AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 8, NUMBER 3 ISSUE NO. 54 JUNE 1961

THE AMERICAN MATHEMATICAL SOCIETY

Edited by GORDON L. WALKER CONTENTS

MEETINGS

Calendar of Meetings ••••• , •• , • , • , ••• , • , ••••••••• , • , • , ••• , • 196 Program of the June Meeting in Seattle • • • • • • • • • • • • • • • • • • • • • • • • • • 197 Abstracts for the Meeting - pages 247-259 PRELIMINARY ANNOUNCEMENT OF MEETING. • • • • • • • • • • • • • • • • • • • • • • 204 ACTIVITIES OF OTHER ASSOCIATIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 207 MATHEMATICS IN CONTINENTAL CHINA, 1949-1960- By Marshall Stone •••••• 209 FROM THE AMS SECRETARY •••••••••••••••••••••••••••••••••••• 216 NEWS ITEMS AND ANNOUNCEMENTS ••••••••••••••••••••••••••• 21.5,217 PERSONAL ITEMS •••••••••••••••••••••••••••••••••••••••••••• 223 LETTERS TO THE EDITOR ...... 226 MEMORANDA TO MEMBERS

The Employment Register • • • • • • • • • • .. • • • • • • • • • • • • • • • • • • • • • • • 230 Reciprocity Agreement with the Edinburgh Mathematical Society •••••••• , 230 Addresses of Authors of Abstracts • • • • • • • • • • • • • • • • • • • • • • • • • • • • . 230 Retired Mathematicians Available for Employment • • • • • • • • • • • • • • • • • • 230

NEW PUBLICATIONS •••••••••••••••••••••••••••••••••••••••••• 231 CATALOG OF LECTURE NOTES •••••••••••••••••••••••••••••••••• 233 SUPPLEMENTARY PROGRAM Number 4. • • • • • • • • • • • • • • • • • • • • • . • • • • • 234 ABSTRACTS OF CONTRIBUTED PAPERS •••••••••••••••••••••••••••• 237 ERRATA ••••••••••••••••••••••••••••••••••••••• o...... 280

RESERVATIONS FORM •••••••••• o ••••••••••••••••••••••••••••• 287 MEETINGS

CALENDAR OF MEETINGS

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

582 August 2 9 - September 1, 1961 (66th Summer Meeting) Stillwater, Oklahoma July 14 583 October 28, 1961 Cambridge, Massachusetts Sept. 14 584 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 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) August, 1965 Ithaca, New Yorlc August, 1 966 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 deadlines dates for by title abstracts are July 7, and September 7.

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, 13 50 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).

Olpyright © 1961 by the American Mathematical Society Printed in the Ulited States of America

196 Five Hundred Eighty-First Meeting University of Washington Seattle, Washington June 13-16, 1961

PROGRAM

The five hundred eighty-first meeting Book Exhibits will be located in the lobby of the American Mathematical Society will of Bagley Hall. be held on Tuesday, Wednesday, Thursday A banquet will be held on Friday and Friday, June 13-16, 1961 at the Uni­ evening, costing approximately $3.00 per versity of Washington inSeattle, Washing­ person. There will be a Social Evening for ton. This meeting will be in conjunction members of any of the five mathematical with meetings of the Mathematical Asso­ organizations on Thursday from 9:00P.M. ciation of America on June 17, the Institute to 11:00 P.M. The location of this event of Mathematical Statistics on June 14-17, will be announced at registration time. A the Institute of Management Sciences on tea at the University of Washington Faculty June 16-17, and the American Statistical Club will be presented on Saturday from Association on June 14-15. For information 4:30 to 6:00 P.M. All persons attending the concerning the programs of the meetings meetings are cordially invited to these of these organizations, see the section on events. Activities of Other Associations, page 207 There are two hotels and one motel of these NOTICES. within walking distance of the campus: By invitation of the Committee to Hotel Edmund Meany, E. 45th and Brook­ Select Hour Speakers for Far Western lyn Avenue, $8.00 - $9.00 (single room), Sectional Meetings, and with the financial $9.50 - $12.50 (double room); Wilsonian support of the Air Force Office of Scienti­ Hotel, E. 47th and University Way, $5.00- fie Research, a Symposium on Convexity $6.00 (single room), $7.00 - $9.00 (double will be held on Tuesday, Wednesday and room); Coach House, 4701 24th North East, Thursday, June 13-15. $10.00 (one bedroom suite), $14.00 (two By invitation of the same Committee, bedroom suite). Hotel reservations should there will be an address at 2:00 P.M. on be sent directly to the chosen hotel, men­ Friday in Room 131, Bagley Hall, byPro­ tioning the mathematics meeting at the feasor T. M. Apostol of the California In­ University of Washington. stitute of Technology on "Some lattice Breakfast and lunch will be available point problems in the theory ofnumbers." Tuesday through Friday in the Cafeteria Sessions for contributed papers will of the Husky Union Building (HUB) on cam­ be held on Saturday at 9:00 A.M. and at pus, beginning at 8:15 A.M. Dinner will 3:30 P.M. Abstracts of the papers to be also be served on Wednesday and Thurs­ presented at these sessions appear on day from 5:00 to 6:30 P.M. Information pages 237- 280 of these NOTICES. There concerning restaurants in Seattle will be are cross references to the abstracts in available at the Registration Desk. the program. For example, the title of Seattle is served by United, Western, paper ( 1) in the program is followed by West Coast and Northwest Airlines, the (581-42) indicating that the abstract can be Northern Pacific, Union Pacific, Great found under the designation 581-42 among Northern and Milwaukee Railroads, and the the published abstracts. There will be a Greyhound and Continental Trailways Bus session for late papers on Friday after­ Lines. To get to the campus from the noon if necessary. Information concerning Seattle-Tacoma Airport, take the Air­ late papers will be available at the Regis­ porter Bus to the Olympic Hotel in down­ tration Desk. town Seattle. Then take a Seattle Transit All sessions of the meeting will be in bus, or a taxi (cost -- about $2.50) to the Bagley Hall. The Registration Desk and University district. Members who drive to

197 the meeting can park on campus for 25 Residence Halls, 1201 Campus Parkway, cents per day. Those who stay in the Men's can park there for 25 cents per day.

PROGRAM OF THE SYMPOSIUM ON CONVEXITY The Organizing Committee for the Symposium consists of Professor Victor K1ee, Chairman, Pro­ fessor David Gale, Professor Branko Grunbaum, and Dr. Merle Andrew (member for liaison with the A.F .O.S.R.). All sessions of the Symposium will be held in Room 140, Bagley Hall.

TUESDAY, 9:00 A.M. First Session. Chairman: Professor H. S. M. Coxeter 9:00 - 9:45 Problem on a circle Professor A. S. Besicovitch, University of Pennsylvania and Cambridge University 10:00 - 10:45 Various notions of convexity for functions defined on matrix spaces Dr. Chandler Davis, American Mathematical Society ,Providence, Rhode Island 11:00 - 11:45

The dual cone and Helly type theorems Professor F. A. Valentine, University of California, Los Angeles

TUESDAY, 1:30 P.M. Second Session, Chairman: Professor A. S. Besicovitch 1:30 - 2:15 Helly's theorem and its relatives Professor Ludwig Danzer, University of Washington and Univer­ sity of Munich 2:30 - 3:15 An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size Professor H. S.M. Coxeter, University of Toronto 3:30- 4:15 Measures of asymmetry for convex sets Professor Branko Grunbaum, University of Washington and The Hebrew University, Jerusalem 4:30 - 5:15 Closedness under a set of linear combinations Professor T. S. Motzkin, University of California, Los Angeles

WEDNESDAY, 9:00 A.M. Third Session. Chairman: Professor T. S. Motzkin 9:00 - 9:45 Cyclic and neighborly polytopes Professor David Gale, Brown University 10:00 - 10:45 Simplifications of linear programs Dr. Alan Hoffman, General Electric Company, New York, New York

198 11:00 - 11:45 Total positivity and convexity Professor Samuel Karlin, The Hebrew University, Jerusalem, and Stanford University

WEDNESDAY, 2:00P.M. Fourth Session, Chairman: Professor Vlastimil Ptak 2:00 - 2:45 Rotundity Professor Mahlon Day, University of Illinois 3:00 - 3:45 Some results on fixed-point and extreme-point properties of com­ pact convex sets Professor Ky Fan, Wayne State University 4:00 - 4:45 Some theorems on extreme points Dr. Robert Phelps, University of California, Berkeley

THURSDAY, 9:00A.M. Fifth Session. Chairman: Professor Mahlon Day 9:00 - 9:45 Convex cones and spectral theory Professor Helmut Schaefer, University of Michigan 10:00 - 10:45 Convexity and weak compactness Professor Vlastimil Ptak, Tulane University and the Czecho­ slovakian Academy of Sciences 11:00 - 11:45 To be announced Professor Aryeh Dvoretzky, The Hebrew University, Jerusalem

THURSDAY, 2:00 P.M. Sixth Session. Chairman: Professor Ky Fan 2:00 - 2:45 Semispaces and the topology of -convexity Professor Preston Hammer, University of Wisconsin 3:00 - 3:45 Topological classification of convex bodies Professor Victor Klee, University of Washington 4:00 - Discussion of unsolved problems

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

FRIDAY, 9:00 A.M. Session on Convexity, Room 237, Bagley Hall 9:00 - 9:10 (1) On the existence of extreme rays in cones Professor R. E. Fullerton, University of Maryland (581-42) 9:15 - 9:25 (2) Extremal elements of the cone of semi-norms on the Euclidean plane Professor E. K. McLachlan, Oklahoma State University (581-28) 9:30 - 9:40 (3) Half rings and convex polytopes in Banach spaces Dr. P. H. Maserick, United States Air Force, Silver Spring, Maryland (581-25) 9:45 - 9:55 (4) Utility theory without the completeness axiom Dr. R. j. Aumann, Princeton University (581-39) 10:00 - 10:10 (5) A generalization of a theorem of Caratheodory Mr. W. E, Bonnice, University of Washington (581-21) 10:15 - 10:25 (6) Maximal convex sets Professor E. G. Straus and Professor F. A. Valentine*, University of California, Los Angeles (581-33) 10:30 - 10:40 (7) Further means of convex bodies. Preliminary report Professor W. J, Firey, Washington State University (581-22) 10:45 - 10:55 (8) The total length of the edges of a non-Euclidean polyhedron Professor H. S. M. Coxeter, University of Toronto (581-36) 11:00 - 11:10 (9) An extremal problem for plane convex curves Dr. Chandler Davis, American Mathematical Society, Providence, Rhode Island (581-10) 11:15 - 11:25 (10) Convex spaces associated with a family of linear inequalities Dr. H. Poritsky, General Electric Company, Schenectady, New York

General Session, Room 211, Bagley Hall 9:00 - 9:10 (11) Representative sets and an orthogonality relation for certain families of sets Mr. john Backus, I. B. M. Research, Yorktown Heights, New York (581-12) (Introduced by Dr. R. E. Gomory) 9:15 - 9:25 (12) Boolean calculi of n categories Professor R. L. Stanley, Washington State University (581-27) * For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

200 9:30 - 9:40 (13) On free alpha-extensions of Boolean algebras. Preliminary report Mr. G. W. Day* and Mr. F. M. Yaqub, Purdue University (581-30) 9:45 - 9:55 (14) On the definition of a free algebra. Preliminary report Professor H. F. J. Lowig, University of Alberta (581-3) 10:00 - 10:10 (15) On f-rings with the ascending chain condition Professor F. W. Anderson, University of Oregon (581-38) 10:15- 10:25 (16) Modules over a Dedekind ring Professor E. G. Straus and Dr. K. Rogers*, University of California, Los Angeles (581-5) 10:30 - 10:40 (1 7) Splitting conditions for abelian subgroups Professor K. H. Hofmann and Professor P. S. Mostert*, Tulane Univer­ sity (581-18) 10:45 - 10:55 (18) Splitting theorems for vector subgroups of topological groups Professor K. H. Hofmann* and Professor P. S. Mostert, Tulane Univer­ sity (581-19) 11:00 - 11:10 (19) Extreme means are homomorphisms Dr. S. P. Lloyd, Bell Telephone Laboratories, Murray Hill, New jersey (581-24) 11:15 - 11:25 (20) The weak continuity of the lattice operations in a vector lattice Mr. A. L. Peressini, Washington State University (581-15) 11:30 - 11:40 (21) Decomposition and homogeneity of continua on a 2-manifold Mr. H. C. Wiser, University of Utah (581-4) 11:45- 11:55 (22) Noncoincident maps of a sphere into the plane Dr. L. P. Neuwirth, Princeton University (581-2)

FRIDAY, 2:00 P.M.

Invited Address, Room 131, Bagley Hall Some lattice point problems in the theory of numbers Professor T. M. Apostol, California Institute of Technology

FRIDAY, 3:30P.M.

Session on Analysis, Room 236, Bagley Hall 3:30 - 3:40 (23) Simplification of turning point problems for systems of linear differential equations Professor Wolfgang Was ow, University of Wisconsin (581-17) 3:45 - 3:55 (24) On a system of integrodifferential equations occurring in reactor dynamics. II Dr. j. j. Levin*, Lincoln Laboratory, MIT, Lexingtor.., Massachusetts, and Professor J. A. Nohel, Georgia Institute of Technology (581-13) 4:00 - 4:10 (25) On the scattering of waves in the infinite bar. Preliminary report Professor J. B. Butler, Jr., University of Arizona (581-1)

201 4:15 - 4:25 (26) Removal of the log factor in the asymptotic estimates of the membrane eigen­ values Mr. P. B. Bailey, University of Washington (581-6) 4:30 - 4:40 (27) The shortest path through points of a small set Dr. J. R. Kinney, Lincoln Laboratory, MIT, Lexington, Massachusetts (581-14) 4:45 - 4:55 (28) Bounds for the capacitance of convex surfaces Dr. W. E. Parr, U.S. Naval Ordnance Laboratory, Silver Spring, Mary­ land (581-7) 5:00 - 5:10 (29) Hilbert decomposition and semicontinuity Dr. Lionello Lombardi, University of California, Los Angeles (581-20) (Introduced by Professor M. R. Hestenes) 5:15 - 5:25 (30) The moment problem in certain general function spaces. Preliminary report Dr. M. S. Ramanujan, University of Michigan (581-32)

Session on Applied Mathematics, Statistics and Probability, Room 237, Bagley Hall 3:30 - 3:40 (31) On quaternary cyclic codes Dr. Gustave Solomon, Lincoln Laboratory, MIT, Lexington, Massachu­ setts (581-29) 3:45 - 3:55 (32) Interval analysis Professor W. L. Strother and Professor R. G. Selfridge*, Miami Uni­ versity (581-37) 4:00 - 4:10 (33) Algorithms for Tchebycheff approximation by abx + c Dr. J. R. Rice, General Motors Research Laboratories, Warren, Michi­ gan (581-26) 4:15- 4:25 (34) An approximate Wiener-Hopf decomposition Dr. Julius Kane, University of Rhode Island (581-23) 4:30 - 4:40 (35) Limiting theorems for a position-dependent branching process Dr. H. E. Conner, Lincoln Laboratory, MIT, Lexington, Massachusetts (581-35) 4:45 - 4:55 (36) On the Wold decomposition of non-Gaussian random processes Dr. R. F. Drenick, Bell Telephone Laboratories, Murray Hill, New Jer­ sey (581-41) 5:00 - 5:10 (37) The characterization of vector-valued Fourier-Stieltjes transforms Professor J. R. Brown, University of Massachusetts (581-40)

Session on Algebra and Theory of Numbers, Room 311, Bagley Hall 3:30 - 3:40 (38) The irrationality of gamma or of sets of similar constants Professor W. E. Briggs, University of Colorado (581-8) 3:45 - 3:55 (39) On Waring's problem for algebraic number fields Professor G. J. Rieger, Purdue University (581-31)

202 4:00- 4!10 (40) A result in the geometry of numbers Dr. L. C. Eggan*, University of Michigan, and Professor E. A. Maier, Pacific Lutheran University (581-11) 4:15 - 4:25 (41) A combinatorial theorem on arithmetic progressions Professor Wolfgang Schmidt, University of Colorado (581-16) 4:30 - 4:40 (42) Certain semigroups Professor Takayuki Tamura, University of California, Davis (581-44) 4:45 - 4:55 (43) Simplicity of the transformation semigroup of a set Professor Naoki Kimura, University of Saskatchewan (581-43) 5:00 - 5:10 (44) (m,n)-distributive ring multiplications on direct sums of cyclic groups Professor Paul Yearout, Knox Colleg;e (581-34) 5:15 - 5:25 (45) Exponential rings Professor B. R. Toskey, Seattle University (581-9) R. S. Pierce Associate Secretary Seattle, Washington

Probability and the Logic INC. CUSHING-MALLOY, Belief 1350 N. Main St. of Rational P. 0. Box ll87 By Henry E. Kyburg, Jr. Ann Arbor, Michigan AN IMPORTANT NEW CONTRIBUTION TO THE LITERATURE ON PROBABILITY LITHOPRINTERS The concept here developed is intended to resolve many of the philosophical puzzles of the NOTICES raised by other definitions and to have prac­ Printers tical applications in scientific inference and in the testing of statistical hypotheses. The study is rigorously reasoned throughout, Known for and readers in the fields of mathematics, symbolic logic, and philosophy will find it a QUALITY- ECONOMY fascinating and basic contribution. Dr. Kyburg, B.S. Yale, A.M. and Ph.D. SERVICE Columbia, is assistant professor of mathema­ tics at Wesleyan. x + 350 pages $10.00 Let us quote on your next Wesleyan University Press printing BOX 360 DEPT. P. I. MIDDLETOWN, CONNECTICUT

203 PRELIMINARY ANNOUNCEMENT OF MEETING

SIXTY-SIXTH SUMMER MEETING AND FORTIETH COLLOQUIUM Oklahoma State University Still water, Oklahoma August 29 - September 1, 1961

The sixty- sixth summer meeting and papers which fail to meet the deadline, the fortieth Colloquium of the American namely, july 14. Society will be held at Okla­ Mathematical REGISTRATION, ROOMS, MEALS homa State University, Stillwater, Okla­ homa, from Tuesday, August 29, to Friday, Registration headquarters will be in September 1, 1961. During the same week the second floor corridor of the Union there will be meetings of the Mathematical Building. All persons attending the meet­ Association of America and the Society for ing are requested to register immediately Industrial and Applied Mathematics. Pro­ on arrival. A directory of all persons fessor R. H. Bing will deliver the Hedrick attending the meetings and an information lectures for the Association and Professor desk will be maintained at registration Mark Kac the von Neumann lecture for headquarters. SIAM. In addition, there will be a session University dormitory accommoda­ for the presentation of contributed papers tions will be available to all attending the to be held jointly with the Econometric meetings and to their families. The dor­ Society. mitories will be in Stout Hall, Murray Professor G. W. Mackey of Harvard Hall, Willard Hall, and North Hall, all lo­ University will deliver the Colloquium cated near the Student Union Building. lectures entitled "Infinite Dimensional Dormitory rooms may be occupied from Group Representations". The first of these 2:00 P.M., Sunday, August 27 to 9:00A.M., lectures will be held on Tuesday, August Saturday, September 2. The early check­ 29, at 2:00 P.M. and will be followed by out on Saturday is due to the fact that there three others on Wednesday, Thursday, and is another large group which starts meet­ Friday at 9:00A.M. All of these lectures ing on the campus at noon. The cost of will be held in the Upper Ballroom of the dormitory housing will be $2 a day per Union Building. person in double rooms with separate beds. The Committee to Select Hour Speak­ Single occupancy in a double room is at ers for Summer and Annual Meetings has the rate of $4 a day. When a single room invited Professor Leon Ehrenpreis of is available, the charge will be $3 a day. Yeshiva University and Professor Stephen No special rates will be available for Smale of the University of California, children requiring adult accommodations. Berkeley, to address the Society. Profes­ Bedding, towels, soap, and limited daily sor Leon Ehrenpreis will speak at 11:30 maid service will be provided, Automatic A. M. on Wednesday, August 30, on "Some laundry facilities and electric irons will be applications of the theory of distributions available in the dormitories. to lacunary theories". Professor Stephen Sixteen stone cabins at Lake Carl Smale's lecture, "A Survey of Some Re­ Blackwell, eleven miles from the campus, cent Developments in Differential Topo­ accommodate from four to six persons. logy", will be delivered at 10:15 A.M. on Each is completely equipped except for Thursday, August 31. Both of these events cooking utensils and dishes and will be will occur in the Upper Ballroom of the available from 6:00 P.M., Sunday, August Union Building. 2 7, to noon Friday, September 1, at rates There will be various sessions for from $6 to $8.50 per cabin. Extra rollaway contributed papers at times to be an­ beds are available at a charge of $2 per nounced in the final Program. There will, bed. There are dining facilities at the lake however, not be any special sessions for area.

204 No camping facilities will be available. RESERVATIONS Cafeteria service will be maintained in the Student Union Cafeteria throughout Reservations for University dormi­ the meetings. Meal hours will be: Break­ tory housing and the cabins on Lake Carl fast 6:15 A. M. to 8:15 A. M., Lunch 11:00 Blackwell should be made by writing A. M. to 1:15 P.M., Dinner 5:00P.M. to directly to Mr. L. C. Thomas, Director, 6:30P.M. Residence Halls, Auxiliary Enterprises, Food service will also be available in Oklahoma State University, Stillwater, the Union Club Coffee Shop and in the Oklahoma, and should be made by August several Student Union Snack Bars. _ll.

HOTELS AND MOTELS Name of Hotel or Motel No. of Units Rates Circle D. Motel 28 $ 5.00 (single) 923 North Main Street $ 6.50 (double) john Diehm, Manager $ 7 .SO (triple) (FRontier 2-5611) Curran-Au-Tel 16 $ 5.00 (single) 315 North Main Street $ 5.50-$6.00(double) Weldon Curran, Manager (FRontier 2-3525) Fifty-One Motel 15 $ 5.00 (single) 1324 East 6th Street $ 8.00 (double) Floyd Hemphill, Manager $10.00 (4) (FRontier 2-8408) HiLo Motel 40 $ 6.00 (single) 2313 West 6th Street $ 7 .SO (double) Mr. Orr, Manager $10.00 (4) (FRontier 2-2425) Town House Motel 8 $ 5.00 (single) 207 North Main Street $ 5. 50- $6.00 (double) B. L. Weeks, Manager $12.00 (4) (FRontier 2-4100) Town Park Motel 22 $ 4.50-$5.00 (single) 310 North Main Street $ 5.50-$6.50 (double) Mr. Bynum, Manager $ 8.00 (4) (FRontier 2-8822) Grand Hotel 35 $ 5.00 (single) 604 South Main Street $ 6.00 (double) Mrs. Ada Stancoff, Manager (FRontier 2-2033) Rains Hotel 12 (with private $ 3.50-$5.00 (single) 1004 South Main Street baths) Up to $6.00 (double) Earl Rains, Manager 21 (with no bath) $ 2.00 for rooms with (FRontier 2-9660) no baths Ralph Milam Hotel 40 (with bath) $ 4.00 (single) 118 West 7th Street $ 5.00 (double) Mr. Ralph Milam, Manager 20 (no bath) $ 2.50 (single) (FRontier 2-0212) $ 3.50 (double) Persons desiring hotel and motel accommodations should make their reservations di­ rectly with the appropriate manager, and under no circumstances write to Mr. Thomas who is in charge only of the Universityfacilities.

205 ENTERTAINMENT AND RECREATION AVAILABLE NOW! Lounges in the Student Union and in the ELEMENTS OF THE four dormitories will be open to members THEORY OF FUNCTIONS and their guests at all times. AND FUNCTIONAL ANALYSIS Athletic facilities include table tennis in the dormitory recreation rooms and VOLUME TWO immediate area. The tennis courts in the MEASURE. THE LEBESGUE INTEGRAL. University indoor swimming pool will be available at posted times. An excellent HILBERT SPACE municipal golf course is available for (WITH SUPPLEMENT AND CORRECTIONS nominal green fees; and Crystal Plunge, a TO VOL. ONE) privately operated, excellent outdoor swimming pool, will be available at special By A, N. KOLMOGOROV and S. V. FOMIN rates. Translated by Hyman Kamel and Horace Komm The Department of Mathematics will A distinctive feature of the Graylock eclition be host at a Tea, Tuesday, August 29, from is a set of exercises (preparecl by H. Kamel) 4:00 P.M. to 6:00 P.M. in the Chinese which not only test the reacler but substan­ Lounge of the Student Union. tially extencl the theory in the text. There will be a Western-style barbe­ Volume 2 ofKolmogorov and Fomin's Functional cue in the Theta Pond area of the campus, Analysis is devoted to an exposition of measure Wednesday, August 30, at 5:00P.M. Tick­ theory with applications to Hilbert space theory ets will be at $2 for adults and $1 for and to integral equations, A detailed account is given of plane Lebesgue measure with indications children. of how one develops linear and general n-dimen­ sional Lebesgue measures, The above serves as TRAVEL a model for the subsequent development in the text of both finitely additive measure (Jordan Stillwater, population 22,000, is lo­ measure) as well as countably additive measure cated in north central Oklahoma, midway (abstract Lebesgue measure), An account is given of measureable functions as well as of the between Oklahoma City and Tulsa, on state Lebesgue integral with respect to a (countably highways 51 and 40. Nearby U.S. highways additive) measure, Spaces of square integrable functions are discussed, supplemented in the ex­ 64 and 77 connect with these state highways. ercises by the spaces of pth power summable Rail passenger service is not avail­ functions. Abstract Hilbert space follows, An able. application is given to linear equations in Hilbert space and to the Fredholm theory of integral MKO bus service to Stillwater is equations, available from Perry, Oklahoma, the near- · ix + 128 pages + index, Cloth $4,00 est rail station, and from Tulsa and Okla­ homa City. THE COMPANION VOLUME ONE: Metric Stillwater is served by Central Air and Normed Spaces is available in cloth Lines, which will arrange special flights at $3.95. from Oklahoma City and Tulsa air termi­ nals during the meetings. The major air­ OTHER NEW TITLES lines serving the southwest serve both P. S. ALEKSANDROV. Combinatorial Topo­ Oklahoma City and Tulsa. logy, Volume Three: Duality, classifica­ Public transportation in Stillwater is tion and fixed point theorems, $6.50. provided by taxicab companies. MAIL AND TELEGRAM A. Y. KHINCHIN. Mathematical Foundations of Quantum Statistics, $10.00. Communications with members of the Society and their guests should be ad­ See the Graylock exhibit at the Seattle meeting, Complete detailed catalog available on request. dressed to them in care of the American Mathematical Society, Oklahoma State Uni­ GRAYLOCK PRESS PUBLISHERS Stillwater, Oklahoma. versity, p, Q, Box 244 Albany, New York J, W. T. Youngs Associate Secretary Bloomington, Indiana

206 ACTIVITIES OF OTHER ASSOCIATIONS

JUNE 14-17 ATSEATTLE invited papers on "Optimization proces­ ses" and "Inventory and renewal proces­ The Institute of Mathematical Statis­ ses". tics will hold its 1961 Annual Meeting in The program of the Section on Physi­ conjunction with the Far Western Sectional cal and Engineering Sciences of the Amer­ Meeting of the American Mathematical So­ ican Statistical Association consists of ciety on June 14-17, 1961 at the University seven sessions for invited half hour ad­ of Washington, Seattle, Washington, There dresses. The speakers at these sessions will also be sectional meetings at the Uni­ are Frank Proschan, George Weiss, Jack versity of Washington of the Mathematical Nadler, S. C. Saunders, Jack Capon, J, E. Association of America on June 17, the Walsh, M. H. DeGroot, Gideon Schwarz, Institute for Management Sciences on A. E. Albert, J. A. Lechner, M. V. Johns, June 16-17, and the American Statistical Jr., Sigeiti Moriguti and Herbert Robbins, Association on June 14-15. All sessions of N. D. Ylvisaker, G. E. P. Box, Sidney these meetings will be in Bagley Hall. Addelman, B. Kurkjian, Lajos Takacs, The program of the Institute of Mathe­ R. Syski, Julian Keilson, David Blackwell, matical Statistics will include the eighth J. L. Kelly, Jr., and Shu-Teh C. Moy. biennial Rietz lecture by Professor David Blackwell of the University of California, with the title "Dynamic programming", and the fourth series of Wald lectures, given by Professor Charles Stein of Stan­ INTERNATIONAL CONGRESS OF ford University on the subject"Estimation MATHEMATICIANS-1962 of many parameters". There will be two special hour addresses, given by Profes­ The International Congress of Mathe­ sor Marek Fisz of the University of Wash­ maticians 1962 will be held in Stockholm ington and the University of Warsaw on from the 12th to the 22nd of August. No "Infinitely divisible distributions", and by special symposia will be arranged in con­ Dr. Tore Dalenius of the University of nection with the congress. In addition to Stockholm on "Recent developments in the short communications given by the sample survey theory and method". In members of the congress there will be addition, the program of the Institute of between 15 and 20 one-hour lectures and Mathematical Statistics includes 30-minute a number of half-hour lectures delivered invited addresses by A. T. Bharucha-Reid, by invited speakers. The one-hour lec­ Patrick Billingsley, Herman Chernoff, R. tures are intended to be surveys of fields J. Buehler, Cuthbert Daniel, Ramanathan of current interest, and the speakers will Gnadadesikan and Martin Wilk, N. R. be chosen with this in view. The Inter­ Goodman, F. A. Graybill, S. S. Gupta, national Mathematical Union has estab­ D. L. Hanson, H. 0. Hartley, Samuel lished an advisory body to assist the Karlin, Jack Kiefer, John Lamperti, C. L. Swedish mathematicians in the choice of Mallows, Harold Ruben, Herman Rubin, invited speakers. G. P. Steck, J. W. Tukey, David Wallace, The Organizing Committee has made and W. H. Williams. arrangements with Thos. Cook and Son to The program of the Mathematical secure hotel accommodations for the Association of America includes hour ad­ period of the congress. dresses by Professor Wolfgang Wasow of More detailed information will be the University of Wisconsin on "Singular given later this year. Until further notice perturbations of differential equations", the address of the Congress is Inter­ and by Professor R. J. Wisner of Michigan national Congress of Mathematicians 1962, State University, Oakland, on the activities Djursholm 1, Sweden. of CUPM. The Institute for Management Sciences For the Organizing Committee: will sponsor jointly with the Institute for Mathematical Statistics two sessions for Otto Frostman Ake Pleijel

207 FUTURE MEETINGS OF RELATED ORGANIZATIONS

This Calendar includes symposia, seminars, and institutes sponsored by the Society, but does not include regular meetings of the AMS or MAA, which are listed elsewhere in the NOTICES.

June, 1961 October, 19 61 American Mathematical Society and Air Force Office of Armour Research Foundation Scientific Research Computer Applications Symposium A Symposium on "Convexity" Hotel: Morrison Hotel Place: University of Washington Location: Chicago, illinois Location: Seattle, Washington Date: October 25-26, 1961 Date: June 14-16, 1961 Program Chairman: Benjamin Mittman, Armour Chairman: Professor Victor Klee, Department of Research Foundation, 10 West 35th Street, Mathematics, University of Washingtonc Seattle 5, Chicago 16, illinois Washington November, 1961 July, 1961 Operations Research Society of America - 20th National Air Force Office of Scientific Research/Aeronautical Meeting Sciences Directorate and RIAS Hotel: Jack Tar A Symposium on "Differential Equations in Non-Linear Location: San Francisco, California Mechanics" Date: November 9-10, 1961 Location: Air Force Academy, Colorado Chairman: Paul Stillson Date: July 31-August 4, 1961 Fall, 1961 Contact: Captain John Gilbert, Air Force Office of 33rd Session of the International Statistical Institute Scientific Research, Washington 25, D. C. , or Location: Paris, France Dr. Joseph LaSalle, RIAS, Baltimore, Maryland Date: Fall, 1961 August, 1961 December, 1961 American Mathematical Society and the National Science International Mathematical Union, United Nations Educa­ Foundation tional, Scientific and Cultural Organization, U. S. 1961 Summer Research Institute on the topic "Applications National Science Foundation of Functional Analysis'' Inter-American Conference on Mathematics Instruction Place: Stanford University at Secondary and University Levels Location: Stanford, California Location:Bogot:i, Columbia Date: August 1-26, 1961 Date: December 4-9, 1961 Chairman: Peter D. Lax, Institute of Mathematical Contact: Professor Marcelo Alonso, Division of Sciences, New York University, New York, Science Development, Pan American Union, New York Washington 6, D. C. September, 1961 International Mathematical Union and Czechoslovak American Association for the Advancement of Science - Academy of Sciences 128th Annual Meeting "Topology and Its Methods in other Mathematical Location: Denver, Colorado Disciplines" Date: December 26-31, 1961 Location: Prague, Poland Date: September 1-9, 1961 American Economic Association, American Statistical Contact: Professor Kazimierz Kuratowski, University Association, The Econometric Society of Warsaw, Warsaw, Poland Location: New York, New York Date: December 27-29, 1961 International Association for Analog Computation Location: Belgrade, Yougoslavia Eastern Joint Computer Conference Date: September 4-9, 1961 Location: Washington, D. C. Comit6 Yougoslave Contact: Dusan Strujic, President, Date: December, 1961 de l'Electronique, des Telecommunications, de l'Automatisme et de la Technique Nucleaire, Date Unknown 1961 Decanska 14/IV, Belgrade, Yougoslavia International Union of Theoretical and Applied Mechanics Colloquium on"Non-Linear Vibrations" Conference of the Association for Computing Machinery Location: Moscow, USSR Hotel: Statler Hilton Hotel Date: 1961 Location: Los Angeles, California Contact: Maurice Roy, Membre, Acad6mie des Date: September 5-8, 1961 Sciences, 29, av., de la Division, Leclerc Contact: E. F. Sherman, Control Data Corporation, Chltillon-sous-Bagneux (Seine), France 8421 Wilshire Boulevard, Beverly Hills, California June, 1962 7th Midwestern Conference Fourth U. S. National Congress of Applied Mechanics "Fluid Mechanics and Solid Mechanics" Location: Berkeley, California Place: Michigan State University Date: June 18-21, 1962 Location: East Lansing, Michigan Date: September 6-8, 1961 Date Unknown 1962 Contact: Professor J. E. Lay, Mechanical Engineering National Scientific Research Center (France) Department, Michigan State University, International Colloquium on Partial Differential Equations East Lansing, Michigan Location: Paris, France International Association for Cybernetics Date: 1962 International Congress on Cybernetics, 3d Contact: Professor Malgrange, Facult6 des Sciences, Location: Namur, Belgium Universit6 de Paris !l.la Sorbonne, 47, rue des Date: September 11-15, 1961 Ecoles, Paris 5e, France 208 MATHEMATICS IN CONTINENTAL CHINA, 1949-1960

by Marshall H. Stone

The present state of mathematics in At the present time only a small continental China can be characterized in a number of documents is needed in order to few words -- and is, at the same time, obtain a fairly complete and detailed pic­ more or less what might be expected a ture of the state of mathematics in con­ priori from a consideration of the circum­ tinental China. The "Bibliography of Math­ stances under which mathematics has de­ ematics Published in Communist China veloped there. While mathematics is more During the Period 1949-1960", compiled actively pursued than ever before in China, by Dr. Chia Kuei Tsao (Wayne State Uni­ it is still largely derivative, is still rather versity, Detroit) lists (with some dupli­ concentrated on the more detailed aspects cations) 1271 titles by Chinese authors. of classical problems, and is still almost It also includes a roster of Chinese peri­ entirely dependent for its scientific in­ odicals containing mathematics and a re­ spiration upon the leadership of a few port by the Mathematics Group of the gifted Chinese mathematicians and politi­ Academia Sinica (Peking) entitled "Re­ cally controlled contacts with the mathe­ search Work in Mathematics in China maticians of other Communist countries, from 1949 to 19 59". To obtain more de­ chiefly the Soviet Union. Some ofthe Math­ tailed information concerning the papers ematics published during the period 1949- recorded in the "Bibliography" it is nec­ 1960 has been original, significant, and essary to consult the reviews which have interesting; but a good deal has not. The appeared in the three reviewing journals-­ government and the Communist Party have Mathematical Reviews, Zentralblatt fur made a strong effort to stimulate re­ Mathematik, and Referatny Zhurnal. Be­ search, publication, and education in math­ tween 19 50 and the early part of 1960 ematics, as in other branches of science. Mathematical Reviews had published some They have also attempted to direct the 371 reviews of articles which appeared in interests of more mathematicians toward continental China. The Russian Referatny fields allied to applications and to the ap­ Zhurnal offers a somewhat wider coverage plications themselves. Following more or of the Chinese literature and generally less closely the Russian model, they have gives somewhat more extensive comment entrusted the task of developing science in on individual articles, but it is published China to a powerful and highly organized entirely in Russian. It may be observed scientific academy, the Academia Sinica, that Mathematical Reviews and Referatny which like the organization of the same Zhurnal have an arrangement for the ex­ name in Taiwan stems from the academy change of material. Thus in theory, at created long before World War II at Peking least, no really significant Chinese paper and later transferred to Nanking. The which reaches Moscow need be over­ activities of the Academia Sinica (Peking) looked by Mathematical Reviews. To com­ and the related activities of the Chinese plete a rapid survey of the contemporary universities have already visibly stimu­ mathematical scene in Communist China lated and broadened the interest in mathe­ one needs access to a general account of matics and have markedly increased the the Academia Sinica (Peking) such as may quantity, if not yet the quality, of mathe­ be found elsewhere in this report of the matical publication in continental China. Symposium on "Science in Communist China." Note: Reprinted by permission of the In giving a more detailed analysis American Association for the Advance­ and appraisal of Chinese achievements in ment of Science from "The Sciences in mathematics during the period 1949-1960, Communist China." we may begin by asking the question, "What

209 outstanding contributions, if any, have been L. C. Hsu (analysis), Loo Keng Hua(num­ made by continental Chinese mathemati­ ber theory, algebra, theory of analytic cians in recent years?" This question is functions), Chao Ho Ku (differential geom­ one which I have put to a number of dis­ etry), Tao Shing Shah (theory of analytic tinguished colleagues known for their functions). Bu Chin Su (differential geom­ wide mathematical interests, among them etry), and Wen Tsun Wu (topology). When a few who have visited China since 1949. the authors of ten or more papers over the On the whole, the answer seems to be that same period are listed, 19 additional the Chinese contributions have created no names appear and the corresponding fields very great stir in mathematical circles of interest then include the theory of inte­ during a period which has been distin­ gral equations, the theory of partial dif­ guished by an extra ordinarily intense re­ ferential equations, functional analysis and search activity and the rapid proliferation mathematical logic. These Z7 authors are of new ideas, new points of view, and new responsible for 474 of the 1Z71 papers methods. A small number of ·continental listed. Thus it is clear that in Communist Chinese are recogni:~::ed everywhere as China, as in other parts of the world, a gifted and accomplished mathematicians, small fraction of the mathematical pro­ and their recent contributions are highly fession not only produces the largest part esteemed. As examples there may be cited of the good mathematics done but also Wen Tsun Wu's introduction of new topo­ stimulates a good deal of the rest. logical invariants and Loo Keng Hua's A closer examination of the sub­ studies on the classical domains in the stance of mathematical publication in Com­ theory of analytic functions of several munist China shows that it is still mainly complex variables. The fields in which concerned with rather detailed studies of Chinese contributions have stood out are problems already posed outside China, the theory of analytic functions, number many of them classical. The distribution theory, differential geometry, and topol­ by fields of the 1Z71 papers listed in the ogy•. The self-appraisals to be found in "Bibliography" is as follows: Theory of official reports of the Academia Sinica analytic functions - Z45; Geometry (al­ (Peking), such as the one cited above, do most entirely differential) - 163; Ordinary not give a satisfactory reply to our question differential equations - 119; Analysis though they are interesting in themselves (mixed classical topics) - 116; Algebra - for obvious reasons. Unfortunately they 113; Topology - 107; Number theory - 84; are somewhat undiscriminating as to the Functional analysis - 58; Partial differ­ achievements they recite. In an effort to ential equations - 46; Statistics - 4Z; deflate individual vanity, they omit refer­ Fourier and similar series - 4Z; Mathe­ ences to the literature and cite no author's matical logic - Z8; Integration and mea­ names; but the generally self-congratula­ sure theory - Z8; Integral equations - ZS; tory tone of these reports shows that and the rest scattered. These figures ac­ vanity banished from one domain creeps quire greater meaning when it is noted back, magnified, into another. how much concentration there is on rather A simple statistical analysis of sharply defined aspects of certain of the Tsao's "Bibliography" throws agreatdeal broad topics listed above. Thus a large of light on the nature of mathematical de­ part of the work on the theory of analytic velopment in continental China. While the functions is concerned with conformal authors listed number 308, it is not sur­ mapping, especially the theory of" schlicht" prising that a much smaller number of functions and the Bieberbach problem, mathematicians is to be credited with an while a good deal of that on ordinary important fraction of the works recorded, differential equations is concerned with the particularly of the more interesting and behavior of the solutions of systems near significant ones. Only eight of the 308 a singular point. From a survey of their authors have published ZO or more papers work, one gains the impression that a each over the period1949-1960. Withtheir large number of Chinese mathematicians principal fields of interest they are: Kien have very narrow interests which they Kwong Chen (theory of analytic functions), pursue with a kind of lapidary zeal and ap­ Yiian Shiin Chin (differential equations). plication. The result is that many of their

210 papers seem to deal with very limited and and the Soviet Union, where significant usually incomplete results of a rather parallels are to be found. In spite of a complicated kind, interesting only to other very early scholarly interest in mathe­ specialists. As already mentioned, the matics and some effective contacts with more outstanding papers fall in the fields foreign mathematical trends a few hundred of theory of analytic functions, number years ago, China remained mathematically theory, differential geometry, and topol­ inert during the entire modern period of ogy. mathematical science until after 1900. The The foregoing account does not give conclusion of the Boxer Rebellion then an adequate survey of the interest shown broke down the relative isolation of China, by Chinese mathematicians in the various opening up broad opportunities for contact branches of applied mathematics. The with the learned and scientific activities "Bibliography" contains only 11 titles in of the rest of the world. These opportuni­ applied mathematics if we except the 4Z ties were at once seized upon, and the titles listed in statistics and some papers modern scientific and mathematical his­ on numerical methods and computational tory of China began. In spite of the politi­ mathematics included under the heading cal instability and turbulence which for "Analysis". On the other hand, Mathemati­ China marked a very large part of the cal Reviews has published abstracts of Z7 period between 1900 and the present time, papers in elasticity or plasticity and of 1Z there was a slow but steady growth of others in scattered fields -- statistical scientific activity in the country. Almost mechanics, fluid mechanics, electromag­ all Chinese scientists received their high­ netic theory, quantum mechanics. There­ er academic training abroad, especially port of the Mathematics Group of the in Western Europe and America, but re­ Academia Sinica (Peking) mentioned above turned to China to take up positions in the lays considerable stress on the common universities. By the mid-thirties mathe­ aim of the Government, the Communist matics had reached a point where it could Party, and the Academy to stimulate work begin to flourish. The number of gifted in applied mathematics, and draws atten., younger mathematicians was on the in­ tion to the active and increasing interest crease, and the creation of the Institute of of Chinese mathematicians in applied Mathematics of the Academia Sinica at fields and in such closely allied branches Nanking gave them a center for higher of mathematics as the theory of differen­ study and research which guaranteed an tial equations and numerical analysis. autonomous future for Chinese mathema­ While the existing literature contains tics. Under the direction of Professor only a small number of papers in func­ S. S. Chern (later of the University of tional analysis -- and relatively few of Chicago, now of the University of Califor­ any great significance -- it is reported nia), it soon had a remarkably stimulating that a lively interest in this direction is effect and began to leave its mark on the also springing up. Plans for the develop­ younger generation of Chinese mathemati­ ment of centers for modern high-speed cians. The chaos into which China was electronic computing are mentioned. From plunged by the Second Sino-Japanese War, such indications it may be inferred that beginning in 1937, and the Second World the growing attention to applied mathe­ War disrupted the life of the country far matics has so far been more apparent at more profoundly than the long series of the relatively routine levels associated disorders and internal struggles which had with the solution of specific practical preceded. Chinese scholars and scientists problems than at the level of fundamental were generally uprooted and their work had mathematical research into the underlying to be carried on under the gravest diffi­ physical phenomena. culties. With the establishment of com­ In order to interpret this bird' s eye munist control, a great many made the view of continental Chinese mathematics decision to quit China for posts in Europe as it is today, we must examine the his­ or America. It was then, for instance, that torical background and make some com­ S. S. Chern left China for America. The parisons with developments in other coun­ loss of a largenumberoftheablestmathe­ tries, such as Japan, the United States, maticians, including many younger men,

211 was a severe blow to the development of levels. Already the universities are be­ mathematics in continental China after ginning to modernize their mathematical 1949. On the other hand, a sufficient num­ programs along lines somewhat resem­ ber remained to provide continuity and to bling those familiar to observers of the initiate the vigorous new development Russian universities. Some of the leading which is now evidently under way. The nu­ Chinese universities are experimenting cleus of the mathematical profession in at the present time with a five-year pro­ present day continental China therefore gram, comprising a three-year curricu­ consists largely of mathematicians who lum of fundamental topics in pure and ap­ have had much training and extensive sci­ plied mathematics followed by a two-year entific contacts outside China, expecially stage devoted to various topics, largely in Western Europe and America. For in­ optional, at a more advanced level. The stance Professor L. K. Hua, the present development of a satisfactory system of Director of the Institute of Mathematics in higher mathematical instruction will in­ the Academia Sinica (Peking), has spent evitably alter the flow of students from a great deal of time abroad in Great China to the Soviet Union. Britain, Russia, and America. In fact he Whatever the motives behind the left a research professorship at the Uni­ prevailing policy of isolation, they are versity of Illinois to return to China in very strong. As a result the policy is 19 50 and eventually to become in effect hardly one which will be lightly or easily the successor to Professor Chern. changed in the immediate future. On the In recent years -- to be more pre­ contrary, one may justifiably surmise cise, since some time in 1957 or 1958 -­ that a political coolness between Commu­ China has severed its mathematical con­ nist China and the Soviet Union might even tacts outside the communist world. Thus in lead to an extension of this policy to scien­ 1960 there are in Paris no students of ad­ tific relations with the latter. The intem­ vanced mathematics from continental China perate language and exaggerated asser­ known to my colleagues there, though tions affected by Chinese mathematical today Paris is probably the world's leading leaders in their published references to center for mathematical research. The "capitalist" mathematics give some mea­ principal foreign scientific contacts for sure of the intensity of their emotional Chinese mathematicians and Chinese stu­ involvement in this shift to scientific iso­ dents of higher mathematics are those lationism. which can be had in the Soviet Union -­ The history of the relations of especially in Moscow, another of the China to the International Mathematical world's leading mathematical centers. Union (IMU) and the International Con­ The number of professors and research gress of Mathematicians (ICM) throws workers who travel as exchange visitors further light on this subject. In the period between China and the European com­ 19 57-19 58 both Communist China and Na­ munist countries does not appear to be tionalist China (Taiwan) applied for mem­ very large though systematic arrange­ bership in IMU. The constitution of IMU, as ments for exchanges are made. On the well as that of the International Council of other hand a great many Chinese students, Scientific Unions, made it possible for the including numerous specialists in mathe­ two applications to be considered simul­ matics, now spend considerable time in taneously and independently of any under­ the Soviet Union. To some extent this un­ lying political considerations. The appli­ doubtedly reflects the inadequacy of the cation from Communist China raised two existing facilities for higher mathematical difficulties which could not be overcome instruction in continental China-- an in­ by negotiation. One was the request to be adequacy which is recognized quite frankly accepted in the highest category of mem­ by the leading Chinese mathematicians and bership, Group V, alongside the USSR· and which, as they state in the report cited the USA but ahead of France, Germany, above, is to be overcome through the Great Britain, and Japan -- all countries instrumentality of the Academia Sinica which clearly lead Communist China in (Peking) and by means of a general re­ mathematical activity but are enrolled in form of mathematical education at all Group IV. The other difficulty was Com-

212 munist China's claim to represent not to establish a truly independent and well­ only the mathematicians of continental rounded mathematical activity of unim­ China but also those of Taiwan. As a re­ peachable quality. In the case of Japan a sult the Executive Committee of IMU was somewhat longer period elapsed between unable to recommend to the General As­ the emergence of the first modern Japan­ sembly the admission of Communist China. ese mathematician of international sta­ Since no such difficulties arose in the case ture, the late Professor Takagi, and of Nationalist China, the Executive Com­ Japan's attainment of mathematical matu­ mittee recommended the admission of the rity, which may be dated in the mid thirties latter, and the General Assembly at its or a little later. The war years 1947-1949 meeting at St. Andrews in 19 58 adopted were for China a more disturbing factor this recommendation. In the vote, the than anything affecting the development of Soviet delegation and some other delega­ Japanese mathematics between 1900 and tions from communist countries abstained. 19 37; and, on the other hand, at this mo­ While these applications for mem­ ment China is already well advanced on its bership in IMU were pending, the Interna­ path toward mathematical maturity. These tional Congress of Mathematicians 19 58 observations must, of course, be kept in was being prepared and invitations to par­ mind in drawing any conclusion from the ticipate were being sent out to individual comparison of the two countries. mathematicians all over the world. The A somewhat less obvious but equally ICM is a self-perpetuating organization of valid parallel may be drawn with the individuals independent of IMU, but assist­ United States. While at all times after 16ZO ed both financially and scientifically by the North America had some contacts with the latter. Not only were such individual invi­ European scientific and mathematical tra­ tations sent to Chinese mathematicians dition, so far as mathematics was con­ both in Communist China and elsewhere, cerned these contacts were relatively but Professor W. T. Wu (now Deputy Di­ feeble and totally inadequate to provide the rector of the Institute of Mathematics in basis for developing a strong independent Peking, and one of China's most gifted native movement until well into the nine­ mathematicians, as mentioned above) was teenth century. Except for a few sporadic invited to deliver one of the principal ad­ noteworthy figures, American mathe­ dresses, of which approximately a score matics hardly got underway before the were being planned. However, no mathe­ 1890's when the gifted young mathemati­ matician from Communist China partici­ cians of the day, fresh from advanced pated in the Congress when it met in Edin­ studies in the universities of Europe, burgh in August of 1958, though mathema­ initiated a development which has since ticians from Russia, Poland, Hungary, and made the United States a leading center of other communist countries were present world mathematics. It was at least fifteen in considerable numbers and took a very years before 0. Veblen and G. D. Birkhoff, active part. among the first outstanding mathemati­ The rather obvious parallel between cians to be trained entirely in the United the development of mathematics in China States, earned their doctorates at the Uni­ ancl that in Japan suggests that we shall versity of Chicago. And it was a good eventually see mathematics in continental many more before the country had attained China reach a very high level of activity, full mathematical independence and ma­ whether we view it in terms of quality or turity. This achievement may properly be quantity. China has already clearly demon­ dated in the 19Z0' s or perhaps even a little strated its intellectual potential in the do­ later. main of modern mathematics and is com­ It may be remarked that the history mitted to providing the material and or­ of mathematics in Russia offers no true ganizational support necessary for reali­ parallel to what has taken place in recent zing this potential in fullest measure. It times in China, Japan, or the United is not easy to estimate the time which may States. For already in the eighteenth cen­ be required for this goal to be attained, tury Russia was participating quite fully but it would probably be conservative to in the European mathematical tradition allow as much as twenty years for China and by the nineteenth century had already

213 reached a high degree of maturity, with threatens to interpose an additional and the production of such outstanding mathe­ truly formidable barrier to communi­ matical figures as Lobachevski, Cheby­ cation. In the long run, this and many other shev, and Liapounov. The point at which problems in the field of scientific com­ the mathematical histories of Russia and munication can be solved only by the de­ China begin to show certain analogies is velopment of adequate translation ser­ the point at which, in each instance, com­ vices. Even if scientists should return to munist doctrines concerning the organi­ the use of some one common language for zation and support of science began to play the purpose of exchanging ideas and results a major role. There is no question that the at the level of higher research, it would Soviet Union, working in accordance with still be necessary to carry on education those doctrines, has succeeded in main­ and local communication in the local lan­ taining and reinforcing the Russian mathe­ guages. Hence some kinds of translation matical tradition to the point where Russia services would remain altogether indis­ now enjoys a position second to none in the pensable. So far as mathematics is con­ domain of mathematics. The similar cerned the problem presented by China is course being followed in Communist China at the moment not an urgent one though it today, we must assume, is capable of win­ might eventually become so if it were neg­ ning a comparable success. At the same lected. The reason is that the major con­ time the Soviet example suggests that the tributions of Chinese mathematicians are isolationist and doctrinaire tendencies ap­ currently being published not only in parently inseparable from the communis­ Chinese but also in some more easily tic guidance of scientific development may learned language -- most often English or produce undesirable effects, more than Russian. Even if the present Russian orien­ likely to be magnified or even exaggerated tation of continental Chinese mathematics in the case of China. Certainly there are should continue and eventually cause publi­ objective and highly qualified observers cation in Russian to predominate, the im­ who believe that Russian mathematics has portant ideas and techniques which may be been unduly narrowed and in some resp­ developed in China would thus remain re­ spects definitely weakened by the attenu­ latively accessible. Of course, the fact that ation of its contacts with Western Europe a fairly adequate exchange of periodical and America as well as by the constant literature is possible not only between pressure from above to emphasize in edu­ Communist China and other Communist cation and in research those mathematical countries but also between Communist subjects most obviously useful for the ap­ China and the rest of the world -- one of plications. Indeed, there are some who the leading European mathematical jour­ think that, mostly because of these effects, nals, for instance, is sending about fifty the post war generation of Russian mathe­ exchange copies to China -- means that maticians may not reach the heights at­ the flow of ideas and information may be tained by their predecessors and teachers, slowed down but can probably be main­ whose achievements made the period 1917- tained at a fairly effective level. Needless 1939 such a brilliant one in the annals of to say, the bulky but less important litera­ Russian mathematics and continue toil­ ture which is published only in Chinese lumine as brightly the present day mathe­ must be scanned and abstracted by mathe­ matical scene in the Soviet Union. maticians able to read Chinese, so that its The emergence of continental China content may be understood in essence if as an important but possibly isolated not in detail. Fortunately the presence of mathematical center is so nearly certain Chinese mathematicians in Europe and that it must be taken into account by mathe­ America currently enables the mathemati­ maticians who seek to accelerate and unify cal world to have a good knowledge of the advances of their science by strength­ mathematical progress in Communist ening their means for collecting and dis­ China without leaning heavily upon the seminating mathematical information. Any Russian abstracting and reviewing ser­ tendency towards isolation restricts the vices. mutually beneficial exchange of ideas, but As time passes, the situation we in the case of China the language barrier have described can easily change in any

214 one of several different directions, some and using a system of writing which is dif­ representing an improvement but others a ficult to master, is in urgent need of some deterioration. Since at this moment China kind of language reform as a fundamental would clearly lose much more than would step towards rebuilding Chinese society. the rest of the world by further steps to­ Consequently the consideration of the lan­ wards scientific isolation, a deterioration guage problem in its present form does not in this sense does not seem to be imme­ necessarily allow us to predict the exact diately in the offing. Even the language nature or extent of the Chinese translation barrier, which might gradually become service which must eventually be intro­ more formidable as more mathematicians duced, if the world is to be kept adequately are trained exclusively in China, may be abreast of the progress of science and affected by the measures which the govern­ mathematics in continental China. The ment and the Communist Party may take in problem is one which needs to be continu­ order to improve the Chinese language as ally reviewed and restudied and should be a medium for internal communication. For dealt with in a deliberate and flexible it must be remembered that China, in manner, as the scientific history of Com­ speaking many widely divergent dialects munist China unfolds.

NEWS ITEMS AND ANNOUNCEMENTS

SMSG GOES WEST- One of the nation's ate curriculum. In addition to the numer­ major centers for the study of public school ous opportunities that will be providedfor mathematics will move from the east to the the discussion and exchange of ideas, there west next fall. Edward G. Begle, now at will be talks by leaders in the field, among Yale University and director of the School whom can now be listed: Kenneth 0. May, Mathematics Study Group, will join Stan­ Robert Norman, Paul C. Rosenbloom and ford University's School of Education as R. L. Wilder. professor of mathematics education. Participants will be selected from The project was createdin 1958under institutions that wish to initiate programs a million dollar grant from the National of undergraduate mathematical research, Science Foundation. The activities of study as well as from those which already have group members from all over the country such programs. Application for participa­ are coordinated in the office of the direc­ tion in the conference should be made by tor. writing to Professor Seymour Schuster, Many of the school tests of experimen­ Conference on Undergraduate Mathemati­ tal materials have been carried out during cal Research, Carleton College, North­ the last two years in California and other field, Minnesota. southwestern states. Mr. Begle also serves as chief consultant to the California State Board of Education for the major overhaul of the state's mathematics curriculum.

LUSIN'S PROBLEM FOR FOURIER A CONFERENCE ON UNDERGRADU­ SERIES. At the meetingofthe Polish Math­ ATE RESEARCH IN MATHEMATICS. A ematical Society April14, 1961, Professor conference on the subject of "Undergradu­ Z. Zahorski answered affirmatively the ate Mathematical Research," sponsored following classical problem posed by Lusin by the National Science Foundation, will in 1877: If f is a square integrable be held at Carleton College, June 19-23, function, does the Fourier series asso­ 1961. The conference proposes to discuss ·ciated with f converge to f almosi. fundamental questions relating to under­ everywhere? graduate research: desirability, aims, Reported from Warsaw by criteria, and role in the basic undergradu- Michael Bleicher (Tulane)

215 From the AMS Secretary

Lowell J. Paige, U.C.L.A. Acting Secretary

tion at the meeting. This limitation as to the TEN-MINUTE PAPERS total will be the only ~requirement im­ posed. Of the otherwise acceptable papers, The communications received from the first M papers will be accepted in order the members of the Society in response to of receipt of the abstracts in the Providence Professor Green's note in these Notices Office of the Society, the order or receipt of concerning ten-minute papers indicate a simultaneously received papers being de­ smoldering dissatisfaction with the scien­ cided, if necessary, by any impersonal de­ tific aspect of the Annual Meetings. Apart vice. The surplus papers (if any) will be of the problem is certainly the growing transferred to a later meeting. Authors number of papers and the availability. of may specify in advance their second choice suitable facilities to present papers at a of a meeting. meeting. The "first come, first served" pro­ The By-Laws of the Society provide cedure is intended to ensure the least pos­ that papers intended for presentation at sible interference by the Society with the any meeting shall be passed upon in ad­ nature and direction of the research of its vance by a Program Committee, and only members. such papers shall be presented as shall For the Winter Meeting 196Z, Mis set have been approved by this Committee. at ZOO. It is to be noted that, at least for However, except for certain minimal re­ the present, no change in procedure is con­ quirements which must be fulfilled to the templated for the regional or Summer satisfaction of the appropriate Associate Meetings. Therefore, authors may wish to Secretary, it has long been the privilege submit their papers directly to the Nov em­ of every member of this Society to present ber meeting in the Mid-West or the Feb­ one ten-minute paper at each and every ruary meeting in New York. Ample time regularly scheduled Meeting of the Society. for ten-minute papers and an audience less The number of such papers has now grown distracted by a heavy program is usually too large, particularly at the Annual Winter available at these meetings. Meetings. There were 168, ZZZ and Z34 The Committee on Invited Addresses ten-minute papers at the Annual Winter at the Annual Meetings has been informed Meetings of 1959, 1960 and 1961, respec­ of the Council's action. This Committee tively, and there is every reason to expect feels that the scientific aspect of any meet­ that, without some change in procedure, ing is a responsibility of everyone pre­ the numbers will continue to grow. senting a paper. Therefore it has suggest­ To meet this difficult problem, the ed to the Council that the American Mathe­ Council of the Society has recommended matical Society urge the prospective the following plan to the Associate Secre­ authors of ten-minute papers to limit them­ taries: Beginning with the 196Z Winter selves to the presentation of definitive and Meeting a maximum number (M) will be set significant results of general interest with (in the Preliminary Announcement of the the aim of improving the quality and attrac­ meeting) for the number of ten-minute tiveness of the scientific program at meet­ papers which can be accepted for presenta- ings.

216 NEWS ITEMS AND ANNOUNCEMENTS

THE NATIONAL ACADEMY OF SCI­ Of 44 NATO Postdoctoral Fellowships, ENCES, at their 98th Annual Meeting on awards in mathematics went to Irwin Fein­ April 25, 1961, elected 35 new members. berg, Herman R. Gluck, Wilbur C. Holland, Election to membership in the Academy is Bruce L. Reinhart, Robert T. Seeley, considered to be one of the highest honors William G. Strang and Andrew Whinston. which can be accorded to an American One of the 19 O.E.E.C. Senior Visit­ scientist. ing Fellowships was awarded in mathe­ Three members of the AMS were in­ matics to Raymond G. Ayoub. cluded among those honored by election to NSF announced 1100 Cooperative the Academy: Graduate Fellowships for 1961-1962 and 625 Summer Fellowships for Graduate Shiing-Shen Chern Teaching Assistants for the summer of Donald C. Spencer 1961. John W. Tukey Of the Cooperative Graduate awards, 201 were made in mathematics, 256 in en­ gineering, 425 in the physical sciences, OTTO E. NEUGEBAUER, Chairman including a number in interdisciplinary of the Department of the History of Math­ fields, 186 in the life sciences, and 32 in ematics at Brown University, has been the social sciences. Fellows were selected named a recipient of a $10,000prizebythe from 3241 applicants representing all 50 American Council of Learned Societies. States, the District of Columbia, and He was one of ten outstanding university Puerto Rico. professors to be honored "for havingcon­ Of the 625 Teaching Assistant awards, tributed most significantly to knowledge 103 were made in mathematics, 61 in and the pursuit of knowledge in their re­ engineering, 259 in the physical sciences, spective fields." Professor Neugebauer including a number in interdisciplinary was cited for his original work in Egyptian fields, 175 in the life sciences, and 27 in and Babylonian mathematics and astron­ the social sciences. Fellows were selected omy. from 1366 applicants representing 47 Brown University Press has an­ States and the District of Columbia. nounced the publication of Egyptian Astro­ nomical Texts 1: The Early Decans,by Otto E. Neugebauer and Richard E. Parker. FELLOWSHIPS IN BIOMETRY FOR MATHEMATICS GRADUATES. Training programs designed to prepare students in FELLOWSHIP ANNOUNCEMENTS. the application of statistical and mathe­ The Alfred P. Sloan Foundation awarded matical methods to biological problems, 70 two-year unrestricted research grants, particularly those related to health and including 15 grants in mathematics to medical sciences, now exist in more than Maurice Auslander, Felix E. Browder, 20 universities throughout the country. Bernard M. Dwork, Paul R. Garabedian, Supported by training grant funds from the Harish-Chandra, Bertram Kostant, Serge Publich Health Service, NIH, these pro­ Lang, Peter D. Lax, David Lowdenslager, grams provide unusual opportunities for Jurgen K. Moser, C. D. Papakyriakopoulos, careers in teaching, research, and con­ Isadore M. Singer, Elias M. Stein, Richard sultation. Employment opportunities for G. Swan, John G. Thompson. biometricians are excellent, with the de• The NAS-NRC Postdoctoral Research mand by governmental and voluntary Fellowship Program supported by AFORS health agencies, medical research and of the Air Force Research Division has awarded fellowships in mathematics to, educational institutions, and industry run­ George Blakley, Herman R. Gluck, Donald ning far in excess of the available supply L. Iglehart. of trained personnel.

217 Programs of study are individually versitat Saarbrucken; Dr. Heinrich Sieden­ designed to lead to doctoral degrees, and topf, Universitat Tiibingen; Dr. Ewald in special instances, to other academic de­ Wicke, Universitii.t Munster; together with grees. Traineeship stipends are provided the editors of Friedrich Vieweg und Sohn. at various levels depending on previous Each manuscript submitted should education and experience of the trainee consist of a minimum of 100 typewritten and include allowances for dependents. pages and should lend itself for publica­ Substantially full economic support or par­ tion in book form. The manuscripts must tial support may be provided, depending be in the German language and may not upon the proportion of time spent in train­ have been published previously. German ing. or foreign scientists may enter the com­ Interested applicants may secure fur­ petition. ther information from Em marie C. Hemp­ The manuscripts must be submitted hill, Advisory Committee on Epidemiology before April 1, 1962 in a sealed, registered and Biometry, Publich Health Service, envelope marked "Vieweg-Jubilaums­ Bethesda 14, Maryland. Preis" and should be sent under a nom de plume. Accompanying the manuscript each competitor must send a separate envelope, also under the nom de plume, which con­ FORTY FELLOWSHIPS FOR AMERI­ tains the name and address of the author. CAN WOMEN are to be awarded by The Manuscript and letter are to be mailed to: American Association of University Wo­ Postfach 185, Verlag Friedrich Vieweg men Educational Foundation for the period und Sohn, Braunschweig, Germany. July 1, 1962, through June 30, 1963. The In the fall of 1962 the selection of the Fellowships may be used abroad or in the prize winners and the distribution of USA. prizes will take place. The Committee is Application forms may be obtained the sole authority to judge and distribute August 1, 1961, and must be filed by De­ the prizes. Should the Committee not cember 1, 1961,attheFellowships Office, award a prize in one or the other of the AAUW Educational Foundation, 2401 Vir­ three fields, it reserves the right to either ginia Avenue, N. W., Washington 7, D. C. award two prizes in one field or use the Notification of awards will be made March money for other scientific purposes. The 1' 1962. decision of the Committee is final, and scientists submitting manuscripts agree to its decision without recourse to law. Fur­ A SCIENTIFIC COMPETITION. On the ther, the applicants will grant Friedrich 1 75th anniversary of their publishing house Vieweg und Sohn the right to publish the Friedrich Vieweg und Sohn, Braunschweig, papers at the usual terms of the publish­ Germany, will award DM 15,000 for scien­ ers. This option expires six months after tific papers in the fields of mathematics, award of the prizes if the publisher does physics and chemistry. Awards of DM not exercise the option. 5,000 will be made in each field. The pa­ pers submitted should cover a subject in which articles have appeared in journals THE U. S. NAVAL RESEARCH LAB O­ but which have not appeared in the German RA TORY announces the creation of an Ap­ language in book form. plied Mathematics Staff as a part of the The Committee awarding the prize Office of the Director of Research. The consists of the following: Dr. Julius Bar­ staff is headed by Horace M. Trent. It tels, Universitat Gottingen: Dr. Walther features research programs on Numerical Gerlach, Universitat Miinchen; Dr. Wolf­ Analysis under the direction of Benjamin gang Haack, Universitii.t Berlin; Dr. Rolf Lepson, Mathematical Physics under Rich­ Huisgen, Universitiit Miinchen; Dr. Josef ard A. Toupin, and Optimization Tech­ Mattauch, Max- Planck - Om stotits fiir niques under Sanford P. Thompson. The Chemie; Dr. Wilhelm Quade, Technische staff includes a Research Computation Hochschule Hannover; Dr. Fritz Sawter, Center headed by Alan B. Bligh. Universitat Koln; Dr. Friedrich Seel, Uni-

218 ACKNOWLEDGMENTS TO REFEREES. Publication in the NOTICES of combined lists of referees has become an annual ceremony. This form of acknowledgment pro­ vides the referees with a measure of recognition, but at the same time protects their anonymity. The indebtedness of the mathematical community to the individuals serving in this capacity was well expressed by Dr. Franz L. Alt when he was Chairman of the Editorial Board of the Association for Computing Machinery: "The publication of scientific jour­ nals would be impossible without the dedicated and unselfish efforts of referees. The job of a referee is a thankless one. He must necessarily remain anonymous, while others participating in various stages of publicationcanatleast receive credit for their contri­ bution. To read manuscript from a referee's viewpoint is often unpleasant and always time consuming. The combined good judgment of the referees is essential to making a journal successful." The following two lists are based on information available at the Headquarters Offices of the Society as of April 30, 1961. Referees of the BULLETIN, NOTICES, the PROCEEDINGS, and the TRANSAC­ TIONS of the American Mathematical Society:

S. Abyankhar, Shmuel Agmon, R. ber, S. Ghurye, L. Gillman, A. Gleason, P. Agnew, A. A. Albert, WarrenAmbrose, I. Glicksberg, j. G. Glimm, G. Goes, C. R. D. Anderson, N. C. Ankeny, T. Apostol, Goffman, R. R. Goldberg, A. W. Goldie, Richard Arens, M. G. Arsove, M. Aus­ j. M. Gonzalez Fernandez, A. W. Good­ lander, William G. Bade, W. L. Baily, man, F. M. C. Goodspeed, D. Gorenstein, F. Sunyer i Balaguer, W. Barcus, Robert M. Goto, W. H. Gottschalk, L. M. Graves, Bartle, H. Bass, P. Bateman, j. D. Baum, L. W. Green, L. Greenberg, U. Grenander, L. Baum, Gilbert Baumslag, G. Baxter, Emil Grosswald, W. Gustin, F. Haas, A. R. A. Beaumont, E. F. Beckenbach, R. Haefliger, M. Hall, Paul Halmos, I. Hal­ Bellman, jerome Berkowitz, P. Billings­ perin, P. C. Hammer, Mary-Elizabeth ley, R. H. Bing, David Blackwell, j. R. Ham strom, F. Harary, D. K. Harrison, Blum, R. P. Boas, Jr., Salomon Bochner, A. Heins, E. Heinz, A. Heller, H. Helson, j. Boen, W. Boone, W. M. Boothby, L. S. E. Hemmingsen, M. Henriksen, R. Her­ Bosanquet, j. L. Brenner, H. j. Bremer­ mann, C. S. Herz, F. Herzog, M. Hetenyi, mann, A. Brown, E. H. Brown, Jr., M. Edwin Hewitt, G. Higman, K. A. Hirsch, Brown, N. G. de Bruijn, D. Buchsbaum, M. W. Hirsch, I. I. Hirschman, Jr., R. C. Buck, Herbert Busemann, R. H. F. Hirzebruch, U. Hochstrasser, K. Hoff­ Cameron, G. Cargo, L. Garlitz, j. W. S. man, K. H. Hofmann, C. C. Hsiung, S. T. Cassels, C. C. Chang, T. S. Chihara, G. Hu, j. A. Hummel, Gilbert Hunt, j. lgusa, Choquet, S. Chowla, A. Church, P. Civin, j. R. Isbell, K. Iwasawa, S. Izumi, N. j. A. Clarkson, A. H. Clifford, N. Coburn, jacobson, W. E. jenner, M. jerison, F. E. A. Coddington, H. Cohen, R. M. Cohn, john, R. E. johnson, B. jones, F. B. jones, P. Connor, A. Copeland, H. H. Corson, B. j6nsson, Mark Kac, R. V. Kadison, C. W. Curtis, M. L. Curtis, D. Darling, S. Kakutani, I. Kaplansky, S. N. Karp, Y. Chandler Davis, H. T. Davis, Martin Davis, Katznelson, j. L. Kelley, L. M. Kelly, P. Davis, R. L. Davis, M. M. Day, j. C. E. j. Kemeny, j. C. Kiefer, j. M. Kister, Dekker, A. Devinatz, D. Dickinson, j. V. Klee, T. Klotz, E. T. Kobayashi, S. Dieudonne, R. P. Dilworth, M. P. Drazin, Kobayashi, R. j. Koch, C. W. Kohls, L. E. Dyer, D. A. Edwards, S. Eilenberg, Kokoris, G. Kolettis, Paul Koosis, j. R. Ellis, D. Epstein, A. Erdelyi, J. P. Korevaar, A. Kosinski, Bertram Kostant, Evans, E. R. Fadell, K. Fan, Herbert H. W. Kuhn, R. E. Lane, C. E. Langenhop, Federer, A. Feldzamen, William Feller, j. P. LaSalle, E. Lehmer, j. Lehner, W. A. Fialkow, Nathan Fine, Harley Flanders, LeVeque, G. R, Livesay, Charles Loewner, W. Fleming, E. E. Floyd, K. Fan, M. K. A. j. Lohwater, L. H. Loomis, L. Lorch, Fort, T. Fort, R. H. Fox, W. C. Fox, T. T. G. Lorentz, D. Lowdenslager, E. Lukacs, Frankel, Avner Friedman, Bernard Fried­ G. Lumer, R. Lyndon, A. j. Macintyre, man, 0. Frink, W. H. j. Fuchs, G. Fuhrken, G. W. Mackey, S. MacLane, Wilhelm Mag­ D. Gale, Paul Garabedian, M.Gerstenha- nus, H. B. Mann, M. Marden, L. Markus,

219 T. Matsusaka, A. Mattuck, L. F .McAuley, D. Scott, W. R. Scott, W. T. Scott, A. Sei­ Charles A. McCarthy, N. H. McCoy, H. P. denberg, S. M. Shah, H. N. Shapiro, Harold McKean, j. E. McLaughlin, E. Mendelson, S. Shapiro, S. Sherman, A. C. Shields, Allan E. A. Michael, W. Mills, H. Mirkil, Shields, 0. Shisha, j. E. Shoenfield, Y. S. Mizohata, E. E. Moise, j. C. Moore, Sibuya, A. j. F. Siegert, E. Silverman, L. Mordell, C. B. Morrey, Harry Moses, A. Simon, j. M. Slye, M. F. Smiley, K. T. P. S. Mostert, G. D. Mostow, T. S. Motz­ Smith, L. j. Snell, E. H. Spanier, E. P. kin, S. Mrowka, F. J. Murray, j. R. Myhill, Specker, C. Spector, F. Spitzer, G. I. Namioka, Zeev Nehari, E. Nelson, A. Springer, I. Stakgold, j. Stallings, E. Stein, Nerode, B. Neumann, Louis Nirenberg, R. L. Steinberg, R. Stemmler, T. E. Stew­ K. Nomizu, R. Z. Norman, E. S. I. Niven, art, W. Stine spring, M. H. Stone, E. Straus, Northam, F. Oberhettinger, j. M. H. Olm­ G. Sunouchi, M. Suzuki, R. G. Swan, R. sted, P. Olum, 0. Ore, J. C.Oxtoby,F. Swann, G. Szego, 0. Taussky-Todd, A. E. Peterson, R. Phelps, Ralph S. Phillips, G. Piranian, R. L. Plunkett, F. Pollaczek, Taylor, W. j. Thron, j. Tits, L. Tornheim, M. H.Protter, F. Quigley, R. Rado, T. Rad6, L. B. Treybig, W. j. Trjitzinsky, J, L. D. Ray, G. E. Raynor, M. Reade, W. T. Ullman, Peter Ungar, C. T. C. Wall, A. D. Reid, I. Reiner, B. L. Reinhart, W. F. Rey­ Wallace, j. L. Walsh, H. C. Wang, L. E. nolds, R. D. Richtmyer, R. K. Ritt, M. S. Ward, Jr., Stefan Warschawski, W. R. Robertson, R. M. Robinson, H. Rogers, Wasow, C. E. Watts, A. Weinzweig, J. G. R. H. Rosen, A. Rosenberg, M. Rosenlicht, Wendel, B. Wendroff, J, Wermer, G. W. J. B. Rosser, H. Rossi, G.-C. Rota, E. H. Whitehead, D. V. Widder, H. Widom, A. Rothe, j. Rotman, L. A. Rubel, M. E. Wilansky, C. H. Wilcox, H. S. Wilf, R. F. Rudin, Walter Rudin, D. E. Rutherford:. Williams, j. Wolfowitz, C. B. Wright, R. Ryser, G. Sabidussi, Han Sah, P. F. B. Wright, H. Yamabe, Ti Yen, B. Yood, Samuel, L.Sario, E. Schenkman, P. Scherk, G. S. Young, L. C. Young, 0. Zariski, D. Hans Schneider, I. j. Schoenberg, L. Zelinsky, A. C. Zitronenbaum, H. Zucker­ Schoenfeld, A. Schwartz, J. T. Schwartz, man, A. Zygmund.

Referees of The Annals of Mathematical Statistics, the Canadian journal of Math­ ematics, the Duke Mathematical journal, the journal of the Association for Computing Machinery, the Mathematics of Computation, and the Michigan Mathematical journal.

j. H. Abbott, R. P. Agnew,FranzL. Conte, A. H. Copeland, F. J, Corbato, Alt, I. Amemiya, S. A. Amitsur, T, W. R. R. Coveyou, H. S. M. Coxeter, C. C. Anderson, Fred Andrews, F .j. Anscombe, Craig, Haskell B. Curry, E. Cuthill, R. J, Arms, F. V. Atkinson, D. D. Aufen• joseph Daly, H. E. Daniels, Herbert T. kamp, Maurice Auslander,Charles Baker, David, Martin Davis, Philip Davis, A. P. B. Banaschewski, E. Bareiss, Michael Dempster, Cyrus Derman, R. deVogelaire, Barnett, R. C. F. Bartels, Donald Bavly, R. P. Dilworth, N. J. Divinsky, J. Dixmier, P. R. Beesack, Robert J,Beeber, Richard W. J. Dixon, M. D. Donsker, W. S. Dorn, Bellman, Paul Benacerraf, D. Bernstein, j. Douglas, F. G. Dressel, G. F. D. Duff, Morton Bernstein, Patrick Billingsley, R. J. Duffin, Meyer Dwass, Eldon Dyer, R. H. Bing, Allan Birnbaum,David Black­ R. P. Eddy, H.P. Edmundson, Albert Edrei, well, G. Blanch, I. E. Block, E. K. Blum, Louis W. Ehrlich, C. C. Elgot, B. E. Elli­ julius R. Blum, R. M. Blumenthal, R. P. son, A. Erdelyi, M. A. Every, R. M. Fano, Boas, Wolfgang Borsch-Supan,Raoul Bott, jack Feldman, J, M. G. Fell, W. Feller, Hermann Bottenbruch, Alfred Brauer, Leo W. Fenchel, C. T. Fike, Evelyn Fix,T.M. Breiman, Patricia Bremer,Morton Brown, Flett, E. E.Floyd, George Forsythe, M. K. B. Brainerd, G. D. Bruce, F. E. Browder, Fort, Jr., F. G. Foster, D. A. S. Fraser, C. M. Bull, D. L. Burkholder, A. P. Cal­ K. 0. Friedrichs, R. H. Fox, Ray Fulker­ deron, j. M. Cameron, L.Carlitz,john W. son, D. Gale, T. M. Gallie, Jr., D. A. Carr, III, john W. Cell, J, H. H. Chalk, Gardiner, S. I. Gass, Walter Gautschi, Douglas G. Chapman, Herman Chernoff, Seymour Geisser, B. R, Gelbaum, J. J, K. L. Chung, Eckford Cohen, W. j. Coles, Gergen, S. G. Ghurye, E. N. Gilbert, P, E. Conner, WilliamS. Connor, Samuel Wallace Givens, C,Goffman,Karl Goldberg,

220 R. R. Goldberg, Alan Goldman, Moise H. Nadler, H. Nakano, E. Nelson, Raymond J. Goldstein, Jr., I. J. Good, R. A. Good, Leo Nelson, Morris Newman, B. Noble, Gott­ A. Goodman, D. Gorenstein, A. A. Grau, fried No ether, P. Obreanu, John 0' Connor, Robert T. Gregory,Julien Green,O. Gross, A. G. Oettinger, Ingram Olkin, Ascher E. Grosswald, Fred Gruenberger, A. P. Opler, W. Orchard-Hays, H. Oser, Donald Guinand, S. S. Gupta, John Gurland, F ,Haas, B. Owen, Emanuel Parzen, George W. Seymour Haber, Marshall Hall,Jr ., P ,Hall, Patterson, E. Paulson, J. Percus, B. J. P. R.Halmos, I. Halperin, Edward Halpern, Pettis, Ralph S. Phillips, K. C. S. Pillai, J. M. Hammersley, R. W. Hamming, James John Pratt, Noah Prywes, Hilary Putnam, Hannan, Morris Hansen, Frank Harary, Ronald Pyke, M. F. Quenouille, P. Rabino­ R. T. Harris, T. E. Harris, H.O.Hartley, witz, Hans Rademacher, E. D. Rainville, Juris Hartmanis, R. M. Hayes, E. V. M. S. Ramanujan, F. Raymond, R. D. Haynsworth, Peter K. Henrici, Leon Her­ Richtmyer, Dock Sang Rim, R. F. Rinehart, bach, R. T. Herbst, J. Heller, P. Henrici, J. H. Roberts, R. C. Roberts, M.S. Robert­ Edwin Hewitt, D. Higman, P. J. Hilton, son, G. de B. Robinson, Saul Rosen, P. G. I. I. Hirschmann, Jr., G. P. Hochschild, Rooney, A. Rosenberg, H. Rosenblatt, J. H. Hodges, Franz Hohn, Anatol W. Holt, Harold Ruben, Morris Rubinoff, Walter A. S. Householder, D. R. Hughes, J. S. Rudin, H. J. Ryser, H. E. Salzer, John Hunter, R. P. Hunter, E. J.Hutton,M.Hy­ Dennis Sargan, S. Saunders, I. R. Savage, man, Peter Z. lngerman, Kenneth Iverson, L. J. Savage, H. H. Schaefer, R. D. Schafer, Allan James, G. S. James, N. L. Johnson, Emil Schell, P. Scherk, C. Schubert, J. T. R. E. Johnson, F. B. Jones, Louis Joseph, Schwartz, H. Schwerdtfeger, I. E. Segal, M. L. Juncosa, R. V. Kadison, Irving Kap­ William W. Seifert, W. T. Sharp, J. C. lansky, S. Karlin, G. Karrer, Leo Katz, H. Shaw, John Sheldon, G. C. Shepard, Peter Keller, J. L. Kelley, R. R. D. Kemp, Oscar B. Sheridan, F. A. Sherk, Joseph R. Shoen­ Kempthorne, M. G. Kendall, Harry Ken­ field,Robert Singleton,Rosedith Sitgreaves, sten, Ellen Kerksieck, Jack Kiefer, R. J. Walter Smith, E. Snapper, E. H. Spanier, Koch, L. A. Kokoris, Charles H. Kraft, D. C. Spencer, Frank Spitzer, G.P.Steck, G. Kuby, H. W. Kuhn, S. Kullback, R. G. Charles M. Stein, B. M. Stewart, P. Swirl­ Laha, John Lamperti, Peter Lax, Lucien ing, William Taylor, Herbert M. Teager, LeCam, C. N. Lee, C. Y. Lee, Chester Lee, Henry Teicher, D. Teichroew, G. H. M. E. L. Lehmann, D. H. Lehmer, Joseph Thomas, J. M. Thomas, Neal Throckmor­ Lehner, W. Leighton, Anne Lester, W. J. ton, G. Tintner, C. J. Titus, Fred Tonge, LeVeque, Melvin H. Lieberstein, J. Leib­ George R. Trimble, Jr., William Turanski, lein, C. Lindholm, D. Livingstone, L. H. W. T. Tutte, Maurice C. K. Tweedie, K. Loomis, A. N. Lowan, Eugene Lukacs, Uncapher, Stefan Vajda, J. R. Vanstone, John McCarthy, J. McCluskey, R. M. Mc­ Richard S. Varga, E. A. Walker, H. S. Leod, Robert McNaughton, A. M. Mac­ Wall, A. D. Wallace, David L. Wallace, heath, C. C. MacDuffee, A. Madansky, Hans Willis Ware, Seth L. Warner, Wolfgang J. Maehly, Benoit Mandelbrot, H. B. Wasow, G. S. Watson, W. J. Webber, Lionel Mann, Marvin Marcus, M. E. Maron, Harry Weiss, B. L. Welch, J. G. Wendell, Peter Markowitz, E. Mason, J. !}. Mauldon, Whipple, H. Whitney, G. T. Whyburn, D. V. George Mealy, M. A. Melkanoff, John P. Widder, Robert A. Wijsman, A. Wilansky, Menard, N. Mendelsohn, J. C. P. Miller, M. B. Wilk, Philip Wolfe, J. Wolfowitz, Rupert Miller, W. E. Milne, Marvin Min­ E. M. Wright, Fred Wright, David M. sky, Edward Moore, P. G. Moore, Lincoln Young, Jr., D. H. Young, Gail Young, 0. E. Moses, Georg Mostow, Thomas H. Mott, Zariski, Marvin Zelen, H. S. Zuckerman, Jr., T. S. Motzkin, F. D. Murnaghan, Jack B. Zumino, Georg Zyskind.

221 THE MATHEMATISCHE FOR- mathematics credits will be spread as SCHUNGSINSTITUT OBER WOLFACH held broadly as possible across the various a colloquium on Abelian groups March 5- fields of advanced mathematics rather than 10, 1961. The organizing committee, con­ largely concentrated in one area, as in the sisting of L. Fuchs (Budapest) and F. W. case for a Ph. D. candidate in mathemati­ Levi (Freiburg), arranged for the partici­ cal research. pation of 15 mathematicians from England, The fifteen required credits will in­ France, Holland, Hungary and Germany. clude courses in thehistoryofmathemati­ In addition, there was a colloquium cal thought, readings in the masterworks March 12-17 on Partial Differential Equa­ of mathematics, methods. and materials in tions under the chairmanship of W. Haack teaching college mathematics, higher edu­ and G. Wellwig of Berlin. the program cation in American society, and adolescent included Z8 addresses; three were given psychology. by the American participan~s. R. Courant, B. The second difference will be the F. john and j. Moser. requirement of field teaching in colleges under the supervision of the faculty of the Department of Mathematics and Science Education. NEW DEGREES FOR TEACHERS. Two C. The third difference will be in the Yeshiva University graduate schools ha nature of the doctoral dissertation to be joined forces in a new program to train submitted. For the doctorate in mathe­ college and high school mathematics teach­ matics, the student must submit a disser­ ers. The program, which will get under­ tation demonstrating ability to conduct way this summer, is designed to alleviate original mathematical research. For the the critical shortage of college teachers of new degree, the dissertation must be a mathematics and to help make possible a serious analytical or expository investi­ nationwide reform of high school mathe­ gation of some important portion of mathe­ matics programs as recently recommended matical thought. The dissertation will be by the governing bodies of MAA and AMS. jointly supervised by a research mathe­ The new Department of Mathematics matician and by a specialist in mathe and Science Education combines the facili­ matic s education. ties of the Yeshiva University Graduate School of Science and Graduate School of Similarly, the requirements for the Education. Dr, Abe Gelbart, dean of the new Master of Science degree in Mathe­ Graduate School of Science, will head the matics Education will require several new program and chair the department. special courses as well as three more cre­ Requirements for the new Ph.D. in dits than the current Master of Science Mathematics Teaching, to be offered by degree from the Yeshiva University Gradu­ the program, will differ from the Ph. D. ate School of Science. in Mathematics offered by the Graduate The required courses include fifteen School of Science and other universities credits in mathematics content and six in three basic respects: credits in professional education. Twelve additional elective credits in mathematics A. The first difference is in the dis­ must be taken, bringing the total to 33 as tribution of required credits. These will compared with 30 in the Graduate School include a special core of required courses of Science. particularly pertinent to the needs of col­ The Master of Science program has lege-level teachers. A total of 90 credits been developed in line with the newest New beyond the bachelor's degree will be re­ York State certification requirements. quired -- 54 in mathematics, 15 in the Prerequisite to admission to the M. S. special core program, 6 in electives, and program is one year of satisfactory study 15 for the dissertation. The fifty-four in calculus.

222 PERSONAL ITEMS

Mr. T. L. AUSTIN of Technical Opera­ accepted a position as mathematician at tions Incorporated, Fort Monroe, Virginia, the Federal Aviation Agency, Atlantic City, has accepted a position as manager at New Jersey. Operations Systems Incorporated, Haw­ Dr. W. C. HOFFMAN of the University thorne, California. of Queensland, Australia, has accepted a Dr. C.-Y. CHAO of the University of position as member of the technical staff Michigan, has accepted a position as re­ at the Boeing Scientific Research Labora­ search mathematician with the Interna­ tories, Seattle, Washington. tional Business Machines Corporation, Professor A. 0. HUBER of the Swiss Yorktown Heights, New York. Federal Institute of Technology, has been Assistant Professor A. J. COLEMAN appointed to a professorship at the Eidge­ of the University of Toronto, has been ap­ noessische Tech. Hochschule, Zurich, pointed to head of the mathematics depart­ Switzerland. ment at Queen's University, Kingston, Mr. R. KASSLER has accepted a po­ Ontario, Canada. sition as mathematical analyst at the Dr. T. B. CURTZ of the University of Chrysler Corporation, Detroit, Michigan. Michigan, has accepted a position as man­ Miss B. C. McKEON of Georgetown ager of the computing laboratory at the University, has been appointed to a re­ Conductron Corporation, Ann Arbor, Mich­ search assistantship at Johns Hopkins igan. University. Assistant Professor A. DAIGNEAULT Dr. L. A. MACCOLL of the Bell Tele­ of the University of Ottawa, has been ap­ phone Laboratories, New York, will retire pointed to an assistant professorship at as of April 1, 1961. After September 1, the University of Montreal, Montreal, 1961 he will be appointed to a professor­ Canada. ship at the Polytechnic Institute of Brook­ Mr. J. R. DEAN of Technical Opera­ lyn. tions Incorporated, Fort Monroe, Virginia, Dr. W. S. MARTINDALE of the Univer­ has accepted a position as mathematical sity of Chicago, has been appointed to an specialist at Lockheed Aircraft Corpora­ assistant professorship at Smith College. tion, Sunnyvale, California. Assistant Professor S. MIZOHATA on Mr. J. R. ENTERLINE of Western leave from Kyoto University, Japan, has Electric Company, Incorporated, New returned after a leave of absence at New York, New York, has accepted a position York University. as senior engineer at the Systems Research Dr. G. W. MORGENTHALER of The Group, Incorporated, Mineola, New York. Martin Company, Denver, Colorado, has Dr. R. E. FAGEN of the Hughes Air­ been appointed to a visiting professor ship craft Company, Culver City, California, at the University of Colorado. has accepted a position as director of the Assistant Professor T. P. MULHERN information sciences division at the Amer­ of Fordham University, has accepted a po­ ican Systems Incorporated, Hawthorne, sition as mathematician with the Shell Oil California. Company, New York, New York. Assistant Professor J. K. GOLD­ Professor L. NACHBIN of Brandeis HABER of Washington University,has been University, has been appointed to a visit­ appointed to a research associate profes­ ing professorship at the Faculte des sorship at the University of Maryland. Sciences de Paris for the academic year Mr. A. J. HALTMAIER of Oceanside 1961-1962. Senior High School, Oceanside, New York, Professor H. NAKANO of Hokkaido has been appointed chairman of the mathe­ University, Sapporo, Japan, has been ap­ matics department at the West Islip Public pointed to a visiting professorship at School, West Islip, New York. Queen's University, Kingston, Ontario, Mr. F. J. HARDY of the Naval Post­ Canada. graduate School, Monterey, California, has Professor P. B. NORMAN of Long

223 Island University, has accepted a position geles, California. as member of the technical staff at the Mr. G. E. SACKS on leave from Cor­ Aerospace Corporation, El Segundo, Cali­ nell University, has been appointed to a fornia. visiting fellowship at Princeton University. Dr. F. C. OGG, research scientist at Mr. A. SEIKEN of Southern niinois johns Hopkins University, mathematician University, has been appointed a temporary at McCoy College, and consultant at the research assistant at the University of Martin Company and Aeronca Manufactur­ Michigan. ing Company, is no longer at McCoy Col­ Professor M. E. SHANKS, on leave lege, the Martin Company, or Aeronca from Purdue University, has been appoint­ Manufacturing Company. ed to a visiting professorship at the Uni­ Mr. P. D. OYER of the Defense De­ versity of North Carolina. partment, Fort Meade, Maryland, has ac­ Mr. C. j. SMITH of SperryRandCor­ cepted a position as data processing spe­ poration, St. Louis, Missouri, has accepted cialist and mathematical statistician at a position as associate engineer with the the Westinghouse Electric Corporation, McDonnell Automation Center, St. Louis, Baltimore, Maryland. Missouri. Dr. N. P ADMA of Annamalai Univer­ Mr. A. S. STANKOVICH oftheSystem sity, India, has been appointed to a tempo­ Development Corporation, Paramus, New rary assistant professorship at Conne.cti­ jersey, has accepted a position as opera­ cut College. tions analyst at the General Electric Com­ Assistant Professor M. PAP ADO­ pany, Philadelphia, Pennsylvania. POULOS of Brown University, has been ap­ Mr. F. SUP NICK has been appointed to pointed a Senior Lecturer at the University an assistant professorship at City College, of Melbourne, Australia. New York. Mr. R. F. PAVLEY of Syracuse Uni­ Dr. S. G. VANDENBERG of the Uni­ versity, has accepted a position as sys­ versity of Michigan, has been appointed tems engineer at the Radio Corporation to an associate professorship in the De­ of America, Moorestown, New jersey. partments of Pediatrics and Psychology, Associate Professor S. PERLIS, on and as associate director of the Twin leave from Purdue University, is serving Study Child Development at the University temporarily as assistant program director of Louisville. for mathematics at the National Science Professor j. V. WEHAUSEN, on leave Foundation, Washington, D. C. from the University of California, Berke­ Dr. j. D.PINCUSofNewYorkUniver­ ley, has been appointed to a visiting pro­ sity, has accepted a position as associate fessorship at the University of Hamburg, mathematician with the Brookhaven Na­ Germany. tional Laboratory, Upton, New York. Dr. G. G. WEILL of the University of Assistant Professor A. F. PIXLEY California, Los Angeles, has accepted a of the College of Notre Dame, has been position as research fellow at Harvard appointed to an assistant professorship at University. San Francisco State College. Dr. I. j. WEINBERG of Massachusetts Dr. A. RALSTON of the American Institute of Technology, has accepted a Cyanamid Company, New York ZO, New position as staff scientist with Avco, Wil­ York, has been appointed to an associate mington, Massachusetts. professorship at Stevens Institute of Tech­ Dr. M. j. WONENBURGER has ac­ nology. cepted a position as research fellow with Mr. D. j. ROSS of Scientific Planning the National Research Council, Ottawa, Associates Corporation, Silver Spring, Ontario, Canada. Maryland, has accepted a position as oper­ Dr. D. A. WOODWARD of the Univer­ ations analyst at General Electric Com­ sity of Minnesota, has accepted a position pany, Philadelphia, Pennsylvania. as assistant mathematician with the Ar­ Mr. D. ROTHMAN of North American gonne National Laboratory, Argonne, Aviation, Incorporated, has accepted a po­ lllinois. sition as member of the technical staff at Electronics Specialty Company, Los An- The following promotions are announced:

224 S. A. AMITSUR, The Hebrew Univer­ Mr. C. C, CONLEY of Massachusetts Insti­ sity of jerusalem, to a professorship. tute of Technology, has been appointed a W. A. CRABTREE, Austin Peay State fellow of the Institute of Mathematical Sci­ College, to an assistant professorship. ences, New York University. A. B. CUNNINGHAM, West Virginia University, to a professorship. The announcement on page 36 of the T. P. DENNEHY, john Carroll Uni­ February issue of the NOTICES concern­ versity, to an assistant professorship. ing Mr. R. B. RICE of the Ohio Oil Com­ E. H. FELLER, UniversityofWiscon­ pany should read as follows: R. B. RICE, sin, to an associate professorship. Ohio Oil Company, to supervisor of the M. J. GREENBERG, University of physics department. California, Berkeley, to an assistant pro­ fessorship. K. HONDA, St. Paul's University, Tokyo, Japan, to a professorship. S. LEE, Western Michigan University, to an associate professorship. P. MALLIA YIN, University of Caen, France, to a professorship. Dr. N. F. G.MARTIN,Universityof Virginia, to an assistant professorship. Dr. R. H. MOORE, University of Wis­ consin, to an assistant professorship. R. D. RIGHTMYER, New York Univer­ sity, to a professorship and to director of the A. E. C. Computing and Applied Mathe­ matics Center. P. G. ROONEY, University of Toronto, to an associate professorship. Y. TOMONAGA, Utsunomiya Univer­ sity, Japan, to a professorship.

The following appointments to instructor­ ships are announced:

University of North Carolina: Dr. J. D. BUCKHOLTZ; University of Chicago: ,, Dr. C. C. MANER!; University of illinois: . ,. WEICHSEL; University of Texas: Dr. P.M. ~-R.flmt.r-HotJ 1 /I.L-F.. ,s Mrs. E. H. PEARSON.

Deaths: ,·'J our , 4111· .J, 5· :J, ~

Professor F. E. ALLEN of Madison, Wisconsin, died on December 31, 1960 at ?tease sePJ J'"vr tJrdlr the age of 84. He had been a member for to : 46 years. Mr. F. MARAN! of Rome, Italy died on March 9, 1961 at the age of 53. Gd wa rtf11 ~rtJthf'r,,J,,, ERRATA The announcement on page 127 of the Clnn tf,. bor,JJI/,ft~411 April issue of the NOTICES concerning Mr. C. C. CONLEY of Massachusetts Institute of Technology, should read as follows:

225 LETTERS TO THE EDITOR

Editor, the NOTICES The great danger is that in the face of the large demand for college teachers of After reading the details of the new mathematics, the de facto requirements proposal for a Doctor of Arts degree in for the new degree will fall rapidly. More mathematics, I was left with a feeling of precisely, these students will probably not incredulity thatthe Council of the Society be given examinations of the same difficulty has endorsed so hastily a program fraught as prospective Ph.D. candidates and there with danger. How can such eminent mathe­ is intense danger thatthe "critical, histori­ maticians endorse such a proposal without cal, or philosophical" dissertation re­ investigating the premises on which it is quirement may degenerate into a grotesque based or looking for safeguards to prevent joke. There is also danger that many stu­ it from degenerating into a crude mecha­ dents capable of writing standard disserta­ nism for giving the title of "Doctor" to un­ tions who are lacking in self-confidence qualified aspirants? I looked in vain for may choose this "safer" route to a Doc­ evidence that careful thought had been tor's degree, thereby decreasing the sup­ given to the social consequences ofhaving ply of Ph.D.'s. Even those who feel that I two different academic credentials with exaggerate the dangers involved will surely the title of "Doctor." Will the search for grant their possibility. Is it safe to turn academic panaceas never end? such a vague proposal loose on so many What evidence is there that lack of universities with such varying standards creative ability is the big bottle-neck in the without any effort at building in safe­ training of Ph.D.'s in mathematics? I guards? would guess that ten times as many stu­ dents have failed to earn a Ph.D. because What about the sociology of the situa­ of inability to pass courses or preliminary tion? Are the holders of the new degree to examinations than students who have be second-class academic citizens who passed these examinations and could not follow orders issued by their big brothers write dissertations. It was not so long ago with Ph.D.'s? If not, are they capable of that financial problems contributed con­ designing curricula to meet the needs of siderably to the decimation of the graduate tomorrow as well as today? Is it not likely student population. While the financial lot that hostility will develop between the two of the graduate student has improved con­ kinds of degree holders? siderably over what it was in the bad old There is a very real shortage of tal­ days, is it really so rosy today? I don't ented teachers of mathematics in this claim that these last named factors ex­ country. This shortage can be overcome plain everything, but may I not ask with only by attracting more talented young fairness that the proponents of the new de­ people into mathematics teaching. There­ gree back up their diagnosis with facts cent improvements in salaries and working and figures? conditions for mathematicians in colleges Almost all the articles of a critical, has just begun to have some effect. A cor­ historical, or philosophical nature found responding improvement in the lot of the in such expository journals as the Ameri­ graduate student would help. (In so many can Mathematical Monthly and Scripta places, he is cheap academic labor first Mathematica are written by mathemati­ and a student second.) Respectabilizing cians who have had the Ph.D. for some substandard products by giving them fancy time. This leads me to believe that writing titles can onlyhinderprogress. Let us stop a dissertation of the nature proposed that looking for panaceas and face the problem is worth the paper it is printed on is con­ realistically. There are a lot of talented siderably more difficult than it is to do the young Americans with creative talent. They research necessary for a Ph.D. disserta­ should be found and trained. tion of average quality. I sincerely hope that the Society will

226 reconsider this matter before real harm is methods is not nearly so simple as the done. above quotation suggests. Following iso­ Melvin Henriksen lated contributions, back as far as the eighteenth century, by various authors, Karl Pearson's initial publication on chi­ square methods (1900) began the first Editor, the NOTICES major chapter in the nonparametric field. In the February 1961 issue of these Soon afterwards, Spearman and others NOTICES the following appeared on page published on rank correlation methods; an 29 of an article by j. P. La Salle: important paper from the United States was that of Hotelling and Pabst (1936), whose " •.• in mathematical statistics, scien­ title may be compared with the above quo­ tists in [the United States] concentrated on tation: "Rank correlation and tests of sig­ the development of so-called parametric nificance involving no assumption of nor­ methods, based on the assumption of a mality." In England, Fisher's text, The normal distribution in the population at Design of Experiments (First Edition, hand. In Russia, however, the main interest 1935) contained a suggestion that initiated was directed towards statistical methods the energetic study of permutation tests, which avoided the assumption of normality a central topic of nonparametric analysis. and this resulted in the development of Russian work in the nonparametric what is now known as nonparametric area might be thought of as beginning with methods. After some tinie the results so the Chebychev-Bienayme inequality and its obtained were also appreciated in the many variations, a little after the middle of United States, and the study of this new the last century. The first clearly statisti­ statistics was undertaken here. branch of cal step in nonparametrics by Russian sci­ But the neglect of the parametric methods entists was the publication of a few very was only recently ended in Russia. Now, important papers by Glivenko (1933), Kol­ however, both parametric and nonpar a­ mogorov (1933), and Smirnov (1936). methods are taught in Russia." metric Since World War II, nonparametric We feel that this statement is mislead­ methods have been actively studied ing, for the following reasons. First, there throughout the world. is a tradition of research on parametric So far as we know,mostRus.sian work methods in Russia, for example in least in mathematical statistics and probability squares. Although the theory of least has been from the viewpoint of pure proba­ squares was primarily developed by Gauss, bility, rather than that of statistical infer­ contributions were made by later writers ence. Of Russian work in nonparametric in many countries; perhaps the most emi­ methods, a larger proportion (but not a nent Russian worker in this field was A. A. larger amount) than in other countries Markov. Another area, one outside the interests the pure mathematician. dichotomy, in parametric-nonparametric William Kruskal exists, which a Russian research tradition Frederick Mosteller surveys (see S. S. is that of sample I. Richard Savage Zarcovic, j. Roy. Stat. Soc. A, Vol. 119 (1956), p. 336). Second, the above quotation wrongly Editor, the NOTICES methods are in­ implies that parametric At the business section of the national trinsically connected with the normal dis­ meeting in Chicago in January, 1960, I tribution. While they are perhaps most asked that the Society return to its former often normally oriented, there are many in late December on parametric methods connected with other policy of meeting remarks were families of distribution: negative exponen­ college campuses. My tial, Poisson, binomial, etc. Methods con­ greeted with applause and there were sev­ nected with the Pearson family of distri­ eral supporting speeches. No one present butions do not assume normality, but they supported the idea of continuing to meet are parametric. in hotels. While it is true that the accom­ Third, the history of nonparametric modations afforded us at the Willard in

227 Washington this year were much better a step in a very wrong direction. than those afforded by the Conrad Hilton William M. Perel in Chicago the previous year, I would like to report that my opinion has not changed. john Green, in the February, 1961 Editor, The NOTICES issue of the NOTICES, p. Z 1, discusses this There are several comments I would matter .All of his arguments are predi­ like to make concerning Professor Perel 's cated on the assumption that our national recent letter concerning the time and place meetings must be held in january. May I of Winter Meetings. At the time of the 1960 remind him that the between semester meeting these became a matter of great break does not come at the same time at concern to both the Society and Associa­ all universities and that those universities tion, and in early 1960 a joint committee which operate on the quarter system are of the two organizations was appointed to not helped at all. study these problems, and in particular, to Several comments of Mr. Green are reassess the decision to hold Winter Meet­ interesting. I agree that it is desirable that ings in late january. This Committee con­ meetings be held in places which are read­ sisted of Professor R. H. Bing, J. W. T. ily accessible, but there are universities in Youngs, and A. D. Hedlund, the latter being our metropolitan areas. The Washington chairman. The Committee reported at the meeting could have been held at the Uni­ January meeting this year in Washington, versity of Maryland and the Chicago meet­ and it was its recommendation that we ing could have been held at the University should continue with the late January date. of Chicago. Not all colleges are inacces­ The Council of the Society and the Board of sible and most are more easily reached Governors of the Association approved the than many resort hotels. report. It is true that the January date Mr. Green's reference to "excellent causes some inconvenience,particularly to blackboard facilities, which are sometimes those of schools on the quarter system, but available in universities" must have been holding meetings during the Christmas intended as a joke. Surely, any university season has disadvantages also. It certainly in the country could provide "excellent cannot be claimed that the january date blackboard facilities" and also provide damages attendance at our meetings. Ifwe better places for us to meet than the aver­ made our meetings much more convenient age hotel. to attend, we would have to hire a conven­ Mr. Green feels that we have not yet tion hall. had sufficient experience with hotel meet­ Now the location of Winter Meetings. ings. We have now had four of them, and I After the Chicago meeting it was clear that feel that is sufficient experience. there was a considerable sentiment in the At the Chicago meeting, Mr. Green Society in favor of campus meetings. Also indicated that he would seek an expression the Committee referred to above urged of opinion from the membership. He has that campus facilities not be overlooked. not done so, I feel that the officers of the At that time the 196Z January meeting had Society have behaved in a most autocratic already been scheduled, however as a and undemocratic manner in this area. Mr. direct result of the sentiments expressed Green had an expression of opinion at the at Chicago, arrangements were shortly Chicago meeting, which he has chosen to ignore. after made to hold the 1963 january meet­ In short, I ask my fellow members of ing on the Campus of the University of the Society to join with me in demanding a California, Berkeley, (though not the living return to meeting during Christmas week accommodations). When I left the U. S. A. and to meeting on a college campus. If it in January, prospects were bright for a rains every day in Berkeley, I would only 1964 January meeting on a college campus. question why our first campus meeting in However I feel that we shall continue to five years had to be held in one of the meet at hotels when we cannot get the right rainier parts of our country. And may I kind of facilities on campuses or when add that the current attempt to move from suitable living accommodations very con­ city hotels to resort hotels strikes me as venient to campuses cannot be arranged.

228 I hope that the above will permit the in the Notices, the failure of wives of reader to judge whether or not Professor Russian mathematicians to go to inter­ Perel's remarks concerning my behavior national meetings, the random walk of Erdos, the social status of programmers, and that of other Society officers (and pre­ and in particular, the lapses of other letter­ sumably those of the Association) are justified. writers. Dr. Bridgland would, I think, have been In conclusion, I would like to add three Dr. comments. One is that I was not joking perfectly in place in controverting for example, when I referred to "excellent blackboard Lorch's "propaganda". I was, about the facilities that are sometimes available in interested in Lorch's remarks Berkeley universities." Another has to do with Pro­ East European statisticians at the fessor Perel' s remark that the Washington symposium, which I had not known about. case meeting could have been held at the Univer­ Now there is a rather reasonable sity of Maryland and the Chicago meeting that can be made for the State Department's else, could have been held at the University of policy. Dr. Bridgland, or someone helped to Chicago. Is Professor Perel absolutely could have made this case, and certain that the University of Chicago, for clarify our understanding of the issues. example, has facilities available for a Continued discussion could conceivably stand against meeting of the size under consideration? lead to the Society's taking a a I am far from certain, myself, and if there the State Department's policy, or taking of help­ is anything I have learned from dealing with stand for it, perhaps as a means Before arrangements for meetings it is to take ing Kazarinoff get to Novosibirsh. nothing for granted. The third comment is I would vote to put the Society on record see one to which I feel I am entitled as a long for either position, I would want to time California resident, and refers to the overwhelming evidence that the position such possibility of rain in Berkeley. My point was correct. But I believe that, given there was that in Berkeley we shall meet evidence, it is the duty of the Society, to take in one place and sleep in another and pro­ mathematics and to the country, to tracted rain is much more of a nuisance such a stand. The more discussion there column, the there than if we were meeting in a single was beforehand in the Letters building such as a hotel. The possibility of happier I would be. much rain would also exist almost any­ It seems to me that I have implicitly where else in the United States. It would evolved a rule that admits Lorch's letters, cover certainly exist in Death Valley, Chicago or and Martin's letter; it does not would New York. Bridgland' s letter or mine. But I like to propose a more general rule that John Green does let our letters in: Any member of the Society can write Editor, the NOTICES a letter to the Notices about anything which has any tenuous connection with mathe­ I looked up the two letters by Lee matical activity. He is entitled to display Lorch that T. F. Bridgland, Jr., complained a quite unreasonable amount of emotion, about in the February, 1961, Notices, since short of libel. Any other member can write they had not seemed out of line to me on my about the first letter, and the first writer original reading. They still don't. can reply, and maybe the second can reply, There has been an astonishing variety and once in a while, the first can reply of material covered in the Letters: The again. The Editor will ex officio have com­ manuscript requirements of certain edi­ mon sense and use it. tors, the desirability of voting in national elections, the danger of insulting printers, the use of the term" Iron Curtain countries" G. S. Young

229 MEMORANDA TO MEMBERS

THE EMPLOYMENT REGISTER by which members of each may become members of the other by The Mathematical Sciences Employ­ paying half the regular dues. The regular ment Register, established by the Ameri­ dues of mem• bers of the Edinburgh can Mathematical Society, the Mathemati­ Mathematical So­ ciety are 21 shillings a year; cal Association of America, and the Society therefore an American Mathematical for Industrial and Applied Mathematics, Society member would pay $1.50 a year. Privileges will be maintained at the Summer Meeting of mem­ bership include at Oklahoma State University, Stillwater, receipt of two parts ofthe Proceedings each year Oklahoma, on August 29, 30 and 31, 1961, (four parts consti­ tuting a volume). Each volume The Register will be conducted from 9:00 carries an issue of the Edinburgh Mathematical A.M. to 1:00 P.M. and 2:00P.M. to 5:00 Notes. Those members of P.M. on each of these three days. the American Mathe­ matical Society wishing There is no charge for registering to take advantage of this arrangement either to job applicants or to employers, should write to The Honorary Secretary, Edinburgh except when the late registration fee for Mathe­ matical Society, Mathematical employers is applicable. Provision will be Institute, 16 Chambers Street, Edinburgh made for anonymity of applicants upon re­ 1, Scot­ land. quest and upon payment of $1 to defray the It is understood that members cost involved in handling anonymous list­ under ings. the reciprocity agreement spending time in the other country should pay Job applicants and employers who wish the regular dues while they are there. to be listed will please write to the Em­ ployment Register, 190 Hope Street, Providence 6, Rhode Island, for application forms and for position description forms, which must be completed and returned to Providence not later than August 4, 1961, ADDRESSES OF AUTHORS in order to be included free of charge in OF ABSTRACTS the listings at the meeting in Stillwater, Oklahoma. Forms which arrive after this For the convenience of those who closing date, but before August 14, will be would like to correspond with the author included in the register at the meetingfor of an abstract, the addresses of authors a late registration fee of $3.00, and will will be included beginning with the August also be included in theprintedlistings, but issue of the NOTICES. The address used not until ten days after the meeting. The will be the Mailing Address that the author printed listings will be available for distri­ (or authors) has listed on the Abstract bution both during and after the meeting. Form. It is essential that applicantsandem­ ployers register at the employment regis­ ter desk promptly upon arrival at the meet­ ing to facilitate the arrangement of appoint­ ments, RETIRED MATHEMATICIANS AVAILABLE FOR EMPLOYMENT The Headquarter Offices of the Ameri­ RECIPROCITY AGREEMENT WITH THE can Mathematical Society announce the EDINBURGH MATHEMATICAL SOCIETY release on May 1, of a supplement to the List of Retired Mathematicians Available The American Mathematical Society for Employment, which was issued Febru­ has entered into a reciprocity agreement ary 1, This list is available upon request with the Edinburgh Mathematical Society to the Special Projects Department.

230 NEW PUBLICATIONS

The section NEW PUBLICATIONS will be discontinued in the NOTICES. Arrange­ ments are being made for distribution of this listing concurrently with future issues of the NOTICES to all Institutional Members of the Society. Individual and Corporate Mem­ bers may obtain copies of the list upon request, which should be addressed to the Special Projects Department of the Headquarters Office. The list of New Publications (which .is derived chiefly from books received by MATHEMATICAL REVIEWS for review) has grown to such an extent that it would have required over thirteen pages in the current issue of the NOTICES. Since it is believed that its principle value lies in its serving as a check list for institutions concerned with acquiring new publications for their library, the Society will undertake to maintain this service to its institutional members by mailing the listing of New Publications to them. A listing of books published by the Society will be carried in these NOTICES and continue in future issues. NEW AMS PUBLICATIONS

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS, Vol, 11 NUCLEAR REACTOR THEORY 339 pages; $8.70 List Price; 25o/o discount problems encountered in this fascinating to members. Edited by Garrett Birkhoff field, As a by-product, it may help to put and Eugene P. Wigner. the design of future nuclear reactors on a The current era has been described as more scientific basis. "the atomic age", and it seems probable The contributors to this volume having that mankind will depend increasingly on already done their part, we hope that both nuclear energy during the next century. In pure and applied mathematicians will ac­ the design of nuclear reactors, mathemati­ cept the challenging invitation offered, cal analysis already plays an important thereby continuing the great tradition of role. Archimedes, Newton, Gauss, Fourier, Nevertheless, very few research Maxwell, Poincare, and many others. In mathematicians have so far devoted seri­ this great tradition, each new major field ous effort to the mathematical problems of physical application has both suggested of nuclear reactor theory. The present fundamental new mathematical concepts, volume is intended to increase the number and has owed much of its deeper develop­ of such mathematicians, by indicating the ment to the rigorous mathematical analy­ great variety of interesting mathematical sis of these concepts.

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS, Vol, 12 THE STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS 283 pages; $7,80 List Price; 25% discount Haskell B. Curry; Yuen Ren Chao, Murray to members. Edited by Roman Jakobsen. Eden; Morris Halle; Robert Abernathy; The twenty articles in this book are Hans G. Herzberger; Anthony G. Oettinger; texts of addresses which were delivered at Victor H. Yngve; Gordon E. Peterson and the symposium held in April, 1960. Frank Harary; Joachim Lambek; H. A. The authors contributing papers to this Gleason, Jr.; Benoit Mandelbrot; Charles book are: W. V. Quine; Noam Chomsky; F. Hockett; Rulon Wells; Roman Jakobsen. Hilary Putnam; H. Hiz; Nelson Goodman;

PROCEEDINGS OF SYMPOSIA.IN PURE MATHEMATICS, Vol, 2 LATTICE THEORY

208 pages; $6.30 List Price; 25% discount The eighteen articles in this volume to members. Edited by R. P. Dilworth, are the papers presented at the Symposium

231 on Partially Ordered Sets and Lattice perin, B. j6nsson, K. D. Fryer, J, E. Mc­ Theory held in April, 19 59. Laughlin, Leon Henkin and Alfred Tarski, The authors contributing papers to P. R. Halmos, C. C. Chang, R. S. Pierce, this book are: R. P. Dilworth, P. M. Philip Dwinger, Garrett Birkhoff, Mar shall Whitman, juris Hartmanis, R. A. Dean, Hall, jr ., L. W. Anderson, F. W. Anderson. C. C. Chang and Alfred Horn, Israel Hal-

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, Vol. 3 DIFFERENTIAL GEOMETRY 201 pages; $7.60 List Price; 250/o discount the tensor analysis of the differential to members. Edited by C. B. Allendoerfer. geometry of the 1930's. The Symposium on Differential Geom­ The papers in this volume give a etry was organized as a focal point for the cross- section of many of the types of dif­ discussion of new trends in research. ferential geometry of major current in­ Modern differential geometry has become terest; differential topology, Lie groups, to a large degree differential topology, and complex manifolds, fiber bundles, and dif­ the methods employed are a far cry from ferential geometry in the large.

SELECTED TRANSLATIONS IN MATHEMATICAL STATISTICS AND PROBABILITY A new Series, published for the Insti­ contains 25 papers. 306 pages; $4.80 List tute of Mathematical Statistics by the Price; 25o/o discount to members of IMS American Mathematical Society. Vol. 1 and AMS.

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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 Assist­ Issue No. 51, December 1960. 25 pp. antships and Fellowships in Mathematics 25 cents; 10 copies or more 21 cents; 100 in 1961-1962; Volume7,Number7,Partll, copies or more 18-1/2 cents.

232 CATALOG OF LECTURE NOTES

INSTITUTO DE MATEMATICA PURA E APLICADA, RIO DE JANEIRO

The following item may be ordered from Livraria Castelo, Avenida Erasmo Braga 227, Rio de Janeiro, Brazil.

P. SAMUEL, Progres nkents d' Algebre locale, 203 pp., 1960.

INSTITUTO DE FISICA E MATEMATICA, RECIFE

The following item may be ordered from Livraria Castelo, Avenida Erasmo Braga 227, Rio de Janeiro, Brazil. L. NACHBIN, Integral de Haar, 238 pp,, 1960.

HARVARD UNIVERSITY

CORRECTION to list given in the December, 1960, Issue.

E. ARTIN and J. TATE, Class field theory, approximately 250 pp. In the U.S. $4.00 In Europe 4.50 In Asia 4.75

In Pre~------Experimental Correlograms and Fourier Transforms International Tracts In Computer Science and Technology and Their Application, Volume 4

N. F. Barber Experimental Correlograms and Fourier Transforms reviews the great variety of electrical, mechanical or optical "analogue" devices that have been invented from time to time for creating fourier transforms correlograms and power spectra. Some 250 references are given to allow the reader to consult original papers for fuller details of those machines ·which may partic­ ularly interest him. $5.00 Fourier Transforms and Convolutions for the Experimentalist R. C. Jennison There are many books on Fourier transforms, but there are none that help to bridge the gap which often sepa~ rates the r:nathematician from the experimentalist. This book seeks to convey the beautiful simplicity and universality of the Fourier transform to those whose mathematical memories have faded, or who have failed to relate the mathematical operations with their applications in everyday problems of the laboratory. $5.00 Frequency Modulation Theory - Application to Microwave Links International Series of Monographs on Electronics and Instrumentation, Volume 11

J. Fagot and Ph. Magne A large number of facts concerning frequency modulation and radio links are collected in this volume, whereas previously it was necessary to study numerous books and articles to gain this information. Although thi! work as a whole is of a very high scientific standard, it never loses sight of the problems of practical application. In addition to the numerous curves, graphs and numerical examples it includes many formulae which can be used directly. This volume will be invaluable to all telecommunication specialists. $15.00

~------P__ E_R __ G_A __ M_O __ N ___ P __ R_E __ S_S ____ ~2_2_Eas_t_sst_h_str_ee_t._Ne_w_Y_ork_2_2._N_.Y. __ ~ NEW YORK OXFORD LONDON PARIS

233 SUPPLEMENTARY PROGRAM NO.4

During the interval from February 14,1961 through April 21, 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. 5 will cover the interval from April 24, 1961 through July 7, 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 or in succeeding issues.

(1) Holmgren Riesz (H-R) transform quence integers equations of Riemannian type Professor Eckford Cohen, Univer­ Professor M. A. Al-Bassam, Texas sity of Tennessee (61T-144) Technological College (61 T-131) ( 10) The kernel of a semigroup of me as­ (2) Concerning path length. Preliminary ures report Professor H. S. Collins, Louisiana Dr. W. D. L. Appling, DukeUniyer­ State University (61 T-142) sity ( 61 T-145) ( 11) Cobordism classes of maps of odd (3) An equilibrium theory for n-person prime period cooperative games. Preliminary re­ Professor P. E. Conner and Pro­ port fessor E. E. Floyd, University of Dr. R. J. Aumann, The Hebrew Uni­ Virginia (61T-103) versity and Dr. M. Maschler, (12) Fixed point sets of maps of odd prime Princeton University (61 T-85) period (4) The Hausdorff- Besicovitch dimension Professor P. E. Conner and Pro­ of the graph of a symmetric stable fessor E. E. Floyd, University of process Virginia (61T-104) Professor R. M. Blumenthal and (13) The computation of bordism groups Professor R. K. Getoor, University Professor P. E. Conner and Pro­ of Washington (61 T-87) fessor E. E. Floyd, University of (5) A theorem on the complex projective Virginia (61 T-107) quadric (14) ~-homogeneous relatively universal Mr. E. Brieskorn, Mathematisches systems lnstitut, Bonn, Germany (61T-111) Professor William Craig, Univer­ (Introduced by Professor sity of California, Berkeley (61 T- F. E. P. Hirzebruch) 102) ( 6) Pairs of cardinals for models of a ( 15) On the specialization of birationally given theory equivalent curves Professor C. C. Chang and Mr. H. Professor Irwin Fischer, Univer­ J. Keisler, University of Califor­ sity of Colorado ( 61 T-150) nia, Berkeley (61 T-134) ( 16) On the oriented plateau problem (7) Discontinuity relations for charged, Professor W. H. Fleming, Brown compressible, relativistic fluids University (61 T-126) (with self induction) ( 17) Some results on order-convexity Professor Nathaniel Coburn, Uni­ Mr. S. P. Franklin, University of versity of Michigan (61T-106) California, Los Angeles (61T-112) (8) Some asymptotic formulas in theory (18) Singularities of three-dimensional of numbers harmonic functions. III Professor Eckford Cohen, Univer­ Professor R. P. Gilbert, Michigan sity of Tennessee (61 T-129) State University (61T-136) (9) On the distribution of certain se- (19) A Schwarz lemma for axially sym-

234 metric potentials Dr. J. M. Kister and Dr. L. N. Professor R. P. Gilbert, Michigan Mann, University of Virginia (61 T- State University ( 61 T-137) 96) (20) Some integral inequalities (33) Construction of a class of modular Professor R. P. Gosselin, Univer­ functions. II sity of Connecticut (61T-119) Professor M. I. Knopp, University (21) Nonexistence of a "theorema egre­ of Wisconsin ( 61 T-90) gium" on 3-manifolds (34) Construction of automorphic forms on Professor H. W. Guggenheimer, H-groups University of Minnesota(61T-108) Professor M. I. Knopp, University (22) The independence of the axiom of of Wisconsin (61T-91) choice from the Boolean prime ideal (35) Flexible nilstable algebras theorem. Preliminary report Professor L. A. Kokoris, illinois Mr. J. D. Halpern, University of Institute of Technology ( 61 T-124) California, Berkeley (61T-151) (36) An integral operator for the four (23) L-Heyting algebras dimensional Laplace equation Professor Alfred Horn, University Professor E. 0. A. Kreyszig, Ohio of California, Los Angeles (61 T-88) State University and Technische (24) The separation theorem of intuitionist Hochschule, Graz, Austria (61 T- propositional calculus 114) Professor Alfred Horn, University (37) A theorem on the coefficientproblem of California, Los Angeles (61 T-89) for harmonic functions of three vari­ (25) Finite groups of quaternion matrices ables Professor J. E. Houle, Georgetown Professor E. 0. A. Kreyszig, Ohio University (61 T-86) State University and Technische (26) A characterization of an analytic n- Hochschule, Graz, Austria (61 T- ball 115) Professor William Huebsch and (38) Bergman operators for generating Professor Marston Morse, Institute solutions of the three-dimensional for Advanced Study (61 T-147) wave equation (27) A Schoenflies extension of a real ana- Professor E. 0. A. Kreyszig, Ohio lytic diffeomorphism of S into E State University and Technische Professor William Huebsch and Hochschule, Graz, Austria (61 T- Professor Marston Morse, Institute 116) for Advanced Study (61T-148) (39) A new derivation of relativistic dy­ (28) Mazur's theorem. I namics Professor J. R. Isbell, University of Dr. Ali Kryala, Arizona State Uni­ Washington (61 T-128) versity (61T-130) (29) A discrete queueing problem (40) On roots of elements of a Banach Professor Mark Kac, Cornell Uni­ algebra versity, Mr. P. E. Boudreau and Dr. Svetozar Kurepa, University of Dr. J. S. Griffin, Jr., International Maryland (61 T-92) Business Machines, Endicott, New (41) Semi-discrete analytic functions York (61 T-122) Dr. G. J. Kurowski, Duke University (30) The second dual of the space of con­ (61 T-123) tinuous functions. III (42) Meromorphic close-to-convex func- Professor Samuel Kaplan, Wayne tions State University (61 T-83) Mr. R. J. Libera and Professor M. (31) A characterization of fields whose S. Robertson, Rutgers, The State multiplicative group is torsion University (61 T-121) Professor Samir Khabbaz, Lehigh (43) Cartesian products of contractible University (61 T-143) open manifolds (32) Isotropy structure of compact Lie Dr. D. R. McMillan, Jr., Louisiana groups on complexes State University (61 T-127)

235 (61 T- (44) Uncountably many divisors of E4. setts Institute of Technology 133) Dr. D. R. McMillan, Jr., Louisiana (56) Coefficients for stepwise integration State University (61 T-146) of y(n) = f(x,y,y' , .•. ,y(n-ll)with cen­ (45) Families of measures and represen­ tral differences tations of algebras of operators Dr. H. E. Salzer, Convair Astro­ Professor E. J. McShane, Univer­ nautics, San Diego,California (61 T- sity of Virginia (61T-120) 117) (46) Investigations of relationology (57) Note on o sculatory rational interpola- Mr. A. A. Mullin, University of tion. lllinois (61T-105) Dr. H. E. Salzer, Convair Astro­ (4 7) Perturbation theory and Lie algebras nautics, San Diego, California(61T- Professor F. J. Murray, Duke Uni­ 152) of some finite versity (61 T-93) (58) Autotopism groups (48) Remarks on a paper of G. Kreisel algebras. Preliminary report Mr. R. J. Parikh, Harvard Univer­ Mr. R. I. Sandler, Institute for De­ sity (61 T-139) fense Analyses ( 61 T-135) (49) An interesting example in the theory (59) On approximation by analytic func­ of stability of finite-difference equa­ tions whose Taylor coefficients lie in tions a sector Professor S. V. Parter, Cornell Dr. Oved Shisha, National Bureau University (61 T-110) of Standards, Washington, D. C. (SO) The Cartesian product of two semi­ (61 T-132) groups (60) Analytic continuation of solutions to Mr. Mario Petrich, University of Poisson's equation Washington (61 T-95) Dr. D. P. Squier, California Re­ (51) Hierarchic algebra search Corporation, La Habra,Cali­ Mr. M. B. Prestrud, The RAND fornia (61 T-109) Corporation ( 61 T-118) (61) Quasi-conformal functions tending to (52) Combinatorial results for processes conformality at the boundary with independent increments Professor D. A. Storvick, Univer­ Professor Edgar Reich, University sity of Minnesota (61 T-149) of Minnesota and Aarhus Univer­ (62) A model-theoretical result concern- sitet, Denmark, (61'1'-84) ing infinitary logics (53) Variational methods for functions with Professor Alfred Tarski, Univer­ positive real part sity of California, Berkeley, (61 T- Professor M. S. Robertson, Rut­ 82) gers, The State University (61 T-94) (63) Geometric ergodicity in denumerable (54) Multiplicative extensions of positive Markov chains linear operators Mr. D. Vere-Jones, Oxford Univer­ Professor G.-c. Rota, Massachu­ sity, England (61 T-97) setts Institute of Technology (61 T- ( 64) Classification of linear associative 125) algebras of dimension two. Prelimin­ (55) On the eigenvalues of modulus one of ary report order-preserving linear operators Professor J. Z. Yao, De Paul Uni­ Professor G.-C. Rota, Massachu- versity (61T-113)

236 ABSTRACTS OF CONTRIBUTED PAPERS

The April Meeting in New York, New York April 5-8, 1961

578-50. Leonard Gross; Inversion formulae for the Fourier transform of probability measures on Hilbert space.

A stochastic process almost all of whose sample functions lie in the real Hilbert space H = L 2(a,b) induces on H a probability measure m. The Fourier transform of m defined by ~(y) = j'H exp(i(x,y))dm(x) determines most of the properties of the stochastic process. Cameron and Donsker, Ann. of Math. vol. 54 (1959) have derived various inversion formulae which are special cases of the following formula: J'HG(s)dm(s) = lim X->- o:Jl(}))(det RA) 112 ExEy [¢(.,\y)G(f(A )Ax) .exp(- iAf()\)(Ax,y))] where Ex denotes expectation with respect to the canonical normal distribution on H, A is a Hilbert-Schmidt operator on H, Ril =I+ )) 2 f(~) 2 AA*, h(})) and f(..-\) are real valued func­ tions and G is an arbitrary bounded continuous function on H. In the present paper necessary and sufficient conditions on A, m, f, and h are given in order for this formula to hold for all such G, thus sharpening and extending the above mentioned results of Cameron and Donsker. (Received February 21, 1961.)

578-51. D. E. Schroer: Set-theoretic formulation of general syntax.

The imbedding of syntax in set-theory is realized neatly as follows: Define an expression

(expr) to be a function on an initial segment (possibly ~) of the set Nn of non-negative integers, and

an occurence (occ) to be a function on a bounded interval (possibly ~) of Nn, For an occ U, define

Index U to be the first element in its domain (with Index ~ = 0), Length !;:! to be the ordinal number of its domain, and uP to be the result of translating the domain of U so that its first element is p. Then uO is the expr of which U is an occ, the occ of one expr as part of another is represented by set-theoretic inclusion, and the equivalence relation u0 = v0 expresses that U, V are occs of the

same expr. Define ~ t> V = V, and, where U # ~. define U....,. V = U u vP where p = Index U + Length

U. Then -f+ is associative with identity~- Define the pair (U, V) to be a consecutive pair iff U = ~

or V = rJ or Index V = Index U + Length U, and the tuple ( U 1, ... , Un) to be consecutive iff for each i, 1 ~ i < n, (U1 +> ... t>Ui'Ui+1) is a consecutive pair. Using appropriate lemmas it is then easy to define for disjoint nonempty parts U1, ... ,Un of an occ W the occ {Repl in W: Ul' ... ,Un by v1, ... ,Vn} and give reasonable proofs of theorems such as the following: If~ i U ~ P ~ W, then {Repl in W: U by V] = [Repl in W: P by {Repl in P: U by Vj}. (Received February 21, 1961.)

578-52. j. D. Reid; A note on torsion free groups of infinite rank.

Let G be a torsion free abelian group. A subgroup H of G is full in G if G/H is a torsion group. The group G is quotient-divisible if G/F is divisible for some full free subgroup F of G. The object of this paper is to prove that if G has infinite rank then G is quotient-divisible and that any group G

237 of infinite rank can be written as the sum (not necessarily direct) of two free subgroups. These state­ ments follow from Theorems B and C below which in turn follow from Theorem A. Let t A vi yEN} be a family of full subgroups of G. For each V EN, let {X(i, v) l(i;J)) E I(-v)J be a set of nonzero ele­ ments of G such that {X(i,).l) + Ay l(i,Jl) E I(Y)} generates G/ Ay. Assume that rank G 2 Zv II(Y) 1. Then there exists a free subgroup F of G such that F + A:v = G for all ).) EN. Theorem B. For each prime p let r(p) be the rank of G/pG. Assume that rank G ~ Lpr(p). Then G is quotient-divisible, Theorem C. Let F be a full free subgroup of G. Then there exists a free subgroup F' of G such that G = F + F' if and only if G/F has a set of generators of cardinality ;a rank G. (Received February 21, 1961.)

578-53, Marvin Marcus, Henryk Mine and B. N. Moyls: Some results on nonnegative matrices,

A matrix is doubly stochastic (d,s.) if every row sum and column sum is 1, Let fln denote the polyhedron of nonnegative d,s. matrices and jl(S) the minimum number of terms necessary to represent S E fln as a convex combination of permutation matrices. The known inequality (3 (S) ~ (n - 1) 2 + 1 is refined: Theorem. !!._S E fln and Sis irreducible then (J(S) ~ h(n/h - 1)2 + 1, where h is the number of characteristic roots of S of absolute value 1. The following results are also obtained: Theorem. !!_S € fln' S is irreducible, and XSX -l E fln then X is a multiple of a d.s. matrix. Moreover, there exists Y E n such that YSY-l = XSX -l. Theorem. Let A be a nonnegative --- n -- matrix with maximal characteristic root r § 1, Then a necessary and sufficient condition that there

exist a permutation matrix P such that PAP' is triangular is 1Tf= 1(1- au)= det(In- A). Finally, an inequality is obtained for the difference between the maximum characteristic roots of two irreducible matrices A, B satisfying b ij G" a ij::;;: 0 for all i,j, (Received February 21, 1961.)

578-54. Abolghassem Ghaffari: Some applications of Hardy's formula in Brownian motion.

The most general solutions of one-dimensional Brownian motion, which is governed by Chapman­ Kolmogoroff functional equation (1) f(x,s;y,t) =rnf(x,s;z,u)f(z,u;y,t)dz, s < u < t, are given by the author (Bull, Amer. Math, Soc. vol, 58 (1952)) in the form (2) f(x,s;y,t) = L:oi'i">(x)~a)(y)en, a>- 1. It is known that, for s and t fixed such that s < t and x,y varying arbitrarily over (O,oo), the series solutions (2) converge uniformly over (O,oo) under certain conditions. Using Hardy's formula :L:or+oo'J = J':f(x,s;y ,t)dy (w is the probability) exists and depends on the initial state, i.e., the probability (U is nonoscillatory. (Received February 21, 1961.)

578-55. WITHDRAWN.

238 578-56. T. K. Boehme: Concerning the finite convolution.

It is shown that a necessary and sufficient condition that the convolution,/otr(t - u)g(u) du is locally absolutely continuous on the half line t $; 0 for every g which is locally integrable is that f be equal almost everywhere to a function which is locally of bounded variation. (Received February ZZ, 1961.)

578-57. R. j. Crittenden: Parallelization of cross-sections in associated bundles. Preliminary report.

Let (P, G,1r, M) be a C00 principal bundle with bundle space P, Lie group G, projection1r, and base space M. Let (W, F, G, 'IT', M) be an associated bundle with bundle space W and fibre F. Recall that any p € P may be viewed as a diffeomorphism p: F --..17'•- 1(?7'(p)). Let X be a cross­ section of 'IT', and define the function fx: P- F by: fx(P) = p- 1(X('/7'(p))). Then fx(pg) = g- 1fx(p). Let Y be any vector field on M, Y the horizontal lift of Y to P with respect to a connexion H on P. Definitions. The H-covariant derivative of X with respect toY is the function df Y: P -T(F), the X tangent bundle of F. X is parallelizable if there exists a connexion on P with respect to which all the covariant derivatives of X are zero. Theorem. X is parallelizable if and only if fx(P) is an orbit of Gin F, say fx(P) = Gf. UK= [g E.G/gf = fJ, then the principal bundle can be reduced to K, and any connexion on the reduced bundle parallelizes X. Corollary. U G is transitive on F, then every cross­

section of 71'1 is parallelizable. The theorem easily extends to the simultaneous parallelization of several cross-sections in associated bundles. Applications are given, principally to tensor fields. (Received February ZZ, 1961.)

578-58. R. L. Vaught: A Lowenheim-Skolem theorem for two cardinals.

Let T be a theory based on a first order language with equality and countably many nonlogical symbols U, U', •••• of which the first is a unary predicate. Theorem. U T has a model (A,V,V',···> in which A and V have different infinite powers, then T has a model (B,W,W', ••• ) in which the powers

of B and Ware ~ 1 and ~ 0 • respectively. A special case was established earlier by the author (in a paper to appear in Proceedings of Symposium, Warsaw, 1959). Its proof can be modified to give the general result by relying on (Z) and (3) in Abstract 61 T-lOZ, Notices Amer. Math. Soc. this issue. (Received February Z3, 1961.)

578-59. A. B. Lehman: A solution to the Shannon switching game.

Let f be a given monotone Boolean function of x l•···•Xn all taking values in the set {_9,!11 }. Let two players, denoted 8 and !II, alternate in assigning values, one per play, to xl'···•xn. After n such plays the winner is given by the value of f. Each f has exactly one of the following properties: (1) The first player can win. (Z) The Ill player can win. (3) The e player can win. A solution of this classification problem yields an optimal strategy. In the Shannon switching game (M. Minsky, Steps toward artificial intelligence, Proc. IRE vol. 49 (1961) p. Z3) f is derived from a two-terminal graph:

For each set of values of x 1, ···•ltn• f has value 6 if and only if those branches i for which xi = 9 con­ tain a path connecting the two terminals. Such a function f has property (3) if and only if the graph

239 contains two disjoint cospanning trees spanning the two terminals, A deletion (¢) of a branch in one tree may be "repaired" by a play (8) on the remaining tree. If the trees are maximal the strategy can be independent of the chosen terminal pair. A dual but more complicated condition characterizes property (2). The solution also holds for the corresponding game played on matroids (W. T. Tutte, Matroids and graphs, Trans. Amer. Math. Soc. vol, 90 (1959) pp. 527-552). It is conjectured that a similar solution can be found for all f, thus supplying a strategy for "hex". (Received February 24, 1961.)

578-60. John Mariani: The group of Pythagorean numbers.

The pseudo-orthogonal linear substitutions carrying the indefinite quadratic form with integral coefficients x 2 + y 2 - z 2 = 0 into itself form a group G of which the subgroup G maps proper Pytha­ gorean triplets x,y,z onto themselves and have determinant± l. G has the following properties: G acts transitively on the proper triplets so that every triplet can be mapped onto another by a trans­

formation of G. G is isomorphic to the group of all 2 X 2 matrices with integral coefficients and determinant± 1, The subgroup of substitutions of G that keeps a particular proper triplet fixed is the direct product of an infinite cyclic group and a group of order 2. G is of finite index in G. G is a representation of order 2 of the transformations of the binary unimodular group a' =..±._a, b' = 2ka + b, where (a, b) = 1 and k is an arbitrary integer. (Received February 28, 1961.)

578-61, Bertram Kostant and A. B. J, Novikoff: A homomorphism in exterior algebra,

Let V be a real n-dimensional vector space with basis vl'"''vn and with dual basis fl' ... ,fn in V*. Let u = v 1/\ ... /\vp and g = f 1/\ ... ;\fp, and e(u) and i(g) denote exterior and interior multiplication by u and g respectively. If I = ideal in /\V generated by u and J = subalgebra of !\V generated by i(g) E(u) . vp+1 .... ,vn' then I-"+J--"I are mverse maps, e.g., v 1A ... /\Vp/\Vp+1/\vp+2 ~vp+ 1 Avp+ 2 • Given a linear transformation A: V--> V it extends uniquely to a graded homomorphism of )\ V, also denoted

A.· Let AI= restriction of E(u) i (g) A to I. Then AI: I ·~I is equivalent to i(g)As(u): J ~J. Denote this last map by B. Theorem: If (g, Au> i 0 then 1/(g, Au> B is a homomorphism of J, Corollary: Using vp+l''"'vn to generate a basis for J in the classical way, the restriction of B to Jk (= elements 1 of kth degree in J) is the kth compound of B restricted to J 1 multiplied by (g, Au} -k. Referring AI and B to equivalent bases results in an identity (due to Sylvester) relating those minors of A which properly contain the first p rows and columns and corresponding minors of B. An equivalent identity is called by Bourbaki the extension of order p of the formula for expanding a determinant of order n- p. (Received March 23, 1961.)

578-62. F. V. Atkinson: On a linear functional equation.

It has recently been shown by Baxter (Pacific J. Math. (1960)) that the equation u = 1 + T(ux),, in a commutative Banach algebra, admits an explicit solution when T sa'i:isfies a generalised "integration by parts" formula (Ty)(Tz) = Tty(Tz) + (Ty)z - yze}. It is shown how this explicit solu­ tion may be obtained without the combinatorial arguments needed when the proof proceeds by the comparison of series expansions. Denoting the solution by E(x), one has the generalised exponential law E(x)E(y) = E(x + y- xy9) (cf., R. Bellman, Research problems, Bull. Amer. Math. Soc. (1961) p. 176). (Received April 7, 1961.)

240 The April Meeting in Chicago, Illinois April 14-15, 1961

579-34. Peter Treuenfels: On upper and lower bounds to solutions of boundary-value problems. 2.

A previous note (Abstract 568-13, Notices Amer. Math. Soc. vol. 7 (1960) p. 247) presented a method for the numerical treatment of boundary-value problems of the type - py" + qy' + ry = s, y(a) = Ya• y(b) = yb, p(x) > 0 and r(x) ~ 0 for a;&: x:;:;; b. The present note discusses the relaxation of the restriction r(x) ;;; 0. A numerical example is given. (Received February 21, 1961.)

579-35. G. J. Rieger: On the prime ideals of smallest norm in an ideal class mod f of an algebraic number field.

Linnik proved the following theorem: There exists an absolute constant c such that in every prime residue class mod k there is a prime number p with p <: kc. This theorem is generalized to prime ideals in ideal classes mod f of an arbitrary algebraic number field. (Received February 23, 1961.)

579-36. Mark Mahowald: On obstructions to extending a map.

Let 'tlf be the secondary operation associated with the relation Sq4Sq1 + scf Sq1 scf+ Sq 1sq4 = 0 in the sense of Adam, On the nonexistence of elements of Hopf invariant one, Ann. of Math. vol. 72

(1960). Let j-J4 and r6 be the Bockstein Coboundary and the Adem operator respectively. Then (*) Sq 1.'1! + Sq2r6 + Sq4ft4 = 0 modulo the total indeterminancy involved for all cohomology classes in the domain of '\jf. Using the method of universal examples one can associate with (*) a tertiary operation ~. This operation has the following property: Let K be a complex obtained by attaching to sn,En+4 by a map of homotopy type 4r where is the generator of 7rn+3(Sn) and r is an integer. Then, if

u E Hn(K; z 2), "f(u) is nonzero whenever r is odd. Combining this proposition with known results we have for K = Sn U En+4, where En+4 is attached by some map f to Sn, that f is nontrivial if (i) Sq4u i 0 or P j u i 0, u E Hn(K) or (ii) if (i) fails, then ·J¥

1 961.)

579-37. Peter Werner: Electromagnetic waves in inhomogeneous media.

Consider the stationary Maxwell equations (1) 'iJ X'{; + ir;;'{: = 0, \!X {! - i Jl YJ· = 0, e, u being positive space functions, twice continuously differentiable and constant for lt'l > R. Set 241 e( 1") = eo and p('f) = JlO for I-t' I;;;; R. {. )- are said to satisfy the electromagnetic radiation condition, if as r ._ oo; (2) 't (r"t~'o) = 0(1/r), (Jlo) 112 'fo x ~ (r "f'o) t (e0)1/2{:€ (r ~) = o(1/r) hold uniformly for all directions "f0 • Theorem. For each constant vector '~>7- there are uniquely deter­ mined vector fields ti("'e' !p' ;'Vl) and f i( y '1 ;-vU (i =. 1 ,2! such that (a) e i' 1 i are continuous for "(' =11' and-as functions of '(-satisfy (1) and (2); (b) [h, J1 have the same singularities for -( ---7·[ as the well-known electric (i = 1) and magnetic (i = 2) dipole solutions in homogeneous media at 1:1- with moment t:t_. By means of these generalized dipole solutions the whole existence theory, concerning stationary reflection and diffraction problems in homogeneous media, can be extended to the case of inhomogeneous media. (Received March 1, 1961.)

579-38. J. R. Boen: On p-groups whose automorphism groups have maximal simple transi­ tivity. Preliminary report.

Let G be a finite p-group (p > 3) such that any two elements of G of the same order are conjugate under the automorphism group of G. It is proved that G is abelian of type (pn, pn,. •. ,pn) in the cases that G is generated by one, two, or three elements. It is suspected that this result may be extended to the case G is ann-generator group. (Received March l, 1961.)

579-39. K. Grant and George Whaples: Abstract class formations.

Let k be any field and 0 any (infinite) normal separable extension of k. Using only the Galois group of n over k, it is possible to construct a system of groups E(K), for all K finite over k and contained in n, which satisfies the axioms for a class formation of Artin-Tate-Kawada. (Received March 1, 1961.)

242 The April Meeting in Stanford, California April 22, 1961

580-Z7. E. 0. Nelson: A solution of the generalized heat flow equation in a bounded region as a Wiener integral.

For (t,!) in a bounded subset of the plane, a solution of the equation aZa;e~ Z - ai}G/8t + 8(t,f)G = 0 with certain boundary conditions is expressed as an integral over a subset of the space C of continuous functions x(t) on lfl,l] with x(O) = o. Some of the results of Cameron (The generalized heat flow equation and a corresponding Poisson formula, Ann. of Math. vol. 59 (1954) pp. 434-46Z)

are modified to include the case where 9(t,~) need not have partial derivatives everywhere. The 9(t,_S') for the equation in the bounded region is then extended to an infinite strip. The solution then follows by applying the results of Cameron, and appropriate limiting operations. Boundary conditions are then verified directly. (Received February Z 1, 1961.)

580-ZS. Bruno Harris: Homotopy groups of classical groups.

Let SO(n), SU(n), Sp(n) denote the special orthogonal, special unitary, and symplectic (or quaternion unitary) groups. We show that the homotopy sequences of the fibrations SU(Zn + 1)

~ SU(Zn + 1)/SO(Zn + 1) and SU(Zn)-+ SU(Zn)/Sp(n) reduce to the following direct~ decompositions modulo the class C of Z-primary groups: (l)lTi(SU(Zn + 1))<=< Tfi(SO(Zn + 1)) ED TTi(SU(Zn + 1)/SQZn + 1)) (mod C); (Z)Tfi(SU(Zn))= Tfi(Sp(n)) 19 Tfi(SU(Zn)/Sp(n)) (mod C), for all i. Using (1) and (Z) we prove the isomorphism (3) lTi(SO(Zn + 1)):::: 1T ~Sp(n)) (mod C), conjectured by Serre. The direct sum decomposition in (1) is the decomposition into the +1, -1 eigenspaces of cr*, where IY denotes the complex conjugation automorphism of SU(Zn + 1), and similarly for (Z). (1) and

(Z) are proved by showing that the natural map q: G/K~ G. (where G = SU(Zn + 1) or SU(Zn), K = SO(Zn + 1) or Sp(n)) given by q(gK) = gcr(g)-l behaves like a cross-section in homology with coefficients ZP' p # z. (See A generalization of H-spaces, Bull. Amer. Math. Soc. vol. 66 (1960) There are similar results for Stiefel manifolds.) (Received February Zl, 1961.)

580-Z9. W. E. Milne, R. R. Reynolds and J. J. Kohfeld: Seventh order methods for numerical solution of ordinary differential equations.

The formulas y(P+)ln = y n-5 + (3h/10)(llyn1 - 14yn-1 1 + Z6yn-1 z - 14yn-1 3 + llyn-1 4) and y(C+)ln = Yn-3 + (Zh/45)(7y~~ 1 + 3Zy;._ + 1Zy;.__ 1 + 3Zy;.__z + 7y;.__ 3) have truncation errors proportional to the

seventh power of the interval length h and have been used to continue the solution of y(i) 1(x)

= f(i)(x,y(l), ••• y(N) ), y(i)(x0) = y&i). i = l,Z,~.N. To insure stability, Yn+l = IY~~~ + Yn_4 + (5h/Z88)(19y~~~ 1 + 75y~ + 50y~-l + 50y~-Z + 75y~_ 3 + 19y~_ 4 )]/Z is applied after every k step, where k is a calculable function of hand Clf(i);ayO>. These results extend the fifth order methods described in Milne and Reynolds (J. Assoc. Comput. Mach. vol.6 (1959) pp. 196-Z03 and vol. 7 (1960) pp. 46-56) (Received February Zl, 1961.) 243 580-30. WITHDRAWN.

580-31. Herman Gluck: Tangled manifolds.

A three-dimensional manifold M 3 is called tangled if it is impossible to find a tame knot, k, in M3 such that the meridian of k (a little loop circling k in M3 - k) represents an element in the center of TI1(M3 - k). Most three-manifolds are tangled. Since M#(S2 X s1) is never tangled, assume that s2 X S 1 is not a 'factor' of M3• Then M3 is untangled if and only if all of its prime factors are. We are thus reduced to the case in which M3 is prime. Here we develop very strong conditions 3 which li1 (M3) must satisfy if M is to be untangled. (Received February 21, 1961.)

580-32. W. j. Firey: Mean cross sectional measures of harmonic means of convex bodies.

In Abstract 570-15, (Notices Amer. Math. Soc. vol. 7 (1960) p. 282), p-dot means of two nondegenerate convex bodies K0, K 1 having a common interior point Q in Euclidean n-space En were • (, ,Jl_-" (I~A A introduced. For p = 1, this is the harmonic mean: K.,51- = L(l -v)K0 + 'c/1.<.. 1] where K is the polar re- ciprocal of K with respect to the unit sphere centred at Q, and 0:;; -{}· ~ 1. Also a dual Brunn­ Minkowski theorem was proved which, for the harmonic mean is vl/n(K-i:l);;:; [(1 -.J )v-l/n (Ko) + ,~y-l/n(K 1 )]- 1 , where V(K) is the volume of K. A more complete result is given here. Theorem. W//(n-p) (Ka) ~ J.(1 -"'19-)Wp -l/(n-p)(Ko) +-3Wp~l/(n-p)(K 1 )T 1 , where Wp(K) is the pth "Quermass­ integral", (cf. Bonnesen and Fenchel, Konvexe Korper, 1934, pp. 48-50), p = 0, 1, ... , n - 1. There is equality if and only if Ko and K 1 are homothetic with center of homothety at Q. The proof uses: (a) The dual Brunn-Minkowski theorem which is the case p = 0; (b) Minkowski's inequality, (Hardy,

Littlewood and P6lya, Inequalities, 1934, Theorem 201), for - 1~ p..::. 0; (c) a projection lemma: .IJ ,., I> /1 • If K* is the projection of K onto an Em through Q, (m""' n), then [(1 -u )K0 +-3kiJ 2.~; (d) Kubota's formulae, cf. Bonnesen and Fenchel, loc. cit. (Received February 21, 1961.)

580-33. Stephen Kulik: A method for the solution of ordinary simultaneous equations.

Let (1) f(x,y) = 0 and (2) g(x,y) = 0 be two simultaneous equations andy= pl(x) is the solution of (2). Then the root x = 11 of the equation F(x) = f(x,plx) = 0 can be approximated by use of an iteration formula of higher order, Xn = Ln(x0), where Ln(x0) depends on F(xo) and its higher derivatives, and lim ~ = 11 as n ~. The method proposed does not require the knowledge of pl(x). Namely F = F(xo) and its derivatives are found from the following equations. F = f, F 1 = f + f y' F • • = f + 2f y' + xy• xx xy 2 fyyY' + fyy'', ••• ; gx+ gyy' = 0, gxx + 2gxY'' + gyyY' 2 + gyy'', ••. ; where f= f(x0,y0),g = g(x0,y0) = 0 and their partial derivatives are calculated at (x0,y0), which is a solution of g(x,y) = 0. (Received February 21, 1961.)

580-34. L. H. Lange: The existence of non-Euclidean cercles de remplissage in certain sub­ sets of the unit disc.

The concept of a "set of fl-discs" for complex-valued functions defined in the open unit disc U was introduced in the author's Sur les cercles de remplissage non euclidiens (Ann. Sci. Ecol. Norm.

244 Sup. vol. 77 (1960) pp. 257-280) and various results concerning the existence of such sets associated with spirals and Stolz angles were derived with the aid of non-Euclidean forms of Schottky's theorem. Along with a theorem which yields new proofs of some results proved by W. Seidel (Holomorphic functions with spiral asymptotic paths, Nagoya Math. J. vol. 14 (1959) pp. 159-171), the present paper supplies additional theorems about Stolz angle phenomena and the existence of (l-discs (analogues to

H. Milloux1s classical cercles de remplissage for entire functions) in more general subsets of U. For example, in the notation of the Annales paper, we have the Theorems: (1) Let f be holomorphic in U and let A C U be any set such that A n (U - U) >F ~. If there exist sequences [ z~t and {z:J in A such that lim d(zn' z~) = p'-< p < oo; lim f(z~) = oo; lf(zn) I~ M, a finite constant; then there exists a set of p-points for f in the set Ap =. {z ld(A,z) ...: pj • (2) Let (J R(f,A) consist of at most one (finite) complex value. If there exists a positive constant p and a sequence lz~} of points in A such that lim f(z~) = oo, and {z ld(z~,z) < pj :> A, then there exists a set of f -points in Ap. (Received February 22, 1961.)

580-35. G. J. Rieger: Generalization of a theorem of Ramanujan to algebraic numbers.

Let K be an arbitrary algebraic number field, and let Nf denote the norm of the ideal f of K and v(~) the number of (different) prime factors of the integer .$ of K. Then the number of integers ~ := ..Ymod f of K with v(s) ~ loglog x + w(loglog x)112 whose conjugates{; (j) satisfy 1/2 )lf'4! -t2/2 -1/4 I~ nJ I~ x 1·; n(j = l, ••. ,n; n-degree ofK) is given by (x/(2.1r) Nf[/~oo e dt + of,"'-'(x(loglog x) ·(logloglog x) 1/ 2) where the constant in the remainder depends on K, f, and w alone. This dependence on f and tu can be given explicitly. (Received February 23, 1961.)

580-36. D. E. Myers: A note on perfect operators.

Let "f. denote the set of complex valued functions of the real variable t, x(t), such that x(t) ...... - 0 as t ...... ;. oo and further each x(t) has derivatives of all orders in J:·. Weston (Proc. Roy. Soc. London. Ser. A vol. 250, p. 460) has called these perfect and an operator on* is perfect if it commutes with the convolution product, i.e., (Ap)*q = A(p*q) for p,q e ):: Weston has ·shown that an operator A, is perfect if and only if Ap has for its unilateral Laplace Trans. ~(z)_E(z) for p EX. Extend the perfect operator as follows. Let "}{ be the algebra of entire functions over the complex field with the ordinary product. The author has shown (Thesis, University of Illinois, 1960) that a vector space of sequences of entire functions is sufficient to imbed Schwartz Distributions. Let A be an operator on as follows, Ap has for its transform the sequence ~nf(z)p(z)J, n f(z) an entire function, and p E" t·. For example if D is a Schwartz Distr. with the represent. D(r6) = r.JOnF n(t)r6(rn)(t) dt when restricted to r6 whose

support [o,n). Let nf(z)/z rn be the unilateral trans. of F n n. Then {nf(z)} maps p*q onto F n * [P•qJ(rn) for 0 ~ t ~ n. This generalizes the differential operator. (Received February 23, 1961.)

580-37. C. B. Bell: The Kramer-Gebelein maximal correlation as a measure of common entropy.

For a probability space (0, S, P) with finite subalgebras 0{,, if. "'"'whose atoms are, resp., fAit• {B j~' {11c! Shannon defines common entropy C'p((1t,"t;-) = _Li,/(Ai () Bj)

245 log(P(Ai n B j)/P(Ai)' P(B j)) and Hp(Ol) = - ZiP (Ai)log P(Ai); H. Kramer, generalizing H. Gebelein

(1941), defines Sp0/.~) = sulX,y R(X,Y) where R is the correlation coefficient; and X andY are respectively, measurable -Ol-and -$. Kramer asks: Are Sp and Cp equivalent? To normalize Cp• let Cp(<-'iC,$) = Cp(d' ,,iY) ·{min I}I(N), H(o:b=)~ -l. One proves that Sp and Cp are similar, e.g. (1) 0 ~ Cp' Sp ;2 1; (2) Cp = 0 iff Sp = 0; Sp = 0 iff(}[ and ;fj are independent with respect toP; (3)

(-{ and :J~ are set independent iff there exists P 0 on S such that Sp0 = Cp0 = 0; and either condition implies Sp and Cp<:: 1 for all PonS; (4) CP({Jf,df} = 1 implies ~(6£,$) =1. However, (5) Sp = 1 iff rJt n i} i [¢,!1} fpJ; and (6) CP = 1 iff a' Cr/f' or d)-- Cc'Jf If]. Using these facts one is lead to the construction of an example which proves the nonequivalence of Sp and Cp. (7) Theorem. There exist (S"l,S,P), <~/.:f. ,f\-CJ~ such that (a) Sp(O£,$) >Sp (,9';$); but Cp (c/ ,'f:')' Cp (r8' ,riJ ). (Received February 23, 1961.)

580-38. John Jones, Jr.: On certain matrix equations.

Some theorems concerning the existence of solutions of matrix equations of the types con­ sidered by W. E. Roth, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 392-396, are extended using tech­ niques of W. E. Roth and M. Rosenblum, Duke Math, J, vol. 23 (1956) pp. 263-269. (Received February 27, 1961.)

580-39, R G. Ellis: Solutions of first order, analytic differential equations. Preliminary report.

Under the hypothesis that (1) a > 0, (2) c is a complex number, (3) for p = O,l, ... ,gp is a complex­ valued function from the sect (O,a], (4) u =Jot g0 on (O,a], (5) for p = 1,2, ... Kp e:, 0, and (6) for p = 1,2, .... /o+ lgpuP I;;; Kp luI on (O,a], the following theorems are established. Theorem 1. lf r ;;;_ 0 and, for some number T~ 0, T = 1 + ~':~KprPTP, then for each complex number z such that lz I ;;:;; r there exists a function f such that f(O +) = c, f = c + ~=~ Jo+ gpzP(f- c)P on (O,a], and If I,;;;; tlul on (O,a] if t_;;; 0 and t = 1 + ~=~ KplzPitP. Theorem 2, lf for some number R :G;.O the series L~~ 0 lg piRP converges uniformly on every interval [d,a] for '1\hich 0 < d < a, then on some sect (O,b] the function f of Theorem 1 satisfies f = c +Jot L~=~ gpzp (f - c)P, and if each gp is continuous on (O,b), then f satisfies f' = ~~O gpzP(f - c)P on the segment (O,b). Theorem 3. If u > 0, the inequalities of (6) are reversed, r ~ 0, and for no number T ~ 0 does T = 1 + ~~f KprPTP, then for no complex number z such that lz I ~ r is there a function f -such that f(Ot) = c and f = c + u + JotL~~flgpzP(f - c)P I on some sect (O,b]. The proof of Theorem 1 yields a series for f. (Received April 20, 1961.)

246 The June Meeting in Seattle, Washington June 13-16, 1961

581-1. J. B. Butler, Jr.: On the scattering of waves in the infinite bar. Preliminary report.

4 4 4 Let (i) L 6 = (d/dx) - k + eq(x) where J1(k ) ~ 0, q(x) is a real positive piecewise continuous function, q(x) = 0, x;;; 0, x > b, and e is a positive number. Solutions u(k,x) of the problem

(ii) L 8 u = 0, u --..eikx + b(k,e)e -ikx, x -7- oo, u ~ a(k,e)eikx, x-.... + oo, are investigated. It is shown that given any real interval 6 and 80 > 0 the reflection coefficient b(k,s) is analytic in s for sufficiently small s and for k4 =) + i,l$, J. E /1, 0 -< & < o0• The results may be used to calculate the reflection coefficient for certain waves in a bar on an elastic foundation. Analogous results are obtained when (d/dx)4 is replaced by Lo = Z.f-oPj(x)(d/dx)i where pj(x) are real differentiable up to jth order on [o,oo), p 4(x) ~ e 0 .>-0, L0 is formally self adjoint, and L0 satisfies certain additional conditions given in detail earlier (cf. Abstract no. 571-12, Notices Amer. Math. Soc. vol. 7 (1960) p. 481). (Received February Z3, 1961.)

581-Z. L. P. Neuwirth: Noncoincident maps of a sphere into the plane.

It has been shown that (E2)n- /::,.where 6.. = {«x1,Y 1), (Xz,Yz) •••• ,(Xn,Yn))l Xi= XjYi = Yji ¥ i\ has trivial homotopy groups above dimension 1. One can interpret this as follows: Theorem. !!!!:J n noncoincident maps of a k-sphere into the plane may be extended to n noncoincident maps of a

(k + 1)-ball into the plane if k > 1. (A set of maps, f 1 ...fn, is noncoincident when fi(X) = fj(X) iff i = j.) The theorem is also true for maps into a compact orientable manifold which is not S 2• (Received February Z8, 1961.)

581-3. H. F. J. Lowig: On the definition of a free algebra. Preliminary report.

If C is a freely generated algebra, an algebra A of the same species is called free with respect to C if, for every two different elements, c' and c", of C, there exists a homomorphism h of C into A with h(c') ¥ h(c"). It can: be proved that if Co is a freely generated algebra with jC ol ;; Z then c 0 is free with respect to every C. It follows that if A is free with respect to such a c 0 , it is free with respect to every C. Therefore, after a special c 0 with jc0 I z has been chosen, a free algebra may be defined as one which is free with respect to c 0 • An algebra can be free without being freely generated. E. g., if f is the unary operator on the set of all integers defined by the equation f(k) = k2 then the algebra of this single operator is free but not freely generated. (Received March 6, 1961.)

581-4. H. C. Wiser: Decomposition and homogeneity of continua on a Z -manifold.

A homogeneous nondegenerate proper subcontinuum X of a Z-manifold M is a simple closed curve if it satisfies any of the following conditions: (1) X is arcwise connected; (Z) X contains a

247 simple closed curve; (3) X is aposyndetic; (4) X contains a noncutpoint. Corresponding results for a homogeneous continuum in the plane have been established by F. B. Jones (Bull. Amer. Math. Soc. vol. 55 (1949) pp. 113-114) and H. J. Cohen (Duke Math. J, vol. 18 (1951) pp. 467-474). In proving (2), some properties of collections of disjoint continua filling a continuum are established. In particular, a continuum on a 2-manifold is either an annulus, a Moebius strip, a torus, or a Klein bottle if it is filled by a collection of disjoint simple closed curves. F. B. Jones' continuous decomposition of a planar, decomposable, homogeneous continuum into a simple closed curve of indecomposable homo­ geneous continua (Proc. A mer. Math. Soc. vol. 6 (1955) pp. 735-740) is extended to homogeneous de­ composable continua which either lie on a 2-manifold or satisfy hereditarily the property of having a finite degree of multicoherence. (Received March 6, 1961.)

581-5. E. G. Straus and K. Rogers: Modules over a Dedekind ring.

Let R be an integral domain, K its quotient field, and A an n-by-n matrix over an overfield of K. It is said that "A has property P relative to R" iff for all nonzero n-by-one vectors !! over R, the vector A!! has a component in R *, the nonzero elements of R. Theorem • .!!._ R is a Dedekind ring and A has property P relative to R, then det A E R *. First, A is replaced by a matrix over K with PR and the same determinant. It is shown that for each maximal prime ideal y of R, A has property P relative to Ry, and that (a) if ·#' (R/y) i:; n, then A is equivalent over

Ry to a triangular matrix with nonzero elements of Ry on the diagonal, but (b) ,W(R/y) ..<: n, only the weaker result that det A E Ry is proved, It follows that det A 1 0 and det A E n..,.y = R. (Received March 13, 1 961.)

581-6. P. B. Bailey: Removal of the log factor in the asymptotic estimates of the membrane eigenvalues.

Let the boundary B of a bounded, open, connected subset D of the Euclidean plane be the dis­ joint union of a finite number of polygons, Let Y~j ~ li)Jj+l be the eigenvalues, repeated according to their multiplicities, of the problem - \/Zu =Au over D, u having continuous second partials there and satisfying the Dirichlet or Neumann condition on B according as "Y = 0 or 1; let Ny(>.) = card { ,})j: y\j :=; A}. We verify a conjecture of Brownell, (Abstract 560, Bull. A mer. Math, Soc. 2 1 2 vol. 63 (1957) p. 284) that <5 1,JI + 2:j~\ exp(-JiAjt) = (41?")- 1At- 1 - (- 1)"(8 11" l/ ) Lt-l/ + w + O(exp(- pt-1)) as t ~ 0+ for some W, p > 0 depending only upon B, where A is the area of D and L is the total length of B. A trivial modification of a Tauberian theorem of Ganelius (Kungl. Fysiog. Sallsk. i Lund. Forh. vol. 24, no. 20 (1954)) then yields the result Ny(A) = (411')- 1 A~ - (-)"(47Tf 1 L)) 112 + 0(A 1/ 2) as})~+ oo, We also prove Z~ 1 exp(- )l})js) = (41r)- 1As-1 - (- 1)">'(8 7r 112)- 1 •Ls- 1/ 2 + J)f 1(s) +yf2(s) over {complex s: R[s] >0} with yf1(s) andyf2 (s) analytic there and satisfying lyf1 (s)l ;sa M r!s 1- 1/ 2 over R [s J > 0 and l:vf 2(s) I~ M 2 over s = reiS, IS I ~ s0 , r > 0 for some finite M 1, M 2, e0 with 0 < s0 ~ 2 - 17r. This last raises hope (Brownell, Bull. A mer. Math. Soc. vol. 66 (1960) p. 275) of replacing 0())1/.2) above by 0())112). (Received March 13, 1961.)

248 581-7. W. E. Parr: Bounds for the capacitance of convex surfaces.

The Dirichlet integral corresponding to the exterior Ilrichlet problem for Laplace's equation has been bounded above by G. P6lya and G; Szegii through application of Dirichlet's principle and the Schwarz inequality. A generalization of the P6lya-Szegii inequality yielding systematic improvement of upper bounds is given in this paper. In particular, using a family of parallel surfaces and a family of similarity transforms, the following inequality is obtained: 2 J':dY/[6 K0 + 6(1 + Y- 6)K1 + (1 +"Y- 6)2K2] ~ 1/D(p,p) for suitable restrictions on 6(y) and where the Ki(S) are analytic functions defined by given integrals. The inequality above includes as special cases, not only the P6lya-Szego estimate but also the upper bounds corresponding to a circumscribed sphere and that corresponding to a family of similar surfaces. Application is made to the convex poly­ hedra after explicitly estimating the Ki(iJ). The classical application to the capacitance of a cube of side 2.0 yields an upper bound of 1.3351, an improvement over all upper bounds previously calculated. (Received March 22, 1961.)

581-8. W. E. Briggs: The irrationality of gamma or of sets of similar constants.

Define or,k for integral k 5; 1 and 0 ..c r §. k by L' 1/n = (1/k). log X+ 8r,k + o(1), where the prime denotes summation over n x with n = r (mod k). If g(8) = 2: fe/n(n + 8), then it is shown that &r,k = (1/k) [?"- log k - g(r/k)] + 1/r and from this follows the Theorem. If?' is rational,

then 82,4 is rational, 8a,2a is irrational for a f. 2, and 8a,a is irrational for a> 1. Next let "Xtc (n) be the principal character modulo k and write 4: (s) = Zf Xk(n) n -s = (¢(k)/k) (s - 1) -l + ))k + •••• It is shown that ~k = ¢(k)/k. [Lpjk(log p)/(p- 1) + r] which implies the Theorem. If 'J/ is rational, then )) k is irrational for k > 1. (Received March 27, 1961.)

581-9. B. R. Toskey: Exponential rings.

A system (R, *) is called an exponential ring over R if R is a ring and • is a binary operation on R satisfying (x*y)(x*z) = x*(y + z), (x*y)*z = x*(yz), and (x*y)(z*y) = (xz)*y for all x,y,z E R. The operation * is called trivial if x*y = e for all x,y E R, where e is a fixed idempotent in R. The system (R,*) is called an E-ring over R if the operation * is nontrivial. After some examples and elementary properties of E -rings are considered, all E -rings are computed over rings with additive group which is either a torsion free group of rank one or is a cyclic group of prime power order. (Received April 3, 1961.)

581-10. Chandler Davis: An extremal problem for plane convex curves.

Given, in the plane, the cross consisting of intersecting segments of lengths b,c. For any

convex region which is divided by the cross into four pieces of equal area, the total area is ~ be; this is attained only in case the two segments bisect each other, and then only by the obvious rectangle solution. (Received April 6, 1961.)

249 581-ll. L. C. Eggan and E. A. Maier: A result in the geometry of numbers.

Let R denote the set of real numbers and Z the set of rational integers. For c E R, c 5; 0, define m(c) = max {min f Ia - u 11{3- u I; u € zJ; a, (3 E R, Ia - ,81 = Zc}. The function m is piece­ wise a quadratic polynomial in the sense that for any real c ~ 0 there is an interval containing c on which m is a quadratic polynomial with rational coefficients; we evaluate m explicitly. By means of the same techniques used in this evaluation, we also show that if a, p E R, then there exists an integer u such that 1;3- ul < 1 and Ia- uii;S- ul ;a 1/4 if Ia- ;s'l < 1/Z and Ia- uii,P'- ul .c.. Ia - ,(31/Z if Ia - {31 ;;: 1/Z, where equality holds iff a= f3 = n + 1/Z, some n E Z. We then apply this latter result to yield a simple proof of a classical theorem of Minkowski in Diophantine approxi­ mation theory. (Received April 10, 1961.)

581-lZ. John Backus: Representative sets and an orthogonality relation for certain families of sets.

Definition. A set x is a representative of a family of sets Y if and only if (a) x has a nonempty intersection with each set y in Y, and (b) every proper subset of x is disjoint from some y in Y. Definition. A family X is completely representable if and only if every set y which has a nonempty intersection with each x in X contains a representative of X. Definition. A family X is normal if no member of X contains any other member. Theorem. If X is a completely represent­ able and normal family and Y is the family of all representatives of X, then the family of all representatives of Y is precisely X itself. Theorem. If X is any family of finite sets, then X is completely representable. Thus if the relation R(X, Y) holds if and only if X is the family of all representatives of Y, then R may be regarded as an orthogonality relation over the class of normal finite families of finite sets. (Received April 17, 1961.)

581-13. J. J. Levin and J. A. Nohel: On a system of integrodifferential equations occurring in reactor dynamics, II. oo oZ Z Z Z A study of the real system (1) du/dt = - j'__00 a(x)T(x,t)dx, ao T/8x = b8 T/8x + >[(x)u, begun in J. Math. Mech. vol. 9 (1960), is continued. It is assumed that a,"'{,f E Lz(- oo,oo) and that a,b > 0. Uniqueness theorems concerning the solution of (1) subject to the initial condition (Z) u(O)

= u0 , T(x,O) = f(x) are obtained. Regarding b as a parameter, the dependence on b, for small b, of a particular solution u(t;b), T(x,t;b) of (1), (Z) is investigated in considerable detail. Under quite stringent conditions power series u(t;b) = l: ukbk, T(x,t;b) = l':, Tk (x,t)bk are obtained. Under less stringent conditions asymptotic expansions are obtained. An approximation procedure is given which enables one to obtain analytic functions which approximate u(t;b), T(x,t;b) and to relate these analytic approximations to the asymptotic series for u(t;b), T(x,t;b). (Received April 18, 1961.)

581-14 J. R. Kinney: The shortest path through points of a small set.

Let Ln(B) = 1.u.b.{P1 , ... ,Pn}€BL(P 1 .... ,Pn)' where L(P 1, ... ,Pn) is the length of the shortest closed polygon through the points Pl' ... 'Pn. Beardwood, Halton, and Hammersley showed that if the Pi are chosen independently with respect to the distribution m * on Ek, with dm * /dm = q, where

250 m is Lebesgue measure, then the limit limn._.ooL(Pl, ... ,Pn)/n1-l/'l8 q1-l/kdm exists, is maximized by m* = m, and is independent of the part of m* singular with respect tom. We show that: Theorem. H 8 is a bounded closed a.-dimensional set in Ek, k ;;:; Z, k.,... a. ::> 1, then for positive u and n sufficiently large nl-l/k+u > Ln(B) > n 1-l/a.-u. There exist bounded closed a.-dimensional sets for which both inequalities are violated infinitely many times for any negative u. (Received April Z1, 1961.)

581-15. A. L. Peressini: The weak continuity of the lattice operations in a vector lattice.

Suppose that (E,F) is a dual system over the real field where E is a vector lattice ordered by

the cone K, F is a vector space ordered by the dual cone K 1 of K, and F = K1 - K1 • The topology on E of uniform convergence on the order bounded subsets ofF is denoted by O(E,F). Theorem 1. H ;]' is a compatible topology on E which is finer than the weak topology o-(E,F) and the lattice operations are 0'" -continuous, then o(E,F) is coarser than (]'. Theorem Z. H the lattice operations in E are o-(E,F)-continuous and F has an order unit, then E and F are finite dimensional. Theorem 3. Suppose

that E is a normed vector lattice and that E 1 = K 1 - K 1 , then the lattice operations in E are weakly continuous if and only if E is finite dimensional. (Received April Z4, 1961.)

581-16. Wolfgang Schmidt: A combinatorial theorem on arithmetic progressions.

A well known theorem of Vander Waerden says that to any integer L i: 3 there exists an m$)) with the tollowing property. H m;;: n~) and if C1, Cz is a division of the integers 1, Z, •••• minto two classes, then there exists an arithmetic progression of_} distinct integers between 1 and m

which all belonf to the same class. We prove the Theorem. For some constant c ::> 0,

m5i-) S;: zl-c/L /Zlog!-. For large} this is an improvement of the estimate mj,l) !li (Z)z1)1/Z given by Erdos and R. Rado. (Received April Z4, 1961.)

581-17. Wolfgang Wasow: Simplification of turning point problems for systems of linear differential equations.

Let A(z,e) be an n-by-n matrix holomorphic in both variables at z = e = 0. It is assumed that A(O,O) has only one jordan block and that [ (d/dz) (det A(z,O)Jtz=O f: D. Consider the differential equa­ tion e dy /dz = A(z,e )y for the vector function y(z,e ). It is shown that corresponding to every positive integer m there can be constructed a transformation y =

holomorphic at z = 0 and independent of m and det P 0 (0) f. 0. (Received April Z4, 1961.)

581-18. K. H. Hofmann and P. S. Mostert: Splitting conditions for abelian subgroups.

Let G be a group. A function {II: G -a is a crossed endomorphism if {ll(xy) = x{ll(y)x- 1{11(x). H {II is a crossed endomorphism (endomorphism), then ; .... defined by {ll_..(x) = {ll(x) -lx is an endomor­

phism (crossed endomorphism). Let G{ll denote the kernel of {II. For an endomorphism, the radical of ;, denoted by R{ll, has been defined to be Rill = U nR!iln. We define the crossed radical of {II as

251 C!ll = G!iln.Lm. (Note: For an endomorphism !6, 91n is an endomorphism, and 91n.L a crossed Un,m endomorphism, but 91n.Lm is not in general either.) Then we prove the following Proposition: U !6 is an endomorphism, and !il.L(G) is contained in an abelian group, then the radical and crossed radical are groups, R!il is normal, and R!ll n C!ll = 1. Techniques to some extent analogous to transfer or homology methods allow us in special cases to construct crossed endomorphisms which lead to splitting theorems of the following type: Theorem 1. Let N be an abelian normal torsion subgroup of index n in the group G. Then there is a crossed endomorphism f such that G = Rf • Cf, Rf C N, and Cf n N = {x EN: order of xis relatively prime ton- 1}. Theorem 2. Let N be an abelian normal subgroup of index n in a group G. U N is uniquely divisible by n, then G splits over N. This generalizes a part of Schur's theorem. (For examples of the application of those techniques to topological groups, see the following abstract.) (Received April 24, 1961.)

581-19. K. H. Hofmann and P. S. Mostert: Splitting theorems for vector subgroups of topological groups.

Theorem 1. Let G be a topological group, N a closed nm;mal subgroup satisfying the following conditions: (i) N is isomorphic to the vector group of a reflexive Banach space: (ii) G/N is compact; (iii) there exists a continuous cross section to the cosets of N. Then there exists a compact subgroup C isomorphic. to G/N such that G = N · C, N n C = 1. Remark. Condition (iii) can be omitted in the following cases: (1) N is isomorphic to a Hilbert space; (2) G is first countable; (3) G is locally compact (Iwasawa). Theorem 2. Let G be a maximally almost periodic group such that G/Go is compact (Go the component of 1). Then there is a normal vector subgroup N of G and a compact sub­ group C such that G = N · C, N n C = 1. This product need not be direct as in the case G = Go (Freudenthal, et al.). The essential technique in Theorem 1 is the construction of a crossed endomorphism (see the preceding abstract) from G to N using the cross section and an invariant measure on G/N. Similar theorems on local splitting are obtained for normal subgroups locally isomorphic to vector spaces of the above types. (Received April 24, 1961.)

581-20. Lionello Lombardi: Hilbert decomposition and semicontinuity.

The integrals of the calculus of variations. are redefined and their Hilbert decomposition is defined in abstract terms. A simple general proof that integrals which admit a Hilbert decomposition are semicontinuous is given. This theorem, which establishes a relation between continuous and semicontinuous integrals and indicates a method of constructing Weierstrass tf -functions for each particular problem, is formal and independent of the topology adopted. (Received April 25, 1961.)

581-21. W. E. Bonnice: A generalization of a theorem of Caratheodory.

For convex X contained in En let the k-dimensional interior of X, denoted intkX' be the set of points which are in the relative interior of some k-dimensional simplex which is contained in X. For k = O,l, ••• ,n let M(k,n) be the least integer such that wheneve11 S is a subset of En and p E intkconv S then p € intkconv T for some at most M(k,n)-membered subset T of S. We prove that M(k,n) = max(2k,n + 1). The fact that M(O,n) = n + 1 is Caratheodory's theorem: if p E conv S

252 then there is a subset T of Shaving at most n t 1 elements such that p E conv T. That M(n,n) = 2n is the well known result: if p is interior to conv S then p is interior to conv T for some at most 2n-membered subset T of S. (Received April 25, 1961.)

581-22, W. J, Firey: Further means of convex bodies. Preliminary report.

In earlier issues of these Notices (Abstracts 569-24, vol, 7 (1960) p, 265 and 570-15, vol. 7

(1960) p. 282), means Mp(K0,Ky8) of convex bodies Ki,(i = 0,1), in Euclidean n-space, sharing a common interior point Q were defined for 0 ~ & & 1, IPI;:;; 1 as follows. Let Hi, Fi be the support and distance functions of Ki with respect to Q. For p ;;; 1, Mp(K0 ,K 1; 13) is that convex body having [(1 - J) Hb t 8 H}J l/p = H(p) its support function; for p ;;;. 1, M -p(K0,K 1; iJ) is that convex body having [(l - J )Fb t t9 FlJl/p = F(p) as its distance function. Now let 0 < p ~ 1 and define Mp(K0 ,K1; J) as the intersection of the half spaces (x,w) ~ H(p)(w). Further, for the same p, define M_p(K0 ,K 1; J) as the convex closure of the star-body F(~) (x) ;;; 1. Letting p ~ 0, we obtain limit bodies Mot and M0 _. Set K~) = Mot 0. (Received April 26, 1961.)

581-23. Julius Kane: An approximate Wiener-Hopf decomposition,

Methods for the solution of two-part boundary value problems usually require a function theoretic decomposition. A factorization which has been studied by Clemmow, Senior, Bazer, Karp, Heins, Feshback, et al, requires a function D"+(u,A) such that O"+(u,~)O"+(- u,))) = 1 -h/i(k2 - u2)1/2 where O"+(u,~) is analytic in the U.H.P. Im u >0 and k = lklei8,8 > 0. By considering O"+(u,~) as an analytic function of ~, one can show that 0"+ (u,h) = 1 - (,1/71) / (k2 - u 2) l/Zlog[(iu t (k2 - u 2) 112)/ik] + Cl-(.1) 2). This representation, although approximate, is of high accuracy for lA I.<: 1, and explicitly exhibits the required behaviour in the complex u-plane. The research described in this report has been sponsored by the Air Force Cambridge Research Center, Electronics Research Directorate, under contract no. AF 19(604)-7983. (Received April 26, 1961.)

581-24. S. P. Lloyd: Extreme means are homomorphisms,

With C(X) the Banach algebra of continuous real or complex functions on compact Hausdorff space X, let (}t be any given closed seflfadjoint subalgebra of C(X) containing constants. Let {P denote the (possibly empty) set of bounded positive projections in C(X) onto i1t:(as spaces), In answer to a question raised by F. B. Wright [Generalized means, Trans. Amer. Math, Soc. Vol, 98 (1961) pp, 187-203], it is shown that an extreme point of if is necessarily an algebraic homomorphism of C(X) onto cJt, Although fiJ is convex, bounded, and closed in the space of operators with the weak operator topology, compactness in the weak operator topology may fail because of incompleteness;

253 an example is given where .@is nonempty but has no extreme points. (Received April 26, 1961.)

581-25. P. H. Maserick: Half rings and convex polytopes in Banach spaces.

A closed bounded convex set P in a Banach space X is defined to be a convex polytope in such a way that if X is finite dimensional, P is a convex polyhedron (i.e. P can be represented as the inter­ section of finitely many half spaces), and if X is not finite dimensional, P enjoys many of the properties of its finite dimensional counterpart. Thus a convex polytope is taken to be a reasonable generalization of a convex polyhedron. An ascending half ring is a collection fi! of sets such that:

(i) lf R,R I E JR then R n R I E -'f. (ii) lf R,R I E 1?· R c R I then there exists a countable chain

[Ri} i of monotonically increasing sets of .q such that R 1 "'R, each R;_ - R;.-1 E q and Ui R;_ "'R 1• This definition is well motivated by the definition of a half ring (von Neumann] of sets in finite dimensional spaces. It can be shown that·if J? is an "absorbing" neighborhood base of convex sets for a separable Banach space X then the closure of R is a convex polytope for each R E" (if, and if X is finite dimensional then .q is an ordinary half ring. Moreover all closed subspaces of the Banach space c 0 are characterized as being the only separable Banach spaces having such neighborhood bases. The space c0 itself is characterized as being the only Banach space having a convex parallelotope as its unit sphere. (Received April 26, 1961.)

581-26. j. R. Rice: Algorithms for Tchebycheff approximation by abx + c,

There are several well known algorithms for Tchebycheff approximations to f(x) by linear combinations of given functions on a finite point set. They do not extend to approximation by abx + c because at some stage a best approximation to f(x) may not exist on a particular set of four points. A pseudo best approximation may be defined, however, which overcomes this difficulty. The proper­ ties of these pseudo-functions are devel.oped. A computer program has been written for one of these algorithms and it works quickly and with high accuracy. (Received April 26, 1961.)

581-27. R. L. Stanley: Boolean calculi of n categories.

An elementary method of natural deduction for quantification theory leads with great ease through the following developments: (1) the propositional section of this method reviews the isomor­ phism which holds from proofs and theorems in the propositional calculus, to those in the algebra of sets, and to those in the abstract boolean algebra which these two systems interpret; (2) the whole method (for the first order functional calculus) reveals a corresponding isomorphism from proofs and theorems in quantification theory, to those in a calculus of set-operators, and to those in an abstract boolean calculus of operators which these two systems interpret; (3) this boolean operator calculus is easily generalized to n-category boolean calculi, which involve n kinds of operators; (4) basal logic (propositional calculus, quantification theory, and the theory of identity), enriched to contain also the algebra and operator calculus of sets, is·seen to be an interpretation of a 2-category boolean calculus, with the first category being set-operators, and the second, quantifiers. That every n-category boolean calculus is consistent relative to quantification theory becomes evident to inspection, and the absolute consistency of quantification theory itself is only slightly less obvious, considered from this standpoint. (Received April 26, 1961.) 254 581-28. E. K. McLachlan: Extremal elements of the cone of semi-norms on the Euclidean

The set of real, absolutely homogeneous and subadditive functions p defined on a real linear space L form a cone (!. That is, e is closed under addition, non-negative scalar multiplication and {!_ n C- = {o}. Those functions p of e such that when p = p 1 + p 2 for p 1 and p 2 in (: imply that p 1 and p 2 are non-negative scalar multiples of pare extremal elements off!. Those functions p = If I where f is a linear functional on L are extremal elements of e. For L = E 2 these are the only extremal elements of e. (Received April 26, 1961.)

581-29. Gustave Solomon: On quaternary cyclic codes.

A (k,n) quaternary cyclic code is a k-dimensional cyclic subspace of ordered n-tuples over the field K of four elements. These codes are defined by linear recursion for divisors f(x) of xn + 1 over K. The number of errors which A can correct is determined by d = n-maximum number of zeros in any nonzero vector of A. It is known that d ;;. do(f(x)) for a certain do depending on f(x). For n = p, where 2 has multiplicative order p - I, we improve the previously obtained general

estimate of Zierler and Gorenstein (A class of cyclic, linear, error-correcting codes in~ symbols, Group Report 55-19, Lincoln Laboratory, 1960). We associate to each code word a=

(a0, a 1, a 2, ••• ap_ 1) the S-M polynomial, ga(x) of degree less than or equal top- 1, such that ga(zi) = ai' where z is a primitive pth root of unity. ga(x) then determines the f(x) of the linear re­ cursion. The choice of ga(x) depends on the quadratic character of- 3 relative to p. Our results are:

For k = p + 1/2, d i: 3. We also give a general number-theoretic algorithm to improve d for

particular p, e.g. we obtain d §;; 5 for the (6, 11) code. The results are clearly extendable to codes, using q 2 symbols, q a prime. (Operated with support from the U. S. Army, Navy and Air Force.) (Received April 24, 1961.)

581-30. G. W. Day and F. M. Yaqub: On free alpha-extensions of Boolean algebras. Preliminary report.

An a-complete Boolean algebra fil is said to be a free a-extension of the Boolean algebra tJt if CiJ has a subalgebra tJt 1 isomorphic to rJt such that CiJ is a-generated by tJt 1 and every homomorphism

of (Jf I into any a-complete Boolean algebra e can be extended to an a-homomorphism of fil into e. (L. Rieger has studied the free a-extensions of free Boolean algebras (Fund. Math. vol. 38 (1951) pp. 35-52.).) The concept of a free complete extension of a Boolean algebra may be defined similarly. A Boolean algebra cJt is super-atomic if every subalgebra and every homomorphic image of tt' is atomic. The following results are obtained: (1) For every cardinal a, every Boolean algebra t7t has a unique free a-extension, t/ta. (2) For every Boolean algebra tJt and every cardinal a, £Za is a-representable if and only if ata is an a-field. In particular, for every Boolean algebra 0t, £1£0"" is acT-field. (3) For every cardinal a ;;=z.~O, IZa is a-representable if and only if CJt is super-atomic.

Furthermore, if a/ is superatomic, then OL has a free complete extension, ot00, and this is a complete field. (Received April 26, 1961.)

255 581-31, G. J, Rieger: On Waring's problem for algebraic number fields.

In this paper, Schnirelman's generalization of Waring's original problem is extended to arbitrary algebraic number fields. (Received April 27, 1961.)

581-32, M.s. Ramanujan: The moment problem in certain general function spaces. Preliminary report,

Let C denote a class of positive integrable functions c(x) on (0,1) satisfying the properties: (i) 1 E C, (ii) C is normal and (iii) Jo 1c(x)dx, c E C are all bounded, Let X(C) denote the Banach space of all measurable functions f for which llfll = supcec/'ot(x)lf(x)l dx is finite. Then we determine a set of necessary and sufficient conditions for the existence of a function f(x) E X(C) so that a given sequence {prJ of constants may be the moment constants generated by f(x) i.e. Jln may have the representation Jln=/olxnf(x)dx, n = 0,1,2,. ... More precisely, let f:(x) K K-n = [(K + l)(K + 2}/(n + 1) J Cn /::,. Jln+l' (n + 1)/(K + 2) < x < (n + 1}/(K + 1). If the space X(C) has rearrangement invariant norm and if the integrals F(e) = ,J'~f(x)dx, f E X(C), llfll < 1 are uniformly absolutely continuous then the representation Pn = /olxllf(x)dx with f(x) E: X(C) and llfll :!iii M is possible if and only if the norms llf~ II ;;; M for all n, The spaces LP (p > 1) and the space of bounded functions are spaces of the type X(C) and therefore the solutions of the moment problems for these spaces will follow as simple corollaries. (cf. also the results of Lorentz: Bernstein polynomials, Toronto, 1953, Chapter III). (Received April 27, 1961.)

581-33, E. G. Straus and F. A. Valentine: Maximal convex sets.

Theorem. Let S be a compact connected set in ann-dimensional topological linear space Ln• and let m be a fixed positive integer i1!; 2. If each point of S is contained in m and only m maximal convex subsets of S of dimension ;:; n - 1, then S is empty, The authors proved in an earlier paper that if m = 1 in the above theorem, then Sis convex (Amer. J, Math, vol, 74, no. 3 (1952)). The proof of the above theorem is based on a category argument, and is not elementary. (Received April 27, 1961,)

581-34, Paul Yearout: (m,n)-distributive ring multiplications on direct sums of cyclic groups.

For an (m,n)-distributive ring. (Pro c. Amer. Math, Soc, vol, 9 (1958) p. 876) whose additive group is a direct sum of cyclic groups all multiplications are exhibited and som.e consequences shown, (Received April 27, 1961.)

581-35, H. E. Conner: Limiting theorems for a position.-dependent branching process.

Let Zn(x) be the size of the nth generation for a Galton-Watson type branching process with the particles diffusing in IJJ,L] and with the zeroth generation particle at x. The transformation probabilities are assumed to be a function of position with generating function h(~ ,y), Is I ::; 1 and 0 ;:;; y ~ L. In the case where E [Zn(x) J -+ oo as n - oo, and with a summability type condition on the first moment of h(§ ,y), the sequence Zn(x)/E [~ (x)) converges in distribution to random variable W(x)

256 with E[.\v(x)]= 1. The Laplace-Stieltjes transform of the distribution of W(x) is characterized as the unique solution to an associated nonlinear equation. (Received April Z7, 1961.)

581-36. H. S. M. Coxeter: The total length of the edges of a non-Euclidean polyhedron.

It was conjectured by L. Fejes T6th, proved by A. S. Besicovitch and H. G. Eggleston (The total length of the edges of a polyhedron, Quart. J. Math. Oxford Ser. Z, vol. 8 (1957) pp. 17Z-190) that, of all polyhedra containing a given sphere, the circumscribed cube has the least total edge-length. It is natural to ask whether this property of the cube in Euclidean space holds also for the regular hexa­ hedron in a non-Euclidean space. A negative answer is established by the following counterexamples. In elliptic space, a sufficiently large sphere admits a circumscribed tetrahedron whose total edge­ length is less than that of the circumscribed hexahedron. In hyperbolic space, a sufficiently large sphere admits a circumscribed dodecahedron whose total edge-length is less than that of the circumscribed hexahedron. (Received April Z7, 1961.)

581-37. W. L. Strother and R. G. Selfridge: Interval analysis.

Let f be a function which maps intervals onto intervals. One can define addition of intervals by A + B = [xlx = a + b, a E A, b E B) and multiplication by AB = [xlx = ab, a E A, b E B]. Integration can be defined as an intersection of cross products of intervals and one then defines differentiation by saying g is the derivative of f if for all intervals (a,b) one has f(a,b) = j~g(x,x)dx. With sub­ traction of intervals suitably defined a mean value theorem for such functions as have derivatives is possible, and this can be extended to unions of disjoint intervals. Necessary and sufficient conditions for the existence of a derivative are given in terms of partial derivatives with respect to an end point of an interval. (Received April Z4, 1961.)

581-38. F. W. Anderson: On f-rings with the ascending chain condition.

Let A be an f -ring. ~ee Birkhoff and Pierce, An. A cad. Brasil. Ci. vol. Z8 (1956) and D. G. Johnson, Acta Math. vol. 104 (1960) for the general theory off-rings.] If A has the ascending chain condition for L-ideals and if N(A) = 0, then A is isomorphic to a subdirect sum of finitely many prime ) -rings. If in addition J(A) = 0, then A is isomorphic to a direct sum of finitely many .!-primitive f-rings. Thus for J-semi-simple f-rings the ascending and descending chain conditions for_l-ideals are equivalent. (Received April Z8, 1961.)

581-39. R. J. Aumann: Utility theory without the completeness axiom.

Let X be a mixture space (with operation xpy) on which there is defined a transitive, irreflexive

and anti-symmetric partial order ~satisfying: if 0 < p < 1, then x r y if and only if xpz >- ypz; and if xpy ;:- z for all p > 0, then not z >- y. Theorem 1. There is a function u from X to the reals such that: u(xpy) = pu(x) + (1 - p)u(y); and x '(-y implies u(x) > u(y). u is called a utility; it is a generalization of the von Neumann-Morgenstern utility. Theorem Z. Let E be a convex polyhedron (i.e. the convex hull of a finite number of points) in X, and let x E E. Then x is maximal in E if and only if there is a

257 utility u ~X such that x maximizes u ~E. Theorem 2 generalizes a result of Shapley (Naval Res, Logist. Quart, vol. 6 pp, 57-61), and can be used to compute solutions for games in which each player's order on the possible outcomes is partial rather than complete, The proof of Theorem 2 follows

Shapley's proof of his more special result. The proof of Theorem 1 makes use of convex cones and separation theorems. (Received April 28, 1961,)

581-40, J, R. Brown: The characterization of vector-valued Fourier-Stieltjes transforms.

The well-known characterization of Fourier-Stieltjes transforms on the real line due to S. Bochner has been extended to locally compact abelian groups G by W. Eberlein (Duke Math, J, vol. 22 (1955) pp. 465-468). In this paper is proved a generalization of Bochner's theorem to Fourier­ Stieltjes transforms of vector-valued measures on the character group 6 taking values in a reflexive Banach space X. The function !6: G -x is the Fourier-Stieltjes transform of a measure Ji: ilJ (G)-+ X if and only if there exists a constant K such that II Z:. antl

581-41, R. F. Drenick: On the Wold decomposition of non-Gaussian random processes.

According to a theorem by Wold (see e.g. J. L. Doob, Stochastic processes, p. 571), a stationary Gaussian random process (over discrete time) can always be decomposed into two canonical com­ ponent processes, one of which is made up of mutually independent random variables and the other of which is deterministic, It has been surmised for some time that a similar decomposition exists for certain non-Gaussian processes. More specifically, let {xn, - oo < n < + oo} be the given process, :fn the Borel field generated by xk' k ~ n. Define the random variables f§n,- oo < n < + oo} and

?> ,- oo < n < + ool by F(x ) and ?J G(x where F and G are conditional distribu- { rn :J sn = n I.Jn- 1 tn = n I~) n tion functions, ~is the Borel field generated by §"k' k-::; n, and ~the one generated by ?j'k' k l!i n,

· The ~ and 7( processes then constitute a Wold-type decomposition of the x process if the transforma­ tions Xn to Sn and xn to ??n are one-one almost surely, and_.#n and »n are independent. (Received April 28, 1961.)

581-42. R. E. Fullerton: On the existence of extreme rays in cones.

Let C be a closed cone in a real locally convex linear topological space X. The extreme sub­ sets of C other than the vertex and C itself are called faces of C. Let C satisfy the following condi­

tions: (1) Every face F of c has a unique complementary face F I such that c = F + F I and LF n LF 1 = 0 where L F denotes the closed linear s•Jbspace determined by F, (2) for every x E C, the set P x = C n (x - C) is compact, (3) every face F of C is a sub cone of C satisfying (1) and (2), Let the faces of C be partially ordered by inverse inclusion and let the faces of each Px be similarly ordered. Then if for some face F there exists an x E C, and a number ex > 0 such that the family of faces of P x

258 contains a maximal linearly ordered subfamily all of diameter greater than ex, the face F contains an extreme ray of C, If this condition is satisfied for all faces F of C, C is the closed convex hull of its set of extreme rays. (Received April 28, 1961,)

581-43. Naoki Kimura: Simplicity of the transformation semigroup of a set.

LetS be the semigroup of all transformations of a set X into itself. We shall determine the greatest indempotent quotient of S and the greatest commutative quotient of S both of which are reduced to a one-element semigroup when the set X is not finite. Some other quotients will be discussed. (Received April 28, 1961.)

581-44. Thkayuki Thmura: Certain semigroups.

In the ]. Gtkugei Thkushima Univ. vol. 5 (1954) the author determined the structure of r -semi­ groups, i.e., a semigroup whose subsemigroups form a chain. He defines a T*-semigroup to be a semigroup whose every proper subsemigroup is a r -semigroup, In the present paper he has deter­ mined all the types of r *-semigroups except groups. (1) A T*-semigroup is unipotent. (2) Every element of a T*-semigroup is of finite order. (3) A f*-semigroup whose unique idempotent is zero is of order < 5. Using these lemmas, any T*-semigroup except groups can be completely determined. (Received January 16, 1961.)

259 ABSTRACTS PRESENTED BY TITLE

61T-82. Alfred Tarski: A model-theoretical result concerning infinitary logics.

For notation see Tarski, Colloq. Math. val. 6 (1958) p. 171. Given a set S of sentences in logic P a.' M(S) denotes the class of all models of S. Thus M(S) is a class of similar relational structures formed by relations with rank < eva.. Given any class K of such structures, K E ECa. means: K is elementary in P a. (i.e., K = M(S) for some set S of sentences in P a.); K 6. UCa. means: K is universal in P a.; if K contains only algebraic structures, K E. EQa. means: K is equational in P a.· S(K) denotes the class of all substructures of structures in K. A cardinal .Ka. is incompact if, for some setS of sentences in P a.' M(S) = 0 while M(T) i 0 whenever T~ SandT has power <~a.; 1\a. is strongly in­ compact if in addition S has power ~a.· All accessible regular cardinals are incompact; by a result of Hanf, most of the (strongly) inaccessible cardinals > *o are strongly incompact. Theorem T. For every regular ~ the following conditions are equivalent: (i) lt; is compact (i.e., not incompact); (]. (]. (ii) lf_K E ECa. and every substructure with power < ,Ilia. of a str~ A is isomorphic to a structure in S(K), then A itself is isomorphic to a structure in S(K); (iii) S(K) E UCa. whenever K E EC .t (iv) S(K) E EC a. whenever K E ECa.. The implication (i)->- (ii) generalizes Henkin's theorem for logic P 0; similarly, (i) ~(iii) generalizes an earlier result of the author. (Received February 20, 1961.)

61 T-83. Samuel Kaplan: The second dual of the space of continuous functions. III.

Let X be a locally compact space. We denote by c<» the vector lattice of the real continuous functions which vanish at infinity, and by C k the (vector lattice) ideal in C'i.l consisting of the functions which have compact support. A study is made of some of the relations between their (vector lattice) duals and between their second duals; also, of the ring C of~ real continuous functions on X as a ring of operators on these various spaces. A large part of the paper is devoted to the following two (still unsolved) problems, upon which a deep study of the above seems to be dependent: (1) If every !:tomic (Radon) measure on X is bounded, does it follow that every (Radon) measure is bounded? (2) Is the property of ck, that every element has a relative strong order unit, also possessed by its second dual? (One element g of a vector lattice is a relative strong order unit of another element f, if every element of the closed ideal generated by f is dominated by some multiple of g.) (Received February 16, 1961.)

61 T-84. Edgar Reich: Combinatorial results for processes with independent increments.

Given a stochastic process with sample function x(t), 0 ~ t :2 T, we are interested, firstly, in

studying (A): m(t)- x(t), where m(t) = sup0 ,r.,t x(L). The following is of a purely combinatorial nature. Suppose (a) x(t± 0) exists, x(t) = x(t - 0) .s x(t + 0), (b) x(t - 0) ~ k ~ x(t + 0) has at most a finite

number of roots. Let ,"'(t;~) bel or 0 depending on whether m(t)- x(t) is ~ t. or >$,and let ? (u,t;$) be the number of 'l:''s, u ~ 'L < t, such that x(1:') = x(t) + s, but such that x does not have a relative maximum at 7:. Then (*) p

260 x(t) - x(O) + 5 is ~ 0 or < 0. If x(t) is a stochastic process with independent (not necessarily station­ ary) increments satisfying certain regularity properties corresponding to (a),(b), then on taking the expectation of(*) one obtains a Volterra equation of the 2nd type for F(t) = Pr£m(t) = x(t)J, and a formula for Prfm(t)- x(t) ~ ~}. /1 >- 0, in terms of F(t). By appropriately modifying the definition of the kernel J- one obtains similar combinatorial results and Volterra equations for (B): the distribu­ tion of the first passage time of x(t) through x(t) = 0, under the additional hypothesis that x(t) is increasing in a neighborhood of each point of continuity. (Received February 9, 1961.)

61T-85. R. J, Aumann and M. Maschler: An equilibrium theory for n-person cooperative games. Preliminary report.

We are concerned with n-person games in characteristic function form, not necessarily super­ additive. A pair (x,£2) = (x1, ... ,xn; B 1, ... ,Bk) consisting of a payoff vector and a partition of the players into coalitions is a payoff configuration (p.c.) if (i) 2:iE.Bxi = v(B) for each B f' ~; and (ii) C C B E (i3 implies LiECxi ~ v(C). Let (x,;;3) be a p.c., let B E ;,-;}, and let K,L C B, K,L # r6, K n L = r6. An objection to (x;.@) by K against L is a p.c. (y; e) for which y i ::>xi when i E K, and Yi;;;;. xi when i is "cooperating" with K, i.e., belongs to a coalition in (!that intersects K; it is also required that no member of L cooperate with K. A counter-objection by L against K is a p.c.

(z;J:';) for which zi .S: xi when i is cooperating with L in the counter-objection, and zi ?;;; Yi when i both cooperates with L in the counter-objection and cooperated with Kin the original objection; also not all members of K may cooperate with L in the counter-objection. A p.c. (x;J./3) is stable if there is a counter -objection to each objection to (x;W). The structure of the set of stable p.c.' s has been found for all 2- and 3-person games, and some others; it is always computable for a given game. (Received February 20, 1961.)

61 T-86. j. E. Houle: Finite groups of quaternion matrices.

Let G be a finite group with r conjugate classes, s of which contain the inverses of all their elements, and let R be the regular representation of G by matrices of zeros and ones. A classification of representations of use in mathematical physics j!.omont, Applications of finite groups, 1959, Chapter III] and criteria for complex-irreducible semigroups of complex matrices to be quaternion­ reducible and for comp1ex-nonsimilar semigroups of complex matrices to be quaternion-similar are used to prove the theorem: There are exactly (r + s)/2 quaternion-irreducible, quaternion-non­ similar constituents of R and every quaternion-irreducible representation of G is similar to one of these. A further consequence is that two matrix representations of a finite group are quaternion­ similar if and only if the real parts of the traces of the corresponding matrices are equal. (Received February 20, 1961.)

61T-87. R. M. Blumenthal and R. K. Getoor: The Hausdorff-Besicovitch dimension of the graph of a symmetric stable process.

Let {X(t); t ~ oJ be the symmetric stable process of index 11, 0 «: 11-:-; 2, in one dimension. We assume X(O) = 0 and that the sample functions are normalized to be right continuous and have

261 left-hand limits everywhere. Let G(C<.!) = {(t,X(t,.v)): t ;;:; 0}, that is, the graph of the sample function X(• ,.v). In the present note we prove: (i) If 1

61T-88, Alfred Horn: L-Heyting algebras,

An L-Heyting (LH) algebra is a Heyting (H) algebra in which (x -+Y) + (y ----7X) = 1 for all x,y, It is shown that an H algebra is an LH algebra if and only if it is a subdirect product of simply ordered H algebras. An H algebra is an LH algebra if and only if the sum of any two incomparable prime ideals is the whole algebra. The normal completion of an LH algebra is not always an LH algebra. It is shown that the set of formulas of the predicate calculus which are valid in every simply ordered complete H algebra is the same as the set of those which are valid in the set of reals in [0, 1], and this set of formulas is identical to the set of theorems of the system obtained from the intuitionist predicate calculus by adding the axiom (p :::> q) v (q ::J p). A similar result holds for the propositional calculus, as shown by Dummett. The present proof is more algebraic. (Received February 20, 1961.)

61T-89. Alfred Horn: The separation theorem of intuitionist propositional calculus.

Consider a standard formulation of the intuitionis~ propositional calculus in which the only rules of inference are modus ponens and substitution. The separation theorem states that any theorem a. of this calculus can be proved from the axioms for implication together with only the axioms containing the connectives actually appearing in a.. The original proof of the separation theorem by Wajsberg contains an error. Other proofs depend on Gentzen's theorem on elimination of cuts, or its equivalent. An algebraic proof is given here. The problem is reduced to that of characterizing subsets of a Heyting algebra which are closed under ~and none or more of the operations •, +,and'· (Received February 20, 1961.)

61T-90. M. I. Knopp: Construction of a class of modular functions. II.

In a previous paper[Abstract 864-203, Notices Amer. Math. Soc. vol, 6 (1959) p. 854] the author constructed modular functions connected with the principal congruence subgroup G(j), of level j, for j ~ 2, Fourier series A,.,('t;j) in e('t/j) were given which define functions analytic in d ('t') >- 0 and satisfying the transformation equations for an abelian integral connected with G(j), By taking a suitable linear combination of the ))y('t';j) we obtained functions f(t') which are invariant under G(j); that is, f(VT) = f('n for all V E: G(j) and o(('t') >- 0. The functions f('t') vanish at m but their behavior at the other parabolic cusps of G(j) was previously undetermined. In this paper we show that, in the appropriate uniformizing variable, )).,,(T;j) has a pole of order 7' at one fixed parabolic cusp and is regular at all the others. Hence )):r('L';j) is an abelian integral connected with G(j). Furthermore we introduce new functions which are abelian integrals connected with G(j). Each of these has a single pole of order)-) at one parabolic cusp and is regular at the others. The functions are such that the choice in placing the pole is fairly wide. It follows that when the genus of the fundamental region

262 of G(j) is zero, i.e. when 2 ;;;_ j ~ 5, the )).f('L;j) are themselves modular functions. In these cases no linear combination is necessary. (Received February 20, 1961.)

61T-91. M. I. Knopp: Construction of automorphic forms on H-groups.

Lehner {Michigan Math.]. vol. 4 (1957) pp. 265-279] has obtained the Fourier series expansions of entire automorphic forms of dimension r > 0 connected with H groups T having exactly one para­ bolic cusp. (The restriction to one cusp is not essential and is lifted by Lehner in a later publication.) The Fourier coefficients are infinite series involving Bessel functions and exponential sums related to Kloosterman sums. While every automorphic form on r of dimension r >0 has a Fourier series of the type given by Lehner, the converse is not true. That is, a Fourier series of the type given may or may not represent an automorphic form on T. However, we show that when r is a positive integer such series satisfy the transformation equations for an automorphic form of dimension r on r, except for the appearance of an additive polynomial of degree at most r. Using this result and the fact that T is finitely generated it is a simple matter to construct automorphic forms on r. The results can be extended to the case r = 0 for those groups for which the resulting exponential sums have a nontrivial asymptotic estimate. The author has previously shown that this is the case for the modular group and two groups closely related to it. (Received February 20, 1961.)

61 T-92. Svetozar Kurepa: On roots of elements of a Banach algebra.

Theorem 1. Let !11 = {u,f3 •...; be a field of real or complex numbers and B = £a,b, ... .J a Banach algebra over the field !11 with a unit e. Then, there exists a Banach algebra B' = [A,C, .. .J over ¢with

a unit E and a mapping (imbedding) 1o: B ---T B' such that: (l) ,O(ua +/b) = "/(a) + ..~ J(b), (2) f(ab) = p(a);"(b), (3) li,V(a)ll = llall and (4) /l(a) and fb have the same spectrum. Furthermore, if a is any element of B and n any natural number, then there exists at least one A E B' such that: (I) A f(a) = p(a) A and (II) An =/(a). By use of this theorem one can imbed B in a normed

algebra B' 1 which has the property that any element of B' ' possesses any root in B' '. Theorem 2. Let f(z) = ~unzn be an entire function and B a Banach algebra of matrices of finite (any) order. Then, the mapping T ---"?f(T) = L~"n Tn of B in B is onto B if and only if for any complex number u the set ru = {z if(z) = u, f'(z) f- OJ is not empty. (Received February 20, 1961.)

61T-93. F.]. Murray: Perturbation theory and Lie algebras.

An approach to perturbation theory is considered based on the formula exp (- i B) A exp (i B) = A exp (i B]), where A B] = A B - B A. The operators, A, form a linear space ,ft and the operators B considered are such that B] take,!{_ into itself. The case where fi: is isomorphic with the set of n dimensional vectors c is considered. Sufficient conditions for the basic formula are given in terms of the "analytic vectors" of E. Nelson. The set,yt, of B's available is a Lie algebra. Exponentiation leads to a Lie group of operators U, and A and A' are said to be B equivalent if A'= u-l AU. Each B is associated with an n X n matrix, b, which specifies the operation B] relative to the vectors c. The matrices b form a Lie algebra with a corresponding Lie group of matrices u such that A and A' are B equivalent if and only if for some u, c' = uc. The set of A' equivalent to a given A is determined by

263 the orbit of a given vector c under the Lie group; i.e., the set of u c. A functionally complete and functionally independent set of invariants for B equivalence are obtained and also global invariants in the form of similarity invariants. Two examples are discussed. (Received February 20, 1961.)

61 T-94, M. S. Robertson: Variational methods for functions with positive real part.

In order to obtain variations of the form W* " W + fir/J(W) for various classes of univalent functions f(z), regular in lz I< 1, when f(z) can be expressed in terms of a regular function P(z), P(O)" 1, with positive real part in lz I< 1, the problem is resolved by introducing a variation formula for the functions P(z), The variation formula is used to give a characterization of the (n - 1) coefficient space En-l' a closed simplex, for the extremal functions P(z) for which ReP(n)(O) is a maximum over the class of functions P(z). Much of the Caratheodory-Toeplitz theory may be derived by this variational method, In addition, a variation formula is derived for regular functions F (z) "f[!..l(z)} which are subordinate to a given univalent function f(z) in lz I< 1. (Received February 20, 1961.)

61T-95, Mario Petrich: The Cartesian product of two semigroups.

Let S and T be semigroups, The semigroup S X T (with coordinatewise multiplication) is called the Cartesian product of S and T. A prime ideal in S is a two-sided ideal whose complement is a semigroup. Theorem 1. L is a prime ideal in S X T if and only if L " (I X T) U (S X J) for some A prime ideals I inS and J in T. LetS be. the set of all semicharacters of S. Theorem 1 is used for ~ ·"" "' proving Theorem 2, S X T" {X:IX(x,u)" r/J(x)~(u) for all (x,u) E S ?< T, for some r/J E. S,ljf E T}, and different pairs r/J and 1jf yield different X-. If S and T are topological semigroups, then S X T is a topological semigroup (under the product topology). In this case, a statement similar to Theorem 2 is valid for continuous semicharacters. There are a number of consequences of the above results. The set of zeros of a continuous semicharacter of a topological semigroup S is called a generating prime ideal in S. A statement similar to Theorem 1 is valid for generating prime ideals, All statements generalize to any finite number of (topological) semigroups, (Received February 15, 1961.)

61 T-96. J. M. Kister and L. N. Mann: Isotropy structure of compact Lie groups on complexes.

The following gives an affirmative solution to a conjecture of Floyd [seminar on transformation groups, Annals of Mathematics Studies, 1960, p,95], Theorem. If G is a compact Lie group operating on a finite complex K, then there are only finitely many distinct conjugate classes of isotropy sub­ groups. The proof makes use of a decomposition of K into a finite number of open invariant manifolds, each of which has an orientable covering manifold whose integral cohomology (with compact supports) is finitely generated. Lifting the action of G to the covering manifolds, an application is then made of a result of L. Mann ~bstract 576-111, Notices Amer, Math, Soc, vol. 7 (1960) p. 932] to establish the above theorem, A simple example is given to show the theorem is false for K a locally-finite complex having finitely generated integral cohomology. (Received February 21, 1961,)

264 61 T-97. D. V~ere- Jones: Geometric ergodicity in denumerable Markov chains.

D. G. Kendall has called a denumerable, irreducible, aperiodic Markov chain (with matrix P) geometrically ergodic if the sequences (pfj>> tend geometrically to their C - 1 limits, and he has shown that a solidarity theorem holds for this type of convergence. Now drop the assumption that P is stochastic, and assume only that it and its iterates have finite, nonnegative elements, and that it is "irreducible" and "aperiodic". Then (i) the generating functions Pij(z) have the same radius of convergence R, (ii) for R > 0, and a suitable "positive recurrence" condition, the value 1/R is associated with essentially unique positive eigenvectors (Ei)' (mj) of P, pT respectively, such that p~j)Rn ___,. ))ij = ~ooEim j' (iii) if any one of the P ii(z) can be extended as a meromorphic function to an

open region covering the disc lz I ~ R, the Pij (z) can all be so extended, and there is a common larger circle within which each has as its only singularity a simple pole at R. Applied to Markov chains, these results imply that whether a geometrically ergodic chain is transient or positive-recurrent, a common value less than unity can be given to the ratios of geometric series dominating the conver­ gence of the sequences (ptj>). (Received February 21, 1961.)

61T-98, 61T-99, 61T-100, 61T-101 WITHDRAWN.

61 T-1a2. William Craig: ~a-homogeneous relatively universal systems.

Let c.1t be ~a-homogeneous if and only if it satisfies (ii) of Abstract 55a-39, Notices A mer. Math. Soc. vol. 5 (1958) p. 78a. Let £. "~ C'i if and only if, for each m <. c:Jand each set 2: of elemen­ tary formulas with v 0, .•• ,vm as only free variables, if 2: is satisfied by some {ba•···•bm,bm+1, ..• ) in df:-: then I! is satisfied by some (aa•···•am,am+1 ••• .) in t9!. (1) If 6:' is ~a-homogeneous, d!f' ~ d::'; and dfis denumerable, then CJt is an elementary extension of an isomorph of rif. (2) If moreover;;;· is .kt0 -homogeneous, c?! ;§ :£.. and ff is denumerable, thencY and £5- are isomorphic. The construction is similar to Cantor's embedding of a linearly ordered denumerable set in the rationals. (3) Any ~·has an ~ 0 -homogeneous elementary extension of the same power. (3) was also known to Vaught and follows easily from a result by Morley (unpublished) and Svenonius (Theoria (1960)). (Received February 23, 1961.)

61T-103. P. E. Conner and E. E. Floyd: Cobordism classes of maps of odd prime period.

We consider the cobordism group fln(B(Zp)), B(Zp) a classifying space for the group Zp of integers mod p, pan odd prime. Elements of fln(B(Zp)) are interpreted as cobordism classes [T,vn] of pairs (T,Vn), where Tis a differentiable map of period p on vn, without fixed points, and yn is a

compact oriented manifold with orientation preserved by T. Now n.(B(Z )) is a right n-module, p with (T, V ~ [Wm] = [T X 1, ym X Wm]. Moreover, n.(B(Zp)) splits into the direct sum of fl (the Thorn ring) and a reduced module n.(B(ZP))' the latter all (T,VnJ with yn/T (equivalently, yn) cobording. A choice of a [T,Sn], one for each odd n, gives a generating set for the fl-module n.(B(Zp)). Each (T,Sn] is of order p a+l, where a(2p - 2) < n <(a + 1) (2p - 2). The additive structure of n.(B(Z )) is as follows. Select manifolds x 4i as in Milnor; that is, the ijc 4i] are a base for the torsion-fre: part of fl. For each 4i1, •.• ,4ik with 4ij 'I 2p - 2, and each odd n there is a cyclic subgroup of

265 ,.., r.; n· r; 4i1 4i~n a+l O*(B(Zp)) generated by 1:r,s J ~ X ••• X X -J· This subgroup is of order p , a as above. Moreover, O*(B(Zp)) splits (additively) into the direct sum of these subgroups. (Received February 23, 1961.)

61T-104. P. E. Conner and E. E. Floyd: Fixed point sets of maps of odd prime period.

Consider pairs (T, vn), where T is a differentiable map of odd prime period p on the compact oriented manifold vn. Denote by Fm the union of the m-dimensional components of the fixed point set F of T. Let q m: B m ~ F m denote the normal (n - m - 1 )-sphere bundle to F m; (T ,B m) is a fixed point free action. Now the (T,Bm) determine (vn] in 0/pO. More precisely, denote by B~ the Whitney join of B m with a trivial circle bundle on F m. Select an orthogonal fixed point free action

(T",S 1). The join ofT and T" induces a fixed point free (T',Bk). Then Lm[T',B'mJ = (T 11 ,S1J[Vn] (see the preceding abstract). Turn now to (T,Vn) having only isolated fixed points, and consider first the case in which the fixed points are all of the same type (have equivariantly equivalent neighborhoods). The number of such fixed points is then a multiple of pa+l, a(2p - 2) < n <(a+ 1)(2p - 2). Moreover,

(vnj = k[Pp_ 1(C)X ••• XPP_1(C)] mod pO. These assertions are "best possible." Turning to the case where the fixed points are isolated, but not of the same type, we furnish some evidence for the following conjecture: [vn] is cobordant, mod pO, to the disjoint union of products of manifolds of dimension:;; 2p - 2. This is true for p = 3. (Received February 23, 1961.)

61T-105. A. A. Mullin: Investigations of relationology.

(*) By a relation (on a nonempty collection C of nonempty sets Si) is meant a nonempty subset T of some cartesian product P of some nonempty subcollection D of the sets of C. By a Postian relation is meant a relation which has, together with the P of (*), a Glide! representation that is re­ cursively enumerable in the sense of Post (Bull. Amer. Math. Soc. vol. 50 (1944) pp. 284-316] and for which every Si (i ED) of (*) has a Godel representation that is recursively enumerable in the sense of Post. By a Kleenean relation is meant a relation that has, together with the P of (*) a Giidel representation which is, respectively, recursive in the sense of Kleene L'rrans. Amer. Math. Soc. vol. 53 (1943) pp. 41-73) and recursively enumerable in the sense of Post and for which every S i (i ED) of (*)has a Geidel representation that is recursively enumerable in the sense of Post. A relation is said to have a recursively solvable decision problem provided it is a Kleenean relation. A relation is Kleenean iff both it and its nonrelation with respect to the P of (*) are Postian. There exists (in a constructive sense) a dyadic relation with an unsolvable decision problem. (Received February 27, 1961.)

61 T-106. Nathaniel Coburn: Discontinuity relations for charged, compressible, relativistic, fluids (with self induction).

In this paper, we extend the theory in a previous paper (Discontinuity relations for charged, compressible, relativistic fluids, J. Math. Mech., to appear May, 1961) to the general self-inductive case. By introducing a new set of four independent vectors, it is shown through use of a decomposition

266 for skew-symmetric second order tensors due to R. H. Abraham (Discontinuities in general relativity, Ph. D. Thesis, University of Michigan, June, 1960) that the equations of Pham Mau Quan (Etude electro­ magnetique et thermodynamique d'un fluide relativistique charge, J. Rational Mech. Anal. vol. 5 (1956) pp. 473 -538) can be analyzed in a manner very similar to that of our previous paper. The key relation is the Maxwell equation involving current. Whereas, in our previous paper, it was shown that the jump of current is tangent to the discontinuity manifold; in our present paper, it is shown that the jump of current is a linear combination of a tangent vector and some of the other independent vector fields which were introduced previously. The additional details of the theory remain to be developed. (Received February 2 7, 1961.)

61T-107. P. E. Conner and E. E. Floyd: The computation of bordism groups.

We defined in Abstract 60T-23 (Notices Amer. Math. Soc. (1960) p. 1000) for each space X a group Qn(X) of cobordism classes of pairs (f, Vn), Vn a compact oriented differentiable manifold and f:Vn --:.-X. These are a special case of Atiyah's bordism groups and are in his notation MSOn(X). If 2 X is a finite complex there was a spectral sequence Epr •q with Ep,q ~ Hp (X;O q ), n q the oriented Thorn cobordism group of closed q-manifolds, and whose E 00-term is associated with a filtration of MSOn(X). For r ~ 2 and all pairs (p,q) the image of dr:Erp,q--7 Erp-r,q+r- 1 is a finite group of odd order. There is a natural homomorphism 1:': MSOn(X) - Hn(X;Z) assigning to each map of a closed oriented manifold into X the image of the orientation class under the induced homomorphism.

The spectral sequence is trivial iffT: MSOn(X)·~Hn(X;Z) is an epimorphism for all n ~ 0. If H. (X;Z) has no odd torsion and all its 2-torsion has order 2 it follows that MSOn(X) ~ 2: Hn-q(X;Oq). The unoriented bordism groups MOn(X) are similarly defined. For any finite complex, MOn(X)

~ ZHn-q(X;Nq)' and MO.(X) is a free graded right module over the unoriented Thorn cobordism ring N •• For every finite complex there is an exact Roblin sequence MSO.(X) ~MSO.(X) ~MO.(X). (Received March 1, 1961.)

61T-108. H. W. Guggenheimer: Nonexistence of a "theorema egregium" in 3-manifolds.

One might ask whether it is possible locally to characterize isometric Riemann spaces by a set of scalar functions, as given by the theorema egregium in 2 dimensions. The resulting set of Pfaffian equations is seen to be contradictory in the case of nonhomogeneous 3 spaces. (Received March 2, 1961.)

61T-109. D.P. Squier: Analytic continuation of solutions to Poisson's equation.

Let Jordan curve C be the boundary of a simply-connected domain D. Let F be an analytic

arc dividing D into D 1 and D2, C into c 1 and c2 • Let f 1, f2 be analytic functions of x, yin D1 and

D 2, resp., and in the neighborhood of F. Then if V 1, V 2 are continuous on D1 + F and 0 2 + F, resp., and satisfy \12V 1 = f 1 in D 1, V 2V2 = f 2 in D2, V1 = V2 on F, K':IV 1/on= rJV2 /dn on F, K>O; and take prescribed continuous values on C 1 and c2, resp., then V 1 and V2 are analytic in a neighborhood of F. The proof is achieved by constructing the solution when D is the unit circle and F is a diameter (which can be done for every K f. -1). (Received March 3, 1961.)

267 61T-110. S. V. Parter: An interesting example in the theory of stability of finite-difference equations.

The mixed boundary-initial value problem for a function u = u(x,t) which satisfies ut + aux

= 0 in 0 -;:: x ~ 1, 0 ""-· t :::, T with u(x,O), u(O,t) and u(l,t) prescribed in advance is over-determined. If a :>0, then u(l,t) is determined by u(x,O) and u(O,t). However the finite-difference equations of Lax and Wendroff (Comm. Pure Appl. Math. vol. 13 (1960) pp. 217-237) u(x,t + ..6.t) = (1/2)(},2 +A> • u(x - ,6.x,t) + (1 - A2 )• u(x,t) + (1/2)(>12 - h) u(x + L':!,x,t), )) = a Ax/b. t, requires this additional information. Theorem. The solutions of the stable ( lA I ~ 1) finite-difference equations converge L 2 to the solution of the partial differential equation determined by u(x,O) and u(O,t) independently of the "smooth" data prescribed on x = 1. Further examination of this problem shows that the heuristic "stability" condition described by J, Douglas, Jr., (Trans. Amer. Math. Soc, vol. 89 (1958) pp. 484-518) gives an incorrect result in this case, (Received March 3, 1961.)

61T-lll. E. Brieskorn: A theorem on the complex projective quadric.

Theorem. Let X be an n-dimensional (n;;;:; 3) compact Kahler manifold which is C 00-differen­ tiably homeomorphic to the nonsingular complex projective quadric Qn with its usual differentiable structure. In case the dimension n of X is odd, X is complex-analytically homeomorphic to Qn· Let g be the positive generator of the second cohomology group H2(X,Z) = Z. In case n is even, X is

complex-analytically homeomorphic to Qn if and only if the first Chern class c 1 of X is not equal to - ng. The proof is similar to that one given by F. Hirzebruch and K. Kodaira for a theorem of the same type on the complex projective spaces (J. Math. Pures Appl. vol, 36 (1957) pp. 201-216). (Received March 7, 1961.)

61T-112. S. P. Franklin: Some results on order-convexity.

Let (X, ~ > be a partially ordered set and o(a,b) = [ x E X Ia-;;;, x ~ b }. A subset A of X is order-convex iff for all a, b E A, cr(a,b) £;.A. The family I' of all order-convex subsets of X is a complete lattice. For A ~X, let {](A)= n{c E:: T lAC c}. Theorem I. For all At;;. X,(!, (A) = Uf o-(a,b) la,b E A}. A point p E A C X is an extreme point of A iff for all a, b E A, p E cr(a,b) implies p = a or p = b. Let (A) be the set of extreme points of A. Theorem II. If p E 8f!. (A) the following are equivalent: (1) p E ~(A), (2) B ~ C(A) and p E C

268 61T-113. J. Z. Yao: Classification of linear associative algebras of dimension two. Preliminary report.

Let A be a linear associative algebra of dimension two over a field F whose characteristic is not two. The Classification Theorem. (1) Let A be of index one. If A is commutative, then A is an algebra of binary numbers, i.e., A has a basis 1 and i such that i 2 = (3. for some fiEF. If A is noncommutative, then A has no identity, and A has a basis e and f such that either: e 2 = e, ef = f, fe = 0, f 2 = 0, or: e 2 = e, fe = f, ef = 0, f 2 = 0. (2) Let A be of index two. Then either A is a null algebra, or A has a basis e and f such that e2 = e, ef = 0, fe = 0, f2 = 0. (3) Let A be of index three. Then A has a basis e and f such that e2 = f, ef = 0, fe = 0, f2 = 0. Corollary. A linear algebra A over F is an algebra of binary numbers, if and only if: (i) A is of dimension two, (ii) A is associative and commutative, (iii) A is of index one. (Received March 7, 1961.)

61 T-114. E. 0. A. Kreyszig: An integral operator for the four dimensional Laplace equation.

Recently the author introduced an integral operator for generating solutions of the three dimensional wave equation. Clearly, this operator may also be used in connection with the Laplace equation in four variables. The present paper is concerned with another such operator which has the advantage that it may be represented by real as well as complex double integrals of a simple form. It is shown that the integrand can be specialized in such a way that the resulting solutions are closely related to Jacobian polynomials. This fact is of practical interest for computational and other pur­ poses, because it can be used for obtaining various relations between those harmonic functions from relations for Jacobian polynomials. (Received March 6, 1961.)

61T-115. E. 0. A. Kreyszig: A theorem on the coefficient problem for harmonic functions of three variables.

The harmonic functions are represented in terms of harmonic polynomials closely related ·to spherical harmonics. The theorem is concerned with conditions for the coefficients of this represen­ tation under which a given harmonic function is the sum of an entire harmonic function and a harmonic function which possesses only finitely many simple singularity curves. The location and type of these curves are obtained in terms of those coefficients. The proof uses a Bergman operator which gen­ erates harmonic functions from complex analytic functions. (Received March 6, 1961.)

61 T-116. E. 0. A. Kreyszig: Bergman operators for generating solutions of the three-dimen­ sional wave equation.

S. Bergman developed the theory of operators which transform analytic functions of two com­ plex variables into harmonic functions of three real variables. It is shown that a similar operator may be defined for generating solutions of the three dimensional wave equation. The inverse of this operator is relatively simple and can be represented as a double integral. By means of the operator one obtains a set of particular solutions such that any solution of the wave equation regular at the origin can be represented in terms of the functions of that set. These functions, which are defined

269 by contour integrals, can also be obtained by means of a generating function. The aforementioned representation can be used as a starting point for investigating the coefficient problem, in particular with respect to the location and nature of the singularities of the solutions. (Received March 6, 1961.)

61T-117. H. E. Salzer: Coefficients for stepwise integration of y(n) = f(x,y,y', ... ,y(n-1)) wi!Q central differences,

The coefficients A~m in the n-fold quadrature formulas for the stepwise integration of (1) y(n) = f(x,y,y', ..• ,y 1, as well as mid-interval tabular arguments, requires only even-order differences, on a single line, and provides great accuracy due to rapid decrease of A~m as m increases. However, the integration may be slowed down by the need to estimate and refine iteratively the later values of y,y', ... ,y(n-1) required in 32m f0 • Reference to earlier collected formulas of Legendre, Oppolzer, Thiele, Lindow, Salzer, Milne and Buckingham, reveals Thiele and Buckingham to come closest to (2), (3) containing schemes that involve just tabular arguments throughout. For n odd, they give formulas that are based upon the series in ~ 2 m for (1/JI)(ll/D)n instead of p(u/D)n as in the present arrangement. (Received March 7, 1961.)

61 T-118. M. B. Prestrud: Hierarchic algebra.

Traditionally algebraic operations connect numbers. Instead, consider that each number has an associated process, determined by a real number, the "hierarchic index," k, defined as: log(k)a =log a iterated k times. The inverse relation is log(-k)a =antilog a iterated k times. The rule of composition for numbers and their associated indices is log(k)a + log(t)b + ... = log(n)d, abbreviated as IJc,i, ... : n]. Exponentiation, ab = c, or log<2>a + log(l)b = log<2>c, becomes (?.,1:2]. Addition, multiplication, and exponentiation are thus seen as elements in a family of related operations. Furthermore, operations with integral indices are the discrete elements of a continuous range whose operations have nonintegral indices. These can be approximated graphically, or from slide-rule type nomograms. Nulls can be indicated graphically for each index, but convenient inverses exist only for certain operations. Ordinary laws of commutation, association, and distribution may be generalized for this enlarged family. The foregoing rule of composition is closely tied to the exponential function, but may be generalized by using instead, f~k)a + f1)b + ••• = f~)d, where the fi are arbitrary functions. Use of such rules of composition with appropriate tables may prove highly useful for computers. (Received March 8, 1961.)

61 T-119. R. P. Gosselin: Some integral inequalities.

The positive, measurable function !D is said to be subadditive in the interval (O,A), 0 --::A & oo, if {D(u + v) ~{D(u) + {D(v) where u, v, and u + v belong to (O,A). Let a. be any real number, let

270 p ?; 1, and let ~ be positive, measurable, and subadditive on (O,A). Then VoA~P(u)/ul+P

61T-1ZO, E. j. McShane: Families of measures and representations of algebras of operators.

Let (2 be a weakly closed commutative .-algebra of bounded operators on a Hilbert space H, and let {Be..~: cuE n} be a subset of (;3 not contained in any weakly closed proper subalgebra of ,til. Then there is a cartesian product T of intervals J0 (w E 0) and a projector-valued measure 1i on a G"-algebra fL of subsets ofT such that the integrals of bounded measurable functions are members of tiJ, each B being such an integral. But not all members of .r:iJ are necessarily thus representable. Let [x1c j E Jy be a cyclic set in H such that the xj are carried by the projectors rr(S) into mutually orthogonal subspaces H(xj). Let mj be the measure on.f[ defined by mjS = (7r'(S)xj,xj). A function f:T X J -+R is a quasi-function on T if for each countable subset Jc of J there

exists f:T ~R such that for each j in Jc• f(t,j) = f(t) except on a set of mfmeasure zero. Then H is isomorphic with the direct product Ho of the Lz(T,mj)• and in Ho the algebra (j] is represented by the multiplications by bounded quasi-functions. Likewise, CiJ coincides with the algebra of integrals of bounded quasi-functions with respect to the measure "Jr. (Received March 13, 1961.)

61T-1Zl, R. j. Libera and M.S. Robertson: Meromorphic close-to-convex functions.

Generalizing the concept of close-to-convex regular functions introduced by W. Kaplan, (Michigan Math. j. vol. 1 (1952) pp. 169-185) we say that f(z) = (1/z) + 2::fanzn, f(z) regular and f'(O);. 0 in 0 < lzl ct: 1, is close-to-convex relative to the function F(z) = (b- 1}/z + 2:_Wbnzn, regular, schlicht and star -like in 0 < lz I < 1, if Re{zf'(z)/F (zu- >- 0 for lz I ...: 1, f(z) need not be schlicht as an example shows; however f(z) is close-to-convex if, and only if, /t2Ref'1 + rei9f"(rei9)/f'(rei6)}d9 ~ ff for 0 < r < 1, a1 ...c.. 62• If f(z) is also schlicht then for n = 1,Z, •••• (nla nl + lb nl>-::;; c, Z ~· c;.:;; Z(Z)1/Z. Sharp coefficient bounds are also obtained for meromorphic functions f(z) which have real coefficients and which are convex in the direction of the imaginary axis. (Received March 15, 1961.)

61T-1ZZ. Mark Kac, P. E. Boudreau and j. S. Griffin, jr,: A discrete queueing problem,

The following problem is considered: Customers arrive at times 0, .:1t, z6t, ... ; the number of arrivals at k.6 t being a random variable Xk such that Prob. {xk = 1J = fli• fo I= 0. The random variables Xk are assumed to be independent. The customers are then processed in order of arrival by one server and the processing time T can assume the values tl.t, z.C. t, 3f:. t, ... with

probabilities p 1, Pz• p3, ... respectively. The processing times of different customers are also assumed to be independent. For simplicity it is assumed that the customers leave the server just before the end of processing time. By a suitable definition of a state the problem can be treated as a Markoff chain and one of the principal results is the following: Let p(s)(J) denote the

271 probability that at time sAt, J customers are waiting while the server is occupied. Then (O~w~l, O~z~l):Z:, 0 wsL:J~ 0 p(s)(J):J' = [(wf(z)- 2:~ 1 p~(wf(z)/)/(l- wf(z) )] •[(f(z) - (1 - wf(z))A(w))/(z - L:~ 1 P_e (wf(z))j)] where f(z) = '£..~ofJZ j and A(w)=f(9(w))/(l-wf(6(w))), 8(w) being the unique real root in 0 ::§ z ~ 1 of the equation z - Lhl pj(wf(z)),f = 0. (Received March 16, 1961.)

61T-123, G. j. Kurowski: Semi-discrete analytic functions.

Of concern are complex-valued functions of one continuous and one discrete variable which are defined on a uniformly spaced sequence of lines parallel to the real axis. Such functions whose real and imaginary parts satisfy a pair of differential-difference equations obtained from the classic Cauchy-Riemann equations on replacing the y-derivative by a difference (either symmetric or non­ symmetric) provide semi-discrete analogues for analytic functions. With path integration defined, analogues for Cauchy's integral theorem and formula are developed and the "singularity function" which corresponds to 1/z is given, The "derivative" and "indefinite integral" of a semi-discrete analytic function are also shown to be semi-discrete analytic. A method similar to analytic continu­ ation is discussed which enables suitable functions to be "continued" on the lines as semi-discrete analytic functions. The family of semi-discrete analytic functions is not closed under the usual multiplication; consequently, a modified "multiplication" operation is discussed. Appropriate analogues for the powers of z, and thus polynomials, are presented. (Received March 17, 1961.)

61 T-124. L. A. Kokoris: Flexible nilstable algebras.

A simple flexible power-associative algebra A over an algebraically closed field of character­ istic # 2,3 is said to have degree 2 if its unity element 1 = u + v where u,v are orthogonal

primitive idempotents. It is known that A has the decomposition A= A 1 + A 12 + A2 where x is in A 1 if xu+ ux = 2x, in A 12 if xu + ux = x and in A2 if xu+ ux = 0, where A 1, A2 are orthog­ onal subalgebras and A12Ai s; A 12 + A3 _i, AiAl2 f: A12 + A3 _i, i = 1,2, It is also known that A1 = uF + G1 and Az = vF + G2 where G 1 and G2 are nil subalgebras under the operation xoy = (xy + yx)/2 of the associated algebra A+. We say that A is nilstable if A 12Ai c;; A12 + G.3-i and Ai A 12 ~A 12 + G 3 _i' i = 1,2 for every idempotent u. By leaning heavily on the proof for the case where A is commutative, we prove that A is a simple Jordan algebra of degree 2. Then all flexible power-associative algebras of degree 2 over an algebraically closed field which have the property that A is the simple Jordan algebra of degree 2 are determined completely. The class of nilstable algebras is a desireable class to study since it can be proved that a simple flexible power­ associative algebra of degree 2 and characteristic zero is nilstable. (Received December 12, 1960.)

61T-l25. G. C. Rota: Multiplicative extensions of positive linear operators.

Let (S,I:., p) be a probability space, and let T be a linear operator defined on bounded real­

valued measurable functions f(s), s E S such that Tf ;;; 0 iff !:;, 0, and Tl = 1, where 1 is the function

identically equal to one. Theorem. There exist: (a) a 0'"-field· I:.' containing ~and a measure }l defined on~· and extending Jl; (b) a measurable transformation Ill: S- S, such that p~- 1 (E) E ~· forE E 1:,';

272 such that for all integers n ;:;; 0 and for all f t L00(S,!:,p) we have ornf = EVnf, where Vis defined by

Vg(s) = g(,S(s)) and E is the conditional expectation operator (projection) defined on L00 (S,'E'.p) and projecting onto L 00 (S,"~::.p). Applications are made to probability theory and ergodic theory. In particular, a proof of the ergodic theorem for general Markov processes of E. Hopf is obtained as a simple consequence of the classical ergodic theorem of G. D. Birkhoff. (Received March Z1, 1961.)

61 T-126. W. H. Fleming: On the oriented Plateau problem.

Let T be an integral current in euclidean R n which is minimal (see Federer and Fleming, Ann. of Math. vol. 72 (1960) pp. 458-5ZO) and A= spt T - spt l?T. For k = Z, n = 3 it is shown that A is a Z-manifold (not necessarily connected) which is locally a minimal surface in the classical sense. The proof uses a theorem of E. Reifenberg (Acta Math. vol. 104 (1960) pp. 1-92, Chapter 4). For 2 < k = n - 1 the partial result is obtained that A is the union of a countable family of k-manifolds and a singular set of zero Hausdorff k-measure. (Received March Zl, 1961.)

61T-127. D. R. McMillan, Jr.: Cartesian products of contractible open manifolds.

Let U be a contractible open 3-manifold ("open" means without boundary and noncompact). It is known that U may not be topologically E 3, but in several known examples U X E 1 is topologically E4 • Is this true in general? Using recent results of Stallings and Brown, one can easily show that U x E 3 is topologically E 6. Suppose now that each finite 2-polyhedron in U can be embedded in E 3

(such is the case if the Poincare Conjecture for dimension 3 is true). Theorem 1. U = T 1 + T z + ••• , where Ti is a cube with handles, TiC lnt Ti+l' and each loop inTi can be shrunk to a point in Ti+l" 1 Theorem 2. U X E is topologically E4 • Theorem 3. If u 1, U2, have the above properties, then U 1 X Uz is topologically E 6• (Received March 27, 1 961.)

61T-128. j. R. Isbell: Mazur's theorem. I.

Following P. S. Aleksandrov, Uspehi Mat. Nauk vol. 15 (1960) p. 74, a linear topological space A is said to satisfy Mazur's theorem if every sequentially continuous linear functional on A is continuous. What Mazur has done, improved by S. Mrowka (loc. cit.), determines for which completely regular spaces X the space C(X) of all continuous functions, in the topology of pointwise convergence, satisfies Mazur's theorem. Generalizing, suppose A is a linear space of functions on a set X, in the topology of pointwise convergence. Suppose for convenience that A separates points on X. Then A induces a weak uniformity on X, which determines a completion [X]. Consider the condition (*): For every point p of [X] - X, there is f in A vanishing nowhere on X but converging to zero at p. Theorem. If A is a vector lattice of functions containing the constants, then (*) is sufficient for A to satisfy Mazur's theorem; if also A is closed under uniform convergence, then(*) is necessary. (Received March Z8, 1961.)

61T-129. Eckford Cohen: Some asymptotic formulas in the theory of numbers.

Let /:!. Z(n) denote the greatest square divisor of the positive integer n, and place I (n)

= n/ 1J. Z(n). Let t denote a real number E; 0 and S an arbitrary set of integers n. In this paper

273 asymptotic estimates for the sum L rt(n) are obtained, subject to the condition that Ll (n) E s. n~x A "unitary" analogue of this problem is also considered, and corresponding asymptotic formulas are proved. These results generalize and at the same time refine estimates of Kanold (Crelle, vol. 193 (1954) pp. 250-252) in the case of the first problem, a.nd Renyi (Acad. Serbe, vol. 8 (1955) pp. 157-162) in the case of the second. The method of the paper is elementary, and is entirely different from the methods used by Kanold and Renyi. (Received March 29, 1961.)

61T-l30. Ali Kryala: A new derivation of relativistic dynamics.

Defining the rest-mass density m 0 to be the integrating factor for the Pfaffian form dP = fic2dt - ftv· dR (withfl(l - (v jc2) 1/2 = 1) derived from ds2.ft= dt(dP) one obtains the related

Pfaffian form m 0dP = Edt - p• dR where E = mc2, p = mv and m =ftm 0 • The condition for unique integrability of modP regardless of path between any two fixed points in space-time yields the differ­ ential equations for the integrating factor in the form 6tp + VE = 0 = V Xp which then constitute the equations of dynamics in the absence of electromagnetic fields and may be combined in the form

(Dp/Dt) ,.8= - VE0. The analogous argument for a general metric tensor proceeds from the observa­ tion that ds2 = gba dxadxb may be rewritten as ds = gbavadxb with va defined by vads = dxa. Then the condition for unique integrability of ds is V[c (gb]ava) = 0. If this cannot be satisfied due to other conditions imposed on gba then an integrating factor M may be introduced subject to the condition V [c(Mgb]ava) = O. In this case the metric has not been chosen a priori as in the special relativistic case. (Received March 31, 1961.)

61 T-131. M. A. Al-Bassam: Holmgren Riesz (H-R) transform equations of Riemannian type.

By applying some of the properties, studied by the author (Dissertation, Texas Univ., 1951), of . xa n l'x a+n-1 the H-R Transform defmed by Ia f = Dx/r(a t n) Ja (x- t) f(t)dt, Rat n > 0 (n = 0,1,2, .•• ), f E en on [a,b], it is shown that if ai• ai are numbers, R(n - w) > 0, z(x) € c 2 on [a,b] and E: -wrr3 . x -1 3 1-a x w-1 I: 1 1 1 (x - ai)al Ia IT 1 (x - ai) i Ia z = 0, then (1) E is the Riemann differential equation in the reduced form with singularities at ai if and only if K: w - L~= 1ai t 1 = 0, a condition which is satisfied by F: w = z a, a1 = a' + .J3 + r. a2 = a +.If' + r. a3 = a + _,8 + ;Y1 , where La + a' = 1 and {a], { a1 , are the indices of Riemann P -function; otherwise E is a differential-integral equation of Riemann-Volterra type. (2) If z = TI(. ) (x - a.) -au, and K and F are satisfied then E becomes l,a. 1 Riemann-Papperitz equation in u(x). (3) E is reduced to the Gauss's equation when for fixed i

(say i = 2) a 2 ---? oo. By the operational properties of the transform, twenty-four solutions (similar to Kummer's) forE have been obtained in each of the two cases: w = t8J• a 1 = :V- f!'r• a 3 = f~J - Y + 1, a 1 = 1, a 2 ~ oo, a 3 = O. (4) Mambriani's form of the Gauss's equation (Boll. Un. Mat. Ital. vol. 3 (1940)) is not of generalized order as it was indicated, but it is a particular case of E, equivalent to the case mentioned in (3), (Received April 3, 1961.)

274 61T-132.. Oved Shisha: On approximation by analytic functions whose Taylor coefficients lie in a sector.

Let S( 31) be a set, all of whose points (except 1) belong to the unit disk lz I< 1, and let f(z) be a (finite) complex function with domain S. Let D. be a closed angular region whose vertex is the origin and whose angular measure pf satisfies 0...:. ¢ <:: T1'. Let Pk(z) =L..~ 0a~k)zn(k = 1,2., •.• ) be a sequence of analytic functions, such that limk--+ooPk(z) = f(z) for every z E S, and such that a~k)E ~ (k = 1,2., ••• ; n = 0,1, ••• ). Then f(z) is a restriction of a function which is regular throughout lz I< 1. (Received April 3, 1961.)

61T-l33. G. C. Rota: On the eigenvalues of modulus one of order-preserving linear operators.

The following conjecture (see e.g. Karlin, J. Math. Mech. val. 8 (1959) pp. 907-938) is proved:

Theorem. LetT be an everywhere defined linear operator in L 00 (S,~p). p(S) = 1, such that Tf iS 0 iff 5; 0, f E L 00 and Tl = 1, where 1 is the function identically equal to one, andj'ITfl \/If I for f E L 00• Let Tf = ..\f, where I)) I= 1, and decompose f in the form f = lflg, where If I, g E L 00 lgl = 1. Then f = lgn is an eigenfunction of the operator T, belonging to the eigenvalue )) n •• The proof is n If based upon the representation Tn = Eyll (n = 0,1,2., ••• ) (See the author's Abstract 61T-12.5, Notices A mer. Math. Soc., this issue) where V is a measurable transformation (V(fg) = VfVg) in an extended probability space of which (S,l:.p) is a subspace, and E is the conditional expectation of the conditional expectation of the extended space onto (S,'E,p). (Received April 4, 1961.)

61T-134. c. C. Chang and H. j. Keisler: Pairs of cardinals for models of a given theory.

Let I denote the power of I, a,f3,r, ••• denote infinite cardinals, and letS= { s ~ IIO"" s < .tt0J. Let r be a set of sentences of a first order predicate logic with identity, 8(v cY be a formula with only Vo free, 8(A) be the set of all elements of A which satisfy 8, and X= i (a.j3) I r has a model A of power a and 6(A) =ji]. Known results: (a) T & 7 and {a.f3)E X imply ('r, ?') E X; (b) T :2$ 'Y and (a,f3"> EX and j3 & ?"'1! a imply (7'.(1) EX (Lowenheim-Skolem-Tarski); (c) f :2$ ~O and (a,f3) EX and f3 EX implies (ar•l') EX. Theorem 4. ( a.f-1} E X implies (L:o"":YaS, La.::rP'8)E X. Several special conclusions can be drawn. For example, assuming the Gen. Cont. Hyp., we have (1) { ~2.' .11\0 ) E X implies ( ~2.' .t\) E X; (2.) ( .lt; w' ~ 1) EX implies (~+I' ~ 1 ) EX; (3) (.lticu+l' ~ 1) EX implies ( jl;,cu+l'.lli"') EX; (4) f (Jt..n+l• Jl; 1) In < cJ} .!: X implies (.1-1\((J+l'JI; 1) Ex. In the negative direction, we have Theorem 5. Let a< aJio, f3 f. 'Y, and f3 be nonmeasurable. Then there exist f, 8 such that: (i) ( a,jB) EX and (a,'Y) rj:x, (ii) (f3.a) EX and(?',a}¢ X. (Received April 6, 1961.)

61 T-135. R. I. Sandler: Autotopism groups of some finite algebras. Preliminary report.

Let K = GF(q II), where q = pr, S: x~xq for x inK. Define an algebra, J:J8, of dimension n over K with basis 1, ;.,, ;.,z, ... ,An-l where multiplication is defined by (a) xo))i = Ai(xfh for x inK,

275 i < n; (b) (Aia1) "(Aja2) = ))i+j(a1Sj)a2 for a 1, a 2 inK; (c) .-li "~j = Ai+j if i + j < n; (d) Ai ",.s..i = ,.s..i+i-ns if i + j ~ n. ?"(Q,P,U) is an autotopism of~ if Q, P, U arenonsingular linear transforma­ tions of ./)8 such that (xP) 0 (yQ) = (x" y)U for all x,y ino8&. Determination of the autotopism group is equivalent to finding the collineation group of the projective plane coordinatized by .ll8 if o8& is a division algebra. Theorem 1. If n = m, .88 is a division algebra if 8 satisfies no polynomial of degree n - 1 over GF (q). Generators for the autotopism groups of these division algebras can be determined; there are six when n = 2, and five when n > 2. Their orders depend on 8, but are easily computable. The groups are all solvable. If m > n, generators for the autotopism groups can still be explicitly determined, although no such algebras are known to be division algebras. (Received April 7, 1961.)

61T-136. R. P. Gilbert: Singularities of three-dimensional harmonic functions. III

Let hn,m(X) be the homogeneous, harmonic polynomials of degree n defined by the relation tn =[- (x1 - ix2) (!:"/2) + x 3 + (x 1 + i:xz)( l/2!::)}n = z::::-=-nhn,m(X)~-m. and let the analytic function of two complex variables f(t,~). and the three-dimensional harmonic function H(X) be defined by the expan­

sions f(t,~) = ~O Liif=-n anmtn!;"m, It I< R, 1 - e < 1~1 < 1 + e, H(X) = L:oL==-nanmhnm(X), IIX II < R, which converge in the indicated regions. Furthermore, let the singularities of f(t,_r) be given in the form t = "lJf(~). where "(jf(~) is analytic in~. Then H(X) is singular at X, providing X does

not lie on the x3 -axis, if and only if X E EfS(X;~) := t - "\lf(t) = 0} n E {

61T-137. R. P. Gilbert: A Schwarz Lemma for axially symmetric potentials.

Axially symmetric potentials (ASP) may be obtained by a modified form of the Whittaker­ Bergman operator (R. Gilbert, Arch. Rat, Mech. Anal. vol. 6 (1960) pp. 171-176); in this case, how­

ever, the operator A3(f,cl':p0) maps functions of one complex variable onto the axially symmetric prove the Theorem: Let V(r,9) = '"" 00 a rnP (cos 9) potentials. By considering this operator we may - .L...n= 0 n n

be an ASP, whose coefficients [an1 (f satisfy the conditions an ~ 0, and lim supn-+ 00 !an ll/n = d < 1. Furthermore, let V(0,9) = 0, and V(r,9) ;;;; 1 for all r < 1, then V(r,9);;; r for r < 1. It may be also shown that under these conditions a Hadamard three-sphere theorem holds. Similar results may be obtained for harmonic functions in three-variables; in this case we use the Whittaker-Bergman

operator, H(X) = B 3(f, £.x0), (which maps functions of two complex variables onto harmonic functions), the Schwarz lemma for functions of several complex variables, and the Hadamard three sphere theorem. (Received April 10, 1 961.)

61T-138. WITHDRAWN.

61T-139. R. j. Parikh: Remarks on a paper of G. Kreisel.

In the following,definitions and notations from a paper by G. Kreisel (Nonunigueness results for

276 transfinite progressions, OOR technical report No.4, Feb. 1961, Stanford) are used. Consider the following conditions on 0*. (a) There is a notation w for c..Jin 0* such that the following holds. Given a primitive recursive (p.r.) predicate P(x) let e be the number of the p.r. function f such that f(O) = 1, f(n t 1} = f(n) t 0 w if P(n) and f(n t 1} = f(n} t 0 2 otherwise. Then c = 3.5e E. 0*. (b) Let Q(y,x) be a p,r. predicate and let P i(x) = Q(i,x). Let ci be associated with P 1(x) as in (a). Let e be the number of p.r. function g such that g(O) = 1, g(n + 1) = g(n} t 0 en. Then 3,5e E 0*, The following results are proved. (i) Let 0* satisfy (a) and be closed under t 0 and 0• Let Py be progressive, recursive and a.-unique for every a.< 2. Then there exists hE' 0*, lhl = c:v2 +;a, such that Ah ~Ad. Hence Ay cannot be «..Z-unique. (ii) Let 0* be closed under < 0 and satisfy (b) and let Cy be progressive and recursive. Then Cy cannot be (J + I)-unique. (Received April 17, 1961.)

61T-140, WITHDRAWN.

61T-141. WITHDRAWN

61T-142, H. S. Collins: The kernel of a semigroup of measures.

LetS be the semigroup of measures on a compact semigroup S, with K its kernel, and K be the kernel of S. Theorem 1. These are equivalent: (1} )l E K, (2} ;us Jl = {JIJ, (3} p 2 = p and carrier }l is a union of maximal groups of the form eSe, with e 2 = e E. K, (4) ;u 2 = p and H X H = H, all x E S, (5) p 2 =}land carrier pis a band of maximal groups of the form eSe, with e 2 = e E. K. Corollary 1. K is the carrier of K. Corollary 2. These are equivalent: (1} K is convex, (2) K is either a minimal left or right ideal, (3) there is either a right or left invariant mean on S, (4) K consists of left or right zeros, (5} K is either a minimal left or right ideal. Corollary 3. Sis simple iff xy = x all x,y E S 2!. xy = y all x,y = S. Corollary 4, K contains every idempotent of S iff e 2 = e E S implies eSe = { e 1. The final result makes use of (1) implies (3) of Theorem 1 to derive a new proof (but similar to Wendel's) of the existence of Haar measure on a compact group. (Received April17, 1961.)

61 T-143, Samir Khabbaz: A characterization of fields whose multiplicative group is torsion.

Theorem. The following statements concerning a ring A are equivalent: (I) Every nonzero subring of A is simple. (II) Every nonzero subring of A is primitive. (III) Every nonzero subring of A is a field, (IV) A is a field whose multiplicative group is torsion. Corollary: Every uncountable ring contains a nonzero commutative at most countable subring which is not simple and therefore not primitive. The proofs are easily carried out by using Herstein's commutativity theorem plus some standard theorems on rings. Once it is shown that I implies that A is a field, the rest is immediate. (Received April 11, 1961.}

61 T-144. Eckford Cohen: On the distribution of certain sequence integers.

Let the integer n > 0 have distinct prime factors p l' .... pr, and place n = p ~ 1 ... p: r. For positive integers a, b, let Sa,b denote the set of integers n whose exponents ek (k = 1, ... , r) occur

277 among the numbers of the progressions: ai +(a+ b)t, bj + (a + b)t, t ;:; 0, 0 ~ i ;;:; b - 1, 1 ~ j & a, Further, let sa,b(x) denote the number of n ~ x contained in sa,b' It is proved in this note that, if 1/a 1/b 1/(a+b) b ::>a ::> 1, (a,b) = 1, then (*) Sa,b(x) = ax +fix + O(x ), where a and f3 are nonzero con- stants depending upon a and b. The proof is elementary. As a simple consequence of (*), a similar result for a related sequence sa,b * is proved. In case a= 2, b = 2k + 1, where k is a positive integer, (*) reduces to the analogous special case of the main result proved in an earlier paper (Abstract 564-249, Notices Amer. Math. Soc. vol. 7 (1960) p. 64). Misprint: In Abstract 573-2 (Notices Amer. Math. Soc. vol. 7 (1960) p. 721), replace O(x1/ 2log x) by O(x1/ 2lolx). (Received April 17, 1961.)

61 T-145. W. D. L. Appling: Concerning path length. Preliminary report.

If each of m and s is a real valued function on the number interval [a,b) with m nondecreas­ ing, then m(q)- m(p) ;:;;;; js(q) - s(p) I for each subinterval jp,q) of [a,b] is necessary and sufficient for the existence of a real valued function g on j!l.,b] such that if a< x ~ b, then m(x) - m(a) = l.u.b. of all 2 2 1 2 sums L: 0 {(.6g) + (L\s) } / for all subdivisions D of [a,x]. The above inequality implies that the function f on [a,b] such that f(a) = 0 and, if a < x ~ b, then f(x) = g.l.b. of all sums Lof< ,1m)2 - 2 (l;.s) } 1/ 2 for all subdivisions D of [a,xJ, has the required property with respect to m and s. This situation is analogous to that encountered with respect to Hellinger integrals. (Received April 17, 1961.)

61T-146. D. R. McMillan, Jr.: Uncountably many divisors of E 4 •

There exists an uncountable collection [W aJ (in fact, with cardinality of the continuum) of spaces with the properties: (1) each is a contractible open subset of the 3 -sphere; (2) each is the union of a properly ascending sequence of unknotted solid tori; and (3) no 2 Wa's are homeomorphic. According to a previous abstract of the author (Cartesian products of contractible open manifolds) in the Notices, each Wa X E 1 is topologically E 4• Hence, there are uncountably many different ways to express E 4 as the product of a 3-manifold and a line. There are contractible open subsets of the 3-sphere which do not enjoy property (2) above. (Received April 17, 1961.)

61 T-147. William Huebsch and Marston Morse: A characterization of an analytic n-ball.

Let A be an open subset of a euclidean n-space E. LetS be an (n - I)-sphere in E. The set 1 A will be said to be interiorly of type C if there exists a sequence M 1' M2, ... of disjoint C 1-diffeo­ morphs of S in E whose open interiors form an increasing sequence of subsets of E whose union is A. The image in E of an open euclidean n-ball under a real analytic diffeomorphism will be called an analytic n-ball. Theorem. A necessary and sufficient condition that A be an analytic n-ball is that A be interiorly of type C 1. As a consequence the interior of a C 1-diffeomorph of S in E is always an analytic n-ball. (Received April 19, 1961.)

278 61T-148, William Huebsch and Marston Morse: A Schoenflies extension of a real analytic diffeomorphism of S into E.

The notation of the preceding abstract is used, Let JS be the closed n-ball in E bounded by S,

Theor~. Let z be an arbitrary point of S. A real analytic diffeomorphism f of S into E admits a homeomorphic extension F defined over a set Z U z where Z is some open neighborhood of JS - z, and F IZ is a real analytic diffeomorphism, This extension F of f defines an analytic diffeomorphism of its domain of definition with z deleted, and a homeomorphism with z included, F has no singularity on the interior of S, or on S, except at most at z. (Received April 19, 1961.)

61T-149, D. A. Storvick: Quasi-conformal functions tending to conformality at the boundary.

It is proved that if w = f(z) is a bounded K-quasi-conformal function in Im(z) ,. 0 and if Q ~(z)] .is its dilatation quotient and if there exists an essentially bounded measurable function )((x) such that Q ~(x + iy)] - 1 !!!- ~(x)y for all z = x + iy then the limit, limy-+Of(x + iy) = f*(x), exists for all x except possibly for a set of linear measure zero·, The proof of this generalization of Fatou's theorem depends on another result which is established for K-quasi-conformal mapping with Q ~(z)] - 1 ;; ..:((x)y. If w = f(z) is a K-quasi-conformal mapping of Im(z) > 0 onto Im(w) :> 0 such that Q~(x + iy)] - 1 ;;; .f{x)y where :((x) is an essentially bounded measurable function then the limit of

~(x + h) - f(x)]/h as h -+ 0, Im(h) ;;: 0, exists for each x and is different from 0 and oo. These results are related to the work of Lehto: On the differentiability of quasiconformal mappings with prescribed complex dilatation. Ann, Acad, Sci, Fenn. Z75, (1960), (Received April 19, 1961,)

61T-150, Irwin Fischer: On the specialization of birationally equivalent curves.

H. Hironaka has proved that if an absolutely irreducible curve of genus g is specialized over a field to a reducible curve r without multiple components then, if gi is the genus of the ith com­ ponent of 'i."', we have g 1 + gz + ... + gh ~g. We apply this result to prove the following theorem. Let r 1' r z be a pair of absolutely irreducible curves which are birationally equivalent and of genus g. Specialize the pair to I\. f z. Then if I\. f z are both absolutely irreducible and of genus g they are birationally equivalent, Chow and Lang have proved this result if f 1 and r z are nonsingular, For the proof, we observe that the graph G of the birational transformation is a curve of genus g. Extend the specialization to G-a. If G is irreducible, we are done. Otherwise, G splits into Z curves of genus g. This is impossible for g :> 0 by Hironaka' s inequality. If G > 1 we can deduce

our theorem from the weaker inequality g 1 + gz + ... + gh- h + 1 :iii g, for which we can give a simpler proof. (Received April 19, 1961.)

61T-151. j. D. Halpern: The independence of the axiom of choice from the Boolean prime ideal theorem. Preliminary report,

The axiom of choice is used to prove the following two lemmas, Lemma l, Let R be the set of rational numbers. Denote by cpT the group of order preserving permutations of R which leave

T k R fixed, LetS £; R be finite and let f be an homomorphism of '~'s onto a group of automorphisms of a Boolean algebra B. If for every x E B there is a finite set Ph R such that f(l!l)(x) = x for every

279 rJ E '~'p then there is a prime ideal I in B such that I is closed under s and x E I). The lemma is proved by showing that every proper nonprime ideal closed under s• and then using Zorn's lemma. The notation in the following is that of Mostowski (Fund. Math. vol. 32 (1939) pp. 201-252). Lemma 2. If B E: ?tlis a Boolean algebra then B has a prime ideal I Em+ such that A(I) = A(B). (Follows from Lemma 1 by takingS = A(B)~ Theorem. If Mostowski's system G of set theory is consistent then the axiom of choice is not deducible in 6 from the statement, "There exists a function ':)! such that for every-Boolean algebra B, f(B) is a prime ideal in B." (Received Apri121, 1961.)

61T-152. H. E. Salzer: Note on osculatory rational interpolation.

In n-point osculatory interpolation of order ri - 1 at points xi, i = 1,2, ••• ,n, by a rational expression N(x)/D(x), where N(x) and D(x) are polynomials L:ajxj and Eblj• we use the lemma that the system (1) N(xi)/D(xi) (m) = f(m)(xi), m = 0,1, ••• ,ri- 1 is equivalent to (2) Jm) (xi) = f(xi)D(xi) (m), m == O,l, •• .,ri - 1, D(xi) -1 0. This equivalence does not require N(x) or D(x) to be a polynomial or even a linear combination of given functions. The lemma implies that (1), super­ ficially nonlinear in aj and bj, being the same as (2), is actually linear. For then-point interpolation problem, the linear system, of order 2:f= 1ri' which might be large, is replaceable by separate linear systems of orders ri (or even ri + ri+1 + ••• + ri+j when conveniently small) by applying the lemma to the continued fraction (3) N(x)/D(x) = a 1,o + (x- x 1)/(al,1 + (x- x1)/(a1,2 + .•• + (x- x1)/(a1,r1-1 + (x- x1)/(a2,0 + (x- x2)/(a2,1 + ••• + (x- x2)/(a2,r2-1 + (x- x2)/(a3,0 + ••• + (x- xn-1)/(an,O + (x - xn)/(an 1 + ••• + (x - xn)/an r ))) ••• ))). In (3), which has the property proven in two ways) that ' ' n-1 the determination of ai,m is independent of all a's that follow, we find ai,m stepwise, but several at a time (instead of singly which is more tedious), retrieving them readily from the solutions of those lower-order linear systems. (Received April 12, 1961.)

ERRATA, Volume 7

Donald Monk: Representation theorems for cylindric algebras. Page 1000, Abstract 60T-22. Line 7. Replace "x d = 0" by "x d # 0",

Line 14. Replace "CA " by "CA 11

ERRATA, Volume 8

j. L. Brenner: Roots and vectors of Mahler matrices. Page 166, Abstract 580-1. Line 5. For "elementary divisions" read "elementary divisors". Line 6. For "scalar matrices" read "square matrices".

280 Creativity can assume infinite form and direction. The ideal formula for creativity has yet to be determined. However, at Amherst Laboratories, progressive leadership in advanced electronic R & D can be attributed to the dynamic stimulus of the Fifth Freedom ... Freedom of the mind, an essential ingredient in keeping the world free. To some, it is "freedom of thought".• :"freedom of investigation" ... whatever the tide ... it is successful at Amherst Laboratories. Results are evidenced in the conclusion of numerous ·projects and the ever increasing backlog of prime assignments in advanced Ground, Air and Space Communications. PROFESSIONAL STAFF AND MANAGEMENT OPPORTUNITffiS are unlimited for Physicists, Mathematicians and Electronic Engineers with advanced degrees and creative desire. All qualified applicants will receive consideration for employment witholft regard to race, creed, color or national origin. You are invited to direct inquiries in confidence to Mr. H. L. Ackerman, Professional Employment, AMHERST LABORATORIES • 1186 WEHRLE DRIVE • WILLIAMSVILLE, NEW YORK

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Cumberland, Maryland PSAM Vol. XII PROCEEDINGS OF THE SYMPOSIUM ON THE STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL OPENINGS FOR ASPECTS MATHEMATICIANS The twenty articles in this book are texts of addresses which were delivered at the PHILCO CORPORATION symposium held in April, 1960. The authors contributing papers to this CALIFORNIA book are: W. V. Quine; Noam Chomsky; Hil­ ary Putnam; H. Hiz; Nelson Goodman; Has­ Openings exist for mathema­ kell 8. Curry; Yuen Ren Chao; Murray Eden; Morris Halle; Robert Abernathy; Hans G. ticians with PhD. who are inter· Herzberger; Anthony G. Oettinger; Victor H. ested in doing mathematical Yngve; Gordon E. Peterson and Frank Harary; research with emphasis in the Joachim Lambek; H. A. Gleason, Jr.; Benoit field of analysis. A Philco Tran­ Mandelbrot; Charles F. Hockett; Rulon Wells; sac S-2000 Is available as a Roman Jakobson. research tool. For further in· 285 pp. approx. $6.60 approx. formation, write to Dr. E. David 25% discount to members Callender, Phllco Corporation, Western Development Labora­ AMERICAN MATHEMATICAL SOCIETY tories, 3875 Fabian Way, Palo 190 Hope Street, Providence 6, Rhode Island Alto, California.

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The total number of pages of the Russian journal to he translated in 1961 will he about 1600. All branches of Pure Mathematics are covered in the DOKLADY in short articles which provide a comprehensive, up-to-date report of what is going on in Soviet mathematics.

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