SOCIETY

THE AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. Walker

CONTENTS

MEETINGS Calendar of Meetings ...... • ...... • ...... • . . . . . 64 Program of the April Meeting in Chicago, Illinois . . . • ...... 65 Abstracts of the Meeting - pages 104- 117 Program of the April Meeting in Atlantic City, New Jersey ...... •..... 71 Abstracts of the Meeting - pages 118-130 Program of the April Meeting in Monterey, California. . • . . . • ...... 78 Abstracts of the Meeting- pages 131-138

PRELIMINARY ANNOUNCEMENT OF MEETING .....•...... 82 A REPORT TO MEMBERS ...... •.....•...... 85 NEWS ITEMS AND ANNOUNCEMENTS .•.....•...... •.•.... 81, 88, 9Z, 94 PERSONAL ITEMS ...... • . • . • ...... • . . . . • . . • ...... 89 NEW AMS PUBLICATIONS ...... • . • ...... 93 LETTERS TO THE EDITOR ...... •. 95 MEMORANDA TO MEMBERS in Computation • ...... • ...... 96 Reprinting of Back Volumes of Mathematical Reviews . • ...... 96 Reciprocity Agreement With the Sociedade de Matematica de Sao Paulo . . . 96 Change of Address Notification Deadlines for Journals ....• , . . . . • . . . 96 Group Travel to Stockholm . . . . • . • ...... • . . . . • . . . . . • . • . . . . 97 CATALOG OF LECTURE NOTES ...... • ...... • . • • . . • . 99 SUPPLEMENTARY PROGRAM -No. 10...... 100 ABSTRACTS OF CONTRIBUTED PAPERS ...... •••...... 103 ERRATA - Volume 9...... • ...... 155 INDEX TO ADVERTISERS...... • ...... • ...... 163 RESERVATION FORM ...... • . . • ...... • . . . . 163 MEETINGS

CALENDAR OF MEETINGS

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

Meet­ Deadline ing Date Place for No. Abstracts*

(june Issue of the NOTICES) April 27 592 August 27-31, 1962 Vancouver, British Columbia july 6 (67th Summer Meeting) 593 October Z7, 1962 Hanover, New Hampshire Sept. U- November 16-17, 1962 Tallahassee, Florida November 17, 1962 Los Angeles, California january 24-28, 1963 Berkeley, California (69th Annual Meeting) August 26-30, 1963 Boulder, Colorado (68th Summer Meeting) january Z0-24, 1964 Miami, Florida (70th Annual Meeting) August, 1964 Ann Arbor, Michigan (69th Summer Meeting) August, 1965 Ithaca, New York August, 1966 New Brunswick, New jersey

* 'the abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for by title abstracts are April 27 and June 29.

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The NOTICES of the .American Mathematical Society is published by the Society six times a year, in February, April, june, August, October and November. 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, 190 Hope Street, Providence 6,Rhode Island. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34. Zl, P. L. and R.). Accepted for mailing at the specialrateofpostagepro­ vided for in section 34.40, paragraph (d).

Copyright© 1962 by the American Mathematical Society Printed in the United States of America

64 Five Hundred and Eighty-Ninth Meeting University of Chicago Chicago, Illinois April 12-14, 1962

PROGRAM

The five hundred eighty-ninth meet­ Mark Kac, (Rockefeller Institute); R. D. ing of the American Mathematical Society Richtmyer, (New York University); and will be held in Chicago, lllinois on April A. H. Taub, (University of lllinois). The 1Z-14, 196Z. A feature of the meeting is objective of the Symposium is to examine that there will be a Symposium on Experi­ ways in which the arithmetic potential of mental Arithmetic supported by the Insti­ modern high speed computing equipment tute for Defense Analyses. can furnish experience which sheds light Meeting headquarters will be at the on outstanding mathematical problems, in­ Shoreland Hotel, 5454 South Shore Drive. cluding those arising in other sciences. The Shoreland is offering a special flat Thus, particular attention will be paid to rate of $8 per single room and $6 per per­ the arithmetic experience which may now son in a twin-bedded double. Members be acquired to give insight into important who wish to stay where most of the mathe­ mathematical problems. The design of maticians will be housed should make re­ arithmetic experiments, the reduction and servations directly with the Shoreland analysis of data obtained through arithme­ mentioning the Society. tic experiments, and the mathematical Registration will be in the Common theories pertaining to arithmetic experi­ Room on the second floor of Eckhart Hall ment, such as the theory of Monte Carlo beginning at 9:00 A.M. on Thursday, April calculations, will concern many of the 1Z. speakers. Table calculations and other By invitation of the Committee to non-experimental arithmetic will not be Select Hour Speakers for Western Sec­ included. tional Meetings, Professor Andrew Wal­ Sessions for the presentation of lace of Indiana University and Professor contributed papers will be held at 10:00 Noboru Ito of the University of lllinois A.M. on Thursday and at9:00 A.M. on Fri­ will address the Society. Professor Wal­ day and Saturday. Due to the crowding of lace's title is "Geometric methods in dif­ the program with the Symposium, there ferential topology," while Professor Ito will probably not be a special session for will speak on "On permutation groups of the presentation of papers which failed to prime degree." Both lectures will be in meet the deadline unless the program of Room 133,Eckhart Hall. Professor Wallace contributed papers is fairly . A Vu­ will speak at 11:00 A.M. on Friday, April Graph will be available for those who wish 13 and Professor Ito at the same hour on to employ it in Eckhart 133, Eckhart Z06, Saturday, April 14. the other room to be used for the presen­ The sessions of the Symposium will tation of contributed papers, is a modern be held on Thursday, Friday, and Saturday classroom with first class blackboard beginning at Z;OO P.M. facilities. However, a Vu-graph will also The Organizing and Invitations be available for those who wish to employ Committee for the Symposium consists of it. N. C. Metropolis (Chairman), (Institute for The facilities of Hutchinson Com­ Computer Research, University of Chi­ mons, a dining room directly across from cago); Marshall Hall, Jr.(Californialnsti­ Eckhart Hall will be available for the So­ tute of Technology); Peter Henrici, (Uni­ ciety and guests for all meals. versity of California at Los Angeles);

65 PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes and no more. It is true that the contributed papers are scheduled at fifteen minute intervals, but the extra five minutes is for the purpose of permitting a listener to go from one session to another and to allow time for discussion of the preceding paper. The schedule in the Program must, therefore, be adhered to and to this end the time limit will be strictly en­ forced. ALL SESSIONS WILL BE HELD IN ECKHART HALL

THURSDAY, 10:00 A.M. Session on Analysis and Applied Mathematics, Room 133 10:00 - 10:10 (1) On extensions of a positive-definite function from an interval Professor Joshua Chover, University of Wisconsin (589-19)

10:15 - 10:25 (2) Some spaces of Fourier coefficients Professor G. W. Goes, DePaul University and University of Western Ontario (589-28) 10:30 - 10:40 (3) The remainder of certain linear approximation formulae for two variables Professor D. D. Stancu, University of Wisconsin (589-11) 10:45 - 10:55 (4) An example of a measureable control function which is not piecewise continu­ ous Professor E. 0. Roxin, RIAS, Baltimore, Maryland (589-30) 11:00 - 11: lO (5) Isoperimetric inequalities of moments of inertia of plane convex sets Professor T. W. Ting, University of Texas (589-3) 11:15 - 11:25 (6) A method of generating integral representations Mr. W. W. Turner* and Professor Alfred Leitner, Michigan State Uni­ versity (589-33) 11:30 - 11:40 (7) Generalized mathematical fundamentals of relativistic theories Professor M. Z. v. Krzywoblocki, Michigan State University (589-9)

THURSDAY, 10:00 A.M. Session on Topology, Room 206 10:00 - 10:10 (8) Improving the intersection of lines with surfaces Professor R. H. Bing, University of Wisconsin (589-13) 10:15 - 10:25 (9) On the sum of two solid Alexander Horned Spheres Mr. B. G. Casler, University of Wisconsin (589-15) (Introduced by Professor R. H. Bing) 10:30 - 10:40 (10) Uncountably many different involutions of the three sphere Mr. W. R. Alford, Tulane University (589-4)

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

66 10:45 - 10:55 (11) Factorization of compact 3- and 4-manifolds. Preliminary report Dr. C. H. Edwards, Jr., University of Wisconsin (589-26) 11:00-11:10 (12) Conditions for tameness of a 2-sphere which is locally tame modulo a tame set Mr. Norman Hosay, University of Wisconsin (589-43) (Introduced by Professor R. H. Bing) 11:15 - 11:25 (13) Joins of topological spaces Professor K. W. Kwun*, Seoul National University and University of Wisconsin and Professor Frank Raymond, University of Wisconsin (589-37) 11:30 - 11:40 (14) On the most general plane closed point set through which it is possible to pass a pseudo-arc Mr. Howard Cook, The University of Texas (589-2)

THURSDAY, 2:00P.M. Symposium on Experimental Arithmetic, First Session, Room 133 Chairman: Professor Marshall Hall, Jr., California Institute of Technology Discussion Leader: Professor Peter Henrici, University of California at Los Angeles 2:00 Unexpected dividends in the theory of prime numbers Professor J. Barkley Rosser, Cornell University 2:40 Methods of successive restrictions in computational problems involving discrete variables Professor C. B. Tompkins, University of California, Los Angeles, California. 3:20 On the use of machines to study sequence space mappings Professor Gustav A. Hedlund, Yale University 4:00 Computer investigation of orthogonal latin squares of order ten Dr. E. T. Parker, Sperry Rand Corporation, St. Paul, Minnesota

4:40 On some computing experiments in linear programming Professor R. E. Quandt and Harold W. Kuhn*, Princeton Univer­ sity, Princeton, New Jersey

FRIDAY, 9:00A.M. Session on Analysis, Room 133 9:00 - 9:10 (15) On partial sums of entire functions. Preliminary rerport Dr. T. L. McCoy, Illinois Institute of Technology (589-31) 9:15 - 9:25 (16) The extension problem for positive -definite functions Professor Walter Rudin, University of Wisconsin (589-29) 9:30 - 9:40 (1 7) On the nilpotent part of a spectral operator Professor J. E. Simpson, Marquette University (589-12) 9:45 - 9:55 (18) Sequence transformations and related continued fractions Professor Robert Heller, Jr., University of Houston (58 9- 20)

67 10:00 - 10:10 (19) On the orthogonality of measures induced by L processes Professor Marek Fisz, Columbia University (589-41) 10:15 - 10:25 (20) A note on hyponormal operators Professor S. K. Berberian, State University of Iowa (589-10) 10:30 - 10:40 (21) A geometric characterization of closable linear transformations on a Mr. L. J. Senechalle, University of Tennessee (589-17)

FRIDAY, 9:00A.M. General Session, Room 206 9:00 - 9:10 (22) Commutativities involving replacement and substitution Mr. David Schroer, University of Rochester (589-40) 9:15 - 9:25 (23) Relations irreducible to classes Professor F. G. Asenjo, Southern Illinois University (589-36) 9:30 - 9:40 (24) The genus of a graph is the sum of the genuses of its blocks Mr. Joseph Battle, Professor Frank Harary and Dr. Yukihiro Kodama*, The University of Michigan (58 9-18) 9:45 - 9:55 (25) A remark on the Nijenhuis tensor Professor E. T. Kobayashi, Northwestern University (589-34) 10:00 - 10: 10 (26) Metric characterization of Gauss surfaces. Preliminary report Professor L. M. Blumenthal and Mr. W. A. Kirk*, University of Missouri (589-25) 10:15 - 10:25 (27) On tensor connexions Mr. Arnold Seiken, University of Michigan (589-35) 10:30 - 10:40 (28) On the tangent bundle of a quotient space. Preliminary report Dr. R. H. Szczarba, Yale University (589-5)

FRIDAY, 11:00 A.M. Invited Address, Room 133 Geometric methods in differential topology Professor Andrew Wallace, Indiana University

FRIDAY, 2:00P.M. Symposium on Experimental Arithmetic Second Session, Room 133 Chairman: Professor Mark Kac, Rockefeller Institute Discussion Leader: Professor Marshall Rosenbluth, University of Califor­ nia at LaJolla 2:00 Few particle experiments in statistical mechanics Professor B. Alder, University of California, Livermore 2:40 An approach to the Ising problem using a large scale fast digital com­ puter Professor Chen-ping Yang, Ohio State University

68 3:20 Adaptive neural networks as brain models Professor H. D. , Cornell University 4:00 Numerical experiments in atmospheric hydrodynamics Professor Jule G. Charney, Massachusetts Institute of Technology

SATURDAY, 9:00A.M. Session on Topology, Room 133 9:00 - 9: 10 (29) Some relations among Stiefel-Whitney classes of manifolds. Preliminary report Professor F. P. Peterson, Massachusetts Institute of Technology (589-22) 9:15 - 9:25 (30) Stable homotopy of projective spaces Dr. Arunas Liulevicius, The Institute for Advanced Study and The Uni­ versity of Chicago (589-32) 9:30 - 9:40 (31) A certain nonmetrizable Hausdorff space Professor L. B. Treybig, Tulane University (589-23) 9:45 - 9:55 (32) A totally nonaposyndetic, compact, Hausdorff space with no cut point Professor E. E. Grace, Emory University and University of Wisconsin (589-38) 10:00 - 10:10 (33) Semigroups on certain continua ruled by arcs Professor R. J. Koch, Louisiana State University and Professor L. F. McAuley*, University of Wisconsin (589-39) 10:15- 10:25 (34) Homeomorphisms on a solid torus Professor D. R. McMillan, Jr., Florida State University (589-21) 10:30 - 10:40 (35) Extended topology: Perfect sets Professor P. C. Hammer, University of Wisconsin (589-16)

SATURDAY, 9:00A.M. Session on Algebra, Room 206 9:00 - 9:10 (3 6) Generalized powers Professor Gloria Olive, Anderson College (589-6) 9:15 - 9:25 (37) On commutators in a simple Lie algebra Mr. G. E. Brown, Cornell University (589-14) 9:30 - 9:40 (38) Certain subdirect products of simple groups. Preliminary report Mr. T. J. Head, The University of Kansas (589-27) 9:45 - 9:55 (39) A condition for a simple, power-associative algebra to be a field. Prelimin­ ary report Mr. R. L. Hemminger, Michigan State University (589-7) 10:00 - 10:10 (40) Outer automorphisms of cyclic extensions of abelian p-groups Professor Charles Godino, Notre Dame University (589-42) (Introduced by Mr. K. M. K ronstein)

69 10:15 - 10:25 (41) Betweenness relations in lattices. Preliminary report Professor L. M. Blumenthal and Mr. R. j. Bumcrot*, University of Missouri (589- 24) 10:30 - 10:40 (42) On a unifying technique of multiplying in various systems Professor Edgar Karst, Evangel College (589-8)

SATURDAY, 11:00 A.M. Invited Address. Room 133 On permutation groups of prime degree Professor Noboru Ito, University of Illinois

SATURDAY, Z:OO P.M. Symposium on Experimental Arithmetic Third Session, Room 133 Chairman: Professor A. H. Taub, University of Illinois Discussion Leader: Dr. Stanislaw M. Ulam, Los Alamos Scientific Labora­ tory, Los Alamos, New Mexico

Z:OO On jacobi rotation patterns Professor Heinz Rutishauser, Eidgenossische Technische Hoch­ schule, Zurich, 2:40 Automatic numerical integration of ordinary differential equations Dr. Arnold T. Nordsieck, General Motors Corporation, Goleta, California 3:20 Eliminating the irrelevant from mechanical proofs Professor M. Davis, Yeshiva University 4:00 The particle-in-cell method for numerical solution of problems in fluid dynamics Dr. F. H. Harlow, Los Alamos Scientific Laboratory, Los Alamos, New Mexico j. W. T. Youngs Bloomington, Indiana Associate Secretary

70 Five Hundred and Ninetieth Meeting Chalfonte-Haddon Hall Hotels Atlantic City, New Jersey April 16 -19, 1962

PROGRAM

The five hundred ninetieth meeting buted papers on the afternoon of Wednes­ of the American Mathematical Society will day, April 16 and on the morning and after­ be held in Atlantic City at the Chalfonte­ noon of April 17. There will be provision Haddon Hall Hotels on April 16-19, 1962. for a limited number of late papers. All sessions will be held in the public The registration desk will be located rooms on the Lounge Floor of Haddon Hall. on the Lounge Floor in the English Lounge. The Lounge Floor is one floor above the It will be open from 9:00A.M. to 5:00 P.M. Office Floor, which is at ground level. on Monday through Wednesday,Aprill6-18, There will be a Symposium on the and from 9:00 A.M. to 3:00 P.M. on Interactions between Mathematical Re­ Thursday, April 19. search and 'Pligh Speed Computing. It is There will be exhibits of books and sponsored jointly by the Association for computing machinery in the English Lounge Computing Machinery and the American during most of the meeting. Mathematical Society, with financial sup­ There will be a meeting of the port from the U. S. Army Research Cen­ Council on Wednesday, April 18, at 5:00 ter, Durham, and the National Science P.M. in the Card Room, lasting into the Foundation. The choice of subject lay with evening with an intermission for dinner. the Committee on Applied Mathematics, There are rooms at various prices consisting then of D. M. Young, Chairman, in Haddon Hall and in the Chalfonte. There V. Bargmann, G.E.Forsythe,P.R.Gara­ is a rate schedule incorporated in the re­ bedian, R. C. Prim and j. j. Stoker. The servation blank on the inside of the back Invitations and Organizing Committee, re­ cover. Reservations will be acknowledged sponsible inter alia for the selection of by the hotels. There are a small number speakers, consists of john Todd, Chair­ of rooms for students in the Chalfonte, man, G. E. Forsythe, P. D. Lax, D. H. without attached bath but with running Lehmer, H. H. Goldstine, C. B. Tompkins water in the room, which are available at and D. M. Young. There will be five ses­ a suitably reduced rate and which will be sions, scheduled in the Vernon Room on allotted on a first-come-first-served ba­ the morning and afternoon of Monday, sis. April 16, and Tuesday, April 17, and on The YMCA hotel and the YWCA the morning of Wednesday, April 18. The hotel are each within easy walking distance full program is given below. of the sessions. By invitation of the Committee to One or more of the dining rooms of Select Hour Speakers for Eastern Section­ the hotels will be available for the service al Meetings, there will be two invited ad­ of meals. dresses. On Wednesday, April 18, at 2:00 Atlantic City can be reached by P.M. in the Vernon Room, Professor public transportation on bus, train or joseph B. Keller of New York University plane. From Philadelphia, both Public Ser­ will speak on "Some problems in wave vice and Trailways supply bus service to propagation." On Thursday, April 19, at Atlantic City. Information is available at 2:00 P.M. in the Vernon Room, Professor the Union Bus Terminal in Philadelphia, David B. Lowdenslager of Princeton Uni­ LOcust 7-4300. From New York, both versity will address the Society on "The Public Service and Lincoln Transit supply theorems of F. and M. Riesz on functions service, with information for Public Ser­ analytic in the unit circle." vice in New York at LOngacre 5-7040 and There will be sessions for contri- for Lincoln Transit at BRyant 9-1000. All

71 these busses go to the Public Service Ter­ Company. The same company operates minal in Atlantic City, which is within door-to-door limousine service between walking distance of Haddon Hall. Philadelphia and Atlantic City. The tele­ Atlantic City is served by trains of phone in Philadelphia is SA 6-9955. the Pennsylvania-Reading Seashore Line Schedules are expanded to meet originating at the 30th Street Station in seasonal requirements, particularly for Philadelphia, with additional service orig­ the bus service. Prospective travelers inating in Camden. are urged to check schedules at a time There is plane service to Atlantic close to the date of the meeting to take City from the Philadelphia International advantage of the expanded schedules. Airport on the Allegheny Airlines and from A booklet on transportation to At­ the Newark Airport on Eastern Airlines. lantic City may be obtained by writing to Arrivals by plane at Philadelphia the office of the American Mathematical International Airport will find connecting Society, 190 Hope Street, Providence 6, limousine service to the door in Atlantic Rhode Island. City, operated by the Salem Transportation

PROGRAM OF THE SYMPOSIUM INTERACTIONS BETWEEN MATHEMATICAL RESEARCH AND HIGH SPEED COMPUTING

MONDAY, 10:00 A.M. First Session, Vernon Room Chairman: Professor john Todd, California Institute of Technology Discussion Leader: Professor j. Barkley Rosser, Cornell University 10:00 Symposium Convenes 10:15 Purposeful and unpurposeful computing Professor Harvey Cohn, University of Arizona 11:15 How programming difficulties can lead to theoretical advances Dr. E. C. Dade, California Institute of Technology and Professor Hans Zassenhaus, University of Notre Dame 11:45 Large and nonconvex problems in linear programming Dr. R. E. Gomory, IBM Research Center, Yorktown Heights, New York

MONDAY, 2:00 P.M. Second Session, Vernon Room Chairman: Professor Garrett Birkhoff, Discussion Leader: Professor C. B. Tompkins, University of California, Los Angeles, California

2:00 Some high s,peed logic Professor D. H. Lehmer, University of California, Berkeley California 2:30 The free oscillations of the earth and its atmosphere Professor G. F. MacDonald, University of California, Los Angeles, California 3:30 Mathematical problems in theoretical chemistry Professor j. 0. Hirschfelder, University of Wisconsin, Madison, Wisconsin

72 TUESDAY, 9:30A.M. Third Session, Vernon Room Chairman: Dr. A. S. Householder, Oak Ridge National Laboratory Discussion Leader: Professor D. M. Young, Jr., University of Texas, Austin, Texas 9:30 New aspects in numerical quadrature Professor F. L. Bauer, University, Mainz, Germany, Professor E. L. Stiefel, ETH, Zurich, Switzerland and Professor H. Rutis­ hauser*, ETH, Zurich, Switzerland 10:30 Error analysis in matrix computations Dr. J. H. Wilkinson, National Physical Laboratory, Teddington, Middlesex, England 11:30 Stability questions for some numerical methods for ordinary differen­ tial equations Professor G. G. Dahlquist, Royal Institute of Technology, Stock­ holm, Sweden and University of California, Los Angeles

TUESDAY, 2:00 P.M. Fourth Session, Vernon Room Chairman: Professor R. Courant, New York University Discussion Leader: Dr. H. H. Goldstine, IBM Research Center, Yorktown Heights, New York 2:00 Information theory and decoding computations Professor Peter Elias, Massachusetts Institute of Technology 3:00 Some applications of the quotient-difference algorithm Professor Peter Henrici, University of California, Los Angeles, California 3:30 Some problems in the stability of numerical procedures Professor Peter D. Lax, New York University

WEDNESDAY, 9:30A.M. Fifth Session, Vernon Room Chairman: Professor J. H. Curtiss, University of Miami Discussion Leader: Professor G. E. Forsythe, Stanford University 9:30 Methods for proving programming algorithms Professor A. J. Perlis, Carnegie Institute of Technology 10:00 Computers as an aid to theorem-proving Professor Hao Wang, Harvard University 10:30 Towards more versatile mechanical translators Mr. E. T. Irons, Institute for Defense Analyses, Princeton, New Jersey 11:00 The automation of science Dr. R. W. Hamming, Bell Telephone Laboratories, Murray Hill, New Jersey

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

73 PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. The contributed pa­ pers are scheduled at fifteen minute inter­ vals so that listeners can circulate between the sessions. To maintain the established schedule, time limits will be strictly en­ forced.

WEDNESDAY, 2:00 P.M. Invited Address, Vernon Room Some problems in wave propagation (One hour) Professor Joseph B. Keller, New York University

WEDNESDAY, 3:15P.M. Session on Algebra, Garden Room 3:15 - 3:25 (1) The L1 (G)(\ L2 (G). Preliminary report Mr. C. R. Warner, University of Rochester (590-31) 3:30 - 3:40 (2) Automorphisms of antisemisimple algebras Professor Murray Gerstenhaber, University of Pennsylvania and Insti tute for Defense Analyses (590-27) 3:45 - 3:55 (3) Decomposability in Abelian groups. Preliminary report Professor S. A. Khabbaz, Lehigh University (590-1) 4:00 - 4:10 (4) Solutions of equations over groups Professor Frank Levin, Rutgers, The State University (590-34) 4:15 - 4:25 (5) On the space of lattices in a Lie group Professor Hsien-Chung Wang, Northwestern University and Institute for Advanced Study (590-8) 4:30 - 4:40 (6) Error-correcting codes: An axiomatic approach Dr. E. F. Assmus, Jr., Columbia University and Sylvania Applied Re­ search Laboratories and Dr. H. F. Mattson*, Sylvania Applied Research Laboratories, Waltham, Massachusetts (590-28) 4:45 - 4:55 (7) On the number of positive entries in the powers of a non-negative matrix. Preliminary report Mr. N.J. Pullman, Syracuse University (590-23)

WEDNESDAY, 3:15P.M. Session on Applied Mathematics, Vernon Room 3:15 - 3:25 (8) Cauchy's method of minimization Dr. A. A. Goldstein, Massachusetts Institute of Technology (590-10) 3:30 - 3:40 (9) Application of the stroboscopic method to a nonlinear equation of nonautono­ mous character Professor Abolghassem Ghaffari, University of Teheran, Iran and Na­ tional Bureau of Standards, Washington, D. C. (590-17)

74 3:45 - 3:55 (10) Bounds on the truncation error in the solution of the characteristic boundary value problem for hyperbolic equations Professor A. K. Aziz*, Georgetown University and Professor B. E. Hubbard, University of Maryland (590- 21) 4:00 - 4:10 (11) A heat flow problem with a multiplicity of steady-state solutions Mr. M. S. Klamkin, AVCO Research and Advanced Development Divi­ sion, Wilmington, Massachusetts (590-11) 4:15- 4:25 (12) Some a priori inequalities with application in the Neumann problem for uni­ formly elliptic operators Professor L. E. Payne* and Professor j. H. Bramble, University of Maryland (590-32) 4:30 - 4:40 (13) Eberlein-measure and mechanical quadrature formulae. Preliminary report Mr. V. L. N. Sarma, University of Rochester (590-20) 4:45 - 4:55 (14) Optimal m-invariant iteration functions Dr. j. F. Traub, Bell Telephone Laboratories, Murray Hill, New jer­ sey (590-14)

THURSDAY, 10:00 A.M. Session on Applied Mathematics, Garden Room 10:00 - 10:10 (15) On the formulation of finite difference analogues of the Dirichlet problem for poisson's equation Professor j. H. Bramble and Professor B. E. Hubbard*, University of Maryland (590-2) 10:15 - 10:25 (16) Error bounds in the finite difference solution of the Dirichlet problem for Poisson's equation Professor j. H. Bramble* and Professor B. E. Hubbard, University of Maryland (590-3) 10:30 - 10:40 (17) The normal bivariate density function and its applications to weapons systems analysis; a review Mr. H. C. Sebring, General Electric Company, Philadelphia, Pennsyl­ vania (590-39) 10:45 - 10:55 (18) Error-distributing codes and rock domains Dr. S. W. Golomb, California Institute of Technology (590-4)

THURSDAY, 10:00 A.M. Session on Analysis, Vernon Room 10:00 - 10: 10 (19) Riccati' s equation with loosely inter-related coefficients Mr. Iwao Sugai, ITT Federal Laboratories, Nutley, New jersey (590-13) (Introduced by Dr. R. B. Kelman) 10:15 - 10:25 (20) Short proof of a theorem of Wintner on the asymptotic behavior of the adiabatic linear oscillator Dr. R. B. Kelman, Remington Rand Univac, Washington, D. C. and Howard University (590-19)

75 10:30 - 10:40 (21) Asymptotic behavior of the period of solutions of certain plane autonomous systems near centers Professor W. S. Loud, University of Minnesota (590-29) 10:45 - 10:55 (22) On the commutation of finite integral operators, with difference kernels, and linear self-adjoint differential operators Dr. J, A, Morrison, Bell Telephone Laboratories, Murray Hill, New Jersey (590-5) 11:00- 11:10 (23) Mixed problems in several variables Professor Reuben Hersh, Fairleigh Dickinson University (590-12) (Introduced by Professor Peter Lax) 11:15- 11:25 (24) Strict approximations Dr. j. R. Rice, General Motors Corporation, Warren, Michigan (590-7) 11:30 - 11:40 (25) Time-domain analysis of a continuous parameter, weakly stationary stochastic process Mr. j. B. Robertson* and Professor Pesi Masani, Indiana University (590-24) 11:45 - 11:55 (26) Isometric flows on Hilbert space Professor P. R. Mas ani, Indiana University (590-25)

THURSDAY, 2:00P.M. Invited Address, Vernon Room The theorems of M. and F. Riesz on functions analytic in the unit circle (One hour) Professor David B. Lowdenslager,

THURSDAY, 3:15P.M. Session on Logic and Foundations, Garden Room 3:15 - 3:25 (27) Behavior of syntactic structure under replacement and substitution Mr. David Schroer, University of Rochester (590-36) 3:30 - 3:40 (28) A reduction to a class of AEA formulas containing one dyadic predicate Professor A. S. Kahr, Massachusetts Institute of Technology (590-37) 3:45 - 3:55 (29) Degrees of finitely axiomatizable theories. Preliminary report Mr. W. P. Hanf, IBM Corporation (590-30) (Introduced by Professor R. L. Vaught) 4:00 - 4: 10 (30) A remark on the reduction problem with an application to the AEA formulas Mr. A. S. Kahr and Professor Hao Wang*, Harvard University (590-38) 4:15- 4:25 (31) Some basic properties of provable recursive functions Mr. P. C. Fischer, Massachusetts Institute of Technology (590-33) 4:30 - 4:40 (32) Recursive enumerability and the jump operator. Preliminary report Dr. G. E. Sacks, Institute for Advanced Study (590-22)

76 THURSDAY, 3:15P.M. Session on Analysis and Topology, Vernon Room 3:15 - 3:25 (33} Equivariant imbeddings of compact abelian Lie groups of transformations Professor j. M. Kister, University of Michigan and Professor L. N. Mann*, University of Virginia (590-9) 3:30 - 3:40 (34) Proteus forms of wild and tame arcs Professor E. E. Posey, Virginia Polytechnic Institute (590-18} 3:45 - 3:55 (35) Extended topology: cardinal spectra and continuity Professor P. C. Hammer, University of Wisconsin (590-26) 4:00 - 4:10 (36} Functions with maxima at each point of a dense set in a linear interval Dr. W. S. Snyder, ORNL, Oak Ridge, Tennessee, Professor E. E. Posey and Professor L. B. Rall*, Virginia Polytechnic Institute (590-16} 4:15 - 4:25 (37) Net motions and quasigroups Professor Rafael Artzy, Rutgers, The State University (590-6) 4:30 - 4:40 (38) Invariant subspaces of linear transformations in Hilbert space Dr. Louis de Branges, New York University (590-15} 4:45 - 4:55 (39) On an inequality involving group characters. Preliminary report Professor J, J, Price, Cornell University (590-35) Everett Pitcher Bethlehem, Pennsylvania Associate Secretary

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77 Five Hundred and Ninety-First Meeting U.S. Naval Postgraduate School Monterey, California April 28, 1962

PROGRAM

The five hundred ninety-first meet­ modern building located near the Sloat ing of the American Mathematical Society Avenue entrance of the U. S, Naval Post­ will be held on Saturday, April Z8, 196Z graduate School. Information concerning at the U. S. Naval Postgraduate School, late papers can be obtained at the Regis­ in Monterey, California. tration Desk. By invitation of the Committee to Luncheon will be served at the Select Hour Speakers for Far Western Officers Club. The Faculty Lounge on the Sectional Meetings, there will be an ad­ second floor, Room ZOO, of Spanagel Hall dress at Z:OO P.M. in King Hall by Pro­ will be open throughout the day, and coffee fessor R. A. Beaumont of the University will be available there. of Washington on "Torsion free rings.• " Monterey is served by United Air­ Sessions for contributed papers lines, Pacific Airlines, Southern Pacific will be held at 10:00 A.M.andat3:30 P.M. Railroad, and Greyhound Bus. There are in Rooms 400 and 4Zl, Spanagel Hall. Ab­ two entrances to the U. S. Naval Post­ stracts of the papers to be presented at graduate School: the Sloat Avenue entrance these sessions appear on pages 131-138 near the intersection of Fremont Avenue of these NOTICES. There are cross refer­ (California Route 1) and Sloat Avenue, and ences to the abstracts in the program. For the Del Monte Avenue entrance off Del example, the title of paper ( 1) in the pro­ Monte Avenue. There will be signs to the gram is followed by ( 591-Z4) indicating parking areas and to the Lobby of Spanagel that the abstract can be found under desig­ Hall from each entrance. nation 591-Z4 among the published ab­ There are numerous motels and stracts. hotels within a few miles of the School. The Registration Desk at this meet­ The following hotels will have reserva­ ing will be located in the first floor Lobby tions available for members attending the of Spanagel Hall, which is the five story meeting: Single Double Twin Beds California Motel, Z04Z Fremont Street $8.00 $8.00 $10,00 Casa Munras, 700 Munras Avenue $8.00-$10.00 $1Z.OO- $14.00 $1Z.OO- $14.00 El Castell Motel, Z10Z Fremont Street $6.00 $ 7 .oo- $ 9 .oo $ 8.oo- $1 o.oo La Fonda Motel, 755 Abrego Street $10.00 $1Z.OO- $14.00 $14.00 Mark Thomas Inn, 1300 Fremont Street $8.00- $26.00 $10,00- $30,00 $1Z.OO- $30.00 Hotel San Carlos Franklin and Calle Principal $6.00-$8.00 $ 8,00-$10.00 $10,00-$14.00 Reservations should be made directly with the desired hotel at the indicated Street address in Monterey.

78 PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. The contributed pa­ pers are scheduled at 15 minute intervals. To maintain the schedule, the time limit will be strictly enforced.

SATURDAY, 10:00 A.M. General Session, Room 400, Spanagel Hall 10:00 - 10:10 (1) Constructive pseudo-well-orderings Professor Solomon F eferman, Stanford University (591-24) 10:15- 10:25 (2) On generalized quantifiers Mr. Gebhard Fuhrken, University of California, Berkeley (591-4) 10:30 - 10:40 {3) Some theorems on the axiom of choice Professor A. H. Kruse, New Mexico State University (591-13) 10:45 - 10:55 (4) Trivial extensions of topological spaces Professor E. A. Michael, University of Washington (591-23) 11:00-11:10 (5) The Euler characteristic in combinatorial geometry Professor V. L. Klee, University of Washington (591-2) 11:15- 11:25 (6) Pseudo-conformal vector fields Professor T. K. Pan, University of Oklahoma (591-18) 11:30- 11:40 (7) A new series of line involutions Professor C. R. Wylie, Jr., University of Utah (591-12)

SATURDAY, 10:00 A.M. Session on Algebra, Room 421, Spanagel Hall 10:00 - 10: 10 {8) Embedding of a commutative semigroup into certain divisible semigroups Professor Takayuki Tamura and Mr. D. G. Burnell*, University of Cali­ fornia, Davis (591-9) 10:15- 10:25 (9) Semigroups, all subsemigroups of which are ideals Professor Takayuki Tamura and Mr. R. B. Merkel*, University of Cali­ fornia, Davis (591-10) 10:30 - 10:40 (10) p-hasic subgroups of abelian groups Professor D. L. Boyer, University of Idaho (591-16) 10:45 - 10:55 (11) On high extensions of abelian groups. Preliminary report Professor J, M. Irwin*, Miss Carol Peercy, and Professor E. A. Walker, New Mexico State University (591-14) 11:00 - 11:10 (12) On pa.-pure sequences of groups Professor J, M. Irwin, Miss Carol Peercy, and Professor E. A. Walker*, New Mexico State University (591-21)

79 11:15- 11:25 (13) Purity and subfunctors of the identity Professor R. J. Nunke, University of Washington (591-15) 11:30 - 11:40 (14) On the coverings of Lie algebras of classical type Professor R. E. Block, California Institute of Technology (591-17)

SATURDAY, 2:00P.M. Invited Address, King Hall Torsion free rings Professor R. A. Beaumont, University of Washington

SATURDAY, 3:30P.M. Session on Algebra and Number Theory, Room 400, Spanagel Hall 3:30 - 3:40 (15) A set of matrices for testing computer programs Dr. J. L. Brenner, Stanford Research Institute, Menlo Park, California (591-1) 3:45- 3:55 (16) Some theorems on periodic decimal fractions Mrs. D. H. Kauffman* and Dr. E. G. McNiel, Lockheed Missiles and Space Corporation, Sunnyvale, California (591-5) (Introduced by Professor C. D. Olds) 4:00 - 4: lO ( 17) Conjugate algebraic integers in real point sets Professor R. M. Robinson, University of California, Berkeley (591-6) 4:15- 4:25 (18) Algebraic integers as sums of polynomial values Dr. Otto Koerner, University of Utah (591-11) (Introduced by Professor Peyerimhoff) 4:30 - 4:40 ( 19) On a cubic congruence in three variables Professor L. J. Mordell, University of Arizona (591-3) 4:45 - 4:55 (20) A Diophantine problem associated with linear recurring series. Preliminary report Professor Morgan Ward, California Institute of Technology (591-20)

SATURDAY, 3:30P.M. Session on Analysis, Room 421, Spanagel Hall 3:30 - 3:40 (21) On second order nonlinear differential equations Mr. John Jones, Jr., United States Air Force, Springfield, Virginia (591-25) 3:45 - 3:55 (22) A note on an imbedding space representation. Preliminary report Dr. D. E. Myers, University of Arizona (591-7) 4:00 - 4:10 (23) The range as range space for compact operators Professor E. 0. Thorp* and Professor Seymour Goldberg, New Mexico State University (591- 8) 4:15- 4:25 (24) Simultaneous automorphisms and the Landau properties Professor M. G. Arsove*, University of Washington and Professor A. Alexiewicz, Poznan, Poland (591-22)

80 4:30 - 4:40 (25) On an absolute mean value theorem for Ces1l.ro methods Professor Alexander Peyerimhoff, University of Utah (591-19)

R. S. Pierce Berkeley, California Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

THE MATHEMATISCHES FORSCH­ THEORY OF ALGORITHMS, by UNGSINSTITUT OBERWOLFACH RE­ A. A. Markov (444 pages; price $4.50), has ports two meetings held in October, 1961. been made available in English translation At the meeting on Ring theory, October 23- through the Office of Technical Services, 28, the following lectures were given: Business and Defense Services Admini­ W. Nobauer (Vienna), Funktionen aufkom­ stration, U. S. Department of Commerce. mutativen Ringen; C. j. Penning (Delft), Published by the USSR Academy of Sci­ Duplikatorringe und Assoziierte Ringe; ences in 1954, Markov's study was trans­ J. Guerindon (Rennes), Sur une clas se de lated for the Department of Commerce modules gradues; D. G. Higman (Ann Ar­ through a cooperative Federal agency bor), Homological conductors; H. Kupisch translations program coordinated by the (Heidelberg), Quasi-Frobenius-Algebren; NSF. The translating of foreign scientific A. Almeida Costa (Lisbon), Sur la theorie and technical material under this program generale des demianneaux; E. A. Behrens is financed through the sale of surplus (Frankfurt au main),Uberden Idealverband U. S. agricultural commodities abroad, eines Ringes; W. Krull (Bonn), Ordnungs­ under the provisions of Public Law 83-480. funktionen und Bewertungen; K. E. Aubert Theory of Algorithms (60-51085) may be (Oslo), A general ideal theory and its ap­ ordered from OTS, U. S. Department of plications; A Kertecz (Debrecen), Commerce, Washington 25, D. C. Artinsche Ringe; and A. W. Goldie (New­ The existence of unsolvable mass castle upon Tyne), Non-commutative prin­ problems, according to Markov, means cipal ideal rings. that "even in such comparatively narrow The Symposium on Group Theory, areas of mathematics as the theory of October 16-21, was held under the chair­ associative calculi and the theory of in­ manship of Professors R. Baer and H. teger matrices, the transfer of the intel­ Wielandt. The participating mathemati­ lectual faculties to a machine is impossible cians from the United States were M. . ... the process of obtaining new knowledge Suzuki, Urbana, illinois; D. G. Higman and in these areas cannot be automatized com­ D. R. Hughes, Ann Arbor, Michigan; j. S. pletely". Frame, East Lansing, Michigan; and W. Holland, Tulane University. Some of the topics were: On a class of double transi­ tive groups, CN-groups and related ques­ tions, Abstract group-theoretical proper­ ties of finite groups, The constructive A CONFERENCE TO REVIEW THE reduction of finite group representations, EXPERIENCE OF SCHOOLS AND COL­ Collineation groups of finite projective LEGES with the Advanced Placement spaces, A class of projective planes, Com­ Program in Mathematics of the College binations and permutation groups, Groups Entrance Examination Board (475 River­ with automorphisms of special kinds, Sub­ side Drive, New York 27, New York) will group theorem for free groups of exponent be held june 21-23, 1962, at Princeton. 3, T-Systems of some finite groups, An For information and registration write to embedding theorem for lattice-ordered A. W. Tucker, 128 Pyne Administration groups, and Semi-direct partition oftopo­ Building, Princeton University, Princeton, logical groups. New jersey.

81 PRELIMINARY ANNOUNCEMENT OF MEETING

SIXTY-SEVENTH SUMMER MEETING University of British Columbia Vancouver, B. C. August 27 31, 1962

The sixty-seventh summer meeting dences containing mostly single rooms. of the American Mathematical Society will The cost of permanent residence housing be held at the University of British Colum­ will be $4 a night per per son. A small bia in Vancouver, B. C.onAugust27to 31, number of double rooms with separate 196Z. To help members who expect to at­ beds are available at $3 a night per person. tend this meeting make their plans early, Towels and bedding will be provided. Auto­ the Committee on Arrangements has col­ matic laundry facilities are available. lected the following information on Van­ Rooms in residence may be occupied from couver's accommodation and travelfacili­ Z:OO P.M. Saturday, August Z5 to Z:OO P.M. ties. Additional information concerning the Saturday, September 1. program and other aspects of the meeting Reservations for University resi­ will be published in the Preliminary An­ dences and dormitories should be made by nouncement section of the June NOTICES. writing directly to the Conference Office, University of British Columbia, Vancouver ROOMS AND MEALS 8, Canada at the earliest possible date and Accommodations will be available before August 9. in the University residences to all attend­ The nearest hotels or motels are ing the meetings and to adult members of located in downtown Vancouver (about 6 their families. Only a few units are avail­ miles from the campus). The nearest pub­ able on campus which are suitable for lic campsites are located at Alouette Lake, families with children under twelve. Chil­ about 40 miles from the campus. dren twelve and over can be accommodated Persons desiring hotel and motel in the regular residences at the regular accommodations should make their reser­ rates. Low cost dormitory space is avail­ vations directly with the appropriate man­ able in converted huts in Fort Camp and ager, and under no circumstances write to Acadia Camp. The cost of dormitory hous­ the Conference Office which is in charge ing will be $Z a night per person in a single only of the University facilities. It is ad­ room. In a double room with separate beds visable to make reservations early because the cost is $1.50 a night per person. The the Seattle World's Fair is expected to University also has a number of new, attract an unusually large number of tour­ modern, three-storey permanent resi- ists to the Pacific Northwest.

Suggested Hotels and Motels Unit or Minimum Rates Name Address Phone .Rooms Single Double Burrard Motel 1100 Burrard MU 1-Z331 60 $ 8.00 $11.00 Devonshire Hotel 894 W.. Georgia MU 1-5481 150 7.50 10.50 Doric Howe Motor Hotel 1061 Howe St. MU Z-3171 103 8.00 10.00 Georgia Hotel 801 W. Georgia MU 3-118Z 314 9.00 1Z.50 Georgian Towers Motor Hotel 1450 W. Georgia MU l-43Z1 105 7.00 13.00 Kam o Apt. Hotel 11ZO Denman St. MU 4-7474 47 7.00 10.00 Travlodge Motel 1304 Howe St. MU Z-Z767 74 7.00 9.00 Bayshore Inn 1601 Georgia St. MU z.-3377 308 1Z.OO 15.00 Sands Motor Hotel 1755 Davie St. MU Z-1831 100 8.50 9.50 Sylvia Hotel 1154 Gilford St. MU 1-93Z1 1Z5 5.00 8.00 Vancouver Hotel 900 W. Georgia MU 4-3131 561 9.00 11.50

82 TRAVEL Pacific and Orient Lines, but their sailing and arrival dates make them completely Vancouver is approximately 140 unsuitable for persons attending the sum­ miles north of Seattle, Washington on High­ mer meetings. There are other lines that way 99. The best route to the University operate freighters with excellent accom­ from the south is on Highway 99-B via modations for a small number of passen­ De as Island tunnel (toll), Oak Street bridge gers but they do not keep very rigid sche­ (toll), West 41st Avenue and Marine Drive. dules. The trip from Los Angeles to Van­ Vancouver is served by Canadian couver takes between 5 and 8 days and the Pacific, Qantas, Trans Canada and United fares vary from about $130 to $180 on the Airlines; the Canadian National, Canadian Holland American Line. Less expensive Pacific and Great Northern Railroads; and accommodations are available on the Am­ the Greyhound Busline. Automobile ferry erican Mail and the Canadian Blue Star service is available from Vancouver Island Lines. Reservations should be made in on British Columbia Government ferries advance. from Nanaimo and Swartz Bay and by Persons who are planning to attend Canadian Pacific from Nanaimo. both the Vancouver meeting and the Inter­ Steamship Lines. The only regularly national Congress of Mathematicians in scheduled ships between the California Stockholm may find the following flight ports and Vancouver are operated by the schedules helpful.

Saturday flight

Lv. Stockholm 9:50 a.m. (Sat.) SAS 565 Ar. Amsterdam 11:05 a.m.

Lv. Amsterdam 1:25 p.m. (Sat. or Sun.) CPA Flight 301 Ar. Vancouver 4:35p.m.

Sunday Flight

Lv. Stockholm 6:05 p.m. (Sun.) SN Flight 174 Ar. Amsterdam 7:20p.m. Lv. Amsterdam 8:30 p.m. (Sun. only) CPA Flight 363 Ar. Vancouver 11:40 p.m.

Saturday or Sunday flight

Lv. Stockholm 8:30 a.m. (daily) SAS Flight 551 Ar. Copenhagen 9:45a.m.

Lv. Copehnhagen 10:30 a.m. SAS Flight 551 Ar. Amsterdam 11:55 a.m.

Lv. Amsterdam 1:25 p.m. Sat. Sun. CPA Flight, 301 (Flies Sat. and Sun. only) Ar. Vancouver 4:35p.m.

83 Thursday, Friday, Saturday

Lv. Stockholm 9:45 p.m. Thurs. Fri. Sat. SASFlight935(via Oslo, Copenhagen and Green­ land) Ar. Los Angeles 5:ZO a.m. Fri. Sat. Sun. Lv. Los Angeles 7:30 a.m. There would be an added Ar. Vancouver lZ:OO noon-Fri. Sat. Sun. cost of about $10 on this flight due to surcharge on jet travel in the U. S. With a Vancouver-Stockholm ticket stopovers are permittedinEuropean cities anywhere west and south of Stockholm at no added cost.

IT IS ADVISABLE TO MAKE FLIGHT RESERVATIONS AS EARLY AS POSSIBLE. R. S. Pierce Berkeley, California Associate Secretary **********************

84 A REPORT TO MEMBERS

The following report was given by Office--than to increase in the number john W. Green, Secretary of the Society, listed. At the time the list was presented at the Business Meeting of the 68th Annual the membership of the Society was 7, 781. Meeting of the Society, in Cincinnati, on The new procedures have had the good re­ january 24, 1962. AreportontheBusiness sult of making the list available a month Meeting itself will, as usual, appear in the or more earlier than usual. May, 1962 issue of the BULLETIN. Following are notes on some Society Activities which may not have come to Ladies and Gentlemen: your attention. I should like to make a brief report The Society has agreed to take over on some of the affairs of the Society, to from the National Research Council the comment on recent Council actions, and to publication of Mathematics of Computation point out some of the problems facing us. (formerly Mathematical Tables and Aids to First a few factual matters. The Computation). The agreement is for an Annual election of officers was concluded initial three year period and is subject to in November with the following results: review afterwards. W. Feller and A. Gleason were elected The Society has also agreed to take Vice Presidents; L. Henkin, A. House­ over, starting in 1963, the role of theRe­ holder, P. Lax, R. C. Lyndon and L. Mar­ gional Development Committee of theN RC kus were elected members-at-large of the in the organization of the Summer Insti­ council; all uncontested candidates were tutes for Graduate Students, which is spon­ elected. sored by the NSF. This work is to be Early each year, the Council elects handled by a Committee under the chair­ by mail ballot two members to its Execu­ manship of Professor R. H. Bing. tive Committee for two year terms. I re­ The 1962 Summer Institute will be mind you that the Executive Committee held at the University of California, Santa consists of the President, President-elect Barbara. The subject will be Differential (or Ex-President), Secretary, and four Geometry and Relativity. elected members. In recent years, the We have received an acceptance Executive Committee has not been very from Professor l. N. Vekua, the Rector of active, mainly because of the fact that it the University of Novosibirsk and one of can only exercise such powers as have the world's leading experts on elliptic been expressly delegated toitbytheCoun­ partial differential equations, to come to cil, but no mechanism has been devised to America under the Society's Visiting Lee­ delegate powers to it. However, under a turers program. He expects to come in proposal adopted last evening by the Coun­ March and/or April, 1962. cil, which will be described later in this You may have noted on the cover of report, it is anticipated that the Executive the Proceedings that the number of Editors Committee will play a more important role was recently raised to six by the addition in Society affairs. This year M. M. Day of Professor George Whaples and Pro­ and G. A. Hedlund were elected to the fessor Fritz john. This increase was made Executive Committee. R. Bott and R. C. necessary by the large number of papers Buck are the other elected members hold­ being handled by the Proceedings. ing over from last year. Some of you may not have seen in Although this is the year the Com­ the book exhibit the 20-year Author index bined Membership list is distributed to of Mathematical Reviews. These two im­ Association members, I am surethatmost pressive volumes (eleven pounds) index of you have seen this impressive volume. the reviews for the period 1940-19 59 and The increase in size is due more to the are available at $35.00, minus member's method of composition and printing--photo­ discount of 25o/o. offset from copy made in the Providence Those who have been eagerly await-

85 ing the appearance of the new Russian­ Cincinnati, the Council directed that a English Dictionary will be pleased to hear maximum of 200 such papers be accepted that it appeared in December. Also, a new (and anti-climactically actually only 129 version of the manual of authors will ap­ were submitted). Although this is probably pear before long. Prepared by a committee only a measure for temporary relief, it is under the chairmanship of Norman Steen­ planned for the present to continue some rod, it will be bound with a forthcoming modest limitation on the number of 10- issue of the Bulletin, and reprints will be minute papers and to use the time thus distributed to new members. freed to introduce other features into the Now to the major problems faced program. by the Society, and proposed solutions to It is anticipated that our programs some of them. I should like to mention will include major invited addresses, as three such problems. The first of these, is traditional. We also plan to hold ses­ and in some sense the greatest, is the sions for short (probably 20 minute) in­ problem of what to do about the enormous vited presentations of recent research. growth of mathematical activities in recent These will be in several fields at each years and how to keep the spirit of re­ meeting, and each will be under the direc­ search and scholarship in mathematics, to tion of a distinguished specialist in the which the Society is dedicated, from being field. suffocated by the great volume of peri­ A second problem is how best to pheral mathematical activities. One often direct and administer the complex activi­ hears the remark that nowadays Mathe­ ties of the Society. In the past the Council matics is Big Business. This is partially has maintained a rather close direction of true, but much of the "big business" in­ the administration which has been by the volves not so much scholarship and re­ Secretary, Treasurer, and President, with search, but other things such as applica­ the able assistance of the Executive Direc­ tion of mathematics, computing, curricu­ tor. At present, however, a very intimate lum and textbook revision and other edu­ direction by the Council seems too clumsy cational activities, providing career infor­ to be practical. The Council meets three mation, public relations, etc. It is impor­ times a year, at meetings of the Society; tant that we allow these other things to the time is short, the agenda long, the distract us as little as possible from the atmosphere hurried; the Council member­ central aims of the Society. These distrac­ ship present varies from meeting to meet­ tions have become particularly obvious in ing (33 at Washington, 9 at Stillwater). connection with the Annual Meeting, the Because of these difficulties, the Council size and bustling activity of which have has just approved a plan by which the noticeably impaired its scholarly atmos­ Executive Committee will play a larger phere. It is difficult to know what to do role in the direction of Society affairs. It about this, but one step would be to ar­ is anticipated that the Executive Commit­ range a less crowded program--one that tee will meet formally a number of times could be attended and enjoyed in somewhat a year, transact much of the routine coun­ less frantic haste than recent ones. To ac­ cil business, and prepare agenda andre­ complish this it appears necessary to do commendations on important matters for something about the large number of con­ Council meetings. It is hoped that in this tributed ( 10 minute) papers now appearing way the Council can devote its time and on the program of the Annual Meeting. You attention to those important matters of may recall that in a recent Chicago meet­ policy and principle which sometimes have ing, we were running four simultaneous been slighted because of the press of im­ sessions of 10-minute papers late into the mediate business. night. Admittedly these are an extremely A third and very pressing problem valuable activity of the Society and should is the question of the continued financing by no means be dropped, but it is natural of Mathematical Reviews. At present the to think of transferring them in large part deficit of Mathematical Reviews is running to the regional meetings. A step in this about a quarter of a million dollars a year. direction was made last April when, con­ We have been very fortunate in obtaining sidering possible crowded conditions at grants from the NSF in recent years to

86 cover much of the deficit, and before that drop. Since then, however, as you undoubt­ from AFOSR; it is however extremely edly know, many people have expressed problematical how long we can hope to interest in continuing the letters, and it receive this support. There is not only the was requested that the matter be brought general problem of keeping the journal up at the present business meeting. The going, but the specific one of maintaining Council has no wish to make an issue of a subscription price low enough to permit this matter and in view of the large and individual subscriptions. I am sure many genuine interest expressed, it voted to of you noted the increase in subscription rescind its previous directive to do away prices this year--something lregretvery with letters-to-the-editor, and to restore much but which seemed unavoidable. this department to the Notices. It was the Various possible remedies have instruction of the Council, however, that been suggested. One is to pare down the letters should be subject to editorial coverage of Mathematical Reviews in selection and supervision, as is custom­ breadth and depth and put out a product of ary for articles published in Society more modest size. Some relief could prob­ journals. ably be realized in this way, but I should like to indicate that almost no program of this sort would really solve the problem. According to our present best projection, the number of articles we will have tore­ view in 1965 or 1966 without increasing the breadth or depth of our coverage, will be approximately three times the number in 1960. Therefore even ifwethinnedthem out to one-third their number, we would still have left for 1965 a Mathematical Reviews of the 1960 size, complete with large deficit. Obviously something must and will be done; the matter is presently under close study by the Mathematical PROCEEDINGS OF SYMPOSIA Reviews Editorial Committee and a Special IN PURE MATHEMATICS Committee under the chairmanship of VOLUME IV Professor W. Feller. PARTIAL DIFFERENTIAL EQUATIONS In conclusion I would like to make some few remarks about the Notices. Like edited by the Reviews, the Notices have swollen to CHARLES B. MORREY, JR. they were unprecedented size. Though Proceedings of the Fourth Symposium originally devised essentially to contain in Pure Mathematics of the American only the programs of our meetings, little Mathematical Society, held at the by little a very large amount of other University of California at Berkeley, material has crept in. It has become the California, April 1960. Papers by feeling of the Council that this extra ma­ Charles B. Morrey, Jr.; James Serrin; terial should be kept to the barest mini­ N. Aronszajn; A. P. Calderon; P. C. Felix E. Browder; Francois mum, and so you will note the absence Rosenbloom; Treves; H. F. Weinberger; Louis Niren­ lately of some features formerly appear­ berg; Martin Schechter; Avner Fried­ ing in the Notices. To administer this man; Fritz John; David Gil borg; Robert policy, the Council appointed an Editorial Finn; Erhard Heinz; and H. 0. Cordes. Committee consisting of the Secretary and 169 pages $8.30 Executive Director, replacing the Execu­ 25% discount to members tive Director alone. This new appointment requires a change in the by-laws on which AMERICAN MATHEMATICAL SOCIETY you will shortly be asked to vote. 190 Hope Street, Providence 6, Rhode Island The letters-to-the-editor column was one feature which the Council voted to

87 NEWS ITEMS AND ANNOUNCEMENTS

THE COMPUTING DEVICES COM­ The Faculte des Sciences has also MITTEE of the American Institute of Elec­ arranged an International Exposition of trical Engineers announces a call for pa­ Calculating Machines from June 3 to 11. pers for the Third Annual Symposium on Cooperating in this Exposition will be the Switching Circuit Theory and Logical De­ Palais de la Decouverte, the Conserva­ sign, to be held at the 196Z Fall General toire National des Arts et Metiers, and Meeting in Chicago, October 7-1Z. the principal manufacturers of the mach­ Authors are invited to submit pa­ ines. pers having theoretical or practical inter­ Anyone wishing to take part in the est and describing new work in all areas Colloquium, and to be kept informed of of switching theory and logical design of new developments as they occur, is invited digital systems. Since it is expected that to send his name to the Secretariat Scien­ one of the sessions will be devoted to a tifique, 3 rue Kessler, Clermont-Ferrand. discussion of significant unsolved prob­ lems in switching theory, well formulated presentations of difficult research prob­ lems are also invited. THE NETHERLANDS UNIVERSI­ April 1 is the deadline for receipt TIES FOUNDATION FOR INTERNATION­ in quadruplicate of a 100-ZOO word ab­ AL COOPERATION (NUFFIC) announces stract and a 500 word summary of the pa­ its 196Z International Summer Session in per. Notification of paper selection will be Science on the topic Asymptotic Distri­ made on May 1, and July 1 is the deadline bution modulo 1, to be held at Nyenrode for receipt of full-length Symposium pa­ Castle, Brukelen, the Nether lands, from pers. August 1 to 11. Lectures will be given in Correspondence relating to the Sym­ English, and fall chiefly into the following posium should be addressed to the Sym­ groups: The number theoretical aspect, posium Chairman: Dr. Warren Semon, the metrical aspect, ergodic problems, Sperry Rand Research Center, North Road, relation to probability theory, and abstract Sudbury, Massachusetts. theories. Lectures by the following have already been announced: J. F. Koksma, E. Hlawka, P. Erdos, J. H. B. Kemperman, THE F ACUL TE DES SCIENCES OF G. Helmberg, and J. W. S. Cassels. CLERMONT-FERRAND, with the coopera­ Applications are invited from math­ tion of the Societe Mathematique de France ematicians with an active interest in the and the Association Fran'iaise de Calcul topic, and should be filed before April 1, et de Traitement de !'information, has 196Z (or as soon thereafter as possible) organized a Mathematics Colloquium to be with:The Registrar, NUFFIC, Z7 Molen­ held June 4-8, 1962, commemorating the straat, the Hague, the Netherlands. The Third Centenary of the death of Blaise number of participants will be limited to Pascal. The Colloquium will be concerned about fifty. with the principal questions raised in the The only charge will be 150 Dutch mathematical work of Pascal. The Pro­ guilders (approx. $40) for each individual gram will consist of one-hour lectures, attending (including wives) to meet the each followed by a discussion period. A costs of accommodation, meals, service, number of mathematicians from many and excursions during the course. countries have already been invited as The North Atlantic Treaty Organi­ lecturers. Those invited from the United zation has contributed a grant towards the States are A. Tarski and R. Courant. cost of organizing this summer session. ++++++++

88 PERSONAL ITEMS

Assistant Professor A. A.-CLARKE tute Technologico de Aeronautica, Brazil on leave from Fordham University has re­ has been appointed a Member of the Insti­ ceived a National ScienceFoundationFac­ tute for Advanced Study during the spring ulty Fellowship at Yeshiva Univerity. session of 1962. Associate Professor R. L. CHITTIM Mr. S. J, CHULAY of the Massa­ on leave from Bowdoin College has re­ chusetts Institute of Technology has been ceived a National ScienceFoundationFac­ appointed a Graduate Resident Engineer at ulty Fellowship at University College, the University of California, Berkeley, London, England. California, Dr. W. V. HOUSTON of William Professor A. H. CLIFFORD of Tu­ Marsh Rice University has been appointed lane University is on leave until June 1 in President of the American Physical So­ Paris. ciety. Professor H. COHN of the Univer­ Dr. C. C. HSIUNG on leave from sity of Arizona has returned from leave Lehigh University has received a National for the fall semester of 1961-1962, during Science Foundation Grant at the University which time he lectured in Switzerland, of California, Berkeley, California. Germany, and Yugoslavia. Dr. F. N. FRENKIEL, a member of W. A. COP PEL of the University of the staff of the David Taylor Model Basin, London has accepted a fellowship at the was elected Chairman of the Division of Australian National University, Canberra, Fluid Dynamics of the American Physical Australia. Society. Dr. M. V.CROSS, JR. of Hughes Air­ craft Company has accepted a position as Mr. N. G. ANTON of Anton Elec­ Research Scientist at the Space and Infor­ tronic Laboratories has accepted a posi­ mation Systems Division of North Ameri­ tion as President and Director of Research can Aviation, Downey, California. at the EON Corporation, Brooklyn, New Dr. L. CSELLE of Shell Oil Com­ York. pany has accepted a position as research Mr. A. A. ARMENDARIZ of William staff member at the International Busines a Marsh Rice University has been appointed Machines Corporation, Yorktown Heights, to an assistant professorship at Tulane New York. University. Dr. R. F. DRENICK of Bell Tele­ Dr. J, R. BLUM of the Sandia Cor­ phone Laboratories, Incorporated has poration has been appointed to a professor­ been appointed to a professorship at the ship at the University of New Mexico. Brooklyn Polytechnic Institute. Mr. T. A. BORDEAUX of the G. 0. Dr. H. P, EDMUNDSON of the Plan­ Noville and Associates has accepted a po­ ning Research Corporation, has accepted sition as Supervisor of Operations Analy­ a position as Senior Staff Member at sis at the Northrop Corporation, Haw­ Thompson Ramo-Wooldridge, Incorporated, thorne, California. Canoga Park, California. Dr. D. BOYANOVITCH of Stevens Dr. D. 0. ELLIS of Litton Industries Institute of Technology has accepted a has accepted a position as Head of the Ad­ position as Senior Mathematician at the vanced Development Section of the Data United Aircraft Company, East Hartford, Systems Division of Litton Systems, Can­ Connecticut. oga Park, California. Mr. R. E. BUBAofSystem Develop­ H. E. FLEMING of the University ment Corporation has accepted a position of Maryland has accepted a position as as Research Associate at Planning Re­ Mathematician with the U. S. Weather search Corporation, Los Angeles, Califor·­ Bureau, Meteorological Satellite Labora­ nia. tory, Suitland, Maryland. Professor K.-T. CHEN of the lnsti- Mr. M.D. FRIEDMAN of Morris D.

89 Friedman, lncorpor a ted has accepted a Laboratory, Washington, D. C. position as Mathematical Specialist at the Mr. G. E. MAHONEY of Radio Cor­ Lockheed Missiles and Systems Division, poration of America, has accepted a posi­ Sunnyvale, California. tion as a Member of the Technical Staff Dr. R. A. FUCHS, of Electro-Opti­ of The Mitre Corporation, Bedford, cal Systems, has accepted a position as Massachusetts. Senior Staff Physicist at the Hughes Air­ Associate Professor M. MARCUS craft Corporation, Space Systems Division, of the University of British Columbia, Culver City, California. Vancouver, Canada, has returned after a Mr. J, GIL DE LAMADRID has been leave of absence at the National Bureau of appointed Research Associate and Lec­ Standards, Washington, D. C. turer at Yale University. Dr. N. M. MARTIN of Space Tech­ Dr. R. M. HAYES of Magnavox Re­ nology Laboratories, Incorporated has ac­ search Laboratories has accepted a posi­ cepted a position as Technical Staff Mem­ tion as President of the Advanced Infor­ ber of Legicon, Incorporated, Redondo mation Systems Company, Los Angeles, Beach, California. California. Dr. W. G. MAY has been appointed Mr. L. HODES has acceptedaposi­ to an assistant professorship at Wake tion as Staff Member, Research Center, Forest College. International Business Machines Corpora­ Dr. D. D. MORRISON of Ramo­ tion, Yorktown Heights, New York. Wooldridge Corporation, has been appoint­ Miss G. M. HYDER of the AVCO ed to an assistant professorship at the Manufacturing Corporation has accepted a University of Texas. position as Programming Analyst at the Professor J. R. MUSSELMAN of System Development Corporation, Bur­ Western Reserve University has retired lington, Massachusetts. with the title of Professor Emeritus. Professor A. JAEGER of the Uni­ Dr. W. NACHBAR of the Lockheed versity of Cincinnati has been appointed Aircraft Corporation has been appointed Director of Graduate Studies. a Research Associate of the Department Mr. S. JAR VIS, JR. of Frankfor.d of Aeronautics and Astronautics at Stan­ Arsenal, has accepted a position as an ford University. Applied Mathematician at the National Mr. S. F. NEUSTADTER of San Bureau of Standards, Boulder, Colorado. Francisco State College has accepted a Mr. J. H. JORDAN of the Univer­ position as Mathematician at Sylvania sity of Colorado has been appointed to an Electric, Waltham, Massachusetts. assistant professorship at Washington Dr. J, W. ODLE of Arthur D. Little State University. Incorporated has accepted a position as Mr. W. D. KAHN of ArmyMapSer­ Scientific Representative at the North vice has accepted a position as Mathemati­ American Aviation, Incorporated, Bedford, cian at the National Aeronautics and Space Massachusetts. Administration, Greenbelt, Maryland. Mr. R. C. O'NEIL of the University Mr. A. KARRASS of New York Uni­ of Chicago has been appointed to an assist­ versity has been appointed to an assistant ant professorship at William Marsh Rice professorship at Adelphi College. University. Assistant Professor D.E.KEARNEY Mr. C. M. PEARCY JR. of William of Mississippi State University has ac­ Marsh Rice University has accepted a cepted a position as Research Assistant position as Research Engineer at the at the Adaptronics Incorporated, Annan­ Humble Oil and Refining Company,Houston, dale, Virginia. Texas. Dr. J. W. LAMPERT!, on leave from Mr. D. M. PETERSON of the Uni­ Stanford University, has been appointed to versity of California, has accepted a posi­ a visiting assistant professorship at Dart­ tion as Technical Staff Member of the mouth College. General Telephone and Electronics Labor­ Mr. R. E. McGILL of the University atory, Incorporated, Menlo Park, Califor­ of Maryland has accepted a position as nia. Mathematician at the U.S. Naval Research Dr. V. S. PLESS has acceptedapo-

90 sition as Mathematician at the Cambridge position as Mathematician at the United Research Center, Bedford, Massachusetts. Technology Corporation, Sunnyvale, Cali­ Mr. C. M. PRICE of North Ameri­ fornia. can Aviation Incorporated has accepted a Mr. D. V. SWARD of Montana State position as Staff Scientist at the Aerospace University has accepted a position as Corporation, Los Angeles, California. Senior Associate Mathematician at the In­ Mr. S. G. REED of Service Bureau ternational Business Machines Corpora­ Corporation has accepted a position as tion, Los Angeles, California. Manager of Advance Systems Program­ Assistant Professor E. J, TAFT, ming Systems at the Product Development on leave from Rutgers, The State Univer­ Laboratories of the International Business sity has been awarded a grant from the Machines Corporation, Poughkeepsie, New Rutgers Research Council to study at Yale York. University. Mr. J. N. ROGERS of the General Dr. S. J, TAYLOR oftheUniversity Electric Company has accepted a position of Birmingham, England has been appoint­ as Staff Member of Sandia Corporation, ed a Reader at the University of London, Livermore, California. England. Associate Professor W. C. ROY­ Professor S. UENO from the Uni­ STER on leave from the University of Ken­ versity of Kyoto, Japan has returned tucky has been appointed a temporary after a leave of absence at the Rand Cor­ member at the Institute for Advanced poration, Santa Monica, California. Study. Mr. V, F. VOLERTAS of Aircraft Professor H. RUND of the Univer­ Armament, Incorporated has accepted a sity of Natal, Durban, South Africa has position as Senior Research Engineer at been appointed to a professorship at the the Radio Corporation of America, Cam­ University of South Africa, Preloria, den, New Jersey. South Africa. Mrs. G. G. WAHBA of Operations Associate Professor W. M. SAN­ Research Incorporated has accepted a DERS on leave from Mississippi Southern position as Senior Associate Analyst with College has been appointed to an assistant­ International Business Machines Corpora­ ship at the University of Illinois. tion, Bethesda, Maryland. Dr. E. M. SCHEUERofSpaceTech­ Mrs. A. K. WEINSTEIN of A. C. nology Laboratories, Incorporated, has Spark Plug Division has accepted a posi­ accepted a position as Mathematician at tion as a Member of the Technical Staff the Rand Corporation, Santa Monica, Cali­ of the Hughes Aircraft, Aerospace Group, fornia. Guidance and Controls Division, Culver Dr. P. SEIBERT on leave from City, California. RIAS, has been appointed to a visiting Mr. E. W. WOMBLE of Wake Forest associate professorship at the University College has been appointed to an assistant of Notre Dame. professorship at Pfeiffer College. Mr. W. T. SLEDD of the University Mr. T. E. WOOD of Thiokol Chemi­ of Kentucky has been appointed to an assist­ cal Corporation has accepted a position as ant professorship at Michigan State Uni­ Associate Scientist at AVCO Research and versity. Advanced Development Division, Wilming­ Mr. R. M. SMULLYANofPrinceton ton, Massachusetts. University has been appointed to an assist­ Mr. S. A. ZADOFF of Radio Recep­ ant professorship at Yeshiva University. tor Company, has accepted a position as a Mrs. R. C. STRODT of Columbia Member, Technical Staff, of the Sperry University has been appointed a Lecturer Gyroscope Company, Great Neck, New at Hunter College. York. Dr. R. R. STRUIK of the University Mr. J. A. ZILBER of the American of British Columbia has been appointed to Mathematical Society has been appointed an acting assistant professorship at the a Research Associate at Yale University. University of Colorado. The following promotions are announced: Mr. R. I. SUTTON of the Advanced Technology Laboratories has accepted a B. BERNHOLTZ, University of To-

91 ronto to an associate professorship. to a professorship. E. COHEN, University of Tennessee, B. E. RHOADES, Lafayette College, to an associate professorship. to a professorship. B. R. DEAL, Oklahoma State Uni­ J. W. SAWYER, Wake Forest Col­ versity, to a professorship. lege, to a professorship. I. F !SCHER, University of Colorado, 0. K. SMITH, Space Technology to an associate professorship. Laboratories, Incorporated, to Manager, R. P. GOSSELIN, University of Applied Mathematics Department. Connecticut, to a professorship. J. P. TULL, Ohio State University, J. L. HATFIELD,College of William to an associate professorship. and Mary, to an associate professorship. B. HORELICK, Lafayette College, The following appointments to instructor­ to an assistant professorship. ships are announced: T. E. HULL, University of British Columbia, Vancouver, Canada, to a pro­ College of St. Rose: SISTER KATH­ fessorship. LEEN ANN; University of California, J. D. HWANG, Sacramento State Berkeley: M. G. ROTHENBERG; Univer­ College, to an associate professorship. sity of Pennsylvania: D. J. OSTROFF; R. F. KELLER, University of Mis­ Wayne State University: C. A. SCHULZ, souri, to an assistant professorship and JR. Director, Computer Research Center. J. H. McKAY, Michigan State Uni­ Deaths: versity to an Associate Dean of Science. Dr. E. KRAHN of Naval Ordnance P. D. MINTON, Southern Methodist Laboratories, Silver Spring, Maryland died University, to a professorship. on March 6, 1961 at the age of 67. H. F. MONT AGUE, University of Professor M. MARSHALL of Provo, Buffalo, to Acting Chairman of the Mathe­ Utah died on September 16, 1961 at the age matics Department. of 66. He had been a member of the Society E. REICH, UniversityofMinnesota, for 17 years.

NEWS ITEMS AND ANNOUNCEMENTS

THE INSTITUTE OF MATHEMA­ Conference should be directed to the TICS OF THE POLISH ACADEMY OF SCI­ Institute of Mathematics of the Polish ENCES is organizing a Conference on Ana­ Academy of Sciences, 30, Solskiego Str., lytic Functions in Cracow from August Cracow, Poland. 30th to September 4th, 1962. The main subjects of the Conference will be A CONFERENCE ON "MATHE­ 1. Extremal problems MATICAL MODELS IN PHYSICAL SCI­ 2. Functions of several complex vari­ ENCES" will be held for invited persons ables at the University of Notre Dame on April 15, 16, and 17, 1962. The object of the Surveys and communications on Conference will be to discuss some con­ other fields of analytic functions are also ceptual aspects of the mathematical treat­ welcome. During the Conference there will ment of physical systems. For information be delivered half-hour lectures and 10 address: Professor Stefan Drobot, Depart­ minute communications. ment of Mathematics, University of Notre All correspondence relating to the Dame, Notre Dame, Indiana.

92 NEW AMS PUBLICATIONS

COLLOQUIUM PUBLICATIONS

Volume 38

THEORY OF GRAPHS Since the manuscript to these lectures was By Oystein Ore not completed for publication at that time it seems appropriate that this book should 280 pages; $9.20 List Price; 25o/o discount appear in the Colloquium Lecture to members. Series of the Society. An effort has been made to The book has grown out of courses present the subject matter in as simple a on graph theory given from time to time at form as possible. Almost all proofs have Yale University. A first set of lectures on been revised; a considerable number of binary relations and graphs was presented new results are also included. A systema­ before the American Mathematical Society tic terminology is introduced which it is at its summer meeting in Chicago, 1942. hoped may prove acceptable.

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Volume IV PARTIAL DIFFERENTIAL EQUATIONS April 1960. Papers by Charles B. Morrey, Edited by Charles B. Morrey, Jr. Jr.; James Serrin; N. Aronszajn; A. P. Calder6n; P. C. Rosenbloom; Felix E. 169 pages; $8.30 List Price; 25o/o discount Browder; Frans;ois Treves; H. F. Wein­ to members. berger; Louis Nirenberg; Martin Schech­ Proceedings of the Fourth Sympo­ ter; Avner Friedman; Fritz John; David sium in Pure Mathematics of the American Gilbarg; Robert Finn; Erhard Heinz; and Mathematical Society, held at the Univer­ H. 0. Cordes. sity of California at Berkeley, California,

TWENTY-VOLUME AUTHOR INDEX OF MATHEMATICAL REVIEWS 1940 - 1959

In Two Volumes:

Part 1, A - L, 1098 pages !CAL REVIEWS, with complete cross­ Part 2, M- Z, 1115 pages references to joint authors, and, in the case of items without a personal author, List Price: by editor's name or by title. Thus, by U.S.A. and Canada: $35.00 means of a single reference work, there­ Foreign: $37.50 search scholar can find complete biblio­ 25o/o discount to members. graphical information about any article or The Index lists every item published book reviewed by MATHEMATICAL RE­ in the first twenty volumes of MATHE MAT- VIEWS between 1940 and 19 59.

93 MATHEMATIDALSURVEYS

Volume VII course at Tulane University during the academic year 1958-1959. The book aims THE ALGEBRAIC 'I:HEOR Y OF at being largely self-contained, but it is SEMIGROUPS assumed that the reader has some famil­ By A. H. Clifford and C. B. Preston iarity with sets, mappings, groups, and 224 pages; $10.60 List Price; 25o/o discount lattices. Only in Chapter 5 will more pre­ to members. liminary knowledge be required, and even there the classical definitions and theo­ The material in this volume was rems on the matrix representations of presented in a second-year graduate algebras and groups are summarized.

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume VI pure Mathematics of the American Mathe­ matical Society. Held at California Insti­ ON FINITE GROUPS 1960 INSTITUTE tute of Technology, Pasadena, California, Hall, Jr. Edited by Marshall August 1960. Papers by: G. Higman, 114 pages; $4.80 List Price, 25o/o discount H. Wielandt and B. Huppert, D. R. Hughes, to members. E. T. Parker, M. Hall, Jr., W.Feit, W. F. Reynolds, J. S. Frame, M. Suzuki and of the Symposium in Proceedings R. Ree.

NEWS ITEMS AND ANNOUNCEMENTS

A SEMINAIRE DE MATHEMATIQUE treal, Introduction a 1' algebre homologique; SUPERIEURE will be held each year during Professor Maurice L'Abbe, University of the summer, under the sponsorship of the Montreal, Theorie des fonctions recursives Canadian Mathematical Congress, starting et applications metamathematiques. This this next summer, at the University of seminar is specially intended for graduate Montreal, Montreal, Canada. The pro­ students. Application may be made for gram will include four courses: P·ro­ financial assistance to cover all or part fessor Jacques L. Lions, University of of travel and living expenses of those re­ Nancy, Problemes aux limites dans les gistered at the seminar. The registration full informa­ equations aux derivees partielles; Profes­ fee is ten dollars. To obtain sor Lucien Waelbroek, University of tion and registration forms, write to: of Brussels, Theorie des algebres de Banach Department of Mathematics, University et des algebres localement convexes; Pro­ Montreal, P. 0. Box 6128, Montreal, fessor Jean Maranda, University of Mon- Quebec, Canada.

94 LETTERS TO THE EDIT-OR

Editor, the NOTICES support. "Je desire remercier vivement les Many American mathematicians collegues americains qui ont signe la may be astonished to learn that among the declaration, et envoye une souscription most recent publications of .Laurent pour l'aide a Madame Audin eta safamille, Schwartz, awarded the Field Prize at the ou pour le Comite Maurice Audin, ou pour 19 50 International Congress of Mathema­ le Comite du Prix Mathematique Maurice ticians and widely known for his work on Audin. Cet argent a ete le bienvenu, mais, The Theory of Distributions, is a paper plus encore que 1' argent, le soutien moral entitled "Le probleme de la torture dans de nos collegues mathematiciens. Nous la France aujourd'hui". Actually, a con­ vivons actuellement en France une epoque siderable group of French mathematicians, tres difficile. Les tortures se sont devel­ of all shades of political opinion, have oppees sur une echelle gigantesque et sont joined in a movement of opposition to tor­ restees impunies, souvent meme les tor­ ture. tionnaires ont ete recompenses. Des atro­ This movement arose out of the cites racistes ont ete perpetrees Paris case of Maurice Audin, a meme, au mois d'octobre. Des centaines Audin was a young mathematician, d' Algeriens ont ete deportes. Plusieurs preparing a doctoral dissertation under dizaines ont ete tues dans les locaux de la Schwartz, when he was taken from his police. On a retrouve des cadavres dans home in Algiers by French parachutists la Seine ou dans les bois des environs de one night in 19 57 and never returned. The Paris. Les membres du Comite Audin qui overwhelming evidence indicates that he luttent pour que soit connue la Verite et was tortured and killed. He was granted que Justice soit faite, les mathematiciens the Ph.D. degree posthumously by The qui ont souscrit pour le prix Maurice Sorbonne, and a prize was created to his Audin, se sont engages pour une cause name to be awarded annually to a promis­ symbolique, qui est la condamnation de la ing young French mathematician. torture et de tous les sevices contre A "Co mite Maurice Audin" was l'homme, oil. que ce soit, sous quelque formed to raise funds for Audin' s widow pretexte que ce so it. Le nom d' Audin est and young children, and to press for an un symbole, mais notre lutte est univer­ official investigation into his disappear­ selle, valable pour tous les pays et toutes ance- -responsibility for which was dis­ les epoques. Nous sommes heureux que claimed by the French military and police. nos collegues et amis mathematiciens des Subsequently this committee broadened its Etats- Unis aient compris notre point de goals to include a general campaign of op­ vue, et nous aient prouve leur solidarite. position to the systematic use of torture in the Algerian struggle, a practice which Laurent Schwartz their documented publications allege to P.S. have received sanction at the level of the "J' allais envoyer ce texte quand highest French ministers. Laurent notre collegue Roger Godement a ete Schwartz is the president of this Commit­ victime d'un attentat, pour sa lutte contre tee. la guerre d' Algerie: une bombe a explose dans sa maison la rendant inhabitable pour In the Spring of 1961 more thanZOO plusieurs mois. Voila dans quel climat American mathematicians contributed politique nous vivons aujourd'hui." money to support the aims of the Comite Audin or to maintain the Prix Audin. Most Those wishing to receive more in­ of them signed a statement requesting an formation about the work of the Comite official investigation of the Audin case and Audin should write to M. Louis Lalande, affirming their opposition to all use of 10 rue Jean Bart, Paris VI. torture. Schwartz has written the following Leon Henkin letter to express his appreciation for this Hazelton Mirkil

95 MEMORANDA TO MEMBERS

MATHEMATICS OF COMPUTATION With the recent reprinting of Vol­ umes 8, 9, and 17, allvolumesofMATHE­ Beginning with the January 1962 MATICAL REVIEWS are now available. issue, the American Mathematical Society The list prices are as follows: Volumes 1 took over publication of the journal Mathe­ through 16 (up to 1956), $42.00; Volumes matics of Computation for the National 17 through 22, $50.00. The discount to Academy of Sciences-National Research members of 25o/o off the list price applies. Council. First published in 1943 by the Issues No. 2 and 3 of Volume 22, National Research Council under the title which ran out last November, have been Mathematical Tables and Other Aids to reprinted to balance our stock for 1961, Computation, this journal has a long tradi­ and will be maUed soon to those sub­ tion of publication in the field of mathemati• scribers for 1961 who ordered but did not cal computation. It was given its current receive them. title in 1960 to reflect the increased scope of its coverage. In addition to reviews and notes, Mathematics of Computation pub­ RECIPROCITY AGREEMENT WITH THE lished original (and survey) papers in the SOCIEDADE DE MATEMATICA fields of , the applica­ DE SAO PAULO tion of computational methods, mathemati­ cal tables, highspeed calculators, and other The American Mathematical Society aids to computation. has entered into a reciprocity agreement Members of the American Mathe­ with the Sociedade de· Matematica de Sao matical Society are invited to submit Paulo, by which members of each may be­ papers for publication in Mathematics of come members of the other by paying half Computation. Contributions should be ad­ the regular dues. The regular dues of the dressed to H. Polachek, Chairman, Edi­ Sociedade de Matematica de Sao Paulo are torial Committee, Mathematics of Compu­ $3.00 a year; therefore an American Math­ tation, Applied Mathematics Laboratory, ematical Society member would pay $1.50. David Taylor Model Basin, Washington 7, Privileges of membership include the pre­ D. C. sentation of papers at meetings of the Mathematics of Computation is pub­ Sociedade de Matematica de Sao Paulo, lished quarterly. The subscription rate is receiving the BOLETIM, and purchasing $8 per year. Requests for subscriptions other publications at a discount. It is should be addressed to the American understood that members under the re­ Mathematical Society, 190 Hope Street, ciprocity agreement spending time in the Providence 6, Rhode Island. other country will pay regular dues while they are there. Those members of the American REPRINTING OF BACK VOLUMES OF Mathematical Society wishing to take ad­ MATHEMATICAL REVIEWS vantage of this arrangement should write to Dr. Omar Catunda, President, Sociedade In view of the steady demand for de Matematica de Sao Paulo, R. Maria back volumes of MATHEMATICAL RE­ Antonia, 258, S. Paulo, Brasil. VIEWS,the American Mathematical Society plans to keep the entire series· in print by reprinting volumes as our original stock CHANGE OF ADDRESS NOTIFICATION becomes depleted. These reprints will be DEADLINES FOR JOURNALS bound in complete volumes, and will be sold only in that form. Some individual We wish to call members' attention back issues are available from stock, and to the necessarily long notification period they will continue to be offered for sale which must be given to change the mailing for as long as we have them on hand. address of the Society's journals. The lead

96 time required varies for the different required by the printer. Thus three weeks journals, depending upon where they are are the absolute minimum required after printed. In general, notice received the receipt of the change. At least another first Friday of the month prior to publica­ week should be allowed for mailing and for tion is adequate, but in the case of those printing and other delays. journals mailed from foreign printers--­ This schedule applies only when we MATHEMATICAL REVIEWS currently, the are informed of a change. In many cases, TRANSACTIONS as of july---more lead we become aware of a change only when a time is needed. Printed below is a sche­ journal is returned because of incorrect dule of deadlines for change of address address. This means deleting the sub­ notification for the remainder of 196Z. scriber from the mailing list, securing Before an address change can go the new address, and back-ordering the into effect it must go through a number of missed issues, all of which adds indefi­ steps. Five days must be allowed in the nitely to the delay. We do the best we can headquarters offices to process changes tokeepupwithourmembers, but continu­ and run the mailing list; at least a week ing service can be given only when new is needed for mail delivery of the list to addresses are made available to us in good the printer; and a lead time of ten days is time.

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BROWN UNIVERSITY IVANAGA, Class field theory, 1961, 65pp. $1.20. The following item may be ordered from the Brown University Bookstore, KAPLANSKY, Rings of operators, 1955, Providence Rhode Island, for $1.50 per 53 pp. $1.00. copy including postage. MACKEY, Theory of group representations, K. NOMIZU, Vector spaces and tensor 19 55, 182 pp. $3.00. algebras, 83 pp. 1961. MacLANE, Simplicial topology, 1959, 42 pp. $1.00. CALIFORNIA INSTITUTE OF , Reports, Seminar in topology, --:-::-::-= TECHNOLOGY 1957, 120 pp. $2.00. The following item is out of print. WElL, Fiber spaces in algebraic geome­ try, 1952, 48 pp. $1,00. A. ERDEL YI, Operational calculus. Discontinuous subgroups of The following lecture notes may be classical groups, 1958, 91 pp. $1.50, ordered from Caltech Bookstore, Pasa­ dena, California, for $1.85 including post­ age. THE INSTITUTE FOR A. ERDEL Yl, Asymptotic solutions of ADVANCED STUDY ordinary linear differential equations. The following items may be ordered from: The Institute for Advanced Study, Attention: Miss Mary Pochyla, Secretary, UNIVERSITY OF CHICAGO Building C, School of Mathematics, Princeton, New Jersey. Remittance should The following mimeographed notes be made payable to The Institute for Ad­ are currently available from the Depart­ vanced Study. ment of Mathematics, University of Chi­ cago. Orders should be addressed and MARSTON MORSE, Introduction to analy­ checks made payable to: Mathematics sis in the large 160 + 31 (Appendix) Lecture Notes, Eckhart Hall, University pp. $3.60. of Chicago, Chicago 37, lllinois. Please OSWALD VEBLEN and JOHN VON NEU­ make payment in advance. The prices in­ MANN, Geometry of complex domains, clude postage. 259 pp. $4.00 plus postage. CHERN, Differentiable manifolds, 1959, ANDRE WElL, Adeles and algebraic 132 pp. $2.00. groups, 121 pp. $3.00 plus postage. ElLENBERG, Foundations of fiber bun­ dles, 1957, 75 pp. $1.25. IMS COMMITTEE ON TABLES GARDING, Cauchy's problem for hyper­ bolic equations, 1957, 106 pp. $1.75. The following item contains informa­ tion about the policies of various journals GRANAS, Introduction to topology offunc­ concerning the publication of tables, the tional spaces, 1961, 94 pp. $1.60. policies of depositaries and distribution GRAVES, The estimates of Schauder and agencies concerning the filing, lending, their application to existence theo­ and selling of statistical tables, and rems for elliptic differential equa­ sources of financial aid. It is available tions, 1959, 67 pp. $1.00, free of charge from the author, at 1904 Idaho Avenue, Las Cruces, New Mexico. HERSTEIN, Theory of rings, 1961, 125pp. $2.00, PAUL C. COX, A Guide for the Prepara­ tion, Publication, and Distribution of HOCHSCHILD, Representation theory of Statistical Tables. 10 pp. typed. No Lie algebras, 1959, 92 pp. $1.75. charge.

99 SUPPLEMENTARY PROGRAM - NO. 10

During the interval from January 10, 196Z through February ZO, 196Z, the papers listed below were accepted by the American Mathematical Society for presentation by title. Readers may wish to refer to Page 713 of the November, 1960 issue (No. 49 of these NOTICES) where it is explained in detail that the presentation of papers by title is now dissociated from meetings of the Society. Supplementary Program No. 11 will cover the interval from February Z1, 196Z through April Z7, 196Z. After each title on this program is an identifying number. The abstract of the paper will be found following the same number in the section on Abstracts of Contributed papers in this issue of these NOTICES. (1) C(S) spaces of the Ph -type (10) Products of contractible open mani­ Professor Dan Amir, Hebrew Uni­ folds versity, Jerusalem, Israel (6ZT-94) Dr. C. H. Edwards, Jr., University (Introduced by Professor Aryeh of Wisconsin (6ZT-9Z) Dvoretsky) (11) Sums of distinct unit fractions (Z) Infinite series and nonnegative valued Dr. Paul Erd6s and Professor S.K. interval functions. Preliminary re­ Stein, University of California, port Davis (6ZT-66) Dr. W. D. L. Appling, Duke Univer­ (lZ) On minimal models of complete theo­ sity (6ZT-51) ries (3) The growth of entire functions of min­ Mr. Gebhard Fuhrken, University of imal exponential type California, Berkeley (6ZT-73) Dr. Louis de Branges, New York (13) Harmonic functions in four variables University (6ZT-99) with rational p4 - associates (4) Trigonometric series expansions of Dr. R. P. Gilbert, University of certain arithmetical functions Maryland (6ZT-85) Professor Eckford Cohen, Univer­ (14) A coloring problem with square tiles. sity of Tennessee (6ZT-90) Preliminary report (5) The number of planes contained in Professor R. E. Greenwood and Mr. the complement of a quadric in an D. R. Stocks, The University of affine Galois space Texas (6ZT-78) Professor Eckford Cohen, Univer­ (15) Curves in Euclidean n-space sity of Tennessee (6ZT-83) Professor H. H. Guggenheimer, (6) Dissertation on Correia's sequence University of Minnesota (6ZT-88) and its relationship with Cramer's (16) Bounds for certain invariants in conjecture pseudo-conformal mappings Dr. F. B. Correia, U.S. Navy and Mr. K. T. Hahn, Stanford University University of Colorado (6ZT-100) (6ZT-81) (7) On relative characterizability in a ( 17) Bounds for certain invariants in con­ language. Preliminary report formal mappings Professor William Craig and Mr. Mr. K. T. Hahn, Stanford University William Hanf, University of Cali­ (6ZT-74) fornia, Berkeley (6ZT-95) (18) Isomorphism in elementary logic. (8) Relative characterizability and gen­ Preliminary report eralized existential quantifiers. Pre­ Mr. W. P. Hanf, IBM Corporation liminary report (6ZT-75) Professor William Craig, Univer­ (19) A reduction class for the Entschei- sity of California, Berkeley f6ZT-96) dungsproblem (9) Concerning the uniform approxima­ Mr. A. S. Kahr, Massachusetts In­ tion of a bounded function stitute of Technology (6ZT-61) Dr. S. I. Drobnies, William Marsh (Introduced by Professor H. P. Rice University (6ZT-68) McKean, Jr.)

100 (20) A new reduction procedure Dr. A. A. Sagle, University of Chi­ Mr. A. S. Kahr, Massachusetts In­ cago (62T-62) stitute of Technology (62T-54) ( 32) On the collineation groups of the free (Introduced by Professor H. P. planes. Preliminary report McKean, Jr.) Dr. Reuben Sandler, Institute for (21) Curves whose affine images are mir­ Defense Analyses, Princeton, New ror-symmetric jersey (62T-55) Professor Fred Krakowski, Univer­ (33) Extension properties of Banach spa­ sity of California, Davis (62T-67) ces (22) On the modulus of smoothness Professor Andrew Sobczyk, Univer­ Professor joram Lindenstrauss, sity of Miami (62T-52) The Hebrew University, jerusalem, (34) Relations among Whitney classes of Israel (62T-97) n-manifolds (Introduced by Professor Aryeh Mr. R. E. Stong, University of Chi­ Dvoretsky) cago (62T-69) (23) The extension of compact operators (35) Some results on Whitney numbers Professor joram Lindenstrauss, Mr. R. E. Stong, University of Chi­ The Hebrew University, jerusalem, cago (62T-70) Israel (62T-98) (36) Introduction to the geometry of the (Introduced by Professor Aryeh real hyperbolic plane Dvoretsky) Mr. j. A. Synowiec, De Paul Uni­ (24) Two remarks concerning productive versity (62T-71) and contraproductive centers (37) Basic algebra of the real hyperbolic Mr. T. G. McLaughlin, University complex number system of California, Los Angeles (62T-84) Mr. j. A. Synowiec, De Paul Uni­ (25) Some remarks about productive and versity (62T-72) contraproductive centers (38) Extreme points in (M)-spaces Mr. T. G. McLaughlin, University Mr. D. M. Topping, Tulane Univer­ of California, Los Angeles (62T-80) sity (62T-79) (26) Inequalities for stochastic nonlinear (39) Medial tetrads programming problems Mr. T. N. Tracewell, University of Dr. 0. L. Mangasarian and Dr. j. B. California, Berkeley (62T-56) Rosen, Shell Development Company, (Introduced by Professor j. L. Emeryville, California (62T-93) Kelley) (27) Cebysev approximation in a Cartesian (40) Note on the reduction of induced re­ product space presentations Professor D. J. Newman, Yeshiva Miss P. A. Tucker, University of University and Professor H. S. illinois (62T-91) Shapiro, New York University (62T- (41) On almost periodic solutions of the 63) Helmholtz equation (28) A quantitative form of Haar' s theorem Professor Alexander Weinstein, Professor D. j. Newman, Yeshiva University of Maryland (62T-57) University and Professor H. S. (42) A necessary and sufficient condition Shapiro, New York University (62T- in the maximum-minimum theory of 64) eigenvalues (29) Solution of the Waring-Goldbach prob- Professor Alexander Weinstein, lem for algebraic number fields University of Maryland (62T-53) Professor G. j. Rieger, Purdue (43) Geodesic spheres in Grassmann man­ University and University of Mun­ ifolds ich, Germany (62T-60) Dr. j. A. Wolf, The Institute for (30) Concerning two questions of S. K. Advanced Study (62T-65) Stein. Preliminary report (44) Locally symmetric homogeneous spa­ Professor D. A. Robinson, Georgia ces. III Institute of Technology (62T-82) Dr. j. A. Wolf, The Institute for (31) On derivations of semi-simple Malcev Advanced Study (62T-89) algebras (45) Space forms of Grassmann manifolds

101 Dr. j. A. Wolf, The Institute for Advanced Study (62T-50) (46) Existence of linear connections with MATHEMATICAL respect to which a given tensor is re­ current or parallel. II SURVEYS Professor Yung-Chow Wong, Uni­ Published by versity of Hong Kong, Hong Kong (62T-58) THE AMERICAN (47) On a formula for preferential ar­ MATHEMATICAL SOCIETY rangements Volume List Price Dr. David Zeitlin, Remington Rand Univac, St. Paul, Minnesota(62T-86) 1. J. A. Shohat and J. (48) On the evaluation of a class of infinite D. Tamarkin, The series problem of moments~ Dr. David Zeitlin, Remington Rand 1943; reprinted 1960, Univac, St. Paul, Minnesota (62T-87) xiv, 144 pp. $5.80 (49) Two-point boundary conditions linear 3. Morris Marden, The in a parameter geometry of the zeros Professor H. j. Zimmerberg, Rut­ of i1 polynomial in a gers, The State University (62T-59) complex variable~ 1949, x, 183 pp. 5.00 4. 0. F. G. Schilling, The theory of valua­ tions~ 1950, viii, 253 pp. 6.00 5. Stefan Bergman, The kernel function and conformal mapping~ 1950, viii, 161 pp. 4.00 Applied Mathematician 6. C. C. Chevalley, the Masters degree and experience. Introduction to theory of algebraic Background in Markov processes and functions of one var­ optimization problems with interest in iable~ 1951, xii, 188 the application of these techniques to pp. 4.00 control system and guidance problems. 7. ( 1 ) A. H. Clifford and C. B. Preston, This is an excellent professional The algebraic theory opportunity in a growing, highly tech­ of semigroups~ 1962, nical group. A qualified mathematician xvi, 224 pp. 10.60 will find good compensation, and a The price to members of the compatable environment within the American Mathematical company and in metropolitan Boston. Society is 25% less than list Contact Mr. S. Herzstein at 438-3900. Order from AMERICAN DYNAMICS RESEARCH CORPORA liON MATHEMATICAL SOCIETY 38 MONTVALE AVENUE 190 Hope Street STONEHAM, MASSACHUSffiS Providence 6, Rhode Island

102 ABSTRACTS OF CONTRIBUTED PAPERS

The February Meeting in New York February 22, 1962 588-30. M. E. MAHOWALD, Syracuse University, Syracuse 10, New York. On plane bundles over real projective spaces.

In this note it is proven that Py (v-dimensional projective space) is immersible in R2V-3

(2"V-3 dimensional Euclidean space) if,.) = 3 mod 4. This is a consequence of the following more general theorems. Theorem 1. If V = 3 mod 8 then every v- 2 plane bundle over Pp with vanishing characteristic class has a nonzero cross section. (Definition. We call a map f:PJ.I ~ Gj (the Grass­ mann manifold of j planes in R00) (k,n)-extensible if there exists a map g:P)J+k ~ Gn such that the composition Py---.. f Gj C Gn agrees with the composition P,.. C P:v+k- g Gn.) Theorem 2. If >' = 7 mod 8 then every Y - 2 plane bundle over P;J, with vanishing characteristic class and whose classify­ ing map is (5,n)-extensible for some n E;. Y + 7, has a nonzero cross section. Theorem 3. For any k there exists anN such that if n ~ N then the normal bundle to an immersion of P.P into R>'+j has a classifying map which is (k,n)-extensible. If )! = 1 mod 4 we have not been as successful but we have this partial result. Theorem 4. If )) = 1 mod 4 then every Y - 2 plane bundle over Pp , with a vanish­ ing characteristic class and whose classifying map is both (2, }I + 1)- and (5,)1 + 7)-extensible, has a nonzero cross section. From this theorem we can prove for example, that P9 is immersible in R 15 if P 11 is immersible in R 17. (Received January 10, 1962.)

103 The April Meeting in Chicago April 12 -14, 1962

589-1. WITHDRAWN.

589-2. HOWARD COOK, University of Texas, Austin 12, Texas. On the most general plane closed point set through which it is possible to pass a pseudo-arc.

R. L. Moore and J. R. Kline proved that, in the plane, a closed and compact point set M is a subset of an arc if and only if every component of M is either a single point or an arc t such that no point of t, except its end points, is a limit point of M - t (On the most general plane closed point set through which it is possible to pass a simple continuous arc, Ann. of Math. 20 (1919), 218-223). In the present paper, it is shown that, in the plane, a closed and compact point set M is a subset of a pseudo-arc if and only if every component of M is either a single point or a pseudo-arc. Indeed, if M is such a closed and compact point set, there exists a pseudo-arc, containing M, no composant of which contains two components of M. These two statements remain true if "pseudo-arc" is replaced by "chainable continuum". Every pseudo-arc T contains a totally disconnected perfect point set C such that no proper subcontinuum of T contains two points of C. (Received January 18, 1962.)

589-3. T. W. TING, Department of Engineering Mechanics, University of Texas, Austin, Texas. Isoperimetric inequalities of moments of inertia of plane convex sets.

Let j( be the class of all closed plane convex sets K with given area v, CK be the centroid of K, and P be a point in the plane of K. Denote by J(K,P) the polar moment of inertia of K about the point P.

If the line )(p,S) lies in the plane of K through the point P with direction 8, 0 ;€, 9 a 2'!1", denote by I(K ,P ,8) the moment of inertia of K about the line )\P ,9). The following two theorems have been

104 proved: Theorem 1, Let l:IP AB be a triangle in .:(with P B > P A, Let A* be a point in segment P B with PA* = PA. If K E .::( which contains PA,PA* in its boundary, then J(L\.PAB,P) ~ J(K,P), Theorem Z, For all K E :J(, minei(K,CK,9);5 vz/6·3 1/Z. The equality sign holds, if, and only if, K is an equilateral triangle, Theorem Z is a conjective of Peter Ungar (Arch, Rat, Mech, Anal. 51 (1960)), (Received January ZZ, 196Z,)

589-4. W. R. ALFORD, Tulane University, New Orleans 18, Louisiana. Uncountably many different involutions of the thre.e sphere,

An involution is a periodic homeomorphism of period Z, If f and g are two homeomorphisms from a space X onto itself, then f and g are topologically equivalent if there is a homeomorphism h of X onto itself such that hfh -l = g. R. H. Bing has shown (A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann, of Math. 56 (195Z), 354-36Z) that there exists an involution on the three sphere that is not equivalent to a differentiable involution. Using methods very similar to those used by Deane Montgomery and Leo Zippin (Examples of transformation groups, Proc, Amer, Math, Soc. 5 (1954), 460-465) a class of nondifferentiable involutions on the three sphere with cardinality of the continuum is constructed so that no two are equivalent, This answers a conjecture proposed by Deane Montgomery. The examples are modifications of those constructed by Ta-Sun Wu (Abstract 61T-6Z8, Notices Amer, Math, Soc. 8 (1961), 518). (Received January 31, 196Z.)

589-5, R, H. SZCZARBA, Box Z 155, Yale Station, New Haven, Connecticut, On the tangent bundle of a quotient space, Preliminary report.

Let 7! = (E,p,B) be a fiber bundle with group G and fiber F. We say that a group r acts on 17 if r acts on both E and B with p(?'x) = ?'p(x), ?' € r and x E E. If suitable restrictions are placed on the action of r, (E/ r, p,B/ r) is again a fiber bundle with fiber F and group G. We denote this bundle by 7f/ r. Now, suppose ~ = (X, 1T",M) is a differentible principal G-bundle and let !6: X- RN be an em­ bedding equivariant relative to a representation a: G -+O(N). Mostow (Ann, of Math, 65 (1957), 43Z-446) has shown that such embedding always exist if G and X are compact, Let r(X) denote the tangent bundle of X, 1:"1 the sub-bundle of vectors along fibers, and v the normal bundle to the embed­ ding !6. Then G acts on each of these bundles (in the above sense) and t'(X)/G, r 1/G, and J)/G are bun­ dles over M. If c:;• denotes the a-extension of ~(see Borel and Hirzebruch, Amer, J, Math, 80 (1958), 477) we have Theorem, Let 'C"(M) denote the tangent bundle to M. Then t"(M) EB 1:VG ED v/G and t;• are associated bundles. Applications are given to manifolds of constant positive curvature and to cyclic products of manifolds, (Received January 31, 196Z.)

589-6, GLORIA OLIVE, Anderson College, Anderson, Indiana. Generalized powers,

After defining O~b) = 1 and orb) = 0 (where p is a positive integer and b is a complex number called "the base"), the generalized power x(b) is defined recursively as ~~=ON: (b) (x - 1)(b) where the "extended binomial coefficients" N ~(b) are functions of generalized factorials. After restricting x and t to non-negative integers, a natural extension of the theory permits them to assume complex values, Results include (q, r non-negative integers; x, y, b complex numbers): (1) 1('b) = 1; (Z) x~b( 1;

105 (3) x(~) = x; (4) (x + y)(~) = .L:i=oN~(b)x(b{Y(b) (which reduces to the binomial theorem when b = 1); . . .IJ.p+r qp r q r 1-p q (5) 1f b has perwd p, x(b) = x(b)"x(b) = x ·X(b) and (1 + x)(b) = 1 when Ix I < 1; (6) (- x)(b) = (- 1)(1) · x{l/b) if b f. 0; (7) x~b) = xtb) ·xtb) implies (b - l)(x - 1)x = 0; (8)((b - 1)/(b + l){b) = o if b.;:.- l; (9) (q + l)(o)= (r + l)(o)= (q (!j r); (10) (- q)(o) = o if r :> q; (11) O(~ =1Tj~ 1 l/(l- b-j) if lb I > 1. An immediate consequence of the development is that when b is a prime, 2(b) represents the number of subgroups in an elementary Abelian group of order br, (Received February 1, 1962,)

589-7, R. L. HEMMINGER, Michigan State University, East Lansing, Michigan, A condition for a simple, power-associative algebra to be a field, Preliminary report.

Let A be a finite-dimensional, power-associative algebra of characteristic p 1= 2,3 such that A. has an idempotent and satisfies the identity x(yz) = z(yx). It is well-known that if e is an idempotent then A is the direct sum of the subspaces Ae(i), i = 0, 1/2, 1; where x E: Ae(i) if and only if xe + ex= 2ix. Theorem 1. (a) Ae(l) is a commutative, associative subalgebra, (b) Ae(O) is an ideal, and (c) Ae(l/2) is a nilpotent ideal of index 2, Defining A to be semi-simple if and only if the nil-radical is zero we have Theorem 2. If A is semi-simple then A is commutative and associative. Thus if A is simple then A is a field, (Received February 12, 1962,)

589-8, EDGAR KARST, Evangel College, Springfield, Missouri. On a unifying technique of multiplying in various systems,

A machine which would be able to calculate in number systems with bases 2,3;4, .. ,, up to 10, exists already in form of any decimal digital computer, but the 4 essential features, (l) Ten-Comple­ ment of Base, (2) Base, (3) Tempora.ry Storage of Multiple of Base, and (4) Temporary Storage of Multiple of Ten-Complement of Base, could be built into the hardware of the machine. Small loss in computer time will be more than compensated by 8 times as much versatility, The author shows first multiplication without intermediate calculation, due to Bhaskara (see J, Tropfke, Geschichte der Elementar-Mathematik in systematischer Darstellung, vol, l, 3d ed,, Berlin, 1930). Then the usual multiplication in various number systems is mentioned, The inconvenience of calculating simultane­ ously in 2 bases is eliminated by using methods where all intermediate calculation is done in base 10, and that the method ofF. L. Parsons (A simple desk-calculator method for checking binary results of digital-computer arithmetic operations, J, Assoc. Comput. Mach, (1955)) for binary, and the method of E. Karst (A simple octal multiplication method for checking computer results, Math, Comput. (1956)) for octal. The author's method is then extended to all integer bases from 2 till 10. (Received February 20, 1962,)

589-9. M. Z. v, KRZYWOBLOCKI, l Olds Hall, Michigan State University, East Lansing, Michigan. Generalized mathematical fundamentals of relativistic theories.

The author proposes to consider three fundamental conservation laws: of momentum, continuity, and energy. The laws of conservation of energy serves as the fundamental four-dimensional metric of the space in question. With this, the Einstein special theory of relativity appears to be only a particular case of the presently proposed formalism. The velocity of light is not assumed to be a

106 constant magnitude. It may be any function of coordinates and time. The invariance of the energy equation in all the system of coordinates serves as the proposed basic fundamental law of the relativity. (Received February 14, 1962.)

589-10. S. K. BERBERIAN, State University of Iowa, Iowa City, Iowa. A note on hyponormal operators.

Tyushi Ando has recently shown that every hyponormal (TT* ;:; T*T) completely continuous operator is necessarily normal. This answers negatively the question raised in the last exercise of [S. K. Berberian, Introduction to Hilbert space, New York, 1961]. In this note another proof is offered, and a number of properties of completely continuous operators are deduced. (Received February 14, 1962.)

589-11. D. D. STANCU, 217 South Mills, Madison, Wisconsin. The remainder of certain linear approximation formulae for two variables.

One constructs a linear approximation formula, for two variables, of the following form n· . (1) T(f) = L.eo Lj~0 cj(f) + R(f), where Tis a given linear (additive and homogeneous) operator, which one approximates by the indicated sum of the linear operators c}. conveniently chosen. In the paper one makes a study of the remainder R(f), which also is a linear operator, defined by the formula (1). One gives general illustration of the established results by obtaining formulae of the following type: interpolation, Taylor, approximation by Bernstein polynomials, numerical partial differentiation and cubature. (Received February 19, 1962.)

589-12. ]. E. SIMPSON, Marquette University, Milwaukee 3, Wisconsin. On the nilpotent part of a spectral operator.

Certain results of McCarthy (Pacific J. Math. 9 (1959), 1223-1231) are here generalized to locally convex spaces. E is assumed to be a separated locally convex linear space over the complex field, which is barreled and quasi-complete. T is a spectral operator with associated spectral mea­ sure ~ which satisfies condition PC o> (see Tulcea, Bull. Amer. Math. 67 (1961), 125-128) so that T has the resolution T = S + Q, S being a scalar operator and Q a quasi-nilpotent one. The following rate of growth condition is equivalent to the one given by McCarthy when E is a . Definition. Let L(E,E) be given the topology, Cb of uniform convergence on the bounded subsets of E, RT(.\) is said to satisfy an mth order rate of growth condition if there is a positive integer m such that the set {d(z,o-)mRTo-(z)PJ' (o-): z ¢ o-, cr compaclj is bounded. Here, d(z,O") is the distance from z to cr. We have the following theorems: Theorem. If RT(A,) satisfies an mth order rate of growth condition, then Qm+2 = 0. Theorem. If also E is weakly complete, then Qm+1= 0. (Received February 19, 1962.)

107 589-13. R. H. BING, University of Wisconsin, Madison 6, Wisconsin. Improving the intersec­ tion of lines with surfaces.

It is shown that if Sis a surface in E 3, ab is a straight line interval, and e ?"0, then there is a b.omeomorphism h of E 3 onto itself such that ab•h(S) is at most finite, h moves no point more thane, and his fixed outside an a-neighborhood of ab. Unfortunately, in proving this theorem we used conse­ IJ.Uences of the Approximation Theorem for Surfaces which states that if S is a surface in E3 and f is a continuous non negative function on E 3, then there is a homeomorphism h of S into E3 such that h moves no point s of S more than f(s) and h(S) is locally polyhedral at h(s) if either f(s) > 0 or s is locally polyhedral at S. Had this theorem been proved without the Approximation Theorem for Surfaces, it would have eliminated the most difficult step in the proof of this Approximation Theorem. (Received E1ebruary 19, 196Z.)

589-14. G. E. BROWN, White Hall, Cornell University, Ithaca, New York. On commutators in a simple Lie algebra.

Theorem. Every element of a simple Lie algebra over the complex numbers can be expressed as a commutator. The main lemma established in proving this result is that for an element a of a simple Lie algebra L, there exists an automorphism of L mapping a into an element of E, the sub­ space of L spanned by the root vectors e,s for all nonzero roots {II. The proof of this lemma employs the nonsingularity of the Killing form and induction on the number of simple roots. The same proof is easily adapted to prove a similar result for analogues of the classical and exceptional Lie algebras over many other fields. A discussion is given showing for what fields this proof is valid. (Received E1ebruary 19, 196Z.)

589-15. B. G. CASLER, University of Wisconsin, Madison, Wisconsin. On the sum of two solid Alexander horned spheres.

Alexander described a simple surface, M, (a set homeomorphic to SZ) in s3 such that one com­ i>lementary domain, U, of M was not simply connected. Bing described a solid horned sphere as M + U, that is Alexander's horned sphere, M, together with the nonsimply connected complementary domain of M, U. Theorem. If a continuum is the sum of three mutually exclusive sets, M, u 1, and UZ such that there is a homeomorphism of M + ui (i = l,Z) onto a solid horned sphere where M is taken onto the boundary of the solid horned sphere and ui is taken onto the interior of the solid horned sphere, then the continuum is s3• (Received February 19, 196Z.)

589-16. P. C. HAMMER, Department of Numerical Analysis, 5534 Sterling Hall, University Jf Wisconsin, Madison 6, Wisconsin. Extended topology: Perfect sets.

If f is the function mapping set X into the set of all its limit points of order at most a, then the f-perfect sets may be defined as the fixed sets off, i.e., fX = X if and only if X is closed and self dense. Any such function f is isotonic (i.e., inclusion preserving). The generation of fixed sets Jf any isotonic function f mapping the class of all subsets of a space into itself is treated here.

[f uX =X() fX and gX = XV fX then for appropriate ordinals p and lu~'"= v, an interior function,

108 and g~ = h, a closure function. There follows: (1) {X: fX;;? XJ = fX: uX =X)= {vX}.

(2) £X: fX ~ X5 ={X: gX = X}= {hX}. (3) IX: fX = X} = fX: uX = gX =X} = !hvX} = £vhXJ. The classes (1), (2), (3) correspond, in the proper context, respectively to self-dense sets, closed sets, and perfect sets. A surprisingly detailed theory includes the pointwise shelling of a closed set to its maximal perfect subset and the dual incrementing of a self-dense set to its minimal perfect superset. An application of the proof of (3) gives Tarski's Theorem: Every isotonic function mapping a complete partially ordered set into itself has at ieast one fixed point. (Received February 21, 1962.)

589-17. L. J. SENECHALLE, University of Tennessee, Knoxville 16, Tennessee. A geometric characterization of closable linear transformations on a Hilbert space.

A linear transformation T from a Hilbert space H into H is continuous if and only if it has the following property: If Sis a closed subset of Hand if 0 is not inS, then 0 is not in T- 1(S). Closable linear transformations (a linear transformation is said to be closable in case it has a closed linear extension) possess a similar characterization. Theorem. A linear transformation T from a linear

subset of H into H is closable if and only if it has the following property: ~ S is a closed and bounde!i convex subset of H such that 0 is not in S, then 0 is not in T-l (S). (Received February 21, 1962.)

589-18. JOSEPH BATTLE, FRANK HARARY, and YUKIHIRO KODAMA, University of Michigan, Ann Arbor, Michigan. The genus of a graph is the sum of the genuses of its blocks.

Let G be a finite graph. A connected subgraph B is called a block if B has no cut point and B is a maximal subgraph with this property. It is well-known that any graph is decomposed uniquely into its blocks. By definition, the genus "Y(G) of G is the minimum number n such that G is imbeddable on a closed orientable 2-manifold Sn with genus n. Theorem 1. If G is a finite graph and {Bi:i = l, ••• ,kj is the block decomposition of G, then the equality 'V(G) = Lf=l'Y(Bi) hold's. An imbedding of Gin Sn is called a 2-cell imbedding if each component of Sn- G is an open 2-cell. For a 2-cell imbedding of G in Sn, we denote by 8 [G,SnJ the number of components of Sn - G. The number 8(G) is defined by J. W. T. Youngs as the minimum of 8 [G,SnJ for all possible 2-cell imbeddings. Theorem 2. If G is a connected finite graph and fBi:i = l, ••• ,k} is the block decomposition of G, then 8(G) = 1 - k +

~~= 18 (Bi). (Received February 23, 1962.)

589-19. JOSHUA CHOVER, University of Wisconsin, Madison, Wisconsin. On extensions of a positive-definite function from an interval.

Let m be a finite nonnegative measure on the line, and ~its Fourier Transform. For b > 0 let Sb be the span in L 2(m) of the functions eitx, it I< b. Theorem. A necessary condition that the positive-definite function~ have a unique positive-definite extension from the interval (-2a, 2a) is that Sa= L2(m). Corollary. If dm = fdx where j'log fdx/1 + x2 > - oo, then~ can have a unique extension from no finite interval. Theorem (W. Rudin). A sufficient condition for uniqueness of

extension from (- 2a,2a) is that Sb = L2(m) for some b < a. Construction. There exist~ which have unique extensions from every interval, but are not quasi-analytic or even smooth, e.g. Iil given by certain lacunary series. (Received February 23, 1962.)

109 589-20. ROBERT HELLER, JR., University of Houston, Houston, Texas. Sequence transforrna- tions and related continued fractions.

If a= {a 1, a 2, a 3, •••J is a complex number sequence, then f(a) = f£ 1, f 2, f 3, ...J denotes the sequence of approxirnants of the continued fraction 1/l + a 1/l + a 2/l + •... Let f*(a) denote the sequence of approxirnants of the continued fraction l + d 1/l + d 2d 3/l + d4d5/1 + ... , where d2p-l = -a2p-l/(l + a2p-l + a2p) and d2p = a2p/(l + a2p-l + azp); p = 1,2,3, .... This latter continued fraction, called the odd part of f(a), has sequence of approxirnants [f 1, f 3, f5, ···3· In this paper, a sequence transformation Tk, involving a nonzero complex number k, is defined such that if a is a complex number sequence and a.= Tk(a), then fp(a.) = l + k - k·f;(a); p = 1,2,3,. ... The sequences a and a. may appear to be of quite different character with respect to certain convergence theorems, but of course the above equation shows that the absolute convergence, convergence, or divergence of f*(a.) implies, respectively, the absolute convergence, convergence, or divergence of f*(a). With appropriate modifications in the definition of the transformation, analogous results are obtainable involving the even parts of f(a) and f(a.). (Received February 23, 1962.)

589-21. D. R. McMILLAN, JR., Florida State University, Tallahaseesee, Florida. Horneornor- phisrns on a solid torus.

Let J = {J 1, ..• , Jn} be a disjoint collection of polygonal simple closed curves in the boundary of a (triangulated) solid torus H of genus n. J is said to generate Tf1 (H) if there is a point p in Bd H and polygonal paths .L1, .•• ,)_n, in Bd H such that _ii joins p to Ji, meeting Ji only at the terminal point of .Li and missing the other J i' s, and the elements of Tf1(H,p) determined by the loops _L 1· j 1 ·]1 , ••. , Jn• in •in generate1T 1(H,p) (his a path corresponding to Ji and the bar denotes the reverse path). Theorem. If J and J* are two such collections each generating~(H), there is a piecewise linear homeomorphism of H onto H taking the elements of J onto those of J*. (Received February 23, 1962.)

589-22. F. P. PETERSON, 2-181, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. Some relations among Stiefel-Whitney classes of manifolds. Preliminary report.

Let H be the cohomology ring mod 2 of a compact differentiable n-dirnensional manifold. Define i i 1. i n-i uiEH andviE H byui•x= X(Sq)xandvi•x=Sqxforx€.H . J.F.Adarns,Onforrnulaeof Thorn and Wu, Proc. London Math. Soc. ll (1961), 741, shows that the Steenrod algebra operates on the right as well as on the left on H and shows that the "universal domain" U is a polynomial algebra on generators u1, u2, .••• Theorem. Let P E U. Then (P) X(Sqi) = -.::::-"'!_ u.•Sqi-j (P). Corollary. -- --- L-J-0 J - ui = Wi. Let Jn C H*(Gn; Z 2), where Gn is the Grassmann manifold of n planes in R 00 , be the ideal over the Steenrod algebra generated by vi for n/2 < i ;:;. n. Jn goes to 0 in any H considered above. Thus the tangent map f: M--->- Gn can be factored through Pn where Pn is a fibre space over Gn with k-invariants vi for n/2 < i ~ n. H*(Pn ;Z2 ) contains H*(Gn; z 2)/ Jn as a subalgebra. Any element of H*(P n; z 2) not in this subalgebra gives rise to a (possibly) new characteristic class of M. Theorem.

There exist such elements corning from the following relations in H*(Gn; z 2): x 2(L_f;01sq2iv4k-l-2i) """k-1 2i+l ~r i = 0 ~n = 4k or 4k + l, X 2 (L-i=OSq v4k-zi) = O~n = 4k + 2 or 4k + 3, and X1(L..i=OSq vn-i) = 0 2 1 for n = 2r + l, where x2 = Sq + v 1 u Sq + v 2 u and x 1 = Sq1 + v 1 u. (Received February 23, 1962.)

Ito 589-23. L. B. TREYBIG, 608 jefferson Park Avenue, New Orleans 21, Louisiana. A certain nonmetrizable Hausdorff space.

In this paper an example is given of a totally ordered, homogeneous, heriditarily separable, compact Hausdorff space which is not metrizable. Two preliminary theorems are proved. Suppose L is a totally ordered, connected topological space such that (1) the topology of L is the order topology, and (2) L satisfies the first axiom of countability. Theorem 1. If M is a compact, totally disconnected subset of L, then M is metrizable if and only if the set of all components of L-M is countable. Theorem 2. If M is a separable subset of L, then M is heriditarily separable. (Received February 23, 1962,)

589-24. L. M. BLUMENTHAL and R. j. BUMCROT, University of Missouri, Columbia, Missouri. Betweenness relations in lattices. Preliminary report.

Let L be a lattice and let M be any subset of L. Denote by L(x,y, ... ) the sublattice of L generated by x,y, ... E L, by PM(x,y, ... ) the set of all elements in M, lattice contained in any element of fx,y .... J. and by IPM(x,y .... ) the set fx,y, ...J together with all elements of M which are covered by some element of fx,y .... J. Pitcher and Smiley [Trans. Amer. Math. Soc. 52 (1942), 95-114} investi­ gated transitivities of an arbitrary betweenness and characterized a lattice betweenness G(a,b,c): ab + be = b = (a + b)(b + c). In this paper a number of other definitions of lattice betweenness are investigated and compared, and a characterization is obtained for the betweenness G*(a,b,c):

ab +be= b = b + ac, and for its dual G**. Denoting by t 1, t 2, t 3 the four-point transitivities studied by Pitcher and Smiley, we prove: A triadic relation R defined on LX LX L .k_G* if and only if (i) R(a,b,c)- R(c,b,a), (ii) R(a,b,c) and R(a,c,b)- b = c, (iii)_!! R(a,b,c) holds, then R has transiti­

.Y.!!Y t 1 in PL'(a,b,c), where L' = L(a,b,c), (iv) R(a,b,c) _,. ac l! b, (v) a§ b;;;. c- R(a,b,c), (vi) R(a, a+ c, c) and R(a, ac, c) hold for all a, c E L, (vii).!!:_ G*(a, b, c) holds and L' is modular

then R has transitivity tz in PL'(a,b,c), while if L' is not modular, then R has transitivity t 3 in IP L '(a,b,c). (Received February 26, 1962.)

589-25. L. M. BLUMENTHAL and W. A. KIRK, University of Missouri, Columbia, Missouri. Metric characterization of Gauss surfaces. Preliminary report.

A metric quadruple Q has an imbedding curvature k(Q), positive, zero, or negative, provided it is congruent with a quadruple of the convex two-sphere, the euclidean plane, or the hyperbolic plane of Gauss curvature k, respectively. According to Wald, a metric space M has at an accumulation point p curvature K (p) provided (i) no neighborhood of p is linear, and (ii) corresponding to each e > 0 there is a S >- 0 such that each quadruple Q of M with distance from p less than S has an im­ bedding curvature k(Q) with IK(p) - k(Q) I< e. Weakening Wald's definition by restricting its appli- cation to quadruples containing a linear triple, it is proved that a Gauss surface is characterized among all metric spaces which are locally compact, convex, two-dimensional manifolds, by the property of possessing at each point a curvature in the weaker sense. At each point this curvature equals the classical Gauss curvature. (Received February 26, 1962.)

Ill 589-26, C. H. EDWARDS, JR., University of Wisconsin, Madison, Wisconsin, Factorization of compact 3- and 4-manifolds. Preliminary report,

R. H. Bing has shown that a set is a 3-cell if its cartesian product with an interval is a 4-cell. This theorem is generalized to prove that, if the product of M and an interval is a compact 4-manifold, then M is a compact 3-manifold, In regard to the question as to when a cartesian factorization of a compact 3- or 4-manifold is unique, the following results are obtained. (1) If the compact 3-manifold M3 is not a cube with handles, then M3 has a unique prime factorization. (2) Let A X C and B X C be two cartesian factorizations of the compact orientable 4-manifold M4 into 2-dimensional factors, If C is not a punctured 2-sphere (a 2-sphere minus the union of the interiors of finitely many mutually disjoint closed 2-cells), then A and B are homeomorphic, (Received February 26, 1962,)

589-27, T. J, HEAD, The University of Kansas, Lawrence, Kansas, Certain subdirect products of simple groups. Preliminary report,

By a boolean ring of sets we mean a family of subsets of a set which is closed under symmetric difference and finite union. Theorem. For the lattice L of normal subgroups of a group G to be isomorphic with the lattice of ideals of a boolean ring of sets it is necessary and sufficient that G have the two properties: any two finite direct decompositions of G have a common refinement, and for each X E G the subgroup X= n{N € L lx E N} is a direct summand. The proof of sufficiency is carried out by means of a representation of Gas a subdirect product of simple groups fSal a fA which is constructed such that L is isomorphic to the lattice of ideals of a boolean ring of subsets of A.

Corollary. If fSa1 a fA is a family of noncyclic simple groups then the lattice of normal subgroups of G = TTa eASa is isomorphic to the lattice of ideals of the boolean ring of all subsets. of the set A if and only if X: is a direct summand of G for each x E: G. (Received February 26, 1962,)

589-28. G. W. GOES, University of Western Ontario, London, Canada. Some spaces of Fourier coefficients,

With the usual notations and f = :::£~ 1aj cos jt, g = L~ 1 aj sin jt, Ak = L:j:ka/j, dV the space of Fourier Stieltjes series and EN the space off E E for which·the partial sums off converge strongly in the E-, it is proved: (i) aj ~0 does not imply f E dV, (ii) aj to or L~ 11aj - aj+ 1 I<. oo with aj -> 0 implies h € dV if and only if h E L, for h = f or h = g, (iii) aj-> 0 and supj j(log j)¥1 Aaj I

< oo for Y' ) 1 imply f E LN' (iv) supj lj aj I < oo implies 2:: 1Ak sin kt E L 00 , (v) jaj ~ 0 implies L:,l Ak sin kt € CN. The proofs are performed mainly with functional analytic methods. (Received February 26, 1962,)

589-29, WALTER RUDIN, University of Wisconsin, Madison 6, Wisconsin. The extension problem for positive-definite functions.

Let Q be ann-dimensional cube in Euclidean n-space R n, putS = Q - Q, and call a continuous complex function f positive-definite on S if~f.j=l.cicj f(xi- xj) Ji: 0 for all constants cl'"''cm and all points xl'"''xm in Q. Can every positive-definite function on S be extended to a positive-definite function on Rn? For n = 1, the answer is affirmative (M. G. Krein), It is shown in the present paper

112 that the answer is negative for all n ~ 2, The proof depends on the theorem of Hilbert which states that there exist positive polynomials in 2 real variables which are not sums of squares of polynomials. The analogous (true) result for trigonometric polynomials shows that the answer to the above question is negative if R n is replaced by the group of all lattice points in the plane, and an interpolation proce­ dure us used to derive the answer for the plane. The case n > 2 follows trivially. (Received February 26, 1962,)

589-30, E. 0. ROXIN, 7212 Bellona Avenue, Baltimore, Maryland. An example of a measure­ able control function which is not piecewise continuous,

We consider the differential equation i = f(x,t,u) where xis a vector in Rn, f(x,t,u) satisfies the conditions for existence and uniqueness of solutions and u = u(t) is a control function taking its values in a compact set U such that the image f(x,t,U) is convex. The basic statements concerning such systems (for example the Caratheodory conditions) suppose measurable functions; on the other hand in control theory it is normally admitted that the control function u(t) is piecewise continuous. Here we construct a measurable control function such that the corresponding trajectory of the differ­ ential equation in the interval 0 ~ t ;!5 1 has the property that it is not differentiable on a dense subset of this interval. The control function is, therefore, discontinuous on that dense subset, This example is interesting because it shows the kind of difficulties encountered in proving differentiability condi­ tions of trajectories, defined in an axiomatic and direct approach to control theory, without direct reference to the control function u(t). (Received February 26, 1962,)

589-31. T. L. McCOY, 7834 S. Coles Avenue, Chicago 49, Illinois. On partial sums of entire functions. Preliminary report.

Some results of the author's thesis (University of Wisconsin, June 1961) are described. In the sequel, f(z) = 1 + Ljajzj is an entire function, Pn(z) the partial sum of degree n, and !lr and 'Y are positive numbers independent of n. Theorem 1. If for all n, Pn(z) is free of zeros in IArg z - !!In I~ 1,6-, then for some A > 0 and n > n0 , there is lan I ~ exp. f- A n(log n)(log log n)j. Theorem 2. If for all n, IPn(z)l;:; rlanznl for IArg z - !ilnllli l/f, then for some A> 0 and n >no, lanl <. exp, r- An log2 n}. If in addition !!In 50 and the aj are positive real numbers satisfying the inequality anan_2 ~ a~_ 1 , then an= O(exp. - Bn2 ), for some B > 0. Definition. The region R is admissible if whenever zj, j = 1,2, ... ,n lie in R, we haveL_j= 1 lzjlm!!! K Maxl1!a!!m1L:.j=1zj lm/a, m and K depending only on R. Theorem 3, If for all n, an admissible region R, and a sequence fcn1 of complex numbers, the zeros of Pn(z) are contained in cnR, then an= O(exp, - Bn2) for some B > 0, Theorem 1 and the first part of Theorem 2 hold under somewhat weaker conditions on the distribution of zeros of the Pn (z). (Work supported in

part by the Office of Naval Research, Contract N 7onr 28507, J. Korevaar, principal investigator.) (Received February 26, 1962,)

113 589-32. ARUNAS LIULEVICIUS, The Institute for Advanced Study, Princeton, New Jersey. Stable homotopy of projective spaces.

Let RP 00 be the infinite-dimensional real projective space. Let snRP 00 be the nth suspension of RP 00• The groups 1Tn+k(SnRP00) for n > > k are as follows fork< 10: k = 0, 0; k = I, z 2; k = 2, z 2 ; k = 3, z 8; k = 4, Zz; k = 5, 0; k = 6, Z2 ; k = 7, z 16 6) Z2 ; k = 8, z2 ED z 2 ED z 2 ; k = 9, z 2 6) z 2 ED z 2 ED z 2. This is obtained from the Adams spectral sequence for RP. It is shown that the E 2 -terms of the Adams spectral sequences for RP00 , cpOO, HP00 (C = complex numbers,

H = quaternions) are themselves related by spectral sequences associated with the homomorphism a.:

A- A if p = 2 (A is the Steenrod algebra over Z 2, a. is the dual of the squaring map in A*). (Received February 26, 1962.)

589-33. W. W. TURNER and ALFRED LEITNER, Michigan State University, East Lansing, Michigan. A method of generating integral representations. Preliminary report.

Imposing the condition that the Schrodinger equation 'i7 2u + !llu = 0 be simultaneously separable in at least two coordinate systems sharing a coordinate, one obtains functional equations whose solution completely determines !II. The special functions obtained by the separated ordinary operators can be related through integral relations by using a well- known integral theorem. {1;ee Ince, Ordinary differential equations, p. 186]. With this theorem one can predict the value of the integral involving special functions. Thus a unified theory of these integral representations is obtained. For example consider the simultaneous separability in rectangular and cylindrical coordinates. The form assumed . 2 2 2 2 2 2 2 2 2 2.2 by Ill 1s !II= k + a 1(x + y) + a 2;x + a 3/y = k + a 1p + a 2/f cos + a3/f sm. The separated solu- tions in this case yield three Whittaker functions and a hypergeometric function [see Leitner and Meixner, Arch. d. Math. 10 (1959), 452-459]. The theorem then gives this hypergeometric function as an integral of three Whittaker functions of arbitrary parameters. In a similar manner pairs of the following coordinates have also been considered for simultaneous separability: rectangular, cylindrical, spherical, parabolic cylindrical, paraboloidal, elliptic cylindrical, prolate spheroidal, oblate spheroidal and spheroconal. (Received February 26, 1962.)

589-34. E. T. KOBAYASHI, Northwestern University, Evanston, Illinois. A remark on the Nijenhuis tensor.

Consider a vector 1-form h on a manifold M, whose classical canonical form over the reals at all points on M is equal to a fixed matrix p.. The frames z at x such that z -lhxz = p. define a sub­ bundle of the frame bundle over M. The subbundle is called the G-structure defined by h. We find, that if each block of p. corresponding to each elementary divisor of p. is semisimple or nonderogatory, then the vanishing of the Nijenhuis tensor of h, i.e., [!t,h] = 0, gives the integrability of the G-structure. (Received February 26, 1962.)

114 589-35, ARNOLD SEIKEN, 111 Texas, Rochester, Michigan. On tensor connexions.

We show that a tensor connexion of order p on a C00 manifold M determines a connexion on a principal fibre bundle B of dimension n2P + n, and conversely. For p = 1, B is the frame bundle, In addition, we obtain (i) curvature and torsion forms for the connexion in B, (ii) structure equations for these forms, and (iii) integrability conditions for the structure equations. (Received February 26, 1962.)

589-36. F. G. ASENJO, Southern Illinois University, Carbondale, Illinois. Relations irreducible to classes.

This paper formalizes Whitehead's concept of internal relations. A formal system is described in which "term variables" and "relation variables" each constitute a different domain for predicate formulae. Formation rules for "terms" and "relations" are given, For example: "If x 1 and y 1 are terms and x 2 is a relation, then x 1x 2y 1 is a term." Equality of terms (and relations) is defined, also the order of a term (or relation), according to the number of formal term variables (or formal relation variables) included. Postulates are given and theorems proved to outline a "term-relation formal number theory." This theory provides an arithmetic of "term-relation numbers" in which elementary operations can be performed, satisfying some of the formal laws of arithmetic. Finally, the consis­ tency and interpretation of this term-relation number theory is discussed, (Received February 26, 1962.)

589-37. K, W. KWUN and FRANK RAYMOND, 210 North Hall, University of Wisconsin, Madison, Wisconsin, Joins of topological spaces.

The Join A o B of two spaces A and B is obtained from A X B X I by identifying each a X b X 0 to a € A and each a X b X 1 to b E B. The well-known typical examples are IP" Iq = Ip+q+l = IP .. Sq and S Po Sq = sP+q+l. It is shown that this is the complete picture of the join constructions at the level of generalized manifolds. Let A and B be any two compact spaces, Theorem. A o B is a generalized manifold if and only if each of A and B is a spherelike generalized manifold, Consequently, A " B must be spherelike, Theorem, A '" B is a generalized manifold with nonempty boundary if and only if either both A and B are generalized cells or one is a generalized cell and the other a sphere­ like generalized manifold. Consequently, A o B must be a generalized cell, As a corollary, one obtains a characterization of the n-sphere in terms of join decomposition. Theorem. For each n !; 1, a combinatorial n-manifold is Sn if and only if it is the join of two spaces. The main tool here is the factorization theorem for generalized manifolds. Examples are constructed to show that for each n il: 7 neither join factor of ann-manifold need be locally euclidean. (Received February 26, 1962.)

589-38. E. E. GRACE, University of Wisconsin, Madison 6, Wisconsin. A totally nonaposynde­ tic, compact, Hausdorff space with no cut point.

This paper gives a negative answer to the following question posed by A. D. Wall ace. Does each totally nonaposyndetic, compact, connected, Hausdorff space contain a weak cut point? In describ­ ing the example it is convenient to assume the continuum hypothesis. Let (1) K be a perfect, totally

115 disconnected, compact, Hausdorff space, consisting of c points, such that each subset of cardinal less than c is nowhere dense; (2) S be the union of two long lines L 1 and L 2 with common end points a and b (on the "long" end); (3) h be a homeomorphism from L 1 onto L 2; (4) d 1,d2 , .•• ,dt•···• fort< n, be a nowhere dense subset of L 1 - a U b (whose closure contains b), such that {du lu ~ t} is closed for t < n, (5) k 0 ,kp ..• ,kt, .... , fort< n, be a most economical well-ordering of K; and (6) Kt = Uu:otku· Let H be the quotient space of K X S determined by the weakest equivalence relation = consistent with (1) (k,b) = (k',b), for k,k' E K, (2) (k,b) =(k 0 ,a), fork E K, (k0 ,d) =(k 0 ,h(d)), ford E L 1, and (3) (k,dt) = (k',dt), fork, k' E Kt and fort< n. (1) alone would make the space a cantor fan of "long circles". Adding (2) makes the space nonaposyndetic at the vertex of the fan. Adding (3) prevents weak cutting by the vertex while retaining the desired nonaposyndesis. (Received February 26, 1962.)

589-39. R. J. KOCH, Louisiana State University, Baton Rouge, Louisiana and L. F. McAULEY, University of Wisconsin, Madison, Wisconsin. Semigroups on certain continua ruled by arcs.

We prove that a dendrite (acyclic Peano continuum - also called a tree) admits the structure of a topological semigroup with zero and unit. Our techniques show that the elements of a larger class of continua admit such a structure. We term this class "ruled continua." These continua are characterized by eight conditions which describe a ruling by simple arcs from a point 0 which satisfy suitable convergence conditions and their end points (different from 0) possess an ordering which may be neither a natural ordering nor a continuous one. The class of "ruled continua" includes such continua as n-cells, the Cantorian swastika, and infinitely many cones with a common vertex whose pairwise intersection is an interval [a,b]. (Received February 27, 1962.)

589-40. DAVID SCHROER, University of Rochester, Rochester 20, New York. Commutativities involving replacement and substitution.

Using definitions, techniques, and results developed as indicated in Abstracts 578-51 and 587-44 (especially, those involving the characteristic sequence of a nested pair), precise expression and smooth proofs are given for about twenty elementary facts about replacement and substitution in languages with quantifiers (conceived set-theoretically), the particular facts chosen being those con­ cerning operative commutativity which prove useful in deeper investigations to be reported at a later date. Simple sample: Let F, A, D, I, J, A', D', I' be well-formed occurrences such that (A,F),

589-41, MAREK FISZ, 220 West 104th Street, Apartment 12, New York 25, New York. On the orthogonality of measures induced by L processes.

Let txi(t), t E (O,l]j (i = 1,2) be centered L processes with no fixed points of discontinuity and with Xi(O) = 0, For any t, Xi(t) has an infinitely divisible distribution. Denote by Hi(t,u) the Levy function of the canonical representation of the characteristic function of Xi(t), and by H{(t,u) the lefthand and righthand derivatives in u of Hi(t,u) for u < 0 and u > 0, respectively. Finally, let

116 Px and Px be probability measures induced in the space of real functions by the process x1 (t) and 1 2 x2(t), respectively. Conditions for Px 1..L Px 2 are given in terms of H{(t,u). It is shown, in particular, that if x 1(t) and x2(t) are stable and, for some t0 E (0,1], H1(to,u) ;f. H2 (to,u), then Px 1 j_ Px 2• (Received February 27, 1962.)

589-42. CHARLES GODINO, Notre Dame University, Notre Dame, Indiana. Outer automor­ phisms of cyclic extensions of abelian p-groups.

The author proves that if a finite p-group G is a cyclic extension of an abelian subgroup N, then G has an outer automorphism. In establishing this result, the main theorems proved for finite p-groups are: (1) If the center Z of G is not contained in the commutator subgroup G', then G has an outer automorphism (cf. Abstract 61T-198, Notices Amer. Math. Soc., August 1961). (2) If G is the semi-direct product of a normal abelian subgroup A and any other subgroup 8, then G has an outer automorphism. (3) Let N be a normal abelian subgroup of G such that G/N is cyclic of order pk k generated by the coset xN. Clearly, xP = z where z E Z n N. Now, if z is ~an element of maximal order in Z n N, then G has an outer automorphism. (For p = 2, the extra condition exp Z n N > 2 is required.) Note: It can be shown (as a simple consequence of (1)) that, if G is a cyclic extension of a non-abelian subgroup N such that Z is not contained in the center of N, then G has an outer auto­ morphism. (Received February 27, 1962.)

589-43. NORMAN HOSAY, University of Wisconsin, Madison 6, Wisconsin. Conditions for tameness of a 2-sphere which is locally tame modulo a tame set.

Theorem. Let S1 be a tame 2-sphere in E 3 and S2 a 2-sphere whose intersection with s1 is connected and nondegenerate such that (i) Sz is locally tame modulo s1 • s2, and (ii) s2 lies in the closure of one complementary domain of s1• Then s2 is tame. Definition. A set A is said to have large components if there exists a positive number b. such that no component of A is of diameter less than A. The same method of proof allows the above theorem to be generalized to include the case where S1 • S2 has large components. Corollary. Let S 1 be a tame 2-sphere in E3 and S2 a 2-sphere which is locally tame modulo s 1 • s2 such that (i) the components of s1 • s2 are large, and (ii) no com· ponent of SI" s2 separates s1. Then s2 is tame. The above results generalize to closed 2-manifold in 3-manifolds. (Received February 27, 1962.)

117 The April Meeting in Atlantic City April 16 -19, 1962

590-1, S. A. KHABBAZ, Lehigh University, Bethlehem, Pennsylvania. Decomposability in abelian groups. Preliminary report.

Theorem. Let G be any abelian group, let T be its maximal torsion subgroup, let B be any basic subgroup of T, let H be any subgroup of G such that the quotient group G/H has the form

G/H ; (tDxE'xfa xl> tD (tDY EYfcy}) Ill M where tD indicates the weak direct sum, each fax} is a nonzero cyclic group of finite order generated by ax, each {cy} is an infinite cyclic group generated by cy and where X andY are index sets and assume in addition that each element of H is divisible in G by each power of each prime relevant to (DxEXfaxt• Then G may be written in the form G; (tDxExfbx}> @ (Eily EY { dy} )@ N where each bx is a nonzero element of B whose order is a power of a prime relevant to @xEX{ax} and each dy is an element of G having infinite order. When G is taken to be a reduced primary group, the above theorem yields a result quite a bit stronger than an affirmative answer to problem 17 in L. Fuchs' book Abelian groups. (Received September 11, 1961.)

590-2, ]. H. BRAMBLE and B. E. HUBBARD, Institute for Fluid Dynamics and Applied Mathe­ matics, University of Maryland, College Park, Maryland, On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation.

In this paper we obtain some estimates of the type given by Gerschgorin. A general theorem is stated which can be used as a guide in the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation. Various examples are given and analyzed, included among these are analogues whose truncation error is shown to be O(h4). (Received November 18, 1961.)

590-3, ]. H. BRAMBLE and B. E. HUBBARD, University of Maryland, Institute for Fluid Dynamics and Applied Mathematics, College Park, Maryland. Error bounds in the finite difference solution of the Dirichlet problem for Poisson's equation.

A certain finite difference analogue of the Dirichlet problem for Poisson's equation in the plane is formulated, Bounds on the difference between the exact solution and the approximate solution are obtained in terms of square integrals of the data and derivatives of the data. If the boundary and the exact solution are sufficiently smooth the bounds are O(h2), where h is the mesh size. (Received November 18, 1961,)

590-4, S. W. GOLOMB, 4800 Oak Grove Drive, Pasadena, California. Error-distributing codes and rook domains.

The cells of a hyper-chessboard of size nk are to be colored in a maximum number of colors w(n,k) such that a rook placed anywhere will cover (attack or occupy) at least one cell of each color. Properties of the combinatorial function w(n,k) are derived. The cells of the hyper-chessboard are

118 coordinatized by k-tuples from ann-symbol alphabet, and all k-tuples (code words) corresponding to a given color are synonyms. The required property is that given any code word, it can be mutated into some member of any specified synonym class by a simple mutation (i.e. a change in only one coordinate). Such a code may be termed error-distributing or error-inducing. The error-inducing property is related to the single-error correcting property by the relation w(n,k) s(n,k) € nk, where s(n,k) is the size of any single-error-correcting dictionary of k-letter words over ann-letter alphabet. More specifically, if one has a single-error-correcting group code of s(n,k) words, its coset space may be taken as the collection of synonym classes of an error-inducing dictionary. The concept of "rook-domain packing" (the extension of "sphere packing" to the nonbinary case) is also appropriate in defining latin squares, and in showing their relationship to single error detecting codes. (Received January 19, 1962.)

590-5. J. A. MORRISON, Room 3D-278, Bell Telephone Laboratories, Murray Hill, New Jersey. On the commutation of finite integral operators, with difference kernels, and linear self- adjoint differential operators.

Let K (u(t)} = J'~ f (t - s)u(s)ds. It is shown that the general analytic kernel for which there

exists a second order linear self-adjoint differential operator which commutes with K is p(t) = (b/c)cosech bt sin ct, omitting a multiplicature constant. The corresponding differential operator is L [u(t)] = (d/dt) IT1-cosech2b sinh2bt)du/dtJ + [X-(b2 + c 2)cosech2b sinh2bt]u. If u(t) is a continuous

solution of Lu = 0 in the closed interval[- 1,1) and u(- t) = ± u(t), then K[u(t)] = A.u(t), for some A.

The equation Lu = 0 may be reduced to Heun's equation. It is also shown that, if u(- t) = ± u(t) and u(l) = 0, there is a fourth order self-adjoint differential operator which commutes with K when p(t) = b cosech bt (q sin at cos ct + r sin ct cos at). A parameter 'Yin the differential operator must

satisfy the condition u 11 (1) = yb- 1 tanh b u'(l). This generalizes the results obtained for the limit­ ing case of b = 0 (see Abstract in Notices Amer. Math. Soc., February 1962). (Received January 26, 1962.)

590-6. RAFAEL AR TZY, Rutgers, The State University, New Brunswick, New Jersey. Net motions and quasigroups.

A loop (L, ) can be considered as one of the loops of a 3 -net. Permutations of the net line families were used for defining new operations over L [Artzy, Proc. Amer. Math. Soc. 11 (1960), 847-851]. The resulting loops are mapped onto each other by isostrophisms which form a group R. The isotopisms that carry a loop of the net into another loop are analogues of euclidean plane trans- lations, while R is an analogue of the group of rotations through 7r/3 about the origin and reflections in lines through the origin. Therefore products of isostrophisms and loop-isotopisms will be called net motions. In analogy with the euclidean plane where the translation group is normal in the motion group, one obtains the Theorem: R is a homomorph of the net motions, the kernel consisting of loop isotopisms. The direct proof is loop theoretic. An alternative proof uses a theorem about conjugates of a quasigroup [for definition cf. Stein, Trans. Amer. Math. Soc. 85 (1957), 228-256) mapped onto each other by parastrophisms. Theorem. The isotopism group of a quasigroup Q is normal in the group generated by the parastrophisms and the isotopisms of Q. (Received January 31, 1962.)

119 590-7. j. R. RICE, Research Laboratories, General Motors Corporation, 12 Mile and Mound Roads, Warren, Michigan. Strict approximations.

There are situations where best Tchebycheff approximations fail to be uniquely determined. This occurs in particular for approximation to functions of several variables. For a function f(x) defined on a finite point set in En one may define the strict approximation. The definition involves too many unfamiliar items to be given here. If the best Tchebycheff approximation is uniquely determined then this is the strict approximation, if not then the strict approximation is one of the best Tchebycheff approximations. One may establish Theorem 1. Let f(x) be a function defined on a nondegenerate finite subset of a Banach space B and let L be an n-dimensional linear subspace of the space of

continuous functions on B. Then (A) f(x) possesses a strict approximation L(A*,x)~L. (B) The strict approximation is unique. (C) L(A*,x) is the strict approximation to f(x) if and only if the set of extremal points of L(A * ,x) - f(x) contains a strict critical.££!nt set of dimension n. A method of ascent algorithm of the 1 for 1 exchange type is described which generates a sequence {L(Ak,x)} of approximations. One may establish Theorem 2. The sequence {L(Ak,x)} converges to the strict approximation L(A * ,x) .!£ f(x) in a finite number of steps. (Received February 5, 1962.)

590-8. H.-C. WANG, Institute for Advanced Study, Princeton, New jersey. On the space of lattices in a Lie group.

Any connected and simply-connected Lie group G takes the form G 1 X G2 with G1 semi-simple and G2 no simple normal subgroup. By a lattice in G, we mean a discrete subgroup of G with compact quotient. Let A be a lattice in G, .Gj((A,G) the space (with c - 0 topology) of all isomorphisms r:

A~ G such that r(.A) is a lattice, and ..Gt'1 (A ,G) the connected component of tJr (.II., G) containing the identity map .A-+ G. Let -n-i: G -ai be the projections (i = 1,2), S a semi-simple part of G2, N the maximal connected nilpotent normal subgroup of G2, x/ the group of inner automorphisms of Gz induced by elements of S, and ?f' the e-component of the group of automorphisms of the nonconnected Lie group AN. Assume that the semi-simple part of G has no compact factor. The following are

proved: "(1) 1?"1 (A) is a lattice in G1. (2) -'(1 (A,G) consists of all r: A -a of the form r(~) = (r11rl(~). i"Cc.l1r'Z(A.)) where ), EA. r1 E Gt\(1r1(.t1), G1), t' E ;/. cuE '!J-r. (3) The product Cr\ (71'"1 (.t\.), G1) x;J X 'Jr is a covering space of .'(1 (A, G) with finite number of leaves." Thus the problem of deformation of lattices in G is reduced to that in the semi-simple G1 which has been solved completely by A. Weil. (Received February 5, 1962.)

590-9. j. M. KISTER, University of Michigan, Ann Arbor, Michigan and L. N. MANN, University of Virginia, Charlottesville, Virginia. Equivariant imbeddings of compact abelian Lie groups of transformations.

Mostow (Annals 57) has shown that every action of a compact Lie group with a finite number of nonconjugate isotropy subgroups on a finite dimensional seharable metrizable space can be equivari­ antly imbedded in a linear action of the group on some Euclidean space. However the required dimen­ sions of the Euclidean spaces are not explicit in Mostow's proof. In this present paper the authors show the following for compact abelian Lie groups: Let G be such a group operating with t distinct

120 nontrivial (G and f e} excepted) isotropy subgroups on a locally compact separable metrizable n-dimensional space X, Suppose G = Tm Ill N1 Ill ... Ell Nr Ell H 1 Ell ... (j) Hs where Tm is an m-dimen­ sional toroid, theN's are cyclic groups of order i 2 and the H's are all of order 2. Then (X,G) can be equivariantly imbedded in a Euclidean space of dimension (t + 1) { (n - m + 1) (m + r + s)} + 2n + 1 for n - m odd and of dimension (t + 1) [

590-10, A. A. GOLDSTEIN, 26-259 Massachusetts Institute of Technology, Cambridge, Massa- chusetts. Cauchy's method of minimization.

In the following, let ~k = ~(xk) denote the gradient of f at xk, and let the norm of a matrix A be given by IIA II = sup liz II= 111Az II. Let Q(x) denote the Hessian matrix of f at X, Theorem. Let f be a real valued function on En. Given x 0 in En, assume f(x0) defined and f in c 2 whenever f(x) ~ f(x0 ).

Assume the setS= fx: f(x);; f(x0 )} is bounded. Choose fio to satisfy suPxES IIQ(x)ll ~ 1/,.0o...:. oo. Assume either that fk satisfies f(xk - flk~k) 2 f(xk - f~k) whenever 0 ~ fi ~ oo, (steepest descent) or that for arbitrary S, 0 "" ll :§ flo• flk satisfies S :;ii fk 2fo - a (method of gradients). Let xk+1 k = xk - fk~k· Then (i) a subsequence x n converges to a point z E S such that ~(z) = 0, (ii) f(xk) con- verges downward to f(z). (iii) If z is unique, xk converges to z. (Received February 12, 1962.)

590-11. M. S. K LAMKIN, Avco Research and Advanced ~velopment Division, Wilmington, Massachusetts. A heat flow problem with a multiplicity of steady-state solutions.

In the problem considered here, i,e., 8T/8t = 82T/fJx2, 0 ""x..:. 1, t > 0; T(1,t) = 0; f BT /3x = - F (T)} x=O' F (T) ~ 0; there is a multiplicity of steady-state solutions, This multiplicity arises due to the sign of F (T). In the usual heat flow conduction problem, F (T) would be negative

indicating that heat is being lost to the surroundings as a function of one end temperature. Here

heat is being gained due to interaction with an electric field, It is immesiately apparent that at least

two steady-state solutions exist (for a wide class of F) and are given by T 1 = 0, and T 2 = ~ (1 - x) where A is a nonzero root of II.= F(>.). A particularly interesting case occurs when F(T) = T. For this case, 1\. can have any value and furthermore the corresponding transient problem can be solved analytically, It is shown that the appropriate value of A is given by 3 J6G(x)(1 - x)dx where G(x) represents the initial temperature distribution in the transient problem, This result has been verified numerically on a digital computer for various functions G. (Received February 9, 1962,)

590-12, REUBEN HERSH, 295 Rutland Avenue, Teaneck, New jersey. Mixed problems in several variables.

We solve the mixed initial-boundary value problem in the quarter-space t :e; 0, x 5; 0, for a first-order system of differential equations with constant, not necessarUy symmetric, matrix co­

efficients, Ut + AUx + 2::BiUyi + CU = 0, det At- 0, The problem is well-posed for smooth initial data

121 if and only if (1) for real ~. 17• the matrix tA + L 'liBi has real (not necessarily distinct) eigenvalues, and (2) on x = 0 the solution is required to lie in a linear space N of dimension equal to the number of negative eigenvalues of A, such that, for all imaginary 1J and for all t' with sufficiently large real

part, N is free of linear combinations of generalized eigenvectors of A -1( '!;' + 2:'7jBj) corresponding to eigenvalues with negative real part, This condition is similar to one used by S. Agmon in a theorem concerning single higher-order equations, The proof is by a Laplace transformation in t, Fourier transformation in y, and then solving an ordinary differential equation in x. Solutions are arbitrarily

smooth if the data are sufficiently smooth, For C = 0, the solution grows at most polynomially in t, An application to Maxwell's equations is given, (Received February 13, 1962,)

590-13. IWAO SUGAI, 500 Washington Avenue, Nutley 10, New Jersey, Riccati's equation with loosely inter-relatell coefficients.

In the past, several transforms which linearize a generalized Riccati's equation having three independent variable coefficients have been reported. In order to solve the linearized differential equation, these transforms required certain inter-relationships among the variables, for example, where one of the three variable coefficients is the derivative of the division between the two remain-

ing variable coefficients, Recently, a simple transform was found. It solves a generalized Riccati's equation when one of the three variable coefficients is either sum or difference of the remaining two variable coefficients. (Received February 16, 1962,)

590-14. J, F. TRAUB, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey, Optimal m-invariant iteration functions.

Let the iteration function !6(x) = G(x,f(x), f'(x), •.• ,t

multiplicity m of f(x). If !6(x) is of order p, then !6(x) E j+1 Ip. If j = p - 1, then {IJ'(x) is optimal. If {IJ'(x) E: j+1Ip form= 1, while !6(x) E j+1I 1 form :> 1, {IJ'(x) ism-linear variant. If {IJ'(x) E j+lip for all m ~ 1, !6(x) ism-invariant. Let !6(x) = PE(x) = x -Zj=lYj(x)uj(x), where Yj(x) =- [(- l)j/j!J ·~'(x)]j g(j)(y), x = g(y), u(x) = f(x)/f'(x), s = p - 1. An explicit expression has been derived for theY j(x). Then pE(x) is (i) optimal for m = 1 only, (ii) m-linear variant, and (iii) (pE(x) - a)/(x - a)

= [(- l)s /(s!m s)] TTJ= 1 (1 - ..fm) + O(x - a). Coefficients f's,j(m) are sought such that pe (x,m) = x- Lj=1f-s,j(m)Yj(x)uj(x) is optimal and m-invariant. The general solution of this problem has been found, The fs,j(m) are polynomials in m of degree s, with coefficients depending on Stirling

numbers of the first and second kind, In particular, 2e (x,m) = x - mu(x), (modified Newton iteration function), which is the only previously known member of the family pG (x,m). The Ps,j(m) have certain interesting properties. (Received February 20, 1962.)

590-15. LOUIS DE BRANGES, 778 Upper Gulph Road, Wayne, Pennsylvania. Invariant sub­ spaces of linear transformations in Hilbert space.

The existence of an invariant subspace ~ is conjectured for a bounded linear transformation T of a Hilbert space Jl into itself. (I) If the origin is the only point in the spectrum ofT, then "f may be chosen so that T restricted to ?1( has arbitrarily small bound. (II) If the spectrum of T contains a

122 point with positive real part and a point with negative real part, then "Jtt may be chosen so that T restricted to '77( has its spectrum in the half -plane x E; 0 and T* restricted to ?l( .L has its spectrum in the half-plane x ;!$ 0, It is well known that (I) is true under the additional hypothesis that T - T* is completely continuous, A proof is now obtained for (II) when T - T* is of Schmidt class, Hilbert spaces of vector valued analytic functions are constructed to make the necessary approximations, Difficulties in the way of a general solution are discussed. (Received February 20, 1962,)

590-16. W, S, SNYDER, Health Physics Division, ORNL, Oak Ridge, Tennessee, E. E, POSEY, and L, B. RALL, Virginia Polytechnic Institute, Blacksburg, Virginia. Functions with maxima at each point of a dense set in a linear interval,

Let f(x) be a continuous real valued function on a real interval I and suppose that f assumes a maximum at each point of an everywhere-dense subset X in I. An example is given to show that such a function may assume a proper maximum at each point x in X and the following theorem is proved: In each subinterval of I there exists an infinite sequence of distinct points fxn} such that f(xn) = a (real constant) for all n, (Received February 20, 1962,)

590-17, ABOLGHASSEM GHAFFARI, National Bureau of Standards, Washington 25, D. C. Application of the stroboscopic method to a nonlinear equation of non autonomous character.

This paper is concerned with the existence of a periodic solution of Duffing' s equation with

viscous damping term (1) X + ax +X + bx3 = F cos

~in the phase plane and trying to satisfy equation (1) by series solutions of the form (2) r(t) oo n "C"""'oo n = Ln=OJl rn(t), ?V(t) = ~n=OJl ?V n(t), one sees that the zero order terms are r 0 (t) = ro, lb'o(t) = !llo - t, and that the first order approximations are given by the system (3) dr 1/dt = - aro sin2 ¥'o - j3r~ sin ¥'c 3 3 cos3 l'ro + -ysin ~ 0 coscvt, d¥"1/dt=- a sin l"o cos ~0 - ;tr0 cos ¥-o + (r/ro)cos <:Lro coswt, Applying the stroboscopic method (developed by N, Minorsky, C. R. Acad, Paris 232 (1951)) one obtaim the corresponding stroboscopic system (4) dr/dT =- 1rar + g(,Jii), dJii/d'T' = - (3/4).,.,.er3 + g(«))Q(.v,Jii)/r, where P(w,Jii), Q(w,Jii) and g(w) are known functions, and a.j3 and 7' are small positive constants, r is called the stroboscopic time, The critical point of (4), which corresponds to a periodic solution of (l), is given by rc = gP hra, ~ = AQ, where A = 47r2a 3 /3j3g2, (Received February 21, 1962.}

590-18, E. E. POSEY, Virginia Polytechnic Institute, Blacksburg, Virginia, Proteus forms of wild and tame arcs.

The main result is the following theorem: Theorem, Let A be an arc in E 3, Let e :> 0 be given, Then there is a space homeomorphism f that maps A onto an arc B, and f and B have the following properties: (i) f is pointwise the identity on the complement of N(A,e}, (ii) ,P(x,f(x)} < e for each x in E 3, (iii) B is locally polygonal at each x that is an interior point of some tame sub arc of B,

(iv) If B fails to be locally polygonal at x, then x = f(x), (v) If there is a point p on B and a neighbor­ hood N (p) that contains no straight line segment of B, then for an arbitrary plane Q there is a plane p

123 parallel to Q such that P n B n N = W has an infinite cardinal but W is not a dense subset of any arc in P. Some relationships between the imbedding of an arc and its parametric representation are given. A rather obvious example is that an arc is tame if one of its parameterizing functions is nondecreas­ ing. (Received February 21, 1962.)

590-19. R. B. KELMAN, Advanced Systems Research, Remington Rand Univac, Washington, D. C. Short proof of a theorem of Wintner on the asymptotic behavior of the adiabatic linear oscillator.

The following theorem was proved by Wintner (Amer. J. Math. 69 (1947), 251-272) and general­ ized by Levinson (Duke Math. J. 15 (1948), 111-126). Theorem. Let g(x) be a function of bounded

variation on 0 ;!iii X< ao such that g(x) > e > - 1 (~ a constant). The differential equation 2 ~ 1/2 - y" = (f') y where f =Jo(l +g) has solutions y 1 and y2 such that y 1 - sin f and y2 -- cos f while yJ. ~ f' cos f and Yz -- f' sin f. The proof is begun by introducing coordinates r and 8 subject to y = r sin (f + 8) and y' = rf' cos(f + 8). Substitution into the given differential equation and some manipulation give 28' = (f" /f')sin 2(f + 8) and r' /r = -(f" /f')cos2(f + 8). These equations have

a solution, say (r1,81), such that r 1 -----+ 1 and 8 1 -+ 0 as x -ao. (Received February 23, 1962.)

590-20. V. L. N. SARMA, University of Rochester, Rochester 20, New York. Eberlein­ measure and mechanical quadrature formulae. Preliminary report.

(! = {(t1•···•tk): -1 ~ tj ~ 1; 1" ;= j :!!! k} C Ek• ./{(e) is the set of real analytic functions x on C, such that the sequence of coefficients fxn n : 0 ;:;; n 1, ••• ,nk < oo} in the power-series expansion 1··· k of x is absolutely summable. jf((J) can be identified with the sequence space, ,/1• C(S ex) is the set of w*-continuous real functions on the unit-sphere, S , of ~ . An integral E can be defined on C(S ) ao 1 ao in such a manner that, when fin C(Sao) is a function of xo ••• o<= x0) alone, E(f) = (1/2) 1'~ f dx0• This integral induces a countably additive measure on Sao. For x E Sao• set I(x) = J- ~ ··/1 x dt1 ••• dtk; and JN(x) = ~~= 1Amx

590-21. A. K. AZIZ, Georgetown University, Washington, D. C. and B. E. HUBBARD, University of Maryland, College Park, Maryland. Bounds on the truncation error in the solution of the characteristic boundary value problem for hyperbolic equations.

This paper is concerned with the approximate solution of the problem u xy= f(x,y,u,ux,uy)•

u(x,O) = !ll(x), u(O,y) = ~(y), by finite differences. For the linear case the analogue of Riemann's formula is developed for a certain class of finite difference analogues. The truncation error is bounded explicitly by a function of the data of the problem which is O(h2) where h is the mesh size. Similar bounds are obtained for the error in approximating the derivatives ux• uy• uxy· Similar re-

124 suits are obtained for the nonlinear case iff satisfies suitable conditions. (Received February 23, 1962.)

590-22. G. E. SACKS, 40 Einstein Drive, Princeton, New Jersey. Recursive enumerability and the jump operator. Preliminary report.

By degree is meant degree of recursive unsolvability. A degree c is said to be recursively enumerable in a degree b if there is a set of degree c which is the range of a function of degree b.

A degree is called recursively enumerable if it is the degree of a recursively enumerable set (i.e., if it is recursively enumerable in 0). Theorem. Let b, c and d be degrees such that b...:: d ~ b' ~ c and c is recursively enumerable in b'; then there exists a degree g such that b ~ g ;:;; b' : c = g', d g and g is recursively enumerable in b. Corollary 1. If cis a degree such that 0' ~ c and cis recursively enumerable in 0', then there exists a recursively enumerable degree g such that c = g'. Thus one application of the jump operator completely obliterates the distinction between recursively enumerable degrees and degrees less than or equal to 0'. Corollary 2. There exists a recursively enumerable degree g such that g < 0' <. 0" = g'. Corollary 3. For each natural number n, there exists a recursively enumerable degree g such that 0 < g < 0' < g' < 0'' <. ••• < o(n) ""'g(n) < o(n+1). (Received February 26, 1962.)

590-23. N.J. PULLMAN, 1412 Madison Street, Syracuse 10, New York. On the number of positive entries in the powers of a non-negative matrix. Preliminary report.

Let A be a real, r by r, non-negative matrix. A is said to be essentially singular if and only if some m by r submatrix of A has at most m - 1 nonzero columns (two equivalent formulations of this property are given). Let W(i,A) be the number of positive entries in the ith row of A and let W(A) be the number of positive entries in A. Define index (i,A) = minfn > 0: W(i,A n) = max {W(i,A m): m > oJt and index (A)= min{n > 0: W(An) = max{W(Am): m >OJ}. A is said to be regular if and only if n 2 W(A ) = r for some n. A result due to Wielandt (see Math. z. 52 (1950), 648) when expressed in these terms is: if A is regular then index (A) ;a (r - 1) 2 + 1 (equivalently; if A is regular then for each

i, index (i,A) l!5 (r - 1)2 + 1). Using lattice theoretic methods, the following results of a similar nature

are obtained: (1) If A is not regular and A is not essentially singular then max {index (i,A): 1 so i !li r}

=index (A) til (r - 2)2 + 1;(2). If A is essentially singular then, for each i .;§ r, index (i,A) lfl! (r - 2) 2 + 2 These inequalities are best possible (for r > 2). Wielandt's result can also be obtained by these methods. (Received February 26, 1962.)

590-24. J. B. ROBERTSON and P.R. MASANI, Indiana University, Bloomington, Indiana. Time-domain analysis of a continuous parameter, weakly stationary stochastic process.

With a weakly stationary, mean-continuous stochastic process (ft = Utf,t real) we associate the I discrete parameter process (fn = Vnf, - oo < n < oo) where V is the Cayley transform of H, iH being the infinitesimal generator of the group of unitary operators Ut. We show that the two processes have the same remote pasts, and the same past and present from the standpoint t = 0 = n. In the non­ deterministic case let h0be the normalised innovation of the f~-process for n = 0, and 5t = Tt(h0),

125 where Tt = 2 - 112 fut- I+ ./~UsdsJ. Using the Wold decomposition for the f~-process and new relations between ut and vn, we show that the increments of the scprocess are differential innova­ tions of our ft-process, and so obtain the Wold decomposition of ft, viz. Jt = llt + v t' ut j_ vt, where ut =J'g:'c(s)ds~t-s' c E L 2 (l>,oo), and the vt-process is deterministic, (Received February 26, 1962.)

590-25, P, R. MASANI, Indiana University, Bloomington, Indiana, Isometric flows on Hilbert space.

An isometry Von a Hilbert space X with range R C X yields a "Wold decomposition" of X into orthogonal sub.spaces: X = 2:~o Vk(R.L) + n~=O yk (X). It is shown that a strongly continuous semigroup (St't ~ 0) of isometries on X to X yields an analogous decomposition of X. The direct sum occurring above is replaced by a "direct integral", in which the measure itself, rather than the inte­ grand, is subspace-valued, This measure is obtained by applying an operator-valued measure to RJ.., where R is the deficiency subspace of the infinitesimal generator of the semigroup, (Received February 26, 1962,)

599-26, P. C. HAMMER, Numerical Analysis Department, 5534 Sterling Hall, University of Wisconsin, Madison 6, Wisconsin, Extended Topology: Cardinal spectra and continuity,

Let Mi be the class of all subsets of spaces Mi' i = 1,2, An expansive function u mapping M1 into itself is defined to have properties uX ;;; X and u(X '-' Y) ~ uX .._,uy (isotonicity). Let v be an expansive function in M: 2• Then a transformation t: M 1-+ M2 is (u,v)-continuous, provided tuX <;;;;, vtX for all X E M 1. The domain cardinal spectrum, CT u' of u is the minimum set of cardinal numbers such that uX = X '--' Uf uY: Y ~ X, IIY II Eo-u J. where IIY II is the cardinal number of Y, Let ei be the identity function in M 1• Theorem. Let t be (u,v)-continuous, Then if c 1 E. a-u' c 2 E. o-v implies c 1 ...::. c 2 always it follows that tis (u,e2)-continuous and hence tuX= tX for all X E. M' 1• Remarks. Usually, if c 2 is the minimum number in a- v and c 1 the minimum number in a-u then a (u,v)-continuous transformation is trivial unless c 2 ~ c 1, (Received February 26, 1962,)

590-27, MURRAY GERSTENHABER, Institute for Defense Analyses, von Neumann Hall, Princeton, New jersey, Automorphisms of antisemisimple algebras,

If A is an antisemisimple algebra which is not a zero algebra, then it has been shown by the author (On nilalgebras and linear varieties of nilpotent matrices IV, to appear in Ann. of Math,) that there exists a naturally defined algebra B having A as radical such that, if B 1 is any other algebra with A as radical, with no simple direct summand, and with B 1 I A separable, then there is an isomor­ phism of B 1 into B which is the identity on A. It is shown here that every automorphism of A can be extended to be inner in B, and that if the automorphism is the identity on A/A2 then it is in fact inner in A in the sense that it isfconjugation by an element 1 +a, with a in A, (Received February 26, 1962,)

126 590-28. E. F. ASSMUS, JR .• Columbia University, New York 27, New York, and H, F.

MATTSON, Sylvania ARL, 40 Sylvan Road, Waltham 54, Massachusetts. Error-correcting codes: An axiomatic approach.

Given a ring K, we define a linear error-correcting code as a K-module A together with encoding functions fa, a E a, which map A homomorphically into K. The following coding axiom is imposed: The common kernel of all the fa's is 0; in other words, the encoding functions must distinguish the code, If B, gf, f3 E tJ is another code, a map from A to B is defined as a homomor­ phism {6: A- B such that g1f:>{6 is an encoding function of A for each f3 E 7.7. The weight w(a) of a in A is defined as the number of a in a such that fa(a) f 0. Under some fairly general restrictions on the encoding functions, the following holds: Theorem. If 16 is a map from A to B, then La EA w(a)w({6(a)) = (q - l)qk- 2(mn(q - 1) + '£..,13 ~ 13n13 ), where q, q k, m, and n are respectively the cardinalities of K, A, Cl, and 15', and for each f3 E 1-7. n j3 is the total number of a E Cl such that 2 k-2 2 r .2 . fa = ~ {6. As a corollary we have LaEA w(a) = (q - l)q (m (q - 1) + z 1 Ji ), where the h' s are the multiplicities of the r distinct.encoding functions of A. (Received February 26, 1962.)

590-29. W. S. LOUD, 119 Folwell Hall, University of Minnesota, Minneapolis 14, Minnesota. Asymptotic behavior of the period of solutions of certain plane autonomous systems near centers.

In the neighborhood of (0,0) let P(x,y) and Q(x,y) be in c3 , let P (x,y) be an odd function of y and Q(x,y) be an even function of y. Then the system x' = P(x,y) = y + Axy + Dx2y + Ey3 + ... , y' = Q(x,y) = - x + Bx2 + Cy2 + Fx3 + Gxy2 + ... has a center at (0,0), and all solutions of the system originating sufficiently close to (0,0) are closed curves surrounding (0,0). By perturbation techniques involving some complication it is found that the period of the solution starting at (a,O) for sufficiently small a has the asymptotic form: T = 27T + (1T/12)(A2 - AB - 5AC + lOB 2 + lOBC + 4C 2 - 3D- 9E

+ 9F + 3G)a2 + o(a2). This is verified for Duffing's equation x" + x + (3x3 = 0 for which the period of solutions is given by T = 21T- (37r/4) f3a 2 + o(a2). The research for this paper was supported in part by the Army Research Office Durham, Contract No. DA-ll-022=0RD-1869. (Received February 26, 1962.)

590-30. W. P. HANF, University of California, Berkeley 4, California. Degrees of finitely axiomatizable theories. Preliminary report.

Theorem. If E is any recursively enumerable set which is not recursive, then there exists a finitely axiomatizable theory T .such that the set of valid sentences of T has the same degree of unsolvability as E. This solves a problem of F eferman (J. Symb. Logic 22, 174). In addition, T may be taken to have either of the following additional properties: (I) T is essentially undecidable. (II) T has no finite extension which is essentially undecidable. (I) strengthens a result of Shoenfield (J. Symb. Logic 23, 398); the proof makes use of his construction of a pair of recursively inseparable sets each having the same degree of unsolvability as a given recursively enumerable but not re- cursive set. (II) solves a problem of Tarski (see problem 4, p. 51 in Vaught, J. Symb. Logic, 25). The axioms of T express the operation of a specially constructed Turing machine of the non-writing variety described by Minski (Ann. of Math. 74, 327). One remaining problem is: Do there exist

127 finitely axiomatizable undecidable theories which have only countably many complete extensions? (Received February 26, 1962.)

590-31. c. R. WARNER, University of Rochester, Rochester 20, New York. The Banach algebra L 1(G) () L 2(G). Preliminary report.

Let G be a locally compact Hausdorff abelian group with character group 6, and let the linear 1 2 1 2 space L (G) (lL (G) be equipped with the norm llxll = llxii 1 + llxft 2 (x E L (G) n L (G)), where llxll p = f./G lx(s) IPds} 11P. If multiplication in L 1(G) (l L2 (G) is given by convolution, then L 1(G) f't L 2(G) ...... , is a commutative Banach algebra whose regular maximal ideal space is G, and the abstract Silov theorem (see page 86 of Introduction to abstract harmonic analysis by L. Loomis) obtains for this algebra. (Received February 26, 1962.)

590-32. L. E. PAYNE and J. H. BRAMBLE, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. Some A priori inequalities with application in the Neumann problem for uniformly elliptic operators.

Under suitable normalization a priori inequalities of the following type are derived for a function u E c 1 in a bounded domain D with boundary C: (1~cu 2 dS 1f K1 .fDigrad ul2dV and 2 (2)/if 2dV li K2 j'D lgrad u 1 dV. The constants K 1 and K 2 are upper bounds for the reciprocals of the first nonzero Stekloff eigenvalue and the first nonzero free membrane eigenvalue for D. They depend on the geometry of D and may be computed explicitly. Using inequalities (1) and (2), point­ wise bounds for the solution (and its derivatives) of the Neumann problem for uniformly elliptic operators may be derived. A simple bound for the energy integral is also possible. (Received February 26, 1962.)

590-33. P. C. FISCHER, 91 University Road, Brookline 46, Massachusetts. Some basic properties of provable recursive functions.

A recursive function f is provable recursive (a p-function) with respect to a conventional axiomatic formulation S of arithmetic if, for some Godel number of f, the fact that f is totally defined is provable in S. (Cf. Rogers, Bull. Amer. Math. Soc. 63 (1957), 140). Similarly, p-one-one, p-(strictly)-increasing, p-non-decreasing, p-onto, p-infinite (range) p-functions, and p-permutations can be defined. Theorem 1. Let f be a p-function that is one-one or increasing or non-decreasing. Then f has the corresponding p-property and a proof for the property can be found in a uniform manner. Theorem 2. There exists a p-function f such that f is onto but, if S is sound, f is not p-onto. Theorem 3. If f is a recursive function which is onto, and e and e' are Godel numbers of f such that (i) there is a proof that the eth partial recursive function is totally defined and (ii) there is a proof that the e'th p.r.f. is an onto relation, then there is a Godel number e" off, which depends effectively on e and e', for which there exists a proof that f is a p-onto p-function. Theorems 2 and 3 hold with "onto" and "p-onto" replaced by "infinite" and "p-'infinite" or by "a permutation" and "a p-permuta- tion". (Received February 26, 1962.)

128 590-34. FRANK LEVIN, Rutgers, The State University, New Brunswick, New Jersey. Solutions of equations over groups.

There exist equations over a group which cannot be solved in any extension of the group; for

example, if a and b are elements of a group G having different orders, then the equation x - 1axb - 1 = has no solution in any extension of G. In the present paper it is shown that for any group G and any n1 n2 nk elements gi E G, the equation x g 1x g 2 ... x gk = 1, ni;:; 0 for all i, has a solution in a suitable extension of G; this result is established by constructing the solution. (Received February 27, 1962.)

590-35. ]. ]. PRICE, Cornell University, Ithaca, New York. On an inequality involving group characters. Preliminary report.

Let G be a compact abelian group and

(1) f =Ll_l 1a .¢ ., a. complex. (2) f(g) !:; 0 for all g E G. (3) /, fdp = 1 where p is Haar measure. J= J J J G For fixed g E G, let Mn(g) = sup f(g), f ETrn· It is easily seen that Mn(g) = Mn independent of g and

that Mn § n. Theorem. Mnk = nk for a sequence of indices 1 = n 1 < n 2 < n 3 < ... if and only if for each k the characters {¢j}~ 1 form a subgroup of . It follows that G must be totally disconnected. (Received February 27, 1962.)

590-36. DAVID SCHROER, University of Rochester, Rochester 20, New York. Behavior of syntactic structure under replacement and substitution.

Using definitions, techniques, and results developed as indicated in Abstracts 578-51 and 587-44 (especially, those involving the characteristic sequence of a nested pair), precise expression and smooth proofs are given for about thirty elementary facts about replacement and substitution in languages with quantifiers (conceived set-theoretically), the particular facts chosen being those con- cerning structural persistence which prove useful in deeper investigations to be reported at a later date. Simple samples: (l) Let F, G, A, J, A', J' be well-formed occurrences such that (A,F),

, are nested pairs and A'= {Repl in A: F by Gj, and suppose that F n J = \J and X(A',J') = )((A,J); then if J E AFree it follows that J' E" A'Free. (2) F, J, X, C, F' be well-formed occurrences such that

C for free X]; then there exists a unique well-formed occurrence J 1 such that (F 1 ,J') is a nested pair and /((F',J') = X(F,J); and J' = [Sub in J: C for FFree X]. (Received February 27, 1962.)

590-37. A. S. KAHR, 71 Mt. Vernon Street, Boston 8, Massachusetts. A reduction to a class of ABA formulas containing one dyadic p:r;edicate.

It is shown that a subset of the set of ABA formulas containing a single dyadic predicate letter in addition to monadics is a reduction class (with respect to satisfiability) for the predicate calculus (including equality and function symbols). (Received February 27, 1962.)

129 590-38, A. S. KAHR and HAO WANG, Computation Laboratory, 33 Oxford Street, Cambridge 38, Massachusetts, A remark on the reduction problem with an application to the ABA formulas.

Definition. Consider classes of formulas of the predicate calculus. For any class X, let N(X), I(X), F (X) be the subclasses of X which contain all formulas in X which have respectively no models, only infinite models, finite models. If R is a reduction procedure which reduces a given class Y to Y * and every subclass Z of Y to Z *, then R is said to be a conservative reduction procedure for Y, if (N(Y))* = N(Y*), (I(Y))* = I(Y*), (F(Y))* = F(Y*). Theorem 1, If K is the class of all formulas of the predicate calculus and R is a conservative reduction procedure for K, then no two of the three classes N (K *), I(K *), F (K *) are recursively separable, There is a simple method by which many of the newer reduction procedures can be made conservative. In particular, when this method is used on Kahr's reduction procedure of K to the class A 1 of all ABA formulas with monadic predicates plus a single dyadic predicate, we obtain: Theorem 2. No two ofthe three classes N(L1. 1), I(.A 1), F(Ll.. 1) are recursively separable. In this and indeed all cases considered, it is not necessary to appeal to the known result that N(K) and F(K) are recursively inseparable, (Received February 27, 1962.)

590-39, H. C. SEBRING, General Electric Company, Department of Advanced Weapons Systems Analysis, Missile and Space Vehicle Department, P. 0. Box 8555, Philadelphia 1, Pennsylvania. The normal bivariate density function and its applications to weapons systems analysis.

The normal bivariate density function is derived from a priori considerations. It is discussed in terms of probability area in a plane, and as a correlation surface. Several numerical methods of solving the normal bivariate distribution double integral are presented, and a curve is included for converting elliptical error distributions to circular probable errors. Regression and correlation coefficients are discussed, Relative to weapon systems analysis, examples are given of uses in studying impact and location errors. Analyses of search and detection for stationary and moving ob­ jects are given specific mathematical treatment, An Appendix examines the elliptical properties of normally correlated distributions. The investigation has resulted in a reference paper for the normal bivariate density function. (Received February 21, 1962,)

130 The April Meeting in Monterey April 28, 1962

591-1. J, L. BRENNER, Stanford Research Institute, Menlo Park, California. A set of matrices for testing computer programs,

To construct matrices of high order which have known roots and vectors, matrices of consider­ ably lower order can be modified in various ways (bordering, formation of Kronecker product, trans­ formation). This paper gives a new method for making such a modification. Let fi be any nonzero ni -vector (i = l, .. ,,t); the matrix A is a partitioned matrix with (k,.J,) box ak). 8 JsLI~ + bk_.Lfkf,£ *, where a, b are constants and Ikk is the nk X nk identity matrix, The vectors, roots, determinants, and inverses of such matrices are easily written down. It can be arranged that all quantities involved are real; vectors of A may be as nearly coincident as desired, or A may be derogatory. A is not necess­ arily symmetric, (Received December 7, 1961.)

591-2, VICTOR KLEE, University of Washington, Seattle 5, Washington, The Euler character­ istic in combinatorial geometry.

Hadwiger has observed that for the class ~ of all finite unions of compact convex sets in En, the existence of an Euler characteristic X. can be established in an elementary way, independent of algebraic topology; X is a valuation on 1L such that X~ = 0 and XC = 1 for each nonempty convex C E 'U. The basic notion is presented here in a way which exhibits its lattice-theoretic nature, and some additional applications are described (for example, to a recent intersection theorem of Ghouila­ Houri). Let L be a lattice with initial element 8, S C L, and s..l. the smallest sublattice of L which contains S U {0}. Then an Euler characteristic for S is a real-valued function ;;( on sL. which satisfies the following two conditions: (E ') ')(8 = 0; Xs = 1 for 8 f: s E S; (E") X a + Xb = 7( (a ~b) + ;x. (a"b) for all a,b € si-. In addition to results paralleling those of Hadwiger, it is proved that for an intersectional subset K of L, the following three assertions are equivalent: (a.) K admits an Euler characteristic; ({J) if A is finite~# A C K, and VA E K, then ;'A= 1; (r) for finite subsets F of K,tF depends only on VF. (Here $F = ~!~d F (- l)i JliF, where JliF is the number of i-membered sets G C F having /\G f. 8.) The paper will appear in the American Mathematical Monthly. (Received February 13, 1962,)

591-3, L. J, MORDELL, University of Arizona, Tucson, Arizona, On a cubic congruence in three variables,

Let f(x,y,z) be a cubic polynomial with integer coefficients and let p be a prime. It is conjec­ tured that in general, the congruence f(x,y,z) =0 (mod p) has p2 + O(p) solutions. This is proved in the special case when f(x,y,z) = z 2 - f(x,y)- k where f(x,y) is a binary cubic not of the form a(bx + cy)3. The paper will appear in the Acta Arithmetica, (Received February 19, 1962,)

131 591-4, GEBHARD FUHRKEN, University of California, Berkeley 4, California. On generalized quantifiers.

Let L be a language obtained from a first-order language by adding the quantifier "there are at least !It a."'" (cf. A. Mostowski in Fund. Math. 44 (1957), 12-31). Call :tti\3 "small" for .p with r:-, < .Ita. for all $ E' wp, TTt~c..>ts "Ct< ~a.· (1) If d!f is the ultra-product of ("tt>tET with respect to the ultra-filter D, then for every sentence !

591-5, D. H. KAUFFMAN, Lockheed Missiles and Space Corporation, Sunnyvale, California and E. G. McNIEL, 3251 Hanover Street, Palo Alto, California. Some theorems on periodic decimal fractions.

Theorem 1. If p is a prime of the form 2n(Zk + 1) + I and 10 is a znth power residue mod p, then the period length e of 1/p is odd and does not exceed Zk + l. If p is a prime of the form Znh + and 10 is a zn- 1th but not a Znth power residue, then the period length of 1/p is even and does not exceed Zh. Theorem 2. Let p be an odd prime =/: 5 and e the number of digits in the recurring period for 1/p. Then the partial remainders, Ri =10 i (mod p), i ~ 1,2,.,,,e, obtained in the decimal expansion of 1/p are the complete set of nth power residues mod p, where n ~ (p - 1)/e. Theorem 3, Let R;_, i ~ 1,2,.,, e, be the partial remainders obtained in the decimal expansion of 1/m, where (m, 10) ~ 1. Then for any positive integer n, Ri =Rni (mod m), where the subscript ni is reduced mod e and e is the order of 10 mod m. Theorem 4. Let p be an odd prime ~10 and e the number of digits in the recurring period for 1/p (e =/: p - 1). Then a necessary and sufficient condition that the digit k be missing from the decimal expansion is that the integers, J, ~p/10] < J < [(k + 1)p/10], be consecutive nth power nonresidues of p, where n ~ (p - 1)/e. (Received February 23, 1962,)

591-6. R. M. ROBINSON, University of California, Berkeley, California, Conjugate algebraic integers in real point sets.

It is shown that any real point set which is the union of a finite number of closed intervals and has transfinite diameter > 1 must contain infinitely many sets of conjugate algebraic integers. This generalizes the result announced in Abstract 556-18 for a single interval. The first step is to show that by a judicious shortening of the intervals, keeping the transfinite diameter of the set > 1, it can be insured that the Chebyshev polynomial for the set, of any degree n having a certain divisor, will

132 oscillate n times between its maximum and minimum in the intervals of the set. It is then shown that for arbitrarily large values of n, this Chebyshev polynomial can be approximated by a polynomial xn + ••. with integer coefficients sufficiently well that the new polynomial will have the same number of roots in each of the intervals as the old. Thus the roots of the new polynomial will be algebraic integers lying in the given set, which yields the stated result. Finally, it is shown that there are real closed point sets with arbitrarily large transfinite diameter not containing any algebraic numbers. Thus the condition that the set consists of a finite number of intervals cannot be omitted. (Received February 23, 1962.)

591-7. D. E. MYERS, University of Arizona, Tucson, Arizona. A note on an imbedding space representation. Preliminary report.

The author has previously constructed an imbedding space (Pacific J. Math. 11, No.4,~ imbedding space for Schwartz distributions) using analytic functionals on the set of functions [e -zt}. -a < R(z) 0}. A is an algebra with convolution for the product and usual addition. There is a 1-1 correspondence between the regular maximal ideals of A and the £z} such that IR(z)l!!! a. This latter correspondence is used to establish a representation of the imbedding space by analytic functionals on the regular maximal ideal space. (Received February 23, 1962.)

591-8. E. 0. THORP and SEYMOUR GOLDBERG, New Mexico State University, Las Cruces, New Mexico. The range as range space for compact operators.

Let X and Y be normed spaces and T a linear operator from X to Y. Let To be the operator from X to R(Y), the range of T. Obviously when T is continuous so is To. When T is compact, To

need not be compact as the example T: co ---»./2 defined by T(xl' x2 •••• ,xn' ••• ) = (x 1, x2/2, •.• ,xn/n, ••• ) shows. Note that the conjugate of To is compact and that T0 , although not compact, is the limit of a sequence of operators with finite dimensional range. When does T compact imply T0 is compact? A partial answer is given in the Theorem: Let T: _lp - .lq• 1 !0 q =< p !! oo, be a compact diagonal operator, i.e. of the form T({xk}) = {ekxk}• where {xk! is in.Jp• such that fekxkl is in .Jq for all {xkf in ,lp. If T is compact, To is compact. Corollary. A compact symmetric operator on separable Hilbert space remains compact when the range space is reduced to the range of the operator. (Received February 23, 1962.)

591-9. TAKA YUKI TAMURA and D. G. BURtlELL, University of California, Davis, California. Embedding of a commutative semigroup into certain divisible semigroups.

Let S be a commutative semigroup, and let A a denote a multiple of an element a i.e.

~ = A.a, where A_ is a positive integer. A commutative semigroup T is called a divisible X semigroup if T satisfies the following condition: for any element x of T and for any positive integer). , there is an element y of T such that A. y = x. If S is a commutative semigroup satisfying the condition that A. a = ~b implies a = b, then S can be embedded into a smallest divisible semigroup T satisfying the same condition. This result can be generalized for a semigroup with operators. (Received February 26, 1962.) 133 591-10. TAKA YUKI TAMURA and R. B. MERKEL, University of California, Davis, California. Semigroups, all subsemigroups of which are ideals.

By an ideal I of a semigroup S we mean a subsemigroup satisfying IS C. I and SIC I. A sub­ semigroup is not necessarily an ideal, but, as special cases, if a semigroup S has the property that any subsemigroup is an ideal, then S contains zero 0 and (1) abc= 0 for every a, b, c E S. (2) ab \ 0 implies ab = a 2 = b 2; and conversely. Furthermore, the authors show the maximum and minimum of the number of ideals when the order n of a finite semigroup of this kind is assigned. (Received February 26, 1962.)

591-11. OTTO KOERNER, 1329 Sunnyside Avenue, Salt Lake City, Utah. Algebraic integers as sums of polynomial values.

Let K be an algebraic number field of degree n, f an integral-valued polynomial of kth degree (k > 1) such that (a) the leading coefficient is totally positive (b) not for all integral arguments f equals f(O) mod p where p is any prime ideal of K. Let J be the set of all integers inK, j(f) its additive sub­ group generated by all numbers f(b) where b E j. Define G(f) to be the smallest natural number s with the property: all "sufficiently large" totally positive 1 E. J(f) have a decomposition 1 = f(m 1) +

••• + f(ms) with~ E j. We prove: G(f) exists if and only if J(f) = J(f- f(b)) for some bE J, in which case G(f) IIi n ((k - 1)2k+l_ 1)+ 1. Obviously all f with a zero in J satisfy the condition of this theorem, but there are pairs (f,K) which violate it. For n = 1 our theorem specializes to a result of Hua's. (Received February 26, 1962.)

591-12. c. R. WYLIE, JR. 201 Mathematics Building, Salt Lake City, Utah. A new series of line involutions.

Let V be a nonsingular quadric in s3 and let r be a curve on V meeting the generators of one regulus, R 1, in a single point and the generators of the other regulus, R2 , inn - 1 points. A general line,_.£, meets V in two points, P and Q, through each of which passes a line of R 1. Let these lines, p and q, meet r in P 1 and Q 1, respectively; and let the lines of R2 which pass through P 1 and Q1 intersect q and p in Q2 and P 2 , respectively. On p and q let P' and Q' be the harmonic conjugates of P and Q with respect to (P 1,P2) and (Q1,Q2) respectively. Then j• = P 'Q' is the image of) in the involution here considered. It is further shown that any line involution with a complex of singular lines containing, but not consisting exclusively of the lines which meet a twisted curve, r, is of this type. (Received February 26, 1962.)

591-13. A. H. KRUSE, New Mexico State University, P.O. Box 756, University Park, New Mexico. Some theorems on the axiom of choice.

For each power p, let T(p) be the statement that for all powers m and n with p < m < 2P and p ""- n ""2P, either m ~ n or n lli m. Theorem 1. The axiom of choice holds iff T(p) for each power p. Theorem 1 includes both the classical theorem ofF. Hartogs that the trichotomy law for powers im­ plies the axiom of choice and the theorem of A. Lindenbaum and A. Tarski that the generalized continuum hypothesis implies the axiom of choice. As in the writer's paper Some developments in the

134 theory of numerations, Trans. Amer. Math. Soc. 97 (1960), 523-553, for each set X let ~(X)= [flf is a one-one function with domain an ordinal number and with image C XJ. Theorem 2. The axiom of choice holds iff for all sets X and Y: !!._ I!~V(X) I "" I"*'(Y) 1. then IX I < IY 1. Theorem 2 substantially generalizes 7.3 of op. cit. Theorem 3. The axiom of choice holds if for all sets X andY: if IX I "" IY 1.

~ I!Jr (X) I < I,._(Y) 1. The writer does not know whether "-"'" may be replaced by ''.p" (for set of all subsets in Theorems 2 and 3. In this connection the routine observation that I?V(X)I = IJ>(X)I for each well-orderable set X should be cited. (Received February 26, 1962.)

591-14. ]. M. IRWIN, CAROL PEERCY, and E. A. WALKER, New Mexico State University, Box 396, University Park, New Mexico. On high extensions of abelian groups. Preliminary report.

In what follows all groups considered are Abelian. Let d be the subgroup of elements of infinite height in G. A subgroup H of a group G is called high in G iff H is maximal with respect to the property H n G 1 = 0. Consider a group H with H 1 = O. An exact sequence 0 - H ---+X ---+ D --+ 0 is called a high extension of H by D iff H is high in X. When H is high in X, D has to be divisible. Con­ cerning the set Hext(D,H) of all high extensions of H by D (regarded as a subset of Ext(D,H)) there is Theorem 1: Let H be an Abelian group with H 1 0. Then Hext(D,H) is a subgroup of Ext(D,H). When His a p-group an interesting fact is Theorem 2: Let H be a p-group with H1 = 0 and D be an arbitrary divisible p-group. Then Hext(D,H) = p Pext(D,H). (Received February 26, 1962.)

591-15. R. ]. NUNKE, University of Washington, Seattle 5, Washington. Purity and subfunctors of the identity.

A subfunctor of the identity in the category of abelian groups assigns to each group A a subgroup

S(A) such that each f: A --+ B sends S(A) into S(B). An extension f: 0 -+ Z ----"> G --> H- 0 of the integers defines a subfunctor of the identity S(fJ") by S(~)A = Im(Hom(G,A) ~A). S has the form S(o/) iff (i) S(A) contains the maximal divisible subgroup of A, (ii) S commutes with direct products,

(iii) there is a cardinal c such that a E S(A) iff a E S(B) for some B !:;; A with power < c. S is called a cotorsion functor if it has the form S(~) with H torsion. If cotorsion functors S l' s2 agree on all cotorsion groups, then s 1 = s2; hence s 1 = S2 if S 1Ext = S2Ext. An S satisfying (i)-(iii) is cotorsi'on if B !:;; A and A/B torsion-free imply S(B) = B f"'\ S(A). We say that B!; A is S-pure in A if the corres­ ponding extension is in SExt(A/B,B). The following properties of S-purity are equivalent: (1) AS-pure in B and B S-pure in C imply A S-pure inC, (2) SExt(A, ) is half exact on S-pure exact sequences, (3) the same for SExt( ,C), (4) S(B) = B n S(A) if B is S-pure in A. If p" assigns to A the subgroup of

elements of p-height il; a (p a prime and a an ordinal), then p" is a cotorsion functor with the above properties. (Received February 26, 1962.)

591-16. D. L. BOYER, University of Idaho, Moscow, Idaho. p-basic subgroups of abelian groups.

L. Fuchs [Acta. Math. A cad. Sci. Hungar. 11, (1960), 117-125) has generalized the concept of Kulikov's basic subgroup of an abelian p-group to p-basic subgroup of an arbitrary abelian group. The present paper generalizes theorems of Baer and Kulikov which describe the embedding of abelian

135 p-groups with ·no elements of infinite height as pure subgroups of the torsion subgroup of a certain complete direct sum of cyclic p-groups to the corresponding theorems for p-hasic subgroups of an abelian group with no elements of infinite p-height. (Received February 26,1962.)

591-17. R. E. BLOCK, California Institute of Technology, Pasadena, California. On the cover­ ings of Lie algebras of classical type.

Let C be a finite dimensional Lie algebra over a field F of characteristic p .,.. 3, and let f be a homomorphism of C onto a Lie algebra L, with kernel K, where L is of classical type (i.e., a semi­ simple analogue over F of one of the complex semisimple Lie algebras -- see Mills and Seligman, J. Math. Mech. 6 (1957)). Some results are obtained on the splittability of the extension. In particular it is proved that if K is contained in some term of the upper central series of C and L is simple, then either K is a direct summand of C or else, for some multiple n of p, L ~ PSMn(F) (then X n matrices of trace 0 modulo scalars) and C is the direct sum of a subalgebra of K and a copy of SMn(F). This result will be used by Zassenhaus and the author in determining, when F is algebraically closed, all Lie algebras over F with a nondegenerate trace form and all Lie algebras over F with a quotient trace form. (Received February 26, 1962.)

591-18. T. K. PAN, University of Oklahoma, Norman, Oklahoma. Pseudo-conformal vector fields.

In a differentiable n-manifold V of class r with positive definite metric gijdxidxj and metric connexion E~k' a vector field Si is defined as a {If-pseudo-conformal vector field if .Sili = 0, where the solidus denotes covariant differentiation with respect to E~k and where {II is a:.n arbitrary scalar. Properties of such vector fields in V with torsion, particularly their global nonexistence, are investi­ gated. Some results are obtained to include as special cases those about pseudo-harmonic and pseudo­ Killing vector fields, as shown for example in the following theorem. In a compact orientable metric manifold whose torsion tensor Si~ satisfies sj/= 0, there exists no ¢-pseudo-conformal vector field _si such thatj'V(Ejk~j~k - 25ilsSjisgi + ~iljs\)dv ~ 0 unless {II= 0. Pseudo-conformal tensor fields and other generalizations are considered. (Received February 26, 1962.)

591-19. ALEXANDER PEYERIMHOFF, University of Utah, Salt Lake City, Utah. On an absolute mean value theorem for Ces'll.ro methods.

The absolute convergence of _ aka.( ) s is proved for a certain class of sequences k(n)! .L-v-~n 0 n ,v v and under the assumption that L~=Oa~vsv is absolutely convergent; (a~v) denotes a Ces'll.ro-matrix of order 0 -;; a. ~ 1. This theorem can be considered as a counterpart to a theorem by Bosanquet in J. London Math. Soc. 16 (1941), 146-148. There are extensions to general matrix methods. (Received February 26, 1962.)

136 591-ZO. MORGAN WARD, California Institute of Technology, Pasadena, California. A diophan­ tine problem associated with linear recurring series. Preliminary report.

Let Ux be the general term of an integral linear recurrence of order k !: Z with a nondegenerate characteristic polynomial. In this paper the bounds for the number of solutions of the diophantine equation Ux = C are studied fork ~ 4. Here C is a given integer. Let these bounds be Mk if C = 0 and Nk if C ¥ 0. Then Nk f. Mk+1 and if C ¥ 0, one may assume that C = ±1. Precise bounds are known for Nz, Mz and M3 if the roots of the characteristic polynomial are real. If the roots of the characteristic polynomial are complex, it is known that M3 !!0 6. No quadratic recurrence is known with Nz = 6. (Received February Z6, 196Z.)

591-Z1. J. M. IRWIN, CAROL PEERCY, and E. A. WALKER, New Mexico State University, Box 396, University Park, New Mexico. On If" -pure sequences of groups.

All groups considered are Abelian. This paper is motivated by the fact that the subgroup of elements of infinite height in Ext(B,A) is Pext(B,A); that is, each element of infinite height in Ext(B,A) is represented by an exact sequence 0 - A --+X- 8- 0 with A pure in X. Let p be a prime, n 11: 0 and A C B. Then A is said to be pn-pure in B if An pm 8 = pm A for all m ~ n. Inductively, pa.­ pure may be defined for all ordinals a.. Also an element g E G hasp-height il'i. a. if g E pa.G. An exact sequence 0 -A-X- 8--+ 0 is pa.-pure if A is pa.-pure in X. Let Ext(B,A,pa.) be the subset of Ext(B,A) represented by pa.-pure sequences. The following hold: (1) pa.Ext(B,A) = Ext(B,A,pa.) for

all a. ;:< ""· That is, the elements in Ext(B,A) of height ;:. a. are precisely those represented by the

pa.-pure short exact sequences for a. 1i c:..>. (Z) pa.Ext(B,A) C Ext(B,A,pa.) for all a.. (3) If 0- A- 8

- C _,. 0 is pa.-pure then 0 _., Hom(X,A)---> Hom(X,B) --t Hom(X,C) -t Ext(X,A, pa.)- Ext(X,B,pa.) - Ext(X,C,pa.) -o, and 0- Hom(C,X)- Hom(B,X)- Hom(A,X)- Ext(C,X,pa.) -Ext(B,X,pa.)

-t Ext(A,X,pa.) _., 0 are exact for a.~ a~. (4) The exact sequence 0 ~A _.,x__.. B -o represents

an element in the divisible part of Ext(B ,A) iff the induced sequence 0 -->At_., Xt -> Bt __,. 0 is splitting exact, where G t denotes the torsion subgroup of G. (Received February Z6, 196Z.)

591-ZZ. A. ALEXIEWICZ, Libelta ZZ mg, Poznan, Poland and M. G. ARSOVE, University of Washington, Seattle 5, Washington. Simultaneous automorphisms and the Landau properties.

Let $be the space of analytic functions on the open unit disc under the topology of uniform convergence on compact sets, and let :t:P (p:;;. 1) be the Lp subspace of .Y. Further, let {1rn} be

a sequence in J', and let Pf = E~oc n 1rrt where f has the Taylor expansion f = ~O en Sn and f7rn} is such that the series for Pf converges pointwise for each f in .Y. The condition

(A) lim supn~oo1111""n11 1 /n <. 1 (LP norm) is then necessary and sufficient for P to be a bounded linear mapping of :Pinto l;P. Suppose that fa.n} and [,<9n} are bases in :f and that the functions 11""n = lofn - a.n satisfy condition (A), Then, for [,Onl to be proper it is necessary and sufficient that f a.nJ be proper. Whenever either basis is proper, there exists a mapping T such that Tan= f3n (n = 0,1, .•• ), T is an automorphism on :J: and T i1Jl is an automorphism on .;t'P. Simultaneous automorphisms of this sort preserve certain properties considered by Landau for the power series basis {8n}. (Received February Z6, 196Z.)

137 591-23. ERNEST MICHAEL, University of Washington, Seattle 5, Washington. Trivial extensions of topological spaces.

Call a subset A of X with dense complement trivial in X if, whenever U is a neighborhood of x E A, U - A does not split into two open sets whose closures both contain x. (Example: E X 0 is trivial in EX [0, 1] but not in EX[- 1,1]). Theorem 1. Let X be a Tychonoff space, and A a subset with dense complement. Then there exist-essentially uniquely-a Tychonoff space X with trivial subset A, and a continuous, closed 7r: X~ X which maps X - A homeomorphically onto X - A and maps A (compact, 0-dimensional) -to-one onto A. (Examples: If X= E X [- 1, 1] and A= E X 0, then X is the disjoint union of EX[- 1,0] and EX [0, 1]. If X= {1/n: n = 1,2, ... } UfO} and A= {OJ, then X is not metrizable. Theorem 2, If X is separable metric, then so is X iff. X - A has a countable col­ lection 13 of open subsets such that, whenever U is a neighborhood of x c A and C is an open-closed

subset of u - A, then X has a neighborhood v such that (V nC) € 13. In case X is compact (making X compact) and A is 0-dimensional (making A 0-dimensional), this reduces to the simpler requirement that X - A has only countably many open-closed subsets, or, equivalently, that the space of quasi­ components of X - A is compact. (Received February 26, 1962.)

591-24. SOLOMON FEFERMAN, Stanford University, Stanford, California. Constructive pseudo-well-orderings.

Reference is to the author's 'Classifications of recursive functions by means of hierarchies' (CRF) to appear in Trans. A mer. Math. Soc. Notation: ?f = class of hyper arithmetic functions and reins., -¢(n) E W*. (By the usual unsecured sqces. argument.) 2, W and the complement of W * are inseparable by sets in .JI. We prove here the corresponding theorems 1', 2' for 0 and 0*. This involves a finer analysis of the operation a IB b

(effective version of ordinal sum) on 0* than given in (CRF). It follows from 1,1 1 and a result of 1 Spector that W*, 0* are both 2: 1 complete (w.r.t. many-one reduc.). Generalizations of 0* are also studied. (Received February 26, 1962.)

591-25. JOHN JONES JR., 6702 Harwood Place, Springfield, Virginia. On second order non- linear differential Equations.

H. A. Antosiewicz (J. London Math. Soc. 30 (1955), 64-67) has shown that ¢(y,y') 0 for all y,y', yg(y) "" 0 for y i 0, JOY g(~)d5 ~ oo as IY I - oo, imply that ly(x) 1, IY' (x) I remain bounded as x- oo, for all solutions of(*) y" + ¢(y,y')y' + g(y) = 0. If the conditions of Antosiewicz above held and ¢(y,y')y' ~ g(y) l!i f(x)y 2n- 1, n ;<; 1,J'irx2n- 1 1f(x)ldx..:. oo, then(*) has no oscillatory solutions as x - oo. (Received February 26, 1962.)

138 Abstracts Presented by Title

62T-50. J. A. WOLF, Institute for Advanced Study, Princeton, New Jersey. Space forms of Grassmann manifolds.

Let F be one of the fields R (real), C (complex) or K (quaternion); let Gq,n (F) denote the Grass­ mann manifold consisting of all q-dimensional subspaces (oriented if F = R) of the left vectorspace Fn,

with its usual structure as a Riemannian symmetric space. In case F = K, in case F = C, and in case F = R and q(n - q) is even, a complete classification (up to global isometry) is given for the Rieman­ nian manifolds which admit a Riemannian covering by Gq,n(F ). In case F = R and q(n - q) is odd, that classification problem is reduced to the special case which is traditionally known as the "spherical space form problem" of Clifford and Klein. (Received January 15, 1962.)

62T-5l. W. D. L. APPLING, Duke University, Durham, North Carolina. Infinite series and nonnegative valued interval functions. Preliminary report.

Theorem 1. If { Akl ~ 1 is a sequence of nonnegative numbers such that Lk= 1Ak -- oo as n---+ oo, then there is a sequence [Rk! ~ 1 of nonnegative numbers such that L~= 1Rk converges as n----. oo, and such that if 0 < p < 1, then Lk= 1 [Ak]P [Rk] 1-P ~ oo as n ---+ oo. All integrals discussed in Theorem 2 are Hellinger type limits of the appropriate sums. Theorem 2. If H is a real nonnegative valued function of subintervals of the number interval [a,b], then/[§t,bJH(I) exists if and only if for

each real valued nondecreasing function m on [a,b] there is a number p such that 0 < p < 1 and /[a,b][H(I)]P[dm]1-P exists. (Received January 15, 1962.)

62T-52. ANDREW SOBCZYK, Box 8052, University of Miami, Coral Gables 46, Florida. Extension properties of Banach spaces.

A Banach space B has the projection property in case that for every superspace C :::>B, where C is also a Banach space, there is a continuous projection of C onto B. Let K (B ,C) = inf IPs I over all continuous projections Pg of C onto 8, and let k(B) = supcK(B,C). It is shown that if B has the projection property, then necessarily k(B) is finite. There is a similar result if B is separable and the superspaces C are restricted to being separable; let h(B) be the quantity which corresponds to k(B) in this case. Then the separable spaces (c) and (co), of respectively convergent sequences and

sequences convergent to zero, have the separable projection property, with h[(c}] = 3, h[(c0)] = 2, although they do not have the projection property. (Goodner, Nachbin, and J. L. Kelley have shown that a space B has the projection property with k(B) = 1 if and only if B is isometric with the space C(E) of all continuous functions on an extremally disconnected compact Hausdorff space E.) (Received January 15, 1962.)

139 62T-53. ALEXANDER WEINSTEIN, University of Maryland, Institute for Fluid Dynamics and Applied Mathematics, College Park, Maryland. A necessary and sufficient condition in the maximum­ minimum theory of eigenvalue.s.

Let Au be a self-adjoint positive operator with discrete spectrum A1 ;a A. 2 ~ ••• and in the usual notation let R(u) = (Au,u)/(u,u) be the Rayleigh quotient. Let pl' p 2, •••• Pn-1' be (n- 1) orthonormal functi~ns and let us denote by A(pl'p2 ..... Pn-l) the minimum of R(u) for u orthogonal to p1,p2, .•.• pn-l· Then as is well known .:\(pl' p 2, .•• , Pn-l) ~An· (For a new proof see A. Weinstein, Abstract 61T-279,

Notices Amer. Math. Soc. 8 (1961), 612.) Let As-l < As= As+l = ••• =An;:;; .-\t+l ;§ •••• Introduce the determinants (A. Weinstein, Memor. Sci. Math., Fasc. 88, Gauthier-Villars, Paris, 1937)

Wm().) = det

62T-54. A. S. KAHR, 71 Mt. Vernon Street, Boston 8, Massachusetts. A new reduction procedure.

Let Dk be the class of formulas of the predicate calculus (without equality or function symbols) of the form: (x)(3y)(z)Mxyz• where Mxyz is quantifier-free and contains only monadic and dyadic predicate letters, of which not more thank are dyadic. Let Gk be the subset of Dk comprising those formulas whose matrix Mxyz is of the form Nxy & Pxyz, where no monadic predicate letter appears in Pxyz and no dyadic predicate letter may appear in Nxy with a subscript other than "xx". Then, it can be shown that the halting problem (for finite tapes and for infinite tapes of finite period) of any Turing machine T may be reduced to the decision problem of Gk(T)' where k(T) is effectively calcul­ able from the description of T. (Roughly speaking, k(T) is a measure of the complexity ofT.) Thus (using a four symbol, seven state universal Turing machine of Minsky), it is possible to prove that

G7 is a reduction class for the Entscheidungsproblem. (Received January 17, 1962.)

62T-55. REUBEN SANDLER, Institute for Defense Analyses, 100 Prospect Avenue, Princeton, New Jersey. On the collineation groups of the free planes. Preliminary report.

Let 1rn (n ;?; Z) be the free projective plane generated by n points on a line and two points off that line (seeM. Hall's Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229-277 for an exposi­ tion of the subject) it is desired to determine the structure of its collineation group. Let n-0be the n + 2 points described above. Then every collineation is determined by its action on the points of 1rg. Thus one need only determine those sets of n + 2 points (n on a line, 2 off) with the property that the subplane they generate is the whole plane. Every such set determines 2(n!) collineations if n > 2, or 4! collineations if n = 2, since obvious permutations in 7ro give collineations. Let 7r~ be the free extension of 1r-(} not on a line, and then by adjoining a point for every pair of lines not intersecting in rr(j. One can ask by inspection what configurations there are in 7r~ which generate all of 7Tn Theorem. If n = 2, the collineations thus obtained in .,.z generate the entire collineation group of 7r 2•

140 The group has three generators. Conjecture. If n :> Z, an analogous result holds. One can still obtain in 'lt~ a large collineation group, but it has yet to be shown that this consists of all collineations. (Received January 17, 196Z.)

6ZT-56. T. N. TRACEWELL, University of California, Berkeley, California. Medial tetrads.

Suppose the set A is a groupoid under each of the four binary operations

Hom a.tJ'Y, then 'Y is commutative. (Received January 17, 196Z.)

6ZT-57. ALEXANDER WEINSTEIN, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. On almost periodic solutions of the Helmholtz equation.

Consider Au + kzu = 0, u = u(x,y) in the strip - oo < x < oo, 0 !i y ~ 1t" with the boundary con­

ditions u(x,O) = u(x,?r) = 0. Assume that there are integers m and n such that kz :> mz, kz :> nz and that the ratio (kz - mz) 1/Z /W - nz )1/Z is irrational. Then, for any such kz, all bounded solutions u are almost periodic vector valued functions of x in the sense of Bochner. Here the Lz norm is used. Similar results hold in more dimensions and for mixed boundary conditions as well as for other elliptic equations which have separable solutions. The proof of this statement as well as an investiga­ tion of unbounded solutions (extended Phragmen-Lindelof principle) is given by an eigenvalue method. This method yields also all values of kz for which the bounded solutions are periodic. See A. Wein­ stein (C. R. Acad. Sci. Paris 184 (19Z7), p. 497; Abh. Math. Sem. Univ. Hamburg 6 (19Z8), p. Z63; Canad. J. Math. 1 (1949), p. Z71) and P. D. Lax (Comm. Pure Appl. Math. 10 (1957), p. 361). (Received January ZZ, 196Z.)

6ZT-58. Y.-c. WONG, University of Hong Kong, Hong Kong. Existence of linear connections with respect to which a given tensor is recurrent or parallel. 11.

It is seen from {1) in Abstract 61T-Z59 these Notices, 8 (1961), 515, that with each recurrent tensor of type (r,r) on a connected n-dimensional C 00 -manifold, there is associated an nr x ns matrix of constants. (4) The problem of finding a criterion for deciding when a recurrent tensor of type (r,r) is almost-parallel is now solved after a thorough study of the matrix equation cr(t!lrK- 1)C(t!lrK) = C, where C is the nr X nr matrix of constants associated with S, cr a scalar parameter, K a non­ singular n X n matrix, and Q!lrK the rth direct power of K. Further results obtained are: (5) For a recurrent tensor of type (r,r) to be almost-parallel, it is necessary that (i) S is nilpotent and its index of nilpotence is at most equal to (n + r - 1)!/(n - 1)! r!, and (ii) S has at least r!/(q!)n(q + l)P,

141 (r = nq + p, 0 ~ p < q), elementary divisors, (6) For a recurrent tensorS of type (r,r) which is not almost-parallel, the covector W in VS = W ® S is globally a gradient. (Received January 23, 1962,)

62T-59, H. J, ZIMMERBERG, Rutgers, The State University, New Brunswick, New jersey, Two-point boundary conditions linear in a parameter.

In considering various classes of differential systems, in vector form y' - A(x)y = ~B(x)y,

s IJ; .>..] = (M 0 + .\M 1)y(a) + (N 0 + A.N1 )y(b) = 0, where M0, M l' N 0 and N 1 are n X n constant matrices such that the n X 2n matrix liMo + AM 1 N0 + AN1 II has rank n for all complex values of A., both Bobonis [Contr. to Calc. of Var., 1938-1941, 99-138] and the author [Illinois j. Math. 4 (1960), 593-608] have

imposed Condition (A): There exist n X n constant matrices M2, N2, P2, Q2 and n X n matrices P(.\),

Q()..) such that for all values of A one has - (M0 + A.M 1)P 2 + (N 0 + A.N 1)Q2 =I, - (M0 + AM 1)P(A.) + (N0 + A.N 1)Q(.\) =0, -M2P 2 + N2Q2 =0, and -M2P(A.) + N2Q(A.) =I, with I the identity matrix. The principal result established is that a necessary and sufficient condition for the existence of an n X n non singular matrix r()..) such that the equivalent set of boundary conditions T (A.)s [y; A.] = 0 remain linear in A and have coefficients satisfying Condition (A) is that there exist adjoint boundary conditions linear in A. In particular, Condition (A) may be eliminated as a restrictive condition for those prob­ lems equivalent to their adjoint under nonSingular transformations. Moreover, the further condition for symmetrizability [Illinois J, Math, 4 (1960) Sec, 5] will be preserved under the change of boundary conditions. (Received January 26, 1962,)

62T-60. G.]. RIEGER, Dahlienstrasse 24, Ottobrunn, Germany, Solution of the Waring­ Goldbach problem for algebraic number fields,

Denote by Pk the sequence consisting of 1 and the kth powers of aU rational prime numbers, by sPk the s-fold Schnirelmann sum Pk + ... + Pk, and by d(sPk) the Schnirelmann density of sPk. The ordinary Waring- Goldbach problem. is equivalent to d(sPk) :>0 for a certains= s(k); it was solved by Hua in 1938 (cf, L. K. Hua, Additive Zahlentheorie, Leipzig, 1959), In the present paper, Hua's approach is generalized to arbitrary algebraic number fields, (Received january 25, 1962,)

62T-61, A. S. KAHR, Massachusetts Institute of Technology, 71 Mt, Vernon Street, Boston 8, Massachusetts. A reduction class for the Entscheidungsproblem.

It has been shown (see Kahr, Moore and Wang, Entscheidungsproblem reduced to the AEA case, to appear in Proc, Nat. Acad, Sci,) that the class F of formulas of the predicate calculus (excluding equality sign and function symbols) of the form: (Ax)(Eu)(Ay)Mxuy• where M is quantifier-free and contains only monadic and dyadic predicates, is a reduction class for the predicate calculus with re­

spect to satisfiability, It is now proved, by a similar argument, that the subclass n 5 ofF comprising those members of F containing not more than five dyadic predicates (in addition to monadic predicates) is also a reduction class. (Received january 25, 1962,)

142 62T-62. A. A. SAGLE, University of Chicago, Chicago 37, Illinois. On derivations of semi­ simple Malcev algebras.

A Malcev algebra is a nonassociative algebra which satisfies the identities suggested by intro­ ducing the commutator of two elements as a new multiplicative operation in an alternative algebra. (See Malcev algebras, Trans. Amer. Math. Soc,, val. 101, Dec, 1961). In this note the derivation algebra .fJ of a finite dimensional semi-simple Malcev algebra is studied and the main result is the following theorem: If A is a finite dimensional semi-simple Malcev algebra of characteristic zero, then every derivation of A is inner and,}) is completely reducible in A. (Received January 25, 1962.)

62T-63. D. J, NEWMAN, Yeshiva University, New York, New York and H. S. SHAPIRO, New York University, New York, New York. Ceby!fev approximation in a Cartesian product space.

Let X andY be compact Hausdorff spaces and !Iii (x), i = O,l, ... ,m and !Vj(y), j = O,l, ... ,n linear!} independent collections of real valued continuous functions on X, Y respectively. Assume moreover

¢0 (x) =: 1, ?Q- 0(y) = 1, and f(x), g(y) are real and continuous on X, Y respectively. Then the best Ceby!fev approximation to f(x) + g(y) on X X Y by "polynomials" of the type Ll= 1 L~ 1aijjl(x) ~j(y) is attained by a "polynomial" of the form L~lbisli(x) + 2:j=lcj¥'j(y). In the special case m = n, X andY are real intervals, and ¢i(x) = xi, ¥'j(y) = yj, this implies that the best Ceby!fev approximation to f(x) + g(y) on the unit square by polynomials of degree n is given by p(x) + q(y), where p and q are best approximations of degree n to f and g respectively. In this case p(x) + q(y) is the unique best approximation to f(x) + g(y) among polynomials of degree :ii n in the two variables (although the uniqueness does not hold in the wider class defined above i.e. polynomials of degree ~ n in each variable separately). The uniqueness result holds also fork variables. (Received January 26, 1962.)

62T-64. D. J, NEWMAN, Yeshiva University, New York, New York and H. S. SHAPIRO, New York University, New York, New York. A quantitative form of Haar's theorem.

Let X be a compact Hausdorff space and f¢i(x)J, i = l, ... ,m real valued continuous functions which form a Ceby!fev system on X. Let f(x) be real and continuous on X, 11'* (x) a "polynomial 11 (i.e. linear combination of the ¢i) of best approximation to f in the sense of Ceby!!ev, and 11"(x) any "poly­ nomial". Then llrr - tr* II;:;; K ( llf - 7r II - llf - 1r* II> where K is a constant depending on the functions l !li(x)} and on f (but not on 1r). In case the f ¢i(x)J are assumed to be a complex-valued (or more generally, taking values in a Hilbert space) Ceby!fev system, and f is a complex (or Hilbert space)

valued function the inequality lin-- 1!"* II :;a K(llf- 11"11 - llf- 1r* 11> 112 holds. All norms are max norms. The stated inequalities, which evidently generalize Haar's theorem, also readily imply certain results of Vallee Poussin and G. Freud. (Received January 26, 1962.)

62T-65. J. A. WOLF, The Institute for Advanced Study, Princeton, New Jersey. Geodesic spheres in Grassmann manifolds.

Let Gn,k(F) denote the Grassmann manifold consisting of all n-dimensional subs paces of a her­ mitian positive-definite left vectorspace Fk over a field F = real numbers, complex numbers or quaternions. Gn,k(F) is viewed as carrying the structure of a Riemannian symmetric space, that

143 structure being defined by the unitary group of pk. Let B be a connected totally geodesic submanifold of Gn,k(F) such that any two distinct elements of B have zero intersection as subspaces of Fk. The main results are that B is a compact Riemannian symmetric space of rank one, that B is isometric to a sphere if and only if every element of B lies in some fixed 2n-dimensional subspace of pk, and that B can be described in a certain way by Clifford algebras if it is isometric to a sphere. The remainder of the paper is devoted to the classification (up to a global isometry of Gn,k(F)) of those submanifolds B which are isometric to spheres. (Received January 29, 1962.)

62T-66. PAUL ERDOS and S. K. STEIN, University of California, Davis, California. Sums of distinct unit fractions.

Two of the questions raised by H. S. Wilf in Bull. Amer. Math. Soc. 67, (1961), 456 concerning the representation of positive integers as the sum of distinct unit fractions are answered in the follow­ ing theorems. Theorem 1. Let f(n) be the smallest number of distinct unit fractions whose sum is the integer n. Then f(n)""' en-Y(where 'Yis Euler's constant). Theorem 2. There exists a subset, R, of the integers such that (a) the density of R is 0 and (b) R can be partitioned into disjoint finite subsets

R 1, R2, •.• , such that each positive rational is the sum of the reciprocals of the elements of precisely one Ri. Theorem 2 can be strengthened to this 'best possible' result: if a 1 < 02 < ... is a sequence of integers with L_ (1/ai) = oo then there is a sequence of integers, r 1 < r 2 < •.•• with ai < ri (i = 1,2, ••• ,), !Ouch that R = U ~ 1 { riJ satisfies the condition of Theorem 2. (Received January 29, 1962.)

62T-67. FRED KRAKOWSKI~ University of California, Davis, California. Curves whose affine images are mirror-symmetric.

Theorem. A Jordan curve J which has the property that all its affine images have an axis of mirror-symmetry is an ellipse. This is proven by showing that J has infinitely many axes of oblique symmetry and hence must be an affine image of a circle. (Received January 31, 1962.)

62T-68. S. I. DROBNIES, William Marsh Rice University, Houston, Texas. Concerning the uniform approximation of a bounded function.

Suppose f(x) is a bounded function on the closed interval [a,b] and c is (b - a). If m(x) is the l.u.b. of the oscillations of f(x) over all intervals of length not greater than (x - a) and m(a) is m(a+) then the interpolated Bernstein polynomials Bnf(x) = 1/cn ~=O (~)(x - a)k(b - x)n-kf(a + ck/n) are such that An= {Bnf(x) - f(x>j f m(a + en - 1/ 2) · 2:, ~~B n] (k + 1)2/211"}2~k+ 1exp(- t2 /2)dtr 1 = O(log n/n)1/ 2 for every x in [a,b]. However, when either (x - a)/c or (b - x)/c = O(log n/n)l/2 then An= O(n -l/3) and An= O(n- 112) elsewhere in case f(x) has a root at each interpolation point (a+ ck/n) for which { ck - n(x - a)} /2n((x - a)(b - x))112 :!l (log n/n)112• m(a+) = 0 in case f(x) is continuous and some Tauberian theorems concerning Hausdorff and Euler summability are obtained by using the fact that /~Bns [nxjdg(x) and Bns jp.x] are the Hausdorff and Euler transformations of the sequence {snJ. (Received January 31, 1962.)

144 62T-69. R. E. STONG, 5442 South Harper, Apartment 101, Chicago 15, Illinois. Relations among Whitney classes of n-manifolds.

The following result has been obtained: Theorem. There is no relation of dimension less than or equal to [n/2] among the Whitney classes of all manifolds of dimension n (i.e. if 0 i' u E: Hm(BO,Z2) and 0 = 7;*(u) E Hm(M, z 2 ) for all n-manifolds M, then m > [n/2]). This result is obtained by showing 2 that 't"*: Hk(BO, z 2 )--. Hk(M k,z2) is monic, where M 2k is the 2k-dimensional manifold obtained by taking the connected sum of the spaces P 2a 1x ... X P 2ar, for all partitions (a1, ... ,ar) of the integer k. (Received February 2, 1962.)

62T-70. R. E. STONG, 5442 South Harper, Apartment 101, Chicago 15, Illinois. Some results on Whitney numbers.

The following result has been obtained: Theorem. Let Mn be a manifold such that every

Whitney number divisible by some w i with 1 & i ;; 2 s is zero. Then: (a) if 0 < n < 2 s+3, n f 2 s+2 , 2s+2 + 2s, or 2s+2 + 2s+l, Mn is cobordic to zero; (b) if n ;f 0 (2s+2), wn(Mn) = 0. This is based on a reduction to a homomorphism of H*(BO;Z2) taking w 1, ... ,w2 s to zero, and the result: Lemma. If !II: H* (B 0; Z 2) ---+ Q is a homomorphism of left a 2 algebras with !li(v i) = 0 for 1 ;;; i ~ 2 s, then (1) !li(vi) = 0 fori F 0 (2s+l), and (2) !li(Sqivj) = 0 for all j and 1;;; i < 2s, where Sq v = w. (Received February 2, 1962.)

62T-7l. j. A. SYNOWIEC, DePaul University, 5248 South Paulina Street, Chicago 9, Illinois. Introduction to the geometry of the real hyperbolic plane.

The system of hyperbolic complex numbers gives rise to an affine space of two dimensions over the field of real numbers, called the real hyperbolic z-plane. The elementary properties of straight lines are developed, and it is found that passing through each point of the hyperbolic z-plane, there are exactly two isotropic lines. A notion of distance is developed, but the resulting space is not a metric space (in the usual sense). Triangles are defined as three points Pl' P 2, P 3 , together with the set of line segments (Pl' P 2), (P2, P 3), (P 3 , P 1). Due to the fact that the distance is not positive definite, three distinct types of triangles must be considered. Formulas are obtained for the area and the perimeter in terms of the sides for each type of triangle. Finally, the differential geo­ metry of the hyperbolic plane is developed. (Received February 2, 1962 .)

62T-72. j. A. SYNOWIEC, DePaul University, 5248 South Paulina Street, Chicago 9, Illinois. Basic algebra of the real hyperbolic complex number system.

The system of hyperbolic complex numbers is defined and is shown to be a commutative ring with unit, possessing divisors of zero. The notions of norm and isotropic components of a hyperbolic complex number are defined, and are utilized in obtaining simple algebraic properties analogous to those of ordinary (elliptic) complex numbers. This system is characterized by the following property. The system of hyperbolic complex numbers is a commutative ring with unit, possessing divisors of zero, which admits an involutorial automorphism, not the identity, such that the self-conjugate elements form a field which is isomorphic to the real number system. Matrix representations of this system

145 are also given. This is fundamental in the study of the projective geometry of a real hyperbolic para­ boloid. (Received February Z, 196Z.)

6ZT-73. GEBHARD FUHRKEN, University of California, Berkeley 4, California. On minimal models of complete theories.

Consider complete first-order theories T with countably many nonlogical constants. By a sub­ system is meant an elementary subsystem. A model <7t of T is (a) minimal iff it has no proper sub­ system; (b) prime iff every model of T has a subsystem isomorphic to a. F 1 (T) is the Boolean alge­ bra of all formulas of T having v0 as only free variable. (1) There is a T such that (i) every model of T has a minimal submodel, but (ii) T has no prime-model. (Z) There is a T some of whose models have a minimal submodel, while some do not. (3) There is aT which has a model containing an element which satisfies neither an atom nor an atomless element of F 1 (T). (4) There is a complete theory T' with uncountably many individual constants such that F 1 (T') has a nonprincipal dual prime­ ideal simultaneously satisfiable in all models of T'. These examples show in particular, that a theorem of Ehrenfeucht's and a theorem of Vaught's (see Theorem Z.l in Proc. Symp. Found. Math. Warsaw 1959, 303-321) cannot be improved by either (i) requiring that the model ofT, whose exist­ ence is asserted in these theorems, can be found among the submodels of a given model of T, or (ii) removing the restriction on the number of nonlogical constants. (Received February 5, 196Z.)

6ZT-74. K. T. HAHN, Stanford University, Stanford, California. Bounds for certain invariants in conformal mappings.

The author considers a domain Din the z-plane (z = x + iy) bounded by a= [zl lzl = 1], a 0 = (z I lz - al = r], - 1 + r < a < 1 - r and a number of closed Jordan curves ;9k. The curves )'k lie in the domain bounded by a and "O· Let A 1 = ~ I lz I < 1, lz - a I > r], Az = [z I lz - b I < f'• lz - a I > r ], r <. fl < 1, be exterior and interior domains of comparison for D, respectively, i.e., A 1:J D? Az. dsD(z) = [KD (z,-zJiiZ ldz I denotes the line element of the Bergman metric of a domain D. (dsD(z) is invariant with respect to conformal mappings). Let C 1 (D; a0) be the collection of all rectifiable closed curves in D which enclose a0 ; here all c in C 1(D;a0) have their (ordinary) lengths less than a fixed finite upper bound. The Bergman invariant length of c is denoted by LD(c). The author proves 1 that there exists a curve em in c (D;a.0) of the smallest invariant length and that E(R1) :; L 0 (cm)/Z ;;o E(R2), where Rk' k = l,Z, are explicit functions of a, b, ,0• r. E(r) = [11";1"(log r;- Z log r, Z7ri) - i ~(1/'"i; - Z log r, Z7ri)] 112, fJ. and Care the Weierstrass elliptic functions (see B. Epstein, J. Math. and Mech. 7 (1950), 9Z 9). (Received February 5, 196Z.)

6ZT-75. W. P. HANF, University of California, Berkeley 4, California. Isomorphism in elementary logic. Preliminary report.

Let L = (z,K,M) be an interpreted language, where :E is the set of all sentences, M is an operation which correlates with each 0" ( :E the class M(o-) of all models of CT, and K = U M(

(i) rd(I:) = ~·. (ii) ~(K) = K', (iii)l1! E: M(<>) iffjb'(Ot) € M(rd(CT)). Let L 0 and L 1 be languages of first

146 order predicate logic with identity; Lo with a binary relation symbol as the only nonlogical constant,

while L 1 has a finite number of constants such that at least one is an n-ary relation with n ;;:; 2, or else at least two are unary operation symbols. Theorem 1. Lo ~ L 1; moreover, !If and !6-l can be to be primitive recursive (under natural enumerations of :E and :E'). Let Bo(B 1) be the Boolean alge­

bra of equivalence classes of sentences of L 0 (L 1). Theorem 2. (i) Bo ~ B 1, (ii) Bo !!!' Bo X Bo, (iii) if s 0 ~ C X D, then Bo ~ C or s0 '!!D. (Received February 7, 1962.)

62T-76. WITHDRAWN.

62T-77. WITHDRAWN

62T-78. R. E. GREENWOOD and D. R. STOCKS, The University of Texas, Austin 12, Texas. A coloring problem with square tiles. Preliminary report.

Squares are divided into 4 equal triangular areas by drawing the two diagonals, and using three colors with repetitions to paint the triangular areas, 24 different square.s are obtained. P. A. MacMahon (New Mathematical Pastimes, (1921), 23) considered the problem of arranging these squares in a 4 X 6 rectangle so that each pair of internal touching edges are of the same color and so that the boundary colors meet certain specifications. For the case where all 20 boundary regions have the same color, MacMahon implies that there are no varieties (save trivial ones arising from permuta­ tions, reflections, and similar "duplications"). Martin Gardner in the "Mathematical Games" section of the March 1961 Scientific American, page 166, states (on the authority of MacMahon) that there is "only one solution pattern." The authors have found several different patterns instead of the "unique" pattern implied by MacMahon and inferred by Gardner. Additional specifications such as

147 the number of "connected" pieces for each of the three colored areas are being made to improve MacMahon's classification, (Received February 12, 1962.)

62T-79, D. M. TOPPING, Tulane University, New Orleans 18, Louisiana, Extreme points in (M)-spaces,

Let V be a pointwise vector lattice of real functions on a set X, v+ = {! E: V: f 0}. If f #: 0 #: g with f 1\ g = 0, f and g (linearly independent) lie on the surface of the positive cone v+ in supporting hyperplanes which are nullspaces of linear lattice functionals. Now suppose V consists of

bounded functions (these V's are just the pre-(M)-spaces of Kakutani) and let U = {f E. V: - 1 ~ f 1}, P = U {') v+. For x ( X, set m(x) = infff(x): fEU}. M(x) = sup{f(x): f E. U} and p(x) = sup{f(x): f E P}. Then (1) f E U is an of U if and only if for each x E X, f(x) = m(x) or f(x) = M(x); and

(2) "f E P is an extreme point of P if and only if for each x E X, f(x) = 0 or f(x) = p(x). The set E of all extreme points of P is a Boolean ring under V and 1\ and lies on the surface of v+ with the excep­ tion of at most one point interior to v+ (the maximum element, if it exists). (Received February 12, 1962.)

62T-80. T. G. McLAUGHLIN, University of California, Los Angeles 24, California. Some remarks about productive and .contraproductive centers,

Let a. be a productive set of natural numbers. Remark 1. If 1r is a productive center of a., and if ,e is any recursive subset of 1r, then rr- f3 is a productive center of a.. Remark 2, If,.. is a

productive center of a., then there is a productive center 1r' of a. such that 11"'

recursively enumerable, Let a. be a contraproductive set of natural numbers. Remark 3, If 1r is a contraproductive center of a., and/} is any recursive subset of 7r, then 1r- f1 is a contraproductive center of a.. Remark 4, If 1i is a contraproductive center of a., then there is a contraproductive center

7r' of a. such that 1r' s;r 1r and 1r - r' is not recursively enumerable. Remarks 1 and 2 sharpen somewhat the content of Proposition D of Dekker's paper Productive sets (Trans, Amer. Math, Soc. (1955)); similarly Remarks 3 and -4, w.r.t. Proposition F of that article. (Received February 12, 1962,)

62T-81. K. T. HAHN, Stanford University, Stanford, California, Bounds for certain invariants in pseudo-conformal mappings.

Let Ak be a product domain in the space of two complex variables z l'z2 bounded by Sk and s 0, k = 1,2; here s 1 = (lzjl = 1, j = 1,2], s 2 = [lzj- bjl = Pj' j = 1,2J, s 0 = (lzj- ajl = rj' j = 1,2], s 0 C [lzjl < 1, j = 1,2) and s 2 C A1• The author considers a domain D of the following type: (1) D has A 1 and A2 as exterior and interior domains of comparison, i.e., A 1 :::> D :JA2; (2) the boundary b of D contains the surface s0. Let S = ~j = fj(s 1,s2), 0 ~ sj 2 21T; j = 1,2] be a differentiable surface in the (zl'z2)-space, dbs = lo(fl'f2)/B(s 1,s2)jds 1ds2 is called the B-area element of Sand dBD(S) = j!

148 8-area, 8D(Sm) is a monotone functional of D. 8 A (S;J can be computed in an explicit form, where S~ is a surface of the smallest invariant area in C~Ak;S 0 ). Using these facts, the author obtains bounds for 8D(Sm). (Received February 19, 1962.)

62T-82, D. A. ROBINSON, Georgia Institute of Technology, Atlanta 13, Georgia, Concerning two que·stions of S, K. Stein. Preliminary report,

If £G, •} is a group, define x o y = xy- 1x for all x, y E G. The resulting system fG,o} is called

the~ of fG,·}. (See Bruck, A survey of binary systems, Springer, 1958, p. 120.) The terms left­ distributive, right-distributive, and medial as applied to binary systems are defined in accordance with Stein (Trans. Amer, Math, Soc, 85 (1957), 228-256). Theorem 1. A group fG, •} is nilpotent of class 2 if and only if its core {G,oJ is medial, Theorem 2, A group {G, •J is an Engel group of type 2 if and only if its core [G,o J is right-distributive. These, and similar, results are used to obtain examples of left-distributive quasigroups which are not right-distributive, The techniques and examples are distinct from those of Stein (Publ. Math. Debrecen. 6 (1959), 10-14), Also examples of quasigroups which are left-distributive, right-distributive, commutative, but not medial are obtained in answer to a question of Stein (Trans. Amer, Math. Soc, 85 (1957), 253). (Received February 16, 1962.)

62T-83, ECKFORD COHEN, University of Tennessee, Knoxville 16, Tennessee. The number of planes contained in the complement of a quadric in an affine Galois space,

Let q denote a power of an odd prime, let F be the Galois field of q elements, and let s3 be the 3-dimensional vector space ove'r F. For elements a. ofF, it is shown that the number of 2-dimen­

sional subspaces of s3 contained in the complement of the quadric, a.= xf + x~ + x~, is equal to 0 or q + 1 according as -a. is a square or a nons quare of F. The general problem of the title can be reduced to this special case, (Received February 14, 1962.)

62T-84. T. G. McLAUGHLIN, University of California, Los Angeles 24, California, Two remarks concerning productive and contraproductive centers.

Remark 1. Let a. be a productive set of natural numbers. If 1r is any productive center of a.,

then (i) if (3 is a recursive subset of Tr, then 1r- fJ is a productive center of a., and (ii) a. has productive centers t",~ such that~ ~ 1:" ~ 1r. 1r- 1:" is not r,e., and -r- > is r.e. but not recursive, Remark 2, Let a. be a contraproductive set of natural numbers. If 7rc is any contraproductive center of a., then

(i) if fJ is a recursive subset of 7rc• then ~ - f3 is a contraproductive center of a., and (ii) a. has con­

traproductive centers X"c• ~c• such that >c ~ t"c £ 7ic• 'Ire - "Lc is not r,e,, and rc - ~c is r,e, but not recursive. These observations sha.rpen somewhat the content of Propositions D and F of Dekker's article Productive sets (Trans, Amer, Math, Soc. 78 (1955), 129-149), (Received February 16, 1962,)

149 62T-85. R. P. GILBERT, Institute for Fluid Dynamics and Applied Mathematics, University of

Maryland, College Park, Maryland. Harmonic functions in four variables with rational p4-associates.

In this paper, properties of solutions of the Laplace equation in four variables are investigated by means of integral operator methods. Results concerning location of singularities and certain re­ presentations of solutions are found. An inverse for the operator p 4(f) also is obtained by using hyper­ spherical harmonics and their orthonormality property on the unit hypersphere. (Received February 12, 1962.)

62T-86. DAVID ZEITLIN, 1650 Vincent Avenue North, Minneapolis 11, Minnesota. On a formula for preferential arrangements.

This paper presents an elaboration and extension of the results cited by 0. A. Gross (Amer. Math. Monthly 69 (1962), 4-8). Identities satisfied by generating functions are exploited to obtain re­ cursion formulas. The principal result is as follows: (1) Let n = 0,1,2, .•• , and let p > 1. Then

L~oknp-k ..v n!/lnn+lp as n ~oo. Gross (see above) proved (1) for p = 2. Additional results are obtained by applying formulas derived in the author's paper (Amer. Math. Monthly 68 (1961), 986-989). The following two limits are of inte~est: (i) for n = 0,1, •.• , (1 - x)n+lL.~oknxk _.. n! as x ---'>1-, (ii) (n!) -lL~n+l {(-l)k(k - l)(k - Z) ••• (k - n)Bkf /k!-. 0 as n- oo, where Bk are Bernoulli numbers. (Received February 14, 1962.)

62T-87, DAVID ZEITLIN, 1650 Vincent Avenue North, Minneapolis 11, Minnesota. On the evaluation of a class of infinite series.

Using differential calculus, a closed form has been obtained for E~ouk+mkpxk, m = 0,1,2,. •• , p = 1,2,3, ..• , where uk, k = 0,1, .•• , satisfies a linear difference equation of order n with real, constant coefficients. A closed form for the above series can also be obtained by a strictly manipulative procedure (i.e., no calculus), and this approach is illustrated for n = 2. The result for the special case, u k :: 1, is well known, and has been proved in the past independently by I. J. Schwatt (1924), R. Stailey (1949), M. S. Klamkin (1957), and D. Zeitlin (1961), each using a different method. (Received February 14, 1962.)

62T-88. H. H. GUGGENHEIMER, University of Minnesota, 119 Folwell Hall, Minneapolis 14, Minnesota. Curves in Euclidean n-space.

The theory of curves in Euclidean n-space is usually based on Frenet formulae that are a straight generalization of those in 3 dimensions. A moving frame of orthonormal vectors ai' i = 1, ..• ,n, satisfies dai/ds = -ki-lai-l + kiai+l" The differential invariant kn_ 1 is of order n. A simple count of dimensions shows that a complete system of invariants must exist of orders l!E [(n + 1)/2] + 1. In the paper, such a system of invariants is constructed. E. g., for n = 4, the fourth order invariant k 3 may be replaced by the integral invariant 6 = j'det.(a1,a2,a3,da3). The corresponding frame may be computed for curves whose third derivatives are of bounded variation. (Received February 16, 1962.)

150 62T-89. J, A. WOLF, The Institute for Advanced Study, Princeton, New Jersey. Locally symmetric homogeneous spaces. III.

Results previously announced (Abstract 61T-162; Notices Amer. Math. Soc. 8 (1961), 356) are sharpened, removing all restrictions concerning E 6/F 4• In particular, if T' is the group of deck transformations of a universal Riemannian covering M - N where M is a Riemannian symmetric manifold, then N is a Riemannian homogeneous manifold if and only if r is a group of Clifford trans­ lations of M. (Received February 19, 1962.)

62T-90. ECKFORD COHEN, University of Tennessee, Knoxville 16, Tennessee. Trigonometric series expansions of certain arithmetical functions.

Let u(n,r) be defined for positive integers n and r by b d ll~(n,d) = r or 0 according as rUn or otherwise. The function u(n,r) is periodic in n with period r'f(r), where 'f(r) denotes the product of the distinct prime divisors of r. In this paper certain functions defined in terms of sums over unitary divisors are expanded into series of the form L~ 1 a(r)u(n,r). These series are analogous to some of Ramanujan's classical trigonometric expansions of the elementary arithmetical functions. Example:

O"*(n)/n = (1t' 2 /6)L:~ 1 (J(r)Jr 4)u(n,r), where o-*(n) is the sum of the unitary divisors of n and J(r) denotes the Jordan totient of second order. (Received February 19, 1962.)

62T-91. P. A. TUCKER, University of Illinois, Urbana, Illinois. Note on the reduction of induced representations.

Let G be a finite group which is an extension of H by B. LetT be an irreducible representation of H over an algebraically closed field K with representation module L. A construction of the components of the induced module L G = K (G) ® K (Hf. is given. This extends earlier results (Abstract 61T-271, these Notices, 8 (1961), 519) in which it was assumed that G is a split extension of H by B. Let B = {1,b2•···•bJ and G = H + b2H + •• , + bnH· Let {(bi,bj)} be the factor set of the extension. Define a subgroupS of B by S = t? E: B: 'f(b) is equivalent to T}. For each b E S, select Db E HomK (L,L) such that DbT(b)(h)Di/ = T(h), for every h ~ H, and D 1 = 1L. Define k(bi,bj) E K by DbiDbj T((bi,bj) - 1) = k(bi,bj)Dbibj" Let A denote the k - 1-twisted group algebra of S, i.e., the crossed product of K and S with factor set k - 1• Then, every left ideal I of A determines a left K (G)- submodule

C(I) of LG such that: (1) dim C(I) = [!3:S] dim L dim I; (2) A= 11 ED ... 9 It implies LG = C(I1) ED ... 6l C(It); (3) Hom A (1 1,12) ~ KHomK(G)(C(I1), C(I2)), (if 11 = 12, this is an algebra isomorphism). Thus, a direct decomposition of A into indecomposable components determines a direct decomposition of L G into indecomposable components. (Received February 19, 1962.)

62T-92. C. H. EDWARDS, JR., University of Wisconsin, Madison, Wisconsin. Products of contractible open manifolds.

McMillan and Zeeman have shown that un X E 2 = Rl+2 if un is a contractible open n-manifold (Notices Amer. Math. Soc. 8 (1961), 506). As corollaries to the proof of this theorem, the following results are obtained: (1) If urn and yn are contractible open manifolds of dimensions m and n respectively, then urn X yn X E 1 = Em+n+l. (2) If urn, vn, wr are contractible open manifolds of

151 dimensions m, n, r respectively, then um X yn x wr = Em+n+r. (3) If un is a contractible open n-manifold and w3 is a contractible open 3-manifold, each of whose compact subsets can be imbedded in E 3, then Un X w 3 = En+3• (4) Let {Hi} be a sequence of n-dimensional cubes with handles such that each Hi is inessential in Int Hi+l• If n > 3, then UHi =En. (Received February 19, 1962.)

62T-93. 0. L. MANGASARIAN and J. B. ROSEN, Shell Development Company, Emeryville, California. Inequalities for stochastic nonlinear programming problems.

The inequalities given by Albert Madansky (Management Science 6 (1960), 200) have been generalized to a class of nonlinear programming problems via the duality theorems of nonlinear pro­ gramming. In particular, the constraints considered are of the type g(x) + h(y) ;~;. b where the com­ ponents of the vectors g and h are nonlinear concave functions of their arguments and satisfy some further restrictions. The right-hand side b is subject to a random variation with an expected value Eb, It is desired to minimize the expected value of the convex objective function !il(x) + j6'(y) subject to the constraints. If ?"(x,b) denotes miny [!6(x) + ~(y)] subject to the constraints, then under certain restric­ tions the following inequalities hold Er(b, i(Eb)) s; minxE?"(b,x) s;. E minx?'(b,x) ;:: minx?"(Eb,x), where x(Eb) denotes the solution of minx?'(Eb,x). It is also shown that the function minx'Y(x,b) is a convex, continuous function of band that the sometimes-sharper upper bound to E minx?'(b,x) given on p. 201 of Madansky also holds if b is defined over a bounded rectangle and has independent elements. (Received February 19, 1962.)

62T-94. DAN AMIR, Hebrew University, Jerusalem, Israel. C(S) spaces of the PA -type.

A Banach space B is called a P~ space if from every Banach space Z :J B there is a projection on B with norm ;:; A. A topological space K is called Stonian if the closure of every open set in K is open. Let C(S) be the Banach space of all the continuous real-valued functions on a compact (Hausdorff) space S with the usual norm, and suppose that C (S) is a P.\ space for a certain A. It is proved that: (1) S contains an open dense subset K which is Stonian. (2) Every convergent sequence in S is eventually constant. (3) Every infinite closed subset of S has a perfect nonvoid subset. (4) If i\ <. 2, Sis Stonian (and hence C(S) is a P 1 space). (5) For 2 ~A<. 3 further results on the structure of S were obtained. Examples are constructed of C(S) spaces that are P;>. spaces and not P}I spaces for any p. <. ~. for each of the numbers A = 3 - 2/n, n = 1,2 ••• and for ~ = 3. Part of the above results were independently obtained by Z. Semadeni .and J. R. Isbell (private communication). (Received February 19, 1962.)

62T-95. WILLIAM CRAIG, Philosophy Department, University of California, Berkeley 4, California and WILLIAM HANF, University of California, Berkeley 4, California. On relative charac­ terizability in a language. Preliminary report.

Let L be any language which includes first-order language with identity, contains an unlimited supply of n-placed predicates for each n E cv, allows relativization of its sentences to 1-placed predicates, and, for convenience, has sentences of finite length only. Let a class K of relational systems be in PC'.,o.(L) iff for some set ::t: of sentences of L: (") K is the set of all relativized reducts

152 (s 0 , R 0!'s 0, R 1ts 0 , .••) of system·s (A, R 0 , R 1, •.•• s0, S 1, ...> satisfying ,E. I: relatively characterizes Oliff (*) holds and .C is in K iff J:, is isomorphic to t:l't-. :E characterizes ~ if, in addition, s0 = A. (I) Suppose tJt has no nontrivial automorphism and some :E in L of power .zty and with a model of power

.:rt: 8 relatively characterizes IX. Consider any .1: whose power B is less tha~ the first strongly inaccessible cardinal, if any, greater than A. Then some I: in L of power 2B + l\",... + J (i3 + .If,.+ '3 assuming the G. C. H. ) and with a model of power 8 + ~.5 relativ!ly characterizes J; where 3 is the order of J:,, (II) For Oi', 8, ~?' as in (I), some I:' in L of power 2B + lt, (8 + .1\";v assuming the G. C. H.) has no model while every subset of power less than B has a model. (III) Suppose :E in L relatively characterizes J:, and has a model of power B. Then some ::1:' in L of power ;E" character­ izes J:. (Received February 19, 1962.)

62T-96. WILLIAM CRAIG, Philosophy Department, University of California, Berkeley 4,

California. Relative characterizability and generalized existential quantifiers. Preliminary report.

The language L and notation is as in abstract 62T-95. Let 3li'J vi··· express: There are at least :fiJ objects vi such that .•. . Quantifiers of this kind are among those considered by Mostowski in Fund. Math. 44 (1957), 12-31. Let L' be a language obtained from first-order language with identity and with an unlimited supply of n-placed predicates for each n E "' by adding quantifiers 3J;t • - J Using (I) of abstract 62T-95 we obtain (1) If 3~"3 is in L' for every strongly inaccessible Jl;J :;t, C, then C is relatively characterizable in L '. By a construction related to that for the Skolem normal form, (2) If (GVJ' <)is relatively characterizable in L for every 3-lt_Jin L', then PC;. (L') <;:;; PCA (L). Thus L' is in some sense minimal. (Received February 19, 1962.)

62T-97. JORAM LINDENSTRAUSS, The Hebrew University, Jerusalem, Israel. On the

modulus of smoothness.

For a Banach space X the modulus of smoothness f(t'), 7: > 0, is defined by 2p(7:) =

sup(llx + Yll + llx- yll- 2), the sup taken over all x EX, y EX with llxll = 1, IIYII = ~- It is known that X is uniformly smooth if and only if p(t') = o(t"). p(t') can be computed exactly from the modulus of

convexity [S (e)] of X*. One has f'(T) = sup(€1:/2 - 8 (e)) the sup taken over 0 ~ e § 1. From this

formula some information on p is obtained by using known results on S ; e.g. p('C) ;;: (1 + ?; 2) l/2 - and equality holds for every r if and only if X is a Hilbert space. A result on series: Let X have modulus of smoothness f such that f(2'"C)/ p(7:) is bounded. Let fxi}~ 1 be a sequence such that Z f( llxi II> ""oo, then one can choose signs ei (i.e. ei = ± l) such that L eixi converges. For the Lp spaces one obtains: Let L: llxdla.(p) converge, then there is a choice of signs e 1 such thatz=. eixi con­

verges, where a.(p) = p for l ~ p ~ 2, a.(p) = 2 for 2 ~ p < oo, a.(oo) = 1 and these are the best possible

constants. Some results of a similar nature were obtained. These results complement, in a certain sense, results on the convergence of series for every choice of signs obtained by W. Orlicz [i)tudia

Math. (1933)] for Lp spaces, and M. I. Kadec [Uspehi Mat. Nauk (1956)] for general uniformly convex spaces. (Received February 19, 1962.)

153 62T-98. JORAM LINDENSTRAUSS, The Hebrew University, Jerusalem, Israel. The extension of compact operators.

Properties of certain Banach spaces (to be called here Ni\- spaces) are studied. A Banach

space X is anN~ -space, if there exists a set B1: of finite-dimensional subspaces of X, directed by

inclusion, such that X = UrB't, and every B-. is a P~ -space. The following are some of the results

obtained: AnN;~. -space is a P;U-space (for a certain )l) if and only if it is complemented in some con­

jugate space. A C(K) space is an N-1 -space for every A. ? 1. If X is an N~+e (or P~+e)-space for every e ;;. 0, then X** is a P)l -space. If X is an N-1 -space then every compact [resp. weakly compact] operator T from X to a Banach space Y has a compact [resp. weakly compact] extension T from any z .::J X to Y with liT II ;;; ,.\liT 11. If Y is a conjugate space a similar result holds for every bounded T. Any compact T from some Y to anNA -space X can be extended (for every e > 0 and Z .::J Y) to a compact T from Z to X with IITII ~(A+ e)IITII. Converse results also hold; e.g. let X be a separable

Banach space with a basis which is not an N~ -space (for any A). Then there exist a conjugate space Y, a space z .::J X, and a compact T from X toY, which has not even a bounded extension from Z toY. (Received February 19, 1962.)

62T- 99. LOUIS DE BRANGES, 778 Upper Gulph Road, Wayne, Pennsylvania. The growth of entire functions of minimal exponential type.

If S(x) ~ 1 is a function of real x such that log S(x) is uniformly continuous and j'(l + t 2) - 1log S(t)dt = oo, there exists a nonconstant entire function F (z) of minimal exponential

type such that IF (x) I ~ S(x) for all real x. Proof depends on the construction of a suitable Hilbert space of entire functions. As a result nonconstant entire functions of minimal type may be construe- ted so as to remain bounded on rather general real sets. Compare with Chapter 9 of N. Levinson, Gap and density theorems, American Mathematical Society, 1940. (Received February 20, 1962.)

62T- 100. F. B. CORREIA, COMBARFORLANT STAFF, cjo F. P. 0. Navy 568, New York,

New York. Dissertation on Correia's sequence and its relationship with Cram~r's conjecture.

This paper is the result of further research on this author's Ph.D. thesis, A theory of primes. One of the results obtained in the thesis is that the difference of consecutive primes is as follows, dn = Pn+1 - Pn = 2(mn - mil. + 0(1))(1 + o(l)) log2pn, where mn and mri_ were functions defined on prime number sets and introduced in the thesis. The sequence (mn - m;'i) became important because

mn - m:J. = 0(1) became a necessary and sufficient condition for Cram~r's conjecture which is that

dn = O(log2pn). The sequence (mn - mti) turns out to have some remarkable properties as follows: (1) its lim inf = 0, (2) every non-negative integer is a limit point of the sequence, (3) it contains every non-negative integer, (4) the largest integer k in the first n terms is k ~ n l/3, (5) the number of terms with value k in the first n terms is asymptotic to n/k(k + 1)(k + 2), (6) the weight function for each integer k in the sequence is 1/k(k + l)(k + 2), (7) the sequence is divergent. The divergence of the

sequence proves Cram~r's conjecture false. Solutions of other related classical problems are in-cluded, the asymptoticity of consecutive primes, and the existence of primes between positive integral powers of consecutive integers. (Received February 12, 1962.)

154 ERRATA- Volume 9

H. G. ELLIS. Solutions of first order, analytic differential equations. Prelim­ inary report. Page Z46, Abstract 580-39.

Line 6, For "R ~ 0" read "R :> 0".

H. G. ELLIS. Solutions of first order differential equations, including singular cases. Preliminary report. Page 579, Abstract 587-36,

Line 3, For 'J3H(x,tlu(x) j)dx ~ K(t) • lu(q) - u(p) I for 0 < p < q ~ a" read Jo+H(x,tlu(x) j)dx ;:; J(t) lui".

AUBERT DAIGNEAULT. Extension of isomorphisms ofpolyadic algebras. Page 609, Abstract 587-130. Line 4. Replace "of obvious reasons" by "for obvious reasons".

Line 13. For "AzA1" read "Az- A1". Line 14, Delete the period immediately after "Svenonius".

K. W. KWUN. Upper semicontinuous decompositions of then-sphere. Page 6Z4, Abstract 61T-317, Line Z, For "any set homeomorphic to the complement of" read "any set whose complement is homeomorphic to that of". Page ZZ5, Line 8. For "n-cells" read "disjoint a-cells".

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Contact the office of your choice: R.K.PATTERSON WILLIAM LOWE REMINGTON RAND UNIVAC REMINGTON RAND UNIVAC Univac Park P.O. Box 6068 St. Paul16, Minnesota San Diego 6, California

R E M N G T 0 N R A N 0 UNIVAC DIVIBIDN DF BPE----V --AND CD--PD--ATIDN

There are also immediate openings T.M.McCABE D. CLAVELOUX in all areas of digital computer Rem. Rand Univac Rem. Rand Univac developmellf at our other laborato­ P.O. Box 500 Wilson Avenue ries. Inquiries should be addressed to: Blue Bell, Penn. So. Norwalk, Conn. (An equal opportunities employer)

162 INDEX TO ADVERTISERS

Name Page Name Page

Academic Press, Jnc. • . • • • • • • • • • • . • • • • • • • 156 Operations Evaluation Group 161 American Mathematical Society •.••.••••. 87, 160 Pergamon Press, Jnc. • ••••.•••••••••••• 159 Cushing-Malloy, Jnc. • • • • • • • • . . . . • . . • . • . . 160 Remington Rand UNIVAC Division of Sperry Rand Corporation •.•..• 162 Dynamics Research Corporation • • • • . . • • • . • 102 The Ronald Press Company • • . • . • • . • • • • • • 98 Gordon and Breach, Science Publishers, Jnc.. 157 Sperry Rand Research Center • • • • . • • . . • • 158 Lockheed Missiles and Space Company • • • • • • 160 U.S. Navy- Bureau of Weapons • . • • . • . • . • 161 Mechanical Enterprises, Jnc. • • . . • • . • • • • . . . 77 University of Wisconsin Press • . • • . . • • • . • 159

RESERVATION FORM

ATLANTIC CITY, NEW JERSEY, MEETING, APRIL 16-19, 1962

To reserve a room at the Chalfonte-Haddon Hall Hotels, fill out the attached reservation card and mail it directly to Chalfonte-Haddon Hall Leeds and Lippincott Company Atlantic City, New Jersey Reservations will be acknowledged by the hotels. The numbers beside the boxes are prices, not including an additional 3% City Tax.

American Mathematical Society Meeting PLEASE RESERVE THE FOLLOWING ACCOMMODATIONS No. of Rooms HADDON HALL Sinsde Bath 1 Person Twin Bedroom Bath 2 Persons Without Ocean View 09 010 012 014 Q15 014 016 017

Side Ocean View 017 019 021 019 021 023

Ocean Front Q25 028 027 Oao

Double and Parlor 034 042 044

Double and Parlor Ocean Front 057 CHALFONTE

Without Ocean View 08 09 on 011 013 Side Ocean View 014 017 016 019 Ocean Front 017 019 021 019 021 023 EACH ADDITIONAL PERSON IN DOUBLE ROOM $4. 00 Student Room (running water, no bath attached) Single D 5 Double D 7

I will share a room with------­ Expect to arrive------Depart ------­ Name Address

163 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE PAID AT 190 Hope Street PROVIDENCE, RHODE ISLAND Providence 6, Rhode Island AND ANN ARBOR, MICHIGAN

Return Requested

MATHEMATICS OF COMPUTATION Beginning with Volume 16, Number 77, January 1962, The American Mathematical Society will publish the journal MATHEMATICS OF COMPUTATION for the National Academy of Sciences - National Research Council. In addition to reviews and notes, MATHEMATICS OF COMPUTATION publishes original papen covering such topics as advances in numerical analysis, the application of numerical methods and high-speed calculator devices, the computation of mathematical tables, and the theory of hi"gh-sp'eed calculating devices and other aids to computation. The Journal is published quarterly in one volume per year. Subscription per year , $8.00 Single copies $2.50 Send Orden to AMERICAN MATHEMATICAL SOCIETY 190 Rope Street Providence 6, Rhode I.land