AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 8, NUMBER 5 ISSUE NO. 56 OCTOBER 1961

THE AMERICAN MATHEMATICAL SOCIETY oticeiJ

Edited by GORDON L. WALKER CONTENTS

MEETINGS Calendar of Meetings . • . • • • • • . • • • . . . • • .. • • • . • • • . • . • . • • • . • . 3 92 Program of the Five Hundred Eighty-third Meeting in Cambridge, Massachusetts •.•.•.•. 393 Abstracts of the Meeting- pages 423-436

PRELIMINARY ANNOUNCEMENTS OF MEETINGS •••••••..••••..••.... 398 ACTIVITIES OF OTHER ASSOCIATIONS .•.•.••••..•....•••..•.... , • 402 FELLOWSHIP AND RESEARCH OPPORTUNITIES •.•••..•....•...••..•. 403 lHE ANNUAL SALARY SURVEY ••.•.••••.•••.•....•.•••.•.•.•.•.. 406 STARTING SALARIES FOR MATHEMATICIANS WITH A PH.D •••..•.••.•••. 409 NEWS ITEMS AND ANNOUNCEMENTS • . • • . . • • • . • . . . • . • . • . . • . . • • • . . 410 PERSONAL ITEMS .•....•.•••..•.•...... •.•••.•.••..•.••.. 412 MEMORANDA TO MEMBERS Australian Mathematical Society Summer Research Institute .•••..•.•.. 401 Proceedings Editorial Committee .••..•..•.•...... •••.•.•.•. 420 Two Volumes of Mathematical Reviews in 1962 •.••••••••.••.•.•.•. 420 Berliner Mathematische Gesellschaft e. V. . .•.•...•••.•••••.... 4ZO Dmtsche Mathematiker Vereinigung ••...•.•••••...•.•...•...•. 420 Union Matematica Argentina ...... ••.•••••.••..••..•.•. , • • . • . 420 SUPPLEMENTARY PROGRAM N0.6 .••••..•••••.•••.•...•...... •• 421 ABSTRACTS OF CONTRIBUTED PAPERS ..•.••..•.....•••.•.•...•.. 423 INDEX OF ADVERTISERS ••••....••.•••.•....••.....•••••••.•.• 459 MEETINGS

CALENDAR OF MEETINGS

Note: This Calendarlists all of the meetings which have been approved by the Council up to the date ·at ·which this issue of the NOTICES 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*

584 November 17-18, 1961 Milwaukee, Wisconsin Oct. 3 585 November 17-18, 1961 Gainesville, Florida Oct. 3 586 November 18, 1961 Santa Barbara, California Oct. 3 587 January 22-26, 1962 Cincinnati, Ohio Nov. 17 (68th Annual Meeting) 588 February 22, 1962 New York, New York Jan. 9 (Emphasis on Finite Groups and Continuous Groups) 589 April 12-14, 1962 Chicago, Illinois Feb. 27 590 April 16-19, 1962 Atlantic City, New Jersey Feb. 27 591 April 25, 1962 Monterey, California Feb. 27 592 August 27-31, 1962 Vancouver, British Columbia (67th Summer Meeting) January 24-28, 1963 Berkeley, California (69th Annual Meeting) August 26-30, 1963 Boulder, Colorado (68th Summer Meeting) January 20-24, 1 964 Miami, Florida (70th Annual Meeting) August, 1965 Ithaca, New York August, 1966 New Brunswick, New Jersey * The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for by title abstracts are November 10 and January 2nd.

The NOTICES of the American Mathematical Society is published by the Society seven times a year, in February, April, June, August, October, November, and December. Price per annual vol­ ume 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 (none available before 1958), and inquiries should be addressed to the American Mathematical Society, 1350 Main Street, Ann Arbor, Michigan, or to 190 Hope Street, Providence 6, Rhode Island. Second-class postage paid at Ann Arbor, Michigan. Authorization is granted under the author­ ity 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 © 1961 by the American Matbematical Society Printed in the United States of America Five Hundred Eighty-Third Meeting Massachusetts Institute of Technology Cambridge, Massachusetts October 28, 1961

PROGRAM

The five hundred eighty-third meet­ Technology is located on the Cambridge ing of the American Mathematical Society side of the Charles River approximately will be held on Saturday, October 28, 1961, one to two miles from the various railway at the Massachusetts Institute of Technol­ stations in Boston. It is easily accessible ogy. All times given below are Eastern by automobile, subway, trackless trolley Standard Time. or taxicab. There will be ample free park­ By invitation of the Committee to ing space in the East Parking Lot of the Select Hour Speakers for Eastern Sec­ Institute grounds for those traveling by tional Meetings, Professor Michel Ker­ automobile. The entrance to this parking vaire of New York University will deliver lot is at the corner of Main and Vassar an address at 2:00 P.M. on Saturday in Streets. The sessions will be held in Room 26-100. His title is "Some results Buildings 2,4,6 and 26. The most conven­ and problems in differential topology." ient entrance for those coming by automo­ The lecture room is the auditorium of the bile or subway is the entrance at the Karl Taylor Compton Laboratories, to the Northwest corner of the Hayden Memo­ north of the main building and easily ac­ rial Library. It is a seven to ten minute cessible by connecting corridors. walk from the Kendall Square station of There will be sessions for contri­ the Cambridge-Dorchester subway. This buted papers on Saturday morning, at Subway may be boarded at various points 10:00 A.M., and on Saturday afternoon at including South Station, Boston, and Har­ 3:15 P.M. in rooms 2-190, 2-390, 4-270 vard Square, Cambridge. Those coming by and 6-120. taxicab or trackless trolley should enter There will be a registration desk on at the main entrance, 77 Massachusetts the first floor of Building 2. The Mathe­ Avenue. Most of the entrances except the matics Common Room, 2-290, and a class­ two mentioned above are closed on Satur­ room, 2-131, near the registration desk, day. will be available for conversation rooms Lunch will be served in an M.l. T. and may be used as coat rooms. cafeteria, and a list of nearby restaurants The Massachusetts Institute of in Boston and Cambridge will be available.

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

SATURDAY, 10:00 A.M.

General Session, Room 2-190 10:00 - 10:10 (l) Geometric properties of polynomial surfaces and hypersurfaces Professor Robert Osserman, Stanford University (583- 44) 393 10:15 - 10:Z5 (Z) Weak:* closure of functions of absolute value one Professor R. R. Goldberg* and Professor A, B. Simon, Northwestern University (583-6) 10:30 - 10:40 (3) Harmonic analysis on a Hilbert space Professor Leonard Gross, Cornell University (583-Z6) 10:45 - 10:55 (4) Analytic continuation of polyharmonic functions Mr. S. R. Kraft, University of Maryland (583 -1) 11:00 - 11:10 (5) A remark concerning the solution of the Dirichlet problem by finite differ­ ences Professor Bernard Epstein, Yeshiva University (583-ZZ) 11:15- 11:Z5 (6) On the global existence of solutions of nonlinear parabolic equations Mr. Stanley Kaplan, New York University (583-Z9) 11:30 - 11:40 (7) A reduction to an integral equation of an inhomogeneous, singular Cauchy problem in harmonic space Mr. B. A. Fusaro, University of Maryland and University of Oklahoma (583-38) 11:45 - 11:55 (8) Convergence of operator averages Professor R. V. Chacon, Brown University (583-30)

SATURDAY, 10:00 A.M. Session on Applied Mathematics, Room Z-390 10:.00 - 10:10 (9) On the geometry of the vorticity tensor and the ·vorticity vector in the rela­ tivistic hydrodynamics Dr. E. R. Suryanarayan, University of Rhode Island (583 -7) 10:15 - 10:Z5 (10) Solution of the space charge problem for a pulsed Townsend discharge Dr. j. A. Morrison* and Dr. D. Edelson, Bell Telephone Laboratories, Incorporated, Murray Hill, New jersey (583-9) 10:30 - 10:40 (11) Holor algebra Professor D. E. Spencer, University of Connecticut (583-15) 10:45 - 10:55 (lZ) The roots of polynomial surfaces Mr. L. j. Cohen, Applied Data Research, Incorporated, Princeton, New jersey (583-ZO) 11:00 - 11:10 (13) On rational Tchebycheff approximation. Preliminary report Dr. E. W. Cheney, Iowa State University and Mr. H. L. Loeb*, Syste·m Development Corporation, Santa Monica, California (583 -19) 11:15- 11:Z5 (14) Solutions of a class of linear partial differential equations with variable co­ efficients Professor W. F·. Ames and Mr. Hernando de la Cuesta*, University of Delaware (583-Z1)

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

394 11:30 - 11:40 (15) Pattern recognition by moments Dr. F. L. Alt, National Bureau of Standards, Washington, D. C. (583-40) 11:45 - 11:55 (16) Channels which transmit letters of unequal duration Professor R. M. Krause, Armour Research Foundation of Illinois Insti­ tute of Technology (583-43)

SATURDAY, 10:00 A.M. General Session, Room 6-120 10:00 - 10: 10 (17) Hierarchic algebra. II Mrs. M. B. Prestrud, The Rand Corporation, Santa Monica, California (583-3) 10:15 - 10:25 (18) Increasing rates of convergence with summability methods Professor G. G. Bilodeau, Boston College (583 -13) 10:30 - 10:40 (19) A theorem on partially ordered sets. Preliminary report Professor Kurt Bing, Rensselaer Polytechnic Institute (583-17) 10:45 - 10:55 (20) Algebraic structure of groups of provable permutations Mr. C. F. Kent, Massachusetts Institute of Technology (583-10) 11:00 - 11:10 (21) A note on bounded-truth-table reducibility. Preliminary report Mr. P. C. Fischer, Massachusetts Institute of Technology (583-5)

11:15- 11:25 (22) WITHDRAWN 11:30- 11:40 (23) The fundamental operator of a stationary Markov process. Preliminary report Professor J, R. Brown, University of Massachusetts (583-41) 11:45 - 11:55 (24) A class of exponential distributions and their associated Minkowski geom­ etries Dr. T. T. Tanimoto, Melpar, Incorporated, Watertown, Massachusetts (583-33)

SATURDAY, 2:00P.M. Invited Address, Room 26-100 Some results and problems in differential topology (One hour) Professor Michel A. Kervaire, New York University

SATURDAY, 3:15P.M. Session on Analysis, Room 2-190 3:15 - 3:25 (25) Asymptotic distribution of eigenvalues of Toeplitz matrices Professor Murray Rosenblatt, Brown University (583-11) 3:30 - 3:40 (26) Note on analytic continuation of functions of two variables Professor R. H. Cameron, University of Minnesota (583-42)

395 3:45 - 3:55 (2 7) The existence of entire functions of zero type bounded on a sequence of real points Dr. Louis de Branges, New York University (583-25) 4:00 - 4:10 (28) Magnitude of the Fourier coefficients of automorphic forms of negative dimension Professor joseph Lehner, Michigan State University and University of Pennsylvania (583-31) 4:15 - 4:25 (29) Triple product integrals of Laguerre functions Professor joseph Gillis, Weizman Institute, Israel, and Institute for Space Studies, New York, New York (583-23) 4:30 - 4:40 (30) On the equivalence of two theorems on the location of roots of polynomials Professor Anthony Ralston, Stevens Institute of Technology (583 -24) 4:45 - 4:55 (31) A decomposition for representations of separable C*-algebras, Preliminary report Mr. E. G. Effros, Columbia University (583-46)

SATURDAY, 3:15P.M. Session on Algebra, Room 2-390 3:15 - 3:25 (32) Elementary divisors in continuous rings Professor I. Halperin, Queen's University, Kingston, Ontario (583-2) 3:30 - 3:40 (33) A new proof of theorems of Perron and Frobenius on non-negative matrices. II Professor A. T. Brauer, University of North Carolina (582-27) 3:45- 3:55 (34) Simple Mal'cev algebras of characteristic zero Dr. A. A. Sagle, University of Chicago (583-8) 4:00 - 4:10 (3 5) On a theorem of Mal' cev Professor Gilbert Baumslag, New York University (583-32) 4:15- 4:25 (36) Analysis of weakly modular lattices. Preliminary report ProfessorS. S. Holland, Jr., Boston College (583-34) 4:30 - 4:40 (37) Translative endovector sets Professor T. S. Motzkin, University of California, Los Angeles (583-36) 4:45 - 4:55 (38) Polynomials dividing infinitely many trinomials. Preliminary report Dr. E. C. Posner and Dr. H. C. Rumsey*, jet Propulsion Laboratory, Pasadena, California (583-39)

SATURDAY, 3:15P.M. Session on Topology, Room 6-120 3:15 - 3:25 (3 9) Remark on the Poincare duality theorem Professor William Browder, Cornell University (583-35) 3:30 - 3:40 (40) The third and fourth obstruction to cross sections in sphere bundles Professor M. E. Mahowald, Syracuse University (583-37)

396 3:45 - 3:55 (41) Transformation groups on cohomology lens spaces and cohomology complex projective spaces Mr. j. C. Su, University of Virginia (583-12) 4:00 - 4:10 (42) Shrinking subsets of n-manifolds Mr. Ross Finney, Massachusetts Institute of Technology (583-16) 4:15- 4:25 (43) Properties of certain function spaces Mr. C. E. Robinson, University of Alabama (583-28) 4:30 - 4:40 (44) On transversals of simply connected regions Professor E. C. Schlesinger, Wesleyan University (583-14)

SATURDAY, 10:00 A.M. and 3:15P.M. Sessions for late papers, Room 4-270 Everett Pitcher Bethlehem, Pennsylvania Associate Secretary

PROFESSIONAL TRAINING IN MATHEMATICS by

F. A. FICKEN AND C. C. MAcDuFFEE and a Selected List of Available Scholarships and Stipends in Mathematics

Reprinted from Special Issue Assistantships and Fellowships in Mathematics in 1961-1962: Volume 7, Number 7, Part II, Issue No. 51, December 1960. 25pp 25¢

AMERICAN MATHEMATICAL SOCIETY 190 HOPE STREET. PROVIDENCE 6. RHODE ISLAND

397 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

FIVE HUNDRED EIGHTY-FOURTH MEETING University of Wisconsin Milwaukee, Wisconsin November 17-18, 1961

The five hundred eighty-fourth $14.00), and the large motel near Lake meeting of the American Mathematical So­ Michigan, the Milwaukee Inn (singles ciety will be held on Friday and Saturday, $11.00, doubles $14.00, twins $15.00). The November 17 and 18, 1961, at the Univer­ first is near the Milwaukee Road depot and sity of Wisconsin, Milwaukee (UW-M), 3 miles from the campus; the second and The sessions will be held in Lapham third are near the Northwestern depot and Hall, a newly constructed science building, Z 1/Z and Z miles respectively from the beginning at 1:30 P.M. on Friday. The campus. All have limousine service to the lounge of the Student Union, diagonally op­ airport, and the Inn also has helicopter posite Lapham Hall, has been reserved for service to the airport. All are on the bus use of the Society. Tea and coffee will be route to the campus. Reservations should served there beginning at 4:00P.M. A din­ be made directly with these hotels at least ner is being arranged at the UW-M about a week .before the meetings, and mention 6:00 P.M. At about 8:00 P ,M., the Society should be made of the Society, will be the guest of the Miller Brewing By invitation of the Committee to Company at a party to be held in the Select Hour Speakers for Western Section­ Miller High Life Inn. Transportation to the al Meetings, Professor Helmut Rohrl of Inn from the UW- M campus is being ar­ the University of Minnesota and Professor ranged, Frederick W. Gehring of the University of Blocks of rooms have been reserved Michigan will address the Society, at Hotels Schroeder (singles $6.50- $10.00, J, W, T. Youngs doubles $10,00 - $15.00, twins $1Z.50 - Associate Secretary $16.00) and Pfister (singles $7,00- $10.00, doubles $9.00 - $1Z,00, twins $10,00 - Bloomington, Indiana

FIVE HUNDRED EIGHTY-FIFTH MEETING University of Florida Gainesville, Florida November 17-18, 1961

The five hundred eighty-fifth meet­ Friday and from 10:30 A.M. to noon on ing of the American Mathematical Society Saturday. will be held on Friday and Saturday, No­ Registration headquarters will be vember 17 and 18, 1961, at the University in Walker Hall and will be open from of Florida. All times given below are East­ 10:00 A.M. Friday until noon on Saturday. ern Standard Time. Mail and telegrams should be sent in care By invitation of the Committee to of the Department of Mathematics, Z05 Select Hour Speakers for Southeastern Walker Hall, University of Florida, Sectional Meetings, Professor R, D. An­ Gainesville, Florida, The University derson of The Louisiana State University Cafeteria will be available for meal ser­ will deliver an address at Z:OO P.M. on vice. Friday in Leigh Z07. The University of Florida is located Sessions for contributed papers will about one mile west of the center of be held from 3:30 P.M. to 5:30 P.M. on Gainesville, about three-quarters of a mile

398 from the railroad and bus stations. Lines. Parking space will be made avail­ Gainesville is on U.S. 441 and state routes able. 26, 24, and 20. It is served by the Atlantic The following partial list of hotels Coast Line Railroad, the Greyhound and and motels with their prices is included Trailway Bus Lines and by Eastern Air for your convenience.

BAMBI MOTEL 1 person $6.50 - $7.50 2119 S. W. 13th Street 2 persons $8.50 - $9.50 Phone FR. 6-2622 2 bedroom suite (connecting bath) $14.00 - $18.00

CASA LOMA LODGE 1 person $7.00 - $8.00 2120 S. W. 13th Street 2 persons 2 beds $8.00 - $9.00 Phone FR. 2-8971 FLORIDA MOTEL 1 person $6.00 2603 S. W. 13th Street 2 persons 1 bed $7.00 Phone FR. 6-3742 2 persons 2 beds $8.00 FRANCISCO MOTEL 1 person $6.00 2307 S. W. 13th Street 2 persons 1 bed $7.00 Phone FR. 2-2045 2 persons 2 beds $8.00 GATOR COURT 1 person $6.50 4170 S. W. 13th Street 2 persons 1 bed $7.50 Phone FR. 6-4667 2 ·persons 2 beds $8.50 HIL-TOP MOTOR COURT 1 person $5.00 - $6.00 3103 N. W. 13th Street 2 persons 1 bed $6.00 - $7.00 Phone FR. 2-4319 2 persons 2 beds $7.00 - $8.00 HOLIDAY INN 1 person $8.50 - $12.50 1900 S. W. 13th Street 2 persons $9.50 - $12.50 Phone FR. 2-3311 HOWARD JOHNSON MOTOR LODGE 1 person $8.50 - $11.00 2820 N. W. 13th Street 2 persons $10.50 - $13.50 Phone FR. 6-1211

MANOR MOTEL single $6.00 - $8.00 2325 N. W. 13th Street double 1 bed $8.00 - $9.00 Phone FR. 6-5212 double twin beds $9.00 - $10.00 extra persons $2.00 each

RICHLAND HEIGHTS MOTOR COURT 1 person $5.00 - $6.00 4155 N. W. 13th Street 2 persons 1 bed $7.00 Phone FR. 6-4368 2 persons 2 beds $7.00 - $8.00 TRA VELODGE MOTEL 1 person $7.00 - $9.00 413 W. University Avenue 2 persons 1 bed $9.00 - $11.00 Phone FR. 6-1224 2 persons 2 beds $10.00 - $12.00 TOM SAWYER MOTOR INN single $6.50 - $7.50 3335 S. W. 13th Street double 2 beds $8.50 - $9.50 Phone FR. 2-1463 UNIVERSITY INN 1 person $8.50 - $10.50 U. S. 441 South 2 persons $10.50 - $12.50 1901 S. W. 13th Street Phone FR. 2-6333

399 MOTEL TABER all doubles with double beds 4041 S. W. 13th Street 2 persons $8.00 Phone FR. 6-4423 $1.50 for each additional person HOTEL THOMAS double with private bath $7.50 615 N. E. 2nd Street single with private bath $4.50 double with private bath $6.00 single with private bath $5.00 (Special Rates) PLAZA HOTEL double with private bath $6.00-$7.00 216 N. Main Street single with private bath $4.00 double with connecting bath $5.00 single with connecting bath $3.00 (2 triples at $3.00 per person with private bath) PRIMROSE INN double with private bath $6.00 214 W. University Avenue single with private bath $4.00 WHITE HOUSE HOTEL double with private bath $7.00 408 N. E. First Street single with private bath $4.00 double with connecting bath $6.00 single with connecting bath $3.50

Further details of the program of October 3, 1961. Blanks for submission of the meeting will appear in the November abstracts, which contain instructions on issue of the NOTICES. Abstracts of con­ acceptable form for abstracts, can be ob­ tributed papers should be sent to the tained by writing to the same address. American Mathematical Society, 190 Hope G. B. Huff Street, Providence 6, Rhode Island, so as Associate Secretary to arrive PRIOR TO THE DEADLINE, University of Georgia

FIVE HUNDRED EIGHTY-SIXTH MEETING University of California, Santa Barbara Goleta, California November 18, 1961

The five hundred eighty-sixth meet­ Registration Desk. ing of the American Mathematical Society All sessions of the meeting will will be held on Saturday, November :18, be in South Hall (Classroom-Auditorium 1961 at the University of California, Santa Building). The Registration Desk will be Barbara, in Goleta, California. in the .. inner courtyard of South Hall. By invitation of the Committee to Registration will begin at 9:30 A.M. Select Hour Speakers for Far Western Coffee and doughnuts will be served Sectional Meetings, there will be an ad­ from 9:30 to 10:30 A.M. near the Registra­ dress at 2:00 P.M. by Professor Henry tion Desk. Since many people at the meet­ Helson of the University of California, ing will have to return to the Los Angeles Berkeley on "New words about Dirichlet area after the meeting, there will be no series". late afternoon tea. Sessions for contributed papers will be held at 10:30 A.M. and at 3:30P.M. on Luncheon on Saturday will be avail­ Saturday. If necessary, there will be a able in Ortega Dining Commons, near session for late papers. Information con­ South Hall. The luncheon tickets will be cerning late papers will be available at the sold at the Registration Desk for $1.7 5.

400 The University of California, Santa through the Chamber of Commerce, or Barbara campus is located in Goleta, Cali­ directly with the chosen hotel or motel. fornia, which is about ten miles north of The Santa Barbara Airport, which Santa Barbara. Bus service is available is adjacent to the University campus, is from Santa Barbara to the campus. Infor­ served by Pacific Airlines and United Air­ mation concerning the names, locations lines. Train service to Santa Barbara is. by and rates of the hotels and motels in Santa Southern Pacific. Barbara can be obtained by writing to the Santa Barbara Chamber of Commerce, R. S. Pierce 11 East De La Guerra Street, Santa Bar­ Associate Secretary bara. Reservations should be made either Berkeley, California

CALL FOR CONTRIBUTED PAPERS

FIVE HUNDRED EIGHTY-EIGHTH MEETING Yeshiva University New York, New York February 22, 1962

The New York Meeting on Thurs­ Brandeis University and is entitled "The day, February 22, 1962atYeshivaUniver­ role of differentiability in the theory of sity will devote special emphasis to con­ transformation groups". tributed papers in the topics of Contributors are reminded of the fact (see these NOTICES, issue no. 54, Finite Groups p. 216) that the number of contributed pa­ Continuous Groups pers at the meeting in january, 1962 in Contributed papers in any area of mathe­ Cincinnati, Ohio is limited. Those submit­ matics will receive the usual considera­ ting papers for the january meeting may tion, but research workers in these two wish to designate the February meeting as areas are specially urged to submit their a second choice. contributions for this meeting. Everett Pitcher The invited address will be de­ Associate Secretary livered by Professor Richard S. Palais of Bethlehem, Pennsylvania

MEMORANDA TO MEMBERS

AUSTRALIAN MATHEMATICAL SOCIETY February 1, 1962. Analysis and Number SUMMER RESEARCH INSTITUTE Theory will be the main fields of research The second Summer Research Insti­ and Professor A. Erdelyi (California) has tute of the Australian Mathematical Society been invited as guest lecturer. Director of will be held at the Australian National Uni­ the second Institute is Professor E. S. versity, Canberra, between january 3 and Barnes (Adelaide).

401 ACTIVITIES OF OTHER ASSOCIATIONS

INTERNATIONAL CONGRESS It would be of considerable assist­ OF MATHEMATICIANS-1962 ance if, where possible, correspondents would use English, French, German or First Communication Russian. At the invitation of the Swedish Na­ Ragnar Thorn tional Committee for Mathematics and the Secretary of the Congress Swedish Mathematical Society, the Inter­ national Congress of Mathematicians will meet in Stockholm from August 15th to NOVEMBER 1961 MEETING OF SIAM August 22nd, 1962. A three-day meeting of the Society The Executive Committee is inviting for Industrial and Applied Mathematics a number of mathematicians to deliver will be held at the National Bureau of one-hour and half-hour addresses; the Standards, Washington, D. C., on Thurs­ former will be in the nature of general day, November 2, Friday, November 3, surveys of recent developments in the and Saturday, November 4. In addition to different fields of mathematics and are in­ being the annual Fall Meeting of the So­ tended also for non- specialists. There will ciety, this meeting is being planned to also be daily sessions devoted to ten­ mark the beginning of SIAM's tenth anni­ minute communications. Those who wish to versary year. The meeting will consist of present such communications will have an invited addresses, a symposium on "Math­ opportunity of offering to do so when they ematical Problems in Control Systems", receive the Second Communication. a symposium on "Applied Mathematics - All fields of mathematics, including What is Needed in Research and Educa­ Probability and Mathematical Statistics, tion", sessions for contributed papers, and Mathematical Physics and Numerical a banquet. Analysis, will be covered. A special sec­ tion for Education will be organized in ORSA-TIMS - JOINT MEETING collaboration with the International Com­ mission for Mathematical Instruction. The first joint national meeting of A program of entertainments and the Operations Research Society of Amer­ excursions is being planned. There will be ica and The Institute of Management Sci­ two categories of membership of the Con­ ences will be held November 8-10, 1961, gress: at the Jack Tar Hotel in San Francisco. Ordinary members, who will be en­ The meeting program will include sessions titled to participate in the scientific and in three broad areas of Operations Re­ social activities of the Congress and tore­ search/Management Science, Tools of ceive the Proceedings of the Congress, and OR/MS, under the chairmanship of Pro­ Associate members accompanying fessor George W. Brown, UCLA, will in­ ordinary members of the Congress, who clude mathematical programming, infor­ will be entitled to many of the privileges mation processing, queueing models, and of ordinary membership, but will not decision theory. Dr. Harold 0, Davidson, participate in the scientific program and Operations Research, Incorporated, is will not receive the Proceedings. chairman for sessions on The Applica­ Those who wish to receive further tions of OR/MS, including management information about the Congress are re­ decision processes, problems in data and quested to communicate their name and model validation, and case studies on full address to the Secretary as soon as military and non-military applications. possible and before the end of October. The Challenge of OR/MS, chairedbyPro­ Towards the end of 19 61 they will receive fessor Gifford H. Symonds, Case Institute, a Second Communication giving more de­ will include use of OR/MS in the area of tailed information and specifying the fees world peace, developments in organization to be paid by members; a form of appli­ theory, and general systems analysis. Both cation for membership of the Congress invited and contributed papers will be in­ will be included. cluded in each session.

402 FELLOWSHIP AND RESEARCH OPPORTUNITIES

The Division of Mathematics, Na­ Science Foundation, Washington 25, D. C. tional Academy of Sciences-National Re­ Closing date for receipt of applications - search Council calls attention to a variety November 1, 1961. Award date- March 15, of fellowships and other support for basic 1962. research in mathematics to be awarded by (3) Summer Fellowships for Graduate agencies of the Federal Government during Teaching Assistants make it pos­ the year 1961-1962. A partial list is given sible for graduate teaching assistants to below; other sources of support are given continue full-time academic study and/or in the bulletin, "A Selected List of Major research at approximately 165 participat­ Fellowship Opportunities and Publications ing institutions. Application materials are for Educational Support," available from available at, and are submitted through, the the Fellowship Office, National Academy of teaching assistants' own institutions. For Sciences-National Research Council, 2101 information and list of participating insti­ Constitution Avenue, Washington 25, D. C. tutions, write to the National Science Foundation. Closing date - December 8, 1. National Science Foundation. 1961. Award date- March 15, 1962. I. Research Grants. The National Sci­ B. Postdoctoral Fellowships ence Foundation also supports basic re­ (4) Postdoctoral fellowships for re­ search in the mathematical sciences by cent recipients of the doctoral degree who means of grants. Proposals for such sup­ desire additional advanced training pre­ port are accepted at any time. They should paratory to specialized scientific work. be submitted about six months before the Application material available from the applicant wishes to receive notification of National Academy of Sciences-National the Foundation's decision. Instructions for Research Council. Two award periods: the preparation of proposals, contained in (1) Autumn: closing date for competition­ a booklet entitled, "Grants for Scientific September 5, 1961. Award date - October Research," may be obtained upon request 16, 1961. (2) Spring: Closingdateforcom­ from the Program Director for Mathemat­ petition-December 18, 1961. Award date - ical Sciences, National Science Foundation. March 15, 1962 II. Fellowships. The various Fellow­ (5) Senior Postdoctoral fellowships ship programs of the National Science are awarded to scientists who have demon­ Foundation are described below. Awards, strated ability and special aptitude for pro­ available only to citizens or nationals of ductive scholarship in the sciences and who the United States, are made solely on the have held the doctoral degree for a mini­ basis of ability. mum of 5 years at time of application. A. Predoctoral Fellowships ~ offered Application material may be obtained from annually in three programs. the National Science Foundation for sub­ (1) Graduate fellowships are awarded mission prior to October 9, 1961. Award at the First Year, Intermediate, and Ter­ date - December 11, 19 61. minal Year levels of study. Applications C. Faculty Fellowships for 1962-1963 will be available in October (6) Science Faculty fellowships are 1961 from the National Academy of Sci­ awarded to college teachers of science ences-National Research Council, 2101 (including mathematics) who plan to con­ Constitution Avenue, N. W., Washington25, tinue teaching and wish to increase their D. C., until the closing date, January 1, competence as teachers. Eligibility re­ 1962. Award date - March 15, 1962. quirements include a baccalaureate degree (2) Cooperative Graduate fellowships and three (3) years of full-time experience are tenable at approximately 165 partici­ at the collegiate level. Application material pating institutions. Application materials may be obtained from the National Science are available at, and are submitted through, Foundation for submission prior to October the selected institution, For information 9, 1961. Award date - December 11, 1961. and list of participating institutions, write to the Fellowships Section, Division of Sci­ (7) Summer Fellowships for Second­ entific Personnel and Education, National ary School Teachers of Science and Mathe-

403 matics provide opportunities for teachers foundations and through contracts with of high ability to pursue individually industrial laboratories. Interested re­ planned programs at the graduate level. search mathematicians are encouraged Tenures of one, two or three summers to submit proposals, through their organi­ are available. Information and application zations, for research in mathematical material will be available in October 1961 fields in which they specialize. Proposals from Secondary School Fellowships, Amer­ should be mailed to the Director of Mathe­ ican Association for the Advancement of matical Sciences, Air Force Office of Sci­ Science, 1515 Massachusetts Avenue,N. W., entific Research, Washington Z5, D. C. Washington 5, D. C. Closing date - janu­ A special AFOSR Postdoctoral Fellowship ary 5, 196Z. Award date- March 15, 196Z. Program has been established with the D. Extramural Fellowships help of the National Academy of Sciences­ (8) NATO Postdoctoral Fellowships National Research Council. Information in Science for persons planning to enter can be obtained from the Fellowship Office, upon or continue postdoctoral study in the NAS-NRC. natural sciences at institutions in aNATO 4, U. S. Army Research Office (Durham). country. Tenure is either one academic Among the functions of the U.S. Army Re­ year or a calendar year. Application search Office (Durham) is the support of materials are available from the National basic research in mathematics in educa­ Academy of Sciences - National Research tional or other nonprofit institutions. Pro­ Council in October 1961 for submission posals for support are made by individual prior to December 18, 1961. Award date­ scientists or groups of scientists through April Z, 196Z. their institutions. Support may be accord­ (9) OECD Senior Visiting Fellowships ed either through grants or contracts with permit institutions in the United States, these institutions. For further information its territories and possessions, to send write to Commanding Officer, U. S, Army senior scientists, mathematicians, or en­ Research Office (Durham), Box CM, Duke gineers on their staffs to study new tech­ Station, Durham, North Carolina. niques and developments at advanced re­ search and educational institutions located 5, Fulbright Awards - Public Law 584 primarily in OECD member countries or (79th Congress). Approximately 600 in countries cooperating with that organi­ awards are offered annually for university zation. Application materials are available lecturing and postdoctoral research in all from the National Academy of Sciences - academic fields. Awards are subject to an National Research Council in October 1961 open competition held approximately a for submission prior to january 5, 196Z. year in advance of the academic year in Award date - April 16, 196Z. the host country during which the award would be effective. The competition for the Office of Naval Reserarch. The Office of z. academic year 196Z-1963 in the following Naval Research, through contracts with countries will close on October 1, 1961: universities and other organizations, sup­ Austria, Belgium-Luxembourg, China ports basic research in broadly selected (Taiwan), Denmark, Finland, France, Ger­ fields of mathematics. Proposals should many, Greece, Iceland, Iran, Ireland, be directed to the Mathematics Branch, Israel, Italy, Korea, japan, the Nether­ Office of Naval Research, Washington Z5, lands, Norway, Portugal, Spain, Sweden, D. C. In addition, postdoctoral research Turkey, the United Arab Republic, and the associateships in pure mathematics are United Kingdom including overseas terri­ established through grants from ONR to tories. Opportunities in the field of mathe­ selected universities. For details and matics are offered in Austria, China application forms write to the above ad­ (Taiwan), Denmark, Ireland, Israel, Italy dress. the Netherlands, the United Arab Republic, 3. Air Force Office of Scientific Research. the United Kingdom, and the UK Overseas The Air Force Office of Scientific Re­ Territories. The next competition, for the search supports research in the mathe­ 1963-1964 academic year in the following matical sciences directly through grants countries, will be open between March 15 and contracts with colleges, universities, and May 1, 196Z: Argentina, Australia,

404 Brazil, Burma, Ceylon, Chile, Colombia, Biological Laboratories at Fort Detrick, Ecuador, India, New Zealand, Pakistan, Maryland. Further information can be ob­ Paraguay, Peru, the Philippines, Thailand, tained from the Fellowship Office, NAS­ and Uruguay. Awards are payable in for­ NRC. eign currency and include round trip 7. Atomic Energy Commission. The Divi­ transportation for the grantee (travel may sion of Research of the Atomic Energy be provided for one dependent of lectur­ Commission through contracts with uni­ ers in certain countries of Asia, Africa, versities and other organizations supports and Latin America); a maintenance allow­ research in the fields of numerical analy­ ance to cover normal living costs of the sis, digital computer design, program­ grantee and his dependents while in resi­ ming research, and related topics. Pro­ dence abroad; a small incidental allowance posals should be submitted to the Division for travel, books, and services essential of Research, Atomic Energy Commission, to the project; and subject to the availa­ Washington 25, D. C. bility of funds, a supplemental dollar grant Brookhaven National Laboratory. to lecturing grantees in specific countries Brookhaven National Laboratory, operated of Asia, Africa, and Latin America. Re­ by Associated Universities, Incorporated, quests for information should be addressed under contract with the Atomic Energy to the Committee on International Ex­ Commission offers postdoctoral research change of Persons, Conference Board of appointments in the fields of numerical Associated Research Councils, 2101 Con­ analysis, digital computing, mathematical stitution Avenue, Washington 25, D. C. physics, differential equations, probability 6. National Bureau of Standards. National and statistics, and various specialized Aeronautics and Space Administration. branches including reactor theory, hydro­ Naval Research Laboratory. Air Research dynamics, and orbit theory. Applications and Development Command. Naval Ord­ should be directed to Head, Applied Mathe­ nance Laboratory. Navy Electronics Lab­ matics Division, Brookhaven National Lab­ oratory. U. S. Naval Weapons Laboratory. oratory, Upton, Long Island, New York. U. S. Army Chemical Corps Biological 8. The Institute for Defense Analyses. The Laboratories. Postdoctoral Resident Re­ Institute for Defense Analyses supports search Associateships are available in a postdoctoral research by individuals with variety of sciences including mathematics doctoral degrees or the equivalent in en­ and are tenable at the Washington, D. C. gineering, physical sciences, or mathe­ and Boulder, Colorado, Laboratories of the matics interested in gaining experience National Bureau of Standards; at the Insti­ and training in operations research and tute for Space Studies in New York City analysis, particularly as related to defense and at the Goddard Space Flight Center, and governmental problems. Applications Greenbelt, Maryland, of the National Aero­ and requests for further information should nautics and Space Administration; at the be directed to Dr. Charles H. Townes, In­ Naval Research Laboratory in Washing­ stitute for Defense Analyses, 1710 H Street, ton, D. C.; at selected development and N. W., Washington 6, D. C. research centers of the Air Research and Development Command; at the Naval Ord­ nance Laboratory in Silver Spring, Mary­ J. BARKLEY ROSSER, Chairman the Navy Electronics Laboratory land; at Division of Mathematics in San Diego, California; at the U.S. Naval Weapons Laboratory, Dahlgren, Virginia, M. H. MARTIN, Executive Secretary and at the U. S. Army Chemical Corps Division of Mathematics

405 THE ANNUAL SALARY SURVEY

The 1961 Salary Survey is the fifth in an annual series of surveys of academic institutions initiated in May 19 57 by the Society's Committee on the Economic Status of Teachers. The value of its data is attested by the response to the questionnaires used in the survey and by the many favorable comments received. This year 294 usable questionnaires were returned, a considerable increase over the 108 received in 1960 and the 99 in 1959. Tabulations in previous years have used a version of the classification of academic institutions introduced in the 1955-1956 SurveyofTrainingand Research Potential in the Mathematical Sciences (the "Albert Survey"), but this classification is no longer con­ sidered to be the most suitable for our purposes. In the interest of securing a classifica­ tion which would reflect more closely the mathematical activity of the institutions sur­ veyed, the Society has developed a system based on institutional membership in the Society. Institutional membership readily provides data on which to base the kind of classification desired, since the dues of a member institution are determined by the number of pages of research results sponsored by the institution and published in the journals published or subsidized by the AMS. The dues paid by a member can therefore be taken as an indication of the amount of research activity in that institution. This survey thus classifies institutions into two major types: Institutional Mem­ bers and Non-Institutional Members. Members are in turn divided into two groups: I. Those which in the three-year period 1958-1960 sponsored 37 1/2 or more pages in journals published or subsidized by the Society, and II, those which contributed less than 37 1/2 pages in the three-year period. In Group I we received 58 questionnaires; in Group II 80 were returned. Non-Institutional Members returned 156 questionnaires. A summary is given for all 294 institutions. The salaries covered by the survey are based on an academic year (9 to 10 months). The number of staff members shown in each rank for 1960-1961 was the actual number employed and serving for that period, whereas the corresponding figures shown for 1961-1962 are those employed as of the time of the survey (June, 1961). It was indi­ cated in the questionnaire that by salary is meant the payment by the school to the individual for full-time work. Grants and contracts were to be included, but sabbatical payments and other part-time salaries were to be excluded. All salary figures are given in hundreds of dollars.

406 INSTITUTIONAL MEMBERS OF THE SOCIETY "I"

Number of usable returns: 58

Total number on the staffs working full time on the campus

RANK 1960-1961 1961-1962 Instructor (only those holding Ph. D.) 106 109 Assistant Professor 350 365 Associate Professor 323 346 Professor 393 429 TOTAL 1172 1249

Sala 1960-1961 1961-1962 RANK Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph. D.) 49-70 51-70 51-80 52-75 57-80 55-80 Assistant Professor 54-89 57-89 60-95 57-90 64-92 66-120 Associate Professor 58-107 66-107 67-130 60-120 75-125 72-130 Professor 72-135 79-155 81-205 72-145 83-200 86-240

INSTITUTIONAL MEMBERS OF THE SOCIETY "II"

Number of usable returns: 80

Total number on the staffs working full time on the campus

RANK 1960-1961 1961-1962 Instructor (only those holding Ph. D.) 15 18 Assistant Professor 236 247 Associate Professor 191 204 Professor 211 222 TOTAL 653 691

Sala 1960-1961 1961-1962 RANK Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph. D.) 50-78 55-78 55-91 51-86 54-86 57-91 Assistant Professor 45-85 48-102 52-104 45-92 48-104 52-104 Associate Professor 49-111 49-117 49-126 50-121 50-121 50-150 Professor 52-135 56-161 56-180 56-145 56-166 56-190

407 NON-INSTITUTIONAL MEMBERS

Number of usable returns: 156

Total number on the staffs working full time on the campus

RANK 1960-1961 1961-1962 Instructor (only those holding Ph. D.) 11 12 Assistant Professor 338 343 Associate Professor 222 233 Professor 231 237 TOTAL 802 825

Sala!l Surve;y 1960-1961 1961-1962 RANK Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph. D.) 45-62 44-72 49-86 45-65 50-73 51-88 Assistant Professor 38-104 43-104 45-104 47-104 52-104 45-104 Associate Professor 48-88 48-115 48-121 53-95 53-107 54-123 Professor 53-166 56-166 56-166 60-166 62-166 62-166

SUMMARY OF ALL INSTITUTIONS SURVEYED

Number of usable returns: 294

Total number on the staffs working full time on the campus

RANK 1960-1961 1961-1962 Instructor (only those holding Ph. D.) 132 139 Assistant Professor 924 955 Associate Professor 736 783 Professor 835 888 TOTAL 2627 2765

1960-1961 1961-1962 RANK Minimum Median Maximum Minimum Median Maximum Instructor (only those holding Ph.D.) 45-78 44-78 49-91 45-86 50-86 51-91 Assistant Professor 38-104 43-104 45-104 45-104 48-104 45-120 Associate Professor 48-111 48-117 48-130 50-121 50-125 50-150 Professor 52-166 56-166 56-205 56-166 56-200 56-240

408 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D This is the second annual Survey of Starting Salaries of graduates with the Ph. D. degree in Mathe­ matics. The figures are based on questionnaires sent to people who received Ph. D.'s in mathemat­ ics during the past year; 304 questionnaires were sent out and 151 usable replies were received.

Minimum starting salaries in academic institutions decreased 8. 2% in the past year, and maximum salaries gained 12. 5%. Those Ph. D.'s who reported research as their primary responsibility received 6. 7% higher salaries in the minimum salary bracket and 9.1% more in the maximum salary group than their colleagues who gave teaching as their primary responsibility.

The corresponding figures in starting salaries in industrial employment are: 11. 5% increase for minimum salaries and 16% increase for maximum salaries.

Government salaries, which had decreased from 1959 to 1960, increased during the past year by 8% in the minimum salary group and by 23% for maximum salaries.

Research Institutes showed a decline of 13.4% in minimum salaries and an increase of 1. 4% in maximum.

Of the 151 Ph. D.'s reporting, 113 took employment in academic institutions: 98 of them in universities and colleges, 10 in technical institutues, 5 in research institutes, and one in high school. Industry and government accounted for 28 and 9 respectively. The Ph.D. recipient who accepted teaching in high school as his first employment receives a salary 5% above the median for Universities and Colleges.

The majority of the Ph. D.'s .reported that they had had more than one year of work experience before receiving the Ph. D. Generally, shorter previous work experience was reported from the group which accepted employment in technical institutes.

Geographically, the heaviest concentration of new appointments of mathematicians is in the North East, with 34.5 percent; the Far West has 24.8 percent; the Mid West 23.4 percent; and the South 17.3 percent.

UNIVERSITIES, COLLEGES AND TECHNICAL INSTITUTES (Nine Month Salary)

TEACIDNG RESEARCH Year Minimum Median Maximum Minimum Median Maximum 1958 $4,800 $ 5,500 $7,100 $ 5, 000 $6,000 $ 7' 000 1959 5,000 6,300 7,600 5,000 6,000 7,700 1960 4,900 6,500 8,000 5,200 6,500 8,000 1961 4,500 6,300 8,250 4,800 6,500 9,000

INDUSTRY ~Twelve Month Sal~) Year Minimum Salary Median Salary Maximum Salary 1958 $ 8,600 $ 10,300 $ 12,700 1959 7,800 10,500 16,000 1960 7,800 11,000 15,000 1961 8,700 11,000 17,400

RESEARCH INSTITUTES ~Twelve Month Salaryl Year Minimum Salary Median Salary Maximum Salary 1958 $4,000 $ 8, 250 $ 11,000 1959 6,900 9,500 14,000 1960 9,700 10,500 14,000 1961 8,400 11,000 14,200

GOVERNMENT {Twelve Month Salary} Year Minimum Salary Median Salary Maximum Salary 1958 $ 7' 500 $ 9, 750 $ 11,600 1959 8,800 10,200 13,000 1960 7,200 9,300 13,000 1961 7,780 8,900 16,000

409 NEWS ITEMS AND ANNOUNCEMENTS

CONTE MP ORAR Y MATHEMATICS U.S. -owned foreign currencies to be used TO BE REPEATED ON CONTINENTAL for scientific research in foreign coun­ CLASSROOM IN 1961-196Z. The National tries. Dr. Franz L. Alt, Coordinator of Broadcasting Company announced on june Special International Programs, will han­ ZB that last season's "Continental Class­ dle the establishment and operation of the room" course in Contemporary Mathemat­ projects. ics will be repeated on color tape record­ This is the first time the National ings from 6:00 to 6:30 A.M. local time, Bureau of Standards has received an ap­ beginning September Z5, 1961. propriation from the foreign currency Contemporary Mathematics, the funds which have accumulated from the new course on "Continental Classroom" in sale of U. S. surplus agricultural products 1960-1961, is taught by john L. Kelley, to foreign governments. The research Professor of Mathematics at the Univer­ programs will be undertaken in India, sity of California, Berkeley, and Fred­ Pakistan, and Israel and will include pro­ erick Mosteller, Professor of Mathemati­ jects which complement and supplement cal Statistics and Chairman of the Depart­ the research programs undertaken in the ment of Statistics at Harvard University. Bureau's laboratories. Professor Kelley teaches Modern Algebra Dr. Alt, who has specialized in during the first semester of Contemporary computing machines and numerical meth­ Mathematics; Professor Mosteller teaches ods, has been at the Bureau since 1948. Probability and Statistics during the sec­ As Assistant Chief of the Applied Mathe­ ond. matics Division, he has worked with com­ The new course on "Continental puting machines and their application to Classroom" during 1961-196Z will be a military tactical problems, mathematical course in American Government. Dr. methods in management, pattern recog­ Peter H. Odegard, one of the nation's nition, and machine translation of lan­ most distinguished political scientists, guages. now Professor of Political Science at the University of California, Berkeley, will teach the two-semester, college-level ACONFERENCEONUNDERGRAD­ course, which will deal with the structure UATE RESEARCH IN MATHEMATICS, and function of the U. S. Government. The supported by a grant from the National course in American Government will be Science Foundation, was held at Carleton televised in color and carriedby approxi·· College june 19 through june Z3, 19 61. mately 170 stations in every part of the Seventy-one aolleges and universities were. country, Monday through Friday, from represented. Invited addresses were given 6:30 to 7:00 A.M. local time, starting by R. L. Wilder, Paul c. Rosenbloom, September Z5. Lloyd B. Williams, Frank L. Griffin, The Conference Board of the Mathe­ Robert z. Norman, and Kenneth 0. May. matical Sciences is one of the sponsors of Lewis Pino and William Rosen spoke on Contemporary Mathematics, along with the behalf of the N.S.F ., Donald Western re­ Learning Resources Institute and the Na­ ported on the work of the Committee on tional Broadcasting Company. Undergraduate Research Participation of G. BALEY PRICE, Executive Secretary the M.A.A., and R. j. Wisner represented Conference Board of the Mathematical the C.U.P.M. Sixty-six institutions re­ Sciences ported a wide variety of activities designed to help students begin to think and work like mathematicians. Some speakers and dis­ SPECIAL INTERNATIONAL PRO­ cussants suggested using the word 're­ GRAMS TO BE ESTABLISHED AT NBS. search' in a broader sense than is now The National Bureau of Standards has re­ customary among the mathematicians and ceived an appropriation of $1,000,000 of in a manner consistent with its use in

410 other fields. Resolutions were passed sible during the academic year 1961-1962, urging continued attention to independent it being expected that the thirty consul­ study and research by undergraduates, tants can serve a total of approximately improved communication between teachers 150 schools. The consultants are: Richard supervising such activity, the provision of D. Anderson, John D. Baum, RoyDubisch, additional opportunities for publication of Lincoln K. Durst, Philip Dwinger, John V. student work by utilizing existingpublica­ Finch, Marion K. Fort, Jr., Leon A. Hen­ tions and by the establishment of a journal kin, James A. Hummel, Bernard Jacobson, directed only to undergraduates and writ­ Paul B. Johnson, Burton W. Jones, Paul ten largely by them, and the increased use J. Kelley, Eugene E. Kohlbecker, Arthur of sectional meetings of the M.A.A. for E. Livingston, Anil Nerode, Robert Z. presentation of undergraduate work of high Norman, William R. Orton, Billy J. Pettis, quality. A complete report of the confer­ Mina Rees, Charles E. Rickart, Hartley ence will be sent to department chairmen Rogers, Jr., Seymour Schuster, LelandL. listed in the Mathematical Sciences Ad­ Scott, William R. Scott, E. Baylis Shanks, ministrative Directory, published by the John Wagner, Elbert A. Walker, James H. A.M.S. Others may request copies by Zant. Further information and applica­ writing to the Department of Mathematics, tions for a visit by a consultant may be Carleton College, Northfield, Minnesota. obtained by writing: Professor Robert J, Wisner Executive Director Committee on the Undergraduate DIRECTORY OF THE WORLD Program in Mathematics Second Edition, 1961, MATHEMATICIANS, Michigan State University, Oakland available from is now ready and is Rochester, Michigan World Directory of Mathematicians Tata Institute of Fundamental Research 5, India Colaba, Bombay THE MATHEMATISCHE FORSCH­ The cost is $1.50 or lOs. 6d. or Rs.7 ; ... UNGSINSTITUT OBER WOLF ACH held a per copy, post free. Cheques, which may Colloquium on Ordered Sets in Aprill961. be in any currency, should be made payable The chairman was L. Lesieur, Paris; the to the Tata Institute of Fundamental Re­ participants were M. L. Dubreil-Jacotin, search, Bombay. Cheques in currencies P. Dubreil, R. Croisot, M. Ego, J. Fort, other than the three above should be for and P. Lefebvre, France; M. Curzio, Italy; the equivalent of $1.50. G. Birkhoff, United States; G. Kurepa, Orders may be placed for single Yugoslavia; E. T. Schmidt, Hungary; M. copies or in bulk. In the case of bulk Novotny and F. Sik, Czechoslovakia; G. orders, copies will be mailed direct to the Bruns, W. Felscher, G. Grimeisen, E. individuals concerned. Harzheim, J. Schmidt, D. Schumacher, and E. A. Behrens, Germany. A Conference on Number Theory THE COMMITTEE ON THE UN­ was held in May 1961, under the chairman­ DERGRADUATE PROGRAM IN MATHE­ ship of Professors H. Hasse and P. MATICS of the Mathematical Association Raquette, Germany. Twenty-three mathe­ of America has established a Consultants maticians took part in the Conference-- 15 Bureau in order to aid colleges in upgrad­ from Germany, 3 from France, 2 from ing and revising their undergraduate offer­ England, and one each from the United ings or with planning new curricula. Upon States, Switzerland, and Denmark. The request, consultants will visit colleges program included a survey of the fields of and universities for a period of two days, algebraic number theory and algebraic the expenses and honoraria being shared geometry. A survey lecture reported on by the host institution and by CUPM. Only recent developments in the arithmetic of a limited number of such visits are pos- absolute abelian number fields.

411 PERSONAL ITEMS

and Com­ Assistant Professor A. AEPPLI of search Group, Arthur Andersen a position Cornell University has been appointed to pany, New York has accepted Staff of Wea­ an assistant professorship at the Univer­ as Member of the Technical Group, a division sity of Minnesota. pons Systems Evaluation The Professor A. A. ALBERT on leave of the Institute for Defense Analyses, C. from the University of Chicago has been Pentagon, Washington, D. of Brown Uni­ appointed Director of the Institute for De­ Professor I. BARSOTTI to a professor­ fense Analyses. versity has been appointed Italy. Dr. G; D. ALLAUD of the University ship at the Universita di Pisa, Pisa, ofWestinghouse of Wisconsin has been appointed Instructor Mr. J. F. BARTRAM accepted a posi­ and Research Associate at the University Electric Corporation has at Raytheon Com­ of Chicago. tion as Staff Engineer Miss K. A. ANDERSON of Computer pany, Newport, Rhode Island. leave from the Sciences Corporation, Palos Verdes, Cali­ Mr. P. T. BATEMAN on been appointed a fornia has accepted a position as Analyst University of illinois has at the Univer­ at C-E-1-R, Incorporated, Beverly Hills, Visiting Research Associate California. sity of Pennsylvania. University of Assistant Professor J. J. ANDREWS Mr. D. BATMAN of the as Senior of the University of Wisconsin has· been Idaho has accepted a position Manufacturing appointed Visiting Lecturer at Florida Scientist with the A vco Massachusetts. State University for the academic year Corporation, Wilmington, leave from the 1961-1962. Dr. L. E. BAUM on the aca­ Dr. K. I. APPEL of the Institute for University of Chicago will spend at the Institute for Defense Analyses, Princeton has been ap­ demic year 1961-1962 pointed to an assistant professorship at the Defense Analyses. University of illinois. Dr. H. P. BEARDofNewcombCollege profes­ Dr. E. L. ARNOFF hasacceptedapo­ has been appointed to an associate sition as Director of Operations Research sorship at Harpur College. the University of at Ernst and Ernst, Cleveland, Ohio. Mr. H. E. BELL of to an assist­ Dr. N. ARTEMIADIS oftheUniversity Wisconsin has been appointed College. of Salonica has been appointed to an asso­ ant professorship at Union F. T. BIRTEL on ciate professorship at the University of Assistant Professor University has been Wisconsin, Milwaukee. leave from Ohio State and Lec­ Dr. G. AVILA, of the Institute of appointed Research Associate for the academic Mathematical Sciences has been appointed turer at Yale University to a professorship at the lnstituto de year 1961-1962. the University of F1sica Te6rica, Sllo Paulo, Brazil. Dr. J. R. BOEN of to the position Dr. A. E. BABBITT, Jr. of the Signal Chicago has been appoin•ed of Michigan. Corps Laboratory, Army Department, Fort of Lecturer at the University of the University of Monmouth, New Jersey has accepted a po­ Dr. S. E. BOHN to an assist­ sition as Advisory Systems Analyst at the Nebraska has been appointed Bowling Green State International Business Machines, Com­ ant professorship at munications Systems Center, Rockville, University. of the Institute for Maryland. Dr. R. A. BONIC appointed to an Miss D. M. BAMBERGER of Cornell Advanced Study has been at Cornell Univer­ University has accepted a position as assistant professorship Mathematician at the Bell Telephone Labo­ sity. BOSE on leave from ratories, Murray Hill, New Jersey. Professor R. C. will spend Dr. L. K. BARRETT of the University the University of North Carolina in Europe to of Utah has been appointed to an associate the academic year 1961-1962 professorship at the University of Ten­ carry on research. J. R. BOYD of nessee. Assistant Professor has been appointed Dr. J. Y. BARRY of Operations Re- Arlington State College

412 to an associate professorship at Guilford cepted a position as Mathematical Analyst College. with Babcock and Wilcox, Lynchburg, Vir­ Mr. R. L. BOYELL of Sperry Gyro­ ginia. scope Company, Sperry Rand Corporation, Mr. D. A. CLARKE of the University has joined the staff of Pennsylvania Re­ of Wisconsin has been appointed Lecturer search Associates, Incorporated, as Re­ at the University of Toronto. search Director. Associate Professor P. E. CONNER, Dr. D. L. BOYER of Fresno State Col­ Jr. on leave from the University of Virginia lege has been appointed to an assistant has been appointed a Visiting Member of professorship at the University of Idaho. the Institute for Advanced Study for the Mr. C. N. BROOKS of the University academic year 1961-196Z. of Wisconsin has accepted a position as Dr. F. F. CONNOR has been appointed Associate Engineer at the Sperry Gyro­ to an assistant professorship at the Loui­ scope Division, Sperry Rand Corporation, siana State University. Great Neck, Long Island, New York. Dr. W. M. CUNNEA of the University Mr. T. A. BROWNofHarvardUniver­ of California has been appointed to an sity has accepted a position as Mathema­ assistant professorship at Washington tician at the RAND Corporation, Santa State University. Monica, California. Visiting Assistant Professor L. W. Dr. W. F. BROWN of CONVAIR, DANZER of the University of Washington General Dynamics Division, San Diego, has accepted an assistantship at the Uni­ California has accepted a position as Head versity of Munich, Germany. of Applied Mathematics at Atomics Inter­ Dr. P. DEDECKER of the University national, Canoga Park, California. of. Liege, Belgium has returned after a Dr. H. D. BRUNK of the Universityof leave of absence at the Universidad Cen­ Missouri has been appointed to a profes­ tral de Venezuela. sorship at the University of California, Dr. C. B. De LYRA of the University Riverside, California. of Silo Paulo, Brazil has returned after a Dr. j. BURLAK on leave from the leave of absence at the Institute for Ad­ University of Glasgow, Scotland has been vanced Study. appointed to a visiting assistant professor­ Professor R. P. DILWORTH on leave ship at North Carolina State College. from California Institute of Technology has Dr. j. W. CARLYLE oftheUniversity been appointed a Member of the Technical of California, Berkeley has been appointed Staff at the Institute for Defense Analyses to an assistant professorship at Princeton. for the academic year 1961-196Z. University. Mr. R. S. DINSMORE of Stanford Uni­ F. W. CARROLL, Jr. ofthe University versity has accepted a position as Systems of Wisconsin has been appointed to an Engineer at the International Business assistant professorship at the Ohio State Machines Corporation, San Francisco, University. California. J. B. CHICCARELLI of Fordham Uni­ Dr. W. DOYLE of Aeronutronic, New­ versity has been appointed to an assistant port Beach, California has accepted a po­ professorship at Stonehill College, sition as Mathematician with the RAND Mr. B. B. CLARK of Thompson Ramo­ Corporation, Santa Monica, California. Wooldridge, Incorporated has accepted a Dr. R. DUBISCH of Fresno State Col­ position as Staff Engineer at the Interna­ lege has been appointed to a professorship tional Business Machines Corporation, at the University of Washington. Rockville, Maryland. Professor B. M. DWORK on leave Dr. C. J. CLARK of Lockheed Air­ from johns Hopkins University has been craft Corporation, Palo Alto, California appointed a Member of the Institute for has accepted a position as Engineering Advanced Study for the academic year Specialist with the Sylvania Electronic 1961-196Z. Defense Laboratory, Mountain View, Cali­ Associate Professor L. EHRENPREIS fornia. has resigned from Yeshiva University and Assistant Professor R. E. CLARK of will spend the coming academic year at the the Virginia Military Institute has ac- Institute for Advanced Study, Princeton,

413 New Jersey. sity of Rennes, France has beenappointed Dr. E.G.EIGEL,Jr.hasbeenappoint­ a Member at the Institute for Advanced ed to an assistant professorship at St. Study, Princeton, New Jersey, for the aca­ Louis University. demic year 1961-196Z. Dr. E. EISENBERG of Hughes Air­ Mr. J. B. GOEBEL of Oregon State craft Company, Culver City, California College has accepted a position as Mathe­ has been appointed Lecturer at the Univer­ matician with General Electric Company, sity of California, Berkeley, California, Richland, Washington. Dr. K. FAN of Wayne State University Professor L. M. GRAVES of the Uni­ has been appointed to a professorship at versity of Chicago has retired with the title Northwestern University. Professor Emeritus. Dr. G. F. FEEMAN of Massachusetts Dr. A. HAEFLIGER of the Institute Institute of Technology has been appointed for Advanced Study has been appointed to a to an assistant professorship at Williams professorship at the University of Geneva, College. Switzerland. Associate Professor M. FISZ of the Dr. C. J. A. HALBERG has been ap­ University of Warsaw,Polandhasbeenap­ pointed to an associate professorship at pointed to a visiting professorship at Col­ the University of California, Riverside. umbia University, for the academic year At present he is on sabbatical leave and 1961-196Z. with an NSF fellowship at the Mathematics Dr. U. FIXMANofYaleUniversityhas Institute, University of Copenhagen. been appointed to an assistant professor­ Mr. J. F. HARDMAN of the Bendix ship at Queen's University, Kingston, On­ Aviation Corporation, Kansas City, Mis­ tario, Canada. souri has accepted a position as Chief Dr. P. F ONG of the University of Mathematician with the Babcock and Wil­ Chicago has been appointed to an assistant cox Company, Alliance, Ohio. professorship at the University of Califor­ Assistant Professor B. HARRIS of nia, Berkeley. Northwestern University has been appoint­ Dr. L. R. FORD, Jr. of the General ed to an associate professorship at Brown Analysis Corporation, Sierra Vista, Ari­ University. zona has accepted a position as Project Dr. M. E. HARRIS has beenappointed Director with C-E-1-R, Incorporated, Be­ to an assistant professorship-at Tufts Uni­ verly Hills, California. versity. Professor Emeritus, C. A. GARABE­ Mr. T. L. HAYDEN of the University DIAN of Wheaton College has been appoint­ of Texas has been appointed to an assist­ ed to a visiting professorship at the Uni­ ant professorship at the University of Ken­ versity of Rhode Island. tucky. Dr. L. D. GATES, Jr. of the Babcock Dr. L. A. HENKIN of the University of and Wilcox Company, Lynchburg, Virginia California is on leave at the Institute for has been appointed to an associate profes­ Advanced Study for the academic term sorship at Southern illinois University. 1961-196Z. Professor M. GERSTENHABER on Assistant Professor D. HER TZIG of leave from the University of Pennsylvania Cornell University has been appointed to has been appointed a Member of the Tech­ a visiting assistant professorship at nical Staff at the Institute for Defense Princeton University. Analyses for the academic year 1961-1962, Professor M. R. HESTENES on leave Dr. R. P. GILBERT of Michigan State from the University of California, Los An­ University has accepted a position as Re­ geles has been appointed a Member of the search Assistant Professor at the Institute Technical Staff at the Institute for Defense for Fluid Dynamics and Applied Mathema­ Analyses for the academic year 1961-196Z. tics, University of Maryland. Dr. W. M. HUEBSCH of the University Professor J. GILLIS of the Weizman of Notre Dame has been appointed to an Institute of Science has accepted a position associate professorship at the Western with the Goddard Institute for Space Studies Reserve University. for the academic year 1961-196Z. Mr. R. W. HUNT of the University of Professor G. GLAESER of the Univer- Utah has accepted a position as Aero-

414 dynamics Engineer and Research Mathe­ Assistant Professor C. W. KOHLS of matician at the Douglas Aircraft Company, the University of Illinois has been appoint­ Santa Monica, California. ed to an assistant professorship at Syra­ Dr. S. A. HUSAIN of the University of cuse University. Saskatchewan has been appointed to a visit­ Assistant Professor J. J. KOHN on ing assistant professorship at the Univer­ leave from Brandeis University has been sity of Idaho. appointed a temporary member at the In­ Dr. T. HUSAIN of Syracuse University stitute for Advanced Study. has been appointed to an assistant profes­ Dr. B. KOSTANT of the University of sorship at the University of Ottawa. California, Berkeley has been appointed to Mr. C. T. IRELAND of Cornell Uni­ a professorship at the Massachusetts versity has accepted a position as Mathe­ Institute of Technology. matician with the National Security Agency, Assistant Professor V. A. KRAMER Fort George G. Meade, Maryland. has been appointed to an associate profes­ Dr. A. R. JACOBY of the Stanford Re­ sorship at the University of California, search Institute, Menlo Park, California Riverside. has been appointed to an associate pro­ Mr. R. S. KUEHNE of the Shell Oil fessorship at the University of New Hamp­ Company, Wood River, Illinois has ac­ shire. cepted a position as Process Analyst with Professor B. A. JESSUP of the Uni­ the T. R. W. Computers Company, Canoga versity of Iowa has been appointed to a Park, California. professorship at the Detroit Institute of Dr. N. KWAK of Worcester Polytech­ Technology. nic Institute has been appointed to an Dr. J. W. JEWETThasbeenappointed assistant professorship 1tt Fordham Uni­ to an associate professorship at the Uni­ versity. versity of Georgia. Dr. C. D. LaBUDDE of Wayne State Dr. H. H. JOHNSON of Princeton Uni­ University has been appointed to an assist­ versity has been appointed to an assistant ant professorship at Northwestern Univer­ professorship at the University of Wash­ sity. ington. Dr,.P.LANZANO ofNortronics, Haw­ Professor W. G. JOHNSON on leave thorne, California has accepted a position from Hiram College has been appointed to as Senior Technical Specialist at North a fellowship at the University of California, American Aviation, Incorporated, Downey, Berkeley. California. Mr. G. S. JONES, Jr. of the General Assistant Professor A. LEVY of the Electric Company, Evendale, Ohio has ac­ Hebrew University has returned after a cepted a position as Staff Mathematician leave of absence at the University of Cali­ with RIAS, Baltimore, Maryland. fornia, Berkeley. Professor M. KAC of Cornell Univer­ Dr. D. J. LEWIS of the University of sity has been appointed to a professorship Notre Dame has been appointed to an at the Rockefeller Institute. associate professorship at the University Professor L. KATZ on leave from of Michigan. Michigan State University has been ap­ Dr. H. A. LINSTONE of Hughes Air­ pointed to a visiting professorship at the craft Company, Culver City, California University of North Carolina. has accepted a position as Research Professor T. KAWATA of the Tokyo Mathematician at the RAND Corporation, Institute of Technology, Tokyo, Japan has Santa Monica, California. been appointed to a visiting professorship Dr. A. J. LOHW ATER of Rice Institute at the Catholic University of America. has accepted a position with the American Dr. H. KESTEN has been appointed to Mathematical Society, Providence, Rhode an assistant professorship at Cornell Uni­ Island as Assistant Executive Editor of versity. "Mathematical Reviews." Dr. M. S. KNEBEL MAN of Washington Dr. M. LOWENGRUB of Duke Univer­ State University has been appointed to a sity has been appointed to an assistant visiting professorship at Allegheny Col­ professorship at North Carolina State Col­ lege. lege.

415 Dr. D. LUBELL of Harvard Univer­ Assistant Professor W. C. NEMITZ sity has been appointed Research Instruc­ of the University of Kansas has been ap­ tor at New York University. pointed to an assistant professorship at Dr. G. S. LUDFORD, VisitingProfes­ Southwestern at Memphis. sor at Brown University, has been ap­ Dr. C. A. NICOL of the University of pointed to a professorship at Cornell Uni­ Oklahoma has been appointed to an asso­ versity. ciate professorship at the University of Dr. D. A. LUDWIG of Princeton Uni­ South Carolina. versity has been appointed to an assistant Professor J. C. C. NITSCHE of the professorship at the University of Califor­ Institute of Technology, University of nia, Berkeley. Minnesota has returned after a leave of Mr. P. J. McCARTHY of Florida State absence at the University of Puerto Rico. University has been appointed to an associ­ Dr. J. A. NOHEL oftheGeorgialnsti­ ate professorship at the University of Kan­ tute of Technology has been appointed to an sas. associate professorship at the University Mr. R. M. McLEOD of Duke University of Wisconsin. has been appointed to an associate profes­ Assistant Professor E. NORMAN of sorship at the American University of Michigan State University has accepted a Beirut. position as Research Mathematician with Dr. D. R. McMILLAN, Jr. of Louisi­ Socony Mobil Oil Company, Paulsboro, ana State University has been appointed to New Jersey. an assistant professorship at Florida Dr. J. L. NOWINSKI of the University State University. of Texas has been appointed Professor of Mr. F. MAEDA has been appointed a Mechanical Engineering at the University Research Associate at the University of of Delaware. Illinois for the academic year 1961-1962. Dr. W. C. OR THWEIN of International Dr. T.MATSUSAKA ofthelnstitutefor Business Machines Corporation, Owego, Advanced Study, has been appointed to a New York has been appointed to an asso­ professorship at Brandeis University. ciate professorship at the University of Dr. E. MENDELSONofColumbiaUni­ Oklahoma. versity has been appointed to an associate Mr. S. P ASIENCIER of Northwestern professorship at Queens College. University has been appointed to an assist­ Dr. A. C. MEWBORN of Yale Univer­ ant professorship at Lake Forest College. sity has been appointed to an assistant Dr. J. W. PETRO of the State Univer­ professorship at the University of North sity of Iowa has been appointed to an Carolina. assistant professorship at Western Michi­ Mr. R. A. MICHELSON of the Univer­ gan University. sity of Washington has been appointed to Dr. J. PONCET of the Institute for the position of Associate Research Engin­ Advanced Study has been appointed to an eer at Boeing Company, Renton, Washing­ assistant professorship at Rutgers, The ton. State University. Mr. S. S. MITRA of the University of Dr. E. C. POSNER of Harvey Mudd Washington has been appointed to an College has accepted a position as Re­ assistant professorship at the University search Specialist at the Jet Propulsion of Idaho. Laboratory, Pasadena, California. Dr. E. F. MOORE of the Bell Tele­ Mr. W. D. POWERS of Oak Ridge Na­ phone Laboratories, Incorporated, Murray tional Laboratory and the University of Hill, New Jersey has been appointed a Tennssee has accepted a position as Visiting Lecturer at Harvard University Mathematician at the Pacific Missile and to a visiting professorship at Massa­ Range, Point Mugu, California. chusetts Institute of Technology for the Dr. M. 0. RABIN of the Hebrew Uni­ academic year 1961-1962. versity will spend the academic year 1961- Mr. H. G. MOORE of Purdue Univer­ 1962 as Visiting Associate Professor at sity has been appointed to an assistant the University of California, Berkeley. professorship at Brigham Young Univer­ Dr. G. RAISBECK of Bell Telephone sity. Laboratories, Incorporated, Murray Hill,

416 New jersey has accepted a position as Carleton College will be a Visiting Asso­ Senior Staff Member at Arthur D. Little, ciate Professor at the University of North Incorporated, Cambridge, Massachusetts. Carolina for the Fall term 1961. Mr. K. E. RALSTON, of Litton Indus­ Associate Professor W. T. SCOTT of tries, Woodland Hills, California has ac­ Northwestern University has been appoint­ cepted a position as Staff Member at ed to a professorship at Arizona State Massachusetts Institute of Technology, University. Lincoln Laboratory, Lexington, Massachu­ Associate Professor B. SCHWARZ of setts. the Israel Institute of Technology has been Assistant Professor M. S. RAMANU­ appointed to a visiting associate profes­ JAN on leave from the University of Michi­ sorship at the United States Army Mathe­ gan will spend the academic year 1961- matics Research Center, University of 1962 at the Ramanujan Institute of Mathe­ Wisconsin. matics, University of Madras, India. Mr. R. G. SELFRIDGE of Miami Uni­ Dr. C. N. LEE on leave from the Uni­ versity has been appointed to an associate versity of Michigan has been appointed to professorship at the University of Florida, an assistant professorship at the Univer­ Gainesville. sity of Western Ontario, Canada. Mr. j. R. SINGLETARY of Fishburne Mr. W. M. RAND has accepted a po­ Military School has accepted a position as sition as Associate Research Engineer Master in Mathematics, Asheville School, with Boeing Airplane Company, Renton, North Carolina. Washington. Assistant Professor L. B. SMITH, jr. Dr. L. j. RATLIFF, Jr. has been ap­ of Elon College has been appointed to an pointed a Lecturer at Indiana University. assistant professorship at Salem College. Mr. R. E. ROBERTS, jr. of Lehigh Dr. F. L. SPITZER of the University University has accepted a position as Re­ of Minnesota has been appointed to a pro­ search Engineer at General Dynamics­ fessorship at Cornell University. Astronautics, San Diego, California. Mr. D. H. STALEY of Oberlin College Dr. R. A. ROBERTS ofOhioWesleyan has been appointed to an assistantprofes­ University has been appointed to an assist­ sorship at Ohio Wesleyan University. ant professorship at Denison University. Dr. P. F. G. STANEK of the Univer­ Dr. W. L. ROOT of Massachusetts In­ sity of Chicago has accepted a position with stitute of Technology has been appointed the Institute for Defense Analyses, Prince­ to a professorship at the University of ton, New jersey. Michigan. Dr. T. H. STARKS oftheE.I.Du Pont Associate Professor D. ROSEN on De Nemours and Company, Wilmington, leave from Swarthmore College has been Delaware has been appointed to an asso­ appointed a Research Associate at the Uni­ ciate professorship at the Southern illinois versity of Glasgow, Scotland for the aca­ University. demic year 1961-1962. Dr. G. A. STENGLE of the University Dr. K. H. ROTH of the University of of Wisconsin has been appointed to an as­ Southern California has been appointed to sistant professorship at Lehigh University. an assistant professorship at Los Angeles Dr. j. K. STERRETT of the Office of State College. Research and Engineering has been ap­ Dr. j. F. SCHELL has beenappointed pointed to a position of Chief Scientist, to an assistant professorship at Florida Des/Communications and Electronics, at State University. Headquarters, North America Air Defense Professor E. V. SCHENKMAN of Command, Ent Air Force Base, Colorado Louisiana State University has been ap­ Springs, Colorado. pointed to a professorship at Purdue Uni­ Dr. j. H. STODDARD of the University versity. of Michigan has been appointed to an as­ Associate Professor 0. F. G. SCHIL­ sistant professor ship at the Michigan State LING of the University of Chicago has been University- Oakland, Rochester. appointed to a professorship at Purdue Dr. A. P. STOKES of RIAS, Baltimore, University. Maryland has been appointed to an assist­ Associate Professor S. SCHUSTER of ant professorship at The Catholic Univer-

417 sity of America. of Washington has been appointed to an Mr. G. W. STRUBLE of the University associate professorship at the University of Wisconsin has been appointed to an as­ of lllinois. sistant professorship at the University of Professor H.-c. WANG onleavefrom Oregon. Northwestern University has been appoint­ Assistant Professor K. SUPRUNO­ ed a Temporary Member at the Institute WICZ of the University of Idaho has been for Advanced Study for the academic year appointed to an associate professorship 1961-1962. at Utah State University. Dr. R. M. WAR TEN, Fellow, National Dr. L. SUYEMOTO has accepted a Science Foundation has accepted a position position as Staff Member with the MITRE as Staff Mathematician at the International Corporation, Bedford, Massachusetts. Business Machines Corporation, Owego, Dr. W. B. SWIFT of the University of New York. Wisconsin has accepted a position as Di­ Dr. P. G. WEBSTER has been ap­ rector of the Engineering Department, pointed to an associate professorship at C-E-1-R, Incorporated, Arlington, Vir­ Mississippi Southern College. ginia. Dr. J. G. WENDEL of the University Mr. M. W. SYVERSON of Stanford of Michigan has been appointed to a visit­ University has accepted a position as Sci­ ing professorship at the Matematisk ln­ entist at the Lockheed Missile and Space stitut, Aarhus Universitet, Aarhus, Den­ Corporation, Palo Alto, California. mark. Dr. S. TAKAHASHI of Tohoku Univer­ Assistant Professor A. M. WHITE of sity, Japan has been appointed a Research Stanford University has been appointed a Associate at Brandeis University. Member of the Mathematics Research Dr. E. THOMA of the University of Center at the University of Wisconsin for Munich has accepted a position as Consul­ the academic year 1961-1962. tant at the University of Heidelberg. Dr. P. M. WHITMAN of Johns Hopkins Dr. A. TRAMP US has accepted a po­ University has been appointed to a pro­ sition as Staff Member with General fessorship and Chairman of the Mathema­ Electric Company, Santa Barbara, Califor­ tics Department at Rhode Island College. nia. Associate Professor H. WIDOM on Dr. P. M. TREUENFELS of Brook­ leave from Cornell University has been haven National Laboratory, Upton, New appointed to a visiting associate profes­ York has accepted a position as Engineer­ sorship at the University of Chicago for Scientist at the Minneapolis-Honeywell the academic year 1961-1962. Regulator Company, Minneapolis, Minne­ Research Professor R. L. WILDER sota. on leave from the University of Michigan Dr. B. A. TROESCH of Space Tech­ has been appointed to a visiting research nology Laboratories, Incorporated, Los professorship at Florida State University Angeles, California has accepted a position during the 1961-1962 academic year. as Manager of the Computing Department, Dr. T. A. WILKE of Ohio State Uni­ Laboratories Division, Aerospace Cor­ versity has accepted a position as Mathe­ poration, Los Angeles, California. matician at the National Bureau of Stan­ Assistant Professor D. A.TRUMPLER dards, Washington, D. C. of Georgia Institute of Technology has been Mr. D. E. WILSON of the University appointed to an assistant professorship at of Kansas has been appointed to an assist­ Rutgers, The State University. ant professorship at the University of Associate Professor J.P. VAN ALST­ Hawaii. YNE of Hamilton College has been appoint­ Assistant Professor P. B. YALE of ed to an associate professorship at Wor­ Oberlin College has been appointed to an cester Polytechnic Institute. assistant professorship at Pomona Col­ Dr. N. R. WAGNER, Jr. has accepted lege. a position as Member of the Technical Mr. R. C. YEAMAN of Thiokol Chem­ Staff with the Bell Telephone Laboratories, ical Corporation has accepted a position as Whippany, New Jersey. Senior Development Engineer at the Her­ Mr. J. H. WALTER of the University cules Powder Company, Magna, Utah.

418 Dr. E. M. ZAUSTINSKY of Santa M. MORDUCHOW, Polytechnic Insti­ Barbara College has been appointed Lec­ tute of Brooklyn, to a professorship. turer at the University of California, M. 0. RABIN, the Hebrew University, Berkeley. to an associate professorship. Dr. M. ZELEN of the University of N. S. SCARRITT, Jr., Purdue Univer­ Maryland has accepted a position as Mem­ sity, to an assistant professorship. ber of the Mathematics Research Center, P. SHANAHAN, College of the Holy United States Army, University of Wiscon­ Cross, to an associate professorship. sin. R. D. STALLEY, Oregon State College, Dr. W. P. ZIEMER of Brown Univer­ to an associate professorship. sity has been appointed Lecturer at In­ W. M. STONE, Oregon State Univer­ diana University. sity, to a professorship. Dr. W. L. ZLOT has been appointed J. L. TRAUB, Wisconsin State Col­ to an assistantprofessorshipattheGradu­ lege, to an assistant professorship. ate School of Science, Yeshiva University. J. V. WHITTAKER, University of British Columbia, to an assistant profes­ The following promotions are announced: sorship. H~ WIDOM, Cornell University, to an S. ARMENTROUT, State University of associate professorship. Iowa, to an associate professorship. R. D. BOSWELL, Jr., Mississippi The following appointments to Instructor­ State University, to a professorship. ships are announced: C. S. COLEMAN, Harvey Mudd Col­ lege, to an associate professorship. Columbia University: E. G. EFFROS: H. J. CUR TIS, University of lllinois, Duke University: Dr. W. R. HARE; Hamp­ to an associate professorship. ton Institute: Mr. J. D. VINEYARD; D. F. DAWSON, North Texas State Harvard University: Mr. N. GREENLEAF; College, to an associate professorship. Lafayette College: Mr. N. R. GRABOIS; B. M. DWORK, Johns Hopkins Univer­ Lehigh University: Mr. RICHARD E. sity, to a professorship. HUGHS; Loyola College: Mr. J. T. JOHN,; E. J. ECKERT, Los Angeles State SON; Massachusetts Institute of Technol­ College, to an assistant professorship. ogy: G. J. MALTESE; North Carolina C. M. FULTON, University of Califor­ State College: D. L. GEORGE; Simmons nia, Davis, to a professorship. College: Mr. J. T. LEVERICH; Stanford M. GERSTENHABER, University of University: Mr. R. J. ROHIT; Tufts Uni­ Pennsylvania, to a professorship. versity: Mr. D. W. JONAH; University of Dr. A. J. GOLDMAN, National Bu­ Bridgeport: Mr. D. G. KUEHNER; Univer­ reau of Standards, Washington, D. C. to sity of California, Berkeley:Mr .D. SHALE, Chief of the Operations Research Section, Dr. L. A. SHEPP; University of Chicago: Applied Mathematics Division. Dr. M. GINSBURG, Mr. G. S. McCARTY, T. HAILPERIN, Lehigh University, to Jr.; University of Connecticut: Mr. H. M. a professorship. ROBERTS; University of Copenhagen: Mr. W. E. HARTNETT, CollegeoftheHoly E. R. TOMER; University of lllinois: Dr. Cross, to an associate professorship. P. A. TUCKER; University of Maryland, N. HEEREMA, Florida State Univer­ R. H. MOUNT JOY; University of Michigan: sity, to a professorship. Mr. R. C. O'NEILL; University of Wash­ D. R.HUGHES, of Michigan University, ington; D. S. NEWMAN. to an associate professorship. Deaths: J. F. JAKOBSEN, State University of Iowa, to an assistant professorship. Associate Professor H. G. BARONE of E. KRISTENSEN, Mathematics Insti­ Pennsylvania State University died on June tute, University of Aarhus, Denmark, to 22, 1961 at the age of 57. He had been a Section Manager. member of the Society for 17 years. L. H. LANGE, San Jose State College, Dr. M. MASON ofthe California Insti­ to an associate professorship. tute of Technology died on March 22, 1961 R. E. MacKENZIE, Indiana University, at the age of 84. He had been a member of to an associate professorship. the Society for 47 years.

419 MEMORANDA TO MEMBERS

PROCEEDINGS BERLINER MATHEMATISCHE EDITORIAL COMMITTEE GESELLSCHAFT e.V. At the Summer Meeting of the So­ The American Mathematical Society ciety at Stillwater, the Council authorized has entered into a reciprocity agreement increasing the size of the PROCEEDINGS with the Berliner Mathematische Gesell­ Editorial Committee to six members. schaft e. V. by which members of each may Therefore, on the ballot to be mailed to become members of the other by paying members on October 10, 1961, there will be ha.If the regular dues. The regular dues four positions to be filled on the PROCEED­ of members of the Berliner Mathematische INGS Editorial Committee, including two Gesellschaft are DM 7,--; therefore an new members to be added to the Committee. American Mathematical Society .member One of the new members on the committee would pay $1.00, Privileges of member­ is being elected for one year and the second ship include SITZUNGSBERICHTE DER new member is being elected for two years, BERLINER MATHEMATISCHEN GESELL• since according to the By-laws, "The term SCHAFT. Those members of the American of office in case of members of the Publi­ Mathematical Society wishing to take ad- cations Committees shall be three years, . vantage of this arrangement should write except that when the size of a Publications to Berliner Mathematische Gesellschaft Committee is changed, the Council may e. V., Berlin-Charlottenburg 2, Harden­ authorize the election of a member for a bergstrasse 34. term of less than three years." It is understood that members un­ der the reciprocity agreement spending time in the other country should pay the regular dues while they are there. TWO VOLUMES OF MATHEMATICAL REVIEWS IN 1962 DEUTSCHE MATHEMATIKER In 1962, two volumes of MATHE­ VEREINIGUNG MATICAL REVIEWS will be published, Volume 23 and Volume 24. Each volume The Report on Reciprocity Agree­ will consist of six issues of Part A, six ments in the April issue of the NOTICES issues of Part B, andtwoindexissues,one lists as privileges of membership under for Part A and one for Part B. Publication the Reciprocity Agreement subscriptions plans for 1962 provide for 4000 pages of to the JAHRESBERICHTE DER DEUT­ reviews compared to 1600 pages in 1960. SCHEN MATHEMATIKER VEREINIGUNG The subscription price per volume at $1.50, The price of $1.50 is for one will be unchanged in 1962. Thus, the list issue; the volume of four issues per year price for a 1962 subscription for two vol­ is $6.00 plus postage. umes is $100, and the price for members of the American Mathematical Society will UNION MATEMATICA ARGENTINA be $32. Subscriptions will not be accepted The Union Matematica Argentina for individual volumes. Subscriptions are requests that the payment of dues of $1.25 accepted on the basis of a calendar year for members of the AMS be made in only. Argentine pesos,

420 SUPPLEMENTARY PROGRAM- NO. 6

During the interval from july 8, 1961 through September 7, 1961 the papers listed below were accepted by the American Mathematical Society for presentation by title. Readers may wish to refer to page 713 of the November, 1960 issue {No. 49) of these NOTICES where it is explained in detail that the presentation of papers by title is now dissociated from meetings of the Society, Supplementary program No. 7 will cover the interval from September 8, 1961 through October 3, 1961. After each title on this program is an identifying number. The abstract of the paper will be found following the same number in the section on Abstracts of Contributed Papers in this issue of these NOTICES. { 1) On a class of Riemann surfaces. Pre­ Professor Nathaniel Coburn, Uni­ liminary report versity of Michigan (6l"T-Z54) Professor R. D. M. Accola, Brown (11) A generalization of Axer's theorem University {61 T-Z14) and some of its applications {Z) On gluing Riemann surfaces Professor Eckford Cohen, Univer­ Professor R. D. M. Accola, Brown sity of Tennessee (61 T-Z51) University {61T-Zl5) (lZ) Note on a class of affine transforma­ {3) Convergence properties of the spline tions fit Professor Eckford Cohen, Univer­ Dr. j. H. Ahlberg and Dr. E. N. sity of Tennessee (61 T-ZZO) Nilson, United Aircraft Corpora­ (13) On the mean parity of arithmetical tion, East Hartford, Connecticut functions (61 T-Z19) Professor Eckford Cohen, Univer­ (4) Hellinger integrals and absolute con­ sity of Tennessee (61 T-Z53) tinuity. Preliminary report (14) Arithmetical notes. IX. On the set of Dr. W. D. L. Appling, Duke Univer­ integers representable as a product of sity (61 T-Z50) a prime and a square ( 5) Concerning non-negative valued in­ Professor Eckford Cohen, Univer­ terval functions. Preliminary report sity of Tennessee (61 T-ZZ8) Dr. W. D. L. Appling, Duke Univer.,. (15) An extension of the Borsuk-Ulam sity (61 T-ZZ5) mapping theorem (6) A norm inequality for a "finite-sec- Professor P. E. Conner and Pro­ tion" Wiener-Hopf equation fessor E. E. Floyd, University of Professor G. E. Baxter, University Virginia (61 T-Z36) of Minnesota and Aarhus Universi­ (16) Actions of elementary Abelian p- tet, Aarhus, Denmark (61 T-Z17) groups (7) Perturbation theory and an atomic Professor P. E. Conner and Pro­ transition model fessor E. E. Floyd, University of Professor F. H. Brownell, Univer­ Virginia (61 T-Z37) sity of Washington (61 T-ZZ9) ( 17) A fluctuation theorem for cyclic ran­ (8) On the perturbation method for deter­ dom variables mining coupled bending and twisting Professor Meyer Dwass, Univer­ motion of a vibrating bar on an elastic sity. of Minnesota (61 T-ZZ7) foundation { 18) Some Markovian optimization prob­ Professor j. B. Butler, Jr., Univer­ lems sity of Arizona (61 T-ZZ6) Professor W. H. Fleming, Brown (9) On the representation of a.-complete University (61 T-Z30) lattices ( 19) A generalization of theorems of Professor C. C. Chang andProfes­ Kirzbraun and Minty sor Alfred Horn, University of Cali­ Professor Branko Griinbaum, Uni­ fornia, Los Angeles (61T-Z3Z) versity of Washington (61T-ZZ4) (10) The limiting speeds of characteris­ (ZO) Classifying inaccessible cardinals. tics in relaxation hydrodynamics Preliminary report

421 Mr. William Hanf and Professor of the Tricomi equation. I Dana Scott, University of California, Mr. Eberhard Lanckau, Konig stein, Berkeley (61 T-240) Germany (61 T-233) (21) Flows on certain solvmanifolds not of (Introduced by Professor Stefan type E Bergman) Professor Louis Auslander, Uni­ (33) A unified representation of solutions versity of Indiana and Dr. Frank of the Tricomi equation. II Hahn, Yale University (61 T-216) Mr. Eberhard Lanckau, Konigstein, (22) Unitary dilations which are orthogonal Germany (61 T-234) bilateral shift operators (Introduced by Professor Stefan Professor Israel Halperin, Queen's Bergman) University, Ontario, Canada (61 T- (34) The constructive existence of extre­ 243) mal conditions on the structure of (23) Unitary dilations for commutingcon- algebraic systems tractions Mr. A. A. Mullin, University of Professor Israel Halperin, Queen's Illinois (61 T-241) University, Ontario, Canada (6i T- (35) Error analysis of Householder's re­ 222) duction. Preliminary report (24) Sequences of elements in a product Mr. J. M. Ortega, Stanford Univer­ space sity (61 T-252) Professor Israel Halperin, Queen's (36) Lp-spaces and their adjoints University, Ontario, Canada (61 T- Professor Pasquale Porcelli, Loui­ 223) siana State University (61 T-218) (25) Transcendental elements in continu- (37) Recursive enumeration of formal ous rings theories Professor Israel Halperin, Queen's Professor M. B. Pour-El, Pennsyl­ University, Ontario, Canada (61 T- vania State University (61 T-255) 210) (38) Idempotent measures on a compact (26) On products of projections topological semigroup Professor Israel Halperin, Queen's Professor Murray Rosenblatt, University, Ontario, Canada (61 T- Brown University (61 T-209) 211) (39) A characterization of contractible (27) Von Neumann's coordinatization theo- compact Hausdorff spaces rem Professor C. W. Saalfrank, Lafay­ Professor Israel Halperin, Queen's ette College (61 T-221) University, Ontario, Canada (61 T- (40) Optimum-point formulas for oscula­ 212) tory and hyperosculatory interpola­ (28) Unitary dilations of contraction oper- tion ators Dr. H. E. Salzer, General Dynamics Professor Israel Halperin, Queen's Corporation, San Diego, California University, Ontario, Canada (61 T- (61 T-249) 213) (41) Automorphisms of Kolmogorov (29) Representations of cyclic groups in Professor Louis Sucheston, Uni­ rings of integers versity of Wisconsin, Milwaukee Professor Alex Heller and Profes­ (61 T-246) sor Irving Reiner, University of (42) Invariant subsp.aces of analytic func­ Illinois (61 T-245) tions (30) Simplexes and vector lattices Mr. Michael Voichick, Brown Uni­ Mr. D. G. Kendall, Magdalen Col­ versity (61 T-231) lege, Oxford, England(61T-242) (43) The location of the zeros of the de­ (31) Automorphism group of the group of rivative of a rational function, re­ recursive permutations visited Dr. C. F. Kent, Massachusetts In­ Professor j. L. Walsh, Harvard stitute of Technology (61 T-248) University (61 T-244) (32) A unified representation of solutions

422 ABSTRACTS OF CONTRIBUTED PAPERS

The October Meeting in Cambridge, Massachusetts October 28, 1961

583-1. RICHARD KRAFT, 662.6 Hillendale Road, Chevy Chase 15, Maryland. Analytic continuation of polyharmonic functions.

Considering harmonic functions in two independent variables, defined in a domain, D, abutting on a portion, CT, of the x-axis; and satisfying certain differential boundary conditions on CT, H. Lewy (Bull. Amer. Math. Soc. vol. 65 (1959) pp. 37-58) showed how to analytically continue these functions into the mirror image 5 of D. We generalize Lewy's method to polyharmonic functions which satisfy boundary conditions on CT of the type given by Agmon, Douglis, and Nirenberg, Comm. Pure Appl. Math. vol. 12. no. 4. Further generalizations to n-dimensions, spherical surfaces, and analytic arcs will be forthcoming. (Received june 2.6, 1961.)

583-2. I. HALPERIN, Queen's University at Kingston, Ontario, Canada. Elementary divisors in continuous rings.

It is a classical theorem that in the discrete ring of all n X n matrices over a commutative field, elements A and B are similar if and only if they have the same elementary divisors, A notion of multiplicity function is now introduced for all discrete or continuous rings which is equivalent to elementary divisors in a discrete ring. It is shown that in a continuous ring two elements are similar

if they have the same multiplicity function and no transcendental parts. (Received July 3, 1961.)

583-3. M. B. PRESTRUD, The RAND Corporation, 1700 Main Street, Santa Monica, California. Hierarchic algebra. II.

In a previous paper the hierarchic index k was defined by the relation a(k) = log a iterated k times, and an additive rule of composition was given for quantities and their indices as a(k) + b(l)

= c(m), abbreviated as (k,l:m). That concept made it possible to extend the family of operations containing addition, multiplication and exponentiation to include new related operations. This paper discusses a generalized rule of composition for the extended family -- [a(f)(p,q:r)b(g))(h)

= (a (k)(s,t:w)b(.[)](m), where f + p = k + s, g + q = l, + t, and h - r = m - w. This new relationship points out that (p,q:r) and (s,t:w) can be considered to be any operators of the family as well as addition. (Received July 17, 1961.)

583-4. WITHDRAWN.

583-5. P. C. FISCHER, Massachusetts Institute of Technology, Cambridge, Massachusetts. A note on bounded-truth-table reducibility. Preliminary report.

Let CT be the truth-table complete, simple set of Post (Bull. Amer, Math. Soc. vol. 50 (1944)

p. 304), and o-n be the Cartesian product of CT with itself ·n times. cr-2 is bounded-truth-table reducible

to a- but not many-one reducible to CT. Thus, bounded-truth-table reducibility differs from many-one

423 reducibility on the recursively enumerable, nonrecursive sets. If m f. n, then O"m and O"n lie in different many-one degrees, for any two positive integers m and n. (Received july 18, 1961.)

583-6. R. R. GOLDBERG and A. B. SIMON, Northwestern University, Evanston, Illinois. Weak* closure of functions of absolute value one.

Theorem. Let p be any positive, continuous, bounded measure on (- oo, oo). Then in L 00()1), the closure in the weak* topology of the set of all p-measurable functions of constant absolute value one is the whole unit ball of L 00(p). (Received August 15, 1961.)

583-7. E. R. SURYANARAYAN, University of Rhode Island, Kingston, Rhode Island. On the geometry of the vorticity tensor and the vorticity vector in the relativistic hydrodynamics.

Definitions: A fluid flow in relativity is totally irrotational if the vorticity tensor vanishes; and a flow is irrotational if the vorticity vector vanishes. Theorem 1. A fluid flow is totally irrotational if and only if the world-lines are geodesics. Theorem 2. In the case of irrotational flow, the vorticity tensor lies in the two-plane formed by the world-line vector and the principal normal vector of the world-line. Theorem 3. In the case of isentropic irrotational flows, the hypersurfaces, pressure = constant are orthogonal to the principal normal vector of the world-lines; in this case, all the thermodynamic quantities are constant along the world-lines. Theorem 4. For steady flows, the Bernoulli hypersurfaces contain the world-lines. They contain also the vorticity vectors, if the entropy does not vary along the vorticity vector. (Received August 15, 1961.)

583-8. A. A. SAGLE, University of Chicago, Chicago 37, Illinois. Simple Malcev algebras of characteristic zero.

Malcev algebras are a natural generalization of Lie algebras obtained by introducing the com­ mutator of two elements as a new multiplicative operator in an alternative algebra. A simple seven dimensional Malcev algebra A* obtained from the eight dimensional split Cayley-Dickson algebra has been discussed by the author (Abstract 582-5, Notices Amer. Math. Soc. vol. 8 (1961) p. 336). In this paper these previous results are considerably extended by the following Theorem. Let A be a finite dimensional simple non- Lie Malcev algebra over an algebraically closed field of characteristic zero. Furthermore assume A contains an element u such that right multiplication by u, Ru: a -au, is not a nilpotent linear transformation. Then A is isomorphic to A*. (Received August 23, 1961.)

583-9. J. A. MORRISON and D. EDELSOJ,\1, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey. Solution of the space charge problem for a pulsed Townsend discharge.

The nonlinear effect of space charge on the one-dimensional motion of ions (produced by a pulse of electrons passing rapidly through an electronegative gas), between the cathode and anode, is analyzed. The third order partial differential equation for the potential is integrated twice, to give a nonlinear first order partial differential equation which involves time derivatives of the position of the ion boundary. This equation is solved implicitly by means of Charpit's method, and application of

424 the initial and boundary conditions leads to a first order nonlinear ordinary differential equation for a parametric function. The current in the external circuit and the position of the ion boundary are expressed in terms of this parametric function. (Received August 24, 1961.)

583-10. C. F. KENT, 204 Tapawingo Road, S. W ., Vienna, Virginia. Algebraic structure of groups. of provable permutations.

This abstract presents results related to those in abstract 570-17 of these Notices, vol. 7, no. 3, There it was stated that the group of recursive permutations, and certain related groups, have parti­ cularly simple normal subgroup lattices, This abstract considers a group, P, of recursive permuta­ tions which can be proved to be permutations (p-permutations) within an axiomatic theory, T. Twill be assumed strong enough to formalize ordinary mathematical argument, to contain elementary num­ ber theory, and to be sound with respect to its formulas. A p-permutation is called oo if it moves infinitely many numbers and p - oo if this fact is provable in T. The principal results are: 1, Any normal subgroup of P which contains a p - oo permutation is all of P. 2, There are oo p-permutations which are not p - oo. (This follows from an unpublished construction of Kreisel.) 3. There are infinitely many nontrivial, incomparable, normal subgroups of P generated by oo but not p - oo p-permutations. 4. P does not possess a composition series. The assumption that T is sound for formulas of number theory is used in 2,3 and 4. (Received August 24, 196I.)

583-II. MURRAY ROSENBLATT, Division of Applied Mathematics, Brown University, Providence I2, Rhode Island. Asymptotic distribution of eigenvalues of Toeplitz matrices.

Let f be an integrable real-valued function on [- 17", 71'} and Rn (f) = fr j-k; j,k = I, ... ,nJ. rk =(l/2~;-eik:XfC~)dA. Here Rn(f) is then X n Toeplitz matrix corresponding to f, Let :>.j,n' j = I,. .. ,n, be the eigenvalues of Rn(f) and Nn the number of eigenvalues of Rn(f) less than or equal to A, It is shown that limn-+ooNn(i\)/n = (I/2'n")m f}ljf()l) ;;;; A} at every continuity point of the right hand side, where m is Lebesgue measure. This is an extension of a corresponding result of G. Szegii (see U. Grenander and G. Szego, Toeplitz forms and their applications, 1958) for essentially bounded functions. A corresponding result for an ns X ns block Toeplitz matrix corresponding to an s X s Hermitian function f E L is also derived, (Received August 24, I96l.)

583-12. J. C. SU, University of Virginia, Charlottesville, Virginia, Transformation groups on cohomology lens spaces and cohomology complex projective spaces,

This paper is the continuation of a previous paper of the author (On a problem of p. A. Smith, Bull. Amer, Math. Soc. vol. 67 (196I) pp. 422-424). Let Zp be the group of integers Z modulo an odd prime p. A compact Hausdorff space Y is said to be a cohomology lens (2n + I)-space mod p if the mod p cohomology structure of Y is the same as that of the lens (2n + I)-space mod p, Y is said to be a cohomology complex projective n-space if the integral cohomology ring of Y is the same as that of the complex projective n-space. Theorem I, If Zp acts on a cohomology lens (2n + I)-space mod p, then the fixed point set F is either empty or it has at most p components CI, ... ,Ck, k:;; p, where each Ci is a cohomology lense (2ni + I)-space mod p, i = l,..,,k, and ni + ... + nk = n - k + I,

425 Theorem 2. If a circle group s1 acts on a cohomology complex projective n-space Y such that Y has finite cohomology dimension (over Z) and the action has finite orbit structure, then the fixed point set F is nonempty, F has at most n + l components, say Cl'···•Ck, k;:;; n + l, where each Ci is a cohomology complex projective ni-space, i = l, ••• ,k, and n 1 + .•. + nk = n - k + l. (Received August 25, 1961.)

583-13. G. G. BILODEAU, Boston College, Chestnut Hill 67, Massachusetts. Increasing rates of convergence with summability methods.

There is a result of Knopp [!:nfinite series, Translation from 2nd German edition, Blackie and Son, p. 263] on the use of an Euler summability method to increase the rate of convergence of certain types of series. Results similar to this are obtained for other kinds of summability methods and for slightly more general types of series. The kind of series studies are of the form 2::_: 0(- l)nun where Z:k=oC:I6m-kuJd & M, m = 0,1, .•• , forM a constant and Ll.un = un- un+l• This is precisely the condition that is necessary and sufficient for fun}~ to be a Hausdorff moment sequence. This fact is the basic tool in the anlysis Special results are obtained for Hausdorff summability methods and, with a properly defined criterion, and "optimum" summability method is found arising from Chebyshev polynomials. (Received August 25, 1961.)

583-14. E. C. SCHLESINGER, Wesleyan University, Box 328, Wesleyan Station, Middletown,

Connecticut. ~transversals of simply connected regions.

A transversal c of a simply connected region G of the extended plane is a homeomorphic image of the open unit interval (0,1) in G with the property that the set-theoretic difference G - c has exactly two components and that these are simply connected. (Every crosscut of G is a transversal, but not conversely, since crosscuts need not have endpoints.) On the one-point compactification of G one can apply a theorem of Kerekjlfrt6 concerning the boundaries of the regions complementary to a finite system of Jordan curves on the 2-sphere. This is the essential step in the proof of the following

result. Th~orem. Let c and d be transversals of a simply connected region G whose boundary (in the extended plane) is nonempty. Then the boundary relative to G of any component of G - (cUd) is one of the following: (i) a Jordan curve, (ii) a transversal of G, or (iii) a pair of such transversals. (Received August 29, 1961.)

583-15. D. E. SPENCER, University of Connecticut, Storrs, Connecticut. Holor algebra.

The behavior of physical systems has been expressed mathematically in various ways using complex numbers, phasors, vectors, matrices, and tensors. A serious fault with all these representa­ tions is their lack of generality. The present paper attempts to eliminate this difficulty by considering a general theory of holors (from Greek ~ AoS, a whole). Associative and commutative properties are accepted for the addition of aU holors; but multiplication need not be associative nor commutative. In the past, products have been limited largely to those of linear associative algebras on the one hand and to tensor products on the other. This does not provide sufficient flexibility to meet the require­ ments of physics and engineering. The paper includes not only the usual products for holors of any

426 valence and any number of elements but also other products that may be of use. Products of holors may have the same valence as multiplier and multiplicand, or they may have a higher or a lower valence than the factors. This flexibility opens up a wide range of possible algebras. Methods of fitting the algebra to the application will be outlined. (Received August 29, 1961.)

583-16, ROSS FINNEY, Massachusetts Institute of Technology, 6 University Road, Cambridge 38, Massachusetts. Shrinking subsets of n-manifolds.

A pseudo-isotopy on a topological space X is a homotopy H of X X I onto X such that for each t < 1 the mapping Ht is a homeomorphism of X X tonto X. A subset B of ann-manifold X is cellular

if there exist closed n-cells Ci (i = 1,2, ... ) in X such that B = 0 Ci and such that Ci+1 C Int(Ci) for each i, Theorem. Let f be a mapping of ann-manifold X onto itself. If there exists a pseudo-isotopy

H ~X such that H 1 = f, then the set r 1(x) is cellular for each point x in X. The theorem may be applied to show (1) that a subset B of ann-manifold X can be shrunk to a point by a pseudo-isotopy on X if and only if B is cellular, and (2) that for a certain monotone mapping p of s3 onto itself there exists no pseudo-isotopy H on s 3 such that H1 = p. (A subset B of a space X is shrunk to a point by

the pseudo-isotopy H on X if H0 is the identity, if the set H1(B) is a point, and if a point x of X belongs to B whenever H 1(x) = H1(B).) (Received August 31, 1961.)

583-17. KURT BING, Rensselaer Polytechnic Institute, Troy, New York. A theorem on partially ordered sets. Preliminary report,

The dual of the following theorem on families of subsets of a set, partially ordered by inclusion, is implicit in Zermelo's so-called second proof of the well-ordering theorem. Let tJ be a function which assigns to every proper subset Y of a set X a subset'Y U fxJ. where x EX- Y, and such that ,P(X) = X. Let :1 be the intersection of all families JJ of subsets of X such that (i) 0 Ex"/, (ii) if y E Jf. then ,P(Y) E.#. (iii) if ?/ b,#. then U {Y: Y E ALJ E .11/. Then .!f is the smallest family.# and is totally ordered by inclusion. The order of Jf is a well-ordering, and :7 has X as greatest element. This theorem has been applied to partially ordered sets arid, by weakening conditions (i), (ii), and (iii), has been generalized in various directions by Hessenberg, Kuratowski, Bourbaki, Kneser, and others.

A theorem on partially ordered sets X, with associated functions ,P: X -x such that x § !l(x), and u: c{-X such that a is a family of nonempty subsets of X and u(Y) is an upper bound of Y, is proved. This theorem includes the results of Bourbaki and Kneser (and hence those of Zermelo, Hessenberg, and Kuratowski) as special cases. The problem of generalizing a related result of Vaughan is raised, (Received August 31, 1961.)

583-18. WITHDRAWN.

583-19. E. W. CHENEY and H. L. LOEB, 9508 Jellico, Northridge, California, On rational Tchebycheff approximation. Preliminary report.

i-1 For c = (c1, .... eN) let ~(c)= supa€xiiblf(x)- P(c,x)/Q(c,x)l where P(c,x) = "'n~i=lcix , Q(c,x) = L~+l cixi-n-1, and f is continuous. The following algorithm generates c 0, c 1, ... such that

427 b. (ck) J. 1::..* = infc ~(c). Choose c 0 on the unit cube, lc~ I -a 1, such that Q(c0 ,x) >- 0 on [a,b]. Given ck, define 8k = maxx {lf(x)Q(c,x) - P(c,x) I - A (ck)Q(c,x)}. Select ck+1 to minimize 8k on the unit cube. This is a "convex programming" problem. If for every c* which minimizes A(c), P(c*,x) and Q(c*,x) are relatively prime, then the sequence fckj. converges to one of these points c*. If the monomials xi are replaced by arbitrary continuous functions and the interval [a,b) by an arbitrary compactum the algorithm is still effective in the sense that ~(ck) converges to the infimum of Ll.(c) for c ranging over the set where the denominators are positive. The best rational approximation over an interval can thus be obtained by calculating best approximations over successively finer discrete sets. (Received September 5, 1961.)

583-20. L. j. COHEN, Applied Data Research, Incorporated, 759 State Highway 206, Princeton, New jersey. The roots of polynomial surfaces.

The function F (x,y) = 2: aijxiyi is considered as a surface and interest is centered on the set of all points of intersection of F(x,y) with the x,y plane. Unique decomposition into the form F(x,y)

= P(y) Q(x) G1(x,y) ••• is demonstrated where P and Q are polynomials and G 1 ••. Gn are surfaces having no further reductions. Theorem. Let Si = (yiF(xi•Y) = 0); A·= (yiP(y) = 0); the degree of F(x,y) in x be u; the degree of Q(x) be m. Then A= S1 S2 ••• Su-m+l· The roots of F(x,y) are separated into three categories, two of which deal with isolated points (xo, Yo) such that F (xo, Yo) = 0. The remaining category contains continuous functions F(x,y) = 0. Certain properties of these curves are discussed with a view toward the simultaneous solution of F1(x,y) = 0 and F2(x,y) = 0. For this problem, the maximum number of isolated solutions is determined, and an algorithm for numerical solution is defined. These techniques have been successfully applied to the numerical solution of certain homo­ geneous integral equations arising in molecular physics. (Received September 7, 1961.)

583-21. W. F. AMES and HERNANDO DE LA CUESTA, Department of Mechanical Engineering, University of Delaware, Newark, Delaware. Solutions of a class of linear partial differential equations with variable coefficients.

The equations of the title are the "energy" type equations of Fluid Mechanics (in vector form) &Tf8t + V• VT = K v 2T where K is constant and V = - VIII. It is shown that this equation reduces to a partial differential equation of the Schrodinger type under the transformation T = h exp [- !li/2K). This equation is separable in any allowable orthogonal coordinate system provided that Ill is a time independent potential and the quantity (VIIi) + Vl¢ is "separate". A function f(9 1, e2, e3) is said to be separate if it can be expressed as Li xi(9i)/g11 where Xi is some function depending on at most one ei and g11 is the covariant component of the metric tensor corresponding to ei, i = 1,2,3. The general form of the potential Ill such that the quantity ( V!li)2 + Vl¢ is separate is established in rec­ tangular and polar coordinates. Applications lie in the area of heat conduction and mass transfer in fluids, and heat conduction and diffusion in nonhomogeneous materials (with Ill =.lnp, where p = function of position only). (Received September 7, 1961.)

428 583-22. BERNARD EPSTEIN, Graduate School of Science, Yeshiva University, New York 33, New York. A remark concerning the solution of the Dirichlet problem by finite differences.

In their classic paper Uber die partiellen differenzengleichungen ••• (Math. Ann. vol. 100 (1928)), Courant, Friedrichs, and Lewy show that the most obvious discrete analogue of the Dirichlet problem furnishes, as the mesh-width approaches zero, a harmonic function which satisfies the boundary con­ dition in the sense that a certain quadratic integral, extended over a "boundary strip", is O(e2), where e denotes the width of the strip. In the present paper it is shown that the argument leading to the above result can be extended so as to prove that the boundary values are assumed in the more conventional sense. First the result O(s2) is used to show that the integral in question is, in fact o(e2);then the o(e Z, condition enables one to use a suitable approximation technique from which the desired result follows. (Received September 7, 1961.)

583-23. JOSEPH GILLIS, Institute for Space Studies, 475 Riverside Drive, New York 27, New

York. Triple product integrals of Laguerre functions.

With the usual definition of the Laguerre function, Ar(x), it has been conjectured that Ar,t

(- l),'l'lf(}lr(x>]2 >-t 0 for r 1:; t ii; 0. This conjecture is proved for the special cases t = 1,2,3,4. For other values oft the integral in question has been evaluated asymptotically for large r, and it is shown that for each t there exists a function R(t) such that Ar,t > 0 for r s:; R(t). Moreover it is shown that, for large r, Ar,t = O(r -1/2) or O(r-312) according as t is even or odd. (Received September 7, 1961.)

583-24. ANTHONY RALSTON, Stevens Institute of Technology, Hoboken, New jersey. On the equivalence of two theorems on the location of roots of polynomials.

The Hurwitz-Routh criterion for the zeros of a polynomial to lie in the left half plane and the Schur-Cohn criterion for the zeros to lie within the unit circle must be formally equivalent by virtue of the mapping relating the two regions. A direct proof of this equivalence is given starting from the familiar matrix formulations of the two criteria. As a corollary of this proof a new formulation of the Hurwitz-Routh criterion is obtained which is useful for computation. (Received September 8, 1961.)

583-25. LOUIS de BRANGES, 778 Upper Gulph Road, Wayne, Pennsylvania. The existence of entire functions of zero type bounded on a sequence of real points.

New results are obtained on a problem of Levinson, Gap and density theorems, Amer. Math. Soc. Colloquium Publications, vol. 26, 1940, Chapters VIII and IX. For example, if (xn) is an in­ creasing sequence of positive numbers ·such that 2:xn - 2< oo and L,

429 583-26, LEONARD GROSS, White Hall, Cornell University, Ithaca, New York. Harmonic analysis on a Hilbert space,

The following two theorems generalize the Bochner representation theorem and Levy continuity theorem to a real separable Hilbert space H. Theorem 1, A positive definite function vJ on H is the Fourier transform of a positive finite measure on H if and only if vJ is continuous at the origin in the topology T whose basic open neighborhoods are of the form U A (xo) = { x: IIA(x - xo) 11 < 1} where A runs over Hilbert-Schmidt operators, Theorem 2, Let Jln be a sequence of probability measures on H with respective characteristic functionals vJn. Let vJ be a uniformly T continuous function on H such that vJ(O) = 1, If fn converges weakly to a probability measure Jl (i.e,,JHG(s)dJln(s) ~/H G(s)d,u(s) for all strongly continuous bounded functions G) whose characteristic functional is vJ then the random variables vJ;; (defined with respect to the normal distribution) converge in probability to vJ-. Conversely if the vJ;:; converge to {6- in probability then the Jln converge weakly to a measure Jl with characteristic functional vJ, (Received September 11, 1961.)

583-27, A, T. BRAUER, University of North Carolina, Chapel Hill, North Carolina, A new proof of theorems of Perron and Frobenius on non-negative matrices. II.

In part I of this paper [Duke Math. J, vol, 24 (1957) pp. 367-378] a new proof of the theorems of Perron and Frobenius on positive matrices was given and some of their results were improved, Moreover, a method was obtained to compute the greatest positive characteristic root of a positive matrix and a characteristic vector belonging to this root as exactly as needed. The same method can be used directly for non-negative matrices, But the proofs are more difficult than for positive matrices. Therefore the results were first published for positive matrices and the publication for the case of non-negative matrices was postponed, In this paper a new method for non-negative matrices is given, The proofs become much simpler and the numerical computation of the greatest root requires a smaller number of steps. (Received September 11, 1961.)

583-28. C. E. ROBINSON, University of Alabama, Box 4153, University, Alabama. Properties of certain function spaces.

Let n be the class of all continuous functions on an arbitrary topological space X into a fully normal space Y. The class of all open covers of Y is used to determine a set of relations in n x n by taking an element of n X n to be in an a-relation if and only if its coordinates are a-near. This forms a base for a uniformity on fl. On replacing a-near by a-homotopic one can define another uniformity on fl. Necessary and sufficient conditions on Y are given in order that each of these uniformities on n be pseudo-metrizable. An exponential law is proved and applications are made to show certain connectivity, retraction, and extension properties of n when topologized by these uniformities. (Received September 11, 1961.)

430 583-29. STANLEY KAPLAN, Institute of Mathematical Sciences, New York University, 25 Waverly Place, New York 3, New York. On the global existence of solutions of nonlinear parabolic equations.

Global a priori bounds are obtained for solutions u(x,t) of the first mixed initial-boundary value problem for the equation 8u/Bt - Lu = F(x,t,u,\]u), where L is uniformly elliptic: ifF = F (x,t,u,w) is dominated, for any fixed w 0 = (w~ , .•• ,w~) and for 0:;; t 'l!j T, by a function HT(u) such thatj+00du/HT(u) = + oo, then an a priori bound for luI is obtained by comparing u with the solution of an ordinary differential equation, via the maximum principle. If F grows no worse than linearly in lw 1, such a bound for luI implies one for IVu 1, and hence an existence theorem. Conversely, by considering the ordinary differential inequality satisfied by the "Fourier coefficient" of u with respect to the first eigenfunction of L with null boundary data, the following is obtained, for self-adjoint L: If H(u) is convex and positive, and /too du/H(u)< + oo, then any solution of the inequality 8u/8t - Lu "E;. H(u) with sufficiently large initial-boundary values, must become infinite as t approaches some finite value. (Received September 11, 1961.)

583-30. R. V. CHACON, Division of A1ppplied Mathematics, Brown University, Providence 12, Rhode Island. Convergence of operator averages.

Let (S,.:1; u) be a IT-finite measure space and L1 the Banach space of complex-valued integrable functions on S. Let {Pnl be a sequence of non-negative and finite valued measurable functions and let

T be a linear transformation on L1 such that (i) liT II~ 1, (ii) If lfi li Pn a.e. then ITfl:;;; Pnt1 a.e., for each fin L 1. The main result is the following: Theorem. _!ff is in L1 then the limit limn-+ooL~=O~f/2:~=0Pk exists and is finite almost everywhere on the set fs:~oPk(s) > 0}. This theorem has as direct consequences the ergodic theorem of Hopf-Dunford and Schwartz as well as an ergodic theorem given by D. S. Ornstein and the author, and thus includes the classical ergodic theorems as well. (Received July 17, 1961.)

583-31. JOSEPH LEHNER, University of Pennsylvania, Philadelphia 4, Pennsylvania. Magnitude of the Fourier coefficients of automorphic forms of negative dimension.

Recently Petersson (Arch. Math. vol. 9 (1958) pp. 176-182) gave estimates for the Fourier coefficients of regular automorphic forms on H-groups (horocyclic groups, defined on the upper half­ plane, which possess translations and a fundamental region bounded by a finite number of sides). Denoting the dimension of the form by -r, the Fourier coefficients by bm, he showed that (1) bm= O(mr/2) for 0 < r < 2, provided r i' 2-h, h = 0,1,2, ..• ; (2) bm = O(mr/2 logr/Zm) when r = z-h. In the present note the author proves that (2) is superfluous; in other words (1) holds for all r in 0 < r < 2. The method is elementary and uses the author's variant of the circle method (Michigan Math. J. vol. 6 (1959) pp. 173-193). (Received September 12, 1961.)

431 583-3Z. GILBERT BAUMSLAG, New York University, Z5 Waverly Place, New York University, New York 3, New York. On a theorem of Mal'cev.

The following theorem will be proved: Every locally nilpotent group can be embedded in a locally nilpotent divisible group. This generalises a theorem of Mal'cev (for torsion-free locally nilpotent groups) and a theorem of the author (for torsion locally nilpotent groups). The proof depends on the fact that the intersection of subgroups of finite index in a finitely generated nilpotent group is the identity, and on Euclid's algorithm. (Received September lZ, 1961.)

583-33. T. T. TANIMOTO, 11 Galen Street, Watertown 7Z, Massachusetts. A class of exponential distributions and their associated Minkowski geometries.

When multicorrelation functions are considered, one is led to ·an obvious generalization of the normal probability density function, viz. P(X) = k exp (-(1/p!)IAij ••. J!CiXj ••• Xrl> where Aij ••• r is a multilinear operator acting on p vectors such that the coefficients of the multilinear form are essen­ tially the multicorrelation functions relating the corresponding variables. The multilinear form de­ fines a metric space whose triangular inequality is exactly Minkowski's inequality. Angles in these geometries are defined by the analogue of Schwartz's inequality. In the one dimensional case, the family is simply P(X) = k exp (-(1/p!)lx!P). Note that p = 1 yields the Laplace distribution, p = Z the Gaussian case and p = oo the uniform distribution where the normalizing factor k =r (1/p)/p. Of all the probability density functions which exist, the ones yielding the maximum entropy when the pth moment is given are exactly those given above. (Received September 13, 1961.)

583-34. S. S. HOLLAND, Jr. Boston College, Chestnut Hill 67, Massachusetts. Analysis of weakly modular lattices. Preliminary report.

If the orthocomplemented lattice L satisfies: a ;;o. b ~ b = a"(b11a.L), Lis called weakly modu­ lar. The following results are generalizations to weakly modular lattices of some portions of Continuous Geometry. All undefined notation is that ofF. Maeda, Kontinuierliche Geometrien or j.

Sci. Hiroshima University Ser. A vol. Z3 (1959) pp. 161-163. If there exists x in L such that a~x = bvx = avb, aAx = bAx = 0, a and bare called strongly perspective (arLb). Theorem. Suppose L is weakly modular. (1) If (aa; a E A), (ba• a E A) are two families of elements from L with the same indexing set A (if A is infinite assume L complete) such that (aav ba)l(ap vbp) for a i fl• and aa...!. ba all a in A, then Vaa ~ Vba· (Z) Strong perspectivity implies perspectivity, but the converse holds if and only if L is modular. ( (1) generalizes Lemma Z9 in Kaplansky, Ann of Math. vol. 61 (1955) p. 5Z4.) Write aTb if a.Lb and all six distributive laws hold for (a,b,x) whatever is x in L. Theorem. In a weakly modular lattice the following are equivalent: (1) aTb, (Z) avb, (3) x = (avx) A (b vx) for all x in L,(4) a..L,b and any eleme.nt x, majorizing no elements (except.O) compatible with a, is compatible with b, (5) p ;11! a, q =' b, and p subperspective to q imply one of p, q = 0. In particular a V b 4=+ bv a and also (4) and (5) are symmetric. (Received September 13, 1961.)

432 583-35, WILLIAM BROWDER, Cornell University, Ithaca, New York. Remark on the Poincar6 duality theorem.

Let X be a space with Poincar6 duality (such as a compact orientable manifold). Then Poincar6 duality holds for the Bockstein spectral sequences of X (for definitions see Ann. of Math, vol. 74

(1961) pp. 24-51). If dim X= Zm + 1, one may then define a nonsingular inner product A~ on I~

=image dr C E~)l (the cohomology Bockstein spectral sequence mod p), by A~(x,y) =(d;1x) uy. If m is odd A~ is symmetric, If m is even A~ is symplectic (A~(x,x) = 0) for p # 2 or p = 2 and r > 1. A~ is symplect.ic if and only if s:: If--+HZm+l(X;Z2) is nonzero. As a consequence one gets: Theorem 1. Let X be a space with Poincar6 duality, dim X= Zm + 1, m even. Then either (1) Hm(X) = F + T + T

or (2) Hm(X) = F + T + T + z 2, where F is free and Tis a torsion group. In case (2) the Stiefel­ Whitney class of X W -F 0. Theorem 2, Let X be as in Theorem 1 and in addition assume H ~X;Zz)

= 0 fori< m, Then (2) holds if and only if m = 2 and w3 # 0, (Received September 13, 1961,)

583-36. T. S. MOTZKIN, University of California, Los Angeles 24, California, Translative endovector sets.

For a point setS, let I\ be the "set of endovectors" >. = f )\1, .... ~r}• 1 ~ r <= oo, such that s 1, ... ,sr E S = 2: >.ksk E S; an endovector set with L:_~k = 1 is "translative". For an integer p #0,1, let 1\ be the smallest endovector set containing f..u· 1 - pJ; then A= {A ; every 1\k: 0,1 •)1 or 1 - Jl ·mod p - p 2} = uJ\ j• where J\o is the set of rows of the unit matrix, and ~+1 = {p'A + (1 - p))\'; )i.~· E Aj'\. The max of 1/\11+ IAzl for (A 1·Az) E Aj is IZp- qj. A set of 1\ with r = 2 is that of all such /) in some A iff the set of the I) 1 is the union of a nonempty multiplicative semigroup of idem­ potent residue classes 'Y'(mod some :V) containing 1 - r with y, The set o'f all r in A is always fl J, t 1,2, ... J. or p,3,5,. ..J; the last only for a single A and characteristic 2, The smallest set of all r whose )\ determine the others can have an arbitrary, or no, maximum. (Received September 13, 1961.)

583-37. M. E. MAHOWALD, Syracuse University, Syracuse 10, New York. The third and fourth obstruction to cross sections in sphere bundles.

Let p:E ._a be a k-sphere bundle. By a bundle cohomology operation is meant a cohomology operation defined on a natural subset of cohomology of the bundle space E which is natural with respect to bundle maps. A theory of secondary and tertiary bundle operations is developed using the method of universal examples. For example, the universal example for a typical secondary operation is a fiber

space over Gk+l' the Grassmann manifold of k + 1 planes in R 00, with the fiber being a space of the same type Adams, (On the nonexistence of elements of Hopf invariant one, Ann. of Math. vol. 72 (1960) pp, 20-109), used as universal examples for secondary operations. In terms of these bundle cohomo­ logy operations the third and fourth obstruction in sphere bundles for the stable range are described, The equation (Sqz + p*cvz • )(Sq2 + p*«.!z • )a(f) = 0 is satisfied in every bundle space with vanishing Euler class, (

tions yields a tertiary operation and the z 2 component of the fourth obstruction is given by such a tertiary operation. (Received September 13, 1961.)

433 583-38. B. A. FUSARO, University of Oklahoma, Norman, Oklahoma. A reduction to an integral equation of an inhomogeneous, singular Cauchy problem in harmonic space.

The Euler-Poissotf-Darboux (EPD) problem (l) Utt + (k/t)Ut - AU= 0; (2) Ut(O,x 1, ••• ,xn) = 0, U(O,x1•···•xn) = f(x1, •.• ,xn>• studied extensively by A. Weinstein and others, has been generalized as follows. Equation (1) has been replaced by (1') U tt+ b(t; k) Ut- b.2u = F, where !::.2 is the Beltrami second differential parameter for a harmonic space, and b(t;k) = k/t + r(t), r is a regular function, its particular form depending on the space in question (in a Euclidean space, r = 0). An integral equation to determine the unknown is found via a method of A. Douglis (1958) for the problem (1' ,2), with EPD parameter k > 2, and can be solved by a method involving iteration, fort sufficiently small. The problem is among those solved by Bureau (1955), using Hadamard's method. (Received September 13, 1961.)

583-39. E. C. POSNER and H. C. RUMSEY, Building 185, 4800 Oak Grove Drive, Pasadena, California. Polynomials dividing infinitely many trinomials. Preliminary report.

Which polynomials with rational coefficients divide infinitely many different trinomials with rational coefficients and nonzero constant term? lf a and b are rational numbers and r is a positive integer, then any rational factor of x 2r - axr - b clearly does. In fact, if u and v are any positive integers then xur, xvr and 1 are linearly dependent over the rationals. It is proved that every poly­ nomial which divides infinitely many different trinomials is of this type, with the possible exception of a rather special class of cubics for which the problem is still unsolved. The principle method used is to reduce the problem to known results in approximability of algebraic numbers (Liouville-Roth type) and of their arguments (Gelfand). (Received September 13, 1961.)

583-40. F. L. AL T, National Bureau of Standards, Washington 25, D. C. Pattern recognition by moments.

It is desired to devise an algorithm for assigning plane point sets to classes of such sets; e.g., for deciding whether a given set "looks like" the letter A. It is proposed to accomplish this by using certain combinations of moments; specifically, the moments of first and second order of a set are used to define an affine transformation which reduces the set to a standard form, and the higher-order moments of the standardized set are used in assigning it to one of a number of given classes. This reduces the problem from classification in the space of all plane point sets to the space of moments, i.e. of sequences of real numbers. Numerical experiments have been carried out using moments up to the sixth order and replacing the integrals occurring in the definition of moments by sums over subsets of a fixed finite grid of points. In these experiments, the only algorithms used for classifying are successions of inequalities involving one moment at a time, i.e. defining regions in moment space bounded by hyperplanes parallel to the coordinate axes. Despite these limitations it has been possible e.g. to differentiate among all alphabetic characters. (Received September 14, 1961.)

583-41. J. R. BROWN, University of Massachusetts, Amherst, Massachusetts. The fundamental operator of a stationary Markov process. Preliminary report.

The concept of a fundamental operator (matrix) for a finite Markov chain introduced by Kemeny

434 and Snell Jrinite Markov chains, 1959, and Teor. Veroyatnost. i Primenen. (1960)) has been used by them to compute various significant quantities related to a Markov chain, e.g. the mean and variance of the first passage times. This concept is here extended to a general state space. Let Tt(Tn) be the semi-group of operators associated with a stationary Markov process and let A be the limiting opera­ tor. Define the fundamental operator by Z = ro

583-42. R. H. CAMERON, University of Minnesota, 119 Folwell Hall, Minneapolis 14, Minnesota. Note on analytic continuation of functions of two variables.

Let f(z,w) be analytic in w throughout lwl < R for each real z between-Rand Rand vice versa; i.e., let f(z,w) be analytic in z throughout lzl < R for each real w between-Rand R. Further, let f be continuous in the two three-dimensional regions just described. It is shown in this paper that f has an analytic continuation to a four-dimensional region where it is analytic in both z and w. (Received September 14. 1961.)

583-43. R. M. KRAUSE, 10 West 35th Street, Chicago, Illinois, Technology Center. Channels which transmit letters of unequal duration.

We consider a I).oiseless discrete channel which transmits letters of unequal durations dl'··· .~, e.g. dots and dashes. Theorem 1. The following three definitions for the capacity C are equivalent.

{1) C = max - LPi log2 Pi /,L:Pj_ di over all probability distribution (p1 , •.• ,pr ). (2) C is the unique real root of 2:::2 -Cdi = 1. (3) C = lim sups~oo (1/s) log m(s) where m(s) is the number of distinct sequences

of letters having total duration s. (lim {1/s) log m (s) is not defined in general, e.g. for d 1 = 1, d2 = 2.) Theorem 2. If a uniquely decipherable encoding has code-durations Dl'···•DR, 2::2 -C 0 i.;;;; 1. The proof follows McMillan's for the equal letter-duration case, again avoiding information theoretic con­ cepts. However McMillan's converse is not true in the general case, as is shown by an example. Finally, a code construction is given, patterned on Shannon's for the binary channel, for matching a specified message space to a channel of the above type. It takes a particularly simple form for the

dot-dash channel with d 1 = 1, d2 = 2. (Received September 14, 1961.)

583-44. ROBERT OSSERMAN, Harvard University, Cambridge 38, Massachusetts. Geometric properties of polynomial surfaces and hypersurfaces.

An elementary argument proves the following: ..!;:!_ P(x,y) be a polynomial of degree n. LetS be the surface z = P(x,y) defined over the whole x,y-plane. Let K be the Gauss curvature and dS the area element on S. Then {1)./s IK IdA;;;; 217'"(n - 1) 2. An analogous result holds for polynomials in any number of variables. By virtue of the criterion of Blanc and Fiala, an immediate consequence is the result: Every polynomial surface z = P(x,y) is conformally equivalent to the plane. This theorem was

435 proved recently by H. Huber using other methods. A number of further statements can be made re­ garding the right-hand side of inequality (1). (Received September 14, 1961.)

583-45. WITHDRAWN

583-46. E. G. EFFROS, 100 Maple Drive, Great Neck, New York. A decomposition for repre­ sentations of separable C*- algebras. Preliminary report.

Let L be a separable Hilbert space representation of a separable C*-algebra A with identity. Lis homogeneous if for all projections E in L(A)' (prime indicates commutant), kernel LE =kernel L. Factor representations are homogeneous. Two homogeneous representations M,N are strongly dis­ joint if kernel M t kernel N. If J is a norm-closed ideal in A, there exists a unique central projection E with L(J) = L(A)E (bar indicates weak closure), The ideal center of L is the abelian ring of opera­ tors generated by such projections. Given a decomposition L = JL x dm(x) over a standard finite measure space (X,m), the L x are almost all homogeneous (resp. homogeneous and strongly disJoint) if and only if the range of the corresponding projection-valued measure contains (resp, coincides with) the projections in the ideal center, If the Lx are almost all factor representations and strongly dis­ joint, the range coincides with the central projections of L(A). If L is homogeneous, kernel L is primitive. The hull-kernel topology generates an analytic Borel structure in the primitive ideal space. Ideal central decompositions may be regarded as integrals of cross-sections to the Borel map L--+ ker L of the concrete homogeneous representations into the primitive ideal space. (Received September 13, 1961.)

436 Abstracts Presented by Title

61T-209. Murray Rosenblatt, Division of Applied Mathematics, Brown University, Providence 12, Rhode Island. Idempotent measures on a compact topological semigroup.

Let S be a compact Hausdorff semigroup. It is shown that p is a regular idempotent finite measure on S if and only if p is a ji measure and the support of p is a completely simple subsemi­ group H of S (seeM. Rosenblatt, J. Math. Mech. vol. 9 (1960) pp. 293-305, for the definition of a ji measure). The incomplete proof of a related result (Theorem 4) in H. S. Collins' Abstract 576-27 (Notices Amer. Math. Soc. vol. 7 (1960) p. 896) was based on a too generous appraisal of Theorem 5 in the above cited paper of M. Rosenblatt. (Received July 12, 1961.)

61 T-210. Israel Halperin, Queen's University, Kingston, Ontario, Canada. Transcendental elements in continuous rings.

If Cf is a continuous ring (in the sense of von Neumann) then x inC? is called transcendental if p(x) has a reciprocal in-'f for every polynomial p(t) = {fi + Zm-1tm- 1 + ••• + ZQ With m i; 1 and all Zi in the centre of t:if. Von Neumann asserted that transcendental elements exist in every continuous ring but left no clue as to how he showed this. A construction is now given which verifies his assertion. (Received July 14, 1961.)

61T-211. Israel Halperin, Queen's University, Kingston, Ontario, Canada. On products of projections.

Let T denote the product T 1 ••• T n of bounded linear operators on a Hilbert space. Let M denote the subspace of all x in H for which Tix = x for i = 1, ••• ,n. It is shown that as n -oo, Tn converges strongly to the projection onto M if each Ti is aprojection. For the case n = 2 this result was found first by von Neumann with a very elementary proof but his proof does not extend to n :>- 2. The proof given here (which is different but also very elementary) is valid for all n and gives a certain result for operators which need not be projections and which act on arbitrary Banach spaces. (Received July 14, 1961.)

61T-212. Israel Halperin, Queen's University, Kingston, Ontario, Canada. Von Neumann's coordinatization theorem.

Von Neumann's celebrated coordinatization theorem asserts: for every complemented modular lattice L possessing a homogeneous basis of order n s; 4 there is a regular ring .q' such that L is isomorphic to the lattice of finitely generated left submodules of qn. Von Neumann first constructs fi( as a ring of L -numbers. Then he verifies the coordinatization theorem by an extension procedure using some brilliant technique. It is now shown that once the ring f1> is constructed, the rest of the theorem can be established by some simple lattice calculations. This simplification includes various extensions of von Neumann's theorem to the case n = 3 (Desarguesian, Moufang, Baer) and also the recent extension by Jonsson to the case that L possesses a large partial frame. (Received July 14, 1961.)

437 61T-Z13. Israel Halperin, Queen's University, Kingston, Ontario, Canada. Unitary dilations of contraction operators.

Let U denote the Sz.-Nagy unitary dilation of a contraction operator T on a Hilbert space H. When is U a bilateral shift operator? Schreiber and Sz.-Nagy showed it is sufficient that T have bound less than 1. De Bruijn proved that it is sufficient that UT~lx II__.,. 0 as n- oo for each x. It is now shown more generally, that it is sufficient if H can be decomposed into L tB M (9 R with (i) TL C L fB M and T(M EB R) C Rand (ii) II(TPL)n xll- O,II(T*PR)nxll-.. 0 as n-oo for each x E H. (PL, PR denote the projections onto L,R respectively.) (Received July 14, 1961.)

61T-Z14. R. D. M. Accola, Brown University, Providence, Rhode Island. On a class of Riemann surfaces. Preliminary report.

Let r h be the Hilbert space of square integrable harmonic differentials on a Riemann surface. Let r he be the exact differentials and let rho= r he *J. • Sarlo's modular criterion for an AD­ removable boundary (L. Sario, Modular criteria on Riemann surfaces, Duke Math. J. vol. ZO (1953)) is sufficient for the condition r he.J.. rhe*• i.e., The C rho· Let 0uQ stand for the class of Riemann surfaces for which r he C rho. Then OuK C OuQ C 0 AD• Examples show that the inclusions are strict and the class OuQ is preserved under quasi-conformal mapping. Let so88(S08 D) be the class of bordered Riemann surfaces for which the maximum principle holds for bounded (Dirichlet bounded) harmonic functions with respect to the border. If WE so88(S08 D) then the double of W is of class OHQ () 0 AB (OHQ). (Received July 14, 1961.)

61 T-Z15. R. D. M. Accola, Brown University, Providence 1Z, Rhode Island. On gluing Riemann surfaces.

Continue notation of the preceding abstract. Let SOg be the class of bordered Riemann sur­ faces whose doubles are parabolic. In the following theorems we consider a (possibly bordered)

Riemann surface W. 8W = A0 U A1 (possibly = {1). W = W0 U W1' W i a bordered surface and BWi = Ai U ai' i = 0,1. All subsurfaces are regularly embedded. Theorem 1: Assume the extremal distance between w - Wo and w- w1 is positive. (a) Assume Wo. w1 E SOuD· lf aw = !I then W E Og. lf OW i !I the W E SOHD· (b) Assume WO• W1 E SOg and aw i {1. Then W E SOg. Theorem Z: Assume W0, W1 E SOHB' Ao i !1. Let u be harmonic on W0 , u = 1 on a0, u = 0 on A0• Assume suppea1 u(p) < 1. Then WE SOHB· Corollary: Ao,A1 i !1. u harmonic on W, u = 0 on A0 , u = 1 on A1• c and d are real numbers such that 0 < c < d < 1. W0 = {o :::0 u ;ad}. W 1 = {c ~ u ~ 1}. (a) If W0 , W 1 €: so88 then W E- SOHB· (b) lf w0 ,W 1 E SOg(SOHD) and if u is Dirichlet bounded then W E S<>g (SOuD>· (Received July 14, 1961.)

61T-Zl6. Louis Auslander and Frank Hahn, Yale University, New Haven, Connecticut. Flows on certain solvmanifolds not of type E.

The terms solvable (nilpotent) Lie group will mean a connected, simply connected solvable (nilpotent) Lie group. lf G is a Lie group and !6: T---? G is a one parameter subgroup and H is a closed subgroup then the flow (G/H, Jll(t)) induced by G on the homogeneous space G/H is defined by

438 gH--. (gH)t= !ll(t)gH. L. Auslander has shown [Amer. J. Math. (1960) pp. 653-660) that if Sis a solvable Lie group with a discrete nilpotent subgroup D such that S/D is compact then S C C•N, where C is torus group, N is the unique nilpotent ".Lie group containing D and dot is the semi-direct product. If S C C •N then we say that a one parameter subgroup !II: T--+ S is in general position if its projection on C is dense. A Lie group is said to be of type E if the exponential map is onto. LetS be a solvable

Lie group with a discrete nilpotent subgroup D such that S/D is compact, and let !II: T ~ S be a one parameter subgroup, then the following statements are true. (1) S is not of exponential type. (Z) 1f the flow (S/D,!Il(t)) has a dense orbit then !II is in general position. (3) If !II is in general position then the flow (S,D,!Il(t)) is isomorphic to (N/D,ljl(t)), where "\jf(t) is a one parameter subgroup of the unique nilpotent Lie group N containing D. (Rec·eived July 17, 1961.)

61 T-217. G. E. Baxter, Matematisk lnstitut, Aarhus Universitet, Aarhus, Denmark. A norm inequality for a "finite-section" Wiener - Hopf equation.

Let f(fl) and log f(fl) be integrable functions on - 'frtii. 9 ~ 1r with Fourier coefficients ck and dk• respectively, and let Dn(f) = det (ci-j)• (i,j= 0,1, ••• ,n). Consider the equation E: (1/Z'Il')./-:h(B)f(9)eik 9 d9 = gk, (0 ;a k ~ n), where h(9) is a polynomial of at most degree n in ei9 and where f(9) satisfies the condition. L: ldk I N satisfying E, Uh II < M llg 11. The sharpness of the theorem is well­ illustrated by the example: f(9) = 1 - aei9, g(9) = 1 and h(9) = E8akeik9 with the two cases Ia I< 1 and Ia I 5::: 1. Applications of the theorem to Toeplitz determinants and Szego polynomials are discussed.

In particular, the limit of Dn(f)/Dn_ 1(f) is found. (Received July 17,1961.)

61T-Z18. Pasquale Porcelli, Louisiana State University, Beton Rouge, Louisiana. Lp-~ and their adjoints.

Let X be a set and S a set algebra of X (i.e. S is not a sigma algebra) and H the set of bounded and finitely additive functions on S. 1f 1 ~ p ;:a co, and -y E H the Lp space of r is the set of all S-simple sequences fin} on X such that limn,m/'1~ - fm IPdr' = 0 (?"' is the variation function of ron S), llf!n}ll = 11mniflfn1Pdr')11P, and flnt = {hn} if llffn} - fhn} II= 0. Using R. B. Darst's generalization of the Lebesgue decomposition we first give a nee. and suff. condition in order that a function 7r E H be such that ?T-(E) = limn/Ef~d?", fin} E Lp and then show that Lp = Lq• (1/p + 1/q)= 1, 1 (ffn} E L 00 ==+ llf~l II= limplimnV1fn1Pdr') 1P). We also develop convergence theorems (weak and strong).,. We justify our definitions by example, by developing isomorphisms-isometries using the Stone-Cech compactification of S, and by recasting the entire theory in terms of Hellinger type inte- grals. (Received July 17, 1961.)

439 61 T-219. J, H. Ahlberg and E. N. Nilson, United Aircraft Corporation, 400 Main Street, East Hartford, Connecticut. Convergence properties of tlie spline fit.

The spline fit is an approximation to a given real-valued function f(x) of a single variable x in which the approximating function consists of segments of cubics. The cubics are joined in such a way that slope and curvature are continuous, and the approximating function interpolates to the given func­ tion at junction points. In a paper to appear in J, Math. Mech., the best approximation properties of the spline fit were examined. The present study is concerned with the convergence of the approximating function and its derivatives as the interval length between interpolation points decreases. It is shown, for example, that if f'(x) satisfies a Lipschitz condition of order 1, the approximating function and its first derivative converge uniformly to f(x) and f'(x), respectively as the interval length approaches zero. If f''(x) satisfies such a condition, then the second derivative of the spline fit converges uniformly to f"(x) provided the distances between subdivision points become asymptotically equal while approach­ ing zero. (Received July 21, 1961.)

61 T-220. Eckford Cohen, The University of Tennessee, Knoxville, Tennessee. Note on a class of affine transformations.

Let F ~ Fq denote the Galois field of order q and let "t' denote an affine transformation of an n-dimensional vector space Vn(F) into an m-dimensional vector space V~ (F). Suppose 1: to be represented with respect to fixed bases 8n and 8Jnof Vn and Vffi, respectively. LetT~ T(8n,8Jn) denote the class of all 1: which fail to preserve the sum of the components of each regular vector of Vn (a vector of Vn is regular with respect to 8n if its components are all #0). This note furnishes a simple characterization of the class T, formulated in terms of the matrices representing the trans­ formations 1:. (Received July 28, 1961.)

61 T-221. C. W. Saalfrank, Lafayette College, Easton, Pennsylvania. A characterization of contractible compact Hausdorff spaces.

For definitions see Saalfrank (Proc. Amer. Math. Soc. vol. 12 (1961) pp. 374-378). Theorem. A necessary and sufficient condition for a set A to be a contractible compact Hausdorff space is that A be an absolute homotopy retract. This result follows from the contractibility of any com,pact parallelotope plus the fact that any homotopy retract of a contractible space is contractible. (Received August 3, 1961.)

61 T-222. Israel Halperin, Queen's University, Kingston, Ontario, Canada. Unitary dilations for commuting contractions,

Notation fT al = given commuting contractions on a Hilbert space H; F =all non-negative integer valued functions of a vanishing for all but a finite number of a; p =p(n,m) has p(a) ~max (0, n(a)- m(a)}, q =: q(n,m) ~ p(m,n); T(n) "= 1TaT~(a). Each minimal unitary dilation [Ua} is uniquely determined (Sz.-Nagy) by the assignment of scalar values to 8(n,x; m,y) =(U(n)xy) (x,y E H; n,m E F) to satisfy; 8 is linear in y, 8(m,y; n,x) ~ 8(n,x; m,y), B(n,x; O,y) ~ (T(n)xy) and

(positivity) .L..tj~l8(ni,xi; nj'xj) ~0 (N;;;;; 1; n 1, ... ,nN E F; x 1, ... ,xN E H). The particular 8 with

440 B(n,x; m,y) = B(p,x; q,y) satisfies these conditions with possible exception of positivity. It is shown: (i) the condition of positivity can be expressed: (T}6 0 for a class of operators Ton spaces H e ... e H and (ii) T = W*PW with W semi-diagonal and P diagonal, Hermitian. Hence T ~ 0 if and only if the diagonal elements of P (operators on H) are ;!!; 0, and hence if and only if, for each finite set of a. (l, .•. ,r, say) L:ua(u)(T 1*) u(l>••• (Tr *)u(r) T 1u(l~ .• T r u(r)sO where a(u) = u(l) + ••• + u(r) and each u(i) can be either 0 or 1. This sharpens previous results of Sz.-Nagy (for doubly commuting Ta.) and of Brehmer (Acta Sci. Math. Szeged (1960)). This proof is valid for real, complex or quaternionic scalars. (Received August 3, 1961.)

61T-ZZ3. Israel Halperin, Queen's University, Kingston, Ontario, Canada. Sequences of elements in a product space.

(i) If llfu II = an and (fn lfm) ~ - .1 for n 'I m then~ nl/(1 + a!)~ 1; (ii) If an :> 0 for all n and Ln 1/(1 +a!)~ 1 there exists fn with llfnll= an and (fn~m) = - 1. Proof uses Gram-Schmidt ortho­ normalization. (i) was proved previously by another method by I. Amemiya for the case (fnlfm) = - 1. (Received August 3, 1961.)

61T-ZZ4. Branko Gl:iinbaum, University of Washington, Seattle 5, Washington. A generalization of theorem·s of Kirzbraun and Minty.

Theorem. Let ai,bi• 1 ;;;i i ~ k, be points in real Euclidean n-space ~ (with inner product

(x;y)) such that (ai- aj; bi- bj> e; 0 for all i,j with 1 ~ i, j ;g k. Let a.,fl be any fixed reals, a. 2 + 19 3 :> 0. Then there exists a point c E En such that (ai + a.c; bi + flc> ~ 0 for all i. Easy corol­ laries of this theorem are the recent result of G. J. Minty (On the simultaneous solution of a certain system of linear inequalities, Proc. Amer. Math. Soc., to appear) and the well-known theorem of Kirzbraun (see I. J. Schoenberg, On a theorem of Kirzbraun and Valentine, Amer. Math. Monthly vol. 60 (1953) pp. 6Z0-6ZZ). (Received August 3, 1961.)

61T-ZZ5. W. D. L. Appling, Il.lke University, Durham, North Carolina. Concerning non­ negative valued interval functions. Preliminary report.

Suppose l}t,bJ is a number interval and each of Hand K is a real non-negative valued function of subintervals of l}t,b}. All integrals referred to are limits for successive refinements of the appropriate sums. Theorem 1. If J[a,b]H(I) exists, then J'f!t,b) [!i(I)Jr' H(J)]1/Z exists and is ./[a,b] H(I). Corollary 1. If each of ./[!t,bJ H(I) and/[a,b] K(I) exists, thenJ(a,b] [H(I)K(I)]l/Z exists and is J'[a,b] [{ ji H(J)}[ Ji KU>}] 1/Z. Theorem z. j'[a,b] H(I) exists if and only if for each real valued nondecreasing function m on [a,b] /[a,b] [H(I)dm) 1/Z exists. (Received August 4, 1961.)

61T-ZZ6. J. B. Butler, Jr., University of Arizona, Tucson, Arizona. On the perturbation

method for determining coupled b~nding and twisting motion of a vibrating bar on an elastic foundation.

Let r be that curve, in a uniform semi-infinite vibrating bar on an elastic foundation, which in the undeformed state coincides with the positive z-axis. Let (r,r6,z) be points on r. The equations

441 of motion of the bar are shown to be: (i) Rzzzz- a. %zzz + sq'11R- sq•12 1J.f + Rrr = 0; (ii) - a.Rzzzz 2 2 + (a.2 + (3>'1fzzzz -lJ'llfzz- sqJ.z R + eqZ,io/+IJ{.-r-= o. In (i), (ii) a.= ~/SP, f!= C1 /AP ,?'= C/AP , 2 2 2 2 q'u = q 11 / A, q]. 2 = q 11 a./ A, q2z = q 11 a. I A + q 22/AP , P = (1 0/S - M~/S ), A = E l)r where 2 f• S, Io, Iy• Mx• E, C, C 1 are standard elastic constants. Also in (i), (ii) R = r + !I!Mx /S, 1jf = P!i!, i flexure = (E Iyl("S)t 2 and sq 11 (z), eq22 (z) are respectively the rigidity of the foundation under pure and pure torsion. Approximate solution of (i), (ii) on the interval 0"' z < oo with boundary conditions 0 is (iii) r(z,O) = r 0(z), f!(z,O) = !i!o(z), rt(z,O} = f!t(z,O) = 0, r(O,t) = rzz(O,t) = !I!(O,t) = !l!zz(O,t) = investigated for the case of waves of very short period setting C = 0. It is shown using previous results (cf. Abstract no. 574-2 Notices Amer. Math. Soc. vol. 7 (I960) p. 733) that a solution of (i), (ii), (iii) may be obtained by the perturbation method provided qii• q22 are in LI[O,oo), s .c.< I. (Received August 7, I96I.)

6IT-227. Meyer Dwass, Johnston 420, Department of Statistics, University of Minnesota, Minneapolis I4, Minnesota. A fluctuation theorem for cyclic random variables.

Random variables X I ..... Xn are called cyclic if their joint distribution law is invariant under cyclic permutations. (Interchangeable sets of random variables are cyclic, but not conversely.) Let Sk =XI+ ••• Xk• k = I, ••• ,n; Mn = max(Sp ••• ,Sn>• M = max(Sl'S2, ••• ), M+ =max (O,M). Theorem. If the Xi's are cyclic, then E(Mn; Mn < 0 ISn} = Sn/n, with probability I. Special case. If the Xi's assign all mass to- I,O,I,2, ••• , then the theorem implies that P(Mn .<:. OISn = u) = - u/n, for u a negative possible value of Sn. Equivalently, if the Xi's assign all mass to O,I,2, ••• , then P(SI < I,. s2 < 2, ••• , Sn < niSn = r) = I - r/n, for r a possible value of Sn between 0 and n. If the Xi's form a sequence of independent and identically distributed random variables, and P(M <.. 0) <: I, then EtM+ = P (M <. 0) [ I - E(tMIM < 0)]/(1 - Etx), t = exp i9, 9 real. In particular this holds if EX exists and is negative. Then setting t = I gives E(M; M < 0) = EX. This can also be obtained directly from the stated theorem. An interesting consequence is that if Xi assigns all its mass to O,I,2, ••• , then if EX < I, then

P(SI

6IT-228. Eckford Cohen, University of Tennessee, Knoxville I6, Tennessee. Arithmetical Notes. IX. On the set of integers representable as a product of a prime and a square.

Let A(x) denote the number of positive integers ~ x that can be written as a product of a prime and a square. It is proved in this note on the basis of the prime number theorem that A(x) ~ (~/6)(x/log x). This result is actually proved in a somewhat more precise form. A related problem is also considered. (Received August 7, 1961.)

6I T-229. F. H. Brownell, University of Washington, Seattle, Washington. Perturbation theory and an atomic transition model.

We give here an exact analysis of an extremely simplified model, which indicates the manner and degree of inadequacy of perturbation theory in describing the emission of radiation by an atom. It is difficult for us to give a more intelligible description of our results within the limits imposed upon an abstract. A five page dittoed summary of this material is available upon writing to the author at the University of Washington, Seattle. (Received February 23, I961.)

442 61T-Z30. W. H. FLEMING, Brown University, Providence 1Z, Rhode Island. Some Markovian optimization problems.

It is shown that a nonlinear parabolic equation of the form: (a) -a u/8s + Llltjozu/Olxi~xj + maxy[F(s,x,y) + L: Gi(s,x,y)au;~hi + H(s,x, y)u].. O, together with given mixed initial-boundary data, has a unique solution u* in any bounded cylindrical domain, provided the coefficients in (a) and the boundary data are sufficiently smooth and y is restricted to a given compact set. For H = 0 this is the basic equation for certain optimal control problems with random error terms. The solution u*(s,x) is the limit of a monotone sequence u(s,x) of solutions of linear parabolic equations, defined by an iterative process of R. Bellman ~Dynamic programming, Princeton, 1957, p. Z7Z]. The method depends heavily on certain a priori estimates of A. Friedman g. Math. Mech. vol. 9 (1960) p. 541] and E. Gagliardo @.ieerche Mat. vol. 5 (1956) p. Z50]. (Received August 11, 1961.)

61T-Z.;1. MICHAEL VOICHICK, Brown University, Providence 1Z, Rhode Island. Invariant subspaces of analytic functions.

Let R be a region on a Riemann surface with boundary r, consisting of simple closed analytic curves, such that R V r is compact and R lies on one side of r. Let Hz be the class of analytic functions F on R such that IF IZ has a harmonic majorant on R. Hz is a Hilbert space where for F E Hz IIF II= (HF(t))1/Z where HF is the least harmonic majorant ofF on R and tis a fixed point on R. Let A be the class of functions continuous on R U T and analytic on R. A closed subspace C of Hz is said to be invariant if WF E C for all W E A and all F E C. A multiple-valued analytic func­ tion cp on R is an inner function if !l = 1 a.e. on r. For .P an inner function Cqo = {F E Hz I IF IZ I I

61 T-Z3Z. C. C. CHANG and ALFRED HORN, University of California, 405 Hilgard Avenue, Los Angeles 24, California. On the represe11-tation of a-complete lattices.

Let a be an infinite cardinal and let L be an a-complete lattice. Definitions: (i) L is a-representable iff it is isomorphic to an a-complete ring of sets modulo an a-complete ideal. (ii) L has property P a iff for any two a-systems ll:ij •htj of elements of L (indexed by I and J), and any two elements x, y E L such that x -j,. y, there exists a prime ideal P such that y E P, x f: p, and for each i E I,2::"j €}iJ E P iff bij E P for each j E J, and TlJEJaij E P iff aij E P for some j E J. Theorem 1. L is a-representable iff L has property P a• Theorem z. L is cu-representable iff L satisfies both (Z,u.>)-distributive laws. Theorem 3. If L is a chain, then L is a-representable iff every densely ordered interval of L has more than a elements. Theorem 4. There exists a complete chain

(therefore, completely distributive) which is not a-representable for each a ~ zG.l. (Received August 15, 1961.)

443 61 T-233. EBERHARD LANCKAU, Am Heideberg 7, Klinigstein, Germany. A unified representa­ tion of solutions of the Tricomi equation. I

The author considers the Tricomi differential equation ~'If + '1'1'-119 = 0. If instead of "l and~ new independent variables z = s + i.S, z = s - i.S, s = (Z/3)7(312 are substituted, then the reduced stream function u(z,z) = ?7 1/ 4 \jf('?,-8) is a solution of .,!;-..(u) 2 (o 2u/O zoz) + a.(a. - l)(z + zf2 u = 0, a.= 1/6. Using Bergman's integral operator (in the form suggested by Eichler), the author determines a family u = !Jf) :jaE(z,z,t)•f(~)d~- f(z) of solution of ,!:..(u) = 0. Here a is an arbitrary point. The generating function E is twice continuously differentiable in z,z,t and for every fixed t, and is holo­ morphic in z and z in a sufficiently large domain. f is an arbitrary analytic function of?;. The use of the original Bergman's integral operator of the second kind yields also a simple expression for u, namely, u = J-)_IE(z,z ,t)f(z/Z(l - t2))dt/(l - t2)112• There exist generating functions E which can be written in the closed form, namely, E = -r-l[c 1 (X- ·n 2a.-l + c2 (X + 'l:"fa.-l J, a. f. 1/2 and E = 1:-1(c 1 + c 2 arc sin 7(.) for a.= 1/2. Here 1:= (1- ;;( 2) 112, 7( 2 = t2z/(z + z). For more details see the paper to appear in ZAMM, 1961. (Received June 21, 1961.)

61 T-234. EBERHARD LANCKAU, Am Heideberg 7, Konigstein, Germany. A unified represen­ tation of solutions of the Tricomi equation. II

Using the theory of Bergman's integral operators, the author determines a generating function E of the operator !.(f) = J'azE(z,z,f;)f(l;)dt - f(z) yielding solutions of ~(u) a uzz + a.(a. + l)(z + zf2 = O. E must satisfy the conditions e3[E(z,z,z)/oz) = 5u(z + z)-2/36 and .!JE(z,z,}>] = o. The function s- tf?(t)/dtJ, t = ( ~- H!)s-1, satisfies these conditions. Here 'P-is the Legendre function of the first kind and of order - 1/6. Thus, u = J1 (f) = s-l./az[c3?=2(t)/dt]f(t)d~ - f(z) represents a family of solu­ tions of the equation .!.:..(u) = 0. The author considers a special case of these solutions. Let a = oo, and let ~be real. If lf(t> I< C ltl(l/6)-e, E: > 0, C < oo, along the path of integration, then T 1 (f) exists. An integration by part yields u = ,I1 (f) =.~ ?2-(t) fi' f(ts + i-a )/dt}dt and '\If= '11/71 (f). One obtains various solutions which are of interest in applications as special cases of the formula for "'If. Choosing f = (z - zo)k and f = ln(z - z 0), one obtains the particular solutions of Guderley, Darboux and Tomotika and Tamada. Also f = exp(kz), f = [(z- zo)/z + z)]k and f = .i'n[(z- z 0)/(z + 7.o>J yield solutions of Tschaplygin, F alkowitsch and Germain, respectively. The formula obtained can be used for the study of singularities of 'lj/. It permits the solution of the initial value problem with data on the sonic line"/= o. There holds 'ljf(O,t9) = T(l/3VQ00f(x + il9)(3x)-6dx/11'112r(5/6), '\lf,(o,t9) = -f(- l/3)j000f'(x + i~) x 3-5/ 6x- 5/ 6dx/17'112r(l/6). (Received June 21, 1961.)

61T-Z35. WITHDRAWN

61 T-236. P. E. CONNOR and E. E. FLOYD, University of Virginia, Charlottesville, Virginia. An extension of the Borsuk-Ulam mapping theorem.

Let Mk be a compact connected C00 -manifold without boundary. For any map f:Sn- Mk let A(f) = {xlx E Sn, f(x) = f(- x)J. If n > k, then dim A(f) ,e: n- k. If n = k and f is inessential, then A(f) is nonvoid. This extends trivially to compact differentiable manifolds with boundary, and then to open

444 differentiable manifolds which can be expressed as the union of an increasing sequence of compact regular submanifolds with boundary. (Received August 16, 1961.)

61T-237. P. E. CONNER and E. E. FLOYD, University of Virginia, Charlottesville, Virginia. Actions of elementary abelian p-groups.

For a prime p an elementary abelian p-group (Zp)k is the k-fold direct sum of Zp with itself. A stationary point of an action is a point left fixed by the entire group. If ((Zzf ,Mn) is a differentiable action with no stationary points on a closed manifold, then [Mn]z = 0. If ((Zp)k, M') is an orientation preserving action on a closed oriented manifold with no stationary points, then all the Pontryagin numbers of the manifold Mn are divisible by the prime p. Finally, if (Zz + Zz, Mn) is a differentiable action with a finite number of stationary points, then l).{n]2 belongs to the subring of ~ generated by the real projective plane, [P2] 2. (Rec;eived August 16, 1961.)

61 T-238. WITHDRAWN

61T-239. WITHDRAWN

61T-240. W. P. HANF and DANA SCOTT, University of California, Berkeley 4, California. Classifying inaccessible cardinals. Preliminary report.

Let ~ be a family of classes of (binary) relational systems. A strongly inaccessible cardinal I{ is e -describable (K E Ds(C)) if for some relation R over K and some K E: e. the system (I(. R) E K. but< A. R t>,} rf.K for all~< 1(. In (e) is the class of e -indescribable inaccessibles. Let n~ (resp.~n) be the family of classes of relational systems characterized by single sentences m of (n + l)st order logic in prenex normal form with m blocks of (n t l)st order quantifiers at the front beginning with universal (resp. existential) quantifiers and with lower order quantifiers follow­ ing. Let?r~ be the least inaccessible-in In(.n~). Results: (i) Ds(IT0+1) = UmDs(II~); (ii) if n > 0 and In(n~) f. 0, then 1T~ E Ds(IT~+l ); (iii) if n > 1, m > 0, and In(n~) 'I 0, then'lr~ E Ds~~); (iv) 17"~= 1r/) = the least inaccessible (if it exists); (v) if 0 <. n 1!i p, In (II~) =f. 0, and either n <. p or m <. q, then -;r~ < '»"~; (vi) In(nb is the class of weakly compact inaccessibles (i.e., not strongly incompact; cf. Tarski, International Congress for Logic, Stanford, 1960); (vii) if I( is measurable (i.e., there is a nontrivial, J.(-additive, two-valued measure on all subsets of!(), then K E In(IT~); (viii) the least measurable cardinal p and hence also the least strongly compact cardinal (if they exist) are strictly greater than 7r·l· Note that p E Ds~~). Whether p = 'lr~ is not known. (Received August 16, 1961.)

445 61 T-241. A. A. MULLIN, 108d EERL, University of Illinois, Urbana, Illinois. The constructive existence of extremal conditions on the structure of algebraic systems.

By an extremal condition on the structure of an algebraic system is meant either a maximal condition or a minimal condition or both on all isomorphic replicas of the algebraic system. Consider the additive algebraic system (N, +)of all natural numbers, N = {1,2,3, •••}. Consider the set A of all poSitive odd integers. Clearly A satisfies a maximal condition inN relative to mutanticity, It can be shown (see the author's paper in Z. Math. Logik Grundlagen Math. vol. 7, No. 3 (1961)) that the property of being a maximal mutant is preserved under isomorphism. For a minimal condition on the structure of (N, +) use isomorphic mappings of the author's minimax mutant given elsewhere (see the author's paper in Amer. Math. Monthly vol. 68 (1961) pp. 487-488) in their corresponding replicas of (N, +). From other results (see the author's Abstract, Notices Amer. Math. Soc. vol. 8 (1961) p. 370) it is now established that N can be represented as the union of disjoint mutants all of which are maximal ones on the structure of (N, +). (Received August 7, 1961.)

61 T-242. D. G. KENDALL, Magdalen College, Oxford, England. Simplexes and vector lattices.

Let B be a nonempty convex set spanning a real vector space X, and suppose that its minimal affine extension does not contain the zero vector. Let C be the pointed positive convex cone

{ Ab: A ~ 0, b E B} having B as base. Then X, partly ordered by C, will be a vector lattice if and only

if both (l 0 ) B is a simplex in the sense of Choquet and (2°) each B -segment i ~ b + (l - ~ )b': 0 :;; A~ 1; b,b' E B} is itself maximal or can be extended to one which is maximal. This appears to be the algebraic result underlying a more special one of G. Choquet (Seminaire Bourbaki, December 1956, 139) in which X is a locally convex Hausdorff t.v.s., B is compact, and condition (2°) is omitted. For convex sets in finite-dimensional spaces (2°) is equivalent to compactness. (Received August 17, 1961.)

61T-243. ISRAEL HALPERIN, Q!een's University, Kingston, Ontario, Canada. Unitary dilations which are orthogonal bilateral shift operators.

Unitary operators U l' U 2 on a Hilbert space K are called O.B.S. if for some subspace S to be called a shifted space, {ufu~s, - oo < m,n < ooJ are mutually orthogonal and spanK. Suppose (a.):

U 1, U 2 are a minimal unitary dilation of commuting contractions acting on H C K and satisfying (U~xiU~y) = (T~xiT~y) for all n, m S; 0 and all x, y E H. Suppose also (fJ): H = Z 6) (Hf; i,j = 1,2,3) and let PI denote the projection onto HI. Suppose T 1 H{ C H{ + H!, T 1 (H! -i- Hh C J3. T 2 H~CHJ +Hf,T2 (Hf t~)CHJ,T 1 *T 2 x=T 2 T 1 *xforxEHj,T 2 *T 1 x=T 1 T 2 *xforxEHiand jn 1n jn 3n suppose li(T1P 1J xll. ii(T2Pj) xll, li(T 1*P3J xll, li(T2*Pj) xll an- 0 as n- oo for all x E H. Theorem.. Then T 1, T 2 are O.B .S. and a shifted space S can be obtained as follows: let A(T) denote [(U- T)A] for any subspace A ([A] denotes linear closure of A). Let A(T1,T2) denote ((U 1U2 -U1T 2 -U2T 1 tT1T 2)A]ifT1T2x= T2 T 1x forx·EA. ThenS=Hi(T1 ,T2 )6lH~(T2 ) Ell Hj (T1*,T2) (!) Hf(T1) 6l H~ 6l H~(T 1 *) Ell Hi(Tl'Tz) 6l H~(T~) 6l H~(Tr*,T 2*). The special case that H = H t gives the theorem: if (a.) holds and IIT~x 11. IIT~x II__,. 0 as n -oo for all x E H, then U l'U 2 are O.B.S. The proofs are valid for real, complex or quaternionic scalars, (Received August 17, 1961.)

446 61T-244. j. L. WALSH, Harvard University, 2 Divinity Avenue, Cambridge 38, Massachusetts. The location of the zeros of the derivative of a rational function, revisited.

Lemma. If la.kl ~ r, l:Ta.k = 0, lzl > r, ZT r 0 , lz - al > r 1 , ,2:f(z- a.k)-l +L:~r. L_~(z - a.k) - 1 = A(# 0); then lz(z - n/A) I ~ r 2. Both (i) and (ii) essentially determine the locus of z as the a.k and {Jk vary. (Received August-18, 1961.)

61 T-245. ALEX HELLER and IRVING REINER, University of Illinois, Urbana, Illinois. Representations of cyclic groups in rings of integers.

It is shown that a cyclic group G of order p 2, where p is prime, has exactly 4p + 1 inequivalent indecomposable representations over the p-adic integers. This result is used to prove that G has only a finite number of inequivalent indecomposable representations over the rational integers Z. On the other hand, if the restriction that the ZG-modules considered be torsion-free over Z is dropped, it is shown that there are infinitely many nonisomorphic indecomposable ZG-modules. It is also es­ tablished that an arbitrary finite group having a noncyclic Sylow subgroup has infinitely many inequiva­ lent indecomposable representations over Z. (Received August 21, 1961.)

61T-246. LOUIS SUCHESTON, University of Wisconsin, Milwaukee 11, Wisconsin. Automorphisms of Kolmogorov.

For definitions, see V. A. Rokhlin, New progress in the theory of transformations with invariant measure, Uspehi Mat. Nauk vol. 15 (1960). ·Theorem.. Automorphisms of Kolmogorov are mixing of all orders. This result has been announced by Rokhlin (loc. cit.) and Ya. G. Sinai, Dynamical systems and stationary Markov chains, Teor. Veryatnost. vol. 5 (1959). The Russian authors assume that the space is countably generated and their proofs seem to be based on spectral theory, while ours is by the martingale convergence theorem, (Received August 21, 1961.)

61T-247. WITHDRAWN

447 61T-248. C. F. KENT, 204 Tapawingo Road, S. W., Vienna, Virginia. Automorphism group of the group of recursive permutations.

Schrier and Ulam, Fundamenta Math. vol. 28 (1937) pp. 258-260, have proved that the group SOD of all permutations of the natural numbers is isomorphic to its automorphism group. Let a subgroup, G, of SOD' have the following two properties: (i) all permutations recursive in permutations in G also belong to G, (ii) any normal subgroup of G containing a permutation which moves infinitely many numbers is all of G. Theorem. Any such group is isomorphic to its automorphism group. From results stated in Abstract 570-17 of these Notices, vol. 7, no. 3, it follows that the group of recursive permutations, the group of arithmetical permutations, and other similar groups, are isomorphic to their automorphism groups. With slight additions the. proof is as given in the paper by Schrier and Ulam, above. Corollary. Every automorphism of the group of recursive permutations is inner, hence effective. (Received August 24, 1961.)

61 T-249. H. E. SALZER, General Dynamics/Astronautics, San Diego, California. Optimum­ point formulas for osculatory and hyperosculatory interpolation.

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for f(x), over (-1,1), in terms of f(xi), f'(xi) and f"(xi) at the irregularly spaced Chebyshev points xi= -cos {(2i - l)Tr/2n}, i = 1, •••• n. The advantage over corresponding formulas for xi equally spaced is in the squaring and cubing, in the respective osculatory and hyperosculatory formu­ las, of the approximate ratio of upper bounds for the remainder in ordinary interpolation using Chebyshev and equal spacing (e.g., for n = 10, the 15o/o ratio for ordinary interpolation becoming 2.4% and 0.37% for osculatory and hyperosculatory interpolation). The upper bounds for the remainders in these optimum n-point r-ply confluent formulas (here r = 1 and 2) are around 2r times those of the optimum (r + 1) n-point nonconfluent formulas. But these present confluent formulas may require fewer computations for irregular arguments when f(x) satisfies a s.imple first or second order differ-

ential equation. To facilitate computation, for n = 2(1)10, auxiliary quantities ai• bi and ci• i = 1, •••• n, independent of x, are tabulated exactly or to 15S, not precisely for the optimum points, but for those Chebyshev arguments rounded to 20 ("near-optimum" points). (Received August 28, 1961.)

61T-250. W. D. L, APPLING, Duke University, Durham, North Carolina. Hellinger integrals and absolute continuity. Preliminary report.

Every function r considered is a real valued function on the number interval [a,bJ such that r(a) = 0. All integrals discussed are Hellinger type limits of the appropriate sums. Theorem. If h is a nondecreasing function and m is an increasing function, then the four following statements are equiva­ lent: (1) there is a function g such that if a ..:. x ~ b, then fa (dg)2 /dm = h(x); (2) if w is a function such that w(x) =fa [dhdmi 12 for a<. x ~ b, then /a (dw)2 /dm =h(x) for a< x ~ b; (3) if u is a function such that for each subinterval [P.q] of [a,b] 0 ·~ u(q)- u(p) ~ h(q) -h(p) andJ~[dudm] 1 1 2 =J~@hdm] 11 2, then u is h; and (4) his absolutely continuous with respect to m, i.e., if c > 0, then there is a number k > 0 such that if D is a finite collection of nonoverlapping subintervals of [a,b] and L 0 11 m <. k, then~ 0 .flh < c. (Received August 28, 1961.)

448 61 T-Z51. ECKFORD COHEN, University of Tennessee, Knoxville, 16, Tennessee. A generali­ zation of Axer's theorem and some of its applications.

Let f(n) and g(n) be arithmetical functions related by f(n) = Ldlng(d). Wintner's generalization of Axer's theorem (Wintner, Theory of measure in arithmetical semi-groups, §5) asserts that limx_,.00(x- 1'2:n£ xf(n) - En~xg(n)/n) = ())- l)M(g), provided the mean value M(g) of g exists and g is of finite norm. In this paper Wintner's theorem is extended so as to include summations over integers n prime to a fixed integer r. The method is elementary and differs considerably from Wintner's methods. Several applications are made, one of which is the determination of the mean value of the number of solutions of certain quadratic _congruences in several unknowns to an odd modulus n(r = Z). (Received August 28, 1961.)

61T-25Z. J. M. ORTEGA, Computation Center, Stanford University, Stanford, California. Error analysis of Householder's reduction. Preliminary report.

The Householder reduction of a real symmetric N X N matrix A to codiagonal form (J. H. Wilkinson, Householder's method for the solution of the algebraic eigenproblem, Computer Journal vol. 3. (1960) pp. Z3-Z7) is the first step of a numerical procedure for calculating the eigenvalues of A. Theorem 1. If this reduction is done on a digital computer in floating point arithmetic an upper bound for the possible eigenvalue error due to rounding is f(N) [1 - f(N)T 1 11A U where the norm is the spectral norm and f(N) = [(Z68.5 + Nl/Z) (N - Z) + 10.1 (N + l)(N + Z) + 3.2(N + 2)5/ 2]e. Here e is the basic relative rounding error. Theorem z. If the computation can be made with an exact accumulated inner­ product, f(N) can be reduced to [(Z8Z.7 + Nl/Z) (N- 2) + 5.4(N +l)3/ 2]e. (Received August 30, 1961.)

61T-Z53. ECKFORD COHEN, The University of Tennessee, Knoxville 16, Tennessee. On the mean parity of arithmetical functions.

Let r, n, k, denote positive integers and man arbitrary integer. Let Pm,r denote the arithmeti­

cal progression of those n 5 m (mod r). In this paper it is shown that the mean value M(m,r) of certain arithmetical functions f(n) over P m,r is an even function of m (mod r). Arithmetical and trigonometric representations of M(m,r) are deduced and are used to obtain trigonometric series expansions of f(n). Much of the discussion of the paper is developed in a generalized form in which the author's "Ramanujan sum of order k", ck(n,r), plays a fundamental part. Particular emphasis is placed on functions of divisor and totient type. The following is a special case of a typical identity proved in this paper: Let T(n) denote the number of square divisors of n; then T(n) = (.,.z/6) 2.:,~ 1 c 2 (n,r)/r 2 • (Received September 5, 1961.)

61T-Z54. NATHANIEL COBURN, University of Michigan, Z74 West Engineering, Ann Arbor, Michigan. The limiting speeds of characteristics in relaxation hydrodynamics.

The purpose of this paper is to obtain two limiting speeds for characteristics in charged (infinite conductive) compressible nonlinear relaxation hydrodynamics. These speeds were shown to exist in the linearized theory by E. V. Stupochenko and I. P. Stakhanov (Relaxation hydrodynamics, Dokl. Akad. Nauk SSSR vol. 134 (1960) pp. 782-785) by small perturbation theory. The theory in this

449 paper is developed from the general discontinuity relations for characteristics. Although the original system of equations consists essentially of nine equations in nine dependent variables, the set of dis­ continuity relations forms an under-determined system. It is shown that this system can be completed by assuming that the normal derivative of q, the rate of change of q, the relaxation variable, along a streamline, is continuous along any discontinuity manifold. Furthermore, it is shown·that in the nonmagnetic case this assumption suffices to determine the limiting speeds given by E. V. Stupochenko and I. P. Stakhanov. Again it is shown that only in the nonlinear case is this assumption consistent with the other equations of the discontinuity system. Both of these speeds imply that the jumps of the derivatives of the various physical quantities approach zero; another limiting speed exists for which these derivatives approach infinity. Finally, the limiting speeds for the magnetic case are determined. (Received September 5, 1961.)

61T-255. M. B. POUR-EL, Pennsylvania State University, University Park, Pennsylvania. Recursive enumeration of formal theories.

The theories considered are formalized in the first order theory of equality with no nonlogical constants. A sequence {Tp} of such theories is k-r.e. if (1) The predicate "n is a Godel number of a theorem of T p" is r .e., (2) each theory T p is repeated at most k times. A family of such theories is k-r.e. if its members can be arranged in a k-r.e. sequence. Theorem l. Given k ~ 2, there exists a family of decidable theories which is k-r.e. but not k - 1-r.e. Corollary 1. There exists a r.e. family of decidable theories which is not recursively enumerable without repetition. The proof employs the following: Theorem 2. Given k E; 2 there exists a class of finite sets which is k-r.e. but not k - 1-r.e. Theorem 2 is a slight generalization of a result due to Putnam. (It obviously holds when "finite" is replaced by "infinite".) Similar results hold for r.e. families of partial recursive functions although every r.e. family of general recursive functions is recursively enumerable without repetition. However, a modification of Friedberg's proof (J. Symb. Logic val. 23 No. 3 p. 313) yields: Theorem 3. _!!. G is any r.e. class there is a r.e. class H such that (1) G ~ H, (2) a E H - G implies a is finite, (3) H is recursively enumerable without repetition. This and other results on r.e. classes have been carried over to obtain results on formal theories of various kinds. (Received September 5, 1961.)

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