Calendar

This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the cJ{oliiriJ 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 fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Abstracts should be submitted on special forms which are available in most departments of mathematics; forms 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 before the deadline for the meeting. Meeting Deadline for Abstracts* Number Date Place and News Items

716 October 26, 1974 Middletown, Connecticut Sept. 3, 1974 717 November 8-9, 1974 Nashville, Tennessee Sept, 25, 1974 718 November 23, 1974 Los Angeles, California Sept. 25, 1974 719 November 23, 1974 Houston, Texas Sept. 25, 1974 720 January 23-27, 1975 Washington, D. C. Nov, 6, 1974 (81st Annual Meeting) March 20-21, 1975 Mobile, Alabama March 23-26, 1975 New York, New York April 18-19, 1975 Monterey, California August 18-22, 1975 Kalamazoo, Michigan November 7-8, 1975 Blacksburg, Virginia November 15, 1975 Los Angeles, California January 22-26, 1976 San Antonio, Texas (82nd Annual Meeting) *Deadline for abstracts not presented at a meeting (by title). October 1974 issue: August 29

OTHER EVENTS January 1975 Symposium on Some Mathematical Questions in Biology New York, New York November 6, 1974 January 1975 Symposium on Theory vs. Practice in the Finite Element Method - New York, New York

The zip code of the Post Office Box of the Society has been changed from 02904 to 02940, Correspondents are requested to note this change in their records,

PI.. affix the peel-ofF label on th- cNOiu:aJ to correspondence with the Society concerning fiscal matters, changes of address, promotioDa, or when placing orders for books and joumals. The ci-/diaiJ of the American Mathematical Society is published by the American Mathematical Society, P. 0. Box 6248, Providence, Rhode laland 02940, in January, February, April, June, Augult, October, November, and December. Subscription per annual volume is $10. Member subscription of $5 is included in annual dues. Price per copy $3. Special price for copiell aold at rqistration dab of meetinga of the Society, $1 per copy. Subscriptions, orders for back numbers (back issu011 of the last two yean only are available), and inquiries should be addreeeed to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Ialand 02940. Second clau postage paid at Providence, Rhode Ialand, and additional mailing offices.

Copyright © 197 4 by tbe American Mathematical Society Printed io tbe United Statea of Amema OF THE AMERICAN MATHEMATICAL SOCIETY

Everett Pitcher and Gordon L. Walker, Editors Wendell H. Fleming, Associate Editor CONTENTS

CALENDAR OF MEETINGS ...... Inside Front Cover PRELIMINARY ANNOUNCEMENTS OF MEETINGS 194 CHAIRMEN AND TOPICS OF SPECIAL SESSIONS 196 INVITED SPEAKERS AT AMS MEETINGS 197 SYMPOSIUM ON SOME MATHEMATICAL QUESTIONS IN BIOLOGY 198 INTERNATIONAL CONGRESS OF MATHEMATICIANS 199 NOMINATIONS FOR VICE-PRESIDENT OR MEMBER-AT-LARGE 204 NONACADEMIC EMPLOYMENT OF PH. D. •S ...... 206 DOCTORAL SCIENTISTS AND ENGINEERS IN THE UNITED STATES 212 ACKNOWLEDGEMENTS . • ...... • . 215 SPECIAL MEETINGS INFORMATION CENTER 218 QUERIES ..•..•.. 220 LETTERS TO THE EDITOR . 222 NEW AMS PUBLICATIONS . 224 PERSONAL ITEMS . . . . . 230 NEWS ITEMS AND ANNOUNCEMENTS 198, 211, 219, 221, 223, 231 ERRATA TO ABSTRACTS 234 ABSTRACTS ..... A-473 SITUATIONS WANTED . . A-511 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Seven Hundred Sixteenth Meeting Wesleyan University Middletown, Connecticut October 26, 1974

The seven hundred sixteenth meeting of the Balcony Suite $35 (2-6) (2 dble beds & American Mathematical Society will be held at 1 sofa bed) Wesleyan University, Middletown, Connecticut, Each additional person $3. OO.Cots, $3. 00. on Saturday, October 26, 1974. Plus 6% Connecticut sales tax. By invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings, MIDDLETOWN MOTOR INN (1 mi) there will be two one-hour addresses. Professor Washington Street Extension (Meriden Road, Philip J. Davis of Brown University will speak Route 66) on "Geometry, computer graphics, and theorems Middletown, CT 06457 of visual type". Professor Mark Kac of Rocke­ Phone: (203) 346-9251 feller University will speak on "Some analytic Single $11-12 problems suggested by statistical mechanics". Double $13-14 (1 bed) Professor Ernest G. Manes of the Uni­ Twin $16-17 (2 beds) versity of Massachusetts, Amherst, will organ­ Each additional person, $2. 00 ize a special session on Category Theory as Plus 6% Connecticut sales tax Applied to Analysis and Topology. Sessions for contributed ten-minute papers Reservations should be made directly with will be scheduled in the morning and the after­ the motel, mention of this meeting being made in noon. No provision will be made for late papers. the correspondence or call. All prices quoted in Abstracts should be submitted to the American this announcement are as of June 1974. Mathematical Society, P.o. Box 6248, Provi­ dence, Rhode Island 02940, so as to arrive prior TRAVEL to the deadline of September 3, 1974. See page 179 of the June 1974 c}(otiui) for new format Middletown, located in central Connecticut on the Connecticut required for abstracts. The final pro~:r:;am will River, is approximately 15 appear in the October issue of these cJIIotiui), miles south of Hartford and 25 miles northeast of New Haven. Midway between New York City ACCOMMODATIONS and Boston, it is about 2-1/4 hours of driving time from the center of each city. State Highways The following motels provide the accom­ 9, 17, and 66 pass thru Middletown, and National modations nearest to the Wesleyan University Interstate Highway I-91 is 5 to 8 miles away. campus, distance in miles being given next to Middletown is served by Continental Trailways the name: and Greyhound Bus Lines. Bradley International Airport, about 32 miles north of Middletown, is CRESTLINE MOTEL (1-1/2 mi) off I-91 in Windsor Locks, and Tweed-New Haven Meriden Road (Route 66) Airport is just outside of New Haven. Direct taxi Middletown, CT 06457 fare between these airports and Middletown is Phone: (203) 347-6955 around $20. The nearest Penn Central passenger Single $10.60 train station (Amtrak) is located in Meriden, Double $12.75 (1 bed) about 10 miles west of Middletown. The taxi Twin $14.85 (2 beds) fare between Meriden and Middletown is approx­ Quadruple $19.10 (2 beds) imately $9. Scheduled limousine service between 6% Connecticut sales tax included Bradley International Airport and Lord Cromwell Motor Inn (see under ACCOMMODATIONS above) LORD CROMWELL MOTOR INN (3 mi) is available at $6. 50 one way and $11. 50 round Berlin Road (Route 72) trip; for information on schedules and fares, Cromwell, CT 06416 contact Central Connecticut Limousine Service, Phone: (203) 347-7427 Inc., 65 Quinnipiac Avenue, North Haven, CT (Numbers in parentheses refer to number of 06473, telephone: (203) 562-3165. persons.) Single $16 (1) $20 (2) (1 dble bed) Walter H. Gottschalk Double $18 (1) $25 (2) (2 dble beds) Associate Secretary King $18 (1) $25 (2) (1 king bed) Middletown, Connecticut

194 The Seven Hundred Seventeenth Meeting Vanderbilt U Diversity Nashville, Tennessee November 8-9,1974

The seven hundred seventeenth meeting of Three cafeterias on campus will be open the American Mathematical Society will be held for all meals. Also a list of local restaurants at Vanderbilt University in Nashville, Tennessee, will be available at the registration desk. on Friday and Saturday, November 8-9, 1974. Three motels near the campus are holding By invitation of the Committee to Select blocks of rooms with a reservation deadline of Hour Speakers for Southeastern Sectional Meet­ October 25. Reservations should be made direct­ ings, there will be three one-hour addresses, ly with them, with mention of this meeting in­ the first being at 1:00 p.m. on Friday in the au­ cluded in that correspondenceo The zip code for ditorium of the Sarratt Commons. All other ses­ each of the following is 37203. sions will be held in the Stevenson Center for the Natural Sciences. Professor Trevor Evans of HOLIDAY INN-VANDERBILT Emory University will give an address entitled (100 rooms reserved) "Word problems". An address entitled "Geom­ 2613 West End Avenue etry of sub-manifolds in Euclidean space" will (five blocks from the Stevenson Center) be given by Professor Robert B. Gardner of the Phone: (615) 383-1147 University of North Carolina, and Professor Single $14. 00 up James R. Retherford of Louisiana State Univer­ Double $18. 50 up (one bed) sity will present an address entitled "Banach TWins $20.50 up (two beds) ideals of operators". There will be three special sessions in ALLEN MOTEL addition to the regular sessions. Professor J. v. (15 rooms reserved) Brawley of Clemson University, Clemson, South 2004 West End Avenue Carolina is arranging a special session on Enu­ (six blocks from the Stevenson Center) merative Combinatorial Theory, Professor Phone: (615) 327-1841 Robert M. McConnel of the University of Ten­ Single $12.00 nessee, Knoxville, Tennessee is arranging a Double $14.00 special session on Number Theory, and Pro­ fessor Richard S. Varga of Kent State Univer­ sity, Kent, Ohio is organizing a special session ANCHOR MOTEL on Approximation Theory. Any member of AMS (40 rooms reserved) who would like to have his or her paper consid­ 1921 West End Avenue ered for inclusion in one of the special sessions (six blocks from the Stevenson Center) Phone: (615) should have his or her abstract so marked and in 327-4581 Providence at least two weeks before the regular Single $13. 00 closing date for contributed papers (by September Double $16. 00 (one bed) Twins $18.00 10, 1974). See page 179 of the June 1974 (two beds) cfloticeiJ for new format required for abstracts. There will be a concurrent meeting of the Accommodations are also available at: Association for Women in Mathematics. The registration desk will be located in HOLIDAY INN-WEST END the lobby of the Mathematics Building in the 1800 West End Avenue Stevenson Center. Registration hours will be (eight blocks from the Stevenson Center) from 12:00 noon to 5:00 p.m. on Friday, Novem­ Phone: (615) 329-3711 ber 8, and from 9:00a.m. to 12:00 noon on Sat­ urday, November 9. There will be a beer party from 8 :00 p. m. until12 :00 midnight on Friday SHERATON -NASHVILLE HOTEL at the University Club; details will be available 920 Broadway at the registration desk. (one mile from the Stevenson Center) Nashville is on Interstates 24, 40, and 65, Phone: (615) 244-0150 is served by Amtrak, and the Greyhound and Trailways Bus Lines. Many airlines have ser­ Emergency messages may be left for delivery vice to Nashville; limousine service from the at (615) 322-6672. airport to the Vanderbilt area is $3o 25. Cars 0. G. Harrold may be rented at the airport through the Avis, Associate Secretary Hertz, and National agencies. Tallahassee, Florida

195 The Seven Hundred Eighteenth Meeting University of Southern California Los Angeles, California November 23,1974

The seven hundred eighteenth meeting of sentation at the meeting, but they will not be the American Mathematical Society will be held listed in the printed program of the meeting. at the University of Southern California in Los Professor William A. Harris, Jr. of the Angeles, California, on Saturday, November 23, University of Southern California is organizing a 1974. special session of twenty-minute papers on Ana­ By invitation of the Committee to Select lytic Theory of Ordinary Differential Equations. Hour Speakers for Far Western Sectional Meet­ The speakers will include Louis J. Grimm, ings, there will be two invited hour addresses. William A. Harris, Jr., Frederick A. Howes, They will be given by Professor C. Edmund Donald A. Lutz, and Yasutaka Sibuya. Most of Burgess of the University of utah and by Profes­ the papers presented at this session will be by sor Paul R. Chernoff of the University of Cali­ invitation, but some will be selected from ab­ fornia, Berkeley, The titles of their addresses stracts submitted to the Society. Anyone who will appear in the October issue of these cNoticeiJ. wishes to have a paper considered for this spe­ As usual, there will be sessions of contri­ cial session should indicate this conspicuously on buted papers . Abstracts for contributed papers the abstract and submit it two weeks earlier than should be sent to the American Mathematical the above deadline, namely by September 10, in Society, P. 0. Box 6248, Providence, Rhode order to allow time for the additional handling. Island 02940, so as to arrive prior to the dead­ line of September 24, 1974. See page 179 of the Kenneth A. Ross June 1974 c}/oticei) for new format required for Associate Secretary abstracts. Late papers will be accepted for pre- Eugene, Oregon

CHAIRMEN AND TOPICS OF SPECIAL SESSIONS

Abstracts of contributed papers to be considered for possible inclusion in special sessions should be submitted to Providence by the deadlines given below and should be clearly marked "For consideration for special session on (title of special session)." Those papers not selected for special sessions will automatically be considered for regular sessions unless the author gives specific in­ structions to the contrary. Middletown, Connecticut, October 1974 August 13, 1974 Ernest G. Manes, Category Theory as Applied to Analysis and Topology

Houston, Texas, November 1974 September 3, 1974 E. Ward Cheney, Approximation Theory Richard D. Sinkhorn, Matrix Theory

Los Angeles, California, November 1974 September 10, 1974 William A. Harris, Jr., Analytic Theory of Ordinary Differential Equations

Nashville, Tennessee, November 1974 September 10, 1974 J. V. Brawley, Enumerative Combinatorial Theory Robert M. McConnel, Number Theory Richard s. Varga, Approximation Theory

Washington, D. C,, January 1975 October 29, 1974 Joseph Auslander and Nelson G. Markley, Topological Dynamics Constantine M. Dafermos, Hyperbolic Conservation Laws Emil Grosswald, Number Theory George G. Lorentz, Interpolation of Operators and Applications Stanislaw M. Ulam, Mathematics and Games

196 The Seven Hundred Nineteenth Meeting University of Houston Houston, Texas November 23, 197 4

The seven hundred nineteenth meeting of tion Theory, and Professor Richard D. Sinkhorn the American Mathematical Society will be held of the University of Houston will organize a at the University of Houston, Houston, Texas, special session on Matrix Theory. Most of the on Saturday, November 23, 1974, All sessions papers to be presented at these two sessions will will be held in the Continuing Education Center be by invitation. However, anyone contributing of the University of Houston. an abstract for the meeting who feels that his By invitation of the Committee to Select paper would be particularly appropriate for one Hour Speakers for Western Sectional Meetings, of those special sessions should indicate this there will be two one-hour addresses. Professor conspicuously on his abstract and submit it three Seymour V. Parter of the University of Wiscon­ weeks earlier than the above deadline, namely by sin will present one lecture; the second speaker September 3, 1974, in order to allow time for will be announced in a later issue of these the additional handling necessary. c}foti.cei) along with titles of the lectures. On Friday, November 22, 1974, the day be­ There will be sessions for contributed ten­ fore the meeting itself, the University of Houston minute papers both morning and afternoon. Ab­ will sponsor a Symposium on Pure and Applied stracts should be submitted to the American Mathematics in memory of Pasquale Porcelli, Mathematical Society, P. 0, Box 6248, Provi­ formerly Professor of Mathematics at Louisiana dence, Rhode Island 02940, so as to arrive prior State University. to the deadline of September 24, 1974, See page Guest rooms for those attending the meet­ 179 of the June 1974 c}foti.cei) for new format re­ ing will be available in the Continuing Education quired for abstracts. Those having time prefer­ Center at the rate of $14 per night for a single ences for the presentation of their papers should room. Detailed information about travel and indicate them clearly on their abstracts. There accommodations will appear in the October issue will be a session for late papers if one is needed, of these cNOticeiJ; the final program of the meet­ but late papers will not be listed in the printed ing will appear in the November c}foticei]. program of the meeting. There will be two special sessions of selected twenty-minute papers. Professor E. Paul T. Bateman Ward Cheney of the University of Texas will Associate Secretary organize such a special session on Approxima- Urbana, illinois

INVITED SPEAKERS AT AMS MEETINGS

This section of these cNOtiaiJ lists regularly the individuals who have agreed to address the Society at the times and places listed below. For some future meetings, the lists of speakers are incomplete.

Middletown, Connecticut, October 1974 Nashville, Tennessee, November 1974 Philip J. Davis Mark Kac Trevor Evans J. R. Retherford R. B. Gardner Houston, Texas, November 1974 Washington, D.C. , January 1975 Seymour V. Parter Donald W. Anderson Wilfried Schmid Sigurdur Helgason Los Angeles, California, November 1974 Nolan Wallach Fritz John (Gibbs Lecturer) C. Edmund Burgess Paul R. Chernoff Linda Keen H, Jerome Keisler (Colloquium Lecturer)

197 Symposium on Some Mathematical Questions in Biology New York, New York January,l975

The ninth annual symposium on Some The third session will be devoted to twenty­ Mathematical Questions in Biology will be held minute short papers selected and refereed in ad­ for one and one-half days during the last week of vance by the committee. Persons wishing to pre­ January 1975 in New York City, in conjunction sent a paper for consideration by the committee with the annual meeting of the American Asso­ should submit an abstract, on a standard AMS ciation for the Advancement of Science. The abstract form, to the American Mathematical symposium will be cosponsored by the American Society, P. 0. Box 6248, Providence, Rhode Mathematical Society and the Society for Indus­ Island 02940. Abstracts should be mailed so as trial and Applied Mathematics. The support of to arrive prior to the deadline of November 5, the National Science Foundation is anticipated. 1974. See page 179 of the June 1974 cJioruw for Registration and local arrangements will be an­ the new format required for abstracts. All ab­ nounced in Science. stracts should be clearly marked "For presenta­ The program is being arranged by the tion at Symposium on Some Mathematical Ques­ AMS-SIAM Committee on Mathematics in the tions in Biology. " There will be no provision for Life Sciences, whose members are Hans J. late papers. A complete program of the sessions Bremermann, Jack D. Cowan, Murray Gersten­ will be included in the January 1975 issue of haber, Alston S. Householder, Simon Levin these cJfoticei). (chairman), and Richard C. Lewontin. The symposium will be divided into three half-day sessions; the main topics of the sym­ posium will be ecology and evolutionary biology and neurobiology. There will be six forty-minute Simon Levin, Chairman lectures presented in two sessions. The names Organizing Committee of the speakers and the titles of their addresses Ninth Annual AMS-SIAM Symposium on will appear in a subsequent issue of these Some Mathematical Questions in Biology c!loruw. Ithaca, New York

NEWS ITEMS AND ANNOUNCEMENTS

NATIONAL ACADEMY 01!' SCIENCES University; Herman Chernoff, Stanford Universi­ ELECTS NEW MEMBERS ty; Richard Duffin, Carnegie-Mellon University; Samuel Eilenberg, Columbia University; Robert The National Academy of Sciences, at its Floyd, Stanford University; John McCarthy, annual meeting April 23, 1974, elected 96 new w. Stanford University; Alan Perlis, Yale Universi­ members. Among them are the following mem­ ty; Franklin P. Peterson, Massachusetts Institute bers of the Society: Herman H. Goldstine, Insti­ of Technology; George Polya, Stanford Uni­ tute for Advanced Study; Leo A. Goodman, Uni­ versity; Alar Toomre, Massachusetts Institute of versity of Chicago; Frederick Mosteller, Hat­ Technology; and Robert w. Zwanzig, University vard University; George D. Mostow, Yale Uni­ of Maryland. versity; Abraham Robinson (posthumously), Yale University; Elias M. Stein, Princeton University; and Jacob Wolfowitz, University of Illinois. Igor MATHEMATICIAN AMONG NATO !§afarevi~, Moscow State University, was elected SENIOR FELLOWS a foreign associate. The National Science Foundation and the AMERICAN ACADEMY OF ARTS AND SCIENCES Department of State have announced that forty ELECTS FELLOWS U.S. scientists have been awarded North Atlantic Treaty Organization Senior Fellowships in Sci­ The following members of the Society have ence. Among the NATO Senior Fellows is Peter .been elected fellows of the American Academy of J. Bickel of the Department of Statistics at the Arts and Sciences: Theodore W. Anderson, Jr., University of California, Berkeley. Professor Stanford University; George E. P. Box, University Bickel will spend the duration of his fellowship at of Wisconsin; Edgar H. Brown, Jr. , Brandeis the Imperial College, London.

198 INTERNATIONAL CONGRESS OF MATHEMATICIANS VANCOUVER, CANADA AUGUST 21- 29,1974 THIRD ANNOUNCEMENT

Those interested in attending the Congress are reminded that ACCOMMODATIONS basic pertinent information and registration forms arc included in the second announcement!' Those planning to come to the Congress are urged to make reservations well in advance. Since accommodations on the uni­ RECEPTION IN CANADA versity campus are filling rapidly, members are advised to send a hotel deposit of CAN $30 with their registration form even if they Most mathematics departments in Canada will hold open house for indicate university residence as their first choice. This will enable visiting mathematicians during the two weeks preceding and the the organizers to reserve a room in a hotel if one on campus is two weeks following the JCM. Those travelling to and from the not available. Congress are cordially invited to visit them. Members with campus reservations should note that if they do RECEPTION IN VANCOUVER not show up on the indicated date of arrival (or by August 21 if no date of arrival is given) their accommodation may be assigned A special reception desk for JCM members will be set up at the to someone else. Vancouver International Airport during August 18-21,1974. The desk will provide information and general assistance, mostly con­ TOURS cerning transporation into town or to the University of British Columbia. Those who register by mail may request the dates for their tours at any time before or after their arrival in Vancouver. If they Those with hotel reservations are advised to go directly to their wish to do it in advance after seeing the programme below, they hotel. From the airport they may take either the regular should indicate date preferences during August 22-29 for each limousine service ($2 per person) which stops near most hotels tour and whether they prefer the morning or afternoon for half­ downtown or a taxi (approximately $7). day tours. Whenever possible, tour tickets for the requested dates will be included in the packet which preregistrants should pick up Those with reservations in the residences on the University campus at the Registration Centre. If no request is made or if the request are advised to go directly to one of the three residence complexes cannot be filled, they will receive tour vouchers to be exchanged at the University of British Columbia to which they have been for tickets. In any case, they will not be sent any notification by assigned: Gage, Totem, or Vanier. If they have not been notified mail concerning the disposition of their requests for specific tour of an assignment, they should go to the Gage residences. A special dates. bus service ($2 per person) between the airport and these com­ plexes will operate on August 18-21 and 29-30. Taxi fare is Those who register upon arrival will have to wait until August 21 approximately $8. to pick up their tour tickets.

Those arriving by train or by bus are advised to take a taxi from Unexpired tickets may be exchanged freely among members. the station to their hotel or to the University of British Columbia. Tour buses which still have empty seats at the announced time of If they wish to use the regular city bus service ( $0.25 per person­ departure will honour any unexpired tickets of comparable value. exact fare only), bus number 10 goes from downtown to the (2 half-day tours= 1 full day tour). university. ACTIVITIES AT THE UNIVERSITY OF VICTORIA AND SIMON FRASER UNIVERSITY

REGISTRATION CENTRE In addition to holding open house for mathematicians during August 21-29, 1974, the mathematics departments at the Uni­ The registration centre for the JCM will be located in Brock Hall versity of Victoria and at Simon Fraser University are planning at the University of British Columbia and will be open August the following activities: 19-30, 1974. It will include various service and information desks, textbook exhibits and the headquarters of the Congress. University of Victoria, August 19-20:

Conference on Probabilistic Methods in Differential Equations

MESSAGES More information may be obtained from Professor C.R. Miers, Dept. of Mathematics, University of Victoria, Victoria, Canada. During August 19-30, telephone messages for members of the Congress will be taken by a telephone operator and may be picked Simon Fraser University up at the mail desk. The number to call to leave a message is (area code 604) 228-1191. Informal seminars on Mathematical Logic (August 23), some For telegrams the cable address is MATHEMATIX VAN­ aspect of Applied Mathematics (August 26), Graph Theory COUVER. (August 27) Mail should be sent in care of International Congress of Mathematicians More information may be obtained from Professor R. Harrop, University of British Columbia Dept. of Mathematics, Simon Fraser University, Burnaby 2, Vancouver 8, Canada Canada. 7*~Se-e~t7h_e_s_e cA(~, June 1974, pages 73-77, for a copy of the second announcement.

199 LIST OF INVITED SPEAKERS- LISTE DES CONFERENCIERS INVITES

and day of their address in August et date de leuv confe'rence au aout

Expository Addresses - Conferences Generales 7. Algebraic and Differential Topology - Top­ 14. Ordinary Diff. Equations and Dynamic ology Algebrique et Differentielle Systems Equations Differentielles et Sys- V.I. Arnold 26 j. L. Lions 28 temes Dynamiques H. Bauer 22 E.C. Milner 25 V.M. Bukhstaber 28 R. j. Milgram 26 E. Bombleri 23 D.G. Quillen 29 T.A. Chapman 2 7 T. Petrie 27 D. V. Anosol' 24 M.M. Peixoto 24 G. Debreu 27 W. Schmidt 1 28 A. T. Fomenko 28 P. Schweitzer 26 R. Bowen 23 A.M. Vershik 26 P. Dellgne 26 I.M. Singer 23 W. Hsiang 23 W. Thurston 27 W. Krieger 26 B. Weiss 25 G.F.D. Duff 23 D.P. Sullivan 29 M.N. Nehoroshev 26 C. Fefferman 22 j, Tits 26 j.G. Gllmm 24 A.G. Vitushkin 27 8. Differential Geometry and Analysis on Mani- 15. Control Theory and Related Optimization folds Geometrie Differentielle et Analyse H.O. Kreiss 24 Problems Theorie du Control et Problemas sur les Variates b'optimisation 1. Mathematical Logic and the Foundations of }. Cheeger 21 j.N. Mather 23 Mathematics Logique et Fondements des A. Bensoussan 27 H.j. Kushner 24 22 Mathematiques B. Lawson 21 V.K. Patodi II. F. Demyanov 28 L. Markus 27 j. Lelong-Ferrand 22 j. Simons 21 A. Friedman 27 A. F. Subbotin 28 G,A. Margulis 22 K.j. Barwise 28 j.H. Silver 27 H.G. Hermes 28 A. V. ·Kuznetsov 26 H. Friedman 28 9. General Topololl¥, Real and Functional Anal­ Y.N. Moschovakis 28 C.E.M. Yates 27 ysis · Topologte Generale, Vanable Reelle S. Shelah 27 16. Mathematical Physics and Mechanics et Analyse Fonctionelle Physique Mathematique et Mecanique 2. Algebra - Algebre Z, Ciesielski 24 H. Herrlich_ 23 /.M. Combes 27 C.W. Misner 28 24 H. Boss 22 B. JOnsson 23 P. Enflo N.P. Korneichuk 24 R.L. Dobrushin 26 E. Nelson 24 27 G.M. Bergman 21 V, Mozurov 26 V. V. F/1/ppov 26 B. Maurey 0. Lanford i4 B. Simon 28 26 M.E. Rudin 26 A. H. Clifford 23 K.M. McCrimmon 24 A. Hdjnal E.H. Lleb 26 A.A. S/avnol1 26 D. Eisenbud 21 W. Scharlau 22 A. Martin 27 V.E. Zakharo•' 27 P, Gabriel 21 M. Sweedler 24 10. Operator Algebras, Harmonic Analysis and S.M. Gersten 22 V.E. Voskres- Representation of Groups Algebras D'op­ 17. Numerical Mathematics Analyse G. Higman 24 enskil 23 erateurs, Analyse Harmonquie et Represent& Numerique tion des Groupes ~· Number Theory - Theorie des Nombres /.Bramble 21 G. Strang 21 A. Connes 21 E. StOrmer 22 E.W. Cheney 21 A.G. Sveshnikov 22 A. Frohllck 21 B. Mazur 21 M. Duf/o 23 j.L.. Taylor 21 A.A. Samarskii 23 }.H. Wilkinson 22 C. Hooley 22 H.L. Montgom- A. Knapp 23 V.S. Varadorojan 22 H.j. Stetter 23 P. Wolfe 22 A.A. Karatsuba 22 ery 22 j.R. Ringrose 21 D. Zelobenko 22 A.F. Lavrik 23 S.A. Stepanov 21 18. Discrete Mathematics and Theory of Com- 11. Probability and Math. Statistics, Potential, putation Mathematiques biscretes et 4. Algebraic Geometry - Geometrie Algebrique Measure and Integration Probabilltes et Theorie du Cal cui Statistique Math, Potential, Mesure et In­ N, A'Campo 24 M. Inoue 24 tegration }.M. Barzdln 24 M.S. Paterson 23 S.j. Arake/ov 26 W. Schmid 25 A.}. Hoffman ·25 R. Rado 23 R. V. Ambortzumian 22 C.R. Roo 22 W. Barth 23 A.N. Varchenko 24 D./. K/eitman 23 V. Strassen 24 F.L. Spitzer 23 C.H. Clemens 23 R.M. Dudley 21 R. Lindner 24 E. Szemeredl 26 V. Statu/eviclus 24 }. Faraut 23 A.R. Meyer 26 j.L. Vaslljev 22 P.j. Huber 23 j.B. Walsh 21 }. Neveu 21 B. Walsh 22 5. Algebraic Groups and Discrete Sub roups - 19. Applied Statistics, Math in the Social and roupes Algebriques et Sou~Groupes Ois­ 12. Complex Analysis - Variable Complexe Biological Sciences Stat1stiques Applrquees, crets Math Dans les Sciences Sociales et. Biologiques M.M. Djrbashian 27 B. Maskit 27 A. Borel 26 H.M. Jacquet 28 F. Gehring 28 H.L. Royden 26 K.j. Arrow 27 P.A.P. Moran 26 E. Freitag 28 D.A. Kazdan 28 G.M. Henkin 27 K. Strebel 26 N. Buslenko 24 j, W. Tukey 24 H. Garland 26 G. Lusztig 27 A. F. Leontiev 26 E.B. Dynkln 27 E,C. Zeeman 26 R. Howe 27 V.P. P/atonov 27 S. Karlin 26 13. Partial Differential Equations - Equations aux Derivees Partielles 20. History and Education Histoire et 6. Geometry - Geometrie W.K. Allard 21 M.G. Crandall 22 Pedagogie C. Baiocchi 24 j.,j. Duistermaat 22 W.j. Firey 21 C.A. Rogers 22 M.S. Baouendi 21 D. Kinder/ehrer 23 B. V. Gnedenko 28 A. Tikhonov 27 V.L. K/ee 21 S.S. Ryskov 22 M.S. Birman 23 L. Nirenberg 21 Th. Hawkins 28 C. Truesdell 28 P, McMullen 21 H. Brezls 22 M.l. Visik 23 G. Matthews 27

200 SCHEDULE- HORAIRE (Number in parenthesis indicates section) (Nombre en parenthese indique to section)

WEDNESDAY, August 21 MERCREDI, 21 aout 14:30-75:15 H. Bass ( 2) Algebraic K-theory 9:30- 12:00 H. Brez/s {13) Non linear semi groups and variational in­ OPENING CEREMONIES including AWARDS OF FIELDS MEDALS equalities. at the QUEEN ELIZABETH THEATRE (in downtown Vancouver) ·· C. Hooley ( 3) On the distribution of sequences In arithmetic CEREMONIES D'OUVERTURE, comprenont /'attribution des progressions. MEDA/LLES FIELDS o QUEEN ELIZABETH THEATRE (centre V.K. Patodi ( 8} Riemannian structures and triangulations of ville), manifolds. E. St~rmer (70} Some aspects of ergodic theory In operator 14:00- 78:00 algebras. Short Communications - Breves communications B. Walsh (11) The theory of harmonic spaces. P. Wolfe (7 7) Difficult problems of optimization. 14:30- 15:15 s.s. Ryskov ( 6} Geometry of positive definite quadratic forms. W,K. Allard {13) On some recent advances in the multl-dimen- {in Russian) slonal co/cu/us of variations, A. Cannes (7 0) Structure theory for type Ill factors. 15:30- 16:15 R.M. Df!d/ey (11) The Gaussian process and how to approflch it. M.G. Crandall (13) Semigroups of nonlinear transformations and D. Eisenbud ( 2) Finite free resolutions. evolution equations. W.f. Firey ( 6} Some open questions on convex surfaces. S.M. Gersten ( 2) K~ Theory and algebraic cycles. A. Frohlich ( 3) Galois module structure and Artln L-functions. f. Le/ong-Ferrand( 8} Some problems In conformal geometry. B. Lawson ( 8) Geometric aspects of the generalized Plateau H.L. Montgomery( 3} The distribution of zeros of the Riemann zeta problem. function. G. Strang {17) The finite element method - linear and non­ C. R. Roo (71) Characterization ofprobability measures through linear applications. properties of statistics. C.A. Rogers ( 6} Probabilistic and combinatorial methods in the 15:30- 16:15 geometry of euclidean spaces. f. Bramble {7 7) Convergence In the maximum~norm of spline A.G. Sveshnikov (1 7) Some numerical problems in diffraction theory. approximations to elliptic boundary value (In Russian) problems. D. Zelobenko (70) Complex harmonic anr;lys/s on semi simple Lie f. Cheeger ( 8) groups, (in Russian) P. Gabriel ( 2) Indecomposable representations of rings, V.L. Klee ( 6) Combinatorial structure of convex polytopes, 16:30- 17:15 f. Neveu {11) Recurrent Markov processes. R. V, Ambartzumian L. Nirenberg (13) (11) The solution of the Buffon-Sylvester problem. f.R. Rlngrose (7 0) Operator algebras and their abel/on suba/gebros. f.}. Dulstermoat (73) S.A. Stepanov ( 3} Elementary method in the theory of equations A.A. Koratsuba ( 3) Trigonometric sums and their applications. {in over finite fields. (in Russian) Russian) G.A. Margulis ( 8) Discrete groups of motions on manifolds with 16:30- 17:75 nonpositive curvature. {In Russian) M.S. Baouendl (7 3} On o class of Fuchs/on type partial differential W. Scharlau ( 2) Topics in hermitian K-theory. operators. V.S. Voradarojon (70} Harmonic analysis on real semi-simple Lie G.M. Bergman ( 2) Some category-theoretic Ideas In algebra. groups. E.W. Cheney {17) A review of recent progress in approximation f. L. Vosi/jev (7 8} theory. }.H. Wilkinson (7 7) Invariant Subspaces. B. Mazur ( 3) The p-adic analytic number theory of elliptic curves. FRIDAY, August 23- VENDREDI, 23 aout P. McMullen ( 6} Metrical and combinatorial properties of con~ vex polytopes. 70:00- 71:00 }. Simons ( 8) E. Bombieri (on P.D.E.) f.L. Taylor {10} Homotopy invariants for Banach algebras. 71:30- 12:30 f.B. Walsh (7 7) Stochastic Integrals In the plane. I.M. Singer Eigenvalues of the Laplacian and Invariants 19:30-21:30 of manifolds ADDRESSES ON THE WORK OF THE FIELDS MEDALS WINNERS CONFERENCES SUR LES TRA VAUX DES REG/PlANTS DES 14:00- 78:00 MEDA/LLES FIELDS Short Communications - Breves communications 14:30-15:15 21:30- (73) The asymptotic behaviour of the discrete spec­ NO HOST PARTY at the PONDEROSA, University of British Columbia M.S. Birman strum of non-smooth elliptic operators. (In Russian) THURSDAY, August 22 -fEUD/, 22 oout j. Farout (77) Semi-groupes d~opirateurs invariants. A.W. Knopp (10) A Szego kernel for discrete series. 70:00- 71:00 A.F. Lavrik ( 3} Application of the density of zeros of Dirichlet H. Bauer Modern Potential Theory functions to the theory of numbers. {In Russian) f.N, Mother ( 8) Foliations and local homology of groups of 71:30- 12:30 diffeomorphlsms. C. Fefferman Recent progress in classical Fourier analysis R. Rado (18) Families of sets. H.}. Stetter (17) Recent progress in the numerical treatment of 14:00- 18:00 ordinary differential equations. Short Communications - Breves communications V,E, Voskresenskii(2} Some questions of blrational geometry of alge­ braic tori.

201 15:30- 16:15 V. Strassen (7 B) The computational complexity of representa­ C.H. Clemens ( 4) Some applications of the theory of prym tions of polynomials and rational functions. varieties. M. Sweedler ( 2} Some ideas related to the Brauer group and M.Duflo {10) Inversion formula and im·ar/ant differential Amitsur cohomology, operators on solvable Ue groups. }. W. Tukey (19) Mflthematics and the picturing of data. W. Hsiang ( 7) Recent developments in cohomology theory of A.N. Vorchenko ( 4) Algebraic equisingu/arlty and local topological transformation groups. classification ofdifferehtloble mops. {in Russian) B.}6nsson ( 2) Varieties of algebras and their congruence varieties. 20:00- D. Kinder/ehrer (13) Elliptic variational inequalities. RECEPT/ON at SIMON FRASER UN/ VERSITY D.}. Kleltman (JB) Extremal properties in partially ordered sets. A.A. Samarskli (17) Theory of difference schemes and operational methods. (in Russian) SUNDAY, August 25 -- DIMANCHE, 25 aout F.L. Spitzer (17) Time evolution of infinite particle systems. 14:00- 15:00 16:30- 17:15 E. C. Milner Transversal theory. W. Barth ( 4) Submanifolds of low cod/mens/on In projective space. 15:30-76:15 R. Bowen (14) Symbolic dynamics for Axiom A flows. A.}. Hoffman (1 B) Spectral functions of a graph. A.H. Clifford ( 2) Orthodox semigroups which are unions of W.Schmid ( 4) Degeneration of algebraic manifolds. groups. B. Weiss (14) The structure of Bernoul/ian systems. H. Herrlich ( 9) Topological structures. P.j. Huber (11) Some mathematical problems arising In robust MONDAY, August 26- LUND!, 26 aout statistics. M. Paterson (JB) 10:00- 11:00 M.l. Visik (13) Analytical solutions of equations with variation­ j. Tits On the theory of buildings and its applications. al derivatives and their applications. 11:30- 12:30 20:30- 21:30 P. Deligne Weights in the cohomology of algebraic varieties. G.F.D. Duff Mathematical Problems of Tidal Energy. 14:00- JB:OO SATURDAY, August 24- SAM ED!, 24 aout Short Communications - Breves communications

10:00- 11:00 14:30- 75:75 }.G. Glimm Analysis over infinite dimensional spaces and S.j. Arake/ov ( 4) Theory of intersections on the arithmetic sur~ applications to quantum field theory. face, 11:30- 12:30 Initial boundary value problems for partial A. Hajnal ( 9) Results and Independence results in set theor- H.O. Kreiss differential equations and their approximate etical topology, solution. W, Krieger (14} On generators In ergodic theory. E.H. Lieb {16} 14:00- JB:OO V. Mozurov ( 2} On solvable subgroups of finite simple groups. Short Communications - Breves communications. (in Russian) H.L. Royden (12) Intrinsic differential metrlcs for Telchmiiller 14:30- 15:15 space, C. Ba/occhi {13} Free boundary problems In the theory of fluid E. Szemeredi (1B) On sets of Integers containing no k elements in flow through porous media. arithmetic progression. j.M. Barzd/n (JB) Inductive Inference of automata functions and E. C. Zeeman (19) Applications of catastrophe theory to the social programs. and biological sciences. P. Enflo ( 9} Factorization of Banach spaces. M.lnoue ( 4) New surfaces with no m.eromorphlc functions. 15:30-76:15 H.j. Kushner (15) Computational methods and problems In sto~ H. Garland ( 5} On the cohomology of arithmetic groups. chostlc control theory. S. Karlin (19} Some problems In mathematical genetics. K.M. McCrimmon( 2} Quadratic operations in nonassociotive algebras. A.R. Meyer (JB) The inherent computational complexity of de- V. Statu/evic/us {11} Limit theorems for dependent random variables cidable theories of ordered sets. under different conditions of regularity. M.E. Rudin ( 9) Properties preserved by products. P. Schweitzer ( 7} Compact leaves of foliations. 15:30- 16:75 A.A. Slovnov {16} Renorma/izatlon of theories with nontriva/ in- D. V. Anosov (14) Geodesics In Finsler geometry. ternal symmetry group, N. A 'Campo ( 4) Le· groupe de monodromie du deploiement K. Strebel (12) Dn quadratic differentials and extremal quasi- verse/ des s/ngu/arltes /solies de courbes planes. conformal mappings, N. Buslenko (19) The systems analysis and automation of the A.M. Vershlk (14) Combinator/a/approximation in measure theory ·simulation of Iorge scale systems. (in Russian) and ergodic theory. (in Russian) Z. Ciesielski ( 9) Bases and approximation by splines. G. Higman ( 2} On p loco/ conditions, for odd primes p, in 16:30-17:15 finite-simple groups. A. Borel ( 5} Cohomology of arithmetic groups. R. Lindner (1 B) Theorie der lnference-Operatoren (in Russian} R.L. Dobrush/n (16) The problem of symmetry breakdown in con- E. Nelson (16) Markov fields. tinuous systems. V. V. Filippov ( 9} A survey of dimension theory. 16:30- 17:/S A. F. Leontiev {12} On the representation of analytic functions by N.P. Korneichuk ( 9) Some extremal problems of approximation R.j. Milgram ( 7) Dirichlet series. theory. P.A.P. Moran {19) The future of stochastic modelling. 0. Lanford {16} Time-evolution of Infinite classical systems. N.N. Nehoroshev(/4) On the behaviour of systems which are close M.M. Peixoto (14} to being integrable. (in Russian) A. V. Kuznetsov ( 1) On superintultionist/c logics. (in Russian)

202 20:30-21:30 WEDNESDAY, August 28- MERCREDI, 28 aout V.I. Arnold 10:00- 71:00 TUESDAY, August 27 - MARDI, 27 aout W. Schmidt Applications of Thue 's method In various branches of number theory, 10:00- 17:00 11:30- 12:30 G. Debreu Four aspects of the mathematical theory of economic equilibrium. j.L. Lions ContrOie Optimal de Systemes Distribues.

17:30- 12:30 14:00- 18:00 Short Communications . Breves communications H.G. Vitushkin Uniform approximation b.v holomorphic func­ tions. 1~:30- 15:15 14:00- 18:00 V.M. Bukhstaber( 7} V.F. Short Communications - Breves communications DemyanO<' (15) Some minima.\· problems in optimization theory. F. Gehring {12} Metric properties of quos/ conformal mappings. 14:30- 15:15 Th. Hawkins {20} The theory of matrices In the 19th century. H.M. Jacquet K.j. Arrow {19] Economic equilibrium In uncom•entionol models. ( 5} Euler product and automorphic forms. A. Bensoussan {I 5) Contro/e impulsionnel et inequations quasi Y.N. Moschovakis{ I) The recent revival of descriptive set theory. voriationnelles. B. Simon {7 6} Approximation of path integrals and Markov M.M. Djrbashjan {I 2} The theory of factorization and boundary Fields by spin systems. properties of functions meromorphic In a disc. 15:30- 16:15 R. Howe ( 5} On representations of p-adic division algebras. K.j. A. Martin {16) Relations between the phase and the modulus Barwise ( I) Admissable sets and the Interaction of model of a scattering amplitude. theory, recursion theory, and set theory. A. T. Fomenko B. Maurey { 9} p-summing operators, Lp-spaces and geometry ( 7} The multi-dimensional Plateau's problem on of Banach spaces. Riemannian manifolds. T. Petrie ( 7} Lie group actions on smooth manifolds. E. Freitag ( 5) Singularities of modular varieties and fields of S. Shelah ( I) Number of non-isomorphic models and struc­ automorphic functions. ture theorems. C.W, Misner {76] Wave equations on finite and infinite dimen­ sional manifolds In the theory of general 15:30-16:15 relativity - Problems and Conjectures. E.B. Dynkin {19] Stochastic models of economic development, A./. Subbotin {15) Control under conditions of conflict and un­ G. Lusztig ( 5) On the discrete series representations of the certainty, classical groups over finite fields. C. Truesdell (20) Rational thermodynamics based on the axioms L. Markus {I 5) Controllability In topological dynamics. ofF. Reech. B. Maskit (12) Classification of klein/an groups. G. Matthews (20) Science os handmaiden to mathematics. 16:30-17:15 /.H. Silver ( I) The number of equivalence classes 0 II equA'a­ B. V. Gnedenko {20} A survey of the research of Russian mathema­ lence relation may have. ticians in the history of mathematics. (In W. Thurston ( 7} On the construction and classification of folia­ Russian) tions. H.G. Hermes {15) Necessary and sufficient conditions for local V. E. Zakharov {16) The method of inverse scattering for problems controllability and time optimality. in nonlinear Wal'e theory. D.A. Kazdan ( 5) H. Friedman ( I} Some systems of second order arithmetic and 16:30-17:15 their use. T.A. Chapman ( 7} Infinite dimensional manifolds. THURSDAY, August 29- ]EUDI, 29 aout ].M. Combes {16) Application of spectral deformation techniques to N-body Schriidinger operators. 10:00- 11:00 A. Friedman (15} Stochastic differential games with stopping D.P. Sullivan times, and variational inequalities. G. Henkin {12) 11:30- 12:30 V.P. Platonov ( 5) Arithmetical and structural problems in the D. G. Quillen Higher algebraic K-theory linear algebraic groups. A.N. Tikhonov {20} 14:30- CE.M. Yates ( I) A general framework for priority arguments. CLOS/NG CEREMONIES- CEREMONIES DE CLOTURE

203 NOMINATIONS FOR VICE-PRESIDENT OR MEMBER-AT-LARGE

One position of vice-president of the So­ 1. To be considered, petitions must be ad­ ciety and member of the Council ex officio for a dressed to Everett Pitcher, Secretary, Box 6248, term of two years is to be filled in the election Providence, Rhode Island 02940, and must arrive of October 1974. The Council has nominated two by September 3, 1974. candidates for the position, namely 2. The name of the candidate must be given John Milnor as it appears in the Combined Membership List. Cathleen Morawetz. If the name does not appear in the list, as in the Additional nominations by petition in the manner case of a new member or by error, it must be described below are acceptable. as it appears in the mailing lists, for example Five positions of member-at-large of the on the mailing label of these c/{oiieu.). Council for a term of three years are to be filled 3. The petition for a single candidate may in the same election. The Council has nominated consist of several sheets each bearing the state­ seven candidates for these positions, namely ment of the petition, including the name of the Joseph L. Taylor position, and signatures. The name of the candi­ Arthur S. Wightman date must be exactly the same on all sheets. Robert Steinberg 4. On the facing page is a sample form for J. Ernest Wilkins petitions. Copies may be obtained from the Sec­ David Gale retary; however, petitioners may make and use Phillip A. Griffiths photocopies or reasonable facsimiles. Jonathan L. Alperin 5. A signature is valid when it is clearly that of the member whose name and address is Additional nominations by petition in the manner given in the left-hand column. At least fifty valid described below are acceptable. The Council in­ signatures are required for a petition to be con­ tends that there shall be at least ten candidates sidered further. for the five positions and will bring the number 6. The signature may be in the style chosen up to ten if the number of nominations by petition by the signer. However, the printed name and is less than three. address will be checked against the Combined Names of these candidates are published to Membership List and the mailing lists. No at­ assist those who may wish to make nominations tempt will be made to match variants of names by petition. Ordinarily this would have been done with the form of name in. the CML. A name not in the June issue of these cJioticeiJ. This year, the in the CML or on the mailing lists is not that of "April Meeting in Chicago" was in fact in May a member. (Example: The name Everett Pitcher and in DeKalb and the release date of the June is that of a member. The name E. Pitcher ap­ issue was early enough for it to contain the pro­ pears not to be. Note that the current mailing gram of the May meeting. Thus the June issue label of these CN'oticeiJ can be peeled off and af­ could not ·present nominations made at the May fixed to the petition as a convenient way of pre­ meeting of the Council in DeKalb. Because of the senting the printed name correctly.) irregularity of the publication schedule this year, 7. When a petition meeting these various the deadline date for receipt of petitions is later requirements appears, the Secretary will ask than usual as well. the candidate whether he is willing to have his The name of a candidate for the position of name on the ballot. His assent is the only other vice-president or of member-at-large of the condition of placing it there. Petitioners can Council may be placed on the ballot by a petition facilitate the procedure by accompanying the that conforms to several rules and operational petitions with a signed statement from the can­ considerations, as follows: didate giving his consent.

TO THE MEMBERS OF THE SOCIETY

Dear Colleagues: Our Society now faces many problems, both mathematical and professional. The officers and the Council of the Society must wrestle with these problems; each of you-no matter how far away you may he­ can contribute by voting in the annual Society elections. This year the Society will elect (in addition to the other posts) a vice-president and five members-at-large of the Council. The ballots will be mailed out to you early in October, no later than October 10. As soon as you get your ballot, please do vote.

Sincerely yours, Saunders Mac Lane, President

204 NOMINATION PETITION FOR 1974 ELECTION

The undersigned members of the American Mathematical Society propose the name of.___ -r as a candidate for the position of * -=-of";;";';thL"e:-.A-:-m:-:e:-:r:;i-:-can~';'M:;-:a:-:;-trhe-m,___a-:-ti.-:c-:-a•l-;:;:Society for a term beginning January=-::-1-, -:;1-:::9-:::7':::'5-.------Printed or typed name and address or cJVoticei) mailing label Signature

iI I i I

iI i I I

I I I

i I j

; ' ' ! ! I ! f ;

1 : ; i *Specify "vice-president" or "member-at-large of the Council".

205 NONACADEMIC EMPLOYMENT OF PH.D.'S

On Wednesday evening, January 16, 1974, the AMS Committee on Employment and Educational Policy sponsored a panel discussion on "Nonacademic Employment of Ph. D.'s" as a part of the an­ nual mathematics meetings in San Francisco. The committee consists of Richard D. Anderson, Loui­ siana State University (chairman); Michael Artin, Massachusetts Institute of Technology; Wendell H. Fleming, Brown University; Calvin C. Moore, University of California, Berkeley; RichardS. Palais, Brandeis University; and Martha Kathleen Smith, University of Texas, Austin. The panel discussion in San Francisco was arranged by Henry 0. Pollak, Bell Telephone Laboratories, who served as mod­ erator. The members of the panel were Edward E. David, Executive Vice President, Gould, Incor­ porated, Chicago, and former Presidential Science Advisor; John M. McQuown, Vice President and Director of Management Sciences, Wells Fargo Bank, San Francisco; and Carroll V. Newsom, former President of New York University. Texts for the first and third talks follow. The second speech was given extemporaneously and is represented here only in summary form.

Edward E. David, Jr. through the efforts of the few, not the many. Indeed, the numbers of Ph. D. mathemati­ As with many situations in the world today, cians in industry are surprisingly small. Only the coupling of Ph. D. mathematicians with in­ about 14 percent of all current Ph. D. mathema­ dustry presents a number of paradoxes. Princi­ ticians are employed in industry. To be exact, pal among these is that though Ph. D. mathema­ the National Register of Scientific and Technical ticians have made major contributions to the in­ Personnel for 1970 shows around 1450 Ph. D. dustrial scene and have been responsible for the mathematicians employed in industry out of 104 creation of major new enterprises, neither math­ in total. A different survey indicates that between ematicians nor industry have any great affinity 1965 and 1970, the number of Ph. D. mathemati­ for each other. Another aspect is that at a time cians in industry grew from 1100 to 1600-an when Ph. D. mathematicians are being graduated overall growth rate of about 8 percent. If that in great numbers, the positions to which they rate has been sustained, there is now the grand have traditionally aspired in the academic world total of some 2000 industrial mathematicians. are drying up. Looking to industry as a refuge in Since the total number of mathematics Ph. D. 's this situation seems not promising, for the major is increasing around 13 percent per year, the expansions and concerns of industry fall in fields fraction in industry is actually declining. I sus­ which do not have the tradition of advanced math­ pect that the decline is really greater than these ematics. Furthermore, mathematics education numbers indicate because of certain industry has not responded to the challenge of this chang­ trends. ing marketplace for mathematicians. To make These trends are related to the distribution progress in this situation requires, in my view, of mathematicians among industries of various a change in both the motivations of graduating sorts. I suspect that as far as Ph. D. mathema­ mathematicians and in the attitudes of industry ticians are concerned, most of industry is an toward mathematicians. As an industrial scien­ underdeveloped area. If we list IBM, Bell Labs, tist, however, I reject the easy answer-namely, some other computer companies, and the aero­ educate many fewer mathematicians. That would space industry including such companies as preclude what can be a highly fruitful association Boeing, McDonald-Douglas, and Rockwell Inter­ and which can generate substantial benefits for national, you will have accounted for most of society at large. Yet, I won• t pretend that the those 2000 Ph. D. mathematicians. After all, necessary changes in attitudes and expectations how many do GM, Proctor & Gamble, and U. .S. will come easily to any of the participants. Steel have? The mathematically-aware compa­ There is little doubt that Ph. D. mathema­ nies are those which deal in high technology and ticians have operated to great effect in and for they are the ones which in large have faced lay­ industry. John Von Neumann comes to mind as a offs and declining R&D budgets since 1970. That prime example. His recognition of the power of is why I believe that there are even fewer Ph. D. recursion in the stored-program context was the mathematicians in industry than we project. stimulus which gave rise to the world• s most in­ Furthermore, now that I have drawn the novative industry. Claude Shannon made many contrast between high technology industry and contributions, but certainly his best known is the what might be called traditional industry, we can existence proof which showed that information go somewhat further. In the coming decade and could be transmitted error-free over a noisy in contrast to the past years, we will see the communication channel. That was the beginning traditional industries growing at a rapid rate, of information theory and of an explosive growth perhaps faster than the so-called high technology of the communications industry. One thinks too industries. This is certainly true if we include of Norbert Wiener, Richard Bellman, Hendrik the service industries in the traditional category. Bode, and you can add to the list. There have Work and commerce in energy, natural resources, been strong synergies between industry and and social systems including housing, environ­ mathematicians. But they have come to fruition ment, and transportation-all of these will burgeon

206 and will fall predominantly in the traditional sec­ that includes some applied activity in disguise. tor of industry. Space, most military work, and Perhaps this alone, however, is not enough aircraft R&D will not grow as rapidly. to account for the tilt against industry. That Thus, if there are to be many more Ph. D. stems probably from the lack of prestige posi­ mathematicians in industry, the base of industries tions in industry as judged from the academic which look upon them as a resource and which side. Also, there is the assumed necessity for hire them must be broadened. How is this to be industrial researchers to identify with the cor­ done? In part by catering to existing needs, but porate goals and objectives. This duality illus­ also and more importantly, by recognizing and trates clearly that there are attitudes to be cor­ developing embryonic needs which may be seen rected on both sides of the fence. only dimly by industry itself. Some of these are With respect to corporate "loyaltY'', I have perfectly obvious; for example, the consumer heard a research director of a large industrial movement, federal regulation of consumer prod­ laboratory say to its board of directors, "Don•t ucts, and liability laws and judgments are placing require company loyalty from our staff-their a premium on reliability and safety in products primary loyalty belongs to their discipline or and services, That will call forth an uncommon profession. That• s necessary for concentrated; effort on failure analysis and mechanisms, and productive research. When company loyalty re­ upon selective testing and evaluation based upon places a man• s discipline as first priority, he is small samples. Data analysis and the structuring no longer worth his salt." That• s strong stuff and of data bases are keys to such efforts as are more such talk is needed on both sides to adjust proofs of defect-free structures such as sub­ expectations. routines in computer software. So, in seeking ways to build coalitions be­ Another societal trend which is having a tween advanced mathematics and industry, we profound effect on traditional industry is the urge come down to the matter of attitudes and perspec­ toward higher productivity and reduced labor con­ tives on both sides. Perhaps this is best put in tent in goods and services. Automatic control al­ economic terms a la Leontiev. According to his gorithms for real-time operations coupled with frame of reference, outputs are the goods and combinatoric analysis is relevant, but seldom services produced by industry, The objective of applied. You can name many other trends, such industry is output. The inputs which give rise to as modelling and associated mathematical dis­ these outputs are raw materials, energy, sci­ ciplines. What is clear, however, is that tradi­ ence, and technology, and with reference to our tional industry is undergoing a transition to discussion, mathematics. The industry leader is greater sophistication-a transition that high rare who can see through the matrix to the me­ technology industry underwent 25 to 30 years ago, chanism by which mathematics contributes to With cultivation, that transition can be eased and output. Furthermore, the mathematician is input­ even accelerated, oriented. His entire education teaches him that However, let us look at the other side of the first-rate mathematics and publication are ends coin. As I have said, I see an intrinsic, develop­ in themselves. Who is to span that gap and how? ing demand in a new market, traditional industry, It can only be done by mathematicians and indus­ for statisticians, computer scientists, and ap­ try people who are adventurous and far-sighted plied mathematicians, Yet, mathematics educa­ enough to focus on each other• s needs by people tion is not responding, There are some 214 aca­ who can reach beyond their disciplines; in our demic departments offering the Ph, D. By name, case, beyond mathematics, The challenges to there is little applied mathematics. Out of the traditional industry in the coming decade are 214, there are 6 applied mathematics depart­ profound-the problems to be solved challenging, ments, 2 applied mathematics and computer sci­ I have no doubt that success in these industries ence, 27 computer science, and 40 statistics de­ will hinge upon scholarly study and creativity, partments. The rest, 139, are called purely leading to deep insights within the realm of the mathematics. possible. Mathematics can have a vital part of Now, I recognize that this is indictment by that action if mathematicians will focus on the name only. Departmental name says nothing opportunities as a chance for worthwhile mathe­ about the qualifications and attitudes of the grad­ matics and worthwhile mathematicians. uates. Fortunately, a different survey gives some added insight. Ph. D. graduates were asked to classify themselves as to specialty. Out of a total John McQuown of 1281 questioned, 151 classified themselves as statisticians, 163 as computer scientists, 119 as John McQuown discussed in general terms applied mathematicians, and 241 as analysts. The some of the problems involved in the applications rest called themselves algebraists, number theo­ of mathematics in areas of the social sciences: rists, topologists, logicians, and so on. While I finance, economics, and business, in particular. hesitate to say that this division represents fairly He said he thought that operations research and the applied-pure dichotomy, it gives a rough management science had been demonstrably un­ gauge, The total of applied mathematicians is at successful, which he cited as a possible reason most half of the graduates. More importantly for the fact that mathematicians have shown rela­ than mere numbers is the attitude of the grad­ tively little interest in these subjects, in spite of uates-only approximately 5-10% indicated a de­ the facts that a lot of progress is possible and sire for a career in industry, Admittedly, there that important new areas of application for math­ is little basic mathematical research in industry. ematics would be opened as a result. According to NSF, a paltry $14 x 106 was spent He singled out, as a main part of the prob­ by industry on basic work in 1971, and I suspect lem of mathematicians in industry, the wide dis-

207 parity in the states of knowledge of mathemati­ some of the sciences has also been diminished by cians on one hand, and the users (or those affec­ the fact that we are living in an age when, in gen­ ted by its use) on the other. He illustrated this eral, students tend to turn to studies that provide point by citing Thomas Hughes• description of the them with special backgrounds for dealing with twentieth century as composed of fourteenth cen­ the problems of modern society. tury farmers, fifteenth century theologians, six­ A close examination of the models employed teenth century politicians, seventeenth century by Cartter and others in making their melancholy economists, eighteenth century bureaucrats, forecasts reveals that they all seem to assume nineteenth century generals, and twenty-first that 50 percent of the Ph. D. recipients in this century scientists. He suggested that those who country will return to academic employment. are worried about the users of mathematics have Evidence seems to indicate that the percent of to realize that their audience is somewhere in the Ph. D. recipients in mathematics who return to fifteenth, sixteenth, or seventeenth century-if academic work may be as high as 75 percent. In they are lucky. He proposed what he termed iiln­ view of such a situation, it appears that a major tellectual outbreeding" as a response to this prob­ purpose of graduate programs in mathematics lem, in analogy to the concept of genetic out­ has been that of replenishing the academic pro­ breeding described by Darlington in his book on fession. It is easy to understand why graduate the evolution of man and society. programs of only a few decades ago, especially in He observed that the scope of the social mathematics, were based on the premise of sciences is so vast, and includes what he des­ likely future academic employment. But now, cribed as the critical dimension of consciousness, when mathematics and mathematical reasoning that the problems of coming to grips with the ap­ have become so fundamental to almost everything plications of mathematics in these areas are very in which man is involved, should such a premise different from those encountered in applying be accepted? mathematics in the natural sciences. In 1930, G. C. Evans wrote,* "The sys­ When he turned specifically to business, tematization which occurs in a theoretical sci­ economics, and finance, he cited as pressing ence, as we may properly call it in order to dis­ problems those classified under the heading tinguish it from a natural or an applied science, "management of systems", for which no satis­ is a process which is apt to come late in the de­ factory theory exists. He listed three contribu­ velopment of a subject. Evidently some fields of tions which mathematics and mathematicians can knowledge are hardly ready for it, for it is typi­ make here, in the resolution of existing hypothe­ fied by a free spirit of making hypotheses and ses and discrimination between them, in the cre­ definitions rather than a mere recognition of ation of new alternatives and new hypotheses, and facts. But when we find this feeling for hypothe­ in the attitudes, thinking habits, and logical ap­ sis and definition and, in addition, become in­ proach which mathematicians can bring to bear. volved in chains of deductive reasoning, we are Relevant specific areas of mathematics which he driven to a characteristic method of construction listed in this context were information theory, and analysis which we may call the mathematical probability theory, statistics, decision theory, method. It is not a question as to whether mathe­ and a great need for general systems theory. matics is desirable or not in such a subject. We His final conclusion was "consciousness is are in fact forced to adopt the mathematical the clue, and the problem is adaptivity." Man is method as a condition of further progress." not adapting today, his understanding of modern The day is here when, as Evans anticipated, institutions is desperately inadequate, and unless all science, social, biological, and managerial ways can be found to change them for the better, as well as physical, accepts the mathematical many of these institutions may well collapse. method as essential to progress. Scientific and professional literature is now full of mathemati­ cal discourses in which the symbols may have Carroll V. Newsom their interpretation in some narrow field of a biological science, a behavioral science, or Recent projections of supply and demand in management. So the immediate implications of the Ph. D. market made by Allan Cartter and the discourse are to be found in the area of others have indicated that the graduate programs knowledge from which the symbols receive their of universities in the United States are presently interpretation. But the discourse itself is mathe­ turning out too many Ph. D.' f!1 in several fields, matical. including that of mathematics. Such an interpre­ It is presently the trend in graduate educa­ tation has received support, some academic peo­ tion for an advanced student in one of the sciences ple believe, from the discouraging fact that a (including the biological, behavioral, and busi­ considerable number of recent Ph. D.'s in math­ ness sciences) to receive at least enough instruc­ ematics and some other academic disciplines tion in mathematics from his science professors have had difficulty in obtaining positions. Such a to follow the literature in his field and possibly circumstance has obviously been a deterrent for to carry out some research in his narrow spe­ many college students who in the past would have oialty. Progress would be much more rapid, specialized iri mathematics. The enrollments of however, if many of these same individuals were students in advanced mathematics courses and in to pursue studies toward the doctorate in mathe-

*G. C. Evans, Mathematical Introduction to Economics, McGraw-Hill, New York, 1930.

208 matics and if, at the same time, they were given divisions attend intensive classes on the develop­ the opportunity, formally or informally, to have ment and use of mathematical models. Actually, both broad and specialized experiences in mathe­ it is a foolish requirement; for in the case of matical application. most of the persons involved, it is too late to The renaissance in our instructional pro­ rectify their serious mathematical deficiencies. gram that I envisage does not anticipate the mere One might ask why experts in mathematical mod­ introduction of assorted well-formulated mathe­ eling are not assigned to work with the division matical problems. Such problems, drawn from heads. That is done, and will be done more fre­ application, are virtually isolated from their quently, but too often the procedure is not feasi­ original contexts. The kind of instructional pro­ ble since, unfortunately, the mathematician who gram in mathematics that I hope will evolve and does the "leg work" must rely too much on others become popular in the future will be of such a for necessary background information. illtimately, nature that students who are products of it will be of course, the division head, even when he has good mathematicians and, in addition, they will competent mathematical assistance, must be think mathematically and really comprehend the capable of raising significant questions about the fact that they live in a world of mathematics. assumptions on which the model is based. More­ Consequently, it would be anticipated that many over, the ability to detect the flaws in the model Ph. D.'s will be able to detect, isolate, formu­ and to interpret its meaning for the particular late, and then bring their mathematical talents to operation, requires that the model be the man­ bear upon the solution of some of the very signif­ ager's own. icant mathematical problems that are inherent in Kemeny of Dartmouth, in an invited address* many puzzling situations man encounters in his before the MAA in August 1972, notes in connec­ activities. tion with a particular mathematical model with John von Neumann uniquely demonstrated which he had been concerned as a part of his such a capability. He would listen to the state­ necessary planning function as a college presi­ ment of some predicament, meditate for a mo­ dent, "The model is not terribly complicated; it ment and then announce, as his eyes brightened, does not use advanced mathematics; and yet it that he saw a mathematical problem at the heart uses the kind of argument that someone not of the difficulty, Shortly he would isolate the trained in mathematics is not likely to come up problem, frame it in mathematical terminology, with or even be able to follow." Much of the and then possibly suggest a solution or approach time, in fact, the actual mathematical techniques to a solution before the conversation was ended. employed in working with the mathematical mod­ The manner in which his mathematical genius els that have become so common would not be enabled him to understand and interpret Nature regarded as advanced by the standards employed could well become the ideal for many prospective by most mathematicians. Nevertheless, fre­ Ph. D.'s. In truth, a person with a Ph. D. in quently the ingenuity required in the creation and mathematics who possesses just a small fraction use of a model poses a real challenge to any of von Neumann• s capability would never be with­ mathematician. Also, it must be noted that a out a challenging job. mathematical discourse developed by an econo­ So the Cartter and similar studies, it seems mist, a biologist, or an anthropologist may em­ to me, do not demonstrate that institutions of ploy assumptions not explicitly or carefully higher learning are producing too many Ph. D.'s stated. In general, such a discourse may not be in mathematics. Instead they indicate how urgent developed with the same rigor that an author of it is that the base of our graduate programs in a paper in a recognized mathematics journal dis­ mathematics be broadened to accommodate the plays. But the work is a creative endeavor in needs of many students now being ignored. Inter­ mathematics and should be regarded as such. esting and challenging positions are readily avail­ There will be improvement in mathematical con­ able for many more Ph. D.'s in mathematics than cept and usage as more well-educated mathema­ we are now producing. But a substantial number ticians become involved. of those Ph. D.'s should have an educational back­ The mathematical models now being em­ ground that is quite different from that of the typ­ ployed by business and industry are often tre­ ical Ph. D. today. mendously complex. Their design requires great One of the greatest opportunities for the mathematical talent along with penetrating study well-educated mathematician at the present time and a minute understanding of what is being mod­ is in the broad area or' planning and decision­ eled. Recently, for example, the General Electric making, as required of persons involved in some Company announced some of the details of its in­ aspect of management or administration. This is tricate computerized mathematical model of the due, in part, to the nature of modern science and American economy. The model is needed to pro­ the practice of management, whether in govern­ vide the company with its own useful projections ment, education, business, industry, or re­ upon many aspects of the economy so that sound search, which recognizes that the development planning for the future can proceed. The comput­ and use of mathematical models has become a er has made possible the development of complex fundamental part ·of sound planning. At least one models which are a considerable improvement large corporation with which I am familiar has over previous models in the way that they char­ in recent years required that the heads of all its acterize those situations they are supposed to

*American Mathematical Monthly, Vol. 80, No. 8, October 1973, pp. 889-901.

209 characterize. Thus the computer has become an matics. I must not fail to report that the speci­ indispensable tool of the mathematical modeler. fied salary is $24, 247-$31, 519. Perhaps pub­ Now the typical well-managed industrial corpora­ lications of the mathematical organizations per­ tion, educational institution, or government taining to employment opportunities for mathe­ agency is continually developing new computer­ maticians should report regularly on the avail­ ized models or refining old ones to provide un­ ability of such positions, both in federal and in derstanding of and control over an area of major state government, including those of civil service concern. Conclusions provided by the model often grade that are to be filled by examination. become the basis for tremendously important de­ The establishment of the Mathematics War cisions. Policy Committee of World War II provided many A mathematical scientist interested in a federal agencies with personnel and recommen­ position in government, either state or federal, dations upon the handling of complex projects. It will find that it is rare that a government position was common practice for Washington represen­ is described as one in mathematics. It is quite tatives to ask if a problem could be subjected to unusual, in fact, except in academic work, for mathematical analysis and, if there was reason any position, even one demanding great mathe­ to believe that it could, to invite the Committee matical capability, to be designated by the title to suggest mathematicians who might be qualified "mathematician". Rather, if a person• s second­ to do the work. Because of the influence and ary specialty is in economics, for example, he prestige of mathematicians such as Mina Rees in may take a position in some form of applied eco­ the ONR and von Neumann and Ulam in the Man­ nomics, possibly even in agricultural economics. hattan Project, the defense agencies became Then if he displays the special abilities on the aware of the importance of using mathematics in job of which he should be capable, he begins to their analyses and in the perfection of their op­ receive recognition and eventually will move to erations. While the defense agencies, along with the forefront in a program that offers his unusual the National Bureau of Standards and the AEC, talents free play. This assertion is not conjec­ continue to use mathematicians today, awareness ture; it is based on observation. of what mathematicians can contribute to their In state government, in which I held an ad­ operations has considerably diminished in most ministrative position for seven years, I tried, federal agencies since the early forties. insofar as possible, to surround myself with peo­ In spite of such a situation, we should an­ ple who seemed to possess the competence nec­ ticipate at an early date a great growth in the de­ essary to deal with difficult problems in policy mand for mathematically-oriented scientists to and program for a state system in higher educa­ play an essential role in projects of great na­ tion undergoing reorganization. The development tional interest supported by government and by of appropriate models (often employing consider­ private industry. A need is developing for math­ able mathematical intuition and skill) became a ematicians that will exceed the need of three way of life. All of those with whom I had an inti­ decades ago. Pressure is increasing on Wash­ mate association became a part of state govern­ ington, both on the executiVe and legislative ment by way of civil service, a common way of branches, for the creation of large information­ entering government work. In fact, most of them gathering projects that provide a basis for the had taken a prescribed civil service examination making of urgent decisions. The public is de­ which, although specifying little in the way of manding solutions to the problems that plague mathematical background, actually screened out them; and industry, alarmed at the rapid and those who did not possess outstanding mathemat­ highly sophisticated technological development of ical qualifications. One of my earlier associates, competing nations, demands solutions as well. a man with a Ph. D. in mathematics, is now a The so-called energy crisis has suddenly become very successful administrative officer in a large a catalyst in awakening both government and in­ city. Positions that should be of interest to some dustry to the critical necessity of adopting new mathematicians with secondary types of special­ objectives and procedures. Of course, govern­ ization open up frequently on the national level as ment must accept the major responsibility for well as on the state level. The high-level govern­ obtaining solutions to a variety of multi-faceted mental jobs usually are not filled on the basis of problems concerning energy, pollution, food competitive examinations. But interested and production, health, aging, transportation, pov­ properly-educated persons, especially those be­ erty, and unemployment. The manner in which ginning their careers, should not ignore those the federal government will ultimately carry out positions that are filled by means of examination. such a mission is unclear. It is possible that While working on these remarks, an an­ much of the necessary massive research effort nouncement came to my desk of a vacancy to be will become a direct responsibility of NSF or of filled in the Office of Naval Research; the title is NSF• s RANN (Research Applied to National "Physical Science Administrator". The announce­ Needs), now in its third year of operation. Most ment specifies that "candidates must have doc­ likely there will be a new effort, hopefully prop­ torate-level training or the equivalent in hydro­ erly coordinated, that will bring together the re­ dynamics, physical oceanography, coastal geog­ sources of government, educational institutions, raphy/geology, applied mathematics or fluid dy­ and industry. namics and a familiarity with the many naval The question that faces us now is, "Are our problem areas with which these various research graduate programs in mathematics-at least some fields are closely related." A study of the job of them-producing individuals who are ready to description reveals that the position involves the make fundamental contributions to such a nation- use of a substantial amount of advanced mathe- al effort, without extensive additional orientation?"

210 Recently, Harvey Morris, a British fuel expert In England, he said, he would not have such a who has become head of Fuel Economy Consul­ problem for, he observed, English educational tants, Inc, in America, initiated a search for institutions follow a philosophy of adjusting to the young staff people. Much to his amazement, he needs of society. It is my judgment that our was unable to find persons with advanced educa­ graduate programs in mathematics are faced with tional background who could assist him with his a critical need to indulge in some penetrating studies pertaining to "the efficient use of fuel. " introspection.

NEWS ITEMS AND ANNOUNCEMENTS

SALE OF MICROFILM EDITIONS OF BACK VOLUMES OF AMS JOURNALS

The American Mathematical Society announces a new service to current subscribers. The following back volumes are now available on microfilm:

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SAUNDERS MAC LANE NOMINATED body of the National Science Foundation. TO NATIONAL SCIENCE BOARD The National Science Board and the Direc­ tor constitute the National Science Foundation. Saunders Mac Lane, President of the So­ The Board consists of twenty-four Members ap­ ciety, is one of eight persons recently nominated pointed by the President, by and with the advice by President Nixon to serve a six-year term on and consent of the Senate, and of the Director the National Science Board, the policy-making ex officio.

211 DOCTORAL SCIENTISTS AND ENGINEERS IN THE UNITED STATES

The Commission on Human Resources of The total labor force involved in this study the National Academy of Sciences-National Re­ consists of 229, 394 people of both sexes, the search Council has released a report of a survey unemployment rate for whom is 1. 2%. 213, 613 conducted last year by the NAS-NRC with the sup­ are employed full time with a median salary of port of the NSF. The survey covers scientists $20, 890. The total labor force includes 15, 289 and engineers in the United States whose doctoral identified under mathematics, for whom the un­ degrees were granted between 1930 and 1972. The employment rate is also 1. 2%. 14, 750 are em­ total population studied is estimated at 244, 900 ployed full time, the median salary being persons, of whom 15, 952 are identified as mathe­ $19,790. maticians, although only 13, 571 received their Included in the total labor force are doctorates in mathematics. The group of those 211, 345 men, with an unemployment rate of "identified as mathematicians" consists both of 0. 9%. Of these, 199, 905 are employed full time those who are employed as mathematicians and with a median salary of $21,170. There were those who are unemployed but earned their doc­ 14, 419 men identified under mathematics, 170 of torates in mathematics. The sample on which the whom were unemployed (and seeking employ­ survey was based included 4, 409 mathematicians, ment), which yields an unemployment rate of and valid responses were obtained from 3, 166 of 1. 2%; the median salary of those employed full them. time was $19, 930. The following tables giving information on Included in the total labor force are 18,046 mathematicians were extracted from the report. women, with an unemployment rate of 3. 9%. Of Table 1 gives general characteristics of the doc­ these, 13, 706 are employed full time with a torates identified as mathematicians. Table 2 median salary of $17,620. There were 871 gives the employment status of those whore­ women identified under mathematics, 15 of ceived their doctorates in mathematics. Table 3 whom were unemployed (and seeking employ­ indicates the type of employer of doctorates em­ ment); the median salary of those employed full ployed as mathematicians; it includes post-doc­ time was $17,180. toral appointees. Table 4 indicates the primary The complete report, Doctoral Scientists work activity of doctorates employed as mathe­ and Engineers in the United States 1973 Profile, maticians, and it also includes post-doctoral is available without charge in limited quantities appointees. Table 5 gives salary data on doc­ from the Commission on Human Resources of torates employed as mathematicians. It should the National Academy of Sciences, 2101 Consti­ be noted that all academic salaries reported tution Avenue, Washington, D. C. 20418. It is have been multiplied by 11/9 to adjust to a full­ excerpted here with the permission of the pub­ year scale. lisher.

TABLE 1: GENERAL CHARACTERISTICS OF THE DOCTORAL POPULATION Total Population: 15, 952 Mathematicians

SEX CALENDAR YEAR OF PH. D. Male 93.8% 1930-34 1.7% Female 6.3 1935-39 2.1 RACIAL-ETHNIC GROUP 1940-44 2.1 1945-49 White/Caucasian 85.4 3.4 1950-54 Minority Group* 5.9 8.1 1955-59 No Report 8.7 9.1 1960-64 16.1 AGE 1965-69 34.5 Under 30 7.3 1970-72 22.9 30-34 29.4 35-39 19.3 CITIZENSHIP 92.6 40-44 15.0 u.s. Foreign 7.2 45-49 10 •. 7 50-54 7.4 No Report 0.1 55-59 4.6 CATEGORY OF PH. D. 60-64 2.7 U.S. Science 89.2 Over 64 3.4 U.S. Non-Science 6.5 No Report 0.1 Foreign 4.3

*Includes Black, American Indian, and Asian.

212 TABLE 2: YEAR OF DOCTORATE AND 1973 EMPLOYMENT STATUS Population: Ph. D. 1 s in Mathematics

1930-72 Doctorates 1968-71 Doctorates 1972 Doctorates Total Population: 13,571 4,333 1, 254 EMPLOYED FULL TIME Science 89.2% 92.5% 88.1% Non-Science 2.2 2.4 o. 9 EMPLOYED PART TIME Science 1.6 1.5 1.7 Non-Science 0.1 o.o 0.1 POSTDOCTORAL 0.7 0.7 4.9 NOT EMPLOYED Seeking* 1.4 1.4 1.7 Not Seeking 0.6 0.4 0.3 RETffiED 2.0 0.2 0.0 OTHER/NO REPORT 2.4 0.9 2.4

*Percentages are not unemployment rates, since the percentages presented here are calculated on the total population,-which includes those retired, those not seeking employment, and those not re­ porting status, all of whom may not be considered part of the labor force.

TABLE 3: YEAR OF DOCTORATE AND TYPE OF 1973 EMPLOYER Population: Ph. D. 1 s Employed as Mathematicians Including Postdoctoral Appointees

1930-72 Doctorates 1968-71 Doctorates 1972 Doctorates Employed Population: 15,104 5,142 1,332 Educational Institutions* 79.2% 79.9% 74.3% Federal Government 4.9 6.1 8.9 State/Local Government 0.4 0.6 0.0 Hospital/Clinic 0.2 0.1 0.0 Other Nonprofit Organizations 1.9 0.3 4.7 Business 12.2 11.4 11.3 Other/No Report 1.2 1.5 0.8

*Includes elementary and secondary schools as well as higher educational institutions.

TABLE 4: YEAR OF DOCTORATE AND 1973 PRIMARY WORK ACTIVITY Population: Ph. D. 1 s Employed as Mathematicians Including Postdoctoral Appointees

1930-72 Doctorates 1968-71 Doctorates 1972 Doctorates Employed Population: 15, 104 5,142 1,332 Teaching 60.2% 64.6% 60.7% Research 17.9 18.9 17.1 Administration of -Research/Development 5.5 3.0 3.3 -other 3.8 1.3 0.8 Consulting/Prof. Services 1.6 1.5 2. 6 Design/Development 4.8 5.7 10.6 Report/Marketing/ Production/Inspection 0.4 0.2 0.1 Other/No Report 5.6 4.8 4.8

213 TABLE 5: YEAR OF DOCTORATE AND 1973 ANNUAL SALARY RANGE* Population: Ph. D. 's Employed as Mathematicians

1930-72 Doctorates 1968-71 Doctorates 1972 Doctorates Full Time Employed Population: 14,750 4,999 1,251 Lower Decile $13,040 $11,680 $10,800 Lower Quartile 16,250 14,560 12,320 Median 19, 790 17,120 14,850 Upper Quartile 24,550 19,480 17,870 Upper Decile 30,610 22, 850 19,700

*Academic year salaries have been multiplied by 11/9 to adjust to a full-year scale.

Commentary by R. D. Anderson The NA8-NRC report, Doctoral Scien- data from respondents (about 75% of those to tists and Engineers in the United States 1973 whom questionnaires were sent and about 20% of Profile, has been interpreted in various ways. the total doctorate-level population). It seems Science (31 May 1974, pp. 967-969) headlined its plausible that a substantially greater non-re­ story "Scientific Manpower: Demand for Ph.D, 's sponse rate should be expected from those who Up, for Rest Uncertain" whereas Science and have left the field, Doctorates who have left Government Report (15 May 1974) headlined its the country were not included in the study. Em­ story "Study Shows Jobless Rate High for Women ployment in the U.S. has been particularly dif­ Ph.D.'s." Both headlines appear to be some­ ficult for non-U, S, citizens recently, and many what misleading. The comments below are have left the country seeking employment. focused on interpreting the report as it concerns (4) Mathematics and earth sciences were mathematicians. the only two of the nine science fields whose (1) In general, the report was consistent numbers of employed doctorates substantially with other sources of information concerning job exceeded the numbers trained in those fields. patterns for doctoral scientists in mathematics There are about 1000 nonscience doctorates (understood to be the mathematical sciences). (many presumably mathematics-education doc­ (2) The report is not concerned with em­ torates), 1000 engineers and 700 physicists em­ ployment prospects as such, but with 1973 em­ ployed at the doctorate level in mathematics, ployment patterns. Specifically the report does (5) While mathematicians• salaries were not analyze the prospective roles of academic slightly lower than f.Verage (~ $1100 per year and nonacademic employment though it does cite less), some or all of this may be attributable to wide variances among disciplines in 1973 em­ the high incidence of academic employment ployment. The social sciences at 83,3%, mathe­ among mathematicians. matics at 79, 4%, and the biological sciences at (6) The unemployment figures cited in 6 7. O% are the only sciences with more than 56% mathematics were 170 unemployed male doctor­ of their employment in educational institutions. ates and 15 unemployed female doctorates. Engineering at 35,6% and chemistry at 37% are Again, these figures are suspect (as discussed at the bottom of that scale, The employment in (2)). The respective unemployment rates (of problems in those disiplines whose employment unemployed to total labor force) are 1. 2% for is tied to academia would appear to be substan­ males and 1. 9% for females, that for males tially different from those whose employment is being exceeded only by the rates for male phy­ largely nonacademic, Unfortunately, national sicists and chemists of the nine fields studied, science policy seems oriented more toward dis­ whereas the rate for females in mathematics is ciplines and problems of disciplines for which the lowest rate for females in any of the fields. nonacademic employment plays a major role. However, the small numbers involved in the un­ (3) The figure (1, 2%) cited for an overall employed female categories give rates of ques­ unemployment rate in mathematics is prone to tionable reliability. misinterpretation and is somewhat suspect. It (7) Whereas the male-female median is based on a technical definition of unemploy­ salary differential for all fields reported was ment and does not include Ph.D. mathematicians $3,550, in mathematics it was $2,740 with only who are employed outside mathematics, e. g. of one field, physchology, showing a lesser dif­ the employed 1968-1971 doctorates in mathe­ ferential, $2, 460. The nonscience category matics, 2.1% were employed outside science (or (for Ph. D. •s trained in science and employed engineering) and 5. 3% were employed in other outside of science) showed a $4, 520 median dif­ science or engineering fields, Also, the study ferential. was reportedly based on linear extrapolations of

214 ACKNOWLEDGEMENTS

The Society acknowledges with gratitude the support rendered by members during the past year. In addition to the contributing members who pay a minimum of $48 per year in dues, mathematicians also contributed to the Mathematical Reviews Fund, the Friends of Mathematics Fund, the AMS Research Fellowship Fund, and made general contributions; many contributors have requested that their names remain anonymous. These extra funds paid by members provide vital support to the work of the Society.

CONTRIBUTING MEMBERS

Abbott, James H. Farrell, Roger H. Lepage, T. H. Husser, Michael Adams, J. Frank Fass, Arnold L. Levinson, Norman Sally, Paul J., Jr. Akemann, Charles A. Feustel, Charles Dana Lewis, Hugh L. Sampson, Charles H. Alexander, Williams. Fine, Nathan J. Lucas, Harry, Jr. sawyer, Stanley A. Amir-Moez, Ali R. Francis, Eugene A. Macy, Josiah, Jr. Schurrer, Augusta L. Anderson, Richard D. Fuller, Leonard E. Madi-Raj, Hagzl-Rao V. Scott, Walter T. Andrews, George E. Garrison, George N. Mamelak, Joseph s. Seligman, George B. Apostol, Tom M. Gillman, Leonard Mansfield, Maynard J. Sexauer, Norman E. Babcock, William W. Gleason, Andrew M. Mansfield, Ralph Shanks, Merrill E. Barry, John Y. Gordon, Hugh Marchand, Margaret 0. Shay, P. Brian Bauer, Frances B. Gottschalk, Walter H. Mattson, H. F., Jr. Shiffman, Max Baumslag, Gilbert Gould, Henry W. Mayor, John R. Shub, Michael Beals, Richard W. Grace, Edward E. McCulloh, Leon R. Silverman, Edward Beck, William A. Graves, Robert L. Mcintosh, William D. Singmaster, David B. Beckenbach, Edwin F. Green, John William McLeod, Robert M. Sinke, Carl J. Beesley, E. Maurice Greif, Stanley J. McNaughton, Robert Sloan, Thomas D. Bennewitz, William C. Guggenbuhl, Laura McNeill, Robert B. Smith, Duane B. Berg, Kenneth R. Hacker, Sidney G. Meder, Albert E., Jr. Smith, Frank Bing, R. H. Hardy, F. Lane Miller, W. F. Smith, Frank A. Botts, Truman A. Harris, Charles D. Mizel, Victor J. Sorgenfrey, Robert H. Bristol, Edgar H. Hart, William L. Moore, Richard A. Starkey, Joel B. Brunswick, Natascha A. Hashisaki, Joseph Morrey, Charles B., Jr. Sternberg, David Bryant, Jack D. Hendrickson, Morris S. Nashed, M. Zuhair Sudler, Culbreth, Jr. Burke, James E. Herwitz, PaulS. Nelson, Eric J. Tellefsen, Carl R. Burt, Howard H. Hilt, Arthur L. Nishiura, Togo Thompson, James M. Carter, Joan Cooley Hochstadt, Harry Nohel, John A. Thompson, Layton 0. Clark, Harry E. Hodges, John H. Norman, Edward Tsao, Sherman Clifford, Alfred H. Hohn, F. E. Offenbacker, Robert E. Turquette, Atwell R. Cohen, Henry B. Horrigan, Timothy J. Olson, Milton P. Vallarta, M.s. Coleman, A. John Howell, James L. Orloff, Leo N. Walker, James R. Colson, Henry D. Huff, Melvyn E. otermat, Scott C. Wallach, Sylvan Conley, Charles C. Hufford, George A. Owens, Owen G. Wantland, Evelyn K. Cooke, Roger Lee Hukle, George W. Paige, Eugene C., Jr. Weiss, Paul Cowan, John C. III Humphreys, M. Gweneth Paige, Lowell J. Wendroff, Burton Coxeter, H. s. MacDonald Hunt, Burrowes Palais , Richard S. Werner, Frederick G. Cullen, Daniel E. Hunt, Richard A. Palmer, Theodore W. Weyl, F. Joachim Cunkle, Charles H. Hutchinson, George A. Pandey, Jagdish N. White, George N., Jr. Dawson, Reed Ingraham, Mark H. Pate, Robert s. Whitmore, William F. De Marr, Ralph E. Jackson, Stanley B. Peabody, Mary K. Whitney, D. DeFacio, Brian James, R. D. Pearson, Robert W. Whitney, Hassler DeFrancesco, Henry F. Jarnagin, Milton P., Jr. Pell, William H. Whittaker, James V. Dinneen, Gerald P. Kaplan, Wilfred Pellicciaro, E. J. Wilkins, J. Ernest, Jr. Donoghue, William F. , Jr. Kauffman, Robert M. Pflaum, C. W. Williamsen, James s. Doty, Charles F. Kelly, John B. Redheffer, Raymond M. Woeppel, James J. Durst, Lincoln K. Kiernan, Bryce M. Rees, Carl J. Wonenburger, Maria J. Eachus, J. J. Kist, Joseph E. Rees, Mina s. Wood, Mary B. Earle, Clifford J. , Jr. Kossack, C. R. Reid, James D. Wright, Robert K. Ellis, James W. Kunen, Kenneth Rich, Ellis J. Wu, Hung-Hsi Embree, Earl 0. Lanczos, Cornelius Riney, John s. Zink, Robert E. Epstein, Irving J. Laning, J. H. Rose, Donald C. Fair, Wyman G. Lemay, William Rosenblum, Marvin Anonymous (5)

215 MATHEMATICAL REVIEWS FUND

Adams, Robert M. Ehlers, Edward F. Leung, Dominic Rickart, Charles Alexander, John R., Jr. Eiss, Joseph J. LeVeque, William J. Rickey, V. Frederick Alspach, Brian R. Ellls, Kathryn P. Levin, Jacob J. Rieffel, Marc A. Anshel, Michael Engle, Jessie Ann Levine, Howard A. Risch, Robert H. Armbrust, Manfred K. Epstein, George Lewis, D. R. Robertson, James B. Arnold, Leslie K. Ernest, Jon A. Lewis, John A. Robinson, G. B. Aroian, Leo A. Fairchild, Lonnie R. Lewis, William J. Rose, Milton E. Askey, Richard A. Fitting, Melvin c. Lima De Sa, Eduardo Ross, Kenneth Bacon, Harold M. Flanders, Harley Linis, Viktors Rothschild, Linda Bailey, G. H. Fletcher, Peter Lipman, William F. Rovnyak, James Bailey, Harold W. Fong, Paul Lowig, H. F. J. Ryan, Peter M. Balser, Arienne s. Friedman, A. Luehr, Charles P. Ryser, Herbert J. Barbeau, Edward J. , Jr. Fritsche, Richard Lurye, Jerome Sarason, 0. Barton, Eric M. Gill, B. P. MacCluer, Charles Sasaki, Shigeo Bauer, Frances B. G<>ldberg, Richard Maneki, Alfred P. Sasso, Leonard P., Jr. Baugh, James R. G<>od, Richard A. Manjarrez, Victor Schaffer, Juan J. Bausch, A. Frank G<>odman, Victor Marden, Morris Schauer, Richard L. Beard, Jacob T. B., Jr. Gorowara, Krishan K. Marsden, Jerrold E. Schreiner, Erik A. Becker, Horst E. H. Gould, Henry A. Mathsen, Ronald M. Sidney, stuart J. Beckman, FrankS. Graham, R. L. Mayer, Raymond A., Jr. Singmaster, David V. Beeber, Robert J. Greenberg, Marvin J. McAdam, Stephen J. Sinkov, Abraham Bell, Kenneth Griffin, Ernest L. McCoy, Thomas L. Slack, Stephen Bennewitz, William C. Gross, George L. McCready, Robert R. Smith, Martha K. Bergman, George M. Guggenbuhl, Laura McCulloh, Leon R. Smythe, Neville F. Berkovitz, Leonard D. Raimo, F. and D. McFarland, Thomas L. Snygg, Charles E. Berlinghoff, William P. Halton, John H. McLeod, Robert M. Speed, T. P. Berman, Elizabeth A. Hayes, David R. McNab, Robert Spielberg, stephen Berri, Manuel P. Heckscher, Stevens Meisner, M. Srivastav, RamP. Birman, Joan s. Heezen, Alan Menzin, Margaret s. Steinberg, Robert Blackmore, Denis L. Heilbronn, H. Metzler, Richard c. steinig, John Blanchette, Ann L. Henrich, Christopher J. Meyer, Burnett Stewart, Gilbert W. Bloom, David M. Hill, David G. B. Millar, Robert F. Stock, John R. Blowers, James V. Hill, Edward L. Miller, John B. strecok, Anthony J. Boyce, Moffat G. Hillam, Bruce Mills, Ann Sudler, C., Jr. Boyce, William M. Holland, Samuels., Jr. ~oil, William H. Sukou, Shoichiro Brook, Robert B. Hooley, C. Mueller-Roemer, Peter R. Szekeres, G. Buehler, Robert J. Horrigan, Timothy J. Muldoon, Martin E. Tangeman, Richard L. Bunce, John Hughes, Joseph A. Mullen, James A. Tellman, Stephen G. Burckel, Robert B. Hunt, Richard A. Myhill, John Tomasovic, Joseph Burns, John T. Hunter, Louise s. Nagaraj, Muppinaiya Tretkoff, Marvin D. Buschman, R. G. Hyman, Morton A. Nash-Williams, C. Turner, Stephen J. Butts, Thomas R. University of Iowa Neuberger, John W. Vamanamurtby, M.K. U. Byrnes, James s. Ito, Takashi Neumann, B. H. Ber Der Vaart, H. Robert Cain, Bryan Jackson, Stanley B. Neville, Charles W. Votaw, Charles I. Calderon, Alberto P. Jaeger, Chester G. Newberry, R. Steve Waerden, B. L. van der Caldwell, Roderick Jockusch, Carl G., Jr. Niederreiter, H. G. D. H. Wagner Associates Cassidy, Phyllis J. Johnson, John N. Okubo, Tanjiro Wagner, Gretchen, B. Chaida, Milton Kallman, Ralph Olson, Lloyd D. Waid, Margaret and Carter Cheney, Charles Alex Kaplan, Wilfred Orgass, Richard J. Walker, Homer F. Ciarlet, Philippe G. Karlovitz, Les A. Osborn, J. Marshall Wall, Curtiss E. Clark, William E. Kaufman, Robert P. Osserman, Robert Webb, .Glenn F. Clary, s. Kiltinen, John Paciorek, Joseph W. Wehrli, Martin H. Cobham, Alan Kinderlehrer, David s. Palmer, Theodore W. Weinstein, Michael L. Conley, Charles C. Knight, Frank B. Palmquist, Janet Fisher Weir, Maurice D. Cooper, J. L. B. Koehler, Don and Anne Parr, James T. Wells, Raymond 0., Jr. Craig, W. Komatsu, Hikosaburo Peabody, Mary K. Wendroff, Burton Crane, George E. Komm, Horace Pears, A. R. Whaples, G. Crapo, Henry Kostinsky, Alan L. Pedersen, Franklin D. White, George K. Cullum, Jane Krabill, James R. Pellicciaro, E. J. · Wiegold, James Currier, Albert W. Kramper, James P. Perlman, Sanford Willcox, Alfred B. D1Arist,otle, Anthony J. Kruse, Arthur Petro, John W. Williamson, Jack Damkohler, Wilhelm Kurate, Yoshiki Pitt, Loren D. Wolfe, Dorothy Davies, Roy 0. Kuroda, s. T. Pixley, Henry H. Wu, Hung-Hsi Davis, Chandler Lacher, Christopher Poole, George D. Yale, Keith Davis, Robert LadermaD., Jack Pour-El, M. B. Zalcman, Lawrence A. Driver, Rodney D. Laible, Jon M. Quinn, Grace S. Zarantonello, Eduardo H. Duren, Peter L. Langenhop, Carl E. Remmers, John H. Zelinsky, Daniel Ecklund, Earl F., Jr. Lapidus, Arnold Ribe, Martin In Memoriam: E;ffros, Edward G. Lau, Yiu..:was Rice, Henry G. W. J. Trjitzinsky Efroymson, 'O:ustave A. Lehner, J. Richards, Ian J. Anonymous (13) FRIENDS OF MATHEMATICS FUND

Armstrong, Thomas E. Freund, Richard F. Luchins, Edith H. Orey, Steven Comenetz, Daniel Fuchs, Wolfgang H. Maurer, Stephen B. Stone, H. Edward Crampton, Theodore H. M. Golomb, Michael Nelson, Larry D. Torrance, Ellen M. Fischer, Patrick c. Gurney, Margaret Nirenberg, Louis Anonymous (1)

216 AMS RESEARCH FELLOWSHIP FUND

Abbott, James H. Epstein, Bernard MacGregor, Thomas H. Rubel, Lee A. Aissen, Michael ErdHs, Paul Mac Lane, Saunders Ryser, Herbert J. Alexander, J. Ralph, Jr. Fan, Ky Mahowald, Mark Saari, Don Alexander, Stephanie Fitting, Melvin Mattuck, Arthur P. Sacks, Jerome Anderson, Richard D. Fuchs, W.H.J. McCoy, Thomas L. Sally, Judith D. Arnold, Leslie K. Galbrith, A. S. McCrimmon, Kevin Sarason, D. E. Arlin, Michael Gasper, George, Jr. McCulloh, Leon R. Schafer, Alice T. and Jean Giever, John. B. McGhee, J. W., Jr. Schafer, Richard D. Askey, Richard A. Gleason, Andrew M. McLeod, Robert M. Sharp, Henry, Jr. {\youb, Raymond G. Gottschalk, Walter H. McWilliams, R. D. Shimura, Goro Bagby, Richard J. Haimo, Franklin. Merriell, D. Shiffman, Max Bateman, Paul and Felice and Deborah Tepper Meyer, William H. Smiley, Malcolm F. Balser, Arienne S. ltall, Marshall, Jr. Miles, E. P. Smith, Duane B. Beals, Richard W. Hammond, William Miles, Philip E. Smith, James C. Berkovitz, Leonard D. Hancock, John Mines, Ray Smith, Martha K. Bers, Lipman Harrold, 0. G. Mitchell, Joseph Springer-Verlag Birman, Joan Hashisaki, Joseph Mitchell, Josephine Srinivasan, T. P. Blanchette, Anne Hilton, Peter Mitre Corporation Stark, Betty Boas, Ralph P., Jr. Hoffmann, Banesh Moise, Edwin E. Steinberg, Maria Botts, Truman A. Hostinsky, L. Aileen Montgomery, Deane Sternberg, David Booth, George W. Howard, William A. Moore, Richard A. Stone, Alexander P. Boyce, M.G. Hughes, Joseph A. Morawetz, Cathleen S. Sullivan, Richard W. Bresinsky, Henrik Humphreys, M. Gweneth Morrey, Charles B. Summers, John Brooks, James o. Hutchinson, George Morse, Marston Suzuki, Michio Browder, William Isaacs, I. Martin Nash, David Swartz, Charles W. Brown, A. B. Iwasawa, Kenkichi Nathanson, Heston I. Tamura, I. Bruck, R. H. Jackson, stanley B. Nesbitt, c. J. Tews., Melvin c. Bryant, Billy F. Jaeger, Chester G. Oakley, Cletus 0. Thomas, John D. Cain, Bryan E. Jones, Eleanor Green Ostrow, Efrem H. Todd, Olga Harish-Chandra Jones, Phillip s. Palais, Richard s. Tretkoff, Marvin D. Cheng, Henry Jonsson, Bjarni Palmer, W. Theodore Tung-Po, Lin Cherkas, Barry M. Julian, William H. Papakyriakopoulos, Christos Turner, Edward c. Chihara, Theodore s. Karush, William Parter, Seymour V. Wantland, Evelyn K. Christie, Dan E. Kesten, Harry Peabody, Mary K. Wasow, Wolfgang R. Comfort, W. W. Kist, Joseph E. Pearson, Robert W. Whittemore, Alice S. Conley, Charles C. Knight, Frank Pitcher, Everett Whitehead, George Conner, Pierre E., Jr. Knight, William J. Pinsky, Mark A. and Kathleen B. Cowling, Vincent F. Kraines, David Potts, Donald H. Wisner, Robert J. Curtis, Herbert J. Krueger, Warren M. Redheffer, Raymond M. Wood, Mary Dade, Everett c. Laderman, Jack Rees, Mina S. Yaqub, Adil D•Aristotle, Anthony Landin, Joseph Reid, W. T. Yood, Bertram DePree, John D. Lapidus, Arnold Richards, J. Ian Zelinsky, Daniel Doob, J. L. Lashof, Richard K. Rickart, Charles E. Zygmund, A. Drasin, David Liebert, Wolfgang Robison, Gerson B. In Memoriam: Duren, Peter L. Lindsay, John W. Rothe, Erick H. Pasquale Porcelli Eachus, Joseph J. Luchins, Edith H. Rothman, Neal J. Anonymous (15) Ellis, Richard MacGillivray, A. Dean

GENERAL CONTRIBUTIONS

Baugh, James R. Faires, Barbara T. Nashed, M. Zuhair Smith, Duane B. Bechtel, Robert G. Fitting, Melvin Nishiura, Togo Steinberg, Robert Burling, James P. Gleason, Andrew M. Otermat, Scott c. Stone, Ellen R. Chafee, Nathaniel Kaplan, Wilfred Palmer, Theodore W. Whitney, D. Ransom Cohn, Richard M. Kinderlehrer, David S. Ribe, Martin G. Wilkins, J. Ernest, Jr. Doty, Charles F. McKeever, Robert Ross, Kenneth A. Anonymous (1) Durst, Lincoln K. McColloh, Leon R. Schauer, Richard L.

217 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. An announcement will be published in these d/oticeiJ if it contains a call for papers, place, date, subject (when applicable), and speakers; a second announcement will be published only if changes to the original announcement are necessary, or if it appears that additional information should be announced. In general, SMIC announcements of meetings held in the United States and Canada carry only date, title of meeting, place of meeting, speakers (or sometimes general statement on the program), deadline dates for abstracts or contributed papers, and name of person to write for further in­ formation. Meetings held outside the North American area may carry slightly more detailed information. 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 In­ formation Center of the American Mathematical Society. Deadlines for particular ~ues of the c}/Oticei) are the same as the deadlines for abstracts which appear on the inside front cover of each issue.

August 11-16, 1974 Support: Travel and lodging will be paid by the National INFINITE GROUP THEORY CONFERENCE Science Foundation for 25 conference members. University of Calgary, Alberta, Canada Information: Jack Clark, University of Massachusetts, Sponsors: York University and University of Calgary Department of Mathematics, Amherst, Massachusetts Information: Infinite Group Theory Conference, Math­ 01002 ematics Department, York University, 4700 Keele Street, Downsview, Ontario, Canada September 11-November 29, 1974 MATHEMATICS COURSE ON CONTROL THEORY AND August 11-24, 1974 TOPICS IN FUNCTIONAL ANALYSIS 1974 CERN SCHOOL OF COMPUTING The International Centre for Theoretical Physics, Trieste, Italy God~sund, Bergen, Norway Information: I. Barnett, CERN Scientific Conference Program: The course Is intended for mathematicians, Secretariat, 1211 Geneva, Switzerland control and computer engineers, and other suitably pre­ pared scientists. The primary purpose of the course is August 12-16, 1974 to introduce lecturers and researchers from the develop­ REGIONAL RESEARCH CONFERENCE ON THE INTER­ ing countries to these fields of applied mathematics, but ACTION OF SET THEORY AND GENERAL TOPOLOGY the course will also be open to graduate students and University of Wyoming, Laramie, Wyoming post-doctoral scientists from advanced countries. Program: There will be ten principal lectures by Mary Information: The Director, UNESCO, Jalan Thamrin 14, Ellen Rudin of the University of Wisconsin. There will Jakarta, Indonesia; or The Deputy Director, International also be provisions for problem sessions and the presen­ Centre for Theoretical Physics, P.O. Box 586, I-34100 tation of contributed papers. Trieste, Italy Support: (anticipated) National Science Foundation October 4-5, 1974 Information: Joseph Martin, Department of Mathematics, CONFERENCE ON DIFFERENTIAL EQUATIONS University of Wyoming, Laramie, Wyoming 82071 Murray State University, Murray, Kentucky Information: Gary Jones, Department of Mathematics, August 26-30, 1974 Murray State University, Murray, Kentucky 42071 CONFERENCE ON OPERATIONS RESEARCH Eger, Hungary October 21-23, 1974 Program: Topics to be discussed are mathematical pro­ SYMPOSIUM ON ADAPTIVE ECONOMICS gramming, network flows, inventory control, game University of Wisconsin-Madison, Madison, Wisconsin theory, reliablllty theory, Information systems theory, Program: Invited addresses and contributed papers deal­ economic planning, industrial application of operations ing with the formal study, using mathematics and computer research, computing methods. methods, of adaptive behavior, adaptive strategies and Organizers: Bolyai J!fuos Mathematical Society evolutionary processes in economics and industrial set­ Information: Bolyai Janos Mathematical Society, Confer­ tings. Several papers on related work !n engineering and ence on Operations Research, H-1368 Budapest, Pf. 240, biological sciences will also be presented. Hungary Sponsor: Mathematics Research Center, University of Wisconsin-Madison September 3 -7, 1974 Information: R. H. Day, Mathematics Research Center, NSF REGIONAL CONFERENCE ON ERGODIC THEORY University of Wisconsin-Madison, 610 Walnut Street, University of Massachusetts, Amherst, Massachusetts Madison, Wisconsin 53706 ~: Donald Ornstein of stanford University, the Pr1iiCiPiil speaker, wlll present two one-hour talks each January 20-22, 1975 day on: A general survey of results and open problems; SECOND ACM SIGACT-BIGPLAN SYMPOSIUM ON The Kolmogorov-Binai theory of entropy in ergodic theory; PRINCIPLES OF PROGRAMMING LANGUAGES The isomorphism theorem for Bernoulli shifts; Extensions Palo Alto, California and applications of the isomorphism theorem; and Deadline for abstracts: August 15, 1974. Papers are Kolmogorov automorphlsms. being solicited on significant developments in the prln-

218 ciples of programming languages, and theoretical studies August 12-14, 1975 with application to, or primarily motivated by, program­ INTERNATIONAL SYMPOSIUM ON OPERA TOR THEORY ming languages. OF NETWORKS AND SYSTEMS Information: John c. Reynolds, Systems and Information Sir George Williams University, Montreal, Canada Science, 313 Link Hall, Syracuse University, Syracuse, Program: Sessions on operators in networks; special New York 13210 sessions on resolution space and cognate topics in sys­ terns. April 21-May 16, 1975 Sponsors: Electrical Engineering Department, Univer­ 1975 SUMMER SCHOOL IN GRAPH THEORY sity of Maryland; Department of Applied Mathematics and Manila, The Philippines Statistics, SUNY, Stony Brook; Systems Science Depart­ ~: Claude Berge has been invited to be the main ment, University of California, Los Angeles; Electrical spe~aker.~ The summer school is to be divided into three Engineering Department, Sir George Williams University parts: (1) Introductory lectures for beginners; (2) lec­ Information: N. Levan, Program Chairman, Systems tures on advanced topics; (3) research seminar. Science Department, University of California, Los Sponsors: Southeast Asian Mathematical Society and the Angeles, Los Angeles, California 90024 Mathematical Society of the Philippines Information: Southeast As ian Mathematical Society, September 1-5, 1975 Department of Mathematics, Nanyang University, Sing­ INTERNATIONAL FEDERATION FOR INFORMATION apore 22, Republic of Singapore PROCESSING WORLD CONFERENCE ON COMPUTERS IN EDUCATION May 21-23, 1975 Marseille, France 1975 INTERNATIONAL SYMPOSIUM ON MULTIPLE­ Program: A significant part of the conference program VALUED LOGIC will be devoted to the introduction of the methodology of Indiana University, Bloomington, Indiana Informatics into the teaching of different disciplines. Deadline for Authors: December 1, 1974. Authors are Another important aspect of the conference will be a invited to submit papers on the theory and applications of consideration of the use of the methods of Informatics multiple -valued logics, or nonclassical logics connected and the applications of computers to aid in the solution with multiple-valued logics, including algebraic aspects. of the problems of education in developing countries. Sponsors: Indiana University, Office of Naval Research, Contributed Papers: Papers describing practical ex­ and IEEE Computer Society periences and evaluation of results are particularly wel­ Information: G. Epstein, Conference Chairman, Com­ come, but papers dealing with new techniques, results or puter Science, Indiana University, Bloomington, Indiana methods of presenting new theoretical advances also will 47401 be welcome. Papers must be written in English or French, but the oral presentation can be in English, French, June 16-21, 1975 Russian or Spanish. Papers are to be submitted before FOURTH INTERNATIONAL SYMPOSIUM ON MULTI­ October 15, 1974. VARIATE ANALYSIS Information: AFCET -Service des Congres, Immeuble Wright State University, Dayton, Ohio centre Dauphine, Avenue de Pologne, 75775 Paris Cedex Program: The symposium will be dedicated to the mem­ 16, France; or Lee Peng Yee, Department of Mathematics, ory of the late Professors H. Hotelling and P. C. Maha­ Nanyang University, Jurong Road, Singapore 22, Republic lanobis. Those who have either accepted invitations or of Singapore indicated tentative acceptance to present papers are A. V. Balakrishnan, R. C. Bose, T. Hida, J. Kiefer, D. Mid­ September 1, 1975 -August 31, 1976 dleton, C. R. Rao, M. Rosenblatt, Yu. A. Rozanov, and INTERDISCIPLINARY RESEARCH PROJECT ON MATH­ M. Siotani. EMATICAL PROBLEMS OF QUANTUM DYNAMICS Sponsor: Aerospace Research Laboratories Centre for Interdisciplinary Research, Universitat Information: P. R. Krishnaiah, Symposium Chairman, Bielefeld, Bielefeld, Germany ARL/LB, Building 450, Aerospace Research Laborato­ Program: There are openings for 12 to 15 mathematicians ries, Wright-Patterson Air Force Base, Ohio 45433 and physicists from outside, plus members of the local mathematics and physics faculties. There are plans to August 11-13, 1975 start the project with an International Symposium and to CONFERENCE ON THE THEORY OF APPROXIMATION have another such meeting in the course of the year. University of Calgary, Alberta, Canada Beyond this, there will be the possibility of extending Program: Invited addresses, contributed papers short-term invitations to other researchers of interest Information: B. N. Sahney, Department of Mathematics, to the group. University of Calgary, Calgary 44, Alberta, Canada or Information: Horst Behncke, Fakultat fiir Mathematik, A. G. Law, Department of Mathematics, University of Univers it!It Bielefeld, D 48 Bielefeld, Kurt-8chumacher­ Regina, Regina, Saskatchewan Str. 6, Germany

NEWS ITEMS AND ANNOUNCEMENTS

PROJECT TO TRAIN MODERN MATHEMATICS operation of public school systems, educational IMPLEMENTATION SPECIALISTS statistics and measurements, and workshops on specific modern mathematics programs. Washington State University, with the sup­ The program will be directed by James H. port of the National Science Foundation, is spon­ Jordan. The staff will include Calvin T. Long, soring a one-year program to train twelve mathe­ Jack R. Robertson and Duane DeTemple of the matical scholars to become modern mathematics Department of Mathematics, Washington State program implementation specialists. The pur­ University. They will be assisted by Dennis pose is to familiarize the participants with re­ Warner, Toshio Akamine, David Shauver and cently developed mathematics curriculum pro­ John Miller of the Department of Education. grams, school systems, teachers and adminis­ Participants may be awarded the master's trators, the teaching of mathematics to children degree in education or the Doctor of Arts degree in grades K - 12, and pre-college educational in mathematics, subject to the requirements of theory. The program consists of two summer the Graduate School. sessions of four weeks each and an academic year Interested persons should contact James H. field study program. The summer sessions will Jordan, Department of Pure and Applied Mathe­ include graduate work in learning theory, ad­ matics, Washington State University, Pullman, vanced arithmetic methods, the organization and Washington 99163.

219 QUERIES Edited by Wendell H. Fleming The QUERIES column is published in each issue of these cJ/oliuiJ . 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 various 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 typewritten if at all possible and sent to Professor Wendell H. Fleming, American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940.

QUERIES let A (o-+it) 41, Eugene M. Norris (Department of Mathematics and m(cr) ~max [lane n lln;;o}. Computer Science, University of South Carolina, Columbia, South Carolina 29208). I should like to locate Then the question is: Does the condition an article, recently published, concerning an application of finite-state automata to the theory of of the theory (2) continued fractions. I should also like references to any ''unusual" applications of automata theory, i.e. appli­ cations to areas other than recursive function theory and imply logic, engineering, computer science or theoretical biology, (3) lim log M(o-)/log m(O') ~ 1? 0' ... +00 42, Roger L. Cooke (Department of Mathematics, Background, College of Technology, University ·of Vermont, Burlington, It is known that (1) together with Vermont 05401). Let f(x) be a C1 function of period 2!T lim sup log n/log An < CD having Fourier series L::cn exp(inx), Let If Ill~ II n~ CD

L: I en I. It is obvious that there is a constant K such that imply (3) [1], and also that (1) together with

Ill fIll ::§ Kd If I lCD + II f• IlCD). In fact the infinity norms lim sup log n/An < CD n~ CD can be replaced by the L 1 and L 2 norms respectively, do not imply (3), and the inequality is an immediate consequence of Counterexample, an~ exp ((1/p)logn log logn), An~ Parseval 1 s equation and the Schwarz inequality. Can one logn [1]. replace the sum of the two infinity norms by the square Remark. Faulty translations as well as some ambiguity in the text of [2]led several authors to assume that (1) and root of their product, on the subspace of functions for (2) imply (3), References, which c0 ~ 0? [1] G. Valiron, Croissance et zeros des fonctions entieres definies par series d• exponentielles, TBhoku 43. David Shelupsky (Department of Physics, The City Math J. 38 (1933), 358-374, College (CUNY), New York, New York 10031). Can any­ one tell me where I can find descriptions, with proofs, of [2] Yu Chia-yung, Surles droites de Borel de certaines the dual of the dual of the Banach space C([O, 1]), more fonctions entieres, Ann. Sci, Ecole Norm, Sup, (3) 68 generally, of the higher dual spaces of any of the classi­ (1951), 65-104, MR 12, 815. cal nonreflexive Banach spaces? 45, S, Zaidman (Department of Mathematics, Universite 44, Alfonso G, Azpeitia (Department of Mathematics, de Montreal, Montreal, Canada), Several problems about College I, Harbor Campus, University of Massachusetts, sequences of numbers appear on page 107 of the book by Boston, Massachusetts 02125), Given the Dirichlet W, Sierpiii.ski, Algebre des Ensembles, Monografie series Matematyczne, Warsawa-Wroclow, 1958 (Chapter III, §21). 00 A s Have problems 1, 2, and 3 of this list appeared elsewhere? f(s) ~n~O ane n (s~o-+it, O~A 0

220 this is 0, the result is trivial, so assume it positive, Say B = B~ 1 B•. Likewise, when a~ b we can perform the a !i b, Note that for any integers y, 0, if we set same reduction taking B0 =(~')I ~ !!i).

Bo =G- Bin and A• = BoAB(). 40, (Zaidman, June 1974) The answer is no, P, Ungar pointed out the following easy counterexample: Let D be then A• satisfies the same hypotheses as A, and has in unit disk, Let E > 0 and let u = ln(r2 + E) - ln(1 +E). place of c + di the term c' + d•i = (c+ay) + (d+aB)i. Then max lui= llnEI while JSDI6ul = 41r/(1+E) < 41r and by Now from the condition 1 = det(A) (and the assumptions letting E .. 0 one sees that there is no inequality of the a i!!i b, c2 + d2 > 0) it is easy to show that lei or ldl is > required kind, a/2, and hence that for appropriate choices of y and 0 Counterexamples were also provided by A, Huber we will get c'2 + d'2 < c2 + d2• Then by inductive and J. Rauch, hypothesis we can write A• = B1B'*. Hence A = BB*, with

NEWS ITEMS AND ANNOUNCEMENTS

DOCTORATE MANPOWER FORECASTS support for graduate students by field and insti­ AND POLICY tution; job placements and salaries of graduates, as well as analysis of unemployment and under­ The National Board on Graduate Education employment; trends in research and development (NBGE) in a report entitled Doctorate Manpower expenditures, and the distribution of these ex­ Forecasts and Policy(Report Number Two, Nov­ penditures by type of institution and source of ember 1973) rejected policies that would "solve" funds, Continuously revised projections of the the problem of the "Ph. D. glut" by reducing fi­ future market for the various types of highly nancial support for graduate education and gradu­ trained manpower are also needed, In addition ate students. The NBGE made the following six to collecting and providing such information, the recommendations to improve the environment for federal government should encourage and support decision and policy-making with respect to gradu­ research and analytical efforts using these data, ate education: including attempts to develop systematic models 1. The federal government must recognize that incorporate the labor market analysis for that rapid changes in policy create serious prob­ highly trained manpower within existing economic lems for students, universities, states, and models of the national economy. other agencies that must ameliorate insofar as 5, In the performance of their planning possible the results of unpredictable fluctuations duties, state governments should examine care­ in federal support. Major changes in federal fully the need for additional degree programs. policy should be based upon careful evaluation of Existing programs should be reviewed in terms their impact and should be implemented over sev­ of need, quality, and output, On the other hand, eral years through a phased process that is coor­ if graduate education is to remain viable and dinated with the affected states and universities. diverse with respect to the types of students en­ 2, We urge the federal government to rolled, if it is to be available in major urban accept responsibility for ensuring that the most areas, and is to serve varied markets for highly academically talented young people in each col­ educated manpower, opportunities for new pro­ lege graduating class have the opportunity to at­ grams and new combinations of talent should re­ tend high-quality graduate institutions. Competi­ main open, New doctoral programs that simply tive federal fellowship programs, such as the duplicate existing programs insofar as access NSF predoctoral science fellowship program, and objectives are concerned should not be ap­ should be maintained and broadened through the proved in the next several years, appropriate federal agencies to cover all aca­ 6, University administrators and faculty demic disciplines-humanities, social sciences, should explore every avenue possible to ensure a life sciences, physical sciences, and engineering, continuing flow of young faculty members into 3. The federal government and the uni­ academic departments. Several alternative versities should accept joint responsibility for means toward this end have been proposed, in­ ensuring access to, and successful completion of, cluding earlier retirements, changes in tenure graduate degree programs for minority group concepts, and reducing the proportion of under­ members, Similar responsibility should be exer­ graduate teaching that is done by graduate stu­ cised to ensure opportunities for women to enroll dents. There is at present insufficient evidence in graduate degree programs in fields where they and too much institutional variation to permit have historically been underrepresented. specific recommendations, except to call atten­ 4, Only the federal government has the tion to the gravity of the problem in a new era capability and authority to collect consistent and of slow (or no) growth, comprehensive data on trends pertinent to the The complete report is available from the labor market for highly educated manpower, and National Board on Graduate Education, 2101 we urge it to exercise this responsibility. At a Constitution Avenue, N. W,, Washington, D. C, minimum, these data should include enrollment 20418. trends by field and institution; trends in financial

221 LETTERS TO THE EDITOR

Editor, the cJ/otiui) the full force and variety of contemporary math­ put to work . . The data given in the January and April ematics were a few semi-practical suggestions on the decline of support for mathematics Here are cJ/oti.ctiJ end this iso­ by NSF and other government agenc.ies is shock­ of beginning steps we could take to collaboration be­ ing. That we should have declined so dramatical­ lation. First, some scientific ly-in comparison with other sciences-at a time tween AMS, MAA, SIAM and the other scientific when some of the largest corporations in the societies. For example, there is a need for a country are profiting handsomely from our sci­ project to improve the teaching of mathematics ence, defies conventional wisdom. When the de­ to physics graduate students. Ironically, theo­ more cline started, I was relatively pessimistic be­ retical engineers now learn (and use) cause I realized how thoroughly (and foolishly) mathematics than theoretical physicists. We are the profession and its leaders had abandoned ties incompetent to tell grade schools how to teach to the applications, but I did not think that things mathematics; I think that organizing (in collabo­ could get this bad-the self-evident importance of ration) the mathematical education of scientists mathematics in the world should have saved us in is something we should know about. spite of ourselves. Second, in addition to symposia on mathe­ Now that such illusions have been shattered matical biology, chemistry, computer science, which mainly serve and attract specialists is the time to face the fact that we are in a des­ etc., not every important conference have a ' cending trajectory back to the traditional "ground could interdis­ state" for mathematics-the situation of the component as large as possible that is of 1972 1930• s. Strangely, this seems to hold for all ciplinary? For example, in the summer there was an AMS conference on Lie group har­ math~matical areas, pure or applied. Perhaps certam small applied fields are financially monic analysis, which is also an important topic there healthy because they were neglected during the in certain parts of physics. Last summer which expansion, but this is obviously not stable. was a conference on differential geometry de­ What are we to do about this as a profes­ is now used in Systems Theory. I could ndt the program sion? In the short term, probably nothing much tect even a trace of such interests in a marvel­ can be done. One might hope that our leaders of either; it certainly would have been together take steps to publicize our plight. For example, ous opportunity to bring mathematics the a c?rporation president recently stated, as one with interesting applications. Certainly and Lie g.roup ratiOnale for technological exchange, that the Founders of differential geometry sort of c_ountry needs access to the mathematical exper­ theory were intensely interested in this is so much tise of the USSR. Where was the obvious state­ application. Do we think that our work such inter­ ment from an official of the AMS? We probably better than theirs that we can forego should take a serious look at the faculty union­ ests? style of ization movement. Perhaps some of our more Third, a change is needed in the by militant types can think of a way we could strike mathematics papers to make them readable on our economic grievances, instead of wasting professionals in other fields. For example, I am everyone• s time with absurd resolutions often asked mathematical questions by physicists For the long term, the only real s~lution is and engineers; I will typically give them a pre­ an end to our isolated position in the academic cise reference in the mathematical literature and scientific worlds. In my own excursions out­ and they will report that they found it imposstble side of mathematics I have found very few people, to read, even though they clearly knew the back­ distinct from even among engineers and scientists who had the ground quite well ! This proposal is slightest idea what we are doing. Thfs sense of the old nostrum of better expository papers-what we consider good exposition is not necessarily my~tery served quite well in the post-Sputnik by such a person. I hate to suggest an­ periOd, but we are now paying the price. Another readable but that might be a way to do consequence of our isolation is that much of the other Journal, work in developing mathematical ideas in other something. I do not believe or claim that any of these disciplines is done by those with no contact with suggestions would have any measureable imme­ professional mathematicians. I have studied the diate effect on our financial dilemma. Perhaps situation in detail in physics and engineering; they can be considered for their own sake with mathematics is used, but in a far from optimal in victorian way. Much time and effort is wasted in redis­ the hope that Virtue is rewarded, as covering ideas, doing things that we would call novels. "trivial", etc. Substantial progress could be Robert Hermann made on many outstanding applied problems if

222 Editor, the cJioticei) Editor, the cJioticei) The recently published exchange of views on The following letter to Professor Igor R. the issue: "The Ph. D. Thesis" omits insistence Safarevic was circulated at the 80th annual meet­ on the most sharply distinctive aspect of mathe­ ing of the Society in San Francisco and was sent matics, the single respect in which it is, per­ with the signatures of 310 members: haps, like no other discipline on earth: its prac­ "We, the undersigned mathematicians at­ titioners must accept differences as to whether tending the annual meeting of the American Math­ a particular mathematical result was worth prov­ ematical Society in San Francisco, send you our ing, -but they cannot tolerate differences as to expression of admiration and respect. whether it was indeed proved. "Not only your great contributions to math­ It seems to me that many conclusions can ematics, but your steadfast and courageous de­ be drawn from this, but the most notable might fense of the rights of all to seek and express the well be that for a brief time at least while he is truth as they see it, makes us proud to be your busy thinking through the rigor of his work, your colleagues." author has to be a modest, even an humble fellow eagerly dedicated to truth. All of the graduate E. G. Straus students I have known, and I myself, have found this a refreshing-even an exhilirating experience. Leo Zippin

NEWS ITEMS AND ANNOUNCEMENTS

EMPLOYMENT INFORMATION FOR either applicant or employer. The first bulletin MATHEMATICIANS, SUMMER SUPPLEMENT will be mailed on August 1; subsequent issues will be mailed on succeeding Thursdays thereafter The Joint Committee on Employment Oppor­ until September 26 each week that any listings tunities has suggested the implementation of a are received. No bulletin will be issued in any supplementary employment service, on an ex­ week in which no listings are received. Listings perimental basis, for the summer of 1974. The previously published (in either ElM or another committee recommends that all positions which supplementary bulletin) will not be repeated. The become available between the dates of July 30 purpose of the service is to disseminate all list­ and September 24, 1974, be submitted to the ElM ings received in Providence but no guarantee can office in Providence. The data will be collected be made that any will be submitted. All informa­ for dissemination to interested persons. These tion that is received will be distributed. dates were selected since there are no publica­ The committee hopes that this supplemen­ tions or listings available for use by applicants tary bulletin will provide a useful service during or employers at this time. Also, because of the the summer months. Any comments or sugges­ International Congress, there will be no summer tions will be most appreciated. Please write to mathematics meetings of MAA or AMS. the ElM office at the address given above and A weekly supplementary bulletin will be your remarks will be conveyed to the committee. mailed to participants whenever information is available for distribution. Those who wish to participate in this service must forward nine PRE-PAYMENT FOR AMS PUBLICATIONS stamped, self-addressed envelopes (standard business envelopes No. 10 [4-1/8 by 9-1/2 At its meeting in June, the Board of Trus­ inches] preferred) to the ElM Summer Supple­ tees of the Society voted to change the previous ment, Post Office Box 6248, Providence, Rhode policy regarding pre-payment for AMS publica­ Island 02940. Persons residing outside of North tions. Effective August 1, 1974, all orders for America must send the self-addressed envelopes journal subscriptions and books must be accom­ and a check or money order for $1.89 which will panied by payment. In the past, individual mem­ cover the cost of airmail postage; Canadian or bers and institutional members were not required Mexican residents should include a check or to pre-pay. This practice has been discontinued money order for $1. 17 along with the pread­ because of the expenses involved in billing and dressed envelopes. collecting. However, the Society continues to Employers will be encouraged to submit in­ honor BankAmericard and Master Charge, which formation on any late-developing positions in es­ may be used for any book or subscription orders. sentially the same format as is now being used for the regular issues of Employment Information for Mathematicians. It is not necessary, how­ LONDON MATHEMATICAL SOCIETY AWARDS ever, to use a post card; a typewritten statement on ordinary stationery will be sufficient. Com­ The Council of the London Mathematical plete instructions for department chairmen will Society has awarded the 1974 De Morgan Medal be included with the mailing for the July issue of (incorporating the Larmor Prize) to Graham ElM; additional copies will be available on re­ Higman of Oxford University and the 1974 Senior quest from the Register office. Berwick Prize to Paul M. Cohn of Bedford Col­ There will be no charge for this service to lege, London University.

223 NEW AMS PUBLICATIONS

TRANSLA liONS OF In dieser wertvollen, M. G. Krein zum 60. MONOGRAPHS Geburtstag gewidmeten Moncgraphie legen MATHEMATICAL die heiden bekannten Verfasser eine umfas­ CONVOLUTION EQUATIONS AND PROJECTION sende Theorie der Projektions-methoden METHODS FOR THEIR SOLUTION by zur Losung verschiedener Klassen von I. C. Gohberg and I. A. Fel• dman Faltungs-Integralgleichungen und den ent­ sprechenden unendlichen Gleichungssys­ Volume 41 temen sowie von singulliren Integralglei­ 262 pages; list price $27. 00; member price chungen vom Cauchyschen Typ dar. Obwohl $20.25;ISBN0-8218-1591-1 fiir viele der betrachteten Gleichungstypen To order, please specify MMON0/41 exakte LOsungsmethoden vorhanden sind, macht deren praktische Durchfiihrung haufig origins in operational This volume has its grosse Schwierigkeiten, so dass Nli.herungs­ the presentation is based on functional calculus; methoden wie die Projektionsmethode fiir to a definite analysis. This route leads naturally Gleichungen von grosser Bedeutung the diese class of convolution equations that contains sind. In der Monographie werden in der original Wiener-Hopf integral equations. Dis­ Literatur vorhandene diesbezUgliche Re­ crete ari.alogues, pair equations, singular inte­ sultate sowie neue Ergebnisse der Autoren gral equations on the circle, etc. , are discussed, in einheitlicher Weise hergeleitet. The present edition differs significantly from the Russian edition, published in 1971, in that the last two sections of Chapter III are re­ DIRECT AND INVERSE IMBEDDING THEOREM placed by three new sections. The new presenta­ by L. D. Kudrjavcev tion is more complete and is distinguished by greater generality and simplicity. In addition, Volume 42 list price $22. 40; member price addenda have been inserted that reflect literature 205 pages; ISBN 0-8218-1592-X appearing after the publication of the Russian $16. 80; MMON0/42 edition. To order, please specify The chapter headings are: This monograph by L. D. Kudrjavcev is based on variational methods in partial differen­ Wiener-Hopf General Theorems Concerning tial equations. The topic has its origins in the Equations works of Gauss, Thomson and Riemann on the of Galerkin• s Method and Projection Methods investigation of Laplace• s equation when con­ Solving Linear Equations sidered as the Euler equation for the Dirichlet Projection Methods of Solving the Wiener-Hopf integral. The basic principle underlying this in­ Analogue Equation and its Discrete vestigation was proved in the works of Hilbert, Wiener-Hopf Equations with Discontinuous which stimulated more investigations by Courant, Functions Lewy, Fubini, Lebesgue, Z~;~.remba, Bogoljubov, Equations Pair Krylov, Kondra~ov, Lavrent•ev, Methods for Solving Pair Equations Keldy~, Projection Nikol• ski!, Sobolev and others. Equations Ljusternik, Wiener-Hopf Integral-Difference The apparatus of the so-called direct im­ Systems of Equations bedding theorems, which was created by Sobolev of Solutions of Homogeneous Asymptotics and his student Kondra~ov, plays a large role in Convolution Equations the application of the variational method to the A section called Remarks on the Literature problems of mathematical physics, For this rea­ is included at the end of the book, listing by son the main part of the present monograph is chapter and section the references cited and devoted to the subsequent development of ques­ giving brief remarks about them. The bibliogra­ tions connected with direct anll inverse imbedding phy contains one hundred twenty-four references, theorems. The investigations of inverse imbed­ To a considerable extent the presentation ding theorems or, what is the same thing, theo­ in this book is based on papers of the authors rems on the extension of functions and systems which were published in 1963-1967 (MR 28 #443, of functions are a continuation of the correspond­ MR 29 #487, MR 31 #3808, MR 32 #8085, MR 34 ing investigations of Nikol• skit. #18i4, MR 34 #6572, MR 35 #4763, MR 37 On the basis of the results obtained on im­ #1915). A preliminary edition entitled Projection bedding theorems the author applies the varia­ methods of solving Wiener-Hopf equations tional method to the investigation of boundary (Kishinev, 1967) was reviewed by L. von value problems for second order partial differ­ Wolfersdorf (Freiberg) in MR 37 #1915, The ential equations of elliptic type, both nondegen­ first paragraph of the review follows: erate and degenerate, on the whole boundary of

224 the domain or on part of it. A new basis for vari­ ly thereafter. Four numbers are expected to be ational principles is given which extends them to published in 1974. The Russian journal is pub­ arbitrary finite domains without any restrictions lished irregularly, whenever an accumulation of on the structure of the domains and their bound­ 25 or 30 papers has been accepted. Most papers aries. Existence and uniqueness theorems are are between 5 and 10 pages, and most represent proved for both the variational problem and the the Kiev school of probabilists and statisticians. corresponding differential problem in the case of Number one, Russian 1(1970), contains a selfadjoint elliptic equation of second order. "Remark on homogeneous Markov processes with The connection between the smoothness of the co­ discrete component" by A. V. Skorohod, MR 43 efficients of an equation and the smoothness of a #1276, "Remark on the determination of the meal!l solution is studied. In addition, proofs of the of Gaussian distributions in Hilbert space" by A. V. existence and uniqueness of the solution of bound­ Skorohod and I. §. Ibramhalilov, MR 43 #1241, ary value problems with sufficiently general "A reimement of Skorohod• s limit theorem" by boundary conditions a.re also given in the pres­ V. V. Baklan, MR 43 #2772, "A limit theorem ence of a power degeneracy with exponent a> 0. for stationary Gaussian processes" by Ju. M. In discussing these questions the author has Ry!ov, MR 43 #8120, and 21 other papers. mainly followed a method whose foundations go List price $140 for first four numbers, price for back to Hilbert and which have been developed members of AMS or IMS $70 and refined by Sobolev as well as Nikol• skiT and Kondra!'iov. The proof of the existence of a solution of Selected Tables boundary value problems for degenerate elliptic in Mathematical Statistics equations requires a study of the question of Edited by H. L. Harter and D. B. Owen with weighted function extension from the boundary to J. M. Davenport as managing editor the whole domain. This is done in Chapter I, Function Extensions. Here an essential role is The Institute of Mathematical Statistics has played by both the methods of Nikol• skit (for ex­ entered into an agreement with the American ample, the method of approximating differential Mathematical Society to publish jointly a series functions by entire functions of exponential type, of volumes entitled SELECTED TABLES IN the methods of function extensions and others) MATHEMATICAL STATISTICS, The volumes of and the methods of Sobolev (for example, the mathematical tables are prepared under the aegis method of function averagings), which naturally of the Institute of Mathematical Statistics by the receive a further development and sharpening. Committee on Mathematical Tables of the Insti­ In this chapter the theory of best extensions (in tute. the sense of the growth of the derivatives under The purpose of the series is to provide the an approach to the boundary of the domain) of most valuable collection of extensive tables of functions from the boundary of the domain onto high quality and utility available in any series. the whole domain is constructed. The best ex­ The material will be of use to both practical and tension problem is solved here to within an ar­ theoretical statisticians and mathematicians. The bitrary E > 0. series will help make these tables readily availa­ In Chapter II, Weighted Imbedding Theo­ ble and provide an outlet for work of moderate rems, the author considers spaces of functions length which is too long for inclusion in a journal having weak derivatives that are summable to yet too short for a book in itself. some power with a weight; he proves theorems Each volume contains several sets of ex­ on the imbedding of these spaces into ordinary tensive tables of interest to statisticians and function spaces, and he investigates complete­ users of statistical methods, Each set of tables ness, compactness, etc. includes, in the introductory material, discus­ Chapter III, Variational Methods for the sions concerning the methods of computation, Solution of Elliptic Equations, is devoted to the accuracy, methods of interpolation, and applica­ presentation of general variational principles tions with numerical examples of the use of the concerning the first boundary value problem for tables, Each table is checked for accuracy at the selfadjoint elliptic equations of second order, Wright-Patterson Air Force Base under the di­ and to the proof of existence and uniqueness rection of Clem Grabner and James P. Hudson, theorems for boundary value problems of elliptic Authors who would like to submit their work differential equations that are degenerate on the should check several current issues of The Insti­ boundary of the domain. tute of Mathematical Statistics Bulletin and The The bibliography contains fifty references American Statistician for any up-to-date an--­ representing the progress made in this field. nouncements about submissions to this series. Volume 1 is a reprint with corrections of the volume first published in 1970 by Markham Theory of Probability Publishing Company. It includes the following and Mathematical Statistics tables: "Tables of the cumulative non-central A translation of Teorija VerojatnosteT i Mate­ chi-square distribution" by G. E. Haynam, z. matHieskaja Statistika of Kiev University Govindarajulu, and F. C. Leone, "Tables of the exact sampling distribution of the two-sample The AMS is publishing for the Institute of Kolmogorov-Smimov criterion Dmn (m ~ n)" by Mathematical Statistics a cover-to-cover trans­ P. J. Kim and R. I. Jennrich, "Critical values lation of THEORY OF PROBABILITY AND MATH­ and probability levels for the Wilcoxon rank sum EMATICAL STATISTICS, The first issue (num­ test and the Wilcoxon signed rank test" by Frank ber 1) will appear in June 1974 and number 2 short- Wilcoxon, S. K. Katti, and Roberta A. Wilcox,

225 "The null distribution of the first three product­ LECTURES IN APPLIED MATHEMATICS moment statistics for exponential, half-gamma, and normal scores" by P.A. W. Lewis and A. S. NONLINEAR WAVE MOTION, Alan C. Newell, Goodman and "Tables to facilitate the use of or­ Editor thogonal polynomials for two types of error structures" by Kirkland B. Stewart. Volume 15 232 Pages; list price $19.10; member price 1973, 403 pages; ISBN 0-8218-1901-1; list price $14.33; ISBN 0-8218-1115-0 $8. 60; price for members of AMS and IMS $6. 45; To order, please specify LAM/15 To order, please specify TABLE/1 This volume contains the proceedings of the Volume 2 contains the following tables: summer seminar held at Clarkson College of "Probability integral of the doubly noncentral Technology, Potsdam, New York, in July 1972, t-distribution with degrees of freedom n and which was supported by the National Science noncentrality parameters ~ and ;\.n by William G. Foundation and the Office of Naval Research, Bulgren, "Doubly noncentral F distribution­ The 1960s produced some major advances Tables and Applications" by M. L. Tiku, "Tables in the mathematical description of and our under­ of ·expected sample size for curtailed fixed sam­ standing of nonlinear wave propagation phenome­ ple size tests of a Bernoulli parameter'' by na. Of particular significance were the methods Colin R. Blyth and David Hutchinson, and "Zonal of Whitham for describing the slow temporal and polynomials of order 1 through 12" by A. M. spatial modulation of fully nonlinear dispersive Parkhurst and A. T. James. waves and of Gardner, Greene, Kruskal and 1974, 388 pages; ISBN 0-8218-1902-X; list price Miura for finding the initial value solution to the Korteweg-de Vries equation. It was the purpose $14. 10; price for members of AMS and IMS $10. 58; To order, please specify TABLE/2 of this seminar to explore these and related theories and to exchange ideas about the most Volume 3 is now in preparation. One of the fruitful avenues of investigation for the immedi­ tables to be included in this volume is "Cumula­ ate future. tive distribution function of the two-factor and The principal lecturers and their topics are three-factor generalized incomplete modified T. B. Benjamin, "Lectures on nonlinear wave Bessel distributions" by Bernard Harris and motion"; D.J. Benney, "Long waves"; M.D. Andrew P. Soms. Nine other tables are now Kruskal, "The Korteweg-de Vries equation and being checked for possible inclusion. related evolution equations"; Peter D. Lax, ISBN 0-8218-1903-0. To order, please specify "Periodic solutions of the KdV equations"; G. B. TABLE/3 Whitham, "Two-timing, variational principles and waves". In addition, there are five other TRANSACTIONS OF THE lectures, an extensive bibliography, and indexes. MOSCOW MATHEMATICAL SOCIETY The reader of this book should be an ad­ vanced graduate student in the mathematical Volume 26 (1972) sciences. Although this book is not expository in 244 pages; list price $35. 50; member price the sense that a textbook is, it is reasonably $26,63; ISBN 0-8218-1626-0 expository for those having a sufficent back­ To order, please specify MOSCOW/26 ground. In the NSF Annual Report 1973 the entire This volume contains the transactions of the section on mathematics is devoted to initial value Moscow Mathematical Society for the year 1972. problems. The report states the following: "The The translation was prepared jointly by the Amer­ solution of the initial value problem for certain ican Mathematical Society and the London Mathe­ classes of non-linear partial differential equa­ matical Society. The following seven papers are tions is of considerable importance in many sci­ contained in the volume: 11 Subdifferentials of con­ entific disciplines. • •. One of the important vex functions" by A.D. Ioffe and V. L. Levin, equations of this type is the Korteweg-deVries "On the absence of continuity and HtHder continu­ (KdV), which was first suggested in 1895 to de­ ity of solutions of quasilinear elliptic equations scribe the development and propagation of mod­ near a nonregular boundary" by I.N.Krol' and erately small amplitude shallow water waves. V. G. Maz•ja, "Ergodic theorems for general ••• The most recent work ••• , by A. Newell, M. dynamical systems" by A.A. Tempel• man, "An Ablowitz, D. Kaup, and H. Segur, was stimulat­ analog of the theory of statistical decisions in ed by an NSF-sponsored Conference on Non­ noncommutative probability theory" by A. s. Linear Wave Motion in July of 1972 at Clarkson Holevo, "Cluster properties of limit density College of Technology. In a series of papers, matrices for one-dimensional continuous quantum they solve the Sine-Gordon equation and gener­ systems" by Ju. M. Suhov, "Action-angle vari­ ate a technique for the solution of a wide class of ables and their generalizations" by N. N. dispersive equations and for extension of this Nehorolev, and "The analytic form of differential technique to problems with higher derivatives, equations. II" by A. D. Brjuno. as well as subsuming many linear problems."

226 PROCEEDINGS OF SYMPOSIA Spitznagel, Jr.; "A new computer oriented IN APPLIED MATHEMATICS (algorithmic) linear algebra course-preliminary report" by Robert Ducharme; "Computer supple­ THE INFLUENCE OF COMPUTING ON MATHE­ mented business oriented mathematics" by MATICAL RESEARCH AND EDUCATION, edited Kenneth L. Hankerson and Gene A. Kemper. by Joseph P. LaSalle Only some college training in mathematics is needed to read most of the volume. It should Volume 20 be of some interest to high school teachers of 205 pages; list price $16. 50; member price mathematics. $12.38 MEMOIRS OF THE AMERICAN ISBN 0-8218-1326-1 To order, please specify PSAM/20 MATHEMATICAL SOOETY THE CATEGORY OF H-MODULES OVER A This volume contains seven of the invited SPECTRUM, by Jack Palmer Sanders addresses and fourteen of the contributed papers that were presented at the joint American Mathe­ Number 141 matical Society and the Mathematical Association 136 pages; list price $3.50; member price $2.63; of America Conference on the Influence of ISBN 0-8218-1841-4 Computing on Mathematical Research and Edu­ To order, please specify MEM0/141 cation held at the University of Montana, August 13-24, 1973. In this Memoir the category of H-modules The invited addresses were directed pri­ over a ring spectrum is introduced. An H-module marily to the influence of the computer on mathe­ N over a ring spectrum M generalizes the idea matical research and the applications of mathe­ of a module over a spectrum by, among other things, matics and, secondarily, on what this means for replacing a single map the teaching of mathematics f :Mi\N-+N and the education of p,q p q p+q mathematicians. The contributed papers de­ scribe more specifically some experiments in with a class of maps, all of which are homotopic. developing courses in mathematics with comput­ Higher homotopies are built into the structure ing and algorithmic orientations and a few re­ and enable one to construct mapping cones in the ports on computer influenced research. category. Various technical results and con­ The titles of the seven invited addresses structions are given, including the construction and their authors follow: "The influence of of a limit H-module. Each of the classical Thorn computing on research in number theory" by spectra is shown to be an H-module over itself, D. H. Lehmer; "The influence of computers on The mapping cone and limit constructions are algebra" by Charles C. Sims; "Computational applied to MU to construct a tower of homology probability and statistics" by illf Grenander; theories. "An introduction to some current research in numerical computational complexity" and "An ON THE THEORY AND APPLICATIONS OF DIF­ introduction to some current research in numer­ FERENTIAL TORSION PRODUCTS, by V.K.A.M. ical computational complexity" by J. F. Traub; Gugenheim and J. Peter May "Applied mathematics and computing" by Peter D. Lax; "The unexpected impact of computers on Number 142 science and mathematics" by Thomas E. 94 pages; list price $3.20; member price $2.40; Cheatham, Jr. ISBN 0-8218-1842-2 The titles of the fourteen contributed pa­ To order, please specify MEM0/142 pers and their authors follow: "Computational In this Memoir a new approach to differen­ complex analysis" by Peter Henrici; "Combina­ tial homological algebra is developed, one which torial games with an annihilation rule" by Aviezri exploits more general types of resolutions than s. Fraenkel; "The integration of computing and the hi-complexes used traditionally. An example mathematics at the Open University" by F. B. of such a generalized resolution is exhibited and I..ovis and R. V. M. Zahar; "Real time computer used to prove, that the differential torsion product graphics techniques in geometry" by Thomas reduces to the classical torsion product in favor­ Banchoff and Charles Strauss; "Visual geometry, able cases. This result is used to compute the computer graphics and theorems of perceived cohomology of various spaces. For example, if type" by Philip J. Davis; "Dual orthogonal series: H is a closed subgroup of a compact connected A case study of the influence of computing upon Lie group G and if H and G have no torsion, mathematical theory" by Robert P. Feinerman, then the integral cohomology H* (G/H) is isomor­ Robert B. Kelman and Chester A. Koper, Jr. ; phic as a graded Abelian group to "The design and use of an undergraduate numer­ ical analysis laboratory" by Myron Ginsberg; TorH*(BG)(Z, H*(BH)). "Statistical and numerical analysis: A computer oriented approach" by Andre R. Brousseau; The paper also includes proofs (within the new "Some problems in computational probability" by framework) of the results of Eilenberg and Marcel F. Neuts; "The influence of computing on Moore which relate differential torsion products generalized inverse applications in statistical to the homology and cohomology of spaces, a analysis" by Cecil R. Hallum; "On using the discussion of the relationship between differential electronic analog computer to illustrate mathe­ torsion products and matric Massey products, matical concepts" by Tyre A. Newton; 11An in­ and a detailed exposition (required for technical expensive computer assist in teaching large reasons) of the Eilenberg-Mac Lane-Cartan enrollment mathematics courses" by Edward L. calculation of the cohomology of K (tl', n) 's.

227 PROCEEDINGS OF THE Chapter II, Certain Basic Structures of the Theo­ ry of Constructive Locally Convex Spaces, the STEKLOV INSTITUTE author discusses products, projective and induc­ tive limits, and dimension of these spaces. In A REGION POLYNOMIALS ORTHOGONAL OVER Chapter Ill, Spaces Adjoint to Constructive Lo­ by P.K. AND BIEBERBACH POLYNOMIALS, cally Convex Spaces, he gives the constructive Suetin version of bounded sets and the adjoint space. The last three chapters are: Chapter IV, Con­ 100(1971) Number structive Generalized Functions; Chapter V, Or­ list price $15. 00; member price 92 pages; der, Support, and Value at a Point, for a Con­ $11.25 structive Generalized Function; Chapter VI, ISBN 0-8218-3000-7 Spaces of Cons6:uctive Infinitely Differentiable To order, please specify STEKL0/100 Functions and Constructive Functionals on Them. An appendix is devoted to two lemmas on con­ expansion in the The idea of orthogonal structive real numbers. A bibliography of fifty­ by various theory of analytic functions is realized seven titles on the subject is included. methods, one of which is Fourier series in poly­ nomials orthogonal with respect to the area of a MATHEMATICAL QUESTIONS IN THE THEORY region. In this volume asymptotic and approxi­ OF WAVE DIFFRACTION. I, edited by V. M. mation properties of polynomials orthogonal with Babi~ respect to area are investigated relative to prop­ erties of the weight function and the degree of Number 115 (1971) smoothness of the boundary of the region. By 168 pages; list price $29,40; member price way of an application, the rate of convergence of $22. 05; ISBN 0-8218-3015-5 Bieberbach polynomials to a mapping function in To order, please specify STEKL0/115 a closed region is examined. The titles of the chapters are "Polynomials This volume contains twelve papers on the that are orthogonal over a region with unit theory of wave diffraction. The primary subject weight, u "Polynomials orthogonal over a region classifications for this book are partial differen­ with analytic weight function, " "Polynomials tial equations, wave diffraction and dispersion, orthogonal over a region with continuous weight waves and radiation, diffraction and scattering. function," "Polynomials orthogonal over a region The titles of the articles and their authors when the weight function has singularities of the follow: "Rayleigh waves in a surface waveguide" simplest kind," and "The rate of convergence of by A. G. Alenicyn; "A formal method of construc­ the Bieberbach polynomials." tion of a short-wave asymptotic expression for Green's function" by V. M. Babi~; "Diffraction of SOME QUESTIONS IN CONSTRUCTIVE FUNC­ a cylindrical elastic wave on a semi-infinite TIONAL ANALYSIS by Fan Din• -Zieu screen" by B. P. BelinskiT and D.P. Kouzov; "On (Phan DiDh mau) a local method of solution of a nonstationary in­ verse problem for a nonhomogeneous string", Number 114 "An inverse boundary problem in the theory of 238 pages; list price $33. 90; member price propagation of waves in an anisotropic medium" $25. 43; ISBN 0-8218-3014-7 and "A quasi-two-dimensional inverse problem To order, please specify STEKL0/114 for the wave equation" by A. s. Blagovescenskil'; This volume is devoted to questions of con­ "On an inverse problem in a medium with absorp­ structive functional analysis. By the construc.,­ tion" by A, A. Buzdin; "The field of a point source tive trend in mathematics the author means the in a waveguide" by V. s. Buldyrev; "Construction variant of constructive mathematics developed of solutions concentrated close to rays for the by A. A. Markov, N. A. Sanin and their school. equations of elasticity theory in an inhomogeneous The author considers only constructively defined isotropic space" and "Propagation of surface objects in his study. Mathematical inferences waves concentrated near rays in an inhomoge­ are understood in the constructive sense on the neous elastic body of arbitrary shape" by N. Ja. basis of rules of constructive evaluation of in­ Kirpi~nikova; "Analytic nature of the wave field ferences. In particular, this volume is devoted in theJleighborhood of a caustic with reflection" to laying the foundations of theories of construc­ by D. s. Mogilevskil'; "On the field of a point tive locally convex linear topological spaces and source in the neighborhood of an opaque circular constructive generalized functions (distributions). cone. I" by B. G. Nikolaev. The first three chapters are devoted to general questions of the theory of constructive locally THEORY AND APPLICATIONS OF DIFFEREN­ convex linear spaces, and the last three toques­ TIABLE FUNCTIONS OF SEVERAL VARIABLES. tions of the theory of constructive generalized IV, edited by S. M. Nikol• skiT functions. In formulating the constructive ana­ Number 117 (1972) correspond­ logues of concepts and results of the vi, 403 pages; list price $39. 50; member price log theories of classical mathematics, the author $29. 63; ISBN 0-8218-3017-1 mathe­ explains the peculiarities of constructive To order, please specify STEKL0/117 matics within the confines of the theories in question. This volume contains eighteen papers on In Chapter I, Constructive Locally Convex differentiable functions of several variables. The Linear Topological Spaces, the author gives primary subject classifications for the book are basic concepts, properties, and definitions. In real functions, partial differential equations, ap-

228 proximations and expansions, Fourier analysis, to the construction of algorithms for machine functional analysis and operator theory. search for inference. Increasing recent interest The titles of the papers and their authors in this domain of mathematical logic is connected follow: "The behavior of differentiable functions with the development of mathematical cybernetics, on a nonsmooth surface", "On traces on a non­ in particular those of its divisions in which the smooth surface of classes of differentiable func­ question of simulation on computers of complex tions" and "Estimates of moduli of smoothness on forms of human intellectual activity is consid­ domains, and imbedding theorems" by 0. V. ered. Besov; "Properties of solutions of differential In the papers of this collection calculi of equations of higher order in terms of weight clas­ constructive logic are considered as well as ses" by Ja. S. Bugrov; "On approximating func­ calculi of classical logic. But here calculi of the tions of the space cr(O) by functions with com­ second type never occur as the logical basis of pact support, for an arbitrary open set 0" by any investigations; they are considered only as V. I. Burenkov; "Boundaries of subdomains, constructively defined objects of study and, like Holder weight classes and solutions in these clas­ constructively defined objects of other types con­ ses of th·e Poisson equation" and "Weighted error sidered in this collection, they are investigated estimates for the mesh method of solving the La­ on the basis of principles of the constructive trend place and Poisson equations" by E. A. Volkov; in mathematics. [The main lines of the construc­ "Imbedding theorems for spaces of functions tive trend in mathematics are formulated, for whose mixed derivatives satisfy a multiple-in­ example in the paper by A. A. Markov, On con­ tegral Holder condition" and "Families of spaces structive mathematics (Trudy Mat. Inst. Steklov. of functions whose mixed derivatives satisfy a 67(1962), 8-14; English transl., Amer. Math. multiple-integral Holder condition" by A. D. Soc. Transl. (2) 98(1971), 1-9).) In this sense D~abrailov; "Asymptotic distribution of the eigen­ it is possible to say that this collection belongs to values and eigenfunctions of a class of singular the constructive trend in mathematics and bor­ elliptic operators" by I. A. Kiprijanov; "On poly­ ders on the series Problems of the constructive nomial traces and moduli of smoothness of func­ trend in mathematics. [See Trudy Mat. Inst. tions of several variables" by L. D. Kudrjavcev; Steklov. 52(1958); 67(1962); 72(1964); 93(1967); "Operators connected with fractional differentia­ 113(1970) (for English translations of most of tion, and classes of differentiable functions" by these papers, see Amer. Math. Soc. Transl. (2) P. I. Lizorkin; "Asymptotic behavior at infinity 23(1963); 29(1963); 57(1966); 64(1967); 94(1970); of differentiable functions of weight classes" by 97(1971); 99(1972); 100(1972); Constructive real Ju. S. Nikol• ski!; "Investigation of certain clas­ numbers and constructive function spaces, Amer. ses of functions by 'angular• approximation" by Math. Soc., Providence, R.I. , 1968; Proc, M. K. Potapov; "On the representation of func­ Steklov Inst. Math. 93(1967); 113(1970)).] tions defined by a class of hypoelliptic operators" All the papers published in this collection by S. V. UspenskiT; "Integral estimates of gen­ were completed and delivered at seminars in eralized derivatives of solutions of second-order 1965-1971. elliptic equations in the Ln-metric, and some re­ The following seven articles are included lated imbedding theoremsfi by A. S. Foht; "Un­ in this volume: "On a bound for the complexity bounded solutions of degenerate elliptic equa­ of terms in the resolution method" by N. K. tions" and "The first boun.dary value problem for Zamov, "The inverse method and tactics for quasilinear elliptic equations of second-order" by establishing deducibility for a calculus with func­ G. N. Jakovlev. tional symbols" by s. Ju. Maslov, "Decidable classes reducing to a one-quantifier class" by LOGICO-MATHEMATICAL CALCULI. 2, edited S. Ju. Maslov and V. P. Orevkov, "The Skolem by V. P. Orebkov method in intuitionistic calculi" by G. E. Mine, "Undecidable classes of formulas for the con­ Number 121(1972) structive predicate calculus. I" by V. P. Orevkov, 183 pages; list price $22.30; member price "A sequential variant of R.M. Robinson's $16.73 arithmetic system not containing cut rules" by ISBN 0-8218-3021-X A. Ju. Plju§kevi!Sene, and "Sequential variants of To order, please specify STEKL0/121 applied predicate calculi without structural deductive rules" by M.G. Rogava. This volume is the second collection of The primary subject classifications under papers from the seminar on mathematical logic Logic and Foundations are predicate calculus, at the Leningrad Branch of the Steklov Mathe­ formalizations of intuitionism, and decidability matical Institute of the Academy of Sciences of and undecidability. Theorem proving under the the USSR. The collection consists of papers on heading of Computer Science is also a primary the theory of logical inference and its application classification.

229 PERSONAL ITEMS

THEODORE W. ANDERSON of Stanford Uni­ JOSEPH L. TAYLOR of the University of versity will be on sabbatical leave for 1974-1975. Utah has been selected to receive a 1974-1975 He will spend the year at the London School of Distinguished Research Award, by that univer­ Economics. sity. JOSEPH ARKIN of Spring Valley, New York GIOVANNI VIGLINO of Wesleyan Univer­ has been appointed an honorary professor at the sity has been appointed to an associate pro­ Sussex College of Technology, Sussex, England. fessorship at the University of Simon Bolivar. H. FREDERIC BOHNENBLUST of the Cal­ WILLES L. WERNER has been appointed ifornia Institute of Technology has been designated a computer programmer with the Department of Professor Emeritus upon his retirement from Public Safety, Phoenix, Arizona. that school. JOHN E. WOLFE of Northwestern Uni­ SUN-YUNG ALICE CHANG of the University versity has been appointed to an assistant pro­ of California, Berkeley, has been appointed to an fessorship at Oklahoma State University. assistant professorship at SUNY at Buffalo. JAMES A. DEDDENS of the University of JD.diana has been appointed to an assistant pro­ PROMOTIONS fessorship at SUNY at Buffalo. To Director, School of Philosophy, Uni­ . JENS ElSING of SUNY at Stony Brook has versidad del Zulia, Venezuela: ERNESTO H. been appointed to an associate professorship at BATTISTELLA. the Technical University of Denmark. HENRY W. GOULD of West Virginia Uni­ To Chairman, Department of Mathematics, versity has been appointed Editor-in-Chief of the Duquesne university: ROBERT G. McDERMOT. Proceedings of the West Virginia Academy of To Chairman, Department of Mathematics, Science. Red Deer College: VED P. MADAN. SEYMOUR HABER of the National Bureau To Manager, Software of Standards is spending Systems, Comsat the current academic Laboratories: HENRY L. PARKER. year as a visiting professor at the Hebrew Uni­ versity, Jerusalem. To Professor, Florida State University: STANLEY KOGELMAN of New York Uni­ EUTIQUIO C. YOUNG. versity has been appointed to an assistant pro­ fessorship at Baruch College (CUNY). To Professor, Loretto Heights College: F. WILLIAM LAWVERE of the Universite SISTER MARGARET GRACE ELSEY. di Perugia, Italy, has been appointed to a pro­ To Professor, SUNY at Buffalo: STUART fessorship at SUNY at Buffalo. P. HASTINGS. ROBERT A. MELTER of Southampton College, Long Island University, has been ap­ To Professor, university of New Orleans: pointed a lecturer at the Centre d'Enseignement PREM K. KULSHRESTHA. Superieur of Niamey, Niger, for the 1974-1975 To Professor, University of Toledo: academic year. HARRIS WESTCOTT VAYO. PIDLIP NANZETTA of St. Mary's College of Maryland has been appointed Dean of the Fac­ To Associate Professor, Missouri ulty of Natural Sciences and Mathematics at Southern College: CHARLES S. ALLEN; P. K. Stockton State College, New Jersey. SUBRAMANIAN. WILLIAM PRAGER, Professor Emeritus To Associate Professor, SUNY at Buffalo: of Brown university, has been elected Corre­ CLIFFORD 0. BLOOM. sponding Member of the Academy of Sciences of the Instltute of France, Paris. He also was To Associate Professor, Indiana Univer­ awarded the degree of Doctor of Sciences honoris sity: CHURL S, KIM. causa by the University of Manchester. To Associate Professor, University of PATRICK J. RYAN of the University of Toledo: HENRY C. WENTE. Notre Dame has been appointed to an associate professorship at Indiana university, South Bend. To Associate Professor, West Virginia SUSANN SHAW of New York University has University: GEORGE E. TRAPP, JR. been appointed to an assistant professorship at To Associate Professor, Worcester Poly­ San Francisco State University. technic Institute: PAUL W. DAVIS; G. CLINTON MARIA W. STEINBERG of California State SORNBERGER. University, Northridge, is retiring with the status of Professor Emeritus. To Assistant Professor, SUNY at Buffalo: MICIDO SUZUKI of the University of Uli­ JOHN GREGORY. nois, Urbana-Champaign received the Academy To Lecturer, University of Malaya: Prize from the Japan Academy, Tokyo. CHONG-KEANG LIM.

230 INSTRUCTORSHIPS at the age of 66. He was a member of the Society for 20 years. SUNY at Buffalo: DAVID N. BOCK. Professor PAUL E. GUENTHER of Case DEATHS Western University died on April 28,1974, at the age of 58. He was a member of the Society for Professor Emeritus BANCROFT H. BROWN 32 years. of Dartmouth College died on May 7, 1974, at the age of 89. He was a member of the Society for Professor CHARLES W. LYTLE of Drew 52 years. University died on January 9, 1974, at the age of 46. He was a member of the Society for 14 Miss UEI FONG of the University of Cin­ years. cinnati died March 1974 at the age of 23. She was a member of the Society for 1 year. Professor HIDEGORO NAKANO of Wayne State University died on March 11, 1974, at the Professor HELEN M. CLARK of North­ age of 64. He was a member of the Society for western University died on March 11, 1974, at 22 years. the age of 65. She was a member of the Society for 28 years. Professor ABRAHAM ROBINSON of Yale 1974, at the age of Professor ALEXANDER DING HAS of the University died on April 11, the Society for 18 years. Freien Universitat Berlin died on April 19,1974, 55. He was a member of

NEWS ITEMS AND ANNOUNCEMENTS STEELE PRIZES Murray H. Protter Department of Mathematics The Committee on Steele Prizes of the University of California, Berkeley American Mathematical Society is soliciting Berkeley, California 94720 nominations of papers that may be worthy of a Steele Prize. Hans F. Weinberger, Chairman Because there is no summer meeting this School of Mathematics year, the 1974 Steele Prizes will be awarded at University of Minnesota the January 1975 meeting. The Committee is Minneapolis, Minnesota 5545.5 now beginning its search for the winners of prizes to be awarded at the 1975 summer meeting. The Steele Prize is to be awarded for a well written paper which makes accessible to CANADIAN SOCIETY FOR HISTORY other mathematicians an area of mathematics to AND PHILOSOPHY OF MATHEMATICS which the author has made substantial contribu­ tions. The Committee is anxious to find excel­ At a founding meeting on June 3, 1974, in lent expository papers which meet these criteria. Toronto, the Canadian Society for History and NOlp.inations for the Steele Prize should be Philosophy of Mathematics adopted a constitutiol!l submitted to one of the following members of the stating that "the aim of the Society is to promote Committee on Steele Prizes: throughout Canada discussion, research, teach­ Chen Chung Chang ing, and publication in the history and philosophy Department of Mathematics of mathematics." The first Executive Council University of California, Los Angeles was elected as follows: President: Charles V. Los Angeles, California 90024 Jones (York University); Vice-President: Tom W. Settle (Guelph University; Secretary-Treasurer: Edward B. Curtis John L. Berggren (Simon Fraser University); Department of Mathematics Council Members: W. S, H. Crawford (Mount University of Washington Allison University), N. T. Gridgemen (National Seattle, Washington 98195 Research Council), and Fred Ustina (University Paul R. Halmos of Alberta at Edmonton). "to any person with Department of Mathematics Membership is open philosophy of mathe­ Indiana University interest in the history and matics." Annual dues of $4.00 were set for Bloomington, Indiana 47401 1974 and 1975, Historia Mathematica was named H. A. Heilbronn the official journal of the Society, and mem­ Department of Mathematics bers were offered the option of paying a single University of Toronto fee of $10. 00 for membership and subscription to Toronto, Ontario, Canada M5S 1Al the journal, starting with 1974 and Volume I. should write to Edwin Hewitt Those interested in membership Professor J, L. Department of Mathematics the Secretary-Treasurer, Department, Simon University of Washington Berggren, Mathematics Seattle, Washington 98195 Fraser University, Burnaby, British Columbia,

231 LEGAL AID COMMITTEE Committee to Select Gibbs Lecturers for 1975 and 1976: Richard Kadison has agreed to The Legal Aid Committee decides whether serve. legal aid in the form of a loan from the Society Nominating Committee for 1974: Steven is to be recommended for mathematicians who Armentrout, Chairman, David M. Goldschmidt, are victims of discrimination in hiring, promo­ JosephJ. Kohn, B. J. Pettis, DavidA. Sanchez. tion, termination, tenure and salary on the basis Committee to Recommend Cole Prize Win­ of race, sex, politics, religion, ethnic origin, ner for 1975: Walter Feit, Alex Heller, Irving age, or other non-professional characteristics. Kaplansky, Chairman. The Legal Aid Committee does not initiate in­ Committee on Relations with Government: vestigations, but only considers cases referred Add Anil Nerode, P. Emery Thomas. to it by the Committee on Academic Freedom, AMS-MAA Committee on Women in Mathe­ Tenure, and Employment Security. It may then matics: Shirley A. Hill, Chairman, Dorothy L. conduct further investigations to decide whether Bernstein, JaneK. Cullum, Mary W. Gray, I. N. legal aid is to be recommended. Herstein, Cathleen S. Morawetz, C. B. Morrey, There are two situations in which the Com­ Jr., Jacqueline C. Moss, Jane Cronin Scanlon, mittee may recommend legal aid: Alice T. Schafer, Gail S. Young. 1. An individual mathematician who has AMS-MAA-NAM Joint Committee on Grad­ suffered an injustice would be unable to pursue uate Programs At Traditionally Black Institu­ legal justice without a loan from the Society. tions: C. B. Bell (AMS) is chairman. Add Ted 2. The judicial decision could be of general Sykes (NAM). benefit to mathematicians. Committee on Legal Aid (A new committee Loans could be recommended for mathe­ authorized at the Business Meeting of January 16, maticians whether or not members of the Society. 1974 in San Francisco): Morton Curtis, Chair­ The loan limit should be $2, 000, except that it man, Murray Gerstenhaber, Jane Cronin may be higher when suitable security is provided. Scanlon, Todd Dupont, Walter Talbot. Loan application forms are to be submitted Committee on Principles of Reciprocity to the Legal Aid Committee. The applicant may Agreements: James A. Donaldson, Kurt Fried­ submit a loan form at any time, but the LAC will richs, John W. Green, Arthur Mattuck, Tilla not act on a loan application until CAFTES rec­ Milnor, Charles B. Morrey, Jr., Gerald Potter. ommends the case to them. Committee on Emergency Employment The members of the Committee on Legal Problems: Charles Curtis, Chairman, David Aid are Morton Curtis, Chairman, Murray Gale, Judy Green, Richard Palais, Judah Rosen­ Gerstenhaber, Jane Cronin Scanlon, Todd Dupont, blatt, William Singer, Moss Sweedler. and Walter Talbot. Committee on Teaching Loads and Class Size: Daniel Finkbeiner, Mary Gray, Meyer Jerison, George Piranian, Chairman, Lance Small. AMS COMMITTEES

The following changes have taken places in Society committees since the publication of the 1974 Administrative Directory. Dates refer to AMS RESEARCH FELLOWSHIPS AWARDED year of expiration when this is applicable. AMS Research Fellowships have been MATHEMATICAL REVIEWS Crisis Com­ awarded to Fred G. Abramson of the Massachu­ mittee: Donald C. Spencer is no longer serving. setts Institute of Technology and James Li-Ming Executive Committee: Add I. M. Singer Wang of Brown University. The fund was es­ (1975), Cathleen S. Morawetz (1975). tablished last year to support a one-year re­ TRANSACTIONS AND MEMOIRS Editorial search fellowship awarded strictly on the basis Committee: Harry Kesten has agreed to be chair- of mathematical merit. Awards were made by man. the Committee on Postdoctoral Fellowships, Committee to Select Hour Speakers for whose members are C. B. Bell, Walter Feit, Summer and Annual Meetings: Add Alexandra Peter Hilton, Edwin Hewitt, Mark Kac, and Ionescu-Tulcea (1975), Richard M. Dudley Alice T. Schafer, Chairman. (1975). The Society contributes an amount, between Committee on Summer Institutes: Louis $9,000 and $20,000, equal to half the funds raised Auslander is the chairman. Add S. S. Chern from other sources. The Board of Trustees at (1976), Richard K. Lashof (1976). its meeting June 20-21, 1974, accepted the Com­ Committee on Employment and Educational mittee's recommendation endorsed by the Council Policy: Add Wendell H. Fleming (1975), Geoffrey that the fellowship program be continued on these S. Watson, consultant. same terms for 1975-1976.

232 1975 APPLIED MATHEMATICS SYMPOSIUM have been published in the series Proceedings of Symposia in Applied Mathematics (PSAM); papers The Council has approved a recommenda­ presented at subsequent symposia have appeared tion of the AMS-SIAM Committee on Applied in the new series SIAM-AMS Proceedings, which Mathematics (Richard DiPrima, Chairman) that supersedes PSAM. Lectures given at the sum­ there be a symposium on nonlinear programming mer seminars are published in the series Lec­ associated with an April meeting in 1975. tures in Applied Mathematics. -- The Committee has been sponsoring sym­ In accord with a request from the Council, posia and summer seminars in applied mathe­ the complete list of topics for past symposia and matics since 1947. Since 1957, nearly all the seminars is given here. The editor of the pro­ symposia have been held in New York City. The ceedings is given in parentheses following the lectures given in the first nineteen symposia topic.

Symposia

1947 Nonlinear Problems in Mechanics of Continua (E. Reissner) 1948 Electromagnetic Theory (A. H. Taub) 1949 Elasticity (R. V. Churchill) 1951 Fluid Dynamics (M. H. Martin) 1952 Wave Motion and Vibration Theory (A. E. Heins) 1953 Numerical Analysis (J. H. Curtiss) 1955 Applied Probability (L. A. MacColl) 1956 Calculus of Variations and Its Applications (L. M. Graves) 1957 Orbit Theory (G. Birkhoff, R. E. Langer) 1958 Combinatorial Analysis (R. Bellman, M. Hall, Jr.) 1959 Nuclear Reactor Theory (G. Birkhoff, E. P. Wigner) 1960 Structure of Language and Its Mathematical Aspects (R. Jakobson) 1960 Hydrodynamic Instability (R. Bellman, G. Birkhoff, c. C. Lin) 1961 Mathematical Problems in Biological Sciences (R. Bellman) 1962 Experimental Arithmetic, High Speed Computing, and Mathematics (N.C. Metropolis, A. H. Taub, J. Todd, C. B. Tompkins) 1963 Stochastic Processes in Mathematical Physics and Engineering (R. Bellman) 1964 Applications of Nonlinear Partial Differential Equations in Mathematical Physics (R. Finn) 1965 Magneto-Fluid and Plasma Dynamics (H. Grad) 1966 Mathematical Aspects of Computer Sciences ( J. T. Schwartz) 1967 Transport Theory (I.K. Abu-Shumays, R. Bellman, G. Birkhoff) 1968 Numerical Solution of Field Problems in Continuum Physics (G. Birkhoff, R.S. Varga) 1969 Mathematical Aspects of Electrical Network Analysis (F. Harary, H. s. Wilf) 1970 Computers in Algebra and Number Theory (G. Birkhoff, M. Hall, Jr.) 1971 Mathematical Aspects of Statistical Mechanics ( J. C. T. Pool) 1972 Stochastic Differential Equations (J. B. Keller, H. P. McKean) 1973 Complexity of Real Time Computational Processes (R. M. Karp) 1974 Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry (D. S. Cohen)

Seminars

1957 Applied Mathematics (four parts) Probability and Related Topics in Physical Sciences Fluid Mechanics Solid Mechanics Hyperbolic and parabolic PDE 1960 Modern Physical Theories and Associated Mathematical Developments 1963 Space Mathematics ( J. B. Rosser) 1965 Relativity Theory and Astrophysics (J. Ehlers) 1967 Mathematics of the Decision Sciences (G. B. Dantzig and A. F. Veinott, Jr.) 1970 Mathematical Problems in the Geophyscial Sciences (W. H. Reid) 1972 Nonlinear Wave Motion (A. C. Newell) 1974 Inverse Problems 1975 Modern Modeling of Continuum Phenomena

233 ERRATA

Volume20

PETER FLETCHER and WILLIAM F. LINDGREN, Generalizations of y·spaces, Abstract 709-G2, Page A-670.

Line 3, for "x € V n+l(x) C V n(x)", read "x € V n+l(x) C V n(x) € r."

DAVID C. MURDOCH and F. VAN OYSTAEYEN, Noncommutative localisation and affine varieties, Abstract 73T-A216, Page A-480.

Lines 2, 3, 4, "the correspondence ... primes.", require an additional restriction on R. ccThe Artin-Rees condition on ideals in R is sufficient for the truth of both statements. Thus a classical ideal theory implies a classical localization theory for symmetric kernel functors with property (T)."

Volume21

FRANCIS E. MASAT, Right simple elements in a semigroup, Abstract 713-A1, Page A-386.

Line 9, for " ... and R is RS then Ry ~ R ~ ... " read uthen ... "

ALAN CANDIOTTI, The order of K/J/(K/3) 3 for quadratic fields, Abstract 713-A9, Page A-388.

Lines 2, 3, for "Let A+ and A-.· -respectively" read "Let A+ and A- be the subgroups of elements of order 3 in the 3-ideal class groups of k + ••• "

DAN MAULDIN, On th-e measurability of the continuum, Abstract 74T-E49, Page A-447.

Line 2, following " ..• Axiom of Choice." should read "Theorem. For each positive integer i, 1.< i <8, statement i implies statement i + 1. 1, 2No = N 1."

DAVID ZEITLIN, Identities for integer sequences involving the greatest integer function. IV. Preliminary report, Abstract 74T-A95, Page A-367.

Line 2, for "W 2 = 1" read "W 1 =I".

DAVID ZEITLIN, Identities for integer sequences involving the greatest integer function. V. Preliminary report, Abstract 74T-A121, Page A-434.

Line 7, for cc4W n+l, read u4W n+l-".

EUGENE M. NORRIS, Relative act ideals, Abstract 74T-Bl36, Page A-444, should have been classified under Algebra and Theory of Numbers.

234 ABSTRACTS PRESENTED TO THE SOCIETY

Preprints are avaUable from the author 1n cases where the abstract number is stllrred. Invited addresses are indicated by • 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 DJis. cellaneous group includes all abstracts for w"hich tbe authors did not indicate a categ_ory. An individual may present only one abstract by titls in any one issue of the cNotiai) 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 bs accepted for an issue. Algebra & Theory of Numbers

* 74T -A137. S. BRENT MORRIS, Duke University, Durham, North Carolina 27706. The generalized faro shuffle in q-dimensions.

Two permutations which arise naturally in considering the rearrangement of playing cards are the faro shuffle, an idealized riffle shuffle, and the simple cut (which takes one card from the top of a deck to the bottom). The group structure obtained from these permutations operating on decks of any size has been determined by Golomb ["Permutations by cutting and shuffling", SIAM Rev. 3(1961), 114-118], A generalization of the faro shuffle is obtained by considering the permutation achieved by n packets of cards "riffled" together, rather than two. The group structure obtained from the generalized faro shuffle and simple cut has been determined by Hartwig and Morris [Abstract 74T-A46, these~ 21(1974), A-292]. A further abstraction of this problem involves a q-dimensional deck, the direct product of q regular decks, and q-dimensional shuffles and cuts, direct products of regular operations. Again, the group structure is known, but only for the regular faro shuffle [Abstract 73T-A215, these~ 20(1973), A-480]. This paper considers the q-dimensional deck with the generalized faro shuffle and simple cut, and derives an explicit form for the group structure. (Received March 21, 1974.)

*74T-A138, W. KEITH NICHOLSON, University of Calgary, Calgary, Alberta T2N 1N4, Canada. Semiregular rings and modules. Preliminary report.

A ring R with Jacobson radical J is called semiregular if R/J is regular and idempotents can be lifted modulo J. Several characterizations of these rings are given. It is shown that, if R is semiregular, so is any homomorphic image of R, any matrix ring over R, any ideal of R and any subring eRe, e 2 =e. A class of semiregular modules is defined which contains all semiperfect modules, all regular modules and all semiregular rings with identity. Several characterizations of these modules are given and some theorems about semiperfect and regular modules are generalized. In parti~ular, a direct sum of semiregular modules is shown to be semiregular and this result is applied to show that a ring with identity is semiregular if and only if every finitely related (cyclic) module has a projective cover. It is also shown that a projective semiregular module is a direct sum of left ideals generated by idempotents. Finally, endomorphism rings are considered. Semiperfect modules are shown to have semiregular endomorphism rings and this result and the corresponding theorem for injective modules are generalized. Other results on the endomorphism ring of a semiregular module are proved and a characterization of left perfect rings is given, (Received March 21, 1974.)

*74T-A139. TORFENCE D. PARSONS, Pennsylvania State University, University Park, Pennsylvania 16802, Ramsey graphs and block designs. Preliminary report.

For each symmetric (v, k, .\)-block design D admitting a polarity map 11 there is a "polarity graph" G(D, rr) with Tetrices the points of D and x adjacent to y if x f, y and x E rr(y). These graphs generalize

A-473 (v, k, .\)-graphs and, like them, may be characterized intrinsically in terms of common neighbor subgraphs. Given

such a graph G, a design D and polarity 1T can be reconstructed such that G = G(D, IT). For each (v, k, .\)­ difference set in a finite abelian group V, there exists a (,., k, .\)-polarity graph with vertex set V. The Singer (q 2 + q + 1, q + 1, 1) difference sets yield polarity graphs with highly symmetric subgraphs [Abstract 73T-A235, these ll:a!i..c-et + 1, v- k + 1) is either 1 + v or 2 + v. In some cases, such as (v, k, .\) = (q 2 + q + 1, q + 1, 1) for q a prime power, we have R = 1 + v, In others, like (v, k, >t) = (11, 5, 2), we have R = 2 + v. This extends results of Parsons announced in Abstract 73T-A157, these :l1aii.c-e- 20(1973), A-562, (Received \larch 25, 1974.)

* 74T-A140. MELVIN C. THORNTON, University of Nebraska, Lincoln, Nebraska (\8508, Semi groups of selfmaps.

The continuous functions from X into itself form a semigroup S(X). A class of spaces is considered

so that S(X) is also the semigroup of isotone functions for a partial order on X. For this class, necessary and sufficient conditions are given for a semigroup to be isomorphic to some S(X). The homomorphisms among these semigroups are studied and the automorphisms of S(X) are determined. For finite X, a converse to a theorem of Howie is proven, providing a characterization in terms of the algebraic structure of S(X) for X being totally ordered. (Received March 25, 1974,)

* 74T-A141. JITENDRA N. MANOCHA, Kent State University, East Liverpool, Ohio 43920, A generalization of finite dimensionality for modules.

Let R be a ring with identity and let :JllR denote the category of right R-modules, Let ~ be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to

be a ~-submodule of M if C E ~. An R-module M is said to be [-finite dimensional if it does not contain an infinite direct sum of nonzero [-submodules of M. Theorem. Let M be an R-module. Then either M is not ~­ finite dimensional or there is a uniform bound (the [-dimension of M) on the number of nonzero ~-submodules in a direct sum of submodules of M, When [ = :JllR, we recover the definition of dimension in the sense of Goldie. When [ is the class of torsion-free modules relative to a kernel functor a, we derive the formula: dim M =a- dim M + dim (a(M)), where for an R-module N, dim N is the dimension of N in the sense of Goldie and a- dim N is the dimension of N relative to the class of a-torsion-free modules, A special case gives a new interpretation of rank of a module as defined in Goldie [J. Algebra 1(1964), 268-287], (Received March 27, 1974,)

* 74T-A142, ALBERT A. MULLIN, 9213 Kristin Lane, Fairfax, Virginia 22030, On specially structured primitive polynomials over UFD's. Preliminary report.

A basic result in the study of the algebraic structure of polynomial domains over unique factorization domains is the so-called Gauss' lemma: the product of two polynomials with integer coefficients is primitive iff both factors are primitive. Definition. A polynomial with integer coefficients is primordial provided the mosaics of all its coefficients have no prime (number) in common. Clearly, every ~rimordial polynomial is primitive, but not conversely, in general. Several sufficient conditions are given for the product of two polynomials with integer coefficients to be primordial. Clearly, a necessary (but not sufficient) condition for the product of two polynomials with integer coefficients to be primordial is that both of the factors be primitive. (Received April 2, 1974,)

74T-Al43. LAWRENCE D. STRONG and CHARLES C. ALEXANDER, University of Mississippi, University, Mississippi 38677, On graphs with unique distance trees. Preliminary report.

Several properties of graphs with unique distance trees are investigated. Chartrand and Stewart

A-474 conjectured ["Isometric graphs", Recent Trends in Graph Theory, Springer-Verlag, ,vol. 186, 1971, pp. 63-67]: if a graph G has a unique distance tree, then G has at most one cyclic block. A proof of this is given. Also proved is the conjecture by Chartrand and Schuster ["Which graphs have unique distance trees?", Amer. Mach. Monthly 81(1974), 53-56] that every connected graph without cut-vertices having a unique distance tree is regular. (Received March 28, 1974.)

* 14T-A144. SOON-KIONG SIM, Facultad de Ciencias, Universidad de Los Andes, \lerida, Venezuela. Prime ideals and symmetric idempotent kernel functors. Preliminary report.

Lee R be a right noetherian ring with unity. lc is shown that, for a prime ideal P of R, the symmetric idempotent kernel functor a R _p [\lurdoch and Van Oystaeyen, Abstract 73T -A216, these Jl,~ 20(1973), A-480] is the largest symmetric idempotent kernel functor for which R/P is torsion free. Let a be a prime idempotent kernel functor [Goldman, J. Algebra 13(1969), 10-47] and P the tertiary radical of a supporting module for a, Then aR-P 2 a. Theorem. An idempotent kernel functor r is symmetric iff r = lnf {aR-PI P Err! where rr is the family of all tertiary radicals of supporting modules for r, Corollary. Every symmetric prime idempotent kernei functor is of the form aR-P for some prime ideal P of R. (Received April 5, 1974.)

74T-A145, HARIHARAN K. IYER, University of Notre Dame, Notre Dame, Indiana 46556. A characterisation of 0 Zn +2(v, q) by the centraliser of an involution of type 2n. Preliminary report.

Let G be a finite group with no subgroup of index 2, Lee t be an involution of G such that C G(t)

satisfies the following conditions: (a) CG(t) has a subgroup H of odd index; H is isomorphic to 0 2n(f, q), qn =

f (mod 4), q odd and n 2:4. (b) The subgroup M of H which is isomorphic to !1 2n(f, q) is normal in CG(t). (c) ICG(t)nCG(M)I' :S (q +a)', where for any natural number m, its largest odd divisor is denoted by m', and a=± 1

with q =a (mod 4). Then G has a normal subgroup G0 , isomorphic to fl 2n+2(v, q) with qn+! =- v (mod 4), and a

cyclic subgroup W of odd order such that G = WG 0 , WnG 0 = 1 and W acts faithfully on G0 by field automorphisms. (Received April 8, 197 4.)

74T -A146. WILLBI J. BLOK, University of Illinois, Chicago, Illinois 60680. Varieties of closure algebras. II. Preliminary report.

For notations and definitions see Abstract 74T-A91, these~~ 21(1974),A-366, Let l and 0

denote the lattices of subvarieties of fl and B c respectively. The map Q~ l is defined by Ki->K0 , the map l~H by Ki->Kc = {L E B c' L 0 E K!, the map 0 _.":, Q by K~->K* = V({L *: L E Kl), where L * denotes the

B c-subalgebra of L generated by L 0 • Theorem 1, (i) Q~ l is a complete D 0 1 -homomorphism; (ii) l c !1 is a

D0 1-embedding; (iii) l--'=.0~2 is the identity map on l; (iv) For K E l, the pre-image of K under the map 0 is

[(Kc)*, Kc]; (v) ::£~ C:L":.,(B;] is a D 0 1-embedding. Theorem 2, (i) B; is generated by its finite members; (ii) B=~ B c" Theorem 3. (i) If L E Q*, L finite, then L = L *; (ii) If K is a class of finite subdirectly irreducibles

in Q, then every finite subdirectly irreducible in V(K) is in HS(K); (iii) (Q 0 ]~(Q]* (Q*] is a D 0 1-isomorphism. Recent results of K. A. Raker ("Primitive satisfaction and equational problems for lattices and other algebras", preprint) are employed to determine the equations characterizing several varieties, e.g., Q:. n 2: 1, (Received April 4, 1974,)

74T-A147, WILLEM J. BLOK and P 1ULIP DWINGER, University of Illinois, Chicago, Illinois 60680, Varieties of closure algebras. III. Preliminary report.

For notations and definitions see Abstract 74T-A91, these :lku~ 21(1974), A-366 and the preceding

abstract. For an equational class K, let K S 1 be the class of subdirectly irreducibles in K and for A E K, let (K:A) = {B E K: A f. S(B)!. For n 2:0, let Cn be the chain of n + 2 elements and let P n be the finite closure algebra with n + 1 atoms for which P~ = Cn. The results stated in the abstracts mentioned above and results obtained by

A-475 A, Day ("Varieties of Heyting algebras. I", preprint) are used to investigate the equational classes (Be: P n) and (B · p )* n > 0, Theore'm 1, (i) (B ; P ) = (H: C )e; (ii) B = V({(B : P ): n > Ol); (iii) (B : P +l)SI = c" n ' - c n n c c n - c n {L E B : L 0 = L 0 EIH, L E (B : P )}; (iv) (B : P ) is locally finite; (v) B* = V(I(Be: Pn)*: n > 0}), lf L E Be, c 1 1 en en c - a EL, let a0 =a, an= (an_ 1)' +(an_ 1)0 for n2: 1, Theorem 2, (I3e: Pn)* is characterized by the equation Xn= 1, n:: 0, (Received April 4, 197 4.)

74T-A148. MICHAEL JOHN CURRAN, University of Notre Dame, Notre Dame, Indiana 46556, Groups with decomposable involution centralizers. Preliminary report,

In a finite group G call an involution t central if t is in the center of a Sylow 2-subgroup of G.

Theorem. Let G be a finite group with a central involution t whose centralizer has the structure C(t) = x F, where F is isomorphic to either a simple alternating group, or a classical or exceptional Lie type simple group

of odd characteristic. Then G has a subgroup of index 2 (in particular G is not simple) except when F'"' A 5 or F'-'PSL(2, 3 2n+l) (n::: 1), Remark. The results of Z, Janko and J. G. Thompson (J. Algebra 3(1966), 147; J. Algebra 4(1966), 274) show if G has no subgroup of index 2, G=]a, the Janko simple group of order 175, 560, when F=A 5, and G is a simple group of Ree type when F=PSL(2, 32n+l) (n::: 1), (Received April 8, 1974,)

*74T-A149. DAVID E. PENNEY and GERALD B. HUFF, University of Georgia, Athens, Georgia 30602, Schedules for testing individuals in tournaments for pairs.

A schedule S for a pairs tournament is a collection of 2 by 2 matrices (matches) with entries from a finite set of players. A player is paired with the other in the same column, matched against the other two, If P is a schedule of k matches in which each player in [1, 4k] is scheduled, let S(P) be the union of subschedules

S .(P) called rounds, where S .(P) is constructed from P by leaving 4k invariant and for x E [1, 4k- 1], 1 1 replacing x by the least positive residue of x + j mod (4k -1}, Define C(P) [R(P), D(P)] by l!lx- yl!: (x, y) is a column [row, diagonal] of a match in PI, where x, y E [1, 4k- 1], llx- Yll =min { lx- Yi, 4k- 1 - lx- Yll and lix- Yli = 2k if 4k E {x, y}. The results are: 1, S(P) is a fair schedule as defined by Scheid [Amer. Math. Monthly 67(1960), 39-41] iff C(P) = [1, 2k] and every element of [I, 2k] occurs twice in R(P)uD(P). 2, S(P) determines a spouse-avoiding mixed doubles tournament for 4k couples as defined by Brayton, Coppersmith, and Hoffman [Bull. Amer. Math. Soc, 80{1974), 116-118] iff C(P) = R(P) = D(P) = [1, 2k]. Computer-produced examples of P

satisfying the first condition are given for k .::; 8, and examples satisfying the second are given for k = 1, 2, 4, and 6, (Received April 8, 1974.)

* 74T-A150. M. BHASKARAN, 91 Carr Street, Perth, Western Australia 6000. A congruence involving residue class degree and class number.

Let k be a normal extension and let e1, f1 and h1 denote the ramification index, residue class degree and the order in the divisor class group of a k-prime lying above the rational prime l. Then it is proved that for any two rational primes p and q, the congruence qfqhq = ± xePmod p has a solution. Though this result follows easily from Hilbert theory, it is apparently not known before, This is an improvement of an earlier result of the author (Abstract 73T·A165, these~ 20(1973), A-421), (Received April 29, 1974.)

*74T-A151. DAVID E. DOBBS and IRA J. PAPICK, Rutgers University, New Brunswick, New Jersey 08903. On going down for simple overrings, III.

Let R be an integral domain with quotient field K. Theorem 1, The following three conditions are equivalent: (a) R C R[u] satisfies going down (GD) for each u in K; (b) R C V satisfies GD for each valuation averring V of R; (c) R C S satisfies GD for each domain S containing R. If (d) is the condition obtained by

restricting the domains S in (c) to be overrings of R, then (a)=(d) has been proved in case R is: pseudo·B~zout (Dawson and Dobbs, "On going dow~ in polynomial rings", Canad. J, Math., to appear); Krull or integrally closed FC (Dobbs, Proc, Amer. Math. Soc, 39(1973), 515-519); Noetherian (Dobbs, "On going down for simple overrings. II"

A-476 Communications in Algebra, to appear). The proof of Theorem 1 uses results of the last-cited paper heavily. Also proved is Theorem 2. Assume that Spec (R), as a poset under inclusion, is a tree. If T is a domain containing R such that R C R[u, v] satisfies GD for each u and v in T, then R C T satisfies GD. (Received April 12, 1974.)

* 74T-A152. EDWARD A. BERTRAM, University of Hawaii, Honolulu, Hawaii 96822. A density theorem on the number of conjugacy classes in finite groups of given orders. Preliminary report.

Using the class equation, E. Landau (1903) showed that the number of (nonisomorphic) finite groups

G with a given number K of conjugacy classes is finite. P. Erdos and P. Tur:!n, and independently M. Newman,

showed (1968) how Landau's method gives K(G) > c log 2log2(jGj). When G is a p-group, P. Hall (and later

J. Poland) gave a parametric equation for K(G), from which it follows that if G is nilpotent then K(G) > log2 jGj;

however the latter inequality is not true for all solvable groups. Theorem. There exist constants c 1 > 0, and

c 2 : ~ < c 2 < log 2 such that for almost all integers g S n, as n -> oo, if G is a group of order g, then K( G) >

c 10og2n)c2. The proof makes use of the following, where v(m) denotes the number of distinct prime divisors of 4 m: Proposition 1. There exist constants c 3, c 4 > 0 such that if pjjGj, p a prime, then K(G) > c 32c v(p-ll.

Proposition 2. Given an arbitrarily small f > 0, almost all integers S n, as n-> ""• are divisible by at least one prime p S n which satisfies v(p- 1) > (1- t) log log(n). The latter follows from a theorem of Erdos (1935) on the normal number of prime factors of p- 1, and known sieve methods. (Received April 12, 1974.)

74T-A153. JACOB MATIJEVIC, University of Kentucky, Lexington, Kentucky 40506. The maximal ideal transform of a local ring. Preliminary report.

Let R be a commutative Noetherian ring with unique maximal ideal M. Let T be the set of all elements of the total quotient ring of R whose conductor to R contains a power of M. If A is any ring such that R C A C T, then A/xA is a finite R module for any nonzero divisor x of R. It follows that if, in addition, R has no nonzero nilpotent elements, then any ring A such that R C A C T is Noetherian. That T is Noetherian when R has no nonzero nilpotent elements was first observed by Ferrand and Raynaud in the case that T is integral over R. Flexor-Mangeney noticed the "Krull-Akizuki" property between a domain R and the ring T in case T is integral over R and the integral closure of R has a unique maximal ideal. These results can be extended to the case of Noetherian rings with more than one maximal ideal. (Received April 15, 1974.)

* 74T-A154. JOEL DAVID BERMAN, University of Illinois, Chicago, Illinois 60680, Distributive lattices with an additional unary operation.

Let K denote the class of algebras of the form < L; 1\, V, 0, 1, f >,where 1\, V, 0, 1 behave as in distributive lattices with 0 and 1, and f is a unary operation satisfying f(xl\y) = f(xVy), f(xVy) = f(x)/\f(y), f(O) = 1 and /(1) = o. Kp,q is the subclass of K in which r

* 74T-A155. EDWARD SPENCE, University of Glasgow, Glasgow Gl2 8QW, Scotland. Hadamard matrices from relative difference sets.

Using relative difference sets to construct supplementary difference sets, the following results are proved. (i) Let n and n - 2 both be prime powers. If n = 1 (mod 4) then there exists a Hadamard matrix of order 4n, while if n = 3 (mod 4) there exists a Hadamard matrix of order 8n. (ii) Let m and (m - 3);2 both be odd prime powers. Then there exists a Hadamard matrix of order 4m. The method of construction yields the following new orders less than 4000: 292, 356, 404, 436, 596, 772, 964, 1016, 1028, ll08, 1208, 1268, 1396, 1556, 1588, 1604, 1732, 1796, 1828, 1844, 2ll6, 2164, 2228, 2264, 2276, 2836, 3076, 3284, 3524, 3704, 3716. (Received April19, 1974.)

A-477 74T-A156. SAMI BERAHA, City University of New York, Queen's College, Flushing, New York 11367. Concerning the chromatic coefficients of the main Lewis form. Preliminary report.

Let "i.~ =0 c iun-i (c0 = 1) stand for the chromial and let Rm be the number of m·rings in the

corresponding map. For BL regularity (valence= 3, R 2 = 0, R3 = 0) it is known (Lewis) that c 1 = 0, c 2 = n and

(Bari) that c 3=- 2(n -1) + R 4• In the context of all the families of RL regular maps and their chromials known to this author, he finds that, but for a single easily explained exception, c i is an expression in n of degree 1 1 [i/2]. More exactly, for n- i large enough, c 2P = nP/p! + O(nP- ) and c 2p+ 1 =-krl' /(p- 1)! + O(nP- ), The value of k is a constant for all but two closely related families. The known values of k are 2/l, 1, 7/4, 2 when k is constant while otherwise k = 1 + 1/p. (More than one family have the same k, whether or not it is a constant.)

The value k = 2 pertains to the BL asymptotic and to the most important families, For R 4 = 0, c 4 = Y,(n - 2) (n + 9)- R 5. (Received April 19, 1974,) (Author introduced by Professor Gian-Carlo Rota.)

74T-Al57. E. M. WRIGHT, University of Aberdeen, Aberdeen, United Kingdom. The structure of thinly-edged large graphs. Preliminary report.

An (n, q) graph has n nodes and q edges, each pair of different nodes being joined by just one edge

or not joined. We take k0 fixed and suppose n and q large and q = o(n). Then Erdos and Renyi (Magyar Tud. Akad, Mat. Kutat6 Int. Kozl 5(1960), p. 36), showed that almost every labelled (n, q) graph is the union of disjoint trees and so contains no cycle. I prove that almost every unlabelled (n, q) graph consists of just one large connected component and a number of isolated nodes. This connected component is not a tree but contains

a cycle of length k for every k such that 3::: k::: k0 • (Received April 22, 1974.)

*74T-Al58. RONALD D. BAKER, Ohio State University, Columbus, Ohio 43210. Partitioning the planes of AG 2 m(2) into 2-designs.

A t-design SA (t, k, v) is an arrangement of v elements in blocks of k elements each such that every element subset is contained in exactly ,\ blocks. A t-design s, (t, k, v) is called t' -resolvable if the blocks can be partitioned into families such that each family is the block system of a s, (t', k, v), It is shown that the S /3, 4, 2 2m) design of planes in an even-dimensional affine space over the field of two elements is 2-resolvable. 2 Each S 1(2, 4, 2 m) given by the resolution is itself I -resolvable. As a corollary it is shown that every odd­ dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation. (Received April 22, 1974.) (Author introduced by Professor Richard M. Wilson.)

*74T-A159. TEMPLE HAROLD FAY, Hendrix College, Conway, Arkansas 72032. A note on when a regular category is exact.

This note slightly extends a theorem of Burgess and Caicedo (Abstract 72T-A253, these j(~ 19(1972), A-687), Theorem. Let C be a regular category (in the sense of Gri!let, Lecture Notes in Math., vol. 236, Springer-Verlag, 1971), Then the following are equivalent: (1) C is exact; i.e., every equivalence relation is a congruence relation. (2) If R and S are commuting congruences on X such that RnS = l'lx, f = Quot R and g = Quot S, then the pushout TJ/ = (;g is also a pullback. (3) If R and S are commuting congruences on X, h: Quot RnS ... Quot R and k: Quot RIIS ... Quot S are the induced morphisms, then the pushout ph= ..\k is also a pullback. Burgess and Caicedo have shown the equivalence of (1) and (2), We show the equivalence of (2) and (3). (Received April 25, 1974,)

* 74T -Al60, DANIEL ZWILLINGER, 5700 Arlington Avenue, Bronx, New York 10471, On those even numbers that can be expressed as the sum of two prime numbers of a certain form.

Historically, mathematicians have sought to prove Goldbach's conjecture. Not yet being able to do so,

A-478 they have investigated those even numbers that can be expressed as the sum of two prime numbers. An immediate consequence of this is the case where only certain types of primes are chosen to be in the decomposition. This paper treats the case where the primes are all "twin" primes. {Feceived April 25, 1974.)

*74T-Al61. NICKOLAS HEEREMA and DAVIDS. TUCKER, Florida State University, Tallahassee, Florida 32306. Structure of modular field extensions. Preliminary report.

Let K 2 lc be fields of characteristic p f, 0. K is said to be a modular extension of lc if Kpn and k are linearly disjoint over Kpn() k for all n > 0. Sweedler [Ann. of Math (2) 87(1968), 401-410] characterized modular extensions for the case K/k purely inseparable of finite exponent by showing that K/ k is modular iff K is a tensor product over lc of simple extensions of k. The authors provide characterizations for certain cases in which K/lc is not purely inseparable. Theorem 1. If K/lc is algebraic such that for some positive integer r,

k(KP')/k is separable algebraic, then K/k is modular iff K = S ® le M, where S / k is separable algebraic and M/k is purely inseparable modular. An extension H of a field h is said to be a regular extension of h if H/h is separable and h is algebraically closed in H. Theorem 2. Let K/k be finitely generated. K lk is modular iff K = (S®,~,M)® 5 R, where S/k is separable algebraic, M/k is purely inseparable, modular, and R/S is regular. Equivalently, K/k is modular iff K = M ®,. R, where M/k is purely inseparable, modular, and R/k is separable. Additional results related to fields of constants of sets of finite (infinite) higher derivations were also obtained. (Received April 29, 1974.)

*74T-A162. ALEXANDER ROSA, McMaster University, Hamilton, Ontario L8S 4K1, Canada. A theorem on the maximum number of disjoint Steiner triple systems.

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. It is proved that D(2v + 1) ~ v + 1 + D(v) for v > 3. Since D(v) S v - 2 it follows that for v > 3, D(2v + 1) = 2v- 1 whenever D(v) = v- 2. Combining this with known results, one obtains that there are only three admissible values of v S 105 for which the exact value of D(v) has not yet been determined: v = 37, 85 and 97. For all other

admissible values of v (9 S v S 105), D(v) = v - 2. (Received May 3, 1974.)

74T-A163. DAVID ZEITLIN, 1650 Vincent Avenue North, Minneapolis, Minnesota 55411. A polynomial identity for the chromatic polynomials of a complete r-partite graph Kp p . . . p • 1' 2' I T

~ is a Stirling number of the 2nd kind. For r = 2, 3, · · ·, (*) F ,(P 1' • • • , p1 , t) =

IJ',:0 j! G~ (~) F 1 _ 1(p 1' ... , p1 _ 1' t-j), F 1(p, t) = tP (see Laskar and Hare, Abstract 709-A36, these ~ 1 20(1973), A-655). Theorem 1. For r= 2, 3,···, (**) F,(1 +P 1• p 2 ,···, P,. t)+F,(pl' 1 +P 2• p 3, ••• ,p,. l)+···+

F,(Pp···,P,_p 1+P,• t):tF,(p 1, p 2 , ••• ,p,, t)+(r-l)tF,(p 1, p 2, ••• ,p,. t-1). Remarks. Subscript r in F,, the polynomial of our title, is used to keep count of the number of parameters P;· Using (*),the proof of (**) is by mathematical induction. For r = 2, (**) gives the result 2. (H) (Abstract 74T-A71, these~ 21(1974), A-298, erratum, A-464). Set F (p , p , ••• , p, t) = "i1 C .ti, where d = "i' p .. Then Cd = 1, -Cd = U = T 1 2 T ] = 1 ] z= 1 1 - 1 T "i~-~ "i' L+ 1PLP·· For each j, j = 1, 2, · · ·, (' ), let P. :ppp

R combinations obtained by choosing 3 numbers from the r numbers p 1, p 2 , ••• ,p1 • Thus, Theorem 2. Cd_ 2 = (~r)- "if =;pj" (Received May 3, 1974.)

*74T-A164. DAVID W. BALLEW and MICHAEL SELZLER, South Dakota School of Mines and Technology, Rapid City, South Dakota 57701. Certain mixed power sums. Preliminary report.

It was proven by R. C. Vaughan (Ph.D. Thesis, London, 1969) that every large number was representable in terms of certain mixed power sums of integers. In particular, large numbers were representable as a sum of two squares, two cubes and two fourth powers. Numerical evidence is presented in this paper to indicate that considerable improvement should be possible in these theorems. Apparently, except for 23 the fourth powers do not appear to be necessary. Our calculation extends those reported in Dickson's "History of the

A-479 Theory of Numbers", Vols. I, II, p. 725. Like results are presented for sums of two squares and four cubes; two squares, one cube and three fourth powers; and one square, five cubes and one fourth power. (Received May 6, 1974.)

*74T-A165. CHARLES C. EDMUNDS, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Products of commutators as products of squares.

Let F be the countably generated free group ( x 1, x 2 ,. • ·; ¢), and denote the reduced words

x~x; ... x; and [x 1, x 2][x 3, x 4] ... [x 2n-l' x 2n], respectively, by s(n) and c(rz), where [x 1, x 2] is the commutator 1 1 x 1x 2x1 x; • Lyndon and Newman (Proc. Amer. Math. Soc. 39(1973), 267-272) have shown that c(1) is an endomorphic image of s(3) in F but not of s(2). This can be generalized as follows: 1·heorem. For each n > 0, c(n) is an endomorphic image of s(2n + 1) in F but not of s(2n). (Received May 6, 1974.)

*74T-A166. JOHN H. SCHULTZ, JR., 177 Aspinwall Avenue, Brookline, Massachusetts 02146. Some properties of the coefficients of the characteristic polynomial of an endomorphism. Preliminary report.

Elementary proofs are presented for the following results. Consider the characteristic polynomial, det(xl- A)= :Sk=O (-1)k • trk(A) • xn-k; then the map from the real n x n matrices M(n, R) to Rn taking A to

(tr 1(A), .•. , trn(A)) is open if M(n, R) and Rn are given their usual topologies. If norms are given, then any uniformly continuous, similarity invariant, homogeneous of degree one function from M(n, R) to R is a constant

multiple of the trace tr 1• As a simple example, the function (27 · det 113) from M(3, R) to R shows the necessiry of including the word "uniformly". Each trk is characterized in terms of similarity invariance, normalization at

the identity: f(I) = (k), and a strong "homogeneity" condition (cohomology): :S7= 1 /(D/c) •A) = (n- k + k • c)· f(A) for each c in R and each A in M(n, R). Here D;Cc) is the elementary diagonal matrix with {it) entry equal to c and all other diagonal entries equal to 1. This "homogeneity" condition of the minimal right ideals implies homogeneity of degree k. For the rationals Q with the subspace topology, the induced map defined above from M(2, Q) to Q2 is open. Recall the map while open when restricted to some proper "subvarieties" (Brouncker­ Wallis (1957)), fails for others (Diophantos (c. 250) · • · ). (Received April 29, 197 4.) (Author introduced by Professor David S. Browder.)

*74T-A167. EZRA BROWN, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. Diophantine equations of the form X2 + D = P".

We study equations of the form X2 + D = pn, where D and P are fixed integers. Among other results, we obtain Theorem 1. The only solution to the diophantine equation X2 + 3 = 7n is n = 1, x = 2. Theorem 2. The only solutions to the diophantine equation X2 + 11 = 3n are n = 3, x = 4, and at most four other values of n. In addition, we obtain some general results when P is a p.rime and Q 6/D) is a U.F.D. (Received May 13, 1974.)

74T-A168. MICHAEL SLATER, University of Bristol, Bristol BSB 1TW, England and University of British Columbia, Vancouver 8, British Columbia, Canada. Free alternative rings. Preliminary report.

Let L = L(F, X) be the free alternative algebra over the field F on the free generating set X. Let D

be its associator ideal, and I the ideal in L of identities of the split Cayley algebra over F. I. If IF\ = oo and \X\ 2: 3, then LA is a Cayley-Dickson (CD) ring without 0-divisors. It may be thought of as the generic CD ring

and as the free CD ring (on X). 2. If \F\ = oo or \XI = oo, then J(L/1) = 0 (] the ] acobson-Kleinfeld radical). In

any case all the classical radicals of L/1 coincide. 3. If \F\ = oo, the strongly semiprime radical of L = J(L) =

the set of nil elements of L = D n/ (.f, 0 if \X\ 2: 4). Hence, for IF\ = oo, 4. L,lf(L), the "free semiprime"

alternative algebra over F on X, is imbeddable in the direct sum of two division algebras; 5. If a 1 ••• an (in some association)= 0 in L, then there exist i, j such that the intersection of the principal ideals and is nil; 6. The word problem for LIJ(L) is recursively soluble. (For L itself this is still a major open question.) (Received May 13, 1974.)

A-480 *74T-Al69. DAVID HANDELMAN, McGill University, Montreal, Quebec H3C 3G1, Canada. When is the maximal ring of quotients projective? Preliminary report.

Let R be a ring with I, and Q its maximal ring of right quotients. If r € R, then a right insulator

for r is a finite subset of R, (r;)7=I• such that the right annihilator of (rri)7= 1 is zero. Then: If Q is projective as a right R-module, then Q is finitely generated; for R nonsingular, Q is projective as a right R-module if and only if R possesses an injective right ideal, of the form eR (where e 2 = e), such that e has a right insulator. For this e, R = Q if and only if e has a left insulator. As corollaries, if Q is projective and R is regular or commutative semiprime, then R = Q. A similar result is given for Q torsionless, and an example is constructed of a prime nonsingular ring R such that R f, Q but Q is a projective cyclic right R-module. (Received April 23, 1974.) (Author introduced by Mr. R. Michael Josephy.)

*74T-Al70. SIEMION FAJTLOWICZ and JURGEN SCHMIDT, University of Houston, Houston, Texas 77004, Join-congruences and algebraic closure families. Preliminary report.

Let S be a semilattice, s* the lattice of all ideals (including¢) of S, e(S) the lattice of congruences " of S, and 9*(S*) the lattice of algebraic closure families C on S contained in S* (with ¢ E C). The latter are exactly the topologically closed subalgebras of the compact bounded meet-semilattice F(S) = (S*; n, ¢, S, 5'), where ~· is the order-topology(= weak product topology) of P(S) restricted to S*. For I E S*, let " 1 = {(x, y) E SxSJx, y El or x, y /.I!. For "Ee(S). let " 1 =II ES*[" s; "r'· For c Ee*

*74T-A171. DAVID W. WALL, University of New Mexico, Albuquerque, New Mexico 87131. Conditions for rp(n) to properly divide (n - 1),

This research problem was suggested by Alter (Amer. Math. Monthly 80(1973), 192-193). (Here rp(n) is the Euler totient.) Lieuwens showed (Nieuw Arch. Wisk. (3) 18(1970), 165-169) that if each prime in n is at least 7, then n is a product of at least 13 primes, and that if p is a prime in n, then n has no prime of the form kp + 1. By considering the merits of using p in n or using several larger kp + 1 (e.g. using 5 or 11, 31, and 41 as divisors of n), we extend this result to show the Theorem. If every prime in n is at least 7, then n is a product of at least 26 primes. An unrelated result is Theorem. Let e(p) be the largest j such that 2; divides p - I, and let m be the minimum value of e(p) over all primes p in n. Then e(p) is equal to m for an even number of primes p in n, Also considered are the numbers of primes in n in the equivalence classes modulo p, where p is a prime in n. (Received May 16, 1974.) (Author introduced by Professor Abraham P. Hillman,)

74T-A172. KIANG-CHUEN YOUNG, McGill University, Montreal, Quebec, Canada. Some simple subgroups of the Rudvalis simple group. Preliminary report.

Using a computer, we have proved the following result: The Rudvalis simple group of order 145, 926, 144,000 = 2 14. 3 35 37.13.29 contains simple subgroups PSL (2, 29) and PSU (3, 5). The latter subgroup is not maximal. (Received May 17, 1974.)

* 74T-A173. ROBERT A. MEYER, University of Nebraska, Lincoln, Nebraska 68508, The number of solutions of a system of congruences.

In their research of admissible parameters for Steiner systems, Mesner and Kramer (personal communication) posed the following problem: For a fixed integer m?. 4, find the number of elements x in a complete residue system modulo m(m- 1)(m- 2) which simultaneously satisfies the three divisiblity conditions (m- 2)[(x- 2), (m- 1)(m- 2)[(x- 1)(x- 2), m(m- 1)(m- 2)[x(x- 1)(x- 2). Denoting the number of such elements

A-481 by N(m) and letting p' s and q' s represent odd primes, we develop a formula for N(m) in terms of the number of

prime divisors of m and m- I, We prove Theorem. (I) If (m, 4) = 1 so that m = q~ 1 q; 2 • •• q~u and m- I= l1p;2 ... p~v, then N(m) = 2v • 3". (II) If (m, 4) = 2 so that m = 2q~ 2 ••• q~u and m- I= p~1 ••• p~v, then 51 N(m) = 2v+1 • 3u- 1• (III) If (m, 4) = 4 so that m = 2 q;2 .. • q~u, where s 1 2: 2 and m- 1 = p~ 1 p; 2 ••• p~v, then N(m) = 2v+Z · 3"- 1• The author is currently investigating the natural generalization of this problem. (Received May I7, 1974.)

74T-AI74. DANIELS. KUBERT, Yale University, New Haven, Connecticut 06520. Torsion of elliptic curves.

Let K be a number field with at most one complex absolute value. Let N E z+ and let E be an elliptic curve defined over K. Let EN be the kernel of multiplication by N over K. EN is said to be N-deficient

if (N, Gal(EN·A<)) =I. Theorem. Given N E z+, N 2: 3, 3 mN E. R+ s.t. if E is an elliptic curve defined over K

with EN N-deficient, then \E10,(K)\ < mN' Moreover mN is effectively computable. Corollary. Given K as above and a prime number p, the universal bound on p-torsion of elliptic curves defined over K is effectively computable. (Received May 13, 197 4.) (Author introduced by Professor Serge Lang.)

* 74T-A175. HAGHDAD MEMAURI, Auburn University, Auburn, Alabama 36830. On restricted right subdirectly irreducible rings. Preliminary report.

M. G. Deshpande and V. K. Deshpande {Pacific J. Math, to appear) have raised the question: "In a r-RSI local ring with maximal right ideal M, does the condition M nil imply M nilpotent?" The following are partial answers to this question: Theorems: I. If R is a local ring such that every proper R-homomorphic image of it is a subdirectly irreducible R-module, and if the maximal right ideal of R is nil, then it is nilpotent. 2. If R is a r-RSI local ring with nil maximal right ideal M such that the index of nilpotency of each element of M is 2, then R is not prime, and hence M is nilpotent. 3. If R is a r-RSI local ring such that the intersection of all of its strictly right ideals is nonzero, then R is not prime and M is nilpotent. In the same paper the authors asked for an example of a primitive r-RSI ring which was not local. It can be shown that a primitive ring is local if and only if it is a division ring. Thus any primitive r-RSI ring which is not a division ring, such as a matrix ring over a division ring, serves as such an example. (Received May 17, 1974.) (Author introduced by Professor Charles C. Lindner.)

74T-AI76, F. THOMAS FARRELL, Pennsylvania State University, University Park, Pennsylvania 16802, Poincar~ duality and groups of type (FP). Preliminary report.

A group r is of type (FP) if the infinite cyclic group with the trivial r-structure has a resolution of finite length by finitely generated, projective Zr-modules. Theorem. Let r be a group of type (FP) and let k be a field. If Hi(r, kr) = 0 for i < n and Hn(r, kr) contains a nonzero sub-r-module whose k-dimension is finite, then Hi(r, kr) = 0 for i f. n and Hn(r, kr) has k-dimension 1. Corollary. Let r be a finitely presented group of type (FP). If Hi(r, Zr) = 0 for i < n and Hn(r, Zr) is a nonzero, finitely-generated abelian group, then r is a n-dimensional Poincare duality group. These results extend some unpublished observations of A. Borel and J-P. Serre. (Received May 20, 1974.)

74T-A177. GEORG]. RIEGER, Technische Universitat, D-3, Hanover, Federal Republic of Germany. On two arithmetical sums.

For a natural number n, let f(n): = IIP\n, p primeP· Theorem 1. There exists a constant C such that 1 1 for x > 3 we have I 2sns/n log f(n))- = log log x + C + O((log x)- log log x). The proof is elementary and compares the given sum with I 2snsx(n log n)- 1 • For a natural number n, let p(n) be the largest prime factor of

n. Theorem 2. For x > 3 we have I 2snsx (n log p(n))- 1 = e 'Y log log x + O(log log log x), where y denotes the Euler constant. The proof uses a well-known result on the numbers having small prime factors only (see, e.g., K. Prachar, "Primzahlverteilung," Springer-Verlag, Berlin, 1957, p. 158). The two problems studied were suggested orally by P. Erdos (January 21, 1974). Theorem 1 with a weaker remainder term and Theorem 2 as it stands were stated independently by P. G. Schmidt in a letter to the author. (Received May 20, I974.)

A-482 *74T-A178. PATRICK J. COSTELLO, Harvey Mudd College, Claremont, California 91711. Four new amicable pairs.

Two numbers m and n are amicable if a(m) = a(n) ,~ m + n, where a(x) denotes the sum of all the divisors of x. The smallest amicable pair, (220, 284), is of the form (apq, ar), where p, q, r are distinct primes not dividing the common factor a. 41 pairs of this form have been published. Using an approach similar to that

employed by Euler, r ~ (p + 1)(q + 1) -1, and p and q were determined to equal (d + (c- b))/(2b- c) and ((b 2/d) + (c- b));(2b- c), respectively, where a/a(a) = b/c, a fraction in its lowest terms and where d\b 2 but d ~b. 13 amicable pairs of this form were found by Euler with b :S 489 (Scripta Math., val. 12, pp. 61-72). Using Euler's method with b :S 3745, E. B. Escott found 8 more (Ibid.). A computer program was developed which generated amicable pairs where a was the product of powers of 3, 5, 7, 11, 13 and a prime < 1000. In addition to generating some of the pairs already published, 4 new pairs were discovered. These pairs were less than the largest known pair of this type. The new pairs are: (3 2.7.11 2.13 2.61.337.122693, 3 2.7.ll 2.132.61.41470571); (3 4.7 2.13.17 .5 3.83537, 3 4 • 7 2.13.17 .4511051); (3 2• 7 .ll 2.13.43.2837 .8513, 32.7.1 I 2.13.43.24162731); (3 3 .5.11 2 .43. 2837.8513, 33.5.11 2 .43.24162731). (Received May 20, 197 4.) (Author introduced by Professor Alvin M. White.)

* 74T-A179. GERALD A. HEUER, Concordia College, Moorhead, Minnesota 56560. Discrete ordered rin~s.

A fully ordered ring is discrete if the positive class has a least element; otherwise it is dense. This paper studies the embedding of a discrete ordered ring in a discrete ordered ring with unity; conditions for existence of a discrete full order; discrete orders in direct sums; nonisomorphic discrete orders for the same ring; embedding discrete ordered rings in dense ones, and vice versa; discrete subrings of ordered rings; rings with well-ordered positive class; Archimedean ordered rings; order in quadratic extensions of ordered rings. Special attention is given to integral domains. Sample results. A discrete ordered ring R is Archimedean = R is isomorphic to a subring of Z, or to Z with trivial multiplication = the positive class of R is well ordered. Every ordered ring

with unity and well-ordered positive class is isomorphic to Z. R 1 ED R 2 may be discrete orderable when neither Ri is. Every discrete ordered ring may be order-embedded in a dense ordered ring. Every ordered ring may be

order-embedded in a discrete ordered ring. If R is an ordered ring with 1, the order extends to S ~ R[x]/(x- a) 2, If R is discrete so is S. If r ~ s then R[x]/(x- r)(x- s) is not orderable. (Received May 21, 1974.)

74T-A180. JON FROEMKE, Oakland University, Rochester, Michigan 48063. Functional completeness of idempotent algebras. Preliminary report.

Suppose ('( = is a finite idempotent algebra with lA \ 2: 3. If the only automorphism )f V: is the identity, and if no proper suba1gebra of ~L contains more than one element, then V: is functionally complete. As a consequence, if 1J1 ~ with :BJ 2: 3 is a simple algebra with no proper subalgebras and no nontrivial automorphisms, then sa is primal if and only if the idempotent reduct of sa has only the identity as an automorphism and no proper subalgebra with more than one element. (Received May 28, 1974.)

* 74T-A181. J. B. SRIVASTAVA and VISHNU GUPTA, Indian Institute of Technology, Delhi-29, India. On semiprimary and restricted semiprimary group rings.

A ring R is said to be restricted semiprimary if for each two-sided ideal I~ (0) of R, R/I is a semi primary ring. We shall denote by R.S.P. and S. P. the restricted semi primary rings and semiprimary rings, respectively. Now let AG denote the discrete group ring of the group G over the ring A. In this paper we prove: Theorem 1. The group ring AG is semiprimary iff A is semiprimary and G is finite. Theorem 2. If AG is R.S.P. but not S.P., then A is simple Artinian, G contains no finite normal subgroup except 1, and G/H is finite for every normal subgroup H (,f. 1) of G. Corollary. AG is R.S.P. but not S.P. implies either the centre of G is trivial or G is an infinite cyclic group. Theorem 3. lf A is simple Artinian, G contains no finite normal subgroup

except 1, GIH is finite for every normal subgroup H ,f. 1, then AG is R.S.P. but not S.P. if !li ~ {g E: G: 1- g E: I! .f. 1 whenever I is a nonzero two-sided ideal of AG. (Received May 29, 1974.) (Authors introduced by Dr. S. K. Bajpai.)

A-483 * 74T-A182. KIM KI-HANG BUTLER, Alabama State University, Montgomery, Alabama 36101. On Hedrlin subsemigroups of B x·

Let X be an n-set. Let Bx be the semigroup of binary relations defined on X. Let E(S) be the set of all idempotents in a set S. For t E E(Bx), let hx(t) ={a. E Bx: O.t =a= tO.}. We call hx(t) the Hedrlin subsemigroup of B X generated by t. In this note we characterize such a subsemigroup. Let Y be an m-set. Then Bx = Bx,x· If a E Bx, y and x EX, then by xa. we mean {y E Y: (x, y) Ea.}. Similarly, if y E Y, then a.y means lx EX: (x, y) Ea.}. xa. (a.y) is called a row (column) of a.. By R(a.) (C(a.)) we mean the collection of all unions of rows (columns) of a.. An element a of Bx is said to be :R'f.-decomposable with respect to t E E(Bx) if there exist two binary relations (3 and y such that each column of (3 is contained in C(t) and each row of y is contained in R(t). Proposition 1. a E hx(t) iff C(a.) C C(t) and R( a.) C R(d. Theorem 2. a E hx(t) iff ·0. is !iti'-decomposable with respect to t. Remark. In general, hx(t) is not regular. (Received May 30, 1974.)

* 74T-Al83. KARL K. NORTON, 2235 Floral Drive, Boulder, Colorado 80302. On the number of restricted prime factors of an integer. II.

Let E be a nonempty set of primes. For each positive intel!;er n, let w(n; E) denote the number of distinct prime factors of n which lie in E. Let O(n; E) denote the total number of prime factors of n which lie in E, counted by multiplicity. Define E(x) = I.{p- 1: p ~ x, p E E}. Suppose g(n) = w(n; E) V n or g(n) = O(n; E) Vn. For real a., define A(x, a; E, g) to be the number of n ~ x with \g(n)- E(x)\ 2: a.E(x). Theorem 1. Suppose 1 2 E(z) -+ + oe as z -+ + 00• Let 0 < a~ (3 < 1. Then for x > x 1 ((3, E), we have c 1 ((3)xE(x )- 1 exp{Q(a.)E(x)J ~ A(x, a; E, g) S c 2((3)a. - 1xE(x)- 1 12 exp {Q(a.)E(x)l, where Q(a.) = a- (1 + a.) log (1 + a.). (c 1 ((3), c 2((3) are positive numbers depending only on (3.) Theorem 2. For each real (3 and x s.t. E(x) 2: 2, let N(x, (3; E, g) be the number of n ~ x with g(n) ~ E(x)+ (3E(x) 11 2 • Then x- 1N(x, (3; E, g)= (277)- 112f~.,. exp l-t2/2l dt + O(E(x)- 11 2), the implied constant being absolute. (This is best possible in the sense that O(E(x)- 11 2 ) cannot be replaced by o(E(x)- 11 2) uniformly in (3.) (Received May 30, 1974.)

*74T-A184. JOHN GILBERT PHILLIPS, University of Notre Dame, Notre Dame, Indiana 46556. Genera and modular decomposition numbers. Preliminary report.

Let G be a finite group, K a p-adic number field, and R the ring of integers in K. If I is an irreducible (right) K[G] module, then an R[G] lattice M is called an integral realization of I if M ® R K is isomorphic to I as a K[G] module. Main results. (1) If Q is a principal indecomposable R[G] module s.t. I is a nontrivial component of Q ® R K, then there is a surjective R[G) homomorphism ¢: Q-+ N s.t. N is an integral realization of I; (2) if Q', cp', and N' are as cf>, Q, and N in (1), and if Q and fj are R[G] inequivalent, then N and N' are R[G) inequivalent. One can then estimate the total number of inequivalent integral realizations of I by counting the inequivalent principal indecomposables of R[G] that satisfy the hypotheses of (1). If K is a splitting field for G, X is the character afforded by I, and ell is the set of Brauer characters of the modular irreducibles of G, then this estimate is given by the number of nonzero dx q, for cp E ell. As an application, Maranda's results (Canad. J. Math. 7(1955), 516-526) are used to consider the case when K is an algebraic number field. (Received May 31, 1974.)

*74T-A185. BRlAN A. DAVEY and IVAN RIVAL, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Finite sublattices of three-generated lattices.

Every lattice with three unordered generators contains a finite sublattice with three unordered generators. In fact, every lattice with three unordered generators contains a finite sublattice with three unordered generators isomorphic to one of twelve lattices. (Received May 31, 1974.)

A-484 *74T-A186. BRIAN A. DAVEY and IVAN RIVAL, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada and W. POGUNTKE, Technische Hochschule, D-61, Darmstadt, Federal Republic of Germany. A characterization of semidistributivity.

A lattice L is said to satisfy (SD V) if, for all a, b, c € L, a V b = a V c implies a V b = a V (b II c).

If L satisfies both (SD V) and its dual (SD 11 ), then it is called semidistributive. Theorem. A finite lattice is semidistributive if and only if it contains no sublattice isomorphic to one of six lattices. (Received June 3, 197 4.)

*74T-AI87. STEVEN M. ROMAN, University of Washington, Seattle, Washington 98195. A problem of Zarankiewicz.

Zarankiewicz (Colloq. Math. 2(1951), 301, Problem P 101) and others have posed the following problem: Determine the least positive integer k Jm, n) so that if a 0, !-matrix of size m by n contains ka (m, n) ones, ~p •13 then it must have an a by f3 submatrix consisting entirely of ones. This paper improves upon previously known 1 1 upper bounds for ka,f3(m, n) by proving that ka,im, n) S I+ (f3- I)(a! 1)- (';:) + (p + I)(a- l)a- n for each integer p 2: a - 1. Each of these inequalities is better than the others for a specific range of values of n. Equality is shown to hold infinitely often for each value of p. Finally, some applications of this result are made to arrangements of lines in the projective plane, generalizing a result of Canham (Israel J. Math. 9( 1969)) on the sizes of any m distinct polygonal regions in an arrangement of n lines. (Received May 28, 1974.)

*74T-Al88. RAYMOND C. HEITMANN, University of Wisconsin, Madison, Wisconsin 53706. Generating ideals in Prilfer domains. Preliminary report.

How many generators does a finitely generated ideal in a Prufer domain require? Sally and Vasconcelos ("Stable rings", preprint) have shown that two generators will suffice if the domain has Krull dimension one. Using certain properties of prime ideal chains in these domains, we have obtained the following generalization. Theorem.

If R is a Prufer domain of Krull dimension n, then every finitely-generated ideal of R can be generated by (n -t I) elements, the first of which may be selected arbitrarily. (Received June 6, 1974.)

74T-Al89. GARY F. BIRKENMEIER, University of Wisconsin, Milwaukee, Wisconsin 53201. A decomposition of self-injective rings. Preliminary report.

Let R be an associative ring with unity. A proper ideal B in R is semicompletely prime if for r € R

s.t. rn € B, for some integer n, then r € B. Theorem. If R is a right self-injective ring, then R = AEDB, where A is a right ideal of R which is a strongly regular self-injective ring with unity, and B is a two-sided ideal of R which is semicompletely prime, and B is the injective hull of the right ideal generated by the nilpotent elements of R. Also if A is a two-sided ideal, then B is a right self-injective ring with unity s.t. every nonzero two-sided ideal of B has a nonzero nilpotent element. Furthermore, these results extend to continuous rings (see Y. Utumi, "On continuous rings and self-injective rings", Trans. Amer. Math. Soc. 118(1965), 158-173). (Received June 6, 1974.) (Author introduced by Professor E. H. Feller.)

*74T-Al90. GERSON LEVIN, City University of New York, Brooklyn College, Brooklyn, New York 11210. A change of rings theorem for Tor.

Let R be local ring with maximal ideal m and residue field k = R/m. The Poincar.; series of R is the formal power series P(R) = ~':" B .xi, where B.= dimkTor .(k, k). Theorem I. There exists an integer r s.t. for 1 = 0 1 1 1 0 r 2: r0, P(R) represents a rational function iff P(R/m') does. Theorem 2. Let R be a local Gorenstein ring of dimension zero s.t. dimk m/m 2 S 3, and let I = (0: m) be its sod e. Then P(R/1) represents a rational function. (Received June 6, 1974.) (Author introduced by Professor Marvin J. Kohn.)

*74T-Al91. M. H. LIM, University of Malaya, Kuala Lumpur, Malaysia. Transformations on symmetry class of tensors. Preliminary report.

Let U be a finite-dimensional vector space over an algebraically closed field of characteristic zero. Let um(G) denote the symmetry class of tensors over U associated with a subgroup G of Sm and the character

A-485 identically 1. Let T be a linear mapping on um(G) which takes nonzero decomposable elements to nonzero decomposable elements. If dim U > m + 1, then T(x 1*· · ·*Xm) = fa( 1) (x CT( 1)) *" · *f o(m )(x a(m)) for some a in Sm and some nonsingular linear mappings f 1' · • • , f m on U s.t. a G = Ga and f; = f T( i) \1 i and Vr in G. The cases when G = lel and G = Sm were studied by Westwick [Pacific J. Math. 23(1967), 613-620] and Cummings [Pacific ]. Math. 42(1972), 603-613]. (Received June 10, 1974.) (Author introduced by Dr. Sydney Bulman·Fleming.)

Analysis

74T-B 137. J. P. SINGH, Kurukshetra University, Kurukshetra, India. Integrability of trigonometric series with coefficients satisfying certain conditions.

Let L(p, a) be the space of functions ¢(x) s.t. J::'\¢(x)\P(sin x)ap d:x: < oo, a real, p > 0, A sequence {an!' an~ 0 belongs to the class Aj if the sequence Iann-i}, for some j ~ 0, is decreasing, and to the class A- j if for some 7' > 0 the sequence Ia ni }is increasing. Consider the trigonometric series '(x) = !.00 a cos nx. Theorem. ' ' n I' n= 1 n Let lanl € Aj or lanl € A- j. If I:=: p <"" and -1 < ap < p- 1, then a necessary and sufficient condition that

L 11P(1/x)f(x) € L(p, a) is that !.;=1np-ap- 2 L(n)a~ < oo, The theorem also holds for sine series and generalizes certain results of AljanCic, Bojanic and Tomic (Acad. Serbe Sci. Publ, Inst. Math. 8(1955), 67-84), lgari (Tohoku Math. J. 12(1960), 139-146), Yong (Math. Z, 91(1966), 280-293), Askey and Wainger (Duke Math. J, 33(1966), 223-228) and Khan (Rend. Mat. 1(1968), 1-13). (Received February 21, 1974.)

74T-B138. MAXWELL 0. READE, University of Michigan, Ann Arbor, Michigan 48104. Generalizations of alpha convex functions. I. Preliminary report.

2 Let G(w) =A 1w + A 2w + .. · be an entire function such that Re w = 0 implies Re G(w) = 0, and let f(z) = z + · · · be analytic in the unit disc I'! such that (/'(z)f(z)/z) f, 0 holds there. For those functions we have the following results. (A) Let a be a real constant such that the inequality Re [(1- a)G(zf'(z)/f(z)) +aG(1 + z('(z)ff'(z))] > 0 holds in l'l. Then f(z) is starlike and univalent in 1'!. (B) If, in addition, Re w > 0 implies that Re G(w) > 0, then the f(z) in (A) is convex and univalent when a ~ 1 there. (C) Moreover, if for some real a, the inequality Re ([G(zf'(z)/f(z))] 1-,G(1 + z('(z)/f'(z))]a) > 0 holds in 1'!, then f(z) is starlike and univalent in/+,., These results extend earlier ones due to Miller, Mocanu and Reade [Rev. Roumaine Math. Pures Appl. 17(1972), 1395- 1397 and Proc. Amer, Math. Soc. 37(1973), 553-554] and Lewandowski, Miller, and Zlotkiewicz ["On gamma starlike functions", submitted for publication]. (Received March 25, 1974.)

* 74T-Bl39. JAIRO IVAN ALVAREZ, University of Valle, Cali, Colombia. Linear functionals and local measures. The classical version of the Riesz-representation theorem is proved in the context of locally compact spaces, the local compactness playing a central role. This paper establishes a natural extension of this theorem in the context of metric spaces or, more generally, in the context of paracompact spaces. (Received March 29, 1974.)

*74T-Bl40, LEONARD GROSS, Cornell University, Ithaca, New York 14850. The free Euclidean Proca and electromagnetic fields.

If V is a finite-dimensional vector space, m ~ 0, and f is a V-valued tempered distribution on R 4, put 1\/1\.:. 1 = JR4\f(k)\ 2(m 2 + \k\ 2)- 1d 4k, where f is the Fourier transform of f. Theorem 1, Let m > 0, Denote by 2 2 Km the Hilbert space of tempered distributions f on R 4 with values in R 4®C s.t. \\f\1 = \\/\\~ 1 + m- 1\div /1\: 1

< ""· If f is in Km and is supported in the hyperplane x 4 = 0, then the fourth component of f necessarily vanishes and f = g®8(x4) for some function g:R 3 -> R 3®C. The subspace K~ consisting of such f is dual to the one·

A-486 particle positive-energy Hilbert space for a free Proca field (i.e. spin 1, mass m) in the pairing <¢, /> = J 3 I.? ¢ .(x)g .(x)dx. Theorem 2 (Maxwell field). Theorem 1 remains valid for the Maxwell R 1=1 1 1 field (m = 0) if the Euclidean norm is taken to be \1/\\ = \\ /\\_ 1 on the space of (R 4 ®C)-valued distributions for which div f = 0, Thus gauge invariance problems for the free electromagnetic field seem to disappear in the Euclidean setting. The analog 2 2 of Theorem 1 holds for higher integer spin s if one uses for the Euclidean norm 1\/1\ = I.j=0m- js!(j!(s-j)f)-l 1\divj/1\:_ 1, where divj/ is the symmetric tensor field of rank s- j naturally associated to the rank s tensor field f. The higher (integer) spin analog of Theorem 2 also holds. (Received April 1, 1974.)

*74T-B141. SHY AM K. BAJPAI, Indian Institute of Technology, Delhi 29, India. Influence of %('(0) on the a-convexity of normalised starlike analytic functions j(z) of order {3.

Let S a(p, f3) denote the class of normalised analytic functions for which any f in S a(p, {3) has Taylor series expansion f(z) = z + pz 2+... in the unit disc D{\z\ < 1} and satisfies (in D)

Re{ a(1 1- zf"(z)/((z)) + (1- a)zj'(z)/f(z)l > (3, 0::: f3 < 1, Theorem. If f E S0 (p, {3), then f E S a(p, f3) in \z\ < R, where R denotes the smallest positive root of (1) (1- (3) 2- p(1- {3)(a- a(3- 2(3}r + (p 2(3 2 - 2(1- (3) 2(1- 2(3 + 2a- 2a(3)}r 2 +(a+ 2{3- a(3- 4{3 2)(1 - {3)pr 3 + (1 - (3) 2(1- 2{3) 2r 4 = 0 if 0::: a and 0:::: (3::: %, (2) 1 + (2 + a)pr + (2 + 4a- 4{3 + p 2 - 4a{3)r2 +(a+ 2- 4{3)pr 3 + (1 - 2f3} 2r 4 = 0 if a:::- 3. All the results are best possible. Complementary cases remain open. (Received April 1, I974,)

74T-BI42, CARL P. McCARTY, La Salle College, Philadelphia, Pennsylvania I9I4I, Some classes of analytic functions. Preliminary report,

Let P(s, t) = {p(z) =(I - sw(z))/(I - tw(z)): w(O) = 0, \ w(z)\

74T-Fl143. HSIUNG TANG, University of Kentucky, Lexington, Kentucky 40506, Summability of Fourier-Bessel series.

Let I•\ I be the sequence of successive zeros of a Bessel function J11 (x) for p 2: 0, Moore showed (Trans. Amer, Math, Soc. 10(1909), 391-435) that if 0 S x::: 1, sin [Anx- a]= sin [(mr + q)x- a]+ t/J(\)x/n, where t/J(An) is bounded, a and q are constant. We extend this result. Theorems. 1, Let An be a sequence of nonincreasing real numbers bounded below, then I.;=IAn] 11 (Xnx) = 0(1/x) as x-+ + 0 for p 2: 0, 2, Let f(x) E L(O, 1) and J~\f(u + x)(u + x)~- f(u- x)(u- x)!--2- C\ du = 0(1) as t-> 0, Let f(x) be periodic with period 1, and let x E (0, I) and C be a constant which may depend on x. Then (1/Z)I-~= 1 !:=1 AJ11 (Anx) = 0(1) as l-+ .. and p =%,where the An's are the Fourier-Bessel coefficients of f(x). 3. Let the Fourier-Bessel series

I;=I An] 11 (.\nx) converge absolutely in a set E, m(E) > 0, Then (i) an/A~-+ 0 as n-+ oe, (ii) I;:= I \An\ < oo, The converse of (ii) also holds. (Received April 11, 1974,) (Author introduced by Professor Swarupchand M. Shah,)

* 74T-B144. RICHARD S. ELLIS and '-IAF.K A. PINSKY, Northwestern University, Evanston, Illinois 60201. Limit theorems for the linearized Boltzmann and Navier-Stolees equations.

Consider the initial value problem for the linearized Boltzmann equation in RN: apjat + ( ·grad p = £- 1Qp, where Q is the linearized collision operator corresponding to a spherically symmetric hard potential and £ > 0, The solution can be expressed by a contraction semigroup p = T/t)f on the Hilbert space H = L 2(R 2N; (2rr)-N 12exp (-\(\ 2 /2)dx d(), Let n = N((t)f be the semigroup solution of the linearized Navier-Stokes equations anjat =An + £Bn; A and B are matrices whose entries are, respectively, first-order and second-order

spatial derivatives. E(t) = N0 (t) denotes the semigroup corresponding to the linearized inviscid Euler equations.

A-487 Theorem 1, For f d{, lim(_,0 T/t)f = E(t)f, lim(_,0 E(-t/E}TE(t/E){ = N(t)f. N(t) is a semigroup which arises from

N/t) as follows. Theorem 2, For f E H, limf->0 E(-t/dN/t/E){ = N(t)f. The semigroup N(t) corresponds to the parabolic system an/at= (17AB)n. The operator 11AB is a matrix whose entries are second-order spatial derivatives and which can be characterized as the orthogonal projection of B onto the class of matrices which commute with A. If in Theorem 1, le-1 2/ E }( and a 2{jax .ax. E H, then the speed of convergence is O(E), strengthening a result of l 1 Grad. The proof depends on a new spectral theory for the linearized Boltzmann equation. (Received April 12, 1974.)

* 74T-B145. GRAH,4ME BENNETT, Indiana University, Bloomington, Indiana 47401. Unconditional convergence and almost everywhere convergence. Preliminary report,

Let p, q be fixed real numbers satisfying 1 ~ p ~ ""• 1 ~ q ~ ... , and let (fi)~= 1 be a sequence of elements of Lq[O, 1), If (J~I~jd a//t)lq dt)llq ~ (~j= 1 la;IP) 11P for each positive integer n and each sequence (ai)~ =1 of scalars, then (f~ sup,.l~j=l a//t)/log (j + Olq dt)llq ~ K(~~=tlal) 11P, where K is a universal constant, When p = q = 2, this gives a strengthening of the fundamental Mensov-Rademacher theorem on the a.e. convergence of a series of orthogonal functions. When p = ""• q = 1, this answers affirmatively a question of Kwapien and Pelczynski [Problem 3, Studia Math. 3"4(1970), 43-68) concerning unconditionally convergent series in L 1[0, 1], (Received April 15, 1974,)

74T-B146, STANLEY J. POREDA, Clark University, Worcester, Massachusetts 01610, A negative result on the continuity of best approximations.

A complex-valued function f not identically zero is said to be in B, the set of badly approximable functions, if f is continuous on U = {lz I = 1! and if its polynomials of best uniform approximation on U are all

identically zero. Also for a function g defined on U let 11811 denote the uniform norm of g on U. For each f E B and 0 < f3 ~ 1, let 8 Jf, n) = inf {,8/liP n Ill where the inf is taken over aU polynomials p n of degree n and of suitably small norm, say IIP,.II ~II/II (for otherwise the inf is always = 0 if f3 < 1), It is known that

8 J!. n) > 0 if 0 < f3 ~ %.· Theorem. lim,._,.,.8 J!. n) = 0 V f E B, and 0 < f3 ~ 1. A question which still appears to be open is whether 8 Jf. n) > 0 for X < p ~ 1 (for fixed f and n). It is easy to see that 8 Jf. 0) > 0, (Received April 15, 1974.)

*74T-B147. LEONARD Y. H. YAP, University of Singapore, Republic of Singapore, Nonfactorization of functions in Banach subspaces of L 1(G), Preliminary report.

We obtain a general theorem which shows the impossibility of factoring functions in certain Banach subspaces of L 1(G), where G is a nondiscrete Iocaily compact Abelian group. One of the corollaries of our theorem states that the ideal in AP(G) generated by Aq(G) * AP(G) is a proper subset of AP(G), where 1 ~ p < q <.,. and AP(G) = {f: f E L 1(G) and f E LP(G)!. This answers a question of R. Larsen ["The algebras of functions with Fourier transforms in LP: A survey", preprint, University of Oslo, 1973), (Received April17, 1974.)

*74T-B148, RAJENDRA PRASAD SINHA, Purdue University, West Lafayette, Indiana 47907, Hardy transformation on subspaces of L "". Preliminary report,

Let f,f E L and f rv ~;=Ian cos nx. It is known that ~;= 1 ((a 1 +•. · + a,.)/n) sin nx will be the Fourier expansion of some function Tf. Kinukawa and Igari (Tohoku Math. J. (2) 13(1961), 274-280) proved that

f E L ""=Tf E L "". A. A. Konyuskov (Izv. Akad. Nauk SSSR. Ser. Mat, 21(1957), 423-448) showed that for 1 ~ p <

oo, a< 1/p, f E Lip (a, p)=Tf E Lip (a, p). We show that (i) f E C and {(0) = O=Tf E C, (ii) f E Lip (a, p),

1 < p < ""• a> 1/p and f(O) = O=Tf E Lip (a, p), (iii) f E Lip (1/p, p)nL "", 1 ~ p < oo=Tf E Lip (1/p, p). (Received April 22, 1974,)

* 74T-B149. CHANDRA \JOHAN JOSHI, University of Jodhpur, Jodhpur (Raj.), India and M. L. PRAJAPAT, Defence Laboratory, P, B. No. 136, Jodhpur (Raj.), India 342001, Triple integral transformations of certain generalized hypergeometric functions. Preliminary report.

The authors consider a generalization of the various known integral transformations that provide

A-488 interesting operational techniques for augumenting the parameters in the generalized hypergeometric function PF q" In the sequel, several applications of the generalized operator that is introduced are made in the derivation of various properties pertaining to classical polynomials and special functions of single and two variables. (Received April I8, I974.)

74T-Bl50. ~AATTS R. ESS~N, Royal Institute of Technology, Stockholm (70), Sweden and FRANK R. KEOGH, University of Kentucky, Lexington, Kentucky 40506. The Schwarzian derivative and estimates of functions analytic on the unit disc.

Let P denote the class of those functions p: [0, I)-> (0, oe) satisfying the conditions: (i) the problem y" + py = 0, y(O) = 1, y'(O) = 0 has a nonnegative, nonincreasing solution in [0, I); (ii) p is the restriction to

[0, I) of a function p 1 analytic and satisfying IP 1(z)l S p 1(izi) for lzl

* 74T-BI5I. ST-JY A~ JOHARI, University of Illinois, Chicago, Illinois 60680. Asymptotic behavior of the solution to a class of nonlinear Volterra integral equations.

The asymptotic behavior of the solution to the nonlinear Volterra integral equation x(t) = f(t) - f~h(t- s)xn(s) ds. t::: 0, n::: 1, is investigated in the limits t _, + oo and t-> o+. The technique developed is applied to equations arising in the determination of the temperature in a nonlinearly radiating semi-infinite solid in nuclear reactor dynamics and in population dynamics. For n = I, the above equation occurs in renewal theory. (Received April 26, 1974.)

74T-B152. JOHN W. BUNCE and JAMES A. DEDDENS, University of Kansas, Lawrence, Kansas 66045. c*-algebras with Hausdorff spectrum. Preliminary report.

Let A be a bounded operator on separable Hilbert space, and c*(A) the C*algebra generated by A and the identity. John Ernest has asked for a characterization of operators A such that C*(A) has Hausdorff spectrum C*(A) ~. We call an operator pure n-normal if A is n-normal but no direct summand is k-normal for any k less than n. Theorems: 1. If c*(A) ~ is Hausdorff, then every irreducible representation of C*(A) is finite dimensional, and A is an infinite direct sum of operators An with each An pure k-normal for some k, and each C*(A ) ~ Hausdorff. 2. If A is a finite direct sum of operators A ., then n z C*(A) ~ is Hausdorff iff each C*(A;) ~ is Hausdorff. 3. If A is pure n-normal, then c*(A) ~ is Hausdorff iff each p(A) is a direct sum of unitarily equivalent irreducible matrices, where pis defined as in Proposition 2 of [Bunce and Deddens, Trans. Amer. Math. Soc. I71(I972), 30I-307]. The above results characterize when c*(A) ~is Hausdorff in the case that c*(A) has bounded representation dimension. The same methods characterize when separable c* -algebras with identity and bounded representation dimension have Hausdorff spectrum. We have some results when C*(A) does not have bounded representation dimension. (Received April 29, 197 4.)

74T-B153. ERWIN 0. KREYSZIG, University of Windsor, Windsor, Ontario, Canada. Explicit representation of an integral operator.

Let if>, t/J E c"'(O), 0 En c C, be such that y E c"'(n X 0), where y(z, z*) = A.cp'(z)t/J'(z*)/(if>(z) + t/J(z*))2, andy# 0. The equation (I) Lu = 0, L = a2/azaz* + y has recently been considered by various authors, in

connection with integral and differential operators T: V(O)-> S11 (L), where V(O) is the vector space of all holomorphic functions on n, and sll (L) is the space of C"'-solutions of (1) on n X n. The existence of such an integral operator follows from S. Bergman's general theory. To obtain its explicit form needed in applications, it is shown that the representation of the kernel of T as a series in powers of the variable of integration has

coefficient functions of the form c 1'- = "i~= 1 h!'-"(¢ + t/J )-v, where the h JJ.V, s are explicitly given linear combinations of products of derivatives of ¢'. (Received April 29, I974.)

A-489 *74T-B154. JAMES N. HAGLER, Catholic University of America, Washington, D. C. 20017, On the structure of S and C(S) for S dyadic.

A dyadic space S is a continuous image of a generalized Cantor set {0, 1}n for some cardinal number n. Theorem. Let S be a dyadic space and m a cardinal number which is not the sum of a countable number of smaller cardinals. Then the following are equivalent: (i) The topological weight of S is 2: m. (ii) C(S) contains a subspace isomorphic to ! 1(r) for card (r) = m. (iii) C(S)* contains a subspace isometrically isomorphic to

L 1{0, l}m. (iv) If ¢: {0, l}n -> S is continuous and onto, then there exists a subset n of {0, l}n, homeomorphic to {0, 1}m, such that ¢1 0 is a homeomorphism. (v) C(S) contains a subspace isometrically isomorphic to C{O, l}m which is the range of a projection in C(S) of norm one. (Received May 2, 1974.)

* 74T-B155. PHILIP J. BOLAND, University College of Dublin, Belfield, Dublin 4, Ireland. Holomorphir functions on nuclear spaces.

Let E be any quasi-complete complex nuclear space, and E' the set of continuous linear forms on E. E' is endowed with the Mackey topology (uniform convergence on convex balanced compact subsets of E). H(E) is the set of all complex valued holomorphic functions on E (/: E _, C is holomorphic if it is continuous and Gateaux analytic) endowed with the compact open topology. Theorem 1. H(E) is nuclear if and only if E' is nuclear. Corollary. If E is the strong dual of a Frechet nuclear space, then H(E) is a Frechet nuclear space. Theorem 2. If E is the strong dual of a Frechet nuclear space and F is a closed subspace of E, then any f E H(F) may be extended holomorphically to all of E. Moreover the restriction mapping H(E) _, H(F) is a surjective strict morphism. (Received May 3, 1974.)

74T-B156. LEONARD SARASON, University of Washington, Seattle, Washington 98195. Hyperbolic mixed problems in a quarter space. Preliminary report.

Let L =a/at- ~f=lA/x, t)a/axi be a real strictly hyperbolic system with the A/s symmetric k x k matrices. In the region G = R! x Rn- 2 x R +• we consider the boundary value problem (*) Lu = f in G with strictly dissipative boundary condition P/x, t)u = gi on aGn{(x, t): xi= 0}, j = 1, 2, and vanishing initial data

u(x, 0) = 0. Theorem. Under small perturbations of the A/s, strong solutions of (*) continue to satisfy an L 2 inequality of the usual sort. The proof hinges on the construction of a symmetrizer r(x, t; .f, r) which is independent

of .f 1 and of .f 2 when .f 3 = .f 4 = ... = .fn = r = 0, and such that Re(r + r') is strictly and uniformly positive. (Received May 6, 1974.)

*74T-B157. HAROLD S. SHAPIRO, Royal Institute of Technology, Stockholm 70, Sweden. Measure-invariant sets and the spectra of indicator functions.

Let u be an integrable function or measure on the circle T with Fourier coefficients {;;(n)}:=-oo' By P(u) we denote the set of positive integers n such that ;;(n) = 0. Theorem 1. Let S be any finite subset of the positive integers z+. There exists a measurable set E C T whose characteristic (=indicator) function 1 E satisfies P(l E)= S. In fact, E can be taken to be a finite union of intervals. The theorem is a consequence of Theorem 2. Let S be any finite subset of z+. There is a measure fl which is a finite sum of unit masses placed at distinct points of T such that P(fL) = S. Theorem 1 is relevant to some questions about invariant sets recently studied by several probabilists. (Received May 6, 1974.)

74T-B158. FRANK J. PAPP, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada. The functional equation f(xy) = ff(x)xyff(y) in a partially ordered monoid. Preliminary report.

Let M be a (not necessarily commutative) monoid with identity denoted by 1. A quasi-order < is defined on M by y :S x iff there is an element z of M such that x = yz. It is easily seen that if f satisfies the

functional equation of the title, is surjective, /(1) = l, and M is commutative, then idempotency of the binary operation of M (i.e. xx = x for all x in M) is a sufficient condition that the quasi-order :S is in fact a partial-

A-490 order. It is shown by means of three examples that :::; may be a partial-order whether or not a function satisfying the functional equation of the title and f(l) = 1 is surjective, and also that the condition that :S be a partial-order is independent of the surjecti'vity of such an f. Finally, idempotency of the binary operation is guaranteed by the following theorem which serves as a "partial" converse of the preceding. Theorem. If M is a monoid, f satisfies the functional equation f(xy) = ff(x)xyff(y), f(l) = 1, f is surjective, and :S is a partial-order, then the binary operation of M is idempotent. (Received May 6, 1974.)

* 74T-B159. MARTIN SCREOITER, Belfer Graduate School of Science, Yeshiva University, New York, New York 10033. Estimates for multiplication operators on LP.

For q(x) a measurable function on En, a, 5 > 0 and 1 ~ p < .. , put M a.,p, 8(q) = sup,Jix-yl

a> n. Put Ma.,p(q) = Ma,p, 1(q), and let Ma.,p be the set of those q such that Ma,p(q) < oe, Let 11 denote the Laplacian in En. Theorem. If 1 ~ p ~ 2, s > 0, n 2: 3, and q E Msp,p• then q(c 2 - /1)- 2/s is a bounded operator

on LP(En) with norm ~ const. (1 + (cB)-n)Msp,p, 8(q). (Received May 6, 1974.)

* 74T-B160. R. S. DAHIY A, Iowa State University, Ames, Iowa 50010 and BHAGAT SINGH, University of Wisconsin Center, Manitowoc, Wisconsin 54220. On the oscillation of a second order delay equation.

The oscillatory nature of two equations (r(t)y'(t))' + p 1(t)y(t) = /(t), (r(t)y'(t))' + p 2(t)y(t- r(t)) = 0 is

compared when positive functions p 1 and p 2 are not "too close" or "too far apart". Our techniques are different than Keener's (Applicable Anal. 1(1971), 57-63) and extend some of his resuits to delay equations. Another main result states that if h(t) is eventually negative and a twice continuously differentiable function which satisfies

(r(t)h'(t))' + p 1(t)h(t) 2: 0, then this inequality is necessary and sufficient for every bounded solution of

(r(t)y'(t))' + p 2(t)y(t- r(t}} = 0 to be nonoscillatory. (Received May 13, 1974.)

74T-B161. LEE A. RUBEL and W. S. McVOY, University of Illinois, Urbana, Illinois 61801. H00 is coherent. Preliminary report.

00 Let (/ 1, • • ·, fn) be an n-ruple of functions in H • Then the module of relations R = {(g 1 ,. • ·, gn):

00 gj E H : '!.g/i = 0} is finitely-generated-indeed a set of :S n generators can be found. As a corollary we prove that if I and J are ideals in H00 generated by m and n elements, respectively, then a set of fewer than m + n generators for In] can be found. By using a suggestion of Helson, the main result is proved by proving and applying a variant of the Beurling-Lax theorem on shift-invariant subspaces. We prove that given a weak-star closed invariant subspace lTl of H';;, where M is a finite-dimensional Hilbert space, there exists a finite-dimensional Hilbert space N with dim N :S dim M, and a measurable analytic function U on the unit circle, whose values are isometries from N into M, such that U • H';j =ln. (Received May 17, 1974.)

*74T-B162. H. M. SRIVASTAVA, University of Victoria, Victoria, British Columbia VBW 2Y2, Canada and ]. L. LAVOIE, Universite Laval, Quebec City, Quebec G1K 7P4, Canada. A certain method of obtaining bilateral generating functions.

This paper presents a systematic introduction to and several applications of a certain method of obtaining bilinear or bilateral generating relations for a large variety of sequences of special functions. The main result, stated as Theorem 1 in this paper, is shown to apply, for instance, to the Bessel, Brafman, Charlier, Gegenbauer (or ultraspherical), Gould-Hopper, Jacobi, Hermite, Konhauser, Laguerre (or the modified Laguerre) and Srivastava-Singhal polynomials, while its generalization, given by Theorem 2, would apply to the Lauricella polynomials in several complex variables and to the familiar Lagrange polynomials which arise in certain problems in statistics. It is also shown how these results can be extended to yield bilateral generating relations for such other special functions as the Bessel functions. (Received May 21, 197 4.)

A-491 * 74T-BI63. JINFU FENG and THOMAS H. MacGREGOR, State University of New York, Albany, New York 12222. Estimates on integral means of the derivatives of univalent functions. Preliminary report.

Let S denote the set of analytic, univalent functions in the open unit disk so that /(0) = 0 and /'(0) = I. We find upper bounds on the integrals (I/27r)f5" lf'nl(re;eW'' dO, where f varies over S or a subclass of S. In Particular if A> 2/5 and f € S the bound has the form D [I - r] -(n+2) X +·l, which is the best order estimate for ' ' n,A each n. Similar estimates are obtained for the convex and close-to-convex functions in S. These bounds improve those obtained for S and have a larger range of A for which they produce the best orders. In particular, the bound mentioned above for S holds for close-to-convex functions if A> I/3 (n = I, 2, • · • ). (Received May :3:1, 1974.)

*74T-B164. CARL DAVID MINDA, University of Cincinnati, Cincinnati, Ohio 45221. Rings of holomorphic and meromorphic functions on subsets o/ Riemann surfaces.

Let X and Y be nonempty subsets of noncompact Riemann surfaces R and S, respectively. Suppose H(X), H(Y) is the ring of holomorphic functions on X, Y. Every C-algebra homomorphism of H(X)-> H(Y) is of the form /-> f 0 ¢ for all / € H(X), where ¢: Y ->X is an analytic function. A similar theorem holds for C-algebra conjugate homomorphisms. These results are used to obtain a representation for R-algebra homomorphisms. In case no component of Y reduces to a single point, a ring homomorphism of ll(X) into H(Y) satisfying certain natural conditions must either fix all complex numbers or conjugate them. Finally, if X and Y are connected and M(X), M(Y) is the field of meromorphic functions on X, Y, then analogous results hold for field homomorphisms of M(X) into M(Y). These results extend theorems of Bers, "Iss'sa", Kra, Nakai and Royden which hold when R =X and S = Y, and a theorem of Su in case R = C = S. (Received May 23, 1974.)

*74T-BI65. H. M. SRIVASTAVA, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada and REKHA PANDA, University of Victoria, Victoria, British Columbia VBW 2Y2, Canada and Ravenshaw College, Cuttack 3, Orissa, India. Some expansions of hypergeometric functions in series of hypergeometric functions. Preliminary report.

The present note derives a class of expansion formulas for hypergeometric functions in series of certain products of geneEalized hypergeometric polynomials. The various results obtained here would provide generalizations, for instance, of the recent works by Wimp and Luke [Rend. Circ. Mat. Palermo Ser. (2) ll(I962), 351-366], and others [cf., e.g., Math. Comp. I9(1965), 664-666 and Glasnik Mat. Ser. Ill 6(26) (1971), 253-264]. The possibility of several further extensions is also indicated briefly. (Received May 28, I97 4.)

74T-B166. ROBERT F. OLIN, Indiana University, Bloomington, Indiana 4740I. Functional relationships between a subnormal and its normal extension. Preliminary report.

If K is a compact set in the plane, set H(K) = 1/: f analytic on some open set G '::>K}. The uniform closures of the polynomials and rational functions with poles off K are denoted by P(K) and R(K), respectively. Let S be a subnormal operator on a Hilbert space L with its m.n.e. (minimal normal extension) N on K. Theorems: 1. If F € H(a(S)) and f ,f. constant on the components of G, then f(N) on K is the m.n.e. of f(S). Therefore given any f in H(a(S)), the m.n.e. of f(S) can be calculated. 2. If f € H(a(S)), f f. constant and /(S) is normal, then S is normal. 3. If f € P(a(S)), and f f. constant on the components of the interior of the polynomial convex hull of a(S), then f(N) is the m. n.e. of f(S). If, in addition, the nontrivial Gleason parts of R(a(S)) are the components of the interior of a(S), the m.n.e. of f(S) can be calculated for any f in P(a{S)). 4. (Putnam) If a(S) lies on a simple closed curve then S is normal. 5. A theorem similar to (3) holds for f in R(a(S)) if the planar measure of iJa(p.) is zero. 6. If S is an isometry, f € H"", and f f. constant, then f(N) is the m.n.e. of f(S). (Received May 28, 1974.)

*74T-BI67. CHUNG LIN, University of Oregon, Eugene, Oregon 97403. Rearranging Fourier transforms on groups. Preliminary report.

Let G denote an infinite locally compact abelian group and X its character group. Let 8. be a Haar measure on X, and I < p :S 2. For a 8-measurable function ¢ on X, we define (} ¢(t) = 8(lx € X: !rf>(x)! > t!) and r:p*(t) = inf{t > 0: 8 q,(t)::; x! for x > 0. ¢* is called the nondecreasing rearr'lngement of ¢. Note that even though

A-492 ¢ is defined for X, the domain of ¢* is (0, oo). A nonnegative function g defined on (0, oo) is called admissible if g is nonincreasing and limx-+oog(x) = 0. Theorems: I. Let G be nondiscrete with a compact open subgroup and

g admissible. Then g\N =?'IN' where N is the set of positive integers, for some f E LP(G) iff

l;dg(k)PkP- 2 < oo. 2. Let G be nondiscrete with no compact open subgroup and g admissible. Then g = (' for

2 00 some f E LP(G) iff f~g(x)PxP- dx is finite. 3. Let G be an infinite discrete group which contains Z, Z(r ) or Z(r) No as a subgroup, g admissible. Then g\co ,l) = f*lco ,l) for some f E LP(G) iff f~g(x)PxP- 2 dx < oo. (Received May 28, 1974.)

74T-B168. T. K. MUKHERJEE, University of Arkansas, Fayetteville, Arkansas 72701. On hypercompletions. Preliminary report.

Kelley [ "Hypercomplete linear topological spaces", Michigan Math. J. 5( 1958), 235-246] raised the problem of hypercompletion. In this note, we define a notion of hyperco111pletion on the category of all linear topological spaces with a total family of continuous linear functionals. We show that if (E, u) is any locally convex space, then any hypercompletion of (E, u) has to be a completion of some topology of the dual pair.

Further, if (E, u) is any locally convex space, then there is a finest topology v in the same dual pair with v ~ u such that (E, v)"' is hypercomplete. We also study some properties of the hypercompletion. (Received May 28, 1974.)

74T-BI69. N. N. KAULGUD, Indian Institute of Technology, Powai, Bombay 400076, India. Sequence of mappings and fixed points. Preliminary report.

A study of sequence of mappings and fixed points is made similar to that of S. B. Nadler. Mappings considered here include k-set-contractions, demicompact maps, mappings satisfying condition A of Kannan, etc.

The problem considered here } is: given a sequence of mappings {(71 with fixed points {x71 }, if f71 -+ f in some sense does {xm I or some subsequence of it converge to a fixed point of f? We do not assume existence of fixed points of f. This gives some new fixed point theorems. Theorem 1. Let E be a notmed linear space and K a closed bounded

subset of E. Let If 71 } be a sequence of k·set-conttactions, with k < I, converging uniformly to continuous f on K. H

), } x71 = f71(x71 then {x71 has a convergent subsequence x71 k-+ x, where x = f(x). Theorem 2. Let E be a Banach space and K a closed, bounded convex subset of E. If f:K-+ K is a continuous !-set-contraction, then f has a fixed point in K. The paper also contains new single-valued fixed point theorems and their extension to set-valued mappings. (Received May 29, 1974.) (Author introduced by Professor P. C. Jain.)

74T-Bl70. PRATIBHA GHAT AGE, University of Toronto, Toronto, Ontario M5S IAI, Canada. Operators with invertible characteristic function. Preliminary report.

Suppose T is a power-bounded operator whose spectrum is contained in the unit circle. Let (JT(>..) = - T] T + AQr•(l- ,\T*)- 1 Qrl~r be its characteristic function defined on {>.., (>..) f, I}. Following the notation of Davis and Foias (Acta Sci. Math. (Szeged), 1971, pp. 127-139), we denote the ]-unitary dilation of T by U and defect spaces ofT and T* by ~T and ~r• respectively. Let 'JT( = E9~d~~il, and 'JT(* =V;:'= 0 un+IT~-;:Il. Propositions: 1. sup I *1(1- \Xi) Jl(>..- T)- 1JI < oo iff supllgJidi(J T(JT 1(X)g, 9 T 1(,\)g)\ ~ k V {>.., \XI < I}. 2. If 1 1 1 sup\>. 1dl 0 T(X)\1 < oo, then sup\>. 1<1(1 - \Xi) I\(>..- T)- \1 < oo iff sup I >-I ..)\\ < oo. 3. If sup\>. 1<1 11 0 T (>..)JI < oo, then (i) P,.lm* has a densely defined inverse. (ii) 3 a densely defined operator Q 1:'lR*-+ ms.dl.a ED Q 1'

P!R*(] Q 1 a)= Plll *(Ja). (iii) !R f, (0). (iv) If m(a(T)) = 0, then T 71 .;. st 0 and T*n .f. st 0. (Received May 31, 1974.)

* 74T-BI71. JOHN G. AIKEN, University of Rochester, Rochester, New York 14627. An application of direct integral theory to a question of Calkin.

Let B(H) denote the c* algebra of all bounded operators on a separable Hilbert space H. J. W. Calkin introduced a class of representations of B(ll) which kill the compact operators on H in his classic paper (Ann. of Math. (2) 42(1941), 839-873). In this paper, Calkin representations are studied as a nonclassical example of Segal's weak direct integrals (Mem. Amer. Math. Soc. No.9 (1951), also see Abstract 711-47-15, these Ykt.<.ee<>.

A-493 21(1974), A-193). We show how any Calkin representation T u may be decomposed into a direct integral of the representations studied by G. A. Reid, "On the Calkin representations", Pro c. London Math. Soc. 23(197 I), 547-

564). This decomposition is applied to construct an operator A E B(ll) such that the eigenvectors of T /A) do not span Lu for any T u= B(H) -> B(Lu), answering a question posed by Calkin on p. 866 of his paper. The concept of a continuous family of vector fields is then introduced and exploited to expose the topological nature of this decomposition. To the author's knowledge, this is the first application of the theory of weak direct integrals to a situation which apparently does not reduce to the classical case of square-integrable vector-valued functions. (Received June 3, 1974.)

74T-B172. CARMEN CASAS, Universidad Central de Venezuela, Caracas, Venezuela. On a theorem of R. Palais and the projection method ll~{P n }.

Let H be a Hilbert space, {e) an orthonormal basis, d' a Banach algebra of operators on H, and G(d') = {1 +A E GL(H), A Ed'}. Palais [Topology 3(1965), 271-279] considered a special class of such algebras,

called "approximately tame" (a.t.), and proved that if d' is a.t. then there exists a map q 1: G(d') -> G(d') such

that q 1(G(d')) = G(oo) and q 1 ~ 1. Setting ~(T) = y-! we shall have then that ~: G(d)-> G(d') and ~ q 1 ~ ~. (1) We consider the relation between a.t. algebras and mononormant ideals, and show that property (I) is related to the theory of projection methods IIIP n} (of the Galerkin type). In fact (1) expresses, in a different form, the known

fact that T E IIIP n }, for all T E G(A), provided d' is a.t. In case of general algebras ( 1) leads to a new projection method ll~{P n}. We give two stability theorems and some other properties of ll~{P n}. (Received May 17, 1974.) (Author introduced by Professor Stephen A. Andrea.)

*74T-B173. HYMAN J. ZIMMERBERG, RutgersUniversiry, New Brunswick, New Jersey 08903. Linear integro­ differential-bounda'ry·parameter problems.

Recent results of the author [Trans. Amer. Math. Soc. 188(1974), 407-417] are extended to develop

necessary and sufficient conditions for a linear problem A 1(x)y' + A0(x)y + H(x)[M 2y(a) + N 2y(b)] + K(x)J!F(t)y dt + L(x)p = AB(x)y, p' = 0, My( a)+ Ny(b) + J!F(t)y dt = 0, with vectors y and p of possibly different dimensional size, to be symmetric (selfadjoint). The integra-boundary conditions are recast to provide maximal

sets, exclusively, of two-point and integral boundary forms: M0y(a) + N0 y(b) = 0, M 1y(a) + N 1y(b) + J!F 1(t)y dt = 0,

J!F 2(t)y dt = 0. Canonical forms of symmetric problems, extending those previously developed for L(x) = 0 in the

above cited reference, are constructed, wherein H(x), L(x), M 2, N 2 and K(x)F(t) appear as F~(x), - F~(x),

-M1A1 (a), N 1A 1(b) and F~(x)[l)l + A]F 1(t), respectively, with 1)1 constant Hermitian and A= ~[- M 1A 1 (a)M~ + N 1A 1 (b)N~] skew-Hermitian, and the rows of the endpoint coefficient matrices of the integra· boundary forms are jointly orthonormed. The equivalence of two such problems under nonsingular transformations is examined, and a relationship between equivalence, of a problem with its adjoint, and symmetry is obtained. (Received June 5, 1974.)

*74T-B174. J. H. BURRY, Memorial University, St. John's, Newfoundland A1C 587, Canada. Measures generated by functions of two real variables. Preliminary report.

Functions of two -real variables are used to generate Munroe type Method I and Method II measures (Munroe, "Introduction to measure and integration", Addison-Wesley, Reading, Mass.) on the plane. The outer measures are defined by using coverings of rectangles with the set functions defined on the rectangles being the "mixed difference" of the "limits" from inside the rectangle at the four corners; namely, the sum of the lower left and upper right corner limits minus the sum of the other two corner limits. Discontinuities of functions which have limits from the four quadrants at each point are proven to exist on a countable number of parallels to the axes; and for functions that are monotone to the extent that the "mixed difference" at the corners is positive, it is shown that the Method I and Method II measures are identical. (Received June 5, 1974.) (Author introduced by Professor Renzo A. Piccinini.)

A-494 74T-B175. HARI SHANKAR, Ohio University, Athens, Ohio 45701 and RICHARD A. BOGDA, E. I. Dupont de Nemours and Co., Washington Works, Parkersburg, West Virginia 26101. On proximate order and zeros of a holomorphic function. Preliminaty report.

This is a continuation of Abstract 73T-B288, these :Jl~.a. 20(1973), A-579. For definitions of order p, lower order A, type T, lower type t, of a holomorphic function, see Abstract 711-30-8, these :J1.,t<...,...,_ 21(1974),

A-120. Theorem 1. Let f(z) = ~;=Oanzn be a holomorphic function with circle of convergence of radius R, 0 < R < 1 1 oo, and let f be of finite positive order p and of type T; then lim sup7 _,Rn(r)(R/(R - r))- p- ~ 2P+ T; and, lim inf,...Rn(r)(R/(R- r))-P·-'~ .:S pT. Theorem 2. Let f be as in Theorem 1, of any order p. Let {rnl;=l denote the sequence of the moduli of the zeros of /, counted according to their multiplicity, and let a be any positive finite 1 number. Then the series (*) ~: = 1(R/(R - rnn-a- and the improper integral J~n(t)(R/(R - t))-a dt converge and diverge together. Theorem 3. Let f be of finite positive order p. For any real number a> p, the series (*) converges. (Received June 7, 1974.)

*74T-Bl76. RICHARD I. LOEBL, Wayne State University, Detroit, Michigan 48202 and PAULS. MUHLY, University of Iowa, Iowa City, Iowa 52242. Flows on von Neumann algebras. Preliminary report.

Let {a1 }1 eR be a a-weakly continuous one-parameter group of *-automorphisms of a von Neumann algebra ~. and assume there is a faithful normal expectation ¢ from ~ onto ~({O l). (Notation as in [Arveson, J.

Functional Analysis 15(1974), 217-244].) Such a ¢ exists, for example, if !a1 } is a-weakly almost periodic; i.e.,

if for each p E ~* and A E ~.the function of t, , is AP(R). Theorem 1. ~([0, oo)) is a maximal,

subdiagonal algebra w.r.t. ¢ [Arveson, Amer. J. Math. 89(1967), 578-642]. Theorem 2. If {a1 } is homogeneous [Takesaki, Acta Math. 131(1973), 79-121] and a-weakly almost periodic, then there is a discrete subgroup r of R s.t. ~is isomorphic to L""(m)®9l({Ol), where m is Haar measure on f. Under this isomorphism, R([O, oo)) is carried to H00(m) ® 9l({O l). In this case 9l[O, oo) is a maximal a-weakly closed subalgebra of ~ iff 9l({O l) is a factor. Our methods also yield generalizations of results of StJD'rmer [J. Functional Analysis 15(197 4), 202-215]. (Received June 7, 197 4.)

*74T-B177. DEVENDRA S. GOEL, ANTHONY S. B. HOLLAND, CYRIL NASIM and BADRI N. SAHNEY, Department of Mathematics, Statistics and Computing Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada. Best approximation by a saturation class of polynomial operators.

The problem of determining a saturation class had been considered by Zamansky [Ann. Sci. Ecole Norm. Sup. 66(1949), 19-93], Sunouchi and Watari [Proc. Japan. Acad. 34(1958), 477-481] and others. Zamansky has

considered the C~saro means of order 1, and Sunouchi and Watari have studied the Riesz means of type n. Extending these results, we prove Theorem. Let {p n l be a se.quence of positive constants satisfying the following

conditions: Pn-k/pn -+ 1 as n ... oo for a fixed k.::; n, and ~~=O \Pn-k- Pn-k- 1\ = 0 (pn) where p _ 1 = 0. Then the Norlund operators Nn are saturated with order pn/P n' and the class of all continuous functions I for which r E Lip 1. (Received June 10, 1974.)

*74T-B178. ZIAD S. ALI, University of Algiers, Alger, Algeria. (N, Pn) summability of Fourier series.

We consider Norlund summability of Fourier series and extend a result of Ali, "Generalization of a theorem of Pati", Sem. Mat. Barcelona, vol. 24, fasc. 2°, 1973, pp. 201-205). Theorem 1. Let (N, Pn) be a regular Norlund method, defined by a real, nonnegative, monotonic nonincreasing sequence of coefficients !pn\, such that Pn-+ oo, and H(n) = Jig(t)dt = O(pn), as n-> oo, where g(t) is continuous and nonincreasing s.t. g(t) = o(l) as

t-+ ""• and H(t) is slowly oscillating; then if ¢(t) = o(g(1/t)/P 7 ), as t -+ + 0, the Fourier series of f(t), at t = x, is summable (N, pn) to f(x). The case g(t) = h(t)/t is the result in 1. Theorem 2. Hypotheses are as in Theorem 1,

and H(n) = f'ig(t) dt = o(P n), as n ... ""• where g(t) and H(t) are as in Theorem 1; then if (t) = O(g(1/t) /P y), as t -+ + 0, the Fourier series of f(t), at t = x, is summable (N, pn) to f(x). (Received March 7, 1974.)

A-495 74T-B179. P. K. KAMTHAN, MANJUL GUPTA and S. K. RAY, Indian Institute of Technology, Kanpur 208016, India. Schauder decomposition in dual and bidual spaces. Preliminary report.

Let ()(, ~) be a locally convex Hausdorff topological vector space, x* the topological dual of ()(. ~) and x** the topological dual of ()(*, f3(x*, x)). In this paper we have interrelated Schauder decompositions (or Sbos) in X• x*. x**, when they are equipped with various locally convex topologies. We show Theorem. Let ()(, ~) be a distinguished bornological space, and suppose x** has a a

"*74T-B180. ATHANASSIOS G. KARTSATOS and JAMES R. WARD, University of South Florida, Tampa, Florida 33620. Boundedness and periodicity of quasi-linear systems. Preliminary report.

Consider the system (*) x' = A(t, x)x + B(t, x) where A is an n x n matrix continuous on R+ X Rn and B is an n-vector continuous on R+ x Rn, and the systems (*,f) x' = A(t, f(t))x + B(t, f(t)) where f: R+-> Rn is continuous and bounded. The system (*) is said to be equiultimately bounded for bound M if 3 a positive

number M with the property: for every a> 0 and t 0 E R+, 3 a T(t0, a)> 0 s.t. for each x 0 ERn with jjx0 jj :Sa there is a solution x(t) of (*), with x(t0) = x0 , satisfying jjx(t)l! :S M V t 2:: t0 + T(t0, a). The systems (*,f) are said to be isobounded for bound M, if for each f: R+-> Rn, continuous and bounded, the resulting linear system is

equiultimately bounded for bound M with M and T(t0, a) independent of f. Theorem 1. If for every f E C[R+' Rn], f bounded, the systems(*, f) are isobounded for bound M, and i!A(t, u)i! :S y(t), i!B(t, u)i! :S B{t) V (t, u) E R+ x Rn, where y, 8 are locally integrable i [0, oo), then the system (*) is equiultimately bounded for bound M. Theorem 2. Assume that the systems (*,f) ar isobounded for bound M and A(t, u), B(t, u) are w-periodic in t (w > 0). Then there exists at least one w-periodic solution of the system (*). (Received June 10, 1974.)

Applied Mathematics

*74T-C25. LOKENATH DEBNATH, East Carolina University, Greenville, North Carolina 27834. Wave energy on a shallow rotating ocean with bottom friction.

An inviscid shallow rotating ocean model with linearized bottom friction is considered to determine the frictional effects on the unsteady wave motions generated in the ocean by arbitrary as well as special atmospheric disturbances. The asymptotic solution of the free surface elevation is calculated due to stationary as well as travelling wind stress distributions acting on the free surface of the ocean. It is shown that the

transient solution decays exponentially as t -> oe and the ultimate steady-state solution describes standing and progressive long waves propagating upstream and downstream with exponentially damping amplitude. The inclusion of the bottom friction introduces the exponential damping factor into the amplitude and makes other significant changes in the wave structure as well as its various properties. The resonance-type effect on the wave motions observed in a frictionless ocean model is no longer present in this frictional model. Several interesting results are found. (Received March 14, 197 4.)

74T-C26. PADAM C. JAIN and C. V. S. PRAKASH, Indian Institute of Technology, Bombay 400076, India. Universal stability criterion for ferro-fluids.

Convective stability of Boussinesq ferro-fluids in a bounded region under the action of a constant magnetic field gradient is discussed in this paper. The magnetic moment is taken to be an arbitrary function of

A-496 the magnetic field and temperature but is assumed to have a definite bound for its first and second partial derivatives. A stability criterion in terms of newly defined Rayleigh numbers is obtained. It is also shown that the subcritical instability in case of a stationary convective flow is eliminated for ferro-fluids. Similar results for the case of magnetic moment varying linearly with temperature and magnetic field are also obtained. (Received March 21, 1974.)

* 74T-C27. GREGORY J. CHAITIN, Rivadavia 3580, Dpto. lOA, Buenos Aires, Argentina, A theory of program size formally identical to information theory.

A new definition of the program-size complexity is made, H(A, B/C, D) is defined to be the size in bits of the shortest self-delimiting program for calculating the strings A and B if one is given a minimal-size self-delimiting program for calculating the strings C and D. This differs from previous definitions: (1) programs are required to be self-delimiting, i.e., no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size, Unlike previous definitions, this one has precisely the formal properties of the entropy concept of information theory. For example, H(A$) =

H(A, B)- H(B) + 0(1). Al~o, if a program of length k is assigned measure 2-AI, then H(A) = -log2 (the probability that the standard universal computer will calculate A) + 0(1). (Received April 8, 1974.)

*74T-C28. B. N. DATTA, Ahmadu Bello University, Zaria, Nigeria. A constructive method for finding Schwarz form of a Hessenberg matrix.

It is well known that the Schwarz form of a matrix [Z, Angew, Math. Phys. 7, 473-500] plays an important role in stabllity analysis of linear control systems. Indeed, Barnett and Storey ["Matrix methods in stability theory, " Nelson, London, 1970, p. 100] have remarked that "once the matrix A has been transformed into Schwarz form the stability problem is solved immediately." There lies, therefore, a considerable amount of interest in obtaining the Schwarz form of a given matrix, A constructive method for computing a class of nonsingular transforming matrices X that transform a given lower Hessenberg matrix to its Schwarz form by similarity is proposed in this paper. The method thus generalizes and is more efficient than those which attempt to find the Schwarz form of a matrix via the companion matrix. (Received April 12, 1974,)

74T-C29. SUDHANSHU KUMAR GHOSHAL and M. ABU MASOOD, Jadavpur University, Calcutta 32, India. Generalized Runge-Kutta formulas for ordinary differential equations with small parameter. Preliminary report.

In physical problems, nonlinear equations have often been solved by the method of perturbations, We here apply 7th order lOth stage Runge-Kutta formulas for an ordinary differential equation with small parameter to suit our present system. These formulas are for computer use; for very accurate results double precision arithmetic must be introduced. Nonlinear systems of algebraic equations involving unknowns or parameters have been calculated with the IBM 1130 by applying the method of Sarafyan and Brown(B.T.T. 7(1967)). (Received May 1, 1974.)

74T-C30. DAVIDS. LAWRENCE, Courant Institute, New York University, New York, New York 10012. Dynamic metagames of static games. Preliminary report.

Two person nonzero sum games G are investigated by considering the infinite metagame Goo obtained

by discrete iteration of the plays of the game. In such a case as "Prisoner's Dilemma" (G = P), distinct noncooperative and cooperative strategies and values of P can be defined for which there exist "safe improvements"

in P 00 over the noncooperative strategy. Safe strategies are those whose minimum payoff is the noncooperative value of P. Safe improvements are safe strategies whose payoff function exceeds that of the noncooperative strategy for at least one opposing safe strategy, A set S of such safe improvements has propetty C if all outcomes in S 2 are consistent in that they pay each player the cooperative value of the game. Although not all pairs of

A-497 self-consistent improvements are mutually consistent, a natural 'landmark' strategy set for opposing players seeking the cooperative-value payoff is the meet n Mm, "D", where M is the class of all maximal elements m E C if D is nonempty. A (necessary) simplification of the strategies allowed in P"" enables us to exhibit a strategy belonging to D. When there is no unique landmark for G"", the only solution is a class of conventions effective in case the paired players subscribe to the same convention. (Received May 13, 1974.)

*74T-C3I. VEENA KAUL, Indian Institute of Technology, Hauz Khas, New Delhi-110029, India. Optimal rules for the numerical integration of periodic analytic functions.

% +p Optimal rules for the numerical evaluation of L(/) = 0 f(z) I , Jzo ds, ds = dz1, where f(z) is periodic with period p and analytic on 1: z 0 + ap,- oo

74T-C32. ALAN L. SELMAN, Florida State University, Tallahassee, Florida 32306. On the structure of NP. Preliminary report.

For notation see Abstract 74T-C8, these Y1a!i.e<>A 21(1974), A-312. Theorem. For every recursive set p - p - A there is a set B such that B "' T A and B S mB; moreover, if A belongs to NP, then so do B and B. Corollary. If NP is not closed under complements, then there exist S ~-complete sets in N P that are not S ~-complete. (Received May 20, 1974.)

74T-C33. REGINALD P. TEWARSON, Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, New York 11790 and JOHN L. STEPHENSON, Section on Theoretical Biophysics, NHLI and Math Research Branch, NIAMDD, Bethesda, Maryland 20014. Numerical solution of transport equations of renal counterflow systems.

The steady state differential equations for solute and water transport in the counterflow system of the renal medulla are approximated by finite difference equations. These are solved globally by a constrained and modified Newton-Raphson method. Singular behavior of the Jacobian matrix of the difference equations is controlled by using a damped least squares solution. The method is stable, rapidly convergent, and has given solutions to the equations for several different models. Equations for the errors in the finite difference approximations are also derived and used in combination with the Newton-Kantorovich theorem to give a theoretical estimate of the error bounds. (Received May 28, 1974.)

74T -C34. REGINALD P. TEWARSON and ANDY S. KYDES, Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, New York 11790 and JOHN L. STEPHENSON and RAYMOND MEJIA, Section on Theoretical Biophysics, NHLI and Math Research Branch, NIAMDD, Bethesda, Maryland 20014. Use of sparse matrix techniques in numerical solution of differential equations for renal counterflow systems.

The differential equations for solute and water transport in the counterflow system of the mammalian kidney when approximated by finite difference equations lead to a system of nonlinear algebraic equations. These equations are solved by a Newton type iterative method. It is shown that the zero-nonzero structure of the Jacobian of the system of equations can be predicted from the physical connectivity of the model. We exploit this zero-nonzero structure to permute the Jacobian to a bordered block triangular form, which is then used to compute the correction to the solution vector of the nonlinear equations. For this computation a modified form of partial pivoting is used. Results are given that show the high accuracy of the computed solutions, the optimum use of the internal computer storage and the fast rate of computation. (Received May 28, 1974.)

A-498 *74T-C35. H. K. VERMA, Punjab Agricultural University, Ludhiana, Punjab, India. Transient heat transfer through grain stored in spherical bin with periodic change of external temperature. Preliminary report.

We consider the phenomenological equation describing the transient heat transfer through a solid in a spherical system under zero initial and periodic surface conditions. Using Laplace transforms, the solution is obtained in the form of convergent infinite series. The results are also specialized in the case when the external temperature is constant. (Received June 10, 197 4.) Geometry

*74T-D17. BANG-YEN CHEN and KOICHI OGIUE, Michigan State University, East Lansing, Michigan 48823. On compact Kaehler manifolds. Preliminary report.

Let M be an n-dimensional Kaehler manifold with Kaehler metric g = "'i.ga.jfdz adz f3 and Ricci tensor S = "'i.R a.J3dzadz (3" Let det(ga.J3+ tRaf3) ~ {1 + "'i.pk/'l · det(ga.d• A Kaehler manifold is called nearly-Einstein of Kaehler­ if it satisfies p~ :S 4p 2• The class of nearly-Einstein Kaehler manifolds is much wider than the class M. have Einstein manifolds. Let 1. If c /M) =

a • [¢ 2] for a :S 0, then M is flat. (Received April 17, 1974.)

*74T-D18. MICHAEL TARSY and ABRAHAM BERMAN, Technion-Israel Institute of Technology, Haifa, Israel. On the intersection of a cone and a subspace.

Let L be a subspace of a complex cone in V such that L + S is closed. It is shown that /(S, L) = SnS*(SnL}\s*n L l.).l. consists only of the origin. This is used to give a new proof of the key theorems of Tucker, Levinson and Abrams, and Ben Israel. Problem. Is "l(S, L) = {0 l" true in general? (Received May 24, 1974.)

* 74T-D19. KINETSU ABE, University of Connecticut, Storrs, Connecticut 06268. Some examples of nome gular almost contact structUTtfiS on exotic spheres. Preliminary report.

Let M be a C00-manifold of dimension 2n + 1. An almost contact sttucture on M is a triple (¢, ~. TJ) 2 of C00-tensor fields of types (1,1), (1,0) and (0,1), respectively, s.t. (1) TJW = 1 and (2) ¢ (X) =-X+ TJ(X)~

for all C00-vector fields X on M. (cf>, ~. TJ) is said to be regular if the foliation generated by ~ is a regular foliation in the usual sense, and otherwise nonregular. Any principal circle bundle over an almost complex manifold can be given a regular almost contact structure in a natural way. Thus there are quite a few examples of regular almost contact structure. It is our objective here to announce that there is a class of exotic spheres which carry nonregular almost contact structures whose associated leat,es are circles. Incidentally, it has been shown that some of the standard spheres carry nonregular almost contact structures whose associated leaves are circles. (Received June 5, 1974.)

Logie and Foundations

*74T-E56. PAUL E. HOWARD, Eastern Michigan University, Ypsilanti, Michigan 48197 and JOHN W. DAWSON, JR., Pennsylvania State University, University Park, Pennsylvania 16802. Factorials of infinite cardinals.

For any infinite set X, let S(X) = the symmetric group on X and define X! = [S(X)[. Theorems. 1. ZFI-[X[ S(X) well-orderable). 3. ZFC I-X!= 2lxl. 4. Each of the following is consistent with ZF: (i) X! and 2[xlare incomparable •• (ii) X!> 2[x[ • .(iii) X!< 2!Xl_ Theorem 4 is proved by exhibiting Fraenkel-Mostowski models in which X is the set of urelements. The transfer to -ZF follows from a result of ]ech and Sochor. (Received March 28, 1974.)

A-499 74T-E57. PETR STEPANEK and BOHUSLAV BALCAR, Charles University, Prague, Sokolovska 83, 186 00 Praha 8, Czechoslovakia. Rigid Boolean algebras of power N 1 can exist independently on the continuum hypothesis.

R. McKenzie and D. J. Monk suggested the problem of whether the existence of rigid Boolean algebras of power N 1 is provable without CH. A partial solution to the problem is given by the following Theorem. If

Con (ZFC) then Con (ZFC + 2 No > N 1 + "there exists a Boolean algebra b of power N 1 such that the completion of b is rigid"). (Received May 13, 197 4.)

* 7 4T-E58. JAMES H. SCHMERL, University of Connecticut, Storrs, Connecticut 06268. On N 0 -categoricity and the theory of trees.

Theorem I. If T is a decidable N 0 -categorical theoty with finite similarity type, then the function An[card(Stn(T))] is recursive. A counterexample shows the necessity of "finite" in the hypothesis. A tree is a

poset (P, <) in which the set of predecessors of any element is linearly ordered by <. If a, b E P, define P ab = {x E P: 'r/y(y .:5 a, b ... y < x)l. For cardinal K call (P, <) K-branching iff Va, bE P, card({{x E P ab: (3 Y E P ab)

(y .:5 x, z)l: z E P ab I) < K. Theorem 2. If (P, <) is N 0 -categorical, then (P, <) is decidable. Theorem 3. If (P, <) is finitely axiomatizable, then (P, <) is n-branching for some n < w. Theorem 4. If (P, <) is N 0 -categorical

and N 0 -branching, then T is finitely axiomatizable. Theorem 5. If 3 N 0 -categorical (P, <) E Mod (a), then 3

N0 -categorical N0 -branching (P, <) E Mod(u). (Received May 30, 1974.)

* 74T-E59. STEPHEN WHITNEY, Universite Laval, Quebec 10, Quebec G1K 7P4, Canada, Model classes of linear theories. II. Preliminary report. For a relation R 5;Sn, the power relation is PR = {(A 0 , ••• ) E (Ps)niVi

74T-E60, Withdrawn

* 74T -E61. MANUEL LERMAN, University of Connecticut, Storrs, Connecticut 06268, Quotients of the lattice of a-recursively enumerable sets by ideals of generalized finite sets.

Let a be an admissible ordinal, and let lb(a) denote the lattice of a-r.e. sets. Let a* be the projectum of a, For f3

A-500 74T-E62. JEFFREY B. REMMEL, Cornell University, Ithaca, New York 14850. Combinatorial functors on co·r. e. structures. Preliminary report.

We sharpen all category arguments in Crossley-Nerode's "Combinatorial functors", Springer-Verlag, 1974, by the priority method to yield Dedekind types X with representative n co-r.e. in 0'. For example, Theorem. Suppose F, G: C(M')-> C(M'') are partial recursive combinatorial functors inducing functions F, G from C(M')·types to C(M')-types. Then F(X) ~ G(X) for all dense Dedekind types XC(A!') with representative Q co-r.e. in 0' iff F and G are uniformly effectively equivalent on some neighborhood. Hay's (1966) framework is generalized to spaces of algebraically closed sets. Using the analogue of her notion of productive function and productiye center, we obtain the following Theorem. The following classes are productive with productive center contained in Ac.s. ~ !Dedekind types X\3U EX, n is cosimple}. (1) IX E C(M)\Hl EX, n is coinfinite and is co-r.e.}. If M is an infinite-dimensional vector space over a finite field and B is an r.e. independent set, then also (2) IX E C(M)\ X is not soundly based}, and (3) IX E C(M)\3 n E X,cl({lnB) ~ 0}. (Received April 19, 1974.)

* 74T-E63. CHARLES C. PINTER, Bucknell University, Lewisburg, Pennsylvania 17837. On the amalgamation property.

Let S and T be first-order theories, and suppose S is model-consistent with T. We say that S has

the T-amalgamation property (or, S amalgamates T) if, whenever ~I= T, !J3 1 , !J3 2 I=S, and f;: ~-> !Bi are embeddings

(i ~ 1, 2), there is a model :I I= S with embeddings g;= llli-> :I such that g 1/ 1 ~ gzi2• Our first theorem is a syntactic characterization of the relation "S amalgamates T". Theorem. The following is a necessary and sufficient condition for S to amalgamate T (the formulas below are all assumed to have their free variables among

v 1' • • ·, v n): For any formulas cp 1' cp 2 E V 1 and /L E 3 1, if S 1- /L -> cp 1 V cp 2, then there are formulas /L 1' /L 2 E 3 1

such that S 1- /L 1 -> cp 1, S 1-h -> cp 2, and T 1- /L -> /L 1 VIll. (Proof is a simple argument on diagrams and omitting

types.) Theorem. S amalgamates Tiff SV amalgamates Tr Theorem. LetS f Vl' T f V 2, and supposeS is model-consistent with T. (i) S amalgamates T iff S amalgamates every component of T. (ii) Let T be irreducible. S amalgamates T iff some component of S amalgamates T. (Received May 3, 1974.)

*74T-E64. MATATY AHU RUBIN, Hebrew University, Jerusalem, Israel. Interpretation of second order logic in the automorphism .group of atomic boolean algebras. Preliminary report.

Let B A be the saturated atomic boolean algebra of cardinality A, At(BA) the set of atoms of B A, G A the automorphism group of BA. If f EGA and x E At(B A), then Or(f, x) =min lk\k = w, or 0 < k and fk(x) = xl. Theorem. (a) Let la;\i

CV x E At(B A)) (n t Or(f, x) +-+ (Vi < A)(x ::5 ai)). This partially characterizes the sets of fixed points in At(B)I ). Theorem. 1 2 3 Let K ~ IG~ \A~~ A, K = I\AA ~A}, K ~ IL 2 (.\)\A~ ~A}, where L 2(A) is the full second order logic on A, then K 1, K2 are explicitly hi-interpretable and K2, K3 are hi-interpretable. For definition of interpretations, see S. Shelah, "First order theory of permutation groups", Israel J. Math. 14( 1973), 149-162. (Received June 3, 1974.) (Author introduced by Professor Saharan Shelah.)

* 74T-E65. J. A. PAULOS, Temple University, Philadelphia, Pennsylvania 19122. A. note on l'l-closures of logics. Preliminary report.

In an unpublished note, Barwise shows that l'l (w-logic) ~LA' A. the first admissible set containing w. A similar proof establishes Theorem. For 'Ill a countable acceptable structure, l'l Oil-logic)= LH • Moreover YP'II! whenever L is strong enough to characterize a set of sentences, l'l(L) contains all sentences cp(K)-definable

(K E l'l(L)) over the set. Hence, for example, the l'l-closure of second-order logic contains "huge" sentences. We also show Theorem. Given any L, there is an LA 2 l'l(L) which is adequate to truth in itself. This is proved by taking A. to be a transitive set closed under pairing, containing B = lcp(K): K E l'l(L)} as an element, and having the cp(K) (K E l'l(L)) as urelements. (Received June 7, 1974.) (Author introduced by Professor Louis Raymon.)

A-501 74T-E66. R. STEVE NEWBERRY, 1415 Bellevue Avenue, Burlingame, California 94010. Diagonal paradoxes. I. Preliminary report.

S is a set of monadic functions (or descriptions of such functions) satisfying a predicate P. (/ E S = P.) The paradox is that S can be 'proven' to be non-r.e. whenever the diagonal function lx + 1 satisfies P (e.g., when it can be shown that lx + 1 is in S). Example. S contains all and only functions (or descriptions, etc.) of the forms: x, x + 1, x + 1 + 1, .•.• Then I E S =>I + 1 E S, and for every value k of the free variable x there are infinitely many orderings of S s.t. f E S =>f occurs as the kth element of the ordering, hence f,. E S=>fk + 1 E S V k E w, and fx +IE S. Theorem. Sis not r.e. Proof. Suppose

not. Then there exists a recursive function L s.t. L = f k V k E cu and f E S. Define d ~df fx + 1. If L enumerates S thenthereexistsa k s.t. d=L=f,. and d=fk. Then for X=k wehaveboth d = f,. and (by definition) d = fk + 1, e.g., fk = h + 1, absurd. Q.E.D. But surely L', defined as

/ 0 = x, fn+l = fn + 1 = x + n + 1 is recursive and does enumerate S. Paradox! (Received June 10, 1974.)

74T-E67. SAHARON SHELAH, Hebrew University, Jerusalem, Israel. Various results in mathematical logic. Preliminary report.

Theorem 1. Let H(>..) be the set of sets which are hereditarily of power < >... Let K be, e.g., a variety defined by a recursive set of identities, FA its free number with >..-generator and Cat A(K) the category of

homomorphisms between members of K which E H(>..). Then H(>..), CatA (K) are hi-interpretable; and if >.. = K+, the semigroup of homomorphisms of FA is hi-interpretable with them. Conclusion 2. FA is (first-order) definable in CatJ.i.(K) iff >.. is (first-order) definable in H(p.). This answers a question of Feferman. Theorem 3. It is

consistent with 2FC that the class of orders of power >X 0 has a finite base. Theorem 4. (V = L, or 0 K 1). If

'¥ E L "'t."'(Q) has < 2 K 1 nonisomorphic models of power X l' then '¥ has a model of power X 2• (Received June 10, 1974.)

Statistics and Probability

74T-Fll, THEODORE E. HARRIS, University of Southern California, Los Angeles, California 90007, Reciprocal {lrocesses. I: Finite case. Preliminary report.

Let Z be a finite set, S the set of subsets of Z, ~~~) a Markov process with state space E having a stochastically continuous semigroup. Another such process {~I on a different probability space is called "reciprocal" to let l if peletn., I=¢ l = p; \~n eI=¢ I. ve . ., E E. If the two processes have the same transition

law, !e1l is called self-reciprocal, In this terminology the simple exclusion process was shown by Spitzer to be self-reciprocal. Let C0 be the set of functions f: S-> R 1 s.t. /(¢) = 0 and M0 the set of signed measures p. on

9'(S) s.t. p.(E.) = 0, Let ¢ be the 1:1 map of M0 onto C0 given by ¢p.(g) = p.({.,: .,ne I=¢}), and let 1/J be the

inverse of ¢ given by an inclusion-exclusion formula. Define 8 e= S-> R 1 by 8 t('T/) = 1 if en11 /: ¢ and 0 if en.,=¢. Let (f be the generator of let l. Then a reciprocal exists iff Ct maps c 0 into itself and l/Jff8 e

Topology

* 74T-G80, ROBERT A. HERRMANN, U.S, Naval Academy, Annapolis, Maryland 21402, H(i) semiregular, U(i) and R(i) extensions. II.

We continue our study of extension§; for a space (X, j), \\here we assume no separation properties for X, Definitions. Let (Y, T) be an extension of (X, :f). Then Y will be called a quasi-Urysohn-extension if for each p EX and q E Y -·X 3 T-open neighborhoods N(p), N(q), respectively, 3 clyN(p)nclyN(q) =¢, An extension (Y, T) of (X, :f) has property UC if for each nonclustering open ultrafilter F X on X which is a

U-filter, the open filter Fy = {UIU E T and UnX E Fxl clusters in Y. If (Y, T) is a UC-extension and for each

P E Y- X 3 some F y 3 F y -> p and F y nX = F X• \\here F X is a nonclustering open ultrafilter on X which is a

A-502 U-filter, then (Y, T) will be called a simple UC-extension. Observe, that a UC-extension is a U(i)-extension. Theorem. If (X, ~) is a non-U(i} space, then 3 an extension (Y, T) 3 Y C .X and (i) (Y, T) is U(i), X E T, Y

is T 2 except for X, X is OCE in Y, (ii) Y is a quasi-Urysohn-extension, (iii) Y is a simple UC-extension, (iv) if (Z, T') is a UC-extension of(X, ~\ 3 a continuous ¢: Y--> Z, fixed on X, 3 ¢[Y] is a simple UC-extension,

and (v) Y is a projective maximum in the class of all T 2 except for X simple UC-extensions of X. Certain other properties are also discussed. (Received February 13, 1974.)

*74T-G81, KRYSTYNA KUPERBERG, Rice University, Houston, Texas 77001, Two Vietoris isomorphism theorems in Borsuk's theory of shape.

Two Vietoris-type isomorphism theorems in shape theory are proved: one for Borsuk's fundamental groups ~ and one for Vietoris-Cech homology groups iin. In both cases a fundamental sequence (not necessarily

generated by a map) from a pointed compactum (X, x 0 ) to another pointed compactum (Y, y 0 ) is considered and an assumption on f , corresponding to Vietoris' assumption on the acyclicity of the inverse images of points is formulated. In both theorems, the statement is that the homomorphism [*,induced by f, is an isomorphism. In the case of a fundamental sequence f generated by a map, the classical isomorphism theorems are obtained as corollaries. (Received "darch 14, 1974.) (Author introduced by Professor William Jaco.)

*74T-G82, ROSS GEOGHEGAN, State University of New York, Binghamton, New York 13901, Hilbert cube manifolds of maps.

The well-known examples of Q-manifolds (i.e., metrizable spaces locally homeomorphic to the Hilbert cube, Q) are products of the form X x Q where X is a manifold or complex. But function spaces can be Q-manifolds. Example. Let X and V be Riemannian manifolds of positive dimension, X compact, V flat; let Y be a locally compact locally convex subset of V each of whose components has positive dimension; let a > 0 be a number; and let D be the set of all maps from X to Y uniformly approximable by C 1 maps whose differential is uniformly less than a, Then D is a Q-manifold. If Y is compact, D is a compact Q-manifold, (F.eceived March 29, 1974.)

*74T-GB3. ROSS GEOGHEGAN, State University of New York, Binghamton, New York 13901 and R. CHRISTOPHER LACHER, Florida State University, Tallahassee, Florida 32306, Compacta with the shape of finite complexes.

The theory of shape, as introduced by Borsuk, is the Cech homotopy theory of compact metric spaces (compacta). It is natural to ask: which compacta have the shape of compact polyhedra? Theorem l, Let X be a finite-dimensional compactum which is 1 - UV. Then X has the shape of a compact polyhedron if and only if its Cech cohomology with integer coefficients is finitely generated, Theorem 2, Let X be a finite-dimensional compactum. Then X has the shape of a compact polyhedron if and only if X can be embedded in some sphere sn in such a way that sn\x is homeomorphic to the interior of a compact topological manifold. (Received March 29, 1974,)

*74T-G84, PETER NICHOLAS, University College of North Wales, Bangor, Caerns LL57 2UW, Wales. Subgroups of the free topological group on [0, 1],

A complete characterization is given of those compact Hausdorff spaces X with the property that F(X), the free topological group on X, can be embedded both algebraically and topologically as a subgroup of F([O, 1]), the free topological group on the closed unit interval [0, 1], In fact, a compact Hausdorff space has this property if and only if it is finite dimensional and metrizable. Two other main results are proved. Firstly, it is shown that if a compact Hausdorff space X possesses a certain kind of continuous multiplication ( in particular, if it is a topological group), then F(X} has subgroups topologically isomorphic to F(Xn), the free topological group on the product space xn, for each positive integer n, Secondly, it is shown that for a compact metric space X, the dimension of F n(X), the subspace of F(X) consisting of words of length at most n, is equal to the dimension of the product space xn, for each n, These last two results are employed in the proof of the first, They are also extended to the case when X is a (Hausdorff) k"'·space. (Received April 2, 1974,) (Author introduced by Dr, Sidney A. Morris,) A-503 74T-G85. C. J. M. RAO, Indian Institute of Technology, Kanpur- 208016, India. On Richardson's compactification for convergence space.

We prove that Richardson's compactification [Proc. Amer. Math. Soc. 25(1970), 403-404] R(S) of a Hausdorff convergence space S is the largest Hausdorff compactification if and only if the following two conditions are satisfied: (a) S is locally compact [Proc, Amer. Math. Soc., to appear], (b) v = i(v)i\ t~, for each nonconvergent ultrafilter v on S, (Received April 8, 1974.) (Author introduced by ProfessorS. A. Naimpally.)

74T-G86, CHARLES E. AULL, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. A semistratifiable space is developable iff it has a 88-base.

A base ~ for a topological space X is a 8-base (88-base) for X if !.B can be written as ~ = Unl~n: n::: II in such a way that given an open set T C X and a point x E T, there is an nx s.t. ~(n) has finite (countable) order at x and there exists B E ~(n) s.t. x E B CT. We modify a proof of Heath [Bull. Acad. Pelon. Sci. ~r. Sci. Math. Astronom. Phys. 13(1965), 393-395] that a semimetric space with a point countable base is developable to show that a semimetric space X with a 88-base is developable. Proof. Set X lr. = lx EX,

X E u~,.. and ~,. is of countable order at xl. For each X Ex,. let R,.(l, x), R,.(2, x), .•• be a simple ordering of· the set of all members of X containing x and, for each n, let N(n, x) =interior {y: d(x, y) < l/nl and h,.(n, x) = N(n, x)nR,.(n, x). Let a be a well-ordering of X and, for each x Ex,. and each natural number n, let y,.[n, x] be the first member z of a s.t. x Eh,.(n, z). Finally, for each x Ex,. and each natural number n, let g,.(n, x) =

N(n, x) n

74T-G87, PANAYOTIS TH. LAMBRlNOS, University of Thessaloniki, Thessaloniki, Hellas, Greece. Generalised boundedness topological concepts. Preliminary report.

Following previous work (the author's thesis and Manuscripta Math. 10(1973), 289-296), the topological and productive concepts of almost (resp. nearly) bounded subsets of a topological space X are defined and studied; i.e, subsets which belong to every local boundedness (S. T. Hu, J. Math. Pures Appl. 28(1949), 287-320) in the space having a closed base (resp. regularly open subbase) or equivalently, subsets which are contained in a finite union of closures (resp. interiors of closures) of members of any open cover of the whole space X. Several characterizations are given by means of ® (resp. B)-convergence of filters defined by N. Velicko (Mat. Sb. 70(112) (1966), 98-112). Extensions of known results on almost compact (H-closed) and nearly compact spaces are obtained, Specifically, product theorems are also proved for the H (resp. N)-subsets, i.e. subsets of X which are contained in a finite union of closures (resp. interiors of closures) of members of any open cover of them. Counterexamples are given in order to distinguish among the concepts of bounded (resp. almost, nearly)-bounded and that of relatively compact (resp. almost, nearly)-compact subsets, Countably almost (nearly) bounded subsets are also studied, (Received March 27, 1974,) (Author introduced by Dr. S. M, Karnik.)

* 74T-G88, LEWIS l!..UM, University of Tennessee, Knoxville, Tennessee 37916, Order preserving and monotone retracts of a dendroid.

This paper is a sequel to Abstract 711-54-25, these Jlati..ee<~. 21(1974), A-215, We show a dendroid is smooth at P if and only if each [p, q] admits a :S P -preserving retraction; and a dendroid is a dendrite if and only if each [p, q] admits a monotone retraction, (:Received April 8, 1974.) (Author introduced by Professor J, H. Carruth.)

*74T-G89. WILLIAM F. LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. Every regular semistratifiable wN-space is a Nagata space.

Lemma 1. If X is a a-refinable topological space which has a G~-diagonal or is regular and has a

G s·diagonal, then X admits a sequence ( V n ) of relations on X satisfying condition (A) of Fletcher and Lindgren

A-504 [Abstract 709-G2, these Yl..t<.:= 20(1973), A-670]. LC'mma 2. If X is a first countable wN-space which admits a sequence of relations satisfying (A), then X is a Nagata space. These two lemmas may be used to show that Hodel's question [Duke 'Aath. J. 39(1972), 252-263], "Is a regular semi- stratifiable wN-space a Nagata space?", has an affirmative answer. (l!eceived April 18, 1974.)

* 74T-G90. V. KANNAN and A. K. VIJAYA KUMAR, \ladurai University, Madurai 625021, India. Rigid subsets of the real line.

A subset A of a metric space is said to be rigid if no two distinct pairs of distinct elements in A have the same distance. Janos (Abstract 73T-658, these Yl..t<"""- 20(1973), A-343) asked whether R can be written as a union of two rigid subsets, and, in general, whether Rn can be written as a union of (n + 1) rigid subsets. We give a negative answer. Theorem. R (and hence Rn for each n = 1, 2, · · ·) cannot be written as the union of a finite number of rigid subsets. Theorem. Every rigid subset of R has inner measure zero. But there exist dense rigid subsets of R. We do not know whether R can be written as a countable union of rigid subsets. (Received April 22, 1974.) (Author introduced by Professor M. Rajagopalan.)

*74T-G91. WILLIAM F. LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057 and PETER FLETCHER, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. wl'! spaces. Preliminary report.

Every a-orthocompact w!! space is a wy-space and a I.# space. (Received June 3, 1974.)

7 4T-G92. ROBERT A. DOOLEY, Summer Institute of Linguistics, 70000 Brasilia-DF, Brazil. On convex metric extens ion. Preliminary report.

A metric D is convex if for every two points x, z there is a third point y such that D(x, y) + D(y, z) = D(x, z). A generalized continuum is a connected, locally compact, metric space. For two topological spaces

M 1 and M 2 whose topologies agree on their intersection, let r 0 designate the topology for M 1 UM 2 consisting of all sets Q such that QnMi is open in Mi for i = 1, 2. The following result includes an earlier one by the

author (Proc. Amer. Math. Soc. 40(1973), 590-596), and much of the same proof is used. Theorem. Let M 1 be a

space with a complete convex metric D 1,and let M2 be a locally connected generalized continuum such that the

topologies of M 1 and M 2 agree on their intersection. In order for there to be a complete convex metric D 3 that

extends D 1 and generates some topology on M1uM 2 for which M2 is a subspace, it is necessary and sufficient

that M1nM2 be closed in M2• Furthermore, if the above metric D 3 can be so obtained, then the following three

statements are equivalent: (i) D3 generates r0 ; (ii) the M2 boundary of M1nM 2 is closed in M1; (iii) M2\M 1

and M 1\M 2 are separated sets in the topology generated by D 3• (Received April 29, 1974.) (Author introduced by Professor John M. Jobe.)

* 74T-G93. GARY F. GRUENHAGE, University of California, Davis, California 95616. Large basis dimension and metrizability. Preliminary report.

A topological space is said to have finite large basis dimension (or a basis of finite big rank) if it has a basis which is the union of a finite number of rank 1 collections of open sets. Theorem. Every compact

T 2-space having finite large basis dimension is metrizable. This answers a question of A. V. Arhangel'skii. Some generalizations of the above result are also obtained. (Received May 2, 1974.)

*74T-G94. MATTHEW G. BRlN and JOHN 0. BERGE, University of Wisconsin, Madison, Wisconsin 53706. A handlebody with one pillbox has no fake 3-cells.

The main result establishes upper bounds on the rank of H 1 of orientable 3-manifolds with fake 3-

cells and on the rank of 11 1 of closed orientable 3-manifolds with fake 3-cells. As a consequence we get that a 3-manifold obtained by sewing one pillbox on a handlebody of arbitrary genus cannot contain a fake 3-cell. (Received May 3, 1974.) (Authors introduced by Professor Daniel R. McMillan, Jr.)

A-505 *74T-G95. ALLAN JAWORSKI, University of Maryland, College Park, Maryland 20742. The Kakutani-Bebutov theorem for groups.

Suppose (X, G) is a transformation group, where X is a locally compact separable metric space and

G is a locally compact topological group. Denote by F 0 the closed G-invariant subset of X consisting of points left fixed by the identity component of G. Consider the shift dynamical system :S(G), the transformation group consisting of the continuous real-valued functions on G equipped with the topology of uniform convergence on compact sets, where G is acting by left translation. We extend results of Kakutani (J. Differential Equations

4(1968), 194-201). Theorem. (X, G) admits an equivariant embedding in :S(G) iff (F 0, G) does. Theorem. If the covering dimension of X is finite and G acts freely, then (X, G) may be equivariantly embedded in ~(G). When G = Z, the additive group of integers, denote by P m the closed Z-invariant set of points of period :5 m. Theorem. Suppose X may be embedded in Euclidean n-space so that Z is a group of locally Lipschitz mappings. Then (X, Z) admits an equivariant embedding in ~(Z) iff (P ( Zn + l)2, Z) does. Included are applications to harmonic analysis and the theory of invariant measures. (Received May 6, 1!J7 4.)

*74T-G96. RICHARD H. WARREN, Aerospace Research Laboratories, Wright Patterson AFB, Ohio 45433 and TERRENCE E. DOOHER, Metropolitan State College, Denver, Colorado 80204. Structure of the set of compatible proximities when the set is finite.

Let the family of all proximity relations on a completely regular (not necessarily T 2) topological space be ordered by set inclusion. Theorem. If the family is finite, then the family is lattice isomorphic to a symmetric partition lattice. Corollary I. If the family is a lattice, then it is modular iff it contains less than 6 members. Corollary 2. If the family is a lattice, then it is distributive iff it contains less than 3 members. For

each positive integer n, we show how to construct a T 2-space and a non-T 2 -space for which the family of all proximity relations is lattice isomorphic to a symmetric partition lattice on n objects. (Received May 13, 1974.)

/ / *74T-G97. TEODOR PRZYMUSINSKI, Polish Academy of &iences,00-656 Warsaw, Sniadeckich 8, Poland. A note on collectionwise normality and product spaces.

Theorem. For every infinite cardinal number A there exists a (perfectly normal, metacompact) A· collectionwise normal space which is not A+ -collectionwise normal. Corollary. For every infinite cardinal number A there exists a (perfectly normal, metacompact) space X A such that for every compact K, the product space

X A x K is normal iff w(K) :5 A. (w(K) denotes the weight of the space K.) The above theorem affirmatively answers the question of Gantner (Fund. Math. 66(1970), 263-281). In fact more general results are obtained. Our proof exploits Engelking's exposition of Bing's example of a normal not collectionwise normal space. W. G. Fleissner gave an example of a normal, collectionwise Hausdorff not collectionwise normal space. We present another description of essentially the same example. Our construction seems to be simpler and more natural. (Received May 13, 197 4.)

74T-G98. ARTHUR THOMAS CHARLESWORTH, Duke University, Durham, North Carolina 27706. Infinite cardinal functions which are minimal on metrizable spaces. Preliminary report.

Corresponding to the properties metrizable, paracompact, developable, point countable base, point countable separating open cover, a-space, strong ~-space, p-space, G~·diagonal, countable base, countable net,

and N 1-compact are the cardinal functions m, pa, D, pw, psw, a, ~. p, S, w, nw, and !'!... (The functions m, pa, psw, and p are defined by Hodel in Abstract 73T-G96, these Yloti..c..e6. 20(1973), A-506; Abstract 74T-G7, ibid.

21(1974), A-16; Abstract 74T-G17, ibid. 21(1974), A-324.) Theorem 1. On the class of regular T 1 spaces, m = pa • D, D = p • a, D S: a • pw, aS: ~ • psw, I::; pa • p, and w = D • /';.. It follows that m = pa • p • psw (Hodel, Abstract 74T-G7), m = pa • p • a, and m = pa • ::£ • pw; these extend metrization theorems of Nagata, Siwiec and Nagata, Michael, Slaughter, and Shiraki. Let n be an infinite cardinal. Theorem 2. If X is the product of :5 n

spaces X 1 for which (pa • ~)(X 1) S: n, then (pa • ~)(X) S: n. Theorem 2 extends a result of Nagami. Theorem 3.

A-506 If X is a Hausdorff space such that every open cover of X of cardinality :<; n has a finite subcover and for which either S(X) :<; n or psw(X) :<; n, then w(X)::; n. The following generalization of Urysohn' s theorem is extended to

higher cardinality: If X is a regular T 1 space with a base ~ such that each point of X has a neighborhood meeting just countably many members of ~. then X is metrizable. Two factorizations of nw due to Hodel are given analogues in the context of Michael's k-networks. (Received May 23, I974.)

*74T-G99. RENZO A. PICCININI, Memorial University, St. John's, Newfoundland AIC 5S7. Canada. Homology theories and Kan extensions.

Let (C', C) be a pair of admissible categories so that C' is C-small, as in C.R. Acad. Sci. Paris

S~r. A-B 274(I972), p. 828. Let h be a homology theory on C. In the previously mentioned paper we described a method of constructing a homology theory on C' which extends h. This new theory is not necessarily a Kan extension; it will be so if, for every X E [C'[, the (homotopy) category HCX of objects of C over X is filtering. Theorem I. Let (C', C) be as before with HCX filtering for every X E [C'[. Let h be a homology theory on C; then, the Kan extension of h to C' is a homology theory. If C' and C are respectively the categories of based, connected CW-complexes and of based, connected, finite CW-complexes, then Theorem 2. Let h' be an extension of the functor h to C'. The following are equivalent: (I) h' is equivalent to a Kan extension of h; (2) (VX E [C'[) h'(X)~ colimh(Y), where Y runs over all finite, based subcomplexes of X; (3) h' is equivalent to a homology theory on C' defined by a spectrum. (Received May 28, I974.)

*74T-G100. URI PRAT, Technion-Israel Institute of Technology, Haifa, Israel. On the weak sum theorem in dimension theory.

A family F of topological spaces is said to satisfy wst (the weak sum theorem) if X E F whenever X=U~=0 X;,where Xi aredisjoinrclosedsubsetsof X and Xi EF, i=O, 1, .... Adimensionfunction dis said to satisfy wst if F = IX[d(X) :=: nl satisfies wst. For metric spaces the following results are obtained: (a)

There exists a metric space X= U ~=OXi with ind X= 1 s.t. X0 = lx0 I is a one point set, ind Xi= 0, Ind Xi= 1, where Xi' i = 1, 2, · · ·, are clopen and ·\· i = 0, 1,. ··, are disjoint subsets of X. (Thus d = ind does not satisfy wst.) (b) The result in (a) is best possible in the following sense: If X= U ~=O Xi is a metric space s.t. Xi are closed and disjoint with ind Xi= 0, i = 0, 1, ... , and if ind X> 0, then for infinitely many indices i, one has Ind Xi> 0, (c) Let S = IX[ ind X= 0!. Then the family Q =IX[ Ind X= 0 I contains every subfamily of S satisfying wst. (d) Let P denote a nonempty topologically closed family of metric spaces s.t. if X E P and A is a closed subset of X, then A E P (i.e. P is closed monotone). If P-ind satisfies wst, then for every metric space X one has P-ind = P-Ind. (a) shows also i.a. that the small inductive dimension of a metric space can be raised by adjoining of a single point, a fact recently proved by Erick Van Douwen and by T. Przymusinski. (Received May 28, 1974.) (Author introduced by Professor Meir Reichaw.)

* 74T-G10 1. HOWARD H. WICKE and JOHN M. WORRELL, JR., Ohio University, Athens, Ohio 45701. Completeness of semicomplete Moore spaces.

Definition. Let X and Y be topological spaces. Then X is (}-refinably embedded in Y iff X f Y, and for every Y-open cover 11 of X there is a countable collection ICn: 11 E Nl s.t. each (\ is an open refinement of 'U covering X, and for each z E nlue : 11 EN I some ek is finite at z [see Abstract 711-54-40 n ' these 1k.ti~ 21(1974), A-219]. A Moore space X is semicomplete (or Rudin-complete) iff X satisfies 1" of [Duke Math. J. 17(1950), 317-327]. An equivalent formulation is that it has a base ~ s.t. for every monotonic subcollection :lJi of 93 with all elements nonernpty, niB: ll E:lJil ,f.¢. A complete Moore space is a space satisfying Axiom 1 of R. L. Moore's "Foundations of point set theory". Theorem. A semicomplete Moore space is a complete Moore space iff it is (}-refinably embedded in its Wallman cornpactification. (Received June 7, 1974.)

A-507 *74T-Gl02. ULRICH KOSCHORKE, Rutger.s University, New Brunswick, New Jersey 08903. Existence and classification of framefields.

Let M be a closed smooth n-manifold. Denote by O,(M. q) the group of bordism classes of triples (S, f: S-+ M x P 1' F: TSED f*(>..®TM) ED Rq-r ~ f*(>..q ED TM)), where S is a closed r-manifold, f continuous, and q- ~ A is the canonical line bundle on projective space P q- 1• Theorem 1. If n(?_) 2q + 1, then there is a well-defined

invariant w q(M) E Oq- 1(M, q) which vanishes iff M admits a q-field. Related invariants measure the precise obstruction to the existence of a q-field with finite singularities (resp. with prescribed first q - 1 vector fields).

Theorem 2. Let n(,?) 2q + 2. Given a fixed q-field, the homotopy (resp. concordance) classes of all q-fields on M correspond bijectively to the elements of Oq(M, q) (resp. of a quotient consisting entirely of 2-primary torsion). Theorem 3. If the Euler number of M is zero, then 2q- 1 • w (M) = 0. The order of the index of any q-field with q finite singularities is also a power of 2. In contrast, e.g., for n(_?) 2q + 3, n = q(2), M has at least as many nonhomotopic q-fields (if any) as there are odd torsion elements in The underlying obstruction theory, based rl.q on nondegenerate singularities, extends to bundle monomorphisms in general, possibly with boundary conditions. (Received June 10, 1974.)

*74T-Gl03. CLIFFORD KOTTMAN and EMILIO GAGLIARDO, Oregon State University, Corvallis, Oregon 97330. Orientation-preserving homeomorphisms of the plane which interchange two points have fixed points.

Let T be an orientation-preserving homeomorphism of the Euclidean plane. Theorem. If there exist two points P = T(Q), Q = T(P) (that is, if T 2 has a fixed point), then T has a fixed point. The proof would become trivial if there exists a closed curve from P through Q to P which is transformed into itself, but this is shown to be not always true. Conjectures. (i) If T" has a fixed point for some n, then T has a fixed point. (ii) If {Tn(P): n = 1, 2, ···I is bounded for some P, then T has a fixed point. (iii) If there exists a compact set K :J T(K), then T has a fixed point. All of (i), (ii), (iii) would generalize the Theorem. (Received June 10, 1974.)

* 7 4T-G 104. STAGG NEWMAN, Cornell University, Ithaca, New York 14850. One-point compactifications of Q-manifold factors and infinite mapping cylinders. II. Preliminary report.

Let Q be the Hilbert cube. Let XUoo be the one-point compactification of a locally compact space X by oo, Let the cone on X be denoted by c(X) =X x [0, 1]/X x 0. Theorem 1. Let X be a noncompact 2° countable space such that X x Q is a Q-manifold. Then the cone on Xu oo reduced over oo is a Q-factor, i.e.

(c(XUoo)/oo x [0, 1]) x Q ~ Q. For all i ~ 1, let Ki be a compact CW complex. Let f;: Ki-+ K;+ 1 be continuous.

Define the infinite mapping cylinder M(f;) = U~ = 1 Ki x [i- 1, i]/"' , where (k, i) "'(f(k), i) V k € Ki Vi. Theorem 2. (M(f;)Uoo) x Q is homeomorphic to Q. (Received May 28, 1974.)

* 74T-Gl05. SHASHI BALA SUD, Indian Institute of Technology, Hauz Khas, New Delhi 110029, India. A new product topology and function spaces.

Given topological spaces X and Y, let X®Y be the largest topology on the set X x Y, s.t. for each

x 0 EX and y 0 E Y, the maps ax0 : Y-+ X x Y and ay0 : X-+ X x Y, defined by ax 0 (y) = (x0, y), ay0(x) = (x, y0), are continuous. It is proved that this new product topology is commutative, associative and ( 1) the projection maps are continuous; ( 2) the finite product of identification maps is again an identification map, and thus the product (finite) of quotient spaces is the quotient space of the product. Finally, it is shown that if we give the topology of pointwise convergence to the function space Yx, then the following are true: (1) The evaluation map

w: yX ® X -+ Y is continuous. (ii) X Y ® z = (XZ) Y. (iii) The map cp 1: yX ® y•X' -+ ( Y ® Y')X ® x' defined by cp N· /') (x, x') = (f(x), f'(x')) is continuous. (iv) The map ¢2: yX ® zX -> yZ defined by ¢z(f. g) =I 0 g is continuous. (v) xX is a topological semigroup. (Received June 5, 1974.) (Author introduced by Professor Paul T. Bateman.)

A-508 The April Meeting in Santa Barbara, California April27, 1974

714-A8. MOHAMMAD ISHAQ, Laval University, Quebec, Quebec GIK 7P4, Canada. On homomorphic relations in abstract al~ebras.

Let [A, !l] and [B, A] be two abstract algebras and p a relation between A and B such that Ya E A, 3 b E B s.t. (a, b) E p, and Vb' E B, 3 a' E A s.t. (a', b') E P· Let u be a relation between n and A s.t. for

w E!l and A E A with rank(w) = n = rank(A), say, (w, A) c:u iff V (ai, hi) E p, ai c:A, hiE B, i =I, 2, •.• , n,

implies ((a 1, • • ·, an)w, (b 1 ,. • ·, bn)A) E p. Then (p, u) is said to be a homomorphic relation from [A, !l] to

[B, A] iffy (j) En s.t. (w, A) E u and VA' E A, 3 (j) I En s.t. (w'' A') E u. It can be seen that the identity relation, the inverse of a homomorphic relation, and the product of two homomorphic relations are all homomorphic. A number of results involving homomorphic relations and congruences have been obtained. (Received May 6, 1974.)

The May Meeting in DeKalb, Illinois May 13-17,1974

* 7I5-A23. FOBERT TIJDEMAN, Mathematical Institute, Wassenaarseweg 80, Leiden, Netherlands. Hilbert's seventh problem: The method of Gel' fond-Baker and its application. • In I748 Euler made a conjecture which can be stated as follows. If a f, 0, 1 is rational and {3 is algebraic and irrational, then af3 is irrational, At the Mathematical Congress in 1900 Hilbert formulated the following natural extension of Euler's problem. If a f. 0, I is algebraic and {3 is algebraic and irrational, then af3 is transcendental, Examples of such numbers are e17 (= z- 2i) and 2'./2. The first partial solution of the Euler-Hilbert problem was given by 'Gel' fond in 1929. He proved the conjecture in case {3 is an imaginary quadratic irrational. Siegel and Kuz'min proved independently that the method can be used also if {3 is a real quadratic irrational. The complete solution of the problem was obtained in 1934 by Gel' fond and, by a related but different technique, by Schneider. This was a very important breakthrough in transcendental number theory and many generalizations and applications have been given since, In 1966 Baker introduced a generalization of the Gel' fond-Schneider theorem of great importance. One of his theorems reads as follows. If al' ··.,an and 1 {3 0 , {31' • • ·, {3n are arbitrary nonzero algebraic numbers, then ef3o af • • • a;:. is transcendental, Refinements and generalizations of this technique have been given by Fe!' dman, Baker, Stark, Coates and several others, The method has proved to be very useful in applications in transcendental number theory, the theory of diophantine equations and approximations, the theory of class numbers of quadratic fields and in problems on rational integers composed of relatively small primes. One of the important features of the Gel' fond-Baker method in these applications is its effectiveness. (Received April 8, I974,)

715-A24. ALBRECHT PFISTER, Johannes Gutenberg University, Saarstrasse 21, Mainz 65, Federal Republic of Germany. Hilbert's seventeenth problem and related problems.

• Hilbert's 17th problem is concerned with the representation of positive definite functions as sums of squares. It has been solved by E. Artin in 19 26: Let K be a real closed field or a subfield of the real numbers

which has only one ordering. Let f(x 1, ••• , xn) E K(x 1' •.• , xn) be a rational function which takes nonnegative

values at all points (a 1, ••• , an) E Kn where f is defined. Then f is a sum of squares in the field K(x 1, • • • , xn). For K real and closed there is a quantitative improvement of this result depending only on the number of variables:

Any positive definite element of K(x 1, ••• , xn) is a sum of 2n squares. (This has been proved by D. Hilbert for n ~ 2, by J. Ax for n ~ 3, by A. Pfister for arbitrary n.) It is not known whether this bound is best possible except

for n ~ 2. Another open problem is the existence of a number P Q (n) such that every' positive definite element of Q(Xp• · ·, xn) is a sum of P Q(n) squares in this field. (Received May 6, I974.)

A-509 715-B9. G. G. LORENTZ, Mathematische Institut A Universitat Stuttgart, 7 Stuttgart, I, Herdweg 23, Federal Republic of Germany. Hilbert's thirteenth problem: Representation of functions in terms of functions of a smaller number of variables. • The 13th problem of Hilbert reads: (I) "prove that the equation of the seventh degree is not solvable with the help of any continuous functions of only two arguments''. It has an algebraic and an analytic part. No wonder that it would better be replaced by rwo problems, one entirely algebraic, the other analytic. We formulate the second one: (II) "prove that there are continuous functions of three variables not representable by continuous functions of two variables". Kolmogorov' s theorem refutes (II). It reads (after improvements of Sprecher,

Lorentz): (III) For each n, there exist continuous, monotone functions ¢1' ·. ·, ¢ 2n+! and constants .\1' · ·., .\n

so that each continuous function f of n variables can be written f(x 1, • • ·, x,) = ~~=~! g(~;=l\¢q(xp))' where

g is some continuous function of one variable. One can take the functions ¢ q in the class Lip I. This fact is obtainable without calculations (Kahane) if one considers the geometric formulation of (III), and selects for each

ofthe curves y = ¢q(xp) the length of arc as its natural independent variable. However, Kolmogorov's theorem (III) does not entirely answer (II). One could hope to save the truth of the conjecture by assuming that all functions involved are smooth, say continuously differentiable. For superpositions linear in the free functions g,., this was indeed proved by Vituskin. The proof may be based on the notions of entropy and capacity, introduced by Kolmogorov. Let G be a closed bounded region in R". Let X be a rearrangement-invariant Banach function space (in the sense of Luxemburg) of integrable functions on G. We consider a certain family of norms II • II 8 on X, defined for 8 > 0. Using for X the €-entropy with respect to these norms, it is possible to obtain the following. The set of all linear superpositions Ik=lplo • glc oqlc' where gk are .arbitrary continuous functions of one variable, pTe, qlo are defined on G, P,. continuous, qlo continuously differentiable, is nowhere dense in X. (Received May 9, 1974.)

*715-G6, JEROME KAMINKER, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46205 and CLAUDE L. SCHOCHET, Indiana University, Bloomington, Indiana 47401, A spectral sequence for &xt(X), Preliminary report.

The functor &xt(X), from compact metric spaces to abelian groups, introduced by Brown, Douglas, and Fillmore, defines a generalized homology theory, denoted &j.X), agreeing with complex K-theory homology on finite complexes (Bull. Amer, Math. Soc, 79(1973), 973-978). We provide methods of calculating &xt(X) via a spectral sequence. Theorem. Let X be a finite-dimensional compact metric space. There is a natural f!rst and

fourth quadrant sequence, converging to with Ep2 equal to 5 HP(X; K (pt)), the reduced Steenrod spect~al ~(X), ,q q homology of X, with coefficients in K-theory homology of a point. (Received March 29, 1974.)

A-510 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 cNoticaJ . 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 0294(). 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. *See temporary exception, page 189 of the June 1974 cf/otiai).

MATHEMATICS TEACHER AND RESEARCHER, Age 28, statistics, and computer science. Soundly versed in Ph.D. Texas Christian University 1974 in topology. One Goldberg's Introduction to Biometrics, G. F. Morrison• s year full-time teaching at TCU, three years half-time at Multivariate Statistical Methods. References upon re­ Tarleton State University, Stephenville, Texas. Effective quest. Will relocate if necessary. Available immediately. teacher. Two papers in preparation. Resume, refer­ Dr. J. V. Moroose II, 1016 N. 19th Street, Allentown, ences, abstract upon request. Available immediately. Pennsylvania 18104. John H. Gresham, Box 1, Bluff Dale, Texas 76433. MATHEMATICS TEACHER AND RESEARCHER. Age 27. MATHEMATICS PROFESSOR, TEACHING AND RE­ Ph.D. May 1974. Specialty: Riemann Surfaces, Modular SEARCH. Ph. D. 1950. Age 51. Specialty: applied mathe­ Forms. Seeking full-time teaching September 1974. matics, functional analysis, topology, abstract algebra Experience: 5 1/2 years lecturer at City College of New and logic. 21 published articles. 24 years experience in York. Resuml'\ and references upon request, Ronald teaching and research. Knowledge of eight foreign lan­ Simenauer, 10 Bennett Avenue, New York, New York guages. References and rt3sume upon request. Available 10033, Telephone (212) 781-9067. immediately. Ludvik Janos, Department of Mathematics, University of Montana, Missoula, Montana 59801.

TEACHING AND/OR RESEARCH. Ph.D, 1974 Southern ANONYMOUS Ulinois University. Age 32. Specialty: Algebraic Num­ ber Theory and Algebra. Three papers in preparation. Available September 1, 1974. Resume and references MATHEMATICIAN, TEACHING AND RESEARCH, OR upon request. S.K. Lo, P.O. Box 185, Energy, illinois RESEARCH. Ph, D., top British University. Specialty: 62933. analysis. Have taught almost all areas of pure mathe­ matics and can teach several areas of applied mathe­ TEACHING AND RESEARCH. Ph. D. 1974. Age 24, matics. Excellent publications, Long teaching experi­ Topology, analysis, and algebra. Three years teaching ence-applied and pure. Currently writing books on Dif­ assistant. Primary lecturer informal graduate research ferential Equations and Operations Research. Widely seminar in topology. Four papers presented past year. travelled. Available immediately. SW36 Rflsume and references upon request, Available immedi­ ately. Maurice Hugh Miller, Jr., 5520 11th. Court South, Birmingham, Alabama 35222. ASSISTANT PROFESSOR . Ph.D. Columbia, 1971, Age 2 9. 3 1/2 years experience teaching full range of courses including externally funded pedagogical projects. 3 papers MATHEMATICS PROFESSOR, TEACHING AND RE­ published in descriptive set theory, several submitted, SEARCH. Ph. D. 1972. Age 32. Specialty: extensive Author of forthcoming remedial book. SW37. Telephone knowledge in applied mathematics including probability, (201)763-7808.

A-511 Mathematics

Stable Homotopy and The Analysis of Generalized Homology Frequency Data }. Frank Adams Shelby }. Haberman Professor Adams demonstrates how Provides a general unified treatment an understanding of stable homotopy of log-linear models for frequency theory provides for the setting up of data by means of a coordinated free the machinery of generalized ho­ method of linear algebra and differ­ mology and cohomology. He makes ential calculus. Examples from the available "Boardman's category of biological and social scit!nces are spectra," and presents a detailed given to illustrate practical applica­ study of complex cobordism. tions of these models. Chicago Lectures in Mathematics Statistical Research Monographs series, edited by Irving Kaplansky series 1974 384 pages Cloth $11.50 1974 432 pages Cloth $20.00 Paper$4.95

The University of Chicago Press Chicago 60637

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XXVI HARMONIC ANALYSIS ON HOMOGENEOUS SPACES edited by Calvin C. Moore This volume constitutes the proceedings of the nineteenth Summer Research Institute of the American Mathematical Society which was held at Williams College on July 31-August 18, 1972, and was supported by a grant from the National Science Foundation. The scientific program of the institute consisted first of all of six major lecture series, .each devoted to relatively broad areas within the subject and consisting of from four to six lectures. These lecturers were C. C. Moore, V. S. Varadarajan, Harish-Chandra, H. Furstenberg, S. Helgeson, and B. Kostant (jointly with R. Blattner). The first part of this volume contains articles prepared by the lecturers and based on these lecture series. These are intended to be surveys of various aspects of the subject of the institute, and it is hoped that they will provide material not only for advanced graduate students and postdoctoral mathe­ maticians who want to get into the subject, but also for more senior mathematicians who work in related areas and who need a survey of what is known in the subject, and what techniques are available. The second part of the program consisted of five research seminars, each under the leadership of an invited chairman. The seminars were devoted to the presentation of talks by participants on current ·research in the various fields: representation theory of solvable groups and harmonic analysis on solvmanifolds; irreducibility and realization of various series of representations of semisimple groups; boundary behavior, special functions, and integral transforms in group representations; representa­ tio~s of p-adic groups; and e(G/f) and automorphic functions. Short summaries of these talks are included in this volume in the format of Research Announcements, and they are arranged under the five seminar headings. ISBN 0-8218-1426-5 Book code: PSPUM/XXVI List Price $34.40, Member Price $25.80 480 Pages merican Mathematical Society P. 0. Box 6248 Providence, R. I. 0294

A-512 \ Booth # 16, International Congress of l\1 athematicians Y ancouver, Canada

:\IARCEL DEKKER, I~C. Publishers in \lathematics and Statistics cordially invites you to peruse the first 26 volumes of Dr. S. Kobayashi's PURE Al\'D APPLIED MATHEMATICS SERIES

Value Distribution Theory (in 2 parts}, ed. by R. Kujala and A. Vitter Cohomology and Differential Forms by I. Vaisman Banach Algebras: An Introduction by R. Larsen Finite Dimensional Multilinear Algebra (in 2 parts) by M. Marcus Harmonic Analysis on Homogeneous Spaces by N. Wallach Geometry of Submanifolds by B.-Y. Chen Introduction to the Theory of Formal Groups by J. Dieudonne Functional Analysis: An Introduction by R. Larsen Geometry, Physics, and Systems by R. Hermann Tangent and Cotangent Bundles: Differential Geometry by K. Yano and S. Ishihara Stochastic Processes and the Wiener Integral by J. Yeh Rings with Polynomial Identities by C. Procasi An Introduction to the Theory of Distributions by J. Barros-Neto Cohomological Methods in Group Theory by A. Babakhanian Multiplicative Ideal Theory by R. Gilmer Functional Analysis and Valuation Theory by L. Narici, E. Beckenstein and G. Bachman Topology: An Outline for a First Course by L. E. Ward, Jr. Differentiable Manifolds by Y. Matsushima Necessary Conditions for an Extremum by B. N. Pshenichnyi Group Representation Theory (in 2 parts) by L. Dornhoff Infinite Group Rings by D. Passman Hyperbolic Manifolds and Holomorphic Mappings by S. Kobayashi Equations of Mathematical Physics by V. Vladimirov Integral Formulas in Riemannian Geometry by K. Yano Symmetric Spaces: Short Courses Presented at Washington University ed. by W. Boothby and G. Weiss The Separable Galois Theory of Commutative Rings by A. Magid

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A-513 LECTURES IN APPLIED MATHEMATICS, Volume 15

"Nonlinear Wave Motion", Alan C. Newell, Editor

This volume contains the proceedings of the summer seminar held at the Clarkson College of Technology, Potsdam, New York, in July 1972, which was supported by the National Science Foundation and the Office of Naval Research. The nineteen-sixties produced some major advances in the mathematical description of and our understanding of nonlinear wave propagation phenomena. Of particular significance were the methods of Whitham for describing the slow temporal and spatial modulation of fully nonlinear dispersive waves and of Gardner, Greene, Kruskal and Miura for finding the initial value solution to the Korteweg-de Vries equation. It was the purpose of this seminar to explore these and related theories and to exchange ideas about the most fruitful avenues of investigation for the immediate future. The principal lecturers and their' topics are T. B. Benjamin Lectures on Nonlinear Wave Motion, D. J. Benney, Long Waves, M. D. Kruskal, The Korteweg-de Vries Equation and Related Evolution Equations, Peter D. Lax, Periodic Solutions of the KdV Equations, G. B. Whitham, Two-timing, Variational Principals and Waves. There are in addition five other lectures, an extensive bibliography, and indexes. The reader of this book should be an advanced graduate student in the mathematical sciences. Although this book is not expository in the sense of a textbook, it is reasonably expository for those having a sufficient background. In the NSF Annual Report 1973 the whole section on Mathematics is devoted to "Initial Value Problems." The report states the following: "The solution of the initial value problem for certain classes of non-linear partial differential equations is of considerable importance in many scientific disciplines." ... "One of the important equations of this type is the Korteweg-de Vries (KdV), which was first suggested in 1895 to describe the development and propagation of moderately small amplitude shallow water waves." ... "The most recent work ... , by A. Newell, M. Ablowitz, D. Kaup, and H. Segur, was stimulated by an NSF-sponsored Conference on Non-Linear Wave Motion in July of 1972 at Clarkson College of Technology. In a series of papers, they solve the Sine-Gordon equation and generate a technique for the solution of a wide class of dispersive equations and for extension of this technique to problems with higher derivatives, as well as subsuming many linear problems."

ISBN: 0-8218-1115-0 BOOKCODE: LAM/15 PRICE: List Price $19.10, Member Price $14.33 PAGES: 240

"""------American Mathematical Society.

A-514 .Publish or Arish.'lnc. 6 Beacon Street, Boston, Massachusetts 02108 (U.S.A.) The prices below are for books ordered directly from Publish or Perish, Inc. They repre­ sent a 20% discount from the retail price. Individual orders must be prepaid; libraries will be billed. Mathematics Lecture Series 1. Cutting and Pasting of Manifolds; SK-groups U. Karras, M. Kreck:l. W. Neumann, E. Ossa, Bonn University and Institute for Advanced Study This volume describes recent research on alteration of closed smooth manifolds by "cutting and pasting." Modulo this operation, the set of oriented singular manifolds in a space X be­ comes a graded group SK*(X), which among other things turns out to contain natural obstructions for multiplicativity of signature for fibre bundles and for Winkelnkemper's "open book decompositions" of manifolds. CONTENTS: Introduction to SK; SK of fibre bundles; equivariant SK; controllable in­ variants; other SK concepts; Winkelnkemper's open book theorem; SK of (B,f)-manifolds. Paperback; 70 pages. $2.50 in U.S.A., Canada, Mexico. $2.75 foreign. 2. Applications of Global Analysis in Mathematical Physics Jerry Marsden, University of California, Berkeley An introduction to methods of global analysis which have been found useful in various problems of mathematical physics. Ideas from geometry are used to shed light on problems in analysis which arise in physics, with emphasis on attacking specific problems. The first three chapters contain background material, assuming some familiarity with Riemannian geometry and functional analysis. CONTENTS: Infinite Dimensional Manifolds. Hamiltonian Systems. Elliptic Operators and Function Spaces. The Motion of an Incompressible Fluid. Turbulence and Chorin's formula. Symmetry Groups in Mechanics. Quantum Mechanical Systems. Completeness Theorems and Nonlinear Wave Equations. General Relativity as a Hamiltonian System. Linearization Stability of the Einstein Equations. Appendix: On the correspondence principle. Bibliography. Paperback; 273 pages. $8.00 in U.S.A., Canada, Mexico. $8.40 foreign. 3. The Atiyah-Singer Theorem and Elementary Number Theory F. Hirzebruch and D. Zagier, Bonn University This book is about the enigmatic relationship of certain corollaries of the Atiyah-Singer index theorem with some rather classical objects from the theory of numbers. CONTENTS: Topological Preliminaries: Background on complex manifolds; Signature theorems; the L-class of a rational homology manifold; The alpha invariant of Atiyah and Singer. Cotangent Sums and Related Number Theory: Elementary properties; Quadratic reciprocity and cotangent sums; modular forms; Markoff triples. Applications: The Signature theorems on low-dimensional manifolds; The action of Tn+l on Pn(C); Briskorn manifolds; The Browder-Livesay invariant of lens spaces. Bibliography. Paperback; 274 pages. $8.00 in U.S.A., Canada, Mexico. $8.40 foreign. 4. The Index Theorem and the Heat Equation Peter B. Gilkey, Princeton University This volume deals with the proof of the index theorem for the DeRham, Dolbeault, and Signature complexes using heat equation methods. The first three chapters develop in detail the necessary calculus of pseudo-differential operators. The fourth chapter proves the classi­ cal Chern-Gauss-Bonnet formula for the Euler characteristic. The fifth and sixth chapters sketch the proof of the corresponding results for the Dolbeault and Signature complexes. Paperback; 130 pages. $5.00 in U.S.A., Canada, Mexico. $5.40 foreign.

Spaces of Constant Curvature, third edition Joseph A. Wolf, University of California, Berkeley This volume, an exposition of Riemannian geometry and of some of the relations between geometry and group theory, gives a systematic account of results on the classification prob­ lem, many of which are due to the author, including new results on symmetric spaces and indefinite metric manifolds of constant curvature. Chapters 1, 2, and the first half of 8 form a concise introduction to modern differential geometry and Riemannian symmetric spaces. The third edition contains many corrections, and some additions, to the first two editions. CONTENTS: Introduction to Differential Geometry (Chapters 1,2); Crystallographic Groups and Flat Riemannian Manifolds (3); Representations of Finite Groups (4); Riemannian Quotient Manifolds· of Spheres (5, 6. 7); Riemannian Symmetric spaces (8); Riemannian Quotients of Symmetric Spaces (9, 10); Space Forms of Indefinite Metric Manifolds (11,12). Paperback; 408 pages. $8.00 in U.S.A., Canada, Mexico. $8.60 foreign. A Comprehensive Introduction to Differential Geometry Michael Spivak Volume 1. Paperback; 656 pages. $12.50 in U.S.A., Canada, Mexico. $13.00 foreign. Volume 2. Paperback; 425 pages. $10.50 in U.S.A., Canada, Mexico. $11.00 foreign. A-515 -·- Selected Tables in Mathematical Statistics Edilt•d h\ lht• lnslilull' ot '\lalht•malit:,II O.,f,Uislin.

A new series ... Each volume contains several sets of extensive tables.

A reprint- with corrections- of the volume first published Volume 1 in 1970 by Markham Publishing Company. (ISBN 0-8218-1901-1) Tables of the cumulative non-central chi-square distribution G. E. HAYNAM, Z. GOVINDARAJULU, and F. C. LEONE (78 pages) 403 pages Tables of the exact sampling distribution of the two-sample $8.60 list Kolmogorov-Smimovcriterion Dmn(m ;1; n) P. J. KIM and JENNRICH (92 pages) $6.45 for members R.I. Criticalualues and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test FRANK WILCOXON, S. K. KATTI, and ROBERTA A. WILCOX (90 pages) The null distribution of the first three product-moment statistics for exponential, half-gamma, and normal scores P. A. W. LEwis and A. S. GooDMAN (62 pages) Tables to facilitate the use of orthogonal polynomials for two types of error structures KIRKLAND B. STEWART (81 pages) Probability integral of the doubly noncentral t-distribution Volume 2 with degrees of freedom nand non-centrality parameters aand A WILLIAM G. BuLGREN (138 pages) (ISBN 0-8218-1902-X) Doubly noncentral F distribution- Tables and Applications M. L. TIKU (38 pages) 388 pages Tables of expected sample size for curtailed fixed sample size $14.10 list tests of a Bernoulli parameter $ 10.58 for members COLIN R. BLYTH and DAVID HUTCHINSON (22 pages) Zonal polynomials of order 1 through 12 A.M. PARKHURST and A. T. JAMES (190 pages)

In preparation. Contributions for future volumes are solicited. Volume 3 I Refer to preface in Volume 1 or Volume 2 for details.

Donald B. Owen, Chairman of the IMS Committee on Tables, is a co-editor with H. Leon Horter. James M. Davenport is managing editor. Order from Atrlfllit.tut. Mot.lt.f.ntof.i. ~ol'.i.d.g P. 0. Box 6248, Providence, R.I. 02940 Standing orders for this series are being solicited.

A-516 Graphs and Hyperoraphs An Introduction to by CLAUDE BERGE Complex Analysis in 1973. 542 pages. Dfl. 80.00 (about US$30.80) Since 1957 when the author published the Several Variables first modern book on graph theory, this field has expanded geometrically in depth and im­ Second revised edition portance. The first part of his new book deals by LARS HORMANDER with the present status of, and new trends in, graph theory from a unifying point of view. 1973. 226 pages. Dfl. 36.00 (about US$13.85) The second part offers a systematic study of Taken entirely from the viewpoint of the hypergraphs which both generalizes and theory of partial differential equations, this greatly simplifies a large part of the theory book develops the theory of analytic functions of finite graphs, at once providing a new line and, conversely, includes some applications of attack on the problems of graph theory. of analytic function theory to the study of over­ determined systems of partial differential equations. It is essentially identical in content Model Theory to a graduate course given by the author at by C. C. CHANG and H. JEROME KEISLER Stanford University in 1964. In this second edition the central existence theorems in 1973. 610 pages. Dfl. 85.00 (about US$32.70) Chapter IV have been further improved and Model theory is the branch of mathematical references have been added to a number of logic that deals with the connection between recent applications. a formal language and its interpretations, or CONTENTS: models. This book presents the model theory Analytic functions of one complex variable. of first order predicate logic, the language in Elementary properties of functions of several which the techniques of the subject were complex variables. Applications to commu­ originally developed and are still best ex­ tative banach algebras. L2 Estimates and plained. The early chapters of the book pre­ existence theorems for the c operator Stein sent the basic methods of constructing models manifolds. Local properties of analytic func­ and some applications. The methods are tions. Coherent analytic sheaves on stein diagrams, elementary chains, Skolem func­ manifolds. Bibliography. Index. tions, indiscernibles, ultraproducts and satu­ rated models. The later chapters contain more advanced topics which combine several of Locally Finite Groups these methods and apply them to algebra and set theory. by 0. H. KEGEL and B. A. F. WEHRFRITZ 1972. 220 pages. Dfl. 40.00 (about US$15.40) CONTENTS: An Introduction to Introduction. Basic Methods, Concepts and Examples. Centralizers in Locally Finite Functional Analysis Groups. Locally Finite Groups with MIN-p. by MISCHA COTLAR and ROBERTO CIGNOLI Locally Finite Simple Groups. Characteriza­ 1974. 600 pages. Dfl. 95.00 (about US$36.50) tions of the Groups PSL (2, F) and of Certain Paperback Dfl. 60.00 (about US$23.10) Locally-Soluble-by-Finite Groups. Universal Groups and Direct Limits of Symmetric A self-contained introduction to functional Groups. Bibliography. Index. analysis, this book was written in an attempt to awaken interest in the subject. Particular emphasis is placed on the didactical aspect, Graphs, Groups and on the motivation of concepts or ideas and on a detailed explanation of the background material. Surfaces by A. T. WHITE CONTENTS: Index of notation. Introduction. Convex sets 1973. 152 pages. Dfl. 25.00 (about US$9.60) and hyperplanes in vector spaces. Normed The theories of graphs, groups and surface spaces. Ordered vector spaces. Integration topology interact in many fascinating ways. theory. LP and C(K) spaces. Approximation This book explores several of these inter­ theory. Linear operators in normed spaces. actions, culminating in a discussion of quo­ Homomorphisms and Fredholm operators. tient manifolds, with applications to graph Banach algebras, spectral formula and Fourier imbeddings, depictions of groups in surfaces transforms. Appendix. References. Index. and map-coloring problems.

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A-517 mE UNIVERSITY OF CALIFORNIA I"iDe Campus

seeks to appoint a Dean of the School of Physical Sciences effective July 1, 1975. The responsibilities of the Dean are that of academic Dean of the School and administrative head of the School, which is of Mathematics and Science comprised of the Departments of Chemistry, Physics, and Mathematics; the Dean admini­ MONTCLAIR STATE COLLEGE strates the School in all its educational research, and business functions, and represents the A college of 7000 undergraduate and 4600 School to the Chancellor of the Irvine Campus graduate students, located 1/2 hour from New and the Vice-Chancellor of Academic Affairs. York City. Applicant should have a terminal The position calls for a person having strong degree in one of the following: biological or educational interests; strong research interests; physical sciences or mathematics. Proven a record of outstanding achievement in the scholarship and administrative experience; concern for students and the quality of teaching. physical sciences and in administration; and Starting salary no higher than mid-point a forward-looking outlook concerning the future development of the School of Physical Sciences. of the range $21,876- $29,534. The position is an academic appointment and Position open after February 1, 1975. the Dean is eligible for a tenure appointment Send applications and supporting documents in the professorial series. Applications or to: Dr. PhilipS. Cohen nominations may be addressed to: Search Committee, Russ Hall Chairman of the Search Committee for the Dean of Physical Sciences MONTCLAIR STATE COLLEGE Department of Chemistry Upper Montclair, N.J. 07043 UNIVERSITY OF CALIFORNIA Equal Opportunity/ Affirmative Action Institution l"ine, California 92664 and should be received no later than October 31, 1974. Applications from or nominations of all qualified candidates are welcome; minorities UNIVERSITY OF NOTTINGHAM and women are encouraged to apply. STATISTICS Applications are invited from probabilists, stotisti· PROCEEDINGS OF SYMPOSIA cians ond econometricians for TWO posts of PROFESSOR, IN READER or SENIOR LECTURER. One appointment will be in Probability or Mathematical Statistics and PURE MATHEMATICS the second in Econometrics. Both appointments will be Volume XXIV within the Mathematics Department. The Mathematical Statistician will be responsible ANALYTIC NUMBER THEORY for teaching and research within the Mathematics edited by Harold G. Diamond Department. The Econometrician vacancy arises from the resignation of Professor C. W. J. Granger and This volume comprises the proceedings of a the appointee will have responsibility for teaching symposium on analytic number theory and and research within the Mathematics ond Economics related parts of analysis held at St. Louis Departments. The salary will be on the scale of £4,707 to University on March 27-30, 1972. The thirty £5,844 per annum for a Reader or Senior Lecturer papers included cover a broad spectrum of and a minimum of £5,973 for Professor, together contemporary work in number theory, and with F.S.S.U. it is hoped that the lively ideas disseminated Furtlier details and forms of application from by this volume will result in new number the Staff Appointments Officer theoretic research. UNIVERSITY OF NOTTINGHAM ISBN-0-8218-1424-1 CODE: PSPUM/XXIV University Park List price $23.00 Member price $17.25 Nottingham, England Ref. No. 361 340Pages American Mathematical Society

A-518 THE FRACTIONAL CALCULUS TREATISE ON ANALYSIS, Theory and Applications of Differentiation and VOLUME4 Integration to Arbitrary Order by J. A. DIEUDONNE by KEITH B. OLDHAM and JEROME SPANIER A Volume in the PURE AND APPLIED A Volume in the MATHEMATICS IN SCIENCE AND MATHEMATICS Series ENGINEERING Series TENTATIVE CONTENTS: Differential Calculus on a This book, which assumes a basic knowledge of Differential Manifold; Elementary Local Theory of classical calculus and elementary differential equa­ Differential Systems; Lie Groups and Lie Algebras; tions, extends the notions of ordinary multiple dif­ Principal Connections and Riemannian Geometry; ferentiation and integration to fractional (in fact, to Appendix: Tensor Products and Formal Power­ arbitrary) orders. It not only presents the theory Series. underlying the properties of such generalized deri­ 1974, 460 pp., vatives and integrals (termed "differintegrals") but $34.001£16.30 also illustrates some of the wide variety of fields to which these ideas are applied with profit-trans­ PROBABILITY THEORY mission line theory, chemical analysis of aqueous A Historical Sketch solutions, design of heat-flux meters, rheology of by L. E. MAISTROV soils, growth of intergranular grooves at metal sur­ translated and edited by SAMUEL KOTZ faces, quantum mechanics, and dissemination of atmospheric pollutants. A Volume in the PROBABILITY AND MATHEMATICAL STATISTICS Series 1974, 248 pp., $19.501£9.35 Because the history of probability theory has been A HANDBOOK on of the least investigated areas in the history of mathematics, this unique book-by a well-known OF INTEGER SEQUENCES Russian scholar-represents an important and long by NEIL J. A. SLOANE overdue addition to the scarce literature in the field. "Field guides are uncommon in mathematics. Here It does not attempt a systematic, detailed coverage is one, identifying with care some 2,300 sequences of all developments in probability theory. Instead, of positive integers, each with an implied-and it concentrates on problem areas which have not named-rule for continuation, world without end, been sufficiently studied, have not been presented unto infinity.... It is not evident how often people appropriately by earlier historians, or, in some in­ wish to identify some integer sequence fleetingly stances, have not been investigated at all-particu­ encountered in reading or shyly hidden in their own larly those areas of probability theory which were mathematical woodlands. They surely will seek this developed in Russia and which appear for the first book. Many another reader with mathematical bent time in English in this book. can profit by it as a starting point into a very wide 1974, 296 pp., $22.50/£10.80 variety of mathematical literature or merely as a challenge, a fresh scent along the trail of math­ SOFTWARE ematical truth. It does not tell too much; what it FOR does is exhibit a specimen, list a name and include NUMERICAL MATHEMATICS a pointer to more. Many an amateur will browse in Proceedings of the Loughborough University Con­ it and most libraries will want it." ference of the Institute of Mathematics and its Ap­ -Scientific American plications held in April 1973 1973, 220 pp., $10.001£4.70 edited by D. J. EVANS CONTRIBUTIONS Here is a book which is intended to promote the TO ANALYSIS exchange of information and ideas on mathemat­ ical software-a new discipline of great A Collection of Papers Dedicated to Lipman Bers importance in computing science. The book contains papers edited by LARS V. AHLFORS, IRWIN KRA, and discussions by internationally known authors BERNARD MASKIT and LOUIS NIRENBERG and research workers on algorithms in the fields of This book covers such topics as: Teichmi.iller linear algebra, sparse matrices, quadrature and in­ spaces and Kleinian groups; theta functions and tegral equations, ordinary and partial differential algebraic geometry; quasiconformal mappings and equations, optimization, curve and surface fitting, function theory; and differential equations and dif­ and special functions. In addition, papers given by ferential topology. Some of the papers are essen­ representatives of the Numerical Algorithm Group, tially expository, giving surveys of important recent Oxford, and the Argonne National Laboratory, Illi­ results in analysis, while others contain new re­ nois, deal with mathematical library organization sults. A number of them answer questions posed and portability, user documentation, and the in­ by Bers, and the influence of his work is evident in teraction between computer user and program almost all of them. libraries. 1974, 456 pp., $36.501£17.50 1974, 460 pp., $28.00/£10.80 ACADEMIC PRESS, INC. A Subsidiary of Harcourt Brace Jovanovich, Publishers 111 FIFTH AVENUE, NEW YORK, NEW YORK 10003 24-28 OVAL ROAD, LONDON NW1 7DX Ill 3 I -,\, / Introducing a new series ... ~ Lecture Ill 9 Notes ~ 3 • ...- w 1n ( Biomathematics Edited by S. Levin Cornell University, Ithaca, New York The new series Lecture Notes in Biomathematics ~__j deals with applications of mathematical methods to the life sciences. Since the development of C£_ mathematical biology has advanced through the introduction of computer techniques, thus making w the simulation and modelling of complex systems =? possible, these methods will be included. ~ Volume 1 Deterministic Threshold Models in the Theory of Epidemics By (:j P. Waltman University of Iowa, Iowa City, Iowa L 1974. v, 101p. 15 illus. paper/$6.20 (OM 16,-) C£_ ISBN 0-387-06652-7 0_ Forthcoming volumes Conference on Some Mathematical Problems ~ in Biology Edited by ?Q P. van den Driessche Conference on Mathematical Decision Problems ~ LL in Ecology Edited by ~ W. Lynn and A. Charnes :;! Summer School on Physics and Mathematics tJ ~ of the Nervous System ~ L Edited by ~ ConradI Gilttingen I Ci n ~ 1 iS - IQ ~ ~~~ ~ \0 • tJ! § rst I Springer-Verlag New York Inc. J ry 175 Fifth Avenue, New York, NY 10010 ~0~ ~ ~~ct J