AMERICAN MATHEMATICAL SOCIETY

Nottces

Edited by GORDON L. WALKER

Contents

MEETINGS Calendar of Meetings • • • . • • • • • • • • . . • • • • • . • • • • • • 4 Program of the February Meeting in Tucson • • • . • . • . • . 5 Abstracts for the Meeting, pp. 80-84 Program of the February Meeting in ••••••••• 10 Abstracts for the Meeting, pp. 85-94 PRELIMINARY ANNOUNCEMENTS OF MEETINGS ••••••.••• 15 ACTIVITIES OF OTHER ASSOCIATIONS ••••••••••••••••• 19 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••••••• 27 FOREIGN SCIENCE INFORMATION •••••••••.•••••••••• 38 PERSONAL ITEMS •••••••.•••••••••.••••••••••••• 47 NEW PUBLICATIONS ••••••••••••••••••••••••••••. 50 CATALOGUE OF LECTURE NOTES - Supplement No.1 . • • • • • 57 ABSTRACTS OF CONTRIBUTED PAPERS • • • • • • . • • • • • • • • 59 RESERVATION FORM . • . • • • • • • • • • • . • • • • • . • . • • • • • • 99

3 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 meeting dates which fall rather far in the future are subject tr change. This is particularly true of the meetings to which no numbers have yet been assigned,

Meet­ Deadline ing Date Place for No, Abstracts*

567 April 14-16, 1960 New York, New York Mar. 1 568 April 22-23, 1960 Chicago, Illinois Mar, 1 569 April 22-23, 1960 Berkeley, California Mar, 570 June 18, 1960 Missoula, Montana May 5 571 August 29-September 3, 1960 East Lansing, Michigan July 15 (65th Summer Meeting) 572 October 22, 1960 Worcester, Sept. 8 January 24-27, 1961 Washington, D. C. (67th Annual Meeting) August, 1961 Stillwater, Oklahoma (66th Summer Meeting) January, 1962 Kansas City, Missouri (68th Annual Meeting) August, 1962 Vancouver, British Columbia (67th Summer Meeting) August, 1963 Boulder, Colorado (68th Summer Meeting)

*The abstracts of papers to be presented at the meetings must be received i. the Headquarters Offices of the Society in Providence, R.I., on or before these deadlines, The deadlines also apply to news items. ------~

The NOTICES of the America~;~ Mathematical Society is published by the Society seven times a year, in February, April, June, August, October, Novem­ ber, and December, Price per annual volume is $7.00, Price per copy, $2.00, Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (none available before 1958), and inquiries should be addressed to the American Mathematical Soci~ty, Ann Arbor, Michigan, or 190 Hope Street, Providence 6, Rhode Island. Second-class postage paid at Ann Arbor, Michigan. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August 4,1947 (Sec, 3421, P. L. and R.). Accepted for mailing at the special rate of postage provided for in section 34-40, paragraph (d),

Copyright, @,American Mathematical Society, 1960 Printed in the of America

4 FIVE HUNDRED SIXTY-FIFTH MEETING

University of Arizona Tucson, Arizona February 18-20, 1960

PROGRAM

The five hundred sixty-fifth meeting of the American Mathe­ matical Society will be held on Thursday, Friday and Saturday, February 18-20, 1960 at the University of Arizona, Tucson, Arizona. This extraordinary meeting of the ar Western Section of the Society as planned as a part of the seventy-fifth anniversary celebration of the University of Arizona. By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, and with the financial support of the National Science Foundation, a Symposium on Will be held on Thursday and Friday in Room 222, Liberal Arts Build­ ing. By invitation of the same Committee, there will be an address at 9:30A.M. on Saturday by Professor T. S. Motzkin of the University of California, Los Angeles on "Convex sets in analysis" in Room 222, Liberal Arts Building. Sessions for contributed papers will be held on Saturday at 11:00 A.M. in Rooms 114, 206 and 222, Liberal Arts Building. Ab­ stracts of the papers to be presented at these sessions appear on pages 80 - 84 of these NOTICES. There are cross references to the abstracts in the program. For example, the title to paper (1) in the program is followed by (565-3) indicating that the abstract can be around under the designation 565-3 among the published abstracts. The registration desk will be located in the Student Union Build­ ing. It will be open from Wednesday evening until Saturday morning. Information concerning late papers can be obtained at the registration desk. Luncheon will be served on Thursday and Friday in the Junior Ball Room of the Student Union Building. Arrangements for lunch on Saturday will be announced later. An extensive program of entertainment is planned for mem­ bers and their families attending this meeting. The events include a odeo Parade at 10:00 A.M. on Thursday, Tea at 5:00P.M. Thursday the Kress Gallaries on the campus, a tour of the State Museum on afternoon, and a banquet with entertainment on Friday evening. The other activities, tours and a fashion show will be available s sufficient interest.

5 Tucson is served by American, Continental, Frontier and Trans-World Airlines, by Southern Pacific Railroad, and by Grey­ hound Bus Lines. The principal highway through Tucson is U.s. 80-89. Special parking permits entitling the bearer to reserved parking will be available at the registration desk for fifty cents for the three days. There is usually adequate parking off campus, how­ ever.

PROGRAM OF THE SYMPOSIUM ON DIFFERENTIAL GEOMETRY The Program Committee for the Symposium consisted of Professor C. B. Allendoerfer, chairman, Professors Herbert Buse­ mann, Hans Samelson and D. C. Spencer. They will also serve as chairmen of the four sessions. All sessions of the Symposium will meet in Room 222, Liberal Arts Building. THURSDAY, 9:00A.M. 9:00 - 10:00 A report on the unitary group Professor Raoul Bott, Harvard University 10:15- 10:45 On compact, Riemannian manifolds with constant, negative curvature Professor Eugenio Calabi, University of Minnesota 10:45- 11:15 Elementary remarks on surfaces with curvature of fixed sign Professor Louis Nirenberg, New York University 11:15 - 11:45 Spaces of non-positive curvature Professor Shoshichi Kobayashi, University of British Columbia THURSDAY, 2:00P.M. 2:00 - 3:00 Some remarks on homological analysis and structures Professor D. C. Spencer, 3:15 - 3:45 Almost product structures Professor A. G. Walker, University of Liverpool and 3:45- 4:15 Differential-geometric methods in the study of families complex structures Professor Albert Nijenhuis, University of V ton

6 4:15- 4:45 Complex homogeneous contact manifolds Professor W. M. Boothby, Washington University

FRIDAY, 9:00A.M. 9:00 - 10:00 Vector bundles and homogeneous spaces Professor F. E. P. Hirzebruch, University of Bonn and Institute for Advanced Study 10:15 - 10:45 Algebraic homogeneous spaces and the cohomology of nil­ potent Lie algebras Professor Bertram Kostant, University of Califor­ nia, Berkeley 10:45- 11:15 Locally affine spaces and uniform subgroups of Lie groups Professor Louis Auslander, Indiana University 11:15 - 11:45 The cohomology of Lie rings and the lifting of group actions to fibre bundles Dr. R. S. P ala is, Institute for Advanced Study FRIDAY, 2:00P.M. 2:00 - 3:00 A procedure of killing homotopy groups of differentiable mani­ folds Professor John Milnor, Princeton University and University of California, Berkeley 3:15 - 3:45 The Alexander-Pontrjagin duality in function space Professor James Eells, Jr., Columbia University 3:45 - 4:15 Total curvature of immersed manifolds -- some problems Dr. R. K. Lashof, 4:15 - 4:45 On immersion of manifolds Professor Hans Samelson, University of Michigan

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

SATURDAY, 9:30A.M. Invited Address, Room ZZZ, Liberal Arts Building Convex sets in analysis Professor T. S. Motzkin, University of California, Los Angeles

SATURDAY, 11:00 A.M. Session on Algebra, Room 114, Liberal Arts Building 11:00 - 11:10 (1) Commutators of unitary matrices which commute with one factor Dr. Olga Taus sky, California Institute of Technology (565-3) 11:15 - 11:Z5 (Z) A theorem on independence of two-element primal alge­ bras Dr. E. S. O'Keefe, Boeing Airplane Company, Seattle, Washington (565-7) 11:30 - 11:40 (3) Reduction mod-p of divisor classes and linear series. Preliminary report Professor E. D. Nering, University of Arizona (565-8)

Session on Geometry and , Room, Z06, Liberal Arts Building 11:00 - 11:10 (4) On axioms of the n dimensional projective space Professor A. Zirakzadeh, University of Colorado (565-6) (Introduced by Professor R. W. McKelvey) 11:15 - ll:Z5 (5) Cyclic order of strictly monotone arcs Professors. B. jackson, University of Maryland and University of Washington (565-5) 11:30 - 11:40 (6) Isotopy classes in En and sn Dr. j. M. Kister, University of Michigan (565-9)

8 Session on Statistics and Applied , Room 222, Liberal Arts Building 11:00 - 11:10 (7) A lower bound for the distribution of the statistic Dri: Professor J. B. Butler, Jr., University of Arizona, and Mr. R. C. McCarty, Boeing Airplane Company, Seattle, Washington (565-2) 11:15 - 11:25 (8) Asymptotic& I. Laplace's method for multiple integrals Professor W. B. Fulks and Mr. J. 0. Sather, University of Minnesota (565-4) 11:30 - 11:40 (9) of the expansion process within a turbine Professor M.A. Dengler, Arizona State University (565-1)

SUPPLEMENTARY PROGRAM (To be presented by title)

(10) Concerning the bounded linear transformations in a cer­ tain function . Preliminary report Dr. W. D. L. Appling, USATRECOM, Fort Eustis, Virginia (11) Chain sequences and orthogonal polynomials Professor T. S. Chihara, Seattle University (12) A universal operator in separable Hilbert space Professor Gian-Carlo Rota, Massachusetts Institute of Technology (565 -10) (13) Topological and measure theoretic properties of analytic sets Professor Maurice Sion, University of California, Berkeley (14) Commutators of matrices. III Mr. R. C. Thompson, California Institute of Technology R. S. Pierce Associate Secretary Seattle, Washington January 6, 1960

9 FIVE HUNDRED SIXTY-SIXTH MEETING

New York University Washington Square Campus New York, New York February Z7, 1 960

PROGRAM

The five hundred sixty-sixth meeting of the American Mathe­ matical Society will take place on Saturday, February Z7, 1960 at the Washington Square Campus of New York University in . All sessions will be in the Main Building. By invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings, Professor Serge Lang of Columbia Uni­ versity will deliver an hour address at Z:OO P.M. in Room 703. The title of his address is "Some conjectures in diophantine equations." There will be sessions for contributed papers at 10:00 A.M. and 3:15P.M. Late papers will be programmed if possible, either by interpolation in the established program or in a special session. Abstracts of most of the papers to be presented in person appear in this issue of these NOTICES. Thus paper number (1) is followed by the number 566-9 which indicates that its abstract can be found under that number in the list of abstracts which begins on page 85 • Ab­ stracts of the remaining papers on this program can be expected in the next issue of these notices. The registration desk will be near Room 703 and will be open from 9:00A.M. till 3:30P.M. Room 707 will be available as a con­ versation room. The Main Building of New York University is at the Northeast corner of Washington Square. Washington Square may be reached by public transportation as follows: Lexington Avenue (Interborough) Subway (IR T) -- Local to Astor Place Station. Walk west on Astor Place to Broadway, then south on Broadway to Waverly Place, and west on Waverly Place to Washington Square. Seventh Avenue (Interborough) Subway (IRT) -- Local to Sheri­ dan Square Station. Walk east on Waverly Place to Washington Square. Broadway (-Mimhattan) Subway (BMT) -- Brighton local or Fourth Avenue local to Eighth Street Station. Walk south on Broadway to Waverly Place, then west on Waverly Place to Washing­ ton Square. Sixth or Eighth Avenue (Independent) Subway (IND) --Express to West Fourth Street-Washington Square Station, Walk east on West Fourth Street or Waverly Place to Washington Square.

10 Fifth Avenue Bus -- Busses numbered 3, and some numbered 5, to last stop, which is Washington Square.

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

SATURDAY, 10:00 A.M. Session on Analysis, Room 708 10:00 - 10: 10 (1) On the recovery of companded signals Dr. H. j. Landau, Institute for Advanced Study, and Dr. W. L. Miranker, IBM Research Center, Yorktown Heights, New York (566-9) 10:15 - 10:Z5 (Z) Asymptotic behavior of a differential-functional system Dr. R. B. Kelman, IBM Corporation, Yorktown Heights, New York (566-8) 10:30 - 10:40 (3) On the order of the difference of two meromorphic func­ tions ProfessorS. M. Shah, Northwestern University (566-14) 10:45 - 10:55 (4) A Tauberian conjecture and polygonal eigenvalue distribu­ tions. I Professor F. H. Brownell, University of Washington (566-4) 11:00 - 11:10 (5) Families of solutions of a perturbation problem Professor jane C. Scanlon, Polytechnic Institute of Brooklyn (566-13) 11:15 - 11:Z5 (6) A Stone-Weierstrass theorem for C*-algebras Dr. J. G. Glimm, Institute for Advanced Study (566-7) 11:30 - 11:40 (7) Invariance of cones by positive definite operators Dr. A. B. J. Novikoff, Stanford Research Institute, Menlo Park, California (566-ZO)

11 11:45 - 11:55 (8) Barrelled spaces and the closed graph theorem Professor M. E. Mahowald, Syracuse University (566-17)

General Session, Room 709 10:00 - 10:10 (9) On the completely semi-associative and non-commutative loops and loop algebras of the fifth order Professor Volodymyr Bohun-Chudyniv, Morgan State College (566-6) 10:15 - 10:25 (10) On the structure of rings without nilpotent ideals and their rings of quotients Professor Joachim Lambek, Institute for Advanced Study (566 -16) 10:30 - 10:40 (11) On a pellian equation conjecture Professor L. j. Mordell, University of Colorado (566-10) 10:45 - 10:55 (12) Sub-orderings of recursive degrees Mr. G. E. Sacks, Cornell University (566-12) 11:00 - 11:10 (13) Extensions to isolic integers. II Professor Anil Nerode, Cornell University (566-11) 11:15 - 11:25 (14) Partitions of mass -distributions and convex bodies by hyperplanes Dr. Branko Grunbaum, Institute for Advanced Study (566 -15) 11:30 - 11:40 (15) The generalized Whitehead product Mr. Martin Arkowitz, Cornell University (566-1)

SATURDAY, 2:00 P.M. Invited Address, Room 703 Some conjectures in diophantine equations (One hour) Professor Serge Lang, Columbia University

SATURDAY, 3:15P.M. Session on Applied Mathematics, Room 708 3:15 - 3:·25 (16) Simplification of Boolean functions Mrs. B. K. Bender, Dr. A. j. Goldman, and Mr. R. B. Thomas, National Bureau of Standards, Washington, D.C. (566-3)

12 3:30 - 3:40 (17) Integral geometric methods in information theory. V. Stationarity and ergodicity Professor W. R, Baum, Syracuse University (566-2) 3:45 - 3:55 (18) The solution of the linearized equations of relative motion of two light bodies in a gravitational field, Preliminary report Mr. J, B. Nance, Jr., Radio Corporation of America, Burlington, Massachusetts (566 -19) (Introduced by Dr. Paul Nesbeda) 4:00 - 4:10 (19) A study of the non-uniform convergence of the solution of a doubly infinite set of equations Mr. R. Mittra, University of Illinois (566-18) (Introduced by Professor Everett Pitcher)

SUPPLEMENTARY PROGRAM (To be presented by title)

(20) On the solution of an arbitrary system of linear equations Professor Smbat Abian, University of Pennsylvania, and Dr. Elliott Mendelson, Columbia University (21) A model for the universe with discontinuities. Prelimin­ ary report Mr. Ralph Abraham, University of Michigan (Introduced by Dr. S. H. Gould) (22) A note on multiple series of positive terms Professor J. W. Andrushkiw, Seton Hall University (23) On a coloring problem Mr. Edgar Asplund and Dr. Branko Griinbaum, Institute for Advanced Study (24) A stability theorem for non-linear equations Dr. V. E. Bene§', Bell Telephone Laboratories and (566-21) (25) A Tauberian conjecture and polygonal eigenvalue distribu­ tions, II Professor F. H. Brownell, University of Washington (566- 5) (26) Note on a theorem of van der Corput Professor Eckford Cohen, University of Tennessee (27) On an elementary method in the asymptotic theory of numbers Professor Eckford Cohen, University of Tennessee

13 (Z8) The quaternion reducibility of semi-groups of complex matrices Professor j. E. Houle, Georgetown University (Z9) Obstructions to imposing differentiable structures. Preliminary report Dr. j. R. Munkres, Princeton University (30) Extensions to isolic integers. I Professor Anil Nerode, Cornell University (31) Estimates of some k-dimensional exponential sums. I Dr. E. M. Stein, University of Chicago (3Z) Estimates of some k-dimensional exponential sums. II. Applications Dr. E. M. Stein, University of Chicago (33) Limits of sequences of operators. I Dr. E. M. Stein, University of Chicago (34) Limits of sequences of operators. II. Applications Dr. E. M. Stein, University of Chicago (35) On harmonic functions in several variables. I Dr. E. M. Stein, University of Chicago (36) On harmonic functions in several variables. II Dr. E. M. Stein, University of Chicago Everett Pitcher Associate Secretary Bethlehem, Pennsylvania january 6, 1960

14 PRELIMIRARY ARROURCEMERT OF MEETIRG

FIVE HUNDRED SIXTY-SEVENTH MEETING Hotel New Yorker New York, New York April 14-16, 1960

The five hundred sixty-seventh meeting of the American Mathe­ matical Society will be held on Thursday through Saturday, April 14-16, 1960 at the Hotel New Yorker, Two symposia will be held in conjunction with the regular meet­ ing. A Symposium on Stability Problems in Hydrodynamics will be held in four sessions on Thursday morning and afternoon and Friday morning and afternoon, This is another in the series of Symposia in Applied Mathematics sponsored by the Office of Ordnance Research. The four sessions will be devoted respectively to (i) Stability of Laminar Flow, (ii) Non-linear Flow Instability, (iii) Statistical Models in Hydrodynamics, and (iv) Control of Instabilities. Hydro­ magnetic instability will be treated in the first session and the gener­ ation of waves by wind in the second session. The final session will deal with the general mathematical theory of stability and control. Professor Garrett Birkhoff of Harvard University is the Chairman of the Organizing Committee, A distinguished panel of speakers and sessions chairmen is being invited for this symposium. A Symposium on The Structure of Language and its Mathemati­ cal Aspects, jointly sponsored by the Association for Symbolic Logic, the Linguistic Society of America, and this Society with financial support from project FOCUS of the Institute for Defense Analyses, will be held on Thursday morning and afternoon and Friday morning. Professor Roman Jakobson of Harvard University is the Chairman of the Program Committee, The other members of the committee are: Noam Chomsky, H. B. Curry, Henry Hiz, Hilary Putnam and W. V. Quine. Besides the members of this committee, the list of speakers includes R. Abernathy, Y. R. Chao, A. Church, H. A. Gleason, N. Goodman, M. Halle, C. F. Hockett, J. Lambek, B. Mandel­ brot, A. G. Oettinger, R. s. Wells, and z. H. Yngze. The discussion will be led by A. A. Hill, M. Black, and J. Whatmough. By invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings, Professor Peter D. Lax of New York University will address the Society on the subject "Harmonic Analy­ sis" at Z:OO P.M. on Friday. By invitation of the same Committee, Professor N. C. Ankeny will deliver an address on "The theory of representations by quad­ ratic forms" at Z:OO P.M. on Saturday,

15 By invitation of the International Business Machines Corpora­ tion, one or more tours of their Product Display Center at IBM World Headquarters, 57th Street and Madison Avenue, will be avail­ able on the morning of Friday, April 15. The tour will include an introduction to electronic data processing machines and demonstra­ tions of the IBM 705 data processing system, the IBM RAMAC 305, and other equipment. There will be a discussion of the application of electronic computers to numerical analysis. There will be sessions for contributed papers on Friday after­ noon and on Saturday morning and afternoon. In particular, there will be a session for contributed papers on Mathematical Problems in Linguistics in the event that suitable papers in this area are sub­ mitted, The Council of the Society will meet at 5:00P.M. on Friday and will continue its meeting after an intermission for dinner. The Hotel New Yorker is on the corner of 34th Street and 8th Avenue, one block west and one block north of the Pennsylvania Station, from which there is a tunnel direct. The hotel is on the 8th Avenue line of the Independent Subway, at the 34th Street or Pennsyl­ vania Station stop. The hotel can be reached by walking underground from the 34th Street station of the IR T subway. The hotel is one block east from the 34th Street exit of the Lincoln Tunnel and is less than half a block west of the Greyhound Bus Terminal. Airline ser­ vice into LaGuardia and Idlewild airports is served by the East Side Terminal. Buses serving the Newark airport come into the West Side Terminal, eight blocks north and two blocks west of the New Yorker, which is somewhat nearer to the hotel than the East Side Terminal is. The hotel has its own garage, which advertises reason­ able rates, A reservation blank will be found on page 99 of this issue of the NOTICES. Persons intending to stay at the Hotel New Yorker should use this blank and send it in as soon as possible. It may not be possible to run another copy of this blank in the April NOTICES, since postal regulations forbid more than one such blank per issue. Further details about the meeting will appear in the April issue of these NOTICES. Abstracts of contributed papers should be sent to the American Mathematical Society, 190 Hope Street, Providence 6, Rhode Island, so as to arrive PRIOR TO THE DEAD­ LINE, March 1,

Everett Pitcher Associate Secretary Bethlehem, Pennsylvania January 6, 1960

16 FIVE HUNDRED SIXTY-EIGHTH MEETING Chicago, Illinois April ZZ-Z3, 1960

The five hundred sixty-eighth meeting of the American Mathe­ matical Society will be held in Chicago, Illinois on Friday and Satur­ day, April ZZ -Z3, 1960. There will probably be four to six sessions for the presentation of contributed papers. The Committee to Select Hour Speakers for Western Sectional Meetings has not yet made its final selection. It should be noted that the deadline for the receipt of abstracts for this meeting is March 1. It is hoped, however, that it will be possible to accommodate those whose abstracts fail to meet the deadline by having a special session for late papers. Complete details of the program will appear in the next issue of the NOTICES. j. W. T. Youngs Associate Secretary Bioomington, Indiana january 6, 1960

FIVE HUNDRED SIXTY -NINTH MEETING University of California Berkeley, California April Z l-Z3, 1960

The five hundred sixty-ninth meeting of the American Mathe­ matical Society will be held on Thursday, Friday and Saturday, April Zl-Z3, 1960 at the University of California, Berkeley, Califor­ nia. By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, a Symposium on Differential Equa­ tions will be held on Thursday and Friday. The Symposium is financially supported by the Air Force Office of Scientific Research. The Program Committee for th-e Symposium consists of Professors C. B. Morrey, Jr., Chairman, David Gilbarg, Louis Nirenberg and P. C. Rosenbloom. By invitation of the same Committee, there will be an address on Saturday by Professor F. B. Fuller of the California Institute of Technology on "Fixed points of multiple-valued transformations". There will also be sessions for contributed papers on Saturday.

17 The meetings will be held in Kroeber Hall at Bancroft and College Avenues, The Registration Desk will be in the Lobby of Kroeber Hall, There will be a tea for people attending the meetings at the F acuity Club on Saturday following the last session for contri­ buted papers, Luncheon will be available on Saturday at the Faculty Club, To reach Berkeley from San Francisco, take the "F" bus from the inter-urban terminal, First and Mission Streets, and get off at Shattuck and University A venues in Berkeley, To reach the campus from Oakland take either Bus No, 51 on Broadway Avenue, or Bus No, 40 on Telegraph Avenue, and get off at Bancroft Avenue in Berke­ ley, Those planning to drive to the campus on Thursday and Friday should notify the Mathematics Department so that special arrange­ ments can be made to admit them to parking spaces, There are many motels and hotels in Berkeley. The Hotel Durant at Z600 Durant Avenue, Berkeley 4, is very close to the cam­ pus, The rates at this hotel range from $6,00 to $10,00 per day for a single room and from $8,00 to $13,50 for a double room, The Bel Air Motel at 1330 University Avenue is also convenient, The rates are $8,00 and $9.00 per day for a room for two people, $1Z,OO for a furnished apartment including kitchen for two people and $14,00 for a furnished apartment for four, Reservations should be sent directly to the hotel or motel, R. s. Pierce Associate Secretary Seattle, Washington january 6, 1960

18 ACTIVITIES OF OTHER ASSOCIATIONS

This Calendar includes symposia, seminars, and institutes sponsored by the American Mathematical Society, but does not in­ clude regular meetings of the Society, which are listed elsewhere in the NOTICES.

FUTURE MEETINGS OF RELATED ORGANIZATIONS

March, 1960 Society of Actuaries Hotel - Mayflower Hotel Location - Washington, D. C. Date -March 24-25, 1960 Secretary - Dr. A. A. McKinnie, Society of Actuaries, 208 South La Salle Street, Chicago 4, Illinois

April, 1960 The Institute of Mathematical Statistics Place - Location - Lafayette, Indiana Date -April 7-9, 1960 Secretary- Dr. George E. Nicholson, jr., De­ partment of Statistics, University of North Carolina, Chapel Hill, North Carolina

American Mathematical Society and The Institute for Defense Analyses "Symposium on Mathematical Problems in the Structure of Language" in conjunction with 1960 meeting of the Society Hotel - Hotel New Yorker Location - New York City Date -April 14-16, 1960

American Mathematical Society and the Office of Ordnance Research, U.S. Army "Symposium on Stability Problems in Hydrody­ namics 11 in conjunction with the 1960 meeting of the Society Hotel - New Yorker Hotel Location - New York Date -April 14-16, 1960

19 The Institute of Mathematical Statistics Place - Columbia University Location - New York City Date - April Z l-Z3, 1960 Secretary - Dr. George E. Nicholson, jr ., De­ partment of Statistics, University of North Carolina, Chapel Hill, North Carolina

American Mathematical Society and the Air Force Office of Scientific Research "Symposium on Differential Equations" in conjunction with 1960 meeting of the Society Place - University of California Location - Berkeley, California Date -April ZZ-Z3, 1960

May, 1960 Western joint Computer Conference Location - San Francisco, California Date -May Z-6, 1960 Chairman - H. M. Zeigler, Technical Program Committee, WSJCC, Stanford Research Insti­ tute, Menlo Park, California

Society of Actuaries Hotel - Roosevelt Hotel Location - New Orleans, Louisiana Date -May 5-6, 1960 Secretary - Dr. A. A. McKinnie, Society of Actuaries, Z08 South LaSalle Street, Chicago 4, Illinois

Operations Research Society of America 17th National (8th Annual Meeting) Hotel - Statler Hilton Hotel Location - New York City Date - May 19-ZO, 1960 Chairman - Professor Max Woodbury, Depart­ ment of Mathematics, New York University, New York 53, New York

zo june, 1960 The American Society of Mechanical Engineers Semi-Annual Meeting and Aviation Conference Hotel - Statler Hilton Location - Dallas, Texas Date -june 5-10, 1960 Director of Public Relations - L. S. Dennegar, The American Society of Mechanical Engineers, Z9 West 39th Street, New York 18, New York

International Conference on Many-Body Problems Location - Utrecht, Netherlands Date - june 7-1Z, 1960 Contact- Professor L. c. P. van Hove, Depart­ ment of Theoretical Physics, State University, Utrecht, Netherlands

Mathematics Research Center, u.S. Army; Uni­ versity of Wisconsin "International Conference on Partial Differential Equations and Continuum Mechanics" Place - University of Wisconsin, Madison, Wis­ consin Housing - Elizabeth Waters Hall - a university dormitory Date - june 7-15, 1960 Contact- Professor R. E. Langer, Mathematics Research Center, U. s. Army, The University of Wisconsin, Madison 6, Wisconsin

International Union of Pure and Applied Physics Commission on Thermodynamics and Statistical Mechanics Meeting Location - Utrecht, Netherlands Date -june 1Z-16, 1960 Contact- Professor L. c. P. van Hove, Depart­ ment of Theoretical Physics, State University, Utrecht, Netherlands

Z1 The American Society of Mechanical Engineers Applied Mechanics Conference Place - Pennsylvania State University Location - University Park, Pennsylvania Date - June ZO -zz, 1960 Director of Public Relations - L. S. Dennegar, The American Society of Mechanical Engineers Z9 West 39th Street, New York 18, New York

International Mathematical Union International Colloquium on "Topology and Differential Geometry" Location - Zurich, Switzerland Date - June ZO-Z6, 1960 Chairman - Professor H. Hopf

International Statistical Institute, Session Location - Tokyo, Japan Date - June, 1960 Contact - E. Lunenberg, Director of the Perma­ nent Office, Z Oostduinlaan, The Hague, Netherlands

International Federation of Automatic Control International Congress for Automatic Control Location - Moscow, USSR Date - June Z5-July 5, 1960 Contact - Secretariat of the Federation of Auto­ matic Control, c/o Verein Deutsche Ingenieure 79 Prinz-George-Strasse, Dusseldorf, German

July, 1960 International Mathematical Union International Symposium on "Linear Spaces - Geometrical Aspects and Applications to Analysis" Location - Jerusalem, Israel Date - July, 1960 Chairman - Professor A. Dvoretzky

zz American Mathematical Society, Summer Seminar "Modern Physical Theories and Associated Mathematical Developments" Sponsored by Atomic Energy Commission; Na­ tional Science Foundation; Office of Naval Re­ search; Office of Ordnance Research, U. S. Army Place - University of Colorado Location - Boulder, Colorado Date - july Z4 - August 19, 1960 Chairman - Professor K. 0. Friedrichs, New York University, Z5 Waverly Place, New York 3, New York

August, 1960 American Mathematical Society Summer Institute on Finite Groups Sponsored by the National Science Foundation Place - California Institute of Technology Location - Pasadena, California Date -August 1-ZB, 1960 Chairman- Marshall Hall, Jr.

Association for Computing Machinery National Association for Computing Machinery Conference Place - Marquette University Location - Milwaukee, Wisconsin Date -August Z3-Z5, 1960

The Institute of Mathematical Statistics and American Statistical Association Place - Stanford University Location - Stanford, California Date - August Z3 -Z6, 1960 Secretary - Dr. George E. Nicholson, jr ., Department of Statistics, University of North Carolina, Chapel Hill, North Carolina

Z3 International Congress for Logic, Methodology and Philosophy of Science Sponsored by the International Union for History and Philosophy of Science Place - Stanford University Location - Stanford, California Date - August 24 - September 2, 1960 Contact - Professor Patrick Suppes, Serra House, Stanford University, Stanford, California

4th London Symposium on Information Theory Location - London, England Date - August 29 - September 3, 1960 Contact - Professor E. C. Cherry, Imperial College, London S.W. 7, England

lOth International Congress of Applied Mechanics Place - Stresa, Italy Date - August 31-September 7, 1960 Chairman - Professor Gustavo Colonnetti, Chairman, Organizing Committee, c/o Con­ siglio Nazionale delle Ricerche, Piazzale delle Scienze 7, Rome, Italy

September, 1960 Second International Conference on Operational Research Location - Aix-en-Provence, France Date - September 5-10, 1960 Contact- john B. Lathrop, ORSA Representative, IF ORS, Operations Research Division, Lock­ heed Aircraft Corporation, Burbank, Califor­ nia

5th Austrian Congress of Mathematicians Place - University of Innsbruck Location - Tyrol, Innsbruck, Austria Date - September 12-18, 1960 Chairman - Professor W. Grabner, Mathemati­ cal Institute of the University, Innrain 52, Innsbruck, Austria

24 The Yugoslav Congress of Mathematicians and Physicists Location - Belgrade, Yugoslavia Date - September 19-24, 1 960 Contact - Savez Dru§'tava Matematil!ara i Fizi~ara, Beograd, pp, 791

October 1, 1960 Society of Actuaries Hotel - Edgewater Beach Hotel Location - Chicago, Illinois Date- September 28-30, 1960 Secretary - Dr. A. A. McKinnie, Society of Actuaries, 208 South La Salle Street, Chicago 4, Illinois

October, 1960 Operations Research Society of America 18th National Meeting Hotel - Statler Hilton Hotel Location - Detroit, Michigan Date - October 10-1 , 1960 Chairman - Mr. George 0' Brien; Touche, Niven, Bailey and Smart; 1292 National Bank Building; Detroit 26, Michigan

November, 1960 The American Society of Mechanical Engineers Annual Meeting Hotel - Statler Hilton Location - New York City Date- November 27-December 2, 1960 Director of Public Relations - L. S. Dennegar, The American Society of Mechanical Engineers, 29 West 39th Street, New York 18, New York

December, 1960 American Association for the Advancement of Science 127th Annual Meeting Location - Philadelphia, Pennsylvania Date - December 26-31, 1960

Eastern Joint Computer Conference Location - New York City Date - December, 1960

25 1960 (date not International Mathematical Union yet determined) International Symposium on "Fonctions de plusieurs variables complexes et analyse fonctionnelle 11 Location - Portugal Date - 1960 Chairman - Professor Sebastiao e Silva

The Statistical Laboratory of the University of California Fourth Berkeley Symposium on Mathematical Statistics and Probability Date - summer, 1960 Contact - Director, Statistical Laboratory, University of California, Berkeley 4, Califor­ nia

Z6 MEWS ITEMS AND AMMOUMCEMERTS

THE ASIA FOUNDATION has made a grant of $2,500 to the American Mathematical Society for the purpose of assisting Asian mathematicians with memberships, journal subscriptions, and travel expenses to meetings of the Society. The funds will be used for: (1) individual grants from the Society to enable distinguished or promising mathematicians now in the United States to attend meetings of the Society, and (2) presentation of memberships in the Society to these scientists and mathematicians, and to select Asian mathe­ maticians in Asia for periods of one to three years. The grantees' country of origin must be Japan, Republic of Korea, Hong Kong, Taiwan (Formosa), Thailand, Burma, Vietnam, Cambodia, Laos, Singapore/Malaya, Indonesia, Pakistan, India, Afghanistan, Ceylon, or The Philippines. There is no stipulation that membership and journal recipients in Asia need have studied in the United States. Within the limits pos­ sible, there will be an understanding that the grantee, if now in resi­ dence in the United States, plans to return to his homeland upon com­ pletion of the course of study or the temporary assignment which has brought him to the United States. The grantees should generally be at least on the graduate student level. Grants will be awarded by the Secretary of the Society, Pro­ fessor John W. Green, Department of Mathematics, University of California, Los Angeles 24, California. Applications for a grant or nominations of prospective grantees should be addressed to Professor Green.

NSF ANNOUNCES SENIOR POSTDOCTORAL FELLOWSHIP A WARDS. The National Science Foundation announced on December 18, 1959, the award of 75 Senior Postdoctoral Fellowships in the sciences for the fiscal year 1960, selected from 2 59 applicants. Senior Postdoctoral Fellowships were awarded to scientists of demonstrated ability and special aptitude for productive scholarship in the sciences. Twenty-eight awards were made in the life sciences, and forty-seven in the physical sciences, including eleven in mathe­ matics. Applications were evaluated by panels of outstanding scien­ tists appointed by the National Academy of Sciences - National Re­ search Council. Selection of awardees was made by the National Science Foundation. Senior Postdoctoral Fellows receive stipends adjusted to ap-

27 proximate their salaried incomes and an allowance to aid in defray­ ing the cost of travel to the fellowship institution, These fellowships are awarded for study and/or research at an appropriate nonprofit American or nonprofit foreign institution. All awards were approved by Dr. Alan T. Waterman, Director of the Foundation, The National Science Foundation expects to reopen the Senior Postdoctoral Fellowship Program in May for awards to be made in December 1960. The following is a list of the awards in mathematics: NATIONAL SCIENCE FOUNDATION SENIOR POSTDOCTORAL FELLOWSHIP AWARDS DECEMBER, 1959 Name Fellowship Institution Buchsbaum, David A, Institute for Advanced Study Devinatz, Allen Institute for Advanced Study Hewitt, Edwin University of Washington, Institute for Advanced Study Karlin, Samuel Hebrew University, University of Oxford Kolchin, Ellis R. University of Paris Lashof, Richard K. Institute for Advanced Study, Uni­ versity of Chicago Marcus, Rudolph A. New York University Shapiro, ArnoldS. University of Paris Spitzer, FrankL. Princeton University Truesdell, Clifford A. University of Bologna, University of Basel Washnitzer, Gerald Institute for Advanced Study

NSF ANNOUNCES SCIENCE FACULTY A WARDS. The Founda­ tion announced on December 18, 1959, the award of Z85 Science F acuity Fellowships in the sciences for the fiscal year 1960. These fellowships are offered as a means of improving the teaching of science, mathematics, and engineering in American colleges and universities, The primary purpose of these awards is to provide an opportunity for college and university science teachers to enhance their effectiveness as teachers. Eligibility requirements include a baccalaureate degree or its

Z8 equivalent, demonstrated ability and special aptitude for science teaching and advanced training, and three years of full-time science teaching at the college level, Science Faculty Fellowship awardees were selected from 76Z applicants from all parts of the United States and its territories on the basis of ability as indic-ated by letters of recommendation, aca­ -demic and professional records, and other evidence of attainment and promise, Applications were evaluated by panels of persons ap­ pointed by the Association of American Colleges who were con­ sidered especially competent to make judgments as to the demon­ strated and potential ability of the applicants as teachers of science, Selection of awardees was made by the National Science Foundation, Science Faculty Fellows receive stipends adjusted to approxi­ mate their salaried incomes and an allowance to aid in defraying the cost of travel to the fellowship institution, These fellowships are awarded for study and/or research at an appropriate nonprofit Ameri­ can or nonprofit foreign institution. All awards were approved by Dr. Alan T. Waterman, Director ot the Foundation, The National Science Foundation expects to reopen the Science Faculty Program in May for awards to be made in December, 1960. There were sixty-three awards in mathematics as follows:

NATIONAL SCIENCE FOUNDATION SCIENCE FACULTY FELLOWSHIP AWARDS DECEMBER, 1959 Name Fellowship Institution Ayer, Miriam C. University of Missourj Aucoin, Clayton V. Stanford University Bauserman, Thomas University of Pittsburgh Beatty, Richard L. Stanford University Blakney, Simmie S, University of Illinois Bretscher, Miriam E. Purdue University Bridgman, Robert J, University of Michigan Clark, Helen M. Columbia University Crowley, Ward State University of Washington Deal, Roy B., jr. University of Paris Dillon, Thaddeus, III University of Pittsburgh Dodge, Clayton W. Princeton University Embree, Earl 0. University of Illinois Freeman, Harold A. Harvard University

Z9 .Nrurul. Fellowship Institution Gateley, Wilson Y. Oklahoma State University Goldberg, Samuel Stanford University Goldman, Oscar Institute for Advanced Study Haas, Violet University of Michigan Harrell, John D., jr. University of North Carolina Heimer, Ralph T. Pennsylvania State University House, Lawrence C. Yale University Imhoff, john L. Stanford University jewett, John W. Yale University Kerwin, M. Alicia, Sr. University of Michigan King, Donald A. Purdue University Landin, joseph University of Illinois Larsson, Robert D. Syracuse University LaRue, james A. University of Pittsburgh Mach, George R. Purdue University Mann, William R. University of California, Berkeley Mark, Abraham M. University of California, Berkeley McBrien, Vincent 0. University of California, Berkeley Moder, joseph J. Iowa State University Monahan, Irene P. Virginia Polytechnic Institute Moreland, Robert A., jr. University of New Mexico Mott, Thomas E. Pennsylvania State University Murphy, Kenneth H. University of Pittsburgh Olson, Lloyd D. University of Minnesota Oschwald, Karamaneh I. University of Missouri Parker, Jean T. Vanderbilt University Prenowitz, Walter Columbia University Seely, Nathan T ., jr. New York University Sheinhart, Edna New York University Shively, Ralph L. University of California, Berkeley Shuford, Florence V. Pennsylvania State University Sigler, Laurence E. Columbia University Silber, jack University of North Carolina Snyder, Robert D. Brown University Stevens, Richard H. Ohio State University Stewart, james C. University of Michigan Thigpen, Charles C. Virginia Polytechnic College Timon, William E., Jr. Oklahoma State University Transue, William R. University of Paris

30 Name Fellowship Institution Vance, Elbridge P. Stanford University Vander lin, Carl J ., jr. University of Wisconsin Westberry, john E. University of Michigan Western, Donald W. University of Michigan White, Myron E. Columbia University Willig, Warren C. University of North Carolina Wisner, Robert J, University of Washington Wolf, Frank L. University of California, Berkeley Youse, Bevan K. University of Notre Dame Yozwiak, Bernard J, University of Pittsburgh

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICA­ TIONS. This new journal will provide a medium for the rapid publi­ cation of carefully selected mathematical papers treating classical analysis and its manifold applications. The refereeing system used in most journals is replaced by a board of Associate Editors, each of whom may accept manuscripts, In this manner, it is hoped that the delay between receipt and publi­ cation of manuscripts will be minimized. Only papers written in a lucid, expository style will be eligible for publication. The use of symbols and preparation of manuscripts should conform to the style followed in publications of the American Mathematical Society. In recognition of the fact that other disciplines contribute new concepts and problems to the continuing growth of mathematics, papers devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering will be encouraged, In these papers the emphasis will be upon the analytical aspects and the novelty of problem and solution, The Editor is Richard Bellman, The RAND Corporation, Santa Monica, California and the Associate Editors are: F, V. Atkinson K.Fan J, Richardson G. Birkhoff M. juncosa P. Rosenbloom R. P, Boas S. Karlin H. N. Shapiro S. Chandrasekhar j. Kemeny S. Ulam C, Dolph P. Lax H. S. Vandiver R. j. Duffin N. Levinson j. W. T. Youngs L. Zadeh The first issue is scheduled to appear early in 1960. A sub­ scription to volume 1, which consists of four numbers, costs $16,00, and should be sent to the publishers: Academic Press, Incorporated, 111 Fifth Avenue, New York 3, New York,

31 SCIENTIFIC RESEARCH AND DEVELOPMENT EXPENDI­ TURES TO REACH $12 BILLION IN 1959-60. The National Science Foundation on December 26, 1959 estimated total funds for scientific research and development in the United States will reach an all-time high of $12 billion for the year 1959-60, up $7 billion from the level of 1953. A Foundation study found an increase of nearly 160 percent in funds used in the performance of research and development by pri­ vate firms and related organizations from 1953-54 to 1959-60. Ex­ penditures by these organizations are expected to rise from $3.6 billion to a projected $9.4 billion during the period. More than half the funds expended by industry were found to have come from the Federal Government, with 70 percent of industry's funds expended on development rather than research. Expenditures by other sectors of the economy for the perfor­ mance of research and development also are expected to demonstrate substantial increases for the 1953-54 to 1959-60 period. The Foun­ dation estimated expenditures by the Federal Government will rise from $970 million for 1953-54 to $1.78 billion for 1959-60, by col­ leges and universities from $450 million to $1 billion, and by other nonprofit institutions such as philanthropic foundations and health agencies from $100 to $250 million. The study, "Funds for Research and Development in the United States, 1953-59," is number 16 of the series Reviews of Data on Research and Development. In the study, hyphenated years represent a 12-month composite of fiscal, academic, and business years. Transfers of research and development funds from the sources to the performers for 1953-54 and 1956-57 are also given. In the 1956-57 survey year, the Federal research and development dollar was divided as follows: industry, 65 cents; Federal scientific instal­ lations, 25 cents; colleges and universities, 8 cents; and nonprofit institutions such as philanthropic foundations, voluntary health agencies, and independent nonprofit research institutes, about 2 cents, according to the study. Copies may be obtained from the Superintendent of Docu­ ments, U.S. Government Printing Office, Washington 25, D. C., for 10 cents.

EMPLOYMENT OF SCIENTISTS AND ENGINEERS IN AMERI­ CAN INDUSTRY. American business firms employed about 780,000 scientists and engineers in January 1959, the National Science Foun­ dation announced on January 1, 1960. About 3 5 percent of the total were engaged in engineering and scientific research and development activities.

32 Scientific and engineering employment data released today are based on a survey conducted for the National Science Foundation by the Bureau of Labor Statistics of the U. S. Department of Labor. Now in preparation, a report on the survey will be released early in 1960. Scientists and engineers in the industry sectors included in the current survey are estimated to represent approximately two­ thirds of the scientists and engineers employed in the Nation. The other third are employed in government agencies, educational insti­ tutions, non-profit organizations, and self-employed. Industries employing the largest groups of scientists and en­ gineers in January 1959 were aircraft and parts, electrical equip­ ment, and chemicals and allied products. Each employed more than 75,000 scientists and engineers; combined, they employed one-third of the total number of scientists and engineers in private industry and approximately half of the scientists and engineers engaged in the performance or administration of research-and-development activities in private industry. Each of two other industries employed more than 50,000 but less than 75,000 scientists and engineers: machinery (except electrical) and engineering and architectural ser­ vices. Next in size, in terms of scientific and engineering employ­ ment, were construction, fabricated metal products and ordnance, primary metals, and motor vehicles and equipment; each of these industry groups employed more than 30,000 but less than 50,000 scientists and engineers. Employment of scientists and engineers in private industry was about 4 percent higher in January 1959 than in January 1958. Engineering employment increased from about 605,000 in January 1958 to 630,000 in January 1959. The number of scientists em­ ployed in private industry increased from about 145,000 in January 1958 to 150,000 in January 1959.

NSF REVISES INDIRECT COST POLICY FOR RESEARCH GRANTS. In connection with its programs for the support of basic research, the National Science Foundation for some years has fol­ lowed the policy of permitting institutions to apply for and receive, as an indirect cost allowance, up to 15o/o of the total direct costs in­ volved in approved grant proposals. Because of rising costs of administration and the adverse effect of such increases upon the ability of institutions to carry on research work, the Foundation has made a revision in its indirect cost policy. Pending completion of a study of the entire problem of indirect costs, the Foundation recently announced that, effective January 1, 1960, it will permit institutions to request up to ZO% of total direct

33 costs as the allowance for indirect costs in research proposals, In no event, however, may such indirect costs exceed the last "audited" or "negotiated" rate approved for the institution by a Federal agency for purposes of Government-sponsored research and development.

NEW PREFIXES FOR UNITS. The National Bureau of Stand­ ards has decided to follow the recommendations of the International Committee on Weights and Measures to use new prefixes for denoting multiples and sub-multiples of units, The Committee adopted the prefixes at its meeting in Paris in the fall of 1958, In addition to the 8 numerical prefixes in common use, which are given in the table be­ low, the Committee expanded the list by adding the 4 prefixes marked with an asterisk, Thus, for example, 10-12 farad is called 1 pico­ farad, and is abbreviated 1 pf,

Multi~les and sub-multieles Prefixes Sx:mbols Pronunciation 1 000 000 000 000 = 1012 tera* T d!'r'a

1 000 000 000 'C 10 9 giga* G jt'ga 1 000 000 = 106 mega M 1 000 = 103 kilo k 100 = 10 2 hecto h 10 = 10 deka dk 0,1 .. 10-1 deci d 0,01 = 10-2 centi c 0,001 = 10-3 milli m 0,000 001 = 10-6 micro u 0 ,000 000 001 = 10 - 9 nano* n na 'nti 0.000 000 000 001 = 10-12 pi co* p pi'ct

E. I, DUPONT de NEMOURS AND COMPANY, INCORPORATED, a Corporate Member of the Society, announced it has awarded $1 ,300,000 to 143 colleges and universities under its annual aid to education program, Of the total, $580,000 was awarded to more than 100 institutions of higher education to support teaching in scientific and related fields. Part of the money may be used to stimulate science students' interest in liberal arts, Funds were also made available to provide scholar­ ships for students interested in becoming high school science and mathematics teachers. Awards totaling $475,000 went to 37 institutions for fundamen­ tal research in the physical sciences, And another 24 grants total­ ing $43,200 were made available to 24 university staff members to do research of their choice during the summer. Capital equipment grants totaling $250,000 were made to 12

34 colleges and universities for aid in financing new buildings, renova­ tions and equipment purchases.

A LIST OF RETIRED MATHEMATICIANS AVAILABLE FOR EMPOYMENT is being maintained by the Headquarters Offices of the American Mathematical Society, 190 Hope Street, Providence 6, Rhode Island. The next list will be issued in March, 1960, and copies of this issue may be obtained by writing to the Headquarters Offices.

DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE FELLOWSHIPS leading to a Ph.D. degree in a Special Program in Applied Mathematics. Five three-year fellowships will be awarded in the Department of Mathematics, Institute of Technology, of the University of Minnesota, to students beginning graduate work. The program consists of nine independent one-quarter courses in gradu­ ate mathematics, over the next three years, designed to introduce doctoral candidates in mathematics to a diverse selection of topics and techniques in applied mathematics. A doctoral candidate in this program is expected to meet high standards in the traditional mathe­ matical fields of analys-is, algebra, and geometry. In addition he would be expected to take about one-third of his graduate mathema­ tics from the courses in the Special Program. For information and application forms write to Professor s. E. Warschawski, Head, Department of Mathematics, Institute of Tech­ nology, University of Minnesota, Minneapolis 14, Minnesota. Completed applications must be received by February Z5, 1960.

35 BACKLOGS OF MATIIEMATICAL RESEARCH JOURNALS

Information on this important matter is being published twice a year, in the February and the August issues of the NOTICES, with the kind cooperation of the respective editorial boards. To broaden the coverage of applied mathematics, we have added four additional journals to the group pre­ viously :Included in the survey.

It is important that the reader should :Interpret the data with full allowance for the wide and sometimes meaningless fluctuations which are characteristic of them. Waiting times in particular are affected by many transient effects, which arise in part from the refereeing system. Extreme waiting times as observed from the published dates of receipt of manuscripts may be very misleading, and for that reason, no data on extremes are pre­ sented in the table on the following page.

Some of the columns in the table are not quite self-explanatory, and here are some further details on how the figures were computed.

Column 2. These numbers are rounded off to the nearest 50.

Column 3. For each journal, this is the estimate as of the in­ dicated dates, of the total number of printed pages which will have been ac­ cepted by the next time that manuscripts are to be sent to the printer, but which nevertheless will not be sent to the printer at that time. (It should be noted that pages received but not yet accepted are being ignored.)

Column 4. Estimated by the editors (or the Editorial Depart­ ment of the American Mathematical Society in the case of the Society's jour­ nals), and based on these factors: Manuscripts accepted, manuscripts re­ ceived and under consideration, manuscripts in galley, and rate of publica­ tion. There is no fixed formula.

Column 5. The first quartile (Q~) and the third quartile (Q 3) are presented to give a measure of the dispers1on which will not be too muCh distorted by meaningless extreme values. The median (Med.) is used as the measure of location. The observations were made from the latest issue re­ ceived in the Headquarters Offices before the deadline date for this issue of the NOTICES. The waiting times were measured by counting the months from receipt of manuscript in final revised form, to month in which the issue was received at the Headquarters Offices (not counting month of receipt of manu­ script but counting month when issue was received at Headquarters Offices). It should be noted that when a paper is revised, the waiting time between re­ ceipt by editors of the final revision and its publication may be much shorter than is the case for a paper which is not revised, so these figures are to that extent distorted on the low side.

36 1 2 3 4 5 Observed Waiting Time Approx. No. pages Backloe: Estimated Current in latest issue ~o.issues published currently 1/30/59 5/31/59 Waiting Time Q Med. Q Journal ner vear ner vear l>IUreS l>IUreS Months Monbts Months Monits American Journal 4 NR(a) NR(a) NR(a) NR(a) 10 13 16 Annals of Math. 6 1,200 NR(a) NR(a) 8 6 8 10 Annals of Math. Statistics 4 1,300 0 0 10 7 8 10 Archive for Rational Me- _(b) chanics and Analysis Unspec. 750 0 4 5 6 7 Canadian Journal 4 672 200 200 13 12 13 17 Duke Journal 4 675 100 250 10 10 11 12 illinois Journal 4 600 250 300 15 12 16 17 Journal of-Math. and Me- _(c) _(c) _(c) chanics 6 1,000 200 450 12 ..., Journal of Math. and -.1 _(b) Physics 4 300 100 9 5 7 11 Michigan Journal 4 400 80 40 11 7 9 13 Pacific Journal 4 1,300 800 750 12 9 11 13 Proceedings 6 1,006 200 280 12 10 11 12 _(b) Quarterly of Applied Math 4 450 NR(a) 10-12 11 12 14 SIAM Journal 4 600 200 300 12 10 11 11 _(b) SIAM Review 4 400 0 9 7 8 8 Transactions 12 2,200 430 550 12 13 13 18

Footnotes: (a) NR means that no response was received to a request for information. (b) Not included in the May 31 request for information. (c) Dates of receipt of manuscripts not indicated in this journal. FOREIGN SCIENCE INFORMATION A Report on Translation Activity in the United States

The complacent assumption that an English-speaking scientist able to read scientific French and German would have access to all the scientific literature of any possible interest to him has in recent years been severely challenged, Spectacular Soviet achievements in science and technology and the growing quantity and quality of scien­ tific research in China have brought home the fact that the transla­ tion of scientific literature from these and other countries whose languages are not generally known to Americans is an important part of research and development in this country. The following report is an attempt to describe what steps are being taken to bring the results of research in foreign countries to American scientists. AMS TRANSLATIONS Two New Projects The AMS program of Russian translation, which is sponsored by the NSF, has recently been expanded by two new projects, one to bring the results of Russian research in mathematics to American mathematicians more rapidly, and the other to make available Rus­ sian books which would not otherwise be published in English. The first of these projects is a new subscription journal, Soviet Mathematics -- Doklady, which will contain translations of the entire pure mathematics section of the Doklady, or Reports, of the Academy of Science of the USSR. The Doklady, which consists of short reports of current research in the USSR, will provide a com­ prehensive, up-to-date survey of what is going on in Soviet mathema­ tics, The AMS journal will appear six times a year, each issue cor­ responding to one volume of the Russian Doklady, which is issued three times a month, six issues constituting a volume, The second new undertaking is the translation and publication by the AMS of Russian books. Three books have been tentatively scheduled for 1960, and several others are under consideration. The three to be undertaken in 1960 are: Some Applications of Functional Analysis in , by S. L. Sobolov, Z55 pages (re­ viewed in MATHEMATICAL REVIEWS, Vol. 14, p, 565); Some v Questions in the Theory of Moments, by N. Akhiezer and M. Krein, Z55 pages; and Theory of Analytic Functions of Several Complex Variables, by B, A. Fuks, 475 pages (reviewed in MATHEMATICAL REVIEWS, Vol, lZ, p. 3Z8).

38 The AMS expects to make available books which, although they are valuable contributions to mathematical research, are too special­ ized to sell widely and are therefore not attractive to commercial publishers. Also, since the AMS is a non-profit organization and receives Government support for the translation projects, it can sell these books more cheaply than a commercial publisher could afford to do. AMS Translations Series Although neither books nor entire journals have previously been undertaken by the AMS, the Society is no newcomer to the field of publishing foreign mathematics. The AMS publication Selected Translations of Russian Mathematical Articles, begun in 1948, was the first in this country containing translations of selected articles. From May, 1948, to February, 1954, during which time the transla­ tion program of the AMS was under the auspices of the Office of Naval Research, 105 paper-bound booklets were issued, each devoted to a single article or to several by the same author. These are now out of print. Sponsorship of the translation project was taken over by the NSF in 1954. Under the NSF program, 1,500 Russian pages of pure mathematics are translated annually, the translated articles being collected into bound volumes which are issued three times a year. So far twelve of these volumes have appeared, under the title Selec­ ted Translations of Russian Mathematical Articles, Series II. A total of 6,251 copies of these twelve volumes have been distributed, 4,533 in this country, and 1,718 abroad. AMS-IMS Probability and Statistics Translation Series Five hundred Russian pages of probability and statistics have been added to the annual quota beginning with the current fiscal year, and will be issued in a new translation series, the publication to be called AMS-IMS Probability and Statistics Translation Series. Scien­ tific direction of this project is vested in a joint committee on Rus­ sian Translations of the AMS and the Institute of Mathematical Statis­ tics. TRANSLATION PROGRAMS FOR SOVIET SCIENTIFIC LITERATURE IN THE UNITED STATES Selective-translation publications have been an exception to the rule, the AMS being the only agency until quite recently to have such a program, In May, 1959, the American Rocket Society, also under NSF sponsorship, began translation of selected Soviet articles in the field of astr-onautics, distributing the translated articles quar­ terly as a supplement to the American Rocket Society Journal, in­ stead of as an independent publication.

39 These programs are only part of the enormous amount of translating being done in the United States, under a variety of pro­ grams, of books, periodicals, separate or collected papers, mono­ graphs, and abstracts. The following descriptions of the more promi­ nent of these programs may give some idea of the scope of trans­ lating activity now going on. Most of the periodical literature translated from the Russian is in the form of cover-to-cover translations, with most of the trans­ lated editions matching the original issue for issue. Among the books and more than seventy complete periodicals which comprise a large part of the translated material, there are three basic patterns of pro­ duction: 1. Under grants made by a Government agency to a non-profit scientific society or research organization. The major grant-making organization of the Government is the NSF. Z. Under direct Government contracts made with translation agencies. 3. By commercial translating agencies without Government or other subsidy. An example of the first method of production is NSF support of certain physics journals. Foundation grants are made to the Ameri­ can Institute of Physics, which then takes full responsibility for the preparation and publication of the journals. The AlP selects the jour­ nals to be translated, and an Institute Advisory Board on Russian Translations maintains a check on the quality of the product. Sub­ scription income is applied against the cost of the project, so that as the circulation of any given translated journal increases, its need for subsidy lessens, freeing grant funds for expansion of the program to other projects. A similar pattern is followed in the Foundation's support of the new AMS journal Soviet Mathematics - Doklady, of translation journals of the American Institute of Biological Sciences, and of translation projects in other fields. The National Institutes of Health follow the second procedure in supporting translation of journals in the medical field. With the ad­ vice of expert panels, individual consultants, and their own scientists, the Institutes select the periodicals to be translated and then contract directly with a translating agency for preparation and publication of the English-language editions. The NIH also take the responsibility for quality control. Again, subscription income is used to reduce contract requirements. The translation journals by commercial publishers, prepared according to the third method, are assumed to be self-supporting. Subscription rates for most periodicals in this category tend to be quite high when calculated on a per -page basis -- prohibitively so,

40 in fact, for many potential subscribers. Consequently, if US scien­ tists are to have adequate access to this material, some Govern­ ment support has been found necessary to permit reduction in sub­ scription prices. Such support, for example, has been given by the NSF to Consultants Bureau, Incorporated, under contract, to reduce the subscription prices of the translated chemical journals which they publish. Well-selected Russian scientific books, on the other hand, are often profitable, and commercial publishers are showing greater interest in them. Because books are so expensive and time-consuming to trans­ late, care in the selection of titles and avoidance of duplication are particularly important. Selection of books, as well as articles, to be translated by the AMS is made by the Joint Committee on Russian Translations of the AMS and IMS, every effort being made to insure that only worthwhile works are chosen. To avoid duplication, a book that is being considered is checked with other translation centers, not only in the United States but in Great Britain and Germany, to determine whether it is being translated by another agency into English, French, or German. Miscellaneous translations. of other Soviet scientific literature include occasional collections of translated papers in special fields, issued from time to time, but on no regular schedule, by various government agencies and some commercial firms, and monographs, which are issued in many different ways -- by commercial publish­ ers, university presses, as supplements to journals, by the Govern­ ment Printing Office, and as Government scientific reports. In ad­ dition to these more or less formal publications, an enormous num­ ber of separate Soviet papers are translated by and for numerous Government and private agencies; these are prepared on an indivi­ dual basis and usually in response to specific agency or laboratory requests. A relatively new feature of the entire translation activity is the translation of abstracts from the Soviet abstracting journal, REFERATIVNYI ZURNAL. Although this journal covers scientific and technical literature from the entire world, only those abstracts are translated that are prepared from papers originally published in the USSR or in other Iron Curtain countries. Some translated abstracts are published as separate journals in the respective fields; others, wherever applicable, are included in the regular United States abstracting journals. NSF OFFICE OF SCIENCE INFORMATION SERVICE In order to provide for more effective dissemination of scien­ tific material, both domestic and foreign, the NSF, in 1958, estab­ lished the Office of Science Information Service, directed by Burton Adkinson. This agency is the American counterpart of the Soviet

41 All-Union Institute of Scientific and Technical Information, which is reported to be very successful in abstracting and citing the world's scientific and technical literature. Since the creation of the Science Information Service, far more translating has been done in all Scien­ tific fields. During the fiscal year 1958-59, for example, the Office of Science Information, in cooperation with the Office of Naval Re­ search and the National Bureau of Standards, supported the efforts of 28 professional societies and academic institutions in cover-to­ cover translations of 35 Russian scientific and technical journals and 18 books and monographs, as well as special articles and col­ lections of scientific papers. This support permitted the translation of approximately 70,000 pages of foreign scientific information, a significant increase over the fiscal year 1957-58 support program when the 28 journals, ten books and monographs, and other material translated totaled 37,000 pages. The Office of Science Information Service supports two activi­ ties of the AMS, under two different programs. MATHEMATICAL REVIEWS is supported by the Publication and Information Services Program, while the translation projects previously described fall under the Support of Translation Publication section of the Foreign Science Information Program, directed by Scott Adams. The For­ eign Science Information Program is concerned not only with trans­ lation but with the broadest aspects of the international exchange of information, its basic goal being the widest possible dissemination in the United States of the published results of foreign research. TRANSLATION PROGRAMS ABROAD Public Law 480 As a supplement to the NSF's dollar-supported translation pro­ gram conducted with the cooperation of American scientific societies, an entirely new program has recently been undertaken under the provisions of Public Law 480. This law, which amended the Agricul­ tural Trade Development and Assistance Act of 1954, authorized the President to negotiate and carry out agreements with friendly nations to provide for the sale of surplus agricultural commodities for for­ eign currencies. On June 30, 1958, Public Law 480 was amended by adding a new subsection, 104K, to authorize the use of foreign cur­ rencies to finance scientific activities. Under this provision, for­ eign currencies can be used to collect, collate, translate, abstract, and disseminate scientific and technical information. A sum of $1 ,ZOO ,000 has been appropriated for translation work in countries where funds are available. The funds will be allocated to the NSF, to be administered on behalf of other Government agencies. The initial effort for fiscal year 1961 has been to establish programs in Israel, Poland, and Yugoslavia. The Israeli contract, signed on

42 April Z4, 1959, provides for the translation of approximately 10,000 pages of Russian scientific literature. Support of Russian transla!ion in other countries Other sources of translated scientific literature, particularly of Soviet journals, are the foreign countries which have recently joined the United States in the effort to translate complete Soviet journals for the benefit of English-speaking scientists and engineers. The major participant at present is the Department of Scientific and Industrial Research in Great Britain; the Scandinavian countries have expressed their serious intent, and other countries have dis­ cussed it as a possibility. SCOPE AND LIMITATIONS OF CURRENT TRANSLATION PROGRAMS Although the translation program covering Russian journals, papers, monographs, books, and abstracts is obviously of substantial size, it is still far from adequate if the goal is to make all of the significant results of Soviet scientific research readily available to United States scientists. At present, for example, only about five per cent of Russian scientific periodicals are being translated, and even to maintain this percentage the volume of translation must be in­ creased, since the number of Russian scientific publications is in­ creasing and the journals currently being translated are themselves growing larger. Furthermore, the USSR has abandoned its once general practice of publishing scientific articles in English, French, or German, with the result that the United States has had to begin translating material that about fifteen years ago was being published in the USSR in languages that American scientists can read. A LOOK AT SCIENTIFIC LITERATURE IN OTHER COUNTRIES Although procurement and translation of scientific information of Soviet origin continued during fiscal year 1959 to be of primary importance, considerable attention was also given to the scientific literature of other countries, such as Japan, Communist China, Poland, Yugoslavia, etc. Scientific literature in the Oriental lan­ guages, in particular, is increasing in volume in all subject fields. Some 3,000 scientific periodicals are published annually as follows: Japan 2,500 Communist China 450 Nationalist China Z5 North Korea 15 South Korea 10 Scientific literature from Nationalist China and North and South Korea, as revealed by this breakdown, is small in amount, andre­ flects the present comparatively low state of scientific development.

43 japanese and Communist Chinese scientific literature, on the other hand, is large and rapidly growing, and is sophisticated in many sub­ ject fields. japan The principal sources of scientific literature in japan are government and quasi-government laboratories, together with their private university and industrial counterparts, and professional societies. A recent survey by the japan Information Center of Sci­ ence and Technology lists 36,000 "abstractable" items from 1,358 japanese language scientific journals exclusive of medicine and agri­ culture. In order to establish and maintain scientific communication with the outside world, japanese scientists use English for approxi­ mately ZO per cent of their publications. This takes the form of an English language journal or abstracts, an English edition of a Japan­ ese publication, or of an English abstract, table of contents, or caption in a japanese language journal. A number of japanese lan­ guage journals, in such fields as astronautics, theoretical and solid state physics, high polymer chemistry, and genetics, deserve publi­ cation in the form of English translations, abstracts or indexes w!th a minimum of delay. Communist China Communist Chinese scientific literature has grown impressive­ ly in the last five years, and is becoming increasingly interesting to American scientists, making a translation program for Chinese Scien­ tific literature highly desirable. The present total of approximately 450 scientific publications in China represents a 500 per cent increase over the total of five years ago. Whereas the 1949-54 publications contained a preponderance of Soviet translations and political exhor­ tations, the primary scientific journals today contain an increasing amount of original Chinese basic and applied research. Publication activities are centered around the Academia Sinica, which has been created in the image of the Soviet Akademia Nauk. A continued in­ crease in scientific publications can be expected. Approximately 80 per cent of the primary journals in such fields as mathematics, physics, chemistry, and biology contain a table of contents and abstracts in English, with the remaining ZO per cent having this information in Russian. The bimonthly English lan­ guage SCIENCE ABSTRACTS OF CHINA, in five sections (Biological Sciences, Earth Sciences, Technical Sciences, Mathematical and Physical Sciences, and Chemistry and Chemical Technology), contain some 450 abstracts from 1ZO journals. Even though a considerable amount of Chinese scientific litera­ ture is published in English, much of it remains inaccessible because of trade restrictions. The United States has laws prohibiting the pur-

44 chase of goods (including scientific publications) from Communist China without a Treasury Department license, and the Chinese them­ selves, while soliciting exchange with American professional so­ cieties, have made the problem more difficult by limiting the num- ber of their journals for sale in such places as Hong Kong and London, U, S, AGENCIES FROM WHICH TRANSLATED PUBLICATIONS MAY BE OBTAINED Translated material is disseminated in this country through such channels as scientific societies, commercial publishers, and other translation depositories, At present there are two main trans­ lation depositories in the United States: the Special Libraries Asso­ ciation Translation Center, located at the john Crerar Library in Chicago, and the Office of Technical Services, Technical Information Division of the Department of Commerce, in Washington, The SLA Translation Center collects translations from non­ Government sources, such as universities, professional societies, and industry, which contribute their translations to the Center for dissemination to the public, The OTS Technical Information Division collects translations from Government agencies and sources in for­ eign countries such as Great Britain, Canada, Australia, and others. These two depositories contribute copies of the translations received to each other. All translations are listed, with abstracts of the arti­ cles translated, in Technical Translations, published bimonthly since january, 1959, by the OTS as a sequel to the previous SLA Transla­ tion Center publication Translation Monthly, Copies of Technical Translations are available individually or on subscription from the Office of Technical Services, U, S, Department of Commerce, Washington 2.5, D. C., at 60¢ a copy or $12.,00 per year. Microfilm copies of the translations listed are available from both the SLA Translation Center and the OTS Technical Information Division at nominal rates. Translations prepared by commercial translating agencies are also listed in Technical Translations. However, although the com­ mercia! translating agencies usually provide the necessary biblio­ graphic information about the translations available, they sell copies of these translations themselves. Information on where such trans­ lations are available is included in the listing. Thus, much of the material currently translated by Govern­ ment agencies, private industries, universities, and other organiza­ tions, including agencies abroad, is consolidated in a single listing and is made available to the public through two large regional de­ positories.

45 CONCLUSION An examination of present programs shows that, assuming that an American scientist has a fluent reading knowledge of French and German, a good start has been made towards providing him with information from the other important languages of science -- in particular Russian and Chinese, the japanese and Polish languages being in a special position because of the extent to which their authors write in English. However, it is clear that only a beginning has been made, as indicated by the fact that only five per cent of Russian scientific periodicals and books are being translated, a far smaller percentage than the Russians are translating in the opposite direction. The present small proportion of foreign literature being trans­ lated means that there is plenty of first-class material, not only in Russian but also in Chinese, from which to choose in expanding our translation programs. Many Russian books are first-rank exposi­ tions of the highest quality and would, if translated, be of great ass is­ tance to American researchers and graduate-school teachers; the usefulness of the primary research material contained in Russian periodicals would in many cases be even greater. Fortunately, the existence of the newly founded Office of Sci­ ence Information Service, and of the two large depositories, the SLA Translation Center in Chicago and the Office of Technical Ser­ vice in Washington, assures us not only of a means whereby the sci­ entific desirability of proposed new translation can be evaluated from a nationwide point of view, but also of places of safe -keeping where the growing volume of translated material will be readily and perma­ nently available to American scientists.

46 PERSONAL ITEMS (This section is restricted to members of the Society)

Dr. I. T. A. C. Adamson of the Queen's University, Belfast, Northern Ireland, has been appointed a lecturer at the Queen's College of the University of St. Andrews, Dundee, Scotland, Professor F. J. H. Burkett of Union College has retired with the title professor emeritus. Dr. Dieter Gaier of the Technische Hochschule, Stuttgart, has been appointed to a professorship at the University of Geissen, Ger­ many. Dr. W. L. Harkness of Michigan State University has been ap­ pointed to an assistant professorship at Pennsylvania State University. Associate Professor Henry Helson of the University of Califor­ nia, Berkeley, will be on leave in the spring of 1960 and will spend the term at the Institute for Advanced Study. Professor F. F. Helton of Central College has been appointed to a professorship at the College of the Pacific. Dr. Robert Hermann of Harvard University has accepted a position as staff member of the Lincoln Laboratory of Massachusetts Institute of Technology, Lexington, Massachusetts. Dr. J. E. Householder of the University of Colorado has been appointed to an assistant professorship at Humboldt State College. Professor G. A. Hunt of Cornell University has been appointed to a professorship at Princeton University. Mr. E. H. Hussey of Cooper Metallurgical Associates has ac­ cepted a position as technician with National Carbon Research Labora­ tory, Parma, Ohio, Dr. C. N. Lee of the University of Virginia has been appointed a research associate at the University of Michigan. Assistant Professor L. F. McAuley, on leave from the Univer­ sity of Wisconsin, has been appointed to a visiting associate profes­ sorship at Louisiana State University. Assistant Professor R. G. Mcintyre of the University of Arkan­ sas has accepted a position as senior technical specialist with the Standard Oil Company of Ohio, Cleveland, Ohio, Assistant Professor J. S. Maybee of the University of Southern California has been appointed to an assistant professorship at the University of Oregon, Dr. M. V. K. Menon of Ohio State University has been appointed a visiting associate statistician at the University of North Carolina, Raleigh, Associate Professor J. E. L. Peck of McGill University has been appointed to an associate professorship at the University of Alberta,

47 Assistant Professor W. D. Peeples, jr. of Alabama Polytech­ nic Institute has been appointed to a professorship at Howard College. Dr. Gordon Raisbeck, on leave for one year from the Bell Telephone Laboratories, will be with the Advanced Research Pro­ jects Division of the Institute for Defense Analyses, Washington, D.C. Assistant Professor J. W. Riner of St. Louis University has been appointed to an assistant professorship at Ohio State University. Mr. B.S. Sackman of the Naval Weapons Laboratory has ac­ cepted a position as mathematician with General Electric Company, Philadelphia, Pennsylvania. Mr. D. R. Schuette of the University of Michigan has been ap­ pointed to an assistant professorship at Drake University. Associate Professor W. S. Snyder of the University of Tennes­ see has accepted a position as assistant director of the health physics division of Oak Ridge National Laboratory, Oak Ridge, Tennessee. Dr. Louis Solomon of Harvard University has been appointed to an assistant professorship at Haverford College. Dr. lvar Stakgold of the Office of Naval Research has been ap­ pointed to an associate professorship at Northwestern University. Assistant Professor D. R. Truax of the University of Kansas has been appointed to an assistant professorship at the University of Oregon. Assistant Professor W. A. 0. Waugh of the University of Western Ontario has been appointed to an assistant professorship at McGill University. Dr. Alan Wilson of Cambridge, Massachusetts has been ap­ pointed to an assistant professorship at the University of Pennsyl­ vania. The following promotions are announced: Edward Halpern, University of Michigan, to an assistant professorship. R. B. johnson, Catawba College, to an associate professorship. M. j. Poliferno, Trinity College, to an assistant professorship~ C. R. Putnam, Purdue University, to a professorship. D. E. Sanderson, Iowa State University, to an associate pro­ fessorship. L. j. Warren, San Diego State College, to an associate pro­ fessorship. The following appointments to instructorships are announced: University of British Columbia: Mr. J. H. Lindsay; University of California, Riverside: Dr. R. E. Haymond; Columbia University: Dr. J. W. Gray; Illinois Institute of Technology: Dr. D. H. Lorenz, Mr. David Sachs; Mohawk Valley Technical Institute: Mr. R. A. Schwabe; University of North Carolina: Dr. G. W. Henderson;

48 Southwestern Louisiana Institute: Mr. A. j. Hulin; Western Michigan University: Mr. R. E. Sechler; Yale University: Dr. H. I. Levine, Dr. s. L. Salas. Deaths: Mr. joseph Canning of Butte, Montana died on june 10, 1959 at the age of 71 years. He had been a member of the Society for Z 7 years. Dr. M.A. Martino, jr. of General Electric Company died on June Z6, 1959 at the age of Z9 years. Mr. F. S. Perryman of Royal Insurance Company, Limited died on November 30, 1959 at the age of 6Z years. Mrs. Bobbie E. j. Sparks of the Naval Air Development Cen­ ter died on November 15, 1959 at the age of 31 years.

49 IIE'W' PUBLICATIOIIS

Abhyankar, S. Ramification theoretic methods in . (Annals of Mathematics Studies, no. 43.) Princeton University Press, 1959. 9 + 96 pp. $2.75. Actes du colloque de calcul numerique, Perigueux, 1957. (Publ. Sci. Tech. Minist~re de l'Air, Notes Tech. no. 80.) Paris, 1959. 7 + 87 pp. 1800 fr. Advances in chemical physics, Vol. z. Ed. by I. Prigogine. New York, Interscience , 1959. 9 + 412 pp. $11.50. Aizerman, M.A. Lekcii po teorii avtomatil!eskogo regulirovaniya. Zd ed., enlarged and rev. Moscow, Gosudarstv. Izdat. Fiz.­ Mat. Lit., 1958. 520 pp. 15.75 rubles. Alexandrow, A. D. Kurven und Fliichen. Berlin, VEB Deutscher Ver­ lag der Wissenschaften, 1959. 82 pp. Paperbound, 4.00 DM. Aseltine, J. A. Transform method in linear system analysis. (McGraw-Hill Electrical and Electronic Engineering Series.) New York, McGraw-Hill, 1958. 16 + 300 pp. $8.50. Bachmann, F. Aufbau der Geometrie aus dem Spiegelungsbegriff. (Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI.) Berlin, Springer, 1959. 13 + 311 pp. 49.80 DM. Bailey, N. T. J. Statistical methods in biology. New York, Wiley, 1959. 9 +zoo pp. $4.50. Batchelor, G. K. See The scientific papers of Sir Geoffrey Ingram Taylor. Batchelor, J. H. Operations research: An annotated bibliography. Zd ed. St. Louis University Press, 1959. 10 + 866 pp. Bateman, H. Partial differential equations of mathematical physics. New York, Cambridge University Press, 1959. 12 + SZZ pp. Paperbound, $4.95. Berge, C. Espaces topologiques: Fonctions multivoques. (Collection Universitaire de Mathematiques, Vol. III.) Paris, Dunod, 1959. 11 + Z 7Z pp. 3400 fr. Bianchi, L. Opere. Vol. 11. Corrispondenza. Ed. by the Unione Matematica Italiana with contributions by the Consiglio Nazionale delle Ricerche. Rome, Cremonese, 1959. 304 pp. 3000 Lire. Bogoliubov, N. N ., and Shirkov, D. V. Introduction to the theory of quantized fields. Authorized English ed. Rev. and enlarged by the authors. Trans. by G. M. Volkoff. (Interscience Mono­ graphs in Physics and Astronomy, Vol. III.) New York, Inter­ science, 1959. 16 +no pp. $17.00. de Borde, A. H. See Chisholm, J. S. R. Born, M., and Wolf, E. Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light. New York, Pergamon, 1959. 26 + 803 pp., 19 plates. $17.50.

50 Bracewell, R.N. See Paris symposium on radio astronomy. Brelot, M. EUments de la th~orie classique du potentiel. Les Cours de Sorbonne. 3d cycle. Paris, Centre de Documentation Uni­ versitaire, 1959. 191 pp. Paperbound. Bush, R. R. See Studies in mathematical learning theory. Le calcul des probabilit~s et ses applications. Paris, 15-20 Juillet 1958. (Colloques Internationaux du Centre National de la Re­ cherche Scientifique, LXXXVII.) Paris, Centre National de la Recherche Scientifique, 1959. 196 pp. Cashwell, E. D., and Everett, C. J. A practical manual on the Monte Carlo method for random walk problems. (International Tracts in Computer Science and Technology and their Application, Vol. 1.) New York, Pergamon, 1959. 9 + 153 pp. $6.00. Cazacu, C. A. See Stoilow, S. Chisholm, J. S. R., and de Borde, A. H. An introduction to statistical mechanics. (International Series of Monographs on Physics, Vol. 2.) New York, Pergamon, 1958. 9 + 160 pp. $6.00. Constructivity in mathematics: Proceedings of the colloquium held at , 1957. Ed. by A. Heyting. (Studies in Logic and the Foundations of Mathematics.) Amsterdam, North-Holland, 1959.) 8 + 297 pp. $8.00. Convegno, internazionale: Reticoli e geometrie proiettive, Palermo, 25-29 ottobre 1957; Messina, 30 ottobre 1957. Ed. by the Unione Matematica Italiana with contributions from the Consiglio Nazionale delle Ricerche. Rome, Cremonese, 1958. 7 + 141 pp. 1800 Lire. Crandall, S. H. See Notes for the M. I. T. special summer program on random vibration. Dennis, J. B. Mathematical programming and electrical networks. Cambridge, Technology Press of the Massachusetts Institute of Technology, 1959. 6 + 186 pp. $4.50. Diamond, S. Information and error: An introduction to statistical analysis. New York, Basic Books, 1959. 12 + 307 pp. $5.00. Dini, U. Opere. Vol. 4. Serie di Fourier. Ed. by the Unione Mate­ matica Italiana with contributions from the Consiglio Nazionale delle Ricerche. Rome, Cremonese, 1959. 11 + 273 pp. 3000 Lire. Duschek, A. Vorlesungen iiber hohere Mathematik. Vol. 2. Integra­ tion und Differentiation der Funktionen von mehreren Verander­ lichen. Lineare Algebra. Tensorfelder. Differentialgeometrie. 2d., rev. ed. Vienna, Springer, 1958. 8 + 401 pp. $10.70. Einige Arbeiten des Lehrstuhles fur Mathematik im Lehrjare 1956- 1957. Budapest, Wissenschaftliche Veroffentlichungen der Technischen Universitat fiir Bau- und Verkehrswesen in Budapest, 1958. 80 pp.

51 Estes, W. K. See Studies in mathematical learning theory. Everett, C. j. See Cashwell, E. D. Gantmacher, F. R. Matrizenrechnung. II. Spezielle F ragen und Anwendungen. (Hochschulbiicher fiir Mathematik, Bd. 37 .) Berlin, VEB Deutscher Verlag der Wissenschaften, 1959. 8 + 244 pp. 26.00 DM. Gantmacher, F. R. The theory of matrices. Vols. 1,2. New York, Chelsea, 1959. Vol. 1, 10 + 374 pp., $6.00; Vol. 2, 9 + 276 pp. $6.00. Gel'fand, I. M ., and Silov, G. E. Prostranstva osnovnyh i obob!l'1!ennyh funkcii. (Obob!rl!ennye funkcii, Vypusk 2.) Moscow, Gosudarstv. lzdat. Fiz.-Mat. Lit., 1958. 307 pp. 10.05 rubles. Gerc, G. Principy mehaniki, izlolennye v novoi svyazi. Trans. by V. F. Kotov and A. V. Sulimo-Samuilo. (Institut istorii estestvoznaniya i tehniki. Klassiki nauki.) Moscow, Izdat. Akad. Nauk SSSR, 1959. 386 pp., 1 plate. 15 rubles. Geronimus, Ya. L. Mnogol!leny, ortogonal'nye na okrulnosti ina otrezke. Ocenki, asimptotil!eskie formuly, ortogonal'nye ryady. (Sovremennye problemy matematiki.) Moscow, Gosudarstv. lzdat. Fiz.-Mat. Lit., 1958. 240 pp. 8.45 rubles. Griffith, B. A. See Synge, J. L. Haar, A. Gesammelte Arbeiten. Ed. by B. Szokefalvi-Nagy. (Part in Hungarian.) Budapest, Verlag der Akademie der Wissen­ schaften, 1959. 661 pp., 1 plate. 250Ft.; $14.00. Herzfeld, K. F., and Litovitz, T. A. Absorption and dispersion of ultrasonic waves. (Pure and Applied Physics, Vol. 7 .) New York, Academic Press. 18 + 535 pp. $14.50. Heymann, W. Trigonometrie der Ebene. Fachbuchverlag Leipzig, 1959. 282 pp. 12.50 DM. Heyting, A. See Constructivity in mathematics. Hinze, J. 0. Turbulence: An introduction to its mechanism and theory. (McGraw-Hill Series in Mechanical Engineering.) New York, McGraw-Hill, 1959. 9 + 586 pp. $15.00. Hua, L. K. Additive Primzahltheorie. Leipzig, Teubner, 1959. 6 + 174 pp. 20.50 DM. Jeans, j. An introduction to the kinetic theory of gases. New York, Cambridge University Press, 1959. 7 + 311 pp. Paperbound, $2.95. Kindler, H. See Solodownikow, W. W. Kitov, A. I., and Krinickii, N. A. Elektronnye vyl!islitel'nye ma!riny. (Nau~no-Populyarnaya Seriya.) Moscow, Izdat. Akad. Nauk SSSR, 1958. 131 pp. 1.90 rubles. von Koppenfels, W ., and Stallmann, F. Praxis der konformen Abbildung. (Die Grundlehren der mathematischen Wissen­ schaften, Bd. 100.) Berlin, Springer, 1959. 13 + 375 pp. 69.00 DM.

52 Kotov, V. F. See Gerc, G. Krasovskii, N. N. Nekotorye zadai:!i teorii ustoTi:!ivosti dvileniya. Moscow, Gosudarstv. Izdat. Fiz.-Mat. Lit., 1959. Zll pp. 8. 70 rubles. Krinickii, N. A. See Kitov, A. I. Krull, W. Elementare und klassische Algebra. II. (Sammlung Goschen, Bd. 933.) Berlin, de Gruyter, 1959. 13Z pp. 3.60 DM. Lee, T. H. See McCracken, D. D. Litovitz, T. A. See Herzfeld, K. F. Livesley, R. K. See Matheson, J. A. L. Luce, R. D. Individual choice behavior: a theoretical analysis. New York, Wiley, 1959. 1Z + 153 pp. $5.95. Lyapunov, A.M. Sobranie soi:!inenii. Vol. 3. Moscow, lzdat. Akad. Nauk SSSR, 1959. 375 pp. 16.90 rubles. McCracken, D. D., Weiss, H., and Lee, T. H. Programming business computers. New York, Wiley, 1959. 17 + 510 pp. $10.2.5. Mamedbeili, G. D. Muhammed Nasir~ddin Tusi o teorii paralle1'nyh linii i teorii otno!fenit. Baku, Izdat Akad. Nauk Azerbaidzansk. SSR, 1959. 100 pp. Z.70 rubles. MarZ!evskii, M. N. Teoriya i:!isel. Kratkii kurs. Kharkov, lzdat. Har'kovsk. Gos. Univ. im A.M. Gorkil, 1958. 144 pp. Paper­ bound, 4 rubles. Marchuk, G. I. Numerical methods for nuclear reactor calculations. Trans. from the Russian. New York, Consultants Bureau, 1959. 6 + Z95 pp. Paperbound, $60.00. Martin, R. M. Toward a systematic pragmatics. (Studies in Logic and the Foundations of Mathematics.) Amsterdam, North­ Holland, 1959. 15 + 107 pp. $3.2.5. Matheson, J. A. L. Hyperstatic structures: an introduction to the theory of statically indeterminate structures. Vol. I. With chapters by N. W. Murray and R. K. Livesley. New York, Academic Press, 1959. 15 + 474 pp., 6 plates. $15.50. Maxfield, j. E. See Selfridge, R. G. Mercier, A. Analytical and canonical formalism in physics. Amster­ dam, North-Holland, 1959. 8 + ZZZ pp. $6.75. Moritz, R. E. On mathematics and mathematicians. (Formerly titled: Memorabilia mathematica or the philomath's quotation­ book.) New York, Dover, 1958. 13 + 410 pp. Paperbound, $1.95. Mott, N. F. Elements of wave mechanics. New York, Cambridge University Press, 1958. 9 + 156 pp. Paperbound, $Z.95. Mullt.r, W. Theorie der elastischen Verformung. (Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Bd. 2.7 .) Leipzig, Geest and Portig, 1959. 11 + 3Z7 pp. 31.50 DM. Murray, N. W. See Matheson, j. A. L. Muses, C. A. See Rothstein, J.

53 Nassif, M. Products and zeros of basic sets. (University of Assiut Monograph Series, No.2.) Assiut, 1958. 12 + 142 pp. Notes for the M. I. T. special summer program on random vibra­ tion. Ed. by S. H. Crandall. Cambridge, Technology Press of Massachusetts Institute of Technology, 1958. 8 + 406 pp. Ogorodnikov, K. F. Dinamika zvezdnyh sistem. Moscow, Gosud­ arstv. Izdat. Fiz.-Mat. Lit., 1958. 627 pp., 16 plates. 17.40 rubles. Paris symposium on radio astronomy. International Astronomical Union symposium No. 9 and International Scientific Radio Union symposium No. 1, held from 30 July to 6 August, 1958. Ed. by R.N. Bracewell. Stanford University Press, 1959, 12 + 612 pp, $15,00. Parkus, H. Instationare Warmespannungen. Vienna, Springer, 1959. 5 + 166 pp. $9.05, Prager, W. An introduction to plasticity. Reading, Addison- Wesley, 1959, 8 + 148 pp. $9.50, Prigogine, I. See Advances in chemical physics. Priwalow, I. I. Einfiihrung in die Funktionentheorie. II. (Mathe­ matisch-Naturwissenschaftliche Bibliothek, 22,) Leipzig, Teubner, 1959. 5 + 194 pp. 8.00 DM. Le raisonnement en mathematiques et en sciences experimentales. Colloques Internationaux du Centre National de la Recherche Scientifique, LXX. Paris, Centre National de la Recherche Scientifique, 1958. 140 pp, 1400 fr. Redei, L. Algebra. Part 1. (Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Bd. 26, Teil 1,) Leipzig, Geest and Portig, 1959. 15 + 797 pp, 48,00 DM. Rothstein, J, Communication, organization, and science, With a foreword by C. A. Muses. Indian Hills, Falcon's Wing, 1958, 96 + 110 pp. $3.50. Sackmann, L. See Salles, F. Salles, F. Lieu des racines d'une equation algebrique dependant d'un paraml!tre. Application illa stabilite et au giudage des fusees. Preface by L. Sackmann. (Publ. Sci. Tech. Ministl!re l'Air, no. 351,) Paris, 1959. 10 + 69 pp. 1810 fr. Schwartz, L. Etude des sommes d'exponentielles. 2d ed. (Publica­ tions de l'Institut de Mathematique de l'Universite de Stras­ bourg, V. Actualites Sci, Ind., no. 959,) Paris, Hermann, 1959. 151 pp. 1800 fr. The scientific papers of Sir Geoffrey Ingram Taylor. Vol, II. Meteorology, oceanography and turbulent flow, Ed. by G. K. Batchelor. New York, Cambridge University Press, 1959. 8 + 512 pp., 3 plates. Seitz, F. See Solid state physics.

54 Selfridge, R. G., and Maxfield, j. E. A table of the incomplete elliptic integral of the third kind. New York, Dover, 1958. 14 + 805 pp. $7.50. Sl!minaire Bourbaki; lOe annl!e: 1957/1958. Textes des confl!rences; Exposl!s 152 A 168. Paris, Secrl!tariat mathl!matique, 1958. 189 pp. mimeographed. Sl!minaire Henri Cartan; 9e annl!e: 1956-1957. Quelques questions de topologie. Paris, Secrl!tariat mathl!matique, 1958. 73 pp. mimeographed. Shirkov, D. V. See Bogoliubov, N. N. Sikorski, R. Funkcje rzeczywiste. [Real functions.] Vol. 2. (Mono­ grafie Matematyczne. Tom 37 .) Warsaw, Panstwowe Wydawn­ ictwo Naukowe, 1959. 262 pp. 35.00 z1. Silov, G. E. See Gel'fand, I. M. Slovar' matematil!eskih terminov. Baku, Izdat. Akad. Nauk Azerbatdlansk. SSR, 1958. 76 pp. 1.15 man. Sokolov, A. A. Vvedenie v kvantovuyu ~lektrodinamiku. Moscow, Gosudarstv. lzdat. Fiz.-Mat. Lit., 1958. 534 pp. 18.05 rubles. Solid state physics: Advances in research and applications. Vol. 6. Ed. by F. Seitz and D. Turnbull. New York, Academic Press, 1958. 14 + 429 pp. $12.00. Solid state physics: Advances in research and applications. Vol. 9. Ed. by F. Seitz and D. Turnbull. New York, Academic Press, 1959. 15 + 548 pp. $14.50. Solodownikow, W. W. Grundlagen der selbsttatigen Regeling. Vol. 2. Einige Probleme a us der Theorie der nichtlinearen Regelungssysteme. Rev. by H. Kindler. Munich, Oldenbourg, 1959. 8 + 452 pp. 52.00 DM. Stallmann, F. See von Koppenfels, W. Stebbing, L. s. Philosophy and the physicists. New York, Dover, 1958. 17 + 295 pp. Paperbound, $1.65. Stoilow, S. Teoria functlilor de o variabila complexa. Vol. II. Func~ii armonice. Suprafese Riemanniene. [Theory of functions of a complex variable. Vol. 2. Harmonic functions. Riemann surfaces.] In collaboration with c. A. Cazacu. Editura Aca­ demiei Republicii populare Rom1'ne, 1958. 378 pp. 25.70 Lei. Struik, D. J. A matematika rovid tortl!nete. (A concise history of mathematics.] Gondolat Kiad6, 1958. 219 pp., 10 plates. 30.00 Ft. Strydom, B. C. Abstract Riemann integration. Getal en figuur, 10. Assen, Van Gorcum, 1959. 5 + 46 pp. 5.25 Hfl. Studies in mathematical learning theory. (Stanford Mathematical Studies in the Social Sciences, III.) Ed. by R. R. Bush and W. K. Estes. Stanford University Press, 1959. 8 + 432 pp. $11.50. Sulimo-Samullo, A. V. See Gerc, G.

55 Synge, J, L., and Griffith, B. A. Principles of mechanics, 3d ed. New York, McGraw-Hill, 1959, 17 + 55Z pp. $9,50. Sziikefalvi-Nagy, B. See Haar, A. Tables of the bivariate normal distribution function and related func­ tions, (National Bureau of Standards Applied Mathematics Series, No. 50,) Washington, Government Printing Office, 1959, 45 + Z58 pp, $3.Z5, Treizi~me congr~s des matht!maticiens Scandinaves, tenu l Helsinki 18-Z3 aout 1957, Helsingfors, Mercators Tryckeri, 1958, Z09 pp., 1 plate, Turnbull, D. See Solid state physics, Volkoff, G. M. See Bogoliubov, N. N, Wait, J, R. Electromagnetic radiation from cylindrical structures, New York, Pergamon, 1959, 9 +ZOO pp. $8.00. Weiss, H. See McCracken, D. D. Whitney, D. R. Elements of mathematical statistics. New York, Holt, 1959. 9 + 148 pp. $4.75. Whittaker, E. T, A treatise on the analytical dynamics of particles and rigid bodies: With an introduction to the problem of three bodies. 4th ed. New York, Cambridge University Press, 1959. 14 + 456 pp. Paperbound, $4.95. Whittaker, E. From Euclid to Eddington: A study of conceptions of the external world. New York, Dover, 1958. 9 + Z1Z pp. Paper­ bound, $1.35. Wolf, E, See Born, M. Yadernye reakcii pri malyh i srednih ~nergiyah. Proceedings of the All-Union conference, November 1957. Moscow, lzdat. Akad, Nauk SSSR, 1958. 614 pp., 1 plate, 1 insert, 3Z.60 rubles.

56 CATALOGUE OF LECTURE IIOTES Supplement No. 1

THE UNIVERSITY OF KANSAS

The following reports are available, and copies can be obtained by ordering them from the Student Union Book Store, The University of Kansas, Lawrence, Kansas. J. L. LIONS, Boundary value problems, 101 pp. $2.00 W. R. SCOTT, CALVIN V. HOLMES, and ELBERT A. WALKER, Contributions to the theory of groups, 129 pp. 2.00 ARTHUR H. KRUSE, Introduction to the theory of block assemblages and related topics in topology, viii+ 306 pp. 3.00 A. GROTHENDIECK, A general theory of fibre spaces with structure sheaf (2nd ed., 1958), 100 pp. 2.50 EBERHARD HOPF, PETER D. LAX, and M. M. SCHIFFER, Lecture series of the symposium on partial differential equations, iii + 149 pp ., (Berkeley, California, Summer 1955). 1.50 Conference on Partial Differential Equations, Summer 1954. 2.50

The following reports are available, and copies can be obtained by ordering them from N. Aronszajn, Department of Mathematics, The University of Kansas, Lawrence, Kansas. N. ARONSZAJN and W. F. DONOGHUE, On exponential representations of analytic functions in the upper half­ with positive imaginary part, 67 pp. (Published journal d'Analyse Math. vol. V, 1956-57, pp. 321-387.) no charge N. ARONSZAJN and K. T. SMITH, Functional Spaces and Functional Completion, 61 pp. (Published Annales de l'Institut Fourier, Vol. VI, 1955-56, pp. 125- 185.) no charge N. ARONSZAJN and K. T. SMITH, Theory of Bessel potentials, Part I. no charge j. SZARSKI, Characteristics and Cauchy problem for nonlinear partial differential equations of first order. no charge

57

ABSTRACTS OF COIITRIBUTED PAPERS THE NOVEMBER MEETING IN WINSTON-SALEM, NORTH CAROLINA November ZO-Zl, 1959

561-ZZ, R, D. Anderson: Multi-valued functions, inverse limits and groups.

Let ((Xi}, {fil) be an inverse system where for each i, Xi is a finite set and fi (single-valued) maps Xi+l onto Xi. Let Gibe the set of all multi-valued functions of Xi onto itself, Let G be the set of all sequences {gi) where for each i, gi E Gi and, for x E. Xi, fi(gi+l)(fil(x)) =gi(x). Set-theoretic conditions are used to define certain subsets of G which are groups under naturally defined operations. (Received October 7, 1959,)

561-Z3. Nathaniel Coburn: Discontinuity relations for charged, self­ gravitating, compressible relativistic fluids.

If characteristic manifolds are defined to be three-dimensional hyper­ surfaces along which the mechanical variables (the density, etc.) and the electro-magnetic field tensor Fjk are continuous with one-sided derivatives, then it is shown that such manifolds are contact manifolds for the current density vector. Further, the jumps of the derivatives of the skew-symmetric tensor Fjk can be expressed as ,JjApk where the non-null vector ,Jj lies along the normal to the characteristic hyper surface and A jk is a skew -symmetric tensor of rank two whose null plane (a principal leaf) lies in the local tangent hyperplane to the characteristic hypersurface at each point. The jump relations yield seven linear homogeneous equations in nine unknowns, Thus, the charac­ teristic manifolds are underdetermined. In the case of "infinite conductivity," the system is determinate and in the Newtonian case furnishes the well-known characteristic equation, Further, it is shown that weak shocks for infinite con­ ductivity propagate along characteristics. In the general case of strong shocks, the jump of Fjk has the same properties as the tensor Ajk' (Received October 5, 1959,)

561-Z4. Eckford Cohen: Unitary products of arithmetical functions.

Let n denote a positive integer, The unitary product (convolution) of two arithmetical functions f(n), g(n) is defined by h(n) = £f(d)g(6), where the sum­ mation if over d,~ such that do = n, (d,6) = 1. A generalized inversion formu-

59 la for unitary products is proved and applied to the distribution of the set sa,b• defined for integers a,b as follows. Let n have distinct prime divisors pi' n =pel .•• pet, and suppose that b 7 a> 1, (a,b) =1. The set Sa b consists of 1 t ' those integers n for which each ei (i :: 1 , .•. ,t) is divisible by either a or b.

Letting Sa b(x) denote the number of integers ~ x contained in Sa b it is proved tha~ sa,b(x) = orxl/a +fS'xi/b+ O(xl/clog x), where c =a+~ and oc,p are constants depending on a and b. Some related problems are also treated. (Received September ZB, 1959 .)

56l-Z5. R. P. Gosselin: A theorem on orthonormal series.

We say a sequence fm~ of distinct, positive integers is of upper density one if lim sup O"(n)/n = 1 where

561-Z6. C. A. Muses: Some applications of equi-infinitesimal analysis.

l'he concept of an equi-infinitesimal domain in a given manifold will be shortly announced (see The geometry of equi-infinitesimals, Abstract 563-ZZ, these Notices vol. 6 (1959) p. 764). Suffice it to say here that each of the 7 kinds of 3-space patterning is capable of its specific line of development to more complex forms with more component infinitesimal units, exact models of which are readily obtainable. Certain final results were obtained first in May, 1958, though two forms -of the simplest 1-core family had been studied since 1941. There is a powerfully effective correspondence between equi­ infinitesimals and the sequence of natural numbers consisting of equal units. This correspondence has been exploited to yield results, an important aspect of which may be summarized as the furnishing of non-arbitrary 3-dimensional models of pure numbers. The model-manipulation and analysis yield valuable data on otherwise cumbersomely or impossibly expressible number-theoretic

60 facts and properties, as well as applications to bio -morphology and crystal microstructure. Another application of the results obtained through equi­ infinitesimal analysis demonstrates the Aleph-null and c are the only two general types of trans -finite numbers, although each has an infinity of sub­ forms. The measure of magnitude of the prototype infinitesimal is taken as k where (1 + k/oo)00 = ek, k being finite, and >D. (Received October 2, 1959,)

561-27, J, R. Rice: The analog of Haar's problem for the existence and characterization of best Tchebycheff approximations.

Let F(a,x) = L:ai(lii(x) where (lii(x) are continuous in (0,1] and the ai are real parameters. The following theorem answers the principal questions of a general nature in the theory of Tchebycheff approximations: Theorem. Let f!lii(x)) form a Tchebycheff set and let f(x) be an arbitrary function continuous on (0 ,1]. Then A. f(x) possesses a best approximation, B. a necessary and sufficient condition that F(a* ,x) be a best approximation to f(x) is that maxjF(a*,x)- f(x)j alternates at least n times in (0,1), C. the best approxi­ mation to f(x) is unique, A. Haar [Math. Ann, vol, 78 (1928) pp. 43-56] posed and answered the following question: What conditions on F are necessary and sufficient for Theorem C to be valid 1 The condition he found is that filii (x)] must form a Tchebycheff set, This paper poses and answers the following three questions: (1) What conditions on F are necessary and sufficient for Theorem A to be valid 1 (2) What conditions on F are necessary and sufficient for Theorem B to be valid 1 (3) What conditions on F are necessary and suf­ ficient for both Theorems A and B to be valid 1 This paper does not tacitly assume that the ai may assume all values. (Received October 7, 1959,)

61 THE ANNUAL MEETING IN CHICAGO, ILLINOIS January 27-29, 1960

564-242. Edgar Reich and S. E. Warschawski: Canonical conformal maps onto a circular slit annulus.

Let Q be a bounded region contained in {lz I :;:.- rJ, with flz I = r) as bound­ ary component. Let F ( Q) be the class of functions f(z) regular and univalent in Q., lf(z)l;:;; 1, such that f maps every curve in a with index+ 1 with respect to z = 0 onto a curve having index+ 1 with respect to w = 0. The following is a generalization of a theorem proved by Rengel for regions of finite connecti­ vity. Theorem 1: There is a unique (within a rotation of the z and w planes) function ¢(z) sF (Q) for which infzeD-1 ¢(z) I - maxf£F (.a) infzen lf(z) 1. ¢(0) is a minimal circular slit annulus in the sense of Grotzsch. (The existence of the mapping onto a circular slit annulus is known [Griitzsch; Jurchescu]. The point of Theorem 1 is the extension of Rengel's extremal problem to regions

of arbitrary connectivity). Let R be a bounded region containing z = 0,

Qe c R n flz I >- e], and let 9le(z) be defined for Qe as ¢ was defined for 0. The following limiting property of ¢e(z), as e -+0, holds: Theorem 2: For

every sequence ~ l 0 as n--> oo, ¢8 (z)- 'f(z), a function regular and uni- n valent for lz I < 1. "f(Q) is a minimal circular slit disk. (Received Decem- ber 3, 1959.)

564-243. W. G. Leavitt: Maximal ideals.

It is known that a ring which has no maximal one-sided ideals must be a radical ring. Such rings are considered in this paper, with the following the most important results: (1) lf a ring A has a maximal right ideal I with A211\ I,

then A is not a radical ring. (2) If A is a radical ring with A2 = A, then it auto­ matically fails to have a one-sided ideal. (3) A ring has a maximal left ideal if and only if it has a maximal right ideal. It is also shown that constructions exist for the following types of rings: (1) A ring with a maximal nonmodular ideal I satisfying A2 ~I. (2) Radical rings satisfying 0 C A2 C A, both with and without maximal one-sided ideals. (3) A radical ring satisfying A2 =A. (Received December 4, 1959.)

62 564-244. G. M. Fisher: On the group of all of a mani- fold.

Let M be a connected n-manifold, compact or not, with boundary or not, triangulable or not, arbitrary n. The group GO(M) of all homeomorphisms h of M such that h = TT~ _1hi, hi the identity outside same internal closed n-cell Ci in M, is the smallest nontrivial normal subgroup of the group G(M) of all homeomorphisms of M, and it is simple. If M is closed and dim(M);::;; 3, then a0 (M) is the group of topological deformations of M, as announced in previous abstracts. For such an M, the identity component of G(M) (F rechet metric topology) is simple. This generalizes results of Ulam and Schreier

(M = s 1), Ulam and von Neumann (M = S2), and Anderson (M = s2 or S 3). (Received December 7, 1959.)

564-245. U. H. R. 0. Christian: On the factors of automorphy for the group of integral modular-substitutions of second degree.

In the paper Ueber die Multiplika.torensysteme zur Gruppe der ganzen Modulsubstitutionen n- ten Grades, Math. Ann. val. 138 (1959) pp. 363-397 the factors of automorphy were classified for the group of Integral Modular-Substi­ tutions of degree n '/- 2 and it turned out that there were only finite many classes.

In the above mentioned paper the case n = 2 is investigated. It is possible to make a complete classification also in this case, but the number of different classes is infinite. In a certain way the different classes can be described by infinite many complex parameters. (Received December 7, 1959.)

564-246. Harold Widom: Inversion of Toeplitz matrices. III

If f(6) ~ LckeikS belongs to L 00(0,21r) then Tf = (cj-k)j,k = 0,1, ... represents a bounded operator on the space of square summable sequences fx0 ,x1, .. .J. Combining a previous result of the author (Theorem I in Inversion of Toeplitz matrices II; to appear in Illinois J. Math. ) with a recent result of Helson and Szego, the following is proved: A necessary and sufficient condition that Tf have a bounded inverse is that f = uei(vtw) with u !;. S > 0, v real and !vI :;: tr/2 - e (e > 0), and w real and bounded. Here w denotes the conjugate function of w. (Received December 7, 1959 .)

63 564-247. A. H. Stone: Hereditarily compact spaces.

A space S is hereditarily compact, or a Zariski space, if every subspace of Sis compact (but not necessarily Hausdorff). For simplicity, assumeS

is T 1 . Characterizations are given; in particular, the kind of compactness con­ sidered does not matter, and it is enough to specify that the countable, or open, subsets of S are compact. To each such S is assigned an ordinal number, its ""; this can be arbitrarily large. All Zariski spaces can be "construc­ ted" by transfinite induction and a modified direct limit process. The product of finitely many Zariski spaces is a Zariski space, but a nontrivial infinite pro­

duct never is. A Zariski space is first countable if and only if it has only countably many open sets. There exists a countable Zariski space of

"dimension" 2 with .1( 1 open sets. (Received December 7, 1959.)

564-248. R. P. Hunter: Results on arcs in semigroups.

Let S be a compact connected normal semigroup with identity 1 and minimal ideal K. Let T denote the hyperspace of the upper semicontinuous decomposition of sets Lx• (By Lx we mean fy!Sy = Sx}). Let h denote the canonical mapping. S contains a compact connected subsemigroup M contain-

ing K and 1 such that h(M) is an arc. Corollaries: If each subgroup of S is totally disconnected then S contains an arc-semigroup from zero to identity. If S is in the plane then S contains a local one parameter semigroup at 1. A particular semigroup is constructed which is compact connected with zero and

identity with the property that Hn(S) is nontrivial for each n ~ 1. (Hn(S) denotes then dimensional singular homology group of S.) There exists a compact connected locally connected semigroup with identity with no local one parameter semigroup at the identity. There exists a compact connected semigroup with zero satisfying the conditionS 2 = S which separates the plane. (Received December 7, 1959.)

564-249. Eckford Cohen: A set of integers.

Let n,a,b denote integers, a,b fixed, b :>a :> 1, (a,b) = 1, and suppose

that n = p~1 ... p~r, where p 1, ... ,pr are the distinct prime divisors of n. Let Ra ,b denote the set of integers n for which each ki (i = l, ... ,r) is either divisible

by a or lies in the progression, ak + b (k ii:" 0). Letting Ra,b(x) denote the num­

ber of integers ~ x contained in Ra,b• it is shown by essentially elementary methods that Ra,b(x) = (~(b/a)/~(2b/a))xl/a + ~(a/b)/_7(2))xl/b + O(xl/(a+b)),

64 where i;(s) is the Riemann !;'-function, Refinements of this estimate are de­ duced on the basis of known results of a nonelementary nature. The problem of this paper generalizes one treated by Erdos and Szekeres in the special case a == 2, b = 3 (Acta Szeged vol. 7 (1934) pp. 95-102). (Received December 7, 1959,)

564-250, J.D. Baum: An equicontinuity condition for transformation groups.

Let T be an abelian ; a set PC T is said to be replete if it contains a translate of every compact set in T. The following is proved: If X is a compact , Tan abelian topological group, and (X,T) a transformation group, then the following statements are equivalent: (1) For each index a: of X there exists a replete semigroup P C T and an index jlof X such that (x,y) E jJ implies (xp,yp) E ((.for each p E P, (2) T is uniformly equi­ continuous. (Received December 7, 1959 .)

564-251, Ralph Abraham: The sound speeds of adiabatic fluid,

The equations of motion, mass and energy conservation of a thermally conducting, inviscid relativistic fluid are expressed using the energy-momen­ tum tensor of Pham Mau Quan (Ann, di Mat. Ser. 4 vol. 38, 1955). Assume the flow parameters are (CO ,C2), with I: a characteristic of this hyperbolic system. Then the O'Brien-Synge jump relations constrain the temperature to be C 1 across l:. The characteristic equation provides that contact surfaces must be adiabatic, and that two sound speeds exist, their difference a function of heat flow through the shock front. This predicts that a weak shock in hydrogen at one atmosphere would split at a rate of about w- 4 millimeters per degree centigrade, As the conductivity tends to zero, the adiabatic case is not obtained, because the O'Brien-Synge conditions no longer constrain the tempera­ ture gradient. Replacing the temperature C 1 condition by entropy C 1, the modified characteristic equation provides a contact surface and the classical adiabatic sound wave. (Received December 14, 1959,)

564-252. Ralph Abraham: The sound speeds of a charged fluid.

The equations of motion, mass and energy conservation for a relativistic, electrically charged, inviscid, adiabatic fluid of finite conductivity and variable dielectric strength and magnetic permeability are expressed using the energy-

65 momentum tensor of Pham Mau Quan (J. Math. Mech. vol. 5(3) (1956)). Assume the flow parameters are (C O,c 2), and !:is a characteristic hypersurface of the hyperbolic system above. Then Ohm's law constrains the electrical current to be continuous, and all Maxwellian terms in the characteristic equation are pre­ ceded by a factor (p - 1), where pis the index of refraction. Hence in the non­ inductive case, p = 1, the Maxwell-mechanical coupling vanishes, and the characteristic equation is that of the neutral adiabatic fluid. If p 'f 1, the characteristic equation provides that contact surfaces must be perpendicular to Poynting's vector, and that two sound speeds exist, their difference a func­ tion of the flow of electro-magnetic energy through the shock front. Waves of Alven type do not occur, and as the conductivity does not appear in the system, the well known results for the perfectly-conducting fluid to not appear as a special case of this model. (Received December 14, 1959.)

564-253. Glen Baxter: Polynomials defined by a difference system. I.

int d -int ( d The system un- un-l "'a:'ne vn an vn- vn_ 1 ""J1ne un uo an vo arbitrary), where f

= eit(-71"" !i t :=.; 11"). A converse theorem is stated in the following abstract. (Received Deaember 11, 1959.)

564-254. Glen Baxter: Polynomials defined by a difference system. II

Let fi, be the class of functions g = [:Cjeijt with L:ICjl...::. oo. Let f(t) =LAjeiit E./{, and let Dn = det(Aj-i) (i,j = O,l, ... ,n). It is known that there exists unique (up to the sign) polynomials !ll'n and s n (n = 0,1, ..• ) of degree n in eit and e -it, respectively, with equal leading coefficients, such that

66 1 /27r I' r f(t)dt = 8 if .l_rr ,P n m~ nm and only if D n I 0 for all n. Let k~ = AoDn_ 1 /Dn . If Dn f 0 for all n, there exist sequences {an} and £6'nl such that un = eintkn ;n

and v = e-intk: rPn satisfy the system in the abstract above. n n Furthermore, k~ = k~ (see above). Theorem: ~ f(t) = Lj~ -kA fijt and if log f(t) is in .J&, then L::l Cl'n I ..::: oo ~ L I.P'n I .., oo. Moreover, f+ .f 0 (see above) for all 2 lz I ::;;. 1, "f -1 0 for all Iz I ~ 1, and f(t) = k 00 /'f +"'f'-. The theorem just stated for polynomials f(t) could be proved generally if we could prove the Conjecture: Let f(t) E./l and Dn i 0. Let En (Fnl be the cofactor of A_n(An) in Dn. Then, (i) log f(t) E: A is equivalent to (ii) LIEn/Dn I and L:IF n/Dnl converge. Heuristically speaking, the above approach to orthogonal polynomials allows a very simple extension to the "continuous" case of a family of orthogonal functions as opposed to a sequence of orthogonal polynomials. We simply replace the difference system by the corresponding differential system du/dx

= a.(x)eitxv and dv/dx =;6'(x)e-itxu. (Received December ll, 1959.)

564-255. V. E. Bene!!': Probabilistic and Tauberian methods for stability analysis of certain integra-differential equations.

The vector equations ,P = c.J + /o00k(x) fq> - ,P} (t - x)dx + h, where '~'i (t) = L:~aijrPj(t- Tij), r i1: 0, c.J i:O, k(•) is a response function, h(·) is a disturb­ ance vector, and (aij) is a stochastic matrix, is considered. Wide conditions

are found under which the components of ~ all converge to a linear combination <;-·A·t.J., and the differences ,P.- ,P. converge. The methods used include -.£....1 1 1 1 J (i) finite ergodic Markov chains, (ii) renewal theory, and (iii) generalizations of Wiener's theorem on the quotients of Fourier transforms. (Received December 9, 1959.)

564-256. A. T. Bharucha-Reid: On random resolvent operators.

Let U be a bounded closed subset in finite dimensional Euclidean space

RK, and let ( Q ,.If. ,p) be a probability space. We consider the random operator

E!quation (T(c.J)- AI)x = y, where T(c.J) is a random endomorphism on a Banach

space ~ of generalized random variables (g.r .v .) , and x(u,t.J) and y(u,t.J) are

g .r .v. with values in~ . In this note we establish the existence and measura­ bility of the random resolvent operator associated with T(c.J). In addition, we show that the random resolvent operator admits a Neumann-type series repre­ sentation, the series converging weakly almost surely in the Banach algebra of random operators. (Received December 14, 1959 .)

67 564-Z57. R. H. Bing: A-)-cell is the only object whose cartesian product with an arc is a 4-cell.

Suppose the cartesian product of a topological object X and an interval [0,1] is ann-cell. It is shown that X is topologically equivalent to a subset M of Euclidean (n - 1)-space En-1 such that Int M is connected, contractable, and ULCn for each integer n. Furthermore, Bd M is connected and locally contractable. Poinaru has given an example for the case n = 5 of a set X for which Bd M is not simply connected. However, if n = 4 it follows that Bd M is separated by each simple closed curve in it but by no pair of its points. Then Bd M is a Z-sphere in E 3 • Since Int M is ULC 1oM is a 3-cell. Similarly, it follows that if n < 4, M is an (n - 1)-cell. (Received December 11, 1959.)

564-Z58. Hsin Chu: A characterization of integer group and real group.

In this paper, the following theorem is proved: "A locally compact group G, which contains a free cyclic group C and a compact subset K such that G - C • K, but contains no compact subgroup other than identity, is nothing but (a) real group, the additive group of all real numbers with the usual topology or (b) integer group, the additive group of all integers with the discrete topology." (Received December 14, 1959.)

564-Z59. Hsin Chu: Some structure theorems in topological dynamics.

In this paper, the enveloping semigroup E of a transformation group (X,T), where X is a compact Hausdorff space, is studied. The conception of relative enveloping semigroup E is introduced. The main results as follows: (1) Let E be abelian. Let a EX. Then the following statements are equivalent (i) (X,T) is a minimal set and (ii) there exists a unique group structure of X which makes

X a compact topological group such that ,J a= T ~X where (t) pia = at for t € T is a continuous homomorphism such that (T) rlla .. X. (Z) Let (X,T) be pointwise almost periodic with T abelian. Then the following statements are equivalent: (i) E is abelian and (ii) the relative enveloping semigroup on each orbit is a group of homeomorphisms. (Received December 14, 19.59.)

564-Z60. Trevor Evans: Embedding incomplete latin squares.

An incomplete latin square of order n based on 1,Z , ••• ,n is a square array of n rows and n columns such that a subset of the nz places are occupied by integers from l,Z, ••• ,n and such that no integer occurs more than once in the

68 same row or column, It is proved that for n t;: 4, an incomplete latin square of

order n can be embedded in a latin square of order t for any t lO Zn, Further­

more, for n i!: 4, there is an incomplete latin square of order n which cannot be embedded in a latin square of order less than Zn - 1, Some consequences of this are (i) an incomplete loop containing n elements can be embedded in a loop of order Zn, (ii) an infinite loop defined by a finite set of generators and relations has a finite homomorphic image of order t for every t greater than or equal to k, where k is a positive integer depending on the generators and relations of the loop, (Received December 14, 1959,)

564-Z61, E. R. Fadell: The equivalence of fiber spaces and bundles.

Theorem. Every Hurewicz fiber space (i.e. the universal covering homo­ topy theorem holds) over a base space which is dominated by a locally finite polyhedron is fiber homotopy equivalent to a Steenrod fiber bundle. The proof makes use of Milnor's universal bundle construction fAnn. of Math, vol, 63 (1956) pp, Z7Z-Z84J and the following extension of a theorem of A. Dold [Math. Z. vol. 6Z (1955) pp. 111-136}. namely that if E and E' are Hurewicz fiber spaces over a common base space which is dominated by a CW-complex and if h: E ___, E' is a fiber-preserving map which when restricted to some fiber is a homotopy equivalence, then his a fiber homotopy equivalence. One application of the above theorem is that every space X which is dominated by a locally finite polyhedron admits for every integer n 5:;. 1, an n-connective Steenrod fiber bundle over it, (Received December 10, 1959,)

564-Z6Z, G. W. Goes: Complementary spaces of Fourier coefficients.

If E is a sequence space of Fourier coefficients aj,bj, we write f = (aj,bj) - L:t~ 1 (a los jt + bjsin jt) E E, and if F is another sequence space of Fourier coefficients, then the F-complementary space of E (denoted byE- F)) is t_he space of elements g - (cj,dj) for which for every f = (aj,bj) E E (1) h = (Aj,Bj) = (f(t), g(x- ~)) •l::J~ 1 { sin Jx} is a element ofF. If E and Fare Banach spaces, then (E ---~>F) is a Banach space with the norm (forgE (E -F)) lliii(E~F)- supllfiiE 111(1(t), g(x- ~>).-.lip and (E -F) has many of the properties of the spaces defined by K&the and Toeplitz [J, Reine Angew, Math. vol. 171 (1934) pp. 193-ZZ6J and of the spaces investigated recently by Chillingworth and Matthews [Neder. Akad. Wetensch. Proc, Ser, A vol. 61 (1958) pp. 307-315, Z98 - 306 and vol, 6Z (1959) pp, 1-10,

69 45-51]. (E -F) includes all these spaces as special cases. (Received December 14, 1959.)

564-Z63. R. P. Gosselin: On the interpolation of LP functions by Jackson polynomials.

Let f belong to IJ> , p ;;;.. 1. Let J n, u(x;f) be the Jackson trigonometric polynomial of order n assuming the values f(u + Z'll"i/(n + 1)) at the points u + Z'll"i/(n + 1), i ""O,l, ... ,n. Theorem (i) For almost every (x,u), lim supiJn,u(x;f)IP/n < oo. (ii) Let 4.1(n) be any increasing function on the positive integers such that e.J(n) - -o(n). There exists a function fin LP such that for almost every (x,u), lim suplJn. u

564-Z64. Johannes de Groot: On the group of a compact Hausdorff space.

What can be said in general about the homeomorphism group H(A) of a compact Hausdorff space A? If the dimension of A is greater than zero, nothing can be said, since every group is the H(A) of some connected A (Math. Ann. vol. 138 (1959) pp. 80-lOZ). However, if dim A=- 0 and A is dense in itself, then either (i) H(A) equals the unit element, or (ii) H(A) contains a (sub) group S which is the restricted direct sum of an uncountable number of cyclic groups of order two. Moreover, this result is best possible (assuming the continuum hypothesis), since there exist such A with either H(A) = 1, or H(A) = S. The case where A is not dense in itself, can also be treated. Similar methods ad­ mit determination of the structure of the center of H(A). Translation into the language of Boolean algebras leads to corresponding statements about their automorphism groups. (Received December 14, 1959.)

564-Z65. H. J. Keisler: Isomorphism of ultraproducts.

For notation see Abstracts 550-7, 550-1, 550-30, these Notices vol. 6 (1959) pp. 673, 671, 776. Let

70 not countably complete. For a given similarity class, let H = {cfl {f( :l!' 1(1L {)($;#'means O{,f.#'are isomorphic, and tJ( =,#means (){,;&are elementarily

equiva~nt, Theorem: Let DE P(I), K E: ET, and

EC' =:- {K n HjK E EC}and EC~ "'tK nHIKE ECc.}· Theorem: Let M be closed under isomorphism and M b H. ME EC' iff M and H - M are closed under countably indexed ultraproducts; ME" EC,d iff M is closed under countably indexed ultraproducts and H - M is closed under countably indexed ultrapowers. (Received December 14, 1959,)

564-266. H. J, Keisler: Properties preserved under reduced products,

For notation see the preceding abstract. Compare with abstract 559-139, Theorem 1: SupposeD is an ideal in 21 which is not countably complete, {)t_ =

0( e L for every Horn class L ;;;;? K' and I = Ito. Theo. ct /D s

71 564-Z67. L, F. McAuley: On uppersemicontinuous decompositions of E 3 into E 3 .

The following theorem is proved, Suppose that G is an u.s.c, collection of straight line intervals and points filling up E 3 and that H is the collection of all nondegenerate elements of G. Furthermore, H=~Hi•·l where Hi* is closed relative to H*, and for each i, there exist two planes Pi and Qi such that (1) each element of Hi intersects each of Pi and Qi in exactly one point and (Z) if K is an open covering of Hi where the elements of K are logical sums of elements of G, then there exists a locally finite refinement Ki of K such that (a) each member of Ki is a straight sided cylinder (also a topological 3-cell), (b) if two elements of Ki intersect, then either their intersection belongs to Ki or their intersection lies on the Qi - side of Pi' and (c) each element of Hi lies in the interior of some element of Ki, Then the decomposition space is homeo­ morphic to E3, (Received December 14, 1959.)

564-Z68, G. H. Meisters: A general form of functional dependence, Preliminary report.

Let X and Y denote two nondegenerate sets. For each x in X define the evaluation function ex: yX -Y by ex(f) - f(x) for each fin yX, IfF C yX, define eF: X -yF by eF(x) ~ exiF for each x in X, For gin yF and F C. yX, g is said to be functionally dependent on F (denoted by g .AF) iff there exists a function h: yF- Y such that g "'hoeF. Theorem 1: Iff E F then f A F. Theorem Z: If kAF, and f .AG for each fin F, then kAG. F distinguishes

points in X iff for each pair of distinct points x 1 and Xz there exists a function fin F such that f(~) .f f(xz>· Theorem 3: F distinguishes points in X iff each g in yX is functionally dependent on F. For each F C yX, DF is defined to be

fCx 1 ,xz) I for some f in F, f(x1) F f(xz)}. DF is called the set of all pairs of points distinguished by F. Some obvious (but useful) properties of DF are:

F c G 9-DF c: Do, Dutt:.F~- Ua:DFoc.• DF na c: DF n Da, CDF c D CF• and CDF is an equivalence relation. Theorem 4: g A. F iff Dfg} C DF. If either Y, or X and Y, are topological spaces, then the choice of the function h is to be restricted, The above theorems correspond to the selection of the trivial (indiscrete) topology for Y. (Received December 14, 1959 .)

7Z 564-Z69. S. T. C. Moy: Asymptotic properties of derivatives of stationary measures.

Let X be a nonempty set andAbe a !>-algebra of subsets of X. We con­ sider the infinite product space 0. = Tii~ l 'S_ with Xi -= X, i = 1,Z , ••• , and the

infinite product IT-algebra of subsets of 0 .!F ~ TT'f;, 1 .Mf with~ = ;;/,

i = 1,Z, ..•• Let .:1n consist of all n-dimensional cylinder sets of :fl. Let Jl• l) be two probability measures defined on :P and Pn• '])n be contractions of p, "on .:1n respectively. Let '))n be absolutely continuous with respect to Pn and fn = d 'J)n/dpn. An iteration formula of fn is obtained for Markovian p and ')). If, in addition, )) is stationary and f has stationary transition probabilities, it

is proved that (1/n) log fn converges with :I)-probability one and also in L 1 (v) as n - oo under some integrability conditions. If :Vis ergodic, the limit is equal to a constant with ]}-probability one. If X is countable and Y is stationary only the same conclusion holds true under some integrability conditions. (Received December 11, 1959.)

564-Z70. c. A. Muses: A new parameter relating hyperbolic and circular functions.

Where N may assume any positive real value in the open interval (0 ,oo), it can be shown that exp flli/Z = (N + i)/(NZ + 1) 1/Z; cos ¢ = tanh9; exp !IIi = (sinh 9 + i)/cosh 9; exp 9 =(cos !II + 1)/sinfll; cosh 9 = (NZ + 1)/ZN; cos !II - (NZ - 1)/(NZ + 1); cot !II - (NZ - 1)/ZN, whence cot(fii/Z) .., N or !II = Z arc cot N. Above, 9 is defined as 1n N. It can also be shown that N =(a+ b)/c, where a, b and c are respectively the hypotenuse, base and altitude of a right triangle wherein b assumes negative values if the base extends left from the altitude (positive, if extending to right). Thus while N varies from 0 to oo, 9 varies from -oo to +oo; and !II, from 1/""to 0. Geometrically, !II is the angle included be­ tween band a for positive b (i.e. N > 1), and the supplement of that angle for negative b (i.e. N < 1). The sign of e follows that of b of course. Projectively !II can be considered as the angle by which an original triangle (of which the given triangle is the projection) has been rotated away from (or toward) the observer on c as axis, with b parallel to the horizontal plane. In some ways !II is more fundamental and less arbitrary than the Gudermannian. If N be an even integer or 0, a =NZ + 1; b =NZ- 1; c =ZN. While if N be odd or non­ integral, we take a - (NZ + 1)/Z; b - (NZ - 1)/Z; and c - N. In the former case, a+ b- ZNZ 2 cZ;z; in the latter, a+ b - NZ = cZ. (Received December 10, 1959.) 73 564-Z71. Anil Nerode: Extensions to Isols. II.

For notation see preceding abstract. Partial functions fin G extend to partial functions fA inA • Call such f eventually recursive combinatorial (ere)

if for some n, f(x 1 + n, xz + n, ••• ) is recursive combinatorial (rc); then f ere 4=* X k( .A- G ) ~domain f.A• Call f: xk ~ - C: almost rc (arc) if specifying values in e for 0 ~,I< k arguments always yields an ere; then f arc ..,f.a.: xk a"""* Q #fA: xk A-.A. The arc are closed under compo­ sition (which commutes with extension) and identities between arc extend from e to 0. If f,g are arc, then fJI.;;: gA. for all X E xk /l+o+ g - f arc. Let u be a k-variable quantifier-free conjunctive normal form formula with function and relation symbols respectively denoting extensions to Q of arc and recursive relations; then all sequences from X k( A - e) satisfy u ... all but one unne­ gated atomic formula can be struck from each conjunct to obtain a Horn U' with (x E xk c; ]x doesn't satisfy U'] not tu. For equations U between arc, [x E xk AJx satisfies UJ = [x € xkE; Jx satisfies U]Jl.; and all extensions to A of recursive relations so arise. (Received December 10, 1959 .)

564-Z7Z, E, C. Posner: Integral closure of linear differential equations,

Let E be a region in the complex plane and R the ring (sic) of functions analytic throughout E and satisfying a linear homogeneous differential equation with polynomial coefficients, Then R is integrally closed in its quotient field, The proof uses differential automorphisms, wronskians, a theorem that certain differentiably simple rings are integrally closed, and the following theorem which may be of independent interest. Let the differential equations for f,g have as many linearly independent solutions as their order, analytic through­ out E. Then the same is true for the equations for f + g and fg. (Received December 14, 1959,)

564-Z73, E. C, Posner: Accumulability and preservation of sums,

Let A (ain>• i,n = l,Z, ... , be a matrix, A sequence fynjis called A-accumulable if the set of LnainYn which are defined has exactly one finite point of accumulation, Theorem: Let A accumulate every convergent sequence, Then a number of rows can be stricken from A such that the new matrix sums every convergent sequence, The same result holds if A is a Toeplitz matrix and convergent sequences are replaced by (c,l)-summable sequences, Theorem: Let A be a Toeplitz matrix and fa function such that{fJ is

74 A-accumulable wheneverfynf is convergent. Then f is continuous. The proofs are straightforward. (Received December 14, 1959.)

564-274. Irma Reiner: Involutory matrices modulo a prime power.

For p an odd prime, all matrix solutions of order m of x 2 = I (mod pa) are determined. It is known that when a = 1, the solutions are exactly the matrices similar modulo p to Jt = diag (It,- Im-t>• with 0 :;::it;:; m. Let a matrix Q similar modulo p to Jt be fixed. Then an induction argument and the use of similarity transformations show that there are exactly p 2t(m -t)(a-1) solutions X of x2 ;;;: I (mod pa), for which X = Q (mod p). These solutions can be obtained recursively by the induction argument. They are also found explic­ itly by use of the binomial theorem. (Received December 11, 1959.)

564-275. J. B. Robinson: The undecidability of exponential Diophantine equations.

The recent work of Martin Davis and Hilary Putnam (see Abstract 560-10 Notices Amer. Math. Soc. vol. 6 (1959) p. 544 and their Air Force Scientific Research Report, October, 1959) has been modified and extended to show that every recursively enumerable relation can be existentially defined in terms of exponentiation. Hence there is no general method of deciding the solvability of exponential Diophantine equations. (Received December 8, 1959.)

564-276. Alex Rosenberg and Daniel Zelinsky: Duality in modulP.s.

If P and Q are R-modules (Ran arbitrary ring with identity) with respec­ tive endomorphism rings U and V, we study HomR (P ,Q) = S as a U -module and as a V-module. Theorem 1: If Pis projective and Tis a finitely generated U-submodule of S then T HomR(P,T(P)}. Theorem 2: Let Q be injective and contain every simple homomorph of P. Let T = Hom R(P ,B) be a U -submodule of S with B = T(P) a R-submodule of Q. Then, if (AnnQAnnyB)/B has finite length, AnnsAnnyT = T. Theorem 3: Let P be R-projective, Q = S(P) and have the property that every proper R-submodule is contained in a maximal one, and letS be a finitely-generated U-module. Then U-rad S = HomR(P,radQ). These theorems all admit duals with the roles of projectives and injectives interchanged and images being exchanged with annihilators. These results are related to those of Curtis [Illinois J. Math. vol. 3 (1959) pp. 134-144],

Dieudonne [Illinois J. Math. vol. 2 (1958) pp. 346-354} and Morita [Sci. Rep.

75 Tokyo Kyoiku Daigaku Sect. A vol. 6 (1958) pp. 83-14Z]. (Received December 14, 1959.)

564-Z77. W. Slowikowski: A generalization of the Banach closed graph theorem.

Definitions: ~ is the class of X;l·l, where X is a linear space and 1·1 is a complete pseudonorm over X (homogeneity not assumed). X 1 .1·11 ~ Xz,l·l z if Xz is a linear subset of x 1, (xn) C Xz, limnlxnlz = 0 implies limnlxnll,.. 0. (:/-) = £He E ~0 : 1'"1, F"z E Re implies F"l' Fz ~ F3 for some F3 E ttj-. ~ 1 IS: Nt'z if f 1 E h't: implies F1 e Fz for some Fz E 1fz. ( 1-o> is the class of all one-element subsets of -1o. For 1(. E <1>, 1fn C Ht'n+l•n = l,Z, ••• , ':t) is the U n nhfn E ( Y inductive limit of ('hfon ). n For Pf. ~ (Xt ,n;!•It ,n) E ( ,)!), with ~n+l c "fnn ~ l,Z, ••. , · (r) :::>

564-Z78. Konrad Suprunowicz: N -universally complete sets of sentences of formalized theories. Preliminary report.

The notation and terminology is as in A. Tarski, Contributions to the theory of models I, Indag. Math., 1954. A consistent set~ of sentences of a formalized theory T(R) is N -universally complete iff for any universal sen­ tence, ,J, of T(R) with at most N quantifiers, either ,J follows from :& or there is m i: 0 such that from I: it follows that ,J is equivalent to a sentence express­ ing that there are at most m elements. (H. Ribeiro). N -universally complete sets of sentences are characterized in terms of condition on corresponding classes K of relational systems and examples are discussed. Set E is N -universally complete if and only if for any set AN of not more than N ele­ ments of any model ct E K the restriction of ()(. to AN and the set of distin­ guished elements of d( is isomorphically embeddable in any model # E K if and only if card.;& > m, for some m. Some examples: The usual set of sen­ tences characterizing lattices which are not chains is Z -universally complete, but not 3-complete. There are sentences characterizing special classes of

76 lattices (and other systems) which are N-universally complete for N ~ 3. (Received December 14, 1959.)

564-Z79. Dov Tamari: "Near-groups" as generalized normal multiplica­ tion tables (g.n.m .t.).

In recent work (not yet published) A. Ginzburg constructed certain IP -loops having an arbitrary normal subgroup of index Z as homomorphic images of square sets which are effectively nonassociative. Such constructions can be described generally as follows: Let G ={l, .•. ,f,g, •.•J be a group, N the set of its elements, B' = B/E .. (N X N)/E(cc G) its full n.m.t. [Tamari, C.R. vol. ZZ9 (1949) p. 1Z91 and vol. Z3Z (1951) p. 133Z], t any inversion (permutation

of order:; Z) of G; augment N by one new element, say x, toN 1 augmenting B' to BJ. by a new row fxg} and corresponding column fxtg} so that multiplication

is extended by xtXg = f(tg) -l, implying fxg - ~g' xtg .. xt(g-ltf). Theorem:

If tl - 1, then Bi is associative if and only if tis an anti-au~omorphism. Corollary: If t .., I (identity mapping), then BJ. is associative if and only if G is commutative. Near-groups with index G = 3 are constructed as homomorphic images of N* X N*, N* = f xgJ Up }Ufy gi , containing Z copies of B', with p a permutation of G and multiplication extended by xtXg =- fg -l, YpfXpg ... C 1g,

~xpg'"' Y r(f,g)' YpfYg = ~(g,f), r(f,g) = g- 1p((p-1r 1)cl), r 1 - r(l, 1). More generally one can construct, similar to the group extension method of Schreier, to given g.n.m.t. of a near-group Q and a full n.m.t. of G a near-group Q* with Q*/G O

564-ZBO. Daniel Waterman and S. A. Husain: Uniform summahility of orthogonal expansions.

Let 11.1\.,}lllg:>_,..}l be an infinite matrix with real elements. L:fa,.!ll,.(x), an o.n. expansion off on [0,1] is said to he uniformly A summable if L:~=O ~v,Jl:J>fll,.(x) converges uniformly for each p to some Ap,Jf) and {hp.,x(f)} converges uniformly asp -ro. Let Kn,u (x,t) =- L:~.o.li.,pfllv(x)~(t). Assuming that the finite linear combinations of the !II,. are dense in C and that 1!11.-(x) I ..:.A,. for all x, we give NASC that A be a uniform summability method for functions of Lp, p !: 1. Clearly Ap,x(f) :aJo1f(t)K)l (x,t)dt, where KP' as a function oft, is hounded and measurable iff E" L, or is in Lq iff e IP (p > 1). The conditions are: (1) lim)J-+IXlAJ)Jl exists for all Y, (Z) f 6 IP (p > 1), .fo1 1Kn,p(x,t)f~t < Mp for all n,p, and x; f E L, ess.sup. 0 #t~ 1 1Kn,p.

77 for all n, p, and x, (3) f E" LP(p ;;;.- 1),,/?/IKp(x,t) lqdt < M for all p and x; f e: L, ess. sup.Ollfts:liliK)l(x,t) I< M for all )1 and x. Results of the same nature can be given for the bounded measurable functions without the condition on the closure of f!l} in C. The summability methods considered here do not, in general, sum continuous functions to their values. (Received December 16, 1959.)

564-ZBl. J. M. Worrell, Jr.: Concerning a question about Vitali coverings.

In Euclidean space of one dimension the point set K is said to have property V if and only if it is true that if G is a set of segments covering K in the Vitali sense then, for every positive number e, K is covered by some subcollection Ge of G such that the sum of the lengths of the segments of Ge is less than e. In 19Z4 (Fund. Math., vol. 5, pp. 3Z8-330) J. Splawa-Neyman showed that every closed and bounded point set of measure 0 has property V and indicated that Sierpinski had raised the question whether this proposition remains true if the stipulation that K be closed is omitted. In November, 19Z4 (Proc. Nat. Acad., vol. 10, no. 11, pp. 464-467) R. L. Moore answered the question in the negative. Noting that Splawa-Neyman's result remains true if the stipulation that K be closed is replaced by the stipulation that K be the sum of countably many closed point sets, Moore raised the question whether every bounded point set of measure 0 which has property V is the sum of countably ;nany closed point sets. In the present paper the author shows that if the hypothesis of the continuum is true then the answer to this question is in the negative. (Received December 9, 1959,)

564-ZBZ. G. S. Young: A fixed-point theorem for arcwise connected continua.

It is proved that if an arcwise connected metric continuum does not have the fixed-point property, then it contains a continuum of one of two types. The first is a continuum M for which there is a continuous map f of M onto S' such that only one set rl(x) is nondegenerate, and such that a nondegenerate point-inverse has no interior points, The second is a continuum containing the continuous one-to-one image of a ray, the image being dense in the continuum. A corollary is the theorem of Borsuk (Bull. Acad. Polan. Sci, Cl. III vol, Z {1954) pp. 17 -ZO) that every arcwise connected hereditarily unicoherent curve

78 has the fixed -point property. There are arcwise connected continua of arbi­ trarily large dimension that contain no subcontinuum of either type. The proof consists in showing that the hypotheses of an earlier fixed-point theorem of the author's (Amer. j. Math. vol. 68 (1946) pp. 479-494) are satisfied. (Received December 14, 1959.)

79 THE FEBRUARY MEETING IN TUSCON, ARIZONA February 19-ZO, 1960

565-1. M.A. Dengler: Mathematical analysis of the expansion process within a turbine.

Steam expansion within a turbine follows the state line which defines the enthalpy or total heat h(T ,p) as a function of the entropy s(T ,p), for given throttle flow. The expansion characteristics of a turbine are correlated to various thermodynamic effects and can be predicted only from performance curves derived from experiments. The key to the analytical formulation of turbine state lines is the observation that individual lines n connecting end­ points An and Bn may be conceived as one specific curve placed in different positions on the Mollier chart (h,s - diagram). Individual state lines are de­ fined by the end point positions A and B and may be distinguished by an expan­ sion parameter e. The position of a state line end point may now be calculated by solving two simultaneous equations !il(s,h,e) - 0 and s - 'l((h,p) "" 0, the first one defining the turbine expansion sheet, the second one giving the entropy of a steam-liquid mixture as a function of the total heat and the pressure. The two equations may be combined into a single one of the final form A (h,p,11) • !il(ll'(h,p),h,ll) = rfuC1!1'(h,p), h,e)} + v(lt(h,p),h,e) ..., 0, where r(u) is the invariant shape of the expansion line. (Received June 15, 1959.)

565-Z. J. B. Butler and R. C. McCarty: A lower bound for the distribu­ tion of the statistic o!.

Let D~ be the statistic defined by D~ = sup -oo< x< 00 (Fn (x) - F (x)) where F (x) is a continuous distribution function and F n (x) is the corresponding empirical distribution function for a sample of n elements. Let Pn (e)

.. Pr(D~ ;§ e). The following inequality is proved: (i) Pn(e) ~ 1 - (1 - e) -3/Z •(I + e)-1/Z exp(- ZnsZ)(l + K(8)n-l/Z) where K(e) is continuous 0 < 8 < l,K(8) = 0(8-3). Using (i) it follows that: (ii) Pn(e) ?;Pn (8) = (1 - exp(- Znez)) for

0 ..::: el ..::: e c:; 1/Z, n ~ N(e 1>· This verifies in part a conjecture of Birnbaum and McCarty (Ann. Math. Stat. val. Z9 (1958) pp. 558-56Z). In particular for .I .c:: e <:. .5 the inequality (ii) holds if n > 106. The proof of (i) is obtained by Z 1/Z Z tz -1/Z approximating Pn\11) by the integral (iii) (Zne hr) exp(- Zne Vt 1 (1 - 8 + t)

80 •(l t e- tfl/Z (l t e t t)-1eJ

565-3. Olga Taussky: Commutators of unitary matrices which commute With one factor.

Theorem 197 in Speiser's Gruppentheorie (based on results of Bieberbach and Frobenius) states: Let A and B be unitary matrices and let A commute with ABA-lB- 1• If the characteristic roots of B cover less than a semi-circle then A commutes with B. What actually does happen to B if A does .!!.!!!._commute with B? The answer is: If all eigenvalues of A are different then the charac­

teristic polynomial of B contain11 a factor x r- e where r 7 l and leI - l. In the general case A and B can be transformed by a unitary similarity transfor­ mation so that A becomes A1 +Az and B becomes the product of a permutation matrix and B 1 + Bz where A 1 and B 1 commute and where Bz has a character­ istic polynomial which splits up into factors xr - e. (Received December 10, 1959 .)

565-4. W. B. Fulks and J. 0. Sather: Asymptotics I: Laplace's method for multiple integrals.

We prove the following theorem: Suppose f and g are measurable func- -hof tlons on a measurable set R in Ep and (l) 3 ho 3 e g € L 1(R), (Z) :3 a neighborhood No of the origin contained in R, (3) for each neighborhood N of the origin '3A > 0 3 f;::; A if x ¢ N, (4) f =jl~ ~fk(OV'k + ovn+"') uniformly as ,fJ- 0, where the fk's are continuous on the unit sphere Sp_ 1,f0 ;.- 0 and ).1 7 0, (5) g = pA -PL:8gk(o.Y,k + oyon+") uniformly as I' -+ 0, where the gk's are continuous on Sp-l• Then I(h) =JR e-hfg dx exists for all h ~ ho and l(h) == L:8ckh-(k+~)/JJ + O(h -(nt..l)/l') where each ck is a constant depending on fj's and gj's for j ;:; k. The main points of interest are (a) the fact that we get a complete expansion, and (b) the lack of smoothness conditions .on f and g. In particular if f and g are differentiable of class cn+Z and en respectively and iff has a second order minimum at the origin the conditions (4) and (5) are fulfilled. (Received january 4, 1960.)

81 565-5. S. B. Jackson: Cyclic order of strictly monotone arcs.

An arc of continuous strictly monotone curvature has finite cyclic order k. Each of the circles through the two endpoints and tangent to the arc at one

of them meets the arc in exactly k - 3 interior points, all intersections. If these two circles coincide in a common bitangent circle through the endpoints, the arc is called maximal of order k, since any arc having a subarc of this character has order at least k + 1. Any strictly monotone arc of cyclic order k (k > 3) has subarcs maximal of order i for i = 3 , ••• , k - 1. Let M denote the minimum number of arcs of order 3 into which a strictly monotone arc may be decomposed. This is called the dissection order of the arc. The cyclic order k and dissection order M of a strictly monotone arc satisfy the inequalities M + 2 :;; k :a 3M. Neither of these inequalities can be improved since for any positive integers k and M satisfying these relations there are constructed strictly monotone arcs of cyclic order k and dissection order M. (Received January 4, 1 960 .)

565-6. A. Zirakzadeh: On axioms of then dimensional projective space.

A k-plane is defined as the locus of all points of all the lines connecting a given point 0 to all the points of a given (k - I)-plane. A 1-plane is an ordinary line. Using this definition, and assuming the existence of k-planes for all values of k (0 < k :3 n), it is proved: "Any line in an n dimensional pro­ jective space is either incident with a given (n - I)-plane of the n-space or intersects the (n - I)-plane in a point." This proposition was accepted by some mathematicians as one of the incidence axioms of n dimensional projec­ tive geometry. In fact, as far as the author has been able to find out, all the incidence properties of the n dimensional projective space is a consequence of either this proposition or other propositions which follow from this propo­ sition, using the axioms of plane projective geometry. Hence a possible con­ jecture is: "Using the above definition of k-planes and assuming the existence of k-planes for all k (0 ""- k ;g n), the only axioms needed to study the incidence properties of the n dimensional projective space are the axioms of plane projective geometry." (Received January 4, 1960.)

82 565-7. E. S, O'Keefe: A theorem on independence of two-element primal algebras.

Let 0(. =(A, 0 1 , o2 , ... ,0t) be a universal algebra. The algebra 07_is said to satisfy the range condition if each primitive operation Oi of CJt. has A as its range. Theorem 1, If two nonisomorphic primal algebras satisfy the range condition, and one algebra has but two elements, then the algebras are strictly

independent, A consequence of this theorem and results of the author to appear in Math, Z. is Theorem 2: Every set of two-element primal algebras in which (i) no pair of algebras is isomorphic, and (ii) each algebra satisfies the range condition, is a primal cluster, [For definitions see A. L. Foster, The identities of-- and unique subdirect factorization within-- classes of universal

algebras, Math, Z. vol, 62 (1955) pp, 171-188]. (Received January 5, 1960,)

565-8, E. D. Nering: Reduction mod-p of divisor classes and linear series. Preliminary report.

The author applies methods of a previous paper [Ann. of Math. vol. 67 (1958) pp. 590-606] for reduction of an algebraic function field modulo a prime

in the constant field to the question of what happens to the divisor classes and linear series under such reductions. Deuring [Math. z. vol. 47 (1942) pp. 643- 654) has discussed these questions under the assumption that the reduction is nondegenerate. These results are extended to the finite number of degenerate cases. (Received January 6, 1960 .)

565-9. J. M. Kister: Isotopy classes in En and Sn.

Denote by .?f(M) the space of all homeomorphisms of M onto itself with the compact open topology. It is known that ~(En) and ?((sn) have precisely two components for n ~ 3. It is shown that there are subgroups ~(En) and ""(Sn), respectively, for all n, containing the piecewise linear homeomorphisms

and which have precisely two components. Using this result and uniform local-

0-connectedness of homeomorphisms in En, G{En) can be taken as closed in the uniform topology. Thus by denseness of piecewise linear homeomorphisms ../?(En) "'~(En), n :a 3, giving new proofs for the above mentioned results. (Received January 6, 1960 .)

83 565-10. Gian-Carlo Rota: A universal operator in separable Hilbert space.

LetS and T be bounded linear operators in separable Hilbert space H. (In inseparable space the following is trivial.) We say that S is a model for T when there exist (a) an invariant subspace M of S and (b) a bounded one-to-one linear operator IT in H whose range is the closed subspace M, such that T coincides with the operator IT -lSIT. This amounts to saying that the restriction of S to some one of its invariant subspace& gives an operator similar to T. Theorem. There exists in separable Hilbert space a single operator S which is a model for every operator of norm less than one. Corollary. There exists in separable Hilbert space an operator S whose lattice of closed invariant sub­ spaces contains as an ideal (in the lattice sense) a lattice isomorphic to the lattice of closed invariant subspaces of every other bounded linear operator. (Received November 1Z, 1959.)

84 THE FEBRUARY MEETING IN NEW YORK, NEW YORK February 2 7, 1960

566-1. Martin Arkowitz: The generalized Whitehead product.

Let 11'(R,S) denote the homotopy classes of maps of (R,r) into (S,s), (r E R, s E S), I: R the suspension of R, and R 'IS the quotient space R X S/r X S U R X s. Following Hilton, two definitions of the generalized

Whitehead product (GWP) are given. Each assigns to ot E.17'(~A,X) and jl E '/T'(~B,X) an element [al,ji] E '17(L:(A t1 B),X) (A,B polyhedra). In definition (1) representatives of d. and ;9 give rise to maps from E(A )(B) to X whose commutator induces a map from l!(A #B) to X. This homotopy class is [a' yeJ. Definition (2) is essentially a direct generalization of the ordinary Whitehead product. It is proved that these two definitions coincide. Using definition (1), it is then shown that the GWP vanishes in an H-space and is "annihilated by suspension". Anti-commutativity, hi-additivity, and a Jacobi identity are also established by generalizing G. W. Whitehead's argument in Comment Math. Helv. vol. 28 (1954) pp. 320-328. The proofs of all of these properties depend on basic facts about commutators and about the group '7t(Z(A X B),X) (or, for the Jacobi identity, 'lr(~(A X B X C),X)). The Jacobi identity states that

[CI(,[,P,r]], [/,[?',Cl']], and [?',[Cl'.jJ]] regarded as elements of 1/'(~(A # B # C),X) •(A #B #C "'(A X B X C)/(A X B X c U A X b XC U a X B X C)) have sum zero if the polyhedra A, B, and Care suspensions. (Received November 16, 1959 .)

566-2. W. R. Baum: Integral geometric methods in information theory. V. Stationarity and ergodicity.

The fact that the concept of source in multi-dimensional information theory depends on a transitive r-parameter Gr of transformations (cf. Abstract 557-12, Notices Amer. Math. Soc. vol. 6 (1959) pp. 179-180) en­ tails the possibility and necessity of considering the notions of stationarity and ergodicity as "relative" properties in the following sense. Let r be a (nonempty) subset of Gr, A the sample space underlying the source (cf. loc. cit.), and ,U\) the probability measure defined on a cr-algebra of subsets of A. If for each M C A and each T E f it is true that p(TM) "'p(M), then the source is said to be stationary relative to r (orr -stationary). r-invariant subsets of

85 A are characterized by the relation TM ~ M for each T E T. If, for a fixed set f, the probability of every r-invariant set M is equal either to 0 or to 1, then the source is r -ergodic. These "relative" concepts are relevant for the study of the source in particular in case that T is a subgroup of Gr, and here, especially, if r has more than one generator. [Sponsored by the Information Systems Branch of ONR (NONR-669(10)).] (Received January 5, 1960.)

566-3. B. K. Bender, A. J. Goldman and R. B. Thomas: Simplification of Boolean functions.

A digital computer technique requiring relatively little internal storage has been devised for the partial simplification of Boolean functions. Functions of as many as 14 variables have been treated on SEAC (Z048 -word internal memory, no external memory, 44-bit word length). Phases 1 and Z of the technique produce the columns of Quine's reduced table of prime implicant& (Amer. Math. Monthly vol. 59, p. 5Z8). Phase 3 selects some prime implicants for inclusion in a minimal normal disjunctive form, and eliminates other prime implicants from further consideration. A validity proof is given for the rules !lmployed in Phase 3. (Received January 4, 1960 .)

566-4. F. H. Brownell: A Tauberian conjecture and polygonal eigen­ value distributions. I

Let F (..\) be a real valued function of real )I ;: 0 which is of bounded variation over every finite interval [O,N), which is continuous at A"" 0 with F(O) = 0, and which hascJlfe-tA ldF()I)I < + oo for real t > 0. With s = t + iv, t and v real, define g(s) by the Lebesgue-Stieltjes integral g(s) .. J~e-s'~~dF(},), analytic in t ::> 0. Let F satisfy the conditions that (i) g(t) - b + O(exp(- c/t)) as t -o+ for some real constants c > 0 and b, and that (ii) F (A) + K.Ji'> be strictly increasing over )) e;; 1 for some real constants K > 0 and V 1;:; 1. Based on the work of G. Freud, Ganelius [Kung, Fys. Salls. Lund F. vol. Z4 (1954) no. ZO] and Korevaar [lndag. Math. vol. 16 (1954)pp.36-45] have shown that (i) and (ii) here imply that as~ - + oo (iii) F(,o\) = O(.IIV-1/Z). Is it true, as conjectured, that (iii) can be strengthened to (iv) F(.-\) = o (}j")'-1/Z) as 1\- + oo if in addi­ tion to (i) and (ii) it is also assumed that g(iv) = limt~o+ g(t + iv) exists finite "for all v F 0, that the resulting g(s) is continuous in t ?; 0 and s f 0, and that over all such s holds (v) jg(s)l ;aM 11si-V+l? + Mz for some finite constants M 1 and Mz and 1't > 0? (Received November 4, 1959.)

86 566-5. F. H. Brownell: A Tauberian conjecture and polygonal eigen- value distributions. II.

The example F (..\) "'J'o}, sin (x 112) x" -ldx, [Korevaar, lndag. Math. val. 16 (1954) p. 43, ex. 3.5)], for integer J) satisfies (i), (ii), and (iii) but not (iv), proving (iii) to be best possible under (i) and (ii) only. Since this example also violates (v), the conjecture is still open. The affirmative proof of this conjecture for J) = 1 would give important information, namely (vi) N(.A)

= (p2 (D)/411"),.\- (- 1)""()'(B)/47T),p12 + o(..\ 112), about the number N(.A) of eigenvalues ;§ >. of the membrane equation for a bounded region D of the plane with polygonal boundary B, ,.u 2(D) being the area of D and .,[(B) the sum of the perimeters of the polygons composing B, and 'Y = 0 or 1 denoting Dirichlet or Neumann boundary conditions respectively. For letting F (A) be the remain­ der in (vi), (ii) with "')) = 1 is automatic and (i) is known [Brownell, J. Math. Mech. val. 6 (1957) pp. 119-166 and Bull. Amer. Math. Soc. val. 63 (1957) p. 284]; also (v) appears obtainable from the same analysis. The formula

(vi) (and more) is well known for rectangles, and its truth for general poly­ gonal regions D would partially confirm a conjecture of Poly a [C. R. A cad. Sci. Paris val. 242 (1956) pp. 708-709]. (Received November 4, 1959.)

566-6. Volodymyr Bohun-Chudyniv: Qn the completely semi-associative and noncomm utative loops and loop algebras of the fifth order.

We call an algebra completely semi-associative if number ,z of non­ associative cases in 6 products of all ordered triples u"' u,.,u....-of the basal units, not belonging either to the center or to the same sub-algebras, is equal to the number jJ of the associative cases (author's papers in the Proceedings of the Fourth Canadian Mathematical Congress, Banfft, 1957, pp. 82-84). In the author's paper, On semi-associative and semi-commutative algebras, presen­ ted at International Congress of Mathematicians in Edinburgh, Scotland, 1958, completely semi-associative algebras were not considered. The aims of this paper are: I. To determine all types of completely semi-associative loops and loop algebras of the fifth order. II. To prove existence among the completely semi-associative loop algebras of fifth order, of two algebras satisfying:

(1) flexible law and the condition that A+ is Jordan algebra. (Received Decem­ ber 31, 1959.)

87 566-7. j. G. Glimm: A Stone-Weierstrass theorem for C*-algebras.

The following results are obtained: Let A and B be C *-algebras with units. Theorem 1. If A contains B and B separates the pure state space P(A) of A (the w* -closure of the pure states of A) then A ,.. B. Theorem Z. If A acts on a Hilbert space H then the w*-closure V(A) of the vector states of A is

{a(·x,x)IA + (1 - a)g: 0 ;5 a~ 1, xis a unit vector in H, g is a state of A and (1 - a)g annihilates the completely continuous operators in A}. If A acts irre­ ducibly, it is known that P(A) = V(A). Theorem 3. If A is weakly closed, then P(A) is the set of states f of A such that f restricted to the center of A is a homomorphism and f = ah + (1 - a)g where 0 ;::; a :::; 1, h and g are states of A, ah(E) = a for some abelian projection E in A and (1 - a)g annihilates the abelian projections in A. In particular if A has no type I part then P(A) is the set of states f of A such that f restricted to the center of A is a homomorphism. Theorem 4, If A is weakly closed and B is weakly dense in A then P(B) is the set of restrictions to B of elements of P(A). The proof of Theorem 1 depends essentially upon I. Kaplansky, The structure of certain operator algebras, Trans Amer, Math, Soc. vol. 70 (1951) pp. Zl9-Z55, If A is commutative then Theorem 1 is implied by the Stone-Weierstrass theorem, (Received September 14, 1959,)

566-8, R. B. Kelman: Asymptotic behaviour of a differential-functional system.

Consider the system(*) YJ.(x) = g 1(x) + gz(x) cos (u1x 1 + v 11y 1(x1) + vzxYz(x1) + w 1) and Yz(x) = g3(x) + g4(x)cos (uzxz + v 1zy1(xz) + vzzYz. + wz>· Here the ui (1= 0), vij and wi are real constants, Further x 1 = px + q and xz = px1 + q with p and q real constants such that p ~ 1 or p iii: 1 and q ;o: 0. Let X ( ~ 0) denote a constant and assume that g}l(x)(p = 1 ,Z ,3 ,4) is a function of bounded variation in (X,co] such that the total variation of g}l in [x,co] is Ox-SJl where Bp (> 0) is constant. We find that the asymptotic behaviour of (*) depends on the algebra-number theoretic properties of u~ouz, and p, For example, if ux/uz is an algebraic unit, then each solution of(*) satisfies an order relation Yk(x) = Kk +J'; gZk-l +ox- 8Z k (k = 1 ,Z) with y IJX) = Kk an arbitrary constant, If uz/u1 is not an algebraic unit, more complicated results are obtained which depend on whether or not p is a member of the algebraic closure of the extension of the rationals formed by adjoining u 1 and uz. This extends previous results of the author (these Notices vol, 5 (1958) p. 836, Abstract 553-117), (Received january 4, 1960,)

88 566-9. H. J. Landau and W. L. Miranker: On the recovery of companded signals.

In this paper we devise a method for recovering band limited signals which have been subjected to a distorting procedure called. companding. A band limited signal is a function in LZ whose Fourier transform has compact support. In practice most real signals are in this class since they have no indefinitely high frequency components. The operation of companding a signal f(t) consists of replacing it by !ll[f(t)], where !ll(x) is a given function. This operation does not in general preserve the band-limitedness of f(t). We reproduce an unpub­ lished uniqueness theorem of A. Beurling which shows that the knowledge of the transform of the companded signal on the support of the transform of the orig­ inal signal uniquely determines the original signal, provided the companding function is monotonic increasing. We then provide an existence theorem show­ ing that the recovery may actually be performed. Viz: To each function in LZ defined only on a compact set there is one and only one band limited function with band contained in this compact set, the Fourier transform of whose com­ panded version coincides with the given function. Our method is constructive and proceeds via a stable iteration scheme (Picard iterations). The conver­ gence of our method requires that the companding function have a derivative which is bounded above and below by two positive numbers. (Received Decem­ ber Z 1 , 1 9 59.)

566-10. L. J. Mordell: On a pellian equation conjecture.

It has been conjectured by Ankeny, Artin and Chowla that if (x,y)"' (u,t) is the fundamental solution of yz - pxz =- 4 where p is a prime := 1 (mod 4), then u ;;!:0 (mod p). Professor 0. Taussky-Todd has had this verified for p < ZOO,OOO. The conjecture is proved true (1) if p =: 1 (mod 4) is a regular prime, i.e. the class number of the cyclotomic field K(e Z11\IP) is not divisible by p. (Z) if p = 5 (mod 8), then u =0 (mod p) if and only if B(p-l)/4 := 0 (mod p). Here the Bernoulli number is defined by t/(et - 1) = l + t/Z + L:";:',. 1(-lf-lsntZn;(Zn)~ It is known from Kummer's work that pis regular if and only if p does not divide any of the numerators of the first (p-3)/Z Bernoulli numbers. This paper will appear in the Acta Arithmetica. (Received December 31, 1959.)

89 566-11. Anil Nerode: Extensions to isolic integers. II

For notation, see abstract above. Let U be a prenex conjunctive normal form elementary statement with function and relation symbols denoting in e * respectively grf and recursive relations, in A* their extensions. If U is Horn and true in c *, then U is true in A* provided that in c * quantifiers can be replaced by recursive Skolem functions. If U is universal, U true in A* #all but one unnegated atomic formula can be struck from each conjunct to obtain a Horn U' true in e•. The zero divisors in A* constitute A* -ce * - (0])A*' the units [ + 1' - IJA* (or the X with X 2 = I). There are uncount­ able subsets of A* algebraically (in fact in a natural sense recursively) independent over e*. Finally, there are no primes in A•. (Method of proof.

Let f,g: x2 e• ---f' e*, f(x,2y) = I a= g(x,2y + I), f(x,2y + 1) -X = g(x,2y) for x, y E ~·. Then f A,.(x,y)g.t\. *(x,y) == x is an identity in A*; while if x E A* is such that for all y € A*, (f./l. * (x,y))2 = I or (gil.* (x,y))2 = I, then x2 = 1.) (Received December 16, 1959.)

566-12. G. E. Sacks: Sub-orderings of recursive degrees.

The fundamental theorem of the paper is: for any recursive degree,!., a partially ordered set S of cardinality less than or equal to that of the first uncountable ordinal is imbeddable in the recursive degrees greater than A if and only if every member of S has at most a countable number of predecessors. The theorem follows from an extension lemma: let T be an upper semi-lattice; let A and 8 be countable subsets of T such that no member of 8 is less than or equal to any member of A and such that the join of any two members of A is a member of A; let ..6:._be a set of recursive degrees order-isomorphic to A; then there is a set of recursive degrees J!.such that the union of A andJi.is order-isomorphic to the union of A and 8 by an extension of the given isomor­ phism between A and A,. (Received December 21, 1959 .)

566-13. J. C. Scanlon: Families of solutions of a perturbation problem.

An in-the-large study is made of the periodic solutions of a system (*) x =Ax + pf(x,t, u), where A is a constant matrix and f has period 211" in t, by using topological degree and the implicit function theorem. It is shown that if the components of f are polynomials in x and if the coefficients in these polynomials and the forcing term in f are varied, however slightly, then there are just a finite number of periodic solutions of the linear system x = Ax with this

90 property: from each such solution there "branches" a unique family x(t,p) of periodic solutions of(*), where x(t,p) is continuous in p. An easily computed topological degree yields a lower bound for the number of these families. (Received January 4, 1960 .)

566-14. S.M. Shah: On the order of the difference of two meromorphic functions.

Let F(z)"' f 1 - f 2 = / 1exp(az1' + ... ) P 1/P3 -l2 exp(bzt" + ..• ) P2/P4 where Pi (i "' l ,2 ,3 ,4) are canonical products of integer order fl and

lim supr--+00T(r,Pi)/rl"= ai..:: oo. It is proved that t(F) =lim infr_. 00T(r,F)/rf' .: (1/7T){Max(jaj,jbj,la- bj)- Min (jaj,jbj,ja- bl}> provided the last expression

is positive. If a 1 = a 3 = o,rr(a2 + a 4) < jb- aj,1iaz ial then t(F),;: lai/17'"- a 2;

and if a 2 = a 4 = 0,7T(a1 + a 3) < jb- aj, 7ia1 ..:: jbj, then t(F);;:; lblhr- a 1 • If -4 ,.., L:.1ai = 0 then t(F);: (1/;r)Max(jal,lbl,la- bj). Let lim supr-.00 T(r,fi)/r = cti..::. oo. Then f =< f~ f 2 - f 1fz is of order jl type >0 provided ct1/ot2 > 1 + 2k3f[(l/log k)log(16 ek2 /(k- 1)) + 2(k + 1)/(k- 1)} for some k > 1. If 8(0,f1) +

8(oo,f1) > 1 and 3G(2 < 0. (Received December 28, 1959.)

566-15. Branko Griinbaum: Partitions of mass-distributions and convex bodies by hyperplanes.

Generalizing results established by B. H. Neumann (J. London Math. Soc. vol. 20 (1945) pp. 226-237; cf. H. G. Eggleston, ibid. vol. 28 (1953) pp. 32- 36, and D. J. Newman Abstract 548-108, these Notices vol. 5 (1958) p. 510) for the Euclidean plane, the following theorems are obtained: (i) For any mass­ distribution in En there exists a point P E: En such that each closed half-space which contains P contains at least 1/(n + 1) of the total mass. (ii) For any convex body K C En there exists a point P E K such that for each closed half­ space H with P E H the volume of K n H is at least [n/(n + l) t ( > e -l) times the volume of K. The estimates in both theorems are the best possible. The first theorem is an easy consequence of Helly's theorem on intersections of convex sets; the proof of the second relies on properties of the "Schwarz spherical symmetrization". (Received January 6, 1960 .)

91 566-16, Joachim Lambek: On the structure of rings without nilpotent ideals and their rings of quotients.

In any associative ring R, let A* denote the intersection of all prime ideals not containing A. A* is also the residual quotient of R * by A on either side, The ideals A such that (A*)*= A form a complete Boolean algebra B(R), with intersection as meet and * as complementation. If R is semi-prime (that is R* = 0) so is Q, the two-sided ring of quotients of R defined by E. T. Wong and R, E. Johnson [Canad. Math, Bull. vol, 2 (1959) pp. 167-173]. More­ over the elements of B(Q) are direct summands of Q. If P ranges over the prime ideals in B(R), one has R == (i:pP* + npP)**, the sum being direct when R is semi-prime. This is similar to a result of R. E. Johnson [Trans.

Amer. Math. Soc, vol. 76 (1954) pp. 375-388]. One deduces that Q == Q 1 ED Q2 , where Q 1 is a complete direct sum of prime rings and B(Q2) has no atoms. (Received January 6, 1960 .)

566-17. M. E. Mahowald: Barrelled spaces and the closed graph theorem.

Our interest is in the following theorem: Iff is linear map from F into E with a closed graph, then f is continuous. Ptak, Completeness a11d the closed graph theorem, Bull. Soc. Math. France vol, 86 (1958) pp. 41-74, proved the theorem under the additional hypothesis that F is a barrelled space and E is Br-complete, We prove that this property characterizes barrelled spaces, that is, Theorem: If .Q is the collection of locally convex topological vector spaces F for which the above property is valid when E is a Banach space, then F is in .Q if and only if F is barrelled, If we restrict our attention to linear maps f which take bounded sets into bounded sets, then the above theorem gives a characterization of quasi-barrelled spaces, (Received January 6, 1960,)

566-18. R. Mittra: A study of the nonuniform convergence of the solu­ tion of a doubly infinite set of equations.

The paper deals with a doubly infinite set of equations obtained in the formulation of a boundary value problem in a waveguide. The infinite matrix associated with the doubly infinite set is first truncated to a finite size of [(p + q) X (p + q)], where p and q are the number of equations from the first and second set, respectively, out of the double set of equations. It is shown that when p and q are increased indefinitely while the ratio R '"" p/q is held constant,

92 the solution tends to a limit which is a function of R. There is therefore a non- uniform convergence of the solution of the doubly infinite matrix. In the second part of the paper it is shown how the Edge Condition prescribed by the finite Energy storage criterion can be imposed to obtain a unique ratio of R and hence a unique solution of the boundary value problem. It is indicated that the method is also useful in connection with a class of boundary value problems which yield a doubly infinite set of equations and when it is not possible to derive the solu­ tion of the infinite set by analytical means. (Received January 6, 1960 .)

566-19. J. B. Nam:e, Jr.: The solution of the linearized equations of relative motion of two light bodies in a gravitational field. Preliminary report.

The equations of motion of a light body in the gravitational field of a large, spherical, homogeneous mass can be given in the form L" = - fl!,./r 3, where!. is the position vector of the light body relative to the center of the more massive body, and pis a positive constant. Let_! be the position vector of another light body relative to the first. The nonlinear system of equations satisfied by 25. can be written in the form:!." = 3 p.(£ •!,)J:/.! 5 - p;!/r3 + !.• where 1.~ I ,.., O( I.!//1.!;.1 4). (This system also describes the error generated in a mech­ anically perfect inertial guidance mechanism in an orbital vehicle). Approxi­ mate solutions of this system, for special cases, have been obtained by various authors; the properties of the solutions to the linear system obtained by neg­ lecting! are discussed by J. J. Levin (MI. T. Lincoln Laboratory, TR No. 182, June 16, 1958). In the present paper, the general solution of the linear system

(~ = 0) is derived in closed form. Furthermore, it is shown that, if !,is repre­ sented in the form.!= x 1,£ + Xz!:' + x3(t X!.'), the solutions assume a simple form in which computations are straightforward. Procedures that permit the solutions to be transformed into any desired representation, as well as a few properties of the solutions not discussed by Levin, are shown and discussed. (Received January 6, 1960 .)

566-20. B. J. Novikoff: Invariance of cones by positive definite operators.

Let P be a closed cone with interior in a real finite dimensional Euclidean space S. The dual cone P* = {x: (x,y);;: 0 ally in PJ is then also inS. A cone is said to be obtuse if P * C P, ~if P * :J P. The cone defines a partial order­ ing inS, in which the elements of P are called nonnegative. Let A be a positive definite operator in P and B the positive definite self-adjoint square root of A.

93 A nsc, that BP be obtuse is that A-lp* C P. Generalizing a construction of Aronszajn-Smith (Amer. J, Math, val, 79 no, 3 (1957)) in this elementary case, we prove the Theorem: A nsc, that a cone C be obtuse is that every vector u admit! a "major ant" u such that (i) ii ,?: ± u (ii) lliill ~ llull. In particular, if P == P*, we derive a nsc. that A-lp C P. The example of the Lorentz cone, which is self-dual but has no set of no dim(S) generators shows that this result is not directly accessible by the reproducing kernel interpretation originally employed by Aronszajn-Smith. (Received January 6, 1960,)

566-Zl. V. E. Bene!!': A stability theorem for nonlinear equations,

Let z( ·) satisfy the vector equation z - Az + u, with A a nonlinear opera­ tor on vector-valued functions of time. Of interest are conditions under which z stays close to the diagonal, on which all components of z are equal, Where dz(t) is a suitable measure of the departure of z(•) from the diagonal in (O,t), we define A ~B to mean llfAz - Bz)(t)ll - o(dz(t)) as dz(t) __,.0, for all z(•). Theorem: If A ""'B, B is linear, and a solution y( •) of y - By + u satisfies dy(t) ;::i }11' then dz(t) ;; Jl• provided that p > Jll' (d/dt - B)-l is bounded, and (roughly) }l is sufficiently small. This result is proved by a modification of Levinson's proof of the Lyapunov-Poincar6 theorem. (Received December 9, 1959 .)

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