric Space X is a Real Number capacity such that if n(e) denotes the minimum number of open sets of diam- eter less than or equal to e, then n(e) is proportional to e~ D as e — > 0. Explicitly, -^capacity ,. miV hm €-►0+ hie (if the limit exists), where N is the number of elements forming a finite Cover of the relevant Metric SPACE and e is a bound on the diameter of the sets involved (informally, e is the size of each element used to cover the set, which is taken to to approach 0). If each ele- ment of a Fractal is equally likely to be visited, then ^capacity = ^information, where ^information is the INFOR- MATION Dimension. The capacity dimension satisfies ^correlation S: ^information S: ^capacity where correlation is the Correlation Dimension, and is conjectured to be equal to the LYAPUNOV DIMENSION. see also CORRELATION EXPONENT, DIMENSION, HAUS- DORFF Dimension, Kaplan- Yorke Dimension References Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 538-541, 1995. Peitgen, H.-O. and Richter, D. H. The Beauty of Frac- tals: Images of Complex Dynamical Systems. New York: Springer- Verlag, 1986. Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: M. Dekker, 1977. Caratheodory Derivative A function / is Caratheodory differentiate at a if there exists a function which is CONTINUOUS at a such that f(x) -/(a) = <t>(x)(x-a). Every function which is Caratheodory differentiable is also FRECHET DIFFERENTIABLE. see also Derivative, Frechet Derivative Caratheodory's Fundamental Theorem Each point in the CONVEX Hull of a set S in R n is in the convex combination of n + 1 or fewer points of 5. see also Convex Hull, Helly's Theorem Cardano's Formula see Cubic Equation Cardinal Number Cardioid 189 Cardinal Number In informal usage, a cardinal number is a number used in counting (a Counting Number), such as 1, 2, 3, Formally, a cardinal number is a type of number defined in such a way that any method of counting SETS using it gives the same result. (This is not true for the ORDINAL Numbers.) In fact, the cardinal numbers are obtained by collecting all ORDINAL NUMBERS which are obtain- able by counting a given set. A set has No (ALEPH-0) members if it can be put into a One-TO-One correspon- dence with the finite ORDINAL NUMBERS. Two sets are said to have the same cardinal number if all the elements in the sets can be paired off One-to- One. An Inaccessible Cardinal cannot be expressed in terms of a smaller number of smaller cardinals. see also Aleph, Aleph-0 (Ho), Aleph-1 (Hi), Can- tor-Dedekind Axiom, Cantor Diagonal Slash, Conttnuum, Continuum Hypothesis, Equipol- lent, Inaccessible Cardinals Axiom, Infinity, Ordinal Number, Power Set, Surreal Number, Uncountable Set References Cantor, G. Uber unendliche, lineare Punktmannigfaltig- keiten, Arbeiten zur Mengenlehre aus dem Jahren 1872- 1884. Leipzig, Germany: Teubner, 1884. Conway, J. H. and Guy, R. K. "Cardinal Numbers." In The Book of Numbers. New York: Springer- Verlag, pp. 277- 282, 1996. Courant, R. and Robbins, H. "Cantor's 'Cardinal Numbers.'" §2.4.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 83-86, 1996. Cardinality see Cardinal Number Cardioid and the parametric equations The curve given by the POLAR equation r = a(l + cos#), sometimes also written r = 26(1 + cos 0), where b = a/2, the Cartestan equation / 2 . 2 n2 2/ 2 . 2\ [x + y -ax) — a (x +y ), (1) (2) (3) x = acost(l + cost) y = asini(l + cost). (4) (5) The cardioid is a degenerate case of the LlMA<JON. It is also a 1-CuSPED EPICYCLOID (with r = R) and is the CAUSTIC formed by rays originating at a point on the circumference of a CIRCLE and reflected by the Circle. The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its Arc Length was found by La Hire in 1708. There are exactly three PARALLEL TANGENTS to the cardioid with any given gradient. Also, the TANGENTS at the ends of any Chord through the Cusp point are at Right Angles. The length of any Chord through the Cusp point is 2a. The cardioid may also be generated as follows. Draw a CIRCLE C and fix a point A on it. Now draw a set of Circles centered on the Circumference of C and passing through A. The ENVELOPE of these Circles is then a cardioid (Pedoe 1995). Let the CIRCLE C be centered at the origin and have RADIUS 1, and let the fixed point be A — (1, 0). Then the RADIUS of a CIRCLE centered at an ANGLE 9 from (1, 0) is r 2 = (0-cos(9) 2 + (l-sin(9) 2 = cos 2 0+l-2sin0 + sin 2 = 2(1- sin 0). (6) J ^ The Arc Length, Curvature, and Tangential An- gle are /' Jo 2|cos(!i)|dt = 4asin(i0) 3|sec(i0)| 4o (7) (8) (9) As usual, care must be taken in the evaluation of s(t) for t > n. Since (7) comes from an integral involving the 190 Cardioid Caustic Cards ABSOLUTE Value of a function, it must be monotonic increasing- Each Quadrant can be treated correctly by defining + 1, (10) Cardioid Evolute l_7T where [a; J is the FLOOR FUNCTION, giving the formula s(t) = (~l) 1+[n (mod 2)] 4sin(|i) + 8 Ll n J ' ( U > The Perimeter of the curve is /»2tt / Jo |2acos(|i9)|d0 = 4a / cos (| Jo 9)dB /•7r/2 / 1 t/2 = 4a / cos <j>(2 d<$>) — 8a / cos (j)d<fi Jo Jo -8a[sin0]o /2 = 8a. (12) The Area is /•27T A= \ I r 2 d6=\a I (1 + 2cos<9 + cos 2 6) dO Jo Jo = 2 a / Jo {1 + 2 cos + | [1 + cos(26>)]} d0 /»27T = |a 2 / [§ + 2cos(9+|cos(26>)]dl9 Jo = \A¥ + 2sin # + \ sin^lo" = 2tt _ 3 2 (13) see also Circle, Cissoid, Conchoid, Equiangular Spiral, Lemniscate, LiMAgoN, Mandelbrot Set References Gray, A. "Cardioids." §3.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 41-42, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 118-121, 1972. Lee, X. "Cardioid." http : //www .best . com/ ~xah/ Special PlaneCurves_dir/Cardioid_dir/cardioid.html. Lee, X. "Cardioid." http://www.best.com/-xah/Special PlaneCurves_dir/Cardioid_dir/cardioidGG.html. Lockwood, E. H. "The Cardioid." Ch. 4 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 34- 43, 1967. MacTutor History of Mathematics Archive. "Cardioid." http : //www-groups . dcs . st-and. ac .uk/ -history/Curves /Cardioid. html. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995. Yates, R. C. "The Cardioid." Math. Teacher 52, 10-14, 1959. Yates, R. C. "Cardioid." A Handbook on Curves and Their Properties. Ann Arbor, Ml: J. W. Edwards, pp. 4-7, 1952. Cardioid Caustic The Catacaustic of a Cardioid for a Radiant Point at the Cusp is a Nephroid. The Catacaustic for Parallel rays crossing a Circle is a Cardioid. y^ ~-^ / V / \ / \ / \ \ /^~^\ \ \ \ \ f \ \ V \ I 1 / \ u ; / / / \ / \ / / N. / *"■' **" x = -a + |a cos 0(1 — cos#) y = |asin#(l — cos#). This is a mirror-image Cardioid with a = a/3. Cardioid Inverse Curve If the Cusp of the cardioid is taken as the Inversion Center, the cardioid inverts to a Parabola. Cardioid Involute x — 2a + 3a cos 9(1 — cos 0) y = 3a sin 0(1 — cos#). This is a mirror-image CARDIOID with a 1 = 3a. Cardioid Pedal Curve / / / / y - - NX V \ / / \ \ \ \ \ // - The Pedal Curve of the Cardioid where the Pedal Point is the Cusp is Cayley's Sextic. Cards Cards are a set of n rectangular pieces of cardboard with markings on one side and a uniform pattern on the other. The collection of all cards is called a "deck," and a normal deck of cards consists of 52 cards of four dif- ferent "suits." The suits are called clubs (Jt), diamonds (<0>), hearts (\?), and spades (♦). Spades and clubs are Carleman's Inequality colored black, while hearts and diamonds are colored red. The cards of each suit are numbered 1 through 13, where the special terms ace (1), jack (11), queen (12), and king (13) are used instead of numbers 1 and 11-13. The randomization of the order of cards in a deck is called Shuffling. Cards are used in many gambling games (such as POKER), and the investigation of the probabilities of various outcomes in card games was one of the original motivations for the development of mod- ern Probability theory. see also Bridge Card Game, Clock Solitaire, Coin, Coin Tossing, Dice, Poker, Shuffle Carleman's Inequality- Let {a,i}™ =1 be a Set of Positive numbers. Then the Geometric Mean and Arithmetic Mean satisfy n n ^J(aia 2 • • • a;) 1/j < - ^J a». Here, the constant e is the best possible, in the sense that counterexamples can be constructed for any stricter Inequality which uses a smaller constant. see also Arithmetic Mean, e, Geometric Mean References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1094, 1979. Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 249-250, 1988. Carlson-Levin Constant N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. Assume that / is a Nonnegative Real function on [0, oo) and that the two integrals Carlyle Circle 191 / Jo x P ~ [f(x)] p dx *'- 1+M [/(aO]' dx (1) (2) exist and are FINITE. If p = q — 2 and A = /x = 1, Carlson (1934) determined / f(x)dx< \M / (I [f(x)] 2 dx 1/4 x / x*[f{x)Ydx\ (3) 1/4 and showed that ^pK is the best constant (in the sense that counterexamples can be constructed for any stricter INEQUALITY which uses a smaller constant). For the general case / f(x)dx<cl x p - 1 - x [f(x)] p dx\ C 9 - 1+ "[/(x)]* dx and Levin (1948) showed that the best constant r(;)r(i) (4) (pa)*(qty (A + / «)r(4±i) where t = ppL + qX A pfi + q\ a = 1 — s — t and T(z) is the GAMMA FUNCTION. (5) (6) (7) (8) References Beckenbach, E. F.; and Bellman, R. Inequalities. New York: Springer- Verlag, 1983. Boas, R. P. Jr. Review of Levin, V. I. "Exact Constants in Inequalities of the Carlson Type." Math. Rev. 9, 415, 1948. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/crlslvn/crlslvn.htnil. Levin, V. L "Exact Constants in Inequalities of the Carlson Type." Doklady Akad. Nauk. SSSR (N. S.) 59, 635-638, 1948. English review in Boas (1948). Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, 1991. Carlson's Theorem If f(z) is regular and of the form <9(e fc '*') where k < tt, for K[z] > 0, and if f(z) = for z = 0, 1, . . . , then f(z) is identically zero. see also Generalized Hypergeometric Function References Bailey, W. N. "Carlson's Theorem." §5.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 36—40, 1935. Carlyle Circle n A = (1,0) B = {s,p) Y=(0,p+l) C=(0,p) H 2 S = (j, 0) Consider a Quadratic Equation x 2 -sx+p = where s and p denote signed lengths. The CIRCLE which has 192 Carmichael Condition Carmichael Number the points A = (0,1) and B — (s,p) as a DIAMETER is then called the Carlyle circle C S>P of the equation. The Center of C SjP is then at the Midpoint of AB, M = (s/2,(l +p)/2), which is also the Midpoint of S = (s, 0) and Y = (0, 1 + p). Call the points at which C SiP crosses the x-AxiS Hi = (2:1,0) and #2 = (#2,0) (with x\ > X2)> Then s = Xi -\- X2 p = X1X2 (# — x\)(x — X2) = x 2 — sx + p, so xi and X2 are the ROOTS of the quadratic equation. see also 257-gon, 65537-gon, Heptadecagon, Pen- tagon References De Temple, D. W. "Carlyle Circles and the Lemoine Simplic- ity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991. Eves, H. An Introduction to the History of Mathematics, 6th ed. Philadelphia, PA: Saunders, 1990. Leslie, J. Elements of Geometry and Plane Trigonome- try with an Appendix and Very Copious Notes and Il- lustrations, J^th ed., improved and exp. Edinburgh: W. & G. Tait, 1820. Carmichael Condition A number n satisfies the Carmichael condition IFF (p — l)\(n/p - 1) for all PRIME DIVISORS p of n. This is equivalent to the condition (p - l)\(n - 1) for all Prime Divisors pofn. see also Carmichael Number References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996. Carmichael's Conjecture CarmichaeFs conjecture asserts that there are an In- finite number of Carmichael Numbers. This was proven by Alford et al. (1994). see also CARMICHAEL NUMBER, CARMICHAEL'S TO- tient Function Conjecture References Alford, W. R.; Granville, A.; and Pomerance, C. "There Are Infinitely Many Carmichael Numbers." Ann. Math. 139, 703-722, 1994. Cipra, B. What's Happening in the Mathematical Sciences, Vol 1. Providence, RI: Amer. Math. Soc, 1993. Guy, R. K. "Carmichael's Conjecture." §B39 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 94, 1994. Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimesto25-10 9 ." Math. Comput. 35,1003-1026, 1980. Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer- Verlag, pp. 29-31, 1989. Schlafly, A. and Wagon, S. "Carmichael's Conjecture on the Euler Function is Valid Below lO 10 - 000 - 000 ." Math. Com- put. 63, 415-419, 1994. Carmichael Function A(n) is the LEAST COMMON MULTIPLE (LCM) of all the Factors of the Totient Function <j>(n), except that if 8|n, then 2 a ~ 2 is a FACTOR instead of 2 a ~ 1 . \{n) = < 0(n) for n = p a ,p = 2 and a < 2, or p > 3 \<t>{n) for n = 2 a and a > 3 LCM[X(jH ai )]i for n = YiiPi ai Some special values are for r > 3, and A(l) = 1 A(2) = 1 A(4) = 2 A(2 r ) - 2 r ~ 2 X(p r ) = 4>tf) for p an ODD PRIME and r > 1. The ORDER of a (mod n) is at most A(n) (Ribenboim 1989). The values of A(n) for the first few n are 1, 1, 2, 2, 4, 2, 6, 4, 10, 2, 12, . . . (Sloane's A011773). see also MODULO MULTIPLICATION GROUP References Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer- Verlag, p. 27, 1989. Riesel, H. "Carmichael's Function." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhauser, pp. 273-275, 1994. Sloane, N. J. A. Sequence A011773 in "An On-Line Version of the Encyclopedia of Integer Sequences." Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addison- Wesley, p. 226, 1991. Carmichael Number A Carmichael number is an Odd Composite Number n which satisfies Fermat's Little Theorem a n_1 -1 = (mod n) for every choice of a satisfying (a,n) = 1 (i.e., a and n are Relatively Prime) with 1 < a < n. A Car- michael number is therefore a PSEUDOPRIMES to any base. Carmichael numbers therefore cannot be found to be Composite using Fermat's Little Theorem. However, if (a,n) ^ 1, the congruence of Fermat's Lit- tle Theorem is sometimes Nonzero, thus identifying a Carmichael number n as COMPOSITES, Carmichael numbers are sometimes called ABSOLUTE PSEUDOPRIMES and also satisfy KORSELT'S CRITERION. R. D. Carmichael first noted the existence of such num- bers in 1910, computed 15 examples, and conjectured that there were infinitely many (a fact finally proved by Alford et al. 1994). Carmichael Number CarmichaeFs Totient Function Conjecture 193 The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ... (Sloane's A002997). Carmichael numbers have at least three PRIME FACTORS. For Carmichael numbers with exactly three PRIME FACTORS, once one of the PRIMES has been specified, there are only a finite number of Car- michael numbers which can be constructed. Numbers of the form (6fc + l)(12fc + l)(18fc + l) are Carmichael num- bers if each of the factors is Prime (Korselt 1899, Ore 1988, Guy 1994). This can be seen since for N = (6fc+l)(12fc+l)(18fc+l) 1296fc 3 +396/c 2 +36£;+l, N - 1 is a multiple of 36k and the LEAST COMMON Multiple of 6fc, 12fc, and 18k is 36fc, so a^" 1 = 1 modulo each of the PRIMES 6A; + 1, 12k + 1, and lSk + 1, hence a N ~ x = 1 modulo their product. The first few such Carmichael numbers correspond to k = 1, 6, 35, 45, 51, 55, 56, ... and are 1729, 294409, 56052361, 118901521, ... (Sloane's A046025). The largest known Carmichael number of this form was found by H. Dubner in 1996 and has 1025 digits. The smallest Carmichael numbers having 3, 4, ... fac- tors are 561 = 3 x 11 x 17, 41041 = 7 x 11 x 13 x 41, 825265, 321197185, ... (Sloane's A006931). In total, there are only 43 Carmichael numbers < 10 6 , 2163 < 2.5 x 10 10 , 105,212 < 10 15 , and 246,683 < 10 16 (Pinch 1993). Let C(n) denote the number of Carmichael num- bers less than n. Then, for sufficiently large n (n ~ 10 7 from numerical evidence), C(n) 2/7 (Alford et al. 1994). The Carmichael numbers have the following properties: 1. If a PRIME p divides the Carmichael number n, then n = 1 (mod p — 1) implies that n = p (mod p(p — 1)). 2. Every Carmichael number is SQUAREFREE. 3. An Odd Composite Squarefree number n is a Carmichael number Iff n divides the DENOMINATOR of the Bernoulli Number B n -\. see also CARMICHAEL CONDITION, PSEUDOPRIME References Alford, W. R.; Granville, A.; and Pomerance, C. "There are Infinitely Many Carmichael Numbers." Ann. Math. 139, 703-722, 1994. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 87, 1987. Guy, R. K. "Carmichael Numbers." §A13 in Unsolved Prob- lems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 30-32, 1994. Korselt, A. "Probleme chinois." L 'intermediate math. 6, 143-143, 1899. Ore, 0. Number Theory and Its History. New York: Dover, 1988. Pinch, R. G. E. "The Carmichael Numbers up to 10 15 ." Math. Comput. 55, 381-391, 1993. Pinch, R. G. E. ftp:// emu . pmms . cam .ac.uk/ pub / Carmichael/. Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimesto25'10 9 ." Math. Corn-put 35, 1003-1026, 1980. Riesel, H. Prime Numbers and Computer Methods for Fac- torization, 2nd ed. Basel: Birkhauser, pp. 89-90 and 94- 95, 1994. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p'. 116, 1993. Sloane, N. J. A. Sequences A002997/M5462 and A006931/ M5463 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Carmichael Sequence A Finite, Increasing Sequence of Integers {a ly . . . , a m } such that (en - l)|(ai •■ -ai-i) for i = 1, . . . , ?n, where m\n indicates that m DIVIDES n. A Carmichael sequence has exclusive EVEN or Odd ele- ments. There are infinitely many Carmichael sequences for every order. see also GlUGA SEQUENCE References Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen- sohn, R. "Giuga's Conjecture on Primality." ^4mer. Math. Monthly 103, 40-50, 1996. CarmichaePs Theorem If a and n are RELATIVELY PRIME so that the GREATEST Common Denominator GCD(a,n) = 1, then a ^ = 1 (mod n) , where A is the Carmichael Function. CarmichaePs Totient Function Conjecture It is thought that the Totient Valence Function AT (m) > 2 (i.e., the TOTIENT VALENCE FUNCTION never takes the value 1). This assertion is called Car- michael's totient function conjecture and is equivalent to the statement that there exists an m ^ n such that <t>{n) = <p(m) (Ribenboim 1996, pp. 39-40). Any counterexample to the conjecture must have more than 10,000 DIGITS (Conway and Guy 1996). Recently, the conjecture was reportedly proven by F. Saidak in November, 1997 with a proof short enough to fit on a postcard. see also Totient Function, Totient Valence Function References Conway, J. H, and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 155, 1996. Ribenboim, P. The New Book of Prime Number Records. New York: Springer- Verlag, 1996. 194 Carnot's Polygon Theorem Cartan Torsion Coefficient Carnot's Polygon Theorem If Pi, P2, . • ■ , are the VERTICES of a finite POLYGON with no "minimal sides" and the side PiPj meets a curve in the POINTS Piji and Pj-,2, then Ui^ P ^Ui P 2P23i--Ui P ^ P ^ = 1, where AB denotes the DISTANCE from POINT A to B. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 190, 1959. Carnot's Theorem Given any TRIANGLE A 1 A2A$ i the signed sum of PER- PENDICULAR distances from the C IRC UM CENTER O to the sides is OOi + OO2 + OO3 = R + r, where r is the INRADIUS and R is the ClRCUMRADIUS. The sign of the distance is chosen to be POSITIVE IFF the entire segment OOi lies outside the TRIANGLE. see also JAPANESE TRIANGULATION THEOREM References Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, pp. 256 and 262, 1972. Honsbergcr, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 25, 1985. Carotid-Kundalini Fractal Fractal Valley Gaussian Mtn. Oscillation Land 0.5 -1 , ,11:., 111 //>;'); V . Mmmm I '''■-jffi/i x m \ i m 0.5 A fractal-like structure is produced for x < by super- posing plots of Carotid-Kundalini Functions CK n of different orders n. The region — 1 < x < is called FRACTAL LAND by Pickover (1995), the central region the Gaussian Mountain Range, and the region x > Oscillation Land. The plot above shows n — 1 to 25. Gaps in FRACTAL LAND occur whenever cos(27rr/<?) for r = 0, 1, ..., [q/2\, where \z\ is the Ceiling Function and L^J is the Floor Function. References Pickover, C. A. "Are Infinite Carotid-Kundalini Functions Fractal?" Ch. 24 in Keys to Infinity. New York: W. H. Freeman, pp. 179-181, 1995. Carotid-Kundalini Function The Function given by CK n (x) = cos(nxcos _1 x), where n is an Integer and — 1 < x < 1. see also Carotid-Kundalini Fractal Carry l 1 1 5 8- H 249 - 407- -carries - addend 1 - addend 2 -sum The operating of shifting the leading DIGITS of an AD- DITION into the next column to the left when the Sum of that column exceeds a single DIGIT (i.e., 9 in base 10). see also ADDEND, ADDITION, BORROW Carrying Capacity see Logistic Growth Curve Cartan Matrix A Matrix used in the presentation of a Lie Algebra. References Jacobson, N. Lie Algebras. New York: Dover, p. 121, 1979. Cartan Relation The relationship Sq*(x ^ y) = Z j+k =iSq j (x) -- Sq k {y) encountered in the definition of the Steenrod Alge- bra. Cartan Subgroup A type of maximal Abelian SUBGROUP. References Knapp, A. W. "Group Representations and Harmonic Anal- ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. Cartan Torsion Coefficient The Antisymmetric parts of the Connection Coef- ficient r A u „. -i p xcos X = 2it- Q for p and q RELATIVELY PRIME INTEGERS. At such points #, the functions assume the \(q + l)/2] values Cartesian Coordinates Cartesian Ovals 195 Cartesian Coordinates 2-axis A The Gradient of the Divergence is y-axis Cartesian coordinates are rectilinear 2-D or 3-D coordi- nates (and therefore a special case of CURVILINEAR CO- ORDINATES) which are also called Rectangular Co- ordinates. The three axes of 3-D Cartesian coordi- nates, conventionally denoted the a>, y-, and z-Axes (a Notation due to Descartes) are chosen to be linear and mutually PERPENDICULAR. In 3-D, the coordinates x, y, and z may lie anywhere in the INTERVAL ( — 00,00). The Scale Factors of Cartesian coordinates are all unity, hi = l. The Line Element is given by ds — dx x + dy y + dz z, and the Volume Element by dV = dx dy dz. The Gradient has a particularly simple form, „J?_ ,d_ ^d_ dx dy dz ' as does the Laplacian dx 2 dy 2 dz 2 * (i) (2) (3) (4) The Laplacian is V 2 F = V-(VF) d 2 F d 2 F dx 2 dy 2 d 2 F dz 2 + y + z The Divergence is V-F - and the CURL is d 2 F x d 2 F 2 dx 2 + d 2 F v dx 2 d 2 F z dx 2 dy 2 d 2 F y dy 2 d 2 F x dz 2 d 2 F, + + d^F z dy 2 + dz 2 d 2 F z dz 2 dF x dx dF v . 8F X dy + dz (5) (6) V x F : x _d_ dx F x y a + (dF z \ dy y dx dy z d_ dz F z dFy dz dF Q x + ( dF x V dz V(V-u) a ( du_x_ 1 du y 1 <t x "I" dy ~r c du x 1 9uy_ du z \ dx ~T~ dy ^~ dz J dy o 1 du x 1 dv-y 1 *~ l dx ~*~ dy ^ r A. % dy _d_ dz du x du v du z ___ _j * _j dx dy dz (8) Laplace's Equation is separable in Cartesian coordi- nates. see also COORDINATES, HELMHOLTZ DIFFERENTIAL Equation— Cartesian Coordinates References Arfken, G. "Special Coordinate Systems— Rectangular Cartesian Coordinates." §2.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94- 95, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, p. 656, 1953. Cartesian Ovals A curve consisting of two ovals which was first studied by Descartes in 1637. It is the locus of a point P whose distances from two FOCI F\ and F2 in two-center BIPO- LAR Coordinates satisfy mr ± nr = k, (i) where m,n are Positive Integers, A; is a Positive real, and r and r are the distances from F\ and F2. If m = n, the oval becomes an an ELLIPSE. In CARTESIAN Coordinates, the Cartesian ovals can be written iy/(x - a) 2 + y 2 + ny/(x + a) 2 + 1 (2) / 2 , 2 . 2w 2 2\ / 2 . 2\ 7 2 (x -\- y + a ){m — n ) — 2ax{m + n ) — k = -2n^{x + a) 2 + y 2 , (3) [(m 2 - n 2 )(x 2 + y 2 + a 2 ) - 2ax(m 2 + n 2 )] 2 = 2(m 2 + n 2 )(n 2 + y 2 + a 2 ) - 4ax(m 2 - n 2 ) - A; 2 . (4) (5) (6) dF z \ „ Now define (7) ,22 — 771 — n _ 2 . 2 c = m +n , 196 Cartesian Product Cassini Ovals and set a = 1. Then [b(x 2 +y 2 )-2cx + bf +Abx + k 2 -2c = 2c(x 2 +y 2 ). (7) If c is the distance between Fi and F2, and the equation r 4- mr = a (8) is used instead, an alternate form is [(l-m 2 )(x 2 +y 2 )+2m 2 c'x+a' 2 -m 2 c 12 } 2 = 4a' 2 (x 2 +y 2 ). (9) The curves possess three Foci. If m — 1, one Cartesian oval is a central CONIC, while if m = a/c % then the curve is a LlMAgON and the inside oval touches the outside one. Cartesian ovals are ANALLAGMATIC CURVES. References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 155-157, 1972. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967. MacTutor History of Mathematics Archive. "Cartesian Oval." http : //www-groups . dcs . st -and . ac . uk/ -history/ Curves /Cart esian.html. Cartesian Product see Direct Product (Set) Cartesian Trident see Trident of Descartes Cartography The study of Map Projections and the making of ge- ographical maps. see also Map Projection Cascade A Z-Action or N- Action. A cascade and a single Map X — ¥ X are essentially the same, but the term "cascade" is preferred by many Russian authors. see also Action, Flow Casey's Theorem Four Circles are Tangent to a fifth Circle or a straight Line Iff £12^34 i £13^42 db £14^23 = 0, where Uj is a common TANGENT to CIRCLES i and j. see also PURSER'S THEOREM References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 121-127, 1929. Casimir Operator An Operator on a representation R of a LIE ALGEBRA. References Jacobson, N. Lie Algebras. New York: Dover, p. 78, 1979. Cassini Ellipses see Cassini Ovals Cassini's Identity For F n the nth FIBONACCI NUMBER, Fn~iF n +i — F n — (— l) n . see also Fibonacci Number References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, p. 12, 1996. Cassini Ovals The curves, also called CASSINI ELLIPSES, described by a point such that the product of its distances from two fixed points a distance 2a apart is a constant b . The shape of the curve depends on b/a. If a < 6, the curve is a single loop with an Oval (left figure above) or dog bone (second figure) shape. The case a = b produces a Lemniscate (third figure). If a > b, then the curve consists of two loops (right figure). The curve was first investigated by Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one FOCUS of the oval. Cassini ovals are Anallagmatic Curves. The Cassini ovals are defined in two-center Bipolar Coordinates by the equation T\T2 = b , (1) with the origin at a FOCUS. Even more incredible curves are produced by the locus of a point the product of whose distances from 3 or more fixed points is a con- stant. The Cassini ovals have the CARTESIAN equation [(x-a) 2 +y 2 ][(x + a) 2 +2/ 2 ] = 6 4 (2) or the equivalent form (x 4- y + a ) — 4a x = b (3) Cassini Ovals Cassini Surface 197 and the polar equation Cassini Projection 4 . 4 r 4- a 2aVcos(2(9) = & 4 . (4) Solving for r 2 using the QUADRATIC Equation gives 2 r = 2a 2 cos(2(9) + ^a 4 cos 2 (20) - 4(a 4 - b 4 ) = a 2 003(20) + V / a 4 cos 2 (2(9) + 6 4 -a 4 = a 2 cos(20) v/a 4 [cos 2 (20) - 1] + fe 4 = a 2 cos(20) + ^b 4 - a 4 sin 2 (20) cos(20) + J(-} -sin 2 (20) (5) If a < 6, the curve has Area A= L r i de = 2 (l) f r 2 c J-tv/4 a J +6^(- ), (6) where the integral has been done over half the curve and then multiplied by two and E(x) is the complete Elliptic Integral of the Second Kind. If a = 6, the curve becomes r 2 = a 2 |cos(20) + >/l-sin 2 0l = 2a 2 cos(2<9), (7) which is a Lemniscate having Area A = 2a 2 (8) (two loops of a curve y/2 the linear scale of the usual lemniscate r 2 — a 2 cos(2#), which has area A = a 2 /2 for each loop). If a > 6, the curve becomes two disjoint ovals with equations r = ±aJ cos(20) ± J (~) -sin 2 (20), (9) where £ [— 0o,9q] and A — 1 * "I t/o = f sin &' (10) see a/so Cassini Surface, Lemniscate, Mandelbrot Set, Oval References Gray, A. "Cassinian Ovals." §4.2 in Modern Differential Ge- ometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 63-65, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 153-155, 1972. Lee, X. "Cassinian Oval," http : // www . best . com / - xah / SpecialPlane Curves _dir/CassinianOval_dir/ cassinian Oval.html. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 187-188, 1967. MacTutor History of Mathematics Archive. "Cassinian Ovals." http: //www-groups .dcs .st-and.ac .uk/ -history /Curves/Cassinian.html. Yates, R. C. "Cassinian Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 8-11, 1952. iCTION. x — sin - B (i) y = tan - tan<£ (2) cos(A — Ao) where B = cos</>sin(A - Ao). The inverse FORMULAS are <t> = sin -1 (sin D cos x) -l ( tan x \ A = Ao + tan ( — 1 , V cos D J where D = y + <f> . (3) (4) (5) (6) References Snyder, J. P. Map Projections — A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 92-95, 1987. Cassini Surface The QUARTIC SURFACE obtained by replacing the con- stant c in the equation of the CASSINI OVALS {(x-a) 2 +y 2 ][(x + af + y 2 ] = c 2 by c = z 2 , obtaining [(x-a) 2 +y 2 }[(x + a) 2 +y 2 ] = z 4 . As can be seen by letting y = to obtain / 2 2\2 4 (x — a ) — z 2.2 2 x + z = a , (i) (2) (3) (4) 198 Castillon's Problem Catalan's Conjecture the intersection of the surface with the y — PLANE is a Circle of Radius a. References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 20, 1986. Fischer, G. (Ed.). Plate 51 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, p. 51, 1986. Castillon's Problem Inscribe a TRIANGLE in a CIRCLE such that the sides of the Triangle pass through three given Points A, B, and C> References Dorrie, H. "Castillon's Problem." §29 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 144-147, 1965. Casting Out Nines An elementary check of a Multiplication which makes use of the CONGRUENCE 10 n = 1 (mod 9) for n > 2. Prom this CONGRUENCE, a MULTIPLICATION ab — c must give a = > a,i = a* bi = b* C = 2~J Ci — c* , so ab = a*b* must be = c* (mod 9). Casting out nines is sometimes also called "the Hindu Check." References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, pp. 28-29, 1996. Cat Map see Arnold's Cat Map Catacaustic The curve which is the ENVELOPE of reflected rays. Curve Source Catacaustic cardioid cusp nephroid circle not on circumf. lima^on circle on circumf. cardioid circle point at oo nephroid cissoid of Diocles focus cardioid 1 arch of a cycloid rays _L axis 2 arches of a cycloid deltoid point at oo astroid In x rays || axis catenary logarithmic spiral origin equal logarithmic spiral parabola rays _L axis Tschirnhausen cubic quadrifolium center astroid Tschirnhausen cubic focus semicubical parabola see also CAUSTIC, DlACAUSTIC References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 60 and 207, 1972. Catalan's Conjecture 8 and 9 (2 3 and 3 2 ) are the only consecutive POWERS (excluding and 1), i.e., the only solution to Cata- lan's Diophantine PROBLEM. Solutions to this prob- lem (Catalan's Diophantine Problem) are equiva- lent to solving the simultaneous Diophantine Equa- tions X 2 - Y s = 1 X 3 -Y 2 = 1. This Conjecture has not yet been proved or refuted, although it has been shown to be decidable in a Fi- nite (but more than astronomical) number of steps. In particular, if n and n H- 1 are POWERS, then n < exp exp exp exp 730 (Guy 1994, p. 155), which follows from R. Tijdeman's proof that there can be only a FI- NITE number of exceptions should the CONJECTURE not hold. Hyyro and Makowski proved that there do not exist three consecutive POWERS (Ribenboim 1996), and it is also known that 8 and 9 are the only consecutive CUBIC and Square Numbers (in either order). see also Catalan's Diophantine Problem References Guy, R. K. "Difference of Two Power." §D9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 155-157, 1994. Ribenboim, P. Catalan's Conjecture. Boston, MA: Academic Press, 1994. Ribenboim, P. "Catalan's Conjecture." Amer. Math. Monthly 103, 529-538, 1996. Ribenboim, P. "Consecutive Powers." Expositiones Mathe- maticae 2, 193-221, 1984. Catalan's Constant Catalan's Constant 199 Catalan's Constant A constant which appears in estimates of combinatorial functions. It is usually denoted K, /3(2), or G. It is not known if K is IRRATIONAL. Numerically, K = 0.915965594177... (1) (Sloane's A006752). The CONTINUED FRACTION for K is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (Sloane's A014538). K can be given analytically by the following expressions, K = /3(2) (-l) fe _ J__jL 1 (2fc-fl) 2 ~ l 2 3 2 + 5 2 + " (2) (3) = 1 + 71 = 1 oo oo ^ (4n + l) 2 ~ 9 ~ ^ (4n + 3) 2 ^ I Jo (4 tan -1 xdx l In xdx (5) (6) where (3(z) is the Dirichlet Beta Function. In terms of the POLYGAMMA FUNCTION *i(as), *=£*iU)-£Mi) (7) = ^*i(A) + ^*i(A)-> 2 (8) = i* 1 (l)-i* 1 (|)-i^. (9) Applying CONVERGENCE IMPROVEMENT to (3) gives ^=^E( TO + 1 )^C(m + 2), (10) where ((z) is the Riemann Zeta Function and the identity 1 1__ _ ^ 3 m -l (l-3^) 2 (I-*) 2 ~ 2^ TO + 1 > 4 „ has been used (Flajolet and Vardi 1996). The Flajolet and Vardi algorithm also gives K - - 1 - n (i - ±-\ W-l V2 11 \ 2»V/3(2*) k^i^/i 2 ^ 1 ) (12) where f3(z) is the Dirichlet Beta Function. Glaisher (1913) gave *-i-E nC(2n + l) 16 n (13) (Vardi 1991, p. 159). W. Gosper used the related FOR- MULA K = where V2 *(2) - 1 n -il/(2 fe+1 ) *(m) -*(2 fc ) -1 K-m^rn _ l)4 m - 1 S m ' (14) (15) where B n is a Bernoulli Number and ip(x) is a Poly- gamma Function (Finch). The Catalan constant may also be defined by Jo K{k) dk, (16) where K(k) (not to be confused with Catalan's constant itself, denoted K) is a complete Elliptic Integral of the First Kind. K = 7rln2 8 +£ at 2L(i+l)/2Ji2> where {o<} = {1,1,1,0,-1,-1,-1,0} (17) (18) is given by the periodic sequence obtained by appending copies of {1, 1, 1, 0, — 1, — 1, — 1, 0} (in other words, en = a[(t-i) (mod 8)]+i for i > 8) and [x\ is the FLOOR Function (Nielsen 1909). see also Dirichlet Beta Function References Abramowitz, M, and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972. Adamchik, V. "32 Representations for Catalan's Con- stant." http://www.wolfram.com/-victor/articles/ catalan/catalan.html. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 551—552, 1985. Fee, G. J. "Computation of Catalan's Constant using Ra- in anuj an' s Formula." ISAAC '90. Proc. Internal. Symp. Symbolic Algebraic Cornp., Aug. 1990. Reading, MA: Addison-Wesley, 1990. Finch, S. "Favorite Mathematical Constants." http://www. mathsof t . com/ asolve/constant/catalan/catalan. html. Flajolet, P. and Vardi, I. "Zeta Function Expan- sions of Classical Constants." Unpublished manu- script. 1996. http://pauillac.inria.fr/algo/flajolet/ Publications/landau. ps. Glaisher, J. W. L. "Numerical Values of the Series 1 - 1/3" + 1/5" - 1/7" + 1/9" - &c for n = 2, 4, 6." Messenger Math. 42, 35-58, 1913. Gosper, R. W. "A Calculus of Series Rearrangements." In Algorithms and Complexity: New Directions and Recent Results (Ed. J. F. Traub). New York: Academic Press, 1976. Nielsen, N. Der Eulersche Dilogarithms. Leipzig, Germany: Halle, pp. 105 and 151, 1909. 200 Catalan's Diophantine Problem Catalan Number Plouffe, S. "PloufiVs Inverter: Table of Current Records for the Computation of Constants." http://lacim.uqam.ca/ pi/records .html. Sloane, N. J. A. Sequences A014538 and A006752/M4593 in "An On-Line Version of the Encyclopedia of Integer Se- quences." Srivastava, H. M. and Miller, E. A. "A Simple Reducible Case of Double Hypergeometric Series involving Catalan's Constant and Riemann's Zeta Function." Int. J. Math. Educ. Sci. Technol. 21, 375-377, 1990. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 159, 1991. Yang, S. "Some Properties of Catalan's Constant G." Int. J. Math. Educ. Sci. Technol 23, 549-556, 1992. Catalan's Diophantine Problem Find consecutive POWERS, i.e., solutions to b d -, a — c = 1, excluding and 1. CATALAN'S CONJECTURE is that the only solution is 3 2 - 2 3 = 1, so 8 and 9 (2 3 and 3 2 ) are the only consecutive POWERS (again excluding and 1). see also CATALAN'S CONJECTURE References Cassels, J. W. S. "On the Equation a x - 6^ = 1. II." Proc. Cambridge Phil Soc. 56, 97-103, 1960. Inkeri, K. "On Catalan's Problem." Acta Arith. 9, 285-290, 1964. Catalan Integrals Special cases of general FORMULAS due to Bessel. Jo(\A 2 -2/ 2 ) = - / e ycosd cos(z sin 0)d6, 77 Jo where J is a BESSEL FUNCTION OF THE FIRST KIND. Now, let z = 1 — z' and y = 1 + z' . Then Jo(2iv^) = - / e (1+z)cos6 cos[(l-z)sm0]d6. n Jo Catalan Number The Catalan numbers are an INTEGER SEQUENCE {C n } which appears in TREE enumeration problems of the type, "In how many ways can a regular n-gon be di- vided into n — 2 TRIANGLES if different orientations are counted separately?" (EULER'S POLYGON DIVI- SION Problem). The solution is the Catalan number Cn-2 (Dorrie 1965, Honsberger 1973), as graphically il- lustrated below (Dickau). The first few Catalan numbers are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (Sloane's A000108). The only Odd Catalan numbers are those of the form c 2 fc_i, and the last DIGIT is five for k = 9 to 15. The only PRIME Catalan numbers for n < 2 15 - 1 are C 2 = 2 and C 3 = 5. The Catalan numbers turn up in many other related types of problems. For instance, the Catalan number C n -i gives the number of BINARY BRACKETINGS of n letters (CATALAN'S Problem). The Catalan numbers also give the solution to the Ballot PROBLEM, the number of trivalent Planted Planar Trees (Dickau), ^J^O^^ the number of states possible in an n-FLEXAGON, the number of different diagonals possible in a FRIEZE PAT- TERN with n+1 rows, the number of ways of forming an n-fold exponential, the number of rooted planar bi- nary trees with n internal nodes, the number of rooted plane bushes with n EDGES, the number of extended Binary Trees with n internal nodes, the number of mountains which can be drawn with n upstrokes and n downstrokes, the number of noncrossing handshakes possible across a round table between n pairs of peo- ple (Conway and Guy 1996), and the number of SE- QUENCES with NONNEGATIVE PARTIAL SUMS which can be formed from n Is and n -Is (Bailey 1996, Buraldi 1992)! An explicit formula for C n is given by '2n\ _ _^_ (2n)! _ (2n)! n C n — 1 n+1 n + 1 n! 2 (n + l)!n!' (1) ■&mQm<s><^ where ( 2 ™) denotes a BINOMIAL COEFFICIENT and n\ is the usual Factorial. A Recurrence Relation for C n is obtained from Cn+i (2n + 2)! (n+l)(n!) 2 C n (n + 2)[(n+l)!] 2 (2n)! __ (2n + 2)(2n + l)(n + l) (n + 2)(n-f-l) 2 _ 2(2n + l)(n + l) 2 _ 2(2n+l) (n+l) 2 (n + 2) ~ n + 2 ' (2) Catalan Number Catalan Number 201 _ 2(2n + l) t-'n+l — T~^ ^n* n + 2 Other forms include C n — 2-6-10---(4n-2) (n + 1)! 2 n (2n~l)!! (n + 1)! (2n)! n!(n+l)f (3) (4) (5) (6) Segner's Recurrence Formula, given by Segner in 1758, gives the solution to Euler's POLYGON DIVISION Problem E n = E^En-x + EsE n -2 + . . . + E n -iE2. (7) With Ei = E 2 = 1, the above RECURRENCE RELATION gives the Catalan number C n _2 = Z2 n . The Generating Function for the Catalan numbers is given by 1 VI 4x = Y CnX " = i + x + 2x 2 + bx s + .... (8) n=0 The asymptotic form for the Catalan numbers is C k v^FP/2 (9) (Vardi 1991, Graham et al. 1994). A generalization of the Catalan numbers is defined by if pk \_ 1 (pk (10) for k > 1 (Klarner 1970, Hilton and Pederson 1991). The usual Catalan numbers Ck = 2<ih are a special case with j) —2. p dk gives the number of p-ary TREES with k source-nodes, the number of ways of associating k appli- cations of a given p-ary OPERATOR, the number of ways of dividing a convex POLYGON into k disjoint (p + 1)- gons with nonintersecting DIAGONALS, and the number of p-GoOD PATHS from (0, -1) to (]fe, (p-l)k-l) (Hilton and Pederson 1991). A further generalization is obtained as follows. Let p be an INTEGER > 1, let P k = (k,(p - l)k - 1) with k > 0, and q < p - 1. Then define p d q o = 1 and let p d q k be the number of p-GoOD PATHS from (1, q — 1) to Pk (Hilton and Pederson 1991). Formulas for p d q i include the generalized JONAH FORMULA k z = l - pi (11) and the explicit formula p^qk p-q (pk - q\ ok — qyk ~ 1 J A Recurrence Relation is given by pd q k - 2_^ p&p — T,i P^Q-^Tyj (12) (13) k + 1 where i,j, r > 1, k > 1, q < p — r, and i 4- j (Hilton and Pederson 1991). see also BALLOT PROBLEM, BINARY BRACKETING, Binary Tree, Catalan's Problem, Catalan's Triangle, Delannoy Number, Euler's Polygon Division Problem, Flexagon, Frieze Pattern, Motzkin Number, p-Good Path, Planted Planar Tree, Schroder Number, Super Catalan Number References Alter, R. "Some Remarks and Results on Catalan Numbers." Proc. 2nd Louisiana Conf. Comb., Graph Th., and Corn- put, 109-132, 1971. Alter, R. and Kubota, K. K. "Prime and Prime Power Divis- ibility of Catalan Numbers." J. Combin. Th. 15,243-256, 1973. Bailey, D. F. "Counting Arrangements of l's and — l's." Math. Mag. 69, 128-131, 1996. Brualdi, R. A. Introductory Combinatorics, 3rd ed. New York: Elsevier, 1997. Campbell, D. "The Computation of Catalan Numbers." Math. Mag. 57, 195-208, 1984, Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration of Certain Sets of Planted Trees." J. Combin. Th. Ser. B 18, 209-221, 1975. Chu, W. "A New Combinatorial Interpretation for General- ized Catalan Numbers." Disc. Math. 65, 91-94, 1987. Conway, J. H. and Guy, R. K. In The Book of Numbers. New- York: Springer- Verlag, pp. 96-106, 1996. Dershowitz, N. and Zaks, S. "Enumeration of Ordered Trees." Disc, Math. 31, 9-28, 1980. Dickau, R. M. "Catalan Numbers." http: //forum. swarthmore.edu/advanced/robertd/catalan.html. Dorrie, H. "Euler's Problem of Polygon Division." §7 in 100 Great Problems of Elementary Mathematics: Their His- tory and Solutions. New York: Dover, pp. 21-27, 1965. Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How Likely is a Function to be Convex?" Math. Mag. 61, 211— 219, 1988. Gardner, M. "Catalan Numbers." Ch. 20 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, 1988. Gardner, M. "Catalan Numbers: An Integer Sequence that Materializes in Unexpected Places." Sci. Amer. 234, 120- 125, June 1976. Gould, H. W. Bell & Catalan Numbers: Research Bibliogra- phy of Two Special Number Sequences, 6th ed. Morgan- town, WV: Math Monongliae, 1985. Graham, R. L.; Knuth, D. E.; and Patashnik, 0. Exercise 9.8 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- Wesley, 1994. Guy, R. K. "Dissecting a Polygon Into Triangles." Bull. Malayan Math. Soc. 5, 57-60, 1958. Hilton, P. and Pederson, J. "Catalan Numbers, Their Gen- eralization, and Their Uses." Math. Int. 13, 64-75, 1991. Honsberger, R. Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 130-134, 1973. 202 Catalan's Problem Catalan's Surface Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 146-150, 1985, Klarner, D. A. "Correspondences Between Plane Trees and Binary Sequences." J. Comb. Th. 9, 401-411, 1970. Rogers, D. G. "Pascal Triangles, Catalan Numbers and Re- newal Arrays." Disc. Math. 22, 301-310, 1978. Sands, A. D. "On Generalized Catalan Numbers." Disc. Math. 21, 218-221, 1978. Singmaster, D. "An Elementary Evaluation of the Catalan Numbers." Amer. Math. Monthly 85, 366-368, 1978. Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 18-20, 1973. Sloane, N. J. A. Sequences A000108/M1459 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Red- wood City, CA: Addis on- Wesley, pp. 187-188 and 198-199, 1991. Wells, D. G. The Penguin Dictionary of Curious and Inter- esting Numbers. London: Penguin, pp. 121-122, 1986. Catalan's Problem The problem of finding the number of different ways in which a PRODUCT of n different ordered FACTORS can be calculated by pairs (i.e., the number of BINARY Brack- ETINGS of n letters). For example, for the four FAC- TORS a, 6, c, and d } there are five possibilities: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d) y and a(b(cd)). The solution was given by Catalan in 1838 as c: = 2 ■ 6 • 10 • (4n - 6) r\ C' and is equal to the CATALAN NUMBER C n -i see also Binary Bracketing, Catalan's Diophan- tine Problem, Euler's Polygon Division Problem References Dorrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 23, 1965. Catalan Solid The Dual Polyhedra of the Archimedean Solids, given in the following table. Archimedean Solid Dual rhombicosidodecahedron small rhombicuboctahedron great rhombicuboctahedron great rhombicosidodecahedron truncated icosahedron snub dodecahedron (laevo) snub cube (laevo) cuboctahedron icosidodecahedron truncated octahedron truncated dodecahedron truncated cube truncated tetrahedron deltoidal hexecontahedron deltoidal icositetrahedron disdyakis dodecahedron disdyakis triacontahedron pentakis dodecahedron pentagonal hexecontahedron (dextro) pentagonal icositetrahedron (dextro) rhombic dodecahedron rhombic triacontahedron tetrakis hexahedron triakis icosahedron triakis octahedron triakis tetrahedron Here are the Archimedean DUALS (Holden 1971, Pearce 1978) displayed in alphabetical order (left to right, then continuing to the next row). Here are the Archimedean solids paired with the corre- sponding Catalan solids. O © Q © € © w see also Archimedean Solid, Dual Polyhedron, Semiregular Polyhedron References Catalan, E. "Memoire sur la Theorie des Polyedres." J. I'Ecole Polytechnique (Paris) 41, 1-71, 1865. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Catalan's Surface A Minimal Surface given by the parametric equations x(u, v) = u — sin u cosh v y(u, v) = 1 — cos u cosh v z(u,v) = 4sin(|w)sinh(|u) (i) (2) (3) Catalan's Triangle Categorical Variable 203 (Gray 1993), or x(r, <j>) = asin(2</>) — 2a<fi + \o>v 2 cos(2<fi) y(r, <j>) = — acos(2<p) — ~av 2 cos(2(p) z(r,(fi) = 2avsin0, where -r + (4) (5) (6) (?) (do Carmo 1986). References Catalan, E. "Memoir sur les surfaces dont les rayons de courburem en chaque point, sont egaux et des signes con- traires." C. R. Acad. Sci. Paris 41, 1019-1023, 1855. do Carmo, M. P. "Catalan's Surface" §3.5D in Mathemati- cal Models from the Collections of Universities and Muse- ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 45-46, 1986. Fischer, G. (Ed.). Plates 94-95 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, pp. 90-91, 1986. Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, pp, 448-449, 1993. Catalan's Triangle A triangle of numbers with entries given by (n + m)\{n — m + 1) Cnrn= m!(n+l)! for < m < n, where each element is equal to the one above plus the one to the left. Furthermore, the sum of each row is equal to the last element of the next row and also equal to the CATALAN NUMBER C n . 5 14 14 14 28 42 42 20 48 90 132 132 (Sloane's A009766). see also Bell Triangle, Clark's Triangle, Eu- ler's Triangle, Leibniz Harmonic Triangle, Num- ber Triangle, Pascal's Triangle, Prime Trian- gle, Seidel-Entringer-Arnold Triangle References Sloane, N. J. A. Sequence A009766 in "An On-Line Version of the Encyclopedia of Integer Sequences." Catalan's Trisectrix see TSCHIRNHAUSEN CUBIC Catastrophe see Butterfly Catastrophe, Catastrophe The- ory, Cusp Catastrophe, Elliptic Umbilic Catas- trophe, Fold Catastrophe, Hyperbolic Umbilic Catastrophe, Parabolic Umbilic Catastrophe, Swallowtail Catastrophe Catastrophe Theory Catastrophe theory studies how the qualitative nature of equation solutions depends on the parameters that appear in the equations. Subspecializations include bi- furcation theory, nonequilibrium thermodynamics, sin- gularity theory, synergetics, and topological dynamics. For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" oc- cur for four or fewer variables: (1) FOLD CATASTROPHE, (2) Cusp Catastrophe, (3) Swallowtail Catastro- phe, (4) Butterfly Catastrophe, (5) Elliptic Um- bilic Catastrophe, (6) Hyperbolic Umbilic Catas- trophe, (7) Parabolic Umbilic Catastrophe. More specifically, for any system with fewer than five control factors and fewer than three behavior axes, these are the only seven catastrophes possible. The following tables gives the possible catastrophes as a function of control factors and behavior axes (Goetz). Control Factors 1 Behavior Axis 2 Behavior Axes fold cusp swallowtail butterfly- hyperbolic umbilic, elliptic umbilic parabolic umbilic References Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin: Springer- Verlag, 1992. Gilmore, R. Catastrophe Theory for Scientists and Engi- neers. New York: Dover, 1993. Goetz, P. "Phil's Good Enough Complexity Dictionary." http ; //www . cs .but f alo . edu/~goetz/dict .html. Saunders, P. T. An Introduction to Catastrophe Theory. Cambridge, England: Cambridge University Press, 1980. Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, p. 211, 1987. Thorn, R. Structural Stability and Morphogenesis: An Out- line of a General Theory of Models. Reading, MA: Read- ing, MA: Addison- Wesley, 1993. Thompson, J. M. T. Instabilities and Catastrophes in Science and Engineering. New York: Wiley, 1982. Woodcock, A. E. R. and Davis, M. Catastrophe Theory. New York: E. P. Dutton, 1978. Zeeman, E. C. Catastrophe Theory — Selected Papers 1972- 1977. Reading, MA: Addis on- Wesley, 1977. Categorical Game A Game in which no draw is possible. Categorical Variable A variable which belongs to exactly one of a finite num- ber of Categories. 204 Category Catenary Category A category consists of two things: an OBJECT and a MORPHISM (sometimes called an "arrow"). An OB- JECT is some mathematical structure (e.g., a GROUP, Vector Space, or Differentiable Manifold) and a Morphism is a Map between two Objects. The Mor- PHISMS are then required to satisfy some fairly natural conditions; for instance, the IDENTITY MAP between any object and itself is always a Morphism, and the composition of two MORPHISMS (if defined) is always a Morphism. One usually requires the MORPHISMS to preserve the mathematical structure of the objects. So if the objects are all groups, a good choice for a MORPHISM would be a group HOMOMORPHISM. Similarly, for vector spaces, one would choose linear maps, and for differentiable manifolds, one would choose differentiable maps. In the category of TOPOLOGICAL SPACES, homomor- phisms are usually continuous maps between topologi- cal spaces. However, there are also other category struc- tures having TOPOLOGICAL SPACES as objects, but they are not nearly as important as the "standard" category of Topological Spaces and continuous maps. see also Abelian Category, Allegory, Eilenberg- Steenrod Axioms, Groupoid, Holonomy, Logos, monodromy, topos References Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster- dam, Netherlands: North-Holland, 1990. Category Theory The branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, sets, topological spaces, etc.) of the same type, subject to the constraint that the collections contain the identity mapping and are closed with respect to compo- sitions of mappings. The objects studied in category theory are called CATEGORIES. see also CATEGORY Catenary The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform grav- itational force. The word catenary is derived from the Latin word for "chain." In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging un- der gravity would be a PARABOLA (MacTutor Archive). The curve is also called the ALYSOID and CHAINETTE. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli. Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690 (MacTutor Archive). If you roll a PARABOLA along a straight line, its FOCUS traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum SURFACE Area (the Catenoid) for the given bounding CIRCLE. The Cartesian equation for the catenary is given by y =l a (e x/a + e- K/a ) = acoshg), (1) and the Cesaro Equation is {s 2 +a 2 )K=-a. (2) The catenary gives the shape of the road over which a regular polygonal "wheel" can travel smoothly. For a regular n-gon, the corresponding catenary is where y = -Acosh I — j , A = R cos (3) (4) The Arc Length, Curvature, and Tangential An- gle are s = asinh ( — ) , (5) n=--sedi 2 (-) y (6) a \a/ <f>= -2 tan" 1 [tanh (^-)1 * (?) The slope is proportional to the Arc Length as mea- sured from the center of symmetry. see also Calculus of Variations, Catenoid, Linde- lof's Theorem, Surface of Revolution References Geometry Center. "The Catenary." http://www.geom.umn. edu/zoo/diffgeom/surf space/catenoid/catenary.html. Gray, A. "The E volute of a Tract rix is a Catenary." §5.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 80-81, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 199-200, 1972. Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A Book of Curves. Cambridge, England: Cambridge Univer- sity Press, pp. 118-124, 1967, MacTutor History of Mathematics Archive. "Catenary." http : //www-groups . dcs . st-and . ac . uk/ -history/Curves /Catenary .html. Pappas, T. "The Catenary & the Parabolic Curves." The Joy of Mathematics. San Carlos, CA: Wide World Publ./ Tetra, p. 34, 1989. Yates, R. C. "Catenary." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 12-14, 1952. Catenary Evolute Catenary Evolute x = a[x — \ sinh(2t)] y = 2a cosh t. Catenary Involute \ \ \ / y / The parametric equation for a Catenary is dx dt dr dt 1 sinh 2 ayl + sinh 2 t = acoshi and dr rpi dt i dt | secht tanhi (1) (2) (3) (4) ds 2 = \dr 2 \ = a 2 (I + sinh 2 t) dt 2 = a 2 cosh 2 <ft 2 (5) dt a cosh i. Therefore, -•/ cosh tdt = a sinh £ and the equation of the INVOLUTE is x = a(t — tanht) y — asechi. This curve is called a TRACTRIX. (6) (7) (8) (9) Catenoid 205 Catenary Radial Curve \ / \ / / ^ "^^v^ S^^***'^ ^^^_ ^^^^ The Kampyle of Eudoxus. Catenoid A Catenary of Revolution. The catenoid and Plane are the only SURFACES OF Revolution which are also Minimal Surfaces. The catenoid can be given by the parametric equations x = ccosh cosu y = c cosh ( - J sin u (i) (2) (3) where u G [0, 2w). The differentials are dx — sinh ( - j cos u dv - cosh ( - J sin u du (4) dy = sinh I - J sin u dv -f cosh [ - j cos u du (5) dz = du, (6) so the Line Element is ds 2 = dx 2 + dy 2 + dz 2 = [sinh 2 Q) + l] dv 2 + cosh 2 Q) du * = cosh 2 f^\ dv 2 + cosh 2 (-) du 2 . (7) The Principal Curvatures are Kl — — sech 2 f - j K2 — - sech 2 ( - ) • The Mean Curvature of the catenoid is (8) (9) (10) 206 Caterpillar Graph Cauchy Distribution and the GAUSSIAN CURVATURE is (i) (ii) The HELICOID can be continuously deformed into a catenoid with c = 1 by the transformation x(u, v) = cos a sinh v sin u + sin a cosh v cos u (12) y(u, v) = — cos a sinh v cos u -f sin a cosh f sin u (13) z(?z, u) = u cos a + v sin a, (14) where a = corresponds to a HELICOID and a = n/2 to a catenoid. see also CATENARY, COSTA MINIMAL SURFACE, HELI- COID, Minimal Surface, Surface of Revolution References do Carmo, M. P. "The Catenoid." §3.5A in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986. Fischer, G. (Ed.). Plate 90 in Mathematische Modelle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, p. 86, 1986. Geometry Center. "The Catenoid." http://www.geom.umn, edu/zoo/diffgeom/surf space/catenoid/. Gray, A. "The Catenoid." §18.4 Modern Differential Geom- etry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 367-369, 1993. Meusnier, J. B. "Memoire sur la courbure des surfaces." Mem. des savans etrangers 10 (lu 1776), 477-510, 1785. Caterpillar Graph A TREE with every NODE on a central stalk or only one EDGE away from the stalk. References Gardner, M. Wheels, Life, and other Mathematical Amuse- ments. New York: W. H. Freeman, p. 160, 1983. Cattle Problem of Archimedes see Archimedes' Cattle Problem Cauchy Binomial Theorem V^ y m q m(m+l)/2 ( tl m=0 ^ J[(l + yq k ), where ( n ) is a Gaussian Coefficient. \m/ q see also g-BlNOMIAL THEOREM Cauchy Boundary Conditions Boundary Conditions of a Partial Differential Equation which are a weighted Average of Dirich- let Boundary Conditions (which specify the value of the function on a surface) and Neumann Boundary CONDITIONS (which specify the normal derivative of the function on a surface). see also Boundary Conditions, Cauchy Prob- lem, Dirichlet Boundary Conditions, Neumann Boundary Conditions References Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 678-679, 1953. Cauchy's Cosine Integral Formula /.tt/2 / ' J-rr/2 a + u-2 ni0(v.-v+2£) dO 7rV(fl + V ~ 1) 2*+"- 2 r( M + 0rV-0' where F(z) is the GAMMA Function. Cauchy Criterion A Necessary and Sufficient condition for a Se- quence Si to CONVERGE. The Cauchy criterion is sat- isfied when, for all e > 0, there is a fixed number N such that \Sj - Si\ < e for all i,j > N. Cauchy Distribution The Cauchy distribution, also called the Lorentzian Distribution, describes resonance behavior. It also de- scribes the distribution of horizontal distances at which a Line Segment tilted at a random Angle cuts the x-AxiS. Let 6 represent the ANGLE that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then tan# : b 6 = tan~ ■(?) dx bdx 1 + fJ 6 b 2 -rx 2 ' so the distribution of ANGLE is given by <W_ _ 1 bdx 7T 7T b 2 + X 2 ' (i) (2) (3) (4) Cauchy Distribution This is normalized over all angles, since /7T/2 ■tt/2 d9 = 1 (5) and f J — c i feds _ i [-,/nr ■K b 2 +X 2 7T L VX/J _oo = i[i w -(-i ff )] = l. (6) The general Cauchy distribution and its cumulative dis- tribution can be written as P(x) 2 X 7r(x- M ) 2 + (|r)2 .(*)=I + i tan -l(^) (7) (8) where T is the FULL WIDTH AT HALF MAXIMUM (r = 26 in the above example) and /x is the MEAN (/x — in the above example). The Characteristic Function is <m dx ~ 7T / 1 t/ — oo _ e -i M t-r|ti/2^ 1 + x 2 cos(Tta/2) + (r^/2) 2 dz The Moments are given by 2 \i2 = cr = oo for ji = M3 .oo for fi / /44 = oo, (9) (10) (11) (12) and the STANDARD DEVIATION, SKEWNESS, and KUR- TOSIS by _ f for fj, = 71 ~ I oo for /x # 72 = oo. (13) (14) (15) If X and Y are variates with a NORMAL DISTRIBUTION, then Z = X/Y has a Cauchy distribution with MEAN fi — and full width (16) Cauchy Inequality 207 see a/so Gaussian Distribution, Normal Distribu- tion References Spiegel, M. R, Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 114-115, 1992. Cauchy Equation see Euler Equation Cauchy's Formula The Geometric Mean is smaller than the Arith- metic Mean, 1/JV n~) <%=■ Cauchy Functional Equation The fifth of HlLBERT'S PROBLEMS is a generalization of this equation. Cauchy-Hadamard Theorem The Radius of Convergence of the Taylor Series ao + cl\z + aiz + . . . is 1 r = lim (Kl) 1 /" n— too see also RADIUS OF CONVERGENCE, TAYLOR SERIES Cauchy Inequality A special case of the HOLDER SUM INEQUALITY with y ^flfc&fc E- 2 E»* 2 • w Ok I < ^^ . k=l / \ k-1 / \ k=l where equality holds for ak = cbk- In 2-D, it becomes (2) It can be proven by writing (a 2 +6 2 )(c 2 + a 2 ) > {ac + bdf. Y^iatx + bi) 2 = f> 2 (x+ ^-) 2 = 0. (3) i=l i=l If bi/di is a constant c, then x = — c. If it is not a constant, then all terms cannot simultaneously vanish for REAL x, so the solution is COMPLEX and can be found using the QUADRATIC EQUATION 2j2a i b i ±^4&a i b i ) -4^a, 2 ^6 i 2 2J>^ • (4) 208 Cauchy Integral Formula In order for this to be COMPLEX, it must be true that $>* <£«.'£« (5) with equality when hi /at is a constant. The VECTOR derivation is much simpler, (a-b) 2 = aV cos 2 6 < ab 2 , yhere = E^ 2 2 _ V^ 2 a = a • a — x (6) (7) and similarly for b. see also Chebyshev Inequality, Holder Sum In- equality References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Cauchy Integral Formula r Yo Y r Given a Contour Integral of the form / /(*) dz Z — Zo (1) define a path 70 as an infinitesimal CIRCLE around the point zo (the dot in the above illustration). Define the path 7 r as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around zq. The total path is then 7 = 7o + It (2) tm±=tm*+tixr=L. (3 ) L z ~ z ° L a z ~ z o L r ~ - Z - Zq Prom the Cauchy Integral Theorem, the Contour Integral along any path not enclosing a Pole is 0. Therefore, the first term in the above equation is since 70 does not enclose the Pole, and we are left with r Hz)dz = r f_(z)dz Cauchy Integral Formula Now, let z = z + re iB , so dz = ire w d9. Then f fWdz = f A z ~ Zo A, -I f{Zo + r / ) ire ig d0 re™ f{zo + re ie )id9. (5) But we are free to allow the radius r to shrink to 0, so f Hz)dz = lim f f f ZQ + re ™\ id0 = f f( ZQ )idO / Z - ZQ r->Q / / = if(zo) [ dd = 2<Kif(z ), (6) J It and /(*>) 2 ™L f(z) dz z — Zq' (7) If multiple loops are made around the POLE, then equa- tion (7) becomes t/ 7 )dz (8) where 71(7, z ) is the WINDING NUMBER. A similar formula holds for the derivatives of f(z), f(zo) = i im n«+h)-m h^t-0 h = ]im J-([ f^ dz - f M*z\ /i^o 2izih \Jz — zo-~h J z — zo I _ y 1 f f(z)[(z - z ) - (z - zo - h)} dz h^o 2nih j lim h im — — - / ■-+0 27Vih I 2 ™ 7 7 ( z - > (z - zo - h)(z - zo) hf(z) dz (z — zo — h)(z — zq) Iterating again, ™-ht£ z) dz zo) 3 (9) (10) Continuing the process and adding the WINDING Num- ber n, see also Morera's Theorem References Arfken, G. "Cauchy's Integral Formula." §6.4 in Mathemati- cal Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 371-376, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 367-372, 1953. Cauchy Integral Test Cauchy Ratio Test 209 Cauchy Integral Test see Integral Test Cauchy Integral Theorem If / is continuous and finite on a simply connected region R and has only finitely many points of nondifferentia- bility in i£, then £ f(z)dz = (1) for any closed CONTOUR 7 completely contained in R. Writing z as z = x + iy (2) and f(z) as f(z)=u + iv (3) then gives (p f(z) dz — \ (u + iv)(dx + idy) = / udx -vdy + i / vdx + udy. (4) Prom Green's Theorem, J f(x J y)dx-g(x J y)dy=- fj (f| + fj) <**<fo / f(x,y)dx+g{x,y)dy^ // ( so (4) becomes <9x % (5) dxdy (6) -h//(£-£')«M* (7) But the Cauchy-Riemann Equations require that du _ <9v dx dy du dv dy dx ' (8) (9) £ f(z)dz = 0, Q. E. D. For a Multiply Connected region, f f(z)dz= f f(z)dz. (10) (11) see also Cauchy Integral Theorem, Morera's Theorem, Residue Theorem (Complex Analysis) References Arfken, G. "Cauchy's Integral Theorem." §6.3 in Mathemati- cal Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 365-371, 1985. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part I. New York: McGraw-Hill, pp. 363-367, 1953. Cauchy-Kovalevskaya Theorem The theorem which proves the existence and uniqueness of solutions to the Cauchy Problem. see also Cauchy Problem Cauchy-Lagrange Identity (ax 2 + a 2 2 + ■ • • + an 2 )(&i 2 + b 2 2 + . . . + b n 2 ) = (aib 2 - a 2 h) 2 + (ai& 3 - a3&i) 2 + • • • +(a n -i&n - a n &n-i) • From this identity, the n-D Cauchy Inequality fol- lows. Cauchy-Maclaurin Theorem see Maclaurin-Cauchy Theorem Cauchy Mean Theorem For numbers > 0, the Geometric Mean < the Arith- metic Mean. Cauchy Principal Value fix) dx = lim -00 / f(x)dx J-R I J a PV I f{x)dx = lim I f{x)dx+ / f(x)dx J a J c+e where e > and a < c <b. References Arfken, G. Mathematical Methods for Physicists f 3rd ed. Or- lando, FL: Academic Press, pp. 401-403, 1985. Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 158, 1991. Cauchy Problem Tf f(x,y) is an ANALYTIC FUNCTION in a NEIGHBOR- HOOD of the point (xo,yo) (i.e., it can be expanded in a series of Nonnegative Integer Powers of (x - x ) and (y — yo)), find a solution y(x) of the DIFFERENTIAL Equation dy dx /(*), with initial conditions y = yo and x = xq. The existence and uniqueness of the solution were proven by Cauchy and Kovalevskaya in the Cauchy-Kovalevskaya The- orem. The Cauchy problem amounts to determining the shape of the boundary and type of equation which yield unique and reasonable solutions for the CAUCHY Boundary Conditions. see also Cauchy Boundary Conditions Cauchy Ratio Test see Ratio Test 210 Cauchy Remainder Form Cauchy Root Test Cauchy Remainder Form The remainder of n terms of a TAYLOR Series is given by (x-c) n_1 (a;-a) Rn — where a < c < x. (n-l)! r'(c), Cauchy- Riemann Equations Let f(x,y) = u(x,y) + iv(x,y) y where z = x + iy, (1) (2) These are known as the Cauchy- Riemann equations. They lead to the condition d 2 u d 2 v dxdy dxdy (14) The Cauchy-Riemann equations may be concisely writ- ten as (du .dv\ . ( du .dv\ \dx dx) \dy dy J df _ df df _ (du ( . dv \ t . ( du t . dv )x dy du dv dx) \dy . . du dv\ + * -^- + -^- =0. dz* dx dy \dx dx) \dy dy du dv dx dy J ' " \dy dx (15) dz = dx -\- i dy. (3) The total derivative of / with respect to z may then be computed as follows. (4) (5) x = z - ty, dy __ 1 dz dx dz and In terms of u and t>, (8) becomes df __ / du .dv\ . I du .dv \dx dx) \dy dy dz \dx dx j (6) (7) V = dldx + dldy = dl_ i dl dz dx dz dy dz dx dy' (du ,dv\ ( .du dv\ ,„, = U + ^) + (-^ + ^J- (9) Along the real, or as- Axis, df /dy = 0, so df _ du .dv . . dz dx dx Along the imaginary, or y-axis, df /dx = 0, so df _ .du dv . . dz dy dy * If / is Complex Differentiable, then the value of the derivative must be the same for a given dz, regardless of its orientation. Therefore, (10) must equal (11), which requires that and dv du dx dy' (13) In Polar Coordinates, f(re ie ) = R(r,0)e i@(r ' e \ so the Cauchy-Riemann equations become dR dr IdR RdQ r d6 — = -*$©. r dd dr (16) (17) (18) If u and v satisfy the Cauchy-Riemann equations, they also satisfy Laplace's Equation in 2-D, since d^u d?u dx 2 dy 2 d_ (dv dx \dy )+ £(-£)- " 9 > d 2 v d 2 v _ d ( du\ d (du\_ { . dx 2 dy 2 dx \ dy J dy \dx) By picking an arbitrary f(z), solutions can be found which automatically satisfy the Cauchy-Riemann equa- tions and Laplace's Equation. This fact is used to find so-called Conformal Solutions to physical prob- lems involving scalar potentials such as fluid flow and electrostatics. see also Cauchy Integral Theorem, Conformal Solution, Monogenic Function, Polygenic Func- tion References Abramowitz, M . and Stegun, C . A . (Eds . ) . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972. Arfken, G. "Cauchy-Riemann Conditions." §6.2 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 3560-365, 1985. Cauchy's Rigidity Theorem see Rigidity Theorem Cauchy Root Test see Root Test Cauchy-Schwarz Integral Inequality Cayley Cubic 211 Cauchy-Schwarz Integral Inequality Let f(x) and g(x) by any two Real integrable functions of [a, 6], then '/"■ x)g(x) dx < nb "I r pb I f 2 (x)dx / g 2 (x)dx yd J Lv a with equality IFF f(x) = kg(x) with k real. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1099, 1993. Cauchy-Schwarz Sum Inequality |a-b|<|ailb|. E , fe = l akbk Equality holds IFF the sequences ai, a2, ... and &i, 62, . . . are proportional. see also Fibonacci Identity References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979. Cauchy Sequence A Sequence ai, 02, . . . such that the Metric d(a m , a n ) satisfies lim d(a m , a n ) = 0. min(m,n)— ^00 Cauchy sequences in the rationals do not necessarily Converge, but they do Converge in the Reals. Real Numbers can be defined using either Dedekind Cuts or Cauchy sequences. see also Dedekind Cut Cauchy Test see Ratio Test Caustic The curve which is the ENVELOPE of reflected (CAT- ACAUSTIC) or refracted (DIACAUSTIC) rays of a given curve for a light source at a given point (known as the Radiant Point). The caustic is the Evolute of the Orthotomic. References Lawrence, J. D. A Catalog of Special Plane Curves. New York; Dover, p. 60, 1972. Lee, X. "Caustics." http://www.best.com/-xah/Special PlaneCurves_dir/Caustics-dir/caustics.html. Lockwood, E. H. "Caustic Curves." Ch. 24 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 182-185, 1967. Yates, R. C. "Caustics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 15-20, 1952. Cavalieri's Principle 1. If the lengths of every one-dimensional slice are equal for two regions, then the regions have equal Areas. 2. If the AREAS of every two-dimensional slice (CROSS- Section) are equal for two SOLIDS, then the SOLIDS have equal Volumes. see also Cross-Section, Pappus's Centroid Theo- rem References Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 126 and 132, 1987. Cayley Algebra The only Nonassociative Division Algebra with REAL SCALARS. There is an 8-square identity corre- sponding to this algebra. The elements of a Cayley al- gebra are called CAYLEY NUMBERS or OCTONIONS. References Kurosh, A. G. General Algebra. New York: Chelsea, pp. 226- 28, 1963. Cayley-Bacharach Theorem Let Xi, X 2 C P 2 be CUBIC plane curves meeting in nine points pi, . . . , pq. If X C P 2 is any CUBIC containing Pi, ■ - ■ , Ps, then X contains pg as well. It is related to GORENSTEIN RINGS, and is a generalization of PAPPUS'S Hexagon Theorem and Pascal's Theorem, References Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 295-324, 1996. Cayley Cubic * **4 A Cubic Ruled Surface (Fischer 1986) in which the director line meets the director CONIC SECTION. Cay- ley's surface is the unique cubic surface having four OR- DINARY Double Points (Hunt), the maximum possible for Cubic Surface (EndraB). The Cayley cubic is in- variant under the TETRAHEDRAL GROUP and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (En- draB). If the Ordinary Double Points in projective 3-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is 1 1 1 1 — + — + — + — =0 Xq X\ X2 X3 212 Cay ley Cubic Cayley -Hamilton Theorem (Hunt). Denning "affine" coordinates with plane at in- finity v — Xq 4- x\ + X2 + 2^3 and Xq v v X 2 then gives the equation -b(x 2 y+x 2 z+y 2 x+y 2 z+z 2 y+z 2 x)+2(xy+xz+yz) = plotted in the left figure above (Hunt). The slightly different form 4(x 3 + y + z 3 + w ) - (x + y'+ z + • is given by Endrafi which, when rewritten in Tetrahe- dral Coordinates, becomes x + y — xz + yz-\-z — 1 = 0, plotted in the right figure above. The Hessian of the Cayley cubic is given by = Z 2 (xia:2 + X 1 X 3 + X2Xz) + X l (x X2 + X0X3 + Z2Z3) +xl(x Xi + XqX 3 + X1X3) + xI(xqX! + X X 2 + X1X2). in homogeneous coordinates xq, #1, x 2 , and X3. Taking the plane at infinity as v = 5(:ro + x\ + #2 + 2xz)j2 and setting a;, y, and 2 as above gives the equation 25[x 3 (y+z)+y 3 (x+z)+z 3 {x+y)]+b0(x 2 y 2 +x 2 z 2 +y 2 z 2 ) — 125(x 2 yz + y xz-\-z xy)-{-60xyz — 4(xy-{-xz-\-yz) = 0, plotted above (Hunt). The Hessian of the Cayley cubic has 14 ORDINARY Double Points, four more than a the general Hessian of a smooth CUBIC SURFACE (Hunt). References Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV- Mitteilungen 4, 17-20, Apr. 1995. Endrafi, S. "The Cayley Cubic." http://www.mathematik. uni-mainz . de/AlgebraischeGeometrie/docs/ Ecayley.shtml. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 14, 1986. Fischer, G. (Ed.). Plate 33 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, p. 33, 1986. Hunt, B. "Algebraic Surfaces." http://www.mathematik. uni-kl . de/-wwwagag/Galerie . html. Hunt, B. The Geometry of Some Special Arithmetic Quo- tients. New York: Springer- Verlag, pp. 115-122, 1996. Nordstrand, T. "The Cayley Cubic." http://www.uib.no/ people/nfytn/cleytxt.htm. Cayley Graph The representation of a GROUP as a network of directed segments, where the vertices correspond to elements and the segments to multiplication by group generators and their inverses. see also Cayley Tree References Grossman, I. and Magnus, W. Groups and Their Graphs. New York: Random House, p. 45, 1964. Cayley's Group Theorem Every Ftntte GROUP of order n can be represented as a Permutation Group on n letters, as first proved by Cayley in 1878 (Rotman 1995). see also Finite Group, Permutation Group References Rotman, J, J. An Introduction to the Theory of Groups, J^th ed. New York: Springer- Verlag, p. 52, 1995. Cayley-Hamilton Theorem Given a>\\ ~ X ai2 aim 0,21 &22 — X ft2m dml dm2 a>mrn X — X ~T~ Cjn — \X 771—1 , • + c , (1) then A m + c m - 1 A m - 1 + ... + c l = 0, (2) where I is the Identity Matrix. Cayley verified this identity for m = 2 and 3 and postulated that it was true for all m. For m = 2, direct verification gives a — x b c d — x = (a — x)(d — x) — be — x 2 — (a + d)x + {ad — be) = x 2 + c\x + C2 (3) Cayley's Hypergeometric Function Theorem Cayley-Klein Parameters 213 A = A 2 = -{a + d)A = (ad — be) I = a b c d a b a b c d c d a 2 + be ab + bd ac + cd be + d 2 —a — ad —ab — bd' —ac — dc —ad — d 2 ad — be ad — be ) \-(ad-bc)\ = "o o" (4) (5) (6) (7) (8) The Cayley-Hamilton theorem states that a n x n MA- TRIX A is annihilated by its Characteristic Poly- nomial det(xl — A), which is monic of degree n. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed, San Diego, CA: Academic Press, p. 1117, 1979. Segercrantz, J. "Improving the Cayley-Hamilton Equation for Low-Rank Transformations." Amer. Math. Monthly 99, 42-44, 1992. Cayley's Hypergeometric Function Theorem If oo (1 - z) a+h ~ c 2 Fi (2a, 26; 2c; z) = VJ a n z n , n = then 2 Fi(a,6;c+ \- 1 z) 2 F 1 (c- a,c- b;e\\z) oo = E ( c )" a ,» where 2 Fi (a, b; c; z) is a HYPERGEOMETRIC FUNCTION. see also Hypergeometric Function Cayley-Klein Parameters The parameters a, f3, 7, and S which, like the three Euler Angles, provide a way to uniquely characterize the orientation of a solid body. These parameters satisfy the identities and aa* + 77* = 1 aa* + 00* = 1 00* + SS* = 1 a*/? + 7*5 = a5 — /?7 = 1 /3 = -7* 5 = a*, (i) (2) (3) (4) (5) (6) (7) where z* denotes the COMPLEX CONJUGATE. In terms of the EULER ANGLES 8, </>, and tj>, the Cayley-Klein parameters are given by a = e *(*+*)/ a OOB(i#) s i(V>-«)/2, d = c -(*+*)/ a cos(itf) /3 = te* l *- w/:, 8in(ie) ■y = le " v ' r ' r " J sin(^) (8) (9) (10) (11) (Goldstein 1960, p. 155). The transformation matrix is given in terms of the Cayley-Klein parameters by A = I (a 2 - 7 2 + S 2 - (3 2 ) |i( 7 2 - a 2 + S 2 - /3 2 ) 7* - ct{3 \i{a 2 + 7 2 - P 2 - 6 2 ) i (a 2 + 7 2 + ^ 2 + <* 2 ) -i(a/3 + 7 tf) /3£ — a7 (Goldstein 1960, p. 153). i(ay + p8) a<5 + /37 (12) The Cayley-Klein parameters may be viewed as param- eters of a matrix (denoted Q for its close relationship with Quaternions) Q = a 7 6 which characterizes the transformations u = au + 0v (13) (14) (15) of a linear space having complex axes. This matrix sat- isfies Q f Q = 0(^ = 1, (16) where I is the IDENTITY MATRIX and A f the MATRIX Transpose, as well as iQriQi = i. (17) In terms of the Euler Parameters a and the Pauli MATRICES cr iy the Q-matrix can be written as Q = e l + z(ei<n + e 2 a 2 + e 3 cr 3 ) (18) (Goldstein 1980, p. 156). see also EULER ANGLES, EULER PARAMETERS, PAULI Matrices, Quaternion References Goldstein, H. "The Cayley-Klein Parameters and Related Quantities." §4-5 in Classical Mechanics, 2nd ed. Read- ing, MA: Addison- Wesley, pp. 148-158, 1980. 214 Cayley-Klein-Hilbert Metric Cayley's Sextic Evolute Cayley-Klein-Hilbert Metric The METRIC of Felix Klein's model for HYPERBOLIC Geometry, 9ii 912 922 a 2 (l-x 2 2 ) (1-Z! 2 -Z 2 2 ) 2 a X\X2 (1-Zl 2 ~X 2 2 ) 2 a 2 (l-X! 2 ) (1-xi 2 -X2 2 ) 2 ' see also HYPERBOLIC GEOMETRY Cayley Number There are two completely different definitions of Cayley numbers. The first type Cayley numbers is one of the eight elements in a Cayley Algebra, also known as an OCTONION. A typical Cayley number is of the form a + bio + ci\ + dii + ei 3 + fU + gh + hi Qi where each of the triples (10,11,13), (n,^,^), (22,^3,25), (z3,i4)*6)) (i4,*5,*o)» (*5»*6j*i), (*e,*o,«2) behaves like the QUATERNIONS (i,j,k). Cayley numbers are not AS- SOCIATIVE. They have been used in the study of 7- and 8-D space, and a general rotation in 8-D space can be written x ' -> {{{{{( xc i)c2)c3)c 4 )c 5 )cq)c 7 . The second type of Cayley number is a quantity which describes a Del Pezzo Surface. see also Complex Number, Del Pezzo Surface, Quaternion, Real Number References Conway, J. H. and Guy, R. K. "Cayley Numbers." In The Book of Numbers. New York: Springer- Ver lag, pp. 234- 235, 1996. Okubo, S. Introduction to Octonion and Other Non- Associative Algebras in Physics. New York: Cambridge University Press, 1995. Cayley's Ruled Surface see Cayley Cubic Cayley's Sextic A plane curve discovered by Maclaurin but first studied in detail by Cayley. The name Cayley's sextic is due to R. C. Archibald, who attempted to classify curves in a paper published in Strasbourg in 1900 (MacTutor Ar- chive). Cayley's sextic is given in POLAR COORDINATES by r = acos 3 (|0), (1) or r = 4&cos 3 (§0), (2) where b = a/4. In the latter case, the CARTESIAN equa- tion is 4(x 2 + y 2 - bxf - 27a 2 (x 2 + y 2 ) 2 . The parametric equations are x(t) = 4a cos 4 (I t) (2 cost - 1) y(t) =4acos 3 (|t)sin(|t). (3) (4) (5) JV_ The Arc Length, Curvature, and Tangential An- gle are s(t) = 3(i + sini), K(i) = !sec 2 (£t), <f>(t) = 2t. (6) (7) (8) References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 and 180, 1972. MacTutor History of Mathematics Archive. "Cayley's Sex- tic." http: //www-groups . dcs . st-and. ac . uk/ -history/ Curves/Cayleys.html. Cayley's Sextic Evolute / \ / \ 1 \ \ \ \ '"""N ^ \ I A \ \ A ) 1 S^T s < / ) j / \ ^J i 1 ^-^ 1 \ / \ / \ / \ / y •^ y The Evolute of Cayley's sextic is x=\a + ^a[3cos(|t) - cos(2<)] y=^a[3sin(|t)-sin(2t)] ) which is a Nephroid. Cayley Tree Cellular Automaton 215 Cayley Tree A Tree in which each NODE has a constant number of branches. The PERCOLATION THRESHOLD for a Cayley tree having z branches is 1 Pc see also CAYLEY GRAPH 1" Cayleyian Curve The Envelope of the lines connecting correspond- ing points on the JACOBIAN CURVE and STEINERIAN CURVE. The Cayleyian curve of a net of curves of or- der n has the same Genus (Curve) as the JACOBIAN Curve and Steinerian Curve and, in general, the class 3n(n— 1). References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 150, 1959. Cech Cohomology The direct limit of the COHOMOLOGY groups with CO- EFFICIENTS in an ABELIAN GROUP of certain coverings of a Topological Space. Ceiling Function 1**1 Ceiling [x] Nint (Round) |jc| Floor -4 -2 Jj_: i JT i i JT i L i _ j u ' -2 The function \x] which gives the smallest INTEGER > as, shown as the thick curve in the above plot. Schroeder (1991) calls the ceiling function symbols the "Gallows" because of the similarity in appearance to the structure used for hangings. The name and symbol for the ceiling function were coined by K. E. Iverson (Graham et al. 1990). It can be implemented as ceil(x)=-int (-x), where int(x) is the INTEGER PART of x. set also Floor Function, Integer Part, Nint References Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer Functions." Ch. 3 in Concrete Mathematics: A Foun- dation for Computer Science. Reading, MA: Addison- Wesley, pp. 67-101, 1990. Iverson, K. E. A Programming Language. New York: Wiley, p. 12, 1962. Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 57, 1991. Cell A finite regular POLYTOPE. see also 16-Cell, 24-Cell, 120-Cell, 600-Cell Cellular Automaton A grid (possibly 1-D) of cells which evolves according to a set of rules based on the states of surrounding cells, von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universal constructor." von Neumann proved that an automaton consisting of cells with four orthogonal neighbors and 29 possible states would be capable of simulating a TURING MACHINE for some configuration of about 200,000 cells (Gardner 1983, p. 227). l-D automata are called "elementary" and are repre- sented by a row of pixels with states either or 1. These can be represented with an 8-bit binary num- ber, as shown by Stephen Wolfram. Wolfram further restricted the number from 2 8 = 256 to 32 by requiring certain symmetry conditions. The most well-known cellular automaton is Conway's game of Life, popularized in Martin Gardner's Scien- tific American columns. Although the computation of successive Life generations was originally done by hand, the computer revolution soon arrived and allowed more extensive patterns to be studied and propagated. see Life, Langton's Ant References Adami, C. Artificial Life. Cambridge, MA: MIT Press, 1998. Buchi, J. R. and Siefkes, D. (Eds.). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Ex- pressions. New York: Springer- Verlag, 1989. Burks, A. W. (Ed.). Essays on Cellular Automata. Urbana- Champaign, IL: University of Illinois Press, 1970. Cipra, B. "Cellular Automata Offer New Outlook on Life, the Universe, and Everything." In What's Happening in the Mathematical Sciences, 1995-1996, Vol 3. Providence, RI: Amer. Math. Soc, pp. 70-81, 1996. Dewdney, A. K. The Armchair Universe: An Exploration of Computer Worlds. New York: W. H. Freeman, 1988. Gardner, M. "The Game of Life, Parts I— III." Chs. 20-22 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 219 and 222, 1983. Gutowitz, H. (Ed.). Cellular Automata: Theory and Exper- iment. Cambridge, MA: MIT Press, 1991. Levy, S. Artificial Life: A Report from the Frontier Where Computers Meet Biology. New York: Vintage, 1993. Martin, O.; Odlyzko, A.; and Wolfram, S. "Algebraic Aspects of Cellular Automata." Communications in Mathematical Physics 93, 219-258, 1984. Mcintosh, H. V. "Cellular Automata." http://www.es. cinvestav.mx/mcintosh/cellular.html. Preston, K. Jr. and Duff, M. J. B. Modern Cellular Au- tomata: Theory and Applications. New York: Plenum, 1985. Sigmund, K. Games of Life: Explorations in Ecology, Evo- lution and Behaviour. New York: Penguin, 1995. Sloane, N. J. A. Sequences A006977/M2497 in "An On-Line Version of the Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency- clopedia of Integer Sequences. San Diego: Academic Press, 1995. 216 Cellular Space Toffoli, T. and Margolus, N. Cellular Automata Machines: A New Environment for Modeling. Cambridge, MA: MIT Press, 1987. Wolfram, S. "Statistical Mechanics of Cellular Automata." Rev. Mod. Phys. 55, 601-644, 1983. Wolfram, S. (Ed.). Theory and Application of Cellular Au- tomata. Reading, MA: Addis on- Wesley, 1986. Wolfram, S. Cellular Automata and Complexity: Collected Papers. Reading, MA: Addison- Wesley, 1994. Wuensche, A. and Lesser, M. The Global Dynamics of Cel- lular Automata: An Atlas of Basin of Attraction Fields of One- Dimensional Cellular Automata. Reading, MA: Addison- Wesley, 1992. Cellular Space A Hausdorff Space which has the structure of a so- called CW-COMPLEX. Center A special POINT which usually has some symmetric placement with respect to points on a curve or in a SOLID. The center of a CIRCLE is equidistant from all points on the CIRCLE and is the intersection of any two distinct DIAMETERS. The same holds true for the center of a Sphere. see also Center (Group), Center of Mass, Cir- CUMCENTER, CURVATURE CENTER, ELLIPSEj EQUI- Brocard Center, Excenter, Homothetic Cen- ter, Incenter, Inversion Center, Isogonic Cen- ters, Major Triangle Center, Nine-Point Cen- ter, Orthocenter, Perspective Center, Point, Radical Center, Similitude Center, Sphere, Spieker Center, Taylor Center, Triangle Cen- ter, Triangle Center Function, Yff Center of Congruence Center Function see Triangle Center Function Center of Gravity see Center of Mass Center (Group) The center of a GROUP is the set of elements which commute with every member of the GROUP. It is equal to the intersection of the Centralizers of the Group elements. see also ISOCLINIC GROUPS, NlLPOTENT GROUP Center of Mass see Centroid (Geometric) Centered Pentagonal Number Centered Cube Number A Figurate Number of the form, CCub n = n +(n- l) 3 = (2n - l)(n 2 - n + 1). The first few are 1, 9, 35, 91, 189, 341, ... (Sloane's A005898). The Generating Function for the cen- tered cube numbers is x(x 3 + 5z 2 + 5x + 1) n 2 * ^ 4 -^ -, -^j [ — - = x + 9x 2 + 35z + 91a? 4 + . . . . (x- l) 4 see also Cubic Number References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 51, 1996. Sloane, N. J. A. Sequence A005898/M4616 in "An On-Line Version of the Encyclopedia of Integer Sequences." Centered Hexagonal Number see Hex Number Centered Pentagonal Number A Centered Polygonal Number consisting of a cen- tral dot with five dots around it, and then additional dots in the gaps between adjacent dots. The general term is (5n 2 - 5n + 2)/2, and the first few such num- bers are 1, 6, 16, 31, 51, 76, ... (Sloane's A005891). The Generating Function of the centered pentago- nal numbers is x(x 2 + Sx + 1) (z-1) 3 x + 6x 2 + 16z 3 + 31z 4 + . . . . see also CENTERED SQUARE NUMBER, CENTERED TRI- ANGULAR Number References Sloane, N. J. A. Sequence A005891/M4112 in "An On-Line Version of the Encyclopedia of Integer Sequences." Centered Polygonal Number Centered Polygonal Number N, / ^"*--~- / ^ — • — • — d A Figurate Number in which layers of Polygons are drawn centered about a point instead of with the point at a Vertex. see also Centered Pentagonal Number, Centered Square Number, Centered Triangular Number References Sloane, N. J. A. and Plouffe, S. Extended entry for sequence M3826 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Centered Square Number A Centered Polygonal Number consisting of a cen- tral dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is n 2 + (n — l) 2 , and the first few such numbers are 1, 5, 13, 25, 41, ... (Sloane's A001844). Centered square numbers are the sum of two consecutive SQUARE Numbers and are congruent to 1 (mod 4). The Gen- erating Function giving the centered square numbers is (1 — x) 6 see also Centered Pentagonal Number, Centered Polygonal Number, Centered Triangular Num- ber, Square Number References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 41, 1996. Sloane, N. J. A. Sequence A001844/M3826 in "An On-Line Version of the Encyclopedia of Integer Sequences." Centered Triangular Number A Centered Polygonal Number consisting of a cen- tral dot with three dots around it, and then additional Central Beta Function 217 dots in the gaps between adjacent dots. The general term is (3n — 3n + 2)/2, and the first few such numbers are 1, 4, 10, 19, 31, 46, 64, . . . (Sloane's A005448). The Generating Function giving the centered triangular numbers is x{x' + x + l) =x + 4x * + 10x > + 19x * + .... (1 — X) 6 see also CENTERED PENTAGONAL NUMBER, CENTERED Square Number References Sloane, N. J. A. Sequence A005448/M3378 in "An On-Line Version of the Encyclopedia of Integer Sequences." Centillion In the American system, 10 303 . see also Large Number Central Angle An Angle having its Vertex at a Circle's center which is formed by two points on the CIRCLE'S Cir- cumference. For angles with the same endpoints, C = 29 i, where 0; is the INSCRIBED ANGLE. References Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., pp. xxi— xxii, 1995. Central Beta Function 10r ;im[zj ;im[z] MzH*? The central beta function is defined by f3(p) = B(p,p), (1) 218 Central Binomial Coefficient Central Conic where B(p,q) is the BETA FUNCTION. It satisfies the identities ^(p) = 2 1 - ap B(p > i) (2) = 2 1 - 2p cos(7rp)B(f-p,p) (3) 1 t p dt _ 2 T-r n(n + 2p) V *1 (n + p)(n + p)" (4) (5) With p = 1/2, the latter gives the WALLIS FORMULA. When p = a/b, b/3(a/b) = 2 1 - 2a/b J(a,b), where a a,b)= f Jo 1 1*- 1 dt The central beta function satisfies (2 -\- 4x)0(l -\- x) = x0(x) (1 - 2x)j8(l - x)f3(x) = 27rcot(7nr) P(\ - x) = 2 4x_1 t<m(7rx)/3(x) (6) (7) (8) (9) (10) P(x)0(x + |) = 2 4 * +1 7r/?(2z)/3(2; C + §). (11) For p an Odd Positive Integer, the central beta func- tion satisfies the identity ^ )= vP n -^ V fc=l fc=0 n>(" + i;J- < 12 > see a/so BETA FUNCTION, REGULARIZED BETA FUNC- TION References Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua- tion of the Gamma Function at Rational Values of Small Denominators." IMA J. Numerical Analysis 12, 519—526, 1992. Central Binomial Coefficient The nth central binomial coefficient is defined as ( i n / 2 i ) > where (™) is a BINOMIAL COEFFICIENT and [n\ is the Floor Function. The first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, . . . (Sloane's A001405). The central binomial coefficients have GENERATING FUNCTION 2(2# 3 - x 2 ) The central binomial coefficients are SQUAREFREE only for n = 1, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, . . . (Sloane's A046098), with no others less than 1500. The above coefficients are a superset of the alternative "central" binomial coefficients CD- (2n)! (n!) 2 ' which have GENERATING FUNCTION v 7 ! - 4z : 1 + 2x + 6x 2 + 20z 3 + 70x 4 + . . . . The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (Sloane's A000984). Erdos and Graham (1980, p. 71) conjectured that the central binomial coefficient ( 2 ^) is never SQUARE- FREE for n > 4, and this is sometimes known as the Erdos Squarefree Conjecture. Sarkozy's The- orem (Sarkozy 1985) provides a partial solution which states that the BINOMIAL COEFFICIENT ( 2 ") is never Squarefree for all sufficiently large n > no (Vardi 1991). Granville and Ramare (1996) proved that the only Squarefree values are n — 2 and 4. Sander (1992) subsequently showed that ( 2n T f d ) are also never SQUAREFREE for sufficiently large n as long as d is not "too big." see also BINOMIAL COEFFICIENT, CENTRAL TRINO- MIAL Coefficient, Erdos Squarefree Conjec- ture, Sarkozy's Theorem, Quota System References Granville, A. and Ramare, O. "Explicit Bounds on Exponen- tial Sums and the Scarcity of Squarefree Binomial Coeffi- cients." Mathematika 43, 73-107, 1996. Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull London Math. Soc. 24, 140-142, 1992. Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985. Sloane, N. J. A. Sequences A046098, A000984/M1645, and A001405/M0769 in "An On-Line Version of the Encyclo- pedia of Integer Sequences." Vardi, I. "Application to Binomial Coefficients," "Binomial Coefficients," "A Class of Solutions," "Computing Bino- mial Coefficients," and "Binomials Modulo and Integer." §2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematica. Redwood City, CA: Addison- Wesley, pp. 25-28 and 63-71, 1991. Central Conic An Ellipse or Hyperbola. see also CONIC Section References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited, Washington, DC: Math. Assoc. Amer., pp. 146-150, 1967. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 77, 1990. Central Difference Central Limit Theorem 219 Central Difference The central difference for a function tabulated at equal intervals fi is defined by ^(/n+l/2) = <Wi/ 2 = $n + l/2 = /n+1 - fn- (1) Higher order differences may be computed for Even and Odd powers, 2fc / \ C +1 /2 =£(-1)' 2 f/n+ fc 2fc+l / \ (2) +fc+i-j- (3) see a/so Backward Difference, Divided Differ- ence, Forward Difference References Abramowitz, M. and Stegun, C. A. (Eds.). "Differences." §25.1 in Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972. Central Limit Theorem Let x\ , X2 , . . . , xn be a set of AT INDEPENDENT random variates and each Xi have an arbitrary probability distri- bution P(a?i, . . . , xn) with MEAN fii and a finite VARI- ANCE cr^ 2 . Then the normal form variate A norra — v^ (1) Vi* has a limiting distribution which is NORMAL (GAUS- SIAN) with Mean \l = and Variance a 2 ~ 1. If conversion to normal form is not performed, then the variate X ^5> (2) is Normally Distributed with fi x = \x x and a x = o~ x /y/N. To prove this, consider the Inverse Fourier Transform of Px{}). /oo e 2 * ifX p(X)dX -OO J —c sr ( 27ri n=Q •J —oo p{X) dx (2^/) n /„Xn E^<*> (3) Now write (X n ) = (AT n (xi +X2 + ... + x N ) n ) /OO N~ n (xi + . .. + xn) u p(xi) - • -p(xN)dxi ---cIxn, •oo (4) so we have (2«/)» *■ — ' n n = «/-oo /*°° y^ r 27rz/(x 1 + ... + x JV ) l" 1 <J — oo _ rt + .., + x^) n x p(xi) • • -p(x N ) dxi ■ • ■ dxj\ x p(#i) • * -p(xjyf) dx\ • • 'dxj\ /oo pix^dx! F V — c w p(xn) dxjsr p(x) dx }' = / 6 a-</-/JVp(a.) da . = / p(x)dx-\ / xp(x) dx L 1 ' — oo <J — oo Now expand ln(l + x) = x-\x 2 + \x 3 + ... (5) (6) w exp < AT N {X) 2N* \ X I + ^<*> 2 + <^- 3 ) : exp J exp (2nf) 2 ((x 2 ) - (x) 2 ) 2iviffi x (27T/)V, 2 2N (7) 220 Central Limit Theorem Centroid (Geometric) Hx = (x) a 2 = (x 2 ) - (x) 2 Taking the FOURIER TRANSFORM, (8) (9) /OO e -wr-i[P x (f)]df -oo = f°° e 2^if(^ x -x)-(2^f) 2 a x 2 /2N d , ^ J — oo This is of the form /CO iaf - e -CO bf df, (11) where a = 2iz(ti x — x) and 6 = (27ro~ x ) 2 /2N. But, from Abramowitz and Stegun (1972, p. 302, equation 7.4.6), /CO e iaf- -oo bf 2 df = e -a 2 /ib /W (12) Therefore, 7T J -[27T(fl x -X)} 2 exp ' — — 2AT 27TJV 47T 2 <7 X 2 exp 4 (2™*) 2 f ^ 2AT J 4tt 2 (^ -x) 2 2iV 4 • 47T 2 cr x 2 ViV But ax = <7 x /VN and //x = Man so P x = \ c -(mx-^) 2 ^x 2 (rx\/27r (13) (14) Central Trinomial Coefficient The nth central binomial coefficient is denned as the co- efficient of x n in the expansion of (l-\-x-\-x 2 ) n . The first few are 1, 3, 7, 19, 51, 141, 393, . . . (Sloane's A002426). This sequence cannot be expressed as a fixed number of hypergeometric terms (Petkovsek et al. 1996, p. 160). The Generating Function is given by /(*) = 1 ^(l + z)(l-3x) = 1 + x + 3z 2 + 7x 3 + . . . . see also Central Binomial Coefficient References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, 1996. Sloane, N. J. A. Sequence A002426/M2673 in "An On-Line Version of the Encyclopedia of Integer Sequences." Centralizer The centralizer of a Finite non-ABELiAN Simple Group G is an element z of order 2 such that C G (z) = {geG:gz = zg}. see also Center (Group), Normauzer Centrode C = rT + kB, where r is the TORSION, k is the CURVATURE, T is the Tangent Vector, and B is the Binormal Vector. Centroid (Function) By analogy with the GEOMETRIC CENTROID, the cen- troid of an arbitrary function f(x) is defined as {x} = IZo f( x ) dx The "fuzzy" central limit theorem says that data which are influenced by many small and unrelated random ef- fects are approximately NORMALLY DISTRIBUTED. see also LlNDEBERG Condition, Lindeberg-Feller Central Limit Theorem, Lyapunov Condition References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972, Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 112-113, 1992. Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483-494, 1995. References Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 139-140 and 156, 1965. Centroid (Geometric) The Center of Mass of a 2-D planar Lamina or a 3-D solid. The mass of a LAMINA with surface density function o~(x,y) is M-- : //' (x ' y)dA. (1) The coordinates of the centroid (also called the CENTER of Gravity) are ff xo~(x,y) dA M (2) Centroid (Orthocentric System) Jfya(x,y)dA y M (3) The centroids of several common laminas along the non- symmetrical axis are summarized in the following table. Figure y parabolic segment |/t 3tt semicircle In 3-D , the mass of a solid with density function p(x,y,z) is Iff**'* M= I I I p(x,y,z)dV, (4) and the coordinates of the center of mass are _ _ fffxp(x,y,z)dV M JJfyp(x,y,z)dV M JfJzp(x,y,z)dV M (5) (6) (7) Figure cone ^ h conical frustum ^Y^t 3 ^ hemisphere paraboloid pyramid 4(R 1 2 +R 1 R 2 +R2 2 ) \h see also Pappus's Centroid Theorem References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987. McLean, W. G. and Nelson, E. W. "First Moments and Cen- troids." Ch. 9 in Schaum's Outline of Theory and Prob- lems of Engineering Mechanics: Statics and Dynamics, 4th ed. New York: McGraw-Hill, pp. 134-162, 1988. Centroid (Orthocentric System) The centroid of the four points constituting an ORTHO- CENTRIC System is the center of the common Nine- Point Circle (Johnson 1929, p. 249). This fact auto- matically guarantees that the centroid of the Incenter and Excenters of a Triangle is located at the Cir- cumcenter. References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Centroid (Triangle) 221 Centroid (Triangle) The centroid (Center of Mass) of the Vertices of a Triangle is the point M (or G) of intersection of the Triangle's three Medians, also called the Median Point (Johnson 1929, p. 249). The centroid is always in the interior of the TRIANGLE, and has TRILINEAR Coordinates csc A : esc B : esc C. If the sides of a TRIANGLE are divided so that A 2 Pi A3P2 A ± P 2 PiA 3 P 2 A X P 3 A 2 P 9 (2) (3) the centroid of the TRIANGLE AP1P2P3 is M (Johnson 1929, p. 250). Pick an interior point X. The TRIANGLES BXC, CXA, and AXB have equal areas IFF X corresponds to the centroid. The centroid is located one third of the way from each Vertex to the Midpoint of the opposite side. Each median divides the triangle into two equal areas; all the medians together divide it into six equal parts, and the lines from the Median Point to the Vertices divide the whole into three equivalent TRIANGLES. In general, for any line in the plane of a Triangle ABC, d= l{d A + d B + d c ), (4) where d } d A , ds, and dc are the distances from the cen- troid and Vertices to the line. A Triangle will bal- ance at the centroid, and along any line passing through the centroid. The Trilinear Polar of the centroid is called the Lemoine Axis. The Perpendiculars from the centroid are proportional to s^ -1 , CL1P2 = CL2P2 = dtps - § A, (5) where A is the Area of the Triangle. Let P be an arbitrary point, the Vertices be Ai, A 2) and A 3) and the centroid M. Then PA X +PA 2 +PA 3 = MA! +Mi 2 +MA Z +3PM . (6) If O is the ClRCUMCENTER of the triangle's centroid, then OM 2 =,R 2 -|(a 2 + 6 2 +c 2 ). (7) The centroid lies on the EULER LINE. The centroid of the PERIMETER of a TRIANGLE is the triangle's Spieker Center (Johnson 1929, p. 249). see also ClRCUMCENTER, EULER LlNE, EXMEDIAN Point, Incenter, Orthocenter References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970. 222 Certificate of Compositeness Ceva's Theorem Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 7, 1967. Dixon, R. Mathographics. New York: Dover, pp. 55-57, 1991. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 173-176 and 249, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Kimberling, C. "Centroid." http : //www . evansville . edu/ -ck6/tcenters/class/centroid.html. Certificate of Compositeness see Compositeness Certificate Certificate of Primality see Primality Certificate Cesaro Equation An Intrinsic Equation which expresses a curve in terms of its ARC LENGTH s and RADIUS OF CURVA- TURE R (or equivalently, the CURVATURE k). see also Arc Length, Intrinsic Equation, Natural Equation, Radius of Curvature, Whewell Equa- tion References Yates, R. C. "Intrinsic Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 123-126, 1952. Cesaro Fractal A Fractal also known as the Torn Square Frac- tal. The base curves and motifs for the two fractals illustrated above are show below. see also Fractal, Koch Snowflake References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, p. 43, 1991. Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 79, 1989. ^ Weisstein, E. W. "Fractals." http://www. astro. Virginia. edu/-eww6n/math/notebooks/Fractal.m. Cesaro Mean see FEJES TOTH'S INTEGRAL Ceva's Theorem Given a Triangle with Vertices A, £?, and C and points along the sides D, E, and F, a NECESSARY and Sufficient condition for the Cevians AD, BE, and CF to be Concurrent (intersect in a single point) is that BDCE-AF^DCEA- FB. (1) Let P = [Vi, . . . , V^] be an arbitrary n-gon, C a given point, and k a Positive Integer such that 1 < k < n/2. For i = 1, . . . , n, let Wi be the intersection of the lines CVi and Vi-kV i+ k, then n Vi-kWi WtVi i+k = 1. Here, AB\\CD and AB VCD\ (2) (3) is the Ratio of the lengths [A, B] and [C, D] with a plus or minus sign depending on whether these segments have the same or opposite directions (Grunbaum and Shepard 1995). Another form of the theorem is that three Concurrent lines from the Vertices of a Triangle divide the op- posite sides in such fashion that the product of three nonadjacent segments equals the product of the other three (Johnson 1929, p. 147). see also Hoehn's Theorem, Menelaus' Theorem References Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967. Grunbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254-268, 1995. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 145-151, 1929. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., p. xx, 1995. Cevian Cevian A line segment which joins a Vertex of a Triangle with a point on the opposite side (or its extension). In the above figure, 6 sin a sin(7 + a') References Thebault, V. "On the Cevians of a Triangle." Amer. Math. Monthly 60, 167-173, 1953. Cevian Conjugate Point see ISOTOMIC CONJUGATE POINT Cevian Transform Vandeghen's (1965) name for the transformation taking points to their ISOTOMIC CONJUGATE POINTS. see also Isotomic Conjugate Point References Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Trian- gle." Amer. Math. Monthly 72, 1091-1094, 1965. Cevian Triangle Given a center a : (3 : 7, the cevian triangle is defined as that with VERTICES : : 7, a : : 7, and a : P : 0. If A'B'C is the CEVIAN TRIANGLE of X and A"B"C" is the Anticevian Triangle, then X and A" are Harmonic Conjugate Points with respect to A and A', see also Anticevian Triangle Chain Rule 223 Chain Let P be a finite Partially Ordered Set. A chain in P is a set of pairwise comparable elements (i.e., a Totally Ordered subset). The Width of P is the maximum CARDINALITY of an Antichain in P. For a Partial Order, the size of the longest Chain is called the Width. see also Addition Chain, Antichain, Brauer Chain, Chain (Graph), Dilworth's Lemma, Hansen Chain Chain Fraction see Continued Fraction Chain (Graph) A chain of a GRAPH is a SEQUENCE {x u z 2 , . . ■ , x n } such that {x u x 2 ), (052,2:3), .--, (z„_i,a:n) are EDGES of the Graph. Chain Rule If g(x) is DlFFERENTlABLE at the point x and f(x) is DlFFERENTIABLE at the point g(x), then / o g is DlF- FERENTlABLE at x. Furthermore, let y = f(g(x)) and u = g(x), then dy _ dy du dx du dx (i) There are a number of related results which also go un- der the name of "chain rules." For example, if z — f(x,y), x = g{t), and y - h(t), then dz _ dz dx dt dx dt dz dy dy dt ' (2) The "general" chain rule applies to two sets of functions yi /1 (ui,..., «p) and :(3) y m - fm{ui i ... J Up) U± = £l(25i, . .. ,X n ) :(4) U P = 0p(#l»- • • j^n)- Defining the m X n JACOBI MATRIX by dyi dx. dvi . dx 2 dyi dx n dxi dy m 8x2 dXn (5) and similarly for (dyi/duj) and (diii/dxj) then gives dyi dx. -(£)(£)■ m 224 Chained Arrow Notation Champernowne Constant In differential form, this becomes dpi du p du p dxi d _ | dy^diH * dui dxi + ^^L ]dxi I dmdu± ^dup\ ^ du\ &X2 ' ' ' du p 8x2 J (Kaplan 1984). see also Derivative, Jacobian, Power Rule, Prod- uct Rule References Anton, H. Calculus with Analytic Geometry, 2nd ed. New York: Wiley, p. 165, 1984. Kaplan, W. "Derivatives and Differentials of Composite Functions" and "The General Chain Rule." §2.8 and 2.9 in Advanced Calculus, 3rd ed. Reading, MA: Addison- Wesley, pp. 101-105 and 106-110, 1984. Chained Arrow Notation A Notation which generalizes Arrow Notation and is defined as a\-<*"\h = a^b^>c. see also Arrow Notation References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 61, 1996. Chainette see Catenary Chair Chaitin's Constant An Irrational Number Q which gives the probability that for any set of instructions, a Universal Turing MACHINE will halt. The digits in are random and cannot be computed ahead of time. see also Halting Problem, Turing Machine, Uni- versal Turing Machine References Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/chaitin/chaitin.html. Gardner, M. "The Random Number Bids Fair to Hold the Mysteries of the Universe." Set. Amer. 241, 20-34, Nov. 1979. Gardner, M. "Chaitin's Omega." Ch. 21 in Fractal Music, HyperCards, and More Mathematical Recreations from Sci- entific American Magazine. New York: W. H. Freeman, 1992. Kobayashi, K. "Sigma(N)0-Complete Properties of Pro- grams and Lartin-Lof Randomness." Information Proc. Let 46, 37-42, 1993. Chaitin's Number see Chaitin's Constant Chaitin's Omega see Chaitin's Constant Champernowne Constant Champernowne's number 0.1234567891011. . . (Sloane's A033307) is the decimal obtained by concatenating the Positive Integers. It is Normal in base 10. In 1961, Mahler showed it to also be TRANSCENDENTAL. The Continued Fraction of the Champernowne con- stant is [0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, A Surface with tetrahedral symmetry which, according to Nordstrand, looks like an inflatable chair from the 1970s. It is given by the implicit equation (x 2 +y 2 + z 2 -ak 2 ) 2 -b[(z-k) 2 -2x 2 ][{z + k) 2 ~2y 2 ] = 0. see also Bride's Chair References Nordstrand, T. "Chair." http://www.uib.no/people/nfytn/ chairtxt.htm. 457540111391031076483646628242956118599603939- • • 710457555000662004393090262659256314937953207- - • 747128656313864120937550355209460718308998457* • * 5801469863148833592141783010987, 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...] (Sloane's A030167). The next term of the Continued Fraction is huge, having 2504 digits. In fact, the coefficients eventually become unbounded, making the continued fraction difficult to calculate for too many more terms. Large terms greater than 10 5 oc- cur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468, 136, . . . digits (Plouffe). Interestingly, the Copeland-Erdos Con- stant, which is the decimal obtained by concatenating the Primes, has a well-behaved Continued Fraction which does not show the "large term" phenomenon. see also COPELAND-ERDOS CONSTANT, SMARANDACHE Sequences Change of Variables Theorem Chaos 225 References Champernowne, D. G. "The Construction of Decimals Nor- mal in the Scale of Ten." J. London Math. Soc. 8, 1933. Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/cntfrc/cntfrc.html. Sloane, N. J. A. Sequences A030167 and A033307 in "An On- Line Version of the Encyclopedia of Integer Sequences." Change of Variables Theorem A theorem which effectively describes how lengths, ar- eas, volumes, and generalized n-dimensional volumes (Contents) are distorted by Differentiable Func- tions. In particular, the change of variables theorem reduces the whole problem of figuring out the distortion of the content to understanding the infinitesimal dis- tortion, i.e., the distortion of the DERIVATIVE (a linear Map), which is given by the linear Map's Determi- nant. So / : R n -► W 1 is an Area-Preserving linear MAP Iff |det(/)| = 1, and in more generality, if S is any subset of MJ 1 , the CONTENT of its image is given by I det(/)| times the CONTENT of the original. The change of variables theorem takes this infinitesimal knowledge, and applies CALCULUS by breaking up the DOMAIN into small pieces and adds up the change in AREA, bit by bit. The change of variable formula persists to the general- ity of Differential Forms on Manifolds, giving the formula / (/*w) = f (u Jm Jw under the conditions that M and W are compact con- nected oriented MANIFOLDS with nonempty boundaries, / : M — > W is a smooth map which is an orientation- preserving DlFFEOMORPHISM of the boundaries. In 2-D, the explicit statement of the theorem is /. f(x,y)dxdy -L f[x(u,v),y(u,v)] d(x,y) d(u,v) dudv and in 3-D, it is / /(a;, y, z) dx dy dz ■ I f[x(u, v,w), y(u, v, w) } z(u, u, J R* W)] d(x,y,z) du dv dw , d(u, v, w) where R = f(R*) is the image of the original region R* , d(u,v,w) is the JACOBIAN, and / is a global orientation-preserving DlFFEOMORPHISM of R and R* (which are open subsets ofM n ). The change of variables theorem is a simple consequence of the Curl Theorem and a little de Rham Cohomol- OGY. The generalization to n-D requires no additional assumptions other than the regularity conditions on the boundary. see also Implicit Function Theorem, Jacobian References Kaplan, W. "Change of Variables in Integrals." §4.6 in Ad- vanced Calculus, 3rd ed. Reading, MA: Addison- Wesley, pp. 238-245, 1984. Chaos A Dynamical System is chaotic if it 1. Has a Dense collection of points with periodic or- bits, 2. Is sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), and 3. Is TOPOLOGICALLY TRANSITIVE. Chaotic systems exhibit irregular, unpredictable behav- ior (the Butterfly Effect). The boundary between linear and chaotic behavior is characterized by PERIOD DOUBLING, following by quadrupling, etc. An example of a simple physical system which displays chaotic behavior is the motion of a magnetic pendulum over a plane containing two or more attractive magnets. The magnet over which the pendulum ultimately comes to rest (due to frictional damping) is highly dependent on the starting position and velocity of the pendulum (Dickau). Another such system is a double pendulum (a pendulum with another pendulum attached to its end). see also Accumulation Point, Attractor, Basin of Attraction, Butterfly Effect, Chaos Game, Feigenbaum Constant, Fractal Dimension, Gin- gerbreadman Map, Henon-Heiles Equation, Henon Map, Limit Cycle, Logistic Equation, Lya- punov Characteristic Exponent, Period Three Theorem, Phase Space, Quantum Chaos, Reso- nance Overlap Method, Sarkovskii's Theorem, Shadowing Theorem, Sink (Map), Strange At- tractor References Bai-Lin, H. Chaos. Singapore: World Scientific, 1984. Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Intro- duction, 2nd ed. Cambridge: Cambridge University Press, 1996. Cvitanovic, P. Universality in Chaos: A Reprint Selection, 2nd ed. Bristol: Adam Hilger, 1989. Dickau, R. M. "Magnetic Pendulum." http:// forum . swarthmore . edu / advanced / robertd / magnetic pendulum . html . Drazin, P. G. Nonlinear Systems. Cambridge, England: Cambridge University Press, 1992. Field, M. and Golubitsky, M. Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature. Oxford, England: Oxford University Press, 1992. Gleick, J. Chaos: Making a New Science. New York: Pen- guin, 1988. 226 Chaos Game Character Table Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer- Verlag, 1997. Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer- Verlag, 1994. Lorenz, E. N. The Essence of Chaos. Seattle, WA: University of Washington Press, 1996. Ott, E, Chaos in Dynamical Systems. New York: Cambridge University Press, 1993. Ott, E.; Sauer, T.; and Yorke, J. A. Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems. New York: Wiley, 1994. Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac- tals: New Frontiers of Science. New York: Sprhiger- Verlag, 1992. Poon, L. "Chaos at Maryland." http://www-chaos.umd.edu. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990. Strogatz, S. H. Nonlinear Dynamics and Chaos, with Appli- cations to Physics, Biology, Chemistry, and Engineering. Reading, MA: Addis on- Wesley, 1994. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989, Tufillaro, N.; Abbott, T. R.; and Reilly, J. An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, CA: Addison-Wesley, 1992. Wiggins, S. Global Bifurcations and Chaos: Analytical Meth- ods. New York: Springer- Verlag, 1988. Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer- Verlag, 1990. Chaos Game Pick a point at random inside a regular n-gon. Then draw the next point a fraction r of the distance between it and a Vertex picked at random. Continue the pro- cess (after throwing out the first few points). The result of this "chaos game" is sometimes, but not always, a Fractal. The case (n,r) = (4,1/2) gives the interior of a SQUARE with all points visited with equal probabil- ity. <Tfc ******** A A 4% Ah; f\ A. A ******** (3,1/2) && pig (5,1/3) & hi (5,3/8) see a/so Barnsley's Fern (6,1/3) References Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993. Dickau, R. M. "The Chaos Game." http:// forum . swarthmore.edu/advanced/robertd/chaos_game.html. Wagon, S. Mathematica in Action. New York: W. H. Free- man, pp. 149-163, 1991. # Weisstein, E. W. "Fractals." http: //www. astro. Virginia. edu/-eww6n/math/notebooks/Fractal.m. Character (Group) The Group Theory term for what is known to physi- cists as the Trace. All members of the same Conju- GACY Class in the same representation have the same character. Members of other Conjugacy Classes may also have the same character, however. An (abstract) Group can be uniquely identified by a listing of the characters of its various representations, known as a Character Table. Some of the Schonflies Sym- bols denote different sets of symmetry operations but correspond to the same abstract GROUP and so have the same Character Tables. Character (Multiplicative) A continuous HOMEOMORPHISM of a GROUP into the Nonzero Complex Numbers. A multiplicative char- acter w gives a REPRESENTATION on the 1-D SPACE C of Complex Numbers, where the Representation ac- tion by g 6 G is multiplication by uj(g). A multiplicative character is UNITARY if it has ABSOLUTE VALUE 1 ev- erywhere. References Knapp, A. W. "Group Representations and Harmonic Anal- ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996. Character (Number Theory) A number theoretic function Xk(n) for POSITIVE integral n is a character modulo k if X*(l) = l Xk{n) = Xk(n + k) Xk(m)xk(n) = Xk(mn) for all m^n, and X*(») = 0- if (fc,n) ^ 1. Xk can only assume values which are (j>{k) Roots of Unity, where <j> is the Totient Function. see also DlRlCHLET L-SERIES Character Table C x E A 1 C 8 E CTh A B 1 1 1 -1 -3 j JXx , •t* j y 2 2 2 x ,y ,z yz,xz xy Character Table Character Table 227 a E i A 9 A u 1 1 1 -1 x,y,z x 2 ,y 2 ,z 2 ,xy,xz yz c 2 E c 2 A B 1 1 1 -1 z,R z x,y,R x ,R y x'\y\z'\xy yz,xz C 3 E C3 Cz e = exp(27rz/3) A E 111 {! I- f } z,R z {x,y)(R x ,R y ) 222 x ,y ,z ,xy (x 2 -y 2 ,xy){yz,xz) c 4 E O3 C 2 C4 A B E 1111 1-1 1-1 ri i -1 -n ll-i 1 i) z,R z (x,y)(R x ,R y ) 2,22 x +y ,z x 2 -y 2 ,xy (yz,xz) D 6 E 2C 6 2O3 O2 3Gj 3G 2 A, 1 1 1111 x 2 +y\z 2 A 2 1 1 1 1-1-1 z, R z B 1 1 -1 1-1 1-1 B 2 1 -1 1-1-1 1 (x^yXR^Ry) E 1 2 1 -1-2 (xz,yz) E 2 2 -1 -12 (x 2 -y 2 ,xy) C2v E C 2 cr v (xz) °'v{yz) A 1 1 1 1 1 z 2 2 x ,y z 2 A 2 1 1 -1 -1 Rz xy 3i 1 -1 1 -1 X, ity xz B 2 1 -1 -1 1 y,Rx yz c$ v Ai A 2 E E 2 C3 3<x v 1 1 1 1 1 -1 2-10 z Rz (x,y)(R x ,R y ) ~^2~, 2 2~~ x +y ,z (x 2 -y 2 ,xy)(xz,yz) c & E C 5 C 5 2 c 5 3 c 5 4 e = exp(27ri/5) A 11 1 1 1 2,H, 2,22 a; 4- y ,z E, fie e 2 tl e * e 2 ' e 2 * e 2 r} (x,*/)^,^) (yz, xz) E 2 (1 £ 2 e* ll e 2 * e e £* ?} (x 2 — y 2 ,xy) c. E c 6 c 3 C 2 Gz <V £ = exp(27rt/6) A 1 1 1 1 1 1 z,R x 2 1 2 2 x + y ,z B 1 _i 1 -1 1 -1 Ei (I £ — £* — e -1 -£ ~1 ~<T I'} (s,y) (R x , Ry) (f,^) E 2 (i — £ — £* — £ — £* 1 -£* 1 -£ -:■} (x 2 - y 2 , xy) Z>2 E C 2 (z) C 2 {y) C 2 (x) A 1 1111 2,22 x +y ,z B 1 1 1-1-1 z,R z xy B 2 1-1 1-1 y,Ry xz B 3 1-1-1 1 z,R z yz D 3 A 1 A 2 E E 2C 3 3C 2 111 1 1 -1 2 -1 z,R z (x,y)(R x ,R y ) ar -\-y,z z xy (x 2 -y 2 ,xy){xz,yz) D 4 E 2C4 C 2 2C 2 2C2 Ai 11111 2,22 x +y ,z A 2 1 11-1-1 z,R z Bi 1-11 1-1 2 2 x -y B 2 1-11-1 1 xy E 2 0-20 (x,y)(R x ,R y ) (xz,yz) D 5 E 2C 5 2C 5 2 5C 2 A x Bi B 2 £3 1 1 1 1 2 2 cos 72° 2 2 cos 144° 1 1 2 cos 144° 2 cos 72° 1 -1 z,R z (x,y)(R x ,R y ) x 2 ^y 2 ,z 2 {xz,yz) {x 2 -y 2 ,xy) Cav E 2C4 O2 2(T V 2(Td A 2 B 1 B 2 E 1 1 1 1 2 1111 1 1-1-1 -11 1-1 -11-1 1 0-200 z Rz (x,y)(R x ,R y ) 2,22 x z +y,z z 2 2 x -y xy (xz,yz) c 5v E 2C 5 2C 5 2 5<7v A x Bi B 2 B3 1 1 1 1 2 2 cos 72° 2 2 cos 144° 1 1 2 cos 144° 2 cos 72° 1 -1 z R z (x,y)(R x ,R y ) x 2 +y 2 ,z 2 (xz.yz) (x 2 -y 2 ,xy) c, v E 2C 6 2C 3 C 2 3cr v 3cr d A! 1 1 1 1 z * 2 \y\z 2 A 2 1 1 -1 -1 Rz B l 1 -1 -1 1 -1 B 2 1 -1 -1 _i 1 E l 2 -1 -2 (x,y)(R x ,R y ) (xz,yz) E 2 2 -1 -1 2 (x 2 - y 2 ,xy) c». £7 Coo* • oocr„ A x = S + 1 1 1 z x 2 +y 2 ,z 2 A 2 = E" 1 1 .. -1 Rz E x = n 2 2 cos <£ (x,y);(R x ,R y ) (xz,yz) £? 2 = A 2 2 cos 2* (x 2 - y 2 ,xy) S 3 =* 2 2 cos 3* References Bishop, D. M. "Character Tables." Appendix 1 in Group Theory and Chemistry. New York: Dover, pp. 279—288, 1993. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990. Iyanaga, S. and Kawada, Y. (Eds.). "Characters of Finite Groups." Appendix B, Table 5 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1496- 1503, 1980. 228 Characteristic Class Characteristic (Field) Characteristic Class Characteristic classes are Cohomology classes in the Base Space of a Vector Bundle, defined through Obstruction theory, which are (perhaps partial) ob- structions to the existence of k everywhere linearly independent vector Fields on the Vector Bundle. The most common examples of characteristic classes are the Chern, Pontryagin, and Stiefel- Whitney Classes. Characteristic (Elliptic Integral) A parameter n used to specify an ELLIPTIC INTEGRAL of the Third Kind. see also AMPLITUDE, ELLIPTIC INTEGRAL, MODULAR Angle, Modulus (Elliptic Integral), Nome, Pa- rameter References Abramowitz, M. and Stegun, C. A, (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972. Characteristic Equation The equation which is solved to find a Matrix's Eigen- values, also called the CHARACTERISTIC POLYNOMIAL. Given a 2 x 2 system of equations with MATRIX M the Matrix Equation is a b c d a b c d X = t X y_ which can be rewritten (i) (2) (3) (4) which contradicts our ability to pick arbitrary x and y. Therefore, M has no inverse, so its Determinant is 0. This gives the characteristic equation a — t b c d — t = t M can have no Matrix Inverse, since otherwise X = M" 1 "o" = V a — t b c d — t = 0, (5) where | A| denotes the Determinant of A. For a general k x k Matrix (6) an ai2 ■ - • aifc 021 ^22 • . . Q>2k afci &k2 . •• a>kh the characteristic equation is an — t a 12 0,21 CL22 — t CLkl ak2 aifc a2fc &kk — t (7) see also Ballieu's Theorem, Cayley-Hamilton Theorem, Parodi's Theorem, Routh-Hurwitz Theorem References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1117-1119, 1979. Characteristic (Euler) see Euler Characteristic Characteristic Factor A characteristic factor is a factor in a particular fac- torization of the Totient Function <j>(n) such that the product of characteristic factors gives the represen- tation of a corresponding abstract Group as a Direct PRODUCT. By computing the characteristic factors, any Abelian Group can be expressed as a Direct Prod- uct of Cyclic Subgroups, for example, Z 2 ® Z 4 or Z2® Z2® Z 2 . There is a simple algorithm for determining the characteristic factors of Modulo Multiplication Groups. see also Cyclic Group, Direct Product (Group), Modulo Multiplication Group, Totient Func- tion References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 94, 1993. Characteristic (Field) For a FIELD K with multiplicative identity 1, consider the numbers 2 = 1 + 1, 3 = 1 + 1 + 1,4 = 1 + 1 + 1 + 1, etc. Either these numbers are all different, in which case we say that K has characteristic 0, or two of them will be equal. In this case, it is straightforward to show that, for some number p, we have 1 + 1 + .. . + 1 = 0. p times If p is chosen to be as small as possible, then p will be a Prime, and we say that K has characteristic p. The Fields Q, E, C, and the /?-adic Numbers Q p have characteristic 0. For p a Prime, the Galois Field GF(p n ) has characteristic p. If H is a Subfield of K, then H and K have the same characteristic. see also Field, Subfield Characteristic Function Chasles's Polars Theorem 229 Characteristic Function The characteristic function <j>(t) is defined as the Four- ier Transform of the Probability Density Func- tion, /CO e iix P{x)dx (1) ■oo /OO /"OO P(x)dx + it / xP(x)dx ■oo J — OO /OO x 2 P(z)dx + ... (2) OO = 1 + ii/i'i - ^2 - ^f« 3 /*3 + ^Vi + . . . , (4) where fi f n (sometimes also denoted i/ n ) is the nth MO- MENT about and {j! = 1. The characteristic function can therefore be used to generate MOMENTS about 0, or the Cumulants « n , OO z — ' n! (5) (6) A Distribution is not uniquely specified by its Mo- ments, but is uniquely specified by its characteristic function. see also Cumulant, Moment, Moment-Generating Function, Probability Density Function References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972. Kenney, J. F. and Keeping, E. S. "Moment-Generating and Characteristic Functions," "Some Examples of Moment- Generating Functions," and "Uniqueness Theorem for Characteristic Functions." §4.6—4.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 72-77, 1951. Characteristic (Partial Differential Equation) Paths in a 2-D plane used to transform Partial Dif- ferential Equations into systems of Ordinary Dif- ferential EQUATIONS. They were invented by Rie- mann. For an example of the use of characteristics, con- sider the equation Ut - 6uu x = 0. Now let u(s) = u(x(s))t(s)). Since it follows that dt/ds = 1, dx/ds = — 6u, and du/ds = 0. Integrating gives t(s) = s, x(s) — -6su (x) J and u(s) = uo(x) 7 where the constants of integration are and Uq(x) = u(x, 0). Characteristic Polynomial The expanded form of the CHARACTERISTIC EQUATION. det(al - A), where A is an n x n MATRIX and I is the IDENTITY Matrix. see also Cayley-Hamilton Theorem Characteristic (Real Number) For a Real Number x, [^J = int(x) is called the char- acteristic. Here, [x\ is the FLOOR FUNCTION. see also MANTISSA, SCIENTIFIC NOTATION Charlier's Check A check which can be used to verify correct computation of Moments. Chasles-Cayley-Brill Formula The number of coincidences of a (i/, i/') correspondence of value 7 on a curve of Genus p is given by v + v + 2^7. see also Zeuthen's Theorem References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 129, 1959. Chasles's Contact Theorem If a one-parameter family of curves has index N and class M, the number tangent to a curve of order m and class mi in general position is mi TV -hm M. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 436, 1959. Chasles's Polars Theorem If the Trilinear Polars of the Vertices of a Tri- angle are distinct from the respectively opposite sides, they meet the sides in three Collinear points. see also COLLINEAR, TRIANGLE, TRILINEAR POLAR du ds dx dt ~ru x + -j-u t , ds ds 230 Chasles's Theorem Chasles's Theorem If two projective PENCILS of curves of orders n and n' have no common curve, the LOCUS of the intersections of corresponding curves of the two is a curve of order n + n f through all the centers of either PENCIL. Conversely, if a curve of order n + n 1 contains all centers of a PENCIL of order n to the multiplicity demanded by Noether'S Fundamental Theorem, then it is the Locus of the intersections of corresponding curves of this PENCIL and one of order n projective therewith. see also Noether's Fundamental Theorem, Pencil References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 33, 1959. Chebyshev Approximation Formula Using a Chebyshev Polynomial of the First Kind T\ define Cj = ^^f{x k )Tj{x k ) k=i N = NZ^ f cos {^v— / cos { k=l L y J J ^ "*i(*-§) jv Then f{x)K^c k T k (x)-\c . It is exact for the TV zeros of T N (x). This type of ap- proximation is important because, when truncated, the error is spread smoothly over [—1,1]. The Chebyshev approximation formula is very close to the MlNIMAX Polynomial. References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Chebyshev Approximation," "Deriva- tives or Integrals of a Chebyshev- Approximated Function," and "Polynomial Approximation from Chebyshev Coeffi- cients." §5.8, 5.9, and 5.10 in Numerical Recipes in FOR- TRAN: The Art of Scientific Computing, 2nd ed. Cam- bridge, England: Cambridge University Press, pp. 184- 188, 189-190, and 191-192, 1992. Chebyshev Constants N.B. A detailed on-line essay by S. Finch was the start- ing point for this entry. The constants where inf sup \e x ~ r(x)\, reRm, n x >o r(x) = P(s) q{xY p and q are mth and nth order POLYNOMIALS, and R mt n is the set all RATIONAL FUNCTIONS with REAL coeffi- cients. Chebyshev Differential Equation see also One-Ninth Constant, Rational Function References Finch, S. "Favorite Mathematical Constants." http://www. mathsoft.com/asolve/constant/onenin/onenin.html. Petrushev, P. P. and Popov, V. A. Rational Approximation of Real Functions. New York: Cambridge University Press, 1987. Varga, R. S. Scientific Computations on Mathematical Prob- lems and Conjectures. Philadelphia, PA: SIAM, 1990. Philadelphia, PA: SIAM, 1990. Chebyshev Deviation max {|/(x) - p(x)\w(x)}. a<x<b References Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI: Amer. Math. Soc, p. 41, 1975. Chebyshev Differential Equation ( 1 -^)S-S+-'» = o W for | x | < 1. The Chebyshev differential equation has reg- ular Singularities at -1, 1, and oo. It can be solved by series solution using the expansions y = ^2a n x n (2) OO oo y = \ na n x n ~ = y na n x n ~ n=0 n=l oo = J^(™ + l)a n +ix n (3) 71 = oo oo y" = ^(n + l)na ri+ ix n " 1 = ^(n + ^na^ix 71 ' 1 n=0 n=l oo = ^(n + 2)(n + l)a n+2 x n . (4) 71 = Now, plug (2-4) into the original equation (1) to obtain oo (1 - x 2 ) ^(n + 2)(n + l)a n+2 x n n-0 oo oo -x "^(n + l)n n+1 x n + rn ^ a n x n = (5) n=0 n=0 oo oo ^(n + 2)(n + l)a n+2 x n - ^(n + 2)(n + l)a n + 2 x n+2 n=0 n=0 oo oo - J](n+l)a n+1 x n+1 +m 2 ^a„x n = (6) Chebyshev Differential Equation OO CO VVn + 2)(n + l)a n+ 2X n - V^ n(n - l)a n x n+2 n=0 n=2 OO OO — > na n x n -\- m /, a nX n = (?) 2 2 2 * la2 + 3 • 2a%x — 1 • ax + m ao + m aiz + y^[(" + 2)(n + l)a n+2 - n(n - l)a„ — na n + m an]/ = (8) n=2 (2a 2 4- m 2 a ) + [(m 2 - l)ai + 6a 3 ]a + ^[(n + 2)(n + l)a n+2 + (m 2 - nVl^ = °> ( 9 ) 2a 2 +771 ao = (m 2 — l)ai + 6a3 = a n +2 2 2 n — m for n = 2, 3, . (10) (11) (12) (n + l)(n + 2) The first two are special cases of the third, so the general recurrence relation is n 2 — m for n = 0, 1, (n+l)(n + 2) Prom this, we obtain for the EVEN COEFFICIENTS a 2 = -|m 2 ao a4 a2n -a 2 = (2 2 - m 2 )(-m 2 ) ao 3*4 ~* 1-2*3*4 [(2n) 2 - m 2 ][(2n - 2) 2 - m 2 } • • • [-m 2 ] (2n)! ao, (13) (14) (15) (16) and for the Odd Coefficients So the general solution is [A,* _ m 2 ][(k - 2) 2 -m 2 ]---[-m 2 ] r V = a 1 + E z + E [{k - 2) 2 - m 2 ][(Jfe - 2) 2 - m 2 ] ■ • • [I 2 - m 2 ] 3 fc! Chebyshev- Gauss Quadrature 231 If n is Even, then y\ terminates and is a Polynomial solution, whereas if n is ODD, then y 2 terminates and is a Polynomial solution. The Polynomial solutions defined here are known as CHEBYSHEV POLYNOMIALS of the First Kind. The definition of the Chebyshev Polynomial of the Second Kind gives a similar, but distinct, recurrence relation , (n+ l) 2 - m 2 , , . fln+2 = ; , w .^ n for n = 0, 1, . . . . (21) (n + 2)(n + 3) Chebyshev Function 0(z) = ^lnp, p<a: where the sum is over PRIMES p, so hm -^-r = 1. Chebyshev-Gauss Quadrature Also called Chebyshev Quadrature. A Gaussian Quadrature over the interval [—1,1] with Weight- ing Function W(x) = l/\/i - z 2 - The Abscissas for quadrature order n are given by the roots of the CHEBY- SHEV Polynomial of the First Kind T n (x), which occur symmetrically about 0. The WEIGHTS are Wi ■ A n +l7n A n 7n-l ' A n Tk(xi)T n +i(xi) A n -! T n - l (x i )T n (x t )' (1) where A n is the COEFFICIENT of x n in T n (x). For HER- mite Polynomials, 1-m 2 o (17) Additionally, 3 2 -m 2 (3 2 -m 2 )(l 2 -m 2 ) a 5 = 4 5 a 3 = 5 , (18) so [(2n - l) 2 - m 2 ][{2n - 3) 2 - m 2 ] ■ ■ ■ [l 2 - -m 2 ] Since a ' n - L ~ (2n + l)! ai- (19) A n = 2 A n+1 A n In = = 2. |tt, WJi = T n+1 (xi)T n (xi)' T n {x) = cos(ncos x), the ABSCISSAS are given explicitly by (2i- 1)tt" Since Xi = cos T' n {Xi) = In (~1)' +1 » (20) T„ + i(o;i) = (-l)'sinai, (2) (3) (4) (5) (6) (7) (8) (9) 232 Chebyshev Inequality where on = (2i - 1)tt 2n ' all the Weights are Wi (10) (11) The explicit Formula is then f(x)dx i: vr Zt'hF^)]*^'™®- < 12 > 11^ 2 ±0.707107 1.5708 3 1.0472 ±0.866025 1.0472 4 ±0.382683 0.785398 ±0.92388 0.785398 5 0.628319 ±0.587785 0.628319 ±0.951057 0.628319 References Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 330-331, 1956. Chebyshev Inequality Apply Markov's Inequality with a = k 2 to obtain P[{x-fxf >k 2 } < ((x-nf) _a 2 k 2 = h- (^ Therefore, if a RANDOM Variable x has a finite Mean H and finite VARIANCE <r 2 , then V ft > 0, P(\x - fi\ > ft) < -^ P(\x - fi\> ka) < (2) (3) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972. Chebyshev Integral x p (l-x) q dx. /■ Chebyshev Polynomial Chebyshev Integral Inequality / fi(x)dx I f 2 (x)dx--- I f n (x)dx «/ a J a J a <{b- J a f(xi)f(x 2 )"-f n (x)dx t where /i, / 2 , . . . , f n are NONNEGATIVE integrable func- tions on [a, 6] which are monotonic increasing or decreas- ing. References Gradshteyn, IS. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979. Chebyshev Phenomenon see Prime Quadratic Effect Chebyshev Polynomial of the First Kind 0.5 -0.5 A set of Orthogonal Polynomials defined as the so- lutions to the Chebyshev Differential Equation and denoted T n (x). They are used as an approxima- tion to a Least Squares Fit, and are a special case of the Ultraspherical Polynomial with a = 0. The Chebyshev polynomials of the first kind T n (x) are illus- trated above for x £ [0, 1] and n = 1, 2, . . . , 5. The Chebyshev polynomials of the first kind can be ob- tained from the generating functions 9i(t>n) \-t z 1 - 2xt + t 2 = T {x) + 2j2T n (x)t n (1) and 9*(t,z)= , \j\^ =Y. T ^ tn ( 2 ) l-2xt + t 2 n=0 for \x\ < 1 and \t\ < 1 (Beeler et al 1972, Item 15). (A closely related Generating Function is the basis for the definition of Chebyshev Polynomial of the Second Kind.) They are normalized such that T„(l) = 1. They can also be written (3) Chebyshev Polynomial Chebyshev Polynomial 233 or in terms of a DETERMINANT X 1 ■ •• 1 2x 1 ■ ■■ 1 2x 1 ■ ■■ 1 2x ■ ■• * •• 1 2x (4) In closed form, L«/2J / v T n (x) = cosmos" 1 z) = ^ I £)* n ~ 2m (* 2 " 1)™ m=0 ^ ' (5) where (™) is a BINOMIAL COEFFICIENT and \_x\ is the Floor Function. Therefore, zeros occur when *(*-§) for k — 1, 2, . . . , n. Extrema occur for X — cos I — J , (6) (7) where k = 0, 1, . . . , n. At maximum, T n (x) = 1, and at minimum, T n (x) = -1. The Chebyshev POLYNOMI- ALS are Orthonormal with respect to the Weighting Function (1 - x 2 )~ 1/2 /', T m (x)T n {x)dx Vl-x 2 {I ir8 n m for m ^ 0, n ^ for m = n = 0, (8) where £ m n is the KRONECKER DELTA. Chebyshev poly- nomials of the first kind satisfy the additional discrete identity m s -£-' m for % = 7 = 0, where Xk for fc = 1, . . . , m are the m zeros of T m (x). They also satisfy the Recurrence Relations T n+1 (x) = 2xT n (x) - T n _i(x) (10) T n+ i(a:) - xT„(x) - ^/(l- x 2){l-[T n (x)}2} (11) for n > 1. They have a Complex integral representa- tion Tn{x) = 4ri I l-2 X z + z> (12) and a Rodrigues representation Using a FAST FIBONACCI TRANSFORM with multiplica- tion law (A, B)(C, D) = (AD + BC + 2xAC, BD - AC) (14) gives (T n+ i(aO,-T n (aO) = (Ti(aO,-T (aO)(l,0) n . (15) Using Gram-Schmidt Orthonormalization in the range (-1,1) with Weighting Function (1-x 2 ) c ~ 1/2) gives Po(x) = pi(x) = p 2 {x) = /^^(1-x 2 )- 1 / 2 ^ /^(l-a: 2 )- 1 ^^ [-(l-* a ) 1/3 ]li =g ' [sin 1 a:]l: 1 /^(l-x 2 )- 1 / 2 ^ /^^(l-o: 2 )- 1 / 2 ^ f\(l - x 2 )- 1 / 2 dx (16) (17) X - • 1 = [x — 0]x — - = x — h, etc. Normalizing such that T n (l) = 1 gives T (x) = 1 Tx(x) = x T 2 (x) = 2x 2 -1 T 3 (x) = 4x 3 -Sx T 4 (x) = 8x 4 -8x 2 + l Ts(x) = 16z 5 -20z 3 + 5z T 6 (x) = 32z 6 - 48a; 4 + 18x 2 - 1. (18) The Chebyshev polynomial of the first kind is related to the Bessel Function of the First Kind J„(x) and Modified Bessel Function of the First Kind I n {x) by the relations J n (x) = i n T n (i-j^j Jo(x) (19) I n {x)=T n (J^)lo(x). (20) Letting x = cos 8 allows the Chebyshev polynomials of the first kind to be written as T n (x) = cos(rz0) = cos(ncos~ x). (21) 234 Chebyshev Polynomial Chebyshev Polynomial The second linearly dependent solution to the trans- formed differential equation d T n t 2 d9 2 + ri T n = (22) is then given by V n (x) = sin(n#) = sin(ncos~ a;), (23) which can also be written V n (x) = Vl-X 2 C/„-i(x), (24) where U n is a Chebyshev Polynomial of the Sec- ond Kind. Note that V n (x) is therefore not a Poly- nomial. The Polynomial x n - 2 L - n T n (x) (25) (of degree n — 2) is the POLYNOMIAL of degree < n which stays closest to x n in the interval (—1,1). The maximum deviation is 2 1 ~ n at the n -+- 1 points where (26) for k = 0, 1, . . . , n (Beeler et al. 1972, Item 15). see also Chebyshev Approximation Formula, Chebyshev Polynomial of the Second Kind References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972. Arfken, G. "Chebyshev (TschebyschefF) Polynomials" and "Chebyshev Polynomials — Numerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985. Beeler, M.; Gosper, R. W.; and Schroeppel, R HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM 239, Feb. 1972. Iyanaga, S. and Kawada, Y. (Eds.). "Cebysev (Tscheby- schefF) Polynomials." Appendix A, Table 20.11 in Encyclo- pedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1478-1479, 1980. Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990. Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi- als T n (x) and U n (x)" Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987. Chebyshev Polynomial of the Second Kind A modified set of Chebyshev Polynomials defined by a slightly different GENERATING FUNCTION. Used to de- velop four- dimensional SPHERICAL HARMONICS in an- gular momentum theory. They are also a special case of the Ultraspherical Polynomial with a = 1. The Chebyshev polynomials of the second kind U n (x) are illustrated above for x 6 [0, 1] and n— 1, 2, ..., 5. The defining GENERATING FUNCTION of the Chebyshev polynomials of the second kind is g<2(t,x) = 1 1 - 2xt + t 2 Y,Un(x)t n (1) for \x\ < 1 and \t\ < 1. To see the relationship to a Chebyshev Polynomial of the First Kind (T), take dg/Ot, ^ = -(1 - 2xt + t 2 )~\~2x + 2t) - 2(t - x){l - 2xt -\- 1 2 )~ 2 oo = \ nC/n(x)£ n-1 . n— Multiply (2) by t, oo {2t 2 -2xt){l-2xt-rt 2 )~ 2 = ^nU n {x)t n n=0 and take (3) -(2), {2t 2 - 2tx) - (1 - 2xt + t 2 ) _ t 2 - 1 (2) (3) (l-2xt + t 2 ) 2 {l-2xt + t) 2 oo = 5> -!)£/„(*)*"• (4) The Rodrigues representation is Un{x) = (-i)> + iysF 2\n+l/2i [(1 _ X *)W} 2»+ 1 (n+ |)!(1 -x 2 y/*dx n The polynomials can also be written u n {x)= X)(-ir( n /)(2xr- a - rv 2 i / x ^ \2m + l/ v } (5) (6) where [a; J is the Floor Function and \x] is the Ceil- ing Function, or in terms of a Determinant U n 2x 1 2x 1 1 2x 1 1 2x (7) Chebyshev Quadrature Chebyshev Quadrature 235 The first few POLYNOMIALS are U (x) = 1 Ui(x) = 2x U 2 {x) = 4x 2 - 1 U 3 (x) = 8x 3 - 4x Ut{x) = 16z 4 - 12z 2 + 1 U 5 (x) = 32a; 5 - 32a; 3 + 6a; U 6 (x) = 64a; 6 - 80a; 4 + 24a; 2 - 1 Letting x = cos 6 allows the Chebyshev polynomials of the second kind to be written as U n (x) = sin[(ra+l)fl] sin# (8) The second linearly dependent solution to the trans- formed differential equation is then given by W n (x) cos[(n+l)fl] sin# which can also be written W n (x) = {l-x 2 )- 1/2 T n + 1 (x), (9) (10) where T n is a CHEBYSHEV POLYNOMIAL OF THE FIRST Kind. Note that W n (x) is therefore not a Polynomial. see also Chebyshev Approximation Formula, Chebyshev Polynomial of the First Kind, Ultra- spherical Polynomial References Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972. Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and "Chebyshev Polynomials — Numerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985. Rivlin, T. J. Chebyshev Polynomials, New York: Wiley, 1990. Spanier, J. and Oldham, K. B. "The Chebyshev Polynomi- als T n (x) and U n [x). n Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987. Chebyshev Quadrature A Gaussian QuADRATURE-like Formula for numeri- cal estimation of integrals. It uses Weighting Func- tion W(x) = 1 in the interval [-1, 1] and forces all the weights to be equal. The general FORMULA is /; f(x)dx = - \ }{xi). n *■ — ^ The ABSCISSAS are found by taking terms up to y n in the MACLAURIN SERIES of Sn(y) = exp < | in -2 + ln(l-y)(l-i) + ln(l + y) H)]} and then defining G n (x) = x n s n (-) The ROOTS o£G n (x) then give the ABSCISSAS. The first few values are G (x) = 1 G\{x) = x G 2 (x) = l(3x 2 ~l) G s {x) = l(2x 3 -x) G*(x) = ^(45z 4 -30:£ 2 + i) G s (x) = ^(72a; 5 - 60x 3 + 7x) G G {x) = ^(105x 6 - 105x 4 + 21z 2 - 1) Gr{x) G 8 (x) G 9 (x) = j^ (6480a; 7 - 7560a; 5 + 2142a; 3 - 149a;) 56700x 6 + 20790a; 4 6480 42k (42525a; 8 - 2220a; 2 - 43) 22^ (22400a; 9 - 33600x 7 + 15120a; 5 2280a; 3 + 53a;). Because the ROOTS are all REAL for n < 7 and n = 9 only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature. The error term is _ I c n (n+1)! n n ~) c f {n+2) U) _ I ° Tl (n+2)! U odd even. where {J_ xG n (x)dx n odd I-i x 2 Gn{x)dx n even. The first few values of c n are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to n = 7 and Hildebrand (1956) up to n = 9. 236 Chebyshev-Radau Quadrature Chebyshev's Theorem cally for small n. n x» 2 ±0.57735 3 ±0.707107 4 ±0.187592 ±0.794654 5 ±0.374541 ±0.832497 6 ±0.266635 ±0.422519 ±0.866247 7 ±0.323912 ±0.529657 ±0.883862 9 ±0.167906 ±0.528762 ±0.601019 ±0.911589 d w eights can be n Xi 2 ±|V3 3 ±|V2 4 5 i ■ 1 y/h-2 ± V sVs ±\^-^F .1 /s+x/TT =C 2 V 3 see a/so Chebyshev Quadrature, Lobatto Quad- rature References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345-351, 1956. Chebyshev-Radau Quadrature A Gaussian QuADRATURE-like Formula over the in- terval [-1, 1] which has Weighting Function W(x) = x. The general FORMULA is /l " xf(x)dx = ^Wilfixt) - f(-Xi)]. 1 i=i n Xi Wi 1 0.7745967 0.4303315 2 0.5002990 0.2393715 0.8922365 0.2393715 3 0.4429861 0.1599145 0.7121545 0.1599145 0.9293066 0.1599145 4 0.3549416 0.1223363 0.6433097 0.1223363 0.7783202 0.1223363 0.9481574 0.1223363 References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987. Chebyshev Sum Inequality If Cb\ > 0,2 > . • • > 0,-n h >b 2 >...>6n, then n z2 akbk - ( Z-s ak } [ z2^ k J ' k^i \ fc=i / \ k=i / This is true for any distribution. see also CAUCHY INEQUALITY, HOLDER SUM INEQUAL- ITY References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979. Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 43-44, 1988. Chebyshev-Sylvester Constant In 1891, Chebyshev and Sylvester showed that for suf- ficiently large x, there exists at least one prime number p satisfying x < p < (1 + a)x, where a = 0.092.... Since the PRIME NUMBER THE- OREM shows the above inequality is true for all a > for sufficiently large x t this constant is only of historical interest. References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 22, 1983. Chebyshev's Theorem see Bertrand's Postulate Checker-Jumping Problem Chern Number 237 Checker-Jumping Problem Seeks the minimum number of checkers placed on a board required to allow pieces to move by a sequence of horizontal or vertical jumps (removing the piece jumped over) n rows beyond the forward-most initial checker. The first few cases are 2, 4, 8, 20. It is, however, impos- sible to reach level 5. References Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 23-28, 1976. Checkerboard see Chessboard Checkers Beeler et al. (1972, Item 93) estimated that there are about 10 12 possible positions. However, this disagrees with the estimate of Jon Schaeffer of 5 x 10 20 plausible positions, with 10 18 reachable under the rules of the game. Because "solving" checkers may require only the Square Root of the number of positions in the search space (i.e., 10 9 ), so there is hope that some day checkers may be solved (i.e., it may be possible to guarantee a win for the first player to move before the game is even started; Dubuque 1996). Depending on how they are counted, the number of Eu- LERIAN CIRCUITS on an n x n checkerboard are either 1, 40, 793, 12800, 193721, ... (Sloane's A006240) or 1, 13, 108, 793, 5611, 39312, . . . (Sloane's A006239). see also Checkerboard, Checker-Jumping Prob- lem References Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Dubuque, W. "Re: number of legal chess positions." math- fun@cs.arizona.edu posting, Aug 15, 1996. Kraitchik, M. "Chess and Checkers" and "Checkers (Draughts)." §12.1.1 and 12.1.10 in Mathematical Recre- ations. New York: W. W. Norton, pp. 267-276 and 284- 287, 1942. Schaeffer, J. One Jump Ahead: Challenging Human Supremacy in Checkers. New York: Springer- Verlag, 1997. Sloane, N. J. A. Sequences A006239/M4909 and A006240/ M5271 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Checksum A sum of the digits in a given transmission modulo some number. The simplest form of checksum is a parity bit appended on to 7-bit numbers (e.g., ASCII characters) such that the total number of Is is always EVEN ("even parity") or Odd ("odd parity"). A significantly more sophisticated checksum is the CYCLIC REDUNDANCY Check (or CRC), which is based on the algebra of poly- nomials over the integers (mod 2). It is substantially more reliable in detecting transmission errors, and is one common error- checking protocol used in modems. see also Cyclic Redundancy Check, Error- Correcting Code References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cyclic Redundancy and Other Checksums." Ch. 20.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cam- bridge University Press, pp. 888-895, 1992. Cheeger's Finiteness Theorem Consider the set of compact n-RlEMANNlAN MANIFOLDS M with diameter(M) < d, Volume(M) > V, and \K\ < k where k is the Sectional Curvature. Then there is a bound on the number of DlFFEOMORPHlSMS classes of this set in terms of the constants n, d, V, and k. References Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994. Chefalo Knot A fake KNOT created by tying a SQUARE Knot, then looping one end twice through the KNOT such that when both ends are pulled, the KNOT vanishes. Chen's Theorem Every "large" EVEN INTEGER may be written as 2n = p -J- m where p is a Prime and m 6 P2 is the Set of Semiprimes (i.e., 2-Almost Primes). see also ALMOST PRIME, PRIME NUMBER, SEMIPRIME References Rivera, C "Problems & Puzzles (Conjectures): Chen's Conjecture." http://www.sci.net .mx/-crivera/ppp/ conj_002.htm. Chern Class A Gadget defined for Complex Vector Bundles. The Chern classes of a Complex Manifold are the Chern classes of its Tangent Bundle. The ith Chern class is an OBSTRUCTION to the existence of (n — i + 1) everywhere COMPLEX linearly independent VECTOR Fields on that Vector Bundle. The zth Chern class is in the (2z)th cohomology group of the base SPACE. see also OBSTRUCTION, PONTRYAGIN CLASS, STIEFEL- Whitney Class Chern Number The Chern number is defined in terms of the Chern Class of a Manifold as follows. For any collection Chern Classes such that their cup product has the same Dimension as the Manifold, this cup product can be evaluated on the Manifold's Fundamental CLASS. The resulting number is called the Chern num- ber for that combination of Chern classes. The most important aspect of Chern numbers is that they are COBORDISM invariant. see also Pontryagin Number, Stiefel-Whitney Number 238 Chemoff Face Chess Chernoff Face A way to display n variables on a 2-D surface. For in- stance, let x be eyebrow slant, y be eye size, z be nose length, etc. References Gonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, p. 212, 1993. Chess Chess is a game played on an 8x8 board, called a CHESS- BOARD, of alternating black and white squares. Pieces with different types of allowed moves are placed on the board, a set of black pieces in the first two rows and a set of white pieces in the last two rows. The pieces are called the bishop (2), king (1), knight (2), pawn (8), queen (1), and rook (2). The object of the game is to capture the opponent's king. It is believed that chess was played in India as early as the sixth century AD. In a game of 40 moves, the number of possible board positions is at least 10 120 according to Peterson (1996). However, this value does not agree with the 10 pos- sible positions given by Beeler et al. (1972, Item 95). This value was obtained by estimating the number of pawn positions (in the no-captures situation, this is 15 ), times all pieces in all positions, dividing by 2 for each of the (rook, knight) which are interchangeable, divid- ing by 2 for each pair of bishops (since half the posi- tions will have the bishops on the same color squares). There are more positions with one or two captures, since the pawns can then switch columns (Schroeppel 1996). Shannon (1950) gave the value P(40) : 64! 32!(8!) 2 (2!) 6 10 4 The number of chess games which end in exactly n plies (including games that mate in fewer than n plies) for n = 1, 2, 3, . . . are 20, 400, 8902, 197742, 4897256, 119060679, 3195913043, ... (K. Thompson, Sloane's A007545). Rex Stout's fictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: "Wolfe grunted. One hundred and sixty-nine million, five hundred and eighteen thousand, eight hun- dred and twenty-nine followed by twenty-one ciphers. The number of ways the first ten moves, both sides, may be played" (Stout 1983). The number of chess positions after n moves for n — 1, 2, . , . are 20, 400, 5362, 71852, 809896?, 9132484?, . . . (Schwarzkopf 1994, Sloane's A019319). Cunningham (1889) incorrectly found 197,299 games and 71,782 positions after the fourth move. C. Flye St. Marie was the first to find the correct number of po- sitions after four moves: 71,852. Dawson (1946) gives the source as Intermediare des Mathematiques (1895), but K. Fabel writes that Flye St. Marie corrected the number 71,870 (which he found in 1895) to 71,852 in 1903. The history of the determination of the chess se- quences is discussed in Schwarzkopf (1994). Two problems in recreational mathematics ask 1. How many pieces of a given type can be placed on a Chessboard without any two attacking. 2. What is the smallest number of pieces needed to oc- cupy or attack every square. The answers are given in the following table (Madachy 1979). Piece Max. Min. bishops 14 8 kings 16 9 knights 32 12 queens 8 5 rooks 8 8 see also BISHOPS PROBLEM, CHECKERBOARD, CHECK- ERS, Fairy Chess, Go, Gomory's Theorem, Hard Hexagon Entropy Constant, Kings Problem, Knight's Tour, Magic Tour, Queens Problem, Rooks Problem, Tour References Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre- ations and Essays, 13th ed. New York: Dover, pp. 124- 127, 1987. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Dawson, T. R. "A Surprise Correction." The Fairy Chess Review 6, 44, 1946. Dickins, A. "A Guide to Fairy Chess." p. 28, 1967/1969/ 1971. Dudeney, H. E. "Chessboard Problems," Amusements in Mathematics. New York: Dover, pp. 84-109, 1970. Fabel, K. "Nusse." Die Schwalbe 84, 196, 1934. Fabel, K. "Weihnachtsniisse." Die Schwalbe 190, 97, 1947. Fabel, K. "Weihnachtsniisse." Die Schwalbe 195, 14, 1948. Fabel, K. "Eroffnungen." Am Rande des Schachbretts, 34— 35, 1947. Fabel, K. "Die ersten Schritte." Rund um das Schachbrett, 107-109, 1955. Fabel, K. "Eroffnungen." Schach und Zahl 8, 1966/1971. Hunter, J. A. H. and Madachy, J. S. Mathematical Diver- sions. New York: Dover, pp. 86-89, 1975. Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathemati- cal Recreations. New York: W. W. Norton, pp. 267-276, 1942. Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in Madachy 's Mathematical Recreations. New York: Dover, pp. 34-54, 1979. Peterson, I. "The Soul of a Chess Machine: Lessons Learned from a Contest Pitting Man Against Computer." Sci. News 149, 200-201, Mar. 30, 1996. Petkovic, M. Mathematics and Chess. New York: Dover, 1997. Schroeppel, R. "Reprise: Number of legal chess positions." tech-news@cs.arizona.edu posting, Aug. 18, 1996. Schwarzkopf, B. "Die ersten Ziige." Problemkiste, 142—143, No. 92, Apr. 1994. Shannon, C. "Programming a Computer for Playing Chess." Phil. Mag. 41, 256-275, 1950. Sloane, N. J. A. Sequences A019319 and A007545/M5100 in "An On-Line Version of the Encyclopedia of Integer Se- quences." Chessboard Chi Distribution 239 Stout, R. "Gambit." In Seven Complete Nero Wolfe Novels. New York: Avenic Books, p. 475, 1983. Chessboard A board containing 8x8 squares alternating in color between black and white on which the game of Chess is played. The checkerboard is identical to the chessboard except that chess's black and white squares are colored red and white in CHECKERS. It is impossible to cover a chessboard from which two opposite corners have been removed with DOMINOES. see also Checkers, Chess, Domino, Gomory's The- orem, Wheat and Chessboard Problem References Pappas, T. "The Checkerboard." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 136 and 232, 1989. Chevalley Groups Finite Simple Groups of Lie-Type. They include four families of linear SIMPLE GROUPS: PSL(n,q), PSU(n,q), PSp(2n,q), or PQ € (n,q). see also Twisted Chevalley Groups References Wilson, R. A. "ATLAS of Finite Group Representation." http : //f or . mat . bham . ac . uk/atlas#chev. Chevalley's Theorem Let f{x) be a member of a Finite Field F[xx, #2, . . • jX n ] and suppose /(0,0,...,0) = and n is greater than the degree of /, then / has at least two zeros in A n {F). References Chevalley, C "Demonstration d'une hypothese de M. Artin." Abhand. Math. Sem. Hamburg 11, 73-75, 1936. Ireland, K. and Rosen, M. "Chevalley's Theorem." §10.2 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, pp. 143-144, 1990. Chevron A 6-Polyiamond. References Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton University Press, p. 92, 1994. Chi ^, ./ s , f Z cosht — 1 . ■ Chi(jz) = 7 + In z + / dt } Jo ^ where 7 is the Euler-Mascheroni Constant. The function is given by the Mathematica® (Wolfram Re- search, Champaign, IL) command CoshlntegralEz] . see also Cosine Integral, Shi, Sine Integral References Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co- sine Integrals." §5.2 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972. Chi Distribution The probability density function and cumulative distri- bution function are Pn{x) 2 l-n/2 x n~l e -x 2 /2 D n {x) = Q{\n,\x 2 ), where Q is the Regularized Gamma Function. v^r(i(n + i)) M= r(in) 2 ^ 2[r(in)r(l + ln)-r 2 (f(n+l))] = 2T*{\{n + 1)) - 3r(|n)r(§(n + l))r(l + \n) 71 [r(In)r(l + in)-r»(i(n + l))] 8 / a (1) (2) (3) (4) 240 72 = Chi Inequality [r(in)r(l + in)-H(I(„ + l))]3/2 -3r*(i(n + 1)) + er(§n) + r 2 (|(n + i))r(i + i n ) (5) [r(|n)r(^)-r»(i(n + i))]» -AT*C-n)T(\(n + i))r(*±=) + r»(±n)r(±±*) [r(^)r(^)-r 2 (i(n + i))] 2 (6) where m is the MEAN, <r 2 the VARIANCE, 71 the Skew- ness, and 72 the Kurtosis. For n = 1, the x distribu- tion is a Half-Normal Distribution with = 1. For n = 2, it is a Rayleigh Distribution with a = 1. see a/50 Chi-Squared Distribution, Half-Normal Distribution, Rayleigh Distribution Chi Inequality The inequality (j + l)aj -V ca> (j + l)i, which is satisfied by all ^-SEQUENCES. References Levine, E. and O'Sullivan, J. "An Upper Estimate for the Reciprocal Sum of a Sum- Free Sequence." Acta Arith. 34, 9-24, 1977. Chi-Squared Distribution A x 2 distribution is a Gamma Distribution with = 2 and a = r/2, where r is the number of DEGREES OF Freedom. If Y» have Normal Independent distribu- tions with MEAN and VARIANCE 1, then -£* 2 (i) is distributed as x* witn n DEGREES OF FREEDOM. If Xi 2 are independently distributed according to a x 2 dis- tribution with m, 712, . . . , n*. DEGREES OF FREEDOM, then Xj (2) is distributed according to x with n = X] n =i n J DE- GREES of Freedom. P n (x) = \ r(|r)2-/2 - (3) for x < 0. The cumulative distribution function is then _ , a , f x t^e-^dt Chi-Squared Distribution where P(a, z) is a REGULARIZED GAMMA FUNCTION. The Confidence Intervals can be found by finding the value of x for which D n (x) equals a given value. The Moment-Generating Function of the x 2 distri- bution is M(t)-- = (1 - 2t)~ T/2 (5) R(t) = Eblj M(t) = - §rln(l- -2t) (6) R'(t) -- 1- r -2t (7) R"(t) = 2r (8) (1 -2ty M = R'(0) = = r (9) 2 = R"(0)-- = 2r (10) 71 12 (11) 72 (12) The nth Moment about zero for a distribution with n Degrees of Freedom is m' n = 2- r( ' 1 1 ^ = r(r + 2) ■ ■ ■ (r + 2n - 2), (13) and the moments about the MEAN are fJL2 = 2r (14) A*3 = 8r (15) p 4 = 12n 2 + 48n. (16) The nth CUMULANT is « n = 2 n r(n)(|r) = 2 n - x (n - l)!r, (17) The Moment-Generating Function is -r/2 9 *\/2A -r/2 As r* — ► 00, so for large r, lim M(t) = e* 2/2 , r/2 ^i/E (x< - /J,) 2 <Ti' (18) (19) (20) Chi-Squared Distribution Chi-Squared Test 241 is approximately a Gaussian Distribution with MEAN y/2r and VARIANCE <t 2 = 1. Fisher showed that X 2 ~r V27--1 (21) is an improved estimate for moderate r. Wilson and Hilferty showed that 1/3 (22) is a nearly GAUSSIAN DISTRIBUTION with MEAN \i = 1 - 2/(9r) and VARIANCE a 2 = 2/(9r). In a Gaussian Distribution, P(x) dx = ~^=e~ (x ~ » )2/2(r2 dx, (23) let Then so But z = (x — fi) I a . dx = — -=dz. 2v^ P(z)dz = 2P(x)dx, r(f)2V2 \/27r (24) dz ^2(x-^ dx= 2^z dx (T z (7 (26) (27) P(x) dx = 2 —^—e-^ 2 dz = -^=e~ z/2 dz. (28) This is a \ 2 distribution with r = 1, since 1/2-1 -z/2 1/2-1/2 P(z) ^ = e d* = -L dz. (29) oFi is the Confluent Hypergeometric Limit Func- tion and T is the GAMMA FUNCTION. The Mean, Variance, Skewness, and Kurtosis are \i = A + n 2 7i 72 2(2A + n) 2y / 2(3A + n) (2A + n)3/2 12(4A + n) (2A + n) 2 * (34) (35) (36) (37) see also Chi Distribution, Snedecor's F-Distribu- tion References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 940-943, 1972. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Incomplete Gamma Function, Error Func- tion, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992. Spiegel, M . R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115-116, 1992. Chi-Squared Test Let the probabilities of various classes in a distribution be pi , p2 , . . . , Pk • The expected frequency £ (mi - Npj) 2 N Pi is a measure of the deviation of a sample from expecta- tion. Karl Pearson proved that the limiting distribution of \s 2 is x 2 (Kenney and Keeping 1951, pp. 114-116). If Xi are independent variates with a NORMAL DISTRI- BUTION having MEANS \i{ and VARIANCES a 2 for i = 1, . . . , n, then i 2 _ v^ (Xi -in) = £ 2 A ~ £^ 2<7i 2 1=1 (30) is a Gamma Distribution variate with a = n/2, r( ? n) (31) The noncentral chi-squared distribution is given by P(x) = 2-" /2 e - (A+l)/2 x n/2 - 1 F(in, f Ax), (32) where F(a,z) = oFi(;a;z) T(a) ' (33) Pr(* 2 >X* 2 )= f^ f(x 2 )d( X 2 ) Jxs 2 2\M)/2 = 1 ~ 2 = 1 f (*) ,V/3 r(ft=i) d(x 2 ) = 1-1 Xs k-3 V^^T)' 2 where I(x i n) is PEARSON'S FUNCTION. There are some subtleties involved in using the x 2 test to fit curves (Ken- ney and Keeping 1951, pp. 118-119). When fitting a one-parameter solution using x 2 > the best-fit parameter value can be found by calculating % 2 242 Child Choose at three points, plotting against the parameter values of these points, then finding the minimum of a PARABOLA fit through the points (Cuzzi 1972, pp. 162-168). References Cuzzi, J. The Subsurface Nature of Mercury and Mars from Thermal Microwave Emission. Ph.D. Thesis. Pasadena, CA: California Institute of Technology, 1972. Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951. Child A node which is one EDGE further away from a given Edge in a Rooted Tree. see also Root (Tree), Rooted Tree, Sibling Chinese Hypothesis A Prime p always satisfies the condition that 2 P — 2 is divisible by p. However, this condition is not true exclusively for PRIME (e.g., 2 341 — 2 is divisible by 341 = 11*31). Composite Numbers n (such as 341) for which 2 n - 2 is divisible by n are called Poulet Numbers, and are a special class of Fermat Pseudoprimes. The Chinese hypothesis is a special case of FERMAT's LITTLE Theorem. see also Carmichael Number, Euler's Theorem, Fermat's Little Theorem, Fermat Pseudoprime, Poulet Number, Pseudoprime References Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 19-20, 1993. Chinese Remainder Theorem Let r and s be Positive Integers which are Rela- tively Prime and let a and b be any two Integers. Then there is an INTEGER N such that and the 6, are determined from M and N = a (mod r) N = b (mod 5) . (i) (2) Moreover, iV is uniquely determined modulo rs. An equivalent statement is that if (r,s) = 1, then every pair of Residue Classes modulo r and s corresponds to a simple RESIDUE CLASS modulo rs. The theorem can also be generalized as follows. Given a set of simultaneous CONGRUENCES x = a,i (mod rrii) (3) for i — 1, . . . , r and for which the rrti are pairwise Rela- tively Prime, the solution of the set of Congruences is x = aibi (- . . . -h a r b r (mod M), (4) mi m r bi — = 1 (mod rrii). TTli (6) where M = m\m2 - - *rn r (5) References Ireland, K. and Rosen, M. "The Chinese Remainder Theo- rem." §3.4 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer- Verlag, pp. 34-38, 1990. Uspensky, J. V. and Heaslet, M. A. Elementary Number The- ory. New York: McGraw-Hill, pp. 189-191, 1939. Wagon, S. "The Chinese Remainder Theorem." §8.4 in Math- ematica in Action. New York: W. H. Freeman, pp. 260- 263, 1991. Chinese Rings see Baguenaudier Chiral Having forms of different HANDEDNESS which are not mirror-symmetric. see also Disymmetric, Enantiomer, Handedness, Mirror Image, Reflexible Choice Axiom see Axiom of Choice Choice Number see Combination Cholesky Decomposition Given a symmetric POSITIVE DEFINITE MATRIX A, the Cholesky decomposition is an upper TRIANGULAR MA- TRIX U such that A-U T U. see also LU Decomposition, QR Decomposition References Nash, J. C. "The Choleski Decomposition." Ch. 7 in Com- pact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 84-93, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Cholesky Decomposition." §2.9 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 89-91, 1992. Choose An alternative term for a BINOMIAL COEFFICIENT, in which C?} is read as "n choose k" R. K. Guy suggested this pronunciation around 1950, when the notations n C r and n C r were commonly used. Leo Moser liked the pro- nunciation and he and others spread it around. It got the final seal of approval from Donald Knuth when he incorporated it into the TeX mathematical typesetting language as {n\choose k}. Choquet Theory Chow Coordinates 243 Choquet Theory Erdos proved that there exist at least one Prime of the form Ak + 1 and at least one Prime of the form 4k -f 3 between n and 2n for all n > 6. see also Equinumerous, Prime Number Chord chord^ The Line Segment joining two points on a curve. The term is often used to describe a LINE Segment whose ends lie on a CIRCLE. In the above figure, r is the RA- DIUS of the CIRCLE, a is called the Apothem, and s the Sagitta. s s_ The shaded region in the left figure is called a Sector, and the shaded region in the right figure is called a SEG- MENT. All ANGLES inscribed in a Circle and subtended by the same chord are equal. The converse is also true: The LOCUS of all points from which a given segment subtends equal ANGLES is a CIRCLE. Let a Circle of Radius R have a Chord at distance r. The Area enclosed by the Chord, shown as the shaded region in the above figure, is then f , v / J? 2„ 7 .2 A = 2 / x(y) dy. Jo But y 2 + (r + x) 2 = R 2 , x(y) = \/R 2 - y 2 - r (1) (2) (3) and A = 2 / (y/R 2 -y 2 Jo r)dy y^R 2 -y 2 +R 2 tan" 1 ■i^ 2ry ■.ry/B? ™r 2 + J^ 2 tan" 1 sfR? :i)'- = i^tan" 1 (f) : - r^R 2 - \ 2r^R? - r 2 (4) Checking the limits, when r = R, A = and when r->0, A=\kR\ (5) see also Annulus, Apothem, Bertrand's Problem, Concentric Circles, Radius, Sagitta, Sector, Segment Chordal see Radical Axis Chordal Theorem The LOCUS of the point at which two given CIRCLES possess the same POWER is a straight line PERPENDIC- ULAR to the line joining the MIDPOINTS of the CIRCLE and is known as the chordal (or RADICAL Axis) of the two Circles. References Dorrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965. Chow Coordinates A generalization of GRASSMANN COORDINATES to m-D varieties of degree d in P n , where P n is an n-D pro- jective space. To define the Chow coordinates, take the intersection of a m-D VARIETY Z of degree d by an (n - m)-D SUBSPACE U of P n . Then the coordi- nates of the d points of intersection are algebraic func- tions of the Grassmann Coordinates of U, and by taking a symmetric function of the algebraic functions, a hHOMOGENEOUS POLYNOMIAL known as the Chow form of Z is obtained. The Chow coordinates are then 244 Chow Ring the Coefficients of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor. References Chow, W.-L. and van der Waerden., B. L. "Zur algebraische Geometrie IX." Math. Ann. 113, 692-704, 1937. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117-1124, 1996. Chow Ring The intersection product for classes of rational equiva- lence between cycles on an Algebraic Variety. References Chow, W.-L. "On Equivalence Classes of Cycles in an Alge- braic Variety." Ann. Math. 64, 450-479, 1956. Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-L "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117-1124, 1996. Chow Variety The set C n ,m,d of all rn-D varieties of degree d in an n-D projective space P n into an M-D projective space P M . References Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. "Wei-Liang Chow." Not. Amer. Math. Soc. 43, 1117-1124, 1996. Christoffel-Darboux Formula For three consecutive ORTHOGONAL POLYNOMIALS Pn(x) = (A n X + B n )p n -lX ~ C n p n -2(x) (l) for n = 2, 3, . . . , where A n > 0, B n , and C n > are constants. Denoting the highest Coefficient of p n (x) by fc n , A n = kn-l •A-n rCn^n — 2 A n -i kn-i 2 (2) (3) Then Po(x)po{y) 4- . . -+p n (x)p n {y) = k n Pn + l(x)p n (y) - Pn(x)p n + l(y) kn+x x-y In the special case of x = y, (4) gives (4) \P0(X)} 2 + . . . + \p n (x)] k kn+l \Pn+l{x)Pn{x) ~ P n ( X )Pn+l(x)}. (5) References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 785, 1972. Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI: Amer. Math. Soc, pp. 42-44, 1975. Christoffel Number Christoffel-Darboux Identity <f> k (x)(f) k (y) _ <p m +l(x)(f>m(y) - (t>m{x)<f>m+l{y) a m ^ m {x -y), k=0 "" ' x (1) where 4>k{x) are ORTHOGONAL POLYNOMIALS with Weighting Function W(x) y 7m= J[cj>r n {x)fW{x)dx, (2) and Q>k — Z±±i A k (3) where A k is the COEFFICIENT of x k in <f>k(x). References Hildebrand, F. B. Introduction to Numerical Analysis. New . York: McGraw-Hill, p. 322, 1956. Christoffel Formula Let {p n {x)} be orthogonal Polynomials associated with the distribution da(x) on the interval [a, 6]. Also let p = c(x — Xi)(x - X2) ' ' • (x — Xi) (for c ^ 0) be a Polynomial of order I which is NONNEGATIVE in this interval. Then the orthogonal Polynomials {q(x)} associated with the distribution p(x) da(x) can be represented in terms of the POLYNO- MIALS p n {x) as p{x)q n {x) = Pn(x) p n + l(x) Pn(xi) Pn + l(xi) Pn(Xl) Pn+l{xi) Pn+l{x) Pn+l(xi) Pn+l{xi) In the case of a zero x k of multiplicity m > 1, we replace the corresponding rows by the derivatives of order 0, 1, 2, . . . , m - 1 of the POLYNOMIALS p n (xi), . . . , p n +l{xi) at x — — x k . References Szego, G. Orthogonal Polynomials, J^.th ed. Providence, RI: Amer. Math. Soc, pp. 29-30, 1975. Christoffel Number One of the quantities Xi appearing in the GAUSS-JACOBI Mechanical Quadrature. They satisfy Ai + A 2 + . . . + A„ = / J a da(x) = a{b) - a(a) (1) Christoffel Symbol of the First Kind Christoffel Symbol of the Second Kind 245 and are given by J a [Pn(x v )(X - X, A„ = &n + 1 1 k n Pn+l(Xv)Pn(Xv) k n 1 da(x) (2) (3) (4) (5) k n -\ p n -r{xu)Pk{x u ) where A; n is the higher COEFFICIENT of p n (x). References Szego, G. Orthogonal Polynomials, ^th ed. Providence, RI: Amer. Math. Soc, pp. 47-48, 1975. Christoffel Symbol of the First Kind Variously denoted [ij,k], [\ J ], r obc , or {ab,c}. [ij, k] : (i) where p mfc is the METRIC TENSOR and But df dq k ~ dq* [€i ' 6j) " a 9 * ' ej ei ■ a<? fc = [»M + b"M, (3) so [ab,c]= \{9ac,b+ 9bc,a- 9ab,c)' (4) References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 160-167, 1985. Christoffel Symbol of the Second Kind Variously denoted { . m . } or rg. -*m C76i kmr • • j i _ 1 fcm ( <9ffi 2 5 \ dgJ rJ d^ dq k (1) where rjj is a CONNECTION COEFFICIENT and {6c, d} is a Christoffel Symbol of the First Kind. \ b a c j =9ad{bc,d}. (2) The Christoffel symbols are given in terms of the first Fundamental Form E, F, and G by r 12 1 _ GE U - 2FF U + FE V 2(EG-F 2 ) GE V — FG U 2{EG - F 2 ) 2GF V — GG U — FG V 2(EG - F 2 ) 2£F U - EE V - FE U 2(EG-F 2 ) EG U — FE V r*22 r 2 - 1 11 — r 2 - 1 12 — 2(£G-F 2 ) ■p2 SGd — 2FF V + FG U 1 22 — 2(£G - F 2 ) (3) (4) (5) (6) (7) (8) and T^ = T\ 2 and T^ = r? 3 . If F = 0, the Christoffel symbols of the second kind simplify to (9) (10) (11) (12) (13) (14) (Gray 1993). The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first Fundamental Form, r 1 1 ii = E u 2E r 1 1 12 = E v 2E r 2 2 = G u 2E r 2 -L 11 = E v 2G r 2 1 12 = G u 2G r 2 1 22 = G v 2G T\ 1 E + T\ 1 F=\E U T 12 E + T 12 F — ^E v ^22^ + 1^22^ ~ -Pw — 2^* u ^nF + T 1X G = F u — -E v r"l2-^ + ^12^? = ^G u 1^22^ + T22G = oG v (15) (16) (17) (18) (19) (20) (21) (22) Fii + T? 2 = (In y/EG - F=> )„ ria + Im = (In yjEG - F* ). (Gray 1993). For a surface given in Monge'S Form 2 = F(x,y), r k - = ZijZk C2S^ see also Christoffel Symbol of the First Kind, Connection Coefficient, Gauss Equations 246 Chromatic Number ci References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 160-167, 1985. Gray, A. "Christoffel Symbols." §20.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 397-400, 1993. Morse, P. M. and Feshbach, H. Methods of Theoretical Phys- ics, Part L New York: McGraw-Hill, pp. 47-48, 1953. Chromatic Number The fewest number of colors j(G) necessary to color a Graph or surface. The chromatic number of a surface of GENUS g is given by the HEAWOOD CONJECTURE, l(9)= §(7+7485 + 1) where [x\ is the Floor Function. j(g) is sometimes also denoted x(p)- For g = 0, 1, ... , the first few values of x(9) are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (Sloane's A000934). The fewest number of colors necessary to color each Edge of a Graph so that no two Edges incident on the same Vertex have the same color is called the "Edge chromatic number." see also Brelaz's Heuristic Algorithm, Chro- matic Polynomial, Edge-Coloring, Euler Char- acteristic, Heawood Conjecture, Map Color- ing, Torus Coloring References Chartrand, G. "A Scheduling Problem: An Introduction to Chromatic Numbers." §9.2 in Introductory Graph Theory. New York: Dover, pp. 202-209, 1985. Eppstein, D. "The Chromatic Number of the Plane." http:// www . ics . uci . edu / - eppstein / junkyard / plane-color/. Sloane, N. J. A. Sequence A000934/M3292 in "An On-Line Version of the Encyclopedia of Integer Sequences." Chromatic Polynomial A Polynomial P(z) of a graph g which counts the number of ways to color g with exactly z colors. Tutte (1970) showed that the chromatic POLYNOMIALS of pla- nar triangular graphs possess a ROOT close to <j> 2 = 2.618033 . . ., where <j> is the GOLDEN Mean. More pre- cisely, if n is the number of VERTICES of G, then (Le Lionnais 1983). References Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Tutte, W. T. "On Chromatic Polynomials and the Golden Ratio." J. Corabin. Th. 9, 289-296, 1970. Chu Space A Chu space is a binary relation from a Set A to an antiset X which is defined as a Set which transforms via converse functions. References Stanford Concurrency Group. "Guide to Papers on Chu Spaces." http : //boole . Stanford . edu/ chuguide .html. Chu-Vandermonde Identity (x + a) n = Y^ Uj(a)fcO*On-fc where (™) is a Binomial Coefficient and (a) n = a(a - 1) • • • (a - n + 1) is the Pochhammer Symbol. A special case gives the identity max(fe,n) £ ( = m k-l i)-\ k )■ see also BINOMIAL THEOREM, UMBRAL CALCULUS References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, pp. 130 and 181-182, 1996. Church's Theorem No decision procedure exists for Arithmetic. Church's Thesis see Church-Turing Thesis Church- Turing Thesis The Turing Machine concept defines what is meant mathematically by an algorithmic procedure. Stated another way, a function / is effectively COMPUTABLE Iff it can be computed by a Turing Machine. see also ALGORITHM, COMPUTABLE FUNCTION, TUR- ING Machine References Penrose, R. The Emperor's New Mind: Concerning Comput- ers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 47-49, 1989. Chvatal's Art Gallery Theorem see Art Gallery Theorem Chvatal's Theorem Let the Graph G have Vertices with Valences di < . . . < d m . If for every i < n/2 we have either efc > i + 1 or d n -i > n - 2, then the Graph is Hamiltonian. Chu Identity see Chu-Vandermonde Identity ci see Cosine Integral a Circle 247 Ci see Cosine Integral Cigarettes It is possible to place 7 cigarettes in such a way that each touches the other if l/d > 7\/3/2 (Gardner 1959, p. 115). References Gardner, M. The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, 1959. Cin see Cosine Integral Circle A circle is the set of points equidistant from a given point O. The distance r from the Center is called the Radius, and the point O is called the Center. Twice the Radius is known as the Diameter d = 2r. The Perimeter C of a circle is called the Circumference, and is given by C = ird = 2tt7\ (1) The circle is a Conic SECTION obtained by the intersec- tion of a Cone with a Plane Perpendicular to the Cone's symmetry axis. A circle is the degenerate case of an Ellipse with equal semimajor and semiminor axes (i.e., with ECCENTRICITY 0). The interior of a circle is called a Disk. The generalization of a circle to 3-D is called a SPHERE, and to n-D for n > 4 a HYPERSPHERE. The region of intersection of two circles is called a LENS. The region of intersection of three symmetrically placed circles (as in a VENN DIAGRAM), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux Triangle. The parametric equations for a circle of RADIUS a are x — a cos t y = a sin t. For a body moving uniformly around the circle, X t y -asint a cost, and x = —a cost y" = —asint. (2) (3) (4) (5) (6) (7) When normalized, the former gives the equation for the unit Tangent Vector of the circle, (-sint,cost). The circle can also be parameterized by the rational func- tions x = y- 2t 1 + t 2 ' (8) (9) but an Elliptic Curve cannot. The following plots show a sequence of NORMAL and TANGENT VECTORS for the circle. The Arc Length s, Curvature k, and Tangential ANGLE <j> of the circle are s(t) = ds= \/x f2 + y' 2 dt = at (10) (j>(t) = I K(t)dt= -. (12) The Cesaro Equation is K=~. (13) a In POLAR COORDINATES, the equation of the circle has a particularly simple form. r = a (14) is a circle of RADIUS a centered at Origin, r = 2acos9 (15) is circle of RADIUS a centered at (a, 0), and r = 2asm6 (16) 248 Circle Circle is a circle of RADIUS a centered on (0, a). In CARTE- SIAN Coordinates, the equation of a circle of Radius a centered on (xo,2/o) is (x - x ) 2 + (y-yo) 2 (17) In Pedal Coordinates with the Pedal Point at the center, the equation is pa = r 2 . (18) The circle having P1P2 as a diameter is given by (x - xi)(x - x 2 ) + (2/ - yi){y - 2/2) = 0. (19) The equation of a circle passing through the three points (xi,yi) for i = 1, 2, 3 (the Circumcircle of the Tri- angle determined by the points) is (20) The Center and Radius of this circle can be identified by assigning coefficients of a Quadratic Curve 2 , 2 x +y X y 1 2 1 2 xi +2/1 Xi 2/1 1 2 1 2 x 2 +t/2 X 2 2/2 1 2 , 2 XZ +J/3 xz 2/3 1 ax 2 + cy 2 + dx + ey + / = 0, (21) where a — c and 6 = (since there is no xy cross term) . Completing the Square gives The Center can then be identified as Xq 2/o 2a e 2a and the Radius as where d 2 + e 2 / a 4a 2 (23) (24) (25) e = xi 2/1 I #2 2/2 1 (26) xz 2/3 1 #i 2 +2/i 2 2/i 1 Z2 2 +2/2 2 2/2 1 (27) £3 2 +2/3 2 2/3 1 zi 2 +2/i 2 X! 1 Z2 2 +2/2 2 £ 2 1 (28) Xz 2 + 2/3 2 #3 1 #i 2 +2/1 2 asi 2/i Z2 2 +2/2 2 Z 2 2/2 (29) #3 2 +2/3^ ! xz 2/3 Four or more points which lie on a circle are said to be Concyclic. Three points are trivially concyclic since three noncollinear points determine a circle. The ClRCUMFERENCE-to-DlAMETER ratio C/d for a cir- cle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor s in- creases its Perimeter by s), and d also scales by s. This ratio is denoted -k (Pi), and has been proved Transcen- dental. With d the Diameter and r the Radius, C == 7rd = 27r?\ (30) Knowing C/d, we can then compute the Area of the circle either geometrically or using CALCULUS. From Calculus, A = p1t\ nr Jo Jo rdr = (27r)(^r ) = irr (31) Now for a few geometrical derivations. Using concentric strips, we have As the number of strips increases to infinity, we are left with a Triangle on the right, so A = \{2nr)r = nr . (32) This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC). If we cut the circle instead into wedges, ^ *+ nr ► As the number of wedges increases to infinity, we are left with a RECTANGLE, so (-Kr)r = nr . (33) see also Arc, Blaschke's Theorem, Brahmagupta's Formula, Brocard Circle, Casey's Theorem, Chord, Circumcircle, Circumference, Clif- ford's Circle Theorem, Closed Disk, Concentric Circles, Cosine Circle, Cotes Circle Property, Diameter, Disk, Droz-Farny Circles, Euler Tri- angle Formula, Excircle, Feuerbach's Theorem, Circles-and-Squares Fractal Circle-Circle Intersection 249 Five Disks Problem, Flower of Life, Ford Cir- cle, Fuhrmann Circle, Gersgorin Circle Theo- rem, Hopf Circle, Incircle, Inversive Distance, Johnson Circle, Kinney's Set, Lemoine Circle, Lens, Magic Circles, Malfatti Circles, McCay Circle, Midcircle, Monge's Theorem, Moser's Circle Problem, Neuberg Circles, Nine-Point Circle, Open Disk, P-Circle, Parry Circle, Pi, Polar Circle, Power (Circle), Prime Circle, Ptolemy's Theorem, Purser's Theorem, Radi- cal Axis, Radius, Reuleaux Triangle, Seed of Life, Seifert Circle, Semicircle, Soddy Circles, Sphere, Taylor Circle, Triangle Inscribing in a Circle, Triplicate-Ratio Circle, Tucker Cir- cles, Unit Circle, Venn Diagram, Villarceau Circles, Yin- Yang References Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 125 and 197, 1987. Casey, J. "The Circle." Ch. 3 in A Treatise on the Analyt- ical Geometry of the Point, Line, Circle, and Conic Sec- tions, Containing an Account of Its Most Recent Exten- sions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 96-150, 1893. Courant, R. and Robbins, H. What is Mathematics?: An El- ementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 74-75, 1996. Dunham, W. "Archimedes' Determination of Circular Area." Ch. 4 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 84-112, 1990. Eppstein, D. "Circles and Spheres." http://www. ics . uci . edu/*eppstein/ junkyard/sphere. html. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 65-66, 1972. MacTutor History of Mathematics Archive. "Circle." http: //www -groups . dcs . st -and .ac.uk/ -history /Curves/ Circle.html. Pappas, T. "Infinity & the Circle" and "Japanese Calculus." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 68 and 139, 1989. Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Yates, R. C "The Circle." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 21-25, 1952. Circles-and-Squares Fractal m A FRACTAL produced by iteration of the equation Zn+i = z n (mod m) which results in a M0IRE-Iike pattern. see also FRACTAL, M0IRE PATTERN Circle Caustic Consider a point light source located at a point (//,0). The CATACAUSTIC of a unit CIRCLE for the light at fi = oo is the Nephroid x = ~ [3 cost - cos(3£)] y = \ [3 sin t — sin(3i)]. a) (2) The CATACAUSTIC for the light at a finite distance fx > 1 is the curve V : fi(l — 3/j, cos t + 2fi cos 3 t) -(l-\-2fi 2 )-r3ficost 2fi 2 sin 3 t 1 + 2// 2 — 3/xcost ' (3) (4) and for the light on the CIRCUMFERENCE of the CIRCLE {i — 1 is the CARDIOID x = | cos t(l + cos t) - | y — | sini(l + cost). (5) (6) If the point is inside the circle, the catacaustic is a dis- continuous two-part curve. These four cases are illus- trated below. The CATACAUSTIC for PARALLEL rays crossing a CIRCLE is a Cardioid. see also CATACAUSTIC, CAUSTIC Circle-Circle Intersection Let two Circles of Radii R and r and centered at (0, 0) and (d, 0) intersect in a LENS-shaped region. The equa- tions of the two circles are 2,2 D 2 x +y — R (x - df +y 2 = r 2 (1) (2) 250 Circle-Circle Intersection Combining (1) and (2) gives (x-d) 2 + (R 2 -x 2 ) = r 2 . Multiplying through and rearranging gives x 2 - 2dx + d 2 - x 2 = r 2 - R 2 . Solving for x results in d 2 - r 2 + R 2 2d (3) (4) (5) The line connecting the cusps of the LENS therefore has half-length given by plugging x back in to obtain 2 D 2 2 D 2 / d - r + R y = R — x = R 2d Ad 2 R 2 -{d 2 -r 2 +R 2 ) 2 Ad? (6) giving a length of a= ^V 4 ^ 1 * 2 ~ ( d2 ~ r2 + R2 ) 2 = h(-d + r-R)(-d-r + R) a x [(-d + r + R){d + r + R)] 1/2 . (7) This same formulation applies directly to the SPHERE- Sphere Intersection problem. To find the AREA of the asymmetric "Lens" in which the Circles intersect, simply use the formula for the circular SEGMENT of radius i^'and triangular height d' A{R!,d') = i^cos" 1 f^\ -d'^R' 2 -d<* (8) twice, one for each half of the "Lens." Noting that the heights of the two segment triangles are di = x ■ d 2 -r 2 + R 2 dz = d — x ■■ 2d d 2 +r 2 - R 2 2d (9) (10) The result is A = A(Ri,d 1 )+A(R 2 ,d 2 ) _i (d 2 + r 2 -R 2 2 r cos 2dr + R* cos / d 2 +E 2 -r 2 \ ^ 2dR ) - \^{d - r - R)(d + r - R){d - r + R)(d + r + R). (11) Circle Cutting The limiting cases of this expression can be checked to give when d — R + r and A = 2R 2 cos" 1 (^) - \d\/AR? - d? (12) = 2A{\d,R) (13) when r = i2, as expected. In order for half the area of two Unit Disks (R = 1) to overlap, set A = irR 2 /2 = 7r/2 in the above equation |tt = 2cos~ l (±d) - \d^J\ - d? (14) and solve numerically, yielding d w 0.807946. see also Lens, Segment, Sphere-Sphere Intersec- tion Circle Cutting 2 4 7 11 Determining the maximum number of pieces in which it is possible to divide a CIRCLE for a given number of cuts is called the circle cutting, or sometimes PANCAKE Cutting, problem. The minimum number is always n + 1, where n is the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the nth cut creates n new regions, so /(l) = 2 (1) /(2) = 2 + /(l) (2) /(n) = n+/(n-l). (3) Therefore, f(n) = n+[(n-l) + f(n-2)} n = n + (n-l) + ... + 2 + /(l) = J^ k fW fc-2 n = ^fc-l + /(l)-in(n+l)-l + 2 k = l = §(n 2 +n + 2). (4) Evaluating for n = 1, 2, . . . gives 2, 4, 7, 11, 16, 22, . . . (Sloane's A000124). OO 12 4 8 A related problem, sometimes called Moser's CIRCLE PROBLEM, is to find the number of pieces into which a Circle is divided if n points on its Circumference Circle Evolute Circle Involute 251 are joined by Chords with no three Concurrent. The answer is »<»>=(:)+©+> = 5j(n 4 - 6n 3 + 23n 2 - 18n + 24), (5) (6) (Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where (£) is a Binomial Coeffi- cient. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (Sloane's A000127). This sequence and problem are an example of the danger in making assumptions based on limited trials. While the series starts off like 2 n ~ 1 , it begins differing from this GEO- METRIC Series at n = 6. see also Cake Cutting, Cylinder Cutting, Ham Sandwich Theorem, Pancake Theorem, Pizza Theorem, Square Cutting, Torus Cutting References Conway, J. H. and Guy, R. K. "How Many Regions." In The Book of Numbers. New York: Springer- Verlag, pp. 76-79, 1996. Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697-712, 1988. Noy, M. "A Short Solution of a Problem in Combinatorial Geometry." Math. Mag. 69, 52-53, 1996. Sloane, N. J. A. Sequences A000124/M1041 and A000127/ M1119 in "An On-Line Version of the Encyclopedia of In- teger Sequences." Yaglom, A. M. and Yaglom, I. M. Problem 47. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, 1987. Circle Evolute x = cos t x = — sin t x ~ — cost (i) y = sin t y = cos t y = - - sin t, (2) so the Radius of Curvature is ^_(x' 2 +y' 2 ) 3/2 y" x' — x"y' (sin 2 t + cos 2 t) 3/2 — i i"*t (— sint)(— sint) — (— cost) cost and the TANGENT VECTOR is — sint cost Therefore, cos r —T • x = — sin t sin r ~T • y = cos t, (4) (5) (6) and the EVOLUTE degenerates to a POINT at the ORI- GIN. see also CIRCLE INVOLUTE References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, p. 77, 1993. Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 55- 59, 1991. Circle Inscribing If r is the Radius of a Circle inscribed in a Right Triangle with sides a and b and Hypotenuse c, then r = i( a + 6-c). see Inscribed, Polygon Circle Involute First studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the cir- cle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a CYCLOID. For a Circle with a = 1, the parametric equations of the circle and their derivatives are given by x = cost x =— sint x =— cost (1) y — sin t y = cos t The Tangent Vector is - sin t. T = — sint cost and the Arc LENGTH along the circle is so the involute is given by (2) (3) (4) n = r - sT = cost sint j -t — sint cost = cos t + t sin t sin t — t cos t (5) £(t) = x — R sin r — cos t — 1 • cos t = (7) >q(t) = y + Rcosr = sint + 1 * (-sint) = 0, (8) x = a(cost -f tsint) y = a(sint — tcost). (6) (7) 252 Circle Involute Pedal Curve Circle Lattice Points The Arc Length, Curvature, and Tangential An- gle are J ds= / ^x' 2 + y' 2 dt = \ 1 K = = i. The Cesaro Equation is Vas' at 2 (8) (9) (10) (11) see also Circle, Circle Evolute, Ellipse Involute, Involute References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, p. 83, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190-191, 1972. MacTutor History of Mathematics Archive. "Involute of a Circle." http://www-groups.dcs.st-and.ac.uk/-history /Curves/Involute. html. Circle Involute Pedal Curve The Pedal Curve of Circle Involute / = cos t + t sin t g = sin t — t cos t with the center as the PEDAL Point is the ARCHIME- DES' Spiral x ~ t sin t y = — tcost. Circle Lattice Points For every POSITIVE INTEGER n, there exists a CIRCLE which contains exactly n lattice points in its interior. H. Steinhaus proved that for every POSITIVE INTEGER n, there exists a Circle of Area n which contains ex- actly n lattice points in its interior. Schinzel's Theorem shows that for every Positive Integer n, there exists a Circle in the Plane hav- ing exactly n LATTICE POINTS on its CIRCUMFERENCE. The theorem also explicitly identifies such "Schinzel Circles" as {x (x l) 2 + y 2 1 cfc-] 4 5 1 r2fc 9 5 for n = 2k for n = 2fc + 1. (1) Note, however, that these solutions do not necessarily have the smallest possible RADIUS, For example, while the Schinzel Circle centered at (1/3, 0) and with RADIUS 625/3 has nine lattice points on its CIRCUM- FERENCE, so does the CIRCLE centered at (1/3, 0) with Radius 65/3. Let r be the smallest INTEGER RADIUS of a CIRCLE cen- tered at the Origin (0, 0) with L(r) Lattice Points. In order to find the number of lattice points of the Cir- cle, it is only necessary to find the number in the first octant, i.e., those with < y < [r/v^J , where [z\ is the Floor Function. Calling this N(r% then for r > 1, L(r) = 8N(r) - 4, so L(r) = 4 (mod 8). The multipli- cation by eight counts all octants, and the subtraction by four eliminates points on the axes which the multi- plication counts twice. (Since ^/2 is IRRATIONAL, the MIDPOINT of a are is never a LATTICE POINT.) Gauss's Circle Problem asks for the number of lat- tice points within a CIRCLE of RADIUS r N(r) = 1 + 4 [rj + 4 ^ ^r 2 - i 2 . Gauss showed that where N(r) = nr 2 + E(r), \E(r)\ < 2V2nr. (2) (3) (4) i The number of lattice points on the CIRCUMFERENCE of circles centered at (0, 0) with radii 0, 1, 2, . . . are 1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, . . . (Sloane's A046109). The following table gives the smallest RADIUS r < 111,000 for a circle centered at (0, 0) having a given number of LATTICE POINTS L(r). Note that the high water mark radii are always multiples of five. Circle Lattice Points Circle Map 253 L(r) r 1 4 1 12 5 20 25 28 125 36 65 44 3,125 52 15,625 60 325 68 < 390,625 76 < 1,953,125 84 1,625 92 < 48,828,125 100 4,225 108 1,105 132 40,625 140 21,125 180 5,525 252 27,625 300 71,825 324 32,045 * If the CIRCLE is instead centered at (1/2, 0), then the Circles of Radii 1/2, 3/2, 5/2, . . . have 2, 2, 6, 2, 2, 2, 6, 6, 6, 2, 2, 2, 10, 2, . . . (Sloane's A046110) on their Circumferences. If the Circle is instead centered at (1/3, 0), then the number of lattice points on the Circumference of the Circles of Radius 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, ... are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 5, 3, . . . (Sloane's A046111). Let 1. a n be the RADIUS of the CIRCLE centered at (0, 0) having 8n + 4 lattice points on its CIRCUMFERENCE, 2. b n /2 be the RADIUS of the Circle centered at (1/2, 0) having 4n + 2 lattice points on its CIRCUMFER- ENCE, 3. c n /3 be the Radius of Circle centered at (1/3, 0) having 2n + 1 lattice points on its CIRCUMFERENCE. Then the sequences {a n }, {&n}, and {c n } are equal, with the exception that b n — if 2|n and c n = if 3|n. How- ever, the sequences of smallest radii having the above numbers of lattice points are equal in the three cases and given by 1, 5, 25, 125, 65, 3125, 15625, 325, ... (Sloane's A046112). Kulikowski's Theorem states that for every Posi- tive Integer n, there exists a 3-D Sphere which has exactly n Lattice Points on its surface. The Sphere is given by the equation (x-a) 2 + {y-b) 2 + (z-^) 2 : C + 2, where a and b are the coordinates of the center of the so-called Schinzel Circle and c is its Radius (Hons- berger 1973). see also CIRCLE, CIRCUMFERENCE, GAUSS'S CIRCLE Problem, Kulikowski's Theorem, Lattice Point, Schinzel Circle, Sciiinzel's Theorem References Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117-127, 1973. Kulikowski, T. "Sur l'existence d'une sphere passant par un nombre donne aux coordonnees entieres." L'Enseignement Math. Ser. 2 5, 89-90, 1959. Schinzel, A. "Sur l'existence d'un cercle passant par un nombre donne de points aux coordonnees entieres." L'Enseignement Math. Ser. 2 4, 71-72, 1958. Sierpiiiski, W. "Sur quelques problemes concernant les points aux coordonnees entieres." L'Enseignement Math. Ser. 2 4, 25-31, 1958. Sierpinski, W. "Sur un probleme de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191-194, 1959. Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964. # Weisstein, E. W. "Circle Lattice Points." http:// www . astro . Virginia . edu/ -eww6n/ math /notebooks /Circle LatticePoints .m. Circle Lattice Theorem see Gauss's Circle Problem Circle Map A 1-D Map which maps a CIRCLE onto itself 0n+i = n + Q-^- sin(27r0„), (1) where # n +i is computed mod 1. Note that the circle map has two parameters: Q and K. Q can be interpreted as an externally applied frequency, and K as a strength of nonlinearity. The 1-D JACOBIAN is d9, n+l d0 n l-ii:cos(27r(9n), (2) so the circle map is not Area-Preserving. It is related to the Standard Map /n+l = Jn + — sin(27r0 n ) @n + l — n + /n + l, (3) (4) 254 Circle Method for / and computed mod 1. Writing 8 n +i as n+ i = n + /„ + ^- sin(27rl9 n ) (5) gives the circle map with I n = Q, and K = —K. The unperturbed circle map has the form 0n + l=0n+fi. (6) If fi is RATIONAL, then it is known as the map WINDING Number, defined by (7) and implies a periodic trajectory, since n will return to the same point (at most) every q ORBITS. If Q is Irrational, then the motion is quasiperiodic. If K is NONZERO, then the motion may be periodic in some finite region surrounding each RATIONAL Q. This exe- cution of periodic motion in response to an IRRATIONAL forcing is known as Mode Locking. If a plot is made of K vs. Q with the regions of pe- riodic MODE-LOCKED parameter space plotted around Rational Q values (Winding Numbers), then the re- gions are seen to widen upward from at K = to some finite width at K = 1. The region surrounding each Ra- tional Number is known as an Arnold Tongue. At K = 0, the Arnold Tongues are an isolated set of Measure zero. At K = 1, they form a Cantor Set of Dimension d « 0.08700. For K > 1, the tongues overlap, and the circle map becomes noninvertible. The circle map has a Feigenbaum Constant 6= lim n—¥oo U n + 1 On — On-1 n 2.833. (8) see also Arnold Tongue, Devil's Staircase, Mode Locking, Winding Number (Map) Circle Method see Partition Function P Circle Negative Pedal Curve The Negative Pedal Curve of a circle is an Ellipse if the Pedal Point is inside the Circle, and a Hy- perbola if the Pedal Point is outside the Circle. Circle Notation A Notation for Large Numbers due to Steinhaus (1983) in which is defined in terms of STEINHAUS- Moser Notation as n in n SQUARES. The particular number known as the MEGA is then defined as follows. ©-E A-\A 4 4 256 see also Mega, Megistron, Steinhaus-Moser No- tation References Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, pp. 28-29, 1983. Circle Packing Circle Order A Poset P is a circle order if it is Isomorphic to a Set of Disks ordered by containment. see also ISOMORPHIC POSETS, PARTIALLY ORDERED Set Circle Orthotomic The Orthotomic of the Circle represented by X = cos t y = sin t with a source at (x, y) is (1) (2) x = x cos(2£) - y sin(2t) + 2 sin t (3) y = ~x sin(2i) - y cos(2t) + 2 cos t. (4) Circle Packing The densest packing of spheres in the PLANE is the hexagonal lattice of the bee's honeycomb (illustrated above), which has a Packing Density of 2\/3 = 0.9068996821.. Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes Toth proved that the hexagonal lattice is indeed the densest of all possible plane packings. Solutions for the smallest diameter CIRCLES into which n Unit Circles can be packed have been proved op- timal for n = 1 through 10 (Kravitz 1967). The best known results are summarized in the following table. Circle Packing Circle-Point Midpoint Theorem 255 n d exact d approx. 1 2 3 4 5 6 7 8 9 10 11 12 1 2 l+fx/3 1 + V2 1.00000 2.00000 2.15470... 2.41421... 2.70130... 3.00000 3.00000 3.30476... 3.61312... 3.82... 4.02... 1 + \/2(l + l/\/5) 3 3 1 + csc(tt/7) 1 + ^/2(2 + ^/2) For Circle packing inside a Square, proofs are known only for n = 1 to 9. n d exact d approx. 1 1 1.000 2 0.58... 3 0.500... 4 i 2 0.500 5 0.41... 6 0.37. . . 7 0.348... 8 0.341... 9 1 3 0.333. . . 10 0.148204... The smallest Square into which two Unit Circles, one of which is split into two pieces by a chord, can be packed is not known (Goldberg 1968, Ogilvy 1990). see also Hypersphere Packing, Malfatti's Right Triangle Problem, Mergelyan-Wesler Theorem, Sphere Packing References Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer- Verlag, 1992. Eppstein, D. "Covering and Packing." http://www.ics.uci . edu/-eppstein/ junkyard/cover, html. Folkman, J. H. and Graham, R. "A Packing Inequality for Compact Convex Subsets of the Plane." Canad. Math, Bull. 12, 745-752, 1969. Gardner, M. "Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another." ScL Amer. 240, 18-28, Jan. 1979. Gardner, M. "Tangent Circles." Ch. 10 in Fractal Music, HyperCards, and More Mathematical Recreations from Sci- entific American Magazine. New York: W. H. Freeman, 1992. Goldberg, M. "Problem E1924." Amer. Math. Monthly 75, 195, 1968. Goldberg, M. "The Packing of Equal Circles in a Square." Math. Mag. 43, 24-30, 1970. Goldberg, M. "Packing of 14, 16, 17, and 20 Circles in a Circle." Math. Mag. 44, 134-139, 1971. Graham, R. L. and Luboachevsky, B, D, "Repeated Patterns of Dense Packings of Equal Disks in a Square." Elec- tronic J. Combinatorics 3, R16, 1-17, 1996. http://www. combinatorics. org/Volume^3/volume3.html#R16. Kravitz, S. "Packing Cylinders into Cylindrical Containers." Math. Mag. 40, 65-70, 1967. McCaughan, F, "Circle Packings." http://www.pmms.cam. ac, uk/ -gj ml i/cpacking/ info. html. Molland, M. and Payan, Charles. "A Better Packing of Ten Equal Circles in a Square." Discrete Math. 84, 303-305, 1990. Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 145, 1990. Reis, G. E. "Dense Packing of Equal Circle within a Circle." Math. Mag. 48, 33-37, 1975. Schaer, J. "The Densest Packing of Nine Circles in a Square." Can. Math. Bui. 8, 273-277, 1965. Schaer, J. "The Densest Packing of Ten Equal Circles in a Square." Math. Mag. 44, 139-140, 1971. Valette, G. "A Better Packing of Ten Equal Circles in a Square." Discrete Math. 76, 57-59, 1989. Circle Pedal Curve / s* \ 1 / / / / \^\ / / \ \ / 1/ 1 y 1 / / / / / / / ^ = ^^; — " ^s^ The Pedal Curve of a Circle is a Cardioid if the Pedal Point is taken on the Circumference, and otherwise a LlMAQON. Circle-Point Midpoint Theorem Taking the locus of MIDPOINTS from a fixed point to a circle of radius r results in a circle of radius r/2. This follows trivially from r(0) —x +K rcosS rsinO - —x ~r cos9 — \x - \ sin References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 17, 1929. 256 Circle Radial Curve Circle Radial Curve The Radial Curve of a unit Circle from a Radial Point (x, 0) is another Circle with parametric equa- tions x(t) = x — cost y(i) = — sint. Circle Squaring Construct a SQUARE equal in Area to a CIRCLE using only a STRAIGHTEDGE and COMPASS. This was one of the three Geometric Problems of Antiquity, and was perhaps first attempted by Anaxagoras. It was fi- nally proved to be an impossible problem when Pi was proven to be TRANSCENDENTAL by Lindemann in 1882. However, approximations to circle squaring are given by constructing lengths close to tt = 3.1415926.... Ramanujan (1913-14) and Olds (1963) give geomet- ric constructions for 355/113 = 3.1415929.... Gard- ner (1966, pp. 92-93) gives a geometric construc- tion for 3+ 16/113 = 3.1415929.... Dixon (1991) gives constructions for 6/5(1 + <fi) = 3.141640... and y / 4+[3-tan(30°)] = 3.141533 . . .. While the circle cannot be squared in EUCLIDEAN Space, it can in Gauss-Bolyai-Lobachevsky Space (Gray 1989). see also GEOMETRIC CONSTRUCTION, QUADRATURE, Squaring References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, pp. 190-191, 1996. Dixon, R. M athographics. New York: Dover, pp. 44-49 and 52-53, 1991. Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 20-26, 1990. Gardner, M. "The Transcendental Number Pi." Ch. 8 in Martin Gardner's New Mathematical Diversions from Sci- entific American. New York: Simon and Schuster, 1966. Gray, J. Ideas of Space. Oxford, England: Oxford University Press, 1989. Meyers, L. F. "Update on William Wernick's 'Triangle Con- structions with Three Located Points,"' Math. Mag. 69, 46-49, 1996. Olds, C. D. Continued Fractions. New York: Random House, pp. 59-60, 1963. Ramanujan, S. "Modular Equations and Approximations to 7T." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914, Circle Tangents Circle Strophoid The Strophoid of a Circle with pole at the center and fixed point on the CIRCUMFERENCE is a FREETH'S Nephroid. Circle Tangents There are four CIRCLES that touch all the sides of a given TRIANGLE. These are all touched by the CIRCLE through the intersection of the ANGLE BISECTORS of the Triangle, known as the Nine-Point Circle. Given the above figure, GE — FH, since AB = AG 4- GB = GE + GF = GE + {GE + EF) = 2G + EF CD = CH + HD = EH + FH = FH + (FH + EF) = EF + 2FH. Because AB = CD, it follows that GE = FH. The line tangent to a CIRCLE of RADIUS a centered at (a,y) x — x + a cos t V — V + o, sin t through (0,0) can be found by solving the equation x + a cos t y 4- a sin t a cost a sint giving t — db cos —ax db y\/x 2 -\- y 2 — a 2 x 2 + y 2 Circuit Two of these four solutions give tangent lines, as illus- trated above. see also KISSING CIRCLES PROBLEM, MlQUEL POINT, Monge's Problem, Pedal Circle, Tangent Line, Triangle References Dixon, R. Mathographics. New York: Dover, p. 21, 1991. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1991. Circuit see Cycle (Graph) Circuit Rank Also known as the Cyclomatic Number. The circuit rank is the smallest number of EDGES 7 which must be removed from a GRAPH of N EDGES and n nodes such that no Circuit remains. 7 = N - n + 1. Circulant Determinant Gradshteyn and Ryzhik (1970) define circulants by Xn X n -1 Xn-2 Circular Functions 257 Xl X 2 X3 Xn Xl x 2 m-1 X n Xl X2 Xz X4 Xl = Y\( Xl + X 2ti>j +X3Wj 2 + .- ■ +Xn(Jj n ), (1) i=i where u>j is the nth ROOT OF Unity. The second-order circulant determinant is Xl X2 X2 Xi and the third order is Xl X2 Xz Xz Xi X2 X2 Xz Xi = (xi -\-x 2 )(xi - x 2 ), (2) = (xi + x 2 + X3)(asi + ujx 2 + oj xz){xi + OJ X2 + UJXz), (3) where u) and u 2 are the COMPLEX CUBE ROOTS of Unity. The Eigenvalues A of the corresponding n x n circulant matrix are \j = xi -f- X20JJ 4- Xz^j + . . . + x n ujj n see also CIRCULANT MATRIX (4) References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1111-1112, 1979. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 114, 1991. Circulant Graph A Graph of n Vertices in which the zth Vertex is adjacent to the (i + j)th and (i - j)th Vertices for each j in a list I. Circulant Matrix An n x n MATRIX C defined as follows, 1 (?) G) - UO L (?) (?) (?)■•• i c = n[(i+u,,r-i], 3 = where u;o = 1, cji, ..., u) n -i are the nth ROOTS OF UNITY. Circulant matrices are examples of LATIN Squares. see also CIRCULANT DETERMINANT References Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994. Stroeker, R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988. Vardi, I. Computational Recreations in Mathematica. Read- ing, MA: Addison-Wesley, p. 114, 1991. Circular Cylindrical Coordinates see Cylindrical Coordinates Circular Functions The functions describing the horizontal and vertical po- sitions of a point on a Circle as a function of Angle (COSINE and Sine) and those functions derived from them: cot a; = tana; = tana; 1 sinx 1 cos a; sinx (i) (2) (3) (4) The study of circular functions is called TRIGONOME- TRY. see also COSECANT, COSINE, COTANGENT, ELLIPTIC Function, Generalized Hyperbolic Functions, Hyperbolic Functions, Secant, Sine, Tangent, Trigonometry References Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func- tions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972. 258 Circular Permutation Circumcircle Circular Permutation The number of ways to arrange n distinct objects along a Circle is P n = (n- 1)1 The number is (n - 1)! instead of the usual FACTORIAL n! since all Cyclic Permutations of objects are equiv- alent because the CIRCLE can be rotated. see also Permutation, Prime Circle Circumcenter The center O of a TRIANGLE'S CIRCUMCIRCLE. It can be found as the intersection of the PERPENDICULAR BI- SECTORS. If the Triangle is Acute, the circumcenter is in the interior of the TRIANGLE. In a RIGHT TRI- ANGLE, the circumcenter is the Midpoint of the Hy- potenuse. OOi + OQ 2 + OOz =R + r, (1) where Oi are the MIDPOINTS of sides Ai, R is the Circumradius, and r is the INRADIUS (Johnson 1929, p. 190), The Trilinear Coordinates of the circum- center are cos A : cos B : cos C, (2) and the exact trilinears are therefore R cos A : R cos B : R cos C. The Areal Coordinates are (^acotA, \bcotB, |ccotC). (3) (4) The distance b etween the Incenter and circumcenter is ^R(R — 2r). Given an interior point, the distances to the Vertices are equal Iff this point is the circum- center. It lies on the BROCARD AXIS. The circumcenter O and ORTHOCENTER H are ISOGO- nal Conjugates. The Orthocenter H of the Pedal Triangle AO1O2O3 formed by the CIRCUMCENTER O concurs with the circumcenter O itself, as illustrated above. The circumcenter also lies on the EULER LINE. see also Brocard Diameter, Carnot's Theorem, Centroid (Triangle), Circle, Euler Line, Incen- ter, Orthocenter References Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p, 623, 1970. Dixon, R. Mathographics. New York: Dover, p. 55, 1991. Eppstein, D. "Circumcenters of Triangles." http://www.ics .uci.edu/-eppstein/junkyard/circumcenter.htnil. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994. Kimberling, C. "Circumcenter." http://vvv.evansville, edu/-ck6/tcenters/class/ccenter.html. Circumcircle Circumcircle Circumradius 259 A Triangle's circumscribed circle. Its center O is called the Circumcenter, and its Radius R the Cir- cumradius. The circumcircle can be specified using Trilinear Coordinates as Pya. + yab + a/3c = 0. (i) The Steiner Point S and Tarry Point T lie on the circumcircle. A Geometric Construction for the circumcircle is given by Pedoe (1995, pp. xii-xiii). The equation for the circumcircle of the Triangle with Vertices (zu, yi) for i = 1, 2, 3 is 2 , 2 x + y X y 1 2 i 2 xi +2/i X\ yi 1 2 , 2 X 2 +V2 X2 2/2 1 2 , 2 xz +2/3 xz 2/3 1 = 0. Expanding the DETERMINANT, a(x 2 + y 2 ) + 2dx + 2/y + 5 = 0, where (2) (3) Xi yi l a — X 2 2/2 1 X3 2/3 1 d=- 1 2 xi 2 +2/ # 2 2 +2/ Z3 2 +2/ 2 1 2 2 2 3 yi 2/2 2/3 Xi 2 +2/i 2 Xi J 2 x 2 2 + 2/2 2 x 2 2 i 2 Xz +2/3 xz 2 , 2 Xl +2/1 Xi 9 = ~ 2 , 2 #2 +2/2 x 2 2 , 2 Z3 +2/3 xz COMPLETING THE SQUARE gives a { x+ lY +a (" + z?- a which is a CIRCLE o : the form 1 1 1 2/1 2/2 2/3 (x - zo) 2 + (y- yo) 2 = r 2 , with ClRCUMCENTER Xq yo a ./ a and Circumradius P±&_9 a 2 a (4) (5) (6) (?) + 5 = (8) (9) (10) (11) (12) see also CIRCLE, ClRCUMCENTER, CIRCUMRADIUS, EX- CIRCLE, INCIRCLE, PARRY POINT, PURSER'S THEOREM, Steiner Points, Tarry Point References Pedoe, D. Circles: A Mathematical View, rev. ed. Washing- ton, DC: Math. Assoc. Amer., 1995. Circumference The Perimeter of a Circle. For Radius r or Diam- eter d = 2r, C = 27vr = ltd, where tv is Pi. see also Circle, Diameter, Perimeter, Pi, Radius Circuminscribed Given two closed curves, the circuminscribed curve is simultaneously INSCRIBED in the outer one and CIR- CUMSCRIBED on the inner one. see also Poncelet's Closure Theorem Circumradius The radius of a TRIANGLE'S CIRCUMCIRCLE or of a Polyhedron's Circumsphere, denoted R. For a Tri- angle, R = abc y/(a + b + c)(b + c - a)(c + a - b)(a + b - c) (1) where the side lengths of the TRIANGLE are a, 6, and c. This equation can also be expressed in terms of the Radii of the three mutually tangent Circles centered at the Triangle's Vertices. Relabeling the diagram for the SODDY CIRCLES with VERTICES Oi, O2, and 3 and the radii 7*1, r 2 , and rz, and using a = T\ + V2 b = V2 + 7"3 c — r\-\-rz (2) (3) (4) then gives R = (n +r 2 )(n + r 3 )(r 2 +r 3 ) 4^/Vir 2 r3(ri + r 2 + rz) If O is the ClRCUMCENTER and M is the triangle Cen- TROID, then OM 2 =R 2 - §(a 2 + 6 2 + c 2 ). Rr = Q1Q2Q3 As (6) (?) 260 Circumscribed Cissoid of Diodes COS CKi + COS Ct2 + cos 0:3 — 1 + R v = 2R cos ai cos 0:2 cos a$ ai 2 + a 2 2 + a 3 2 = 4r# + 8iZ 2 (8) (9) (10) (Johnson 1929, pp. 189-191). Let d be the_distance between INRADIUS r and circumradius R, d = rR. Then = 2Rr 1 1 R- d R+d (11) (12) (Mackay 1886-87). These and many other identities are given in Johnson (1929, pp. 186-190). For an ARCHIMEDEAN SOLID, expressing the circumra- dius in terms of the INRADIUS r and MlDRADIUS p gives tf =±(r + xA 2 +a 2 ) s> (13) (14) for an Archimedean Solid. see also Carnot's Theorem, Circumcircle, Cir- CUMSPHERE References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929. Mackay, J. S. "Historical Notes on a Geometrical Theorem and its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5, 62-78, 1886-1887. Circumscribed A geometric figure which touches only the VERTICES (or other extremities) of another figure. see also ClRCUMCENTER, CIRCUMCIRCLE, ClRCUMIN- scribed, Circumradius, Inscribed Circumsphere A Sphere circumscribed in a given solid. Its radius is called the CIRCUMRADIUS. see also Insphere Cis Cis x = e 1 ' ■ cosx 4- i since. Cissoid Given two curves C\ and C2 and a fixed point O, let a line from O cut C at Q and C at R. Then the LOCUS of a point P such that OP = QR is the cissoid. The word cissoid means "ivy shaped." Curve 1 Curve 2 Pole Cissoid line line circle circle circle circle circle parallel line circle tangent line tangent line radial line concentric circle same circle any point center on C on C opp. tangent on C center (0A0) line conchoid of Nicomedes oblique cissoid cissoid of Diocles strophoid circle lemniscate see also Cissoid of Diocles References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 53-56 and 205, 1972. Lee, X. "Cissoid." http : //www . best . com/~xah/Special PlaneCurves^dir/Cissoid_dir/c issoid.html. Lockwood, E. H. "Cissoids." Ch. 15 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130-133, 1967. Yates, R. C. "Cissoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30, 1952. Cissoid of Diocles A curve invented by Diocles in about 180 BC in con- nection with his attempt to duplicate the cube by geo- metrical methods. The name "cissoid" first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the Area between the curve and its asymptote was 3a (MacTutor Archive). From a given point there are either one or three TANGENTS to the cissoid. Given an origin O and a point P on the curve, let S be the point where the extension of the line OP intersects the line x — 2a and R be the intersection of the CIRCLE of RADIUS a and center (a, 0) with the extension of OP. Then the cissoid of Diocles is the curve which satisfies OP = RS. Cissoid of Diodes Clark's Triangle 261 The cissoid of Diodes is the Roulette of the Vertex of a Parabola rolling on an equal Parabola. Newton gave a method of drawing the cissoid of Diocles using two line segments of equal length at RIGHT ANGLES. If they are moved so that one line always passes through a fixed point and the end of the other line segment slides along a straight line, then the MIDPOINT of the sliding line segment traces out a cissoid of Diocles. The cissoid of Diocles is given by the parametric equa- tions x = 2a sin 6 _ 2a sin 3 V ~ cos 6 Converting these to POLAR COORDINATES gives sin 6 ' (1) (2) 2 2.2 r = x + y : 4a 2 [ sin 4 + cos 2 (9, : 4a 2 sin 4 0(1 + tan 2 6) = 4a 2 sin 4 6 sec 2 0, (3) so r = 2a sin 2 sec = 2a sin 6 tan 0. In Cartesian Coordinates, ,3 Qrt 3 • 6/1 (4) X 2a -x 2a — 2a sin 2 . 2 sin 6 2 = 4a — = y . = 4a* sin 1 - sin 2 8 s 2 9 An equivalent form is x(x 2 -\-y 2 ) = 2ay . Using the alternative parametric form *(*) = y(t) 2at 2 1 + i 2 2at 3 1 + t 2 (Gray 1993), gives the Curvature as *<*)= a \t\{t* + 4)3/2- (5) (6) (7) (8) (9) References Gray, A. "The Cissoid of Diocles." §3.4 in Modern Differ- ential Geometry of Curves and Surf 'aces. Roca Raton, FL: CRC Press, pp. 43-46, 1993. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 98-100, 1972. Lee, X. "Cissoid of Diocles." http://www.best.com/-xah/ SpecialPlaneCurvesjdir/CissoidOf Diocles jdir/cissoid OfDiocles.html. Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130-133, 1967. MacTutor History of Mathematics Archive. "Cissoid of Dio- cles." http: //www-groups . dcs . st-and.ac.uk/-history/ Curves/Cissoid.html. Yates, R. C. "Cissoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30, 1952. Cissoid of Diocles Caustic The Caustic of the cissoid where the Radiant Point is taken as (8a, 0) is a CARDIOID. Cissoid of Diocles Inverse Curve If the cusp of the CISSOID OF DIOCLES is taken as the Inversion Center, then the cissoid inverts to a PARABOLA. Cissoid of Diocles Pedal Curve \ \ \ The Pedal Curve of the cissoid, when the Pedal Point is on the axis beyond the Asymptote at a dis- tance from the cusp which is four times that of the Asymptote is a Cardioid. Clairaut's Differential Equation - x ^y + f f^M.\ dx V dx ) or y = px + f(p), where / is a Function of one variable and p = dy/dx. The general solution is y — ex + /(c). The singular solution ENVELOPES are x — ~f f (c) and y = f(c) - cf'(c). see also d'Alembert's Equation References Boyer, C B. A History of Mathematics. New York: Wiley, p. 494, 1968. Clarity The Ratio of a measure of the size of a "fit" to the size of a "residual." References Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison- Wesley, p. 667, 1977. Clark's Triangle (m-l)3 12 7 1 t> 18 19 8 1 24 37 27 9 1 30 61 64 36 10 1 36 91 125 100 46 11 1 // 262 Clark's Triangle A Number Triangle created by setting the Vertex equal to 0, filling one diagonal with Is, the other diag- onal with multiples of an INTEGER /, and rilling in the remaining entries by summing the elements on either side from one row above. Call the first column n = and the last column m = nso that c(m, 0) = fm c(rri) m) = 1, (1) (2) then use the Recurrence Relation c(m, n) = c(m — 1, n — 1) + c(m — 1, n) (3) to compute the rest of the entries. For n = 1, we have c(m, 1) = c(m -1,0) + c(m - 1, 1) (4) c(m, 1) - c(m - 1, 1) = c(m -1,0) = f(m - 1). (5) For arbitrary m, the value can be computed by Sum- ming this Recurrence, c(m, 1) = / j J2 k I + X = l/ m ( m - 1) + 1. (6) Now, for n = 2 we have c(m, 2) = c(m - 1, 1) + c(m - 1, 2) (7) c(m,2)-c(m-l,2) = c(m-l,l) = |/(m-l)m+l, (8) so Summing the Recurrence gives c(m, 2) = 5}±/*(* - 1) + 1] = ]T(§/fc 2 " 3** + X ) fc=i fc=i = \f[\m{m + l)(2m + 1)] - \f[\m{m + 1)] + m = ±(m-l)(/m 2 -2/m + 6). (9) Similarly, for n = 3 we have c(m, 3) - c(m -1,3) = c(m - 1, 2) = |/m 3 -/m 2 + (^/ + l)m-(/ + 2). (10) Taking the Sum, m c(m,3) = ^ i/fc 3 - /fc 2 + (ff + l)k - (/ + 2). (11) fc = 2 Evaluating the Sum gives c(m,3) = ^(m- l)(m-2)(/m 2 -3/m+12). (12) Ciass Number So far, this has just been relatively boring Algebra. But the amazing part is that if / = 6 is chosen as the Integer, then c(m, 2) and c(tm, 3) simplify to c(m, 2) = \{m - l)(6m 2 - 12m + 6) -(m-1) 3 (13) c(m,3)=|(m-l) 2 (m-2) 2 , (14) which are consecutive Cubes (m — l) 3 and nonconsecu- tive Squares n 2 = [(m - l)(m - 2)/2] 2 . see a/so Bell Triangle, Catalan's Triangle, Euler's Triangle, Leibniz Harmonic Triangle, Number Triangle, Pascal's Triangle, Seidel- Entringer-Arnold Triangle, Sum References Clark, J. E. "Clark's Triangle." Math. Student 26, No. 2, p. 4, Nov. 1978. Class see Characteristic Class, Class Interval, Class (Multiply Perfect Number), Class Number, Class (Set), Conjugacy Class Class (Group) see Conjugacy Class Class Interval The constant bin size in a HISTOGRAM, see also Sheppard's Correction Class (Map) A Map u : R n -► R n from a Domain G is called a map of class C r if each component of u(x) - (ui(zi,...,Xn),...,u m (a;i J ...,x„)) is of class C r (0 < r < 00 or r — w) in G, where C d denotes a continuous function which is differentiable d times. Class (Multiply Perfect Number) The number k in the expression s(n) — kn for a Mul- tiply Perfect Number is called its class. Class Number For any IDEAL 7, there is an IDEAL 7* such that Hi = z, (1) where z is a Principal IDEAL, (i.e., an IDEAL of rank 1). Moreover, there is a finite list of ideals h such that this equation may be satisfied for every I. The size of this list is known as the class number. When the class number is 1, the Ring corresponding to a given IDEAL has unique factorization and, in a sense, the class Class Number Class Number 263 number is a measure of the failure of unique factorization in the original number ring. A finite series giving exactly the class number of a Ring is known as a CLASS NUMBER FORMULA. A CLASS Number Formula is known for the full ring of cyclo- tomic integers, as well as for any subring of the cyclo- tomic integers. Finding the class number is a computa- tionally difficult problem. Let h(d) denote the class number of a quadratic ring, corresponding to the Binary Quadratic Form ax + bxy + cy , with Discriminant d = b — 4ac. (2) (3) Then the class number h(d) for DISCRIMINANT d gives the number of possible factorizations of ax 2 + bxy + cy 2 in the QUADRATIC Field Q(y/d). Here, the factors are of the form x 4- yVd, with x and y half INTEGERS. Some fairly sophisticated mathematics shows that the class number for discriminant d can be given by the Class Number Formula ,, f-^E^VWlnsin(^) ford>0 /x mElt\d\r)r for d < 0, where (d\r) is the Kronecker Symbol, 77(d) is the Fundamental Unit, w(d) is the number of substitu- tions which leave the Binary Quadratic Form un- changed ( 6 for d = -3 w(d) ^<4 for d = -4 (5) [ 2 otherwise, and the sums are taken over all terms where the Kron- ecker SYMBOL is defined (Cohn 1980). The class num- ber for d > can also be written ^M-) = TJ Bin -(-|r)^^ (6) for d > 0, where the PRODUCT is taken over terms for which the Kronecker Symbol is defined. The class number is related to the DlRlCHLET L-Series by L„(l) h(d) = K{d) (7) where /c(d) is the DlRlCHLET STRUCTURE CONSTANT. Wagner (1996) shows that class number h(—d) satisfies the Inequality -»^(>-M) lnd, (8) for -d < 0, where [x] is the Floor Function, the product is over PRIMES dividing d, and the * indicates that the Greatest Prime Factor of d is omitted from the product. The Mathematica® (Wolfram Research, Champaign, IL) function NumberTheory'NumberTheoryFunct ions' ClassNumber [n] gives the class number h{d) for d a Negative Squarefree number of the form 4k -f 1, Gauss's Class Number Problem asks to determine a complete list of fundamental DISCRIMINANTS — d such that the CLASS Number is given by h(—d) = m for a given m. This problem has been solved for n < 7 and Odd n < 23. Gauss conjectured that the class number h(—d) of an IMAGINARY quadratic field with Discriminant —d tends to infinity with d, an assertion now known as Gauss's Class Number Conjecture. The discriminants d having h(~d) = 1, 2, 3, 4, 5, ... are Sloane's A014602 (Cohen 1993, p. 229; Cox 1997, p. 271), Sloane's A014603 (Cohen 1993, p. 229), Sloane's A006203 (Cohen 1993, p. 504), Sloane's A013658 (Co- hen 1993, p. 229), Sloane's A046002, Sloane's A046003, The complete set of negative discriminants hav- ing class numbers 1-5 and Odd 7-23 are known. Buell (1977) gives the smallest and largest fundamental class numbers for d < 4, 000, 000, partitioned into EVEN dis- criminants, discriminants 1 (mod 8), and discriminants 5 (mod 8). Arno et al. (1993) give complete lists of val- ues of d with h{-d) = k for ODD k = 5, 7, 9, . . . , 23. Wagner gives complete lists of values for k = 5, 6, and 7. Lists of NEGATIVE discriminants co rrespon ding to Imaginary Quadratic Fields Q(y/—d(n) ) having small class numbers h{—d) are given in the table below. In the table, N is the number of "fundamental" values of — d with a given class number h{—d)^ where "funda- mental" means that — d is not divisible by any SQUARE Number s 2 such that h(—d/s 2 ) < h(—d). For example, although h(— 63) = 2, —63 is not a fundamental dis- criminant since 63 = 3 2 • 7 and h(-63/3 2 ) = h(-7) = 1 < h(-63). Even values 8 < h(-d) < 18 have been computed by Weisstein. The number of negative dis- criminants having class number 1, 2, 3, . . . are 9, 18, 16, 54, 25, 51, 31, ... (Sloane's A046125). The largest negative discriminants having class numbers 1, 2, 3, . . . are 163, 427, 907, 1555, 2683, . . . (Sloane's A038552). The following table lists the numbers with small class numbers < 11. Lists including larger class numbers are given by Weisstein. h(-d) N d 1 9 3, 4, 7, 8, 11, 19, 43, 67, 163 2 18 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427 3 16 23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907 264 Class Number Class Number h(-d) N d 4 54 39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555 5 25 47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683 6 51 87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763 7 31 71, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787, 1987, 2011, 2083, 2179, 2251, 2467, 2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923 8 131 95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651, 660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240, 1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528, 1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780, 1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995, 2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308, 2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715, 2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315, 3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195, 4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707, 5947, 6307 9 34 199, 367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 2003, 2803, 3163, 3259, 3307, 3547, 3643, 4027, 4243, 4363, 4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563, 8803, 9067, 10627 10 87 119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492, 1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923, 2152, 2164, h(~d) N d 2363, 2452, 2643, 2776, 2836, 2899, 3028, 3091, 3139, 3147, 3291, 3412, 3508, 3635, 3667, 3683, 3811, 3859, 3928, 4083, 4227, 4372, 4435, 4579, 4627, 4852, 4915, 5131, 5163, 5272, 5515, 5611, 5667, 5803, 6115, 6259, 6403, 6667, 7123, 7363, 7387, 7435, 7483, 7627, 8227, 8947, 9307, 10147, 10483, 13843 11 41 167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203, 3347, 3499, 3739, 3931, 4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443, 9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683, 15667 The table below gives lists of Positive fundamental discriminants d having small class numbers h(d), cor- responding to Real quadratic fields. All Positive SQUAREFREE values of d < 97 (for which the KRON- ECKER SYMBOL is defined) are included. h(d) d 1 5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77 2 65 The POSITIVE d for which h(d) = 1 is given by Sloane's A014539. see also Class Number Formula, Dirichlet L- Series, Discriminant (Binary Quadratic Form), Gauss's Class Number Conjecture, Gauss's Class Number Problem, Heegner Number, Ideal, j-FUNCTION References Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 40, 321-334, 1992. Arno, S.; Robinson, M. L«; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http:// www.math.uiuc . edu/Algebraic -Number-Theory/ 0009/. Buell, D. A. "Small Class Numbers and Extreme Values of //-Functions of Quadratic Fields." Math. Comput. 139, 786-796, 1977. Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer- Verlag, 1993. Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 and 234, 1980. Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997. Davenport, H. "Dirichlet's Class Number Formula." Ch. 6 in Multiplicative Number Theory, 2nd ed. New York: Springer- Verlag, pp. 43-53, 1980. Iyanaga, S. and Kawada, Y. (Eds.). "Class Numbers of Al- gebraic Number Fields." Appendix B, Table 4 in Encyclo- pedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1494-1496, 1980. Montgomery, H. and Weinberger, P. "Notes on Small Class Numbers." Acta. Arith. 24, 529-542, 1974. Sloane, N. J. A. Sequences A014539, A038552, A046125, and A003657/M2332 in "An On-Line Version of the Encyclo- pedia of Integer Sequences." Class Number Formula Clausen Formula 265 Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14, 1-27, 1967. Stark, H, M. "On Complex Quadratic Fields with Class Num- ber Two." Math. Comput. 29, 289-302, 1975. Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 785-800, 1996. # Weisstein, E. W. "Class Numbers." http: //www. astro . Virginia. edu/~eww6n/math/notebooks/ClassNumbers .m. Class Number Formula A class number formula is a finite series giving exactly the Class Number of a Ring. For a Ring of quadratic integers, the class number is denoted h(d) y where d is the discriminant. A class number formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. This formula includes the quadratic case as well as many cubic and higher-order rings. see also Class Number Class Representative A set of class representatives is a SUBSET of X which contains exactly one element from each Equivalence Class. Class (Set) A class is a special kind of Set invented to get around RUSSELL'S PARADOX while retaining the arbitrary cri- teria for membership which leads to difficulty for Sets. The members of classes are Sets, but it is possible to have the class C of "all Sets which are not members of themselves" without producing a paradox (since C is a proper class (and not a Set), it is not a candidate for membership in C). see also Aggregate, Russell's Paradox, Set Classical Groups The four following types of GROUPS, 1. Linear Groups, 2. Orthogonal Groups, 3. Symplectic Groups, and 4. Unitary Groups, which were studied before more exotic types of groups (such as the SPORADIC GROUPS) were discovered. see also GROUP, LINEAR GROUP, ORTHOGONAL Group, Symplectic Group, Unitary Group Classification The classification of a collection of objects generally means that a list has been constructed with exactly one member from each ISOMORPHISM type among the ob- jects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of math- ematical objects which have been classified include the finite Simple Groups and 2-Manifolds but not, for example, Knots. Classification Theorem The classification theorem of FINITE Simple GROUPS, also known as the ENORMOUS THEOREM, which states that the Finite Simple Groups can be classified com- pletely into 1. Cyclic Groups Z p of Prime Order, 2. Alternating Groups A n of degree at least five, 3. Lie-Type Chevalley Groups PSL(n,q), PSU(n,q), PsP(2n,g), and Pft € (n,g), 4. Lie-Type (Twisted Chevalley Groups or the Tits Group) s D 4 (q) y E Q (q) y E 7 (q), E s (q), F 4 (g), 2 F 4 (2*% G 2 (q), 2 G 2 (3 n ), 2 B(2 n ), 5. Sporadic Groups Mu, M i2 , M 22 , M23, M 24 , Ji = HJ, Suz, HS, McL, Co 3 , Co 2 , C01, He, Fi 22} ^'23, Fi' 24 , HN, Th, B, M, J u OW, J 3 , Ly, Ru, J 4 . The "Proof" of this theorem is spread throughout the mathematical literature and is estimated to be approx- imately 15,000 pages in length. see also FINITE GROUP, GROUP, j-FUNCTION, SIMPLE Group References Cartwright, M. "Ten Thousand Pages to Prove Simplicity." New Scientist 109, 26-30, 1985. Cipra, B. "Are Group Theorists Simpleminded?" What's Happening in the Mathematical Sciences, 1995-1996, Vol 3. Providence, RJ: Amer. Math. Soc, pp. 82-99, 1996. Cipra, B. "Slimming an Outsized Theorem." Science 267, 794-795, 1995. Gorenstein, D. "The Enormous Theorem," Set Amer, 253, 104-115, Dec. 1985. Solomon, R. "On Finite Simple Groups and Their Classifica- tion." Not Amer. Math. Soc. 42, 231-239, 1995. Clausen Formula Clausen's 4^3 identity / 9 (2a)| d |(a + %|(26)| d | (2a + 2b)\d\a\ d \b\d\ holds for a + b + c- d= 1/2, e = a + 6 + 1/2, a + / = d+l = 6 + p, da nonpositive integer, and (a) n is the POCHHAMMER Symbol (Petkovsek tt al. 1996). Another identity ascribed to Clausen which in- volves the Hypergeometric Function 2 i*i(a, b\c\z) and the GENERALIZED HYPERGEOMETRIC FUNCTION 3F2 (a, 6, c; d, e; z) is given by a, 6 a + b+k'' X = 3-^2 (• 2a, a + b, 2b + 6+|,2a + 26 ;:C see also GENERALIZED HYPERGEOMETRIC FUNCTION, HYPERGEOMETRIC FUNCTION References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K, Peters, pp. 43 and 127, 1996. 266 Clausen Function Clausen Function sin(kx) *.(*>- £n£ C n {x) = J2 cos(kx) (i) (2) and write cl " (x) = \c„(x)=Er=i ££ ^ 1 "° dd - Then the Clausen function Cl n (x) can be given symbol- ically in terms of the Polylogarithm as /ii[Li n (e-")-Li n (e-)] r. Ol nW - | i [Lin(e -i*) + Li n (e-)] n even odd. For n = l, the function takes on the special form Cli(x) = Ci(x) = -ln|2sin(|x)| and for n = 2, it becomes Clausen's Integral Cl 2 (a:) - S 2 (x) = - / ln[2sin(ft)]dt. (4) (5) The symbolic sums of opposite parity are summable symbolically, and the first few are given by i~ 2 1_ 4 48^ C 2 (ac) = ±tt - ±ttx+±x C 4 (z) = ^ - T^ 2 ^ 2 + T2 7 ™ 3 ~ -h* 5i(x)=§(7T-x) (6) (7) (8) (9) 5 5 (x) = i7r 4 x-^7rV + ^7rx 4 -^x 5 (10) for < x < 27r (Abramowitz and Stegun 1972). see also CLAUSEN'S INTEGRAL, POLYGAMMA FUNC- TION, Polylogarithm CLEAN Algorithm References Abramowitz, M. and Stegun, C. A. (Eds.). "Clausen's Inte- gral and Related Summations" §27.8 in Handbook of Math- ematical Functions with Formulas, Graphs, and Mathe- matical Tables, 9th printing. New York: Dover, pp. 1005- 1006, 1972. Arfken, G. Mathematical Methods {or Physicists, 3rd ed. Or- lando, FL: Academic Press, p. 783, 1985. Clausen, R. "Uber die Zerlegung reeller gebrochener . Funk- tionen." J. reine angew. Math. 8, 298-300, 1832. Grosjean, C. C. "Formulae Concerning the Computation of the Clausen Integral Cl 2 (a)." J. Comput. Appl. Math. 11, 331-342, 1984. Jolley, L. B. W. Summation of Series. London: Chapman, 1925. Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953. Clausen's Integral 0.5 -1- The Clausen Function C1 2 (0) = - / \n[2sm(lt)]dt t/0 see also CLAUSEN FUNCTION References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972. Ashour, A. and Sabri, A. "Tabulation of the Function ip(0) = V°° -i£l»£i.» Math. Tables Aids Comp. 10, 54 and 57- 65, 1956. Clausen, R. "Uber die Zerlegung reeller gebrochener Funk- tionen." J. reine angew. Math. 8, 298-300, 1832. CLEAN Algorithm An iterative algorithm which DECONVOLVES a sampling function (the "Dirty Beam") from an observed bright- ness ("DIRTY Map") of a radio source. This algorithm is of fundamental importance in radio astronomy, where it is used to create images of astronomical sources which are observed using arrays of radio telescopes ( "synthesis imaging"). As a result of the algorithm's importance to synthesis imaging, a great deal of effort has gone into optimizing and adjusting the ALGORITHM. CLEAN is a nonlinear algorithm, since linear DECONVOLUTION algo- rithms such as Wiener Filtering and inverse filtering CLEAN Algorithm CLEAN Algorithm 267 are inapplicable to applications with invisible distribu- tions (i.e., incomplete sampling of the spatial frequency plane) such as map obtained in synthesis imaging. The basic CLEAN method was developed by Hogbom (1974). It was originally designed for point sources, but it has been found to work well for extended sources as well when given a reasonable starting model. The Hogbom CLEAN constructs discrete approximations I n to the CLEAN Map in the (£,77) plane from the CON- VOLUTION equation b' *I = /', (1) where b' is the Dirty Beam, I' is the Dirty Map (both in the (£>r?) Plane), and f*g denotes a Convolution. The CLEAN algorithm starts with an initial approxi- mation Jo = 0. At the nth iteration, it then searches for the largest value in the residual map I n ^ I' - b' * I n -1. (2) A Delta Function is then centered at the location of the largest residual flux and given an amplitude /x (the so-called "Loop Gain") times this value. An antenna's response to the Delta FUNCTION, the DlRTY Beam, is then subtracted from I n -i to yield I n . Iteration con- tinues until a specified iteration limit N is reached, or until the peak residual or Root-Mean-Square resid- ual decreases to some level. The resulting final map is denoted In, and the position of each Delta Function is saved in a "CLEAN component" table in the CLEAN Map file. At the point where component subtraction is stopped, it is assumed that the residual brightness dis- tribution consists mainly of NOISE. To diminish high spatial frequency features which may be spuriously extrapolated from the measured data, each CLEAN component is convolved with the so-called CLEAN Beam 6, which is simply a suitably smoothed version of the sampling function ("Dirty Beam"). Usu- ally, a Gaussian is used. A good CLEAN Beam should: 1. Have a unity FOURIER TRANSFORM inside the sam- pled region of (u, v) space, 2. Have a FOURIER TRANSFORM which tends to out- side the sampled (u, v) region as quickly as possible, and 3. Not have any effects produced by NEGATIVE side- lobes larger than the NOISE level. A CLEAN Map is produced when the final residual map is added to the the approximate solution, [clean map] = In * b -\- [I — b * In] in order to include the NOISE. (3) CLEAN will always converge to one (of possibly many) solutions if the following three conditions are satisfied (Schwarz 1978): 1. The beam must be symmetric. 2. The Fourier Transform of the Dirty Beam is NONNEGATIVE (positive definite or positive semidef- inite). 3. There must be no spatial frequencies present in the dirty image which are not also present in the Dirty Beam. These conditions are almost always satisfied in practice. If the number of CLEAN components does not exceed the number of independent (u,v) points, CLEAN con- verges to a solution which is the least squares fit of the Fourier Transforms of the Delta Function com- ponents to the measured visibility (Thompson et al. 1986, p. 347). Schwarz claims that the CLEAN algo- rithm is equivalent to a least squares fitting of cosine and sine parts in the (u, v) plane of the visibility data. Schwab has produced a NOISE analysis of the CLEAN algorithm in the case of least squares minimization of a noiseless image which involves am N x M MATRIX. However, no NOISE analysis has been performed for a Real image. Poor modulation of short spacings results in an under- estimation of the flux, which is manifested in a bowl of negative surface brightness surrounding an object. Pro- viding an estimate of the "zero spacing" flux (the to- tal flux of the source, which cannot be directly mea- sured by an interferometer) can considerably reduce this effect. Modulations or stripes can occur at spa- tial frequencies corresponding to undersampled parts of the (u,v) plane. This can result in a golf ball-like mottling for disk sources such as planets, or a corru- gated pattern of parallel lines of peaks and troughs ("stripes"). A more accurate model can be used to sup- press the "golf ball" modulations, but may not elimi- nate the corrugations. A tapering function which de- emphasizes data near (u, v) = (0,0) can also be used. Stripes can sometimes be eliminated using the Cornwell smoothness-stabilized CLEAN (a.k.a. Prussian helmet algorithm; Thompson et al 1986). CLEANing part way, then restarting the CLEAN also seems to eliminate the stripes, although this fact is more disturbing than reas- suring. Stability the the CLEAN algorithm is discussed by Tan (1986). In order to CLEAN a map of a given dimension, it is nec- essary to have a beam pattern twice as large so a point source can be subtracted from any point in the map. Because the CLEAN algorithm uses a Fast FOURIER Transform, the size must also be a Power of 2. There are many variants of the basic Hogbom CLEAN which extend the method to achieve greater speed and produce more realistic maps. Alternate nonlinear De- convolution methods, such as the Maximum En- tropy Method, may also be used, but are gener- ally slower than the CLEAN technique. The Astro- nomical Image Processing Software (AIPS) of the Na- tional Radio Astronomical Observatory includes 2-D 268 CLEAN Algorithm CLEAN Algorithm DECONVOLUTION algorithms in the tasks DCONV and UVMAP. Among the variants of the basic Hogbom CLEAN are Clark, Cornwell smoothness stabilized (Prussian helmet), Cotton-Schwab, Gerchberg-Saxton (Fienup), Steer, Steer-Dewdney-Ito, and van Cittert iteration. In the Clark (1980) modification, CLEAN picks out only the largest residual points, and subtracts approximate point source responses in the (£,77) plane during minor (Hogbom CLEAN) cycles. It only occasionally (dur- ing major cycles) computes the full /„, residual map by subtracting the identified point source responses in the (ujv) plane using a Fast Fourier Transform for the Convolution. The Algorithm then returns to a mi- nor cycle. This algorithm modifies the Hogbom method to take advantage of the array processor (although it also works without one). It is therefore a factor of 2-10 faster than the simple Hogbom routine. It is implemented as the AIPS task APCLN. The Cornwell smoothness stabilized variant was devel- oped because, when dealing with two-dimensional ex- tended structures, CLEAN can produce artifacts in the form of low-level high frequency stripes running through the brighter structure. These stripes derive from poor interpolations into unsampled or poorly sampled re- gions of the (u, v) plane. When dealing with quasi-one- dimensional sources (i.e., jets), the artifacts resemble knots (which may not be so readily recognized as spuri- ous). APCLN can invoke a modification of CLEAN that is intended to bias it toward generating smoother solu- tions to the deconvolution problem while preserving the requirement that the transform of the CLEAN compo- nents list fits the data. The mechanism for introducing this bias is the addition to the Dirty Beam of a Delta FUNCTION (or "spike") of small amplitude (PHAT) while searching for the CLEAN components. The beam used for the deconvolution resembles the helmet worn by Ger- man military officers in World War I, hence the name "Prussian helmet" CLEAN. The theory underlying the Cornwell smoothness stabi- lized algorithm is given by Cornwell (1982, 1983), where it is described as the smoothness stabilized CLEAN. It is implemented in the AIPS tasks APCLN and MX. The spike performs a NEGATIVE feedback into the dirty im- age, thus suppressing features not required by the data. Spike heights of a few percent and lower than usual loop gains are usually needed. Also according to the MX doc- umentation, PHAT ; (noise) 1 2(signal) 2 ~ 2(SNR) 2 Unfortunately, the addition of a Prussian helmet gen- erally has "limited success," so resorting to another de- convolution method such as the MAXIMUM ENTROPY METHOD is sometimes required. The Cotton-Schwab uses the Clark method, but the major cycle subtractions of CLEAN components are performed on ungridded visibility data. The Cotton- Schwab technique is often faster than the Clark variant. It is also capable of including the w baseline term, thus removing distortions from noncoplanar baselines. It is often faster than the Clark method. The Cotton-Schwab technique is implemented as the AIPS task MX. The Gerchberg-Saxton variant, also called the Fienup variant, is a technique originally introduced for solv- ing the phase problem in electron microscopy. It was subsequently adapted for visibility amplitude measure- ments only. A Gerchberg-Saxton map is constrained to be Nonzero, and positive. Data and image plane con- straints are imposed alternately while transforming to and from the image plane. If the boxes to CLEAN are chosen to surround the source snugly, then the algorithm will converge faster and will have more chance of finding a unique image. The algorithm is slow, but should be comparable to the Clark technique (APCLN) if the map contains many picture elements. However, the resolu- tion is data dependent and varies across the map. It is implemented as the AIPS task APGS (Pearson 1984). The Steer variant is a modification of the Clark variant (Cornwell 1982). It is slow, but should be comparable to the Clark algorithm if the map contains many pic- ture elements. The algorithm used in the program is due to David Steer. The principle is similar to Barry Clark's CLEAN except that in the minor cycle only points above the (trim level) x (peak in the residual map) are selected. In the major cycle these are removed us- ing a Fast Fourier Transform. If boxes are chosen to surround the source snugly, then the algorithm will converge faster and will have more chance of finding a unique image. It is implemented in AIPS as the exper- imental program STEER and as the Steer-Dewdney-Ito variant combined with the Clark algorithm as SDCLN. The Steer-Dewdney-Ito variant is similar to the Clark variant, but the components are taken as all pixels having residual flux greater than a cutoff value times the current peak residual. This method should avoid the "ripples" produced by the standard CLEAN on ex- tended emission. The AIPS task SDCLN does an AP- based CLEAN of the the Clark type, but differs from APCLN in that it offers the option to switch to the Steer- Dewdney-Ito method. Finally, van Cittert iteration consists of two steps: 1. Estimate a correction to add to the current map es- timate by multiplying the residuals by some weight. In the classical van Cittert algorithm, this weight is a constant, where as in CLEAN the weight is zero everywhere except at the peak of the residuals. 2. Add the step to the current estimate, and subtract the estimate, convolved with the DIRTY BEAM, from the residuals. CLEAN Beam Clebsch Diagonal Cubic 269 Though it is a simple algorithm, it works well (if slowly) for cases where the DlRTY BEAM is positive semidefmite (as it is in astronomical observations). The basic idea is that the DlRTY MAP is a reasonably good estimate of the deconvolved map. The different iterations vary only in the weight to apply to each residual in determining the correction step, van Cittert iteration is implemented as the AIPS task APVC, which is a rather experimental and ad hoc procedure. In some limiting cases, it reduces to the standard CLEAN algorithm (though it would be unpractically slow). see also CLEAN Beam, CLEAN Map, Dirty Beam, Dirty Map References Christiansen, W. N. and Hogbom, J. A. Radiotelescopes, 2nd ed. Cambridge, England: Cambridge University Press, pp. 214-216, 1985, Clark, B, G. "An Efficient Implementation of the Algorithm 'CLEAN'." Astron. Astrophys, 89, 377-378, 1980. Cornwell, T. J. "Can CLEAN be Improved?" VLA Scientific Memorandum No. 141, 1982. Cornwell, T\ J. "Image Restoration (and the CLEAN Tech- nique)." Lecture 9. NRAO VLA Workshop on Synthesis Mapping, p. 113, 1982, Cornwell, T. J. "A Method of Stabilizing the CLEAN Algo- rithm." Astron. Astrophys. 121, 281-285, 1983. Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Syn- thesis Imaging in Radio Astronomy: Third NRAO Sum- mer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and A. H. Bridle). San Francisco, CA: Astronomical Society of the Pacific, pp. 178-179, 1989. Hogbom, J. A. "Aperture Synthesis with a Non-Regular Dis- tribution of Interferometric Baselines." Astron. Astrophys. Supp. 15, 417-426, 1974. National Radio Astronomical Observatory. Astronomical Im- age Processing Software (AIPS) software package. APCLN, MX, and UVMAP tasks. Pearson, T. J. and Readhead, A. C. S. "Image Formation by Self-Calibration in Radio Astronomy." Ann. Rev. Astron. Astrophys. 22, 97-130, 1984. Schwarz, U. J. "Mathematical-Statistical Description of the Iterative Beam Removing Technique (Method CLEAN)." Astron. Astrophys. 65, 345-356, 1978. Tan, S. M. "An Analysis of the Properties of CLEAN and Smoothness Stabilized CLEAN — Some Warnings." Mon. Not. Royal Astron. Soc. 220, 971-1001, 1986. Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Inter jerometry and Synthesis in Radio Astronomy. New York: Wiley, p. 348, 1986. CLEAN Beam An Elliptical Gaussian fit to the Dirty Beam in order to remove sidelobes. The CLEAN beam is con- volved with the final CLEAN iteration to diminish spu- rious high spatial frequencies. see also CLEAN Algorithm, CLEAN Map, Decon- volution, Dirty Beam, Dirty Map CLEAN Map The deconvolved map extracted from a finitely sampled Dirty Map by the CLEAN Algorithm, Maximum Entropy Method, or any other Deconvolution pro- cedure. see also CLEAN Algorithm, CLEAN Beam, Decon- volution, Dirty Beam, Dirty Map Clebsch- Aronhold Notation A notation used to describe curves. The fundamen- tal principle of Clebsch-Aronhold notation states that if each of a number of forms be replaced by a POWER of a linear form in the same number of variables equal to the order of the given form, and if a sufficient number of equivalent symbols are introduced by the ARONHOLD Process so that no actual Coefficient appears except to the first degree, then every identical relation holding for the new specialized forms holds for the general ones. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 79, 1959. Clebsch Diagonal Cubic A Cubic Algebraic Surface given by the equation xo 3 + xi S + x 2 3 + x 3 3 + X4 3 = 0, (1) with the added constraint xo + Xi + X2 + £3 + X4 0. (2) The implicit equation obtained by taking the plane at infinity as xq + x\ + x 2 + x$/2 is 81(x -hy -\-z ) — 189(x y-\-x z + y x-\-y z + z x + z y) +54xyz + 126(xy + xz + yz) - 9(x 2 + y 2 + z 2 ) -9(x + y + z) + 1 = (3) (Hunt, Nordstrand). On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's Seal Lines) present on a general smooth CUBIC SURFACE are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called ECKARDT POINTS (Fischer 1986, Hunt), and the Clebsch diago- nal surface is the unique CUBIC SURFACE containing 10 such points (Hunt). If one of the variables describing Clebsch's diagonal sur- face is dropped, leaving the equations xq 3 + xi 3 + x 2 3 + #3 3 = 0, (4) 270 Clebsch-Gordon Coefficient x + xi + x 2 + xz = 0, (5) the equations degenerate into two intersecting Planes given by the equation {x + y)(x + z){y + z) = Q. (6) see also Cubic Surface, Eckardt Point References Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 9-11, 1986. Fischer, G. (Ed.). Plates 10-12 in Mathematische Mod- elle/ Mathematical Models, Bildband/ Photograph Volume. Braunschweig, Germany: Vieweg, pp. 13-15, 1986. Hunt, B. The Geometry of Some Special Arithmetic Quo- tients. New York: Springer- Verlag, pp. 122-128, 1996. Nordstrand, T. "Clebsch Diagonal Surface." http://www. uib , no/people/nf ytn/clebtxt . htm. Clebsch-Gordon Coefficient A mathematical symbol used to integrate products of three SPHERICAL HARMONICS. Clebsch-Gordon coeffi- cients commonly arise in applications involving the ad- dition of angular momentum in quantum mechanics. If products of more than three SPHERICAL HARMONICS are desired, then a generalization known as WlGNER 6J-SYMBOLS or WlGNER 9?'-Symb0LS is used. The Clebsch-Gordon coefficients are written C J mim2 = UiJ2mim 2 \jiJ2Jm) (1) and are denned by ^jm = 2_^ Cm 1 m 2 ^m 1 m 2 , ( 2 ) M=Mi+M 2 where J = Ji 4- J 2 - The Clebsch-Gordon coefficients are sometimes expressed using the related RACAH V- COEFFICIENTS V(jiJ2J;m 1 7n 2 7n) (3) or Wigner 3 j- Symbols. Connections among the three are (jiJ2mim2\jiJ2m) (jiJ2m 1 m 2 \jiJ2Jm) 3i mi 32 7712 (4) V(ji32J;rn 1 m 2 m) = (-1)" -h+32+3 I 3i 32 3i m 2 mi m 2 Clenshaw Recurrence Formula They have the symmetry (jiJ2mim 2 \jij 2 jm) = (-iyi+w (j 2 j 1 Tn 2 m 1 \j 2 jijm), (7) and obey the orthogonality relationships "y y j (jiJ2Tn 1 m2\jiJ2Jm)(jiJ2Jm\j 1 J2Tn' 1 tn' 2 ) = S, >6„ Tn, l TTl i Tri 2 Tn { (8) ^ (ji J2mim 2 \jiJ2Jm)(jiJ2J'm'\jiJ2mim2) see also Racah ^-Coefficient, Racah ^-Coef- ficient, Wigner 3j-Symbol, Wigner 6j-Symbol, WlGNER 9J-SYMBOL References Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition Coefficients." §27.9 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1006-1010, 1972. Cohen- Tannoudji, C; Diu, B.; and Laloe, F. "Clebsch- Gordon Coefficients." Complement B x in Quantum Me- chanics, Vol 2. New York: Wiley, pp. 1035-1047, 1977. Condon, E. U. and Shortley, G. §3.6-3.14 in The Theory of Atomic Spectra. Cambridge, England: Cambridge Univer- sity Press, pp. 56-78, 1951. Fano, U. and Fano, L. Basic Physics of Atoms and Molecules. New York: Wiley, p. 240, 1959. Messiah, A. "Clebsch-Gordon (C.-G.) Coefficients and 'Sf Symbols." Appendix C.I in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1054-1060, 1962. Shore, B. W. and Menzel, D. H. "Coupling and Clebsch- Gordon Coefficients." §6.2 in Principles of Atomic Spectra. New York: Wiley, pp. 268-276, 1968. Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec- tra and Radiative Transitions, 2nd ed. Berlin: Springer- Verlag, 1992. Clement Matrix see Kac Matrix Clenshaw Recurrence Formula The downward Clenshaw recurrence formula evaluates a sum of products of indexed COEFFICIENTS by functions which obey a recurrence relation. If f( X ) = Y, ckFk ^ fc-0 (-l) i+ "V2j + lV{jij 2 j;mim 2 - m) (5) and F n +i(x) = a(n,x)F n (x) + f3(n,x)F n -i(x), (6) Cliff Random Number Generator where the CfcS are known, then define VN+2 = Vn+i = y k = a(/c, x)y k +i + 0{k + 1, x)y k+2 + c k for k ~ N, N - 1, . . . and solve backwards to obtain y 2 and yi. Cfc = J/* - a(fe, ^)y fc+ i - /?(fc + 1, x)y fc +2 N f(x) = ^2c k F k (x) fc=0 - coFo(x) + [t/i - a(l,x)y 2 - /3(2,x)y 3 ]F 1 (x) + [y 2 - a(2,x)y<i - (3(3,x)y4]F 2 (x) + [ys - a(3,x)y 4 - j3(4,x)y5]F s {x) + [y 4 - a(4, a) 3/5 - /3(5, x)y 6 ]i ? 4(x) + . . . = c Fo(x) + yi Fi (a:) + y 2 [F 2 (x) - a(l, z)Pi(z)] + ys[F 3 (x) - a(2, z)P 2 (:r) - 0(2, a)] + 2/ 4 [F 4 (x) - a(3,z)F 3 (x) - 0(3, x)] + . . . = c Fo(x) + y2[{a(l,x)F 1 ( : r)+/?(l,x)Fo( : r)} ~a{l t x)F 1 (x)]+yiF 1 (x) = c F Q {x) + yiFi(a) + 0(l,x)F o (x)y 2 . The upward Clenshaw recurrence formula is y-2 = y-i = _ 1 y *~/?(fc+l,x) for fe = 0, 1,..., N - 1. [y fc _ 2 - a(k 1 x)y k -i - c k ] f(x) = c N F N {x) - P(N ) x)F N -i(x)y N -i - F N (x)y N - 2 . References Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter- ling, W. T. "Recurrence Relations and Clenshaw's Recur- rence Formula." §5.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, Eng- land: Cambridge University Press, pp. 172-178, 1992. Cliff Random Number Generator A Random Number generator produced by iterating X n+ i = 1 100 In X„ (mod 1)| for a Seed X = 0.1. This simple generator passes the NOISE SPHERE test for randomness by showing no structure. see also RANDOM NUMBER, SEED References Pickover, C. A. "Computers, Randomness, Mind, and In- finity." Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233-247, 1995. Clique Number 271 Clifford Algebra Let V be an n-D linear Space over a Field K, and let Q be a Quadratic Form on V. A Clifford algebra is then defined over the T{V)/I(Q), where T(V) is the tensor algebra over V and I is a particular Ideal of T(V). References Iyanaga, S. and Kawada, Y. (Eds.). "Clifford Algebras." §64 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 220-222, 1980. Lounesto, P. "Counterexamples to Theorems Published and Proved in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior Algebra." http://www.hit. f i/~lounesto/counterexamples .htm. Clifford's Circle Theorem Let Ci, <7 2 , C 3 , and C 4 be four CIRCLES of GENERAL POSITION through a point P. Let Pij be the second intersection of the CIRCLES C» and Cj. Let dj k be the Circle PijP ik Pjk- Then the four Circles P234, Pi34, P124, and P123 all pass through the point P1234. Similarly, let C 5 be a fifth CIRCLE through P. Then the five points P2345, P1345, P1245, A235 and P1234 all lie on one Circle C12345. And so on. see also CIRCLE, Cox's THEOREM Clifford's Curve Theorem The dimension of a special series can never exceed half its order. References Coolidge, J. L. A Treatise on Algebraic Plane Curves. New- York: Dover, p. 263, 1959. Clique In a Graph of N Vertices, a subset of pairwise ad- jacent Vertices is known as a clique. A clique is a fully connected subgraph of a given graph. The prob- lem of finding the size of a clique for a given GRAPH is an NP-Complete Problem. The number of graphs on n nodes having 3 cliques are 0, 0, 1, 4, 12, 31, 67, ... (Sloane's A005289). see also Clique Number, Maximum Clique Prob- lem, Ramsey Number, Turan's Theorem References Sloane, N. J. A. Sequence A005289/M3440 in "An On-Line Version of the Encyclopedia of Integer Sequences." Clique Number The number of VERTICES in the largest CLIQUE of G, denoted u)(G). For an arbitrary GRAPH, ^— ' n-di where di is the DEGREE of VERTEX i. References Aigner, M. "Turan's Graph Theorem." Amer. Math. Monthly 102, 808-816, 1995. 272 Clock Solitaire Closure Clock Solitaire A solitaire game played with Cards. The chance of winning is 1/13, and the AVERAGE number of CARDS turned up is 42.4. References Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight- of- Mind from Scientific American. New York: Vintage, pp. 244-247, 1978. Close Packing see Sphere Packing Closed Curve closed curves open curves A CURVE with no endpoints which completely encloses an AREA. A closed curve is formally denned as the con- tinuous Image of a Closed Set. see also SIMPLE CURVE Closed Curve Problem Find Necessary and Sufficient conditions that de- termine when the integral curve of two periodic func- tions k(s) and t(s) with the same period L is a CLOSED Curve. Closed Disk An n-D closed disk of Radius r is the collection of points of distance < r from a fixed point in EUCLIDEAN n- space. see also Disk, Open Disk Closed Form A discrete FUNCTION A(n,k) is called closed form (or sometimes "hypergeometric" ) in two variables if the ra- tios A(n-rl,k)/A(n, k) and A(n,k-\-l)/A(n i k) are both Rational Functions. A pair of closed form functions (F, G) is said to be a Wilf-Zeilberger Pair if F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k). see also Rational Function, Wilf-Zeilberger Pair References Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles- ley, MA: A. K. Peters, p. 141, 1996. Zeilberger, D. "Closed Form (Pun Intended!)." Contempo- rary Math. 143, 579-607, 1993. Closed Graph Theorem A linear Operator between two Banach Spaces is continuous IFF it has a "closed" GRAPH. see also Banach SPACE References Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer- Verlag, 1995. Closed Interval An Interval which includes its Limit Points. If the endpoints of the interval are Finite numbers a and b, then the Interval is denoted [a, 6]. If one of the end- points is ±oo, then the interval still contains all of its Limit Points, so [a, oo) and ( — 00,6] are also closed intervals. see also Half-Closed Interval, Open Interval Closed Set There are several equivalent definitions of a closed Set. A Set S is closed if 1. The Complement of S is an Open Set, 2. S is its own CLOSURE, 3. Sequences/nets/filters in S which converge do so within 5, 4. Every point outside S has a NEIGHBORHOOD disjoint from 5. The Point-Set Topological definition of a closed set is a set which contains all of its Limit POINTS. There- fore, a closed set C is one for which, whatever point x is picked outside of C, x can always be isolated in some Open Set which doesn't touch C. see also CLOSED INTERVAL Closure A Set S and a Binary Operator * are said to ex- hibit closure if applying the Binary Operator to two elements S returns a value which is itself a member of S. The term "closure" is also used to refer to a "closed" version of a given set. The closure of a Set can be denned in several equivalent ways, including 1. The Set plus its Limit Points, also called "bound- ary" points, the union of which is also called the "frontier," 2. The unique smallest CLOSED Set containing the given Set, 3. The Complement of the interior of the Comple- ment of the set, 4. The collection of all points such that every NEIGH- BORHOOD of them intersects the original Set in a nonempty SET. In topologies where the T2-Separation Axiom is as- sumed, the closure of a finite Set S is S itself. Clothoid Cobordism 273 see also Binary Operator, Existential Closure, Reflexive Closure, Tight Closure, Transitive Closure Clothoid see also CORNU SPIRAL Clove Hitch A Hitch also called the Boatman's Knot or Peg Knot. References Owen, P. Knots. Philadelphia, PA: Courage, pp. 24-27, 1993. Clump see Run Cluster Given a lattice, a cluster is a group of filled cells which are all connected to their neighbors vertically or hori- zontally. see also Cluster Perimeter, Percolation Theory, s-Cluster, s-Run References StaufFer, D. and Aharony, A. Introduction to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992. Cluster Perimeter The number of empty neighbors of a CLUSTER. see also PERIMETER POLYNOMIAL Coanalytic Set A Definable Set which is the complement of an An- alytic Set. see also Analytic Set Coastline Paradox Determining the length of a country's coastline is not as simple as it first appears, as first considered by L. F. Richardson (1881-1953). In fact, the answer de- pends on the length of the RULER you use for the mea- surements. A shorter RULER measures more of the sin- uosity of bays and inlets than a larger one, so the esti- mated length continues to increase as the Ruler length decreases. In fact, a coastline is an example of a Fractal, and plotting the length of the Ruler versus the measured length of the coastline on a log-log plot gives a straight line, the slope of which is the FRACTAL DIMENSION of the coastline (and will be a number between 1 and 2). References Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig- ures. Princeton, NJ: Princeton University Press, pp. 29- 31, 1991. Coates- Wiles Theorem In 1976, Coates and Wiles showed that Elliptic Curves with Complex Multiplication having an in- finite number of solutions have //-functions which are zero at the relevant fixed point. This is a special case of the Swinnerton-Dyer Conjecture. References Cipra, B. "Fermat Prover Points to Next Challenges." Sci- ence 271, 1668-1669, 1996. Coaxal Circles Circles which share a Radical Line with a given cir- cle are said to be coaxal. The centers of coaxal circles are COLLINEAR. It is possible to combine the two types of coaxal systems illustrated above such that the sets are orthogonal. see also Circle, Coaxaloid System, Gauss- Bodenmiller Theorem, Radical Line References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 35-36 and 122, 1967. Dixon, R. Mathographics. New York: Dover, pp, 68-72, 1991. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 34-37, 199, and 279, 1929. Coaxal System A system of COAXAL CIRCLES. Coaxaloid System A system of circles obtained by multiplying each Radius in a Coaxal System by a constant. References Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 276-277, 1929. Cobordant Manifold Two open MANIFOLDS M and M' are cobordant if there exists a MANIFOLD with boundary W n+1 such that an acceptable restrictive relationship holds. see also COBORDISM, /i-COBORDISM THEOREM, MORSE Theory Cobordism see Bordism, /i-Cobordism 274 Cobordism Group Code Cobordism Group see Bordism Group Cobordism Ring see Bordism Group with Inversion Center at the Origin and inversion radius k is the QuADRATRIX OF HlPPIAS. x = kt cot y = kt. (2) (3) Cochleoid The cochleoid, whose name means "snail-form" in Latin, was first discussed by J. Peck in 1700 (MacTutor Ar- chive). The points of contact of PARALLEL TANGENTS to the cochleoid lie on a Strophoid. In Polar Coordinates, asin# In Cartesian Coordinates, (x 2 + 2/ 2 )tan- 1 (|) The Curvature is _ 2y / 2l9 3 [2l9-sin(2fl)] ay. [1 + 20 2 - cos(2(9) - 2(9 sin(2<9)] 3 / 2 ' (1) (2) (3) see also QUADRATRIX OF HlPPIAS References Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 196, 1972. MacTutor History of Mathematics Archive. "Cochleoid." http: //www-groups . dcs . st-and.ac . uk/ -history /Curves /Cochleoid. html. Cochleoid Inverse Curve The Inverse Curve of the Cochleoid Cochloid see Conchoid of Nicomedes Cochran's Theorem The converse of FISHER'S THEOREM. Cocked Hat Curve The Plane Curve (x 2 + 2ay - a 2 ) 2 = y 2 (a 2 - x 2 ), which is similar to the BlCORN. References Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989. Cocktail Party Graph (i) A Graph consisting of two rows of paired nodes in which all nodes but the paired ones are connected with an Edge. It is the complement of the Ladder Graph. Coconut see Monkey and Coconut Problem Codazzi Equations see MAINARDI-CODAZZI EQUATIONS Code A code is a set of n-tuples of elements ("WORDS") taken from an ALPHABET. see also Alphabet, Coding Theory, Encoding, Error-Correcting Code, Gray Code, Huffman Coding, ISBN, Linear Code, Word Codimension Coin 275 Codimension The minimum number of parameters needed to fully de- scribe all possible behaviors near a nonstructurally sta- ble element. see also BIFURCATION Coding Theory Coding theory, sometimes called ALGEBRAIC CODING THEORY, deals with the design of ERROR-CORRECTING CODES for the reliable transmission of information across noisy channels. It makes use of classical and modern algebraic techniques involving Finite Fields, Group Theory, and polynomial algebra. It has con- nections with other areas of DISCRETE MATHEMATICS, especially Number Theory and the theory of experi- mental designs. see also Encoding, Error-Correcting Code, Ga- lois Field, Hadamard Matrix References Alexander, B. "At the Dawn of the Theory of Codes." Math. Intel 15, 20-26, 1993. Golomb, S. W.; Peile, R. E.; and Scholtz, R. A. Basic Con- cepts in Information Theory and Coding: The Adventures of Secret Agent 00111. New York: Plenum, 1994. Humphreys, O. F. and Prest, M. Y. Numbers, Groups, and Codes. New York: Cambridge University Press, 1990. MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error- Correcting Codes. New York: Elsevier, 1978. Roman, S. Coding and Information Theory. New York: Springer- Verlag, 1992. Coefficient A multiplicative factor (usually indexed) such as one of the constants ai in the Polynomial a n x n + a n -i£ n_1 4- . . . + aix 2 -f- a\x + a . see also Binomial Coefficient, Cartan Tor- sion Coefficient, Central Binomial Coeffi- cient, Clebsch-Gordon Coefficient, Coeffi- cient Field, Commutation Coefficient, Con- nection Coefficient, Correlation Coefficient, Cross-Correlation Coefficient, Excess Coef- ficient, Gaussian Coefficient, Lagrangian Co- efficient, Multinomial Coefficient, Pearson's Skewness Coefficients, Product-Moment Co- efficient of Correlation, Quartile Skewness Coefficient, Quartile Variation Coefficient, Racah V-Coefficient, Racah ^-Coefficient, Re- gression Coefficient, Roman Coefficient, Tri- angle Coefficient, Undetermined Coefficients Method, Variation Coefficient Coercive Functional A bilinear FUNCTIONAL <j> on a normed SPACE E is called coercive (or sometimes Elliptic) if there exists a POS- ITIVE constant K such that <i>(x,x)>K\\x\\ 2 for all x £ E. see also Lax-Milgram Theorem References Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990. Cofactor The Minor of a Determinant is another Determi- nant |C| formed by omitting the ith row and jth col- umn of the original DETERMINANT |M|. dj = (-l) i+J 'oiM y . see also Determinant Expansion by Minors, Minor Cohen-Kung Theorem Guarantees that the trajectory of Langton's Ant is unbounded. Cohomology Cohomology is an invariant of a TOPOLOGICAL SPACE, formally "dual" to HOMOLOGY, and so it detects "holes'* in a SPACE. Cohomology has more algebraic structure than Homology, making it into a graded ring (multi- plication given by "cup product"), whereas HOMOLOGY is just a graded Abelian Group invariant of a Space. A generalized homology or cohomology theory must sat- isfy all of the Eilenberg-Steenrod Axioms with the exception of the dimension axiom. see also Aleksandrov-Cech Cohomology, Alexan- der-Spanier Cohomology, Cech Cohomology, de Rham Cohomology, Homology (Topology) Cohomotopy Group Cohomotopy groups are similar to HOMOTOPY GROUPS. A cohomotopy group is a Group related to the Homo- topy classes of Maps from a Space X into a Sphere see also HOMOTOPY GROUP Coefficient Field Let V be a Vector Space over a Field K, and let A be a nonempty Set. For an appropriately defined Affine Space A, K is called the Coefficient field. Coin A flat disk which acts as a two-sided Die. see Bernoulli Trial, Cards, Coin Paradox, Coin Tossing, Dice, Feller's Coin-Tossing Constants, Four Coins Problem, Gambler's Ruin References Brooke, M. Fun for the Money. New York: Scribner's, 1963. 276 Coin Flipping Coin Tossing Coin Flipping see Coin Tossing Coin Paradox After a half rotation of the coin on the left around the central coin (of the same RADIUS), the coin undergoes a complete rotation. References Pappas, T. "The Coin Paradox." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 220, 1989. Coin Problem Let there be n > 2 INTEGERS < a\ < . . . < a n with (ai,a 2 ,...,a n ) = 1 (all Relatively Prime). For large enough N = X^-i a i x ii there is a solution in NoNNEG- ATIVE INTEGERS xi. The greatest N — g(ai,a 2 , ...a n ) for which there is no solution is called the coin problem. Sylvester showed g(ai,a 2 ) - {a\ - l)(o 3 - 1) - 1, and an explicit solution is known for n — 3, but no closed form solution is known for larger N. References Guy, R. K. "The Money- Changing Problem." §C7 in Un- solved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 113-114, 1994. Coin Tossing An idealized coin consists of a circular disk of zero thick- ness which, when thrown in the air and allowed to fall, will rest with either side face up ("heads" H or "tails" T) with equal probability. A coin is therefore a two-sided Die. A coin toss corresponds to a Bernoulli Distri- bution with p = 1/2. Despite slight differences between the sides and NONZERO thickness of actual coins, the distribution of their tosses makes a good approximation to a p = 1/2 Bernoulli Distribution. There are, however, some rather counterintuitive prop- erties of coin tossing. For example, it is twice as likely that the triple TTH will be encountered before THT than after it, and three times as likely that THH will precede HTT. Furthermore, it is six times as likely that HTT will be the first of HTT, TTH, and TTT to oc- cur (Honsberger 1979). More amazingly still, spinning a penny instead of tossing it results in heads only about 30% of the time (Paulos 1995). Let w(n) be the probability that no RUN of three consec- utive heads appears in n independent tosses of a Coin. The following table gives the first few values of w{n). n w(n) 1 1 1 2 1 3 7 8 4 13 16 5 3 4 Feller (1968, pp. 278-279) proved that lim w(n)a n + l ■0, (1) vhere a = f [(136 + 24v / 33) 1/3 - 8(136 + 24v / 33)~ 1/3 - 2] - 1.087378025. and = ^ — — = 1.236839845 . . 4 — 3a (2) (3) The corresponding constants for a RUN of k > 1 heads are a*, the smallest Positive Root of and i -* + (!*) k = k + 1 o, k + 1 — kak (4) (5) These arc modified for unfair coins with P(H) = p and P(T) = q = 1 - p to a' k , the smallest Positive Root of l-z + <2pV +1 -0, (6) and & = P&k (7) (k + 1 -ka' k )p (Feller 1968, pp. 322-325). see also BERNOULLI DISTRIBUTION, CARDS, COIN, Dice, Gambler's Ruin, Martingale, Run, Saint Petersburg Paradox References Feller, W. An Introduction to Probability Theory and Its Ap- plication, Vol. 1, 3rd ed. New York: Wiley, 1968. Finch, S. u Favorite Mathematical Constants.' 1 http://www. mathsoft.com/asolve/constant/feller/feller.htnil. Ford, J. "How Random is a Coin Toss?" Physics Today 36, 40-47, 1983. Honsberger, R. "Some Surprises in Probability." Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 100-103, 1979. Keller, J. B. "The Probability of Heads." Amer. Math. Monthly 93, 191-197, 1986. Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 75, 1995. Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 238-239, 1990. Spencer, J. "Combinatorics by Coin Flipping." Coll. Math. J., 17, 407-412, 1986. Coincidence Collatz Problem 277 Coincidence A coincidence is a surprising concurrence of events, per- ceived as meaningfully related, with no apparent causal connection (Diaconis and Mosteller 1989). see also Birthday Problem, Law of Truly Large Numbers, Odds, Probability, Random Number References Bogomolny, A. "Coincidence." http://www.cut— the-knot . com/ do_you_know/coincidence. html. Falk, R. "On Coincidences." Skeptical Inquirer 6, 18—31, 1981-82. Falk, R. "The Judgment of Coincidences: Mine Versus Yours." Amer. J. Psych. 102, 477-493, 1989. Falk, R. and MacGregor, D. "The Surprisingness of Coinci- dences." In Analysing and Aiding Decision Processes (Ed. P. Humphreys, O. Svenson, and A. Vari). New York: El- sevier, pp. 489-502, 1984. Diaconis, P. and Mosteller, F. "Methods of Studying Coinci- dences." J. Amer. Statist. Assoc. 84, 853-861, 1989. Jung, C. G. Synchronicity: An Acausal Connecting Princi- ple. Princeton, NJ: Princeton University Press, 1973. Kammerer, P. Das Gesetz der Serie: Eine Lehre von den Wiederholungen im Lebens — und im Weltgeschehen. Stuttgart, Germany: Deutsche Verlags-Anstahlt, 1919. Stewart, I. "What a Coincidence!" Sci. Amer. 278, 95-96, June 1998. Colatitude The polar angle on a SPHERE measured from the North Pole instead of the equator. The angle <j> in SPHERICAL Coordinates is the Colatitude. It is related to the Latitude 5 by <p = 90° - S. see also LATITUDE, LONGITUDE, SPHERICAL COORDI- NATES Colinear see COLLINEAR Collatz Problem A problem posed by L. Collatz in 1937, also called the 3x + 1 Mapping, Hasse's Algorithm, Kakutani's Problem, Syracuse Algorithm, Syracuse Prob- lem, Thwaites Conjecture, and Ulam's Problem (Lagarias 1985). Thwaites (1996) has offered a £1000 reward for resolving the Conjecture. Let n be an In- teger. Then the Collatz problem asks if iterating fin) i 1 I 3 3n+l for n even for n odd (i) always returns to 1 for POSITIVE n. This question has been tested and found to be true for all numbers < 5.6 x 10 13 (Leavens and Vermeulen 1992), and more recently, 10 15 (Vardi 1991, p. 129). The members of the SEQUENCE produced by the Collatz are sometimes known as Hailstone NUMBERS. Because of the dif- ficulty in solving this problem, Erdos commented that "mathematics is not yet ready for such problems" (La- garias 1985). If NEGATIVE numbers are included, there are four known cycles (excluding the trivial cycle): (4, 2, 1), (-2, -1), (-5, -7, -10), and (-17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34). The num- ber of tripling steps needed to reach 1 for n = 1, 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, . . . (Sloane's A006667). The Collatz problem was modified by Terras (1976, 1979), who asked if iterating T(x) -{I X (Sx + 1) for x even for x odd (2) always returns to 1. If NEGATIVE numbers are included, there are 4 known cycles: (1, 2), (-1), (-5, -7, -10), and (-17, -25, -37, -55, -82, -41, -61, -91, -136, —68, —34). It is a special case of the "generalized Collatz problem" with d = 2, mo = 1, mi = 3, ro — 0, and ri = -1. Terras (1976, 1979) also proved that the set of Integers Sk = {n : n has stopping time < k} has a limiting asymptotic density F(h), so the limit F(k)= lim -, a:-»oo X (3) for {n : n < x and cr(n) < k} exists. Furthermore, F(k) — >- 1 as k -4 oo, so almost all INTEGERS have a finite stopping time. Finally, for all k > 1, 1 - F(k) lim - < 2 £->00 X -T]k where 7] = 1-H(0) = 0.05004... H (x) = —x lg x — (1 — x) lg(l — x) "Si (4) (5) (6) (7) (Lagarias 1985). Conway proved that the original Collatz problem has no nontrivial cycles of length < 400. Lagarias (1985) showed that there are no nontrivial cycles with length < 275,000. Conway (1972) also proved that Collatz- type problems can be formally Undecidable. A generalization of the COLLATZ PROBLEM lets d > 2 be a Positive Integer and mo, . . . , md-i be Nonzero Integers. Also let r»eZ satisfy n = irfii (mod d) . Then T(x) = mix — Ti (8) (9) for x = i (mod d) defines a generalized Collatz mapping. An equivalent form is w-L'r + x t (10) 278 Collatz Problem Collineation for x = i (mod d) where Xo, . . . , Xd-\ are INTEGERS and [r\ is the FLOOR FUNCTION. The problem is con- nected with Ergodic Theory and Markov Chains (Matthews 1995). Matthews (1995) obtained the fol- lowing table for the mapping Tk(x) \i(3x for x = (mod 2) + k) for x = 1 (mod 2), (11) where k = T*\ k # Cycles Max. Cycle Length 5 27 1 10 34 2 13 118 3 17 118 4 19 118 5 21 165 6 23 433 Matthews and Watts (1984) proposed the following con- jectures. 1. If | mo ■ ■ -rrid-il < d d , then all trajectories {T K (n)} for n € Z eventually cycle. 2. If |mo---md-i| > <2 d , then almost all trajectories {T K (n)} for n € Z are divergent, except for an ex- ceptional set of Integers n satisfying #{n £S\-X<n<X} = o(X). 3. The number of cycles is finite. 4. If the trajectory {T K (n)} for n 6 Z is not eventually cyclic, then the iterates are uniformly distribution mod d a for each a > 1, with 1 lim iv^oo AT+ 1 card{if < N\T K (n) = j (mod d a )} (12) for < j < d a - 1. Matthews believes that the map T(x) "{i v 3 7a: + 3 (7a: + 2) 3^-2) for x = (mod 3) for x = 1 (mod 3) for x = 2 (mod 3) (13) will either reach (mod 3) or will enter one of the cycles ( — 1) or (-2,-4), and offers a $100 (Australian?) prize for a proof. see also HAILSTONE Number References Applegate, D. and Lagarias, J. C. "Density Bounds for the 3z + 1 Problem 1. Tree-Search Method." Math. Comput 64, 411-426, 1995. Applegate, D. and Lagarias, J. C. "Density Bounds for the Sx + 1 Problem 2. Krasikov Inequalities." Math. Comput. 64, 427-438, 1995. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Burckel, S. "Functional Equations Associated with Congru- ential Functions." Theor. Comp. Set. 123, 397-406, 1994. Conway, J. H. "Unpredictable Iterations." Proc. 1972 Num- ber Th. Conf., University of Colorado, Boulder, Colorado, pp. 49-52, 1972. Crandall, R. "On the ( 3z + 1' Problem." Math. Comput 32, 1281-1292, 1978. Everett, C. "Iteration of the Number Theoretic Function f(2n) = n, f(2n + 1) = f(3n + 2)." Adv. Math. 25, 42-45, 1977. Guy, R. K. "Collatz's Sequence." §E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, pp. 215-218, 1994. Lagarias, J. C. "The 3x + l Problem and Its Generalizations." Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm, sfu. ca/organics/papers/lagarias/. Leavens, G. T. and Vermeulen, M. "3x + l Search Programs." Comput. Math. Appl. 24, 79-99, 1992. Matthews, K. R. "The Generalized 3x+l Mapping." http:// www.maths.uq.oz.au/-krm/survey.dvi. Rev. Sept. 10, 1995. Matthews, K. R. "A Generalized 3z + 1 Conjecture." [$100 Reward for a Proof.] ftp://www.maths.uq.edu.au/pub/ krm/gnubc/challenge. Matthews, K. R. and Watts, A. M. "A Generalization of Hasses's Generalization of the Syracuse Algorithm." Acta Arith. 43, 167-175, 1984. Sloane, N. J. A. Sequence A006667/M0019 in "An On-Line Version of the Encyclopedia of Integer Sequences." Terras, R. "A Stopping Time Problem on the Positive Inte- gers." Acta Arith. 30, 241-252, 1976. Terras, R. "On the Existence of a Density." Acta Arith. 35, 101-102, 1979. Thwaites, B. "Two Conjectures, or How to win £1100." Math.Gaz. 80, 35-36, 1996. Vardi, I. "The 3# + 1 Problem." Ch. 7 in Computational Recreations in Mathematica. Redwood City, CA: Addison- Wesley, pp. 129-137, 1991. Collinear Three or more points Pi, P2, P3, . .., are said to be collinear if they lie on a single straight LINE L. (Two points are always collinear.) This will be true IFF the ratios of distances satisfy X2 - xi : y 2 - yi : Z2 - zi = x 3 - xi : y 3 — yi : zs - zi. Two points are trivially collinear since two points deter- mine a Line. see also Concyclic, Directed Angle, N-Cluster, Sylvester's Line Problem Collineation A transformation of the plane which transforms COL- LINEAR points into COLLINEAR points. A projective collineation transforms every 1-D form projectively, and a perspective collineation is a collineation which leaves all lines through a point and points through a line invari- ant. In an ELATION, the center and axis are incident; in Cologarithm Combination 279 a HOMOLOGY they are not. For further discussion, see Coxeter (1969, p. 248). see also Affinity, Correlation, Elation, Equi- affinity, Homology (Geometry), Perspective Collineation, Projective Collineation References Coxeter, H. S. M. "Collineations and Correlations." §14.6 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 247-251, 1969. Cologarithm The Logarithm of the Reciprocal of a number, equal to the Negative of the Logarithm of the number it- self, colog x = log ( — J — — log x. see also Antilogarithm, Logarithm Colon Product Let AB and CD be Dyads. Their colon product is defined by AB : CD = C AB D = (A C)(B D). Colorable Color each segment of a KNOT DIAGRAM using one of three colors. If 1. at any crossing, either the colors are all different or all the same, and 2. at least two colors are used, then a KNOT is said to be colorable (or more specif- ically, Three- Colorable). Color ability is invariant under REIDEMEISTER Moves, and can be generalized. For instance, for five colors 0, 1, 2, 3, and 4, a KNOT is five-colorable if 1. at any crossing, three segments meet. If the overpass is numbered a and the two underpasses B and C, then 2a = b -f c (mod 5), and 2. at least two colors are used. Colorability cannot alway distinguish HANDEDNESS. For instance, three-colorability can distinguish the mir- ror images of the TREFOIL KNOT but not the FlGURE- OF-ElGHT KNOT. Five-colorability, on the other hand, distinguishes the MIRROR Images of the FlGURE-OF- Eight Knot but not the Trefoil Knot. see also Coloring, Three-Colorable Coloring A coloring of plane regions, Link segments, etc., is an assignment of a distinct labelling (which could be a number, letter, color, etc.) to each component. Col- oring problems generally involve TOPOLOGICAL consid- erations (i.e., they depend on the abstract study of the arrangement of objects), and theorems about colorings, such as the famous Four-Color THEOREM, can be ex- tremely difficult to prove. see also COLORABLE, EDGE-COLORING, FOUR-COLOR Theorem, ^-Coloring, Polyhedron Coloring, Six-Color Theorem, Three-Colorable, Vertex Coloring References Eppstein, D. "Coloring," http://vvv . ics . uci . edu / - eppstein/ junkyard/color. html. Saaty, T. L. and Kainen, P. C The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. Columbian Number see Self Number Colunar Triangle Given a SCHWARZ TRIANGLE (p q r), replacing each Vertex with its antipodes gives the three colunar Spherical Triangles (p q r'),(p q r f ),(p q r), where P P q q' r r see also SCHWARZ TRIANGLE, SPHERICAL TRIANGLE References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 112, 1973. Comb Function see Shah Function Combination The number of ways of picking r unordered outcomes from n possibilities. Also known as the Binomial Co- efficient or Choice Number and read "n choose r." t.Ct. = rl(n - where n\ is a FACTORIAL. see also Binomial Coefficient, Derangement, Fac- torial, Permutation, Subfactorial References Conway, J. H. and Guy, R. K. "Choice Numbers." In The Book of Numbers. New York: Springer- Verlag, pp. 67-68, 1996. Ruskey, F. "Information on Combinations of a Set." http://sue . esc . uvic . ca/~cos/inf /comb/Combinations Info.html. 280 Combination Lock Combinatorics Combination Lock Let a combination of n buttons be a SEQUENCE of dis- joint nonempty Subsets of the Set {1, 2, . . . , n}. If the number of possible combinations is denoted a n , then a n satisfies the RECURRENCE RELATION i— n \ / with ao = 1. This can also be written 2 / , 2 k ' k=0 (1) (2) where the definition 0° = 1 has been used. Furthermore, a n = 2^i n ,fe2 n = ^^^4n,fc2 , (3) fc = l where A n ,k are EULERIAN NUMBERS. In terms of the Stirling Numbers of the Second Kind s{n,k), a n = \, k\s(n,k). k = l a n can also be given in closed form as a n — 2 Ll -n(2)> (4) (5) where Li n (z) is the POLYLOGARITHM. The first few values of a n for n = 1, 2, ... are 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (Sloane's A000670). The quantity b n = satisfies the inequality 1 2(ln2) n <b n < (ln2) n * (6) (7) References Sloane, N. J. A. Sequence A000670/M2952 in "An On-Line Version of the Encyclopedia of Integer Sequences." Velleman, D. J. and Call, G. S. "Permutations and Combi- nation Locks." Math. Mag. 68, 243-253, 1995. Combinatorial Species see Species Combinatorial Topology Combinatorial topology is a special type of Algebraic Topology that uses Combinatorial methods. For example, Simplicial Homology is a combinatorial construction in ALGEBRAIC TOPOLOGY, so it belongs to combinatorial topology. see also ALGEBRAIC TOPOLOGY, SlMPLICIAL HOMO- LOGY, Topology Combinatorics The branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations which characterize these properties. see also Antichain, Chain, Dilworth's Lemma, Diversity Condition, Erdos-Szekeres Theo- rem, Inclusion-Exclusion Principle, Kirkman's Schoolgirl Problem, Kirkman Triple System, Length (Partial Order), Partial Order, Pigeon- hole Principle, Ramsey's Theorem, Schroder- Bernstein Theorem, Schur's Lemma, Sperner's Theorem, Total Order, van der Waerden's The- orem, Width (Partial Order) References Abramowitz, M. and Stegun, C A. (Eds.). "Combinatorial Analysis." Ch. 24 in Handbook of Mathematical Func- tions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 821-8827, 1972. Aigner, M. Combinatorial Theory. New York: Springer- Verlag, 1997. Bellman, R. and Hall, M. Combinatorial Analysis. Amer. Math. Soc, 1979. Biggs, N. L. "The Roots of Combinatorics." Historia Math- ematica 6, 109-136, 1979. Bose, R. C. and Manvel, B. Introduction to Combinatorial Theory. New York: Wiley, 1984. Brown, K. S. "Combinatorics." http://www.seanet.com/ -ksbrown/icombina.htm. Cameron, P. J. Combinatorics: Topics, Techniques, Algo- rithms. New York: Cambridge University Press, 1994. Cohen, D. Basic Techniques of Combinatorial Theory. New York: Wiley, 1978. Cohen, D. E. Combinatorial Group Theory: A Topological Approach. New York: Cambridge University Press, 1989. Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi- natorial Designs. Boca Raton, FL: CRC Press, 1996. Comtet, L. Advanced Combinatorics. Dordrecht, Nether- lands: Reidel, 1974. Coolsaet, K. "Index of Combinatorial Objects." http://www. hogent.be/~kc/ico/. Dinitz, J. H. and Stinson, D. R. (Eds.). Contemporary De- sign Theory: A Collection of Surveys. New York: Wiley, 1992. Electronic Journal of Combinatorics. http : //www . combinatorics.org/previousjvolumes.html. Eppstein, D. "Combinatorial Geometry." http://www.ics. uci.edu/-eppstein/junkyard/combinatorial.html. Erickson, M. J. Introduction to Combinatorics. New York: Wiley, 1996. Fields, J. "On-Line Dictionary of Combinatorics." http:// math.uic.edu/-fields/dic/. Godsil, C. D. "Problems in Algebraic Combinatorics." Elec- tronic J. Combinatorics 2, Fl, 1-20, 1995. http: //www. combinatorics. org/Volume_2/volume2.html#Fl. Graham, R. L.; Grotschel, M.; and Lovasz, L. (Eds.). Hand- book of Combinatorics, 2 vols. Cambridge, MA: MIT Press, 1996. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Add is on- Wesley, 1994. Hall, M. Jr. Combinatorial Theory, 2nd ed. New York: Wi- ley, 1986. Knuth, D. E. (Ed.). Stable Marriage and Its Relation to Other Combinatorial Problems. Providence, RI: Amer. Math. Soc, 1997. Comma Derivative Commutation Coefficient 281 Kucera, L. Combinatorial Algorithms. Bristol, England: Adam Hilger, 1989. Liu, C. L. Introduction to Combinatorial Mathematics. New- York: McGraw-Hill, 1968. MacMahon, P. A. Combinatory Analysis. New York: Chelsea, 1960. Nijenhuis, A. and Wilf, H. Combinatorial Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press, 1978. Riordan, J. Combinatorial Identities, reprint ed. with correc- tions. Huntington, NY: Krieger, 1979. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980. Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Rota, G.-C. (Ed.). Studies in Combinatorics. Providence, RI: Math. Assoc. Amer., 1978. Ruskey, F. "The (Combinatorial) Object Server." http:// sue.csc.uvic.ca/-cos. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., 1963. Skiena, S. S. Implementing Discrete Mathematics: Combi- natorics and Graph Theory with Mathematica. Reading, MA: Addison- Wesley, 1990. Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/ -njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995. Street, A. P. and Wallis, W. D. Combinatorial Theory: An Introduction. Winnipeg, Manitoba: Charles Babbage Re- search Center, 1977. Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley, 1995. van Lint, J. H. and Wilson, R. M. A Course in Combina- torics. New York: Cambridge University Press, 1992. Wilf, H. S. Combinatorial Algorithms: An Update. Philadel- phia, PA: SIAM, 1989. Comma Derivative 9k dx k see also COVARIANT DERIVATIVE, SEMICOLON DERIV- ATIVE Comma of Didymus The musical interval by which four fifths exceed a sev- enteenth (i.e., two octaves and a major third), (I) 2 2(|) 2^.5 81 80 1.0125, also called a Syntonic Comma. see also COMMA OF PYTHAGORAS, DlESIS, SCHISMA Comma of Pythagoras The musical interval by which twelve fifths exceed seven octaves, ill 2 7 3^ 2 19 531441 524288 1.013643265. Successive CONTINUED FRACTION CONVERGENTS to log 2/ log (3/2) give increasingly close approximations m/n of m fifths by n octaves as 1, 2, 5/3, 12/7, 41/24, 53/31, 306/179, 665/389, ... (Sloane's A005664 and A046102; Jeans 1968, p. 188), shown in bold in the ta- ble below. All near-equalities of m fifths and n octaves having R. (§r 2^ Om+n with \R — 1| < 0.02 are given in the following table. m n Ratio m n Ratio 12 7 1.013643265 265 155 1.010495356 41 24 0.9886025477 294 172 0.9855324037 53 31 1.002090314 306 179 0.9989782832 65 38 1.015762098 318 186 1.012607608 94 55 0.9906690375 347 203 0.9875924759 106 62 1.004184997 359 210 1.001066462 118 69 1.017885359 371 217 1.014724276 147 86 0.9927398469 400 234 0.9896568543 159 93 1.006284059 412 241 1.003159005 188 110 0.9814251419 424 248 1.016845369 200 117 0.994814985 453 265 0.9917255479 212 124 1.008387509 465 272 1.005255922 241 141 0.9834766286 477 279 1.018970895 253 148 0.9968944607 494 289 0.9804224033 see also COMMA OF DlDYMUS, DlESIS, SCHISMA References Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer- Verlag, p. 257, 1995. Guy, R. K. "Small Differences Between Powers of 2 and 3." §F23 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer- Verlag, p. 261, 1994. Sloane, N. J. A. Sequences A005664 and A046102 in "An On- Line Version of the Encyclopedia of Integer Sequences." Common Cycloid see Cycloid Common Residue The value of fr, where a = b (mod m), taken to be NON- NEGATIVE and smaller than m. see also Minimal Residue, Residue (Congruence) Commutation Coefficient A coefficient which gives the difference between partial derivatives of two coordinates with respect to the other coordinate, c ap^ — [^cn^a] = V^e/3 - V^e a . see also CONNECTION COEFFICIENT 282 Commutative Compactness Theorem Commutative Let A denote an M- algebra, so that A is a VECTOR Space over R and A x A ->■ A (x,y) M- x-y. Now define Z = {x e a ; x • y foi some y 6 A / 0}, where € Z. An ASSOCIATIVE R-algebra is commuta- tive if x • y = y * x for all x, y € A. Similarly, a Ring is commutative if the MULTIPLICATION operation is com- mutative, and a LIE ALGEBRA is commutative if the Commutator [A, B] is for every A and B in the LIE Algebra. see also Abelian, Associative, Transitive References Finch, S. "Zero Structures in Real Algebras." http://www. mathsoft.com/asolve/zerodiv/zerodiv.html. MacDonald, I. G. and Atiyah, M. F. Introduction to Com- mutative Algebra. Reading, MA: Addison- Wesley, 1969. Commutative Algebra An Algebra in which the + operators and x are Com- mutative. see also Algebraic Geometry, Grobner Basis References MacDonald, I. G. and Atiyah, M. F. Introduction to Com- mutative Algebra. Reading, MA: Addison-Wesley, 1969. Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer- Verlag, 1996. Samuel, P. and Zariski, O, Commutative Algebra, Vol. 2. New York: Springer- Verlag, 1997. Commutator Let A, £, . . .be Operators. Then the commutator of A and B is defined as [A,B] = AB-BA. (1) Let a, 6, ... be constants. Identities include [/(*),*] = (2) [A,A]=0 (3) [A,B] = -[B,A] (4) [A,BC] = [A,B]C + B[A,C] (5) [AB, C] = [A, C]B + A[B, C] (6) [a + A,b + B] = [A,B] (7) [A + B,C + D} = [A,C] + [A,D] + [B,C] + [B,D]. (8) The commutator can be interpreted as the "infinitesi- mal" of the commutator of a Lie Group. Let A and B be Tensors. Then [A,B]=X? a B-VbA. (9) see also Anticommutator, Jacobi Identities Compact Group If the parameters of a LIE GROUP vary over a CLOSED Interval, the GROUP is compact. Every representation of a compact group is equivalent to a UNITARY repre- sentation. Compact Manifold A Manifold which can be "charted" with finitely many Euclidean Space charts. The Circle is the only com- pact l-D Manifold. The Sphere and n-ToRUS are the only compact 2-D MANIFOLDS. It is an open ques- tion if the known compact MANIFOLDS in 3-D are com- plete, and it is not even known what a complete list in 4-D should look like. The following terse table there- fore summarizes current knowledge about the number of compact manifolds N(D) of D dimensions. D N(D) see also Tychonof Compactness Theorem Compact Set The Set S is compact if, from any Sequence of ele- ments Xi, X 2y ...of S, a subsequence can always be extracted which tends to some limit element X of S. Compact sets are therefore closed and bounded. Compact Space A Topological Space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X. A subset A of a Topological Space X is compact if it is compact as a TOPOLOGICAL Space with the relative topology (i.e., every family of open sets of X whose union contains A has a finite subfamily whose union contains A). Compact Surface A surface with a finite number of TRIANGLES in its TRI- angulation. The Sphere and TORUS are compact, but the PLANE and TORUS minus a Disk are not. Compactness Theorem Inside a Ball B in R 3 , {rectifiable currents 5 in BL Area S < c, length dS < c} is compact under the Flat Norm. References Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103, 369-376, 1996. Companion Knot Complete Axiomatic Theory 283 Companion Knot Let Ki be a knot inside a TORUS. Now knot the TORUS in the shape of a second knot (called the companion knot) K2. Then the new knot resulting from K\ is called the Satellite Knot K 3 . References Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, New York: W. H. Freeman, pp. 115-118, 1994. Comparability Graph The comparability graph of a POSET P = (X, <) is the Graph with vertex set X for which vertices x and y are adjacent IFF either x < y or y < x in P. see also INTERVAL GRAPH, PARTIALLY ORDERED SET Comparison Test Let J2 ak and J2^ k be a Series with Positive terms and suppose a\ < &i, 02 < ta, 1. If the bigger series CONVERGES, then the smaller series also Converges. 2. If the smaller series DIVERGES, then the bigger series also Diverges. see also Convergence Tests References Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 280-281, 1985. Compass A tool with two arms joined at their ends which can be used to draw Circles. In Geometric Construc- tions, the classical Greek rules stipulate that the com- pass cannot be used to mark off distances, so it must "collapse" whenever one of its arms is removed from the page. This results in significant complication in the complexity of GEOMETRIC CONSTRUCTIONS, see also Constructible Polygon, Geometric Con- struction, Geometrography, Mascheroni Con- struction, Plane Geometry, Polygon, Poncelet- Steiner Theorem, Ruler, Simplicity, Steiner Construction, Straightedge References Dixon, R. "Compass Drawings." Ch. 1 in Mathographics. New York: Dover, pp. 1-78, 1991. Compatible Let 1 1 A 1 1 be the MATRIX NORM associated with the MA- TRIX A and ||x|| be the Vector Norm associated with a Vector x. Let the product Ax be defined, then ||A|| and ||x|| are said to be compatible if l|Ax||<||A||||x||. References Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se- ries, and Products, 5th ed. San Diego, CA: Academic Press, p. 1115, 1980. Complement Graph The complement Graph G of G has the same Vertices as G but contains precisely those two-element SUBSETS which are not in G. Complement Knot see Knot Complement Complement Set Given a set S with a subset F, the complement of E is defined as E' = {F:FeS,F^E}. (1) If E = 5, then E' = S' = 0, (2) where is the EMPTY SET. Given a single Set, the second Probability Axiom gives 1 = P(S) = P(EUE'). Using the fact that E n E f = 0, 1 = P(E) + P(E') P(E') = 1-P(E). (3) (4) (5) This demonstrates that P(S') = P{0) = 1 - P(S) = 1-1-0. (6) Given two Sets, P(E O F') = P(E) - P(E F) (7) P(E' r\F f ) = l- P(E) - P{F) + P(E O F). (8) Complementary Angle Two ANGLES a and 7r/2 - a are said to be complemen- tary. see also ANGLE, SUPPLEMENTARY ANGLE Complete see Complete Axiomatic Theory, Complete Bi- graph, Complete Functions, Complete Graph, Complete Quadrangle, Complete Quadrilat- eral, Complete Sequence, Complete Space, Completeness Property, Weakly Complete Se- quence Complete Axiomatic Theory An axiomatic theory (such as a Geometry) is said to be complete if each valid statement in the theory is capable of being proven true or false. see also CONSISTENCY 284 Complete Bigraph Complete Graph Complete Bigraph see Complete Bipartite Graph Complete Bipartite Graph Complete Graph A Bipartite Graph (i.e., a set of Vertices decom- posed into two disjoint sets such that there are no two VERTICES within the same set are adjacent) such that every pair of VERTICES in the two sets are adjacent. If there are p and q VERTICES in the two sets, the complete bipartite graph (sometimes also called a COMPLETE Bl- GRAPH) is denoted K p , q . The above figures show K^^ and i^2,5* see also Bipartite Graph, Complete Graph, Complete ^-Partite Graph, ^-Partite Graph, Thomassen Graph, Utility Graph References Saaty, T. L. and Kainen, P. C. The Four-Color Problem; Assaults and Conquest. New York: Dover, p. 12, 1986. Complete Functions A set of Orthonormal Functions </> n (x) is termed complete in the CLOSED INTERVAL x € [a, b] if, for every piecewise CONTINUOUS Function f(x) in the interval, the minimum square error E n = ||/-(ci0i + ... + c n n )|| 2 (where || denotes the Norm) converges to zero as n be- comes infinite. Symbolically, a set of functions is com- plete if lim 771— »-00 f f(x) - y^an4>n(x) n=Q w(x) dx — 0, where w(x) is a Weighting Function and the above is a Lebesgue Integral. see also BESSEL'S INEQUALITY, HlLBERT SPACE References Arfken, G. "Completeness of Eigenfunctions." §9.4 in Mathe- matical Methods for Physicists, 3rd ed. Orlando, FL: Aca- demic Press, pp. 523-538, 1985. A Graph in which each pair of VERTICES is connected by an EDGE. The complete graph with n VERTICES is denoted K n . In older literature, complete GRAPHS are called UNIVERSAL GRAPHS. K 4 is the Tetrahedral Graph and is therefore a PLA- NAR GRAPH. K$ is nonplanar. Conway and Gordon (1983) proved that every embedding of K G is INTRINSI- CALLY Linked with at least one pair of linked triangles. They also showed that any embedding of Kj contains a knotted Hamiltonian Cycle. The number of Edges in K v is v(v — l)/2, and the Genus is (v — 3)(v — 4)/12 for v > 3. The number of dis- tinct variations for K n (GRAPHS which cannot be trans- formed into each other without passing nodes through an EDGE or another node) for n — 1, 2, . . . are 1, 1, 1, 1, 1, 1, 6, 3, 411, 37, The Adjacency Matrix of the complete graph takes the particularly simple form of all Is with Os on the diagonal. It is not known in general if a set of Trees with 1,2,..., n — 1 Edges can always be packed into K n . However, if the choice of TREES is restricted to either the path or star from each family, then the packing can always be done (Zaks and Liu 1977, Honsberger 1985). References Chartrand, G. Introductory Graph Theory. New York: Dover, pp. 29-30, 1985. Conway, J. H. and Gordon, C. M. "Knots and Links in Spatial Graphs." J. Graph Th. 7, 445-453, 1983. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 60-63, 1985. Saaty, T. L. and Kainen, P. C. The Four-Color Problem,: Assaults and Conquest. New York: Dover, p. 12, 1986. Zaks, S. and Liu, C. L. "Decomposition of Graphs into Trees." Proc. Eighth Southeastern Conference on Com- binatorics, Graph Theory, and Computing, pp. 643-654, 1977. Complete k-Partite Graph Complete fc-Partite Graph A A;-Partite Graph (i.e., a set of Vertices decom- posed into k disjoint sets such that no two VERTICES within the same set are adjacent) such that every pair of Vertices in the k sets are adjacent. If there are p, q, . . . , r Vertices in the k sets, the complete bi- partite graph is denoted i^ P) ^,...,r- The above figure Shows 1^2,3,5- see also COMPLETE GRAPH, COMPLETE fc-PARTITE Graph, ^-Partite Graph References Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest, New York: Dover, p. 12, 1986. Complete Metric Space A complete metric space is a METRIC SPACE in which every CAUCHY SEQUENCE is CONVERGENT. Examples include the Real Numbers with the usual metric and the p-ADic Numbers. Complete Permutation see Derangement Complete Quadrangle If the four points making up a Quadrilateral are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. Note that a complete quadrilateral is defined differently from a COMPLETE Quadrangle. The midpoints of the sides of any complete quadrangle and the three diagonal points all lie on a CONIC known as the Nine-Point Conic If it is an Orthocentric Quadrilateral, the Conic reduces to a Circle. The Orthocenters of the four Triangles of a complete quadrangle are COLLINEAR on the RADICAL Line of the Circles on the diameters of a Quadrilateral. see also Complete Quadrangle, Ptolemy's Theo- rem References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230-231, 1969. Demir, H. "The Compleat [sic] Cyclic Quadrilateral." Amer. Math. Monthly 79, 777-778, 1972. Complete Sequence 285 Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 61-62, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 101-104, 1990. Complete Quadrilateral The figure determined by four lines and their six points of intersection (Johnson 1929, pp. 61-62). Note that this is different from a COMPLETE QUADRANGLE. The midpoints of the diagonals of a complete quadrilateral are COLLINEAR (Johnson 1929, pp. 152-153). A theorem due to Steiner (Mention 1862, Johnson 1929, Steiner 1971) states that in a complete quadrilateral, the bisectors of angles are CONCURRENT at 16 points which are the incenters and EXCENTERS of the four TRIAN- GLES. Furthermore, these points are the intersections of two sets of four CIRCLES each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the ClRCUMCIRCLES of the quadrilateral. see also COMPLETE QUADRANGLE, GAUSS-BODENMIL- ler Theorem, Polar Circle References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230-231, 1969. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 61-62, 149, 152-153, and 255- 256, 1929. Mention, M. J. "Demonstration d'un Theoreme de M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 16-20, 1862. Mention, M. J. "Demonstration d'un Theoreme de M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 65-67, 1862. Steiner, J. Gesammelte Werke, 2nd ed, Vol. 1. New York: Chelsea, p. 223, 1971. Complete Residue System A set of numbers clq, cli, ..., a m -i (mod m) form a complete set of residues if they satisfy ai = i (mod m) for i = 0, 1, . . . , m — 1. In other words, a complete system of residues is formed by a base and a modulus if the residues r; in b l = Vi (mod m) for i = 1, . . . , m - 1 run through the values 1, 2, ..., m — 1. see also Haupt-Exponent Complete Sequence A Sequence of numbers V — {u n } is complete if every Positive Integer n is the sum of some subsequence of V, i.e., there exist a; = or 1 such that / v aM (Honsberger 1985, pp. 123-126). The Fibonacci Num- bers are complete. In fact, dropping one number still 286 Complete Space Complex Analysis leaves a complete sequence, although dropping two num- bers does not (Honsberger 1985, pp. 123 and 126). The Sequence of Primes with the element {1} prepended, {1,2,3,5,7,11,13,17,19,23,...} is complete, even if any number of Primes each > 7 are dropped, as long as the dropped terms do not include two consecutive PRIMES (Honsberger 1985, pp. 127— 128). This is a consequence of BERTRAND'S POSTU- LATE. see also Bertrand's Postulate, Brown's Cri- terion, Fibonacci Dual Theorem, Greedy Al- gorithm, Weakly Complete Sequence, Zeck- endorf's Theorem References Brown, J. L. Jr. "Unique Representations of Integers as Sums of Distinct Lucas Numbers." Fib. Quart. 7,243-252,1969. Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for Fibonacci Numbers. XIL" Fib. Quart. 11, 317-331, 1973. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985. Complete Space A Space of Complete Functions. see also COMPLETE METRIC SPACE Completely Regular Graph A POLYHEDRAL Graph is completely regular if the Dual Graph is also Regular. There are only five types. Let p be the number of EDGES at each node, p* the number of EDGES at each node of the DUAL GRAPH, V the number of VERTICES, E the number of EDGES, and F the number of faces in the Platonic Solid cor- responding to the given graph. The following table sum- marizes the completely regular graphs. Type 9 P* V E F Tetrahedral 3 3 4 6 4 Cubical 3 4 8 12 6 Dodecahedral 3 5 20 39 12 Octahedral 4 3 6 12 8 Icosahedral 5 3 12 30 20 Completeness Property All lengths can be expressed as Real Numbers. Completing the Square The conversion of an equation of the form ax 2 + bx + c to the form a { x + ^) + ic -4-al' which, defining B = b/2a and C = c — b 2 /4a, simplifies to a(x + B) 2 + C. Complex A finite Set of SlMPLEXES such that no two have a common point. A 1-D complex is called a GRAPH. see also CW-Complex, Simplicial Complex Complex Analysis The study of Complex NUMBERS, their DERIVATIVES, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of phys- ical problems. CONTOUR INTEGRATION, for example, provides a method of computing difficult INTEGRALS by investigating the singularities of the function in regions of the Complex Plane near and between the limits of integration. The most fundamental result of complex analysis is the Cauchy-Riemann Equations, which give the condi- tions a Function must satisfy in order for a com- plex generalization of the Derivative, the so-called Complex Derivative, to exist. When the Complex Derivative is defined "everywhere," the function is said to be ANALYTIC. A single example of the unex- pected power of complex analysis is PlCARD'S Theo- rem, which states that an Analytic Function as- sumes every Complex Number, with possibly one ex- ception, infinitely often in any NEIGHBORHOOD of an Essential Singularity! see also ANALYTIC CONTINUATION, BRANCH CUT, Branch Point, Cauchy Integral Formula, Cau- chy Integral Theorem, Cauchy Principal Value, Cauchy-Riemann Equations, Complex Number, Conformal Map, Contour Integration, de Moivre's Identity, Euler Formula, Inside- Outside Theorem, Jordan's Lemma, Laurent Se- ries, Liouville's Conformality Theorem, Mono- genic Function, Morera's Theorem, Permanence of Algebraic Form, Picard's Theorem, Pole, Polygenic Function, Residue (Complex Analy- sis) References Arfken, G. "Functions of a Complex Variable I: Analytic Properties, Mapping" and "Functions of a Complex Vari- able II: Calculus of Residues." Chs. 6—7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 352-395 and 396-436, 1985. Boas, R. P. Invitation to Complex Analysis. New York: Ran- dom House, 1987. Churchill, R. V. and Brown, J. W. Complex Variables and Applications, 6th ed. New York: McGraw-Hill, 1995. Conway, J. B. Functions of One Complex Variable, 2nd ed. New York: Springer- Verlag, 1995. Forsyth, A. R. Theory of Functions of a Complex Variable, 3rd ed, Cambridge, England: Cambridge University Press, 1918. . Lang, S. Complex Analysis, 3rd ed. New York: Springer- Verlag, 1993. Morse, P. M. and Feshbach, H. "Functions of a Complex Vari- able" and "Tabulation of Properties of Functions of Com- plex Variables." Ch. 4 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 348-491 and 480-485, 1953. Complex Conjugate Complex Number 287 Complex Conjugate The complex conjugate of a Complex Number z = a+bi is defined to be z* = a— hi. The complex conjugate is Associative, (zi + z 2 )* = zi* + z 2 *, since (ai H- M)* + (a 2 + M)* — ai - ibi + a 2 - i&2 = (ai - ibi) + (a 2 - ib 2 ) = (ai+6i)* + (a 2 + b 2 )*, and Distributive, (ziz 2 ) m = zi*z 2 *, since [(ai + bii)(a 2 + 62*)]* = [( a i a 2 - 6162) + i(ai& 2 + 0261)]* = (ai(X2 — &ifr 2 ) — i(ai6 2 + a 2 6i) = (ai - z6i)(a 2 - i6 2 ) — (ai + i6i)*(a2 + 162)*. References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972. Complex Derivative A Derivative of a Complex function, which must sat- isfy the Cauchy-Riemann Equations in order to be Complex Differentiable. see also Cauchy-Riemann Equations, Differentiable, Derivative Complex Complex Differentiable If the Cauchy-Riemann Equations are satisfied for a function f(x) = u(x) + iv(x) and the PARTIAL DERIVA- TIVES of u(x) and v(x) are Continuous, then the Com- plex Derivative df/dz exists. see also Analytic Function, Cauchy-Riemann Equations, Complex Derivative, Pseudoanalytic Function Complex Function A Function whose Range is in the Complex Num- bers is said to be a complex function. see also Real Function, Scalar Function, Vector Function Complex Matrix A Matrix whose elements may contain Complex Num- bers. The Matrix Product of two 2x2 complex matrices is given by xu + 2/i 1* Z12 + y 12 i £21 + V2ii £22 + 2/22* uu -\-Vni U12 + V121 U21 -\-v21i 1*22 + ^22^ R11 R12 -H 111 1 12 R21 R22 hi 1 22 where R11 — u\\x\\ + u 2 ixi 2 — viij/11 — v 2 iyi 2 Rl2 — Wl2Xll + ^22^12 - V122/11 - U222/12 R 2 1 = U11X21 + U21X22 - Ul 12/21 - V21J/22 R 22 = Ui 2 X 2 ± + u 22 x 22 — vi 2 y 2 i — V 222/22 In = vnxii + ^21X12 + wnyii + U21IJ12 111 = V12X11 + ^22^12 + U122/11 + ^222/12 ^21 = ^113521 + ^21^22 + U112/21 + ^212/22 i~22 = V\ 2 X 2 1 + V 22 #22 + ^122/21 + ^222/22- see a/so Real Matrix Complex Multiplication Two Complex Numbers x = a + ib and y = c + id are multiplied as follows: xy — (a + i&)(c + zd) = ac + ibc + zad — 6d = (ac - bd) + i(ad -f 6c). However, the multiplication can be carried out using only three REAL multiplications, ac, bd, and (a+b)(c-\-d) as R[(a + ib)(c + id)] = ac - bd 9f[(a + ifc)(c + id)] = (a + 6)(c + d) - ac - bd. Complex multiplication has a special meaning for EL- LIPTIC Curves. see also Complex Number, Elliptic Curve, Imagi- nary Part, Multiplication, Real Part References Cox, D. A. Primes of the Form x 2 +ny 2 : Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997. Complex Number The complex numbers are the Field C of numbers of the form x + iy, where x and y are REAL NUMBERS and i is the Imaginary Number equal to >/-!• When a single letter z - x + iy is used to denote a complex number, it is sometimes called an "AFFIX." The FIELD of complex numbers includes the Field of Real Numbers as a Subfield. Through the Euler FORMULA, a complex number z = x -f iy (1) may be written in "PHASOR" form z = \z\ (cos + i sin 6) = \z\e ie . (2) Here, \z\ is known as the Modulus and 9 is known as the Argument or Phase. The Absolute Square of 288 Complex Number Complex Structure z is defined by \z\ 2 — zz* , and the argument may be computed from Complex Plane arg(z) — = tan I — J (3) de Moivre's Identity relates Powers of complex numbers z n = |z| n [cos(n#) + zsin(n#)]. (4) Finally, the Real R(z) and Imaginary Parts $s(z) are given by »w = i(^+o (5) *(*> = ^^ = ~W - O = 5*(** " *)■ ( 6 ) 2z The Powers of complex numbers can be written in closed form as follows: -0 ri-2 2 . I n \ n-4 4 x y + 1 4 p y + i >~v 3 F y +.. (7) The first few are explicitly z 2 = (x 2 - y 2 ) -{- i(2xy) z = (x — 3xy ) + i(3x y — y ) z 4 = (x 4 - 6x 2 y 2 + y 4 ) 4- i(4z 3 y - 4xy 3 ) z 5 = ( x 5 - I0x 3 y 2 + 5zy 4 ) + i{$x A y - 10xV + y 5 ) (8) (9) (10) (11) (Abramowitz and Stegun 1972). see also Absolute Square, Argument (Complex Number), Complex Plane, i, Imaginary Number, Modulus, Phase, Phasor, Real Number, Surreal Number References Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16-17, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or- lando, FL: Academic Press, pp. 353-357, 1985. Courant, R. and Robbins, H. "Complex Numbers." §2.5 in What is Mathematics? : An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University- Press, pp. 88-103, 1996. Morse, P. M. and Feshbach, H. "Complex Numbers and Vari- ables." §4.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 349-356, 1953. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet- terling, W. T. "Complex Arithmetic." §5.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 171-172, 1992. Imaginary The plane of COMPLEX Numbers spanned by the vec- tors 1 and i, where i is the IMAGINARY NUMBER. Every Complex Number corresponds to a unique Point in the complex plane. The LINE in the plane with i = is the Real Line. The complex plane is sometimes called the Argand Plane or Gauss Plane, and a plot of Complex Numbers in the plane is sometimes called an Argand Diagram. see also AFFINE COMPLEX PLANE, ARGAND DIAGRAM, Argand Plane, Bergman Space, Complex Projec- tive Plane References Courant, R. and Robbins, H. "The Geometric Interpretation of Complex Numbers." §5.2 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Ox- ford, England: Oxford University Press, pp. 92-97, 1996. Complex Projective Plane The set P 2 is the set of all Equivalence Classes [a, 6,c] of ordered triples (a, 6, c) E C 3 \(0,0,0) under the equivalence relation (a, 6, c) ~ (a', &', c') if (a, 6, c) = (Aa', A6',Ac') for some Nonzero Complex Number A. Complex Representation see Phasor Complex Structure The complex structure of a point x = PLANE is defined by the linear MAP J : '. J{Xi,X 2 ) - (-Z2,Zl), X\ , X2 in the and corresponds to a clockwise rotation by rr/2. This map satisfies J 2 = -I (Jx).(Jy) = x.y ( Jx) • x = 0, where / is the IDENTITY MAP. More generally, if V is a 2-D Vector SPACE, a linear map J : V — > V such that J 2 = — I is called a complex structure on7. If V = M. , this collapses to the previous definition. References Gray, A. Modern Differential Geometry of Curves and Sur- faces. Boca Raton, FL: CRC Press, pp. 3 and 229, 1993. Complexity (Number) Complexity (Number) The number of Is needed to represent an INTEGER us- ing only additions, multiplications, and parentheses are called the integer's complexity. For example, 1 = 1 2 = 1 + 1 3=1+1+1 4=(1 + 1)(1 + 1) = 1 + 1 + 1 + 1 5 = (1 + 1)(1 + 1) + 1 = 1 + 1 + 1 + 1 + 1 6 = (1 + 1)(1 + 1 + 1) 7 = (1 + 1)(1 + 1 + 1) + 1 8 = (1 + 1)(1 + 1)(1 + 1) 9=(1 + 1 + 1)(1 + 1 + 1) 10 = (1 + 1 + 1)(1 + 1 + 1) + 1 = (1 + 1)(1 + 1 + 1 + 1 + 1) So, for the first few n, the complexity is 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, . . . (Sloane's A005245). Referen