AMERICAN MATHEMATICAL SOCIETY

Notices Edited by J. H. CURTISS

.......................................................................... '" .................................................................................. " VOLUME 6, NUMBER 2 ISSUE NO. 37 APRIL 1959 •IHHIIIIIIIIIItllllltlllllllllllllllllllllllllltllltiiiiiiiiiiiiiiiiiiiiiiiiiJIIIIIIIIIAIIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIItlllllllll

Contents MEETINGS

Calendar of Meetings . . . . • . • . . . . . . . . . . . • . . . . 90 Program of the April Meeting in Chicago . . . • . . . . . . 91

Abstracts for Meeting, pp. 147-161 Program of the April Meeting in Monterey . . . . . . • . • 98

Abstracts for Meeting, pp. 162-174 Program of the April Meeting in New York . . .• 105

Abstracts for Meeting, pp. 17 5-196

PRELIMINARY ANNOUNCEMENT OF MEETING .. 116

NEWS ITEMS AND ANNOUNCEMENTS .....•.......... 117

PERSONAL ITEMS . . . • . . . . . . .........•......... 121

LETTERS TO THE EDITOR ......•.........•.

MEMORANDA TO MEMBERS

129

New Policy Concerning Authors' Royalties ......... 132 Preliminary Information Concerning Summer Meeting

in Salt Lake City •.........•.......•....•.. 133 1958-1959 Combined Membership List ...........• 135

ABSTRACTS OF CONTRIBUTED PAPERS .........•.•.• 137

RESERVATION FORM ............••.•.•......... 199

Please send in abstracts of papers to be presented in person well in advance of the deadline.

Published by the Society ANN ARBOR, MICHIGAN and PROVIDENCE, RHODE ISLAND

Printed in tile United States of America

MEETINGS

CALENDAR OF MEETINGS

NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press, The meeting dates which fall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned.

Meet ing No.

Date Place

558 June 20, 1959 Eugene, Oregon 559 September 2-5, 1959 Salt Lake City, Utah

(64th Summer Meeting) 560 October 31, 1959 Cambridge, Massachusetts

November 20-21, 1959 Winston-Salem, North Carolina November 27-28, 1959 Detroit, Michigan January 27-29, 1960 Chicago, Illinois

(66th Annual Meeting)

Deadline for

Abstracts*

May 7 July 20

Sept. 17

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

The NOTICES of the American Mathematical Society is published seven times a year, in February ,April, June,August, October, November, and Decem­ber. Price per annual volume is $7.00. Price per copy, $2.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (none available before 1958), and in­quiries should be addressed to the American Mathematical Society, Ann Arbor, Michigan, or 190 Hope Street, Providence 6, R, I.

Second-class postage paid at Ann Arbor, Michigan. Authorization is granted under the authority of the act .of August 24, 1912, as amended by the act of August 4, 194 7 (Sec. 3421, P. L. and R .) • Accepted for mailing at the special rate of postage provided for in section 34-40, paragraph (d).

Copyright, American Mathematical Society, 1959.

90

FIVE HUNDRED FIFTY-FIFTH MEETING

University of Chicago Chicago, Illinois April17-18, 1959

PROGRAM

The five hundred fifty-fifth meeting of the American Mathemati­cal Society will be held at the University of Chicago, Chicago, Illinois, on Friday and Saturday, April 17-18, 1959. All sessions will be held in Eckhart Hall.

Registration will be in the Common Room on the second floor of Eckhart Hall, beginning at 9:00A.M., Friday.

By invitation of the Committee to Select Hour Speakers for Western Sectional Meetings, Professor I. I. Hirschman of Washing­ton University and Professor S. A. Amitsur visiting the University of Notre Dame from Hebrew University will address the Society. Pro­fessor Amitsur's lecture will be held in Room 133 at 11:00 A. M., Friday, April 17. His title is "Pivotal monomials and polynomial identities of rings". Professor Hirschman will speak on "Multiplier ransformations" in Room 133 at 2:00P.M. on Friday, April 17.

Sessions for the presentation of contributed papers will be held at 9:45A.M. and 3:15P.M. on Friday and 10:00 A.M. on Satur­day. Abstracts of the papers to be presented at these sessions ap­pear on pages 137-198 of these NOTiCES. There are cross refer­ences to the abstracts in the program. For example the title to paper (1) in the program is followed by (555-30) indicating that the abstract can be found under the designation "555-30" among the published ab­stracts.

If necessary there will be a special session Saturday afternoon for the presentation of papers which failed to meet the deadline. Fur­ther details will be available at the registration desk.

There will be a tea in the Common Room of Eckhart Hall start­ing at 4:15P.M. on Friday.

The facilities of Hutchinson Commons, a dining hall directly across from Eckhart Hall, will be available to members of the So­ciety and guests for all meals.

The following hotels have agreed to accommodate those mem­bers of the Society making reservations in advance:

91

In the University District

Single Double Shoreland Hotel

5454 South Shore Drive

Del Prado Hotel 5307 South Hyde Park Blvd.

Hotels Windermere 1642 East 56th Street

Hotel Broadview 5400 South Hyde Park Blvd.

Hotel Miramar 6218 South Woodlawn

Hyde Park Y. M. C. A. 1400 East 53rd Street

$7.00 up $9.00 up

7 .00- 11.00 9 .oo- 13 .oo

6.50- 8.50 8,50-11.00

4.00- 6.00 6.00- 8.00

3,50- 5.50 5,00- 7.00

2,25

In the Loop District

The Conrad Hilton 6.50-10.00 12.00 up

The prices listed above are subject to change. Reservations should be made directly with the hotel.

PROGRAM OF THE SESSIONS

The time limit for each contributed paper, as in the past, is ten minutes. However, the Society is trying an experiment at this meeting to make it easier for people to get to the papers they wish to hear. Thus, it will be noted that in the program below papers are planned at fifteen-minute intervals; ten minutes for the presenta­tion and five minutes for discussion and intermission. For this plan to have any chance of success, speakers must be prepared to adhere rigidly to the ten-minute limit.

FRIDAY, 9:45A.M.

Session on Analysis, Room 133 9:45 - 9:55

(1) Generalization of a theorem of Loewner Mr. Adam Koranyi, University of Chicago (555-30)

10:00 - 10:10 (2) A note on metric density of sets of real numbers

Dr. N. F. G, Martin, Iowa State College (555-6)

92

10:15 - 10:25 (3) On the convergence of infinite exponentials

10:30 - 10:40

Mr. D. L. Shell, University of Cincinnati (Introduced by Professor j. W. T. Youngs)

(4) On absolutely convergent exponential sums. II Dr. Leon Brown, Wayne State University, Professor A. L. Shields, University of Michigan, and Dr. Karl Zeller, University of Tubingen (555-25)

Session on Topology, Room 206 9:45 - 9:55

(5) Some compact product spaces which cannot be imbedded in Euclidean n-space

10:00 - 10:10

Mr. Beauregard Stubblefield, University of Michigan (Introduced by Professor G. S. Young)

(6) A decomposition of 3-space that cannot be imbedded in 4-space

Mr. R. H. Rosen, University of Wisconsin (555-5) 10:15- 10:25

(7) On uniqueness of representation of 3 -manifolds Professor D. E. Sanderson, Iowa State College (555-26)

10:30 - 10:40 (8) On involutions of the 3- sphere

Professor M. W. Hirsch and Dr. Stephen Smale, Institute for Advanced Study (555-4).

Session on Geometry and Logic, Room 207 9:45 - 9:55

(9) On groups which act without fixed points. Preliminary report

Mr. j. A. Wolf, University of Chicago (555-12) 10:00 - 10:10

(10) Relatively constant breadth curves

10:15- 10:25

Professor P. C. Hammer, University of Wisconsin (555-29)

(11) Fundamental groups of compact solvmanifolds Professor Louis Auslander, Indiana University (555-20)

10:30 - 10:40 (12) A characterization of theories with isomorphic denumerable

models Professor Erwin Engeler, University of Minnesota (555-19)

93

FRIDAY, 11:00 A.M.

General Session, Room 133 11:00 - 12:00

Pivotal monomials and polynomial identities of rings (One hour) Professor S. A. Amitsur, Hebrew University and University of Notre Dame

FRIDAY, 2:00P.M.

General Session, Room 133 2:00 - 3:00

Multiplier transformations (One hour) Professor I. I. Hirschman, Washington University

FRIDAY, 3:15P.M.

Session on Algebra, Room 113 3:15 - 3:25

(13) Elements of a theory of intrinsic functions on algebras Professor R. F. Rinehart, Duke University (555-2)

3:30 - 3:40 (14) On methods in class field theory

Professor G. W. Whapl'es, Indiana University (555-28) 3:45 - 3:55

(15) On cubic forms permitting composition

4:00 - 4:10

Professor R. D. Schafer, University of Connecticut and Institute for Advanced Study (555-24)

(16) On residue class rings Professor D. W. Wall, University of Michigan and University of North Carolina (555-21)

Session on Topology, Room 206 3:15 - 3:25

(17) Conceptual proofs of two theorems of J. C. Moore and H. Cartan. Preliminary report

Mr. J. z. Yao, University of Chicago (555-7) 3:30 - 3:40

(18) On the decomposition of topological vector spaces into direct sums of closed subspaces. Preliminary report

Professor Jesus Gil de Lamadrid, University of Minnesota (555-33)

3:45 - 3:55 (19) Uniform continuity and compactness in topological groups

Mr. J. M. Kister, University of Wisconsin (555-18)

94

4:00 - 4:10 (20) Proof that if a decomposable compact continuum is topo­

logically equivalent to each of its nondegenerate subcon­tinua, it is an arc

Mr. G. W, Henderson, University of Texas (555-10) (Introduced by Professor R, L, Moore)

Session on Analysis, Room 207 3:15 - 3:25

(21) Regular sequences and frequency distributions Professor E. B. Leach, Case Institute of Technology (555- 32)

3:30 - 3:40 (22) Weak convergence and compactness in spaces of additive

type functions

3:45 - 3:55

Professor Pasquale Porcelli, University of Wisconsin (555-11)

(23) On the reducibility of certain differential operators Professor H. E. Stelson, Michigan State University (555-17)

4:00 - 4:10 (24) Concerning convergence of continued fractions

Professor D. F. Dawson, University of Missouri (555-8)

SATURDAY, 10:00 A.M.

Session on Applied Mathematics and Statistics, Room 133 10:00 - 10:10

(25) The end problem of cylinders. The variational solution of a boundary layer problem in elasticity

10:15 - 10:25

Dr. Gabriel Horvay, General Electric Research Labora­tory, Schenectady, New York

(26) Singularity occurrence and stably posed problems for elliptic equations

10:30 - 10:40

Dr. H. M. Lieberstein, Jr., University of Wisconsin (555-27)

(2 7) A skew five by five matrix and general relativity Professor G. Y, Rainich, University of Michigan and University of Notre Dame (555-31)

10:45 - 10:55 (28) On the uniqueness of compressible fluid motions. I. Vis­

cous fluids Professor J, B. Serrin, University of Minnesota (555-26)

95

11:00- 11:10 (29) Optimal burning programs for an idealized upward-direc

ted missile

11:15- 11:25

Dr. G. M. Ewing, University of Oklahoma and U, S. Army Artillery and Missile School (555-3)

(30) Quadrature formulas involving derivatives of the integrand Professor P, C. Hammer, University of Wisconsin, and Dr. H. H. Wicke, Sandia Corporation, Albuquerque, New Mexico (555-1)

11:30 - 11:40 (31) On stochastic linear programming

Professor J. V. Talacko, Marquette University (555-15)

Session on Algebra, Room 206 10:00 - 10:10

(32) A structure theory for a class of lattice-ordered rings. III. Preliminary report

Mr. D. G. Johnson, Purdue University (555-13) 10:15- 10:25

(33) Derivations and embeddings of a field in its power series ring

10:30 - 10:40

Professor Nickolas Heerema, Florida State University (555-9)

(34) Convex ideals and prime ideals in rings of continuous functions. Preliminary report

Professor C. W. Kohls, University of Illinois (555-19) 10:45 - 10:55

(35) Prime dual ideals in Boolean algebras Professor L, J, Heider, Marquette University (555-22)

11:00 - 11:10 (36) A note on Boolean operations

11:15- 11:25

Dr. C. H. Cunkle, Cornell Aeronautical Laboratory, Inc., Buffalo, New York (555-14)

(37) Prime polynomial identity rings Dr. E. C. Posner, University of Wisconsin (555-34)

11:30 - 11:40 (38) The abstract algebra of linear programming

Mr. R. T, Rockafellar, Marquette University (555-16) (Introduced by Professor J, V. Talacko)

96

SUPPLEMENTARY PROGRAM (To be presented by title)

(39) On a class of completely primary rings. Preliminary re­port

Professor E. H. Batho, University of Rochester (40) Each homogeneous plane continuum that contains an arc

is a simple closed curve Professor R. H. Bing, University of Wisconsin

(41) Inscribed orthogonal n tuples in convex bodies Professor D. G. Bourgin, University of Illinois

(42) Special points under sphere maps Professor D. G. Bourgin, University of Illinois

(43) Concerning Stieltjes integrals Mr. R. C. Bzoch, University of Minnesota

(44) Intrinsic form of the characteristic relations for a perfect compressible fluid in general relativity and non-steady Newtonian mechanics

Professor Nathaniel Coburn, University of Michigan (45) The average order of certain types of arithmetical func­

tions Professor Eckford Cohen, University of Tennessee

(46) The universal curve of Sierpinski is not a minimal set Professor W. H. Gottschalk, University of Pennsylvania

(47) Permutations with restricted position Professor Frank Harary, Princeton University and Institute for Advanced Study

(48) Relations between a closed operator and its adjoint Mr. J. T. Joichi, University of Illinois

(49) Integral closure of differential rings Dr. E. C. Posner, University of Wisconsin

(50) Arithmetical extensions with prescribed cardinality Dr. M. 0. Rabin, Hebrew University

(51) Computional algorithm of the "dual-symmetric" method Mr. R. T. Rockafellar and Professor j. V. Talacko, Marquette University

(52) On the uniqueness of compressible fluid motions. II. Non­viscous fluids

Professor J. B. Serrin, University of Minnesota (53) On additive functors

Dr. C. E. Watts, University of Chicago (54) Linear transformations on Grassmann product spaces

Mr. Roy Westwick, University of British Columbia (Introduced by Dr. M.D. Marcus)

Bloomington, Indiana March 4, 1959

97

J. W. T. Youngs Associate Secretary

FIVE HUNDRED FIFTY-SIXTH MEETING

U, S. Naval Postgraduate School Monterey, California

April 16-18, 1959

PROGRAM

The five hundred fifty-sixth meeting of the American Mathe­matical Society will be held on Thursday, Friday and Saturday, April 16-18, 1959 at the U, S. Naval Postgraduate School, Monterey, Cali­fo-rnia,

By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, a Symposium on Lattice Theory will be held on Thursday and Friday in Room 400, Spanagel Hall.

By invitation of the same Committee, there will be an address at 1:30 P.M. on Saturday by Dr. Olga Taus sky Todd of the California Institute of Technology. Her subject is "Matrices of rational inte­gers".

Sessions for contributed papers will be held on Saturday at 10:00 A.M. and at 3:00P.M. Abstracts of the papers to be presen ted at these sessions appear on pages 137-198 of these NOTICES. There are cross references to the abstracts in the program. For example, the title to paper (1) in the program is followed by (556-28) indicating that the abstract can be found under the designation 556-28 among the published abstracts. A Symposium on Lattice Theory, sponsored by the Society with the financial aid of the National Science Foundation, will have sessions on Thursday morning and afternoon and on Friday morning and afternoon in Room 400, Spanagel Hall. Registration will begin at 9:00A.M. on Thursday and Friday in Room 400, Spanagel Hall. After the symposium sessions begin, the registration desk will be moved to the doorway of Room 400. On Saturday, registration will be in the lobby of Spanagel Hall, begin­ning at 9:00 A.M. A session for late papers will be held starting at 3:00 P. M. on Saturday in Room 321, Spanagel Hall.

Luncheon on Thursday and Friday will be available in the cafe­teria in King Hall. On Saturday, there will be a luncheon at the officers' club in Hermann Hall. Coffee will be available during the day in the Faculty Lounge, Room 200, Spanagel Hall,

Monterey is served by United Air Lines, Pacific Airlines, Southern Pacific Railroad and Greyhound Bus. It may be reached by following California Route 1, U. S, 10 1, and other roads. There are

98

two entrances to the U. s. Naval Postgraduate School: the Sloat Avenue entrance near the intersection of Fremont Avenue (California Route 1) and Sloat Avenue, and the Del Monte Avenue entrance off Del Monte Avenue. There will be signs to the parking areas and to the lobby of Spanagel Hall from each entrance. All sessions of the meeting will be held in Spanagel Hall, the five story modern building near the Sloat Avenue entrance.

There are numerous motels and hotels within a few miles of the school. The following hotels will have reservations available for members attending the meeting.

Casa Munras Mission Inn San Carlos Hotel Mark Thomas Inn

Single $6.60-$14.00 6.00- 8.00 7.00 8.00- 12.00

Double $10.50-$16.50

7.00-10.00 9.00

PROGRAM OF THE SYMPOSIUM ON LATTICE THEORY

Twin Beds $11.50-$18.00

11,00- 12.00 10.00 12.00- 16.00

The Program Committee of the Symposium consisted of Pro­fessor R. P. Dilworth, chairman, Professors Garrett Birkhoff, Alfred Tarski and R. S. Pierce. All sessions of the symposium will meet in Room 400, Spanagel Hall. The following program is tenta­tive: there may be additional speakers and the subjects of the talks may be revised.

THURSDAY, 9:30A.M.

Boolean algebras with operators Cylindrical algebras

Professor Alfred Tarski Injective and projective Boolean algebras

Professor P. R. Halmos

45 minutes

20 minutes Cardinal and ordinal multiplication of relation types

Professor C. C. Chang 20 minutes Boolean retracts

Professor Philip Dwinger 20 minutes

THURSDAY, 2:00P.M.

Complemented modular lattices Complemented modular lattices

Professor I. Halperin Coordinates in non-Desarguesian planes

Professor K. D. Fryer

99

45 minutes

20 minutes

Coordinatization problems Professor Bjarni Jonsson

Complete Boolean algebras Professor R, S, Pierce

FRIDAY, 9:30A.M.

Structure of lattices Structure and decomposition theory of lattices

ZO minutes

ZO minutes

Professor R, P. Dilworth 45 minutes Sublattices of free lattices

Professor R, A. Dean ZO minutes The status of word problems especially in modular lattices

Professor P, M, Whitman ZO minutes Lattice theory of generalized partitions

Professor Juris Hartmanis ZO minutes

FRIDAY, Z:OO P, M.

Applications of lattice theory Applications of lattice theory

Professor Garrett Birkhoff 45 minutes On the lattice of normal subgroups of a group

Professor Marshall Hall ZO minutes Topological lattices

Professor L, W, Anderson ZO minutes

PROGRAM OF THE SESSIONS (Time limit for each contributed paper, 10 minutes)

(As an experiment, the contributed papers are scheduled at 15 minute intervals so that listeners can circulate between different sessions, To maintain this schedule, the ten minute time limit will be strictly enforced.)

SATURDAY, 10:00 A.M.

Session on analysis and topology, Room 400, Spanagel Hall 10:00 - 10:10

(1) The zeros of entire functions of exponential type Professor Paul Malliavin, University of Caen, France, and Professor L, A. Rubel, University of Illinois (556-28)

10:15 - 10:25 (Z) Generalized bases in topological linear spaces

Professor M.G. Arsove, University of Washington, and Dr. R, E, Edwards, London University (556-13)

100

10:30 - 10:40 (3) On Hadamard's variation formula for Green's function

Professor S. E. Warschawski, University of Minnesota and University of California, Los Angeles (556-26)

10:45 - 10:55 (4) Singular integrals on compact manifolds

Dr. R, T. Seeley, Harvey Mudd College (556-3) 11:00 - 11:10

(5) Harmonics asymptotic to a particular indefinite integral Professor D. E. Edmondson, Southern Methodist Univer­sity (556-16)

11:15 - 11:25 (6) A decomposition of the Haar integral in compact groups

Professor G. M, Helmberg, Tulane University (556-9) 11:30 - 11:40

(7) A group algebra without a real involution

11:45- 11:55

Mr. R, A. Sonic, University of Southern California (556 -20)

(8) A note on generalized fundamental groups and generalized covering spaces

Professor J, H. Case and Professor R. E. Chamberlin, University of Utah (556-27)

Session on Topology, Algebra and the Theory of Numbers, Room 428 Spanagel Hall 10:00 - 10:10

(9) On the existence of continuous lattice homomorphisms, Preliminary report

Professor L. W, Anderson, University of Oregon (556-6) 10:15 - 10:25

(10) A combinatorial result and its application in proving cer­tain metrization theorems

Professor M. B, Smith, Jr. University of Utah (556-19) 10:30 - 10:40

(11) The ring of number-theoretic functions

10:45 - 10:55

Dr. E, D. Cashwell and Dr. C, J, Everett, Los Alamos Scientific Laboratory (556-2)

(12) A class of irreducible systems of generators for infinite symmetric groups

11:00 - 11:10

Professor R, B, Crouch, New Mexico College of Agri­culture and Mechanical Arts (556-21)

(13) On a theorem arising from a study of matrix commutators. Preliminary report

Professor D. W, Robinson, Brigham Young University (556-12)

101

11:15- 11:25 (14) On matrix commutators

11:30- 11:40

Mr. R. C. Thompson, California Institute of Technology (556-7)

(15) Intervals containing infinitely many sets of conjugate alge­braic integers

Professor R. M. Robinson, University of California, Berkeley (556-18)

SATURDAY, 1:30 P.M.

Invited Address, King Hall Matrices of rational integers

Dr. Olga Taus sky Todd, California Institute of Technology

SATURDAY, 3:00P.M.

Session on Algebra, Logic and Foundations, Room 400, Spanagel Hall 3:00 - 3:10

(16) Aleph-irreducible cardinals and decompositions of cardi­nals

Professor Herman Rubin, University of Oregon (556-17) 3:15 - 3:25

(17) Models of arithmetic through function rings Professor Solomon Feferman, Stanford University,

3:30 - 3:40

Dr. D. S. Scott, University of Chicago, and Mr. S. Ten­nenbaum, Illinois Institute of Technology (556-31)

(18) On unions of relational systems

3:45 - 3:55

Mr. H. J. Keisler, California Institute of Technology (556-14)

(Introduced by Dr. Olga Taus sky Todd)

(19) Cancellation of unary algebras

4:00 - 4:10

Mr. Steven Bryant and Professor J. G. Marica, Fresno State College (556-30)

(20} A characterization of abelian groups which are automor-phism groups of simply ordered set

4:15- 4:25

Professor C. c. Chang, University of California, Los Angeles, and Mr. Andrzej Ehrenfeucht, University of California, Berkeley (556-23}

(21} Simply ordered sets with non-abelian automorphism groups Professor Anne C. Morel, University of California, Davis (556-24)

102

4:30 - 4:40 (22) Non-linear recursive sequences

4:45 - 4:55

Professor E. A. Walker, New Mexico College of Agri­culture and Mechanical Arts (556-22)

(23) The join irreducible excess function in finite lattices Professor S. P. Avann, University of Washington (556-5)

5:00 - 5:10 (24) The dimensionality of finite free distributive lattices

Professor Randolph Church, U. S. Naval Postgraduate School, Monterey (556-29)

Session on Applied Mathematics, Room 428, Spanagel Hall 3:00 - 3:10

(25) A note on Kato's uniqueness criterion for Schrodinger operator self-adjoint extensions

3:15 - 3:25

Professor F. H. Brownell, University of Washington (556-1)

(26) Computation using Kronecker indices

3:30 - 3:40

Professor C. B. Tompkins, University of California, Los Angeles (556-4)

(27) A bivariate Tchebycheff inequality for symmetric convex polygons

3:45 - 3:55

Professor A, W. Marshall and Professor Ingram Olkin, Stanford University (556-10)

(28) On some games with moves in the unit interval Mr. Martin Fox, Stanford University

(Introduced by O.r. S. H. Gould) 4:00 - 4:10

(29) The calculation of the complete elliptic integral of the third kind

4:15 - 4:25

Professor Morgan Ward, California Institute of Tech­nology (556-25)

(30) Natural sorting. II

4:30 - 4:40

Dr. R, M. Baer, Cal Research Corporation, Richmond, California, and Dr. Paul Brock, Willow Run Labora­tories, Ann Arbor, Michigan (556-8)

(31) A computer technique for breaking ciphers. Preliminary report

Mr. B. L, Schwartz, Technical Operations, Inc., Monterey, California (556-11)

103

4:45 - 4:55 (32) Secant method for simultaneous equations

Dr. Philip Wolfe, RAND Corporation, Santa Monica, California (556-15)

SUPPLEMENTARY PROGRAM (To be presented by title)

(33) The Runge-Kutta method in Banach spaces Professor M. Altman, California Institute of Technology and Polish Academy of Sciences

(Introduced by Mr. John Todd) (34) An arithmetical inversion principle

Professor Eckford Cohen, University of Tennessee (35) Decidability at the theory of one function

Mr. Andrzej Ehrenfeucht, University of California, Berkeley

(36) Decidability of the theory of linear ordering relation Mr. Andrzej Ehrenfeucht, University of California, Berkeley

(3 7) A decidable theory which has exactly one undecidable com­plete extension

Mr. Andrzej Ehrenfeucht, University of California, Berkeley

(38) Non-archimedean models for arithmetic Mr. S. Tennenbaum, Illinois Institute of Technology

(Introduced by Dr. D. S. Scott) (39) Characterization of singularities of a harmonic function

in three variables in terms of the coefficients of its series development

Professor A. M. White, University of Santa Clara (Introduced by Professor Irving Sussman)

Seattle, Washington March 8, 1959

104

R. S. Pierce Acting Associate Secretary

FIVE HUNDRED FIFTY-SEVENTH MEETING

Hotel New Yorker New York, New York April 23-25, 1959

PROGRAM

The five hundred fifty-seventh meeting of the American Mathe­rn a tical Society will be held on Thursday through Saturday, April 23-25, 1959, at the Hotel New Yorker in New York, New York. All ses­sions will be held in assembly rooms of the hotel.

Two Symposia will be held in conjunction with the meeting. A Symposium on Nuclear Reactor Theory, sponsored by the Society with the financial aid of the Office of Ordnance Research, will have sessions on Thursday morning and afternoon and on Friday morning and afternoon in the Grand Ballroom on the second floor. The Pro­gram Committee consists of Professor E. P. Wigner, Chairman, and Professor Garrett Birkhoff, Dr. H. L. Garabedian, Dr. S.M. Ulam, and Dr. J. E. Wilkins.

A Symposium on Finite Groups, sponsored by the Society with the financial aid of Project FOCUS of the Institute for Defense Analy-es, will have sessions on Thursday morning and afternoon and Fri­

.1ay morning in the North Ballroom on the second floor. The Pro­gram Committee consists of Professor A. A. Albert, Chairman, and Professors Walter Feit, Marshall Hall, I. N. Herstein, and Irving Kaplansky.

By invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings, Professor Jun-ichi Igusa of The Johns Hopkins University will deliver an hour address entitled "On the Kroneckerian model of fields of elliptic modular functions" on Satur­day at 10:45 A. M. in the Grand Ballroom.

By invitation of the same committee, Professor J, W, Milnor of Princeton University will address the Society on the subject "Differentiable manifolds which are homotopy spheres" on Saturday at 2:00P.M. in the Grand Ballroom.

Sessions for contributed papers will be held on Friday after­noon and Saturday morning and afternoon. Abstracts of the papers to be presented at these sessions appear on pages 137-198 of these NOTICES. There are cross references to the abstracts in the pro­gram. For example, the title to paper (1) in the program is followed by (557-5), indicating that the abstract can be found under the desig­nation 557-2 among the published abstracts. Late contributed pa­pers will be scheduled in the regular sessions for contributed papers

105

so far as this can be done. If the number of late papers is too large, there will be a session for late papers on Saturday afternoon.

The Council will meet in the Boston Room on the fourth floor at 5:00P.M. and will reconvene after dinner.

REGISTRATION

The registration desk in the North Ballroom Foyer on the second floor will be open from 9:00A.M. to 5:00P.M. on Thurs­day and Friday and from 9:00A.M. to 3:30P.M. on Saturday.

THE HOTEL

The New Yorker is the official headquarters hotel. Persons desiring to stay there should communicate directly with the hotel. The request should be addressed to Miss Betty C. Pace, Hotel New Yorker, 34th Street and 8th Avenue, New York 1, New York, and should mention the meeting of this Society. Single rooms are $8.00 per day and up; rooms for two are $11.50 per day and up. On page 87 of the February NOTICES (Issue No. 36) is a more complete list­ing of prices of rooms and a room reservation form.

The hotel has its own garage, which advertises reasonable rates. American Express Credit Cards are honored by the hotel. Diners' Club Credit Cards are honored only in the Golden Thread Cafe in the hotel.

MAIL AND TELEGRAMS

Correspondence for those known to be registered at the Hotel New Yorker may be addressed there directly. Mail and telegrams for others attending the meeting may be addressed in care of the American Mathematical Society at the Hotel New Yorker. The hotel mailing address is 34th Street and 8th Avenue, New York 1, New York.

TRANSPORTATION

The Hotel New Yorker is on the corner of 34th Street and 8th Avenue, one block west and one block north of the Pennsylvania Sta­tion, from which there is a tunnel direct. The hotel is on the 8th A venue line of the Independent Subway, at the 34th Street or Pennsyl­vania Station stop. The hotel can be reached by walking underground from the 34th Street station of the IR T subway. The hotel is one block east from the 34th Street exit of the Lincoln Tunnel and is less than half a block west of the Greyhound Bus Terminal. Airline ser­vice into LaGuardia and Idlewild airports is served by the East Side Terminal. Buses serving the Newark airport come into the West Side Terminal, eight blocks north and two blocks west of the New Yorker, which is somewhat nearer to the hotel than the East Side terminal is.

106

PROGRAM OF THE SYMPOSIUM ON NUCLEAR REACTOR THEORY

THURSDAY 9:30A.M.

Session on REACTOR PHYSICS, Grand Ballroom Chairman: Mr. Jack Chernick, Brookhaven National Laboratory

Types of reactors Dr. A.M. Weinberg, Oak Ridge National Laboratory

Neutron thermalization Dr. M, S. Nelkin, General Atomic Division of General Dynamics Corporation

Deep penetration of radiation Dr. Ugo Fano and Dr. M. J, Berger, National Bureau of Standards

Resonance escape probability Dr. L, W. Nordheim, General Atomic Division of General Dynamics Corporation

THURSDAY 2:00P.M.

Session on CRITICALITY, Grand Ballroom Chairman: Dr. J, C. Mark, Los Alamos Scientific Laboratory

Mathematical problems of nuclear reactor theory Professor E. P. Wigner, Princeton University

Diffusion approximation to transport equation Dr. J. E. Wilkins, Nuclear Development Corporation of America

Criticality and positivity Professor Garrett Birkhoff, Harvard University

Existence theorems and spectral theory for the m ultigroup diffusion model

Dr. G. J. Habetler and Dr. M. A. Martino, Jr., Knolls Atomic Power Laboratory

Transport theory and spectral problems Professor G, M, Wing, University of New Mexico

FRIDAY 9:30A.M.

Session on NUMERICAL CALCULATIONS OF CRITICALITY, Grand Ballroom

Chairman: Dr. S, M. Ulam, Los Alamos Scientific Laboratory

107

One-dimensional multigroups calculations: estimation of con­stants

Dr. Richard Ehrlich, Knolls Atomic Power Laboratory

Multi-dimensional multigroup calculations Dr. R, S, Varga, Westinghouse Atomic Power Division

Monte Carlo methods Professor R. D. Richtmyer, New York University

Transport calculations Mr. Bengt Carlson, Los Alamos Scientific Laboratory

FRIDAY 2:00 P, M,

Session on REACTOR DYNAMICS, Grand Ballroom Chairman: Professor B. Davison, University of Toronto

Problems of reactor kinetics Professor Harry Soodak, City College of the College of the City of New York

Core kinetics Dr. H. L, Garabedian, General Motors Corporation

Effects of power on reactivity Dean Harvey Brooks, Harvard University

System kinetics Dr. T, A. Welton, Oak Ridge National Laboratory

PROGRAM OF THE SYMPOSIUM ON FINITE GROUPS

THURSDAY 10:00 A. M.

First Session, North Ballroom Chairman:· Professor A. A. Albert, University of Chicago

Proof of a conjecture of Frobenius Professor J, G. Thompson, De Paul University

Burnside groups and Engel rings Professor R. C. Lyndon, University of Michigan

On the structure of certain solvable groups Professor Daniel Gorenstein, Clark University

On groups which contain Frobenius groups as subgroups Professor Walter Feit, Institute for Advanced Study

Discussion leaders: Dr. Olga Taussky and Professor L. J, Paige

108

THURSDAY 2:00P.M.

Second Session, North Ballroom Chairman: Professor I. N. Her stein, Cornell University

Current studies on permutation groups Professor Marshall Hall, jr ., Ohio State University

Collineation groups Professor Daniel Hughes, University of Chicago

On finite groups with geometrical properties Professor Wilhelm Magnus, New York University

Symmetrical definitions for the binary polyhedral groups Professor H. S. M. Coxeter, University of Toronto

Discussion leaders: Professor G. de B. Robinson and Profes­sor D. G. Higman

FRIDAY 10:00 A.M.

Third Session, North Ballroom Chairman: Professor Irving Kaplansky, University of Chicago

Applications of groups characters Professor M1chio Suzuki, University of Illinois

On maximal subgroups Professor W. E. Deskins, Michigan State University

Some applications of the theory of Lie algebras to finite groups Professor Hans Zassenhaus, California Institute of Tech­nology

Discussion leaders: Professor Charles Curtis and Professor Irving Reiner

PROGRAM OF CONTRIBUTED PAPERS AND INVITED ADDRESSES

(Time limit for each contributed paper, 10 minutes)

FRIDAY 1:30 P.M.

Session on Analysis, Panel Room, Third Floor (1) Spectral multiplicity of singular integral operators

Mr. j. D. Pincus, New York University (557-5) (2) Linear transformations of a functional integral

Mr. T. I. Seidman, University of California, Livermore (557-16)

(3) Unique Hahn-Banach extensions and unique best approxi­mation

Dr. R. R. Phelps, Institute for Advanced Study (557-19)

109

(4) The bilinear relation on open Riemann surfaces Mr. R. D. M. Accola, Harvard University (557-21)

(5) Convex ideals in ordered group algebras and the unique­ness of the Haar measure

Dr. K. E. Aubert, Institute for Advanced Study (557-23) (6) Isomorphism of function spaces

Professor G. G. Lorentz, Syracuse University (557-30) (7) The Wedderburn principal theorem for certain commuta-

tive Banach algebras Professor W. G. Bade, University of California, Berkeley, and Yale University, and Professor P. C. Curtis, Jr., University of California, Los Angeles, and Yale University (557-35)

Session on Algebra, Washington Room, Fourth Floor (8) Note on Parker's method of constructing pairwise ortho­

gonal sets of Latin squares. Preliminary report Professor R. C. Bose, University of North Carolina (557-11)

(9) Group divisible designs and the construction of pairwise orthogonal sets of Latin squares. Preliminary report

Dr. S. S. Shrikhande, University of North Carolina (557-17)

(10) On the minimization problem for Boolean functions. Pre-liminary report

Mr. J, A, Riley, Parke Mathematical Laboratories, Inc., Carlisle, Massachusetts, and Brandeis University (557-15)

(11) Partition rings of finite Abelian groups. Preliminary report

Mr. K. I. Appel, University of Michigan (557-22) (12) Reduction formulae for partitioned matrices

Dr. Emilie V. Haynesworth, National Bureau of Stand­ards, Washington, D. C. (5757-51)

(13) On GL(2,K(x]) Dr. Hirosi Nagao, University of Michigan (557-13)

(14) A generalization of the Riemann-Roch theorem Dr. H. F. Mattson, Air Force Cambridge Research Center, Bedford, Massachusetts (557-49)

Session on Applied Mathematics, Empire Room, Fourth Floor (15) A necessary and sufficient condition for locally stable

numerical integration Mr. H. S. Wilf, Nuclear Development Corporation of America, White Plains, New York (557-28)

(16) On the non-linear diffusion equation Professor I. I. Kolodner, University of New Mexico (557-26)

110

(17) Type-insensitive finite difference methods for symmetric positive equations

Dr. C.-K. Chu, New York University and Pratt Institute (557-38)

{18) On some separable solutions of the two-dimensional heat conduction equation with temperature-varying thermal properties

Mr. M.S. Klamkin, Avco Research and Advanced De­velopment Division, Wilmington, Massachusetts (557-47)

(19) A Runge- Kutta process for hyperbolic partial differential equations

Mr. R. H. Moore, University of Michigan (20) Integral geometric methods in information theory. III.

Sources Professor W. R. Baum, Syracuse University (557-12)

{21) Retarded vector theorems Professor Domina E. Spencer, University of Connecti­cut (557-39)

FRIDAY 3:30P.M.

Session on Analysis, Panel Room, Third Floor (22) Diffraction by convex objects

Professor Harry Hochstadt, Polytechnic Institute of Brooklyn (557-1)

{23) Maximal homomorphisms and almost periodic functions Professor B. A. Rattray, McGill University (557-14)

(24) Some examples in potential theory. Preliminary report Professor Makoto Ohtsuka, University of Kansas and Hiroshima University (557-20)

{25) Approximation of functions of two real variables Dr. T. J. Rivlin, Fairchild Engine Division, Deer Park, New York, and Professor H. S. Shapiro, New York Uni­versity (557-33)

{26) Orthonormal sets with non-negative Dirichlet kernels Professor J. j. Price, Cornell University (557-37)

{27) Existence and stability theorems for periodic solutions of nonlinear Lipschitzian differential systems and fixed point theorems

Professor Lamberto Cesari, Purdue University and RIAS, Baltimore, Maryland

·session on Topology, Washington Room, Fourth Floor. (28) Embedding topological semigroups in topological groups

Dr. N.j. Rothman, University of Rochester (557-45)

111

(29) Arcs in certain semigroups Professor R. P. Hunter, University of Georgia

(Introduced by Professor Everett Pitcher) (30) On imbedding decompositions of 3-space in 4-space

Dr. R. P. Goblirsch, University of Virginia (557-44) (31) The extension of the set on which mappings are homotopic

Professor M. K. Fort, Jr., University of Georgia (557-3) (32) Hyperspaces of the inverse limit space

Mr. Jack Segal, University of Georgia (557-4)

General Session, Empire Room, Fourth Floor (33) A coefficient of stochastic dependence

Dr. S. P. Lloyd, Bell Telephone Laboratories, Murray Hill, New Jersey (557-7)

(34) Equidistributed sequences of events Professor Louis Sucheston, University of Rochester (557-32)

(35) Extended combinatory formulation of standard theories Dr. D. E. Schroer, University of Rochester (557-6)

SATURDAY 9:30A.M.

Session on Analysis, Panel Room, Third Floor (36) On the rank of an essential singularity for a system of

differential equations Professor Jiirgen Moser, Massachusetts Institute of Technology

(37) Asymptotic stability in three-space Dr. C. S. Coleman, RIAS, Baltimore, Maryland (557-2)

(38) Partial Holder continuity and elliptic partial differential equations

Mr. P. C. Fife, New York University (557-41) (39) Some generalized Cauchy problems

Mr. Robert Carroll, University of Maryland (557-46)

Session on Topology and Geometry, Washington Room, Fourth Floor (40) Embedding of Riemann surfaces of genus 1 in 3-space

Mr. E. R. Rodemich and Mr. A.M. Garsia, Massachu­setts Institute of Technology (557-18)

(41) On the Schottky uniformization of compact canal surfaces Mr. E. R. Rodemich and Mr. A.M. Garsia, Massachu­setts Institute of Technology (557-40)

(42) An improved counterexample for Kempe's "proof" of the four-color theorem

Professor H. S. M. Coxeter, University of Toronto (557-34)

(43) Generalisations of the Poincare- Birkhoff fixed point theo­rem

Dr. Robert Hermann, Harvard University (557-9)

112

SATURDAY 10:45 A.M.

Invited Address, Grand Ballroom, Second Floor On the Kroneckerian model of elliptic modular functions(One hour)

Professor Jun-ichi Igusa, Johns Hopkins University

SATURDAY 2:00P.M.

Invited Address, Grand Ballroom, Second Floor Differentiable manifolds which are homotopy spheres (One hour)

Professor John Milnor, Princeton University

SATURDAY 3:15P.M.

Session on Analysis, Panel Room, Third Floor (44) Length of gaps and size of region of overconvergence.

Preliminary report Professor A. J. Macintyre, University of Cincinnati (557-27)

(45) Hausdorff transforms of bounded sequences Dr. J, H. Wells, University of North Carolina (557-36)

(46) On the completeness of sets of convolutions, and a class of schlicht functions defined by Fourier transforms

Professor H. S. Shapiro, New York University (557-52) (47) Intrinsic operators in three-space

Professor V. L. Shapiro, Institute for Advanced Study and Rutgers, The State University (557-48)

(48) Lower bounds for eigenvalues of self-adjoint operators Mr. N. W. Bazley, National Bureau of Standards, Washington, D. C. (557-29)

(49) Comparison of the method of averages with the method of least squares. Fitting a parabola

Professor Morris Morduchow and Mr. Lionel Levin, Polytechnic Institute of Brooklyn (557-8)

Session on Algebra, Washington Room, Fourth Floor (50) Groups having the same group characters

Professor D. W. Wall, University of Michigan and University of North Carolina (557-50)

(51) Group-generated incidence algebras Mr. E. C. Dade and Dr. Karl Goldberg, National Bureau of Standards, Washington, D. C. (557-45)

(52) On a certain Silov boundary Dr. W. W. Comfort, Harvard University (557 -42)

(53) A class of hyper-FC-groups Dr. A. M. Duguid, Brown University (557-24)

113

(54) Reduction of algebraic varieties modulo arbitrary powers of a prime divisor in the ground field

Mr. M. j. Greenberg, Rutgers, The State University (557-25)

SUPPLEMENTARY PROGRAM (To be presented by title)

(55) On the solution of an implicit first order partial differ­ential equation

Dr. Smbat Abian and Professor A. B. Brown, Queens College

(56) A property of commutators in free products Professor B. J. Ball, University of Virginia

(57) On the average number of direct factors of a finite abelian group

Professor Eckford Cohen, University of Tennessee (58) The elementary arithmetical functions

Professor Eckford Cohen, University of Tennessee (59) The behavior of a semi-infinite rigid-ideally plastic beam

under a dynamic loading at the finite end Dr. M. F. Conroy, Parke Mathematical Laboratories, Inc., Carlisle, Massachusetts

(60) A categorical theory of completions Professor G. D. Findlay, McGill University, Professor J. M. Maranda, University of Montreal, and Professor Joachim Lambek, McGill University

(61) Weak limits of chains Professor W. H. Fleming, Brown University

(62) Solutions of first order differential equations which are solutions of linear equations of higher order

Professor Lawrence Goldman, Stevens Institute of Technology

(63) A matrix algorithm for solutions and r-bases of a finite irreflexive relation

Professor Frank Harary, Princeton University and Institute for Advanced Study, and Professor Moses Richardson, Brooklyn College

(64) On the normal bundle to a sphere imbedded in Euclidean space

Professor W. S. Massey, Brown University (65) Approximate solutions for certain nonlinear hyperbolic

partial differential equations Mr. R. H. Moore, University of Michigan

(66) Weak and strong solutions of general boundary value problems

Dr. Martin Schechter, New York University

114

(67) On generalized infrapolynomials of a certain type Dr. Oved Shish a, Harvard University and Technion­Israel Institute of Technology

(Introduced by Professor j. L. Walsh) (68) On higher homotopy associativity

Mr. J. D. Stasheff, Brasenose College, Oxford (69) On degree of approximation by bounded harmonic func­

tions Professor j. L. Walsh, Harvard University

Bethlehem, Pennsylvania March 4, 1959

115

Everett Pitcher Associate Secretary

PRELIMINARY ANNOUNCEMENT OF MEETING FIVE HUNDRED FIFTY-EIGHTH MEETING

University of Oregon Eugene, Oregon June 20, 19S9

The five hundred fifty-eighth meeting of the American Mathe­matical Society will be held on Saturday, June 20, 19S9, at the Uni­versity of Oregon, in Eugene, Oregon. There will be a meeting of the Mathematical Association of America on Friday, June 19, and a meeting of the Society for Industrial and Applied Mathematics on Friday and Saturday.

By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, the Society will be addressed on Saturday afternoon by Professor Ernest Michael of the University of Washington, Seattle. His talk is entitled "Abstract Topological Spaces." Sessions for contributed papers will be held on Saturday morning and afternoon.

Dormitory accommodations will be available for the nights of June 18, 19 and 20 at the rate of $3.00 for one person, $S.OO for a married couple, $1.00 for each child under twelve. A banquet will be held on Friday evening, costing $2.2S per person. Dormitory and banquet reservations may be obtained through Professor Kenneth S. Ghent, Mathematics Department, University of Oregon, Eugene, Oregon. The application for a reservation should include the appli­cants name, institution, his expected times of arrival and departure, and the name and relationship of each member of his party. Cafe­teria service for all meals will be available in the dormitory.

There are numerous hotels and motels in Eugene:

Eugene Hotel Eugene Travlodge Flags tone Motel Rose Motel

Single Double $6.00 $9.00 - $10.00

6.SO 7.SO - 9.00 6.SO s.so

7 .so -6.SO -

10.00 7 .so

Reservations can be obtained by writing to the Eugene Chamber of Commerce, indicating a first and second choice, or by writing di­rectly to the hotel or motel.

Eugene is about 120 miles south of Portland, Oregon on U.S. Highway 99. It is served by Southern Pacific Railway, United Air Lines, West Coast Airlines and Greyhound Bus. Members who drive to the meeting will find ample free parking on the campus of the University.

Seattle, Washington March 1, 19S9

116

R. S. Pierce Acting Associate Secretary

HEWS ITEMS AHD AHHOURCEMERTS

THE 1962 INTERNATIONAL CONGRESS OF MATHEMA­TICIANS. The Secretary of the International Mathematical Union has announced that the Committee authorized by the Congress in Edin­burgh has accepted an invitation from the Swedish National Commit­tee for Mathematics and the Swedish Mathematical Society to hold the 1962 International Congress in Stockholm. The invitation was signed by Ake Pleijel for the Swedish National Committee for Mathematics and by Goran Borg for the Swedish Mathematical Society. Further information concerning the Stockholm Congress will be published in the NOTICES as rapidly as it becomes available to the Editor.

THE INTERNATIONAL MATHEMATICAL UNION has an­nounced that the Republic of China (Taiwan) has been admitted to the International Mathematical Union through the Chinese Mathematical Society (Taipei, Taiwan) as the adhering organization.

THE FOURTEENTH ANNUAL MEETING OF THE ASSOCI­ATION FOR COMPUTING MACHINERY will be held at the Massa­chusetts Institute of Technology, Cambridge, Massachusetts, on Sep­tember 1-3, 1959. Local arrangements will be under the direction of Professor F. M. Verzuh, Massachusetts Institute of Technology.

A CONFERENCE ON MACHINE SEARCHING AND TRANSLA­TION. Western Reserve University and the RAND Corporation will sponsor a three-day international conference on "Standards on a Common Language For Machine Searching and Translation" from September 6 to 12, 1959, at the Tudor Arms Hotel in Cleveland, Ohio. The primary purpose of the Conference will be to encourage the de­velopment of a common machine language or a series of compatible machine languages to prepare scientific and technical literature for searching, selecting, correlating, and translating by automatic equip­ment. Efforts will be made to foster agreements for cooperative processing and exchange of research materials to machine search scientific literature on a world-wide basis. Complete information on the Conference is available from the Secretariat: Center for Docu­mentation and Communication Research, Western Reserve University, Cleveland 6, Ohio.

THE FOURTH CONGRESS ON THEORETICAL AND APPLIED MECHANICS was held at the Bengal Engineering College, Howrah from December 28 to December 31, 1958, under the Presidentship

117

of Dr. S. R. Sen Gupta, Director, Indian Institute of Technology, Kharagpur. About two hundred and fifty scientists and engineers registered themselves for the Congress. These included members from Australia, Burma, Czechoslovakia, Egypt, Hungary, Italy, Japan, Poland, U.S. A. and the U, S. S. R. Sri A, C. Roy, Principal, Bengal Engineering College, Howrah welcomed the delegates. Messages were read from prominent workers all over the world including those from Von Karman, H. L. Dryden, S. Goldstein, L, Rosenhead, K. S. Krishnan, J, C. Ghosh and M. S, Thacker.

The Congress received 75 original papers of which 40 were read. The subjects dealt with included finite deformation, visco­elasticity, stress waves, stresses in strips, columns and discs, plasticity, elasto-porous problems, vibration and stability, fluid flow, ballistics and statistics.

In his Presidential Address Dr. S. R. Sen Gupta stressed the importance of experimental methods in the solution of engineering problems. He considered in detail the causes of cracking of some capitals of stone columns in the Arts F acuity building of the Osm ania University. By rejecting a few of the plausible hypotheses he con­cluded that the cracks might be due to the localized concentration of stress due to unevenness of bearing between the capital and the column.

Half hour addresses were delivered by S. K, Chakravarty, S. I, Pai and M, Lunc. Dr. Chakravarty spoke on propagation of elastic waves across continents and oceans. He dwelt on the existence of small disturbances, called microseisms superposed on Rayleigh waves through multi-layered media, Professor Pai discussed the propagation of a cylindrical shock wave produced by the instantane­ous release of energy from infinite wire into the surrounding med­ium. Professor Lunc outlined the molecular aspects of gas flows.

Popular lectures on "The present concept of the universe" and "Mechanics and the engineer" were respectively delivered by Pro­fessor N. R. Sen and Dr. J, C. Morrison of Glasgow University.

Dr. A, N. Khosla was elected President for the next two years. Dr. S. K. Chakravarty was elected Vice-president in the vacancy created by the retirement of Sri V. Cadambe. Professor B. R, Seth was re-elected Secretary-Treasurer for the ne·xt three years. Dr. Jai Kishan, Professor B. Sen Gupta, and Professor D. Banerjee were elected new members of the Executive Committee.

The Executive Committee accepted the invitation of University of Roorkee to hold the Fifth Congress in Roorkee in December 1959.

118

THE GOLDEN JUBILEE CELEBRATION OF THE CALCUTTA MATHEMATICAL SOCIETY. An announcement received in the Edi­torial Offices of the NOTICES on January 19, 1959, describes the Calcutta Mathematical Society and says that "in December, 1958 the Society proposes to celebrate its Golden Jubilee." (Editorial Note: From the wording of the announcement it was not clear to the Editor whether the celebration was held last December, or whether it is now going on, or whether it is something for the future. Perhaps readers who are interested in the organization will know the answer.) It is stated that the plans include the holding of seminars and sym­posia on the study of mathematics in India during the last fifty years, the holding of an exhibition, the invitation of outstanding foreign and Indian mathematicians to deliver lectures, and the publication of a Commemoration Volume containing contributions from outstanding mathematicians all over the world. The estimated expenses for the Golden Jubilee Celebration are Rs. 20,000/-. An appeal is being made to educational institutions, universities, and other organizations to contribute to the Golden Jubilee Fund. Donations may be sent to: The Treasurer, Calcutta Mathematical Society Golden Jubilee Fund, 92, Upper Circular Road, Calcutta 9, India.

NEW EDITORIAL STAFF FOR THE JOURNAL "MATHEMATI­CAL TABLES AND OTHER AIDS TO COMPUTATION". The Division of Mathematics of the National Academy of Sciences - National Re­search Council announces that Harry Polachek, Technical Director of the Applied Mathematics Laboratory of the David Taylor Model Basin, has been appointed Chairman of the Editorial Committee for the quarterly journal Mathematical Tables and Other Aids to Compu­~ effective January 1959. He succeeds C. B. Tompkins of the University of California at Los Angeles, who held the post since No­vember 1954. The other members of the Editorial Committee are: C. C. Craig, A. Fletcher, E. Isaacson, D. Shanks, C. V. L. Smith, A. H. Taub, C. B. Tompkins and J. W. Wrench, Jr.

Mathematical Tables and Other Aids to Computation was founded in 1943 by R. C. Archibald with the aid of a grant from the Rocke­feller Foundation and is published by the Division of Mathematics of the National Academy of Sciences - National Research Council. It features original papers in numerical analysis, high speed computer methods and other aids to computation as well as short articles re­porting upon the latest developments in these fields. It serves as an information center on tables and other aids to computation appearing in the current literature not only in the fields of mathematics, physics, statistics, astronomy and navigation, but also in such areas as actu­arial science, aeronautics, chemistry, engineering, geodesy, medi­cine and meteorology. A file of unpublished mathematical tables is

119

maintained for use by subscribers. Articles for publication in Mathf' matical Tables and Other Aids to Computation should be addressed to: Harry Polachek, Editor, Mathematical Tables and Other Aids to Computation, David Taylor Model Basin, Washington 7, D. C. Infor­mation on subscriptions may be obtained from National Academy of Sciences, Printing and Publishing Office, Z 10 l Constitution A venue, Washington Z5, D. C.

A NEW MEMOIR. MEMOIR No. 3Z, "Translation Lattices", by Richard Scott Pierce, is now available. The price is $1.70 list and $1.28 to members and agents. The author has furnished the following description of this paper:

The translation lattices considered in this Memoir are systems of bounded, real valued functions on a set, closed under the addition of constants and the pointwise greatest lower bound operation. An example is the family of all bounded, superharmonic functions on a domain of Euclidean space. Any translation lattice can be realized by continuous functions on a compact Hausdorff space. The principal results in the Memoir are concerned with the uniqueness of such representations.

THE AWARD OF THE BOCHER PRIZE. Dr. Louis Nirenberg, professor of mathematics at the Institute of Mathematical Sciences at New York University, received the Society's Bacher Memorial prize for outstanding contributions to mathematical analysis at the annual meeting of the Society in Philadelphia on January ZO -zz, 1959. Past recipients of the prize have been G. D. Birkhoff, E. T. Bell and Solomon Lefschetz (jointly), j. W. Alexander, Marston Morse and Norbert Weiner (jointly), John von Neumann, Jesse Douglas, A. C. Schaeffer and D. C. Spencer (jointly), and Norman Levinson.

THE NEW PRESIDENT OF THE AMERICAN PHYSICAL SO­CIETY is Professor George E. Uhlenbeck, Professor of Theoretical Physics at the University of Michigan. The announcement was made at the annual business meeting of the American Physical Society in New York on January Z8. It was also announced that Professor Victor F. Weisskopf was elected to the Vice Presidency, Dr. Karl K. Darrow of Columbia was re-elected Secretary, and Dr. S. L. Quimby was re-elected Treasurer.

lZO

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

Professor V, G. Grove of Michigan State University has retired with the title, Professor Emeritus.

Associate Professor W. J. LeVeque, on leave from the Univer­sity of Michigan, is at the University of Gottingen, Germany, on a Sloan Research Fellowship.

Associate Professor Janet McDonald of Vassar College has been awarded a National Science Foundation Faculty Fellowship for the academic year 1959-60. She has been granted a leave of absence from Vassar College and will spend the year at Indiana University.

Professor Abba V. Newton, on leave from Vassar College, is studying at the University of Michigan under a National Science Foundation Faculty Fellowship for the academic year 1958-59.

Dr. V. K, Balachandran of the Indian Statistical Institute has been appointed a reader in mathematics at the University of Madras, Madras, India.

Professor Emeritus A. A. Bennett of Brown University has been appointed to a visiting professorship at Southern Illinois University.

Mr. R. B. Block of Wright Patterson Air Force Base has ac­cepted a position as mathematician with Eglin Air Force Base, Florida.

Dr. B. P. Bogert of Bendix Aviation Corporation has accepted a position as member of the technical staff of Bell Telephone Labora­tories, Inc., Murray Hill, New Jersey.

Professor Volodymyr Bohun-Chudyniv of Atlanta University has been appointed to a professorship at Morgan State College.

Mr. H. H. Brown of Lockheed Aircraft Corporation has accepted a position as senior mathematician with the Corporation for Economic and Industrial Research, Arlington, Virginia.

Mr. W. H. Burgin, Jr. of Princeton University has been ap­pointed a mathematics instructor at Mercersburg Academy, Mercers­burg, Pennsylvania.

Associate Professor J, M. Calloway, on leave from Carleton College, will be at the Institute for Advanced Study for the spring term.

Associate Professor J, W. Carr, III, of the University of Michi­gan has been appointed director of the research computation center and associate professor of mathematics at the University of North Carolina,

Mr. C. J. Cillay of American General Life Insurance Company has accepted a position as policy analyst with the Texas State Board of Insurance, Austin, Texas.

121

Dr. C. E. Clark of B ooz, Allen and Hamilton has accepted a position as mathematician with the System Development Corporation, Santa Monica, California.

Mr. A. W. Coutris has accepted a position as engineer with Ammann and Whitney, New York, New York,

Mr. M. V. Cross, Jr. of Lockheed Aircraft Corporation has accepted a position as staff engineer with the Hughes Aircraft Company, Culver City, California,

Dr. D. B. DeLury of the Ontario Research Foundation has been appointed to a professorship at the University of Toronto.

Dr. Bernard Dimsdale of Sperry Rand Corporation has accepted a position as senior mathematician with the International Business Machines Corporation, Los Angeles, California.

Mr. R. K. Froyd of the University of California, Los Angeles, has been appointed to an assistant professorship at Long Beach State College.

Professor H. A. Giddings of New York University has been mtmed director of the New York University Graduate Center at the Bell Telephone Laboratories, Murray Hill, New Jersey.

Mr. G. E. Goode of Duke University has accepted a position as mathematician with the National Security Agency, Department of Defense, Fort George G. Meade, Maryland.

Mr. G. R. Grainger of CONVAIR has accepted a position as associate with the Planning Research Corporation, Los Angeles, California,

Professor Ulf Grenander of Brown University has been ap­pointed to a professorship at the University of Stockholm, Stockholm, Sweden.

Dr. T. N. E. Greville of the Department of Health, Education and Welfare, Washington, D. C. has accepted a position as mathema­tician with the office of the quartermaster general, Department of the Army, Washington, D. C.

Mr. Milton Halem of New York University has accepted a po­sition as senior computing engineer with Republic Aviation Corpo­ration, Brooklyn, New York.

Mr. R. T. Heimer of Pennsylvania State University has been appointed to an assistant professorship at Lock Haven State Teachers College.

Dr. J, L, Holley of the United States Air Force Department has been appointed principal mathematician at Johns Hopkins University, Silver Spring, Maryland.

Professor Eberhard Hopf of Indiana University has begun a visiting appointment at the mathematics research center, University of Wisconsin,

Assistant Professor B. M. Ingersoll of San Diego State College has been appointed to an associate professorship at Lamar State College of Technology.

122

Mr. B. Y. C. Koo of the University of Maryland has accepted a position as mathematician with the Electromagnetic Research Corpo­ration, Washington, D. C,

Mr. W. E. Kopka of Syracuse University has accepted a position as staff mathematician with International Business Machines Corpo­ration, Poughkeepsie, New York.

Mr. G. F. Kottler of General Astonautics Corporation has ac­cepted a position as mathematics- systems engineer with the Radio Corporation of America, Moorestown, New jersey.

Dr. Ralph M. Krause of Harvard University has been appointed to an assistant professorship at the University of Illinois.

Dr. H. M, Lieber stein of CONVAIR has been appointed a mem­ber of the Army Research Center and an assistant professor at the University of Wisconsin, Madison, Wisconsin,

Mr. H. H. Love, jr. of the University of Delaware has accepted a position as mathematician with E. I. Dupont de Nemours and Com­pany, Inc., Wilmington, Delaware.

Dr. Edith H. Luchins of the University of Oregon has been ap­pointed a research associate at the University of Miami,

Mr. R. G. Mcintyre of Sohio Petroleum Company has been ap­pointed to an assistant professorship at the University of Arkansas.

Dr. A. W. Marshall of the University of Washington has been appointed to an acting assistant professorship at Stanford University.

Colonel J. D. Matheson of Melpar, Inc., has accepted a po.sition as associate mathematician with Analytic Services, Inc., Alexandria, Virginia.

Dr. C. N. Maxwell of the University of Illinois has been ap­pointed to an associate professorship at the University of Alabama,

Dr.]. S, Maybee of the University of Southern California has been promoted to an assistant professorship. He is on leave at the Institute for Mathematical Sciences, New York University.

Dr.]. G. Meiler has accepted a position as technical advisor with Bowaters Engineering and Development, Calhoun, Tennessee.

Dr. C. E. Miller of the California Research Corporation has accepted a position as staff mathematician with the Standard Oil Company of California, San Francisco, California.

Dr. W. L, Miranker of Bell Telephone Laboratories, Inc. has accepted a position as staff mathematician with International Busi­ness Machines Corporation, Yorktown Heights, New York.

Assistant Professor V. J. Mizel of the University of Tennessee has been appointed to an assistant professorship at Carnegie Institute of Technology.

Dr. W. D. Montgomery of CONVAIR has been appointed to an assistant professorship at Washington Missionary College.

Mr. R, A, Moore of Northrop Aircraft, Inc, has accepted a position as member of the technical staff of Space Technology Laboratories, Inc., Los Angeles, California.

123

Mr. D. E. Morrill of Western Electric Company has accepted a position as computing unit supervisor with Thiokol Chemical Corpo ration, Denville, New jersey.

Mr. M. M. Nanda of the University of Wisconsin has been ap­pointed a lecturer at Revenshaw College, Cuttack, Orissa, India,

Professor Fritz Oberhettinger of American University has been appointed to a professorship at Oregon State College.

Dr. R. R, O'Brien of Massachusetts Institute of Technology has accepted a position as advanced research engineer with Sylvania Electric Products, Inc., Waltham, Massachusetts.

Mr. R, E. Offenbaeker has been appointed to an associate pro­fessorship at Randolph-Macon College.

Dr. Daniel Orloff of Cornell Aeronautical Laboratory has ac­cepted a position as director of the information processing division of Dasol Corporation, New York, New York.

Mr. H. I. Ottoson of Northrop Aeronautical Institute has ac­cepted a position as mathematician with Bendix Aviation Corporation, North Hollywood, California.

Dr. Mary H. Payne of Fairchild Guided Missiles Division, Inertial Systems Research Laboratory has accepted a position as principal computing engineer with Republic Aviation Corporation, Farmingdale, New York.

Dr. P, A. Penzo of CONVAIR has accepted a position as mem­ber of the technical staff with Space Technology Laboratories, Inc., Los Angeles, California,

Dr. M. 0, Rabin of the Institute for Advanced Study has been appointed a lecturer at The Hebrew University, jerusalem, Israel.

Mr. George Rabinowitz of New York University has accepted a position as program analyst with International Electric Corporation, Paramus, New jersey.

Mr. G. E. H. Reuter of the University of Manchester, England, has been appointed to a professorship at the University of Durham, England.

Mr. N. S. Rosenfeld of Syracuse University has been appointed a lecturer at City College, New York, New York.

Mr. D. J. Ross of Operations Research, Inc. has been appointed vice president of the Scientific Planning Associates Corporation, Silver Spring, Maryland.

Reverend C. H. Rust of Xavier University has been appointed to an associate professorship at Loyola University, Chicago, Illinois.

Miss jean E. Sammet of Sperry-Rand Corporation has accepted a position as supervisor, MOBIDIC Programming Section of Sylvania Electric Products, Needham, Massachusetts.

Mr. W. M, Sanders of the University of Illinois has been ap­pointed to an associate professorship at Mississippi Southern College.

124

Mr. A. B. Schacknow of Republic Aviation Corporation has ac­cepted a position as head of the computational laboratory of the Arma Division of American Bosch Arma Corporation, Garden City, New York.

Mr. E. M, Scheuer of the Naval Ordnance Test Station has ac­cepted a position as member of the technical staff of Space Tech­nology Laboratories, Inc., Los Angeles, California.

Dr. S, H. Schot of the University of Maryland has been appointed to an assistant professorship at American University.

Mr. B. L. Schwartz of the Battelle Memorial Institute has ac­cepted a position as mathematician with Technical Operations, Inc., Monterey, California.

Dr. Sol Schwartzman of Johns Hopkins University has accepted a position as mathematician with RIAS, Inc., Baltimore, Maryland.

Assistant Professor N. E. Sexauer of Ohio University has been appointed to an assistant professorship at the University of Utah.

Dr. S.M. Shah of Muslim University, India, who has been a visiting professor at the University of Wisconsin for the first sem­ester of this academic year, will be a member of the Mathematics Research Center, University of Wisconsin, for the second semester.

Assistant Professor Abe Shenitzer of Rutgers, The State Uni­versity has been appointed to an associate professorship at Adelphi College.

Dr. A, H. Smith of the University of Southern California has been appointed to an assistant professorship at Long Beach State College.

Dr. R. C. Spencer of Sylvania Electric Products, Inc, has ac­cepted a position as principal staff engineer with The Martin Company, Baltimore, Maryland.

Dr. T. H. Starks of Virginia Polytechnic Institute has accepted a position as statistician with E. I. du Pont de Nemours and Company, Wilmington, Delaware.

Dr. Fritz Steinhardt of Barnard College has been appointed to an assistant professorship at City College, New York, New York.

Mr. Irwin Stoner of Raytheon Manufacturing Company has ac­cepted a position as senior engineer with the American Bosch Arma Corporation, Garden City, New York.

Dr. Louis Sucheston of Wayne State University has been ap­pointed to an assistant professorship at the University of Rochester.

Mr. H. W. Sullivan of David Bogen Company has accepted a position as engineering department head of Olympic Radio and Tele­vision, Long Island City, New York.

Mr. Melvin Tainiter of Fair child Guided Missile Division has accepted a position as associate engineer with the American Bosch Arma Corporation, Garden City, New York,

125

Miss Eileen J, Theisen of North American Aviation, Inc. has accepted a position as applied mathematician with Bell and Howell, Chicago, Illinois.

Dr. D. E. Thoro of the University of Florida has been appointed to an assistant professorship at San Jose State College.

Dr. S. F. Tuan of the University of California has been appointed a research associate and director of the Enrico Fermi Institute for Nuclear Studies at the University of Chicago.

Professor J. W. Tukey of Princeton University and Bell Tele­phone Laboratories has been appointed director of research in com­munications principles at the Laboratories.

Professor J, G. van der Corput of the University of California has been appointed a professor and staff member of the mathematics research center at the University of Wisconsin.

Assistant Professor L. M. Weiner of De Paul University has accepted a position as research engineer with American Machine and Foundry Company, Chicago, Illinois,

Mrs. Edith S. Windsor of New York University has accepted a position as mathematician-programmer with International Business Machines Corporation, New York, New York.

Dr. Michael Yanowitch of New York University has been ap­pointed to an associate professorship at Adelphi College.

Mr. Herman Zabransky of Ford Instrument Company has ac­cepted a position as senior engineer with the Radio Corporation of America, New York, New York.

Dr. R. A. Zemlin of Remington Rand Corporation has accepted a position as senior mathematician with Standard Oil Company of California, San Francisco, California.

Mr. Abraham Zukerman of the California Institute of Technology has accepted a position as research engineer with Aerojet General Corporation, Azusa, California.

The following promotions are announced:

Winifred A. As prey, Vassar College, to a professorship. J, E. Bearman, University of Minnesota, to a professorship. S. D. Bernardi, New York University, to an associate pro-

fessorship. H. J, Cohen, City College, New York, New York, to an assistant

professor ship, Ellen Correl, University of Maryland, to an assistant professor-

ship. Jane Cronin, Polytechnic Institute of Brooklyn, to an associate

professorship. Arwel Evans, University of Western Ontario, to an assistant

professorship. I. L. Glicksberg, University of Notre Dame, to an associate

professorship.

126

Dr. J, H. Griesmer, International Business Machines Corpo­ration, to staff mathematician in the Information Research Depart­ment at the Lamb Estate Research Center.

Frank Harary, University of Michigan, to an associate pro­fessorship.

E. R. Immel, Georgia Institute of Technology, to an associate professorship.

Florence D. Jacobson, Albertus Magnus College, to an assist­ant professorship.

Constantine Kassimatis, Cornell University, to an assistant professorship.

ship.

M. W. Keller, Purdue University, to a professorship. Jacob Korevaar, University of Wisconsin, to a professorship. L. L, Lassen, Arlington State College, to a professorship. Howard Levene, Columbia University, to an associate professor-

Dr. D.P. Ling, Bell Telephone Laboratories, Inc., to director of military analysis, Whippany, New Jersey

R. G. Long, Wesleyan University, to an assistant professorship. R. M. McLeod, Duke University, to an assistant professorship. Angelo Mar gar is, Ohio State University, to an assistant pro-

fessorship. Lawrence Markus, University of Minnesota, to an associate

professorship. Samuel Melamed, McGill University, to an assistant professor-

ship. z. A. Melzak, McGill University, to an assistant professorship. Dr. H. D. Mills of Market Research Corporation of America to

president of MATHEMATICA., a subsidiary of Market Research Corpo­ration of America, Princeton, New Jersey.

C. C. Gehring, University of Tennessee, Knoxville, to an assist­ant professorship.

G. W. Patterson, University of Pennsylvania, to an associate professorship.

ship.

George Piranian, University of Michigan, to a professorship. F azlollah Reza, Syracuse University, to a professorship. T. R. Richards, Wilkes College, to an associate professorship. W. G. Rosen, University of Maryland, to an assistant professor-

Leo Sario, University of California, Los Angeles, to a pro­fessorship.

J. A. Steketee, University of Toronto, to an assistant professor-ship.

Doris S. Stockton, University of Massachusetts, to an assistant professorship.

127

W. H. Warner, University of Minnesota, to an associate pro­fessorship.

Robert Weinstock, University of Notre Dame, to an associate professorship.

The following appointments to instructorships are announced:

University of British Columbia: Dr. Rimhak Ree; City College, New York: Dr. Bernard Sohmer; Foothill College, Mountain View, California: Mr. R. B. Merkel; Lehigh University: Mr. F. c. Oglesby; Los Angeles State College: Mr. E. J. Eckert; Massachusetts Insti­tute of Technology: Mr. J, R. Tessmer; University of Michigan: Mr. W. B. Woolf; Michigan State University: Mr. D. R, Lick; University of Minnesota: Dr. W, A. Harris, Jr.; Pasadena City College: Mr. T. E. Sydnor; Reed College: Mr. M. L. Weiss; Rut­gers, The State University: Mr. T. H. MacGregor; Salem College, West Virginia: Mr. M.G. Sperry; University of The South: Mr. J, P. McAllister; Southern Illinois University: Mr. Arnold Seiken; United States Air Force Academy: Captain R. L. Eisenman: Wheaton College, Wheaton, Illinois: Mr. S. F. Ebey.

Deaths:

Mr. H. C. Arnold of the Federal Enameling and Stamping Com­pany died on September 27, 1958 at the age of 66 years. He had been a member of the Society for 12 years.

Dr. F. G. Fisher of the Bureau of Ordnance, Department of the Navy, Washington, D. C. died on June 7, 1958 at the age of 48 years. He had been a member of the Society for 19 years.

Associate Professor 0. M. Rasmussen of the University of Denver died on June 20, 1958 at the age of 45 years. He had been a member of the Society for 11 years.

Professor Wilhelm Suss ·of the University of Freiburg, Germany, died in May, 1958 at the age of 63 years. He had been a member of the Society for 5 years.

128

LETTERS TO THE EDITOR

Editor's Note: Authors of letters to be published in this department of the NOTICES are asked please to limit themselves to 1,000 words. Longer letters will be published only by direction of the Council.

Editor, the NOTICES.

The much conforming non- conform ism of Dr. Helson in his February 1959 letter to the Editor has amused me sufficiently to write the following note.

I hope it was only the spirit of the times that made Mr. Helson think that his knowledge of both politics and the structure of national defense exceeds that of Congress.

In addition may I say that his "psychological requirements for research" seem to be quite meaningless in light of the great strides that Russian scientists are making, in spite of definitive action upon them for being suspected of any anti-communist activities. Their .!_ife depends upon signing an anti-non-communist affidavit -- but evidently little can they complain.

We should be proud to have a Constitution to defend. At least we are in a position to do something about our grievances (like pub­lishing letters to the Editor).

Instead of complaining about "the system", may I suggest to Mr. Helson to contact his Congressman and give him some sugges­tions, and put in no uncertain terms that he will not receive Mr. Helson's vote unless some action is taken on the suggestions.

Editor, the NOTICES.

A. A. Mullin Urbana, Illinois

One of the aspects of research to which too little thought has been devoted is the matter of titles for papers. There are doubtless no principles to which everyone will subscribe, but I propose to give the advice that I should like to give to many of the authors whose papers pass through my own editorial hands, and should give individu­ally except that I am too lazy.

In my view, the purpose of a title is to appear on the cover of the journal, in the volume index, and in abstracting journals, and to give the potential reader some idea of whether or not he should look at the paper. The title should be as short as is consistent with this aim, preferably capable of fitting into a single line of type, and should not attempt to serve as an abstract. A long title is wasteful of space and wasteful of editorial time, both in the original journal and in ab-

129

stracting journals. The following examples, illustrating various points in title-theory, are (for obvious reasons) made up ad hoc.

"A certain property of the empty set." "A certain" usually means "A". English is fortunate in having an indefinite article: use it, Unfortunately English has no plural indefinite article, but "some" will usually serve, if indeed anything is needed at all. In "The exis­tence of certain measures" the author is just too lazy to be specific. "Measures in non-Archimedean rings" would be more informative; whether the measures exist or not will appear in the paper.

"On a theorem of G. H. Hardy." Hardy has many theorems. Is this one about Dirichlet series, number theory, inequalities, or what? If it is about Dirichlet series, "A theorem of G. H. Hardy on Dirichlet series" would be better; but why include Hardy's name, except for the sake of fame by association? "A theorem on Dirichlet series" is still better, because shorter and equally informative. "The absolute convergence of Dirichlet series" is even better. (Pre­sumably all papers are about theorems.) However, "Schwarz's lemma" or "the Riemann hypothesis" are admissible in titles, since there is only one of each of these things.

"On ..•. " Some editors deprecate the use of "on" at the begin­ning of titles. However, "On the Riemann hypothesis" suggests that you have something to say about it, while "The Riemann hypothesis" might suggest that you have proved it, so there is a legitimate place for "on." If you have proved the Riemann hypothesis, on the other hand, it is false modesty to hide behind "On the zeros of the zeta function," and "Proof of the Riemann hypothesis" is the proper title. In any case, if you feel th'at you must say "on," say "on", and do not indulge in periphrases like "concerning" or "notes on." When in doubt, leave it out.

"On a paper of Bourbaki." Which paper, and what was it about? If it was about minimal left ideals in quasi-rings, call your paper "Minimal left ideals in quasi-rings." If you are trying to imply either that your contribution wouldn't be worth while if it weren't connected with something already done, or that Bourbaki 'is incompetent and that you are going to set him right, this is best left to the body of the paper.

"On a problem of Bourbaki." What is the problem about? You can justify your choice of problem by referring to Bourbaki in the text. "Bourbaki's problem on multi-additive functions" is better; "The characterization of multi-additive functions" is better still.

"The enumeration of the delicate points of ankylosed mani­folds." This would be good if it didn't introduce two unfamiliar terms in the title. As it stands, it doesn't say more than "Manifolds." Similarly, "A.p. functions on B-local subsets of HOLC spaces" is apt to be baffling. Avoid abbreviations that are meaningful only to the cognoscenti.

130

"Distributions." This is nice and short; but are they probability distributions or Schwartz distributions? Brevity can be overdone.

"A brief remark concerning the fundamental theorem of topo­logical loops." If you concoct a title like this you are presumably saying, "I don't know whether this is worth publishing but anyway it isn't very long and it does have to do with an important subject." If you want to publish it, and the referee will let you, then go ahead, but don't apologize. Call it "The fundamental theorem of topological loops."

Continued stories. There is no reason for not having a series of papers, e.g. "Fourier integrals, I;" Fourier integrals, 11;" ... There is usually no valid reason for using subtitles too. Either the subtitle is redundant or the series title is.

Avoid formulas in titles. They are difficult to get printed or copied correctly.

Finally, there may be valid reasons in particular cases for violating any or all of the principles I have advocated; all that I am really asking is that they should be violated only intentionally.

131

R. P. Boas, Jr. Northwestern University

MEMORANDA TO MEMBERS

A NEW POLICY CONCERNING AUTHORS' ROYALTIES

The Council and Trustees have decided that in the future the Society will pay royalties to authors of books in the COLLOQUIUM SERIES and the MATHEMATICAL SURVEYS SERIES; and also in the PROCEEDINGS OF SUMMER SEMINARS IN APPLIED MATHEMA­TICS when the proceedings are published by the Society. The royalty rate will be a flat 15 per cent of list price, including foreign sales. The action will apply to all books in these series that are printed or reprinted after january 1, 1959, including those books which are in process at that time but have not been priced.

It is hoped that this action will in particular provide a new stimulus to the SURVEYS SERIES, which has been dormant for some time. It is also hoped that potential authors for the COLLOQUIUM SERIES will also be encouraged to accelerate their work on their manuscripts. The Society now has ample working capital to invest in worthy book publication projects.

Prospective authors will doubtless notice that the Society's royalty rate is comparable with ordinary commercial practice for advanced technical monographs and is perhaps even more generous than usual because of the inclusion of foreign sales in the 15 per cent rate. A question might arise in a prospective author's mind as to whether the Society can sell as many copies of a book over a given length of time as the top commercial publishers do. Valid compari­sons are somewhat elusive because of a certain lack of parallelism between the Society's books and commercially published advanced mathematical books. (The Society of course is not consciously in the textbook business.) However, some data are available from the PRO­CEEDINGS OF THE SYMPOSIA IN APPLIED MATHEMATICS, some of which were published by the Society and some by a well- known and highly regarded commercial publisher. These data all seem to indi­cate that in the case of an advanced monograph of interest mainly to mathematicians and applied mathematicians, the Society will sell about as many copies as a commercial publisher. (Of course, this probably would not be the case for an advanced monograph written to appeal mainly to, say, engineers or physicists.) In the case of some of the COLLOQUIUM numbers, the Society has sold as many as four to six thousand copies over ten or fifteen years, which is considered to be very good for an advanced technical monograph.

In any case, a step-up in the Society's book promotion is now taking place. Exchange agreements permitting the advertising of books as well as journals have been concluded with nine outside scientific journals; new catalogues are being designed; and a very

132

large mailing list for direct mail advertising is being compiled. The list prices of the Society, on which royalties will be com­

puted, are now roughly comparable to those of the university presses or the more price-conscious commercial publishers of mathematical treatises. Many of the Society's sales are made at the members 1

prices, which are 25% off of list, but of course authors will get the full royalties on these sales.

For the convenience of prospective authors, the names of the members of the current SURVEYS and COLLOQUIUM editors are listed below:

COLLOQUIUM Committee

Salomon Bochner, Chairman Nathan Jacobson E. J. McShane

MATHEMATICAL SURVEYS Committee

I. J. Schoenberg, Chairman Irving Kaplansky S.M. Ulam

PRELIMINARY INFORMATION CONCERNING THE SUMMER MEETING IN SALT LAKE CITY

At the request of the Chairman of the Local Arrangements Committee, the following advance information is being published concerning the 64th Summer Meeting of the Society, to be held in Salt Lake City, Utah, on September 2-5, 1959. A more complete pre­liminary announcement will appear over the signature of the cogni­zant Associate Secretary as usual in the June issue of the NOTICES.

Registration headquarters will be in the lobby of the Union building and will be open Sunday afternoon and evening August 30, from 2:00P.M. to 10:00 P.M. and Monday through Friday from 8:30 A.M. to 5:00P.M. Those arriving after the registration desk is closed and having dormitory reservations may go directly to Ballif Hall. As usual, a directory of all persons registered, an information desk, a mail desk, and a textbook exhibit will be located in the regis­tration area.

Meals will be served on the campus throughout the week of the meetings. Dormitory accommodations will be available; rooms may be occupied from Sunday evening, August 30, until the following Satur­day. There are 525 spaces available, mostly in double rooms. The prices are $3.50 for a single room and $2.50 per person for a double room.

There are no hotels or motels within easy walking distance of the campus. Those listed below are within ten minutes by car. The

133

campus is about twelve minutes by bus from the business district. It is advisable that hotel and motel reservations be made by August 1. In general the motels require no deposit if arrival is to be prior to 6:00 P.M.

HOTELS (business district)

Newhouse DA 8-8366

Temple Square EL 5-2961

Utah DA8-9114

4th South and Main

75 West South Temple

South Temple and Main

* s $6.50 D 8.00-$10.00

s 5.00- 7.50 D 6.50- 8.50

S 8.00 up D 10.00 up

MOTELS (business district)

Covey's New America 522 South Main EM 3-6781

Utah Motor Park EL 5-2987

972 South State

D 8.50-12.00

s 6.00- 7.00 D 7.00-12.50

MOTELS (SE of campus, on or near U.S. 40)

Bonneville 1325 Foothill s 5.00 HU 7-7551 D 6.00-10.00

Free 2665 Parley's Way s 6.00 HU 4-3431 D 7.00-10.00

Scenic 1345 Foothill D 5.00-12.00 IN 7-1528

Skyline 1635 Foothill *S 6.00 IN 6-0728 D 7.00-10.00

*Rates apply to people attending meetings

The reader's attention is invited at this point to the reservation blank on page 199 of this issue of the NOTICES. It would assist the Local Arrangements Committee if all those having firm plans to at­tend would fill out the registration form and return it now to Professor W. ]. Coles, Department of Mathematics, University of Utah, Salt Lake City, Utah, whether or not they wish dormitory accommodations. It is particularly desirable for those who do wish dormitory accom­modations to apply for it as early as possible. In any case the reser­vation forms are returnable by August l.

There are 85 to 100 Camp sites within 30 to 40 minutes by car from the campus, located in various canyons at altitudes ranging from 5000 to 9000 feet; between 20 and 25 of these can be reserved. The

134

average m1mmum temperature during the first week of September is 59 degrees at 5000 feet and 37 degrees at 9000 feet; the maxima are 87 degrees and 67 degrees. Further information will be sent to those expressing interest on the reservation blank.

There will be a smorgasbord picnic at Brighton on Wednesday afternoon. The charge will be about $3.00 per plate. For a small charge, individuals may ride the ski lift to the top of Mt. Millicent. At this altitude, warm clothing is needed in the late afternoon.

On Thursday evening there will be a program at Temple square especially for persons attending the meetings. There will be a talk by Richard Evans and a few selections by the Mormon Tabernacle Choir, with Dr. Alexander Schreiner at the organ.

Baby sitters will be available. Salt Lake City is served by Western and United Airlines from

the east and west, Bonanza and Frontier Airlines from the north and south, and the Rio Grande and Union Pacific Railroads. The railway and bus depots are near the business district; there is limousine service from the airport.

The principal highways through Salt Lake City are U.S. 40, 90, and 89. Members may wish to plan their trips to include some of the following: Yellowstone, Grand Teton, Grand Canyon, Rocky Moun­tain, Bryce Canyon, and Mesa Verde National Parks; Zion, Cedar Breaks, Natural Bridges, Arches, Dinosaur, and Capitol Reef National Monuments; and Monument Valley. From Salt Lake City the extreme distances of the above places are 174 miles (Dinosaur) and 447 miles (Mesa Verde); the mean is 310 miles, with driving time of 6-7 hours. More information can be obtained from the Utah Tourist Council, Dept. 133, State Capitol, Salt Lake City, Utah.

The University of Utah is about 2 miles due east of the bus­iness district. City busses 4 and 5 pass within a block of the campus; the fare is 15¢ or fwo tokens for 25¢.

THE 1958-1959 COMBINED MEMBERSHIP LIST

The 1958-1959 Combined Membership List is extremely late this year, and the Executive Director feels that an explanation and an apology are due members. It is hoped that the distribution will have taken place by the time this issue of the NOTICES reaches the readers.

A combined directory of the MAA and AMS was first published in 1952. This directory listed 7,304 names. It was printed by what is known as the Flexoprint method. In this method, the names and relevant pieces of information are typed onto paper slips which are slipped into slots in metal frames. The information so exhibited is printed by the photo offset process.

The Flexoprint method was used for the succeeding directories up to the 1957-58 Combined Membership List. The method is

135

satisfactory provided that there are not too many names to handle. The bottleneck in the method is the final setting up of the frames, which can be done only after the deadline for address changes had been reached and (conveniently) only by one operator. Because of this bottleneck, the directory has been coming out later and later. Actually the only reason that the directory came out as early as it did several years ago was that Miss Kellar, then in the Editorial De­partment, was willing to work 80 hours a week on the frames during the entire month of November~

The current directory, which of course includes the SIAM membership as well as the MAA and the AMS membership, will list ll, 717 names. We decided last fall that the time had come to make a change, and after some investigation we found an English printer, William Clowes and Sons, who was willing to print the directory by letter press directly from a standard card file.

It was at this point that we made a miscalculation. It consisted in underestimating the time it would require to prepare a clean copy of the entire consolidated card file of ll, 717 names for the printer to use The trouble was that so many of the cards had to be retyped because they were full of repeated scratch-outs, style deviations, and irrelevant information.

The problem is a change- over "bug", and it is expected that next year's directory will appear at least in time for the annual winter meeting. Once again, apologies to the AMS members, and also to members of SIAM and MAA.

136

J. H. Curtiss Executive Director

ABSTRACTS OF COIITRIBUTED PAPERS

THE ANNUAL MEETING IN PHILADELPHIA, PENNSYLVANIA

January zo-zz, 1959

553-Z47. T. H. Southard: On the Chebyshev polynomials for the unit

square in the complex plane. Preliminary report.

Using results of Motzkin, the Chebyshev polynomials of low degree for

the set S • fz = x + iy;lxl ;;; l,lyl ~ lJ are determined, along with some of their

properties. Theorems useful in the determination of higher degree Chebyshev

polynomials for S are proven. (Received January 19, 1959.)

137

THE FEBRUARY MEETING IN NEW YORK, NEW YORK

February Z8, 1959

554-17. K. W. Kwun: A characterization of the upper semicontinuous

decompositions of sn having a countable number of nondegenerate elements

whose decomposition spaces are n-spheres.

Suppose G is an upper semicontinuous decomposition of sn such that the

decomposition space is ann-sphere. Contrary to the general case, if G has

only a countable number of nondegenerate elements, then each nondegenerate

element must be acyclic (Cech homology with coefficients in a field) and if

n ~ 3, it must have the open 3-cell complement. In this sense, the case of G

having a countable number of nondegenerate elements is more restrictive than

the other case. It is natural to ask how restrictive the former is. In this paper,

it is shown that there is essentially only one way to get such G. To show this,

a modified (extended) version of a result in the author's dissertation is used

(University of Michigan, 1958). It is also pointed out that there is a class of

such G whose nondegenerate elements are dense in G (hence, their union is not

a G8 set) and whose quotient maps are obtainable as the final stages of pseudo­

isotopies in the sense of Bing. (Received january 9, 1959 .)

554-18. Rafael Artzy: Relations between loop identities.

In a 3 -net, the effect of permutations of the three line families on a loop

(L, +),obtained from this net, is studied. Using equi-coordinate ("koordinaten­

gleich") mappings (Pickert, Projektive Ebenen, Springer, 1955, p. 54), a loop

(L,t9) ;;s Lp is set up for each permutation P. The assumption Lp~ (L, +)

leads to the validity of a loop identity r(P) for each P. Identities obtained, among

others, are respectively: right I. P.,left I. P., crossed-inverse property, auto­

morphic-inverse property, commutativity. If P 1 and Pz are two of the permu­

tations, simultaneous validity of r(P1 ) and r(Pz) implies r(P 1P 2), and this

yields a method for studying the relations between loop identities. The same

method, using 4;nets, is applied to the investigation of certain identities in

double-loops. (Received January lZ, 1959.)

138

554-19. R. C. Buck: A complete characterization of extreme functionals.

Let E be a real linear Sf>ace with a (semi) norm 1\ \1 • Let S be the set

of linear functionals of norm l. Then, the extreme points of Scan be charac­

terized as follows. With any L E S, we associate a nested sequence of sets

F 1 "::> F 2 "::> ••• ; then, L is extreme if and only if E = F k - Fk for every k. The

set F k is the collection of all x E E such that 1\x\1 - L(x) a 1/k. Immediate

application. Let M be a closed subspace of E, and let SM be those functionals

of norm l that vanish on M. Then, L in SM is extreme in SM if and only if

(Fk - Fk) + M = E for every k. (Received January 14, 1959 .)

554-20. Eckford Cohen: Asymptotic averages in a class of arithmetical

functions of two variables, II. An alternative approach.

A simpler and quite different method is used to obtain new proofs for the

asymptotic estimates proved in Part I for certain types of arithmetical func-

tions of the form f((m ,n)). However, it is necessary to consider separately the

case O"ol((m,n)), OC ~ l, where O"Oi!(n) denotes the sum of the OL-th powers of the

divisors of n. The estimates for the more general functions considered are

then reduced to those obtained for O"ol((m,n)). The error term obtained in I for

functions corresponding to the case 01. = l is improved from O(x312log x) to

O(x312). (Received January 12, 1959.)

554-21. Eckford Cohen: Partitions in homogeneous, finite abelian groups.

The concepts of relatively even and totally even function (mod r) are

reformulated in group-theoretical terminology. Applications are made to the

determination of the number N(OI.) of binary decompositions, OL = x + y, of an

element 0: in an abelian group Gr = Cr E9 Cr, where Cr is cyclic of order r,

and x ,y are subjected to various restrictions. (Received January 12, 1959 .)

554-22. L. I. Deverall: Fundamental frequency of clamped vibrating

plate by modified method of collocations.

The method of collocations requires that an approximating solution

satisfy the partial differential equation of vibration and the boundary conditions

at suitably chosen points; a modification of this method would be to take approxi­

mating functions which are solutions of the differential equation and then satisfy

the boundary conditions at points. In this study, the complex operator method

of S. Bergman (Duke Math. J. vol. 11 (1944) pp. 617-649) was used to generate

139

particular solutions to the vibrating plate equation; the approximating solution

was then taken to be a linear combination of these particular solutions (using

particular solutions with required symmetry for the particular mode of

vibration under consideration). Satisfaction of the boundary conditions at

points gives a system of linear equations. In order to assess accuracy of the

method, a calculation of the fundamental frequency of a clamped square plate

(length of side = a) was done in which the relevant boundary conditions were

satisfied at eighteen points on the boundary. This gave a value for the funda­

mental frequency £..l=35.96 oc,(ct:" = (l/a2(D/}l) 1 /~ D.flexural rigidity, p =mass

per unit area) which can be compared to the value obtained by D. Young

G) =35.99oC (J. Appl. Mech. vol. 17 (1950) pp. 448-453). (Received December

28, 1958.)

554-23. I. s. Gal: Proximity relations and precompact structures.

A proximity relation 1\ for a set X is an anti-reflexive and symmetric

binary relation on the power set of X such that A 1 A A2 U A 3 if and only if

A1 1\ A2 and A 1 1\ A3, and if A 1 1\ A2 then there exist sets Bi (i - 1 ,2) such

that Ai 1\ cBi and B 1 1\ B 2. If for disjoint finite sets Ai (i = 1 ,2) the relation

A 1 /\ A2 holds then 1\ is called separated. These relations were introduced by

Efremovic. We have the following: There is a natural order preserving one­

to-one correspondence between the proximity relations and precompact struc­

tures of a given set. This correspondence maps the set of separated relations

onto the set of separated structures. This result is an extension of Smirnov's

theorem. The proof is quite different from Smirnov's. As further results we

have: .!!... " corresponds to the pre compact structure 'U and xi is the closure

of At in the compactification with respect to U then A1 1\ A:t, if and only if

A1 and A2 are disjoint. In the next theorem the relations corresponding to the

structures 7L and ZJ- are both denoted by 1\ : The map f: X - Y is uniformly

continuous with respect to the pre compact structures U and Q< if and only if -1 -1

B 1 A Bz implies f (B 1> 1\ f (B2) for every B 1 ,B2 ~ Y. This result can be

applied to prime ends and Freudenthal compactifications. (Received January

15, 1959.)

140

554-24. M. L. Glasser: Some addition formulas for periodic differential

equations.

Certain addition formulas have been derived for solutions of one dimen-

sional Schroedinger type equations with even periodic potential functions. For

example, if X andY are the even and odd solutions of a fundamental set for

such an equation, then X(2a) = X(a)Y'(a) + X'(a)Y(a) where the prime denotes

differentiation with respect to the argument and 2a is the period. The formulas

greatly simplify the computation of Mathieu dispersion curves and one dimen­

sional energy bands. (Received January 12, 1959.)

554-25. R. R. Goldberg: Averages of Fourier coefficients.

Up~ 1 and an =/.;'"f(t) cos nt dt, n = 1,2, .•• , for some f E LP(0,11')

then an will be called a p-sequence. A well-known theorem of Hardy states

that if an is a p-sequence and bn = n- 1(a1 + ... +an) then bn is also a

p-sequence. The following generalization of Hardy's theorem is proved: Let

Y..(x) be of bounded variation on 0 ~ x ~ 1. Then if an is a p-sequence and bn

= n -IE!= 1 Y'(m/n) am, then bn is also a p-sequence. (Received December 22,

1958.)

554-26. Edward Halpern: A theorem on Hopf algebras with divided

powers.

By a monogenic twisted polynomial algebra of binomial type (with integer

coefficients) is meant the free group generated by a sequence of elements

x 0 ,x 1,x2 , ••• ,xi•··· with multiplication defined by xmxn = ((m + n~)/(m~n~))x.m+n"

LetT be a torsion-free H-space with homotopy-associative and homotopy­

commutative multiplication. The following theorem is proved: If the Hopf

algebra H*(T) (integer coefficients) has divided powers (in the sense of H. Car­

tan) then it is isomorphic to a tensor product of monogenic exterior algebras

and monogenic twisted polynomial algebras of binomial type. (Received January

14, 1959.)

554-27. Simon Kochen: Reduced powers and completeness. I.

For the definition of the reduced power A I /D (or A I I := D) of a relational

system A see Abstracts 550-7 and 549-24. The ideal Din 2I is subsequently

assumed to be nonprincipal and prime. Two relational systems A and B are

power equivalent, in symbols A ~ B, if there exist reduced powers A I;D and

141

si' /D' of A and B such that A I /D ~ si';n•. A::::: B implies that A and B are

arithmetically equivalent. Thus, to prove the completeness of an (elementary)

theory T with no finite models it is sufficient to show that all models of T of a

fixed cardinality are power equivalent. By this method the theories of (i) real

closed fields, (ii) divisible ordered abelian groups, (iii) densely ordered sys­

terns without first or last elements, and (iv} infinite discretely ordered systems

without first or last elements are shown to be complete. These results were

previously obtained by the method of elimination of quantifiers (Tarski, Lang­

ford) or by the method of diagrams (A. Robinson). As a byproduct of the proof

of (iv), an arithmetical extension of the natural numbers is exhibited, which,

assuming the continuum hypothesis, has order type 6.l+ (w* + W) '1 1• (Received

January 15, 1959.)

554-28. Simon Kochen: Reduced powers and completeness. II.

For terminology see previous abstract. (i) Let A and B be two real

closed fields of cardinality ~ c. Any two reduced powers AI /D and BI /D',

where I is denumerable, are real closed fields of cardinality c which are

'Y!1-sets. Hence, c = i't1 implies by a known isomorphism theorem (See Ann. of

Math. vol. 61, p. 543, Theorem 2.1) that AI;n ~ si;n• i.e. A::::: B, and hence

that the theory of real closed fields is complete. The continuum hypothesis can

then be eliminated by considering Go del's model b. for set theory, as pointed

out to the writer by D. Scott. The proof of (ii) is similar to the above, and

requires the result obtained by the writer that any two divisible ordered abelian

groups of cardinality st'O< (01. > 0) which are l'\0(. -sets are order isomorphic.

Case (iii) is similar to (i) but simpler. For (iv) the reduced power AI/D, where

A is an infinite discretely ordered system of cardinality ~ c and I is denumer­

able, is isomorphic to one of the four systems of order type (w* +W)ft1•

W+ (w* +1..))'1(1, (w* +W)'I'!1 +w*, or W+ (w* + 1..)))1,1 +W* (assuming c = ">t1).

Thus, every discretely ordered system is arithmetically equivalent to one of

type 1,2,3, ..• ,1.)* +w,w,tJ', or W+W*. (Received January 15, 1959.)

554-29. Anthony Mardellis: The monodromic group and the Picard­

Vessiot theory. II. Preliminary report.

Ludwig Schlesinger in Handbuch der Theorie der Linearen Differential­

gleichungen, Leipzig, 1897, makes the following two statements: (A) "The

Picard- Vessiot theorem remains completely correct when one replaces in it

the group of transformations g by its denumerable subgroup M, if and only if

142

the differential equation belongs to the Fuchs class." (B) "The monodromic

group M is in a position to replace the group of transformations g, in the case

of questions related to reducibility, only for differential equations of the Fuchs

class." These statements should read respectively as follows: (A') "The

f'icard- Vessiot theorem remains completely correct when one replaces in it

the group of transformations g by its denumerable subgroup M, if and only if

;I = g: 1". (8 ') "The monodromic group M is in a position to replace the

group of transformations g, in the case of questions related to reducibility, if

and only if J = 1o 1 ". The following question arises: What differential

equations, besides those of the Fuchs class, are such that ~ = :Jr 1 ? The author

shows that Hamburger equations with· n normal solutions: wi = ezz 't'i (deter­

mining factor ez, exponent 'Z'i) where 'l:'1, 'l:'2, ... , 'Ln are nonrationallinearly

independent over the rationals, have this property. (Received January 5, 1959,)

554-30, W. S. Massey: On th.e Stiefel- Whitney classes of a manifold,

Let Mn be a compact manifold and let Wi and Wi denote its ith Stiefel­

Whitney class and its ith dual Stiefel- Whitney class respectively (both mod 2).

Theorem 1. If Wn-q :j: 0, then there exist integers h~o ... ,hq such that

h ~ h ;;;:- ~ h 2: 0 d 2hl h2 hq . n 1 - 2 - ... _ q - an n = + 2 + ... + 2 • Moreover, 1f M is

orientable, the following 3 cases are excluded: (a) q = 1, (b) n s 2 mod 4 and

hq = 1, and (c) hi= hq + 1 for an odd number of the indices i. Corollary 1.

If Wn -1 'f: 0, then n is a power of 2 and Mn is nonorientable, Corollary 2. If

wn_ 2 :f:. 0, then n = 2k(2h + 1) for nonnegative integers hand k; if Mn is orien-

h k table the cases n = 2(2 + 1) for h :> 0 and n = 3 • 2 are excluded, Corollary 3,

If n = 2r- 1, then Wi = 0 fori> n- r. Theorem 2, If n is even and Mn is

orientable, then Wn-1 = 0, Theorem 3, If n;; 3 mod 4 and Mn is orientable,

then Wn= Wn-1 = Wn-2 = 0, In the proofs of these theorems, one uses results

of Wu (C. R, Acad, Sci. Paris val, 230 (1950) pp. 508-511), Serre (Comm. Math.

Helv. val. 27 (1953) pp. 198-232) and J, Adem's relations on Steenrod squares,

(Received December 17, 1958 .)

554-31. Elliott Mendelson: On a proposed nonstandard model.

If M is a nonstandard model for Peano's Postulates (with recursion equa­

tions for addition and multiplication), let M' be the sub-semiring consisting

of all elements which differ from an infinitely divisible element by an integer,

Kemeny (Math, Ann. val. 135 (1958) pp. 160-169) shows that Goldbach's con-

143

jecture and several other open arithmetic problems are to be answered in the

negative in M', and asks whether M' is a nonstandard model, A negative answer

to this question results from the following proposition, Theorem. Let M be a

nonstandard model for Peano's Postulates, Let R be the ring obtained by ad­

joining "negative" elements toM. Then there is no ring homomorphism from

R onto the ring of integers. (Received January 5, 1959.)

554-32, Edgar Reich: The entropy of the sum of random variables,

Preliminary report.

If X is, say, a one-dimensional random variable with density f(x) and CX>

standard deviation a-x, we define the functional jJ(X) by logp(X) = -./-00 f(x)

log f(x)dx- log((21Te) 1/ 2a-x), p(X) is a measure of the "Gaussianness" of X, as

0;!! }l(X) :i 1, and p(X) = 1 if and only if X is Gaussian, and fJ(~ +P) = _.P(X)

if 0: =F 0, Theorem: p(X + Y) ii; p(X).fl(Y) if X, Y are independent, Conjecture:

sup ~(X + Y) jl(X) = a, jl(Y) = b} is a strictly increasing function of a and b.

If the conjecture is correct it provides a generalization of Cramer's unique­

tactorization theorem, (Received january 5, 1959.)

554-33. Oved Shisha: A characterization of certain nearest functions.

For the term "nearest function", see Abstract 553-158, these Notices,

vol, 5 (1958) p. 853, Theorem: LetS= {z 1,z2 , ... ,zn+l1 (n :1;' 1) be a set of

n + 1 (distinct) complex numbers and p 1 (z),Pz (z), ... ,pn+l (z) complex functions,

defined and linearly independent on S, Let p 1(z),p2(z), ... ,pn(z) be linearly

independent on every subset of S whose number of elements is n, Let Tf

denote the set of all complex functions expressible throughout S in the form

2::= 1 ').. Py (z) (c'V complex), and let V j(a1 ,a2 , ... ,aj) denote, for every j-tuple

(a1 ,az, ... ,aj) of elements of S (j = n, n + 1), the j-by- j determinant whose )) -th

row is p 1(ay) pz(aj)) ... p1(a11 ). A necessary and sufficient condition for a com­

plex function p(z) to belong to 1T, to be a nearest function to Pn+l (z) on S with

respect to 1T and to be 'I' Ih+l (z) throughout S, is the existence of positive

.>.1'~2 , ... ,f1n+l with sum 1, such that p(z) = Pn+l(z) + L:~11.>.'1J(- l)nH'vn+l(z 1,

z 2 , ... ,Zu-1 ,z, zLl+l , ... ,zn+l)/V n<z 1 ,z2 , ... ,z..,_ 1 ,zV+l , ... ,zn+a> throughout S.

(Received January 14, 1959.)

144

554-34, W, R. Utz: Boundedness of solutions of a linear equation,

This paper considers the boundedness of solutions, as t __.. oo, of the real

differential equation x" + (a/t)x' + q(t)x = 0, t :> 0, for certain differentiable

functions q(t) and real constants a, Although the equation is generally not

integrable in closed form it includes several interesting integrable equations

such as a well-known Euler equation, The following two theorems are the

principal results of the paper. If 0 :!i a ;;; 2 and if for all large t, q'(t) ;;!! 0

and taq(t) > o > 0, then each solution of the equation valid for all large tis

bounded as t-oo. If x = x(t) is a solution of the equation valid for all large t

and if a iii; 0, q'(t) 0!: 0 for all t ~ t 0 and q(t) > 0 for some value oft;;;: t0 , then

limt.,.00x(t) = 0, (Received January 14, 1959.)

554-35, J, G. Wendel: Order statistics of partial sums, II.

The methods sketched in a previous abstract (552-5, NAMS vol. 5 (1958)

p. 693) are applied to obtain two probabilistic identities, slightly generalizing

known ones that had been obtained combinatorially. They are both of the form

A(n,k) = A(n - k,O) A(k,k). (1) A(a; n,k) = E(I(n,k)exp iO"Sn), where I(n,k) is 1

or 0 according as the number of positive terms in the sequence of partial sums

S1, ... ,Sn is k or not; IT= 0 recovers an identity due toE, Sparre-Andersen

(Skand, Aktuarietidskr. (1953) pp. 123-138), (2) A<p,a; n,k) = E(exp i[,PZn,k

+ aSnJ), in which Zn,k = kth largest among 0 = So,S1, ... ,Sn(Zn,n = max);

Bohnenblust, Spitzer and Welch discovered the case !T == 0, but do not seem to

have published the result; it was brought to the author's attention by Spitzer.

(Received January 2, 1959,)

554-36. H. E. Salzer: Hermite's general osculatory interpolation formu­

la and a finite difference analogue,

Expressions for Hermite's general osculatory interpolation polynomial,

or P(x) of degree [:f. 1 ri- 1, for a function f(x) which with its first ri- 1 • tr·-1)

derivatives assumes the values fi,fi, ... , ft 1 at the generally unequally spaced

points Xi, i c 1,2, ... ,n, have been given by C, Hermite, P. Johansen, W. Simon­

sen, T. N. E, Greville and J, Kuntzmann, A formula for P(x), whose derivation

is more direct and motivated, is presented here for the special case where

every ri - 1 ,. r - 1. The essential coefficients are found from a simple

triangular system of linear equations. A decomposed and rearranged form leads

to an algorithm better suited for calculation. The finite difference analogue of

145

P (x) is the polynomial Q(x) of degree rn - 1 having prescribed values f1 and

advancing differences ~jfi at xi, j - l,Z, ••. ,r- 1, i = 1,Z, .•. , n. All prescribed

differences are for the same intervals in x, even though the xi may be irregular­

ly located. The derivation of the expression for Q(x) is suggested by the given

development of P(x). From the uniqueness of Q(x) one obtains also a decom­

posed form which is computationally shorter. (Received February 19, 1959.)

146

THE APRIL MEETING IN CHICAGO, ILLINOIS

April 17-18, 1959

555-1, P. C. Hammer and H. H. Wicke: Quadrature formulas involving

derivatives of the integrand.

The existence of quadrature formulas of the following type is demon

strated. Fork an odd positive integer: .Ji f(x)dx = 2L(f~6)/2f(2i)(0)/(2i + 1)~ + L:j, 1aJf(k)(xj)- f(k)(- xj)) + Rm,k(f). A similar formula is developed for

even k. The formulas are exact for polynomials of degree at most 4m + k for

odd and 4m + k - 1 for even k, The formulas are established by considering

the even and odd components off and reducing the problem to one to which the

theo:r:y of orthogonal polynomials can be applied, From this the existence,

reality, and distinctness of the x j• and the positiveness of the a j are obtained.

The specific polynomials used are orthogonal over [0, l] with respect to the

weight functions w(x) = (1 - x 112)k (k odd);w(x) = (1 - x 1 /~k;x1 12 (k even),

which have not been identified with any treated in detail in the literature. Ex­

pressions for the remainders Rm ,k(f) are also derived. (Received January 19,

1959.)

555-2, R, F, Rinehart: Elements of a theory of intrinsic functions on

algebras.

To provide a function theory on a linear algebra which is more sensitive

to the ring character of the algebra, the concept of intrinsic function is pro­

posed. A function ..E on a ring~ is called intrinsic on .R if for every automor­

phism or antiautomorphism D of Ji, the relation F(Q5) = Q F( ~)is satisfied

for each admissible s, Such functions are shown to fulfill a set of F antappill

conditions. Further, if Ji is a finite dimensional associative algebra with identity

over the real or complex field and .E is a primary function on A (the extension

of a scalar function to A), then .E. is intrinsic, The special case of the algebra

Q. of real quaternions yields: (1) Every intrinsic function on g_ is primary (2) If

the quaternion argument is written as s = x0 + p p., where p. is a unit three-dim­

ensional vector, then an intrinsic function.E. has the form u(x0 ,p) + v(x0 ,p)p..

147

Thus for arguments s which lie in a fixed plane through the real axis of the

four-dimensional space g_, intrinsic function theory becomes ordinary complex

function theory in that plane. Among the consequences are simplified points of

view on several problems treated in the literature. (Received February 9, 1959.)

555-3. G. M. Ewing: Optimal burning programs for an idealized upward­

directed missile,

Given the maximal height Y and payload mass M 1> it is desired to mini­

mize initial mass Mo. Formulation of a model including definitions of classes

of admissible burning programs M and admissible trajectories Y yields a

Problem of Bolza not meeting classical hypotheses. A fairly complete theory

(believed to be the first such for even this simple case) is developed, Among

questions not answerable by treatments which (without explicit hypotheses or

precise statement of the problem) confine attention to the formalism of the first

variation are: Can an optimal M have intervals of constancy? Is such M neces­

sarily discontinuous at t - 0? Can such M have discontinuities for 0 < t < t 1 =

burnout time? Or at t1 1 For the present model the last question remains un­

settled. Respective answers to the others are no, yes, no. The paper applies

with minor changes to the related problem: Given Y and Mo to maximize M1.

Given Mo and M 1 to maximize Y. The author is indebted to H. S. Tsien and

R. C. Evans, Optimum thrust programming for a sounding rocket, journal

American Rocket Society vol. Zl (1951) pp. 97-107, which is very suggestive yet

inconclusive because of a variety of defects. (Received February 13, 1959.)

555-4. M. W. Hirsch and Stephen Smale: On involutions of the 3-sphere.

Let s 3 be the unit sphere in Euclidean 4-space E4 with coordinates

x1, ... ,x4. An involution is a homeomorphism of period two, It is shown in the

work of P. A. Smith that the set of fixed points of an involution of s3 form a

sphere of dimension d, -1 ~ d ::5, 3. Theorem 1. If T:S3 - s 3 is an involution

with exactly two fixed points, then there is a homeomorphism g:S3 -+ s 3 such -1 that g Tg(x1 ,xz,x3 ,x4) = (- x 1,- xz,- x 3,x4). In the course of the proof we

obtain the following results: Theorem Z. Let M be a nonorientable triangulated

3-manifold, If there exists an element ft in the fundamental group of M which

reverses orientation and such that tf = 1, then the projective plane can be

embedded in M piecewise-linearly. Theorem 3, LetT be an involution of a

3-manifold M with an isolated fixed pointy, There is an open neighborhood

148

V of y and a homeomorphism h of E 3 onto V such that h- 1Th(x) = -x. The

proofs depend strongly on the results and methods of Papakyriakopoulos (Ann.

of Math. (1957) pp. 1-26) and Shapiro-Whitehead (Bull. Amer. Math. Soc. (1958)

pp. 174-178). (Received February 19, 1959.)

555-5. R. H. Rosen: A decomposition of 3-space that cannot be imbedded

in 4-space.

R. H. Bing and M. L. Curtis have given an example of a decomposition of

E 3 whose only nondegenerate elements are twelve circles (subsequently the

number of circles has been reduced to nine) so that the decomposition space

cannot be imbedded in E 4• (Abstract 551-9 Notices Amer. Math. Soc. vol. 5

(1958) p. 685). By modifying their example and slightly strengthening a result

of Flores (Ueber n-dimensiona1e Komplexe, die im R2n+1 absolut selbst-ver­

schlungen sind, Ergebnisse Bines Mathematischen Kolloquiums, vol. 6, 1933-

1934, pp. 4-6) the author constructs a decomposition of E 3 whose only nonde­

generate elements are six circles such that the decomposition space cannot be

imbedded in E4 • The six circles consist of a triple J 1 , J 2 , and J 3 each pair of

which are linked and another triple Kl' K2 , and K3 , Ki linking Ji (i • 1,2,3) but

no other element of the decomposition. (Received February 26, 1959.)

555-6. N. F. G. Martin: A note on metric density of sets of real numbers.

Casper Goffman (Proc. Amer. Math. Soc. vol. 1 (1950)) proved that the set

of points at which the metric density of a measurable set of real numbers exists

but is not zero or one is a set of the first category. As a partial converse he

then constructed a set which had density of 1/2 at every point of an arbitrary

Fa- of measure zero. By modifying the method of Goffman's construction a set

is obtained whose metric density exists and is equal to a given 8, 0 <8 < 1,

at every point of an arbitrary Fll'" of measure zero. (Received February 25,

1959.)

555-7. J. Z, Yao: Conceptual proofs of two theorems of J, C. Moore and

H. Cartan.

In the theory of S. S. complexes, the definition of a fiber map p: E - B by

the extension property of Kan is equivalent to that by the covering homotopy

property, explicitly: given a pair of complexes Y c X with the injection i:

X X (e) U Y X I -+X X I, e = 0 or 1, and maps f: X X (e) U Y X I -+ E and g:

149

X X I ~ B so that p• f - g • i, then there is a map h: X X I-E which extends f

and covers g. Also for any triple of complexes U, V, and W, there is a natural

isomorphism between Map (U X V,W) and Map (U,Hom(V,W)). The definition

of a fiber map by the covering homotopy property and the natural isomorphism

are used to prove: (1) The Moore-Cartan Theorem- If p: E - B is a fiber map,

and C any complex, then p*: Hom(C,E) --*Hom(C,B) is a fiber map, (2) The con­

verse theorem-p: E -a is a fiber map if p*: Hom(C,E) ~Hom(C,B) is a fiber

map for some complex C. (3) A theorem of J, Moore-If K is a Kan complex and

DCC is any pair of complexes, then i*: Hom(C,K)-+ Hom(D,K) is a fiber map.

(Received February 23, 1959.)

555-8, D. F, Dawson: Concerning convergence of continued fractions.

"The continued fraction d 0 + c 0/d 1 + crfd2 + c 21CIJ+= converges ab­

solutely at least in the wider sense" means there exists a positive integer n

such that the continued fraction dn-l + cn_ 1;a;+c;;:!dn+l + ... converges

absolutely, where the ci and di are complex numbers. Let f(a) denote the

continued fraction l/T+ili/'f""+ii2/T+a3/I+-::: and fp denote the pth approxi­

mant of f(a). Theorem. If the even (odd) part of f(a) converges absolutely at

least in the wider sense, the odd (even) part converges, and lim infjf2p1 = oo

(lim infjf2p-ll = oo), then the series [] b2pl (the series Ll b 2p _1p con­

verges and lim suplb 1 + b3 + ... + b2P_ 11 < oo (lim suplb2 + b4 + ... + b2pl4!: oo).

As an application of this theorem it is shown that if there exists a sequence

f rp}~~ 1 of nonnegative numbers such that r 111 + a 11 ~ I a 1 I, r 211 + a 1 + a 2 I ~ 1a2 ,

rpP + ap-l + apl ;;: rprp- 21 ap-ll + 1 ap I• p = 3 ,4,5, ... , with actual inequality

holding either in the first relation or in the second, then either some ap = 0

and f(a) converges, or else no ap = 0 and the divergence of the series l:J bpI is

necessary and sufficient for the convergence of f(a). (Received February 27,

1959.)

555-9, Nickolas Heerema: Derivations and embeddings of a field in its

power series ring,

Let {7ri}f be a sequence of derivations on a field F of characteristic zero.

Let [77Jt 1, ... ,in represent the sum of aU the distinct products of the derivations

:7ri 1'"''7lln and let (1) Tn = L ([-rr]/k~)i 1 , ... ,ik where the sum is taken over all

sets of indices i1 , ... ,ik such that i 1 + ... + ik = n. The following theorem is

proved, Let Fftri\ denote the set of all power series of the form a+ ii'1(a)x

150

+ .•. + ffn(a)xn + •••. Then, given any automorphism a-+ a' on F, the mapping

a-+ a' + 7F1(a')x + •.• + tTn(a')xn + .•. is an isomorphism ofF onto F'. Con­

versely, every subring F' of the power series ring F[[x]] isomorphic to F is

an Ft11'i}for some sequence of derivations ftril· The correspondence Ff.,.iJ

-+{nil is biunique. This result is obtained by observing that every isomor­

phism ofF into F[[x]] is of the form a~ a'+ 171(a')x + .•. + ?Tn(a')xn + ...

where a -+a' is an arbitrary automorphism on F and the sequence of mappings

flT if is an arbitrary set of "embedding mappings" on F, the latter being defined

by the relations 7T1(a +b) =7ri(a) + 77i(b),7ri(ab) =L~=O"J(a)7Ti-j(b) (7trf.a) =a). Moreover, Relation (1) provides a 1 to 1 correspondence between

sequences of derivations f7ri} on F and sequences of embedding mappings on F.

Certain implications of this result regarding the automorphism group of F [[x]]

are discussed. (Received February 27, 1959.)

555-10. G. W. Henderson: Proof that if a decomposable compact con­

tinuum is topologically equivalent to each of its nondegenerate subcontinua,

it is an arc.

In 1921, S. Mazurkiewicz (Fund. Math. vol. 2 (1921) p. 285, problem 14)

raised the question whether it is true that if a continuum in space of m

dimensions is topologically equivalent to every one of its nondegenerate sub­

continua, then it is an arc. In 1948, making use of a certain plane indecom­

eosable continuum (Trans. Amer. Math. Soc. vol. 63 (1948) pp. 581-594),

E. E. Moise showed that this question may be answered in the negative. In the

present paper it is shown that for compact decomposable continua in a metric

space, the answer is affirmative. (Received February 27, 1959.)

555-11. Pasquale Porcelli: Weak convergence and compactness in

spaces of additive type functions.

Let X be a set, CT an algebra of subsets of X (i.e. cr contains X and is

closed under complementation and finite union), H(X,CT) the NLC space of

bounded additive (not necessarily completely additive) functions on CT, where

the norm is the total variation, and r the collections of all sequence {Eplp l: 1,

finite or infinite, such that Ep E cr and Ei • Ej = 0, i j j. Theorem 1. If

fn € H(cr,X), n = 1,2, ••• , then [fn}g:',. 1 is weakly convergent ~limnLp;:lfn(Ep)

exists for each sequence in r ; moreover, H(cr ,X) is weakly complete.

fheorem 2. If Ho is subset of H(X ,.r), then Ho is sequentially weakly compact

151

*'*. H0 is a bounded set limm,nLp~mfn(Ep) = 0 for every sequence

{fn}~=l E H0 and every sequence fEp}p~l E r; an equivalent way of formulat­

ing this is that every sequence {fn} E Ho contains a subsequencefgnJ such that

limnLp! gn(Ep)J exists for all {Ep} E r. (Received February 27, 1959 .)

555-12. J. A. Wolf: On groups which act without fixed points.

Preliminary report.

Let G be a compact connected Lie group, K a closed connected subgroup.

By a (p)-group, a finite direct product of cyclic groups of order p is meant. If

a finite subgroup r of G acts without fixed points on G/K, and, for every prime

p dividing the order of r, every (p)-subgroup of r lies in a torus of G, then

every abelian subgroup of r is a direct product of cyclic subgroups whos~ num­

ber is ::;; rank.G - rank.K. Let r be a given finite subgroup of G, TG a maxi­

mal torus of G which contains a maximal torus TK of K. Then certain number­

theoretic criteria involving the Weyl group of G and the way T K lies in T G

determine whether r acts without fixed points on G/K. These methods yield

specific results when, say, G = U(n) and rank.K = n- 1. This particular case

generalizes some of Vincent's work (Comm. Math. Helv. vol. 20). (Received

February 27, 1959.)

555-13. D. G. Johnson: A structure theory for a class of lattice-ordered

rings III. Preliminary report.

Notation and terminology are as in Abstracts 553-64 and 553-203. The

main question considered here is the following: "If A is an f-ring without

identity, can A be imbedded as an 1-ideal in an f-ring with identity?" Theorem

1. If A contains an element x such that Jxal /\ JaJ =lax!/\ JaJ = JaJ for every

a E A, then A cannot be imbedded as an _..!'-ideal in an f-ring with identity.

Theorem 2. A can be imbedded as an 1-ideal in an f-ring with identity if and

only if the ring A'= [(n,a): nan integer, a E AJ (where, as usual, (n,a) + (m,b)

= (n + m ,a + b) and (n,a)(m ,b) = (nm,nb + rna + ab)) is an f-ring under the order

given by: (n,a) ?; 0 if and only if (nd + ad) 1\ 0 = (nd + da) 1\ 0 = 0 for every

d ?; 0 in A. Theorem 3. If A is a (totally) ordered ring, then A can be imbedded

as an 1-idea1 in an f-ring with identity if and only if [abJ & Jail\ JbJ for all

a, b E A, in which case A can be imbedded as an 1-ideal in an ordered ring

with identity. The following more general question is considered: "If A is an

f-ring without identity, can A be imbedded as a right 1-ideal in an f-ring with

152

identity?"; and also the same question for imbedding as a sub-f-ring.

(Received March 2, 1959 .)

555-14. C. H. Cunkle: A note on Boolean operations.

A Boolean operation is a binary operation of the form x * y = axy + bxy'

+ cx'y + dx'y', where a,b,c,d are constant elements of a Boolean algebra, The

complement of x• y is seen to be a'xy + b'xy' + cx'y + dx'y', and by proper

assignment of a,b,c,d, OeMorgan's formulas follow as special cases. If any two

elements commute, then so do their complements. In case any single element

commutes with its complement, then the operation is commutative. If x•y

denotes the symmetric difference, all group operations are of the form e•x•y

with identity e, which may be any element of the Boolean algebra, If a,b,c are

any elements, then a unique group operation satisfies a•b = c. An isomorphism

between any two such groups is exhibited. (Received March 2, 1959,)

555-15, J, V. Talacko: On stochastic linear programming,

This paper is a generalization of the paper on stochastic linear inequali­

ties (see Abstract 552-15). Given a standard L. Pr. System: Them • n matrix

A, m-tuple [bi] and n-tuple [cj); xj ~ 0. Let all the elements of A, band c be

statistics .x/, with known confidence intervals Si ~ Si ~st. Without any further

restrictions (non-negativity, known distributions) on these Sj_ involved, in only

two solutions, we find the range of deterministic optimum functional Fo(x):

F (}(x) ;;;. F0 (x) ~ F6 (x), whenever Fo (x) exists. This range of boundaries may

even be found without any knowledge of real value of Fo (x). The single L. Pr.

problem (min. or max,) may be extended to dual L, Pr. problems. One may

prove the uniqueness of the result and demonstrate that dual L, l>r. problems

are statistically dual, The extension of this "range-method" to two person-zero

sum game and other stochastic matrix games is obvious. The probability of

P (F 0 ) to be in the range (F 0 - F ~) is possible to estimate by the inequality:

7Tr-lp(Sk) ~ P(Fo), where N is the number of statistics Sk involved and p(Sk)

that they are in the interval (S S+). (Received March 2, 1959,)

555-16. R, T, Rockafellar: The abstract algebra of linear programming.

The problem of maximizing or minimizing a linear functional subject to a

system of linear inequalities has been investigated by G. W, Oantzig and many

others. In the current paper, the author attempts an algebraic theory of the

153

situation which does not resort to converting the problem's inequalities into a

system of linear equations. The familiar theorems of linear programming are

reproved in this context, and several new results are obtained which point the

way to a more general solution algorithm than has previously been suggested.

Given an (m + 1) - dimensional real vector space, U and its dual space U*, an

(n + 1) - space V and its dual V*, each m X n linear programming problem is

associated with a set fJ of (m + 1) + (n + 1) vectors in U*. With each partition of

fr into two subsets, there is a corresponding bilinear form connecting U and V*.

This bilinear form is given a canonical representation, which will always have

certain characteristics if the linear programming problem is solvable. Solu­

tions of the problem are shown to be in one-to-one correspondence with certain

"resolvent" partitions of 3f • The author attempts to find a partial ordering of

the class of all partitions of !f, such that every partition belongs to a linearly

ordered chain terminating in a resolvent partition. (Received March 2, 1959.)

555-17. H. E. Stelson: On the reducibility of certain differential

operators.

Some nth order linear differential operators have been shown to be

reducible (See Klamkin and Newman Bull. Amer. Math. Soc. Abstract 63-6-703

also Bull. by Avco. Research Div. Lawrence, Mass., March, 1958). This paper

considers the reducibility of all operators of the form xkonxm in r factors which

are repeated n times. This is best accomplished by use of the differential

operator, D, where xm om = D(D- 1) ••. (D - m + 1). The polynomial shift

!ii(D)xmy = xm!ii(D + m)y is used. Differential operators of the form,

[x~<'(D + A1)(D + A2) ... (i5 + Ar)J, where one or more of the factors (D +At) are

used, are considered and they are expressed in terms of the ordinary operator,

D. By means of these reduction formulas, certain general types of differential

equations are solved, (Received March 2, 1959 .)

555-18. J, M. Kister: Uniform continuity and compactness in topological

groups.

Question: Is a nondiscrete topological group G compact if each real­

valued continuous function on G is uniformly continuous? An example is given

to show the answer in general is no but if G is locally compact in addition to the

other hypotheses then G is compact. The latter follows as a corollary to a

theorem (too long to state here) for uniform spaces, as does a result of Samuel

154

(Theorem 15 in Trans. Amer, Math. Soc. vol. 64 (1948)). The generalization is

effected by introducing a suitable notion of local boundedness with respect to

the uniformity. (Received March 2, 1959.)

555-19. C. W. Kohls: Convex ideals and prime ideals in rings of

continuous functions. Preliminary report.

Let C = C(X) denote the ring of all continuous real-valued functions on a

completely regular Hausdorff space X. The following results are similar to an

earlier theorem of the author on .B -ideals in C(X) (Prime ideals in rings of

continuous functions, Illinois J. Math, vol. 2 (1958) p. 522). Theorem 1. Let I

be any convex ideal in C. The following statements are equivalent. (1) The

prime ideals of C containing I form a chain. (2) The convex ideals of C con­

taining I form a chain, (3) C/I is a totally ordered ring with prime radical,

(4) C/I is a totally ordered ring. (5) For f,g E C, if fg = 0, then f E I or g E I.

Theorem 2, Let I be any ideal in C. Then the statements (1), (2) and (5) of

Theorem 1 are equivalent, It is evident that if an ideal I contains a prime ideal,

then (5) holds. The validity of the converse, although strongly indicated, is still

an open question. (Received February 27, 1959 .)

555-20, Louis Auslander: Fundamental groups of compact solvmanifolds.

We have proven that any group which is the fundamental group of a solv­

m anifold is the fundamental group of a compact solvmanifold, (Received

March 3, 1959.)

555-21. D. W. Wall: On residue class rings.

For any ring A and any type of (two-sided) ideal Q let Q(A) = [zj Z is an

ideal of type Q of A}. For any class of rings X and any type of ideal Q let

[X,Q) = fAiZ E Q(A)----> A/Z E Xj. Let X and Q have property (0) if A E X

implies 0 E Q(A). Let X and Q have property(*) if A EX, Z E Q(A),

ZoE Q(A/Z) imply Z* E Q(A) where Z* = [x E Ajx + Z E Zol· For classes of

rings Xl'X2 and types of ideals Ql'Q2 , a study is made of the relationships

among the inclusion relations among the classes of rings Xl' x2 , [Xl'Q 1),

[X 1 ,Q2 ), [X2,Q 1) and [x2 ,Q2 ). Examples of these inclusion relations are:

Xi C Xi, Xi C [Xj,Qk), [:lS. ,Qk) C Xi, [Xi,Qk) C [Xj,Qh); where i,j,k,h = 1,2. The

approach here is to study the conditions under which such inclusions hold for

any classes rather than to study specific classes for which they hold, Some of

155

the results obtained are listed here. If X1 C X2 and Ql C Q2 then [X 1,Q2]

C [X2 , Q1]. If x 1 and Q1 satisfy (0) then [Xl'Q1] C x 1. If x2 and Q 1 satisfy

(*)then [x2 ,Q 1] C x 1 implies [x2 ,Q 1] c [X 1,Q 1]. If x2 and Q1 satisfy(*) then

[X2 ,Q 1]-= X1 implies X1 C [X 1,Q 1]. (Received March 3, 1959.)

555-22. L, J, Heider: Prime dual ideals in Boolean algebras.

Presented here is a triple of characterizations of Boolean algebras re­

lated to fields of subsets, either as isomorphic to such a field, or as posses­

sing analogous distributivity properties, or as is om orphic to a quotient field

of subsets. Let m denote a cardinal number, while 'I, j' indicate the cardinality

of index sets I, J and ao, aj, aij denote elements of a given Boolean algebra.

Let ao -= VjEJaj, j E J, j;:;;; 1ffl be called a ,.._representation of ao, while ao

= VjEJi aij• j;:; m.-, Ji;;; WI' is called a W-family of :oW-representations of a 0 .

Then: (A) The algebras isomorphic to :m-fields of sets are the :m--complete

algebras having for each element a prime dual ideal containing a component of

each :m--representation of that element. (B) The ffl-complete and 1'11-distribu­

tive algebras are the $-complete algebras having for each element and for

each "JW-family of :ffl--representations of that element a principal dual ideal con­

taining a component of each member of that family. (C) The -m- complete and

1f1.-representable algebras are the ~-complete algebras having for each ele­

ment and for each ffl-family of~ -representations of that element a prime dual

ideal containing a component of each member of that family. (Received March 3,

1959 .)

555-23. J, B. Serrin: On the uniqueness of compressible fluid motions, I.

Viscous fluids.

The author considers the general equations of fluid mechanics governing

the motion of a compressible, viscous fluid with equation of state p = f(~,T) and

constitutive equations T = (-p +>ldivv)I + 2pD,q ~ -x. grad T. It is shown

that for a certain wide class of boundary conditions (involving in general the

specification of the velocity and temperature at all points of the boundary of the

flow region V = V(t) together with prescription of the density at points where

fluid is entering V), there cannot be more than one flow with given initial

velocity, temperature, and density distribution. The proof uses the classical

"ener!JY method", in a setting appropriate to the problem at hand. It is also

possible to treat heterogeneous inert fluids, and fluids in which the viscosity

anti heat-conduction coefficients are nonconstant. (Received March 3, 1959.)

156

555-24. R. D. Schafer: On cubic forms permitting composition.

Let A be a finite-dimensional nonassociative algebra with 1 over a field

F of characteristic .f 2,3. There is a nondegenerate cubic form N(x) on A

which permits composition (that is, N(xy) = N(x)N(y)) if and only if A is one of

the following: F 1, a cubic field over F, a central simple associative algebra of

degree 3 over F, or a direct sum F Ell B where there is a nondegenerate quad­

ratic form on B permitting composition. Thus A is alternative, and the possible

dimensions for A over F are 1 ,2 ,3 ,5 ,9. (Received March 3, 1959 .)

555-25. Leon Brown, A. L. Shields and Karl Zeller: On absolutely con­

vergent exponential sums. II.

Let E denote the set of all entire functions h admitting a representation

(1) h(z) = L:anexp(ctnz), Llan I< oo, \ trn I< 1. E is a Banach space with the

norm: llhll = inf L~nl, taken over all representations (1) of h. The conjugate

space is H 00 , the bounded analytic functions in the unit disc with the supremum

norm. Let }J be a finite Borel measure on the open unit disc, then the function

h(z) =./exp(zw)dp(w) is in E, and Var(p) $;\\hll. Let F be the set of all entire

functions h admitting a representation (2) h(z) =Jexp(zw)r6(w)dw, where r6

is an integrable function defined on the bou~dary of the unit disc, and the inte­

gral is taken around the boundary of the unit disc. Define the norm by:

Uhll = inf(1/217"i)j'lr6(w)\\dw\, taken over all representations (2) of h. Then

E = F and the norms are identical. Corollary. The complex sequence {cnj

(n = 0,1, •.• ) is the sequence of Fourier coefficients on one side of an integrable

function if and only if there exist complex sequences fan}, fll( nl• such that L \ani

<OO, \cznl<1, and[:ana.~= ck (k = 0,1, .•• ). See also Abstract 553-80, Notices

Amer. Math. Soc. (1958) p. 820. (Received March 3, 1959.)

555-26. D. E. Sanderson: On uniqueness of representation of 3-manifolds.

The representation of an arbitrary orientable closed (i.e., compact with­

out boundary) 3-manifold M by identifying the boundaries of two solid tori of

the same genus h (i.e., 3-cells each with h :;;; 0 solid handles) is well-known.

In this note, using results from the author's paper Isotopy in 3-manifolds. I.

(Proc. Amer. Math. Soc. vol. 8 (1957) pp. 912-922), it is proved that if M = s3

there is essentially only one way of carrying out the identification. That is,

.!f_ T and T' are two closed surfaces in s3 of the same genus and both separate

s3 into two components whose closures are solid tori, then there is an isotopy

157

~ s3 onto itself deforming T onto T' (everything considered from a semilinear

point of view). This answers a question raised by Papakyriakopoulos (16.4 in

Some problems on 3-dimensional manifolds, Bull. Amer. Math. Soc. vol. 64

(1958) pp. 317-335). The proof shows possibilities of extension or modification

to cover more general 3-manifolds. (Received March 3, 1959.)

555-27. H. M. Lieberstein, Jr.: Singularity occurrence and stably posed

problems for elliptic equations.

In a two dimensional Cauchy problem for a second order elliptic partial

differential equation with analytic data, interpreting one canonical coordinate

as complex (see Garabedian, P. R., and Lieber stein, H. M., On the numerical

calculation of detached bow shock waves in hypersonic flow, Journ. Aero. Sci.

vol. 25 (1958) pp. 109-118), singularities in the complex extended data are ex­

pected to propagate down characteristics to the real plane causing isolated

singularity occurrence in the real plane. A relation between primitive and

derivative data for the equation Uxx + uyy = h(x,y,u,ux,uy), h appropriately re­

stricted, is derived for which an infinite singularity in the data propagates for­

ward only and not aft of the data line. When this concept is applied to perturba­

tions of initial data, it reveals that the class of finite perturbations of data say

on the y-axis which give rise to finite displacements of the solution in the right

half plane is not the class of perturbations which give rise to finite displace­

ments in the left half plane. Then insofar as the concept of well-posed can be

replaced by what is here called stably posed, a new idea of direction is intro­

duced into studies of well-posed problems for elliptic equations. (Received

February 25, 1959.)

555-28. G. W. Whaples: On methods in class field theory.

To define a norm residue symbol over a local field k, it suffices to define

a pairing of k and the character group of the Galois group of the abelian

algebraic closure of k (Whaples, Duke Math. J. vol. 19 (1952) pp. 505-517;

Chevalley, Ann. of Math. vol. 41 (1940) pp. 394-418). Construction of a suitable

pairing requires only the lemma: If G/H - F, and H1(H,A) - 0 then the sequence

0 ~ H2(F ,A H) - H 2(G,A) ---. H2(H,A) (under lift and restriction) is exact. This

lemma can be stated as a lemma about group extensions. Starting with the

intrinsic definition of a group extension given by Eilenberg (Bull. Amer. Math.

Soc. vol. 55 (1949) pp. 3-27) one can define lift and restriction intrinsically and

158

simplify the proof of the lemma. Thus one can prove the basic theorems of

class field theory without using cohomology of finite groups or theory of simple

algebras, and without making the local theory depend on the global. (Received

March 4, 1959.)

555-29. P. C. Hammer: Relatively constant breadth curves.

Let C be a convex body in the plane and let C* ~ 0.5(C - C) be its sym­

metroid. Then C (or its boundary) is of constant breadth relative to the

Menkowski metric generated by C*. Then there exists a unique point xo in­

terior to C and two positive numbers r 1 and ro such that C 1 = x 0 + r 1 C* con­

tains C and Co = xo +roC* is contained inC where r1 is the smallest positive

number r such that a translate of rC* contains C and ro is the largest number

r such that C contains a translate of rC*. The common boundary points of C 1

and C and of c 0 and C are paired to lie on lines through x 0 . The following

conditions also hold: r 0 + r 1 = 2 and 1 ~ r1 ;; 4/3. If r 1 = 4/3 then C is nec­

essarily a triangle. If B is a convex body with center at the origin and if

C * = B implies that C is a translate of B then B is a parallelogram. (Received

March 4, 1959.)

555-30. Adam Koranyi: Generalization of a theorem of Loewner.

The following is a generalization to functions of two variables of a funda­

mental theorem of Loewner about monotone matrix functions. The real-valued

function f of the two real variables x 1,x2 in(- 1,1) can be represented in the

form f(x 1,x2) =J-1if_11x 1(1- sx1)- 1x2(1- tx2)- 1d,u(s,t) with a bounded positive

measure }ll if and only if (1) f(x 1 ,0) = f(O ,x2) = 0, (2) f(x 1 ,x2) is twice continu­

ously differentiable, (3) for k(x 1 ,x2;y 1 ,y2) = [f(x 1 ,x2)- f(x1 ,y2)- f(y 1 ,x2)f(y1 ,yz)]

-<x1- n><xz- Y2f1 the quadratic form LmL.n k<xim>,x~m>;x<f>,x~n>>a:mcrn is

positive definite for any choice of xp>,x~l), ... ,xiN),x~N) in(- 1,1). The proof

is an extension of the author's proof of Loewner's original theorem (Acta Sci.

Math. Szeged val. 17 (1956) pp. 63-70), based on Hilbert space methods. An

analogous theorem holds also for functions of n variables. (Received March 4,

1959.)

555-31. G. Y. Rainich: A skew five by five matrix and general relativity.

The fundamental assumption of the ("already unified") general relativity

theory is the identification with the contracted corrected curvature tensor of

159

the so called complete tensor which is a combination of the tensor of matter

and the electro-magnetic stress-energy tensor both of which were introduced

in Einstein's basic paper of 1916. Here the matrix of the complete tensor is

obtained by deleting one row and one column from (untrimming) the square of

a five by five skew matrix whose elements are interpreted as (1) the components

of the electric vector (2) the components of the magnetic vector and (3) the com­

ponents of the modified mom en tum vector of matter. A new proof of the unique­

ness of decomposition of the complete tensor into material and electromagnetic

parts is given. Only the nonsingular case is considered in lhis paper. (Received

March 4, 1959.)

555-32. E. B. Leach: Regular sequences and frequency distributions.

From a sequence f of digits we may form a sequence g whose terms are

finite sequences of digits, or words, of length m, where g(k) = (f(k),f(k + 1), ..• ,

f(k + m - 1)). Such sequences of words are called regular. The frequency with

which words appear in a sequence g is described in terms of its limit frequency

distribution, which is a point, or a subset of a frequency distribution space.

This set is a nonempty compact connected set of regular frequency distributions,

the adjective regular being a condition related to regularity of the sequence g.

The main theorem to be proved here is sufficiency of these conditions; given a

set of frequency distribution with these properties, there is a regular sequence

g having that limit frequency distribution. (Received March 4, 1959.)

555-33. Jesus Gil de Lamadrid: On the decomposition of topological

vector spaces into direct sums of closed subspaces. Preliminary report.

An example of Murray (Trans. Amer. Math. Soc. vol. 41, pp. 138-152)

shows that a closed vector subspace of a topological vector space does not

necessarily have a closed algebraic supplement. One might further ask if an

algebraic supplement, assuming it exists, is necessarily topological. The

answer is yes for Banach spaces, and, in fact, for any space F = F1 6) F2

(algebraic), F 1 and F 2 closed, for which the open mapping theorem is appli­

cable with domain F 1 X F 2 (topological) and range F. However, for arbitrary

locally convex topological vector spaces, an algebraic decomposition is not

always topological. A counterexample is the decomposition of the space F of all

bound~d real valued functions on [ -1, 1] as direct sum of the spaces F 1 and F 2 of

even and odd functions, respectively, if we give F a suitable .0-topology, where

160

11 is a family of subsets of [-l,l]. The treatment of this example is made pos­

sible by a theorem stating a necessary and sufficient condition for the continuity,

under the D-topology, of certain transformations of the space of all bounded

mappings of a given set into a topological vector space, (Received March 4,

1959 .)

555-34, E. C. Posner: Prime polynomial identity rings.

If R is a prime ring satisfying a polynomial identity, then R has a 2-sided

quotient Mn(D) where D is a division ring finite dimensional over its center.

(The converse is obvious.) For one verifies directly that R satisfies the

ascending chain condition on left and right annihilator ideals, and that every

direct sum of left or right ideals of R is finite, Then Goldie's Theorem (Proc.

London Math. Soc. vol, 8 (1958) pp. 589-608) is applicable, and the finite dimen­

sionality of the division ring follows because it can be shown to satisfy any

multilinear homogeneous identity that R satisfies. (Received March 4, 1959.)

555-35. Erwin Engeler: A characterization of theories with isomorphic

denumerable models.

Let T be a first-order theory with at most denumerably many constants,

having infinite models. Consider equivalence-classes of n-tuplets of elements

of a model M of T with respect to first-order formulas, Call an equivalence­

class "characterizable" if there exists a closed formula (Ex 1) .. ,(Exn)G(x1, ... ;xn)

such that a tuplet (a l''"'an) belongs to the class iff G(a 1, ... ,an) holds in M.

Lemma: If an equivalence-class of n-tuplets is represented in all denumerable

models ofT, then it is characterizable, Necessary and sufficient for the

l( 0 - categoricity of T are: (i) T is complete; (ii) For every n there exist at

most finitely many equivalence-classes of n-tuplets in any denumerable model

of T. Condition (ii) can be replaced by: (iii) For any adjunction of finitely many

new individual constants to the language of T there are at most finitely many

corresponding complete extensions ofT. (Received February 16, 1959,)

161

THE APRIL MEETING IN MONTEREY, CALIFORNIA

April 17-18, 1959

556-1. F. H. Brownell: A note on Kato's uniqueness criterion for

Schrodinger operator self-adjoint extensions.

Kato (Trans. Amer. Math. Soc. vol. 70 (1951) p. 195-211) has shown square

integrability over a finite region of n-space Rn with boundedness at oo to be a

sufficient condition on the real measurable potential coefficient function V with

n = 3, or more generally n ~ 3(1 with similar separate conditions for each R3

factor, for the Schrodinger operator in L 2(Rn) to have a unique self-adjoint

extension. We generalize this result to general n ;:;,. 1, with no separation into

R3 factors, by showing sufficientJixl ~ blv(x) jl/2(n+ I') dJ!n(x) < + oo with

p :> 0 and (n + f)/2;: 2 and IV(i)j essentially bounded on fxiiXl ~ bl. If

(n + fl)/2 ~ 2 is relaxed to (n + p)/2;;;, 1, which is significant only for n 1,2,

or 3, we show this condition to imply our earlier one (condition I), top p. 558,

Ann. of Math. vol. 54 (1951) pp. 554-594), and hence the existence of a possibl}

not unique self-adjoint extension of the Schrodinger differential operator

- V2 + V. With this relaxed condition we also obtain the expected variational

characterization of the spectrum. (Received November 26, 1958.)

556-2. E. D. Cashwell and c. J. Everett: The ring of number-theoretic

functions.

The set .Q of all functions ~t(n) on N ,. {1,2,3, •• .] to the complex field C

forms a domain of integrity under addition and arithmetic product: (« ·,B)(n)

= l:oC(d)p(n/d), summed over all din, dEN. The group of units (functions

with <or(1) fo> of .Q contains as subgroup the set of all multiplicative functions.

Against this background the "inversion theorems" of number theory appear as

trivial consequences of ring operations, and generalizations of the standard

functions arise in a natural way. The domain 0 is isomorphic to the ring P of

formal power series over C in a countable number of indeterminates x 1 ,x2 , ••• 't"'.. a1 a2 a1 a2

under the correspondence « --. kt£(n)x1 x2 ..• where n -= P 1 p2 ••• , P 1, P2••••

being the primes in any fixed order. The theorem of unique factorization into

primes, up to order and units, is shown to hold in P and hence in n. (Receive

January 23, 1959.) 162

556-3. R. T. Seeley: Singular integrals on compact manifolds.

Let D" be the partial derivative defined by the k-tuple of non-negative

integers, a: =(1X1, ... ,ttk), 1~1 ='Loci, lid = (./Ek JfJp)l/p, and llf~m = <2:::: 1 ... 1 §miJDctdP) 1/P. L~ is the closure of the C 00 functions with compact

support in Ek, with respect to ilf~m. Then, for 1 < p <: ro, L~ is shown to be

isomorphic to Lp. Corresponding spaces are defined on any compact manifold

M of class Cn, n ,;:;; 3, and used to give a definition of singular integral operators

on LP(M). The approximate calculus of "symbols" developed by Calderon and

Zygmund for such operators on Ek (Amer. J. Math. vol. 79 (1957) pp. 901-921)

has its parallel here. In order to apply these operators to representation of

vector fields on an orientable Riemannian M, a semi-group of operators con­

taining ( L':.- Lf 1 ( Ll the Laplacian, L a real multiple of the identity) is con-

structed by means of a parametrix and contour integration. The domain of the

closure of lJ. on L 2 (M) is shown to be L~(M). A vector field V is represented

as HA., where A2 = L - /J., and His a singula~ integral operator whose symbol

is essentially the function on the cbsphere bundle defined by taking inner pro­

ducts with V. The semi-group constructed is related to the fractional integrals

of M. Riesz. (Received January 27, 1959.)

556-4. C. B. Tompkins: Computation using Kronecker indices.

Let M be a compact, oriented, triangulated n-manifold, let L be a line,

a ray or a segment, and let v be a smooth function on M XL to vectors in En+l•

Suppose that for some point on L, v does not vanish for any point on M; then

the Kronecker index at this point of L is the (integral) number of spheres swept

out by v in En+l as its argument covers M. If M is triangulated this index may

be expressed as the coefficient of a cycle on the faces of a fixed simplex S in

En+1; this cycle is an integral multiple of the cycle of all faces of S. It is con­

structed by a linear extension of an approximation to v on the vertices of the

triangulation by vectors from the origin to the vertices of S. The mappings of

simplexes of the triangulation onto one chosen face of S define the index. The

index is constant along segments of L on which it is defined; a change in value

indicates a point of L at which v must have a zero on M. This property permits

solution of some equations. (Received January 29, 1959.)

163

556-5. S. P. Avann: The join irreducible excess function in finite lattice!'

In a finite lattice :t' let r(a] be the number of join irreducibles contained

in a, S[a] be the dimension of a, and the join-irreducible excess function be

11 [a]= r(a] - 8 [a]. In any ascending connected chain 7'and 8 are strictly

increasing, while lJ is nondecreasing. In any lattice r is a lower-semi-modular

functional, and in any lattice with Jordan-Dedekind chain condition vis also

lower semi-modular. Lower semi-modularity of 6 is a sufficient condition that

;t: be lower-semi-modular. Upper semi-modularity of 8 plus the Jordan­

Dedekind chain condition together are sufficient for :c' to be upper-semi­

modular. If .;rQ is a sublattice of .z"and c ='din .;co,>.[c]- >.(d] is compared

with ~o[c] - ~o[d], the corresponding evaluation of ~ in .<:Q for the three cases

>. = 1T, )\ = 8, )\ = v. Finally, the values of the three functions are considered

in a direct product of lattices. (Received February 4, 1959.)

556-6. L. W. Anderson: On the existence of continuous lattice homomor­

phisms. Preliminary report.

Let L be a compact, connected, finite-dimensional, metrizable distribu­

tive topological lattice. Theorem: If x andy are distinct elements of L then

there is a continuous homomorphism, f, on L into the closed unit interval so

that f(x) f f(y). COROLLARY: L is the iseomorphic image of a sublattice of

a cube. (Received February 2, 1959.)

556-7. R. C. Thompson: On matrix commutators.

Let A be ann row square matrix with coefficients in a field K. Theorem:

If determinant A ""1 and if K has at least n + 1 elements, then A- CDC-lD-l

where C and D are matrices with coefficients in K. This improves and simpli­

fies results due to Shoda (K. Shoda, Einige Siitze iiber Matrizen, Japanese J.

Math. vol. 13 (1937) pp. 361-365; Uber den Koinmutator der Matrizen, J. Math.

Soc. of Japan vol. 3 (1951) pp. 78-80.) The theorem is not valid for two row

matrices with coefficients in the field of two elements. (Received February 5,

1959 .)

556-8. R. M. Baer and Paul Brock: Natural sorting, II.

In the previous paper of the same title, M X M matrices were developed

whose (i,j) elements were the totality of permutation sequences of length M

containing maximal increasing subsequences of length i, and maximal decreasi

164

subsequences of length j. Using digital computing equipment for mathematical

experimentation purposes, matrices were constructed forM ':! 9. For the cal­

culated cases, the matrix elements are sums of squares of the characters in

the decomposition of [1M] for the symmetric group on M elements. The paper

reports this and other work done to date. (Received February 9, 1959.)

556-9. G. M. Helmberg: A decomposition of the Haar integral in

compact groups.

Let C(G) be the space of all complex valued continuous functions on

the compact group G. Let {H 1, ••• ,Hs}be a system of subgroups of G having

the property that (l) for every nontrivial irreducible representation R (.>..)

of G there is at least one index k).. such that R (.>.), restricted to Hk >., does

not contain the trivial representation of Hk.\. Then (Z)/af(g)dg

=/Hs""/'H/H 1f(h 1h 2 ••• hs)dh 1dh2 •.. dh 8 for all f E C(G) and G = H 1H2 •.• Hs (3).

(All integrals are the normalized Haar integrals with respect to the

indicated groups.) The condition (1) is also necessary for (2) and for (3)

if (s - 1) of the subgroups are normal subgroups. If G is abelian, then (1),

(2) and (3) are equivalent. In the non-abelian case let P be any choice func­

rion assigning to every left coset g' of the subgroup H an element p(g')

thereof. If G' denotes the corresponding left coset space, then it is possible

to define a linear functional Ig' on C(G') such that/of(g)dg = Ig'{../Hf(")V(g')h)dh]

for all f E. C(G). If His a normal subgroup then I is the Haar integral of the

factor group G/H. (Received February 23, 1959.)

556-10. A. W. Marshall and Ingram Olkin: A bivariate Tchebycheff

inequality for sym·metric conve:t polygons.

Let X = (X 1,X2) be a random vector with (EXiXj) == E, and let T be the

convex polygon fx:fwilx 1 + wi2x2 I:!! 1, i = l, ... ,n). A bound for P{X E T J is

obtained. If the 2 X 2 matrix A satisfies (1) xAx' ~ 1 for x E T, (2) xAx' iii;;. 0

all x, then P{X E 'i'} ~ E XAX' = tr A~. The bound is obtained by minimizing

tr A~ for A satisfying (1) and (2). Examples attaining equality are given for

certain polygons, and it is conjectured that the bound is sharp in general.

(Received February 23, 1959.)

165

556-11. B. L. Schwartz: A computer technique for breaking ciphers.

Preliminary report.

The use of automatic high-speed computers for "breaking" of ciphers

appears worthy of investigation, since a machine has the capability of trying a

large number of possible codes in comparatively short times. A special case

considered in this paper is the arithmetical type cipher in which alphabetic

letters represent digits in an arithmetic problem. This type problem is emi­

nently suited to computer solution, since the correctness of a solution can be

verified with certainty. A method has been developed for attacking these

cryptarithms. Trial codes are systematically assigned and tested by the

machine for correctness. As soon as an impossibility is detected, the code is

rejected and another tried. The distinguishing feature of the method is that a

relatively small number of trials (by computing machine standards) may suffice

to determine the solutions. For example, the famous cipherS END+ M 0 R E

= M 0 N E Y could be solved for the unique solution with a total number of

trials of about one thousand contrasted with the three-and-one-half million pos­

sible assignments of digits to the ten letters present. Detailed computer pro­

gramming is in progress. (Received February 27, 1959.)

556-12, D. W. Robinson: On a theorem arising from a study of matrix

commutators. Preliminary report.

Let D be an n-by-n matrix over a field F of characteristic p. Let F' be

an extension of F in which the characteristic polynomial of D is completely

reducible. It is shown that Dis nonsingular with D - I similar to I - D- 1 if

and only if the similarity invariants of D are each expressible over F' in the

form (*) (x - 1)tTT(= 1Tik=l (x - Aik) Si, where t\k == (ki\il - (k - 1))

• ((k- 1)Au - (k- 2))- 1, fork= 1, ... ,p and i = 1, ... ,q. (By convention, it is

understood that (*) is (x - l)t whenever p = 0 or q = 0.) This result applies to

m ultplicative commutators D = ABA-1 B-l satisfying A(AB - BA) = (AB - BA)A.

In particular, it is shown that a theorem of Putnam and Wintner, and Her stein

[see I. N. Her stein, On a theorem of Putnam and Wintner, Pro c. Amer. Math.

Soc. vol. 9 (1958) pp. 363-364] follows as an immediate corollary, and, further­

more, that the condition imposed on the characteristic of the field in their

theorem cannot be relaxed. Sufficient conditions are also given for represent­

ing a nonsingular D with D - I similar to I - I) 1 as a commutator ABA- 1 B -1

with A(AB- BA) ::= (AB- BA)A. (Received February 27, 1959.)

166

556-13, M.G. Arsove and R. E. Edwards: Generalized bases in topo­

logical linear spaces.

A generalized basis in a topological linear space is a family of points to

which there corresponds a biorthogonal separating family of continuous linear

functionals. (Total generalized bases have been studied by Markushevich.) In

terms of increasing generality there is the strict ordering: Schauder basis

-< Markushevich basis ~generalized basis-< maximal biorthogonal system.

It is shown that the translates of a function x in L2 (I), where I is a discrete

abelian group, form a total generalized basis in L2 (I) if and only if 1/~ belongs

to L2 (l). An isomorphism theorem (corresponding to that for similar bases in

Frechet spaces) is derived for similar generalized bases in complete metric

linear spaces, and a theorem is given simultaneously generalizing a theorem of

Newns (on continuity of the coefficient functionals in Frechet spaces) and a

theorem of Day (connecting weak and strong bases in Banach spaces). (Received

February 27, 1959.)

556-14. H. j. Keisler: On unions of relational systems.

The results which follow are proved by applying the methods introduced

in Abstract 550-14. For notation see Tarski-Vaught, Compositio Mathematica,

vol. 13, p. 85 ff. For a fixed natural number n, a sentence of a first order

predicate calculus is called V n 3 if it is of the form Vx1 , ... ,Xn_3 Y1,. .. ,ymM,

where M is quantifier-free. Theorem: Given an elementary class K and a

relational system Q, the following statements (i) and (ii) are equivalent. (i)

Every 'V n 3 sentence which holds throughout K also holds in 0(, (ii) Or:. has an

elementary extension 0{' such that every collection of at most n elements of()(..'

is contained in a model of K which is a subsystem of 0!'. Examples show that

if K is not elementary the conclusion of the theorem is not in general true.

Ot. is a union of the sequence <;6-i \ i € 1) of relational systems if each ;fi is a

subsystem of Oland every element of OC is an element of some Aj_. For the

case n = l, (ii) is equivalent to the statement (ii'): ot has an elementary

extension ()(.' which is a union of a sequence of models of K. (Received March 1,

1959 .)

556-15. Philip Wolfe: Secant method for simultaneous equations.

The following procedure for solving the simultaneous equations fi (x) = 0

(i = l, ... ,n) in the variables (x 1 , ... ,xn) =xis the direct generalization of the

167

secant method for one function of one variable: Given n + 1 trial solutions

l n+1 d · -,1 -.n+l h h '\'ntl ~if ( i) 0 '\' j x , .•. ,x , eterm1ne" , ... ,A sue t atLj=l" i x = •wjA = 1;

replace xr for which .L:'dfi(xr)l is maximal by X:= L/\ixi, and repeat. Con­

vergence is of order greater than one. Since each iteration requires 4n2 oper­

ations besides evaluation of n functions (rather than gradients), the process can

be more efficient than Newton's method. (Received March 2, 1959.)

556-16, D. E. Edmondson: Harmonics asymptotic to a particular in­

definite integral.

The indefinite integral(«+ pt..>sinrllt)exp(-oct + ~cost.Jt)/exp(oct- ,scos&Jt)dt

arises in the solution of the mathematical model of a particular problem in

space charge theory. It is proved to be asymptotic to a periodic function, this

function is expressed as definite integral, and the fourier coefficients expressed

in closed form as the product of two Bessel functions. (Research sponsored by

Texas Instruments, Inc,) (Received March 1, 1959,)

556-17. Herman Rubin: Aleph-irreducible cardinals and decompositions

of cardinals.

We call an infinite cardinal aleph-irreducible if it is not the sum of a

smaller cardinal and an aleph. Any infinite cardinal is either an aleph, aleph­

irreducible or the sum of an aleph and an aleph-irreducible cardinal. In the

latter case, we call the two summands the aleph and aleph-irreducible campo-

nents, respectively. If the aleph-irreducible component is not transfinite, it

is determined to within a finite cardinal; otherwise the decomposition is unique,

The sum of two aleph-irreducible cardinals and the product of any two infinite

non-alephs are aleph-irreducible; the product of the aleph m and an infinite

non-aleph p is aleph-irreducible unless the aleph component of p exceeds m.

If m is a non-aleph, the aleph-irreducible component of 2m exceeds m.

(Received March 1, 1959.)

556-18, R. M. Robinson: Intervals containing infinitely many sets of

conjugate algebraic integers.

In 1857, Kronecker determined the algebraic integers which lie together

with their conjugates in the interval [- 2 ,2]. This interval contains infinitely

many sets of conjugate algebraic integers, but no proper subinterval does. The

same conclusions evidently apply to any interval of length 4 whose end points

168

are rational integers. The negative result was extended in 1918 by Schur and

P6lya to an arbitrary interval of length less than 4. In this paper, it is shown

that there are infinitely many sets of conjugate algebraic integers contained in

any interval of length greater than 4. The problem is still unsolved for inter­

vals of length 4 whose end points are not rational integers. (Received March 1,

1959 .)

556-19. M. B. Smith, Jr.: A combinatorial result and its application in

proving certain metrization theorems.

In this paper a certain type of triangular array of real numbers is

examined, and an inequality pertaining to such arrays is established. Using

this result, certain metrization theorems are then proved, in particular the

Urysohn-Alexandroff Metrization Theorem. (Received March 1, 1959.)

556-20. R. A. Sonic: A group algebra without a real involution.

Let G be the free group on generators a and b, with a 2 ,.. 1 the only re­

lation. Elements of the group algebra L 1(G) of G over the complex numbers can

be written in the form E x(g)g where E I x(g)l < oo. An element E x(g)g of

~ 1 (G) is said to be hermitian if x(g- 1) = x(g) for all gin G. By considering

the elements of L 1(G) as left multiplication operators on L 1(G) it can be shown

that the hermitian element a + b + b-1 contains i as an element of its residual

spectrum. This is done by writing down the matrix representation for

i + a + b + b -1 and then finding a linear functional on L 1 (G) that is "orthogonal"

to all the rows of this matrix. For the case when the group is not discrete, an

example of this phenomenon was given by M.A. Neumark in 1948, the group

being the 2 X 2 complex matrices of determinant one. (Received March 3, 1959.)

556-21. R. B. Crouch: A class of irreducible syste.ms of generators for

infinite symmetric groups.

Let N be the set of positive integers; d the cardinal number of N; d+ the

successor of d; S(d,d+) the group of all one-to-one mappings of N onto itself;

A(d,d) the alternating subgroup of S(d,d+); S(d,d) the symmetric subgroup of

S(d,d+) consisting of all mappings which move only a finite number of elements

of N. If G is a group and M a subset of G then {M} is the smallest subgroup of

G containing M. If {M} = G then M is a system of generators for G. If no

roper subset of M is a system of generators for G then M is irreducible. It is

169

proved that: (1) the sequence of odd length cycles, (1,2, ••• ,n1Hnl'nl + l, .•• ,n2),

(ni,ni + l, ..• ,nit1) .•• , with order greater than or equal to three is an irreducible

system of generators for A(d,d). (2) The sequence of even length cycles

(1,2, .•. ,n 1), (n 1,n 1 + l, ..• ,n2), ••• ,(ni,Z\ + l, ..• ,ni+l), ••• is an irreducible system

of generators for S(d,d). (Received March 2, 1959.)

556-22. E. A. Walker: Nonlinear recursive sequences.

Let J be the Cartesian product of n copies of the Galois field GF(2).

Let f be a mapping from_.! into GF(2). If S = (al,a2····•an) E.£, let F(S)

=(az,a3•·•·•an-l• f(a 1, ... ,an)). F and f uniquely determine one another. IfF is

one-to-one, f is called nonsingular, and F decomposes J into disjoint cycles.

f is called maximal if .,l, is decomposed into just one cycle. Assume f is non­

singular. Taking each cycle generated by F in reverse order determines a

mapping Rf of k into GF(2) called the reverse of f. If S = (a1 , .•. ,an), the dual

of Sis S = (a 1 + 1 •···•ll:n + 1). If (S1 , ••• ,Sk) is a cycle generated by F, the dual

of that cycle is the cycle (S 1,. •• ,Sk). The duals of the cycles generated by F

determine a mapping Df of ,J.. into GF(2) called the dual of f. It is proved that

if F generates an odd number of cycles and if n > 2, then f 'f Df. In particular

iff is maximal and n > 2, then f :/: Df. This settles a question asked by Rosse

Also if n > 2, if n is even, and if F generates an odd number of cycles, then

Rf :f: Df, and hence f :f: RDf. Each f can be represented uniquely by a polynomial

in GF(2) [x1 , .•• ,xnJ• These results are obtained by examining the structure of

the polynomial representing f. (Received March 3, 1959.)

556-23. C. C. Chang and Andrzej Ehrenfeucht: A characterization of

abelian groups which are automorphism groups of simply ordered set.

Theorem: An abelian group H is isomorphic to the group of all automor­

phisms of a simply ordered set if and only if H = 1TtHt, where Ht is isomorphic

to a subgroup of the additive group of real numbers. The proof is based upon

the results of abstract 549-59 (32) and following lemmas: Lemma 1. If X is a

linearily ordered set and the group G(X) of all automorphisms of X is abelian

then G(X) = lTtGt, where Gt admits an archimedean ordering. Lemma 2. If a

group G admits an archimedean ordering then G is isomorphic to a subgroup of

the additive group of real numbers. (Received March 3, 1959.)

170

556-24. A. C. Morel: Simply ordered sets with non-Abelian automor­

phism groups.

LetS be a simply ordered set, let ocbe the order type of S, and let G(S)

be the group of all automorphisms of S. Theorem. (G(S) is non-Abelian iff oc

has an interval of type (/H e•(c.>* + (..))) (t..>* + (..))withe f 0. The theorem

solves Problem (b) of Goffman's BAMS Research Problem 60-3-10. A partial

solution was previously obtained by the author in BAMS abstract 62-4-553.

Corollary I. Every non-Abelian G(S) has at least the power of the continuum.

Corollary II. No non-Abelian G(S) is an orderable group. (II was previously

obtained by Cohn, Mathematika 4, p. 47 .). (Received March 3, 1959.)

555-25. Morgan Ward: The calculation of the complete elliptic integral

of the third kind.

A rapidly convergent iterative procedure is given for calculating the

numerical value of any real complete elliptic integral of the third kind, The

procedure is well adapted to a digital computer or hand computation with a desk

computer. (Received March 4, 1959.)

556-26. S. E. Warschawski: On Hadamard's variation formula for

Green's function.

The paper aims to establish Hadamard's variation formula under mini­

mal smoothness assumptions regarding the boundary. Theorem: (1) Suppose

C = U ~ .. 1 Ck, where Ck are closed jordan curves forming the boundary of a

region Il of connectivity n, c 1 being the outer, c 2 , ... ,Cn, the inner boundaries.

Let C be represented by z"' ~(t), 0' t ~ 1, where 0 < ¥ 1 ~I d;/dtl ~ Y2•

(2) There is a constant c > 1 such that, for any two points z 1 and z 2 on Ck, the

(shorter) arc /:); s of Ck between z 1 and z 2 satisfies the condition

b.s "clz2- z 11. (3) Suppose C*:= U ~=l C_k is obtained from C by means of

the displacement z = "Z;* (t) = ~ (t) + e h(t), where \ h(t) I ~ 1 and I h' (t) I ~ 1, such

that C* is the boundary of .n.•, .(}.*c. .fl. (4) Let C and C* be of class .A2 , c•

uniformly in e. (C e Aa if for functions 161 (z) and +k(z) which map int(C 1) and

ext(Ck) conformally onto lwl < 1, fc 11 t6 ~ (z)\ldz\ < oo and/ ck\ +~(z~\dz\ < oo

(k !5:' 2). Then, if G(z,z0), G*(z,z 0) are the Green's functions of .!1, .0.*, respec­

tively, G(z 1z 0)- G"(z 1z0) = l/21Tfc Im{(~*- '?;)ld~l/d';}(liG(~(t), z0 )/0n)

((IG(~(\:), z)/bn)\d';\ + o(E) where o(e) holds uniformly for z,z0 in any compact

;ubset of fl., as e - 0. Condition (4) is satisfied e.g. if ~' (t) is sectionally

171

continuous and C has only corners (no cusps). Other simple sufficient conditior

for C to be of class 112 are given for curves of bounded rotation. (Received

March 4, 1959,)

556-27. J, H. Case and R, E. Chamberlin: A note on generalized funda­

mental groups and generalized covering spaces.

Call a space covering simply connected (esc) if it has no nontrivial

connected generalized covering spaces. Theorem: A compact connected

Hausdorff space X of finite covering dimension is esc. if and only if every map­

ping of X into a finite connected complex Y can be lifted to the universal cover­

ing space of Y. Theorem: Any generalization of the definition of the fundamen­

tal group to the category of finite-dimensional compact metric spaces which

agrees with the usual definition on the category of complexes cannot simultane­

ously satisfy the continuity axiom and the following proposition: "A connected

space X is esc. if and only if the generalized fundamental group of X is trivial".

These results depend upon the methods used by the authors in Abstract 553-164.

(Received March 4, 1959.)

556-28. Paul Malliavin and L.A. Rubel: The zeros of entire functions

of exponential type.

New methods are introduced that solve a number of hitherto inaccessible

problems of uniqueness for entire functions of exponential type. The applications

include best-possible results of the Szasz-Miintz type, necessary and sufficient

conditions for the continuability of certain Dirichlet series through "channels",

best possible gap theorems for power series, etc, A representative result is

Theorem: Let {><n1 be any sequence of positive real numbers. In order that

there exist an entire function f(z) of exponential type satisfying f(An) = 0 for

each n, but f(z) r/= 0, and I f(iy)l ~ exp('ll'b\y\), it is necessary and sufficient that

A(t) - ,).(s) ;li. b log (t/s) + 0(1) uniformly for t :> s :> 0, where>. (t) is defined

by .Mt) = L )ln'itAn - 1• Necessity is proved via the theorems of Carleman and

jensen, sufficiency by appropriately modifying the solutions of a set of integral

equations involving the Weierstrass product W(z) = TTU - z 2;l.!>. The desired

function f(z) has the form W(z)K(z) where K(z) has only imaginary zeros.

(Received March 4, 1959.)

17l

556-29, Randolph Church: The dimensionality of finite free distributive

~·

The fact that Lln, the free distributive lattice generated by n independent

elements (without 0 and I adjoined), is isomorphic with a sublattice of the direct n

product of zn- Z chains of length one (i.e. the Boolean lattice of order z 2 -z)

raises a question: Is Lln isomorphic with a sublattice of the direct product of

fewer chains some of which would then be of greater length than one? It is

shown in this paper that Lln is isomorphic with a sublattice of the direct pro­

duct of the following: nCl chains of length (n- 1), nCz - nCl chains of length

(n - 3), nc3 - nCz chains of length (n - 5), etc., where the nCi are binomial

coefficients. Lln is not isomorphic with a sublattice of the direct product of

fewer chains than the total number given above (i.e, nCi, i = [n/Z]) and the

length of no chain in the direct product can be decreased unless the total num­

ber of factors is increased, (Received March 4, 1959.)

556-30. Steven Bryant and J. G. Marica: Cancellation of unary algebras.

U (A,:} is a unary algebra then it will be called basic if: it is connected,

i.t has exactly one element x with the property x' = x, and no element has more

~han a finite number of immediate predecessors. Theorem: If A, B, C are basic

unary algebras and A X B :l! A XC o~; B2 :a! c 2 then B ~ C. This is shown by

ordering the collection of basic algebras, and the result is then extended to a

larger class of algebras. With any finite unary algebra a finite collection of

infinite basic algebras is associated, the associated collection being used to

prove, Theorem: U A ,B are finite unary algebras and A 2 ;;: B 2 then A 5: B.

(Received March 4, 1959,)

556-31. Solomon Feferman, D. s. Scott and S. Tennenbaum. Models of

arithmetic through function rings,

The set 3" of general recursive functions becomes a semi-ring with the

natural definitions of point-wise addition and multiplication. Theorem, 1i2_

homomorphic image of the semi-ring ~ that is not isomorphic to the semi­

ring of integers can be a model for all provable sentences of first-order arith­

metic. Indeed a provable sentence +can be found such that if an image of 7i satisfies + , then it results by projection, i.e. by evaluating the functions in

~at a point. This is in marked contrast to the well-known result of Skolem

173

whereby the homomorphic images of the semi-ring of all arithmetically de­

finable functions yield many nonisomorphic models for all~ sentences of

arithmetic, (Received March 4, 1959.)

174

THE APRIL MEETING IN NEW YORK, NEW YORK

April Z3-Z5, 1959

557-1. Harry Hochstadt: Diffraction by convex objects.

A method is developed for finding the diffracted wave in the shadow zone

of an illuminated object. The outgoing wave is represented as a surface integral

over the object. The integrand is the product of the free space Green's function

and a kernel depending on the incoming wave. In separable coordinate systems

the kernel is easy to find. Asymptotic results are then obtained by applying the

Watson Transform to the kernel and evaluating the surface integral by station­

ary phase techniques. (Received January 6, 1959.)

557-Z. C. S. Coleman: Asymptotic stability in three space.

Let (1), dx/dt =fm(x) + g(x,t), be a system for which x,fm, and g are

"-vectors, fm(x) is a homogeneous function of x of degree m, g(x,t) is o~xlm)

niformly in t, fm(O) = g(O,t) = 0, and fm and g satisfy appropriate Lipschitz

and continuity conditions. Massera (Ann. of Math. vol. 64 (1956) pp. 18Z-Z06)

extending a result of Malkin, proved that the asymptotic stability of the trivial

solution of the system (Z), dx/dt- fm(x), implies the asymptotic stability of the

trivial solution of (1). For n = 3 the following theorem gives sufficient con­

ditions for the application of Massera's theorem. Theorem. Let y = x/lxl. T

Suppose (a) (y ,f(y)) < 0 on all roots of f(y) - (y ,f(y))y and (b)/0 (y ,f(y))dt < 0 on

any closed curve y(t) which is a solution of the system, dy /dt - f(y) - (y ,f(y))y

(where ( , ) denotes scalar product and Tis the period of y(t)). Then the trivial

solution of (Z), n = 3, is asymptotically stable. (Received February 16, 1959.)

557-3. M. K. Fort, Jr.: The extension of the set on which mappings are

homotopic.

The following theorem is the principle result obtained: Iff and g are map­

pings of a metric s.pace M into an absolute neighborhood retract Y, and f and g

are homotopic on a subset M 0 of M, then there exists an open set G containing

M0 on which f and g are homotopic. This result answers a question raised by

Granas. (Received February ZO, 1959 .)

175

557-4. Jack Segal: Hyperspaces of the inverse limit space.

Let X be a metric continuum and C(X) the set of all nonempty subcontinua

of X with the topology induced by the Hausdorff metric. It is the purpose of

this paper to answer questions raised by J. L. Kelley [see Hyperspaces of a

continuum, Trans. Amer. Math •. Soc. vol. 5Z (194Z) pp. ZZ-36) about dimension

and homological properties of C(X) where X is non-Peanian. In section 1 C(X)

is shown to be acyclic in all dimensions and in section Z sufficient conditions

for the finite dimensionality of C(X) are obtained. (Received February ZO, 1959.)

557-5. J.D. Pincus: Spectral multiplicity of singular integral operators.

We study the one-parameter family of operators L ,.x(>.) ~ A()l)x(~)

+ E(?rif 1 /~K 1 (.!1)Kz(u) (u- ~f 1 X(u)du defined over Lz(a,b). We take e ~ 0,

K1(u),Kz(UJ i=O on [a,b] and require that A(u), K1(u), and Kz(u) be continuously

differentiable. The functions R:t(u) - A(u) ± eK1 (u)Kz(u) are required to

change direction of increase in [a,b] at only a finite number of points. We

partition [a,b] into subintervals a "' lltto··:r"'i•"'i+l; •••• ~n • b such that in

[C(i,«i+l] one of the four mutually exclusive cases obtains: R+t, R-t; R+ J., R- J.;

R + 1', R- ! ; R+ l, R- 1. This is done by starting at a with the appropriate case

and proceeding to the right until we come to the point at which we enter anotht~

case. For each subinterval we form Et(l,z,e)"' exp{(zmf 1.f...i+Icln R-(u)

- ln R+-(u)] (u- z)- 1duJ and define Ht(.i,~,E)=limn,..O[E1 (.l,i'l + i~,e)-

EtC~,)\- i"t,e)]. The functions\\\(~,)\) • limn-.0 [Hi( s + iyt,..\ - e)

- Hi( J- i1t,~l - e)]/Kz()l) are independent generalized eigenfunctions of L 6

and generate its spectral representation. The spectrum of ~ is the union:

tT(Le) • Uf.l (min"E{oC"i;t!Ci+1] (A()I)- eKi (..\)Kz(~)), max)\€(DCt,«i+1)(A()I)

+ eK1 ()I)KzC\))]. The spectrum is purely continuous and the multiplicity of any

of its points is the number of times it appears in the above urlion. (Received

January 30, 1959.)

557-6. D. E. Schroer: Extended combinatory formulation of standard

theories.

(See Abstracts 546-41, 547-34, 553-ZZ5, and 548-44.) In Abstract 548-44,

all combinatory logics (resp. combinatory structures) are to be considered in

association with fixed (finite) sequences of distinguished symbols (resp. dis­

tinguished objects) corresponding to those of the classical ones. The unique

occurrence of the word "consistent" in Abstract 548-44 may be omitted.) Tht.

176

following extensions of the theorem (last sentence) of Abstract 548-44 are

Jbtained: For standard theories with finitely-many primitive symbols, there­

quirement of finite axiomatizability may be relaxed while the effective character

of the translation into combinatory form is retained, provided that formulation

of a standard theory is considered to involve a basis for Godelization and a

fixed system of recursion equations for the strictly increasing enumeration of

numbers of nonlogical axioms; without such provision the translation can still

be specified, but the construction is set-theoretic rather than effective. In

either case the concepts of combinatory formulation and combinatory diagram

are modified so as to involve the combinatory integers. By further modification

of these concepts so as to involve representation of sequences of distinguished

entities, the theorem is obtained for arbitrary finitistic standard theories.

(Received February 23, 1959.)

557-7. S. P. Lloyd: A coefficient of stochastic dependence.

In the probability space [Q ,:7 ,P], let tJt. denote an indexed collection

(}(. = [Jl-y. v E NJ of Borel subfields A.)) C :T, with liN I\ ;;-- 1. In the product

[QN, .7N, pN] of JINft copies of [ Q ,:7 ,P], let TT denote product measure pN,

'ld let 11 denote the set function whose value for cylinders Z CoN with base

A11_ X ... X A'1! (AI{ E :J) is Ll{z} = P{Av1 n ... n Avn}· It is a theorem

that ..11 extends uniquely to a measure on a certain Borel field containing .;rN.

Define I(O() = j'log (dJo /d TT o)d /J.., ( = + oo unless /J..o << TTo), where subscript

0 indicates restriction of domain to XvGNA.v ( C .JN). Some properties are

(1) I(O() ~ 0, with equality if and only if the members of every finite subcollec­

tion of 0( are mutually stochastically independent. (2) If tl( 1 <: 0(2 means N 1 c N2

and At C fl.~ far every ll EN 1 , then I( ) is increasing and continuous from

below on any upward directed set of O(•s.(3) If .fl.v0 ~[¢,Q] then I({ A.v• VENj)

I(f ./4, v EN0}) where N0 = N -[v0 J. (4) If N 1 n N2 = ¢, then I([ Av, v EN 1 U N2})

= I(f ./{v,1.1 E Nl}) + I(f Av, V EN2}) + I(fVvENlfi. v' VpEN2AJl}>• (5) When N

= 2 (as in the last member of the preceding equation), I([ A., ;;8) is the shared

entropy of Jl and ;:(i of Kolmogorov [Dokl. Alcad. Nauk vol. 111 {1956) pp. 745-

748]. (Received February 23, 1959.)

177

557-8. Morris Morduchow and Lionel Levin; Comparison of the method of

averages with the method of least squares: Fitting a parabola.

In a previous paper (Journal of Applied Physics, October 1954) the first

Author showed that if the method of averages is applied in the customary manner

to fitting a straight line ton points with uniformly spaced abscissas, then the

standard deviation (CT a> of the residuals of this line will be at most 2/3 112

= 1.155 times as great as that (CT s> of the least-squares straight line, The

present paper is an investigation of this type of result for fitting a parabola.

The following are among the res1,1lts obtained, If the method of averages is

applied with the averaging intervals chosen so that the first interval contains

essentially the first n/5 points, the second the next (3/5)n points, and the last

the last n/5 points, then the ratio (CT alcrs) can be at most 6/5 = 1.200, For

n ~ 20, this result is valid with the factor 6/5 replaced practically by the

corresponding asymptotic factor for n -+ oo, namely (5/4)(5/6)1/2 = 1.142. If

the averaging intervals are taken as all of essentially equal size, then the

maximum possible value of (CTa/CTs) is found to be higher, namely 9/(2(10) 112)

1.422, (Received February 16, 1959.)

557-9. Robert Hermann: Generalisations of the Poincare-Birkhoff fix<

point theorem.

Let A be a closed region in the plane, with outer boundary a smooth

Jordan curve C, and inner boundary composed of smooth Jordan curves

Cl'"''Cm. Consider a sufficiently differentiable one parameter group of trans­

formations t -+ Tt : A ---> A such that each T t preserves areas, and rna ps the

boundary curves on themselves without fixed points, and moves C in the op­

posite sense to the way it moves the inner boundary curves, Then, the group

has at least m + 1 fixed points inside A. There are generalisations to real

symplectic manifolds with boundary, i.e. manifolds with a closed 2-differential

form of maximal rank. The proofs involve applying the Morse theory of critical

points of functions on manifolds with boundary. (Received February 26, 1959.)

178

557-10 WITHDRAWN

557-11. R. C. Bose: Note on Parkers method of constructing pairwise

orthogonal sets of Latin squares. Preliminary report.

The object of this note is to generalize a recent result of Parker (Notices

Amer. Math. Soc. vol. 5 (1958) p. 815) on the existence of set of pairwise orthog­

onal Latin squares (p.o.f.s.). It is shown that if there exists a set of t-1

.o.£.s. of order k and also a balanced incomplete block (BIB) design with v

rreatments, A = 1 and block size k, then there exists a set of t-2 p.o.f.s. of

order v and the number in the set can be increased by unity under certain con­

ditions, which are in particular fulfilled when the BIB design is symmetric or

resolvable. Parker's theorem (with an improvement under appropriate con­

ditions) follows by noting that when k is a prime power there exists a set of

k-1 p.o.i.s. The results are illustrated by considering certain series of BIB

designs, in particular the resolvable series v = k(Zk- 1), k =2m, >. = 1,

which shows that there are an infinity of values of v for which the previously

known best result due to Mann can be improved. (Received February 20, 1959.)

557-12. W. R. Baum: Integral geometric methods in information theory.

II. Sources.

Positive sources and negative sources (sinks, receivers) are introduced

in multidimensional information theory in the following way: Given an n-dimen­

sional space En, a set of geometric elements X, a r-parametric Lie group Gr

of transformations of En, and a set a (an "alphabet") of indeterminates. With

any point P of En shall be associated an element X EX such that Gr, acting on

179

En, transforms the X's transitively. Introducing in En a (fixed) coordinate

system K and considering at any point P a moving frame k, rigidly connected

with X at P, the position of X is given by a "vector" 5 whose components are

determined by the position of k relative to K. Thus ~ depends upon Gr, and the

pair (X,_Lo) is varying over the homogeneous space of Gr. Consider now func­

tions ~mapping the pairs (X,$) into Cl. A defined set A of certain functions <!::

("elementary events") constitutes the sample space underlying the definition of

the source, (An element T E Gr induces a transformation T ("shift") in the

space A.) By defining a probability measure on a <r-algebra of subsets of A one

obtains a probability space which represents a source, [Sponsored by the Infor­

mation Systems Branch of ONR (NONR-669(10).] (Received February 27, 1959.)

557-13. Hirosi Nagao: On GL(2,K[x]).

It was pointed out by D. Livingstone that GL(n,K[x]) was finitely generated

if n 3 and K was a finite field, We shall show that this is not the case for

n 2, and we shall determine the type of the group GL(2,K[x]). The crucial

(0 1 !"Xi fi(X)) lemma is as follows: Let W = 1 0 , Ti = 0 fti ~ c:rio fti E" K. M

deg f 1(x) > 0, then there is no relation of the form Ti W 1 T 2 W 2 ... TrW r

= E, ~ E is the identity. From this lemma we can also prove

that any free product of two free abelian groups or two elementary

abelian groups with the same prime exponent has a matric represen­

tation in GL(2,K). (Received February 27, 1959.

557-14. B. A. Rattray: Maximal homomorphisms and almost periodic

functions.

Let G be a topological semigroup. We define an associated compact

group G and a homomorphism 1J:G- G such that; (a) V(G) is dense in G; (b) if His any compact group and)l a homomorphism of G into H there is a

unique homomorphism p:G __,. H such that p = pv. Let f be a continuous map

of G into a topological space X. Then f is called almost periodic if there is a

continuous map f: G -+X such that f = f v. If X is the space of real numbers

this is equivalent to the definition given by Maak (Acta Mathematica vol. 87

(1952) pp. 33 -58) who, however, considers only discrete semigroups. The

method of definition of G can also be used to give new constructions of: (a) the

Cech compactification of a topological space; (b) the completion of a uniform

space, (Received February 27, 1959.)

180

557-15. j. A. Riley: On the minimization problem for Boolean functions.

Preliminary report.

A Boolean formula, in the variables x,y, .•• is a "word" in the letters

x,y , ••• , their complements x' ,y' , ... , and the operations U and n. x and x' are

different letters, but the same variable. Complement signs are allowed only

over single letters. Let f,g be formulas. f implies g, f =t g, if this implication

holds for the corresponding Boolean functions. f is equivalent tog, f = g, if

f o+g and g o+f. The length, L(f), of a formula, f, is the number of letters

(counting repetitions) in f. f is minimal if whenever g = f, L(g) ~ L(f). xis

inessential to f if f(x = 1) =+ f(x = 0) (here f(x = 1} denotes the value of the

corresponding Boolean function when x = 1, x' = 0). Denote by fx the formula

obtained by replacing by 1 each appearance of the letter x in f. Theorem: xis

inessential to f if and only iff = fx· Some consequences of this theorem for the

theory of minimal formulas are: (1) the letters in a minimal formula are

essential; (2) iff is a formula in which each letter appears just once, then f is

minimal if and only if its letters are essential; (3) iff, g are formulas with no

variables in common, and if xis a variable occurring neither in f nor in g,

then xf U x'g is minimal. This research was done under contract with the Air

orce Cambridge Research Center. (Received February 2 7, 1959 .)

557-16. T. I. Seidman: Linear transformations of a functional integral.

Let I(•) be the integral over the space of all sequences taken with respect

to the Kolmogoroff extension of then-dimensional normal distribution as

applied to sets invariant under projections with finite dimensional range. Sym-

f "" 2 2 1/2 bolically, writing x = s1,s2•···l• I(f) ~./ f(x)Tfkexp(-~k/20" ]d~k/a-(211') .

Let CJr. be the set of operators fA} on the space of square-summable sequences

such that IIAII0 = 2(TA • A)112 +I r AI (T- trace) is finite and the determinant of

(I + A) is different from 0. Then for any A E 0(. and any integrable functional

f one may define g(x) = f(x + Ax)ldet(I + A~ exp( -(x, rx)/2o-2] where r = A + A*

+A* A and g is defined a.e., integrable, and I(g) - I(f). (Received March 2,

1959.)

557-17. s. S. Shrikhande: Group divisible designs and the construction of

pairwise orthogonal sets of Latin squares. Preliminary report.

If there exists a set oft - 1 pair-wise orthogonal Latin squares (p.o.i.s)

Jf order k, and s - 1 p.o . .l.s. of order n, and also a Group Divisible design (Bose

181

and Connor, Ann. Math. Stat. vol. 23 (1952) pp. 367-383) with parameters

v,. mn,b,r,k, .11 1 = 0, >1 2 = 1, where there are m groups of n treatments each

and..\ 1 and A2 refer to the number of times any within-group and between­

group pair occurs in the design, then a set of q = min(s - 2,t- 2) p.o.-/.s. of

order v can be shown to exist. This num her can be increased by one, if the de-

sign satisfies certain conditions which are fulfilled in particular for symmetric

or resolvable designs. Thus from the design v = b = 80, r = k = 9, A1 =0,

)\ 2 = 1, m = 10, n = 8 one gets 7 p.o.-R.s. of order 80, Mann's result [Ann.

Math. Stat. vol. 13 (1942) pp. 418 - 423) gives the number 4. Results of Parker

(Notices Amer. Math. Soc. vol. 5 {1958) p .815) and Bose (These Notices, Abstract

557-11) do not apply here. (Received March 2, 1959 .)

557-18. E. R. Rodemich and A.M. Garsia: Embedding of Riemann

surfaces of genus 1 in 3- space.

The following theorem is proved: Every closed Riemann surface of genus

can be C 00 embedded in Euclidean 3-space. The method of proof uses an

auxiliary embedding with singulari{ies which is an isometry of the natural

Euclidean metric of the surface. (Received March 1, 1959.)

557-19. R. R. Phelps: Uniqueness of Hahn-Banach extensions and unique

best approximation.

Let M be a closed linear subspace of a normed space E; M has property

U if each continuous linear functional on M has a unique extension (of same

norm) to E. Call M a Haar subspace if for each x in E there exists a unique

y in M such that llx - Yll = inf{lx - zll: z E M J. Theorem 1. M has property U if

and only if MJ. (the annihilator of M in E *) is a Haar subspace. Definition:

If X E E(f £ E*) let A(x) = fg E E*: IJg~ = 1 and g(x) = nxu} (B(f) = {Y E E: IIYII

= 1 and f(v) =!Ifill]. Theorem 2. If M is of deficiency n 0: 1 and if the dimen­

sion of B(f) is ~ n for some f :j: ¢ in MI.., then M is not a Haar subspace.

Theorem 3. If M is n-dimensional and if the dimension of A(x) is ~ n for some

x f ¢ in M, then M is non- U. Theorem 1 is used to deduce Theorem 3 from

Theorem 2, and the latter two are. applied to special spaces to obtain results of

the following sort: An n-dimensional subspace M of .£\has property U if and

only if each nontrivial sequence in M has at most n - 1 zeros-since this is

equivalent to dim A(x) < n for each x :f: ¢ in M, the condition in Theorem 3 is

necessary and sufficient. An n-{iimensional subspace M of c 0[L 1(0,1)] has

property U if and only if for each x 'f ¢in M, llxiJ = lxil for at most n integers i

182

[x vanishes only on a set of measure zero]. The space L 1 (0, 1) admits no Haar

subspaces of finite deficiency; the same is true of c 0. (Received March 1, 1959.)

557-2.0. Makoto Ohtsuka: Some examples in potential theory. Prelimin-

ary report.

Let i(z, ~) be a positive symmetric kernel in E 2 X E 2 which is finite for

z 'f ~ and whose reciprocal is continuous. For p, 0 !!ii )1 < oo, with compact

support S)l in E 2 , UP(z) is defined by /i(z,~)dp(~). Let 6 be [p; (p,p)

s /UJl(z)dp (z) < oo}. t(z,~) satisfies (E) ((P) reap.) if, for every different

p, \lEG, (p-lJ ,p -v) = (p,p) + (lJ,'II)- 2(p,v):;:. 0 (~ 0 resp.). +(z,~) satisfies

(M,\) ()\ ~ 1) if, for every p and at each z E Ez, uP(z) ~ >.sup uJicr> taken on

Sp. + (z, 7.;) satisfies (C) if the continuity of the restriction of ul-l(z) to Sp implies

that of UP(z) inEz. It is known that, for any>.. ~ 1, (MA)- (C) and (M 1)- (P).

Examples are: (1) Given A? 1, there exists a positive decreasing function

{ll'(r), 0 < r < 1, such that +(z,'{) = !11(\z -~\) satisfies (M;>) but not (P). (2.)

There exists +(z ,~) which satisfies (E) but not (C). Next for a compact set K

in E 2 , let C 1(K) be supfp(K); Sp C K, U)((z) ~ 1 on Sp}and Cz(K) be

supfp(K); S}lC K, UJl(z) :l 1 in some open set containing sp1· An example of

+(z,~) is given for which there exists K with c 1 (K):;:. Cz(K) = 0. Since it is

Known that c2 (K) = 0 implies C 1 (K) = 0 under (C), the example does not

satisfy (C). (Received March 2., 1959.)

557-2.1. R. D. M. Accola: The bilinear relation on open Riemann surfaces.

Let Hh be the Hilbert space of square integrable harmonic differentials

on an open Riemann surface W. If fwnJ is an exhaustion of W, let A1 ,B 1 , .•. ,

Ap(n)•Bp(n)•··· be a canonical homology basis, where A1 ,B 1 , ... ,Ap(n)•Bp(n) is

a basis modulo dividing cycles of Wn• Let Hhe be exact, Hhse be semi-exact, •L •L

and Ha be analytic differentials. Let Hho= Hhe• and Hhm = Hhse• Let O"(c)

be the unique differential in Hho such that /ct.>=(cll,<T*(c)), denoted c(c:.>), where

cis a cycle. For c~>E Hhse let Tnl.l=E~iiif£k(cll)<T(Ak)- Ak(c.J)O"(Bk). We say

the generalized bilinear relation (GBR) holds for {,) E Hhse if ('Z" ,(T nell)*)- (7,(.)*)

for all T" in Hho• The GBR holds for w if and only if the projections of T nl.l on

Hho are bounded. U WE OHD let !Ilk € Ha satisfy Ak({ll'j) = Skj' The validity of

the GBR for all c..> E Hh implies that L~~n:>Ak(cx){ll'k- "'- strongly for CIC. E Ha.

Parabolic surfaces exist where the GBR holds and does not hold. Let H A be the

span of the 0"(-'\:)'s, H8 the span of the O"(Bk)'s. U W G OHD and HA + H8 is

183

closed then ("l:',(TnCJ)*)- ('Z',Gi") whenever 7: and ~have a finite number of non­

zero A (or B) periods. Then the -k's span Ha. Examples of parabolic surface

exist where HA + HB is not closed. A surface exists where Hhm is a proper

subset of Hhorl Hhe· On this latter surface the GBR can hold for no canonical

homology basis. (Received March Z, 1959.)

557-ZZ. K. I. Appel: Partition rings of finite abelian groups. Prelimin­

ary report.

Let G be a finite abelian group. A subring P of the group ring of G over

the integers is a partition ring provided there exists a partition of G into sub­

sets Go= {1), G1, ... ,Gr, and provided, moreover, that there exist positive

integers IDg (one for each gin G) such that the elements Ak = Eg€Gkmgg form

a basis for P. We may suppose that for each Gk the greatest common divisor

of the mg, for gin Gk, is 1. Theorem 1. If Ak = LgEGkmgg is a basis element

of the partition ring p' and q is prime to the order of G, then Lg€Gk mzgq is a

basis element of P. Corollary. All mg are 1. Theorem 2.. If G is a group of

odd prime power order and P is a partition ring of G, then each basis element

of P is either a sum of equivalence classes under the full automorphism group

of G or else the sum of a single equivalence class under a subgroup of the auto­

morphism group of G. A classification of all partition rings of the cyclic groul­

of odd prime power order pe is obtained. (Received Martch 1, 1959 .)

557-2.3. K. E. Aubert: Convex ideals in ordered group algebras and

the uniqueness of the Haar measure.

If LJ denotes the real group algebra of a locally compact abelian group

G we shall consider Li_ as an ordered ring with respect to pointwise ordering

a.e. The Haar measure on G then gives rise to an orderpreserving ring homo­

morphism of LJ_ onto the reals which has a closed regular convex maximal

ideal of codimension one as kernel. The uniqueness of the Haar measure tells

us that there is exactly one such maximal ideal which contains the set

T = {f - fa} where a E G and f E L J_. This result can be improved by establishing

the following sharpening supplement to the uniqueness of the Haar measure: _!!_

~ is an orderpreserving ring homomorphism of LJ onto an ordered field F then

F is the field of real numbers and p is the Haar measure of G. We also give a

further refinement of this result as well as a couple of other simple facts which

are related to the scarcity of convex ideals in Li_; in particular that Li_ can

184

never (i.e. for G =/= fe}) be embedded with preservation of algebraic and order

structure in a direct product of totally ordered rings. (Received March 2, 1959 .)

557-24, A.M. Duguid: A class of hyper-FC-groups.

The FC-subgroup H of a group G is the set of those elements of G which

have a finite number of conjugate elements in G. The upper FC-series {1}

= Ho~ H1 ~ ... is defined by the rule Hit 1/Hi = H(G/Hi), and is thus analogous

to the upper central series. Like the latter, it may be continued transfinitely

by defining HO£. = U,s < Cl H,s, when a is a limit ordinal. G is said to have F C­

class OL if H« = G, but H,s f. G, for all p ~IlL. In the present paper a construc­

tion, using transfinite induction, is given for groups of arbitrary FC-class. If

oc is a limit ordinal, and G/8 has FC-class p, for all (3 ~ot., then the (restricted)

direct product 1T,s.::otG,s has FC-class a. If GO( has FC-class 01., then a group

G of F C -class OL + 1 is obtained by wreathing the regular representation of G«

with an FC-group of permutations of the positive integers, which is chosen in

such a way that all FC-factors of the constructed groups are infinite, but have

the unit subgroup for centre. (Received March 2, 1959 .)

557-25, M, J, Greenberg: Reduction of algebraic varieties modulo

arbitrary powers of a prime divisor in the ground field.

Let V be an algebraic variety defined over a complete discrete valuation

ring (R, Oj) (and what follows is susceptible of being generalized to complete

local rings). Then there is a functor associating to V a sequence of (possibly

reducible) varieties and morphisms Vo4-llioy1..__1li 1 ... Vv ..._9111 ... defined

over the perfect residue field k of R. V)) is obtained from V when V is affine

by reducing the ~quations of V l:nod o/'v+1 and then introducing a suitable co­

ordinatization (e.g., that given by the Witt vectors in the unramified character­

istic-unequal case). The points of V rational over R can then be identified with

the projective limit of the points of the V')) rational over k. If V is a group­

variety with nondegenerate reduction mod tg:(e.g., the general linear group), then

each Vl) is a group-variety and each lliv a homomorphism. As applications, we

obtain another proof of Chow's recent generalization of Hensel's Lemma, and

the following Theorem: Assume R unramified and k finite. If A is an abelian

variety over R with nondegenerate reduction mod OJ'• then any principal homo­

geneous space of A which has a rational point in an unramified extension of the

otient field K of R has a rational point inK. (Received March 2, 1959.)

185

557-2.6. I. I. Kolodner: On the nonlinear diffusion equation.

The initial Value Problem (*)(DE): (D(c)cx)x = Ct, (x,t) € ( -oo,.,..)x(O,ca),

(IC): c(x,O) =- q if x < 0, c(x,O) = cz. if x > 0, lei~ A, where 0 < c1 ~ cz.,

DE Cp[c 1,cz.], and 0 < 0£ ~ D(c) ~ fJ• can be reduced to the Boundary Value

Problem(**) (DE): w" + Z.zf(w)w' = 0, z E (-oo,oo), (BC): w(-oo) = 0, w(+oo) = 1,

for w(z) = v(c(x,t)) with z = x/Z.at1/Z., where v(c) = b /c~ D('t)d 't', f(v)

= a2/D(c(v)), F(v) =/ovf('t')d'l', and a and bare constants such that v(c2) = 1,

F(1) = 1. Together with(**) one considers an associated Integral Equation

(***,k): w(z) = (Z.k)- 1(erf(k 1/Zz) + 1) +/-:H(z,'t',k) [w('t')- k- 1F(w(t'))]d't' where

k > 0 and H(z;r,k) = [2k't'G(z,"Z:,k)]t", while G is the Green's Function for the

operator dz./dzz. + Z.zk(d/dz) with zero data at lzl = oo. Let m = inf f, M =sup f.

Theorem: (i) For every k :> 0, (***,k) has exactly one solution w which is

independent of k. (ii) This w is the unique solution of(**). (iii). If k > M/2.,

w can be constructed by iterations. (iv) If k ~ M/2 the iterations will in

general diverge. Accurate bounds for w are obtained by exploiting the isotone

(antitone) property of the operator in (*** ,k) for k !!!!; M(k ~ m). Extensions, in

particular that to an associated Free Boundary Problem, are also considered.

(Received March 2, 1959.)

557-2.7. A. j. Macintyre: Length of gaps and size of region of over­

convergence. Preliminary report.

For an overconvergent gap power series it is a natural conjecture that

the longer the gaps the larger the region of overconvergence and indeed that if

the gaps are sufficiently long the region of overconvergence will include a

prescribed point of the region of regularity. These conjectures are justified

for power series in the whole plane less a radial slit by considering the inter­

polation of the coefficients by an entire function. This method suggests general­

ized overconvergence theorems requiring only that almost all coefficients in the

gap vanish. In this way Fabry's gap theorem as well as that of Hadamard is

exhibited as a corollary of an over convergence theorem. (Received March Z.,

1959.)

186

557-28. H. S. Wilf: A necessary and sufficient condition for locally stable

numerical integration.

The stability properties of a numerical integration formula for the equa­

tion y' = f(x,y) are usually found by assuming that af/ay is locally constant,

deriving the difference equations satisfied by the error, and requiring that the

roots of the associated polynomial equation lie interior to the unit circle. It is

shown that a necessary and sufficient condition for this to happen is that a

matrix A of the form A= Ao + A 1>. + A2 1\2 be positive definite, where the Aj

can be computed directly from the coefficients of the integration formula, and

'). = of/Ciy. This permits the question of stability to be settled, for any given

)\without solving any algebraic equations. (Received March 3, 1959.)

557-29. N. W. Bazley: Lower bounds for eigenvalues of self-adjoint

operators.

The procedure given here for finding lower bounds is an essential modifi­

cation of a method which goes back to A. Weinstein and was extended by

Aronszajn. Let A be a self-adjoint operator with domain D in a Hilbert space

Suppose A= A' +A where A is self-adjoint and A' is symmetric, positive

.. efinite, and has domain D';! D; the eigenvalue problem for A, whose solution

we assume known, is called the base problem and gives rough lower bounds.

If iii(i = l, .•. ,k) are k discrete eigenvectors of A and if pi= (A'f 1iii(i = l, .•. ,k)

exist, then one can improve the lower bounds by explicitly solving the inter­

mediate eigenvalue problem with opera tor A (k) = A + A 1 p(k) ; here P (k) is the

projection on the subspace generated by the Pi's in the Hilbert space 'J1• with

norm (A'u,u) 112• Because of the choice of Pi's one need only solve a polynomial

equation instead of the transcendental equation required in the general theory.

Furthermore, the above procedure can be combined with the original Weinstein

method to permit base problems obtained by changing both the operator and the

boundary conditions. As an application, lower bounds to the 1ls and 2ls energy

eigenvalues of the helium atom are computed. (Received February 27, 1959.)

557-30. G. G. Lorentz: Isomorphism of function spaces.

Let X be an Orlicz space L ll! of functions defined on a measure space

and Y = ll.. W .a A-space of such functions f(x) with the. norm lfl = .fo00W(f- 1(y))dy,

·vhere W(e) is a concave increasing set function, and f- 1(y) is the set of points

,,.ith lf(x~ ~ y. If X is given, does there exist a W(e) such that X is isomor-

187

phic (algebraically and topologically) to the A -space Aw? Using results of

B, Eisenstadt and the author, this is shown to hold only if the inverse ir (u) of

t'(u) increases exceptionally rapidly, and precisely if and only if

/o00~&u)"i'(u)-2d'i'(u) < oo for some !i > 0. The same results probably re­

main true for the "modulared spaces" of Nakano instead of the Orlicz spaces.

(Received March 3, 1959 .)

557-31 WITHDRAWN.

557-32. Louis Sucheston: Equi.distributed sequences· of events.

(See Volume 5, Abstract 553-159, where page 854,line 5 limni-+oo

should be replaced by lim infn·-+OO') It is shown that a sequence equidistributed 1

of some order r is equidistributed of all orders and with the same density. A

different situation prevails with respect to mixing defined as follows: a sequence

of events A1 is called mixing of order r and with density p if for each non-null

event M,limn1_. 00PM(An 1 ... Anr> = pr, n1-.:: ... <nr· It is shown that if Ai is

mixing of some order r, then it is mixing with the same density of all orders

s < r, but not necessarily s > r. (Received March 3, 1959 .)

557-33, T. J, Rivlin and H. S. Shapiro: Approximation of functions of

two real variables.

Haar gave a condition necessary and sufficient that every real continuous

function on a compact set B in euclidean n-space have a unique polynomial of

best approximation on B. (cf, Achieser, Theory of Approximation, p. 67)

Mairhuber has shown (Proc. Amer. Math. Soc. vol. 7 (1956) p. 609) that Haar's

condition can be satisfied, if, and only if, B is homeomorphic to a closed subset

of the circumference of a circle, hence, for n e; 2, in no interesting point sets.

Recently, however, Collatz noted (Z. Angew. Math. Mech. vol. 36 (1956) p. 198)

that if f(x,y) has continuous first partial derivatives in a strictly convex closed

region B of the plane then there exists a unique linear polynomial (ax + by + c)

of best approximation to f on B. The authors show by explicit construction that

Collatz' result has no analogue, even for quadratic polynomials, even if f(x,y)

is a C 00 function on an arbitrary region of the plane. (Received March 3, 1959.)

188

557-34. H. s. M. Coxeter: An improved counterexample for Kempe's

·•proof" of the four-color theorem.

In 1880, A. B. Kempe published a plausible argument which was accepted

for ten years as a proof that every map on a sphere can be colored with 4

colors. In 1890, P. J. Heawood drew attention to the fallacy in Kempe's argu­

ment, using for a counterexample a map having 18 regions. This counterexample

is now reduced to an agreeably symmetrical map of 9 regions, which reveals the

fallacy far more clearly. (Received March 3, 1959.)

557-35. W. G. Bade and P. C. Curtis, Jr.: The Wedderburn principal

theorem for certain commutative Banach algebras.

Theorem: Let A be a commutative Banach algebra with unit and totally

disconnected maximal ideal space X A. Let R denote the radical of A. Then

(i) A/Rr;;; C(XA) and (ii) there exists a subalgebra B of A, B a! A/R, and

A = B 8l R if and only if the idempotents of A form a bounded set. If the latter

condition holds then B may be taken to be closed and the"isomorphism is then a

homeomorphism. Furthermore if R is a nil ideal, then B is unique. IfF is

nite dimensional then condition (i) alone implies that the idempotents form a

... ounded set. It can also be shown that if XA is a totally disconnected F space

[Gilman-Hendrikson, Trans. Amer. Math. Soc. vol. 77 pp. 340-362] then the

idempotents of A are automatically bounded, and hence if B is any semi-simple

Banach algebra with such a space XA as maximal ideal space, then B = C(XA).

Lastly it is shown that an algebra constructed by Feldman [Proc. Amer. Math.

Soc. vol. 2,pp. 771-777] gives an example in which (ii) holds but, as is shown

by Feldman, the subalgebra B cannot be closed. (Received March 3, 1959.)

557-36. J. H. Wells: Hausdorff transforms of bounded sequences.

It has been shown by Barone [Duke Math. J. vol. 5 (1939) pp. 740-752]

that the special Hausdorff sequence to sequence transformations of Holder,

CesAro and Euler have the property that the set of limit points of the transform

of each bounded complex number sequence is connected. In this paper we prove

the following extension: Let 91 be a complex valued function of bounded variation

on (0 ,1] such that 91(0+) = 91(0) and 91(1) ~ 91(0) "' 1 and let H(91) be the regular

Hausdorff transformation generated by 91. The set of limit points of the H(91)

transform of each bounded complex number sequence is connected only in case

.-) = 91 (1). (Received March 3, 1959.)

189

557-37. J. J. Price: Orthonormal sets with non-negative Dirichlet

kernels.

Let (S,p) be a measure space with )l a positive, nonatomic measure. A

class of orthonormal sets in L 2 (S,}l) is defined. The sets are composed of real­

valued "step" functions analogous to the classical Haar functions and are called

Haar systems with respect to Jl· Theorem. Let Vn(s)}~=O be an orthonormal

set in L2 (S,p). The associated Dirichlet kernels .L:j~5fj(s)fjm are non­

negative iff, for n ii!; 0, fn(s) = JO (s){lln(s) where f!iln(s)}:=o is a Haar system with

respect to the measure dV'=f~(s)d)l. (Received March 3, 1959.)

557-38. C. K. Chu: Type-insensitive finite difference methods for

symmetric positive equations.

Friedrichs (Comm. Pure Appl. Math. vol. 11 (1958) pp. 333-418) intro­

duced the notion of symmetric positive partial differential equations, to study,

in a unified way, equations of different types, elliptic, hyperbolic, or mixed.

This paper studies the corresponding finite difference methods. The basic dif­

ference scheme is applicable to rectangular domains, and to more general do­

mains which map into rectangles with a positive jacobian in the closure. Fc-­

each fixed mesh, a unique solution to the difference equations exists. As the

mesh size tends to zero, the finite difference solutions uh converge strongly

in Lz to the solution u of the differential equation, if it is assumed u € C 1 in the

closure of the domain. The difference scheme, after modification, is solvable

by iterations. If the differential equation satisfies the additional conditions

(Friedrichs p. 367), and another restriction on the boundary, the existence of

a strong solution u to the differential equation problem , and its differentiability,

are proved by finite differences. Moreover, uh converge to u weakly. A class

of mixed elliptic-hyperbolic equations is shown to satisfy all such restrictions.

(Received March 3, 1959.)

557-39. D. E. Spencer: Retarded vector theorems.

The vector theorems of Gauss, Stokes and Green are ordinarily obtained

for a section t' = const of the 4-space (u 1' ,u2 ' ,u3 ' ,t'). But, in dealing with the

retarded potentials of electromagnetic theory, a different section is needed

with 1 held constant at a field point I? and with different values of!.' associated

with each element of volume (t' = t - r/c). The paper derives divergence and

190

11 Z' 3' url theorems for retarded vectors [~] = l!:(u ,u ,u ,t'), where the integrals

are evaluated at t = const. An extension of Green's theorems to the retarded

case is also included. (Received March 3, 1959.)

557-40. A. M, Garsia and E, R. Rodemich: The Schottky uniformization

of compact canal surfaces.

Some conformal properties of compact complete canal surfaces will be

discussed in this paper together with an extension and geometric interpretation

of the results announced in Abstract No. 546-16 of these Notices. The Schottky

model corresponding to a marked Riemann surface of genus one, can be defined

by a Moebius transformation of the type (t"z - ot)/(t"z - fJ)= fei9 (z - 0(.)/(z - (-i).

In the case of a Canal surface of genus one it can be shown that e and 9 have an

interesting geometric characterization: e represents the rate at which a family

of orthogonal trajectories to the generating spheres approaches a fixed closed

curve A which is also an orthogonal trajectory; 9 represents the winding rate

of these trajectories around A. Both e and 9 can be computed by quadratures.

These considerations are used to exhibit explicit examples of analytic surfaces

of genus one which are not conformally equivalent to any of the tori of revolu­

tion. (Received March 3, 1959 .)

557-41. P. C, Fife: Partial Holder continuity and elliptic partial differ­

ential equations.

Consider the boundary-value problem Lu(P) = f(P), P in t1!J.; u(P) = 0,

P on i:J, where L is a linear Znd order elliptic differential operator in Z

variables with smooth coefficients, ,Sis a bounded domain, and f E Cot... A

standard result (Korn, Schauder) is that there exists a solution u E Cz+cx. with

lulz+at. <: K\flel.' where the norms are the pointwise norms appropriate to

CZ+ot and C"'. Suppose suitable local coordinates n,s are introduced, regular

in each of a finite number of subdomains, one of which contains the boundary

.iJ, and such that i7-is given by n =const. Let C~+ocbe a class more general

than C>.+oc..in that }1-th derivatives of its functions are Holder continuous with

respect to s, but need be only piecewise continuous in n. Then the above result

is still valid if the superscripts is adjoined to C everywhere. This may be

generalized to certain equations of higher order and probably to higher

dimensions. (Received March 3, 1959.)

191

557-42, W. W. Comfort: On a certain Silov boundary.

A • Let G, the space of semi-characters on the commutative semi-group G,

A distinguish points. For each x E.G, let Ax= {z € G \if 'X.._E G and ?((z) = 0,

then:l((x) = o]. For ?(EG, let SO() =fz E G\x_(z) f. o}. For J(E G, let

Y(;() = [z E G\S(J()jz = z 2 or az = z 2 for some a E S(l() or a 1z = a2z for some

ai E S(l() with ?(_(a1) ";t ;(_(a2)}. Let E = t?(E G\S(~ =Ax for some x E G}, so

that E is w*-dense in G. Let D = {?(E.GIS(>i')= S(i() whenever \fl EG and S(Y')

:::l S('X) and S(lf') n Y("() =A} n{-x: E o\x € G implies l((x) = 0 or h((x) I = 1}.

Then, with o denoting the Silov boundary induced in G by the algebra (.i,(G)]",

we have the following pair of theorems. 1. DC o. 2, If S(l(_) =Ax and if there

is a finite Z C G\S(i() for which S(lf) ~ S(;() implies \fi(Z) q: {o}, and if I( Eo,

then -!,_ED-. Our results depend heavily on the structural decomposition

theorems for G given by Hewitt and Zuckerman (Trans. Amer. Math. Soc.

vol. 83 (1956) pp. 70-97). (Received March 4, 1959.)

557-43. N. J. Rothman: Embedding topological semigroups in topological

groups.

It is well known that a commutative semigroup with cancellation is

embeddable in its quotient group. A commutative topological semigroup S is

said to have property (F) if for x andy inS, with x in an open set U, there is an

open set V containing y such that xy € Uy' for all y' in V. Theorem. Let S be

a commutative topological semigroup with cancellation. The quotient group

Q(S) is a topological group and S is embeddable as a subset with nonempty

interior in Q(S) if and only if S contains an ideal I such that I is a semigroup

with property (F). Further, if S has property (F) then Swill be open in Q(S).

(Received March 4, 1959.)

557-44. R. P. Goblirsch: On imbedding decompositions of 3-space in

4-space.

Let K be a proper closed subset of the 1- sphere S 1 . Then S 1 X K can be

imbedded in Euclidean 3-space so that if treK, t2E K and trf= t2, then S1 X tr

is a circle which links the circle s 1 X t2. The decomposition space associated

with the decomposition of 3-space whose only nondegenerate elements are the

circles s1 X t forte K is imbeddable in 4-space. The case where K is a set of

three points answers a question of R. H. Bing and M. L. Curtis· in Abstract

551-9, Notices Amer. Math. Soc. vol. 5. (Received March 4, 1959.)

192

557-45. E. C. Dade and Karl Goldberg: Group-generated incidence

.gebras.

Let G be a permutation group on S = f1,2, ... ,n}. Let P(G) be the algebra

(over an arbitrary ground field K) spanned by the set of permutation matrices

of order n which represent G faithfully. Then the centralizer of P (G) in Kn is

an algebra A(G) spanned by a unique set of 0,1 matrices {Ap}, whose sum is

J = (1). We call this algebra a group-generated incidence algebra. (For an

equivalent definition in terms of an equivalence on S X S see Abstract 538-46,

Notices Amer. Math. Soc. (1958) p. 41). The centralizer of A(G) in Kn is P(G).

Several properties of A(G) are determined from the properties of G. For

example, if G is doubly transitive then A(G) is spanned by {I,J - r}whose

centralizer is D, the set of all matrices in Kn with row and column sums equal

to each other. Thus Dis spanned by the permutation matrices representing G.

As previously noted, we can use these algebras to find a sufficient condition for

the construction of Hadamard matrices based on properties of G. (Received

March 4, 1959.)

557-46. Robert Carroll: Some generalized Cauchy problems.

Some generalized Cauchy problems related to the Euler-Poisson-Darboux

(EPD) equation are studied from the point of view of L. Schwartz (Annales

Fourier 1950). The convexity and growth theorems of Weinstein (Annali di Mat.

1957) for the EPD equation are generalized and extended to some other equations.

As an example of these results it may be stated that the Cauchy problem for the

EPD equation of index k E;. 0, ('"02 /ot2)t..lx(t) + (k/t)(o/dt)Qx(t) - t::. * t.:lx(t) = 0,

with Ulx(O) = T E r/:J~ ; ("b/ot)G.lx(t)lt~O = 0 is well posed in GO~ for any finite

time interval. If 6 "T is a measure ~ 0 and k ~ n - 1 then, almost everywhere

in x, Ulx(t) is increasing in t, and is convex in t 2 -n where n is the dimension.

If 0:::?. k < n- 1 and Ai•T ;;p 0 fori:::?. 1 + [(p + 1)/2] where k + 2p:;;:;: n- 1, then,

almost everywhere, G.>x(t) is convex in t 1-k. (Received March 4, 1959.)

557-4 7. M. S. Klamkin: Some separable solutions of the two-dimensional

heat conduction equation with temperature-varying thermal properties.

Some separable solutions are derived for the following cases: 1. Steady­

state, with the thermal conductivity a linear function of the temperature.

2. Unsteady- state, with the thermal conductivity a linear function of the

te~·nperature and the thermal diffusivity a constant. These solutions can be

193

used to check out digital programs for the numerical solution of the two-dimen­

sional heat flow equation subject to temperature-varying thermal properties.

(Received March 4, 1959 .)

557-48. V. L. Shapiro: Intrinsic operators in three-space.

Let v(x)= [v 1(x),v2(x),v3(x)] be a continuous vector field defined in a

neighborhood of the point xo in Euclidean three-space, E 3, with x = (x 1,x2 ,x3).

Define the intrinsic curl of vat x 0 to be the vector w(xo) = [w 1(xo),wz(xo),w3(xo)]

provided the following three limits exist: wj (ll()) = liffit -o('ll't2) - 1/c j(XO ,t) (v dx)

j • 1 ,Z ,3 with C j(x0 ,t) the circumference of the circle of radius t and center x0 in

the plane through x0 normal to the xj-axis where C j(x0 ,t) is oriented in the

counter-clockwise direction when seen from the side in which the xraxis points.

Define the upper intrinsic divergence of vat xo, div*v(xo), to be the following

expression: lim supt_.o3(4'11't3) - 1/s(xo,t) (v ,n)dS where n is the outward point­

ing unit normal on S(x0 ,t), the spherical surface with center x 0 and radius t.

Define the lower intrinsic divergence, div * v(x0), similarly using lim inf. The

following theorem then holds: Theorem: Let D be a bounded domain in

Euclidean three-space, and let v(x) be a continuous vector field defined in D.

Then a necessary and sufficient condition that v(x) be locally in D the gradh'nt

of a Newtonian potential due to a bounded density is that (i) intrinsic curl v = 0

in D and (ii) div*v(x) and div.v(x) be locally bounded in D. (Received March 4,

1959.)

557-49. H. F, Mattson: A generalization of the Riemann-Roch theorem.

A generalization of the Riemann-Roch theorem is proved for a finite­

dimensional left A-, right B -module over the function field K (of one variable),

where A and B are semi-simple algebras of finite rank over the center K. This

theorem includes the Riemann-Roch theorem for simple algebras of E, Witt

(Math. Ann. vol. 110 {1935) pp. 1Z-Z8) and that for matrices over function fields

of A. Weil (J. Math. Pures Appl. vol. 17 (1938) pp. 47-87) in case Weil's

"signature" is taken to be identically 1. The constant field is arbitrary in each

case, however. The methods use·d are those of linear topology and duality, as

first explained by K. Iwasawa (Annals of Math. vol. 57 (1953) pp. 331-356).

The linear topology leads to some simplification in the arithmetic of simple

algebras over function fields. (Received March 4, 1959.)

194

557-50. D. W. Wall: Groups having the same group characters.

Although isomorphic finite groups necessanly have the same (ordinary)

group characters, groups having the same group characters need not be isomor­

phic. The purpose of this paper is to give examples of nonisomorphic groups

which have the same group characters. Let fa,b) be the free group with two

generators. For any integers m,n,k and p let G(m,n,k,p) be the subgroup

(if it exists) of {a,b) defined by the relations: am = 1, bn = ak, ab = baP (The

choices of m ,n,k and p are not independent.) Given here are classes of non-

isomorphic groups of this type such that all of the groups in any class have the

same table of group characters. For example, for any positive integer s,

G(4s ,2 ,2 s ,4s - l) and G(4s ,2 ,4s ,4s - l) are nonisom orphic groups with the

same group characters. The methods employed are computational and elemen-

tary. (Received March 4, 1959.)

557-51. E. V. Haynsworth: Reduction formulae for partitioned matrices.

A theorem of L. Goddard and H. Schneider, (Pairs of M1atrices with a

Non-Zero Commutator, Proc. Camb. Phil. Soc., 51,4 (1955) pp. 551-553)

concerning square matrices A and B, of orders nand m respectively, which

sat',sfy an equation AX= XB for some n X m matrix X, is generalized here for

rectangular matrices A and B, with dimensions n 1 X n 2 , m 1 X m 2 , which satisfy

AX2 = X 1 B, where Xi has dimensions niX mi for i = 1,2. This result is used

to find reduction formulae for partitioned matrices with sub-matrices, Aij•

having dimensions niX nj, and satisfying equations AijXj = XiBij· The reduc­

tion formulae given here are also generalizations of a theorem by j. Williamson

(The Latent Roots of a Matrix of Special Type, Bull. Amer. Math. Soc. vol. 37

(1931) pp. 585-590) concerning partitioned matrices whose submatrices are all

square and satisfy AX= XB, ·where B is triangular and X is square. (Received

March 4, 1959.)

557-52. H. S. Shapiro: On the completeness of sets of convolutions, and

a class of Schlicht functions defined by Fourier transforms.

For real functions f(t), g(t) belonging to the class L 1(0,oo) we define

f * g =/otf(u)g(t- u)du. Let fr=f, fntl = fn * f for n > 1. Let F(z)

= /S0f(t)e itzdt (z = x t iy ,y ~ 0). Theorem: The fn are complete in L 1 (0 ,oo)

if and only if F(z) is schlicht in y > 0. (Because of the special nature ofF this

equivalent to F(xr) =f F(x2) for all real x 1 'f= x 2 .) A sufficient condition for

195

this is that both f(t) and its Fourier cosine transform be positive and decreasing

in (O,oo), for instance f(t) = /o00e -tuda-(u), where a- is any positive mass distri

bution for which this function is integrable. Moreover, if f E. L 2 n L 1, the

fn are also complete in L2 in the case that F is schlicht. It is also shown that

if f(t) is in L 1 (-oo,oo), a sufficient condition that the sub-algebra of the group

algebra on (-oo,oo) generated by the two elements f(t), f(-t) is dense in

L 1(-oo,oo) is that f vanish fort < 0 and, outside some x-set of measure zero,

F(x) (x real) assumes no value twice (e.g. the characteristic function of (0,1)

is a suitable f). The proof involves an analogue of the Stone-Weierstrass

theorem for L 2 approximation. Application is made of the first mentioned

theorem to the completeness of the functions {ti) with respect to the weight

function e-t in (O,oo). (Received March 4, 1959.)

196

MEMOIRS

OF THE

AMERICAN MATHEMATICAL SOCIETY

1. G. T. Whyburn, Open mappings on locally compact spaces. ii, 25 pp. 1950. $0.75

2. Jean Dieudonne, On the automorphisms of the classical groups, with a supplement by L. K. Hua. viii, 122 pp. 1951. 1.80

3. H. D. Ursell and L. C. Young, Remarks on the theory of prime ends. 29 pp. 1951. 1.00

4. Kiyosi Ito, On stochastic differential equations. 51 pp. 1951. 1.00

5. Oscar Zariski, Theory and applications of holomorphic functions on algebraic varities over arbitrary ground fields. 90 pp. 19 51. 1.40

6. K. L. Chung, M. D. Donsker, P. Erdos, W. H. J. Fuchs, and M. Kac, Four papers on probability. ii, 12 + 19 + 11 + 12 pp. 1951. 1.20

7. R. P. Kadison, A representation theory for commutative topological algebra. 39 pp. 1951. 1.00

8. J. W. T. Youngs, The representation problem for Frechet surfaces. 143 pp. 1951. 1.80

9. I. E. Segal, Decompositions of operator algebras. I and II. 67 + 66 pp. 1951. 1.80

10. S.C. Kleene, Two papers on the predicate calculus. 68 pp. 1952. 1.30

11. E. A. Michael, Locally multiplicatively-convex topological algebras. 79 pp. 1952. 1.40

12. S. Karlin and L. S. Shapley, Geometry of moment spaces. 93 pp. 195 3. 1.50

13. Walter Strodt, Contributions to the asymptotic theory of ordinary differential equa-tions in the complex domain. 81 pp. 19 54. 1. 50

14. Lie algebras and Lie groups. Five papers prepared in connection with the First Summer Mathematical Institute. vi, 54 pp. 1955. 1.30

15. I. I. Hirschman, Jr., Tbe decomposition of Walsh and Fourier series. 65 pp. 195 5. 1.40

16. Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucleaires. 191 + 140 pp. 195 5. 2.80

The price to members of the American Mathematical Society is 2 5% less than list.

NATIONAL ACHIEVEMENT TESTS

.Algebra Test for Engineering and Sci­ence, First Algebra, Plane Geometry, Solid Geometry and Plane Trigonome­try Tests. Specimen Set of these Tests

.Send for Test Catalog ............. $1.00

Acorn Publishing Company Rockville Centre, New York

COMING SOON Colloquium Publications

Volume xxiii

ORTHOGONAL POLYNOMIALS

by GABOR SZEGO

With corrections and new material.

AMERICAN MATHEMATICAL SOCIETY 190 Hope Street

Providence, Rhode Island

MATHEMATICAL REVIEWS

A Journal Containing Reviews of the Mathematical Litera­ture of the World, with full Subject and Author Indices

Sponsored by

The American Mathematical Society The Mathematical Association of

America The Society tor Industrial and

Applied Mathmnatics The Institute of MathenLatical Statistics

The Edinburgh Mathematical Society Societe Mathematique de France

Dansk Matematisk Forening Het Wiskundig Genootschap te

Amsterdam The London Mathematical Society Polskie Towarzystow Matematycne

Uni6n Matemd.tica Argentina Indian Mathematical Society Union Matematica Italiana

Subseriptions accepted to eover the calendar year only. Issues appear monthly except July. $50.00 per year. $25.00 to members of spon­soring organizations. An edition printed on one side, for biblio­graphical purposes, is available at an additional charge of $1.00 per year. Uneseo Book Coupons may be used in payment.

Send subscription orders to

AMERICAN MATHEMATICAL SOCIETY

190 Hope Street Providence 6, R. I.

SALT LAKE CITY MEETING AT THE UNIVERSITY OF UTAH

September 2-5, 1959

RESERVATION FORM

Detach on the dotted line below and mail to Professor W. j. Coles, Department of Mathematics, University of Utah, Salt Lake City, Utah, as soon as possible. No reservations accepted after August 1.

Name (Last)

AMERICAN MATHEMATICAL SOCIETY

RESERVATION FORM Salt Lake City Meeting

(First) (Please print)

Institution •.•. a.m.

Arrival date .•••...... p.m. Departure date a.m.

. p.m.

Dormitory accommodations desired for ... men, ..••. women, ..•.. girls

.•••.•. (ages), •.•...•• boys, •.•••... (ages). Indicate relationship to you.

(Rates in program; payable on registration). Number of rooms ••...•

Number desiring to attend Picnic •.....•..

Interested in Camping .....••••

(Those who wish to stay at a hotel or motel should make their reser­vations directly with the hotel or motel.)

199

AM

l!.n.1\...AN

MA

TH

EM

AT

ICA

L S

OC

IET

Y

190 Hope S

treet P

rovidence 6, R

hode Island

Return P

ostag

e Guaranteed

SEC

ON

D-cL

ASS PO

STAG

E PA

ID A

T A

NN

ARBO

R, M

ICH

lGA

N