OF THE

AMERICAN

MATH-EMATICAL

·soCIETY

VOLUME 12, NUMBER 8 ISSUE NO. 86 DECEMBER, 1965

SPECIAL ISSUE ASSISTANTSHIPS AND FELLOWSHIPS IN MATHEMATICS IN 1966-1967

cNotiaiJ OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. \Yalker

CONTENTS

MEETINGS

Calendar of Meetings o 0 ••••••••••• 0 •••• 0 ••• o •••• o o ••••••• 660 Program of the Meeting in Cambridge, Massachusetts • 0 •••• 0 •• 0 0 ••• 661 Abstracts for the Meeting- Pages 690-699

PRELIMINARY ANNOUNCEMENTS OF MEETINGS •• o o •• o o o. • • • • • • • • • • 665

NEWS ITEMS AND ANNOUNCEMENTS ••••• o •• o o •••• o •••• 664, 683, 685, 689

1965 SUMMER RESEARCH INSTITUTE •• o • o •••• o •••••••••••••••••• 669 1965 SUMMER SEMINAR. • • • • • • • • • • • • . • • • • • • • • • • • . • • • • • • • • • • • 670 MEMORANDA TO MEMBERS

List of Graduate Assistantships to be Published in December •• 0 • • • • • 666 ANNUAL SALARY SURVEY •••••••••••••••••••••••••••••••••••• 672 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D. • • • • • • . • • • • • 674 DOCTORATES CONFERRED IN 1964. • • • • • • • • • • • • • • • • • • • • • • • • • • • • 675

PERSONAL ITEMS •••••.••••••••••.•••••••••••••••••• o • • • • • 676

NEW AMS PUBLICATIONS •••. o ••••••••••••••••••••••••••••••• 684

SUPPLEMENTARY PROGRAM- Number 34 •••••••••••••••••••••• o • 686

ABSTRACTS OF CONTRIBUTED PAPERS o o •••• o ••••••••••••••••••• 690

INDEX TO ADVERTISERS ••••••• o ••••••••••••••••••••••••••••• 741

RESERVATION FORMS •••••••••••••••••••••••••••••••••• o •• o 742 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 c}(otit:tJJ was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned.

Meet­ Deadline ing Date Place for No. Abstracts*

6Z7 November 1Z-13, 1965 Lexington, Kentucky Sept. ZB 6Z8 November Z6-Z7, 1965 Iowa City, Iowa Sept. ZB NOVEMBER, 1965 Southern California Meeting None scheduled 6Z9 December Z9, 1965 Berkeley, California Sept. ZB 630 January Z4-Z8, 1966 (7Znd Annual Meeting) Chicago, illinois Dec. 3

April 9, 1966 Honolulu, Hawaii August Z9-September Z, 1966 (7lst Summer Meeting) New Brunswick, New Jersey January Z4-Z8, 1967 (73rd Annual Meeting) Houston, Texas August ZB-September 1, 1967 ( 7Znd Summer Meeting) Toronto, Ontario, Canada January, 1968 San Francisco, California

* 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 deadline date for the by title abstracts is November Z6, 1965.

ThecN0ticeiJof the American Mathematical Society is published by the Society in January, February, April, June, August, October, November and December. Price per annual volume is $7.00. Price per copy $2.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, Box 6248, Providence, Rhode laland 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority 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 mailing at the special rate of Postage provided for in section 34,40, paragraph (d).

Copyright«:)), 1965 by the American Mathematical Society Printed in the United States of America

660 Six Hundred Twenty-Sixth Meeting Massachusetts Institute of Technology Cambridge, Massachusetts October 30, 1965

PROGRAM

The six hundred twenty-sixth m,~et­ Parking space will be available in ing of the American Mathematical Society the East Parking Lot and the East Parking will be held at the Massachusetts Institute Garage on the Institute grounds for those of Technology on Saturday, October 30, traveling by automobile, The entrance, 1965, both to the parking lot and to this parking By invitation of the Committee to garage, is at the corner of Main and Vassar Select Hour Speakers for Eastern Sectional Streets. If needed, additional parking space Meetings, there will be an address by Pro­ in the West Parking Garage (entrance on fessor Max Koecher of the University of Vassar Street, west of Massachusetts Munich, who is Visiting Professor at Yale Avenue) will also be available. University. The title of his lecture is M. I. T. is a seven to ten minute "On homogeneous (non-associative) alge­ walk from the Kendall Square station of the bras," It will be presented at 2:00 P.M. in Cambridge-Dorchester subway. This sub­ the Compton Auditorium, room 26-100, way may be boarded at various points, in­ There will be sessions for contrib­ cluding South Station, Boston, and Harvard uted papers at 9:30 A.M. and at 3:15P.M. Square, Cambridge. The most convenient in rooms 2-190 and 6-120. There will be entrance for those coming by subway is at provision for a limited number of late the Northwest corner of the Hayden Mem­ papers. orial Library. The registration desk wiil be on the Those coming by taxicab or track­ first floor of Building 2, It will open at less trolley will find it convenient to use 9:00A.M. the main entrance, 77 Massachusetts Ave­ The Massachusetts Institute of Tech­ nue, Most of the entrances except the two nology is located on the Cambridge side of mentioned above are closed on Saturday. the Charles River approximately one to two Lunch will be served in an M.I. T. miles from the various railway stations in cafeteria, and a list of nearby restaurants Boston, It is easily accessible by automo­ in Boston and Cambridge will be available. bile, subway, trackless trolley, or taxicab.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. The papers are scheduled at 15 minute intervals in order that the listeners can circulate between sessions. To maintain the schedule, the time limit will be strictly enforced.

661 SATURDAY, 9:30A.M. Session on Analysis and Applied Mathematics, Room 2-190 9:30 - 9:40 ( 1) Quasi-interior points and the extension of linear functionals Professor R. E. Fullerton and Professor C. C. Braunschweiger*, Univer­ sity of Delaware (626-1) 9:45 - 9:55 (2) On finite metric sets I: Imbe'dding in n-dimensional Minkowski space Professor Dorothy Wolfe, Pennsylvania Military College (626-12) 10:00 - 10:10 (3) A uniqueness theorem for conformal maps Mr. W. J. Schneider, Syracuse University (626-27) 10:15 - 10:25 (4) How the 23 functional-equations problems of 1963 stand now Professor J. D. Aczel, University of Massachusetts (626-7) 10:30 - 10:40 (5) Uniqueness in Cauchy's problem for elliptic equations with double character­ istics Professor R.N. Pederson, Carnegie Institute of Technology (626-11) 10:45 - 10:55 (6) Approximation with vector valued norms Mr. A. C. Bacopoulos, University of Wisconsin and SUNY at Buffalo (626-3) 11:00 - 11:10 (7) A method of determining the Riesz kernel for the EPD operator Dr. E. C. Young, Florida State University (626-4) 11:15 - 11:25 (8) Formulation of the integral equation for interflections in helical coils Professor D. E. Spencer*, University of Connecticut and Mr. J. F. Fitz­ gerald, Sylvania Electric Products, Inc., Danvers, Massachusetts (626-25) 11:30 - 11:40 (9) On the geometry of steady screw motions Mr. E. R. Suryanarayan, University of Rhode Island (626-26)

SATURDAY, 9:30 A.M. General Session, Room 6-120 9:30 - 9:40 ( 1 O) Two recursively enumerable sets Mr. R. W. Robinson, Cornell University (626-8) 9:45 - 9:55 (11) Marking automata and a basis for the r.e. sets Professor D. L. Kreider, Dartmouth College and Professor R. W. Ritchie*, University of Washington (626-9) 10:00- 10:10 ( 12) Dialectic logic Professor F. G. Asenjo, University of Pittsburgh (626-15) 10:15 - 10:25 ( 13) A characterization of Markov processes in terms of their hitting character­ istics. Preliminary report Mr. Eugene Denzel, Dartmouth College (626-2) 10:30 - 10:40 ( 14) Distribution-generated spaces as Menger spaces Mr. Howard Sherwood, Illinois Institute of Technology ( 626-19)

* For papers with more than one-author-:-an-asterisk follows the name of the author whj lans to resent t:he paper at the meeting

662 10:45 - 10:55 ( 15) Markov processes whose hitting distributions are dominated by those of a given process Mr. C. T. Shih, Cornell University (626-30) 11:00- 11:10 ( 16) On the uniform continuity of the probabilistic distance Professor Berthold Schweizer, University of Massachusetts (626-10) 11:15- 11:25 ( 17) Postulation formulas for flag and Grassmann varieties Professor Federico Gaeta, SUNY at Buffalo (626-28) (Introduced by Dr. S. H. Gould) 11:30 - 11:40 (18) The projective invariants of the configuration of two lines and a third order differential element Professor Rodney Angotti, SUNY at Buffalo (626-20)

SATURDAY, 2:00P.M. Invited Address, Compton Auditorium, Room 26-100 On homogeneous (non-associative) algebras Professor Max Koecher, University of Munich and Yale University

SATURDAY, 3:15P.M. Session on Algebra and Theory of Numbers, Room 6-120 3:15 - 3:25 ( 19) Structure of certain finite groups Mr. F. P. Callahan, General Electric Company, Philadelphia, Pennsyl­ vania (626-29) 3:30 - 3:40 (20) On the Herbrand quotient Mr. J. H. Smith, The University of Michigan (626-18) 3:45 - 3:55 (21) Invariant splitting in Jordan and alternative algebras Professor E. J. Taft, Rutgers, The State University (626-16) 4:00 - 4:10 (22) Extensions and retractions of rings Professor C. W. Kohls* and Professor L. J. Lardy, Syracuse University (626-5) 4:15 - 4:25 (23) A method for the computation of the greatest root of a non-negative matrix Professor Alfred Brauer, University of North Carolina and Wake Forest College (626-14) 4:30 - 4:40 (24) Partition theorems related to the Rogers-Ramanujan identities Professor G. E. Andrews, The Pennsylvania State University (626-13)

SATURDAY, 3:15P.M. Session on Topology, Room 2-190 3:15 - 3:25 (25) On the realcompactification of a product space Professor W. W. Comfort*, University of Massachusetts and Professor Stelios Negrepontis, University of Indiana (626-6) 3:30 - 3:40 (26) Some remarks on heavy points in countable spaces Dr. A. K. Snyder, Massachusetts Institute of Technology (626-17)

663 3:45 - 3:55 (Z 7) Concerning complete amonotonic collections of subcontinua of a compact con­ tinuum Professor H. L. Baker, Jr., University of Massachusetts {6Z6-Z4) 4:00 - 4:10 {ZS) Abstract convergence theory Dr. D. A. Mattson, Trinity College ( 6Z6-Z 1) 4:15 - 4:Z5 (Z9) Uniformities on locally compact topological groups. Preliminary report Professor R. W. Bagley, University of Miami and Professor T. S. Wu*, University of Massachusetts {6Z6-ZZ) 4:30 - 4:40 (30) Homomorphisms of lattices of continuous functions Professor S. D. Shore, University of New Hampshire {6Z6-Z3) Everett Pitcher Bethlehem, Pennsylvania Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

FOURTH ANNUAL SCIENCE CONFERENCE OF YESHIVA UNIVERSITY November 15-16, 1965

The Fourth Annual Science Confer­ P. W. Anderson, E .E. Sal peter, and G.C. Wick, ence sponsored by the Belfer Graduate in physics; Bernard M. Dwork, Claude E. School of Science of Yeshiva University Shannon, Paul A. Smith, Kenkichi Iwasawa, will be held on November 15 and 16, 1965 G. D. Mostow, and Richard Brauer, in at the Hotel Astor in New York City. The mathematics. sessions are open to all who wish to attend, Of additional interest will be the and admission is free. presentation of the annual award for dis­ Participants in the Program will be: tinguished service to science by the Belfer Feza Gursey, T. D. Lee, A. S. Wightman, Graduate School of Science.

ADVANCED SCIENCE SEMINAR ON METHODS OF SOLUTION FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

The University of Delaware and the 1965, Some room and board support is National Science Foundation are jointly available. Applications are invited. sponsoring an advanced science seminar For further information write to on Methods of Solution for Nonlinear Par­ Professor W. F. Ames, Department of tial Differential Equations. This meeting Statistics and Computer Science, Univer­ will be held at the University of Delaware, sity of Delaware, Newark, Delaware 19711. Newark, Delaware, December Z7, Z8, Z9,

664 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Six Hundred Twenty-Seventh Meeting University of Kentucky Lexington, Kentucky November 12-13, 1965

The six hundred twenty-seventh Following is a list of hotels and meeting of the American Mathematical So­ motels within walking distance of the Uni­ ciety will be held at the University of Ken­ versity campus: tucky in Lexington, Kentucky on November 12-13, 1965. By invitation of the Committee to Phoenix Hotel Select Hour Speakers for Southeastern Sec­ 120 East Main Street tional Meetings, the Society will be ad­ Single Double Twin dressed at 2:00 P.M. Friday, November 12, $7.00 $10.00 $13.00 by Professor Tatsuo Homma of Florida State University. Professor P. S. Mostert Center Motel of Tulane University will address the So­ 900 South Limestone ciety Saturday, November 13 at 9:00A.M. $8.00 $12.88 The title of Professor Homma's talk will be "Piecewise linear approximations of Town House Motel embeddings of manifolds". The title of Pro­ South Limestone and Rose Streets fessor Mostert's talk will be "The Struc­ us 27 ture of Compact Connected Semi-groups". $7.00 $8.00 ~.oo All sessions of the meeting will be held in the Student Center. The registration desk Suburban hotels and motels near the Uni­ located in the Student Center will be open versity: from 11:00 A.M. to 4:00P.M. Friday and Campbell House Motor Hotel from 8:30 A.M. to 10:00 A.M. Saturday US 68 and Waller A venue morning. By invitation o.f the Committee to $8.00 $1?.00 $18.00 Select Hour Speakers for Southeastern Sec­ Imperial House tional Meetings there will be a special US 68 and Waller A venue session of 15-minute papers on "Semi­ groups and Topological Algebra" organized $12.00 $16.00 $18.00 by R. j. Koch and chaired by Wayman Howard Johnson Strother. The speakers will be.L. W. Ander­ US 27 South son, Haskell Cohen, R. j. Koch, Pierre $8.00 $14.00 $14.00 Conner, C. T. Yang and L. E. Ward. This session will be held on Saturday afternoon Reservations should be made directly with starting at 2:00 P.M. the hotels and motels. Lexington is served There will be a no-host buffet dinner by Southern Railway, L & N Railroad; American, Pied­ at Spindletop Hall Friday, November 12 at mont, Eastern and Delta Arilines 6:30 P.M. The facilities are limited to ap­ and the Greyhound Bus Company. proximately 150 persons. Reservations There will be ample should be made by November 5 by writing parking facilities on campus. Lex­ ington is in the Eastern Standard time to the Secretary of the Mathematics De­ zone. partment. Tickets at approximately $3.50 0. G. Harrold per person can be purchased at the Regis­ Associate Secretary tration Desk. Tallahassee, Florida 665 Six Hundred Twenty-Eighth Meeting University of Iowa Iowa City, Iowa November 26-27, 1965

The six hundred and twenty-eighth A tea will be held on Friday after­ meeting of the American Mathematical noon at 5:00 P.M. at the Iowa Memorial Society will be held at the University of Union, one block north of the Physics Iowa on Friday, November Z6 andSaturday, Building. November Z7. In conjunction, there will be Iowa City may be reached by car on a meeting of the Iowa section of SIAM on Interstate 80, five hours from Chicago, by Saturday. Registration and all sessions will bus via Continental or Greyhound, by rail be in the Physics Building. on the Rock Island Lines, and by air on Sessions for contributed papers will Ozark Airlines to Iowa City and on Ozark be held on Friday afternoon and on Saturday and United to Cedar Rapids, which is morning. eighteen miles north of Iowa City. By invitation of the Committee to Mail for those attending the meeting Select Hour Speakers for Western Sectional may be addressed to the Department of Meetings, Professor F. V. Atkinson of the Mathematics, Room 110 Physics Building. University of Kentucky will speak on Multi­ New and excellent accommodations parameter spectral theory on Friday at are available at the Iowa Memorial Union Z:OO P.M. and Professor Jim Douglas Jr. on campus. Rates are $9.50 for a single of Rice University will speak: on Unstable and $13.00 for a double. A reservation physical problems and their numerical blank can be found on the inside back cover approximation on Saturday at 11:00 A.M. of these c){oticei). In addition, information In addition, Professor R. H. Bing on housing, restaurants and parking will of the University of Wisconsin will chair be available at the Registration Desk in a Special Session of twenty-minute papers the Physics Building. The Amana Colonies on Topology at 9:00A.M. on Saturday. The and the Herbert Hoover Birthplace and speakers will be Professors Steve Armen­ Library are points of interest within fif­ trout, Edmund Burgess, Wolfgang Haken, teen miles of Iowa City. and Russell McMillan. On Saturday afternoon, Dr. M. Papa­ S. Sherman dopoulos of the Mathematics Research Cen­ Associate Secretary ter of the University of Wisconsin will ad­ Bloomington, Indiana dress SIAM on Generalized functions.

MEMORANDA TO MEMBERS

LIST OF GRADUATE ASSISTANTSHIPS TO BE PUBLISHED IN DECEMBER

In the past, part two of the January the early closing dates for applications to is sue of .cAfot:iceD has been devoted to some institutions. graduate assistantships. However, this list The addition of this special issue will henceforth be published as a separate will bring the total to eight issues of the issue in December to allow for the dis­ c){oticei) per year instead of seven. semination of this information in advance of

666 Six Hundred Twenty-Ninth Meeting U niversi ty of California Berkeley, California December 29, 1965

The six hundred twenty-ninth meet­ found in recent issues of Science magazine. ing of the American Mathematical Society The main registration desk for the will be held on Wednesday, December 29, meeting will be located on the ground level 1965 at the University of California at of the Student Union Building. The regis­ Berkeley. This meeting will be a part of tration desk will be open from 8:00 A M. the Annual Meeting of the American Asso­ to 8:00 P.M. throughout the meeting. The ciation for the Advancement of Science. AAAS has a registration fee of $5.00. The By invitation of the Committee to return on this investment includes a copy Select Hour Speakers for Far Western of the General Program of the AAAS meet­ Sectional Meetings, an hour address will ing, admission to the exhibits and the be presented by Professor T. M. Cherry Science Theater, and the warm feeling of of the University of Melbourne and the having helped pay a share of the expenses University of Washington, Professor of the meeting. Nonmembers of the AAAS Cherry's talk, entitled "Biharmonic bound­ can register in advance or at the meeting ary value problems", will be given at on the same basis as members. Payment 1:30 P.M. in Room 155 Dwinelle Hall, of the $5.00 registration fee will be wel­ There will be sessions for contributed comed, although this is not mandatory. papers at 9:30 A.M. and at 3:00 P.M. in Accommodations near the campus California Hall. are available either in hotels and motels,. The AAAS meeting will begin on or else in the University of California Sunday, December 26, and will end on Residence Halls. A $5.00 deposit is re­ Friday, December 31. Other events at this quired by all hotels and motels. Rooms in meeting which may be of special interest the Residence Halls are available for one to mathematicians include sessions of Sec­ or two persons per room, for couples, tion A (Mathematics) of AAAS on Monday, and for children 14 years or older. The December 2 7, and a meeting of the Society full amount for room rental will be col­ for Industrial and Applied Mathematics on lected in advance. The following table Thursday, December 30. Further informa­ gives the rates at the hotels, motels and tion concerning the AAAS program can be the Residence Halls. HOTELS SINGLE DOUBLE .IffiN_ Claremont (300) $11.00 $15,00 $15.00 Durant (200) 8,50* 12.00* Shattuck (250) 8,50 11,00 14.00 *A few single rooms at $5.50, twins at $7,50 MOTELS Berkeley House (112) $10.50 $14.50 $14.50 Berkeley Plaza (52) 7,00 8,50 9.50 Berkeley Travelodge (46) 8,00 10.00 11.00 California Motel (42) 6,50 7 .oo 8.00 Golden Bear (44) 7,00- 8,00 8,00 - 10,00 10.00-12.00 RESIDENCE HALLS Single occupancy-- $7.50 without meals; $8.50 with breakfast and lunch Two in a room-- $6.50 each without meals; $7,50 each with breakfast and lunch

667 Housing arrangements will be handled by and the Western Pacific Railroads. Most the AAAS Housing Bureau. A reservation major transcontinental airlines have flights form will be found on the inside of the back to the San Francisco International Airport. cover of these Notices • Reservations The Oakland International Airport is some­ for housing should be sent on or before what closer to Berkeley, but is served by December 5. fewer direct flights. There is helicopter Meal service will be available service from the San Francisco and Oak­ throughout the meetings in the Dining Com­ land Airports to Berkeley. Reservations mons adjacent to the Student Union Building. for helicopter transportation should be Persons who stay in the Residence Halls made along with flight reservations, since can obtain breakfast and lunch where they this generally brings a reduction in the are living by paying $1.00 each day. Of cost of the helicopter ticket. course, the San Francisco Bay area is R. S. Pierce noted for its many excellent restaurants. Associate Secretary Berkeley and San Francisco are served by the Santa Fe, the Southern Pacific,

Seventy-Second Annual Meeting Chicago Illinois January 24-28, 1966

SPECIAL SESSIONS AND CONTRIBUTED PAPERS

There will be a number of special Those contributing papers to the sessions for twenty-minute papers at the Annual Meeting who feel that they would Annual Meeting at Chicago, similar to those be appropriate to one of these sessions held at the three preceding Annual Meetings. should submit their abstracts a week earlier Papers will be presented at these sessions than the ordinary deadline, to allow time for by invitation and by selection from ten­ the additional handling- -that is, by Novem­ minute papers submitted for the meeting-­ ber 26. authors of those selected will have the As at recent Annual Meetings there opportunity of expanding their presentations will be at most two hundred ten-minute to twenty minutes. Topics presently under papers accepted for presentation at the consideration for these sessions and their regular sessions for contributed papers. chairmen are: Partial Differential Equations Felix Browder John W. Green Combinatorial Mathematics Secretary Herbert Ryser Los Angeles, California Algebraic Groups Robert Steinberg Seymour Sherman Markov Processes Associate Secretary Wendell Fleming Bloomington, Indiana

668 1965 SUMMER RESEARCH INSTITUTE ON ALGEBRAIC GROUPS AND DISCONTINUOUS SUBGROUPS

The Twelfth Summer Research In­ Groups and Discontinuous Subgroups, en­ stitute of the American Mathematical So­ compasses a broad and quite active field ciety was held from July 5 to August 6, that has attracted a large school of re­ 1965, at the University of Colorado in searchers. Therefore, the program was Boulder. The Society was fortunate in organized in such a manner as to feature securing the financial support of the Na­ new developments along with current re­ tional Science Foundation and the Office of search work and to allow for a maximum Naval Research for this program. The topic exchange of views among participants. under consideration was selected by the Formal lectures, informal conversations Society's Committee on Summer Institutes and seminars presented ample opportunity in an effort to survey the principal develop­ for speakers and participants to take ad­ ments in the arithmetic aspects of algebraic vantage of the Institute. The speakers and groups. their topics are listed below in the com­ plete scientific program. Scientific Program The subject of study, Algebraic

The Institute consisted mainly of survey lectures and informal seminars limited to five major themes. Subjects and their speakers are as follows: I. Algebraic linear groups and arithmetic groups N. Allan,A. Borel, F. Bruhat, P. Cartier, N. Iwahori, B. Kostant, H. Matsumoto, J. Tits.

II. Adelization, Tamagawa numbers, the Siegel-Weil formula, Galois cohomology, and approximation theorems P. Cartier, M. Kneser, R. P. Langlands, J. G. M. Mars, T. Ono, T. Springer, T. Tamagawa.

III. Automorphic forms A. Borel, R. Godement, Y. Ihara, R. P. Langlands, I. Sa: take

IV. The compactification and projective imbedding of arithmetically defined quotients of bounded symmetric domains, partial desingularization of the compactification, fiber ~ stems of olarized abelian varieties fields of moduli. f'-functions, 8-functions, -functions, Fourier-Jacobi series, symplectic re_p~.!!.t~iOJ!~_,. W. Baily, W. Hammond, J. -I. Igusa, M. Kuga, D. Mumford,!. Satake, G. Shimura.

V, Vector-valued cohomology and rigidity theorems H. Garland, S. Murakami

669 Lecture Notes Acknowledgements Copies of the lecture notes were The success of the 1965 Summer made available to members of the Institute Institute was due, in a large degree, to the in order to familiarize participants with intelligent and efficient planning of Dr. material. Copies were also sent to each of Gordon L. Walker, Executive Director of the supporting agencies and to individuals the American Mathematical Society, and upon request. However, due to plans for his office staff. Credit for the scientific formal publications, further copies ofnotes program presented by the Institute goes are no longer available for distribution. to the Joint Invitations and Organizing The Society will publish the lectures in Committee, consisting of W. Bailey of the 1966ina volume of the series, Proceedings University of Chicago, A. Borel (co-chai;r­ of Symposia in Pure Mathematics, to be man) of the Institute for Advanced Study, edited by A. Borel and D. Mostow. G. D. Mostow (co-chairman) of Yale Uni­ versity, A. Selberg of the Institute for Participants Advanced Study and T. Tamagawa of Yale Sixty-two mathematicians attended University. the Institute, in addition to their thirty wives and thirty-two children. Eighteen of A. Borel and G. D. Mostow, Co-Chairmen the participants were from foreign coun­ Joint Invitations and Organizing Com­ tries including Belgium, France, Germany, mittee Japan and the Netherlands. 1965 Institute.

1965 SUMMER SEMINAR ON RELATIVITY AND ASTROPHYSICS

The Fourth Summer Seminar on with basic information in both the fields applied mathematics was held from July 26 of selected study, Relativity and Astro­ to August 20, 1965, at Cornell University, physics, the Seminar organized its pro­ with the financial support of the Air Force gram around five series of basic lectures. Office of Scientific Research, the Atomic In addition, less general lectures were Energy Commission, the National Aero­ given on topics relating to the main sub­ nautics and Space Administration, the Na­ jects of the Seminar. Participants had tional Science Foundation, and the Office opportunity to hold smaller seminars and of Naval Research. Members of the sem­ to speak informally with the lecturers and inar are exceedingly grateful to the above major participants. The speakers and their agencies for contributing to the program. topics are listed below in the complete Scientific Program scientific pro,g-ram of the Seminar. Because of participants unacquainted

The series of lectures provided expressly to acquaint participants with the Seminar topics were as follows: I. General Relativity - A. Schild II. Galactic Dynamics and Galactic Structure - L. Woltjer III. Stellar Structure - E. E. Salpeter IV. Stability Problems - S. Chandrasekhar

670 Additional lectures relating to the basic series were presented by the following speakers: I. Relativity and Cosmology L. Schiff, M. Schmidt, W. Bonner, E. Schucking, R. Sachs, R. Kerr, J. Weber, R. Penrose, G. McVittie, F. Dyson, A. H. Taub, P. Peebles, D. Layzer, I. Robinson II. Glactic Structures E. M. Burbidge, D. Lynden-Bell, J. Linsley, I. King, C. C. Lin, G. Conto­ poulos, K. Prendergast, C. Hunter. III. Stellar Structure M. May, C. Misner

Lecture Notes degree prior to 19 60, 2 6 who received Mimeographed copies of the lecture their Ph.D. degree after 1960, 1 without a notes were made available to participants of degree, 4 major participants and 28 lec­ the Seminar. Additional copies were also turers. Thirteen of the participants came sent to each of the supporting agencies and from various foreign countries including to individuals upon request. However, there France, England, Canada, Mexico and Den­ are none left for further distribution due to mark. plans for formal publication of the proceed­ ings of the Seminar. Professor Jurgen Acknowledgements Ehlers has undertaken the responsibility of Although many individuals contrib­ editing three volumes based on lectures-. uted to the success of the 1965 session, Titles of the volumes are as follows{ Dr. Gordon L. Walker, Executive Director Lectures in Applied Mathematics of the of the American Mathematical Society and 1965 American Mathematical Society Dr. William Smith, Director of Summer Summer Seminar on Relativity and Session and Extramural Courses atCornell, Astrophysics were especially effective in planning the Volume I. Relativity and Cosmology Seminar so that it functioned smoothly. Volume II. Galactic Structure The scientific program was directed by the Volume III. Stellar Structure. Joint Invitations and Organizing Committee, consisting of S. Chandrasekhar, C. C. Lin, C. Misner, A. Schild, and A. l!· Taub.

Participants A. H. Taub, Chairman Eighty individuals attended the 1965 Joint Invitations and Organizing Summer Seminar. Of these, 45 were gradu­ Committee ate students, 8 who received their Ph.D. 1965 Seminar

671 THE ANNUAL SALARY SURVEY

This year's Annual Salary Survey is based on returns from 383 departments in mathematics and the mathematical sciences, covering 3888 academic positions held in 1964-1965 and 4315 positions in 1965-1966. A comparison between the academic years 1964-1965 and 1965-1966 shows a general increase in the salaries of all ranks in each group and an overall increase in staff size. Institutional non-members showed the largest percent increases in staff size in all ranks with a total increase of 17%. The greatest increase in any category was 38, 7%, occurring in the instructor rank of institutional non-members. Groups I and II, defined below, showed general increases in staff of 6.7% and 11.% respectively, although, interestingly, both of these groups showed a decrease in the instructor rank. The fastest growing rank in Group I was professor while the fastest growing rank in Group II was associate professor. The basis of the classification of institutions remains the same as in previous Annual Salary Surveys. Institutions are divided into two classes, Institutional Members and Institutional Non­ Members. Institutional Members are further divided into Group I and Group II according to the volume of their mathematical publications in the years 1959 through 1961. Group I is composed of institutions which during that time sponsored 37 1/2. or more pages in journals published or subsidized by the Society. Group II comprises those institutions which sponsored fewer than 37 1/2. pages during the same period. Each institution submitted a minimum, median and maximum salary figure for each of the four academic ranks, creating 2.4 categories of salary figures. The data presented here in each of the categories are the range of the middle 50% of all salary figure's received for that category. For example, the data in the following report shows the minimum salary during 1964-1965 for a Group I instructor ranges from $6000 to $7900, indicating that salaries in this category are greater than $7900 at 2.5% of the institutions reporting, and less than $6000 at 2.5% of the institutions. All salaries refer to an academic yearof9 or 10 months. Grants and contracts are included but sabbatical payments and other part-time salaries are excluded. All salary figures are given in hundreds of dollars. This survey is the ninth in an annual series begun in May, 1957 by the Society's Committee on the Economic Status of Teachers.

JN8TlfPTIONAL :MEMBEBS OF THE SOCIETY. GROUP I Number of usable returns: 89 Total number of the staffs working full tJme on the campus ~ 1984-1985 1965-1966 IDstructor (only those holc:Uug Ph. D.) 141 140 Asslatant Professor 802 827 Associate Professor 424 457 Professor 585 824 TOTAL M "'ii48 SALARY SURVEY 1964-1985 1985-1986 ~ Minimum ~ Maxlmum lWDimum Median :Maximum Instructor (only those holding Ph.D.) 80- 79 70- 80 70- 81 70- 80 72- 85 78- 90 Assistant Professor 78- 88 84- 92 93-107 81- 90 90- 97 98-115 Associate Professor 95-110 108-118 120-144 100-117 112-129 126-145 Professor 114-150 144-175 180-220 124-153 150-180 190-240

672 INSTITUTIONAL MEMBEHI OF THE SOCIETY. GROUP ll Number of usable returns: 84 Total number of the staffs working full time on the campus RANE: 1964-1965 1965-1966 Instructor (only those holding Ph. D.) 28 24 Assistant Professor 372 411 Associate Professor 267 311 Professor 268 293 TOTAL 935 1039 SALARY SURVEY 1964-1965 1965-1966 Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph.D,) 60- 75 65- 76 70- 85 61- 80 68- 85 74- 85 Assistant Professor 69- 83 76- 88 83- 96 72- 87 80- 94 89-102 Associate Professor 85-100 93-108 100-120 90-105 98-116 103-130 Professor 100-130 113-140 125-160 107-136 120-152 130-170

INSTITUTIONS WIDCH ARE NOT MEMBERS OF THE SOCIETY Number of usable returns: 230 Total number of the staffs working full time on the campus RANE: 1964-1965 1965-1966 Instructor (only those holdlng Ph. D.) 31 43 Assistant Professor 520 616 Associate Professor 319 379 Professor 351 390 TOTAL 1221 1428 SALARY SURVEY 1964-1965 1965-1966 Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph. D.) 54- 64 61- 71 65- 77 55- 70 66- 75 70- 85 Assistant Professor 65- 76 70- 84 76- 90 68- 82 75- 89 81- 96 Associate Professor 75- 95 80-100 86-105 80- 98 87-109 95-115 Professor 90-113 94-125 106-136 95-126 101-133 115-146

SUMMARY OF ALL INSTITUTIONS SURVEYED Number of usable returns: 383 Total number on the staffs worldng full time on the campus RANE: 1964-1965 1965-1966 Instructor (only those holding Ph. D.) 200 207 Assistant Professor 1494 1654 Associate Professor 1010 1147 Professor 1184 1307 TOTAL 3888 4315 SALARY SURVEY 1964-1965 1965-1966 Minimum Median Maximum Minimum Median Maximum

Instructor (only those holding Ph. D.) 57- 72 65- 76 70- 80 60- 78 67- 80 74- 85 Assistant Professor 66- 82 73- 88 80- .96 70- 86 77- 93 85-103 Associate Professor 81-101 88-110 95-120 85-107 94-117 101-130 Professor 95-130 106-145 120-180 102-136 115-155 126-195

673 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D.

This survey is compiled from questionnaires sent to individuals who received their Ph.D. in mathematics during 1964. There were 319 usable returns. The academic life again attracted the largest proportion of new Ph.D.'s in mathematics, 83.7"/o of the total reporting. Of these, 73.4% were primarily engaged in teaching, 10.1% in research, 11.6% in a combination of these, and 4.9% were on fellowships. Universities, rather than colleges, attracted the greater number of new Ph.D. appointments, taking almost three-quarfers of the new Ph.D.'s in teaching and all in research and fellowship appointments. Industry attracted the next largest number of new Ph.D. mathematicians; however, even with its significantly higher salaries it managed to attract only 8.1% of those reporting. Research institutions and government employment, both of which also offered generally higher beginning salaries than academic institutions, attracted a small 3.7% and 4.1%, respectively. This year three significant new groups under the academic category emerged from the data. One, teaching and research, was mentioned earlier. Another, totalling 5.2% of the academic category, was teaching and receiving a salary on a yearly basis. The third new group was comprised of those engaged in research at an academic institution on a yearly basis and constituted 3. 7% of the academic category. The comparison between teaching and :)."esearch appointments is of interest: while most teaching appointments require the academic year oniy, 37% of resea.rch appointments require a full year. Again the Northeast attracted the greatest number of new Ph.D.'s, 37.3% of the total. The Midwest attracted 22.9% and the South was next with 15.7o/o. The Far West, which last year was second place in popularity, attracted only 13.8% this year, and 5.6% went to the Southwest. A small 2.8% were employed abroad. An interesting correlation occurs between this general geographic distribution and the geographic distribution of the largest academic group, those teaching for 9/10 months. In that group, 35.0% went to the Northeast; 26.6% to the Midwest; 16.1% to the South; 13.9% to the Far West; 5.0% to the Southwest; and 1.7% abroad. The great majority of Ph.D.'s reporting had had some degree of experience before re­ ceiving their doctorate. 56.1% had had more than one year of experience and 14.1% had had between 1/2 and 1 year, while 26.6% had had less than 1/2 of a year's experience in their field prior to their first postdoctoral appointment. All salaries listed below are in hundreds of dollars. Universities, Colleges and Technical Institutes TEACHING RESEARCH FELLOWSHIP (Nine Month Salary) (Nine Month Salary) (Yearly Stipend)

Year Min. ~ Max. Min. Median ,Ms Year Min. Median Max. 1961 45 63 82 48 65 90 1962 43 70 92 45 65 90 1963 45 72 95 45 68 98 1963 45 65. 90 1964 41 79 110 60 72 105 1964 40 60 85 1965 54 82 115 71 81 90 1965 55 65 91

NEW CATEGORIES ADDED lN 1965 Min. ~ .Ms TEACHING AND RESEARCH (Nine Month Salary) 70 80 105 TEACHING (Twelve Month Salary) 78 104 121 RESEARCH (Yearly Salary) 81 93 107 Industry, Research Institutes and Government Employment lNDUSTRY RESEARCH INSTITUTES GOVERNMENT (Twelve Month Salary) (Twelve Month Salary) (Twelve Month Salary) Year Min. Median Max. Min. Median Max. Year Min. Median Max. 1961 87 110 174 84 110 142 1961 78 89 160 1962 90 115 162 60 100 145 1962 88 107 143 1963 105 120 185 55 117 135 1963 101 112 150 1964 104 132 168 90 118 170 1964 70 99 167 1965 100 136 180 75 94 121 1965 70 126 160

674 DOCTORATES CONFERRED IN 1964 (Supplementary List)

The following additions and corrections are supplementary to the list of doctorates appearing in the June, 1965 issue of these cJVoticti), pages 419-442.

UNIVERSITY OF COLORADO UNIVERSITY OF OREGON Drake, David Dimitroff, George Ernest On the representation of an abstract lattice Partially ordered spaces and local trees as the family of closed sets of a topologi­ (L. E. Ward) cal space Lee, Yu-Lee (W. J. Thron) Topologies with the same class of homeo­ Hage Gordon Berton morphisms On the class number function of certain (L. E. Ward) ternary quadratic forms (B. W. Jones) Hursch, Jack Lionel, Jr. The ordering of uniformities (W. J. Thron) CHANGES IN LISTING OF ADVISORS Porter, A. Duane Some systems of polynomial equations in a AUBURN UNIVERSITY finite field Ford, Jo Wharton; advisor (J. H. Hodges) B. J.Ball Wunderlich, Marvin Charles 1ULANE UNIVERSITY Sieve-generated sequences of natural num­ bers Bell, Harold; advisors (W. E. Briggs) G. S. Young and L. B. Treybig

675 PERSONAL ITEMS

Associate Professor J. W. ADDISON, has been appointed to the MacLaurin Chair JR. of the University of California, Berke­ of Mathematics at the University of Edin­ ley will be on sabbatical leave for the fall burgh, Edinburgh, England. semester, 1965. He will be in residence in Associate Professor G. E. BREDON Berkeley. of the University of California, Berkeley Dean A. A. ALBERT of the University has been awarded a Sloan Fellowship for of Chicago has received anhonoraryDoctor 1965-1967. He will be on leave from teach­ of Laws degree from the University of ing duties for the fall semester, 1965. Notre Dame. Associate Professor H. J. BREMER­ Mr. J. U. ALVES of the University of MANN of the University of California, California at Berkeley has been appointed Berkeley will be on sabbatical leave for the to an associate professorship at the Uni­ spring semester 1966. He will be in re­ versidade de Brasilia, Brasilia, Brazil. sidence in Berkeley. Professor A. G. ANDERSON of Western Mr. R. N. BRYAN of the University of Kentucky State College has been appointed Utah has been appointed to an assistant a Professor and Chairman of the Mathe­ professorship at Ithaca College. matics Department at Uppsala College. Dr. J. L. BRYANT of the University Dr. D. W. ANDERSON of Van Nuys, of Georgia has been appointed to an assist­ California has been appointed to an assist­ ant professorship at the University of ant professorship at the Massachusetts Mississippi. Institute of Technology. Assistant Professor D. H. CARLSON Mr. S. K. ATIYAH of Colorado Uni­ of Oregon State University will be on a versity has been appointed to an assistant Fulbright grant at the Institute de Mate­ professorship at Old Dominion College. matica y Estadistian, Montevideo, Uruguay Assistant Professor G. S. S. A VILA of until March 1966. the University of Wisconsin has been ap­ Professor Y. W. CHEN of Wayne State pointed to an associate professorship atthe University will be on leave for the academic Georgetown University. year 1965-1966 at the University ofMassa­ Professor W. G. BADE of the Univer­ chusetts. sity of California, Berkeley has been ap­ Assistant Professor J. A. CIMA of the pointed Research Professor in the Miller University of North Carolina has been ap­ Institute for Basic Research, Berkeley, for pointed to a professorship at the University the academic year 1965-1966. of Arizona. Dr. I. A. BARNETT of the University Dr. D. S. COHEN of the Rensselaer of North Carolina has been appointed to a Polytechnic Institute has been appointed to visiting professorship at Fairleigh Dickin­ an assistant professorship at the California son University, Teaneck Campus. Institute of Technology. Assistant Professor R. J. BEAN of the Assistant Professor S. H. COLEMAN University of Wisconsin has been appointed of the University of Wisconsin has been to an assistant professorship at the Uni­ appointed to an associate professorship at versity of Tennessee Georgia Institute of Technology. Dr. N. P. BHATIA of Berla College, Dr. MIKLOS CSORGO of Princeton P ilani, India has been appointed to an University has been appointed to an assist­ associate professorship at the Western ant professorship at McGill University, Reserve University for the academic year Montreal, Quebec, Canada. 1965-1966. Dr. B. V. DEAN has been appointed Professor F. F. BONSALL of Pure Professor of Organizational Sciences and Mathematics at the University ofNewcastle Chairman of the Operations Research Group upon Tyne, Newcastle upon Tyne, England at Case Institute of Technology.

676 Dr. F. R. DEUTSCH of Brown Univer .. the Department of Sociology at Princeton sity has been appointed to an assistant pro­ University. fessorship of Mathematics and Computer Assistant Professor A. C. FLECK of Science at the Pennsylvania State University. Michigan State University has been appointed Professor PHILIP DWINGER of the an AssistantProfessorandDirectorof Pro­ Technological University of Delft, Delft, gramming at the University of Iowa. Netherlands has been appointed to a pro­ Dr. W. W. FLEXNER, having reached fessorship at the University of Illinois in the retirement age at the United Nations, Chicago. has been appointed to an associate pro­ Professor ELDON DYER of Rice Uni­ fessorship at the Cooper Union School of versity has been appointed to a visiting Engineering. professorship at Columbia University for Dr. S. P. FRANKLIN of the University the Fall Semester. of Florida has been appointed to an assist­ Dr. C. J. EARLE, JR. of the Institute ant professorship at the Carnegie Institute for Advanced Study has been appointed to of Technology. an assistant professorship at Cornell Uni­ Assistant Professor S. M. GERSTEN versity. of Rice University has received a National Professor E. G. EFFROS of the Uni­ Science Foundation Postdoctoral Fellow­ versity of Pennsylvania is on leave during ship award and will be at the Mathematics the 1965-1966 academic year at the Math­ Institute at Oxford University, Oxford, ematics Institute, Aarhus, Denmark. England. Dr. L. C. EGGAN of the University of Professor MURRAY GERSTENHABER Michigan has been appointed an associate of the University of Pennsylvania is on professor and Chairman at the Pacific Luth­ leave during the Fall semester of 1965- eran University. 1966 at the Institute for Advanced Study. Dr. J. A. ERNEST of the University of Dr. ABRAHAM GINZBURG of Technion­ Rochester has been appointed to a visiting Israel Institute of Technology, Haifa, Israel assistant professorship at the University has been appointed to a visiting associate of California, Berkeley for the academic professorship at the Carnegie Institute of year 1965-1966. Technology. Dr. E. B. FABES of the University of Professor A. M. GLEASON of Harvard Chicago has been appointed to an assistant University has been appointed to a visiting professorship at Rice University. professorship at the Massachusetts Institute Professor CARL FAITH of Rutgers, of Technology for the fall term 1965-1966. The State University has been awarded a Dr. C. F. GODINO of Columbia Uni­ Rutgers Faculty Fellowship at the Univer­ versity has been appointed to an assistant sity of California at Berkeley as a visiting professorship at Brooklyn College. scholar for the academic year 1965-1966. Dr. W. J. GORDON of Brown Univer­ Professor KY FAN of Northwestern sity has accepted a position as an Asso­ University has been appointed to a profes­ ciate Senior Research Mathematician with sorship at the University of California at the General Motors Research Laboratories, Santa Barbara. Warren, Michigan. Professor J. M. G. FELL of the Uni­ Mr. C. M. GRAM of the Royal Tech­ versity of Washington has been appointed nical University, Copenhagen, Denmark to a professorship at the University of has been appointed a Lecturer and assist­ Pennsylvania. Dr. Fell is on a leave of ant research mathematician in the Com­ absence at the University of California, puter Center at the University of California, Berkeley during the academic year 1965- Berkeley for the academic year 1965-1966. 1966. Assistant Professor ALFRED GRAY Mr. j. C. FERRAR of Yale University of the University of California, Berkeley has been appointed to an assistant profes­ has been awarded a National Science Found­ sorship at Ohio State University. ation Postdoctoral Fellowship for the spring Dr. C. S. FISHER of the University of semester, 1966. He will be on leave from California at Berkeley has been awarded teaching duties for the spring semester. a National Science Foundation Postdoctoral He will be in residence in Berkeley. Fellowship and will be a visiting Fellow in Assistant Professor F. I. GROSS of

677 Occidental College has been appointed to in charge of the graduate program in en­ an assistant professorship at the University gineering. of Alberta, Alberta, Canada. Mr. S. M. ISAAK of Wisconsin State Professor EMIL GROSSWALD of the University, Whitewater has been appointed University of Pennsylvania has just returned to an assistant professorship at Wayne after a leave of absence at the University State University. of Paris, Paris, France. Professor J. R. ISBELL of the Uni­ Dr. H. R. HALKIN of the Bell Tele­ versity of Washington has been appointed phone Laboratories Incorporated, Whippany, to a professorship at the Case Institute of New Jersey has been appointed to an asso­ Technology. ciate professorship at the University of Dr. NAGA YOSHI IWAHORI of the Uni­ California, San Diego. versity of Tokyo, Tokyo, Japan has been Professor P. R. HALMOS of the Uni­ appointed visiting Professor and Research versity of Michigan has been appointed to Mathematician at the University of Cali­ a visiting professorship at the University fornia, Berkeley for the academic year of Miami for the academic year 1965-1966. 1965-1966. Dr. J. W. HAMBLEN, former Director Associate Professor GUY JOHNSON, of the Data Processing and Computer Cen­ JR. of Rice University has been appointed ter of Southern Illinois University, has to a visiting professorship at Syracuse joined the staff of the Southern Regional University for the academic year 1965- Education Board as Project Director for 1966. the Regional Development Program for Associate Professor B. F. JONES, JR. Computers and Computer Sciences. of Rice University has been appointed to a Dr. P. C. HAMMER of the University Membership at The Institute of Advanced of Wisconsin has been appointed to a pro­ Study for the academic year 1965-1966. fessorship of Mathematics and Computer Assistant Professor R. E. D. JONES Science at the Pennsylvania State Univer­ of the University of Wichita has been ap­ sity. Professor Hammer is also Head of pointed to an associate professorship at the Computer Science Department. the University of Missouri at Rolla. Assistant Professor M. E. HARRIS of Professor R. V. KADISON of the Uni­ Tufts University has been appointed to an versity of Pennsylvania is on leave during assistant professorship at the University the academic year 1965-1966 at the Math­ of Illinois, Chicago Center. ematics Institute, Aarhus, Denmark. Dr. LOUIS HAY of Smith College has Associate Professor COSTAS KASSI­ been appointed to an assistant professor­ MATIS of the University of Windsor has ship at Mount Holyoke College. been appointed to a professorship at Wayne Dr. MELVIN HENRIKSEN of Purdue State University. University has been appointed to a pro­ Professor TOSIO KATO of the Univer­ fessorship at the Case Institute of Tech­ sity of California, Berkeley will be on nology. sabbatical leave for the fall semester, 1965. Associate Professor A. P. HILLMAN He will be in residence in Berkeley with of the University of Santa Clara has been plans to visit Japan. appointed to an associate professorship at Assistant Professor J. A. KELINGOS the University of New Mexico. of Duke University has been appointed to Professor M. W. HIRSCH of the Uni­ an assistant professorship at the University versity of California, Berkeley has been of Minnesota. awarded a Sloan Fellowship for 1964-1966. Professor J. L. KELLEY of the Uni­ He will be on leave from teaching duties versity of California, Berkeley will con­ for the fall semester 1965. tinue his leave for the fall semester, 1965, Dr. P. F. HUL TQUIS'I: of Ball Brothers at Kanpur, India, taking part in the Kanpur Research Corporation, Boulder, Colorado Indo-American Project, of which the Uni­ has been appointed to a professorship of versity of California is a joint sponsor. Applied Mathematics and Electrical En­ Assistant Professor STANLEY KERT­ gineering at the Colorado Springs Center ZNER of the University of Massachusetts of the University of Colorado. He will be has been appointed to an assistant pro-

678 fessorship at Hofstra University. Science degree from the University. Associate Professor NAOKI KIMURA, Mr. TENG-SUN LIU of the University of the University of Oklahoma has been of Pennsylvania has been appointed to an appointed to a professorship at the Univer­ assistant professorship at the University sity of Arkansas. of Massachusetts. Mr. GABRIEL KLAMBAUER of Mc­ Professor J. S. LOMONT of the Poly­ Master University has been appointed to an technic Institute of Brooklyn has been ap­ assistant professorship at the University pointed to a professorship at the University of Ottawa. of Arizona. Associate Professor SHOSHICHI KO­ Dr. W. F. LUCAS of Princeton Univer­ BAYASHI of the University of California, sity has been awarded a Fulbright grant as Berkeley has been awarded a Sloan Fellow­ Lecturer at the Middle East Technical ship for 1964-1966. He will be on leave University in Ankara, Turkey during the from teaching duties for the spring semes­ academic year 1965-1966. ter, 1966. Assistant Professor J. E. MACK of Ohio Mr. DANIEL KOCAN of Columbia Uni­ University has been appointed to an asso­ versity has been appointed to an assistant ciate professorship at the University of professorship at the Stevens Institute of Kentucky. Technology. Dr. BERNARD MAS KIT of the Institute Dr. ADAM KORANYI of Princeton Uni­ for Advanced Study has been appointed to versity has been appointed to an associate an assistant professorship at the Massa­ professorship at the Belfer Graduate School chusetts Institute of Technology. of Science at Yeshiva University. Dr. j. J. MASTERSON of Purdue Uni­ Mr. j. D. KUELBS of the University of versity has been appointed to an assistant Minnesota has been appointed to an assist­ professorship at Michigan State University. ant professorship at the University of Wis­ Mr. D. E. MCLEOD of the University consin. of California at Riverside has been ap­ Assistant Professor W. B. LAFFER II pointed a Lecturer at the California State of the Western Washington State College College at San Bernardino. has been appointed to an associate pro­ Dr. C. K. MEGIBBEN of the University fessorship at Armstrong College. of Washington has been appointed to an Assistant Professor YU-LEE LEE of assistant professorship at the University the University of Connecticut has been ap­ of Houston. pointed to an assistant professorship at Dr. C. K. MILLER of the University of the University of Florida. Genoa, Genoa, Italy has been appointed to Dr. MILTON LEES of the California an assistant professorship atthe University Institute of Technology has been appointed of California, Berkeley. to a professorship at the Case Institute of Professor J. M. MITCHELL of Penn­ Technology. sylvania State University has returned after Professor D. H. LEHMER of the Uni­ a year's leave of absence attheMathemati­ versity of California, Berkeley will be on cal Research Center of the University of sabbatical leave for the academic year Wisconsin. 1965-1966. He will carry on research in Associate Professor C. C. MOORE of Berkeley and at the Australian National the University of California, Berkeley has University, Camberra, Australia. been awarded a Sloan Fellowship for 1965- Dr. S. A. LEVIN of the University of 1967. He will be on leave from teaching California at Berkeley has been appointed duties for the spring semester 1966. to an assistant professorship at Cornell Dr. R. E. MOORE of Lockheed Air­ University. craft Corporation, Palo Alto, California Dr. JOSEPH LEWITTES of Harvard has been appointed to an a.ssociate pro­ University has been appointed to an assist­ fessorship in the Computer Sciences De­ ant professorship at the Belfer Graduate partment at the University of Wisconsin. School of Science at Yeshiva University. Mr. C. R. NICOLA YSEN of the Con­ Professor j. E. LITTLEWOOD of the tinental Oil Company, Ponca City, Oklahoma University of Cambridge, Cambridge, Eng­ has been appointed to an assistant pro­ land has received an honorary Doctor of fessorship at the United States Naval Acad-

679 emy, Annapolis, Maryland. 1966. He will remain in residence during Assistant Professor E. A. NORDGREN the academic year. of the University of New Hampshire has Professor LAJOS PUKANSZKY of the been appointed to a visiting assistant pro­ University of Pennsylvania has just re­ fessorship at the University of Miami. turned after a leave of absence at the Uni­ Assistant Professor A. P. OGG of the versity of Paris, Paris, France. University of California, Berkeley will be Dr. K. W. REED, JR.ofTexasSouthern on sabbatical leave for the fall semester, University has been appointed to a visiting 1965. He plans to carry on research at assistant professorship at Rice University. Cambridge University, Cambridge, England. Dr. CHOON-JAI RHEE of the University Professor NORMAN OLER of the Uni­ of Georgia has been appointed to an assist­ versity of Pennsylvania is on leave during ant professorship at Randolph-Macon Wo­ the academic year 1965-1966 at the Mathe­ man's College. matics Institute, Aarhus, Denmark. Professor PAULO RIBENBOIM of Assistant Professor RICHARD O'NEIL Queen's University has been appointed to a of Rice University has been appointed visiting professorship at Northeastern Uni­ "Maitre de Conferences Associe" at the versity for the academic year 1965-1966. University of Montpellier, Montpellier, Associate Professor D. S. RIM of France for the academic year 1965-1966. Brandeis University has been appointed to Associate Professor J. M. OSBORN a professorship at the University of Penn­ of the University of Wisconsin will be on sylvania. leave for the academic year 1965-1966 at Dr. R. F. RINEHART of Case Institute Yale University. of Technology has been appointed to the Dr. B. N. PARLETThasbeenappointed ranking academic position of Senior Pro­ Assistant Professor and Assistant Research fessor at the U. S. Naval Postgraduate Mathematician in the Computer Center at School, Monterey, California. the University of California, Berkeley. Mr. T. J. ROBERTSON of the Univer­ Assistant Professor C. M. PEARCY sity of Missouri has been appointed to an of the University of Michigan has been ap­ assistant professorship at the University pointed to a visiting associate professor­ of Iowa. ship at the University of Miami for the Dr. C. D. ROBINSON of Arizona State academic year 1965-1966. University has been appointed to an asso­ Assistant Professor R. E. PEINADO ciate professorship at the University of of the University of Iowa has been appointed Mississippi. to a visiting professorship at the Univer­ 1st Lt. E. H. ROGERS of the U. S. sidad de Puerto Rico, Mayaguez, Puerto Army Ballistic Research Laboratories, Rico for the academic year 1965-1966. Aberdeen Proving Ground, Maryland has Assistant Professor J. A. PFALTZ­ been appointed to an assistant professor­ GRAFF of the University of Kansas has ship at the Rensselaer Polytechnic Institute. been appointed to a visiting assistant pro­ Professor MURRAY ROSENBLATT of fessorship at Indiana University. the University of California at San Diego Professor EDMUND PINNEY of the will be on a leave of absence at the Uni­ University of California, Berkeley will be versity College London, during the aca­ on sabbatical leave for the spring semes­ demic year 1965-1966. ter, 1966. He will be in residence in Dr. S. I. ROSENCRANS of the Massa­ Berkeley. chusetts Institute of Technology has been Dr. G. J. PORTER of the Massachu­ appointed to an assistant professorship at setts Institute of Technology has been ap­ Tulane University. pointed to an assistant professorship at Associate Professor A. A. SAGLE of the University of Pennsylvania. He is on Syracuse University is on leave at Yale leave during the academic year 1965-1966 University for the academic year 1965- at Brandeis University as an ONR Post­ 1966. doctoral Research Associate. Dr. MARTIN SCHECHTER ofNewYork Professor M. H. PROTTER of the University has been appointed Professor University of California,Berkeley is on sab­ and Chairman of the Mathematics Depart­ batical leave for the academic year 1965- ment at the Belfer Graduate School of Sci-

680 ence at Yeshiva University. University has been appointed to a visiting Professor I. J. SCHOENBERG of the professorship in engineering at the Uni­ University of Pennsylvania is continuing versity of California, Los Angeles for the his leave during the academic year 1965- academic year 1965-1966. 1966 at the U.S. Army, Mathematical Re­ Dr. J. A. THORPE of the Massachu­ search Center at the University of Wis­ setts Institute of Technology has been ap­ consin. pointed to an assistant professorship at Professor LOWELL SCHOENFELD of Haverford College. Pennsylvania State University has returned Dr. SHIGEAKI TOGO of the Hiroshima after a year's leave of absence at the University, Hiroshima, Japan has been Mathematical Research Center of the Uni­ appointed Visiting Professor and Visiting versity of Wisconsin. Research Mathematician at the University Dr. M. H. SCHULTZ of Harvard Uni­ of California, Berkeley for the academic versity has been appointed to an assistant year 1965-1966. professorship at the Case Institute of Tech­ Dr. D. B. J. TOMIUK of Paris, France nology. has been appointed to an associate pro­ Professor ABRAHAM SEIDENBERG of fessorship at the University of Ottawa. the University of California, Berkeley will Dr. PHILIPPE TONDEUR of the Uni­ be on sabbatical leave for the spring sem­ versity of Zurich, Zurich, Switzerland has ester 1966. He will carry on research in been appointed Lecturer and assistant re­ Rome, Italy. search mathematician at the University of Mr. H. S. SHANK of the University of California, Berkeley. Chicago has joined the Center for Naval Dr. D. M. TOPPING of the University Analyses of Franklin Institute. of Chicago has been appointed to an assist­ Dr. A. S. SKIDMORE of Western Re­ ant professorship at the University of Wash­ serve University has been appointed to an ington. assistant professorship at Rollins College. Dr. C. T. C. WALL, Reader in Mathe­ Dr. L. W. SMALL of the University of matics at Oxford University, has been ap­ Chicago has been appointed to an assistant pointed to a second Chair of Pure Mathe­ professorship at the University of Califor­ matics at the University of Liverpool. nia, Berkeley. Professor D. W. WALL of the Uni­ Dr. R. M. SOLOVAY of Princeton versity of Iowa has been named a Research University has been appointed to an assist­ Professor for the Fall Semester 1965-1966 ant professorship at the University of at the University of Newcastle, Newcastle California, Berkeley. upon Tyne, England. Mr. D. z. SPICER of the University of Professor A. H. WALLACE of Indiana Minnesota has been appointed to an assist­ University has been appointed to a pro­ ant professorship at the University of fessorship at the University of Pennsylvania. California, Los Angeles. Associate Professor GEORGE WHAP­ Assistant Professor A. P. STOKES of LES of Indiana University will be on sab­ the Catholic University of America has batical leave for the first semester of the been appointed to a professorship at George­ academic year 1965-1966 at the University town University. of Pennsylvania. Dr. KONDAGUNTA SUNDARESAN of Dr. HAROLD WIDOM of Cornell Uni­ the University of Washington has been ap­ versity has been appointed to a visiting pointed to an assistant professorship at professorship at the University of Califor­ the Carnegie Institute of Technology. nia, Berkeley for the academic year 1965- Dr. G. D. TAYLOR of the University 1966. of Michigan has been appointed to an assist­ Dr. N. M. WIGLEY of the Los Alamos ant professorship at the University of Scientific Laboratory, Los Alamos, New Arizona. Mexico has been appointed to an assistant Assistant Professor S. D. TELLMAN professorship at the University of Arizona. of Pomona College has been appointed to Professor R. L. WILDER of the Uni­ an assistant professorship at the University versity of Michigan has been appointed to of Arizona. membership on the Committee on Science Professor T. Y. THOMAS of Indiana and Public Policy of the National Academy

681 of Sciences-National Research Council. SRISAKDI CHARMONMAN, McMaster Dr. R. K. WILLIAMS of Vanderbilt University, to an assistant professorship. University has been appointed to an assist­ EDWIN DUDA, University of Miami, ant professorship at the Southern Metho­ to an associate professorship. dist University. ALFRED GRAY, University of Cali­ Dr. T. J. WILLMORE of the University fornia, Berkeley, to an assistant professor­ of Liverpool, has been appointed a Profes­ ship. sor of Pure'Mathematics at the University F. P. GREENLEAF, University of Cali­ of Durham. fornia, Berkeley, to an assistant professor­ Associate Professor j. A. WOLF of ship. the University of California at Berkeley H. G. HELFENSTEIN, University of has been appointed to a professorship at Ottawa, to a professorship. the Institute for Advanced Study. SIGURDUR HELGASON, Massachusetts Dr. HUNG HSI WU of Princeton Uni­ Institute of Technology, on leave during the versity has been appointed to an assistant academic year 1965-1966, to a professor­ professorship at the University of Cali­ ship. fornia, Berkeley. W. N. HUFF, University of Oklahoma, Dr. OSWALD WYLER of the University to a professorship. of New Mexico has been appointed to a pro­ j. P. LEVINE, University of California, fessorship at the Carnegie Institute of Berkeley, to an associate professorship. Technology. Dr. M. W. LODATO of the Mitre Cor­ Dr. P. R. YOUNG of Reed College has poration, Bedford, Massachusetts, to Head been awarded a National Science Foundation of the Operations Analysis Subdepartment Postdoctoral Fellowship at Stanford Uni­ of the Applied Mathematics Department. versity for the academic year 1965-1966. ARNE MAGNUS, University of Colo­ Dr. E. H. ZARANTONELLO of the rado, to a professorship. University of Cordoba, Cordoba, Argentina A. P. MA TTUCK, Massachusetts Insti­ has been appointed to a visiting professor­ tute of Technology, to a professorship. ship at the University of California, Berke­ THEODORE MITCHELL, SUNY atBuf­ ley for the spring semester 1966. falo, to an associate professorship. Professor N. R. ZITRON of Purdue C. C. MOORE, UniversityofCalifornia, University, on a Fulbright Research Grant Berkeley, to an associate professorship. at the Technical University of Denmark, R. T. MOORE, University of California, Lyngby, Denmark has been appointedAsso­ Berkeley, to an assistant professorship. ciate Research Mathematician at the Radia­ TAKASHI ONO, University of Pennsyl­ tion Laboratory of the University of Michi­ vania, to a professorship. gan, on leave of absence from Purdue Uni­ C. R. PATT, Belfer Graduate School versity. of Science at Yeshiva University, to an assistant professorship. The following promotions are announced: F. P. PETERSON, Massachusetts In­ stitute of Technology, to a professorship. W. F. AMES, University of Delaware, C. C. PUGH, University of California, to a professorship. Berkeley, to an assistant professorship. R. W. BAGLEY, University of Miami, D. G. QUILLEN, Massachusetts Insti­ to a professorship. tute of Technology, to an assistant pro­ PIERRE BERTHIAUME, University fessorship. of Ottawa, to an assistant professorship. MARC RIEFFEL, University of Cali­ MANUEL BLUM, Massachusetts Insti­ fornia, Berkeley, to an assistant professor­ tute of Technology, to an assistant profes­ ship. sorship. WOLFGANG SCHMIDT, University of G. E. BREDON, University of Califor­ Colorado, to a professorship. nia, Berkeley, to an associate professor­ j. M. SHAPIRO, Ohio State University, ship. to an associate professorship. LUTZ BUNGART, University of Cali­ YASUTAKA SIBUYA, University of fornia, Berkeley, to an assistant professor­ Minnesota, to a professorship. ship. A. F. STREHLER, Carnegie Institute

682 of Technology, to Associate Dean of Gradu­ ASTER, A. K. SNYDER, L. H. THARP; ate Studies. University of Michigan: T. F. STORER, R. R. STRUIK, University of Colorado, B. A. TAYLOR; Northeastern University: to an associate professorship. V. Y. KRAINES; Ohio Wesleyan University: R. H. SZCZARBA, Yale University, to J. S. BIDDLE; Pasadena College: J. E. an associate professorship. GROVES; University of Pennsylvania: JOHN J. W. TOOLE, University of Maine, V. LEAHY; University of Rochester: SAM­ to an associate professorship. UEL MERRILL III; Technische Hogeschool, F. W. WARNER III, University of Cali­ Delft, Netherlands: L. A. M. VERBEEK; fornia, Berkeley, to an assistant professor­ Temple University: H. C. WASSERMAN; ship. University of Washington: SRINIVASA H. S. WILF, University of Pennsylvania, RAMANUJAM; Wellesley College: B. L. to a professorship. AUSLANDER.

The following appointments to Instructor­ Deaths: ships are announced: Dr. C. E. CLARK of the Systems De­ Alabama Christian College: W. A. velopment Corporation, Santa Monica, Cali­ CRABTREE, JR.; Arkansas State College: fornia died June 16, 1965. He was a mem­ W. R. LIVINGSTON; Brown University: ber of the Society for 28 years. HENRY POHLMANN; UniversityofCalifor­ Professor EVAN JOHNSON, JR. of nia, Berkeley: DAVID BELL, M. G. CRAN­ Pennsylvania State University died on July DALL, FRED GALVIN, L. J. LIPKIN, 0. C. 13, 1965 at the age of 55. He was a member McGEHEE, A. V. PHILLIPS; Harvard Uni­ of the Society for 34 years. versity: W. B. ARVESON; Lafayette College: Professor R. H. MOUNT JOY of the J. D. BADGER; Lincoln University: J. A. University of Maryland died May 23, 1965 SANDERS; Massachusetts lnstitute of Tech­ at the age of 34. He was a member of the nology: H. P. ALLEN, T. F. BICKEL, Society for 10 years. THOMAS BLOOM, EGBERT BRIESKORN, Associate Professor G. E. SCHWEI­ HARRY DYM, BRIAN HARTLEY, A. W. GERT of the University of Pennsylvania KNAPP, D. P. KRAINES, R. G. LARSON, died on July 13, 1965 at the age of 58. He E. G. K. LOPEZ-ESCOBAR, A. B. MAN- was a member of the Society for 33 years.

NEWS ITEMS AND ANNOUNCEMENTS

THE BRITISH MATHEMATICAL COLLOQUIUM

The 18th Annual Meeting of the Invited addresses will also be given British Mathematical Colloquium will be by: R. H. Bott, D. A. Burgess, A. L. S. held at Imperial College, London in the Corner, G. Dirac, J. F. P. Hudson, J. F. C. University of London, from March 29th to Kingman, D. G. Northcott, C. Pommerenke, April 2nd, 1966. and J. E. Roseblade. The following lectures will be given: In addition there will be meetings of splinter groups' for shorter papers. Ac­ R. V. Kadison (New York): A Survey of commodation will be provided in University the Theory of Algebras of Hilbert Halls of Residence. Space Operators. Further information, and application I. Kaplansky (Chicago): Recent advances forms, may be obtained from the Secretary: in commutative algebra. Dr. T. Kovari, Department ofMathematics, B. Segre (Rome): Galois Geometries and Imperial College, London, S. W. 7. Combinatorial Analysis.

683 NEW AMS PUBLICATIONS

INDEX TO AMS SELECTED TRANSLATIONS

A Selected Translations Index to be include complete bibliographic information published in 1966 by the American Mathe­ of the original article as well as the AMS matical Society will be a cumulative index series title, volume, and pages. There will which will include Selected Translations in also be other types of listings; for example, Mathematical Statistics and Probability and a listing according to the Russian journal Selected Translations Series I and Series II. from which the article was translated. The major listing will be alphabetical, The date for publication for this by authors, of all articles translated in Index is March 1, 1966. Pre-publication Selected Translations, Series I, Volumes prices for orders received before January 1-50 of Series II, and Volumes 1-5 of 1, 1965 are: $5.30 List, $3.98 Members. Selected Translations of Mathematical After January 1, the price will be at Statistics and Probability. Each entry will least $5.80 List, $4.35 Members.

5-YEAR MATHEMATICAL REVIEWS INDEX

In 1966, the American Mathematical including all current systems of trans­ Society will publish a 5-Year Mathematical literations of Chinese names. Also included Reviews Index in 2. volumes which will will be errata applicable to the zo- Year cover the years 1960-1964 and volumes Index. 2.1-2.8. This is a continuation ofthe2.0-Year The date of publication for the 5- Mathematical Reviews Index and follows the Year Mathematical Reviews Index is March same format. However, even though this 1, 1966. Pre-publication prices for orders Index will cover only 5 years instead of 2.0, received before January 1, 1966 are: $35.40 it will include over 6,000 reviews and will List, $2.6.55 Members. After this date the be almost as large as the 2.0- Year Index. price will be at least $39.30 List, $2.9.48 The new Index will present a list of Members. common Chinese characters and a table

MEMOIR Number 35 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATIONS By Richard Bellman and K. L. Cooke

96 pages; List Price ~.90; Member Price the asymptotic series for .the solutions of $2.18. linear differential-difference equations whose coefficients possess asymptotic Reprint of a Memoir providing the first series. detailed discussion of the deterlll-ination of

684 Recent Reprints

COLLOQUIUM PUBLICATIONS Volume 3Z TOPOLOGY OF MANIFOLDS By R. L. Wilder 4ZO pages; List Price $J .40; Member Price ology, and the book as a whole will give $7 .os. the general reader a very good idea of how these ·two branches of topology have gradu­ Since its original publication in 1949 ally come, to a certain extent, to be amal­ (with some slight additional material for gamated. The remaining nine chapters. 1963 reprinting regardingnew proof meth­ carry out the avowed program of the book, ods and references to solutions of some namely to investig.ate the properties (in of the unsolved problems), this standard particular, the "positional invariants" -gen­ work has always been of great interest eralizing the Jordan curve theorem and its both to specialists and non-specialists in converse) of higher-dimensional manifolds topology. The first five chapters pxesent defined by a suitable generalization of the all the basic information in expository classical definition (which was topologically form, Chapters I-IV for general point set satisfactory only for dimensions 1 and Z). topology and Chapter V for algebraic top-

Errata

On page 537 in the August issue of Partial Differential Equations in Mathe­ these cfloW:tiJ the prices of Proceedings matical Physics," were misquoted. They of Symposia in Applied Mathematics, Vol­ should be List Price, $8.00; Member price ume 17, entitled "Applications of Nonlinear $6.00.

NEWS ITEMS AND ANNOUNCEMENTS

VISITS TO U.S.S.R. SUPPORTED BY GRANTS

American scientists wishing to visit after its expiration at the end of this year. the U.S.S.R. during the 1966-1967 academic Generous financial support is avail­ year may submit applications to the National able through the National Academy. Partic­ Academy of Sciences. Approximately five ipants can receive transportation to and persons may make one month visits for the from the U.S.S.R. and per diem allowance purpose of familiarization with Soviet sci­ for meals. Persons staying for three or ence, and about 13 may make more exten­ more months will be reimbursed for salary sive visits of three to ten months, according lost during that time. Further financial aid to an agreement between the NAS and the can be obtained by persons who remain five Academy of Sciences of the U.S.S.R. The or more months and wish to take their agreement requires that applicants be U.S. families. The deadline for application to citizens and have a doctoral degree or its this program is November ZZ. Further in­ equivalent in physical, biological, or be­ formation can be obtained by writing to havioral science, mathematics, or engin­ the Office of the Foreign Secretary, NAS, eering. This program is contingent upon Washington Z0418. the renewal of the NAS- USSR agreement

685 SUPPLEMENTARY PROGRAM-Number 34

During the interval from July 3, 1965 through September 3, 1965 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Contributed Papers in this issue of these dfotitxJJ. One abstract presented by title may be accepted per person per issue of the c}(oticti). Joint authors are treated as a separate category; thus in addition to abstracts from two authors individually one joint abstract by them may be accepted for a particular issue. ( 1) Representation of simply ordered sets (9) Sieves with generalized intervals and the generalized continuum hypo­ Professor R. G. Buschman and Pro­ thesis. I. fessor M. C. Wunderlich, SUNY at Professor Alexander Abian and Mr. Buffalo (65T-424) David Deever, The Ohio State Uni­ (10) The means of order t, and the laws of versity (65T-428) thermodynamics (2) Refinement-unbounded set functions Dr. E. D. Cashwell and Dr. C. J. and absolute continuity Everett, Los Alamos Scientific Labo­ Professor W. D. L. Appling, North ratory, Los Alamos, New Mexico Texas State University ( 65T-412) (65T-395) ( 3) Convergence of lacunary random power ( 11) Flows with continuous spectra series Professor R. V. Chacon, Ohio State Mr. L. Arnold, Technische Hoch­ University (65T-399) schule Stuttgart, Germany (65T-408) (12) Continuity of generalized rational ap- (Introduced by Professor A. T. proximation Bharucha- Reid) Professor E. W. Cheney, University (4) The Vitali approximation of automor­ of Texas and Professor H. L. Loeb, phic forms of Fuchsian groups by Aerospace Corporation, El Segundo, Poincare series of rational functions California (65T-449) Mr. David Bell, Brown University (13) Enumeration by rank of the element (65T-368) of the free distributive lattice with (5) An iterative method for computing the seven generators generalized inverse of an arbitrary Professor Randolph Church, U. S. matrix Naval Postgraduate Schgol, Mon­ Dr. Adi Ben-Israel, Technion, Israel terey, California (65T-447) Institute of Technology, Haifa, Israel ( 14) The intrinsic characteristic forms (65T-396) of the steady state equations of non­ (6) On the distortion of a pseudo-con­ equilibrium hydrodynamics formally invariant metric under hom• Professor Nathaniel Coburn, Univer­ eomorphism sity of Michigan (65T-402) Professor Stefan Bergman, Stanford ( 15) A Vietoris theorem for simple homo­ University (65T-398) topy type (7) Cubes in cubes Mr. M. M. Cohen, University of Professor D. G. Bourgin and Pro­ Michigan (65T-405) fessor C. W. Mendel, University of (16) SU-cobordism and KO(X). Illinois ( 65T- 376) Professor P. E. Conner and Pro­ (8) Interior-like elements of the positive fessor E. E. Floyd, University of cone Lp Virginia ((~5T-401) Professor C. C. Braunschweiger, ( 17) The Mobius function of a lattice University of Delaware (65T-377) Professor H. H. Crapo, University

686 of Waterloo (65T-426) ring. Preliminary report (18) An integral for Cesaro summable Professor R. W. Gilmer, Jr., Florida series State University ( 65T-423) Professor George Cross, University ( 30) Uncountably many contractible open of Waterloo {65T-389) 4-manifolds ( 19) Bounds for functions holomorphic in Professor L. C. Glaser, Rice Uni­ analytic polyhedra versity (65T-400) Mr. E. B. Davis, Stanford University {31) Finite sets on smooth curves and sur­ ( 65T-419) faces {20) A characterization of the quaternion Professor H. W. Guggenheimer, Uni­ algebra Q versity of Minnesota (65T-441) Professor John DeCicco, Illinois In­ (32) Inversion and representation of the stitute of Technology and Professor reduced Poisson-Hankel transform R. V. Anderson, Chicago Teachers Professor D. T. Haimo, Southern College South (65T-388) Illinois University and Harvard Uni­ (21) On (k - I)-connected (2k + I)-mani­ versity (65T-383) folds. I. Preliminary report (33) Quasi regular elements and Dorroh Mr. Rodolfo DeSapio, Stanford Uni­ extensions versity (65T-375) Professor Franklin Haimo, Wash­ (22) Distinctness and strong distinctness ington University (65T-382) of certain semigroups of operators. (34) Random matrices and graphs Preliminary report Dr. B. R. Heap, National Physical Mr. R. P. Dickinson, Jr ·• Lawrence Laboratory, Teddington, England Radiation Laboratory, Livermore, (65T-417) California (65T-427) (35) Transfer and compactness properties (Introduced by Professor Takazuki Tamura) of some generalized quantifiers {23) On sets represented by formulas of Mr. Martin Helling, University of consistent Rosser theories California, Berkeley (65T-443) Professor Robert DiPaola, Univer­ (Introduced by Mr. J. H. Silver) sity of California, Los Angeles (65T- (36) On a theorem of Kac and Achiezer 450) Professor I. I. Hirschman, Jr., {24) Imbeddings of a graph and the Betti Washington University (65T-435) number (37) An imbedding theorem for countable Dr. R. A. Duke, University of Vir­ ZF models ginia and University of Washington Dr. Ronald Jensen, Bonn University, (65T-390) Bad Godesberg, Germany (65T-434) (25) A divisibility property of Euler's phi- (Introduced by Professor Theodore function Hail per in) Dr. C. J. Everett, Los Alamos Sci­ ( 38) Cylinders and positive reducibility. entific Laboratory, Los Alamos, New Preliminary report Mexico (65T-407) Mr. C. G. J ockusch, Jr., Mas sachu­ (26) A construction of cardinally maximal setts Institute of Technology ( 65T- classes of nonequivalent order types 433) Mr. Charles F efferman, University of (Introduced by Professor Hartley Rogers) Maryland (65T-409) (39) Riemann functions for systems of (Introduced by Professor Carol Karp) linear hyperbolic equations (27) Approximating dominant character­ Dr. Richard Kraft, Negev Institute, istic roots of a matrix Beer Sheva, Israel (65T-420) Dr. L. E. Fuller, Kansas State Uni­ ( 40) Variations of finite independent sets in versity (65T-442) a Banach space (28) An asymptotic phase for a Sturm­ Professor Svetozar Kurepa, Univer­ Liouville operator with unbounded po­ sity of Zagreb, Yugoslavia ( 65T- 380) tential ( 41) Totally ordered partitions of a partly Professor R. C. Gilbert, California ordered set State College at Fullerton (65T-411) Professor C. W. Leininger, Arlington (29) The pseudo-radical of a commutative State College ( 65T-43 7)

687 (42) Rational approximations Professor B. L. Osofsky, Rutgers, Mr. H. L. Loeb, Aerospace Cor­ The State University (65T-430) poration, El Segundo, California (56) Piercing points of homeomorphisms. (65T-394) Preliminary report (43) On defining well-orderings. II Mr. J. L. Paul, Western Reserve Dr. E. G. K. Lopez-Escobar, Massa­ University ( 65T-404) chusetts Institute of Technology (57) Topological aspects of function semi- (65T-374) groups (44) A note on potential recursiveness of Professor Sidney Penner, The City regressing functions College of New York and Mr. K. J. Professor T. G. McLaughlin, Uni­ Schroeder, SUNY at Buffalo (65T- versity of Illinois (65T-436) 425) (45) Computable classes of real numbers. (58) Some results on n-books, in E 3 Preliminary report Mr. C. A. Persinger, Air Force Mr. B. H. Mayoh, University of Institute of Technology, Wright-Pat­ Oslo, Blindern, Norway ( 65T-41 0) terson Air Force Base, Ohio (65T- ( 46) Distributions associated with the 403) quadrivariate normal (59) K*BG studied via the Moore-Eilenberg Dr. K. S. Miller and Mr. Harold spectral sequence Sackrowitz, Columbia University Mr. T. E. Petrie, Institute for De­ (65T-448) fense Analyses (65T-406) (4 7) Extension of the Minkowski-CaratMo- (60) A note on subsolvable groups dory theorem on convex hulls Professor R. E. Phillips, Wisconsin Professor T. S. Motzkin, University State University and Mr. C. R. Com­ of California, Los Angeles (65T- brink, The University of Kansas 385) (65T-429) (48) Mean approximation on an interval (61) Extrema in space-time for an exponent less than one Professor L. V. Quintas, St. John's Professor T. S. Motzkin, University University and Professor Fred Sup­ of California, Los Angeles and Pro­ nick, The City College of New York fessor J. L. Walsh, Harvard Uni­ (65T-440) versity (65T-381) (62) I. On some problems of Erdos and (49) Some new estimates in prime-number Hajnal theory. Preliminary report Mr. W. N. Reinhardt and Mr. J. H. Mr. A. A. Mullin, University of Silver, University of California, California, Livermore (65T-371) Berkeley (65T-445) (50) Determinateness, measurability and ( 63) The eigenproblem for displacement the property of Baire integral equations Mr. Jan Mycielski, Polish Academy Mr. A. L. Roark and Dr. L. F. of Sciences, Poland (65T-392) Shampine, Sandia Laboratory, Albu­ (51) Survey of quasi-uniform spaces querque, New Mexico (65T-393) Professor S. A. Naimpally and Pro­ (64) Elliptic regularization for symmetric fessor M. G. Murdeshwar, Univer­ positive systems sity of Alberta (65T-413) Professor Leonard Sarason, Stan­ (52) Non-nuli implication ford University (65T-416) Professor David Nelson, George (65) The conjugacy problem Washington University ( 65T-414) Mr. P. E. Schupp, Queen Mary Col­ (53) On the index of first order pseudo­ lege, England (65T-397) deficiencies of mero­ differential operators ( 66) On the Valiron Mr. Umberto Neri, University of morphic functions of finite order Chicago (65T-386) Mr. D. F. Shea, Syracuse Univer­ (54) The lack of self-adjointness for three sity ( 65T- 378) point boundary value problem (Introduced by Professor AlbertEdrei) Professor J. W. Neuberger, Emory (67) Metamathematical properties of cer­ University (65T-422) tain large cardinals (55) Global dimension of valuation rings Mr. J. H. Silver, University of Cali-

688 fornia, Berkeley (65T-444) University (65T-415) (68) Differential-integral calculus for ab­ (75) A class of bisimple inverse semi­ stract algebraic-topological struc­ groups. II tures. IV Professor R. J. Wa!rne, West Vir­ Mr. R. M. Sorensen, Wolf Research ginia University (65T-372) and Development Corporation, (76) Asymptotic simplification of self­ Hyattsville, Maryland (65T-431) adjoint differential equations with a (69) A new composition of arithmetic func­ parameter tions Professor Wolfgang Wasow, The Professor M. V. Subbarao, Univer­ University of Wisconsin (65T-438) sity of Alberta (65T-379) (77) Strong differentials in LP {70) On the existence of large sets of Professor Mary Weiss, De Paul Dedekind cardinals University ( 65T-439) Professor Alfred Tarski, University (78) Finiteness theorems for pseudo­ of California, Berkeley (65T-432) riemannian symmetric spaces (71) Completely plastic torsion Professor J. A. Wolf, University of Professor Tsuan Wu Ting, North California, Berkeley (65T-373) Carolina State University at Raleigh ( 79) On finite metric sets II. Separating (65T-369) points (72) A characterization of the double point Professor Dorothy Wolfe, Pennsyl­ structure of the projection of a poly­ vania Military College (65T-446) gonal knot in regular position (80) On pseudo-creative sets, splinters, Professor L. B. Treybig, Tulane and bounded-truth-table reducibility University ( 65T-421) Dr. P. R. Young, Stanford University (73) Relative ideals in semigroups. III (65T-391) Professor A. D. Wallace, University {81) The Hankel transformation of certain of Florida (65T-384) distributions of rapid growth (74) An extension of the generalized Bern­ Professor A. H. Zemanian, SUNY, stein lemma at Stony Brook (65T-418) Professor J. L. Walsh, Harvard

NEWS ITEM

SUMMER INSTITUTE IN MATH FOR LIFE SCIENTISTS

Mathematics for life scientists is plans for a one-to-one student-faculty ratio. the subject of a summer institute planned Participation in this program is limited, by the University of Michigan for next therefore, to ten persons. Stipends will year, June 1 to August 23. The program will vary, but will include travel allowances. cover the equivalent of about three semes­ November 30 is the deadline for receipt ters of college mathematics. Topics include of applications. Further information can be foundations of mathematics, calculus, linear obtained by writing R. M. Thrall, Depart­ algebra, probability, and digital computer ment of Mathematics, West Engineering instruction. There is no mathematics re­ Building, University of Michigan, Ann Arbor, quirement. Presently, the University has Michigan 48104.

689 ABSTRACTS OF CONTRIBUTED PAPERS October Meeting in Cambridge October 30, 1965.

6Z6-l. R. E. FULLER TON and C. C. BRAUNSCHWEIGER, University of Delaware, Newark Delaware. Quasi-interior points and the extension of linear functionals.

A~ F of the unit ball U in a real normed linear space X is a maximal convex subset of the surface S of U, the closure of whose affine hull, V(F), has codimension 1. A convex subset E C U is an extremal subset of U if x, y E U and E 3 (1 - a)x + ay for some a, 0 < a < 1, imply x, y E E. xis a quasi-interior (q.i.) point of F if, when K = fay: a ;;; 0, y E F}. K n(x - K) is total in X. Among observations of a geometrical nature are: (1) J!.x ~ q.i. point of a face F of U then F is the minimal closed extremal subset of U containing x. (Z) ~ f be a functional defined and bounded on a subspace M..2f X. If there exists a point x !!!_S nM which is a q.i. point of a face of U and for which f(x) =

Jlffl(rel. M)~ f has a unique norm preserving extension to all of X. Particular attention is given to the function spaces C(U) and L(T,l:,~.t). The faces of U and the q.i. points of these faces are com­ pletely characterized in these function spaces. (Received July 9, 1965.)

6Z6-Z. EUGENE DENZEL, Dartmouth College, Hanover, New Hampshire. A characterization of Markov processes in terms of their hitting characteristics. Preliminary report.

Paul Levy lfrocessus stochastiques et mouvement Brownien, Gauthier-Villars, Paris, 19~8] established a classical characterization of one-dimensional Brownian motion in terms of martingale functions of the process, which was generalized by M. Arbib [M.I. T. Tech. Rep. 419, 1964] to one­ dimensional diffusions. The author has shown that in order for a process to be a Markov process it is sufficient that the mean hitting times and hitting distributions of all open sets have the Markov property starting at random times. He has used this result to obtain a simpler proof of Levy's theorem which has a valid generalization to n dimensions and to obtain a similar characterization for a class of birth and death processes on the integers. The same techniques yield generalizations of Arbib's results to higher dimensions. (Received August lZ, 1965.)

6Z6-3. A. C. BACOPOULOS, University of Wisconsin, Madison, Wisconsin. Approximation with vector valued norms.

Let X be a real linear space, Y an n-dimensionallinear subspac·e of X, fl Jl 1, II flz two norms on X and UJ Ill a vector valued norm defined as follows: For x, xl'xZ EX, lllxlll = (Jixfl 1, flxflz>• and Ill Ill is partially ordered by: lllx 1 JJI ~ lllxziii~Jix 1 11 1 ~ flxz 11 1 and Jlx 1 flz ~ flxzllz· Definition. Given x E X, F x (A 1 •••• ,An) = lllx - :Ef= 1 Ai yi Ill where {Yi} is a basis for Y. Definition. -4 (x) is the set of points in the range of Fx which are minimal relative to the above partial order, where (A 1,Az•••••An) varies over K CEn. Theorem 1. Given x EX, there exists ann-cube In such that..£ (x) = ..£ (x). In En Theorem z. F (:f) is simply connected. Theorem 3. Characterization of Jl = F-l (..£ (x)), where X X En Fx is induced by two inner products ( )1' ( )z on X, i.e., Fx(Al' .... ~) = lllx- DiYilllz = (llziJ 1,1JziJz) = (v(z,z) , v(z,z) ). (Received August 13, 1965.) 1 z 690 6Z6-4. E. C. YOUNG, Florida State University, Tallahassee, Florida. A method of determining the Riesz kernel for the EPD operator. •

Consider the Riesz integral solution of the Cauchy problem for the nonhomogeneous wave equation (1) u - u - ••• - u = f(y2, ... ,y , y~ - y~)e.\(j with initial conditions (Z) u(y) = 0, YoYo Y1Y1 YmYm m z z z (iJu/ iln) = 0, y = (y0 ,. .. ,ym) prescribed on the space-like part of the surface S: y0 - y 1 = t 0, t 0 > 0 a constant. By introducing the new coordinates t,xl'x2, ... ,xm-l' 8 defined by the transformation Yo = t cosh 8, y 1 = t sinh 8, yi = xi-l' i = Z,3, ... ,m and then integrating with respect to the variable 8, one obtains a solution in the sense of Riesz of a corresponding Cauchy problem for the transform of equation (1), which is closely related to the Euler-Poisson-Darboux operator. From the result obtained the Riesz kernel for the EPD operator is deduced and verified. *(This method is a special case of a general method of embedding of Professor Stellmacher of the University of Maryland.) (Received August 11, 1965.)

6Z6-5. C. W, KOHLS and L. J. LARDY, 15 Smith Hall, Syracuse University, Syracuse, New York 13Zl0. Extensions and retractions of rings.

For terminology see [Mac Lane, Illinois J. Math. Z (1958), 316-45] and [Gillman and Jerison, Rings g1 continuous functions, Princeton, 1960]. A splitting extension is one with h(x,y) = h(xly) = 0. (This includes adjunction of an identity.) It seems natural to apply Mac Lane's general theory of ring extensions to the study of splitting extensions and extensions of special classes of rings. Theorem 1. A homomorphism {3: E ->A is a retraction of E onto A iff E is equivalent to a splitting extension of the kernel of {3 by A. Theorem z. Let F be a topological field, X and Y compact spaces (if F = R, real­ compact), E = C(X,F) and A= C(Y,F). If either F is formally real or E and A are von Neumann regu­ lar, then A is a retract of E iffY is a retract of X. Theorem 3. Let A be a subdirect sum of the fields {F-y:1' Ex}. Then MA is the largest subring of IT-yF-y containing A as an ideal. IfF is a topological field, X a topological space, and A CC(X,F) then MA CC(X,F). Theorem 4. Let Y be a closed subset of a compact space X, Z = X - Y and A a Silov regular subalgebra with identity of C(Y). There is a 1-1 correspondence between classes of equivalent extensions of c0 (Z) by A and continuous maps of {3Z - Z into Y. (Received August 9, 1965.)

6Z6-6. W. W. COMFORT University of Massachusetts, Amherst, Massachusetts and STELIOS NEGREPONTIS, University of Indiana, Bloomington, Indiana. On the realcompactification of a product space.

We consider only completely reguiar Hausdorff spaces. Theorem. If the metric space C*(Y) is realcompact, then the identity v(X X {3Y) = vX X {3Y holds for each space Xi. Corollary. Let card Y be nonmeasurable, and suppose that either Y is compact, or the projection from X X Y to X is closed, or X XY is C*-embedded in X X {3Y. Then v(X X Y) = vX X vY. Example. Let n be the smallest measurable car.dinal, and let D be the discrete space of cardinality n. Then the relation v(D X {3D) = vD X /3D fails. ,(Received August 16, 1965.)

691 626-7. J. D. ACZEL, University of Massachusetts, Amherst, Massachusetts. How the 23 functional-equations problems of 1963 stand now.

(Numbering as in Arch, Math. 15 (1964), 435-444) - Ad 23, a new general theorem. (Aczel):

All nonconstant continuous solutions f of functional equations of the form f(x + y) = F(x,f1 (x), f(y), f 2(G(x,f3 (x),f(y))), ••• ) are 'strictly monotonic. Ad 21, partial result by W. Eichhorn (J. Reine Angew. Math, 1966). Ad 19, solution resp. reduction in the case m :;,; n by M. Kucharzewski-A. Zajts (Ann. Polon. Math. 1967). Ad 14: Proved by W. B. Jurkat (Proc. Amer, Math. Soc. 16, (1965)) and N. G. de Bruijn (unpublished). Ad 13: Proved by W. B. Jurkat (ibidem), S. Kurepa (Glasnik Mat.­ Fiz. Astronom. 19 (1964), 23-36) and R. 0. Davies (unpublished). Ad 5, partial result by A. Smajdor (Arch, Math. 1966). Ad 4, solved by J, Aczt!l - G. Pickert (Arch. Math. 1966): There exist monotonic noncommutative groups on subsets of the set of real numbers, but there do not exist such groups on the set of..!!l!.real numbers. Further problems arising from these. (Received August 3, 1965.)

626-8. R. W. ROBINSON, Cornell University, Ithaca, New York. Two recursively enumerable

A set X is cohesive (r-cohesive) if for every r.e. (recursive) set S one of S r-1X and S n X is finite. A set M is maximal (r-maxmimal) if M is r.e. and M is coheslVe (r-cohesive). A set Y is quasicohesive if Y is a finite union of cohesive sets, and a setH is quasimaximal if H is r.e. and H is quasicohesive. Several of these definitions are due to Ullian, and to McLaughlin (Mich. Math, J. 11 (1964), 83-87). It is trivial that every quasimaximal set is hyperhypersimple, and that every maximal set is r-maximal. In order to disprove the converses of these trivialities an elaboration on Yates' version of the maximal set construction is used (see C.E.M. Yates, Three theorems on the degrees of recursively enumerable sets, Duke Math. J., to appear). Theorem 1. There is a hyperhyper­ simple set which is not quasimaximal. Theorem 2. There is an r-maximal set which is not hyper­ hyper simple. The r-maximal set of Theorem 2 is contained in a maximal set, and its construction appears to require this, (Received August 19, 1965.)

626-9. D. L. KREIDER, Dartmouth College, Hanover, New Hampshire, and R. W. RITCHIE, University of Washington, Seattle, Washington 98105. Marking automata and a basis for the r.e. sets.

Marking automata (M.A.) are defined to be Turing machines which cannot extend their tape (and hence are special cases of the two-way automata of [Abstract 597-142, these c}/oticei) 10 (1963), 99]) and which are further restricted to the mere placement and removal of a preassigned finite number of "place markers. 11 More precisely, with each M.A. there is a pair of integers n > k > 0 such that at no point in any computation on any k-tuple input tape (a tape of the form Bt1Bt2B ... BtkB where B is the blank symbol and each ti is a tape containing no B 's) are there more than n B 's on the tape. Further, any square which contains a symbol x in the input can contain only x or B at any step of a computation. These M.A. are a mild generalization of the two-way (nonprinting) finite automata of Rabin and Scott (Finite automata and their decision problems, IBM J. 3 (1959), 114-125], yet they can accept many more sets of tapes. Let us say that an M.A. accepts a set of (k-tuple input) tapes if it stops in a final state on exactly these (k-tuple input) tapes. Theorem. Every S-rudimentary set

692 as defined by Smullyan (Theory of formal systems, Princeton Univ. Press, 1961, p. 89] is accepted by some M.A. Corollary. The sets acceptable by M.A. form a basis for the recursively enumerable sets. (Received August 23, 1965.)

626-10. BERTHOLD SCHWEIZER, University of Massachusetts, Amherst, Massachusetts. On the uniform continuity of the probabilistic distance.

It is a well-known fact that if (S,d} is a metric space then the distance function dis a uniformly continuous mapping from S X S into the non-negative reals. The extension of this result to probabilistic metric spaces is given by the following Theorem. Let (S,ff.T) be a Menger space under at-norm T which is such that lirnx ~ 1 T(a,x) = a uniformly on the closed interval [0,1]; let D be the set of one­ dimensional distribution functions; letS be endowed with the f, A-topology, S X S with the induced product topology, and D with the topology induced by the L~vy metric. Then the probabilistic distance function ff is a uniformly continuous mapping from S X S into D. (Received August 23, 1965.)

626-ll. R. N. PEDERSON, Carnegie Institute of Technology, Pittsburgh 13, Pennsylvania. Uniqueness in Cauchy's problem for elliptic equations with double characteristics,

Let P (x, ~) be a homogeneous rnth degree elliptic polynomial in ~=( ~ 1 • ~2 .... , ~nl with complex coefficients which are of class c 2 in a neighborhood N of the origin in Rn. Assume that P, considered as a polynomial in ~ 1 • has no roots of multiplicity greater than two for any ~· = (0, ~ 2 ..... ~n) i 0. Assume further that the roots \ (x, ~'}, A2(x, ~'}, ••• ,Am (x, ~') are, after proper identifications, analytic for ~ E R - tO} and of class C' on N x(R - fOI). Theorem. If Sis any surface of class Crn which n-1 n- 1 - - 2 is tangent to the hyperplane x 1 = 0 at the origin, then any solution of the inequality IP (x, D)u 1 2 ;;;; K Lja.j;i;m- 1 1Da.uj which has vanishing Cauchy Data on S is identically zero in a neighborhood of the origin. This generalizes a result of Hormander, Math, Scand. 7 ( 19 59), 177-190, for the case where P is the product of two polynomials with simple roots. An example of an irreducible polynomial 2 which satisfies our conditions is given by P (~) = ( ~ ~ + ~ ~ + ~~) 2 + ( ~ 2 - ~ 3) ( ~~ + ~ ~). Plis, C PAM 14 ( 1961), 599-617, has given an example of a fourth order elliptic polynomial which fails to have the uniqueness property, but which satisties all of our conditions except for the smoothness of the roots. (Received September 10, 1965.)

626-12, DOROTHY WOLFE, Pennsylvania Military College, Chester, Pennsylvania 19013, On finite metric sets 1: Imbedding inn-dimensional Minkowski space,

The Minkowski n-dimensional space Mn is the set of points x = (x 1, ... ,xnl with the metric xy = maxtlxi- Yil· This paper extends the known result that any set of n + 1 points for which a metric is defined can be isometrically imbedded in Mn• Lemma. Given a metric set of n + 1 points, if some n of these points can be imbedded isometrically in Mk' then the n + 1 points can be imbedded isometric­

ally in Mk+1• Theorem. Any metric set of n + 2, where n ;?; 2, points can be imbedded isometrically in Mn. (Received September 2, 1965.)

693 626-13. G. E. ANDREWS, The Pennsylvania State University, University Park, Pennsylvania. Partition theorems related to the Rogers-Ramanujan identities.

Theorem. Let X > 0, 0 < a ;;;; k be integers. Let AX,k,a(N) denote the number of partitions of N into parts not of the forms X(2k + 1)m, X(2k + 1)m + >..a, X(2k + 1)m + X(2k + 1 - a). Let B X,k,a(N) denote the number of partitions of N of the form :E:1 fi • i (~ denotes the number of times the sum­ mand i appears in the partition), where (1) fi ;;;; Xa - 1; (2) if fi = 11 (mod X) (0 ;;;; a. < X), then fi + fi+1 ;;;; Xk +a.- 1. Then AX,k,a(N) = BX,k,a (N). If X= 1, this theorem reduces to Gordon's generalization of the Rogers-Ramanujan identities (Amer. J. Math. 83 (1961), 393-39). If k = a= 1, the theorem reduces to Glaisher's generalization of Euler's theorem (Messenger of Math. 12 (1883), 158-170). (Received September 3, 1965.)

626-14. ALFRED BRAUER, University of North Carolina, Durham, North Carolina. A method for the computation of the greatest root of a non-negative matrix.

In 1957, it was shown IJ. Soc. Indust. App1. Math. 5, pp. 250-253] that the greatest root of a positive matrix can be computed as exactly as needed by multiplying the rows and columns of the given matrix by certain constants. It could be expected that the method works for non-negative matrices, too. But the proof for the convergency seemed to be difficult in this case. In order to handle non-negative matrices their case was reduced to the case of positive matrices [Studies in mathematical analysis and related topics, published in honor of George Pdlya by Stanford University, 1962, pp. 48-55]. But this method often requires the computation of a high power of the given matrix before the greatest root can be obtained. It is shown in this paper that this can be avoided since the original method is convergent for non-negative matrices. (Received September 87, 1965.)

626-15. F. G. ASENJO, University of Pittsburgh, Pittsburgh, Pennsylvania. Dialectic logic.

If the dialectic principle is taken as a rule of inference, the formalization of dialectic requires the framework of an inconsistent logic. To go beyond this formal limitation, the dialectic principle is taken instead as a rule of formation and a formalization of dialectic is outlined as an applied higher order predicate calculus with a finite number of synthesizing predicate constants. In addition, as a special interpretation, a dialectic number theory is described. (Received September 9, 1965.)

626-16. E. J. TAFT, 14 Vandeventer Avenue, Princeton, New Jersey 08540. Invariant splitting in Jordan and alternative algebras.

Let A be a finite-dimensional jordan or alternative algebra over a field of characteristic zero. Let G be a finite group of automorphisms and antiautomorphisms of A. Let S and T be two G-invariant maximal semi-simple subalgebras of A (such subalgebras exist). Theorem. S and T are strictly conjugate via an automorphism of A which commutes pointwise with G and which is of the form exp D, D a nilpotent inner derivation in the radical of the enveloping associative algebra of A. Conjecture. D can be expressed in terms of right and left multiplication by fixed points of G. The conjecture has an affirmative answer for associative and Lie algebras (in the associative case, one obtains adjoints of elements which are fixed points of the automorphisms in G and are sent into their negatives by the antiautomorphisms in G). (Received September 10, 1965.)

694 626-17. A. K. SNYDER, Massachusetts Institute of Technology, Cambridge, Massachusetts. Some remarks on heavy points in countable spaces.

Let N be the positive integers with a T 3 topology. After Henriksen and Isbell (Bull. Amer. Math. Soc. 70 (1964), 287-290) a point n inN is called a heavy point if there exists a regular matrix which sums every bounded real function on N continuous at n. A heavy point is in the lower topo­ logical limit of a disjoint sequence of finite sets. Assume the matrix A sums C*(N). The functional x-->limAx on C*(N) corresponds to a Radon measure !Lon {JN. Now !LWill be supported by N. Henriksen and Isbell state the following theorem without proof: The measures !Lon {JN supported by the set of heavy points in N and satisfying !L(/IN) = 1 are precisely those which arise from regular matrices summing C *(N). The present writer shows that A may be assumed positive in the definition of heavy points. Using this fact a proof (perhaps similar to that envisioned by Henriksen and Isbell) of the above theorem is obtained. Let A = (ank) be a positive regular matrix. Let N have the topology generated by {U: 1 E U and lim Ax(U) = 1J \J ffnl: n > 1}. Call the space N A" Then 1 is a heavy point in NA. The following are equivalent: 1 is not a sequential limit point in NA; N A is pseudo-finite; supkank---+ 0. (Received September 10, 1965.)

626-18. J. H. SMITH, The University of Michigan, Ann Arbor, Michigan. On the Herbrand quotient.

If G is cyclic of order n, a formula is derived for the Herbrand quotient h 211 (A) of a finitely generated G-module A in terms of the Z-rank of AH for the various subgroups H of G. This is used to obtain a new derivation of h 2/ 1 (Us)• where Us is the group of S-units of a cyclic extension L of a number field K, and to obtain some criteria for an infinite group to have an outer automorphism of a special type. (Received Septe-mber 9, 1965.)

626-19. HOWARD SHERWOOD, Illinois Institute of Technology, Chicago, Illinois 60616. Distribution-generated spaces as Menger spaces.

A probabilistic metric space (S,ff) is called a distribution-generated space over En if:

(1) With every set of k points pl'p2, ••• ,pk in the setS, there is associated a kn-dimensional distribu­ tion function H such that HP p p (Wl'w2, ••• ,Wk) = HP P P (Wu(1) ,W..~ 2 ) , ••• ,Wu(k)) P1P2·""Pk 1 2""" k u(1) o(2)••• u(k) u' where u is a permutation on the first k positive integers and wi = (w (i- 1)n+1,w (i- 1)n+2'"""' w(i-l)n+n>• and HP P P (Wl'W2, ••• ,wk_ 1,(+ oo, + oo, ••• , + oo)) = HP P P (wl'W2, ••• ,wk_ 1). (2) The mapping 1 2••• k 1 2··· k-1 ffis given by ff(p,q) = Fpq where Fpq(x) = J... flu-vl· Theorem. Every distribution-generated space over En is a Menger space under the t-norm T m defined by T m (a,b) = Max (a + b - 1,0). This generalizes a result of Schweizer and Sklar who showed that any space of a certain subclass of the class of distribution-generated spaces is a Menger space under the weaker t-norm Tw where Tw(a,b) is equal to a when b = 1, is equal to b when a= 1, and is zero otherwise. (Received September 10, 1965.)

695 626-20. RODNEY ANGOTTI, Michael Hall, State University of New York at Buffalo, Buffalo, New York 14214. The projective invariants of the configuration of two lines, and a third order differen­ tial element.

Bompiani (J. Math. Pures Appl. 41 (1962), 193-200) has shown that the configuration of two lines and a third order differential element in the ordinary projective space P 3 has two projective invariants. This present paper discusses the invariant properties of this configuration in the event that the lines are incident; in particular, it is shown that the configuration of two incident lines and a third order differential element has a third invariant and, in addition, that each of the invariants which Bompiani discusses is not geometric when the two lines are incident, i.e., each has a constant value for every pair of incident lines. (Received September 13, 1965.)

626-21. D. A. MATTSON, 86 Vernon Street, Apartment 1, Hartford, Connecticut. Abstract convergence theory.

In this paper set-valued set functions are used to classify abstract spaces. An operator is defined which determines the product space for a finite number of spaces and which retains the structure of the component spaces. A space (M,g) is called a Frechet space if g is expansive. A convergence theory is introduced which generalizes nets and characterizes limit points in (M,g) and which may be used to develop a theory of product spaces for an infinite collection of component spaces. Although certain classes of abstract spaces retain many of the properties of topological spaces, it is shown that it is possible to characterize limit points by means of nets only in spaces (M,g) where g is additive (Cech spaces). Theorem. If (M,g) is a Frechet space, the following are equivalent: (A) The space (M,g) is a Cech space. (B) A point p is in g(X) iff there is a directed system in X which converges top. (Received September 13, 1965.)

626-22. R. W. BAGLEY, University of Miami, Miami, Florida and T. S. WU, University of Massachusetts, Amherst, Massachusetts. Uniformities on locally compact topological groups. Preliminary report.

Let G be a topological group and a be a neighborhood base of G at identity e of G. Let G* denote the topological group defined on the abstract group G with the neighborhood base {J at e, where V* E {J, iff V* = (I gEG fgvg - 1 for some V E a f· The following statements are obtained: (1) Examples are given which show that G* is not necessarily locally compact. (2) If G is a Lie group, then G* is also a Lie group. (3) Let R be the group of reals with usual topology. If R ~ G*, as an invariant sub­ group, then R is contained in the center of G. (4) Suppose G = HK, where K is a compact subgroup. Let r = {V': V = n hElPvh - 1, for some V E a J, and G' be the topologtcal group obtained from G by taking r* as the neighborhood base at e on abstract group G, then G' = G*. (5) Some structure theo­ rems are obtained when G* is locally compact. (Received September 13, 1965.)

626-23. S. D. SHORE, University of New Hampshire, Durham, New Hampshire. Homomorphisms of lattices of continuous functions.

C(X,R) is the lattice of all real-valued continuous functions defined on a completely regular Hausdorff space X with the lattice operations defined in C(X,R) in the usual way. If X' is a compacti-

696 fication of X (i.e. a compact Hausdorff space which contains X as a dense subspace), then a lattice homomorphism cP: C(X,R) --+ R is associated with a point p in X' iff f(x) ;:!! g(x) for each x in some neighborhood of p implies that cP(f) ~ c/J(g). Theorem 1. Every lattice homomorphism cP: C(X,R) -->R is associated with a point of f3X (the Stone-Cech compactification of X). Theorem 2. Every lattice homomorphism c/J: C(X,R) --+R is associated with a point of X iff X is compact. A lattice homomor­

phism c/J: C(X,R) --+R is countably cofinal iff there is a sequence f l'f2, ... of functions in C(X,R) such that for any g E C(X,R) and any r E cP (C(X,R)], g ~ (fk V f') for some integer k and some f' E C(X,R) with c/J(f') ;:!! r. Theorem 3. If X is realcompact, then a lattice homomorphism c/J: C(X,R)--+R is

associated with a point of X iff c{J is countably cofinal. Corollary 3.1 (Shirota). If X is realcompact, then C(X,R) determines X. (Received September 13, 1965.)

626•24. H. L. BAKER, JR., University of Massachusetts, Amherst, Massachusetts. Concerning complete amonotonic collections of subcontinua of a compact continuum.

ln Abstract 619-119, these cAfoficei) 12 (1965), 91, the notions of an amonotonic collection of sets and of a complete amonotonic subcollection of a collection of sets were introduced. The theorem was stated that a compact metric continuum has cyclatomic subsets of type I if and only if there is a countable complete amonotonic subcollection of the collection of all subcontinua of it, each element of which has a connected complement. Theorem. A compact metric continuum is a simple closed curve if and only if there is a countable complete amonotonic subcollection of the collection of all subcontinua of it each element of which is an arc. Theorem. If M is a compact metric continuum which is the sum of a finite number of simple closed curves but which does not contain infinitely many such sets, then there exists a countable complete a monotonic subcollection of the collection of all subcontinua of M. (Received September 13, 1965.)

626-25. D. E. SPENCER, University of Connecticut, Storrs, Connecticut and J. F. FITZGERALD, Sylvania Electric Products, Inc., 100 Endicott Street, Danvers, Massachusetts. Formulation of the integral equation for interflections in helical coils.

Nearly every incandescent lamp contains a helical coil. Yet exact design of incandescent lamp filaments is not possible today. A first step toward the complete solution of the filament design problem is the formulation of the integral equation for interflections in helical coils. Noteworthy features of the formulation are the introduction of a new oblique system of coordinates and the formu­ lation and solution through computer techniques of the problem of intra-coil blocking. In general, the solution is a function of two variables but for infinitely long coils becomes one-dimensional. (Received September 13, 1965.)

626-26. E. R, SURYANARAYAN, University of Rhode Island, Kingston, Rhode Island 02881. On the geometry of steady screw motions.

For screw motions the vorticity vector is along the velocity vector. Screw flows and complex­ lamellar flows form mutually exclusive types of flows. Screw flows are rotational flows. The equations of motion, the equation of continuity and the energy relation are expressed along the tangent vector, along the binormal vector and along the principal normal vector of the streamline, From

W7 these intrinsic equations, it is found that the magnitude of the velocity vector q, does not vary along the binormal vector and the derivative of log q along the principal normal vector is equal to the cur­ vature of the streamline. It is found that for steady screw motions the strong form of the Bernoulli law holds. Moreover, the screw flows admit only isentropic flows. The equations of motion, continu­ ity and energy are obtained in the case of circular helical screw motions, and from these equations one finds that the flow quantities depend only on the radius of the cylinders. The equations are inte­ grated to obtain two classes of flows. (Received September 13, 1965.)

626-27. W. J. SCHNEIDER, Syracuse University, Syracuse, New York 13210. A uniqueness theorem for conformal maps.

A well-known result of Radd' states that if f(z) univalently takes a plane domain (not the whole plane) containing the origin onto itself with f(O) = 0 and f' (O) > 0 then f(z) = z. One might ask to what extent conformal maps are determined if one increases the number of points whose images are known but has no information about the derivatives at these points. Theorem. Let a,b and c be three distinct points of a plane Dirichlet domain D (i.e. each point of the boundary of D is a regular point for the Dirichlet Problem) and let f(z) and g(z) univalently map D into D with f(a) = g(a), f(b) = g(b) and f(c) = g(c) then f(z) = g(z). The proof follows from: (i) If z 0 is an element of a Dirichlet domain D and g(z,z0 ,D) is the Green's function forD with pole at z 0 then {z: g(z,z0 ,D) < A< 0} is a finitely multiply connected domain, (ii) the Lindelof Principle, (iii) every finitely multiply connected domain is conformally equivalent to a circular domain, (iv) Koebe's Theorem on conformal maps from cir­ cular domains to circular domains. The same method also yields an alternate proof of Radd''s result for Dirichlet domains. An example can be constructed of a conformal map from one plane domain into another, with infinitely many fixed points, which is not the identity map. (Received September 13, 1965.)

626-28. FEDERICO GAETA, Michael Hall, State University of New York at Buffalo, Buffalo, New York 14214. Postulation formulas for flag and Grassmann varieties.

It is shown that the polynomial form of X(g) is valid for any g = 1,2, ... , and for any given flag 2 variety F(d 1 ;;;:; d2 ;;;:; ••• ;;;:; dk;m). This variety represents all the vector flags v< 1> ~ v< >2 ... 2 y(k) belonging to a fixed m-dimensional vector space ((dim y(i)) = ~) over the commutative ground field K (of char. zero). X(g) is obtained as a consequence of the character formulas for irreducible tensor representations of GL(V). Fork= 1, d 1 = d, F(d;m) is the Grassmannian of the Vd in V; then X(g) may be written in the following rectangular form X(g) = ((d- 1)!!(m - d - 1)!!/(m - 1)!!) ((g + 1)(g + 2) ••• (g +d); (g + 2)(g + 3)••• (g + d + 1); ••• ;(g + m - d)(g + m - d + 1) ••• (g + m - 1)] where (h!! = 1!2! ••• h!). This formula emphasizes the symmetry in d and m - d corresponding to the natural duality between G(d;m) and G(m - d;m). (Received Septemer 13, 1965.)

626-29. F. P. CALLAHAN, General Electric Company, P. 0. Box 8555, Philadelphia, Pennsylvania. Structure of certain finite groups.

Let c/J be a function from Z/N, the ring of rational integers, into itself. Let a product 0 be defined by i o j = i + 1/l(i)j. (Z/N, o) is known to be a group, J(c/1) for certain choices of 1/l, (1). This

698 paper studies the structure of such groups, an

626-30, C. T. SHIH,Cornell University, Ithaca, New York 14850, Markov processes whose hitting distributions are dominated by those of a gi"l!en process,

Suppose X and Y are two Markov processes with the same state space such that the hitting distributions of X dominate those of Y. It is natural to ask whether there is a subprocess (in some sense) of X that has identical hitting distributions as Y. Sur (Soviet Math. 3 (1962), 1626-1629) proved that this is the case if X is a Brownian motion and Y a standard process. A different approach yields

the following result: if X and Y are Hunt processes with a locally compact separable metric space as their common state space, then, under the additional hypothesis that the hitting time of the com­ plement of any compact set is finite almost everywhere for both processes, there is a subprocess of X that has identical hitting distributions as Y. The proof is by constructing a multiplicative functional of X. (Received September 13, 1965,)

699 ABSTRACTS PRESENTED BY TITLE

65T-368. DAVID BELL, Brown University, Providence, Rhode Island 02912. The Vitali approximation of automorphic forms of Fuchsian groups bz Poincare! series of rational functions.

Using recent results of Berson the L 1-approximation of analytic functions by rational functions one can show: Theorem. Let D be a bounded simply-connected domain in the plane, G a properly­ discontinuous group of biholomorphic transformations of D, and q an integer > 1. Let f be a mereo­ morphic function on D such that fog• jJ = f for each g in G, where jg is the complex Jacobian of g. Let 0 < m ;;;; oo such that f has no poles of order > m. Then one can choose a sequence Rj of rational functions with no pole of order > m such that the sequence Fj = Lg in 0 (Rjog•jJ) is Vitali convergent to fonD as j approaches oo. (E.eceived May 18, 1965.)

65T-369. TSUAN WU TING, North Carolina State University at Raleigh, Raleigh, North Carolina 27607. Completelz plastic torsion.

Let G be a Jordan domain with piecewise twice continuously differentiable boundary, ao. The problem is to find the non-negative function t{;(x,y) such as to satisfy the following conditions: (i) if; is continuous in G, piecewise smooth in G and vanishes on ao, (ii) jgrad 1/1 .z = k 2 > 0 in G,

(iii) 1/; maximizes the integral J0 t{;dx dy among the class of all functions

65T-370. WITHDRAWN.

700 65T-37l. A. A. MULLIN, University of California Lawrence Radiation Laboratory, Box 808, Livermore, California 94551. Some new estimates in prime-number theory. Preliminary report.

For definitions of arithmetic functions IL* and A* see Abstract 64T-450, these cNoticeiJ ll (1964), 680. Lemma 1. Lm;:;;; n I~J.(m) I ::£ Lm;:;;; ni!L*(m) I = o(n), where the implicit constant k satisfies 6hr2 < k < 1, !L is the ordinary Mobius function and IL* is its modification. Lemma 2. I r

65T-372. R. J. WARNE, 428 Cedar Street, Morgantown, West Virginia 26505. A class of bisimple inverse semigrours. II.

Let S be a bisimple inverse semigroup and Es denote the set of idempotents of S. If Es under its natural order is order isomorphic to r0 X r0 under the order (n,m) < (s,t) if n > s or n = s and m > t, we say ES is lexicographically ordered. ES is lexicographically ordered if and only if H (Green's relation) is a congruence on S and S/H"" C " C (Abstract 65T-336, these cNoticeiJ 12 (1965), 614). We determine the multiplication for a certain class of bisimple inverse semigroups for which Es is lexiocographically ordered in terms of a group multiplication, a single endomorphism of this group, and certain operations on the non-negative integers. These results generalize to arbitrary finite dimensions. (Received August 30, 1965.)

65T-373. J. A. WOLF, University of California, Berkeley, California 94720. Finiteness theorems for pseudo-Riemannian symmetric spaces.

The result is: Let M = M~ be a pseudo-Riemannian symmetric space where the metric has s negative squares, n - s positive squares. Let G be the largest connected group of isometries, suppose that G is a simple Lie group with finite center, and represent M~ = G/H. Suppose that the metric on Mn is induced by a negative multiple of the killing form. Let u be a Cartan involution of s G which preserves H and let K be the fixed point set of u. Then the following conditions are equiva- lent. (1) Every discontinuous group of isometries of M~ is finite. (2) If a subgroup r C G is properly discontinuous on Mn, then r is finite. (3) The Lie algebra of H contains a Cartan sub- s algebra of the symmetric pair (G,H). (4) 2s ::£ n. Except for the proof that (4) implies one of the others, the hypotheses can be weakened considerably in each implication. This extends the author's extension to isotropic spaces [Comment Math. Helv. 39 (1964), 21-63; seep. 46] of the author's extension to spaces of constant nonzero curvature [Ann. of Math. 75 (1962), 77-80; seep. 78] of a result of Calabi and Markus on Lorentz manifolds of constant positive curvature [Ann. of Math. 75 (1962), 63-76; see p. 69}. (Received July 1, 1965.)

701 65T-374. E. G. K. LOPEZ-ESCOBAR, Massachusetts Institute of Technology, Room 2-I55A, Cambridge, Massachusetts. On defining well-orderings. II.

For undefined notation see Abstract 65T-295, these cNoticeiJ I2 (I965), 60I. If U = (A,R), 2 R c,; A 2, 5.8 = ( B ,A,S~) ~ < 0, B 2 A, and R = s 0 t A (the restriction of S0 to A) then Qt. is the relativized ~of 5.8. Theorem. There does not exist a cardinal a. such that for some set T of sentences of La.w'lffi (the class of well-orderings) is the class of relativized reducts of models of T. (Received July I, I965).

65T-375. RODOLFO DeSAPIO, Stanford University, Stanford, California• ..£!:_ (k - I)-connected (2k + I)-manifolds. I. Preliminary report.

Theorem. Let M2k+I be an almost parallelizable, (k - I)-connected (2k + I)-manifold, where k > 2, and suppose that IT I•···, up' TI•···, Tq is a set of generators for a direct sum decomposition of Hk(M), ui free and Ti finite order with q minimal. Then there is a diffeomorphism h:

(Sk x sk)1 # ••• # (Sk x sk) ~ (Sk x sk) # ••. # (Sk x sk) of the connected sum of p + q copies p+q I l+q of sk X sk such that M is the union of two copies of #f:((S X Dk+ I, Sk X Si)i with points identified under the diffeomorphism h of the boundary. If k ""' 3,5,6, 7 (mod 8), then the above decomposition is valid for any (k - I)-connected (2k + I)-manifold. This result generalizes the well-known Heegaard decomposition for 3-manifolds. Applications of this theorem have been made for even k. For example: (I) For k = 2,4,6 (mod 8) there are at most 4 • [II ] nondiffeomorphic almost parallelizable 2k+I (k- I)-connected (2k + I)-manifolds M2k+I with Hk(M) cyclic and not zero, and they all have the same homotopy type. Hence H k(M) is infinite cyclic. If k = 2,6 (mod 8), k i 2, then they are all 7r-manifolds and, in particular, if k = 6 (mod 8) then these include all (k - I)-connected (2k + I)-manifolds with

Hk (M) cyclic and not zero. (2) Fork ""'0 (mod 8) there are at most 8 • [112k+J nondiffeomorphic (k - I)-connected almost parallelizable (2k + I)-manifolds M2k+I with Hk(M) cyclic and not zero, and they all have the same homotopy type. (Received July I3, I965.)

65T-376. D. G. BOURGIN and C. W. MENDEL, University of Illinois, Urbana, Illinois. Cubes in cubes.

A direct proof is presented that up to reflections and rotations a cube can be circumscribed in one way only by another not too much larger. Variants are discussed. (Received July 6, 1965.)

65T-377. C. C. BRAUNSCHWEIGER, University of Delaware, Newark, Delaware. Interior-like elements of the positive cone in Lp.

The following theorem is proved. (For definitions of the underlined terms see Schaefer, Halbgeordnete lokalkonvexe Vektorraume. III, Math. Ann. I4I (I960), 113-142). Theorem. Let (T,::!:,.U) be a positive measure space containing no atoms of infinite measure. If Lp(T,::!:,,u),

I ~ p < oo, has a weak order unit then ,u is u-finite. If ,u is u-finite then for X= Lp(T,::!:,.U) the follow­ ing statements are equivalent: (i) xis a quasi-interior point of the positive cone K in X, (ii) xis a nonsupport point of K, (iii) x is a weak order unit of X, and (iv) x(t) > 0 a. e. on T. If, furthermore, X is infinite dimensional it contains no order units. (Received July 9, 1965.)

702 65T-378. D. F. SHEA, Syracuse University, Syracuse 10, New York. On the Valiron deficiencies of meromorphic functions of finite order.

The value c of the meromorphic function f(z) is deficient, in the sense of Valiron, if liminfN(r,c)/T(r,f) < 1. Put X= liminfN(r,O)/T(r,f), Y = liminfN(r,oo)/T(r,f). Theorem 1. Let f(z) be an entire function of finite order A ;;; 1 and lower order IJ., all of whose zeros lie on the negative

real axis. Then necessarily X ~ min/L ~p~Aisin 'II"PI/(1 +I sin 1rpl). (ln particular, X = 0 when A is a positive integer; this fact completes a result of I. V. Ostrovskii [Zap. Mat. Otd. Fiz.-Mat. i Khar'kov (4) 28 (1961), 23-32] on entire functions of nonintegral order.) Theorem 2. Let f(z) be meromorphic, with order 1/2 ~ A ~ 1. If f(z) has negative zeros and positive poles, then necessarily (*) x2 + Y2 - 2XY cos 1rA ~ sin2 11"A, Combining(*) with a result of Edrei [Duke Math. J. 31 (1964), 1-21] leads to information on the growth of certain classes of functions: Corollary. Let f(z) be meromorphic, with

negative zeros and positive poles. If f(z) has order A (1/2 < A ~ 1) and lower order ,.,., and if the limits lim N(r,O)/T(r,f), lim N(r,oo)/T(r,f) exist, then 1J. = A .and equality holds in (*). (Received July 9, 1965.)

65T-379. M. V. SUBBARAO, University of Alberta, Edmonton, Alberta, Canada. ~ composition of arithmetic functions.

a1 ar . In what follows let n = p 1 ••• Pr be the canonical form of a posit1ve integer n > 1. In the course of some other investigations the author was led to consider an apparently new operation "·" on the set S of arithmetic functions, which might be called exponential composition, defined for any two arithmetic functions f(n) and g(n) by: (f • g)(1) = 1, and for n > 1, (f • g)(n) = Lt(d)g(d'), summed b b over all "exponential divisors" d of n, i.e. divisors of n of the form d = p 1 ••• p r where a /b a /b 1 r bilai (i = 1, ••• ,r), and d 1 = p1 1 1••• Prr r. This paper examines the properties of the commutative semigroup (S, •) and of certain arithmetic functions which naturally arise out of this process. Thus (S, •) has the identity element IIL(n) I (where IJ.(n) is the usual Mobius function and its units are functions f(n) for which f(n) of 0 whenever n is square free). There is an analogue of the Mobius inversion, in

which the corresponding "Mobius" function ILe(n) is= 1 for n = 1 and= IJ.(a 1) ••• IJ.(ar) for n > 1. The order of the function denoting the number of exponential divisors of n is also considered. (Received July 12, 1965.)

65T-380. SVETOZAR KUREPA, University of Zagreb, Marulic!ev trg 19/I p.p. 314 Yugoslavia. Variations of finite independent sets in a Banach space.

Motivated by the problem of D. Travis (Amer. Math. Monthly (B) 70 (1963), 899) we prove

Theorem. Let X be a Banach space; x 1, ••• ,xn independent set of vectors in X, X~ the dual of the subspace Xn ~X spanned by x 1, ••• ,xn and xl Ex; such that xl(xk) = Oik" Then the number K = 1/max, IlL~ >.ixi*ll, (maxi taken over all A= (A , .•• ,>.) such that IA 1 = ••• =IX I= 1) has the " 1= 1 1 n 1 n following properties: (a) if A, .. ., /n are vectors such that llxi - /iII < K (i = 1, ••• ,n) then A are

independent and (b) there is a system of dependent vectors /', such that llx - ~·11 = K (i = 1, ••• ,n). 1 i .,.,.. i (Received July 12, 1965.)

703 65T-381. T. S. MOTZKIN, University of California at Los Angeles, Los Angeles, California 90024 and J. L. WALSH, Harvard University, Cambridge, Massachusetts 02138. Mean approximation on an interval for an exponent less than one.

Expressions for 5(G) = J~ lc - xalp dx, a > 0, in particular for real c near 0, different according as 1,1/(1 - p),2/(1 - p) in turn surpasses, equals or is smaller than a are obtained by use of proper­ ties of the incomplete beta-function in the complex domain. There follow (piecewise) analyticity, monotonicity, convexity and extremal properties of B(c) in certain cases. These results for the pth power (p < 1) measure of approximation to xa by a polynomial c of degree zero are applied and extended to more general real approximees that are piecewise of the form gk(x)lx- akiak, '1c > 0, with well-behaved gk. (Received July 12, 1965.)

65T-382. FRANKLIN HAIMO, 77 Snake Hill Road, Belmont, Massachusetts. Quasi regular elements and Dorroh extensions.

For an S-algebra U (S, a commutative ring), let the Dorroh extension V of U by S consist of all (s,u) under direct-sum addition and with multiplication (s,u)(s',u') = (ss', su' + s'u + uu'). Let Q(T) be the group of quasi regular elements of the ring T. One shows that Q(S) EB Q(U) can be represented by a group of nonsingular linear transformations on u+. Given an abelian group A, there is essentially only one way to turns+ EB A into a ring V extending a ring on A by S with S central in V, and this extension is a Dorroh extension. If an abelian extension G of A by s+ is not necessarily splitting, G can be turned into a ring V with Dorroh-like multiplication if and only if, basically, the cohomology class of the group extension is in the kernel of a certain element of Hom(Ext(S+ ,A), Ext(S+, (End u+)+)). Change the multiplication xy on U to xy - txy where t E Q(S), now calling U, Ut• a!nd form Q'(U) = (') Q(Ut) over all such t. The set Q' (U) extends the radical of U and can be, under some circumstances, a subring of U; Q'(S) is a subring of S if S has a unity. If Sis an integral domain and if Q'(U) is a subring of U, then Q' (V) is a Dorroh extension of Q' (U) by Q' (S) where V is a Dorroh extension of U by S. (Received July 12, 1965.)

65T-383. D. T. HAIMO, 77 Snake Hill Road, Belmont, Massachusetts 02178. Inversion and representation of the reduced Poisson-Hankel transform.

The reduced Poisson-Hankel transform is defined by ]~G(y;t)l/>(y)d!L(y), where G(x;t) = (1/2t)v+l/2e-Y2 /4t, d!L(y) = [2v- 112r(v + 1/2)r1 lv dy, v a fixed positive number. Inversion theorem. Let 1/> be a function integrable in every finite interval, and let f(t) = J~G(y;t)l/>(y)d!L(y), the integral converging absolutely for 0 < t <.b. If limh _,0+ 1/hJ~+h [ip(y) - 1/>(x)]d!L(y) = 0, then 1/>(y) = lim ~ r(v + 1/2) (2 2 ~! r(v + 1/2 + n)r 1y2nf(n)(t). Representation theorem. If f(t) is analytic t--O+.L..n=O at t = 0, f(t) = :E:of(n)(O)tn/n!, ltl < b, then f(t) = f~G(y;t)l/>(y)d!L(y), 0 < t <. b, where I/> is an even entire function of growth (1,1/4b), 1/>(y) = :E:or(v + 1/2) [2 2nn! r(v + 1/2 + n)r 1f(n)(o)ln. 0 ;::; y < oo. When v = 0, these theorems reduce to results derived by D. V. Widder in a paper entitled The inversion of a transform related to the Laplace transform and to heat conduction, J. Austral. Math. Soc. 4 (1964), 1 - 14. (Received July 12, 1965.)

704 65T-384. A. D. WALLACE, University of Florida, Gainesville, Florida. Relative ideals in semigroups. III.

In two papers (published under the same title) attention has been given to the properties of relative ideals of (topological) semigroups. If S is a semigroup and if T is a subset of S then A is a left T-ideal if 0 Y. TA CA. In the papers mentioned above, results of Clifford, Faucett and Schutzen­ berger (among others) were extended. In this paper we consider, in particular, properties of relative ideals which are also subsemigroups and show that many of their properties are analogous to those of ideals and that the structure of S <;an be more easily explicated than by using ideals alone. (Received July 19, 1965.)

65T-385. ·T. S. MOTZKIN, University of California, Los Angeles, California 90024. Extension of the Minkowski-Carathe'odory theorem on convex hulls.

For a convex polyhedron P, let dim P + 1 be the sum of the positive integers d l'"""'dk. Then P is the union of the convex hulls of all F = 11 n ... n F k' where F j' j = 1, ••• ,k, is any (d j - 1)- dim en-· sional face of P. (The special case d 1 = ••• = dk = 1 is the-Minkowski-Carathe'odory theorem.) If d1 = 1 only all F with a fixed, arbitrary, F 1 need be included; if d1 > 1, there may be no F 1 with this property. There follow similar results for arbitrary closed convex sets. (Received July 15, 1965.)

65T-386. UMBER TO NERI, University of Chicago, Eckhart Hall, Chicago 37, Illinois. On the index of first order pseudo-differential operators.

Let !:zf2 denote the space of functions u E L 2 (R n) with values in C m = C X ••• X C, C the complex field. Consider a densely defin~d closed linear operator L: !:zf2 ---> !:zf2 of the form L = A + B, where A and B are m X m matrices of pseudo-differential operators, in the sense of Kohn and Nirenberg, of orders 0 and 1, respectively. If u(D) = Lkdk is the full symbol of a pseudo-differential operator D

on Rn, we write uk(D) = ~· Theorem. Let L = A+ B be as above and assume that u0 (A) = a0 is

Hermitian, u1 (B)= b 1 is skew, a0 commutes with b 1 and with b0 = u0(B), and that b 1 commutes with a_ 1 = u_ 1(A). If for all x and all ~ t- 0 the matrices a~± Lla.l= 1 f[

.A'(L) is finite-dimensional and ~(L) is closed and has finite-codimension. (Received July 16, 1965.)

65T-387. WITHDRAWN.

705 65T-388. JOHN De CICCO, Illinois Institute of Technology, 330 South Federal Street, Chicago, Illinois 60616, and R. V. ANDERSEN, Chicago Teachers College South, Chicago, Illinois 60621. A characterization of the quaternion algebra Q.

A quasi-real fieldS, is a commutative field for which the following two additional properties hold, namely: (I) Any x of S has one of the three forms x = 0, x = a 2, or x =- a 2, where a¥ 0 is inS. (II) The quadratic form oa,f3aa.a(3, with aa.,a(3, inS, is either zero or is of the form oa(3aaa{3 = y 2, where y E S. Moreover it is zero if and only if every a a= 0. Every quasi-real field S is ordered, and hence of characteristic zero. The Gaussian nth roots of unity are elements of the smallest such quasi-real field s0 S s. A noncommutative field Q* is isomorphic to a quaternion algebra constructed on a quasi-real fieldS if and only if it admits an involutorial reverse automorphism T, not the identity I, such that the invariant elements under T form the quasi-real fieldS. In particular, if the ground field S is a complete linearly ordered field R#, then the noncommutative field Q* is isomorphic to the ordinary quaternion algebra of Hamilton. The preceding results are analogues of characterizations of commutative quadratic rings Q constructed on a commutative field K. (Received July 19, 1965.)

65T-389. GEORGE CROSS, University of Waterloo, Waterloo, Ontario, Canada. An integral for Cesllro summable series.

The method of Taylor (S. J. Taylor, An integral of Perron's type, Quart. J, Math. Oxford (2) 6 (1955), 255-274) and properties of (C,k) summable series with coefficients o(n) are used to construct an integral which permits a Fourier representation for such series. This integral is less general than Taylor's but the conditions that are imposed on the trigonometric series i:o obtain the main result are similar to the conditions imposed by James (R. D. James, Summable trigono­ metric series, Pacific J. Math. 6 (1956), 99-110) and reveal to some extent the connection between the work of James and Taylor. (Received July 20, 1965.)

65T-390. R. A. DUKE, University of Washington, Seattle, Washington. Imbeddings of a graph and the Betti number.

Using a technique of J, R. Edmonds for obtaining 2-cell imbeddings of graphs in orientable 2-manifolds (cf. Abstract 572-1, these cNoticei) 7 (1960), 646), a sufficient condition for the non- minimality of such imbeddings is obtained and a procedure, called a reduction, is described for transforming an imbedding satisfying this condition into a 2-cell imbedding of the same graph in an orientable manifold of lower genus. This reduction procedure is used to establish the following, where d(G) denotes the regional number of the graph G and (3(G) its !-dimensional Be~i number. Theorem 1. For a finite, connected graph G, d(G) = 1 (resp., 2) if and only if (3(G) = 0 (resp., 1). Corollary 1. If, for a 2-cell imbedding of a graph G in an orientable 2-manifold M, the number of components of M - G is less than three, then M is a 2-sphere or the imbedding is not minimal.

706 Corollary 2. If the genus of G is n, n > 0, then {3(G) ~ 2n + 2. Another application of the reduction technique is Theorem 2. If there exist 2-cell imbeddings of a graph Gin orientable 2-manifolds of genera m and n, then for each integer k, m < k < n, there exists a 2-cell imbedding of Gin the orientable 2-manifold of genus k. (Received July 21, 1965.)

65T-391. P.R. YOUNG, Stanford, University, Stanford, California. On pseudo-creative sets, splinters, and bounded-truth-table reducibility.

The new result of this paper is the existence of a pseudo-creative set which is not a splinter. Using the methods of On semicylinders, splinters, and bounded-truth-table reducibility, Part I, (P.R. Young, Trans. Amer. Math. Soc. 115 (1965), 329) we prove the following Theorem. If K is a creative set, then K has a subset Q satisfying: (1) Q is not a splinter, (2) Q is bounded-truth-table complete, and (3) Q is pseudo-creative. (Received July 21, 1965.)

65T-392. JAN MYCIELSKI, Polish Academy of Sciences, ul. Sniadeckich 8, Warszawa, Poland. Determinateness, measurability and the propertx of Baire.

R denotes the set of real numbers and H the Hilbert cube ( 0,1 )w. For other notations see my paper On the axiom of determinateness, Fund. Math. 53 (1964), 205-224. The first two of the fol­ lowing theorems are proved without using the axiom of choice: (1) NR_ ---->.9 & ..!¥R_*; (2)Ni* ---->:C &..-It&~; (3)NH(P) holds true for every analytic set P ~ Hw. (But recall that, by I.e. Theorem 3 (ii), it is consistent to suppose that NH(P) fails for some set P which is complement of analytic.) Thus .!4fR has also the principal consequences which ..!¥2 had and by (3) it seems that the consistency of NR is a much better founded conjecture than that of~· (Received July 23, 1965.)

65T-393. A. L. ROARK and L. F. SHAMPINE, Sandia Laboratory, Sandia Base, Albuquerque, New Mexico 87115. The eigenproblem for displacement integral equations.

Roark and Wing (Numer. Math. (7) 2 (1965), 159-170) give a procedure for obtaining the eigen- values, but not the eigenfunctions, of displacement integral equations when the kernel is a cosine transform of a non-negative L 1 function. They are obtained as limits of the eigenvalues of certain finite matrices. By the use of rather different methods, it can be shown that the procedure works when the sign restriction is removed with, however, the strengthening of the integrability requirements to L 2• Moreover, in both cases the limits of the eigenvectors of the matrices are identified as the eigenfunctions evaluated at certain (known) points. The eigenfunctions may be fully described by a uniformly and absolutely convergent series obtained from these limits. (Received July 27, 1965.)

65T-394. H. L. LOEB, 9508 Jellico, Northridge, California, Rational approximations.

Let R~ = {r = p/q: degree of the polynomial p = ap ~ n; degree of the polynomial q = aq ~ m; Let rn (f) = p* /q* be the Chebyshev approximation to f E C (!t,b) from R n where q > 0 over [a,b]J. m m p* and q* are relatively prime and llq*ll = 1 where !I II designates the Chebyshev norm over [a,b). Let d~(f) = min{n- ap•, m- aq•} and H = [f E Cj!t,b]: d~(f) = 0 for all couples (n,m)j. Theorem. The members of H which are analytic are dense in C [a,b], Definition. f E C [a:,b] has "k variations

707 in sign" iff there exists a set of k + 1 points fxi f such that a ;;; xi < xi+1 ;;; b and sgnf(xi) = - sgn f(x. + ) # 0, and there does not exist k + 2 points with this property. Theorem. lim !If - rn (f) II 1 1 m-oo , m = 0 iff f has at most "n variations in sign." This extends some results of Walsh and Boehm. (Received July 28, 1965,)

65T-395. E. D. CASHWELL and C. J, EVERETT. Los Alamos Scientific Laboratory, Los Alamos, New Mexico, The means of order t, and the laws of thermodynamics,

A mean M = M(T., wj) of n numbers Tj > 0, relative to a set of functions wj(T), positive, J I T continuous on (O,oo), is uniquely defined by e(M) = 0, M > 0, where e(T) = e(T;'lj ,wj) = L JTjwjdT is increasing, continuous on (O,oo). If f(T) is positive, continuous, decreasing for T > 0, it is trivial that (I): e(M;Tj,wl) ?; 0, equivalently, M(Tj ,wl) ;;;! M(Tj ,wj)' with (=) if and only if all Tj are equal. (For wj = Tt- 1mj, where m j > 0 have ,Lmj = 1, M(Tj,w/ is the classical mean M(t) of order t, and the monotonicity of M(t) follows.) If Fj > 0 satisfy I;JT.J w .dT = 0, then the Tj and Fj have the same F· J J M mean M relative to the wj' and the inequality L:JT~wldT;;! L:fTfldT (the latter non-negative) is an immediate consequence, This is specialized to yield the solution of a simple heat conduction problem, the above relations arising from energy-conservation and entropy-maximization. General­ ization of these results to the case of a continuous function T(x) is straightforward, with the expected corollaries for integral means and continuous media. (Received August 2, 1965.)

65T-396. ADI BEN-ISRAEL, Israel Institute of Technology, Haifa, Israel, An iterative method for computing the generalized inverse of an arbitrary matrix.

Let A be a (nonzero) complex m X n matrix; A •, A+ its conjugate transpose resp. generalized inverse. Let >.. 1(A • A) be the largest eigenvalue of A* A. Theorem. Let x0 = a.A * where 0 ..1(A*A)). Then the sequence Xk+1 = Xk(2I- AX~ (k = 0,1, ...) converges to A+ ask--+ oo. (This simplifies the results of a paper by the same author and title in Math, Comp, 19 (1965), 452-455,) Examples and applications are given, (Received August 2, 1965,)

65T-397. P. E. SCHUPP, Queen Mary College, University of London, London, England, The conjugacy problem.

Let the group G be the quotient of F, free with given basis, by the normal closure of a subset

R of F. Conjugacy problem: given wl' w2 in F, are their images conjugate in G? We assume R finite, and that r in R implies r cyclically reduced with every cyclically reduced conjugate of r±1 in R.

Condition C(>..): r 1, r 2 in R with r 1r 2 # 1 implies the part of r 1 that cancels in the product r 1r 2 has length less than >..times that of r 1• Triangle condition T: r1 ,r2,r3 in R implies one of r 1r 2, r 2r 3, r 3r 1 is reduced without cancellation. Greendlinger (Dokl. Akad, Nauk SSSR 154 (1964), 507-509 = Soviet Math, 5 (1964), 110) solved the conjugacy problem for R satisfying C(1/6), or C(1/4) and T. We provide a simpler and unified solution for these cases, together with those where R satisfies C(1/5), or C(1/3) and T, or C(1/2) and a certain pentagon condition. The proofs use the graphical methods and results of Lyndon (Abstract 65T-175, these cNOficeiJ 12 (1965), 370.) Greendlinger, and Lipschutz (Comm. Pure Appl. Math. 1960), obtained results under similar hypotheses on centralizers

708 of elements and on torsion elements. Results in these directions are also obtained graphically. (Received August 3, 1965.)

65T-398. STEFAN BERGMAN, Stanford University, StanJ;ord, California. On the distortion of a pseudo-conformally invariant metric under homeomorphism.

Using the kernel function of the domain _e, one defines the length dsb(du,du) = r~:::T m:ii(z,z)dumdlinJ1/ 2 of a metric invariant under pseudo-conformal tr;:;sformations of_£. (see Memor. Sci, Math., vols. 106 and 108, p. 52.] du = fdu1,du2 } is a line element at the point z = f z I ,z 2I. Suppose the domain£_ has the property that to every boundary point p of E. there exists a hyper sphere l of radius P1 which lies in..!;! and a hyper sphere..!!. of radius P2 which contains~ Further, the boundaries 0_!, oE_ and OJ have the point p and only p in common. ~is an exterior andJ, an interior domain of comparison for.!?, at p.) Let W be a homeomorphism of ..2, onto the domain _!L possessing the above indicated properties. Suppose further that a constant e exists, such that 0 < e-I ~ [L(q1,q2)/L(QI,Q2)] ~ e, where L(q1,q2) is the Euclidean distance between q I and q2 and

c:i' = W(qk). Under some additional hypotheses there exists a c = c(p1,p2,e) such that 0. < c -I ~ (ds!!.(z,z,du,dii)/ds!! (Z,Z, dU,dV)] ~c. (Received August 3, 1965.)

65T-399. R. V. CHACON, Ohio State University, Columbus, Ohio. Flows with continuous spectra,

We obtain discrete and continuous versions of the following result. Theorem. Let frt} be an ergodic flow on a nonatomic separable measure space of finite measure. Then the speed on each orbit can be changed to yield another flow [rj) having continuous spectrum. (Received August 4, 1965.)

65T-400. L. C. GLASER, Rice University, Houston, Texas. Uncountably many contractible open 4-manifolds.

The main results are: Theorem I. There exist countably many different contractible 2-complexes P. with regular neighborhoods Nf C s 4 such that for every i: (I) N~ xI ""'IS, I l 1 4 (2) ri(BdNt} "1 I, (3) 1ri(S - Pi) f 1; and if i i j, (4) r 1(BdN;) i r 1(BdNf) and hence N; i N: and 4 4 4 int N; "1 int ~ , (5) ri(S - Pi) i r 1(S - P .). Theorem 2, For n <:: 4 there exist countably many different contractible (n - 2)-complexes with regular neighborhoods C !f such that for P~-zl M~l 2 every i: (I) Mf XI""' In+I, (2) ri(BdM~) i I, (3) r 1(Sn- P~- ) f. 1; and if if. j, (4) ri(Sn- P~-~ f. r1(sn - Pj-2). Theorem 3. There exist uncountably many contractible open 4-manifolds, Theorem 4, There exist uncountably many different involutions of E4 any two distinguished by the fact that their point sets are manifolds having nonisomorphic fundamental groups. Theorems I and 2 are obtained by generalizing results of Zeeman (Topology 2 (1964), 34I-358) and Glaser (Proc. Amer. Math. Soc, (to appear)). Theorems 3 and 4 follow by applying the techniques and results of M. L. Curtis and K. W, Kwun (Topology 3 (1965), 31-42), (Received August 4, 1965,)

709 65T-40l. P. E. CONNER and E. E. FLOYD, University of Virginia, Charlottesville, Virginia. SU-cobordism and KO(X).

An SU-cobordism theory I'*(X,A} may be defined from the MSU-spectrum. For a simplectic n-plane bundle 11 ->X, SU-cobordism characteristic classes pi(v) E 1'4i(X), 0 ~ i ~ n are defined. 4 The correspondence KSp(X) --> 1' (X) given by 11--> p 1 (v) is an additive homomorphism. By means of Thorn classes S E KSp(MSU(4n + 2}} a natural homomorphism f.1. : 1''\x) --> KSp(X) is defined. n s If X is connected then 11 = f.1. p (11) in Ksp(X}, thus in this case KSP(X) is a direct summand of 1'4(X). s l By means of the isomorphism KO(X,A) ~ KSP(S4 A (X/ A)) it follows that for every finite CW -pair, KO(X,A) is a natural direct summand of r 0 (X,A}. (Received August 4, 1965.)

65T-402. NATHANIEL COBURN, University of Michigan, Ann Arbor, Michigan. The intrinsic characteristic forms of the steady state equations of nonequilibrium hydrodynamics.

In this paper, three topics are discussed: (l) the determination of the intrinsic forms (in terms of characteristic variables) of the basic equations of three-dimensional steady state nonequili­ brium (or relaxation) hydrodynamics; (2) the definition of simple waves and a discussion of their properties; (3} determination of sufficient conditions for a chemically reacting fluid to possess simple waves of a special type (Case I). Using a coordinate system introduced by F. Tan (Generalized Prandtl-Meyer flow Technical Report DA-20-0 18-0. R. D. -17213, University of Michigan Research Institute, 1959) two classes of simple waves are defined. It is shown that the waves are of the generalized Prandtl-Meyer type and belong to the Cases I and 11 introduced by the author in the one-dimensional nonsteady case (General theory of simple waves, to appear in J. Math. Anal. Appl.) (Received August 4, 1965.)

65T-403. C. Ao PERSINGER, Air Force Institute of Technology-Air University, Wright­ Patterson Air Force Base, Ohio. Some results on n-books in E .

Let A be an arc in ~ which is locally tame except perhaps at an endpoint p. If U is an open set containing A-p, then U Up is a tapered neighborhood of A. Theorem 1. If A is an arc in E 3 which is locally tame except perhaps at an endpoint p, then A is wild if and only if there exists a tapered neighborhood U U p of A and every arc in U U p with p as an endpoint is wild. Theorem 2. Let Bn be ann-book in E 3 which is locally polyhedral except at a point p on the back of Bn where it is wild. Then each arc A in Bn which contains p is wild. Theorem 3. Let Bn be an n-book in E 3 which consists of k tame leaves and (n - k) leaves which contain only tame arcs. Then each arc in Bn lies in a tame 3-book in E 3• Theorem 4. Let Bn be an arbitrary n-book in E 3. If C is any compact set in Bn, then C lies in an n-book B~ in E 3 which is locally tame except possibly on C and on the back of the book. (Received August 4, 1965.)

65T-404. J. L. PAUL, Western Reserve University, Cleveland, Ohio. Piercing points of homeomorphisms. Preliminary report.

Definition. Let f: U ->En be a homeomorphism, where U is an open subset of euclidean n-space En. A point x E U is called a piercing point off if there exist cP-diffeomorphisms,

710 (p > 0), g: (- I,I]--> U,h: En--> En of En onto itself, and an (n - I)-hyperplane P in En such that (i) g(O) = x, (ii) hfg([- I, I]> n P = hf(x), (iii) hfg(- I) and hfg(I) lie in opposite components of En - P. Iff is a cP-diffeomorphism, (p > 0), then every point of U is a piercing point of f. Theorem. There exist homeomorphisms of En onto itself having a dense set of nonpiercing points (and leaving an (n - I)-hyperplane pointwise invariant). (Received August 5, 1965.)

65T-405. M. M. COHEN, Institute for Advanced Study, Princeton, New Jersey. A Vietoris theorem for simple homotopy type.

Theorem. Suppose that K and L are finite simplicial complexes and that there exists a sim­ plicial map f of K onto L such that C I(x) is contractible for each x in L. Then K and L have the same simple homotopy type. (Received August 5, I965.)

65T-406. T. E. PETRIE, Institute for Defense Analyses, von Neumann Hall, Princeton, New Jersey. K*[BaJ studied via the Moore-Eilenberg spectral sequence.

Let U C G be compact connected Lie groups with K * (U] and K * [G] torsion free. The Milnor construction of the classifying space of U yields a spectral sequence for Bu and a related one for G/U. We prove Theorem 1. 3 a spectral sequence Ei'q such that (1) E~,q = Cotor~~[u][Z,Z],

(2) Er =>K*[Bu]• (3) E 2 = E 00 • Corollary. If K*(U] is torsion free, K*[BU) = R[U] is a power series ring. Theorem 2. 3 a spectral sequence Ei'q such that (1) E~,q = Cotort~tu][K* [G],Z], (2) Er => K * [G/U]. (Received August 5, I965.)

65T-41J7. C. J. EVERETT, 1334 43rd Street, Los Alamos, New Mexico. A divisibility property of Euler's Phi-function. . j For z > y > 0, (z,y) = 1, p prime, Nj = zPl - yP, Qj = NlN0 , denote cases I. pfN0; IIA. p!N0 with p odd, or p = 2, N0 = 0 (4); liB. p!N0, p = 2, N0 = 2 (4), where z + y = 2bu, b !1: 2, u odd. Then the exponent of p in t/>(Qj) is at least: Bj = j(j + 1)/2 in case I, j - I + Bj in IIA, and b + j- 3 + Bj in liB. If a !1: 0 is theexponentofp in N0, its exponent int/>(Nj) exceeds that int/>(Qj) by a. This extends and generalizes results obtained for the case y = 1 by H. Gupte, A congruence property of Euler's t/>-function, J. London Math. Soc., 39 (1964), 303-306. An incomplete result is indicated for the case of a general exponent. (Received August 6, I965.)

65T-408. L. ARNOLD, Technische Hochschule Stuttgart, Stuttgart, Germany. Convergence of lacunary random power series.

Let ak(w), k = O,I, ••• , be complex-valued random variables for which the moduli lak(w)l are independent and identically distributed with distribution function F (x). It is known (cf. L. Arnold, Thesis, Technische Hochschule Stuttgart, 1965) that the radius of convergence r of the random power series I;ak ( w)zk is I or 0 according as Jf'log x dF < oo or = oo. For the lacunary random power series I;ak(w)znk the radius of convergence r = r(F,{nk}) is in the interval !l>,I), with r = 1 when f~logxdF < oo. Let J~logxdF = oo. If limjlimkinfnk/nk > I, then r = 1 or 0. Necessary for

711 r > 0 is that limksup nk/k = oo. For nk = qkP (q,p = l,Z, ••• ) r = 1 or 0 according as J~(logx) 1 /PdF < oo or = oo. (Received July 30, 1965.)

65T-409. CHARLES FEFFERMAN, 9006 Linton Street, Silver Spring, Maryland. A construction of cardinally maximal classes of nonequivalent order types.

If .I$= (A, < ) is an ordered system, and f a 1, ••• ,an} is a sequence (possibly A, the empty sequence) of po'ints of A, define the~· T X(.Q(; a 1, •• .,an) by induction over the ordinal X as follows: (1) T0 (..!4(; ~, ••• ,an)= £(i,j )lai ~ aj ~ (Z) TX+l (..!4(; a 1, ••• ,an) = {Tx(J4(; a 1, ••• ,an, x)lx E A} (3) T 71 ~ a 1, ••• ,an) = {T#l(J

65T~410. B. H. MAYOH, Universitetet i Oslo, Blindern, Norway. Computable classes of real numbers. Preliminary report.

Let C~ be the class of real numbers defined in A. H. Lachlan, Recursive real numbers,

J. Symb. Logic Z8 (1963), 1~ 16. Let P A be the class given by replacing "recursive" by "partial recursive" in the definition of C~. Then (1) P A and C~ include all reals that are not noncomputable limits from above and below (nclab) of A. (Z) an nclab of an effectively enumerable subset of A--e.g. an nclab of A that is not a limit point of the complement of A~~is not inC~ or P A' (3) there is a set A such that ci and P include an nclab of A, (4) there are sets A and B such that I A A I c A U c~ f. c~nB' (5) there are sets A and B such that cAn C~ f. C~ U B" It is not known whether (6) PA = C~ for all A, (7) for all A, pAis the set of reals that are not nclabs of an effectively enumerable subset of A. (Received August 9, 1965.)

65T~411. R. C. GILBERT, California State College at Fullerton, 800 North State College

Boulevard, Fullerton, California. An asymptotic phase for a Sturm~Liouville operator with unbounded potential.

Let Ri (z), i = 1,2, be the resolvent and pi the spectral function of the operator in LZ(O,oo) defined by the differential operator Liy = - y" + [~ x + (i ~ 1)p(x)]y and the boundary condition y(O) = 0, where pis real, p E c 2 !9,A], p(x) = 0 for x ~A, J~p(x)dx = 0. If cpi(x,X) is the solution of Liy = 1 1 Xy ,y(O) = 0, y' (0) = ~ 1, then for x ~A, cf>1(x, X) (pj(X)] /Z = [(x + X )/3] /Z[J 113(X)sin a.i(X) ~ 3 Y 1; 3(X)cos a.i(X)], where X= (Z/3)(x + X) /Z, and limX-d:ro[a.z(X) ~ a. 1(X)) = 0. o(X) = a.z(X) ~ a. 1(X) is called the asymptotic phase. It is absolutely integrable, and the following equations are true: S fRz (iT) -R1 (ir)} = ~ (1/'11") J~00 (X ~ iT)~z /i(X)dX, where S stands for trace, and (1/'11") f~ o(X)dX = - f-C:f~p(x)cpi(x,>.)dxdp 1 (X). (Received August 9, i965.)

712 65T-412., W. D. L. APPLING, North Texas State University, Denton, Texas, Refinement-­ unbounded set functions and absolute continuity,

Suppose F is a field of subsets of a set U, R+ is the set of all real-nonnegative-valued functions defined on F, and R~ is the set of all finitely additive elements of R+, Suppose each of hand m is if in R ~· Definition, If Q is in R +, then the statement that Q is m-refinement-unbounded means that if 0 < K, then there is a subdivision D of U such that if I is in a refinement of D and m(I) # 0, then Q(I) > K. Lemma. If, for each V in F, g(V) =sup fvmin{Km(I), h(I)} for 0 < K, and K* ~ 1, then for each V in F, fvminfK*[h(I)- g(l)], h(I)} = h(V)- g(V). Theorem, If there is an m-refinement­ unbounded element of R +, then the following two statements are equivalent: (1) If P is a,.!l m -refinement­ unbounded element of R+, then fumin{P(I)m(I), h(I)j = h(U), (2.) his absolutely continuous with respect to m. (Received August 9, 1965,)

65T-413, S. A. NAIMPALLY and M. G. MURDESHWAR, University of Alberta, Edmonton, Alberta, Canada, Survey of quasi-uniform spaces.

The purpose of this paper is threefold, Firstly, to point out known results of uniform spaces which are also true in quasi-uniform spaces (e.g., Theorem 6.4, Kelley, General topology, Van Nostrand, Princeton, 1955). Secondly, when this is not so (e.g., Theorem 6, 7, ibid), to supply counter-examples, Thirdly, to extend such results with additional hypothesis. E.g., it can be proved that Theorem 6. 7, ibid, is true for R0-spaces, The paper also includes characterizations of R0 - spaces and regular spaces, some properties of quasi-uniformly continuous functions, and function spaces and hyperspaces of quasi-uniform spaces. Some of the main results are: "A quasi-uniform space X is regular iff for each entourage U and x E X, there exists a symmetric entourage V such that V oV[x] CU(x]" and "A quasi-uniform space X is pre-compact iff every 0-net in X is Cauchy." (Received August 11, 1965,)

65T-414. DAVID NELSON, George Washington University, Washington 6, D. C. Non-null implication.

Three kinds of negationless constructive systems for both predicate logic and arithmetic are given by sequent calculi of the Gentzen type and are shown to satisfy certain realizability definitions.

Central to all the systems is an implication A(X) :::>xB(x) which, if true, has A(x) satisfiable for some choice of numerals for the variables x. In systems P 1 and A1, every formula which is part of a formula occurring in a provable sequent has a closure with existential quantifiers which is provable. In the less strict systems P 2. and Az this property holds only for formulas which are part of a provable formula, In P 1, AI' P 2.' Az, every formula which is part of a realizable formula has a realizable substitution instance, In P3 and ~ this property holds only for formulas not within the scope of a disjunction, but in a disjunction formula at least one of the component formulas has this property, (Received August 12., 1965,)

713 65T-415. J. L. WALSH, Harvard University, Cambridge, Massachusetts. An extension of the generalized Bernstein lemma.

Let E be a closed bounded point set whose complement is connected, and regular in the sense that it possesses a Green's function G(z) with pole at infinity. We denote generically by ru the locus G(z) ; log u ( > 0) and its interior by EO". Let rational functions rnv(z) of respective types (n, v}, 1 where vis constant, satisfy lim supn ~ 00 [max lrnv(z) j, z on E] /n ~ 1/ p 1, 1 < p1 ~ oo. Let S be a closed set in the closed interior of EO", 1 < u < p1, and containing no limit point of the poles of the rnv(z). Then the sequence rnv(z) converges uniformly to zero on S, and we have lim supn~ 00 1 [maxlrnv(z}j, z on SJ /n ~ u/p1• Thernv need notbe definedfor every n. (Received August 13, 1965.}

65T-416. LEONARD SARASON, Stanford University, Stanford, California. Elliptic regularization for symmetric positive systems.

Let (*) be the boundary value problem Lu ; f in G, u E Yon the boundary S of G, where

G C Rn is given by xi > 0, i; 1, ••• ,k, - oo < xj < oo, k < j ~ n. The boundary space Yis defined on each smooth component S.£ of S by equating certain components of u to zero, i.e. sets If of positive integers are given such that if j E Ii, then uj ; 0 on S p,. Suppose that S is noncharacteristic, Y is maximal (strictly) positive with respect to the natural boundary form JuBu dS, and an a priori estimate in L 2 (G) for 'Vu can be found in terms off and 'Vf, as in Sarason (Comm. Pure Appl. Math. XV (1962}, 237-288). Then weak solutions of(*) are strong, and iff and 'Vf are in L2(G}, (*)has a strong solution u with 'Vu in L 2(G}, and the restriction of 'Vu to Sin L 2 (S). The proof uses elliptic regularization as suggested by Kohn and Nirenberg, Noncoercive boundary value problems, (Comm. Pure Appl. Math. XVIII, (1965)). (Received August 16, 1965.)

65T-417. B. R. HEAP, National Physical Laboratory, Teddington, Middlesex, England. Random matrices and graphs.

Let A be anN X N matrix whose off-diagonal elements are assigned nonzero or zero values randomly with probabilities p and q (; 1 - p) respectively. Let PN, QN denote the probabilities that A can be reduced by permutation of its rows and columns into block-triangular or block-diagonal form. Then P N' QN are also the probabilities that the random graph associated with A should contain more than one strong component or more than one component respectively. By enumerating graphs having these properties, asymptotic expressions for PN and QN can be derived in the form of the first few terms in their series developments in powers of q. The leading terms give PN ~2NqN-1 and QN ~Nq 2 N- 2 • The final expression for PN is correct to O(q 4 N- 1 ~ and that for QN is correct to O(q 12N- 7'). (Received August 16, 1965.)

65T-418. A. H. ZEMANIAN, State University of New York, Stony Brook, New York. The Hankel transformatioa of certain distributions of rapid growth.

The Hankel transformation is extended to certain distributions having no restriction on their rate of growth as x --> oo. These distributions comprise the dual space B~ of the space BJL, defined as follows. ti>(x) E BJL(JL ;;:; - 1/2) iff (1} tjJ is infinitely differentiable on 0 < x < oo, (2} tjJ(x); 0 for

714 x >X for some X depending on ¢, and (3} (x -lD)kx -ll-l/'1p is bounded on 0 < x < oo, where D = d/dx.

The Hankel transformation T f.L of B11 is a certain space Yf.L of analytic functions of 1J having branch points at 1J = 0 and 1J = oo. With suitable topologies assigned to Bf.L and Y11 , TIL is a topological isomor­ phism from Bf.L onto Y!L. Let Y~ be the dual of Y!L' and assign the weak topologies to B~ andY~. For f E B~, define F = Tf.Lf by ( F, ) = ( f, ¢) where E Yf.L and ¢ E B JL• Then, Tf.L is a topological isomorphism from B~ onto Y~. (Received August 16, 1965.}

65T-419. E. B. DAVIS, Stanford University, Box 2966, Stanford, California. Bounds for functions holomorphic in analytic polyhedra.

Let f(z l'z2) be a function of two complex variables which is holomorphic in an analytic poly­ hedron M. Let F be the distinguished boundary surface of M. Then F = U~,s=lFks' and Fks = Pks(Vks)' where k f. s,Vks is a domain in an auxiliary Zks -plane, and Pks is a one-to-one map. See Bergman, Uber eine Integraldarstellung von Funktionen 21weier komplexer Veriinderlichen, Mat. Sb. (43} 1 (1936}, 851-861 for a description of such domains and of the maps Pks" Using the Bergman-Wei! integral formula, the following bounds for If I in M in terms of If I on F are obtained:

lf(z 1,z2)1 ~ (1/411"1~(1/a.ka.Jrks ffvkslfldAks' and lf(z 1,z2)1 ~ (l/41Tl~)1/a.ka.)Qkscks" Here (z 1,z2) is an interior point of M; each sum is over k = l, ••• ,n; s = l, ••• ,n; k f. s; the functions ~ = ~(z l'z t and the constants rks' Cks' and n depend only on the domain M and not on the function f; and Qks = max If I on Fks" At least one of the functions ~ (z 1,z2) tends to zero as (z 1,z2) tends to any boundary point of M. The first bound utilizes more information about If I on F, but it is valid only for a more restricted class of domains than the second one is. (Received August 16, 1965.}

65T-420. RICHARD KRAFT, Negev Institute, P. 0. B. 1025, Beer Sheva, Israel. Riemann functions for systems of linear hyperbolic equations.

By developing a suggestion of Avron Douglis, Riemann functions of several species are defined for linear, hyperbolic systems of partial differential equations in two independent variables. From these different species a fittest species is naturally singled out and shown to exist by two methods one of which is very constructable. (Received August 16, 1965.)

65T-421. L. B. TREYBIG, Tulane University, New Orleans, Louisiana 70118. A characterization of the double point structure of the projection of a polygonal knot in regular position.

Suppose 0 < a 1 < a 2 < .•• < a 2n < 1 (n ~ 1) and that W is a decomposition of An= fa 1,. .. ,a2n} into two element sets. Let f: [O,l)-->E 2 be a map such that (1) f(t) = f(t') fort< t' if and only if {t,t'} E W (2) Im f = "m B.C. where (a) each B.C. is a straight line interval, and (b) f(An) C E 2 - L-.1 1 1 1 1 {B 1, ... ,Bm,C 1, ... ,CmJ' and (3) f(t) -->f(O) as t -->1. W is said to determine the double point structure of f. For such a decomposition W a condition is given which is both necessary Jnd sufficient that there exist an f whose double point structure is determined by W. (Received August 16, 1965.)

715 65T-422, J, W. NEUBERGER, Emory University, Atlanta, Georgia 30322, The lack of self­ adjointness for three point boundary value problema.

Suppose that a < c < b and each of p and q is a continuous function on [a,bJ so that p(x) > 0 for all x in ~,b ]. Suppose furthermore that each of P, Q and S is a 2 X 2 real number matrix and Q is not a 0-matrix, Theorem, If there is a unique solution f.!£. (pf')' - qf = 0, PF(a) + QF(c) + SF(b) = 0 (F(x) is the ordered pair (f(x), p(x)f'(x)) for x in [a,b]), then the Green's function K for the corres­ ponding inhomogeneous problem is not symmetric, i.e., it is not true that K(x,t) = K(t,x) for all x and t ~ jjl.,b]. A similar result holds for n-point boundary value problems for n > 3. (Received August 18, 1965,)

65T-423, R, W. GILMER, JR., Florida State University, Tallahassee, Florida, The pseudo­ radical of a commutative ring, Preliminary report.

If D is an integral domain with identity having quotient field K, the pseudo-radical of D is defined to be the intersection of all nonzero prime ideals of D. The pseudo-radical arises naturally in considering the relation between the statements "D has Jacobson radical zero" and "D [u] has Jacobson radical zero, where u E K". Theorem 4 proves that the first statement implies the second,

From this it follows that if M is a prime ideal of the polynomial ring R (t] over a commutative ring R and if P = M nR, then M is an intersection of maximal ideals of R[X] if P is an intersection of maximal ideals of R, This latter theorem implies the following result of Goldman on Hilbert rings:

If R is a Hilbert ring then R [X] is also a Hilbert ring, (Received August 18, 1965,)

65T-424, R. G. BUSCHMAN and M. C, WUNDERLICH, State University of New York at Buffalo, Buffalo, New York 14214, Sieves with generalized intervals,

Parallel to the development of W. E. Briggs [Duke Math. J. 30 (1963), 297-312] we begin with A (1) = {k + 1} and construct A (n+ 1) from A (n) by sieving out one element from each interval I~n) = (n + (k - 1) .Un, n + k,un], where .Un denotes the interval length, (The sieving terminates unless .Un > 1.) The ultimate sequence A= { ak J = {aik)}, If we assume J1clk ~ a > 0, then an can be bounded below, Feedback is introduced by setting ,uk = Akak or avoided by defining .Uk explicity, Continuing as M, C, Wunderlich [Canad. J, Math, (to appear)] we obtain analogous conditions for a ~ cnlogn. Let i.(n) denote the number of stages at which exactly one element less than a is n n sieved out, Set d(n) = n/(n + l(n)), If an > c n L(n) (L(n) an iterated logarithm) and 1Ln = Anan,An ~A> 0, then an~ nlogniff 1 + L~= 1 /nd(k)/(kAk) ~ d(n)logn/n. For An"" A, an~ (nlogr);A iff I:~= 1 d(k)/k ~ d(n) logn, An example with no feedback, !Ln = Anlogn, yields an~ cn(logn)1/A (1 + i(n)/n)--A entering in a quite different manner, (Received August 12, 1965.)

716 65T-425. SIDNEY PENNER, The City College, New York, New York 10031 and K. J. SCHROEDER, State University of New York, at Buffalo, Buffalo, New York 14214. Topological aspects of function semigroups.

Let (S, ~, 0 , n, R, L) be a function semigroup [B. Schweizer and A. Sklar, Math. Ann. 143 (1961), 440-447]. Let F be some fixed set of restrictions of n. The elements ofF correspond to

"closed sets". Definition. An element f E S is continuous if F o f is a subset of f o F. This definition corresponds to the continuity of a function on its domain. Several equivalent forms of "continuity" are given and a "property C" corresponding to the property of being a closed map is defined. Theorems concerning these concepts are presented along with a definition of "homeomorphism". A definition of "compactness" is stated for elements less than n. It is proven that "the continuous image of a compact element is compact". A definition of "complemented" function semigroup and examples of complemented and noncomplemented function semigroups are given. For complemented semigroups another characterization of continuity is made. Also, for these systems, a definition of "connected" is given and it is shown that "the continuous image of a connected element is connected". (Received August 19, 1965.)

65T-426. H. H. CRAPO, University of Waterloo, Waterloo, Ontario, Canada. The Mobius function of a lattice.

Let L be a finite lattice, with Mobius function !Land zeta function .\, and let x be any element of L. Then JL(0,1) =LJL(O,y).\(y,z)JL(z,1), a double summation over all pairs (y,z) of complements of x in L. The proof proceeds by an analysis of chains, and an application of Rota's cross-cut theorem. (Received August 19, 1965.)

65T-427. R. P. DICKINSON, JR., Lawrence Radiation Laboratory, Box 808, Livermore, California. Distinctness and strong distinctness of certain semigroups of operators. Preliminary report.

Let E be a set with n elements. Let SE be the set of all relations on E. Let R, S, T and C be operations on SE defined respectively by (4.1), (4.2), (4.5) and (4.13) of [1]. Let G* be the semigroup of Theorem 4.1 of [1]. Let N* be the semigroup of Theorem 4.3 of [1]. Gn' a homomorphic image of G*, is the semigroup generated by R, S, and T. Similarly, Nn is the semigroup generated by R, S, T, and C. If for any X, Y E G* 3 X¥ Y, one can find a p ESE 3 pX ¥ pY, then we say Gn is distinct on SE. If 3 a p ESE 3 for all pairs X, Y E G* and X i Y and pX f. pY, then we say Gn is strongly distinct on SE. Similar definitions apply to Nn• It is known that: (1) Gn' n ~ 2 is not strongly distinct on SE, and if n ;;; 5, Gn is strongly distinct on SE; (2) 3 a semigroup E of order

6 3 N 6 is strongly distinct on SE" Results. Theorem 1. G 3 and G 4 are not strongly distince on SE" Theorem 2. N 5 is not strongly distinct on SE; E a groupoid. Theorem 3. 3 a semigroup E of order n ;;; 3 3 Nn is distinct on SE" Reference. [1], T. Tamura, Operations on binary relations and their applications, Bull. Amer. Math. Soc. 70 (1964), 113-120,. (Received August 20, 1965.)

717 65T-428. ALEXANDER ABIAN and DAVID DEEVER, The Ohio State University, 231 West 18th Avem.1e, Columbus, Ohio 43210. Representation of simply ordered sets and the generalized continuum Hypothesis. I.

Definition 1. A subset D of a partially ordered set (P, ;;;; ) is said to be weakly dense in P if for every two elements p and q of P with p < q there exists a d ED such that p < d ;;;; q. Furthermore, D is said to be Hausdorff-dense in P if there exists two elements a and b of D such that p ;;;; a < b ;;;; q. Theorem 1. Let (P, ;;;; ) be a simply ordered set with a weakly dense subset of power ;;;; N;\: Then (P, ;;;; ) is isomorphic to a set of sequences of type wA made up of 0,1 and ordered by the principle of first differences. Theorem 2. Let S be a set of sequences of type wA made up of 0,1 and ordered N· by the principle of first differences. Then S has a Hausdorff-dense subset of power ;;;; limi

65T-429. C. R. COMBRINK, The University of Kansas, Lawrence, Kansas 66045 and R. E. PHILLIPS, Wisconsin State University, Eau Claire, Wisconsin. A note on subsolvable groups.

Definition. A group G is subsolvable if every non-E homomorphic image of G contains a non-E subnormal Abelian subgroup. Definition. A group G is an SJ*-group if G has an ascending solvable normal series such that each term of the series is subnormal in G. If each term of the series is normal in G, G is called an SI*-group. Theorem. The class of SJ*-groups coincides with the class of subsolvable groups. Using this result and an example of P. Hall (see The Frattini subgroups of finitely generated groups, Proc. London Math. Soc. (3} 11 (1961}, 350-351}, it is shown that the class of subsolvable groups strictly contains the class of SI*-groups. This gives a negative answer to a question raised by R. Baer (see Nilgruppen, Math. z. 62 (1955), 422}. Theorem. The class of SJ*­ groups is a radical class in the sense of A. G. Kurash (see Radicals in group theory, Soviet Math. (1) 3 (1962}, 260-263}. Theorem. Let G be a group, B the Baer radical of G, and S the SJ*-radical of G. Then B and S are simultaneously finite or simultaneously infinite. (Received August 23, 1965.)

65T-430. B. L. OSOFSKY, Rutgers, The State University, New Brunswick, New Jersey. Global dimension of valuation rings.

Let nn be the first ordinal of cardinality Nn. Theorem A. Let R be a valuation ring, I an ideal of R, S = {xRjx E I} linearly ordered by inclusion. Then the projective dimension of I= n + 1 if and only if nn is cofinal inS. Corollary. There exists a ring R such that sup{inj dim (I)ji a right ideal of R} = sup {inj dim (R/I) II a right ideal of R} = 1, yet the global dimension of R = n, where n is any pre-assigned value, 1 ""'n ;;;; oo. Hence there is no analog of the Global Dimension Theorem in terms of injective dimensions of cyclic modules or ideals. However, if R is a right perfect ring: Theorem B. sup{inj dim (I)ji a right ideal of R} =global dimension (R} ;;;; sup{inj dim (R/I)ji a right ideal of R} + 1. Theorem A is proved by construction of a projective resolution of I. The corollary

718 is based on a theorem of Matlis (Nagoya Math, J, 15 (1959), 60) which implies that, for an almost maximal valuation ring, the given supremums are 1. Theorem B is based on a theorem of Sandomierski (Ph.D. thesis, Pennsylvania State University, 1964} which implies that, if R is right perfect, then R is a test module for projectivity. (Received August 23, 1965.)

65T-43l. R. M. SORENSEN, 3400 Toledo Terrace, Apt. J-1, Hyattsville, Maryland. Differential­ integral calculus for abstract algebraic-topological structures. IV.

Let G be a multiplicative Abelian topological group with regular topology T, Let Q be a fundamental system of closed neighborhoods, and let F, f be mappings from G into G.- Definition.

F has derivative f on N in Q iff given N 0 in Q, such that l"o C N, and any distinct points x,y in N0, 1 1 there is a third point z in N0 such that F(x)F(yf = f(z)xy- • Theorem 1. If F has derivative f on N in Q and g0 is any fixed element of G, then g0F has derivative f on N. Theorem 2, If F, F 1,F 2 have respective derivatives f, f1 , f2 on N in Q, and if F(g) = F 1 (g)F 2 (g), then f(z) = f 1 (z 1}f2 (z2)xy -l. Theorem 3, If F has derivative f on N in Q, then F defined by F(g) = F(gf 1 has a derivative f on N 1 1 with f(z) = xy - xy -lf(z 1) - • Theorem 4. If F has derivative f on N in Q and if N = N 1 U N 2 with Nl'N2 in Q, then setting F(g) = X 1(F(g))X2(F(g}}, in which Xi is the "Multiplicative" Characteristic Function of Ni' yields F(x}F(y)- 1 = [X1(x)(X1(y))-lt1(z)] [X2(x)(X2(y))- 1f 2(z)] in which fi(z) = f(z) if z E N and equals e otherwise, 'theorem 5, If F has derivative f on N in Q, then for g0 such that 1 1 g0x, g0y belong toN, we have F(g0x)F(g0y)- = g0x(g0y)- f(z 1) = xy-1f(z/ (Received August 9, 1965.)

65T-432, ALFRED TARSKI, University of California, Berkeley, California 94720, On the existence of large sets of Dedekind cardinals.

IAI denotes the cardinality of the set A; lA I= w, or lA I < w. iff A is denumerable, or finite, respectively. Af denotes the set of finite sequences without repetitions, with terms from A • .6. is the class of Dedekind cardinals, i.e., infinite cardinals n with n 'f n + 1. As is known, the axiom of choice implies .6. = 0. Without assuming this axiom the following theorems hold: I. _.!!.. .6.1 0, then there is a

~ };£:_.6, ~III = 2w such that any two a., {3 E .6. are comparable (a. ;:; {3 or {3 ;:; a.); I is ordered similarly to the set of reals in natural order; for any A,B with lA 1. IB I E}; there is a mapping of A

~B. In fact, let D be a set with ID I E .6. and g a mapping of the set {J.L:IL < w} onto the set of rationals; for every real x let Sx = [ tr: tr E Df,g( I tri} < x}. Then we can let }; = {ISx l:x any. real}. (If Dis a set of reals, then the sets Sx can also be constructed as sets of reals.) II. If there are two noncomoarable a.', /3' E .6., then there is a set I' ~ .6. ~II' I= w in which no two distinct elements are comparable. In fact, we let };' = [a.' • 'Y + {3 '· J.L: 'Y + IL = a.', IL < w}. In proving I and II we apply the following lemmas (independent of the axiom of choice): a. E .6. iff neither a. < w nor W

65T-434. RONALD JENSEN, Annaberger Strasse 400, Bad Godesberg, Germany. An imbedding theorem for countable ZF models.

Let M be a countable complete standard model of ZF and AC. Let

(c) If N

(iii) (Na.)N = (N

65T-43·5. I. I. HIRSCHMAN, JR., Washington University, St. Louis, Missouri 63130. On a theorem of Kac and Achiezer.

Let £a(0 be a measurable function on (-oo,oo) which satisfies the conditions:

00 2 (i) jc W jd < oo, 1£ (0 1 (1 + I~ I> < oo; (ii) c(t) =I 0 for -oo < t < oo, and [arg c(t)]:000 = 0 Joo-oo-a ~ J-IX! a (here c(t) = 1- J~oo£a(~)e 1t~ d~); (iii) the limits limh~O+h- 1 J~£a(~)d~ and limh~Oth -lJ~h£a(0d~ exist. We also suppose that £a (0) = limh~0+(2h)- 1 J~h&a (Oct t this, however, is merely a normaliza­ tion condition. Let D(r) be the Fredholm determinant corresponding to the kernel £a a -11), 0 ~ t 1J ~ r. Then it is shown that lim D(r)G-r = E where G = exp {(1/21r)J00 log c(t)cttJ, the integral being r~ oo -oo evaluated by a suitable summability method. In general E is given by a rather complicated formula, but at least in the case .£a(0 is Hermitian E = expfJ~&.;(~)l:<~(- ~)~ d~} where J-:£~(0eit~d~ = logc(t). This extends results of Kac, Duke Math. J., 21 (1954), and Achiezer, Ukrainian Math. J., 16 (1964). (Received August 26, 1965.)

720 65T-436. T. G. McLAUGHLIN, University of Illinois, Urbana, Illinois. A note on potential recursiveness of regressing functions.

Consider the following properties of an infinite set a. C N: (1) a. is retraceable; (2) a. has an infinite subset (3 such that (3 is retraced by a general recursive function; (3) some ordering

of a. can be regressed by a function with general recursive extension. Using an attractive charac­ terization of property (2) due to D. A. Martin, it is easy to prove the existence of an infinite a. for

which (1) is true and such that (3) fails for all infinite subsets of a.. Again making use of Martin 1 s characterization, we further prove the Theorem. (1) &(2) ==/==> (3). (Received August 27, 1965.)

65T-437. C. W. LEININGER, Arlington State College, Arlington, Texas 76010. Totally ordered partitions of a partly ordered set.

If K is a set partly ordered by the relation R, let K* denote a partition of K such that if C is a subchain of K, no two elements of C belong to one set of K *, and let R * denote the collection of pairs (A, B) of sets of K* such that the relation A X B n R is from A onto B. Theorem 1. If K is partly ordered by R, the following two statements are equivalent: (1) K * is maximal and

U (A,B) ER *A X B n R = R. (2) K* is a chain with respect toR*. Theorem 2. If with respect toR each maximal subchain of K is well-ordered and K* is a chain with respect toR*, then K* is unique and well-ordered. (Received August 27, 1965.)

65T-438. WOLFGANG WASOW, University of Wisconsin, Madison, Wisconsin 53706. Asymptotic simplification of self-adjoint differential equations with a parameter.

Let A(x, ~) be an n-by-n matrix function holomorphic in both variables for - x 0 "" x ""x0 , 0 < ~ ""~ 0 • and admitting a uniform asymptotic expansion A(x,~) ~L~oAr(x)~r. as ~---> + 0. It is known [Y. Sibuya, J. Fac. Sci. Univ. Tokyo Sect. I, 7 (1958), 527-540] that A0 (0) has more than one distinct eigenvalue the system of differential equations ~hdy /dx = A(x, f)y (h a positive integer) can be decomposed into several systems of the same type but of lower order by performing a change of dependent variables involving asymptotic series. In this paper it is assumed that the system is self-adjoint, i.e., that A *(x, ~) = - A(x,q, where A *(x, ~) = AT (x,f), and it is shown that a transforma­ tion somewhat different from Sibuya's preserves the self-adjointness in the decomposition process.

An analogous theorem holds if A(x,f) E C 00• If A(x.~) is analytic a slightly weaker result is proved

to be valid in open sectors of the x and ~-planes. In that case the decomposed system may differ from self-adjointness by terms that are asymptotically zero. (Received August 27, 1965.)

65T-439. MARY WEISS, c/o Professor Al. Buccino, 5707 Kenwood Avenue, Chicago 37, Illinois. Strong differentials in LP.

If x = (xl' ••• ,xn) is a point of then-dimensional space and f(x) E LP, 1 ""p < oo, we say that f has at x a kth differential in LP--briefly, a (k,p) differential--if there is a polynomial P(t) of degree "" k such that (*) flQI-l JQ jf(x + t) - P(t) JP dt] 1/p = o(hk), where Q is a cube of edge h containing the origin. We say that f has a strong differential in LP--briefly, differential (k,p)'--if (**)fiR 1-l JRif(x + t)- P(t)jPdt l/p = o(wk), where R is a rectangle with sides parallel to the axes containing the

721 origin and of largest edge length w. (1) If n = 2, k ;?: 1, and iff has a (k,p) differential in a set E, then it has a (k,p)' differential almost everywhere in E. (2) For n > 2, (k,p) differential in E implies

(k,p- E)' differential a.e. in E, if £ > 0. (3) This is false for E = 0, but(*) in E implies (**)a. e. in E, provided R has two different edge lengths at most. (4) There are analogous results for k = 0, p = ·1 (the case p > 1 is very well known) with hypotheses slightly stronger than (*). (Received August 30, 1965.)

65T-440. L. V. QUINTAS, St. John's University, Jamaica 32, New York and FRED SUPNICK, The City College of New York, New York 31, New York. Extrema in space-time.

Criteria are established, which, if satisfied by a finite setS of events inn-dimensional space-time, then one is able to determine immediately, from among all possible polygonal circuits with vertex set S, circuits having the least and the greatest Minkowskian length. Various classes of events which are realizations of the criteria established are presented and studied. An explicit statement of these results will appear in Bull. Amer. Math. Soc. and the paper will appear in Canad. J. Math. (Received August 30, 1965.)

65T-441. H. W. GUGGENHEIMER, University of Minnesota, 400 Ford Hall, Minneapolis, Minnesota 55455. Finite sets on smooth curves and surfaces.

A gap is filled in Schnirelmann's proof of his theorem on the existence of a square in smooth Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface in Rn, c 3 - diffeomorphic to sn- 1, there exist 2n points which are the vertices of a

regular 2n-cell en. 2. Every plane C' Jordan curve can be C 1 approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (angles 1r not excluded). 3. On every plane c 2 Jordan curve there exist 5 distinct points which are the vertices

of an axially symmetric pentagon with given ·base angle a., 1r/2 ~ a.< 1r, and three equal edges. (It cannot be asserted that the angle at the vertex on the axis of symmetry is f 1r.) (Received August 30, 1965.)

65T-442. L. E. FULLER, Kansas State University, Manhattan, Kansas 66504. Approximating dominant characteristic roots of a matrix.

If the dominant characteristic root of a matrix has multiplicity one in the minimum equation of the matrix, it can be approximated by normalizing the product of the matrix and an arbitrary vector. The product of the matrix and the normalized vector is normalized repeatedly until the normalizing factors converge. It is shown that the normalizing factors can also be used to determine dominant characteristic roots when there is more than one in the minimum equation of the matrix. A set of equations is formed that have normalizing factors as coefficients. Denote the product of the elementary divisors of the dominant roots in the minimum polynomial as g(x). Any fixed component of the vector Akg(A)Y yields the required equation in the dominant roots. As many equations as needed can be formed by using successive values of_!\;. The solutions for sets with increasing .!s are compared until there is convergence for each root to a specified accuracy. At present, in order to use this process, it is necessary to know the nature of the dominant roots unless they are a complex

722 conjugate pair. An investigation is now underway to determine if the behavior of the normalizing factors can give a clue as to the nature of the dominant characteristic roots. (Received August 30, 1965.)

65T-443. MARTIN HELLING, 2719 Webster Street, Berkeley, California 94705. Transfer and compactness properties of some generalized quantifiers.

Let T be a set of closed formulas in a formal language L obtained by adding a quantifier symbol Q to a first-order language L0• Let a.,{3 denote infinite cardinals. Definition. a. is weakly compact iff a. is not in the class c 0 of Keisler-Tarski, From accessible to inaccessible cardinals, Fund. Math. 53 (1964), 227. A structure A for Lo is a (3-model ofT iff all formulas ofT become true of A when quantifier expressions Qx are read: "for at least f3x". Transfer theorem. If w, T < fJ and a. is weakly compact, then T has an a.-model implies T has an a.•model. Compactness theorem. If w ::;;; T < a. and a. is weakly compact, then T has an a-model iff every finite subset of T has an a-model. These results partly confirm conjectures of G. Fuhrken. (Received September 1, 1965.)

65T-444. J. H. SILVER, International House, Berkeley 4, California. Metamathematical properties of certain large cardinals. < Xo If ,\and v are cardinal numbers, we say ,\->(V} if for any equivalence relationE on the set of finite subsets of ,\, having two equivalence classes, there is a subset X~,\ having cardinality v such that any two finite subsets of X with the same number of elements are E-equivalent (a notion of Erdos and Hajnal). Suppose K--> (K) < Xo and ~ = (K, <, R) where < is the usual ordering on K. Using methods of Mostowski and Ehrenfeucht (Fund. Math. 1956), which show how to generate models from orderings, one can construct an elementary tower of structures ((B , < , R ) : a is an ordinal) a a a elementarily equivalent to ~ with the following properties: (1} Each substructure of (B a' < a.'Ra) having cardinality < K can be imbedded isomorphically into ~. (2) If a.= U a. ::;;; fJ, then B a is an initial segment of BfJ with respect to < fJ• (3) If a. is an uncountable cardinal, the order type of (B a' (N1) < OJ. (Received September I, 1965.)

65T-445. W. N. REINHARDT, University of California, Berkeley 4, California, and ]. H. SILVER, International House, Berkeley 4, California. I. On some problems of Erdllis and Hajnal.

For terminology see th~ preceding abstract and Hanf-Scott, (Abstract 61 T-240, these cJfoticei), 8 (1961), 445}. Ra will denote the set of all sets having rank < a. If K is a cardinal such that K--> (K) < No, then, using rneth :>ds of the preceding abstract, we show that there is some cardinal ,\ < K which satisfies the formda ('v'u)(li(R n u)--> ( 3y < x)li(R n u)) in the structure ( RK, f) for X y every formula 11, with one free variable, not involving x, y, or u. Thus the first cardinal K0 for

723 (K0) exceeds the first IT~-indescribable inaccessible cardinal for each integer n, generalizing a result of Vaught (Abstract 63T-12, these cJ.foticei), 10 (1963}, 126}. In particular, not in (Tarski-Keisler, K0 exceeds the first weakly compact inaccessible, i.e. the first inaccessible c 0 From accessible to inaccessible cardinals, Fund. Math. 1963}, settling a problem of Erdos-Hajnal, Some remarks concerning our paper On the structure of set mappings , Acta Math. Acad. Sci. Hungar. 1962. (This last result was obtained by Reinhardt, using methods of the preceding abstract.) (Received September 1, 1965.)

65T-446. DOROTHY WOLFE, Pennsylvania Military College, Chester, Pennsylvania 19013. On finite metric sets II: Separatin& points.

Definition. Given a finite setS= {p1, ... ,p2m}' and a metric pjpk defined on S, the point p0 is a separating point if S U fp0) is a metric space and if S can be partitioned into m disjoint pairs of points such that if Pa and pb are paired, then PaPa + pbpO = papb. If S CM2 (the space of points x = and are produced in a (x1,x2) with metric xy = maxilxi- yii)' then all separating points also lie in M 2 construction by I. ]. Schoenberg. Any sum of 2m distances formed in such a way that each point of S appears exactly twice is called a sum of 2m linked distances. Theorem. Given a metric set of 2m points, a necessary and sufficient condition for the existence of a separating point is that the maximal sum of 2m linked distances be equal to 2(p 1p 2 + p 3p4 + ••• + p 2m-l Pzm ), with appropriate numbering of the points. (Received September 2, 1965.)

65T-447. RANDOLPH CHURCH, U. S. Naval Postgraduate School, Monterey, California 93940. Enumeration by rank of the elements of the free distributive lattice with seven generators.

A method has been devised by which it has been feasible to calculate, using a high speed digital computer (CDC 1604] as time on it was available, the number of isotone functions from B ~he Boolean lattice having two elements, 0 and 1] to FDL(6). The method of enumeration yields a table in which the entry in the ith row and jth column, 6Lij' is the number of these isotone functions for which the rank in FDL(6) of the image of 0 is i and the rank of the image of 1 is j [i,j = 0,1, ••• ,26]. This table is triangular since nLij = 0 fori > j. The number of elements of each rank of FDL(7}, 7Lr' r =

0,1, ••• ,27, results from summation of the 6Lij for which i + j = r. The maximum of these, f 64, is 81,203,064,840 and the total, the order of FDL(7} with 0 and I adjoined, is 2,414,682,040,998. The 5Lij were an early by-product of the present investigation. Their sum was published by Morgan Ward in 1946. His result was confirmed by R. P. Agnew (J. Indian Math. Soc. 24 (1~60), 1-21). The principal feature of the method here employed is that it only requires storage in the computer of two relatively short lists: (A) a list of the 210 quotient polynomials of FDL(5} [each having 33 coefficients], with the number of elements of FDL(5) belonging to each; (B) a list arranged by rank of the 7581 elements of FDL(5) as elements of the Boolean lattice B 5, with the serial number of the quotient polynomial belonging to each. (Received September 3, 1965.)

724 65T-448, K. S, MILLER and HAROLD SACKROWITZ, 632 West 125 Street, New York, New York 10027. Distributions associated with the quadrivariate normal.

Let X= {xl'x2,. • .,xpJ be a p-dimensional Gaussian random vector with arbitrary mean vector and arbitrary positive definite covariance matrix, Let yk = gk(xk,xk+l'""xp)' l ~ k ~ p - l, where the gk are homogeneous [in the sense that gk(Axk,. .. ,>.xp) = gk(xk""'xp) for X >OJ. Then the joint density function of Y = [yl'y2, ... ,yp-lf may be expressed in closed form in terms of parabolic cylinder functions. Many t, F, and (3 type random vectors may be subsumed in the above class of Y vectors. Similarly, if X is a mean zero Gaussian vector and yk = gk(xk,xk+l''"'xp-l'x~ 1 ), 1 ""k "" p - l, and the gk are homogeneous, then the joint frequency function of Y may be expressed in closed form in terms of modified Bessel functions of the second kind, Additional results have also been obtained when the gk functions enjoy other types of special properties, Also included are vari­ ous bi- and tri-variate (closed form) density functions based on four-dimensional Gaussian-Markoff processes, (Received September 3, 1965,)

65T-449. E. W. CHENEY, University of Texas, Austin, Texas and H. L. LOEB, Aerospace Corporation, El Segundo, California. Continuity of generalized rational approximation.

Let P and Q be finite dimensional linear subspaces of C [a,b]. Assume that each f E C [!l,b] possesses a unique best approximation Tf in the set R = (p/q: p E P ,q E Q,q > 0 J. Theorem. The operator T is continuous at f if and only iff E R or dim [P + (Tf)Q] = dim P + dim Q - l. Here "dim" means "dimension of" and P + rQ is fp + rq: p E P ,q E Q}. Theorem. If ~-> f uniformly then Tfn ->Tf in measure. Various parts and special cases of these theorems were known: Maehly and Witzgall [Numer. Math. 2 (1960), 142-150], Cheney and Loeb [SIAM j. SerB l (1964)], Cheney J}'roc. General Motors Symposium, 1964], and Werner [Math. Z. 86 (1964)]. (Received September 3, 1965.)

65T-450. RaBER T DiPAOLA, University of California, Los Angeles, California 90024. On sets represented by formulas of consistent Rosser theories.

Let T be a consistent axiomatizable theory in which all recursive functions of one argument are definable, and in which some EI (effectively inseparable) pair of re sets is separable; T(S) designates the theory obtained from T by adjoining S as a new axiom, where S is ..!!!!Y.. sentence undecidable in T. Theorem 1. If (A, B) is any pair of re sets with A C R C B, where R is recursive, there is a formula

which represents A in T and B in T(S). Corollary. If (dl'd2) is any pair of re degrees, there is a formula which represents a set of degree d 1 in T and of degree d 2 in T(S). In terms of Abstract 611-5, these cNoticeiJ , 11 (1964), 317: Theorem 2. If T is a consistent re extension of Peano arithmetic, Sis a II~ sentence undecidable in T, and Na.' Nf3 are pseudo-complement functions ofT and T(S) respectively, then for any A, R, B as above, there is a number n such that {Na.(n)} = A,

~,a(n)} =B. (Received by September 2, 1965.)

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8638 Georgia Avenue, Silver Spring, Maryland (Suburb of Washington, D. C.) An equal opportunity employer

734 GROWTH OPPORTUNITIES FOR COMPUTER-ORIENTED SPECIALISTS WITH TRW SYSTEMS in Los Angeles and Houston

GROWTH has been the by-word of TRW Systems (formerly TRW Space Technology Laboratories - STL) since its founding over a decade ago. Having established standards of TECHNICAL EXCELLENCE in aero· space and related fields, TRW now has new and important opportunities for computer-oriented specialists. AT TRW Systems in Redondo Beach, California. and in Houston, Texas, specialists in scientific and busi· ness programing are invited to join our staff in advancing the application of computer technology to underwater defense systems, Gemini and Apollo mission planning and analysis, advanced ballistic mis­ sile studies, advanced space probes, Mars studies and space communications systems. Vitally involved in the nation's major space programs, TRW Systems provides its personnel in scientific and business programing with two IBM 7094 systems. Insta.lla.tion of an advanced GE 635 system is scheduled for early 1966.

We urge you to investigate these ground-floor opportunities NOW!

Computer Software Project Engineers Real-Time Programers

Background should include technical or business Software development for small or medium-size digital degree, scientific or information systems programing, computers on real-time or near real-time applications. leadership experience, and ability to direct the devel· Prefer experience with real-time for handling instru­ opment of complex software projects. mentation or telemetry data. BS in Engineering or Math preferred. Computer Programing Engineers Systems Programers Requires technical degree and experience in analysis, programing and debugging of computer solutions of Experienced in development and/or modification of advanced engineering problems encountered in space control programs, i.e., supervisors, monitors, loaders, and undersea research. etc., for large-scale computing systems. Background in multiprocessors and multiprograming very desirable Scientific Programers but not required. Experience in programing high speed digital com­ Data Management Systems Programers puters. Will assist in solution of problems arising in missile and space vehicle engineering, with responsi­ Design, develop and implement computer-supported bility for analysis, programing and debugging of com­ management information/data processing systems for puter solutions. Requires BS or MS in Math, Physics engineering data management, computer center admin­ or Engineering. istration, aerospace project management and govern· ment. BS/BA degree and two years' experience on Test Evaluation Programers large-scale computers required. BS or advanced degree in Math or the Physical • Sciences, with programing experience on high speed Please submit resume to G. 0. Deshler, TRW Professional digital computers, and experience with scientific test Placement, Dept. AMS-10, One Space Park, Redondo Beach, data. Responsibilities will include mathematical and California 90278. TRW is an equal opportunify employer. computational aspects of physical problems, and the formulation and programing of test evaluation com­ sYSTEMS puter problems, employing data obtained from various TRW test facilities and systems, including flight test ONE SPACE PARK, REDONDO, BEACH, CALIFORNIA telemetry. Formerly TRW Space Technology Laboratories- STL

735 The problem may be to recognize MATHEMATICIANS the problem Early in World War II, arming mer­ chant ships with anti-aircraft guns seemed useless: they weren't destroy­ ing planes. Then an operations analyst pointed out armed ships suffered fewer sink· ings than did unarmed ones because Nazi pilots, seeing the guns, did not press their attacks. Arming the ships, PROGRAMMERS therefore, was the answer to the real problem - delivering cargoes, not downing planes. Today, the same emphasis on defini­ tion of problems is a major concern of the Center for Naval Analyses of The Franklin Institute, a private sci­ entific organization engaged in oper­ ations research and analysis. CNA's mathematicians, analysts, physical scientists, economists, and research engineers submit their re­ ports to Navy planners and decision makers for their guidance. with exceptional abilities are invited to investi· For superior scientists with the ability to define problems, there are gate opportunities with the Research Labora­ still a few staff vacancies. tories of Brown Engineering Company, Inc. For further information, write: Positions are available in • numerical analy­ Director sis • probability theory • stochastic pro­ CENTER FOR NAVAL ANALYSES cesses • information theory • operations Dept. AM, 1401 Wilson Blvd. research • data processing • computer pro­ Arlington, Va. 22209 gramming. Openings normally require ad­ vanced training (30% of the staff hold PhD degrees) but inquiries are invited from recent honor graduates at the BS level. Submit CENTER FOR NAVAL ANALYSES your resume in confidence to: Raymond C. OF THE FRANKLIN INSTITUTE Watson, Jr., Director Of Research. INS • Institute of Naval Studies SEG • Systems Evaluation Group OEG • Operations Evaluation Group NAVWAG • Naval Warfare Analysis Group MCOAG • Marine Corps Operations Analysis Group An equal opportunity employer

BROWNCOMPANY, INC. ENGINEERING 300 Sparkman Drive, N.W. Huntsville, Alabama 35807

An Equal Opportunity Employer

736 Mathematicians and Operations Research Analysts Senior Research Mathematician M.A. or Ph.D. for analytical studies in engineering and procedural systems. Develop mathematical models to simulate behavior of systems for space and military Provide methods of solutions and sufficient information for programmers to code for computer development models of transportation and logistic systems. 2-3 years experience in military operations research very desirable. Statistical Consultant Ph.D. or equivalent with at least 5 years' experience in application of mathematical statistics, as part of interdisciplinary team, with knowledge of computers EDP and simulation. Will apply, adapt or develop statistical techniques for broad spectrum of systems research: Industrial and military operations, transportation, communication and reliability. Will direct and/or coordinate research projects; aid in proposal preparation. Operations Research Analyst M.S. or Ph.D., with 2-3 years' experience in industrial engineering or military operations research, for studies in plant operations, research laboratories planning, transportation, logistics, and/ or com­ puter applications. Will develop simulation models, provide input for EDP programmers, structure data formats and gather data. Will analyze results, make recommendations, and write reports. Electrical Engineer-Analyst B.S., with 3 years' experience in the area of navigation and guidance, for engineering analysis of systems, trade off studies, state-of-the-art surveys, cost effectiveness evaluations, design and development.

* * * Battelle, world's largest research institute, offers fine facilities, distinguished company, stability with refreshing variety of projects and fine working atmosphere. Very desirable home city; an educa­ tional and cultural center. Send resume to Mr. L. G. Hill,

BATTELLE MEMORIAL INSTITUTE COLUMBUS LABORATORIES 505 King Avenue Columbus, Ohio43201

Several opportunities in other fields at Battelle. Write and tell us of your qualifications.

An Equal Opportunity Employer

737 Himmelsmechanik Career Appointments (Celestial Mechanics) by K. Stumpff Instruction manuals for physics, university level, volumes 32 and 40 Volume I: DasZweikorperproblem und die Methoden der Bahnbestimmung der Planeten und Kometen (The laws of planetary motion governing two bodies and the methods to determine the path Booz ·Allen of planets and comets) 508 pages, 60 illus., 10 tables, 1 plate, 8oo, bound in artificial Applied Research Inc. leather, price: 56,80 MDN Has Program Responsibility for Volume II: Das Dreikorperproblem (The laws of planetary motion The Combined Arms governing three bodies) 684 pages, 90 illus., 8oo, bound in artificial leather, price: Research Office 90,-MDN Can be ordered through leading bookdealers Ground-floor career opportunities of uncom­ mon potential exist at CARO for electrical VEB DEUTSCHER VERLAG DER engineers, communications engineers, math­ WISSENSCHAFTEN • 108 Berlin ematicians and physicists who have advanced degrees, preferably Ph.D. or equivalent, and who have the technical proficiency and the operational understanding to work within an operations research and systems analy­ sis environment. The Combined Arms Research Office (CAROl, located at Fort Leavenworth near CUSHING-MALLOY, Kansas City, Missouri, is a continuing mili­ tary operations research program charged INC. with the responsiblility of performing ad­ vanced studies in the fields of ... Tactical Field Communicatior.s ... Command and 1350 North Main St., P. 0. Box 632 Control Communications Systems ... Elec­ trical and Nuclear Power Requirements ... Transportation Logistics and Combat Ma­ neuvers . . . Advanced Weapons Systems Ann Arbor, Michigan 48107 ... Cost Effectiveness Studies and Programs. Pertinent and advanced techniques of LITHOPRINTERS operations research and systems analysis, gaming, design of experiments, model build­ ing, cost effectiveness, etc., are being con­ • stantly applied and related to the evaluation of present and future military operational requirements. Known for Your inquiry is invited. If you wish to know more about the outstanding career opportunities now available at CARO, con­ QUALITY- ECONOMY tact: Mr. Robert H. Flint, Director of Profes­ sional Appointments. SERVICE BOOZ •ALLEN APPLIED RESEARCH Inc. • 135 South LaSalle Street Chicago, Illinois 60603 We specialize in reproducing out-of­ Phone: (312) FR 2-1728 Washington • Cleveland print books, as well as typewriter com­ New York • Chicago • Los Angeles position. Let us quote on your next An equal opportunity employer printing.

738 strategy: the science and art of interrelating political, economic, psychological and military forces of a nation to alford maximum support for its adopted policies

The Institute for Defense Analyses contributes to our nation's deliberations on strategy by advising on the weapons systems, processes, and economics of defense. In this activity, IDA's guiding belief is that meaningful advice can best be developed by bringing together highly qualified experts from a variety of fields, and providing stimulating guidance and a creative environment. IDA conducts studies for elements of the Department of Defense such as the Joint Chiefs of Staff and the Director of Defense Research and Engineering, as well as other executive agencies of the Government. To enhance our response to their requests, we wish now to add as professional staff members a few outstanding people at the level of Ph.D., with backgrounds in physics, engineering, operations research and systems analysis. Write toT. J. Shirhall, Institute for Defense Analyses, 400 Army-Navy Drive, Arlington, Virginia (near the Pentagon). An equal opportunity employer sponsored by twelve of the nation's leading universities. Ab IDA

739 TRANSACTIONS OF THE PROGRAMMER MOSCOW MATHEMATICAL SOCIETY This volume of the Trudy Moskov­ PROGRAMMER - Parts - skogo Matematiceskogo Obscestva is Familiar with APT, ADAPT and the first to be published in a cover­ AUTOMAP. Machine shop experi­ to-cover English translation. Volume 12 contains twelve articles on original ence desired. Salary open. Job lo­ research in pure mathematics by F. A. cation, Bronx, New York. Send Berezin, I. M. Gel'fand, S. G. Gindi­ resume to: kin, 0. N. Golovin, G. I. Kac, A. N. Kolmogorov, V. K. Mel nikov, I. I. Pjateckii-Sapiro, I. A. Svedov, N. Ja. Personnel Manager Vilenkin, :g. B. Vinberg, M. I. Visik, FARRAND CONTROLS, INC. and D.P. Zelobenko. 99 Wall Street Volume12 524pages Valhalla, New York 10595 Tel. 914·761-2600 List Price $5.30 Member Price $3.98 (An Equal Opportunity Employer) Please order from AMERICAN MATHEMATICAL SOCIETY P. 0. Box 6248, Providence, Rhode Island 02904

MANAGEMENT SCIENCE

Major petroleum company has opening with unusual potential for self-development and growth.

RESEARCH ANALYST - Graduate degree in operations research, or mathematics. Several years experi­ ence in mathematical programming, related com­ puter handling of operating systems, identifica­ tion of operations susceptible to treatment by O.R. methods, and application of mathematical­ statistical techniques.

Attractive salary and employee benefits program. Send full resume, including salary history, in stri~~ confid9nce. All resumes will be acknowl­ edged.

H. M. Overley, P.O. Box 7258, Phila., Pa. 19101

AN EQUAL OPPORTUNITY EMPLOYER

740 INDEX TO ADVERTISERS Allyn and Bacon, Inc...... 7'29 American Mathematical Society . . . 740,741 Battelle Memorial Institute ...... 737 Booz . Allen Applied Research Inc.. . . 738 Brown Engineering Company Inc. . . 736 Center for Naval Analyses...... 736 Cushing-Malloy, Inc...... 738 V eb Deutscher Verlag der W issenschaften ...... 73:>, 738 Farrand Controls, Inc...... 740 Holt, Rinehart and Winston, Inc...... 700 Institute for Defense Analyses ...... 739 International Business Machines Corporation. . . 733 The Johns Hopkins University, Applied Physics Laboratory 734 McGraw-Hill Book Company...... 7'lB New York University, Office of Special Services to Business and Industry . . 741 H. M. Overley...... 740 Springer-Verlag New York Inc...... 'l'Zl, 732 Stechert-Hafner Inc...... 7'lB TRW Systems ...... 735 D. VanNostrand Company, Inc. . 731 Wayne State University Press...... 700 University of Wisconsin Press...... 7'28 Wolf Research and Development Corporation ...... inside back cover

LIFE TESTING and SYSTEMS RELIABILITY and MAINTAINABILITY October 18-29, 1965 Course conducted by Dr. Benjamin Epstein Contact: Office of Special Services to Business and Industry New York University New York, New York 10003

COLLOQUIUM PUBLICATIONS-RECENT REPRINT Theory of Graphs by Oystein Ore New fields of application, such as game theory and programming, com­ munications theory, electrical networks and switching circuits, and problems from biology and psychology, have given an intense stimulus to the develop­ ment of graph theory in the past twenty years. The present volume, the first of a projected two-volume work, gives an almost complete treatment of the basic concepts and the results of particular systematic interest; the second volume will deal with such topics as the four-color conjecture, the theory of flow, electrical networks, and games. The fifteen chapters of the first volume present, in a well-organized setting, the results of Cayley, Ramsey, Frucht, Hall, Mann, Ryser, Dirac, and many other outstanding researchers in the subject. This volume is reprinted without changes from the first edition of 1962. Volume 38 280 pages List Price $9.20 Member Price $6.90 Please order from AMERICAN MATHEMATICAL SOCIETY P. 0. Box 6248, Providence, Rhode Island 02904

741 RESERVATION FORMS

Iowa City, Iowa University of California, Berkeley November 26-27, 1965 December 29, 1965

New and excellent accommodations are Accommodations near the campus are available at the Iowa Memorial Union on listed in the Preliminary Announcement on Campus. Rates are $9.50 for a single and page 667 of these cNoticeiJ . A $5.00 de- $13.00 for a double. Please direct thefol- posit is required by all hotels and motels. lowing reservation blank to: Please send reservations to: Mr. William D. Coder AAAS Housing Bureau Director of Conferences P. 0. Box 210 University of Iowa Berkeley, California 94701 Iowa City, Iowa Enclose hotel or motel room deposit. Should you check a commercial accommo­ Make checks payable to AAAS Housing dation, your reservation will be made at Bureau. All rooms will be assigned and the most desirable place available. confirmed in order of receipt of reserva­ tion.

(Please detach on this line) (Please detach on this line)

I wish housing reserved for the AMS Meeting Please reserve the following accommodations for the 132ndmeeting of the AAAS in Berkeley, as follows: 26-31 December, 1965, I. M. U_,•.__ __ _ First Choice of Hotel, Motel, or Residence Hotel.___ _ Motel----- Hall.______Single.___ _ Double---- Second Choice______for the night(s) of______Single.______Double______Double Twin Bed______Suite ____ _ Rate: Desired _____Maximum rate___ _ Name______Number in party______Address ______Sharing this room will be:______City and State______

Dates: Arrival______A. M. __P. M. Departure Name______Address______City and State.______

742 SECOND-CLASS POSTAGE AMERICAN MATHEMATICAL SOCIETY PAID AT P.O. Box6248 PROVIDENCE, RHODE ISLAND AND Providence, Rhode Island 02904 ANN ARBOR, MICHIGAN

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