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Main CRC Concise Encyclopedia of Mathematics, Second Edition
CRC Concise Encyclopedia of Mathematics, Second EditionEric W. Weisstein
Upon publication, the first edition of the CRC Concise Encyclopedia of Mathematics received overwhelming accolades for its unparalleled scope, readability, and utility. It soon took its place among the top selling books in the history of Chapman & Hall/CRC, and its popularity continues unabated. Yet also unabated has been the dedication of author Eric Weisstein to collecting, cataloging, and referencing mathematical facts, formulas, and definitions. He has now updated most of the original entries and expanded the Encyclopedia to include 1000 additional pages of illustrated entries. The accessibility of the Encyclopedia along with its broad coverage and economical price make it attractive to the widest possible range of readers and certainly a must for libraries, from the secondary to the professional and research levels. For mathematical definitions, formulas, figures, tabulations, and references, this is simply the most impressive compendium available.
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SECOND EDITION CRC CONCISE ENCYCLOPEDIA OF MATHEMATICS SECOND EDITION CRC CONCISE ENCYCLOPEDIA OF MATHEMATICS ERIC W. WEISSTEIN CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. THE COVER The cover of this book consists of a collage of images from the second edition of Alfred Gray's "Modern Differential Geometry of Curves and Surfaces with Mathematica" published in 1998 by CRC Press LLC. Thanks go to Jonathan Pennell for his patience and help with the cover for the new edition. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress 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 authors 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 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-347-2 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Introduction to the First Edition 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. The entries are extensively crossreferenced, not only to related entries but also to many external sites on the Internet. This makes locating information very convenient. It also provides a highly efficient way to "navigate" from one related concept to another. 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 Dictionary). At the same time, the descriptions are more accessible than in "technical" mathematical encyclopedias (e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of Mathematics). While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate, research scientist, or recreational mathematician. In this work, the most useful, interesting, and entertaining (at least to my mind) aspects of topics are discussed in addition to their technical definitions. For example, in my entry for pi (π), 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 OzTEX program available for the Mac; I began the task of converting all my documents to TEX resulting in a vast improvement in readability. While undertaking the Word to TEX 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 TEX to HTML converter, LATEX2HTML. After some additional effort, I was able to post an HTML version of my encyclopedia to the World Wide Web. 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 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. 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 Preface to the New Edition The long awaited second edition of this Encyclopedia is finished, and it is now more complete than ever. Heavily revised by the author Eric Weisstein over the past three years, it contains well over 3,000 pages. Mr. Weisstein has updated all of the original material, added approximately 3,600 new entries and many illustrations, and updated the bibliographies that follow each entry to include the most recent references. As yet another enhancement, this edition integrates the use of the Mathematica software into many of its entries, presenting the precise commands that allow you to implement the formulas presented, perform many different calculations, construct graphical displays of your results, and generate remarkable mathematical illustrations. This is a unique touch and to our knowledge, a first for an encyclopedia. With definitions, formulas, and facts presented in clear, engaging prose along with a multitude of illustrations, extensive cross-references, and even links to the Internet, this new and improved edition remains one of the most readable and accessible references in mathematics. This is truly a unique book written by an individual who is clearly dedicated to the study and field of mathematics. Users of the first edition of the Encyclopedia have described it as "extraordinary," "impressive," and "fascinating, " and report spending hours browsing its pages simply for pleasure. We hope you will do the same. Acknowledgments Although I alone have compiled and typeset this work, many people have contributed indirectly and directly to its creation. I have not yet had the good fortune to meet Donald Knuth of Stanford University, but he is unquestionably the person most directly responsible for making this work possible. Before his mathematical typesetting program TEX, it would have been impossible for a single individual to compile such a work as this. Had Prof. Bateman owned a personal computer equipped with TEX, 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 TEX for the Macintosh, OzTEX (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 LATEX2HTML 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 mathematical constants (www.mathsoft.com/asolve/constant/constant.html), and also for his kind permission to reproduce excerpts from some of these essays. I hope that Steven will someday publish his detailed essays in book form. Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed and 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 Budney for sharing some of his expertise, to Charles Walkden for his helpful comments about dynamical systems theory, and to Lambros Lambrou for his contributions. Thanks to David W. Wilson for a number of helpful comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permission to reproduce their beautiful knot and link diagrams. Thanks to Gavin Theobald for providing diagrams of his masterful polygonal dissections. Thanks to Wolfram Research, not only for creating an indispensable mathematical tool in Mathematica®, but also for permission to include figures from the Mathematica® 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. (1, 0, 1)-Matrix Numerals (1, 0, 1)-Matrix The number of distinct (1; 0; 1)/-/nn matrices (counting row and column permutations, the transpose, and multiplication by 1 as equivalent) having 2n different row and column sums for n 2, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (Kleber). For example, the 22 matrix is given by 1 1 ; 0 1 To get the total number from these counts (assuming that 0 is not the missing sum, which is true for n5 10); multiply by (2n!)2 : In general, if an -matrix which has different column and row sums (collectively called line sums), then 1. n is even, 2. The number in fn; 1n; 2n; . . . ; ng that does not appear as a line sum is either n or , and 3. Of the largest line sums, half are column sums and half are row sums (Bodendiek and Burosch 1995, F. Galvin). See also ALTERNATING SIGN MATRIX, C -MATRIX, INTEGER MATRIX References Bodendiek, R. and Burosch, G. "Solution to the Antimagic 0; 1; 1 Matrix Problem." Aufgabe 5.30 in Streifzüge durch die Kombinatorik: Aufgaben und Lösungen aus dem Schatz der Mathematik-Olympiaden. Heidelberg, Germany: Spektrum Akademischer Verlag, pp. 250 /253, 1995. (1, 1)-Matrix See also HADAMARD MATRIX, INTEGER MATRIX References Kahn, J.; Komlós, J.; and Szemeredi, E. "On the Probability that a Random 91 Matrix is Singular." J. Amer. Math. Soc. 8, 223 /240, 1995. 0-Free ZEROFREE 0 DIVISION BY ZERO, FALLACY, NAUGHT, ZERO, ZERO DIVISOR, ZERO-FORM, ZERO MATRIX, ZERO-SUM GAME, ZEROFREE 0 1 FALLACY 1 1 (0, 1)-Matrix A (0; 1)/-INTEGER MATRIX, i.e., a matrix each of whose elements is 0 or 1, also called a binary matrix. The numbers of binary matrices with no adjacent 1s (in either columns or rows) for n 1, 2, ..., are given by 2, 7, 63, 1234, ... (Sloane's A006506). For example, the binary matrices with no adjacent 1s are 0 1 0 0 0 0 0 0 ; ; ; 0 0 1 0 0 1 0 0 1 0 1 0 0 1 ; ; ; 0 1 0 0 1 0 These numbers are closely related to the HARD The numbers of binary matrices with no three adjacent 1s for , 2, ..., are given by 2, 16, 265, 16561, ... (Sloane's A050974). SQUARE ENTROPY CONSTANT. Wilf (1997) considers the complexity of transforming an mn binary matrix A into a TRIANGULAR MATRIX by permutations of the rows and columns of , and concludes that the problem falls in difficulty between a known easy case and a known hard case of the general NP-COMPLETE PROBLEM. See also ADJACENCY MATRIX, FROBENIUS-KÖNIG THEOREM, GALE-RYSER THEOREM, HADAMARD'S MAXIMUM DETERMINANT PROBLEM, HARD SQUARE ENTROPY CONSTANT, IDENTITY MATRIX, INCIDENCE MATRIX, INTEGER MATRIX, LAM'S PROBLEM, S -CLUSTER, S -RUN References Brualdi, R. A. "Discrepancy of Matrices of Zeros and Ones." Electronic J. Combinatorics 6, No. 1, R15, 1 /12, 1999. http://www.combinatorics.org/Volume_6/v6i1toc.html. Ehrlich, H. "Determinantenabschätzungen für binäre Matrizen." Math. Z. 83, 123 /132, 1964. Ehrlich, H. and Zeller, K. "Binäre Matrizen." Z. angew. Math. Mechanik 42, T20 /21, 1962. Komlós, J. "On the Determinant of -Matrices." Studia Math. Hungarica 2, 7 /21 1967. Metropolis, N. and Stein, P. R. "On a Class of Matrices with Vanishing Determinants." J. Combin Th. 3, 191 /198, 1967. Ryser, H. J. "Combinatorial Properties of Matrices of Zeros and Ones." Canad. J. Math. 9, 371 /377, 1957. Sloane, N. J. A. Sequences A006506/M1816 and A050974 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdos. Papers from the Conference in Honor of Erdos' 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bollobás and A. Thomason). Cambridge, England: Cambridge University Press, pp. 557 /562, 1997. Williamson, J. "Determinants Whose Elements Are 0 and 1." Amer. Math. Monthly 53, 427 /434, 1946. 1 The number one (1), also called "unity" is the first POSITIVE INTEGER. It is an ODD NUMBER. Although the number 1 used to be considered a PRIME NUMBER, it 2 2 3 requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own (Wells 1986, p. 31). The number 1 is sometimes also called "unity," so the th roots of 1 are often called the th ROOTS OF UNITY. FRACTIONS having 1 as a NUMERATOR are called UNIT FRACTIONS. If only one root, solution, etc., exists to a given problem, the solution is called UNIQUE. The GENERATING 1 is given by FUNCTION having all COEFFICIENTS 1 1xx2 x3 x4 . . . : 1x References Daiev, V. "Problem 636: Greatest Divisors of Even Integers." Math. Mag. 40, 164 /165, 1967. Guy, R. K. "Residues of Powers of Two." §F10 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p. 250, 1994. Montgomery, P.-L. "New solution to 2^n 3 (mod n)." firstname.lastname@example.com posting, 24 Jun 1999. Sloane, N. J. A. Sequences A036236 and A050259 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/ eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 41 / 44, 1986. 2x mod 1 Map See also FALLACY, ONE-FORM, ONE-MOUTH THEOREM, ONE-NINTH CONSTANT, ONE-SHEETED HYPERBOLOID, ONE-TO-ONE, ONE-WAY FUNCTION, 2, 3, COMPLEXITY (NUMBER), EXACTLY ONE, ROOT OF UNITY, UNIQUE, UNIT FRACTION, ZERO Let x0 be a RATIONAL NUMBER in the CLOSED INTERVAL [0; 1]; and generate a SEQUENCE using the MAP xn1 2xn (mod 1): Then the number of periodic PRIME) is given by References Np Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 30 / 32, 1986. The only known solutions to the CONGRUENCE n 2 3 (mod n) are n 4700063497 (Sloane's A050259; Guy 1994) and 63130707451134435989380140059866138830623361447484274774099906755 (P.-L. Montgomery 1999). In general, the least satisfying 2n k (mod n) for k 2, 3, ... are n 3, 4700063497, 6, 19147, 10669, 25, 9, 2228071, ... (Sloane's A036236). See also 1, BINARY, 3, RULER FUNCTION, SQUARED, TWO-EARS THEOREM, TWO-FORM, TWO-GRAPH, TWOSCALE EXPANSION, TWO- S HEETED HYPERBOLOID , ZERO ORBITS 2p 2 p of period p (for (2) (i.e, the number of period- repeating bit strings, modulo shifts). Since a typical ORBIT visits each point with equal probability, the NATURAL INVARIANT is given by 2 The number two (2) is the second POSITIVE INTEGER and the first PRIME NUMBER. It is EVEN, and is the only EVEN PRIME (the PRIMES other than 2 are called the ODD PRIMES). The number 2 is also equal to its FACTORIAL since 2!2: A quantity taken to the POWER 2 is said to be SQUARED. The number of times k a given BINARY number bn b2 b1 b0 is divisible by 2 is given by the position of the first bk 1; counting from the right. For example, 12 1100 is divisible by 2 twice, and 13 1101 is divisible by 2 zero times. (1) (3) r(x)1: See also TENT MAP References Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 26 /31, 1993. 3 3 is the only INTEGER which is the sum of the preceding POSITIVE INTEGERS (12 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. The sequence 1, 31, 331, 3331, 33331, ... (Sloane's A033175) consisting of n 0, 1, ... 3s followed by a 1. The th tern is given by a(n) 10n1 7 3 : The result is prime for , 2, 3, 4, 5, 6, 7, 17, 39, ... (Sloane's A055520); i.e., for 3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, ... (Sloane's A051200), a fact which Gardner (1997) calls "a remarkable pattern that is entirely accidental and leads nowhere." 3x1 Mapping 6 EQUILATERAL See also 1, 2, 3X1 MAPPING, CUBED, PERIOD THREE T HEOREM , T ERNARY , T HREE- C HOICE P OLYGON , THREE-CHOICE WALK, THREE-COLORABLE, THREE CONICS THEOREM, THREE JUG PROBLEM, THREEVALUED LOGIC, TREFOIL KNOT, WIGNER 3J -SYMBOL, ZERO 3 TETRAHEDRON PENTATOPE SIMPLEX POLYGON POLYHEDRON POLYCHORON POLYTOPE LINE SEG- PLANE HYPERPLANE HYPERPLANE OCTAHEDRON 16-CELL CROSS POLY- TRIANGLE MENT SQUARE References TOPE Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, p. 194, 1997. Sloane, N. J. A. Sequences A033175, A051200, and A055520 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Smarandache, F. Properties of Numbers. University of Craiova, 1973. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 46 / 48, 1986. EDGE FACE FACET FACET AREA VOLUME CONTENT CONTENT The SURFACE AREA of a HYPERSPHERE in -D is given by 2pn=2 Sn ; G 12 n and the VOLUME by 3x1 Mapping pn=2 Rn ; Vn G 1 12 n COLLATZ PROBLEM where G(n) is the 4 GAMMA FUNCTION. See also DIMENSION, HYPERCUBE, HYPERSPHERE See also FOUR COINS PROBLEM, FOUR-COLOR THEOFOUR CONICS THEOREM, FOUR EXPONENTIALS CONJECTURE, FOUR TRAVELERS PROBLEM, FOUR-VECTOR, FOUR-VERTEX THEOREM, LAGRANGE'S FOURSQUARE THEOREM REM, References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 55 / 58, 1986. 4-D Geometry 4-DIMENSIONAL GEOMETRY References Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health Research, 1993. Manning, H. The Fourth Dimension Simply Explained. Magnolia, MA: Peter Smith, 1990. Manning, H. Geometry of Four Dimensions. New York: Dover, 1956. Neville, E. H. The Fourth Dimension. Cambridge, England: Cambridge University Press, 1921. Rucker, R. von Bitter. The Fourth Dimension: A Guided Tour of the Higher Universes. Boston, MA: Houghton Mifflin, 1984. Sommerville, D. M. Y. An Introduction to the Geometry of Dimensions. New York: Dover, 1958. 5 4-Dimensional Geometry 4-dimensional geometry is Euclidean geometry extended into one additional DIMENSION. The prefix "hyper-" is usually used to refer to the 4- (and higher-) dimensional analogs of 3-dimensional objects, e.g. HYPERCUBE, HYPERPLANE, HYPERSPHERE. -dimensional POLYHEDRA are called POLYTOPES. the 4-dimensional cases of general -dimensional objects are often given special names, such as those summarized in the following table. See also FIVE DISKS PROBLEM, MIQUEL FIVE CIRCLES THEOREM, PENTAGON, PENTAGRAM, PENTAHEDRON, TETRAHEDRON 5-COMPOUND References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 58 / 67, 1986. 5-Cell PENTATOPE 2-D 3-D 4-D General CIRCLE SPHERE GLOME HYPERSPHERE SQUARE CUBE TESSERACT HYPERCUBE 6 See also 6-SPHERE COORDINATES, HEXAGON, HEXAHE- 4 6-Sphere Coordinates 10 DRON, SIX CIRCLES THEOREM, SIX-COLOR THEOREM, SIX EXPONENTIALS THEOREM, WIGNER 6J -SYMBOL which gives spheres tangent to the xy -plane at the origin for w constant. The metric coefficients are References guu gvv gww Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 67 / 69, 1986. ðu2 1 : v2 w2 Þ2 (7) See also CARTESIAN COORDINATES, INVERSION 6-Sphere Coordinates References Moon, P. and Spencer, D. E. "6-Sphere Coordinates (u; v; w):/" Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 122 / 123, 1988. 7 See also SEVEN CIRCLES THEOREM References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 70 / 71, 1986. 8 The coordinate system obtained by INVERSION of CARTESIAN COORDINATES, with u; v; w (; ): The transformation equations are References x u u2 v2 w2 (1) y v v2 w2 (2) z u2 TESSERACT (3) 9 The equations of the surfaces of constant coordinates are given by x 1 2u y2 z2 1 ; 4u2 References !2 1 1 x y z2 ; 2v 4v2 2 (5) which gives spheres tangent to xz -plane at the origin for v constant, and 2 x y z 1 !2 2w 1 4w2 See also NINE-POINT CENTER, NINE-POINT CIRCLE, NINE-POINT CONIC, WIGNER 9J -SYMBOL (4) which gives spheres tangent to the yz -plane at the origin for u constant, 2 Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 71 / 73, 1986. 8-Cell w : u2 v2 w2 !2 See also EIGHT CURVE, EIGHT-POINT CIRCLE THEOEIGHT SURFACE REM, : (6) Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 73 / 76, 1986. 10 The number 10 (ten) is the basis for the DECIMAL system 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 100 1/s place. For example, the number 1234.56 specifies 11 15 Puzzle 1103 2102 3101 4100 5101 2 610 The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, ... (Sloane's A011557), called one, ten, HUNDRED, THOUSAND, ten thousand, hundred thousand, MILLION, 10 million, 100 million, and so on. The names of subsequent decimal places for LARGE NUMBERS differ depending on country. Any 12 One : of 10 which can be written as the of two numbers not containing 0s must be n n n OF THE FORM 2 × 5 10 for an INTEGER such that n n neither 2 nor 5 contains any ZEROS. The largest known such number is 5 DOZEN, or a twelfth of a GROSS. See also DOZEN, GROSS References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 13 POWER PRODUCT 1023 233 × 533 8; 589; 934; 592 × 116; 415; 321; 826; 934; 814; 453; 125: 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. There are 13 ARCHIMEDEAN SOLIDS. Mazur and Tate (1973/74) proved that there is no ELLIPTIC CURVE over the rationals Q having a RATIONAL POINT of order 13. See also BAKER'S DOZEN, TRISKAIDEKAPHOBIA A complete list of known such numbers is 101 21 × 51 102 22 × 52 103 23 × 53 104 24 × 54 105 25 × 55 106 26 × 56 107 27 × 57 109 29 × 59 1018 218 × 518 1033 233 × 533 (Madachy 1979). Since all POWERS of 2 with exponents 86Bn54:6107 contain at least one ZERO (M. Cook), no other POWER of ten less than 46 million can be written as the PRODUCT of two numbers not containing 0s. See also BILLION, DECIMAL, HUNDRED, LARGE NUMBER, MILLIARD, MILLION, THOUSAND, TRILLION, ZERO References Mazur, B. and Tate, J. "Points of Order 13 on Elliptic Curves." Invent. Math. 22, 41 /49, 1973/74. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 14 References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. 15 See also 15 PUZZLE, FIFTEEN THEOREM 15 Puzzle References Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 127 /128, 1979. Pickover, C. A. Keys to Infinity. New York: Wiley, p. 135, 1995. Sloane, N. J. A. Sequences A011557 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 76 / 82, 1986. 11 References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a 44 box leaving one position out of the 16 empty. The goal is to reposition the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements, this rearrangement is possible, but for others, it is not. To address the solubility of a given initial arrangement, proceed as follows. If the SQUARE containing the number i appears "before" (reading the squares in the box from left to right and top to bottom) numbers which are less than , then call it an inversion of order , and denote it ni : Then define 6 15 Puzzle N 16-Cell 15 X i1 ni 15 X ni ; i2 where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so n1 must equal 0). If N is EVEN, the position is possible, otherwise it is not. This can be formally proved using ALTERNATING GROUPS. For example, in the following arrangement n2 1 (2 precedes 1) and all other ni 0; so N 1 and the puzzle cannot be solved. / Johnson (1879) proved that odd permutations of the puzzle are impossible, which Story (1879) proved that all even permutations are possible. While Herstein and Kaplansky (1978) wrote that "no really easy proof seems to be known," Archer (1999) presented a simple proof. A more general result due to Wilson (1974) showed that for any CONNECTED GRAPH on nodes, with the exception of CYCLE GRAPHS Cn and the THETA-0 GRAPH, either exactly half or all of the n! possible labelings are obtainable by sliding labels, depending on whether the graph is BIPARTITE (Archer 1999). u0 has six inequivalent labelings, which has (n2)! inequivalent labelings. Reversing the order of the "8 Puzzle" made on a 33 board can be proved to require at least 26 moves, although the best solution requires 30 moves (Gardner 1984, pp. 200 and 206 /207). The number of distinct solutions in 28, 30, 32, ... moves are 0, 10, 112, 512, ... (Sloane's A046164), giving 634 solutions better than the 36-move solution given by Dudeney (1949). Hurd, S. and Trautman, D. "The Knight's Tour on the 15Puzzle." Math. Mag. 66, 159 /166, 1993. 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 Imagination. 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. Liebeck, H. "Some Generalizations of the 14 /15 Puzzle." Math. Mag. 44, 185 /189, 1971. Loyd, S. Mathematical Puzzles of Sam Loyd, Vol. 1. New York: Dover, pp. 19 /20, 1959. Loyd, S. Jr. Sam Loyd's Cyclopedia of 5,000 Puzzles, Tricks, and Conundrums. Lamb Pub., 1993. Mallison, H. V. "An Array of Squares." Math. Gaz. 24, 119 / 121, 1940. Sloane, N. J. A. Sequences A046164 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Spitznagel, E. L. Jr. Selected Topics in Mathematics. New York: Holt, Rinehart and Winston, pp. 143 /148, 1971. Spitznagel, E. L. Jr. "A New Look at the Fifteen Puzzle." Math. Mag. 40, 171 /174, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 14 /16, 1999. Story, W. E. "Notes on the '15 Puzzle. II."' Amer. J. Math. 2, 399 /404, 1879. Whipple, F. J. W. "The Sign of a Term in the Expansion of a Determinant." Math. Gaz. 13, 126, 1926. Wilson, R. M. "Graph Puzzles, Homotopy, and the Alternating Group." J. Combin. Th. Ser. B 16, 86 /96, 1974. 15 Schoolgirl Problem KIRKMAN'S SCHOOLGIRL PROBLEM 16-Cell References Archer, A. F. "A Modern Treatment of the 15 Puzzle." Amer. Math. Monthly 106, 793 /799, 1999. Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 312 /316, 1987. Beasley, J. D. The Mathematics of Games. Oxford, England: Oxford University Press, pp. 80 /81, 1990. Bogomolny, A. "Sam Loyd's Fifteen." http://www.cut-theknot.com/pythagoras/fifteen.html. Bogomolny, A. "Sam Loyd's Fifteen [History]." http:// www.cut-the-knot.com/pythagoras/history15.html. Davies, A. L. "Rotating the 15 Puzzle." Math. Gaz. 54, 237 / 240, 1970. Dudeney, H. E. Problem 253 in The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949. Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 64 /65, 200 /201, and 206 /207, 1984. Herstein, I. N. and Kaplansky, I. Matters Mathematical, 2nd ed. New York: Chelsea, pp. 114 /115, 1978. The finite regular 4-D CROSS POLYTOPE with SCHLÄf3; 3; 4g and VERTICES which are the PERMUTATIONS of (, 0, 0, 0). The 16-cell is the dual of the TESSERACT. Its graph is isomorphic to the CIRCULANT GRAPH Ci1; 2; 3 (8):/ FLI SYMBOL See also 24-CELL, 120-CELL, 600-CELL, CELL, CROSS POLYTOPE, HYPERCUBE, PENTATOPE, POLYCHORON, POLYTOPE, TESSERACT References Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 17 36 Officer Problem 17 17 is a FERMAT which means that the 17-sided REGULAR POLYGON (the HEPTADECAGON) is CONSTRUCTIBLE using COMPASS and STRAIGHTEDGE (as proved by Gauss). PRIME 7 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 See also CONSTRUCTIBLE POLYGON , FERMAT PRIME, HEPTADECAGON References Lefevre, V. "Properties of 17." http://www.ens-lyon.fr/~vlefevre/d17_eng.html. 17-gon HEPTADECAGON 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. Continue, placing every th point so that all points are on different (1=N)/th of the LINE SEGMENT. Formally, for a given , does there exist a sequence of real numbers x1 ; x2 ; ..., xN such that for every n f1; . . . ; Ng and every k f1; . . . ; ng; the inequality k1 k 5xi B n n holds for some i f1; . . . ; ng/? 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 solution 4 7 2 5 16 1 1 5x1 B 12 ; 7 5x2 B 17 ; 17 5x3 B1; 14 5x4 B 13 ; 7 8 6 1 2 14 5x5 B 11 ; 5 5x6 B 13 ; 7 5x7 B 13 ; 17 5x8 B 56; 11 15 11 3 5 11 3 3 5x9 B 13 ; 17 5x10 B 23; 14 5x11 B 13 ; 8 15 9 5x12 B 11 ; 1 5x12 B 17 ; 17 12 2 1 05x14 B 17 ; 13 5 6 10 5x15 B 45; 16 5x16 B 17 ; 17 5x17 B 11 ; 17 17 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 "Generalization." Problems 6 and 7 in One Hundred Problems in A finite regular 4-D POLYTOPE with SCHLÄFLI SYMBOL f3; 4; 3g: Coxeter (1969) gives a list of the VERTEX positions. The EVEN coefficients of the /D4/ 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). The 24-cell is self-dual, and is the unique regular convex POLYCHORON which has no direct 3-D analog. One construction for the 24-cell evokes comparison with the RHOMBIC DODECAHEDRON. Given two equal cubes, we construct this dodecahedron by cutting one cube into six congruent square pyramids, and attaching these to the six squares bounding the other cube. Similarly, given two equal tesseracts, we can construct the 24-cell by cutting one tesseract into eight congruent cubic pyramids, and attaching these to the eight cubes bounding the other tesseract (Towle). See also 16-CELL, 120-CELL, 600-CELL, CELL, HYPERCUBE, PENTATOPE, POLYCHORON, POLYTOPE 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." http:// www.research.att.com/~njas/sequences/eisonline.html. Sloane, N. J. A. and Plouffe, S. Figure M5150 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 36 Officer Problem How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, and major, a captain, a lieutenant, and a sub-lieutenant be ar- 8 42 ranged in a regular 66 array such that no row or column duplicates a rank or a regiment? The answer is that no such arrangement is possible. See also EULER'S GRAECO-ROMAN SQUARES CONJECTURE, LATIN SQUARE References Bose, R. C.; Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture." Canad. J. Math. 12, 189, 1960. Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1, 88 /93, 1949. Parker, E. T. "Orthogonal Latin Squares." Not. Amer. Math. Soc. 6, 276, 1959. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 31, 1999. Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 1, 122 /123, 1900. Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 2, 170 /203, 1901. 196-Algorithm Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 144 A DOZEN DOZEN, NUMBER and a References RULE 144 is a SQUARE 163 The number 163 is very important in number theory, since d 163 is the largest number that the pffiffiffisuch IMAGINARY QUADRATIC FIELD Q d has CLASS NUMBER h(d)1: It also satisfies the curious identities 163 4 X 8 i i0 1 4 4 2 " 4 1 4 X 4 2 i0 8 4 4 i (1) (2) 2 # ; (3) where nk is a BINOMIAL COEFFICIENT (Stoschek). An approximation due to Stoschek is given by Adams, D. The Hitchhiker's Guide to the Galaxy. New York: Ballantine Books, 1997. 72 Rule GROSS. See also DOZEN 42 According to Adams (1997), 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. also called a SUM-PRODUCT NUMBER. p: 29 512 :3:1411043; 163 163 (4) which is good to 3 digits. OF 72 120-Cell See also RAMANUJAN CONSTANT References Stoschek, E. "Modul 33: Algames with Numbers." http:// marvin.sn.schule.de/~inftreff/modul33/task33.htm. 196-Algorithm A finite regular 4-D POLYTOPE with SCHLÄFLI SYMBOL f5; 3; 3g: The 120-cell has 600 vertices (Coxeter 1969), and consists of 120 DODECAHEDRA and 720 PENTAGONS (Coxeter 1973, p. 264). In the plate following p. 176, Coxeter (1973) illustrates the polytope. The dual of the 120-cell is the 600-CELL. See also 16-CELL, 24-CELL, 600-CELL, CELL, HYPERCUBE, PENTATOPE, POLYCHORON, POLYTOPE, SIMPLEX References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. "Stellating ." §14.2 in Regular Polytopes, 3rd ed. New York: Dover, pp. 136 /137, 157, 264 /267, and 292, 1973. Take any POSITIVE INTEGER of two DIGITS or more, reverse the DIGITS, and add to the original number. Now repeat the procedure with the SUM so obtained. This procedure quickly produces PALINDROMIC NUMBERS for most INTEGERS. For example, starting with the number 5280 produces (5280, 6105, 11121, 23232). The end results of applying the algorithm to 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, ... (Sloane's A033865). The value for 89 is especially large, being 8813200023188. The first few numbers not known to produce PALINare 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. In 1990, John Walker computed 2,415,836 iterations of the algorithm on 196 and obtained a number having 1,000,000 digits. This was extended in 1995 by Tim Irvin, who obtained a DROMES 196-Algorithm number having 2,000,000 digits. The rec.puzzles archive states that a 3,924,257-digit nonpalindromic number is obtained after 9,480,000 iterations. The number of terms a(n) in the iteration sequence required to produce a PALINDROMIC NUMBER from (i.e., a(n)1 for a PALINDROMIC NUMBER, a(n)2 if a PALINDROMIC NUMBER is produced after a single iteration of the 196-algorithm, etc.) for , 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, 1, 2, ... iterations to reach a palindrome are 0, 10, 19, 59, 69, 166, 79, 188, ... (Sloane's A023109). The 196-algorithm can be implemented in Mathematica as PalindromicQ[n_Integer?Positive]: Module[ {sn ToString[n]}, sn StringReverse[sn] ] Algorithm196[n_Integer?PalindromicQ,it_:0]: {n} Algorithm196[n_Integer?Positive, it_:Infinity]: FixedPointList[# ToExpression[StringReverse[ToString[#]]]&, n, it, SameTest- (PalindromicQ[#2]&) ] M. Sofroniou gives an efficient Mathematica implementation which has complexity Oðk2 Þ for steps, requiring approximately 10.6 hours on a 450 MHz Pentium II to compute 250,000 iterations. Extrapolating the timing data suggests that approximately 42 days would be needed on this same machine to match Walker's 2,415,836 iterations. See also ADDITIVE PERSISTENCE, DIGITADDITION, MULTIPLICATIVE PERSISTENCE, PALINDROMIC NUMBER , P ALINDROMIC N UMBER C ONJECTURE , RATS SEQUENCE, RECURRING DIGITAL INVARIANT References Brown, K. S. "Digit Reversal Sums Leading to Palindromes." http://www.seanet.com/~ksbrown/kmath004.htm. De Geest, P. "Websources about '196' Becoming Palindromic by Using Reversal Sums." http://www.ping.be/~ping6758/ weblinks.htm. Eddins, S. "The Palindromic Order of a Number." IMSA Math. J. 4, Spring 1996. http://www.imsa.edu/edu/math/ journal/volume4/webver/palinord.html. 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. Irving, T. "About Two Months of Computing, or, An Addendum to Mr. Walker's Three Years of Computing" http://www.fourmilab.ch/documents/threeyears/two_months_more.html. Math Forum. "Ask Dr. Math: Making Numbers into Palindromic Numbers." http://forum.swarthmore.edu/dr.math/ problems/barnes10.11.html. Peters, I. J. "Search for the Biggest Numeric Palindrome." http://www.floot.demon.co.uk/palindromes.html. 243 9 rec.puzzles archive. 1996. ftp://rtfm.mit.edu/pub/usenet/ news.answers/puzzles/archive/arithmetic/part1. Safroniou, M. "Palindromic Numbers: The 196-Algorithm." MATHEMATICA NOTEBOOK ALGORITHM196.NB. Sloane, N. J. A. Sequences A006960/M5410, A023109, A030547, and A033865 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Walker, J. "Three Years of Computing: Final Report on the Palindrome Quest." http://www.fourmilab.ch/documents/ threeyears/threeyears.html. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTEBOOK INTEGERSEQUENCES.M. 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 1 1 1 1 1 p4 tan tan ; 4 5 239 which is related to the fact that 2 × 134 12392 ; pffiffiffi which is why 239/169 is the 7th CONVERGENT of 2: Another pair of INVERSE TANGENT FORMULAS involving 239 is 1 1 1 tan1 70 tan1 99 tan1 239 tan1 1 408 tan1 1 577 : 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 Schroeppel, R. Item 63 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 24, Feb. 1972. 243 Feynman (1997) noticed the curious fact that the decimal expansion 1 0:004115226337448559 . . . 243 repeats pairs of the digits 0, 1, 2, 3, ... separated by the digits 4, 5, 6, 7, .... Just after this point, the pattern breaks, since the fraction is given exactly by the repeating decimal 1 0:004115226337448559670781893: 243 This pattern is related to the fact that 1 0:1̄ 9 10 257-gon and 2187 600-Cell 1 0:0123456789: 81 References Feynman, R. P. and Leighton, R. 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, p. 99, 1997. 257-gon 257 is a FERMAT PRIME, and the 257-gon is therefore a CONSTRUCTIBLE POLYGON using COMPASS and STRAIGHTEDGE, as proved by Gauss. An illustration of the 257-gon is not included here, since its 257 segments so closely resemble a CIRCLE. Richelot and Schwendenwein found constructions for the 257-gon in 1832 (Coxeter 1969). De Temple (1991) gives a construction using 150 CIRCLES (24 of which are CARLYLE CIRCLES) which has GEOMETROGRAPHY symbol 94S1 47S2 275C1 0C2 150C3 and SIMPLICITY 566. See also 65537-GON, CONSTRUCTIBLE POLYGON, FERMAT PRIME, HEPTADECAGON, PENTAGON A finite regular 4-D POLYTOPE with SCHLÄFLI SYMBOL f3; 3; 5g: The 600-cell has 120 VERTICES (Coxeter 1969). In the plate following p. 160, Coxeter (1973) gives two illustrations of the polytope. The dual of the 600-cell is the 120-CELL. See also 16-CELL, 24-CELL, 120-CELL, CELL, HYPERCUBE, PENTATOPE, POLYCHORON, POLYTOPE, SIMPLEX References Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. "Gosset's Construction for . §8.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 136 /137, 153 / 154, and 157, 1973. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991. 666 References Bachmann, P. Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Leipzig, Germany: Teubner, 1872. Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97 /108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /386, 1955. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Klein, F. "The Construction of the Regular Polygon of 17 Sides." Part I, Ch. 4 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 24 /41, 1980. Pascal, E. "Sulla costruzione del poligono regolare di 257 lati." Rendiconto dell Accad. della scienze fisiche e matemat. sezione della Soc. a reale di Napoli, Ser. 2 1, 33 /39, 1887. Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964. Richelot, F. J. "De resolutione algebraica aequationis X 257 1; sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata." J. reine angew. Math. 9, 1 /26, 146 /161, 209 / 230, and 337 /358, 1832. Trott, M. " cos(2p=257) à la Gauss." Mathematica Educ. Res. 4, 31 /36, 1995. A number known as the BEAST NUMBER appearing in the Bible and ascribed various numerological properties. See also APOCALYPTIC NUMBER, BEAST NUMBER, LEVIATHAN NUMBER References De Geest, P. "The Number of the Best 666." http:// www.ping.be/~ping6758/weblinks.htm. Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 96, 1993. 1729 1729 is sometimes called the HARDY-RAMANUJAN It is the smallest TAXICAB NUMBER, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways: NUMBER. 172913 123 93 103 : See also HARDY-RAMANUJAN NUMBER, TAXICAB NUMBER 2187 The digits in the number 2187 form the two VAMPIRE 21871827 and 21872781: 2187 is also given by 37. NUMBERS: 65537-gon See also VAMPIRE NUMBER 65537-gon 11 De Temple (1991) notes that a GEOMETRIC CONSTRUCcan be done using 1332 or fewer CARLYLE CIRCLES. TION References Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19, 26 /29, Spring 1997. See also 257-GON, DECAGON, CONSTRUCTIBLE POLYGON, HEPTAPENTAGON References 65537-gon 65537 is the largest known FERMAT PRIME, and the 65537-gon is therefore a CONSTRUCTIBLE POLYGON using COMPASS and STRAIGHTEDGE, as proved by Gauss. The 65537-gon has so many sides that it is, for all intents and purposes, indistinguishable from a CIRCLE using any reasonable printing or display methods. Hermes spent 10 years on the construction of the 65537-gon at Königsberg around (1900). After the Second World War, his manuscripts were moved to the Mathematical Institute in Göttingen, where they can now be viewed (Coxeter 1969). Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97 /108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 /386, 1955. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Hermes, J. "Ueber die Teilung des Kreises in 65537 gleiche Teile." Nachr. Königl. Gesellsch. Wissensch. Göttingen, Math.-Phys. Klasse , pp. 170 /186, 1894. AAA Theorem abc Conjecture 13 Abacus A 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 1808 (/p RADIANS), i.e., CpAB: See also AAS THEOREM, ASA THEOREM, ASS THEOSAS THEOREM, SSS THEOREM, TRIANGLE REM, A mechanical counting device consisting of a frame holding a series of parallel rods on each of which beads are strung. Each bead represents a counting unit, and each 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 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(p A B) : 2 sin A 2 sin A (1) Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal Fractions." A History of Mathematics, 2nd ed. New York: Wiley, pp. 199 /01, 1991. Fernandes, L. "The Abacus: The Art of Calculating with Beads." http://www.ee.ryerson.ca/~elf/abacus/. Gardner, M. "The Abacus." Ch. 18 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 232 /41, 1979. Pappas, T. "The Abacus." In The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 209, 1989. Pullan, J. M. The History of the Abacus. New York: Prager, 1968. Smith, D. E. "Mechanical Aids to Calculation: The Abacus." Ch. 3 §1 in History of Mathematics, Vol. 2. New York: Dover, pp. 156 /96, 1958. Yoshino, Y. The Japanese Abacus Explained. New York: Dover, 1963. The third angle is given by CpAB; (2) since the sum of angles of a TRIANGLE is 1808 (/p RADIANS). Solving the LAW OF SINES a b sin A sin B (3) A CONJECTURE due to J. Oesterlé and D. W. Masser. It states that, for any INFINITESIMAL e > 0; there exists a CONSTANT Ce such that for any three RELATIVELY PRIME INTEGERS a , b , c satisfying abc; for b gives the sin B ba : sin A (4) cb cos Aa cos Ba(sin B cot Acos B) a sin B(cot Acot B): (1) INEQUALITY max(½a½; ½b½; ½c½) 5Ce Y p 1e (2) p½abc Finally, (5) (6) See also AAA THEOREM, ASA THEOREM, ASS THEOSAS THEOREM, SSS THEOREM, TRIANGLE REM, abc Conjecture holds, where p½abc 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 ln x WIEFERICH PRIMES 5 x for some constant C (Silverman 1988, Vardi 1991). 14 abc Conjecture Abel Transform (3) Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 66, 1991. Vojta, P. Diophantine Approximations and Value Distribution Theory. Berlin: Springer-Verlag, p. 84, 1987. (4) Abel Polynomial The conjecture can also be stated by defining the height and radical of the sum P : a b c as h(P) maxfln½a½; ln½b½; ln½c½g r(P) X ln p; p½abc where p runs over all prime divisors of a , b , and c . Then the abc conjecture states that for all e > 0; there exists a constant K such that for all P : abc; h(P)5r(P)eh(P)K with 2p e !1=4 > 1:517 (7) for l0:5990; improving a result of Stewart and Tijdeman (1986). See also FERMAT'S LAST THEOREM, MASON'S THEOMORDELL CONJECTURE, ROTH'S THEOREM, WIEFERICH PRIME REM, f (t)te at ; (1) An (x; a)x(xan)n1 : (2) given by (5) (van Frankenhuysen 2000). van Frankenhuysen (2000) has shown that there exists an infinite sequence of sums P : abc or RATIONAL INTEGERS with large height compared to the radical, pffiffiffiffiffiffiffiffiffiffi h(P) h(p)]r(P)4Kl ; (6) ln[h(P)] Kl 2 l=2 A polynomial An (x; a) given by the associated SHEFwith FER SEQUENCE The GENERATING FUNCTION X Ak (x; a) k t e xW(at)=a ; k! k0 (3) where W(x) is LAMBERT'S W -FUNCTION. The associated BINOMIAL IDENTITY is (xy)(xyan)n1 n X n (4) xy(xak)k1 [ya(nk)]nk1 ; k k0 where nk is a BINOMIAL COEFFICIENT, a formula originally due to Abel (Riordan 1979, p. 18; Roman 1984, pp. 30 and 73). The first few Abel polynomials are A0 (x; A1 (x; A2 (x; A3 (x; A4 (x; References Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. Math. Monthly 101, 3 /4, 1994. Elkies, N. D. "ABC Implies Mordell." Internat. Math. Res. Not. 7, 99 /09, 1991. Goldfeld, D. "Beyond the Last Theorem." The Sciences 36, 34 /0, March/April 1996. Goldfeld, D. "Beyond the Last Theorem." Math. Horizons , 26 /1 and 24, Sept. 1996. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 75 /6, 1994. Lang, S. "Old and New Conjectures in Diophantine Inequalities." Bull. Amer. Math. Soc. 23, 37 /5, 1990. Lang, S. Number Theory III: Diophantine Geometry. New York: Springer-Verlag, pp. 63 /7, 1991. Mason, R. C. Diophantine Equations over Functions Fields. Cambridge, England: Cambridge University Press, 1984. Mauldin, R. D. "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436 /437, 1997. Nitaq, A. "The abc Conjecture Home Page." http:// www.math.unicaen.fr/~nitaj/abc.html. Silverman, J. "Wieferich's Criterion and the abc Conjecture." J. Number Th. 30, 226 /37, 1988. Stewart, C. L. and Tijdeman, R. "On the Oesterlé-Masser Conjecture." Mh. Math. 102, 251 /57, 1986. Stewart, C. L. and Yu, K. "On the ABC Conjecture." Math. Ann. 291, 225 /30, 1991. van Frankenhuysen, M. "The ABC Conjecture Implies Roth's Theorem and Mordell's Conjecture." Mat. Contemp. 16, 45 /2, 1999. van Frankenhuysen, M. "A Lower Bound in the abc Conjecture." J. Number Th. 82, 91 /5, 2000. is a)1 a)x a)x(x2a) a)x(x3a)2 a)x(x4a)3 : References Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. Roman, S. "The Abel Polynomials." §4.1.5 in The Umbral Calculus. New York: Academic Press, pp. 29 /0 and 72 /5, 1984. Abel Transform The following INTEGRAL TRANSFORM relationship, known as the Abel transform, exists between two functions f (x) and g(t) for 0BaB1; x f (x) g (x t) g(t) dt a (1) 1a (2) 0 g(t) t sin(pa) d p sin(pa) p dt " g g (x t) f (x) dx 0 # df dx f (0) 1a 1a : t 0 dx (t x) t (3) The Abel transform is used in calculating the radial Abel Transform Abel's Convergence Theorem mass distribution of galaxies (Binney and Tremaine 1987) and inverting planetary radio occultation data to obtain atmospheric information as a function of height. Bracewell (1999, p. 262) defines a slightly different form of the Abel transform given by g g(x)A[f (r)]2 x f (r)r dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r2 x2 (4) The following table gives a number of common Abel transform pairs (Bracewell 1999, p. 264). Here, ! x 1 1 for 0BxB0 Pa (x)P (5) 0 otherwise 2a 2 where P(x) is the RECTANGLE FUNCTION, M(x)2p x 3 g and x J0 (x) dxx 2 J0 (x) (6) 0 References Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 11 and 97, 1988. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 875 /76, 1985. Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987. Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 262 /66, 1999. Hilfer, R. (Ed.). Applications of Fractional Calculus in Physics. Singapore: World Scientific, pp. 3 /, 2000. Liouville, J. "Memoire sur quelques quéstions de géométrie et de mécanique, et sur un nouveau genre pour réspondre ces quéstions." J. École Polytech. 13, 1 /9, 1832. Lützen, J. Joseph Liouville, 1809 /882. Master of Pure and Applied Mathematics. New York: Springer-Verlag, p. 314, 1990. Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 376 /77, 1967. Abel's Binomial Theorem The identity p2 [J1 (x)H0 (x)J0 (x)H1 (x)]; x2 where Jn (x) is a BESSEL FUNCTION and Hn (x) is a STRUVE FUNCTION. (7) OF THE FIRST KIND m X m (wy)my1 (zy)y w 1 (zwm)m y y0 (Bhatnagar 1995, p. 51). There are a host of other such BINOMIAL IDENTITIES. See also BINOMIAL IDENTITY, f (r)/ / / Pa (r)/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /2 a 2 x 2/ / / g(x)/ (a 2 r 2 )1=2 Pa (r)/ /p/ / a 2 > x 2/ / pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 r 2 Pa (r)/ / 1 p(a 2 x 2 )/ 2 / a 2 > x 2/ / (a 2 r 2 )Pa (r)/ / 4 2 (a x 2 )3=2/ 3 / a 2 > x 2/ / (a 2 r 2 )3=2 Pa (r)/ / / a 2 > x 2/ / / / e r 2 / r 2 e r 2 =s 2 / J0 (vr)/ M(r)/ Abel's Convergence Theorem Given a TAYLOR / SERIES ax/ / f (z) pffiffiffi x 2 =s 2 1 2 2 /s(x s ) pe / 2 X Cn z n n0 / s > 0/ / s > 0/ / s > 0/ X Cn r n e inu ; (1) n0 where the COMPLEX NUMBER z has been written in the polar form zre iu ; examine the REAL and IMAGINARY PARTS 2 e r =s 2 1 2 pffiffiffi (r 2s )/ / s p 1 / / b2 r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a 2 x 2 x 2 cosh 1 / a / x Abel, N. H. "Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist." J. reine angew. Math. 1, 159 60, 1826. Reprinted in Euvres Complètes, 2nd ed., Vol. 1. pp. 102 03, 1881. Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U (n ) Extensions. Ph.D. thesis. Ohio State University, p. 51, 1995. Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. / =s 2 2 3 p(a 2 x 2 )2/ 8 2a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pa (x)/ a2 x2 pffiffiffi x 2 =s 2 /s pe / d(ra)/ THEOREM References a 2 > x 2/ (a r)Pa (r)/ 1 1 a / cosh / p r Q -ABEL'S conditions / / 15 x 2 e x / 2 =s 2 / p /pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/ b2 r2 2 cos(vx) / / v 4 8p xv / sin 2 / 2p v2x2 u(r; u) / b 2 x 2 > 0/ / v > 0/ v > 0/ Cn r n cos(nu) (2) Cn r n sin(nu): (3) n0 v(r; u) / X X n0 Abel's theorem states that, if u(1; u) and v(1; u) are CONVERGENT, then See also FOURIER TRANSFORM, HILBERT TRANSFORM, INTEGRAL EQUATION u(1; u)iv(1; u)lim f (re iu ): r01 (4) 16 Abel's Curve Theorem Abel's Duplication Formula 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 CONVERGENCE 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. Orlando, FL: Academic Press, p. 773, 1985. Now, take y1 (3) minus y2 (2), y1 [yƒ2 P(x)y?2 Q(x)y2 ]y2 [yƒ1 P(x)y?1 Q(x)y1 ]0 (4) (y1 yƒ2 y2 yƒ1 )P(y1 y?2 y?1 y2 )Q(y1 y2 y1 y2 )0 (5) (y1 yƒ2 y2 yƒ1 )P(y1 y?2 y?1 y2 )0: (6) Now, use the definition of the WRONSKIAN and take its DERIVATIVE, W y1 y?2 y?1 y2 Abel's Curve Theorem The sum of the values of an INTEGRAL of the "first" or "second" sort g x1 ; y1 x0 ; y0 P dx . . . Q g xN ; yN x0 ; y0 P dx F(z) Q and (7) W?(y?y?2 y1 yƒ2 )(y?1 y?2 yƒ1 y2 ) y1 yƒ2 yƒ1 y2 : Plugging W and W? into (6) gives (8) W?PW 0: This can be rearranged to yield (9) P(x1 ; y1 ) dx1 P(xN ; yN ) dxN dF ; . . . Q(x1 ; y1 ) dz Q(xN ; yN ) dz dz dW P(x) dx W (10) from a FIXED POINT to the points of intersection with a curve depending rationally upon any number of parameters is a RATIONAL FUNCTION of those parameters. which can then be directly integrated to " # W(x) ln P(x) dx; W0 References where lnx is the NATURAL LOGARITHM. Exponentiating then yields Abel's identity Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959. Abel's Differential Equation g W(x)W0 eg P(x) dx ; (11) (12) where W0 is a constant of integration. The Abel equation of the first kind is given by See also ORDINARY DIFFERENTIAL EQUATION–SECONDORDER y?f0 (x)f1 (x)yf2 (x)y 2 f3 (x)y 3 . . . (Murphy 1960, p. 23; Zwillinger 1997, p. 120), and the Abel equation of the second kind by References Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 118, 262, 277, and 355, 1986. [g0 (x)g1 (x)y]y?f0 (x)f1 (x)yf2 (x)y 2 f3 (x)y 3 (Murphy 1960, p. 25; Zwillinger 1997, p. 120). References Murphy, G. M. Ordinary Differential Equations and Their Solution. Princeton, NJ: Van Nostrand, 1960. Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997. Abel's Differential Equation Identity Given a homogeneous linear SECOND-ORDER ORDIN- ARY DIFFERENTIAL EQUATION, yƒ P(x)y? Q(x)y 0; (1) call the two linearly independent solutions y1 (x) and y2 (x): Then Abel's Duplication Formula The duplication formula for ROGERS L -FUNCTION follows from ABEL'S FUNCTIONAL EQUATION and is given by ! x 2 1 : L(x )L(x)L 2 1x See also ABEL'S FUNCTIONAL EQUATION, DILOGARITHM yƒ1 P(x)y?1 Q(x)y1 0 (2) References yƒ2 P(x)y?2 Q(x)y2 0: (3) Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /48, 1997. Abel's Functional Equation Abel's Functional Equation Let L(x) denote the ROGERS L -FUNCTION defined in terms of the usual DILOGARITHM by i 6 h Li2 (x) 12 ln x ln(1x) 2 p " # 6 X xn 1 2 ln x ln(1x) ; p 2 n1 n 2 L(x) then L(x) satisfies the functional equation ! ! x(1 y) y(1 x) L(x)L(y)L(xy)L L : 1 xy 1 xy ABEL'S tity. DUPLICATION FORMULA follows from this iden- See also ABEL'S DUPLICATION FORMULA, DILOGARITHM, F UNCTIONAL E QUATION, POLYLOGARITHM , RIEMANN ZETA FUNCTION, ROGERS L -FUNCTION References Abel, N. H. Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 189 /92, 1988. Bytsko, A. G. Two-Term Dilogarithm Identities Related to Conformal Field Theory. 9 Nov 1999. http://xxx.lanl.gov/ abs/math-ph/9911012/. Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431 /48, 1997. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 14 and 21, 1999. Rogers, L. J. "On Function Sum Theorems Connected with n 2: the Series a 1 x =n /" Proc. London Math. Soc. 4, 169 89, 1907. 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, SUBTRACTIONS, MULTIPLICATIONS, DIVISIONS, and ROOT EXTRACTIONS. This was also shown by Ruffini in 1813 (Wells 1986, p. 59). See also CUBIC EQUATION, GALOIS'S THEOREM, POLYNOMIAL, QUADRATIC EQUATION, QUARTIC EQUATION, QUINTIC EQUATION References Abel, N. H. "Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen." J. reine angew. Math. 1, 65, 1826. Reprinted in Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 66 7, 1988. Artin, E. Galois Theory, 2nd ed. Notre Dame, IN: Edwards Brothers, 1944. Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103, 51 7, 1996. Fraleigh, J. B. A First Course in Abstract Algebra. Reading, MA: Addison-Wesley, 1982. Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975. Abel's Lemma 17 Hungerford, T. W. Algebra. New York: Springer-Verlag, 1980. van der Waerden, B. L. A History of Algebra: From alKhwarizmi to Emmy Noether. New York: Springer-Verlag, pp. 85 8, 1985. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 59, 1986. Abel's Inequality Let ffn g and fan g be SEQUENCES with fn ]fn1 > 0 for n 1, 2, ..., then j m X n1 j an fn 5Af1 ; where Amaxf½a1 ½; ½a1 a2 ½; . . . ; ½a1 a2 . . .am ½g: Abel's Irreducibility Theorem If one ROOT of the equation f (x)0; which is irreducible over a FIELD K , is also a ROOT of the equation F(x)0 in K , then all the ROOTS of the irreducible equation f (x)0 are ROOTS of F(x)0: Equivalently, F(x) can be divided by f (x) without a REMAINDER, F(x)f (x)F1 (x); where F1 (x) is also a POLYNOMIAL over K . See also ABEL'S LEMMA, KRONECKER'S POLYNOMIAL THEOREM, SCHÖNEMANN'S THEOREM References Abel, N. H. "Mémoire sur une classe particulière d'équations résolubles algébriquement." J. reine angew. Math. 4, 1829. Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 120, 1965. 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 p th POWER of an element of the FIELD. Jeffreys and Jeffreys (1988) use the term "Abel's lemma" for another LEMMA related to ABEL'S UNIFORM CONVERGENCE TEST. See also ABEL'S IRREDUCIBILITY THEOREM, GAUSS'S POLYNOMIAL THEOREM, KRONECKER'S POLYNOMIAL THEOREM, SCHÖNEMANN'S THEOREM References Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965. 18 Abel's Test Abelian Group Jeffreys, H. and Jeffreys, B. S. "Abel's Lemma." §1.1153 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 41 /2, 1988. Abel's Test ABEL'S UNIFORM CONVERGENCE TEST References Freyd, P. Abelian Categories: An Introduction to the Theory of Functors. New York: Harper & Row, 1964. Grothendieck, A. "Sur quelques points d'algèbre homologique." Tôhoku Math. J. 9, 119 /21, 1957. Mac Lane, S. and Gehring, F. W. Categories for the Working Mathematician, 2nd ed. New York: Springer-Verlag, 1998. Abel's Theorem ABEL'S BINOMIAL THEOREM, ABEL'S CONVERGENCE THEOREM, ABEL'S CURVE THEOREM, ABEL'S IMPOSSIBILITY THEOREM, ABEL'S IRREDUCIBILITY THEOREM, ABELIAN THEOREM, Q -ABEL'S THEOREM Abelian Differential An Abelian differential is an DIFFERENTIAL on a RIEMANN SURFACE. MORPHIC ANALYTIC COMPACT or MEROor closed Abelian Extension This entry contributed by NICOLAS BRAY Abel's Uniform Convergence Test Let fun (x)g be a SEQUENCE of functions. If 1. un (x) can be written un (x)an fn (x);/ 2. aan is CONVERGENT, 3. fn (x) is a MONOTONIC DECREASING SEQUENCE (i.e., fn1 (x)5fn (x)) for all n , and 4. fn (x) is BOUNDED in some region (i.e., 05fn (x)5 M for all x e [a; b])/ then, for all x [a; b]; the SERIES aun (x) CONVERGES UNIFORMLY. See also CONVERGENCE TESTS, CONVERGENT SERIES, UNIFORM CONVERGENCE References Bromwich, T. J. I'a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 59, 1991. Jeffreys, H. and Jeffreys, B. S. "Abel's Lemma" and "Abel's Test." §1.1153 /.1154 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 41 /2, 1988. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 17, 1990. Abelian A group or other algebraic object is said to be Abelian is the law of commutativity always holds. If an algebraic object is not Abelian, it is said to be NONABELIAN. If F is an ALGEBRAIC GALOIS EXTENSION of K such that the GALOIS GROUP of the extension is ABELIAN, then F is said to be an Abelian extension of K . See also ALGEBRAIC EXTENSION, GALOIS EXTENSION, GALOIS GROUP Abelian Function An INVERSE FUNCTION of an ABELIAN INTEGRAL. Abelian functions have two variables and four periods, and can be defined by X 2 q? U y; t; 2 2piy(lq?)pit(lq?) 2piq(lq?) q l Baker (1907, p. 21). Abelian functions are a generalization of ELLIPTIC FUNCTIONS, and are also called hyperelliptic functions. See also ABELIAN INTEGRAL, ELLIPTIC FUNCTION, THETA FUNCTIONS References Baker, H. F. Abelian Functions: Abel's Theorem and the Allied Theory, Including the Theory of the Theta Functions. New York: Cambridge University Press, 1995. Baker, H. F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge University Press, 1907. Weisstein, E. W. "Books about Abelian Functions." http:// www.treasure-troves.com/books/AbelianFunctions.html. See also ABELIAN CATEGORY, ABELIAN DIFFERENTIAL, ABELIAN FUNCTION, ABELIAN GROUP, ABELIAN INTEGRAL, ABELIAN VARIETY, COMMUTATIVE, NON-ABE- Abelian Group LIAN 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. Abelian Category An Abelian category is an abstract mathematical CATEGORY which displays some of the characteristic properties of the CATEGORY of all ABELIAN GROUPS. See also ABELIAN GROUP, CATEGORY N.B. A detailed online essay by S. Finch was the starting point for this entry. Abelian Group Abelian Integral 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 a pi i ; (1) ! 8 < 2:294856591 . . . Y j Ak 14:6475663 . . . z : k j1 118:6924619 . . . for k1 for k2 for k3; 19 (5) j"k and z(s) is again the RIEMANN ZETA FUNCTION. [Richert (1952) incorrectly gave A3 114:/] DeKoninck and Ivic (1980) showed that i where the pi are distinct a(n) PRIME FACTORS, Y N X pffiffiffiffiffi 1 BN O[ N (ln N)1=2 ]; n1 a(n) then P(ai ); (2) where i where P(k) 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 0 denotes an impossible number (i.e., not a product of partition numbers) of nonisomorphic Abelian, groups. The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, ... (Sloane's A046064). The incrementally largest numbers of Abelian groups as a function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (Sloane's A046054), which occur for orders 1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... (Sloane's A046055). The KRONECKER DECOMPOSITION THEOREM states that every FINITE Abelian group can be written as a GROUP DIRECT PRODUCT of CYCLIC GROUPS of PRIME POWER ORDER. If the ORDER of a FINITE GROUP is a PRIME p , then there exists a single Abelian group of order p (denoted Zp ) and no non-Abelian groups. If the ORDER is a prime squared p2 then there are two Abelian groups (denoted Zp 2 and Zp Zp : If the ORDER is a prime cubed p 3; then there are three Abelian groups (denoted Zp Zp Zp ; Zp Zp 2 ; and Zp 3 ); and five groups total. If the order is a PRODUCT of two primes p and q , then there exists exactly one Abelian group of ORDER pq (denoted Zp Zq ):/ Another interesting result is that if a(n) denotes the number of nonisomorphic Abelian groups of ORDER n , then X a(n)n s z(s)z(2s)z(3s) ; (3) n1 where z(s) is the RIEMANN ZETA FUNCTION. Srinivasan (1973) has also shown that N X a(n)A1 N A2 N A3 N 1=3 O[x 105=407 (ln x)2 ]; Y ( 1 " X k2 # ) 1 1 1 0:752 . . . P(k 2) P(k) p k (4) (7) is a product over PRIMES. Bounds for the number of nonisomorphic non-Abelian groups are given by Neumann (1969) and Pyber (1993). See also FINITE GROUP, GROUP THEORY, KRONECKER DECOMPOSITION THEOREM, PARTITION FUNCTION P , RING References Arnold, D. M. and Rangaswamy, K. M. (Eds.). Abelian Groups and Modules. New York: Dekker, 1996. DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980. Erdos, P. and Szekeres, G. "Über die Anzahl abelscher Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem." Acta Sci. Math. (Szeged) 7, 95 /02, 1935. Finch, S. "Favorite Mathematical Constants." http:// www.mathsoft.com/asolve/constant/abel/abel.html. Fuchs, L. and Göbel, R. (Eds.). Abelian Groups. New York: Dekker, 1993. Kendall, D. G. and Rankin, R. A. "On the Number of Abelian Groups of a Given Order." Quart. J. Oxford 18, 197 /08, 1947. Kolesnik, G. "On the Number of Abelian Groups of a Given Order." J. reine angew. Math. 329, 164 /75, 1981. Neumann, P. M. "An Enumeration Theorem for Finite Groups." Quart. J. Math. Ser. 2 20, 395 /01, 1969. Pyber, L. "Enumerating Finite Groups of Given Order." Ann. Math. 137, 203 /20, 1993. Richert, H.-E. "Über die Anzahl abelscher Gruppen gegebener Ordnung I." Math. Zeitschr. 56, 21 /2, 1952. Sloane, N. J. A. Sequences A000688/M0064 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Srinivasan, B. R. "On the Number of Abelian Groups of a Given Order." Acta Arith. 23, 195 /05, 1973. Abelian Integral INTEGRAL OF THE FORM x n1 where B An 1=2 (6) ffi; g pffiffiffiffiffiffiffiffi R(t) dt 0 where R(t) is a POLYNOMIAL of degree > 4: They are also called HYPERELLIPTIC INTEGRALS. 20 Abelian Theorem See also ABELIAN FUNCTION, ELLIPTIC INTEGRAL References Siegel, C. L. Topics in Complex Function Theory, Vol. 2: Automorphic Functions and Abelian Integrals. New York: Wiley, 1988. Abhyankar's Conjecture Z2 Z2 ; and for instance, i?j?j?i? in the Abelianization. See also ABELIAN, GROUP, HOMOMORPHISM Abel-Plana Formula This entry contributed by DAVID ANDERSON Abelian Theorem A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly. For example, The Abel-Plana formula gives an expression for the difference between a discrete sum and the corresponding integral. The formula can be derived from the ARGUMENT PRINCIPLE A(x)x G implies A1 (x) g x 0 A(t) dt 12x 2 for any A(x): The converse is false, but can be made into a correct TAUBERIAN THEOREM if A(x) is subjected to an appropriate additional condition (Hardy 1999, p. 46). See also TAUBERIAN THEOREM References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 46, 1999. 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 f (z) g X X g?(z) dz f (mn ) f (nm ); g(z) n m (1) where mn are the zeros of g(z) and nm are the poles contained within the CONTOUR g: An appropriate choice of g and g then yields X f (n) n0 g f (x) dx 0 12 f (0) 12 g [f (it)f (it)][cot(pit)i] dt; (2) 0 or equivalently X n0 f (n) g f (x) dx 0 12 f (0)i g 0 f (it) f (it) dt: e 2pt 1 (3) The formula is particularly useful in Casimir effect calculations involving differences between quantized modes and free modes. See also ARGUMENT PRINCIPLE References Murty, V. K. Introduction to Abelian Varieties. Providence, RI: Amer. Math. Soc., 1993. Shimura, G. Abelian Varieties With Complex Multiplication and Modular Functions. Princeton, NJ: Princeton University Press, 1999. Shimura, G. and Taniyama, Y. Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Tokyo: Mathematical Society of Japan, 1961. References Abelianization Abhyankar's Conjecture In general, groups are not ABELIAN. However, there is always a GROUP HOMOMORPHISM h : G 0 G? to an ABELIAN GROUP, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the COMMUTATOR SUBGROUP [G, G ]. So G? G=[G; G]: Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal expressions may become equal, or even represent the IDENTITY ELEMENT. For a FINITE GROUP G , let p(G) be the SUBGROUP generated by all the SYLOW P -SUBGROUPS of G . If X is a projective curve in characteristic p 0, and if x0 ; ..., xt are points of X (for t 0), then a NECESSARY and SUFFICIENT condition that G occur as the GALOIS GROUP of a finite covering Y of X , branched only at the points x0 ; ..., xt ; is that the QUOTIENT GROUP G=p(G) has 2gt generators. For example, in the eight-element QUATERNION GROUP /G f91; 9i; 9j; 9kg/, the COMMUTATOR SUBGROUP is f91g: The Abelianization of G is a copy of Mostepanenko, V. M. and Trunov, N. N. §2.2 in The Casimir Effect and Its Applications. Oxford, England: Clarendon Press, 1997. Saharian, A. A. "The Generalized Abel-Plana Formula. Applications to Bessel Functions and Casimir Effect." http://www.ictp.trieste.it/~pub_off/preprints-sources/2000/ IC2000014P.pdf. 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. Ablowitz-Ramani-Segur Conjecture Absolute Geometry 21 See also FINITE GROUP, GALOIS GROUP, QUOTIENT GROUP, SYLOW P -SUBGROUP multiplied by another absolutely convergent series, the product series will also converge absolutely. References See also CONDITIONAL CONVERGENCE, CONVERGENT SERIES, RIEMANN SERIES THEOREM Abhyankar, S. "Coverings of Algebraic Curves." Amer. J. Math. 79, 825 /56, 1957. American Mathematical Society. "Notices of the AMS, April 1995, 1995 Frank Nelson Cole Prize in Algebra." http:// www.ams.org/notices/199504/prize-cole.pdf. Harbater, D. "Abhyankar's Conjecture on Galois Groups Over Curves." Invent. Math. 117, 1 /5, 1994. Raynaud, M. "Revêtements de la droite affine en caractéristique p 0 et conjecture d'Abhyankar." Invent. Math. 116, 425 /62, 1994. Ablowitz-Ramani-Segur Conjecture The Ablowitz-Ramani-Segur conjecture states that a nonlinear PARTIAL DIFFERENTIAL EQUATION is solvable by the INVERSE SCATTERING METHOD only if every nonlinear ORDINARY DIFFERENTIAL EQUATION obtained by exact reduction has the PAINLEVÉ PROPERTY. See also INVERSE SCATTERING METHOD References Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 351, 1989. References Bromwich, T. J. I'a. and MacRobert, T. M. "Absolute Convergence." Ch. 4 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 69 /7, 1991. Jeffreys, H. and Jeffreys, B. S. "Absolute Convergence." §1.051 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 16, 1988. Absolute Deviation Let ū denote the MEAN of a SET of quantities ui ; then the absolute deviation is defined by Dui jui ūj: See also DEVIATION, MEAN DEVIATION, SIGNED DESTANDARD DEVIATION VIATION, Absolute Error The DIFFERENCE between the measured or inferred value of a quantity x0 and its actual value x , given by Abnormal Number A hypothetical number which can be factored into primes in more than one way. Hardy and Wright (1979) prove the FUNDAMENTAL THEOREM OF ARITHMETIC by showing that no abnormal numbers exist. See also FUNDAMENTAL THEOREM OF ARITHMETIC References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 21, 1979. Abs Dxx0 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 VALUE Absolute Frequency Abscissa The x - (horizontal) coordinate of a point in a two dimensional coordinate system. Physicists and astronomers sometimes use the term to refer to the axis itself instead of the distance along it. See also AXIS, ORDINATE, REAL LINE, Z -AXIS X -AXIS, Y -AXIS, Absolute Convergence an un is said to CONVERGE absolutely if the an jun j CONVERGES, where jun j 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 A The number of data points which fall within a given CLASS in a FREQUENCY DISTRIBUTION. See also CUMULATIVE FREQUENCY, FREQUENCY DISTRIBUTION, RELATIVE FREQUENCY, RELATIVE CUMULATIVE FREQUENCY References Kenney, J. F. and Keeping, E. S. "Frequency Distributions." §1.8 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 12 /9, 1962. SERIES SERIES Absolute Geometry GEOMETRY which depends only on the first four of EUCLID'S POSTULATES and not on the PARALLEL POSTULATE. Euclid himself used only the first four 22 Absolute Moment Absolute Value postulates for the first 28 propositions of the ELEbut was forced to invoke the PARALLEL POSTULATE on the 29th. MENTS , a9be id 2(a9be id )(a9be id ) a 2 b 2 9ab(e id e id )a 2 b 2 92ab cos d: (4) If a 1, then (4) becomes See also AFFINE GEOMETRY, ELEMENTS , EUCLID'S POSTULATES, GEOMETRY, ORDERED GEOMETRY, PARALLEL POSTULATE 19be id 21b 2 92b cos d (19b)2 4b sin 2 (12 d): (5) If a 1, and b 1, then References Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 90 /1, 1989. 1e id 24 sin 2 (1 d): 2 (6) Finally, Absolute Moment The absolute moment of Mn of a probability function P(x) taken about a point a is defined by g ½e if 1 e if 2 ½ 2 (e if 1 e if 2 )(e if 1 e if 2 ) 2[1cos(f2 f1 )] 4 cos 2 [12(f2 f1 )]: (7) n Mn j xaj P(x) dx: See also CENTRAL MOMENT, MOMENT, RAW MOMENT See also ARGUMENT (COMPLEX NUMBER), COMPLEX NUMBER, MODULUS (COMPLEX NUMBER) References Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 146, 1984. Absolute Value Absolute Monotonic Sequence See also ABSOLUTELY MONOTONIC SEQUENCE References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 224, 1971. Absolute Pseudoprime CARMICHAEL NUMBER Absolute Square Also known as the squared norm. The absolute square of a COMPLEX NUMBER z is written j zj2 ; where j zj is the MODULUS and is defined as j zj2zz̄; where z̄ denotes the COMPLEX CONJUGATE REAL NUMBER, (1) simplifies to (1) of z . For a (2) j zj2z 2 : If the COMPLEX NUMBER is written zxiy; then the absolute square can be written (3) j xiyj2x 2 y 2 : An absolute square can be computed in terms of x and y using the Mathematica command ComplexExpand[Abs[z ]2, TargetFunctions- {Conjugate}]. An important identity involving the absolute square is given by The absolute value of a REAL NUMBER x is denoted j xj and given by the "unsigned" portion of x , j xjx sgn(x) x for x50 x for x]0; where sgn x is the sign function SGN. The absolute value is therefore always greater than or equal to 0. The same notation is used to denote MODULUS of a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pthe COMPLEX NUMBER zxiy; j zj x 2 y 2 ; a P -ADIC NORM, or a general VALUATION. The NORM of a VECTOR x is also denoted jxj; although jxj is more commonly used. Other NOTATIONS similar to the absolute value are the FLOOR FUNCTION b xc; NINT function [x]; and CEILING FUNCTION d xe:/ The integral of the absolute value of the different of two variables is given by Absolutely Continuous 1 1 Abstract Algebra g g jxyj dx dy (n 1)(n 2) ; 2 n 0 0 which has values 1/3, 1/6, 1/10, 1/15, 1/21, ... for n 1, 2, ..., i.e., the inverses of the TRIANGULAR NUMBERS (Sloane's A000217). 23 Absolutely Monotonic Function This entry contributed by RONALD M. AARTS A function f (x) is absolutely monotonic in the interval aBxBb if it has nonnegative derivatives of all orders in the region, i.e., f (k) (x)]0 See also ABSOLUTE SQUARE, CEILING FUNCTION, FLOOR FUNCTION, MODULUS (COMPLEX NUMBER), NINT, RECTANGLE FUNCTION, SGN, TRIANGLE FUNCTION, VALUATION for aBxBb and k 0, 1, 2, .... For example, the functions References and f (x)ln(x) Sloane, N. J. A. Sequences A000217/M2535 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. A MEASURE l is absolutely continuous with respect to another measure m if l(E)0 for every set with m(E)0: This makes sense as long as m is a POSITIVE MEASURE, such as LEBESGUE MEASURE, but l can be any measure, possibly a COMPLEX MEASURE. THEOREM, l(E) g (2) (3) f (x)sin 1 x (05x51) are absolutely monotonic functions (Widder 1941). See also ABSOLUTELY MONOTONIC SEQUENCE Absolutely Continuous By the RADON-NIKODYM to saying that (15xB0) (1) this is equivalent f dm References Widder, D. V. Ch. 4 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. Absolutely Monotonic Sequence See also ABSOLUTE MONOTONIC SEQUENCE, ABSOMONOTONIC FUNCTION LUTELY E where the integral is the LEBESGUE INTEGRAL, for some INTEGRABLE function f . The function f is like a derivative, and is called the RADON-NIKODYM DERIVATIVE dl=dm:/ The measure supported at 0 (/m (E) 1 iff 0 E) is not absolutely continuous with respect to LEBESGUE MEASURE, and is a SINGULAR MEASURE. See also COMPLEX MEASURE, CONCENTRATED, HAAR MEASURE, LEBESGUE DECOMPOSITION (MEASURE), LEBESGUE MEASURE, MUTUALLY SINGULAR, POLAR REPRESENTATION (MEASURE), SINGULAR MEASURE References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 224, 1971. Absorption Law The law appearing in the definition of a BOOLEAN which states ALGEBRA affl(a b)a (afflb)a for binary operators and ffl (which most commonly are logical OR and logical AND). References See also BOOLEAN ALGEBRA, LATTICE Rudin, W. Functional Analysis, 2nd ed. New York: McGrawHill, pp. 121 /25, 1991. References Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, p. 317, 1996. Absolutely Fair A sequence of random variates X0 ; X1 ; ... is called absolutely fair if for n 1, 2, ..., Abstract Algebra (Feller 1971, p. 210). That portion of ALGEBRA dealing with theoretical as opposed to applied topics. Ash (1998) includes the following areas in his definite of abstract algebra: logic and foundations, counting, elementary NUMBER THEORY, informal SET THEORY, LINEAR ALGEBRA, and the theory of linear operators. See also MARTINGALE See also ALGEBRA References References Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971. Ash, R. B. A Primer of Abstract Mathematics. Washington, DC: Math. Assoc. Amer., 1998. (X1 )0 and (Xn1 ½X1 ; . . . ; Xn )0 24 Abstract Manifold Abstract Manifold Abundant Number A(n)/ Number / An abstract manifold is a MANIFOLD in the context of an abstract space with no particular embedding, or representation in mind. It is a TOPOLOGICAL SPACE with an ATLAS of COORDINATE CHARTS. B0 / / -1 For example, the SPHERE S2 can be considered a 3 SUBMANIFOLD of R or a QUOTIENT SPACE O(3)=O(2): But as an abstract manifold, it is just a MANIFOLD, which can be covered by two coordinate charts / f1 : R2 0 S2/ and /f2 : R2 0 S2/, with the single TRANSITION FUNCTION, 2 2 f1 2 (f1 : R (0; 0) 0 R (0; 0) / DEFICIENT NUMBER ALMOST PERFECT NUMBER 0 PERFECT NUMBER 1 QUASIPERFECT NUMBER > 0/ ABUNDANT NUMBER See also ABUNDANCY, DEFICIENCY References defined by 2 2 f1 2 (f1 (x; y)(x=r ; y=r ) where /r2 x2 y2/. It can also be thought of as two disks glued together at their boundary. See also A LGEBRAIC M ANIFOLD , H OMOGENEOUS S PACE , M ANIFOLD , S UBMANIFOLD , T OPOLOGICAL SPACE Abstract Mathematics Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45 /6, 1994. Abundancy The ratio s(n)=n; where s(n) is the DIVISOR FUNCTION. See also ABUNDANCE, ABUNDANT NUMBER References Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63, 3 /0, 1990. ABSTRACT ALGEBRA Abundant Number Abstract Simplicial Complex An abstract simplicial complex is a collection S of finite nonempty sets such that if A is an element of S , then so is every nonempty subset of A (Munkres 1993, p. 15). See also SIMPLICIAL COMPLEX References Munkres, J. R. Elements of Algebraic Topology. Perseus Press, 1993. Abstract Vector Space An abundant number is an INTEGER n which is not a PERFECT NUMBER and for which s(n)s(n)n > n; (1) where s(n) is the DIVISOR FUNCTION. The quantity s(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 9453 3 × 7 × 5: (2) That 945 is abundant can be seen by computing See also QUOTIENT VECTOR SPACE, VECTOR SPACE Abstraction Operator LAMBDA CALCULUS Abundance 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 numbers. Define the density function The abundance of a number n is the quantity A(n)s(n)2n; where s(n) is the DIVISOR FUNCTION. Kravitz has conjectured 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(x) lim n0 ½fn : s(n) ] xng½ n (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 al. (1977) showed that 0:2441BA(2)B0:2909; (5) Acceleration Acceleration and Deléglise (1998) showed that 0:2474BA(2)B0:2480: (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). References Acceleration (3) dt 2 dx d 2 x dy d 2 y dz d 2 z dtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt 2 dt dt 2 dt dt 2 v !2 !2 !2 u u dx dy dz t dt dt dt See also ALIQUOT SEQUENCE, DEFICIENT NUMBER, HIGHLY ABUNDANT NUMBER, MULTIAMICABLE NUMBERS, PERFECT NUMBER, PRACTICAL NUMBER, PRIMITIVE ABUNDANT NUMBER, WEIRD NUMBER Deléglise, M. "Bounds for the Density of Abundant Integers." Exp. Math. 7, 137 /43, 1998. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3 /3, 1952. Erdos, P. "On the Density of the Abundant Numbers." J. London Math. Soc. 9, 278 /82, 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 /6, 1994. Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, pp. 11 and 13, 1997. Sloane, N. J. A. Sequences A005101/M4825 in "An On-Line Version of the Encyclopedia of Integer Sequences." http:// www.research.att.com/~njas/sequences/eisonline.html. Souissi, M. Un Texte Manuscrit d'Ibn Al-Banna' Al-Marrakusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Wall, C. R. "Density Bounds for the Sum of Divisors Function." In The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29-May 1, 1971. (Ed. A. A. Gioia and D. L. Goldsmith). New York: Springer-Verlag, pp. 283 /87, 1971. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 26, 773 /77, 1972. Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 31, 616, 1977. d2s VECTOR (4) dx d 2 x dy d 2 y dz d 2 z ds dt 2 ds dt 2 ds dt 2 (5) dr d 2 r × : ds dt 2 (6) The 25 acceleration is given by dv d 2 r d 2 s ds T̂k a 2 2 dt dt dt dt !2 N̂; (7) where T̂ is the UNIT TANGENT VECTOR, k the CURVATURE, s the ARC LENGTH, and N̂ the UNIT NORMAL VECTOR. Let a particle move along a straight LINE so that the positions at times t1 ; t2 ; and t3 are s1 ; s2 ; and s3 ; respectively. Then the particle is uniformly accelerated with acceleration a IFF " # (s2 s3 )t1 (s3 s1 )t2 (s1 s2 )t3 a2 (t1 t2 )(t2 t3 )(t3 t1 ) (8) is a constant (Klamkin 1995, 1996). Consider the measurement of acceleration in a rotating reference frame. Apply the ROTATION OPERATOR ! d R̃ v dt body twice to the notation, RADIUS VECTOR (9) r and suppress the body !2 d aspace R̃ r v dt ! ! d dr v vr r dt dt 2 Let a particle travel a distance s(t) as a function of time t (here, s can be thought of as the ARC LENGTH of the curve traced out by the particle). The SPEED (the SCALAR NORM of the VECTOR VELOCITY) is then given