AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 7, NUMBER 6 ISSUE NO. 49 NOVEMBER 1960

AMERICAN MATHEMATICAL SOCIETY Nottces

Edited by GORDON L. WALKER Contents MEETINGS Calendar of Meetings ••••••••••.••••••••••••.••• 660 Program of the November Meeting in Nashville, Tennessee . 661 Abstracts for Meeting on Pages 721-732 Program of the November Meeting in Pasadena, California • 666 Abstracts for Meeting on Pages 733-746 Program of the November Meeting in Evanston, Illinois . • • 672 Abstracts for Meeting on Pages 747-760 PRELIMINARY ANNOUNCEMENT OF MEETINGS •.•.••••• 677 NEWS AND COMMENT FROM THE CONFERENCE BOARD OF THE MATHEMATICAL SCIENCES ••••••••••••.••• 680 FROM THE AMS SECRETARY •...•••.••••••••.•••.• 682 NEWS ITEMS AND ANNOUNCEMENTS .•••••.•••••••••• 683 FEATURE ARTICLES The Sino-American Conference on Intellectual Cooperation .• 689 National Academy of Sciences -National Research Council .• 693 International Congress of Applied Mechanics ..••••••••. 700 PERSONAL ITEMS ••.•.••.•••••••••••••••••••••• 702 LETTERS TO THE EDITOR.. . . . • • • • • • • . • . . . • . 710 MEMORANDA TO MEMBERS The Employment Register . • . . • • . . • • • • • . • • • • 712 Employment of Retired Mathematicians ••••••.••••..• 712 Abstracts of Papers by Title • • . • . • • • . • . • • . . • . • • . • 713 Reciprocity Agreement with the Societe Mathematique de Belgique .••.•••.••.••.•.•••••..•••••.•. 713 NEW PUBLICATIONS ..•..•••••••••.•••.•••••••••• 714 ABSTRACTS OF CONTRIBUTED PAPERS •.•••••••.••..• 715 RESERVATION FORM .•...•.•.•••••.••••..•••...• 767 MEETINGS

CALENDAR OF MEETINGS Note: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press. The summer and annual meetings are joint meetings of the Mathematical Asso­ ciation of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned.

Meet­ Deadline ing Date Place for No. Abstracts*

576 January 24-27, 1961 Washington, D. C. Dec. 9 (67 Annual Meeting) 577 February 25, 1961 New York, New York Jan. 12 578 April 6-8, 1961 N'ew York, New York Feb. 22 579 April 14-15, 1961 Chicago, Illinois Feb, 22 580 April 22, 1961 Stanford, California Feb. 22 581 June 14-16, 1961 Seattle, Washington May 1 582 August, 1961 Stillwater, Oklahoma (66th Summer Meeting) November 17-18, 1961 Milwaukee, Wisconsin January, 1962 Cincinnati, Ohio (68th Anual Meeting) August, 1962 Vancouver, British Columbia (67th Summer Meeting) January, 1963 Berkeley, California (69th Annual Meeting) August, 1963 Boulder, Colorado (68th Summer Meeting) *The abstracts of papers to be presented in person at the meetings must be re­ ceived in the Headquarters Offices of the Society in Providence, Rhode Island on or before these deadlines. The deadlines also apply to news items. The next two deadline dates for by title abstracts are December 2 and January 3.

The NOTICES of the American Mathematical Society is published by the Society seven times a year, in February, April, June, August, October, November, and December. Price per annual 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 Society, 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. 34.21, P. L. and R.). Accepted for mailing at the special rate of postage provided for in section 34.. 40, paragraph (d).

Copyright © 1960 by the American Mathematical Society Printed in tbe United States of America

660 FIVE HUNDRED SEVENTY-THIRD MEETING

Vanderbilt University Nashville, Tennessee November 18-19, 1960

PROGRAM

The five hundred seventy-third meeting of the American Mathe­ matical Society will be held at Vanderbilt University, Nashville, Ten­ nessee, on November 18-19, 1960, in Furman Hall, By invitation of the Committee to Select Hour Speakers for the Southeastern Sectional Meetings, Professor GayleS. Young, Jr. will address the Society on "Topological Methods for Complex Functions in the Disk" at 2:00P.M. Friday in Room 108, Furman Hall. Sessions for contributed papers will be held from 3:30 P.M. to 5:30P.M. on Friday and from 10:30 A.M. to noon on Saturday. Registration headquarters will be in Furman Hall and will be open from 10:00 A.M. Friday until noon on Saturday. The Rand Hall Cafeteria will be available for meal service. In addition most of the hotels and motels listed below provide dining service, and the others are in the immediate vicinity of such service. Reservations should be made directly with the chosen hotel or motel, Be sure to mention the American Mathematical Society in your correspondence as some of the rates quoted are somewhat be­ low the regular rates. Downtown hotels approximately two miles from the meeting: Single Twin Hermitage $5.oo -=---$IT.oo $10.00 - $18.00 Noel 5.50 - 10.50 9.00 - 13.00 james Robertson 4.50 - 6.50 8.50 - 10.00 Maxwell House 5.00 - 6.00 9.00 Andrew jackson 6.00 - 9.00 10.50 - 14.00 The following are within two blocks of the meeting place: Double Allen Hotel 2004 West End Avenue $6.00 $8.00 Anchor Motel West End at 20th Avenue 8.00 10.00 Medical Arts Building (Very limited number rooms available) 5.00 8.00

661 The motels listed below are within 10 minutes drive of the cam- pus: Single Double York Hotel Court 2501 Franklin Road $5.50 $7.00 Drake Motel Highway 41-70S 7.00 8.00 Biltmore Hotel Court 2400 Franklin Road 6.00 7.50 Vanderbilt University is about two miles west of downtown Nashville and is on West End Avenue at 21st Avenue, South. Mail and other messages for those attending the meeting may be sent in care of the Math em a tics Department, Vanderbilt University, Nashville, Tennessee.

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

FRIDAY, 3:30P.M. Session on Topology, Room 202, Furman Hall 3:30 - 3:40 (1) Real commutative semigroups on the plane Professor J. G. Horne, Jr., University of Georgia (573-5) 3:45 - 3:55 (2) Cartesian products with intervals Professor Morton L. Curtis, Florida State University (573-11) 4:00 - 4:10 (3) The fundamental group of certain deleted product spaces Mr. C. W. Patty, University of North Carolina (573-12) 4:15- 4:25 (4) Shrinking continua in the n-sphere Mr. C. H. Edwards, Jr., University of Tennessee (573 -14) 4:30 - 4:40 (5) Local connectedness and inverse limit spaces Professor M. K. Fort, Jr., and Mr. Jack Segal, Univer­ sity of Georgia (573-4)

662 4:45 - 4:55 (6) Cut points in totally nonsemi-locally-connected continua Professor E. E. Grace, Emory University (573-22)

FRIDAY, 3:30P.M. Session on Analysis, Room 208, Furman Hall 3:30 - 3:40 (7) A note on subseries convergence Professor Charles W. McArthur, Florida State Univer­ sity (573-7) 3:45 - 3:55 (8) Local linear dependence and the vanishing of the Wronskian Dr. Gary Meisters, RIAS, Baltimore, Maryland and the University of Nebraska (573-10) 4:00 - 4:10 (9) Facto ring merom orphic functions Professor William C. Fox, Tulane University (573-15) 4:15 - 4:25 (10) Generalized differentiation Professor Costas Kassimatis, North Carolina State Col­ lege (573-16) 4:30 - 4:40 (11) Generalized condensation points Professor N. F. G. Martin, University of Virginia (573 -17)

SATURDAY, 10:00 A.M. General Session, Room 202, Furman Hall 10:00 - 10:10 (12) An axiom system for set theory Professor E. Baylis Shanks, Vanderbilt University (573-26) 10:15 - 10:25 (13) The secular characteristics of the motion of an earth satel­ lite Professor G. C. Caldwell and Professor R. A. Struble, North Carolina State College (573-18) 10:30 - 10:40 (14) General perturbational solutions of the Mathieu equation Professor Raimond A. Struble, North Carolina State Col­ lege (573-27) 10:45 - 10:55 (15) On the inverses of finite Toeplitz matrices Dr. Roy Leipnik, Naval Ordnance Test Station, China Lake, California and Dr. John E. Maxfield, University of Florida (573-24)

663 11:00- 11:10 (16) Properties of algebras almost equivalent to identities Professor Trevor Evans, Emory University (573-21)

SATURDAY, 10:00 A.M. Session on Topology, Room 208, Furman Hall 10:00 - 10:10 (17) A linearly ordered continuum of arbitrarily high cardinality Professor Ben Fitzpatrick, Jr ,, Auburn University (573 -3) 10:15- 10:25 (18) Indecomposability and T(p) sets Professor H. S, Davis, Mr. D.P. Stadtlander and Pro­ fessor P. M. Swingle, University of Miami (573-13) 10:30 - 10:40 (19) Irreducible continua and T(p) sets Professor H. S. Davis, Mr. D.P. Stadtlander and Pro­ fessor P.M. Swingle, University of Miami (573-20) 10:45 - 10:55 (20) A clan with zero and without the fixed point property Professor Haskell Cohen, Louisiana State University (573-19) 11:00- 11:10 (21) Semigroups on trees Professor R. J, Koch, Louisiana State University and Professor L. F. McAuley, University of Wisconsin (573-23) 11:15 - 11:25 (22) A certain contractible open 3 -manifold, Preliminary report. Dr. D. R. McMillan, Jr ,, Louisiana State University (573 -25)

SUPPLEMENTARY PROGRAM (To be presented by title) (23) Asymptotic behavior of general queues with one server. II. Dr. V. E. Beneff, Bell Telephone Laboratories, Incorpo­ rated, Murray Hill, New Jersey (573-1) (24) Some operational equations for symmetric polynomials Professor Leonard Garlitz, Duke University (25) Some congruences for the Bell polynomials Professor Leonard Garlitz, Duke University (26) A theorem on "ordered" polynomials in a finite field Professor Leonard Garlitz, Duke University

664 (27) Criteria for generalized Kummer's congruences Professor Leonard Carlitz and Mr. Harlan Stevens, Duke University (28) On the singular Cauchy problem Professor Robert W. Carroll, Rutgers, The State Univer­ sity (29) Arithmetical notes, V. A divisibility property of the divisor function Professor Eckford Cohen, University of Tennessee (573 -2) (30) A characterization of tame curves in the 3-sphere Mr. C. H. Edwards, Jr., University of Tennessee (31) Conditions for endomorphism closure in an algebra Professor Trevor Evans, Emory University (32) On the existence of outer automorphisms of some finite P-groups Professor Ossie J. Huval, University of Southwestern Louisiana (573-6) (33) On the equation ax - xb = c in division rings Dr. Gary Meisters, RIAS, Baltimore, Maryland and Uni­ versity of Nebraska (573-8) (34) Negative norms and boundary problems. IV Professor Martin Schechter, Institute of Mathematical Sciences, New York University (573-10) Gerald B. Huff Associate Secretary Athens, Georgia October 15, 1960

665 FIVE HUNDRED SEVENTY-FOURTH MEETING

California Institute of Technology Pasadena, California November 19, 1960

PROGRAM

The five hundred seventy-fourth meeting of the American Ma­ thematical Society will be held on Saturday, November 19, 19 60 at the California Institute of Technology in Pasadena, California, By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, there will be an address at Z:OO P.M. in Room ZZ Gates Laboratory, by Professor Halsey Royden of Stanford University. Professor Royden's subject is "Function algebras". Sessions for contributed papers will be held at 10:00 A.M. and 3:30 P.M. in Rooms 151, 153 and 155 of the Sloan Laboratory of Ma­ thern atics and Physics. Abstracts of the papers to be presented at these sessions appear on pages 733-746 of these NOTICES. There are cross references to the abstracts in the program. For example the title of paper (1) in the program is followed by (574-18) indicating that the abstract can be found under the designation 574-18 among the pub­ lished abstracts. Late papers may be added to the program. For infor­ mation concerning late papers, inquire at the registration desk. Registration for the meeting will begin at 9:00A.M. The regis­ tration desk will be located in the Sloan Laboratory of Mathematics and Physics. A tea for persons attending the meeting will be served in the Athenaeum following the afternoon sessions for contributed papers. Luncheon will be available at the Athenaeum at a cost of $2.00 per person. The California Institute of Technology is located on California Street, between Hill Avenue and Wilson Avenue. It is easily reached by automobile from downtown Los Angeles, taking the Pasadena Freeway (Arroyo Parkway) into Pasadena, then turning east on California Street for approximately one mile, There is a parking lot on the south side of California Street, directly opposite the Sloan Laboratory of Mathema­ tics and Physics. Hotels in Pasadena include the Green and the Huntington-Shera­ ton. There is frequent limousine service between these hotels and the Los Angeles International Airport. Reservations for rooms should be made directly with the chosen hotel.

666 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 listen­ ers can circulate between different sessions. To l!l_~!~!~_in thi~_!jche­ dule, the time li_l!l!! "!'_~!l_b_~_ strictly enforced.

SATURDAY, 10:00 A. M. Session on Analysis, Room 151, Sloan Laboratory of Mathematics and Physics 10:00 - 10:10 (1) Derivations in commutative Banach algebras Professor P. C. Curtis, Jr., University of California, Los Angeles (574-18) 10:15 - 10:25 (2) Fourier transforms of certain classes of integrable func­ tions Mr. Robert Ryan, California Institute of Technology (574-17) 10:30 - 10:40 (3) m-independence, m-measures, and m-homomorphisms Professor C. B. Bell, San Diego State College (574-30) 10:45 - 10:55 (4) On normed linear spaces of measurable functions Dr. W.A. j. Luxemburg, California Institute of Technol­ ogy (574-16) 11:00- 11:10 (5) Two spectral theorems for a pair of singular first order dif- ferential equations Dr. B. W.Roos and Dr. W.C. Sangren, General Dynamics Corporation, General Atomic Division, San Diego, Cali­ fornia (574-9) 11:15- 11:25 (6) On certain two point boundary value problems of sublinear and asymptotically linear type Dr. G. H. Pimbley, University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico. 11:30 - 11:40 (7) Band-limited functions and improper boundary value prob- lems for a class of nonlinear partial differential equations Mr.]. M. Zimmerman, University of Southern California and Rocketdyne, A Division of North American Aviation, Los Angeles, California (547-14)

667 Session on Topology, Geometry and Applied Mathematics, Room 153, Sloan Laboratory of Mathematics and Physics 10:00- 10:10 (8) Characterization of the fixed point property for a class of set-valued mappings Professor L. E. Ward, Jr., University of Oregon (574-12) 10:15 - 10:25 (9) G-spaces with curvature > K > 0 Mr. E. D. Kann, University of Southern California (574-21) 10:30 - 10:40 (10) Cohomology on Lorentz manifolds. Preliminary report Dr. Y. H. Clifton and Dr. J. W. Smith, Massachusetts In­ stitute of Technology (574-15) 10:45 - 10:55 (11) New possibilities for a unified field theory Professor W. Israel and Mr. J. R. Trollope, University of Alberta (574-20) 11:00- 11:10 (12) Trigonometric interpolation and predictor-corrector for­ mulas for numerical integration Dr. H. E. Salzer, Convair, General Dynamics Corpora­ tion, San Diego, California (574-10) Session on Algebra, Room 155, Sloan Laboratory of Mathematics and Physics 10:00 - 10:10 (13) Concerning the little projective group Professor T. G. Ostrom, Washington State University (574-7) 10:15 - 10:25 (14) The purity of high subgroups of Abelian groups Dr. J. M. Irwin and Professor E. A. Walker, New Mexico State University (574-25) 10:30 - 10:40 (15) Pure extensions of Abelian groups Professor E. A. Walker, New Mexico State University (574-29) 10:45 - 10:55 (16) Representation of a semigroup by transformations acting transitively on a set Mr. E. J. Tully, Jr., California Institute of Technology (574-28) 11:00 - 11: 10 (17) The common fixed-point property in semigroups. Preliminary report Dr. Roy Leipnik, U. S. Naval Ordnance Testing Station, China Lake, California (574-27)

668 11:15- 11:25 (18) A theorem on the structure of certain finite semigroups Professor Takayuki Tamura, University of California, Davis 11:30 - 11:40 (19) Some remarks on a certain class of rings Professor Adil Yaqub, University of California, Goleta, and Purdue University (574-24)

SATURDAY, 2:00P.M. Invited Address, Room 22, Gates Laboratory Function algebras Professor Halsey Royden, Stanford University

SATURDAY, 3:30P.M. Session on Analysis, Room 151, SloanLaboratoryofMathematics and Physics 3:30 - 3:40 (20) Equatorial orbits around an oblate primary Dr. J. L. Brenner, Stanford Research Institute, Menlo Park, California (574-1) 3:45 - 3:55 (21) On the existence of periodic solutions of f'(x) = -a. f(x - 1) {1 + f(x)} Mr. G.S.Jones, Jr., General Electric Company, Cincin­ nati, Ohio 4:00 - 4:10 (22) Existence of solutions of an n'th order hyperbolic partial differential equation Dr. James Conlan, U. S. Naval Ordnance Laboratory, White Oak, Maryland and Professor J. B. Diaz, Univer­ sity of Maryland (574-4) 4:15- 4:25 (23) Solutions of a system of two nonlinear pa.rtial differential equations of the first order Professor E. I. Deaton, San Diego State College (574-19) Session on Algebra, Number Theory and Foundations, Room 155, Sloan Laboratory of Mathematics and Physics 3:30 - 3:40 (24) Stable matrices Dr. Olga Taussky, California Institute of Technology (574-23)

669 3:45 - 3:55 (25) On the distribution of consecutive quadratic and cubic resi­ dues Professor D. H. Lehmer and Mrs. Emma Lehmer, Uni­ versity of California, Berkeley (574-22) 4:00 - 4:10 (26) Recurrent divisibility sequences Professor Morgan Ward, California Institute of Technol­ ogy 4:15 - 4:25 (27) Concerning the generalized continuum hypothesis and the axiom of choice. Preliminary report Professor A. H. Kruse, Research Center, New Mexico State University and University of Kansas (574-26)

SUPPLEMENTARY PROGRAM (To be presented by title)

(28) Concerning weak solutions of one dimensional vibration problems by the perturbation method. Preliminary report Professor J. B. Butler, Jr., University of Arizona (574-2) (29) On the spectral measure of a perturbed operator whose re­ solvent has a Carleman kernel Professor J. B. Butler, Jr., University of Arizona (574-3) (30) Proof that Weber's normal unit is no perfect power Professor Harvey Cohn, University of Arizona (31) On the class of limit ultrapowers of a relational system Mr. H. J. Keisler, University of California, Berkeley (32) Cardinalities of ultrapowers and a theorem of Rabin Mr. H. J. Keisler, University of California, Berkeley (33) Nonrepresentable polyadic algebras of finite degree Mr. Donald Monk, University of California, Berkeley (574-5) (34) Polyadic Heyting algebras Mr. Donald Monk, University of California, Berkeley (574-6) (35) A transformation for a pair of first order differential equa- tions Dr. B. W.Roos and Dr. W.C. Sangren, General Dynamics Corporation, General Atomic Division, San Diego, Cali­ fornia (574-8) (36) Infinite radicals in the complex plane Mrs. Georgellen Schuske and Professor W. J. Thron, University of Colorado (574-11)

670 (3 7) Singularities of harmonic functions of three variables gen­ erated by Whitaker-Bergman operators Professor A.M. White, University of Santa Clara (574-13) R. S. Pierce Associate Secretary Seattle, Washington October 10, 1960

THE AMERICAN MATHEMATICAL SOCIETY respectfully invites and deeply appreciates BEQUESTS AND OTHER CAPITAL GIFTS

A legacy or other capital gift to the American Mathematical Society is an effective contribution to the advancement of mathematical research. Investment income provides essential support for the activities of the Society, and most of it is derived from gifts and bequests which have been made over the years by persons interested in mathematics. It is the policy of the Society to recognize a donor by attaching his name, or any other name specified by him as a memorial, to the fund given by him or to the activity supported by the fund. Gifts and bequests to the Society are deductible for income, gift, and state tax purposes.

AMERICAN MATHEMATICAL SOCIETY 190 Hope Street, Providence 6, Rhode Island

671 FIVE HUNDRED SEVENTY-FIFTH MEETING

Northwestern University Evanston, lllinois November 25-26, 1960

PROGRAM

The five hundred seventy-fifth meeting of the American Mathe­ matical Society will be held at Northwestern University, Evanston, lllinois, on Friday and Saturday, November 25-26, 1960. All sessions will be held in the Technological Institute. The Committee to Select Hour Speakers for Western Sectional Meetings has invited Professor Graham Higman of Oxford and the University of Chicago to address the Society. He will speak at 2:30 P.M. Friday in Lecture Room 1 on the subject "Recursive functions in group theory". Sessions for the presentation of contributed papers will be held at 10:30 A.M. on Friday and 10:00 A.M. on Saturday. If necessary, there will be special sessions for the presentationof contributed pa­ pers which fail to meet the deadline of October 5, 1960. If such ses­ sions are held, they will be publicized by a special program available at the registration desk. Rooms 162, 165, 167, 186, and 187 will be available for discus­ sions. There will be a tea at 4:00 o'clock on Friday in the Faculty Lounge on the third floor of the Technological Institute. The registration desk will be located in the front lobby of the Northwestern Technological Institute Building located on Sheridan Road at Noyes Street. Those who attend the meetings are requested to register at any time from 9:30 A. M. to 5:00 P. M. on Friday, and from 9:30 A. M. to noon Saturday. Mem hers of the Society may use the parking lot directly across the street from the Technological Institute. Meals will be served in Sargeant Hall north of the Institute Building. The Orrington Hotel, 1710 Orrington Avenue, Evanston, has agreed to hold a block of rooms for Society members until November 15, 1960. Members should write directly to the Orrington. Single rooms range from $7.00 to $12.00 and double rooms from $10.00 to $14.00. The North Shore Hotel in Evanston has comparable prices.

672 PROGRAM OF THE SESSIONS The time limitfor each contributed paper is ten minutes; how­ ever, the papers are scheduled at 15 minute intervals. Thus the audi­ ence has an opportunity to circulate between the various sessions, and those who are interested in a particular paper are certain of the exact time at which it will be presented. To achieve these objectives, the time limit will be strictly enforced. All sessions will be held in the Northwestern Technological Institute Building. FRIDAY, 10:30 A.M. Session on Algebra, Lecture Room 1 10:30 - 10:40 (1} Rings with semi-simple quotient rings. Preliminary report Mr. Lawrence S. Levy, University of Illinois (575-17} 10:45 - 10:55 (2) The space of minimal prime ideals of commutative ring Professor Melvin Henriksen, Wayne State University and Professor Meyer Jerison, Purdue University (575-14} 11:00- 11:10 (3) A decomposition theory for finite groups Dr. Paul M. Weichsel, University of Illinois (575-20} 11:15- 11:25 (4) Homological dimension of ideals in Noetherian domains Professor Irving Kaplansky and Mr. Robert E. MacRae, University of Chicago (575-25} 11:30-11:40 (5) Abstract mean values Professor Trevor Evans, Emory University (575-22} 11:45 - 11:55 (6) Imbeddings of partial (incomplete) multiplicative systems (monoids}, associativity and wordproblem Dr. Dov Tamari, Institute for Advanced Study and Israel Institute of Technology, (575-30} 12:00 - 12:10 (7} On the characteristic kernel of two-person zero-sum games Professor joseph V. Talacko, Marquette University (575-19}

FRIDAY, 10:30 A.M. Session on Applied Mathematics and Topology, Lecture Room 2 10:30 - 10:40 (8) On components in some families of sets Dr. B. Griinbaum and Professor T. S. Motzkin, University of California, Los Angeles (575-3}

673 10:45 - 10:55 (9) The density theorem for two-dimensional structural stability Dr. M. M. Peixoto, University of Brazil (575-4) 11:00 - 11:10 (10) Sharpening of Sperner's lemma applied to homology groups Professor A. B. Brown, Queens College and Professor S. S. Cairns, University of Illinois (575-21) 11:15- 11:25 (11) Two theorems on metrization of Moore spaces Professor james N. Younglove, University of Missouri (575-8) 11:30 - 11:40 (12) Reduction of limit functions Professor P. C. Hammer, University of Wisconsin (575-11) 11:45 - 11:55 (13) Some relations on symmetric products Professor Arthur Mattuck, Massachusetts Institute of Technology (575-28)

FRIDAY, 2:30P.M. General Session, Lecture Room 1 Recursive functions in group theory (one hour) Professor Graham Higman, Oxford University and the University of Chicago

SATURDAY, 10:30 A.M. Session on Analysis, Lecture Room 10:00 - 10: 10 (14) Optimal control for nonlinear systems Professor Lawrence Markus, Yale University (575-27) 10:15- 10:25 (15) Connectedness of limit points of transforms of sequences Dr. M.S. Ramanujan, University of Michigan (575-5) 10:30 - 10:40 (16) Turningpointproblemsfor certain systems of linear differ­ ential equations. I Professor Wolfgang Wasow, University of Wisconsin (575-7) 10:45 - 10:55 (17) On uniform convergence of orthonormal expansion Professor Syed A. Husain, University of Saskatchewan (575-24)

674 11:00 - 11:10 (18) Functions of BVC type Professor Casper Goffman and Mr. Richard E. Hughs, Purdue University (575-18) 11:15 - 11:25 (19) On approximate derivatives Professor Casper Goffman and Dr. C. J. Neugebauer, Purdue University (575-23) 11:30 - 11:40 (20) On the number of prime factors of the natural numbers in arithmetic progressions Professor G. J. Rieger, Purdue University (575-29) 11:45 - 11:55 (21) Estimation of certain exponential sums in the theory of auto­ morphic forms Dr. Marvin Isadore Knopp, University of Wisconsin (575-26)

SUPPLEMENTARY PROGRAM (To be presented by title)

(22) A pair of reciprocal matrices with integer valued elements for test purposes Mr. Edgar Asplund, University of Stockholm (575-1) (23) Asymptotic behavior of general queues with one server. III Dr. V. E. Bene§', Bell Telephone Laboratories, Incorpo­ rated, Murray Hill, New Jersey (575-2) (24) Bounds on the product of an infinitesimal generator and its semi-group operator Mr. Norman Fritz, University of Minnesota (575-10) (25) Domain finite expansive functions Professor P. C. Hammer, University of Wisconsin (575-12) (26) Neighborhoods for extended topology Professor P. C. Hammer, University of Wisconsin (575-13) (27) A classification of rings by module type Professor W. G. Leavitt, University of Nebraska (575-15) (28) An extension of the concept of torsion. Preliminary report Mr. Lawrence S. Levy, University of Illinois (575-16) (29) An explicitformulationfor the value of a fractional factorial Dr. C. A. Muses, Barth Foundation

675 (30) Properties of double chain sequences Professor Robert Seall, Illinois Institute of Technology and Professor Marion Wetzel, Denison University (575-6) (31) Equivalence of representations under extensions of local ground rings Professor H. J. Zassenhaus, University of Notre Dame and Professor Irving Reiner, University of Illinois (5 75- 9) J. W. T. Youngs Associate Secretary Bloomington, Indiana October 5, 1960

G. B . Price, H. L. Alder, C. B . Allendoerfer and G. L. Walker at the East Lansing Meeting

676 PRELIMINARY ANNOUNCEMENT OF MEETING

SIXTY-SEVENTH ANNUAL MEETING Washington, D. C. January 23-26, 1961

The sixty- seventh Annual Meeting of the American Mathemati­ cal Society will be held at the Hotel Willard in Washington, D. C., from Monday, January 23 to Thursday, January 26, 1961. All sessions will be held in public rooms of the hotel. During the same week with headquarters at the same hotel there will be meetings of the Associ­ ation for Symbolic Logic, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. The fortieth Josiah Willard Gibbs Lecture will be delivered by Professor J. J. Stoker of New York University. Professor Stoker will speak at 8:00 P.M. on Tuesday, January 24, in the Grand Ballroom on "Problems in nonlinear elasticity." By invitation of the Committee to Select Hour Speakers for Summer and Annual Meetings, Professor Lars Hormander of the Uni­ versity of Stockholm and The Institute for Advanced Study and Pro­ fessor Helmut Wielandt of the University of Tubingen and the Califor­ nia Institute of Technology will address the Society. Professor Wie­ landt' s subject is "The structure of finite groups." Sessions for contributed papers will be held on the morning and afternoon of each day that the Society is meeting. Abstracts of contributed papers should be sent to the American Mathematical So­ ciety, 190 Hope Street, Providence 6, Rhode Island, so as to arrive prior to the deadline of December 9, 1960. The sessions will not in­ clude any provision for late papers. This deadline does not apply to abstracts of papers by title, which are now received, judged and pub­ lished on a schedule which is independent of meetings of the Society. The council of the Society will meet at 5:00 P.M. on Wednesday, January 25, in the Executive Room and will reconvene after an inter­ mission for dinner. There will be a business meeting of the Society at 2:00 P.M. on Thursday, January 26, in the Grand Ballroom. The Employment Register will function on Tuesday, Wednesday, and Thursday, January 24, 25, and 26, from 9:00 A.M. till 5:00 P.M.

REGISTRATION Registration headquarters will be located in the Caucus Room on the first floor (up one flight) of the Hotel Willard. The Registration Desk will be open on Sunday, January 22, from 2:00 P.M. till 8:00 P.M. and on Monday through Friday, January 23-27, from 9:00A.M. till 5:00 P.M. All members attending the meetings are requested t<:>_r-_:_::

677 gister at the headquarters on arrival. The directory of registration and an information service will be maintained at these headquarters. The registration fee will be $2.00 for each member of any participating organization and $.50 for each accompanying adult.

EXHIBITS Various publishers and purveyors of items of interest to the profession will have exhibits in the jackson Room on the first floor.

ACCOMMODATIONS The Willard Hotel is the official hotel for this meeting. The Willard, the Washington Hotel (next door to the Willard), and the Raleigh Hotel (two blocks away) are cooperating in reserving blocks of rooms. There is a reservation card in the back of this issue of the NOTICES. It should be mailed to the Hotel Willard for a reservation at any of the three cooperating hotels. It is important both to the So­ ciety and to the individual that the card be used, first because the hotel room rates have been negotiated and second because the poten­ tial of the Society to make suitable future arrangements with other hotels depends on the records of attendance and related hotel reserva­ tions. It may be impossible to reprint the reservation card in the De­ cember issue of the NOTICES. The Y.M.C.A., 15th and G Streets, N. W., and the Y.W.C.A., 17th and K Streets, N. W. are not far from the Hotel Willard. The Marriott Motor Hotels, Twin Bridges on U.S, 1, Washington 1, D. C. is also near the Willard and will have rooms available. For reserva­ tions with any of these three institutions, one should write directly.

TRAVEL INFORMATION The Willard Hotel is located at 14th and Pennsylvania Avenue, Northwest. Taxicab and limousine service is available from Washing­ ton National Airport directlytothehotel. Thelimousine fare is $1.20. Any street car serving northwest Pennsylvania Avenue from Union Station will stop opposite the hotel. It is best to take a taxicab from the bus terminal. The fare is $.50. Thosearrivingby auto can follow U.S. 1 until it joins 14th Street near the Washington Monument, which is three blocks from the Willard Hotel. The Washington Hotel is located at15thandPennsylvania Ave­ nue, Northwest, adjoining the Willard. The Raleigh Hotel is located at 12th and Pennsylvania Avenue, Northwest, two blocks from the Willard.

678 COMMUNICATIONS Mail and telegrams for those attending the meetings can be addressed in care of the American Mathematical Society, Hotel Wil­ ard, Washington 4, D.C. Committee on Arrangements H. L. Alder M. W. Oliphant J, W. Brace Everett Pitcher, Chairman G. L. Walker

THE EIGHTH AMERICAN MATHEMATICAL SOCIETY SUM­ MER INSTITUTE will be held at Stanford University, Stanford, Cali­ fornia in August, 1961. Sponsored by the American Mathematical So­ ciety with the support of the National Science Foundation, the Sum­ mer Institute will be held onthetopicAPPLICATIONSOF FUNCTION­ AL ANALYSIS. An indication of the timeliness and importance of this topic, submitted by Professor Peter D. Lax, follows: "Some of the most im­ portant recent research in analysis is characterized by a certain de­ gree of abstraction, whereby the concepts and methods of the theory of Hilbert and Banach spaces, and of operators acting on such spaces, is used to formulate and solve concrete problems. The purpose of this Summer Institute on Applications of Functional Analysis is to have a number of leading analysts deliver series of lectures on their own recent researches and related developments. Thus the attendants would have a chance to learn about modern problems and modern methods." There will be a series of lectures on the following subjects: Recent developments in harmonic analysis. Theory and applications of singular integral operators. Recent developments in the theory of partial differential equa- tions. Applications of methods of several complex variables to Banach algebras. Several Russian scientists have been invited to participate. The Program Committee consists of Professor Ralph Phillips of the University of California at Los Angeles; Professor Henry Hel­ son of the University of California at Berkeley, and Professor Peter D. Lax of the Institute of Mathematical Sciences, New York Univer­ sity, Chairman. In view of the unusual amount of interest in this Institute, both for its subject and for the quality and diversity of lectures, a larger number of participants than usual will be invited.

679 lfews and comment lrom the CORFERERC£ BOARD OF THE MATHEMATICAL SCIERCES

G. Baley Price Washington. D. C. A STUDY OF THE DESIGN OF FACILITIES FOR MATHEMATICS As its first project, the Conference Board will conduct a study of the design of buildings and facilities for mathematics; Educational Facilities Laboratories, established by the Ford Foundation in New York City, has agreed to provide the necessary funds. There are many reasons why it is desirable to make a study of the design of facilities for the mathematical sciences at the present time. In the first place, mathematics has been very poorly housed in the past. In the second place, enrollments are now expanding rapidly. Many colleges and universities have five times as many majors in ma­ thematics as they had only four or five years ago, and the great in­ creases in enrollment for the nation as a whole are still to come. In the third place, a study of the design of mathematics facili­ ties is appropriate because of the many changes that have taken place in the mathematical sciences, The project will undertake a study of the design of facilities to support the total activities of the mathemati­ cal sciences. These activities include research and instruction in pure mathematics, applied mathematics, and statistics; preparation ofthe manuscripts of research papers; preparation of the manuscripts of textbooks and expository manuscripts for instructional purposes; teacher training; instruction in the operation of desk calculators and electronic digital computers; and the operation of summer and aca­ demic year institutes. Modern facilities for the mathematical sciences must provide headquarters space; classrooms; seminar rooms; of­ fices for the staff; library space; a statistics laboratory with desk calculators; a computation center for the electronic digital computer; facilities for the use of films, television, and other teaching aids; and a common room. The construction of appropriately designed facilities for mathe­ matics is important for a special reason at this time. There is a great shortage of mathematics teachers, and it is probable that this shortage will continue for many years. Under these conditions it is imperative that we make the teachers we do have more efficient than they have been in the past. Some universities are teaching elementary courses in sections of one to two hundred students; others would like to do so, but lecture rooms for classes of this size are not available. In many cases several staff members--even senior staff members-· are crowded into one office. When one has a visitor, the others stop work. There are important universities that have never been able to provide 680 one chair and desk per staff member. Better classrooms, better of­ fices, and better facilities of all kinds will certainly make our mathe­ matics staffs more efficient- -will enable our mathematics teachers to teach more students and to teach them better. In the study of the design of facilities for mathematics, it will be kept in mind that mathematics is a peculiarly human and personal activity. Some academic subjects involve work with laboratory equip­ ment or machines, but the typical mathematician sits at a desk and works with pencil and paper. If he is to be productive of ideas, he must be in pleasant, comfortable, and congenial surroundings. The project will begin by collecting information about good features of buildings already in existence, both in the United States and abroad. It will be necessary to hold a small conference at the be­ ginning of the projectto consi~er the nature of the activities to be con­ ducted in a center for the mathematical sciences, and to explore the design of facilities. The end result of the project will be a report which canbepublishedcommercially. The staff will consist of a math­ ematician and a secretary, and architectural services will be pro­ vided by an architectural firm in Washington, D. C. The project, ex­ pected to be completed in one year, will begin as soon as the staff has been organized. The project will include a study of mathematics facilities for high schools, and the final report will include a section or chapter on mathematics classrooms and other facilities for high schools. This part of the report will be published separately so that it will be easily available to high school administrators and others interested only in the high school field.

THOMAS E. CLEMMONS has been appointed director of ad­ vanced studies by International Business Machines Corporation. He was director of the company's executive development program. Mr. Clemmons will continue to direct all executive develop­ ment activities of the company. In addition, he will be responsible for the administration of IBM's Systems Research Institute scheduled to open this fall.

681 From The AMS Secretary John W. Green. UCLA

I would like to draw the attention of members of the Society to the opportunities they possess of taking advantage of the various re­ ciprocity agreements that have been made with other mathematical organizations. For some time the Society has had reciprocity agree­ ments with The London Mathematical Society, Unione Matematica Italiana, Deutsche Mathematiker Vereinigung, Norsk Matematisk Forening, Schweizerische Mathematische Gesellschaft, Societe Ma­ thematique de France, Dansk Matematisk Forening, Wiskundig Genootschap te Amsterdam, Polskie Towarzystwo M atematyczne, Svenska Matematikersamfundet, Suomen Matemaattinen Yhdistys, Islenzka Staerofraeoafelagio, Sociedad Matematics Peruana, and the Indian Mathematical Society. In 1959 to this list hasbeenaddedthe Australian Mathematical Society and the Mathematical Society of Japan, and this month the Societe Mathematique de Belgique, as maybe seen from an announce­ ment in another part of these NOTICES. The details of the agreement with the Australian Mathematical Society may be seen in the Decem­ ber, 1959, issue of the NOTICES, and those with the Mathematical Society of Japan in the November, 1959, issue. In addition it is anticipated that agreements will shortly be made with the Union Matematica Argentina, the Gesellschaft fiir Angewandte Mathematik und Mechanik, the Turkish Society of Pure and Applied Mathematics, the Glasgow Mathematical Association, and the Osterreichische Mathematische Gesellschaft. These agreements provide that members of one organization may join the other for one-half the regular dues. I will be glad to assist members of the Society wishing to join one of these recipro­ cating organizations by locating the correct person to write and the correct amount of dues. We do not publish a current list of these because of the difficulty of keeping up with frequent changes. A forthcoming Society publication that may be expected to be received with enthusiasm is the Russian-English Dictionary of the Mathematical Sciences prepared by A. J. Lohwater with the collabora­ tion of S. H. Gould. A similar English-Russian dictionary is expected to be published in Russia. The Society will have completed its seventy-fifth year in 1963. A committee, consisting of former Presidents G. T. Whyburn, R. L. Wilder, and Richard Brauer, has been appointed to consider the de­ sirability and feasibility of having a suitable "gaudy". John W•. Green

682 NEWS ITEMS AND ANNOUNCEMENTS

THE SCHOOL OF MATHEMATICS OF THE INSTITUTE FOR ADVANCED STUDY, Princeton, New jersey, will grant a limited num­ ber of memberships, in some cases with financial support, for re­ search in mathematics at the Institute during the academic year 1961- 1962. Candidates must have given evidence of ability in research com­ parable at least with that expected for the Ph.D. degree. Application blanks may be obtained from the Secretary of the School of Mathema­ tics, and should be returned by January 15 (whether or not funds are expected from some other source).

THE MIT DEPARTMENT OF MATHEMATICS wishes to an­ nounce the availability of C. L. E. Moore Instructorships in Mathe­ matics for 1961-1962, open to young mathematicians with doctorates who show definite promise in research. The base salary for these in­ structorships will be at least $6,900 and the teaching load will be six hours per week. The salary can be supplemented by summer work on research contracts or by teaching in the summer session. The ap­ pointments are annual but are renewable for one additional year. Applications should be filed not later than january 9, 1961 on forms obtained from the Department.

A BORIS A. BAKHMETEFF RESEARCH FELLOWSHIP will be available for the 1961-1962 academic year in an amount up to $3000. It is intended to be a specific contribution for a definite research project of an original and creative nature in the general field of mech­ anics of fluids. The recipient shall be a full-time graduate student who is a candidate for the master's or doctoral degree. He shall not hold, or expect to hold, any other fellowship or major income-producing com­ mitment that will interfere with his research work and study on a full­ time basis. It is expected that the stipend will be adequate to cover tuition, subsistence, and, if necessary, a portion of the research ex­ penses. The study and research may be undertaken at an institution of the Fellow's choice. In the judgment of the Committee, the adequacy of the facilities of the institution will have substantial weight in the se­ lection of the Fellow. Applications should be filed by February 15, 1961 with Dean William Allan, School of Technology, The City College of New York, New York 31, New York.

683 THE 1961 ONR POSTDOCTORAL ASSOCIATESHIPS IN MA­ THEMATICS have been announced. A number of awards will be made for research at any one of the following institutions: Brown, Califor­ nia Institute of Technology, Columbia, Cornell, Illinois, Johns Hopkins, Kansas, Michigan, Northwestern, Notre Dame, Ohio State, Virginia, University of Washington, Yale. The associates will receive $6,000 for the academic year, an additional $1,300 for the summer, if desired, and an allowance of $500 for incidental expenses, including travel. They may also teach a course of an intermediate or advanced nature, with an additional stipend not to exceed $1,500. Applications must be submitted by January 15, 1961. For further information and application forms, write to: Mathematics Branch, Office of Naval Research, Washington 25, D. C.

THE ANNUAL LIST OF CHAIRMEN OF DEPARTMENTS has been compiled. Copies may be obtained free of charge by writing to the Headquarters Offices, 190 Hope Street, Providence 6, Rhode Island.

THE ASIA FOUNDATION. With reference to the announcement published in the February issue of the NOTICES, we should again like to call your attention to the grant awarded the American Mathematical Society for the purpose of assisting Asian mathematicians with mem­ berships, 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 scien­ tists and mathematicians, and to select Asian mathematicians 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, Af­ ghanistan, Ceylon, or The Philippines. There is no stipulation that member ship 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 Cali-

684 fornia, Los Angeles 24, California. Applications for a grant or nomi­ nations of prospective grantees should be addressed to Professor Green.

THE FORD FOUNDATION has announced grants to three inter­ national scientific bodies to help speed up the task of defining, meas­ uring, and limiting hazards resulting from man-made radiation. The agencies and their grants are as follows: International Commission on Radiological Units and Measure­ ments -- $185,000 grant, to assist in improving the measurement of radiation received by living matter. International Commission on Radiological Protection-- $250,000 grant, to assist in determining objective safety standards for human exposure to radiation. International Bureau of Weights and Measures-- $32,500 grant, to accelerate planning of a major effort to establish world-wide stan­ dards of radiation measurement. "While man has adapted himself to a certain level of natural background radiation," William McPeak, a Foundation vice president, said, "recent developments in both nuclear weapons and the peaceful use of nuclear energy have added greatly to man's exposure. It is of extreme importance that judgments about the possible effects on hu­ man life be based on sound scientific knowledge. The Foundation's new grants will enable three experienced international agencies to broaden their consideration of radiation problems independent of the policies of governments."

A SYMPOSIUM ON CONVEXITY, sponsored by the American Mathematical Society with the financial support of the Air Force Office of Scientific Research, will be held in connection with the June 1961 meeting of the Society in Seattle, Washington. The Symposium will precede the meeting and will be scheduled so that no sessions of the Symposium will conflict with any sessions of the meeting. The Program Committee for this Symposium consists of Pro­ fessor David Gale of Brown University; Professor Branko Griinbaum of the University of Jerusalem, the Institute for Advanced Study and the University of Washington; and Professor Victor Klee of the Uni­ versity of Washington, Chairman. The Symposium will last for two days and will have four main sessions. Approximately sixteen speakers, experts in convexity and allied fields, will be invited to present papers of varying lengths. Emphasis will be laid upon talks which can be understood by non­ specialists, and upon presentation of unsolved problems.

685 Since it would seem impossible to cover all important aspects of the subject in a two-day symposium, the program will be devoted chiefly to the following qualitative aspects of the theory: "I - Infinite-dimentional convexity. Though quiescent in Banach's book of 193Z, the notion of convexity has since assumed a dominant role in some of the central parts offunctional analysis. An important function of the Symposium will be to survey these areas, and to serve as a meeting ground for those whose interest in convexity stems from functional analysis and those whose approach has been more tradi­ tional. "II - Combinatorial aspects. Two combinatorial aspects seem of spec­ ial importance. One is the study of the facial structure of convex poly­ hedra, which has been stimulated by the requirements of game theory and linear inequalities. (The study of infinite-dimensional "polyhedra" is in its infancy, but promises to have important applications in analy­ sis.) The other is the study of intersection properties of convex sets, originating with Helly's theorem around l9ZO. There have appeared more than sixty papers in the general area, containing many isolated results but no unifying principles. An overall appraisal of this area is called for, and the Symposium may be able to supply it." The program of the Symposium will appear in the February 1961 issue of the NOTICES of the American Mathematical Society, along with the program of the Society meeting.

A SYMPOSIUM ON MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES, co-sponsored bytheOfficeofOrdnance Re­ search and the National Science Foundation, will be held in connection with the April Meeting of the American Mathematical Society at the Hotel New Yorker in New York City, on April 6, 7, and 8. The subject was chosen by the Committee on Applied Mathe­ matics, consisting at present of S. M. Ulam, Chairman, P. R. Gara­ bedian, Brockway McMillan, R. C. Prim, J. j. Stoker, and David M. Young. In its reportto the Council of the Society, the Committee wrote as follows: "It is the objective of this symposium to inform and interest mathematicians in the problems of biology and medicine, and to stimu­ late investigations of topics in both pure and applied domains. "Many of the problems suggested by the title are not generally known by professional mathematicians. Indeed many of the problems have not been suitably formulated. Furthermore, the conceptual bases of biological theories appear to be undergoingrapidchanges as are­ sult of recent experimental discoveries. "The time now seems ripe for mathematicians to become better

686 acquainted with the challenging problems presented by the biological and medical fields, something beyond the realm of strict statistics. The effort von Neumann put into this area in the closing years of his life attests to the challenging nature of this field. "Mathematically a symposium on this general topic might in­ clude problems which belong to hydrodynamics, information theory, combinatory analysis, probability and statistical theories, and the like. However, such a program could easily result in merely another set of purely mathematical papers, and would be rather meaningless and even perhaps futile unless a portion of the program was concerned primarily with biological and medical problems of mathematical in­ terest. To insure a healthy balance to the program, it is therefore considered that a number of the speakers should be selected from professional biologists and medical men. However, no attempt should be made on the part of these speakers to teach biology or medicine. Moreover, the balance between the biological and medical approaches with those of the mathematical approaches must be tempered by the inclusion of material which seems mathematically respectable to an audience composed primarily of mathematicians, many of whom may not be much interested in applications. "Because the mathematics useful in biology does not always appear to have the mathematical unity that is characteristic of some of the more highly developed fields of physical science, the program will need to be limited a bit more strongly than would be true of some fields to avoid objections which stem from this unity problem. Perhaps this may be accomplished by organizing the sections of the program using the spectrum of branches of mathematics as a basis. "Possible topics for subsections of the symposium are: Mathematical problems of growth and form. Hydrodynamical problems of circulation, hormones, en­ zymes, etc. Geometry of spatial objects as reconstructed from x-ray diffraction patterns, etc. Differential equations and the problems of physiological and physical chemistry. Probability theory and the statistical problems of biology and medicine. Information theory, mathematical logic, and combinatories arising in: Problems in genetics. Statistical theory. Classical theory. Genetic 'code'. Operation of the nervous system. Analysis of the actual brain. Synthesis of computers to stimulate the brain.

687 Reliability problems. How design reliable machines (or organisms) using unreliable components? Self-reproduction problems. How can machines be designed that are capable of reproducing themselves?" The selection of speakers for the Symposium has been delegated to an Invitations and Steering Committee consisting of Dr. S. M. Ulam (Los Alamos Scientific Laboratory), Chairman; Dr. Richard E. Bell­ man (Rand Corporation), Secretary; Dr. John Jacquez (Sloan Ketter­ ing Institute for Cancer Research); Professor Claud E. Shannon (Massachusetts Institute of Technology); Professor Anthony Bartholo­ may (Biophysics Research Laboratory, Peter Bent Brigham Hospital, Harvard Medical School); Professor Theodore T. Puck (University of Colorado Medical Center).

THE SYMPOSIUM ON RECURSIVE FUNCTIONS, jointly spon­ sored by the Association for Symbolic Logic, the Association for Com­ puting Machinery, and this Society, with financial support from project FOCUS of the Institute for Defense Analysis, will be held on the first two days of the April meeting of the American Mathematical Society. Three sessions are planned, the first two on the notion of recursive functions and traditional applications, and the third on such newer applications as to computing, automata, problem solving, etc. The sessions will be chaired as follows: Thursday morning, April 6: Professor S. C. Kleene, University of Wisconsin; Thursday afternoon: Professor Frederic Fitch, Yale University; Friday morn­ ing, April 7: Professor J. B. Rosser, Cornell University. The Organizing Committee of the Symposium consists of Pro­ fessors S. C. Kleene (University of Wisconsin), Chairman; J. B. Rosser (Cornell University), J. C. E. Dekker (Rutgers, the State Uni­ versity), Joseph R. Shoenfield (Duke University) and John McCarthy (Massachusetts Institute of Technology). The Symposium will contri­ bute to and be a continuation of the reports in the same general field as last year's Symposium on the "Structure of Language." The programs of the Symposia will appear in the December 1960 issue of the NOTICES of the American Mathematical Society, along with the program of the Society meeting.

688 FEATURE ARTICLES

THE SINO-AMERICAN CONFERENCE ON INTELLECTUAL COOPERATION

A Sino-American Conference on Intellectual Cooperation was held in Seattle july 10-15, 1960. The Conference was called on the initiative of two academic institutions of the Republic of China and eight American universities. The purpose of the Conference was to discuss the scholarly and professional problems of intellectual com­ munication and cooperation, and in particular to investigate in detail the intellectual relations between Taiwan and America. Approximate­ ly 100 delegates and observers attended the Conference, about equally divided between Chinese and American delegates. Dr. S. H. Gould, executive editor of THE MATHEMATICAL REVIEWS, was a guest and representative from the Society. Some 50 papers on humanities, social sciences and natural sciences were read and discussed at the confer­ ence. A short note on "Mathern atics and Physics Education and Re­ search in Taiwan of the Republic of China" was published in the Octo­ ber issue of the NOTICES, pp. 582-585. At the end of the Conference's deliberations, three committees met to report conclusions and recommendations. There were commit­ tees for the humanities, social sciences, and natural sciences. It is the report of the latter that is printed below. REPORT OF THE COMMITTEE ON THE NATURAL SCIENCES The principal problems in intellectual cooperation between the natural scientists of the Republic of China and of the United States arise from the obstacles which have lain in the path of scientific and educational development in Taiwan in the last fifteen years. Great progress has been made in overcoming these obstacles, and there are many areas of science in which work now being conducted in the Re­ public of China is fully comparable with that in other parts of the world. Much still remains to be done, however, and it is to this that the present report is principally addressed. The chief needs of the natural sciences in China are in the areas of manpower and of research facilities. In the first of these areas, we recommend that steps be taken under the following cate­ gories: (a) The sending of faculty members in Chinese universities to the United States for visits and study. There are many promising young faculty members whose training should be supplemented

689 by periods of residence in American institutions where they could conduct research and receive the stimulation of contacts in this wider sphere. Many such visits have been made in the past and the benefits both to the individuals and to their own uni­ versities have been great. It is very desirable to continue and to expand this operation. In addition, senior professors can be greatly helped by spending leaves for research and study at ap­ propriate institutions in the United States. (b) Reciprocally, visits of American research scientists to Taiwan have, in the past, proved to be both stimulating and productive, and have in many cases made essential contributions to the training of Chinese scientists. The number and scope of these should be increased. (c) Steps to hasten the creation in Taiwan of conditions and facilities which will induce Chinese scientists now abroad to return. A large number of Chinese graduate students who have come to the United States in the past ten years have, after the completion of doctoral or post-doctoral training here, taken posts in this coun­ try instead of returning to China. Among the causes of this are: (i) the demand here for their services, and their knowledge that their work here is useful, and (ii) the fact that there are, at present, very few places in Tai­ wan where they could do work which would fully utilize their training. Their case is different from that of faculty members in Chinese universities, who have maintained a high average of returning to their posts upon completion of studies in America, and they represent a resource upon which it would be highly de­ sirable for science in China to be able to draw. It is our feeling that if and when the work in centers in Taiwan can have reached a high enough standard in program and facilities in any given field, many Chinese scientists now in the United States will be ready to return to Taiwan either permanently or for limited periods. For them to do so would further accelerate the develop­ ments which we desire to see. For this purpose, consideration should be given to the creation in Taiwan of a few foci of excep­ tional strength where such a process could begin. (d) The establishment of graduate-student and post-doctoral fellow­ ships tenable in Chinese universities. This would enable a larger proportion of Chinese students to receive advanced training in China, and would also create opportunities for American gradu­ ate or post-doctoral students interested in appropriate subjects (e.g. certain areas in biology, geology and the chemistry of natu­ ral products) to conduct research in Taiwan.

690 (e) The strengthening of programs of subsidies to research workers in Taiwan. The exceedingly low level of academic salaries in Taiwan has been a source of grave concern, by the National Council of Science Development, created by the Chinese Govern­ ment, which provides subsidies to research workers to enable them to devote their attention more fully to research. This pro­ gram has been exceedingly valuable and should be assisted so as to make possible the inclusion oflarger numbers of persons. (f) The rating of outgoing students from Taiwan. The difficulty which admissions officers in American graduate schools have experi­ enced in evaluating the credentials of applicants from Taiwan has been perplexing and has in some places even led to the dis­ continuance of admission of such students. This is most unfor­ tunate since in many cases the students are highly qualified and in all cases their qualifications are known to the administrations of the schools from which they have graduated. We consider that great benefit would follow from the setting up of a procedure by which admissions officers could receive from Chinese university administrations evaluations of individual students upon which they could base their actions. In the area of research facilities, we recommend that steps be taken toward: (a) Provision of laboratory equipment and supplies needed by com­ petent people engaged in research. (b) The improvement of library facilities needed for research. This would include provision of microfilm facilities so that, for ex­ ample, rare or expensive journals available in one university need not be duplicated in another. (c) Exchange of research materials between China and the United States. This would be particularly valuable in certain biological fields, where exchange of specimens could greatly facilitate re­ search on both sides of the ocean. (d) Provision of an electronic computer for the Computing Center of the Mathematical Institute of the Academia Sinica. (e) Assistance to research publication. Whereas in some fields of science publication of research results in journals is relatively easy, it is occasionally desirable to publish monographs and there are some fields in which journal space is very limited. In such cases, the expenditure of relatively modest sums to as­ sist publication would bring important benefits. In meeting the needs identified in the preceding paragraphs, we re­ commend that the following principles be followed:

691 (a) Encouraging realistic basic research projects and programs in the natural sciences. (b) Preserving a proper relationship between the provision of equip­ ment and the need for its current use. (c) Encouraging the establishment of reciprocal relations between individuals and institutions in China and the United States. (d) Identifying projects in the field of the natural sciences which have particular relevance in the Taiwan context, but also (e) Maintaining the principle of supporting free inquiry by outstand­ ing individuals rather than emphasizing projects. (f) Exploring available sources of support from foundations and the United States government. In order to implement the foregoing recommendations, we con­ sider it essential that the organizers of this conference create a standing committee to take certain practical steps.One of these should be aimed at the creation of a permanent organization which might, for example, be called a "Sino-American Association for Intellectual Co­ operation." This would need to have solid institutional sponsorship on both sides of the Pacific but should be so set up that committees of active scholars would determine its policies. Since the natural sciences constitute only one of three areas in which this conference was organized, we can speak only for it, but would feel that the poli­ cies of the association as they relate to the natural sciences should be determined by persons who are fully conversant with the nature and practice of creative scientific work and who are also, or can be­ come, familiar with the situation in Taiwan as it exists at the time. We assume that the policies of the association would be implemented through appropriate small secretariats which would have offices in Taiwan and in the United States and would need, of course, to work in close association with each other. Since the establishment of such a permanent organization must necessarily take considerable time, we also recommend that the standing committee from this conference proceed in the meantime with such steps as may be possible for the interim implementation of such of our recommendations as can be acted on at once. This report was drafted by a Committee consisting of: Chien Shih-Liang (President, National Taiwan University), Yen Chen­ Hsing (President, Provincial ChengKungUniversity), LingChih-Bang (Academia Sinica), Alfred E. Emerson (University of Chicago), Lynne L. Merritt (Indiana University), HenryS. Frank, Chairman (University of Pittsburgh).

692 NATIONAL ACADEMY OF SCIENCES NATIONAL RESEARCH COUNCIL DIVISION OF MATHEMATICS 1959-1960

It is a pleasure to announce that J, Barkley Rosser, now Chair­ man Designate, will serve as Chairman of the Division for two years beginning July 1, 1960, During this period, E. J. McShane will be Chairman Designate of the Division. He has also accepted appointment by President Bronk asChairmanfortwoyearsbeginning July 1, 196Z. The Division is very fortunate that these two distinguished mathema­ ticians have consented to assume these duties. The Nominating Committee of the Division, consisting of J, Barkley Rosser, Chairman, Mina Rees, and J, L. Walsh, met in New York City on December 1Z, 1959 and prepared recommendations for membership on committees of the Division, In its deliberations, the committee benefited from the advice of almost all the past and present committee chairmen. Several replacements due to resignations were made during the year. Thornton Page of the Van Vleck Observatory, Wesleyan Univer­ sity, replaced B. 0. Koopman as representative ofthe Operations Re­ search Society of America. R. M. Thrall, University of Michigan, re­ placed R. F. Rinehart on the Committee Advisory to the Office of Ord­ nance Research. P. S. Jones, University of Michigan, replaced E. P. Northrup as a representative of the National Council of Teachers of Mathematics on the U, S. Commission on Mathematical Instruction, a subcommittee of the U. S. National Committee on Mathematics. As in the past, one of the most important activities of the Divi­ sion has been to render advice in the selection of candidates for fel­ lowship awards. In its January meeting, the Committee on Postdoctor­ al Fellowships considered sixty-two applications for National Science Foundation Postdoctoral Fellowships and thirteen applications for Air Research and Development Command Postdoctoral University Re­ search Associateships. This represents a substantial increase over last year. The Committee on Mathematics Advisory to the Committee on International Exchange of Persons evaluated twenty-four applica­ tions for Fulbright grants for 1960-1961. At the September 19, 1959 meeting of the Committee on Mathe­ matics Advisory to the Office of Naval Research, the following modi­ fications were recommended in the Research Associate ship Program: 1. Addition of the University of Kansas to the eligible list of univer­ sities for 1960-1961, z. Inclusion of the following sentence in the associateship poster: "At the request of the university, the successful candidate will

693 teach a mutually acceptable course of an intermediate or ad­ vanced nature, for which he will receive an additional stipend not to exceed $1,500". The February 26, 1960 meeting of this Committee was devoted to evaluating forty-six applications for Research Associateships. From this group an ordered list of sixteen candidates was drawn up to assist the Office of Naval Research in making the six awards which were available. In reviewing the work of these committees, it is noted with re­ gret that there was a large number of duplicate applications from highly qualified candidates. This produced an unfortunate result in the Regular Postdoctoral Fellowship program of the NSF, where twenty-one fellowships were offered hut only thirteen awards were made, due primarily to the acceptance by the candidates of other awards. This has resulted ina significantreduction in the total num­ ber of postdoctoral fellowships in mathematics to be held during the coming year. H. F. Fehr, a memberoftheU.S. Commission on Mathematical Instruction, spent several months in the summer andfall of 1959 as­ sisting the Organization for European Economic Cooperation (OEEC) to organize a seminar on mathematical instruction in Western Europe. This seminar was· held in November 1959 with M. H. Stone, President of the International Commission on Mathematical Instruction (ICMI), as General Chairman. The U. S. National Committee forM athematics met in Princeton in May to discuss a proposal for a survey-conference for the Ameri­ can states similar in spirit and organization to the OEEC project described above. Tentative plans call for sponsorship by the ICMI and the Organization of American States. Further progress toward the implementation of the Bronk-Nes­ meyanov agreement between the U. S. National Academy of Sciences and the U. S. S. R. Academy of Sciences for exchange of scientists has been made during the year. Theagreementprovides for exchanges of twenty scientists in unspecified fields to give lectures and to take part in seminars and conferences; of twenty scientists in specific fields for short term visits (approximately one month); and of ten scientists in specified fields for longer term visits (six - twelve months). R. Courant, j. L. Doob, and S. Lefschetz have been nominated by Presi­ dent Bronk as scientists in the first category. No nominations have yet been made in the two mathematical fields specified for visits in the second and third categories: probability and stochastic processes (for approximately one month), and nonlinear mechanics and differen­ tial equations with applications to electrical and mechanical systems (for approximately six months). In another area of the agreement, the first semi-annual sche-

694 dule of scientific meetings in the United States to which Soviet scien­ tists are invited was sent to the Soviet Academy in early january. This list of twenty-three meetings contained the following six mathe­ matical conferences selected by the Division after consultation with the societies represented in the Division of Mathematics: 1960 Meeting, Place Organizer Feb. 19- Symposium on Differential Professor Carl B. Allen- 20 Geometry - Tucson, Arizona doerfer (Academicians A. D. Department of Mathematics Aleksandrov and A. V. University of Washington Pogorelov) Seattle 5, Washington April 5-7 Symposium on Plasticity - Professor E. H. Lee Providence, Rhode Island Graduate Division of Applied (Academicians A. A. Gvozdev, Mathematics Yu. N. Rabotnov, V. V. Brown University Sokolovskii, V. S. LenskiT Providence, Rhode Island April 12- Symposium on Combinatorial Dr. H. j. Griesmer 13 Problems - Princeton, I. B. M. Research Center New jersey P. 0. Box 218 Yorktown Heights, New York june 27- 4th Berkeley Symposium on Professor jerzy Neyman Aug. 6 Statistics and Probability - Statistical Laboratory Berkeley, California Department of Mathematics University of California Berkeley, California Aug. 1-28 Summer Institute on Finite Professor Marshall Hall, jr. Groups - Pasadena, Department of Mathematics California Ohio State University (Academician P. S. Novikov, Columbus 10, Ohio Professor A. G. Kuros) Summer Summer Conference on Professor john W. Carr, III Frontier Research and Computation Center Digital Computers - University of North Carolina Chapel Hill, North Carolina Chapel Hill, North Carolina (Candidate A. D. Edov) The responses to these invitations have been disappointing thus far. However, the Bronk-Nesmeyanov agreement by no means covers all of the possible avenues of exchange of scientific personnel between the United States and the U. S. S. R. The Division and the Office of International Relations of the Academy stand ready to assist other ex­ changes in any way possible. In particular, reports on any recent scientific contacts with Russian mathematicians are welcome. Such

695 information can be very helpful to future visitors to Russia. The joint American-Russian project to produce a Russian­ English and English-Russian mathematical dictionary has made good progress during the year. The American part of the effort in produc­ ing this dictionary is being done under National Science Foundation grants to the American Mathematical Society and the Rice Institute. Draft versions of the two parts have been exchanged and are currently under critical review by the American and Russian committees. It is expected that the final version will be ready for a printer during this year. The Committee on Travel Grants screened over thirty propo­ sals to the National Science Foundation for supportfor travel to for­ eign scientific meetings. Although most of this work was done by mail, the Committee met in Chicago in January to review a group of pending applications. On the basis of a unanimous recommendation from the Editorial Committee of "Mathematical Tables and Other Aids to Computation," the name of the journal was changed to" Mathematics of Computation" as of January, 1960. Noticeofthename change was sent to more than one hundred editors of scientific journals. This change in no way re­ flects a diminished interest in tables but rather a wider coverage of the field of the mathematics of computation. A strong promotional effort, supported by approximately $2,000 contributed by IBM, Bur­ roughs, and Remington-Rand, was made during the year to increase the number of subscriptions to the journal. Over 20,000 copies of a leaflet describing the aims of the quarterly have been mailed to scien­ tists and institutions most likely to be interested in computational pro­ blems. The number of subscriptions has been increased thus far from less than 1,200 to approximately 1,800. For the first time, all book dealers, both foreign and domestic, have been offered a uniform 10o/o discount on the subscriptions which they handle. Copies of the new cover together with an announcement of the name change were printed and distributed at the winter meetings of the Association for Com pu­ ting Machinery, Institute of Mathematical Statistics, Mathematical Association of America, American Mathematical Society, Society for Industrial and Applied Mathematics and American Physical Society. Finally, efforts are being made to establish a regular exchange of tables of contents with similar journals; one such exchange has been arranged with the "SIAM Review." The most effective support for "Mathematics of Computation" has been the editor, H. Polachek, and his Editorial Committee. Through their vigorous efforts, the journal is now appearing on schedule. The Committee on Applications of Mathematics has completed the preparation of the pamphlet on mathematical careers designed for high school students and guidance counselors. A new feature which has been introduced is a list of about forty mathematicians in a wide

696 variety of key positions who have agreed to serve as references for students seeking information about careers in their respective fields. A first printing of approximately 250,000 copies of the booklet is planned for the fall, supported by grants of $10,000 from IBM and $21,000 from NSF. The distribution will be made with the cooperation of the National Council of Teachers of Mathematics. The Committee on Statistics received a request for advice from the Army Chemical Warfare Laboratory asking for ways and means by which its statistical support could be improved. Several possible methods of providing the laboratory with better statistical advice are now being explored. It has been necessary to accept with regret the resignation of the Chairman of the Committee, J. W. Tukey, because of the heavy burden placed on him by his appointment to the Presi­ dent's Science Advisory Committee. In early May, 1959, the Division organized a small conference of thirty leading meteorologists and statisticians on the de sign and conduct of experiments in cloud physics and weather modification, The conference was supported by a grant from the National Science Foundation. A report on this conference has now been published under the title "The Skyline Conference on the De sign and Conduct of Experi­ ments in Weather Modification." This report, which emphasized the need for more careful statistical design in experiments on weather modification, was given wide circulation. After consultation with the Chairman of the Division's Committee on Statistics, with individual members of the International Statistical Institute, and with the Executive Committee of the Division, it was decided to recommend to the President of the Academy that he accept the invitation of the Japanese Government to designate a delegation to represent the Academy at the 32nd Session of the International Statis­ tical Institute to beheldinTokyo,May30 to June 9. The membership of the Academy delegation, which was appointed by President Bronk, follows: S. S. Wilks, Chairman (Princeton University), R. C. Bose (University of North Carolina), W. G. Cochran (Harvard University), Gertrude M. Cox (North Carolina State College), M .H. Hansen (Bureau of the Census), Jerzy Neyman (University of California-Berkeley), Frank W. Notestein (Population Research Council), W. J. Youden (National Bureau of Standards). With the exception of M. H. Hansen, all of themembersattendedthemeeting. The Committee on Revision of Mathematical Tables held a meet­ ing this spring. The Committee is in the process of reviewing the various chapters in the volume "Tables for the Occasional Computer", which has been prepared at the National Bureau of Standards. The Committee on Scientific Conferences explored the need for an international conference on Algebraic Groups which had been sug­ gested about a year ago, andfoundthatthe time did not seem ripe for such a meeting. In view of the expressed willingness of the Executive

697 Committee of the International Mathematical Union to entertain a pro­ posal for an international symposium in the United States in 1961, all members and former members ofthe Division were invited to suggest suitable topics. No suggestions were received; the matter of a pos­ sible international symposium at a later date is still under review, however. The Committee on Regional Development held a meeting during the Chicago meeting of the American Mathematical Society in January. The principal item discussed by this Committee at its meeting was a proposal for the establishment of two regional summer institutes in 1961 for graduate students in two differentfields of mathematics. The purpose of each institute would be to bring together twenty to twenty­ five of the best advanced graduate students in a given region of the country particularly interested in a given branch of mathematics and give them eight to ten weeks of intensive participation in courses and seminars in that field under the guidance of three or four of the best mathematicians in the country in the field. This proposal is still under study and discussion by the Committee. No use was made of the Revolving Funds for Publication of Ma­ thematical Books or Tables during the year. As of june 30, 1960, these funds stood at $3,388.61 and $16,476.45, respectively. Of the latter, $10,000 is invested and the remainder is available for immedi­ ate use by the Committee. In 1958, the Division participated in the drafting of a proposal by the Mathematical Association of America to the National Science Foundation to make a survey of mathematical employment in govern­ ment, industry and business. This survey is being conducted by the Bureau of Labor Statistics, working with an advisory committee which was nominated by the Chairman of the Division. Questionnaires have been prepared and these are now being mailed to the smaller com­ panies and laboratories, while the larger industries are being investi­ gated by personal interviews. Thedatagatheringphase of the survey will be completed by September and the approximately 10,000 replies will then be tabulated and analyzed. Preliminary findings may be available early next year. At the request of Lloyd V. Berkner, Chairman of the Space Science Board, an Ad Hoc Committee on Space Mathematics has been appointed with the following membership: j. Barkley Rosser, Chair­ man (Institute for Defense Analyses), R. B. Kershner (Johns Hopkins University), Brockway McMillan (Bell Telephone Laboratories), Marston Morse (Institute for Advanced Study), F. J. Murray (Office of Ordnance Research), C. V. L. Smith (National Aeronautics and Space Administration). The purpose ofthis committee is to survey the cur­ rent and potential contributions of the mathematical sciences to the solution of problems in space science. It will also recommend pro­ cedures by which the Space Science Board might have continuing ad-

698 vice and participation from the mathematical sciences. The commit­ tee met for the first time in Washington in january 1960. After a careful survey of the present status of the theory of traffic flow, the Academy - Research Council has authorized a study and summary appraisal of available information on analytical ap­ proaches to the problem of traffic flow and a publication of the find­ ings. This study will be supported by a grant from the Automotive Safety Foundation and will be under the joint direction of the Highway Research Board's Committee on the Theory of Traffic Flow and the Division of Mathematics. An Ad Hoc Committee on Traffic Flow has been appointed to undertake this study, with three members nominated by the Highway Research Board and threebythe Division. The com­ plete membership, with the representatives of the Division marked by an asterisk, is as follows: D. L. Gerlough, Chairman (Thompson Ramo Wooldridge, Incorporated), D. E. Cleveland (Texas A and M College), *H. P. Galliher (Massachusetts Institute of Technology), *E. W. Montroll (University of Maryland), *G. F. Newell (Brown Uni­ versity), C. C. Robinson (Automotive Safety Foundation). For the third consecutive year, the Division of Mathematics held its annual meeting as part of the program of the March meeting of the National Research Council. Sixteen members and six liaison representatives were able to attend. In an afternoon session, M. H. Trytten, Director of the Office of Scientific Personnel spoke on the fellowship programs under the aegis of the Academy- Research Coun­ cil, while B. Dees, Assistant Director for Scientific Personnel and Education, National Science Foundation, described the fellowship pro­ grams supported by the National Science Foundation. They were fol­ lowed by E. Hille andj. W. Carr, III, who presented short accounts of their recent trips to the Soviet Union. In an evening session, the Divi­ sion heard presentations from the following individuals representing various organizations concerned with mathematical education and cur­ riculum revision: M. H. Stone (International Commission on Mathe­ matical Instruction), A. W. Tucker (Commission on Mathematics, College Entrance Examination Board), E. G. Begle (School Mathema­ tics Study Group), M. Beberman (University of illinois Committee on School Mathematics), G. B. Price (Committee on the Undergraduate Program in Mathematics, Mathematical Association of America), J. R. Mayor (Science Teaching Improvement Program, American As­ sociation for the Advancement of Science). Following its customary practice, the Division distributed the announcement of fellowship and research opportunities and the list of visiting foreign mathematicians. Short announcements concerning these bulletins placed in several journals brought in many additional individual requests. S. S. Wilks, Chairman H. W. Kuhn, Executive Secretary

699 INTERNATIONAL CONGRESS OF APPLIED MECHANICS

The tenth International Congress of Applied Mechanics met at Stresa (Italy) from August 31 to September 7, 1960. The program of the Technical Sessions contained four invited general lectures and two hundred contributed papers. The general lecturers were G. Colonetti ("Deformations non-lineaires des solides"), A. Ferri ("Selected topics in hypersonic flow"), Yu. A. Mitropol'ski ("Methodes analy­ tiques dans la theorie des oscillations non-lineaires", in collaboration with N. Bogolyubov), and I. J. Stuart ("Non-linear effects in hydrody­ namic stability.") The repeated appearance of the term "non-linear" in these titles is worth noting, as it indicates the main emphasis of current research in applied mechanics. The contributed papers were presented in four simultaneous sessions, two each for mechanics of fluids and mechanics of solids. Compared to earlier Congresses, both the number ofpapers and that of simultaneous sessions had been drastically reduced. To achieve this, the Executive Committee of the Congress had established quotas for the maximum number of papers that would be accepted from the various countries. Abstracts of contributed papers had to be submitted before january 1, 1960. The number of abstracts received by the Executive Committee forced a revision of the original quotas. For example, the original quota of 35 for the United States was increased to 65, when over 300 abstracts had been submitted by authors from this country. Contrary to what had been expected, the severe limitation of the number of papers accepted for presentation at the Congress did not lead to a corresponding reduction of the Congress membership. While the advance registration indicated only 562 full members, the actual membership was stated to be about 750 full and 150 associate members. Unfortunately, the increase in the numbers of papers and participants made it necessary to hold the Technical Sessions in two buildings that were about one third of a mile apart. As is usual at meetings of this kind, the attempt to anticipate the distribution of in­ terests of the congress members and schedule papers to minimize commuting time proved futile. By splitting the membership into two ill-defined groups, this separation of meeting rooms made it difficult to locate members and thus defeated one of the purposes of the re­ duction in the number of simultaneous sessions. Since the large number of contributed papers at recent Con­ gresses of Applied Mechanics had led to an increase in the number of simultaneous sessions and a curtailment of presentation time that were manifestly absurd, a new acceptance procedure had to be de­ vised. It appears doubtful, however, that the one tried for the Stresa Congress achieves its goal. The wide variety in the quality of accepted

700 papers indicates that screening by abstracts is inadequate. For the U. S. National Congresses of Applied Mechanics, papers have to be submitted in full at least six months in advance. This facilitates screening, but excludes the presentation of up-to-date results. It will not be easy to devise a scheme that meets with general approval. Of the many recipes that were informally discussed at Stresa, the following is particularly noteworthy: no contributed papers; fifteen to twenty invited lectures reporting on significant achievements since the last Congress; and combination of the Congress with the two or three Symposia which the International Union of Theoretical and Applied Mechanics would normally organize in the congress year. W. Prager

THE OPTICAL SOCIETY OF AMERICA has been awarded a $50,000 grant by the National Science Foundation for the support of the translation and publication of the 1960 is sues of the leading sci en­ tific journal of the USSR Academy of Sciences, OPTIKA I SPEKTROS­ KOPIYA. OPTICS AND SPECTROSCOPY - the English title - is now in its second year of translation by working physicists and chemists who are also first rate Russian linguists. The Scientific Editors a:~;e Carl W. Garland of the Massachu­ setts Institute of Technology, Mathew Gibbons of the Naval Research and Development Laboratory and Cyrus Feldman of Oak Ridge Na­ tional Laboratory. The entire translation program is under the administrative supervision of Miss Patricia R. Wakeling, the Assistant Secretary of the Society. OPTICS AND SPECTROSCOPY has a circulation of nearly 6,000 and is published within three months of receipt of the Russian original.

701 PERSOIIAL ITEMS (This section is reserved for members of the Society)

Professor V. 0. McBRIEN of the College ofthe Holy Cross bas received a National Science Foundation Science Faculty Fellowship for the University of California, Berkeley. Professor WILLIAM PRAGER of Brown University was the first recipiant of the Theodore von Karman Award of the American Society of Civil Engineers on October 1Z, 1960. Associate Professor W. C. WILLIG of San Fernando Valley State College has received a National Science Foundation Fellowship for the academic year 1960-1961 at the University of North Carolina. Mr. J. P. ANDERSEN of the University of Nebraska has been awarded a National Science Foundation Fellowship at Harvard Uni­ versity. Dr. J. H. ABBOTT of Purdue University has been appointed to an associate professorship at the University of New Mexico. Associate Professor SHREERAM ABHYANKAR, on leave from johns Hopkins University, has been appointed a visiting lecturer at Harvard University. Dr. J. ABRAMOWICH of the University of British Columbia has been appointed to an assistant professorship at Assumption University of Windsor, Windsor, Ontario, Canada. Dr. R. D. M. ACCOLA of Harvard University has been appointed to an assistant professorship at Brown University. Mr. R. L. ADLERofYaleUniversityhasaccepted a position as associate mathematician at International Business Machines Corpo­ ration, Yorktown Heights, New York. Dr. D. B. AMES of Hughes Aircraft Company has been appointed to a professorship and department head at Orange County State Col­ lege. Assistant Professor N. AR TEMIADIS of the University of Wis­ consin has been appointed to a professorship at the University of Salonika, Salonika, Greece. Assistant Professor L. AUSLANDER, on leave from Indiana University, has been appointed to a research associateship at Yale University. Miss D. M. BAMBERGER has been appointed to a teaching assistantship at Cornell University. Dr. R. E. BARLOW of Sylvania Electric Products Incorporated has accepted a position with the Institute for Defense Analyses. Assistant Professor E. H. BATHOoftheUniversityof Roches­ ter has been appointed to an associate professorship at the University of New Hampshire.

70Z Assistant Professor B. j. BEECHLER of Wilson College has been appointed to an associate professorship at Wheaton College. Assistant Professor P. R. BEESACK of McMaster University has been appointed to an associate professorship at Carleton Univer­ sity. Mr. H. BEIM AN of the Operations Evaluation Group, Pentagon, has accepted a position as mathematician with International Business Machines Corporation, Bethesda, Maryland. Assistant Professor D. C. BENSON of Chico State College has been appointed to an associate professorship at South Dakota School of Mines and Technology. Dr. G. G. BILODEAU of Sylvania Electric Products Incorpo­ rated has been appointed to anassistantprofessorshipat Boston Col­ lege. Dr. F. T. BIRTEL of the University of Notre Dame has been appointed to an assistant professorship at Ohio State University. Dr. J. F. BLACKBURN of International Business Machines Corporation has accepted a position as manager, intermediate pro­ grammer training, at the International Business Machines Corpora­ tion, New York, New York branch. Dr. W. A. BLANKINSHIP of the National Security Agency, De­ fense Department, has accepted a position as mathematician at the Defense Department, Ft. George G. Meade, Maryland. Dr. W. W. BLEDSOE of Sandia Corporation has accepted a po­ sition as mathematician with Advanced Research, Palo Alto, Califor­ nia. Miss A. E. M. BODE of the University of Saskatchewan has been appointed a lecturer at Huron College, London, Ontario, Canada. Mr. W. D. BOUWSMA of the University of Michigan has been appointed to an assistant professorship at Pennsylvania State Univer­ sity. Dr. L. D. BRAM has accepted a position as mathematician at the Office of Naval Research, Washington, D. C. Reverend E. W. BRANDE of St. Louis Universityhas been ap­ pointed research associate at Woodstock College. Dr. G. E. BREDON ofthelnstituteforAdvanced Study has been appointed to an assistant professorship at the University of California, Berkeley. Dr. T. F. BRIDGLAND, Jr. of Boeing Airplane Company has accepted a position as senior research analyst with Radiation Incorpo­ rated, Orlando, Florida. Dr. J. L. BRITTON, on leave from the University of Glasgow, has been appointed to a visiting assistant professorship at the Univer­ sity of illinois. Assistant Professor R. C. BZOCH of the Universityof Minne­ sota has been appointed to an assistant professorship at the University of Utah.

703 Professor Emeritus C. C. CAMP of the University of Nebraska has been appointed to a visiting professorship at the University of Puerto Rico, M ayaguez, Puerto Rico. Dr. C. E. CAPEL of Westinghouse Electric Corporation has been appointed to a professorship at Miami University, Oxford, Ohio. Dr. F. S. CATER of the University of Southern California has been appointed to an assistant professorship at the University of Oregon. Dr. R. CHALKLEY has been appointed to an assistant profes­ sorship at Knox College. Dr. C. H. CHICKS of the University of Oregon has accepted a position as advanced research engineer at Sylvania Electric Products Incorporated, Mountain View, California. Assistant Professor j. CHRISTOPHER of Sacramento State College has been appointed associate professor and director of the computing center at the University of Nebraska. Associate Professor E. COMEGYS of the University of Maine has been appointed a lecturer at Wellesley College. Dr. F. F. CONNOR of lllinois Institute of Technology has been appointed to a research instructorship at Louisiana State University. Professor j. CORNFIELD of johns Hopkins University has ac­ cepted a position as mathematical statistician at the National Institute of Health, Bethesda, Maryland. Assistant Professor L. j. COTE of Syracuse University has been appointed to an associate professorship at Purdue University. Mr. j. D. COWIE of Rutgers, The State University, has accepted a position as member of the technical staff at the MITRE Corporation, Bedford, Massachusetts. Dr. W. F. DAVIS ON of the University of Michigan has accepted a position as senior mathematician at the Alpha Corporation, Richard­ son, Texas. Assistant Professor M. H. De GROTT, on leave from Carnegie Institute of Technology, has accepted a position at the Western Man­ agement Science Institute at the University of California, Los Angeles. Brother L. De La SALLE of St. Mary's College has been ap­ pointed dean at Lewis College. Professor A. H. DIAMOND of Webb Institute of Naval Architec­ ture has been appointed to a professorship at Stevens Institute of Technology. Mr. R. B. DiFRANCO of Rutgers, The State University, has been appointed to a teaching assistantship at Indiana University. Dr. A. E. DOLD of the University of Heidelberg has been ap­ pointed to an assistant professorship at Columbia University. Dr. A. M. DUGUID of Brown University has accepted a position with Esso Petroleum Company, London, England.

704 Mr. A. L. DUQUETTE has been appointed to an assistant pro­ fessorship at the University of Kentucky. Associate Professor H. A. DYE of the State University of Iowa has been appointed to a professorship at the University of California, Los Angeles. Associate Professor j. M.G. FELL, on leave from the Univer­ sity of Washington, Seattle, has been appointed a research associate at Harvard University. Professor H. FREUDENTHAL of the University of Utrecht, will be on leave for the academic year 1960-1961 at Yale University. Associate Professor W. B. FULKS of the University of Min­ nesota has been appointed to a professorship at Oregon State College. Dr. I. S. GAL of Yale University has been appointed to a visit­ ing associate professorship at the University of Minnesota. Dr. E. D. GAUGHAN of the University of Kansas has been ap­ pointed to an assistant professorship at New Mexico State University. Dr. W. M. GILBERT of Princeton University has been appoint­ ed to an associate professorship at Iowa State University. Dr. G. W. GOES of Northwestern University has been appointed to an assistant professorship at DePaul University. Mr. R. GOLDBERG of the University of Michigan has accepted a position as mathematician programmer at International Business Machines Corporation, White Plains, New York. Dr. LEON GREENBERG, on leave from Brown University, has been appointed to a fellowship at the University of Copenhagen, Den­ mark. Associate Professor M. HENRIKSEN, on leave from Purdue University, has been appointed to a professorship at Wayne State Uni­ versity. Associate Professor G. A. HEUER of Concordia College has been appointed to a visiting assistant professorship at the University of Nebraska. Professor S. B. JACKSON of the University of Maryland has returned after spending the academic year 1959-1960 atthe University of Washington, Seattle. Professor j. H. B. KEMPERMAN, on leave from Purdue Uni­ versity, will spend the academic year 1960-1961 at the University of Wisconsin. Dr. E. R. KEOWN of Aero jet General Corporation has been ap­ pointed to a professorship at the Agricultural and Mechanical College of Texas. Dr. M. I. KNOP of the Institute for Advanced Study has been appointed to an assistant professorship at the University of Wisconsin. Dr. E. T. KOBAYASHI of the University of Washington has been appointed to a visiting assistant professorship at Northwestern University.

705 Assistant Professor C. W. KOHLS of the University of illinois will spend the academic year 1960-1961 as a visiting assistant pro­ fessor at the University of Rochester. Mr. J. C. KOKENofSyracuse, New York, has been appointed to an associate professorship at the State University of New York. Dr. K. KREITH of the University of California, Berkeley, has been appointed to an assistant professorship atthe University of Cali­ fornia, Davis. Professor KUO-TSAI CHEN of the Instituto Tecnologico de Aeronautica, Brazil, will spend the months of December, 1960, through February, 1961, at the Institute for Advanced Study. Associate Professor JOACHIM LAMBEK, on leave from McGill University, has returned after spending the academic year 1959-1960 at the Institute for Advanced Study. Associate Professor C. E. LANGENHOP of Iowa State Univer­ sity has accepted a position as director of research at Mathematica, Princeton, New Jersey. Mr. L. L. LAYTON of Auburn University has accepted a posi­ tion as mathematician with David Taylor Model Basin, Washington, D. C. Assistant Professor G. F. LEGER, Jr. of the University of Pittsburgh has been appointed to an associate professorship at West­ ern Reserve University. Mr. BERNARD LEVENSON of Columbia University has been appointed to a lecturer and research associate at Johns Hopkins Uni­ versity. Assistant Professor F. LEVIN of the University of Kentucky has been appointed to an assistantship at Manchester College of Science and Technology, Manchester, England. Mr. H. W. LEW of Hughes Aircraft Company has accepted a position as staff scientist at System Development Corporation, Santa Monica, California. Assistant Professor W. LITTMAN of the University of Wiscon­ sin has been appointed to an assistant professorship at the University of Minnesota. Dr. J. E. McFARLAND of Oregon State College has been ap­ pointed to an assistant professorship at Western Washington College. Dr. A. W. McKINNEY of General Electric Company has accept­ ed a position as staff member at Sandia Corporation, Livermore, California. Dr. J. J. McMAHON of the National University has been ap­ pointed to a professorship at St. Patricks College, Maynooth, Ireland. Dr. D. G. MALM of Royal Holloway College has been appointed to an assistant professorship at the College on Long Island, State Uni­ versity of New York.

706 Associate Professor A. V. MARTIN of the University of New Mexico has been appointed to a professorship at the State University of New York. Mr. JOHN S. MOORE, jr. has recently been appointed Director of the Division of Terminal Electronics at Florida Christian College. He continues his position as head of the divisions of Physics and Ma­ thematics at the same institution. Associate Professor G. W. MORGANTHALER of the University of illinois has accepted a position as manager, operations research department, The Martin Company. Professor J. L. NOWINSKI of the University of Wisconsin has been appointed to a professorship at the University of Texas. Assistant Professor T, D. OXLEY, jr. of Kansas State Univer­ sity has been appointed to an assistant professorship at Drake Univer­ sity. Professor j. C. OXTOBY of Bryn Mawr College will spend the academic year 1960-1961 as a research associate at Yale University. Mr. S. T. PAINE of Stanford University has accepted a position as senior research engineer at CONVAIR, San Diego. Dr. A. RALSTON of Bell Telephone Laboratory has accepted a position as manager of Technical Computing at American Cyanamid Company, New York, New York. Mr. P. REAL of Westinghouse Electric Corporation has accept­ ed a position as manager, operations analysis department at the Cook Electric Company, Franklin Park, illinois. Dr. j. R. RICE of the National Bureau of Standards has accept­ ed a position with General Motors Corporation, Warren, Michigan. D. F. RIDDLE of the UniversityofTexashas been appointed to an assistant professorship at Texas Lutheran College. Dr. D. S. RIM of Columbia University has been appointed to an assistant professorship at Brandeis University. Mr. R. W. RITCHIE of Princeton University has been appointed a lecturer at Dartmouth College. Dr. E. R. RODEMICH of Massachusetts Institute of Technology has been appointed to an assistant professorship at the University of Minnesota. Mrs. J, ROSENSTARK of New York Universityhas received a fellowship at New York University for the academic year 1960-1961. Dr. G. 0. SABIDUSSI of Tulane University has been appointed to an associate professorship at McMaster University. Associate Professor I. SA TAKE, on leave from the University of Tokyo for the academic year 1959-1960, at the Institute for Ad­ vanced Study has returned to the University of Tokyo. Professor E. V. SCHENKMAN, on leave from Louisiana State University at University College of North Staffordshire, has returned to Louisiana State University.

707 Assistant Professor E. SEIDEN of Northwestern University has been appointed to an associate professorship at Michigan State University. Dr. T. I. SEIDMAN of the University of California, Livermore, has been appointed lecturer at the University of California, Los Angeles, for the academic year 1960-1961. Dr. TETSUNDO SEKIGUCHI of Oklahoma State University has been appointed to an assistant professorship at the University of Ar­ kansas. Dr. B. SHERMAN of Westinghouse Electric Corporation has ac­ cepted a position as mathematician at Space Technology Laboratory, Los Angeles, California. Associate Professor W. A. SMALL of Grinnell College has been appointed to a professorship at Tennessee Polytechnic Institute. Mr. S. SRIVASTAVA of Purdue University has been appointed to an assistant professorship at the University of Lucknow, Lucknow, India. Assistant Professor T. TAMURA of Tokushima University has received a visiting associate professorship at the University of Cali­ fornia, Davis. Assistant Professor H. G. TUCKER, on leave from the Univer­ sity of California, Riverside, at the University of California, Berkeley, has returned to the University of California, Riverside. Mr. R. S. WASSERMAN of Princeton University has accepted a position as programmer at International Business Machines Corpora­ tion, Wappingers Falls, New York. Dr. C. E. WATTS of the University of Chicago has been ap­ pointed to a fellowship at the Institute for Advanced Study. Miss L. M. WEBB of Sandberg-Serrell Corporation has accept­ ed a position as mathematician at the National Engineering Science Company, Pasadena, California. Mrs. A. K. WEINSTEIN of Litton Industries has accepted a po­ sition as research engineer at A. C. Spark Plug Division of General Motors Corporation, El Segundo, California. Associate Professor F. L. WOLF, on leave from Carleton College, has been appointed to a research associate at the University of California, Berkeley. Assistant Professor T. YEN of Illinois Institute of Technology has been appointed to an assistant professorship at Michigan State University. The following promotions are announced: MAX BEBERMAN, University of illinois, to a professorship. E. CALABI, University of Minnesota, to a professorship. D. J. CHRISTENSEN, Reed College, to anassistantprofessor- ship.

708 N. COBURN, University of Michigan, to a professorship. R. C. Di PRIMA, Rensselaer Polytechnic Institute, to an asso­ ciate professorship. M. L. EARLY, Eastern Michigan University, to an associate professorship. D. H. ERKILETIAN, University of Missouri, to a professorship. J. M. G. FELL, University of Washington, to an associate professorship. R. M. GUNDERSEN, Illinois Institute of Technology, to an as­ sociate professorship. T. E. HARRIS, Rand Corporation, Santa Monica, California, to head, mathematical analysis department. H. H. HARTZLER, Mankato State College, to a professorship. F. S. HOLT, Tufts University, to an associate professorship. R. JUBERG, University of Minnesota, to an assistant profes- sorship. W. S. LOUD, University of Minnesota, to associate chairman. MARY LISTER McCAMMON, Pennsylvania State University, to an associate professorship. L. MARKUS, University of Minnesota, to a professorship. B. C. MEYER, Univer!'!ity of Colorado, to an associate pro­ fessorship. W. F. REYNOLDS, Tufts University, to an associate professor- ship. W. C. ROYSTER, University of Kentucky, to anassociate pro- fessorship. I. SAT AKE, University of Tokyo, to an associate professorship. J. B. SERRIN, University of Minnesota, to a professorship. F. L. WOLF, Carleton College, to an associate professorship. The following appointments to instructorships are announced: Baghdad University: Mr. M. J. SAADALDIN; University of Buffalo: Mr. R. J. BENICE, Dr. JUDITH R. BLANKFIELD; California Institute of Technology: Dr. G. D. CHAKERIAN; California State Poly­ technic College: Mr. M. L. CLINNICK; University of Chicago: Dr. PAUL PONG; Clark University: Mr. B. L. BEVELHEIMER; University of Dayton: Dr. T. A. DWYER; Lafayette College: Mr. B. HORELICK; Louisiana State University: Dr. E. H. CONNELL; University of Michi­ gan: Dr. L. C. EGGAN; Swarthmore College: Dr. STEVENS HECK­ SCHER; College of Wooster: Mr. R. G. TOBEY; Yale University: Dr. J. A. LINDBERG, Jr.

709 LETTERS TO THE EDITOR

Editor, the NOTICES One of us recently submitted to a Society publication a manu­ script that had been produced by utilizing the services of a ditto­ machine operator. In acknowledging its receipt, the editor: ( 1) re­ quested a written guarantee that the same mathematical production was not being concurrently submitted to any other journal; (Z) alleged that printers might not accept ditto copy. Concerning (1): His request has been refused--itisan attempt by an officer of the Society to enforce professional integrity by means of a disclaimer. It would be pleasant to learn that every member of the Society has sufficient intelligence to see that any device resem­ bling a loyalty oath is useless as a safeguard against malpractices; apparently, however, this cannot be affirmed even of every officer of the Society. In a letter that appeared in the june, 1958 issue of these NO­ TICES, an equally useless safeguard was proposed: that only type­ scripts be deemed acceptable. The three replies that appeared in the succeeding issue seemed adequate in pointing out the worthlessness of such a rule as well as the inconvenience it would cause, and in em­ phasizing the many advantages in producing a paper by ditto or other duplicating processes. Nevertheless, the problem has arisen again, and in a considerably nastier form than before. Why should it still be necessary to defend ditto? Our goal is to exchange mathematical information; ditto wins over carbon copies on all counts in speed, accuracy, and economy. In preparation of a manu­ script for publication, time is saved, copying errors reduced, and needless correspondence avoided because handwritten symbols (in­ cluding instructions to the printer) are inserted once and for all; the author's copy contains a given mistake if and only if the referee's and printer's copy does. Furthermore, interested specialists are supplied with preprints immediately--usually a year or more before the ap­ pearance of the published paper. To discourage ditto is to obstruct the communication of mathematical results. Concerning (Z): It is conceivable that because some ditto copy is unskillfully prepared, an editor or printer might falsely equate purple with illegible and refuse to read any purple copy. The correct issue is legible vs. illegible. We have seen many dittoed manuscripts that were neat and legible; we have seen equally many typescripts that were messy, lightfaced and scrawny-lettered. If a printer actually turns up who refuses to read neat, legible copy, then don't hire him. Leonard Gillman Carl W. Kohls Ralph A. Raimi

710 Editor, the NOTICES Budapest, 1960 IX 29 Received your letter of July 15 a few days ago.• I am very glad that Chandler is back at the Reviews. Kac writes that his working conditions are excellent. Gould should have a nice time in Hongkong, a very beautiful place. Now about recent happenings. From Canberra I went to Bris­ bane, Sydney, Melbourne, Adelaide, Perth; everywhere the univer­ sities try to expand and new univs are founded but everywhere there is a shortage of mathematicians. From Perth I went to Singapore for two days and then to Budapest via Istanbul. In Australia I could get no Turkish visa since there is no Turkish consulate there; at first the authorities did not want to let me leave the airport but the BOAC ar­ ranged it and I could visit the Bosporus. I arrived to Budapest on July 29. The math congress here was held Aug 24-31, there were more than 200 foreigners about 50 from the West, many Russians were pre­ sent, unfortunately no Chinese. The congress was very pleasant + successful; one new+ useful feature was that some of the meetings did not have a fixed program, but problems were raised. In some of the subjects this worked very well. Then I went to Oberwolfach where there was a symposium on number theory, about 40 maths were present from Germany, Austria, England, Hungary; Lewis and Leveque from the US. A small meeting has the great obvious advantage of personal contact, the lectures in Oberwolfach were 20 minutes + there was plenty of time for discus­ sions. From Oberwolfach I went to Innsbruck, the Austrian math meet­ ing; Innsbruck is one of the beauty spots of the world, more than 300 foreigners were present, relatively few from the east (except from Hungary). In Poland there was a meeting in memory of Banach (it overlapped with Oberwolfach) and e.g. Klee visited Budapest, Poland + lnnsbruck. At the moment there is a Jugoslav math congress at Belgrade, but I had no time to go there (I will visit there in Novem­ ber). I will be in Hungary for 6 weeks, then go to the meeting in Ber­ lin ( 150 anniversary of the Humboldt Univ) and later in November to Haifa for 3 months. Szego spent in Hungary two months + gave several lectures, he left for Austria, Greece and Israel, we will meet again in Berlin. There are several scientist here at present, Bernal, Blackett, Cockroft from England (all of them preached), Sabin from US + Frank from USSR. Kind regards to all P. E. (Paul Erdos)

711 MEMORANDA TO MEMBERS

THE EMPLOYMENT REGISTER The Mathematical Sciences Employment Register, established by the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics, will be maintained atthe Annual Meeting atthe Willard Hotel in Wash­ ington, D. C., on january Z4, Z5, and Z6, 1961. The Register will be conducted from 9:00 A.M. to 5:00 P.M. on each of these three days. The Employment Register Desk will be located in room ZZB. There is no charge for registering either to job applicants or to em­ ployers, except when the Late Registration Fee for employers is ap­ plicable. Provision will be made for anonymity of applicants upon re­ quest and upon payment of $1 to defray the cost involved in handling anonymous listings. job applicants and employers who wish to be listed will please write to the Employment Register, 190 Hope Street, Providence 6, Rhode Island, for application forms and for position description forms, which must be completed and returned to Providence not later than january 4, 1961, in order to be included free of charge in the listings at the meeting in Washington, D. C. Forms which arrive after this closing date, but before January 17, will be included in the listings at the meeting for a Late Registration Fee of $3.00, and will also be in­ cluded in the printed listings, but not until ten days after the meeting. The printed listings will be available for distribution both during and after the meeting. The prices are as follows: Position descriptions, $Z.OO; listing of applicants, academic only, $5.00; comprehensive listing of applicants, academic, industrial, and government, $ZO.OO. It is essential that applicants and employers register at the Employment Register Desk promptly upon arrival at the meeting to facilitate the arrangement of appointments.

EMPLOYMENT OF RETIRED MATHEMATICIANS The Headquarters Offices of the American Mathematical So­ ciety, 190 Hope Street, Providence 6, Rhode Island, again plans to is­ sue the List of Retired Mathematicians Available for Employment, which it has maintained yearly. Mathematicians who are retiring this year and who are inter­ ested in being included in the list are asked to send the following in­ formation to the Headquarters Offices: name; date of birth; highest degree and where obtained; most recent employment; present address; date available; preferences, including preference for academic or in­ dustrial employment.

71Z Data thus received will be incorporated into the next issue of the List, the issuance date of which will be announced in the February issue of the NOTICES.

ABSTRACTS OF PAPERS BY TITLE A revision of the procedure for handling papers by title has been instituted. Under the new procedure the author will offer his pa­ per to the Society by submitting the usual abstract marked "by title." Such abstracts will be checked editorially and judged for suitability for publication just as in the past. The change of procedure lies in the fact that the paper will not be placed on the supplementary program of any meeting. Instead there will be a program of papers by title in each number of the NOTICES (with exceptions when the number threatens to become unwieldy). It will be assembled following a convenient dead­ line, not ordinarily the same as that for meetings, and will contain all suitable papers received since the previous deadline. The next two deadline dates following the publication of an issue will appear on the back of the title page. Abstracts will be published in the earliest pos­ sible issue just as at present.

RECIPROCITY AGREEMENT WITH THE SOCIETE MATHEMATIQUE DE BELGIQUE The American Mathematical Society has entered into a reci· procity agreement with the Societe Mathematique de Belgique by which members of each may become members of the other by paying half the regular dues. The regular dues of members of the Societe Mathema­ tique de Belgique living abroad are 100 Belgian francs; therefore an American Mathematical Society member would pay 50 Belgian francs, or one U. S. dollar. Privileges of membership include free attendance at meetings (usually in Brussels) of the Societe Mathematique de Belgique and receiving the Bulletin de la Societe Mathematique de Belgique, now in its twelfth year. Those members of the American Mathematical Society wishing to take advantage of this arrangement should write to: Mr. Guy Hirsch, Secretaire, Societe Mathematique de Belgique, Secretariat: 317, Avenue Ch. Woeste, Bruxelles 9, Bel­ gium. It is understood that members under the reciprocity agreement spending time in the other country should pay the regular dues while they are there.

713 NEW PUBLICATIONS

Abbott, I. H. and von Doenhoff, A. E. Theory of wing sections: Includ­ ing a summary of airfoil data. New York, Dover, 1959. x + 693 pp. Paperbound, $2.95. deBroglie, Louis. E:Iements de theorie des quanta et de mecanique ondulatoire. Traite de Physique Theorique et de Physique Ma­ thematique, III. Paris, Gauthier-Villars, 1959. viii+ 302 pp. 30.00 NF. Dahuti, D. G. See Esbi, u. Ross. Dickson, L. E. Algebras and their arithmetics, New York, Dover, 1960. xii + 241 pp. Paperbound, $1.35. von Doenhoff, A. E. See Abbott, I. H. Eddington, A. S. The internal constitution of the stars. With a new in­ troduction by Lloyd Motz. New York, Dover, 1960. xvi + 407 pp. Paperbound, $2.25, Esbi, U. Ross=Ashby, W. Ross. Vvedenie v kibernetiku. [An introduc­ tion to cybernetics.] Translated from the English by D. G. Dahuti; edited by V. A. U spenskit; with a preface by A. N. Kol­ mogorov. Moscow, Izdat, Inostr. Lit., 1959.432 pp. 17.40 r. Gale, David. The theory of linear economic models. New York, Mc­ Graw-Hill, 1960. xxi + 330 pp. $9.50. Goertzel, Gerald and Tralli, Nunzio. Some mathematical methods of physics. New York, McGraw-Hill, 1960. xiii + 300 pp. $8.50, Hartley, E. M. Cartesian geometry of the plane. New York, Cambridge University Press, 1960, xi+ 324 pp. $3.75, Hohn, F. E. Applied Boolean algebra: An elementary introduction. New York, Macmillan, 1960. xx + 137 pp. Paperbound, $2.50. Jackson, J. D. The physics of elementary particles, Investigations in Physics, Vol. 9. Princeton, Princeton University Press, 1958. X + 135 pp. $4.50, Kolmogorov, A. N. See Esbi, U. Ross. Lorenzen, Paul. Die Entstehung der exakten Wissenschaften. Ver­ standliche Wissenschaft, Bd. 72. Berlin, Springer-Verlag, 1960. v + 163 pp. DM 8,80. Mandl, F. Introduction to quantum field theory, New York, Inter sci­ ence, 1959. vii+ 202 pp, $6.00, Maxwell, E. A. Advanced algebra. Part I. New York, Cambridge Uni­ versity Press, 1960, xii + 311 pp. $2.75. Motz, Lloyd. See Eddington, A. S. Roark, R. J. Formulas for stress and strain. 3rd ed. New York, Mc­ Graw-Hill, 1954. xiii + 381 pp. $8.25. Robinson, E. A. An introduction to infinitely manyvariates. Griffin's Statistical Monographs and Courses, No, 6. New York, Hafner, 1959. 132 pp. Paperbound, $4.75, Tralli, Nunzio. See Goertzel, Gerald. Uspenskii, V. A. See E!:lbi, U. Ross. 714 ABSTRACTS OF CONTRIBUTED PAPERS

THE OCTOBER MEETING IN WORCESTER, MASSACHUSETTS October 22, 1960

572-12. WITHDRAWN.

572-13. B. K. Bender and A.]. Goldman: Maximal cellular Boolean functions.

A cellular Boolean function of.!! variables is a proper subset S of the vertices of the unit ,!!-cube In, such that every component (= maximal connected subset) of S is a face of In. Let Mn be the maximum possible number of ver­ tices in such a function. Theorem 1: Mn = 2n-l. Theorem 2: For each S,

0 ;ad .c: n, and each d-face Fd of In, there is a unique cellular function of !!: variables with Mn vertices, Fd as one component, and every component a d-face; the complement of this function also has 2n-d- 1 components, all d-faces.

Theorem 3: Every cellular function of.!! variables with Mn vertices is of the type described in Theorem 2. (Received August 1, 1960.)

572-14. V. E. Be'leif: Asymptotic behavior of general queues with one server. I

Let K( •) be a nondecreasing stochastic step-function, and W(·) the solu­ tion of W(t) = K(t) - t +,/otU(- W(u))du, where U(·) is the rt-continuous unit step 0, with K(O) = W(O) = 0. K(·) represents the cumulative load, and W(•) the virtual delay in a single-server queue with order or arrival service. An earlier paper reduced the study of distr [W( • >} to finding P(.) = Pr [W(·) = 0} from the equation F(t) = /oR(t,u)P(u)du, with F(t) = E{maxi!J,t - K(t)JJ and R(t,u)

= Pr {K(t) - K(u) ~ t - u IW(u) = 0}. Tauberian methods are now used to study the asymptotic behavior of P(·) under the "stationarity" assumption R(t,u) = R(t - u). The principal results give conditions under which P(t)----? F '(oo)/R(oo) as t -+oo, obtained by the Wiener-Pitt theorem on reciprocals of Fourier­ Stieltjes transforms, and a variant of Ikehara's theorem. (Received August 26, 1960.)

715 57Z-15, F. T. Birtel: Isomorphisms and isometric mutipliers.

Notation and terminology are as in Abstract 571-69, Algebras of multi­ pliers are a natural generalization of the measure algebra of a locally compact abelian group, In analogy with measure algebras, it is shown that an isomor­ phism of commutative semi-simple Banach algebras extends to an isomorphism of their associated algebras of multipliers and that an isomorphism of algebras of multipliers vanishing at infinity of two regular commutative semi-simple Banach algebras which are tauberian induces an isomorphism of the underlying algebras, Also if A is a commutative semi-simple Banach algebra with dis­ crete maximal regular ideal space, then every automorphism of Am induces an automorphism of A provided that A is spanned by its idempotents, Under the assumption that the unit ball of Am is weak operator compact the isometric onto multipliers are shown to form a compact topological group G. Using me­ thods of vector-valued integration, A is shown to be strongly spanned by the common eigenvectors of G which have eigen-values of modulus 1, Furthermore, if G separates points ofo/1(A), then ?'i((A) is discrete and A is spanned by its idempotents. (Received August 23, 1960.)

57Z-16. J. L. Brenner: A new property of the Jacobi symbol.

Let m, n be positive integers with no common factor; n odd; n = n 0s 2, where n0 is either 1 or the product of distinct primes p l'"''Pt• and s is ah integer. The Jacobi symbol (m/n) is defined as +1 if n 0 is 1, or if m is a quadratic nonresidue of an even number of p 1, .. ,pt; otherwise (m/n)= -1. The new property is the following. Define the equivalence relation r 1 ""r2 among the residue classes mod h to mean that there is a positive integer x such that mXr 1 i:.r2 (mod n). The number of equivalence classes under this relation is even if (m/n)= -1, and odd if (m/n)= +1. (Received July 26, 1960.)

572-17. R. C. Buck: Zero sets for continuous functions.

Let V be a linear subspace with dimension N in the space C [DJ of real valued continuous functions defined on a domain D. Theorem: There exist

716 disjoint nonempty open sets 0 1, o 2 , ... ,DN in D and a constant B such that for any f E V, and any points p 1 € D1, pz E Dz•···•PN E DN, maxpeoif(p)l ~ B{if(p 1)i + if(p2)i + ••• +if(pN)I}. In particular, the sets Dj are uniqueness domains for V; if f and g are in V and if f(q) = g(q) for at least one choice of q in each of the sets Dj' then f = g. For the latter statement to hold, there need be no other restriction on the sets Dj; any collection of disjoint open sets can be uniqueness domains for a space V C C fp]. (Received July 18, 1960.)

572-18. Nathaniel Coburn: Intrinsic characteristic equations for relativistic compressible hydromagnetic flows.

Two topics are discussed in this paper. First, intrinsic forms of the basic equations (the conservation of mass, stress-energy, entropy along a streamline and the Maxwell relation which does not involve the unknown current) are determined by expressing the unit world line vector, uj, and the relativistic magnetic field, Hj, in terms of a unit normal space-like vector, jlij, and its corresponding unit time-like bicharacteristic, tl. The basic equations can then be expressed in terms of directional derivatives of the density, p, and the magnitude of the relativistic magnetic field vector, b, along tl, jlii, and vi, wl (two mutually orthogonal unit space-like vectors which are perpendicular to jlii, tl) and the curvatures of these four congruences, Secondly, the case when the characteristic surfaces are oo 1 parallel hyperplanes of special relativity is studied. Explicit expressions for f• b, HJ, ul, are determined. Further, explicit solutions for the density, the velocity, and Newtonian magnetic field are determined for the corresponding Newtonian flow. (Received August 1, 1960.)

572-19. Eckford Cohen: Arithmetical notes, IV. A set of integers related to the divisor function.

Let q be an odd prime and let t"(n) denote the number of divisors of the positive integer n. Further, let Sq(x) denote the number of n ~ x such that qit:(n). A result of Pillai asserts that Sq(x) = (1 - ~(q)/~(q - l))x + O(xl/q-llog x), where (s) is the Riemann t-function. The object of this note is to improve the estimate for Sq(x) by reducing the order of the remainder to O(xl/q-1), This is achieved in an elementary way on the basis of a simple esti­ mate for the average order of the number t'k(n) of k-th power divisors of n, (Received August 3, 1960 .) 717 572-20. Eckford Cohen: Averages of completely even arithmetical functions over certain types of plane regions.

Let n, a, b denote integers, n :> 0, (a,b) the g.c,d. of a and b, and let f(n) denote a suitably restricted arithmetical function. This paper is concerned with the average order F i(x) ::= Lf((a,b)) of the function f((a,b)) over the lattice

points fa,bJ of the following regions Ri of the X, Y plane, R 1: X+ Y ~ x, X ~ 0, Y <:; 0 (triangular); R2: X2 + Y2 ;::; x (circular); R 3 : XY & x, X:> 0, Y :> 0 (hyperbolic). For the case in which f(n) is the Euler function ¢(n), (

the following estimates are obtained: '!' 1(x) "'1Sx2log x/11..4, 2 (x) "'-' l8x log x/tr3 , ct> 3(x) "-' 9x log2x/1r4 as x --7 oo. These results are corollaries of much more precise eStimates proved for F i (x) in each of the three cases. The treatment is elementary. (Received August 25, 1960.)

572-21. E. A. Maier: A class of error-correcting codes over a finite field.

A code over a finite field, Fq, of order q is a mapping, ¢, of a vector space, Vk(F q), of dimension k over Fq into a vector space, V n k over Fq. If A E Vk (F q), let matrix from F q such that any k X k submatrix of C is nonsingular; It is shown that these codes are e-error-correcting and, given D E

572-22, J. B. Rosen: Partition programming.

A method is given for solving large linear programming problems, which possess a block diagonal structure, by a partition of the original matrix

718 and an iterative solution of the resulting subproblems. This method which is based on the solution of the subproblems using gradient projection (J. B. Rosen, J, Soc. lndust. Appl. Math. vol. 8 (1960) pp. 181-21 7), differs from the de com­ position principle of Dantzig and Wolfe (Operations Res. vol. t! (1960) pp. 101- 111), It is shown that the optimality condition for the original complete problem is given in terms of the optimal solutions, inverse bases, and shadow prices for the subproblems. Under certain conditions the method can be extended by an iterative linearization to the nonlinear constraint programming problem in which the matrix coefficients and right-hand sides are (linear or nonlinear) functions of a set of auxiliary variables. (Received September 6, 1960.)

572-23. Martin Schechter: A representation theorem for certain function spaces.

Let G be a bounded domain in En with boundary 8G and closure G. C 00(G) is to be the set of complex functions infinitely differentiable in G and C~(G) the set of those u E C 00(G) which vanish near 8G. Let V be any linear space of functions satisfying clf(G) C V s C00 (G). For any integer s ~ 0 and real number p ;> 1, let !lull be the sum of the Lp(G) norms of u and all its s,p derivatives of orders ~ s, and set In 1-s,p = lubvEvl(u, v) I llv u;;p.. where (u,v) =,/auvdx and p' = p(p- 1)-1• Let VS•P(G) (resp. y-s,p(G)) be the com­ pletion of V with respect to the norm II lls,p (resp. I 1-s,p>· Clearly, (u,v) may be defined for u E vs,p (G) and v E v-s,p' (G), satisfying l(u,v) I

§ llnlls,plvLs,p'· Theorem. For every bounded linear functional F(v) on vS•P(G) (resp. v-s,p(G)) there is au E v-s,p' (G) (resp. ys.p' (G)) such that

F(v) = (v,u) for all v E yS•P(G) (resp. v-s,p(G)). (Received July 22, 1960.)

572-24. Martin Schechter: On the Visik-Sobolev problems.

Problems considered by Visik-Sobolev (Dokl. Akad. Nauk. SSSR vol. 111 (1956) pp. 521-523) and Lions (C. R. Acad. Sci. Paris vol. 244 (1957) pp. 1126- 1128) are generalized and necessary and sufficient conditions are obtained for solutions to exist. (Received July 22, 1960.)

719 572-25. Martin Schechter: Negative norms and boundary problems, III.

The notation of the preceding abstract is employed. Let A be a linear partial differential operator of order m with coefficients in C00 (G). V' is the set of all v E C 00(G) such that (Au, v) = (u,A *v) for all u E V, where A* is the formal adjoint of A. Let Hs,p (G) and H-s,p (G) be the completions of C 00(G) with respect to the norms llulls,p and Uull-s,p = luhvecx> (G)I(u,v)lllvll~~p'' respectively. Let r,s,t be arbitrary integers with s ;s:; 0. Theorem 1. Given f E Ht,q(G), there is au E Hr•P(G) such that (1) (u,A *v) = (f,v) for all v E V' if, and only if, l(f,v)l ~ ciiA*viLr,p' for all v E V'. There is au E yS•P(G) satisfying (1) if, and only if, l(f,v)l ~CIA*vl-r,p for v E V'. Now assume that N 1 , the set of v E V' such that A *v = 0, is finite dimensional. Theorem 2. A necessary and sufficient condition that for each f E Ht,q(G)/N' there exist au EHr,p(G) satisfying (1) is that llvll-t,q' ~ ciiA*vll_r,p'' for all v E V'/N'. A necessary and sufficient condition that a u E yS•P(G) satisfying (1) exist for each such f is that llvll-t,q ~ ciA*vl-r,p' for v E V'/N'. These theorems lead to new results even for elliptic equations. In particular, if A is an elliptic operator and V is determined by a normal set of boundary operators which covers A, then Uull-s,p '&. c(IIAuiLm-s,p + lluLm-s,p> for u E. V. (Received July 22, 1960 .)

720 THE NOVEMBER MEETING IN NASHVILLE, TENNESSEE November 18-19, 1960

573-l. V, E. Bene~: Asymptotic behavior of general queues with one server. II

As in the first paper, K(·) is a nondecreasing stochastic step-function, and W( ·)is the solution of W(t) = K(t) - t +JdU(- W(u))du, where U(·) is the rt-continuous unit step at 0, with K(O) = W(O) = 0, K(·) is the cumulative load, and W(·) the virtual delay in a queue with service in order of arrival, We assume the system is stationary in the weak sense that Efexpf- sK(t) + sK(u)

+ s(t- u)J} = ¢(s,t - u), t > u. Let a(s) =lim sup f 1 log ¢(s,t) as t --7 oo, s > 0. Under weak conditions a' (0) = 7? exists, and for T > U there is unique solution s(T) of (l) 7'- s = a(s). Equation (l) is closely analogous to the "fundamental equation" of branching processes. The Laplace transform of Pr{W(•) = o} is 1/s(r). If lim Pr {W(t) = o} as t-.oo exists, it is positive if 7? >- 1 and zero otherwise; is 17 ..c- l, then Pr [W(•) = 0} is integrable. (Received August 26, 1960,)

573-2, Eckford Cohen: Arithmetical notes, V. A divisibility property of the divisor function.

Let k denote a fixed odd integer > 1 and n an arbitrary positive integer.

Let T(n) denote the number of divisors of n and Sk(x) the number of n ;;i x with the property that klr(n), Sathe has proved (Amer. ]. Math. vol. 67 (1945) pp. 397-406) that Sk(x)"" ukx as x -+oo, where uk is a complicated positive constant depending upon k. The proof of this result was based upon a deep estimate in the theory of primes, In the present note, the following refinement of Sathe's result is proved by elementary means: Sk(x) =

573-3. Ben Fitzpatrick, Jr.: A linearly ordered continuum of arbitrarily high cardinality.

Suppose a is a transfinite cardinal number. A simply ordered Hausdorff continuum of cardinality 2" is constructed, It is shown that if the generalized

721 continuum hypothesis is true, then the space is a-separable in the sense of Lubben (Separabilities of arbitrary orders and related properties, Bull. A mer.

Math. Soc, vol. 46 (1940) pp. 913-919). If a~ ~ 0 • the space is not homogeneous. (Received September 13, 1960,)

573-4. M. K. Fort, Jr. and Jack Segal: Local connectedness and inverse limit spaces. f f f Let X be the inverse limit of the system X 1..] X2..,..? x 3 ~ ••• where each Xi is a metrizable continuum, and let 7Ti be the projection mapping of X onto Xi.

If di is a metric for Xi for each i, the sequence d pd2,d3, ..• is called admissible if there exists a metric d for X such that limi~ 00di(7ri(u),?li (v)) = d(u,v) uniform­ ly on X X X. It is shown that an admissible sequence of metrics always exists.

Moreover, if d 1,d2,d3, ... is an admissible sequence of metrics, then X is locally connected if and only if the collection {

For e~ample, if each Xi is a 2-sphere and X is locally connected and 2-dimen­ sional, then X is a 2-sphere. (Received September 13, 1960.)

573-5, J. G. Horne, Jr.: Real commutative semigroups on the plane.

This is a correction and extension of Commutative multiplications on the plane (abstract 564-204). A real semigroup on the plane E is a commutative topological semigroup on E which contains a sub-semigroup R isomorphic to the multiplicative semigroup of real numbers so that 0 and 1 play their usual roles. A crude classification of these can be given in terms of the number and arrangement of the components of the maximal subgroup H(l). If there is only one component, E is isomorphic to the multiplicative semigroup of complex numbers. There can be two components of H(l) and in this case, H(l) may or may not be dense. If E has no nilpotent elements then H(l) must have either one, two or four components and all of the ways in which these can be arranged can exist. If H(l) has four components and G is the component of the identity, H(l)/G is the four group. Theorem. If E is a real semigroup on the plane

722 which has no nilpotent elements, precisely four idempotents and X f-0 implies -x f X, then E is isomorphic to R x R. Other characterizations of R x R are given. Examples show that none of the conditions in the theorem can be dropped. (Received August 8, 1960.)

573-6. 0. ]. Huval: On the existence of outer E ttomorphisms of some finite p-groups.

Suppose G is a finite p-group and t1t is the automorphism group of G. Theorem 1. Suppose M and N are normal subgroups of G whose factor groups

are cyclic and suppose M and N generate G. If Z(M n N) = Z(M)Z(N) and if exp Z(M n N) § mim fG:M,G:NJ, then (G: 1)/(£1t.: 1). Theorem 2. Suppose the factor group of the normal subgroup M of the group G is cyclic. If M is com­ plemented in G and exp G ~ exp C where C is the complement of M in G, then (G: 1)/(a: 1) Corollary: If every element of G is of order p and the order of G is greater than p2, then (G:1)/(0't :1). R. Ree, On the existence of outer auto­ morphisms of some groups, Proc. Amer. Math. Soc. vol. 12 (1956) pp. 962-964. E. Schenkman, The existence of outer automorphisms of some class 2 groups, Pro c. Amer. Math. Soc. vol. 6 (1955) pp. 6-11. (Received September 26, 1960.)

573-7. C. W. McArthur: A note on subseries convergence.

Let L~f'i denote a series in a Banach space X and let X* denote its normed conjugate space. Consider the following conditions: (A) Each sub­ series of Z~ 1 xi converges; (B) Each subseries of I:~ 1 xi converges weakly to an element; (C) limn-+oo L~Ii~(xi)l = 0 for each sequence {fn} in X* such that fn(x) ~ 0 for all x in the closed linear span of the xi' i = 1,2, ••. ; (D) {LiEo-xi:

of the Orlicz-Pettis theorem obtained by showing that (A)~ (B)=~ (C)~ (D)

~(A). This approach leads to simple proofs of some other equivalences, e.g. the equivalence due to Gelfand of (D) with the condition: limn...,oo Li=n lf(xi) I = 0 uniformly for all f E x• such that llfll:;;;;; 1. A corollary of the main theorem is: if L..~ 1 xi is subseries convergent then for each bounded sequence a = fai} of reals the series 2:::~ 1aixi is subseries convergent and the linear operator L defined from (m) to X by L(a) = 2::~ 1 aixi for each a E (m) is compact. (Received September 20, 1960.)

723 573-8, Gary Meisters: On the equation ax-xb = c in division rings.

Consider the condition (C): a,b,c E R and a .f b implies ax - xb = c has a solution x in R, for a ring R. Lemma: If a ring R satisfies (C) then R is a division ring. Question 1: Can a noncommutative division ring satisfy (C)? The theorems of this paper .give a partial negative answer to this question. Theorem 1: If a division ring .6 is finite dimensional over its center F and if ,6. satisfies (C), then b. is commutative. Theorem 2.: If 6. is a division ring algebraic and separable over its center F and if A satisfies (C), then A is commutative. Theorem 3: If b. is a division ring satisfying (C), then every nonzero inner-derivation of .6 maps LJ. onto itself. Question 2.: Does there exist a division ring LJ. which possesses an inner-derivation mapping !J. onto itself? And if so, Question 3: Does there exist a noncommutative division ring b such that every nonzero inner-derivation of b. maps /l onto itself? Theorem 4: Let F be a field and let s be an automorphism of F other than the identity automorphism. Then the Hilbert division ring F (t;s) always possesses a nonzero inner-derivation which does not map F {t;s) onto itself, (Received September 2.6, 1960,)

573-9. Gary Meisters: Local linear dependence and the vanishing of the Wronskian.

It is well known that n complex-valued functions f 1, ... ,fn, which are either analytic or solutions of a normal linear differential equation on an interval I, are linearly dependent on I iff their Wronskian W is identically zero on I. That W =. 0 is not sufficient to guarantee linear dependence in the class D00 of infinitely differentiable functions was pointed out by B6cher in 1900, However, the author offers the following definition and attendant theorems.

Definition: n functions f 1, ... ,fn are said to be locally linearly dependent (l.l.d.) on I iff for every nonempty open subinterval J of I there exists a nonempty open interval K contained in J and on which f 1, ... ,fn are linearly dependent. Theorem 1: (n- 1)- times differentiable functions f 1, ... ,fn are l.l.d. on I iff their Wronskian W vanishes on some open dense subset G of I. It is possible to con­ struct examples to show that this theorem is "best possible". Theorem 2.:

If f 1, ... ,fn are l,l,d, on I and are (n- 1)- times differentiable on I, then W vanishes at every point of I at which it is quasicontinuous ( ::=:: neighborly).

Corollary: In D00 , 1,1,d, on I is equivalent to W '=" 0 on I. An improvement on

72.4 Theorem 2 is Theorem 3: If f 1, ... ,fn are l.l.d. on I and are (n - 1)-times differentiable on I, and if W is quasicontinuous almost everywhere on I, then W =0 on I. (Received September 26, 1960.)

573-10. Martin Schechter: Negative norms and boundary problems. IV

Let G be a domain in Euclidean n-space and let A be a linear partial differential operator of order m. Let V be any linear subspace of C 00(G) con­ taining those C 00 functions with compact support in G. Let V' be the set of those v E C 00(G) which satisfy (Au,v) = (u,A'v) for all u E V, where A' is the

formal adjoint of A and (u,v) = JGuvdx. For s ,?; 0 let lin II s,p be the sum of the LP(G) norms of u and all its derivatives of orders & s(p :> 1), and set llull-s,p

= l.u.b·vECffi(G)I(u,v)lllvll=~.p'• where p' = p/(p- 1). Let N (resp. N') be the set of all u E V (resp. v E V') satisfying Au = 0 (resp. A'v = 0). Assume: (l) for every f E C 00(G)/N' (=the set of those f E C 00(G) such that (f,w) = 0 for all wEN') there is au E V such that Au= f, and for every g E C 00(G)/N there is a v E V' such that A'v = g; (2) for each s ;;;; 0 there is a constant Cs such

that llullm+s,p;;;; Cs(IIAnlls,p + llnlls,p) for all u E: V and llvllm+s,p' & Cs •(IIA'vlls,p' + llvlls,p') for all v E V'. Conc!!!sion: Under mild assumptions on G, for every t (- ro < t < ro) there is a constant Ct such that llu llt,p ;s Ct

·(IIAullt-m,p + llullt-m,p) for all u E V and llvllt,p & Ct(IIA'vllt-m,p + llvllt-m,p) for all v E V'. (Assumptions (l) and (2) can be weakened.) (Received September 21, 1960.)

573-ll. M. L. Curtis: Cartesian products with intervals.

It is shown that for each n ~ 5 there is a combinatorial n-manifold Mn

· h b n n+l _J_ • Wlt oundary B such that M X I = I whereas ?r1 (B) T l. The constructwn given here is based on the affirmative solution of the Poincare Conjecture (for n S: 7 by John Stallings and for n = 5,6 by E. C. Zeeman), and the Morton Brown version of the Schoenflies theorem. Poenaru has given such a manifold for n = 4, and Bing has proved no such manifold can exist for n = 1,2,3. (Received September 29, 1960.)

725 573-12. C. W. Patty: The fundamental group of certain deleted product spaces.

The deleted product space X* of a topological space X is the space X X X - Dx with the relative topology, where Dx is the diagonal of X X X. In this paper, we compute the minimal number of elements in a system of genera­ tors for the fundamental group of the deleted product space of any connected, finite, !-dimensional polyhedron which is not an arc. First, we compute the fundamental group of the deleted product space of: (1) a triod, and (2) a simple closed curve. Then, we show that: (1) If X is obtained from A by adding a !-simplex incident to one vertex, then 7ri(X*) is the free product of 11J.(A*) and the free group on 2n - 4 generators, where n is the order in X of the incident vertex. (2) If X is obtained from A by addition of a 1-simplex in such a way that it is then incident to two vertices of order 2 and if a system of generators for 17i (A*) consists of the minimal number p of elements, then there is a system of generators for 77J. (X*) ·consisting of the minimal number p - 2n + 2 of elements, where n is the deleted product number of A with respect to the two incident vertices. An important fact about this deleted product number is that it is obtained by considering only the structure of A. (Received September 30, 1960.)

573-13. H. S. Davis, D.P. Stadtlander and P.M. Swingle: lndecomposa­ bility and T(p) sets.

LetS be a compact continuum. We say that a continuum W is one with n-fold set U Zi (i = 1,2, •.• ,n) of probable indecomposability if and only if every subcontinuum V, which contains a region w .r .t. S, contains some Zi; it is with minimal n, if there does not exist an m-fold set U Zl form< n. For AC S, T(A) = {xI There does not exist a subcontinuum W and open set Q such that S - A :J W .::::> Q ::J x J. Then each of the following are necessary and sufficient conditions that S be a continuum with n-fold set U Zi of probable indecomposa­ bility with minimal n, although (2) has the further condition that S is heredit· arily unicoherent: (1) For piE Zi and some qi ¢ U Zi, T(U Pi)= S and T(U qi) f S; under the necessary condition in (2), each essential part E(T(pi)) is connected. (Received October 3, 1960.)

726 573-14, C. H. Edwards, Jr.: Shrinking continua in then-sphere.

Let M be the decomposition space associated with an upper semi-continu­ ous collection of continua in then-sphere Sn, and denote by Y the set of points in M corresponding to non-degenerate elements. It is proved that, if M is a manifold andY is a discrete set in M, then M is homeomorphic with sn. Let f be the natural monotone continuous map of Sn onto M. With Y = { Yd and Xi= f- 1(yj_) for each i, the generalized Schoenflies theorem of M. Brown is used to show that there exists a sequence [Aii of mutually disjoint n-cells in, Sn and a sequence {Bd of mutually disjoint n-cells in M such that, for each i, (1) Xi C Int Ai, (2) Yi E lnt Bi, (3) f(Ai) = Bi, and (4) diam Bi < 1/i. For each i, let hi be a homeomorphism of Ai onto Bi agreeing with f on Bd Ai. If the map h is defined by h(x) = f(x) for x E sn - U Ai and h(x) = hi(x) for x E Ai, it can be shown that his a homeomorphism of sn onto M. In the case n = 3, this result is a corollary to a theorem recently announced by M. L. Curtis. (Received October 3, 1960 .)

573-15. W. C. Fox: Factoring meromorphic functions.

Every continuation manifold (i.e,, Riemann surface consisting of all continuations and algebraic singularities of some given power series) admits a pair of meromorphic functions called c and v (c sends a power series onto its center, v sends it onto its value at the center). IfF and G are meromorphic on the Riemann surface X, then there exists an analytic map cr- of X into some continuation manifold Y so that F and G factor in the forms c(o-) and v(o-). The map rr is onto Y when F and G cannot be jointly extended, or equivalently, when paths in Y may be lifted under o- with prescribed initial point. In that event cr- is also 1-1 provided there is a point near which (F, G) is 1-1 and takes the same value elsewhere only with F and G having unequal multiplicities. Main application: If X is a continuation manifold, take F = v and G = c and it follows that cr- is a conformal homeomorphism between X and its "inverse manifold" Y. Thus, e.g., the surfaces w =log z and w = z 1/n are conformally equivalent with those of ew and wn respectively, the plane and the sphere. Similarly for the inverses of all transcendental and rational functions. (Received October 3, 1960.)

727 573-16. Costas Kassimatis: Generalized differentiation.

Let ~ + a 1h, ••• , x + an+lh; f(x)J be the nth divided difference of the con­ tinuous function f(x). If the limh-+on! llr + a1 h, ••• ,x + an+lh; f(x)] exists (finite), it is called the nth generalized derivative fn(x) of the function f(x). Theorem: If fn (x) is identically zero in the interval (a, b), the set A of the points of (a,b) in the neighborhood of which f(x) is not a polynomial of degree at most (n - 1) is nondense and closed. The author has obtained the sufficient conditions under which f(x) doesn't reduce to a polynomial of degree at most (n - 1) in the neigh­ borhood of a point of the set A. (Received October 3, 1960.)

573-17. N. F. G. Martin: Generalized condensation points.

Let (X,'U) be a topological space and let :J be a family of subsets of X satisfying the following: (a) every set in .!7 contains at least two points, (b) if E t: :f and E C F then F E :f. and (c) if E U F E .:f then either E E J or FEY. If E C X, xis called an ;{limit point of E if for every open set U E ~. U n (E - x) C. §. A set is J'-closed if it contains all its .1-limit points. The class of all subsets of X whose complements are J-closed forms a topology, 'Z/·, for X. The topology ris finer than ~. Iff is a function on X to a regular space Y and ':"{, C .Y. then f is continuous with respect to the topology '2l if and only iff is continuous with respect to'::< If (X,'~{) is a second countable space and (c) above is replaced by (c') U~ 1 Ei E$ implies at least one Ei E :f, then a necessary and sufficient condition that a set V be '2f-open is that V = U - Z where U E P/L and Z ~:f. The space (X,V') is T1, not regular, not first countable at any point, contains no nontrivial countably compact subsets, and is connected if and only if (X;>tt) is connected. The (o/) discrete subspaces of X are exactly the sets not in .Jf. If (X,'?{) is the real line with the usual topology and .:/is the family of sets with positive outer Lebesgue measure then a function is differentiable (in usual sense) iff it is v:differen­ tiable. (Received October 3, 1960.)

573-18. G. C. Caldwell and R. A. Struble: The secular characteristics of the motion of an earth satellite.

In a previous study (R. A. Struble, An application of the method of averag­ ing in the theory of satellite motion, J. Math. Mech. (1961)), the method of averaging was employed in the calculation of the secular characteristics of the

728 motion of a satellite of an oblate spheroid. The reduced nonlinear equations were integrated in closed form in terms of standard elliptic integrals. The potential function which was considered represented the main effect of the oblateness of the earth, but did not include the fourth harmonic, which is known to be of some importance in the secular portion of the motion of an earth satellite. In the present paper, the theory has been extended to include the effects of this term in the solution. Analogous closed-form expressions lead to untabulated hyperelliptic integrals. In order to provide for the practical application of the results, a numerical theory has been developed for the approximate evaluation of these integrals. Therefore, accurate predictions of the secular characteristics of the motion of earth satellites are possible with­ out recourse to perturbational procedures or lengthy machine computations. (Received October 4, 1960.)

573-19. Haskell Cohen: A clan with zero and without the fixed point property.

There is an example of Kinoshita (Fund. Math. vol. 40 (1953) pp. 96-98) of a contractible continuum without the fixed point property. A multiplication is exhibited which makes Kinoshita's example into a clan (i.e. a compact con- nected Hausdorff topological semigroup with identity) with zero. This provides a counterexample to a conjecture of Wallace. (Received October 5, 1960.)

573-20. H. S. Davis, D.P. Stadtlander and P.M. Swingle: Irreducible continua and T(p) sets.

Let S be a compact continuum irreducible between the points a and b.

For A C: S, T(A) = [ x J There does not exist a subcontinuum W and open set Q such that S -A -::JW ::J Q :J xJ; and T(Tn(A)) = Tn+1(A). Among other theorems, we obtain the following: If Tn+1(p) f Tn(p), then either Tn+1(p) = Tn+Z(p) or there exists an indecomposable subcontinuum V such that V n Tn(p) ~ (.1; If x, y E S and n is finite, then x E Tn(y) implies y E Tn(x); T 2 (a) f T(a) implies that T(a) is an indecomposable continuum; If there exists ann such that, for every p E S, T(Tn(p)) = Tn(p), then the Tn(p) are an equivalence class, i.e. for p, q E S, Tn(p) n Tn(q) f (.1 implies Tn(p) = Tn(q); and S is an n-indecom · posable continuum if and only if Tm(a) = S form = n but not for m .<::. n. (Received October 3, 1960 .)

729 573-21. Trevor Evans: Properties of algebras almost equivalent to identities.

The ideas developed in this paper stem from a solution to a problem discussed by Etherington and Frink concerning sums of endomorphisms in an additive groupoid. Frink has said essentially that the identity (x + y) + (z + w) = (x + z) + (y + w) in an additive groupoid is a necessary as well as a sufficient condition for the sum of two endomorphism s to be an endomorphism and Etherington has given examples to show that the condition is not necessary. A proof is given here that Frink is "almost" correct in the sense that the above identity is the weakest which implies additive closure of the set of endomor­ phisms and that if properties of varieties (equationally defined classes) of group­ oids are considered rather than properties of individual algebras, then the two properties are equivalent. Other theorems of this type are (i) the commutative law for groups and the associative and commutative laws for loops are the weak- est laws which imply the Hamiltonian property that every subgroup (subloop) is normal or which imply that the sum of two suggroups (subloops) is a subgroup (subloop), (ii) for varieties of groups and loops these identities are equivalent to the Hamiltonian property and to the sum of subgroups (subloops) property. (Received October 4, 1960 .)

573-22. E. E. Grace: Cut points in totally nonsemi-locally-connected continua.

Theorem. Let T be a connected topo1ogical space, D an open set such that each open subset of D contains an open set the boundary of which is bicom­ pact and on which T is Baire topological and let I be a dense-domain intersection subset of D such that T is not semi-locally-connected at any point of I. Then the set of weak cut points of T is dense in D and for each point z, there is a set A, dense in D, such that each point x of A either cuts z weakly from each point at which T is nonaposyndetic with respect to x or is cut weakly from z by each point y such that T is nonaposyndetic at x with respect to y. Corollary: If the connected, locally compact, metric space M is not semi-locally-connected at any point of a dense G0 subset, then the set of weak cut points of M is dense in M. [For definitions and related work using similar methods see the author's paper Cut sets in totally nonasposyndetic continua, Proc. A mer. Math. Soc. vol. 9 (1958) pp. 98 -104]. (Received October 4, 1960.)

730 573-23. R. ]. Koch and L. F. McAuley: Semigroups on trees.

Let T be a tree (= locally connected acyclic compactum) whose set of endpoints I is compact. It is shown that I can be ordered so that x·y = min(x,y) is continuous for x,y E I, and a multiplication can be introduced in T so that T becomes the homomorphic image of the "fan" over I. The "fan" over I denotes the semigroup formed from I X [01] by shrinking I X {0} to a point (here I!J1] denotes the unit interval of real numbers provided with any continuous associa- tive multiplication in which 0 acts as a zero and 1 acts as a unit.) (Received October 5, 1960.)

573-24. Roy Leipnik and]. E. Maxfield: On the inverses of finite Toeplitz matrices.

In stationary process theory it is necessary in the development of finite record theory to invert Toeplitz matrices of arbitrary finite size. In this paper a process is described that produces the inverse of finite Toeplitz matrices that posess a linear recurrence relation among the rows, as the product of at most five matrices. Since frequently the quantity of interest is the inverse pre- and post-multiplied by a vector there is little loss in having the inverse so expressed. (Received October 5, 1960.)

573-25. D. R. McMillan, Jr.: A certain contractible open 3-manifold. Preliminary report.

]. H. C. Whitehead has shown that there are contractible open 3 -manifolds which are not homeomorphic to E 3. R. H. Bing has raised the question as to whether each such 3 -manifold can be embedded in s3 • The author provides a partial solution to this problem by showing that a certain open 3-manifo1d M cannot be embedded in s3 in a very "nice" way. Specifically, if there is an em­ bedding, h, of M into s3 , then h(M) cannot be the union of a properly ascending sequence of unknotted multiple tori. It can then be shown that M is not homeo­ morphic to the complement in s3 of any closed set of dimension less than two. (Received October 5, 1960.)

573-26. E. B. Shanks: An axiom system for set theory.

It is shown how to modify Godel's axiom system for set theory so that the use of the word "set" is avoided and the class of ordinals is assumed as an 731 axiom (the ordinal axiom). Then a set is defined as a class for which there exists a one-one correspondence between it and some ordinal. The ordinal axiom in effect assumes the ordinals not greater than w + I and a one-one correspondence relating the class of ordinals to the class of ordinals that are cardinals (the catapulting axiom). The definition of set and the ordinal axiom imply both the axiom of infinity and the axiom of choice. Although this procedure leaves unanswered such questions as whether the class of real numbers is a "set", it seems to place such conjectures in a better perspective as well as to make the class of ordinals central in set theory. (Received October 4, 1960.)

573-27. R. A. Struble: General perturbational solutions of the Mathieu equation.

General perturbational solutions of the Mathieu equation are obtained in the form of asymptotic series. The principal part of a solution is obtained by a modified variation of parameters technique which admits only slow (long period) variations in the amplitude and phase, All rapid (short period) pertur­ bations are obtained as additive corrections to the principal part. The explicit solutions are either almost-periodic functions or diverging or decaying oscil­ lations. The resonance phenomena associated with critical values of the equation parameters are manifest in the variational equations and the familiar periodic solutions (Mathieu Functions) emerge merely as singular solutions of these variational equations. Also, both divergent and decaying oscillations emerge near the critical parameter values, thus revealing the well known stability properties of the Mathieu equation. The techniques employed were originally developed for the solution of the artificial satellite problem and so reflect somewhat the notions of celestial mechanics. However, even in that setting they are novel and it now appears that these techniques will be applicable to linear and nonlinear vibration problems of very general forms. (Received October 4, 1960.)

732 THE NOVEMBER MEETING IN PASADENA, CALIFORNIA November 19, 1960

574-1. J. L. Brenner: Equatorial orbits around an oblate primary.

This paper contains a rigorous classification of the cases which the perturbation procedures commonly used represent an equatorial orbit accurately. 2 For the differential equation U" + U - 1 = J 1u , no periodic solutions exist if Jr > 1/4; periodic solutions exist if Jr > 1/4. For the differential equation U" + U - 1 = Jr u2 + (3/7)D1u\ no periodic solutions exist if Jr > 1/4, or if D1 > 63/256. Necessary and sufficient conditions for the existence of periodic solutions of this last equation are obtained also. (Received June 6, 1960.)

574-2. J. B. Butler, Jr.: Concerning weak solutions of one dimensional vibration problems by the perturbation method. Preliminary report.

Let Le = Lo + eq(x) where Lo is an ordinary differential operator of order n = 2V on 0-;;;_ x < oo, e is a real parameter, and q(x) is a continuous function. Suppose the problem 1. 0u =Au, [91 0 j'u] (o) = 0, j = l, ••• ,v is self adjoint (for notation and definitions seeK. Kodaira, Amer. J. Math. vol. 72 (1950) pp. 502-543). The inhomogeneous equation (i) Leu+ utt = PLl. (x,t) has a weak solution (ii) u = ~ slx,i)uk(t,.i)djJjk(.l), where A= [a,,.eJ. -oo""' a -< ,8 ..:: oo, Uk = fk(i')cos(.f'112t) + (l/..l)11yJPk(T,i)sinl112( t-T) d'; fk =Cskfdx,

Pk = J/fskP dx. Theorem. If L 0, q are such that for l E /;;., 0 .c. o <: 80, j,k = l, .•. ,n, 0 ~ x < oo, (iii)Jo004> 2(x)lq(x)ldx;;;. 'Y. 4> = suplsj(x,l)l, (iv) IMjk(l + icS)I;;;; K, (v)~CIMjkl 2 1s .1 2 lql !/otskfdjl2dx dl & P 2 11t11 2• I' f'OO jk 2 2 I ~ 2 J2 2 2 -1 IJ..:lv'O IM llsjiJqlyt s/dffl dxdl~P llfll,thenfor lei-<(Kn ?') the equation (ii) can be expanded into a norm convergent power series in e which represents a weak solution of (i). The result is extended to the case where

Lois a matrix differential operator. In (i) P4 (x,t) =J';,. sjPkdfjk(L). (Received May 17, 1960.)

733 574-3. J. B. Butler Jr.: On the spectral measure of a perturbed operator whose resolvent has a Carleman kernel.

Let H 6 = Ho + eq where H0,q are operators on L 2(En) and e is a real parameter. It is assumed that (i) H 0 is self adjoint, q closed positive sym­ metric, and for A = J. + i8, j E A, 4 = [a.,f'J,O < S

(iii) JG 0(x,$,-\)J ~ C(x,.f"), (iv).8(Ho)Ct:iJ(q) and JlquiJ tf aiJH0uiJ + blJuiJ, u tdJ(H0), (v) the operator q 1/ 2g 0 (.1))q 112 has a Hilbert-Schmidt kernel and is uniformly bounded by KO' (vi) the operator q 112_& 0(,\) has a Carleman kernel and t.t. Jlq 1 1:J 0 ()1)u1J 2 d~J 112 is a uniformly bounded linear functional. It is well known that these assumptions imply the H6 is a self adjoint operator (for exampleS. T. Kuroda, J. Math. Soc. Japan vol. 11 (1959) pp. 247-262). Here it is proved that the spectral measure E 6 (L1 '),A' s A, corresponding to H6 is analytic in e for Je J < K{j 1. The proof uses the identity (E e ( .1)u, v)

= (1/7r)limg--..O+L ( -e)vLp+X~ v~ (,R(#0 )(q_,J' 0)Pu,(iJ0{ v)di. The result is applied to perturbed ordinary differential operators. (Received August 24, 1960.)

574-4, James Conlan and J. B. Diaz: Existence of solutions of an n'th order hyperbolic partial differential equation.

The existence of a solution of the classical initial value problem; y' = f(x,y), y(O) = y 0 , can be proved by: (1) Picard's method of successive approximations, (2) the Euler-Cauchy polygon method, and (3) the method of Weierstrass (see, e.g., M. Muller, Math. Z. vol. 26, pp. 619-645). In method (3) the function f(x,y) is assumed to be only continuous in (x,y) (there is no Lipschitz condition in y). Patrick N. Leehey (Ph.D. Thesis, Brown University, 1950) devised a generalization of method (3) and obtained an existence for the problem uxy = f(x,y,u,ux,uy), u(x,y0) = g(x), u(x0 ,y) = h(y), g(x0) = h(x0), where f is required to be Lipschitzian only in ux and uy. In his proof, Leehey used an integral inequality ofT. H. Gronwall, (Ann. of Math. (2) vol. 20 (1919) pp. 292- 296). By means of a suitable generalization of Gronwall's inequality to any number of variables, a corresponding existence theorem is proved for the equation ux1 ... xn = f, where f is required to be Lipschitzian only in the pure mixed partial derivatives ux1 •... ,uxn•··••ux1x 2 •... , up to order n - 1. (Received September 13, 1960.)

734 574-5. Donald Monk: Nonrepresentable polyadic algebras of finite degree.

Leon Henkin has constructed a nonrepresentable three dimensional cylindric algebra (Henkin, La structure algebrique des theories mathematiques, pp. 37-39) and subsequently generalized the construction to any finite dimension

~ 3. There is a natural way of defining substitution operators S(T) on Henkin's exampies, and it is then just a (somewhat lengthy) computation to show that a polyadic equality algebra is obtained. Call a polyadic algebra A representable if it is isomorphic to a sub-direct product of 0-valued functional polyadic alge­ b_ras. There are equations not involving equality which hold in all representable

polyadic algebras but not in Henkin's examp1.es. We conclude: Theorem, There are nonrepresentable polyadic algebras of each finite degree ;;; 3. (Received September 26, 1960.)

574-6. Donald Monk: Polyadic Heyting algebras.

!jeyting algelJ_,r_~ are understood in the sense of Rasiowa-Sikorski, Fund. Math. vol. 40, pp. 62ff. A quantifier on a Heyting algebra is understood in the sense of Halmos, Compositio Math. vol. 12, pp. 217ff, with "Boole" replaced by "Heyting". Making the same replacement in Halmos, Fund, Math. vol. 43, pp. 255ft, we arrive at the notion of a functional polyadic Heyting alge­ bra and a polyadic Heyting algebra, and we take over the definition of a support of an element of the algebra. A polyadic Heyting algebra A is l'IY-small (where IW is a cardinal number) if every element of A has a support of cardinality

< J#', The local dimension of the algebra A is the least cardinal W such that A is I#'-small. Theorem. Every polyadic Heyting algebra of infinite degree is isomorphic to a functional polyadic Heyting algebra whose domain has any

specified power ~ ~(degree, local dimension). (Received September 26, 1960 .)

574-7. T. G. Ostrom: Concerning the little projective group.

Let G be the projective group of collineations in the plane coordinatised by the field F, and let H be the little projective group over F. Let M be the multiplicative group in F, and let M' be the subgroup of M consisting of the cubes in M. Theorem (a) if M' = M, then G = H (b) If M' is of index 3 in M, then H is of index 1 or 3 in G. Summary of the proof: Using nonhomogeneous coordinates, let R(a) and S(a) denote the elations (x,y)-+ (x,xa + y) and (x,y)

735 -+ (x + ya -1 ,y) respectively. If R(a - l)S(a(a - 1)-1) is transformed by S(- 1)R(a(a + 1)-1), we obtain the collineation (x,y)--.. (xa - 1 ,ya). Transforming again by the involution (x,y) ._ (x-1 ,x - 1y), we get (x,y)~ (xa,ya2). Using the product, we get (x,y)-+(x,ya3). (Received August 23, 1960.)

574-8, B. W. Roos and W. C. Sangren: A transformation for a pair of first order differential equations.

In order to deal with the singular cases of the pair of first order differential equations (1) u'(x) - [Aa(x) + b(x)], v(x) = 0; v'(x) + [,\c(x) + d(x)), u(x) = 0 when the coefficients are not bounded at the singularity it is desirable to transform these equations. A transformation involving a change of the two dependent variables and the independent variables has been found. This trans­ formation allows one to conveniently find the asymptotic expression for the solutions of the differential equations, (Received September 22, 1960,)

574-9. B. W. Roos and W. C. Sangren: Two spectral theorems for a pair of singular first order differential equations.

The spectra for the pair of two singular first order differential equations

(1) u'(x) - [,\a(x) + b(x)], v(x) = 0; v'(x) + [~c(x) + d(x)], u(x) = 0 for the case when a(x) = x2k and c(x) = x-2k is investigated for the interval from 0 to oo. For the system (1) it can be shown that for the semi-infinite interval

(1 ~ x < oo) and under linear boundary conditions at x = 1 there exists a continu­ ous spectrum over the real )) -axis provided b(x) and d(x) are real-valued con­ tinuous functions and b(x)x-2k and d(x)x2k belong to the class L(1;oo). It can also be shown for system (1) over the interval 0 & x ;:>; 1 and under linear boundary conditions at x = 1 there exists a discrete spectrum over the real ,1-axis provided b(x) and d(x) satisfy suitable integrability conditions. (Received September 22, 1960,)

574-10, H. E. Salzer: Trigonometric interpolation and predictor­ corrector formulas for numerical integration.

Formulas are derived for n/2-point osculatory trigonometric interpola­ tion, which give the unique trigonometric sum Sn (x) = ao + Z"j: 1 (a j cos jx + b j sin jx) satisfying Sn(xi) = f(xi) := fi' i = 1,2, ... ,n, and S~ (xi)= f'(xi) = if•

736 i = 1,2, ... ,n t 1. In conjunction with previously derived formulas for n-point osculatory interpolation applied to extrapolation, we have predictor -corrector formulas for stepwise numerical integration of y' = i

574-11. Georgellen Schuske and W. J. Thron: Infinite radicals in the complex plane.

Given a sequence {an] of complex numbers, an infinite radical is defined as the sequence fun!• where un = {ii 1 t (a2 t •.. t an1/ 2) 11 2]112• A convergence criterion given by Herschfeld (Amer. Math. Monthly vol. 42 (1935) pp. 419-429) for infinite radicals with positive real elements is extended by means of the Stieltjes-Vitali Theorem to furnish sufficient conditions for convergence in the case where the elements are complex numbers. A typical result is: A sufficient condition for convergence of an infinite radical is that both

(i) limn-.oolanlz-n <. oo, and (ii) jarg ani< 271'/3 - il for all n, hold, where e is an arbitrarily small fixed positive number. (Received August 5, 1960.)

574-12. L. E. Ward, Jr.: Characterization of the fixed point property for a class of set-valued mappings.

Let us say that a (metric) continuum has property (C) if each continuous, closed set-valued mapping has a fixed point, (U) if each upper semi-continuous, continuum -valued mapping has a fixed point, (H) if the continuum is hereditarily unicoherent. Theorem: For arcwise connected continua, (H) implies (C) and

737 (H) is equivalent to (U). This result, together with a theorem of R. L. Plunkett, yields the Corollary: For Peano continua, (H), (C) and (U) are equivalent. (Received September Z6, 1960.)

574-13. A.M. White: Singularities of harmonic functions of three variables generated by Whittaker-Bergman operators.

One considers harmonic functions of three real variables x,y,z generated by the Whittaker-Bergman integral operator F(x,y,z) = (l/Z1ri)

·.;;~I= 1 f(u,£")d~, u =x + Z -l(iy + z>t + 2 -l(iy - z)(l/t). (see Bergman, Integral operators ... , Ergebnisse der Math. Wiss. vol. Z3, Springer-Verlag (1960) Chapter II), This integral operator defines, in some open set, one-valued harmonic functions which can be extended by analytic continuation to multi­ valued harmonic functions, The notion of regular and singular points of these functions is introduced, Then in the first two parts of the paper one character­ izes the locus of possible singular points in the case where the associate func­ tion f(u,~) is rational, In the third part of the work, based on the previous results, an arbitrary harmonic element of x,y,z given by a series development in a neighborhood of (0,0,0) is considered, and one proves a theorem concerning the relation between the coefficients of the series development and the locus of possible singularities of the multi-valued harmonic function defined by the element. The locus of singularities is, in general, an algebraic curve. Some examples and problems complete the work. (Received September ZO, 1960.)

574-14. J. M. Zimmerman: Band-limited functions and improper boundary value problems for a class of nonlinear partial differential equations.

In this paper the equation (*)L (u(x,y)] = f(u(x,y)) is studied (where L = + a (y)(aP+q/axP yq)) under the initial conditions oqu(x,O)/Clyq L p q:;;m pq 0 = vq(x), q = 0,1, ... ,m - 1. In some cases (e.g., if Lis elliptic) such problems are "improperly posed" in the sense of Hadamard. That is, a solution to an initial value problem of this type may not depend continuously on the given data.

To remedy this situation a class, B(.v0,x), of Lz functions is considered each member of which possesses a Fourier transform with support contained in the interval [- (()0' CUOJ and it is assumed that v q(x) E B(410,x) for all q. In addition, the right member of (*) is replaced with the "filtered" version, f(u(x,y)) 1 poo "o ieu(x-~) = (l/Z'IIU-oo/-41of(u(s,y))e d4>d¢'. Under suitable restrictions on f(u) and

738 the coefficients apq(y), the existence of a unique B(4{j,x) solution is proven

(for -oo <. x < oo and sufficiently small y) by reducing the Cauchy problem to an equivalent pair of integral equations and subsequently employing the method of successive approximations. It is shown that the solution depends continuously on the initial data and the function f(u). The method can be generalized to the Cauchy problem for the equation L[uJ = f(x,y,u,ou/c3x,;;lu/0 y,Q2u/axoy, ••• , om -lu/<7:P -1 ,om -1 u/ayffi -1 ); the cases L j!IJ = f(u,ouox) and L [uJ = f(u;ouoy) are treated in some detail. (Received August 16, 1960.)

574-15. Y. H. Clifton and J. W. Smith: Cohomology on Lorentz manifolds. Preliminary report.

A cohomology theory has been devised which is capable of measuring the global properties of a Lorentz manifold M. Let B denote the bundle of time­ like vectors on M. Each point x in B determines a geodesic in M, which can be lifted canonically to a curve through x. This defines a foliation of B. Every x in B has a system of special neighborhoods on which the foliation is a fibering. For every such neighborhood U one can construct a base space V which is a differentiable manifold. A differential form on B is said to be invariant if its restriction to every special neighborhood U is the lifting of a form on V. The invariant forms on B constitute a differential ring R. Various calculations have been carried out to show that the cohomology of R measures global proper­ ties of M, such as the existence and distribution of closed time-like paths. (Received September 28, 1960.)

574-16. W. A. J, Luxemburg: On normed linear spaces of measurable functions.

Let (X,A.p) be a nontrivial measure space and let P denote the set of all p-measurable nonnegative functions on X. A mapping )) of P into the set of all nonnegative real numbers is called a function norm if it satisfies the following conditions: (P 1) 0-;;;; }j (f)§ oo; },(f) = 0 if and only iff = 0 a.e.; )\(fl + fz) 3 }l(f 1) + )\(fz); },(a.f) = a})(f) for all a~ 0. Furthermore for some f E P, 0 <~(f)< oo. (Pz) o & f 1 13 f2 implies ,1(f1 );:;; ~(fz). By LA(X,A,p) we denote the normed linear space of all p-measurable complex f(x) on X satisfying )\(jfj) ""- oo. Recently, in a joint paper with A. C. Zaanen, it was shown that if the measure space in question is of totally cr-finite measure there exists an in-

739 creasing sequence of measurable sets [Xn} (n = 1,2, ••• ) such that (i) ))(ex ) < + oo (n = 1,2, .•. ), where Cx is the characteristic function of Xn; n n (ii) for all f E P,Jxn fdp ~ An))(f) (n = 1,2, ••• ) where fAn! (n = 1,2, ••• ) in some sequence of finite constants; (iii) f E L)) implies f(x) = 0 a.e. on the set X - UXn. In this paper the author gives among other things a generalization of this result for the case of an arbitrary measure space. (Received September 30, 1960.)

574-17. Robert Ryan: Fourier transforms of certain classes of integrable functions.

Let G be a locally compact abelian group with character group G."' P(G) denotes the set of continuous positive definite functions defined on G and (x,x) denotes the value of the character ~ E Gat the point x E G. For a bounded Radon measure p write p(x) = Jo (- x,x)dp(x) and for f E L 1 (G) write f(x) =A (x,x)f(x)dx(dx,dx = Haar Measures) [L 1 (G) n P(G)] denotes the space span­ ned by L 1(G) nP(G). For f E L 1(G) define Jlf(E) =,Ai f(x)dx for all Borel sets E. Assume that )lis a bounded Radon measure. Theorem 1. p. = Pf for some f E L 1(G) n Lp(G), 1 .c p ~ oo, if and only if there exists a constant K > 0 such that Jib g(x)p(x)dxl ~ K!lgllq for all g E [L 1((}) nP(C})] where 1/p + 1/q = 1. Theorem 2. p = )lf for some f E L 1 (G) nLP(G), 1 < p :s oo, if and only if ji • g E (L 1 (G) nLP (G))" for g E L 1 (G). Theorem 3. )l = }If for some f E L 1 (G) if and only if there exists some p, 1 ~ p < oo, such that for every s > 0 there exists a 8 > 0 with the property that J,ta g(x)j1(x)dx I ~ s llg lip whenever g E [L l(G)nP(G>] and llgll ~oWl()()" Taking G =the unit circle and p = 2 in Theorem 3 gives a theorem by R. Salem (Comptes Rendus vol. 192 (1931)). The theorem for G = the additive group of reals and p = 1 was proved by A. C. Berry (Ann. of Math. (2) vol. 32 (1931)). (Received September 29, 1960.)

574-18. P. C. Curtis, Jr.: Derivations in commutative Banach algebras.

By a derivation in an algebra A is meant a linear mapping D of A into itself satisfying D(ab) = a(Db) + (Da)b. In 1955 Singer and Wermer proved that a bounded derivation in a commutative Banach algebra necessarily maps the algebra into the radical. They conjectured that the assumption of boundedness could be dropped. In support of this conjecture, the following result is announced. Let A be a commutative regular Banach algebra. If A is semi-simple then any derivation in A is trivial. If A has nil-potent radical then any derivation

740 in A maps A into the radical. This result is essentially an application of results appearing in Homomorphisms of commutative Banach algebras, Amer. j. Math. vol. 82 (1960) pp. 589-608 by W. G. Bade and the author. (Received October 3, 1960.)

574-19. E. I. Deaton: Solutions of a system of two nonlinear partial differential equations of the first order.

If fi(xl,x2,ul,u2), i = 1,2, is continuous in all of its arguments for

0 ~ x 1 ;§ a, 0 ~ x2 ~ b, all u 1 and all u 2, and if fi satisfies a Lipschitz condition in u 1 and u 2, and if g(x2) is continuous for 0 § x 2 ;:;;; band h(x 1) is continuous for

0 § x 1 § a, then there is one and only one pair u i(x 1 ,x2) = ui(x), of functions continuous in x defined over [O,a; O,b] such that u}(x) = f 1(x1,x2,u1(x),u2(x)), u~(x) = f 2(x1,x2,u1(x), u 2 (x)), and u 1(0,x2) = g(x2 ), u2 (x1,0) = h(x1). Under cer- tain additional conditions this system is equivalent to one second-order non­ linear hyperbolic par:,;.al differential equation with accessory boundary con­ ditions on a restricted region. (Received October 3, 1960 .)

574-20. Werner Israel and R. Trollope: New possibilities for a unified field theory.

The introduction of nonsymmetric gik in unified field theories of the Einstein-Schrodinger type is open to the objection, on group-theoretical grounds, that the symmetric and anti-symmetric parts transform independently. This objection does not apply to the use of nonsymmetric r 1tc' since these quantities are irreducible under the "extended group", consisting of the point transforma­ tions and the Einstein A-transformations. We consider a theory based on sym­ metric gik and nonsymmetric T{k. The Lagrangian L is assumed to depend only on gik and the contracted curvature tensor Rik (this ensures the >l-invariance and transposition - invariance of the theory), For simplicity, we suppose further that L involves Rik rationally and, at most, quadratically. The resulting theory is able to account satisfactorily for the main features of gravitation, electromagnetism and their interaction. In particular, the theory yields the correct equations of motion for charged masses. The electromagnetic tensor is associated with the skew part of Rik• and the A-transformations correspond roughly to the gauge-transformations of electrodynamics. (Received October 3, 1960 .)

741 574-21. E. D. Kann: G-spaces with curvature >K :;;.- 0.

This paper gives a definition of curvature > K > 0 for G-spaces (cf. Herbert Busemann, The geometry of geodesics) and proves that 2 -dimensional G-spaces with such curvature and with locally convex circles are bounded. G-spaces are metric spaces which are defined by a set of axioms which insure the existence of segments and geodesics. A segment (written T(a,b)) joining a and b is a curve whose length equals the distance ab of its endpoints. It is a shortest join of its endpoints and is isometric to a closed interval of length ab of the real line. If the endpoints are close enough the segment is unique, A segment may be prolonged indefinitely in both directions to a unique geodesic, A geodesic is a locally isometric mapping of the entire real line into the G-space and hence is locally a segment. A G-space is said to have curvature

::> K ::> 0 if every point has a neighborhood such that for three points a, b, c in that neighborhood and not on one segment 2 a'b' ;;- ab + Kh 2a'b' where a' and b' are defined by aa 1 + a' c = ac, aa 1 = ac and bb' + b' c = be, bb' = b' c respectively and h is the altitude from c on T(a', b' ). The neighborhood is taken small enough so that all these elements are uniquely determined by a,b,c. A finite difference inequality is deduced which is a close analogue of Jacobi's differen­ tial equation used in the classical proof of the present result. (Received October 3, 1960.)

574-22. D. H. Lehmer and Emma Lehmer: On the distribution of consecutive quadratic and cubic residues.

It can be easily seen that every prime greater than 5 has a pair of con­ secutive quadratic residues not exceeding the pair (9, 10). Similarly, M. Dunton has recently shown that every prime greater than 13 has a pair of consecutive cubic residues not exceeding the pair (77, 78). It is the purpose of this note to show that these phenomena do not persist for triplets of consecutive quadratic or quadruplets of consecutive cubic residues. On the contrary, given any arbitrarily large integer N, there exists a prime whose first set of three con­ secutive quadratic residues lies beyond N. Similarly, another prime can be found whose first set of four consecutive cubic residues lies beyond N. The proof consists in imposing character patterns (mod p) on the integers less than N and prime to 3 and makes use of the quadratic and cubic reciprocity laws and the theorems of Dirichlet on the existence of primes represented by

742 certain linear and quadratic forms, depending on N. The corresponding questions for three consecutive cubic residues (or any odd-power residues) or for pairs of consecutive kth power residues for k ::>- 3 remain open. (Received October 3, 1960.)

574-23. Olga Taussky: Stable matrices.

A matrix A is called stable if all its characteristic roots have negative real parts. Lyapunov showed that real stable A are characterized by the existence of a positive definite G such that AG + GA' = - I (where I is the unit matrix). It is now shown that every real stable matrix is orthogonally similar to (- I+ S)O where Sis skew symmetric and 0 is a diagonal matrix with positive diagonal elements. It is further shown that Lyapunov's theorem is equivalent with the fact that - I + S is 0-stable. A stable matrix X is called

0-stable if XO (O diagonal) is stable if and only if 0 has positive diagonal elements (see A. C. Enthoven and K. J. Arrow, Econometrica vol. 24 (1956) pp. 288-293 and K. J. Arrow and M. McManus, Econometrica vol. 26 (1958) pp. 448-454). It follows trivially that every real stable matrix is similar to a 0-stable matrix. (Received October 3, 1960 .)

574-24. Adil Yaqub: Some remarks on a certain class of rings.

Let n be a fixed integer. A ring R with unit is called ann-like ring if, and only if, it satisfies (1) (ab)n - abn - anb + ab = 0, and (2) na = 0, for all a, bin R. Some properties of n-like rings are derived and, in particular, it is shown that an n-like ring is commutative. Furthermore, a simple characteri­ ation of n-like rings is given. (Received October 3, 1960.)

574-25. J. M. Irwin and E. A. Walker: The purity of high subgroups of Abelian groups.

Let G be an Abelian group and G1 be its subgroup of elements of infinite height. A subgroup H of G is called a high subgroup of G iff H is maximal with respect to disjointness from G1 in G. It has recently been shown (See Abstract 568-19, Notices Amer. Math. Soc. vol. 7 (1960) p. 249) that high subgroups of Abelian torsion groups are pure. As a generalization of this, we have the following Theorem 1: Let H be a high subgroup of an Abelian group G. Then

743 H is pure in G. As a corollary to this we obtain Theorem 2: Let S be any in­ finite subgroup of G with S n G l = 0. Then there exists a subgroup K pure in G with K n G l = 0 and IS I = IK I. Theorem 2 provides the best answer for Abelian groups to L. Fuch's Problem 4 in his book, Abelian groups. (Received October 4, 1960.)

574-26. A, H. Kruse: Concerning the generalized continuum hypothesis and the axiom of choice. Preliminary report.

Let A be the statement that for each infinite cardinal m, there is no cardinal between m and 2m. Let B be the statement that for each aleph m, there is no cardinal between m and 2m. Let C be the axiom of choice (for sets, not classes). It is well known that A is equivalent to the conjunction of B and C.

Assume one is working in the axiomatic set theory of K. Godel's Consistency of the continuum hypothesis, Ann. of Math. Studies, No. 3, including the restrictive axiom but excluding the axiom of choice. Then B implies C, and thus A is equi­ valent to B. This is a consequence of known results and the theorem that C follows from the statement that 2'r.t:a is an aleph for each ordinal a. Another consequence: C is equivalent to the statement that every set totally ordered by some relation is well-ordered by some relation. (Received October 5, 1960.)

574-27. Roy Leipnik: The common fixed-point property in semi-groups. Preliminary report.

The property of fix-points can be generalized to abstract semi-groups as follows: xis a fixed element for yin case xy = yx = x (in Malcev's terminology, xis a zero for y). A semi-group is said to have the common fixed-point property (CFP) if Z1, Zz f 0 and YlY2 = Y2Y1 implies Z1 n Zz f 0, where Zj is the set of fixed elements of Yj· Malcev has characterized semi-groups isomor­ phic to semi-groups of transformations, where the fixed element is equivalent to a fixed-point. Examples are given of semi-groups which are CFP but not Malcev-type, and vice-versa. (Received October 5, 1960.)

744 574-28. E.]. Tully, Jr.: Representation of a semigroup by transforma- tions acting transitively on a set.

A representation of a semigroup S will mean here a homomorphism 1r of S into the semigroup of all transformations on a set M. We write xa for x(a1r)

(x EM, a E S), and call M an operand over S. M (or 7T) is called faithful if 7T is 1-1, and transitive if for all x, y E M there exists a E S with xa = y. Theorem 1. All transitive operands arise (essentially) by letting S act by right multiplication on the set of c:r-classes, where cJ is a right congruence on S, eacra (some e E S, all a E S), and each ()-class meets each right ideal. M is faithful if and only if CT contains no congruence except the identity relation. Example. The free semigroup of rank n has a faithful transitive representation if and only if n;;;:: 2. Theorem 2. A semigroup having a minimal right ideal R has such a representation if and only if for all a,b E S either a = b or ca ~ cb (some c E R). (Received October 5, 1960.)

574-29. E. A. Walker: Pure extensions of Abelian groups.

For Abelian groups X andY, let Pext(X, Y) be the group of all pure exten­ sions of Y by X, let X 1 be the subgroup of all elements of infinite height in X, and let Q and Z denote the additive groups of rationals and integers, respective­ ly. Suppose G is a reduced Abelian group, and T is the torsion subgroup of G.

Then the following hold. (1) T 1 = 0 if and only if Pext(Q/Z ,G) is torsion free.

(2) If G is torsion or torsion free, then G1 = 0 if and only if Pext(Q/Z,G) is torsion free. (3) There exists a reduced group G such that G 1 is torsion free "f 0, T 1 = 0, and hence by (1), Pext(Q/Z,G) is torsion free. The methods are homological. (Received October 4, 1960.)

574-30. C. B. Bell: m-independence, m-measures, and m-homomor- phisms.

Form > ~0• let least m-algebra, Fm (K); m-independence; m-homomor­ phism; and probability m -measures be extensions of the corresponding concepts form = ~ 0 • Further, K( j ¢,Y) is stochastically m-independent with respect to (w .r .t.) a set function A (0 ~ A ;:;; 1) if there exists a probability m -measure iv iv }l on Fm (K) such that p( n vBv ) = 1T v [,.\(Bv)J for each m -sequence fBv ]. Then the Marczewslci-Sikorski (1950-1951) analogies between measures and

745 homomorphisms can be extended as follows: Theorem 1. The following condi­ tions are equivalent: (1) K = { Av} is an m -independent class; (2) for every class L, every mapping J6 of K onto L can be extended to an m-homomorphism of Fm(K) onto Fm(L); (3) the mapping J6(Av) = Ay can be extended to an m­ isomorphism of Fm(I() onto Fm(K) where Avis the cylinder set of XvY corres­ ponding to Av. Theorem 2. If K is stochastically m-independent w.r.t. each set function X (0 ;§! )) & 1), then K is an m-independent class. Theorem 3, If there exists a two-valued m-measure (on S) vanishing for single points, then there exists an m-homomorphism, h, (on S) which vanishes for every one-point set, Conjecture: The converses of Theorems 2 and 3 are valid. (Received October 5, 1960,)

746 THE NOVEMBER MEETING IN EVANSTON, ILLINOIS November 25-26, 1960

575-l. Edgar Asplund: A pair of reciprocal matrices with integer valued elements for test purposes.

Let the n X n symmetric unimodular matrix A = (aik) be defined by aii = 6 fori:;;;;; n- 2, an-l,n-l = 5, ann= l, ai,i+l = -4 fori"§ n- 2, an-l,n = -2, ai,i+2 = land aik = 0 fork;;;;; i + 2. Then the inverse A -l = B = (bik) of this matrix is defined by bik = 6 -li(i + l)(3k - i + l) for k -;;;; i - l and by bik = 6 -lk (k + l){3i - k + 1) fork~ i + l. The condition number (spectral ra­ dius of A times spectral radius of B) is approximately equal ton . Among the wealth of published pairs of explicitly given reciprocal matrices this example has the following two advantages for test purposes. First, all elements of both A and its inverse A - 1 are integer valued. Second, both matrices are given by the "same formula" for all orders n. The matrix A is a band matrix of order 2 whose continuous analog is the differential equation and boundary conditions for the uniform beam with one end clamped and the other end free. The matrix B =A - 1 is a Green's matrix of order 2 corresponding to the Green's function of the above mentioned boundary value problem. For a general theory of band matrices and their inversion see: Edgar Asplund, Inverses of matrices (aij) which satisfy aij = 0 for j > i + p, (Math. Scand. vol. 7 (1959) pp. 57-60). (Received September 15, 1960.)

575-2. V. E. Bene!l': Asymptotic behavior of general queues with one

~.III

As in papers I, II of the same title, K(•) is a nondecreasing step-function, and W(·) is the solution of W(t) = K(t) - t + /otU(- W(u))du, where U(·) is the rt-continuous unit step at 0, with K(O) = W(O) = 0. K( •) is the cumulative load, and W( ·) the virtual delay in a queue with service in order of arrival. The "stationarity" condition E [ exp[- sK(t) + sK(u) + s(t - u)} IW(u) = 0} = «s ,t - u) is used. Once the asymptotic behavior of Pr£W(·) = 0} has been determined

(see I, II,) that of Pr [W( •) ~ w} (w ::> 0) can be studied by Abelian methods with only weak additional hypotheses; e.g., the limiting characteristic function

747 of W(oo)(if any) is expressed in terms of

575-3 .. Btanko Grunbaum and T. S. Motzkin: On components in some families of sets.

A family S of sets is intersectional if for every subfamily S* C S the intersection of all members of S* belongs to S; it is nonadditive if no union of (more than one) pairwise disjoint members of S belongs to S; it has the Helly property of order h if for every subfamily S* the intersection of all members of S* is nonempty provided the intersection of all members of each subfamily S** CS*, containing less than h members, is nonempty. LetS be intersectional

and nonadditive, and U the family whose members u are unions of a set of pair­ wise disjoint members si of S (the "components" of u over S). For u E U let c(u) be the cardinal number of components of u over S. Theorem l. Let V C U and suppose c( n V) = n + l (n finite or infinite). Then there exists a subfamily V' C V of k sets, with k ~ n, such that c(nv•);;:; n + l. Theorem 2. Let Shave the Helly property of (finite) order h; if c(w n w') -g 2 for all w, w' E W C U, then W has the Helly property of order 2h - l. Related results apply to the family of all unions of two disjoint closed convex sets. (Received August 18, 1960 .)

575-4. M. M. Peixoto: The density theorem for two-dimensional structural stability.

Let M be a two dimensional compact manifold of class c 2 endowed with a metric. Call,0' the space of all vector fields on M with the C 1-topology. A vector field X is said to be structurally stable if given arbitrarily e > 0 there

is a neighborhood ~of X such that whenever a vector field Y E L:, then there exists a homeomorphism of M onto itself transforming trajectories of X onto trajectories of Y and moving each point by less than e. Theorem: the set ~of all structurally stable vector fields is open and dense in Ji3. The case where M is the sphere s2 follows immediately from a previous result of the author [!\nn. of Math. vol. 69 (1959) p. 199-222]. This theorem shows that for dimension 2 structural stability is the rule rather than the exception .. (Received September 13, 1960.)

748 575-5. M. S. Ramanujan: Connectedness of limit points of transforms of sequences.

Let ftnJ denote the transform of the sequence fsn} by the matrix A. Suitable order conditions on {SnJ are determined so that the set of limit points of ftn~may be connected. In the cases of the Holder transformation (H,r) (r > 0), and of the Ces11.ro and Riesz transformations of order r .Eo 1, if

Sn = o(n) then the set of limit points of ftn~ is connected. If 0 <:::: r < 1 we require Sn = o(nr) in the case of Ces11.ro and Riesz transformations of order r. Also, for the Euler transform (E,p), the Taylor method (T,p) and the Laurent series method (S,p), with 0 < p < 1, if Sn = o(n 112) we have a similar result. Consequently, if A denotes either a regular Hausdorff method (H,Jln) or a regular quasi-Hausdorff method (H*•Jln+1) generated by a normalised function X(t) of B.V. in [0,1] with X(+ 0) = X(O) = 0 then, for the class of sequences s with sn = o(n 112) the limit points of ftnt form a connected set if X(1) = X(1 - 0). Thus in each of these cases there certainly exists a function c(n) such that whenever sn = o(c(n)), the limit points of [tnJ are connected. The nature of the matrices involved also enables us to assert that in each case there exists another c 1 (n), such that c 1 (n)/c(n)']' + oo and if sn = o(c1(n)) then the same conclusion on the set of limit points of [tzJ holds. (Received July 18, 1960.)

575-6. Robert Seall and Marion Wetzel: Properties of double chain sequences.

Double chain sequences arise in the study of the moment problem for the interval(- 1,1) (R. Seall and M. Wetzel, Some connections between continued fractions and convex sets, Pacific J. Math. vol. 9 (1959) pp. 861-873). These are sequences of real numbers of the forms a~= 4g2n(l- g2n-2)g2n-1(1 - g2n-1>• ho = 1 - 2g2n-2(1 - g2n-3) - 2g2n-1(1 - g2n-2> where 0 ~ gn-1 ~ l. Analogues of several properties of ordinary chain sequences are obtained. These include connections between double chain sequences and quadratic forms. In particular, fa~J. fbpJ form a double chain sequence if and only if the quadratic forms h~=1(1 + bp)x~- 2z~:iapxpxp+1 are positive semidefinite. It is shown that a double chain sequence has maximal parameters Lp,Mp; g 2p_ 1 !!iii Lp ;:;O 1, g 2p l!2 Mp ~ 1; and minimal parameters ,tp,mp; 0 :;::;;; -4 ~ g2p-l, 0 ;:§ mp & g 2p. Using methods of H. S. Wall, (Continued fractions and totally monotone sequences, Trans. Amer. Math. Soc. vol. 48 (1940) pp. 165-184) the following

749 theorem is proved: The sequence fcnJ is a moment sequence for the interval (- 1, 1) if and only if L~O cpxP, co = 1, has a J -fraction expansion in which the elements form a double chain sequence. A number of applications of double chain seq!iences are given. (Received September 7, 1960.)

575-7. Wolfgang Wasow: Turning point problems for certain systems of linear differential equations. I

The asymptotic expansions for the solutions of vectorial analytic differ­ ential equations of the form sh du/dt = A(t,e)u, as e ~o. are developed in a manner that reveals the nature of the singularities of the coefficients of these expansions at a turning point, assumed to be located at t = 0. Under certain conditions the ratio of consecutive terms of the asymptotic series is shown to be O(t-h(n+l)/ne), as t ~0. e ____,.0, If h = 1, a suitable combination of stretching and shearing transformations changes the system into the form dv/ds = H(s,el/(n+l))V; which admits series solutions in powers of el/(n+l). The ratio of consecutive terms of this power series is shown to be O(s(n+2/n e 1/(n+l)), as s ~ oo, e ~ 0. These results are essential for a proof that the various series expansions have asymptotic character in regions which together cover a full neighborhood of the turning point. (Received September 23, 1960,)

575-8. J. N. Younglove: Two theorems on metrization of Moore spaces.

A space is said to have Property P provided that if G is an open covering of S, the set of all points, there is a refinement H of G which covers S such that no point is in each set of an uncountable subcollection of H. L. F. McAuley_ has shown 1}\ note on complete collectionwise normality and paracompactness, Proc. Amer. Math. Soc. vol. 9 (1958) pp. 796-799] that a normal, separable and pointwise paracompact Moore space is metrizable. This is modified as follows. Every separable Moore space which has Property P is metrizable. R. H. Bing has shown jMetrization of topological spaces, Canad. J. Math. vol. 3 {1951) pp, 175-186] that a normal screenable Moore space is metrizable. This is modified in the following theorem. A normal complete Moore space is metrizable provided that the boundary of each domain which has a boundary is screenable. A complete Moore space is one which satisfies all of R. L. Moore's Axioms 0 and 1. (Received August 22, 1960.)

750 575-9. H. ]. Zassenhaus and Irving Reiner: Equivalence of representa­ tions under extensions of local ground rings.

Let R be a valuation ring in an algebraic number field K, let K' be a finite extension field over K, and let R' be a valuation ring of K' containing R. For a finite group H, let M and N be left RH -modules having finite R-bases. Generalizing a result of one of the authors (Behavior of integral group repre­ sentations under ground ring extension, to appear in Illinois ]. Math.), it is shown that R' @ RM;;;; R' 0 RN as R'H- modules if and only if M ~N as RH­ modules. This result also holds for modules over R-orders in K-algebras provided some natural restrictions are imposed. The proof uses the Krull­ Schmidt Theorem together with a result due to Maranda (On p-adic integral representations of finite groups, Canad. ]. Math. val. 5 (1953) pp. 344-355.) (Received September 21, 1960.)

575-10. Norman Fritz: Bounds on the product of an infinitesimal generator and its semi-group operator.

Let T(t) be the semi-group generated by a closed linear operator A with dense domain in a Hilbert space and satisfying Re(Ax,x) :§ 0. If A is self-adjoint, one proves by elementary means that A T(t) is bounded and has a norm not greater than (et) - 1. In general, A T(t) will not be bounded. But suppose that for some m ~ 0 and all x in the domain of A, m Re(Ax,x) ~ - IIm(Ax,x) 1. the num­ berm measuring the operator's deviation from self-adjointness. Then using a contour integral representation, it is shown that A T(t) is bounded with norm not exceeding 2(7/'t)- 1(1 + m)((1 + m 2)1/ 2 + m). If A is normal, one obtains a "best" bound (et)-1(1 + m 2) 112 • (Received September 29, 1960.)

575-11. P. C. Hammer: Reduction of limit functions.

Let M be the space and N its null set. Let M be the class of all subsets of M. A function f mapping M into M is called a limit function provided X nfx = N for all X. The primitive function, f', of a limit function f is defined by f'X = U[fY: Y S. X}. The set f'X is the set of primary f-limit points of X.

Iff is an arbitrary function mapping Minto M then the limit function f 1 off is given by f 1X = fX - X. Two limit functions are equivalent provided their primitive functions are identical. Let f be a limit function and let f 1 be the

751 limit function off', the primitive function of f. Define two functions u,v as follows where nx II is the cardinal number of X. uX = f 1 X - u[f'Y: y ~ X, IIYII < nxn}. vX = pX- U{f'Y: Yc;;: X, IIYH = uxn. llf'YII < llf'XII}. Then u and v are limit functions equivalent to f, v <;_ u k fl • uX = vX for nx II finite, and if uX,;. N implies !IX II is finite then u is the minimal limit function equivalent to f. The class {X: vX 7'- N} is a convergence class for the extended topology generated by f. (Received September 29, 1960.)

575-12. P. C. Hammer: Domain finite expansive functions.

Let M be a set and N its null set. Let M be the class of all subsets of M.

Let f: M ·--? M be a function and consider three possible properties of f. 1. fX 2 X (enlarging). 2. f(X U Y) = fX U fY (isotonicity). 3. f(fX) = fX

(idem potence). A function satisfying (1) and (2) for all X, Y c;: M is called~ pansive and a function satisfying (1), (2), (3) is called a closure function. An expansive function g such that for each X, gX = U [gY: Y £ X, Y finite} is domain finite. Theorem: Let g be an expansive function. 1. If g is a domain finite expansive function then h = gc.> = Ugn is the minimal closure function containing g. 2. The closure function h of g is domain finite if g is domain finite. 3. If gX =: UtgY: Y ~X, IIYII & nJ where IIYII is the cardinal number of Y then gkX E U[gkY: Y ~X, IIY II;;; nkJ. 4. If n = 1 in (3) it follows that the closure function h of g is universally additive i.e. for X f N,hX = U {h fP}: p E XJ. 5. The union function of a family of domain finite expansive functions is domain finite. 6. The composition function of a sequence of domain finite expansive functions is domain finite. (Received September 29, 1960.)

575-13. P. C. Hammer: Neighborhoods for extended topology.

Let M be the class of all subsets of a set M with null set N. Let g be an expansive function mapping M into itself and let r = cgc be the contractive function dual to g (c is the complement function). Then a set Y is a g-neighbor­ hood of a point p if and only if p E rY. Theorem A. Let g,r = cgc be given. Let Np be the class of all g-neighborhoods of p. Then 1. ~ is empty if and only if p E gN. 2. Np is ancestral i.e. X 2 Y, Y E ~ implies X E Np. 3. If Y E Np then p E Y. 4. p is a primary g-limit point of X if and only if Y II (X - {P}) = N for each Y E NP. Theorem B. To each point p E Z ~ M let

752 N be a nonempty class of subsets of M satisfying (2) and (3) of Theorem A. p Then Np is the class of g - neighborhoods of p for an expansive function g = ere where rX = [p: X E Np}. The set gN is comprised of points without neighbor­ hoods. (Received September 29, 1960.)

575-14. Melvin Henriksen and Meyer jerison: The space of minimal prime ideals of a commutative ring.

GJ = lf(A) denotes the space of minimal prime ideals of a commutative ring A (with the hull-kernel topology). lf B is an ideal of A, there is a homeo­ morphism of {?(B) onto an open subspace of J:T (A). If N is the set of nilpotents of A, then lf(A) and V(A/N) are homeomorphic; so we assume henceforth

that N = (0). if is a T 2 -space that has a base of open and closed sets. If B, I CA, let h(I) = {P E /?:I C P} and let O[(B) =fa EA: aB = 0}. If for every a,b E A, there is a c E A such that h(c) = h(a) n h(b), then {?is compact iff {h(a): a E: A} is a base for the open sets of .@and iff for every x E A, there is a y E A such that h(y) = h( CX(x)). Let C = C(X) denote the ring of all continuous real-valued functions on a space X and let C* denote the subring of bounded elements of C. Then .(9(C) and Cfl(C*) are homeomorphic. If X is metrizable, then ,@(C) is compact. For any space X, .lf(C) is countaply compact but need not be locally compact. If j2/> is locally compact, then C(j?) is a conditionally a--complete lattice. (Received September 29, 1960.)

575-15. W. G. Leavitt: A classification of rings by module type.

All rings considered are with unit. A ring is dimensional (type d) if all finitely based A-modules have invariant length of basis. It is known that all

Artinian, Noetherian, and commutative rings are dimensional. It is proved that for any nondimensional ring A there exist unique integers (n,k) such that

(i) an A-module with basis of length<:: n has invariant basis length, (ii) ~ other finitely based A-module has bases of length h + mk for some fixed h,

n ~ h < n + k and arbitrary m. Such a ring is said to have module type (n,k), and rings of arbitrary type are constructed. Module types form a lattice under: (n,k) "::! d; (n,k) '2 (n',k') if n"" n' and k lk'. The lattice operations are shown to be connected with certain ring constructions. The following is proved: Let A,A' have types a,a' respectively; _!!_A-A' is a unit preserving homomor­

phism of A into A' then a'~ a. Additional dimensional rings are the complete

753 matrix rings Am when A is dimensional, and the PI-rings. It is shown that if A

is of type (n,k) then Am has type (n + r/m,k/g), where 0 ;;ii r < m and g = G.C.D.(k,m). (Received September 30, 1960.)

575-16. L. S. Levy: An extension of the concept of torsion. Preliminary report.

Definition: An element m of a right module is a torsion element if md = 0 for some regular element d of the ring of operators. Theorem 1: The torsion elements of each right R module form a submodule if and only if R has

a right ring of quotients. Theorem 2: If R has a two sided ring of quotients which is semi-simple, then every finitely generated module is isomorphic to a submodule of a free module. If, in addition, R has an identity and every finitely generated right ideal is projective, then every finitely generated right module is isomorphic to the external direct sum of its torsion submodule and a finite number of right ideals. A side result of the methods used in proving the above theorem is a new proof that a ring has a right quotient ring if and only if: it contains a regular element and for every pair of elements x,d with

d regular there exist a pair d t, y with d 1 regular such that xd 1 = dy. (Received September 29, 1960.)

575-17. L. S. Levy: Rings with semi-simple quotient rings. Preliminary report.

Definitions: A ring is quotient-(semi-) simple if it has a two sided quotient ring which is (semi-) simple with minimum condition. A subdirect sum R of rings Ri is irredundant if for each j the mapping r - Lij jri (r E R, ri E Ri) has a nonzero kernel. A right ideal J is pure if xd E J and d regular =} x E J. Theorem 1: A ring is quotient-semi-simple if and only if it is a finite subdirect sum of quotient-simple rings. The sum can be taken to be irredundant, and then the kernels of the projections onto the summands are uniquely determined ideals. Hence the summands themselves are unique up to isomorphism. Theorem 2: A ring R with a two sided quotient ring is quotient­ semi-simple if and only if: (1) R satisfies the DCC for pure right ideals; and (2) the intersection of the maximal pure right ideals is 0. Theorem 1 relates the main theorems of two papers of A. W. Goldie [i=>roc. London Math. Soc. ser.3 vols. 8,10): Theorem 3: A ring is semi-prime with the left and right

754 quotient conditions if and only if it is a finite subdirect sum of prime rings with the left and right quotient conditions. (Received September 28, 1960.)

575-18. Casper Goffman and R. E. Hughs: Functions of BVC type.

A real function f, defined on the unit square Q, is BVC if there are g and h, equivalent to f, whose sections gx and hy are BV a.e. and.fo1V(gx) .<:: oo, J'Jv(hy) < oo. It is shown that iff is BVC, there is a single k whose sections kx and ky are BV a.e. andJ01V(kx) < oo, .folv

575-19. J. V. Ta1acko: On the characteristic kernel of two-person zero-sum games.

The two-person zero-sum game is a special case of linear programs, equivalent to a compound matrix, an abstract structure, called "Standard Form." (See Rockafellar-Talacko, Abstract 555-52, Notices Amer. Math. Soc. val. 6 (1959) p. 261.) For every (2P-OS) game (y,x; A; ln,- lm; 0), explicitly written, the Resolvents { x,y; A; a.,ft; v} may be found in a minimum number of iterations, without use of determinants by pivotal, direct, computational algorithm of linear systems called "Symmetric Method." To every distinct basic solution exists not only a distinct nucleous square matrix (Shapley-Snow Kernel) H s; A, but also a distinct corresponding H k A, called "Characteristic Kernel," invariant with respect to the value of the game. It has this property: If H = fpi1J; L.Jtij = L,jliij = 0. Every completely mixed pay-off matrix has a unique resolvent with the unique characteristic kernel H = [_liij]. This is an extension of the saddle point principle to any order of nucleous matrices. The character­ istic kernel is an important characteristic, an inseparable property of the two­ person zero-sum game. (Received October 3, 1960.)

575-20. P. M. Weichsel: A decomposition theory for finite groups.

Let {Ga.} be a set of finite groups. Define the closure of {Ga.J• {a;,.}. to be the intersection of all sets of finite groups which contain [Ga.\ and are closed under the operations of taking subgroups, factor groups and finite direct products. A finite group G is called decomposable if G E {A a.~, with f Aa.s the set of proper subgroups and factor groups of G. Some general proper-

755 ties of decomposable groups are studied and the indecomposable p-groups of class two are characterized in several ways. For example, a p-group of class two is indecomposable iff it has two generators and its center is cyclic. In the case p f Z the defining relations of the indecomposable groups are compiled. It is also shown that if G is a finite p-group of exponent p and class greater than two then G is decomposable if G/Z(G) is a direct product. The equivalence relation G =H if fGJ = [fiJ is compared with P. Hall's notion of isoclinism and it is shown that for a certain class of p-groups and a modified form of the closure defined above these relations are equivalent. (Received October 3, 1960.)

575-Zl. A. B. Brown and S. S. Cairns: Sharpening of Sperner's lemma applied to homology groups.

Let sn be an n-simplex in En subdivided into a linear simplicial complex Kn and fa mapping of the 0-cells of Kn into the vertices of sn which carries any given 0-cell t0 into a vertex of the face of sn on which t 0 lies. Then the number of n-simplexes of Kn whose vertices are carried into all n + I vertices of sn with preservation of orientation exceeds by one the number for which orientation is reversed. The proof is by mathematical induction, using the property that the image of the boundary of a chain is the boundary of the image. This sharpening of Sperner's Lemma makes possible a proof of the topological invariance of the k-dimensional combinatorial homology group of a complex

without introducing the (k + Z)-dimensional 'prism 1 complex, thus greatly shortening the proof. The sharpening of Sperner's lemma was obtained by Brown while reading the manuscript of Cairns' forthcoming Introductory topology, Ronald Press , and, together with the application to homology theory, will appear in the book. (Received October 4, 1960 .)

575-ZZ. Trevor Evans: Abstract mean values.

A number of authors have studied conditions on continuous real functions which characterize mean value functions. In this paper it is shown that in some of the characterizations the continuity and order conditions may be dropped as well as the assumption that the underlying set is the real numbers and that the remaining algebraic conditions plus a weak assumption of reversibility of the functions still characterize mean values. For example, Kolmogorov has shown that if Mn, n = 1,2,3,. •. is an infinite sequence of real continuous functions each

756 of which is symmetric, idempotent, increasing, and which satisfy, for all nand k;;:; n, Mn(xr,x2•···•xn) = Mn(Mk•···•Mk,xk+l"··•xn>• where Mk stands for Mk(xr,x2•···•xk), then Mn(xr,x2,··•xn> = f- 1 {{f(xr) ••. f(xn))/nj-. It is proved in this paper that if the real number, order and continuity conditions are replaced by the assumption that, for each Mn, the equation Mn(x, a2,a3, ... ,an) = b has a unique solution, then the underlying set of elements is a vector space over the rationals and Mn is the usual arithmetic mean. (Received October 4, 1960.)

575-23. Casper Goffman and C. J. Neugebauer: On approximate derivatives.

Let I= [0,1], R be the real line, and f:l -R be a function which posses­ ses everywhere on I an approximate derivative f' • G. Tolstoff [Rec. Mat. ap (Mat. Sbornik) N. S. vol. 4 (1938) pp. 499-504] has shown that f~p is of Baire class one, and A. Khintchine [Fund. Math. val. 9 (1927) pp. 217-279] has proven the mean-value theorem for approximate differentiation. The proofs of Baire class one and the mean-value theorem in the above sources are long and difficult. The present paper gives a short and simple treatment of these pro­ perties. Baire class one is established by showing that f~p is the limit of a convergent interval function. It is shown that the mean-value theorem is equi­ valent to the Darboux property of f~p' and the Darboux property of f~p is established by an easy argument. (Received October 4, 1960 .)

575-24. S. A. Husain: On uniform convergence of orthonormal expansion.

Let C denote the class of continuous periodic functions of Period 21r, and CF the class of functions f(x) .0: C having uniformly convergent Fourier Series. Let W(8) be the modulus of continuity of f(x), and let fl(o) be a continuous func­ tion of 8 ,0(8) J. 0, and fl(x + y) ;§. fl(x) + fl(y), then there exists a function f(x) E C having as its modulus of Continuity fl(8) [N. K. Bari and S. B. Steckin, Trudy Moskov. Mat. Obl!'l!. vol. 5 (1956)]. A function f(x) E C belongs to the class c0 if W(8) = 0(0(8)), 8 ~o. Let fi!l'))(x)J- be an orthonormal system in [0,1]. Then f(x) "'L~=oC 11 i!l:v(x) denotes that L~OCyi!ly(x) is the orthonormal expansion of f(x) in the system {111-v(x)J. We note that Sn(f;x) = Zff=oAvCvi!lv(x) =Jo1f(t) Kn(x,t) dt, where Kn(x,t) =L:~=o)jvi!l:v(x) i!l:v(t). Now an analogue of

Bojanic's theorem ~. Bojanic, Publ. lnst. Math. Acad. Serbe Sci. val. X (1956)], is given by the following: Theorem: Let fi!ly(x)} be an 0. N. S. in [0,1], closed

757 in C. If given an E-;.- 0; 3 anN and a~ :.>O,E SJ(8)j~/1Kn(x,t)- Km(x,t)ldt ~ E for all x, and n, m > N, then [}jv3 belongs to the class (Cn,Cp) i.e., for every f(x) E cfl the Series I:.:oAvCvlll:>~(x) is uniformly convergent. (Received October 3, 1960.)

575-25. Irving Kaplansky and R. E. MacRae: Homological dimension of ideals in Noetherian domains.

The authors show: let R be a Noetherian domain and I an ideal generated by two elements. If the homological dimension of I is finite then it is either 0 or 1. The pattern of proof involves the observation that if a,b is an R-sequence then a+ bx generates a prime ideal in Rflc). One then transfers matters to the quotient ring, R[x]s, where S consists of the powers of a+ bx. Among the con­ sequences of the theorem is the assertion that every regular local ring is a unique facto·rization domain. (Received October 5, 1960.)

575-26. M. I. Knopp. Estimation of certain exponential sums in the theory of automorphic forms.

Let G(.l11Z,, (J! = 1,2,3), denote the group of transformations of J} fl;) > 0 onto itself, generated by S(7:) = 7:+ j}/2, T(·z;) = - 1/7:. G(l) is the modular group. In constructing automorphic forms of dimension zero, with arbitrary multiplier systems for G(L1/2), by means of Fourier series, certain exponential sums occur which are related to the well-known Kloosterman sum. In order to carry out this construction of forms it is necessary to obtain an asymptotic estimate for these exponential sums. This is accomplished by showing that each of these sums can be expressed as a constant times a Klooster­ man sum. Since the needed asymptotic estimate is known for the Kloosterman sum, the estimate also follows for the exponential sums in question. The proof that the sums can be reduced to Kloosterman sums makes use of a determination of all possible multiplier systems connected with the dimension zero for G(2 1/ 2) and 0(31/2) previously given by the author [Duke Math. j., to appear]. Also, use is made of a procedure used by Lehner,(Michigan Math. j. vol. 6 (1959) pp. 71-88] in treating such sums connected with the dimension -2 for G(l). (Received October 4, 1960.)

758 575-27. Lawrence Markus: Optimal control for nonlinear systems.

The author proves that he exists, optimal control (for nonlinear pro­ cesses) exists, extremal control exists, and that this is the best of all possible worlds. (Received October 4, 1960.)

575-28. Arthur Mattuck: Some relations on symmetric products.

Let C(g) be the g-fold symmetric product of an algebraic curve C. On it is the positive canonical divisor S, the simple coincidence locus L., and the locus Xtp] of g-tuples containing p. Then we have (i) 2S + 6 ~zig-Zx[Pi_] where LPi is a canonical divisor on C, and (ii) 1T- 1(0) = S +X tp0], if rr: C(g) --+], J =Jacobian. These relations simplify Weil's proof of the Castel­ nuovo-Severi inequality, Matsusaka's proof of er:; r~ wg-r and give the Weier­ strass points. And a generalization of (ii) gives the Chern classes of the bundle (J, C(n}, 7r} for n > > 1, and thus the homology groups of high symmetric products of C. (Received October 4, 1960.)

575-29. G. ]. Rieger: On the number of prime factors of the natural numbers in arithmetic progressions. n For the natural number n with the canonical representation n = TfplnP P let V(n) = v I n . We denote by N(n,x; k,{} the number of natural numbers ~p n p - m ;;;;_ n with m = J mod k and V(m) - loglog n-< x(loglog n) 112 . Theorem. There exists a positive constant c(k} only depending on k with IN(n,x; k~}/n 2 - (l/k(2?r) 112>_,toox e -u 12du I-< c(k}(loglog n)112 for arbitrary real x. This generalizes a result of Renyi and Turan (Acta Arith.4, 71-84, 1958) where k = 1.

The method here is similar: evaluate the characteristic function of the corres- ponding probability distribution analytically by complex integration and apply a theorem of Esseen (Acta. Mathern. 77, l-125, 1945, Theorem 2a). c(k) can be given explicitly. Extending the problem, Tanaka (Jap. J. Math. vol. 2 7 (1957} pp. 103-127) considers simultaneously Vln) = LpJn,pC.PjnP (j = 1, ... ,t}, where

P 1, ... ,Pt are certain disjoint sets of primes, and proves error terms o(1) (n--+oo) essentially by Brun's elementary method. These error terms are improved here to Ox(logloglog n/(loglog n) 1/ 2) by refining the elementary method and to O((loglog n)112) analytically as in the Theorem above. Similar results are obtained for other additive number-theoretic functions and for arbitrary algebraic number fields. (Received October 5, 1960.)

759 575-30. Dov Tamari: Imbeddings of partial (incomplete) multiplicative systems (monoids), associativity and word problem,

Earlier work of the author is expanded, clarified and related to work of Lambek [Canad. J, Math. vol. 3 (1951) p. 34], Bates [Amer. J. Math. vol. 69 (1947) p. 499], Evans [J. London Math. Soc. vol. 26 {1951) p, 64; vol. 28 {1953) p.76], Baer [Amer. J. Math. vol. 71 {1949) p. 706; vol. 72 {1950) pp. 625,647], H. and B. H. Neumann and others. Let M = (M,•,p) be a ;-monoid (jJ a regular, reflexive relation), and Ms a symmetrical M, i.e. with 1: (m)(1·m = m • 1 = m) and I. P. (inverse property): (m)(3m)(m•(m·x) = x, (y·m)·ffi = y if m•x, re~Jp. y•m, exist). Certain simple Ms -properties are n.a.s.c. for symmetrizing M (imbedding in an Ms>· The symmetrizations of M generated by M form a lattice with smallest el. M U M (M anti-isomorphic to M; obvious multiplication), followed by several quotient constructions and a greatest el. - an I. P. -loop. Ms totalizes to a group G if and only if Ms is ,o-associative; hence a n.a.s.c. for imbedding an M into a group is the existence of p-asso. symmetrization. All Ms generating G being ;J-asso., the most convenient might be tested. Applications: (1) Wordproblem: a system of generators and defining rel. can be considered an asso. M; are its symmetrizations asso. (2) Imbedding problem of semigroups M into a group: requiring a certain simple Ms to be J-asso., yields immediately the author's circular model of generalized Malcev conditions and new insights in their arithmetic and prototype theory. (Received September 13, 1960.)

760 Experienced AP P Ll ED mATHEmATIIIADS Preferably With Ph D

You are invited to consider positions now available in a new applied mathematics group being formed within General Electric's Heavy Military Electronics Department. The Department's activities encompass design and manufacture of land-based and seaborne military electronics equipment including: radar, sonar, data processors, communication systems and guidance equipment. Areas for mathematical investigation include:

ORBITS AND TRAJECTORIES- Developing new methods of integrating equations of motion and optimizing guidance system parameters in the study of earth satellite launcher trajectories and paths of lunar probes.

ANTENNAE- Re-examination of classical equations of electromagnetic theory and developing new methods of solution.

BOOLEAN ALGEBRA- Developing new methods for eliminating circ)lit components, investigating the use of multi-valued logics and study of new computer organizations made possible by recent advances in solid state components.

PROBABILITY- Research in extraction of signals from noise, digital smoothing and anti-jam characteristics.

GENERAL NUMERICAL ANALYSIS-Solution of a variety of problems in integration, interpolation, statistics, non-linear equations, gradient methods, eigenvalue problems, etc.

MANAGEMENT SCIENCES- Applications to problems in reliability, manufacturing scheduling, spare parts stocking policy, applied game theory and the transportation problem. An IBM 7090 and programming services are available for probl01ms requiring machine solution.

Write in full confidence to Mr. W. J. Eschenfelder, Dept. 123-MK HEAVY MILITARY ELECTRONICS DEPARTMENT GENERAL ELECTRIC CouRT STREET • SYRACUSE, NEw YoRK

GE-2056 BY APPLYING TOMORROW'S CONCEPTS TODAY .. . AT AMHERST LABORATORIES

Man's vision of future exploration in the infinity of outer space is predicated on pertinent data that can be gathered today's dynamic by each space probe today. As the probe distances increase ... our universe becomes smaller and communications are R&D mission the vital link that proves success or failure of each venture into the unknown. It is imperative that new systems of com­ also includes creative munications be developed well in advance of other facets of space technology. At Amherst Laboratories new solutions opportunities in are sought to communications problems even before they are encountered. Creative ideation is not compromised by con­ Navigation, Microwave ventionally accepted conclusions. Amherst Laboratories is currently anticipating tomorrow's electronics needs in areas Data Handling, of advanced Ground, Air and Space Communication. PROFESSIONAL STAFF AND MANAGEMENT OPPOR­ ECM and Radar TUNITIES are unlimited for Electronics Engineers, Physicists atzd Mathematiciam with advanced degrees and creative desire. Systems You are invited to direct inquiries itz confidence to Mr. W. Serniuk, Director, Amherst Laboratories . .. or call N F 3-8315 for infor­ matiotz. All inquiries will be ack11owledged promptly.

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