Calendar

NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the cJfo!aW was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which tall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned.

Meeting Deadline for Abstracts* Number Date Place and News I terns

702 April 14, 1973 Stanford, California Feb. 26, 1973 703 April 18-21, 1973 New York, New York Feb. 26, 1973 704 April 27-28, 1973 Evanston, Illinois Feb. 26, 1973 705 June 16, 1973 Bellingham, Washington May 3, 1973 706 August 20-24, 1973 Missoula, Montana June 28, 1973 (78th Summer Meeting) 707 October 27, 1973 Cambridge, Massachusetts Sept. 6, 1973 708 November 3, 1973 Minneapolis, Minnesota Sept. 6, 1973 709 November 16-17, 1973 Atlanta, Georgia Oct. 1, 1973 710 November 24, 1973 Tucson, Arizona Oct. 1, 1973 711 January 15-19, 1974 San Francisco, California (80th Annual Meeting) January 23-27, 1975 Washington, D. C. (81st Annual Meeting) January 22-26, 1976 San Antonio, Texas (82nd Annual Meeting)

*Deadline for abstracts not presented at a meeting (by title). June 1973 issue: April 26 August 1973 issue: June 21

OTHER EVENTS

June 30, 1973 Symposium on Some Mathematical Questions in Biology Mexico City, Mexico September 3-15, 1973 International Meeting on Combinatorial Theory Rome, Italy August 21-29, 1974 International Congress of Mathematicians Vancouver, B. C., Canada

Abstracts should be submitted on special forms which are available in most departments of mathematics; f(Jrms can also be obtained by writing to the headquarters of the Society. Abstracts to be presented at the meeting in person must be received at the headquarters of the Society in Providence, Rhode Island, on or be­ f(Jre the deadline for the meeting.

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The cJ,fotiaiJ of the American Mathematical Society is published by the American Mathematical Society, 321 South Main Street, P. 0. Box 6248, Providence, Rhode Island 02904 in January, February, April, June, August, October, November and December. Price per annual volume is $10.00. Price per copy $3.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904. Second class postage paid at Providence, Rhode Island, and additional mailing offices.

Copyright© 1973 by the American Mathematical Society Printed in the United States of America OF THE AMERICAN MATHEMATICAL SOCIETY

Everett Pitcher and Gordon L. Walker, Editors Wend ell H. Fleming, Associate Editor

CONTENTS

MEETINGS

Calendar of Meetings ....•...... •.....•....•....•...... •.... Inside Front Cover

PRELIMINARY ANNOUNCEMENTS OF MEETINGS...... • . . . • • . . . 70 REVIEWING CHARGE FOR MATHEMATICAL REVIEWS . • ...... • . . • . . . . • . . . . . • • . • . • . . • 75 QUERIES . • ...... • ...... • . . . . . • ...... • . . . . • ...... • ...... • . . 76 SPECIAL MEETINGS INFORMATION CENTER . • . . . . • ...... • . . • • ...... • . . • 77 BACKLOG OF MATHEMATICS RESEARCH JOURNALS . . . . . • . • ...... • . . • . . . . • ...... 80 LETTERS TO THE EDITOR . • ...... • . • . • . . . . . • ...... • • . • ...... • • . • . • ...... 81 NEWS ITEMS AND ANNOUNCEMENTS ...... • . • . . . . • . . • . . . . • • . . • . • . . . . • . . . . • . • . . . • . . • 81 SUMMER GRADUATE COURSES . . . • ...... • . . • . . . . • . . • . . . • . . . • ...... • . . . . • . • 82 ASSISTANTSHIPS AND FELLOWSHIPS IN MATHEMATICS IN 1973-1974 (Supplementary List) ...... • . • • . . • . • . . • . • • • . . • • . • . . • . . • • • • . • . . . . . • . • . . • • . • . . . . 83 DOCTORATES CONFERRED IN 1971-1972 (Supplementary List)...... 84 NEW AMS PUBLICATIONS •..•...... •.....•.....•.•..••...•....•...•.•...• 75,85 PERSONAL ITEMS • • . . • . . • . . . . . • ...... • ...... • . . . . • . . • • . • . . • . • • . • • • • . • • . • . • • . • . • • 86 ABSTRACTS . . . • • . . . • ...... • . . • • ...... • ...... • . . . . • . • • . . . . • . . • • . . . • . . • . A-249-A-294 ERRATA TO ABSTRACTS • . • . . . . • . . . . . • . . • ...... • . . • • . • . • • . . . . • . • • . • . . • . . . . • . . . . A-295 SITUATIONS WANTED A-296 INDEX TO ADVERTISERS .•.•.•..•...••.••.•..•...•.•.••....•..•..•..•.•.••.•..• A-311 RESERVATION FORM .•.•.....•.....•.•..•...... •..••.•.••.•....••..••..••.•.. A-312 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Seven Hundred Seeond Meeting Stanford UDiversity Stanford, California Aprill4, 1973

The seven hundred second meeting of the Single $18.00 up American Mathematical Society will be held at Double 25.00 up Stanford University, Stanford, California, on Saturday, April 14, 1973. RIVIERA MOTOR LODGE By invitation of the Committee to Select 15 El Camino, Menlo Park 94025 Hour Speakers for Far Western Sectional Meet­ Phone: (415) 321-8772 ings, there will be two one-hour addresses. Pro­ Single $12.00 up fessor Theodore W. Gamelin of the University of Double 15. 00 up California, Los Angeles, will lecture at 11:00 TIKI INN MOTEL a.m. on "The algebra of bounded analytic func­ 531 Stanford Avenue, Palo Alto 94304 tions. " Professor Robert M. Blumenthal of the Phone: (415) 327-3550 University of Washington will address the Society Single $13. 00 up at 2:00 p.m. His talk will be entitled "Stopping Double 18. 00 up time constructions." Both lectures will be given in the Bishop Auditorium located in the School of The Riviera Motor Lodge and the Tiki Inn Motel Business. are nearest to campus, but they are a thirty- There will be two Special Sessions on Par­ or forty- minute walk away. Reservations should tial Differential Equations consisting of invited be made directly with the desired motel or hotel. thirty-minute talks. These sessions have been Stanford University is located approximately arranged by Professor David Gilbarg of Stanford twenty miles from the San Francisco Interna­ University. There will be three speakers at each tional Airport and approximately fifteen miles session, including Michael G. Crandall, C. Den­ from the San Jose Airport. Taxf fare from these son Hill, Mario Miranda, and Keith Miller. airports to the university area is approximately Sessions for contributed papers will be $15 from the San Francisco Airport and $13 from scheduled in the morning and the afternoon. Ab­ the San Jose Airport. Limousine service is stracts should be submitted to the American available from either airport for $8 per person Mathematical Society, P. 0. Box 6248, Provi­ with a maximum of $26 for five or more. There dence, Rhode Island 02904, so as to arrive prior is occasional bus service from the airports to to the deadline of February 26, 1973. Late papers the bus terminal in downtown Palo Alto as fol­ will be accepted for presentation at the meeting, lows: from San Francisco Airport (6:00 a.m., but late papers will not be listed in the printed 2:00 p.m., 6:00p.m.; $1. 85), from San Jose program of the meeting. Airport (3:30p.m., 7:35p.m.; $1. 60). The registration desk will be located in the Persons driving to the meeting on the Bay­ main lobby of the Mathematics Department. Reg­ shore Freeway (U.S. 101) from either San Fran­ istration will begin at 8:30a.m. on Saturday. cisco or San Jose should take the off-ramp at The following are among the motels and University Avenue West, which leads to the hotels in the Stanford area: campus. University Avenue becomes Palm Drive on campus, and Palm Drive ends in an oval, with FLAMINGO MOTOR LODGE parking spaces on two sides. The Department 3398 El Camino, Palo Alto 94304 of Mathematics is located at the right-hand cor­ Phone: (415) 493-2411 ner of the Outer Quadrangle, the row of buildings Single $10. 50 up that one faces when one reaches the end of Palm Double 13. 50 up Drive. No permits for parking on campus are MOTEL 6 required. If parking spaces in the oval are in­ 4309 El Camino, Palo Alto 94306 sufficient, a map showing the location of nearby Phone: (415) 941-0220 additional spaces will be available at the regis­ Single $ 6. 60 up tration desk. Double 7. 70 up The only place within walking distance where lunch is available is Tressider Union, which PALO ALTO HOTEL serves cafeteria-style meals. There are several 425 High Street, Palo Alto 94301 good restaurants in nearby Palo Alto and Menlo Phone: (415) 328-9803 Park. A partial list of restaurants will be avail­ Single $ 8. 00 up able at the registration desk. Double 9. 50 up RICKEY• S HYATT HOUSE Kenneth A. Ross 4219 El Camino, Palo Alto 94306 Associate Secretary Phone: (415) 493-8000 Eugene, Oregon

70 The Seven Hundred Third Meeting Biltmore Hotel New York, New York April18- 21, 1973

The seven hundred third meeting of the Martin Schultz (Yale University), Joseph Traub American Mathematical Society will be held at (Carnegie-Mellon University), and Ronald Fagin the Biltmore Hotel, Madison Avenue at 43rd (University of California, Berkeley). The com­ Street New York, New York, from Wednesday, plete list of speakers with the titles of their ad­ April is, through Saturday, April 21, 1973. dresses will appear in the April issue of these By invitation of the Committee to Select c){oticei) • Hour Speakers for Eastern Sectional Meetings, there will be four one-hour addresses. Profes­ REGISTRATION sor Clifford J. Earle, Jr., of Cornell Univer­ The registration desk will be located in the sity will present an address entitled "Some re­ Key Room of the Biltmore Hotel on the nineteenth cent results in Teichmiiller theory." Professor floor adjacent to the Grand Ballroom. The desk Teruhisa Matsusaka of Brandeis University will will be open from 8:30 a.m. to 4:30p.m. on present an address entitled 11 0n global deforma­ Wednesday, April 18, through Friday, April 20; tions of algebraic varieties." Professor Paul ,A. and from 8:30 a.m. to 3:30p.m. on Saturday, Schweitzer of the Pontificia Universidade Cato­ April 21. lica, Rio de Janeiro, will speak on "Vector The registration fees for the meeting are fields with no closed trajectories." The title of as follows: the address to be given by Professor W. Gilbert Strang of the Massachusetts Institute of Tech­ Member $3 nology will appear in the April c){oticei) . Student and unemployed 1 By invitation of the same committee, there Nonmember 5 will be a special session of invited twenty-minute ACCOMMODATIONS papers in Hopf Algebras which will be arranged by Professor Richard G. Larson, University of Persons intending to stay at the Biltmore Hotel Illinois at Chicago Circle, and Professor Earl J. should make their own reservations with the Taft, Rutgers University. hotel. A reservation form and a listing of Sessions for contributed ten-minute papers room rates will be found on the last page of these will be scheduled in the morning and afternoon c){oticei). The deadline for receipt of res­ ervations is on Friday and Saturday. No provision will be April 11, 1973. made for late papers. Abstracts should be sub­ TRAVEL mitted to the American Mathematical Society, The Biltmore Hotel is located on P. 0. Box 6248, Providence, Rhode Island Madison Avenue at 43rd Street on the east side of New 02904 so as to arrive prior to the deadline of York City. Walkways to Grand Central Station 26, 1973. Febr~ary are located under the hotel and signs are posted The Council of the Society will meet on Fri­ directing persons to the lobby of the hotel. day, April 20, 1973, at an hour to be announced. Those arriving by bus may take the Inde­ pendent Subway System from the Port Authority SYMPOSIUM ON THE COMPLEXITY OF REAL Bus Terminal. There is shuttle bus service COMPUTATIONAL PROCESSES from LaGuardia and Kennedy Airports directly to Grand Central Station. Starters can direct With the anticipated support of the Nation­ participants to the correct bus. al Science Foundation, a symposium on The Air passengers arriving at Newark Airport Complexity of Real Computational Processes is can take a shuttle bus to the West Side Terminal to be held on Wednesday and Thursday, April 18 and take either a subway or taxi to the hotel. and 19. The topic of the symposium was select­ Those arriving by car will find many park­ ed by the AMS-SIAM Committee on Applied ing facilities in the neighborhood in addition to Mathematics whose members are Donald G. M. those at the hotel. Parking service can be ar­ Anderson, Hirsh G. Cohen, Joaquin B. Diaz, ranged through the hotel doorman at a cost of Harold Grad, Stanislaw M. Ulam, and Richard $6 for a 24-hour period. There will be an ad­ S. Varga (chairman). The Organizing Commit­ ditional charge for extra pickup and delivery tee includes Stephen Cook, John Hopcroft, service if it is required. The parking fee is sub­ Richard M. Karp (chairman), and Shmuel Wino­ ject to New York City taxes. grad. The tentative program is set up to include MAIL ADDRESS eight one-hour talks and four half-hour talks. Registrants at the meeting may receive The list of speakers includes St~ Aanderaa mail addressed in care of the American Mathe­ (IBM T. J. Watson Research Center), Michael matical Society, The Biltmore Hotel, Madison Fischer (Massachusetts Institute of Technology), Avenue at 43rd Street, New York, New York Juris Hartmanis (Cornell University), Albert 10017. Meyer (Massachusetts Institute of Technology), Walter H. Gottschalk Michael Paterson (University of Warwick, Eng­ Associate Secretary land), Michael Rabin (Hebrew University, Israel), Middletown, Connecticut

71 The Seven Hundred Fourth Meeting Northwestern University Evanston, Illinois April27 - 28, 1973

The seven hundred fourth meeting of the be held all day Friday and probably part of Sat­ American Mathematical Society will be held at urday; the tentative list of speakers includes Northwestern University, Evanston, Illinois, on John E. Gilbert, Henry Helson, Richard A. Hunt, April 27 and 28, 1973. Northwestern University Robert P. Kaafman, 0. Carruth McGehee, is located near Lake Michigan, about twelve Donald E. Ramirez, and Sadihiro Saeki. Profes­ miles north of downtown Chicago. All sessions sor Kenneth R. Mount of Northwestern Univer­ will be held in the Norris Center, the new cam­ sity is arranging a special session for Friday pus center of Northwestern University. on Singularities of Varieties and Mappings; the By invitation of the Committee to Select tentative list of speakers includes Shreeram Hour Speakers for Western Sectional Meetings, Abhyankar, Joseph Lipman, Tzuong-Tsieng Moh, there will be four one-hour addresses. Professor Peter P. Orlik, and Joel L. Roberts. Professor Howard A. Osborn of the University of illinois at Steven Orey of the University of Minnesota is Urbana-Champaign will speak on Friday, April arranging a special session on Sample Functions 27, at 11 :00 a. m. ; his topic will be "Some dif­ of Stochastic Processes for Saturday morning; ferential geometry in PL." Professor Norman the tentative list of speakers includes Bert E. Blackburn of the University of nlinois at Chicago Fristedt, Joseph Horowitz, Robert P. Kaufman, Circle will address the Society on Friday, April Michael B. Marcus, and Michael Wichura. 27, at 1:45 p.m.; his subject will be "Theorems There will be three of the so-called in­ on counting subgroups of finite p-groups." Pro­ formal sessions. Professor Richard A. Askey fessor Alexandra Ionescu-Tulcea of Northwest­ of the University of Wisconsin will conduct an ern University will speak on Saturday, April 28, informal session Saturday afternoon on Hand­ at 11:00 a.m.; the title of her talk will be "On books of Special Functions. Professor Wolfgang measurability, pointwise convergence, and com­ R. G. Haken of the University of Illinois at Ur­ pactness." Professor Carl M. Pearcy of the bana-Champaign will conduct an informal ses­ University of Michigan will address the Society sion Saturday afternoon on the Four-Color Prob­ on Saturday, April 28, at 1:45 p.m. ; his topic lem. Professor Kenneth R. Mount of Northwest­ will be "Quasitriangular operators and the in­ ern University will conduct an informal session variant subspace problem: some recent pro­ Friday on Singularities of Varieties and Map­ gress." pings; this will be mixed in with the special ses­ There will be sessions for contributed pa­ sion of twenty-minute papers on the same sub­ pers on both Friday and Saturday, April 27 and ject, the two enterprises taking place during 28. Abstracts should be submitted to the Amer­ alternate half-hours. ican Mathematical Society, P.O. Box 6248, ACCOMMODATIONS Providence, Rhode Island 02904, so as to arrive prior to the deadline of February 26, 1973. Thost The following hotels and motels are holding having time preferences for the presentation of blocks of rooms. Those planning to stay in one their paper should indicate them on their ab­ of these should make reservations directly. stracts. There will be a session for late papers if one is needed, but late papers will not be listed in the printed program of the meeting. HOWARD JOHNSON• S There will be five special sessions of se­ 9333 Skokie Boulevard lected twenty-minute papers. Professor Richard Skokie, Illinois 60077 A. Askey of the University of Wisconsin is ar­ Phone: (312) 679-4200 ranging one such session for Friday afternoon (30-minute drive) and Saturday morning on the subject of Special (Reservations by March 31) Functions; the tentative list of speakers includes Single $17 - $19 Richard A. Askey, Leonard Carlitz, Loyal Durand Ill, Charles F. Dunkl, George Gasper, Jr., Walter Gautschi, Willard Miller, Jr., and HYATT HOUSE HOTEL Frank W. J. Oliver. Professor George K. Fran­ 4500 Touhy Avenue cis of the University of nlinois at Urbana-Cham­ Lincolnwood, Illinois 60646 paign is arranging a special session on Closed Phone: (312) 677-5400 Curves on Surfaces and in Space, to be held Fri­ (30-minute drive) day morning, Friday afternoon, and Saturday (Reservations by April 12) morning; the tentative list of speakers includes Single $26 Keith D. Bailey, Thomas F. Banchoff, S. Blank, George K. Francis, Benjamin R. Halpern, Mor­ ORRINGTON HOTEL ris L. Marx, Michael Menn, Victor T. Norton, 1710 Orrington Avenue Jr., W.illiam F. Pohl, Charles J. Titus, and Evanston, Illinois 60201 James H. White. Professor Colin C. Graham of Phone: (312) 864-8700 Northwestern University is arranging a special (walking distance) session on Commutative Harmonic Analysis, to Single $10 - $27

72 SHERIDAN -CHASE MOTEL FOOD SERVICE 7300 North Sheridan A cafeteria will be open in the Norris Cen­ Chicago, nlinois 60626 ter during the meeting. Phone: (312) 761-5750 (15-minute drive) Single $14 - $16 - $18 TRAVEL

Other accommodations in the area are the fol­ Amtrak offers direct service from many lowing: points to Union Station in Chicago. Those coming by air should use O•Hare International Airport. HOMESTEAD HOTEL The Evanston Airport Bus leaves O•Hare Airport 1625 Hinman Avenue forty-five minutes after the hour and takes pas­ Evanston, lllinois 60201 sengers to the Orrington Hotel. Those coming by Phone: (312) 475-3300 car should use the Dempster Street Exit of the (walking distance) Edens Expressway.

LIBRARY PLAZA HOTEL 1637 Orrington Avenue Evanston, nlinois 60201 Paul T. Bateman Phone: (312) 864-8000 Associate Secretary (walking distance) Urbana, Illinois

The Seven Hundred Fifth Meeting Western Washington State College Bellingham, Washington June 16, 1973

The seven hundred fifth meeting of the Sessions for contributed papers will be held American Mathematical Society will be held at on Saturday. Abstracts should be submitted to Western Washington State College in Bellingham, the American Mathematical Society, P. 0. Box Washington, on Saturday, June 16, 1973. The 6248, Providence, Rhode Island 02904, so as to Mathematical Association of America and the arrive prior to the deadline of May 3, 1973. Late Society for Industrial and Applied Mathematics papers will be accepted for presentation at the will hold Northwest Sectional Meetings in con­ meeting, but will not appear in the printed pro­ junction with this meeting of the Society. The gram of the meeting. Information on travel and Association will have sessions on Friday and accommodations will appear in the April c/'foticRiJ. Saturday, June 15 and 16. Aspects of the pro­ gram will emphasize numerical analysis and its role in the mathematics curriculum. There will be two invited one-hour addres­ Kenneth A. Ross ses. The names of the speakers and titles of Associate Secretary their addresses will be given in the April issue Eugene, Oregon of these cJ/oticRiJ.

Symposium on Some Mathematical Questions in Biology Mexico, D. F. June 30, 1973

The seventh annual symposium on Some meetings of the American Association for the Mathematical Questions in Biology will be held Advancement of Science and the Consejo Nacion­ on June 30, 1973, at the Unidad de Congresos al de Ciencia y Technologia. It is anticipated del Centro Medico Nacional, Avenida Cuauhte­ that the symposium will be supported by the Na­ moc 330, Mexico, D. F. This symposium will be tional Science Foundation. Registration and hotel cosponsored by the American Mathematical So­ arrangements will be announced in SCIENCE. ciety and the Society for Industrial and Applied The program is being arranged by Hans J. Mathematics and is being held in conjunction with Bremermann, Hirsh Cohen, Jack D. Cowan, and

73 Murray Gerstenhaber, all of whom are members of Chicago), and Jack D. Cowan (University of of the AMS-SIAM Committee on Mathematics in Chicago). Titles of the lectures and the time they the Life Sciences. The speakers at the sympo­ will be given will appear in the April issue of sium will include Rene Thom (Institut des Hautes these C}/oticti). Etudes, Paris), R. May (Princeton University), Stephen Smale (University of California, Berke­ ley), S. Papert (Massachusetts Institute of Tech­ Jack D. Cowan nology), L. Wolpert (Middlesex Hospital Medical Chairman School, London), A. D. J. Robertson (University Chicago, Illinois

1973 Summer Institute on Di:tJerential Geometry

The twentieth Summer Research Institute Lawson, "Minimal varieties"; James Simons, of the American Mathematical Society will be "Geometry of characteristic classes" ; I. M. devoted to the topic of Differential Geometry, Singer "Geometry of the spectra. II''. In addi­ and will be held at Stanford University, Stanford, tion to these series of lectures, there will be California, for a period of three weeks from seminars devoted to talks on current research, July 30 through August 17, 1973. The Organizing each under the supervision of an invited chair­ Committee consists of Professors Raoul H. Bott, man. The seminars scheduled so far are "Sub­ Eugenio Calabi, S. S. Chern (cochairman), Leon manifolds," "Differential equations related to W. Green, Shoshichi Kobayashi, Tilla K. Milnor, differential geometry," "Integral geometry," Barrett O•Neill, Robert Osserman (cochairman), "Complex differential geometry." Additional James Simons, and I. M. Singer. It is expected seminars will be spontaneously generated, de­ that the institute will be supported by a grant pending on the interest of the participants. from the National Science Foundation. The main Inforn;J.ation on travel and accommodations emphasis will be on global aspects of differential will be available in subsequent announcements. geometry. Funds for participant support will be lim­ The scientific program of the institute will ited, and it is hoped that a number of partici­ consist first of all of series of lectures, each pants will find their own sources of support. including about two to four survey talks of a The institute is open to all mathematicians spe­ general nature devoted to various subjects en­ cializing in differential geometry and related compassed by the institute. The speakers with topics, and to advanced graduate students in the titles of their lectures are Marcel Berger, this field. Those wishing to participate are in­ "Geometry of the spectra. I"; Raoul Bott, "Fo­ vited to write to Dr. Gordon L. Walker, Amer­ liations and Gelfand-Fuks cohomology"; Robert ican Mathematical Society, P. 0. Box 6248, Geroch, "General theory of relativity"; Phillip Providence, Rhode Island 02904. Recent Ph. D.'s A. Griffiths, "Differential geometry and com­ and advanced graduate students who wish to be plex function theory"; D. Gromoll, "Curvature considered for support should write before and topology in Riemannian geometry" ; Blaine March 1, 1973.

Conference on the Influence of Computing on Mathematical Research and Education Contingent upon the support of the National panelists; thirty or forty persons who will pre­ Science Foundation, the American Mathematical sent papers (selected and refereed in advance); Society and the Mathematical Association of and approximately three or four hundred other America will sponsor a Conference on the Influ­ participants. ence of Computing on Mathematical Research The topic of the conference was selected by and Education. The conference will be held in the AMS-SIAM Committee on Applied Mathemat­ mid-August, just preceding the summer meeting ics, composed of Donald G. M. Anderson, Hirsh of the two organizations, and the site has not yet G. Cohen, Joaquin B. Diaz, Harold Grad, Stan­ been determined. Further announcements will islaw M. Ulam, and Richard s. Varga (chair­ appear in the c}/oticti) and the MONTHLY when man), in conjunction with the officers of the all of the details of the meeting have been de­ American Mathematical Society and the Mathe­ cided upon. matical Association of America. The Organizing The conference will be devoted to two major Committee includes William S. Dorn (University areas of interest: (1) recent developments in of Denver), Stephen J. Garland (Dartmouth Col­ mathematical research in which modern compu­ lege), Thomas E. Hull (University of Toronto), tational methods and concepts play a critical Donald E. Knuth (Stanford University), and role, and (2) implications for instruction in Joseph P. LaSalle (Brown University), chair­ mathematics stemming from the development of man. Anyone wishing to participate in this con­ modern techniques of calculation and information ference should request further information as it processing. It is expected that the program will becomes available from Meeting Arrangements consist of eight to ten invited speakers who will Department, American Mathematical Society, present major addresses; four or five invited P. 0. Box 6248, Providence, Rhode Island 02904.

74 REVIEWING CHARGE FOR MATHEMATICAL REVIEWS

In the August 1972 issue of the cJfotiai) , fraction of the cost of the services provided by pages 242-243, Professor Nathan Jacobson, then MATHEMATICAL REVIEWS. This charge will be president of the American Mathematical Society, added to the publication charges currently as­ published an open letter to the membership of the sessed, beginning with articles accepted in July Society discussing some of the difficulties faced 1973. The officers of the Society hope that the by MATHEMATICAL REVIEWS in the presence authors of mathematical papers published in of rising production costs and the growing body other journals will feel it appropriate to assist of mathematical literature. The Trustees of the MATHEMATICAL REVIEWS in carrying out its Society have appointed a committee, under the work in the coverage of world literature by mak­ chairmanship of Professor Ralph P. Boas, to ing voluntary contributions corresponding to the study ways of alleviating this financial crisis. reviewing charge which is being introduced for The committee has been working with the articles in AMS journals. The reviewing charge MATHEMATICAL REVIEWS Editorial Commit­ for articles appearing in AMS journals will be tee, and among the things proposed has been the collected at the same time and in the same way recent decision to make individual sections of that the publication charges are currently col­ the MATHEMATICAL REVIEWS available by lected. Individual authors are not held respon­ subscription. In addition, experiments are to be sible for these charges, but their institutions conducted on economies in production, including are requested to honor them either by a charge the use of less expensive paper and binding. against institutional membership dues or by the At the meeting in Dallas, Texas, in Jan­ transferring of funds. Such expenses are ordi­ uary 1973, the Council of the Society voted to narily covered by research grants in the case of introduce a reviewing charge of $25 for each papers produced under such grants and contracts. article published in AMS journals to defray a

NEW AMS PUBLICATIONS

ADMINISTRATIVE DIRECTORY -1973 mathematics. A special section contains a list of 161 pages; list price $5; member price $1 approximately 2, 500 names, addresses, and To order, please specify ADM telephone numbers of chairmen of departments of mathematics and heads of nonacademic re­ The Mathematical Sciences Administrative search groups. Directory-1973 lists the officers, editors, and The Administrative Directory is an invalu­ committee chairmen of twenty-two professional able reference book for mathematical scientists, societies for mathematical scientists; addresses, professional societies, editors, and administra­ personnel, and telephone numbers of ten govern­ tive personnel of universities and other institu­ ment agencies; and the names and addresses of tions. the editors of approximately eighty journals of

75 QUERIES edited by Wend ell H. Fleming

The QUERIES column is published in each issue of these cJVoliui) This column welcomes questions from AMS members regarding mathematical matters such as details of, or references to, vaguely remembered theorems, sources of exposition of folk theorems, or the state of current knowledge concerning published conjectures. When appropriate, replies from readers will be edited into a definitive composite answer and published in a subsequent column. All answers received to QUERIES will ultimately be forwarded to the questioner. Consequently, all items submitted for consideration for possible publication in this column should include the name and complete mailing address of the person who is to receive the replies. The queries themselves, and responses to such queries, should be addressed to Professor Wendell H. Fleming, American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02904.

11, Richard J. Bonneau (Project MAC, Room 15, K. K. Lee (Department of Physics, Syracuse 839, Massachusetts Institute of Technology, University, Syracuse, New York 13210). I have Cambridge, Massachusetts 02139), Let p be an heard the following statement: "For any non­ odd prime number. We know that there are compact, connected, orientable, differentiable primitive roots modulo p, Let r be the minimum 4-manifold M, if w2 (M) = 0 then M is paral­ such primitive root. We also know that if r is lelizable," Any references, proof, and exam­ any primitive root satisfying !:P-1:;1:1 (modiiio ples, if any) please. p2), then r is a primitive root modulo pn for alln>O.- Is it known whether the minimum positive root, r, modulo p always satisfies the above RESPONSES TO QUERIES "in-congruence" and, hence, is a primitive root Several responses have been received to modulo pn for all n > 0? (I conjecture that this queries published in recent issues of these is true, based on numerous computations.) C/{otiu.i). The following summarizes information therein, arranged according to query number 12. A, Wilansky (University of Reading, Read­ and name of the questioner. ing, England, and Lehigh University, Bethle­ hem, Pennsylvania 18015). Let X be an infinite 1. (Boas, Oct. 1972) The following references dimensional Banach space. Must there exist a should be added to those listed: sequence {fnl in X' with llfnll = 1, fn-+0 c. Z, Semadeni, Generalizations of Bohr• s weak*? (Remark. It would be sufficient to find theorem on Fourier series with independent a dense subspace Y of X such that Y has a characters, Studia Math. 23 (1963), 159-179, larger complete norm. ) MR 29 #6256, 13, Alan H. Schoenfeld (Department of Mathe­ 3. (Polek, Oct. 1972) The following expository matics, Stanford University, Stanford, Cali­ article deals with algebraic deformation theory. fornia 94305) Suppose IL is a finite nonatomic It includes bibliographical notes and references. Borel measure with compact support in i-2. Is a. Albert Nijenhuis, On a class of common there a vector V and a line L = !kv: kER I properties of some different types of algebras, such that, for every hyperplane M in .£2. Nieuw Archie£ voor Wiskunde (3) 17 (1969), M 1. L implies IL(M) = 0? Is there an ortho­ 17-46; Part II, ibid. 87-108, MR 41 #3302; normal basis {Vi: i = 1, 2, ••. j of .£2 such 41 # 5466b, that every line Li = {kv1j has this property? 4. (Moler, Nov, 1972) It has been suggested Analogous properties hold in Rn. that the journals Photochemistry and Photo­ 14. K. K, Lee (Department of Physics, Syracuse biology and Biochemica et Biophysica Acta University, Syracuse, New York 13210). It is would be good places to look for mathematical well known that for every compact, connected, treatments of photosynthesis and other such orientable, differential 3-manifold N, N is biological processes. Anyone wishing to work parallelizable, Can this theorem be extended to in this area should devote considerable time to the noncompact case? References, proof and understanding the biological process of which examples (or- counterexamples, if any) please. the mathematics is a model.

76 SPECIAL MEETINGS INFORMATION CENTER

The purpose of this center is to maintain a file on prospective symposia, colloquia, institutes, seminars, special years, meetings of other associations, and to notify the organizers if conflicts in subject matter, dates or geographical area become apparent. A first announcement will be published in the c}/oticei) if it contains a call for papers, place, date, and subject (when applicable); a second announcement must contain reasonably complete details of the meeting in order for it to be published. Information on the pre­ preliminary planning will be stored in the files, and will be available to anyone desiring information on prospective conferences. All communications on special meetings should be sent to the Special Meetings Information Center of the American Mathematical Society.

March 1973 March 26 - April 7, 1973 COLLOQUIUM ON CARDIOVASCULAR AND NATO ADVANCED STUDY INSTITUTE ON RESPIRATORY MECHANISMS FOUNDATIONS OF QUANTUM MECHANICS AND London, England ORDERED LINEAR SPACES Information: Dr. D. G. Caro, Physiological F1ow Studies Marburg, Federal Republic of Germany Unit, Department of Aeronautics, Imperial College, Speakers: C. M. Edwards (Oxford), A. J. Ellis (Swan­ Prince Consort Road, London, SW7 2BY, England sea), G. Ludwig (Marburg), G. M. Prosperi (Milan), H. H. Schafer (Titbtngen) March 2-3, 1973 Information: Professors Alois Hartkamper and Bolger SYMPOSIUM ON DIFFERENTIAL EQUATIONS Neumann, Fachbereich Physik der PhUipps-Universitat, Arkansas State University, State University, Arkansas 355 Marburg, Renthof 7, Federal Republic of Germany Speakers: Hour speakers include F. Brauer, A. M. Fink, R. K. Miller, W. T. Reid; half-hour speakers include C. D. Ahlbrandt, J. S. Bradley, S. B. Eliason, K. W. March 28-30, 1973 Schrader; and several quarter-hour speakers . JOHN H. BARRETT MEMORIAL LECTURES Information: Professor J. L. Linnstaedter or J. B. University of Tennessee, Knoxville, Tennessee Bennett, Drawer F, State University, Arkansas 72467 Speaker: W. N. Everitt, University of Dundee, ·Scotland, "Singular boundary value problems for ordinary differen­ March 5 - May 25, 1973 tial equations and related topics" VON KARMAN INSTITUTE FOR FLUID DYNAMICS Abstracts: For 30-minute contributed papers on March Information: von Karman Institute for F1uid Dynamics, 29; submit to Professor Bradley Rhode-Sant-Genese, Belgium Information: Professor JohnS. Bradley, Department of March 5-9 Mathematics, University of Tennessee, Knoxville, Ten­ Advances in numerical fluid dynamics nessee 37916 Director: E. Krause March 19-23 April 2, 1973 Advances in turbulent shear flows CONFERENCE ON MATHEMATICAL ANALYSIS OF Director: B. E. Richards FUNDAMENTAL BIOLOGICAL PHENOMENA April 2-6 Barbizon-Plaza Hotel, New York Helicopter aerodynamics and dynamics Sponsor: The New York Academy of Sciences Director: P. F. Yaggy Program: Sessions on neural biology, development and May 14-18 evolution, molecular and cellular dynamics, and be­ Atmospheric turbulence and diffusion havioral dynamics Director: W. Frost Information: ·The New York Academy of Sciences, 2 East May 21-25 63rd Street, New York, New York 10021 Transonic flows in turbomachinery Director: J. Chauvin April 7, 1973 THIRD SEMI-ANNUAL ILLINOIS NUMBER THEORY March 22-24, 1973 CONFERENCE CONFERENCE ON LATTICE THEORY Ulinois State University, Normal, lllinois Shamrock Hilton Hotel, Houston, Texas Program: Twenty-minute papers by illinois number Speakers: Kirby A. Baker, G. Bruns, George A. Grat­ theorists on current research interests zer, K. H. Hofmann, Bjami Jonsson, H. Lakser, W. A. Abstracts: Should be submitted by March 1, 1973 Lampe, J. D. Lawson, R. N. McKenzie, Gian-Carlo Information: Professor L. C. Eggan, Department of Rota, Jurgen Schmidt Mathematics, Ulinois State University, Normal, illinois Contributed papers: Participants -invited to submit papers 61761 for inclusion in two sessions for contributed papers Information: Professors D. R. Brown, M. Friedberg, April 18, 1973 or Jurgen Schmidt, Department of Mathematics, Univer­ TWENTIETH ISRAEL CONFERENCE ON sity of Houston, Houston, Texas 77004 THEORETICAL AND APPLIED MECHANICS Tel-Aviv, Israel March 26-30, 1973 Information: Professor z. Hashin, Department of Mate­ REGIONAL CONFERENCE ON LINEAR GROUPS rial Engineering, Technion City, Haifa, Israel Arizona state University, Tempe, Arizona Program: Lecture series by 0. Timothy O•Meara, Uni­ May 7-10, 1973 versity of Notre Dame, on "Automorphisms of linear CONFERENCE ON SOME MATHEMATICAL PROBLEMS groups"; contributed papers IN BIOLOGY Support: NSF (for limited number of invited participants) University of Victoria, British Columbia, Canada Information: Professor Ronald Jacobowitz or Thomas L. Speakers: J. J. Blum (Duke University), C. W. Clark Sherman, Department of Mathematics, Arizona State (University of British Columbia), J. D. Cowan (Univer­ University, Tempe, Arizona 85281 sity of Chicago), A. Friedman (Northwestern University),

77 C. S. Holling (University of British Columbia), N. D. June 3-8, 1973 Kazarinoff (SUNY at Buffalo), D. A. Ludwig (New York REGIONAL CONFERENCE ON EXTERIOR INITIAL­ University), z. A. Melzak (University of British Colum­ BOUNDARY VALUE PROBLEMS FOR HYPERBOLIC bia), E. C. Pielou (Dalhousie University), A. T. Winfree PARTIAL DIFFERENTIAL EQUATIONS (Purdue University) State University of New York at Buffalo, Amherst, Abstracts: For twenty-minute papers; deadline March 15, New York 1973 Program: Cathleen S. Morawetz, principal speaker; Information and applications: Professor P. van den sessions for contributed papers Driessche, Department of Mathematics, University of Support: NSF (pending); travel and subsistence allowances Victoria, Victoria, British Columbia, Canada for twenty-five participants Information: Professor Nicholas D. Kazarinoff, Depart­ May 7-11, 1973 ment of Mathematics, State University of New York at FIFTH IFIP CONFERENCE ON OPTIMIZATION Buffalo, 4246 Ridge Lea Road, Amherst, New York 14226 TECHNIQUES Rome, Italy June 3-30, 1973 Information: Professor Luigi Grippo, Istituto di Auto­ SEMINAR ON FIXED POINT THEORY AND ITS matica, Universita di Roma, 18 via Eudossiana, APPLICATIONS TO ANALYSIS I-00184 Rome, Italy Universite de Montreal, Montreal, Quebec, Canada Speakers: R. Bott, H. Brezis, F. Browder, A. Dold, May 16-18, 1973 E. Fadell, K. Geba, A. Granas, S. Kakutani, Ky Fan, SYMPOSIUM ON COMPLEXITY OF SEQUENTIAL AND J. Leray, L. Nirenberg, Shih Weishu PARALLEL NUMERICAL ALGORITHMS Program: Topological fixed point theorems, fixed points Computer Science Department, Carnegie-Mellon Uni­ in functional analysis, applications to partial differential versity, Pittsburgh, Pennsylvania equations, multivalued maps (applications to optimal Program: Fourteen invited speakers on topics such as control theory), common fixed points for families of the influence of machine organization on algorithms, the transformations, fixed points in global analysis· influence of specific problems on machine organization, Information and applications for financial assistance: algebraic and analytic computational complexity, se­ Seminaire de Mathematiques Superieures, Uni versite quential and parallel algorithms, and empirical results de Montreal, c. P. 6128, Montreal101, Quebec, Canada on algebraic efficiency. Contributed papers are solicited. ~:Office of Naval Research June 11-29, 1973 :rn.rorm:a:tion: Professor J. F. Traub, Computer Science NSF SHORT COURSE IN CATEGORY THEORY (FOR Department, Carnegie-Mellon University, Pittsburgh, TEACHERS IN FOUR-YEAR INSTITUTIONS) Pennsylvania 15213 Colgate University, Hamilton, New York Application deadline: March 1, 1973 May 21-25, 1973 Information: Dr. Malcolm W. Pownall, Department of INTERNATIONAL CONFERENCE ON DISCONTINUOUS Mathematics, Colgate University, Hamilton, New York GROUPS AND RIEMANN SURFACES 13346 University of Maryland, College Park, Maryland .• Topics: Fuchsian groups, Kleinian groups, Teichmiiller June 13-15, 1973 spaces, quasi-conformal mappings, Jacobian varieties CONFERENCE ON THE APPLICATION OF UNDER­ Speakers: L. Ahlfors, A. Beardon, L. Bers, J. Bir­ GRADUATE MATHEMATICS IN THE LIFE, MANAGE­ man, C. Earle, L. Ehrenpreis, H. Farkas, F. Gehring, RIAL, SOCIAL, AND ENGINEERING SCIENCES L. Greenberg, J. Lehner, A. Macbeath, A. Marden, Georgia Institute of Technology, Atlanta, Georgia B. Maskit, H. Royden, and others as yet unannounced Sponsors: Committee on the Undergraduate Program in Information: Professor Leon Greenberg, Department of Mathematics, Georgia Institute of Technology Mathematics, University of Maryland, College Park, Program: Invited speakers will survey latest use of Maryland 20742 undergraduate mathematics in their disciplines and par­ ticipate with mathematicians in panel discussions on May 29 - June 2, 1973 desirable directions for undergraduate mathematics INTERNATIONAL SYMPOSIUM ON DIFFERENTIAL, education. INTEGRAL, AND FUNCTIONAL EQUATIONS Information and registration forms: Professor Gunter H. illed, Yugoslavia Meyer, School of Mathematics, Georgia Institute of Information: Professor Josef Plemolj, Institut za Mate­ Technology, Atlanta, Georgia 30332 matika, P. P. 543, Jandransa 19, Ljubljana, Yugoslavia May 30 - June 1, 1973 June 18 - August, 1973 EASTERN REGIONAL MEETING OF THE INSTITUTE OF SUMMER INSTITUTE FOR TEACHERS OF MATHEMATICAL STATISTICS MATHEMATICS IN DEVELOPING COLLEGES (Jointly with the American Statistical Association and the University of Montana, Missoula, Montana Biometric Society) Application deadline: April 9, 1973 Ithaca, New York Information: Dr. Gloria C. Hewitt, Department of Speakers: Naresh Jain (University of Ulinois, University Mathematics, University of Montana, Missoula, Montana of Minnesota), Simeon M. Berman (New York University), 59801 Clifford Qualls (University of New Mexico), Milton Sobel June 25-29, 1973 (University of Minnesota), Shanti S. Gupta (Purdue Uni­ INTERNATIONAL CONFERENCE ON BANACH SPACES versity), Khursheed Alam (Clemson University), B. Wabash College, Crawfordsville, Indiana Kleiner (Bell Laboratories), C. Daniel (private consul­ Program: Approximately twenty invited talks tant), K. C. S. Pillai (Purdue University), G. S. Watson Sponsors: University of illinois, Indiana University, (Princeton University), L. D. Brown (Cornell University); Purdue University, Wabash College, and the National presidential address featuring George E. P. Box, G. C. Science Foundation Tiao, W. J. Hamming lnformation: Professor Earl R. Berkson, Department of Abstracts: Deadline March 19, 1973; send to Executive Mathematics, University of Ulinois at Urbana-Champaign, Secretary, Institute of Mathematical Statistics, 426A Urbana, Ulinois 61801 Wells Hall, Michigan State University, East Lansing, ·Michigan 48823 June 25-30, 1973 Information: Professor R. J. Serfling, Department of CONFERENCE ON INFINITE AND FINITE SETS IN Statistics, Florida state University, Tallahassee, Flori­ HONOR OF PAUL ERDOS da 32306 (program committee); Professor DavidS. Ador­ Keszthely, Hungary no, Division of Business Administration, Ithaca College, Information: Professor A. Hajnal, Mathematical Institute, Ithaca, New York 14850 (local arrangements) Hungarian Academy of Sciences, Budapest, Hungary

78 July 2-4, 1973 September 3-5, 1973 COLLOQUIUM ON DYNAMICS OF RAREFIED GASES COLLOQUIUM ON GYRODYNAMICS Gottingen, Federal Republic of Germany Louvain, Belgium Information: Professor W. Wuest, DFVLR-AVA, Information: Professor F. Buckens, Institut de Meca­ 3400 Gtlttingen, Bunsenstrasse 10, Federal Republic of nique et Mathematiques Appliquees, Universite Catho­ Germany lique de Louvain, 300 Celestignenlaan, Heverlee, Belgium July 2-27. 1973 SUMMER INSTITUTE ON MATHEMATICAL MODELS September 3-6, 1973 AND STOCHASTIC PROCESSES IN ENVIRONMENTAL COLLOQUIUM ON TRANSONIC AERODYNAMICS SCIENCE Stockholm, Sweden Cornell University, Ithaca, New York Information: Dr. G. Drougge, Aerodynamics Depart­ Support: NSF ment, FAA, P. 0. Box 11021, S-16111 Bromma 11, Participation: Limited to twenty-five college teachers Sweden Speakers: M. Brown, W. F. Lucas, N. U. Prabhu of Cornell University; F. S. Roberts of Rutgers University September 3-15, 1973 Information: Professor W. F. Lucas, Upson Hall, Cor­ INTERNATIONAL MEETING ON COMBINATORIAL nell University, Ithaca, New York 14850 THEORY Rome, Italy July 9-27. 1973 Sponsors: Accademia Nazionale dei Lincei and the CONFERENCE ON RELIABILITY AND BIOMETRY American Mathematical Society Florida State University, Tallahassee, Florida Support: In part by NATO as an Advanced Study Institute Speakers: R. E. Barlow, Z. W. Birnbaum, R. Cornell, Participant support: Available for a few outstanding stu­ H. A. David, J. D. Esary, F. Grubbs, S. Gupta, D. G. dents who do not have any other source of support Hoel, P. A. W. Lewis, G. Marsaglia, A. W. Marshall, Information: Secretariat, Accademia Nazionale dei Lin­ J. Neyman, M. Zelen cei, Via della Lungara, 10-00165, Roma, Italy, or Pro­ Sponsors: Air Force Office of Scientific Research, Na­ fessor Marshall Hall, Jr. (representative of the AMS), tionai Cancer Institute, Florida State University California Institute of Technology, Pasadena, California Participant support: Limited support for biometry grad­ 91109 uate students Information: Professor Frank Proschan, Conference October 1973 Director, Department of Statistics, Florida State Uni­ COLLOQUIUM ON THE MECHANICAL PROPERTIES OF versity, Tallahassee, Florida 32306 INTERFACES BETWEEN TWO-PHASE FLUIDS Palermo, Italy July 23-27. 1973 Information: Professor G. Marrucci, Facolta di SYMPOSIUM ON ALGEBRAIC AND GEOMETRIC Ingegneria, Universita di Palermo, Pale:rmo, Italy TOPOLOGY University of California, Santa Barbara, California Ootober 15-17, 1973 Information: Professor Kenneth C. Millett, Department FOURTEENTH ANNUAL SYMPOSIUM ON SWITCHING of Mathematics, University of California, Santa Barbara, AND AUTOMATA THEORY California 93106 University of Iowa, Iowa City, Iowa Sponsor: Switching and Automata Theory Committee of August 22-24, 1973 the IEEE Computer Group and the University of Iowa PROGRAMMING LANGUAGE CONGRESS 73 Abstracts: Seven copies by May 7, 1973, to Dr. H. R. Northern Europe University Computing Center, Lyngby, Strong, IBM Thomas J. Watson Research Center, P.O. Denmark Box 218, Yorktown Heights, New York 10598 Organizing Committee: Hans J orgen Helms (president), Information: Dr. Sheldon B. Akers, Secretary, IEEE Philip S. Abrams, Frank Anscombe, Giovanni Bartoli, Technic3l Committee on Switching and Automata Theory, Per Gjerl~v Building 3, Room 223, General Electric Company, Elec­ Registration deadline: May 1, 1973 tronics Park, Syracuse, New York 13201 Information: Mrs. Helle Hornung, APL Congress 73, Br!llndby\6ster Boulevard 22, 2650 Hvidovre, Denmark

September 1973 COLLOQUIUM ON THE DYNAMICS OF MACHINE NOTES ON PREVIOUS ANNOUNCEMENTS FOUNDATIONS Bucharest, Romania Information: Professor Gh. Buzdugan, Institutul Poli­ tehnic Bucaresti, Catedra de Rezistenta materialelor I, The International Conference on Manifolds and Splaiul Independentei 313, Bucharest, Romania Related Topics in Topology, April 10-17, 1973, will take place at the Keidanren-Kaikan, not the University September 1973 of Tokyo as originally announced in the October 1972 COLLOQUIUM ON THE EXCHANGES AT THE AIR/SEA cNotitW. BOUNDARY Marseille, France Information: Professor A. Favre, Institut de Mecanique Statistique de la Turbulence, 12 Av. d. General Leclerc, The Summer Seminar in Mathematics and Behav­ Marseille 3e, France ioral Science, June 18-July 27, 1973.._ Williams College, announced in the January 1973 (j{orictJ_] is open~ to September 1973 teachers of mathematics. Support is being provided by COLLOQUIUM ON FLOWS WITH CONCENTRATED the NSF. VORTICITY Norwich, England Information: Professor N. Riley, School of Mathematics and Physics, University of East Anglia, University Village, Norwich, NOR 88C, England

79 BACKLOG OF MATHEMATICS RESEARCH JOURNALS

Information on the backlog of papers for re- the result of unusual circumstances arising in part search journals is published in the February and Au- from the refereeing system. gust issues of these cJVoticaJ with the cooperation of The observations are made from the latest the respective editorial boards. Since all columns in issue of each journal received at the Headquarters the table are not self-explanatory, we include further Offices before the deadline for the appropriate issue details on their meaning. of these c/{otica) • Waiting times are measured in Column 3. This is an estimate of the number months from receipt of manuscript in final form to of printed pages which have been accepted but are not receipt of final publication at the Headquarters Offices. necessary to maintain copy editing and printing ached- When a paper is revised, the waiting time between an ules. editor's receipt of the final revision and its publication Column 5. The first (Q1) and third (Q3 ) quar- may be much shorter than is the case otherwise, so tiles are presented to give a measure of normill dis- these figures are low to that extent. persian. They do not include misleading extremes,

1 2 3 4 5 Observed waiting Est. time for paper time in latest JOURNAL No. Approx. no. BACKLOG submitted currently published issue issues pages per to be published (in months) per year year 12/31/72 6/30/72 (in months) Q1 Med. Q3 Acta Informatica 4 388 376 0 10 *** American J. of Math. 4 1255 1042 NR* 8 15 18 21 Annals of Math. 6 1200 NR* NR* 12 11 12 14 Annals of Probability** 6 1000 0 12} 9 10 11 Annals of Statistics** 6 1100 0 9-12 Arch. Rational Mech. 23 1852 0 0 7-8 8 8 9 Canad. J. of Math. 6 200 1200 700 13 9 13 15 Comm. Math. Physics 21 1779 425 0 5-6 6 7 7 Duke Math. J. 4 800 0 0 8 6 7 8 illinois J. of Math. 4 700 700 1000 18 29 30 30 Indiana Univ. Math. J. 12 1200 300 400 8 7 8 9 Inventiones Math. 15 1181 NR* 0 6-7 6 8 10 J. Amer. Stat. Assoc. 4 1000 0 70 12.6 9 11 13 J. Assoc. for Comp. Mach. 4 NR* NR* NR* NR* 9 10 11 J. of Comp. and Sys. Sets. 6 600 300 200 10 8 9 11 J. Diff. Geometry 4 600 750 780 15 If J. Math. Physics 12 2000 NR* 0 6 7 9 18 J. Symbolic Logic 4 900 0 0 10 17 18 24 Linear Algebra and Appl. 4 NR* NR* 120 NR* 14 28 29 Math. Systems Theory 4 400 NR* NR* 12 17 17 18 Math. of Comp. 4 NR* 0 0 8 10 12 14 Math. Annalen 19 1629 261 NR* 11-12 12 16 19 Math. Zeitschrift 20 1872 0 0 7-8 8 8 10 Michigan Math. J. 4 400 120 150 12 9 11 16 Numerische Math. 9 846 826 0 10-11 9 11 19 Operations Research 6 1400 200 0 13 15 20 30 Pacific J. of Math. 12 3500 NR* NR* 18 15 18 20 Proceedings of AMS 12 3250 50 150 9-11 10 12 15 Proc. Nat•l Acad. Sci. 12 4000 0 0 2 2 3 3 Quarterly of Appl. Math. 4 620 620 600 12 13 15 17 Semigroup Forum 8 752 0 0 6-8 *** SIAM J. of Appl. Math. 8 1600 0 0 8-10 11 12 14 SIAM J. on Computing 4 500 0 0 4-6 *** SIAM J. on Control 4 800 0 0 8-10 11 14 17 SIAM J. on Math. Anal. 4 700 0 0 8-10 12 13 16 SIAM J. on Numer. Anal. 6 1200 0 0 9-11 14 16 18 SIAM Review 4 700 0 0 8-10 11 13 14 Transactions of AMS 12 6000 1000 400 10-15 15 17 19 z. Wahrscheinlichkeitstheorie 13 1092 NR* 0 11-12 9 10 12 *NR means no response was received to a request for information. **Formerly the Annals of Mathematical Statistics. ***Not computable for this journal. #No new issue received since last backlog survey.

80 LETTERS TO THE EDITOR

Editor, the cJ{oticei) Editor, the cJ{oticei) In spite of my general enthusiasm for the During the recent meeting of the Screening outgoing spirit recommended for mathematicians Committee, we, the undersigned members, noted by Dale W. Lick in your October, 1972 issue I that of the 60-odd applications for a Fulbright must, with all due respect, express my intense Award, only one had been made by a woman. We disappointment that you give publication space to are drawing this statistic to the attention of the so shallow a treatise. The denunciation of a vague mathematics community in the hopes that more establishment, the repetition of outworn euphe­ of our women colleagues will be encouraged to misms and truisms, the coarse naming of fields apply for future awards. The deadline for ap­ of endeavours {with grossly inadequate descrip­ plications for the 1974-75 awards is July 1, tions of them), and the uncritical refeyencing of 1973. Applications may be made after July 1 for particular works on uncertain application areas travel-only grants for a number of countries. where serious study is needed-how can these {Inquiries are welcome. ) For further details, help our cause? How can we face our colleagues write to: Committee on International Exchange in other fields and other countries when you so of Persons, 2101 Constitution Avenue, Washing­ represent us? ton, D. c. 20418. Some months ago you published a similar article to which I and others took sharp excep­ Carl Faith tion, but apparently your policies have not been Peter Lax affected. Ralph Phillips George Weiss H. M. Lieberstein

NEWS ITEMS AND ANNOUNCEMENTS

COLLOQUIUM LECTURES MAA FILMS One series of colloquium lectures was pre­ The Committee on Educational Media and sented at the AMS annual meeting in Dallas, the Film Projects Advisory Committee of the Texas, in January 1973: "The index of elliptic Mathematical Association of America are co­ operators" by Professor Michael F. Atiyah of operating in the collection of information on the the Institute for Advanced Study. A limited num­ quality of service provided by the distributors of ber of Professor Atiyah' s lecture notes are still MAA films. Letters requesting data on the qual­ available. Requests for copies should be accom­ ity of service have been sent to those institutions panied by a check for one dollar to cover the cost known to have rented MAA films in the past two of handling; requests should be mailed to the So­ years. Anyone who can contribute any first-hand ciety, P. 0. Box 6248, Providence, Rhode Island information on the film rental service but who 02904. Informally distributed manuscripts and has not received a questionnaire is requested to articles should be treated as a personal com­ send such data {favorable or unfavorable) to Pro­ munication and are not for library use. Reference fessor J. D. E. Konhauser, Department of Math­ to the contents in any publication should have the ematics, Macalester College, St. Paul, Minne­ prior approval of the author. sota 55105.

81 SUMMER GRADUATE COURSES

The following is a list of graduate courses being offered in the mathematical sciences during the summer of 1973. Another list will appear in the April issue of these CJVotiui) • ARIZONA UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Urbana, lllinois 61801 NORTHERN ARIZONA UNIVERSITY Application deadline: June 12 Flagstaff, Arizona 86001 Information: James W. Armstrong, Graduate Super­ Application deadline: April 30. visor, Department of Mathematics Information: Richard D. Meyer, Chairman, Depart­ ment of Mathematics, Box 5717 June 18 - August 11 Ring Theory June 11 - July 14 Foundations of Geometry Math 504 Elements of Algebraic Systems Banach Spaces Math 511 Introduction to Higher Algebra Mathematical Methods of Physics I, II Math 563 Numerical Analysis Partial Differential Equations Math 641 Point Set Topology Math 665 Ordinary Differential Equations KENTUCKY July 16 - August 18 UNIVERSITY OF LOUISVILLE Math 505 Elements of Analysis Louisville, Kentucky 40208 Math 516 Finite Dimensional Vector Spaces Application deadline: June 11 Math 531 Advanced Calculus I Information: Roger H. Geeslin, Chairman, Depart­ Math 609 Seminar in Mathematical Education ment of Mathematics

June 11 - July 27 FLORIDA 421 Theory of Numbers (M.A. T. only) FLORIDA STATE UNIVERSlTY 521 Modern Algebra I Tallahassee, Florida 32306 550 Advanced Euclidean Geometry Application deadline: June 1 633 Fields, Rings, and Ideals Information: F. W. Leysieffer, Department of Statistics MICHIGAN June 14 - August 18 STS 510-511 Statistical Procedures for the Behavioral ANDREWS UNIVERSITY Sciences I, II Berrien Springs, Michigan 49104 STS 512-513 Statistical Procedures for the Natural Application deadline: June 18 Sciences I, II Information: Harold T. Jones, Chairman, Depart­ STS 514 Computational Methods in Statistics ment of Mathematics STS 517 Applied Regression Methods STS 520 Applied Nonparametric Statistics June 18 - August 9 STS 521 Advanced Biometry Real An8iysis STS 545 Sequential Analysis Topology STS 548 Linear Statistical Models STS 552 Operations Research: Queuing Theory STS 561-562 Reliability Theory and Life Testing I, II MISSISSIPPI (offered at doubled pace) STS 647 Advanced Topics in Multivariate Analysis UNIVERSITY OF MISSISSlPPI STS 661 Time Series Analysis University, Mississippi 38677 Application deadline: May 21 Information: Roy D. Sheffield, Department of Mathe­ ILLINOIS matics NORTHEASTERN ILLINOIS UNIVERSITY Chicago, lllinois 60091 June 10 - July 14 Application deadline: April 1 Math 513 Theory of Numbers I Inform·ation: Graduate College Math 525 Modern Algebra I Math 569 Theory of Integrals April 30 - June 22 Math 575 Mathematical Statistics I Number Theory for the Elementary School Teacher Math 631 Foundations of Geometry* Modern Geometry for the Elementary School Teacher Modern Geometry July 17 - August 19 Math 514 Theory of Numbers II June 25 - August 17 Math 526 Modern Algebra II Modern Algebra for the Elementary School Teacher Math 576 Mathematical Statistics II Theory of Fields Math 640 History of Mathematics* Advanced Numerical Analysis

*Graduate credit only

82 June 19 - August 7 OKLAHOMA Computer Augmented Learning OKLAHOMA STATE UNIVERSITY Stillwater, Oklahoma 74074 VIRGINIA Application deadline: June 4 Information: Department of Computer and Information VIRGINIA POLYTECHNIC INSTITUTE AND STATE Sciences UNIVERSITY Blacksburg, Virginia 24061 June 4 - July 27 Application deadline: April 15 COMSC 4343 Data Structures and Programming Information: Bruce E. Reed, Department of Languages Mathematics COMSC 5323 Computer Operating Systems COMSC 5712 Computer Operations Laboratory June 18 - July 25 405 College Geometry UNIVERSITY OF OKLAHOMA 411 Fourier Series and Partial Differential Equations Norman, Oklahoma 73069 415 Introduction to Numerical Analysis Application deadline: May 1 (preferable) 424 Elementary Topology Information: Office of Admissions and Records 511 Matrix Theory 6110 Topics in Applied Mathematics June 4 - July 28 Advanced Calculus II July 31 - September 3 Theory of Numbers I 421 Fourier Series and Partial Differential Equations Topics in Foundations & Logic 425 Introduction to Numerical Analysis Seminar in Algebra 434 Elementary Topology Topics in Topology I 521 Matrix Theory 6210 Topics in Applied Mathematics PENNSYLVANIA WISCONSIN UNIVERSITY OF PITTSBURGH MARQUETTE UNIVERSITY Pittsburgh, Pennsylvania 15213 Milwaukee, Wisconsin 53233 Application deadline: April 15 Information: E. W. Swokowski, Chairman, Information: Orrin E. Taulbee, Department of Graduate Studies Committee Computer Science June 22 - August 3 April 25 - June 15 Algebraic Structures I Social Implications of the Computer Algebraic Structures II Compiler Design and Construction Mathematical Analysis II Seminar in Curriculum Development I April 25 - August 7 Advanced Geometry I Automata Theory Probability and Statistics Advanced Computer Operating Systems Theory of Games and Mathematical Programming

ASSISTANTSHIPS AND FELLOWSHIPS IN MATHEMATICS IN 1973-1974 Supplementary List I. FOR GRADUATE STUDY AT UNIVERSITIES

TYPE STIPEND TUITION SERVICE REQUIRED DEGREES AWARDED of flnancial assistance amount 9 or 12 if not included houn type Academic year {with number anticipated 1973-197 4} in dollars months in stipend { dollan) per week of service 1971-1972

PENNSYLVANIA Clarion State College, Clarion 16214 DEPARTMENT OF MATHEMATICS Applications due: 7/15/73 Bachelor• s by inst. 1272 S. Gendler, Chairman Bachelor• s by dept. 96 Master's by dept. 22 Teaching Assistantship (4) 1350-2700 9 10 Teaching, Assisting WEST VffiGINIA West Virginia University, Morgantown 25606 DEPARTMENT OF MATHEMATICS Applications due: 3/1/73 Bachelor's by inst. 2303 I. D. Peters, Chairman Bachelor's by dept. 77 Master• s by dept. 22 Teaching Assistantship (16) 2200-2700 9 * 20

*Tuition is waived. Special fees are assessed.

83 DOCTORATES CONFERRED IN 1971-1972 Supplementary List

!!'he following are among those who received doctorates in the mathematical sciences and related subjects from universities in the United States and Canada during 1971-1972. This is a supplement to the list printed in the October 1972 and January 1973 issues of these cNofilxiJ •

CALIFORNIA

UNIVERSITY OF SOUTHERN CALIFORNIA modity with fixed lifetime (1 in Pure Mathematics, 1 in Applied Mathematics) Green, Jeffrey J. Two runs tests of randomness against trend Dewar, James A. Marathe, Vijay P. On converging sets of congruences Priority queuing systems with simultaneous server Kuzanek, Jerry Frank requirements Isoperimetric problems with nonlinear field side Miller, Douglas R. conditions in fluid sloshing On the asymptotic behavior of regenerative pro­ cesses and functionals of regenerative processes DISTRICT OF COLUMBIA Soden, John V. Sequential decision making under uncertainty in AMERICAN UNIVERSITY chance-constrained programming and bid-pricing (2 in Statistics) environments Department of Mathematics and Statistics Turnbull, Bruce W. Bounds and optimal strategies for stochastic Bailar, Barbara Ann The effect of measurement error on discriminant systems function analysis OKLAHOMA Lynch, Cornelius J. A method for computing regression coefficients UNIVERSITY OF OKLAHOMA utilizing incomplete observations (1 in Biostatistics) NEW YORK Department of Biostatistics and Epidemiology Stewart, Raymond Doyle CORNELL UNIVERSITY An algebraic approach to blocking and confounding (6 in Operations Research) in factorial arrangements Fries, Brant E. Optimal ordering policy for a perishable com-

84 NEW AMS PUBLICATIONS

MEMOIRS OF THE AMERICAN TRANSLATIONS OF MATHEMATICAL SOCIETY MATHEMATICAL MONOGRAPHS

COMPACT ZERO-DIMENSIONAL METRIC EXTRINSIC GEOMETRY OF CONVEX SPACES OF FINITE TYPE by SURFACES by R. S. Pierce A. V. Pogorelov Number 130 Volume 35 66 + ii pages; list price $2.40; member price 669 + vi pages; list price $40. 50; member price $1. 80; ISBN 0-8218-1830-1 $37. 38; ISBN 0-8218-1585-7 To order, please specify MEM0/130 To order, please specify MMON0/35 Motivated by the problem of classifying This volume is devoted to a fairly complete countable Boolean algebras, a special class S of solution of several problems which arise natur­ compact, zero-dimensional, metrizable spaces ally on the road to a theory of convex surfaces. is defined and studied in this Memoir. It is shown The title, "Extrinsic geometry of convex sur­ in Part One that the homeomorphism types of faces," both reflects its content and stresses its spaces in S are completely classified by an in­ relation to Aleksandrov• s monograph, "Intrinsic variant consisting of a finite Boolean algebra B geometry of convex surfaces," of which it is, in with two unary operations (the analogues of topo­ a certain sense, a sequel. No specialized geo­ logical closure and derivative), and an integer­ metric prerequisites are demanded of the reader. valued function on a certain subset of the atoms Nevertheless, the diversity of the methods em­ of B. The homeomorphism types of the spaces in ployed necessitates a certain familiarity with S form a countably infinite semi-ring, with addi­ neighboring fields of mathematics. This book is tion and multiplication induced by topological aimed primarily at geometers: advanced students, union and product respectively. The last two parts graduate students, and scientific workers. Some of the Memoir use the invariant developed in Part chapters may hold interest for mathematicians One to explore the structure of this semi-ring. in other fields. A number of unsolved problems which might be attacked by young geometers are formulated in the appendix. These problems dif­ CBMS REGIONAL CONFERENCE SERIES fer in difficulty, but in almost all cases a prom­ IN MATHEMATICS ising approach to the solution is indicated.

HOMOLOGICAL DIMENSIONS OF MODULES by Barbara L. Osofsky PROCEEDINGS OF THE Number 12 STEKLOV INSTITUTE 92 +viii pages; list price $4.40; member price $3. 30; ISBN 0-8218-1662-4 To order, please specify CBMS/12 PARAMETRIC-NORMED SPACES AND NORMED MASSIVES by These notes were prepared for a series of K. K. Golovkin ten lectures given at a Regional Conference of the Conference Board of the Mathematical Sci­ Number 106 ences in June 1971. The bulk of the lectures were 124 + iv pages; list price $17. 60; member price on projective dimensions of "very large" modules. $13. 20; ISBN 0-8218-3006-6 Although the material in these notes is not new, To order, please specify STEKL0/106 there are several places where existing work has Taking into account the already existing at­ been simplified. For example, a commutative tempts to introduce "families" of Banach spaces local nondomain of global dimension 3 is de­ into functional analysis, the author presents a scribed without reference to analysis, and the di­ new constructive version of this idea. The notion mension of a quotient field of a polynomial ring of parametric norm is defined axiomatically, and rather than a regular local ring is calculated. A this leads to the concept of normed massive, derivation of Tor, one step at a time without the which is a class of equivalent compatible sys­ usual derived functor machinery, is also included. tems of parametric norms.

85 PERSONAL ITEMS

ADRIANO BARLOTTI of the University of to an assistant professorship at the University of Perugia has been appointed to a professorship at Maryland. the University of Bologna, Italy. EDWARD C. HOOK of the Massachusetts ANTHONY D. BERARD, JR. , of the Air Institute of Technology has been appointed to an Force Institute of Technology has been appointed assistant professorship at Fordham University. to an associate professorship at Kings College. LEONARD R. HOWELL, JR., of the U. S. CHRISTOPHER BINGHAM of the University Air Force has been appointed to an assistant of Chicago has been appointed to an associate professorship at Valdosta State College, Georgia. professorship in applied statistics at the Univer­ WU-CHUNG HSIANG of Yale University sity of Minnesota, St. Paul. has been appointed to a professorship at Prince­ JAMES V. BLOWERS of Northwestern Uni­ ton University. versity has been appointed a laboratory mathe­ SIMON C. HSIEH of the University of South matician with the U. S. Air Force, Elgin Air Carolina has been appointed to an associate pro­ Force Base, Florida. fessorship at the National Tsing Hua University, TERRENCE J. BROWN of the University of Hsinchu, Taiwan, Republic of China. Missouri, Kansas City, has been appointed to a GARY S. ITZKOWITZ of the University of visiting assistant professorship at Kansas State California, Irvine, has been appointed to an as­ University. sistant professorship at Glassboro State College. PAUL-JEAN CAHEN of Queen's University BENIGNO B. JORQUE of the University of has been appointed a lecturer at McGill Univer­ Oklahoma has been appointed an engineer at sity. W & W Steel Co. , Oklahoma City. STEPHEN ALAN DOBLIN of the University HOUSTON T. KARNES of Louisiana State of Alabama has been appointed to an assistant University, Baton Rouge, has become National professorship at the University of Southern Mis­ President of Pi Mu Epsilon. sissippi. LEE L. KEENER of Rensselaer Polytech­ ROGER T. DOUGLASS of the University of nic Institute has been appointed to an assistant Massachusetts has been appointed to an assistant professorship at Dalhousie University. professorship at Alfred University. JOYCE S. KIM of the New Jersey Educa­ THOMAS A. DOWLING of the University of tion Association has been appointed to the chair­ North Carolina has been appointed to an associate manship of the Department of Mathematics at professorship at Ohio State University. Dynamic Springs Institute, Wayne, Pennsylvania. JOHN C. DRUMMOND, JR., of Texas Tech GARO K. KIREMIDJIAN of SUNY at Stony University has been appointed a senior project Brook has been appointed to an assistant profes­ scientist with Mason & Hanger, Silas Mason Co., sorship at Stanford University. Inc., Amarillo, Texas. JOHN R. KNUDSEN of New York University THOMAS A. W. DWYER III of Northern has been appointed a member of the Technical lllinois University has been appointed a visiting Staff at Bell Telephone Laboratories, Holmdel, lecturer at University College, Dublin, Ireland. New Jersey. SIEMION FAJTLOWICZ of the University GREGORY KOURILSKY of California Insti­ of Colorado has been appointed to an associate tute of Technology has been appointed a research professorship at the University of Houston. scientist with Teledyne Systems Co., North­ BERNARD A. FLEISHMAN of Rensselaer ridge, California. Polytechnic Institute has been selected to be a RICHARD LATTER of the Rand Corpora­ Senior Fulbright-Hays Scholar for the year 1972- tion has been appointed vice president of the 1973. He will spend the spring semester at the R & D Associates, Santa Monica, California. Technical University of Delft, Netherlands. JOSEPH LEHNER, now Mellon professor A. M. FOLEY of Mobil Sekiyu, Tokyo, at the University of Pittsburgh, has been ap­ Japan, has been appointed manager, Data Pro­ pointed professor emeritus of Mathematics at cessing at Dodwell & Co., Akasaka, Tokyo, the University of Maryland. Japan. KENNETH 0. LELAND of the Illinois In­ Kill BY W. FONG has been appointed a stitute of Technology has been appointed senior staff member with the Computing Science and resident research associate at the Aerospace Services Division of the Los Alamos Scientific Research Laboratory, Wright Patterson Air Laboratory. Force Base, Ohio. WILLIAM H. FORD of the University of CHARLES H. C. LITTLE of the University lllinois has been appointed to an assistant pro­ of Waterloo has been appointed a lecturer at the fessorship at Clemson University. Royal Melbourne Institute of Technology. GEORGE F. HADDIX of the University of DAVID L. LOVELADY of the University of Nebraska, Omaha, is on leave to Henningsen, South Carolina has been appointed to an assistant Durham and Richardson, an architectural firm. professorship at Florida State University. MIRIAM P. HALPERIN of Bryn Mawr Col­ LAWRENCE A. MACHTINGER of the Illi­ lege and Brandeis University has been appointed nois Institute of Technology has been appointed

86 to an associate professorship at Purdue Univer­ HAROLD B. REITER of the University of sity-North Central Campus. Hawaii has been appointed to an assistant pro­ ANDREA MAGGIOLO-SCHETTINI of IBM fessorship at the University of North Carolina has been appointed a researcher at the Labora­ at Charlotte. torio de Cibernetica del Consiglio Nazionale Ri­ MURRAY B, RITTERMAN of GTE Labora­ cerche, Arco Felice, Italy. tories, Inc., has been appointed to an assistant ANDY R. MAGID of Columbia University professorship at York College, CUNY, has been appointed to an assistant professorship DESMOND A. ROBBIE of Western illinois at the University of Oklahoma. University has been appointed a senior lecturer PIERRE J. MALRAISON, JR., of Cornell at the University of Papua and New Guinea, University has been appointed to an assistant pro­ GARRY RODRIGUE of the University of fessorship at Carleton College. Southern California has been appointed to an as­ MICHAEL MASCHLER of Hebrew Univer­ sistant professorship at Kent state University, sity has been promoted to a professorship at that J, ELI ROSENFIELD of the University of university. He will also be visiting at Stanford Minnesota has been appointed to an associate University, Departments of Operations Research professorship at Washington Technical Institute. and Economics, for the winter and spring 1973 JOSEPH G. ROTHSCHILD of the New York quarters. Institute of Technology has been appointed to an SAMUEL MERRILL of the University of assistant professorship at the Bronx Community Rochester has been appointed to an associate College. professorship at Wilkes College. ED RUNNION of the U, S. Army has been RUSSELL MERRIS of California State Uni­ appointed to an assistant professorship at the versity, Hayward, has been awarded a Senior. University of Pittsburgh at Johnstown. Fulbright-Hays Grant to lecture at the University W, M. RUST, JR., oftheHumbleOil & of Islamabad, Pakistan, from January to August, Refining Company has been appointed to an ad­ 1973. junct professorship at the University of Texas, GLEN MEYERS of SUNY at Albany has Austin, been appointed to an assistant professorship at CHRISTINE A, SHANNON of Purdue Uni­ the University of Rhode Island. versity has been appointed to an assistant pro­ DONALD R. MILLER of the University of fessorship at Georgetown College, Georgetown, Florida has been appointed a biomathematician Kentucky, with the National Research Council of Canada, HARRY SHERMAN of General Research Ottawa, Canada. Corporation has been appointed to an assistant JAMES R. MILLER of the University of professorship at Fairleigh Dickinson University. Maryland has been appointed a presidential in­ WERNER W, SHOULTZ of Creighton Uni­ tern at the National Center for Atmospheric Re­ versity has been appointed to an assistant pro­ search, Boulder, Colorado. fessorship at the University of Nebraska, Omaha. ROBERT W. MILLER of the University of DAVID C. SHREVE of the University of Wisconsin, Madison, has been appointed to an Minnesota has been appointed to an assistant assistant professorship at the College of William professorship at the University of Wisconsin, and Mary. Milwaukee. LOTHROP MITTENTHAL of the U. S. Ar­ BRUCE H. STEPHAN of Manhattan College my Materiel Command has been appointed com­ has been appointed to an associate professorship mander at the U. s. Army Research Office-Dur­ at the Webb Institute of Naval Architecture, ham. D, STROOCK of New York University, CONRAD E. MUELLER of the 3M Company Courant Institute of Mathematical Sciences has has been appointed principal research scientist been appointed to an associate professorship at with the Research Department, Honeywell, St, the University of Colorado, Paul. HECTOR J. SUSSMANN of the University RICHARD J. O•MALLEY of Purdue Uni­ of Chicago has been appointed to an assistant versity has been appointed to an assistant pro­ professorship at Rutgers University, fessorship at the University of Wisconsin, Mil­ J, W, THOMAS of the University of Wyo­ waukee, ming has been appointed to an associate profes­ KENNETH PACHOLKE of the University of sorship at Colorado State University, Colorado has been appointed to an assistant pro­ EDWARD C, TURNER of the Massachusetts fessorship and to the chairmanship of the De­ Institute of Technology has been appointed to an partment of Mathematics at Northland College. assistant professorship at SUNY at Albany, DWIGHT MILTON PAINE of Wells College MARTIN T, WECHSLER of Wayne State has been appointed to an associate professorship University has been appointed to a visiting pro­ at Messiah College, fessorship at Westfield College, University of GORDON PRICHETT of San Diego State London. University has been appointed to an assistant DELANO P. WEGENER of Ohio University professorship at Hamilton College, has been appointed to an assistant professorship MADAN L, PURl of Indiana University has at Central Michigan University. been elected to membership in the International DONALD R, WEIDMAN of the U. S. Naval statistical Institute. He was recently the guest Weapons Laboratory has been appointed to the professor of Statistics at the Universities of senior research staff at the Urban Institute, Gottingen and Dortmund, Federal Republic of Washington, D. C, Germany, W, ROY WESSEL of the Argonne National

87 Laboratory has been appointed a research asso­ BENFELD; Simon Fraser University: J. J. SEM­ ciate at the Geophysical Fluid Dynamics Institute BER; University of Southern California: ZDENEK of Florida State University. VOREL; Wichita state University: PREM N. ROGER A. WIEGAND of the University of BAJAJ; University of Wisconsin, Milwaukee: Wisconsin has been appointed to an associate ERNST SCHWANDT. professorship at the University of Nebraska. SAMUEL ZAIDMAN of the University of To Assistant Professor. Rutgers Univer­ Montreal is visiting the University of Florence, sity, Newark Campus: PHILIP T. GUZA. Italy, from May 1972 until August 1973. INSTRUCTORSHIPS Bennett College: DOROTHY M. HELLER; PROMOTIONS Brooklyn College: CECILE S. FEDER; Fordham To Dean, Institute of Astronomy and Aero­ University: MICHAEL M. TSUJI; Loop College, physics. University of Sao Paulo: GIORGIO E. City Colleges of Chicago: JOHN C. WENGER; o. New Hampshire College: CHRISTOPHER J. TOY; GIACAGLIA. Pennsylvania State University: RICHARD J. To Assistant Dean. College of Charleston: KRAMER; Westark Community College: DIXIE GEORGE E. HABORAK. SILVERS. To Director of Evaluation, Teacher Educa­ tion. Far West Laboratory for Educational Re­ DEATHS search & Development, San Francisco: MORRIS Professor RALPH B. BENNETT of Auburn K. LAI. University died on December 9, 1972, at the age of 32. He was a member of the Society for 11 To Chairman and Professor, Department years. of Mathematics. College of Idaho: ROGER HIG­ Mr. DAVID EVAN COOPER of Hampton, DEM. Virginia, died on December 3, 1972, at the age To Chairman and Associate Professor, of 26. He was a member of the Society for 8 Department of Mathematics. University of Rhode years. Island: GERASIMOS LADAS. Professor HOWARD B. CURTIS, JR., of the University of Texas at Austin died on De­ To Chairman, Department of Mathematics. cember 4, 1972, at the age of 48. He was a mem­ Aurora College: STEVEN R. LAY; Clarion State ber of the Society for 16 years, College: S. I. GENDLER; Long Island University, Professor Emeritus JAMES M. EARL of C. W. Post Center: JOHN C. STEVENSON; Van­ the University of Nebraska at Omaha died on couver City College: NORMAN BARTON; West November 26, 1972, at the age of 76. He was a Virginia University: I. DEE PETERS. member of the Society for 44 years. To Professor. Hendrix College: CECIL W. Professor JUDITH R. GUMERMAN of McDERMOTT; New Mexico State University: West Chester State College died on August 28, JOSEPH D. ZUND; New York University: ED­ 1972, at the age of 41. She was a member of the WARD M. CARROLL; University of North Dako­ Society for 14 years. ta: THOMAS J. ROBINSON; University of Sas­ Mr. RONALD J. LORING of the State Col­ katchewan, Regina: W. DUANE MONTGOMERY. lege at Boston died on November 13, 1972, at the age of 33. He was a member of the Society To Senior Lecturer. Univ~rsity of Otago, for 10 years. New Zealand: DAVID G. B. HILL. Professor Emeritus H. S. VANDIVER of To Associate Professor. Ball State Uni­ the University of Texas at Austin died on Jan­ versity: T. K. PUTTASWAMY; City College uary 4, 1973, at the age of 90. He was a mem­ (CUNY): JONATHAN D. SONDOW; Herbert H. ber of the Society for 59 years. Lehman College (CUNY): ROBERT FEINER­ Professor DENNIS P. VYTHOULKAS of MAN; University of Detroit: MICHAELS. the Ministry of Education, Athens, Greece, died SKAFF; University of Iowa: PAULS. MUHLY; on January 19, 1972, at the age of 65, He was a Marquette University: THOMAS A. BRONIKOW­ member of the Society for 19 years. SKI; University of Nebraska, Omaha: J. SCOTT Mr. FLOYD E, YOUNG of the Western DOWNING, MARGARET P. GESSAMAN; Univer­ Life Insurance Company, Helena, Montana, died sity of Oklahoma: BERNARD R. McDONALD; on October 3, 1972, at the age of 73. He was a Rensselaer Polytechnic Institute: LESTER RU- member of the Society for 15 years.

88 ABSTRACTS PRESENTED TO THE SOCIETY

Preprints are available from the author in cases where the abstract number is starred.

The papers printed below were accepted by the American Mathematical Society for presentation by title. The ab­ stracts are grouped according to subjects chosen by the author from categories listed on the abstract form. The mis­ cellaneous group includes all abstracts for which the authors did not indicate a category. An individual may present only one abstract by title in any one issue of the cfloticti] but joint authors are treated as a separate category. Thus, in addition to abstracts from two individual authors, one joint abstract by them may also be accepted for an issue.

Algebra & Theory of Numbers

73T-A33. KIM KI-HANG BUTLER, Pembroke State University, Pembroke, North Carolina 28372. New representation of posets.

For terminology see Abstract 72T-A122, these cAfutiaiJ 19(1972), A-502. Let N(Bn) denote the set of all nonsingular matrices of B . Let V = \A(adj +lA\"") :A E N(B )\where adj +IAI+ denotes the (permanent) n n n adjoint of A. Theorem 1. A E N(Bn) iff A= adj(adj +lA I+). Theorem 2. If G*(n) denotes the posets defined on a finite set containing n elements, then G*(n) is isomorphic to Vn. (Received October 16, 1972.)

73T-A34. RONSON J. WARNE, University of Alabama, Birmingham, Alabama 35233. wY-£-unipotent semigroups.

Let I• denote the nonnegative integers and let Y be a semilattice with greatest element 60• We term A~ I' X Y, under the order (k, 6);:;; (s, f/) if k > s or k = s and 5;:;; fl, an WY-semilattice. An

WY-.t-unipotent semigroup is a semigroup S such that E(S), the set of idempotents of S, is an WY-semilattice

A= I• x Y of right zero semigroups (E(n, 6): n E I•, 6 E Y), and E(D0) = U (E(n, 6): n E I•) where (D0 : 6 E Y) is the collection of .&-classes of S (.&is Green's relation). Let T be an WY-semilattice of right groups

(T (k, 0) : k E I •, 6 E Y) and let (n, r) ~ a (n, r) be a homomorphism of C, the bicyclic semigroup, into End T, the semigroup of endomorphisms of T, such that: (1) For each k E I•, there exists e(k, 00 ) E E(T(k, 60)) such that ga(k k) = ge(k 6 ) for all gET. (2) For each k,r,s E I•, T(k • a >" T( • ) if r > k; T(k+s-r 6) if k"' r. ' '0 ,u) (r,s s,u0 , We denote (((n,k), gk0): gko E T (k, 6) ,n,k E I', 6 E ·y) under the multiplication ((n,k), gk6) ((r, s) ,hsf/) = ((n,k) (r, s), gko a (r, s)hSfl) where jmctaposition denotes multiplication in C and T by (T, a (r, s)). Theorem. S is an

WY-£-unipotent semigroup if and only if S is isomorphic to (T, a(r, s)) for some collection T, a(r, s).

Theorem. S is a regular simple (bisimple) semigroup with E(S) an w-semilattice of right zero semigroups iff S is

WY-£-unipotent with Y a finite chain (a single element). (Received October 16, 1972.)

73T-A35. J. L. BRENNER, 10 Phillips Road, Palo Alto, California 94303, Maximal ideals in the near ring of polynomials modulo 2.

The two maximal ideals in the near ring of polynomials modulo 2 are: V, the set of all polynomials

p such that p(O) = p(1); and T, the set of all polynomials p1 such that p1 (9) is either 0 or 1; 8 is the imaginary

solution of 83 = 1. (The first ideal had already been discovered by D. Doi Watkins as a student; the point of this 2 3 paper is that T is also a maximal ideal, and that there are no others.) T is the additive closure of 1, x + x, x ,

x4 + x, x5 + x, x6, x 7 + x, x8 + x, x9 , •... This article will appear in Pacific J, Math. (Received November 8, 1972.)

A-249 73T-A36. GEORG J. RIEGER, University of Munich, D-8000 Munich, Federal Republic of Germany. On M-void numbers.

Let M be a set of natural numbers > 1. A natural number a is called M-void if and only if all the exponents in the canonical prime factorisation of a are outside M. Theorem. For any finite set M 1- If of natural numbers with minimal element m > 1, the number of M-void numbers below x > 0 is equal to xiTP(1- (p -1)~rEMP -r-1)

+ OM(x1/m) where the product is extended over all primes p and where the constant in the error term depends on M only. This generalizes a result of Suryanarayana (Elem. Math. 26(1971), 39-40). A similar formula is proved for the number of representations of a natural number as the sum of a prime and an M-void number. Related distribution questions are discussed. (Received November 9, 1972.)

*73T-A37. TOSHIHIKO YAMADA, Tokyo Metropolitan University, Tokyo, Japan. Schur subgroups of 2-adic fields.

Let k be a cyclotomic extension of the rational 2-adic field Q2. In this paper the Schur subgroup

S(k) of the Brauer group Br(k) is completely determined. Suppose that k(l;: 4)/k is ramified. Let h be the smallest integer such that h ;;; 2 and that k is contained in Q2 ( ~ h) for some n, (2, n) ~ 1. Set L ~ Q2 ( ~ J) and let F be the n2 n2 maximal unramified extension of k contained in L. It turns out that L = F•k(~ 4), F n k(~ 4) ~ k and ~ 4 ~ F. Let

Gal(L/F) ~ (w) (w2 ~1), ~ ~ = ~zh (zE Z). It can be shown that z mod 2h is uniquely determined by k. Now there 2 2 are only two possibilities: (i) z o= -1 (mod 2h), and (ii) z = -1 + 2h-1 (mod 2~ (h;;; 3). For the case (i), S(k) is the

subgroup of order 2 of Br(k). Example. k ~ Q2(,fi) c Q 2 (~ 8 ). For the case (ii), S(k) ~ 1. Example. k '" Q2(,;=2) c Q2 (~ 8 ) (h~ 3). Suppose next that k(~ 4 )/k is unramified (including the case ~ 4 Ek). Then, S(k) ~ 1. Example. k = Q2(/i), where k(~ 4 ) (=Q 2 (~ 12 )) is unramified over k of degree 2. (Received November 9, 1972.)

*73T-A38. IVAN RIVAL, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Finite modular lattices with sublattices of all orders.

The spectrum sp(L) of a lattice L is the set of all integers n such that L has an n-element

sublattice; the spectrum of L is complete if sp(L) ~ {nlo;;; n;;; ILl!. Let .I,(L) denote the~!!!_ of L, that is the

order of a maximum-sized chain in L minus one. It is shown that every modular lattice L of finite length satisfying

ILl;;; (1/3)(5./,(L) + 7) has a complete spectrum. (Received November 10, 1972.)

73T-A39. TAO-CHENG YIT, Carleton University, Ottawa, Ontario K1S 5B6, Canada. On subdirect product of rings without symmetric divisors of zero. Preliminary report.

A theorem of Andrunakievic and Rjabuhin asserts that any ring R (1) without nilpotents is a subdirect

product of Ekewdomains (Soviet Math. Dokl. 9(1968)). Consequently such a ring is (2) compressible in the sense of

G. Thierrin, that is, for every a, b, and x in R, if ax~~ 0 then axb ~ 0. Now let R be a ring with involution *

and suppose that (1) holds for the symmetric elements x ~ x* of R. Does this imply that R is a subdirect product

of rings where nonzero norms (xx*) and traces (x+x*) are regular, which rings were recently studied by C.

Lanski ("Rings with involution whose symmetric elements are regular," to appear in Proc. Amer. Math. Soc.)? The

2 x 2 matrices over the integers modulo 3 is a simple ring without symmetric nilpotents but with nontrivial symmetric

idempotents. This shows that (1) does not generalize to rings with involution. But if we require (2) from norms and

traces of R (that is, ax~= 0 with x = tt* or x = t + t* implies axb ~ 0), we show here that R modulo its prime

radical has the desired subdirect representation and conversely that every ring which has the desired subdirect

representation meets the requirement above. (Received November 10, 1972.) (Author introduced by Professor

Maurice Chacron.)

A-250 73T-A40. ROGER H. HOU, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. The functor Qxt on the category End@). Preliminary report.

Let R be a ring with 1 f. 0, End(R) be the category of endomorphisms of R-modules, and Idm(R) be the full subcategory of idempotents of End(R). We have the characterization of the functor Qsp on Idm(R) (Canad. J.

Math. (1971), 503-506). Define Qxt(o:, m= Ext(A,B)/Qsp(o:, fJ), where o:, fJ E End(R), o:: A ... A and fJ: B ... B. Then Qxt is a bifunctor on End(R), takes short exact sequencestolongexactsequences. End(R) has enough projectives

and injectives. We have a complete characterization of Qxt on End(R), similar to that of Ext. In particular,

Qxt(O, 0) = 0 and equals Ext on automorphisms. (Received November 13, 1972.)

73T-A41. MELVIN HENRIKSEN, Harvey Mudd College, Claremont, California 91711. Rings generated by their units. Preliminary report.

Throughout, R denotes a ring with identity, U(R) the subring generated by the units of R, and n a

positive integer. The following generalize results announced by R. M. Raphael. (See Abstract 72T-A205, these

c/'lolic£i) 19(1972), A-574.) Theorem 1. If n > 1, then every element of Rn is a sum of 4 units. Theorem 2. If R

is commutative and contains prime ideals P 1, P2, neither of which contains the other, and (P1 + P2) n U(R) = {ol, then R2 contains elements which are not the sum of less than 4 units. (The hypothesis holds, in particular, if

R = F [x, y] for any field F.) Theorem 3. If n > 1, and for every A ERn' there are units P, Q of Rn such that PAQ

is diagonal, then every element of Rn is a sum of two units. Theorem 4. If (i) for every a E R, there is a y E R

such that aya = a and any= yan, and (ii) max[2, (n -1)!] is a unit, then every element of R is a sum of 4(n- 1) units.

If R is primitive, satisfies (i) and n > 1 or 2 is a unit, then every element of R is a sum of 2 units. (Received November 13, 1972 .. )

73T-A42. SYED A. HUQ, Institute of Advanced Studies, Australian National University, Canberra, ACT 2600, Australia. Properties of push outs. Preliminary report.

Given two morphisms A ':, B, A ~ C in a category C-, we consider the class J; of pairs of coterminal

morphisms issuing from B and C. We say the pair (o:, fJ) has push out with respect to J;, if we find two coterminal

morphisms y: B ... E and o : C ... E, in J; such that yo: = ofJ and for any other pair ( y 1 , 6 1) in J;, with y 1 o: =

0 1 {3, wemusthaveauniquemap 1J. inC-with 0 1 =1J.6 and y 1 =1J.y. Forthecategoryofgroups, ifwetake (o:,fl)

a pair of monomorphisms and (i) J; = class of all coterminal monomorphisms or (ii) J, = class of all commuting

morphisms in the sense of S. A. Huq (Quart. J. Math. Oxford Ser. 19(1968), 363-389), then (o:, fJ) has push outs

(i) free product with amalgamated subgroups or (ii) central product in the usual group theoretic sense. Various

properties of push outs of a pair, with respect to a class of morphisms or a class of pairs which have push outs with

respect to a class of morphisms, has been studied. (Received November 21, 1972.)

*73T-A43. LOUIS HALLE ROWEN, Yale University, New Haven, Connecticut 06520. Rings with central polynomial.

Let R be an associative ring with 1 and with center C. R has a central polynomial g(X1, •.• ,Xm)

if g is not a polynomial identity but Xm+ 1g- gXm+1 is a polynomial identity (of R). Central polynomials which are homogeneous in all variables and linear in the last variable are called regular. Formanek has shown (J. Algebra, to

appear) that matrix algebras have regular central polynomials; it follows that any semiprime ring with polynomial

identity has a regular central polynomial. Let I be the ideal generated by specializations of all regular central

polynomials. Theorem 1. If I f. 0 and if C is a field, then R is simple. Now let P be a prime ideal of C. Form -1 -1 -1 Rp viewing R as C-module; RP has multiplication (r1s1 )(r2s 2 ) = (r1r 2)(s1s 2) , s 1,s2 inC- P.

Theorem 2. Let P be prime, I¢, P. Then cent RP = CP; PR n C = P; there is a unique prime ideal P 1 of R whose

A-251 intersection with C is P; Rp/Pp is the classical ring of quotients of R/P'; (PR)p is maximal in Rp. If P is maximal

in C then PR is maximal in R. The condition I¢ P is seen to be natural. Applying these and other results gives a

proof of Artin's theorem on Azumaya algebras, based on central polynomials. (Received November 14, 1972.)

*73T-A44. DAVID K. HALEY, Mannheim University, 68 Mannheim A, 5, West Germany. Note on compactifying Artinian rings.

Theorem. The following are equivalent conditions on an Artinian ring R: (i) R is equationally

compact. (ii) R+ is the direct sum of a finite group B and a finite sum of Prnfer groups P, and R•P = P•R = \0}.

(iii) R is a (algebraic) retract of a compact topological ring. (iv) R is a (algebraic) subring of a compact topological

ring. (v) R has an equational compactification. (vi) R has a quasi-compactification. The Theorem extends

Theorem 2 of Abstract 71 T-A261, these cN'ow 18(1971), 1096. (Received November 14, 1972.)

73T-A45. WILFRID A. HODGES, Bedford College, Regent's Park, London NW1, England. Algebraic closures need Zorn's lemma. Preliminary report.

Each of (a), (b) below is consistent with (Zermelo-Fraenkel) set theory minus Zorn's lemma.

(a) Let p be a prime or 0; then there is a field of characteristic p which has no separable closure, hence no algebraic closure. (b) The field of rationals has two nonisomorphic algebraic closures. From the consistency of (a) it follows that no explicit construction by generators and congruence relations can be given for algebraic closures of fields.

From the consistency of (b) it follows that there is no effective proof of the van der Waerden-Zariski-Chevalley theorem on extension of valuations. (Received November 15, 1972.)

73T-A46. MAURICE FRECHETTE, McMaster University, Hamilton, Ontario, Canada. Equivalence of sesquilinear forms. Preliminary report.

Let f1 and f2 be J sesquilinear forms with asymmetries A1 and A2 over division ring D, where [D: cen D] < oo. In case A1 and A2 are unipotent, ch D = 2, and J = id on cen D, there are finite sequences of hermitian and pseudo-quadratic forms attached to f1 and f2 such that f1 is equivalent to f2 if and only if A1 and A2 are similar and corresponding forms in the sequences are equivalent. Similar techniques apply to the other cases, previously solved by G. E. Wall ["On the conjugacy classes in the unitary, symplectic and orthogonal groups,"

J. Austral. Math. Soc. 3(1963), 1-62]. (Received November 20, 1972.) (Author introduced by Professor Carl Riehm,)

*73T-A47. SHARI L. LAWRENCE and TORRENCE D. PARSONS, Pennsylvania State University, University Park, Pennsylvania 16802. Path-cycle Ramsey numbers.

Let f(h, k) be the least integer N such that any graph on N vertices satisfies at least one of the following: (1) the graph contains a path on h vertices as a subgraph, (2) the complement of the graph contains a cycle on k vertices as a subgraph. Then f(h, k) is (a) h + (k/2) - 1, if k is even and k ;;;:; h, (b) k + [h/2] - 1, if

4;;;; h;;;:; k and either k is even or k 5; (3/2)h, (c) 2h - 1, if k is odd, (h, k) ol (3, 3), and either k ;;;; h or 4;;;; h ~ k ~

(3/2)h, (d) 6, if (h,k) = (3, 3), (e) k, if h = 3 and k 5; 4, or if h = 2, (f) 1, if h = 1. (Received December 11, 1972.)

*73T-A48. NAI-CHAO HSU, Eastern Ulinois University, Charleston, Ulinois 61920, Quasi-homology and universal coefficients.

For an arbitrary sequence C = \dn: en ... Cn_1} of modules en over a ring with 1 and homomorphisms dn' the n-dimensional quasi-homology module "n(C) is defined to be the factor module of Ker dn + Im dn+1 by Ker dn n Im dn+ 1• Analogues of the ordinary universal coeffi.cient theorems are sought by means of an obvious decomposition "n = Z£n Ell wn where !!'n(C) is the factor module of Ker dn by Ker dn n Im dn+1 and Wn(C) is the factor module of Im dn+l by Ker dn n Im dn+ 1• The decomposition induces the decompositions O!n =~nEll O!n and

A-252 cx.n ; cx.n Ell where ex. is a mapping of 11 (C) 181 G into 11 (C 181 G) and cx.n is a mapping of 1/n(Hom(C, G)) into - 7Jl n n n Hom(11n(C),G) defined in the same way as the familiar mappings in the ordinary universal coefficient theorems. The mappings ex. , 'ii , cx.n and an are examined. A corollary states that ex. is an isomorphism if C is a sequence of -n n - n vector spaces and G is a vector space over the same field, but no parallel statement about cx.n can be made.

(Received November 24, 1972.)

*73T-A49. GEORGE A. GRATZER and H. LAKSER, University ofManitoba, Winnipeg, ManitobaR3T 2N2, Canada. Three remarks on the arguesian identity. I.

Let p(x1, ••• ,~) !0 q(x1, ••• ,x6) be the arguesian identity of B. Jonsson (Math. Scand. 2(1954), 295-314). (See G. Birkhoff, "Lattice theory," 3rd ed. , Ex. 7, p. 109.) Note that p has no repetition of variables.

In the subspace lattice of a projective geometry p !0 q holds iff Desargues' theorem is true (B. Jonsson, loc. cit.).

This identity is obvious for points of a projective space in which Desargues' theorem holds. The usual proof of p !0 q uses the coordinatization theorem of projective spaces. This can be avoided however by using the following result: Theorem 1. Let p' and q' be lattice polynomials subject to the condition that no variable occurs twice in p'.

Let L be an algebraic atomic modular lattice. If p' !0 q' holds in L whenever the variables are substituted by the atoms of L, then p'::;; q' holds in L. Let x,y,z be variables other than x1, ••• ,x6, define u; (xVy) A (xV z) A (yV z), v; (xAy) v (xAz) v (YAz), and ii; xi A ((u Ax) v v) for i; 1,2, ••• ,6; set p; p(x1, ••• ,x6) v v, q; q(x1, ••• ,i6)

V v. Theorem 2. The identity p !0 q holds in the equational class generated by complemented modular lattices but there is a modular lattice (of length four) in which p !Oq fails. Corollary (R. P. Dilworth and M. Hall, Ann. of Math. 45(1944), 450-456). There is a modular lattice that cannot be embedded in a complemented modular lattice.

(Received November 27, 1972.)

*73T-A50. HUGO D'ALARCAO and THOMAS E. MOORE, Bridgewater State College, Bridgewater, Massachusetts 02324. Order as a subgroup-lattice homomorphism.

Let G be a finite group of order n, D(n) the lattice of positive integral divisors of n, X(G) one of the lattices: S, of all subgroups; N, of all normal subgroups; SN, of all subnormal subgroups; C, of all characteristic subgroups, of G. Let o: X(G) .. D(u) be the mapping that assigns to each HE X(G) its order o(H). Theorem 1. The following are equivalent: (1) o is a lattice homomorphism on S(G); (2) o is a lattice isomorphism; (3) S(G) is a distributive lattice; (4) G is cyclic. Theorem 2. If G is nilpotent and o is a lattice homomorphism on N(G) then G is cyclic. Theorem 3. If G is solvable and o is a lattice homomorphism on SN(G) then G is metacyclic.

Theorem 4. If G is an abelian p-group then o is a lattice homomorphism on C(G) iff whenever pi is an invariant of

G, i ; jJ. or i ; jJ. - 1, where IJ. is the exponent of G. (Received November 21, 1972.)

73T-A51. DAVID W. BALLEW, South Dakota School of Mines and Technology, Rapid City, South Dakota 57701. Hereditary orders and their idempotents.

Let A be a Dedekind domain, K its quotient field and 2:: a finite dimensional central K-algebra.

Assume A, r are A-orders in 2:: and that r :2 A. The purpose of this paper is to investigate the A-projectivity of modules of the form M ; 8~4 A IJ.i where the IJ.i are in A but are not necessarily units in 2:: and where 8~; 1 r IJ.i is r-projective. Several criteria for the A-projectivity of M are obtained in terms of idempotents in r. These theorems are then applied to hereditary orders. Sample theorem. If A is an hereditary A-order and ex. is an idempotent in any order r containing A, then there is an idempotent (3 in A such that rex.; r (3. (Received

November 30, 1972.)

A-253 *73T-A52. G. McNULTY and ROBERT WILIJS QUACKENBUSH, Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. The ascending and descending varietal chains of a variety.

Let V be a variety (equational class of algebras); Vn is the variety generated by the free algebra over V on n free generators and -tf1 is the variety of all algebras such that every n-generated subalgebra belongs to

V. Clearly v1 s;; v2 s;; ••• and ~ :2 ~ :2 ••• ; the first chain is the ascending varietal chain of V while the second is the descending varietal chain of V. Theorem 1. Given any S s;; !1, 2, ••• ! there is a variety V such that, for n !!i 0,

Vn -# Vn+ 1 iff n + 1 E S. Theorem 2. Given any S s;; \2, 3, ••• l with 2, 3 E S there is a variety of semigroups V such that, for n!!; 1, -tf1-# vn+1 iff n + 1 E s. Note. The asserted theorem in Abstract 72T-A212, these c/{otictiJ 19(1972),

A-576, claims more than has been proven; given an S not containing both 2 and 3, it is an open question whether there is a variety V with the appropriate descending varietal chain. (Received November 17, 1972.)

*73T-A53. DAVIDA.DRAKE and MARK P. HALE, University of Florida, Gainesville, Florida 32601. Half counts in Latin quarters. Preliminary report. --

If L is a matrix of order 2m or 2m+ 1, the m x m submatrices of L are called quarters of L.

If B is a quarter of a Latin square of order 2m or 2m+ 1, one defines h(B) to be the number of cells of B occupied by the m numbers which occur in B with greatest frequency. One sets H(L) equal to the maximum h(B) as B varies 2 over the quarters of L. Theorem. Let L be a Latin square of order k ~ 2m or 2m + 1. Then H(L) !!; (m /2) + m - 1 if k ~2m; H(L) !!; (m2/2) + (3m/4)- 3/4 if k ~2m+ 1 with m odd; H(L) !!; (m2/2) + (3m/4) + 1/2 if k ~2m+ 1 with m even and m !!; 10. (Received November 13, 1972.)

*73T-A54. MICHAEL MAKKAI, Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. A proof of K. A. Baker's finite-base theorem.

A proof is given of the following theorem announced by K. A. Baker: Theorem. Every finite algebra of an arbitrary congruence-distributive variety of finite type generates a finitely based variety. The proof uses the main result of Baker's preprint "Primitive satisfaction and equational problems for lattices and other algebras." The key of the proof is the use of certain first order sentences which are preserved in subdirect products (but they are not, e.g., universal Horn formulas). The proof uses the compactness theorem of first order logic, so it is

"nonconstructive." But it does not require any algebraic analysis beyond Baker's theorem of his preprint. (Received

November 15, 1972.) (Author introduced by Professor George A. Gratzer.)

73T-A55. RENU LASKAR, Clemson University, Clemson, South Carolina 29631 and HENRY ARTHUR PELLERIN, University of North Carolina, Charlotte, North Carolina 28205. On r-lattice graphs. Preliminary report.

An r-latticegraphis defined as a graph G, whose vertices are identified with the ordered r-tuples on n symbols, such that two vertices are adjacent if and only if the corresponding r-tuples differ in exactly one coordinate position. Characterizations of such graphs for r ~ 2, 3, and 4 have been studied by different authors. In this paper it is shown that similar properties characterize such graphs for any r provided n > 1 + ~ r(r+ 1). (Received December 6, 1972.)

*73T-A56. STEPHEN IL Y. HUNG and N. S. MENDOLSOHN, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Handcuffed designs.

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parameters v,k, A consists of a system of ordered k-subsets of a v-set, called handcuffed blocks. In a block

!A1,A2, ••. ,Akl each element is assumed to be handcuffed to its neighbors and the block contains k- 1 handcuffed pairs (A 1,A2), (A2,A3), ••• , (Ak_1,Ak). These pairs are considered unordered. The collection of handcuffed blocks

A-254 constitutes a handcuffed design if the following are satisfied: (1) each element of the v-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of the v-set are handcuffed in exactly >.. of the blocks. If the total number of blocks is b and each element appears in r blocks the following conditions are necessary for the handcuffed design to exist. (1) >..v(v -1) = (k -1)b. (2) rv = kb. In this paper it is shown that the necessary conditions are also sufficient. (Received December 8, 1972.)

73T-A57. FRANK J, OLES, Cornell University, Ithaca, New York 14850. A characterization of bounded Dedekind prime rings. Preliminary report.

It can be shown that a bounded Noetherian prime ring R is Dedekind if and only if R is a maximal order and the Krull dimension of R is one. From this one can easily derive the following result which is basically in

Auslander and Goldman, Trans. Amer. Math. Soc. 97(1960), 1-24: If Dis a commutative Dedekind domain with quotient field K, and if R is a maximal D-order in a semisimple K-algebra, then R is hereditary. (Received

December 11, 1972.)

*73T-A58. B. N. DATTA, University of ottawa, Ottawa, Ontario, Canada. An algorithm for computing a symmetrizer of an arbitrary matrix.

The knowledge of a symmetric solution X of the matrix equation XA =A TX, called the symmetrizer of A, makes possible the reduction of an apparently nonsymmetric matrix eigenvalue problem, Ay = >.,y, to a symmetric one of the form Cy- >..Xy = 0 where C and X are both symmetric. It is, therefore, of some interest to compute such a solution numerically. Several algorithms are known to exist in case the given matrix A is a diagonal or a companion matrix of a polynomial. In case of an arbitrary matrix, a method was proposed by Howland and Farrel

(A. C. M. National Conference, Denver, Colorado, 1963) but found to be numerically unstable. A method proposed by

Howland and Senez (Numer. Math. 16(1970), 1-7) for solving the Lyapunov equation XA +A TX =-I can, however, be adapted to compute the symmetrizer of an arbitrary real matrix. Another constructive procedure, believed to be numerically stable, is proposed in this paper. (Received December 11, 1972,) (Author introduced by Professor Remi Vaillancourt.)

73T-A59. THEODORE A. SUNDSTROM, University of Massachusetts, Amherst, Massachusetts 01002. Groups of automorphisms of simple rings. Preliminary report.

Let R be a direct sum of n simple rings, and let G be a finite solvable group of automorphisms of

R. Assume that \G\x = 0 implies that x = 0 for x E R. Let T = \x E R \XT = x, for all T E G}. The following results have been obtained: (1) T is a finite direct sum of simple rings. Moreover, if G leaves the centroid of R elementwise fixed, then T is a direct sum of k simple rings, where n/\G\;;; k;;; n\G\. (2) If \G\ = 2r and if R has an identity, then the simple components of R are finite dimensional over their centers if and only if the simple components of T are finite dimensional over their centers. Now, let H be a finite solvable group of automorphisms and anti-automorphisms of R, and assume that \H\x = 0 implies that x = 0 for x E R. Let G = \T E H\r is an automorphism}, and let S =

\x E R\ xcr = x, for all cr E H}. The above results were used to obtain: S is a finite direct sum of simple Jordan rings, and if G leaves the centroid of R elementwise fixed, then S is a direct sum of m simple Jordan rings, where n/\H\;;; m;;; n\G\. (Received December 11, 1972.) (Author introduced by Professor Wallace S. Martindale III.)

*73T-A60. RICHARD L. MYERS, 429 Fletcher Road, Wayne, Pennsylvania 19087. Super magic cubes. Preliminary report.

The numbers 1 through 512 are dispersed in the cells of an 8 by 8 by 8 cube so as to produce a number cube having the following properties: (1) Every square section of the cube is a magic square having total 2052.

A-255 (2) Taking any rectangular solid of cells centered in the cube, the 8 corners total 2052. (3) Dicing the cube into 64 equal cubes produces 2 by 2 by 2 cubes whose 8 cells total 2052. (4) A symmetry property: Any two numbers equidistant from the ends of the list 1,2, 3, ••• , 512 appear in cells that are symmetric with respect to the center of the cube. The algorithm for constructing such a cube (there are many such) is currently being refined. (Received

December 12, 1972.) (Author introduced by Professor Herman E. Gollwitzer.)

*73T-A61. R. PADMANABHAN and ROBERT WILLIS QUACKENBUSH, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Minimal bases for equational classes with distributive congruences.

Let ~ be an equational class of algebras with the distributive congruence property. Then B. Jonsson has shown (Math. Scand, 21(1967), 110-121) that there exists a natural number n!!! 2 such that ~ has n ternary polynomials satisfying certain identities (type n). Theorem 1. If ~ is a finitely based equational class of algebras with the distributive congruence property of type n then ,!5 is n-based. Theorem 2. If, in addition, ~ has permutable congruences then ~ is one-based. Corollary. The equational theory of any quasi-primal algebra is one-based. The proofs make use of results of K. A. Baker ("Equational problems for lattices and other algebras," preprint), A. F.

Pixley (Proc. Amer. Math. &>c. 14(1963), 105-109) and Padmanabhan (Algebra Universalis 1(1972), 57-61).

(Received December 20, 1972.)

73T-A62. HUDSON V. E. KRONK, State University of New York, Binghamton, New York 13901 and JOHN A. MITCHEM, California State University, San Jose, California 95192. Critical point-arboritic graphs.

The point-arboricity of a graph G, denoted p(G), is the minimum number of subsets in any partition of the point set of G so that each subset induces an acyclic graph. Graph G is k-critical if p(G) ~ k and each proper subgraph H of G has p(H) < k. Theorem. For every k !!! 3 and every odd integer n ;;; 2k - 1, there exists a k-critical graph of order n. Theorem. If G is k-critical then G is (2k-2)-line connected. Theorem, If G is not complete and p(G) = k;;; 3, then the maximum degree of G is at least 2k- 1. Corollary. If G is triangle-free with genus 1, then p(G) = 2. Other properties of k-critical graphs are given including a method of constructing a k-critical graph from two others. (Received December 18, 1972.)

*73T-A63. MARK BLONDEAU HEDRICK, University of Houston, Houston, Texas 77004. Two examples of the behavior of the permanent function.

The author has an example of a segment in the set of 3 X 3 doubly stochastic matrices on which the permanent function is constant. In addition, he has an example of a 3 x 3 generalized doubly stochastic matrix whose permanent is 3!/33, (Received December 11, 1972.) (Author introduced by Professor Richard D. Sinkhorn.)

*73T-A64. JOE W. FISHER, University of Texas, Austin, Texas 78712. Finiteness conditions for projective and injective modules.

Let R be an associative ring with unity. We consider the following Question. Does Hopkins' theorem extend to projective R-modules, i.e., are projective Artinian R-modules Noetherian? An example is given to answer this question in the negative. However, in each one of the following cases, the answer is affirmative:

(a) R is commutatuve, (b) R is hereditary, or (c) the projective Artinian R-module is a generator in the category of R-modules. Dually, are injective Noetherian R-modules Artinian? An example given in Abstract 72T-A193, these cJVotit:Li) 19(1972), A-571, answers this question in the negative. In the affirmative we have the following: Theorem.

If M is an injective Noetherian R-module where R is a right hereditary Noetherian ring with polynomial identity, then

M is Artinian. Corollary. If M is an injective Noetherian R-module where R is commutative, then M is Artinian. Also see Fisher, Abstract 71T-A85, these cJVotit:Li) 18(1971), 619. (Received December 21, 1972.)

A-256 *73T-A65. MICHAEL RICH, Temple University, Philadelphia, Pennsylvania 19122, A commutativity theorem for algebras.

Let A be an algebra over a field E satisfying the condition that to each pair of elements x,y in F there is an element O!(x,y) in F such that xy ~ O!(x,y)yx. Then the following theorem holds. Theorem. A is either

a commutative or an anti-commutative algebra. (Received December 26, 1972.)

*73T-A66. DONALD E. McCLURE, Brown University, .Providence, Rhode Island 02912. Asymptotic eigenvalue distributions for closed algebras of finite Hermitian Toeplitz matrices. Preliminary report.

The approximate distribution of eigenvalues is characterized for finite matrices contained in

norm-closed algebras generated by finite sections of infinite Hermitian Toeplitz matrices, Consider n X n principal

sections Rn of an infinite Toeplitz matrix R associated with a real-valued function fin L00[-'IT, 'IT]; the entries of R

are Rv~ = (1/2'1T)Jei(V-~)xf(x)dx. Consider sequences r of finite sections, r ~ (R1,R2, ••• ), and let B denote the algebra over the reals generated by all such sequences; addition and multiplication of sequences are term-by-term

extensions of finite-matrix addition and multiplication. Limiting eigenvalue distributions of terms of a member p of

B are characterized by U. Grenander and G. Szegl) ("Toeplitz forms and their applications," Univ. of California

Press, 1958). The limiting distributions are described in terms of the multinomial P that determines the sequence

p from the generating sequences r and in terms of the functions f associated with the generating sequences. B is

normed by 1\r \1 ~ sup\1\Rn\\1:1• where 1\Rn \\ denotes the spectral norm of Rn. The results of Grenander and Szego are extended to sequences in the norm-closure of B with respect to the Banach algebra X of sequences x of

n X n matrices for which 1\x\lls finite. This result yields asymptotic expressions for error probabilities in a binary

signal detection problem. (Received December 28, 1972.)

*73T-A67. STEVE LIGH, University of Southwestern Louisiana, Lafayette, Louisiana 70501. Near rings on certain groups. Let (R,+) be a group and H a subset of R such that 0 is in H. Define a multiplication * on R as

follows: h * g ~ 0, x * g ~ g for each h in H, x in R- H and g in G. It has been shown that (R,+, *) is a near ring

(J. J. Malone, "Near rings with trivial multiplications," Amer. Math. Monthly 74(1967), 1111-1112). In this note, it

is shown that (R,+, *) is a d. g. near ring if and only if H is a subgroup of index two and (R,+) is generated by

elements of order two. Consequently (R,+, *) is a ring if and only if each element of (R,+) is of order two. Define

another multiplication • on R as follows: h • g ~ 0 for all g in R, x • r ~ w, where w + w ~ 0, for all x, r in R- H,

and x • h ~ 0 for all h in H. Then (R,+, •) is a commutative near ring if and only if H is a subgroup of index two.

It is also shown that there is a nontrivial multiplication defined on a finite noncommutative simple group. (Received December 26, 1972.)

*73T-A68. JUDITH Q. LONGYEAR, Dartmouth College, Hanover, New Hampshire 03755. Tactical constructions.

The purpose of this paper is to begin the systematic investigation of large tactical configurations,

with a view toward applications. Several new constructions are given, as well as descriptions of the limitations of

current proof techniques. The restriction to pathlike configurations is suggested as a temporary evasion of the major

unsolved construction problems, and it is noted that most of the known configurations are pathlike, particularly those

most used in applications. (Received December 26, 1972.)

A-257 73T-A69. HANSRAJ GUPTA, Punjab University, Chandigarh, Allahabad 211002, India. The combinatorial recurrence. Preliminary report.

Given c(n,O) = a(n), c(1,k) = b(k), n,k"' 1; the recurrence relation c(n+1,k) = c(n,k) + c(n,k-1)

defines c(n,k) for all natural numbers n and integers k"' 0. In this paper, two simple generating functions are "n-1 r-1 "'k-1 n-1 obtainedfor c(n,k). Thesegivetheelegantresult c(n,k)=LJr='k(k_1 )a(n-r)+ LJr=o< r )b(k-r), k?;l. A particular case has recently been considered by C. C. Cadogan and L. Carlitz (Fibonacci Quart. 9(1971), 329-336;

10(1972), 157-162). The results in this paper are very much simpler than those of Garlitz. (Received December 26,

1972.)

*73T-A70. ROBERT GILMER and TOM PARKER, Florida State University, Tallahassee, Florida 32306. Conditions under which the semigroup ring has an identity. Preliminary report.

Let R be a commutative ring, S an additive abelian semigroup, and let R[X;S] denote the

semigroup ring of S over R. Theorem. The following three conditions are equivalent: (1) R[X;S] has an identity.

(2) R has an identity and there exists a finite subset T of S such that s E S implies s + t = s for some t E T.

(3) R has an identity and S contains a finite subs emigroup T of idempotents such that s E S implies s + t = s for

some t E T. Corollary 1. Suppose that R [X;S] has an identity. Let 1r be the characteristic sub ring of R. Then

the identity of R[X;S] belongs to 'Tr[X;S]. Corollary 2. If S contains a cancellative element, then R[X;S] has an

identity if and only if R has an identity and S has a zero. For each positive integer n, there is an abelian semigroup

Sn with n elements such that (1) Sn has no zero element, and (2) R[X;Sn] has an identity for each commutative ring

R with identity. (Received December 26, 1972.)

73T-A71. V. S. RAMAMURTHI, De la Salle College, Karumathur 626514, India. On splitting cotorsion radicals.

Let R be a ring with identity and MR, the category of unitary right R-modules. A subfunctor p of

the identity functor on MR has been called a cotorsion radical if the dual of the functor 1/ p: M ~ M/ p(M) in the dual

category MR* is a torsion radical for MR* [J. A. Beachy, Bull. Austral. Math. Soc. 5(1971), 241-253]. Let p be called hereditary (resp. splitting) if, M, N E MR, M c N and p(N) = N implies p(M) = M (resp., the exact sequence

0 ~ p(M) ~ M ~ M/ p(M) ~ 0 splits for every ME MR). Theorem 1. A cotorsion radical p is hereditary<> R/p(R) is flat as a left R-module. Theorem 2. If p is a hereditary cotorsion radical and p(R) contains no nonzero nilpoint ideals of R, then p is splitting <> p(R) is finitely generated as a left ideal. Theorem 3. If each cyclic, flat, left R-module is projective, then all hereditary cotorsion radicals are splitting <> any ideal of R, which is a direct summand of RR is also a direct summand of RR. These theorems are used to determine the cotorsion radicals of

MR which are hereditary and/or splitting, when R is a semiperfect ring, left Noetherian ring, von Neumann regular ring, quasi-Frobenius ring, integral domain, commutative semilocal ring, etc. (Received December 27, 1972.)

(Author introduced by Professor Edgar A. Rutter.)

73T-A72. C. J. EVERETT, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544. A greatest integer theorem for Fibonacci spaces.

Let S = \snl be an arbitrary integral sequence of the Fibonacci space C(f) associated with the polynomial f(x) = -a - a x-••. -aN xN-1 + xN = 'Tr(X- r.), a. E Z, r. distinct, jr.j < 1 for i "'2. Then S is a linear 0 1 - 1 1 I 1 1 combination L:: 1Nc.R. of the geometric sequences R. = \1, r.,r~, .. -l of C(f); namely s = L:: Nc.r~, n;; 0, and s = 11 1 11 n 1 11 n c 1 r~ + (n' (n ~ 0 ("Fibonacci spaces," Fibonacci Quart. 4(1966), 100, Theorem IE). For fixed k;; 0, and~ fixed F on (0, 1), it follows that (**) [r~sn + F] = ~+n for n sufficiently large. This is a broad generalization (in an asymptotic sense) of a conjecture by D. Zeitlin (Abstract 72T-A282, these c}fotictiJ 19(1972)) where N = 2, a0 = 1, a1 = M"' 1, S = \0, 1,M, ••. l, F = M/(M+1), and (**) is asserted to hold for n;; 2, k = 1, and all n;; k;; 2.

A-258 Unfortunately this appears to be false in the single case n odd !!o k odd !!o 3. Here, [r~sn + F] is ~+n only when -r;sk < 1/(M + 1), otherwise (and always for n = k) it is sk+n + 1. (Received December 27, 1972.)

*73T-A73. JORGE MARTINEZ, University of Florida, Gainesville, Florida 32601. Structure of Archimedean lattices. Preliminary report.

An algebraic lattice is a complete lattice in which every element is the join of compact elements in the lattice. If a E L is compact there exist elements lmx\X E AI maximal in the interval [0, a) (with each mX

L. Archimedean lattices can also be characterized as follows: if 0 < c < d E L are compact elements of the algebraic lattice L, then there is a q E L maximal w. r. t. not exceeding both c and d; this q is necessarily meet-irreducible.

For an Archimedean, Brouwer lattice we prove: if a E L is polar-closed (i.e. a= a", where x l!O a' if and only if a 1\x= 0) then La= [a, 1) is also an Archimedean, Brouwer lattice. We need no more than the assumption that L be

a complete Brouwer lattice to show the following: call a E L basic if 0 ..: a and 0 is prime in [0, a]; then the following

are equivalent: (1) a is basic, (2) a" is basic, (3) if 0 < b;;;; a then a' = b 1 , (4) a' is prime, (5) a' is a minimal prime, (6) a' is a maximal polar, (7) a" is a minimal polar, and (8) a'' is a maximal basic element. As a corollary we obtain that in a complete Brouwer lattice L the Boolean algebra of polars is atomic if and only if each

0 < x E L exceeds a basic element. (Received December 27, 1972.)

*73T-A74. w. D. WALLIS, University of Newcastle, New South Wales, Australia 2308 and RONALD C. MULLIN, University of Waterloo, Waterloo, Ontario N2L 3GI, Canada. On the number of pairwise Room quasi.groups. Preliminary report.

It is well known that a Room square of side r is equivalent to a pair of idempotent symmetric

quasigroups of order r which satisfy an "orthogonality" condition. Such a pair is a Room pair. Write v(r) for the

size of the largest possible set of pairwise Room quasigroups of order r. Theorem. Given a positive integer m,

there exists an integer v(m) such that if r is odd and r !!o v(m), then ll(r) ~ m. The proof uses R. M. Wilson's

theorems on PBD closure. (Received January 2, 1973.)

73T-A75. WITHDRAWN.

*73T-A76. W. D. WALLIS, University of Newcastle, New South Wales, Australia 2308. Properties of Room squares. Preliminary report.

Theorem 1. Given a Room square R, there exists a number N(R) such that, for every odd integer

n greater than N(R), there is a Room square of side n with a subsquare isomorphic to R. Theorem 2. There is a

number N such that, if n is an odd integer greater than N, there is a skew Room square of side n, Theorem 3. Let

G be a finite abelian group of odd order whose 3-sylow subgroup is cyclic. Then there can be no adder for the

patterned starter in G which produces a skew Room square. Notation. Room squares of side r with common

leading diagonal 01, 02, 03, •.. , Or are called complementary if it is possible to superimpose one on the other without

producing any multiple elements off the diagonal; it has been asked whether there are complementary Room squares

which are not essentially a skew Room square and its transpose. Theorem 4. There are complementary Room

squares, R and S say, of side 13, such that neither R nor S is isomorpic to a skew Room square. This cannot occur

at side 7; the situation for sides 9 and 11 is undecided. (Received January 2, 1973.)

A-259 *73T-A77. JAMES W. BREWER and PHILIP R. MONTGOMERY, University of Kansas, Lawrence, Kansas 66044. The finiteness of I when R [X]/ I is projective.

Let R be a commutative unitary ring, X an indeterminate over R and I an ideal of R[X].

Theorem. In order that R [X]/ I be a free R-module it is necessary and sufficient that I= fR fX], f a monic polynomial of R[X]. Noting that R[X]/I can be R-projective without I being a finitely generated ideal of R[X] we have

Theorem. Suppose that R [X]/I is a projective R-module. The following are equivalent: (1) I is finitely generated as an ideal of R[X], (2) I= fR[X], where there exist pairwise orthogonal idempotents e0, e1, ••. , en of R such that e/ is monic in e.R[X] for 0 ~ i ~ n and if e = L;~ e., (1- e)f = o. Finally: Theorem. For a commutative unitary ring R, 1 1=0 1 ---- the following are equivalent: (1) Finitely generated flat R-modules are projective. (2) If I is an ideal of R[X], then

R[X]/I R-projective implies I is a finitely generated ideal. (Received January 2, 1973.)

73T-A78. S. K. JAIN, Ohio University, Athens, Ohio 45701 and SURJEET SINGH, Bedford College, London, and Aligarh University, Aligarh, India. Rings in which every left ideal is quasi-projective.

A ring R (with unity 1 ,;, 0) is called left qp-ring (q-ring) if each of its left ideals is quasi-projective

(quasi-injective). q-rings have been studied earlier by Jain, Koehler, Mohamed, Singh and others. Some of the results proved in this paper are: (1) Let R be a local ring which is also left as well as right perfect. Then R is a left qp-ring iff (i) R is left artinian, left uniserial, or (ii) N2 = (0) where N is the radical of R. (2) Let R be a

QF-ring. Then R is left qp-ring iff every homomorphic image of R is a q-ring. (3) Let R be left artinian and left qp-ring and N be its Jacobson radical. If R = Re1 + ••• + Ren where 1 = e 1 + ••• +en and {eil is a set of primitive orthogonal idempotents then T = DReiNei is an ideal of R and R/T is left hereditary. Further R is left hereditary iff T = (0). (Received December 26, 1972.)

*73T-A79. C. R. JOHNSON, National Bureau of Standards, Washington, D. C. 20234 and C~RLES R. DePRIMA, California Institute of Technology, Pasadena, California 91109. The range of A- A*.

Let G be the class of nonsingular n X n complex matrices and let F(A) = {x*Ax: x E en, x*x o ll, the field of values of a complex n X n matrix. Theorem 1. There is an A E G such that T =A-lA* if and only if

T-l is similar to T*. Theorem 2. For A E G the following are equivalent: (a) A - 1A * is similar to a unitary;

(b) A is congruent to a normal; and (c) there is aBE G such that A-1A* = B-1B* and 0 ~ F(B). Theorem 3. For T E G, the following are equivalent: (a) T is similar to a unitary; (b) T -ls = ST*, 0 f F(S); (c) T-lQ = QT*,

Q Hermitian positive definite; and (d) T = XY-l where XX*= yy* are positive definite. Theorem 4. If T E G and

T* = u*T-1U where U is unitary and 0 f F(U), then Tis unitary. Theorem 5. If U E G is unitary, then A-lA*= U if and only if A= PV where P is positive definite Hermitian, V unitary, ~ = u* and P and V commute. Some consequences of A - 1A * = B-1B* are studied and related to certain commutators. (Received January 3, 1973.)

73T-A80. SUKHAMAY KUNDU, Thomas J. Watson Research Center, Yorktown Heights, New York 10598. Factorization of multi-graphs.

In this note, we generalize the results in the paper "A factorization theorem for a certain class of graphs" [Abstract 698-A4, these c/{otiai) 19(1972), A-772] to multi-graphs and multi-digraphs. One of our theorems is stated as follows. Let a degree sequence ( di) be called a-realizable if there exists an s-graph (i.e. , a graph with at most s lines joining a pair of vertices and no loops) whose degree sequence is ( di). The degree sequence of G is denoted by (G) and (di) - (di) stands for the sequence (di- df). Theorem. Let s ;;:: r + 1 and F be an r-graph.

There exists an s-graph G with degree sequence (d) and containing F if and only if the sequence ( di) - ( F') is a-realizable for all subgraphs F' ~ F. For s = r the necessary and sufficient condition holds true provided the graph

F belongs to a certain class of r-graphs :I, which generalizes the class of graphs described in the Abstract 698-A4.

A-260 In the case of digraphs, a similar theorem is true for s i!O r and arbitrary r-digraph F. (Received January 5, 1973.)

(Author introduced by Dr. Alan J. Hoffman.)

*73T-A81. ROBERT EDWARD LEWAND, Windham College, Putney, Vermont 05346. Extending a Jordan ring homomorphism.

In this paper a homomorphism from an ideal 18 of a quadratic Jordan algebra 3 without 2-torsion over a ring .P onto a unital quadratic Jordan algebra 3' without 2-torsion is extended to a homomorphism from 3 to 3'· We then show if ;Q is any class of quadratic Jordan algebras without 2-torsion, then the upper radical property determined by ;Q is hereditary. (Received January 8, 1973.)

*73T-A82. THOMAS BETH, Jnstitut fiir Math. Statistik, Universita"t GO"ttingen, 34 Gb"ttingen, Federal Republic of Germany. Algebraic resolution algorithms for some infinite families of 3-designs. Preliminary report.

In connection with Kirkman's School Girl Problem, J. J. Sylvester conjectured in 1875 that a

(3n, 3,1, 3)-design is resolvable for each positive integer n. This was first proved by R. Peltesohn (Diss. a. d.

Fr. -Wilh. -Univ. Berlin) in 1936. From Peltesohn's methods it is not clear that one can efficiently calculate a resolution for a given design. Here effective algorithms for large families of n's are developed: Theorem 1. For n E I let Vn be the n-dim. euclidian vector space over GF(3). There are (3;-1) base blocks of cardinality 3, which under suitable translations generate (3;-l disjoint parallel classes and hence a resolution of the (3n, 3, 1, 3)-design (Vn,93(Vn)). Definition. Let B = {a,b,c} be a 3-subset of GF(q), where q is a primepower "':!:_1 (6). Let liB= [b- a, c-b, a-c] be the equivalence class of all 6 difference triples of B. Let He be the subgroups of the eth roots

* d: -1 _J_ of unity, Se a SR of the cosets mod He. Theorem 2. Let a E GF(q) s. th. a 'f H2 U H3, af- -2, a r- -2. Let A:= {a,-(a+1)a-1, -(a+1)-1}, B: = {a-1,-a(a+1)-l,-(a+1)}. Then: An B =(I; 6A = -6B; :!I~ E GF(q)* s.th. 6A =

~[1,a,-(a+ 1)]. Corollary 1. If q = -1 (6) there is a partition !£> of X:= GF(q) U {oo} into 3-subsets such that 9 3(X)

= (S2 ot>) + GF(q). Corollary 2. If q = 1 (6) there are two partitions 9 and tl of X:= GF(q) U {oo1,oo2} into

3-subsets s. th. @3(X) = ((S3 • 9) + GF(q)) U ((S2 •tl) + GF(q)). (Received January 8, 1973. )(Author introduced by Dr. U. Krengel.)

73T-A83. E. M. WRIGHT, University of Aberdeen, Aberdeen, United Kingdom. Hamiltonian graphs and digraphs. Preliminary report. Take any Hamiltonian cycle (lL c.) in the complete graph on n nodes and let llh (r) denote the

number of lL c. which have just r edges in common with the original H. c. Let ~ (r) be the corresponding number

defined with respect to the complete digraph. We write B(n,r) = nl/{r!(n- r)!}. We find that ~(r) = B(n,r) Kn-r(O) -1 and Kn (0) + Kn_1 (0) = Dn-1' where Dn is Euler's rencontre number. Hence Kn(O) ~ (n -1) le for large n and -1 -2 r-1 -2 2 .Kh(r)~(n-1)1e /rlifr=o(n). AgainHn(O)~(n-1)!e and:sn(r)~(n-1)12 e /r!ifr =o(n). These

results enable us to apply the method which O'Neil (Proc. Amer. Math. Soc. 25(1970), 39-45) used for (0,1) matrices. An (n,q) graph (digraph) is one with~ nodes and q edges. We write 1/J(n) = qn-312• If 1/J(n) ~ oo as

n .. oo, almost all (n,q) graphs are Hamiltonian and almost all (n,q) digraphs are Hamiltonian. If 1/J(n) ~a>

(log 2) - 1/ 2, then the proportion of (n, q) digraphs which are non-Hamiltonian does not exceed exp(a- 2) - 1 in the limit.

In the other direction, if q = n(log n +log log n + c)/2, then the proportion of non-Hamiltonian (n,q) graphs is at least

1 - exp(-e-') in the limit. (Received January 8, 1973.)

A-261 73T-A84, HENRY W. GOULD, West Virginia University, Morgantown, West Virginia 26506 and L. c. HSU, Jilin University, Changchun, People's Republic of China. A new pair of inverse series relations.

The main object of this paper is to present the following: Theorem. Put >J! (x, n) = IT~ (a. + b.x), 1=1 1 1 with >IJ(x,O) = 1. Define a series transform by f(n) = :B~= 0 (-1)k(~)>IJ(k,n)g(k), Then inversely g(n) = :B~= 0 (-1)k(~)("k+ 1 +kbk+l)l/>(n,k+ 1)-1f(k). Conversely, if g is given as above, then the first formula is the inversion giving f in terms of g. This theorem generalizes a result of Gould (Duke Math. J. 28(1961), 193-202) and was communicated to Gould by Hsu in 1965. lh this paper we also present some corresponding extensions of infinite series transforms first given by Gould. (Received January 8, 1973.)

*73T-A85. ALBERT A. MULLIN, U, S, A. Research Office, 3045 Columbia Pike, Arlington, Virginia 22204. On geometric number-theory. Preliminary report.

Definition 1. By analogy with results of F. Herzog and B. M. stewart (Amer. Math. Monthly

78(1971), 487-496), an _!!-dimensional lattice point (x1, x2, ••• ,xn) is said to be highly visible iff the mosaics (see, e. g., the author's research problem in (Amer. Math. Monthly 79(1972), 1118-1119)) of x1 ,x2, ••• , xn have no prime number in common. Prelemma. Every highly visible point is visible (in the sense of Herzog and Stewart, loc. cit.), but not conversely, in general. Definition 2. Let F(r, n) be the number of highly visible lattice points in the n-dimensional ball of radius r. Lemma. Let r ;;; 2. Then F(r, 3) = k•r3 + O(r2), where the absolute constant k satisfies the inequalities 0 < k < 41!/31::(3), where ~ is Riemann's zeta-function. Problem 1. Estimate F for any finite-dimensional real Euclidean space. (Clearly, F is related to Riemann's zeta-function.) Problem 2. Estimate

F for any real Hilbert space. Problem 3. Estimate F for any real Banach space. Problem 4, Estimate F for any real NLS. (Received January 9, 1973.)

73T-A86. CHRISTOPHER R. HOWLETT, McMaster University, Hamilton, Ontario, Canada. Universal algebra in a topos. Preliminary report.

Sheaves of (finitary) algebras, i.e. classical universal algebras modelled in a Grothendieck topos,

are studied. The notions of polynomial and equation are introduced through formal categorical techniques and are

shown to be equivalent to systems of polynomials (resp. equations) parametrized by the site of definition. For a fixed finitary theory T the following theorems are proved. Theorem 1. If injectivity is well-behaved in the base variety

(models of T in sets), it is well-behaved in the category of sheaves of T-algebras provided the site of definition has enough points. Theorem 2. Any topologically compact sheaf of algebras is equationally compact. Theorem 3. In the

category of sheaves of modules over a sheaf of rings, homological purity is equivalent to a local type of equational purity. lh connection with Theorem 1 it is shown that if C is the site associated with the double negation sheaves on a topological space X (\ui} covers U iff U = r Int(UiUi)) then the category of points of C is isomorphic to the category of nonempty irreducible regular closed subsets of X. Hence for X Hausdorff, for example, this topos has no points.

(Received January 9, 1973.)

*73T-A87. RONALD J. EVANS, University of illinois, Urbana, lllinois 61801. Discrete free products of two complex cyclic matrix groups. Preliminary report.

View all matrices as linear fractional transformations on the extended complex plane. Let L+, L-

be the open half-planes to the right, left of the extended imaginary axis L, respectively. Let J be the set of complex

2 X 2 matrices M with real trace and determinant:!:. 1 such that M(L+) c L -. Let K be the set of M E J which

satisfy one of the following conditions: (1) \tr M\;;; 2, det M = 1, (2) \tr M\ = 2cos(rr/q) (q an integer;;; 2), det M = 1,

(3) M2 E J, det M = -1. We prove that whenever A E K and B E K do not both have a fixed point on L, then the group

generated by A and Bt (B transpose) is the discrete free product of the cyclic groups (A) and (B\ (Received

January 9, 1973.) A-262 *73T-A88, ALLAN B. CRUSE, University of San Francisco, San Francisco, California 94117. On embedding incomplete symmetric latin squares.

Let T be an r x r symmetric latin rectangle based on symbols 1, 2,, •• , n. Denote by N(i) the number of occurrences of the symbol i in T, Theorem. In order for T to be extendible to an n X n symmetric latin square based on 1, 2, •• , , n, it is both necessary and sufficient that (1) N(i) ~ 2r - n for every symbol i = 1, 2, ••• , n, and (2) N(i) =n (mod 2) for at least r of the symbols i. The proof is by induction on r and is based on the theorem of Mann and Ryser on systems of distinct representatives with "marginal elements" (Amer. Math. Monthly 60(1953),

397-401). Corollaries. Any incomplete n x n symmetric latin square can be embedded in a complete symmetric latin square of order 2n; any incomplete n X n symmetric diagonal latin square can be embedded In a complete symmetric diagonal latin square of order 2n + 1. (Received January 11, 1973,)

*73T-A89. STUART A. STEINBERG, University of Toledo, Toledo, Ohio 43606. On semiprlme P. L rings and their maximal rings of quotients. Preliminary report.

Let R be a ring with center CR and maximal right quotient ring Q = Q(R). Theorem 1. If Q is a regular right injective ring, then CQ is regular and self-injective, and Q is an injective nonsingular CQ -module.

Assume that R is either a semiprime P.I. ring or is strongly regular. Theorem 2. Then (a) CQ = Q(CR)' (b) R is an essential CR-submodule of Q, and (c) R is right-injective if and only if R is an injective CR-module.

Theorem 3. Q = RCQ if and only if for each q E Q and each (essential) ideal A of CR, qA !:: RCQ implies qCQ £: RCQ.

Theorem 4. R is a Goldie ring if and only if CR is Goldie, and then Q = RCQ' These results are a consequence of a theorem of W. S. Martindale III and a theorem of L. Rowen which in turn depend heavily upon the nonvanishing central polynomials of E. Formanek. Theorems 1 and 2 imply Theorem 5 (Martindale), Q is a P. I. ring and is the maximal left quotient ring of R, provided R is semiprime P. I. (Received January 11, 1973.)

73T-A90. DAVID ZEITLIN, 1650 Vmcent Avenue North, Minneapolis, Minnesota 55411. Identities for integer sequences involving the greatest integer function. Preliminary report.

Let M be a positive integer and define the integer sequence Wn by the relation Wn+ 2 = M Wn+ 1 + Wn' 2 where w 0 and w 1 are integers; if w 0 = o, w 1 = 1, then Wn = un. Let P > 0 and N < 0 be the roots of x - Mx- 1

= o; set F = M/(M+ 1), A(x) = jxw0 - w1j, D(j)= (M+ 1) Uk+j' and [x] the greatest integer function. Our results are:

(R1) If A(P) < pk/D(O), then ~Wn +F] = Wn+ k' n ~ k ~ 0. (R2) If A(N) < ~/D(O), then rJfw -n +F] = Wk -n , n ~ k;;; o. (R3) Let i = 0,1, .... If A(P) < pk/D(i), then [Pk+fwn + F] = Wn+k+i' n ~ k ~ o. (R4) Let C = 2 2 2 (PW0 -W1)/(P-N). If (P (F-j -1)/(P + 1)) < C !'!i F-j, then for n ~0, [P n+lw2n+1 + F] =W4n+ 2+ j, where j = 1,2, ... , 2 2 2 2 [F+ P ]. (R5) Define Cas in (R4). If F + j < C !'!i (P (F+ j+ 1)/(P + 1)), then for n ~ 0, [P n+lw2n+1 + F] = w4n+2 - j - 1, where j = 0,1, ... , [P2 - F]. (R6) Define C as in (R4). If (P4 (j- F)/(P4 -1)) !'!i C < j + 1- F, then for n ~ 0, 2 4 [P ~2 n + F] = W4n + j, where j = 1, 2, ... , [P + F- 1]. (Received January 11, 1973.)

Analysis

73T-B40. SUDHANSHU KUMAR GHOSHAL and BARADA KINKAR RAY, Regional Engineering College, Durgapur-9, West Bengal, India, On local fixed points. Preliminary report.

Let X denote a complete metric space. We prove the following theorems. Theorem 1. Let T1 and

T2 be two operators mapping X into itself such that p(T1x, T2y) !'!i a[p(x, T1x) + p(y, T2y)], 0

r-neighbourhood S(x,r) of the point x EX. Then T1 and T2 have a unique common fixed point. Theorem 2. Let T1

and T2 be two mappings of X into itself such that p(T~x, T~) !'!i ap(x, T~x) + tl p(y, TiY) for all x,y E S(x,r), an r-neighbourhood of the point x E X, p,q > 0 are integers, a> 0, 8 > 0, a+ tl < 1. Then T1 and T2 have a unique

A-263 common fixed point. Theorem 3. Let T 1 and T 2 be two continuous mappings of X into itself and let M be an everywhere dense subset of X such that for every pair of points x,y E M, p(T1x, T2y) ~o:p(x, T1x)+ {Jp(y, T2y),

0! > 0, {3 > 0, 0! + {3 < 1. Then T1 and T2 have a unique common fixed point in X. (Received September 14, 1972.)

73T-B41. SU-SIDNG CHEN, University of Florida, Gainesville, Florida 32601. B-convexity of complex manifolds.

We denote by B = B(X) the algebra of bounded holomorphic functions on a complex manifold X.

Theorem 1. Let D be a Riemann domain. Jf D is a domain of holomorphy, then D = Ukl\' where 1\ is

B(l\)-convex and 'i\ c Dk+1 for each k. Theorem 2. Let x 1 and x 2 be two complex manifolds, ~ be B(X2)­ convex and cl> be a proper holomorphic mapping from ~ onto x 2• Then ~ is B(X1)-convex. (Repeived September 18, 1972.)

*73T-B42. DAVID J. HALLENBECK, University of Delaware, Newark, Delaware 19711. Convex hulls and extreme points of some families of univalent functions. Preliminary report.

Let fl.= \z: lzl < 1l and let A be the set of functions analytic in fl.. Let S denote the subset of A consisting of the functions f that are univalent in fl. and satisfy f(O) = 0 and F' (0) = 1. Let F R denote the set of functions in S which are convex in the direction of the imaginary axis and real on (-1, 1). Let A have the topology of uniform convergence on compact subsets. We determine that the closed convex hull of FR denoted by W FR consists of the functions represented by f(z) = Jx(l/(x-x))log((1-xz)/(1-xz))dj.I(X) where X is the unit circle and 1o1 is a probability measure on X. The set of extreme points ofthis hull is {f:f(z) = (1/(x-i)) log((1-iz)/(1-xz)), lxl = 1j.

Let P(o:) denote the set of functions satisfying f(O) = 0, f' (0) = 1 and Re f' (z) > 0! for z in fl. where 0 ~ 0! < 1. It is known that P(O!) c S. We determine the closed convex hull of P(o:) denoted by WP(O!) and its extreme points precisely. We prove that if f E P(O!) then f(z)/z is subordinate to the function 20! - 1 + (20!- 2)log(1 - z)/z for z in fl.,

As· a corollary we show that if f E P(O!) then Re f(z)/z > (20! -1) + (2-20!)(1/lzl) log (1+ lzb for lzl < 1. Further we

show that if f(z) = r:: 1anzn is subordinate to any function in P(O!) then \ani~ 1 for n =1, 2, 3, ••• and the result is sharp for each n. Finally we determine the radius of convexity of P(1/2) to be 1/.j2. (Received October 2, 1972.)

73T-B43. HWANG-WEN PU, Texas A & M University, College Station, Texas 77843, A monotony criterion for arbitrary functions.

Wazewski proved that a necessary and sufficient condition for a real-valued continuous function to be nonincreasing is mf(Q) = 0, where Q = {x:Q+f(x) > oj and m is the Lebesgue measure (Ann. Polon. Math. 24(1951)).

This proof is very elementary and elegant. The only disadvantage of his proof is that it calls for continuity of the

function f several times and this condition is too strong for the result. In the present paper, the above mentioned

disadvantage is overcome and a criterion for arbitrary functions is obtained. (Received October 27, 1972.)

73T-B44. WILLIAM D. L. APPLI;NG, North Texas State University, Denton, Texas 76203. Continuity and upper and lower integrals.

U, F, p, p~, the notion of Integral and the notions of sum supremum functional, L, and sum infimum functional, G, are as in previous abstracts of the author. Theorem 1. If each of h and m is in p~, h is absolutely continuous with respect to m, 0 ~ M and 0 < c, then there is d > 0 such that if A is in p, lxl ~ M for all x in the range union of A and Su[L(Am)(I) - G(Am)(I)J < d, then Su[L(Ah)(I) - G(Ah)(I)J 0 such that if \Ail~= 1 is a sequence of elements of p such that for i = 1, ••• ,n, lxl ~ M for all x in the range union of Ai and Ju[L(Aim)(I)- G(Aim)(l)J < d, then Su£L(g(A1, ••• ,An)m)(I)- G(g(A1, ••• ,An)m)(I)]

A-264 hypothesis of Theorem 2 the condition "h is in p~ and absolutely continuous with respect to m" is added, then the conclusion of Theorem 2 remains valid if, in the final inequality, m is replaced by h. (Received October 24, 1972.)

73T-B45. CARL P. McCARTY, LaSalle College, Philadelphia, Pennsylvania 19141. Functions with real part greater than a.

Let P(a) denote the class of functions p(z) = 1 + b1z +••• which are analytic and satisfy Re p(z)

>a for lzl < 1 where a E [0, 1). A sharp lower bound is found for Re\zp'(z)/p(z)} as a function of \b1 \ and a. Also, the exact radius of convexity is found for functions starlike of order a and functions whose derivative has real part greater than a as a function of a and the second coefficient. This extends the results of the author found in

[Proc. Amer. Math. Soc. 35(1972), 211-217]. (Received November 8, 1972.)

*73T-B46. T. K. PUTTASWAMY, Department of Mathematical Sciences, Ball state University, Muncie, indiana 47306. The solution of a certain third order ordinary differential equation in the large.

In this paper, the author has solved in the large, the linear homogeneous differential equation of 3 3 3 2 2 2 third order, (1) z (a0 + a1 z) (d y/dz ) + z (b0 + b1 z)(d y/dz ) + z(c0 + c1z)(dy/dz) + (d0 + d1 z + d2z)y = o. The variable z is regarded as complex, as are the constants ai, bi' ci, ~ (i=O, 1, k=O, 1, 2) with a0 ,;, 0, a1 f. 0 and d2 f. 0. If

1.1 = -(aofa1), t.'len (1) will have two regular singular points, namely z = 0 and z = II.• and an irregular singular point, z = ro. The indicia! equation at z = 0 is found to be (2) a0h(h- 1) (h- 2) + b0h(h- 1) + c0h + d0 = 0. It is assumed that the roots hi (i=1,2,3) of (2) are such that no two of them differ by an integer. This paper is a sequel to an earlier paper by the author (Abstract 691-34-8, these cilotiuiJ 19(1972), A-127). (Received November 8, 1972.)

73T-B47. CHARLES K. CIWI and CHIN-IWNG CHING, Texas A & M University, College station, Texas 77843. A uniqueness theorem for analytic functions. Preliminary report.

Let U be the open unit disc and T the unit circle in the complex plane. We obtain the following result. Theorem. Let f be holomorphic in U and continuous on U such that its restriction on T is a function of bounded variation. If E~= 1f(ei 2 7Tk/n) = 0 for n = 1, 2, ••• , then f is the zero function. (Received November 9, 1972.)

*73T-B48. ALDO J. LAZAR, Tel Aviv University, Tel Aviv, Israel and DONALD C. TAYLOR, University of Missouri, Columbia, Missouri 65201. Derivations on Pedersen's ideal of a c*-algebra. Preliminary report.

Let A be a c*-algebra, K its Pedersen's ideal, and r(K) the double centralizers of K. Let K denote the locally convex Hausdorff topology for r(K) generated by the family of seminorms >-x:[(S, T)] = l!S(x) ll and

Px[(S, T)] = \\T(x) ll for all x E K. Proposition 1. Let B be a It-closed, selfadjoint commutative subalgebra of r(K). If 6 is a derivation on B, then 6 = o. Theorem 2. For each derivation 6 on K there is a unique derivation "'6 on

r (K) that extends 6; moreover, 6 is II -continuous. Theorem 3. If A is a two sided ideal in some w*-algebra, then every derivation on r(K) is inner. (Received November 13, 1972.)

*73T-B49. ROGER T. LEWIS, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. Oscillation criteria for fourth order linear differential operators.

The operator L4y = (ry")"- (qy')' + py is oscillatory on (O,ro) if for any a> 0 there is a b >a and a function y I= 0 such that y(k) (a) = 0 = y(k) (b) for k = 0, 1 and L4 y = 0. Theorem 1. If 0 < r ;!0 M, j 00q = -ro, and j 00i\PI < ro then L4 is oscillatory on (O,ro). Theorem 2. If .f 00p = -ro, .f00q = -ro, and 0 < r ~ M then L4 is oscillatory on (O,ro). Theorem 3. Let P1(x) = .f:p. If 0 < r;!S M, JP1 = -ro, j 00\q\/x < ro, and q(x)-+ 0 as x ... ro then L4 is oscillatory on (O,ro). Theorem 4. Suppose 0 < r ~ M, .f00p < ro, q ;!0 0, and P 1 ~ Cx-4 for x > o.

If lim infX""

A-265 73T-B50. EVELYN MARIE SILVIA, Clark University, Worcester, Massachusetts 01610. On a class of spirallike functions. Let a= 0, 0 1!5 fJ < 1,1 AI < '11/2 and suppose that f(z) = z + ~: 2anzn is holomorphic in U = lz/lzl < 1}. If Re[eiAzf'(z)/f(z) + a(zf"(z)/f'(z) + 1 - zf' (z)/f(z))] > fJ cos A for z E U, then f(z) is said to be a-A-Spirallike of order fJ and we write f(z) E s!(fJ). The class S~({J) is the class of A.-spiral functions of order S defined by R. Libera [Canad. J. Math. 19(1967), 449-456]. Also, S~(O) is the class of a-starlike functions introduced by P. Mocanu [Mathematica (Cluj) 11(1969), 127-133]. The author shows that for each a !!: 0, a-A-spirallike functions of order fJ are N-spirallike of order fJ. The following representation theorem is obtained: The function f(z) E s!(fJ)

(a> 0, 0 1!5 fJ < 1, IAI < '11/2), if and only if there exists a function F( 1;:) A-spirallike of order fJ such that f(z) = iA z eiA/a -1 ae-iA 1 [e /a Jo [F(I;:)] C di;:J . A distortion theorem for loglf(z)/z and a rotation theorem for argf(z)/z are also proved for functions in this class. (Received November 13, 1972.)

73T-B51. ERWIN 0. KREYSZIG, University of Karlsruhe, Karlsruhe, Federal Republic of Germany. On a differential operator by E. Pescbl and K. W. Bauer. --

The equation (1) uzz* + Am(m+ 1)T-2u = 0, T = 1 + AZZ*, m integral, appears in connection with the wave equation and minimal surfaces. E. Pescbl and K. W. Bauer developed a corresponding function theory, using a differential operator from the space of complex analytic functions into the space of solutions of (1) (cf. K. W. Bauer,

J. Reine Angew. Math. 221(1966), 48-84, 176-196), and M. Kracht and the author (Manuscripta Math. 1(1969),

369-376) proved that the operator can be converted to a Bergman integral operator B defined by (Bf) (z, z*) =

J_~ g(z, z*, t)f(l/1)1,0 - 1/ 2 dt, cp = 1 - t2, 1/> = zcp/2. It is shown that for (1) one even has the elementary representation g(z, z*, t) = (1- a) - 1/ 2 cos [(2m+ 1)sin- 1a1/ 2], a= kAzz*t2/ T, k any constant. This representation can be used in characterizing the behavior of the classes of solutions corresponding to associates f that are meromorphic or have isolated essential singularities. (Received November 14, 1972.)

73T-B52, RIDGLEY LANGE, Temple University, Philadelphia, Pennsylvania 19122. Complemented subspaces of decomposable operators. Preliminary report.

Let T be a bounded linear operator on the complex Banach space X. Call T refined if T has the single-valued extension property and, for each closed set F in the complex plane 'IT, XT(F) is a closed subspace of X such that X= ~(F) E9 ~('IT- F). (For terminology see Foia~, "Spectral maximal spaces and decomposable operators­ in Banach space", Arch. Math. 14(1963), 341-349.) Theorem. Tis refined if and only if T is decomposable.

Answers to two questions posed by Foias are given; viz, Theorem. Let Y be a spectral maximal space of the decomposable operator T. Then (1) the restriction TIY is decomposable and (2) the operator T induced on the quotient XIY by T is also decomposable. (Received November 15, 1972.)

*73T-B53. MYRON S. HENRY, Montana State University, Bozeman, Montana 59715. Approximate solutions of functional differential equations. Preliminary report.

Consider the initial value problem (*) X(t) + a(t)X(g(t)) = h(t), X(O) = a 0, X(O) = a 1• Suppose that A =max 1a(t)l, G=max T T]lg(t)l. Then if A max(2G,G2) < 2, (*)has a unique solution y(t) on I= [-T,T]. [-T,T1 [- , .. I "'lll-1 i I . Let L(X,g) = X(t) + a(t)X(g(t)). Then sup1 LJi=2 ciL(t ,g)+ a 0L(1,g) + a 1L(t,g)- h(t) J.S minimized for some 2 (c2, ... ,c;+1), and if Plll-1(t) = a0 + a 1t+c2t + ... +cn+1tlll-l, then limn_.00 IIP~l 1 (t)- yg~11 = 0, j = o, 1, 2. If [L(t2,g),.,,, L(tn+l,g)} forms a Chebyshev set, then the above best approximations are completely characterized by known properties of Chebyshev sets and may be computed by the second algorithm of Remes. Error analysis is considered, and examples of operators that yield Chebyshev sets are given. (Received November 16, 1972.)

A-266 73T-B54. LAWRENCE A. ZALCMAN, University of Maryland, College Park, Maryland 20742. Mean values and differential equations.

Let P((1, ••• , ( 0 ) be a homogeneous polynomial. There exists a measure j.l such that, for D a domain in JR0 , u E C(D) is a weak solution of P(o/ox1, ••• , o/oxJu = 0 if and only if Ju(x+rt)dj.l(t) = 0 for each xED and 0 < r < dist(x, oD). Whlle the measure j.l is by no means unique, it is easy to characterize the collection of measures with this property and (in certain cases) to determine an "optimal" choice of measure. The case

P((1, ( 2) = (~ + (~ yields the mean-value theorem for harmonic functions and its converse; the choice P((1, ( 2) =

( 1 + i(2 gives rise to versions of the theorems of Cauchy and Morera. Our results extend to functions in rioc(D) and, in the case D = lRn, yield "two-circle" theorems of the sort discussed in our paper "Analyticity and the Pompeiu problem" (Arch. Rational Mech. Anal., in press). (Received November 17, 1972.)

73T-B55. WITHDRAWN.

*73T-B56. JOHN G. AIKEN, Louisiana State University, Baton Rouge, Louisiana 70803. A perturbation property of w* algebras.

Let M be a Banach algebra with identity. Theorem A. If an element A of M is contained in a proper two sided ideal of M, then A does not perturb any T in M, i. e. the spectrum of T + A intersects the spectrum ofT for all T in M. Dyer, Porcelli, and Rosenfeld proved the converse of this result forM= ~H). the algebra of all bounded operators on a HUbert space H, and they conjectured it for a rr1 factor. The author proves that the converse of Theorem A is true for any w* algebra but that it is false for the shift algebra, the c* subalgebra of ~(H) generated by a unilateral shift. The problem now is to classify all Banach algebras with identity where the converse of

Theorem A is true. (Received November 17, 1972. )

*73T-B57. GERD H. FRICKE, University of Kentucky, Lexington, Kentucky 40506. A note on multivalence of a function of bounded index. Preliminary report.

The relationship between bounded index and multivalence of an entire function is examined.

Theorem 1. Let f be an entire function. Then f' is of bounded index if and only if there exists an integer N > 0 and a constant 6 > 0 such that, for any point in the plane, f or one of the first N derivatives has radius of univalence of at least 6. Theorem 2. Let f be an entire function. Then f' is of bounded index if and only if there exists an integer p > 0 such that f is p-valent in any disc of radius 1. (Received November 20, 1972.)

73T-B58. MADAN MOHAN CHATTERJEE, Regional Engineering College, Durgapur, West Bengal, India. Some results on common fixed point theorems for sets of operators.

A generalised result, in the case of sets of operators on a complete metric space, has been deduced.

If Ti and Tj are two sets of operators mapping a complete metric space X into itself such that (1) d(Tfx, TjY) !!! + where x,y EX; p,q are positive integers, a> 0, f3 > 0 and a+ < 1. ad(x,T~x)1 ,.."d(y,T~) J fJ (2) A sequence 1~1 is defined in general as x2n+l = T~nt 1x2n and x2nt2 = Tint2x2n+l where x E X is arbitrary and the x~s are distinct.

Then Ti and Tj have a unique common fixed point for i = 1, 3, ••• , 2n + 1, ••• , and j = 2,4, ••• , 2n + 2, •••

(Received November 21, 1972.) (Author introduced by Professor Sudhanshu Kumar Ghoshal.)

73T-B59. DAVID G. COBrA, Brown University, Providence, Rhode Island 02912. Asymptotic behavior of solutions of symmetric hyperbolic systems. Preliminary report.

Let u(x, t), x E lRn, n odd ~ 3, t ~ 0, be a solution of a first order symmetric hyperbolic system of

partial differential equations, with smooth initial data f of compact support. Theorem. Assume the system is strictly

A-267 hyperbolic with the characteristic roots being convex functions. Then, \\u(•,t)\\r"' Cqtn/q-(n-1)/2 \\f\\W(n+l)/2,p for all t > 0, where 1/r= 1/p+ 1/q-1;;; 0, 2n/(n-l) < q;!; ro; in particular, sup \u(x,t)\ =O(t-(n-1)/2) as t ... +ro. X (Received November 21, 1972.)

*73T-B60. SANFORD S. MILLER, state University of New York, Brockport, New York 14420, PETRU T. MOCANU, Babes-Bolyai University, Cluj, Romania and MAXWELL 0. READE, University of Michigan, Ann Arbor, Michigan 48104. The radius of a-convexity of univalent functions, 1 1E a;;;; ro. Preliminary report.

The authors prove the following theorem. Let f(z) = z+ ••• be analytic and univalent in the unit disc

\z\ < 1, and let a be a constant, 1;!; a;;,; ro. Then the inequality Re[(1-a)zf'/f + a(1 + zf"/f')] > 0 holds for \z\ <

(1+ a) -J(1+ a)2 -1. This radius (of a-convexity) is sharp for the class of univalent functions, for 1;!; a;;;; ro. This

complements a result due to Cernikov [Mat. Zametki 11(1972), 227 -232] who obtained the result for coth 1 - 1;!; a ;;:; 1.

(Received November 24, 1972.)

73T-B61. THOMAS H. MacGREGOR, State University of New York, Albany, New York 12222. Hull subordination and extremal problems for starlike and spirallike mappings. --

Let a denote the set of functions analytic in /:;. = {z: \z\ < 1], and let E(HF) denote the extreme points

of the closed convex hull of a subset F of a. We show that ifF is compact and if E(HF) = {f:f(z) =f0(xz),\x\ ~ 1],

then L(f) is hull subordinate to L(f0) in /:;. for each f in F, where L is any continuous linear operator of order zero. This generalizes results proved by R. M. Robinson in [Trans. Amer. Math. Soc. 16(1947), 1-35]. We indicate

families F to which this theorem is applicable and also identify each such operator L with a suitable sequence of

complex numbers. Let .P be a nonconstant entire function and suppose that 0 < \z0 J < 1. Then the maximum of

Re{.P [log f(z0)/z0J] over the class of starlike mappings of order a is attained only for a function of the form f(z) ~ z/(l-xz)2- 2a,\xl = 1. A similar result bolds for spirallike functions. These theorems generalize the corresponding

result of G. M. Golusin for starlike mappings proved in [Amer. Math. Soc. Transl. (2) 18(1961), 1-14]. (Received

November 27, 1972.)

*73T-B62. S. K. BAJPAI, Clark University, Worcester, Massachusetts 01610. Two convexity theorems for certain classes of analytic functions. Preliminary report.

Let f(z) = z + D:2anzn be an analytic function in the unit disc Dl\z\ < 1]. Define F(z) = 2z - 1 J;f(t) dt. If F(z) is starlike of order {3, then in Theorem 1 it has been found in what disc f(z) will be starlike of order 0!. This settles a question raised by Libera and Livingston (see: Proc. Amer. Math. Soc. 30(1971), 331).

Theorem 2 extends a Theorem 6 of Libera and Livingston (see: Czechoslovak Math. J. 22(1972)) to a wider class

which also includes a result of the author (Indian J. Pure Appl. Math. (1971)). (Received November 28, 1972.)

(Author introduced by Professor Herb Silverman.)

*73T-B63. N. K. SHARMA, Pahlavi University, Shiraz, Iran. Isolated points of the spectra of conservative matrices. Preliminary report.

Let /:;. denote the Banach algebra of conservative lower triangular matrices. Let A =lank] be a

member of this algebra. Then one can prove the following: Theorem 1. The only possible isolated points of the

spectrum of A are ann's. In case of a conservative Hausdorff method the following stronger theorem can be proved:

Theorem 2. If spectrum of H is disconnected then it consists of only two components one of which is the singleton set

{!Jol where 1-'o is the first diagonal element of H. Theorem 1 answers the Question 2 of "Spectra of conservative matrices," Proc. A mer. Math. Soc. (1972) in the negative. (Received November 17, 1972. )(Author introduced by

Professor Hossein Zand.)

A-268 73T-B64. BRUCE P. PALKA, Brown University, Providence, Rhode Island 02912. On the uniform convergence of quasi-conformal mapPings, Preliminary report.

Let D be a domain in extended Euclidean n-space with "smooth" boundary and let I f.l be a sequence J of K-quasiconformal mappings of D which converges uniformly on compact subsets of D to a quasiconformal mapping. This paper considers the Question. When does the sequence I fj l converge uniformly on all of D? Geometric conditions on the domains Dj = fj(D) are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in Rn is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane. (Received November 16, 1972,)

73T-B65. ROBERT W. CARROLL, University of Maryland, College Park, Maryland 20742 and University of lllinois, Urbana, lllinois 61801. Some control problems with differentiably constrained controls.

Some control problems are considered with cost functionals of the form (i = 1, 2) J i (v) = Kr (v) +

:E~ 1 ajGj(v)wbere aj=O or 1, ~(v) = 1\Cy(v) -z 1 1\~2 (F), ~(v)= 1\Dy(T,v) -z 2 1\~, G1 (v)= \\v-g1 1\~. G2(v)= 1\v(T) - g2 1\i, G3(v) = ((Mv(T), v(T)))E,and G4(v) = ((Nv, v))U. The state variable y(v) = y(t, v) satisfies y' + Ay = 2 2 2 Bv + f E L (V'); y(O, v) = y 0 E H where y E L (V) and y' E L (V') with adjoint states pi (v) determined as in the book by J. L. Lions, "Contr~e optimal de syst~mes gouverne's par des e'quations aux de'rive'es partielles", Dunod, Paris, 1968. In order to have well defined values v(T) a control space of "smooth" functions of the form U = ~(E) = {vEL2 (E); v' E L 2(E); v(O) = ol is used where E, F, V, and H are real separable Hilbert spaces with V c H c V'

and Vis dense in H with continuous injection; A, B, C, D, M, and N are suitable bounded operators and the zm' gm,

and f are arbitrary, It is shown that the unconstrained background cases are characterized by differential equations A~1 B*(•)pi(•,u) + (1-D2)g = 0 (where (e,~) = (AEe,~)) which involves additional smoothness of optimal controls in 0 the sense that g' = a1(u' -g]_) + a4Nu' E C (E). Terminal conditions also hold in the form (a1 + a4N)u'(T) +

(a2 + a3M)u(T) = a1g]_ (T) + ~g2 • (Received November 13, 1972.)

73T-B66, PAUL ROSENTHAL, Apt. 103, 90 Heather Avenue, San Francisco, California 94118 and Stanford University, stanford, California 94305, On the location of the singularities of the function generated by the Bergman operator of the second kind. Preliminary report.

Let 1/l(z, z*) = P2(f) = f.~,E(z, z*, t)f(~z(l-t2 ))dt/ ,Jl7" be Bergman's operator of the second kind.

P 2(f) maps functions f(q) analytic at q = 0 into solutions of the partial differential equation 1/lzz* + N((z+ z*)/2) •(1/lz + 1/JzJ = O, z = X+ iy, z* = X- iy, N(X) is analytic for - oo oj, N(X) = -1/12X (Tricomi case), z* = z (conjugate of z). In a new paper we generalize our results and also obtain an order relation for certain

fixed X, y ... oo, where now f(q) is a rational function with certain prescribed poles. (Received November 16, 1972.)

(Author introduced by Professor Halsey L, Royden.)

*73T-B67. JAN MYCIELSKI, University of Colorado, Boulder, Colorado 80302. Monte Carlo interpolation over a measure-space.

Let (X, j.l) be a probability measure space, C be a finite covering of X with no more than c

j.l-measurable sets, and f:X ... lo, 1j be a j.l-measurable function. Let us choose n + 1 points x1, ... ,xn and x at random in X. Theorem, The probability that there exists an A E C such that x E A and f(xi) # f(x) for all

x1 E lx1, ••• ,xnJ n A is not larger than (2c/(n+ 1))(1+ 1/n) -n. Thus if we find a set A E C such that x E A and f(xi) = v for all xi E 1~, ... ,xnJ n A then the probability that f(x) = v is 1- o(n). This theorem can be applied to the theory of learning. It improves some theorems in [A. Ehrenfeucht and J. Mycielski, "Interpolation of functions,,.,"

A-269 J. Approximation Theory (to appear)] in the respect that the probability estimate does not depend on 1-1• (Received

December 5, 1972.)

*73T-B68. ATHANASSIOS G. KARTSATOS, University of South Florida, Tampa, Florida 33620. Bounded solutions to perturbed nonlinear systems and asymptotic relationships. Preliminary report.

system: (*) x' A(t,x)x + B(t,x) and let S x ERn; 1\x\\ ~ r). Consider the differential = r =! Theorem. Assume that (i) A E C (R x Sr' L(Rn, Rl)]; (ii) B E C (R x Sr' R~; (iii) for every m = 1, 2, ..• , and every f E C([-m, m]) with norm \\film = sup llf(t) \1 ~ r, there exists a fundamental matrix Xlt) of solutions of the system x' = A(t,f(t))x and a projection matrix Pf such that \1 J'~m Xf(t)PrX;\s)B(s,f(s)) ds- .ftmXf(t)(I- Pf)x;1(s)B(s,f(s))ds 11m

~ r. Then there exists at least one bounded solution of ( *) defined on R. This result is used to obtain almost periodic, or periodic solutions for the system (*). Asymptotic relationships between (*) and its associated unperturbed system are considered as well as convergence of the solutions of (*)· (Received December 6, 1972.)

73T-B69. WITHDRAWN.

73T-B70. YIN-CHEONG MA, University of Toronto, Toronto 181, Ontario, Canada. Covariance algebras of operator-valued measures. Preliminary report.

A w*-covariance triple is a system (G,A, a) consisting of a locally compact group G, a w* -algebra

A, and a a-weakly continuous action a: G ... Aut(A), of G on A. The Mackey-Takesaki notion of induced covariant representation is extended to the context of W*-covariance triplets, and then one can introduce in M(G,A), the Banach space of A-valued regular Borel measures on G with bounded variation, a convolution and an isometric involution, depending on a, with respect to which it becomes a Banach *-algebra M(G,A, a) = M. G and A have natural top-algebraic embeddings in M, equipped with an appropriate weak topology cr, and together generate M. A natural bijection exists between unitary cr -weakly continuous *-representations of M and covariant representations of

(G, A, a), corresponding representations having the same commutant. The projective tensor product M(G) ®A is embedded in M(G,A); it consists of all measures defined by densities. A c*-covariance triplet with units, (G, C, a), consists of a locally compact group G, a c*-algebra C with unit, and a norm-continuous action a: G ... Aut(C), of G on c. Wendel's result on centralizers of L 1-group algebras is generalized to the context of c* -covariance triplets with unit. (Received December 8, 1972.) (Author introduced by Professor L. Terrell Gardner.)

73T-B71. ZIAD S. ALI, IBM Corporation, Kingston, New York 12401. @Uln) summability of Fourier series.

This paper extends some results ofT. Pati (Indian J. Math. 1-3(1958-61), 85-90). Let f(t) be a periodic function with period 2rr, which is integrable in the sense of Lebesgue over the interval (-rr, rr). Let the

00 sin nt -j, a +I: A (t). Let cp(t) = (t) f(x+t) + f(x-t) - 2f(x); Fourier series of f(t) be -j,a0 + L:: nna cos nt + b n = 0 n .,..x,~ = .P(t) = j'~ \cp (u)l du, and PT = [1/t], the integral part of 1/t. Theorem 1. Let (N, pn) be a regular Norlund method, defined by real, nonnegative, monotonic nonincreasing sequence of coefficients (Pnl• such that P n ... ro, and H(n) =

J;(h(t)/t) dt = O(Pn)' as n ... ro, where h(t) is any slowly oscillating function; then if .P(t) = o(t•h(1/t)/PT)' as t ... +0, then the Fourier series of f(t), at t = x, is summable (N,pn) to f(x). The case h(t) equals a constant is Pati's theorem. Theorem 2. Let (N,pn) be a regular Norlund method, defined by real, nonnegative, monotonic nonincreasing sequence of coefficients (p ) , such that P ... ro, and g(n) = o(P ) , as n ... ro, where g(n) = n(h(t )/t) dt, n n n J1 and h(t) is any slowly oscillating function; then if .P(t) = O(t • h(l/t))/PT as t ... +0, then the Fourier series of f(t) at t = x, is summable (N,pn) to f(x). Further similar results for "Conjugate Fourier series" (H. P. Dikshit, Rend.

A-270 Circ. Mat. Palermo 11(1962), 217-224), and for "A series associated with the derived Fourier series'' (V. P. Kataria,

Rend. Circ. Mat. Palermo 16(1967), 121-128) are obtained. (Received December 8, 1972.) (Author introduced by

S.M. Shah.)

*73T-B72. ROBERT G. BUSCHMAN and H. M. SRIVASTAVA, University of Victoria, Victoria, British Columbia, Canada. Inversion formulas for the integral transformation with the H-function as kernel.

In this note the authors present a systematic discussion of the inversion problem involving the integral

transformation (*) j'rox- 1 n[xy] f(x) dx f(y), which is a Mellin convolution 0 Ifl•p,q ~ with the H-function of C. Fox [Trans. Amer. Math. Soc. 98(1961), 395-429]. In particular, relations between some of the known forms of inversion of (*)

are considered, the difficulties in the inversion given recently by R. Singh [Proc. Nat. Acad. Sci. India Sect. A 40

(1970), 57-64] are explained, and certain new forms of the inversion are obtained. (Received December 11, 1972.)

73T-B73. DOUGLAS N. HAWLEY, Morehouse College, Atlanta, Georgia 30314. Idempotent multipliers. Preliminary report.

This paper examines subsets of G which do not support idempotent multipliers on L1 (G) for G a compact connected group. A finite union of intervals S in G does not support an idempotent multiplier on L1 (G) unless S is finite or co-finite, but any such S supports an idempotent multiplier on Lp (G) if 1 < p < ro. If

A t_; B c G, A is infinite and well-ordered and every element in B is greater than or equal to any element in A, then

A :.; B does not support an idempotent multiplier on L1 (G). Throughout this abstract G has denoted the topological dual of G. (Received December 11, 1972.)

73T-B74, MOUR.<\D EL-HOG"SSIE}<1 ISMAIL, University of Alberta, Edmonton 61, Alberta, Canada. A sequence-to-function analogue of the Hausdorff means for double sequences. The [J, f(x, y)] means.

In this paper we extend the Jakimovski [J,f(x)] means to double sequences. We call the new means

the [J,f(x,y)] means. We characterize such f's that give rise to regular and to totally regular [J,f(x,y)] means. We

also give a necessary and sufficient condition to represent a function f(x, y) as a double Laplace transform with a

determining function of bounded variation in two variables. (Received December 13, 1972.)

*73T-B75. STEVEN ELDON MOSIMAN, University of Missouri, Columbia, Missouri 65201. The strict topology and the ordered vector space C(X). Preliminary report.

Characterizations are given of spaces of measures which are duals of C(X) when C(X) is equipped

with a strict topology, f3L' determined by a collection, L, of compact subsets of f3X- X. An example of such a characterization is the following theorem which relates f3L to the natural order on C(X). Theorem. Let M 6 be a sublattice of i\!(X) containing the collection of all point measures of points in X. Then M 6 is the dual of a strict topology iff i\1 E l\!(X): 6 ~ (C!l for any net lfo:)o:EA ~ C(X) with fo:--. 0 and fo: ... 0 in the topology cr(C(X),M 6), one has '-'(fo:) ... 0). The strict topology, f3u is shown to be closely tied to Dini's theorem and is used to study spaces of

measures from an order theoretic point of view. (Received December 14, 1972.)

*73T-B76. ANDRE de KORVIN and LAURENCE E. KUNES, Indiana State University, Terre Haute, Indiana 47809. Operators from X into 1p where X is reflexive.

For notation see Abstract 72T-B277, these cJYotiai) 19(1972), A-701. Proposition. Assume that X

is a reflexive space. The following conditions are then equivalent: (1) cr is Hausdorff in the topology generated by

Pm* as m*E Vp. (2) The set of all finite sums of the form ~simf(Ei)/IJ(Ei) where si are scalars, Ei E :1::: and

m~ E V is norm dense in x*. (3) The topology on cr is generated by P * as m* E V is stronger than 1 p m p the weak topology of cr. Theorem. Let X be reflexive. Assume that p has property (J). Let G* be the measure

A-271 corresponding to T. If T is compact, cr is compact in the P G* topology. Conversely if cr is compact in the P G* topology then Tis compact. Define ll!u!ll~ =sup{:B(y*, U(f.•x.))} where the sup is over y*E cr* and p(:Bf.•x.);<;; 1. 1 1 1 1 operator from ?It (X) into Y with !!lull!~< oo then there exists some Proposition. Let U be a bounded linear p measure G such that U(f • x) = T(f) • x = Jf dG(x) and G generates a family of seminorms on Y* such that whenever U is compact then cr* is compact. (Received December 15, 1972.)

*73T-B77. RICHARD A. ALO, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 and ANDRE de KORVIN and LAURENCE E. KUNES, Indiana State University, Terre Haute, Indiana 47809. Operators with range in L(X, Y).

For notation see Abstract 72T-B277, these cJ{oliaJ) 19(1972), A-701. Lp ®A X will denote the least cross norm product of L with X; assume ?It = L , let = SP{f: 0 ... X/f is weakly measurable and w (f) < oo} p p p p vr p where w (f)= sup p(x*f). If A E!: define W (A)= sup w (!:XE • x ) where En c A, En E c'J, En are disjoint, and p p p n n p(!:XE • x ) ;;; 1. p will be c. a. if W (0) < oo and for A '>{A, W (A ) ... 0. Let T* be the unit ball in X*. If T* n n p n p n is compact in some appropriate topology this implies that p c. a. is equivalent to some elements of Lp ®A X is also equivalent to a family of compact operators of L(L *,X) converging uniformly in the A norm to 0 and this p converging to 0. Finally a Darboux type of property is shown for some topologies i. e. , a topology TopX is defined and wk* < TopX < N where N is a norm topology; then one can pick x1 and x2 such that TopX = wk* and TopX = N. 1 2 (Received December 15, 1972.)

73T-B78. JOHN J, SWETITS, Old Dominion University, Norfolk, Virginia 23508. Almost summable sequences of O's and 1's. Preliminary report.

A summability matrix, T = (amn>• almost has the Borel property, T E (ABP), if T transforms almost all sequences of O's and 1's into sequences which are almost convergent to 1a. We investigate the consequences of assuming T E (ABP), and seek conditions sufficient to guarantee T E (ABP). The results are analogues of theorems of J. D. Hill for summability matrices which possess the Borel property [Ann. of Math.

46(1945), 556-562; Pacific J. Math. 1(1951), 399-409]. Theorem. If T E (ABP), then (i) l:B:0amnl:=o is almost convergent to 1, (H) for each n, {amnl:=O is almost convergent to O, (iii) :B: 0 a~n < ro for each m, and 2 (iv) limp... ro :B:0((1/ (p+ 1)) :Bf=oam+j, n) = 0 uniformly in m. (Received December 18, 1972.)

*73T-B79. NATHANIEL A. FRIEDMAN, State University of New York, Albany, New York 12222. Induced mixing of all orders. Preliminary report.

The author and D. S. Ornstein have proven that each invertible ergodic measure preserving transportation on the unit interval with Lebesgue measure induces mixing transformations on a dense class of subsets

(to appear in Advances in Math.). The construction is refined to prove Theorem. Each invertible ergodic measure preserving transformation on the unit interval with Lebesgue measure induces transformations that are mixing of all orders on a dense class of subsets. (Received December 18, 1972.)

*73T-B80. H. M. SRIVASTAVA and REKHA PANDA, University of Victoria, Victoria, British Columbia, Canada. Some operational techniques in the theory of special functions.

In the literature on special functions one often finds applications of operational techniques involving various integral transforms. The Laplace transform and its inverse, which have indeed been exploited a great deal, are capable of yielding fruitful results. In the problem of augmentation of parameters of the generalized hypergeometric function PFq [z], however, they obviously lack in the sense that any single application of the Laplace transform (or its inverse) can augment only one numerator (or denominator) parameter. With this point in view,

A-272 several workers, for instance, H. M. Srivastava and J. P. Singhal [Proc. Cambridge Philos. Soc. 64(1968), 425-430],

have introduced more effective tools for augmenting parameters in the PFq function. In the present paper the authors

consider a number of generalizations of the Srivastava-Singhal transform and show how these transformations lead to

interesting operational techniques to derive linear, bilinear and bilateral generating functions, and certain reduction

formulas, involving a large variety of special functions. (Received December 11, 1972.)

*73T-B81. R. E. EDWARDS, Australian National University, Canberra, A. C. T., Australia 2600 and KENNETH A. ROSS, University of Oregon, Eugene, Oregon 97403, p-Sidon sets.

Let G be an infinite compact Abelian group with character group X. Consider 1 ;!ii p < 2 and let

q =p/ (P - 1), r = 2p/ (2- p), s = 2p/ (3p - 2). A set E c X is defined to be p-Sidon iff fA E p,P for every E-spectral

function f E C(G). Theorem. E is p-Sidon iff MAlE=> J,q(E) iff (L 1) AlE=> J.q(E). Numerous conditions, each

necessary for E to be p-Sidon, are given. Among these are (i) 11 A E ~,r for every E-spectral measure 11 E M(G);

and (what is equivalent) (ii) C AlE::::> J.s(E); (iii) card(E n ci?) = O((log N)r/2) for N-element sets ci? ranging over

suitable "test families" (as in Indiana Univ. Math. J. 21(1972), 790). If E is p-Sidon and p < 4/3, then

min(card(A), card(B)) is bounded for finite subsets A, B of X satisfying ABc E. Not all p-Sidon sets are (1-)Sidon

sets. For example, X necessarily contains infinite sets A and B such that AB is p-Sidon iff p ;;:; 4/3; it suffices to

select disjoint A and B so that A U B is dissociate. (Received December 19, 1972.)

*73T-B82. ANTOINE DERIGHETTI, Universite"de Lausanne, College prope"deutique de Dorigny, 1015 Lausanne, Switzerland. On wesk containment. III.

Let !;G be the set of all continuous unitary representations of an arbitrary locally compact group G

·with a left invariant measure dx. Let H be a closed subgroup of G with a left invariant measure dh and let q be a

strictly positive continuous solution of q(xh) = q(x)AH(h)AG(h- 1) for x E G, h E H; dx is the corresponding quasi­

invariant measure on G/H. We define aH to be the convex hull of IAxlxE H} where Ai(y) = f(yx)AG(x) for f E L1(G),

x, y EG. Proposition 1. For every 1T E !;G and f E L1(G) we have infll\lT(Af)\IIA E aH};!ii\ITifl\1' where !T(f) is the bounded operator SGf(x) !T(x)dx and T If is the function of L 1 (G/H) defined by T JCx) = SJCxh) (q(xh)) - 1dh.

Proposition 2. Let 1T be an element of !:G. There is no lT' E !;G weskly contained in 1T of which restriction to H

werJdy contains the one-dimensional identity representation iH of H if and only if infl 1\ lT(Af) Ill A E a Hl = 0 for every

f E L \a). Proposition 3. Let 1T be an element of :r.H. The representation 1T does not weskly contain iH if and only

if infl SG/H\IT lTAf(x)\1 ctX\A E a Hl = 0 for every f E L1(G), where T !Tf(x) is the bounded operator defined by

~· HlT(h)f(xh)(q(xh)) -ldh, The representation 1T weskly contains iH if and only if Sa;H\\TlTAf(x)\\ dx\A E aH = \ITJ\1 1 for every f E L 1(G). These results are related to those of Abstract 71T-G135, these cA~W 18(1971), 834. (Received December 20, 1972.)

*73T-B83. ANTHONY G. O'FARRELL, Brown University, Providence, Rhode Island 02912. Density of parts of plane function algebras. Preliminary report.

Let X c

If E A : f extends analytically to a neighborhood of x} is uniformly dense in A in the uniform norm on X. Let C denote

Newtonian capacity, An(x) denote the annulus I z E

the Gleason norm of R(X). Theorem. If x is not a pesk point for A and a > 0 then ~:~ 2nC(An (x)\P(x, a)) < +oo.

Theorem. If A admits a pth order bounded point derivation at x (for some positive integer p) and a> 0, then

~:~2(p+ 1 )nC(An(x)\P(x, a)) < +oo. These strengthen and extend Browder's Metric Density Theorem for R(X). Similar results hold for sub algebras of Ff> (U), for a bounded open set U c a:. (Received December 21, 1972.)

A-273 73T-B84. MARTIN SCHECHTER, Belfer Graduate School of Science, Yeshiva University, New York, New Yorl< 10033. Boundary points of the numerical range. Preliminary report.

Let T be a bounded operator on a complex Hilbert space, and let >.. be a point on the boundary of its

numerical range. Theorem. (a) N(T- X.) ~ N(T*-"I). (b) (T- X.)xn ""' 0 iff (T*-l )xn ""' O. (c) If >.. is also a nonisolated point of O'(T), then there is an orthonormal sequence I

*73T-B85. JON c. HELTON, Arizona State University, Tempe, Arizona 85281. Product integrals and inverses in normed rings. I.

Functions are from S x S to N, where S denotes a linearly ordered set and N denotes a normed

complete ring. See B. W. Helton [Pacific J. Math. 16(1966), 297-322] for additional background. Theorem. If G is

a function from S x S toN such that J~\G2 \ ~ 0, allb(l+G) exists, G E OP0 on (a,b} and (b,a) and G(x,y) ~ -G(y,x)

for each subdivision \a,x,y,b) of \a, b), then (1) blla(1+G) exists, and (2) [allb(1+G)f1 exists and is blla(1+G).

Theorem. If G is a function from S X S to N such that s:la21~ 0, G E OP0 on (a,b} and {b,a), G E OM0 on (a, b) and G(x,y) ~ -G(y,x) for each subdivision \a,x,y,b) of \a,b}, then G E OM0 on \b,a). (Received December 26, 1972.)

*73T-B86. PAUL WILLIG, Stevens Institute of Technology, Hoboken, New Jersey 07030. A certain convex set in W-* algebras.

If R is a W-* algebra on separable Hilbert space H and Z is the center of R, define for any T E R the set C(T) as the intersection of Z and the weakly closed convex hull of the set \UTU*\U unitary in R}.

Theorem 1. Let R ~SAG R(A)/ol(dX.) be the direct integral decomposition of R into factors. For any T E R, C(T) ~

(A E R\ A(A.) E C(T(X.))~J.-a. e.}. Using Theorem 1 and a theorem of Conway (J. Functional Analysis 5(1970), 428-435) various consequences about the properties of Schwartz maps as defined by de Korvin(Trans. Amer. Math. Soc.

148(1970), 283-291) can be derived. As an example, one can prove Proposition 2. If R is type I and R1 is a continuous W-* subalgebra of R, then every Schwartz map P of R onto R1 annihilates the compact operators of R. (Received December 26, 1972.)

73T-B87. CHARLES L. BYRNE, JR., Catholic University of America, Washington, D. C. 20017. Isometries of arbitrary LP spaces that commute with Stone's operation. Preliminary report.

A linear isometry, U, on an arbitrary LP space, LP(X, !:,m), !:, a delta-ring, and m, a positive measure on !: , 1 ;>;; p

We show that U has this property iff there is a measure-preserving isomorphism, T, of the delta-ring, !:, of integrable sets, and U(cA) = cT(A)' for all A in !:, where cA denotes the characteristic function of the set A. This generalizes the result obtained by Lamperti (Pacific J. Math. (1958)) for totally finite Lp spaces; he showed that an isometry, U, is induced by a measure-preserving isomorphism of the sigma-algebra, via the formula U(cA) ~ cT(A)' iff U(1) = 1.

For the totally finite Lp spaces, U(1) ~ 1 iff U commutes with Stone's operation. (Received December 27, 1972.)

*73T-B88. EDMOND E. GRANIRER, University of British Columbia, Vancouver, British Columbia, Canada. Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group.

In the notations of P. Eymard, "L'algebre de Fourier d'un groupe localement compact," Bull. Soc.

Math. France 92(1964), 181-236, it has been proved by R. Burkel and later differently by Dunkl-Ramirez that if G is locally compact abelian then G is discrete iff C(G) ~ W(G) (thew. a. p. bded. fens. on G). This result has been improved (in our Mem. Amer. Math. Soc. No. 123(1972), "Exposed points of convex sets and weak sequential convergence••• "). Dunkl-Ramirez ("W. a. p. functionals on the Fourier algebra," to appear in Trans Amer. Math. Soc.)

A-274 prove that if G = n~Gn is the complete direct product of a countable collection of nontrivial compact groups then W(Ci:) 'iii VN(G). We define the subspace UCB(G) c VN(G) (of bded. unif. cont. functionals on G) for any locally compact group G and improve the above results in the Theorem. Let G be !!EX locally compact group. If for some !!2!:!!!_

separable subspace X c VN(G), UCB(G) c W(G) +X then G is discrete. If G is discrete then UCB(Ci) c AP(G) c

W(G). The results take simpler form for amenable locally compact G. (Received December 26, 1972.)

73T-B89. LYNDA HAMILTON, Syracuse University, Syracuse, New York 13210 and S. J. PERLMAN, Wayne State University, Detroit, Michigan 48202. On some types of derivatives. I. Preliminary report.

Waterman [Bull. Amer. Math. Soc. 73(1967), 109-112] says that a set E in Rn is a-dense, a ~ 1,

at a point x in Rn if \E0 n S(x;r)\ 1\S(x;r)\a -+ 0 as r -+ o. Using this definition of density in place of the classical definition he defines an a-approximate derivative. Neugebauer [Indiana Univ. Math. J. 22(1972)] has proved theorems

relating a-approximate derivatives (a) with harmonic (H), Borel (B), and LP-derivatives (LP). Theorem 1. (a) and

ess. bdd. -+ (Lp) for a ;:.; (n+p)/n. This is best possible in the sense that neither the ess. bdd. nor the condition on a

can be omitted. Theorem 2. (a) and ess. bdd. -+ (H) for a ;:.; (n+ 1)/n. This is also best possible in the sense that

neither the ess. bdd. nor the condition on a can be omitted. Theorem 3. (a) f (B) a. e. for a ~ 1. Theorem 4. (a) f

(Lp) a. e. for a, p ;:.; 1. Theorem 5. (a) f (H) a. e. for a ;:.; 1. (Received December 26, 1972.)

73T-B90. S. J. PERLMAN, Wayne State University, Detroit, Michigan 48202. On some types of derivatives. II. Preliminary report.

In a previous notice, the classes (a), (H), (B) and (Lp) were defined. Theorem 1. (Lp) -+ (B),

Theorem 2. (H) f (Lp) a. e. for p > 1. Theorem 3. (H) f (a) a. e. for a > 2. Theorem 4. (B) f (Lp) a. e. for

p > 1. Theorem 5. (B) f (a) a. e. for a > 2. (Received December 26, 1972.)

73T-B91. HUGO D. JUNGHENN and CHOY-TAK TAAM, George Washington University, Washington, D. C. 20006. Arcs defined by one-parameter semigroups of operators.

Let T(t) = etA (t i!i 0) be a one-parameter semigroup of continuous linear operators on a locally

convex space X. If x EX, [a,b] a finite subinterval of [O,ro) and p a continuous seminorm on X, the~

L(x;a, b;p) of the arc T [a, b]x is the supremum over all partitions a= t0 < t1 < •. , < tn = b of the sums 2::~= 1p(T(ti)x- T(ti_1)x). The function T(•)x is of bounded variation on [a,b] if L(x;a,b;p) 0, then x E D(A) if and only if

T(•)x is of bounded variation on each compact subinterval of [O,ro); in this case L(x;a,b;p) = J~P(T(t)Ax) dt.

Corollary. If D(A) "X (e.g., if X is an F-space and A is not continuous), then there exist continuous seminorms p on X

such that each neighborhood of any point y in X contains points x for which L(x;a,b;p) is arbitrarily large (finite and infinite). Examples can be given to show that in general the theorem is false if either X is not reflexive or T(t) is

not 1-1 and relatively open for no t > 0. (Received January 2, 1973.)

*73T-B92. SUSHEEL CHANDRA, Indian Institute of Technology, Kanpur, India. On certain subclasses of regu!ar and p-valent in the unit disc.

A. W. Goodman [Trans. Amer. Math. Soc. 68(1950), 204-223] defined the classes S(p) and C(p) of

functions regular and p-valent in D = !z: \zl < 1}. In this paper we solved some extremal problems for the subclasses

of S(p) and C(p) in which each element is of the form f(z) = zP + :B:1ank+1znk+p, k = 1,2, ••• , similar to those obtained by B. Pinchuk [Duke. Math. J. 35(1968), 721-734]. (Received January 4, 1973. )(Author introduced by Dr. o. P. Kapoor.)

A-275 73T-B93. RICHARD C. BROWN, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin 53706. The adjoint and generalized inverse of a linear system with general boundary conditions.

Conditions for the existence of the adjoint of a first order vector valued linear differential system with

boundary conditions represented by a matrix valued measure are derived when the system is viewed as an operator with

domain and range in L~[0,1], 12 p < ro. The adjoint is then constructed when these conditions are satisfied. Both the

operator and its adjoint are shown to be normally solvable Fredholm operators with mutually orthogonal ranges and

kernels. This implies some elementary results concerning their states and spectra. Regardless of the index of the

problem it is shown that every bounded mapping from the range to the domain ("quasi-inverse") can be represented

by a generalized Green's matrix structurally similar to the ordinary Green's matrix. The analogue of the principal

generalized matrix of Reid is constructed and shown to determine a generalized inverse via certain natural projections

which includes the Hilbert space generalized inverse as a special case. These results will appear as MRC Technical

Summary Reports 1287 and 1312. (Received January 8, 1973.)

*73T-B94. STEPHEN A. McGRATH, U.s. Naval Academy, Annapolis, Maryland 21402. An ergodic theorem for convex combinations of isometries induced by point transformations of the unit interval.

Let (X, 3', j.i) be a a-finite measure space and T a linear operator of LP(X, 3', j.i), p fixed, 1 < p <

ro. If there exists a constant c such that, for each f E LP (X, 3', j.l), Jsupn\f, (f+ Tf)/2, ••• , (f+ Tf+ ••• +Tn-1f)/n\Pdj.l 2 cp J\f\pdj.i, then we say that T admits of a dominated estimate with constant c. In this paper, isometries of Lp(0,1)

induced by point transformations a: (0,1) .... (0,1) of the form ax= J', k > 0, are considered. Theorem. Let n be

a positive integer and let T be a convex combination of n isometries of Lp (0, 1) induced by point transformations of

the form ax= xk, k > 0. Then T admits of a dominated estimate with constant p/(p -1). (Received January 8, 1973.)

*73T-B95. JOHN B. BUTLER, JR., Portland State University, Portland, Oregon 97207. A kernel calculus on a regular rigged Hilbert space. Preliminary report.

Let .P ~ S';) ~ .P' be a rigged Hilbert space with corresponding representation space £ = J:,S) H(1)dj.1(1) as a direct integral of Hilbert spaces, and isomorphism F: S';) .... £: x>-> x(1) = T(1)x. We assume that there exists a selfadjoint operator A on S';) corresponding to multiplication by 1 on H(1), JJ(A) :;;, .P. Define a convolution on

A A A A roo A -1 A L(S';), S';)) by (p •q)(t, s) = Lro J(D-roP(/l, s) T (/l)T(

formula (p• q) A = p * q. Extend this multiplcation to certain p E L(.P, .P'), q E L(S';), S';)) by p• q = w-lim(pn • q), p ~

w-lim pn, pn E L(S'd, S';l). Let 6 E L(.P, .P ') denote the kernel of the delta distribution and 1), 1)* solutions of nA ~ io,

A 1)* = -io. S';) will be called regular iff (1) + 1)*) is a multiplicative identity. Theorem. If .P ~ S';) ~ .P' is a regular

rigged Hilbert space and AipAj are nuclear, 1) • AipAj E L(S';), S';)), i + j = o, 1, then (i) (1) • p)A = io•p + 11 • pA,

(ii) A*(1J•p) = -iO• p + 1)"Ap, (iii) A*(o•p) + (O•p)A = O•(Ap+pA). Using (i), (ii), (iii) one obtains a kernel calculus

such as that employed in the theory for inverse scattering by a potential in L2(Rn) spaces. (Received January 8, 1973.)

73T-B96. MARGARET C. WAID, University of Delaware, Newark, Delaware 19711. The first initial-boundary value problem for a nonlinear time degenerate parabolic operator. Preliminary report.

Consider, for simplicity, the operator Lu = uxx- c(x, t, u)ut where c(x, t, u) is a bounded, real­

valued function on a domain D = 0 x (0, T] c Rn+ 1• Assume that c(x, t, u) is Holder continuous and nonnegative, but

not necessarily bounded away from zero, on D. Sufficient conditions on c are found which guarantee existence of a

solution u E c2+a to the first initial-boundary value problem Lu = f(x,t), u = !/> on the normal boundary of D, where

!/> E c2+a" Existence is proved by direct application of a fixed point theorem due to Schauder and uses the fact that there exists a solution to the linear problem Mu = uxx- c(x, t)ut = f(x, t), u = !/> on the normal boundary of D, as well

as a priori estimates obtained for the solution to the linear problem. The author has already proved existence of a

A-276 unique solution to the linear problem. The techniques obviously apply to a more general class of operators.

(Received November 16, 1972.)

*73T-B97. ARVJND B. BUCHE, Panjab University, Chandigarh-14, India. On an exponential-cosine operator­ valued functional equation.

Let I be a Banach space, and let 8(I) denote the Banach algebra of endomorphisms of I. A

one-parameter family of operator-valued functions \S(t ), t E R+\, S: R+ ... 8(I), S(O) ; I, is said to be an exponential­

cosine operator family if (1) S(s+t) - 2S(s)S(t) ; \S(2s) - 2~ (s) l S(t- s), s, t E R+, s < t. Let A ; li~ ... 0(S(h)- I)/h, and B ; limh... O(S(2h) - 2S(h) + I)/h2 exist in the uniform operator topology. Then A and B are called the first and

second infinitesimal generators respectively of the family \S(t) \. If A and B commute, then S(t) ; T(t) C (t) is a

solution of the equation (1), where \T(t), t E R+\, T: R+ ... 8(I), T(O); I, is an operator semigroup with A as the

infinitesimal generator (of. E. Hille and R. S. Phillips, "Functional analysis and semi-groups," Amer. Math. Soc.

Colloq. Publ., vol. 31, Providence, R.I., 1957), and \C(t), tER+\, C:R+ ... 8(I), C(O); I, is a regular cosine operator function generated by B -A 2 (cf. M. Sova, Rozprawy Mat. 49(1966)). (Received January 10, 1973.)

*73T-B98. WILLIAM PAUL WAKE, 12407 Chelton Lane, Bowie, Maryland 20715. Ideal boundary theory for harmonic functions. Preliminary report.

Using the axiomatic framework of Brelot ["Lectures on potential theory," Tata Institute of Fundamental

Research, Bombay, 1960], Loeb and Walsh r•A maximal regular boundary solution of elliptic differential equations,"

Ann. Jnst. Fourier (Grenoble) 18(1968), 283-308] studied the behavior of bounded harmonic functions at the ideal

boundaries of their domains. We show that, with appropriate modifications, the results of Loeb and Walsh hold in the

more general axiomatic framework of Boboc, Constantinescu, and Cornea p•on the Dirichlet problem in the axiomatic

theory of harmonic functions," Nagoya Math. J. 23(1963), 73-96]. Also, we extend certain results of Loeb ["An

axiomatic treatment of elliptic differential equations," Ann. Jnst. Fourier (Grenoble) 16 (1966), 167 -208] (based on

Brelot's axioms) to the more general setting. (Received January 10, 1973.)

73T-B99. WILLIAM P. NOVINGER, Florida State University, Tallahassee, Florida 32303. Continuous mappings induced by linear isometries of certain subspaces of C (X) into C (Y). Preliminary report.

Let X and Y be compact Hausdorff spaces, A be a closed subspace of C (X) which separates points

and contains the constant functions, and T a linear isometry of A into C(Y). Theorem. If the Choquet boundary

Ch(A) for A is closed (which means that Ch(A) is the same as the Shilov boundary for A) then there is a closed

subset Q of Y and continuous functions ( and 11, (: Q ... unit circle and fT: Q .... Ch(A) such that Tf(y) ; f(y)f(fT(y))

(y E Q) (f E A). This theorem generalizes a result due toW. Holsztynski (Studia Math. 26(1966), 133-136). (Received January 10, 1973.)

*73T-B100. HERMANN G. BURCHARD and OOUGLAS F. HALE, Department of Mathematics and Statistics, Oklahoma State University, Stillwater, Oklahoma 74074. Direct and converse theorems for piecewise polynomial approximation on optimal partitions. Preliminary report. Let .1: and ~~ denote the spline and piecewise polynomial functions of order n with k knots. For 1 ~ p < oo and given mesh u, let B (f, u) = :B E distO'(f,~ (u. , u.)) with ~ the polynomials of degree at most p, n ui u p n-1 1-1 1 n-1 n - 1 and 0'; 1/ (n+ 1/p). Let B (f) ; lim sup\B (f, u): u\ and p,n p,n Np,n (f) ; sup\Bp,n (f, u): u\. IY• n is the F-space of p-integrable f with N (f) finite and LP• n p,n 0 the closure in this space of those f with absolutely continuous (n -1)st derivative. The nth order splines on any simple mesh are in Lp,n and Lp,n[O, b] contains such functions with

nonintegrable nth derivative as x 01 , rx > -1/p. Theorem 1. f E Lp'n, k 5;: 1 => kndist (f,.@kn) --- p ~ ~/O'(f).p,n

A-277 Theorem2, fELp,n and fl(a,f3) ~9 for any a

73T-B101. GlhiTHER W. GOES, illinois Institute of Technology, Chicago, illinois 60616. Banach sequence algebras. Preliminary report. Let w be the space of complex valued sequences. A Banach algebra A c: w in which the product is the pointwise product xy = (¥k)1 is called a Banach sequence algebra (BSA). If B, C c: w then (B-+ C) = !x E w: xy E C for every y E Bl. Theorems. (i) A is a BSA if and only if A c: (A-+ A) and A is a BK-space i.e.

a Banach space of complex sequences x on which the functionals x -+llk: (k= 1, 2, ••• ) are continuous. (U) If A is a

BSA and B is a BK-space such that (A-+ B) c: A, then (A -+B) is a BSA. (iii) If A is a BSA with AD, i.e. if the

sections of x form a dense set in A, or If lllkl, where 6k =(liT) with 6f= 6jk the Kronecker 6, is a Toeplitz basis for A, then A c: c0, the space of null sequences. (iv) If A is a BSA with AD or with !tiki as a Toeplitz-basis, then every multiplicative linear functional T on A is of the form T(X) = Tk(x)"' llk: for some k E 11, 2, ••• 1. (Received January 11, 1973.)

Applied Mathematics

*73T-C10. KAILASH CHANDRA, Division of Physical Sciences, Institute of Advanced studies, Meerut University, Meerut, U. P., India. Thermal instabili1;y of a viscosity stratified fluid layer heated from below.

The stability of a fluid layer (under the Boussinesq's approximation) with viscosity jJ(Z) (z is the vertical coordinate) heated from below between two parallel planes (at z = 0 and 1) is investigated under linear theory

by the normal mode technique. It is shown that the transition from stability to instability must occur via a stationary

state, and the solution of the characteristic value problem at the marginal state can be expressed in terms of

variational principles. In case 1.1 = 1 +liz (6 is real), and the boundaries are nondeformable free, the critical Rayleigh

number for the onset of instability is found to be (1+ 6/2)657•5, which shows that the critical Rayleigh number for the

onset of instability is increased or decreased according as viscosity is increasing or decreasing upwards, Further,

the characteristic value problem at the marginal state is solved for two rigid boundaries, neglecting the second and higher powers of 6. The critical Rayleigh number is found to be (1+ 6/2)1715. (Received November 9, 1972. )(Author introduced by Professor J. N. Kapur.)

*73T-Cll. F. H. HSU, Case Western Reserve University, Cleveland, Ohio 44106. Reachable set at infinite time. Preliminary report. Consider a controllable autonomous linear system x= Ax + p in real n-space with compact restraint set P, zero in the relative interior of convex hull of P. Theorem. The reachable set R = Ut;:;OR(t) has R = M + L

where L is the linear span of the real and imaginary parts of those elements of a jordan basis of A which correspond

to eigenvalues with nonpositive real parts, and M is a bounded convex set containing zero. (Received November 9,

i972. )(Author introduced by Professor Otomar Hajek.)

73T-C12. MARTIN M. LIPSCHUTZ, William Paterson College, Wayne, New Jersey 07470. Radiation conditions for the elastic body.

A regular radiating function for the elastic body in a regular exterior domain D is a complex-valued

function U belonging to c2 in D and c1 in the closure i5' and (i) 1.1 AU+ (j.l + >..)V(V• U) + wO!pU = 0 in D;

A·278 (ii) limrr(ll /\ u- ik1~ /\ U) =limrr(ll •U-ik2~· U) = 0, uniformly in~= (x/r), ki = W(XPIIJ,. k; = Wrxp/(CXjJ+ A)· Theorem. Prescribed data for either the displacement U or the stress vector T(U) on the boundary of D uniquely determines a regular radiating function U in D. Theorem. Every regular solution W of L(W) = 0 in D is uniquely expressed as a sum W = u + V, where U is a regular radiating function in D and V is a solution of L(V) = 0 which is everywhere regular. An asymptotic representation for regular radiating functions is obtained in terms of simple progressive waves of the form exp(ikr)/r. (Received November 16, 1972.)

*73T-C13. w. JOHN WILBUR, Andrews University, Berrien Springs, Michigan 49104. Orthocomplementatioll.s on the closed subspaces of topological vector spaces. Preliminary report.

This is an extension of results announced in Abstract 72T-C65, these c}/ofiai) 19(1972), A-761. Let

A be the lattice of closed subspaces of a topological vector space E. If there is an orthocomplementation on A under which A is a logic and if the associated field automorphism is continuous, then A is isomorphic to the lattice of

closed subspaces of a Hilbert space and if E is a Mackey space E is a Hilbert space and the orthocomplementation on

A is the usual one. If there is an orthocomplementation on A and E is a complete Mackey space, then E is a Hilbert

space and the orthocomplementation on A is the usual one. These results hold over the real and complex fields.

(Received December 13, 1972.)

*73T-Cl4. W. BAUER and LES A. KARLOVITZ, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland 20742. An alternate version of the Kuhn-Tucker theorem.

For the optimization problem (P): minimize h(x) subject to gj(x) ;§; 0, j = 1, •.• ,m and x E En

with h,g1, ••• ,gm convex on En, we state the following alternate version of the Kuhn-Tucker theorem. Theorem.

Assume there exists y so that gj(y) < 0, all j. Then x' is a solution of (P) if and only if x' is a solution of the

unconstrained problem: minimize [h(x) + MlJ.(g.(x)) ] over all x E En, where M is any number satisfYing M > J J + (h(y) -m)/inf{-g.(y): j} and where m is any lower bound for the value of problem (P). This theorem has a very simple J direct proof. Thus, without calculating Lagrange multipliers, a constrained problem is replaced by an explicit

unconstrained problem, Further, this alternate version suggests extensions of the K-T theorem, of which the

following are typical examples: Theorem. Suppose that the feasible region of (P) is bounded (alternatively, suppose

that h(x) ~ ro as \xl ~ ro). Then x' is a solution of (P) if and only if x' is a solution for:

Maximize [minimum [h(x) + ML:. (g.(x))+' x E En] M >OJ, Algorithm. Using an approximate solution for each M, J J one can construct a sequence which converges to a solution of (P). (Received December 13, 1972.)

*73T-C15. R. K. RATHY and KAILASH CHANDRA, Division of Physical Sciences, Institute of Advanced Studies, Meerut University, Meerut, U. P., India. Thermal instability of a viscosity stratified fluid layer in the presence of a vertical magnetic field.

The stability of a fluid layer (under the Boussinesq's approximation) with viscosity IJ(Z) (z is the

vertical coordinate) heated from below between two horizontal parallel planes in the presence of a vertical magnetic

field is investigated under linear theory by the normal mode technique. A variational principle for the critical

Rayleigh number at which the nonoscillatory disturbances are marginally stable is established. An approximate

solution of the characteristic value problem is obtained for two nondeformable free boundaries when instability sets

in as stationary convection, assuming IJ = 1 + oz (0 is real), by applying Chandrasekhar's variational technique.

The critical Rayleigh number R is found to be given by R = lT4(1+x-1)(1+ 5/2)[(1+ x) 2 + Q/rr2], where xis the 2 2 solution of 2i + 3x = 1+ Q/i, and Q1 = 1Jo~d /1Jc1)(1+ o/2). Here 1J0,H,d,1Jc and 11 respectively stand for the magnetic permeability, applied magnetic field, thickness of the fluid layer, characteristic viscosity and electrical

resistivity of the fluid. (Received December 19, 1972. )(Authors introduced by Vice Chancellor J. N. Kapur.)

A-279 *73T-C16. OTOMAR HAJEK, Case Western Reserve University, Cleveland, Ohio 44106. Duality for pursuit games with constraints. Consider a game in n-space, x= Ax - p + q; with state, pursuer and quarry constraint sets C, P, Q; termination condition Mx(t) = 0 for all t 5!0 some 9(x); and rule: pursuer control choice p(t) depends on q(t) only (and initial position). Also consider the associated control problem x=Ax - u, same state constraint and termination condition, and control constraint set U = ((P+ T A)!. Q) n P where T A is the largest A-invariant subspace of T =

\x: Mx=Oj and!. is the Pontrjagin difference (xE V!!..W iff x+ VcW). Duality theorem. The game is equivalent to the control problem in respect to time-optimality if: P is closed, 0 E Q, C + T A c C. If u steers x to target, then p(t) = u(t) + q(t) is a pursuer choice. (Received December 1~, 1972.)

*73T-C17. JAMES D. FABREY, University of North Carolina, Chapel Hill, North Carolina 27514. Weyl systems for the (cp 4) 3 model. Two Weyl systems have been developed independently by Hepp ("Theorie de la renormalization,"

Springer-Verlag, Berlin, 1969) and the author (Comm. Math. Phys. 19(1970), 1-30, and Abstract 69T-B108, these cNoticeiJ 16(1969), 672) for the (cp 4)3 model. The representations are respectively defined by "weak limits" on a Hilbert space enlarged by Hepp via a Gelfand-Naimark-Segal construction and by the author via inductive limits.

These systems are shown to be equal and a cyclic vector is exhibited. (Received December 15, 1972.)

73T-C18. B. SPEELPENNING, University of Technology, Delft, The Netherlands. The generalized element method. Preliminary report.

The generalized element method is aimed at improving the wavefront method for solving large sparse sets of linear equations derived from networks (Melosh and Bamford, J. Struct. Div. ASCE 95, April1969). A triangular storage scheme is exploited which is well suited both to superposition of element matrices and elimination of rows. Eliminated rows are transferred to backing store to be used later for forward and backward substitution.

The element matrices are suitably organized in an "element pool" on backing store. The triangle in core is interpreted as containing in each stage a "generalized element", which also belongs to the element pool. Superposition consists of superposing an element from backing store on the element in core and deleting it from the element pool. Elimination of a row replaces the element in core by a smaller one. Permitting the element in core to be returned to backing store and clearing core, means that active rows can become passive again. This is used if the next row to be eliminated is not in the triangle. With this method, the number of operations is essentially the same as with sparse storage schemes; it does have the advantage, however, that there are far fewer address calculations. (Received

January 5, 1973. )(Author introduced by Professor L.A. M. Verbeek.)

*73T-C19. STEPHEN GROSSBERG, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Limiting distributions of reverberating neural networks. Preliminary report.

The system xi= -[A+ L:k#f(1.:)]x1 + (B -xi)f(xi)' i = 1, 2, ••. , n, describes a recurrent on-center off-surrouna network of cell populations. Let x = L:kx., y. = x.x- 1, E = limt... x(t), Q. = limt y.(t), M = max\.y.j, 1<: 1 1 ro 1 -oro1 1 m = min\yij, and g(w) = w -1f(w). g(w) is continuous and nonnegative throughout. Theorem. Let g(w) be strictly increasing for 0 iii w !!0 x<1>, equal to C for x<1) ~ w ~ x<2), and strictly decreasing for x<2) !!0 w !!0 B. Define x(O) by A -1 (1) -1 (2) g(x0) = g(B), x0 , x(t) 5!0 f > 0 for some f > 0, and y1(0) < M(O), then Q1 = o. If for some K, 0 < K < n, (n-K+1)x( ) 5!0

A-280 max(B-AC-1, x(O)) and yi(O)min[B-A(L:~ 1yi(O)C)- 1 , x(O)] !!!: x<1> then m(t) is increasing, M(t) is decreasing, but m(ro) < 1/n < M(ro). Various other results exist showing how monotonicity properties of g(w) influence the limits

Qi and E. In particular, bifurcations exist. (Received November 17, 1972.)

Geometry

73T-Dl THOMAS E. CECIL, Brown University, Providence, Rhode Island 02912. Geometric applications of critical point theory to submanifolds of cpn, ·. m+1 ..,.m - 2m+1 m+1 . For z = (z0, ... , zm), w = (w0, ... ,wm) m C let (z,w) = .uk=O ~wk; S c C 1s the set of z in em+1 such that (z, z) = 1. CPm is given the Fubini-Study metric via the Hopf fibration with projection P from s2m+1 to CPm. Mn is a complex n-dimensional manifold holomorphically immersed in CPm. For p in CPm, x in Mn, and z, win s2m+1 such that P(z) = p, P(w) = x define L (x) = cos-1(\ (z,w)l2). With a suitable definition p of focal point, an index theorem for Lp (analogous to the index theorem in Milnor, "Morse theory," p. 37) is proven and applied to prove: Theorem. Let Mn (n!!!: 2). be a connected, compact, complex n-manifold holomorphically immersed in CPm. Suppose there is a dense subset D of C~ such that every nondegenerate critical point of every function of the form Lp' pin D, has index 0 or n, then Mn is either CPn or r;f. Here CPn is a totally geodesic submanifold of CPm; Qn is the complex quadric hypersurface of some totally geodesic CPnt1• To begin the proof, one shows that A~ = X2I for any nornial 1;: to Mn at any x in ~. This implies Mn is a hypersurface of a totally geodesic CPnt1• The result then follows from B. Smyth (Ann. of Math. (2) 85(1967), 246-265). For the case of Qn c CPnt1, the set of focal points is RPn+1 naturally embedded in cp0+1• (Received November 20, 1972.)

73T-D2. BANG-YEN CHEN, Michigan state University, East Lansing, Michigan 48823. Surfaces with nontrivial normal connection. Preliminary report.

Let M be a surface in a space form Rm with the induced normal connection D. A normal vector field e is said to be parallel if De = 0. The normal connection is said to be nontrivial if the curvature tensor of the normal connection is not identically zero. A unit normal vector field e is called an isoperimetric section if Trace A (e) is constant, where A(e) is the second fundamental tensor at e, The following theorem is proved. Theorem. Let M be a surface in an m-dimensional space form Rm(c) of curvature c with nontrivial normal connection. Then M is contained in a (small or great) hypersphere of Rm(c) if and only if there exists a parallel isoperimetric section on M.

(Received December 11, 1972,)

73T-D3, PAUL EWING EHRLICH, state University of New York, Stony Brook, New York 11790, Deformations of Ricci curvature. IT. Preliminary report.

We extend our results of "Deformations of Ricci curvature on Riemannian manifolds" in Abstract 72T-D17, these cNotiuiJ 19(1972), A -596. Given a sufficiently small disk with positive or negative Ricci curvature, we can extend the positive or negative curvature using a conformal variation to a larger disk. Let (M, g) be a Riemannian manifold and let c(g) = i.nf\1, convexity radius at p; pin M}. With the c 2 topology on the space of Riemannian structures, c is lower semicontinuous. Then, Theorem 1. Let (M,g) be complete with c(g) 1!; c > 0, and with nonpositive Ricci curvature and negative Ricci curvature at some point. Then M admits a complete metric of negative

Ricci curvature. Theorem 2. Let (Mn,g) be compact with kAg(v, v);;; Ric(v, v);;; kg(v, v) for all v with constants k,A with 0 ;!5 A < 1/(n-1) and k > 0. Suppose there is a point p where kAg(v, v) < Ric(v, v) ;!5 kg(v, v) or kAg(v, v);;;

Ric(v, v) < kg(v, v) for all v at p. Then it is possible to improve the Ricci pinching. Given a disk D, we study Ric' for arbitrary variations g(t) of g with g(t) = g in M\D, and show that without additional geometric information, the bound obtained from conformal variation in Theorem 2 cannot be improved, A similar theorem holds for negative

Ricci pinching. (Received December 11, 1972.) A-281 *73T-D4. KENNETH B. STOLARSKY, University of Ulinois, Urbana, illinois 61801. Sums of distances between points on a sphere. II. Preliminary report.

Let p1, ••• ,pN be a set of N points on ~, the unit sphere of m dimensional Euclidean space Em.

Let d(p1.,pJ.) denote the Euclidean distance from p. top .. Let S(N,m) be the maximum of L::.< .d(p.,p.) over all 1 J 1 J 1 J 2 2 possible choices of p1, ••• , pN. Let c(m) ~ limN_,roS(N,m)N- • Theorem. S(N,m) < c(m)N - c1NS(m)-t for any 2 -2 ( < 0 where c1 > 0 depends only on m and (, and 9(m) ~ (m - 5m + 2)(m- 1) • Our proof uses results of

W. M. Schmidt on irregularities of distribution. J. R. Alexander has proved a lower bound of this type with (, c1, -1 and 9(m) replaced by 0, c2, and 1- (m -1) • (Received December 18, 1972.)

Logic and Foundations

*73T-Ell. c. J. ASH, Monash University, Clayton, Victoria 3168, Australia. Sentences with finite models.

A first-order sentence is said to have property F if either it has a finite model or it has no model at all. Theorem 1. In a language with only unary predicate letters, constant letters and equality, every sentence has property F. Theorem 2. In a language with only unary predicate letters, unary function letters and constant letters, but without equality, every sentence has property F. Examples are given to show that Theorems 1 and 2 give all such languages. Theorem 3. In a language with only unary predicate letters, ~function letter and equality, every uniyersal sentence has property F. Examples again show that Theorems 2 and 3 give all such languages apart from those without function letters. (Received November 13, 1972.) (Author introduced by Professor Neil H. Williams.)

*73T-E12. SAHARON SHELAH, Hebrew University, Jerusalem, Israel. The monadic theory of order. II.

Theorem 1. (CH) The monadic theory of the real order is undecidable. Remarks. (1) We can weaken CH. (2) The methods give some other similar results. Theorem 2. Let P be a dense subset of the reals -~-- such that for any closed nowhere-dense set Q of reals \P n Q\ < 2 °. Then the monadic theory of (P, <) is equal to + the monadic theory of the rational order. Theorem 3. The monadic theory of \a: ex

(A,<) such that (1) \AI;;; ~ 1 , (2) for every countable B ~A, the number of Dedekind cuts of B which elements of A realize is ;;; ~ 0 , (3) the number of Dedekind cuts of A which has cofinality w1 from above and from below is ;;; ~ 1 • Theorem 4. The monadic theory of K is decidable. (Received November 15, 1972.)

*73T-E13. MARTIN SEBASTIAN GERSON, Simon Fraser University, Burnaby 2, British Columbia, Canada. The inadequacy of the neighbourhood semantics for modal logic.

Two modal logics which are incomplete with respect to the neighbourhood or Scott-Montague semantics are presented. They are exactly those shown by S. K. Thomason and K. Fine respectively to be incomplete with respect to the relational or Kripke semantics. The first does not include S4 but every neighbourhood frame of it models 84; the second is an extension of 84. The results separately show the inadequacy of the neighbourhood semantics for modal logic, or in other words, that the Scott-Montague semantics does not have maximal depth.

(Received November 20, 1972.)

*73T-E14. EMERSON C. MITCHELL, University of Wisconsin, Milwaukee, Wisconsin 53201. Class theory axioms with full abstraction, complemented classes, and small sets. Preliminary report.

The term-forming operator T (see Bourbaki,"Set theory") is used to form an object for every abstract \x: P(x)j. Each xis in \x: P(x)) if P(x) holds, but not in general conversely. An abstract is called well-behaved in case xis in {x: P(x)) iff P(x). "Class" and "set" are undefined terms. The axioms given lead to the

A-282 following results: If an abstract is a class it is well-behaved; the abstracts and the classes each form Boolean algebras; the Zermelo-Fraenkel axioms, except for regularity, hold for sets; the sets form an ideal of the Boolean algebra of classes; finally, the theory contains a model of von Neumann-Bernays-Godel class theory. Ad hoc hypotheses, e.g. that {x: x is a group\ is well-behaved, can be explored. The paper is currently in handwritten form, but Xerox duplicates are available for 50 cents duplicating cost. (Received November 20, 1972.)

*73T-E15. JULIA F. KNIGHT, Pennsylvania State University, University Park, Pennsylvania 16802. Theories with finitely many CAl-models. Preliminary report.

Let L be a language with a unary relation symbol U and constants !! for all n E w. An L-structure

!ll is called an w-model if U!ll = l!!!ll: nEw\. It can be shown that for each n E w, there is a countable, complete theory Tn such that Tn has exactly n w-models. The proof uses forcing, with the models of Tn being constructed by means of successive generic expansions. (Received November 21, 1972.)

73T-E16. LEO A. HARRINGTON, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. The ordinals recursive in some objects of finite type. Preliminary report. The superjump, S, and equality for type 2 objects, 3E, are two type 3 objects which may both be viewed as analogues of the Turing jump. w~ is the sup of the order types of well ordered reals which are recursive in S. 1t~E is the sup of the order types of pre-well-orderings of the continuum which are recursive in 3E.

Theorem. (a) w~ is the first recursively Mahlo ordinal. (b) For any ordinal a it is consistent with ZF for Ct. < k~E (or to be precise, if ~ is a card of L, corL (~) > W, aL !!0 ~, and if fl is the ordinal of t.'le 1st admissible set containing (Ci.w)L, then there is a Boolean extension of L, which preserves cardinals, such that Ct.< 1t~E < fl, and

2w = ~). (Received November 24, 1972.)

*73T-E17. PETER IL KRAUSS, University of Heidelberg, 69 Heidelberg, Im Neuenheimer Feld, West Germany. On the embedding and amalgamation properties. Preliminary report.

Let M E EC A and N ~ M. N is called a component of M if N E EC A and N has the embedding property, and for every K E EC A , if N ~ K ~ M and K has the embedding property then N = K. Theorem 1. M is the union of its components. Theorem 2. M has the amalgamation property iff distinct components of M are disjoint and every component of M has the amalgamation property. Theorem 3. M has the embedding property iff there exists !ll E M such that M = M n Mod Th1f!ll. (Received November 27, 1972.)

*73T-E18. MELVIN c. FITTING, Lehman College, City University of New York, Bronx, New York 10468. Model existence theorems for modal and intuitionistic logics. Preliminary report.

We define notions of consistency properties for the first order modal logics 84, T and K (without the

Barcan formula) and for intuitionistic logic. Model existence theorems are proved: any member of one of these consistency properties is satisfiable in the appropriate Kripke model. The completeness of axiomatic, tableau, and

Gentzen-sequent formulations of these logics, compactness theorems, Skolem-Lcwenheim theorems, and Craig interpolation theorems are all easy corollaries. Methods used are simple generalizations of classical first order analogs. (Received November 27, 1972.)

*73T-E19. E. W. MADISON, University of Iowa, Iowa City, Iowa 52240. The existence of nonsimple constructive extensions of the Boolean algebra of clopen sets of the Cantor space.

Theorem. There exist nonsimple constructive extensions of the Boolean algebra of clopen sets of the

Cantor space. (Received December 4, 1972. )(Author introduced by Professor W. A. Kirk.)

A-283 73T-E20. TELIS K. MENAS, University of California, Berkeley, California 94720. On the partition property. Preliminary report.

For notation see Abstract 72T-E18, these c}(otiui] 19(1972), A-331. In the following let K be a supercompact cardinal. A measure 11. on p11 X is fine if it is !!.-complete, nonprincipal, and for all a < X,

11. dx E p11 X I a E xh = 1. A fine measure on p11 >.. has the partition property if every partition of p11 >.. into two disjoint sets has a homogeneous set of measure one. Let Q(l!., )..) assert that there is a normal measure on p 11 >.. not having the partition property. Solovay has shown that if 11 i!i >.. > 1!. are cardinals s. t. there is a normal measure on p >..II, then Q(l!.,v). Kunenthenprovedthattheleastcardinal >.. > 1!. s.t. Q(K,II) is l!'~ indescribable. Theorem 1. Let>.. be the least cardinal > 1!. s. t. for all 11 i!i >.., Q(l!., II). Then >.. is supercompact. Theorem 2. If >.. is the least cardinal > 1!. s. t. Q(K, 2 ~, >.. is measurable. Theorem 3. There is a regular cardinal >.. > K s. t. if 11. is normal on p11 >.. having the partition property, there is a fine measure 11.* on p11 >.. isomorphic to 11. but not having the partition property. (Received December 4, 1972. )(A:1thor introduced by Professor Jack H. Silver.)

73T-E21. ANDRZEJ EHRENFEUCHT, Department of Computer Science, University of Colorltdo, Boulder, Colorado 80302. A finitely axiomatisable decidable theory without finitely axiomatisable complete extensions.

Such a theory can be obtained from a theory T constructed by G. Fuhrken and the author in 11A finitely axiomatisable complete theory with atomless F1 (T)", Arch. Math. Logik Grundlagenforsch. 14(1971), 162-166, where

F 1 (T) is the Boolean algebra of formulas of T, with one free variable v 0, modulo equivalence in T. Let T* be obtained from T by adding one individual constant. Thus by a well-known theorem (see Tarski, Mostowski, Robinson,

"Undecidable theories," North Holland, 1953) T* is also decidable, and obviously has no consistent complete finitely axiomatisable extensions. (Received December 11, 1972. )(Author introduced by Professor Jan Mycielski.)

73T-E22. PIDLIP OLIN, York University, Downsview 463, Ontario, Canada. Free products and elementary equivalence.

Feferman and Vaught (Fund. Math. 47(1959), 57-103; see footnote p. 76) ask whether free products preserve elementary equivalence. We answer this question negatively, with a counterexample involving free products of semigroups. Denumerable semigroups m1 and m2 are constructed such that m1 -< w2 and if m3 is the 1-element semigroup then there is a :E 6-sentence tp (With no constants) such that ~ 3 * m2 ~ tp and !!!3 * !!!1 ~ ~ cp. A groupoid is a set with a binary function. We show that the free product operation on two groupoids preserves both elementary equivalence and elementary subsystem. (Received December 15, 1972.)

73T-E23. KENNETH A. BOWEN, Syracuse University, Syracuse, New York 13210. Elementary notations for ordinals and formulas. Preliminary report.

The system o, is a system of notations for ordinal numbers which uses only elementary functions for determining fundamental sequences. (Substantial modifications of Kleene1s construction of 0 are necessary since some of the original auxiliary functions are evidently not elementary.) Theorem. The least ordinal not receiving a notation in o, is the first nonrecursive ordinal. Let Ff be the system of notations for infinitary formulas which bears the same relation to the Kino-Takeuti system F as Ot does to 0. Theorem. For every formula A receiving a notation in F there is a formula A 1 receiving a notation in Ft such that A and A 1 are logically equivalent, and conversely. (Received December 13, 1972.)

A-284 73T-E24. ROBERT I. SOARE, University of Ulinois at Chicago Circle, Chicago, Ulinois 60680. Automorphisms of the lattice of recursively enumerable sets. III: recursive and nonrecursive sets.

Let 8* denote the lattice 8 of r. e. sets \W xlxEN under inclusion modulo the ideal 3' of finite sets. An r. e. set A is low if its jump 0 ~ \x: xE ~ l has degree 9.,:.-.. , and weakly low if t ~ \x: W x n A f iil has degree R,:.V. (It is obvious that t ~T 0 and easy to show that there exists a weakly.low set in every r. e. Turing degree.)

Recursive sets are low as are most incomplete nonrecursive r. e. sets which are constructed by a finite-injury priority argument, such as in the Friedberg-Muchnik theorem. For each r.e. set A, Lachlan considers the principal filter

£*(A) ~ \ Wx: W x;;, A l modulo 3' and has asked whether £*(A) "' -L *(B) if A and B are both low and simple.

Theorem. If A is weakly low and coinfinite then -L *(A) "" a*. Corollary 1 (Lachlan). If A is low and coinfinite and

H is any hh-simple set (e.g., a maximal set) then for some r. e. B;;, A, -L*(B) "'-L*(H). Corollary 2. In every

Turing degree there exists an r. e. set A such that -L *(A)'=' a*. Corollary 3. -L *(A) "'-L *(B) does not imply that there is an automorphism of a* mapping A to B. (We are grateful to Carl Jockusch for pointing out that our original hypothesis "low" could be weakened to "weakly low.") (Received December 18, 1972.)

73T-E25. ALAN H. MEKLER, stanford University, Stanford, California 94305. Infinitary equivalence of groups. Preliminary report.

A criterion for L00X-equivalence to free groups is developed and with the aid of analogous criteria of Eklof ("Infinitary equivalence of abelian groups," preprint), the following theorems are proved. In each case

"free" can be replaced by ''free-abelian" or by "direct sum of cyclic groups." Lemma. If G is (2X)+ -free, G is LooX+ -equivalent to a free group. Hence Theorem 1. The class of free groups is axiomatizable in some L00x iff

J! X such that if G is x-free, G is free. Examples of Griffith's ("~ -free abelian groups," preprint) yield ~ n Theorem 2. If 2 n-1 < ~ then free abelian groups are not axiomatizable in L Also Theorem 3. If ). is a ---- W COWn• ---- strong limit cardinal > ~ 0 , G is ).-free iff G is L00 A-equivalent to a free group. This gives Corollary 1 (Higman; Hill for free-abelian). If cf(A) ~ w and ). is a strong limit cardinal > ~O then if G is ).-free, G is A+ -free. By ultrapowers one has Theorem 4. If x is strongly compact, then if G is x-free, G is x +-free. (Received January 1, 1973.)

*73T-E26. LARRY J. STOCKMEYER, Electrical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Polynomial space is reducible to the first order theory of equality. Preliminary report.

Let L1 ;;; !:;:, L2 ;;; !:; be languages. Define !Qg_-space reducibility, L1 ~log L2 if there is a Turing machine transducer !In which, when given any x E !:~ of length n, uses at most log n auxiliary space, and produces a y E !:; such that x E L1 "' y E L2• L2 is p-space complete if L2 is recognizable in polynomial space and if for all

L1 recognizable in polynomial space, L1 ~log L2• Theorem 1. The first order theory of equality is p-space complete. Corollary. Any decision procedure for the first order theory of equality requires space at least c• (length of s)1/ 2- 6 for some c > O, all 6 > 0, and infinitely many sentences s. Note that this theory can be decided in space O(n log n).

Theorem 2. The first order predicate calculus (without equality) with a single monadic predicate is p-space complete.

Corollary. If there is a context sensitive language which cannot be recognized in polynomial time, then there is no polynomial time decision procedure for the above theories. (Received December 19, 1972.) (Author introduced by Professor Albert R. Meyer.)

73T-E27. ALEXANDER ABIAN, Iowa State University, Ames, Iowa 50010. Convergent subsequences of generalized sequences of sets,

Let A be a set of ordinals and (Si)iE A be a sequence of type A of sets Si. The limit superior

A-285 limiEA Si and limit inferior limiEA Si of (Si)iEA are defined respectively by \xl U \ v\ v EA and x E Sv} = UA} and

\xl U \v\v E A and x ~ Sv} < UA}. The sequence (Si)iEA is called convergent if and only if limiE A Si = limiEA Si" Theorem. Let 0 be a cardinal number which is cofinal to w, i.e., cf(O) = w. Let (Si)iEO be a sequence of type

0 of subsets Si of w. Then (Si)iEO has a convergent subsequence of type 0. Lemma. Let c be an infinite cardinal and (Du)uEc be a sequence of subsets Du of the power set of c such that Du = \H\H,:; c and u E H) for every u E c. Then (Du)uEc has no convergent subsequence of any infinite cardinal type a with a:S c. (Received December

21, 1972.)

73T-E28. LEON C. BRUER, Computer Sciences Corporation, 650 North Sepulveda Boulevard, El Segundo, California 90245. Towards a more rigorous logical foundation for mathematical analysis.

The primary purpose of the paper is to develop a foundation for mathematical analysis which presents

a dialectic combination of some traditionally opposing points of view into a unified approach which retains the firm

basis made possible by the constructive philosophy. This approach is shown to clarify some well-known points of

logical confusion such as Russell's paradox and the diagonal proof of the uncountability of the real numbers. The

paper defines a class of C-functions from n-tuples of natural numbers into the natural numbers which is designed to

represent the intuitive notion of a constructively defined function. The C-functions are found to be constructively

enumerable although they avoid any possibility of beiug criticized as incomplete by means of a diagonal argument.

It is then shown how the C-functions can be used as a basis for a logically more satisfactory definition of the real number system. (Received December 26, 1972.)

73T-E29. EGON BORGER, Universit& di Salerno, Istituto di Scienze dell'Informazione, 84100 Salerno, Italy: Reduction classes of Krom formulae with only one predicate symbol and function symbols. Prehmmary report.

Let the class ofKrom formulae Kr be the class ofprenex formulae of the first order predicate calculus

with a matrix in conjunctive normal form and binary disjunctions. Let P((n1, n2, ••• ) , [m1 , m2, ••• ]) be the class of all closed prenex first order formulae with a prefix of the form P, at most ni resp. mi predicate resp. function symbols of

rank i and without equality. We identify a finite sequence (n1, ••. ,nr) with the infinite sequence (n1, ••• ,nr,O,O,O, .•• ) and especially (0,0, 0, ••• ) with the empty sequence. Extending Theorems 8 and 9 in Yu. Sh. Gurevich, "The decision

problem for the logic of predicates and operations" (Algebra and Logic 8 (1969), 160-173) we show that the classes AA((0,1,), [2]) n Kr, AA((1), [1,1]) n Kr and AAA((1), [0,1]) n Kr are reduction types and that A00((ro,ro), [1]) n Kr has a recursively solvable decision problem with respect to satisfiability. We note furthermore that EAA ((0, 0, 1,), [1])

n Kr is a conservative reduction type. The reduction classes are obtained by convenient axiomatization of stop- problems of 2-register machines, whereas the required decision procedure consists in a reduction to the Aanderaa case

AEA((ro,ro)) n Kr (seeS. Aanderaa, "On the decision problem for formulae in which all disjunctions are binary",

Proc. Second Scand. Logic Sympos., 1971). (Received December 27, 1972.)

*73T-E30. ROBIN GILES, Queen's University, Kingston, Ontario, Canada. A nonclassical logic for physics.

A formalized physical theory is set up by analogy with a first-order mathematical theory. The main differences are as follows. (1) The usual interpretation in terms of a model is replaced by a physical interpretation, largely formalized in terms of "documents". (2) Each prime proposition is thus associated with an elemcntary(=yes-no)

experiment. Since such experiments usually show dispersion (repetition need not give the same result) a proposition

is not in general "either true or false" so that (3) classical logic must be rejected. To obtain an alternative, each proposition is given a meaning in terms of an obligation incurred by the speaker. An assertion now initiates a debate

A-286 in which this obligation and its byproducts are discharged. The special case in which each speaker assigns to each elementary experiment a definite (subjective) probability leads to a structure incorporating the infinite-valued logic

L 00 of Lukasiewicz. (In the dispersion-free case classical logic occurs in place of t 00.) Finally, it is observed that when quantifiers are present an analogue of the intuitionistic critique is applicable, and the arguments leading to L

(or to classical logic) are not justified. (Received January 2, 1973.)

Statistics and Probability

*73T-F4. R. SHANTARAM, University of Michigan, Flint, Michigan 48503. Two generalizations of a result of Wang.

Consider a sequence of n Bernoulli trials with probability of success (S) and failure (F), respectively,

p and q ~ 1 - p, 0 < p < 1. Define the random variable Nn as the number of times the combination SF occurs in the

sequence. In the symmetric case (p=~) Wang rPeter C. Wang, "A binomial identity with an application to sequence

of symmetric Bernoulli trials," Rend. Sem. Fac, Sci. Univ. Cagliari 39(1969), 153-155 (MR 41 #2734)] has proved that

(*) P(N 2-n( n+ 1), n;;; 2r, r;; 0. We generalize this result in the following: Theorem 1. P(N =r) = n ~r) ~ 2r+ 1 ---- n t::f;; (~) (~-j)pjqn-j. Theorem 2. If the trials are Markov dependent with transition probability p of remaining in the same state and probability q of changing states, then for the initial distribution P(S) = p, P(F) = q, we have n-2r+l 2r-1 n-1 n-2r 2r n-1 n-2r-1 2r+l n-1 n-2r-2 2r+2 n-1 P(Nn=r) ~ P q

reduce to (>) in the symmetric case. (Received November 9, 1972,)

73T-F5. ROY N. M. N. TAKENAGA, 912 South Hastings Avenue, Fullerton, California 92633. A redefinition of the fiducial. Preliminary report.

For simplicity, let the distributions of interest here have but one parameter 9, be monotone in 9,

and have all properties convenient in what follows. Let F(x; 9) denote one. Given sample values x1, x2, x3, .•. , xn' its point fiducial density function and corresponding distribution are defined, respectively, by the equations

p(9;x1, ... ,xn) =ciT\ F9(xi; 9)\ and p(9;x1, ••• ,xn) = J~00p(cp;x1 , ••. ,xn) • dcp, where c is such that P(oo;x1, ••• ,xn) = 1. Let G(P( ); 9) denote the probability, given 9, of all points (t1, t2, ... , tJ in the sample space such that

P( 9;t1, ••. , tn)"' P(9;x1, ••. ,xn). As a function of 9 it would redefine the classical fiducial if Theorems 3 and 4 below are true. Theorem 1. G(P; 9) = P(9; ) if 9 is a location parameter. Given 9, estimation procedures determine

a partitioning of the sample space. The partitioning by G(P; 9) has the advantage of being unbiased. Also:

Theorem 2. It is invariant of functional changes in 9. Theorem 3. It agrees with the classical fiducial partitioning

if x and 9 satisfy Theorem 1, or can be functionally transformed so that the new x and 9 do. Theorem 4

(currently unproved). It agrees with the maximum likelihood partitioning if sufficiency holds. (Received January 2,

1973.)

*73T-F6. GUNNAR A. BROSAMLER, University of British Columbia, Vancouver 8, British Columbia, Canada. The asymptotic behaviour of certain additive functionals of 1- and 2-dimensional Brownian motion.

The author has proved pathwise ''stability" for certain nonnegative additive functionals of 1- and

2-dimensional Brownian motion. Theorem L If Ts is the sojourn time of !-dimensional Brownian motion in [O,oo)

up to time s, then P \limt (1/log t) j'\ds/s)(T /s) = 1/2) = 1, x ER. If A is a nonnegative additive functional of x ~m 1 s s !-dimensional Brownian motion, with measure a, then P \limt (1/log t) t (ds/ s)(A I /8) = J2/rr a(R)) = 1, x E R. x -oo J1 s Theorem 2. If As is a nonnegative additive functional with measure a, of Brownian motion on a cylinder C with

circumference U, then P \limt (1/log t) Jt (ds/ s)(A / JS) = Ma(C)/U) = 1, x E C. If A is a nonnegative additive x ~m 1 s s

A-287 the plane, then P llimt (1/log log t) rt(ds/s log s)(A /logs) = functional with measure a, of Brownian motion in x .... oo .Je s (l/2'1T) • a(R2ll = 1, x ER2• The preceding theorems supplement the arc sine-law and the Kallianpur-Robbins laws.

Moreover, they refine the property of null-recurrence in the case of 1- and 2-dimensional Brownian motion.

(Received January 9, 1973.)

Topology

*73T-G23. LUDVIK JANOS and F. R. CORNING, Universidad de Buenos Aires, Facultad de Cs. Exactas, Departamento de Matematica, Buenos Aires, Argentina. Autohomeomorphism groups of locally connected spaces. Preliminary report.

Let G be a group, A a set and P(A) the full permutation group of A. By fG;A] we understand the group defined on the set GA X P(A) by the formula (f1,rr1) * (f2, rr2) = (f1• f2 • rr 1,rr2 • 'IT 1) where f1, f 2 EGA and rr1,

'IT 2 E P(A). If G carries a topology we topologize [G;A] by the product topology. If X is a topological space we denote by G(X) the group of all autohomeomorphisms of X. If X is locally connected it can be represented as a free sum L::i E J "0 a EA. (Yi X I alJ where Yi' i E J are the components of X which are mutually not homeomorphic. 1 Theorem. The group G(X) is isomorphic to the direct product lliEJ[G(Yi);Ai]. If X is also locally compact and G(X) and G(Yi) (i E J) carry the compact-open topology, then these groups are also isomorphic as topological groups.

(Received October 13, 1972.)

*73T-G24. JOHN M. ATKINS and RAYMOND F. GITTINGS, University of Pittsburgh, Pittsburgh, Pennsylvania 15213. e-refinabUity and local properties.

We assume all spaces are Tl" Theorem. Let X be 8-refinable and have a local property P where

P is any of cr, cr*, quasi-complete, wA, -z;*, cr*, fJ, semistratifiable, G0 diagonal; then X has property P. If further we assume X is completely regular, then the theorem is true for p and strict p spaces. There is an example of a locally metrizable metacompact Moore space which is not metrizable and hence not wM, M*, M or stratifiable.

(Received November 20, 1972.)

*73T-G25. NEAL E. FOLAND and RONALD B. KIRK, Southern Illinois University, Carbondale, Illinois 62901. Products of spaces with m-dense subsets.

Let X be a topological space and m a fixed infinite cardinal number. An m-density pair for X is a pair (D, I) where D is a subset of X with D;;; m, I is a totally-ordered set with '[;;; m, and every point of X is the limit of a net in D directed by I. A space with an m-density pair is said to have an m-dense subset. Professor

A. Wilansky has conjectured (Math. Monthly 79(1972), 764-765) that if {Xa: a E A I is a family of spaces each of which has an m-dense subset, then the product of these spaces has an m-dense subset if A < 2m and has no m-dense subset, in general, if A= 2m. The following partial solution is given: Theorem. Assume the generalized continuum hypothesis. Let m be a regular cardinal and let I Xa: a E AI be a family of Hausdorff spaces each of which has an m-dense subset. Then the product has an m-dense subset if A < 2m and may have no m-dense subset if A =2m.

Let cf(m) denote the least ordinal similar to a cofinal subset of m. It is proved assuming the generalized continuum hypothesis that for the class of Hausdorff spaces, Wilansky's conjecture is equivalent to the following statement. If m is a nonregular cardinal and if X has an m-dense subset, then there is a set D c X such that (D, cf(m)) is an m-density pair for X. (Received November 20, 1972.)

*73T-G26. FRANK A. CHIMENTI, State University College of New York, Fredonia, New York 14063. Tychonoff's theorem for hyperspaces. Preliminary report.

To say that a topology for the collection of nonempty subsets of a topological space preserves the

A-288 Hausdorff convergence structure for this collection means that every net of nonempty sets that converges to a nonempty set in the sense of Hausdorff also converges in the topology to the same set. A result of Y. -F. Lin (see "Tychonoff's theorem for the space of multifunctions," Amer. Math. Monthly (1967), 399-400) can be improved to: The Tychonoff product of the family of collections of nonempty subsets of each member of a family of spaces is compact if and only if each member of the family of spaces is compact. This result is valid so long as each collection in the family is equipped with a topology that preserves the Hausdorff convergence structure for the collection. A class of topologies, called topologies of finite type, is identified which always preserves the Hausdorff convergence structure. The more common topologies for the collection of nonempty subsets of a space are topologies of finite type. A similar result concerning sequential compactness is valid for a countable family of spaces. (Received November 13, 1972.)

73T-G27. JACK B. BROWN, Auburn University, Auburn, Alabama 36830. A topological Baire space in which Blumberg's theorem fails. Let R be the reals, T be the ordinary topology on R, and T' = !o I for some Q E T and subset Q• of Q of cardinality less than c, 0 = Q- Q•l. (R, T') is a topological Baire space. Let f: R-+ R be the function of

Sierpinski and Zygmund which has no (T, T)-continuous restriction to a set having cardinality c. f has no

(T', T)-continuous restriction to a T'-dense subset of R. (Received December 1, 1972.)

73T-G28, PETER FLETCHER, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 and WILLIAM F. LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. Spaces that admit a coarser quasi-metric topology. Preliminary report.

A topological space (X, 7') is subguasi-metrizable provided. that there is a coarser quasi-metrizable topology for X. A sub-nonarchimedean quasi-metrizable space is defined analogously. Theorem. Every orthocompact space with a G6-diagonal is sub-nonarchimedean quasi-metrizable. Theorem. Every regular wt::.. space that is subquasi-metrizable is a Moore space. Corollary. A countably compact regular space with a

G 6 -diagonal is compact if and only if it is orthocompact. (Received December 4, 1972.)

*73T-G29, JOEL GIBBONS, Ch~cago State University, Chicago, niinois 60628, The nonexistence of some hyperbolic sets in S •

We define a restricted class, called uniform, of 1-dimensional basic sets of Anosov Smale systems and prove Proposition. If A is a uniform basic set on s3, it is either a source or a sink. These have been classified by Williams [Topology 6(1967)]. (Received December 4, 1972.)

*73T-G30. c. L. BANDY, University of Houston, Houston, Texas 77004 and University of Saskatchewan, Saskatoon, Saskatchewan, Canada. A characterization of Hurewicz spaces. Preliminary report.

According to A. Lelek, a Hurewicz space is a topological space having the property that each

sequence of open covers, G1,G2,.,, has a subcollection H that covers the space such that H = ~ U H2U • • • where each H is a finite subcollection of G , Theorem. In a regular Lindelof space the Hurewicz property is equivalent N N ------to each normal sequence having a locally-finite subcollection covering the space. (Received December 4, 1972.)

*73T-G31. ULRICH KOSCHORKE, Rutgers University, New Brunswick, New Jersey 08903. Bordism of manifolds with line element fields, Preliminary report.

A line bundle ( over a (connected) closed C00 -manifold M gives rise to a line element field

(i.e. imbeds into TM) iff 9(M, ~) = Ewi(M)wi(()n-i E z 2 vanishes (when n =dim M is odd) resp. iff the Euler number of M vanishes (when n is even). Define two manifolds with line element fields to be bordant if there is a

A-289 bordism between them extending the line element fields appropriately. According to what orientation conditions are

imposed on the manifolds resp. the line bundles (or their normal bundles) one gets the bordism groups On(1), O~r(1), or coor . . !lln(1), !lln (1), !lln (1). For each kind there lS a long exact sequence (involving natural forgetful maps) which leads to

the determination of these groups, e.g., On(1) ""On(B0(1)) for n = 1,2 (4), and for n = 3,4 (4), On(1) = ker a

(a subgroup of On(B0(1)) of index 2); !lln (1) ""!lln (B0(1))/7ll: 2; !ll~or (1) "" 7ll:4 Ell (!ll4k/:111 2 EB 7ll: 2), !ll~oor (1) ""!lln for n ¥ 0 (4). It follows that every line element field is bordant in !ll*(1) to one which is transversal to a foliation. The same

holds in 0*(1) at least if the underlying manifold has zero signature. (Received December 5, 1972.)

*73T-G32. JAMES R. BOONE, Texas A & M University, College Station, Texas 77843. A characterization of metacompactness in the class of 9-refinable spaces.

A space X is said to have property (p), if for each discrete collection of closed subsets of X,

\Fa : a E A l, there exists a point-finite collection of open sets \G a : a E A l such that Fa c Ga, for each a E A and

Fan G/3 =II, if a i' (3. Theorem. A space is metacompact if and only if it is 9-refinable and has property (p).

Property (p) is hereditary in spaces with closed sets G0• A semistratifiable space is hereditarily metacompact if and only if it has property (p). (Received December 11, 1972.)

73T-G33, C. J. M. RAO, Indian Institute of Technology, Kanpur, India. A minimal T2 compactification for convergence spaces.

Let (X, q) be a convergence space. [Fischer, Math. Ann. 137(1959), 269-303]. Let Y =XU

\x*\, where x* ~X. A convergence structure q' on Y is defined as: 3' E F[Y], X E 3', 3'/X E q(x), then 3' E q'(x), 3' E F[Y], 3' ;;;: n~= 1 3'i' where 3'i E F[Y] such that if trace of any 3'i on X exists, then such a trace has no q-adherence points: then 3' E q(x*). Theorem 1. If (X,q) is a T 2, noncompact convergence space, then (Y,q') is

a T 2 compactification of (X, q). Theorem 2. If (X, q) is also locally compact [i.e. , (X, q) is open in each of its T 2

compactifications], then (Y, q') is a minimal T2 compactification. (Received December 13, 1972. )(Author introduced by Professor S. A. Naimpally.)

*73T-G34. H. E. WHITE, JR., 251 North Blackburn Road, Athens, Ohio 45701. More on Blumberg's theorem. ~ There is a completely regular, Hausdorff, Baire space for which, if 2 °= ~ 1 , Blumberg's theorem does not hold. This space is cocompact, strongly a-favorable, and pseudo-complete. It is not normal and not Cech

complete, (Received December 18, 1972.)

73T-G35. JUN-ITI NAGATA, University of Pittsburgh, Pittsburgh, Pennsylvania 15213. A property of Hyman's M-space.

Theorem. Every M-space in the sense of D. M. Hyman ("A category slightly larger than the metric

and CW-categories", Michigan Math. J. 15(1968), 193-214) is an M1-space in the sense of J. G. Ceder ("Some generalizations of metric spaces", Pacific J. Math. 11(1961), 105-126). This theorem generalizes Ceder's theorem

that every CW-complex is M1, as well as Hyman's theorem (essentially a corollary to Borge's theorems) that every M-space is stratifiable (C. J. R. Borges, "On stratifiable spaces", Pacific J. Math. 17(1966), 1-16), and it also

excludes any possibility to settle Ceder's problem by finding out a non-M1-space among M-spaces. (Ceder's problem:

is every stratifiable space M1 ?) (Received December 18, 1972.)

A-290 73T-G36. RAYMOND F. DICKMAN, JR., Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 and L. R. RUBIN, University of Oklahoma, Norman, Oklahoma 73069. Examples of unicoherent spaces not having property c. Preliminary report.

A connected space X has Property C if for each closed, disconnected subset A of X there are disjoint closed and connected subsets H, K of X both meeting A and with A c H UK. Property C has been used in characterizing unicoherence in certain spaces. The authors have shown that the connected spaces having Property C include all polyhedra and all locally compact, locally connected, Lindelof Hausdorff spaces. Two examples have been constructed. In both examples the space is connected, locally connected, Lindel.Of and unicoherent, but does not have Property c. One of the examples admits a metric under which it is uniformly locally connected; the other is a CW-complex. (Received December 19, 1972.)

73T-G37. GARY F. GRUENHAGE, University of California, Davis, California 95616. A continuously perfectly normal space which is not first countable. Preliminary report.

Phillip Zenor has introduced the concepts of continuously normal and continuously perfectly normal spaces, and states that the question which motivated their study is whether every continuously normal space is metrizable. (He conjectures a negative answer.) In this paper an example of a continuously perfectly normal space which is not first countable is given. The space has only one nonisolated point. (Received December 19, 1972.)

73T-G38. ANTHONY J. D'ARISTOTLE, State University College of New York, Geneseo, New York 14454. On stone Wierstrass spaces.

N. Velicko generalized the well known result of A. D. Tajmanov on the extension of continuous functions by showing that Tajmanov's theorem holds when Y (the image space) is H-closed and Urysohn and the mapping f is a-continuous. We obtain, in a more direct fashion, an even stronger generalization of this theorem. We proceed to show that the class of all SW spaces is not reflective in the category of all completely Hausdorff spaces and continuous mappings. However, an epi-reflective situation is achieved by suitably enlarging the class of admissible morphisms. Some related results are also obtained. We conclude by establishing a mapping theorem for almost realcompact spaces with SW Katetov extensions. (Received December 22, 1972.)

73T-G39. ROBERT A. DOOLEY, Department of Mathematics and Statistics, Oklahoma State University, Stillwater, Oklahoma 74074. Local compactness in strongly convex metric spaces.

In a metric space, a subset that is isometric to a closed interval [0, r] of real numbers is known as a segment; if every two points of the space are joined by a unique segment, then the space is said to be strongly convex.

For a strongly convex metric space (X, p), a function fp :X X X X [0, 1] _, X is uniquely determined if it satisfies p(x,fp(x,y,t)) = tp(x,y) and p(y,fp(x,y,t)) = (1- t)p(x,y). It is known that if fp is continuous, then (X, p) is contractible; also, the compactness of the space implies that fp is continuous, while the completeness of p is not enough to imply the continuity of f • The following is also true. Theorem. If (X, p) is a locally compact, strongly p -- convex, complete metric space, then ~is continuous. (Received December 26, 1972.) (Author introduced by Professor

John M. Jobe.)

73T-G40. ROY CHRISTIAN OLSON, University of Washington, Seattle, Washington 98195. On the product of compact Frlchet spaces. Preliminary report.

Recently, T. K. Boehme and M. Rosenfeld ("An example of two compact Hausdorff Frlchet spaces

whose product is not Frlchet") used the continuum hypothesis to obtain an example of two compact Hausdorff Frlchet

spaces whose product is not Frlchet. The example can be modified so that instead of the continuum hypothesis, the

axiom of Martin and Solovay r•rnternal Cohen extensions", Ann. Math. Logic 2(1970), 143-178) suffices. (Received December 26, 1972.)

A-291 *73T-G41. ROBERT A. HERRMANN, U.S. Naval Academy, Annapolis, Maryland 21402. Nonstandard topological extensions. L Preliminary report.

Let (X, T) be a topological space. In this paper we use nonstandard topology and our theory of

A space (Y, T ) will be called a nonstandard extension of X iff (i) X c y c X V-filters, where V ~ T. Definition. y and X f Y, (ii) X is a dense subspace in Y. Observe that if q EX- X, then q ~ Pn(X), where Pn(X) is a finite power

set iteration. Theorem 1. If there exists a free T-ultrafilter on X, then there exists a nonstandard Baire extension

(bX, Tb) of X. Theorem 2. Let (Z, Tz) be a T 3-space and f: X~ Z continuously such that f[X) ~ Z and f[X] f z.

Then (bX, Tb) exists. Moreover, there exists a subspace (Y, T ~ of (bX, Tb) and a continuous map h from y onto z

such that h \ X~ f. For the following theorems, we assume that (X, T) is not quasi-H-closed. Theorem 3. There

exists a T' ::o Tb such that (bX, T') is a nonstandard quasi-H-closed extension of X and X E T'. Theorem 4. There

exists a nonstandard quasi-H-closed extension (hX,Th) such that X E Th and if X is T 2, then (hX,Th) is T2 except

for hX- X. Theorem 5. There exists a nonstandard quasi-H-closed extension (kX, Tk) of X such that X E Tk'

hX - kX is an infinite set and if X is T 2, then kX is essentially the same as (isomorphic to) the Katetov extension. Theorem 6. If X c L c kX and L is a proper subspace in (kX, Tk)' then L is not quasi-H-closed. (Received

December 26, 1972.)

73T-G42. RONNIE FRED LEVY, Washington University, St. Louis, Missouri 63130. Baire spaces and Blumberg functions. Preliminary report.

If X and Y are topological spaces, a function f: X ~ Y is called Blumberg if there is a dense subset

D of X such that the restriction of f to D is continuous. A space X is called Blumberg if every real-valued function

f: X~ R is Blumberg. Bradford and Gcfftnan p•Metric spaces in which Blumberg's theorem holds,'' Proc. A mer.

Math. Soc. 11(1960), 667-670) proved that a metrizable space is Blumberg if and only if it is Baire. Example. If

space which is X is an 171-set with the interval topology such that card(X) ~ c, then X is a zero-dimensional Baire not Blumberg. Theorem. For an arbitrary topological space X, the following are equivalent: (i) Every f: X ~ R

such that card(f(X));;;;; ~O is Blumberg. (ii) For every Y, every f:X ~ Y such that card(f(x));;;;; ~O is Blumberg.

(iii) Every f: X~ N is Blumberg. (iv) X is Baire. (Received December 29, 1972.)

*73T-G43. WLODZIMIERZ HOLSZTYNSKI, University of Texas, Austin, Texas 78712. Mappings into spheres and cubes and applications to 0'-Cantorian manifolds, reductions and compactifications.

All sets A, x1 , x2, ••• and spaces are assumed to be compact. Definition. A sequence is said to be an n-decomposition of (X,A) if X~ AU U~ X. and dim X. n X.< n -1 for every 1~ 1 1 1 J i ,P j. Remark. n could be an infinite cardinal number (except of Theorem 1). Theorem 1. Let (X1,x2, ••• ) be an

(n + 1)-decomposition of (X, A) and let Y be an ARn -space (the main case Y ~ Sn) and n < ro. Then every map

A ~ Y which can be extended onto each A U Xi (i~ 1,2, ••• ) admits an extension onto whole X. Theorem 2. Let

(X1,x2, ••• ) be an n-decomposition of (X, A), and let f: X~ f and i(x)); then f is not a universal map. Remark. Theorem 2 generalizes the main case, n 1 when Y ~ S , of Theorem 1. Corollary 1. Let (X1,x2, •.• ) be ann-decomposition of X. Then for every universal map f: X ~ f there exists k such that f\~: ~ ~ f is universal. Corollary 2 (Hadziivanov). f is an n-dimensional

0'-Cantorian manifold. Theorem 3. Every n-dimensional compact space contains an n-dimensional 0'-Cantorian

manifold. (Received November 30, 1972.)

A-292 *73T-G44. RICHARD C. DET:J'iER, University of Tennessee, Chattanooga, Tennessee 37401. Sets which are tame in arcs in E •

Definition. Suppose that X is a compact subset of an arc A which is topologically embedded in

E3; X is said to be untangled iff for each r > 0 there is an s > 0 such that if J is a loop in E3 - X which bounds

(homologically) on an s-set in E3 - X, then J shrinks (homotopically) on an r-set in E3 - X. Theorem. Suppose that X is a compact subset of an arc A which is topologically embedded in E3 and that X is untangled; then for each q > 0 there is a homeomorphism f:A .. E3 such that f(x) = x for each x in X, d(x,f(x)) < q for each x in A, and f(A) is tame. (Received December 11, 1972.)

73T-G45. MICHAEL D. RICE, Ohio University, Athens, Ohio 45701. Strengthening of theM-fine condition.

The purpose of this note is to announce three new theorems dealing with M -fine uniform spaces. By a cozero set we mean \ x: f(x) " 0 l for some real valued uniformly continuous map, while a ULUC map f: X .. Y is one whose restriction to each member of some uniform cover of X is uniform. Theorem. X is M-fine and each real valued ULUC map on X is uniformly continuous precisely when each 0'-uniformly discrete cozero cover is uniform.

Theorem. If X is M-fine and each metric valued ULUC map on X is uniformly continuous, then each uniformly locally finite cozero cover is uniform. Theorem. Assume that X is finest in its proximity class. Then the following are equivalent. (1) X is M-fine. (2) X is M-fine and locally fine. (3) Each 0'-uniformly discrete cozero cover is uniform and each uniformly locally finite cozero cover is uniform. (Received January 8, 1973.)

*73T-G46. ROBERT D. EDWARDS, University of California, Los Angeles, California 90024. The topological invariance of simple homotopy tyPe for polyhedra.

A new proof is given of the following theorem, originally proved by Chapman as a corollary to some

more general wom (Abstract 699-G27, these c#ofiai) 19(1972), A-810). Theorem. Suppose X and Y are (locally compact but not necessarily finite dimensional) polyhedra and f: X .. Y is a topological homeomorphism. Then f is a

simple homotopy equivalence. The main tools of the proof are Kirby-Siebenmann•s notion of simple homotopy type for

TOP manifolds (Bull. Amer, Math. Soc, 75(1969), 742-749) and Siebenmann's result concerning approximating cellular

maps between manifolds by homeomorphisms (Topology 11(1972), 271-294). (Received January 8, 1973.)

*73T-G47. LYLE E. PURSELL, University of Missouri, Rolla, Missouri 65401. Compactifications from subrings of C*(X).

It is well known that if X is a completely regular, Hausdorff space X, then the structure space of

c*(X), the ring of all real, continuous functions on X, is homeomorphic to flX, the Stone-Cech compactification of X

[I. Gelfand and A. Kolmogoroff, Dokl. Akad. Nauk SSSR 22(1939), 11-15], We show: Theorem. Any Hausdorff

compactificatlon of a completely regular, Hausdorff space X is a homeomorphic copy of the structure space of some

subring of C*(X). Remark. Let T be any compact, Hausdorff space containing X as a dense subset. Denote CT (X) =

\fiX: f E c*(T)j. Then CT (X) is a subring of c*(X) which contains the unity function. Using results by L. Gillman

and M. Jerison p•Rings of continuous functions," D. Van Nostrand, 1960, §6. 5], it follows that CT (X) is isomorphic

to C*(T). Hence the structure space of CT (X) is homeomorphic to the structure space of C*(T) which is

homeomorphic to T. (Received January 9, 1973.)

*73T-G48. B. J. BALL and RICHARD B. SHER, University of Georgia, Athens, Georgia 30602, A theory of proper shape for locally compact metric spaces.

The notion of the topological shape of a compactum was introduced by K. Borsuk in 1968; it may be

A-293 considered a generalization of homotopy type in the sense that (1) any two compact metric spaces of the same homotopy type have the same shape and (2) any two compact ANR's which have the same shape are of the same homotopy type.

Of the several extensions of Borsuk's shape theory which have been suggested, all retain the applicable versions of (1) and (2) and hence generalize the notion of homotopy type. For noncompact spaces, however, a more geometric approach might be to generalize proper homotopy type instead, and one way of doing this, for locally compact metric spaces, is given here. This notion, called "proper shape", agrees with Borsuk's definition of shape in the case of compacta and satisfies (1) and (2) with "compact" replaced by "locally compact" and "homotopy type" by "proper homotopy type".

In addition, spaces having the same proper shape share a number of geometric properties which are not invariant under any of the other applicable notions of shape; for example, if two locally compact connected metric spaces have the same proper shape, then their one-point compactifications have the same shape, as do their Freudenthal endpoint compactifications. (Received January 11, 1973.)

73T-G49. DOUGLAS MOREMAN, Auburn University, Auburn, Alabama 36830. Convex product topology. Preliminary report.

This paper represents a continuation of work referred to in Abstract 699-B23, these cilotit..V 19(1972), A -797. Suppose that S, B, C is a convex product space in the following sense: S is a set and there exists a set A and a transformation T from Ax S such that (1) for each member x of A and each point P of S, T(x, P) is a point of a space Sx with a T 1 topology Bx and an intersectional convexity ex' (2) S is the Cartesian product of the sets Sx, (3) B is the product topology of the spaces Sx, Bx and (4) C is the convexity for S to which M belongs provided M is a subset of S such that for each member x of A, the set T(x, M) is an element of ex. Theorem. Suppose that the statement that S (alternately Sx) has Property Q has one of the following meanings: (1) S (S::J is convexly perfectly compact, (2) S (S::J is locally convex, (3) S (S::J is convexly Hausdorff, (4) S (S::J is convexly regular, (5) each point set with only one member is convex, (6) each point set that is the closure of a convex point set is itself convex. Then,

S has Property Q if and only if for each member x of A, Sx has Property Q. (Received January 11, 1973.)

Miscellaneous Fields

*73T-Hl. A. BROWN and A. SCHREMMER, Community College of Philadelphia, Philadelphia, Pennsylvania 19107. Model theory as an introduction to mathematics. Preliminary report.

Elementary model theory is investigated as a possible alternative to the usual first semester of finite mathematics. Simple relational and/or operational structures are drawn from the real world. Elementary languages (i.e. involving only names, relation and/or operation symbols) are first introduced and truth is defined semantically. The structures remain simple thronghout but the languages are progressively enriched to first order languages with equality. Consequence and the Galois connection between theories and sets of models are first defined with respect to ontologies consisting of small numbers of interpretations, and then made logical by passing to the universal ontology. Finally, J. Corcoran's natural deductive system is introduced via strong completeness. A

"Student directed learning" approach is used. This is followed, the second semester, by a course where small groups are looked upon as models of equational axioms and/or by a course where small geometries are seen as models of various sets of 2-sorted axioms. More follow-up courses are possible. (Received December 18, l972.)(Authors introduced by Professor F. Schremmer.)

A-294 ERRATA

Volume 19

JAMES W. CARTER. Banach, Hilbert spaces and manifolds modeled thereon, Abstract 72T-Gl90, Page A-771.

WITHDRAWN.

NABIL A. KHABBAZ. An uncountable number of infinite hierarchies. Preliminary report, Abstract 72T-C48,

Page A-708.

On line 5, the phrase "while L(.J-) ~ \x E !;*I CJ :!;, x, 7r E A}" should be replaced by "while L(.J-) ~

{xE !;*icr ~ x, 1r E A}". 7r W. SHEFFIELD OWEN. The Rees theorem for locally compact semigroups, Abstract 699-Al5, Page A-785.

The statement was made that if S is a locally compact completely 0-simple semigroup with no

zero-divisors, the set of nonzero idempotents of S is compact (provided 0 is not isolated). This

is not true. However, if one adds the hypothesis that S - {0} be connected, the desired result

follows.

UPADHYAYULA V. SATANARAYANA. Distribution of supports of representing measures for H, Abstract 698-B8,

Page A-775.

The title should read "Distribution of supports of representing measures for J!Dn.

ARUN K. SRIVASTAVA. Stable and local adjunctions, Abstract 72T-Al86, Page A-569.

In the fourth line "J. J. Kaput" appears as "J. J. Kapur" and the result (ii) should read as

follows: A faithful functor T: !!._ ~ ~ is right adjunctable if and only if it is both locally right

adjunctable and stably right adjunctable.

MANFRED STOLL. Properties of the space hP (0 < p;;o 1) of harmonic functions on the unit disc. Preliminary

report, Abstract 72T-B216, Page A-586.

Theorem 3 should read "Every uniformly bounded family

closure in the topology of uniform convergence on compact subsets".

CHANDAN S, VORA. Fixed point theorems for certain symmetric product mappings of A-ANR and a manifold,

Abstract 72T-Gl03, Page A-543.

On line 7, "M a topological manifold is defined" should read "M, a metric topological manifold,

exists". On line 8, "M be an m-manifold" should read "M be a metric manifold". On line 11,

"there exist C- -Lefshetz maps" should read "there exist ~o~- Lefshetz maps". At the end of the

abstract add the following: "The author has also proved the general case in his dissertation."

H. E. WHITE, JR. The approximation of one-one measurable transformations by measure preserving homeomorphisms,

Abstract 72T-B265, Page A-698.

On line 7, "m<\x: T(x) of cp(x)} < (" should be "~o~<\x: T(x) of cp(x)} < (".

A-295 SITUATIONS WANTED

Unemployed mathematicians, or those under notice of involuntary unemployment, are allowed two free advertisements during the calendar year; retired mathematicians, one advertisement. The service is not available to professionals in other disciplines, nor to graduate students seeking their first postdoctoral positions; however, veterans recently released from service will qualify. Applicants must provide (1) name of institution where last employed; (2) date of termination of service; (3) highest degree; (4) field. Applications from nonmembers must carry the signature of a member. Free advertisements may not exceed fifty words (not more than six 65-space lines) , including address of advertiser; excess words are charged at the rate of $0.15 per word (minimum charge $1). Anonymous listings are carried for an additional fee of $5; correspondence for such applicants will be forwarded to them. Employed members of the Society may advertise at the rate of $0.15 per word; nonmembers, currently employed, will be charged $0.50 per word (minimum charge $15). Deadline for receipt of advertisements is the same as that for abstracts; date appears on the inside front cover of each issue of the c}/oticeiJ . Application forms may be obtained from, and all correspondence should be directed to, the Editorial Depart­ ment, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904. Correspondence to applicants listed anonymously should be directed to the Editorial Department; the code which is printed at the end of the listing should appear on an inside envelope in order that correspondence can be forwarded.

MATHEMATICAL SCIENCES, Ph, D, 1968, Age 38, Cur­ MATHEMATICS TEACHING, RESEARCH, OR ADMINIS­ rently an exchange scientist with the Institute of Mathe­ TRATION, Age 30, Duke Ph, D. 1967 in topological al­ matics, Academy of the Socialist Republic of Romania. gebra. Four published articles, one submitted, Six years Specialities include numerical analysis, control theory, experience, four at the University of Minnesota, two at operations research, and differential equations, Four Northern Michigan University, teaching wide range of years teaching experience, consultant with NASA, and undergraduate and graduate courses. Some administra­ five years working experience with NASA. Publications. tive experience, Available September, 1973, J. 0, Kil­ Available July, 1973, Bernard A. Asner, AmConGen tinen, 806 W, College Ave,, Marquette, Minnesota 49855, (BUCH), APO New York 09757, MATHEMATICS PROFESSOR, TEACHING AND RE­ MATHEMATICS PROFESSOR. Ph, D. 1967; post-doctoral SEARCH, Ph, D. 1965 (London). Speciality: analysis. study 1967-68, Age 36, Speciality: topology and category Five published articles. Twenty years experience in theory, Three publications. Five years teaching expe­ graduate and undergraduate teaching including three in rienc~. Vita and references on request. S, Baron, Clark America, Subjects: complex variable, differential equa­ University, Worcester, Massachusetts 01610, tions, topology, algebra_ Available immediately, Refer­ MATHEMATICS INSTRUCTOR, Age 27, Four years ences and resume on request. S, Mukhoti, F-1919-10, teaching experience, Prefer tw&-year or junior college, University Drive, Cedar Falls, Iowa 50613, References and resume available upon request. Available MATHEMATICIAN, TEACHING AND RESEARCH, Ph, D, June, 1973, William R. Beversdorf, 312 Ohio Street, 1968, complex variables, Age 36, Three articles pub­ Decorah, Iowa 52101, lished, two submitted. Seven years college and university experience plus assistantship teaching, Interested in MATHEMATICS PROFESSOR, TEACHING & RESEARCH, Ph, D. 1954, Age 49, Specialty: applied mathematics. 7 interdisciplinary program, Available spring or fall of published articles, 20 years teaching, several years in 1973. Reprints, references and resume upon request to industrial research. Resume upon request. Available Joseph Warren, 370 Riverside Dr. 1F, New York, New immediately, Harvey J, Fletcher, c/o Harvey Fletcher, York 10025, 1615 N, Willow Lane, Provo, Utsh 84601,

A-296 I~•••· s••••e•·i•••· scllttln••sllilt a111tl ... NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Second Edition Richard W. Hamming, Bell Telephone Laboratories I International Series in Pure and Applied Mathematics 1973, 612 pages (tent.), (025887-2), $14.95 (tent.) Designed for advanced undergraduate and graduate students with calculus backgrounds, this text concentrates on broad fundamentals rather than operational "tricks". The idea of the invariant algorithm, for example, is used to unify the field of algorithms, and classes of formulas are studied rather than isolated ones. Extensively rearranged and rewritten, the new edition continues to offer a unique treatment of the modern frequency approach to computing, and new chapters have been added, including ones on Laplace transforms, optimization, eigenvalues and eigenvectors, quantization, Chebyshev practice, and linear independence. MATRIX THEORY AND FINITE MATHEMATICS Martin Pearl, University of Maryland I International Series in Pure and Applied Mathematics 1973, 480 pages (tent.), (049027-9}, $14.95 This text presents a unified treatment of matrix algebra coupled with an introduction to three of the most important modern applications of matrices: the theory of games, finite Markov chains, and the theory of graphs. Designed for serious upper division students in operations research, information science, statistics, and mathematics, the book shows the student how relatively abstract mathematics can be made relevant to the important problems of today's business. FUNCTIONAL ANALYSIS Walter Rudin, University of Wisconsin, Madison I McGraw-Hill Series in Higher Mathematics 1973, 397 pages, (054225-2), $14.95 This excellent introductory text presents the axiomatics of the field (i.e., the general theory of topological vector spaces), treats several topics in some depth, and contains some interesting applications to other branches of mathematics, such as the study of distributions and Fourier transforms, Banach algebras, and the symbolic calculus. CONFORMAL INVARIANTS Lars Ahlfors, Harvard University I McGraw-Hill Series in Higher Mathematics 1973, 192 pages (tent.), (000659-8}, $10.95 (tent.) This is a textbook primarily intended for students with approximately a year's background in complex variable theory. It emphasizes the geometric approach as well as classical and semi-classical results which the author feels every student of complex analysis should know before embarking on independent research.

A-297 CALCULUS AND ANALYTIC GEOMETRY Sherman K. Stein, University of California, Davis 1973, 920 pages (tent.), (061 006-1 ), $14.95 (tent.) A Solutions Manual will be available. Designed for use in a general undergraduate calculus course, this completely new text offers several distinctive features: (1) incorporation of all necessary analytic geometry into the flow of the course; (2) early treatment of the derivatives of all elementary functions; (3) the separation of integration over plane regions from integration over solid regions by six chapters; (4) approximately 4000 exercises divided into three levels, ranging from the very simple and elementary (the most numerous) through the extremely complex and difficult; and (5) a student-oriented approach, with a summary and study guide (including lists of terms, symbols, key facts, formulas, and theorems) and a guide quiz at the end of each chapter.

CALCULUS: AN INTRODUCTION TO APPLIED MATHEMATICS Harvey P. Greenspan and David Benney, both of Massachusetts Institute of Technology 1973, 800 pages (tent.), (024342-5), $13.95 (tent.) A Solutions Manual will be available. As the authors have stated, "The difficult task of dealing with nature and the social sciences requires a special approach ... in which methods and procedures are judged and vindicated by the scientific comparison of theory with experiments and observations." This text emphasizes the topics of calculus which are of greatest importance in applied mathematics, science, and technology. Special attention is given to the formulation of problems, numerical analysis, approximation methods, perturbation theory, limits, differentiation, integration, series, vectors, and vector calculus. A great variety of methods and techniques actually used in science today are presented for the first time as part of the calculus, where they properly belong.

HANDBOOK OF MATHEMATICAL TABLES AND FORMULAS, Fifth Edition Richard S. Burington 1973, 500 pages, (009015-7), $5.50 This text has been designed to aid those in academic, professional, scientific, engineering, and business fields in which mathematical reasoning, processes, or computations are required. A serious effort has been made to retain information of a more traditional nature while incorporating those definitions, theorems, formulas, and tables needed for contemporary applications. The text has been carefully and accurately compiled; each subject treated is developed in a logical manner, to enable the user to interpret the information easily and properly.

A-298 ftt ~lass•·••••••• 11eetls ... COLLEGE ALGEBRA E. Richard Heineman, Texas Tech University 1973, 332 pages (tent.), (027936-5), ($9.95 tent.) An Instructor's Manual will be available. Designed for a freshman level course, this text seeks to instill in the student a realization that mathematics is a logical science and to develop his capability and understanding of those concepts which have traditionally constituted college algebra. The author achieves these goals by using modern terminology and the postulational approach to the properties of real numbers-together with clarity of presentation and carefully graded sets of diversified problems. Common student errors are anticipated and discussed. CONTEMPORARY TRIGONOMETRY Howard E. Taylor, West Georgia College, and Thomas L. Wade, Florida State University 1973, 264 pages, (067640-2), $7.95 This text presents a concise, modern introduction to the field with emphasis on the trigonometric functions, their inverses, and their properties. The approach utilizes the basic concepts of the real number and rectangular coordinate systems, together with a small amount of set notation and a few elementary concepts of sets. Trigonometric functions and their inverses are viewed as non-empty sets of ordered pairs, no two of which have the same first entry. Historical notes are also provided. MODERN ALGEBRA AND TRIGONOMETRY, Second Edition J. Vincent Robison, Emeritus, Oklahoma State University 1973, 431 pages, (053330-X), $9.50 An Instructor's Manual will be available. This text develops and integrates traditional algebra and trigonometry through the use of concepts and techniques of set theory. Influenced by the recommendations of both the Committee on the Undergraduate Program in Mathematics and the School Mathematics Study Group, it is designed for students having no more than three semesters of high school algebra. The exposition is based on a judicious blend of mathematical rigor and intuition, and a large number of worked examples illustrate concepts and provide methods of attack. ~'J~ .. ·l•itjl( a• ,\\~t;•·a\\'·Hill text. •tnri

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A-299 for many students, studying mathematics can be a difficult task ...

NERING ELEMENTARY LINEAR ALGEBRA Following recommendations of the CUPM in introducing linear algebra earlier in the mathematics curriculum, Professor Nering provides a well-organized presen­ tation for a one-semester or one-quarter course-at the sophomore level. While it is expected that the student will have taken some calculus, such study is not pre­ requisite. The needs of the student are met not by offering a diluted or trivial presentation, but by a careful integration of conceptual material with concrete computational methods. Numerous problems are offered. By Evar D. Nering, Arizona State University. About 305 pages, illustrated. Ready April, 1973. Order Code 6755.

SHAMPINE & ALLEN NUMERICAL COMPUTING: AN INTRODUCTION This text examines the solution of the more common problems in numerical com­ putation. It is written for a one-semester course. Prerequisites are an understand­ ing of calculus and a modest acquaintance with FORTRAN programming. The authors focus on a single effective method for solving each problem and by so doing are able to give a detailed development of the theory. In addition, they show the art of computing as well as the science of numerical analysis. Codes (written in FORTRAN) implementing the various up-to-date algorithms are provided. These are production quality codes which reduce the time otherwise required for programming and debugging. Examples and problems are also both realistic and plentiful. By L. F. Shampine, Sandia Laboratories; and R. C. Allen, Jr., University of New Mexico. About 200 pages. Ready May, 1973. Order Code 8150.

CAIN ELEMENTARY STATISTICAL CONCEPTS Is one of your problems teaching statistics to those whose mathematical back­ ground is weak - yet who need a meaningful one-semester coverage of the basic concepts? Instead of the usual straight exposition, Dr. Cain provides dozens of simple experiments for the student to perform. Each experiment discusses the principles involved, shows how they are applied in common research situations, and asks the student to actively participate in gathering and analyzing experi­ mental data. The student's accomplishments are immediately rewarded by a sim­ ple task well done and, most important, by a thorough understanding of some aspect of statistical theory. By Rolene B. Cain, Morehead State University. 268 pages, illustrated. $5.50. July, 1972. Order Code 2238.

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BLUM & ROSENBLATT PROBABILITY AND STATISTICS Relevance to today's social and scientific problems is the hallmark of this out­ standing, highly-readable text in probability and statistics. Written for students at the junior, senior, or first-year graduate level who are enrolled in either a one­ semester course in probability or a two-semester sequence in probability and sta­ tistics, this text treats the problems encountered in the physical and social sci­ ences and in engineering. The authors cover discrete and continuous probability as well as statistical inference. A Student's Solutions Manual is available at $3.50 per copy from W. B. Saunders Company. A complete Solutions Manual for the Instructor is available gratis from Dr. Rosenblatt, Case Western Reserve Uni­ versity, Cleveland, Ohio 44106. By Julius R. Blum, University of New Mexico; and Judah I. Rosenblatt, Case Western Reserve University. 549 pages, illustrated. $13.95. March, 1972. Order Code 1763.

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OWEN & MUNROE FINITE MATHEMATICS AND CALCULUS Professors Owen and Munroe present the elementary mathematical concepts essential to students pursuing programs in the social and management sciences. Conceptual material is carefully motivated through the use of intuitively easy-to­ grasp, practical examples. These numerous examples of quantitative methods have been drawn from the professional literature. Among the many topics care­ fully developed for this book's particular audience are the first steps of the cal­ culus of several variables-with applications. Another equally important feature of the book is its more than 625 problems. Answers to a majority of them are provided in the text. By Guillermo Owen, Rice University; and M. Evans Munroe, University of New Hampshire. 598 pages, 120 illustrations. $10.50. May, 1971. Order Code 7040.

FAIRCHILD & IONESCU TULCEA SETS This text is outstanding from both the standpoint of mathematical precision and of elegance. It is ideal for sophomore or junior-level courses on set theory. Selected materials are appropriate for freshman courses. Formal systems are avoided, although references are made to various axioms of set theory. Definitions and theorems are carefully stated and proofs meticulously written out. Many examples are presented to illustrate concepts as they are defined, and to motivate new concepts or to make plausible new results. Modern terminology is used throughout the text. By William W. Fairchild, Union College; and C. Ionescu Tulcea, Northwestern University. 121 pages, illustrated. Soft Cover. $4.50. July, 1970. Order Code 3540.

FAIRCHILD & IONESCU TULCEA TOPOLOGY The fundamental notions of topology are presented in this text for the first year graduate student or advanced undergraduate. Central theorems in the study of topology, for example those of Tychonov, Urysohn, Tietze, and Stone-Weierstrass, are covered. 271 pages. $12.00. April, 1971. Order Code 3543.

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A-302 Saunders texts help make your job and theirs a good bit easier

MAXFIELD & MAXFIELD ABSTRACT ALGEBRA AND SOLUTION BY RADICALS In this junior-level text for a one-semester course in abstract algebra, the authors achieve an unusually strong sense of motivation by focusing the entire study on a goal result, i.e., Abel's theorem that equations of degree higher than four are not solvable by radicals. This approach permits the natural introduction and development of the basic concepts of modern algebra: groups, rings, fields, and the Galois theory connecting groups and fields. By John E. Maxfield, Kansas State University; and Margaret W. Maxfield. 203 pages, illustrated. $9.75. April, 1971. Order Code 6187. KASRIEL UNDERGRADUATE TOPOLOGY Written specifically for junior and senior mathematics majors, this text provides an introduction to set-theoretic topology built pedagogically upon a firm founda­ tion in metric spaces. In addition, this volume places greater-than-usual emphasis on the study of open, closed, compact, and quasi-compact mappings. By Robert H. Kasriel, Georgia Institute of Technology. 285 pages, illustrated. $10.50. January, 1971. Order Code 5298. MUNROE CALCULUS Outstanding for a three semester sequence, this basic text presents the calculus of one variable, linear algebra, and multidimensional calculus in an elegant, easy­ going manner (what might be called the "soft approach"). Broad coverage of ideas and techniques is emphasized rather than the rigorous development of any one topic. A significant number of exercises are offered. A Teacher's Guide is avail­ able. By M. Evans Munroe, University of New Hampshire. 763 pages, 362 illustrations. $12.75. April, 1970. Order Code 6608. OWEN GAME THEORY This book provides an excellent introduction to both two-person (including multi­ stage) and n-person game theory-from a mathematical point of view. Included are superb chapters on infinite games (including games of timing), multi-stage games (including games of exhaustion, stochastic games, and differential games), and modifications of the game concept (including games with a continuum of players). By Guillermo Owen, Rice University. 228 pages, illustrated. $9.00. January, 1968. Order Code 7028.

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A-303 ADDISON ·WESLEY IN MATH IS LIKE HAXALL IN FIELD GOALS*

Other publishers offer many good math books, but when you look at Addison­ Wesley's, we think you'll find they are as outstanding in their field as Haxall was in his. Have you seen these books, for example? The Elementary Functions, Second Edition by Charles R. Fleenor, Merrill E. Shanks, and Charles F. Brumfiel. This CUPM-in­ spired text is designed to provide the neces­ sary foundation for a standard calculus

* Among the great college and professional place­ kickers, James Haxall stands out because he set a record no one has equalled in 91 years: he kicked a 65-yard field goal for Princeton in 1882. (The N.F. L. record of 63 yards was set in 1970 by Tom Dempsey of the New Orleans Saints.)

A-304 course, treating those functions that are later differentiated and integrated in the calculus. Among the changes in the revi­ I sion are two new chapters on sequences and real and complex numbers. Addison-Wesley Analytic Geometry, Fourth Edition by PUBLISHING COMPANY Gordon Fuller. This book is intended as a Reading, Massachusetts 01867 one-semester bridge between algebra and ATTN: Mary Clare McEwing calculus. The revision includes new materi­ al on relations and functions, families of send me the Addison-Wesley circles, and surfaces of revolution. Please catalog for 1973-74. A First Course in Calculus, Third Edition Please send me the following book(s) by Serge Lang. This self-contained text to consider for adoption in the presents the basics of derivative and inte­ course(s) described below: gral calculus and the techniques and appli­ cations which accompany them. It also re­ views analytic geometry and includes the first four chapters of Calculus of Several Variables. Calculus of Several Variables by Serge Lang. A revision of A Second Course in Calculus, Second Edition, this text is designed to al­ low maximum flexibility in structuring a second-year calculus course. Topics in lin­ ear algebra are introduced only as they are needed for the calculus.

Calculus with Probability: For the Life c.. N and Management Sciences by Willard E. Baxter and Clifford W. Slayer. Motivated by probability throughout, this text pre­ sents single and multivariate calculus with an introduction to differential equations. w ~ w To meet the needs of students in the life § ...J 1- 0 <( and management sciences, the authors 0 1- 1- 1- Ill z i= U) stress mathematical modeling and applica­ w :2: tions. ...J ...J 0 Simplified BASIC Programming, with Com­ a: z panion Problems and Simplified FORTRAN w Programming, with Companion Problems 0 z by Lisa and Judah Rosenblatt. These intro­ <( ductory paperback texts are designed to (/) supplement pre-calculus, calculus, and in­ w ...J troductory computer science courses. 0 1- w i= (/) ::J ~ >- a: ...J w 1- Ill z w :2: (/) ::J w z a: (/) a: w c.. (/) 0 (/) w Addison-Wesley THE SIGN OF (/) w a: EXCELLENCE ::c a: 1- PUBLISHING COMPANY, INC. 1- ::J X :2: 0 >- ::J 0 w <( 0 1- Reading, Massachusetts 01867 <( u 1- z <( u

A-305 New Texts for1973 Math Courses FINITE MATHEMATICS WITH APPLICATIONS By Abe Mizrahi, Indiana University, Northwest; and Michael Sullivan, Chicago State University An introduction to probability, the mathematics of finance, Markov chains, linear programming, matrices, directed graphs, game theory, and statistics utilizing real-life applications from the business, social and behavioral sciences. Features a chapter on mathematical models. Fall 1972 Approx. 512 pages $10.95 (tent.) ELEMENTARY LINEAR ALGEBRA By Howard Anton, Drexel University. Presents the fundamentals of linear algebra in the clearest possible way, using over 200 computational examples and geometrical interpretation. Moves from systems of linea( equa­ tions and matrices to determinants, vectors in 2-space and 3-space, vector spaces, linear transformations, eigenvalues, eigenvectors, and quadratic forms, ending with an introduction to numerical methods. Jan. 1973 Approx. 352 pages $10.25 (tent.)

ELEMENTARY ALGEBRA: Structure and Skills Third Edition By Irving Drooyan, Walter Hadel, and Frank Fleming, all of Los Angeles Pierce College. The third edition of this modern, structure oriented beginning algebra text reflects seven years of classroom experience. As a result, additional emphasis has been given to techniques of graphing and new sections on order of operations, numerical evaluation, and the distance between two points have been added. Jan. 1973 Approx. 512 pages $9.95 (tent.)

ELEMENTS OF STATISTICS By E. W. Averill, Clarion State College. Basic statistics for students with no mathematical background beyond high school algebra. Includes easy-to-understand treatments of descriptive statistics, probability, sampling, regres­ sions and correlation, non parametric statistics, chi-square, and analysis of variance. Fall1972 Approx. 288 pages $10.95 (tent.)

INTRODUCTION TO THE THEORY OF STATISTICS By Harold J. Larson, Naval Postgraduate School. Covers the commonly used statistical methods in a clear and mathematically precise way. Offers students in mathematical statistics courses an unusual number of realistic examples from applied fields. 1972 242 pages $11.95

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A-306 Data Structures and Programming Malcolm C. Harrison, Courant Institute, New York University © 1973, 352 pages, hardbound Concerns the properties of the data-objects which are manipulated by programs, and the data-structures used to represent them in memory. Linear Programming with Fortran Carvel S. Wolfe, United States Naval Academy © 1973,240 pages, softbound Presents the principal topics of linear programming and simultaneously introduces computer routines to aid in computations. Foundations of Differentiable Manifolds and Lie Groups Frank W. Warner, University of Pennsylvania © 1971,270pages,hardbound Provides a proof of the strong form of the de Rham theorem via axiomatic sheaf cohomology theory and a complete, self-contained proof of the Hodge theorem, in addition to basic material on manifold theory and Lie groups. The Lefschetz Fixed Point Theorem Robert F. Brown, University of California, Los Angeles © 1970, 186 pages, hardbound Deals with topological theory and results, the Lefschetz Fixed Point Theorem and its converse in particular. Lecture Notes on Elementary Topology and Geometry I. M. Singer, Massachusetts Institute of Technology John A. Thorpe, State University of New York, Stony Brook © 1967,214 pages, softbound Uses the reader's knowledge of algebra and advanced calculus to point out results relating geometry, topology, and group theory. I Price includes postage and handling. Payment must accompany order. Add sales I ,.------~tax where applicable. I Qty Code Title Price I 5964 Data Structures and Programming $11-95 1 I 7797 Linear Programming with Fortran $ 6.95 I 5737 Foundations of Differentiable $10.50 I Manifolds and Lie Groups 1 I 5395 The Lefschetz Fixed Point Theorem $12.95 1 5163 Lecture Notes on Elementary Topology $ 7.25 I and Geometry I Send books to: I I Name I 1 Institution I 1 Address I I City State Zip 1 I Send order to: I I James Shorr, Advertising Department 1 Scott, Foresman College Division I, ______1900 East Lake Avenue Glenview, Illinois 60025 ,I

A-307 Macmillan's Outstanding New Math Texts

Coming in March ALGORITHMIC COMBINATORIES By Shimon Even, The Weizmann Institute ELEMENTARY MATRIX ALGEBRA of Science, Rehovot, Israel Third Edition This text on combinatorial mathematics and graph theory for upper-level undergraduate or graduate By Franz E. Hohn, students in computer science, applied mathematics, University of Illinois, Urbana and electrical engineering presents the varied aspects of the subject from a computer-oriented viewpoint An introductory matrix or linear algebra text easy The only assumed background is basic college alge­ for non-mathematically oriented students to under­ bra. Covering such wide ranging subjects as permu­ stand. No prior knowledge of matrix or linear alge­ tations, combination, partitions, graphs, trees, cir­ bra is assumed, and all explanations are full and cuits, and networks, this text offers more comprehen­ clear. This edition features an early introduction of sive treatment than competing texts. The chapters substantial geometric material, simplification of are almost entirely independent of each other, en­ notation, a more coherent arrangement of the mate­ abling students to concentrate on the topics they are rial, expansion of the lists of exercises, and reloca­ most interested in. A wide variety of algorithms for tion of the chapter on determinants in order to place many graph theoretic problems are included. Pro­ maximum emphasis on the basic concepts and meth­ gramming language is not used. ods of linear algebra. 1973 400 pages $13.95 1973 approx. 416 pages, prob. $10.95

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AN INTRODUCTION TO INTRODUCTION TO LINEAR ALGEBRA MATHEMATICAL REASONING By Franz E. Hohn, By Boris lglewicz and Judith Stoyle, University of Illinois, Urbana both, Temple University A simple, mathematically sound, and reasonably This text helps students develop an understanding of comprehensive introduction to linear algebra for mathematical logic and thought by exposing them to students with no calculus background. There are the methods and logical basis of the mathematical many illustrative examples and exercises, explana­ proof. Assuming only a knowledge of basic high­ tions and proofs are presented in full, and the nota­ school mathematics, the book provides extensive tion has been kept as simple as possible. Standard coverage of the indirect method of proof, counter­ topics are arranged from simple to abstract. In addi­ examples, the use of analogy in mathematics, and tion, the book features strong emphasis on geometry many other important topics. The problem-oriented as an intuitive foundation for abstract concepts, approach devotes a minimum of space to discussion basic solutions of systems of linear equations, anc and a maximum to many illustrative examples and special attention to geometrical applications and end-of-chapter exercises. There is an exceptionally interpretation of linear equations. A Solutions Man­ useful bibliography and appendix. ual is available, gratis. 1973 256 pages $5.95 1972 321 pages $10.75

For further information write to: MACMILLAN PUBLISHING CO., INC. 100 A Brown Street Riverside, New Jersey 08075 In Canada, write to Collier-Macmillan Canada, Ltd., 1125B Leslie Street, Don Mills, Ontario

A-308 Math professors tell why they like: Willerding & Mathematics: The Alphabet of Science, 2nd Edition ($9.95) Instructor's Manual. Haywara: "I like the fact that each chapter continues long enough to get into some of the interesting, advanced problems. This gives the better students a challenge. Also, if the whole class actually gets enthused over a section, it then becomes possible to pursue the matter further. I believe this last point is why we chose to adopt this text over the !]I) current one we are using." -J. E. Koehler, Seattle Univ. Drooyan& Elementary Algebra for College Students, 3rd Edi­ tion ($8.95) Study Guide by Charles Carico. Wooton: "Very well written. Clear exposition. Chapter material presented in a logical manner. Sample problems pre­ sented in a manner which the student can understand." -M.A. Chmielewski, Virginia State College

Elementary Statistics, 3rd Edition ($1 0.25) Already in use at 150 colleges and universities! "In the elementary statistics book, Hoel blends statistical theory with applications in an easy, straight-forward manner which is appealing to the concerned statistics student." -R. M. Johnston, Midland Lutheran College

For more information about these 3 successful math texts. contact your local Wiley represen­ tative, or write to Ben Bean. Dept.1139, N.Y. office. Please include your course title, enroll­ ment, and present text.

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A-309 ~------~etV Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Edited by Lucien LeCam, Jerzy Neyman, and Elizabeth L. Scott Vol. I: Contributions to the Theory of Statistics, 776 p., $28.50 Vol. II: Contributions to Probability Theory, Part I, 620 p., $22.50 Vol. Ill: Contributions to Probability Theory, Part II, 690 p., $25.50 The first three volumes of the Berkeley Symposium series are the result of the conference concerned with the classical fields of statistical research. Vol. IV: Contributions to Biology and Problems of Health, 300 p., $13.50 Volume Four focuses on the uses of statistical work in biology and medicine. It discusses the stochas­ tic processes designed to represent biological phenomena, theory and practice of clinical trials, and the problems of demography. Vol. V: Contributions to Problems of Molecular Evolution, 370 p., $13.50 The fifth volume concerns the work of the conference on the participation of mathematical statis­ ticians in the field of Non-Darwinian or Neo-Darwinian evolution with its roots in the discovery of DNA molecules. Vol. VI: Contributions to Effects of Pollution on Health, 610 p., $22.50 f (~The sixth, and last, volume deals with the work of the conference on environmental pollution and ~~-j~'h/;~~u~:Uj;;;;:··~ '" ,h~ tl~

University of California Press • Berkeley 94720

BMS Regional Conference Series in Mathematics

UNIVERSITY OF NEW BRUNSWICK Fredericton, N. B., Canada ANALYSIS ON LIE GROUPS AND HOMOGENEOUS SPACES is actively searching for a new chairman by Sigurdur Helgeson of the department of mathematics. Candi­ Number 14 dates should have the Ph.D. or its equiva­ vi + 66 pages; list price $4.00; individual price lent in any appropriate field and should $3.00; ISBN 0-82 7 8-1664-9 Book code CBMS/14 also have a strong interest in research The theme of this volume is a treatment of differ­ and in teaching. Prior administrative ential equations on a Coo manifold V by separation experience desirable. The term as Chairman of variables techniques. More specifically, given is for three years beginning July 1, 1973 a Lie transformation group L of V and a Lie sub­ and is renewable by mutual agreement. group H C L, if D(V) is the set of differential Applications should be received by operators on Vinvariant under L, then the principal is the set of distributions T on V February 28, 1973 and should be sent to object of study satisfying the following two conditions: (Q T is an Dr. Thomas J. Condon eigendistribution of each DE D(V}; (ii) T is Dean of Arts and Chairman invariant under H. of the Search Committee

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A-310 ROYAL IRISH ACADEMY Summer School on Group Theory and Computation

The National Committee for Mathematics of the Royal Irish Academy will hold a Summer School on Group Theory and Computation at University College, Galway, Ireland from 16th to 21st July 1973. Courses will be given by the following: Dr. J. H. Conway (Cambridge) Professor Marshall Hall (Jr.) (California Institute of Technology) Professor K. Hirsch (Queen Mary College, London) Dr. P. Neumann (The Queen's College, Oxford) Further details and registration forms may be obtained from the Director, Mr. M.P. J. Curran Department of Mathematics UNIVERSITY COLLEGE Galway, Ireland

INDEX UNIVERSITY OF OF ADVERTISERS MIAMI Academic Press ...... cover 3 Addison-Wesley invites applications for Publishing Co ...... A-304, A-305 Graduate Teaching Assistantships American Mathematical Society ..... A-310 for fall 1973 The Macmillan Co ...... A-308 McGraw-Hill Book Co ...... A-297, A-298, A-299 Applications will be accepted until Royal Irish Academy ...... A -311 Stipends $2700- September 1, 1973. W. B. Saunders Co. $3150 plus tuition remission...... A-300, A-301, A-302, A-303 Scott, Foresman and Co...... A-307 Springer-Verlag Professor David Hertzig New York, Inc ...... cover 4 University of California Press ...... A-310 University of Miami University of Miami ...... A-311 P. 0. Box 9085 The University of Coral Gables, Florida 33124 New Brunswick ...... A-310 John Wiley & Sons, Inc ...... A-306, A-309

A-311 RESERVATION FORM NEW YORK MEETING THE BILTMORE HOTEL APRIL 18-21, 1973

The Biltmore Hotel of New York is the official headquarters hotel. Your room reservation should be sent directly to the Reservations Office, The Biltmore Hotel, Madison Avenue at 43rd Street, New York, New York 10017. Be sure to specify the names of persons for whom reservations are being made. BRING THE CONFIRMATION SLIP WITH YOU AS PROOF OF YOUR RESERVATION. Unless otherwise requested, the hotel will hold reservations until 6:00 p.m. on the day of your arrival. Checkout time is 1:00 p.m. In order to secure the special rates listed, you must use the form below. Please copy or tear off the attached coupon and use as your room reservation blank.

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A-312 TREATISE ON ANALYSIS FOURIER ANALYSIS IN Volume 3 PROBABILITY THEORY by J. A. DIEUDONNE by TATSUO KAWATA, Keio University, Yoko­ A Volume In the hama, Japan PURE AND APPLIED MATHEMATICS Series A Volume in the Dleudonne's TREATISE ON ANALYSIS pro­ PROBABILITY AND MATHEMATICAL vides, for the mathematician of the 1970's, the STATISTICS Series equivalent of a cours d'analyse In the tradition This book consolidates the most useful results of Jordan, Picard, and Goursat. Volume three from the theories of Fourier series, Fourier contains chapters sixteen and seventeen of the transforms, and Laplace transforms-and clari­ Treatise. Chapter sixteen concentrates on fies the interactions and analogies among these topological and geometrical foundations, and theories-in a way that enables the reader to Introduces differential manifolds, fibration, and easily find the results and proofs he must have the Lie group. Other topics covered in this before he can proceed to more thorough In­ chapter Include: orientation; Integration of vestigations. The book falls roughly into two differential forms; Sard's theory; embedding parts. The first section presents the elements and approximation theorems; a thorough treat­ of classical Fourier analysis in the context of ment of covering spaces of a manifold. Chapter their applications to probability theory. The seventeen covers the notion of distributions (on second section covers basic results from the a differential manifold) In an elementary man­ theory of characteristic functions of probability ner, and then moves on to the definition of distributions, the convergence of distribution differential operators (of any order), an elemen­ functions In terms of characteristic functions, tary treatment of vector fields (as operators), and series of independent random variables. exterior differentiation, and the differential 1972, 684 pp., $32.00. operators associated with a linear connection on a manifold. 1972, 404 pp., $18.50. DIFFERENTIAL ALGEBRA AND REALIZABILITY THEORY FOR ALGEBRAIC GROUPS by E. R. KOLCHIN, Columbia University CONTINUOUS LINEAR A Volume In the SYSTEMS PURE AND APPLIED MATHEMATICS Series by A. H. ZEMANIAN, State University of New This book provides a unified and self-contained York at Stony Brook exposition of present-day differential algebra A Volume In the in a purely algebraic setting. He develops the MATHEMATICS IN SCIENCE AND basic facts about differential rings, differential ENGINEERING Series polynomials, and differential fields and applies Every physical system defines a relation be­ them to the Galois theory of differential fields. tween Its input and output signals. Various Features include: the only comprehensive physically significant properties of the system treatment of the Ritt theory of differential equa­ -linearity, time-invariance, causality, passivity, tions published since Rltt's book of 1950; a etc.-induce analytical descriptions for that unique, unified account of the Galois theory of relation; conversely, particular classes of an­ differential fields. alytical relations characterize the physical 1973, about 450 pp., In preparation. properties. This book covers the latest and most general results In the field. A notable feature of SYMMETRY GROUPS the work Is Its generalization of the axiomatic AND THEIR APPLICATIONS treatment of electrical n-ports-achieved by by WILLARD MILLER, Jr., University of allowing the Input and output signals to take their instantaneous values in an arbitrary com­ Minnesota plex Hilbert space, rather than In n-dimensional A Volume In the Euclidean space. 1972, 242 pp., $14.50. PURE AND APPLIED MATHEMATICS Series This book treats those aspects of group theory INFINITE ABELIAN GROUPS that are useful in the physical sciences. It Volume 2 presents the mathematical apparatus under­ lying the applications with a high degree of by LASZLO FUCHS, Tulane University rigor. Distinguishing features of the book In­ A Volume in the clude: a systematic derivation of crystallo­ PURE AND APPLIED MATHEMATICS Series graphic point and space groups; a simplified This two-volume work pro- yet precise presentation of vides an Introduction to the the theory of local linear Lie study of Abelian (I.e., com- groups; a construction of the mutative) groups and a com­ representations of the classi­ prehensive summary of the ~ cal groups using both weights material on which research in and Young diagrams; an inte­ Abelian groups can be based. ACADEMIC grated theory which Includes Primary emphasis Is on struc­ applications not only to classi­ tural problems, with attention PRESS cal and quantum physics but given to homological ques­ also to geometry and special 11: FIFTH AVFNUE tions and some topological NEW YORK N Y 10003 function theory. 1972, about considerations. 1973, about 425 pp., In preparation. 375 pp., In preparation. Ill Horst Schubert 3 ~ Ill Categories ~ -.... Translation from the German by E. Gray

XI, 385 pages. 1972 ISBN 0-387-05783-8 Cloth $29.50 This comprehensive textbook is of great inter­ est not only for topologists and algebraists, but also for applied mathematicians and for logi­ cians, for whom the mastery of categorical con­ cepts is now increasingly important. Hans Hermes Introduction to

Mathematical Logic ~ -~ ...:I ~ Translation from the German by Diana Schmidt 0\ < N u 0 '0 ~ CP Approx. 200 pages 1972 ISBN 0-387-05819-2 Soft cover $8.90 -< '0 CP ~ a fil - This book gives an introduction to classical ~ """'... :c CP ::s predicate logic and requires as prerequisite ~ -'0 c.!)"' only an elementary knowledge of fundamental mathematical concepts. ~~~ ~ ~>D.; -"'0 u ~ ~ c.. et=-8 =... ~ . ·~ ::s Springer-Verlag New York Inc. 175 Fifth Avenue • New York, NY 10010 ~ ~ c..~ c..e ~-