THE AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. Walker

CONTENTS

MEETINGS

Calendar of Meetings • . • • • • • • • • . • • • • • • • • • • • • • • • • • • • . • • 608 The November Meeting in Atlanta, Georgia • • • • . • • • • • • • . . • • • • 609 Abstracts of the Meeting - pages 643-650 The November Meeting in Pasadena, California • • • • • . • • • • • . • • • • 613 Abstracts of the Meeting - pages 651-660 The November Meeting in Madison, Wisconsin • • • • • • • . • • • • • • • • 619 Abstracts of the Meeting - pages 661-667

PRELIMINARY ANNOUNCEMENT OF MEETING. . • • • • • • . . • • • • • • • . • • • • 622 NATIONAL ACADEMY OF SCIENCES - NATIONAL RESEARCH COUNCIL.. • . • 625

NEW NSF POLICIES AND THEIR IMPLEMENTATION. 0 ••• 0 ••••••••• 0 • • • 627

THE VEBLEN PRIZE •.•.•••••.•••••• o o ••• o •••••• 0 • • • • • • • • • • • • 629

NOTES FOR SPEAKERS ••••••.••••..••••••• 0 ••••••••• 0 • • • • • • • • 630 VISITING FOREIGN MATHEMATICIANS • • • • • • . . • . • . • • • • • • • • • • • • • • • . 631

NEW AMS PUBLICATIONS •••••.••• 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • 636 PERSONAL ITEMS • • • • • . • • • . • • • • • • • . . • . • . • • • . • • • • . • • • • • • • • . • 638 NEWS ITEMS AND ANNOUNCEMENTS •••••••..••.•••••••••••••• QL2,6"18,624 MEMORANDA TO MEMBERS

The Employment Register ••••••••••••.•.•.•••. 0 • • • • • • •• 621

Retired Mathematicians • • • • • • • . •••.•••••••••••• 0 • • • • • • 62 9 Sum:mer Employment Opportunities • • . . . • • • . • • • • • . • • • . • • • . • 642

SUPPLEMENTARY PROGRAM- No. 21 •••••••...••• 0 ••••••••••••••• 641

ABSTRACTS OF CONTRIBUTED PAPERS ••• 0 ••••••• 0 ••••••• 0 ••••• 0 643

INDEX TO ABSTRACTS - Volume 10 •••••••••••••••••••••••••••.. 0 676 INDEX - Volume 10 • • . • • • • • • • • • • • • • • • • • • • • . • . . • • • • • • • • • • • • • . • 697

INDEX TO ADVERTISERS •••••••••••••.•.••••.• 0 • • • • • • • • • • • • • •• 703

RESERVATION FORM .•••. 0 ••••••••••••••••••••• 0 •••••••••••• 703 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*

608 January 23-27, 1964 (70th Annual Meeting) Miami, Florida Nov. 26 609 February 29, 1964 New York, New York Jan. 16 610 April 18, 1964 Reno, Nevada Mar. 5 611 April 20-23, 1964 New York, New York Mar. 5 612 April 24-25, 1964 Chicago, illinois Mar. 5 613 June 20, 1964 Pullman, Washington May 7 614 August 24-28, 1964 (69th Summer Meeting) Amherst, Massachusetts July 3 January 25-29, 1965 (71st Annual Meeting) Denver, Colorado August 30 - September 3, 1965 (70th Summer Meeting) Ithaca, New York August 1966 (71st Summer Meeting) New Brunswick, New Jersey August, 1967 (72nd Summer Meeting) Toronto, Canada * 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 January 9, and February 27, 1964.

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The NOTICES of the American Mathematical Society is published by the Society in January, 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 (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, 190 Hope Street, Providence, Rhode Island 02906. 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, 21, P.L. and R.). Accepted for mailing at the special rate of postage provided for in section 34,40, paragraph (d).

Copyright© 1963 by the American Mathematical Society Printed in the United States of America Six Hundred Fifth Meeting Georgia Institute of Technology Atlanta, Georgia November 15-16, 1963

PROGRAM

The six hundred and fifth meeting ACCOMMODATIONS of the American Mathematical Society will Since there are other meetings and be held at the Georgia Institute of Technol­ conferences in Atlanta at that time, all ogy on Friday and Saturday, November 15 requests for accommodations should be and 16, 1963. The Georgia Institute of Tech­ mailed to: Convention Housing Bureau, nology is at 225 North Avenue in Atlanta, American Mathematical Society, 1102 Georgia. Atlanta is on Eastern Standard Commerce Building, Atlanta, Georgia, Time. 30303. All requests will receive prompt By invitation of the Committee to confirmaton. Following is the list of Select Hour Speakers for Southeastern hotels and motels: Sectional Meetings, Professor Fred B. Wright of the Tulane University of Louisi­ ana will speak on "Invertible elements in Banach Algebras" at 2:00 P.M., Friday, Atlanta Americana November 15. Professor Wright's address Motor Hotel Single $11.00 $14.00 will be in the Auditorium of the A. French Double 14.00 18.00 Textile School. Twin 16.00 20.00 be a session for contrib­ There will Atlanta Cabana afternoon at 3:30 uted papers on Friday Motel Single 11.00 13,00 at P.M. and Saturday morning beginning Double 14.00 16.00 the papers to be 10:00 A.M. Abstracts of Twin 16.00 17.00 presented appear on pages 643-650 of these NOTICES. The titles with corres­ Atlantan Hotel Single 6.00 8.50 ponding numbers are listed in this Pro­ Double 8.50 10.00 gram. Twin 10.00 11.50 Desk will be lo­ The Registration Dinkier Plaza cated in the lower lobby of Price Gilbert Hotel Single 7.00 15.00 Library and will open at 12:30 P.M. Friday. Double 10.00 15.00 Following the sessions there will be Twin 14.00 18.00 a Cocktail Party from 5:00 until 7:00P.M., at the Progressive Club 1050 Techwood Howard Johnson Drive, N. W. A" cash bar" will be operated. Motor Lodge, Single 9.00 12.00 It is recommended that public trans­ N.W. Double 11.00 13.00 portation be used from the hotels to the Twin 12.00 15.00 Georgia Tech campus. Anyone desiring to Peachtree Manor park on the campus should apply ahead of Motel Single 6.00 8.00 time to Professor Bertram M. Drucker, Double 9.00 11.00 Department of Mathematics, Georgia Insti­ Twin 9.00 12,00 tute of Technology, 30332, for a permit to park on the campus. Parking without a Piedmont Hotel Single 6.50 10.50 permit may cause the off-hauling of the Double 10.50 13.50 car. Twin 12.50 16.00

609 TRAVEL Southern, T. W .A., and United Airlines, and Atlanta is served by Delta, Eastern, by the Central of Georgia, Georgia, L and Northwest Orient, Piedmont, Southeastern, N, Seaboard, and Southern Railroads.

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

FRIDAY, 2.:00 P.M. Invited Address. Auditorium of the A. French Textile School, Georgia Institute of Tech­ nology Campus Invertible elements in Banach Algebras Professor Fred B. Wright, Tulane University of Louisiana

FRIDAY, 3:30 P.M. Session on Analysis and Applied Mathematics, Classroom 2.49, Annex to Library 3:30 - 3:40 (1) Characterization of regular Hausdorff moment sequences Professor J. S. Mac Nerney, University of North Carolina {605-1) 3:45 - 3:55 (2.) Fixed points in gouged convex sets Dr. G. S. Jones, RIAS, Baltimore, Maryland {605-4) 4:00 - 4:10 {3) Uniform bases and the equicontinuity of projections associated with Schauder decompositions Professor C. W. McArthur and Mr. J. R. Retherford*, Florida State Uni­ versity, (605-5) 4:15 - 4:2.5 (4) On the representation of bilinear functionals Professor R. C. Bzoch, Louisiana State University (605-13) 4:30 - 4:40 (5) Some relationships between stability and truncation error for a class of nine­ point analogues of the one-dimensional heat equation Professor J. M. Gwynn, Jr., Georgia Institute of Technology (605-14)

FRIDAY, 3:30P.M. Session on Topology and Algebra, Wilby Room 3:30 - 3:40 (6) Representations of a semigroup Mrs. R. S. Cox, University of North Carolina (605-2.) 3:45 - 3:55 (7) Homomorphisms of d-simple inverse semigroups with identity. II Professor R. J. Warne, Virginia Polytechnic Institute (605-3)

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

610 4:00 - 4:10 (8) Inverse dimension type. I. Types in the real line Professor Jack Segal, Institute for Advanced Study (605-6) 4:15 - 4:25 (9) Upper semi-continuous collections filling up hereditarily indecomposable continua Professor Howard Cook, Auburn University (605-7) 4:30 - 4:40 ( 1 0) Primary ideals and valuation ideals Professor R. W. Gilmer, Jr. *• Florida State University and Professor J. E. Olm, University of Wisconsin (605-8) 4:45 - 4:55 ( 11) A note on torsion-free rings Professor F. L. Hardy, Emory University (605-9) 5:00 - 5:10 (12) Two functions related to the k-free integers Professor Eckford Cohen, University of Tennessee (605-1 0)

SATURDAY, 10:00 A.M. Session on Topology and Geometry, Wilby Room 10:00 - 10:10 (13) Idempotents in semigroups on a half-space Professor J. G. Horne, Jr., University of Georgia (605-11) 10:15- 10:25 ( 14) Common fixed points of commuting continuous functions on the unit interval. Preliminary report Professor J. E. Maxfield, and Professor W. J. Howe*, University of Florida (605-12) 10:30 - 10:40 ( 15) On Topological translations in En Professor R. D. Anderson, Louisiana State University (605-15) 10:45 - 10:55 (16) Area in a non-euclidean geometry. Preliminary report Professor R. G. Vinson, Huntington College (605-18) 11:00- 11:10 (17) Slicing a contractible 3-manifold with boundary Dr. B. G. Casler, Louisiana State University (605-19) 11:15- 11:25 (18) A non-pointwise paracompact Moore space with a point-countable base. Pre­ liminary report Professor R. W. Heath, University of Georgia (605-22) 11:30 - 11 :40 ( 19) Homotopy groups of compact abelian groups Professor E. E. Enochs, University of South Carolina (605-23)

SATURDAY, 10:00 A.M.

Session on Analysis and Algebra, Classroom 249, Annex to Library 10:00 - 10:10 (20) Level sets of continuous functions. Preliminary report Professor M. K. Fort, Jr., University of Georgia (605-17) 10:15- 10:25 (21) Iterated w*-sequential closure of a Banach space of functions in its second conjugate space Professor R. D. McWilliams, Florida State University (605-20)

611 10:30 - 10:40 (22) Functions satisfying a weighted average property Mr. A. K. Bose, St. Augustine's College (605-24) 10:45 - 10:55 (23) The kernel of the semigroup of subsets of a s~migroup Mr. J. H. Carruth, Louisiana State University (605-16) (Introduced by Professor R. J. Koch) 11:00 - 11:10 (24) Identities in multiplicative systems. II. Anti-finite and anti-associative laws Professor Trevor Evans, Emory University (605-21) Morton L. Curtis Tallahassee, Florida Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

AN APPEAL FROM CUPM manuscripts or sets of notes which re­ flect these recommendations. It also As part of its effort to improve issues an appeal to the mathematical mathematics education in the colleges and community at large for clues leading to any universities of our nation, the Committee existing notes for experimental under­ on the Undergraduate Program in Mathe­ graduate mathematics courses which ap­ matics (CUPM) has produced proposals pear to be promising and merit wider cir­ for recommended curricula and outlines culation. Such notes may be extremely for specific courses in several broad helpful to the Committee and might even areas. Some of the CUPM recommended effect major changes in some of its views. courses are of such a nature that they can Any person who is reached by this be taught using existing textbooks or com­ appeal and who knows of the existence of binations of available books. Many of them, any such manuscript or rough set of notes however, either contain material not pre­ will contribute significantly to the work of sently available in texts at the undergradu­ CUP M by sending this information to: ate level or are organized in such a way as to make them difficult or impossible to CUPM Central Office teach without the preparation of new texts. P.O. Box 1024 In some areas, where present publi­ Berkeley 1, California cations and experience are particularly Such assistance will be greatly ap­ thin, CUPM has directly sponsored the preciated by CUPM and may indeed be of writing of text materials. However, the great benefit to mathematics education now Committee does not exist for the purpQtSe and in the future. of producing textbooks; it is determined, in fact to do as little of this as possible. It exists for the purpose of proposing FELLOWSHIP AND RESEARCH changes in mathematical curricula and OPPORTUNITIES stimulating the production, by individual authors, of texts reflecting its recommen­ The Division of Mathematics, Na­ dations. At the same time, the Committee tional Academy of Sciences-National Re­ and its Panels are always eager to learn search Council, calls attention to a variety of forward-looking ideas in mathematics of fellowship and other support for basic education which may be at large today and research in mathematics at both the pre­ may influence their deliberations and con­ doctoral and postdoctoral levels to be clusions. awarded during the year 1963-1964. Copies For this reason, the Committee of the complete announcement are available issues in this note a strong appeal to all from the Division of Mathematics, National persons associated with its work or ac­ Academy of Sciences-National Research quainted with its recommendations for Council, 2101 Constitution Avenue, Wash­ information leading to the discovery of ington, D. C. 20418.

612 Six Hundred Sixth Meeting California Institute of Technology Pasadena, California November 21-23, 1963

PROGRAM

The six hundred sixth meeting of $1.75 per person. On Thursday and Friday, the American Mathematical Society will be lunch can be obtained at the Chandler Cafe­ held on Thursday, Friday, and Saturday, teria on campus. November 21-23, 1963 at the California The following hotels and motels are Institute of Technology in Pasadena, Cali­ some of the nearest ones to the California fornia. Institute of Technology campus. The near­ By invitation of the Committee to est motel (the Imperial 400 Motel) is ap­ Select Hour Speakers for Far Western proximately one mile from the campus. Sectional Meetings, and· with the financial Green Hotel Single $8.00 support of the National Science Foundation, 50 East Green Double 11.00 a Symposium on Recent Developments in the Theory of Numbers will be held on Huntington-Sheraton Thursday and Friday, November 21, 22, in Hotel Single 8,50-16.50 conjunction with this meeting. All ses­ 1401 S. Oak Knoll Double 13.00-19.50 sions of the Symposium will be in Room Chalet Motel Single 7.50 151 of the Sloan Laboratory of Mathemat­ 2800 E. Colorado Double 9.00 ics and Physics, Twin 9.50 By invitation of the same Commit­ tee, Professor Kai Lai Chung of Stanford Imperial 400 Motel Single 8.00 University will present an hour address 1203 E. Colorado Double 10.00 at 11:00 A.M. on Saturday in Room 22, Twin 11.00 Gates Laboratory. The title of Professor Saga Motor Hotel Single 9.00-11.00 Chung's talk is "Boundary theory for 1633 E. Colorado Double 11,00-13.00 Markov chains." Twin 12.00-13.00 There will be sessions for contrib­ uted papers at 9:00 A.M. and at 2:00 P.M. Travel Lodge Single 6,00- 7.00 on Saturday, November 23 in Rooms 151 Pasadena Double 10,00 and 153 of the Sloan Laboratory of Mathe­ 2767 E. Colorado Twin 8.00- 9.00 matics and Physics, Abstracts of the pa­ The California Institute of Technol­ pers to be pre-sehted at these sessions ap­ ogy is located on California Street, between pear on pages' 651-660 of these NOTICES. Hill Avenue and Wilson Avenue. It is easily Late papers may be added to the program. reached by automobile from downtown Los For information concerning late papers, Angeles, taking the Pasadena Freeway inquire at the Registration Desk. (Arroyo Parkway) into Pasadena, then The Registration Desk for the meet­ turning east on California Street for ap­ ing will be located in the Sloan Laboratory proximately one mile. There is a parking of Mathematics and Physics. lot on the south side of California Street, There will be a tea at the Athen­ directly opposite the Sloan Laboratory of aeum for persons attending the meetings. Mathematics and Physics. Frequent lim­ It will take place on Saturday afternoon ousine service is available from the Los following the sessions for contributed pa­ Angeles International Airport to Pas adena, pers. A luncheon will be held on Saturday and there are buses from the Los Angeles in the Athenaeum at a cost of approximately railroad terminal to Pasadena.

613 PROGRAM OF THE SYMPOSIUM ON NUMBER THEORY

The Program Committee for the sions of the Symposium will be in Room Symposium consists of Professors Leo­ 151 of the Sloan Laboratory of Mathematics nard Carlitz,D.H.Lehmer, W.j.LeVeque, and Physics. and A. L. Whiteman, Chairman. All ses-

THURSDAY, 9:00 A.M. Principal Address The estimation of Fourier coefficients of modular forms and Kloostermann sums and their generalizations Atle Selberg, The Institute for Advanced Study Session on Diophantine Analysis and Algebraic Number Theory. Chairman: Professor Leonard Carlitz, Duke University p-adic forms Professor D. J. Lewis, University of Michigan Uniform distribution Professor W. j. LeVeque, University of Michigan and University of Colorado Characters and cyclotomy Professor Marshall Hall, jr., California Institute of Technology Theorems on Brewer and jacobsthal sums Professor A. L. Whiteman, University of Southern California On the lack of unique factorization in quadratic fields Professor Ivan Niven, University of Oregon, and Professor Herbert S. Zuckerman, University of Washington On the degrees of polynomials irreducible over a field Professor E. G. Straus and Professor B. Gordon, University of Califor­ nia, Los Angeles

THURSDAY, 2:00 P.M. Principal Address Some results on cyclotomic fields Professor Kenkichi Iwasawa, Massachusetts Institute of Technology Session on Matrices and Quadratic Forms Chairman: Professor D. H. Lehmer, University of California, Berkeley Matrices modulo n and groups of matrices Dr. Morris Newman, National Bureau of Standards Some new results connected with matrices of rational integers Dr. E. C. Dade and Professor Olga Taussky Todd, California Institute of Technology Block designs and quadratic forms Dr. E. T. Parker, Sperry Rand Corporation On the weight of a genus of positive n-ary quadratic forms Professor Gordon Pall, Louisiana State University

614 FRIDAY, 9:30 A.M. Principal Address Conjectures on elliptic curves Professor Bryan J. Birch, The University, Manchester, England Session on Analytic Number Theory Chairman: Professor W. J. LeVeque, University of Michigan and University of Colorado The sieve method Professor N. C. Ankeny, Massachusetts Institute of Technology Primes represented by irreducible polynomials in one variable Professor Paul T. Bateman, University of illinois Sets of values taken by Dirichlet's L-series Professor Tom M. Apostol, California Institute of Technology On the divisor problem ProfessorS. Chowla, University of Colorado, and Professor H. Walum, Harvey Mudd College On p-adic analysis Professor Bernard M. Dwork, Johns Hopkins University

FRIDAY, 2:00 P.M. Principal Address Generating functions Professor Leonard Garlitz, Duke University Session on Analytic Number Theory and Modular Functions Chairman: Professor A. L. Whiteman, University of Southern California Bounded consecutive residues and related problems Professor W. H. Mills, Yale University Second moments in additive number theory Professor Leo Moser, University of Alberta Extremal problems in number theory Paul Erdos, University of Alberta, Canada On the shape of the fundamental domain for certain Hilbert modular functions Professor Harvey Cohn, University of Arizona Representations of discrete groups Professor Joseph Lehner, University of Maryland

615 PROGRAM OF THE SESSIONS The time limit for contributed papers is ten min­ utes • The contributed papers are scheduled at 15 minute intervals so that listeners can circulate be­ tween the different sessions. To maintain this sche­ dule, the time limit will be strictly enforced. SATURDAY, 9:00A.M. Session on Geometry and Topology, Room 151, Sloan Laboratory of Mathematics and Physics 9:00 - 9:10 (1) Completions of quadrangles in singly-generated planes Professor R. B. Killgrove, San Diego State College (606-12) 9:15 - 9:25 (2) On the number of vertices of a convex polytope Professor Victor L. Klee, University of Washington and Boeing Scientific Research Laboratories, Seattle, Washington (606-6) 9:30 - 9:40 (3) A note on a theorem of Bonnice-Klee Professor j. R. Reay, Western Washington State College (606-4) 9:45 - 9:55 (4) Minimal surfaces in En Professor Robert Osserman, Stanford University (606-24) 10:00 - 10:10 (5) On grouplike manifolds. Preliminary report Professor R. F. Brown, University of California, Los Angeles (606-9) 10:15 - 10:25 (6) Homotopy normality of subgroups Professor G. S. McCarty, Jr., Harvey Mudd College (606-17) 10:.30 - 10:40 (7) A new proof of the Tychonoff theorem Mr. Peter A. Loeb, Stanford University (606-3)

SATURDAY, 9:00A.M. General Session, Room 153, Sloan Laboratory of Mathematics and Physics 9:00 - 9:10 (8) Non-parametric estimation and consistency Dr. G. B. Crawford* and Dr. S. C. Saunders, Boeing Scientific Research Laboratories, Seattle, Washingtori (606-16) 9:15 - 9:25 (9) Local times for Markov processes Professor R. M. Blumenthal and Professor R. K. Getoor*, University of Washington (606-8) 9:30 - 9:40 (10) Asymptotic expansions at a corner of solutions of mixed boundary value problems Dr. Neil M. Wigley, Los Alamos Scientific Laboratory, Los Alamos, New Mexico (606-5) 9:45 - 9:55 ( 11) Divided differences for functions of two variables for irregularly spaced arguments Dr. Herbert E. Salzer, General Dynamics/Astronautics, San Diego, Cali­ fornia (606-31) *For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

616 10:00 - 10:10 (1Z) A class of solvable differential games Professor M. P. Drazin, Purdue University (606-Z1) 10:15 - 10:Z5 (13) Permanents of (0,1)-circulants Professor Henryk Mine, University of California, Santa Barbara (606-19) 10:30 - 10:40 ( 14) Inequalities for subpermanents Professor M.D. Marcus and Mr. W. R. Gordon*, University of California, Santa Barbara (606-18)

SATURDAY, 11:00 A.M. Invited Address, Room ZZ, Gates Laboratory Boundary theory for Markov chains Professor Kai Lai Chung, Stanford University

SATURDAY, Z:OO P.M. Session on Analysis, Room 151 Sloan Laboratory of Mathematics and Physics Z:OO - Z:10 ( 15) A property of Hausdorff measure Professor H. G. Eggleston, University of London, England, and the Uni­ versity of Washington (606-Z7) Z:15 - Z:Z5 (16) Some properties of integrals on Riesz spaces Professor W. A. J. Luxemburg, California Institute of Technology ( 606-14) Z:30 - Z:40 ( 1 7) Topological groups and measure problems in distribution-free statistics Professor C. B. Bell, San Diego State College (606-Z) Z:45 - Z:55 (18) Pointwise summability of Fourier transforms on LCA groups Professor R. E. Edwards, Australian National University, and Professor Edwin Hewitt*, University of Washington (606-11) 3:00- 3:10 (19) An elementary proof of Mautner's lemma Professor Leon W. Green, University of Minnesota (606-ZO) 3:15 - 3:Z5 (ZO) Weakly-continuous representations of the multiplicative algebra (BV) Professor G. L. Krabbe, Purdue University (606-Z8) 3:30 - 3:40 (Z1) Arens multiplication and spectral theory Professor W. H. Sills, University of Oregon ( 606-1 O) 3:45 - 3:55 (ZZ) On sublinear operators in Hilbert space with uniformly symmetrizable derivatives Dr. G. H. Pimbley, Los Alamos, Scientific Laboratory, Los Alamos, New Mexico (606-7) 4:00 - 4:10 (Z3) Branched regular curve families and finite asymptotic values of analytic functions Professor D. V. V. Wend, University of Utah (606-13)

617 SATURDAY, 2:00P.M.

Session on Algebra and Theory of Numbers, Room 153 Sloan Laboratory of Mathematics and Physics 2:00 - 2:10 (24) Certain embedding problems of semigroups Professor Takayuki Tamura, University of California, Davis (606-25) 2:15 - 2:25 (25) Equivalence classes of sets of functions over a finite field Dr. S. R. Cavior, State University of New York, Buffalo (606-30) 2:30 - 2:40 (26) On a relation between the period and the restricted period of a linear re­ current sequence. Preliminary report Professor D. W. Robinson, Brigham Young University (606-23) 2:45 - 2:55 (27) Hadamard matrices and constant distance codes Professor Richard Block, California Institute of Technology (606-26) 3:00 - 3:10 (28) On the expression of a number as the sum of two square in totally real alge­ braic number fields Dr. W. G. Schaal, Massachusetts Institute of Technology (606-29) 3:15- 3:25 (29) On Weyl's criterion Professor W. J. LeVeque, University of Michigan (606-1) 3:30 - 3:40 ( 30) Power series with restricted coefficients Professor D. G. Cantor, University of Washington (606-22) 3:45 - 3:55 ( 31) Unique factorization Mr. D. J. Deckard and Professor L. K. Durst*, Rice University (606-15) R. S. Pierce Seattle, Washington Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

A CONFERENCE ON and P. Malliavin, France; H. Behnke, APPROXIMATION THEORY H. Berens, H, Brab, P. L. Butzer, K. Endl, D. Ernst, E. Gorlich, H. Gunzler, H. Hil­ A conference on approximation the­ gers, R. J. Nessel, W. Quade, P. 0. Runck ory was held at THE MATHEMATICAL and H. Schulte, Germany; J. L. B. Cooper, RESEARCH INSTITUTE OBER WOLF ACH Great Britain; G. Alexits and G. Freud, from August 4 to August 9, 1963. The con­ Hungary; G. Sunouchi, Japan; R. A. Hirsch­ ference, which was by invitation, was di­ feld and F. Schurer, Netherlands; J. rected by P. L. Butzer, of the Technical Korevaar, G. G. Lorentz, I. J. Schoenberg University of Aachen, section chairmen and H. S. Shapiro, United States. being: The program included a special J. L. B. Cooper, seminar on new and old unsolved .problems J. Favard in approximation theory. The proceedings J.· Korevaar of the conference will be published in the G. G. Lorentz near future by the Birkhauser Verlag, I. J. Schoenberg Basel. These will be dedicated to the Following is the list of twenty-seven parti­ memory of Ch. de la Vallee Poussin, with cipants from nine countries: G. Goes, an article giving a description of the sig­ Canada; Th.S.V.Bang, Denmark; J. Favard nificance of the latter's work by J. Favard.

618 Six Hundred Seventh Meeting University of Wisconsin Madison, Wisconsin November 29-30, 1963

PROGRAM

The six hundred seventh meeting of Name Single Rate Double Rate the American Mathematical Society will be Madison Inn Motel held at the University of Wisconsin on Sat­ 601 Langdon Street urday, November 30, Registration and all Madison, Wisconsin sessions will be in Van Vleck Hall, ALpine 7-4391 $10,00 $14,00 By invitation of the Committee to Select Hour Speakers for Western Sec­ Ivy Inn Motel tional Meetings, Professor Louis de 2355 University Avenue Branges of Purdue University will address Madison, Wisconsin the Society. He will speak on "New and CEdar 3-9717 9.00-10,00 12,00-18,00 old problems for entire functions" in Town Campus Motel Room B-102 at 2:00P.M. 441 North Frances Street Sessions for the presentation of Madison, Wisconsin contributed papers will be held at 10:30 ALpine 7-4881 8,50-12,00 13,00-15,00 A.M. and 3:15P.M. A special session for papers which failed to meet the deadline The Edgewater Hotel will be held at 3:15P.M. Details will be 666 Wisconsin Avenue a vail able at the registration desk. Madison, Wisconsin Madison may be reached by car, ALpine 6-907110.50-12.50 15,50-18,75 bus (Greyhound), rail (Milwaukee Road) Loraine Hotel and air (North Central, North West and 123 West Washington Avenue Ozark Air Lines). For complete details Madison, Wisconsin members should consult a travel agent. ALpine 6-0231 8,50 8,50-12.50 Mail for those attending the meeting should be addressed c/o Department of All of these hotels and motels are Mathematics, 213 Van Vleck HalL within a mile of Van Vleck Hall. The following hotel and motel in­ There are a number of good restau­ formation may be useful in planning the rants in Madison. Details will be available trip: at the registration desk.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. To maintain the schedule, the time limit will be strictly enforced.

SATURDAY, 10:30 A.M.

Session on Topology, Room B-102 10:30 - 10:40 ( 1) Stable homeomorphisms on E 5 can be approximated by piecewise linear ones Professor R. H. Bing, University of Wisconsin (607-16)

619 10:45 - 10:55 (2) Cellular decompositions of E3 Professor Thomas Price, University of Wisconsin (607-2) 11:00- 11:10 (3) The sum of a cube is defined to be the closure of the interior of a 2-sphere in euclidean 3-space Mr. Norman Hosay, University of Wisconsin (607-17) (Introduced by Dr. S, H. Gould) 11:15- 11:25 (4) Uncountably many inequivalent nearly tame arcs Mr. C. H. Giffen, Princeton University (607-19) 11:30- 11:40 (5) Extended topology: Foundations of approximation theory Professor P. C. Hammer, University of Wisconsin (607-8) 11:45 - 11:55 (6) Subplane counts in projective planes Professor R. B. Killgrove, San Diego State College (607-7)

SATURDAY, 10:30 A.M.

Session on Algebra and Theory of Numbers, Room B-130 10:30 - 10:40 (7) Nil rings satisfying certain chain conditions Professor I. N. Herstein and Mr. L. W. Small*, University of Chicago (607-4) 10:45 - 10:55 (8) Abstract characterization of an algebra of multi-place functions. I Mr. H. I, Whitlock, Illinois Institute of Technology (607-11) (Introduced by Professor Haim Reingold) 11:00 - 11:10 (9) Axiomatics for partially ordered systems of multiplace functions Miss V. J, Kafka, Illinois Institute of Technology (607-12) (Introduced by Professor Haim Reingold) 11:15- 11:25 (10) On rings with composition Mr. R, C. Reilly, Illinois Institute of Technology ( 607-1 0) (Introduced by Professor Haim Reingold) 11:30- 11:40 (11) Ideals in two-place tri-operational algebras Professor E. F. Stueben, Illinois Institute of Technology (607-9) (Introduced by Professor Haim Reingold) 11:45- 11:55 (12) Component congruences for a class of divisors Professor M. V. Subbarao, University of Alberta, Canada (607-3) 12:00 - 12:10 (13) On a formal product over the conjugate classes of a free group Professor M, P. Schutzenberger, University of Poitiers, Antony, France and Professor Seymour Sherman*, Wayne State University (607-1)

SATURDAY, 2:00P.M. Invited Address, Room B-102 New and old problems for entire functions Professor Louis de Branges, Purdue University *For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

620 SATURDAY, 3:15P.M.

Session on Analysis and Probability Theory, Room B-102 3:15 - 3:25 ( 14) Construction of a Markov process from hitting probabilities Professor Frank Knight and Professor Steven Orey*, University of Min­ nesota (607-18) 3:30 - 3:40 ( 15) An elementary proof of Beurling' s theorem. Preliminary report Professor j. L. Rovnyak, Purdue University (607-5) 3:45 - 3:55 (16) Quasi-bounded linear lattices Professor Hidegoro Nakano and Professor B. j. Eisenstadt*, Wayne State University (607-6) 4:00 - 4:10 (17) A convexity condition on Banach spaces invariant under conjugation. Prelim­ inary report. Mr. D.P. Giesy, University o.f Wisconsin (607-15) 4:15 - 4:25 (18) Maximum properties of Cauchy's problem in three-dimensional space-time Mr. Duane Sather, University of Minnesota (607-14) (Introduced by Professor H, F. Weinberger) 4:30 - 4:40 ( 19) Continued fraction expansions for nth roots and other functions Professor Evelyn Frank, University of Illinois (607-13) j. W. T. Youngs Princeton, New jersey Associate Secretary

MEMORANDA TO MEMBERS

THE EMPLOYMENT REGISTER job applicants and employers who wish to be listed will please write to the The Mathematical Sciences Em­ Employment Register, 190 Hope Street, ployment Register, established by the Providence 6, Rhode Island, for applica­ American Mathematical Society, the Math­ tion forms or for position description ematical Association of America, and the forms. These forms must be completed Society for Industrial and Applied Mathe­ and returned to Providence not later than matics, will be maintained at the Annual December 15, 1963, in order to be included Meeting at the University of Miami, Coral in the listings at the Annual Meeting in Gables, Florida, on january 24, 25, and 26, Miami. Position description forms which 1964. The Register will be conducted from arrive after this closing date, but before 9:00 A.M. to 5:00 P.M. on each of these january 5, will be included in the register three days. at the meeting for a late registration fee There is no charge for registration, of $3.00. The printed listings will be either to job applicants or to employers, available for distribution both during and except when the late registration fee for after the meeting. employers is applicable. Provision will be It is essential that applicants and made for anonymity of applicants upon employers register at the Employment request and upon payment of $3.00 to defray Register Desk promptly upon arrival at the the cost involved in handling anonymous meeting to facilitate the arrangement of listings. appointments.

621 PRELIMINARY ANNOUNCEMENT OF MEETING

Seventieth Annual Meeting University of Miami, Coral Gables, and Miami, Florida January 23-27, 1964

The seventieth annual meeting of the building number. It is a girl's dormito: American Mathematical Society will be and derives its name from the number held in Miami, Florida in conjunction with girls residing there.) The registrati1 the annual meeting of the Mathematical desk will be open from 2:00 P.M. to 8:1 Association of America. All sessions will P.M. on Wednesday, January 22, from9:1 be held on the campus of the University of A.M. to 5:00 P.M. on Thursday throu1 Miami, except as noted below. Saturday, and from 9:00 A.M. to 2:00 P .1 The thirty- seventh Josiah Willard on Monday. All members attending tl Gibbs Lecture will be delivered by Pro­ meeting are requested to register. fessor Lars Onsager at 8:00 P.M. on The schedule for registration fee; Friday, January 24, in the Everglades as follows: $2.00 for members of partie Room of the Everglades Hotel. By invita­ pating organizations. Another 50 cents tion of the Committee to Select Hour charged for the first nonmember of Speakers for the Annual and Summer Meet­ member's family. Additional nonmembe: ings, Professor Heisuke Hironaka will in a member's family pay no registrati< address the Society at 2:00P.M. on Friday, fee. Students are also exempt from t January 24. registration fee. For nonmembers who a Both the first and second awards of not in any of these categories the fee the Veblen Prize in Geometry will be made $5.00. at 2:00 P.M. on Thursday, January 23. At The Employment Register will 2:00 P.M. on Saturday, January 25, the maintained from 9:00 A.M. to 5:00 P .l EScher Prize will be awarded. on Friday, Saturday, and Sunday. Bo The Society is reminded that by exhibits will be located in the Lounge action of the Council the number of contrib­ the 730 building on the University of Mia! uted papers will be limited to ZOO on a Campus. "first come, first served" basis, and there will be no sessions for late papers. ACCOMMODATIONS The Annual Business Meeting will be held at 1:30 P.M. on Thursday, January Accommodations for the meeti 23. will be handled by the Housing Bureau The Council of the Society will meet the Convention Bureau of the City at 4:00 P.M. on Wednesday, January 22, Miami. The reservation form on the i at the Everglades Hotel. side back cover of these NOTICES shou be used in requesting accommodation REGISTRATION The Housing Bureau will make reserv tions as nearly as possible in accordan Registration headquarters will be in with the member's request at one oft the lobby of the 730 building on the Univer­ hotels in the list below. All reservatio sity of Miami campus. {This is not the will be confirmed by the Housing Burea

622 Hotel Singles Doubles Twins Suites

Alcazar $ 7.00-$ 8.00 $ 9.00-$10.00 $12.00-$16.00 $24.00 Biscayne Terrace 10.00-12.00 12.00 -14.00 12.00- 14.00 Columbus 11.00 16.00 16.00-18.00 40.00 Dupont Plaza 12.00 -15.00 17.00-20.00 39.00 Everglades 9.00- 12.00 15.00-16.00 28.00 -36.00 Leamington 10.00 McAllister 9.00-11.00 15.00 -16.00 Miami Colonial 9.00-10.00 12.00 -14.00 Ponce de Leon 12.00 Patricia 8.50 10.50 Triples 12.50 Robert Clay 7.00 9.00 9.00 Towers 10.00 10.00- 12.00 10.00 -12.00 18.00

Dallas Park 12.00 12.00 20.00

Shuttle busses will be provided to trans­ Canada Airlines. Pan American and BOAC port persons to and from the University is sue seventeen-day round-trip tickets of Miami Campus. The boarding areas in from Miami to Nassau, tourist class, at a downtown Miami will be the Everglades, 10 percent discount. DuPont Plaza and Patricia Hotels. All Some airlines offer San Francisco other hotels on this list are within a few to Miami passengers a detour through New minutes walk from one of these boarding York for only $11.10 extra. areas. Most Miami Beach hotels are an The following railroad lines con­ additional ten or fifteen miles away from nect with the Seaboard Airline Railroad the University of Miami Campus and and offer direct pullman service to Miami: shuttle bus service is not available to Baltimore and Ohio, Chesapeake and Ohio, Miami Beach. Louisiana and Nashville, New York Cen­ tral and New Haven, Santa Fe, Seaboard, TRAVEL INFORMATION Southern Pacific and Texas Pacific. Miami is served by Greyhound and There is regular airline service to Trailways bus lines. Miami International Airport by the follow­ ing airlines: Braniff, Delta, Eastern, Na­ ENTERTAINMENT tional, Northeast, Northwest, Pan Ameri­ can, Trans-World and United. If members It is anticipated that a travel rep­ plan a stop-over in Nassau before or after resentative will be available near the the meetings, it is advisable to make registration desk to arrange tours. A bird­ travel and hotel reservations early as watchers tour is being planned. A tour to this is the peak season in Nassau. the International Design Center will leave Nassau can be reached from Miami: the Everglades Hotel at 9:00 A.M., Satur­ By Air - Bahamas Airways, Cunard-Eagle, day, January 25. There will be a $1.00 BOAC, and Pan American. By Sea - S. S. transportation charge for each lady making Bahama Star, S. S. Florida. From St. this tour. Other entertainments will be Petersburg, Tampa, Ft. Lauderdale and listed in the program of the meeting. West Palm Beach: Mackey Airlines, sev­ eral flights daily. From New York: By MISCELLANEOUS Air - BOAC and P AA, 2 1/2 hours non­ stop. By Sea - M. V. Italia, weekly sail­ Miami is on Eastern Standard Time. ings. From Montreal and Toronto: Trans- The expected diurnal temperature

623 variation for january is between 55 and 75 A. T. Butson degrees. Rain is unlikely at this time of M. L. Curtis, ex officio the year. Mrs. Georgia K. Del Franco Mail and telegrams for those at­ Edwin Duda tending the meeting should be addressed E. F. Low, Jr. in care of the Mathematics Department at Herman Meyer the University of Miami. Andrew Sobczyk The committee on arrangements G. L. Walker, ex officio consists of: j. H. Curtiss, Chairman M. L. Curtis H. L. Alder, ex officio Associate Secretary R. W. Bagley Tallahassee, Florida

NEWS ITEMS AND ANNOUNCEMENTS

CONFERENCE ON mathematicians with doctorates who show ARITHMETICAL ALGEBRAIC GEOMETRY definite promise in research. The base December 5-7, 1963 salary for these instructorships will be at least $8000 and the teaching load will Purdue University is scheduling a be six hours per week. The salary c~n be conference on Arithmetical Algebraic supplemented by summer work on a re­ Geometry, December 5-7, 1963. Invited search contract, sponsored by the Air addresses will be given by: Force Office of Scientific Research, or by S. Abhyanker (Purdue University and teaching in the summer session. The ap­ Harvard University) pointments are annual but are renewable W. Bailey (The University of Chicago) for one additional year. B. Dwork (Thejohns Hopkins Univer­ Applications should be filed not sity) later than january 6, 1964, on forms ob­ H. Hironaka (Columbia University and tained from the Department. Brandeis University) D. Mumford (Harvard University) T. Tamagawa (Yale University) j. Tate (Harvard University) A. W eil (The Institute for Advanced Study) INSTITUTE FORADVANCEDSTUDY 1964-1965 Memberships For information write to anymem­ ber of the Organizing Committee: Pro­ The School of Mathematics of the fessors David Hertzig, Eugene Schenkman, Institute for Advanced Study, Princeton, Otto F. G. Schilling, Division of Mathe­ New jersey, will grantalimitednumberof matical Sciences, Purdue University, La­ memberships, in some cases with financial fayette, Indiana. support, for research in mathematics at the Institute during the academic year 1964-1965. Candidates must have given evidence of ability in research comparable THE M.I.T. at least with that expected for the Ph.D. DEPARTMENT OF MATHEMATICS degree. Application blanks may be ob­ The M.I. T. Department of Mathe­ tained from the Secretary of the School of matics wishes to announce the availability Mathematics, and shQuld be returned by of C. L. E. Moore Instructorships in january 15 (whether or not funds are ex­ Mathematics for 1964-1965, open to young pected from some other source).

624 NATIONAL ACADEMY OF SCIENCES NATIONAL RESEARCH COUNCIL 2101 Constitution A venue, Washington, D.C.

DIVISION OF MATHEMATICS

Chairman: E. J. McShane Past Chairman: J. Barkley Rosser Chairman Designate: G. A. Hedlund Executive Secretary: M. H. Martin Abstract of Annual Report for 1962- 1963 This year committees of the Division advised on the selection of candidates for fellowships and other awards as follows: Applications Awards NSF Postdoctoral (Regular) Fellowships 64 29 NAS-NRC Postdoctoral Research Fellowships 3 NSF Senior Postdoctoral Fellowships 5 NSF Graduate Fellowships l, 131 332 NSF Cooperative Graduate Fellowships 601 227 NSF Summer Fellowships for Graduate Teaching Assistants 251 137 Postdoctoral Resident Research Associateships 6 5 Fulbright Fellowships 16 ONR Postdoctoral Research Associateships 16 5 The U. S. National Committee for projects: one was a report on the connec­ Mathematics reported on the work of the tion between arithmetic and algebra in ele­ International Mathematical Union. Begin­ mentary instruction, the other was the ning with the Stockholm Congress, scien­ selection and arrangement of a textbook tific programs for International Congres­ exhibition at the Stockholm Congress. The ses will be based on collaboration between U. S.Commission on Mathematical Instruc­ the Local Organizing Committee and a Con­ tion has recommended to the International sultative Committee appointed jointly by Commission on Mathematical Instruction the Organizing Committee and the Inter­ that the following studies should be under­ national Mathematical Union. The most taken: a total investigation of the combined active commission of the Union has been teaching of mathematics and science; com­ the International Commission on Mathe­ puter education at the secondary and uni­ matical Instruction which recently has ar­ versity levels; and means and methods for ranged symposia and seminars on the the international interchange of persons teaching of mathematics in the following interested in mathematical education. countries: Denmark, Yugoslavia, Switzer­ The Committee on Applications of land, Italy, and Colombia, in addition to a Mathematics is concerned with arousing colloquium at the Stockholm Congress. In interest in the mathematical education of its report the Commission recommends: engineers and physicists among depart­ that at least three meetings on mathemati­ ments of mathematics throughout the coun­ cal instruction be held each year, at least try. In view of the rising tendency towards one of which should be outside Europe; the creation of separate departments of that consideration be given to the estab­ applied mathematics in U. S. universities, lishment of an international bibliographical the committee felt: that proper textbooks and informational service in the field of are a key to the problem, that undergradu­ mathematical education; and that the Com­ ate courses should be taught to show that mission extend its activities to new areas, the applications of mathematics are ex­ such as Africa. The U. S. Commission on citing and to maintain communication be­ Mathematical Instruction completed two tween mathematicians and those who apply

625 mathematics, that the first two years of a enabled over 130 mathematicians to attend mathematics curriculum should provide a the Stockholm International Congress. suitable foundation both for those who plan The Committee on Revision of Math­ to major in pure mathematics and for those ematical Tables reports that the "Hand­ who plan to major in applied mathematics, book of Mathematical Functions" is near­ and that a textbook writing group of mathe­ ing completion with publication expected maticians of stature knowledgeable in the in the fall of 1963. applications should be set up to work with The manuscript of the publication the Committee on the Undergraduate Pro­ on Theory of Traffic Flow reached the gram in Mathematics, possibly as a sum­ printer in june 19 63 and the ad hoc Com­ mer study group. The report also contains mittee on Traffic Flow responsible for an outline of the plans of the National Aero­ this publication will remain in force until nautics and Space Administration for sup­ the galley proofs have been checked. port of university research and training. The ad hoc Committee on Space The Editorial Committee for Mathe­ Mathematics has submitted its final report matics of Computation reports that over which recommends: the preparation of a the last year the number of pages published list of mathematical problems in space increased from 450 to 514 and the number of science with suitable bibliographical and subscribers from approximately 2,000 to supporting information, a program of 2,500. fellowships for study in space mathemat­ The Committee on Revolving Fund ics, summer employment of undergradu­ for Publication of Mathematical Tables ates in National Aeronautics and Space sponsored publication of" Table ofF actor­ Administration laboratories, summer ials from 0! to 9999!" by ]. B. Reid and seminars in space mathematics for faculty G. Montpetit (Publication 1039 of the Na­ members and advanced students, and the tional Academy of Sciences- -National Re­ publication of a series of expository mono­ search council) and was instrumental in graphs in space science. The report con­ arranging for the publication of a table of tains an initial list of mathematical prob­ groups of orders 2, 4, 8, 16, 32 and 64 by lems in space research and a summary of Marshall Hall, jr. and J. K. Senior. mathematical research activity in space The Committee on Regional Devel­ science in various U. S. universities. opment has considered the problem of A new Committee on Uses of Com­ undergraduates who are talented in mathe­ puters was created in the Division to: matics, who attend small colleges without enumerate those field of science, natural special mathematics facilities, and who and social, in which significant advances plan to go on to graduate work in mathe­ are being made, or can be foreseen, matics. Such students frequently arrive at through the use of computers; to estimate graduate schools poorly prepared in math­ the scale on which such use might reason­ ematics, and to remedy this situation the ably be developed; and evaluate the signifi­ committee has proposed experimental cance of such use in the advancement of study plans whereby students in small col­ research and education in those fields; and leges can attend regular advanced under­ to estimate the total present and foresee­ graduate courses in a large university able national need of colleges and univer­ nearby while still being enrolled in their sities for computing facilities in support of own colleges. The committee proposes research and education. that the program be instituted on a trial basis in perhaps five to ten states where * * * the need is obvious and where the large Copies of the complete annual re­ centers and the small colleges are eager port of the Division of Mathematics may to give the program a trial, with expenses be obtained by writing to: met possibly by a National Science Found­ ation grant. Division of Mathematics The Committee on Travel Grants National Academy of Sciences reported that the $70,000 grant from the National Research Council National Science Foundation and an addi­ 2101 Constitution Avenue tional fund of $7,622 subscribedbyindustry Washington, D. C. 20418

626 NEW NSF POLICIES AND THEIR IMPLEMENTATION By Robert H. Owens

Several years have elapsed since a (v) The present staff of the Mathemati­ description of NSF Research Programs of cal Sciences Section, listed below, interest to the mathematical community is entirely new. has been made available in these Notices Dr. Robert H. Owens, Acting Head* (see Notices Aug. 1960whereinDr.Arthur Dr. Milton E. Rose, Program Direc- Grad provided a compxehensive summary tor for Mathematics of NSF activities in Mathematics which is Dr. Donald T. Laird, Program Di­ still current). For reasons listed below in rector for Computer Science chronological order it is particularly ap­ Dr. Ralph M. Krause, Acting Pro­ propriate that new information be made gram Director for Probability and available at this time. Statistics and Associate Program Director for Mathematics. Note: Inquiries by mail should nor­ (i) The growth and scope of the original mally be sent to the appropriate Mathematical Sciences Program Program Director. However, the had increased to such an extent that Section Staff works closely together it was necessary in January 1963 and therefore each member can to replace it by the Mathematical handle all im tiries. Since staff Sciences Section which currently members are rrequently on travel, contains three programs, The Math­ inquiries by phone can always be ematics Program, The Probability taken by any member available. In and Statistics Program, and the particular the formal status of a Computer Science Program. proposal can be provided by Mr. J. (ii) A new set of Instructions for the Hamilton Andrews, Administrative Preparation of proposals for com­ Assistant. puter facilities and for research in computer science was prepared in The Mathematics Program accepts May 1963. proposals for research in any branch of pure and applied mathematics. (iii) Foundation policies regarding the The Probability and Statistics Pro­ submission and administration of gram also accepts proposals in both pure basic research proposals are made and applied areas of research. by the National Science Board and The Computer Science Program ac­ the Director of the Foundation. cepts proposals for the support of com-, These policies were made available puter facilities for multidisciplinary re­ to the Scientific Community in Janu­ search needs and for research in computer ary 1960 when the brochure "Grants science. This research includes artificial for Scientific Research" last went intelligence, mechanical translation, heur­ to press. A new and extensively re­ istic programming, pattern recognition, vised "Grants" brochure was pre­ time sharing systems (and other new con­ pared during the first pa:.:-t of this tributions to system programming), the year and was issued in June 1963. use of computers directly connected with (iv) Dr. Leland J.Haworth was appointed scientific experiment, numerical analysis, Director of the Foundation by the and almost any computer oriented or com­ President effective July 1, 1963. ·puter induced research which does not Dr. John T. Wilson was appointed *Dr. Arthur Grad, Head of the Math­ Deputy Director of the Foundation ematical Sciences Section, is on by Dr. Haworth effective July 1, leave at Stanford University for 1963. the year 1963-1964.

627 fall into the jurisdiction of other programs 6. Information is given that "under within the Foundation. certain circumstances the stipend of a In addition to these programs, a research assistant who is a candidate for separate appropriation is available for an advanced degree may be considered to international travel. This Section accepts be a fellowship and excluded from gross applications for foreign travel from mem­ income (for income tax purposes)," and bers of the mathematical community and the following statement appears: "Inter­ provides, for qualified applicants, round pretation of the U.S. income tax laws is, trip fare from the home institution to the of course, a responsibility of the Internal designated place in the foreign country. Revenue Service and the courts and the Foreign travel should generally be made foregoing statements are merely intended with these funds and not from grant funds to be informative. Requests for answers (see pages 12 and 17 of" Grants" brochure). to specific questions regarding this issue It is emphasized that the only state­ should be made to the local Internal Re­ ment of Foundation policies regarding venue Office" (p.25). grants for research available to the Scien­ Items 3 and 5 need elaboration. tific Community appears in the new bro­ When 2/9 academic salary is provided for chure "Grants for Scientific Research." summer research it is generally not per­ Copies of this brochure, of the instructions missible for senior personnel to obtain an for computer proposals, and application additional 1/9 salary contribution from forms for international travel may be ob­ another NSF supported project at any tained by writing to: institution. In figuring the summer salary The Mathematical Sciences Section contribution to be made in the cases of the National Science Foundation quarter system, the trimester system or Washington, D. C. 20550 any other system (not yet invented) built around an academic year the "academic" In the "Grants" brochure particular salary is divided by 9 in order to deter­ changes to be noted are: mine the summer monthly contribution. 1. A provision is made for submitting In the case of calendar year salary a sum­ privileged information (such as individual mer contribution is no longer meaningful. salaries) in a separate statement accom­ In this case it is possible for an institution panying a proposal (p.2). to claim (and receive) reimbursement for a faculty member who works for say three 2. More detail is required about those months during the summer. It should be who will participate in the proposed in­ implied, of course, that when a faculty vestigation (p. 7). member is paid under the calendar year 3. The Foundation's contribution to system that no portion of his salary should summer salaries generally will not exceed be directly contingent on Government sup­ two-ninths of the regular academic year port of his research; if there is a question salary for senior personnel. Exceptions in this regard program members may may be made in those rare occasions make appropriate inquiries. when it is unwise to undertake the summer The detailed budget which accom­ research program unless a three months panies the grant letter must be adhered period can be made available and where to. In accordance with the policy stated on the institution is unable to make a contrib­ page 22 one may send a written request to ution of 1/9 of the investigator's salary the appropriate Program Director for per­ (p.S). mission to modify this budget. Reasonable requests will be granted provided they are Normally, overl;lead of 25 per cent 4: within the limits of the budget that appeared will be paid on direct costs (p.l3). in the original proposal. Requests not in­ 5. A detailed budget will be included cluded in the original proposal must first with the grant letter (p.21). In administer­ pass through the same official institution ing a grant, deviations from the original channels as required for the original pro­ proposal are more restricted than for­ posal before being sent to the Program merly (p.22). Director.

628 The Foundation encourages maxi­ amount of academic year support given to mum financial participation by each insti­ senior investigators. Such support may be tution in the research programs receiving given only in those cases where the inves­ Foundation support. For this reason and tigator states that the available time will because the available funds are always in­ be used to carry out the proposed research adequate for the requests received the and particularly where a genuine reduction Foundation is compelled to restrict the in teaching load is provided.

The American Mathematical Society Announces the Establishment of THE VEBLEN PRIZE IN GEOMETRY

The Veblen Prize in Geometry has Society and not more than fifty years old been established in honor of Professor at the time of the publication of his mem­ Oswald Veblen and in recognition of his oir. However, these rules are not always contribution to geometry and to American strictly applied; in recent years, a number Mathematics. The fund to endow the prize of awards made by the Society have not was raised by a group of former students specified a particular paper, but have been and associates of Professor Veblen, with made on the basis of a man's work in the aid of Mrs. Veblen. The Prize will be general. awarded for a notable research memoir in The other prizes awarded by the geometry, which in this connection is to be Society are the B6cher Memorial Prize, interpreted in a broad sense and in parti­ established in 1923, and awarded for work cular is to include topology. in analysis, and the Frank Nelson Cole In 1964, the first two Prizes will be Prizes in Algebra and in the Theory of awarded at the Annual Meeting of the So­ Numbers, established in 1928. ciety. Thereafter the Prize will be awarded Prizes to be given in January of at five-year intervals, beginning in 1966. each year by the American Mathematical Conditions governing the Veblen Society are scheduled as follows: Prize will be similar to those of the other prizes given by the Society: The winning memoir should have appeared during the 1964 B8cher Prize, Veblen Prize five years preceding the award in a recog­ 1965 Cole Prize in Algebra nized journal published in the United 1966 Veblen Prize States or Canada. The recipient must be 1967 Cole Prize in Number Theory a member of the American Mathematical 1968 None

MEMORANDA TO MEMBERS

RETIRED MATHEMATICIANS included in the list are asked to send the following information to the Headquarters The Mathematical Sciences Em­ Offices: name; date of birth; highest degree ployment Register, 19 0 Hope Street, P rov­ and where obtained; most recent employ­ idence 6, Rhode Island, again plans to issue ment; present address; date available; the List of Retired Mathematicians Avail­ preferences, including preference for aca­ able for Employment, which has been demic or industrial employment. maintained yearly. Data thus received will be incorpo­ Mathematicians who are retiring rated into the next issue of the list which this year and who are interested in being will be published in February, 19 64.

629 NOTES FOR SPEAKERS

Editor's Note: The following notes were found among the effects of a veteran member of the Society, and are reprinted here to show how much the standard of presenting papers has been raised in a generation or two.

Half your audience are older than {c) Speak slowly, clearly and fairly you, and half are smarter; each class in­ loudly The old high school rule of addres­ trinsically deserves respect. Your other sing your remarks to someone in the back hearers are there because they chose you, row is very good. Don't be conversational; and therefore have by this choice earned it's fun for your friends in the front row, your most considerate treatment. but not for strangers in the twelfth. If a You show this respect and consider­ microphone is provided, the intensity of ation by making your talk as clear as you your voice may be that of conversation but can, as interesting as you can, and no the speed should be considerably less. If longer than the allotted time. {Your suc­ you have to leave or turn away from the cessor is a member of your audience, microphone, increase your volume. and deserves at least this much consid­ {d) If you use slides, make them by eration.) hand, using large letters, or use a type­ The mechanics of doing these things writer that makes letters twice as high as are simple, but take some effort: the ordinary, and very black. A slide made {a) Unless you are an experienced on an ordinary typewriter is almost use­ speaker, write out every word of your less and very irritating. Try your slides at talk. {Experienced speakers do this with­ your home institution beforehand. If you out being told.) Deliver it to the mirror, use a Vu-Graph, or a similar device, write slowly, allowing time for writing formu­ in a big round hand. The messages on these las. Add ten per cent to your measured devices should be short, and set off by time; if this makes the total greater than plenty of white space. If you use a black­ your allotment, cut something out. Prac­ board, write large, and with a wide, firm tice several times. stroke. Stand on the right, as seen from {b) Omit details. Spend your time on the audience {if you are right-handed), so the ideas you are developing, and on a clear that your body {graceful and elegant, but statement of your most important theo­ opaque) does not obscure the writing. rem or two. Omit proofs; if you have plenty This takes practice, but you will learn of time you can indicate briefly the general how to do it some time, in order to be a line of reasoning. Omit almost all refer­ good teacher. ences; nobody believes you thought it all {e) Practice is necessary to learn up yourself, and unless a reference will how to express your ideas clearly and suggest an idea or a course of reasoning briefly, so practise. The popular phrase to your audience, save time by omitting it. "and uh" is not an ornament of style but {Of course, if you have thirty minutes a sign that you haven't done your home­ instead of ten, you can safely relax these work. restrictions.) Peccavi Quoque

630 VISITING FOREIGN MATHEMATICIANS

The following list contains the names of foreign mathematicians who are visiting at various institutions in the United States this year. The list is compiled from responses received on or before October 10 to requests sent out by the Society to academic institutions.

Home Country Host Institutions Period of Visit

Amir, Dan Israel University of California, Berkeley Sept. 1963-June 1964 Anderson, J. Scotland University of Michigan Sept. 1963-.June 1964 Ando, Tsuyoshi Japan California Institute of Technology Sept. 1963-Sept. 1964 Asplund, Edgar Sweden University of California, Berkeley Jan. 1963-Feb. 1964 Avila, G. Brazil Mathematics Research Center, Sept. 1963-June 1964 U. S. Army, University of Wisconsin Baer, Reinhold Germany University of California, Berkeley Sept. 1963-0ct. 1963 Barratt, M. England University of Chicago Oct. 1963-June 1964 Bhargave, T. N. India Kent State University Sept. 1963-June 1964 Bhatia, N. P. India Western Reserve University Sept. 1963-June 1964 Brickell, F. England Northwestern University Sept. 1963-June 1964 Brooker, R. A. England T. J. Watson Research Center, Oct. 1962-Sept. 1963 IBM Carnian, P. Italy T. J. Watson Research Center, Sept. 1963-June 1964 IBM Chakravarti, I. M. India University of North Carolina Nov. 1963-June 1964 Challifour, J. L. England Princeton University Sept. 1963-June 1964 Chatterji, S. D. India Mathematics Research Center, Sept. 1963-Aug. 1964 U. S. Army, University of Wisconsin Choquet, Gustave France Cornell University Sept. 1963-Feb. 1964 Choquet, Yvonne France Cornell University Sept. 1963-Feb. 1964 Clement, M. France University of California, Berkeley Sept. 1963-July 1964 Conti, Roberto Italy RIAS Sept. 1963-Feb. 1964 University of Maryland March 1964-May 1964 Cryer, Colin W. South Africa California Institute of Technology Oct. 1963-0ct. 1964 Daigneault, Aubert Canada University of California, Berkeley Sept. 1963-June 1964 Das, Anadi India Carnegie Institute of Technology Sept. 1963-June 1965

Dekker, Theodorus J. Netherlands University of California, Berkeley Sept. 1963-June 1964 Dinges, Hermann Germany Cornell University Sept. 1963-June 1964 Dou, A. Spain Mathematics Research Center, Sept. 1963-June 1964 U. S. Army, University of Wisconsin Duby, J. J. France T. J. Watson Research Center, Oct. 1963-0ct. 1964 IBM Duff, G. F. D. Canada Institute for Advanced Study Jan. 1964-April 1964 Dvoretzky, Aryeh Israel Columbia University June 1963-Nov. 1963 Edelstein, M. Israel Michigan State University Sept. 1963-July 1964 Eggleston, H. G. England University of Washington July 1963-June 1964 Engeli, M. Switzerland University of Minnesota Sept. 1963-June 1964 Ezeilo, J. Nigeria University of Michigan Sept. 1963-June 1964

631 Nrune Home Country Host Institutions Period of visit

Foguel, S. R. Israel Northwestern University Sept. 1963-June 1964 Foster, F. G. li:ngland Columbia University Feb. 1964-June 1964 Gaschutz, Wolfgang Germany Michigan State University Sept. 1963-Sept. 1964 Gilbert, J. E. England University of California, Berkeley Sept. 1963-June 1964 Gill, A. E. Australia Massachusetts Institute of Oct. 1963-June 1964 Technology Giri, N. C. India Cornell University Sept. 1963-June 1964 Graham, Roland England Massachusetts Institute of Sept. 1963-June 1964 Technology de Groot, Johannes Netherlands Washington University Sept. 1963-Jan. 1964 Grunbaum, Branko Israel University of Washington Aug. 1963-0ct. 1963 Haefliger, A. Switzerland Columbia University Oct. 1963-Jan. 1964 Hain, Klaus W. Germany National Bureau of Standards Jan. 1964-Aug. 1964 Haken, W. R. G. Germany Institute for Advanced Study Sept. 1963-April 1964 Hallin, Shlomo Israel University of California, Berkeley Sept. 1963-June 1964 Hardie, K. A. South Africa University of California, Berkeley July 1963-Jan. 1964 Hasumi, Morisuke Japan University of California, Berkeley Sept. 1962-June 1964 Hirzebruch, Friedrich Germany University of California, Berkeley Sept. 1963-0ct. 1963 Hirzebruch, Ulrich Germany Massachusetts Institute of July 1962-June 1964 Technology Hodgkin, L. H. England Institute for Advanced Study Sept. 1963-April 1964 Holenweg, Werner Switzerland University of California, Los Angeles Sept. 1963-June 1964 Horadam, A. F. Australia University of North Carolina Feb. 1964-May 1964 Horadam, E. M. Australia University of North Carolina Feb. 1964-May 1964 Huber, P. J. Switzerland Cornell University Sept. 1962-June 1964 Huckemann, Friedrich Germany University of Tennessee Sept. 1963-June 1964 Huppert, B. Germany University of illinois Sept. 1963-June 1964 Hustad, Otte Norway University of California, Berkeley Sept. 1963-July 1964 Ifram, A. F. Lebanon Northwestern University Sept. 1963-June 1964 Ikebe, Teruo Japan University of Washington Sept. 1963-June 1964 Ise, Mikio Japan Institute for Advanced Study Sept. 1963-April 1964 Itoh, Makoto Japan North Carolina State University Sept. 1962-Sept. 1964 Jain, P. C. India Mathematics Research Center, March 1963,... U. S. Army, University of Wisconsin March 1964 Jussila, 0. K. Finland Princeton University Sept. 1963-June 1964 Kahane, Jean-Pierre France Institute for Advanced Study Sept. 1963-Dec. 1963 Kanno, Tsuneo Japan University of California, Berkeley Sept. 1963-June 1964 Kappos, D. A. Greece Catholic University Sept. 1963-May 1964 Karamata, J. Switzerland Mathematics Research Center, June 1963-0ct. 1963 U. S. Army, University of Wisconsin Katz, Paul Israel University of Washington Sept. 1963-June 1964 Kegel, Otto Germany University of Chicago Oct. 1963-June 1964 Kneser, Martin Germany Institute for Advanced Study Sept. 1963-April1964

632 Name Home Country Host Institutions Perlod of Visit

Koeth, G. M. Germany University of Maryland Sept. 1963-Sept. 1964 Kovari, Thomas England University of Maryland Sept. 1963-June 1964 Kreisel, Georg France Institute for Advanced Study Sept. 1963-April 1964 Kreiss, Heinz-Otto Sweden California Institute of Technology Dec. 1963-Aug. 1964 Kristensen, Leif Denmark University of California, Berkeley Sept. 1963-June 1964 Kubota, Tomio Japan Institute for Advanced Study Sept. 1963-April 1964 Kuga, Michio Japan University of Chicago June 1963-March 1964 K.ur.atowski, Kazimierz Poland Ohio State University Sept. 19 63 -Sept. 19 63 Kurth, R. England Michigan State University Sept. 1963-Aug. 1964 Laxton, R. England University of Michigan Sept. 1963-June 1964 Learner, Arnold England University of Illinois Sept. 1963-June 1964 Leopoldt, Heinrich Germany Johns Hopkins University Sept. 1963-May 1964 Lerner, Jean Marc France University of California, Berkeley Sept. 1963-June 1964 Lindenstrauss, Joram Israel Yale University Sept. 1963-June 1964 Linder, Arthur Switzerland University of North Carolina Sept. 1963-June 1964 Lindley, D. V. Wales Harvard Business School June 1963-Dec. 1963 Lintz, R. Brazil University of Michigan Sept. 1963-June 1964 Manohar, R. India Mathematics Research Center, Sept. 1962-June 1964 U. S. Army, University of Wisconsin Matsumura, Hideyuki Japan Johns Hopkins University Sept. 1963-Jan. 1964 Harvard University Feb. 1964-April 1964 Milne-Thomsen, L. M. England University of Arizona Sept. 1961 Misra, M. India University of Texas Sept. 1963-Sept. 1964 Mordell, L. J. England University of Arizona Sept. 1961 Morris, Grainger Australia RIAS June 1963-March 1964 Moyls, B. W. Canada Harvard University Sept. 1963-June 1964 Murakami, Haruo Japan University of Kansas Sept. 1963-May 1964 Murasugi, K. Japan Princeton University Sept. 1963-June 1964 Nagahara, T. Japan Northwestern University Sept. 1963-June 1964 Nagami, Keio Japan Duke University Oct. 1963-July 1964 Nagano, Tadashi Japan University of California, Berkeley Sept. 1963-June 1964 Nagata, Junichi Japan Institute for Advanced Study Sept. 1963-April 1964 Nagy, Bela Hungary Columbia University Feb. 1964-May 1964 Narasimhan, M. N. L. India Mathematics Research Center, July 1962-June 1964 U. S. Army, University of Wisconsin Nariboli, G. A. India Iowa State University Sept. 1963-May 1964 Noble, B. Scotland Mathematics Research Center, Sept. 1962-June 1964 U. S. Army, University of Wisconsin Okubo, T. Japan University of California, Los Angeles Sept. 1963-June 1964 Opial, Zdzistaw Poland RIAS Sept. 1963-Sept. 1964 Pan, J. B. Formosa University of California, Berkeley Sept. 1963-July 1964 Papadopoulous, M. United Kingdom Mathematics Research Center, Oct. 1961-June 1964 U. S. Army, University of Wisconsin Papangelou, Fredos Greece Catholic University Sept. 1963-May 1964

Parameswaran, M. R. India Washington State University Sept. 1963-June 1964 633 Home Country Host Institutions Period of Visit

Pathak, P. K. India University of Illinois Sept. 1963-June 1964 Penrose, Roger England University of Texas Sept. 19 63 -Sept. 19 64 Persson, Arne Sweden University of Kansas Sept. 1963-May 1964 Pinl, Maximilian Germany University of Idaho Sept. 1963-June 1964 Pitman, E. J. G. Tasmania The Johns Hopkins University Sept. 1963-June 1964 Poi3naru, V. Rumania Harvard University Sept. 19 63 -Sept. 19 65 Pumpliin, H. Germany University of Minnesota Sept. 1963-June 1964 Rahman, Q. I. India Northwestern University Jan. 1963-0ct. 1963 Rao, C. Radhakrishna India Johns Hopkins University Sept. 1963-Sept. 1964 Rao, K. M. India Princeton University Sept. 1963-June 1964 Rastall, P. Canada University of Texas Sept. 1963-June 1964 Reichaw, Meir Israel Cornell University Sept. 1963-June 1964 Riccia, G. D. Italy Massachusetts Institute of March 1963-Jan. 1964 Technology Rosenblat, Simon England Massachusetts Institute of Sept. 1963-March 1964 Technology Roxin, Emilio Argentina RIAS Aug. 1963-indefinite Saini, G. L. India Mathematics Research Center, Sept. 1962-March 1964 U. S. Army, University of Wisconsin Sakai, Shoichiro Japan Yale University Sept. 1963-June 1964 Sakaguchi, Minoru Japan George Washington University Sept. 1963-Sept. 1964 Sakurai, A. Japan Mathematics Research Center, April1963-March 1964 U. S. Army, University of Wisconsin Sanderson, Brian J. England Yale University Sept. 1963-June 1964 Scarpellini, B. Switzerland Mathematics Research Center, July 1963-June 1964 U. S. Army, University of Wisconsin Schaal, W. G. H. Germany Massachusetts Institute of July 1962-June 1964 Technology Schinzel, Andrzej Poland Ohio State University Oct. 1963-June 1964 Selberg, Henrik L. Norway Institute for Advanced Study Sept. 1963-April 1964 Shamir, Eliahu Israel University of California, Berkeley July 1963-June 1964 Shih, W eishu France Institute for Advanced Study Sept. 1963-April 1964 Shikata, Yoshihiro Japan University of California, Berkeley Sept. 1963-June 1964 Shimizu, Hideo Japan Institute for Advanced Study Sept. 19 62 -April 19 64 Shimura, G. Japan Princeton University Sept. 1962-June 1964 Singal, M. K. India University of Illinois Sept. 1963-June 1964 Sinha, I. India Michigan State University Sept. 1963-Aug. 1964 Sneddon, Ian N Scotland North Carolina State University March 1963-Apr. 1963 Stein, K. Germany University of Minnesota March 1963-Apr. 1964 Suschowk, Dietrich Germany University of California, Los Angeles Sept. 1963-Aug. 1964 Sutherland, W. A. England Massachusetts Institute of July 1963-June 1964 Technology Takeuti, Gaisi Japan University of Illinois Sept. 1963-June 1964 Tamari, Dov France State University of New York at Sept. 1963-June 1964 Buffalo

634 Name Home Country Host Institutions Perigd of Visit

Thomee, v. c. Sweden University of Maryland Sept. 19 -Jan. 1964 Tomita, Minoru Japan University of Washington June 1963-Dec. 1963 Tranquilli, G. Italy University of North Carolina Sept. 1963-June 1964 Urabe, M. Japan Mathematics Research Center, Sept. 1963-Aug. 1964 U. S. Army, University of Wisconsin Varma, H. 0. Netherlands Harvard University Sept. 1963-June 1964 Veldkamp, F. D. Netherlands Yale University Sept. 1963-June 1964 Vorel, Zdenek Czechoslovakia RIAS Jan. 1964-Feb. 1964 Vyborny, Rudolf Czechoslovakia University of Maryland Sept. 1963-Jan. 1964 Waadeland, Haakon Norway University of Colorado Sept. 1963-July 1964 Wagner, A England University of Michigan Sept. 1963-June 1964 Wilker, Peter Switzerland State University of New York at Sept. 1963-June 1964 Buffalo Wormleighton, Ralph Canada The Johns Hopkins University Sept. 1963-June 1964 Yates, M. C. E. England Cornell University Sept. 1963-June 1964 Yoeli, M. Israel Mathematics Research Center, Sept. 1963-Aug. 1964 U. S. Army, University of Wisconsin Yoshizawa, T. Japan Iowa State University Sept. 1963-May 1964 Zelazke, W. T. Poland Yale University Sept. 1963-June 1964

2 New Texts for 1964 . .. ORDINARY DIFFERENTIAL EQUATIONS: AModern Approach Harry Hochstadt, Professor of Mathematics, An introduction to the Polytechnic Institute of Brooklyn This new text presents the modern viewpoint HISTORY of MATHEMATICS, on differential equations within a framework requiring only a calculus prerequisite. It Revised Edition emphasizes mathematics, rather than its ap­ plications, but provides the basic background Howard Eves, University of Maine required for an understanding of applica­ tions. Professor Hochstadt uses vector nota­ As in the highly successful first edition, the tion consistently, and devotes an entire student is not merely taught that the Greeks to boundary value problems. He He chapter solved quadratic equations geometrically. treats in detail equations with constant co­ is presented with problems to be solved by efficients and thoroughly investigates series the Greek method. He achieves not only a methods for cases where coefficients are appreciation mastery of the method, but an either analytic or have Fuchsion singularities. of Greek mathematical accomplishment. Designed for an undergraduate course in the ]an., 1964 300 pp. $6.50 tentative history of mathematics, this new edition has been expanded and updated, with chapters arranged flexibly to accommodate students' varying mathematical backgrounds. March, 1964 460 pp. $7.50 tentative

635 NEW AMS PUBLICATIONS

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS Volume XV HIGH SPEED COMPUTING AND EXPERIMENTAL ARITHMETIC Edited by N. C. Metropolis, A. H. Taub, John Todd, C. B. Tompkins 408 pages; List Price $9.10; 25% discount The second symposium was held at to members. Atlantic City, New Jersey, April 16-18 on the subject of Interactions between Mathe­ This volume contains all but two of matical Research and High-Speed Com­ the papers which were presented at two puting. This symposium was sponsored by symposia sponsored by the American the American Mathematical Society and Mathematical Society and other co-spon­ the Association for Computing Machinery sors in the spring of 1962. and was supported financially by the U.S. The first symposium was held in Army Research Office (Durham) and the Chicago, lllinois, April 12-14, on the sub­ National Science Foundation. Its organiz­ ject of Experimental Arithmetic. This ing and invitations committee consisted of symposium was sponsored by the Ameri­ John Todd (Chairman), G. E. Forsythe, can Mathematical Society and it was sup­ P. D. Lax, D. H. Lehmer, H. H. Goldstein, ported financially by the Institute for De­ C, B. Tompkins and D. M. Young, Jr. fense Analyses. The organizing and in­ The objective of this symposium vitations committee consisted of N. Met­ was to enable mathematicians to become ropolis (Chairman), Marshall Hall, Jr., familiar with the potentialities of compu­ Peter Henrici, Mark Kac, R. D. Richtmyer, ters of types currently available and with and A. H. Taub, the problems involved in the proper and The objective of this symposium effective exploitation of these computers. was to examine ways in which the arith­ The close relationship between the metical potential of modern high· speed subject matters of the two symposia computers can furnish experience which prompted the organizing committees to sheds light on outstanding problems in merge the proceedings into this single mathematics and other sciences, volume,

SELECTED TRANSLATIONS, SERIES II Volume 32 400 pages; List Price $5.50; 25% discount Seventeen papers on Functions of to members. Complex Variables.

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Volume VI HARMONIC ANALYSIS OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES IN CLASSICAL REGIONS by K. L. Hua 168 pages; List Price $6.90; 25% discount systematization of results of the author to members. published in numerous articles over the past fifteen years. This work is an exposition and a

636 Volume X THE THEORY OF IRRATIONALITIES OF THE THIRD DEGREE by B. N. Delane and D. K. Faddeev 490 pages; Prepublication Price $9.30 an exhaustive treatment of cubic irration­ (before December 15). List Price notless alities and the problems in number theory than $10.30. connected with them. It is distinguished by its extensive use of geometrical arguments This outstanding monograph gives and its valuable numerical tables.

Volume XI PROBABILISTIC METHODS IN THE THEORY OF NUMBERS by I. Kubilius ZOO pages; Prepublication Price $7.80 many new and important results on the dis­ (before December 15). List Price notless tribution of the values taken by an additive than $8.60. function. The second section gives a for­ mal account of how the general theory of This work, in which the theory of stochastic processes can be applied to the probability is used as one of the standard study of number theoretic functions. tools of the theory of numbers, contains

Volume XII QUALITATIVE METHODS IN MATHEMATICAL ANALYSIS by L. E. El'sgo'c 280 pages; Prepublication Price $13.20 differential equations and particularly of (before January 15, 1964). List Price not differential-difference systems, are pre­ less than $14.60. sented on the basis of the work of Morse, Birkhoff, Schauder, Ljapunov, and Poin­ The important properties of exist­ care. ence, uniqueness, periodicity, asymptotic behavior and stability, of the solutions of

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THE PROBLEM OF MOMENTS by j. A. Shohat and j. D. Tamarkin 144 pages; List Price $5.80; 25"/o discount A reprinting of the edition pub­ to members. lished in 1960.

637 PERSONAL ITEMS

Dr. H. I. ANSOFF of Lockheed Elec­ Wisconsin has been appointed to an tronics Company has been appointed to a associate professorship at Orange State professorship at the Carnegie Institute of College. Technology. Dr. G. W. GOES of the University of Associate Professor F. G. ASENJO Western Ontario,London, Ontario, Canada, of Southern Illinois University has been has been appointed to an assistant pro­ appointed to an associate professorship fessorship at the University of Kansas. at the University of Pittsburgh. Professor W. A. HIJAB of the Ameri­ Dr. F. W. ASHLEY, JR. of Southwest can University of Beirut has been appointed Missouri State College has been appointed Visiting Lecturer at the California State to an assistant professorship at the Uni­ Polytechnic College for the academic year versity of Arkansas. 1963-1964. Mr. T. I. BARTHA of the University Professor P. D. HILL of Auburn Uni­ of Illinois has been appointed to an assist­ versity has been appointed to an associate ant professorship at Pratt Institute. professorship at Emory University. Mr. G. W. BATTEN, JR.of William Dr. HUNG-TA HO of Hydronautics, Marsh Rice University has accepted a Incorporated has been appointed to an position as Research Associate at the assistant professorship at the University Digital Computer Laboratory, University of Oklahoma. of Illinois. Dr. R. F. JACKSON of the University Dr. H. F. BECHTELL of Lebanon of Delaware has been appointed to a pro­ Valley College has been appointed to an fessorship at the University of Toledo. associate professorship at Bucknell Uni­ Dr. V. L. KANTHAM of RIAS has versity. been appointed to an associate professor­ Dr. JOHN BEEKMAN of the University ship at the University of Alberta, Calgary, of Minnesota has been appointed to an Alberta, Canada. assistant professorship at Ball State Dr. U. R. KODRESoftheinternational Teachers College. Business Machines Corporation, has been Dr. E. J. BELTRAMI of Lerici, Italy, appointed to an associate professorship at has accepted a position as Research Math­ the U. S. Naval Postgraduate School, ematician with the Grumman Aircraft Monterey, California. Engineering Corporation, Bethpage, New Dr. OTTO KOERNER of the University York. of Marburg, Germany has been appointed Dr. J. R. BUCHI of the University of to an assistant professorship at the Uni­ Michigan has been appointed to a pro­ versity of Utah. fessorship in Mathematics and Computer Professor ALI KYRALA of Tempe, Sciences at Purdue University. Arizona has received a Fulbright Lecture­ Associate Professor W. B. CATON ship as Professor of Physics and Applied of DePaul University has been appointed Mathematics at the University of Alexan­ to an associate professorship at the Illinois dria, Egypt, for the academic year 1963- Institute of Technology. 1964. Dr. J. S. DENTON, JR. of Harvard Mr. GEORGE LANGBERG of the Uni­ University has been appointed to an assist­ versity of Illinois has been appointed to an ant professorship at McGill University, assistant professorship at Orange State Montreal, Canada. College. Professor J. J. GERGEN of Duke Dr. E. S. LANGFORD of Rutgers, The University will be on leave for the aca­ State University, has accepted a position as demic year 1963-1964 in Glasgow, Scot­ Specialist-Research with Autonetics, a land. Division of North American Aviation, Dr. R. C. GILBERT of the Mathe­ Anaheim, California. matics Research Center, University of Professor HARRY LANGMAN of Ball

638 State Teachers College has been appointed Mr. R. M. VOGT of the University of to a visiting professorship at Clarkson illinois has been appointed to an assistant College of Technology. pFofessorship at the University of Arizona. Dr. W. j. LEVEQUE of the University Assistant Professor R. J. WARNE of of Michigan has been appointed to a visiting Louisiana State University has been ap­ professorship atthe University of Colorado pointed to an associate professorship at for the academic year 1963-1964. the Virginia Polytechnic Institute. Dr. R. G. MciNTYRE of Texas Instru­ Professor S. E. WARSCHAWSKI of ments, Incorporated has been appointed to the University of Minnesota has been ap­ an assistant professorship at Oklahoma pointed to a professorship at the University State University. of California, San Diego. Professor C. G. MAPLE oflowaState Dr. E. M. ZAUSTINSKY of the Uni­ University has been named Director of versity of California, Berkeley has been the Computation Center. appointed to an associate professorship Associate Professor D. G. MEAD of at the State University of New York at Pratt Institute has been appointed to an Stony Brook. associate professorship at the University of Santa Clara. The following promotions are announced: Mr. R. B. MERKEL of the Univer­ sity of California, Davis has been appointed R. L. ALVES, Sacramento State Col­ to an assistant professorship at the Sacra­ lege, to an assistant professorship. mento State College. D. F. ATKINS, Eastern Illinois Uni­ Assistant Professor R. A. MOORE of versity, to a professorship. St. Mary's University has been appointed R. T. BARNES, Ohio State University, to an associate professorship at the State to an assistant professorship. University College at Oswego. W. A. BECK, Chatham College, to an Dr. M. G. OSSESIA of Slippery Rock associate professorship. State College has been appointed to an R. j. BLATTNER, UniversityofCali­ associate professor ship at Seton Hall Uni­ fornia, Los Angeles, to an associate pro­ versity. fessorship. Dr. I. R. SAVAGE of the University of RICHARD BLOCK, University of Wis­ Minnesota has been appointed to a pro­ consin-Milwaukee, to an assistant pro­ fessorship at Florida State University. fessorship. Dr. L. J. SENECHALLE of the Illinois G. T. CARGO, Syracuse University, Institute of Technology has been appointed to an associate professorship. to an assistant professorship at the Uni­ A. G. DAVIS, Clarkson College of versity of Arizona. Technology, to an associate professorship. Dr. S. S. SHRIKHANDE of Banaras PLATON DELIYANNIS, Illinois Insti­ Hindu University, India has been appointed tute of Technology, to an assistant pro­ to a professorship at the Bombay Univer­ fessorship. sity, Bombay, India. WALTER EHRENPREIS, Trenton Professor j. M. STARK on leave of State College, to an associate professor­ absence from Lamar State College of Tech­ ship. nology for the academic year 19 63-19 64 E. E. ENOCHS, University of South has received a Science Faculty Fell ow ship Carolina, to an associate professorship. sponsored by the National Science Found­ ARTHUR FINE, Illinois Institute of ation at Stanford University. Technology, to an assistant professor ship. Dr. ROBERT VERMES of the Univer­ STEVENSON HECKSCHER, Swarth­ sity of Wisconsin has been appointed more College, to an assistant professor­ Lecturer at McGill University, Montreal, ship. Quebec, Canada. WU-CHUNG HSIANG, Yale University, Dr. BURTON WENDROFF of the Los to an assistant professorship. Alamos Scientific Laboratory, Los Alamos, J. M. IRWIN, New Mexico State Uni­ New Mexico has been appointed to a visiting versity, to an associate professorship. associate professorship at Brown Univer­ H. C. KRANZER, Adelphia College, sity. to a professorship.

639 K. M. McMILLIN, Michigan College of E. A. WALKER, New Mexico State Mining and Technology, to a professorship. University, to a professorship. ABE SHENITZER, Adelphi College, G. G. WALTER, University of Wis­ to a professorship. consin-Milwaukee, to an assistant pro­ R. H. SORGENFREY, University of fessorship. California, Los Angeles, to a professor­ WILLIAM WITTHOFT, Illinois Insti­ ship. tute of Technology, to an assistant pro­ W. F. STEELE, Heidelberg College, to fessorship. a professorship. EDWIN STUEBEN, illinois Institute of The following appointments to Instructor­ Technology, to an assistant professorship •. ships are announced: J. W. SUMMERS, California State College at Hayward, to an associate pro­ Berea College: F. D. JOHNSON; Uni­ fessorship. versity of North Carolina: M. E. WATKINS; P. E. THOMAS, University of Califor­ University of Pittsburgh: JOSEPH BAR­ nia, Berkeley, to a professorship. BACK; Princeton University: C. E. WElL; E. O. THORP, New Mexico State Uni­ Seton Hall University: RICHARD GRATH­ versity, to an associate professorship. OFF; Washington University: P. J. SALLY, F. D. TRINE, Wisconsin State College, JR.; University of Washington: T. W. to an associate professorship. HUNGERFORD; Wisconsin State College: D. H. TUCKER, University of Utah, R. 0. SOMMER. to an associate professorship.

CONSULTING MATHEMATICIAN To lead staff of applied mathematicians in solving complex problems in the nuclear reactor field. • A wide range of interesting mathematical problems is encountered in the nuclear reactor field.* Independent mathematical research is currently being conducted in numerical analysis with emphasis on partial differential equation soluton by means of finite difference techniques and Monte Carlo methods. In the area of physics, typical analysis problems arise in connection with the solution of multigroup neutron diffusion and neutron transport equations, as well as from problems in reactor dynamics. In the area of engineering, diverse problems are encountered in heat transfer, fluid flow, and mechanics; for example, the solution of equations for non-steady state conduction for a variety of boundary conditions, and the solution of elasticity and plasticity problems. • Consultation in mathematics is provided throughout the Laboratory and there is opportunity for the mathematicians to work closely with theoretical physicists and analytical engineering specialists on problems of mutual interest. • Extensive supporting facilities include digital and analog computers with associated staffs of programmers. • Candidate's qualifications include PhD in Mathematics plus pertinent experience. *See e.g. Proceedings of Symposia in App'lied Math, Vol. 11, Nuclear Reactor Theory, American Mathematical Society 1961

Room 157K To apply or ~ain additional information, write fully in strict confidence to Mr. Richard Bouton, K~A~E~L~ GENERALfj ELECTRIC U.S. Citizenship Required • Schenectady, New York • An Equal Opportunity Employer

640 SUPPLEMENTARY PROGRAM NO. 21

During the interval from September 5, 1963 through September 24, 1963 the papers listed below were accepted by the American Mathematical Society for presenta­ tion by title. After each title on this program is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Con­ tributed Papers in this issue of these NOTICES,

( 1) Sets of infinite limit (10) On surfaces in E 3 with constant Professor R. J, Bumcrot, Ohio negative curvature. Preliminary re­ State University and Mr. P, D. port Zvengrowski, University of Chicago, Professor Tilla Klotz, University (63T-407) of California, Los Angeles (63T-408) (2) The determination of an ordinary dif­ ( 11) Some algebraic characterizations of ferential operator from its spectral topological properties. Preliminary density report Professor J, B. Butler, Jr., Port­ Professor K. D. Magill, Jr., State land State College (63T-406) University of New York College at (3) An extremum problem in the space of Buffalo (63T-418) matrices. Preliminary report ( 12) Completions of abelian groups Mr. R. L. Causey, Lockheed Mis­ Mr. C. K. Megibben, Texas Tech­ siles and Space Company, Palo Alto, nological College (63T-414) California (63T-404) (13) Multiples of pure subgroups (Introduced by Professor G. E. For­ Mr. C. K. Megibben, Texas Tech­ sythe) nological College (63T-415) (4) On embedding stably parallelizable (14) Kaplan sky's test problems for alge­ manifolds in euclidean space. I. braically compact groups Preliminary report Mr. C. K. Megibben, Texas Tech­ Mr. R. V. DeSapio, University of nological College (63T-416) Chicago (63T-409) ( 15) Recursive unsolvability of the de­ ( 5) On embedding stably parallelizable cision problem for models of .the manifolds in euclidean space, II. fundamental theorem of arithmetic Preliminary report Mr. A. A. Mullin, University of Mr. R. V, DeSapio, University of Illinois (63T-412) Chicago (63T-410) ( 16) The fundamental theorem of arithme­ (6) The insufficiency of barycentric sub­ tic has only three inequivalent forms division Mr. A. A. Mullin, University of Dr. R. L. Finney, Massachusetts Illinois (63T-413) Institute of Technology (63T-399) ( 17) Groups or order 1 (7) Continued fraction expansions for Professor E. S. Rapaport, Poly­ v'C + L and iterations of Newton's technic Institute of Brooklyn (63T- formula 401) Professor Evelyn Frank, University ( 18) Extremal harmonic functions of of Illinois and Professor Ambikesh­ several variables war Sharma, University of Chicago Professor Leo Sario, University of (63T-411) California, Los Angeles (63T-397) (8) Generalized solutions of boundary ( 19) Smoothness of Fourier transforms value problems Professor M. H. Taibleson, Wash­ Professor M. D. George, Univer­ intgon University (63T-400) sity of Missouri (63T-419) (20) Concerning continua which are con­ (9) A property of d-polyhedral graphs tinuous images of compact ordered Professor V, L. Klee, University spaces of Washington (63T-403) Professor L. 8, Treybig, Tulane

f:i41 (Introduced by Professor RADIOACTIVITY Nathaniel Coburn) Problem: A convoy has been tacked by a thermonuclear wea): What maneuvers will permit ships to best avoid the radioacth that will follow? This is an example of the challe ing tasks assigned to the Center Naval Analyses of The Franl Institute. Several possible tactics have b evaluated by CNA analysts. E has its advantages and disadv tages. If the convoy stays toget MEMO TO MEMBERS and maintains course, possibilit~ collision is minimized, but swiftest escape from contaminat SUMMER EMPLOYMENT may not be attained. While ot OPPORTUNITIES maneuvers may reduce the po bility of contamination, they 11 The Mathematical Sciences Em· lead to confusion, minimizing ployment Register, 190 Hope Street, Prov­ convoy's over-all progress, 1 idence 6, Rhode Island announces that the increasing the danger of rep attack. The conclusion list of Opportunities for Summer Employ­ is that of the intermediate tactics is b ment for Mathematicians and College Mathematics Students will again be com­ CAREER OPPORTUNITIES in this. piled. other problem areas are now av able with CN A for Operati· Institutions which have summer Analysts, Mathematicians, Ph: openings and would welcome applications cists, and Engineers. For additic for summer employment for mathemati­ information, write: cians, applied mathematicians, theoretical Director physicists, and engineers ·.vith advanced CENTER FOR NAVAL ANALYSES mathematical training should write to the Dept. XX Mathematical Sciences Employment Reg­ 1710 H St., N. W., Washington, D ister, 190 Hope Street, Providence 6, Rhode Island, requesting forms for the listing of summer jobs. There is no charge for this list, CENTER FOR NAVAL ANAL' which will be available at the Annual OF THE FRANKLIN INSTITUTE Meeting in Miami in January, 1964 and on OEG • OPERATIONS EVALUATION GROU request from the Mathematical Sciences INS • INSTITUTE FOR NAVAL STUDIES Employment Register. It will be issued NAVWAG • NAVAL WARFARE ANALYSIS I early in January, 1964. An equal opportunity employer

642 ABSTRACTS _OF CONTRIBUTED PAPERS

The 1\" ovember J\'1 eeting in Atlanta, Georgia November 15-16, 1963

6a5-l. J. S. MAC NERNEY, University of North Carolina, Chapel Hill, North Carolina.

Characterization of regular Hausdorff moment sequences.

Let C [0,1] denote the class of continuous complex-valued functions from the interval (a,I].

Theorem. In order that the infinite complex-number sequence p., with P.a = 1, be a regular Hausdorff moment sequence, it is necessary and sufficient that, for each f in C [a,l], the limit L(f) =

Lilllu--->ooL~=a (- l)n-kf(k/n)Cn,kAn-kp.k exist, and in this case, for each fin C[a,l], L(f) = f(l)•Lim p.. This theorem was suggested to the author by H. S. Shapiro's proposal [Amer. Math. Monthly 69 (1962), Ial3] to the effect that, for each fin c[a,l), Limn_,ooL~=a<- l)kf(k/n)Cn,k/2n =a. It follows from the Theorem that this proposal is precisely the assertion that {l/2n} ~ is a regular Hausdorff moment sequence with limit zero. (Received June 1a, 1963.)

6a5-2. R. S. COX, University of North Carolina, Chapel Hill, North Carolina. Representations of a semigroup.

LetS be a semigroup with zero. The radical of S, N(S), is the set js C S: if M is a a-transitive operand of S then Ms is the zero element of M}. Thus N (S) = n j x:x is a zero of S/~ where ~ is a maximal right congruence on S such that there is a left identity modulo ~}. If e C S then e is a right

quasi-regular element of S if there does not exist a right congruence ~on S such that S/~ is a

a-transitive operand of Sand e is a left identity modulo o An ideal of Sis quasi-regular if each element in it is right quasi-regular inS. Then N(S) is a quasi-regular ideal of S which contains each quasi-regular right ideal of Sand N(S) contains each nil right ideal of S. If S satisfies the minimum condition on right ideals and T is a right ideal of S which is not nilpotent then there exists an element ofT which is not right quasi-regular; hence a nil right ideal of S is nilpotent. If Sn is the

semigroup of all row-monomial n X n matrices over S then N(Sn) = [N(S)]n. If s, t C S let spt provided that if m is in some a-transitive operand of S then ms = mt. Then pis a two-sided con­

gruence called the radical congruence of S. If p is the identity relation then S is semisimple. With these definitions, there are semigroup analogs of many results from the representation theory for rings. (Received June 1a, 1963.)

6a5-3. R. J. WARNE, Virginia Polytechnic Institute, Blacksburg, Virginia. Homomorphisms of d- simple inverse semigroups with identity. II.

The terminology and notations of Redei are foilowed (Acta. Sci. Math. Szeged 14 (1952), 252- 2 73 ). A B- semigroup is a right cancellative semigroup with identity. Theorem 1. Let U be a group and let S be a B-semigroup with trivial unit group. Then every Schreier extension P of U by S is a

643 B- semigroup whose unit group is isomorphic to U. L is a congruence relation on P and P /L ~ S. The intersection of principle left ideals in P is a principle! left ideal in P iff this is the case in S. Conversely, let P be a B-semigr:mp on which Lis a congruence relation. Let U be the unit group of p. Then P is a Schreier extension of U by P /L. P /L is a B- semigroup and has a trivial unit group. Theorem 2. Let P and P • be B- semigroups on which L and L • are congruence relations. Let U and U* be their respective unit groups. Let f be a homomorphism of U into U*, g a homomorphism of P/L into P*/L*, and h be a function of P/L into U* such that (ah)(bh)ag(ag)bg = (abf)(ab)h and (bh)(A)bg = (A bf)(bh) (a,b in P /L, A in U). For each (A,a) in P define (A,a)M = [(Af), (ah)ag] where square brackets denote elements of P • Then M is a homomorphism P into P • Conversely every such homo- morphism is obtained in this fashion. N is an isomorphism iff f and g are isomorphisms. In im­ portant special cases, the above conditions simplify greatly. (Received June 24, 1963.)

605-4. G. S. JONES, RIAS, 7212 Bellona Avenue, Baltimore 12, Maryland. Fixed points in gouged convex sets.

A commonly encountered situation in applications of fixed point theory is the problem of "weeding out" an undesired fixed point from a set in such a way as to allow one to conclude the existence of a fixed point still remaining in the set. For example, suppose in a linear space X that 0 is contained in a convex subset A and is a fixed point under a mapping F defined on A. The problem might be to verify the existence of a nontrivial fixed point under T in A. A technique for obtaining such results is provided by the following theorem. Theorem. Let X be a complete locally convex linear topological space, A a closed convex subset of X, and F a compact mapping of A into itself.

Let B be a closed convex body in the linear extension Y of A, and let x 1 be a point in the interior of A defined relative toY. If the cone with vertex at x 1 and generated by the intersection of B with the boundary of A in Y is convex and F(A---=--8) c A - B, then F has a fixed point in A - B. (Received June 21, 1963.)

605-5. C. W. McARTHUR and J. R. RETHERFORD, Florida State University, Tallahassee, Florida. Uniform bases and the equicontinuity of projections associated with Schauder decompositions.

A sequence {Mil of nontrivial subspaces of a linear topological space X is a decomposition of

X if and only if each x E:: X can be uniquely represented x = l.:';oxi, xi E:: M. A decomposition {Mi} is

Schauder if the projections {Ei}, where Ei(x) =xi if x = l.:fxi, are all continuous and {Mi} is an e-Schauder decomposition if 0 is a point of equicontinuity of the projections { Sn} where Sn(x) = l.:~Ei (x). If X is a Banach space which is also a linear topological space with a topology T then a decomposition {Mi} is T-uniform if x = limnL~Ei(x) uniformly for l!xll $1, where convergence is relative to the topology T. Among the theorems proved are the following: (1) Every Schauder de com­ position of a second category linear topological space is an e-Schauder decomposition. (2) An infinite dimensional Banach space (conjugate space) does not admit an e-Schauder basis relative to its weak (its w*) topology. (3) A Banach space X has a Schauder decomposition in the norm topology if and only if X* has a Schauder decomposition in the w*-topology. (4) Every w*-Schauder decomposition of a conjugate Banach space is w*-uniform. (5) A weak Schauder decomposition {Mi,Ei} of a

644 Banach space is weak-uniform if and only if {R(Et), E{} is a Schauder decomposition of X* where

E{ is the adjoint of Ei and R(~*) is the range of Et. (Received August 9, 1963.)

605-6. JACK SEGAL, Institute for Advanced Study, Princeton, New Jersey. Inverse dimension

~ I. Types in the real line.

In this paper a new dimension type is defined which is analogous to Frckhet dimension type, but under which many more spaces are comparable. Definition. The inverse dimension type of a topological space X is less than or equal to the inverse dimension type of a topological space Y iff, for each covering a. of X, there is a closed a.-map of X into Y. Now there are 2c Fr~chet dimension types represented by subsets of the line. It is shown that there are only denumerably many inverse dimension types represented by subsets of the line. Furthermore, the partial ordering of these types is completely determined and a topological characterization is given of the linear sets having a given type. (Received September 5, 1963.)

605-7. HOWARD COOK, Auburn University, Auburn, Alabama. Upper semi-continuous collections filling up hereditarily indecomposable continua.

Suppose that M is a compact metric hereditarily indecomposable continuum and x is a positive number smaller than the diameter of M. Then there exists an upper semi-continuous collection G of mutually exclusive continua filling up M such that each element of G has diameter x. If M is a

subset of the plane, G is nearly continuous in the sense that it is upper semi- continuous and, if a. and fJ are sequences of elements of G having sequential limiting sets Hand K, respectively, and Hand K are subsets of the same element of G, then one of the two continua H and K is a subset of the other. (Received September 13, 1963.)

605-B. R. W. GILMER, JR., Florida State University, Tallahassee, Florida, and J. E. OHM, University of Wisconsin, Madison, Wisconsin. Primary ideals and valuation ideals.

Suppose D is an integral domain with identity having quotient field K ~D. An ideal A of Dis a valuation ideal if there exists a valuation ring R with D <; R S K such that AR n D = A. Let U denote the set of valuation ideals of D and let Q denote the set of primary ideals of D. The following theo­ rems partially characterize domains in which a containment relation exists between U and Q.

Theorem 1. U = Q if and only if D is a one-dimensional Priifer domain. Theorem 2. U c::::; Q if and only if Dis one-dimensional. Theorem 3. If the ascending chain condition holds for prime ideals of

D, Q c::::; U if and only if Dis a Priifer domain. (Received September 16, 1963.)

605-9. F. L. HARDY, Emory University, Atlanta 22, Georgia. A note on torsion-free rings.

If A= [a 1, a 2, .•• , an] is a torsion-free Abelian group of finite rank n, ITi(a) = ri for i = 1,2, ••. , n where a= L~ 1 riai. If Ai is defined to be the group generated by all ITi(x)y for x, y C A and a.i is an endomorphism of Ai, call the set of endomorphisms a.l' a. 2, •.• , a.n compatible with respect to A provided :Lf=1[ITi(x)y]a.i CA for all x, y CA. Theorem. Let A be a torsion-free Abelian group of finite rank n with basis a 1, a 2, .•• , an. Then A(+, ·) is a ring if and only if there are endo- 645 morphisms a.i : Ai--+ Ai, i = 1,2, •••• n, compatible with respect to A, such that x•y = I;~= dili(x)y]a.i for all x, y E: A. (Received September 20, 1963.)

605-10. ECKFORD COHEN, University of Tennessee, Knoxville 16, Tennessee. Two functions related to the k-free integers.

Suppose that k ~ 2; a residue class (mod n) will be called admissible if it contains at least one k-free integer. Define 3Jc (n) to be the maximum relative density of the k-free integers in an admissiple residue class (mod n), and define bk(n) as the probability that the relative density of the k-free integers in an admissible class (mod n) assumes its maximum. Asymptotic expressions for the averages of ak and bk are proved in this paper by elementary means. A single result is cited:

The mean value of b2(n) is Il(l - 1/p2 + 1/p3 ) where the product ranges over the primes. (Received September 23, 1963.)

605-11. J. G. HORNE, Jr. University of Georgia, Athens, Georgia. ldempotents in semigroups o.n a half-space.

By a semigroup on a half-space is meant a semigroup S with the following properties: (a) S has an identity; (b) the underlying space of S is homeomorphic to {(x,y,z): z ;;;:; 0;} (c) the maxi­ mal subgroup G occupies the set with z > 0 (thus the boundary L of G occupies the plane z = 0). A number of properties are obtained about idempotents which hold promise of yielding an early determination of the possible multiplications on L and of connections between the particular group G and such multiplications. The most crucial results are the following: (1) L must contain an idempotent; (2) if e E:: L is idempotent then the left isotropy group of e is a connected group H and e E:: H-. Furthermore, if His a line, H- = H U {e} and if H is a plane, H-is a half-plane; (3) if e E:: L is idempotent and dim Ge = 1 then Ge is a line with at most one endpoint in L. If dim Ge = dim eG = 1 then Ge is a group and Ge = eG. If dim Ge = 1 and dim eG = 2 then G is isomorphic to the cartesian product of the reals and the noncommutative group on the plane. (Received September 26, 1963.)

605-12. J. E. MAXFIELD and W. j. HOWE, University of Florida, Gainesville, Florida. Common fixed points of commuting continuous functions on the unit interval. Preliminary report.

Let f and g be continuous functions on (0,1] into itself which commute. Definition. If c E:: (0,1) and .J an interval (a,b) such that for an infinite number of values of n, fn(c) E:: (a,b) where (1) a and b are consecutive fixed points of f or (2) one of a and b is an endpoint of [0 ,1] and the other is the fixed point of f nearest that endpoint, then f is said to circulate with c. Theorem. If f 2 has exactly the same fixed points as f or if there is a fixed point of g with which f does not circulate, then f and g have a common fixed point. The proof of this theorem rests on the following lemma: If fn has a fixed point which is not a fixed point of f then i has a fixed point which is not a fixed point of f. (Received September 27, 1963.)

646 605-13. R. C. BZOCH, Louisiana State University, Baton Rouge 3, Louisiana. On the repre­ sentation of bilinear functionals.

Let Qh[a,b] be the Banach space of quasi-continuous functions on [a,b] to the reals normalized by the real number h; i.e., f E:: Qh [a ,b] iff f is continuous at a and b, f(x+) and f(x-) exist for a< x

Qh[a,b] X Qk[c,d] such that for each f E::Qh[a,b] the functional Tf(g) = T(f,g) on Qk[c,d] is linear,

homogeneous, and bounded, and for each g E::Qk[c,d] the functional Tg(f) = T(f,g) on Qh[a,b] is linear, homogeneous, and bounded. Let R be the rectangle [a,b] X [c,d]. Theorem. There exists a function u(x, t) of bounded variation on R such that T(f,g) = M I IR f(x)g(t)du(x, t), where the integral is the double mean Stieltjes integral defined by the refinement limit. (Received September 27, 1963.)

60 5- 14. J. M. GWYNN, JR., Georgia Institute of Technology, Engineering Experiment Station, Atlanta 13, Georgia. Some relationships between stability and truncation error for a class of nine-point analogues of the one-dimensional heat equation.

In approximating the solution of the system ut(x,t) = uxx(x,t), 0 < x < 1, 0 < t, u(O,t) = u(l,t) 0, 0 < t, u(x,O) f(x), 0 :;; x:;; 1, use is made of the nine-point analogue a(r) •(U .. + = = l,J 1 - U l,J...) + (l- a(r))·(Ui,j- ui,j- 1) = ra(r)·(Ui+1,j+l- 2Ui,j+l + ui-l,j+ 1) + rb(r)·(Ui+ 1,j- 2Ui,j + ui- 1} + r(l- a(r)- b(r))·(Ui+1,j_ 1 - 2Ui,j- 1 + Ui- 1,j_ 1), where r is the mesh ratio k/h2 and a, A, and b are real-valued functions on the positive numbers. Study of the truncation error and the mesh ratios for which the analogue is stable lead to a sequence of interesting relationships between truncation error and stability. As truncation error is reduced, the set of stable mesh ratios diminishes; moreover, explicit and implicit analogues have different sets of stable mesh ratios for a given order of truncation error. (Received September 27, 1963.)

605-15. R. D. ANDERSON, Louisiana State University, Baton Rouge, Louisiana, On topological translations in En.

Let G be one of the following three groups of homeomorphisms of En, n > 1, onto itself: (a) the group of all orientation-preserving diffeomorphisms; (b) the group of all orientation­ preserving piecewise-linear homeomorphisms; (c) the group of all homeomorphisms each of which is the product of a homeomorphism supported on a cell by a homeomorphism supported outside a cell. Theorem I. Each element of G is the product of at most 8 topological translations, i.e., conjugates of a given geometric translation where the conjugating homeomorphisms are of type (a), (b) .5!! (c) as appropriate. Corollary, Each element of g is isotopic to the identity with the levels of the isotopy being elements of (a), (b) or (c) as appropriate, Theorem II. Let h be any element of G without compact support. Each element of G is the product of at most 48 conjugates of h and h - 1 with conjugating homeomorphisms as above, Corollary. The group G has exactly one nontrivial normal subgroup, namely the group of elements of G with compact support. (Received September 30, 1963.)

647 605-16, ]. H. CARRUTH, Louisiana State University, Baton Rouge 3, Louisiana. The kernel of the semigroup of subsets of a semigroup,

Let S be a semigroup and let 2S be the semigroup of nonvoid subsets of Sunder the operation AB = { ab: a E:: A and b E:: B}. Let E be the set of idempotents of S and let H(e) be the maximal group containing an idempotent e. Theorem 1. If S has a nonvoid completely simple kernel K, then 2S has a non void completely simple kernel K * and each element of K * is idempotent. Corollary. If S has a non void completely simple kernel K, then K * = 2SH(e)2S for each e E:: E n K. Theorem 2. For a semigroup with nonvoid completely simple kernel K, the following are equivalent: (I) A E:: K * (the kernel of 2s). (II) A= As A for all s E:: S. (III) A is a semigroup contained in K which has the property that A n H(e) ¥ iJ for e E:: E n K implies A :J H(e) (i.e., A is saturated with respect to H-classes). (IV) A is a set consisting of a union of maximal groups inK with the property that x,y E:: A implies [xS n Sy) n A¥ iJ (i.e., A is symmetric). (Received September 30, 1963.)

605- 17. M. K. FORT, JR., University of Georgia, Athens, Georgia. Level sets of continuous functions. Preliminary report.

Let C be the set of all continuous real-valued functions on [0,1]. Iff E::C, let [af'bf] denote the range off, and let Ef be the set of all points of [0, 1] at which f has either a relative maximum or a relative minimum. Metrize C by the uniform metric, It is shown that there exists a first category subset F of C such that iff E:: C - F, then: (i) r 1(af) and f- 1(bf) are each sets which have exactly one member; (ii) Ef is a countable dense subset of (0, 1), and if x E:: Ef then f- 1f(x) = {pxf UP x where P x is a Cantor set (perfect and nowhere dense), Px t Px, f has a strict relative maximum or minimum at Px, and P x n Ef = iJ; and (iii) if x 1:- Ef and af < f(x) < bf, then f- 1f(x) is a Cantor set. (Received September 30, 1963,)

605-18, R. G. VINSON, Box 71, Huntingdon College, Montgomery 6, Alabama. Area in a noneuclidian geometry. Preliminary report.

Let ,. be a real projective plane less the convex interior of a conic, Then distances can be defined between pairs of points in ,. by means of cross ratios with the ideal points on the line deter­ mined by the given pair, There will be three distinct such distances, depending on whether the line containing the pair intersects the conic in none, one, or two points. Let D(A,B) be taken as the absolute value of the cross ratio distance. Then ,., with the distance function D, is not a metric, nor even a topological, space. However, an area function, m, can be defined having the properties that for any polygons p,q,r: (1) m(p) is defined and is a positive finite number, (2) m(p) = m(q) whenever p and q are congruent, (3) if pis the union of polygons q and r, such that qxno intersect r in a polygon, then m(p) = m (q) + m(r). This is done by showing that all orthohexagons have the same area and defining areas of triangles in terms of this standard area. (Received September 30, 1963.)

648 605-19, B. G, CASLER, Louisiana State University, Baton Rouge, Louisiana, Slicing a contractible 3-manifold with boundary,

Theorem, Suppose M is a contractible 3-manifold with boundary, D and D' are two disjoint

disks in Bd(M), and • > 0. Then there is a finite set of mutually exclusive disks D = D 1, D 2, ... , Dn = D' such that' Bd(Di) is contained in Bd(M), Int(Di) is contained in Int(M), and Di _ 1 + Di+ 1 is contained in an • neighborhood of Di fori= 2,3,,.,, (n - 1), This theorem is proved using a homotopy form D to D' and an application of the Loop Theorem, (Received September 30, 1963,)

605-20, R, D. McWILLIAMS, Florida State University, Tallahassee, Florida, Iterated w*- sequential closure of a Banach space of functions in its second conjugate space,

Let T be a compact Hausdorff space and X the B- space of all continuous real functions on T. Let Kobe the canonical image of X in X**, and for each countable ordinal a ;;:; 1 let Ka be the w*-sequential closure of U {K/1 : 11

space a of bounded Baire functions of class a on T with supremum of absolute value as norm, Now

let ?Jra be the ath multiplicative class of Borel subsets ofT, starting from ?Jr0 = { n~= 1 Fk: {Fk} S .10}, where ~is the class of all sets F such that F = x- 1(H) for some x E:: X and some closed H. Then it is shown (by generalizing an argument in Kuratowski, Topologie, pp. 293- - 1 299) that a= 'I'd where 'Ira is the class of all bounded real functions y on T such that y (H) E:: ~a

for every closed H. If T is also metrizable and uncountable, then each Kais a proper subset of Ka+ 1, whereas K 1 = K 2 if Tis countable, (Received September 30, 1963,)

605-21, TREVOR EVANS, Emory University, Atlanta 22, Georgia, Identities in multiplicative systems, II. Anti-finite and anti-associative laws,

A study is made of identities in loops and of the structure of the lattice of subvarieties of the variety of loops. Although any set of group identities satisfied by a nontrivial group always has commutative and finite models, the corresponding situation does not hold for loops, There are un­ countably many nontrivial varieties of loops containing no nontrivial associative or finite models, Other results include (i) if a set of loop identities has no nontrivial finite models then it has no nontrivial associative models; (ii) a loop identity may be written in the form {xn•p(x)} •c(x,y, ... ) = where p(x) is a loop word which is 1 on any power-associative loop and c(x, y, .. ,) is a loop word which is 1 on any commutative, associative loop; (iii) a loop belongs to an anti-associative variety (i.e, one

satisfying identities incompatible with associativity) if and only if it satisfies an identity of the form x•p(x) = 1; (iv) the lattice of loop varieties is modular and has countably many associative atoms, uncountably many anti -associative atoms, An example is constructed of a finite loop satisfying an identity incompatible with the associative law, (Received September 30, 1963,)

605-22, R, W. HEATH, University of Georgia, Athens, Georgia, A non -pointwise paracompact Moore space with a point-countable base. Preliminary report,

An example is given of a complete Moore space which is not pointwise paracompact but has a point-countable base (no point belongs to more than countably many elements of the base; hence

649 every open covering has a point-countable open refinement). It is noted that the above space is not strongly complete, and it is shown that every pointwise paracompact complete Moore space is strongly complete. Also it is shown that, if every normal Moore space with a point-countable base is pointwise paracompact, then every separable normal Moore space is metrizable. (Note that a

T 3 - space is a pointwise paracompact Moore space if and only if it has a uniform base (defined by Aleksandrov in Russian Math. Surveys 15 (1960), 40).) (Received October 1, 1963.)

605-23. E. E. ENOCHS, University of South Carolina, Columbia, South Carolina. Homotopy groups of compact Abelian groups.

Using the notion of a free compact Abelian topological group on a topological space it can be shown that for any compact Abelian group G the fundamental group of G is isomorphic to Hom(G*,Z) where G* is the Pontrjagin dual of G and Z the additive group of integers. It follows that the funda­ mental group of G is torsion free and is isomorphic to a subgroup of a product of copies of z. It can also be shown that the higher homotopy groups of G are all 0. (Received October 1, 1963.)

605-24. A. K. BOSE, St. Augustine's College, Raleigh, North Carolina. Functions satisfying a weighted average property.

In this paper consider a region R in En together with a locally integrable weight function, w, from R to the positive real numbers. A real-valued function f defined in R will be said to satisfy the weighted average property with respect to w if fw is locally integrable in R, and if for each point x of R, f(x).jB(x,:q wdu = fB(x,r) fWdll for each ball B(x,r) of radius r centered at x and having its closure contained in R. For each weight function, w, let CZ(w) denote the class of functions satisfying the weighted average property with respect tow. If w E:: C(l)(R) and f E:: tl!(w), then f satisfies the

equation w a f + 2Lf= 1 ((cJf/r'lxi)(cJw /<'lxi)) = 0. It is proved that for all weight functions satisfying A.w + >.w = 0, there are non-constant functions belonging to tl(w) and these are precisely the class

of solutions of wM + 2 :Ef= 1 ((cJf/rlxi)(cJw /Clxi)) = 0. If the weight function w be of the form w = aoexp(L:r=1aixi)• where the ai 's are constants, and iff E::tl(w), then every derivative off also belongs to a (w) and these are the only weight functions satisfying Aw + >.w = 0 which have this property. The maximum principle and various convergence theorems are established. (Received October 1, 1963.)

650 The i\ m-ember i\leeting in Pasadena, California "i\ ovember 21-23, 1963

606-l. W. j, LeVEQUE, University of Michigan, Ann Arbor, Michigan, On Weyl's criterion,

Let {xk} be a sequence of real numbers. For a E:: [0, 1), let Nn (a) be the number of k ~ n for which xk- [xk]

606-2, C. B. BELL, San Diego State College, San Diego, California. Topological groups and measure problems in distribution-free statistics,

Let {"2* = {str, incr, cont. fens with F(- oo) = 0, F( + oo) = l}; .b(S) = {g: g str, incr, cont. mappings of S onto s}. Conjecture (A). Jh(x 1, ... ,xn' y 1, .... ym) IldF(xi) IldG(yj) = 0 for all F,G in rl* implies h = 0 wrt each PW) X Pbm). Conjecture (B). P~N) {T(z 1, ... ,zN) ¥ T(g(z 1), ... , g(zN))} = 0 for Fin rl* all gin _g(R 1), Fin rl* implies P~N) {J* if there is appropriate measurability and there exists on ft(R 1) a u-finite nontrivial

measure with X(·g) < < X for all gin _ir(R 1). Using Hausdorff measure with p = l, one finds Theorem 4. (B) is valid for majorized subclasses of rl* if either (a) spheres are measurable wrt Munroe's x;:, or (/3) Munroe's x0is u-finite, (Received October l, 1963.)

606-3. P. A. LOEB, Stanford University, 305 Stanford Avenue, Palo Alto, California. A new

proof of the Tychonoff theorem.

To prove Y = IlvE::I Xv is compact it is only necessary to assume the existence of a well­

ordering ~ on the index set I and (if I is infinite) a choice function F on the collection { C IC closed, C 'f ¢, C c Xv for some v }. Thus the axiom of choice is not needed to prove the com­

pactness of such product spaces as the countable product of unit intervals. Let 6 be a subcollection of the base 1J for the product topology of Y, j a subset of I, and P J the projection of Y onto IIvE::]Xv. a, xa is deter­ mined so that = {o E:: ()- lx E:: P (0)} is admissible on II X . (Received August 22, 1963.) cJ-•a a a. a. v>a. 651 606-4. ]. R. REAY, Western Washington State College, Bellingham, Washington. A note on a theorem of Bonnice-Klee.

Given a set X in an n -dimensional real linear space E, say that p C intdX if and only if

there exists a d-dimensional flat F through p such that p is relatively interior to F (I X. This may be considered a measure of how well p is embedded in the boundary of X. If p C con X then it is clear that p C intmcon X for some maximal value of m, 0 :;; m:;; n. Let Fm be the associated m-dimensional flat. Lemma. Fm is uniquely determined by p, and Fm n con X= con(Fm n X). Bonnice-Klee prove (The generation of convex hulls, Math. Ann. 149 (1963)) that if p C intdcon X, then p C intdcon U for some subset U of X of cardinality at most max (n + I, 2d). From the above

lemma it follows that if p C intdcon X and p ¢ int2dcon X then p C intdcon U for some subset U of X of cardinality at most 2d. (Received August 26, 1963.)

606-5. N. M. WIGLEY, Los Alamos Scientific Laboratory, P. 0. Box 1663, Los Alamos, New Mexico. Asymptotic expansions at a corner of solutions of mixed boundary value problems.

Let u(x,y) be a solution in a domain D of the partial differential equation 1\.U + ku = f. Suppose that part of the boundary of D consists of two analytic arcs which meet at the origin and form there an interior angle .-a > 0. If on each of the two arcs u(x,y) satisfies a linear boundary condition of order zero or one, and if all the data are CN, then u has an asymptotic expansion in powers of z, z, z l/a, z I/a, log z and log z, with an error term of order approximately N. Furthermore this expansion may be formally differentiated N times. The proof is based on an integral representation of u in terms of the Green's function for the upper half plane. (Received August 26, I963.)

606-6. V. L. KLEE, University of Washington and Boeing Scientific Research Laboratories, Seattle 5, Washington. On the number of vertices of a convex polytope.

For n ~ d + I, let !'(d,n) denote the maximum number of vertices of those d-dimensional convex polytopes that have n(d 0 I)-faces, or, equivalently, the maximum number of (d - I)-faces of those

that have n vertices. The cyclic polytopes (Gale) show that l'(d,n) ~ C(n - [(d + I)/2], n - d) + C(n - [(d + 2)/2], n - d), and several authors (Jacobs and Schell, Gale, Fieldhouse, Motzkin) have conjectured that equality holds. This has been proved by Fieldhouse when d ;£ 6 and by Gale when n ;:;; d + 3. Here it is proved when n i:; (d/2)2 - 1, and the upper bound on the number of (d - I)-faces is established not only for d-dimensional convex polytopes but also for an arbitrary Eulerian (d - I)-manifold of Euler characteristic I - (- I)d. The paper also contains a number of other results on the facial structure of convex polytopes. (For a relevant definition, see the author's paper A combinatorial analogue of Poincare's duality theorem, these Notices IO (1963), 353, whose results are the main tools of the present paper.) (Received September II, I963.)

652 606-7. G. H. PIMBLEY, Los Alamos Scientific Laboratory, P. 0. Box 1663, Los Alamos, New Mexico. On sublinear operators in Hilbert space with uniformly symmetrizable derivatives.

The nonlinear eigenvalue problem T(x) = Ax. is studied in real Hilbert space H. The c.c. odd operator T(x) has four derivatives. T' (x) = U(x)P where U(x), P are positive definite, and PT"(x) x is negative definite, all for given x E:: H. The latter is the sublinearity condition. There is always the trivial solution x = 6. The branch points are the eigenvalues A = ~-'p > 0, p = 0, 1,2, •.. , of the linear problem T' (8)h = Ah. The branches of solutions x p(A) starting simple branch points are continued into the large. Suppose there is a linear bounded operator B such that llx r 111T(x) - Bx II-0 as llxll------..oo. B has eigenvalues A= 'Yp < ~-'p• If the sequences jt

If A <; 0 or A > ~'0• there exists only the trivial solution. (Received September 16, 1963.)

606-8. R. M. BLUMENTHAL, and R. K. GETOOR, University of Washington, Seattle 5, Washington. Local times for Markov processes.

Let X be a temporally homogeneous Markov process satisfying Hunt's hypothesis (A) with

state space E. Let x0 be a fixed point in E such that Xo is regular for {x0 }. Let rf>(x) = Ex(e -T) where T is the hitting time of xo, and set y,A = rf> - (A - 1)UAq, for A > 0. Each .yA is uniformly A-excessive and there exists a continuous additive functional A such that y,A(x) = ExJcfe-AtdA(t) for all A > 0. A is the local time of X at x0 and is unique up to a multiplicative constant. Let I be the points of right increase of A,I' the points of increase of A, and Z = j t: X(t) = x0 }, then I c Z c I' almost surely. Iff is the characteristic function of jx0 } then f(Xt)dA(t) = dA(t) almost surely. If r is the functional inverse to A, then(*) Ex(e-Ar(t)) = [Y,\x)/.yA(xo)l exp [- tNA(x0)]. The process r(t) under Pxo essentially has independent increments and its stochastic structure is completely determined by (*). This construction can be generalized to obtain "local times" for sets much more general than singletons, e.g. to compact sets K such that the set of points in K but not regular for K is polar. In this case X[r(t)] is a strong Markov process that "lives" on K, i.e. an "imbedded" Markov process. (Received September 23, 1963.)

606-9. R. F. BROWN, University of California, Los Angeles 24, California. On grouplike manifolds. Preliminary report.

Let M be a compact topological manifold with empty boundary and let G(M) be the topological group (under composition) of all homeomorphisms of M onto itself (compact-open topology). Pick

e E:: M and define q: G(M) -+ M by q(h) = h(e); then (G(M), q, M) is a locally trivial fibre space as in (N,qiN,M) where N is any nontrivial normal subgroup of G(M). When (G(M), q, M) is trivial, M is called grouplike. A compact manifold is grouplike iff there is a continuous multiplication M X M----+ M such that (1) xe = x for all x E:: M; (2) given x, y E:: M there exists z E:: M with xz = y; (3) xy = xz implies y = z. A grouplike manifold is an H- space and for all x there exists a unique x- 1 E:: M with x£ 1 = e. The function x----+ x -l is continuous. Let H(M) be the subgroup of G(M) consisting of homeomorphisms homotopic to the identity; then if M is grouplike, H(M) is open in G(M). If M is differentiable, the concept of a grouplike differentiable manifold can be defined in a natural way. A grouplike differentiable manifold is parallelizable. (Received September 23, 1963.) 653 606-10. W. H. SILLS, University of Oregon, Eugene, Oregon. Arens multiplication and spectral theory.

Let T be a bounded linear operator in a reflexive Banach space X. If the operational calculus for polynomials p can be raised to a bounded homomorphism Po of a Banach algebra f{, it can be shown that Po• in turn, has a unique extension to a bounded homomorphism p of et** into the closure of Po(~) in the weak operator topology in B (X), where et• * is the second conjugate space of ij ·provided with Arens multiplication. The author considers an operator satisfying llp(T) II ~ K Var1(p), where p(O) = 0 and I= [0,1]. Such operators were introduced by D. R, Smart (J. Austral. Math. Soc. 1 (1960), 319-333}. Here, et is replaced by ACo, the Banach algebra of absolutely continuous functions on I, zero at 0, with total variation norm. The algebra ACo* is non-commutative and non-semi­ simple and is investigated for spectral data by explicitly computing the formulas for the Arens multi­ plication. Smart's projections are images of certain idempotents under p and a spectral theorem is derived. (Received September 24, 1963,)

606-11. R. E. EDWARDS, Australian National University and EDWIN HEWITT, University of Washington, Seattle 5, Washington. Pointwise summability of Fourier transforms on LCA groups.

Let G be a LCA group with character group X and let Y be a u-compact open subgroup of X. There is a double sequence (Km,n>:,n= 1 of nonnegative, integrable, positive definite functions on G such that each Km,n has compact support, with the following properties. Iff E: .!! 1(G) and f vanishes on)", then limm _.. 00 {limn-+oo fxKm,n(x)f(;l()x(x)dx} = f(x) a,e. on G; if u is a measure on G singular with respect to Haar measure, then limm-+ 00 jlimn -+oofXKm,n(X).~(X)iX(x)dx} = 0 a.. e. on G. (Received September 26, 1963,)

606-12, R. B, KILLGROVE, Box 423, San Diego State College, San Diego 15, California. Completions of quadrangles in singly-generated planes.

The studies originated by Professors Hall and Lombardo-Radice on free planes and identifica­ tions therein is continued by a study of the possible completions of various quadrangles to the whole plane. In particular, categories are defined in order to quantify the amount of identification present. The theorem of the paper states that in any singly-generated plane there is a quadrangle which com­ pletes to the whole plane in a way which lies in some category smaller than the 2,0,0,0 category, or equivalently there is no vertex of the quadrangle, the first partial plane, which is collinear with three points adjoined in the fifth partial plane. This theorem in terms of Hall's work says there exists a quadrangle for which V = 0 and in terms of Lombardo-Radice's work it says there exists a quadrangle for which the theorem 3 = 0 does not hold at all. Three lemmas are used; each lemma shows that the existence of a quadrangle in a given category implies the existence of a second quadrangle forced into a smaller category. This is accomplished by suitable choice of the second quadrangle and how identifications in this quadrangle affect identifications in the first quadrangle. (Received September 26, 1963,)

654 606- 13_ D. V. V. WEND, University of Utah, Salt Lake City, Utah. Branched regular curve families and finite asymptotic valued of analytic functions.

Let F be a branched regular curve family filling a simpy connected domain and f an analytic function such that the family of level curves of Re(f) is homeomorphic to F. An investigation is made of the restrictions imposed on F when f has at most a finite number of finite asymptotic values. Also, conditions on F are obtained which guarantee that there exists a corresponding analytic function f with at most a finite number of finite asymptotic values. (Received September 26, 1963.)

606-14. W. A.]. LUXEMBURG, California Institute of Technology, Pasadena, California. Some properties of integrals on Riesz spaces.

A positive linear functional defined on a Riesz space L is said to be an integral (normal

integral) whenever unl and inf un = 0 (uT directed downwards and inf uT = 0) implies infnq,(un) = 0 (infT(uT) = 0). In view of the measure problem the relationship between integrals and normal inte­ grals is studied. It is shown among other things that if L is u-Dedekind complete, then every positive integral on L is normal on some order-dense ideal. If, moreover, L has sufficiently many integrals, then every positive linear functional is normal on some order-dense ideal. Conditions are given in order that on a u- Dedekind complete Riesz space every integral is normal. An important tool in the present investigation is a simple characterization for the smallest normal subspace containing a given ideal. Examples are given to show that the results are in a sense best possible. These results will be contained in a paper which is part of a series of notes on Banach function spaces written in cooperation with A. C. Zaanen and which are to appear in the Proceedings of the Academy of Science Amsterdam. (Received September 26, 1963.)

606-15. D. ]. DECKARD and L. K. DURST, Rice University, Houston, Texas. Unique factorization.

Recently Cashweli and Everett (Pacific J. Math. 13 (1963), 45-64) proved that the ring of power series in an arbitrary set of indeterminates over an integral domain R is a unique factorization ring if and only if all power series rings over R in finitely many indeterminates are unique factoriza­ tion rings. In this paper a shorter and simpler proof is given for this theorem and its analogue in terms of semigroups. (Received September 26, 1963.)

606-16. G. B. CRAWFORD and S. C. SAUNDERS, Boeing Scientific Research Laboratories, P .O.Box 3981, Seattle 24, Washington. Nonparametric estimation and consistency.

Given that a distribution function is a member of a subclass of absolutely continuous measures, consider the problem of nonparameter estimation, with the method of maximum likelihood, of the underlying probability measure of a given sample of independent identically distributed random variables. Sufficient conditions on the space of probability measures and its topology are given for the consistency of such an estimate. Generalizations are given to the case where the likelihood func­ tion may take an unbounded maximum on the boundary of the space under consideration. Applications

655 of this result to the problems of estimating a hazard rate or its distribution function given that the hazard rate has certain regular behavior are given. (Received September 27, 1963.)

606-17. G. S. McCARTY, JR., Harvey Mudd College, Claremont, California. Homotopy normality of subgroups.

If A is a subgroup of the topological group X, let A be homotopy-normal in X if there exists a - 1 homotopy "t of v0: (X X A, A X A)----+ (X, A) : (x,a) ----+xax such that v1 (X X A) c A. The linear sub- group of the 2-dimensional affine group is an example which is homotopy-normal yet not normal, Proposition. The orthogonal group On is not homotopy-normal in On+l' n > 1, The proof rests heavily on the author's results described in Abstract 597-138, these Notices 10 (1963), 98. Partial results are also given for the unitary and symplectic cases. (Received September 30, 1963.)

606-18. M. D. MARCUS and W. R. GORDON, University of California, Santa Barbara, California. Inequalities for subpermanents.

Let A be ann-square complex matrix. Let sr denote the sum of the squares of the absolute values of all r-square subpermanents of A, and let Lr denote the sum of the n!/r!(n - r)! = N largest homogeneous products of degree r of the eigenvalues of AAT. Then sr::;; Lr. If r = 1 equality holds. If r > 1 and A has no zero row (or column) then equality holds if and only if A = oR where o > 0 and R = P diag (z l'"''zn) where P is a permutation matrix and z l'"''zn are complex numbers of modulus 1. If A is doubly- stochastic then sr ::;; 1 + (N - 1)>.2 where A2 is the second largest eigenvalue of AAT. Equality holds for r > 1 if and only if A is a permutation matrix. Thus if A is doubly-stochastic, then sr ~ N with equality for r > 1 if and only if A is a permutation matrix. If A is the incidence matrix of a (v,k, A)-configuration, then sr ;$ k 2r + (Cv,r - 1)k2(r- 1)(k - A) with equality holding for r = 1 and strict inequality holding for r > 1. (Received September 30, 1963.)

606- 19, HENR YK MINC, University of California, Santa Barbara, California. Permanents of (0, 1)-circulants.

Let P denote then-square permutation matrix with 1's in the super-diagonal and per (Q(n,r)) denote the permanent of the (0,1)-circulant Q(n,r) =I+ P + ... + Pr- 1• Then per (Q(n,r)) ;:;_ Lj::-[cja.j r-1 r-2 + cr where a. 1,. .. ,a.r_ 1 are the roots of x - x - ... - x - 1 = 0 (a. 1 > 1; Ia. jl < 1, j = 2, ... ,r - 1) and c 1, ... ,cr depend on r only, It is shown that for r = 2,3,4 and suitable fixed c 1, ... ,cr the equality holds for all n and that it cannot hold for all n if r ;:::; 5. In fact even the formula per (Q(n,r)) ~ ca.n is false for r ~ 5, (Received September 3, 1963.)

606-20. L. W. GREEN, Institute of Technology, University of Minnesota, Minneapolis 14, Minnesota. An elementary proof of Mautner's lemma.

In recent work on ergodic flows on homogeneous spaces done in collaboration with L. Auslander and F. Hahn, an important tool for reducing the general case to a nilflow is what we have called

"Mautner's lemma:" Let U be a unitary representation of the affine group of the line, x~ax +b.

656 Then every eigenvector of the one parameter group U(a,O) is fixed under U(l,b) for all b. By using an idea of Segal and von Neumann (Ann. of Math. 52 (1950), 509-517), a proof of this result is given without using any deep group representation theory. The three-dimensional solvable group for which a similar result is needed can also be handled by this method. However, the detailed nature of the spectrum of U(a) has not yet been obtained by this method. (Received September 30, 1963.)

606-21. M.P. DRAZIN, Purdue University, Lafayette, Indiana. A class of solvable differential

Consider two particles, with position vectors x(t), y(t), moving under a field of force G which depends on both space and time, the initial conditions being such that the paths of the particles intersect for some t. Let the motion of the first particle be modified by a perturbing force F(t) = f(t)d(t), where f is an (unknown) scalar function vanishing for all sufficiently large t and d is an arbitrary unit vector function; and consider how best to choose d (serially, i.e., depending only on the past history of f) so as to minimize the distance of closest approach of the particles. This is a two­ person game in which the players' (one being "Nature") strategies are their choices of the functions f,d respectively, with pay-off W(f,d) = mintlxF+G(t)- y 0 (t)l. Being a 'differential game' in Pontryagin's sense, it could in principle be studied by his techniques, but would be very hard to solve by such an approach, even in simple special cases. However, the problem can be attakced directly, and the somewhat surprising conclusion is that in fact the game is always solvable with value zero; i.e., if motion along a collision path is perturbed by (eventually vanishing) forces of arbitrary magnitude, then, by controlling only the direction of these forces, the particle can always (subject to sufficient differentiability and the nonvanishing of certain Jacobians) be maintained on a collision path. (Received September 30, 1963.)

606-22. D. G. CANTOR, University of Washington, Seattle 5, Washington. Power series with restricted coefficients.

A series of theorems concerning power series whose coefficients are polynomials evaluated on certain sequences of integers are proved. For example, let 8 > 1 and A f- 0. Let f be a polynomial with complex coefficients of degree d ~ 1. Let {an} be a sequence of integers satisfying ian - AOn I is bounded. Put g(z) = L:;~ 0 f(an)zn. If g(z) is meromorphic in a region which properly includes the disc lz I < 181 1-d, then g(z) is rational. These theorems generalize results of L. J. Mordell and M. Newman. (Received September 30, 1963.)

606-23. D. W. ROBINSON, Brigham Young University, Provo, Utah. On a relation between the period and the restricted period of a linear recurrent sequence. Preliminary report.

Let u0, ul' ... 'un•··· be the sequence of integers that satisfy the linear recurrence with characteristic polynomial f(x) = xr- a 1xr-l - .•. - ar• where u0 = ••• = ur-Z = 0, ur-l = 1, and ar f- 0. Let m be a positive integer that is relatively prime to ar• Let 5(m) be the period and a(m) the restricted period of the sequence reduced modulo m, and let ,B(m) and 'Y(m) be the respective exponents r-1 of ua(m)+r-l and (- l) ar modulo m. Under the assumption that the roots of f(x) are distinct,

657 Professor Morgan Ward (unpublished notes, 1962) obtained the relation 5(m) = a.(m){j(m) = (r, tl(m)) [a.(m), 'Y(m)]. Without requiring this assumption on the roots of f(x), this relation is extended to certain linear recurrent sequences in a commutative ring with an identity, Some special cases and corollaries are also considered, In particular, it is shown that for the sequence above, if m = p is a prime, then o(pe) = o(p) implies a.(pe) = a.(p), which generalizes the results of S. E. Mamangakis (Amer, Math. Monthly 68 (1961), 649-649) for the ordinary Fibonacci sequence, (Received September 30, 1963,)

606-24, ROBERT OSSERMAN, Stanford University, Stanford, California, Minimal surfaces in E~

In a recent paper Chern showed that the generalized Gauss map: S -Qn-2 c pn-1 (where Sis a 2-dimensional surface in En, pn- 1 is complex projective space, and Qn- 2 is a hyperquadric which is complex-analytically equivalent to the Grassmannian of oriented 2-planes in En) is an antiholomor­ phic map if S is a minimal surface, and he derived properties of this map in the case that S is complete, In the present paper a more explicit description is given of the generalized Gauss map and

its relation to the normals to S, and the following results are obtained, Theorem, If the generalized Gauss map of S omits an •-neighborhood of hyperplanes in pn- 1, and if p is a point of S whose distance to the boundary is d, then the Gauss curvature K of Sat p satisfies IKI :$ 4(n- l)/d2<2• Corollary 1, If S is complete and not a plane, then the image of S under the generalized Gauss map intersects a dense set of hyperplanes in Pn -l, Corollary 2, If all normals to S make an angle of at least a. with some fixed dire~tion, then the above inequality holds with • = (1/2) sin2 a.. Corollary 3, The normals to a complete minimal surface S in En, not a plane, are everywhere dense. (Received September 30, 1963,)

606-25, TAKAYUKI TAMURA, University of California, Davis, California, Certain embedding problems of semigroups.

The main theorem: A semigroup S can be embedded in a semigroup T with the properties (1) S is an ideal ofT, and (2) every left translation of S is induced by some inner left translation ofT and every right translation of S is induced by some inner right translation ofT if and only if (3) each left translation of S is linked with some right translation of S and each right translation of S is linked with some left translation of S, and (4) every left translation of S commutes with every right translation of S, This is a solution to the open problem which Clifford and Preston proposed on page 12 in their book, The algebraic theory of semigroups, Math. Surveys, Amer. Math. Soc,, Providence, Rhode Island, 1961, (Received September 30, 1963,)

606-26, RICHARD BLOCK, California Institute of Technology, Pasadena, California, Hadamard matrices and constant distance codes.

A new kind of configuration is considered which gives a type of decomposition of certain

Hadamard matrices. Let M be a w X f matrix of 1 1 s and -1 1 s such that the inner product of each pair of rows is f- 2d (d > 0), i,e,, the set C of rows of M forms a w-word binary constant-Hamming­ distance code (CDC) of length f and distance d. Theorem, If C is closed under a nilpotent transitive

658 group G of permutations of the columns of M, then G operates transitively on C, with the possible exception of a trivial (i.e., all 1 or all - 1) word. In particular, CDC's which are closed under the cyclic shift are characterized as iterated difference set codes. Call C (or M) homogeneous if C = C1 U ••• U Ct where there are ri• ki such that each row in Ci has exactly ri1's and each column in the submatrix of M formed by Ci has exactly ki l's, i.e., each Ci gives a BIB design (e.g., with G as above, but not assumed nilpotent, the Ci are the orbits). If moreover Ci has more than one word for at least two i, call C (or M) decomposable. Theorem. If C is decomposable then t = 2, C 1 n c2 = !l. w is even, and there is an integer a such that f = a(w - 1), 2d = aw; if C is of Hadamard type, i.e., a= 1, then there are integers c,e with 4c I e 2 - 1 such that w = 4c2 - e 2 + 1. Decomposable CDC of Hadamard type are shown to exist for w = 4c2 with c = 2,3,4,5,6,7. (Received Septemer ber 30, 1963.)

606-27. H. G. EGGLESTON, University of Washington, Seattle, Washington. A property of Hausdorff measure.

The function h(x) generating the Hausdorff measure m(X,h) of the subset of Euclidean space Ep where m(x,h) = lim 0 --+o(infUXn::JX, d(Xn)< o:E: 1 h(d(Xn)) and d(Xn) is the diameter of Xn, is equivalent to g(x) iff m(X,h) = m(X,g) for all sets X. Then (i) if p = 1 to any h(x) corresponds an equivalent continuous g(x), (ii) if p > 1 (i) is false, (iii) if p > 1 to any h(x) corresponds a continuous g(x) such that m(X,h) = 0 ~ m(X,g) = 0, 0 < m(X,h) ~ 0 < m(X,g) < oo, m(X,h) = oo ~ m(X,g) = oo. (Received September 30, 1963.)

606-28. G. L. KRABBE, Purdue University, Lafayette, Indiana. Weakly-continuous representa­ tions of the multiplicative algebra (BV).

Let f belong to the space 5 of all left-continuous regulated functions on some finite interval ~.b] with values in a locally-convex algebra a (the term "regulated" implies discontinuities of the first kind only); if g belongs to the space (BV) of distribution-functions, the Stieltjes mean integral fg·df defines an element of a. The bilinear mapping j(g,f)-fg·dff sets (BV) in duality with J, and the weak (BV, .Y) topology makes (BV) into a locally convex Hausdorff space G. When f E: .P, the operator uf = {g --+ Jg·df} belongs to the space ~G,tl) of all continuous linear mappings of G into tl; if f is in the class F0 of all f E: .f such that f(a) = o and f(a)f(B) = f(max {a, Bf> whenever a, BE: [a,b], then uf is a "weakly-continuous representation" (i.e., u f is a homomorphism of the

algebra (BV)). In fact, the mapping {f-uff identifies the class F0 with the class of all weakly­ continuous representatj.ons of (BV). If U E:X(G,tl), there is a unique f E: .7 such that Ug = Jg•df for any g E: G. All members of ~(G, Cl) are continuous operators on the Banach space (BV). In the case where A is the complex plane, .;r'(G, Cl) c C** (here C = C [a,b]); however, £(G,d) contains the class of functionals constructed by C. R. Adams - A. P. Morse (Trans. Amer. Math. Soc. 48 (1940), 82- 100). (Received September 20, 1963.)

606-29. W. G. SCHAAL, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. On the expression of a number as the sum of two squares in totally real algebraic number fields.

Let K be a totally real algebraic number field of degree n and with discriminant d. Let 1!1 be an

ideal of K. The number of solutions of the equation ~ = 1'2 + v2 ( ~ E: &) in numbers Jl, v E: is denoted

659 by f(t~). For x 1, ••• ,xn being positive real numbers the theorem is proved: Loo, x 1, ••• xn-oo' then

R(x 1, •.• ,xJ = O((x1 ••• xn)(n/(n+l)+ll) holds. An identity which was given by C. L. Siegel (Trans. Amer. Math. Soc. 39 (1936), 219-224) for real quadratic number fields is generalized to totally real algebraic number fields for the proof of the theorem. This identity will be applied to the problem in a similar way as it was done by the author in a former paper (Math. Ann. 145 (1962), 273-284). (Received October 1, 1963.)

606-30. S. R. CAYlOR, State University of New York, Buffalo 14, New York. Equivalence classes of sets of functions over a finite field.

Let GF(q) denote a finite field and suppose [g]:Pi = gi(a.1, •.• ,a.r) (i = 1, ••• ,r) is a set of r functions in r variables over GF (q). By a transformation one means a set of functions of the above form

possessing an inverse. One says fg] and ~] are right- equivalent if there exists a transformation such that [g].P = [h]. This equation is in fact an equivalence relation, separating the sets of r functions

in GF [q, x 1.... ,x r] into right- equivalence classes. The following equations are also equivalence relations, where 1/>, 1/>1, and 1/>2 are transformations: .P[g] = [h), left-equivalence; q,1fg] q,2 = [h), weak­ equivalence; 1/>-l j$)1/> = h, similarity; and .P1[g] [g] <1>2 = [h), strong-equivalence. The chief object of this paper is to study these five different families of equivalence classes. For each family one seeks the number of equivalence classes, the number of sets of functions in a class, and the number of auto­ morphisms of a set of functions, where an automorphism of [g), say with respect to left-equivalence, is a transformation such that [g)= [g]. (Received October 1, 1963.)

606-31.H. E. SALZER, General Dynamics/Astronautics, P. 0. Box 166, San Diego 92112, California. Divided differences for functions of two variables for irregularly spaced arguments.

Divided differences for f(x,y) for completely irregular spacing of pointe (xi•Yi) are developed here by a natural generalization of Newton's scheme. The generalized nth divided difference is defined by (1) [0 l ••. n] = L:f=ol\ f(xi,yi) where (2) L~=ol\ :xiiy~ = 0, and 1 for the last or (n + l)th equation, for every (j,k) where j + k = 0,1,2, ... in the usual ascending order. The gen. div. diff. [Ol ••. n] is symmetric in (xi•Yi)• unchanged under translation, 0 for f(x,y) an ascending binary poly­ nomial as far as n terms, degree-lowering with respect to (X,Y) when f(x,y) is any polynomial P(X + x, Y + y), and satisfies the 3-term recurrence relation (3) [Ol ••• n] = A([1 ••• n] - [O ••• n - 1]}, where (4) >.= l1 ••• ni·IOI .•• n- 1I/I01 •.• nl·l1 ••• n- 11, the l••. i ••• l denoting determinants in xfylC. The generalization of Newton's div. diff. formula is (5) f(x,y) = f(x0,y0)- (la.OI/IOI}[01] + (la.011/IOli}[012] - (I0.012I/I012i}[Ol23] + ·•• + (- 1)n(la.Ol. •. n- li/IOI ••• n- li)[Ol ••• n] + (- 1f+1(ia.01 .•. ni/I01 •.• ni}[a.Ol. .. n), the a. denoting xiyk terms. The right member of (5) without the last or remainder term, = f(xi'yi) for (x,y) = (xi•Yi)• i = 0,1,···· n. Gen. div. diffs. having most of the preceding properties may be defined for any number of dimensions, and other than polynomial approximation, by replacing xi~ in (2) by other functions. (Received June 27, 1963.)

660 The November :Meeting in Madison, Wisconsin November 30, 1963

607-1, M. P. SCHUTZENBERGER, 74 rue de Vepeau, Antony (Seine), France, and SEYMOUR SHERMAN, Wayne State University, Detroit 2, Michigan, On a formal product over the conjugate classes of a free group.

The following non-commutative version of the Witt identity is established, Let X with elements x 1 < x 2 < ... be a totally ordered set generating a free monoid F. Let F+ ~ jf E:: F: f i 1f, Assign lexicographic order to the words of F+. Let f E:: F+ be a standard word iff~ f'f" and f', f" E:: F- imply f < f' '. Let H be the collection of standard words and let the formal infinite product IIj 1 - h: h E:: H; < } be taken with the factors arranged according to increasing h. Then TI{r - h: hE:: H; < } ~ 1 - L{x:x E:: X}. If one considers the obvious homomorphism from F to the free abelian monoid generated by X then one gets the classical Witt identity [M, Hall, Theory of groups, (1959), 170]. A non-commutative analogue of the Witt identity [S. Sherman, Combinatorial aspects of the Ising model for ferromagnetism, II, A analogue to the Witt identity, Bull, Amer. Math, Soc, 68 (1962) 225-229] is established where a free group plays the r6le of the free monoid and where the right-hand side enumerates the 1-cycles over Z 2 in a certain class of planar diagrams, (Received June 28, 1963,)

607-2. THOMAS PRICE, Van Vleck Hall, University of Wisconsin, Madison 6, Wisconsin, Cellular decompositions of E 3 .

Let G be a cellular (i.e,, pointlike) upper semi-continuous decomposition of E 3 , be the decomposition map of E 3 onto G, and H the collection of non-degenerate elements of G. It is shown that G is homeomorphic to E 3 if the elements of H have the following property: for each h E:: H, h is cellular with 3-cells whose boundaries miss U.h' E::Hh'. Furthermore, it is shown that if H is a countable set, then this property is also necessary. It is shown that if U is an open 3- cell in G then

cf) 1(U) is also an open 3-cell. This provides the following necessary condition for G to be a 3-mani­ fold, For each element g of G, g must have arbitrarily small open 3-cell neighborhoods which are the sum of elements of the decomposition, Finally it is shown that if G is a 3- manifold each of whose compact subsets can be embedded in E3 , then G is homeomorphic to E3. The main technique used is that of cutting handles off cells. In order to accomplish this, it is first necessary to relate the ability to shrink loops in E 3 and G, while missing certain sets. Once this is done the proofs are quite easy, (Received August 5, 1963,)

607-3, M. V. SUBBARAO, University of Alberta, Edmonton, Alberta, Canada, Component congruences for a class of divisors.

Let "r(N,s) denote the sum of the rth powers of those positive divisors d of a positive integer

N for which either d or d', where dd' ~ N, is an sth power integer, where rands are non-negative, A component of "r(N,s) is defined as dr + d'r or dr according as 1 ~ d < VN or d = VN provided

661 either d or d' in the first case, or din the second, is an sth power, Let 11r(N,s) ~a (mod K) mean that every component of 11r(N,s) is = a (mod K). Let r, K, L, a be fixed positive integers where K > 2, (L, K) = 1, and a is a non-negative integer. Theorem I. A necessary and sufficient condition that 11r(nK + L, s) ~a (mod K) for all integral values of n > 0 is (1) L is not a (2s)th power residue of K; (2) I + Lr ::a (mod K); (3) (wrs - I)(wrs + I - a) = 0 (mod K) for all w such that (w,k) = I.

Theorem II, For all positive n, 11I (mK + L, 2) ~ a (mod K) holds for suitable L,a if and only if K is one of 3,4, 5,6,8,10, 12, 15, 16,20,24,30,40,48,60,80, 120,240. The corresponding values of L and a are also determined, The complete solution of 111 (nK + L, 1) ~a (mod K) is already given by the author and V. C. Harris (Pacific J. of Math. 12 (1962), 925-928). (Received August 13, 1963,)

607-4. I. N. HERSTEIN and L. W. SMALL, University of Chicago, Chicago 37, Illinois. Nil rings satfsfying certain chain conditions.

The following two theorems are proved: Theorem 1: .!f. R is a ring which satisfies the ascending chain condition on right and left annihilators, then every nil subring of R is nilpotent. Theorem 2: J!.R is a nil ring which satisfies the ascending chain condition on left annihilators, and any direct sum of left ideals has only a finite number of terms, then R is nilpotent. These two theorems contain or subsume many of the previously known results in this direction. The chain conditions of Theorem 2 are those which arise in the recent work of Goldie, It is also conjectured that Theorem 1 is true with the chain condition holding only on one side. (Received August 22, 1963.)

607-5. J, L. ROVNYAK, Purdue University, Lafayette, Indiana. An elementary proof of Beurling's theorem. Preliminary report.

If e is a Hilbert space, let C (z) be the Hilbert space of formal power series f(z) = Lanzn with coeffic~ents an in (!_such that llfll2 = Llan12 < oo. A famous result of Beurling (Acta Math. 8I (1949), 239-255) characterizes the closed invariant subspaces of multiplication by z in C (z) when dim C = 1. Characterizations for the general vector case were given by Halmos (J. Reine Angew. Math. 208 (1961), 102-112) and Rovnyak (Proc. Amer, Math. Soc. 13 (1962), 360-365). Both of these proofs in­ volve analysis, whereas a simple new proof uses only formal and geometric arguments. (Received August 26, 1963.)

607-6. HIDEGORO NAKANO, and B. J. EISENSTADT, Wayne State University, Detroit 2, Michigan. Quasi-bounded linear lattices.

A spectral theory for sequentially continuous linear lattices is given in Modulared semi­ ordered linear spaces by H. Nakano. To each such lattice S there is associated a topological space E and for certain pairs e in S and p in E there is defined a map from S to the extended reals. The point p is bounded if there exists an e such that the image of S under this map is contained in the reals. In this paper, the lattice S is defined to be quasi-bounded if all the points of E are bounded. Three types of quasi-bounded lattices are defined, depending on the topological properties of certain subsets of E. Under the additional hypothesis of universal continuity, it is shown that such a lattice is the orthogonal direct sum of lattices of these three types; a characterization of each of these types is given; and the relationship to function lattices detailed, (Received July 8, 1963.) 662 607-7, R. B, KILLGROVE, Box 423, San Diego State College, San Diego 15, California, Subplane counts in projective planes,

A program has been written for the IBM 7090 to find every quadrangle of a projective plane of order nine and classify this quadrangle as one which completes to a subplane of order 2, or a subplane of order 3, or to the entire plane, A second program written for SW AC accepts the plane in the form of a digraph complete set of 8 Latin squares and converts this to the canonical incidence matrix and also computes the transpose of this incidence matrix, The 7090 program uses these matrices as input; the output consists of the three counts. The two interesting cases are: the Veblen-Wedderburn plane (1080 subplanes of order 3, 51840 subplanes of order 2); the Carmichael-Hughes plane (1080 subplanes of order 3, 33696 subplanes of order 2), A theorem can be proved stating that any Veblen- Wedderburn plane of order pa will have at least p2a(p2a - l}(p 2a - pa)/p2(l - 1}(p2 - p) subplanes of order p, The inequality is an equality in this example; there are planes in which there is a strict inequality involved. (Received September 26, 1963.}

607-8, P, C. HAMMER, Numerical Analysis Department, 5534 Sterling Hall, University of Wisconsin, Madison, Wisconsin, Extended topology: foundations of approximation theory,

Approximation theory is not an adjunct of normed linear space theory, being too general in principle to be imbedded even in a framework of general topology. It is shown that metric and pseudometric spaces are also inadequate to describe goodness-of-fit criteria, Dropping one axiom and extending another from topology is shown to improve the applicability, but even the resulting extended topology does not cover simple approximation descriptions, Examples which illustrate these points as well as examples (some from the theses of Messrs, D. G, Moursund, Wm, J. Kam­ merer, and William F. Lynch) which demonstrate the fruitfulness of this general approach to approximation theory are given, (Received September 27, 1963,)

607-9, E. F. STUEBEN, Illinois Institute of Technology, Chicago 16, Illinois. Ideals in two­ place tri-operational algebras.

Let A be a two-place tri-operational algebra (K. Menger, Rep. Math, Coll. Notre Dame 7 (1946}, 46- 60) whose elements are polynomials in two indeterminates, x and y, A T-ideal I of A is a ring ideal with the additional property that F(G(x,y}, H(x,y)) E:: I for every F(x,y) E I and G(x,y), H(x,y) E:: A, If the set K of constants of A forms an infinite field, then A ~ K [x,y], the set of all poly­

nomials in x andy. If K is finite, then there exists aT-ideal I of K[x,y] such that A~ K[x,y]/1, Every T-ideal I of K [x,y] contains a Milgram polynomial p(x) (A, Milgram, Rep, Math, Coll. Notre Dame 7 (1946}, 65- 67} such that (p(x), p(y)) c I c (p(x), p(y}, k(x) ·k(y}), where k(z) is the monic poly­ nomial in z of least degree such that k(c) = 0 for every c in K. (Received September 2 7, 1963,}

607-10, R. C. REILLY, Illinois Institute of Technology, Chicago 16, Illinois, On rings with composition.

Adler (Dumeke Math. J, 29 (1962}, 607-623} shows that any commutative ring R can be made a

663 tri-operational algebra (Menger, Rep. Math. Coli. Notre Dame 5-6 (1945), 3-10) by introducing a composition with respect to which the ring operations are distributive. By the foundation K of R, Adler means a subring defined by K = [x E: R; xO = x] where juxtaposition indicates composition. Under what conditions can the foundation be prescribed? Theorem: A necessary and sufficient condition for a given subring S of R to be the foundation of a composition in R is the existence of an ideal N in R such that R is the direct sum of N and S. Corollary 1. If R is the ring of rational integers, then any composition in R has either R or { 0} as its foundation. Corollary 2. Let P be a f field and let R = P [z] be the ring of polynomials over P in the indeterminate z. The only possible foundations for a composition in RareR, P, and {o }. (Received September 27, 1963.)

607-11. H. I. WHITLOCK, Illinois Institute of Technology, Chicago 16, Illinois. Abstract characterization of an algebra of multi-place functions. I.

Ann-place Menger algebra <71(.<1>) means a set ~ with a super-associative (n + 1)-ary operation

!/>provided there are I 1, ••• ,In E: 11( such that (Ii,G 1, ••• ,Gn) = Gi (i = 1, ••• ,n) and (F ,I1 •.•• ,In) = F for all F,G1, ••• ,Gn E:??f. Superassociativity of !/>means that <1>(4>(F,G 1 •••• ,Gn>• Hp •••• Hn) = 4>(F,4>(G 1,Hp ... ,Hn), ••• ,.P(Gn,H1, ... ,Hn)) (cf. Menger, Algebra of analysis, Notre Dame, 1944, p. 45). For any setS, a system ?J( of n-place functions over S (mapping sn into S), closed under composition, is an(?!(,!/>) if (F,G 1, ••• ,Gn) is the function F(Gp ••• ,Gn) assuming for x 1, .•• ,xn the value F(G1(x 1, .•• ,xn) •···•'tt (x 1, ••. ,xn)) and if ?r( includes then-place selector functions. Theorem I. The mapping of F E: ('h(,ti>) on the n-place function F* over ?Je such that F*(G1, •.• ,Gn) = (F ,G1, ••• ,Gn) for all Gl'···•Gn in 71? is an isomorphism. An element F- 1 is called the inverse of F if F(F -I, ... ,F- 1) -1 -1 . -1 -1 = F (F, ••. ,F) = I 1• For any F such that F exists the mappmg G ...... ,F(G, ••• ,G)(F (Il'···•I1) •••• ,F • (In•···•In)) is an automorphism which is called an inner automorphism. Theorem 2: By the mapping of Theorem 1, ann-place Menger Algebra (-:?!,4>) may be imbedded in a system of n-place functions over:?( so that every automorphism of (~,4>) corresponds isomorphically to a restriction of some inner wutomorphism of the function system. (Received September 27, 1963.)

607- 12. V. J. KAFKA, Illinois Institute of Technology, Chicago 16, Illinois. Axiomatics for partially ordered systems of multiplace functions.

Let 7J{ be a set with (1) a superassociative n-place substitution including elements r1, ••• ,In such that F (Il'···•In) = F for each F; (2) a partial order relation S such that Hk S Ik implies Hk (F l'... ,F n) S F k and F(H1, ••• ,Hn) S F; (3) n operators Jf1, .•• , .)pn such that (a) /fkF S Ik for k = l •••• ,n and every F; (b) if HkS Ik then Jt'kHk S Hk; (c) i'ff(F(G 1•••• ,GJ) = J'fi(( i'ijF)(G1, ••• ,Gn)); (d)_ ifF S G then F = G( ";f1F, ••• , .)pnF); (e) Ik(F1, ••• ,Fn) = Fk if and only if /f1Fk = /f1F ifor i = 1, ••• ,n. These assumptions (based on postulates by Menger, Schweizer and Sklar) are satisfied, for any set S, by the system of all mappings of subsets of sn into S provided F S G means that F is a restriction of G, and /fkF is the restriction of Ik having the same domain as F. For every G E: 7J? set a.(G) = {(F 1, ... ,Fn)• G(F 1, ... ,Fn)i( l'fiG)(Flr··•Fn) = F 1, i = l, ••• ,n}. The mapping of G to then-place function a.(G), which generalizes the Schweizer-Sklar representation of systems with 1-place substitu­ tion by !-place functions (Math. Ann. 143 (1961), 440-44 7), is an isomorphism with respect to substi­ tution and partial order. Moreover, Ji'ka.(G) = a.(l/\G). (Received September 27, 1963.)

664 607-13, EVELYN FRANK, P. 0, Box 361, Evanston, Illinois, Continued fraction expansions for nth roots and other functions,

New continued fraction expansions are obtained for ny!X"and for other functions, (Received September 30, 1963.)

607-14. DUANE SATHER, University of Minnesota, Institute of Technology, Minneapolis 14, Minnesota. Maximum properties of Cauchy's problem in three-dimensional space-time.

Let Lu = a2 u/at2 - au where au = g- 112L:Lj=la(g112gijau;a:x;i) I axi and g = [det(gij>r 1. The matrix (gij) is uniformly positive definite and its elements are independent of t. Let K be the

Gaussian curvature of the Riemannian metric defined by the inverse of (gij) and I'\7K 12 = 2 . . . . Li,j=lg11 (aK;ax1)(aK;ax1). Theorem 1. Let k,£,p and q be non-negative constants such that -£2 $. K S, k2 , IVK I ;:;; p and .:lK ~- q, Suppose u(O,x) = 0, a u(O,x)/at ~ O(x ~ (x1,x2)) and Lu :a 0. Then t?b.ere is a positive constant T 0 depending only on k,R,p and q such that u ~ 0 when 0 :;> t -:;;; T0 :;;;; 11"/2k. Corollary. Let K satisfy the hypotheses of Theorem 1, Let B ~ 0 be a constant such that Lu + a[U(O,x) + tau(O,x)/ atJ:;;;;B, Then u:;;; u(O,x) + tau(O,x)/at + Bt2/2 when 0 ~ t~ T0 • Theorem 2, Suppose -j 2 :5 K :$ k2 and 3a2K - 4a IVK I - ¥f + aK ;;:; - 5,} for all a -;;; k cot kT where 0 ~ T ;i T 0• The constants To and 5 f; 0 are such that 0 < To < ,. k- 1 and k -l sin kr - 5 cf- 2 sin h2£r ;;:; 0 when 0 ~ r ~ T 0 ; the constant C depends only on£ and k, Suppose u(O,x) = au(O,x)/dt = 0 and

Lu :> 0 when 0 ~ t ~ T. Then u ~ 0 when 0 :li t l!i T. This theorem contains a result of Weinberger (Proc. Sympos. Pure Math. Vol. 4, Amer, Math. Soc,, Providence, R, I,, Vol, 4(1961), 91-99) when f = li = 0 and To = 11"k -l. (Received September 30, 1963,)

607-15, D. P. GlESY, University of Wisconsin, Madison 6, Wisconsin, A convexity condition on Banach spaces invariant under conjugation, Preliminary report,

For each integer k ;;:; 2 and real number • > 0, a Banach space Xis calles k, •-~ if for each choice of elements xi' 1 ~ i :$. k, from the unit ball of~ there exists a sequence ei,

1 ~ i ~ k, of+ l's and- l's such that lle 1x 1 + ... + ekxkll < k(l- •). J: is B-convex if Xis k,•-convex for some k f; 2, • > 0, James' uniformly nonsquare spaces (these Notices 10 (1963), 483) are spaces which are 2,5-convex for some li > 0, B-convexity is known to be invariant under equivalent re­ norming (Beck, Proc. of Ergodic Theory Symposium, Tulane U. forthcoming). All finite dimensional

spaces and all uniformly convex spaces are B-convex. f1 and £00 are not, Theorems, :;risk, •-convex if and only if ;(• * is. :;(is B-convex if and only if J"• is. The former theorem depends on the fact that x'Jf is dense in :;(• * in the;(* topology of J(• *, where x is the natural imbedding of:;( in ;(• *. The latter result depends on the former result and on a lemma which states in effect that J( is not B-convex exactly in case for each integer k there are arbitrarily good approximations of i~ in X For each integer k this gives arbitrarily good approximatons of£~ in :;(• to show :;!• is not B-convex. (Received September 30, 1963.)

665 607-16. R. H. BING, University of Wisconsin, Madison, Wisconsin. Stable homeomorphisms ~ E5 can be approximated by piecewise linear ones.

A stable homeomorphism of Ff onto itself is one which is the composition of homeomorphisms of En onto itself each of which leaves some open set fixed. Connell has shown (Stable homeomorphisms can be approximated by piecewise linear ones, Bull. Amer. Math. Soc. 69 (January, 1963), 87-90) that if T is an arbitrary piecewise linear structure on En (n ~ 7), then any stable homeomorphism of En onto itself can be approximated by a piecewise linear (wrt T) homeomorphism of If' onto itself. Connell's result is extended by showing that the theorem is also true for n = 5,6. Corresponding results hold for Sn. The extension is accomplished by strengthening his pushing out engulfing lemma. (Received September 30, 1963.)

607-17. N. HOSAY, University of Wisconsin, Madison, Wisconsin. The sum of a cube is defined to be the closure of the interior of a 2-sphere in euclidean 3-space.

Theorem. The sum of a crumpled cube and a real cube formed by identifying the points of the boundary of each is topologically a 3-sphere. (Received October 1, 1963.)

607-18. FRANK KNIGHT and STEVEN OREY, University of Minnesota, Ford Hall 400, Minneapolis, Minnesota, 55455. Construction of a Markov process from hitting probabilities.

K is a compact separable metric space with metric P, ()tis some not too small sub-class of the analytic sets, t> is a point of K. Given a positive function g on K and for every A E:: t>i- and x E:: K a probability measure HA (x,· ) on the Borel sets of the closure of A, the problem is to show that under suitable conditions there exists a Markov process with stationary transition probabilities having K for state space such that (i) the H A (x,-) are the hitting probabilities of the process, (ii) g(x) = expected lifetime of the process started at x = expected first passage time to { t>} of the process started at x. Let E(x, •) = {y E:: K jp(x,y) ~ • }. Sufficient conditions: (0) HA(x,{x}> = 1, x E:: K, A E:: /Jt; (I) for • > 0, limx__,~HE(x,•)U {t>}(x, {t>}) = 1; (2) HA(x, •) = JF!f B(x,dy)HA(y, ·), x E:: K, A S B, A E:: (Jt, B E:: /)!; (3) continuity conditions on HA(x,· ); (4) g is non-negative and continuous, vanishing on t>; (5) g(x) > JKHK _ G(x,dy)g(y), K - G E:: CJl, Ll I x E:: G, G open; (6) weaker condition than the following: for every x E: K there exist arbitrarily small neighborhoods G of x such that K - G E:: tX and (g(y)- JKHK-G(y,dz)g(z)) attains its maximum at y = x. (Received October 1, 1963.)

607-19. C. H. GIFFEN, Princeton University, Princeton, New jersey. Uncountably many inequivalent nearly tame arcs.

Let X 1\B Y denote the union of those components of X n Y that intersect B, and let c(X) denote the number of components of X. Let (M,A,B) be a tripe and let V'be a family of arbitrarily small "admissible" open neighborhoods of B in M. Let p(A,B;V), where V E:'Y." be the minimum value of c(U n A) over all U E: ?/such that IT c V and U n A c V n 8 A; also, let p(A,B;v) be the lim inf p(A,B;V) as V-B, V E: ?./. For each natural number r let p (A,B;V) be the highest power of r that divides p(A,B;V), and let Pr(A,B;?/) be the lim inf Pr(A,B;V) as V-->-B, V E:: (/. The numbers p(A,B;'b') and Pr (A,B;21) are called the (peripheral) penetration and rth penetration indices,

666 respectively, of the pair (A,B) relative to the family?;: For the case where M is a 3-manifold, A an arc locally polyhedral except possibly at the endpoint x = B, and if 1fthe family of tame 3-cell neighborhoods of x whose boundaries are pierced by A at every point of their intersections with A, then the number of inequivalent embeddings of an arc in this manner (nearly tame) has power of continuum. In particular, for any specified values (finite or infinite) of Pr(A,x; 21}, where r is an odd prime, a nearly tame arc (A,x) is constructed. There are many variations on this theme in 3 and more dimensions. (Received October 1, 1963.)

667 ABSTRACTS PRESENTED BY TITLE

63T-397. LEO SARlO, 5.21 Georgina Avenue, Santa Monica, California. Extremal harmonic functions of several variables.

Let Yo be a compact bordered subregion of a region Y c R n, iJV 0 = a. U P. For f E: C on a. let ui E: Con Yo (i = 0,1), ui E: H on Y0 , uila. = f, c'luo/ilniP = 0, udP = const, fa.

{i c v0 , an = a. u i3n• u in on n as above, and limn__, y 0uin = ui = Lif· Given .,. E: H on Yo , fa (c'la/c'ln)dS = 0, there is a principle function piE: H ~n Y such that pi IYo = .,. + Li (pi - a). Let w = the area 21rn/2 /r(n/2) of lz I = 1 and set s = r 2-n /(w(n - 2)). For a,b E: Y let a= s at a, a= -s at b. Consider P = jp} with p E: H in Y\a U b, p = s +hat a, p = -s +kat b, h, k E: H, k(b) = 0. For

B(p) = limn_,y0 ffJnp(ap/iln)dS ~ 0, we have h 1 (a)~ h(a) ~ h0 (a), S = h0 (a) - h 1 (a). Then (Po + p 1)/2 gives minpB(p) = -S/4, and Po- p 1 gives minH(D(u)- 2u(a)) = -S for u E: H, u(b) = 0. Moreover, OG C OHP C OHB C OHD and the same holds for A, K c H defined by j(c'lu/iln)dS = 0 across all com­ ponents or dividing components of every Pn respectively. For subboundary y c c'IY,fl c Y, an=

'Yn Uon•'Yn surrounding 'Y, consider T'Y = jt}, t = -s +hat a, h(a) = o,J-yn (c'lt/iln)dS = 1, tyn E: T y!fl, tyni'Y n= k-yfl' tynl on= cfl" Then for ky = limn_,yk'Yfl' the capacity of 1' is Cy = e -k'Y; ty = limn_,ytyngives minTyB(t) = k'Y, and tfJ gives minTfJ supyt = kfJ• (Received September 5, 1963.)

63T-398. E. C. WEINBERG, 265 Altgeld Hall, University of Illinois, Urbana, Illinois. Free lattice-ordered abelian groups. II.

This discussion of the free abelian i-group with a. free generators, A a., is based on the description and results obtained in the first paper with this title (Math. Ann. 151 (1963), 187-199). Theorem l. Each Aa. is £-isomorphic to a subdirect sum of a family of copies of the ordered group of integers. Theorem 2. For a. > 1, Aa. has no nontrivial summands. Theorem 3. For each a., every family of pairwise disjoint elements of Aa. is countable. Theorem 1 can be used to prove that, for each equationally closed class E of .£-groups, and for each distributive lattice D, there is in E a free £-group over D. In the case where E is the class of abelian £-groups, Theorem 1 may be generalized to the free abelian £-group over an arbitrary distributive lattice. (Received September 5, 1963.)

63T-399. R. L. FINNEY, Fine Hall, Princeton University, Princeton, New Jersey. The insufficiency of barycentric subdivision.

The insufficiency of barycentric subdivision is demonstrated by the following theorem. Theorem. Let K and L be connected, locally-finite, simplicial complexes and let BK and BL be the complexes of their first barycentric subdivisions. If BK and BLare isomorphic then K and L are

isomorphic. If IK I is not a !-manifold, and if K is neither a simplex nor the boundary of a simplex, an isomorphism of B K onto BL maps the vertices of K to the vertices of L. This vertex mapping can be extended to an isomorphism of K onto L. (Received September 9, 1963.)

668 63T-400, M. H. TAIBLESON, Washington University, St. Louis 30, Missouri. Smoothness of Fourier transforms.

Let Lp(En} be the usual Lebesgue spaces on Euclidean n-space, En, and A(u; r, s; En) the Lipschitz spaces of distributions described in the author's announcement (Lipschitz spaces of functions and distributions over En, Bull. Amer. Math. Soc, 69 (1963}, 487-493}, Theorem. If

1 ~ p ;:; 2, 1/p + 1/q = 1, the Fourier transform is a continuous map of (a) Lp(En) into A(O; q, p, En); (b) A(O; p,q,En) into Lq(En); (c) Lq(En) into A(n/q - n/2; 2,q,En); (d) A(n/p - n/2; 2,p,En) into Lp(En}, From the fact that A(O; q,p} is continuously embedded in Lq and that Lp is continuously embedded in A(O; p,q) as proper closed subspaces, we see that part (a) or (b) of the theorem contains and somewhat sharpens the Hausdorff- Young theorem. The theorem contains, as special cases, extensions of many well-known results relating integrability (and/or differentiability) to smoothness. The proof proceeds by application of extensions of classical arguments to the end points (p = 1, q = oo; p = q = 2) and fills the gaps by application of an interpolation argument coupled with a duality argument. (Received September 9, 1963.}

63T-401. E. S. RAPAPORT, Box 175, 333 Jay Street, Brooklyn 1, New York. Groups of order 1.

Let G = (x 1,. .. ,xn; r 1, ... ,rn) = 1, I. Nielsen's method of finding free generators for a subgroup of a free group is proved via Grushko's theorem in a form useful to characterize r 1, ... ,rn. II. As the result is not constructive, techniques of deciding whether G is the identity are of interest. Three are discussed; one is taken from the literature. (Received September 10, 1963,}

63T-402_ HAO WANG, 33 Oxford Street, Cambridge 38, Massachusetts. A genetic definition of the class On of ordinals.

According to the Zermelo-Neumann definition of ordinals, 0 is the empty set, the successor x' of xis x U {x}, the limit ordinal of a set w of ordinals (with no maximum) is Lw (the union of w)_ Usual definitions of the class On of these ordinals do not directly reveal this intuitive picture. DI.

On(x) iff x belongs to every set u, such that (i) v' C u if v C u and v' C x'; (ii) 2:w C u if w ~ u and 1:.w C x'. In particular, the finite ordinals are obtained when (ii) is replaced by the special case when w is 0. DF. Nn(x) iff x belongs to every set u, such that (i) 0 C u if 0 C x', (ii) v' C u if v C u and v' C x'. Th~. If the axioms of extensionality are assumed, Aussonderung, and successor (i.e., (x}(Ey)(y = x' )), then the usual theory of ordinals can be developed (or finite ordinals) on the basis of DI (or DF); in order to get also definitions by transfinite induction, Aussonderung has to be strengthened to replacement; in order to get just recursive definitions for finite ordinals, it is sufficient to strengthen the axiom of successor to (x)(y}(Ez)(z = x U {Y}>. (Received September 11, 1963,}

63T-403. VICTOR KLEE, University of Washington, Seattle 5, Washington, A property of d- polyhedral graphs.

The d-polyhedral graphs are those isomorphic with the 1- skeleton of a d-dimensional convex polytope; for d ~ 3, they have been characterized in graph-theoretic terms, but for d ~ 4 only

669 necessary conditions are known (Steinitz, Balinski, Griinbaum-Motzkin). Previous necessary condi­ tions for d-polyhedrality were also necessary for e-polyhedrality (when e :5? d ;;:- 4); in conjunction with the fact that the complete k-graph is 4-polyhedral for all k !;:' 5, this led to a conjecture that for e ~ 4, every e-polyhedral graph is also 4-polyhedral. It is proved here that for every d there is a graph that is d-polyhedral but is note-polyhedral for any e 'I d. For a graph G, let un(G) be tile largest integer m such that a set of m nodes of G is totally separated by a set of n nodes of G. Construction of the desired graphs is based on the fact that as Granges over all d-polyhedral graphs, max un (G) is equal to 1 if n ;;£ d - 1, 2 if n = d, and IL(d,n) if n 2:: d + 1, where IL(d,n) is the maximum number of (d - I)-faces of a d-polytope having n vertices. (Received September 11, 1963.)

63T-404. R. L. CAUSEY, 1267 Marilyn Drive, Mountain View, California. An extremum problem in the space of matrices. Preliminary report.

Let ?Jrn• 'f/n, and 'lin denote respectively the sets of all arbitrary, normal, and unitary n by n complex matrices. For A= (\j) E:: 'h(n' with eigenvalues\, let fl(A) = diag ( >.1,. •• , >.J and let E denote the Euclidean norm: i(A) =:L.. Ia .. 12• For any unitarily invariant norm v on ?J?,n define l,J 1] dv(A) = Min v(A - X) (minimum taken over X E:: 1/n>· Mirsky (Quart. J. Math. Oxford 11 (1960), 50- 59) proved (1) [dE(A)]2 ;;£ (l/2) ( E2(A)- ltr A2 1) and conjectured that [d.(A)f = (1/2) (v 2(A) - v2 (n(A))). Theorem l sharpens (l): [dE (A)]2 :ii (l/2) (E 2( A) - I tr A 12 /n - ltr A2 - (tr A/- /n j). Theorem 2 proves Mirsky's conjecture for n = 2 and v = E, Here N = (l/2) (A + rA *) + (l/4) tr (A - tA *)I is a closest normal matrix, where A* is the hermitian conjugate of A and I rl = 1. Theorem 3, Let A E:: ?J?n, N0 = u 0D 0 u 0 , where u 0 E:: Un and D0 E:: ?l?n is diagonal. Then ~ satisfies dE (A) = E(A - N0 ) if .and only if E2 (dg(U0Au0)) = Max E2(dg(UAU*)) (maximum taken over u E:: 'Uu> and D0 = dg(Lb AU0) where, if M = (mij), dg M = diag (m 11 , ... , mnn>· (Received September 11, 1963.)

63T-405. C. YUAN, The University of Michigan, Ann Arbor, Michigan. Nonequilibrium hydrodynamics of a chemically reacting fluid.

In this paper, two topics are studied: (l) the thermodynamics (and hence the energy relations) for a chemically reacting fluid with heat of reaction effects; (2) the upper and lower limiting speeds of "linearly perturbed motions" in the fluid. The first of the above leads to the study of two types of relations: (l) the usual extent of reaction equation; (2) linear phenomenological equations for the heat flux and rate of reaction in terms of the affinity and temperature (DeGroot, S. R. and Mazur, P .,

Nonequilibrium thermodynamics, North-Holland Publishing Company (1962)). It is shown that if heat of reaction is neglected and proper assumptions are made on the affinity then the nonequilibrium energy relations of L. J. F. Broer (Characteristics of the equations of motion of a chemically reacting

~· J. Fluid Mech. 4 (1958), 276-282) are obtained. The energy relations of E. V. Stupochenko and I. P. Stakhanov (The equations of relaxation hydrodynamics, Soviet Physics Dokl. 4 (1960), 782-785) are obtained by neglecting the heat of reaction; the variable K is shown to be a function of the thermo­ dynamic variables. For the limit speeds, the following results are obtained: (l) in the low frequency case, the limit speed coincides with that of Broer, Stupochenko-Stakhanov and is independent of heat of reaction; (2) in the high frequency case, the limit speed is independent of the chemical reaction.

670 An explanation of this result has been given by Broer in the above cited paper. (Received September ll, 1963.)

63T-406. J. B. BUTLER, JR. Portland State College, Division of Science, Portland, 1, Oregon. The determination of an ordinary differential operator from its spectral density. Preliminary report.

Let L 0 = L~=OPi(x)(d/dx)i be an ordinary differential operator whose coefficients are (71,71) matrices on the interval - oo < x < oo, h71 = n = l v. Let q(x) be a bounded, positive, symmetric, matrix multiplication operator with elements piecewise continuous and vanishing x ;5 0, • a real para­ meter, and assume that L = L 0 + •q determines a self adjoint operator H, with no point spectrum, on .t'z,11 (-oo,oo), and with spectral measure E(~) given by (E(~)u,v) = J~

"h£.- i=O (( -1 )i- 1ti - P iL..tj="i 1 rij ) , where the vertical bar means evaluate ,t at ,t = x. lf ~contains the spectrum of H then k is the solution of the Gelfand-Levitan equation (cf. I. Kay, H. E. Moses, Nuovo Cimento 3 (1956), Z76). Theorems: (iv) If t(x) is defined by (iii) then E 0 (~)q = E 0 (~)t for x in some interval 0 ~ x ~ N. (v) If ~contains the spectrum of H then t = q, 0 ~ x ~ N. When Pi are constant t = (- iP· )r· ., j = i - 1, as shown by L, A. for L O = (d/dx)h, h l (Mat. Sb. (N.S.) 88 L..t"~ 1= 0 1 1J Sahnovi~ > (1958), 61). (Received September 16, 1963.)

63T-407. ROBERT BUMCROT, Ohio State University, Columbus, Ohio and P. D. ZVENGROWSKI, University of Chicago, Chicago, Illinois. Sets of infinite limit.

A subset A of a complete separable metric space X is of infinite limit if there exists a real­ valued function f defined~ X such that limx -+af(x) exists and is + oo for all a in A. Remark. If A is of infinite limit, f can be chosen so as to approach + oo only on A and to be continuous at least on the complement of X. Theorem. A is of infinite limit if and only if every subset of A contains an isolated point. Call A strong infinite limit iff can be chosen so that limx-+tf(x) exists for all tin X. Theorem. A is of strong infinite limit if and only if A is countable. Thus, for example, the set of midpoints of the component intervals of the complement of the Cantor set is of infinite limit, but not of strong infinite limit. (Received September 16, 1963.)

63T-408. TILLA KLOTZ, University of California, Los Angeles Z4, California. On surfaces in E 3 with constant negative curvature. Preliminary report.

Let S denote a surface with K < 0 immersed smoothly in E 3• Then the positive definite form II' given by H'll' = KI - HII with (H')z = Hz - K and H' < 0 defines a Riemann surface Rz on s. Also, the expression f! = {L - N - Zi M}dzz is a quadratic differential on Rz· Theorem. K s constant < 0 on S iff H'f! is holomorphic on Rz. Remark. It is rather easy to show that an S with K s constant <; 0 can not be complete relative to the II' metric. Remark. The Tchebychef net of asymptotic coordinates x,y which exists on any S with K s constant < 0 is the net of trajectories and orthogonal trajectories of the holomorphic quadratic differential iH'n. In particular, z = x + iy is a conformal parameter on Rz. This is clear, since (if K = - 1 for instance) I= dxz + dyz + l cos w dx dy and II= l sin w dx dy,

671 so that II' = sin w(dx2 + dy2). (Here w is the usual angle between coordinate curves, while 0 < w <; 71" and wxy = sin ware the only immediate restrictions on w.) (Received September 16, 1963.)

63T-409. R. V. DESAPIO, Eckhart Hall, University of Chicago, Chicago 37, Illinois. On embedding stably parallelizable manifolds in euclidean space. I. Preliminary report.

All manifolds are to be compact and oriented. A closed n-manifold Mn is almost differentiably embeddable in euclidean space R n+k if there is a homotopy n-sphere Vn such that Mn # Vn is differ­ entiably embeddable in R n+k (Mn # Vn is the "connected sum'' of Mn and Vn; for any homotopy sphere Vn the manifolds Mn and Mn # Vn are homeomorphic via a homeomorphism which is a diffeomorphism up to a point). An n-manifold is said to be r-coconnected if it is (n - r)-connected. Theorem. Let Mn be an r-coconnected, closed n-dimensional '~~"-manifold, n ~ 5, r ~ [n/2] + 2 if n is odd and r ;;:: n/2 + 1 if n is even. If n is of the form n = 4k + 2 then assume further that by a sequence of framed spherical modifications Mn can be reduced to an (n/2 - !)-connected manifold with Arf invariant zero. Then Mn is almost differentiably embeddable in euclidean space E 2r-l. If Mn bounds a ,--manifold then "almost differentiably" can be replaced by "differentiably." This has the following Corollary I. A (k - !)-connected, closed (2k + 1)-manifold, k ~ 2, is almost differentiably embeddable in R 2k+3 if and only if it is a ,--manifold. The proofs rely heavily on results of J. Minkus and the method of "framed spherical modifications" as developed by Kervaire and Milnor. (Received September 16, 1963.)

63T-410. R. V. DESAPIO, Eckhart Hall, University of Chicago, Chicago 37, Illinois. On embedding stably parallelizable manifolds in euclidean space. II. Preliminary report.

Implications of a previous abstract (On embedding stably parallelizable manifolds in euclidean space. I, these Notices) are obtained. Corollary 2. An ( [n/2] - !)-connected, closed n-manifold, n G; 5, is differentiably embeddable euclidean space R 2 [(n+l)/2]+l if and only if it bounds a 71"-manifold. The following is a known result of Penrose, Whitehead, and Zeeman.

Corollary 3. A closed, almost parallelizable n-manifold which is r-coconnected, n ~ 5 and r 1:: [(n + 1)/2] + 1, can be combinatorially embedded in R 2r. Corollary 4. Let m(n) be the least dimension in which all homotopy n-spheres are differentiably embeddable, n :;?; 5. Then a closed n-dimensional 11"-manifold Mn (assume that Mn is ,--cobordant to a 2k-connected manifold with Arf invariant zero if n is of the form n = 4k + 2) which is at least (n- [(m(n) + 1)/2]}-connected is differentiably embeddable in R m(n). According to Haefliger m(n) ~ [3n/2] + 2. (Received September 16, 1963.)

63T-411. EVELYN FRANK, P .0. Box 361, Evanston, Illinois, and AMBIKESHWAR SHARMA, University of Calgary, Calgary, Canada. Continued fraction expansions for yC + L and iterations of Newton's formula.

Approximants for continued fraction expansions for binomial quadratic surds yC + L are related to approximations by Newton's formula and the rule of regula falsi. The present study deals with similar relations and iterations of Newton's formula. (Rece.ived September 16, 1963.)

672 63T-412. A, A. MULLIN, l08d EERL, University of Illinois, Urbana, Illinois. Recursive un­ solvability of the decision problem for models of the fundamental theorem of arithmetic.

Definition l. By a pure model of FT A is meant a model of FT A determined, except possibly for entries equivalent to l, by primes alone; e.g., Euclid's model. By a mixed model of F T A is meant a model of FTA determined, except possibly for entries equivalent to l, by both primes and composites; e.g., Gauss' model. Theorem l. Every model is either pure or mixed, and the pure models form an infinite class as do the mixed models. Definition 2. A model of FT A is called _!educed if no entry in it is equivalent to l. E.g., Euclid's model is reduced, but the model n = p 1 a.l.p2 a.2 p~ ... , where pm runs through all primes and a.i > 0 for only finitely many i is not reduced. Theorem 2. The reduced pure models of FT A form a countably infinite class as do the reduced mixed models. Theorem 3. The decision problem as to pureness of an arbitrary _Eeduced model of FTA is recursively unsolvable, and a fortiori the decision problem as to pureness of an arbitrary model of FT A is recursively unsolvable. (Received September 17, 1963.)

63T-413. A. A. MULLIN, l08d EERL, University of Illinois, Urbana, Illinois. The fundamental theorem of arithmetic has only three inequivalent forms,

Any model of FT A (with interest for a multiplicative theory) is either pure (determined, except possibly for entries equivalent to natural number l, by primes alone) or mixed (determined, except possibly for entries equivalent to natural number l, by both primes and composites). A pure model cannot be the basis for generating, by multiplicative means, any new model since a pure model con­ sists of multiplicatively indecomposible primes alone. Thus new models, obtained by multiplicative means, can originate only from a mixed model. Fundamental Metatheorem of Arithmetic: the three inequivalent forms of models of FTA are (l) the classical Euclidean form, (2) a mixed form obtained from (l) by an effective method (e.g., Gauss' model), and (3) a pure non-Euclidean form obtained from (2) by an effective method for applying (l) and/or (2) to the composites of (2). At present it would seem that the author's "mosaic model" is the simplest concrete model of form (3). (Received September 17, 1963.)

63T-414. C. K. MEGIBBEN, Texas Technological College, Lubbock, Texas. Completions of abelian groups.

Definition. A group K is said to be a completion of the abelian group G if (i) K is algebraically compact, (ii) G is a pure subgroup of K, (iii) K/G is divisible and (iv) no proper subgroup K' of K satisfies conditions (i)- (iii). Theorem l. If G is a pure subgroup of an algebraically compact group C, then some direct summand of C is a completion of G. Consequently, all abelian groups have completions. Theorem 2. Between any two completions of G there is an isomorphism leaving the elements of G fixed. The topology on the group G defined by taking the subgroups nG as neighborhoods of 0 is called the n-adic topology on G. If G1 = 0, where G1 is the intersection of all the nG's, then G can be imbedded in the usual manner as a dense subgroup of a group G* which is complete in its n-adic topology. A subgroup H of G is said to be a high subgroup of G if H is maximal with respect to H n G1 = 0. Theorem 3. If H is a high subgroup of G and if H* is the n-adic completion of H,

673 then the group generated by the elements of H* and G subject only to H* n G = H is a completion of G. Theorem 4. If E = Ext(Q/Z,G) and Dis minimal divisible containing G 1, then each completion of G is isomorphic to E/E 1 + D. (Received September 18, 1963.)

63T-415. C. K. MEGIBBEN, Texas Technological College, Lubbock, Texas. Multiples of pure subgroups.

Theorem 1. If H is a direct summand of the abelian group G and if K is a pure subgroup of G such that nK = nH for some integer n, then K is a direct summand of G. First the proof is reduced to the case where Hand K have non-bounded direct summands. Next the proof is reduced further to the torsion case, which is handled by a theorem of Irwin and Walker (Pacific J. Math. 11 (1960), 1363-1374). Definition. Two abelian groups Hand K are said to be almost isomorphic if H = A+ H' and K = B + K' where H' ~ K' and A and B are bounded. Theorem 2. If Hand K are pure subgroups of the abelian group G such that nH = nK for some integer n, then H and K are almost isomorphic. The proof involves the use of p-hasic subgroups (see L. Fuchs, Acta. Math. Acad. Sci. Hungar. 11 (1960), 117-125) and is particularly simple if n = pi for some prime p. (Received September 18, 1963.)

63T-416. C. K. MEGIBBEN, Texas Technological College, Lubbock, Texas. Kaplansky's test problems for algebraically compact groups.

Kaplansky has asked the following question: If the abelian groups G and H are isomorphic to direct summands of one another, are G and H isomorphic? In general, the answer is no, though, as Kaplansky observed, an affirmative answer can be given for certain classes of groups (e.g., divisible groups, countable torsion groups, closed p-groups). Theorem 1. If the algebraically compact groups

G and Hare isomorphic to direct summands of one another, then G ~H. By a standard argument, the problem is reduced to the case where G and H are reduced groups. A similar argument further reduces the problem to the separate consideration of the adjusted cotorsion and torsion free cotorsion cases. If G is adjusted, then G ~ Ext(Q/Z,LGp) where Gp, the p-primary component of the torsion subgroup of G, is a closed p-group for each prime p; and the adjusted case is easily seen to reduce to the case of closed p-groups. On the other hand, the torsion free case reduces to the divisible case, since if G is torsion free, G ~ Ext(Q/Z,G) :;;< Hom(Q/Z, 0/G) where D is minimal divisible containing G. A similar analysis establishes Theorem 2. If G + G :;;;: H + H and G and H are alge­ braically compact, then G :;;;: H. (Received September 18, 1963.)

63T-417. L. B. TREYBIG, Tulane University, New Orleans, Louisiana. Concerning continua which are continuous images of compact ordered spaces.

Suppose S is a Hausdorff continuum which is the image of a compact ordered space I under a continuous transformation f. Theorem 1. If S is separable, then S is metrizable. Theorem 2. If no set of less than three points separates S, then S is metrizable. Theorem 2 generalizes in another way (see Abstract 62T-182, these Notices 9 (1962)) the following theorem due to Marde!Hc and Papic:

If each of X and Y is a nondegenerate Hausdorff continuum and X X Y is the continuous image of a compact ordered continuum, then both X and Y are metrizable. Also, it is easy to see by means of simple examples that the number three is "best possible." (Received September 20, 1963.) 674 63T-418. K. D. MAGILL, JR., State University of New York, College at Buffalo, 3435 Main Street, Buffalo 14, New York. Some algebraic characterizations of topological properties. Preliminary report.

All spaces here are Q-spaces. Inter-real ideals and I-dense ideals are defined in these Notices, 10 No. 2 issue 66, 63T-86. An investigation of some properties of these ideals has produced the following results. Theorem 1. Every ideal of C(X) is an inter-real ideal iff X is finite. Theorem 2. Every inter-real ideal of C(X) is a direct summand iff X is discrete. Theorem 3. The sum of every pair of inter-real ideal or C(X) iff X is normal. Theorem 4. No inter-real ideal is the union of all inter-real ideals properly contained in it iff X is perfectly normal. Theorem 5. C(X) contains I-dense ideals iff X consists of only one point or is not compact. Theorem 6. Either the intersection of all I-dense ideals of C(X) is an !-dense ideal of C(X) or C(X) contains no !-dense ideals iff X is locally compact. (Received September 20, 1963.)

63T-419. M. D. GEORGE, University of Missouri, Columbia, Missouri. Generalized solutions of boundary value problems.

Let G be a domain in En with piecewise smooth boundary S. L. Sarason (Comm. Pure Appl. Math. 15, p. 237) has defined several types of generalized solutions of a symmetric positive system of first order partial differential equations Ku = f in G, satisfying the boundary condition Mu = g in S. Here M is a boundary matrix satisfying the three conditions listed in Sarason, p. 246. Strong and weak solutions are certain elements of L 2(G) XC (S); M-strong and M-weak solutions are elements of L 2 (G). Theorem. If u is an M- strong solution, then there exists an element uS of L 2 (S) such that (u,uS) is a weak solution. This theorem completes the ordering of types of solutions, since it is easy to show that the L2 (G) component of a strong (weak) solution is an M-strong (M-weak) solution. (Received September 20, 1963.)

675 INDEX OF ABSTRACTS Volume 10, 1963

Aanderaa, Stal. See Dreben, B. S. Arkowitz, M. A. On the number of multiplications Abbot, J, C. Isomorphism theorems in implica­ of an H-space. Preliminary report, 275. tion algebra, 445. Arkowitz, M. A. and Curjel, C. R. Commutators Abian, Alexander. On the equipollence of sets, in groups of homotopy classes, 570. 191, Armentrout, Steve. U~per semi-continuous de­ Abian, Alexander and McWorter, W. A. On the compositions of E with at most countably index of nilpotency of some nil algebras, 252; many non degenerate elements, 1 04; On upper On the structure of pre-p-rings, 290. semicontinuous decompositions of Euclidean 3-space into tame 3-cells and one-point sets, Adams, J, M. Abstract homotopy for categories. 461. Preliminary report, 476, Arms, R. J, On the limit behavior of steepest Adams, R, D., Aronszajn, Nachman and Smith, descent, 82. K. T. Bessel potentials in a domain, 63. Aronson, D. G. On correct partial differential Adler, R, L., Konheim, A. G, and McAndrew, operators and stable finite difference opera­ M. H. Entropy of continuous functions, 189. tors, 103,

Ahlberg, J, H. See Walsh, J, L. Aronszajn, Nachman, Extension of a theorem of Alling, Norman. Valuations on meromorphic func­ Hartogs to analytic functions of n real vari­ tions fields, 64; On relative compactifications ables, 63. with application to algebra. Preliminary re­ -----· See Adams, R. D. port, 277; A proof of the Corona conjecture for finite open Riemann surfaces. Prelimin­ Aronszajn, Nachman, Mulla, F. J, and Szeptycki, ary report, 443, Powel. Spaces of potentials connected with Lp-classes, 64. Alperin, J, L. System normalizers and Carter subgroups, 252. Arsove, M. G. The Lusin-Privalov theorem for subharmonic functions, 462. Amitsur, S. A. Generalized polynomial identities, 440. Artemiadis, Nicolas. Inequalities, 79.

Anderson, R. D. On topological translations in E~ Artzy, Rafael, Ternary rings and''motions" in an 647 affine plane, 455. Anderson, R. V. Some theorems on polygenic functions of a hyperbolic complex variable, Asenjo, F. G. The arithmetic of the term-relation 359. number theory, 90.

Andrews, J, J, The Minkowski unit of slice knots, Axt, Paul. Iteration of primitive recursion, 112. 253, Aziz, A. K. and Bogdanowicz, Witold, On a mixed Andrews, J, J, and Dristy, F. E. The Minkowski boundary value for nonlinear hyperbolic equa­ units of ribbon knots, 113. tions, 102.

Anselone, P. M. and Moore, R. H. Solution of Babbitt, D. G, A summation procedure for certain nonlinear integral equations from elasticity Feynman integrals. II, 136; Wiener integrals by Newton's method, 252, and singular self-adjoint differential opera­ Anselone, P. M. and Rall, L, B. Newton's method tors, 580. for characteristic value-vector problems, 489. Bade, W. G, and Curtis, P. C., Jr. On Banach Appel, K. I. and Djorup, F. M. On the group gen­ algebras of continuous functions, 4 73. erated by a free semigroup, 300; The equation Barlow, R. E. and Marshall, A. W. Inequalities a!a~ ... a~ = bn in a free semigroup, 505, for distributions with monotone hazard rate. I, Appling, W .D.L. Interval functions and continuity, 62. 129; Interval functions and absolute continuity. Barrett, J, H. A Prufer transformation for third­ Preliminary report, 302, order differential equations, 485. Arens, R. F. and Curtis, P, C., Jr. Commutative Barrett, L, K. A simple closed curve that lies on Banach algebras which are direct summands, the surface of a two sphere and pierces no 116, disk, 485. 676 Basavappa, P. and Kimura, Naoki. On some ma­ sphere, 295. trix equations. II, 266. Birtel, F. T. Singly-generated Liouville algebras, Baumslag, Gilbert. On a problem of Plotkin con­ 514. . cerning locally nilpotent groups, 466. Bishop, R. L. A relation between volume, mean Baxter, G. E. On fixed points of the composite of curvature, and diameter, 364. commuting functions, 297. Blattner, R. J, On induced representations. III,91. Baxter, W. E. and Pellicciaro, E. J, Cyclicly re­ Block, Richard. The Lie algebras with a quotient lated differential equations, 197. trace form, 110; Hadamard matrices and con­ Bean, R. J. Disks which" almost" lie on 2-spheres stant distance codes, 658. in E 3 , 253, Bloom, D. M. The subgroups of SL(3,q). I, 128. Bear, H. S. An abstract Dirichlet problem, 446. Blumenthal, R. M. and Getoor, R. K. Local times Beaumont, R. A. and Pierce, R. S, Isomorphic for Markov processes, 653, direct summands of abelian groups, 107. Boehme, T. K, On the continuity ofperfectopera­ Beckenbach, E. F. On the inequality of Kantoro­ tors, 277; Approximate solutions to the con­ vich, 440, volution equation on the half-line, 4 78. Bednarek, A. R. On some set-theoretic maximal­ Bogdanowicz, Witold, On existence and unique­ ity principles, 66. ness of almost periodic solutions for nonlinear differential equations, 87. Beekman, J, A. Solutions to generalized Schroe­ dinger equations via Gaussian Markov inte­ Bonnice, W. E. and Klee, V. L. A note on positive grals, 501. independence, 269; Barycenters and hull-for­ mation, 441, Bell, C. B. Topological groups and measure prob­ lems in distribution-free statistics, 651. Boothby, W. M. and Wang, H. C. On the finite sub­ groups of a connected Lie group, 483. Bell, C. B. and Doksum, Kjell. Randomized dis­ tribution-free statistics, 52. Borgess, C. R. On continuous maps and M3-spaces, 468, Ben- Israel, Adi. See Charnes, Abraham. Bose, A K. Functions satisfying a weighted aver­ Ben-Israel, Adi and Wersan, S. J, A least-squares age property, 650. method for computing the generalized inverse of an arbitrary matrix, 57. Bradford, R. E. Decomposition of linear forms in cardinal algebras, 120; On the decision Bergman, Stefan. On critical sets of an invariant problem for cardinal algebras, 355; Decompo­ in space of two complex variables, 361; On an sition of linear forms in cardinal algebras, invariant with respect to pseudo-conformal 492. transformations in the space of two complex variables, 361; Bounds for a class of functions Bramble, J, H. and Hubbard, B. E. On the conver­ of two complex variables, 574, gence of iterative methods for the solution of systems of non-positive type, 108; Finite dif­ Berin, A. Transformation of power series, 269, 516, ference analogues for elliptic boundary value problems and the maximum principle, l 08. Berman, S. M. A general arc sine law and its ap­ Brauer, F. G. Nonlinear differential equations plication to diffusion processes, 567. with forcing terms, 483, B erri, M. P. Categories of certain minimal top­ Brenner, J, L. The Jordan normal form: decom­ ological spaces, 71. position theorem for modules, 110, Besicovitch, A. S. Fundamental geometric prop­ Briggs, W. E. Prime-like sequences generated erties of sets of lines, 77. by a sieve process, 113, Bhatia, N. P. On attractors in dynamical systems, Brillhart, J, D., Lehmer, D. H. and Lehmer, 2 53; On stability of sets of dynamical systems. Emma. Bounds for pairs of consecutive seventh Preliminary report, 570, and higher power residues, 467. Bing, R. H. Improving the side approximation Brooks, R. M. A ring of analytic functions, 574, theorem, 453; Stable homeomorphisms on E5 can be approximated by piecewise linear ones, Browder, Andrew and Wermer, John. A method 666. of constructing Dirichlet algebras, 302.

Bing, R. H. and Klee, V. L. Every simple closed curve in ~ is unknotted in E 4, 132. Brown, Leon, and Nakano, Hidegoro. Outer meas­ ures on linear lattices, 89. Bing, R. H. and Kister, J, M. Uncountably many fixed-point-free periodic actions on a 3- Brown, Morton and Rosen, R. H. Triangulated

677 manifolds, 460, cients, 190; Power series with restricted coefficients, 657, Brown, R, F. Path fields on manifolds, 449; On grouplike manifolds. Preliminary report, 653. Cantrell, J, C. Non flat embeddings of sn- 1 in sn. Preliminary report, 301; n-frames in eulcidean Brown, T. A. The existence of periodic solutions k-space, 450, to nonlinear differential-difference equations x(t) = g2 (x(t))g2 (x(t- 1)), 106. Carlitz, Leonard. Recurrences for Bernoulli and Euler numbers, 473; Extended Bernoulli and Bruck, R. H. Modular exponentials and logar­ Eulerian numbers, 575; Functions and poly­ ithms, 254. nomials (mod pn), 575; Some inversion formu­ Brunk, H. D. Note on distributions of functions las, 575. of interchangeable random variables, 121, Carlson, D. H. Bounds on the rank of H under Buchanan, M. L. A necessary and sufficient con­ R(AH) ~ 0, of rank r, 254; Inertial bounds for dition for stability of difference schemes for matrices, 477; Eigenvalue criteria for A when initial value problems, 457. AH is Hermitian, 589. Bumby, R. T. Modules which are isomorphic to Carroll, F. W. On bounded functions with almost submodules of each other, 273. periodic differences, 2 54. Bumcrot, R. J. Algebraic and metric convexity in Carroll, R. W. On the structure of the Green's normed linear spaces, 249. operator, 248; On the spectral determination of the Green's operator, 377, Bumcrot, R. J. and Zvengrowski, P. D. Sets of infinite limit, 671. Carroll, R. W. and Glick, A. J. On the Ginzburg Landau equations, 513, Burgess, C. E. A characterization of tame sur­ faces in E 3, 185; Some properties of certain Carroll, R. W. and Wang, C. L. On the degenerate types of wild surfaces in E 3 , 457. Cauchy problem, 497.

Burlak, Jacob. Dual integral equations with Four­ Carruth, J. H. The kernel of the semigroup of ier kernels, 478, subsets of a semigroup, 648. Burr, S. A. On the occurence of squares in Lucas Carter, D. S. Asymptotic behavior of infinite sequences, 367; On a result of Gordon in the fluid jets under gravity, 106, theory of partitions, 367, Casler, B. G. Slicing a contractable 3-manifold Burrow, M. D. Some properties of unrestricted with boundary, 649. products, 453, free Causey, R. L. An extremum problem in the Buschman, R. G, Integrals involving Legendre space of matrices. Preliminary report, 670. 194; Integrals involving hypergeo­ functions, Cavior, S. R, Equivalence classes of functions equations metric functions, 203; Convolution over a finite field, 44 7; Equivalence classes ker­ with generalized Laguerre polynomial of sets of functions over a finite field, 660. nels, 374. Chacon, R. V. A class of linear transformations, B., Jr. On a problem of unitary equiv­ Butler, J, 583, alence of ordinary differential operators of even order, 53; A representation of the wave Chakerian, G. D. and Stein, S. K. The centroid of operator associated with certain ordinary dif­ a homogeneous wire, 587. 289; The ferential operators of even order, Chandler, Bruce. The representation of a gen­ op­ determination of an ordinary differential eralized free product in an associative ring, erator from its spectral density. Preliminary 476. report, 671. Charlap, L. S. The classification of Zp-manifolds, Terence and Martin, A. V. On a method Butler, 275. of Courant for minimizing functionals, 51. Charlap, L. S. and Vasquez, A. T. The cohomology Butson, A. T. Essentially cancellative rings, 105. of Z p-manifolds, 27 5. of bilinear Bzoch, R. C. On the representation Charnes, Abraham. On some fundamental theo­ func tiona! s, 64 7. rems of perceptron theory and their geometry, 60. Calabi, Eugenio. Matsushima's theorem in Rie­ mannian and Kahlerian geometry, 505, Charnes, Abraham and Ben-Israel, Adi. A new Euclidean representation for the generalized sets, Calder, J, R. Concerning completely convex inverse of an arbitrary matrix, 135, 107. Cantor, D. G. Power series with integral coeffi- Chase, S. U. and Rosenberg, Alex. On quasi-

678 Frobenius algebras and complete cohomology Comfort, W. W. and Gordon, Hugh. Disjoint open of maximal orders, 65, subsets of f3X\x, 430, Chase, S. U., Rosenberg, Alex and Harrison, D.K. Comfort, W. W. and Ross, K, A. Topologies in­ Galois-theory of commutative rings, 515. duced by groups of characters, 435. Cheema, M. S. Number theory and integral linear Comstock, Craig, On the autocorrelation of ran­ programming, 76. dom inhomogenitics, 278. Cheema, M. S. and Pall, Gordon. Quadratic Conley, C. C. On a disk mapping associated with forms of certain determinants, 471, the satellite problem, 465, Chen, K.-T. Decomposition and equivalence of Conley, C. C. and Rejto, P. A. On spectral con­ local diffeomorphisms, 444, centration, 187. Cheney, E. W. and Loeb, H. L. Tchebycheff ap­ Conner, H. E. Extinction probabilities for age and proximation by rational forms, 300, position dependent branching processes, 490. Chover, Joshua. First passage probability for a Conte, S. D. and Lees, Milton, The numerical Gaussian process in the Karhunen representa­ solution of singular biharmonic difference tion, 432. equations, 74, Chu, Hsin. Compactification and duality of topo­ Cook, Howard. A note on subsets of indecompo­ logical groups, 187. sable continua, 255; Upper semi-continuous collections filling up hereditarily indecom­ Chu, Hsin and Geraghty, M. A. The first cohomol­ posable continua, 645. ogy group of a minimal set, 80, Cooper, R. M. A new Riemann function, 591. Church, Alonzo. A revision of Mange's method. Preliminary report,457;Some generalizations Corson, H. H. and Michael, E. A. On metrizabil­ of Laplace's transformation. Preliminary re­ ity, 76, port, 515. Courter, R. C. Maximal commutative algebras Church, P, T. On points of Jacobian rank k, 84, with unlawfully small dimension. Preliminary report, 567. Chwe, B.-S Relative homological algebra and homological dimensions of Lie algebras, 278, Cox, R. S, Representations of a semigroup, 643. Cima, J. A. An analogue of the Eisenstein theorem Coxeter, H. S, M. A noninductive definition for a for the harmonic functions. Preliminary re­ regular polytope. Preliminary report, 468, port, 359, Coxeter, H. S. M. and Fejes T6th, L. The total ----' See Rung, D. C. length of the edges of a non-Euclidean polyhe­ dron with triangular faces, 80, Clark, Edwin, Finite dimensional affine semi­ groups, 130, Crawford, G. B. and Saunders, S.C. Nonparamet­ ric estimation and consistency, 655. Clark, W. E. A class of simple semigroups of non-negative matrices. Preliminary report, Curjel, C. R. See Arkowitz, M. A. 582. Curtis, C. W, On algebras of bounded representa­ Clarkson, J, M. Transformations of regular poly­ tion type, 99. hedrons. Preliminary report, 277. Curtis, P. C. Jr. See Arens, R. F. Clay, J, P. Invariant attractors in transformation ____. See Bade, W, G. groups, 58, Curtiss, J, H. Harmonic interpolation in Fejc§r Coburn, Nathaniel. General theory of simple points with Faber polynomials, 88; The be­ waves in relaxation hydrodynamics, 583, havior of complex interpolation polynomials Cohen, Eckford. An analogue of the totient func­ on the locus of the interpolation points, 484, tion, 138; Pairs of relatively prime integers Czarnecki, A. Z, The surfaces of revolution for in residue classes, 200; The number of repre­ dynamical trajectories of Newtonian inverse sentations of an integer as a sum of two square law, 286; Physical systems Sk as geo­ square-free numbers, 201; The number of desics of a surface with constant gaussian zeroid elements (mod n), 204; Two functions curvature, 286. related to the k-free integers, 646. Cohen, H. B. The k-extremally disconnected Davis, E. D. Rings of algebraic numbers and spaces as projectives, 263, functions, 247, Cohn, J, A. and Livingstone, Donald, Group rings Davis, J, R. On the distribution of eigenvalues of of finite groups. I, 569. generalized Toeplitz forms, 583.

679 Davis, S. H. and Segel, L. A. Surface elevation in Diaz, J. B. and Metcalf, F. T. On an inequality Benard cells, 496. for finite sums, and generalizations of an inequality of Kantorovich, 91; On an inequality Dawson, D. F. Some rate invariant sequences complementary to Holder's inequality, 510. transformations, 503; On the decomposition of sequences, 503; On the extension of some re­ Diran, Sarafyan. The use of the Runge-Kutta type sults of Lane and Wall, 590. methods at singular points, 4 71. Day, G. W. and Schubert, S. R. On perfect map­ Dixon, E. D. Nilpotent matrices and commutators, pings, Z49. 47Z. Pay, M. M. Convolutions, means and spectra, 461. Djorup, F. M. See Appel, K. I. Deal, R. B. An axiomatic approach to the geome­ Doksum, Kjell. See Bell, C. B. try of manifolds. I. Preliminary report, 190. Dolph, C. L. Positive real resolvents and linear Dean, W. C. and Thielman, H. P. Generalized passive Hilbert systems, 85. Laguerre functions, 466. Dolph, C. L. and Lomax, R. J. On the linear Decell, H. P., Jr. Ideals ofmeasuresovera com­ theory of multi-stream electron devices, 139. pact topological semigroup. I, Z6Z; Ideals of Douglas, R. G. A uniqueness property of invariant measures over a compact topological semi­ means, 84. group. II, 360. Drazin, M. P. A class of solvable differential Deckard, D. J. Complete sets of unitary invari­ games, 657. ants for compact and trace class operators, Dreben, B. S. Corrections to Herbrand, Z85. 476. Dreben, B. S. and Aanderaa, Stal. Errors in Deckard, D. J. and Durst, L. K. Unique factoriza­ Herbrand, Z85. tion, 65.7 Dreben, B. S. and Denton, J. S., Jr. Three solva- ble cases, 590. Deckard, D. J. and Pearcy, C.M. On matrices over the ring of continuous complexvalued Dressel, F. G. See Gergen, J. J. functions on a Stonian space, 83. Dristy, F. E. See Andrews, J. J.

Deeter, C. R. and Springer, George. Discrete Dubois, D. W. A note on direct sums of cotorsion harmonic kernels, 431. groups, 133. DeMar, R. F. Uniqueness classes for difference Duren, P. L. Distortion in certain conformal functionals, 91. mappings of an annulus, 71; On the Marx con­ jecture for starlike functions, 587. DeMarr, R. A. Partially ordered linear spaces and locally convex linear topological spaces, Durst, L. K. See Deckard, D. J. 353; An iteration procedure for inverses in a Dwinger, Philip. The dual space of the limit of Banach algebra. Preliminary report, 373; an inverse limit system of Boolean algebras, Partially ordered linear spaces and locally 37Z. convex linear topological spaces, 468; A com­ mon fixed point theorem. Preliminary report, Dwinger, Philip and Yaqub, F. M. Generalized 496; Martingale theory in vector lattices. Pre­ free products of Boolean algebras with an liminary report, 591. amalgamated subalgebra, 135. DeMarr, R. A. and Fleischer, Isidore. General­ Dyson, V. H. A decidable theory for which the ized metric spaces, 114. theory of infinite models is undecidable, 453; A decidable theory for which the theory of Dengler, M. A. The temperature distribution of a finite models is undecidable, 491. heat source between infinite parallel bound­ aries, 53. Edmonds, Jack. On the surface duality of linear DeNoya, L. E. Purity in simple systems. Pre­ graphs, 101. liminary report, 570. Edrei, Albert. On the deficiencies of meromor­ Denton, J. S. Jr. A reduction class with a single phic functions of finite lower order, Z78. binary predicate, 1Z4; A false "decision pro­ Edwards, C. H. Jr. Concentricity in 3-manifolds, cedure" for the halting problem, 1Z5. Z55. ____• See Dreben, B. s. Edwards, R. E. and Hewitt, Edwin. Pointwise DeSapio, R. V. Unknotting spheres, 365; On em­ summability of Fourier transforms on LCA bedding stably parallelizable manifolds in groups, 654. euclidean space. I, II. Preliminary report, Egbert, R. J. Cartesian products of statistical 67Z, metric spaces, Z66. Eggleston, H. G. Minimal universal covers in Foulis, D. J. Semigroups coordinatizing ortho­ real Euclidean space, 438; A property of mocular geometries, 443. Hausdorff measure, 659. Foulser, D. A. The flag transitive collineation Eisenberger, Isidore and Posner, E. C. Estima­ groups of the finite Desarguesian affine tion and goodness of fit tests from quantiles, planes, Z94; A generalization of the Andre Z68. translation planes. Preliminary report, Z94. Eisenstadt, B. J, See Nakano, Hidegoro. Fox, Charles. Matrix integral transforms, Z70; Integral transforms based upon fractional Enochs, E. E. Homotopy groups of compact Abe­ integration, 301. lian groups, 650. Fox, W. C. A positional characterization of sub­ Etter, D. 0. Vector-valued analytic functions, 91. manifolds, 436. Evans, Trevor. Identities in multiplicative sys­ Fraenkel, A. S, Exact numerical solution of tems. II. Anti-finite and anti-associative laws, linear equations with rational coefficients. 649. Preliminary report, 585, Fabian, R. J, and Kent, C. F. General recursive Frame, J, S. The rectangular matrix equation functions by restricted ordinal recursion, 71. A Y + B = YC, 566. Faith, Carl and Utumi, Yozo. On noetherian prime Frank, Evelyn. On continued fraction expansions rings, 468. for binomial quadratic surds, 464; Continued Feferman, Solomon and Levy,Azriel. Independence fraction expansions for nth roots and other results in set theory by Cohen's method. II, functions 665. 593. Frank, Evelyn and Sharma, Ambikeshwar. Con­ Fejes T6th, L. See Coxeter, H. S. M. tinued fraction expansions for VC + L and iterations of Newton's formula, 673, Fell, J, M. G. A topology for the Banach space representations of a group, 479. Frank, Stanley. See Hutcherson, W. R. Fenstad, J, E. On £-groups of uniformly continu­ Franklin, S, P. A "Kre!n Milman" theorem for a ous functions, Z85, p.o.t.s. and some applications, 196; Quotient topologies from power topologies, Z89. Ferguson, D. C. Infinite products of isols. Pre­ liminary report, 43Z. Franklin, S. P. and Sorgenfrey, R. H. Character­ ization of topological properties in terms of de Figueiredo, R. P. A theorem on the existence upper-semicontinuous relations, 440, of n stable limit cycles for Lienard' s equa­ tions, llZ; On a class of damped nonlinear Freilich, Gerald. Caratheodory measure of cylin­ systems with discrete periodic solutions of ders, 361. conservative systems, Z55, Freyd, P. J, The construction of relative injective Finney, R. L. The insufficiency of barycentric modules, 60, subdivision, 668. Fuchs, W. H. J, On the eigenvalues of an integral Firey, W. J, A characterization of certain means equation, 352. of convex bodies, 107; Generalized Loewner Fulks, W. B. Debye' s series in the transitional ellipsoids, 189; Arithmetic and harmonic region, 73; An approximate Gauss mean value means of positive matrices, 190, theorem, 464. Fischer, P. C. Elementary properties of provable Fullerton, R. E. On the coincidence of natural recursive sets, 119. order and the order defined by a basis on a Flatto, Leopold. Functions with a mean value linear space, 103, property. II, 351. Furcht, Roberto and Rota, G.-c. A Mobius-type Fleischer, Isidore. See DeMarr, R. E. inversion formula for partitions, 495. Ford, D. A. A representation theorem, 127. Gabai, Hyman. On the discrepancy of certain Ford, R. M. Concerning a certain definition of sequences mod 1, Z70. dimensions. Preliminary report, Z56. Gagliardo, Emilio. On bounded integral transform­ Forslund, D. W. See Hillman, A. P. ations with positive kernel, 64. Fort, M. K., Jr. Level sets of continuous func­ Ganea, T. On cocategory. Preliminary report, 263. tions. Preliminary report, 648. Gangolli, R. A.lsotropic infinitely divisible meas­ Foster, L. L. A theorem concerning character­ ures on symmetric spaces, 461, istic roots of integral matrices, Z03,

681 Garland, Howard. On certain sheaf cohomology 497; The restricted cancellation law for ideals groups. Preliminary report, 504. in a commutative ring, 510. Gaughan, E. D. The index problem for infinite Gilmer, R. W., Jr. and Mott, J. L. Multiplication symmetric groups. Preliminary report, 264. rings as rings in which ideals with prime radical are primary, 581. Gautschi, Walter. On error reducing multistep methods, 95. Gilmer, R. W. and Ohm, J. E. Integral domains with quotient overrings, 203; Primary ideals Gelbaurn, B. R. Notes on a spectral synthesis, 92. and valuation ideals, 645. George, M. D. Generalized solutions of boundary Ginsburg, Michael. Some immersions of projec­ value problems, 675. tive space in Euclidean space, 85. Geraghty, M. A. See Chu, Hsin. Ginsburg, Seymour and Spanier, E. H. Quotients Gergen, J. J., Dressel, F. G. and Parrish, G. B. of context free languages, 55. Eigenvalues of a Bessel difference system of Ginsburg, Seymour and Ullian, Joseph. Some order zero, 569. remarks about sequences in context free Gershenson, H. H. Composition products, 136; languages, 113. Higher composition products. Preliminary Glauberrnan, George. Finite groups and operator report, 256. groups of relatively prime order, 2 51. Gerstenhaber, Murray. Cohomology and extensions Glick, A. J. See Carroll, R. W. of algebras, 568. Glicksberg, I. L. Some uncornplernented function Getoor, R. K. See Blumenthal, R. M. algebras, 93. Giaccai, G. J. See Hillman, A. P. Goldberg, Michael and Stewart, B. M. A dissec­ Giesy, D. P. A convexity condition on Banach tion problem for triangles, 455. spaces invariant under conjugation. Prelimin­ ary reportJ 665. Gonzalez-Fernandez, J. M. Tauberian theorems, 279. Giffen, C. H. Uncountably many inequivalent nearly tame arcs, 666. Goodman, R. W. One-sided invariant sub spaces and domains of dependence for hyperbolic Gilbert, R. C. Extremal spectral functions of a equations, 273. symmetric operator with unequal deficiency indices, 513. Gordon, Hugh. The space Baire functions and the bidual of the space of continuous functions, 589. Gilbert, R. C. and Kramer, V. A. Traceforrnulas for powers of a Sturm-Liouville operator, 114, ____ . See Comfort, W. W. Gilbert, R. P. Composition formulas in general­ Gordon, W. R. See Marcus, M. D. ized axially symmetric potential theory, 287; Gorowara, K. K. Director curves of a ruled sur­ Operators which generate harmonic functions face, 438. in three-variables, 288; On generalized hi­ axially symmetric potential theory, 506; In­ Grace, E. E. A characterization of the pseudo­ equalities for generalized axially symmetric arc, 85; On rnetrization and covering proper­ potential with entire and rnerornorphic asso­ ties on nowhere dense sets. Preliminary re­ ciates, 506. port, 256. Gil de Larnadrid, Jesus. Measures and tensors, Graham, Ronald. On a conjecture of Erdos, 55; 7 2.; Nuclear transformations and Schauder A theorem on partitions, 124; A property of bases, 447, Fibonacci numbers, 124. Gillman, D. S. and Martin, J. M. Countable de­ Granas, Andrzej. A fixed point theorem in linear compositions of E 3 into points and pointlike topological spaces, 83. arcs, 74. Granirer, Edmond. A theorem on amenable semi­ Gilmer, R. W. Jr. Integral domains which are groups, 76. almost Dedekind, 130; Rings in which every Gray, Alfred and Shah, S. M. A class of entire semi-primary ideal is primary, 130; Corn­ functions and a conjecture of Erdos, 77. mutative rings containing at most two prime ideals, 199; Extension of results concerning Gray, J. W. A special case of existence of adjoint rings in which semi-primary ideals are prim­ functors, 262. ary, 199; On a classical theorem of Noether in ideal theory, 295; The cancellation law for Greathouse, C. A. A theorem on tame ernbeddings ideals in an integral domain, 359; Finite rings of locally flat manifolds, 257; Tame cells in having a cyclic multiplicative group of units, locally flat manifolds, 509.

682 Green, L. W. An elementary proof of Mautnel''s Harris, J. K. Order structures for certain lemma, 656. acyclic topological spaces, 67. Greenblatt, Robert. Twisted characteristic classes Harris, W. A., Jr. and Sibuya, Yasutaka. Asymp­ of real n-plane bundles, 279. totic solutions of systems of nonlinear differ­ ence equations, 462; Note on linear difference Greenspan, Donald. On the numerical solution of equations, 574. problems allowing mixed boundary conditions, 92; The approximate solution of axially sym­ Harrison, D. K. See Chase, S. U. metric problems, 184; Approximate solution of Hartmanis, Juris and Stearns, R. E. On the com­ elliptic boundary value problems in three di­ putational complexity of algorithms, 504. mensions, 454. Hattemer, J. R. Boundary behaviour of solutions Greibach, S. A. Inverses of phrase structure to the heat equation, 510. generators, 369. Hayashi, Yoshio. Integral equations for electro­ Greiner, J. W. A special Pellian equation, 441. magnetic fields in anisotropic inhomogeneous Greville, T. N. E. A product characterization of media, 257. the generalized inverse of a singular square Hayes, D. R. The distribution of irreducibles in matrix, 4 72. the ring of polynomials over a finite field, 448. Griffiths, P. A. See Wolf, J, A. Heacock, L. D. On the numbers of one-"sided zeros Gross, Herbert. Normal subgroups of the ortho­ and one- sided identities of semigroups, 117. gonal group of an infinite dimensional vector­ Heath, R. W. Metrizability, compactness and space with a bilinear form, 441. paracompactness in Moore spaces, lOS; On Griinbaum, Branko. Projective graphs, 290; A metrization of normal separable Moore spaces proof of Rogers' conjecture on pairs of con­ and screenability and pointwise paracompact­ vex domains, 457. ness, 474; A non-pointwise paracompact Moore space with point-countable base. Preliminary Griinbaum, Branko and Motzkin, T. S. On polyhe­ report, 649. dral graphs, 134; The number of hexagons and the simplicity of geodesics on certain polyhe­ Heller, Alex and Reiner, Irving. On Grothendiek dra, 293. group of orders, 365. Gunn, J. E. and Lees, Milton. Linearly solvable Heller, Isidore. Representation of unimodular difference methods for quasilinear parabolic sets. I. Preliminary report, 442; Representa­ differential equations, 134. tion of unimodular sets. II. Preliminary re­ port, 507. Gwynn, J. M., Jr. Some relationships between stability and truncation error for a class of Heller stein, Simon and Rubel, L. A. A family of nine-point analogues of the one-dimensional subfield algebraically closed in the field of all heat equation. 647. meromorphic functions, 112. Helms, L. L. The Peron-Wiener method, 577. Haas, V. B. A stability result for a third order nonlinear differential equation, 200. Hempel, J. P. Extending a surface in E 3 to a closed surface, 191. Hahn, K. T. Bounds for the invariant B -areas of a family of surfaces in the space of two com­ Henderson, G. W. Proof that there does not exist plex variables, 119. an exactly two to one map on any n-ball, 349.

Hammer, P. C. Extended topology: Structure of Henriksen, Melvin and Jerison, Meyer. Minimal isotonic functions, 118; Extended topology: projective extensions of compact spaces, 68. Relative Wallace functions of a separation, 347; Extended topology: structure of associa­ Hermann, Robert. Vanishing of invariant differen­ tions, 473; Extended topology: Foundations of tial operators, 272. approximation theory, 663. Hersh, Reuben. Mixed problems of higher order Hamstrom, M.-E. The space ofhomeomorphisms in several variables, 184;Boundary conditions on a torus, 27 5; Regular mappings and homo­ for Lame's equations of elasticity, 467. topy in homeomorphism spaces, 366; Regular Herstein, I. N. and Small, L. W. Nil rings satis­ mappings and decompositions of product fying certain chain conditions, 662. spaces, 492. Hano, Jun-Ichi. On compact complex coset spaces Heuer, C. V. See Miller, D. W. of reductive Lie groups, 116. Hewitt, Edwin. See Edwards, R. E. Hardy, F. L. A note on torsion-free rings, 645. Hicks, N.J. Submanifolds, 137.

683 Hildebrand, S. K. and Sanderson, D. E. Connec­ 137; B(.7) -spaces and the closed graph theo­ tivity functions and retracts, 464. rem, 494.

Hillman, A. P., Forslund, D. W. and Giaccai, Hutcherson, W. R. and Frank, Stanley. Seven one G. J. Determinantal ideals with application to seven correspondence, 433. differential algebra, 104. Hyers, D. H. and Kyner, W. T. The existence of Himmelberg, C. J. On equiconnected spaces, 480. symmetric steady flows near critical speed, 456. Hinton, D. B. Two Stieltjes-Volterra integral equations, 438. Hyman, M. A. Data recovery. Preliminary report, 349. Hirschman, I. I., Jr. Extreme eigenvalues of cer­ tain Toeplitz operators, 514. Ingelstam, L. E. Real Banach algebras and their Hirzebruch, Ulrich. Jordan algebras and bounded regular groups, 458. symmetric domains, 572. Irwin, J. M., Peercy, Carol and Walker, E. A. Hobby, C. R. and Pyke, Ronald. A combinatorial p-heights of elements of Ext, 114. lemma for sequences of partial sums, 100. Irwin, R. C. Integral lattices in quaternion alge­ Hodges, J. H. Simultaneous pairs of linear and bras, 456. quadratic matrix equations over a finite field, Isbell, J. R. Envelope and boundary in injective 467. metric spaces, 450. Holland, S. S. Jr. A note on distributivity in Israel, Werner. Relativistic kinetic theory of a orthomodular lattices, 279. simple gas, 1!2. Hooper, P. K. Some small multi -tape universal Jacobowitz, Ronald. Multiplicativity of the local Turing machines, 584. Hilbert symbol, 78. Horne, J. G., Jr. ldempotents in semigroups on Jacobson, Bernard and Wisner, R. J. Factoriza­ a half-space, 646. tion of 2 X 2 unimodular matrices, 264; Fac­ Hosay, Norman. The sum of a cube is defined to torization of 2 X 2 singular matrices, 264. be the closure of the interior of a 2-sphere in Jaffa, R. E. Extremal doubly stochastic operators euclidean 3-space, 666. and measure preserving transformations. Pre­ Houh, C. S. Submanifolds in a Riemannian mani­ liminary report, 592. fold with general connections, 67. James, R. C. Weak compactness and separation, Howard, C. M. Finite dimensional analogues to 357; Weakly compact sets. 368; Nonsquare boolean algebras, 582. Banach spaces, 483

Howe, W. J. A new approach to Wittvectors, 117. Janowitz, M. F. Erratum to "A decomposition theorem for quantifiers on an orthomodular ____. See Maxfield, J. E. lattice", 303. Hsiang, W. C. and Szczarba, R. H. On the embed­ Jenner, W. E. On simple algebras with regular dability and nonembeddability of certain paral­ automorphisms, 445. lelizable manifolds, 2 57. Jerison, Meyer. See Henriksen, Melvin. Hsu, L. C. On a kind of extended Fejer-Hermite interpolation polynomials, 590; A kind of sto­ Johnson, B. C. Note on an inversion integral, 362. chastic approximating polynomials, 591. Johnson, D. G. The completion of an archimedean Hubbard, B. E. See Bramble, J. H. f-ring, 481.

Huebsch, W. M. and Morse, Marston. Topological Johnson, H. H. Bracket and exponential for a new groups of diffeomorphisms, 292. type of vector field, 95.

Hummel, J. A. The Grunsky coefficients of a Johnson, R. E. Distinguished rings of linear schlicht function, 82. transformations, 109.

Hunt, R. W. Boundary-value, separation, and Jolly, R. F. Concerning periodic subadditive oscillation properties of ordinary self-adjoint functions, 449. differential equations of arbitrary even order, Jones, B. F., Jr. Singular integrals and parabolic 68. equations, 257. Hurd, A. E. Counterexample to a proof of Shilov, Jones, B. W. Matrices associated with perfect 105; Measure space homomorphisms and point difference sets, 86. transformations, 280.

Husain, Taqdir. On almost continuous mappings, Jones, F. B. A fixed point free mapping of a plane

684 connected set, 437. Kesten, Harry. On the number of self-avoiding walks. Preliminary report, Z91; The discrep­ Jones, G. S. Variable translation operators and ancy of random sequences jkx}. 505. periodic solutions of functional equations, 191; A fixed point theorem for mapping on Killgrove, R. B. Completions of quadrangles in convex sets with deleted boundary points, projective planes, 54; Completions of quad­ 191; A fundamental inequality for generalized rangles in singly-generated planes} Volterra integral equations, 445; Fixed points Subplane counts in projective planes, 654, in gouged convex sets, 644. Kimura, Naoki. On projective semilattices, 484. Kafka, V. J. Axiomatics for partially ordered ____• See Basavappa, P. systems of multiplace functions. 664. Kincaid, W. M. A set of numerical methods for Kahn, P. J. Cobordism class and homotopy type. solving simultaneous equations, 75. Preliminary report, 508. Kinney, J. R. The convex hull of plane Brownian Kahr, A. S. and Wang, Hao.DegreesofRE models motion, 87. of AEA formulas. Preliminary report, 19Z. Kirchner, R. B. Composition and decomposition Kaiser, H. F. A method for determining eigen­ results for Fuchsian groups, 463. values, 3 7 I. Kister, J. M. Differentiable periodic actions on Kaniel, Shmuel. Dirichlet problem for constant­ FJ!3 without fixed points, Z95. coefficient homogeneous operators, 571. ----· See Bing, R. H. Kaniel, Shmuel and Schechter, Martin. The index Klamkin, M. S. On a characterization of circles for elliptic boundary value problems, 357; and spheres, 436. Spectral theory for Fredholm operators, 365. Klee, V. L. Rearrangements of series ofvectors, Kantorowitz, Shmuel. On the characterization of 77; On a theorem of Dubins, Z49; On a question spectral operators. II, 79; On the characteri­ of Bishop and Phelps, Z96; On a conjecture of zation of spectral operators of type k. Prelim­ Lindenstrauss, Z96; On a problem of Hirsch­ inary report, Z89. feld, Z96; A combinatorial analogue of Poin­ Kaplan, Samuel. The second dual of the space of care's theorem, 353; On the number of ver­ continuous functions. IV, 496. tices of a convex polytope 65Z; A property of d-polyhedral graphs, 669. Karst, Edgar. On the art of adding and subtract­ ing the same amount, Z4 7. ____. See Bing, R. H. Kasch, Friedrich. The recent development of -----·See Bonnice, W. E. Frobenius-extensions, 101. Kleisli, H. On injective sheaves, 460. Katti, S. K. Comparison of zero frequency and Kleppner, Adam. The topological algebra of mean in estimation for uniparameter families, infinitely differentiable functions, 104. 51. Klotz, T. S. More on the geometry of Teichmuller Katz, Marjorie, See Sklar, Abe. mappings, 1Z4; On the determination of con­ Kegley, J. C. A new example of a nonnilpotent formal imbeddings. Preliminary report, 585; nil ring, 509. The dilation of some standard mappings. Preliminary report, 585; On the harmonic Keisler, H. J. Intersections of PCa classes, immersion of surfaces in E 3• Preliminary 137, 377; Properties preserved under direct report, 586; On surfaces in E3 with constant factors, 30Z;Properties preserved under re­ negative curvature. Preliminary report, 671. tracts and strong homomorphisms, 37Z. Knight, F. B. A normal convergence theorem, 469. Kelman, R. B. Steady-state diffusion thru a cylinder into a reservoir: An exact solution, Knight, Frank and Orey, Steven. Construction of a 93. Markov process from hitting probabilities, 666 Kent, C. F. Functions which preserve recursive­ Knobloch, H: W. On periodic solutions of non­ ness of sets, 58. linear second-order differential equations, ____• See Fabian, R. J. 111. Knopp, M. I. and Smart, J. R. On Kloosterman. Keown, E. R. Preliminary results on Hartree­ sums associated with modular forms of half Fock calculations, 458. integral dimension, 199. Kerr-Lawson, A. A filter description of the Koehler, John. The type set of a torsion free homomorphisms of H

685 Kolodner, I. I. On some nonlinear boundary value Lancaster, Peter. On eigenvalues of matrices problems, 347. dependent on a parameter, 437. Konheim, A. G. See Adler, R. L. Lanckau, Eberhard. On the singularities of the solutions of elliptic differential equations, Konheim, A. G. and Rivlin, T. J. Extreme points 297. of the unit ball in the space of real poly­ nomials, 460. Lange, J. E. Limits of polynomials with zeros in a radial set, 62. Koppelman, Walter. Spectral multiplicity theory for a class of singular integral operators,l27; Lange, L. H. and White, C. M. Boundarybehavior On the index of elliptic operators on closed of arbitrary functions defined on the open unit surfaces, 303. disc, 430. Koranyi, Adam and Wolf, J. A. Realization of Langenhop, C. E. On the stabilization of linear Hermitian symmetric spaces as generalized systems, 251. half-planes, 458. Lakshmikantham, V. Properties of solutions of Korevaar, Jacob. Power series whose partial abstract differential inequalities, 97; On the sums have few zeros in an angle, 258. comparison between the solutions of ordinary differential systems, 258. Krabbe, G. L. Spectral permanence of scalar operators, 98; Weakly-continuous representa­ Lappan, P. A. Some sequential properties of tions of the multiplicative algebra (BV), 659. normal and non-normal functions with ap­ plications to automorphic functions, 506. Krajkiewicz, Paul. An abstraction of certain theorems on bases and dimension for vector Lawson, C. L. Characteristic properties of seg­ spaces to more general settings, 358. mented rational minimax approximation, 259. Krause, E. F. On the collection process, 250; Lawvere, F. W. The "group ring" of a small cate­ Groups of exponent eight satisfy the fourteenth gory. Preliminary report, 280, 516; Functorial Engel congruence, 301. automata theory, 477. Kreider, D. L. and Ritchie, R. W. Two theorems Leach, E. B. On a related function theorem, 186. on computability by two-way automata, 99. Lee, R. H. Optimum estimate for incremental Kruse, A. H. A notion of random sequence, 88. data and its electrical analogue, 270. Lees, Milton. See Conte, S. D. v. Krzywoblocki, M. Z. Bergman's method in linear ordinary differential equation of second ----· See Gunn, J. E. order with arbitrary coefficients, 59; Energy Lehmer, D. H. See Brillhart, J. D. principle in place of the Mach principle in new mathematical fundamentals of the relativistic Lehmer, Emma. See Brillhart, J. D. theories, 250. Leininger, C. W. Concerning a mean integral. Kulik, Stephen. Solution of two simultaneous equa­ Preliminary report, 594. tions, 441. Leland, K. 0. Topological analysis of analytic Kumpel, P. G., Jr. On the homotopy groups of the functions, 576; Unboundedness point sets of exceptional groups, 138. pointwise bounded collections of analytic func­ tions, 576. Kwun, K. W. Product of euclidean spaces modulo an arc, 502. Lepson, Benjamin. Note on the invariants of the Kwun, K. W. and Raymond, Frank. Sufficient Weierstrassian elliptic functions, 485. conditions that maps be acyclic, 435; Mapping LeVeque, W. J. On Weyl's criterion, 651. cylinder neighborhoods, 502. Levin, J. J. The asymptotic behavior of the Kyner, W. T. Orbits about an oblate planet. I, 478. solution of a Volterra equation, 73. See Hyers, D. H. Levin, J. J. and Nohel, J. A. Note on a nonlinear Volterra equation, 69; On a nonlinear delay LaBudde, C. D. A new algorithm for finding the equation, 446. eigenvectors of a real symmetric matrix, 446. Levine, J. P. Imbedding and isotopy of simply­ Lacey, Elton. A generalization of precompact and connected manifolds. Preliminary report, 370. compact operators in locally convex spaces. Preliminary report, 34 7. Levy, Azriel. Transfinite computability, 286; Independence results in set theory by Cohen's Lacey, Elton and Whitley, R. J. Conditions under method. I, 592; Independence results in set which all the bounded linear maps are compact. theory by Cohen's method. Ill, IV, 593. Preliminary report, 44 7.

686 See Feferman, Solomon. McAndrew, M. H. See Adler, R. L. Levy, L. S. Double direct decompositions over McArthur, C. W. and Retherford, J. R. Uniform Dedekind domains, 115. bases and the equicontinuity of projections associated with Schauder decompositions, 644. Libermann, Paulette. Example of a manifold which admits homogeneous complex and non­ McAuley, L. F. Lifting disks and certain light integrable almost complex structures, 194. open mappings, 120.

Lick, D. R. Sets of non-uniform convergence of McCarthy, C. A. An example in perturbation of Taylor series, 54. operators, 479.

Lin, B.-L. On a theorem of Klee, 491; Some topo­ McCarty, G. S., Jr. A relative Samelson product logical properties of infinite-dimensional Ban­ which yields a generalized Whitehead product, ach spaces, 492. 98; Homotopy normality of subgroups, 656. Lindenstrauss, Joram. Extension of operators McCord, M. C. Inverse limit sequences with with range in a C(K) space, 133; On norm covering maps as bonding maps, 499. preserving extension of operators. Prelimin­ McLaughlin, T. G. A semicreative weak decom­ ary report, 186; On the extension of operators position for certain r .e. sets, 132; On disjoint­ with a finite dimensional range, 373. ness of contraproductive centers, 201; Some Linton, F. E. J. Tensor products of Boolean remarks on semiproductive sets, 205; A note u-rings, 271; The dual of L 1 . Preliminary on contraproduction domains, 373; Remarks on report, 376. hyperimmune sets and Turing degrees, 432; On relative coimmunity. I, II. Preliminary Lipschutz, Seymour. An extension of Greenlinger' s report, 501; A theorem or two on quasi­ results on the word problem, 127; The two­ cohesive sets, 509; Answers to two questions generator metabelian group of exponent four, of Myhill, 592. 292. McMillan, D. R., Jr. A criterion for cellularity Lissner, David and Szczarba, R. H. Vector in a manifold, 499; The singular points of a bundles on lens spaces, 121. topological embedding, 588. Littman, Walter. See Weinberger, H. F. McWilliams, R. D. Iteratedw*-sequentialclosure Liverman, T. P. G. Vector-vector solutions of of a Banach space of functions in its second differential and implicit function equations. conjugate space, 649. Preliminary report, 274. MeW orter, W. A. See Abian, Alexander. Livesay, G. R. Involutions on the three-sphere, 182. Mack, J. E. The order dual of the space of Radon measures, 463. Livingston, A. E. Regularity of the generalized Lototsky transform, 184; Gibbs' phenomenon Mac Lane, G. R. On asymptotic values, 482. for some Hausdorff means, 204. MacLaren, M. D. Horizontal sums of orthocom­ Livingstone, Donald. See Cohn, J. A. plemented lattices, 63. Loeb, H. L. See Cheney, E. W. Mac Nerney, J. S. Note on successive approxima­ tions, 430; An integration-by-parts formula, Loeb, P. A. A new proof of the Tychonoff theo­ 581; Characterization of regular Hausdorff rem, 651. moment sequences, 643. Lomax, R. J. See Dolph, C. L. Magill, K. D., Jr. Methods of finding algebraic Lopez-Escobar, E. G. K. Universal classes in counterparts of topological statements. Pre­ the infinitary language Lw1w, 375. liminary report, 197; Some algebraic charac­ terizations of topological properties. Pre­ Lorch, Lee and Szego, P. A. Monotonicity of the liminary report, 67 5. difference of zeros of Bessel functions as a function of order, 96. Magill, K. D., Jr. and Mrowka, S. G. Subrings of C(X), 283. Loud, W. S. Isochronous oscillations in certain plane autonomous systems, 89. Maharam, Dorothy. Incompressible transforma­ tions, 584; Derivatives of incompressible Ludwig, D. A. On superpositions of distributions, transformations. Preliminary report, 584. 459. Mahowald, M. E. On the non stable homotopy of Luxemburg, W. A. J. Some properties of integrals SO{n), 360. on Riesz spaces, 655,

Lynn, I. L. Linearly orderable spaces, 261. Makky, S. M. Plastic flow and fracture in rotating,

687 circular disks of uniform thickness, 466. ----·F. T. See Diaz, J. B. Maltese, G. J. Spectral representations for some Michael, E. A. The product of a normal space unbounded normal operators, 97. and a metric space need not be normal, 258.

Mangasarian, 0. L. Equilibrium points of hi­ -----· See Corson, H. H. matrix games, 578. Michel ow, James. On the topological group of Mangasarian, 0. L. and Stone, H. Two-person a-adic numbers, 514; The topological ring of nonzero-sum games and quadratic program­ a-adic numbers, 514. ming, 511. Miller, D. W. and Heuer, C. V. An extension prob­ Marcus, Marvin and Gordon, W. R. Inequalities lem for cancellative semigroups, 452. for sub-permanents, 656. Miller, K. S. Distributions involving norms of Marcus, Marvin and Yaqub, A. M. Compound correlated Gaussian vectors, 500. matrix equations, 65. Miller, Keith. Three circle theorems for har­ Marie, Vojislav. On a method of analytic continu­ monic continuation, 109. ation of solutions of certain linear partial dif­ Mills, H. D. An error-free digital algorithm for ferential equations of the. second order, 198; systems of linear equations, 125; The analysis On a type of integral operators, 581. of round off errors in digital computation, 193. Marshall, A. W. See Barlow, R. E. Mills, W. H. Polynomials with minimal value sets, Martin, A. D. and Mizel, V. J, Characterization 280. of certain nonlinear functionals, 281. Mine, Henryk. On matrices with positive inverses, Martin, A. V. See Butler, Terence. 75; Permanents of (0,1)-circulants, 656. Martin, D. A. Simple sets with no maximal super­ Minty, G. J. An existence theorem foranonlinear set, 506. functional equation in Hilbert space, 55; On the resolvent of an antitonic operator in Hil­ Martin, Joseph. A rigid sphere, 189; Tame arcs bert space, 125; Another nonlinear form of the on disks, 434. Lax-Milgram lemma, 371; "Monotonicity" ----· See Gillman, D. S. methods in Banach spaces, 588. Martin, N. F. G. A topology for certain measure Mitchell, Theodore. Constant functions and left spaces, 351. invariant means on semigroups. Preliminary report, 508. Martindale, W. S. Lie isomorphisms of primitive rings, 282. Mizel, V. J, See Martin, A. D. Massey, W. S. Normal vector fields on manifolds. Moore, R. H. See Anselone, P. M. II, 362. Mordell, L. J. Equal products of two and three Mathews, H. T. Left and right boundary cluster consecutive integers, 198. sets inn-space, 250. Mordeson, J, N. and Vinograde, Bernard. Split­ Mattuck, A. P. and Mayer, A. P. The Riemann­ ting of commutative algebras with respect to Roch theorem for algebraic curves, 116. prime ideals, 97. Maxfield, J. E. and Howe, W. J. Common fixed Morel, A. C. Group-structural criteria for group points of commuting continuous functions on orderings, 266. the unit interval. Preliminary report, 646. Morgan, D. L. and Ostrom, T. G. Coordinate sys­ Mayer, A. P. See Mattuck, A. P. tems of some semi-translation planes, 69, Mayoh, B. H. Problems of arbitrary degree of Morris, G. R, An analyst's fixed-point theorem unsolvability in the theory of computable (with an application to differential equations), numbers, 507; Pseudo-Markov properties of 572, computable numbers, 507. Morrison, J. A. On the damping of a satellite Megibben, C. K. Completions of Abelian groups, motion, 437. 673; Multiples of pure subgroups, 674; Kaplan­ Morse, Marston, See Huebsch, W. M. sky's test problems for algebraically com­ pact groups, 674. Moser, J, K. A Harnack inequality for parabolic equations, 349. Merkel, R. B. The structure of finite semigroups and the associated system of left ideals, 117. Moser, Leo, On the additive completion of sets ofintegers, 139. Metcalf, F. T. A generalized Schwarz inequality, 83. Mott, J, L. Equivalent conditions for a ring to be

688 a multiplication ring, 431. Myers, D. E. On a matrix equation, 69; Quasi­ independent trials. Preliminary report, 451. ____.See Gilmer, R. W. Mott, T. E. Some generalizations of the Cantor Nakano, Hidegoro. See Brown, Leon. Lebesgue theorem, 59. Nakano, Hidegoro and Eisenstadt, B. J, Quasi­ Motzkin, T. S. Remarkable points of the triangle, bounded linear lattices, 66Z. 371. Nakano, Kazumi. Orthogonal spaces, 439, -----· See Griinbaum, Branko. N avot, Israel. The Euler-Maclaurin functionalfor Motzkin, T. S. and Straus, E. G. Coverings of sets functions with certain types of irregularities, by linear combinations of boundary points, 1Z9; 451. Coverings of sets by sums of transforms of Nerode, Anil. A decision method for p-adic inte­ boundary points, 1Z9. gral zeros of Diophantine equations, Z69. Mount, K. R. Characteristic classes and Fitting's Neuberger, J, W. A quasi-analyticity condition, invariants. Preliminary report, 187. 53, Moursund, D. G. Chebyshev approximations of a Neuwirth, L. P, A topological classification of function and its derivatives. Preliminary re­ certain 3-manifolds, 18Z. port, Z59. Nilson, E. N. See Walsh, J, L. Mrowka, S. G. On normal metrics, Z81; Rings of integer-valued continuous functions, 481; Lat­ Nishiura, Togo and Waterman, Daniel. Reflex­ tice-homomorphisms of lattices of continuous ivity and summability, 1ZO. functions, 571. Niven, Ivan and Zuckerman, H. S. On the cover­ Mull a, F. J. See Aronszajn, Nachman. ing of lattice points, 354. Mullin, A. A. Topics on mutation, V. Anti-ideals Nohel, J, A. See Levin, J, J, of semigroups, 1Z8; Structure of anti-ideals Norris, D. 0, A topology for Mikusinski opera­ of semigroups, 131; Topics on mutation. Vll. tors, 364. Group structure, 19Z; Subdirect product of certain semigroups, ZOO; Topics on mutation. van Norton, R. N. A corollary on rapidly in­ Vlll. Logical matters, Z04; Upper semi­ creasing kernels, 586. lattice of degrees of properness, Z05; Nunke, R, J. On the structure ofTor, 100. Another look at the fundamental theorem of arithmetic. Preliminary report, Z87; Remarks on another use for the fundamental theorem Oehmke, R. H. and Sandler, Reuben. Thecollinea­ of arithmetic, Z87; Topics on mutation. IX. The tion groups of division ring planes. I. Jordan logic of mutation, Z88; On the psi-function of division algebras, 374. elementary number theory, Z94; On conjugate Ogawa, Hajimu. The singular Cauchy problem pairs of number-theoretic functions. Prelim­ for a quasi-linear hyperbolic equation, 86. inary report, 357; Bounds with the distribution of square-free integers, 358; A simple new Ohm, J, E. See Gilmer, R. W. theorem of analytical number theory, 363; Ohtsuka, Makato. Boundary limits of Dirichlet Remarks on mosaic law for unique factoriza­ finite functions, 350, tion domains, 367; Simply tessellated mosaics, 376; Topics on mutation. X. Proofs for unique O'Keefe, E. S. On strict independence in the weak factorization systems, 375; The fundamental sense, 100. theorem of arithmetic and the theorem of Olive, Gloria, Erratum to "Polynomials defined Schnirelmann, 573; The fundamental theorem by generalized powers", 303; Some relation­ of arithmetic has only three inequivalent forms, ships between generalized powers and Stirling 673; Recursive unsolvability of the decision numbers, 433. problem for models of the fundamental theo­ rem of arithmetic, 673. O'Neil, P, V, The number of trees in a certain network, 569. Mundt, Marvin. See Wright, F. M. Orey, Steven. A ratio limit theorem for Markov Muses, C. A. A new property of the Bernoulli chains, 78. numbers, Z67, 377; Erratum to "The dimen­ sional continuum in Hilbert space" and "Hy­ ____• See Knight, Frank. perspheres and dimensionality", 377. Osofsky, B. L. A characterization of Noetherian Muskat, J. B. Integral criteria for prime power rings, Z73; A semiprimitive ring with nonzero residuacity, 94. singular, 357; Rings all of whose finitely gen-

689 erated modules are injective, 5 80. Pollak, Barth. Transitivity in the commutator subgroup of the orthogonal group. 103. Osserman, Robert. On the total curvature of complete minimal surfaces, 487; On complete Port, S. C. Some asymptotic properties of sums minimal surfaces, 111, 595; Minimal surfaces of random variables, 82; On random walko with in En, 658. a reflecting barrier, 443. Ostrom, T. G. See Morgan, D. L. Porter, G. J. Higher order Whitehead products distinguished by cohomology, 448. Outcalt, D. L. An extension of the class of alter­ native rings, 440. Posner, E. C. See Rumsey, H. C. Owens, 0. G. Uniqueness of the integral value Pour-el, M. B. Godel numberings versus Fried­ problem for the Helmholtz equation, 93. berg numberings, 201. Preston, G. B. Matrix representations of inverse Paine, D. M. Some grouplike properties of semigroups, 369. partitionsystems. Preliminary report, 259. Price, Thomas. Cellular decompositions of E 3, Pall, Gordon. Integral transformations of binary 661. quadratic forms, 52. Prosser, R. T. On the existence of certain quan- tum fields. II, 86; A new formulationofpartial ____. Cheema, M.S. mechanics, 486. Papadopoulos, Michael. Radiative singularities Putnam, Hilary. A hierarchy containing all prov­ and wave propagation, 106;Diffraction by a able ~~sets, 352. dielectric wedge, 194. Pyke, Ronald. See Hobby, C. R. Parrish, G. B. See Gergen, J. J. Passman, D. S. The group algebras of groups of Quintas, L. V. See Supnick, Fred. order p4 over a modular field, 490. Rabin, M. 0. and Wang, Hao. Words in the history Peake, E. J., Jr. On the determination of Serre of a Turing machine with a fixed input, 288. classes of Abelian groups. Preliminary re­ port, 266. Radlow, James. A two-dimensional version of Laplace's integral equation, 474. Pearcy, C. M. See Deckard, D. ].. Rahman, Q. I. Interpolation of entire functions, Peercy, Carol, See Irwin, J. M. 282. Pellicciaro, E. J. See Baxter, W. E. Rajagopalan, Minakshisundaram. The LP-conjec­ Penico, A. J. A matrix reduction method for ture for locally compact groups. Preliminary Maxwell's equations in anisotropic media, 94. report, 470.

P etryshn, W. F. On a class of k-p.d. and non­ Rall, L. B. See Anselone, P. M. K-p.d. operators, 283. Ralston, Anthony. Rational Chebyshev approxi­ Peyser, Gideon. Energy inequalities for hyper­ mation by Remez' second algorithm, 444. bolic equations with multiple characteristics, Ramsay, A. B. Dimension theory in complete 248. orthocomplemented weakly modular lattices, Pflugfelder, Hala. On ascending chains of 1r-loops, 465. 434. Rankin, Bayard. Inner products and countably in­ Phelps, R. R. Cebysev subspaces of finite co­ versed random variables, 188. dimension in C(X), 80; Extreme homomor­ Rapaport, E. S. The Poincare conjecture, 127; phisms of certain function algebras, 486. On the presentation of a group, 572; Groups of Pierce, R. S. See Beaumont, R. A. order 1, 669. Pilgrim, D. H. Engel conditions on groups. Pre­ Ratliff, L. J., Jr. Separably generated spots and liminary report, 260. affine rings over regular rings, 200. Pimbley, G. H. On sublinear operators in Hilbert Raymond, Frank. Local triviality for Hurewicz space with uniformly symmetrizable deriva­ fiberings of manifold, 100. tives, 653. ____.See Kwun, K. W. Pirani, F. A. E. Rigid motion in a gravitation field, 65. Reay, J. R. A note on positive independence, 354; A note on a theorem of Bonnice-Klee, 652. Pless, V. S. On Witt's theorem fornonalternating symmetric bilinear forms over a field of Reddy, William. An open cell of even dimension characteristic 2, 579. admits an expansive autohomeomorphism.

690 Preliminary report, 581. Rivlin, T. ]. See Konheim, A. G. Redheffer, R. M. Stability for mixed boundary­ Rivlin, T. ]. and Sibner, R. ]. The degree of ap­ problems, 363; The theorems of Bihari and proximation of certain functions of two vari­ Langenhop by Freshman methods, 490. ables by a sum of functions of one variable, 202. Redheffer, R. M. and Straus, E. G. On degenerate elliptic equations, 363. Roberts, H. M. On the Pontrjagin classes of a manifold, 129. Reich, Edgar. Sharpened distortion theorems for quasiconformal mappings, 81. Robertson, J. B. On the concordance of the Wold and Lebesgue-Cramer decompositions for a Reid, W. T. Principal solutions of nonoscillatory multivariate stationary stochastic processes. linear differential systems, 486. Preliminary report, 251. Reilly, R. C. On rings with composition 663. Robertson, M. S. Some radius of convexity prob­ Reiner, Irving, See Heller, Alex. lems, 135. Reinhardt, H. E. The relevance of observation in Robinson, D. W. On a relation between the period simple dichotomies, 96. and the restricted period of a linear recurrent sequence. Preliminary report, 657. Reinhart, B. L. Cobordism and the Euler number, 274. Robinson, T. T. Disjunction under implication in the intuitionistic predicate calculus Pd, 197; Rejto, P. A. On gentle perturbations. I, 246; On The condition AlA and recursive realizability gentle perturbations. Preliminary report,350. -(A 1-) in intuitionistic arithmetic N, 198; ____ . See Conley, C. C. The condition rl A and recursive realizability -( rj-) in intuitionistic arithmetic N, 288; The Retherford, ]. R. Orthogonal projections and un­ condition I' I A and recursive realizability-( !'f-) conditional bases, 360. in intuitionistic arithmetic N. II, 293; Wffs A ____.See McArthur, C. W. such that not AlA, 298; Kleene' s condition AlA is nontrivially weaker than Harrop's con­ Reynolds, W. F. Principal indecomposable charac­ dition RH(A) in intuitionistic arithmetic N, 299. ters and normal subgroups, 512; A generaliza­ tion of Brauer characters, 512; Projective re­ Rockoff, M. L. On the numerical solution of finite presentations of finite groups in cyclotomic difference approximations which are not of fields, 513. positive type, 108. Rice, ]. R. A moment problem, 368; On the exist­ Rodabaugh, D. ]. A generalization of antiflexi­ ence of best Tchebycheff approximations by bility. Preliminary report, 194. general rational functions, 576. Rolwing, R. H. On a system of quadratic equations Rice, P. M. Homotopically homogeneous spaces. and its integral analogue, 465. Preliminary report, 260. Rosen, R. H. Neighborhoods of trees in triangu­ Riedl, J. 0., Jr. An extension property of some lated manifolds, 131; A class of pathological partially ordered spaces, 493; Isomorphisms euclidean involutions, 192. of partially ordered topological linear spaces, _____ . See Brown, Morton. 493. Rosen, M. I. Representations of twisted group Rieger, G. ]. On the representation of square­ rings, 502; Projective module:; over non­ free numbers as the sum of a prime and the Abelian groups of order pq, 502. square of a square-free number, 196; On the number S(n) of representations of a natural Rosenberg, Alex. See Chase, S. U. number n as the sum of a square-free number Rosenberg, Milton. The square-integrability of and the square of an odd prime, 196; On a matrix-valued functions with respect to a system of linear equations involving primes, non-negative hermitian measure, 24 7. 205; On the differences of three consecutive primes, 292; On the sum of arbitrary and the Rosenfeld, Azriel. A note on two special types of difference of consecutive primes, 295; A note rings, 123; A note on matrix quadratic resi­ on Eckford Cohen's corollary of the Goldbach dues, 123. conjecture, 366; On the representation of Rosenzweig, S. M. On a version of Witt's theo­ algebraic integers as the sum of three squares, rem, 201. 370; On linked representations of pairs of integers as sums of squares, 373; On there­ Ross, K. A. See Comfort, W. W. presentation of integers as the sum of two Rota, G.-C. A cross-cut theorem for Mobius squares and a small kth power, 511. functions, 494; Mobius systems, 495; Mobius Ritchie, R. W. See Kreider, D. L. systems defined by maps and partitions, 495.

691 Rota, G.-c. See Furcht, Roberto. Sanchis, L. E. A converse of the elimination theorem, 131. Rothenberg, . Melvin. The J functor and the non­ stable homotopy groups of the unity groups, Sandberg, R. T. On the compatibility of the uni­ 121. form integral, 265, Rovnyak, J, L. An elementary proof ofBeurling's Sanders, B. L. A characterization of reflexivity, theorem. Preliminary report, 662. 72. Roy, Prabir. Failure of equivalence of dimension Sanderson, D. E. A-metrizability: A peculiar concepts for metric spaces, 81. generalization of semi-metrizability, 59. Rozycki, E. P. On Egoroff's theorem, 442. See Hildebrand, S. K. Rubel, L. A. A Fourier series method for entire Sansome, F. J, Combinatorial functions andre­ functions, 577. gressive isols, 297; Infinite sums of isolic integ.er s, 50 3. _____ . See Hellerstein, Simon. Sarason, Leonard. On weak and strong solutions Rubel, L. A. and Shields, A. L. Bounded approxi­ of boundary value problems, 102. mation by polynomials, 488, Sa:r;d, Arthur. Highly critical sets, 571. Rubin, Herman, A generating function for trees and connected graphs, 90; Almost Lindelof Sario, Leo, Complex analytic mappings, 181; measures on paracompact spaces, 472, An integral equation and a general existence theorem of harmonic functions, 370; Principal Rudin, M. E. Interval topology in subsets of totally functions and tangent bundles in general value orderable spaces, 449. distribution theory, 435; Proximity functions Rudin, Walter. ldempotents in groups algebras, on Riemann surfaces of infinite genus, 498; 137; Prime numbers and measurable func­ Gaussian mapping of arbitrary minimal sur­ tions, 300, faces, 498; Extremal harmonic functions of several variables, 668. _____ .See Schneider, Hans. Sather, Duane. Maximum properties of Cauchy's Rudin, Walter and Schneider, Hans, Idempotents problem in three-dimensional space-time, in group rings. I, 459. 665 Sato, Daihachiro. On the rate of growth of an en- Ruh, E. A. On the automorphism group of a geo­ tire function of infinite order, 348; Transcen­ metric structure. Preliminary report, 497. dental entire function which together with all Rumsey, H. C, A decision procedure for a class its higher derivatives assumes algebraic of diophantine equations, 264. values at all algebraic points, 579. Rung, D. C. The order of certain real-valued Saunders, S. C. See Crawford, G, B. functions defined in the unit disk, 57. Saworotnow, P. P, On existence of involutionona Rung, D. C. and Cima, J, A. Some theorems on certain Banach algebra,l90; On the condition cluster sets for normal functions. Prelimin­ of continuity of multiplication in an H*-algebra, ary report, 488. 195. Ryff, J, V. On the representation of doubly sto­ Scanlon, J, C. Lyapunov stability and periodic chastic operators, 89. solutions, 570, Schaal, W. G, On the expression of a number as Sacks, G, E. A simple set which is not effectively the sum of two squares in totally real alge­ simple, 61; Minimal upper bounds for se­ braic number fields, 660. quences of degrees, 268, Schafer, R. D. On forms of degree n permitting Sacksteder, R. C. Foliations of co-dimension one, compositions, 55. 298. Schechter, Martin. On estimating partial differ en­ Sade, Albert. Semi -automorphismes de groupo ides tial operators. V, 578. et de quasigroupes, 126; Isotopies d'un ____ • See Kaniel, Shmuel. groupoide avec son conjoint, 364. Schlesinger, E. C. Notes on Daniell integration, Sagle, A. A. On simple extended Lie algebras 120, over fields of characteristic zero, 7 5; Re­ marks on simple extended Lie algebras, 568. Schneider, Hans and Rudin, Walter. ldempotents in group rings. II, 460. Salzer, H. E. Numerical integration employing "overdifferentiation," Some simple experi­ and spectral ments, 271; Divided differences for functions Schreiber, Morris, Numerical range of two variables for irregularly spaced argu­ sets, 105. ments, 660. Schubert, S. R. See Day, G. W. 692 Schiitzenberger, M. P. and Sherman, Seymour. On Sibner, Lesley. A Tricomi problem for an equa­ a formal product over the conjugate classes tion of abruptly changing type. Preliminary of a free group, 661. report, 587. Schwartz, A. J. Omega-limit sets without singular Sibner, R. J. See Rivlin, T. J. points on closed 2-manifolds, 186; The Poin­ Sibuya, Yasutaka. Simplification of a linear or­ care-Bendixson theorem for two manifolds, dinary differential equation of the nth order 358. at a turning point, 102. Schweizer, Berthold. Equivalence relations in statistical metric spaces, 265. ----· See Harris, W. A., Jr. Sills, W. H. Arens multiplication and spectral Scroggs, J. E. A Frckhet space associated with theory, 6-54, vector valued functions, 487. Simon, Hermann. Characterization of the class of Segal, Jack. Perivarieties, 73; Inverse dimension finitely generated groups with finite hyper­ type. I. Types in the real line 645 center group, 202. Segel, L. A. See Davis, S. H. Sklar, Abe and Katz, Marjorie. Approximate func­ Seibert, Peter. On prolongations in dynamical tional equations for a class of Dirichlet series, systems, 281. 115. Seiden, Esther. On a method of construction of a Small, L. W. See Herstein, I. N. class of finite Bolyai-Lobochevsky planes, Smart, J. R. See Knopp, M. I. 577. Smith, K. T. See Adams, R. D. Seidman, T. I. Two-level difference scheme, 90. Smithson, R. E. Change of topology and fixed Selden, John, Jr. A note on compact semirings. points for a class of multi-valued functions, Preliminary report, 260; Multiplicatively left 444. zero-simple semirings. Preliminary report, 471. Smuckler, A. M. On the algebraic elements of a commutative Banach algebra, 498. Selfridge, J. L. Pairings of the first 2n integers so that sums and differences are all distinct, Snapper, Ernst. Cohomology of permutation re­ 195; Covering sets of congruences, 348. presentations. I. Spectral sequences, 454. Sell, G. R. Topological classification of bounded Snyder, A. K. On a definition for conull and core­ solutions. Preliminary report. 459. gular FK spaces, 183. Senechalle, L. J. A functional calculus for uniform Snyder, L. E. OnfunctionsofBaireclassone, 283. operators on a reflexive Banach space, 133. Sobczyk, Andrew. Functional characterization of Seshu, Lily. A Taylor expansion for functions with retracts, 78; Projections, retractions, and real part positive in the right-half plane, 260. C*-embedding, 134. Shah, S. M. See Gray, Alfred. Solomon, A. D. The minimal surface problem of three sheets, 472. Shanks, Daniel. The second-order term in the asymptotic expansion of B{x), 261, 377. Solomon, Louis, A fixed point formula for the classical groups over a finite field, 452. Shanks, E. B. Solutions of differential equations by evaluations of functions, 480. Solovay, Robert. Independence results in the theory of cardinals. I, II. Preliminary report, Sharma, Ambikeshwar. See Frank, Evelyn. 595. ____ . See Walsh, J. L. Sorgenfrey, R. H. See Franklin, S. P. Sharma, Ambikeshwar and Varma, A. K. On Soysal, Selma. Characterization of a class of trigonometric interpolation, 434. linear operators by generalized character­ Sherman, Seymour. A third-order nonlinear sys­ istic vectors, 203. tem arising from a nuclear spin generator, 185. Spanier, E. H. See Ginsburg, Seymour. ____.See Schiitzenberger, M.P. Spangler, C. B. Theorems on the core of a Sherman, T. L. Properties of solution of Nth order bounded generalized sequence on a Banach linear differential equations, 442. space, 488. Shields, A. L. See Rubel, L. A. Spencer, D. E. The Holor representation of alternating currents, 475. Shockley, J. E. On the best bound for the solva­ bility of a linear diophantine equation, 271. Springer, George. See Deeter, C. R.

693 Stackelberg, 0. P. A new law of the iterated Stubblefield, Beauregard. en curves in Euc logarithm, 252. n-space, 60. Stampacchia, G. See Weinberger, H. F. Stueben, E. F. Ideals in two-place tri-opera algebras, 663. Stampfli, J. G. Sums of projections, 493; A hypo­ normal operator with spectrum on an arc is Su, J. C. Periodic transformations on the p: normal, 582. of two spheres, 276. Stanek, P. F. G. A class of groups related to Subbarao, M. V. On Cohen's trigonometric LF(2,q), 246. 119; Component congruences for a cl: divisors, 661. Stanley, R. L. Another condition for groupoid factorability, 354; A note on axioms for some Sucheston, Louis. A condition for weak mb right. algebras, 37 5; Natural deduction, infer­ transformations, 79. ence, and consistency, 469. Sugar, A. C. A new law of universal gravi: Starr, Norton. A limit theorem for positive con­ 272. tractive linear operators. Preliminary report, Sullivan, Michael. Physical systems Sk 372; A convergence theorem on spaces of elastic field of force, 298. infinite measure. Preliminary report, 376. Supnick, Fred and Quintas, L. V. Extreme lc Stearns, R. E. See Hartmanis, Juris. tonian circuits. Resolution of the conv• Stein, S. J. Discrete full sets, 136. case, 185. Stein, S. K. See Chakerian, G. D. Suryanarayan, E. R. The geometry of Be: flows, 477. Steinberg, Arthur. On equations in a free group, 84; On free nilpotent quotient images of Swann, D. W. A canonical form for an arb single defining relation groups, 13 8. square complex matrix, 456. Stengle, G. A. Asymptotic solution of a class of Szczarba, R. H. See Hsiang, W. C. second order differential equations containing ____.See Lissner, David. a parameter, 572. Szego, P. A. See Lorch, Lee. Sterling, D. J. On a covering group defined by C. W. Curtis, Preliminary report, 50; Cover­ Szeptycki, Powel. See Aronszajn, Nach ings of algebraic groups and Lie algebras of classical type. 433. Taft, E. J. Orthogonal conjugacies in assoc Stevens, H. R. Some congruence properties ofthe algebras, 182; Symmetries of Lie alg• Hermite polynomials, 594. 439. Stewart, B. M. See Goldberg, Michael. Taibleson, M. H. Smoothness of Fourier : Stokes, R. A. An integral of a function with re­ forms, 669. spect to a function in a rectangular interval, Tamura, Takayuki. Commutative divisible 444. groups, 87; Operations on relations anc Stoll, W. F. The area of an algebraic set, 439. applications to groupoids, 482;Certain e ding problems of semigroups, 658. Stone, H. See Mangasarian, 0. L. Tarski, Alfred. The elementary undecidabi Stoneham, R. G. The reciprocals of integral pure transcendental extensions of real ' powers of primes and normal numbers, 183. fields, 355. Storvick, D. A. Analytic functions locally of class Tauer, R. J. Maximal abelian subalgeb1 (U*), 274. finite factors of type II, 511. Straus, E. G. See Motzkin, T. S. Taulbee, 0. E. A generalized binary re: ----·See Redheffer, R. M. Preliminary report, 475. Strodt, Walter. Remark on partial orders under Taussky, Olga. Matrices whose powers coi which differentiation is stable, 589. to zero, 454. Stromberg, K. R. Absolutely continuous meas­ Taylor, J. L. The Tomita decomposition of ures on semigroups, 109. of operators, 470. Strother, W. L. Simple separable switching sets, Taylor, W. C. A generalization of the v 94. integral, 461.

Struble, G. W. Orthogonal polynomials: Variable­ Thampuran, D. V. Extended topology: 1\, signed weight functions, 73. Smith convergence, 60.

694 Thielman, H. P. See Dean, W. C. for systems of conjugate harmonic functions. Preliminary report, 474. Thompson, R. C. Unimodular group matrices, 368. Walker, C. P. Properties of Ext and quasi­ Thoro, Dmitri. A property of consecutive integers, splitting of Abelian groups, 267. 512. Walker, E. A. See Irwin, J. M. Thorp, E. 0. On the conjugate of L 1, 88. Walkup, D. W. On a result of Heineken. Prelimin­ Tillmann, H. G. Vectorvalued distributions and ary report, 578. spectral decomposition of unbounded opera­ tors, 195; Remarks on the multiplication of Walsh, J. L. Extremal polynomials and the zeros distributions, 246. of the derivative of a rational function, 359; The convergence of sequences of rational Todd, 0. T. See Taussky, Olga. functions of best approximation, 513; The con­ Trampus, Anthony. On linear matrix equations, vergence of sequences of rational functions of 481. best approximation. II, 588. Traub, J. F. An iteration function of incommen­ Walsh, J. L., Ahlberg, J. H. and Nilson, E. N. surate order, 4 71. Best approximation and convergence proper­ ties of higher-order spline fits, 202. Traylor, D. R. On matrizability of locally com­ pact normal Moore spaces, 455. Walsh, J. L. and Sharma, Ambikeshwar. Least­ square approximation and interpolation in Treybig, L. B. Concerning continua which are roots of unity, 491. continuous images of compact ordered spaces. 674. Walter, G. G. Some orthogonal expansions of Tully, E. J ., Jr. A class of commutative semi- distributions, 70. groups having a group-like property, 57. Wang, C. L. See Carroll, R. W. Turyn, R. J. Character sums and difference sets, 482. Wang, Hao, Remarks on implicational calculi, 193; A universal axiom of conditional set Tyndall, W. F. A duality theorem for a class of existence, 588; Natural hulls and set exist­ continuous linear programming problems, 362. ence, 594; A genetic definition of the class On of ordinals, 669. Ullian, Joseph. See Ginsburg, Seymour. See Kahr, A. S. Ullman, J. L. The zero distribution problem for Faber polynomials and Tchebycheff quadra­ See Rabin, M. 0. ture, 462. Wang, H. C. See Boothby, W. M. Unni, K. R. Hankel transforms and entire func­ Warne, R. J. Homomorphisms of d-simple in­ tions, 261; Weighted entire functions, 282. verse semigroups with identity. I, 567; Homo­ Utumi, Yozo. See Faith, Carl. morphisms of d-simple inverse semigroups with identity. II, 643. Varma, A. K. See Sharma, Ambikeshwar. Warten, R. M. Automatic step size controls for Vasquez, A. T. See Charlap, L. S. Runge Kutta integration, 438. Vaught, R. L. Elementary classes of models Was ow, W. R. A singular perturbation problem closed under descending intersection, 126; for systems of two analytic differential equa­ Indescribable cardinals, 126. tions, 66. Vermes, Paul. Multiplicative groups of row-and/ Waterman, Daniel. See Nishiura, Togo. or column-finite matrices, 123. Webb, D. L. A set of postulates for an algebra on Vesley, R. E. The intuitionistic continuum, 81. three values, 480. Vinograde, Bernard. See Mordeson, J. N. Weil, C. E. On properties of derivatives, 67. Vinson, R. G. Area in a noneuclidean geometry. Weill, G. G. Geodesic circles on complete C00 Preliminary report, 648. two dimensional Riemannian manifolds, 117. Vobach, A. R. On two-dimensional continua struc­ Weinberg, E. C. Free lattice-ordered abelian tured by finite families of simple closed groups. II, 668. curves, 486. Weinberger, H. F., Littman, Walter and Stam­ Volkmann, B. W. A metric problem on transcen­ paccia, G. Regular points for elliptic equa­ dental numbers, 94. tions with discontinuous coefficients, 99. Wainger, Stephen. A variant of Schwarz's lemma Weinstein, Alexander. On lower bounds for the

695 eigenvalues in the Sturm-Liouville theory, 500. Wilker, Peter. On maximal decompositions of linear maps, 72. Weinzweig, A. I. Homotopy classification of fibre bundles, 115. Williams, R. F. Anamolous group actions and dimensionally deficient spaces, 61. Weisfeld, Morris. The convergence of an approxi­ mation used in the relativistic mass-center Williams V. C. On conformal maps of regions of problem, 350. infinite connectivity, 101. Weiss, M. C. On symmetric differentiation, 580. Williamson, J. A. A renewal theorem for inde­ pendent random variables, 70. Weiss, M. L. Cluster sets of bounded analytic functions, 96. Wing, G. M. Invariant imbedding and the phase shift problem, 128. Weiss, M. L. and Woolf, W. B. Boundary behavior of pseudo-meromorphic functions, 85. Winter, D. L. Finite groups with a faithful re­ presentation. Preliminary report, 300. Weiss, Paul. An estimate of the error arising from misapplication of the sampling theorem, Wisner, R. J. See Jacobson, Bernard. 351;An integration formula for interpolation Witz, K. G. A theorem on right amenable sub­ polynomials, 484. spaces. Preliminary report, 290. Wells, James. On function pairs related by a Wolf, J. A. Isotropic manifolds of indefinite Fourier-Stieltjes transform, 98; Multipliers metric, 56;The differentiable fibre bundle of ideals in function algebras, 469. associated to a complete map, 291, 516; Curva­ Wend, D. V. V. Branched regular curve families ture in nilpotent Lie groups, 291. and finite asymptotic values of analytic func­ _____• See Koranyi, Adam. tions, 655. Wolf, J. A. and Griffiths, P. A. Complete maps Wenel, J. G. Zero-free intervals of tied-down and differential coverings, 131. stable processes, 366. Woolf, W. B. See Weiss, M. L. Wermer, John. See Browder, Andrew. Wright, F. M. On the Stieltjes mean sigma inte­ Wersan, S. J. See Ben-Israel, Adi. gral and the Lebesgue-Stieltjes integral, 356. Westman, J. J. An inverse limit of Banach alge­ Wright, F. M. and Mundt, Marvin. On summability bras of continuous functions reducing to the methods ·for improper Lebesgue-Stieltjes inte­ constants, 139. grals. I, 188. White, C. M. See Lange, L. H. Wyler, Oswald. Categories of algebraic struc­ White, J. T. A representation theorem for the tures, 265. Laplace transform, 110. Yamamoto, Koichi, A p-adic identity on Bernoulli Whitley, R. J. Strictly singular operators and numbers, 451. their conjugates, 263. Yaqub, A. M. See Marcus, Marvin. _____ • See Lacey, Elton Yaqub, F. M. Extensions of Boolean algebras, 62. Whitlock, H. I. Abstract characterization of an algebra of multi-place functions. I., 664. See Dwinger, Philip. Whyburn, K. G. A representation theorem for Yoshizawa, Taro. Eventual boundedness of solu­ Mikusinski operators, 431. tions of a perturbed system, 86. Wiegmann, N. A. On irreducible representations Young, P. R. On reducibility by recursive func­ of finite groups of the second kind, 192. tions, 268; A bounded truth-table complete, pseudo-creative, noncylinder, 436; Notes on Wigley, N. M. Asymptotic expansions at a corner the structure of r.e. sets, 586. of solutions of mixed boundary value prob­ lems, 652. Yuan, C. Non-equilibrium hydrodynamics of a chemically reacting fluid,670. Wilansky, Albert, Summability of bounded se­ quences, 183; Divisors of zero and Tauberian theorems, 276; Distinguished subsets and Zimmerman, J. M. A class of improper boundary summability invariants, 463. value problems with" damped" Cauchy data, 56. Wilcox, C. H. An energy inequality for the reflec­ Zink, R. E. A classification of measure spaces, tion coefficient matrix of a pair ofnonuniform 479. coupled transmission lines, 118. Zuckerman, H. S. See Niven, Ivan. Wilf, H. S. On the zeros of Riesz' function in the Zvengrowski, P. D. See Bumcrot, R. ]. analytic theory of numbers, 474.

696 INDEX Volume 10, 1963

ABSTRACTS OF CONTRIBUTED PAPERS, 50, 181, Programs of Meetings 246, 347, 430, 566, 643 January: Berkeley, California, 5 February: New York, New York, 165 Index of Abstracts in Volume 10, 676 April: Chicago, lllinois, 217; University Errata 303, 377, 516, 595 Park, New Mexico, 222; New York, New York, 225 ACTIVITIES OF OTHER ASSOCIATIONS, 32, 171, June: Bellingham, Washington, 321 235, 415, 550 August: Boulder, Colorado, 393 ADVERTISERS, INDEX TO, 159, 211, 315, 387, October: Brooklyn, New York, 541 535, 603, 703 November: Atlanta, Georgia, 609; Pasadena, California, 613; Madison, Wisconsin, 619 DOCTORATES CONFERRED IN 1962, 327, 424 Supplementary Programs, 47, 178, 243, 344, 426, FEATURE ARTICLES 563, 641 Annual Salary Survey, 559 National Academy of Sciences-National Research MEMORANDA TO MEMBERS Council, 625 New NSF Policies and Their·Implementation, 627 Backlog of Mathematical Research Journals, 173, 423 New Translation Program, 36 Chairmen, 39 Notes for Speakers, 630 Repart of the Affairs of the Society, 233, 548 Combined Membership List, 414 Corporate Members, 172, 420 Starting Salaries for Mathematicians with a Ph.D., 562 Employment of Retired Mathematicians 172, 420, 629 ' Veblen Prize in Geometry, 629 Employment Register, 49, 239, 323, 429, 621 LETTERS TO THE EDITOR Mathematical Sciences Professional Directory, 323 J. L. Brenner, 176 National Register of Scientific and Technical E. T. Parker, 176 Personnel, 31, 429 A. D. Wallace, 242 Summer Employment Opportunities, 49, 642 Henryk Mine, 242 Summer Meeting-1964, 556 R. D. Purinton lll, 557 Hyman Gabai, 557 NEW AMS PUBLICATIONS, 38, 245, 342, 421, 551, David Gale, 558 636 MEETINGS OF THE AMERICAN MATHEMATICAL NEWS ITEMS AND ANNOUNCEMENTS, 37, 40, 167, SOCIETY 180, 221, 231, 238, 326, 341, 417,425, 547, 549, 556, 612, 618, 624 Abstracts of Contributed Papers, 50, 181, 246, 347, 430, 566, 643 Calendar of Meetings, 4, 164, 216, 320, 392, PERSONAL ITEMS, 42, 167, 174, 240, 340, 418, 552, 540, 608 638 Preliminary Announcements SYMPOSIA AND INSTITUTES February: New York, 30 April: Chicago, 168; University Park, 168; Summer Institute on Differential and Algebraic New York, 169 Geometry, 30 June: Bellingham, 232 Symposium in Applied Mathematics on Stochastic August: Boulder, 324 Processes in Mathematical Physics and October: Brooklyn, 413 Engineering, 31 November: Atlanta, 544; Pasadena, 545; Symposium on Recent Developments in the Theory Madison, 546 of Numbers, 413 January: Coral Gables and Miami, 622 VISITING FOREIGN MATHEMATICIANS, 37, 631 Another project in space kinetics at STL

How can the motion in space of satellites and interplan­ Computer Programmers, Numerical Analysts and etary vehicles be planned? STL is helping answer this Applied Mathematicians will find interesting assignments question. Its Computation and Data Reduction Center in space kinetics at STL's Computation and Data Reduc­ and Systems Research Laboratory arc performing space tion Center. For more information on Southern Califor­ kinetics work on most of the nation's major space pro­ nia openings, you are invited to contact Dr.. R. C. Potter, grams. Current projects include: Mercury, Gemini, One Space Park, Redondo Beach, California, Dept. 0-11. Ranger, Mariner, OGO, Relay and ICBM programs. STL is an equal opportunity employer.

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PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME XV HIGH SPEED COMPUTING AND EXPERIMENTAL ARITHMETIC Edited by N. C. Metropolis, A. H. Taub, John Todd, C. B. Tompkins

This volume contains all but two of the papers which were presented at two symposia sponsored by the American Mathematical Society and other co-sponsors in the Spring of 1962. The first symposium was held in Chicago, Illinois, on the subject of Experimental Arithmetic. The second symposium was held at Atlantic City, New Jersey, on the subject of Interactions between Mathematical Research and High-Speed Computing. The close relationship between the subject matters of the two symposia prompted the organizing committees to merge the proceedings into this single volume.

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