ENCYCLOPAEDIA OF MATHEMATICS Volume 6 ENCYCLOPAEDIA OF MATHEMATICS Managing Editor M. Hazewinkel Scieniific Board S. Albeverio, J. B. Alblas, S. A. Amitsur, I. J. Bakelman, G. Bakker, J. W. de Bakker, C. Bardos, H. Bart, H. Bass, A. Bensoussan, M. Bercovier, L. Berkovitz, M. Berger, E. A. Bergshoeff, E. Bertin, F. Beukers, A. Beutelspacher, H. P. Boas, J. Bochnak, H. J. M. Bos, B. L. J. Braaksma, T. P. Branson, D. S. Bridges, A. E. Brouwer, M. G. de Bruin, R. G. Bums, H. Capel, P. Cartier, C. Cercignani, J. M. C. Clark, Ph. Clement, A. M. Cohen, J. W. Cohen, P. Conrad, H. S. M. Coxeter, R. F. Curtain, M. H. A. Davis, M. V. Dekster, C. Dellacherie, G. van Dijk, H. C. Doets, I. Dolgachev, A. Dress, J. J. Duistermaat, D. van Dulst, H. van Duyn, H. Dym, A. Dynin, M. L. Eaton, W. Eckhaus, P. van Emde Boas, H. Engl, G. Ewald, V. I. Fabrikant, A. Fasano, M. Fliess, R. M. Fossum, B. Fuchssteiner, G. B. M. van der Geer, R. D. Gill, V. V. Goldberg, J. de Graaf, J. Grasman, P. A. Griffith, A. W. Grootendorst, L. Gross, P. Gruber, K. P. Hart, G. Heckman, A. J. Hermans, W. H. Hesselink, C. C. Heyde, M. W. Hirsch, K. H. Hofmann, A. T. de Hoop, P. J. van der Houwen, N. M. Hugenholtz, J. R. Isbell, A. Isidori, E. M. de Jager, D. Johnson, P. T. Johnstone, D. Jungnickel, M. A. Kaashoek, V. Kac, W. L. J. van der Kallen, D. Kanevsky, Y. Kannai, H. Kaul, E. A. de Kerf, W. Klingenberg, T. Kloek, J. A. C. Kolk, G. Komen, T. H. Koomwinder, L. Krop, B. Kuperschmidt, H. A. Lauwerier, J. van Leeuwen, H. W. Lenstra Jr., J. K. Lenstra, H. Lenz, M. Levi, J. Lin denstrauss, J. H. van Lint, F. Linton, A. Liulevicius, M. Livshits, W. A. J. Luxemburg, R. M. M. Mattheij, L. G. T. Meertens, I. Moerdijk, J. P. Murre, H. Neunzert, G. Y. Nieuwland, G. J. Olsder, B. 0rsted, F. van Oystaeyen, B. Pareigis, K. R. Parthasarathy, K. R. Parthasarathy, I. I. Piatetskil-Shapiro, H. G. J. Pijls, N. U. Prabhu, E. Primrose, A. Ramm, C. M. Ringel, J. B. T. M. Roerdink, K. W. Roggenkamp, G. Rozenberg, W. Rudin, S. N. M. Ruysenaars, A. Salam, A. Salomaa, P. Saunders, J. P. M. Schalkwijk, C. L. Scheffer, R. Schneider, J. A. Schouten, F. Schurer, J. J. Seidel, A. Shenitzer, V. Snaith, T. A. Springer, 1. H. M. Steenbrink, J. D. Stegeman, F. W. Steutel, P. Stevenhagen, I. Stewart, R. Stong, L. Streit, K. Stromberg, L. G. Suttorp, D. Tabak, F. Takens, R. J. Takens, N. M. Temme, S. H. Tijs, B. Trakhtenbrot, N. S. Trudinger, L. N. Vaserstein, M. L. J. van de Vel, F. D. Veldkamp, W. Vervaat, P. M. B. Vitanyi, N. J. Vlaar, H. A. van der Vorst, 1. de Vries, F. Waldhausen, B. Wegner, J. J. O. O. Wiegerinck, J. C. Willems, J. M. Wills, B. de Wit, S. A. Wouthuysen, S. Yuzvinskil, L. Za1cman ENCYCLOPAEDIA OF MATHEMATICS Volume 6 Lobachevskir Criterion (for Convergence) - Optional Sigma-Algebra An updated and annotated translation of the Soviet 'Mathematical Encyclopaedia' KLUWER ACADEMIC PUBLISHERS Dordrecht / Boston / London Library of Congress Cataloging-in-Publication Data Matematicheskaia entsiklopediia. English. Encyclopaedia of mathematics. 1. Mathematics--Dictionaries. I. Hazewinkel, Michiel. II. Title. QA5.M3713 1987 510'.3'21 87-26437 ISBN-13: 978-94-009-5993-4 e-ISBN-13: 978-94-009-5991-0 DOI:10.1007/978-94-009-5991-0 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. All Rights Reserved © 1990 by Kluwer Academic Publishers Softcover reprint of the hardcover 15t edition 1990 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner SOVIET MATHEMATICAL ENCYCLOPAEDIA Editor-in-Chief I. M. Vinogradov Editorial Board S. 1. Adyan, P. S. Aleksandrov, N. S. Bakhvalov, A. V. Bitsadze, V. 1. Bityutskov (Deputy Editor-in-Chief), L. N. Bol'shev, A. A. Gonchar, N. V. Efimov, V. A. Il'in, A. A. Karatsuba, L. D. Kudryavtsev, B. M. Levitan, K. K. Mardzhanishvili, E. F. Mishchenko, S. P. Novikov, E. G. Poznyak, Yu. V. Prokhorov (Deputy Editor-in-Chief), A. 1. Shirshov, A. G. Sveshnikov, A. N. Tikhonov, P. L. UI'yanov, S. V. Yablonskii Translation Arrangements Committee V. I. Bityutskov, R. V. Gamkrelidze, Yu. V. Prokhorov 'Soviet Encyclopaedia' Publishing House PREFACE This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques. These are aimed at a smaller group of readers and require more back ground expertise. Often these articles contain more precise and refined accounts of topics and results touched upon in a general way in the first kind of article. Finally, there is a third kind of article: short (reference) definitions. Practically all articles (all except a few of the third kind) contain a list of references by means of which more details and more material on the topic can be found. Most articles were specially written for the encyclopaedia and in such cases the names of the original Soviet authors are mentioned. Some articles have another origin such as the Great Soviet Ency clopaedia (Bol'shaya Sovetskaya Entsiklopediya or BSE). Communication between mathematicians in various parts of the world has certainly greatly improved in the last decennia. However, this does not mean that there are so-to-speak 'one-to one onto' translations of the terminology, concepts and tools used by one mathematical school to those of another. There also are varying traditions of which questions are important and which not, and what is considered a central problem in one tradition may well be besides the point from the point of view of another. Even for well-established areas of mathematical inquiry, terminology varies across languages and even within a given language domain. Fur ther, a concept, theorem, algorithm, ... , which is associated with one proper name within one tradition may well have another one in another, especially if the result or idea in question was indeed discovered independently and more-or-Iess simultaneously. Finally, mathematics is a very dynamic science and much has happened since the original articles were finalized (mostly around 1977). This made updates desirable (when needed). All this, as well as providing vii PREFACE additional references to Western literature when needed, meant an enormous amount of work for the board of experts as a whole; some indeed have done a truly impressive amount of work. I must stress though that I am totally responsible for what is finally included and what is not of all the material provided by the members of the board of experts. Many articles are thus provided with an editorial comment section in a different and some what smaller typeface. In particular, these annotations contain additional material, amplifica tions, alternative names, additional references, .... Modifications, updates and other extra material provided by the original Soviet authors (not a rare occurrence) have been incorporated in the articles themselves. The final (lO-th) volume of the ENCYCLOPAEDIA OF MATHEMATICS will be an index volume. This index will contain all the titles of the articles (some 6600) and in addition the names of all the definitions, named theorems, algorithms, lemmas, scholia, constructions, ... , which occur in the various articles. This includes, but is by no means limited to, all items which are printed in bold or italic. Bold words or phrases, by the way, always refer to another article with (precisely) that title. All articles have been provided with one or more AMS classification numbers according to the 1980 classification scheme (not, for various reasons, the 1985 revision), as have all items occurring in the index. A phrase or word from an article which is included in the index always inherits all the classification numbers of the article in question. In addition, it may have been provided with its own classification numbers. In the index volume these numbers will be listed with the phrase in question. Thus e.g. the Quillen - Suslin theorem of algebraic K-theory will have its own main classification numbers (these are printed in bold; in this case that number is 18F25) as well as a number of others, often from totally different fields, pointing e.g. to parts of mathematics where the theorem is applied, or where there occurs a problem related to it (in this case e.g. 93D15). The index volume will also contain the inversion of this list which will, for each number, provide a list of words and phrases which may serve as an initial description of the 'content' of that classification number (as far as this ENCYCLOPAEDIA is concerned). For more detai~s on the index volume, its structure and organisation, and what kind of things can be done with it, cf. the (future) special preface to that volume. Classifying articles is a subjective matter. Opinions vary greatly as to what belongs where and thus this attempt will certainly reflect the tastes and opinions of those who did the clas sification work. One feature of the present classification attempt is that the general basic concepts and definitions of an area like e.g. 55N (Homology and Cohomology theories) or 601 (Markov processes) have been assigned classification numbers like 55NXX and 60JXX if there was no finer classification number different from ... 99 to which it clearly completely belongs. Different parts of mathematics tend to have differences in notation. As a rule, in this ENCYCLOPAEDIA in a given article a notation is used which is traditional in the corresponding field. Thus for example the (repeated index) summation convention is used in articles about topics in fields where that is traditional (such as in certain parts of differential geometry (tensor geometry» and it is not used in other articles (e.g. on summation of series). This pertains especially to the more technical articles. For proper names in Cyrillic the British Standards Institute transcription system has been used (cf. Mathematical Reviews). This makes well known names like S. N. Bernstein come out as Bernshtein. In such cases, especially in names of theorems and article titles, the traditional spelling has been retained and the standard transcription version is given between brackets. Ideally an encyclopaedia should be complete up to a certain more-or-Iess well defined level viii PREFACE of detail. In the present case I would like to aim at the completeness level whereby every theorem, concept, definition, lemma, construction which has a more-or-Iess constant and accepted name by which it is referred to by a recognizable group of mathematicians occurs somewhere, and can be found via the index. It is unlikely that this completeness ideal will be reached with this present ENCYCLOPAEDIA OF MATHEMATICS, but it certainly takes substantial steps in this direction. Everyone who uses this ENCYCLOPAEDIA and finds items which are not covered, which, he feels, should have been included, is invited to inform me about it. When enough material has come in this way supplementary volumes will be put together. The ENCYCLOPAEDIA is alphabetical. Many titles consist of several words. Thus the problem arises how to order them. There are several systematic ways of doing this of course, for in stance using the first noun. All are unsatisfactory in one way or another. Here an attempt has been made to order things according to words or natural groups of words as they are daily used in practice. Some sample titles may serve to illustrate this: Statistical mechanics, mathemati cal problems in; Lie algebra; Free algebra; Associative algebra; Absolute continuity; Abstract algebraic geometry; Boolean functions, normal forms of. Here again taste plays a role (and usages vary). The index will contain all permutations. Meanwhile it will be advisable for the reader to tryout an occasional transposition himself. Titles like K-theory are to be found under K, more precisely its lexicographic place is identical with 'K theory', i.e. '-' = 'space' and comes before all other symbols. Greek letters come before the corresponding Latin ones, using the standard transcriptions. Thus X2-distribution (chi-squared distribution) is at the beginning of the letter C. A * as in C·-algebra and *-regular ring is ignored lexicographically. Some titles involve Greek letters spelled out in Latin. These are of course ordered just like any other 'ordinary' title. This volume has been computer typeset using the (Unix-based) system of the CWI, Amster dam. The technical (mark-up-Ianguage) keyboarding was done by Rosemary Daniels, Chahrzade van 't Hoff and Joke Pesch. To meet the data-base and typesetting requirements of this ENCYCLOPAEDIA substantial amounts of additional programming had to be done. This was done by 10han Wolleswinkel. Checking the translations against the original texts, and a lot of desk editing and daily coordination was in the hands of Rob Hoksbergen. All these persons, the members of the board of experts, and numerous others who provided information, remarks and material for the editorial comments, I thank most cordially for their past and continuing efforts. The original Soviet version had a printrun of 150,000 and is completely sold out. I hope that this annotated and updated translation will turn out to be comparably useful. Bussum. August 1987 MICHIEL HAZEWINKEL IX LoBACllEvsKIi CRITERION (FOR CONYER- It was introduced by N.!. Lobachevskii in 1829. GENCE) - A series ~:=Ian with positive terms an References [I] RyZHlK, I.M. and GRADsHTEiN, I.S.: Tables of integrals, series, tending monotonically to zero converges or diverges and products, Acad. Press, 1980 (translated from the Russian). according as the series A.B. Ivanov Editorial comments. For the LobachevskiT function in the m=O sense of 1) (Le. the angle of parallelism) see also [A 1] - converges or diverges, where pm is the largest of the [A4]. indices of the terms an that satisfy the inequality For LobachevskiT's function as defined in 2) see also an;;;..2-m, n=l, ... 'Pm' [A5]. It was proposed by N.!. Lobachevskii in 1834 - 1836. References v.l. Bityutskov [A1] GREENBERG, M.: Euclidean and non-Euclidean geometries, Freeman, 1974. AMS 1980 Subject Classification: 40A05 [A2] CoXETER, H.S.M.: 'Parallel lines', Ganad. Math. Bull. 21 (1978), 385-397. LoBACllEvsKIi FUNCflON - 1) The angle ofp ara 1- [A3] COXETER, H.S.M.: Non-Euclidean geometry, Toronto Univ. Press, 1957. lelism in Lobachevskll geometry is a function that [A4] BONOLA, R.: Non-Euclidean geometry, Dover, reprint, 1955. expresses the angle a between the line u 1 (or u 2) (see [A5] COXETER, H.S.M.: Twelve geometric esays, Carbondale, Fig.) and the segment OA perpendicular to a line a 1968, Chapt. 1. parallel to UI (or U2) in terms of the length I of the seg AMS 1980 Subject Classification: 51 M1 0, 33A70 ment OA: = = a II(I) 2arctane-IIR, LoBACllEvsKIi GEOMETRY - A geometry based where R is a positive constant that corresponds to the on the same fundamental premises as Euclidean scale of measurement of distances. geometry, except for the axiom of parallelism (see Fifth A postulate). In Euclidean geometry, according to this axiom, in a plane through a point P not lying on a ~ straight line A' A there passes precisely one line B' B that does not intersect A'A . The line B' B is called a o parallel to A' A. It is sufficient to require that there is at Q The Lobachevskii function is a continuous monotone most one straight line, since the existence of a non decreasing function with values between 7T /2 and 0: intersecting line can be proved by successively drawing lines PQ l.A 'A and PBl.PQ. lim II(l) = !!.., lim II(I) = O. 1....0 2 1_<Xl T' IJ. It was introduced by N.!. Lobachevskii in 1826. B' B. References u' T [I] KAGAN, V.F.: The foundatiOns of geometry, 1, Moscow Leningrad, 1949 (in Russian). [2] EFlMOV, N.V.: HOhere Geometrie, Deutsch. Verlag Wissen schaft., 1960 (translated from the Russian). A' o A 2) The special function (cf. Special functions) defined Fig. 1. for real x by x = - In Lobachevskii geometry the axiom of parallelism L(x) !lncostdt. o requires that through a point P (Fig. 1) there passes The Lobachevskii function can be represented as a more than one line not intersecting A'A. The non senes intersecting lines fill the part of the pencil with vertex L(x) = x 1n2-1. ~ (-It-I sin~kx. P lying inside a pair of vertically opposite angles TPU 2k=1 k and u'rF situated symmetrically with respect to the The main relations are: perpendicular PQ. The lines that form the sides of the = vertically opposite angles separate the intersecting lines L(-x) -L(x), from the non-intersecting lines, and they themselves are = L('IT-x) 'lT1n2-L(x), also non-intersecting. These limiting lines are called the = parallels at P to the line A'A in its two directions: T'T L('IT+x) 'lT1n2+L(x).