OF THE

AMERICAN

MATHEMATICAL SOCIETY

VOLUME 15, NUMBER 6 ISSUE NO. 108

OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by Everett Pitcher and Gordon L. Walker

CONTENTS MEETINGS Calendar of Meetings ..•..•....•..•....•...... •..•.. 846 Program for the October Meeting in Baltimore, Maryland ..••...•... 847 Abstracts for the Meeting- Pages 903-908

PRELIMINARY ANNOUNCEMENTS OF MEETINGS ••••..••••••.•••••.• 850 ACTIVITIES OF OTHER ASSOCIATIONS ••••••.•.•..••••••••••••••. 854 POLITICAL PRESSURE VS SCIENTIFIC UNITY .•••••..•••••.•••••••. 855 MEMORANDA TO MEMBERS Mathematical Sciences Employment Register ...... •...•.... 857 Retired Mathematicians ..•.•.••.•••••...... •.•..•.•.... 857 Summer Employment Opportunities •••••••••••••••.....•.••••. 858 Periodicals Published by the London Mathematical Society ...•...... 858 DEFENSE RESEARCH: QUESTIONS FOR VIETNAM DISSENTERS •••••.•••• 859 LETTERS TO THE EDITOR •••••••••••••.•••••••••••••..•••••• 861 NEW AMS PUBLICATIONS ••••••••••••••••••••••.•••••.••••.•. 863 NEWS ITEMS AND ANNOUNCEMENTS • • • • • • • • • • • • • • 854, 856, 862, 865, 876

ANNUAL SALARY SURVEY ••••••••••••.••••••..•.•••••••••••. 869 PERSONAL ITEMS ••••••••••••••••••••••••.••••••...••••... 872 DOCTORATES CONFERRED IN 1967-1968 ••••••••••..••.•.•••••••• 877 ABSTRACTS OF CONTRIBUTED PAPERS ••••••••••••••••••••••••• 903

ERRATA • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • . • • • • • • . • • • • • • 949 INDEX TO ADVERTISERS • • • • • • • • • • • • • • . • • • • • . ••••••••••••.• 963 RESERVATION FORM .•••••••••••••••••.••••.•••••.•.•••••• 964 MEETINGS Calendar of Meetings NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the cA~t~

Meet­ Deadline ing Date Place for No. Abstracts*

66() November 8-9, 1968 Clemson, South Carolina Sept, 25, 1968 661 November 16, 1968 Riverside, California Sept. 25, 1968

662 November 29-30, 1968 Evanston, Illinois Sept. 25, 1968 663 January 23-27, 1969 (75th Annual Meeting) New Orleans, Louisiana Nov. 8, 1968

664 April 2-5, 1969 New York, New York Feb. 14, 1969 665 A;>ril 18-19, 1969 Chicago, Illinois Feb. 14, 1969

666 April 26, 1969 Santa Cruz, California Feb. 14, 1969 August 25-29, 1969 (74th Summer Meeting) Eugene, Oregon

January 22-26, 1970 (76th Annual Meeting) Miami, Florida August 24-28, 1970 (75th Sum mer Meeting) Laramie, Wyoming

January 21-25, 1971 (77th Annual M-eeting) Atlantic City, New Jersey

*The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadlines for by-title abstracts will be November I, 1968.

··~~···

The cNotiaiJ of the American Mathematical Society is published by the Society in January, February, April, June, August, Oc~ober, November ~nd December. ~rice per annual volume is $12.00. Price per copy $3.00. Special price for copies sold at reg1stratwn desks of meetmgs of the Soc1ety, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the_ ~uthority of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34,21, P. L. and R.). Accepted for mallmg at the spec1al rate of Postage provided for in section 34,40, paragraph (d).

Copyright©. 1968 by the American Mathematical Society Printed in the United States of America

846 Six Hundred Fifty-Ninth Meeting The Johns Hopkins University Baltimore, Maryland October 26, 1968

The six hundred fifty-ninth meeting buted ten-minute papers at 9:30a.m. and of the American Mathematical Society will at 3:15p.m. be held at The Johns Hopkins University in All sessions and registration will Baltimore, Maryland, on October 26, 1968. take place in Shaffer Hall. By invitation of the Committee to A section of the student cafeteria Select Hour Speakers for Eastern Sectional in Levering Hall serving soups, sand­ Meetings the following two lectures will be wiches, a small selection of salads, and presented: Professor Wilhelm Magnus of beverages will be open and can accommo­ New York University will speak on "Re­ date those attending the meeting. Restau­ sidually finite groups" at 11:00 a.m. Pro­ rants are also con.veniently located off fessor A vron Douglis of the University of campus. Maryland will speak on "The existence of Parking space will be available at weak solutions of first order partial dif­ the locations marked C, D, G, J, K, 4, and 7 ferential equations" at 2:00p.m. on the following map. There will be sessions for contri-

847 PROGRAM OF THE SESSIONS The time limit for each contributed paper is 10 minutes. The contributed papers are scheduled at 15 minute intervals. To maintain this schedule, the time limit will be strictly enforced.

SATURDAY, 9:30 A. M. Session on Applied Mathematics and Analysis, Shaffer Hall, Room 1 9:30-9:40 ( 1) Modified Runge-Kutta-Nystrom methods Professor Srisakdi Charmonman* and Mr. D. G. Barry, University of Alberta (659-5) 9:45-9:55 (2) Holor representation of nonsinusoidal quantities Professor Parry Moon, Massachusetts Institute of Technology, and Pro­ fessor D. E. Spencer*, University of Connecticut (659-15) 10:00-10:10 ( 3) Identification and confidence regions Mrs. F. M. Carroll, Raytheon Company, Sudbury, Massachusetts (659-16) 10:15-10:25 (4) A Fubini-,jessen theorem for the generalized Lebesgue-Bochner integral Professor Witold Bogdanowicz, Catholic University of America, and Pro­ fessor Vernon Zander*, West Georgia College (659-13) 10:30-10:40 ( 5) An algebraic characterization of distributions Professor R. A. Struble, North Carolina State University, Raleigh (659-9)

SATURDAY, 11:00 A.M. Invited Address, Shaffer Hall, Room 3 Residually finite groups Professor Wilhelm Magnus, New York University

SATURDAY, 2:00P.M. Invited Address, Shaffer Hall, Room 3 The existence of weak solutions of first order partial differential equations Professor Avron Douglis, University of Maryland

SA TU RDA Y, 3: 15 P. M. Session on Analysis, Shaffer Hall, Room 1 3:15-3:25 ( 6) On a problem of C. Renyi concerning Julia lines Professor K. F. Barth and Professor W. J. Schneider*, Syracuse Univer­ sity (659-17) 3:30-3:40 (7) Analytic continuation of local functional equations. I Mr. Bernard Berlowitz, University of California, Berkeley ( 659 -12) 3:45-3:55 (8) Two renorming constructions related to a question of Anselone Professor Victor Klee, University of Washington (659-14)

*For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

848 4:00-4:10 ( 9) Some applications of Hewitt's factorization theorem Professor H. S. Collins and Mr. W. H. Summers*, Louisiana State Uni­ versity (659-1) 4:15-4:25 ( 10) Products of representations of discrete groups Mr. J. C. Bradley, Westinghouse Electric Corporation, Baltimore, Maryland (659-4) 4:30-4:40 ( 11) On semi-invariant measure on a locally compact semigroup Professor S. G. Bourne, University of California, Berkeley ( 659-11)

SATURDAY, 3:15 P.M. Session on Topology, Algebra and Logic, Shaffer Hall, Room 2 3:15-3:25 (12) Products of ideals. Preliminary report Professor C. E. Aull, Virginia Polytechnic Institute (659-3) 3:30-3:40 (13) !-regular semigroups. I Professor R. J. Warne, West Virginia University (659-8) 3:45-3:55 (14) A configuration related to symmetric block designs. Preliminary report Professor A. T. Butson, University of Miami (659-10) 4:00-4:10 (15) On permanents of v-k- X configurations Mr. R. B. Levow, University of Pennsylvania (659-2) (Introduced by Professor H. S. Wilf) 4:15-4:25 (16) On the richness of subtheories of ZF + GCH Professor J. H. Harris, Stevens Institute of Technology (659-7) Herbert Federer Providence, Rhode Island Associate Secretary

849 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Six Hundred Sixtieth Meeting Clemson University Clemson, South Carolina November 8-9,1968

The six hundred sixtieth meeting new University Dining Hall, both on camp­ of the American Mathematical Society us. Coffee and doughnuts will be served will be held at the Clemson University, each morning in the lounge of the Mathe­ Clemson, South Carolina, November 8-9, matics Building. An informal social is 1968. planned for Friday evening in lieu of a By invitation of the Committee to banquet. Select Hour Speakers, there will be three The Clemson House is a completely invited addresses. Professor C. H. Ed­ modern, university owned, hotel located wards, Jr. of the University of Georgia on campus within a five minutes' walk will present an address entitled "Piece­ from the Mathematics Building. The hotel wise linear embedding and unknotting will honor all requests for accommodations problems"; Professor Malcolm Robert­ up to 300. Rates are as follows: son of the University of Delaware will Single $1 address the meeting, the title of his talk o.oo 14.00 will be "Quasi-subordination and coeffi­ Double cient conjectures"; Professor Laszio Additional accommodations within Fuchs of Tulane University will present the Clemson area, convenient to the Uni­ an address entitled "Some recent develop­ versity are: ments in abelian groups." The three SENCONEE invited addresses are scheduled to be MOTEL Seneca, S.C. - By-pass held in the Chemistry Auditorium in U.S. 123 (6 miles west of Clemson Brackett Hall. on U.S. 123) Single $ 8.00 ( 1 bed) Abstracts of contributed papers Double 14.00 (2 beds) should be sent to the American Mathe­ matical Society, Providence, Rhode Island, HOWARD JOHNSON'S MOTOR LODGE so as to arrive prior to the deadline date Anderson, S.C.- Junction U.S. 76andi-85 of September 25. ( 11 miles south of Clemson) The Registration Desk will be lo­ Single $ 9.00 cated in the lobby of Martin Hall, Mathe­ Double 14.00 ( 1 bed) matics Building. Registration hours will 16.00 (2 beds) be 9:00-5:00 Friday, November 8, and HOLIDAY 9:00-12:00 noon Saturday, November 9. INN Anderson, S.C. - 3025 North Clemson, South Carolina, is located Main Street (16 miles south of Clemson on U.S. approximately 30 miles southwest of 76) Single Greenville, South Carolina, on U.S. 123 $ 8.50 Double 13.00 (1 bed) and 11 miles northwest of Interstate 85 18.00 (2 beds) on U.S. 76. The Greenville airport is served by Eastern Southern, and Piedmont Reservations should be made .!!!.: airlines. Transportation will be furnished rectly with the Clemson House or one of from the Greenville airport. Clemson is the motels. served directly by the Southern Railway 0. G. Harrold and by Greyhound and Trailways buses. Associate Secretary Luncheon and snacks will be avail­ able at the Clemson House hotel and the Tallahassee, Florida

850 Six Hundred Sixty-First Meeting University of California Riverside, California November 16, 1968

The six hundred sixty-first meet­ CARAVAN MOTEL ing of the American Mathematical Society 1860 East University Avenue will be held at the University of California, Single $ 9,00 - $11.00 Riverside, in Riverside, California, on Double 12,00 - 14,00 November 16, 1968, FARM HOUSE MOTEL By invitation of the Committee to 1393 East University Avenue Select Hour Speakers for Far Western Single $ 6,00 Sectional Meetings, there will be two hour Double 8,00 addresses at this meeting. Professor joseph A. Wolf of the University of Cali· MISSION INN fornia, Berkeley, will lecture at 11:00 a.m. 3649 7th Street on Saturday. The title of his talk is Single $ 8,00 - $14,00 "Geometry and analysis on complex flag Double 12,00 - 20,00 manifolds." Professor Robert Osserman RAMADA INN of Stanford University will speak at 2:00 1150 East University A venue p.m. on Saturday. His lecture is entitled Single $10,00 - $12,00 "Minimal varieties," There will be ses­ Double 11,00 - 13,00 sions for contributed papers on Saturday morning and afternoon. All sessions of TRAVELODGE MOTEL the meeting will be held in Watkins Hall, 1911 East University Avenue The deadline for abstracts of contributed Single $ 7,00 - $ 8,00 papers to be published with the program Double 10,00 - 12,00 of the meeting is September 25, 1968, With the exception of the Mission Inn, the However, late papers will be accepted for above motels are within walking distance presentation at the meeting. of the University campus. Reservations Registration for the meeting will should be made directly with the pre· begin at 8:30 a.m. on Saturday. The ferred motel or hotel. registration desk will be located adjacent Air West has flights to the River­ to Watkins Hall, which is just west of the side Municipal Airport, which is approxi· carillon tower on the Riverside campus. mately six miles west of the University. Riverside has a good selection of Helicopter service is available from the motels and hotels, including the folloWiJ1g: Los Angeles International Airport to the Riverside Heliport via Los Angeles Air­ AMERICAN MOTEL LODGE ways 1358 East University Avenue R. S. Pierce Single $ 8,00 up Associate Secretary Double 12,00 up Seattle, Washington

851 Six Hundred Sixty-Second Meeting Northwestern University Evanston, Illinois November 30, 1968

The six hundred sixty-second meet­ The registration desk for the meet­ ing of the American Mathematical Society ing will be located at the main entrance will be held on Saturday, November 30, of the Technological Institute Building. 1968, at Northwestern University in This desk will be open from 8:30 a.m. to Evanston, Illinois. All sessions will be 4:00 p.m. on Friday, November 29, and held in the lecture rooms of the Techno­ from 8:30 a.m. to 3:00 p.m. on Saturday, logical Institute of Northwestern Univer­ November 30. The Technological Institute sity. is located on Sheridan Road at Noyes By invitation of the Committee to Street, in Evanston. Parking will be avail­ Select Speakers for Western Sectional able in two nearby parking lots. Meetings, there will be two one-hour ad­ The hotel headquarters for the dresses. Professor Donald G. Higman of meeting will be North Shore Hotel, 1611 the University of Michigan will speak at Chicago A venue, Evanston, Illinois 60201. 11:00 a.m. on the topic" Finite permutation A form for requesting accommodations groups of small rank." Professor Wu- Yi may be found on page 964 of these c}/oficei). Hsiang of the University of Chicago, the Those desiring accommodations should Institute for Advanced Study, and the Uni­ complete this reservation form or a versity of California, Berkeley, will ad­ reasonable fa.csimile thereof and send it dress the Society at 1:45 p.m. His subject to the North Shore Hotel at the above ad­ will be "Recent developments in compact dress. The rates are $8.50 for single differentiable transformation groups." By occupancy and $10.50 for double occupancy. invitation of the same committee, there On Friday, November 29, 1968, will be two special sessions of twenty­ Northwestern University will sponsor a minute papers. Professor Antoni Zyg­ symposium on Wiener-Hop! Equations and mund of the University of Chicago will Toeplitz Operators. This symposium will arrange one such session on the subject be in honor of the work of Professor Gabor of differentiation theory, this session to Szego of Stanford University. The list of be held at 8:45 a.m. Professor Hans speakers will include Professor Ronald G. Zassenhaus of the Ohio State University Douglas of the University of Michigan, will arrange the other special session on Professor Albert E. Heins of the Univer­ the subject of Computational Methods in sity of Michigan, Professor Henry Helson Abstract Algebra, this special session to of the University of California, Berkeley, be held at 3:00 p.m. Professor I. I. Hirschman, jr ., of Wash­ There will also be sessions for the ington University and Duke University, presentation of contributed ten-minute Professor Marvin Rosenblum of the Uni­ papers both morning and afternoon. Ab­ versity of Virginia, and possibly one or stracts of contributed papers should be two others. sent to the American Mathematical Society, Paul T. Bateman P .0. Box 6248, Providence, Rhode Island Associate Secretary 02904, so as to arrive prior to the dead­ line date of September 25, 1968. Urbana, Illinois

852 Seventy-Fifth Annual Meeting Jung Hotel New Orleans, Louisiana January 23- 26, 1969

The seventy-fifth annual meeting of in the Providence office by October 15. the American Mathematical Society will be The abstract should be plainly marked for held at the Jung Hotel in New Orleans, consideration for twenty-minute papers. Louisiana, in conjunction with the Mathe­ The Council of the American Mathematical matical Association of America and the Society has set no limit on the number of Association for Symbolic Logic. Hour ad­ ten-minute papers that will be accepted dresses will be given by Professor Jurgen for presentation in the regular sessions for Moser of the Courant Institute of Mathe­ contributed papers at the annual meeting. matical Sciences, New York University,and The deadline for abstracts to be received Professor C. C. Moore of the University in the Providence office is November 8, of California, Berkeley, California. 1968. The general program is outlined The Registration Desk for this as follows: meeting will be in the Convention Lobby. The Desk will be open from 2:00 to 8:00 January 22-23 ASL Sessions p.m. on Wednesday, January 22; from January 23-26 AMS contributed papers 8:00 a.m. to 5:00 p.m. on Thursday, and invited addresses January 23; from 9:00 a.m. to 5:00p.m. January 25-26 MAA-NCTM joint ses­ on Friday through Sunday, January 24-26; sions and from 9:00a.m. to 3:00p.m. on Monday, January 25-27 MAA sessions January 2 7. The registration staff will There will be two special sessions register those attending the sessions of the of twenty-minute papers. One will be on Association for Symbolic Logic only on Geometry in Mathematics, organized by Wednesday from 9:00 a.m. to 2:00 p.m. Professor Preston C. Hammer of Penn­ The Mathematical Sciences Em­ sylvania State University. Professor W. H. ployment Register will be maintained from J. Fuchs of Cornell University will or­ 9:00 a.m. to 5;00 p.m. on Friday, Saturday, ganize the other session on Complex Sunday, and Monday in the Tulane Room. Analysis. In the past a few contributed papers have been chosen by the organizing 0. G. Harrold chairman to be added to the program. Associate Secretary Anyone desiring to have a paper considered in this capacity should have his abstract Tallahassee, Florida

853 ACTIVITIES OF OTHER ASSOCIATIONS

TWELFTH BIENNIAL INTERNATIONAL SEMINAR OF THE CANADIAN MATHEMATICAL CONGRESS

The 12th Biennial International will be "Time series and stochastic pro­ Seminar of the Canadian Mathematical cesses," although other areas of mathe­ Congress will be held at the University of matics will be represented. Traditionally, British Columbia in Vancouver, Canada, the Canadian Biennial Seminars embrace from August 11 through August 27, 1969. two series of lectures: This Seminar will be followed by the an­ 1. A series of research lectures nual meeting of the Canadian Mathemati­ given at the post-doctoral level by four cal Congress at the University of Vic­ internationally recognized research math­ toria, in Victoria, Canada, from August 28 ematicians. through August 30. Members of the Pro­ 2. A series of instructional lectures gram Committee are S. A. Jennings, A. given at the pre-doctoral level. Joffe, J. G. McGregor, R. Pyke (chairman), Seminar participants, both graduate and D. A. Sprott. Members of the Arrange­ students and post-doctoral fellows, will ments Committee are R. Harrop, S. A. be chosen primarily from Europe, the Jennings (chairman), E. Kennedy, B. N. United States, and Canada. Moyls, and J. J. McNamee. Further details will be announced The main theme of the Seminar at a later date. ~ NEWS ITEMS AND ANNOUNCEMENTS

INTERNATIONAL CONFERENCE ON Ergodic Theory, especially in Entropy FUNCTIONAL ANALYSIS AND of Dynamical Systems (as developed by A. RELATED TOPICS, 1969 Kolmogorov and V. I. Arnold). As of June, 1968, the following mathe­ An International Conference on Func­ maticians have tentatively agreed to give tional Analysis and Related Topics will be hour talks: S. Agmon, M. F. Atiyah, J. held in Tokyo, Japan, from Aprill-AprilS, Delsarte, J. L. Doob, C. Foia~, L. Gard­ 1969. The Conference has been organized ing, K. Ito, L. Hormander, S. Karlin, T. by the Mathematical Society of Japan, co­ Kato, P. D. Lax, H. Lewy, J. L. Lions, sponsored by the International Mathemati- A. Martineau, C. B. Morrey, J. K. Moser, cal Union and the Japan Science Council. M. S. Narasimhan, R. S. Phillips, E. T. A selection from the scheduled talks Poulsen, L. Schwartz, L. Nirenberg, G. follows: Stampacchia. Partial Differential Equations (includ­ In addition, there will be approximate­ ing the Control Theory as developed by ly twenty lectures by Japanese mathe­ J. L. Lions and G. Stampacchia). maticians. Differential Equations on Manifolds The registration fee is $10.00, which (Atiyah-Singer Theory). includes a subscription to the proceedings. Theory of Hyperfunctions (of M. Sato Communications should be addressed to as developed recently by A. Martineau Miss N. Oita, Department of Mathematics, and H. Komatsu). F acuity of Science, University of Tokyo, Markov Processes and Potentials. Hongo, Tokyo, Japan.

854 POLITICAL PRESSURE vs SCIENTIFIC UNITY

A POINT OF VIEW* on debt, veterans' payments, etc.) and takes Excerpt from the address by Donald it for granted that the question is "what are F. Hornig, Presidential Science Adviser, the needs of science?" rather than "what upon receipt of the first Mellon Institute are the nation's needs for science?" This Award given at the Carnegie-Mellon Uni­ blithe spirit leads mathematicians to seri­ versity, Pittsburgh, Pennsylvania on 10 ously propose that the common man who May. pays the taxes ought to feel that mathe­ matical creation should be supported with Why then do I ask, "Is there a public funds on the beaches of Rio de cns1s in science?" To judge by my cor­ Janeiro or in the Aegean Islands. I don't respondence, there is surely a financial doubt for a minute that mathematical crisis in our universities as science bud­ creation is possible under those circum­ gets have leveled off in the last few years stances; it may even be improved. But the and are now threatened with decline, But public which pays the bill is not in tune the question I'd like to ask is whether with such colossal intellectual conceit, there is something deeper? After two de­ In short, the scientific community cades of adventurous progress, the Con­ has done much to alienate itself from the gress and the public ask whether we can society which supports it. afford it after all. We now find ourselves pulling back from the exploration of space, slowing down on the development of uni­ versities, and even holding back on health research .... The Council of The American Math­ Congressmen look at this violent ematical Society at their August meeting expansion of the last two decades and ask empowered Charles B. Morrey, Jr., to "Where will it ever end?" I should mention communicate this letter to the editor of that at the rate of expansion of the 1950's Science Magazine. and early 1960's one could have predicted To the Editor of SCIENCE** that it would have ended with every man, woman and child in the United States doing Many members of the mathemati­ research. But now many Congressmen and cal community were dismayed and shocked members of the public are more immedi­ by reading the excerpts of the speech by ately concerned with possible overlap, Dr. Donald Hornig,the Presidential Science duplication and proper accounting proce­ Adviser, which were published in the dures, than they are with the quality and July 19 issue of SCIENCE. The invidious vitality with which the frontier is ex­ comments therein about mathematics and plored, They see a scientific community mathematicians are completely uncaUed which, insisting on its purity, will not for. Implicit in Dr. Hornig's remarks about deign to communicate with the public and vacations on the beaches of Rio or the justify itself, but prefers to believe that its Aegean Islands was a thinly veiled attack virtues are so self-evident that a right­ on Dr. Stephen Smale. The allegations. minded society must necessarily support against Smale were adequately disproved it on its own terms. Theyseeacommunity by Daniel Greenberg in his articles in which, in industry, government labora­ SCIENCE on the Smale-NSF controversy. tories and the universities, consumes 1/4 Dr .Hornig singles out mathematics of the disposable Federal budget (the in suggesting that the scientific community manageable part, not counting the interest * Reprinted from the July 19th issue of ** This letter has been reprinted from Science Magazine with the permission of Science Magazine with the permission of the American Association for the Advance­ the American Association for the Ad­ ment of Science. vancement of Science.

855 is one "which, insisting on its purity, will government agencies has in ten years not deign to communicate with the public tripled the annual number of Ph.D.'s in and justify itself ... " Quite on the contrary, mathematics. These young mathematicians many of the branches of science have re­ have gone on to make a variety of profound cently prepared extensive reports on their and original discoveries ranging from disciplines. In particular, the mathemati­ those with direct practical applications cal community through the medium of the (such as the unexpected use of logic in COSRIMS committee has just completed a computing machines), to theoretical re­ comprehensive report, designed for the sults which have attracted international public and Congress, on the current prob­ acclaim (such as the work of Smale in lems of mathematical research and their dynamics and of Cohen in foundations). relations to the national goals. Aware of The future does involve a financial crisis the ever increasing need and utility of which may have tragic consequences for mathematics in everyday life, this same the next generation of scientists and math­ community embarked on large programs ematicians. Dr. Hornig's statement is (beginning with the "new math") to improve hardly a responsible contribution to the mathematical education in the elementary public discussion of this crisis or to the and secondary schools. Surely Dr. Hornig closing of the gap between working scien­ is aware of these--and other--steps to tists and those in government concerned communicate with the public. with science. Government support of science has been a vital element in the rapid Charles B. Morrey, Jr. development of American science. This President is particularly the case in mathematics American Mathematical Society where support by the NSF and other August 29, 1968. ~ NEWS ITEMS AND ANNOUNCEMENTS

COMPUTER SCIENCE AT THE remain to be filled. Those interested in UNIVERSITY OF KANSAS such positions or other information about the program should write to Professor Professor E. J. Schweppe has been E. J. Schweppe, Chairman, Computer Sci­ named chairman of the new computer ence, Strong Hall, The University of science department at the University of Kansas, Lawrence, Kansas 66044. Kansas. This will be an independent depart­ ment responsible only to the Dean of F acuities and will be a part of no other school within the University. Until formal approval has been received from the INDIAN SEMINAR ON Board of Regents, the department will be STOCHASTIC THEORY administered by the Committee on Comput­ er Science. The Institute of Mathematical Sci­ Looking beyond the eighteen courses ences, Adyar, Madras, India, will hold a to be offered this fall, future plans include seminar entitled "The Methods of Stochas­ a Master of Science degree in Computer tic Theory and Their Applications." The Science (which is expected to be approved seminar will run for three weeks from this coming academic year) and a doctoral September through October, 1968, and will program (which is seen for the future, be divided into three sections with a main when a more adequate faculty can be lecturer in each field. assembled). Interdepartmental coopera­ Professor Alladi Ramakrishnan will tion with the departments of mathematics speak on Mathematical Methods. and engineering and with the School of Professor I. Prigogine will speak on Business are slated for immediate consid­ Mechanics. eration. ProfessorS. K. Srinivasen will speak Additional positions at several levels on Applications of Stochastic Theory.

856 MEMORANDA TO MEMBERS

MATHEMATICAL SCIENCES EMPLOYMENT REGISTER

The Tulane Room in the Jung Hotel made for anonymity of applicants upon in New Orleans, Louisiana, will be the lo­ payment of $5.00 to defray the cost in­ cation of the Mathematical Sciences Em­ volved in handling such a listing. Appli­ ployment Register during the annual meet­ cants and employers who wish to be listed ing. The Employment Register will be should write to the Mathematical Sciences open for four days, January 24 through Employment Register, Post Office Box January 27, 1969, from 9:00a.m. to 5:00 6248, Providence, Rhode Island 02904, for p.m. applicant qualification forms or position The Employment Register is spon­ description forms. These forms must be sored by the American Mathematical So­ completed and returned to the Employ­ ciety, the Mathematical Association of ment Register not later than December 15, America, and the Society for Industrial and 1968, in order to be included in the Janu­ Applied Mathematics for the purpose of ary lists. establishing communication between math­ Those forms which arrive too late ematical scientists available for employ­ to be included in the printed lists are ment and employers with positions to fill. taken to the meeting where they may be As part of the service, interviews between seen by applicants and/or employers who applicants and employers are arranged. are interested in them. The printed lists Registration for the Employment will be mailed to subscribers during the Register is separate and apart from meet­ first week in January. Lists can be ordered ing registration, and it is, therefore, most from the Employment Register office in important that both applicants and em­ Providence. They will also be available at ployers sign in at the Employment Regis­ the meeting. ter desk as early as they can on Friday A subscription to the lists, which morning. No appointments will be sche­ includes three issues (January, May, and duled for Friday, however. A separate August) of both the applicants list and the visual index will be maintained for Em­ positions list is available for $25.00 a ployment Register use only. Appointments year; the individual issues of both lists will be scheduled only for people who have may be purchased in January, May, and actually signed in at the Register. Re­ August for $12.50. A subscription to the quests for appointments can be submitted applicants list alone or single copies of on any or all of the days the Employment that list are not available. Copies of the Register is open. positions list only may be purchased for There is no charge for registration $3.00. Checks should be made payable to except when the late registration fee of the American Mathematical Society and $5.00 is applicable. Provision will be sent to the address given above.

RETIRED MATHEMATICIANS

The List of Retired Mathematicians annual meeting in New Orleans, Louisiana, A vail able for Employment will once again January 23-27, 1969. Retired mathemati­ be published in January 1969 and will be cians who are interested in being included distributed to subscribers to the Employ­ in the list may either request a form ment Register lists when the January from the Employment Register office or issue is mailed. Besides being available send the following information: name, to subscribers, the list is available on date of birth, highest degree earned and request from the Employment Register where it was obtained, most recent em­ office. Copies will also be available atthe ployment, present address, date available,

857 references, preference for academic or ceipt of either the completed form or the industrial employment, and geographic above information is January 1, 1969. location preferred. The deadline for re-

SUMMER EMPLOYMENT OPPORTUNITIES

The 1969 List of Opportunities for no charge for the list. Academic institu­ Summer Employment for Mathematical tions, industrial concerns, and government Scientists and College Mathematics Stu­ agencies that have summer openings and dents will be available in January 1969 would welcome applications from mathe­ and will be sent to subscribers to the maticians and students of mathematics Employment Register lists. Copies will may request forms for listing from the also be available on request from the Em­ Employment Register office. The deadline ployment Register office and during the for receipt of forms is January 1, 1969. annual meeting in New Orleans. There is

PERIODICALS PUBLISHED BY THE LONDON MATHEMATICAL SOCIETY

To alleviate increasing pressure on Taking into account the changes in publication space, the London Mathemati­ publications, the By-laws of the Society cal Society is founding a new periodical, have recently been altered. In the future, to be called the Bulletin of the London members will be entitled to receive free Mathematical Society. Volumes of the of charge, The Bulletin of the LMS, and Bulletin will be published annually in three any one of the following: The Journal of parts, beginning with volume 1 in 1969. the LMS, The Proceedings of the LMS, or The Bulletin will contain survey articles, the Journal of Applied Probability. Mem­ short research papers of exceptional in­ bers may obtain extra publications for a terest, as well as obituaries, reviews, subscription fee of $5.20 for each addi­ and other material of interest to mathe­ tional periodical. maticians. By reciprocity agreement members The present series of the Journal of the American Mathematical Society, of the LMS will terminate with volume 44, not residing in the United Kingdom or which will be published in 1969. The Ireland, will be admitted to full member­ Journal of the London Mathematical So­ ship in the LMS by payment of an annual ciety, second series, will commence pub­ subscription fee of $7. 75. Persons desir­ lication in 1969. In its new form the ing such membership should write to Journal will contain only research papers Dr. D. E. Cohen, Honorary Secretary of of short to medium length. There will be the LMS, Queen Mary College, London, no change in the Proceedings of the Lon­ E. 1., and indicate that they are members don Mathematical Society, third series. of the American Mathematical Society.

858 DEFENSE RESEARCH: QUESTIONS FOR VIETNAM DISSENTERS Defense Department agencies which man of the department of statistics at the support basic research in universities are University of California, Berkeley. Ney­ beginning to question renewal of contracts man is also understood to be principal with investigators who are publicly criti­ investigator in an ONR-supported atmo­ cal of the Vietnam war. spheric research project and to be an In recent weeks letters have gone addressee of one of the ONRletters. out from the Army Research Office and At the time of the Science deadline Office of Naval Research {ONR) raising Neyman was out of the country and LeCam the matter with several mathematicians declined to release the text of the Army who are principal investigators on un­ letter in his colleague's absence. However, classified basic research projects. No news of the letter, sent early in September, projects have been cancelled, but, in one has been circulating in the mathematics case which is apparently unique, the community in Washington and beyond, as Army Research Office has told investiga­ has word of the ONR letters. tors that a decision to terminate the re­ The LeCam-Neyman project is lationship when the contract expires funded at $33,000 for the currejlt year. "appears to be consistent with both our The relationship with the Army is now 16 positions." The ONR letters reportedly years old, and the funds normally pay are less specific, but ONR is asking a summer salaries for LeCam and Neyman few of its contractors who have taken a and support four or five graduate students public position against the war whether engaged in work on the project. From time they feel they should continue under ONR to time junior faculty members have also auspices. been partly supported by the federal funds. Agency action was apparently trig­ The Army letter pointedly notes that gered by a paid announcement, with about the results of the research done under the 345 signatures, in the August cffoticeiJ of contract have been used in various activi­ the American Mathematical Society {AMS). ties related to the Vietnam war and that The text of the announcement was as consultation with the principal investiga­ follows: "MATHEMATICIANS: Job oppor­ tors has also been useful. tunities are announced in the c}/oticei) of Then the letter goes on to say that the AMS~ in the Employment Register, and the mathematicians have every right to elsewhere. We urge you to regard your­ their own opinions and convictions, but selves as responsible for the uses to that their relationship with the Defense which your talents are put. We believe Department must be an "embarrassment," this responsibility forbids putting mathe­ and that, in view of the "unfortunate cir­ matics in the service of this cruel war." cumstances," a mutually acceptable de­ The same announcement appeared cision to end the contract when it expires in the cffoticei] twice in 1967 and again in next June appears consistent. january of this year, when there were Spokesmen for both the Army and some 94 signers. As far as SCIENCE can the ONR say that the researchers affected now determine, only signers of the an­ are first-rate mathematicians and should nouncement are involved in the ARO and have no difficulty in gaining support. from ONR initiative. nondefense agencies like the National Sci­ Affected by the Army action is a ence Foundation, an assumption that might contract for mathematics research held be viewed as optimistic in this tight-bud­ by professors Lucien LeCam and j. Ney- get year. The Army says it stands ready

* Reprinted from the September 20, 1968, issue of SCIENCE with the permission of the American Association for the Advancement of Science.

859 to do what it reasonably can to help if There are signs within the Pentagon termination of the project creates hard­ of reluctance to push for a confrontation. ships for graduate students. The tone of the Army letter, which might Sources in ONR say that four letters be described as overconsiderate, suggests have been sent to mathematicians who this. Civilian scientists in military agen­ signed the c}/oticri) protest, but these cies have close ties with the scientific sources declined to name the persons community and are reluctant to alienate involved or make public the text of the their university colleagues. letter, since the letters were mailed only And there is a tradition, within the last week and might not have reached the military, of supporting basic research and addressees. of dealing with the "best people." In practi­ ONR Chief Scientist Peter King cal terms, the military derives important said the letters represent an "isolated in­ secondary benefits from supporting aca­ cident" and not a new policy. King, like demic research. Military applications of other officials in research-supporting contract research results are brought to agencies, declined to elaborate on policy Pentagon attention by the investigators, implications of the actions. The Air Force who also are available as consultants. Office of Scientific Research supports re­ Contract work provides an excellent re­ search by some signers of the c}/oticri) ad, cruiting ground for young scientists, math­ but says it has no letters in the works. ematicians, and engineers. And there is It is known that the question of what always an awareness that clos~ liaison course to follow in responding to public with the universities is desirable against criticism of the war by contract re­ the day of possible total mobilization. For searchers has been under discussion in the these reasons and others, responsible Pentagon for some time. It is understood officials might regret being pushed to­ that a number of university researchers ward a policy which would increase ten­ who are contractors or consultants to sions between the military and university Defense agencies have signed published researchers. petitions of the "Stop-the-Bombing" vari­ There can be little doubt, however, ety with apparent impunity. The ad in the that the letters introduce a new strain into AMS c}/oticri) however, seems to have the relationship. It is too early to gauge struck a nerve in the Pentagon, perhaps be­ the reaction of mathematicians at large. cause it appeared in a professional publi­ SCIENCE talked to several mathemati­ cation and amounted to an appeal to mathe­ cians who signed the c}/oticri) announce­ maticians not to do defense-related re­ ment, and few expressed surprise. Not search. untypical was the remark of one that, Agency action is probably partly "if people criticize the Army, Navy and explicable in terms of congressional dis­ Air Force, it's perfectly well for the pleasure with campus critics of the war, Army, Navy and Air Force to reconsider reflected in legislative sanctions against doing business with them." But there was those who receive federal funds and engage little evidence of sympathy for those re­ in protest against the war. Protests against sponsible for managing military- supported classified research in the universities have university research during an undeclared also roused the ire of some legislators. war which is unpopular in the universities. Within the Pentagon the view seems ONR and ARO had the candor to put their to be that basic research is supported question in writing. Now quite a few people because of its ultimate value for military are wondering whether this "isolated in­ operations, and that, if researchers protest cident" could escalate. current operations, the Department of Defense is justified in raising the question of conscience. John Walsh

860 LETTERS TO THE EDITOR

Editor, the cJ{oticei) speech as given at Carnegie-Mellon Uni­ versity on May 1 O, 1968, we find that the R. P. Boas' letter (these cJ{oticei) 15, excerpts printed by SCIENCE (though on a 711) should not go without comment. Boas higher rhetorical level than the rest of opposes acknowledgements of thanks to the speech) fit into the basic pattern of [printed "from"] the referee, as benefit­ thought of the speech. This basic pattern, ing no one: taking credit from the author which is relatively common among Fer­ without giving it to the anonymous referee. era! science administrators, is to urge By the same reasoning, a newspaper item scientists to abandon the ethos and goals on an occurrence reflecting credit on no of science and to accept a framework of one, or only on an anonymous passer-by, thought and action more pleasing to the should not be printed. This is wrong. lowest common denominator of Congress­ men (and presumably of administrators). j. R. Isbell There is little doubt what a profoundly negative effect such a transfor~ation Editor, the cJ{oticei) would have upon the vitality of science and the meaningfulness of a scientific career Like many of our colleagues in the to coming generations of potential sci­ mathematical community, we were amazed entists. and shocked by the excerpts from the We regard this administrative at­ speech by Dr. Donald Hornig, the Presi­ titude as a great danger which promises dent's Science Adviser, which were pub­ a good crop of future disasters for Ameri­ lished in the july 19 issue of SCIENCE. can science, some of which have already In these excerpts, Dr. Hornig clearly im­ been delivered. In reading Hornig's speech plies that a large part of the responsi­ and statements like it, one is struck by the bility for the current crisis in Federal indifference and complacency which they financing of science may be laid to an exhibit toward such major problems as: alleged desire of the mathematicians as a group to take Federally financed vaca­ ( 1) The effect of the Vietnam war tions on the beaches of Rio de janeiro and and current draft policy on the develop­ the Aegean. It is difficult to connect up ment of the next generation of scientists this accusation with the real world. The in the graduate schools. only plausible interpretation of this line (2) The drastic mutilation of the of argument is that it refers to the con­ Federal fellowshipandgrantprograms for troversy which was waged last Fall be­ graduate students in the sciences. tween the NSF and Stephen Smale. If this (3) The growing emphasis on pork­ is indeed the case, then what Hornig is barrel and administrative criteria in the doing is trying to revive by innuendo the allocation of funds for scientific research, NSF charges against Smale, charges as opposed to criteria of scientific which were discredited by Daniel Green­ achievement and the encouragement of berg in his final news article on the sub­ higher standards of excellence. ject in SCIENCE which the NSF never (4) The destructive anddemoraliz­ answered. ing effects of current cuts in Federal sup­ Aside from the Smale case, more­ port of science upon the universities and on over, Hornig's statement is a remark­ whole areas of scientific research. ably irresponsible and unconstructive Instead of supporting the basic contribution to the discussion of public needs of scientific development before the science policy, especially by a man in so public, some administrators seem to be high an official position in the Federal happiest in devoting their energies to at­ machinery dealing with the scientific com­ tacking one of the most cherished princi­ m unity. In studying the full text of Hornig's ples of the scientific ethos: the autonomy

861 of the individual scientist in determining ing and warding off the demagogic attacks for himself how he works and what he of a single Congressman, is not a policy works upon. that will cause anyone (and least of all The record of this group of ad­ Congress) to have any great respect for ministrators in being able to convince the principles and integrity of those who Congress of the value of Federal support adopt it. The conspicuous public silence of science is not so wonderful as to pre­ of the whole class of Federal science clude believing that the situation might administrators with regard to the future be a good deal better (and very likely effects of current draft policy and the would not be worse) if the argument for Vietnam war upon American science has such support were based upon a whole­ been positively deafening. In this context, hearted commitment to the scientific Hornig's apparent attempt to turn the dis­ ethos, rather than the lipservice ten­ cussion of the current crisis in Federal dered by Hornig. No such casual acknowl­ support for basic science into a hunt for edgements can take the place of the right scapegoats in the form of "mathemati­ actions and right attitudes at the appro­ cians on the beaches" would be ludicrous priate time. if it were not so destructive. Right actions and right attitudes (as our list of problems above indicates) Hyman Bass E. R. Kolchin have recently been in short supply. In F. E. Browder S. Lang particular, the policy of creating a major William Browder M. Loeve scandal involving the implied application S. S. Chern R. S. Palais of political criteria in grant administra­ Robert A. Herrmann M. H. Protter tion, apparently for the purpose of placat- I. N. Her stein G. Washnitzer

NEWS ITEMS AND ANNOUNCEMENTS

SUMMER RENCONTRES IN at Princeton University; Professor L. MATHEMATICS AND PHYSICS, 1969 Michel of the Institute des Hautes Etudes Scientifique, Bures sur- Yvette, France, Plans for the 1969 "Summer Ren­ and presently at the Institute for Ad­ contres in Mathematics and Physics" have vanced Study; and Professor L. O'Raifear­ been announced by the Battelle Memorial taigh of the Dublin Institute for Advanced Institute. Studies. Professor Valentine Bargmann, The conference will begin July 21, Professor of Mathematical Physics at 1969. About thirty participants will con­ Princeton University, New Jersey, will be sider the topic "Group Representations in the chairman of the six-week conference Mathematics and Physics". in theoretical mathematics and physics. It For additional information regard­ will be held at the Battelle Memorial ing the 1969 Rencontres please direct Institute's Seattle Research Center. inquiries to Professor V. Bargmann, Three of four lecturers for the Princeton University, Princeton, New Jer­ 1969 conference have been named by Pro­ sey 08540, or to the Battelle Memorial fessor Bargmann. They are Professor Institute, 4000 N. E. 41st Street, Seattle, E. M. Stein, Department of Mathematics Washington 98105, attention Dr. R. S. Paul.

862 NEW AMS PUBLICATIONS

LECTURES IN APPLIED MATHEMATICS Volume 22 SPECIAL FUNCTIONS AND THE THEORY Volume 11 OF GROUP REPRESENTATIONS MATHEMATICS IN THE DECISION SCI­ By N. J. Vilenkin ENCES, PART I 613 pages; List Price $28.00; Member Edited by G. Dantzig and A. Veinott Price $21.00. 436 pages; List Price $16.40; Member By means of the theory of group Price $12.30. representations, this book establishes This volume and volume II which order in the hitherto chaotic mass of for­ will follow soon contain the proceedings mulas (differential equations, integral re­ of the Summer Seminar on the Mathemat­ presentations, recurrence formulas, ad­ ics of the Decision Sciences jointly spon­ dition theorems) in the theory of the sored by the American Mathematical So­ special functions of mathematical physics; ciety and the Society for Industrial and namely, the hypergeometric function and Applied Mathematics. The Seminar was its various special and degenerate case.s; held at Stanford University from July 10 functions of Bessel and Legendra, the to August 11, 1967. gamma function, orthogonal polynomials of In general, the topics under dis­ Jacobi, ~eby~ev, Laguerre, Hermite and cussion were involved with the scope of others. applications of mathematics, and how this scope has broadened greatly in recent MEMOIRS OF THE years from physics to nearly every form AMERICAN MATHEMATICAL SOCIETY of human endeavor. Consequently, that as Number 79 the computer revolution gr·ows, mathe­ matical theory will become more and UNITARY REPRESENTATION FOR SOL­ more stimulated and enriched by these VABLE LIE GROUPS new sources of problems. By Jonathan Paul Brezin 128 pages; List Price $1.80; Member Price $1.35. TRANSLATIONS 10F MATHEMATICAL MONOGRAPHS This Memoir continues the study begun in the Memoir by L. Auslander Volume 21 and C. C. Moore (Memoir 62). The basic CONSTRUCTIVE REAL NUMBERS AND objective is to reduce the study of unitary CONSTRUCTIVE FUNCTION SPACES representations of solvable Lie groups to By N. A. Sanin the study of a single linear representation of these groups, the coadjoint representa­ 325 pages; List Price $17 .60; Member tion: Price $13.20. Let S be a connected, simply con­ The first of the three chapters in nected, solvable Lie group, and let S be this book presents the necessary infor­ the Lie algebra of S. The coadjoint re­ mation from the theory of constructive presentation of S (denoted: Ad*) is the real numbers in a form closely related to linear representation of S contragredient Markov's concept of a "regulator of con­ to the adjoint representation of S on S. vergence". Then Chapters 2 and 3 dis­ Let S* be the space of linear functionals cuss the theory of constructive analogues on S. The coadjoint representation is the of such classical concepts as Hilbert "natural" representation of S on S*. space, Lebesgue integral, absolutely con­ The principal result of the Memoir tinuous function. There is an appendix on is the unitary type of S is entirely deter­ criticisii). of classical mathematics. mined by the action of Ad* S on S*. Thus,

863 for example, it is proved that in order integrals is investigated and a numerical for S to be type I, the quotient space experiment by Herron and Heiles compared S*/Ad* S must be at least a To topological with the theory. space. The second paper by Kyner contains In addition to considering the a treatment of the satellite motion under unitary-type problem, the Memoir studies the influence of the oblate planet earth. the coadjoint representation in detail, in It is shown that the typical orbits lie on order to provide the machinery needed to invariant tori in a three dimensional compute the unitary dual of type I, solva­ phase space determined by the inter­ ble Lie groups. This machinery is applied section of two integral manifolds, the to compute the dual of those type I solva­ total energy and the polar component of ble Lie groups all of whose irreducible angular momentum. Their stability is unitary representations are induced from studied as well as the stability of the characters. periodic orbits generated by the circular 'orbits of the Kepler problem. In the con­ Number 80 cluding section, the theory is applied to a special case, the integrable problem of ENDOMORPHISMS OF LINEAR ALGE­ Vinti. BRAIC GROUPS By Robert Steinberg Number 82 112 pages; List Price $1.70; Member COHOMOLOGY OF COALGEBRAS Price $1.28 By David W. Jonah The Memoir is divided roughly 76 pages; List Price $1.60; Member into three parts, one dealing with various Price $1.20. properties, connectedness, number of components, finiteness, of the group of This Memoir gives a development fixed points of an endomorphism of a of a relative cohomology theory for asso­ linear algebraic group, a second with a ciative coalgebras A defined over a cate­ uniform, somewhat expository, treatment gory with multiplication. The second co­ of the known families of finite simple homology group is related to singular groups, and a third with preparatory ma­ co algebra extensions. To interpret the terial on reflection groups, containing, third cohomology group the coannihilator among other things, a1;1 elementary deriva­ N of a coalgebra M and a type of coalgebra tion of a formula due to Bott. extension E: 0 ... A .... r .... M .... 0, where r need not be associative and M has a speci­ Number 81 fic coannihilator N, are introduced. To each equivalence class of such quasi­ LECTURES ON HAMILTONIAN SYSTEMS extensions E there is a 3-cocycle O(E) AND RIGOROUS AND FORMAL STABIL­ representing an element of H3(N,.t\); this ITY OF ORBITS ABOUT AN OBLATE obstruction cocycle O(E) = 0 if and only if PLANET r is associative. Each 3-cocycle repre­ By Jurgen Moser and Walter Kyner senting an element of H3(N,A) is the ob­ struction to associ,ativity O(E) for some List Price $1.60; Member 88 pages; quasi-extension E. The proof of this last Price $1.20. result, which involves an elaborate con­ These articles represent a series struction, uses a generalized effaceability of lectures held by the authors in 1967 and theorem. This result itself involves a are of expository nature. They deal pri­ careful examination of a relationship be­ marily with the stability of Hamiltonian tween two canonical injective coresolu­ Systems and various applications of the tions arising from a pair of adjoint func­ new results of Kolmogorov, Arnold and tors. The work is done categorically; Moser to a number of problems with twe results proven for coalgebras and co­ degrees of freedom. The stability of La­ modules give valid results for algebras grange's equilateral solutions for the merely by categorical duality and the restricted three-body problem is proven, proper choice of the underlying category as well as the question of containment in with multiplication. The corresponding magnetic bottles, is discussed. Further­ (dual) interpretation of H3 (A ,N) for K more, the problem of the existence of algebras is related to Hochschild's. 864 NEWS ITEMS AND ANNOUNCEMENTS

SIAM 1968-1969 Gene H. Golub, Stanford University VISITING LECTURESHIP PROGRAM Hubert Halkin, University of California John H. Halton, University of Wisconsin SIAM has announced its Tenth An­ E. F. Infante, Brown University nual Visiting Lectureship Program for the George Karreman, University of Penn- 1968-1969 academic year. The program sylvania has been broadened in scope and subjects Gordon Eric Latta, University of Vir­ with a number of lecturers now offering ginia lectures at the general and undergraduate George Leitmann, University of Cali­ levels. Thus smaller institutions or those fornia having no doctoral program in mathe­ Frank Proschan, Boeing Scientific Re­ matics will now be able to realistically search Laboratories benefit from a SIAM Visiting Lecturer Werner C. Rheinboldt, University of and are encouraged to apply. Maryland Lecture topics in the graduate and Robert Rosen, State University of New research fields will continue to be offered. York Additionally, graduate and research fields Feodor Theilheimer, Applied Mathe­ will continue to be offered, and where matics Laboratory possible these have now been categorized G. Milton Wing, University of New so that they are readily identifiable, Mexico Upon request, SIAM also hopes to Arthur Wouk, Northwestern University be able to provide a limited number of The brochure for the new program panels consisting of two or more lectures will be mailed in September to Heads of to discuss such areas as numerical analy­ Mathematics Departments and Research sis and biomathematics. Centers, as well as a list of smaller The distinguished group of lecturers colleges and universities. The brochure participating in the 1968-1969 program are contains information on applications, as follows: financial requirements, and also lists the John H. Barrett, University of Ten­ lecturers and their topics. Brochures and nesee application forms are available upon re­ Z. W. Birnbaum, University of Wash­ quest from the SIAM office. ington G. R. Blakley, State University of New York FORTHCOMING IUTAM SYMPOSIA Bruce A. Chartres, University of Virginia On August Z9, 1968, Professor Ward Cheney, University of Texas William Prager of the University of Cali­ Frank M. Cholewinski, Clemson Uni­ fornia attended a meeting of the General versity Assembly of the International Union of Hirsh Cohen, IBM Corporation Theoretical and Applied Mechanics at Charles C. Conley, University of Wis­ Stanford University. consin Professor Prager represents the Richard C. DiPrima, Rensselaer Poly­ American Mathematical Society on the technic Institute United States National Committee on Theo­ William S. Dorn, University of Denver retical and Applied Mechanics, and was James D. Esary, Boeing Scientific Re­ chosen to serve as a delegate to the search Laboratories General Assembly of IUTAM by the Na­ Solomon W. Golomb, University of tional Academy of Sciences. Southern California The following symposia, which are

865 to be organized under'the sponsorship of spring of 1969, will contain a discussion IUTAM, have been scheduled. of the role of mathematical scientists in industry and government, patterns of or­ Flow of Fluid - Solid Mixtures ganizational development of the mathe­ Cambridge, England matical sciences in universities, and future March 24-29, 1969 informational needs regarding the supply Chairman: Professor G. K. Batchelor and demand of mathematical manpower. Cambridge, England The recently published Volume I Instability of Continuous Systems was written by John Jewett, Executive Karlsruhe, Germany Director of the Survey Committee, with Late August or early September, 1969 the assistance of Clarence B. Lindquist, Chairman: Professor H. Leipholz who conducted a somewhat similar study Karlsruhe, Germany for the U. S. Office of Education several years ago. Cosmical Gas Dynamics This volume is divided into two Crimean Astrophysical Observatory, parts. The first part, dealing with univer­ USSR sities and four-year colleges, presents September 1-11, 1969 and analyzes the results of a detailed Chairman: Professor L. I. Sedov questionnaire sent to a representative Moscow, USSR sample of the more than 1,000 such insti­ Creep in Structures tutions which grant bachelor's degrees Gothenburg, Sweden in the mathematical sciences. Tabulations August 16-20, 1970 and discussions present numerous other Chairman: Professor I. Hult facts concerning the mathematical science Gothenburg, Sweden faculty in four-year colleges and universi­ ties and how it has been changing over Unsteady Flow of Water at High Speeds recent years. Leningrad, USSR The second part of Volume I pre­ June 22-2.6, 1971 sents a detailed study of mathematical Chairman: Professor L. I. Sedov science programs in junior colleges, based Moscow, USSR on an extensive questionnaire for the aca­ Other symposia on electrohydro­ demic year 1966-1967. dynamics, dynamics of ionized gases, flow Volume I concludes with extensive of multi-phase fluids in porous media, and appendices giving detailed tables of data, high-speed computing in analysis of elastic both for four-year colleges and universi­ structures are under consideration. ties and for two-year institutions. This volume is available at $1.75 postpaid from the Conference Board of the Mathematical Sciences, 834 Joseph Henry Building, A NEW STUDY OF HIGHER EDUCATION 2100 Pennsylvania Avenue, N. W ., Wash­ IN THE MATHEMATICAL SCIENCES ington, D. C. 2.0037. Also available, at $2.00 postpaid from the same address, is Aspects of Undergraduate Training an earlier Conference Board study, Build­ in the Mathematical Sciences, Volume I of ings and Facilities for the Mathematical a three-volume study being prepared by the Sciences, by J. Sutherland Frame and Survey Committee of the Conference Board John W. McLeod. of the Mathematical Sciences, is now available. The Survey Committee, chaired by Professor Gail S. Young of Tulane Univer­ sity and founded by the Ford Foundation, CATEGORY THEORY has been engaged in this study during the ADVANCED SCIENCE SEMINAR past two years. Volume II, on graduate education in A Category Theory Advanced Sci­ the mathematical sciences and associated ence Seminar for graduate and postgradu­ questions of research and government sup­ ate students of mathematics will be held port, is expected to be published in the at Bowdoin College in Brunswick, Maine, fall of 1968. Volume III, to appear in the from June 24 to August 14, 1969. 866 It is expected that the Seminar will SYMPOSIUM ON APPROXIMATION be financed by the National Science Founda­ THEORY AND ITS APPLICATIONS tion. The aim of this Seminar is to enrich A residential symposium on Approxi­ mathematics education and stimulate re­ mation Theory and its Applications will be search by providing opportunities for held at the University of Lancaster from July 21 to July 25, 1969. Support has al­ ( 1) graduate students to acquire, in ready been promised by a number of mathe­ a research environment, an understanding maticians from the United States and of category theory, and its applications, Europe. The list includes E. W. Cheney, and to discover new topics for research University of Texas; P. J. Davis, Brown as well as new weapons for research in University; C. Lanczos, Dublin Institute progress; for Advanced Studies; J. R. Rice, Purdue (2) postdoctoral mathematicians to University; H. Werner, University of assist and direct graduate students in Munster. studies and investigations, while pursuing It is hoped that the Soviet Union will their own work amidst mathematicians also participate. of similar interests; to be submitted are most {3) experienced research mathe­ Contributions welcome, however authors should send maticians to exchange ideas and to work abstractsbyJanuary 31,1969, to thesym­ in an environment pleasant for their secretary. The mailing address is: families and lively for themselves. posium Dr. A. Talbot, Mathematics Department, The program contains a course University of Lancaster, Bailrigg, Lan­ entitled Categories and Functors, given caster, England. by Saunders Mac Lane, and assisted by Participants will be accom odated in Ulrich Oberst; a research colloquium in one of the Colleges of this new university, Category Theory and Applications, and which lies near the sea, the P ennine hills, seminars at various levels on appropriate and the Lake District. There will be a topics. registration of £6, and a charge of £2 lOs. Graduate student participants will per day. This fee includes meals. There is be selected primarily from the nominees no registration fee for wives. Additional of departments at Ph.D. granting institu­ information on the symposium and regis­ tions. An interested graduate student tration forms may be obtained by writing should ask his chairman or probable dis­ to Dr. Talbot at the above address. sertation advisor to write a letter of en­ dorsement. Postdoctoral students will assist in the seminar program for graduate stu­ dents and also engage in research. Some will be nominees of the senior staff. Each candidate should submit, in addition to a FELLOWSHIP AND RESEARCH letter of endorsement from an established OPPORTUNITIES IN MATHEMATICS mathematician in the field, an academic vita, a list of publications and of research The Division of Mathematical Sciences in progress, and a statement indicating of the National Research Council calls his objectives in this program. attention to a number of fellowships and Senior research mathematicians other support for research in the mathe­ applying for subsistence or travel grants matical sciences at both the predoctoral should write to the Director providing the and postdoctoral levels, to be awarded same information requested from post­ during the year 1968-1969. Copies of the doctoral candidates. complete announcement are available Please address all requests for from The Division of Mathematical Sci­ information to Director, Advanced Alge­ ences, National Research Council, 2101 bra Seminar, Bowdoin College, Brunswick, Constitution Avenue, N. W ., Washington, Maine 04011. D. C. 20418.

867 NSF SEEKS PROPOSALS FOR posals by prospective host institutions REGIONAL CONFERENCES IN THE should be sent to the Mathematical Sci­ MATHEMATICAL SCIENCES ences Section, National Science Founda­ tion, 1800 G Street, N. W ., Washington, The National Science Foundation is D. C. 20550. seeking proposals for five-day regional conferences on subjects of current re­ search interest in the mathematical sci­ RENSSELAER POLYTECHNIC ences. The goal of these conferences lies INSTITUTE AWARDED NSF in the stimulation and broadening of mathe­ SCIENCE DEVELOPMENT matical research activity. GRANT The Conference Board of the Mathe­ matical Sciences, under contract with the The Department of Mathematics at National Science Foundation, will establish Rensselaer Polytechnic Institute has been a panel to evaluate the proposals. Specifi­ awarded a National Science Foundation cally, the panel will evaluate the proposals, Departmental Science Development Grant. visit and report on the individual confer­ The grant, which will be used over a ences, and arrange for the editing and period of approximately three years, will publication of a series of expository aid several of the department's activities, papers based on the lectures. Awards of involving both teaching and research. Conference grants will be made by the The grant will support a special National Science Foundation with the ad­ professorship, as well as visiting pro­ vice of the Board. fessors, post-doctoral associates, sum­ At present ten conferences are pro­ mer research for new members of the jected. Each conference will take place at department, and research leaves. Special a host academic institution during a sum­ awards will be made available to doctoral mer week in 1969 or 1970, or possibly students. These a wards will provide a com­ within a recess of the intervening academic bined program of full-time study, part­ year. time teaching, and research. Topics may be concerned with cine or Funds will be used for experimental more of the various disciplines of the work in teaching basic mathematics cour­ mathematical sciences, including, in ad­ ses, including the use of audio-visual aids. dition to pure mathematics, fields such as The grant will also aid in facility renova­ applied mathematics, statistics, computer tion, purchase of additional library materi­ science, operations research, and manage­ als, and assistance in developing computer ment science. terminal capability. Each conference is to have a lec­ turer, and approximately twenty-five other participants. It is expected that the lec­ turer would give two lectures a day during BRAZILIAN COLLOQUIUM the five days of the conference, with the OF MATHEMATICS remainder of the time available for study, informal discussion, and exchange of ideas. The seventh Brazilian Colloquium of All participants in a conference will Mathematics will be held during three receive allowances for travel and sub­ weeks of July 1969 in Pocos de Caldas, sistence. The lecturer will receive, in ad­ Minas Gerais, Brazil. It will consist of dition, a fee for delivering his lectures post-doctoral and graduate courses, one and for organizing these into a substantial hour survey lectures, short research expository paper. The CBMS will subse­ announcements and sessions on mathema­ quently arrange for the editing and publica­ tics education. Financial support for living tion of these expository papers. expenses while attending the meeting may Preliminary inqmnes regarding be requested and considered. For informa­ details of these regional conferences may tion concerning the scientific program and be addressed to the Conference Board support, write to the organizing institution: of the Mathematical Sciences, 834 Joseph Instituto de Matematica Pura e Applicada, Henry Building, 2100 Pennsylvania Ave­ Rua Luiz de Camoes 68, Rio de Janeiro 58, nue, N. W ., Was"hington, D. C. 20037. Pro- GB, Brazil.

868 THE ANNUAL SALARY SURVEY

This year's Annual Salary Survey ing of institutions that have granted two is based on returns from 508 departments or less doctorates during the last three in mathematics and the mathematical sci­ years. The number of doctorates granted ences, covering 6,653 academic positions by universities was determined from the held in 19 6 7-1968 and 7,456 positions held lists published in these cHo!icei). in 1968-1969. All Canadian universities maintain­ The basis for classifying institu­ ing Ph.D. granting programs are included tions has changed this year. Institutions in Groups III and IV. have been divided into four groups ac­ Each institution submitted a mini­ cording to the highest degree offered in mum, median, and maximum salary figure mathematics. for each of four academic ranks, both for staff members with master's degrees and ( 1) Ph.D. Granting Institutions for those with doctorates, creating 48 (2) Institutions granting a master's de­ categories of salary figures. In some in­ gree in mathematics as its highest stances relatively few universities or col­ degree. leges reported, and in as much as there ( 3) Institutions granting a bachelor's were no significant figures available, degree in mathematics as its highest salaries could not be listed. degree. The data presented in the following ( 4) Junior colleges and other institutions two pages gives the minimum, and maxi­ offering courses but not degrees in mum salary in each of the categories. In mathematics. addition, it presents the middle 50o/o of all The Ph.D. granting universities the salary figures that were received for have been subdivided into four groups. that category. In each set of four figures, Groups I and II include the universities the first figure is the minim urn salary re­ with leading mathematics departments ac­ ported in that group. The two figures in cording to the findings of a survey made parentheses represent the range of the by the American Council of Education in middle 50% of salaries reported, and the 1964* in which departments were rated fourth and final figure gives the maximum according to the quality of their graduate salary in that group. Salaries are given faculty. Group I is composed of institu- in "hundreds of dollars." tions that were considered" Distinguished" All salaries refer to an academic and "Strong"; Group II are those considered year of 9 or 10 months. Grants and con­ "Good" and "Adequate Plus." Groups III tracts are included but sabbatical pay­ and IV consist of all of the remaining ments and other part-time salaries are Ph.D. granting institutions; Group III in­ not. cluding those universities that have This survey is the twelfth in an granted three or more doctorates during annual series begun in 1957 by the Society's the last three years; and Group IV consist- Committee on the Economic Status of Teachers.

*The findings were published in a book by Allan Cartter entitled "An Assessment of Qualfty in Graduate Education." The information on mathematics was reprinted by the Society and can be found on pages 978-980 of the December 1966 issue of these cJfo!icei),

869 PH. D. GRANTING INSTITUTIONS. Group I Number of Usable Returns: 23

Total Number SALARIES(hundreds of dollars) of Staff Members 1967-1968 1968-1969 DOCTORALDEGREE 1967-1968 1968-1969 Minimum Median Maximum Minimum Median Maximum Instructor 79 80 72(82-87)90 78(85-90)95 82(85-93)95 79(85-90)98 85(88-92)98 80(87-96)99 Asst. Professor 272 268 75(90-98)120 92(97-102)110 102(109-118)139 79(95-100)125 99(102-109)125 102(115-120)138 Assoc. Professor 162 183 99(114-130)140 118(125-131)150 127(142-157)175 108(119-130)150 120(130-140)164 132(141-155)180 Professor 318 331 130(145-164)180 160(185-200)230 215(230-270)322 134(150-170)195 169(193-210)240 230(245-290)338 "ll3l Ob2

PH. D. GRANTING INSTITUTIONS. Group II Number of Usable Returns: 24

MASTER'S DEGREE Instructor 31 36 I 60(68-80)87 68(75-84)113 68(80-89)110 I 68(75-90)100 68(75-93)105 68(93-105)125 DOCTORALDEGREE Instructor 12 22 Asst. Professor 262 293 I 85(92-98)115 100(104-110)130 105(112-127)143 I 90(96-105)130 100(105-115)125 104(120-133)180 Assoc. Professor 163 191 96(115-128)145 120(130-138)150 135(143-165)186 104(120-130)170 125(135-144)170 140(148-170)185 Professor 220 241 105(150-170)183 170(185-200)245 200(233-255)282 135(150-180)206 171(195-206)258 191(245-271)290 b57 747

00 ...:! PH. D. GRANTING INSTITUTIONS. Group Ill Number of Usable Returns: 57 0 MASTER' S DEGREE

Instructor 183 180 40(60-73)90 52(68-84)100 65(73-88)116 40(65-77)98 40~66-87)10465(80-93)121 Asst. Professor 46 45 67(80-92)111 70(80-100)111 78(96-107)116 I 70(185-95)123 75 85-110)120 85(91-111)141 Assoc. Professor 35 33 83(92-113)115 83(100-125)169 91(110-123)143 91(110-130)180 110(123-143)180 Professor 6 6 ------100:==~=135)180 270 20!1

DOCTORALDEGREE Instructor 9 7 Asst. Professor 407 481 82(95-100)135 90(100-106)120 93(110-117)145 I 83(97-105)145 90(105-114)129 98(117-125)155 Assoc. Professor 294 308 85(107-124)175 95(123-135)177 112(138-160)1.96 40(114-1.33)1.55 109(1.31-1.45)170 116(1.41.-1.65)205 Professor 311 351 I 1.08(139-1.57)1.90 124(163-1.87)218 124(200-225)320 117(145-169)221 132(173-195)243 132(210-240)340 1021 i4IT

PH. LGRAI'ItliNg_~INSTITUTIONS. Group IV Number of Usable Returns: 29

MASTER'S DEGREE Instructor 91. 80 50(60-74)78 50(72-83)95 65(75-87)108 I 54(67-80)1.09 60(70-84)111 65(80-95)112 Asst. Professor 56 58 I 66(81.-90)99 79(85-98)1.09 82(94-99)1.20 75(85-95)1.08 79(89-100)117 82(99-1.08)127 Assoc. Professor 31 29 I 70(93-93)125 86(lll-120)144 90(120-136)167 83(103-1.22)136 86(1.22-1.27)152 90(130-150)176 Professor 13 11 1.91. 1.78

89 89

97 97

189 189

Returns: Returns:

Returns: Returns:

Returns: Returns:

60(96-114)143 60(96-114)143

69(86-105)151 69(86-105)151

83(ll5-153)193 83(ll5-153)193

85(113-133)169 85(113-133)169

74(100-122)140 74(100-122)140

69(90-105)135 69(90-105)135

83(ll0-130)200 83(ll0-130)200

99(128-146)210 99(128-146)210 62(76-87)106 62(76-87)106

82(97-110)151 82(97-110)151

89(115-137)188 89(115-137)188

85(95-ll0)135 85(95-ll0)135

65(80-95)120 65(80-95)120

ll5(132-164)200 ll5(132-164)200

130(185-235)280 130(185-235)280

102(111-125)160 102(111-125)160

120(132-155)229 120(132-155)229

102(110-125)148 102(110-125)148

135(160-182)268 135(160-182)268

125(140-158)193 125(140-158)193

Usable Usable

Usable Usable

Usable Usable

of of

of of

of of

Number Number

Number Number

Number Number

57(89-104)118 57(89-104)118

70(89-153)175 70(89-153)175

79(92-123)156 79(92-123)156

70(95-115)152 70(95-115)152

60(76-92)136 60(76-92)136

65(82-96)150 65(82-96)150

76(95-105)128 76(95-105)128

83(107-125)154 83(107-125)154

99(120-134)180 99(120-134)180

60(73-82)96 60(73-82)96

90(104-115)131 90(104-115)131 68(105-140)156 68(105-140)156

95(104-120)138 95(104-120)138

77(90-108)165 77(90-108)165

60(75-85)99 60(75-85)99

84(111-128)150 84(111-128)150

105(120-147)175 105(120-147)175

110(148-167)247 110(148-167)247

119(155-187)240 119(155-187)240

118(127-150)186 118(127-150)186

143 143

(125-14_1) (125-14_1)

52(70-83)134 52(70-83)134

59(84-142)175 59(84-142)175

68(120-138)156 68(120-138)156

70(97-ll2)131 70(97-ll2)131

57(70-80)95 57(70-80)95 64(87-110)156 64(87-110)156 49(81-93)112 49(81-93)112

79(121-150)239 79(121-150)239 79(122-158)239 79(135-164)239 92 92

60(79-92)ll5 60(79-92)ll5

70(90-104)117 70(90-104)117

81(106-122'138 81(106-122'138 72(84-97)151 72(84-97)151 99(113-128)156 99(113-128)156

60(70-80)98 60(70-80)98 88(97-110)127 88(97-110)127

88(95-110)130 88(95-110)130

80(100-122)144 80(100-122)144

ll0(131-159)187 ll0(131-159)187

106(148-170)190 106(148-170)190

100(115-131)181 100(115-131)181

I I

I I

I I

I I

78(109-149)175 78(109-149)175

57(71-82)140 57(71-82)140

80(102-130)149 80(102-130)149

56(82-98)125 56(82-98)125

95(105-117)142 95(105-117)142 94(108-150)180 94(108-150)180

76(129-155)231 76(129-155)231

65(95-109)127 65(95-109)127

98(120-155)198 98(120-155)198

90(106-117)136 90(106-117)136 75(105-124)183 75(105-124)183 60(80-100)147 60(80-100)147

80(89-106)125 80(89-106)125

64(96-118)144 64(96-118)144

84(110-128)156 84(110-128)156

82(92-107)142 82(92-107)142 63(76-90)114 63(76-90)114

127(150-170)233 127(150-170)233

101(122-141)187 101(122-141)187

108(130-150)181 108(130-150)181

130(176-220)270 130(176-220)270

55(80-95)120 55(80-95)120

58(73-87)135 58(73-87)135

57(70-77)125 57(70-77)125

56(72-80)96 56(72-80)96

56(79-90)140 56(79-90)140

76(88-110)149 76(88-110)149

90(112-141)167 90(112-141)167

75(99-120)140 75(99-120)140

97(140-158)209 97(140-158)209

65(80-149)155 65(80-149)155

76(87-100)115 76(87-100)115 73(87-100)128 73(87-100)128 99(112-131)166 99(112-131)166

90(98-111)127 90(98-111)127 64(103-130)145 64(103-130)145

61(90-108)144 61(90-108)144

82(122-150)231 82(122-150)231

79(102-120)140 79(102-120)140 84(96-108)130 84(96-108)130

119(140-177)220 119(140-177)220

105(120-131)171 105(120-131)171

55(77-89)103 55(77-89)103

52(65-80)130 52(65-80)130

53(84-134)155 53(84-134)155

54(66-75)115 54(66-75)115

58(91-105)144 58(91-105)144

56(75-86)115 56(75-86)115

64(80-102)149 64(80-102)149

75(102-115)133 75(102-115)133

97(130-149)198 97(130-149)198

92(105-121)140 92(105-121)140

64(103-130)145 64(103-130)145 54(66-75)94 54(66-75)94

80(118-145)231 80(118-145)231

75(94-118)129 75(94-118)129

80(93-100)125 80(93-100)125

65(85-100)122 65(85-100)122

65(80-92)108 65(80-92)108

87(112-133)167 87(112-133)167

106(137-157)173 106(137-157)173

INSTITUTIONS INSTITUTIONS

102(113-120)166 102(113-120)166

I I

176(92-105)127 176(92-105)127

I I

INSTITUTIONS INSTITUTIONS

2 2

2 2

29 29

43 43

36 36

61 61

16 16

226 226

397 397 257 257

298 298

370 370

334 334 359 359

267 267

132 132

162 162

160 160

133 133

112 112

108 108

124 124

140 140

4liE 4liE

190 190

E9ti E9ti

1i55 1i55 Bbo Bbo

ffi ffi

1i'75 1i'75

INSTITUTIONS INSTITUTIONS

GRANTING GRANTING

4 4

4 4

GRANTING GRANTING

50 50

43 43

38 38

27 27

78 78

11 11

374 374

371 371

303 303

920 920 397 397

222 222

207 207

238 238

117 117

107 107 237 237

116 116

122 122

157 157

156 156

162 162

111 111

4l6 4l6

ffi ffi

34ti 34ti ti3b ti3b

Till Till

GRANTING GRANTING

DEGREE DEGREE

DEGREE DEGREE

DEGREE DEGREE

DEGREE DEGREE

DEGREE DEGREE DEGREE DEGREE

DEGREE DEGREE

DEGREE DEGREE

S S

S S

S S

Professor Professor

Professor Professor

Professor Professor

Professor Professor

Professor Professor

Professor Professor

' '

Professor Professor

Professor Professor

Professor Professor

Professor Professor

Professor Professor

Professor Professor

Instructor Instructor

Instructor Instructor

Instructor Instructor

Instructor Instructor Instructor Instructor

Instructor Instructor

Professor Professor

Professor Professor

Professor Professor

Professor Professor

MASTER MASTER Assoc. Assoc.

Assoc. Assoc.

DOCTORAL DOCTORAL

Professor Professor

Assoc. Assoc.

DOCTORAL DOCTORAL Professor Professor

Asst. Asst.

MASTER' MASTER'

Asst. Asst. DOCTORAL DOCTORAL

MASTER'S MASTER'S

Asst. Asst.

Asst. Asst. NONDEGREE NONDEGREE

Assoc. Assoc. MASTER' MASTER' Asst. Asst.

Assoc. Assoc.

BACHELOR'S BACHELOR'S Assoc. Assoc.

Asst. Asst.

-l -l

00 00 ...... PERSONAL ITEMS

Professor W. W. ADAMS of the Uni­ Professor SRISAKDI CHARMONMAN versity of California, Berkeley, will be of the University of Alberta has been on sabbatical leave during the fall quarter appointed to an associate professorship in 1968. He will spend the time at the Uni­ Computer Science at the University of versity of California, Los Angeles. Missouri, Columbia. Dr. ETHAN AKIN of Princeton Uni­ Dr. P. R. CHERNOFF of Harvard Uni­ versity has been appointed Lecturer at the versity has been awarded a postdoctoral University of California, Berkeley, for the National Science Foundation Fellowship academic year 1968-1969. for the year 1968-1969. He will spend the Dr. W. B. ARVESON of Harvard Uni­ year at the University of California, versity has been appointed Lecturer at the Berkeley. University of California, Berkeley, for the Dr. J. M. COHEN of the University academic year 1968-1969. of Chicago has been appointed to an assis­ Professor IACOPO BARSOTTI of the tant professorship at the University of University of Pisa, Italy, has been appoint­ Pennsylvania. ed to a professorship at the University of Dr. G. E. COOKE of the Institute for Padova, Italy. Advanced Study, Princeton, has been ap­ Professor ZAMIR BA VEL of Southern pointed Lecturer at the University of Illinois University has been appointed to an California, Berkeley, for the academic associate professorship in Computer Sci­ year 1968-1969. ence at the University of Kansas. Dr. W. J. ECKERT of Leonia, New Professor RICHARD BEALS of the Jersey, was awarded the honorary Doctor University of Chicago has been appointed of Science degree by Oberlin College at to a visiting associate professorship at the 135th anniversary commencement ex­ Yale University. ercise on June 10, 1968. Professor GORAN BJORCK of the Professor L. C. EGGAN of Pacific University of Stockholm, Sweden, has been Lutheran University has been appointed to appointed to a visiting professorship atthe an associate professorship at Illinois State University of California, Riverside, for the University. academic year 1968-1969. Dr. F. T. FARRELL of Yale Univer­ Professor H. J. BREMERMANN of the sity has been awarded a postdoctoral Na­ University of California, Berkeley, will tional Science Foundation Fellowship for be on sabbatical leave during the spring of the year 1968-1969. He will spend the year 1969. He will spend his leave in Germany at the University of California, Berkeley. and Switzerland, and has been invited to Dr. A. V. FIGA-TALAMANCA of the visit the Computer Center and Mathemati­ University of Genova, Italy, has been cal Institute at Novosibirsk, Russia. appointed Visiting Lecturer at the Univer­ Professor R. B. BROWN of the Univer­ sity of California, Berkeley, for the aca­ sity of California, Berkeley, will be on demic year 1968-1969. sabbatical leave during the fall quarter Professor LEOPOLD FLATTO of 1968. He will be in the Netherlands. Yeshiva University has been appointed to Dr. EUGENIO CALABI of the Univer­ a visiting professorship at the Hebrew sity of Pennsylvania will be on leave of University, Jerusalem, for the academic absence for the academic year 19 68-1969, year 1968-1969. visiting at the lnstitut des Hautes Etudes Professor A. L. FOSTER of the Uni­ Scientifiques, France. versity of California, Berkeley, will be on Dr. L. S. CHARLAP of the University sabbatical leave during the academic year of Pennsylvania will be on leave of absence 1968-1969. He will spend the year in west­ for the academic year 1968-1969, visiting ern Europe. at the University of London, England. Dr. P. J. FREYD of the University of

872 Pennsylvania will be on leave of absence Professor M. W. HIRSCH of the Uni­ for the fall term 1968, visiting at Pahlavi versity of California, Berkeley, will be on University, Iran. leave during the academic year 1968-1969. Professor DAVID GALE of the Univer­ He will spend the year at the Mathematical sity of California, Berkeley, will be on Institute of the University of Geneva, leave during the academic year 1968-1969. Switzerland. He will spend the year at the University Professor G. P. HOCHSCHILD of the of Copenhagen, Denmark. University of California, Berkeley, has Professor KURT GODEL of the Insti­ been appointed to a research professorship tute for Advanced Study, Princeton, and a in the Miller Institute for Basic Research member of the National Academy of Sci­ in Science, Berkeley, for the academic ences, has been elected a foreign member year 1968-1969. of The Royal Society. Mr. A. H. HOLMES of the University Professor L. W. GOODWYN of the of Illinois has been appointed to an assis­ University of Maryland has been appointed tant professorship at the University of to an assistant professorship at the Univer­ Missouri, St. Louis. sity of Kentucky. Professor W. H. HOSKEN of Purdue Professor R. F. GUNDY of Rutgers, University has been appointed to an asso­ The State University, has been appointed ciate professorship in Computer Science to a visiting professorship at the Hebrew at the University of Nebraska. University, Jerusalem, for the academic Professor L. A. HOSTINSKY of Con­ year 1968-1969. necticut College has been awarded aNa­ Professor A. W. HAGER of the Univer­ tional Science Foundation Science Faculty sity of Rochester has been appointed to an Fellowship and will be on sabbatical leave assistant professorship at Wesleyan Uni­ for the academic year 1968-1969 at the versity. University of Oklahoma. Dr. D. T. HAIMO of Southern Illinois Professor WU- YI HSIANG of the Uni­ University, Edwardsville, has been ap­ versity of Chicago has been appointed to pointed to a professorship at the Univer­ an associate professorship at the Univer­ sity of Missouri, St. Louis. sity of California, Berkeley. He has been Professor J. K. HALE of Brown Uni­ awarded a Sloan Foundation fellowship for versity will be on leave for the academic the period 1968-1970 and will spend the year 1968-1969. He will spend two quarters fall quarter 1968 at the Institute for Ad­ at the \University of California, Los Ange­ vanced Study in Princeton. les, and one quarter at the University of Professor J. H. JUSTICE of the Uni­ Southern California. versity of Maryland has been appointed to Dr. W. P. HANF of the International an assistant professorship at the Univer­ Business Machines Corporation, Endicott, sity of Tulsa. New York, has been appointed Visiting Mr. ALAN KAPLAN of Syracuse Uni­ Lecturer and Associate Research Mathe­ versity has been appointed to an assistant matician at the University of California, professorship at Worcester Polytechnic Berkeley, for the summer, fall and spring Institute. quarters 1968-1969. Professor TOSIO KATO of the Uni­ Dr. W. E. HARTNETT of the Parke versity of California, Berkeley, will be on Mathematical Laboratories, Incorporated, leave during the winter and spring quarters Carlisle, Massachusetts, has been appoint­ 1969. During the winter he will be at New ed to a professorship at the State Univer­ York University. sity of New York, College at Plattsburgh. Professor JOHN KENELL Y of Clem­ Professor L. A. HENKIN of the Uni­ son University has been appointed to a versity of California, Berkeley, will be on professorship at Louisiana State Univer­ sabbatical leave during the academic year sity in New Orleans. 1968-1969. He will spend the year at Professor H. B. KEYNES of the Uni­ Oxford University, England. versity of California, Santa Barbara, has Professor H.-H. W. HERDA of Salem been appointed to a visiting assistant State College has been appointed to an as­ professorship at the University of Minne­ sistant l?rofessorship at Tufts University. sota, School of Mathematics.

873 Professor B. R. KRIPKE of the Uni­ tician and Computer Scientist in the De­ versity of California, Berkeley, will be on partment of Statistics, Princeton Univer­ leave for the academic year 19 68-1969. sity. He will spend the time at the Hebrew Uni­ Professor KIYOSHI NOSHIRO of Nago­ versity, Jerusalem. ya University, Japan, will be visiting at Dr. TSIT- YUEN LAM of the University Wayne State University for the period of Chicago has been appointed Lecturer at September to December 1968. the University of California, Berkeley, for Professor HAJIMU OGAWA of the the academic year 1968-1969. University of California, Riverside, has Professor SERGE LANG of Columbia been appointed to a professorship at the University has been appointed to a visiting State University of New York at Albany. professorship at the University of Cali­ Professor A. P. OGG of the Univer­ fornia, Berkeley, for the fall and winter sity of California, Berkeley, will be on quarters 1968-1969. leave for the academic year 1968-1969. Dr. 0. A. LAUDAL of the University He has been appointed a Member of the of Illinois has been appointed Lecturer and Institute for Advanced Study, Princeton. Associate Research Mathematician at the Professor GLORIA OLIVE of Ander­ University of California, Berkeley, for the son College has been appointed to a pro­ academic year 1968-1969. fessorship at Wisconsin State University, Professor D. H. LEHMER of the Uni­ Superior. versity of California, Berkeley, will be on Dr. TAKASHI ONO of the University of sabbatical leave during the fall quarter Pennsylvania will be on leave of absence 1968. for the academic year 1968-1969, visiting Dr. L. E. LEVINE of the University at the Institute for Advanced Study, Prince­ of Maryland has been appointed to an as­ ton. sistant professorship at Stevens Institute Dr. S. J. OSHER of the Brookhaven of Technology. National Laboratory, Upton, New York, has Dr. JORAM LINDENSTRAUSS of the been appointed to an assistant professor­ Hebrew University, Jerusalem, has been ship at the University of California, Berke­ appointed to a visiting associate profes­ ley. sorship at the University of California, J Dr. W. F. PFEFFER of the University Berkeley, for the academic year 1968- of California, Davis, will teach during the 1969. winter and spring quarters, 1969, on the Mr. J. R. MCCORD, III, of Esso Math­ Berkeley Campus. ematics and Systems, Incorporated, Flor­ Professor C. C. PUGH of the Univer­ ham Park, New Jersey, has been appointed sity of California, Berkeley, will be on to an assistant professorship at Emory leave during the spring quarter 1969. He University in the Department of Statistics will spend the time at the University of and Biometry. Warwick, England. Professor W. L. MIRANKER of the Dr. LAJOS PUKANSZKY of the Uni­ International Business Machines Corpora­ versity of Pennsylvania will be on leave tion and the City University of New York of absence for the academic year 1968- has been appointed to a visiting professor­ 1969, visiting at the Massachusetts Insti­ ship at the Hebrew University, Jerusalem, tute of Technology. for the academic year 1968-1969. Professor ARLAN RAMSAY of the Professor C. B. MORREY, Jr., ofthe University of Rochester has been appointed University of California, Berkeley, will be to an associate professorship at the Uni­ on sabbatical leave during the academic versity of Colorado. year 1968-1969. He will spend the winter Professor W. L. REDDY of the State and spring quarters at the Courant Insti­ University of New York at Albany has been tute of Mathematical Sciences and the appointed to an assistant professorship University of Maryland. at Wesleyan University. Professor M. E. MULLER of the Uni­ Dr. MINA REES of the City University versity of Wisconsin will be on leave for of New York has been named provost and the academic year 1968-1969. He has been chief administrative officer of the Univer­ appointed Visiting Senior Research Statis- sity's Graduate Division.

874 Dr. DOCK SANG RIM of the University University of Montana. of Pennsylvania will be on leave of absence Professor ABRAHAM SEIDENBERG for the academic year 1968-19 69, visiting of the University of California, Berkeley, at the Institut des Hautes Etudes Scientifi­ will be on sabbatical leave for the fall ques, France. quarter 1968. He will be in Rome, Italy. Professor R. M. ROBINSON of the Dr. N. A. SHENK, II, of the University University of California, Berkeley, has of California, San Diego, will teach during been appointed to a research professorship the spring quarter 19 69 on the Berkeley in the Miller Institute for Basic Research Campus. in Science, Berkeley, for the academic Professor STEPHEN SMALE of the year 1968-1969. University of California, Berkeley, has Dr. G. F. ROSE of the System Develop­ been elected to the American Academy of ment Corporation, Santa Monica, Cali­ Arts and Science. fornia, has been appointed to a Professor­ Professor KENNAN SMITH ofthe Uni­ ship in Mathematics and Information Sci­ versity of Wisconsin has been appointed ences at Case Western Reserve University. to a professorship at Oregon State Uni­ Professor M. A. ROSENLICHT of the versity. University of Car'ifornia, Berkeley, will be Professor ROBERT SOL OVA Y of the on leave during the academic year 1968- University of California, Berkeley, has 1969. He has been appointed to a visiting been awarded a SloanFellowshipfor1968- professorship at Northwestern University. 1970. Professor H. P. ROSENTHAL of the Professor GUIDO STAMPACCHIA of University of California, Berkeley, will the University of Pisa, Italy, has been be on sabbatical leave during the fall appointed to a professorship at the Uni­ quarter 1968. He will be in residence at versity of Rome, Italy. Berkeley. Dr. C. M. STANTON of Stanford Uni­ Professor D. E. SARASON of the versity has been appointed to an assistant University of California, Berkeley, will be professorship at Wesleyan University. on sabbatical leave during the fall and Dr. ERLING ST¢'RMER of the Uni­ winter quarters 1968-1969. He will spend versity of Oslo, Norway, has been appoint­ the time at Berkeley. ed to a visiting assistant professorship at Professor ICHIRO SA TAKE of the Uni­ the University of Pennsylvania for the versity of Chicago has been appointed to academic year 1968-1969. a professorship at the University of Cali­ Professor A. H. TAUB of the Univer­ fornia, Berkeley. sity of California, Berkeley, has been Dr. R. L. SAYLOR of the University appointed Research Professor in the Mil­ of Chicago has been appointed Lecturer ler Institute for Basic Research in Science, at the University of California, Berkeley, Berkeley, for the academic year 1968- for the academic year 1968-1969. 1969. Professor WILFRIED SCHMID of the Dr. S. K. THOMASON of Simon Fraser University of California, Berkeley, will be Universiy has been appointed to a visit­ on leave for the academic year 1968-1969. ing assistant professorship at the Univer­ He has been appointed to a visiting asso­ sity of California, Berkeley, for the aca­ ciate professorship at Columbia Univer­ demic year 1968-1969. sity. Professor J. A. THORPE of Haver­ Dr. KENNETH SCHOEN of the Uni­ ford College has been appointed to an versity of Pittsburgh has been appointed associate professorship at the State Uni­ to an assistant professorship at Worcester versity of New York at Stony Brook. Polytechnic Institute. Professor K. W. TOLO of the Uni­ Dr. ULRICH SCHOENWAELDER of versity of Nebraska has been appointed New Mexico State University has been to an assistant professorship at the Uni­ appointed to an assistant professorship at versity of Tennessee. the University of Missouri, St. Louis. Dr. GEORGE VAN ZWALENBERG of Professor PHILLIP SCHULTZ of the the University of California, Berkeley, University of Washington has been appoint­ and Fresno State College has been appoint­ ed to an assistant professorship at the ed to a professorship at Calvin College.

875 Professor W. A. VEECH of the Uni­ PROMOTIONS versity of California, Berkeley, will be on leave during the academic year 1968-1969. To Professor. University of Cali­ He has been appointed a Member of the fornia, Berkeley: J. W. ADDISON, Jr., Institute for Advanced Study, Princeton. R. M. SOLOVA Y; University of Georgia: Dr. P. D. WAGREICH of Brandeis Uni­ C. H. EDWARDS, Jr.; Hebrew University, versity has been appointed to an assistant Jerusalem: AZRIEL LEVY; University professorship at the University of Pennsyl­ of Pennsylvania: P. J. FRE YD, WALTER vania. KOPPELMAN; Worcester Polytechnic In­ Dr. F. W. WARNER of the University stitute: L. N. ZACCARO. of California, Berkeley, has been appointed To Associate Professor. University to an associate professorship at the Uni­ of Calgary: B. N. SAHNE Y; University of versity of Pennsylvania. California, Berkeley: M. A. RIEFFEL, Professor M. E. WATKINSoftheUni­ HUNG-HSI WU; Hebrew University, Jeru­ versity of Waterloo has been appointed to salem: HAIM GAIFMAN; Wesleyan Uni­ an associate professorship at Syracuse versity: F. E. J. LINTON; West Virginia University. University: A. M. CHAK. Dr. BENJAMIN WEISS of the Inter­ national Business Machines Corporation, INSTRUCTORSHIPS T. J. Watson Research Center, Yorktown Heights, New York, has been appointed University of California, Berkeley: Senior Lecturer at the Hebrew University, L. D. OLSON; University of Pennsylvania: Jerusalem. F. D. GROSSHANS; Wisconsin State Uni­ Professor R. J. WEISS of Bridge­ versity- Whitewater: L. J. DICKEY. water College has been appointed to an DEATHS associate professorship at Mary Baldwin College. Dr. J. A. JENSEN of the University Professor B. R. WENNER of the Uni­ of Wyoming died on June 22, 1968, at the versity of Vermont has been appointed to age of 36. an assistant professorship at the Univer­ Mr. L. M. KLAUBER of San Diego, sity of Missouri, Kansas City. California, died on May 8, 1968, at the Professor A. C. ZAANEN of the Uni­ age of 84. He was a member of the Society versity of Leiden, Netherlands, has been for 4 7 years. appointed to· a visiting professorship at Professor G. K. OVERHOLTZER of the California Institute of Technology for the California State College at Los Angeles the academic year 1968-1969. died on April 12, 1968, at the age of 47. Professor OSCAR ZARISKI of Harvard He was a member of the Society for 25 University has been appointed to a visiting years. professorship at the University of Cali­ Mr. D. R. URBAN of Altadena, Cali­ fornia, Berkeley, for the spring quarter fornia, died on May 7, 1968, at the age of 1969. 70. ~ NEWS ITEMS AND ANNOUNCEMENTS

BORIS A. BAKHMETEFF ate student who is a candidate for either RESEARCH FELLOWSHIP IN a master's or a doctor's degree with no MECHANICS OF FLUIDS commitment which will interfere with his research work and who will study on a full­ A Boris A. Bakhmeteff Research time basis. The study and research may be Fellowship will be available for the 1969- undertaken at an institution of the Fellow's 1970 academic year in an amount up to choice. $3600. It is intended to be a specific con­ Application forms should be filled by tribution for a definite research project February 15, 1969, and sent to Dean of an original and creative nature in the William Allan, School of Engineering, The general field of mechanics of fluids. City College of The City University of New A qualified applicant must be a gradu- York, New York, New York 10031.

876 DOCTORATES CONFERRED IN 1967-1968

The following are among those who received doctorates in the mathematical sciences and related subjects from universities in the United States and Canada during 1967-1968, The number appearing after each university is the number of doctorates listed for that institution; when appli­ cable, this number includes the doctorates in both the department of mathematics and its related de­ partments. Each entry includes the dissertation title and the minor if nonmathematical. lZl univer­ sities are listed with a total of 949 individual names,

ALABAMA Hagopian, Charles L. A classification of nonlocally connected con­ AUBURN UNIVERSITY (3) tinua Hodson, William T., III Piacun, Norma Mary Effective solutions of a system of linear in­ Ordinary differential equations in Banach equalities arising in bargaining theory spaces Konkle, Gail S. Rautenstrauch, Carl Peter Achievement as affected by teaching for Evaluation of the Boltzman collision integral generalization in mathematics C( C n) as a sum of spherical moments Lustfield, Charles D. Van Cleave, Albert Ray, Jr, Mass functions of bounded variation and Concerning expansive point sets and the starlikeness equivalence of covering dimension CALIFORNIA ARIZONA UNIVERSITY OF CALIFORNIA, BERKELEY (32) UNIVERSITY OF ARIZONA (6) Allen, Dennis Patrick, Jr. Carlstedt, Linda Marie Some relationships between local and global Connected packings of compact sets structure of finite semigroups Fritsche, Richard Thomas Chazin, Robert Louis Topologies for probabilistic metric spaces Differentiable links in manifolds Grossfield, Andrew Donald, John Dustin A free boundary problem for the flow of a Quotient varieties for algebraic equivalence heavy liquid through an unobstructed ori­ relations fice Froemke, Jon Schulenberger, John Richard Independence of universal algebras Wave propagation in inhomogeneous aniso­ Garland, Stephen Jay tropic media Second-order cardinal characterizability Schuster, Eugene Francis Garraway, Annie Watkins Estimation of a probability density function Structure of some cocycles in analysis with applications in statistical inference Hechler, Stephen Herman Smith, Jimmie Baker Generalized scales in model theory Some problems in the theory of approxima­ Jackson, Shirley Lynn, Jr. tion Operators and inner functions Jean, Michel ARIZONA STATE UNIVERSITY (9) Pure structures Kopell, Nancy Jane Bennett, Harold R. Commuting diffeomorphisms Quasi-developable spaces Lu, Yung-Chen Bustoz, Joaquin, Jr, On holomorphic mappings of complex mani­ Gibbs sets and the generalized Gibbs phenom­ folds en on O'Neill, John Duane Creede, Geoffrey Applications of complex cobordism to equi­ Semi-stratifiable spaces and a factorization variant maps of a metrization theorem due to Bing Ozols, Vilnis Finkelstein, Raphael P, Critical points of the displacement function On the units in a quartic field with applica­ of an isometry tions to Mordell' s equation P alis, Jacob, Jr. Gustafson, Grant B. On Morse-Smale diffeomorphisms Conjugate point properties for Nth order Reinhardt, William Nelson linear differential equations Topics in the metamathematics of set theory

877 Risch, Robert Henry Buchman, Edwin Ogilvie The problem of integration in finite terms Generalizations of arcwise convex sets Sankowsky, Daniel Aaron Chang, John Shou-Ming Relative homological dimension of rings and Royden's compactification of Riemannian modules spaces Schmid, Wilfried Councilman, Samuel George Homogeneous complex manifolds and repre­ The class of k-functions on Riemannian sentations of semisimple Lie groups spaces Shub, Michael Ira Farrell, Gerald Paul Endomorphisms of compact differentiable Starshaped sets manifolds Gordon, Carl Edward Spring, David Harold A comparison of abstract computability theo­ Proper embeddings of open manifolds in ries euclidean space Johnson, Gary Don Strand, John Lester On a nonlinear vibrating string Stochastic ordinary differential equations Kaminker, Jerome Alan Sudderth, William David Multiplications on manifolds On the existence of good stationary strategies McCullough, Thomas Anthony Wells, Benjamin Baxter, Jr. Generalization of Mergelyan's theorem on Uncomplemented subspaces of continuous rational approximation functions and weak compactness of mea­ Miers, Charles Robert sures Lie isomorphisms of operator algebras Mikami, Edward Yoshio Group in Logic and the Methodology of Science Quadratic optimal control problems Reyes, Gonzalo Edmundo Motzkin, Joseph John Typical and generic relations in a Baire space P-adic domains of analyticity for models Phillips, Robert Gibson Some contributions to nonstandard analysis Department of Statistics Russak, Ira Bert Bhattacharjee, Manish Chandra On problems with bounded state variables Optimal random walk via dynamic program­ Skaff, Michael Samuel ming Vector valued Orlicz spaces Chao, Min-te Slobko, Thomas Anthony Nonsequential optimal solutions of sequential Norms on commutative algebras decision problems Tapia, Richard Alfred Heilbron, David Carl A generalization of Newton's method with an Estimating the variability of systematic sam­ application to the Euler-Lagrange equation pling Tripodes, Peter Gust Koul, Hira Lal Structural properties of certain classes of Estimation by method of ranks in regression sentences models Walker, Leroy Harold Kulkarni, Shridhar Ramarao Stopping rules for Brownian motion and Optimal C(a) and C zo(a) tests for the presence random walks of variable effects of a treatment Ruiz-Moncayo, Alberto UNIVERSITY OF CALIFORNIA, RIVERSIDE (10) Optimal stopping for functions of Markov chains Bownds, John Marvin Stigler, Stephen Mack Some uniqueness theorems for nonlinear Linear functions of order statistics differential and integral equations van der Muelen, Edward Cornelis DeKleine, Herbert Arthur Transmission of information in aT-terminal Boundedness and asymptotic behavior of discrete memoryless channel some second order equations Huff, Barthel Wayne Subordination of stochastic processes UNIVERSITY OF CALIFORNIA, DAVIS ( 1) Hughes, Robert Byers Boundary behavior of random valued heat Wilbur, Willy John polynomial expansions On nonabsolute integration in topological Johnson, Johnny Albert spaces A-adic completions of Noetherian lattice modules Mason, Jesse David UNIVERSITY OF CALIFORNIA, LOS ANGELES A necessary and sufficient condition that the (19) limit distribution of a sequence of normed, centered sums of independent random vari­ Andrews, Angus Percy ables involving r distributions be a con­ Differential inequalities in Hilbert space volution of r stable distributions 878 Rogers, James Ted, Jr. Schmulian, Robert Jay Pseudo-circles and universal circularly Nearly free molecular heat transfer from a chainable continua sphere Stevenson, Robert Allan Seliger, Robert L. Jacobson structure theory for Hestenes tern­ Part I. On the breaking of nonlinear dis­ ary rings persive waves; Part II. Variational princi­ Wade, William Raymond, II ples in continuum mechanics Uniqueness theory of the Haar and Walsh Yeh, Tyan systems I. Nonexistence of looping trajectories in Whitman, David Gilbert hydromagnetic waves of finite amplitude; Ring theoretic lattice modules and the Hilbert II. Breaking of waves in a cold collision­ polynomial free plasma in a magnetic field; III. On stabilities of periodic waves in a cold UNIVERSITY OF CALIFORNIA, SANTA BARBARA collision-free plasma in a magnetic field {3) STANFORD UNIVERSITY (34) McEnteggart, Peter Principal submatrices Barrett, Eamon Boyd Pierce, Stephen Jay Contributions to the theory of surfaces of con­ Generalized isometries stant mean curvature Roeder, David William Barwise, Kenneth Jon Duality of locally compact groups Infinitary logic and admissible sets Clark, David Charles Asymptotic structure of solutions of the Navier-Stokes equations in an CALIFORNIA INSTITUTE OF TECHNOLOGY(l4) stationary exterior domain Bogart, Kenneth Paul Fink, James Paul behavior of Feynman integrals Structure theorems for local Noether lattices Asymptotic Spencer Buck, Robert Jay Goodman, Gerald and optimal control A generalized Hausdorff dimension for func­ Univalent functions tions and sets Heywood, John Groves problems for the Navier­ Hem stead, Robert Jack On nonstationary and the stability of sta­ Minor: Philosophy Stokes equations, tionary flows Stationary absolute distributions for chains of Marshall infinite order Karon, John sign-regularity properties of a class of Kirk, Ronald Brian The Green's functions for ordinary differential Measures in topological spaces equations and some related results Lane, Richard Neil Loh, Peter Ching-Ming Minor: English Orbits of near earth satellites Normal structures and automorphism groups Angus John of t-designs Macintyre, pairs of real-closed fields Lu, Kau-Un Classifying Thomas Arthur Some properties of the coefficients of cyclo­ McCready, Neumann problem for stationary tomic polynomials The interior solutions of the Navier-Stokes equations Taylor, Richard Forsythe Morrow, James Allen, Jr. Invariant subspaces in Hilbert and normed The topological type of nonsingular deforma­ spaces tions of singular surfaces Pittinger, Arthur Orr Department of Applied Mathematics Boundary decomposition of Markov processes Bluman, George Wallace Roberts, Fred Stephen Construction of solutions to partial differen­ Representations of indifference relations tial equations by the use of transformation Schneider, Robert Bruce groups Fourier analysis on thin sets Laetsch, Theodore Willis Taylor, Laird Edwin Eigenvalue problems for positive monotonic Spaces of analytic functions and associated nonlinear operators cohomologies Mac Dowell, Thomas William Thompson, David Bruce Boundary value problems for stochastic dif­ Subrecursiveness and finite computers ferential equations Mortell, Michael Philip Computer Science Department Some approximate solutions of dynamic prob­ Shaw, Alan Cary lems in the linear theory of thin elastic The formal description and parsing of pictures shells

879 Department of Operations Research Wickes, Harry E. Callero, Monti Don Pre-service mathematics preparation of ele­ An adaptive command and control system mentary teachers: the relative effective­ utilizing heuristic learning processes ness of two programs in determining atti­ Gray, Paul tudes toward, and achievement in, mathe­ Mixed integer programming algorithms for matics site selection and other fixed charge prob­ lems having capacity constraints COLORADO STATE UNIVERSITY (2.) Lippman, Steven Arthur Optimal inventory policy with multiple set-up Bowden, David Clark costs Simultaneous confidence bands Miercort, Frederic Alan Henry, Myron Stanley Some effects of advertising and prices on Best approximate solutions of the Riccati optimal inventory policy matrix equation Rolfe, Alan John The control of a multiple facility, multiple UNIVERSITY OF COLORADO (6) channel queueing system with parallel input streams Boland, Willard Robert Sobel, Matthew Jacob Some convexity theorems for the eigenvalues Production smoothing with stochastic demand of certain Fredholm -type operators and related inventory problems Comer, Stephen Daniel Yuan, John Shang-Chia Some representation theorems and the amal­ Algorithms and multi-product model in pro­ gamation property in algebraic logic duction scheduling and employment smooth­ Duke, John Walter ing Matrices over division rings and EPr matrices Department of Statistics Gaines, Robert Earl Brown, Mark Nonlinear ordinary differential equations with Convergence in distribution of stochastic linear constraints integrals P assi, Harsh Anand Feder, Paul Ira Some asymptotic formulae in partition theory On the likelihood ratio statistic with applica­ Snell, Robert Isaac tions to broken line regression Some convergence criteria for continued Hamilton, Martin Alva fractions based on the nested set property The Robbins-Monro stochastic approximation approach to a discrimination problem Perlman, Michael David CONNECTICUT One-sided problems in multivariate analysis Ross, Sheldon Mark UNIVERSITY OF CONNECTICUT (2.) Denumerable Markovian decision models Singer, Burton Herbert Tsokos, Chris Peter Some limit theorems for Markov chains and Stochastic integral equation applications to related occupancy problems stochastic automatic systems Stroud, Thomas William Woods, Jimmie Dale Comparing conditional distributions under Fourier series and Chebyshev polynomials measurement errors of known variance in statistical distribution theory Tafeen, Stephen Michael A Markov decision process with costly in­ YALE UNIVERSITY (10) spection Wheeler, Alan Clement Blumenthal, Robert George Multiproduct inventory models with set-up The geometric structure of the spectrum of cost a function algebra Zidek, James Victor Burckel, Robert 8 ruce On the admissibility of formal Bayes esti­ Weak almost periodic functions on semigroups mators Cole, Brian James One-point parts and the peak point conjecture COLORADO Coven, Ethan Locally compact transformation groups hav­ COLORADO STATE COLLEGE, GREELEY (2.) ing an equicontinuous replete semigroup Glassman, Neal David Platt, John L. Cohomology of non-associative algebras The effect of the use of mathematical logic Jenkins, Thomas Morton in high school geometry: an experimental Banach spaces of Lipschitz functions on an study abstract metric space

880 Lin, Tsau-young Sanderson, Lee Bjork A spectral sequence in the stable homotopy Statistical decision systems: theoretical and category applied results Sanders, jon H. A generalization of Schur's theorem FLORIDA Scott, Leonard L ., jr. Uniprimitive groups of degree kp FLORIDA STATE UNIVERSITY (9) Smith, Mi-Soo Bae (Mrs. Larry) On the structure of Banach algebras Brewer, james W. Some properties of divisibility in fields and DISTRICT OF COLUMBIA integral domains Magarian, Elizabeth Ann AMERICAN UNIVERSITY (1) Complete tensor products and formal power series rings Dunlap, Paul R. Tindell, Ralph S. Posterior interval estimation for the unequal Piecewise linear and almost piecewise linear variance case imbeddings Wood, Craig A. THE CATHOLIC UNIVERSITY OF AMERICA (6) On general Z.P .I. rings

Bessman, john Department of Statistics The adjoint linear functional differential Arnold, jesse Charles equation and some of its applications Estimation through preliminary tests Bruckner, Lawrence Chitgopekar, Sharad Shanker Some contributions to the theory of homo­ Continuous time Markovian sequential control geneous random fields on locally compact processes Abelian groups Lever, William Edwin Diestel, joseph Part l. The limiting distribution of the likeli­ Orlicz spaces of Lebesgue- Bochner measur­ hood ratio statistic -2ln >-n under a class of able functions and integral representations local alternatives; Part 2. Minimum aver­ of linear and multilinear continuous opera­ age risk decision procedures for the non­ tors on them central chi-square distribution Kissin, Sidney Scheaffer, Richard Lewis The study of the class of random sequences Statistical procedures for some problems in that are strictly stationary of order k inverse sampling of bulk materials McCabe, Bernard Waikar, Vasant Balkrishna Some contributions to the theory of positive Inference with tested priors definite functions on groups Murphy, Catherine UNIVERSITY OF MIAMI (4) Noetherian decompositions in a residuated lattice with an associative, isotone multi­ Keesling, james Edgar plication Mappings and dimension in metric spaces O'Hara, Patrick joseph, jr. GEORGETOWN UNIVERSITY (4) A study of interpolation by complex poly­ nomials Cherkas, Barry M. Ortiz-Suarez, Augusto Herminia Nonlinear diffusion equations in Lz(R m) Amitsur radicals Knight, Ridgway Brewster Storey, Albert jennings On the existence and uniqueness of generali­ Equidivisible semigroups zed solutions of quasi-linear elliptic equa­ tions in two independent variables GEORGIA Manning, john B. Representations of pseudo p-normed algebras EMORY UNIVERSITY (1) Meyers, Ann Marie On periodi~ solutions of hyperbolic partial Spellmann, john Winston differential equations Solutions to u12 = FU in a Banach space

THE GEORGE WASHINGTON UNIVERSITY (3) UNIVERSITY OF GEORGIA (4)

Ku, Hsien Hsiang Duvall, Paul Frazier The problem of interaction in multidimen­ Neighborhoods of !-connected Anr' s in high sional contingency tables, an information dimensional piecewise linear manifolds theoretic approach Leslie, Robert Arthur Minichiello, john Kent The zeros of solutions to certain linear homo­ Negationless intuitionistic mathematics geneous differential equations of even order

881 Machusko, Andrew Joseph Ziomek, Leo Frank End point compactifications of piecewise On the boundary behaviour in the metric Lp linear manifolds of subharmonic functions

Department of Statistics Department of Statistics Clark, Frank Colden Bhattacharya, Rabindra The role of interaction in irregular experi­ Berry-Esseen bounds for the multi-dimen­ ments sional central limit theorem Causey, Beverly Douglas GEORGIA INSTITUTE OF TECHNOLOGY (1) Some multi-dimensional incomplete block designs Law, Alan Greenwell Land, Charles Even Solutions of some countable systems of Tests and confidence intervals from trans­ ordinary differential equations formed data Yarnold, James K. IDAHO The accuracy of seven approximations for the null distribution of the chi-square of fit UNIVERSITY OF IDAHO (1) statistic

Stern, James Leroy Contractible open manifold and cellular map UNIVERSITY OF ILLINOIS (26)

ILLINOIS Alexander, Stephanie Minor: Electrical Engineering UNIVERSITY OF CHICAGO (20) Reducibility of Euclidean immersions of low codimensions Burroughs, John Edward Bergman, John Grant The splitting principle and operations in alge­ Minor: Philosophy braic K-theory Harmonic analysis on algebraically irreduci­ Day, Mahlon Michael ble semigroups Infinite dimensional representations of A 1 Corbett, Joseph Warren de Carvalho, Joao Bosco Pitombeira Fernandes Minor: Botany On the meta-stable homotopy of the unitary On isotopy types of complexes with applica­ groups tion to knots de Guzman, Miguel Crenshaw, James Allen Singular integral operations with generalized Minor: Philosophy homogeneity Extreme positive linear operators Gorman, Howard Edwin Drobot, Vladimir Invertibility of modules over P ruffer rings Minor: Physics Grosshans, Frank David Quasi-dynamical systems Orthogonal representations of absolutely sim­ Fraser, Marshall Monteith ple algebraic groups Minor: Philosophy Harper, John Robb Multiplicities and Grothendieck groups Stable secondary cohomology operations Hall, Peter Robin Iyanaga, Kenichi Minor: Philosophy Arithmetic of special unitary groups and Local connectivity of the space of homeomor­ their symplectic representations phisms of a non-compact manifold Kock, Anders Jungersen Hall, Richard Samuel Limit monads in categories Minor: Electrical Engineering Lasher, Sim Theorems about Beurlings generalized On derivatives and derivates in L CJ primes and the associated zeta function Sagher, Yoram Harty, Sister M. Raymond Capua On hypersingular integrals with complex Minor: Philosophy homogeneity Automorphisms of the 4-dimensional uni­ Segovia-Fernandez, Carlos modular unitary group On the area function of Lusin Kearns, Thomas John Shaker, Richard J. Minor: Philosophy Abstract quadratic forms On representations of Lie algebras of clas si­ Tucker, Todd Stephen ca! type Nonlinear P -compact operator equations and Khurana, Surjit Singh problems of stability Minor: Physics Whitcomb, Albert Measures and barycenters of measures on The group ring problem convex sets in locally convex spaces

882 McDowell, Leland Kitchin Carey, John Joseph Minor: Philosophy Finite deformation analysis of the Euler Variable successive over-relaxation instability of an elastic circular tube Meeden, Glen Dale Chao, Ping- Wen Minor: Philosophy The concept of expansions of points of sets Some problems in prediction theory Chiapetta, Richard Louis Montague, John Stephen Large plastic deformation of thin spherical Minor: Philosophy shells with reinforced openings On transitive extensions of finite permutation Gordon, Milton A. groups On a class of degree one algebras Reid, Kenneth Brooks Harkin, Joseph Bernard Structure in finite graphs Uniform Stieltjes integral Roberts, Charles Emery, Jr. Herakovich, Carl Thomas Minor: Physics Elastic-plastic torsion of multiply-connected On the behaviour of the integral curves of a cylinders by quadratic programming third order autonomous system of differen­ Marcus, Philip S. tial equations in a neighborhood of a criti­ Probabilistic metric spaces and stochastic cal point processes Showalter, Ralph Edwin Neath, Walter James Pseudo-parabolic partial differential equa­ Generalizations of Runge's theorem for tions Banach-valued multidimensional complex Staudte, Robert George functions Minor: Electrical Engi-neering Invariance theory and sequential estimation Department of Mechanics Sward, Gilbert Leinbaugh Saric, William S. Minor: Philosophy Hydrodynamic stability of heterogeneous Transfinite sequences of axiom systems for couette flow set theory Sward, Marcia Peterson NORTHWESTERN UNIVERSITY (9) Minor: Philosophy The mixed boundary value problem along the Chalice, Donald line of parabolicity for a certain class of Function algebras and supports of measures hyperbolic partial differential equations Clough, Robert Thomas, Charles Gomer The cohomology of an H-space realizing Minor: Physics Im(J) and of its classifying space A characterization of the Chevalley groups Goes, Sigrun (G2(2n)) Some compactness criteria for Kothe spaces Trott, William Richard Hill, Richard Oscar, Jr. Minor: Electrical Engineering On the geometric dimension of bundles and Density theorems relative to mean regular cohomology operations coverings and a transformation group in Huff, Melvyn abstract measure space c' conformal mappings and the theory of Weber, Waldemar Carl connections Minor: Physics Salter, Kathleen Conformal vector fields on compact Rieman­ On G-algebra extensions nian manifolds Seebach, J. Arthur Weger, Ronald Clyde Cones and homotopy in categories Minor: Philosophy Tai, Shin Sheng A condition for positive definiteness groups On minimum imbeddings of compact symme­ Wilson, Paul Robert tric spaces of rank one Minor: Philosophy Wichman, Michael Linear algebra over skewfields Modules of finite width

Department of Computer Science INDIANA Richardson, Fontaine Graphical specification of computation INDIANA UNIVERSITY (7)

ILLINOIS INSTITUTE OF TECHNOLOGY (10) Beachy, John Allen Some homological classes of rings and mod­ Calabrese, Philip George ules Triangle inequalities in statistical metric Brady, Walter Foster spaces On the local reciprocity law

883 Elcrat, Alan Ross Vogt, Ernest Doyle Existence theorems for a nonlinear partial The perturbation effects of a magnetic dipole differential equation of viscous, incom­ on a charged satellite pressible flow Wen, Shih-liang Koehler, Anne Bramble Some problems of wave propagation A study of quasi-projective modules Koehler, Donald Otto Department of Computer Sciences Generalized inner product spaces Evans, Bernard B. Kroener, Christian Wilhelm Explicit asymmetric difference algorithms Minor: Physics for the approximate solution of parabolic On radical rings partial differential equations Powers, Michael Jerome Hoff, John Clifford Multi-valued mappings and fixed point theo­ Linear operators and approximation theory rems Hosken, William Henry Certain extended post canonical systems UNIVERSITY OF NOTRE DAME (6) related to context-sensitive languages Kerr, Douglas Santee Gerstein, Larry Joel On some iterative methods for solving a class Splitting quadratic forms over integers of of nonlinear boundary value problems global fields Landweber, Lawrence Hugh Hirschfelder, John Joseph A design algorithm for sequential machines The first main theorem of value distribution and definability in monadic second order­ in several variables arithmetic Kujala, Robert Oakes Sweet, Roland Andrew Meromorphic and entire functions of finite Operators leaving invariant a cone in a Banach A-type in several complex variables space O'Donnell, Stephen Joseph Zahar, Ramsy Vincent Michel Relative cohomology of finite groups done Computational algorithms for line!).r differ­ functorially ence equations O'Toole, James William Manifolds of mappings and group actions Department of Statistics Payne, Thomas Henry Al-Ani, Sabri Precomplete enumerations Some distribution problems concerning char­ acteristic roots and vectors in multivariate PURDUE UNIVERSITY (24) analysis Brown, Bruce M. Azevedo, Alberto Moments of a stopping rule related to the The Jacobian ideal of a plane algebroid curve central limit theorem Brumley, Wilson Edward Gupta, Arjun K. The asymptotic behavior of solutions of a Some central and non-central problems in certain class of differential-difference multivariate analysis equations of neutral type Jain, Aridaman K. Haggard, Gary Martin A statistical study of book use Embedding of graphs in surfaces Jayachandran, Kanta G. Heard, Melvin L. Non-central distributions of some multivari­ Linear functional differential equations of ate test criteria and associated powers of neutral type test Huang, Men -Fon Stout, William F. On the algebraic closure of the field of mero­ Some results on almost sure and complete m orphic functions over an algebraically convergence in the independent and martin­ closed field of characteristic P gale cases Kuczkowski, Joseph Edward On subsemigroups of nilpotent groups IOWA Lin, Ta-Feng A discussion on Brownian motion of !-dimen­ UNIVERSITY OF lOW A ( 1 0) sional, continuous mechanical system in a viscous medium Arendt, Billy Dean Sabbagh, Lian David Semisimple bands Variational problems with lags Grimmer, Ronald Calvin Stager, James Aaron Almost autonomous differential equations The condition of regular degeneration for Neuhaus, Ralph James singularly perturbed linear functional dif­ The derivation algebra of an algebra of degree ferential equations two

884 Real, Sister Cathleen Clare Ruhl, Donna Jean Brogan Rings of quotients and quotient modules Preliminary test procedures and Bayesian Schlunt, Richard Stanley procedures for pooling correlated data Some boundary value problems for nonlinear Sampson, Charles B. second order differential equations Contributions to the study of sequential Swick, Kenneth Eugene covariance analysis Stability and boundedness theorems for third and fourth order nonlinear differential KANSAS equations Vandervelde, Richard Allen UNIVERSITY OF KANSAS (7) Group-like structures in categories Voxman, William Lloyd Greenlee, Wilfred Martin On the shrinkability of decompositions of Rate of convergence in singular perturbations 3-manifolds Hutchinson, John Joseph Intrinsic extensions of rings Department of Statistics Johnson, Robert LeRoy Sievers, Gerald Lester Rings of quotients of topological rings On the probability of large deviations and Maness, William Dale asymptotic efficiency Minor: Physics Stoline, Michael Ross On the contact of Cartan varieties with alge­ A test of the equality of means of normal braic hypersurfaces inN-dimensional pro­ distributions with ordered variances jective space Pirtle, Elbert M., Jr. IOWA STATE UNIVERSITY (15) Integral domains which are almost Krull Shores, Thomas Stephen Curtis, Douglas Warren On groups with category Deficiency and stability in infinite dimension­ Taylor, Howard Lawrence al linear topology The continuity of integral transformations Haeder, Paul Albert between LP spaces with mixed norms On the zeros of solutions of elliptic partial differential equations KANSAS STATE UNIVERSITY (3) Kopecky, Kenneth Joseph Nonlinear sloshing in elliptical tanks Poinsignon, Mireille 0. McCoy, Robert Allan Contributions to semi-ring theory Cells and cellularity in infinite-dimensional normed linear spaces Department of Statistics and Computer Sci­ Riess, Ronald Dean ence Power series eigenvalue analysis Johnston, James William Simpson, David Lenz A distribution free test of inactivation for A numerical method of characteristics for RXC factorial experience with one obliga­ solving hyperbolic partial differentl:al equa­ tion per cell tions Lee, Sung- Wook Watson, James Douglas Non-parametric sequential method A numerical technique for solution of the linear second order elliptic equation in the KENTUCKY plane Winrich, Lonny Bee UNIVERSITY OF KENTUCKY (8) An explicit method for the numerical solution of nonlinear diffusion equations Calys, Emanuel G. Univalence and starlikeness of functions Department of Statistics Dykes, Donald Campbell Ahmad, Munir Generalized Frattini subgroups of a finite Truncated multivariate Poisson distributions group Giesbrecht, Francis Dykes, Nancy Rodgers Study of residuals in planned experiments Mappings and realcompact spaces Johnson, John P. Miller, James Edward Pooling regressions and a statistical outlier A numerical determination of the absorption methodology for lines coefficient of the negative hydrogen ion Lund, Richard Edgar Miller, James Ernest Factors affecting consumer demand for meat, Extremal functions for meromorphic uni­ Webster County, Iowa valent functions Memon, Ahmed Nebb, Jack Z statistic in discriminant analysis Some applications of an integral transform

885 Stepp, James Wilson Gordon, William Two types of semigroups On the method of steepest descent applied Wright, Donald Joseph on the Sobolev spaces Hk( X, Y) On p-valent starlike functions and the uni­ Orlow, Sanford valence of some integrals Symmetric products of algebraic curves and an extension of Torelli's theorem LOUISIANA Department of Biostatistics LOUISIANA STATE UNIVERSITY (9) Corey, Paul N. Statistical inference in Markov chains with Gilbert, John R., Jr. particular application to genetics Rings satisfying the three Noether axioms Leyton, Morley Lofquist, George W. Stochastic models of life cycles of helminthic Class-numbers and sums of three squares parasites in quadratic fields Thorslund, Todd Pierce, Alan C. Computationally simple and 'efficient' esti­ Periodic and almost periodic vectors mators for a one parameter exponential Ristroph, Robert M. model Operators on Hilbert space Vander Zwagg, Roger Sentilles, Francis D., Jr. A stochastic process model for estimating Kernels and operators on the space of con­ the dependence of the probability of a di­ tinuous function sease on several regressors Tramel, Sammy J. Factorization of principal ideals in the sense Department of Statistics of quasi-equality Beran, Rudolph Jaroslav Wenceslaus Vaughan, Nick H. Minor: Operations Research Some containment relations between classes Tests for uniformity of a distribution on a of ideals in an integral domain compact homogeneous space Waid, Charles Carter Davis, Herbert Thaddeus, Ill Ideals and modules of quaternion orders and Minor: Economics their related quadratic forms Inference based on the periodgram for station­ West, James E. ary Gaussian time series Fixed point sets of transformation groups on infinite product spaces

TULANE UNIVERSITY (6) UNIVERSITY OF MARYLAND ( 14)

Cannon, Raymond J. Balakrishnan, Rangaswami Quasi-conformal structures and the metriza­ Topics in (v,k, A)-configurations tion of 2-m anifolds Basmaji, Butrus George Crofts, George On monomial representations of finite groups Echelon spaces, co-echelon spaces and steps Cushing, Jim Michael Flanders, Mary Catherine (Berglund) Local uniqueness for harmonic functions Ideal C*-algebras under nonlinear boundary conditions Riley, John P. Dunkel, Gregory Martin Decompositions of E3 with a compact 0-di­ Some mathematical models for population mensional set of non-degenerate elements growth with lags Schnare, Paul S. Elkin, Richard Michael Infinite complementation of topologies in the Convergence theorems for Gauss-Seidel and lattice of topologies other minimization algorithms Tong, Alfred Hooper, Robert Clark Tensor products of Banach spaces with un­ A study of topological abelian groups based conditional bases on normed space theory Horn, William A. MARYLAND Fixed points determined by conditions on the iterates of a mapping in Banach space: THE JOHNS HOPKINS UNIVERSITY (10) An extension of some work by F. E. Brow­ der Alexander, James Justice, James Horace Minor: Physics Elements isometric under the Gelfand trans­ Mod P connected K-theory and the stable form: Applications to Hilbert algebras, homotopy groups of spheres group algebras, projections and convexity Currier, Albert Lakein, Richard Bruce On the fibering of non-compact solvmanifolds Quadratic extensions of imaginary quadratic which have low dimension or low rank number fields

886 Levine, Lawrence Elliott Josephson, Keith Benjamin Self-similar solutions of the equations gov­ The structure of a Banach algebra invariant erning the two-dimensional, unsteady mo­ for measure preserving automorphisms tion of a polytropic gas Kulkarni, Ravindra S. Saff, Edward Barry Curvature and metric Selected topics in interpolation and approxi­ McCabe, John P. mation by polynomials, rational functions, P -adic theta functions and meromorphic functions Roberts, Lawrence Gordon Shapiro, Louis Welles On the flat cohomology of finite group schemes Group representations and algebraic number Roberts, Leslie Gordon fields Algebraic K1 of vector bundles Solomon, Jay Murrie Taylor, Walter Fuller The solution of some singular Cauchy prob­ Compactness in relational structures lems Waterhouse, William Charles Ullom, Stephen Virgil Ab·=lian varieties over finite fields Normal bases of ambiguous ideals as Galois modules Department of Statistics D' Agostino, Ralph \t\ASSACHUSETTS Estimation of percentiles of continuous popu­ lations 30STON UNIVERSITY { 1) Fairley, William Roots of random polynomials Rosman, Bernard F ienberg, Stephen Chebyshev approximation of discontinuous The estimation of cell probabilities in two­ functions way contingency tables Kleyle, Robert 3RANDEIS UNIVERSITY {5) An application of the direct probability m eth­ od of inference Coe {Nevulis), Penelope Ann Tsao, Rhett Reducing the number of fold curves and cusps A second order exponential model for multi­ of generic maps dimensional dichotomous contingency tables Hertz, Douglas Ambient surgery and tangential homotopy Department of Engineering and Applied quaternionic projective spaces Physics Katz, Victor Amazigo, John Chukwuemeka Brauer group of a regular local ring Buckling under axial compression of cylinders Koschorke, Ulrich with random axisymmetric imperfections Infinite dimensional K-theory and character­ Anderson, Robert Helms istic classes of Fredholm bundle maps Syntax-directed recognization of hand printed Uhlenbeck, Karen two-dimensional mathematics The calculus of variation and global analysis Danielson, Donald Alfred Buckling and initial postbuckling behavior of C:LARK UNIVERSITY {I) spheroidal shells under pres sure deRis, John Norval Kaput, James Joseph The spread of a diffusion flame over a com­ Categories and bicategories bustible surface Fischer, Ferdinand HARVARD UNIVERSITY {32) Stress diffusion from axially loaded stiffeners into cylindrical shells Brown, Lawrence Gerald Fischer, Michael John On the structure of locally compact groups Grammers with macro-like productions Chernoff, Paul Robert Gierasch, Peter Jay Semigroup product formulas and addition of A study of a radiative heating in a Martian unbounded operators atmosphere Corwin, Lawrence Jay Hughes, Cara Joy Measures on locally compact groups with Non-thermal radio radiations from the ter­ certain transformation properties restial exosphere Freifeld, Charles Jay Lautenbacher, Conrad Charles, Jr. The cohomology theory of transitive modules Gravity wave refraction by islands over the primitive infinite Lie algebras Marma, Victor John Henrich, Christopher John Application of decision theory to a binomial Nonunitary representations and harmonic gambling problem and a sampling budget analysis of some solvable Lie groups allocation problem

887 Moore, Dennis Wilson Petersen, Bent Edvard Planetary-gravity waves in an equatorial On the calculus of symbols for pseudo­ ocean differential operators Neuman, Charles Paul Priddy, Stewart Beauregard Frequency domain stability criteria in non­ Primary cohomology operations for simpli­ linear automatic control cial Lie algebras O'Neill, Elizabeth jean Rallis, Stephen james A quasilinear theory for axially symmetric Lie group representations associated to sym­ flows in a stratified rotating fluid metric spaces Pizer, Stephen Murray Rogers, Joel Carle Whitehouse Production and processing radioisotope scans Minor: Physics Woods, William Aaron Existence and uniqueness theorems for in vis­ Semantics for a question-answering system cid incompressible flow Rosenthal, John William UNIVERSITY OF MASSACHUSETTS (2) Addition of relations in model theory; Truth in all of certain well-founded countable Dacey, james Charles, jr. models arising in set theory Orthomodular spaces Segal, David Mordecai Marsden, Edwin Leroy, jr. Symplectic cobordism of manifolds The commutator and irreducibility conditions Walker, Robert P au! on orthomodular lattices Whitehead torsion for free simplicial groups Zalcman, Lawrence Allen MASSACHUSETTS INSTITUTE OF TECHNOLOGY Bounded analytic functions on domains of (23) infinite connectivity

Band, Melvin On the integral extensions of quadratic forms MICHIGAN Bergeron, Robert Francis, Jr. Minor: Mechanical Engineering MICHIGAN STATE UNIVERSITY (8) A nonlinear critical layer theory for shear flow stability Boals, Alfred John Bloom, Stephen Louis Non-manifold factors of euclidean spaces Minor: Philosophy Johnson, Lee Warren The hyperprojective hierarchy Approximation of vector-valued functions Brown, Earl Douglas Kapoor, Shashichand F atehchand Some mathematical models of inspection Inverting and monotone properties of com­ along a production line plexes Ebin, David Gregory McDonald, Bernard Robert On the space of Riemannian metrics Class equation for matrices over integers Feiner, Lawrence mod p 0 Orderings and Boolean algebras not isomor­ Skerry, Herbert Bancroft phic to recursive ones On a generalization of the Lototsky summa­ Friedman, Harvey Martin bility method Subsystems of set theory and analysis Weinstein, Stanley Edwin Gold, Robert Approximations of functions of several vari­ On r-extensions of imaginary quadratic fields ables: Product Tchebycheff approximations Graham, Colin Cloete Symbolic calculus for subalgebras of Fourier­ Department of Statistics and Probability Stieltjes transforms Perng, Shian Koong Israel, Jay Elliot Inadmissibility of various "good" statistical An entropic approach to approximation procedures which are translation invariant Kisilevsky, Hershy Harry Rohatgi, Vijay Kumar On ramification and the ideal class group of Some results on convergence rates and relatively cyclic number fields asymptotic behaviour for the laws of large Knighten, Carol Susan Marians numbers Differentials on quotients of algebraic varie­ ties UNIVERSITY OF MICHIGAN (32) Macintyre, John Morrison Contributions to metarecursion theory Abrahamse, Allan Francis Moses, Joel The tail field of a Markov chain Symbolic integration Barbee, Willard Louise Packel, Edward Wesler On the representations of groups of order p0 Some results on (C0 ) semi-groups and the by matrices over the integers taken modulo Cauchy problem a power of p

888 Case, James Howard Meyer, John Frederick Equilibrium points of n-person differential Minor: Biological Systems games Algebraic isomorphism invariants for transi­ Erickson, R. V. tion graphs Functions of finite Markov chains Mowshowitz, Abraham Abbey Fried, Michael David Entropy and the complexity of graphs Value sets of polynomials Norman, Richard Stephen Glover, Henry Hatfield Signal extraction of symmetric functions from On embedding and immersing manifolds in noise euclidean space Putzolu, Gianfranco Raimondo Gopal, Mangalam Ramaswamy Minor: Physics Some extremal problems in the theory of Probabilistic aspects of machine structure analytic functions theory Gregory, David Allan Reiter, Raymond Vector sequence spaces A study of a model for parallel computations Hedlund, James Howard Roosen-Runge, Peter Hermann Multipliers of HP spaces Minor: Behavioral Systems Hestenes, Marshall Didrick An algebraic description of access and con­ Singer groups trol in information-processing systems Holmes, Charles Shannon Rosenberg, Richard Stuart Pro jecti vities of free products Minor: Biological Systems Horowitz, Joseph Simulation of genetic populations with bio­ Semilinear Markov processes chemical properties Johnson, Jeffrey Lee Global continuous solutions of hyperbolic WAYNE STATE UNIVERSITY (5) systems of quasi-linear equations Kaczynski, Theodore John Benabdallah, Khalid Boundary functions On quasi-essential subgroups of abelian Krom, Myren Laurance groups Invariant means Herda, Hans-Heinrich Wolfgang LaPalm, James Robert Modularized spaces of generalized variation Existence analysis for optimal control prob­ Parnes, Milton lems with exceptional sets Symmetrization and conformal mapping Latil, Bertrand Tse, Kam-Fook Singular perturbations of Cauchy's problem Some sequential properties of meromorphic Pinkerton, Tad Brian functions and compact families of normal Program behavior and control in virtual functions storage computer systems Warren, Joseph Pollatsek, Harriet Suzanne On the growth of unbounded holomorphic Groups generated by transvections over per­ functions in a disc which are bounded in a fect fields of characteristic two spiral Rosenthal, Peter Michael On lattices of invariant subspaces MINNESOTA Tape, Walter Richard Commuting flows and vector fields UNIVERSITY OF MINNESOTA (10)

Akiyama, Yoshio Smith theory and Steenrod powers for Zp Aoki, Masahiko Department of Computer and Communication Increasing returns to scale and market Sciences mechanisms Bagley, John Daniel Bardeen, William Allan The behavior of adaptive systems which em­ A covariant formulation of spin and repre­ ploy genetic and correlation algorithms sentations of the local current algebra Dale, Philip Scott Hager, Richard Allen Children's color categories and the problem Minor: Physics of language and cognition Local behavior of solutions of quasilinear Houde, Robert Alphonse parabolic equations Minor: Electronics and Signal Theory Laursen, Kjeld Bagger A study of tongue body motion during selected Minor: Philosophy speech sounds Tensor products of Banach-* algebras Jump, J. Robert Little, John Alvord Minor: Electrical Engineering Minor: Philosophy Iterative network realization of sequential On singularities of submanifolds of higher machines dimensional euclidean spaces

889 Park, Chull White, Albert George, Jr. Minor: Aeronautical Engineering Two-Banach spaces Generalized Riemann-Stieltjes integral over the K. Y. W. space of functions of two vari­ WASHINGTON UNIVERSITY {7) ables Timsans, Edward Andrew Liu, Mu-Chou On automorphic functions defined on locally Heat equations on Riemannian connected compact groups bundles White, James Harris Phelps, Dean Gifko Minor: Linguistics and Comparative Philology On a coefficient problem in univalent functions Self-linking and the Gauss integral in higher dimensions Department of Applied Mathematics and Computer Science Department of Statistics Hesse, Richard M., Jr. Patel, Jagdish K. A suboptimal method for the global solution Selecting a subset containing the best one of of the nonlinear programming problem several IFRA populations lnselberg, David A. SAP: A model for the syntactic analysis of MISSOURI pictures Krolak, Patrick D. UNIVERSITY OF MISSOURI {5) Bounded variable algorithm for solving inte­ ger linear problems Anderson, Clifford Harold Steinberg, David I. Differential-difference equations of advanced The fixed charge problem type Walsh, Robert A. Koos, Russell Lee Initial value problems associated with Burg­ The initial value problem for non-linear ers' equation evolution equations Valentine, Joseph Earl Metric deve1.opment of hyperbolic geometry MONT ANA

Department of Statistics MONTANA STATE UNIVERSITY (1) Patton, Robert Allen A parameterization of a form of the logistic Crump, Kenny Sherman and a Monte Carlo study of the estimators Some generalized age-dependent branching from small samples processes Wright, Farrell Tim Some results giving rates of convergence of NEBRASKA infinite v-adic forms in independent random variables UNIVERSITY OF NEBRASKA (8)

ST. LOUIS UNIVERSITY (8) Alin, John Suemper Structure of torsion modules Andria, George Daniel Anderson, Donald Alfred Minor: Philosophy On the construction and analysis of multi­ Integral polynomial approximation factorial experimental designs De Conge, Sister Mary Sylvester Chopra, Dharam Vir Minor: French Investigations on the construction and exist­ Two-normed lattices and two-metric spaces ence of balanced factorial fractions of 2m Grant, Kerry Elmer series Minor: Philosophy Fox, D. Frank A two-metric lattice structure Natural deduction and universal validity Gruttke, William Bromley St. Mary, Donald Frank Fifth order pseudo-Runge-Kutta methods Oscillation and comparison theorems for Harshany, Gerald Paul second order linear differential equations Minor: Philosophy Shreve, Warren Eugene An optimizing distribution function utilizing The equation y" = f(x,y,X) on [a,oo) with queueing theory to determine the reliability boundary conditions involving A of a system Tangeman, Richard Louis Ludwig, Hubert Joseph Rings which are radical extensions of sub­ Boolean two-geometry rings Uribe, Carlos Eduardo Vasco Tolo, Kenneth William Homogeneous identities on algebraic loop Factorizable semigroups

890 NEW HAMPSHIRE Rabinowitz, Larry Reachability and connectedness in random DARTMOUTH COLLEGE {5) graphs and digraphs Wolf, Edward Henry Feustel, Charles Dana A test for randomness on the line and a rela­ Isotopic unknotting in M 2 X I ted k-sample test for homogeneity Gross, Jonathan Light The decomposition of a 3-manifold by disks STEVENS INSTITUTE OF TECHNOLOGY{6) Jantosciak, James Samuel Splitting fields of orthogonal geometries Bronson, Richard D. Packard, Robert Warren Singularities of the gain of a fixed endpoint, On a generalized second order linear differ­ fixed time regulation problem ential operator Hilden, Hugh M. Weidenhofer, Neal On the ext. points of some convex sets Simplified language for abstract mathematical of analytic functions structures Kahana, David K. Equal and almost equal weight quadrature UNIVERSITY OF NEW HAMPSHIRE {l) formulas Keeping, Anthony J. Cohen, David Warren Numerical solution of a third-order partial The extension of a locally compact local differential equation group to a global group Larsen, Bruce 0. An abstract comp. with a language related NEW JERSEY to combinatory logic Leibiger, Tusta ve A. PRINCETON UNIVERSITY {9) Wave propagation in an inhomogeneous me­ dium with slow spatial variation Anderson, Robert F. V. A functional calculus for non-commutative NEW MEXICO self-adjoint operators Burr, Stefan Andrus UNIVERSITY OF NEW MEXICO {6) An elementary solution of the Waring-Gold­ bach problem Allen, Richard Crenshaw Cannon, John Tucker Functional relationships for Fredholm inte­ Field theoretic properties of Nelson's model gral equations arising from pseudo-trans­ Collins, Donald James port problems Recursively enumerable degrees and the Crown, Gary conjugacy problem On some categories of partially ordered sets Levine, Daniel Alan with residuated mappings Singular integral operators on spheres Moore, Marion Marsden, Jerrold E. Some results on completability in commuta­ Hamiltonian one parameter groups and gen­ tive rings eralized Hamiltonian mechanics Teufel, Hugo, Jr. Mather, John Norman The qualitative behavior of solutions of non­ Stability of C 00 mappings: The division theo­ linear second order differential inequalities rem Thorne, Billy Joe Singer, William M. Right congruences on Baer semigroups Connected fiberings over BU and U Warren, Anthony Tromba, Anthony Stochastic linear estimation in Hilbert space Degree theory on Banach manifolds NEW MEXICO STATE UNIVERSITY {6) RUTGERS, THE STATE UNIVERSITY {6) Duncan, Louis Dean Bauer, Richard Channel capacity and coding The algebra and geometry of polyadic quasi­ Hardy, Dare! Waldo groups and loops Free semigroups and matrix number theory Cole, Floyd Benton, III Khus, Chiev Extensions of isolic arithmetic Bilateral Laplace transformation and genera­ Fraser, Robert lized functions On Lipschitz functions Parker, Larry Dale An extension of the Ulm-Kolettis theorem Statistics Center Schlette, Annemarie Joiner, Brian Lyon Artinian almost abelian groups and their Some properties of four Cochran-type tests groups of automorphisms

891 Sember, John Joseph Repsher, Sister Marilyn F K spaces and variation matrices Comparison of a survey approach to mathe­ matics for college freshmen with an ap­ NEW YORK proach based on Galois theory

ADELPHI UNIVERSITY (2) Department of Mathematical Statistics Dick, Ronald Stewart Dick, Stanley On single server queues with balking and Dual-hopfian abelian groups single or hatched service Finberg, Harvey Fleiss, Joseph Leon A numerical investigation of the motion of a Minor: Biostatistics non -linear string Analysis of variance methods in assessing errors in interview data ciTY UNIVERSITY OF NEW YORK (5) Kantor, Michael Estimating the mean of a multivariate normal Billera, Louis J. distribution with applications to time series On cores and bargaining sets for n-person and empirical Bayes estimation cooperative games without side payments Mustafi, Chandan Kumar Falley, Peter On problems of inference about time-invariant Ideals of operators parameters Masiello, Charles D. Nadas, Arthur Joseph The average of a gauge On the asymptotic theory of estimating the Meskin, Stephen mean by sequential confidence intervals The isomorphism problem for some classes of prescribed accuracy of one-relator groups Wichura, Michael John Sanders, David On the weak convergence of non-Bore! prob­ On extremal circuits abilities on a metric space

COLUMBIA UNIVERSITY (18) CORNELL UNIVERSITY (9)

Altman, Allen Burchard Anger, Frank David Transcendental and algebraic points on group Algebraic K-theory for projective modules varieties with quadratic form Barton, Charles Marshall, III Atwood, Corwin Lee Some results on ample vector bundles Optimal and efficient designs of experiments Bennett, Bruce Michael Belkin, Barry On the characteristic functions of a local Some results in the theory of recurrent ran­ ring dom walk Curran, Peter Michael Demers, Luc Some theorems on cohomology of groups with Homology theories and duality functors applications Feit, Sidnie Dresher Cellar, Ralph Allen K-mersions of manifolds Shift operators in Banach space Garfinkel, Gerald S. Holme, Audun Amitsur cohomology and an exact sequence Some embedding and projection theorems in involving Pic and the Brauer group formal geometry Guterman, Martin Mayr Johnson, Stephen Curtis On ABA-groups of finite order Categorical decompositions Lerman, Manuel Kovacic, Jerald Joseph Recursive functions modulo co-maximal sets The inverse problem in differential Galois Plotkin, Jacob Manuel theory Generic embeddings Mivata, Takehiko Invariants of certain groups NEW YORK UNIVERSITY (21) Olson, Loren David The group Ck/!TkD k and the period-index Bernstein, Herbert Jacob problem in we groups Extension of Haj6s' factorization theorem to some non-abelian groups Birman, Joan Sylvia Braid groups and their relationship to map­ Department of Mathematical Education ping class groups Fey, James T. Cardoso da Silva, Fernando Antonio Figueiredo Patterns on verbal communication in mathe­ The scar problem for pseudo-differential matics classes operators

892 Chein, Orin Nathaniel Maron, Melvin Some IA automorphisms of a free group Non-archimedean Hilbert spaces Ciment, Melvyn Shanholt, Gerald Stable difference schemes with uneven mesh Stability of periodic motion spacings Shilkret, N iel Daggit, Edward A. Non-archimedian Banach algebra On the use of infinitesimal transformations to find the Riemann (-Green) function Gerber, Porter Dean RENSSELAER POLYTECHNIC INSTITUTE (8) Riemann' s3 probl~m for the equation vt (v /3)x = ( vxxx/3 - ((vz>xx/2 Grannick, Robert N. Gevirtz, julian Nonlinear investigation of the stability of Metric conditions that imply local inverti­ flow between counter-rotating cylinders bility Levy, Paul Halem, Milton The interaction of a time harmonic acoustic Difference approximations for boundary-value wave with a thin circular cylindrical elastic problems of conservative equations shell Kalme, Charles Ivars Seifer, Arnold D. Contributions to the theory of discontinuous Ray transmissions in underwater acoustic groups of Mobius transformations ducts having random bottoms La Vita, james Anthony Sidman, Robert D. Perturbation of the poles of the scattering Scattering of acoustic and elastic waves by matrix a spherical inclusion Mike, Valerie Rainey, john W. Contributions to robust estimation Tridiagonalization methods and eigenvalues Nelson, David Brian of tridiagonal matrices Low fJ stability of the guiding center plasma van der Linde, R. H. Prollo, joa Bosco Eigenfunctions of random eigenvalue prob­ Weighted approximation and operator alge­ lems and their statistical properties bras Wachspress, Eugene L. Ramalho de Azevedo, Roberto Figueiredo Solution of the ADI minimax problem On existence theorems for nonlinear integral Zaretzki, Philip M. equations Initial value problems and periodic solutions Rappeport, Michael A. for a class of nonlinear telegraph equations Algorithms and computational procedures for the application of order statistics to queu­ ing problems UNIVERSITY OF ROCHESTER (7) Roitberg, joseph P L invariants on a smooth manifold Chern, Li-Ching Stebe, Peter F. Homogeneous morphisms in additive cate­ On the groups of certain knots and linkages gories Winograd, Shmuel Chou, Ching On the time required to perform computer Structure and ergodic properties of the set operations of invariant means Yhap, Ernesto F. jategaonkar, A run Vinayak An asymptotic optimally robust linear un­ Principal left ideal rings biased estimator of location for symmetric Krishna Sastry, Munukutuha Sitarama shapes Meta-analytic functions and spin spherical Zuckerman, P au! Robert harmonics Inequalities for derivatives of Green's func­ McCaffrey, james Bernard tions of general coercive elliptic boundary Powers in polynomial rings value problems Michaels, john G. Bimeasurable functions POLYTECHNIC INSTITUTE OF BROOKLYN (7) Ribes, Luis A cohomology theory for pairs of groups Anton, Howard Borel measures on the maximal ideals of a Banach algebra Chai, Winchung Alvin STATE UNIVERSITY OF NEW YORK AT An investigation of biorthogonal polynomials BUFFALO (I) Frechen, joseph On permutation graphs Department of Statistics Goldwasser, Eric Slivka, john joseph Generalizations of perfect measures On the Strong law of large numbers

893 STATE UNIVERSITY OF NEW YORK AT UNIVERSITY OF NORTH CAROLINA (12) STONY BROOK (6) Bitzer, Carl Wilfrid Beaucage, David Robert Stieltjes-Volterra integral equations The mod p cohomology structure of certain Clemens, Stanley Ray fibre spaces The isotopy type of certain finite polyhedra Edelson, Allan L. Dotson, William Grady, Jr. Real vector bundles and spaces with free Some iterative methods for approximation of involutions fixed points and solution of functional equations in Banach spaces Department of Applied Analysis Hoyle, Hughes Bayne, Ill Koh, Eusebio Legarda Connectivity maps and almost continuous The Hankel transformation of generalized functions functions Huneycutt, James Ernest, Jr. Liu, Pan Tai Extensions of abstract valued set functions On a problem of stochastic differential games Page, Nelson Franklin Pandey, Jagdish Narayan Kernel systems of analytic functions in the Complex inversion for the generalized con­ unit disc volution transformations Queen, William Charles Department of Statistics The n-dimensional generalized Weierstrass Hunter, Jeffrey Joseph transformation On the renewal density matrix of semi­ Markov process SYRACUSE UNIVERSITY (7) Koch, Gary Grove The design of combinatorial information Blackwell, Paul K. retrieval systems for files with multiple­ The classification of logic functions valued attributes Darden, Geraldine C. Meyer, Richard Martin On direct sums of cyclic groups Some Poisson-type limit theorems for se­ Geramita, Anthony quences of dependent rare events, with Decomposability in the exterior algebra of a applications free module Smith, Kempton John Cameron Nathan, William Majority decodable codes derived from finite Open mappings on manifolds geometries Oum, Ki-Choul Wilkinson, William Edwin Deficiencies of functions whose zeros have an Branching processes in stochastic environ­ angular density ments Roulier, John A. Poole, William Kenneth Monotone and weighted approximation Minor: Public Health Weitsman, Allen W. Some aspects of linear prediction in station­ Asymptotic behavior of meromorphic func­ ary time series tions with external deficiencies NORTH CAROLINA STATE UNIVERSITY (12) YESHIVA UNIVERSITY (2) Bland, Samuel Russell Fitting, Melvin Minor: Physics lntuitionistic logic model theory and forcing The two-dimensional oscillating airfoil in a Linden, Theodore wind tunnel in subsonic compressible flow Tree procedures for infinitary logic Davis, Charles Alfred Minor: Chemical Engineering Boundary value problems for wedges and cones under heat conduction NORTH CAROLINA Eargle, George Marvin Minor: Engineering Mechanics DUKE UNIVERSITY (2) A nonlinear integra-differential equaticn associated with a class of wedge problems Carmichael, Richard Dudley Perry, Lynn Mciver, Jr. Minor: Philosophy Unitary convolution subalgebras of incidence The Paley-Wiener-Schwartz theorem for algebras functions analytic in a half plane Toney, Fred, Jr. Garthright, Wallace Edward, Jr. Involutory matrix representatons of recipro­ Minor: Philosophy cal automorphisms of modules over residue The Markov channel-joining queue class rings of integers

894 Warmbrod, Grover Karl Morse, James T. The distributional finite Fourier transform Ideal spaces and related topologies for func­ tion rings Department of Experimental Statistics Nelson, George C. Abou- El-F ittouh, Hosni Topics in foundations Genotype by environment interactions in cot­ Perrin, Francois M. ton: their nature, related environmental Applications of monotone operators to dif­ factors and implications in zoning ferential equations Allen, David Mitchell Schwartz, Melvyn H. Multivariate analysis of non-linear models Transition semigroups of sequential devices Fretwell, Stephen DeWitt Minor: Zoology UNIVERSITY OF CINCINNATI (2) On territorial behavior and other factors influencing habitat distribution in birds Cohen, Edgar Allan, jr. johnston, Bruce Owen Simultaneous approximation of a function and Spacing of information in the simple exponen­ its derivatives tial model Sastry, M. A. Sundaram Lee, Sung Won Statistical topological structures Minor: Textiles A probability model for random fiber break­ OHIO STATE UNIVERSITY (8) age Mason, Frank Duff Bonar, Daniel Donald Minor: Physics On annular functions Simulation of a multi-station system of Brown, john Kevin queues with delayed exponential service Finite groups having automorphisms oflarge order Caulfield, Patrick joseph OHIO Multi-valued functions in uniform spaces Klimko, Lawrence Andrew CASE WESTERN RESERVE UNIVERSITY (16) Limit theorems for non-negative matrices Koehl, Frederick Stephen Ahmad, Shair Difference properties for Banach-valued Permutation groups functions Anderson, Larry R. Nachman, Louis jack On the asymptotic behavior of certain dif­ Weak and strong constructions in proximity ferential equations spaces Bajaj, Prem N. Plybon, Benjamin Francis Some aspects of semi-dynamical systems Conservation laws for electromagnetic fields theory Pu, Huay-min Huoh Berard, Anthony D., Jr. Contributions to Hayes integral Characterizations of metric spaces by the use of their midsets OKLAHOMA Beus, Hiram L. The use of information in sorting UNIVERSITY OF OKLAHOMA (7) Campbell, Richard L. Induced morphisms on the semigroups of Ahlbrandt, Calvin Dale automata: A view of automata theory from Disconjugacy criteria for self-adjoint differ­ category theory ential systems Haller, Harold S., jr. Berry, Paul McClellan Optimal c- sample rank-order procedures for Mathematics of electrocardiography selection and tests Cole, jon Kieth Hammer, Richard E. Optimization problems in Banach space Structure grammars Fain, Charles Gilbert Leach, Don E. Some structure theorems for nearrings Periodic solutions of certain second order Gazik, Raymond joseph differential equations Fixed point theorems for spaces represented Lozier, Frank W. by partially ordered systems Internal compactifications of topological Lomanitz, Merry Morgan spaces Non-commutative local rings and quotient McCann, Roger C. rings Planar dynamical systems without critical Shafer, Dale Marks points The development and testing of subject mat­ Miller, Paul G. ter for a course in methods of teaching A class of holomorphic functions secondary school mathematics

895 OKLAHOMA STATE UNIVERSITY ( 9) OREGON STATE UNIVERSITY {10)

Brenneman, Franklin Stauffer Aalto, Sergei Kalvin Derived functors in relative homological An iterative procedure for the solution of non­ algebra linear equations in a Banach space Douglas, Samuel Horace Boles, John Amos Convexity lattices related to topological lat­ The logical design of the NEBULA Computer tices and incidence geometries Driessel, Kenneth Richard Easterling, Robert Gene Significance and invariance in mathematical Sample size determination for Bayesian toler­ structures ance intervals Farlow, Stanley James Moon, Wilchor David Periodic solutions of parabolic partial dif­ A study of orbits and transitive continuous ferential equations functions Hill, Richard David Morris, Peter Craig Generalized inertia theory for complex ma­ Singular extensions and cohomology of Lie trices algebras Hwang, John Dzen Tidmore, Freddie Eugene Minor: Integrated Engineering Extremal structure of star- shaped sets On numerical solutions of the general Navier­ Trail, Stanley M. Stokes equations for two-layered strati­ Displaced balanced incomplete block design fied flows Walls, Robert Clarence Lindquist, Norman Fred Optimum symmetric designs for mixtures of Representations of central convex bodies three components McCoy, Michael Franklin Whitfield, Forrest Dale Optimal stationary ( s,S) inventory policies Properties of connectivity mappings and for stochastically convergent demand se­ certain other non-continuous functions quences Shen, Chung- Yi Minor: Mechanical Engineering OREGON Probability densities and correlation func- tions in statistical mechanics UNIVERSITY OF OREGON ( 12) Shiflett, Ray Calvin Loops and points of density in doubly stochas- Ferguson, Edward Neil tic measures Semigroups on the two-cell with idempotent boundary PENNSYLVANIA Ford, Charles Edwin The Schur index and splitting in metacyclic CARNEGIE -MELLON UNIVERSITY ( 12) groups Franklin, James Myron Anderson, William N. Star representations of pseudonormed alge­ Series and parallel addition of operators bras with involutions Caviness, Bobby Forrester Honerlah, Raymond William On canonical forms and simplification Uniform algebras and maximal closed pre­ Day, William Alan primes Thermodynamics of dissipative systems Kelker, Douglas Herson . Kim, Woo Jong Distribution theory of spherical distributions Disconjugacy and disfocality of differential and some characterization theorems systems Kreiss, Robert Allen Mumaw, Hazel Jo Places on commutative rings On finite linear groups of degree p- 1 Leinbach, L. Carl Niccoli, Nilo Order properties in Banach algebras Exceptional characters and 77-induction Macissac, Sister Clare Rosenberg, Robert Permutations and i/- sequences Compactness in Orlicz spaces based on sets Schneider, Joel Elmer of probability measures A categorical setting for the local theory of Taranto, Donald H. primes Algorithms and recursive functions Tomlinson, Michael Bangs Van Buren, Wayne Topological semi valuations on a commutative On local existence and uniqueness of solutions Banach algebra in finite elasticity Tymchatyn, Edward Dmytro The structure of partially ordered spaces and Department of Computer Science a problem in semigroups Haney, Frederick Marion Warner, Hoyt Dinsmore Using a computer to design computer instruc­ A pot-pourri of results on primes tion sets

896 Manna, Zohar UNIVERSITY OF PITTSBURGH (7) Termination of algorithms Sikl6ssy, Laurent Gavala, George William Natural language learning by computer Continuous methods of summation for double series LEHIGH UNIVERSITY (9) Giudici-Espinozo, Reinaldo H. On the evaluation of character sums DuBois, Joseph Jay Markley, William Aaron Linear transformations on function spaces The zrs(p,q) and related transformations Eisenberg, Sheldon Merven defined on sets of bounded double sequences Some linear operators and associated sum­ O'Neill, John Cornelius mability transforms Some numerical solutions of Volterra's inte­ Fleischman, William Meylin gral equation On certain properties related to open cover­ Reichert, Walter Herman ings of topological spaces A concrete embedding of the set of finite Hartmann, Frederick William m-measures defined on a Boolean algebra A generalization of the Sonnenschein summa­ into an m-saturated linear lattice bility transform Schoen, Kenneth Kirch, Murray Robert New coefficients for fifth and sixth-order Topological expansion and maximal spaces predictor-corrector algorithms Landau, Martin David Sullivan, Francis Edward Transfinite ordinal and cardinal dimension A norm characterization of real Lp-spaces for infinite-dimensional non-metrizable topological spaces RHODE ISLAND Larson, Kenneth Edward A characterization of optimal tests of hypo­ BROWN UNIVERSITY (10) theses Mugridge, Larry Robert Allard, William Kenneth Conformal maps between compact Riemannian On boundary regularity for the Plateau prob­ manifolds lem Weston, James Harley Brennan, James Edward Some results concerning m-compactness Point evaluations and invariant subspaces Hallstrom, Alfred Philip Bounded point derivations and other topics UNIVERSITY OF PENNSYLVANIA (4) concerning rational approximation Leonard, Peter Carney Cherenack, Paul Francis Abstract of G-structures on spheres Homotopy groups of algebraic arcs found of Ryan, Patrick Joseph smooth approximations Homogeneity and some curvature conditions Gordon, P au! for hypersurfaces Pass connecting elementary critical points of dynamical systems Division of Applied Mathematics Wood, James Thornton Bancroft, Stephen Algebraic reduction theory of W*-algebras Boundary-value problems for functional dif­ Young, Frank Hood ferential equations Abelian approximate automorphisms Hall, William Spencer Periodic solutions of a class of weakly non­ linear evolution equations PENNSYLVANIA STATE UNIVERSITY (5) Owen, David Ross Rate-independence and continuum thermo­ D' Aristotle, Anthony Joseph dynamics On generalized Wallman spaces Wilson, Melvin Wayne Gould, Matthew Israel Geometric aspects of quadratures with non­ Multiplicity type and subalgebra structure in negative weights universal algebras Wilson, Raymond Bruce McNeill, Robert Bradley Stability of nonlinear waves Differential systems Singh, Sahib SOUTH CAROLINA Bounds of power residues in arithmetic progression CLEMSON UNIVERSITY (1) Stralka, Albert Robert The Green equivalences and homomorphisms Prochaska, Bobby Joseph on compact semigroups On random normed spaces

897 UNIVERSITY OF SOUTH CAROLINA (2) Zenor, Phillip Lee An investigation of countably paracompact McLaurin, Sandra spaces and related topics Local categories Van De Water, Arthur M., Jr. Properties of modules over rings with a left field of quotients RICE UNIVERSITY (5)

TENNESSEE Bagby, Richard Julian Lebesgue spaces of parabolic potentials GEORGE PEABODY COLLEGE FOR TEACHERS Dupont, Todd (2) A factorization procedure for the solution of elliptic difference equations Hughes, Eugene Morgan Morava, Jack Minor: Higher Education Algebraic topology of Fredholm maps The impact of selected curriculum projects Sampson, Charles Howard on commercially published elementary A characterization of parabolic Lebesgue school mathematics textbooks spaces King, Donald Albert Saylor, Paul Edward Minor: Higher Education A priori estimates for continuation problems A history of infinite series for elliptic and principally normal differ­ ential equations UNIVERSITY OF TENNESSEE (6)

Dolan, James Michael Minor: Physics UNIVERSITY OF TEXAS ( 14) Oscillatory behavior of solutions of linear ordinary differential equations of third Bailey, Johnny Matt order A study of left prime ideals in topological Lawson, Jimmie Don rings Minor: Philosophy Bednar, Jonnie Bee Vietoris mappings and embeddings of topo­ Compact convex sets and continuous affine logical semilattices functions Morris, William Lewis Clark, William Dean Minor: Philosophy Infinite series transformations and their Minimal Weinstein discs in a subspace applications Rodriguez, Rene Sergio, Jr. Cobb, Gary Wayne Minor: Physics Inner product-type transformations Angular and one-sided oricyclic cluster sets Coppin, Charles Arthur for meromorphic functions normal at a Concerning an integral and number set dense boundary point in an interval Tittle, Roger Lee Farmer, Charles Henry Minor: Physics Pseudo-mixing sequences of random varia­ Information theory in sequential analysis bles Wright, Harvel Amos Green, John William Minor: Physics Concerning the separation of certain plane­ Differentiability of set functions like spaces by compact dendrons Guseman, Lawrence F., Jr. VANDERBILT UNIVERSITY (2) Spaces of affine continuous functions Hardy, John Wellden, Jr. Bateman, James Myron Regular Borel measures on compact spaces Minor: Philosophy Johnson, Lee Murphy Finite group algebras with solvable unit An operation calculus on real valued set groups functions Whaley, Thomas Paul Muecke, Herbert Oscar Minor: Philosophy Some normal families of rational functions Algebras satisfying the descending chain Northcutt, Robert Allan condition for subalgebras Perron type differential inequalities Soucheck, Julianne TEXAS Rings of analytic functions

UNIVERSITY OF HOUSTON (2) Department of Computer Sciences Griffith, Phillip Alan Jonas, Ronald William On the structure of abelian groups Translation of programming languages

898 TEXAS A & M UNIVERSITY ( 1) Department of Applied Mathematics and Computer Science Department of Industrial Engineering Wulf, William Allan Lamberson, Leonard R. Minor: Electrical Engineering A theory for time compression in network­ A notation for digital systems based scheduling systems

TEXAS CHRISTIAN UNIVERSITY (4) VIRGINIA POLYTECHNIC INSTITUTE ( 14)

Allen, Paul Jentry, Jr. Garren, Kenneth Ross Ideal theory in semirings Unitary equivalence of spectral measures Boone, James Robert on Baer*-semigroups Mesocompact and sequentially mesocompact spaces Department of Statistics Lord, Michael Erie Brock, Archie Doyle Further theory of operational calculus on The distributions of zY and Z* when the dis­ discrete analytic functions tribution of the complex variable Z is a Sconyers, Woodlea Bernard member of a particular family of distribu­ Characterization of certain topological struc­ tions tures by means of well-ordered open Carter, Walter Hansbrough, Jr. coverings Grouping in item demand problems Denby, Daniel Caleb TEXAS TECHNOLOGICAL COLLEGE (1) A race to the origin between n random walks Hadidi, Nasser Morris, Gerald Lavoy Correlation between arrival and service Characterization of generalized inverses for patterns as a means of queue regulation matrices Howe, Richard Burton On tests involving linear combinations of UTAH variance components Jones, Roger Daniel Hewart UNIVERSITY OF UTAH (4) Sequential analysis of variance with unequal numbers in experimental units Dinerstein, Nelson Terry Larsen, Wayne Ammon Finiteness conditions in groups with systems The analysis of variance for the two-way of complemented subgroups classification fixed effects model with Pettey, Dix Hayes dependent observations One-to-one continuous mappings into the Martz, Harry Franklin, Jr. plane Empirical Bayes estimation in multiple lin­ Wayment, Stanley Glen ear regression Absolute continuity and the Radon theorem Periman, Eugene Avon Wilhelmsen, Russell Kent, Jr. Specification orientated limits Boundary value problems for a class of non­ Philpot, John Wesley linear second order ordinary differential Probability forecasts of 30-day precipitation equations Sheehan, Daniel Michael The computational approach to sampling VIRGINIA methods: A study of certain estimators for the negative binomial distribution UNIVERSITY OF VIRGINIA (8) Springall, Anthony Contributions to Lanchester combat theory Allen, Roger William Thompson, William Oxley, II Conditional Daroczy entropy Optimum response surface designs for esti­ Garcia-Maynez, Ade!berto Cervantes mating polynomial models in the moisture Concerning partially continuous functions problem McMillan, Evelyn Rupard On continuity conditions for functions Mitchell, George Edward WASHINGTON Relative bordism theories Schwarzkopf, Albert Beauregard, III UNIVERSITY OF WASHINGTON ( 17) Optimal curves with trajectories in a restrict­ ed set Arganbright, Deane Eugene Vick, James Whitfield The power-commutator structure of finite On K*(BG) for G a finite abelian group P-groups Ward, James Edward Castagna, Frank Automorphism groups of simple Jordan alge­ Sums of automorphisms of a primary abelian bras group

899 Clapp, Michael Howard Leslie, George Cleve A generalization of absolute neighborhood Minor: Physics retracts Connectedness in the scale of a uniform space Converse, George Arthur Ramsey, Obie C., Jr. Extreme commuting and invariant positive Some properties of the scale of a uniform operators space Cramer, Timothy Edwin Sullivan, Hugh Daniel Characterizations of certain classes of Boo­ Minor: Physics lean algebra Time and trajectories in abstract polysys­ Csontos, Peter tems Lattices of subalgebras Dovan, Robert Stuart Representations of C*-algebras by uniform WISCONSIN CT-bundles and operator theory Jones, James Parks UNIVERSITY OF WISCONSIN (23) Degrees of undecidability of arithmetical theories Constable, Robert Lientz, Benet Price Minor: Computer Sciences On the distribution of Renyi-type statistics Extending and refining hierarchies of compu­ and power of the corresponding tests table functions Root, David Harley Dennin, Joseph Construction of almost surely convergent Some results on the linear groups random processes Fitch, James Runnion, Clark Edward Integral representations of Jacobi-poly­ Some results on the uniqueness of trigono­ nomials and some applications metric series in a class of compact abelian Hannsgen, Kenneth groups An application of the Laplace transform to Schreiber, Bertram Manuel a linear Volterra equation Bounded iterates in Banach algebras Hunter, Kenneth Schulzer, Michael Integral orders in algebras Contributions to the K-sample problem: A Jaco, William H. symmetric spacings statistic Constructing three-manifolds from group Springsteel, Frederick Neil homomorphisms Context--free languages and marking automa­ Langston, Stephen ta Replacement and extension theorems in the Strong, Hovey Raymond theory of Hurewicz fiber spaces An algebraic approach through uniformly Madell, Robert reflexive structures to generalized re­ Topological lattice ordered groups cursive function theory Marsden, Martin Tews, Melvin Carl An identity for spline functions with applica­ On maps which preserve almost periodic tions to variation diminishing spline ap­ functions proximation Wiegand, Roger Allen Morgan, David Sheaf cohomology of locally compact totally Jordan algebras with minimum condition disconnected spaces Schwartz, Alan Local properties of Hankel transforms WASHINGTON STATE UNIVERSITY ( 7) Steinberg, Melvin The homology of configuration spaces of Beck, James Arthur acyclic linear graphs Minor: Physics Underwood, Douglas H. A transform method applied to coupled har­ Ideal theory in non-noetherian rings monic oscillator type problems Wattenberg, Frank Hansen, Rodney Thor Differentiable and topological braids Minor: Economics Complementing pairs of subsets of the plane Department of Computer Sciences Johnson, Norman Lloyd Cirina, Marco Minor: Physics Controllability of hyperbolic systems A classification of semi-translation planes Linz, Peter Kelley, Sara Jeanne Trumbull Minor: Physics Minor: Education Numerical methods for Volterra integral Distribution of quartic and quintic non-resi­ equations with applications to certain dues boundary-value problems

900 Loscalzo, Frank R. CARLETON UNIVERSITY (1) An application of spline functions to the numerical integration of ordinary differ­ Chang, Shao-Chien ential equations Summabilility: projections, closure proper­ Meyer, Robert R. ties and consistency problems Solution of some nonconvex problems by iterative convex programming McGILL UNIVERSITY (7) Talbot, Thomas D. Solutions for ordinary differential equations Burgess, Walter On group rings and their rings of quotients Department of Statistics Byers, Victor Mehta, Jatinder Singh Non-archimedian norms and bounds Preliminary testing and continuous weight Fieldhouse, David functions as an approach to problems in Covering modules statistical inference Gordon, Florence Park, Chong Jin Characterizations of univariate and multi­ The distribution of weighted linear combina­ variate distributions using regression pro­ tions of cell frequency counts perties Pierce, David Alan Hardy, Kenneth Studies in the modeling of stochastic and Rings of normal functions dynamic systems with special reference Kleiner, Israel to the distribution of residual correlations Lie modules and rings of quotients Pollard, Ralph Donald Teitlebaum, Albert The estimation of derivatives and parameters On an F-connection of a manifold admitting from discrete and continuous data F-structure, F3 + F = 0

UNIVERSITY OF WISCONSIN-MILWAUKEE (2) McMASTER UNIVERSITY (4)

Dragosh, Stephen Vincent Bean, Donald Winston Trevor Horocyclic cluster sets of functions defined Infinite exchange systems in the unit disc Chung, In Young Kleiber, Martin Universal algebra complexes: Extensions and Semifields and semifield metrics integral elements Doctor, Hoshang Pesotan CANADA Extensions of a partially ordered set Miller, Donald John UNIVERSITY OF ALBERTA (3) Products and factorizations of graphs

Pandya, Gautam Nileshchandra UNIVERSITY OF SASKATCHEWAN (2) On automorphisms of finite simple groups Prasad, Jagdish Lancaster, George Maurice Some poised and non-poised problems of On conform ally euclidean spaces of class one interpolation O'Shaughnessy, Charles Dennis Willis, Vernon Joseph Quartic designs Polyadic Post algebras SIMON FRASER UNIVERSITY (1) UNIVERSITY OF BRITISH COLUMBIA (3) Florence, John Blair Cockayne, Ernest Three contributions to the theory of recur­ On the Steiner problem sively enumerable classes Lim, Marion-Josephine Sui Sim Characterization of subspaces of rank two UNIVERSITY OF TORONTO (16) Grassmann vectors of order two Mallory, Donald James Adler, Andrew Limits of inverse systems of measures Existential formulas in arithmetic Andrews, David Francis UNIVERSITY OF CALGARY ( 1) Transformations of structural models Barr, Hugh Pattabhiraman, Mangalam V. A general theory of functional calculus A spectral decomposition for a polynomial Ching, Wai-Mee operator Non-isomorphic non-hyperfinite factors

901 Hsu, Hsing- Yuan UNIVERSITY OF WATERLOO (5) Watson and Hilbert transformations of distri­ bution Mowat, Father David Kalotay, Andrew A Galois problem for mappings Transformation models and statistical in­ Thompson, Brian ference Nonabsolutely convergent integrals Levenbach, Hans Thomson, Brian Sheriff Reductions in the unitary structural model Non-absolutely convergent integrals Madan, Ved Perkash Impulsive excitation of viscoelastic beams Department of Applied Analysis and Manickam, S. Computer Science Structure of unsteady symmetric shock waves Baker, John Alexander Mukherjee, Jayati Some functional equations in topological Steady viscous flow of an electrically con­ groups and vector spaces ducting fluid under a transverse magnetic field Department of Combinatorics and Stewart, James Drewry Optimization Positive definite functions and generalizations Haff, Charles Edward Tan, Peter Ching- Yao Classes of graphical representations of par­ R-regular statistical models: sufficiencyand tially ordered sets conditional sufficiency Whitney, James Bedford UNIVERSITY OF WESTERN ONTARIO (l) An analysis of probability distributions deriv­ ed from the structural models Cass, Frank Peter Anthony Wilker, John Brian Strong NBrlund summability Almost perfect packings UNIVERSITY OF WINDSOR (2)

Chandna, Om Parkash Department of Computer Science Further contributions to the theory of steady Kumar, Shyam rotational flow of gasses Semantic clustering of index terms Closs, Michael Paul Rolfson, Cordell B. On certain G-structure of differential mani­ Scheduling work flow in a computer system folds

902 ABSTRACTS OF CONTRIBUTED PAPERS The October Meeting in Baltimore, Maryland October 26, 1968

659-l. H. S. COLLINS and W. H. SUM.VIERS, Louisiana State University, Baton Rouge, Louisiana 70803. Some applications of Hewitt's factorization theorem.

Let A be a Banach algebra, let V b~ a Banach left A-module, and assume A has a bounded left approximate identity (aj 1 which also acts as a left approximate identity for V. Introduction of an auxiliary A-module for which (aj} also acts as a left approximate identity and an application of Hewitt's factorization theorem [Math. Scand. 15 (1964), 147-155] enables us to obtain several new results as well as significant extensions of known results in a simple and elegant way. Typical results follow. Theorem. A Banach space V has the metric approximation property (m.a.p.) if and only if the Banach algebra A of compact operators on V contains a bounded left approximate identity consisting of operators of finite rank which also acts as a left approximate identity for V. Theorem (extending Raimi's result [Proc. Amer. Math. Soc. 6 (1955), 643-646]). lf V is a Banach space with m.a.p. and if Z is a totally bounded set in V, then there is a compact operator T on V and a totally bounded set Yin V so that Z = TY. We also apply our techniques to the Banach space M(X) of bounded Radon

measures on a locally compact Hausdorff space X which is a left module over c 0(X) to obtain several results on C(Xl,s• including a characterization of when C(X),!I is a strong Mackey space. (Received May 24, 1968.)

659-2. R. B. LEVOW, University of Pennsylvania, Philadelphia, Pennsylvania 19104. On permanents of v-k-X configurations.

Using methods similar to those which have been used in the proof of the Birkhoff-von Neumann Theorem it is shown that if A is a (0,1)-matrix representation of a v-k-X configuration, then ldet AI< Per A except in the cases of 1-1-1, 3-2-1, and 7-3-1 configurations. A lower bound for the

difference between Per A and ldet AI is obtained as a function of k and A. (R~ceived Ma.y 24, 1968.)

659-3. C. E. AULL, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. Products of ideals. Preliminary report.

Let A and B be ideals of a commutative semigroup S and let A 9 be the product of the ideals using the semigroup operation. If A is idem::JOtent and B is prime, then AB =A n B. If A is an f-ideal (see

author's Ideals and filters, Compositio Math. (1967), 79-86) and B is prime, A ljl B, then AB =An B. Under these conditions A(B n C)= AB n AC where Cis any ideal. If Sis cancellative it is sufficient for A to be an f-ideal for A(B n C)= AB n AC. Let A be an ideal generated by An where c E An if c = sn for somes E Sand c EA. Let B = {s E S: sn E An}. B is an ideal. lf B is an f-ideal,

Bn =A. lf Sis a ring and A and Bare ring ideals, then AB is a ring ideal if A is an f-ideal. (Received june 14, 1968.)

903 659-4. j. C. BRADLEY, Westinghouse Electric Corporation, Aerospace Division, Baltimore, Maryland 21203. Products of representations on discrete groups.

Let (JVi)iEI be an arbitrary family of complex Hilbert spaces. Guichardet's development (Ann. Sci. Ecole Norm. Sup. 83 (1966), 1-52) is used where@(AilJVi is a "tensor product" of these spaces and (Ail is an object in the set N of vectors from nJVi where each Ai is or norm one. JV(Ai) is the completion of'g;(Ai)JV i• The spaces@(Ail Iii and;&(bilJVi are isometrically isomorphic if ( Jl - (ai/bi)J)iEI is summable, and this gives an equivalence among the elements from N. If the set of equivalence classes for this relation is C., then the Hilbert sum of (~A) A. E C is labeled JV where each JV A is identified with an JV(A.) for (Ail EA. Let (Ti: Gi - 'I..{(JV i )liE 1 be a family of representations l . and each Gi discrete. Then: (1) (g;T1 : nwG- 'I..{(JV) is a representation (simple examples illustrate . . """" . that the spaces JVA are not stable forZT1 ); (2l®T1 extends to a representationZ)T1 on DGi; (3) if Ti is equivalent to si on Gi for every i, thenZ)Ti is equivalent to~Si; (4) almost all Gi's compact and Ti primary, irreducible =>@Ti is primary, irreducible, resp.; (5l® Ti is primary, ""'-' . irreducible"' g)T1 is primary, irreducible, resp. (Received june 13, 1968.)

659-5. SRISAKDI CHARMONMAN and D. G. BARRY, University of Alberta, Edmonton, Alberta, Canada. Modified Runge-Kutta-Nystrom methods.

A second order ordinary differential equation (ODE) may be solved either by applying a Runge­ Kutta (RK) method repeatedly to the equivalent system of two first order ODE's or by applying the classical Runge-Kutta-Nystrom (RKN) method directly to the second order equation. This paper presents several new RKN methods modified from existing RK methods. For identification purpose the new RKN method obtained from any X RK method will be called the modified X RKN method. It is shown that the modified Ralston RKN method has smaller truncation error bound than the repeated Ralston RK method. Furthermore, the method with the smallest truncation error bound in a certain sense among the various new ones obtained is (a) modified Kuntzmann RKN when fin y" = f is a function of y' and (b) modified Boulton RKN when f is not a function of y'. It was found from numerical experimentation on several ODE's with known analytical solutions that the modified RKN methods may not, in a few cases, be as stable as the RK method but they are generally more accurate when used as a starting pro.::edure. For example, for y" = y with y'(O) = y(O) = 1 and step size h = .2, the modified Boulton RKN method as well as the modified Ralston RKN method give at least two signi­ ficant digits more than the repeated Ralston method. (Received june 19, 1968.)

659-6. WITHDRAWN.

659-7. j. H. HARRIS, Stevens Institute of Technology, Hoboken, New jersey 07030. On the richness of subtheories of ZF + GCH.

Let T be any first-order theory with equality containing as its only nonlogical symbol the binary predicate letter '(' and satisfying the axiom of extensionality. Let F and G denote wff' s of one free variable. Definition. Tis rich iff (AFJ[!-T!l!xF(x) => (VGJ!-T(:ilx)(Vy(G(y) .. y E x)F(x)j]. Intuitively, Tis rich iff (AFJU-T:i!xF(x)- (VG)J·?({y: G(y)}JJ. Lemma. Tis rich iff (AF)(VG)f-T(:ilxF(x) - :ilx(Vy(G(y) .. y Ex) F(x))). Theorem. If Tis a subtheory ofT* = ZF + GCH (i.e., every theorem of

904 Tis a theorem ofT*), then Tis not rich. The proof follows directly from the lemma and the result that it is relatively consistent with ZF t GC H to assume that there is no definable well-ordering of

the continuum, proved by Feferman, Some applications of the notions of forcing and generic sets, Fund. Math, 56 (1965). (Received August 22, 1968.)

659-8. R. j. WARNE, West Virginia University, Morgantown, West Virginia 26506. _!_-regular semigroups. I.

Let S be a regular semigroup, let E S denote the set of idempotents of S, and let I denote the set of all integers. Sis said to be !-regular if ES is order isomorphic to I under the reverse of the usual order. Theorem~. Sis a simple !-regular semigroup if and only if S = (U(Gi: 0;;; i;;; d- l))

XI XI, where dis a positive integer and (G1 : i = O,l, ... ,d- 1} is a collection of pairwise disjoint -l the multiplication (g ,m,n)(h ,p,q) = (g (f 6 )(h 6 d d )(f 6 ), m,ntq-p) groups, under s r s n-p,p 0 ,s r p tr,n ts n-p,q 0 ,s if n > p, ((C 1 6 l(g 6 d+ d )(f 6 )h ,mtp-n,q) if p > n, or ((g 6 )(h 6 J.m,q) if n = p p-n,m 0 ,r' s n s,p +r p-n,n 0 ,r r s s,v r r, where g E G ,g E G and t =max (s,r). If m,n E 1°, the nonnegative integers, and m 0, fm,n = m-1 m-2 li(ntl)d60,d li(n+2)d60,d ... u(nt(m-l))d6o,du(m+n)d where (und: n E I} is a collection of elements of

G0 with und = k0 for n > 0. In the cased= 1, we obtain the structure theorem for 1-bisimple semi­ groups [R. j. Warne, 1-bisimple semigroups, Trans. Amer. Math. Soc. 130 (1968), 367-386]. (Re­ ceived August 26, 1968.)

659-9. R. A. STRUBLE, North Carolina State University, Raleigh, North Carolina 27607. An algebraic characterization of distributions.

Let C 00 denote the vector space of infinitely differentiable, complex valued functions on the real line and let./} denote the subspace consisting of those functions with compact support, Theorem. Each (Schwartzian) distribution of one variable is uniquely characterized as a mapping from ./}into C 00 which commutes with convolution. (Received August 28, 1968.)

659-10. A. T. BUTSON, University of Miami, Coral Gables, Florida 33124. _:!\_~onfigura.!i:_

~elated_i_~mll.:...e_Eri~_!:lock~t:_signs__, Preliminary report.

LetS= [a 1' a2, ... , av} be a set of v elements and s 1, s2 , ... , Sv-l be subsets of S each containing k elements. Suppose that each S i and Sj with i" j intersect in exactly X elements of S, and that v > k >X >0. Let ai be contained in exactly ri of the subsets s 1, s2, ... , Sv-l and call r 1, r 2 , ... , rv the

.':_ep!._i~~!._~~um~ers of the configuration. Some of the results concerning this configuration are the following. A quadratic equation is obtained the roots of which are the replication numbers. When X 1 or k = 2 A the existence of the above configuration is equivalent to the existence of a related sym­ metrical balanced incomplete block design, (Received September 3, 1968,)

905 659- II. S. G. BOURNE, Hotel Durant, Berkeley, California 94704. On semi- invariant

LetS be a locally compact semigroup and IJ. a regular Borel measure on S. Let Ba -I= [x ES!xa E B1 and I" is said to be semi-invariant if IJ.(Ba -I)= IJ.(B), all a and Borel sets BinS.

Let F be the support ofF. !heor~m I. IfF posses'ses an idempotent e, then F is a left group which is a right ideal in S. Letj.lxiJ.(E) = J siJ.(Ex- 1)j.i(dx), then IJ. is idempotent if IJ.XI" = '-'· Theo_E_!!~ 2. A necessary and sufficient condition that the support F of an idempotent measure I" be compact is that the semigroup of idempotents E, which is assumed to be not empty, be compact. (Received September 5, I968.)

659-12, BERNARD BERLOWITZ, 1728 California Street, Berkeley, California 94703. Analyti_£

~~!!~E!lii_!.!_on 9.!.J.~Eii!!EEE_~!_o_~l!!_~g!ll!tioE~. I.

By a local functional equation for a meromorphic function f(s) in the plane we shall mean a relation f(s) = f(p(s)), where p(s) is holomorphic in a domain D, this relation holding for all s in D. Putting f(s) = 77-s/2 r(s/2)C(s), where C(s) is the Riemann zeta function, we obtain an analytic con­ tinuation of p(s) along "most" analytic curves in the plane. Suppose PI (s) and p2(s) denote two analytic arcs tending to+ oo, indexed by the same s. While f(pi(s)) = f(p 2(s)) does not imply p 1 (s) = p2(s), let log f(z) denote a single branch defined in Re z >I, and suppose that using this same branch, log f(p 1 (s)) =log f(p2(s)) as pi (s), p2(s)""' + oo. Then PI (s) = p2(s). This is an analytic consequence of the Euler product for C(s). (Received September 6, I698.)

659-I3. WITOLD BOGDANOWICZ, Catholic University of America, Washington, D. C. 200I7, and VERNON 'lANDER, West Georgia College, Carrollton, Georgia 30 Il 7. -~_!_ubini ::_J.esse~eore_~

!'_~~!!_~~~e_!_I!!.!_~~

Let W, Y, Z be Banach spaces and u a bilinear continuous operator from Y X Z into W. For a volume space (X, V,v) let L(v, Y) be the space of Bochner summable functions defined as in Bogdanowicz, Proc. Nat. Acad. Sci. USA 53 (1965), 492-498. Let K(v,Z) be the space of vector­ valued volumes dominated by the positive volume vas in the above paper. The triple (X,V,v) is a probability volume space if X E V and v(X) = I. If (Xt• Vt• vt) is a family of probability volume spaces (t E T) define Xt = X -0t• V T = {A =X T~ : At E Vt and At 'I Xt for at most a finite number of t E T), vT(A) = n T vt

'-'s(A) = IJ.(A xxs,) for all A EVS and the operator Js by (JS(f))(x) = Ju(f(xs,x5r),llg(dxs)) when meaningful, and z~ro otherwise. Theorell!.. Iff E L(vT'Y) and 1J. E K(vT'Z), then the operator Js maps L(vT' Y) into L(vT' W) and Js1 f converges to a constant function having the value Ju(f,diJ.) and Jsf converges to the function u(f(•),IJ.(X)) in the vT-means, where S runs through all finite subsets of T ordered by inclusion. (Received September 5, 1968.)

906 659- 14. VIC TOR K LEE, University of Washington, Seattle, Washington 98105. :!'~~en_<:>_:_mi~&:_

~'?.ll~t_r_~c_tio!l~_r_e_l_?!_~ct t() __ a _q_~e_s_ti_()Il_o_l'_~_!!~~!~ne.

For any point x of a normed space X, let xc denote the set of all points of X • conjugate to x.

The space X is said to have the A-(l_:r:~(l_

659-15. PARRY MOON, Massachusetts Institute of Technology, Cambridge, Massachusetts, and D. E. SPENCER, University of Connecticut, Storrs, Connecticut 06268. !:l_<:>_!or_r:_e_p_r:_e_s_entatio~_

1_1~ns ~n-~s_o_idal _g_~a_n_t_i0~~.

The complex- number representation of sinusoidally varying quantities developed by A. E. Kennelly and C. P. Steinmetz in 1893 is widely used. Since the original introduction of complex numbers to represent current, voltage, impedance and power in sinusoidal electric circuits, many authors have recognized the imperfections of this representation. The development of an adequate holor representaton of sinusoidal electric circuits is a comparatively recent occurrence (Moon and

Spencer, ~_n_e_..v__ma!_il_~ma_t~~i!_l_~pr~~en!_i!_0.9E_()~_?_!t_e_r:_na_t_i!l~ curr_<:_ll!__B· Tensor 14 (1963), 110). The present paper shows how the holor method can be extended to nonsinusoidal systems. The method is applied to wave forms described by Fourier series, to amplitude modulated systems, and to damped sinusoidal waves. The holor method reduces circuit theory with nonsinusoidal applied voltages to a simple procedure and unifies much of circuit theory. (Received September 6, 1968.)

659-16. F. M. CARROLL, Annursnac Hill Road, Concord, Massachusetts 01742. !_

and confidence r<:_!0on~.

A sampling of data on several attributes of object classes is obtained from a variety of sensing devices and is noted in an (m X n) rectangular array. The observed data is tested for consistency with known limiting values for the respective attributes of the object classes. Suppose the screening process leaves k object classes. Suppose then attribute observations are recorded in p samples. The maximum likelihood estimate of the mean (a) of each attribute of one of the k object classes is

computed. The sample mean x = !J,. Let li' be any estimate of the corresponding population mean. A selection of p attributes is made to obtain a (p X p) data matrix. The maximum likelihood estimate

~ of the corresponding covariance matrix S is calculated. Let q (preferably small) satisfy 0 < q < 1.

j.i* which satisfy the equation C p(x - #i')'S- 1 (x - j.j*) 2(q) form a confidence Then the set of all = ~ xp region of J.l at the confidence level (1 - q). The computation can be made for all practical combinations of p attributes. Numerical examples illustrate the method of identification and the calculation of

confidence regions. (Received September 6, 1968.)

907 659-17, K. F. BARTH and W. J, SCHNEIDER, Syracuse University, Syracuse, New York 13210,

Q!_!_~_p_r_o_l:>~~-o_£_~_,_-~~!l_y_i_c .9.!l£.E'!.!l!!l.!LJ~!~J ~!!_~~.

In Hayman's book ~-e~_E'~_!_E_!?_e_!:_~I:J.!_e_!!!~in fl:l_nC!i_9_!l_!_h_e_Q_ry_ (p. 10, P rob1em 2 ,4) the following problem (attributed to C. Renyi) is posed: "Can an entire function E(z) have one finite exceptional value for one Julia line and another finite exceptional value for some other Julia line?" This question

is answered in the affirmative by the following: :T,~or_E'~. There exists an entire function E(z) which has the positive imaginary axis as a Julia line with exceptional value one and the negative imaginary axis as a Julia line with exceptional value zero. The proof consists of first mapping the domain D (= [z: iz I < 1} U [z: 77/4 < arg z < 377/4} U [z: 5rr/4 < arg z < ?rr/4}) conform ally, by the map c(z) (with c(O) = 0, c'(O) > 0), onto the set [z = x + iy: - oo < y < + oo, lxl < f(y)} where f(y) is a positive even continuous function which goes slowly to + oo as y goes to ± oo. Let h(z) conformally map D onto [z: iz - 1/21 < 1/2} with h(O) = 1/2, ih'(O) < 0, Now by an approximation theorem of Keldy!l and of Mergelyan (Amer. Math. Soc, Trans!. (1) 3 (1962)1 335-336, Theorem 1,3) we can find an entire function E(z) which approximates the function w(z) (= ec(z) + h(z)) so closely in D that it has the desired properties, (Received September 6, 1968.)

908 ABSTRACTS PRESENTED BY TITLE

Algebra & Theory of Numbers

68T-A25. M. 0. POINSIGNON, Kansas State University, Manhattan, Kansas 66502. Green's relations on a semiring.

On a semiring R we define the Green's relations thus: a .l b (a R b) if and only if a and b generate the same principal left (right) ideal; .e = .l v R and 11 = .t n R. The relations .l and R commute for a large class of semirings, but not for all semirings. Theorem. Any £-class of R is either a multiplicative £-class or a union of multiplicative £-classes reduced to one element. Similar results hold for R and 11. Under the assumption that each inner left translation of R commutes with every inner right translation of R [Abstract 68T-A10, these cJiotiaiJ 15 (1968), 784], we obtain results similar to Green's Lemma and its applications in semigroup theory; in particular to each 11 -class is associated a Schutzenberger group which depends only on its .e-class. (Received May 23, 1968.)

68T-A26.A. A. MULLIN, Aber<:leen Proving Ground, Forest Hill, Maryland 21050. On new analogues of Lucas' totient and Thacker's totient.

This note establishes new lemmata for variations of two more classical generalizations of Euler's totient by exploiting the simple result: If the mosaics [A mer. Math. Monthly 74 (1967), 1100-1102] of integers in a sequence have no prim·e in common, then those integers are relatively prime, but not conversely, in general. By analogy to E. Lucas' (multiplicative) totient L k [Theorie

des nombres (1891 ), 402] define Lk (modified Lucas' totient) as follows: Let (a1, ••• ,a JJ be a sequence of integers and put Lk(n) equal to the number of those integers b E {0,1, .•• , n - 1} for which the

mosaic of each and every member of the sequence (b - a 1 , ••• ,b - ak) has no prime in common with the mosaic of n. Prelemma. Lucas' totient dominates Lk. Similarly, define Tk (modified Thacker's

!_otient) as follows: Put Tk(n) equal to the sum of the kth powers of the natural numbers ;!; n whose mosaics have no prime in common with the mosaic of n. Prelemma 2. Thacker's totient dominates Tk. By judicious selection of the ak, the modified Lucas' totient yields new estimates in the arith­ metic subtheory of consecutive integers (e.g., bearing on Schemmel's generalization of Euler's totient). (Received May 27, 1968.)

68T-A27. WITHDRAWN.

909 68T-A28. F. A. FICKEN, New York University, Bronx, New York 10453. Generalized inverses which are left symmetrizers.

Let L be a complex (or real) matrix consisting, in block form, of A(r X r) and B(r X c) in the first row, and C(s X r) and D(s X c) in the second row. We assume that A is regular and of rank equal to the rank of L; for this it is necessary and sufficient that D = CA-1B. Let(-)* denote the conjugate transpose (or transpose) of(-). Since AA* > o, G = (AA* + BB*)-l = G* exists. With P(r X s) and T(c X s) arbitrary, define M, in block form, to have first row A*G- PCA -l and P, and second row B*G - TCA -l and T. It is easy to verify that LML = L (M is a generalized inverse of L) and (ML )* = ML (M is a left symmetrizer of L). The rank of M may be increased (from r) by adjusting P and T; if Lis square, then M may be made regular. (Received July 1, 1968.)

68T-A29.I. F. RITTER, New York University, Bronx, New York 10453. An algorithmic definition of the characteristic polynomial of a matrix.

The synthetic division scheme for computing values of a polynomial P(x) is adapted into a synthetic algorithm yielding P(A)'E for a matrix A and a column c. This is modified in turn into a characteristic algorithm (c.a.) which defines by induction on n the monic unique minimum polynomial 1Pn(x) of degree n for the nth order matrix An with indeterminate elements. In proving this and de­ riving from it the properties of numerical matrices, for which tpn(x) becomes the characteristic poly­ nomial, such as inverse, rank, spectral expansion, Jordan form, no reference is made to determinants, the indispensable '~'n(O) passing incognito as the last term in tpn(x). The precedence of" matrix" over "determinant" stressed by Cayley is fully realized by finally deriving all properties of determinants from the definition det A (~ 1 )n~ (0) as furnished by the c.a. Numerically, the c.a. is an efficient n = n direct method for computing tpn(x) and the last column of adg(xln ~An)' for which then the synthetic algorithm supplies any needed column or row. As shown by many years of classroom use, the two algorithms and the banishment of determinants smooth the approach to "a subject that has often failed to win affection" (Turnbull and Aitken, Canonical matrices, preface). (Received June 28, 1968.)

68T~A30. J. M. KELLETT, University of Florida, Gainesville, Florida 32601. A torsion theory for modules over rings without identities.

Let R be an associative ring and let !Ill denote the category of all (left) R~modules. Let F denote the full subcategory of M consisting of those modules M which contain no nonzero element m such that Rm = 0. F is the class of torsion free modules in the smallest torsion theory (see Dickson [MR 33, /f 162 ]) for which all modules M such that RM = 0 are torsion. If R has an identity, F is just the category of unital R-modules. Assume henceforth that RR is in F. F is closed under images iff R has an identity. €all M in F projective if Hom(M,a.) is onto for all a. in F which are epimorphisms in !Ill. Theorem. lf every module in F is projective, then R R = J $ S where S is a semisimple Artinian subring of Rand J is the Jacobson radical of R. (The corresponding theorem for injectives is trivial.) Call a left ideal L of R closed if R/L is in F. The torsion theory ofF coincides with those considered by Goldie (MR 29, #2282) and by Jans (MR 33, #163) iff R has no proper essential closed left ideal. (Received June 28, 1968.) (Author introduced by Professor W. E. Clark.)

910 68T-A3l. E. T. PARKER, University of Illinois, Urbana, Illinois 61801. Disproof of a conjecture of Harary and Palmer. Preliminary report.

A tournament of order n is a complete graph of n nodes with each arc assigned a unique direction. Harary and Palmer (Monatsh. Math. 71, H. 1 (1967), 14-2.3) conjectured at the end of their paper that two tournaments of order n !; 5 must be isomorphic if the sets of n subtournaments obtained by deleting each node are in turn isomorphic in pairs. A counterexample of order 6 is given by two tournaments both having a transitive subtournament of order 5 with nodes A, B, C, D, E, with each arc directed into the node later in alphabetical sequence. The sixth node in one tournament is joined by arcs into A and B, out of the other nodes; in the other tournament the sixth node is joined by arcs into exactly A, B, and C. The two tournaments are not isomorphic. For each has a unique transitive subtourna­ ment of order 5. They are, however, converses (Harary's terminology) of one another--reversing directions of all arcs interchanges them. Thus it is natural now to ask whether this phenomenon

occurs for any pair of nonisomorphic, nonconverse tournaments for any n ~ 5. It is hoped also that further study will shed light on whether this example (together with one of order 4 noted by Harary and Palmer) are isolated cases or admit of generalization. (Received july 8, 1968.)

68T-A32..A. 0. L. ATKIN, Atlas Computer Laboratory, England, and j. LEHNER, University of Maryland, College Park, Maryland. The Hecke-Petersson theory for certain subgroups of the modular group.

Let V N be the Peters son Hilbert space of modular cusp forms of dimension-k (k an even

positive integer) on the subgroup r 0(N) of the modular group defined by Nlc in (a bl c d). Call p a

~ood prime if it is prime toN; call q a bad prime if it divides N. Define for f(T) E VN the Heeke operators fiUq = q- 1 r;~=lf((T t m)/q) and fiTp = fiUp + pk-lf(pT). Define also an involutory operator f IWq = q ak/2. (yNT + q a )-k f((xq a Tt 1)/(yNT + q a )), where q all Nand xqa - yNq -a = 1, x, y E Z. T he known (Petersson) simultaneous diagonalization of V N gives rise to forms F i(T) that are simultaneous eigen­ forms of Tp for all good p. Among the Fi(T) are the old forms, those that are spanned by forms g(d'T)

for g E V d' d < N, and dd' IN. Define the remaining F i(T) to be~ forms. The authors prove the following results for new forms F(T): (1) If qlJN, then F IUq = ~ qk/2.-l, F IW q = t F, the signs corres­ ponding. (2.) If q2 IN, then F IU q = 0, F IW q =.±.F. (3) No two new forms have the same eigenvalues for every Tp' (4) If the Fourier series ofF is F(T) = x + r:r;: c(m)xm, x = exp(2.!Ti.T) and its zeta function ""oo -s -s k-l-2.s -1 -s is C(s) = 1 + L.Jn=2. a(m)m • then C(s) = n %N(l - a(p)p + p ) nqiN(l - a(q)q ), where a(q) = k/Z-1 2. P 2: q if qlJN and 0 if q IN. (Received june 2.6, 1968.)

68T-A33. M. F. NEWMAN, Australian National University, Canberra, A. C. T, 2.600, Australia. The group-law [x,y]2 = [x,y,xJ.

The claim (Gandhi, these cJ{otiai) 15 (1968), 349) that a group satisfying the above law is abelian has been shown to be false (Gupta, these cJ{otiai) 15 (1968), 637). A quick calculation leads to a description, in familiar terms, of the variety of groups defined by the law. Replacing x by yx

in the law yields [x·,y,y] = 1 and hence [x,y]2 = 1; from these follows [x,y,z] = 1. In other words,

911 the variety is that of groups which are nilpotent of class 2 and have commutator subgroup of exponent 2. It is easy to check that this is the variety generated by all abelian groups and the dihedral group of order 8. The laws [x,y]n = [x,y,x] (for integer n) behave similarly. If 3 does not divide n, the variety defined is that of groups of nilpotency class 2 which have commutator subgroup of exponent n.

If 3 divides n, then groups of nilpotency class precisely 3, but of no higher class, occur. ( R~ceived July 16, 1968.)

68T-A34. OLGA TAUSSKY, California Institute of Technology, Pasadena, California 91100, and H. J. ZASSENHAUS, Ohio State University, Columbus, Ohio 43210. On the !-cohomology of the general and special linear groups. I.

The functional equation f(ab) = f(a) + af(b) is discussed in connection with the !-cohomology of the general linear group or the special linear group over a field F acting as inner automorphisms on the full matrix algebra of the same degree or the stable subspace of matrices of trace zero (see also D. Hertzig, J. Algebra 6 (1967), 317-334). The problem breaks down into determining the solution of the logarithmic functional equation for F and in solving the problem for the special linear group acting on the space of matrices of trace zero. The cases n = 2 and characteristic of F equal to two or F = GF(3) are exceptional. In all other cases, every solution for the special linear group can be uniquely presented as a sum of a 'derivation' cocycle and a coboundary. The method is based on a translation of the problems into Lie algebra terms and lroup extension terms. (Received July 17, 1968.)

68T-A35. P. L. ROSENTHAL, Stanford University, Stanford, California 94305. On fields which ar,, plus and dot related. Preliminary report.

A field F {D 1 , +1, •1) is plus and dot related if there is a field T I (D2 , +2, • 2 ) such that D2 = D1, +2 =+I' but •2 '! •I and a field T 2(D3 , +3 , • 3 ) such that D3 = D1 , •3 = •1, but +3 F + 1 •

Theorem 1. Every field of finite order p ~ 5 is plus and dot related. We also give some examples of fields of infinite order which are plus and dot related. The fact that every field of infinite order is plus and dot related remains open to this writer. If, in the above definition, the word related is replaced by isomorphic, then another open question to this writer is: Is every field of infinite order plus and dot isom·:>rphic? (T1 and T 2 are now required to be isomorphic to F,) We give some examples of fields of infinite order which are plus and dot isomorphic. (Received July 8, 1968.)

68T -A36. W. J. HEll\ Z ER and J. F. OHM, Louisiana State University, Baton Rouge, Louisiana 70803. Some remarks on explicit overrings.

Let D be an integral domain with quotient field K and let D1 be an overring of D, i.e. D ~:;; D1 ~:;; K. D1 is said to be well centered on D if for x E DI one has xo1 = yD1 for some y E D; and D1 is called explicit for D if DI = Ds for some multiplicative system S of D. Theorem. A well­ centered overring D 1 of D is explicit for D if there exists an overring D2. of D such that D1 n D2 = D and D1 D2 = {di d2 jdi E Di} is an explicit overring of D. In particular, if DI is a rank one well-centered valuation overring of D, then D1 is explicit for D if there exists a proper overring D2 of D such that DI n D2 =D. The above theorem is obtained from more general results concerning preorders on

912 the divisibility group of D. Griffin in Some results on v-multiplication rings [Canad. j. Math. 19 (1967), 710-722] calls an integral domain an essential ring if there exists a set {Va} of explicit valuation over rings of D such that D = n a V a and conjectures the existence of an essential ring that is not a v-multiplication ring. An example is given proving this conjecture. (Received july !7, 1968.)

68T-A37. C. F. WELLS, Case Western Reserve University, Cleveland, Ohio 44!06. Split translations of groups.

Let G be a group with normal subgroup H. The centralizer in Sym(G) of the group of left multiplications by elements of His the group ST(G;H) of (right) H-split translations of G (see my Abstract 68T-386, these c}/otiai) 15 (1968), 553). Theorem I. A representation p of Gin ST(G;H) is conjugate in ST(G;H) to the right regular representation of G if and only if (i) His the stabilizer of some coset ~ of 1-1, and (ii) the map h- h -l(hg)hp g- 1, where g is a fixed element of ~ and h E H, is an inner automorphism of H. Theorem ll. Let G = H X r. The group AutH(G) of automorphisms of G that fix H elementwise is isomorphic to the split extension of Hom(r,Z(H)) by Aut(r) determined by defining IP" = C1-l ¢for CJ E Aut(r) and ljJ E Hom(r,Z(H)). Theorem ll strengthens the theorem in the Abstract referred to above. The proof uses the fact that AutH(G) = ST(G;H) n Aut(G). (Received july 22, 1968.)

68T-A38. D. F. DAWSON, North Texas State University, Denton, Texas 76203. Some decompo- sition theorems for semigroups.

All definitions can be found in [Clifforrl and Preston, Algebraic theory of semigroups, Vol. 1, Amer. Math. Soc., Providence, R.I., 1961 J. Throughout, S denotes a semigroup containing a left zeroid, L denotes the set of all left zeroids of S, and Q (which might be empty) denotes the set of all idempotents in L. The following theorem improves a result of .C. W. Leininger [Boll. Un. Math. I tal. (3) 21 (1966), 147-149]. Theorem 1. If Q is not empty, then L = U {eS: e E Q}. where the eS, e E Q, are disjoint isomorphic groups. Theorem 2. If there is an element 1-1 E L and an integer n > 1 such

that 1-1 = J.lnx has a solution x E S, then L = U { eS: e E Q} and each eS, e E Q, contains exactly one x

such that 1-1 = J.lnx. If, in particular, 1-1 = J.lnx and xis unique, then L is a group. Theorem 3. If Q is not empty, then L is a maximal regular subsemigroup of S if and only if S - L contains no idempotent.

Theorem 4. If Se is regular and e is the only idempotent in Se - L, then Se is a union of groups and L is a prime ideal of Se (in fact Se - L is a group). Theorem 5. If there is a 1-1 E L such that x~-t2 = 1-1 for a unique x E S, then L = U { eS: e E Q }. Theorem 6. If L contains a normal element of S, then L is a group. (Received July 24, 1968.)

68T-A39. RAYMOND BALBES, University of Missouri, St. Louis, Missouri 63121, and ALFRED HORN, University of California, Los Angeles, California 90024. Injective and projective Heyting algebras.

In the category of Heyting algebras (also called pseudo-Boolean algebras), it is shown that an algebra is injective if and only if it is a complete Boolean algebra. A finite algebra is projective

if and only if it is an ordinal sum of the form P 1 E!l ••• Ell P n' where for each i, Pi is a totally unordered

913 set with at most two elements; and if Pi has two elements, then 2 ~ i ~ n- 2 and Pi- I and Pi+l are singletons. (Received july 26, 1968.)

68T-A40. SAMI BERAHA, Ball State University, Muncie, Indiana 47306. A matrix problem of combinatorial approach and analytic aim.

Given an n X n matrix whose every row and column has p of its elements equal to A and q of its elements equal to B, p + q = n. Find the conditions for this matrix to have an inverse whose every row and column has r of its elements equal to a and s of its elements equal to b, r + s = n, and state the number of independent solutions (i.e. that do not result from transpositions and/ or permatations of rows and/or columns). A necessary an~ sufficient condition for the existence of a solution is that (p,n) =I; in which case there are exactly tp(n)/2 independent solutions. We solve first for four auxiliary constants satisfying k11 + k21 = p, k 12 + k22 = q, k11 + k12 = r, k21 + k22 = s, from which r and s are retrieved diophantically on the basis that the determinant lkij I = -1. The relations between

A,B and a, bare given by (A - B) (a -b)= I and k11Aa + k21 Ab + k12 Ba + k22 Bb = 0. The problem aims at generalizing K or M matrices. (Received August 5, 1968.)

68T-A41. W. ]. HEINZER, Louisiana State University, Baton Rouge, Louisiana 70803. On Krull overrings of a Noetherian domain.

By an civerring of an integral domain A we mean a ring containing A and contained in the quotient field of A. The following theorem is an improvement of results previously announced by the author, Abstract 655-42, these cN'oticei) 15 (1968), 478. Theorem. Let A be a two-dimensional Noetherian domain. If D is a Krull overring of A, then D is Noetherian. (Received August 8, 1968.)

68T -A42. M. T. WEST and R. ]. PLEMMONS, University of Tennessee, Knoxville, Tennessee 37916. On the maximal subgroups of the semigroup of binary relations.

It is well known that the. maximal subgroups of 3'x• the full transformation semigroup on a set X, are symmetric groups. In fact, a similar result holds for the semigroup of partial transforma­ tions on X. It has been conjectured that the maximal subgroups of the semigroup Bx of binary relations on X, for finite X, are also the symmetric groups. However, the following example shows this conjecture to be false. Let X = {1,2,3,4} and f = t U {(1,3 ),(1,4),(2,3 ),(2,4)}. Then Hf, the maximal subgroup of 8X containing£, is V 4, the four group. (Received August 7, 1968.)

68T-A43. D. R. CECIL, North Texas State University, Denton, Texas 76203. Dihedral products. Preliminary report.

A group E is a dihedral product of H by K, denoted H • D K if H and K are proper subgroups of E with H4 E, E = HK, H n K = {e}, and there exist sets of generators {ha. .... } and {k,s•···} for Hand K, respectively, such that ho.k,Bho. = k,s for all o.,{j. Dihedral groups, many metacyclic groups, and

all non-abelian groups of order ~ 14, except A4 , are dihedral products. A dihedral product is also a split extension. If C is the cyclic group of order n and D the dihedral group of order 2 , then n n n Cn "D C2 ""Dn for all nand (Cm X C2 ) •D c2 ""D2m ""(em • D C2 ) X c 2 for odd n. Theorem I.

914 (C 2 X ••• X c2 ) • 0 K .. c2 X ... X c 2 x K for any group K and any finite number of copies of c 2 • Theorem 2. Dn • D C2 .. Dn X C2 for all positive integers n. The dihedrator subgroup of a group E 1 is that subgroup generated by all dihedrators of E, i.e. all a 1 a 2a 1 a~ with a 1 , a 2 E E. Th,eorem 3. The dihedrator subgroups of Dn, of Cn k X c 2 (k any positive integer), and of Cn are isomorphic for all n, the subgroups being cyclic with n (n/2.) elements if n is odd (even). (Received August 7, 1968.)

68T-A44. ROBERT GILMER, Florida State University, Tallahassee, Florida 32306. Com­ mutative rings in which each prime ideal is principal.

Call the commutative ring Ran F -ring if each prime ideal of R is principal. The question arises: Must each ideal of an F-ring R be principal? If R contains an identity, the answer is affirmative, but in general, the answer is negative. In fact, if R is an F-ring containing a regular element, if T is the total quotient ring of R, and if R* is the subring of T generated by R and the identity element of T, then these conditions are equivalent: ( 1) R is a PI R, (2.) R* is a PI R, (3) each maximal ideal of R* properly containing R is principal, (4) each R-primary ideal of R is principal. It follows that if S is a commutative ring containing a regular element and if each primary ideal of S is principal, then S is a PIR. (Received August 14, 1968.)

68T-A45. L. J. MORDELL, St. Johns College, Cambridge, England. On some sextic diophantine equations of genus 2.. 2. 2. 2. 2 Equations of the form y = (a1x + b 1)(a2x + b2}(a3x + b3 ) are given which have no rational solutions, but are everywhere locally solvable. (Received July 1, 1968.)

68T-A46. M. B. SLATER, University of Bristol, Bristol 8, England. Upper radicals and Andrunakievic classes.

Let a "f ll be a class of rings which is (H) closed under homomorphism and (I) closed under ideals (i.e. I ;; A E a"' I E a). '1.( is an H-subclass of a, and 'I.< a. is the a.th step in the construction of the Kurash radical class '1.<* generated by '1.( (Mat. Sb. 33, p. 19). In each of the results below a must

satisfy suitable restrictions "of Andrunakievic type", i.e. concerned with the situation V ;; I ~A Ea.

The restrictions vary but are all satisfied by the class a 1 of all alternative rings. Each result gives a {j such that 'l.

U tf.Il ~In U tf.Al. Hence (ibid., p.ll8), '1.<* = '1.<{1+ 1 and also satisfies (E). In 1,2.,3, the {j given is smallest possible even for a• s consisting of associative commutative rings. '1.< is Z iff

'1.( ~ (REa: RR = o}; w is the first infinite ordinal. 1. If '1.( is I and Z, {j = l. 2.. If '1.( is I, {j = 2.

3. If '1.( is Z, {j = w. 4. If '1.( is arbitrary, {j = w Cal. (Received August 19, 1968.)

68T-A47. R. J, WARNE, West Virginia University, Morgantown, West Virginia 2.6506. !-regular semigroups. II.

See I- regular semigroups. I [Abstract 659-8, these cJfoticei) 15 ( 1968 ), this issue J for definitions. Let (Gi: i E I, i;; 0} be a collection of pairwise disjoint groups. For each i E I (i

let')'i denote a homomorphism of Gi into Gi+ 1• For m,n EI, m

915 and let a.m,m denote the identity automorphism of Gm. Theorem 2. Sis an !-regular semigroup

with a proper kernel K if and only if S = (U(Gi : i E I, i ;;; 0)) U K, where K is a simple regular w-semigroup [W, D. Mnnn, Regular w-semigroups, Glasgow Math. J, 9 (1968), 46-66] disjoint from

each Gi (i < 0) and with group of units G0, under the multiplication xuo yv = (xua.u,n)(yva.v,n) where

n =max (u,v), xu E Gu (u < 0), YvE Gv (v < 0), n = max[u,v}; k o yv = k(yva.v,ol fork E K; Y.v o k = (yva.v,olk fork E K; k o s = ks fork, s E K. Theorem 3. Let S be an !-regular semigroup. Then one of the following is valid: (1) Sis a simple !-regular semigroup. (2) Sis a semilattice Y of groups where Y is order isomorphic to I under the reverse of the usual orrler. {3) Sis an !-regular semi­ group with proper kernel K. Thus, we have completely determ'.ned the structure of !-regular semi­ groups and mod groups. (Received August 26, 1968,)

68T-A4fl. TSIT-YUEN LAM, University of California, I3erkeley, California, and IRVING REINER, University of Illinois, Urbana, Illinois. Reduction theorems for relative Grothendieck rings.

For a finite group G and a subgroup H of G, let a(G,H) be the relative Grothendieck ring (over a field S1 of characteristic p) introduced in Abstract 68T-491, these cJioticu.J 15 (1968), 647-648. Theorem. Let G be a direct product B X K, and let H c: K. If n is a splitting field for B and all its

subgroups, then the map '1': a(B,l) ® a(K,H)- a(G,H) defined by [X]® [M],.. [X# M] is a ring isomorphism. (Here # denotes the outer tensor product,) One first proves that the theorem is true in the special cases: (1) B is a p-group, (2) B is p-free, and then p:coves the theorem in full generality by applying Brauer's Induction Theorem. (Received August 21, 1968.)

68T-A49. MARK MANDELKER, University of Kansas, Lawrence, Kansas 66044. Relative annihilators in lattices. I.

A natural generalization of the pseudo-complement of an element a of a lattice relative to an element b is the ~nnihilator (a,b) of a relative to b; it is the family of all elements x such that a n x ;;; b. Thus a lattice is relatively pseudo-complemented if and only if each annihilator is a principal ideal, It is shown that each annihilator is an ideal if and only if the lattice is distributive. Also, a weaker condition on annihilators is given that is equivalent to modularity, The main results concern distributive lattices Lin which (a,b) U (b,a) = L, identically, i.e, for any elements a and b of L, the join of the annihilator ideals (a,b) and (b,a) in the lattice of ideals of Lis always the improper ideal L itself. It is shown that in a distributive lattice this condition on annihilator ideals is satisfied if and ·Jnly if the filters of L containing any given prime filter form a chain, In an arbitrary lattice, a generalized condition on annihilators is given that is equivalent to this prime filter condition. A characterization of relative Stone lattices (cf, Birkhoff's problem 70) equivalent to Varlet's is obtained directly by means of annihilators. In part II examples will be given in lattices of zero-sets, lattices of closed sets, and lattices of cozero-sets in topological spaces. (Received August 28, 1968.)

916 Analysis

68T-B39. MARTIN SCHECHTER, Yeshiva University, New York, New York 10033. Semi­ Fredholm perturbations.

Let X be a Banach space, and let 4> +(X) denote the set of bounded linear operators A on X having closed range and such that the dimension a.( A) of the null space of A is finite. Let K(X) denote the set of compact operators on X, and let F +(X) denote the set of those operators E such that A t E E ~+(X} for all A E (>+(X). ~fheorem 1. A E '~>t(X) if and only if a.(A - K) < oo for all

K E K(X). Theorem 2. E E F +(X) if and only if a.(A - E) < oo for all A E II>+ (X). Theorem 1 states that A E '~>t(X) if and only if it does not agree with a compact operator on any infinite-dimensional subspace of X. Theorem 2 says that E E F +(X) if and only if it does not agree with any operator in '~>t(X) on any infinite-dimensional subspace. (Received May 16, 1968.)

68T -B40. N. A. TSERPES, Wayne State University, Detroit, Michigan 48202. Invariant measures on semigroups.

We consider Borel measures m with support F on a Hausdorff semigroup S that satisfy (i) m(Bx -l) = m(B ), V Borel B c S and x E S, or (ii) m is regular and m{Cx) ~ m(C ), m(xC) ~ m(C ), V compact C and xEs, or (iii) m(B) = iii(Bx), B =Borel, ffi =the completion of m. We call m atomic if :ii a Borel set A such that m(A) > 0 and for B c A, B = Borel, either m(B) = 0 or m(A - B) = 0. Theorem 1. Let (S,m) satisfy (i) with m atomic; then F is a left group [see Abstract 68T-191, these cJVoticei) 15 (1968), 3561 Theorem 2. A locally compact S admits a fin1te m satisfying (ii) iff S has a kernel which is a compact group. Moreover if S has an infinite such m and also a semigroup of positive measure, then F is loc. compact group. Theorem 3. S admits an m satisfying (iii) if S has a right ideal F which is left group and such that (B n F )x = Bx n Fx, VB =Borel and x E S. Theorem 4. In order for (S,m) with (i) to have F a left group it suffices: (1) Shave a compact subsemigroup of positive measure. (2) S have a compact right ideal. (3) S have a quasi- regular subsemigroup A of positive measure [i.e. for each x E A, :iln with xn=regular ]. (4) m(Q) > 0, where

Q = (x E S: :ii y such that yx =idempotent. Moreover, if Sis Ist countable normal and admits a complete uniform structure, then F is left group iff the right translations are closed. (Received june 24, 1968.)

68T -B41. F. E. SULLIVAN, University of Pittsburgh, Pittsburgh, Pennsylvania 15213. VP spaces are LP spaces.

Let A be a Boolean algebra, and J be a normalized jordan content on A, S. Leader [Ann. of Math. 58 (1953), 528-543] introduced the Banach spaces Vp(A,J) of all finitely additive functions of p-bounded variation on A. It is known [Bull. Amer. Math. Soc. 74 (1968), 153-154] that a real Banach space X is linearly isometric to a real Lp over a totally finite measure algebra iff the norm

917 in A satisfies Clarkson's inequality, and :i!u EX such that Cl(span(Eu)) =X, where E ranges over all Lp -projections on X. The vector u is called an Lp -unit for X. Theorem. Every vP(A,J) is linearly isometric to an Lp and J is an Lp -unit for vP(A,J). (Received June 10, 1968.) (Author introduced by Professor H. B. Cohen.)

68T-B42. WITOLD BOGDOWICZ, Catholic University of America, Washington, D. C. 20017, and JOSEPH DIESTEL and V. E. ZANDER, West Georgia College, Carrollton, Georgia 30117. Generalized Lebesgue-Bochner integral for Orlicz spaces.

Let Y,Z,W be Banach spaces and u a bilinear continuous operator from Y X Z into W. Let qJ, 1/! be complementary continuous Young's functions. Define the Orlicz space Ltp(v, Y) to consist of functions f from X into Y such that, for every set A E V, the function c A f belongs to the space L(v, Y) of sum­ mabie functions generated by the volume space (X,V ,v) in the sense of Bogdanowicz, Proc. Nat. Acad. Sci. USA 53 (1965), 493-498; moreover, the support of the function f is contained in a set BE V a and the seminorm IJfiJtp = sup [JJsfjdv : s E S(V ,R), J 1/! o Jsjdv ;!;; I} is finite, where S(V, Y) denotes

V -simple Y -valued functions. Define the space KIPU(v,Z) to consist of additive set functions j.l on V to Z such that j.I(A) = 0 if v(A) = 0 and the seminorm IJj.!IJqJu = sup{JI:iu(yi'j.I(Ai))J: Yi E Y, {Ai} is a finite disjoint family of sets from the pre ring V, and IJI:i yicAi II ;!;; I} is finite. Theorem I. The Orlicz space is a complete seminormed space. If tp satisfies A2 condition: tP(2r);!;; c tP(r) for r!!; 0, then the set S(V, Y) is dense in it. Define the trilinear integral on simple functions as in the above paper and extend it by continuity onto the closure S(V ,Y). Theorem 2. The integral Ju(f,dj.i) is a bilinear operator from S(V,Y) XK (v,Z) into Wand JJu(f,dj.I)J;!;; IJfiJ llf.IIJ for all functions f E S(V,Y) fPU t(J f{JU and j.l EK (v,Z). (Received June 25, 1968.) t(Ju

68T-B43. FRANZ PITTNAUER, University of Wisconsin, Madison, Wisconsin 53706. Holomor- phic functions with prescribed asymptotic expansions.

This paper gives a construction of functions which are holomorphic in a convex and bounded domain D of the complex plane and which have prescribed asymptotic expansions for an isolated set I of boundary points of D. In case of asymptotic power series and a finite set I, the corresponding result was given by Carleman. Later on it was generalized by Van Der Corput to a more general class of asymptotic series. On the other hand, Franklin gave a proof of the theorem in case of asymptotic power series and a general isolated set I of boundary points. The announced paper establishes the theorem for an isolated set I and for a class of asymptotic series more general than asymptotic power series. The proof combines the methods of Franklin and Van Der Corput. (Received June 10, 1968.) (Author introduced by Professor B. D. Arendt.)

68T-B44. E. J. PELLICCIARO, University of Delaware, Newark, Delaware 19711. An ascent existence and uniqueness proof for linear ordinary differential equations.

This paper gives a method of ascent existence and uniqueness proof for the initial value pro.Qlem for the homogeneous nth order linear ordinary differential equation in normal form with continuous coefficients. The proof provides a representation formula which, starting with the solutions of the first order equation, describes inductively the solutions of the nth order equation

918 in terms of those of the (n - 1 )th order. An explicit Volterra form of the solution is included. Basic to the proof is an equivalent integral equation involving inductively the solutions of the (n - 1 )th order differential equation. (Received July 1, 1968.)

68T-B45. J. E. BRENNAN, Brown University, Providence, Rhode Island 02912. Boundedness of point evaluations. ll.

Let E be a compact set in the complex plane C with positive two-dimensional Lebesgue measure and let R(E) denote the set of all rational functions having no poles on E. For 1 ;:; p < oo, Lp(E) = Lp(E,dxdy) has its usual meaning, RP(E) is the closure of R(E) in LP(E) and

Hp(E) is the set of functions in LP(E) which are analytic in the interior of E. If X c: C, Yp(X) denotes the analytic p-capacity of X. It has been shown by V. P. Havin and L. Bers that Rp(E) = Hp(E) if 1 ;:; p < 2. This is not necessarily the case when p = 2, however (see e.g. Abstract 67T-660, these c}/oticei) 14 (1967), 854). Theorem 1. If p > 2 and if for almost all C E oE the map f ~ f(Cl is unbounded on R(E) as a linear functional in the LP(E)-norm, then RP(E) = Hp(E). Theoren:: 2.. lf p > 2 and lim supr~o(Yp(ll(l;;r)\E)/r) > 0 for almost all C E oE, then Rp(E) = Hp(E). Corollary 1. If the inner boundary of E has zero Lebesgue measure, then Rp(E) = Hp(E) for p > 2. Corollary 2. If E has no interior and p > 2, the following are equivalent: (1) Rp(E) = Lp(E); (2) the map f ~ f(C) is unbounded on R(E) in the Lp(E)-norm for almost all C E E; (3) lim sup (y (6(C;r)\E)/r) > 0 for almost all r~ 0 p C E E. Theorem 2 and Corollary 2 improve earlier theorems of S. 0. Sinanjan (Mat. Sb. (N.S.) 69(111) (1966), 546-578). (Received August 19, 1968.)

68T-B46. E. G. CALYS, Washburn University, Topeka, Kansas 66621. A class of regular functions.

Let S* denote the class of functions f which are regular, univalent and starlike in lzl < 1 and which satisfy (l) f(O) = 0, f'(O) = l. Let :J0 denote the class of functions f, regular in lzl < 1, and satisfying (1) and (2) Re(f(z)/g(z)} > 0 for some g E S* and all z, lzl < 1. Theorem 1. f(z) E 30 if and only if F(z) = J~(f(t)/t)dt is close-to-convex. Theorem 2. lf f(z) E 30, then G(z) = J~(2/t 2 )[J~f(i;)dC]dt is close-to-convex. Theorem 3. If f(z) E :J0 , then H(z) = (2/z)J~f(t)dt is in 3 0• Theorem 4. If f(z) 92 i9 is univalent in lzl < 1, then f(z) E :J0 if and only if J 9 Re(zf'(z)/f(z)}d9 > -TT, z = re , 0 < r < 1, 1 h( 1 211: ·9 i9 0;!! 9 1 < 9 2 ;!! 2TT. Theorem 5. Let f(z) E :J 0 • Then f(z) = ze z (J O ((e1 + z)/(e -z))d¢(9) + icl}, where h'(z) = (l/z)J~11 ((ei 9 + z)/(ei9 - z))dx(9) +if!- 1 }, o., {J real numbers, and ¢(9), x(9) monotone nondecreasing functions. (Received July 1, 1968.)

68T-B47. B. N. SAHNEY, University of Calgary, Calgary-44, Alberta, Canada. A relation between absolute Abel summability and convergence.

Under Dini' s, Jordan's and de la Vallee Poussin' s convergence criterion for Fourier series, J. M. Whittaker, B. N. Prasad and M. L. Misra proved absolute Abel summability of Fourier series. This approach was encouraged by Whittaker, by giving an example in which the Fourier series of a given function was convergent but not absolutely Abel summable. The question of whether absolute Abel summability implies convergence remained unanswered. In an attempt to settle this problem, the following theorem is proved. Theorem. Absolute Abel summability of a series ~~0 an does not necessarily imply its convergence to the same value in the range (0,1 ). (Received June 26, 1968.)

919 68T-B48, H. I. BROWN, and H. H. STRATTON, State University of New York, Albany, New York 12206. The structure of B [c] and extensions of the concept of conull matrix.

The set 'it of conull matrices and the set of co regular matrices partition the algebra r of conservative matrices in such a way that the conull matrices share many properties with the zero matrix and the coregular matrices share many properties with the identity matrix. We consider here the problem of finding a similar partition of the algebra B [c] of bounded operators on the con­ vergent sequences. Let X designate the usual scalar homomorphism on r whose kernel X,~. is llt. Wilansky (Topological divisors of zero and Tauberian theorems, Trans. Amer. Math. Soc. 113 (1964), 240-251) suggests two extensions, X and p, of X and observes that X fails to by multiplicative on B [c] while p fails to be defined on all of B [c]. By investigating the structure of B [c] we establish that pis the only extension of X to a scalar homomorphism on a subalgebra of B [c ], that P.1. is the only subalgebra of B[c] except lltwhose intersection with ris llt, and that r, p,~., and the domain of p are the only proper subalgebras of B[c] which properly contain llt. It is proved that rand the multipli­ cative operators (lim Tx =(lim Te) lim x)are maximal sets on which Xis multiplicative. Furthermore, we prove that X is the only nonzero scalar homomorphism on r. (Received July 9, 1968.)

68T-B49. J. N. PANDEY, University of Alberta, Edmonton, Alberta, Canada. Complex inver­ sion for the generalized Hankel convolution transformation.

Let l!.x stand for the differentiation operator (D~ + (2')1/x)Dx) where x > 0, and let I stand for the open interval (0, oo). lt is also assumed that 0 < y, 0 l! a 1 l! a 2 l! a 3 .•• , and L ~=l aj;? < oo with limk_,cJk/ak) = n (0 < n < oo). We further add that a smooth function 1/J(x) defined over I belongs to the testing function space H(I) iff ,Bk(I/J(x)} ~ supO

68T- BSO. A. G. KA RTSA TOS, Wayne State University, Detroit, Michigan 48202. Existence of bounded monotone solutions of nth-orrler differential equations.

The following theorem is given: For the differential equation (*) x(n) + P(t)g(x,x' , ... ,xln-l )) =

Q(t,x,x', •.• ,x(n-l)), n > 1, suppose that (i) Pis defined and continuous on the interval I= [t0, + oo), t 0 ;;:; 0, and J~ tn-lJP(t)Jdt < + oo; (ii) g : Rn .... R = (- oo, + oo), continuous; (iii1 ) Q : I X Rn - R, continuous, and such that for every constant b > 0 with llxlll! b, x E R, !llxll = max(JxiJ}), there is a function Q0(t,b);;:; 0, continuous, JQ(t,x)J l!Q0(t,b), and J~ tn-lQ0(t,b)dt < + oo; (iii2 ) for every closed sphereS c Rn, and every x, yES, JQ(t,X)-Q(t,y)Jl! Q1(t)X(IIx-VIIJ, where Ql :I"" Rt = [o, + oo), contin­ uous, Jfc, tn-lQ1(t)dt < + oo, and X: Rt .... Rt• X(O) = o, continuous. Then given a constant c f 0, there is a solution xc(t) of(*) which is valid for all large t, and moreover, limt_ooxc(t) =c. Under additional

920 assumptions uniqueness is guaranteed. (Received July II, 1968.) (Author introduced by Professor A, T. Bharucha- Reid.)

68T- BSI. D. D. BONAR, Wayne State University, Detroit, Michigan 48202, An annular function related to Weierstrass's nondifferentiable function.

Let f(z) =Z::nc-l/ 2ei/ln log nzn (n = 1,2, ... ), where 0 < c

Z::nc-l/ 2ei/ln log nzn (n E I(9)), then for each

(pei max (2, 3/c }, ll = 211/b, then there exists a sequence of Jordan curves [Jn} (not circles) in (z: Jzl < l} converging outward to (z: Jzl = l} such that min(jf(z)J: z E Jn}- oo as n- oo, (Received July 29, 1968.)

68T-B52. W, D. L, APPLING, North Texas State University, Denton, Texas 76203. Summability and smallest range-including closed sets,

F, U, R, R~, Cq, Wq and sq for q in R1 are as in Abstract 67T-l59, these cNoticeiJ 14 (1967), 262; and RA is the set of all bounded finitely additive elements of R. All integrals considered are Hellinger-type limits of the appropriate sums. Theorem. Suppose q is in R_t, q(U) > 0, and g is an element of RA whose variation function is in Cq. Then there is a closed number set M such that: ( l) M is a subset of every closed number set which includes the range of an element P of Wq such that g is sq(P), and (2) there is a function T from F into M such that J U Jg(I) - T(I)q(I)J = 0. (Received July 26, 1968.)

68T-853. R. A, MAYER, Courant Institute, New York University, New York, New York 10012. Deceptive convergence of Fourier series.

Let V be a faithful finite-dimensional representation of the group SU(2) =G. Let (v1 , ... , vN} be the coordinate functions of V. Let T m be the space of complex functions on G generated by all functions P(v 1 .... ,vN ), where P(X1 ,.,,,XN) is a polynomial of degree m in N variables. Iff E L 2(G ), let V mf be the orthogonal projection off on T m. Theorem. Let f be a bounded function on G such that f(e) = 0, f is continuous at e, and the set of discontinuities off has Hausdorff dimension < 2. Then the sequence (V mf(e)} either diverges or converges to zero. If V has an irreducible subrepresentation of dimension> 4, then there exists a bounded function g on G such that g vanishes on a neighborhood of e, the set of discontinuities of g has Hausdorff dimension = 2, and (V mf(e)} converges to 1. If every irreducible subrepresentation of V has dimension 1 or 2, then (V mf(e)} converges to zero or diverges for every fin L 2{G) such that f(e) = 0 and f is continuous at e. Iff E L 2 (G), then the set of x in G, such that f is continuous at x but (V f{x)} converges to a value f f(x), has measure zero. m {Received July 19, 1968.)

921 68T-B54. J, W, JENKINS, University of Illinois, Urbana, Illinois 61801, and the State University of New York, Albany, New York. An amenable discrete group with a _nonsymmtltric group algebra.

Theorem. Let L( R) be the group of ordered pairs of real numbers (a,b), a f= D, with m:;ltipli­ cation defined by (a,b)(a' ,b') = (aa', ab' + b). Let s = (a,l) and t = ( -a,l) for some a such that

0

68T-B55. F. E. SULLIVAN and H. B. COHEN, University of Pittsburgh, Pittsburgh, Pennsyl­ vania 15213. Von Neumann algebras in smooth spaces.

A type of orthogonality due to R. C. james is used to show that in a Banach space of the type US (UR), an increasing net of norm-1 projections converges in the weak (strong) operator topology.

In a US or in a smooth U R, for every strictly increasing continuous function :J of the nonnegative reals into themselves, the set of" :!'-projections" gives rise to operator algebras equal to their second commutants. Nontrivial examples can be found among the Orlicz spaces. (Received july 22, 1968.)

68T-B56. B. J, WALSH, Rutgers, The State University, New Brunswick, New jersey 08903. E'lux in axiomatic potential theory. II: Cohomology of Brelot sheaves.

Let W be a locally compact Hausdorff space with a countable base, and J( be a (complete pre-)

sheaf of functions on W such that the pair (W ,J() satisfies the Brelot axioms and such that L E J(.

_!heorem. There exist fine sheaves Rand ,f. over W and an exact sequence of sheaves 0 .... Jl' .... R .... ,f. .... 0. Consequently, the cohomology groups Hq(W ,J() = 0 for q;;; 2. Theorem. If W is compact, then H1(W ,JI') is a one-dimensional vector space whose dual can, be identified with the space of flux functionals (cf. Abstract 655-82, these cifotice0 15 (1968), 489). Other conditions under which dim H1(W,JI') can be determined, and analogous theorems for noncompact spaces W equipped with families of normal operators, are also given. (Received july 24, 1968.)

68T-B57. S. E. HAYES, University of Texas, Austin, Texas. A theorem on Stieltjes integrals.

The integrals considered in the following theorem are defined by a refinement type limit. Theorem. Suppose each off and g is a real-valued function over the interval [a,b], and lfl is the function such that for each number x in [a,b], jfj(x) = jf(x)J, Iff is integrable with respect tog over [a,b], then ifl is integrable with respect tog over [a,b]. (Received july 29, 1968.)

68T-B58. W. A. KIRK, University of Iowa, Iowa City, Iowa 52240. On a nonlinear functional equation in a Banach space.

Let X be a reflexive Banach space and f : X .... X a mapping which has a Gateaux derivative f'(x) at each point x EX, Let T : X .... X be a linear mapping of X into X having an inverse, and let

F = 1 - Tf. Define IFJ = inf11 > 0 (supiJxiJs; 11 IIFxiJ/IIxiJJ. and note that F has a Gateaux derivative given by F'(x) = 1 - Tf'(x), x EX. The principal result is the following Theorem. Assume x .... F'(x) is a continuous mapping. If IJF'(x)IJ < 1 for each x EX, and if JFJ < 1, then the equation f(x) = y has

922 exactly one solution for each y E X, This theorem provides an extension (with stronger hypotheses on F) of a result of Srinivasacharyulu [On some non-linear problems, Canad. J, Math. 2.0 (1968), 394-397]. (Received August 5, 1968.)

68T-BS9. WITHDRAWN.

68T-B60. D. E. MEYERS, University of Arizona, Tucson, Arizona 8572.1. A note on different!- able functions.

Let D(G) be the set of (weakly) differentiable functions on a region of the complex-plane (Heins, Complex function theory, p. 79). Theorem. D(G) is a complete complex algebra with compact­ open topology. Theorem, I is a maximal principal ideal in O(G) if and only if:B:z 0 E G 31 I= (fjf E D(G), f(z 0) = 0}. Let O(G) be the usual complex algebra of functions analytic in G with compact­ open topology and O(G) the complex-conjugates of the elements of O(G) (resp. O(G)) if and only if I is the intersection of a maximal principal ideal, of D(G), with O(G) (resp. O(G)). Let A(G) denote the algebra generated by O(G), O(G). Theorem. If G is the complex plane and f is bounded, f E A(G), then f is a constant, (Received August 5, 1968.)

68T-B61. W. L. WOODWORTH, Drake University, Des Moines, Iowa 50311. Integrals over proto-rings. Preliminary report.

Let T be a nonempty set and P be a nonempty collection of subsets of T. Definition, The collection Pis a proto-ring if for each pair of sets E and F in P, E - F and E n F can be written as finite unions of pairwise disjoint elements of P. An integral is developed for finitely and countable additive scalar-valued set functions defined on a proto-ring of sets, It is shown that a proto-ring is a minimal collection of sets for purposes of most integration theory. Most results obtained do not depend on the additivity of the set functions nor an "algebra-like" structure on the domain of the set functions. Necessary and sufficient conditions are obtained for a function to be integrable, and the analogues to the classical dominated convergence theorem and Radon-Nikodym theorem are shown to hold. The Lp spaces are defined in essentially the same manner as by P. Porcelli. (Portugal, Math, 2.5 (1966), 105-12.2.). It is shown that the integral developed generalizes that of W, Bogdanowicz (Nat, Acad, Sci. Proc, 53 {1965), 492.-498) for countable additive set functions on a prering of sets and that of Dunford and Schwartz (Linear operators, Part I, Interscience, 1958) for finitely additive measures on an algebra of sets. (Received August 16, 1968,) (Author introduced by Professor James A. Dyer,)

68T-B6Z. HANS-HEINRICH KORLE, Universitat Marburg, Mathematisches Institut, 355 Mar­ burg, W. Germany. On the equivalence of Riesz and generalized Ced.ro (absolute) summability.

Let X= !Xn>• 0 ~ X0 < X1 < ... < Xn .... oo, and x O; 0. A series !::nO;O an is called summable

I R,X,xl if the function E:=o {1 - X11 /x)xaJJ, An< x ~X +l' n ~ 0, is of bounded variation on (X0,oo), and . . ~ {X) (X) (xf ..--D. lS called summable ~~~~1 Xj ifLJ jtn - tn+1 j <00 1 tn =LJJJ=O {1- XIJ/An+l) .. ,(l- XIJ/An+[XJ) • (1 - \1>-n_+[xJ+l )K- x a~.~. Theorem. If xis an integer, a series is summable jR,X,xl if and only if it is summab1e jc,X,xj. This is the analogue of a result by D. C. Russell (T8hoku Math. J, {2.) 17 (1965), 410 442., Corrigenda (2.) 18 (1966), 454-455) and A. Meir (Canad. J, Math. 2.0 (1968), 735-738)

923 concerning ordinary se~m nability. For x >I not an integer (the case 0 < x

68T-863. JAGDISH PRASAD and A. VERMA, University of Alberta, Edmonton, Alberta,

Canada. An analogue of a problem -:>f j. Balazs an~ Turan. lll.

Let (1.1) -I= xn + 2 < xn +I< •.• < x2 < x 1 =I be the zeros of P~a)(x) (a>- I) where P~a)(x) is nth ultraspherical polynomial. We are interested in finding the interpolatory P·~lynomials Rn(x) such that (1.2) Rn(x 11 ) = y 11 (11 = 1,2, .•• ,n+2), R~(x 11 ) = Yt (II= 2,3, .•. , n + I), where y11 andy~ are arbitrary prescribed values. We prove the following Theorem 1. If n is even, x 11's are given by

(1.1), -I< a< I, a!- 1/3, and y 11 , Yt are any arbitrary given values, then there exists one and only one polynomial Rn(x) satisfying (1.2). For the explicit form of the polynomial we have (1.3) "'ntl '<"'n+l . Rn(x) =L.JII=I y 11 r 11(x) + L.J 11 ; 2 Yj;,P 11(x) where r 11(x) and P 11(x) are the fundamental polynomrals of (0,2) interpolation. Theorem 2. Let f(x) be a continuous function satisfying Zygmund f(x + h) - 2f(x) + --2 f(x- h) = O(h) in [-1,1] and let y 11 = f(x 11 ) and YjJ = O(n)/./1 - x 11• The Rn(x;f) converges to f(x) uniformly for -I+ f ~ x ~I-f (O < f < i) if -1/2

68T-864. W. C. RiDGE, Indiana University, 8!oomington,lndiana 47401. Com?osition operators. Preliminary report.

Choksi's characterization of unitary composition operators on a separable Hilbert space (J. Math. Mech. (I) 16 (1966), 83-100) is extendecl to arbitrary bounded linear composition operators on same. Also, for a bounded linear operator on a complex Hilbert space of arbitrary clim,,nsion to be a composition operator, it is necessary that its point, approximate point, anrl residual spectra each contain, along with any complex number of modulus different from 0 or I, the entire complex circle of the same m-:>dulus. (Received August 22, 1968.)

68T- 865. j. E. COURY, University of Washington, Seattle, Washington 98105. On the measure of zero S

Let G be a compact topological group (not necessarily Abelian), and let V be a continuous unitary irreducible representation of G, with V ! l. Define the set C = {x E G : (Vx a,{J) = 0}. where a and f3 are arbitrary elements in an orthonormal basis for the representation space of V. Theorem I. If Ker V has (Haar) measure zero, then C has measure zero. In particular, if G is connected, then

C has m-~asure zero. Theorem ll. If G is not connected and non-Abelian, then a sufficient condition that there exist such a representation V for which C has positive measure is that either G be finite or, if G is infinite, that either (i) the commutator subgroup of G be all of G, or (ii) G be totally dis­ connected. (Received August 23, 1968.)

68T -866. H. M. SRI VASTA VA, West Virginia University, Morgantown, West Virginia 26506.

A multiplication formula association with Lauricella's hypergeom·~tric function FA.

In a couple of recent papers [cf. Proc. Cambridge Philos. Soc. 63 (1967), 1087-1089; ibid.

924 65 (1969), Part I (to appear)] the author has proved certain finite and infinite summation formulas

involving the Appel function F 2 • The object of the present note is to derive the multiplication * (n)[ . . ] _...,ml+ ••• +mn(a.+k) . . ] formula( )FA cr,-m1 , ... ,-mn' a. 1 + l, .•. ,a.n + l,X1x, .•• ,Xnx -L.Jk=O k 2F 1 [-k,cr, a.+ 1, x (n+l)[ J (n) . , • FA a. +l,-m1, ... ,-mn,-k; a. 1 + l, ... ,a.n + 1, a.+ 1; X1, .... ~. 1 , where FA is one of Launcella s hypergeometric functions in n arguments. Amcngst several specialized cases of (* ), an interesting multiplication theorem for the generalized hypergeometric PF q-function is also obtained by making use of the Laplace and the inverse Laplace transform techniques and the method of multidimensional mathematical induction. (Received August 26, 1968.)

68T-B67. j. W. BAKER, Florida State University, Tallahassee, Florida 32306. Some uncomplemented subspaces of C(X) of the type C(Y).

Let C(X) denote the space of bounded, continuous, real-valued functions on a T 4 -space X. Theorem. If the nth derived set X(n) is nonempty for some positive integer n, then for each £ > 0

there is a T 4- space Y which is a closed, continuous image of X and there is a subspace S which is linearly isometric to C(Y) such that each projection of C(X) onto S has norm at least 2n- 1 - £.

Theorem. If X(n) is nonempty for each positive integer n, then there is a T 4 - space Y which is a closed, continuous image of X and there is a subspace S which is linearly isometric to C(Y) such that Sis not complemented in C(X). If X is compact and metrizable, this condition is also necessary. _!heorem. Let X be a compact metric space. Then X(n)= fJ for some positive integer n iff for each compact Hausdorff space Y and for each continuous function f of X onto Y there is a linear averaging operator for f. This result answers a question raised by A. Pe}:'czyfiski (see Dissertationes Math. (Rozprowy Mat.) 58 (1968), 74). A generalization of a theorem of R. Arens (see Duke Math. j. 32 (1965), 475, Theorem 3.1) is established which gives a lower bound for the norms of projections of C(X) onto certain subspaces of the type C(Y). (Received june 3, 1968.)

68T -B68. R. B. HOLMES, Mathematical Sciences Building, Purdue University, Lafayette, Indiana 47907. Smoothness and roughness indices defined by convex functions.

Let g be a continuous convex function on a (real) locally convex space X, Let g'(x,y) =

limt ~o(g(x + ty) - g(x))/t and Dx = (y: g'(x,y) exists}. Let Kx = n (kerf: f E og(x) }, where ilg(x) is the

(nonempty) subdifferential of gat x. Then either Dx = Kx Ql" :i!z 3 g'(x,z) F 0 and then Dx = z al Kx. For x E X define the g-smoothness index of x by g-sm(x) =dim Dx and the g-roughness index of x by g-rgh(x) = codim Kx· We have g-sm(x) = codim aff og(x) (w*-closure) and g-rgh(x) = dimo g(x) [ + 1] if xis [not] a g-minimum. When g is sublinear these indices are related to the size of sub­ spaces M containing x which have the unique Hahn-Banach extension property for functionals in M* majorized by gJM--property (g-UHB) of M. For certain x E X, we have the inequalities g-rgh(x)

~ inf (dim \1: x E M = (g-U HB)} and g- sm(x) ;; 1 + sup (co dim M: x E M = (g-U HB)}. With additional hypotheses on g and X these formulas become equalities. In particular, certain results of Phelps, Garkavi, and Poulsen can be generalized and unified. (Received August 28, 1968.)

68T -B69. A RSHAG HAJIAN, Northeastern University, Boston, Massachusetts 02115, and JUYI ITO, Brown University, Providence, Rhode Island 02912. Weakly wandering sets and invariant measures for a group of transformations.

Let (X, 8,m) be a cr-finite nonatomic measure space, and let G (g} be an arbitrary group of

925 nonsingular measurable transformations acting on X, i.e. B E 13 .. g -l(B) E 13 for every g E G, and m(B) =0 .. m(g -l(B)) = 0 for every g E G. A set Win 13 is called weakly wandering under G if there exists an infinite sequence {gi: i = 1,2,3, ... ) in G such that m(g~ 1 (W)n gj1(W)) = 0 fori 1' j. A measure J.l. defined on the measurable space (X,5) is called invariant under G if j.l.(g(B )) = J.l.(g-l (B)) = j.L(B) for all g in G and all B in 15. Two measures m and I" defined on the measurable space (X,/3) are said to be equivalent if m(B) = 0 .. IJ(B) = 0. Theorem. There exists a finite invariant measure J.l. equivalent with m if and only if there exists no weakly wandering set of positive measure under G. (Received August 29, 1968.)

68T -B70. E. H. OSTROW, San Fernando Valley State College, Northridge, California 91324. A generalization of theorems of Plessner and Zygmund.

Suppose I" is a positive measure on (0,21f) satisfying the condition J~diJ(s) ;!!; At for all t (A an absolute constant). Suppose that F is a function having, at each point of a set E, a generalized (Peano) derivative F (k) of order k ~ 1. Thus on E, F (x + t) =r; f;~ F (j)(x)~ I 1: + Wk (x,t)tk/k: where Wk (x,t) = F(k)(x) t o(t) as t .. 0. Thea~ 1. Suppose that F E L(0,21f) and is extended,. periodically. If F(k) exists at each point of a set E of positive measure, then the integral J0 ((W k(x,t) - Wk(x, - t))/t) dj.L(t) 2 (defined as lim( ..0 J :) exists and is finite a.e. on E. Theorem 2. Suppose that FE L (0,1f) and is again extended periodically. IfF (k) exists at each point of a set E of positive measure, the integral J ~ jW k(x,t) - Wk(x, - t)j2 /t)dj.L(t) is finite a.e. on E. When I" is Lebesgue measure, Theorem 1 is due to Plessner (J. Reine Angew. Math. 158 (1927)) and Theorem 2 to Zygmund (Bull, Amer. Math. Soc, 49 (1943), 917-923). The proofs depend on a decomposition theorem of Marcinkiewicz (Zygmund, Trigonometric series, II, Cambridge, 1959, pp. 73-76) and a lemma of Ostrow and Stein (Ann. Scuola Norm. Sup. Pisa (3) 11 (1957), 117-135) and follow the real variable techniques of the theory of singular integral operators. (Received August 10, 1968.) (Author introduced by Dr. William Karush.)

68T-B71. E. E. GRANIRER and ANTHONY LAU, University of British Columbia, Vancouver, Canada. A characterisation of locally compact amenable groups.

Let G be a locally compact group and C(G) the space of bounded real continuous functions on G with sup norm. For f E C(G), y, x E G, (lyf)(x) = f(yx), (ryf)(x) = f(xy). Let LUC(G) [RUC(GJ) be the space of all f E C(G) for which the map y .. I f [y .. r f), from G to C(G) is continuous, y y UC(G) = LUC(G) n RUC(G). Denote by T'c the topology of uniform convergence on compacta; by Co, convex hull, and by RO(f) = {r f; y E G). Theorem. Let G be a locally compact group. Then LUC(G) y admits a left invariant mean (LIM) if and only if: (*)for any f E LUC(G) the T'c closure of CoRO(f)

contains some constant function. If K(f) denotes the set of all such constants then K(f) = {tp(f); tp ranges over all LIM's). (It is sufficient that for such G (*)holds only for f E UC(G),) In fact we show this theorem to hold true for any semigroup G with a completely regular Hausdorff topology in which multiplication is separately continuous, This theorem is a generalisation of a theorem ofT. Mitchell [Trans. A mer. Math, Soc. 119 (1965), 244-261] who proved it for discrete semigroups. (Received August 30, 1968,)

926 Applied Mathematics

68T-C6, M.A. HYMAN, Mathematics Research Center, U.S. Army, University of Wisconsin, Madison, Wisconsin 53 706. Image coding.

"Images" or "multi-dimensional signals" (such as photos, film sequences, etc.) are notoriously redundant; that is, the same "intelligence" could be expressed using much less detail. An important problem in the applications is to find where the intelligence -- which depends on the use intended -- re­ sides in the image ("pattern recognition") and "recode" the image into a more condensed form. This paper considers in particular two-dimensional images, such as pictures, but the methods are in principle more general. First one discretizes the picture, and then considers two complementary types of coding: "magnitude coding" and" position coding". It is shown how one can automatically generate an optimal or near-optimal basis of" sub-pictures" for any class of pictures, in terms of which any picture of the class can be conveniently and economically expressed. The methods used are an extension of the Karhunen-Loeve techniques suggested for one-dimensional signals by ·M. S. Watanabe plus the author's "weighted information theory" (WIT). (Received July 1, 1968.)

68T-C7, H. E. GOLLWITZER, University of Tennessee, Knoxville, Tennessee 37916. Oscilla­ tion theorems for a delay equation.

Consider the delay equation y" (t) + q(t)y(t - T)y = O, where q(t) is continuous, eventually nonnegative, and Tis a variable delay. ')I is the ratio of odd integers, 0 <')I< 1 or 1 < ')1. With appropriate definitions of oscillation and nonoscillation we prove two results, Theorem 1. Let

1 < ')1. The above equation is oscillatory iff J00 sq = oo. Theorem 2. Let 0 <')I< 1. The above equa­ tion is oscillatory iff J 00 s'Y q = oo. (Received July 8, 1968.)

68T-G8 •. JOHN DE CICCO, Illinois Institu.te of Technology, Chicago, Illinois, and ARUN WALVEK:AR, Northeastern Illtilois State College, Chicago, Illinois. The four necessary conditions for a higher order variation problem.

A higher order variation problem for a real space of dimension two is I( E) = Ji f dx = minimum, where f is a function of a differential element of order n !!!: 1, such that it is of at least class (n + 1)!!!: 2. Extensions to this higher order variation problem of the four necessary conditions when n!!!: 1 are studied. The Property (I) is given by the Dubois-Reymond integral equation, namely, a linear combination of rept;ated integrals of orders 0 to n, inclusive, equal to a polynomial of degree

(n - 1) !!!: 0, at most. The Hilbert integral I is a line integral involving the differentials dx, dy, dy1, •.• ,dy(n-1). By use of this Hilbert integral I*, an extension of the Property (II) of Weierstrass is obtained. This involves a Weierstrass E function of the form E(x,y,y 1 , ••. ,yin), yl , ... , y(n)). By the Taylor theorem and a substitution y(~)- f.~l = U8/Un+l' for~= 1,2, .•• ,n, a generalized form of the Property (III) of Legendre is found. Finally, the Property (IV) of Jacobi that an admissible arc

927 has no points conjugate to its initial point is developed for this higher order variation problem. (Received July 15, 1968.)

68T-C9. S. SRIVASTAVA and K. A. ZISCHKA, University of Windsor, Windsor, Ontario, Canada. Radiation condition and uniqueness theorem for n-dimensional reduced wave equation in an infinite domain.

The present work is intended to be a comprehensive and systematic treatment of the radiation conditions (a particular case being Sommerfeld's radiation condition) which guarantee the unique solution of the exterior boundary' value problems for the second-order linear elliptic differential equation (which one can also consider as the reduced general wave equation) (1) L(u) = 2 L;~,j a1j(x)o u/Cixi11xj + ll~b1 (x)Ciu/Cixi + c(x)u = 0 inn-dimensional euclidean space En. First of all, authors generalize Sobolev's integral formula. In order to do so, they introduce the concept of retarded argument and auxiliary functions ern and r (in an appendix). Furthermore, some additional restrictions are imposed on ern and r. Secondly, using this generalized integral formula, they find conditions which are a generUization of the classical Sommerfeld's radiation condition. Th~n maximum principle for.tb.e solution in an Unbounded domain is established which finally leads to a uniqueness theorem for the exterior boundary value problem. Special cases of (1) such as 2 2 L!lu + k u = 0 and Au+. , k .·(x)u=.O . . . .llav!\1 . . aJ~to" be.en. considered. inn-dimensional. space E n • (Received July 16, 1968.)

68T-C 10. P.·. J. !<:ALMAN, Stl!{e UJliVer<y of New York, Stony Brook, New York. The existence of a globally diUereJltiaJ:>~e demand.fu:jictll>l!.

In a class of problemli tn the area df mathematical allocation theory one is concerned with differentiable properties of demand function$. Conditions for local differentiablity are well known. This paper establishes conditions for glopal differentiaPiUty. Let E l:!e a Euclidean space of dimension n + 1. Define the l:!un.Qle set E = (x E. E : xi !l O, i = 1,,.,., n+ 1}. Introduce the binary preference relation~ on E0 which is a complete ordering. IX.= (y. E E 0 : y ~ x and x ;:_y, x,y E E0 } is the indifference set of x. L~t ex= {y E E0 : y ~xJ and Cx = {z E E0 :, x ~z}. We call u(x) a utility function if u(x) ;;; u(y) iff x Z. y, x,y E E0• Let tt = ((p,y) : Pi > 0, i = 1,. •. , n, y ;;; 0 }. Define the~­ straint set r = {(p,y) E n: :ix E E0 :I p • X l! y }. Let ll(p.y) be a vector valued function from rto E0• A demand function is defined as er(p,y) = {ll(p,y) = x: ll(p,y) is a greatest element of E0 for,:;::}. 0 . · ., ;. ·; X 0 Theorem. Let E be connected, convex, compact, l:x: (strongly) convex, Cx and C closed in E • Let u(x) be twice differentiable on the open convex ~et E' = {x E E ; xi > 0, i = 1, ... , n + 1} and satisfies max u(cr(p,y)). Then cr(p,y) is differentiable on r. (Received July 19, 1968.) (Author intro­ duced by Dr. L. E. Schwartz.)

928 68T-Cll. R. B. BARRAR and H. L. LOEB, University of Oregon, Eugene, Oregon 97403. Nonlinear uniform approximation with interpolation.

Let F be a n-parameter family of functions in C [a,b ], and let a ~ x 1 < x 2 < ••. < xk ~ b be a fixed set of k interpolating points where 1 ~ k ~ n. If G = [f E F : f(xi ) = 0, i = 1 ,, •. ,k J. then E =min [/lf/1: f E G} where /lf/1 =max [/f(x)/: x E [a,b]}. Theorem 1. f E G has the property

1/f/1 = E iff there are n- k + 1 points, a ~y 1

68T-Cl2. S, I. ROSENCRANS, Tulane University, New Orleans, Louisiana 70118. On Schwarzschild' s criterion.

Schwarzschild' s criterion for the stability of a vertically stratified gas is given a very short proof, using only elementary calculus. (Previous proofs had to use eigenfunction expansions and the accompanying completeness theorems.) Simultaneously an upper bound is derived for the growth rate of perturbations in the unstable case. Finally this upper bound is shown to be "sharp", i.e. there exists a (generalized) solution having this growth rate. (Received August 15, 1968.)

Geometry

68T-D5. JOHN DE CICCO, Illinois Institute of Technology, Chicago, Illinois 606!6, and R. V. ANDERSON, Chicago State College, Chicago, Illinois 60633. Concerning the relative tensor and connections of a conformal space rn.

Two Riemannian spaces Vn and Vn are conformally equivalent if and only if there exists a conformal cartogram T between them. This conformal equivalence relation decomposes the totality of Riemannian spaces V n into disjoint equivalence classes, each one of which is termed a conformal space r n· If V n is an element of rn, then the relative conformal covariant metric tensor of r n is hij = g -l/ngij· Conformal Christoffel symbols of the first kind Aij,k' and the second kind A~k' are introduced and their laws of transformation obtained. The process of conformal covariant differentia­ tion is discussed and its properties studied. The Weyl curvature tensor Wj~l is expressed in terms of the conformal affine connections A jk. The conformal group G of r n is studied. Its invariants lead to conditions for the conformal equivalence of Riemannian spaces. (Received August 14, 1968.)

68T-D6. A. H. SCHOEN, NASA-Electronics Research Center, Cambridge, Massachusetts 02139. Regular sad rile polyhedra ( RSP ).

Let a RSP be any unbounded surface which has regular skew faces [p}, each spanned by a minimal surface, with regular plane vertex figures (q}. There are 12 RSP for which the face and vertex symmetrices Rand S [(i) Generators and relations for discrete groups, by H. S. M. Coxeter

929 and W. 0. Moser] are rotary reflections and rotations, respectively. To each of the 9 finite regular polyhedra (p,q}h (('il} =Petrie polygon of (p,q}h [(i)]), there corresponds a RSP (h,q} p with the same edges, vertices, and vertex figures [cf. (i), p. 112 ]. The faces (h} and Petrie polygons (p 1 of

(ii,q} pare the Petrie polygons and faces, respectively, of (p,q}h. (4,3}3 , (6,3}4 , (6,4}3 are limiting transforms (11 ~ oo) of infinite regular warped polyhedra [(ii) Abstracts 68T-Dl and 658-30, these 15 (1968), 801, 727]. The 9 quasi-regular polyhedra t(h,q} = (h • q) are identical to the c#oticei) p 9 q qlr derived from (3 • 3), (3 • 4), (3 • 5), ((5/2) • 3), and ((5/2) • 5) by H. S.M. Coxeter et al [(iii) Phil. Trans. Roy. Soc. London A 246 (1954), 401-450]. P. Pearce [(iv) Graham foundation research report ( 1966)] has described (4,4} and (6,3}, RSP analogs of regular p~ane tessellations. The third analog,((b;2) • 6}, which was found in collaboration with N. W. Johnson, is an infinite Riemann surface of 2 sheets (cf. H. S. M. Coxeter, Regular polytopes, p. 113 ). (Received August 22, 1968.) (Author introduced by Mr. J. C. Kotelly.)

Logic and Foundations

68T- E 16. ALBERT SA DE, 364 Cours de la Republique, Pertuis (84 ), F ranee. Lacune dans !'argument qui etablit la non-equivalence d'un ensemble et de son Booleien.

Soit E un ensemble infini, F =8- E !'ensemble de ses parties pro pres; Card (F) =Card( 8).

Soit !'application 1-1, f = E ~ F;B = {xlx E E, xf. f(x)}; C = (xlx E E, x E f(x)}; on a en C =fJ/\8 t C = E On trouve dans taus les traites !'argument suivant: "8 =f(x)=> [x EB =xf/.f(x)=>x f/.B]v[x EC=xEf(x) => x E 8], deux contradictions. Done 8 n'a pas de preimage par f dans E;f n'est pas bijective, Card (E) I' Card ( 8) ." La validite de cette conclusion exige C I' fJ; car si C = fJ, 8 = E, or E fl. F. L 'argument ci-dessus prouve alors seulement que V x E E,f(x) I' E; c'est une tautologie puis que E n'est pas dans F et on ne peut plus en tirer argument pour conclure que f n'est pas bijective. Si fest bijective, on demontre que 8 est necessairement egal a E. En effet, soit f bijective et B F E; alors 8 = f(z), z E E =>(z E 8 => z fl. f(z)= z f/.B)V (z fl. 8 => z fl. f(z)=> z E 8), deux contradictions; done 8 =E. Pour que l'egalite Card (E) I' Card (8) soit demontree il faudrait avoir prouve que dans toute applica­ tion 1-1 de E dans F il existe un element deE qui appartient a son image, ce qui n'a pas ete etabli. Dans la demonstration usuelle la lacune est masquee par le fait que l'on raisonne sur IS et non surF. (Received March 26, 1968.)

68T- El7. SA HA RON SHELAH, The Hebrew University of Jerusalem, Jerusalem, Israel. Categoricity in power.

This notice gives a partial answer to question 3 in Morley's Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514. The findings improve on certain results of F. Rowbottom (these c#oticei) 11 (1964). 248) and J. P, Ressayre (unpublished article). Notations will be as in Vaught's The Lowenheim-Skolem theorem (Proc. Internat. Congress for Logic, Methodology, and Philosophy of Science, 1964, p. 81) where upper bounds for IJ.(X) are mentioned. S(A) is defined in Morley's article. T will denote a first-order theory of an arbitrary infinite cardinality IT I. X will be inf(X: X;; IT I, xtto >X}. Theorem 1. T satisfies just one of the possibilities: (1) For every A, jS(AJI ;!;lA It 2ITI

930 and if IS(AJI > lA I> ITI then there exists B c A, IBI = ITI, and IS(BJI> lA I> IBI. (2) For every A, IS(AJI~IAIITI,andthereexistsanAforeveryA suchthatiAI=A,IS(AJI!!; A~o. (3) Forevery A there exists an A, IS(A) I > lA I = A. Theorem 2. If T is categorical in a power greater than IT I and different from X, then there exists a power A, IT I< A < IJ. ( ITI ), such that T is categorical in every power not smaller than A, and is not categorical in every power smaller than A and greater than IT 1. (Received May 31, 1968.) (Author introduced by Professor Michael Rabin.)

68T-El8.j. R. GEISER, Dartmouth College, Hanover, New Hampshire 03755. A result on limit ultra powers.

Notation and terminology is from Keisler's Lim:t ultra powers, Trans. A mer. Math. Soc. 107 (1963), 382-408. Definition. Let G be a filter on I XI and Dan ultrafilter on I. G is nontrivial w.r.t. E_ iff there is a disjoint partition (En: n < w} of I such that U n <£C En X En E G, but for all n < w, En (/. D. Theorem. (Assume li'l is not a measurable cardinal.) Let D be an w-incomplete ultrafilter on I and let G be a filter on I X I. G is nontrivial w .r .t. D iff the canonical embedding j: !1 ~ ubiG is not onto. Theorem. If G is nontrivial w.r.t. D, then there is an ultrafilter E on wand an elementary embedding IJ! : '!J~ ~ ub IG such that IJ! o j 1 = j where j1 and j are the canonical embeddings from !1 into !1~ and !!~IG respectively. Corollary. If I~ I is an infinite, nonmeasurable cardinal, then j: !1 ~ ub IG is not onto iff there exists an ultrafilter E on w and an elementary embedding

1J!: u; ~ ubiG such that 1J! o j 1 = j. (Received june 27, 1968.)

68T- E 19. A. R. D. MATHIAS, Seminar fur Logik, University of Bonn, 53 Bonn, West Germany. On a generalization of Ramsey's theorem.

Variable x,y ,z denote infinite subsets of w. Definition. A set P of subsets of £C is a Scott

family (SF) iff A x V y ~ x (x E P iff y (/. P ). Silver has proved that ZF 1- no SF is L; ~ (in the Cantor 1 space 2 w) and that ZF + there is a measurable cardinal 1- no L; 2 SF's. ""There is no SF"" is the Erdos-Rado property w ~(w)~, and is refutable in ZFC. Theorem, Con(ZFC +there is an inaccessible

cardinal) implies Con(ZF + DC + £C ~ (w)~) and Con(ZFC + no SF is ordinal definable from a real). The proof uses the following notion of forcing: a condition is a pair ( s,S) where s US ~ w, s is finite

and S infinite, and n E s Am E S ~n

y ~ x (y need not be in M[x]). Then y is P-generic over M. The lemma also yields easy proofs of Silver's results above. By an argument of Cohen and of Soare (Abstract 68T-478, these c/lfotiaiJ 15 (1968), 644) and jensen's observation that P does not collapse cardinals, any two disjoint infinite sub­ sets of an x P-generic over L are of incomparable L-degree. This last is still true if Sis restricted to lie in a certain ultrafilter on w. and P then satisfies c.c.c. (Received july 18, 1968.)

68T- E20. F. B. CANNONITO, University of California, Irvine, California 92664. A hierarchy of conjugacy problems. n n.n n n n ) Let Gn+l: (a,b,c,d; (a b 1 = (c d ) , n E Un+l , 4 ~ n ~ W• be the groups constructed in the author's Hierarchies of computable groups and the word problem, j. Symbolic Logic 31 (1966), 376-392. Then for each nit is possible to solve the conjugacy problem for G ntl by an aa+l function (Grzegorczyk's

931 hierarchy of primitive recursive functions, extended) but not by an c$a. function. This answers affirma­ tively a conjecture of the author made at the annual meeting of the Association for Symbolic Logic in Denver, 1965. The method of proof consists in extending Rabin's computable algebra techniques to the conjugacy problem sharpened to include the "&a." as in the author's paper, above. Thus, one shows a.+1 . that the G a.+l have c$ decidable conjugacy problem by an analysis of the preservation of decidability of the conjugacy problem in free products with amalgamation. The result then follows from the restricted c$a.+l decidability of the word problem of the Ga.+ 1• (Received June 26, 1968.)

68T-E21. DOV GABBAY, The Hebrew University of Jerusalem, Jerusalem, Israel. Ultra­ products of Kripke's models of intuitionistic logic. Preliminary report.

For terminology and notation see Kripke, "Semantic analysis of intuitionistic logic. I" in Formal systems and recursive functions, Amsterdam, 1965. A theory is a pair (A,B) of sets of sentences. The structure G = (G K R) is a model of (A,B) if the sentences of A are trueinG and the sentences of B are false in G. Many notions of classical model theory are carried over, in particular, ultraproducts, ultralimits, EC, ECL:l. Among the theorems proved are: Compactness theorem and upward and downward Skolem-Uiwenheim theorems. If~ and_!:! are elementary-equivalent, then~ is embeddable in an ultra power of_!:!. Robinson's consistency theorem holds for theories (A ,j'J) but fails for general theories. (Received July 1, 1968.) (Author introduced by Professor Azriel Levy.)

68T-E22. J. R. BUCHI and K. J. DANHOF, Purdue University, Lafayette, Indiana 47907. A strong .form of Beth's definability theorem.

A theory A(p,cr) is complete relative (categorical relative) p if any two models (D,R,S1 ) and

(D,R,S2 ) of A are elementarily equivalent (isomorphic). Using the definability theorems of Svenonius and Beth, one can show Theorem 1. Let A(p,cr) be a PC ll which is complete relative p. If C(p,cr) is an elementary concept which, in all models of A, is invariant under p-automorphisms, then, in A, C is elementarily definable from p. Note that this becomes Beth's (Svenonius') theorem if" complete relative p" is replaced by the stronger assumption" categorical relative p" (complete). (Received july 29, 1968.)

68T-E23. J. R. BUCHI, Purdue University, Lafayette, Indiana 47907. Affine definability of affine invariants of Euclidean geometry.

Let r(II,J.) be an elementary concept of Euclidean geometry (of n-dimensions). It is easy to show: Lemma. The proposition" r is invariant under all affine transformations" is elementary, i.e. expressible by an elementary sentence I( II, J.). Corollary. If r is invariant under affine transfor­ mations in some Euclidean space (Sl'lll'J.l)' then the same holds for any elementarily equivalent

space (S2 ,11 2,J. 2). Using the definability theorem of Svenonius (or Theorem 1 of the preceding abstract [Abstract68T-E22]) one obtains, Theorem. Let R be a relation on the real Euclidean space En = (S,C,J.) which is elementarily definable from C and J.. If R is invariant under affine transfor­ mations of En, then R admits an elementary affine definition (i.e. one in terms of C alone). Similar results hold for other pairs of geometries in Klein's hierarchy. (Received July 29, 1968.)

932 68T-E24, R, I. SOARE, University of Illinois, Chicago, Illinois 60680, and 147 Chestnut Street, Apartment E-11, Ithaca, New York 14850. Degrees of subsets.

In Abstract 656-37, these ci/oticeiJ 15 (1968), 513, the author announced the existence of an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are infinite sets (e.g. sets of minimal degree) with no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. Theorem. Every infinite set of natural numbers contains a nonrecursive infinite subset which is either of higher or lower (Turing) degree, The proof is direct and straightforward. (Received July 22., 1968,)

68T-E25. C. W. LEININGER, State University of New York, Cortland, New York 13045. Concerning som·2 proposals for quantum logic.

The proposal made in 1936 by Birkhoff and von Neumann that an orthocomplemented lattice has the structure appropriate for a propositional algebra with quantum-theoretical significance is well known. A m:>re recent proposal is that of a partial Boolean algebra suggested by Kochen and Specker (Logical structures arising in quantum theory, Symposium on the Theory of Models, Amsterdam, 1965). A partial Boolean algebra B is said to be complete if each Boolean subalgebra

of B is complete. Theorem. If (M, V, 1. 0, 1) is a complete partial Boolean algebra, then there is an extension U of the partial function V such that (M, U, 1, 0, 1) is an orthocomplemented lattice. (Received August 13, 1968.)

68T- E26. MANUEL LERMAN, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Some finite nondistributive lattices and the degrees of unsolvability. ll.

Let Ln' n > 2, denote the lattice of n + 2 elements O,l,a1, ... ,an such that 0 < ai < 1 for = 1,2, ... ,n, and i,j E (1,2, ... ,n} antl iF j implies ai and ai are incomparable. Theorem. If n - 1 = pk, p a prime number, k > 0, then Ln is embeddable as an initial segment of the degrees of unsolvability. This theorem extends the results announced in Abstract 68T-E5, these cifoticei] 15 (1968), 804. (Received August 15, 1968.)

68T-E27. BARUCH GERSHUNl, Frans van Mierisstraat 37h, Amsterdam, Nederland. The three kinds of totalities.

In order to arrive at a better understanding and a more justified judgement over the theory of the author, it is indispensable to notice the following: There are three kinds of totalities: plural classes, singular classes, and sets. They are written respectively in the form a,b,c, ... ; (a,b,c, ... ), and (a,b,c, ••• }. The plural classes are plurals, the singular classes and the sets are singulars. The plural classes are n:>t subject to any restriction apart from the possibility of writing them in the formal theory concerned. But when they are proper plural classes, they cannot be elements of any totality. Only singulars may be elements of a totality. These are, besides the singular classes and sets, the individuals. But an individual may be also grasped as an improper plural class, having only one unique element, namely itself. The singular classes obey only to the analogue of the Neumann- Bernays-Quine-axiom of comprehension :3: MV x(x E M"" sing(x)AH(x)). (S. Schmidt,

933 Mengenlahre. I, p. 33.) Here the predicate sing(x) means that xis a singular. H(x) means an arbitr­ ary predicate writable in the corresponding formal system. The (abstract) sets obey to the set-axioms of von Neumann or Bernays. They are not grasped now as systems and are not associated with functions. Because a singular class contains also itself as element, the Russell-class is emp~y. (Received August 15, 1968.)

68T-E28. C. C. CHANG andY. N. MOSCHOVAKIS, University of California, Los Angeles, California 90024. On E~ -relations on special models.

Let .l be a first-order language with identity, let P 1 .... ,P m be finitary predicate or function symbols not occurring in .l, and let cp(P 1, .•. ,P m;x1, .•• ,xn) be a formula of .l U ( P 1 ••. P m} in the free variables xl'···•xn. Theorem. Th·~re is a countable primitive recursive set (crk(xl' .•. ,xn): k E w} of

formulas of .l in the free variables x1, •.• ,xn (containing only the symbols jointly in tp and in .l and the identity symbol) such that: (i) for every k, 1- tp- crk, and (ii) on every special model U for .l,

U I= 1fx1 ••• xn(:!IP 1 ••• Pmlfl- "kcrk). Some corollaries: Corollary 1. Let T be a theory of .e. Then T U (II'} is consistent iff T U (<\:: k E w} is consistent. Corollary 2. In particular, given any

sentence tp of .l, there is a primitive recursive set (crk : k E w} of pure identity sentences such that IP has a model iff (crk: k E w} has a model. The theorem is motivated by a result of Moschovakis (On the Suslin-Kleene theorem, to appear) identifying n~ and inductively defined relations on countable

models. There are analogs of these results where tp is replaced by a set ~p 1 •••• ,P m;x1, .•• ,xn) of formulas. These results help explain why many model-theoretical results on pseudoelementary classes are true. See also the following abstract for an application. (Received August 26, 1968.)

6BT-E29. C. C. CHANG, University of California, Los Angeles, California 90024. A generali­ zation of the Craig interpolation theorem.

Let .l be a first-order language with identity, and let cp(S;P 1, .•• ,P m;x1 , .•• ,xn) and ljl(S;P1, ... ,Pm;xp ... ,xnl be formulas of .l U (S,P1, ... ,PmJ in the free va:dables x 1, ... ,xn. LetT be a theory of .e. The Craig interpolation theorem is the equivalence of: (i) T I=V P 1 ... Pm 1fx1 ... ~ (:B:S!p- 1fSijl); (ii) there is a formula 6(P1 , ... ,Pm;xp ... •xnl of .l U (P1 ... P mJ such that T I= 1f P 1 ... P m1f xl ... Xn [(:B:S!p- 6) " (a - 1fSijl) J. Kueker has obtained (in his thesis, unpublished) an

improvement of the above where 1fx1 ... xn is replaced in both (i) and (ii) by :B:x 1 ... xj1fxj+l ... xn• Using the result of the preceding abstract (Abstract 68T-E28), the following is proved: Theorem. Let .l,cp, 1/;, T be as above, and let Q be any sequence of quantifiers on the variables Pl'"''Pm; x 1, ... ,xn (in any order) provided that all set variables are quantified universally. Then (i) and (ii) are still equivalent with Q replacing 1fP1 ... P m 1fx 1 ... xn. Actually, the method of the above abstract gives a mt:ch better idea of the interpolant a, even in the original Craig's theorem. Some conse­ quences of the original Craig's theorem, e.g. Beth's theorem, have analogs with respect to the new theorem. (Received August 26, 1968.)

6BT-E30. RICHARD LAVER, University of California, Berkeley, California 94720, Fra1sse's conjecture.

An order type is scattered if it does not embed 71, the ordering of the rationals. Let M be the class of all order types which can be partitioned into countably many scattered types; in Erdos'

934 notation, M = [cp: lfl ~ (1J)~}. Theorem. Under the relation of embeddability, M is better-quasi­ ordered (terminology of Nash- Williams, Proc. Cambridge Philos. Soc., 1965 ). Corollary. There is no strictly decreasing infinite sequences of members of M, and every totally unordered subset of M is finite. In the case of scattered types, this was conjectured by Fra'isse (C.R. Acad. Sci., 1948). Methorls and results of Nash-Williams were used in the proof. (Received August!, 1968.)

68T-E31. R. B. JE!'-:SEN, Seminar fur Logik, University of Bonn, Lennestrasse 33a,

53 Bonn, West Germany. Souslin's hypothesis is incompatible w;th V = L.

I. The statement "If V is the constructible closure of a subset of w1 , then SH fails" is a theorem of ZF. Define a f3 Souslin tree to be a well-founded tree of cardinality 8, all of whose chains

and antichains have lesser cardinality. Write R({3) to mean: f3;;; w 1\ f3 v ~ 8 for v < f3. The statement "If R({3) and Vis the constructible closure of a subset of .8+, then there is a {3+ Souslin tree" is a theorem of ZF; as is "If R( {3) and V =Lit A], where A c {3+ and 1-1 is a normal measure on some x, then there is a {3+ Souslin tree." (The first of these generalizations was noticed independently by Drake.) II. Let M be a countable transitive model of ZF + GCH. Then there is a Cohen extension N such that cardinals are absolute in N, N contains no new countable subsets of M, and no So•Jslin tree of M retains the Souslin property in N. (Solovay and Tennenbaum have already produced a Cohen ex­

tension in which SH--but also 2w = w2 --holds. The question whether SH is consistent with GCH is still open.) (Author introduced by Dr. Gisbert Hasenjaeger.) (Received August 20, 1968.)

68T-E32. C. G. JOCKUSCH, JR., University of Illinois, Urbana, Illinois 61801. Degrees of h.h.i. sets.

A Turing degree~ is called h.h.i. if some hyperhyperimmune set has degree~· It is shown

that: ( 1) If~ ~~and~ is h.h.i., then_!: is also h.h.i. (2) Each h.h.i. degree~ satisfies~·?_ ~·.

(3) Each degree! such that~~~· is h.h.i. (4) There exist h.h.i. degrees!. satisfying a" = 0". (Received August 29,!968.)

68T-E33. JAMES BAUMGARTNER, University of California, Berkelely, California 94620. _9n the cardinality of dense subsets of linear orderings. I.

Theorem 1. Let A be a cardinal. If A < cf yand y < cf2A for all cardinals y with A< y ~ 2);_, then there is a linear ordering of carrlinality 2A with a dense subordering of cardinality A. Let r 0 be the order type of the set of functions f : 0-2 ordered lexicographically. Theorem 1 is a special

case of the following theorem. Theorem 2. For any cardinals x and 0, if Yo is either (a) the least cardinal y such that there is a subtype of ro of cardinality x having a dense subtype of cardinality y or (b) the least cardinal y such that there is a set X ,;:; P( 0) such that X = x and card(x n a: x E X and a< 0} = y, then either 0<::: cf Yo or Yo!::: cf x. (Received August 30, 1968.)

68T-E34. WILLIAM MITCHELL, University of California, Berkeley, California 94620.

_9n the c~dinality of dense subsets of linear orderings. II.

Theorem 1. If ZF is consistent, so is ZF t AC together with "there is no linear ordering of

cardin~lity 21:\ 1 with a dense subset of cardinality l:\ 1 ." This follows from the following general

935 result: Theorem 2. Let M be a countable standard monel of ZF t GCH and let X and x be cardinals of M such that in M, X is regular, 2~ = X cf X> X and X < x. Then for any cardinal {3 of M, if N is the result of adjoining a sequence of length f3 of generic subsets of x, then the following statement holds

in !'\: If X .;; P(x), X> x + and card [x n a: x E X and a< x} = y, then X '§ y if cf y > x and X'§ y+

if cf y '§ x. This result partially limits the possible extensions of Baumgartner's theorem in the preceding abstract (Abstract 68T- E33 ). Theorem 1 follows from Theorem 2 with

X= ~ 0 • f3 = ~"'r' and X= ~ 1 . (Received August 30, 1968.)

Statistics and Probability

68T-F4. RICHARD SHACHTMAN, University of Maryland, College Park, Maryland 20742, Sample paths of bounded variation.

Let x = [x(t) : t E T} be a random function on a probability space rl. Necessary and sufficient conditions for the sample paths of x to be of bounded variation on Tare given in terms of the distribu­ tion of the increments x(t) - x(s), s < t, when the latter are independent. Under these conditions the (Stieltjes) stochastic integral may be defined pointwise via the paths. (Received May 16, 1968.)

68T-F5. ARUNAVA MUKHERJEA, Eastern Michigan University, Ypsilanti, Michigan 48197. A Stone-Weirstrass type theorem for random functions.

In this note we consider only real random functions indexed in T, a compact set in a linear topological Hausdorff space, on a probability measure space (rl,!:,

68T-F6. JACK KIEFER, Cornell University, Ithaca, New York 14850. The deviations in Skorokhod-Strassen approximation.

V. Strassen used A. V. Skorokhod' s representation of sums of independent random variables in terms of Brownian motion, and Martingale extensions thereof, in the proof of his strong invariance principles and applications thereof. In particular, Strassen showed [Proc. Fifth Berkeley Sympos., Vol. II (Part I), pp. 315-343] that if the univariate d.f. F has first, second, and fourth moments 0, I, and {3 < oo, respectively, so that there is a probability space on which are defined a standard Brownian motion [~(t), t;;; 0} and a sequence of nonnegative i.i.d. Skorokhod random variables [Ti, i > 0} such that [~(:0~+l 1i)- ~(:0~ Ti)' n;;; 0} are i.i.d. with d.f.F, then (Strassen, Theorem 1.5) ~(:0~ Ti)- ~(n) = O([n(log n)2 (log log n)] 114) wp las n ~ oo. He asked whether replacing 0 by o implies

936 that F is Gaussian. This is answered affirmatively in the Theorem. If f3 > 0, then lim sup + [ ~(~nl T.)- ~(n)]/[2{3n(log n)2log log n] l/4 = 1 wp 1. Analogous results hold for n- oo - 1 cases where ~(L~ Ti) is not a sum ::>f independent random variables. (Received August 23, 1968.)

68T-F7. ALI KYRALA, Arizona State University, Tem')e, Arizona 85281. A generalization of the Gaussian distribution.

By appropriate limiting processes on the equations of a discrete Markov chain, one obtains the integrocHfferential equation for the probability density p(x,t) of occupying location (state) x at

timet. c\p(x,t) = p(x,tlJ:x'00(b(x,y)-b(y,x))dy + J~b(x,y)(p(y,t)-p(x,t))dy. If p(x,t) is analytic in x and only transitions across a small number of states are allowed, one has the differential equation

o t p(x,t) = ~ ~=O Ako~p(x,t) where the coefficients are A 0 = J~~~ (b(x,y)-b(y,x))dy and k:Ak = ;~~~ b(x,y)(y-x)kdy. From this time rates of change of the moments Mk of the distribution corresp"nd- "oo n n k k J oo k ing to p(x,t) are dtMk = LJn=O (-1) (ox(x An)) where Mk = (x ) = _00 x p(x,t)dx. If Ak = 0 fork f 2 one has the (time-dependent) Gaussian distribution and thus the cases not so restricted form generali­

zations of the Gaussian, For M1 = 0, sgn(dtM2 ) can be used as a criterion to distinguish between distributions tending to spread out or become concentrated with time. In particular, the Gaussian should be replaced by a distribution satisfying a higher (than second) order differential equation in all physical applications involving contracting systems (e.g. galaxies) rather than expanding systems. In most applications it will be unnecessary to include x-derivatives of very high order. (Received August 26, 1968.)

68T-F8. K. S. MILLER, Riverside Research Institute, New York, New York 10027. Non- stationary autoregressive processes.

Let Ryt = ut be a stochastic difference equation. Various relations between the input and output covariances and spectral densities are deduced under the hypotheses that R is time-dependent and that ut is a member of a nonstationary random process. In particular, we express the (generally nonstationary) covariance function of the Yt process as an integral involving the spectral density of the ut process and the system function H(;\,t), where H(;\,t) is defined as the ratio of Yt tout evaluated at ut = eitA. (Received August 23, 1968,)

Topology

68T-Gl2. C. J, MOZZOCHI, University of Connecticut, Storrs, Connecticut. On symmetric generalized uniform spaces. I,

Theorem 1. Every p-correct (Abstract 68T-2, these c#(;tiai) 15 (1968), 188) and every p-correct of degree n (Abstract 68T -161, ibid., 34 7) symmetric generalized uniform space (Abstract 67T-612, these c#(;ficei) 14 (1967), 839) has an open base. Theorem 2. Every proximity class of symmetric generalized uniform spaces has a maximum element and a minimum element. Theorem 3, Let (X,'l.<) be a symmetric generalized uniform space. If (x,y) E V for every V in 'l.

937 Theorem 5. Let (X,'!.<) be a symmetric generalized uniform space. Let R = n ( U JU in'!.<}. Then R is an equivalence relation on X. X/ R is an upper semicontinuous decomposition of (X,.1('1J)), and the projection P from X onto X/ R is a closed map where .1('1.<) is the topology on X and X/ R has the quotient topology relative to P. Problem 1. Does every symmetric generalized uniform space have an open base? Problem 2. Characterize the totally bounded spaces in a proximity class of symmetric generalized uniform spaces ( cf, Abstract 68T-2). (Received March 13, 1968.)

68T-Gl3. P. F. LAM, Wesleyan University, Mid

Let X be a uniform space. A transformation semigroup (X,S) is said to be expansive if there is a uniform index a. such that if x F y then there is an s = s(x,y) E S such that (xs,ys) ~a.. Theorem 1. Let X be a locally compact and a-compact uniform space. lf there exists a separable topological semigroup S such that (X,S) is an expansive transformation semigroup, then X is metrizable and separable. Theorem 2. There is no expansive self-homeomorphism on any 2-manifold whose boundary is compact and nonempty. Theorem 3. There is no expansive self-homeomorphism on any compact graph. Theorem ·4. Let X be a noncountable a-compact metric space. Let f be an expansive self­ homeomorphism on X such that every orbit has compact closure. Then the set of nonwandering points of X is infinite. Corollary. There is no orientation preserving expansive self-homeomorphism on a 2-sphere with exactly one fixed point. (Received June 25, 1968.)

68T-Gl4. K. L. SINGH and S. P. SINGH, Memorial University of Newfoundland, St. John's, Newfoundland, Canada. Contractive mappings in uniform spaces.

Theorem 1. Let f be a contractive mapping of a totally bounded Hausdorff uniform space (X,u) into itself with respect to an open ample base {1 for the uniformity u. Iff is also {J-contractive at x, then there exists a periodic point for f, i.e. there exists a positive integer k such that fk{x) = x. Theorem 2. Let f be a {j-contractive mapping of a totally bounded Hausdorff uniform space (X,u) into itself with respect to an open ample base {1. Then the set of periodic points is finite; and this set is closed. (Received July 1, 1968.)

68T-Gl5.J. R. BOONE, Texas Christian University, Fort Worth, Texas 76129. Some characteri­ zations of paracompactness in k- spaces.

A family K* = (Ka. : a. E A} in a space X is said to be a k-collection (RJinnow, Math. Scand. 10 (1962), 217-220) provided: (i) K* is a covering of X, and (ii) for each closed set F c X, F n Ka. E K* for each a. EA. A :f;lmily F* = (F {1 : {1 E B} in a space X with a k-collection K* is said to be K*-finite if for each a. E A, Ka. n F {1 F fJ for at most finitely many {1 E B. Lemma. If a space X has the weak topology in the sense of Whitehead with respect to a k-collection K* in X, then every K*-finite closed family in X is locally finite. Applications of this lemma to paracompactness and the weak topology in the sense of Morita with respect to closed coverings are given. These notions describe the general setting in which the paracompact property in k-spaces is characterized. Four generalizations of paracompactness are introduced. As an application of these concepts, recall the sequentially

938 mesocompact spaces which were defined previously [Abstract 68T-100, these cNotiaiJ 15 (1968), 220J. Theorem. A developable space is metrizable if and only if it is sequentially masocompact. (Received june 26, 1968.)

68T-Gl6. TUDOR GANEA, University of Washington and Universite de Paris, Paris 8, France. Cotriples and Lusternik-Schnirelmann category.

Define a sequence of fibrations pK : EK- B, K ;;; 0, as follows: E0 is the space of based paths in B, and EKtl results by converting into a fibre map the extension EK U CF K- B of pK which maps the cone over the fibre of pK to the base-point. A natural cross-section jK: EK- EKEK may be defined, and (EK,pK,jK) forms a cotriple in the sense of Eilenberg-Moore [Illinois j. Math. 9 (1965), 3 81-398]. The strong Lusternik-Schnirelmann category, Cat B, of B is the least integer k ;;; 0 such that B has the homotopy type of a CW -complex, which may be covered by K t 1 self-contractible sub­ complexes. Theorem. Suppose B is (n - 1 )-connected and let g : B - EK be a cross-section such that

(B,g) is a coalgebra over (EK,pK,jK); then Cat B ;§ k provided dim B:!! (3K t l)n- 5. The converse holds without connectivity or dimension assumptions: If Cat B ;:s k, then (B,g) is a coalgebra over (EK,pK,jK) for some cross-section g. In particular, an (n- I)-connected co-H-space of dimension ;:s 4n - 5 has the (primitive) homotopy type of a suspension if and only if it is a coalgebra over I: n. (Received july 3, 1968.)

68T-Gl7.C. W. MASTIN, Texas Christian University, Fort Worth, Texas 76109. Productively pseudo-compact spaces. Preliminary report.

All spaces considered are infinite Tychonoff topological spaces. A space X is productively

£Seudo-compact if X X Y is pseudo-compact for every pseudo-compact space Y. Theorem. If X is pseudo-compact and has the weak topology with respect to a collection of productively pseudo­ compact subspaces, then X is productively pseudo-compact. A space X is k-countable if for each x EX, there is a sequence (Ui} of open neighborhoods of x such that cl(nui) is compact. Theorem. lf X is a pseudo-compact k-countable space, then X is a k-space. Corollary. If X is a pseudo-compact k-countable space, then X is productively pseudo-compact. The product of two k-countable spaces is also a k-countable space. Theorem. If X is pseudo-compact and Lindelof at infinity, then X is productively pseudo-compact. These results are extensions on some results of H. Tamano (J. Math. Kyoto Univ. 33 (1960), 225-230) and I. Glicksberg (Trans. Amer. Math. Soc. 90 (1959), 369-382). (Received july 3, 1968.)

68T-Gl8. j. S. BIRMAN, Courant Institute, New York University, New York, New York 10012. Nonconjugate braids can define isomorphic knots.

It is well known that braids which are conjugate in the braid group 13n define isomorphic knots or linkages when their ends are identified to form a closed braid. It has been shown by F. A. Garside (Ph.D. thesis, Corpus Christi College, Oxford, 1965) that the converse is not true for linkages. We extend this result to knots, and also show that certain other quantities which had been conjectured to

be knot invariants are in fact not invariants. The braids aia2a3 and a 1 a~a3 and a1 ~a:/ in B4 all define the trefoil knot. It is proved that they are not all conjugate in B4 by using a representation

939 for the n-string braid group in terms of (n - 1) X (n - 1) matrices (Magnus and Pelluso, On knot groups, Comm. Pure Appl. Math. 20 (1967), 749-770) in which the traces and/or determinants of the corresponding matrices differ. If 1/J is a braid, andqJ its matrix in this representation, and Ll(v) the Alexander polynomial of the knot associated with ¢,then (1 - v-n )LI(v) = ( 1 - v-1 )det(~j- I). It then follows that if'¢1 and ~2 define the same knot, then det(¢'1 - I) = det(~2 - I). It had been conjectured that trace f and det ~were invariants of knot type, but both of these are now seen to be false. (Received july 8, 1968.)

68T-Gl9. SHAIR AHMAD, Oklahoma State University, Stillwater, Oklahoma 74074. Dynamical systems of characteristic 0+.

Let R be the reals, (X,R,!T) a dynamical system on the phase space X. Definition. The flow (X,R,!T) is said to have characteristic 0+ if every closed positively invariant set is positively stable in Ura's sense (see T. Ura, Sur le courant exterieur a une region invariante, Funkcial. Ekvac. 9 (1966), 171-179). Theorem 1. (X,R,!T) of char. 0+ and X locally compact (Hausdorff) implies that Vx E X, the positive limit set L + (x) is a positively minimal compact set. Theorem 2. If (X,R,!T) is of char. o+, X is locally compact and M is a compact invariant set which is a p·:>Sitive attractor, then M is globally+ asymptotically stable. The:Jrem 3. If X= R2 in Theorem 1, we have the following classification. Case I. The set of critical p:>ints K = fJ and the flow is parallelizable. Case 2. If K is compact, then either K is a singleton and a Poincare center, or K is globally + asymptotically stable. Case 3. Knot compact implies that K = R2, or the follooving hold. (a) R2 - K is unbounded. (b) K is+ asymptotically stable. (c) K and the region of positive attraction A~(K) of K both have a countable number of components. Components of A+(K) are homeomorphic to R2 • (d) Vx EX,

L+(x) = fj, or L+(x) consists of a single critical point. Further, Vx E R2 - A+(K), L+(x) = fJ, and Vx E R2 - K, L- (x) = fJ. ( Re~eived july 1, 1968.)

68T-G20. G. P. WELLER, University of Chicago, Chicago, Illinois 60637. Stronger results on locally flat imbeddings of topological manifolds.

The following results generalize and strengthen those announced by the author in Abstract 68T-502, these cN'oticei) 15 (1968), 633. The proofs are similar but more intricate. Let Mn be a compact, orientable topological manifold with boundary M, n !!; 2, and let Q q be a simply connected topological manifold with boundary Q. If 2q > 3(n + 1 ), M is 2n - q connected and Q is 2n - q ~ I connected, then any map f: (M,M)- (Q,Q), such that fiM is a locally flat imbedding, is homotopic relative to M to a proper locally flat imbedding g : M - Q. From this result it easily follows that if

2q > 3(n + 1) + 1, M is closed and 2n - q + I connected and Q is 2n - q t 2 connected, then any two locally flat imbeddings f,g : M- Q which are hom:Jtopic are concorrlant; that is, there is a locally flat proper imbed

68T-G21. R. j. TONDRA, Iowa State University, Ames, Iowa 50010. The domain rank of an n-sphere and ann-cell.

For n !!; 2, let S n-l, En, and Hn denote any space which is homeomorphic to the subspace of Euclidean n-space Rn defined by (x E Rnl JlxJI = 1}. (x E Rnj JlxJI ;; 1}. and (x E Rnj xn "0, where

940 x = (x1 , ... ,Xu)} respectively. Theorem. The domain rank (see Abstract 68T-G5, these cJiotic£iJ 15 (1968), 810, for definition) of Sn, Hn, and En, n;!; 2, is 1,2 and 3 respectively. (Received August 13,

1968.)

68T-G22. C. C. ALEXANDER, University of Virginia, Charlottesville, Virginia. Quasi­ developable spaces. II. Preliminary report.

Quasi-developable spaces are defined in [Alexander, Abstract 68T -G4, these cJiotic£iJ 15 (1968), 8101 Pseudo-open and 1T-maps are defined in [Arhangel'skii, Russian Math. Surveys 21 (1966), ll5J. Definition. A 11-map f from a metric space X into a topological space Y is an Int 1T-map iff for each -1 £ > 0 and each pointy in Y, f[G((y)] is a neighborhood of y, where G£(y) = (x E Xld(x,f (y)) < £}.

Theorem 1. A T 0-space is quasi-developable iff it is an Int 11-image of a metric space. Theorem 2. A T0-space is strongly quasi-developable iff it is a pseudo-open 1T-image of a metric space. Theorem 3, A T 0-space is point-finite strongly quasi-developable iff it is a pseudo-open compact image of a metric space, Each of the above theorems may be restated for zero-dimensional complete metric spaces. Theorem 4. A collectionwise normal T 0-space is a locally compact (resp., locally separable) metric space iff it is a pseudo-open compact image of a locally compact (resp,, locally separable) metric space. (Received August 19, 1968.)

68T-G23. KONDAGUNTA SUNDA RESAN, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. Smooth Banach spaces.

The theorem stated below provides the best possible characterization of Hilbert spaces up to isomorphism in terms of smoothness of the norm, thus in particular subsuming a characterization stated in Rao [MR No. 2131, Vol. 35], Definition. If B is a smooth Banach space and if llxll = 1, let G(x) denote the Gateux derivative of the norm of B at x. Theorem. If B is a smooth Banach space and if for some xES the norms of Band B* (the dual of B) are twice weakly, direction­ ally differentiable at x and G(x) respectively, then B is isomorphic to a Hilbert space. In the paper

cited above the author assumes the norms of B and B* are twice Fr~khet differentiable. Such spaces form a small subclass of the class of Banach spaces satisfying the hypothesis in the theorem presented here. (Received August 8, 1968.)

68T-G24. A. D. BERARD, JR., AF Institute of Technology, /.FIT-SEC, Wright-Patterson AFB, Ohio 45433. Characterizations of metric spaces by the use of their midsets.

Let (X,p) be a metric space. We will say that (X,p) has the unique [double] midp:>int property provided for each pair of distinct points x, y EX, (q E Xlp(x,q) = p(y,q)} consists of exactly one

point [consists of exactly two distinct points], Theo~. If (X,p) is connected and has the unique midpoint property, then X is homeomorphic to an interval of the reals. Theorem. If (X,p) is complete, convex, and has the double midpoint property, then X is a simple closed curve. Theorem. If (X,p) is diameterized and has the property that for each pair of distinct points x, y E X, (q E X lp(x,q) = p(y,q)} is a complete convex metric space with the double midpoint property, then X is a 2-sphere. (Received August 13, 1968.)

941 68T-G25. LAWRENCE FEARNLEY, Brigham Young University, Provo, Utah 84601. The pseudo-circle is not homogeneous.

Two well-known problems concerning circularly chainable continua are those raised by R. H. Bing regarding the uniqueness and homogeneity questions for pseudo-circles. ln a recent paper [Abstract 68T-342, these cJ.foticei) 15 (1968), 401] the author has established that the pseudo-circle is topologically unique. In this present paper an extension of the techniques used in the earlier paper is used in showing that the pseudo-circle is not homogeneous. Specifically, let x be called a point of left-normality of a pseudo-circle M if there is a sequence tPi} of circular chains and an associated sequence tfi} of cyclic r-patterns such that xis contained in the link of Pi with subscript zero and fi is left-normal, i = 1,2,3, •••• Using the unboundedness of the collection of lengths of the canonical linear representations of cyclic r-patterns in a defining system for a pseudo-circle and results on compositions of non-left-normal cyclic r-patterns, it is proved that some but not all points of a pseudo-circle are points of left-normality. It is similarly shown that n-pseudo-circles and other hereditarily indecomposable non-snake-like circularly chainable continua are not homogeneous. (Received August 26, 1968.)

68T-G.Z6. R. L. FINNEY, University of Illinois, Urbana, Illinois. Uniform limits of compact cell-like ~aps.

S. Armentrout has defined a subspace B of a topological space X to have proparty UV 00 in X if each neighborhood U of B contains a neighborhood V of B that is contractible in U. Theorem l. Let f be a uniform limit of homeomorphisms of a locally compact metric space X onto itself. Let B c X be compact. Then B has property UV 00 in X if and only if f- 1B has property UV 00 in X. If X is a manifold the proof is easily modified to show that B is cellular if and only if f- 1B is cellular (one chooses V to be a cell). Thus if X is a manifold f is seen to be cellular without assuming the com­ pactness of pre-images of points. A space is cell-like if it is homeomorphic to a cellular subspace of some manifold. Let ANR mean "neighborhood retract in Rn". R. C. Lacher has shown that a compact, finite-dimensional metric space is cell-like if and only if it can be embedded in an AN R so as to have property UV 00 there. Thus, if X is an ANR, then B is cell-like if and only if C 1B is cell-like. A theorem of Lacher on compact maps (pre-images of compact sets are compact) of AN R' s allows the proof of Theorem 1 to be changed to give: Theorem 2. Let X and Y be AN R' s and let f be a uniform limit of compact, cell-like maps of X onto Y. Th·en f is compact, and a subspace B of Y is cell-like if and only if f- 1B is cell-like. (Received August 26, 1968.)

Miscellaneous Fields

68T-Hl9. ERIC MENDELSOHN, McGill University, Montreal, Quebec, Canada. Graph with given semigroup and V-A products.

V-A (Ventex-Arrow) products are defined and used to show that there exist graphs with given semigroup and the following graph theoretical properties (a) without loops, (b) acyclic, (c) directed

942 and having no underlying undirected (forget the orientation) cycles shorter than k, for any k ;;; 3, (d) undirected and having no cycles shorter thank, k ;;; 7, (e) having given chromatic number u, u ;;; u, u a cardinal. (c)- (e) are best possible. (Received August 2, 1968.) (Author introduced by Professor j. Lambek.)

68T-H20. W. j. SAVITCH, University of California, Berkeley, California 94720. Deterministic simulat_ion of nondeterministic Turing machines. Preliminary report.

A special case of the following theorem is: Every context sensitive language is recognizable by a deterministic n2 tape bounded Turing machine, where n is the length of the input. The()rem. Let A be a set of words over a finite alphabet. Let L be a monotone increasing function from the natural numbers to themselves, such that L(m);;; log2 m. Suppose L(m) is computable, in dyadic notation, on an L(2n-l) tape bounded off line Turing machine, where n is the length of the input. If A is recog­ nizable by a nondeterministic L(n) tape bounded off line Turing machine, then A is recognizable by a deterministic (L(n))2 tape bounded off line Turing ma.chine, where n is the length of the input. Th'~ 2 proof uses a back tracking algorithm to simulate the L(n) nondeterministic machine on an(L(n)) deterministic machine. The conditions on L are satisfied by a large class of storage functions including log n, nand, more generally, all polynomials in log nand n. (Received August 8, 1968.) (Author introduced by ProfessorS. A. Cook.)

68T-H21. ALI KYRALA and j. W. KYRALA, Arizona State University, Tempe, Arizona 85281. Quantum differential equations from Markov chains.

For a Markov process characterized by a complex transition matrix U which transforms a complex state vector lj;- lj;+ according to lj;+ = Ulj;, a sufficient condition to ensure conservation of probability normalization lftl/ot = I = li!*IP is that the transition matrix be unitary. In the case of state vectors of a classical Hamiltonian operator H, it suffices to take U = n exp(il-IT/-1'1) where Tis the transition time. Then, approximately for small T, fl(lj;+ - lj;) = -iTH~·. and as T- 0 one has the Schrodin­ ger wave equation -fl<\1/; = iH~·. Mutatis mutandis it suffices in the relativistic case to take U n exp(iY • pa2cT/Il) where y = (y+,y) is the Clifford 4-vector and pis the 4-vector operator ifi(oct' -17) while Tis the proper-time of transition and a is the relativistic factor (l-(v/c)2 )·l/Z For small T 2 = - 2 one has approximately -l'loT!J; = ia cy • plj; but -fioTis the operator identified with ia E 0; hence the Dirac equation Eo!/; = cy • pi/J. Apparently both the latter and the Schrodinger wave equation correspond to instantaneous transitions exactly and to transitions of finite duration only approximately. If, unlike here, transitions between remote states are excluded, one obtains a different generalization of the

Dirac equation E 0 = cY • pg(R • p/fi) where g(O) =I and g is analytic. For g not even, this implies an asymmetrical distribution of p states for particles and antiparticles. R = (ct,R). (Received August 26, I 968.)

68T-H22. j. T. ROGERS, JR., Tulane University, New Orleans, Louisiana. The pseudo-circle is not homogeneous.

Theorem. The pseuclo-arc is the only homogeneous, hereditarily indecomposable, circularly chainable, nondegenerate, compact, metric continuum. Corollary. No pseudo-circle is homogeneous.

(Received August 26, 1968.)

943 68T-H23. C. E. AULL, Virginia Polytechnic Institute, Blacksburg, Virginia. Submetrization.

A Moore space is screenable iff it has a sigma disjoint base. A topological space with a sigma

disjoint base is metrizable iff it is perfectly T 4 . A Moore space is metacompact iff it has a sigma point-finite base. Heath's example of a metacompact, completely regular, nonnormal, nonsc reenable

T 1 Moore space has a sigma point-finite base without having a sigma disjoint base. A sigma point finite base of a hereditary Lindelllf space is countable. Hence Miscenko' s example of a hereditary

Lindel~f non regular T 2 space with a point-countable base does not have a sigma point finite base. (Received August 26, 1968.)

68T-H24. D. E. CAMERON, Virginia Polytechnic Institute, Blacksburg, Virginia. Maximally Lindelof spaces. Preliminary report.

Definition l. Let T and T' be two topologies on the set X. Then T is strictly stronger than T' if and only if T:::: T' and T t- T. Definition 2. A Lindelof topological space (X,T) is maximally Linclelof if and only if for any strictly stronger topology T', the topological space (X, T') is not Lindelof. Theorem 1. A LindeHif space is maximally Lindelof if and only if for every Lindelof subspace G (j T, the complement of G is not a Linrlelof subspace if and only if the Lindelof subsets are

precisely the closed sets. Definition 3. A topological space is a quasi-P space if and only if the

space is T 1 and every Gd is open. Theorem 2. Every maximally Lindelof space is qua si-P but not conversely. Theorem 3. Every Lindelof T 2 qua si-P space is maximally LindelOf, but such a space is not necessarily minimumly T 2 or minimumly qua si-P. Theorem 4. Every Lindelof subset of a maximally Lindelof space is maximally Lindelof. We observe that not every Lindelof space has a strictly stronger topology on the space which is maximally Lindelof; for example, the reals with the usual topology cannot have a strictly stronger, maximally LindelOf topology since every point is a Gd. (Received August 25, 1968.) (Author introduced by Professor C. E. Aull.)

68T- H25. j. A. WILLIAMSON, University of Colorado, Boulder, Colorado 80302. Expected time in the half-line.

Let (Xk} be a sequence of independent and identically distributed random variables. Set

Hx(W) =L::,0 I(Sn(W)~x} where Sn =L~=O Xk and X0 = O. Let H(x) = EX(H(w)). It follows from the 2 work of several that if E((XiJ ) < oo and if 0 < E(X 1) ~ + oo, then H(x) < oo for each x. We offer the following solution to the problem "when is H(x) < oo if E((Xi:J2) = oo." Proposition l. If there exists a - 2a C > 0, x0 > 0, ann a< 1 such that for all x;; x0, P (X1 > x} ;; C/x , and if E({X 1) ) < oo, then for each x, Hx(w) < oo with probability 1 and H(x) O, then H(x) < oo for each x if and only if E((Xi ll < oo. If there exists C > 0, x0 > 0, and a< 1 such that a - a - 2a for all x;; x0, P(X1 > x} ~ C/x , if H(O) < oo, and if E((X1) ) < oo, then E((X1 ) ) < oo. (Received August 27, 1968.)

68T-H26. j. C. CANTRELL, University of Georgia, Athens, Georgia 30601. Some locally flat embedding theorems in codimensions one and two. Preliminary report.

Theorem l. Let D be an n-cell in Rn and let E be an (n - 1 )-cell in Bd D. If D - E is locally flat in Rn and E is locally flat in both Bd D and Rn, then D is locally flat in R n. Theorem 2. Let D1

944 n and D2 be (n- 2)-cells in R such that D1 n Dz = Bd 0 1 n Bd 0 2 = E i.s an (n- 3)-cell that is locally flat in each of Bd D1 and Bd 0 2 , and suppose that D1 - E and D2 are locally flat in Rn. If D1 U TJ 2 is locally flat in the boundary of some (n- I )-cell Fin Rn and F - E is locally flat in Rn, then

D1 U Dz is locally flat in Rn. Theorem 3. Let M be a compact (n- I)-manifold that is topologically embedded in the interior of ann-manifold N, and suppose that M supports a combinatorial structure.

If, under some combinatorial triangulation of M, each closed simplex of M is locally flat inN, then M is locally flat inN. (Received July I, 1968.)

68T- H27. M. G. MI RDESHWA R, University of Alberta, Edmonton, Alberta, Canada, and KISI-IORE TI-IECKEDATI-1, Wilson College, Bombay, India. Boundedness in a topological space.

The paper generalizes Bourbaki's definition of boundedness to quasiuniform structures. This provides a definition of boundedness in a general topological space, since every topological space is quasi-uniformizable. Many properties of boundedness are proved; counterexamples are also given to prove some 'negative' properties. (Received July 5, 1968.)

68T-I-128. A. B. NETTO, University of Campinas, Campinas, Sao Paulo, Brazil. Fuzzy classes. Preliminary report.

In (Information and Control 8 (1965), 338-353), Zadeh has introduced the n•Jtion of a fuzzy set, which is defined as follows: Let X be a set of elements; a fuzzy set A in X is defined by a" mem­ bership function" fA, which associates with each element x E X a real number fA (x) E [O; I]; the value fA(x) of fA at x represents the "grade of membership" of x in A and if A is an "orrlinary" set, its membership function fA can take only the values 0 and 1. Hence, the theory of fuzzy sets is based on the usual set theory and such" sets" are real valued functions. In this work, starting from the notion of fuzzy class as a primitive one, we give an axiomatic theory 3' which contains the usual set theory as a proper part. The underlying logic of 3' is the first order predicate calculus with equality and the customary concepts of usual set theory are not employed. (Received July 11, 1968.) (Author introduced by Professor N.C. A. da Costa.)

68T-H2':l. R. M. RAPHAEL, McGill University, Montreal, Quebec, Canada. An algebraic closure for commutative regular rings with 1.

All rings are commutative with 1. Definition. S => R is an essential extension of R if I n R =

(0) => I = (O) for all ideals, l of S. An example shows that an essential extension of a regular ring need not be regular. Definition. S => R is an algebraic extension of R, if it is essential, and integral in the sense of Zarisky-Samuel. Proposition. An algebraic extension of a regular ring is regular.

_p~finition. An embedding m: R- Sis algebraic, if Sis an algebraic extension of m(R). Theorem 1. Let R be a regular ring. Then t.f.a.e. (1) every algebraic embedding with domain R is an isomor­ phism, (2) R is Baer, and every monic polynomial equation over R has a root in R, {3) R is Baer, and all of its quotient fields are algebraically closed. Examples show that the Baer condition cannot be dropped in (2) and (3). Definition. A regular ring satisfying the conditions of Theorem 1 is called algebraically closed. Theorem 2. Let R be a regular ring. Then R can be embedded algebraically

945 into an algebraically closed ring, which is unique up to isomorphism. Proposition. Every quotient ring of an algebraically closed ring R is algebraically closed if and only if R is completely reducible. (Received july 15, 1968.)

68T-H30. C. F. WIDGER, University of Washington, Seattle, Washington 98105. Multiplicative perturbations of infinitesimal generators of one-parameter semlgroup~. Preliminary rep::~rt.

Let A be the (infinitesimal) generator of a semigroup S(t) (of class C 0 ) on a Banach space X. Let B be a bounded operator on X. We seek conditions that BA generate a semigroup and formulae to rlesc ribe the new semigroup. Thea~ 1. Suppose X is a Hilbert space, and B is semipositive oo - rlefinite: B = J0 AdEA. (i) If B commutes with A, then BA (the closure of BA) generates the semi- oo - group T(t) = J0 S(At)dEA. (ii) If(*) range (A - BA) is dense in X for large A, the BA generates a semigroup. (iii) (G. Lumer, unpublished) If B is invertible, then BA generates a semigroup. For arbitrary Banach spaces, we have Theorem 2. Suppose S(t) is of class I-!(4> 1 ,~2 ), and a(B) lies in the domain of holomorphy of S(t). Then (i) if B commutes with A, BA generates the semigroup T(t) = Jr S(At)R{A - A)dA, where r is a rectifiable curve of index one enclosing the spe•;trum of B and lying in the domain of holomorphy of S(t); (ii) if condition (*)holds, then BA generates a semigroup. Th~~_:em 3. Suppose I!S(t)l! ~ 1 for all t, B is a hermitian in the sense of Vidav, and a( B) c R t (the strictly positive reals). If there is an integer k ~ 1 such that

O"k-l 00 I!R(a t i'T", AJI!kd'T" M, a constant, for all a E R+, then (i) under condition(*), BA generates J-oo ~ a semigroup, and (ii) if B commutes with A, that semigroup is of class (C 0 )u. (Received july 18, 1968.)

68T-H31. N. A. FRIEDMAN, Westfield College, University of London, London, N. W. 3., England, and D. S. ORNSTEII'\, Stanford University, Stanford, California. Partially mixing transformations.

Let (X,O,m) be a measure space with m(X) = 1 and let 'T" be an ergodic measure preserving transformation on X. If lim m{~A n B)= m(A)m(B) for each pair of measurable sets A and B, then n 'T" is mi.xing. Let us say 'T" is partially mixing if there exists a.> 0 such that lim infn m('T"nA n B)~ a.m{A)m(B) for each pair of measurable sets A and B. We!. give an example on the unit interval of a transformation 'T" that is partially mixing for a. = 1/8, but not mixing. This implies a negative answer to a conjecture of]. England and N. F. G. Martin, Bull. Amer. Math. Soc. 74 {1968), 5()5-507. (Received july 31, 1968.)

68T-H32. W. W. SAFFERN, 382 Wadsworth Avenue, New York, New York 1004. Rational approximation. III.

See Abstract 644-48, these ci/oticeiJ 14 (1967), 372; and Abstract 655-110, 15 (1968), 498, for terminology. Theorem. If X is a compact subset of the plane with dense, connected interior satis­ fying (1) the boundary of X has finitely many (n) components and (2) the inner boundary of X is denumerable, then (a) harmonic measure with respect to a point of the interior is a strongly dominant (Glicksberg, Trans. Amer. Math. Soc. 130 (1968)) representing measure for A(X), (b) A(X) is a hypo­ dirichlet algebra with n - 1 equal to the defect of Re A(X) in C R(o X), (c) the minimal boundary of A(X) is the boundary of X, (d) A(X) = B(X), (e) A(X) is a maximal uniformly closed subalgebra of

946 C(oX). (c) is of particular interest in relation to McCullough's theorem (Abstract 66T-22, these cNoticeiJ 15 (1968), 342). This result is definitive in a certain sense since there exist compact sub­ sets of the plane with dense, connected interior and satisfying (-1,2,- b,- c,d) and (1,-2,- d). ( R"'ceived August 6, 1968.)

68T-H33. HAIM GAIFMAN, Hebrew University, and University of California, Berkeley, California 94720. Two results concerning extensions of models of set theory.

Notation. ZFC = Zermelo-Fraenkel's s:=t theory with the axiom of choice. ZF t UW =the set theory obtained by adding a new 2-place predicate to ZF and an axiom stating that it is a wdl-ordering of the whole universe; the replacement schema includes also all formulas in which the new predicate

occurs. Theorem 1. If M is any model of ZFC and the order type of the ordinals of M is cofinal

with w, then there exists R ~;;; M X M such that (M,R) is a model of ZF t UW. The proof is by direct construction and does not resort to forcing methods. Theorem 2. If M is a countable model of ZF +there are arbitrary large inaccessibles, then there is an end-extension N of M such that N is a

model of ZF t V = L. (1\: is an end-extension of M if a E M, b E N and bENa imply b E M where EN is the £-relation of N.) (Received August 5, 1968.)

68T-H34. D. H. PETTEY, University of Missouri, Columbia, Missouri 65201. Mappings onto the plane. Preliminary report.

In a previous report (see Abstract 653-293, these cNoticeiJ 15 (1968), 165) the author announced a proof that if M 2 is a connected 2-manifold with boundary then every 1-1 mapping of M2 onto E2 is a homeomorphism. R. F. Dickman has recently extended this result to the following: If X is a locally connected generalized continuum having no local separating point, then every 1-1 mapping of X onto E 2 is a homeomorphism. In the present paper, use is made of Dickman's result in proving the more general Theorem l which follows. Theorem 2 is a further generalization. Theorem 1. If X is a locally compact, locally connected, connected topological space and f is a 1-1 mapping of X onto E 2 , then f is a homeomorphism. Theorem 2. If X is a locally compact, locally connected, connected topological space and f is a monotone mapping of X onto E2 , then f is a compact mapping. ( Re;:eived August 19, 1968.)

68T- H35. WITHDR.'\. WN.

68T- H3 S. RAPHAEL FiNKELSTEIN, Bowling Green State University, Bowling Green, Ohio, and HYMIE LO:-.IDON, McGill University, M·Jntreal 29, Quebec, Canada. Fibonacci and Lucas cubes.

Let F n be the nth term in the sequence of Fibonacci numbers defined by F nt2 = F nt I + F n, F 1

F 2 = 1, and let Ln be the nth term in the sequence of Lucas numbers defined by Lnt2 = Ln+ 1 + Ln• L 1 = 1, L 2 = 3. Theorem. The only Fibonacci numbers F n• n > 0, which are perfect cubes are F 1 = F 2 = 1 and F 6 = 8; the ::mly Lucas number Ln• n > 0, which is a perfect cube is L 1 = 1. (Received August 22, 1968.)

947 68T-H37. E. W. CHENEY, University of Texas, Austin, Texas 78712; C. R. H03BY and D. E. WU LB E RT, University of Washington, Seattle, Washington 981 05; P. D. MORRIS, Pennsylvania State University, University Park, Pennslyvania !6802; and F. SC HU RE R, Tee hnical University, Eindhoven, Netherlands. Unicity of the Fourier projection.

Let P be a linear projection of c211 (furnished with supremum norm:} onto the trigonometric

p::>lyno mials of degree :§ n. Let F n be the Fourier projection. If II P II = II F n II then P = F n. (Received August 28, 1968.}

S8T-H38. JOHN HAYS, University of Maine, Orono, Maine. Introduction to hyperboolean

~ystems. IV: Generative hyperbooleans; polynomial and rational forms; density and incompleteness.

Bibooleans form rings and quabooleans form (finite, transfinite} fields (Abstracts 648-202, 67T-575, 68T-42, these cNotiaiJ 14 (1967}, 691, 827; 15 (1968), 201), but (like booleans, unlike numbers} are nongenerative. Let U- ffi, for boolean EB!U AB = \EflAB nAB), in definition of bibool­ eans to produce (nonnumerical} generative biboolean rings (" biadbooleans" }; quaboolean rules are k k (explicitly} unchanged, but now imply generative fields (" quadbooleans" }: U B(Q} = kB(Q), n B(Q} = Bk(Qk}; kB(Q) = jB(Q}, Bk(Qk) = Bj(Qj} "'k = j. Theorem II. Quadboolean polynomial forms mani­ fest in rings, rational forms in fields. Theorem 12. "Fermat's Little Theorem" holds for quadbo,Jleans,

hence they can be expanded in cyclic unions of" basel fractions" (denominators: powers of atoms}. Lemma I. From (A,B)-< (C,D) "'nAD -< nac, we have (A,B) -< (UAC, UBD)-< (C,D). _!:-emma 2 ("Pythagorean paradox"}. There is an aleph-null set of quadbooleans in the same lattice, but not in the same chain as their components. Corollary. Noncyclic basal forms cannot be written

as" quadboolean fractions". Th~rem 13. The field of quadbooleans is dense but incomplete. (We shall next define limits and develop hyperboolean analysis.} (Received August 29, !968.}

68T- H39. J. M. WORRELL, JR., Sandia Laboratories, Albuquerq.1e, New Mexico 87115. On conrlitions of absolute _9- refinability.

The topic of paracompactness-like properties has a content pertaining to embedding invariances which can be approached analogously to the way Cech approached topological completeness. For formulating certain of these invariances for the Tychonoff cases, regular T 0 locally F rechet compact embeddings may replace what may be taken as definitional T 2 -bicompactifications. Within the frame­ work of compactification and appropriate relaxations of local paracompactness, Cech completeness and Arhangel'skil's p-space condition may be formulated with respect to appropriate positionings in regular T 0 locally Frechet compact spaces. An illustrative result follows. Theorem. For a Tychonoff spaceS, these conditions are equivalent: (I} Sis absolutely 9-refinable. (2} For some regular T 0 locally Frechet compact space E in which Sis (densely} embedded, it is true that any collection of open sets of E covering Sis refined by each term of a sequence H1 , H 2 .... of collections of open sets of E covering S such that if for each n the point P belongs to the sum of the sets of Hn, then some Hn has only finitely many members containing P. This work was supported by the United States Atomic Energy Commission. (Received August 30, 1968.}

948 Errata-Volume 15

WITOLD BOGDANOWICZ. Existence and characterization of the smallest and the greatest measures generating the same Lebesgue integral. Abstract 656-18, Page 507. Line 5: Replace A U B by A n B.

]. M. GANDI':!l. Som.~ group laws equivalent to the commutative law. Abstract 68T-169, Page 349. 2 Line 4: Part {2) of the Theorem should read "The law [x,yJX = [x,y,xJin a group implies the law [x,y]= 1."

G. F. KOHLMA YR. On the exponential moment transform and a discrete Banach algebra theory of integra-differential equations. Abstract 68T -333, Page 398.

Line 3: "Let Re(s0 ) > 0 •.. " should read" Let Re(s0 ) > 1 •• .''.

H. LAKSER. Free lattices generated by a partially ordered set. ll. Abstract 68T-411, Page 623.

Line 4: "(iii) given x0 , x1, y0 , y 1 E P .• .'' should read "(iii) given x 0 , x 1, y 0 , y 1 E L

W. A. LAMPE. Representations of some transformation semigroups as endomorphism semigroups of universal algebras. Abstract 68T -417, Page 625. The following sentence should be added: "The results in this abstract are closely related to some theorems in a paper by P. Goralclk, Z. Hedrli'n and J. Sichler (to appear)."

PHILIP OLIN. -.t.os subsystems of reduced powers. Abstract 68T-E7, Page 805.

Line 5: Replace a.+ by a..

B. L. OSOFSKY. Homological dimension in rings of continuous functions. Abstract 68T-516, Page 656. Line 2: Replace "Let f E R" by "Let f be a proper zero divisor of R". Line 3: Replace "hd(fR t gR) = 0 or 3" by "hd(fR) = 0 or 3 and hd(fR t gR) ~ 4".

On the postulate of separation, Abstract 68T-336, Page 399, was written jointly by Dr. A. I. Arruda and Professor N.C. A. da Costa of the University of Parana, Curitiba, Parana, Brazil.

On complem·~nted and quasi-complemented subspaces of C(S) for Stonian S, Abstract 68T-489, Page 647, was written by H. P. Rosenthal.

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951 ELEMENTS OF LINEAR ALGEBRA AND MATRIX THEORY John T. Moore, University of Western Ontario. International Series in Pure and Applied Mathematics. 352 pages. $8.95 • A study of linear transformations-and their matrix representation-of finite dimensional vector spaces, with special emphasis placed on the invariants and canonical matrices under equivalence, congruence and similarity.

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SHORT NOTES Sufficient Statistics for the Optimal Stopping Problem - E. B. DYNKIN The Stochastic Field of P. Levy - -E. A. MoRASOVA and N. N. CENCOV Some Inequalities for Sums of Mu~tidimensionul Random Variables - A. A. BoROVKOV An Estimation of a Convergence Rate for the Absorption Probability in Case of a Null Expectation NAGAEV On the Multidimensional Central Limit Theorem - - S. M. SADIKOVA On the Martin Boundary for a Class of Markov Processes M. G. SuR On Connection Between the Local and Integral Theorems for Latticed Distributions N. G. GAMKRELIDZE On Service Processes in Multi-Channel Systems - - A. A. BoROVKov The Local Limit Theorem for Markov Chains and Regularity Conditions -B. M. GuREVIC On the Speed of Convergence in the Multi-Dimensional Central .Limit Theorem V. V. SAZONOV

954 Recently Published Forthcoming ANALYSIS I ANALYSIS II Bv SERGE LA:-.:G, Columbia University. Bv SERGE LA='IG, Columbia University. Designed for a first course in advanced calculus, this This book is intended as a text for a first year gradu­ book has been written to be neither too computational ate course in analysis, and forms a natural sequence to nor too abstract. At the same time, the author has the author's ANALYSIS I. For the convenience of the attempted to exhibit sufficiently the connection be­ reader, and be-cause of the varied background of stu­ tween linear algebra and advanced calculus. Although dents, some material has been repeate-d here. In par­ the text is logically self-contained, it pre-suppost>s the ticular, the differt>ntial calculus in Banach spaces is mathematical maturity acquired by students who will redone rapidly since most readers will at best be ordinarily have had two ye-ars of basic calculus. When acquainted with it only in Euclidean space, and then used in this context, most of the first part may be incompletely. Howeve-r, it is assumed that the reader omitted or reviewed rapidly. is reasonably familiar with notions of uniform conver­ 460 pp. $10.75 gence and the like. In Press (1969) LECTURES ON ORDINARY ELEMENTARY DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Bv DoNALD L. KREIDER, Dartmouth Colle[!,e, RoBERT Bv EINAR HILLE, Yale University. G. KuLLER AND DoNALD R. OsTBERG, Northern Intended for usc in courses at the graduate or Illinois University. advanced-graduate level, this book stresses the theory The aim of this book is to present its subject in the of ordinary differential e-quations. All the topics context of lin<:ar algebra in a styk suitable for math<:­ treate-d havt> figured in lectures given by the author matics majors and non-majors alike. Applications art> over a pt>riod of forty-five years at Harvard, Princeton, emphasized and much of the th<:ory included may Yale and other leading universities. Using this text, be omitted if desired. a variety of courses may be taught at the graduate Content areas include ve-ctor spaces and linear level. transformations, matrice-s, first ord<:r diffnential equa­ This is the first book to give the analytic theory of tions, n1h order linear differ<:ntial equations, Gre-en's lint>ar diffe-re-ntial equations in Banach algebras. In functions for initial-value- problems and Laplace trans­ addition, several chapters deal with mate-rial not to be forms, systems, separation and comparison theorems, found in any other modern treatment. serie-s solutions, existe-nce theorems, and an introduc­ In Press (1969) tion to stability theory. 492 pp., 99 illus. $10.95 INTRODUCTION TO COMPLEX ANALYSIS Bv RoLF NEVANLINNA, Academy of Finland, AND VEIKKO PATERNO, University of Helsinki. Translated ELEMENTS OF THE THEORY OF from the German by T. KovAR! Al'D G. GooDMAN, ALGEBRAIC CURVES Imperial College, University of London. Bv A. SEIDE;o.;BERG, University of California, Berkele)'. Designed for matht>matics majors at junior, senior, This book treats the theory of algebraic curve-s in or graduate levels, this book is an authoritative first such a manner as to bring th<: subject, normally taught course in compkx analysis. The- only prerequisite is to graduate students, within the scope of an under­ advanced calculus. graduate- audie-nce. It was written in th<: belief that a The authors have strive-d for an exact, easily under­ course in algebraic curves is no more difficult than the­ standable presentation. The book notably differs from much more widely offered course in diffe-rential geo­ other expositions in the following sections: introduc­ metry, and can compete with it in use-fulness and tion of complex numbers, fundamental theorems of interest. The only prerequisite is an undergraduate integration, theory and applications of harmonic func­ course in algebra. However, it would be helpful if the tions, especially of the harmonic measure, and bound­ student has takt>n a course in projective gt>omctry. ary correspondence in conformal mapping. (1968) In Press (1969)

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958 " ... designed to provide expositions of the basic areas of mathe­ matics at the upper undergraduate and beginning graduate levels. It .is ·at this point that the student's acquaintance with modern mathematics really begins. A good text will, of course, supply the usual'indispensable features: a high level of exposition, plenty of exerdses ranging from routine ones to ingenious challenges, and stimulbting glimpses of where the subject goes when you pene­ trate deeper. A superb text does even more: it lets the student share. the excitement that accompanied the great discoveries of the last century. It is to the development of superb texts that the Series is dedicated."Consulting Editors, Irving Kaplansky, University of Chicago, and Charles DePrima, California Institute of Technology. THE ALLYN &BACON SERIES IN

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ROTMAN/ THE THEORY OF GROUPS. An Introduction. B>t;;Joseph J.' Rotman, University of Illinois. This introduction to group theory is o ~ff-contoin'ed unit for students who have had an introductory course in modern dl!;!ebra and ore familiar with matrices and vector spaces. Separate chapters cover siri'lple gr6jjps, homological algebra, and word problems. 1965. 6 x 9. 305 pp. ,o,; ,-, ' 1 ,'i-.'-~-~1-"<>:-~~ J:)UGUNDJIJTQPOLOGY •By James Dugundji, University of Southern Colifornia. ltri$>toll'tti•tutes an introduction to set theory and a comprehensive gen­ eral topology, In p4ditJonto the usual topics, it includes: identification and weak ~pologies, conv.~rQen.ce.in terms of filter-bases, homotopy and homotopy type, 'r'aps into spheres, tlie Jordan separation theorem in E", path spaces, H-spaces, direct and inverseli""itspaces. 1966. 6 x 9. 447 pp. MABCU~~. f\ .. ~URVEY OF MATI\IX IHEORY .. , , X lNEQUALITIES. By Marvin Marcus opd He , e'\Ji'l;v~.r~l~ 9f}:aUfornia at Santa Barbara. Beginning CJt.Pn e ...· .... ri ..· ...... , (luthors' proceed qllickly to subjects of current research ln~restfor t.ise qt the ~enior•graduate level or as a source of reference to working rtTCf:tl-u~mptt<;i9rtfl"' A largl!l Jwmber of the sectrons are devoted to examples. 1964. 6 .X 9.18~ :pp~

959 New from the AMS Contents of Contemporary Mathematical Journals On January 1Oth a new journal entitled Contents of Contemporary Mathematical Journals will become available on a bi-weekly basis from the American Mathematical Society. This publication offers the interested individual a reproduction of the tables of contents of about 250 of the major mathematical journals published today. Twenty-six issues will be printed each year at a subscription rate of $18.00 list; $15.30 agents, and $9.00 to individuals . Subscriptions are now being accepted and should be sent to the address below. Mathen1atical Offprint Service The Mathematical Offprint Service is an entirely new concept in in­ formation distribution established by the American Mathematical Society to provide a single source of current information for the research mathematician and to add a new dimension to mathematics publishing. This service will offer to the individual offprints or title listings of articles in all areas of mathematics of specific interest to him. By initially submitting a detailed interest profile, a subscriber may specify the types of articles of which he wishes to receive offprints or title listings. Articles which satisfy the criteria specified by the subscriber will be mailed to him on a continuing basis, along with titles of articles in which he has indicated a peripheral interest. If you are interested in finding out more about the Mathematical Offprint Service it is suggested that you write to the American Mathe­ matical Society for additional information. We will then send you a booklet explaining all the details and answer any questions which you may have.

P. 0. Box 6248, Providence, Rhode Island 02904

960 HANDBOOK OF NONP ARAMETRIC STATISTICS Vol Ill- Analysis of Variance John E. Walsh, s'£uthern Methodist University. Fall 1968, 880 pages, about $20.00 Volume Ill completes the three-volume set of books giving all the important experimental nonparametric procedures developed to 1958. Chapters 2 through 13 are concerned with analysis of variance (ANOVA) situations where the outcomes for the subexperiments are quantitative and are not converted to categorical form. The outcomes are univariate for almost all of the cases that occur, but multivariate ANOVA is also considered. For most of the material in Chapters 14-17, the data are categorical, or have been converted to categorical. In general, the material is oriented to the experimental statistician. The book will find use in all fields that involve experimentation. The reader is presumed to have a basic understanding of the field of mathematical statistics. Use of a concise presentation format allows the pertinent details of a statistical procedure to be stated in a reasonably small space. An extensive bibliography is included. Volume 1: Investigation of Randomness, Moments, Percentiles, and Distributions 575 pages, $15.00 Volume II: Results for two and Severa/ Sample Problems, Symmetry, and Extremes 712 pages, $17.50

Do mathematicians need a mathematics dictionary? New Multilingual Third Edition MATHEMATICS DICTIONARY Glenn James and Robert C. James. 1968,518 pages, $17.50 Yes . .. to define 8,000 terms (also concepts, relationships, theories, rules) indispensable to mathematics studies. because four expert scientific translators have prepared foreign language indexes of mathematics terminology in Russian, German, French, and Spanish- with a total of 6,500 foreign language terms defined so that you have as a reference a great amount of international scientific information. as a reference to terms in currently growing areas of interest such as category theory, measure theory, linear and dynamic programming- as well as in such fields as arithmetic and calculus, differential geometry, the theory of groups and matrices, the theory of potentials, game theory, statistics, modern algebra, and number theory. to have readily available: mathematics tables, integral tables, differentiation for­ mulas, and a dictionary of symbols. to save time- you'll want to keep it on your desk as a convenient reference volume. The Mathematics Dictionary in the New Multilingual Third Edition could be the most important volume in your personal library, and it is certainly a welcome addition to depart­ mental and college libraries. (A regular edition without Multilingual Indexes is a/so available- $13.50)

PRINCETON, N.J. 08540

961 CNA ... and Where You Fit in

The Center for Naval Analyses is a private • Determining optimal replacement ages non-profit scientific institution pursuing a for aircraft broad program of operations research and Problem solving of this scope requires systems analysis for the U.S. Navy. Our pri­ individuals with advanced degrees, and mary research tool is knowledge-particu­ with the imagination and ability to use larly the application of new and advanced sophisticated quantitative techniques ef­ knowledge in science, mathematics, and fectively. Individuals, moreover, who are economics to current and projected prob­ challenged by contributing to the solutions lem areas. of critical national problems and who want Today's research program offers an op­ to see the results of their efforts take tangi­ portunity to come to grips with such prob­ ble form. lems as: To such professionals-physical and so­ • Determining the least cost mix of land­ cial scientists, engineers, mathematicians, and sea-based tactical air that the U.S. economists-CNA offers attractive career should procure and operate opportunities. Salaries are excellent, of course, but the factors that set CNA apart • Deriving the lessons learned from com­ are its independence, environment, and plex fleet exercises involving dozens of a professional development program of ships and thousands of men breadth and generosity. Our relationship • Explaining why ship costs have risen rel­ with the Navy gives us wide latitude in ative to consumer goods choosing research projects and complete freedom in carrying out analysis. In ad­ • Assessing Soviet options for Cuba dition, CNA staff members may pursue advanced study, perform independent • Analyzing a follow-on system for Naval research, and publish-at CNA expense. strategic weapons systems For additional data on the potential of • Developing a model to estimate direct a CNA career, please send your resume to: and indirect costs for U.S. Marine Corps Mr. james F. Armistead, Director of Pro­ units fessional Staffing.

t4iJt WILSON BOULEVARD ARLINGTON. VIRGINIA 22209 An equal opportunity employer

962 CAREER OPPORTUNITIES MATHEMATICAL, BEHAVIORAL AND SOCIAL SCIENTISTS

The Michigan Center for Health Statistics is seeking qualified candidates with education and background in Mathematics, Anthropology, Sociology, Demography, Economics, Political Science, Medicine, Statistics, Psychology, and Business Adminis­ tration to bring their competence to bear on current research in statistical design, analytic and interpretive aspects of health concerns and medic" I care.

To the selected candidates we offer competitive salaries, new and exciting profes· sional challenges, the opportunity for participation in academic affairs, liberal fringe benefits under Michigan Civil Service, and a highly desirable urban location.

Applications from professional teams encouraged.

If you would like to look further into this opportunity, please send a letter or resume to: Robert Lewis, Ph.D., Chief Center for Health Statistics Michigan Department of Public Health 3500 North Logan Lansing, Michigan 48914

An Equa I Opportunity Employer

INDEX TO ADVERTISERS MONASH UNIVERSITY Academic Press Inc ...... cover 4 Melbourne, Australia Addison-Wesley Publishing CHAIR OF MATHEMATICS Company, Inc ...... 955 Allyn and Bacon, Inc ...... 959 Full professorship with responsibility for organisation and development of graduate American Mathematical Society ...... 960 studies in the department; over forty higher W. A. Benjamin, Inc ...... cover3 degree candidates expected in 1969. There Birkhiiuser Verlag Basel ...... 957 are six other chairs of mathematics. Cambridge University Press ...... 956 Salary (Australian currency) $12,000 per Center for Naval Analyses ...... 962 annum. Cushing-Malloy, Inc ...... 958 Information on benefits, application etc. from Four Continent Book Corporation ..... 958 Academic Registrar, Monash University, Kinokuniya Book-Store Clayton, Victoria 3168. Company, Ltd ...... 958 Enquiries on the department to Professor McGraw-Hill Book Company ...... 952 P. D. Finch, Chairman, (Statistics) Professor Michigan Department G. B. Preston (semigroups), Professor J. B. of Public Health ...... 963 Miller (junctional analysis) Professor B. R. Monash University ...... 963 Morton (geophysical fluid dynamics) Professor Prentice-Hall, Inc ...... 953 R. Van der Borght (Applied). Society for Industrial Applications by 31st October 1968. and Applied Mathematics ...... 954 J.D. Butchart, Springer-Verlag New York, Inc ...... 951 Academic Registrar D. VanNostrand Company, Inc ...... 961 W ilea Press, Inc ...... 950

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964 0 Topological Dynamics Algebraic Geometry: An International Symposium Introduction to Schemes by Edited I. G. Macdonald, Oxford University Joseph Auslander, University of Maryland 128 Pages Walter H. Gottschalk, Wesleyan University $3.95 Paper/$9.50 Cloth About 384 Pages Approx: $11.50 Cloth Based on a series of lectures delivered at This volume contains the proceedin~s of an the University of Sussex in 1964-65 this international symposium m topological dy­ namics held at Colorado State University, book presents a brief introduction to the Fort Collins, in August 1967, and sponsored language of schemes. Of primary interest to by the Air Force Office of Scientific Re­ postgraduates in classical geometry, it is search. The papers presented deal with cur­ useful as well to pure mathematicians. An rent research in topological dynamics, re­ elementary knowledge of algebra and topol­ lated fields such as ergodic theory and the ogy is assumed. qualitative theory of ordinary differential equations. This volume is an excellent refer­ ence for research workers, as well as gradu­ ate students. Ergodic Problems of Induced Representations Classical Mechanics V. I. Arnold, University of Moscow of Groups and A. Avez, University of Paris 288 Pages $6.95 Paper/$14.75 Cloth Quantum Mechanics Translated from the French, this mono­ Georze W. Mackey, Harvard University graph is designed to supplement a first- or About 136 Pages second-year graduate course in classical Approx: $3.95 Paper/$8.50 Cloth mechanics. It constitutes a report of re­ This text contains a set of lectures that cent applications of ergodic theory to dif­ were given at the Scuola Normale, Pisa in ferential geometry and classical mechanics, April 1967. Addressed to mathematicians and is written for physicists, astronomers, and physicists, these lectures deal with the mathematicians, graduate students, and nature of the theory of induced representa­ tions and its application to quantum research workers in these fields. mechanics. Science: Battelle Rencontres Men, Methods, Goals Edited by Edited by Boruch A. Brady C•cilll OcWitt, uruversity of North Carolina Massachusetts Institute of rech<~o!cay John A. Wheeler, Princeton University ft~tbolas Capaldi 640 Pages $14.75 Cloth The City University of New York These lectures are based on a meeting held 288 Pages in Seattle, in the summer of 1967, spon­ Approx: $2.95 Paper/$8.00 Cloth sored by the Battelle Memorial Institute. By making readily available some of the The purpose of this meeting was to enlarge most important philosophical statements the lines of communication between mathe­ about science, th1s new anthology will help maticians and physicists. The papers pre­ the scientist and science student reflect topics, upon the nature, methods, and goals of the sented deal with a wide variety of scientific enterprise. Among the readings ranging from Lie groups to the structure of included in the volume are selections from space-time, from homotopy theory to the the writings of CoP.ernicus, Herschel, Kant, quantization of the gravitational field, and Mach, Maxwell, M1ll, Newton, and Poincare. from differential geometry to S-matrix theory. W.A. BENJAMIN, INC. TWOPARKAVENUE•NEWYORK,N.Y.10016 SECOND-CLASS POSTAGE AMERICAN MATHEMATICAL SOCIETY PAID AT P.O. Box 6248 PROVIDENCE, RHODE ISLAND AND Providence, Rhode Island 02904 ADDITIONAL MAILING OFFICES

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THE FUTURE OF STATISTICS Edited by DONALD G. WATTS, Department of Statistics, University of Wisconsin, Madison, Wisconsin Considers the communication problem in statistics ties in their respective fields, deal with the prob- and discusses such areas as the future of statistics lems of transforming data, budget analysis, feed- departments, statistical inference, and statistics forward and feedback control, nonlinear design of in engineering and in the biomedical and social experiments, and measurement in social sciences. sciences. The technical papers, written by authori- 1968, about 325 pp., $12.50

FOUNDATIONS OF MODERN ANALYSIS ENLARGED AND CORRECTED PRINTING By J. DIEUDONNE, lnstitut des Hautes Etudes Scientifiques, Paris This book is the first volume of a treatise which REVIEW OF FIRST EDITION: will eventually consist of four volumes. It is also a "I recommend this book to those who want to be profes- second edition, in which the whole structure of sional mathematicians and who already have had some analysis is built up from the foundations. The only experience with abstract methods ... " -SCIENCE things assumed are the rules of logic and the usual 1968, 385 pp. properties of the natural numbers.

ADVANCES IN MATHEMATICS Edited by GIAN-CARLO ROTA, Massachusetts Institute of Technology Commencing with Volume 3, 1969, ADVANCES IN lections of papers will, subject to sufficient de- MATHEMATICS will be published as a journal un- mand, be issued in hardcover for individual pur- der the chief editorship of Professor Gian-Carlo chase. Supplemental monographs may be published Rota. This new journal will publish papers in any from time to time and these will be offered to sub- area of pure and applied mathematics. scribers at a special price. Volume 2 is also being published in journal format Subscription price for Volume 2, all four fascicles: $20.00 in 1968, but the individual fascicles comprising the Fascicle 1, $3.95 volume may be purchased separately or on a sub- Fascicle 2, $7.00 scription basis. Beginning with Volume 3, sales will j~~g;g:~ ~: f~:~8 be by subscription only. Individual papers and col- Volume 3, 1969, about 600 pp., $28.00

INTERACTIVE SYSTEMS FOR EXPERIMENTAL APPliED MATHEMATICS Edited by MELVIN KLERER, Department of Industrial Engineering, New York University, New York, New York and JURIS REINFELDS, Computer Center, University of Georgia, Athens, Georgia November 1968, about 500 pp.

( ~~NEW YORK AND LONDON ACADEMIC PRESS\ 1" 111 FIFTH AVENlJE NEW YORK N Y 10003