AMERICAN

MATHEMATICAL

SOCIETY

V OLUME 8 , NUMBER 2 ISSUE NO. 5 3 APRIL 1961

THE AMERICAN MATHEMATICAL SOCIETY

Edited by GORDON L. WALKER CONTENTS

MEETINGS Calendar of Meetings . • • • • • . • • • • • • • • • • • • . • • . • . • • • . . . • • • • • • • 84 Program of the April Meeting in New York . • • • . • . • • • . • • • . • . . . • . • . 85 Abstracts for the Meeting- pages 139-155 Program of the April Meeting in Chicago • . • • • • • • • • . • • . . • • . • . • • • • 94 Abstracts for the Meeting- pages 156-165 Program of the April Meeting in Stanford • • • • . • • • • . . • • . • • • • • • . • • 98 Abstracts for the Meeting - pages 166-173 PRELIMINARY ANNOUNCEMENT OF MEETINGS • • • • • • • • . . • • • • . • . • • • . 101 ACTIVITIES OF OTHER ASSOCIATIONS • • • • . • • • • • • • . • . • • . • • • . • • • • • . 102 NEWS AND COMMENT FROM THE CONFERENCE BOARD OF THE MATHEMATICAL SCIENCES • • • • • • • • • . • • • . • • • . • • . • • . • • 103 MATHEMATICS IN TRANSLATION • . • • • . • . • • . • • • • • • . . • • • • . • • • • • . • 111 THE PROPOSED DOCTOR OF ARTS DEGREE - B.y E. E. Moise • • • • • • • • • • • • 112 RUSSIAN-ENGLISH DICTIONARY OF MATHEMATICAL TERMS...... • • • • 116 REPORT ON RECIPROCITY AGREEMENTS. • . • . • • • • • . • . • • • • • . . • • • • • 119 NEWS ITEMS AND ANNOUNCEMENTS , , •• , , , •••••••••• , ••• 115,137,155,125 PERSONAL ITEMS • • . . • • • . . • • . • . . • • • • • . . • • • • • . • . • • • • . . • . • . • . 127 ERRATA. • • • • • . • • • • • • • • • • . • . • . • . . • • . • . . • • • . • • . • . • • • • • • 132,130 LETTERS TO THE EDITOR ••••••...••...••••••. , ••..••••••••••. 131 NEW PUBLICATIONS ..•••.•••.••••••••••.•.••••.••••..••••••• 133 SUPPLEMENTARY PROGRAM -No. 3 • . . . . • • . • . • . • . • . • • • . • • • • • • • • • 138 ABSTRACTS OF CONTRIBUTED PAPERS . . . • . • . • . • . • . • . . • • • . . • • • . • • 139 INDEX OF ADVERTISERS •..••••••.•••.•••.•.•••.•••.•.•.•••••• 190 RESERVATION FORM •••.•••.•.•..••.••.••.•.•••••.•••••••.•. 191 MEETINGS

CALENDAR OF MEETINGS

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

Meet­ Deadline ing Date Place for No. Abstracts*

581 June 13-16, 1961 Seattle, Washington April 28 582 August 29 - September 1, 1961 (66th Summer Meeting) Stillwater, Oklahoma July 14 583 October 28, 1961 Cambridge, Massachusetts Sept. 14 584 November 17-18, 1961 Milwaukee, Wisconsin Oct. 3 585 November 17-18, 1961 Gainesville, Florida Oct. 3 586 November 18, 1961 Santa Barbara, California Oct. 3 January, 1962 Cincinnati, Ohio (68th Annual Meeting) August, 1962 Vancouver, British Columbia (67th Summer Meeting) January, 1963 Berkeley, California (69th Annual Meeting) August, 1963 Boulder, Colorado (68th Summer Meeting)

* The abstracts of papers to be presented in person at the meetings must be received in the Head· guarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for by title abstracts are April 21 and July 7.

The NOTICES of the American Mathematical Society is published by the Society seven times a year, in February, April, June, August, October, November, and December. Price per annual vol­ ume 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, 13 50 Main Street, Ann Arbor, Michigan, or to 190 Hope Street, Providence 6, Rhode Island. Second-class postage paid at Ann Arbor, Michigan. Authorization is granted under the author­ ity of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34.21, P. L. and R.). Accepted for mailing at the special rate of postage provided for in section 34.40, paragraph (d).

Copyright© 1961 by the American Mathematical Society Printed in the United States of America

84 Five Hundred Seventy-Eighth Meeting Hotel New Yorker New York, New York April 5 - 8, 1961

PROGRAM

The five hundred seventy-eighth address the Society on "Homotopy groups" meeting of the American Mathematical at 2:00 P.M. on Saturday in the Grand Ball­ Society will be held on Wednesdaythrough room. Saturday, April 5-8, 1961 at the Hotel New There will be sessions for contri­ Yorker. All sessions will be held in public buted papers on Friday afternoon and on rooms of the hotel. Saturday morning and afternoon, including Two symposia will be held in con­ provision for a limited number oflate pa­ junction with the meeting. A symposium on pers. Abstracts of papers listed on this Mathematical Problems in the Biological program appear on pages 139 to 155 of Sciences will be held during the morning these NOTICES (or, exceptionally, in a and afternoon on Wednesday, Thursday, subsequent issue). There are cross refer­ and Friday. This is another in the series ences to the abstracts in the program. For of Symposia in Applied Mathematics spon­ example, the number ( 578- 5) following sored by the Office of Ordnance Research the title to appear ( 1) in the program indi­ and is sponsored also by the National Sci­ cates that its abstract is so designated ence Foundation. The Invitations and Steer­ among the published abstracts. ing Committee includes Dr. S. M. Ulam, The Council will meet at 5:00 P.M. Chairman, Dr. R. E. Bellman, Secretary, on Friday in Parlor C. Dinner has been Professor Anthony Bartholomay, Dr. John arranged during the intermission in the Jacquez, Professor Theodore Puck, and meeting. Professor C. E. Shannon. A Symposium on Recursive Func­ REGISTRATION tion Theory, with financial support from The registration desk will be open project FOCUS of the Institute for Defense from 9:00 A.M. to 5:00 P.M. each day of Analyses, will be held on Thursday morn­ the meeting except Saturday, when it will ing and afternoon and Friday morning. The close at 3:00P.M. On Wednesday only, the symposium is under the auspices of the desk will be located on the mezzanine op­ Association for Symbolic Logic and of the posite the elevators. During the remainder Association for Computing Machinery as of the meeting it will be located in the well as of this Society. The Organizing Ballroom Foyer on the secondfloor. Committee consists of Professor S. C. Kleene, Chairman, Professor J, c_ E •. EXHIBITS Dekker, Professor John McCarthy, Pro­ fessor J. B. Rosser, and Professor J. R. There will be a number of exhibits, Shoenfield. of both books and computing machines, on By invitation of the Committee to view on the mezzanine on Thursday, Fri­ Select Hour Speakers for Eastern Sec­ day and Saturday. tional Meetings, Professor Jacob Wolfo­ witz of Cornell University will address THE HOTEL the Society on "Coding theorems of infor­ The New Yorker is the official head­ mation theory" at 2:00 P. M. on Friday in quarters hotel. Persons desiring to stay the Grand Ballroom. there should communicate directly with the By invitation of the same Commit­ hotel. On page 191 there is a table of room tee, Professor Daniel M. Kan of the rates and a reservation blank, It is im­ Massachusetts Institute of Technology will portant to the Society with respect to future

85 hotel meetings that the reservation blank block west and oneblocknorthofthePenn­ be used so that the Society will get credit sylvania Station, from which there is a tun­ for having generated the reservatioJl•• nel direct. The hotel is on the 8th Avenue line of the Independent Subway, at the 34th MAIL AND TELEGRAMS Street or Pennsylvania Station stop. The Correspondence for those known to hotel can be reached by walking under­ be registered at the Hotel New Yorker may ground from the 34th Street station of the be sent there directly. The mailing address IR T subway. The hotel is one block east is Hotel New Yorker, 34th Street and 8th from the 34th Street exit of the Lincoln Avenue, New York 1, New York. Mail and Tunnel and is less thanhalfablockwest of telegrams for others attending the meeting the Greyhound Bus Terminal. Airline ser­ may be sent in care of the American Math­ vice into LaGuardia and Idlewild airports ematical Society at the hotel. is served by the West Side Terminal, eight blocks north and two blocks west of the New Yorker, which is somewhat nearer to TRANSPORTATION the hotel than the East Side Terminal ls. The Hotel New Yorker is on the The hotel has its own garage, which adver­ corner of 34th Street and 8th Avenue, one tises reasonable rates.

PROGRAM OF THE SYMPOSIUM ON MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES

WEDNESDAY, 9:ZO A.M. First Session: Role of information theory, nervous systems and computers Terrace Room, off Main Lobby Chairman: Professor Arthur W. Burks, University of Michigan Discussion Leader: Professor Norbert Wiener, Massachusetts Institute of Technology 9:ZO Symposium convenes 9:30 - lO:ZO Mathematical models in the behavior of the central nervous sys­ tem Professor Herbert D. Landahl, Committee on Mathematical Biology, University of Chicago 10:30 - ll:ZO Machine models of self-reproduction Dr. Edward F. Moore, Bell Telephone Laboratories ll:30 - lZ:ZO Mathematical problems in heuristic programming Professor Marvin L. Minsky, Massachusetts Institute of Tech­ nology

WEDNESDAY, Z:OO P.M. Second Session: Biochemistry, Terrace Room, off Main Lobby Chairman: Dr. Stanislaus Ulam, Los Alamos Scientific Laboratory Z:OO - Z:SO Protein synthesis Professor Paul C. Zamecnik, Harvard Medical School 3:00 - 3:50 [Title to be announced] Professor Max Delbruck, California Institute of Technology 4:00 - 4:50 Bio-chemistry: sterile or virgin for mathematicians? Professor Arthur B. Pardee, Virus Laboratory, University of California, Berkeley

86 THURSDAY, 10:00 A.M. Third Session: Short papers, Terrace Room, off Main Lobby Chairman: Dr. John Jacquez, Sloan-Kettering Institute for Cancer Research 10:00 - 10:2.0 Analysis of chemical reaction occurring with migration Dr. Norman Shapiro, National Institutes of Health 10:30 - 10:50 Efficient coding for the desoxyribonucleic channel Dr. Solomon W. Golomb, Jet Propulsion Laboratory, California Institute of Technology 11:00 - 11 :Z.O A study of Lerman models for diffusion and captures of excitons in DNA Dr. Walter Goad, Los Alamos Scientific Laboratory 11:30 - 11:50 Mathematics of pattern recognition for the ECG Professor Max A. Woodbury, New York University, University Heights

THURSDAY, Z:OO P.M. Fourth Session: Enzymes, hormone circulation, and medical diagnosis by com­ puters - Terrace Room, off Main Lobby Chairman: Dr. Richard E. Bellman, The RAND Corporation Z:OO - 2.:50 Medical diagnosis and modern decision making Dr. Robert Ledley, National Biomedical Research Foundation 3:00 - 3:50 Mathematical problems in modern physiology Dr. John Jacquez, Sloan-Kettering Institute for Cancer Re­ search 4:00 - 4:50 Mathematical aspects of adaptive control Dr. Robert Kalaba, The RAND Corporation

FRIDAY, 10:00 A.M. Fifth Session: Enzymology and kinetics, intracellular information transfer Terrace Room, off Main Lobby Chairman: Professor Anthony F. Bartholomay, Harvard Medical School 10:00 - 10:50 Enzymology and kinetics Professor Irwin W. Sizer, Chairman of the Department of Bio­ logy, Massachusetts Institute of Technology 11:00- 11:50 Intracellular information transfer Professor Alexander Rich, Department of Biology, Massachu­ setts Institute of Technology

FRIDAY, 3:15P.M. Sixth Session: Short papers, Grand Ballroom, Second Floor Chairman: Professor Claude E. Shannon, Massachusetts Institute of Tech­ nology 3:15 - 3:35 A study of combinatorial problems in patterns of growth Dr. Stanislaus Ulam, Los Alamos Scientific Laboratory 3:45 - 4:05 From chemotherapy to computers to trajectories Dr. Richard E. Bellman, The RAND Corporation

87 4:15 - 4:35 Growth of monodisperse populations of mammalian cells in vitro Dr. Donald Petersen, Los Alamos Scientific Laboratory

PROGRAM OF THE SYMPOSIUM ON RECURSIVE FUNCTION THEORY

THURSDAY, 9:30A.M • .First Session, Grand Ballroom, Second Floor Chairman: Professor Alfred Tarski, University of California, Berkeley 9:30 Infinite series of isola (25 minutes) Professor j. C. E. Dekker, Rutgers, The State University 1o:oo n - A (2s minutes) Professor john Myhill, University of California, Berkeley 10:30 Non-standard models and arithmetically isolated sets (25 minutes) Professor Anil Nerode, Cornell University 11:00 Representability of sets in formal sets (50 minutes) Professor Andrzej Mostowski, University of Warsaw

THURSDAY, 1:45 P.M. Second Session, Grand Ballroom, Second Floor Chairman: Professor Frederick B. Fitch, Yale University 1:45 Algebras of sets binumerable in complete extensions of arithmetic (25 minutes) Professor Dana S. Scott, University of California 2:15 Some problems in hierarchy theory (25 minutes) Professor john W. Addison, University of Michigan Provably recursive functionals in analysis (25 minutes) Professor Clifford Spector, Ohio State University Recursive functions of higher types (25 minutes) Professor joseph R. Shoenfield, Duke University [Title to be announced] Professor A. A. Markov, V. A. SteklovinstituteofMathema­ tics, Moscow

FRIDAY, 9:15A.M. Third Session, Grand Ballroom, Second Floor Chairman: Professor j. Barkley Rosser, Cornell University and Institute for De­ fense Analyses 9:15 Applications of recursive function theory to number theory (25 minutes) Professor Martin Davis, Yeshiva University 9:45 Sequence generators and digital computers (25 minutes) Professor Arthur Burks, University of Michigan 10:15 The treatment of ambiguity and paradox in mechanical languages (25 minutes) Professor Saul Gorn, University of Pennsylvania

88 10:45 Checking proofs by computer (25 minutes) Professor John McCarthy, Massachusetts Institute of Tech­ nology 11:15 Solvable subclasses of the Suranyi reduction class (25 minutes) Professor Burton Dreben, Harvard University 11:45 Discussion of above papers led by Professor Marvin Minsky, Massachusetts Institute of Technology

PROGRAM OF CONTRIBUTED PAPERS AND INVITED ADDRESSES

The time limit for each contributed paper is ten min­ utes. The papers are scheduled at fifteen minute in­ tervals for the convenience of all. In order that each speaker and listener may know the precise time at which each scheduled paper will be heard, the time limit will be strictly enforced.

FRIDAY, 2:00P.M. Invited Address, Grand Ballroom, Second Floor Coding theorems of information theory (One hour) Professor Jacob Wolfowitz, Cornell University

FRIDAY, 3:15P.M. Session on Logic and Foundations, Panel Room, Third Floor 3:15 - 3:25 (1) The independence of transfinite induction up to ww in recursive arithmetic Mr. J. R. Guard, Princeton University (578-5) 3:30 - 3:40 (2) Theory of models of infinite valued logic. IV Professor C. C. Chang, University of California, Berkeley (578 -9) 3:45 - 3:55 (3) Two families of languages related to Algol Dr. Seymour Ginsburg,* System Development Corporation, Santa Monica, California and Dr. H. G. Ric·e, Computer Sciences Corporation, Palos Verdes, California (578-11) 4:00 - 4:10 (4) The functions computable by linear bounded automata Mr. R. W. Ritchie, Dartmouth College (578 -20) 4:15 - 4:25 (5) Permutations with confined displacements Professor N. S. Mendelsohn, University of Manitoba (578-12) 4:30 - 4:40 (6) Representable Boolean algebras and infinitary logics Professor Alfred Tarski, University of California, Berkeley (578-49)

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

89 FRIDAY, 3:15P.M. Session on Algebra, Washington Room, Fourth Floor 3:15- 3:25 (7) The determinant of the adjacency matrix o.f a graph Professor Frank Harary, University of Michigan (578-23) 3:30 - 3:40 (8) A new treatment of Bose-Chaudhuri codes Dr. H. F. Mattson,* Sylvania Applied Research Laboratory, Waltham, Massachusetts, and Dr. Gustave Salomon, Lincoln Laboratory, Massa­ chusetts Institute of Technology (578-19) 3:45 - 3:55 (9) Binary group codes with even redundancy which correct consecutive errors in bursts of three or less Professor R. C. Bose,* University of North Carolina and Professor I.M. Chakravarti, Case Institute of Technology (578 -25) 4:00 - 4:10 (10) Group codes with odd redundancy which correct consecutive errors in bursts of three or less Mr. A. J. Gross, University of North Carolina (578-26) 4:15- 4:25 (11) On the relation between the permanent and the determinant Professor Marvin Marcus,* National Bureau of Standards, Washington D. C. and Professor Henryk Mine, University of Florida (578-31)

SATURDAY, 10:00 A.M. Session on Analysis, Panel Room, Third Floor 10:00 - 10:10 (12) Vector spaces of harmonic mappings Professor W. V. Caldwell, University of Delaware (578-18) 10:15 - 10:25 (13) On the generalized axially symmetric Helmholtz equation Mr. R. J. Weinacht, University of Maryland (578-24) 10:30 - 10:40 (14) A direct operational method for some Cauchy problems Professor T. P. G. Liverman, The George Washington University (578-39) 10:45 - 10:55 (15) Bernstein's theorem and partial difference operators Dr. W. G. Strang, Massachusetts Institute of Technology (578 -46) 11:00 - 11:10 (16) Product integral solutions of linear, nonhomogeneous systems of first order differential equations LCDR J. H. Billings, Portsmouth Naval Shipyard, Portsmouth, New Hamp­ shire (578-6) 11:15 - 11:25 (17) On the transformation of a class of boundary value problems into initial value problems for ordinary differential equations Mr. M.S. Klamkin, A VCO Research and Advanced Development Division, Wilmington, Massachusetts (5 78-3 5) 11:30 - 11:40 (18) Ultimately periodic solutions to a nonlinear integrodifferential equation Dr. V. E. Bene!t, Bell Telephone Laboratories, Murray Hill, New Jersey (578-10) 11:45 - 11:55 (19) On global stability of two-dimensional autonomous systems. Preliminary re­ port Dr. Czeslaw Olech, RIAS, Baltimore, Maryland (Introduced by Dr. J. P. LaSalle) 90 SATURDAY, 10:00 A.M. Session on Geometry and Topology. Washington Room, Fourth Floor 10:00 - 10:10 (ZO) On minimal sets in the plane Professor Russell Remage, Jr., University of Delaware (578-13) 10:15- 10:Z5 (Zl) A theorem on families of acyclic sets and its applications Dr. Antoni Kosinski, University of California, Berkeley (578-7) 10:30 - 10:40 (ZZ) Local one-one maps on manifolds. Preliminary report Dr. G. H. Meisters* and Dr. Czes.la,w Olech, RIAS, Baltimore, Maryland (578-3Z) 10:45 - 10:55 (Z3) Some imbedding and nonimbedding theorems for n-manifolds Mr. Beauregard Stubblefield, International Electric Corporation, Para­ mus, New jersey (578-33) 11:00 - 11:10 (Z4) Similarities and conformal transformations Professor H. S. M. Coxeter, University of Toronto (578-ZZ) 11:15 - 11:Z5 (Z5) The geometry of extremal quasi-conformal mappings. Preliminary report Dr. T. S. Klotz, University of California, Los Angeles (578-15) 11:30 - 11:40 (Z6) On minimum element-covering from a finite class of weighted sets Mr. j. R. Edmonds, National Bureau of Standards, Washington, D. C. (578-4Z)

SATURDAY, 10:00 A.M. Session on Algebra, Empire Suite, Fourth Floor 10:00 - 10:10 (Z 7) Certain semigroups Professor Takayuki Tamura, University of California, Davis (578-3) 10~15 - 10:Z5 (Z8) On the extension of an ordered ring without identity Professor j. W. Andrushkiw, Seton Hall University (578-14) 10:30 - 10:40 (49) A characterization of Abelian 7( a.-groups in terms of their natural valuation Professor N. L. Alling, Purdue University (578-Z7) 10:45 - 10:55 (30) On isomorphism of groups Mr. George Glauberman, Polytechnic Institute of Brooklyn (578-30) (Introduced by G. Bachman) 11:00 - 11:10 (31) Spectral functions and prime ideals in vector lattices Professor D. G. johnson and Professor J. E. Kist*, The Pennsylvania State University (578-40) 11:15 - 11:Z5 (3Z) On complemented ideals and extremally disconnected spaces Professor D. G. johnson* and Professor j. E. Kist, The Pennsylvania State University (578-41) 11:30 - 11:40 (33) Three-dimensional subalgebras of the classical algebras Professor Murray Gerstenhaber, University of Pennsylvania, (578-43) 11:45 - 11:55 (34) On a class of quadratic algebras Professor R. D. Schafer, Massachusetts Institute of Technology (578-45)

91 SATURDAY, 2:00 P.M. Invited Address, Grand Ballroom, Second Floor Homotopy groups (One hour) Professor D. M. Kan, Massachusetts Institute of Technology

SATURDAY, 3:15P.M. Session on Analysis, Panel Room, Third Floor 3:15- 3:25 (35) Monotone (nonlinear) operators in Hilbert space Dr. G. j. Minty, University of Michigan (578-2) 3;30 - 3:40 (36) The balanced core of a set Professor J. Horvath, University of Maryland (578-44) 3:45 - 3:55 (37) A theorem concerning plane point sets with an application to function theory Professor G. T. Cargo, Syracuse University (578-48) 4:00 - 4:10 (38) On the geometry of Stieltjes transforms Professor D. S. Greenstein, Northwestern University (578-16) 4:15 - 4:25 (39) Prolongations, perturbations, and stability in dynamical systems. Prelimin­ ary report Dr. Peter Seibert* and Professor joseph Auslander, RIAS, Baltimore, Maryland (578-47) 4:30 - 4:40 (40) On the weighted polynomial approximation on a locally compact space Professor Leopolda Nachbin, Brandeis University and Institute for Pure and Applied Mathematics, Rio de janeiro, Brazil (578 -17) 4:45 - 4:55 (41) A linear transformation between function spaces Professor Ernest Michael, Institute for Advanced Study (578-34)

SATURDAY, 3:15P.M. Session on Applied Mathematics, Washington Room, Fourth Floor 3:15 - 3:25 (42) Nonlinear oscillations in a cold plasma Professor C. L. Dolph, University of Michigan (578-37) 3:30 - 3:40 (43) On bounds to eigenvalues of non-selfadjoint differential equations Dr. Peter Treuenfels, Brookhaven National Laboratory, Upton, New York (578-1) 3:45 - 3;55 (44) On eigenmatrices and partial difference equations Mr. B. S. Sackman, The MITRE Corporation, Bedford, Massachusetts (578-21) 4:00 - 4:10 (45) Pointwise bounds for the solution of parabolic differential equations. Part II Lieutenant F. j. Bellar, jr., U.S. Naval Postgraduate School, Beltsville, Maryland and University of Maryland (578-28) 4:15- 4:25 (46) A procedure for obtaining lower bounds to eigenvalues Mr. Norman Bazley, National Bureau of Standards, Washington, D. C. and Mr. David Fox,* Applied Physics Laboratory, johns Hopkins Univer­ sity (578-38)

92 4:30 - 4:40 (47) A solution of Chandrasekhar's integral equation Professor Charles Fox, McGill University (578-29)

SATURDAY, 3:15P.M.

Session on Probability and Statistics, Empire Suite, Fourth Floor 3:15- 3:25 (48) Transformation of probabilities Professor j. H. Blau, Antioch College (578-8) 3:30 - 3:40 (49) Testing hypotheses in human genetics. Preliminary report Dr. Edward Batschelet, The Catholic University of America (578 -4) (Introduced by Professor Eugene Lukacs)

These papers will be followed by late papers.

Everett Pitcher Associate Secretary Bethlehem, Pennsylvania

• • careers in mathematics

Vitro's increased activities in the field of Research & Development have created career oppor. tunities for men with these interests and qualifications: MATHEMATICAL STATIS­ TICIANS MS or PhD for conducting and consulting on analytical studies in a wide variety of applications, including information theory, weapons systems analysis, experimental design, data treatment. Should be familiar with some of the following- Monte Carlo procedures, Markov processes, decision theory, sample data systems, and have had 3-5 years industrial experience in implementing these techniques. Position is in the Information Analysis Group.. • OPERATIONS RESEARCH ANALYSTS MA or PhD in Mathematics, Statistics or Physics. Conduct and direct operations research studies, principally in the areas of weapons systems evaluation, ballistic missile defense, anti-submarine warfare and electronic counter­ measures. Should have experience in some of the following areas: applications of game theory, linear programming, Monte .Carlo techniques, queueing theory and model construction. • Our modern laboratory is located in a S1tburban area with easy access to the cultural and educational facilities of metropolitan New York and New Jersey. Liberal bene­ fits include a tuition refund plan and relocation allowances. • Send resume to Mr. S. Roberts. lhfrl LABORATORIES Division of Vitro Corporation of America 200 Pleasant Valley Way, West Orange, New Jersey • •

93 Five Hundred Seventy-Ninth Meeting University of Chicago Chicago, Illinois April 14-15, 1981

PROGRAM

The five hundred seventy-ninth Eckhart Hall will be available to members meeting of the American Mathematical of the Society and guests for all meals. Society will be held at the University of The following hotels agreed to ac­ Chicago, Chicago, lllinois on Friday and commodate those members of the Society Saturday, April 14-15, 1961. All sessions making reservations in advance: will be held in Eckhart and Kent Halls. In the University district Registration will be in the Common Room on the second floor of Eckhart begin­ Hotel Broadview ning at 9:00 A.M., Friday. 5400 South Hyde Park Boulevard By invitation of the Committee to $ 5.50- $ 6.50 (Single) Select Hour Speakers for Western Sec­ $ 7 .so-$ 9 .oo (Double) tional Meetings, Professor Israel N. Her­ stein of Cornell University and the Uni­ versity of Chicago, and Professor james Del Prado Hotel A. jenkins of Washington University will 5307 South Hyde Park Boulevard address the Society. Professor Her stein's $ s.oo- $1z.oo (Single) title is "Lie and jordan structures in $11.00-$16.00 (Double) simple, associative rings" and Professor jenkins' title is "The general coefficient theorem and certain applications." Both Shoreland Hotel lectures will be in Room 133, Eckhart 5454 South Shore Drive Hall. Professor jenkins will speak atZ:OO $ 6.00 (Single) P.M. on Friday and Professor Her stein at $10.00 (Double) 10:00 A.M. on Saturday. Sessions for the presentation of Hotel Windermere contributed papers will be held at 10:00 164Z East 56th Street A.M. and 3:15 P.M. on Friday and 11:15 $ 7 .so- $10.50 (Single) A.M., Saturday morning. If necessary $11.00- $14.00 (Double) there will be a special session Saturday for the presentation of papers which failed to meet the deadline. Further details will Hyde Park YMCA be available at the registration desk. 1400 East 33rd Street The Department of Mathematics of $ Z.50 (Single) the University of Chicago will entertain --- (Double) the Society at Tea in the Common Room of Eckhart Hall starting at 4:30 P.M. on Mail for all members of the Society Friday. should be addressed c/o The Department The facilities of Hutchinson Com­ of Mathematics, Eckhart Hall, University mons, a dining hall directly across from of Chicago, Chicago 37, Illinois

94 PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes, however, the papers are scheduled at 15 minute intervals. Thus the audience has an oppor­ tunity to circulate between the various sessions, and those who are interested in a particular paper are certain of the exact time at which it will be present­ ed. To achieve these objectives, the time limit will be strictly enforced.

FRIDAY, 10:00 A.M. Session on Geometry and Topology, Eckhart 133 10:00 - 10:10 (1) An instrument in hyperbolic geometry Dr. M. W. Al-Dhahir, NationalResearchCounciland University of Tor­ onto (579-6) 10:15- 10:25 (2) On Ostrom's separation relation in nonordered planes Professor Seymour Schuster, Carleton College (579-32) 10:30 - 10:40 (3) Almost cartesian product neighborhoods of 2-spheres Professor R. H. Bing, University of Wisconsin (579-23) 10:45 - 10:55 (4) A simple method for subdividing smooth manifolds ProfessorS. S. Cairns, University of Illinois (579-7) 11:00 - 11:10 (5) Regular neighborhoods. Preliminary report Professor M. L. Curtis, Florl.da State University (579-27) 11:15 - 11:25 (6) Primitive skew curves as factors of certain compact product spaces in n­ manifolds Dr. Beauregard Stubblefield, International Electric Corporation, Para­ mus, New jersey (579-20) 11:30 - 11:40 (7) Linearly orderable subspaces of the real line. Preliminary report Mr. I. L. Lynn, Wayne State University and Purdue University (579-18) 11:45 - 11:55 (8) Concentric surfaces in E3 Mr. D. S. Gillman, University of Wisconsin (579-17)

FRIDAY, 10:00 A.M. Session on Analysis, Kent 107 10:00 - 10:10 (9) Bicubic "spline" interpolation Mr. Carl de Boor, General Motors Research Laboratories, Warren, Michigan (579-24) (Introduced by Dr. H. L. Garabedian) 10:15 - 10:25 (10) Asymptotic equivalence of differential equations Professor Fred Brauer, University of Wisconsin (579-25)

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

95 10:30 - 10:40 (11) Perturbation of a one-parameter family of autonomous differential equations Professor R. G. Bilyeu, North Texas State College (579-1Z) 10:45 - 10:55 (1Z) A construction for solutions of an nth order linear differential equation con­ taining a parameter in the neighborhood of a turning point Mr. G. A. Stengle, Lehigh University (579-9) 11:00 - 11:10 (13) Approximate proper vectors Professor S. K. Berberian, State University of Iowa (579-4) 11:15- ll:Z5 (14) On roots of normal operators Dr. Svetozar Kurepa, University of Maryland (579-3) 11:30 - 11:40 (15) On the spectra of some translation invariant spaces Mr. J. W. Moeller, Bell Telephone Laboratories, Whippany, New Jersey (579-19) 11:45 - 11:55 (16) A Lebesgue decomposition for complete normed abelian groups Dr. R. B. Darst, Massachusetts Institute of Technology (579-33)

FRIDAY, Z:OO P.M. Invited Address, Eckhart 133 The general coefficient theorem and certain applications (One hour) Professor James A. Jenkins, Washington University

FRiDAY, 3:15P.M. Session on Algebra and Logic, Eckhart 133 3:15- 3:Z5 (1 7) A minimal degree less than O'. Mr. G. E. Sacks, Cornell University (579-5) 3:30 - 3:40 (18) On exponentially closed fields Professor N. L. Alling, Purdue University (579-14) 3:45 - 3:55 ( 19) Ideals generated by the subdeterminants of a matrix Mr. J. A. Eagon, University of Chicago (579-1) 4:00 - 4:10 (ZO) Two p-group counterexamples Mr. J. L. Alperin, University of Chicago (579-15) 4:15 - 4:Z5 (Z 1) Derivations and generations of finite extensions Professor Carl Faith, Pennsylvania State University and Institute for Advanced Study (579-16) 4:30 - 4:40 (ZZ) On the expression of polynomials with real coefficients as symmetric deter­ minants (579-ZZ) Professor J. L. Howland, University of Ottawa (579-ZZ)

FRIDAY, 3:15P.M. General Session, Kent 107 3:15 - 3:Z5 (Z3) The distribution of zeroes of successive derivatives of bounded analytic func­ tions Professor Pasquale Porcelli, Louisiana State University (579-10)

96 3:30 - 3:40 (24) On generalising the Lebesgue decomposition to non Cf -finite measures (579-31) Professor C. A. Rogers, London University and Professor S. j. Taylor,* Cornell University (579-31) 3:45 - 3:55 (25) Further developments of the method of associating partial differential equa­ tions with ordinary ones Professor M. z. v. Krzywoblocki, Michigan State University (579-8) 4:00 - 4:10 (26) Physical systems Sk of curves in a central field of force Mr. A. z. Czarnecki, De Paul University (579-2) 4:15- 4:25 (27) A theorem on continued fractions and the fundamental inequalities Professor D. F. Dawson, North Texas State College (579-11) 4:30 - 4:40 (28) A theorem on rank order statistics Dr. J. S. White, General Motors Research Laboratories, Warren, Michi­ gan (579-13)

SATURDAY, 10:00 A.M. Invited Address, Eckhart 133 Lie and jordan structures in simple, associative rings (One hour) Professor Israel N. Herstein, The University of Chicago and Cornell University

SATURDAY, 11:15 A.M. Session on Analysis, Eckhart 133 11:15- 11:25 (29) An integral representation for the continuous linear functionals on a space of quasi-continuous functions Professor J. R. Webb, Louisiana State University (579-21) 11:30 - 11:40 (30) On equivalent measures Professor Pasquale Porcelli and Dr. F. F. Connor,* Louisiana State University (579-30) 11:45 - 11:55 (31) On the extreme eigenvalues of Toeplitz .matrices ProfessorS. V. Parter, Cornell University (579-29) 12:00- 12:10 (32) Modular forms of positive dimension and Poincare series Professor M. I. Knopp, University of Wisconsin (579-28) 12:15 - 12:25 (33) Metric results for noncontinuable analytic functions Professor F. W. Carroll,* University of Wisconsin-Milwaukee and Pro­ fessor J. H. B. Kemperman, University of Wisconsin (579-26)

SATURDAY, 11:15 A.M. Special Session for Late Papers, Kent 107 j. W. T. Youngs Associate Secretary Bloomington, Indiana

97 Five Hundred Eightieth Meeting Stanford University Stanford, California April 22, 1981

PROGRAM

The five hundred eightieth meeting A tea for persons attending the of the American Mathematical Society will meeting will be given at the Bowman Alum­ be held on Saturday, April ZZ, 1961 at ni House following the afternoon sessions. Stanford University, Stanford, California, The Lounge in Building 70 (Room 7ZA) will By invitation of the Committee to be open throughout the day for informal Select Hour Speakers for Far Western gatherings. Luncheon on Saturday will be Sectional Meetings, there will be an ad­ available at the Stanford Union. dress at 11:00 A.M. in the Little Theater Stanford University is about thirty by Professor Hans Zassenhaus of the miles south of San Francisco, adjacent to the town of Palo Alto. Southern Pacific University of Notre D.ame on "Modern Railway and the Greyhound Bus Line serve developments in number geometry". P a1o Alto. Limousine service from the San Sessions for contributed papers Francisco International Airport to Palo will be held at 9:30 A.M. in the Little Alto is available for about $l,50, The dis­ Theater and at Z:OO P.M. in Rooms E.lH, tance from Palo Alto to the central part 61G and 61A of the Mathematics Building of the Stanford campus is approximately (Building 60). Abstracts of the papers to one mile. Taxi cabs can be hired for this be presented at these sessions appear on part of the trip. pages 166- 173 of these NOTICES. There There are numerous hotels and are cross references to the abstracts in motels within easy driving distance of the the program. For example, the title of campus. A detailed list of them can be ob­ paper (1) in the program is followed by tained from the Palo Alto Chamber of (580-1) indicating that the abstract can Commerce. The hotels in Palo Alto closest be found under the designation 580.. 1 to the campus are the Cardinal Hotel, Z35 among the published abstracts. Hamilton Street, and the President Hotel, During the morning a Registration 488 University Avenue. The prices atthese Desk will be located in the Little Theater, hotels range from $5,00 to $8,50 at the which is beside the Memorial Auditorium. Cardinal, and from $6,00 to $13,00 at the Information concerning late papers can be President. Reservations should be made obtained at the Registration Desk. directly with the chosen hotel or motel.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten min­ utes, The contributed papers are scheduled at 15 minute intervals. To maintain this schedule, the time limit will be strictly enforced,

SATURDAY, 9:30A.M. General Session, Little Theater 9:30 - 9:40 (1) Eigenvalues of sums of Hermitian matrices Professor Alfred Horn, University of California, Los Angeles (580-ZO)

98 9:45 - 9:55 (Z) Roots and vectors of Mahler matrices Dr. J. L. Brenner, Stanford Research Institute, Menlo Park, California (580 -1) 10:00 - 10:10 (3) Two counting problems for bicolored graphs Professor Frank Harary*, University of Michigan, and Dr. G. C. E. Prins, Amsterdam, The Netherlands (580-11) 10:15 - 10:Z5 (4) The relationship between the modular rank and period of a second order recurrence Professor Morgan Ward, California Institute of Technology (580-10) 10:35 - 10:45 (5) Pairs and triples of consecutive power residues Professor D. H. Lehmer* and Mrs. Emma Lehmer, University of Cali­ fornia, Berkeley, and Professor W. H. Mills, Yale University (580-Z6)

SATURDAY, 11:00 A.M. Invited Address, Little Theater Modern developments in number geometry Professor Hans Zassenhaus, University of Notre Dame

SATURDAY, Z:OO P.M. Session on Topology, Analysis and Probability, Room 61H Z:OO - Z:10 (6) Fixed point theorems for pseudo monotone mappings Professor L. E. Ward, Jr., University of Oregon (580-9) Z:15 - Z:Z5 (7) Infinitesimally induced representations Professor R. J. Blattner, University of California, LosAngeles (580-17) Z:30 - Z:40 (8) Finite dimensionality of the Schrodinger operator bottom Professor F. H. Brownell, University of Washington {580-6) Z:45 - Z:55 (9) The Hausdorff-Besicovitch dimension of the zeros of a symmetric stable process Professor R. M. Blumenthal and Professor R. K. Getoor*, University of Washington (580-7) 3:00 - 3:10 {10) Generalized arcsin laws Professor Malcolm Goldman, Reed College (580-19) 3:15 - 3:Z5 (11) A real inversion formula for a class of convolution transforms Professor C. J. Standish, North Carolina State College (580-8)

SATURDAY, Z:OO P.M. Session on Analysis and Applied Mathematics, Room 61G Z:OO - Z:10 (lZ) Direct proof that every dominant matrix is paramount Professor Paul Slepian, University of Arizona (580-Z) Z:l5- Z:Z5 (13) The behavior of solutions -of ordinary, self-adjoint differential equations of arbitrary even order Mr. R. W. Hunt, University of Utah (580-lZ)

99 Z:30 - Z:40 (14) Transcendental equations related to differential-difference equations Professor K. L. Cooke, Pomona College (580-18) Z:45 - Z:55 (15) On difference methods for the solution of a Tricomi problem Mr. Hajimi Ogawa, University of California, Riverside (580-3) 3:00 - 3:10 (16) Variable spacing finite difference analogues. Preliminary report Professor W. R. Mann*, University of North Carolina and University of California, Berkeley, and Mr. j. M. Gwynn, Jr., Georgia Institute of Technology (580-13) 3:15- 3:Z5 (17) A multi-point generalization of Newton's divided difference formula Dr. H. E. Salzer, Convair-Astronautics, San Diego, California (580-Z5) 3:30 - 3:40 ( 18) Fifth order methods for numerical solution of ordinary differential equations Professor R. R. Reynolds and Professor W. E. Milne*, Oregon State College (580-Z4)

SATURDAY, Z:OO P.M.

Session on Algebra and Theory of Numbers, Room 61A Z:OO - Z:10 ( 19) A note on k-commutative matrices Professor D. W. Robinson, Brigham Young University (580-14) Z:15- Z:Z5 (ZO) Slender groups Professor R. j. Nunke, University of Washington (580-Z1) Z:30 - Z:40 (Z1) Characterization of singular semigroups Professor Takayuki Tamura, University of California, Davis (580-15) Z:45 - Z:55 (ZZ) On the direct product of certain groupoids Mr. L. L. Berryman, Mr. J. F. Latimer, Mr. R. B. Merkel* and Pro­ fessor Takayuki Tamura, University of California, Davis (580-16) 3:00 - 3:10 (Z3) Planar half loops Professor T. G. Ostrom, Washington State University (580-Z3) 3:15 - 3:Z5 (Z4) Vector partitions and permutation vectors Mr. M.S. Cheema* and Professor T. S. Motzkin, University of Califor­ nia, Los Angeles (580-5) 3:30 - 3:40 (Z5) Verification of a conjecture of Th. Skolem Dr. E. S. 0' Keefe, Boeing Airplane Company, Seattle, Washington ( 580-ZZ) R. S. Pierce Associate Secretary Seattle, Washington

100 PRELIMINARY ANNOUNCEMENT OF MEETINGS FIVE HUNDRED EIGHTY-FIRST MEETING University of Washington Seattle, Washington June 13-16, 1961

The five hundred eighty-first meet­ pose will be found on page 191 of these ing of the American Mathematical Society NOTICES. Reservations should be mailed will be held on Tuesday, Wednesday, so that they will be received at least three Thursday and Friday, June 13-16, 1961 at weeks prior to the meeting. the University of Washington in Seattle, There are two hotels and one motel Washington. This meeting will be in con­ within walking distance of the campus: Ho­ junction with meetings of the Mathematical tel Edmund Meany, E. 45th and Brooklyn Association of America on June 17, the Avenue, $8.00-$9.00 (single room), $9.50- Institute of Mathematical Statistics on $12,50 (double room); Wilsonian Hotel, June 14-17, the Institute for Management E. 47th and University Way, $5.00-$6.00 Sciences, the Biometric Society, the Amer­ (single room), $7.00-$9.00 {double room); ican Statistical Association and the Society Coach House, 4701 24th North East, $10.00 for Industrial and Applied Mathematics. (one bedroom suite), $14.00 (two bedroom By invitation of the Committee to suite). Hotel reservations should be sent Select Hour Speakers for Far Western directly to the chosen hotel, mentioning Sectional Meetings, and with the financial the mathematics meetings at theUnivers­ support of the Air Force Office of Scienti­ sity of Washington. fic Research, a Symposium on Convexity Registration and all sessions of the will be held on Tuesday, Wednesday and meeting will be held in Bagley Hall. The Thursday, June 13-15. The Program Com­ dormitory accommodations will be in the mittee for the Symposium consists of Men's Residence Halls, located at 1201 Professor V. L. Klee, Chairman, Pro­ Campus Parkway. The Residence Hall of­ fessor Branko Grunbaum, Professor fice closes each night at 11:00 P.M. Break­ David Gale, and Dr. Merle M. Andrew. fast will be available in the Residence Hall By invitation of the same Commit­ under the conditions noted above, and in tee, there will be an address on Friday by the Cafeteria of the Husky Union Building Professor T. M. Apostol of the California (HUB) on campus, beginning at 8:15 A. M. Institute of Technology on Some lattice Luncheon will also be available at the HUB P.Oint problems in the theory of numbers. Tuesday through Saturday and dinner will There will also be sessions for contributed be served there Monday through Friday. papers on Friday, including a special ses­ Seattle is served by United, West­ sion for papers on convexity. ern, West Coast and Northwest Airlines, Dormitory accommodations will be the Northern Pacific, Union Pacific, Great available for the nights of June 12 through Northern and Milwaukee Railroads, and June 17 at the rate of $2.50 per person per the Greyhound and Continental Trailways night, double, and $3.50 per night, single. Bus Lines. To get to the campus from the There will be no special rates for children. Seattle-Tacoma airport, take the Airporter If demand justifies it, breakfasts will be Bus .to the Olympic Hotel in downtown available in the dormitory at $1.00 each. Seattle. Then take a Seattle Transit Bus, To obtain these breakfasts, it will be or a taxi (cost-- about $2.50) to the Uni­ necessary to reserve them in advance and versity district. Members who drive to to take them during the entire stay in the the meeting can park on campus for 25 dormitory. A banquet will be held on Friday cents per day. Parking at the Men's Resi­ evening, costing approximately $3.00 per dence Halls is also available at the rate of person. Dormitory banquet reservations 25 cents per day. may be obtained by writing to Professor R. S. Pierce S. P. Avann, Department of Mathematics, Associate Secretary University of Washington, Seattle 5, Wash­ ington. A reservation blank for this pur- Seattle, Washington

101 ACTIVITIES OF OTHER ASSOCIATIOHS

APRIL 1961 MEETING OF SIAM

A two-day meeting of the Society for Industrial and Applied Mathematics will be held at the U. S. Naval Postgraduate School in Monterey, California on Thursday, April ZO, and Friday, April Zl, 1961. It is timed to immediately precede the meeting of the American Mathematical Society at Stanford, California on Saturday, April ZZ, 1961. The registration desk in the lobby of King Hall will be open from 8:00 A. M. till 4:00P.M. on Thursday and from 9:00A.M. till 1:00 P.M. on Friday. The registration fee is $Z.OO for each member of the Society, and $3.00 for a nonmember. Information concerning hotel or motel accommodations can be obtained by writing: Monterey Peninsula Convention Bureau, P. 0. Box 1571, Monterey, California. Monterey is situated approximately lZO miles south of San Francisco on California Highway No. 1., and 3ZO miles north of Los Angeles. Monterey Airport is serviced by United Airlines and Pacific Airlines. Southern Pacific trains depart from San Francisco for Monterey once daily. Monterey can also be reached via Greyhound buses. The U. S. Naval Postgraduate School is bounded on the south by Fremont Street (California Highway No. 1), on the west by Sloat Avenue, and on the north by Del Monte A venue. King Hall and Spanagel Hall are within short walking distance of the entrance at 9th Street and Sloat Avenue. Parking will be available a short distance beyond the en­ trance. Public transportation from the center of Monterey (Bus Route No. 9) is available one-half block from the Casa Munras Hotel, leaving every 30 minutes on the quarter hour, and stopping at the 9th Street and Sloat Avenue Gate. The running time is approxi­ mately 5 minutes.

PROGRAM OF THE SESSIONS THURSDAY, 9:30 A.M. FRIDAY, 9:30A.M. King Hall King Hall Address of welcome and reply Symposium on planetary probes

Earth conductivity measurements 1. Planetary probe trajectories through a transiently excited dipole Dr. A. B. Mickelwait, Space Tech­ Dr. T. P. Higgins, Boeing Scienti­ nology Laboratories fic Research Laboratory Steady states of linear integral equa­ Space navigation tions z. Dr. R. H. Battin, Instrumentation Professor W. B. Fulks, Oregon State College Laboratory, Massachusetts Insti­ tute of Technology

THURSDAY, Z:OO P.M. 3. Planetary ephemerides King Hall Dr. R. L. Duncombe, Nautical Al­ Address manac Office Solid state devices Dr. j. H. Rowen, Bell Telephone FRIDAY, Z:OO P.M. Laboratories King Hall THURSDAY, 3:30 P.M. Address Spanegel Hall Equivalence of compound matrices Professor Bernard Friedman, Uni­ Sessions for contributed papers versity of California, Berkeley

lOZ News and comment from the be extended for five years. It is recom­ mended that the requirement for a student CONFERENCE BOARD OF THE disclaimer affidavit be eliminated, and MATHEMATICAL SCIENCES that the student loan program be expanded and strengthened in various ways. It is G. Baley Price Washiagtoa. D. C. recommended that the provisions of Title Ill, which provides financial assistance for strengthening science, mathematics, and modern foreign language instruction, be Proposals for Strengthening Education continued, and expanded to include English The Federal Government proposes to and History. (The inclusion of English and strengthen its support of education, and History suggests that more and more the many detailed recommendations concern­ Federal Government will support educa .. ing the form of this increased support have tion for its own sake rather than as a de­ been provided in committee reports and fense measure,) The following detailed re­ other publications (see the references at commendations for the revision and ex­ the end of this article). This article con­ tension of Title IV of NDEA, which pro­ tains a summary of the principal proposals; vides for graduate fellowships, are of many of them concern mathematics and special interest to the mathematicians. higher education. (1) "That this title be continued and Report of the Task Force Committee the amount of money appropriated on Education (the Hovde Committee), This be substantially increased over report (Z) contains three legi.slative pro­ that of previous years because of posals and four recommendations for the acute shortage of college teach­ Presidential administrative action. ers. First, the Committee recommends {Z) "That provision be made for a sub­ that the President support legislation to stitute to fill the term of afellow­ provide, for all states, $30 per annum per ship vacated by a student who does pupil, and to provide additional payments not complete the prescribed tenure for certain low income states and for large of an award. cities. The funds should be sent to the {3) "That the methodofpaymentbased states for transmission to local Boards of on actual calculated costs be re­ Education; these "Boards of Education placed by a system of fixed pay­ should be authorized to use the funds for ment of $Z,500 per year for each construction, salaries, or other purposes fellow. related to the improvement of education." (4) "That the provision under this title Second, the Committee recommends be made more flexible to permit that the President urge Congress to enact fellowships ranging from one to legislation providing for a combined pro­ three years for work toward the gram of loans and grants for the construc­ doctorate degree. tion of academic facilities. The Committee {5) "That a program equivalent in recommends that at least $500 million be size to the present program bees­ provided for the first year, of which $350 tablished for additional fellowships million (70 per cent) should be for match­ which do not require new teaching ing grants and $150 million {30 per cent) personnel or expanded facilities so for loans on the same basis as the College that unused capacities in colleges Housing Loan Program. The Committee and universities can be more fully recommends also that the College Housing utilized. Loan Program, which has been outstand­ (6) "That the fellowship program be ingly successful in its ten years of opera­ extended to include teachers now tion, be continued. in service in order to upgrade the Third, the Committee recommends present teaching staff," that the National Defense Education Act be strengthened, and that authority for There are other important recom­ NDEA programs, with certain exceptions, mendations for the extension of Title IV,

103 and also numerous other recommenda­ direct ("overhead") costs on spon­ tions for the revision of NDEA, but they sored research. are of less interest to the mathematicians, President Kennedy's message on edu­ The Hovde Committee recommends cation to Congress had been expected on four actions by President Kennedy as February 14; it was delayed to permit ad­ follows. ditional time for planning, and it is now I. "The Task Force Committee re­ expected on February 20, As this is writ­ commends that the President take ten (February 18), Kennedy is expected to immediate action to establish a recommend a program similar to, but President's Advisory Committee smaller than, that recommended by the on Education." (This recommen­ Hovde Committee. The President is ex­ dation is an effort to give educa­ pected to recommend a program that tion the same recognition and amounts to about $1 billion per year for prestige accorded science and five years, with $600 million of the total economics by the existence of for elementary and secondary schools, and Presidential advisory committees $400 million for colleges and universities, in these fields.) Report of the Consultants to the Secre­ II. "The Task Force Committee re­ tary of Health, Education, and Welfare, The commends that the President following paragraphs are quoted from the specifically request all Federal foreword of this report (3), which was pre­ agencies, including the Federal pared by a Panel of twenty-one Consul­ Council for Science and Techno­ tants during a period of several months' logy, the National Science Founda• study and released on january lZ, 1961, tion, the Atomic Energy Commis­ sion, the Department of Defense, "The National Defense Education Act, and the Department of Health, within its present scope, has been ef­ Education, and Welfare, to take all fective in meeting certain needs. possible steps within existing There remain, nonetheless, some fur­ statutory and legislative authori­ ther needs which require extension zations to support and implement and revision of the Act and fall in two the recommendations of the Pres­ areas: fulfillment of the Nation's man­ ident's Science Advisory Commit­ power requirements and maintaining tee is sued under date of 15 Novem­ the quality of instruction. The needs ber 1960 in a report entitled in these areas are so urgent that cer­ 'Scientific Progress, the Univer­ tain extensions and revisions of the sities, and the Federal Govern­ Act should be made immediately. ment.'" (The recommendations of "As a Nation, we are losing too much the President's Science Advisory of the potential of our able youth be­ Committee ( 1) are given in full cause of underachievement, drop-outs, below.) uncorrected deficiencies, and finan­ III. The Task Force Committee re­ cial need; because of inadequate teach­ commends that the President take ing staffs and facilities, weaknesses action to rescind a recent ruling in curriculum, and the paucity of re­ of the Internal Revenue Service search in many areas affecting the which discourages gifts by indi­ learning process. viduals to both public and private colleges and universities. "Basic to our consideration of the Na­ IV. The Task Force Committee re­ tional Defense Education Act is our commends that the President take belief that the Federal Government action to arrange for the payment has an obligation to help identify and of '-'full costs" of sponsoredre­ bring to fruition the full potential of search at colleges and universi­ every youth. Further, it is our belief ties. The purpose of this recom­ that failure to do this will imperil not mendation is to remove limita• only the individual, but the Nation and tions on the payment of full in- the free world.

104 "Our recommendations therefore are among the Consultants on the question of designed to strengthen guidance and how the Federal scholarship program counseling services, studentfinancial should be administered. assistance, staff, facilities, and cur­ Ten-Year Objectives in Education. riculum, as well as to contribute fur­ Higher Education Staffing and Physical ther to an appropriate balance in fields Facilities. This report (4) was prepared of study. One of our major recommen­ by the Department of Health, Education, dations is to increase the supply of and Welfare, and it was released on Janu­ qualified teachers for all subjects at ary 19, 1961. Thereportcontainsacareful all levels -- elementary, secondary, analysis of the requirements for staffing and higher education. But we must and for physical facilities for higher edu­ emphasize that these efforts to deal cation during the period 1960-1961 through with specific issues will not adequate­ 1969-1970. The basis for the estimation of ly strengthen our total system of edu­ needs is contained in the following para­ cation unless our schools and colleges graph, quoted from the Introduction and are provided with increased financial Summary. support from the Federal Government in addition to what is here proposed. "Projections of population growth and college entrance rates indicate that "It is not within the scope of our as­ approximately 6,006,600 students will signment to delineate the exact me­ attend higher education institutions in thods by which this broader task shall 1970, and that about 4,504,500 of this be accomplished. Nevertheless, the number will attend full time. This es­ time has now come, in our opinion, for timated increase of to Z,600,000 stu­ the Federal Government to address dents, of whom 1,95Z, 777 are expected itself to this problem." to attend full time, forms the basis for The Consultants recommend that the the projections, in this paper, of bQth provisions of Title III be expanded to in­ the requirements for staffing the Na­ clude English along with science, mathe­ tion's colleges and the cost of provid­ matics, and modern foreign languages. ing them with needed physical facili­ The recommendations for the revision ties." of Title IV of NDEA, which deals with The following two paragrahs describe graduate fellowships, are similar to those the nature and extent of the problem of of the Hovde Committee. staffing. The Consultants recommend that sev­ eral additional titles be established in the "During the next 10 years about 336, revised National Defense Education Act; 000 new, trained professional staff one of the most important deals with a members must be recruited by the Na­ Federal scholarship program. The Con­ tion's colleges and universities (at the sultants recommend "that a Federal schol­ master's and doctor's degree level) to arship program be established to assist replace those lost by death, retire­ additional numbers of able but needy youth ment and resignation, and to accom­ to continue their education beyond the high modate increases in enrollment. The school. The program should provide an Nation's graduate schools must train initial amount of $25 million, increasing in not only these persons, but also the the next four years to a total of $100 mil­ far larger numbers of similarly lion, to establish and continue support for trained people required for business, at least ZS,OOO four-year undergraduate industry, government and other areas. scholarship awards each year." Scholar­ They must produce persons at the ship awards up to $1,000, based on merit level of the doctorate at a greatly in­ and need, would be provided, and there creased rate in order to avert serious would be a payment of $500 to the institu­ deterioration in the level of academic tion at which the person receiving the training of the staff of higher educa­ award is enrolled. It appears from there­ tion, as it grows in size to keep pace port that there was strong disagreement with mounting enrollments. The per-

105 cent of doctor's degree holders among 1961-1969, the gap described above entering staff members in institutions will be cut in half, approximately, of higher education declined from 31.4 amounting to $1,5 billion in 1965 and percent in 1953-1954 to Z3,8 percent $Z,5 billion in 1970. In order to meet in 1958-1959. To re-establish this this deficit, or at least reduce it, two proportion at 30 percent for the period things seem to be desirable: ( 1) great­ 1960-1970, it is estimated that the er support from traditional sources of Nation's graduate schools must award financing, and (Z) new sources of re­ an average of 16,800 doctor's degrees venue. annually during the decade, compared with the current rate of about 9,600, "Although traditional sources of re­ and a projected ten-year average of venue applicable to construction of 13,300 indicated by present trends, physical facilities may yield more Even the goal of 16,800 new doctor's than now appears likely, theywillcer­ degree holders per year will be in­ tainly be inadequate to meet increas­ sufficient unless improved balance ing needs in full. Accordingly, new between supply and need by fields is sources of revenue must be explored, achieved, and unless at least 60 per­ along with efforts to increase revenue cent of those receiving the doctor'e from the usual sources. While it is not degree accept positions in higher edu­ the purpose of this paper to develop cation. specific programs of action or to re­ commend specific allocations of re­ "The problem of recruitment of staff sponsibility among the respective for higher education will become in­ sources of support, the facts we shall creasingly serious, at both master•·s present point strongly toward Federal and doctor's degree levels, unless assistance as a necessity. The Fed­ salaries in colleges and universities eral Government has a tremendous are increased during the next decade stake in the Nation's higher education at an unprecedented rate. The in­ enterprise." creases in salary of professional staff in higher education during the Recommendations of the American last five years fall far short of re­ Council on Education. The report (6) en­ storing the economic ground lost by titled "A Proposed Program of Federal the profession since 1940," Action to Strengthen Higher Education" was approved by the Committee on Rela­ The following two paragraphs deal tionships of Higher Education to the Fed­ with the problem of providing the physical eral Government on January Z3, 1961 and facilities required by higher education. by the Executive Committe.e on January "In the past, the physical facilities Z7. The report seems to have been released costs have accounted for approxi­ to the public about February 10, 1961. The mately ZO percent of the total annual principal recommendations are the follow­ expenditures for higher education. ing three. Current annual expenditures for high­ I. "The Federal Government can and er education facilities approximate should provide greater financial as­ one billion dollars. Allowing for only sistance to approved institutions of normal growth in the current pattern higher learning for expansion and im­ of expenditures, a cumulative gap of provement of facilities." $Z.9 billion between needs and expen­ ditures is estimated for 1965, and a gap of $5.Z billion for 1970, If it is The report recommends that "the Col­ assumed that funds from the Housing lege Housing Loan Program, which has and Home Finance Agency under the proved extraordinarily effective during the provisions of the College Housing past ten years, not only should be continued, Loan Program (or the equivalent but also expanded on a basis thatwillper­ from some other Federal source) will mit institutions to plan further ahead." The be available in the amount of approxi­ report also recommends a new program as mately $Z50 million annually during follows.

106 "A new program of federally adminis­ programs of instruction now offered tered assistance to institutions of by graduate schools. Consequently, higher learning, for construction of a new fellowship program for these academic facilities of the kinds they candidates should be established require to meet their needs in im­ under the NDEA, in which there is provement of quality as well as ex­ no requirement that an institution pansion of enrollment, should be es­ have a 'new or expanded program' tablished, offering each institution the of graduate instruction. following options on each separate 4. "To improvefacultycompetence, project: (a) a grant to defray up to 50 another program should be estab­ percent of the cost of construction, or lished under the NDEA which would (b) a low-interest forty-year loan to provide one-year fellowships for finance such construction, the interest college teachers who are within one rate to be determined under the same year of completing the requirements formula approved for the College for the doctorate. An experimental Housing Loan Program. program of 500 such fellowships a "A long-range program is recom­ year would be a significant begin­ mended, with initial authorization for ning. four years and with funds supplied at 5. "There should be a substantial the average rate of $1 billion a year. increase in funds appropriated for exchanges A survey of member institutions by of faculty members with the American Council on Education foreign countries and for grants to indicates that of this annual amount actual and prospective college approximately 70 percent should be teachers to study abroad. Such ex­ changes appropriated for matching grants and help American institutions 30 percent authorized for loans." to solve one of their major academ­ ic problems -- broadening their in­ The second major recommendation of struction to include the cultures of the American Council on Education is the the entire world." following. The third major recommendation of II. "The Federal Government can and the American Council on Education is the should provide greater assistance in following. increasing the supply and improving the quality of college teachers." III. "The Federal Government can and 1. "Such programs as thegraduate, should provide greater assistance in postdoctoral, and faculty fellowship removing financial barriers to higher programs of the National Science education for qualified students." Foundation should be expanded as One of the important parts of this re- rapidly as qualified candidates can commendation calls for the establishment be discovered. of a new program of four-year undergradu­ Z. "The graduate fellowship pro­ ate scholarships, with the objective of gram under the National Defense seeking out and assisting students of high Education Act should be extended academic promise in greatest financial for at least four more years, with need. The following detailed recommenda­ immediate provision for awarding tions are made concerning this scholar ship fellow ships to alternates if the orig­ program. nal recipients withdraw. Each insti­ "Stipends should tution should receive a grant of * be variable, ac­ cording to need, $3,000 a year for each graduate stu­ with a maximum of $1,000 a year. dent enrolled under this program. 3. "According to reliable estimates * "The scholarships should be award­ by those who have administered the ed by the institutions themselves, present NDEA fellowship program, using funds allocated in approximately at least 1, 500 more doctoral candi­ the same manner as Federal student dates couid be accommodated in loan funds.

107 * "There should be an appropriation of scientific research to national security of $25 million for scholarships the and welfare, all parts of the national com­ first year, with the annual total nsmg munity should assume a greater responsi­ to $100 million in the fourth year. bility for supporting, strengthening, and These sums would provide a minimum expanding basic research and graduate of Z5,000 new scholarships each year. education. * "Each scholarship should be ac­ (Z) In science the excellent is not just companied by a grant to the institu­ better than the ordinary; it is almost all tion which the scholarship holder at­ that matters. It is therefore fundamental tends. A reasonable amount to defray that this country should energetically sus­ part of the educational costs beyond tain and strongly reinforce first-rate work tuition and fees would be half the where it now exists. amount of the scholarship, Additional (3) It is of equal importance to in­ appropriations will be needed for this crease support for rising centers of sci­ purpose and for the expenses of ad­ ence. Over the next fifteen years the ministration." United States should seek to double the number of universities doing generally ex­ Recommendations of the President's cellent work in basic research andgradu­ Science Advisory Committee, Part Three, ate education. pages ZS-31, of the report (1) of the (4) It should be a general basis of President's Science Advisory Committee policy and action that basic research and contains the recommendations. These re­ the education of scientists go best together; commendations are so important that the that they are inseparable functions ofuni­ Hovde Committee recommended that Pres­ versities; that in graduate education the ident Kennedy request all Federal agencies training of scientists involves research; to take all possible steps to support and and that the strength of scientific research implement them (see the first section of grows out of research training in institu­ this article above). For this reason, the tions of higher education. recommendations in this report are repro­ (5) To attract more talented young duced here in full. people to science as a career, both under­ graduate colleges and the Federal Govern­ RECOMMENDATIONS ment should give urgent attention to the The following recommendations grow quality of collegiate instruction in the out of the preceding parts of this report; sciences. Here again research and teach­ they indicate the lines of development ing need to be connected wherever possible, which we think urgent in the immediate so that both teachers and students may future. But for the reader who may turn have the opportunity for learning by scien­ only to the recommendations, we wish to tific inquiry. Better salaries, increased emphasize that the record of this country time for research, rising support for in basic research and graduate education facilities and equipment, and moderniza­ is not one of failure. American science is tion of curriculum -- all are needed in second to none in the world, and the Fed­ undergraduate colleges. eral Government, on balance, has played (6) Both the universities and the Fed­ a highly constructive role in supporting it. eral Government should be energetic and Most of our specific recommendations are imaginative in seeking effective ways of based on our respect for the best existing identifying and supporting new fields of practices of particular agencies of govern­ basic research and in supporting the train­ ment or particular universities. These re­ ing of scientists in such fields. Many re­ commendations are not intended as criti­ search opportunities are emerging in new cisms of what has been accomplished, but fields that are essentially interdisciplin­ as proposals for still greater accomplish­ ary; these require special efforts by uni­ ment in the future. versities to encourage new programs. The Federal Government should stimulate and General Recommendations: support such programs where there is clear evidence that the institutions are (1) In view of the growing importance prepared to establish programs of gradu-

108 ate education fully connected with the new cation. There is urgent need for strength­ research. ening the quantity and quality of represen­ (7) State, local and private resources tation of universities before both the Con­ are needed on a large and growing scale to gress and the Executive Branch. meet the needs and opportunities in basic (8) Universities should accept pri­ research and graduate education. While mary responsibility for ensuring that their this report emphasizes the responsibili­ growing partner ship with the government ties of universities and the Federal Gov­ reinforces their freedom and excellence, ernment, the very fact of growing Federal activity makes it urgent that state, local Recommendations for the Federal Govern- and private efforts also be increased, for, ~ especially as concerns private efforts, (1) Federal support for basic research there will always be much that the govern­ and graduate education in the sciences ment either does not or cannot do. should be continued and flexibly increased so as to support excellence where it al­ Recommendations for Universities: ready exists and to encourage new centers (1) Universities must continue and ex­ of outstanding work, pand their efforts to pay proper salaries, (Z) The Federal Government should provide adequate time and opportunity for continue and enlarge the practice, now research, and maintain an atmosphere of followed with great success ina few agen­ free learning and investigation. cies, of providing research support over (Z) Universities should recognize that long terms and for broad objectives, graduate education in the sciences needs {3) Once support is granted, the Fed­ constant modernization. eral Government should not seek to super­ (3) University programs in graduate vise technical operations directly. Com­ education should ordinarily include ex­ plete scientific responsibility for all perience in both research and teaching, phases of a research operation should re­ whether the student is headed for academic main with the universities. Here again the work or for industrial or governmental best practice of the most effective agen­ research. Such experience should be of a cies is a good model for the government sort which advances the scientific effec­ as a whole. tiveness of the graduate student; it should (4) We repeat the recommendation of not be limited to drudgery in support of an earlier report (see "Strengthening the research or teaching of senior faculty. American Science", p. 34) that "Govern­ (4) Universities should give increased ment departments and agencies concerned recognition to postdoctoral opportunities should uniformly modify the grant and con• for promising students. Appropriate bud­ tract provisions to permit universities and getary arrangements should be made for non-profit research institutions to charge this form of education. full cost of research performed for the (5) Universities should make full edu­ government -- including overhead -- and dational use of affiliated research instal­ to amortize capital expenditures as an lations. These installations should always allowable cost." be available to learning students, and mem­ This recommendation has been imple­ bers of research staffs should, wherever mented to some extent, but still requires possible, be associated in the teaching further attention if we are not to under­ processes of the university itself. mine the strength of the institutions which (6) Universities should strengthen perform the needed research. Unless re­ their faculties for both research and grad• search is to be cut back, the recommenda­ uate teaching by accepting and using Fed­ tion does imply increased expense; as eral as well as non-Federal support for funds increase, the further implementation faculty salaries. of this recommendation should have very (7) The university community as a high priority. whole has a duty to inform the government (5) Since the Federal Governmenthas clearly and in detail of the nature and a deep interest in a rapid increase in the needs o·f basic research and graduate edu- quality and quantity of the nation's teach-

109 ing scientists, its agencies should in graduate education and to university scien­ general seek forms of support for basic tists. research and graduate education which (9) The government should strengthen will permit universities to enlarge their its ability to establish general policies permanent faculties. In particular, the governing its support of basic research government should allow charges against and graduate education at universities. all Federal grants and contracts for time These policies should be formulated under spent by faculty members on work so sup­ the leader ship of the office of the President ported, (However, no such charges against utilizing appropriate advisory machinery. grants and contracts should be permitted The planning of Federal programs in these for extra compensation to individual facul­ areas should allow for early and careful ty members during the regular academic discussion with university leaders. year.) (6) Federally supported fellowship REFERENCES programs should be expanded when truly promising candidates can be found. Fel­ 1. Scientific Progress, the Universities, lowships should be provided both directly and the Federal Government. Statement to talented graduate and postdoctoral stu­ by the President's Science Advisory dents, and also to selected universities Committee. Washington, D. C. Novem­ for allocation to promising applicants. ber 15, 1960. For sale by the Superin­ They should include a supplementary grant tendent of Documents, U.S. Government based on the full cost of such education. Printing Office, Washington Z5, D. C. Such programs should not exclude the stu­ Price 15 cents. dent from part-time assignments in re­ Z. Report of the Task Force Committee on search or teaching or from pay·ment for Education. Multilithed; unbound. 15 pp. such services when it is appropriate. President-elect Kennedy released the (7) Federal support of facilities and report of this committee, known as the equipment should be provided for both Hovde Committee after its chairman, basic research and graduate education so about january 7, 1961. as to increase the quality and quantity of research results and the number of trained 3, Report of the Consultants to the Secre­ scientists. Since the need for buildings and tary of Health, Education, and Welfare equipment is urgent, these should have high and the U. S. Commissioner of Educa­ priority for the present. When the Federal tion on areas related to The National Government has a particular interest in an Defense Education Act of 1958. Multi­ installation of more than local importance, lithed; unbound. iii+ 16 pp. january 1Z, it should expect to meet all the cost of the 1961. undertaking. In other cases, the practice 4. Ten-year Objectives in Education. of sharing the costs offacilities and equip­ Higher Education Staffing and Physical ment between the Federal Government and Facilities, 1960-1961 through 1969- other sources should be encouraged, for 1970. Department of Health, Education, the present at least, since it stimulates and Welfare, OfficeofEducation, Wash­ other sources of support and ensures that ington Z5, D. C. ii + 48 pp. January 19, the receiving institutions have a serious 1961. commitment in the field concerned. (8) In the assignment of funds for 5. Report of a Mathematicians' Conference basic research, the government should on the Support of Higher Education by seek to promote the essential connection the Federal Government, Conference between the conduct of research and the Board of the Mathematical Sciences, training of scientists. Where it is feasible, 1515 Massachusetts Avenue, N. W ~· new undertakings should be established in, Washington 5, D. C. 37 pp. january Z5, or in close association with universities, 1961. Copies are available on request. and the great influence and effectiveness 6. A Proposed Program of Federal Action of basic research in existing government to Strengthen Higher Education. Ameri­ installations should be increased where can Council on Education. ii + 11 pp. possible by improving its connection to january, 1961.

110 MATHEMATICS IN TRANSLATION

The following Russian books are should specify OTS or AEC identification currently being translated into English by numbers. Among translations completed the American Mathematical Society under is: grants from the National Science Found­ Golubev, V. V. Lectures on integration ation: of the equations of motion of a rigid body about a fixed point. ZZ9 pp. $Z.50. Sobolev, S. L. Some applications of func­ OTS 60-Zll63. Russian original: Gos. tional analysis in mathematical physics. lzdat. Tehn-Teor. Lit., Moscow, 1953. Izdat. Leningrad. Gos. Univ., Leningrad, Z87 pp. MR 15, 904. 1950. Z55 pp. MR 14, 565. Among translations announced as in Ahiezer, N.; Kre'i'n, M. G. Some ques­ process are two recent Polish books: tions in the theory of moments. Gos. Nauc.-Tehn. lzdat. Ukrain., Kharkov, Steinhaus, Hugo. 100 problems. 1938. Z55 pp. Straszewicz, S. Problems from the Aleksandrov, A. D.; etal. Mathematics: mathematical Olympiads. its content, methods, and meaning. Izdat. A translation of the following Rus­ Akad. NaukSSSR, Moscow, 1956. Vol. 1, sian book has been prepared for publica­ Z96 pp.; Vol. Z, 395 pp.; Vol. 3, 336 pp. tion by the VEB Deutscher Verlag der MR 19, 5ZO. Wissenschaften, Berlin W 8: Na!mark, M. A. Linear representations Gel'fand, I. M.; Silov, G. E. Generalized of the Lorentz group. Gos. Izdat. Fiz­ functions and operations on them. (Gen­ Mat. Lit., Moscow, 1958, 376pp. MR Zl eralized functions, Part 1.) Gos. Izdat. No. 4995. Fiz.-Mat. Lit., Moscow, 1958. 440 pp. Vekua, I. N. Generalized analytic func­ MR ZO No. 418Z. tions. Gos. Izdat. Fiz.-Mat. Lit., Mos­ To be added to the list with which cow, 1959. 6Z8 pp. MR Zl No. 7Z88. the AMS checks all its translation plans is Translation of another bt:>.ok has the firm of been approved, but deferred so it can be Oliver and Boyd, Ltd. based on the forthcoming second Russian 39A Welbeck Street edition: London W. 1, England Fuks, B. A. Theory of analytic functions which is starting a program of translations of several complex variables. of Russian scientific and mathematical books. The Society's Committee on Trans­ lations has approved the recommendation to translate three additional books: SOVIET PHYSICS-DOKLADY, the American Institute of Physics English Delone, B. N.; Faddeev, D. K. Theory of translation of the physics sections of Dok­ irrationalities of the third degree. lady Akademii Nauk SSR,now published Lyapin, E. S. Semigroups. every other month, will appear monthly, starting with the July, 1961 (Vol. 7, No.1) Levin, B. Ya. Distribution of zeros of issue. Inquiries concerning the journal entire functions. should be sent to: Translations made under the P. L. American Institute of Physics 480 program, administered by the National 335 East 45th Street Science Foundation on behalf of partici­ New York 17, New York. pating Federal agencies, may be ordered from the Office of Technical Services, De­ The next article in this series in partment of Commerce, Washington Z5, the NOTICES will report on the extensive D. C.; the price is about 1 cent per page translation activities of the Pergamon In­ for longer articles and books; orders stitute, London

111 THE PROPOSED DOCTOR OF ARTS DEGREE By Edwin Moise

At the annual meetings in Washing­ mum, with ZBZ.) Five men for every state ton, this january, the A. M. S. Council and of the union would represent a pitiful rate the M. A, A. Board of Governors voted to of recruitment, even if all of them went approve, in principle, a report which re­ into teaching, but of course they don't: commended the establishment of a new about half of them go into government and graduate degree in mathematics. The pro­ industry. In 1953, over a third of the new posal was described to both ofthese bodies mathematics teachers held the Ph. D. By at the summer meetings in East Lansing, 19 59, this proportion had fallen to less than last August; and an ad hoc committee was one fifth. From time to time it is reported asked to study the matter and report in that the growth of college faculties is not january. The committee consisted of M.M. keeping up with rising enrollments. But to Day, P. R. Halmos, A. D. Wallace and the say this in connection with mathematics is author. a misleadingly inadequate description of The present article is an attempt to our troubles: it has recently been reported explain what the proposal is, and how it that the number of mathematics teachers forms a response to the situation in which with Ph.D.'s has declined in the past five the mathematical world now finds itself. years, not merely relative to enrollment The case as I shall put it will be rather but absolutely. close to the spirit of the ad hoc committee's Meanwhile, the task of college math­ report, but it should be understood that in ematics departments has been growing in matters of detail I make no claim to speak ways that are not as widely recognized as for anyone but myself. they deserve to be. The developments have It is plain that the mathematics de­ been peculiar and there is evidence to in­ partments of American colleges need to be dicate that people have not been keeping staffed by people whose scholarly training proper track of it. While total college en­ makes them competent to teach under­ rollments have grown, enrollments in graduate mathematics. There may, how­ mathematics departments have grown ever, be considerable debate over what much faster. And while the size of the constitutes competence. The leading uni­ mathematicians' job has grown, the shape versities, in our generation, have taken the of their job has also changed: the enroll­ position that only a mathematician can ment in advanced courses is a higher teach mathematics; and the word mathe­ proportion of the total than it used to be. matician has been interpreted to mean, at This, of course, is a very gQod thing for best, a currently active research worker, the physical scientists and engineers: they and, at least, a person who has written one seem to be aware that they need mathe­ research paper, under supervision, at matics and plenty of it. It means, however, some time in the past. This is the sub­ that in the past decade the need for highly stance of the policy which demands that trained college mathematics teachers has college mathematics teachers hold the increased even faster than the need for Ph. D. college mathematics teachers in general. Plainly the goal of this policy is de­ There seems to be no reason at all sirable. The trouble is that the goal is im­ to believe that the demand for Ph.D.'s is possible to attain. For mathematical going to decline; it appears, rather, that Ph.D.'s the supply and the demand differ by the demand will continue to rise for the order of magnitude. foreseeable future. Meanwhile, annual re­ At present, America produces about cruitment of Ph.D.'s into the teaching pro­ Z50 mathematical Ph.D.'s per year. From fession has not even been enough to balance 1948 to 19 53, the number of new Ph.D's in­ annual losses. Under these conditions, to creased rather sharply, from lZB to Z41, speak of "holding the line," by insisting but since then the number has been approxi­ that college mathematics teachers hold the mately constant. (1959 was a local maxi- Ph.D., is utterly unrealistic. Indeed, this

llZ idea is so remote from the real possibili­ The difference between dangerous ties that to speak in such terms amounts degrees and merely difficult degrees is in effect to a refusal to face the problem. important; these two properties are large­ The problem is very real, and we can ly independent of one another. For ex­ hardly regard it as beneath our notice. ample, the Harvard B. A., while difficult, It is rather easy, at present, to see is "safe;" a freshman who has once been why the demand for highly trained mathe­ admitted is very likely to graduate in four maticians is so large, It is not, however, years. The same is true of the M. D. at quite so easy to sfi!e why the supply of most of the best medical schools; and it Ph.D.'s is so small. The conditions of seems even to be true of the Ph.D. in math­ mathematical study are rather peculiar. ematics at Princeton, because the admis­ One indication of this is the fact that in the sion standards are so high. On the other period from 1948 to 19 59, the number of hand, the B.A. at many state universities B. A.'s in mathematics was about twice the is dangerous without being difficult: the number of B. A.'s in physics, while the graduation standards are low, but the ad­ number of Ph.D.'s in mathematics was only mission standards are so muchlower that about half the number of Ph.D.'s in physics. the mortality is heavy. The Ph.D. is a research degree, and The dangerousness of the Ph,D. the conditions of research in mathematics creates morale problems over and above are almost unique. In most of the natural the difficulties due to high intellectual sciences, group research work is not only standards. Soon, we shall discuss the idea possible but usual. For this reason, it was of a "safe" degree for college mathematics quite sensible for a government agency to teachers. The foregoing remarks are de­ offer stipends for undergraduate research signed to make it plain that a" safe" degree assistants in science. The applications of need not involve -- and, of course, should this idea to mathematics, however, were not involve -- a collapse ofintellectual unclear, and the government has been standards, It ought to be a course of study asked to give stipends to worthy under­ which is difficult and demanding, in which graduate mathematics students on some success is, however, reasonably predict­ other basis, It is true, of course,that able for suitably qualified people. mathematicians sometimes collaborate, Basically, degrees are a form of usually in groups of two, as novelists language. The primary purpose in giving sometimes collaborate; but essentially, a person a degree is to communicate in­ creative work in mathematics is done by formation, thereby, to various third par• a man alone. ties. A scheme of degrees ought to be a (The above remarks are not taken language adequate to describe and classify, from the ad hoc committee's report.) with at least rough justice, the class of The number of people who can sit people to whom the language might apply. down alone and do creative work in mathe­ The degree-language in mathematics,fran matics is quite small. We might describe this viewpoint,has serious shortcomings. the situation by first saying that as a gen­ In the first place, the standards eral rule, nobody can do bona fide mathe­ for the Ph.D. vary so much, from one uni­ matical research, and then admitting that versity to another, and from one super­ there are scattered exceptions to the gen­ vising professor to another, that the mere eral rule. Moreover, it is hard, except in possession of the degree is not, in itself, extreme cases one way or the other, to a guarantee of anything in particular. This predict which individuals will turn out to is not a serious problem because the num­ be among the exceptions. For these rea­ ber of people involved is --unfortunately-­ sons, doctoral study in mathematics is not so small that personal correspondence can only difficult but dangerous: a student who be used, and commonly is used, as the looks well-qualified may work well for main source of information. years, and then find that he is a "Failed A more serious trouble is that the Ph.D." because he couldn't write a crea­ present degree- scheme divides former tive thesis. (Recently, som~::one at a large graduate students into only two classes: state university coined the phrase "Teach­ those who did creative dissertations and ing Fellow Emeritus.") those who did not. Under the present

113 scheme, there is no way to describe sound higher degree that is worth giving or re­ mathematical scholarship except in those ceiving. cases where it is accompanied by creative (2) It has been proposed that the talent. M. A. be upgraded as a degree for college For approximately those reasons -­ teachers. But the idea of increasing the with one exception, as noted-- the Council prestige of an old degree appears to be a and the Board of Governors approved a pscho-political impossibility. proposal for a new degree, tentatively (3) It has beensaidthatmostpeople called the Doctor of Arts. (The name of the who can pass preli1;t1inary examinations degree is not part of the substance of the can also write dissertations. If this is proposal.) The essential idea is that the is true -- and at most places, I suspect new degree would represent scholarly that it is not -- then it means merely that achievement, but not necessarily creative· the contribution made by the new degree talent. would not be quantitatively as large as one Nobody has attempted to describe, might have hoped. with any exactitude, the sort of program (4) Some have expressed the fear that would lead to this degree. In the that some institutions would reflect dis­ country at large there is no general agree­ credit on the Doctor of Arts degree by ment on exactly what the Ph.D. ought to giving it to inferior students. In fact, this mean, and there is no reason to suppose is surely going to happen. At any rate, it that the Doctor of Arts degree can be, or has already happened to the B. A., the should be, standardized on a national basis. M. A. and the Ph.D. No matter what the The report, however, went on to idea behind a degree may be, it is normal clarify the spirit of the proposal with gen­ for the standard of performance to vary, eral remarks about courses and disserta­ in a wide range, from one institution to tions. In the first place, the idea was for another. the new degree to be granted only by insti­ (5) It has been suggested that the tutions qualified to grant the Ph.D. Few, Doctor of Arts degree may displace the if any, new courses ought to be needed. In Ph.D. as a degree for college teachers, to fact, the report proposed that the course­ the detriment of both research and teach­ work for the Ph.D. and the Doctor of Arts ing. The present attitudes of the colleges, be essentially the same, so as to make it however, suggest exactly the opposite dif­ easy for a student to alter his intentions, ficulty: the colleges are now seeking not one way or the other, at any stage short of simply Ph.D.'s as such, but prospective the dissertation. research mathematicians, with an eager­ In the second place, the creative ness bordering almost on desperation. dissertation should be replaced by a schol­ There seems to be no reason to expect -­ arly dissertation which could be historical, either in fear or in hope -- that these critical or philosophical in nature. Here widely held attitudes will be turned upside the idea of historical dissertations is in­ down merely by some legal or administra­ tended by all means to include studies of tive device. It seems, on the contrary, that recent mathematical developments. What­ considerable persuasion will be required ever the thesis topic may be, the course­ to get the colleges to accept the Doctor of work should be adequate to enable the stu­ Arts degree as evidence of the sort of dent to read current research literature. qualifications that it really would repre­ This describes the spirit of the pro­ sent. (Some have proposed that new jour­ posal. It may now be worthwhile to discuss nals be established, for publication of the various criticisms that have been made, sort of scholar ship that would be involved.) and various alternative proposals that have Finally, it shouldbeunderstoodthat been brought up. the question whether college teachers ( 1) I have heard the Doctor of Arts should be creative mathematicians, and degree described as "the negation of the whether graduate students ought to have Ph.D." This, I think, is a misunderstand­ creative intellects, are not really issues ing: the new degree would negate merely in connection with this proposal. Non­ the idea that the Ph.D. is the only sort of creative people are already teaching in the

114 colleges, not only in large numbers, but for periods which often seem interminable. forming a large majority; and this will The real question at issue is whether the continue to be true unless and until a way best of the noncreative mathematicians is found to increase, by order of magni­ should be encouraged to develop their own tude, the supply of creative mathemati­ capacities in the ways that are possible, cians. Non-creative students are now and rewarded and recognized for the very studying in graduate mathematics depart­ real and very useful intellectual achieve­ ments, not only in large numbers, but also ments of which they are capable,

NEWS ITEMS AND ANNOUNCEMENTS

TRAVEL GRANTS FOR ATTEN­ THE Xth INTERNATIONAL CON­ DANCE AT THE INTERNATIONAL CON­ GRESS OF THE HISTORY OF SCIENCES GRESS OF MATHEMATICIANS. Travel will be held at Cornell University, Ithaca, grants will be made to a number of mathe­ New York, from August26-31, 1962,and at maticians who wish to attend the Inter­ Th.e American Philosophical Society, national Congress of Mathematicians in Philadelphia, Pennsylvania, from August Stockholm, on August 15-22, 1962. It is 31 to September 2, 1962. hoped that funds available through various Communications on the following sources may provide travel assistance subjects are invited, and will be assigned for a considerable number of mathemati­ to the appropriate sections: cians, I. General problems in the history of There will be a greater effort than science; methods, philosophy and historio­ in the past to give aid to younger people. graphy of science As grants will be made only to those who II. History of technology and applied have filed applications, it is urgent that science any who wish to receive a grant should III. Science in antiquity fill out and file an application. Younger IV. Science in the middle ages and the people are urged to file applications so renaissance that their cases can be considered, Ap­ V. Mathematics and the exact sciences plications can be obtained from the Divi­ after 1600 sion of Mathematics, National Academy of 1,. History of mathematics Sciences - National Research Council, 2. History of physics and astronomy Washington 25, D. C., by requesting an ap­ 3. History of chemistry (including plication for a travel grant to the 1962 pharmacy) International Congress. The deadline for filing of applica­ VI. Biological and earth sciences after tions is November 1, 1961, and an attempt 1600 will be made to announce the grants by 1. Natural history and biology (in­ January 1, 1962. cluding medical biology) Awarding of grants will be made 2. Geography, exploration, geology only to those persons whose applications and oceanography have been received, in good order, by VII. Sciences of man (phsychology, anthro­ November 1. The selection will be made pology, sociology, linguistics). by a committee consisting of the regular Committee on Travel Grants of the Divi­ Oral presentation of these papers sion of Mathematics of the National Aca­ will be strictly limited to 15 minutes. An demy of Sciences - National Research abstract, not to exceed 200 words, must be Council enlarged to include representa· submitted in a clean typescript (suitable tives of societies affiliated with the· Divi­ for reproduction) to the Secretary of the sion and representatives of various gov­ Congress, Dr. C. Doris Hellman, Cornell ernmental agencies. University, no later than May 1, 1962,

llS RUSSIAN-ENGLISH DICTIONARY of MATHEMATICAL TERMS

matics, no effort was made to limit the vocabulary to special areas of mathemat­ ics, and the vocabulary carries a useful but not necessarily complete coverage of various branches of theoretical physics along with those of the mathematical sci­ ences: Algebra Algebraic geometry Analysis Astronomy Astrophysics Atom spectra Biophysics Celestial mechanics Classical mechanics Computer science Computers Cosmic ray theory Cybernetics Electromagnetic theory Electron optics General and elementary terms Geometrical optics Geometry Geophysics Heat and thermodynamics The American Mathematical Society Linear programming will publish, late this summer, a Russian­ Magnetism English Dictionary of Mathematical Terms Mathematical economics compiled by A. j. Lohwater, with the col­ Mathematical logic laboration of S. H. Gould. The compilation Mechanics was under the auspices of the American Molecular spectra Mathematical Society, the National Aca­ Numerical methods demy of Sciences of the USA and the Aca­ Particle accelerators demy of Sciences of the USSR. At the same Physical optics time, the Academy of Sciences of the USSR Plasticity will publish an English-Russian Dictionary Probability and statistics of Mathematical Terms essentially identi­ Quantum mechanics cal in size and character. Solid state physics The dictionary will contain over Sound and acoustics 15,000 terms, from pure mathematics as Statistical mechanics well as from all the mathematical sci­ Theory of functions of a complex vari .. ences. The present volume is intendedfor able use in reading papers and books at any Theory of functions of a real variable level in mathematics and theoretical phy­ Theory of numbers sics, and a short grammar precedes the Theory of games vocabulary of the dictionary so that the Topology and theory of sets book is practically self contained. Since there is no clear dividing line between There are two prefaces, describing the "pure" mathematics and "applied" mathe- three -year collaboration between Ameri-

116 can and Russian scientists, one of the pre­ either in Referativnii Zhurnal Mathematika faces describing the American activity, or Mathematical Reviews without the aid and the other the Russian. of any other dictionary. It was agreed that Quoting from the foreword to the before publication the manuscripts of the English edition of the dictionary: two parts of the dictionary would be ex­ changed and commentary made on each July ofl956Professor "In June and by the committee of the other side. By of Princeton University and N. E. Steenrod January 1960 the compilation of both parts Lohwater} comprised the the author [A. J. of the dictionary had been completed and to the Third All­ American delegation the manuscripts had been exchanged for Soviet Mathematicians Union Congress of mutual criticism. The work of incorpo­ In the friendly atmos­ held in Moscow. rating the Soviet suggestions for improve­ prevailed at that Congress, it phere which ment in the American part of the diction­ of exchange and was natural that questions ary was completed ten months later in mathematicians should contact between November. ideas brought up be discussed. Among the "Grateful acknowledgment is made collaboration there, the question of joint to the Guggenheim Foundation, the National and English-Russian on Russian-English Science Foundation and the Office of Air seemed to hold the scientific dictionaries Force Research, for their support of vari­ tangible results to the most promise of ous aspects of the dictionary project. The community. Following American scientific author also wishes to express his grati­ exchange of letters between Detlov an tude to Professor A. I. Markushevich, the of the National Academy Bronk, President Russian First Deputy Minister of Educa­ U.S. A. and A. N. Nes­ of Sciences of the tion, who, on a visit to this country inspec­ the Soviet Academy meyanov, President of ted the project, making helpful suggestions of matjemati­ of Sciences, committees and varifying many idomatic expressions." the two cians were appointed to represent Following are quotations from the committee con­ academies. The Soviet foreword to the Russian edition of the (Chairman), sisted of P. S. Alexandrov dictionary: I. R. Shafarievich, M. M. Postnikov, A. F. Leontiev and A. A. Dezin, while the Amer­ "In order to keep up with the litera­ ican committee consisted of S. H. Gould, tuni a.s well as to maintain scientific con­ A. J. Lohwater and G. I. Rainich. Later the tact between mathematicians of different Soviet committee was augmented by the countries who are publishing in their own appointment of L. N. Bolshiev, S. M. languages, special dictionaries of mathe­ Nikolskit, E. D. Solomentsev and L. D. matical terms have become a necessity. Kudryavtsev, who acted as executive vice­ The highest priority has been accorded to chairmen. a unified English-Russian and Russian­ "On 10 September 1958 representa­ English mathematical dictionary, which is tives of the two committees met in the now placed before the reader. Steklov Institute in Moscow to discuss the "This is the first dictionary of its preparation of the dictionary. At this meet­ kind; it is intended to serve everyone con­ ing essential agreement was reached on nected directly or indirectly with mathe­ the areas of coverage, the means of com­ matics. Basically, the dictionary gives munication and collaboration between the translations from one language to the other two committees, and the eventual publica­ of mathematical terms in all areas of tion of the dictionary. The Russian-Eng­ modern mathematics. Whenever an equi­ lish half of the dictionary would be pub­ valent term has not yet been established lished in the United States by the Ameri­ in the other language, an event not seldom can Mathematical Society while the publi­ encountered, the dictionary gives a short cation of the English-Russian half would description of the term." be undertaken by the Soviet Academy of "During the entire course of the Sciences. The guiding principle in the pre­ work, a regular liaison and a constant ex­ paration of the dictionary was that a per­ change of information were maintained be­ son using the dictionary would be able to tween Soviet and American specialists read virtually any publication reviewed working on the dictionary. Much invaluable

117 information was received from Dr. S. H. ever completed by the joint action of So­ Gould and Professor A. j. Lohwater (who viet and American mathematicians. It is directed the compilation in the United our hope that a subsequent development States). The word-lists which we compiled of such cooperation and joint activities were sent to the United States for exam­ between Soviet and American colleagues ination by Professor Lohwater who re­ will contribute to the further growth of turned to us a detailed critique. We ex­ world science, to an improvement of mu­ press to Professor Lohwater our deep tual understanding and to friendship be­ gratitude for his extensive work on our tween the peoples of our countries." word-lists, an effort which was of substan­ Publication of the Russian-English tial help to us and which served to increase Dictionary is scheduled for late summer. the quality of our Dictionary. For our part After publication, the list price will be we received from Professor Lohwater a not less than $7. 70, Before placing the preliminary draft of the Russian-English print order for the book, we will accept part of the Dictionary and returned to him prepublication orders for the dictionary our critique, which was made by L. D. at the reduced list price of $7,00 and Kudryavtsev, Yu. K. Solntsev, and E. D. AMS members price of $5,25. To reserve Solomentsev." your copy of the dictionary at the prepubli­ "In conclusion it must be pointed cation price, orders must be received be­ out that the project of publishing an Eng­ fore june 1, 1961 at the headquarters lish-Russian and Russian-English mathe­ offices, 190 Hope Street, Providence 6, matical dictionary is the first project Rhode Island.

Just published MATHEMATICAL METHODS FOR ENGINEERS AND TECHNOLOGISTS by P. I. Romanovskii This invaluable book is intended for students of technical colleges and other technical education institutes who require a textbook for more advanced courses of mathematics on the following subjects: Fourier series and the Fourier integral. Analytical Field Theory. The Theory of Analytic Functions. Some Special Functions. Operational Calculus. The Laplace Transform. $8.50 CYBERNETICS WITHOUT MATHEMATICS by H. Greniewski The book opens with a discussion of fundamental concepts as well as the subject matter and method of cybernetics, as understood by the author. This is followed by chapters on modelling and simulation, in which four kinds of models are discussed-biological, praxio­ logical, logical and economic. $6.00 International Series of Monographs on Pure and Applied Mathematics Vol. 13 INTRODUCTION TO SET THEORY AND TOPOLOGY byK.Kuratowski As the title indicates, this work is divided into. two parts: the first covering basic material from the theory of sets and functions, and the second taking up fundamental topics in topology, including the elements of algebraic topology. It provides a textbook for the student who is trying to understand courses in set theory and topology as presented in the classroom, or who may be studying this material on his own. in press Vol. 15 THE THEORY OF MEASURE AND LEBESGUE INTEGRATION by S. Hartman and 1. Mikusinski This text on the theory of measure and Lebesgue integration contains material which constitutes a good introduction to the theory of real functions. The subject matter com­ prises concepts and theorems now considered classical, which every young mathematician

0 e)ght; ; ; G A M0 N p R ES s Ill .... 55th 5treot, N- :::::.,:. y

118 REPORT ON RECIPROCITY AGREEMENTS

During the past fifteen years, the American Mathematical Society has entered into reciprocity agreements with twenty-five foreign mathematical organizations. An interchange of information within the international mathematical community is more desirable now than ever before. So that our members may be aware of all the existing agreements and J:na.Y make use of the privileges offered under the reciprocity agreements, we are giving below a comprehensive report of all agreements now in effect.

NAME OF RECIPROCATING SOCIETY

Wiskundig Genootschap te Amsterdam Apply for Membership to: Dr. J. G. Dijkman, Secretary Ged. Oude Gracht 121 Haarlem, Holland Amount of Dues: $2.25 Pay Dues to: Professor Dr. F. Loonstra Mathematisch Instituut Technische Hogeschool Julianalaan 132 Delft, Holland Privileges of Membership: Nieuw Archief voor Wiskunde Indigationes Mathematicae - $2. 50

Union Matematica Argentina Apply for Membership to: Secretary of the Union Matematica Argentina Casilla de Correo 3588 Buenos Aires, Argentina Amount of Dues: $1.25 Pay Dues to: Secretary of the Union Matematica Ar.gentina Privileges of Membership: Review of the Union MatematicaArgentina Asociacion Fisica Argentina

Australian Mathematical Society Apply for Membership to: Professor H. 0. Lancaster General Secretary Australian Mathematical Society Department of Mathematical Statistics University of Sidney Sidney, Australia Amount of Dues: $7. 50 Pay Dues to: Professor C. S. Davis University of Queensland St. Lucia, Brisbane Queensland, Australia Privileges of Membership: Journal of the Australian Mathematical Society

119 NAME OF RECIPROCATING socmTY "Osterreichische Mathema- tische Gesellschaft Apply for Membership to: P.rofessor Dr. Hans Hornich C5sterreichische Mathematische Gesellschaft Karlsplatz 13 Vienna IV, Austria Amount of Dues: $1.00 Pay Dues to: Professor Dr. Hans Hornich Privileges of Membership: lnternationalen Mathematischen Nachrichten

Societe Mathematique de Belgique Apply for Membership to: Mr. Guy Hirsch, Secretaire Societe Mathematique de Belgique Secretariat: 317, Avenue Ch. Woeste Bruxelles 9, Belgium Amount of Dues: $1.00 Pay Dues to: Mr. Guy Hirsch, Secretaire Privileges of Membership: Bulletin de la Societe Mathematique de Belgique

Chinese Mathematical Society Apply for Membership to: The Secretary The Chinese Mathematical Society P. 0. Box 143, Taipei Taiwan, China Amount of Dues: $.50 Pay Dues to: The Secretary The Chinese Mathematical Society Privileges of Membership: Information Circulars Proposed publications of the Society

Dansk Matematisk Forening Apply for Membership to: Dansk Matematisk Forening Blegdamsvej 15 Copenhagen, Denmark. Amount of Dues: $1.00 Pay Dues to: Dansk Matematisk Forening Privileges of Membership: Mathematica Scandinavica - $3.75 Nordisk Matematisk Tidskrift - $1. 90

The London Mathematical Society Apply for Membership to: Dr. H. Halberstam, Secretary Royal Holloway College Englefield Green Surrey, England Dr. J. A. Todd, Secretary Downing College Cambridge, E11gland Amount of Dues: $6.00 Pay Dues to: The Treasurer London Mathematical Society Burlington House, Piccadilly London W.1, England Privileges of Membership: Proceedings of the London Mathematical Society Journal of the London Mathematical Society

120 NAME OF RECIPROCATING SOCIETY

Suomen Matemaattinen Yhdistys Apply for Membership to: Suomen Matemaattinen Yhdistys University of Helsinki Hallituskatu 11-13 Helsinki, Finland Amount of Dues: $.50 Pay Dues to: Suomen Matemaattinen Yhdistys (Bank: Helsingin Osakepankk:i Helsinki) Privileges of Membership: Arkhimedes

Societe Mathematique de France Apply for Membership to: American Mathematical Society 190 Hope Street Providence 6, Rhode Island Amount of Dues: $3.00 Pay Dues to: American Mathematical Society Privileges of Membership: Bulletin of the Societe Mathematique de France Deutsche Mathematiker Vereinigung Apply for Membership to: Professor Dr. K. P. Grotemeyer HUttenweg 9-11 Berlin-Dahlem, Germany Amount of Dues: $1. 00 Pay Dues to: Girokonto: Kreissparkasse TUbingen 1626, Germany Postscheckkonto: Stuttgart 18517, Germany Privileges of Membership: Satzungen der Deutschen Mathematiker Vereinigung Jahresberichte der Deutschen Mathe­ matiker Vereinigung- $1.50 Gesellschaft fUr Angewandte Mathematik und Mechanik Apply for Membership to: Professor Henry Gortler Mathematisches Institut der UniversiUlt Freiburg Hebelstrasse 60 Freiburg i. Br. , Germany Amount of Dues: $1.50 Pay Dues to: Professor Henry Gortler (Do not send with application. ) Privileges of Membership: Regular publications of the Gesellschaft fUr Angewandte Mathematik und Mechanik

Islenzka Staerdfraedaf~l­ agid Apply for l.iembership to: J. E. Thorlllksson Hagstofa Islands Icelandic Statistical Bureau Reykjavik, Iceland Amount of Dues: $1.50 Pay Dues to: J. E. Thorlliksson Privileges of Membership: Mathematica Scandinavica- $3.75 Nordisk Matematisk Tidskrift - $1. 90

121 NAME OF RECIPROCATING SOCIETY

Calcutta Mathematical Society Apply for Membership to: Dr. P. P. Chattarji, Secretary Calcutta Mathematical Society 92, Acharya Prafulla Chandra Road Calcutta 9, India Amount of Dues: $1. 50 plus $2. 50 admission fee Pay Dues to: Dr. P. P. Chattarji, Secretary Privileges of Membership: Bulletin

Indian Mathematical Society Apply for Membership to: American Mathematical Society 190 Hope Street Providence 6, Rhode Island Amount of Dues: $1.75 Pay Dues to: American Mathematical Society Privileges of Membership: Journal of the Indian Mathematical Society Mathematics Student

Unione Matematica Italiana Apply for Membership to: Professor Mario Villa Unione Matematica Italiana Istituto Matematico Dell 'Universita Largo Trombetti, 4 Bologna, Italy Amount of Dues: $3. 00 Pay Dues to: Professor Mario Villa Privileges ofMembership: Bolletino Unione Matematica Italiana

Mathematical Society of Japan Apply for Membership to: President, Mathematical Society of Japan c/o Faculty of Science University of Tokyo Toyko, Japan Amount of Dues: $2.50 Pay Dues to: President, Mathematical Society of Japan Privileges of Membership: Journal of the Mathematical Society of Japan Sagaku

Sociedad Matematica Mexicana Apply for Membership to: Dr. Alfonso Napoles Gandara, President, Sociedad Matematica Mexicana Tacuba No.5 Mexico 1, D. F., Mexico Amount of Dues: $2.00 Pay Dues to: Dr. Alfonso Napoles Gandara Privileges of Membership: Revista Matematica Boletin de la Sociedad Matematica Mexicana

122 NAME OF RECIPROCATING SOCIETY

Norsk Matematisk Forening Apply for Membership to: Norsk Matematisk Forening Matematisk Institutt Universitetet Blindern Oslo, Norway Amount of Dues: $1.00 Pay Dues to: Norsk Matematisk Forening Privileges of Membership: Mathematica Scandinavica- $3.75 Nordisk Matematisk Tidskrift - $1.90 Sociedad Matematico Peruana Apply for Membership to: Secretario de la Sociedad Matematica Peruana Apartado 4963 Lima, Peru Amount of Dues: $4.00 Pay Dues to: Tesorero de la Sociedad Matematica Peruana Apartado 4963 Lima, Peru Privileges of Membership: Boletin de la Sociedad Matematica Peruana

Polskie Towarzystwo Matematyczne Apply for Membership to: Polskie Towarzystwo Matematyczne ul. Sniadeckich 8 Warszawa, Poland Amount of Dues: $1.00 Pay Dues to: Narodowy Bank Polski IV Oddzial Miejski A/C No. 1528-9-4244 Warszawa, Poland Privileges of Membership: Prace Matematyczne or Wiadomosci Matematyczne

Glasgow Mathematical Association Apply for Membership to: Secretary of the Glasgow Mathematical Association Department of Mathematics The University Glasgow, W. 2, Scotland Amount of Dues: $1.10 Pay Dues to: Treasurer of the Glasgow Mathematical Association Department of Mathematics The University Glasgow W. 2, Scotland Privileges of Membership: Proceedings of the Glasgow Mathematical Association

123 NAME OF RECIPROCATING SOCIETY

Svenska Matematiker­ samfundet Apply for Membership to: Dr. Hans Riesel Box 6131 Stockholm 6, Sweden Amount of Dues: $1.00 Pay Dues to: Bo Kjellberg K.T.H. Stockholm 70, Sweden Privileges of Membership: Mathematica Scandinavica- $3.75 Nordisk Matematisk Tidskrift - $1. 90

Schweizerische Mathe­ matische Gesellschaft Apply for Membership to: Professor B. Eckmann Eidgentlssiche Technische Hochschule Zurich, Switzerland Amount of Dues: $1.00 Pay Dues to: Secretary, Schweizerische Mathe­ matische Gesellschaft Postcheck-Konto VITI 16483 Zurich, Switzerland Privileges of Membership: Commentarii Mathematici Helvetici - $5.75

Turkish Society of Pure and Applied Mathematics Data not available.

RECIPROCITY AGREEMENT WITH THE the other country should pay the regular GLASGOW MATHEMATICAL dues while they are there. ASSOCIATION The American Mathematical Society RECIPROCITY AGREEMENT WITH THE has entered into a reciprocity agreement CHINESE MATHEMATICAL SOCIETY, with the Glasgow Mathematical Associa­ FORMOSA tion by which members of each may be· The American Mathematical Society come members of the other by paying half has entered into a reciprocity agreement the regular dues. The regular dues of with the Chinese Mathematical Society by members of the Glasgow Mathematical which members of each may become mem­ Association are fifteen shillings; there­ ber s of the other by paying half the regular fore an American Mathematical Society dues. The regular dues of members ofthe member would pay $1.10 yearly dues to Chinese Mathematical Society are $1; Treasurer of the Glasgow Mathematical therefore an American Mathematical So­ Association, Department of Mathematics, ciety member would pay 50 cents yearly The University, Glasgow, W. 2, Scotland. dues. Privileges of membership include Privileges of membership include all the all the usual privileges of membership in usual privileges of membership in the the Chinese Mathematical Society such as Glasgow Mathematical Association such receiving the notices of meetings, the IN­ as receiving the syllabus of meetings, the FORMATION CIRCULARS and the pro­ PROCEEDINGS of the Glasgow Mathemati­ posed publications of the Society. Those cal Association, appearing twice a year. members of the American Mathematical Those members of the American Mathe­ Society wishing to take advantage of this matical Society wishing to take advantage arrangement should write to The Secretary, of this arrangement should write to the The Chinese Mathematical Society, P. 0. Secretary of the Glasgow Mathematical Box 143, Taipei, Taiwan, China. It is un­ Association, Department of Mathematics, derstood that members under the recipro­ The University, Glasgow, W. 2, Scotland. city agreement spending time in the other It is understood that members under the country should pay the regular dues while reciprocity agreement spending time in they are there.

124 HEWS ITEMS AND AHHOUHCEMEHTS

THE THIRD BRAZILIAN COLLO­ the IMU), M. Morse (Delegate of the IMU), QUIUM OF MATHEMATICS will be held K. K6utsky, V. Ptak, S. Schwarz, Z. Fro­ at the University of Ceara, in the city of Ilk (Secretary). The address of the Organi • Fortaleza, State of Ceara, Brazil, from zing Committee is Topological Symposium, July 2 to 15, 1961. The colloquia of this Praha 2, Ke Karlovu 3. series are held bi-annually. The 1961 The following program has been set meeting is being organized by the Institute up: for Pure and Applied Mathematics, of Rio September 1: Opening of the Symposium, de Janeiro, and will be sponsored by commemoration of Pro· CAPES, the National Research Council of fessor E. Cech Brazil, the Organization of American September 1-2: Working days States and the University of Ceara. It will September 3: Free day, sightseeing of consist of four instructional courses on Prague Algebraic Topology, Functional Analysis, September 4-5: Working days Ordinary Differential Equations and Rie­ September 6: Excursion mannian Geometry; about ten lectures in September 7-8: Working days the "mise au point" style; some sessions September 8: Closing of the Symposium for presentation of recent research work and a session devoted to a discussion of mathematics teaching at universities. It AIR FORCE OFFICE OF SCIENT!· is expected that this meeting will be at­ FIC RESEARCH APPLIED MATHEMA­ tended by a number of mathematicians TICS PROGRAM. The AFOSR has an­ from Asia, Europe, Latin America and the nounced a broad expansion of its support United States especially invited or who for research projects in the area of ap­ happen to be in Brazil as visiting profes­ plied mathematics. This program which sors at the time of the colloquium, besides will emphasize both the traditional and about sixty Brazilian mathematicians. Fi­ modern aspects of applied mathematics nancial support will be available to some will be a part of the program of the Direc­ participants. Other information can be ob­ torate of Mathematical Sciences of AFOSR. tained from the Director, Instituto de It is the hope of the Air Force that this Matematica Pura e Aplicada, Rua Silo new program will help fill the increasing Clemente 265, Rio de Janeiro, Brazil. need for more mathematical methods that are required to solve today' s complex problems in the physical and engineering THE CZECHOSLOVAK ACADEMY sciences. Present plans call for the pro­ OF SCIENCES AND THE INTERNATION­ gram to start in the fall of 1961. Interested AL MATHEMATICAL UNION have an­ applied mathematicians are encouraged to nounced a Symposium on General Topology submit proposals for support under this and its relations to modern Analysis and program to the Director of Mathematical Algebra, to be held September 1-9, 1961, Sciences, Air Force Office of Scientific in Prague. The object of the Symposium Research, Washington 25, D. C. will be to discuss the present state of the theory of topological spaces, uniform and similar structures together with related questions of analysis and algebra and also SHELLCOMP ANIES FOUNDATION, applications of the methods of general INCORPORATED, announced that it will topology in functional analysis, topological contribute $1,578,000 in 1961 'for educa­ algebra, etc. It is not, however, proposed tional, charitable, religious, and public to include algebraic topology as such in the service organizations. program of the Symposium. The amount, one of the largest in The Organizing Committee is as industry and a new high for the Founda­ follows: J. Novak (Chairman), M. Kura­ tion, is made possible by substantial gifts towski (Vice-chairman), M. Katetov (Vice­ which the Foundation has received from chairman), P. S. Alexandrov (Delegate of Shell Oil Company and its subsidiaries.

125 The largest share of the Founda­ program contributes about $150,000 for tion's 1961 budget, approximately $1,002, 100 four-year scholarships to high school 000, will go towards aid-to-education pro­ students who plan teaching careers in high grams, an increase of more than $100,000 school science and mathematics. over 1960. The Foundation's Shell Fellowships Grants for aid-to-education include will go to 61 outstanding students or young a new program of Shell Merit Residencies teachers doing postgraduate academic for High School Science and Mathematics year or summer work. Single students and Teachers. Through this program, six top married students without children get teachers will spend a minimum of 12 $1, BOO as a personal stipend in connection months' postgraduate study gaining ex­ with academic year fellowships. Married perience in new teaching techniques and students with one child or more will re­ curricula. ceive $2, 100, The student's tuition and Under the new Shell Residencies fees are paid and an additional fund will be program, which will begin this fall, six_ supplied as a cost-of-education supple­ selected teachers will attend Cornell and ment to the department in which he studies. Stanford, three at each. The Residents Colleges and universities not supported by will do advanced work in science and taxes receive a further grant for general mathematics, become acquainted with new administrative use. curricula and teaching materials produced The Foundation's 25 research by national committees on science and grants, each worth $7,500, consist of a mathematics teaching and gain experience $5,000 grant for expanding fundamental in research techniques and supervision. research and $2,500 for any uses the 19 The Foundation will award two institutions receiving grants desire. Insti­ senior Resident awards at $6,000 andfour tutions are given full responsibility and junior Resident awards at $4,600 to cover latitude in selecting regular term and living expenses and will also pay tuition summer fellows and in using funds for and fees. In addition, the Foundation will basic research. One of the research grants, make a grant to Stanford and to Cornell. to the Institute of Mathematical Sciences In addition to these new programs, of New York University, is for research in the 1961 budget will continue from pre­ mathematics. vious years the Foundation's other aid-to­ The Shell Assists consist of $1,500 education programs. These include the grants to each of the approximately 100 Shell Merit Fellowships providing summer participating colleges and universities. seminars at Stanford and Cornell for 100 This money is used by the institutions to outstanding high sc:hool teachers of science help with the "little things" in professional and mathematics; Shell Merit Scholarships development of college faculty members providing four-year college scholarships that are seldom covered by major grants. through the National Merit Scholarship This unique program is now in its third Corporation for 100 students who plan year. careers as teachers of high school chem­ Among other education programs istry, physics, general science, and mathe­ to receive support from Shell Companies matics; 61 graduate-level Shell Fellow­ Foundation, Incorporated, are those ofthe ships and 25 Shell Research Grants at 5.6 Association of American Colleges, the Na­ colleges and universities; the Shell Assists, tional Fund for Medical Education, the Na- a program for furthering the professional tional Science Teachers Association, the development of college teachers at approxi­ Institute of International Education, Incor­ mately 100 institutions. porated, the Council for the Advancement Under the Merit Fellowsh"ip pro­ of Small Colleges, the Council for Inde­ gram, 100 top high school teachers of sci­ pendent School Aid, Incorporated, the ence and mathematics attend summer Council on Social Work Education, the seminars to learn of new advances in their United Negro College Fund, Incorporated, fields and in teaching methods. The Foun­ and the joint Council on Educational Tele­ dation pays tuition and campus living ex­ vision. penses, provides a travel allowance and gives an additional grant of $500 to each The Shell Development Company teacher to offset the loss of potential sum­ supports the activities of the American mer earnings. Mathematic

Dr. M. I.AISSEN,onleavefrom Johns Mr. P. L. CRAWLEY of California Hopkins University, has been appointed a Institute of Technology has been appointed temporary member at the Institute of to a research assistant professorship at Mathematical Sciences, at New York Uni­ the University of Washington. versity. Dr. E. H. CRISLERofBendixAviation Assistant Professor M. W. AL­ Corporation, South Bend, Indiana, has ac­ DHAHIR, on leave from the University of cepted a position as head of the mathemat­ Baghdad, will spend the academic year ics staff ballistic missile laboratories at 1960-1961 as a post-doctorate fellow of Hughes Aircraft Company, Culver City, the National Research Council of Canada. California. Dr. B. H. BARNES of Michigan State Assistant Professor R. S. DEZUR of University has accepted a position as the University of Wyoming has been ap­ senior research engineer at the Jet Pro­ pointed to an assistant professorship at pulsion Laboratory, Pasadena, California. San Diego State College. Dr. J. S. BENDAT, on leave from the Mr. R. B. DOGGETT of the Air Force Thompson Ramo Wooldridge Corporation, Cambridge Research Center, Bedford, Los Angeles, California, has returned Massachusetts, has accepted a position as after spending the academic year 1959· research specialist at Boeing Airplane 1960 at the University of Southern Califor~ Company, Seattle, Washington. nia. Miss C. A. FLANAGAN, on leave Dr. N. A. BRIGHAM of Baird-Atomic from the Naval Ordnance Test Station, Incorporated, Cambridge, Massachusetts, China Lake, California, will spend the aca~ has accepted a position as staff member demic year 1960-1961 at the Universityof at Massachusetts Institute of Technology, Oregon. Lincoln Laboratory. Mr. G. M. FLEISCHNER of Brooklyn, Mr. J. R. BRUGGEMAN of the Thiokol New York, has accepted a position as com­ Chemical Corporation, Brigham City, Utah, puter programmer and research engineer has accepted a position as design special... at the General Dynamics Corporation, San ist at the Martin Company, Denver, Colo­ Diego, California. rado. Associate Professor A. FRIEDMAN Dr. D. R. CHILDS of Allied Research of the University of Minnesota has been Associates, Boston, Massachusetts, has appointed to a visiting associate profes­ accepted a position as principal research sorship at Stanford University for the aca­ scientist at Avco Everett Research Labo­ demic year 1960-1961. ratories, Everett, Massachusetts. Associa.te Professor J. R. GARRETT Assistant Professor P.M. CHIRLIAN of Georgia Institute of Technology has ac .. of New York University has been appointed cepted a position as manager of mathe­ to an associate profe·ssorship at Stevens matical services at the Radio Corporation Institute of Technology. of America Service Company, Patrick Air Professor S. CHOWLA, on leave from Force Base, Florida. the University of Colorado, has been ap­ Dr. D. C. GERNETH of Temco Air­ pointed to a visiting professorship at the craft Corporation, Dallas, Texas, has ac­ University of Notre Dame for the academic cepted a position as assistant technical year 1960-1961. manager at Schlumber Limited, Houston, Dr. C. W. CLARK of the University of Texas. California, Berkeley, has been appointed Mr. H. A. GINDLER of the University to an assistant professorship at the Uni .. of California, Los Angeles, has been ap­ versity of British Columbia. pointed to an assistant professorship at Mr. C. C. CONLEY ofMassachusetts San Diego State College. Institute of Technology has been a{lpointed Dr. G. H. GOLUB of the Universityof a temporary member of the Institute of California, Berkeley, has accepted a posi­ Mathematical Sciences, New York Univer• tion as member of the technical staff at sity. the Space Technology Laboratories, Los

1Z7 Angeles, California. Dr. M. W. KATZ of the University of Professor M. 0. GONZALEZ, on leave British Columbia has been appointed to an from the University of Cuba, has been ap­ assistant professorship at the University pointed to a visiting professorship at the of Wisconsin. University of Alabama. Dr. M. L. KEEDY of the University of Dr. B. GRUNEBAUM of Shell Oil Maryland has accepted a position as super­ Company, Houston, Texas, has accepted a visor of Junior High School mathematics position as consulting mathematician at and science at the Baltimore County Board Boeing Airplane Company, Renton, Wash­ of Education, Towson, Maryland. ington. Dr. E. P. KELLY, JR. of Oklahoma Mr. J. L. HAMMERSMITH of the Na­ State University has been appointed to an val Research Laboratory, Washington, D. assistant professorship at Stephen F. C., has accepted a position as mathemati­ Austin State College. cian at the National Aeronautics and Space Associate Professor M. A. KER­ Administration, Greenbelt, Maryland. VAIRE, on leave from New York Univer­ Dr. M. E. HARRIS of Harvard Univer­ sity, has returned after spending the aca­ sity has been appointed to an assistant demic year 1959-1960 at thelnstitut des professorship at Clark University, Hautes, Paris, France. Dr. C. A. HARVEY of the University Dr. J. M. KISTER, on leave from the of Minnesota has accepted a position as University of Michigan, will spend the aca­ senior research scientist at Minneapolis demic year 1960-1961 at the University of Honeywell Regulator Company, Minnea­ Virginia as a research fellow. polis, Minnesota. Dr. A. KORANYI of Harvard Univer­ Dr. L. 0. HEFLINGER of the Space sity has been appointed to an assistant Technology Laboratories, Los Angeles, professorship at the University of Califor­ California, has accepted a position as nia, Berkeley. mathematical scientist at the Clauser Assistant Professor D. KOVAL of Technology Corporation, Torrance, Cali­ Atlantic Union College has been appointed fornia. to an associate professorship at Washing­ Assistant Professor S. HELGASON, ton Missionary College. on leave from Massachusetts Institute Professor G. L. KRABBE, on leave of Technology has returned after spending from Purdue University, will spend the the academic year 19 59-19 60 at Columbia academic year 1960-1961 at Yale Univer­ imiversity. sity. Assistant Professor B. W. HELTON Professor M. Z. KRZYWOBLOCKI of of the University of Utah has been appoint­ the University of Illinois has been appoint­ ed to an associate professorship at South­ ed to a professorship at Michigan State west Texas State College. University. Professor L. A. HENKIN of the Uni­ Assistant Professor R. S. LEDLEY of versity of California, Berkeley, has been George Washington University has ac­ appointed to a visiting professorship at cepted a position as president of the Na­ Dartmouth College for the academic year tional Biometrical Research Foundation, 1960-1961. Silver Spring, Maryland. Dr. H.-T. HOof Brown University has Mr. S. R. LENIHAN of the University accepted a position as research scientist of California, Livermore, has accepted a at Hydronautic s Incorporated, Rockville, position as mathematician at General Maryland. Atomic, San Diego, California. Dr. G. KARRER of the University of Dr. H.-c. LIU of Los Angeles, Califor­ Michigan has been appointed to an assist­ nia, has accepted a position as senior pro­ ant professorship at the University of grammer at the Minneapolis-Honeywell Bonn, Bonn, Germany. Regulator Company, Wellesley Hills, Mr. I. J. KATZ of Operations Re­ Massachusetts. search Incorporated, Silver Spring, Mary­ Professor G. S. S. LUDFORD of the land, has been appointed a graduate assist­ University of Maryland has been appointed ant at the University of Maryland. to a visiting professorship at Brown Uni-

128 versity. .!l fellow at the University of Leiden, Dr. j. McKENNA of Princeton Uni­ Holland. versity has accepted a position as member Professor G. PALL, on leave from of the technical staff at Bell Telephone lllinois Institute of Technology has been Laboratories, Murray Hill, New jersey. appointed to a visitingprofessorship at Mrs. A. K. MACEDO of johns Hopkins the University of Arizona for the spring University has accepted a position as semester of 1961. mathematician at the Naval Research Dr. S. PARK of Columbia University Laboratory, Washington, D. C. has been appointed to an assistant pro­ Mr. W. D. MAURER of the University fessorship at Rutgers, The State Univer­ of California, Berkeley, has accepted a sity. position as associate scientist for Lock­ Dr. R. L. PESKIN of Princeton Uni­ heed Aircraft Corporation, Palo Alto, versity has been appointed to an assistant California. professorship at Rutgers, The State Uni­ Dr. Z. A. MELZAK of Bell Telephone versity. Laboratories, Murray Hill, New jersey, Professor A. A. M. Rodrigues, on has been appointed to an assistant pro­ leave from the University of Sao Paulo, fessorship at the University of British has been appointed a visiting member at Columbia, Vancouver, British Columbia, the Institute for Advanced Study. Canada. Dr. D. RUTOVITZ of the University Dr. M. V. K. MEN ON of the University of California, Berkeley, has been appoint­ of North Carolina has accepted a position ed a lecturer at the University, Man­ as associate mathematician with Inter­ chester, England. national Business Machines Corporation, Mr. B. L. SCHWARTZ of Technical San jose, California. Operations Incorporated, Oahu, Hawaii, Dr. J, M. MILLER of Esso Research has accepted a position as manager of sys­ and Engineering Company, Madison, New tems analysis at the Laboratory for Elec­ jersey, has accepted a position as re­ tronics, Monterey, California. search engineer with the International Dr. R. J. SEMPLE of Carleton Uni­ Business Machines Corporation, New York, versity has been appointed to an associate New York. professorship at the University of Water­ Dr. J, W. MOELLER of New York loo, Waterloo, Ontario, Canada, University has accepted a position as Dr. L. H. SESHU of Toronto, Canada, member of the technical staff at Bell has been appointed to an assistant pro­ Telephone Laboratories, Whippany, New fessorship at Harpur College. jersey. Professor K-s. SHIH, on leave from Associate Professor J, MYHILL of the National Taiwan University, Formosa, the University of California, Berkeley, has returned after spending the academic has been appointed to a professorship at year 1959-1960 at the California Institute Stanford University. of Technology. Associate Professor E. 0. NELSON of Associate Professor M. C. SHOLAN­ the University of North Dakota has been DER of Carnegie Institute of Technology appointed to an assistant professorship at has been appointed to a professorship at the University of Utah. Western Reserve University. Associate Professor E. D. NERING of Associate Professor D. R. SHREVE the University of Arizona has been ap­ of Oklahoma State University has been pointed to a professorship at Arizona State appointed to an associate professorship University. in charge of the International Business Assistant Professor R. S. NOVOSAD Machines Laboratory at North Carolina of Pennsylvania State University has ac­ State College. cepted a position as associate research Mr. H. B. STANTON of Ginn and Com­ scientist, at the Martin Company, Denver, pany, Boston, Massachusetts, has accepted Colorado, a position as senior editor at Addison­ Associate Professor R. s. B. ONG, on Wesley Publishing Company, Incorporated, leave from the University of Michigan, Reading, Massachusetts. will spend the academic year 1960-1961 as Dr. R. H. SZCZARBA of De Paul Uni-

129 versity has been appointed a research W. E. KOSS, Louisiana Polytechnic associate at Yale University. Institute, to a professorship. Assistant Professor R. N. TOWN­ J. W. MILNOR, Princeton University, SEND of San Jose State College has been to a professorship. appointed to an assistant professorship P. E. NEY, Cornell University, to an at Bowling Green State University. assistant professorship. Dr. T. TSANG of the University of F. J, PALAS, Southern Methodist Uni­ Chicago has accepted a position as assist­ versity, to an associate professorship. ant scientist for the Argonne National G. S. QUINN, American University, Laboratory, Argonne, lllinois. to an associate professorship. Mr. R. J, TUR YN of Newton Centre, H. ROSSI, Princeton University, to an Massachusetts, has accepted a position as assistant professorship. branch leader for Litton Systems Incorpo­ A. TULLER, Hunter College, to an rated, Newton Centre, Massachusetts. associate professorship. J, P. VAN ALSTYNE, Hamilton Col­ The following promotions are announced: lege, Syracuse University, to an associate R. W. BALL, Auburn University, to professorship. a professorship. The following appointments to instructor­ D. R. BAY, illinois State Normal Uni­ ships are announced: versity, to a professorship. D. J. BENNEY, Massachusetts Insti­ Columbia University: Dr. J. C. TAY­ tute of Technology, to an assistant pro­ LOR; Middlebury College: Mr. E. W. fessorship. CHAMBERLAIN; Rutgers, The State Uni­ J. A. BERTON, Indiana State Teach­ versity: Mr. R. T. BUMBY, Dr. D. R. ers College, toanassistantprofessorship. OSTBERG; University of British Colum­ R. J. BICKEL, Drexel Institute of bia: Dr. D. R. REARICK; University of Technology, to a professorship. California, Berkeley: Dr. G. I. HELLER. A. D. BRADLEY, Hunter College, to Deaths: an associate professorship. J. L. BROWN, Pennsylvania State Uni­ Professor emeritus WILLIAM E.AN­ versity, to a professorship. DERSON, died on December 3, 1960, at the J. C. E. DEKKER, Rutgers, The State age of 85, He had been a member of the University, to an associate professorship. Society for 37 years. P. DWINGER, Purdue University, to Professor emeritus ERIC T. BELL, a professorship. died on December 21, 1960, at the age of J. E. EATON, Queens College, to a 77. He had been a member of the Society professorship. for 44 years. G. FREILICH, City College, to an Mr. K. F. SIMPSON, Jr., died in associate professorship. October 1960, at the age of 32, He had E. GROSSWALD, University of Penn­ been a member of the Society for 4 years. sylvania, to a professorship. P. HENRICI, University of California, ERRATUM Los Angeles, to a professorship. The announcement on page 843 of the S. P. HOFFMAN, Trinity College, to December 1960 issue of the NOTICES con­ an associate professorship. cerning Associate Professor S. I. GOLD­ E. R. JOHNSTON, Purdue University, BERG of Wayne State University should to a professorship. read as follows: Associate Professor S. R. R. D. KEMP, Queens University, I. GOLDBERG, on leave from Wayne State Kingston, Ontario, Canada, to an associ­ University, has been appointed to a visit­ ate professorship. ing associate professorship at the Univer­ E. KLEINFELD, Ohio State Univer­ sity of lllinois for the academic year 1960- sity, to a professorship. 1961.

130 LETTERS TO THE EDITOR

Editor, the NOTICES rhage. The loss has been a severe shock to all mathematicians and friends who have As a recent letter (NOTICES, Novem~ known and worked with him. ber 1960, p. 710) pointed out, modern The all too early death of H. Yamabe methods of reproducing manuscripts are has had the additional tragic feature, that, cheap and efficient. We suggest taking owing to his recent change of employment, fuller advantage of this fact. Specifically, and to a technicality in some group in­ we suggest that when an author is pre­ surance policies, his family has found it­ pared to supply copies of an unpublished self deprived of any life insurance bene­ paper on request, the abstract should be fits. This circumstance will inevitably accompanied by an appropriate symbol create some difficulties for his family, to indicate availability of copies. More­ especially at the time when his two daugh­ over, since the annual list of members ters, Kimiko and Noriko, now 6 and 3 years now appears so late as to be useless, we old, will reach college age. For this reason suggest that abstracts in the NOTICES of a group of people at the two universities papers for which copies are available with which H. Yamabe had been associated (and preferably all abstracts) should carry in the last year of his life, Minnesota and the author's current mailing address. Northwestern, have decided, with Mrs. Finally, we suggest that, before submitting Yamabe' s consent, to ask all mathemati­ a paper for publication, the author should cians who loved Yamabe as a person and wait to receive comments on the prelimi­ who admire his achievements in our field nary version. In this way many improve­ of work to contribute to a fund now estab­ ments in the papers might develop and, lished for the education of Kimiko and in extrem~ cases, the need for publication Noriko Yamabe. might even evaporate. All contributions may be sent to the R. P. Boas Yamabe Memorial Fund in care of either A. Rosenberg Professor R. P. Boas, Chairman, Mathe­ matics Department, Northwestern Univer­ sity, Evanston, Illinois, or Professor Ediror, the NOTICES S. E. Warschawski, Head, Department of For some months I have unsuccess­ Mathematics, Institute of Technology, Uni­ fully sought, through the usual used book versity of Minnesota, Minneapolis 14, sources, copies of Volumes I and II of Minnesota, Hilbert and Bernays, "Grundlagen der Eugenio Calabi Mathematik". I should like to suggest that R. P. Boas the NOTICES could perform a useful ser­ vice for the mathematical community by publishing requests of this type. Many of us tend to collect out-of-print books which Editor, the NOTICES have ceased to serve our purposes and T. F. Bridgland's objections to the would be happy to see them go to someone handling of the letters to the editor depart­ with a current need. ment is based on a very narrow interpreta­ Clement F. Kent tion of "interest to the membership." One cannot quarrel with his presumption that the interest referred to is professional in­ Editor, the NOTICES terest, but what is professional interest? It is of course possible to interpret it to TO ALL FRIENDS AND ADMIRERS OF include only technical matters without any THE LATE HIDEHIKO YAMABE regard to the place of the profession in On November ZO, 1960, Professor society and its social responsibilities, or Hidehiko Yam abe, age 37, died suddenly in even without regard to the solidarity of the Evanston, Illinois, of a cerebral hemor- members of the profession. The experi-

131 ence of the last few decades, and especially the experience of the German mathemati­ cians, showed how dangerous this attitude CUSHING-MALLOY, is. The mathematician who cares nothing for his fellow scientists or for his country INC. or the world, and t:he scientist who cares but who puts these concerns in a separate 1350 N. Main St., compartment unconnected with his scien­ P. 0. Box 1187 tific work, does not: in my opinion deserve Ann Arbor, Michigan the name of professional. Mathematics is playing an increasingly important role in political affairs in t:he broadest sense, and LITHO PRINTERS there is considerable evidence that the members of the American Mathematical Printers of the NOTICES Society are aware of their responsibilities and realize that in order to discharge them provision must be made for free discus­ sion. Of course the price of freedom of Known for speech is the expression of opinions that QUALITY - ECONOMY one finds personally distasteful. But if Mr. Bridgland wishes to have available a SERVICE means by which he can express his opin­ ions, he must support the availability of the same means to those with whom he Let us quote on your next does not agree. I hope that the editors will printing continue their policy, which is in the best traditions of American mathematics and American democracy. Kenneth 0. May

Editor's Note: Besides the above letter, 6eeme7l-t~ of­ there were letters from Arnold G. Rawling and Patrick Shanahan concurring with the opinion expressed by T. F. Bridgland, and a letter from H. W. Gould opposing this ;;/ ,·near J,a.cqs opinion. by fl. R. Ami.r-MoeJ', A·L·F•ss

a ,ery a"ttra.c:r,· v e

boiJJ( J oney .$ 5". 00 CORRECTION TO "CATALOGUE OF LEC­ TURE NOTES" ~,. L·, fo,. ma.t ,;,,., On page 870 of the December 1960 issue (volume 7, no. 7) the first book listed wr i. t e to 4. ,, r" 0,. s 0 ,. under University of Washington should read CJ...,tJt·J.s f3rotlter-1 1 Jnc. EDWIN HEWITT, Theory of Functions of a Complex Variable, 405 pp. $3.50. ann. d.,." 0 r/ lJt t.'ct, cj4.n.

132 NEW PUBLICATIONS

Adler, R. B. See Haus, H. A. Davenport, V. B., and Rut, V. L. [Root, Amelin, V. M. See Zongolovic, I. D. W. L.] Vvedenie v teoriyu slucatnyh Analytic functions. (Princeton Mathemati- signalov i sumov. (An intro uction to cal Series, No. 24.) Princeton Univer­ the theory of random signals and noise.] sity Press, 1960. 7 + 197 pp. $5.00, Trans. by B. G. Belkin; ed. by R. L. Andreeva, I. M. See Bellman, R. Dobrusin, Moscow, Izdat. lnostr. Lit., Artin, E. See van der Waerden, B. L. 1960. 468 pp. 20.25 r. Arzelies, H. Milieux conducteurs ou po- Dec, G. [Doetsch, G.] Rukovodstvo k larisables en mouvement. With the prakticeskomu primeneniyu preobra­ assistance of j. Henry, (Etudes Re­ zovaniya Laplasa. [Anleitung zum lativistes.) Paris, Gauthier- Villars, praktischen Gebrauch der Laplace­ 1959. 44 + 347 pp. 58 NF; $12.12, Transformation.] Trans. by G. A. Baghdady, E. j. See Lectures on communi­ Vol'pert. 2d ed. Moscow, Gosudarstv. cation system theory. lzdat. Fiz.- Mat. Lit., 1960. 207 pp. Bakut, P. A. See Val'd, A. 5.60 r. Easson, A. H., and O'Connor, D. j. Intro­ Deming, W. E. Sample design in business duction to symbolic logic. Glencoe, research. (A Wiley Publication in Ap­ The Free Press, 1960. 7 + 17 5 pp. plied Statistics.) New York, Wiley, $3.00, 1960. 20 + 517 pp. $12,00. Belkin, B. G. See Davenport, V. B. Deuxiemes journees internationales de Bellman, R. Adaptive control processes: calcul analogique, Strasbourg 1-6 A guided tour. Princeton University September 1958, Actes. Brussells, Press, 1961. 16 + 255 pp. $6.50. Presses Academiques Europeennes, Bellman, R. Dinamiceskoe programmi­ 1959. 12 + 501 pp. 158 s.; $22.00, rovanie. [Dynamic programming.] Dicke, R. H., and Wittke, J, P. Trans. by I. M. Andreeva,A. A. Korbut, Introduction to quantum mechanics. I. V. Romanovskit, I. N. Sokolova; ed. Reading, Addison-Wesley, 1960. 11 + by N. N. Vorob'ev. Moscow, lzdat. 369 pp. $8.7 s. Inostr. Lit., 1960. 400 pp. 17,60 r. Dobrusin, R. L. See Davenport, V. B. Bol'sev, L. N. See van der Varden, B. L. Downs, B. W. See Lectures in theoretical Booth, A. D. See Progress in automation. physics. Borsuk, K., and Szmielew, W. Foundations Durell, C. V. Readable relativity. Fore­ of geometry: Euclidean and Bolyai­ word by F. j. Dyson. (Harper Torch­ Lobachevskian geometry; projective books/The Science Library.) New York, geometry. Rev. English translation. Harper, 1960. 11 + 146 pp. $1.25. Amsterdam, North-Holland; New York, Dyson, F. j. See Durell, C. V. lnterscience; 1960. 14 +444pp. $12.00, Ellenberg, S. See Kartan, A. Boruvka, 0. Grundlagen der Gruppoid­ Electronic business machines. Ed. by und Gruppentheorie. (Hochschulbiicher j. H. Leveson. New York Philosophical fiir Mathematik, Bd. 46,) Berlin, VEB Library, 1960. 272 pp. $15,00. Deutscher Verlag der Wissenschaften, Euler, L. Vollstandige Anleitungzur Alge­ 1960. 12 + 198 pp. DM 28,20, bra, With the cooperation of J, Brinch, H. j., and Lundgren, H. Haupt­ Niessner in a rev. ed. newly ed. by probleme der Bodenmechanik. Berlin, j. E. Hofmann. Stuttgart, Reclam­ Springer, 1960. 12 + 282 pp. DM36.00, Verlag, 1959. 571 pp. DM 18,50. Brittin, W. E. See Lectures in theoretical Favar, Z. [Favard, J.) Kurs lokal'nor physics. differencial'no'r geometrii. [Cours de Cartan, H. See Kartan, A. geometrie differentielle locale.]Trans. Cheema, M. S. See Representations of by Yu. A. Rozanskaya and S. P. Finikov; primes •.. preface by S. P. Finikov. Moscow, Collin, R. E. Field theory of guided waves. Izdat. lnostr. Lit., 1960. 559 pp. 25.2 5 r. (International Series in Pure and Ap­ Fierz, M. See Theoretical physics •. plied Physics.) New York, McGraw­ Finikov, S. P. See Favar, Z. See also Hill, 1960. 13 + 606 pp. $16,50. Lihnerovic, A. 133 Fizika tverdogo tela. [Solid state physics.] of articles ed. by A. I. Markullevic. Collection of articles. Moscow, Akad. Moscow, Gosudarstv. lzdat. Fiz,-Mat. Nauk SSSR, Otd. Fiz.-Mat. Nauk. Izdat. Lit., 1960. 544 pp. 18.05 r. Akad. Nauk SSSR, 1959. Vol. I: Z98 pp. Ivall, T. E. Electronic computers: Princi­ ZO r. Vol. II: 3Z8 pp. ZZ r. ples and applications. London, Iliffe; Flugge, s. See Handbuch der Physik. New York, Philosophical Library; 1960, Freudenthal, H. Lincos: Design of a lan­ 8 + Z63 pp. $15,00, guage for cosmic intercourse. Part I. jones, H. The theory of Brillouin zones (Studies in Logic and the Foundations and electronic states in crystals. (Ser­ of Mathematics.) Amsterdam, North­ ies in Physics.) Amsterdam, North­ Holland, 1960, 5 + ZZ4 pp. $6,00. Holland; New York, Inter science; 1960, Gerasimov, B. M. See Val'd, A. 9 + Z68 pp. $9.50. Gibbs, J. W. Elementary principles in Kammerer, W. Ziffernrechenautomaten. statistical mechanics: developed with Zd ed. (Elektronisches Rechnen und especial reference to the ration~! Regeln, Vol, 1.) Berlin, Akademie• foundation of thermodynamics. New Verlag, 1960. 8 + 303 pp. DM Z9 .00. York, Dover, 1960. 18 + Z07 pp. $1.45. Kartan, A. and Enenberg, S. [Cartan, H.] Glass, I. 1., and Hall, j. G. Handbook of Gomologiceskaya algebra. [Homologi­ supersonic aerodynamics. Section 18. cal algebra.] Trans. by E. G. Sul'ge'i.'fer; Shock tubes. (Navord Report 1488, ed, by M. M. Postnikov. Moscow, lzdat. Vol. 6.) Washington, D. C., U. S. lnostr. Lit., 1960. 510 pp. ZZ r. Government PrintingOffice, 1959. 38 + Kash, S. W. See Plasma acceleration. 604 pp. (not consecutively paged). $3.7 5. Kelley, J. L. Introduction to modernalge­ Golubev-Novo:lilov, Yu. S. See Vil'yams, bra. (The University Series in Under­ D:l. D. graduate Mathematics.) Princeton, Van Gupta, H. See Representations of primes. Nostrand, 1960, 10 + 338 pp. $Z.75. Gupta, 0. P. See Representations of King, R. W. P., and Wu, T. T. The scat­ primes •.• tering and diffraction of waves. (Har­ Hall, j. G. See Glass, I. I. vard Monographs in Applied Science, Handbuch der Physik. Vol. III/1. Prinzi­ No. 7 .) Cambridge, Harvard University pien der klassischen Mechanik und Press, 1959. Z18 pp. $6.00, Feldtheorie. Ed. by S. Fliigge. Berlin, Korbut, A. A. See Bellman, R. Springer, 1960, 7 + 90Z pp. DM 198.00. Kuriksa, A. A. See Val'd, A. Haselgrove, C. B., and Miller, J. C •. P .. Kuznecov, I. N. See Val'd, A. Tables of the Riemann zeta function. Lamb, Horace, Dynamics. New York, (Royal Society Mathematical Tables, Cambridge University Press, 1960, Vol. 6.) New York, Cambridge Univer­ 11 + 351 pp. $3.75. sity Press, 1960. Z3 + 80 pp. $9,50. Lebedev, S. A., and Mel'nikov, V. A. Haus, H. A., and Adler, R. B. Circuit ObScee opisanie BESM i metodika theory of linear noisy networks. (Tech­ vypolneniya operacir. (General descrip­ nology Press Research Monographs.) tion of the BESM and methods of carry­ Cambridge, The Technology Press of ing out operations.] (Elektronnaya Ci­ M. I. T.; New York, Wiley; 1959. frovaya Vycislitel'naya Mallina BESM, 1Z + 79 pp. $4.50. I.) Moscow, Gosudarstv. lzdat. Fiz.­ Hausmann, B. A. From an ivory tower: Mat. Lit., 1959, ZIO pp. 7 r. A discussion of philosophical problems Lectures in theoretical physics. Vol, II. originating in modern mathematics. Lectures delivered at the Summer In­ Milwaukee, Bruce, 1960. 7 + 1ZZ pp. stitute for Theoretical Physics, Uni­ $3.50. versity of Colorado, Boulder, 1959. Ed. Henry, J, See Arzeli~s, H. by W. E. Brittin and B. W. Downs. New Hofmann, j. E. See Euler, L. York, Interscience, 1960, 7 + 483 pp. Issledovaniya po sovremennym proble- $9,00, mam teorii funkcii kompleksnogo pere­ Lectures on communication system theory. mennogo. [Investigations in contempo­ Ed. by E. J, Baghdady. New York, Mc­ rary problems of the theory of func­ Graw-Hill, 1961, 1Z + 617 pp. $1Z,50. tions of a complex variable.] Collection Ledley, R. S. Digital computer and control

134 engineering. Written with the assist­ Miller, J, C. P. See Haselgrove, C. B. ance of L. S. Rotolo and J. B. Wilson. See also Representations of primes ... (McGraw-Hill Electrical and Electron­ Millman, J., and Taub, H. Pulseanddigital ic Engineering Series.) New York, Mc­ circuits. (McGraw-Hill Electrical and Graw-Hill, 1960. 24 + 835 pp. $14.50. Electronic Engineering Series.) New Lee, Y. W. Statistical theory ofcommuni­ York, McGraw-Hill, 19 56, 16 + 687 pp. cation. New York, Wiley, 1960, 17 + $12,50. 509 pp. $16.75. Nalimov, V. V. Primenenie matematices­ Leveson, J, H. See Electronic business ko1 statistiki pri analize ve!ICestva. machines. [Application of mathematical statistics Levi, E. E. Opere. With the assistance to analysis of matter.J (Fiziko-Mate­ of the Unione Matematica Italiana and maticeskaya Biblioteka In::tenera.) with contributions from the Consiglio Moscow, Gosudarstv. lzdat. Fiz,-Mat, Nazionale delle Ricerche. 2 Vols. Lit., 1960,430pp.12,95r. Rome, Edizioni Cremonese, 1959, 1960. Nemirov, Yu. G. See Meter cur Kapellen, Vol. 1: 19 + 418 pp. {1 plate) L. 4500. v. Vol. II: 448 pp. L. 4500. Newman, J. R. See The world of mathe­ Lihnerovic, A. Teoriya svyaznostel' v matics. celom i gruppy golonomir. [Theorie Newton, I. The correspondence of Isaac globale des connexions et des groupes Newton. Vol. II: 1676-1687. Ed, by d'holonomie.] Trans. by S. P. Finikov; H. W. Turnbull. Pub. for the Royal So­ ed. by V. V. Ry:lkov. Moscow, Izdat. ciety. New York, Cambridge University lnostr. Lit., 1960. 216 pp. 9.20 r. Press, 1960. 13 + 552 pp. (7 plates) Linnik, Yu. V. Raz1o:leniya veroyatnostnyh $25,00. zakonov. Q?' actorization of probability Nidditch, P. H. Introductory formal logic laws.] Leningrad, lzdat. Leningrad. of mathematics. Glencoe, The Free Univ., 1960, 263 pp. 1,38 r. Press, 1960. 7 + 188pp. $3.00. Lucra1rile ce1ui de al IV-lea Congres al Niessner, J. See Euler, L. Matematicienilor Romlni. [Proceed­ Noether, E. See van der Waerden, B. L. ings of the 4th Congress of Romanian O'Connor, D. j. See Easson, A. H. Mathematicians.] Bucharest, Editura Pal'tov, I. P. See Popov, E. P. Academiei Republicii Populare Romine, Pauli, W. See Theoretical physics ... 1960. 227 pp. Lei 8,55. Peano, G. Formulario mathematico. Re- Lundgren, H. See Brinch, H. J, productions in facsimile from the ori­ Lyapin, E. S. Polugruppy. [Semigroups.] ginal edition. With introduction and Moscow, Gosudarstv. lzdat. Fiz,-Mat. notes by U. Cassina and with contribu­ Lit., 1960. 59Z pp. 17.40 r. tions from the commune di Cuneo. Marku!levie!, A. I. See Issledovaniya .. . Rome, Edizioni Cremonese, 1960, 48 + Mehta, A. See Representations of primes .. . 36 + 465 pp. {16 plates) L. 5000. Meier cur Kapellen, V. [Meyer zur Cap- Pearson, K. See Todhunter, I. ellen, W.] lnstrumental'naya matema­ Plackett, R. L. Principles of regression tika dlya in:lenerov. [Instrumentelle analysis. Oxford, Clarendon, 1960. 9 t Mathematik fur den lngenieur .] Trans. 173 pp. $5.60. by K. P. Vi!lnevskil and Yu. G. Nemirov; Plasma acceleration. Ed. by S. W. Kash. ed. by V. V. Vasmanov. (Fiziko-Mate­ 4th Lockheed Symposium on Magneto­ maticeskaya Biblioteka In:lenera.) hydrodynamics. Stanford University Moscow, Gosudarstv. lzdat. Fiz.-Mat. Press, 1960, 7 + 117pp. $4.25, Lit., 1959. 379 pp. 14.15 r. Pogorelov, A. V. Nekotorye voprosy teorii Mel'nikov, V. A. See Lebedev, S. A. poverhnostei v ellipticeskom prostran­ Mesarovic, M. D. The control of mul-ti-· stve. [Topics in the theory of sur­ variable systems. (Technology Press faces in an elliptic space.] Kharkov, Research Monographs.) Cambridge, lzdat. Har'kov. Gos. Univ., 1960. 9Z pp. The Technology Press of M. I. T .; 4,25 r. New York, Wiley; 1960. 11 + 112 pp. .Poletaev, I. A. See Vil'yams, Dz. D. $3.50, Popov, E. P., and Pal'tov, I. P. Pribli-

135 :lennye metody issledovaniya nelinei:nyh Berlin, Springer, 19 59. 15 + 455 pp. avtomati~eskih sistem. [Approximate DM 52.50, methods of investigation of nonlinear Schlichting, H., and Truckenbrodt, E. automatic systems.] Moscow, Gosud­ Aerodynamik des Flugzeuges. Vol. 2: arstv. Izdat. Fiz.-Mat. Lit., 1960. 792 Aerodynamik des Tragfliigels (Part ll), pp. 25.20 r. des Rumpfes, der Fliigel-Rumpf~nord• Postnikov, M. M. See Kartan, A. nungen und der Leitwerke. Berlin, Proceedings of the 1959 Computer Appli­ Springer, 1960. 16 + 485 pp. DM 61.50, cations Symposium. Chicago, Armour Seitz, F. See Solid state physics. Research Foundation of illinois Insti­ Sevast'yanov, B. A. See Val'd, A. tute of Technology, 1960, 10 + 155 pp. Sierpinski, W. 0 rozwiazywaniu r6wnan $3.00. w liczbach ca):'kowitych. [Solutions of Progress in automation. Vol. l. Ed. by equations in integers.] (Monografie A. D. Boot)l. New York, Academic Popularnonaukowe: Matematyka.) War­ Press, 1960, 8 + 231 pp. $8.50. saw, Panstwowe Wydawnictwo Naukowe, Puga~ev, V. S. Teoriya slu~a1nyh funkci1, 1956. 109 pp, z):'. 3.40. i ee primenie k zadacam avtomati~es­ Smirnov, N. V. See van der Varden, B. L. kogo upravleniya. [fheory of random Sokolova, I. M. See Bellman, R. functions and its application to prob­ Solid state physics: Advances in research lems of automatic control.] 2d ed. and applications. Ed. by F. Seitz and rev. and supplemented. Moscow, Go sud· D. Turnbull. New York, Academic arstv. Izdat. Fiz.-Mat. Lit., 1960, 883 Press. Vol. 7: 1958. 14 + 525 pp. $12,00, pp. 32.40 r. Vol. 8: 1960. 14 + 519 pp. $13.50, Vol. Representations of primes by quadratic 11: 1960. 16 + 438 pp. $12.50, forms: displaying solutions of the Dio­ Sul'geifer, E. G. See Kartan, A. phantine equation kp = a + Db • Part Synge, J. L. Relativity: The general theory. I: D = 5,6,10, and 13, (Royal Society (Series in Physics.)Amsterdam,North­ Mathematical Tables, Vol. 5,) Ed. by Holland; New York,lnterscience; 1960, J. C. P. Miller. Prepared by H. Gupta, 15 + 505 pp. $16,50, M. S. Cheema, A. Mehta and 0. P. Szmielew, W. See Borsuk, K. Gupta, New York, Cambridge Univer­ Takacs, L. Stochastic processes: Prob­ sity Press, 1960. 24 + 135pp, $8.50, lems and solutions. Trans. by P. Zador. Rey Pastor, J, Apuntes de teoria de los (Methuen's Monographs on Applied conjuntos abstractos. [Notes on ab­ Probability and Statistics.) London, stract set theory.] San Luis, Instituto Methuen; New York, Wiley; 1960. 11 + de Matematicas, 1957. 2 + 55 pp. 137 pp. 18 s. Rindler, W. Special relativity. (University Taub, H. See Millman, J, Mathematical Texts.) Edinburgh,Oliver Theoretical physics in the twentieth cen­ and Boyd; New York, Interscience; tury: A memorial volume to Wolfgang 1960. 10 + 186 pp. $2.25. Pauli. Ed. by M. Fierz and V. F. Romanovskil, I. V. See Bellman, R. Weisskopf. New York, Interscience, Root, W. L. See Davenport, V. B. 1960, 10 + 328 pp. (1 plate) $10,00, Rotolo, L. S. See Ledley, R. S. Tietjens, 0. Stromungslehre. Physikal­ Rozanskaya, Yu. A. See Favar, Z. ische Grundlagen vom technischen Ryzkov, V. V. See Lihnerovi~. A. Standpunkt. Vol. I: Hydro- und Aero­ Samuel, P. See Zariski, 0. statik. Bewegung der idealen FH:issig­ Sbornik standartnyh i tipovyh programm keit. Berlin, Springer, 1960. 16 + 536 dlya BESM. [Collection of standard and pp. DM 66,00, typical programs for the BESM.] Mos­ Timan, A. F. Teoriya priblizeniya funkci'r cow, Izdat. Akad. Nauk SSSR, 1960. 75 de1stvitel'nogo peremennogo. [Theory pp. 4.10 r. of approximation of functions of a real Schlichting, H., and Truckenbrodt, E. variable.] Moscow, Gosudarstv. Izdat. Aerodynamik des Flugzeuges. Vol. 1: Fiz.-Mat. Lit., 1960. 624 pp. 23.05 r, Grundlagen aus der Stromungsmechanik. Tintner, G. Handbuch der Okonometrie, Aerodynamik des Tragfliigels (Part I). (Enzyklopadie der Rechts- und Staats-

136 wissenschaft.) Berlin, Springer, 1960. Vorob'ev, N. N. See Bellman, R. 11 + 3Z8 pp. DM 48.00. II vsesoyuznaya topologiceskaya konfer­ Todhunter, I. A history of the theory of enciya. [Second all-Union topological elasticity and of the strength of mate­ conference.] 5-10 October, 1959. Ab­ rials, from Galilei to Lord Kelvin, Ed. stracts of reports. Tbilisi, Izdat. Akad. and completed by K. Pearson. New Nauk GSSR, 1959. 53 pp. York, Dover, 1960. Vol. I: 16t936pp.; Wald, A. See Val'd, A. Vol. II, Part I: 13 + 76Z pp. (4 plates); Weisskopf, V. F. See Theoretical physics ..• Vol. II, Part II: 4 + 546 pp. $1Z.50. Williams, J. D. See Vil'yams, Dz. D. Tonelli, L. Opere scelte. Vol. I: Funzioni Wilson, J. B. See Ledley, R. S. di variabile reale. With the assistance Wittke, J. P. See Dicke, R. H. of the Unione Matematica Italiana and The world of mathematics: A small li­ with a contribution from the Consiglio brary of the literature of mathematics Nazionale delle Ricerche. Rome, from A'h-mose the Scribe to Albert Edizioni Cremonese, 1960. 6 + 604 pp. Einstein, presented with commentaries (1 plate) L. 6000. and notes. Ed. by J. R. Newman. 4 vols. Truckenbrodt, E. See Schlichting, H. New York, Simon and Schuster, 1956. Turnbull, H. W. See Newton, I. See also 18 + 7 + 7 + 7+Z, 535 pp. $9.95. Solid state physics •.. Wu, T. T. See King, R. W. P. Val'd, A. Cwald, A.] Posledovatel nyr Zador, P. See Takacs, L. analiz. (Sequential analysis.] Trans. by Zariski, 0., and Samuel, P. Commutative P. A. Bakut, B. M. Gerasimov, I. N. algebra. Vol. II. (The University Series Kuznecov, A. A. Kuriksa; ed. by B. A. iii Higher Mathematics.) Princeton, Sevast'yanov. Moscow, Gosudarstv. Van Nostrand, 1960. 10 +414pp. $7.75. lzdat. Fiz.-Mat. Lit., 1960. 3Z8 pp. Zongolovic, I. D., and Amelin, V. M. 10.15 r. Sbornik tablic i nomogramm dlya obra­ van der Varden, B. L. lVan der Waerden, botki nablyudenit' iskusstvennyh sputni­ B. L. ] Matematiceskaya statistika. kov zemli. [Collection of tables and [Mathematische Statistik.] Trans. by nomograms for processing observa­ L. N. Bol'sev; ed. by N. V. Smirnov. tions of artificial earth satellites.] Moscow, Izdat. Inostr. Lit., 1960. 434 (Institut Teoreticesko'i' Astronomii pp. 19.ZO r. Akademii Nauk SSSR.) Moscow, Izdat. van der Waerden, B. L. Algebra. Part I. Akad. Nauk SSSR,l960.196pp.Z3.40r. Utilizing lectures by E. Artin and E. Noether. 5th ed. of Modernen Algebra. (Die Grundlehren der Mathematischen Wissenschaften, Vol. 33.) Berlin, Springer, 1960. 8 + Z9Z pp. DM ZZ.OO. Vasmanov, V. V. See Meter cur Kapellen, v. Dr. HARRY POLACHEK, Technical Vil'yams, Dz. D. [Williams, J.D.] Sover­ Director of the Applied Mathematics Labo­ senny'i' strateg: ili bukvar' po teorii ratory at the David Taylor Model Basin, strategiceskih igr. [The compleat received the Distinguished Civilian Ser­ strategyst: being a primer on the theory vice Award on November Z3, 1960. This of games of strategy.] Trans. byYu. S. honorary award, the highest that the Sec­ Golubev-Novo~ilov; ed. by I. A. Pole­ retary of the Navy may confer on a civilian taev. Moscow, Izdat. "Sovet. Radio", employee of the Navy, was given in recog~ 1960. Z69 pp. 1.0Z r. nition of Dr. Polachek's achievements in VisnevskH, K. P. See Mel'er cur Kapellen, organizing and directing the Applied Math­ v. ematic Laboratory, in demonstrating the Vol'pert, G. A. See Dec, G. potential of high speed computers and in von Neumann, J. The computer and the his contributions to scientific, technical brain. (Silliman Memorial Lectures, and management fields such as nuclear Vol. 36.) New Haven, Yale University reactor design, advanced programming Press, 1958. 14 + 8Z pp. $4.00. systems, and analysis of acoustic signals.

137 SUPPLEMENTARY PROGRAM No. 3

During the intervalfrom January 4, 1961 through February 13, 1961 the pa­ pers listed below were accepted by the American Mathematical Society for presentation by title. Readers may wish to refer to page 713 of the November, 1960 issue (No. 49) of these NOTICES where it is explained in detail that the presentation of papers by title is now dissociated from meetings of the society. Supplementary program No. 4 will cover the interval from February 14, 1961 through April 21, 1961. After each title on this program is an identifying number. The abstract of the paper will be found following the same number in the section on Abstracts of Con­ tributed Papers in this issue of these NOTICES or in succeeding issues.

(1) On the asymptotic behavior of Toep- (10) Locally compact transformation litz determinants, II groups Professor Glen Baxter, University Professor j. G. Glimm, Massachu­ of Minnesota and Aarhus Universi­ setts Institute of Technology ( 61 T- tet, Denmark (61 T-58) 80) (2) Diffraction of scalar waves by a cir- (11) Semigroups whose set of invariant cular aperture in a planar screen means has finite dimension Mr. E. Granirer, Hebrew Univer­ Professor jack Bazer, New York sity, jerusalem, Israel (61 T-64) University and Professor Harry (Introduced by H. Kesten) Hochstadt, Polytechnic Institute of (12) Integral curves for Pfaffian systems. Brooklyn (61 T-68) Preliminary report (3) On the existence of a bound for a pair Dr. Robert Hermann, Lincoln Lab­ of consecutive quartic residues mod­ oratory, Massachusetts Institute of ulo a prime Technology (61 T-74) Mr. R. Bierstedt, University of (13) On the Lagrange variational problem Notre Dame (61T-61) Dr. Robert Hermann, Lincoln Lab­ (Introduced by S. Chowla) oratory, Massachusetts Institute of (4) A new decomposition formula in the Technology (61 T-73) theory of elasticity ( 14) The trace in finite operator algebras Professor j. H. Bramble and Pro­ Professor R. V. Kadison, Columbia fessor L. E. Payne, University of University (61 T-65) Maryland (61 T-70) ( 15) Real cohomology of fibre spaces (5) An analogue of the spherical har- Mr. D. W. Kahn, Yale University monics for the equations of elasticity (61T-81) Professor j. H. Bramble and Pro­ (16) Occupation times for Markov and fessor L. E. Payne, Universityof semi Markov chains Maryland (61 T-71) Dr. Harry Kesten, Hebrew Univer­ (6) The existence of uncountably many sity, jerusalem, Israel (61 T-77) nonisomorphic nilpotent Lie algebras (17) A problem of Samuel concerning divi­ Mr. Chong- Yun Chao, University of sors of second kind Michigan (61 T-63) Professor H. T. Muhly, University (7) Arithmetical notes, VII. Some classes of Iowa (61T-59) of even functions (mod r) ( 18) The lexicographic product of graphs Professor Eckford Cohen, Univer­ Dr. G. 0. Sabidussi, McMaster Uni­ sity of Tennessee (61 T-72) versity (61 T-66) (8) Fixed point sets of differentiable in- ( 19) On the absolute harmonic summability volutions of fourier series Professor P. E. Conner and Pro­ Professor S. M. Shah, Kansas Uni­ fessor E. E. Floyd, Universityof versity (61 T-62) Virginia (61T-67) (20) A relation between singularities of a (9) On some cubic Diophantine equations three-dimensional harmonic function with no rational solutions and the coefficients of its develop­ Mr. H. M. Edgar, University of ment in spherical harmonics Notre Dame (61 T-60) Dr. Alvin White, University of Santa (Introduced by S. Chowla) Clara (61 T-69)

138 ABSTRACTS OF CONTRIBUTED PAPERS

The February Meeting in New York, New York February 25, 1961

577-14. Leopold Flatto, and A. G. Konheim: The random division of an interval and a problem of Dvoretzky.

Let the unit interval I = (9, 1] be divided into n subintervals by n - 1 points {Pi} placed at ran- dom on I (the {Pi! being independent random variables each uniformly distributed on I). Using geo­ metrical arguments we obtain the distribution functions (d.f.'s) for the kth largest and smallest sub­ intervals as well as the corresponding limiting d.f.' s. These results were previously obtained by Darling (Ann. of Math. Stat., 1953) by the method of characteristic functions. The following problem has been proposed by Dvoretzky (Proc. Nat. Acad. Sci,, 1956). Let arcs LpLz, ••• , of lengths ll,lz, .•• , be placed at random (uniformly and independently) on the boundary of a circle C of unit circumference.

Find necessary and sufficient conditions on [li} such that C is covered by U~ 1 Li (with probability 1). Using the results reported on above we obtain such conditions and an explicit formula for the proba­ bility that Cis covered by Ui~lLr (Received January 13, 1961.)

The April Meeting in New York, New York April 5-8, 1961

578-1. Peter Treuenfels: On bounds to eigenvalues of non-selfadjoint differential equations.

For the eigenvalue problem -pu" + qu' + ru = Asu, u(a) = u(b) = 0, p(x) >0, s(x) ::>0, it is shown that the lowest eigenvalue )I and the lowest eigenvalue })h of a corresponding finite-difference equation

satisfy I)) - Ah I ~ C h, where h is the mesh size and C is a constant independent of h. An upper bound to C is given in terms of b - a, p(x), q(x), r(x), s(x). This extends a previous result !Notices Amer .Math, Soc. val. 7 (1960) p. 358, Abstract 568-39]. (Received December lZ, 1960.)

578-Z. G. J. Minty: Monotone (nonlinear) operators in Hilbert space •

.3( is a (real or complex) Hilbert space, with inner product

139 Kirszbraun's Theorem; see li:ichoenberg, Amer, Math, Monthly val, 60 (1953) pp, 620-622],) (Received J;muary 9, 1961.)

578-3, Thkayuki Thmura: Certain semigroups,

In the journal of Gakugei, Tokushima University, val, 5 (1954) the author determined the struc­ ture of r-semigroups, i,e,, a semigroup whose subsemigroups form a chain. He defines a r-semi­ group to be a semigroup whose every proper subsemigroup is a C-semigroup. In the present paper he has determined all the types of r-semigroups except groups, (1) A r-semigroup is unipotent, (2) Every element of a C*-semigroup is of finite order, (3) A T-semigroup whose unique idempotent is zero is of order < 5. Using these lemmas, any T-semigroup except groups can be completely determined. (Received january 16, 1961,)

578-4, Edward Batschelet: Testing hypotheses in human genetics, Preliminary report,

Assume the disposition for a human disease is inheritable, but the age of onset is not restricted to childhood, Then a statistical method to test a genetic hypothesis is difficult. Earlier attempts failed. A new method is presented to prove an incomplete dominant or an incomplete recessive hypothesis. The number of persons having the trait is dependent on the age of registration. Assuming a certain probability for the pathogen gene and a certain value for the penetrance, the probability is calculated that k (k = 0, 1,2) of the parents and r out of s children in each family have the trait, The family data are selected in such a way that there exists at least one child with the disease (proband). Either single or complete ascertainment is assumed, The unknown parameters are estimated by the maxi­ mum likelihood method. As an application the law of heredity for certain atopic diseases is studied, (Received January 23, 1961,)

578-5, J, R. Guard: The independence of transfinite induction up to wW in recursive arithmetic,

Primitive recursive arithmetic (PRA) is a quantifier-free formal system which has the usual starting functors for primitive recursion, schemata for primitive recursion and composition, the numeral 0, numerical variables, =, ::::::>, ,....,, axiom schemata for the above, modus ponens, substitution, and simple induction, Consider the well-ordering of type ww induced on the numerals by lexicograph­

ically ordering the exponents of their prime decomposition (letting 0 be the first element), Let < 0 be the primitive recursive relation which mirrors this well-ordering, Rule of ww -induction: From P(O),

P(f(x)) ::::l P(x), and x =F 0 ~ f(x) <::: 0x to infer P(x). It is known that analogous rules of wn-induction for finite n are derived rules of PRA. Church conjectured that ww -induction is not a derived rule.

Theorem, The formula th0(y) ¥ 'b = i 1 -which says intuitively that y is not the number of a proof of 0 = 1- is valid but unprovable in PRA. Theorem. tho(Y) ¥ rO = ? is provable in PRA augmented by ww -induction, Combining these two theorems we obtain Theorem. ww -induction is not a derived rule of PRA. (Received January 23, 1961.)

140 578-6. J. H. Billings: Product integral solutions of linear, nonhomogeneous systems of first order differential equations.

Consider the system of equations i du/dx = 2:f= 1ajiuj + fi' ui(b) = uiO' i = l,Z, ••• ,n, where the aji and the fi are given continuous, single-valued, bounded functions of the real variable x, on some nonempty interval of the real line, b ~ x ~ c, and the uiO are n given constants. Under the assumption that the matrix A of the coefficients aji is nonsingular, we show by a limiting process that the solution can be written in the matrix form U(x) = -li!J0 +fll !A)F (§')d.H where Ib A =/:(A('! )

578-7. Antoni Kosinski: A theorem on families of acyclic sets and its applications.

Theorem. LetT be a fixed k-plane in En· Let Gn-k,r-k be the grassmannian of all r-planes, r > k, containing T and for every x E Gn-k,r-k let H(x) be the r-plane in En represented by x. If

<1': ~-k,r-k----+ En is a multivalent upper semi-continuous acyclic mapping then there exists an x E Gn-k,r-k such that H(x) n~P(x) i O. The following corollary gives a typical application of the above theorem: Corollary. Suppose that to every unoriented k-plane H through a point p E En there

is upper semi-continuously assigned a compact and acyclic set (H) C H. Then for some plane H0 , p E I, intersection of every k-plane with M is either empty or acyclic. Then M is convex. (Received January 19, 1961.)

578-8. J. H. Blau: Transformation of probabilities.

Let R be a o--algebra (= Borel field) of subsets of X, and .f the set of all probability measures

on Ji. LetT transform g into itself. For certain sets E E R· knowledge of P throughout E (i.e. ~or all subsets of E belonging to B,) determines TP throughout E. Denote by .!l the class of such sets E. Assuming I& I > 4, it is shown that if .!l is sufficiently large (for example if .!l = B. or if .!l contains a semiring of which X is a member and whose Borel extension is )1), then T is a linear combination of the identity and a constant. There are applications to the theory of learning and to political theory. Related theorems which concern the formation of political parties and the methodology of psychological experiments give results on when, for a partition (Xl, ... ,Xn) of X, then-tuple [?(Xl) •.•• ,P(Xn)] uniquely determines [TP(Xl), ... ,TP(Xn)]. (Received February 8, 1961.)

578-9. Chen Chung Chang: Theory of models of infinite valued logic. IV.

For notation see Abstracts 61T-49, 50, 51, Notices Amer. Math. Soc. vol. 8 (1961) p. 68. !?~finition 1. Let D be an ideal in zi. The reduced productTfiEIIl''/D = (A,Rt• =>tET of a family of systems O{i, i E I, is defined as follows: for f,g in neiAi' a (f,g) = sup XED inf i EX[ a (f(i),g(i))];

f/:= ={g;:: (f,g) = lJ; A= {f/ a; f E1Tu;t\J; fort E T, Rt(f1/=. , ... ,fs(t/=) =sup XED inf i EX [Rt(f1 (i),. .• ,f_$(t)(i))). Definition z. A formula 16 is conditional if 16 = Q(l61/\ ••• l\16n) where (i) Q is a string

141 of quantifiers, (ii) for each i :ii n, !111 = !lin v... V !llimi, and (iii) for each i !iii> n, !lin= ~ r 11

The equation ( 1) :X: = IAJ - k * F (x), t ;;; 0, with c.J a constant, F ( •) of period t', k( •) an impulse response, and * = convolution is considered, and solutions x(•) which are of ultimate period 0 are sought, with 0 a given number, not necessarily the fundamental period. The topology of uniform con­ vergence. on compacta, modified at oo in a manner depending on 0, can be used with Schauder's fixed point theorem to study such solutions, provided that F(•) is bounded and Lipschitzian of order 1 with constant fJ• and a certain integral inequality involving (J. O, and the rate of approach .ft!Xk(u)du to zero are satisfied, Sample result: If T:/0 > ZJrok(u)du I supu IF (u) I = a., then for every integer n there is an CcJ in the interval (n t'/0) ±a. .and an associated solution x( •) of (1) which has ultimate period 0. The argument gives a bound on the rate of convergence to periodicity; indeed, this bound is used to prove the compactness condition needed for use of Schauder's theorem, (Received February 13, 1961.)

578-11. Seymour Ginsburg and H. G. Rice: Two families of languages related to Algol,

Let e be the set of all finite strings of letters of a given finite alphabet ~. Two families of subsets of 6, called the definable sets and sequentially definable sets, are described, Definable sets are obtained from a system of set equations$(i) = fi(s(l>, ... ,_t), i = l, ... ,n. Here each fi(~(l>, .... ~(n)) is a polynomial in the variables $< 1>, ... ,§;(n), each _s(j) ranging over the subsets of 8; plus is set union; multiplication is complex product; and the coefficients are subsets of e. This system is an abstraction of the way the international algorithmic language Algol (Communications of the Association for Computing Machines vol. 3 (1960) pp. Z99-314) is defined. Sequentially definable sets are obtained when s(i) = fi(!;'(l)'"'' ~(i)) for each i. The two families of sets are not identical. A necessary condition for a set to be (sequentially) definable is given, A homomorphic image of a (sequentially) definable set is (sequentially) ·definable. A number of simple sets which are not definable are exhibited. If, in a system of equations, all the coefficients commute, then the derived set is sequentially definable, The definable sets are equivalent to the type Z languages of Chomsky (Informa­ tion and Control vol. Z (1959) pp. 137-167). (Received February 13, 1961.)

14Z 578-12. N. S. Mendelsohn: Permutations with confined displacements.

Let P be a permutation of 1, 2, •.• nand let the image of i under P be P(i). The displacement of i is defined to be the number P(i) - i (mod n). Let !il'(n,r) be the number of permutations of 1, 2, ••• n for which all displacements are in the set of numbers 0, 1, 2, 3, ••• r, and let r6(m, r, s) be the number of permutations of 1, 2, ••• n in which all buts of the displacements are amongst the set 0, 1, 2, •.• r. Exact formulas are obtained in the following cases: fil'(n,r) for r = 1, 2, 3, 4; r6(n,n - 1), fil'(n, n - 2), !il'(n, n - 1, s), !il'(n, n - 2, s). Asymptotic formulas are obtained for fil'(n,r), r6(n, n - 1, s), r6(n, n - 2, s), fil'(n, n - 3, s). (Received February 13, 1961.)

578-13. Russell Remage, Jr.: On minimal sets in the plane.

F. B. Jones has given an example of a continuum in the plane which is locally connected at some points and not at others, and is a minimal orbit-closure under a homeomorphism of the con­ tinuum onto itself. In response to a question of M. L. Cartwright, this homeomorphism is extended to an orientation-preserving homeomorphism of the plane onto itself. An example is also given of an orientation preserving homeomorphism of the plane onto itself with a minimal orbit-closure having an infinite number of nondegenerate components, as well as point components. (Received December 12, 1960.)

578-14. J. W. Andrushkiw: On the extension of an ordered ring without identity.

Let R be an ordered ring without identity. Theorem. There exists a ring R * such that (1) R * contains R. (2) R * possesses the identity, (3) R • can be ordered in such a way that the ordering of R is preserved. (Received February 14, 1961.)

578-15. T. S. Klotz: The geometry of extremal quasi-conformal mappings. Preliminary report.

LetS and S be c 3 surfaces in E 3• Let f: S -s be an extremal quasi-conformal map, i.e., f is a Teichmiiller map defined by a holomorphic quadratic differential;/= 0 on S, and a constant K. ii:" 1.

For K = 1, f is conformal. The following results are proved. If I(> 1 and f preserves lines of curva­ ture, then S and S are isothermal. If I£ > 1 and f preserves lines of curvature and normals, then the isothermal surfaces SandS satisfy a "joint Weingarten relation," W(k 1,k2; 'k 1,'k2) = 0 at corresponding points. W( k1, k2; k1,'k2) = 0 also holds when I{ = 1 for any normal-preserving f. Iff is the standard map of S onto a parallel surface ~. then S and S are isothermal Weingarten surfaces, which reduces to the old fact that S and Shave constant H if l{ = 1. If f is the Gauss map of S, and S the unit sphere, then S is isothermal, satisfying k 1 = .i l(k2• For I(= 1, this is the old fact that only spheres and mini­ mal surfaces have conformal Gauss maps. As a trivial corollary, if H;: 0 on S, and f preserves

normals, then Sis isothermal with k 1 =.± J(k2. (Received February 14, 1961.)

143 578-16. D. S, Greenstein: On the geometry of Stieltjes transforms.

Let {Jln} be a positive definite Stieltjes moment sequence. Let d"ljfn(t) be any mass distribution on the non-negative real axis such that Jo00tkd'llfn(t) ~ Jlk (k = O,.,.,n) and let fn(z) be the corresponding Stieltjes transform jo00 (z - tf 1 d'ifn (t). 1n this paper a precise determination is made of the convex domain Dn(z) determined by all values of fn(z) for given fixed z in the upper half plane. These domains are lens shaped, the bounding circles being easily related to the circles of the Hamburger moment problem obtained by regarding .PO• p. 1 .... and p 1.pz .... as Hamburger moment sequences. The proof of these results depends crucially on Theorem 1. fo (z) = (z/p0 + w)- 1, the range of all possible w being zero plus the wedge arg z < arg w l!ii 7r, The proof of Theorem 1 depends on the following characteriza­ tion of f 0(z) among the larger class of functions obtained by allowing dljf0(t) to have mass on the· negative real axis: zf(z) must map the upper half plane into the lower half plane. (Received February 14, 1961.)

578-17. Leopolda Nachbin: On the weighted polynomial aeproximation on a locally compact space.

Let E be a locally compact space, C(E,R) the algebra of all continuous real functions on E and c0(E,R) the Banach algebra of all continuous real functions onE vanishing at infinity. lf A is a sub­ algebra of C(E,R) containing 1 and M is a vector subspace of c 0(E,R) such that AM M, then M is said to be of finite type under A iff in c0(E,R) belongs to the closure of M in c0(E,R) if and only if the re­ striction fIX of f to X is in the closure in Co(X ,R) of the vector space MIX of restrictions of functions in M to X, for every component X of the partition of E determined by A, if we define x 1 "'"'Xz whenever x 1, xz in E are such that g(x1) = g(xz) for any gin A. Conditions for the rate of growth at infinity with respect to M of functions in A are discussed in order to insure the finite type case, thus extending results on the Bernstein approximation problem by P. Malliavin (A mer. j. Math. vol. 81 (1959)). (Received February 15, 1961.)

578-18. W. V. Caldwell: Vector spaces of harmonic mappings.

Let .,8 be a domain in the complex plane and let :t:'be the system of first-order partial differen­ tial equations Ux = a(x,y)Vx + b(x,y)Vy, - Uy = c(x,y)Vx + d(x,y)Vy where a, b, c, and d have Holder­ continuous partial derivatives and 4bc - (a + d)Z ;::::- 0 in lJ. (We will assume that the system is normal­ ized so that b ::> 0 in lJ .) It is shown that if the set ?r of solutions of£: consist of harmonic mappings, then Lis a Bers system (i.e., b = c and a = - d) and a + ib is analytic in lJ. (Received February 16, 1961.)

578-19. H. F. Mattson and Gustave Solomon: A new treatment of Bose-Chaudhuri codes.

A (k,n) group code A is defined as a k-dimensional subspace of the vector space of all ordered n -tuples over the field F of two elements 0, 1. The number of errors which A can correct is determined by d, the minimum total number of l's in any nonzero vector of A. For any Bose-Chaudhuri (group) code A, defined via linear recursion for a divisor f(x) of xn + 1 over F, n odd, it is known that d e; do for a certain do depending on f(x), Viewing recursions as difference equations, and letting z be a

144 primitive nth root of unity, the authors obtain for each a= (ao•···•an-I> in A a polynomial ga(x) (of degree less than n) such that each ai = ga(zi). From this lemma follows a new and simple proof that d ;;;;- d0 , Some improvements on this lower bound do when n is a prime p = 2h + 1 such that 2 has multiplicative order h modulo pare: do< h implies d il:do + 1; if p =1 (mod 8), then positive even weights are at least 2d0; if p =: - 1 (mod 8), then odd weights m, satisfy m 2 - m + 1 ;;:; p. In particular, that d ~ 7 for the (12,23) Golay code is easy to prove. (Received February 16, 1961.)

578-20. R. W. Ritchie: The functions computable by linear bounded automata.

In the article Linear bounded automata in the University of Michigan summer course notes of 1960, John Myhill defined a class of machines which he calls linear bounded automata. It is straight­ forward to show that the class of functions computable by these machines is precisely the class of functions computable by Turing machines on which the amount of tape used is a function of the lengths of the arguments which can be computed on a finite automaton. Thus it is a proper subclass of each of the classes defined in Abstract 576-210, Notices Amer. Math. Soc. vol. 7 (1960) p. 976. Theorem 1:

The clas~;~ of functions computable by linear bounded automata ( in n-ary notation for any n > 1) is the class e 2 of functions as defined by Grzegorczyk, (Some classes of recursive functions, Rozprawy Mat. (1953)). Hence the class e~ of relations definable by linear bounded automata (that is, relations whose characteristic functions are in e 2) contains a basis for the recursively enumerable sets. In fact, a stronger result is Theorem 2: e; contains the constructive arithmetic relations as defined and studied by Smullyan (Theory of formal systems, Annals of Mathematics Studies, 1961). Here e2 is the subclass of e; defined by Grzegorczyk. (Received February 16, 1961.)

578-21. B.S. Sackman: On eigenmatrices and partial difference equations.

The notion of eigenmatrices is defined and applied to the construction of a direct method for solving systems of partial difference equations. The method is applicable to the nonsymmetric matrix forms obtained from the nonself-adjoint forms of the differential equations and reduces significantly the order of the linear system. Since the method produces only tridiagonal matrices upon which to per­ form elimination, the speed of the method is considerable. Let A = (Aij) be an m X m block matrix each of whose block elements Aij are n X n matrices. Let U = col (U l'u2 .... ,Um), Ui = X or 0, be a mn X 1 vector each of whose m block components Ui is an n X 1 vector and U f. 0. Then if there exists

ann X n matrix M such that AU= diag (M,M, .•• ,M)Um and MX = pX, the matrix M is called an eigen­ matrix of A and U is the corresponding eigenvector of A. Two main theorems are proved: That the eigenvalues of an eigenmatrix are eigenvalues of the matrix itself; and the eigenmatrices of the tri­ diagonal block matrix A= (C,T,B) are Mj = T + ajs 112c 112 where aj = 2cos j'lT/(m + 1), j = 1,2,. .. ,m. It is shown how to apply the method of eigenmatrices to the solution of elliptic partial difference equa­ tions. (Received February 16, 1961.)

145 578-22. H. S. M. Coxeter: Similarities ilnd conformal transformations.

In ordinary Euclidean space, every isometry that leaves no point invariant is either a screw displacement (including a translation as a special case) or a glide reflection. Every other kind of similarity is a spiral similarity: the product of a rotation about a line and a dilatation whose center lies on this line. In real inversive space (i.e., Euclidean space plus a single point at infinity), every conformal transformation is either a similarity or the product of an inversion and an isometry. This last remark remains valid when the number of dimensions is increased. In fact, every conformal transformation of inversive n-space (n :> 2) is expressible as the product of r reflections and s inver­ sions, where r ;i! n + I, s a 2, r + s ~ n + 2. This is implicit in the work of Rene Lagrange, Produits d'inversions et metrique conforme (Paris, 1957), pp._ 21-22, ll2; but the present treatment is simpler. (Received February 17, 196I.)

578-23. Frank Harary: The determinant of the adjacency matrix of a graph.

Let A be the adjacency matrix of digraph D, so that aij = I if there is a (directed) line in D from vertex vi to vj and aij = 0 otherwise. Let xk be a (numerical) value assigned to the k'th line ~k of D and let A(x) be the matrix whose i,j entry is 0 if aij = 0 and is xk if ))k is the line in D from vi to vj. A closed formula is found for the determinant IA(x) I in terms of the distribution and parity of the directed cycles of D. This is specialized at once to a formula for the determinant lA I and is readily modified to yield the determinant of an ordinary (symmetric) graph. (Received February 17, I96I.)

578-24. R. J. Weinacht: On the generalized axially symmetric Helmholtz equation.

Let L[u] = uXlXI + ux2x2 + ••• + uxmxm + uyy + Ky-1Uy +.'\ 2u, K > 0, y S:. 0. Form= I the equation L[u] = 0 has been considered by P. Henrici (Comment. Math. Helv. vol. 27 (I953) pp. 235-293; see also Boundary problems in differential equations, University of Wisconsin Press, I960, pp. 19-34). In the present paper it is shown that form= I,2, ••• , uab = Sk-Jj'oN(..\r)r(I-m-k)/2sink-IIjdlj is a fundamental solution of L with the singular point at (a,b) = (ap a 2, ••• ,am,b), b 6:0. This fundamental solution has no other singularity in the upper half spacey S';O. Here r 2 =(XI - a 1)2 + ... + (Xm- am)2 + y 2 + b 2 - 2by cos,," N is the Neumann funct10n. of order (m + k- 1)/2 and Sk-l- I =..;of'"Tr sin k -1 ljdlj. The formula has been obtained in a way similar to the case )) = 0 which corresponds to generalized axially symmetric potential theory; see Weinstein (Trans. Amer. Math. Soc. vol. 63 (I948) pp. 342-354) and Diaz and Weinstein (Studies in mathematics and mechanics presented to R. von Mises, Academic Press, (1954) pp. 97-102). (Received February I7, 196I.)

578-25. R. C. Bose and I. M. Chakravarti: Binary group codes with even redundancy which correct consecutive errors in bursts of three or less.

Let A be any n X r matrix of rank r, whose row vectors are the r-place binary vectors a1, a2•···• an• Using the main result proved in an earlier paper Alias sets of error vectors in the theory of error correcting group codes (R. C. Bose, Ann. Math. Statist. vol. 31, p. 813) it follows that the necessary and sufficient condition for A to be a parity check matrix for a binary group code, which

146 corrects any burst of consecutive errors of length 3 or less is that the vectors a. 1, a.2 , ..• , a.n; a.1 + a.2'"""' a.n_ 1 + a.n; a.1 + a.3•···•a.n-2 + a.n = a.1 + a.2 + a.3'"""'a.n-2 + a.n_ 1 + a.n are all distinct. Thus one would like to choose for A a matrix for which the above condition holds and subject to this n attains its maximum value for a given r. When r =2m+ 2, m;:;:; 2, we can find a matrix A with n =2m - 1 as follows: Let x be a primitive element of GF(22m) such that 1 + x = x 6, where 8 f. 2 (mod 3). Also let z be a primitive element of GF(2 2). Then a.i = (8i•/i) where 8i is the coefficient vector of the (2m - 1)-th degree polynomial in x, which represents xi and fi has a similar meaning with respect to zi, i = 1,2, •.• ,2 2m- 2. (Received February 20, 1961.)

578-26. A. J. Gross: Group codes with odd redundancy which correct consecutive errors in bursts of three or less.

Bose and Chakravarti have given an optimum binary group code for correcting consecutive errors in bursts of length three or less, when the redundancy is even (Notices Amer. Math. Soc., this volume, Abstract 578-25). In this paper the corresponding problem when the redundancy is odd is considered. Let r = 2m + 1, m ~ 4, and let A be ann X r matrix whose rows are the r-place binary vectors a. 1, a.2•···•a.n· We show that A satisfies the necessary and sufficient condition for being the parity check matrix of the required code if it is obtained in the following manner: Let x be a primitive element of GF (2m), y be a primitive element of GF (2 m-l) and z be a primitive element of GF(22). Further suppose 1 + x = x 6, 1 + y = yll where 6 f. 2(mod 3) if m is even and !l f. 2 (mod 3) if m is odd. Let fBi be the coefficient vector of the (m - 1)-th degree polynomial in x, which represent xi and let hi and f'i have similar meanings with reference to yi and zi. Then a.. = ((3.,)\.,p.), i = O,l, •.• ,(2m - 1)(2m-l - 1) - 1. Although this method does not yield a maximum n, l l l l it is easy to show that as m increases n = (2m - 1)(2m-l - 1) approaches the optimal value. (Received February 20, 1961.)

578-27. N. L. Alling: A characterization of Abelian 'l a.-groups in terms of their natural valuation.

A valuation group is a.-maximal, a. an ordinal >0, if every pseudo-convergent sequence of length < eva.(= short sequence) has a pseudo-limit. An ordered Abelian group G is an ??a.-set iff (i) the value set, sans least element, is an ??a.-set, (ii) the factors of G are either real or integral, (iii) G is a.-maximal. Assume G is an ??a.-set. (i) and (ii) are proved by choosing pre-images in various ways and invoking the ?7a. -condition in G. (iii) is proved by factoring G by the breadth B of a short sequence and by proving that G/B is Cauchy complete. Assume (i), (ii), and (iii) hold. Clearly the upper, lower, and point characters of G are S; eva.. Let :E be the Dedekind completion of G. The

natural valuation on G extends to ~. Let a E ~have characters < .:t.Ja.• S = !.V(a - g): g E G} has a

limit lower character <. wa.. Thus a.-maximality can be invoked to show that no such point a exists, proving the theorem. (Received February 20, 1961.)

147 578-28. F. J. Bellar, Jr.: Pointwise bounds for the solution of parabolic differential equations. Part ll.

In Part I, a method was presented for obtaining theoretically small bounds for the solution of a boundary value problem where the solution is initially given and thereafter satisfies a parabolic differential equation with Neumann type boundary conditions (F. Bellar; Notices Amer. Math. Soc. for Meeting 577). Here the Neumann conditions are replaced by Dirichlet conditions. V is an (N + 1) dimensional domain bounded by hyperplanes y = 0 and y = y 0 > 0 (denoted by D(O) and D(y 0)) and a surfaceS between them. Let the unit normal, (nl•···• nn, ny). satisfy minsLJ.~ 1 n~ = m 1 > O. A pointwise solution bound is presented for the uniformly parabolic problem: (aiiw,i)'j- zw,y = f(x,y,w, vw) in V, w ( ) = g(x) on D(O) and w(x,y) = h(x,y) on S where wand the coefficients of the differential operator are sufficiently smooth. The boundary function f satisfies a Lipschitz condition in its last N + 1 arguments and is integrable over V for continuous functions w and vw. The bound, which is obtained, is in terms of the integral of the square of known functions and hence lends itself to improvement by means of the Rayleigh-Ritz technique. (Received February 20, 1961.)

578-29. Charles Fox: A solution of Chandrasekhar's integral eguation.

Chandrasekhar has shown that many problems of radiative transfer in the theory of light can be reduced to that of solving the integral equation 1 = H(t) - tH(t) .fo1K(u,t)H(u) du, where H(t) is the function to be determined and K(u,t) (u + t) = ljf(u); 11/(u) being given. In this paper it is shown that the equation above can be reduced to a singular integral equation of a well-known type. A solution is then obtained by the standard methods of the theory of singular integral equations. (Received February 20, 1961.)

578-30. George Glauberman: On isomorphism of groups.

As is well known, H 2(B ,A) is the group of all nonequivalent group extensions of A by B corresponding to a given mode of action of the operator group B, i.e., to a given homomorphism of

B into ~(A), the group of automorphisms of A. In the case of finite Abelian p-groups, representation of ~ (A) by a matrix group assists in determination of criteria for isomorphism of nonequivalent groups. These results are applied in conjunction with the general cohomology theory of groups to solve a number of enumeration problems for finite groups. (Received February 20, 1961.)

578-31. Marvin Marcus and Henryk Mine: On the relation between the permanent and the determinant.

Let M(m,n) be the set of m X n rectangular matrices with complex entries. Let Pr(X)

(Cr(X)) denote the C(m,r) X C(n,r) matrix whose entries are the r-square subpermanents (subdeter­ minants) of X E M(m,n) arranged in doubly lexicographic order. Generalizing results of PcSlya and Szegii it is proved that except for m = n = 2 there is no linear map T of M(m,n) into itself such that Pr(T(X)) = SrCr(X). for all X E M(m,n) where Sr is a nonsingular linear map of M(C(m,r), C(n,r)) into itself. Thus there is no linear map T of M(n,n), n > 2, into itself such that per (T(X)) = k det (X) for all X and fixed k # O. This result will appear in the Illinois Journal of Mathematics. (Received February 20, 1961.) 148 578-32, G. H. Meisters and Czes)aw Olech: Local one-one maps on manifolds. Preliminary report.

A mapping f: X - Y of a topological space X into a topological space Y is said to be locally 1 - 1 provided each point x in X possesses a neighborhood N such that fIN is a 1 - 1 map, Some properties of such mappings are established for the case when X and Y are certain subsets of n-manifolds, Mn (i.e, connected Hausdorff spaces which are locally homeomorphic to euclidean n-space). In particular the following two theorems are established: Theorem 1. .!!_X is a compact, proper subset of ann-manifold M~, f is a continuous, locally 1 - 1 mapping of X into M~ and if f lbd X is 1 - 1, then a necessary and sufficient condition that f is a homeomorphism is that (a) f(bd X) C bd f(X), or (b) f is an open mapping. Theorem 2, Let X be a compact, proper subset of M~, and f a continuous, locally 1 - 1 mapping of X into M~ such that f(X) t- M~ and f lbd X is 1 - 1, Then if bd X is connected, f is a homeomorphism. Simple examples are given to show that these (and other) theorems are "best possible" in various senses, The authors have restricted their attention ton-manifolds in order to be able to use Brouwer's theorem on the invariance of domain, (Received February 20, 1961.)

578-33, Beauregard Stubblefield: Some imbedding and nonimbedding theorems for n-manifolds,

The arguments used in the third theorem contained in Abstract 555-51 (Notices Amer, Math, Soc, Vol, 6 (1959) p, 261) are valid for the following three theorems: Theorem 1. LetS be a Claytor curve. Then S X In-2, n s;; 2, cannot be imbedded in ann-manifold, Theorem 2, A necessary and sufficient condition that a Peano continuum P containing no local cut points be imbeddable in a 2-manifold is that P does not contain one of the Claytor curves. Theorem 3, A necessary and sufficient condition that the Cartesian product of In-2 , n .s; 2, and a Peano continuum S which contains no local cut points be imbeddable in ann-manifold is that S can be imbedded in a 2-manifold, Theorem 1 is an extension of the author's theorem to generalized manifolds. The sufficiency of Theorem 2 was proved by Moise and Young (Bull. Amer, Math. Soc, vol, 54 (1948) p. 77). (Received February 20, 1961,)

578-34. Ernest Michael: A linear transformation between function spaces.

Let X andY be metric spaces, and pan open map from X onto Y; with p- 1(y) complete for all y E Y. Let B be a Banach space, and let the function spaces C(X,B) and C(Y ,B) carry the compact­ open topology. Then there exists a continuous linear transformation u fr.om C(X,B) onto C(Y,B) such that, for all f E C(X,B) and ally E Y, jp:(f)](y) E (closed convex hull of f(p- 1 (y))). Note 1: This seems to be new even with X and Y compact, and B the reals, Note 2·: This result is trivial if p has a con­ tinuous cross section q: Y ~X, for one can then define u by ljl(f)](y) = f(q(y)), (Received February 20, 1961,)

149 578-35. M. S. Klamkin: On the transformation of a class of boundary value problems into initial value problems for ordinary differential equations.

A classical example of the transformation of a boundary value problem into an initial value one is the one given by Blasius for the steady two-dimensional flow along a flat plate placed edgewise to the stream. After making a similarity transformation, the relevant partial differential equation reduces to the boundary value problem: y"' + yy" = 0 where y(O) = 0, y' (0) = 0, y' (oo) = 2.. By letting y"(O) =a, the solution can be given in the form y = a 1/ 3F(a113x) where F(x) is the solution corres­ ponding to a= 1. It then follows that y'(oo) = a2./3F'(oo) = 2.. Consequently, the given boundary value problem can be solved by solving two initial value problems. This is very advantageous for numerical computation. This idea can be extended to apply to a wide class of differential equations of any order. In particular, the second order problem in which the procedure may be used is given by y" = Lr,sarsxrys(y')t where y(O) = 0, y'(oo) = a, and t = 1 + m + mr + ms - s. (Received February 2.0, 1961.)

578-36. Czes,law Olech: On global stability of two-dimensional autonomous systems. Preliminary report.

Consider on the plane, R z, the real autonomous system (S): x' = f(x,y), y' = g(x,y), where f and g are of class C 1 in R 2., and the origin is a singular point. Let J denote the Jacobian matrix of f an,d g. Problem 1: _.!!. tr J <. 0 and det J > 0 throughout· R2., does it follow that the origin is globally asymptotically stable? By generalizing a result of Krasowski the author proves the Theorem: !!_ tr J < 0 and det J > 0 ~ R 2. and if for some 'Y > 0 and f > 0, f2. + g2. ;;:- fl whenever x2. + y2.;;;: .y; then the origin is globally asymptotically stable. It follows from this theorem that Problem 1 is equivalent to Problem 2.: Does tr J < 0 and det J > 0 imply that the ·mapping g = f(x,y), 'f = g(x,y) ~ R 2. into R2. is globally one-to-one? The author has solved these problems in the affirmative in several special cases. For example, (a) iff and g are polynomials or (b) if either fy or gx has a constant signum throughout R 2.. The last case is a generalization of a recent result of L. Markus and H. Yamabe concerning Problem 1. (Received February 2.0, 1961.)

578-37. C. L. Dolph: Nonlinear oscillations in a cold plasma.

Exact nonlinear oscillations in a plasma have been discussed by Akhiezer and Lyubarskii (Dokl. Akad. Nauk. SSSR vol. 80 (1951) p. 80), Sen (Phys. Rev. vol. 97 (1955) p. 849), Smerd (Nature vol. 175 (1955) p. 2.97), Bernstein, Greene and Kruskal (Phys. Rev. vol. 103 (1957) p. 546), Vedenov (Zhur. Ekep. Teor. Fiz. vol.- 33 (1957) p. 1509), Sturrock (Proc. Roy. Soc. London Ser. A vol. 2.42. (1957) p. 2.77), Dawson (Phys. Rev. (1959) p. 383), Wyld (Space Tech. Lab. R-60-0000-GR 303 (1960)) and Jackson (Phys. Fluids vol. 5 (1960) p. 831). In this paper a method is developed which unifies many of the results obtained above. It is based upon a decomposition given by Wyld and the theory of first order systems of partial differential equations with equal principal parts. In particular, it yields the travelling wave solutions found and published without proof by Vedenov. In addition, it generalizes the results of Dawson for the cylindrical and spherical geometries to an arbitrary orthogonal Euclidean geometry in which all quantities depend only on time and one spatial coordinate. It is

150 also slightly more general in that it permits two velocity componl!nts and a constant magnetic field perpendicular to their plane. This work was partially sponsored by APRA order No. 120-60 and USASC R and D Contract DA 36-039-SC 75041. (Received February 20, 1961.)

578-38. Norman Bazley and David Fox: A procedure for obtaining lower bounds to eigenvalues.

This abstract extends work of N. Aronszajn, A. Weinstein, and the authors. Let A be a self­ adjoint operator in a separable Hilbert space with inner product (u,v). Let A have ordered eigenvalues

~ 1 ;a Az ...• with corresponding eigenvectors u 1, u2, •••• which precede all limit points of its spectrum. Suppose A= A 0 +A' where A' is positive definite, A 0 is self-adjoint, and the initial spectrum of A 0 consists of known eigenvalues A~::;; Ag •···• corresponding to known eigenvectors u~, ug, •.•• To obtain lmprovable lower bounds to the eigenvalues of A introduce operators A k,,E = A 0 - ?"I + fA 'Pk + :riJ QJ.., where 'Y' is an arbitrary positive constant, ·pk is the projection on the span of arbitrary vectors Pi (i = 1 ••.•• k), with respect to the inner product j!!,v] = (A'u,v), and Q.£ is the projection on the span of the vectors {A'Pk + nJ- 1 u~ (i = l, •••• ,.l), with respect to the inner product

Ak,) has ordered eigenvalues }) ~,.£ ;3 ))~·.£ .•. with eigenvectors u~..t, u~.k ••• ; they are determined by the solution of two matrix eigenvalue problems of respective orders k and"'· These matrix problems involve only the terms (A 'Ps• u~ ), (A 'Ps•Pt>• and (A 'Ps• A'Pt>• (i = 1, •••• ); s,t = l, ••• ,k). The JJf•,l give the lower bounds ))f•t ~ ))i• i = 1,2, •••• (Received February 20, 1961.)

578-39. T. P. G. Liverman: A direct operational method for some Cauchy problems.

(t,x) E R X Rn. Pl)(Dx) =polynomial of order nh in a/axl•···•8/6xn· Q = Dt1(Dt = 8/8t). Operands: the generalized functions of class (J2 e .e> -l. As consequence of result of Abstract 577-13 follows Theorem: If P(Q,D) = ~~l Q)IP11 (Dx) is of Q/D order p = min)j(Mn_,), and if 1 < p < 1/(1 - p). the equation(*) (1 + P(Q,Dx))Uy = Fy• where Fy E- (d Gt>'• has a solution Uy E (JG $>' that is unique in ~C.~)' and is given explicitly by Uy = I:f:o

578-40. D. G. Johnson and J. E. Kist: Spectral functions and prime ideals in vector lattices.

A mapping f with domain a vector lattice E and range contained in the three element chain

{- oo,O, + oo} is called a spectral function provided tha~ f(x) i- 0 for at least one x; f(a.x) = a.f(x); and f(x Y y) = max [f(x), f(y)}. The set F of all spectral functions onE, equipped with a certain partial order, and a certain topology, is called the spectral space of E. Using the concept of spectral function, Amemiya [J. Sci. Fac. Hokkaido Univ. vol. 12 (1953)) developed a spectral theory for vector lattices. An ideal I in E is called prime if lx II\IY I E I implies that x or y is in I. The present authors show that

151 the set of all prime ideals in E, equipped with the hull-kernel topology, and partially ordered by set inclusion, is homeomorphic and order anti-isomorphic with F. It follows that Amemiya's theory can be obtained by ideal-theoretic methods, The authors make a study of the set M of all minimal prime ideals in E, and, in particular, they show that the carrier space of E is isomorphic to a basis for the open sets in M, when the latter set is equipped with the hull-kernel topology. Moreover, it is shown that a result of Ribenboim f?acific J, Math. vol, 10 (1960)] is a consequence of a characteristic prop­ erty of minimal prime ideals. (Received February 20, 1961,)

578-41. D. G. Johnson and J, E. Kist: On complemented ideals and extremally disconnected spaces.

Let A be a -algebra A, the following are equivalent: (i) /1t(A) is extremally disconnected; (ii) A is complete; (iii) every weakly closed ,L -ideal in A is complemented. Theorem 2, For a uniformly closed

578-42, J, R, Edmonds: On minimum element-covering from a finite class of weighted sets,

The Norman-Rabin theorem (Proc, Amer, Math, Soc, vol. 10 (1959) p. 315) on minimum covers of a graph is extended with simplification of proof: In a graph with a real additive set function f on the edges, if C is a vertex-covering set of edges but not one for which f is minimum then there is a path P with edges alternately in C and C, simple except possibly at the end vertices, and such that C'

= (C - P) U (P - C) is a cover for which f(C') <. f(C). A further generalization gives an "algorithm 11 for the "best switching circuit problem" which prompted the N - R work: In a finite class S of sets with a real additive set function f on the subclasses, if C is a cover (C C S, U fC} = U {S}) but not one for which f is minimum then there is a subclass T C S: (1) whose C sets and S - C sets correspond hi-uniquely to alternate vertices of a tree the edges of which correspond to distinct elements of U {S! so that incidence implies inclusion, (2) such that C' = (C - T) U (T - C) is a cover for which f(C') <. f(C). (Received February 20, 1961.)

578-43. Murray Gerstenhaber: Three-dimensional subalgebras of the classical algebras.

If L is a simple Lie algebra over the complex numbers, then it is known that given any nonzero nilpotent element e of L there exist elements f and h of L such that [h,e] = e, j!I,f] = - f, and [e,f] = h (Malcev, Kostant), and the elements h which can so be embedded in a simple three-dimensional sub-

152 algebra of L are known (Dynkin). The corresponding results are derived here for the classical alge­ bras Am• Bm• Cm• and Dm when the ground field is of arbitrary characteristic-;. Z, by bringing elements of these algebras into canonical form. In the case of characteristic-;. 0, however, e does not always determine the class of h. (Received February ZO, 1961.)

578-44. J. Horvl!th: The balanced core of a set.

A set B in a vector space E over the real or complex numbers is balanced if ~B C B for all

I-'ll~ 1. The balanced core B of a set A C E is the union of all balanced sets contained in A. B -;. !l if and only if 0 EA. If B "I jl then B = nl..!li! 1))A. If A is absorbing or convex then B is absorbing or convex respectively. If E is a topological vector space and A is closed or a neighborhood of 0 then B is closed or a neighborhood of 0 respectively. This concept simplifies somewhat the proofs of some known results in the theory of topological vector spaces. (Received February ZO, 1961.)

578-45. R. D. Schafer: On a class of quadratic algebras.

In recent work concerning cubic forms which permit a new type of composition IJ. Math. Mech. vol. 10 (1961) pp. 159-174], the following class of nonassociative algebras A with 1 over F of characteristic -;. Z is encountered: central simple flexible quadratic algebras satisfying the identity (*) [x,lx,y]] = 0. The commutative algebras in this class are the well-known central simple Jordan algebras of degree two. In this paper it is shown that (*) may be replaced by (**) ~.y]Z = 0. For any dimension ;;: 7, there is an algebra A over F in this class which is not commutative. If also F is of characteristic -;. 3, then every algebra of dimension ;:::; 6 in the class is commutative. (Received February ZO, 1961.)

578-46. W. G. Strang: Bernstein's theorem and partial difference operators.

We point out the connection between two widely-known necessary conditions for the convergence of difference approximations to the solution of ut = cuX" A consistent difference analogue is given by U(x,(n + 1) 6t) = L~ ajU(x + j ,6x, n ~t), provided only that L aj = 1, L,Jaj Ax = c b.t. A further re­ quirement, always necessary and in this case sufficient for convergence, is that of von Neumann 12::ajeij61 ;3: 1 for real a. From these conditions and Bernstein's theorem on the derivative of a

trigonometric polynomial we infer m ll x < c .6 t <. p ~x. This is precisely the Courant-Friedrichs­ Lewy condition, that the domain of dependence of U contain that of u(x,t) i, i.e. the point (x + ct,O). Applying the Bernstein theorem to analogues of the heat equation ut= uxx yields Z t.t ~- mp(.t.x)Z, a familiar restriction when m = - 1, p = 1. For a first-order hyperbolic system, we reverse the reasoning to obtain a matrix theorem of Bernstein type. (Received February ZO, 1961.)

578-47. Peter Seibert and joseph Auslander: Prolongations, perturbations, and stability in dynamical systems. Preliminary report.

Let (X,d) be a locally compact metric space. If P: X ~ zX, then,/) P: X- Z X is defined by q E: .l)P(p) if there exist sequences fPnl• fqJ such that Pn- p, 9n--.. q, 9n E P(Pn>· Define..e/P(p)

153 = LJn'!\Pn(p). Let 11"'o be a dynamical system defined on X, and let T(p) denote the positive semiorbit of p. Define the prolongation 0 1 (p) = ,8T(p). If o. is any ordinal number, define inductively D 11 (p)

= .8

D11 (p) is compact, for all o., then D11 (p) is absolutely stable, for some o.. Let F be the set of all dynamical systems defined on X. If ?r, 11'' E F, define jJ(T,T') = supxEX,O:&t~ld(?T(x,t),T'(x,t)). If ?J > 0, let G~(p) = G~(p; 1To) = Ump.t), with t ~ 0 and 1J""such that ,.,0(11"'.1/Q) < "!· Let G(p) = u?J > oG.,(p). A compact 11Q invariant set M is called totally stable or stable under perturbations if, for any e > 0, there exists 8 > 0 such that d(x,M) < 8 and jJ(1r.7TQ) < 8 imply d('T(x,t),M) < e (t <= 0). Theorem 4. M is totally stable if and only if G(M) = M. Theorem 5. If X is a manifold then 0 1 (p) C G(p). (Received February 20, 1961.)

578-48. G. T. Cargo: A theorem concerning plane point sets with an application to function theory.

Let Jo be a Jordan curve situated in the open upper half-plane except for one point at z = 0, and let Jx = £z + x: z E JoJ. Given a point set Z, we say that x is approachable provided x is a limit point of Z nJx· Theorem 1. Let Z be a set in the' open upper half-plane, and let I be a closed interval. Then the set of approachable points is residual in I if, and only if, the derived set of Z contains I. Let ?"o be an arc (not necessarily simple) situated in {z: lz I < 1} except for one end­ point at z = 1, and let ?'a = fzei9: z E 'l0 ~ • Then { r 9: 0 ~ 9 < 271J is called a family of boundary arcs of fixed form. Theorem z. There exists a bounded holomorphic function f such that, for every 9 family of boundary arcs of fixed form fro}• the cluster set c.,..9(f,ei ) is nondegenerate on a residllal subset of fz: lz I= 1}. In particular, the Fatou points off form a set of the first category. The proof of Theorem 2 is accessible to first-year graduate students. Theorems similar to Theorem Z have been proved in the last decade by Ba.gemihl-Seidel, Collingwood, and Eggleston; but the proofs are nonelementary. (Received February 20, 1961.)

578-49. Alfred Tarski: Representable Boolean algebras and infinitary logics.

For notation see Abstract 61 T-82, which will be published in the June 1961 issue of the

Notices. A Boolean algebra B is ~ 11 -complete iff, for every sequence of type (A)_e,fJ < o., of elements

X~ of B, the sum "L.. <{JJ X exists; in addition to usual finitary operations, these sum operations z

> t ~ are included among fundamental operations of B. AB 11 denotes the class of atomistic ~ 11 -complete

Boolean algebras; RBa. is the class of .lj'11-representable Boolean algebras, i.e., algebras isomorphic

to ,lt11 -complete fields of sets. As is known, RB 11 = S(AB 11). Related to Theorem T, Abstract T-14 is:

Theorem 1. For every regular ~ 11 the conditions: .kf'11 is compact, RB 11 E EC11 , and RB 11 E EQ11 ~ equivalent. Theorem 1 is used to derive the implication (iv) ._ (i) of T. The next two theorems

generalize Theorem 1: Theorem z. For every regular~·~ Jt.,..with o. ~ ?"~f3iS compact,

then RB 11 E UC,e, whence RB 11 E EC,13. Theorem 3. For every regular ~.8' if l't,s is incompact, and

154 every inaccessible ,H" ?'with ~ 'Si .y "'-,B is strongly fncompact, then RB~ fJ: EC,s· Open problem: can one omit "strongly" in Theorem 3 7 Theorem 3 greatly improves the author's result that for no accessible ~~are all ~~-complete and ~~-distributive Boolean algebras .K~-representable.

Theorems 1-3, with references to EQ~ and UCpr omitted, remain true if algebras of RB~ are assumed to be provided with finitary operations only. (Received February 20, 1961.)

HEWS ITEMS AND ANNOUNCEMENTS

A SYSTEMS SCIENCE PROGRAM the school can also be accommodated at has been established in the San jose Re­ West Park Hall, at the same rate, pro­ search Laboratory of International Bus­ vided that early application is made, and iness Machines Corporation. Dr. Harwood room is available. G. Kolsky is manager of the new program. The arrangement of courses will be The initial assignment is to study the or­ as follows: ganization of business systems, the search­ 1. Professor Hilton ing of large collections of documents, and Introductory Course on Algebraic numerical analysis. The program also will Topology (Three lectures). include the development of effective tech­ Z. Dr. Atiyah niqUIJS for the analysis and evaluation of Sheaf Theory and Complex Mani­ proposed small commercial and scientific folds (Seven lectures). data processing machines. 3, Dr. Griffiths Morse Theory of Real Differen­ tiable Functions (Seven lectures). 4. Professor Hilton A SUMMER SCHOOL IN GEOME­ Homotopy Theory (Six lectures). TRY AND TOPOLOGY will be held, pro­ 5. Professor MacBeath vided there is a sufficient enrollment, in Discontinuous Groups and Bira­ Queen's College, Dundee, july 10-ZZ, 1961. tional Transformations (Seven lec­ The fee for the school will be £ Z. tures). Accommodation will be available in West For further information write to the Park Hall, Dundee, for £15 for the entire Department of Mathematics, Queen's Col­ period. Wives and friends of members of lege, Dundee, Scotland.

155 The April Meeting in Chicago, Illinois April 14- 15, 1961

.579-l. j. A. Eagon: Ideals generated by the subdeterminants of a matrix.

Generalizations of the Principal Ideal Theorem and the Unmixedness Theorem exist for ideals generated by the subdeterminants of a matrix. Specifically, let R be a Noetherian ring, (commutative,

with unit). Then if (aij) is an r by s matrix. r ~ s ;:; 1, the rank of any prime ideal minimal over the ideal generated by the t by t subdeterminants is less than or equal to (r - t + l}(s - t + 1). If the grade of the ideal generated by the s by s subdeterminants equals r - s + 1, then the homological dimension of this ideal is r - s, (i.e. the ideal is perfect), (For related results see D. G. Northcott, Semi-regular rings and semi-regular ideals, Quart. J, Math, Oxford Ser. (2) vol, 11 (1960) pp. 81-104,) (Received December 16, 1960,)

579-2, Adam Czarnecki: Physical systems Sk of curves in a central field of force.

This is an extension to central fields of force of some of the results contained in the paper by Kasner and DeCicco, Physical families in the gravitational field of force, Amer. Math. Monthly, 1951, Every trajectory of a physical system Sk of a central field of force, is a plane curve whose plane passes through the center 0 of the field of force. The converse is that such a field of force is either central or parallel, A generalization is obtained of Kepler's second law of planetary motion, for a central physical system Sk and for a central velocity system S00• This result characterizes a central field of force, conservative or not, For the conservative case, the explicit equation of a central physi­ cal system Sk including the case k = oo, is found in terms of two quadratures. If a potential function V of a conservative central field of force, is known, the explicit equation of such an Sk, is obtained in terms of a single quadrature, The case where the potential V varies inversely as the (n - 1) power of the radius vector r > 0, from the center 0, where n f. 1, is studied in detail. Further amplifications of these results will be given in succeeding papers. (Received December 3, 1960.)

579-3. Svetozar Kurepa: On roots of normal operators.

Theorem 1. Let X be a Hilbert space, N: X-+ X a bounded and regular normal operator and T: X- X a bounded operator such that Tn = N where n is a natural number, Then, there exists a bounded positive definite, regular selfadjoint operator Q: X --!>X such that 1, QN = NQ, 2. R = Q- 1TQ is a regular normal operator and 3. Rn = N, i.e. every nth root of N is similar to a normal nth root of N. Theorem 2. Let n,N, T and X be the same as in Theorem 1. Suppose that there exists a real number a such that for every x EX, llxll = l,arg (Tx,x) E (a,a + 2'/r/n) holds where arg b is the argu­ ment of a number b. Then T is a normal operator. Theorem 3, If N is a compact normal operator and T a bounded nth root of N then arg (Tx,x) E (it,a + 27r/n] for some real number a implies that T is a normal operator. In the case n = 2, Theorem 3 holds without the assumption that N is a compact operator. This was proved by C. R. Putnam: On square roots of normal operators, Proc, Amer. Math. Soc, vol. 8 (1957) pp. 768-769.) (Received January 12, 1961,)

156 579-4. S. K. Berberian: Approximate proper vectors.

Let T be a (bounded) normal operator in the Hilbert space H. It is well-known that T has an approximate proper value p such that lpl = liT II; a proof for this is given, based on the Hahn-Banach theorem and the Cayley transform. Incidentally, His enlarged to a Hilbert space K, and a representa­ tion S - s0, from operators in H to operators in K, is constructed, in such a way that the point spec­ trum of s0 coincides (as a set of complex numbers) with its approximate point spectrum. (Received January 17, 1961.)

579-5. G. E. Sacks: A minimal degree less than 0'.

Spector has shown (Ann. of Math. vol. 64 (1956) pp. 581-592) that there is a minimal degree less than 0". Shoenfield (Ann. of Math. vol. 69 (1959) pp. 644-652) has asked: is there a minimal degree less than 0'? Theorem. For each degree d there is a degree g such that g is greater than d and less than d' and such that d is less than b is less than g for no degree b. Corollary. There is a minimal degree less than 0'. The proof consists of the simultaneous construction of the members of a sequence of sets of partial functions in such a manner that the members of the sequence have a unique function common to all of them which is of minimal degree. (Received January 25, 1961.)

579-6. M. W. Al-Dhahir: An instrument in hyperbolic geometry.

The following instrument is introduced in hyperbolic geometry: for every pair of distinct lines, a line may be drawn parallel to either and perpendicular to the other. Using this 'hyperbolic ruler' as an ordinary ruler also, the following theorem is proved. Any construction that can be performed by a straight edge and any one of the three types of compasses (ordinary compasses, homocompass or hypercompass) can be performed by the hyperbolic ruler. In proving this theorem, a use is made of the result obtained in [Canadian J. Math. vol. 8 (1956) pp. 389-394]. (Received February 2, 1961.)

579-7. S. S. Cairns: A simple method for subdividing smooth manifolds.

Let En be euclidean n-space with metric d. Let (a1, .. ~,ar) be a set of r > 1 distinct points in En. For each k E (l, ... ,r), the inequalities d(ak,x);;; d(ai,r) (i = l,. •• ,r) define a convex polyhedral n-cell, s~, of finite or infinite diameter, having ak in its interior. The set of all intersections of finite subsets of (sl•···•s~) is a cellular subdivision K of En into convex polyhedral cells. Consider a compact differentiable m-manifold M C En without boundary. Given e > 0, let (a l•··.,ar) be chosen on M so

that d(y,~)

157 579-8. M. z. v. Krzywoblocki: Further developments of the method of associating partial differential equations with ordinary ones.

The author discusses further developments of the methods of associating partial differential equations with ordinary ones, based upon the existence proof of the late Professor Michal. In particu­ lar, it is shown that the technique of the "vanishing-coefficients," although not the most general one, leads in several cases to fundamental solutions which are new, and in some cases satisfy the super­ imposed boundary conditions. (Received February 6, 1961.)

579-9. Gilbert Stengle: A construction for solutions of an nth order linear differential equation containing a parameter in the neighborhood of a turning point.

The e·quation (e D)nx + L~=l (s Df-k [ak(t,e:)x) = 0 is considered in the case that the roots .t\k(t,s) of the associated characteristic equation An+ L~-lakAn-k = 0 all coalesce at t = s = 0. In this cir­ cumstance classical techniques of asymptotic solution yield results only outsidt;! fixed t-intervals con­

taining t = 0. Under certain additional hypotheses concerning the~ k solutions are constructed having a larger domain of validity. This is accomplished by reformulating the problem as a nonlinear integral equation which can be solved by successive substitution to .yield a series possessing strong asymptotic properties. The main result asserts that given the ~k• for any p :> 0, rational operations, exponentia­ tions and integrations can be used to construct an approximate solution differing from an actual solu­ tion by an error that is O(eP) uniformly in t and s on a t-interval whose lower limit approaches 0 as e-0. For a large subclass of problems this limit decreases rapidly enough to permit construction of solutions on a complete t-interval containing 0. (Received February 10, 1961.)

579-10. Pasquale Porcelli: The distribution of zeroes of successive derivatives of bounded analytic functions.

The aim of this paper is to prove theorems of the following type. Theorem. Let g(x) be analytic and bounded for lxl < 1, 0 < lzol < 1, C be the circular region lx- zol < lzo - ei arg zo1, and p a non­ negative integer. Then either g(P)(x) = 0 or 1Ts lxs Ins converges, where the product is taken over all zeroes of g(p)(x) in C and ns is the multiplicity of Xs• We present various generalizations and limita­ tions of the theorem. The proofs are reduced to closure arguments in the various Hp spaces via a representation of g and its derivatives in a canonical form as a Laplace transform. We give some related examples in closure theory and raise some conjectures. (Received February 10, 1961.)

579-11. D. F. Dawson: A theorem on continued fractions and the fundamental inequalities.

Let f(a) denote the continued fraction 1~/1 + az~ where the ap are complex numbers, and consider the systems of inequalities (1) r 1 11 + a 1 1 ;s; (1 + r _1) Ia 11. r Zp+lll + aZp + aZp+ll S. rzp+Irzp-llazpl + lazp+ll and (Z) rzpll + azp-l + azpl Sl::rzprZp-zlazp-ll +I azpl' p = l,Z,3, •••• where the rp are nonnegative numbers. Theorem 1. If the inequalities (1) and (Z) hold, rPi < 1, i = l,Z,3, •••• and L(l - rPi) diverges, then f(a) converges at least in the wider sense, and if either r_ 1 > 0 or r 0 > 0, then f(a) converges. Scott and Wall, Trans. Amer, Math. Sov. vol. 47 (1940) required that

158 lim inf r 1r 2 ••• rp = 0. Theorem z. If the inequalities (Z) hold, no azp = 0, and lim inf r 2r 4 ••. rzp = 0, then the odd part of f(a) converges at least in the wider sense and there exists a point v in the extended complex plane, every neighborhood of which contains infinitely many even and infinitely many odd approximants of f(a). A similar result holds if the inequalities (1) hold, etc. Theorem 3. If the inequalities (1) hold, no azp-l = 0, rzpi-l < 1, i = l,Z,3, ••• , and 2:(1 - r 2Pi_1) diverges, then the even part of f(a) converges at least in the wider sense and there exists a point v as described in Theorem z.

A similar result holds for the other case if r 0 > 0, a1;1d so in this case v '# oo. (Received February 16, 1961.)

579-lZ. R. G. Bilyeu: Perturbation of a one-parameter family of autonomous differential equations.

If the one-parameter family of real n-dimensional autonomous differential equations x' = F(x.p) has, for each p in a compact interval C, a periodic solution p(t,p) for which the linear variational equation has only one characteristic multiplier of unit modulus, it is known that (under certain smooth­ ness conditions) for each )l the perturbed equation (1) x' = F(x,,u) + G(x,t,p,e), with G a periodic func­ tion of t and G = 0 for s = 0, has, for sufficiently small e, an invariant torus near the cylinder generated

by p{t, p). It is shown that there is 11 > 0 such that for e < 11 there is a continuous family of invariant tori for the family (1). The rotation number of solutions on the tori is a continuous function of (p.,e) and, if the period 'lr(JI) of p(t, p) is strongly monotone in the sense that 11''(]1) # 0 in C, then there exists {3 >0 such that for e

579-13. j. S. White: A theorem on rank order statistics.

Consider a sample of m(n) independently and identically distributed X's (Y's) with continuous

distribution functions F (x) (G(y)). P m,Jz) is the probability of the rank order z = (z 1, ... ,zm+n> where zi = 0(1) if the ith smallest observed value is an X(Y). The following theorem is proved: If there exists a distribution function H(X) such that F(x) = H(x +a), G(x) = H(x +b), and H(x) = 1 - H(- x), then Pm+k,n(m O's, n l's, k O's) = Pn,m+k(kl's, nO's, m l's), As special cases it is noted that Pz1(001) = P 1z(Oll) = P 21(010) = P 1z(l01) and Pzz(OllO) = Pzz(lOOl) when X andY are normal with common variances. (Received February ZO, 1961.)

579-14. N. L. Alling: On exponentially closed fields.

An ordered field K is exponentially-closed (= e.c.) if there exists an order preserving isomor­ phism f of the additive group of K onto the multiplicative group of positive elements of K and a positive integer n such that 1 + 1/n < f(l) < n. The exponential closure of an Archimedean field (a subfield of the reals) exists and does not increase cardinality but is in many cases a complicated transcendental extension (e.g., for the rationals). In the non-Archimedean case the residue class field is e.c., the value group divisible, and, perhaps most interesting, the value set of the value group is isomorphic to

the positive elements of G. C(X)/M is e.c., as are real-closed '111-fields of power ,lli 11 , 11 > 0.

(Received February ZO, 1961.) 159 579-15. J. L. Alperin: Two p-group counterexamples.

1n a recent paper (Proc. London Math. Soc, (1956) pp; 1-42) P. Hall and G. Higman defined for each positive integer n the class Xn of finite p-groups, where p is any prime. They asked whether the exponent and derived length of members of Xk tended to infinity as n did. A positive answer would give the known result that the exponent and derived length of p-Sylow subgroups of p-solvable groups G both tend to infinity with the p-length of G. However, such a positive answer would give this result by consideration of p-groups only. But the author shows that the answer is negative, in fact proves the Theorem: For each prime p, the class Xn of p-groups contains metabelian groups of exponent at most p2. Another conjecture of some interest has been the following: If G is a finite p-group of order pn then G possesses a normal abelian subgroup of order pn/2• This also is false. 1n fact, the author constructs, for each prime p:;:: 5, a group of order ps, where s = 6p2 + p + 1, all of whose maximal abelian subgroups have order at most pt, where t = 3p2. (Received February 20, 1961.)

579-16. Carl Faith: Derivations and generations of finite extensions.

Let_!<: be a given ground field, let Fr denote the class of finite field extensions of ~ of tr. d. ~ !:• let F = U~r, and let n(L) = the minimal number of generators of L/k for any L E F. Definition, ,! is an (,!)-field in case no pure transcendental extension of_!<: of tr .d • .! contains an impure sub­ extension of tr .d. !. over!£.. Clearly, if,! is a non (!)-field for some !.• then 1! is not monotone on F r•

Our main result establishes a converse: Theorem 1. ~ k is an (r)-field, and if L/K E F has tr.d. r, then for any subextension L7k of L/k, n(L) ~ n(L'). By theorems of Liiroth and Castelnuovo-Zariski

(see Illinois J. Math. vol. 2 (1958) pp. 303-315) one obtains: Corollary. !!: is monotone on F1 defined over an arbitrary field k; 1! is monotone on F2 defined over an algebraically closed field k of character­ istic 0. The next result, used in the proof of Theorem 1, is of independent interest: Theorem 2. For any field k, dim D is monotone in F, where, for any L E F, D(L) is the vector space over L of k-derivations of L. The connection between.!! and dim Q is given: Lemma. Let L/k be a finite exten­ sion of tr.d. r, lets= dim D(L),~n = n(L). Then: (1) s =rand n #- r ~ n = s + 1 = r + 1;

(2) s > r ~ n = s. (Received February 20, 1961.)

579-17. D. S. Gillman: Concentric surfaces in E3.

A collection {Mal aEA of disjoint surfaces in E 3 is called simultaneously concentric if there is a homeomorphism f of M X [0,1] into E 3 and a collection {tal aE"A' t 11e(0,1], such that f(M X t 11) = M 11 • Edwards (Abstract 573-32, Notices Amer •. Math. Soc. Vol. ·;(1960) p. 85) has shown that given uncount­ ably many disjoint tori in E3, two of them must be concentric. Using heavily a result of Bing, (~­ tions under which a surface in E3 is tame, Fund. Math. vol. 47 (1959)), it is shown that a countable num­ ber of these tori must be simultaneously concentric; moreover, this countable collection will be dense in a cantor set of tori. The methods used immediately generalise to compact 2-manifolds of arbitrary genus. Examples are given showing (1) An uncountable number of tori in E 3 such that no uncountable subset is simultaneously concentric (2) An uncountable number of noncompact 2-manifolds in E3 , no two of which are concentric. (Received February 20, 1·961.)

160 579-18, I. L. Lynn: Linearly orderable subspaces of the real line. Preliminary report.

A topological space (X, T) is called linearly order able if there is a linear ordering < on X such that the interval topology induced on X by< coincides with T. Theorem 1, Every zero-dimensional, separable metrizable space is linearly orderable, Theorem 2, If A is a subset of the real line R which is one-dimensional at every point and no component closed in R is isolated, then A is linearly orderable, The subspace of R consisting of either a component open in R and the cantor set or a component open in R and a closed interval, are counterexamples, if either one of the conditions on A is deleted, More­ over, not every subspace of R consisting of closed intervals is linearly orderable. (Received February 20, 1961.)

579-19, J, W. Moeller: On the spectra of some translation invariant spaces,

Let H denote the subspace of all functions in L 2 (- oo, oo) which vanish for negative values of their argument, let L be any subspace of H which is invariant with respect to left translation (i.e,, h(x) E L implies that the projection of h(x + t') on H belongs to L for all positive 7:'), and let R denote the Fourier transform of r1 n H. Using results due to A. Beurling (Acta Math, val. 81 (1949)) and P, D. Lax (Acta Math, val, 101 (1959)), it is possible to writeR = G( 0 such that IG(2n?r -in)..,*) I > 8 for all n = 0, ± 1, .... (Received February 20, 1961.)

579-20, Beauregard Stubblefield: Primitive skew curves as factors of certain compact product spaces in n-manifolds.

Let P be a Peano continuum which contains an infinite sequence of primitive skew curves with diameters converging to a point p in P. Then P X In-2 , n ;;:: 2, cannot be imbedded in an n-manifold, The preceding lemma is useful in proving the following theorem: A sufficient condition that a Peano continuum be locally imbeddable in the plane is that P X In-2, n ;;;:;- 2, be imbeddable in ann-manifold. (If P contains no local cut points, then the condition is also necessary,) (Received February 20, 1961,)

579-21, J, R. Webb: An integral representation for the continuous linear functionals on a space of quasi-continuous functions,

Let [a,b] denote an interval and Q[a,b] denote the normed linear space of all quasi-continuous, complex-valued functions from [a,b] such that the norm of each function x in Q[a,b] is lublx(t)l fort in [a,b], Theorem: Suppose F is a functional from Q[a,bJ into the complex numbers, F is a continuous linear functional if and only if there exist a continuous nondecreasing function u from [a,b] onto [a,b] and a complex-valued function v of bounded variation on [a,b] such that F(x) = jb~[u]dv for each function x in Q[a,b] and the norm ofF is the variation of v over [a,b]. The indicated integral is a mean Stieltjes integral under refinement and the function x[u] is the composite of x with u. (Received February 20, 1961.)

161 579-22. J. L. Howland: On the expression of polynomials with real coefficients as symmetric determinants.

The polynomial Pn(A) =)In- a 1J\n-l- a2-'ln-2 ••• -an may be written in determinantal form Pn(}l) = (- l)ndet(M- )\I) by constructing the companion matrix M. The process of matrix symmetri­ zation [Kolmogoroff, Math. Ann. vol. 112 (1936) pp. 155-160] may be applied to derive expressions of the form Pn())) =,! det(A- )\B) where Band A= BM = MTB are real, symmetric matrices. In one such case, the structures of A and B are essentially the direct sums of 2 X 2 matrices (0 - 1; - 1 an-2j) and (an-l-2j - 1; - 1 0) respectively, so that the characteristic value-vector problem

Ax-+ = ~Bx-+ is in jacobi or triple diagonal form. The characteristic vectors of this problem are simply expressed in terms of the roots and coefficients of P n(A). In another case, the elements of A and Bare the sums Sp = L~=l,\~of the pth powers of the roots of Pn(})), which may be obtained by means of Newton's Identities. In this case, the matrix B is positive definite if, and only if, Pn(}j) has real roots. Moreover, if R is an arbitrary real matrix, it is known that Sp = Tr [RP] so that the determinantal representation of the characteristic polynomial of R may be obtained directly from R and Rp (1 ~p ~2n). (Received February 20, 1961.)

579-23. R. H. Bing: Almost cartesian product neighborhoods of 2-spheres.

It is shown that 2-spheres (even wild ones) in E 3 almost have cartesian product neighborhoods in the following sense: Suppose h' is an arbitrary homeomorphism of a polyhedral 2-sphere S into s3 and~ >0. Then there is a piecewise linear homeomorphism h of S X[- 1,1] into E 3 such that f' (h'(p), h(p X t)) < e, h'(S) lies except for a finite family of mutually exclusive e-disks between h(S X - 1) and h(S X 1), h(S X - 1) lies except for a finite family of mutually exclusive e-disks in the bounded component of E 3 - h'(S), and h(S X 1) lies except for a finite family of mutually exclusive c-disks in the unbounded component of E3 - h'(S). If we replace e by a non-negative continuous function f on S, we get the following extension of this result. If V is the set of points of S on which f is positive, there is a homeomorphism h of V X [- 1,1] into E 3 such that ,P(h'(p), h(p X t)) < f(p), h'(V) lies except for a family of mutually exclusive small disks (of diameter less than the minimal value off on the disk) in h(V X(- 1,1)), h(V X- 1) lies except for small disks in one component of

E 3 - h'(S) and h(V X 1) lies except for small disks in the other component. The proof of these results are modifications of arguments given in Approximating surfaces with polyhedral ones, Ann. of Math. vol. 65 (1957) pp. 456-483. (Received February 20, 1961.)

579-24. Carl de Boor: Bicubic "spline" interpolation.

A method is described for interpolating a piecewise bicubic polynomial function u(x,y) of class C 2 through a given set of values u(Xi>Yj) = lltj on a rectangular mesh of points (xi•Yj) where au/3n is prescribed at the boundary mesh-points of the grid, and a2u/3x:3y at the four corners of the grid. The procedure is based on linearized "spline" interpolation as described in a paper entitled Smooth surface interpolation, by G. Birkhoff and H. L. Garabedian (cf. Abstract 571-155, Notices Amer. Math. Soc. vol. 7 (1960) p. 633). The method is particularly amenable to rapid machine computation. (Received February 20, 1961.) 162 579-25. Fred Brauer: Asymptotic equivalence of differential. equations.

The systems of differential equations (1) x' = A(t)x, and (2) y' = A(t)y t f(t,y) are said to be asymptotically equivalent if, corresponding to any solution x(t) of (1) there is a solution y(t) of (2) such that limt-Hn [y(t) - x(t)] = 0. Suppose all solutions of (1) are bounded, and let X(t) be a fundamental matrix for ( 1). Then, if for any vector c there is a solution u(t) of u' = X - 1(t)f(t,X(t)u) such that limt_,.oo u(t) = c, the systems (1) and (2) are asymptotically equivalent. The problem of finding solu­ tions of differential equations which tend to specified limits as t-+ oo has been studied by Wintner If\ mer. J. Math. vol. 68 (1946) pp. 451-454], and it is possible to give some explicit conditions for asymptotic equivalence, which include those of Levinson [Amer. J. Math. vol. 66 (1946) pp. 1-6] and Wintner 1!\mer. J. Math. vol. 68 (1946) pp. 185-213]. (Received February 20, 1961.)

579-26. F. W. Carroll and J. H. B. Kemperman; Metric results for noncontinuable analytic functions.

The following results are obtained using a lemma of M, Riesz and the theory of distribution of numbers (mod 1). Let ((y)) = y - fy]. (i) Let p be a fixed integer, [an(Y)} unif. bdd, and meas. such that limN-. 00 supkl(l/N)'L~!~J'J an(y)e211'ipydyl > O. Then for almost all real A, F A(z), given for lz I < 1 by L~ {an( ((An)) ) + sn} zn, is singular at z 0 = e 211'iAp for all null sequences [snL Hence, if a(y). is bdd. and meas. on (0.1) and its F. S. is not a trig. polynomial, then I::, a( ((An)) ) zn has lz I= 1 as a natural boundary for almost all A. (ii) Let {lan!J be bounded, and (*) limNinfk~!~ !ani= oo. If {fnl is real and if, V integer N > 1, ] integers 0 < kl <. k 2 < ... 3, for each choice of integers t 1, ... ,tN not all 0, lim:v_..oo ll:_~ 1 ~fk:v+nl = oo, (e.g., fn = 9n, 9 > 1 and transcendental), then for almost all real A, FA(z) = L:g:> [ian ie 211"iAfn + enr zn has lz I = 1 as N. B. for all null sequences fen}. Example: if (*)holds, then L~ 1anle211'iAn 0, j9;;:; 1) has

iz I = 1 as N. B. for almost all A. (iii) For a large class of functions g(y), including all polynomials in y and [y] of deg. ~ 1 in fy], the function L~g(An)zn has iz I= 1 as N. B. for all irrational A. This generalizes results of Mordell and of M. Newman, (Abstract 571-93, Notices Amer. Math. Soc. vol. 7 (1960) p. 515 and Proc. Amer. Math. Soc. vol. 11 (1960) pp. 699-702). (Received February 20, 1960.)

579·-27. M. L. Curtis: Regular neighborhoods. Preliminary report.

The following two theorems are easy consequences of recent results in topology. Theorem 1; .Let P be a contractible subcomplex of dimension § n - 3 in Sn. Then a regular neighborhood of P is ann-cell. Theorem 2: For each n S;: 4 there exists a contractible (n - 1)-subcomplex P of sn which has a regular neighborhood N with non-simply-connected boundary (and hence N is not a cell). Hence it remains to investigate (n - 2)-dimensional contractible subcomplexes of sn. Mazur (Ann. of Math. (1961)) has shown that there is a contractible 2-dimensional subcomplex P of s 4 with a regular neigh­ borhood M such that bdry M is not simply connected. The present paper is devoted to a scheme for 4 calculating TI 1 (bdry M) for M a regular neighborhood of a contractible 2 -complex P in s • The calculation is feasible because it is possible to reduce consideration to the case in which P is a bouquet of circles with disks attached. We also prove the (unrelated) result: Let G be a compact k-dimensional Lie group embedded (possibly wildly) in R n. Then G has a neighborhood N in R ntk such that N is homeomorphic with G X In. (Received February 20, 1961.)

163 579-ZS. M. I. Knopp: Modular forms of positive dimension and Poincare series.

We here restrict our attention to modular forms of positive even integral dimension r, with multiplier system identically one. (The only essential restriction here is that r be an integer. All of these results have been obtained for forms of positive integral dimension, with arbitrary multiplier system, on an H group.) Let ;u be an integer;;::; 1. We ·start with a Fourier series ~ (?:') = t-;< + Loamtm, t = e(£), J1. (t") > 0, where the coefficients am are those given by Lehner (see Construction of automorphic forms on H groups, this vol.); here am depends upon r and p. By using the fact that F,.tl (i) - ?:'rF(- 1/'t') = P.f<('r'), for cf(7:) ::> 0, where (~/f) is a polynomial in 'r of degree '§! r (see previous reference), we prove: Theorem. F ('r) is a modular form of dimension r if and ~ --- only if the Poincare series G.f<(r) of dimension - r - Z vanl,shes identically. This result is also con- nected with expansions of zero in cf (1:) < 0. Lehner has proved one-half of this theorem by different methods without the assumption that r is an integer. Namely, he has shown that if l)t (7:) is a modular form, then G)"(t")== 0. (Received February ZO, 1961.)

579-Z9. S. V. Parter: On the extreme eigenvalues of Toeplitz matrices.

Earlier results of Kac, Murdock and Szego; H. Widom and S. V. Parter are extended as follows. Let f(a) be a real continuous function and periodic with period Z7f". Let min. f(a) =f(O) = 0 and let a = 0 be the only point (mod Z7r) at which f(B) assumes this minimum value. For any positive real number u let k = k(u) be the smallest integer greater than or equal to u/Z. Let g(a) = ~(BJZk/u and let g(a) have Zk continuous derivatives in some neighborhood of a = 0. Finally let g(Zk)(O) = O'Z > 0 be the first l!lOnvanishing derivative of g(B) at a = o. Note: This means f(a) ,._..,.alB IQ(l + r(B)) for a near 0 and~= [0'2 /(Zk)~]u/Zk. Let T n ~] be the Toeplitz matrix of order n + 1 associated with f(a). Let)) be eigenvalues of T l.f) ordered in nonincreasing order. Theorem. As n- oo, v,n n (n + Z)u .il,.,n __, (3Av where A,. is the Y'th eigenvalue of a particular Integral operator Ko(x,y;u), defined on [0,1). In particular if u = Zk the Ko(x,y;u) is the Green's function of [- (d/dx)2l subject to the boundary conditions U(j)(O) = U(j)(l) = 0, j = 0,1, ••• k - 1. (Received February ZO, 1961.)

579-30. P. Porcelli and F. F. Connor: On equivalent measures.

Using an earlier result of Connor (Ph.D. thesis, I.I.T.), an embedding theorem of Porcelli (J. Math. Mech. vol. 9, p. Z75), and a decomposition theorem of Darst (to appear, Crelle's Jour.) we prove the following Theorem: If X is a set, r: an algebra of subsets of X, and each of $ and r a measure on ~(i.e. E contains X and need not be a sigma algebra; also,.; and r need not be countably additive or positive or complete relative to Z), then s and ?' have the same summability set if, and only if, there exists a sequence fn of I:-simple functions on X and M ::> 0 such that (1) M-l..:: Ifni< M,

(Z) fn converges in ~ measure, (3) 1/fn converges in r measure, (4) "7' (E) = limn./'Efnd;, and (5) ; (E)= limnj'E (1/fn)d?'. Our proof shows that$ and r give rise to the same L 1 space upon their completion on ~ if, and only if, they have the same summability set on a suitably selected and denumerable sub-algebra of I:. We define the summability set (e.g.) of 5 to be the set of all sequences £fJ of !:-simple functions such that fn converges in ~measure andJ'.x lfn -J}n l•ld ~I~ 0. We give an example to show that our definitions and hypotheses are appropriate. (Received February ZO, 1961.)

164 579-31. C. A. Rogers and S. ]. Taylor: On generalising the Lebesgue decomposition to non o -finite measures.

Suppose F(E) is a finite completely additive set function defined on B, the field of Borel sets in Euclidean n-space, and ).) is any measure also defined on B. Then there is a unique decomposition of F into three finite completely additive set functi-ons F 1, F 2, F 3 such that (i) F 1(E) = 0 whenever the Y-measure of E is a--finite; (ii) F 2(E) can be expressed as the integral over E with respect to Y-measure of a point function f(t) which vanishes outside a Borel set S of a--finite J)-measure; and (iii) F 3(E) is singular w.r.t. )!-measure. Whenever Yis the Hausdorff measure defined by the measure function h(t), the above decomposition can be expressed as a restriction of F to the sets where the upper density ofF w.r.t. h(t) is respectively zero, positive and finite, and infinite. This generalises the decomposition of De la Vallee Poussin. (Received February 20, 1961.)

579-32. Seymour Schuster: On Ostrom's separation relation in nonordered planes.

Let F be a field not of characteristic two, such that (1) F contains non-squares, (2) the product of every two non-squares is a square, and (3) - 1 is not a square. T. G. Ostrom has shown (Arch. Math. vol. 10 (1959) pp. 200-205) that a separation relation (lacking only one of the usual separation properties) can be introduced in projective and affine planes coordinatized by F, even though F may be unorderable. His method relies on considering a conic C through two points of a tetrad, and evaluating the quadratic form, which defines C, at the remaining two points of the tetrad. The present paper concerns itself with introducing separation in such planes by means of the cross-ratio, and showing that this procedure yields a relation equivalent to Ostrom 1 s. Each property of separation is then seen as a direct consequence of specific algebraic conditions on the field. (Received February 20, 1961.)

579-33. R. B. Darst: A Lebesgue decomposition for complete normed abelian groups.

Let G denote a generalized complete normed abelian group, where generalized means that the norm (llxll) of the nonzero elements x of G may be infinite (i.e., if x E G and x # 0, then

0 < llxll ~ oo). LetT denote a set of endomorphisms on G such that (1) T is an algebra of endomor­ phisms: if a, b E T, then (i) a • b = b • a E T, (ii) a • a= a, (iii) a' = e - a E T, where e(x) = x for x E G, and (2) if x E G, llxll < oo, and 8 > 0, then there exists e > 0 such that if a, b E T and

lla'b(x)ll >8, then ll(a t a'b)(x)ll > lla(x)ll t e. Theorem. Suppose x, y E G and IIYII < oo. Then there exists uniquely h and s in G such that (1) y = h t s, (2) h is absolutely continuous with respect to x(mod T): i.e., if e > 0, then there exists 8 > 0 such that if a E T and lla(x) II<. 8, then lla(h) II<. s, and (3) s is singular with respect to x (mod T): i.e., if e > 0, then there exists a E T such that lla(x)ll < e and lla'(s)ll <.e. Example. LetS be an algebra of subsets of a set X, G the space of finitely additive set functions on S (if x E G, then llxll = V(x,X)), and T = lj>E ;PE (x)(F) = x(E n F) for E,F E S,x E G]. Then our theorem generalizes the Lebesgue decomposition theorem for bounded and finitely additive set functions on S. (Received February 20, 1961.)

165 The April Meeting in Stanford, California April 22, 1961

580-1. J. L. Brenner: Roots and vectors of Mahler matrices.

In the Proceedings of this Society, vol. 8 (1957) pp. 525-531, K. Mahler defined certain n X n matrices with elements 0,1, -1. D. H. Lehmer found the roots and determinants of these matrices (these Notices, vol. 7, p. 365, Abstract 569-50). In the present paper the matrices are completely solved by a new method so that the elementary divisions, vectors, and roots are found. Moreover Lehmer's re­ sults are generalized in various directions. For example, the elements may be scalar matrices instead of numbers; the generalized Mahler matrix A is then a composite m-circulant (see these Notices, vol. 7, p. 256, Abstract 569-1). The starting point of the investigation is the matrix equation QA = AQm, which defines the Mahler matrix A if Q is suitably chosen. (Received November 29, 1960.)

580-2. Paul Slepian: Direct proof that every dominant matrix is paramount.

It is well known that (1) if A is an nth order real symmetric dominant matrix there exists an n-port resistive network whose short circuit admittance matrix is A, and (2) if Y is the short circuit admittance matrix of a resistive networ.k then Y is paramount. Thus, (1) and (2) show that every dominant matrix is paramount. However, (2) is a rather deep result and involves some topological concepts. This paper gives the first direct algebraic proof that every dominant matrix is paramount. Although notationally complicated the proof is not difficult and depends upon various classical in­ equalities involving adjugate compounds of matrices. (Received January 4, 1961.)

580-3. Hajimu Ogawa: On difference methods for the solution of a Tricomi problem.

Let L = K(y)(J2/ox2 + a2/f.!)y 2 + a(x,y){}/9x + b(x,y)9/oy + c(x,y), with K continuous, K positive for y > 0, and K negative and increasing for y < 0. Let A and B be two points on the x-axis with XA < xa· and let D be the bounded domain bounded by a simple curve r+ in the upper half-plane with endpoints at A and Band by the two characteristic segments r 1 and r 2 passing through A and B, respectively. Consider the problem of finding the solution of Lu =fonD subject to the boundary conditions u = 9i 1 on r+ and u = !1i2 on r 1· The differential equation is approximated by a difference equation in a function U defined on a mesh domain Dh, and it is proved that U satisfies a maximum principle under certain conditions on the coefficients. Using the maximum principle, it is shown that as the mesh size tends to zero, the solution of the difference equation converges to the solution u of the differential equation, provided that the latter exists and is twice continuously differentiable on D. If further restrictions are imposed on r+ and K, it is proved that the regularity conditions on the derivatives of u at the boundary of D may be weakened. (Received January 13, 1961.)

166 580-4. Edgar Karst: On the nonexistence of multiplicity in prime factors of Mersenne numbers.

Based on remarks by A. Ferrier (Etude de zn .±.I, Mathesis LXIV, mal 1955, p. 41, et LXIX, juillet 1960, p. 141) and his latest manuscripts of September 18 and October 5, 1960, the author incorpo­ rated all pk< 20000 in Ferrier's list. The following laws were found or clarified: (I) All prime num­ bers P (except 2) are contained either in zn - I with n odd or in zn + I, but not in both of them. (2) The first occurrence of P is at n = (P - 1)/q where q is even. D = 2n is than the smallest difference in n between neighboring P of the same amount. (3) If P (of zn - I) = zP - 1 then q = (P - 1)/p, for example: P = 7, 31, 127, 8191. (4) If P (of zn +I)= zn +I then q = (P - 1)/n, for example: P = 3, 5, 17, 257. (5) If first occurrence of P is at n = (P - 1)/q then first occurrence of P 2 is at n = P (P - 1)/q, for example: 32 at n = 3, 5 2 at n = 10, 72 at n = 21, 11 2 at n =55. (6) If first occurrence of pk is at n = Pk-l(P - 1)/q then first occurrence of pk+l is at n = Pk(P - 1)/q, for example: 33 at n = 9, 53 at n = 50, 7 3 at n = 147, 34 at n = 27. The last two laws prove the Nonexistence of Multiplicity in Prime Factors of M-ersenne Numbers, that is: zP incongruent I (mod Pk). There is also: zP incongru­ ent -1 (mod Pk) except P = 3. (Received January 31, 1961.)

580-5. M.S. Cheema and T. S. Motzkin: Vect.or partitions and permutation vectors.

Recurrence relations are obtained for b(n) and B(n), the numbers of partitions of n = (n l•···•ns) into parts (nonzero a-vectors with nonnegative components) and into parts with positive components, and for b' and B', where the parts are distinct. The number qr(n) of partitions into at most r parts is

expressed as L:Ar

characteristic numbers equal ml'm2• There follows the asymptotic behavior, for n1;r3 --'> oo,

of qr,q~,Qr,Q~.Pr (the number of partitions into exactly r parts), p~,Pr• P.f: and (implying results of M. Robertson (1960)) of Band B'; also, by a result of G. Rieger (1959), r!(r- I)!sqr =1T'nr 1(1 + Cr,Z ·2:nj'1 + ... )for s >1, ni ~oo and constant r. The qr are easily computable by rqr(n)

= LjZn qr-s(n - jm). Properties of A and q for s = 2 are established (e.g. ,1 <(2M)! for m 1 < M,m2 < M and all r) and explicit formulas given for)) (s = 2, m 2 :;; 3) and for q,q',Q,Q',p,p',P,P' (r &. 4). For (s = 2, r ~ 9) the)\ were obtained by examining all permutations on the SWAC and hence the

q(n1 ~ 29, n 2 ~ 29) on an IBM-709 as coefficients in a product of two polynomials in two variables. (Received February 2, 1961.)

580-6. F. H. Brownell: Finite dimensionality of the Schrodinger operator bottom.

For euclidean n space En• consider [Houl(x) = - 'V 2u(x) + V(x)u(x) over x E En for u €. ;!}0 , the set of all u E L 2(En) such that Vu E L 2(En>• that all order ~ 2 partial derivatives of u exist continuous 2 throughout En' and that all such are 0([1 + rm]e-r 12) with r = lxl and m depending on u. We assume that V is a real valued, measurable function on E for some Ro E (0, + oo) such that V(x) is essentially bounded over lx IE;; R and satisfies Jlxi~Ro IV(x) l.,..d}ln(x) < + oo for some r ~ I satisfying 'J" > n/2, Pn being n dimensional Lebesgue measure. We assume also that V(x) 5: - g( lx I> almost (pn) everywhere 2 over lxl ;;:; R0 , with continuous g(r) ;;;:; 0 on [Ro, + oo) being o(r- ) as r --+ + oo. If In - 21 = I, then

167 J~ r g(r) dr < + oo is also assumed; if n = 2., then g(r) !!!! 0 over r il!:; R for some R ;;;;, Ro is assumed, Let the self-adjoint extension H of Ho be given by H = G-;] - cu2I for sufficiently large cu ::>- 0 ('Y;;: 2. assures a unique self-adjoint extension), where q., is a constructed bounded Hermitian integral opera­ tor with Green function kernel. Let E be the spectral measure of Hand let N(.-\) be the dimension of the eigenspace E(( - oo,-'!]>Lz(En) for real ,\. Then our conclusion is that N(J\);:;; No< + oo over )) < 0, (Received February 6, 1961,)

580-7. R, M. Blumenthal and R. K. Getoor: The Hausdorff-Besicovitch dimension of the zeros of a symmetric stable process,

Let {x(t), t ,s: 0} be the symmetric stable process of index a, 0 < a~ 2., in one dimension. We assume x(O) = 0 and that the sample functions are normalized to be right continuous and have left-hand limits everywhere. Define Z(cu) = ft: x(t,w) = 0 or x(t-,cu) = 0}. It is known that Z(c::..>) is empty with probability one if 0

580-8, C. J, Standish: A real inversion formula for a class of convolution transforms.

The following theorem is established, Theorem. Let f(x) = JC: G(x - t)ll(t)dt, where j'_~G(t) exp (-ist) dt = ~ 1 (1 + x2/ai>J-l, L:~ 1 (a~)- 1 .c:.oo, 0 < lakl ;:o lak+d• k = 1,2., .. ,, converge for a single real x then it converges for all real x and for almost every x is inverted by: lime-+O+ limN~oo j'f(:e")d~.J'TT~=l(l + u2 /a~) exp (- eu2 - iu(x- S')du = ll(x). Other inversion formulas for this transform have been obtained by Pollard, Ann, of Math. val, 49 (1948) pp. 956-965; and by Hirschmann and Widder, Duke Math, J, val. 17 (1950) pp. 667-674. (Received February 13, 1961.)

580-9, L. E. Ward, Jr.: Fixed point theorems for pseudo monotone mappings,

A continuous mapping f:X- Y is pseudo monotone if for each closed, connected A and B with B C f(A), some component of A(\ r 1 (B) maps onto B. Any monotone mapping defined on an hereditarily unicoherent continuum is pseudo monotone. We generalize a theorem of Kelley's (for homeomorphisms) as follows: !f X is a continuum and f:X- X is pseudo monotone, then X contains a cutpoint-free f-fixed subcontinuum. Corollary. If X is a continuum each of whose subcontinua has no cutpoints then X has the fixed point property for monotone mappings. It is not known whether this corollary holds for arbitrary continuous mappings. (Received February 14, 1961,)

168 580-10. Morgan Ward: The relationship between the modular rank and period of a second order recurrence.

If (L) is the Lucas recurrence associated with f(x) = x 2 - Px + Q and m is any number prime to Q, the modular rank of m is its rank of apparition fin (L) and its modular period is the period 'TT'of (L) modulo m. f divides 'lr; 1(' = T'f. This papei> studies T as a function of p.m and f(x); m may assume to be a prime p. The value ofT is explicitly given except when ,Pand the exponent of Q modulo are both odd. This critical case may be reduced to the case when Q = 1 so that t: is one or two. Criteria are given in terms of the 2kic character modulo p of the roots and discriminant of f(x). Each choice of k determines a class of primes for which the criteria are valid, and these classes are given explicitly for k = 1,2. The actual criteria are similar to those given for a related problem in a paper "The prime divisors of Fibonacci numbers", to appear in the Pacific J. Math. (Received February 16, 1961.)

580-11. Frank Harary and Geert Prins: Two counting problems for bicolored graphs.

A bicolored graph is one whose points have been colored using two colors so that each line joins two points of different colors. The number of (nonisomorphic) bicolored graphs having a given number of points and lines has been found If. Harary, Pacific J. Math. vol. 8 (1958) pp. 743-755]. In this note, that result is extended to the counting of connected bicolored graphs and also of bicolored graphs with no isolated points, again using the enumeration method of Polya [Acta Math. vol. 68 (1937) pp. 145-254]. (Received February 17, 1961.)

580-12. R, W. Hunt: The behavior of solutions of ordinary, self-adjoint differential equations of arbitrary even order.

The differential equation (r(x)y(n))(n) + p(x)y = 0, r(x) :>0, p(x) t- 0 on [!t,oo) is studied with re­ gard. to the existence of various types of zeros of its solutions. Of chief interest in the first part of the paper are solutions with two nth-order zeros and solutions y(x) with an nth-order zero followed by an nth-order zero of r(x)y(n)(x). This parallels J. H. Barrett's work for the case n = 2 I!J. s. Army Math. Res. Center Report No. 150). Some of the results obtained in the first part by matrix differential equations techniques, particularly matrix Riccati equations, are special cases of results obtained by different (variational) methods by W. T. Reid (Abstract 576-197, Notices Amer. Math. Soc. vol. 7 (1960) p. 970). In the latter part of the paper, zeros of types which do not seem to lend themselves to varia­ tional methods are considered; and separation and oscillation properties are studied for the case p(x) :> 0. Some of the results of the paper generalize properties obtained by W. Leighton and z. Nehari for the case n = 2 [Trans. Amer. Math. Soc. vol. 89 (1958) pp. 325-377). (Received February 20, 1961.)

580-13. W. R. Mann and j. M. Gwynn, Jr.: Variable spacing finite difference analogues. Preliminary report.

This report consists of the results obtained in a study of finite difference analogues using vari­ able spacing. The first part of the report considers one aspect of the use of variable spacing in approxi-

169 mations to the laws of exponential growth and decay. This leads to'a well-known generalization of the definition of e, and to the conclusion that constant spacing is. in one sense, optimum. The second part of the report considers variable spacing in finite difference analogues of the heat equation. It is found that variable spacing in the x-direction does permit a larger time step, in the explicit, 4-point stepping ahead procedure, than is permitted by the stability requirement for constant spacing. The gain seems to be rather small however, and no general directions have been found for determining spacing except in a few very special cases. (Received February 20, 1961.)

580-14. D. W. Robinson: A note on k-commutative matrices.

The purpose of this note is to provide an elementary proof of the following characterization of k-commutative matrices. Let A and B be n-!!I_-n matrices over an algebraically closed field. Then A is k-commutative with respect to B if and only if B commutes with every principal idempotent of A. The proof of this result not only simplifies much of the previous study of these matrices (see W. E. Rotb, Trans. Amer. Matb. Soc. vol. 39 (1936) pp. 483-495), but also provides several interesting corollaries on matrix commutators. (Received Februax:y 20, 1961.)

5l)0-15. Takayuki Tamura: Characterization of singular semigroups.

A semigroup S defined by xy = y (for all x,y) is called a right singular semigroup, and a semi­ group, xy = y for all x,y, is called a left singular semigroup. Such semigroups have very particular properties even in the algebraic systems. The author will speak of the several characteristics, _for example: (1) Any transformation of S into S is an endomorphism of S. (2) Any transformation of S into S is a right translation of S. etc. Also, he will speak of the relation between singular semigroups and the following properties: (1) Any partition gives a factor semigroup. (2) Any subset is a sub­ semigroup. (Received February 20, 1961.)

580-16. L. L. Berryman, J. F. Latimer, R. B. Merkel and T. Tamura: On the direct product of certain groupoids.

Let S be an algebraic system with a binary operation (i.e., a groupoid in Bruck's sense) satisfy­ ing the following conditions. (1) There is at least one left identity e: ex = x for all x. (2) (xy)z = x(yz) holds if at least one of y and z is a left identity. (3) For any element a, there is a unique left identity e (generally depending on a) such that ae = a. Such an S is called an }1t-groupoid. Then an 11't-groupoid is the direct product of a right singular semigroup and an "'t'groupoid with a two-sided identity. This is a generalization of multiple group in Clifford's sense. As is well known, there are several equivalent conditions for a multiple group. What are conditions for an Jft::.groupoid? (Received February 20, 1961.)

580-17. R. J. Blattner: Infinitesimally indu_ced representations.

Let G be a Lie group, g the Lie algebra of G, gc the complexification of g, and + the conjugate linear anti-automorphism of gc which= - I on g. Let H be a closed subgroup of G and n a complex sub­ algebra of gc such that n n n+ is the complexification of the Lie algebra h of H, n + n+ is a subalgebra

170 of gc, and (ad H) n = n. Let /land 8 be the modular functions of G and H, respectively. Let M and jJ be representations of Hand n, respectively, on the finite dimensional Hilbert space V satisfying:

(1) 6,1/22)-1/2 M is unitary; (2) dM = ;Oih; (3) jJ(adx X)= M(x)f(X)M(x) -1, x ~ H, X E. n. In terms of M and p. a unitary representation uM•,.O of G is defined, which, in case n = n+, is an induced repre­ sentation in the sense of Mackey. The intertwining number of two such representations is estimated

in terms of the dimension of a space of distributions on G. Finally, in case n + n + = gc, it is shown that M irreducible implies uM•f irreducible or= O. (Received February 20, 1961.)

580-18. K. L. Cooke: Transcendental equations related to differential-difference equations.

In a recent report, W. L. Miranker discussed existence, uniqueness, and stability theorems for a system of nonlinear differential-difference equations of a quite general form. Associated with this system is the characteristic matrix H(s) = LjAjs exp (- tjs) + LjBj exp (- tjs). Here the Aj and Bj are constant N by N matrices and the tj and tj are constant "lags "• Miranker establishes asymptotic stability of all roots of the characteristic equation det H(s) = 0 satisfy (l) Re s ,;: - c < 0. He poses

the question of the precise relation between the conditions (l) and (2) inf tj ~ inf t1• In this paper we show that (2) is not sufficient to insure that all roots satisfy (1), nor even that all roots lie in some left half-plane. Under the additional hypothesis that det Ao f. 0, (2) is a necessary and sufficient con­ dition in order that the roots lie in some left half-plane. (Received February 20, 1961.)

580-19. Malcolm Goldman: Generalized arcsin laws.

A modification of the moment method of Darling and Kac (Trans. Amer. Math. Soc. vol. 84 (1957) pp. 444-458) is used to find the asymptotic distribution of time which the one dimensional Wiener process spends in certain subsets S of the positive reals. These subsets are characterized

by the existence of limT- 00 T-pJOT c(t)dt, where c is the characteristic function of S and 0 < p ~ l. The limiting distribution is not found explicitly, but the moments can be found by solving a certain singular integral equation which can be easily solved only when p = l in which case it is a Cauchy type and the Arcsin Law is obtained. (Received February 20, 1961.)

580-20. Alfred Horn: Eigenvalues of sums of Hermitian matrices.

Let a. and (3 be nondecreasing n-tuples of real numbers. Let E be the set of nondecreasing n-tuples ?"such that there exist Hermitian matrices A and B such that A, B, and A + B have a.,;S,

and 'Y respectively as their sequences of eigenvalues. The set E is determined for n ~ 8. It is a convex set on the hyperplane determined by the trace condition, and is determined by inequalities of

the form ?'k1 + ••• + ?"kr ~ a.i 1 + ••• + a.ir + J'Jt + .•• + ftjr· A conjecture is stated for the general case in terms of the sets of triples k,i,j of sequences of integers. A method of determining inequalities of the. above form is given for arbitrary n. A result on the boundary of E for arbitrary n is proved. (Received February 20, 1961.)

171 580-21, R. J. Nunke: Slender groups,

Let TTbe the direct product of countably many copies of the integers Z. J, Los calls an (abelian) group slender if it is torsion-free and if every homomorphism of TT into it sends all but a finite number of the factors of TT into 0, I show here that a torsion-free group is slender if and only if it contains no divisible subgroup, no copy of the p-adic integers for any prime f• and no copy of TT. The proof uses the following theorem: a homomorphic image of Tf is the direct sum of a divisible group, a cotorsion group, and the direct product of at most countably many copies of the integers. A special case of this theorem, namely the structure of TilL: where L, is the direct sum, was proved by S. Balcerzyk. This special case is used in the proof of the theorem. A group G is a B -group if Ext(G, T) : 0 for every torsion group T and is a W -group if Ext(G,Z) : 0, These groups were named by J, Rotman. He proved assuming the continuum hypothesis, that every B-group satisfying another condition and every W-group is slender. Using the above description of the slender groups Rotman's result can be proven without the continuum hypothesis and without the additional condition on the 8-groups. (Received February 20, 1961.)

580-22, E. S. O'Keefe: Verification of a conjecture of Th. Skolem,

Theorem. If n : 2 (mod 4) or n: 3 (mod 4), the integers (1,2,3,. .. ,2n - 1, 2n + 1) can be arranged inn disjoint pairs (ar,br), (r: l, . .,,n) so that br- ar: r. (Received February 20, 1961.)

580-23, T. G. Ostrom: Planar half loops,

A planar half loop L is a set of elements together with certain specified subsets called 0-lines and binary operations satisfying certain conditions, including the condition that the 0-lines have well defined cosets. The 0 -lines and their cosets form the lines of an affine plane, The plane is Veblen­ Wedderburn if L is associative. Some aspects of the algebraic structure of L (and the geometrical interpretations) are investigated. (Received February 20, 1961.)

580-24, W. E. Milne and R, R. Reynolds: Fifth order methods for numerical solution of ordinary differential equations,

The term "fifth order methods" is here applied to predict-correct methods using open and closed quadrature formulas having truncation errors proportional to the fifth power of the step length, The first part of the paper continues the investigation of R. W. Hamming (J. of ACM, January 1959) and of Milne and Reynolds (J. of ACM, April 1959 and January 1960) relative to stability. Improved .methods are presented, one using the stabilization idea of Milne and Reynolds, another using stable formulas as proposed by Hamming. These methods use "four-point" formulas in contrast to "three­ point" formulas derived by T. E. Hull and A. C. R. Newbery (J. SIAM, December 1959). The second part of the paper treats the case where only one substitution is performed per step. The question of stability for this case is investigated and compared with the case of two substitutions per step, It is shown that for some methods and with suitable restrictions on step length it is much better to use only one substitution per step than to double the step length and retain two substitutions per step. (Received February 20, 1961.)

172 580-25, H. E. Salzer: A multi-point generalization of Newton's divided difference formula,

Newton's divided difference formula has the following multi-point generalization: The unique interpolation polynomial P(x), of degree Li=,1ri- l, which is determined by f(xi) =fi =£> 0fi and the first ri- 1 divided differences, iJmf. = [x.x .... x. ], m = 1, ... ,ri- l, at then points xi' i = l, ... ,n, \"n sn~ 1 11 1m ,t{ q-1 is expressible as (l) P(x) = .L-1=ll... j=1,j•H(x- Xj)(X- xh) ... (x- xJrj_ 1>1 Lm=O [xixi1 ... xim] q-m-1 _ _ • L [xi Xi ... xi + J /TT (x - xi)(x - xi ) ... (x - xi )} , where Xio::::. Xi, fi• s= 0 m m+ 1 m s i 1 1 m+s-1 ~i] = the sth order divided difference [x1· xi +1... x1 + ] l/ is for the function l/TI~ ,,..(x- x.,}(x - xJ· ) ... (x- x · . ), m m m S· 71'1 ]- 1 •JT1 J 1 1rr 1 and [ximhhri is multiplied by l when m = 0, Formula (1) generalizes, and is also more explicit than, a previous similar formula of the author, in terms of advancing differences at equal intervals around each xi• Some possible computational advantages of (l) are indicated, The proof of (1) employs induction and the Popoviciu-Steffensen theorem for the divided difference of the product of two functions. (Received February 20, 1961.)

580~26, D. H. Lehmer, Emma Lehmer and W. H. Mills: Pairs and triples of consecutive power residues.

It is readily seen that every prime > 5 has a pair of consecutive quadratic residues not exceeding 10, and that an infinity of primes have no such pair less than (9,10), Similarly for cubic residues the limit 78 was obtained by M. Dunton, and for quartic residues the limit 1225 was established by W. H. Mills, [As yet unpublished]. In this paper it is proved that every prime > 101 has a pair of consecutive quintic residues not exceeding 7889, and that every prime :::>- 277 has a pair of consecutive sextic residues not exceeding 202125. Both limits are best possible and an infinite number of primes requiring these limits are characterized. It has been shown (Noti-ces Amer. Math. Soc. vol, 7 (1960) p. 242] that no such limit exists for three consicutive quadratic residues or four consecutive cubic residues. In this paper we show that every prime > 283 has three consecutive cubic residues not exceeding 63426, but this limit is probably not the best possible. The proofs require the consideration of thousands of cases -- in fact the last one required the consideration of over 100,000 separate cases. The calcula­ tions were carried out by the IBM 701 and 704 computers. (Received February 20, 1961.)

173 Abstracts Presented by Title

61 T-58. Glen Baxter: On the asymptotic behavior of Toeplitz determinants. II

Let f(8) > 0 be an integrable function on - 7r a 8 a trwith Fourier coefficients ck, i.e. f(O) ,..., L ckexp(ik8). ,..Let Dn (f)= det(c j-k>• (j,k = 0, 1, ..• ,n), set ~ = Dn-l (f)/Du (f) and (if it exists) G(f) = exp{(l/211)./-trlog f(8) d8}. Szego has defined a polynomial !1in(z) of degree n for each n by means of the formula (l/27r)J-;!1in(z)r6m(z) f(8) d6 = 8mn, (z = exp(i9)). We use the conjugate function 'lin (z) = znj1{n(z- 1). Theorem: J!.L:Ickl...:::. oo and if~ is a concave sequence, i.e. 2~;;; ~+l + k~_ 1 , then Dn(f)/G(f)n+l ""exp [(1!11")//l'ljl~(x)/llfn(z) 12dcr'"}, where both sides have a finite limit, and where the integration on the right is with respect to two-dimensional Lebesgue measure extended over the circle lzl ~ 1. (Received January 9, 1961.)

61T-59. H. T. Muhly: A problem of Samuel concerning divisors of second kind.

Under certain restrictions on the local domain tJ-, P. Samuel has shown (Multiplicites de certaines composantes singulieres, Illinois J. Math. vol. 30 (1957)) that if w is a prime divisor of the quotient fieid K of rJ that dominates D-, then the function w•(!?) = inf [w(x)/v6"(x); x E f?. xi o}, on the set I of rank one primes £Pin (J, has the property that w(x) = L:w•((?)v*'(x), xi 0, where the sum is extended over I. He raised the question as to whether or not the range of w' is a subset of the rational numbers. This question is answered in the affirmative if tJ.. is an equi-characteristic analytically irreducible local domain. In fact, it is shown that there is a complete ideal W such that Wn is a v-ideal for w for all n, and that for this ideal, w•(i1) = ely/.{?}, where e is an integer and ly/..fi» is the function defined by Samuel in (Ann. of Math. vol. 56 (1952))and proved rational by Nagata in (Mem. Coll. Sci. Kyoto, Ser. A, vol. 30 (1957). (Received January 6, 1961.)

61T-60. H. M. Edgar: On some cubic Diophantine equations with no rational solutions.

Cubic Diophantine equations of the form y2 = x 3 + k, k a rational integer, are treated. It is shown by elementary arguments that certain congruence conditions contained in a theorem of Mordell (Arch. Math. Naturvid. vol. 6 (1947) pp. 143-150), respectively Chang (Quart. J, Math. Oxford Ser. (2) vol. 19 (1948) pp. 181-188) can be replaced by other congruence conditions which are sufficient to force the corresponding equations y2 = x 3 + k to have no rational solutions. (Received January 9, 1961.)

61T-61. R. Bierstedt: On the existence of a bound for a pair of consecutive quartic residues modulo a prime.

Let p be a prime=: 1 (mod 4). The author proves that for every p > 1217 there exists a pair, n + 1 (consecutive positive quartic residues) (mod p) with n ~ 1224. This result extends a result of M. Dunton for cubic residues, and is linked to recent work of D. H. and E. Lehmer. (Received January 9, 1961.)

174 61 T-6Z. S. M. Shah: On the absolute harmonic summability of Fourier series.

Let f(x) be a continuous periodic function, period Z 7r, and of bounded variation, and let w(S) denote the modulus of continuity of f. It is known that, if Lf(1/n)(w(1/n))1/Zj is convergent, then the Fourier series off is absolutely convergent. This result is best possible (R. Salem, On a theorem of Zygmund, Duke Math. j. val. 10 (1943) pp. Z3-31). In a recent paper (Proc. Amer. Math. Soc. val. 11 (1960) pp. 588-595) Varshney proved that if (i) lf(x +h)- f(x)l ;iA[log(1/hr1-e , (e > 0, 0 :Iii x ~ z 11"), then the Fourier series off is absolutely harmonic summable. It is shown here that if a continuous periodic function f, of period Z 11', is of bounded variation and (ii) L:3((loglog n)/n) w(1/n) is convergent, then the Fourier series of f is absolutely harmonic summable. It is easily seen that the condition (i) implies the convergence of the series (ii). (Received january 10, 1961.)

61 T-63. Chong-Yun Chao: The existence of uncountably many nonisomorphic nilpotent Lie algebras.

The following theorem has been proved: There exist uncountably many nonisomorphic real nil­ potent Lie algebras of length Z (i.e. L ;::) L 1 = (J..,LJ ~ Lz = [L,LrJ = 0) for any given dimension n ;r; 10. Corollary. There exist uncountably many nonisontorphic real solvable but not nilpotent Lie algebras of index 3 (i.e. M ::> M 1 = (M,MJ ::J MZ = 1}.1: 1 ,M 1] ~ M3 = [Mz ,Mz] = 0) for any given dimension n ?>. 11. (Received january 1Z, 1961.)

61 T-64. E. Granirer: Semigroups whose set of invariant means has finite dimension.

In the following the definitions and notations of (1] M. Day, Ameanable semigroups, Illinois j.

Math. (1) val. 4 (1957) pp. 509-544, are used. G is a semigroup. A ~ G is a left ideal with left can­ cellation (1.i.l.c.) if GA ~A and ga' = ga", g E G, a',a" E A implies a'= a". If A~ G is a finite (1.i.1.c.) then -A will be the left invariant mean -A(f) = (N(A)f1L:a.e:Af(ai) for f Em(G) and N(A) =number of 1 elements in A. The following theorems are proved: If G is a left ameanable countable semigroup and

K0(G) is the set of left invariant means on G, then dimension of K0(G) = n <-oo if and only if there are exactly n finite groups A1 s;;;_ G (i = 1, .•. ,n) which are (.l.i.1.c.). The existence of the above Als implies that each ¢ E Ko(G) is of the form ¢ = L:~ a1- A (ai !;, 0) L~ at= 1. Remarks. The Ai as left ideals and i groups are minimal left ideals and therefore disjoint. In the "if" part "countable" can be dropped. If G is a group one obtains from Corollary 1 of (1, p. 534] and the above theorem that the dimension of

K0 (G) is one or not finite. It is one if and only if G is finite. It is also proved that if G is a left amean­ able semigroup, which has a left invariant countable mean, then there exists a sequence {A1} of finite groups, which are (1.i.1.c.) such that- =I:~ 1 ai-A-•ai s:;O, L:~ ai = 1. (The sequence A1 can depend 1 1 on {if G is not countable.) (Received January 1Z, 1961.)

61T-65. R. V. Kadison: The trace in finite operator algebras.

The short proof of the additivity of the trace in finite factors given in R. Kadison, On the addi­ tivity of the trace in finite factors, Proc. Nat. Acad. Sci. U.S.A. val. 41 (1955) pp. 385-387, is extended

175 to cover the case of finite von Neumann algebras. The extension to the general case (J. Dixmier, Les anneaux d'operateurs de classe finie, Ann. Ecole Norm. Sup. vol. 66 (1949) pp. 209-261) has been effect­ ed by use of local traces and the Theorit!:me d'approximation (J. Dixmier, Les algit!:bres d'operateurs dans l'espace hilbertien, Paris, Gauthier-Villars, 1957, see especially p. 269 and p. 318). The present proof avoids these considerations, passing directly from the global dimension to the global trace by the use of center-valued, positive linear functionals which are the identity on the center. The following extension of Alaoglu's result (Weak topologies of normed linear spaces, Ann. of Math. vol. 41 (1940) pp. 252-267) establishes the compactness of such 'center states' in the appropriate weak topology . .!!._ V and V' are real or complex linear spaces, F is a separating family of linear functionals on V', S is a subset of V which spans it linearly, and S' a weak-F compact subset of V', then the set of all linear transformations of V into V' which map S into S' is compact in the point-open topology, where V' is taken in the weak-F topology. (Received January 13, 1961.)

61 T-66. G. 0. Sabidussi: The lexicographic product of graphs.

If X and Y are graphs with vertex sets V(X), V(Y) and edge sets E(X), E(Y), respectively, we define the lexicographic product X o Y to be the graph with V(X o Y) = V(X) X V(Y), E(X o Y) = i((x,y),(x',y')]: [x,x'] EX, or x = x' and [y,y'] EY}. By G(X), G(Y), G(X o Y) denote the automorphism groups of X,Y,X o Y, respectively; let G(X) o G(Y) denote the wreath product. For any X andY, G(X) o G(Y) C G(X o Y). A n.a.s.c. is given in order that this inclusion be an equality. This condition involves certain equivalence relations on X, and the connectedness of Y. (Received January 9, 1961.)

61T-67. P. E. Conner and E. E. Floyd: Fixed point sets of differentiable involutions.

Consider a compact differentiable manifold Mn upon which a differentiable involution T acts. Denote by F m the union of all m -dimensional components of the fixed point set F. It is shown that the F m together with their normal bundles determine the (nonoriented) cobordism type of Mn. More pre­ cisely, let 8~ 1 denote the normal sphere bundle to Fm, and 8~ the Whitney join of 8~ 1 and the trivial 0-sphere bundle. There is then the projective space bundle P~ corresponding to the sphere bundle 8~. Then Mn is cobordant to Urn P~. It is proved that if each normal bundle is trivial, then each pm cobords and so does Mn. If F is the disjoint union of a point and a k-sphere, then k = 1,2,4,8, also n = 2k, and Mn has the cobordism type of the appropriate projective plane. Finally, it is shown that there exists an integer-valued function (li(n), approaching infinity with n, such that if

Mn fails to cobord, then dim F &:; (li(n). The proofs are via a cobordism theory for fixed point free in­ volutions. (Received January 16, 1961.)

61 T-68. J. Bazer and H. Hochstadt: Diffraction of scalar waves by a circular aperature in a planar screen.

New integral representations of the solution of v2v + k 2V = 0 for this geometry are presented. On the shadow side of the screen (z > 0) these take the form v<:r:-1 = cos m(~ - ~&m>)rmLm1f'_\ exp(ikR+)

176 • (R+)- 1 ft~>(t)dt) where m = 0,1,2, ... and Lis the operator r- 1a/ar. The subscripts 1 and 2 here refer to the solutions of first and second boundary-value problems; k is the wave number. With the z-axis chosen as the axis of symmetry, r, !1 and z denote the cylindrical coordinates of an arbitrary space point and R +=(r 2 + (z + iat)2)l/Z. The screen is at z = 0; 'a' is the aperture radius, Excitation of the form cos m(!l- !1(/f~u(m)(r,z) in z < 0 is assumed. The functions f 1(t) and f 2(t) are odd and even func­ tions oft respectively, regular in the disc 0 ;§;It I ~ 1 + e, e > 0, and have zeros of order m at t = ±1, The f's are obtained as (Neumann series) solutions of Fredholm integral equations of the second kind whose kernels are small when ka is small, The results are applied to various types of excitations. (Received January 17, 1961.)

61T-69, Alvin White: A relation between singularities of a three-dimensional harmonic function and the coefficients of its development in spherical harmonics.

The author establishes various relations between the coefficients Sm+k,m of the development in a series of modified Legendre polynomials Tm,m+k = ikm~ [{m + k)~r 1R mp m,k(cos 9)eik(6 of a harmonic function H(x,y ,z) = L:=oL: -m Smtk,m r m,mtk and the location of the singularities of H (see Bergman, Ergebnisse der Mathematik val, 23, pp, 45-54), Let D~?~ be the symmetric determinant of p + 1 rows whose element in the ith row, jth column is Sm-2+ko+i+j,m-2+i+j (i,j = l,2, ... ,p + 1), and define4ko) =lim supm=oolo!!.% 111m. lfi'~ko) = l/ri+1, i = 0,1,2,,.,,p- 1, andl~ko) <1/rp+l, and if £H(x,y,z) - L:=osm+ko,m r m,m+k J is an entire harmonic function, then H hasp branch line singu­ larities on the sphere x 2 + y 2 + z 2 = r-p of the type described in Bergman, lac, cit,, p. 46, The branch lines lie in the planes x = xy, ")> = 1, .. ,,p; lx,..l :> lx,.,+ 1 1: x 1 = r cos :51, where g1 can be determined from the coefficients Sm+ko,m· For x~, V > 1, one obtains similar expressions. Further analogous results are discussed, The proofs apply results of Hadamard and Mandelbrojt (Comptes Rendus val. 204, p. 1456) on polar singularities in the theory of functions of one complex variable to Bergman's integral operator theory. (Received January 5, 1961,)

61T-70, J, H. Bramble and L. E. Payne: A new decomposition formula in the theory of elasticity.

In this paper a new representation formula for the spherical components of displacement in the theory of elasticity is obtained. This formula may be used to reduce mixed boundary value problems for the elastic sphere to standard problems in potential theory. (Received January 24, 1961.)

61T-71. J, H. Bramble and L. E. Payne: An analogue of the spherical harmonics for the equations of elasticity.

In this paper two sets of solutions of the displacement equations of classical elasticity are ob­ tained, These solutions are analogous to the interior and exterior spherical harmonics and form com­ plete orthogonal systems on the surface of a sphere. As a consequence of this work it is possible to exhibit nonuniqueness of the solutions to certain boundary value problems in elasticity for values of the elastic constants for which the uniqueness question has up to now remained unsettled, The system

177 of vectors should prove useful for obtaining upper and lower bounds for solutions of physically interest­ ing problems in elasticity. (Received January 24, 1961.)

61 T-72. Eckford Cohen: Arithmetical notes. VII. Some classes of even functions (mod r).

Let r denote a positive integer and o a divisor of r. An even function (mod r) is defined to be a member of the class Br(8) if it is even (mod S) but is not even (mod e) for any proper divisor e of b. A number of functions important in the additive theory of congruences are classified according to this scheme. Example: Let 6s(n,r) denote the number of solutions of n =x 1 + ••• + xs (mod r), (xi,r) = 1; then 9s(n,r) E Br(r(r)), where r'(r) denotes the greatest square-free divisor of r. Some generalizations of this congruence problem are considered in detail. (Received January 30, 1961.)

61 T-73. Robert Hermann: On the Lagrange variational problem.

Let B be a differentiable manifold, M = T(B) X R. (T(B) = tangent bundle, R = real numbers). In a Lagrange variational problem, a real-valued function L on M and a submanifold N C Mare given. Let M' = normal vector bundle toN. Under suitable regularity conditions, the extremals of the problem defined via the Lagrange multiplier device are the projections in B of the integral curves of a vector field on M'. This vector field admits an integral invariant in the sense of E. Cartan, hence its integral curves are given locally as solutions of a set of Hamilton equations, a classical result. Let X be a vector field on B and let~ be its first order prolongation to M, a vector field on M. Suppose that the

Lie derivative of L with respect to ~is zero and that ~is tangent to N. Let A be the subset of b E B such that the value of X at b belongs to N. Theorem. Given b E A, the integral curve of X starting at b is an extremal for the Lagrange problem if and only if b is a critical point for the function L(X) restricted to A. (Received February 1, 1961.)

61T-74. Robert Hermann: Integral curves for Pfaffian systems. Preliminary report.

Let M be a C00 manifold, V(M) be the set of C 00 vector fields on M, considered both as a module over the ring C(M) of C 00 real functions on M and as a Lie algebra over the real numbers. A Pfaffian system is a C(M)-submodule, P, of V(M) whose dimension is constant over M. An integral curve for P is a piecewise C 00 curve in M whose tangent vector is everywhere the restriction of an element of P. For x I, M, let Hx be the set of points of M which can be joined to x by an integral curve. Let D(P) be the smallest Lie subalgebra and C(M)-submodule of V(M) containing P. Theorem. Under suitable regularity conditions, Hx is the leaf of the foliation of M defined by D(P). In particular, it is a sub­ manifold of M. The proof applies variational techniques due to Caratheodory (Ges. Schriften, Bd. I, pp. 270-389). Special cases arise in Caratheodory's classic work on thermodynamics and in the work showing how the holonomy group of an affine connection is determined by the curvature. (Received February 1, 1961.)

61 T-75. WITHDRAWN 61T-76. WITHDRAWN

178 61 T-77. Harry Kesten: Occupation times for Markov and semi Markov chains.

Let Y 0,y 1, .•. be an irreducible, recurrent Markov chain with state space the integers and n step transition probabilities P~j>. Given a sequence Hj(t) of distribution functions with Hj(O -) = 0 we con­ sider the sequence T 0, T 1, •.• of nonnegative random variables whose conditional distribution, given k k+l y0 ,Y1 •••• satisfies PfTo ~ t0, •.• , Tk >lii ~J = Hj0(t0) •.• Hjk(tk). Put Y(t) = Yk if Li=OTi < t ~ Li=O Ti. Y(t) is a semi Markov chain and Ti represents the time it spends in the ith state. Let V(j) be a function different from zero for finitely many j only. Limiting distributions for "[:t, 0 V(Yi) and/okV(Y(t))dt are found, similar to those in [1], Takacs, Trans. Amer. Math. Soc. val. 93 (1959) p. 531. This generalizes [1] where Y possessed two states only. Also a completion of the results in Darling and Kac, Trans. oo (n) n a -1 Amer. Math. Soc. val. 84 (1957) p. 444 is found. If (1) h(z) = l:n=OPoo z "'-'0 - z) L((l - z) ) as z j 1 (0 ~ a ~ 1, L slowly varying) and (2) (h(z))- 1L.n2:j P 1~)z nV(j) ._ 0 as z 1' 1 then the limiting dis­ tribution of (h(k))-l/Z z:f:: 0v(Yi) is of the formj000 fli(xy-l~\dG(y) where fli(x) is the standard normal distribution and G(y) the Mittag-Leffler distribution of order a. For a bonafide Markov chain Y(t) the same limiting distribution applies to (h(k))- 112/ov(Y(t))dt under the continuous time analogues of (1) and (2). (Received February 7, 1961.)

61T-78 and 61T-79 WITHDRAWN

61T-80. J. G. Glimm: Locally compact transformation groups.

The following theorem is proved. Let G be a locally compact T 2 topological transformation group with a countable base for its topology acting on a space M which is locally compact, Tz and which has a countable base for its topology, or more generally, let G be acting on the structure space M of a separable type I C*-algebra. Then the following are equivalent: (1) Each orbit in M is relatively open in its closure. (2) M/G is T 0. (3) M/G is countably separated. (4) If f3 is a (finite) Borel measure on M and if G acts ergodicly on (M.f3) then there is an orbit Gm in M/G such thatp(M'"'-'Gm) = 0. (5) For each min M the map gGm -+gm from G/Gm onto Gm is homeomorphism. (6) M/G con­ tains a dense open T2 subset u1, M/G "-'U1 contains a dense open T 2 subset u2, etc. The motivating example for this theorem is the case in which M is the structure space of the C*-group algebra of a closed normal type I subgroup K of G. In this case, (3) is the statement that K is regularly imbedded in G. (Received February 9, 1961.)

61T-81. D. W. Kahn: Real cohomology of fibre spaces.

It is shown that one may associate, to any countable simplicial complex, a real, anti-commuta­ tive cochain algebra, whose derived algebra is isomorphic to the real singular cohomology algebra of the space of the complex. Then, for a large category of fibre spaces, say (E, F, 8 .p), one may describe J the real cohomology algebra H*(E) as the derived algebra of C*(B) ~ H*(F), with respect to a differen- tial defined via the transgression. If the cohomology of the base is a tensor product of polynomial and exterior algebras, on even and odd generators (resp.), then H*(E) is the derived algebra of H*(B) ~ H*(F). The results apply to the following situations: (1) sphere bundles with odd-dimensional fibres, (2) principal bundles, (3) Postnikov fibrations. If X is (n - I)-connected, one may compute the real cohomology algebras of the first n terms of Postnikov system for X. These results are applied to secondary operations, in several variables, in real cohomology. (Received February 10, 1961.)

179 Hotel New Yorker BOOTH 2, A. M. S. Meeting April 6-8, 1961 Journal of Mathematical Analysis and Applications Edited by RICHARD BELLMAN, The RAND Corporation, Santa Monica, California Volume 2, Number 1, February 1961 V. E. BENES, Extensions of Wiener's Tauberian Theorem for Positive Measures. • GEORGE A. BAKER, ]R. and J. L. GAMMBL, The Pade Approximant. • ROBERT H. ASENDORF, Application of Group Theory to the Band Theory of Solids. • L. TAKACS, The Time Dependence of Palm's Loss Formula. • J. H. BRAMBLE, Continuation of Solutions of the Equations of Elasticiry Across a Spher­ ical Boundary. • P. L. BUTZER, On Dirichlet's Problem for the Half-Space and the Behavior of its Solution on the Boundary. • H. J. LANDAU and W. L. MIRANKER, The Recovery of Distorted Band­ Limited Signals. • OLGA TAUSSKY, A Remark on a Theorem of Liapunov. • RICHARD BELLMAN, On the Computational Solution of Differential-Difference Equations. • I. NORMAN KATZ, On the Existence of Weak Solutions to Linear Partial Differential Equations. FRED BRAUER, Global Behavior of Solutions of Ordinary Differential Equations. Mathematics in Science and Engineering A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, The RAND Corporation, Santa Monica, California Volume 1, Concepts from Tensor Analysis and Differential Geometry By TRACY Y. THOMAS, Indiana University January 1961, 119 pp., illus., $5.00 Forthcoming Volumes: Stability by Liapunov's Direct Method with Applications By ]OSEPH LASALLE and SOLOMON LEFSCHETZ The Optimal Design of Chemical Reactors By RUTHERFORD ARIS Pure and Applied Mathematics A Series of Monographs and Textbooks Edited by PAUL A. SMITH and SAMUEL ElLENBERG, Columbia University Volume 8, Homotopy Theory By S. T. Hu, Wayne State University 1959, 347 pp., illus., $11.00 Volume 9, Solution of Equations and Systems of Equations By ALEXANDER M. OSTROWSKI, University of Basel October 1960, 202 pp., illus., $6.80 Volume 10, Foundations of Modern Analysis By J. DIEUDONNE, Institut des Hautes Etudes Scientifiques, Paris August 1960, 361 pp., $8.50 Advances in Mathematics Edited by HERBERT BUSEMANN, University of Southern California In order to shorten publication time of individual contributions, each volume of this new serial publication will be issued in paper-bound parts at irregular intervals. When sufficient parts have been published, the complete volume will be available as a hard-bound book of about 350 pages. Volume 1, Part 1, 1961, about 100 pp.; contents: ]OHN WERMER, Banach Algebras and Analytic Functions KATSUMI NOMIZU, Recent Development in the Theory of Connections and Holonomy Groups

ACAD E MI C PRESS New York and London 111 Fifth Avenue, New York 3 17 Old Queen Street, London, S.W.1 OUTSTANDING ASSIGNMENTS MATHEMATICIANS for OUTSTANDING SCIENTISTS & at the OPERATIONS EVALUATION STATISTICIANS GROUP We invite your investigation of M. I. T. of these career positions Southern California Seeking scientists who have the in ability and imagination to apply their broad knowledge with origi­ Mathematician to develop programs that trans­ the nality in the field of research, the definitions required Operations Evaluation Group of form design intent into the Massachusetts Institute for numerically controlled tools. Will mathemat­ of Technology offers stimulating ically define rocket engine components. A degree career opportunities to scientists is required. with advanced degrees in mathe­ in Mathematics matics and the physical sciences. Research Statistician to participate in plan­ For over 18 years, OEG, has served as advisor to the Office of ning long range rocket research and development Chief of Naval Operations and programs, with the majority of time spent in sta­ the operating fleet. Engaged in tistics research. Will take part in the development re­ both conventional operations methods for opti­ search and in the solution of com­ of statistical and mathematical plex problems far out of the mizing a set of experimental designs for large, realm of the ordinary, OEG has complex programs. An advanced degree in Sta­ the responsibility for conducting research that cuts laterally across tistics is preferred. disciplines. many scientific Programmer experienced in the application of If you have the interest and the creative ability to apply your high speed digital computers to scientific and basic research findings to the solu­ engineering problems. Rocketdyne currently tion of problems that are vital to possesses an IBM 7090. and the company is rec­ secur­ the Navy and the national developing ity, you are invited to write to ognized nationwide as a leader in OEG. Working in a professional new techniques in the digital computing field. A atmosphere and exchanging stim­ degree in Mathematics is preferred. ulating ideas with colleagues of the same discipline, you will ··find Reliability Statistician to perform analytical for in­ a rewarding opportunity use of reliability creased scientific stature and per­ investigations requiring the sonal growth. techniques of measurement, synthesis, and pre­ diction. Will conduct studies to determine OPERATIONS relationships between reliability and other parameters. A degree in Statistics or Engineer­ EVALUATION ing with statistical training is required. GROUP Send inquiries to Mr. M.D. Stoetzel An Activity of the Massachusetts Institute of Technology Professional Employment Office Calif. Department E 6633 Canoga Ave., Canoga Park, Washington 25, D. C. Physicists • Physical Chemists Mathematicians • Economists ROCKETDYNE It Electronics Engineers DIVISION OF NORTH AMERICAN AVIATION Boeing openings for SYSTEMS ANALYSTS

The Aero-Space Division of Boeing, training in Operations Research. Ex­ holder of major contract responsi­ perience should include simulation, bility for the BOMARC and Minute­ and the definition of systems, starting man missiles and the Dyna-Soar space from operational requirements. glider, has a number of outstanding Broad technical curiosity and an in­ openings for Systems Analysts, at terest in developing original models both technical personnel and super­ are desirable. visory levels. The men assigned to Salaries are open and will match your these positions will be expected to educational and experience qualifica­ advance the knowledge of systems tions. Assignments are in Seattle in analysis and design techniques. They the uncongested Pacific Northwest, will be responsible for the creation, noted for mild year-round climate and development and study of models of outstanding educational and outdoor systems selected for intensive and recreational facilities. penetrating evaluation. Requirements include Bachelor or Send your resume, today, to: Mr. W. graduate degrees in mathematics, B. Evans, Boeing Airplane Company, engineering or physics, plus formal P. 0. Box 3707 ·!MD, Seattle, Wash.

BOEING ••• environment for dynamic career growtk THE OHIO STATE UNIVERSITY The Numerical Computation Lab­ oratory has immediate openings for programmers and analysts at all levels OPENINGS FOR of education and experience. The Laboratory serves the University in MATHEMATICIANS teaching and research. It deals with a wide variety of problems, including PHILCO CORPORATION statistical analyses, real-time simula­ tion, and business applications. Its CALIFORNIA present IBM 704 is soon to be replaced by a 709, followed by a Openings exist for mathema­ 7090, and supplemented by a 1620. ticians with PhD. who are inter­ is competitive; academic year Pay ested in doing mathematical and calendar year appointments are possible. Part-time appointments are research with emphasis in the available to those who wish to devote field of analysis. A Philco Tran· a large portion of their time to study sac S-2000 Is available as a in the University's Graduate School. research tool. For further in­ to Appointments may be arranged formation, write to Dr. E. David include teaching duties in the appli­ cant's field of interest. Callender, Philco Corporation, Send resume to: Western Development Labora­ PROF. RoY F. REEVES, Director tories, 3875 Fabian Way, Palo Numerical Computation Laboratory Alto, California. The Ohio State University 1314 Kinnear Road, Columbus 12, Ohio

MATHEMATICIANS ELIGIBLE * * * * * * * * * SAVE M~~H $30 E~: $100 * * * * * * * * * * * * * * * * * * * * * * YOU NOW * NOW OVER 700,000 POLICYHOLDERS * Founded in 1936, GEICO is now one of the largest auto insurers in America because SPEND ON * GEICO offers ••• quick and fair claims settlement ... equal protection at lower cost * ••• proven reliability and experience. 97% of GEICO's policyholders renew their * expiring policies every year. * * Here is How GEICO Can Offer You Such Big Savings * AUTO GEICO provides· the hii!hesl quality insurance protection ."direct-to-the-policyholder" * to effectively eliminate a large percentage of the premium costs usually required for * * :~;~~~__:~~aT~~~i~~~e~~~s~;i~:~sc~~~s~~!se~~ilfie~ri~0e~r~~:;~!df~~~=~· risk'' occupational * * * * * * * * * * * * * * * * * * * * * * * INSURANCE GEICO SAVINGS charged, dividends of 20% on liability and 30% on other coverages are being MAIL COUPON TODAY FOR YOUR EXACT RATE You save 30% from Basic Bureau Rates paid on currently expiring policies. NO OBLIGATION • NO SALESMAN WILL CALL ...... on Collision and Comprehensive cover­ Must be over age 21 ages in all states. Here is the Protection You Get-You get and under. 65 exactly the same Standard Family You save 25% from Basic Bureau Rates Policy used by most other leading Name ...... 126 on Liability coverageS in most states companies, and you are fully protected .. (exceptions: 22¥2% in Kentucky and wherever you drive in the U.S. and its Residence Address ...... New Jersey-15% in New York). possessions. Your GEICO policy can City ...... Zone ...... County ...... State ...... comply with the Financial Responsi­ 0 Single D Married D Male D Female GEICO is licensed in all states (except bility laws of all states, including the Age...... Massachusetts) and offers its insu~ance compulsory insurance requirements of Occupation ...... services under the authority granted New York and North Carolina. by the respective State Insurance location of car (if different from residence address) ...... Departments. GEICO is rated A+ (Excellent) by Best's Is car principally kept on farm or ranch? 0 Yes D No Insurance Reports. Compact Cars; GEICO rates are reduced an additional 10% in most states. Country-Wide Personal Claim Service- More-Than-One-Car: GEICO rates are re­ ~~~~ :~:2y f2°se~J!i~ourT;Ye~enndta~J~~~ duced an additional 25% in most in­ wherever you drive in the U.S. and its Days per week car driven to work? ...... One way distance is ...... miles. stances. possessions. Settlements are prompt ~ ~:~ uo~~n any occupation or business? (Excluding to and from work) and courteous. In Texas, where rates prescribed by the State Board of Insurance must be You May Pay in Installments Additional male operators under age 25 in household at p'resent time: Age Relation Married or Single % of Use GovERNMENT EMPLOYEES INSURANCE COMPANY Capital Stock Company not affiliated u•ith U.S. Government GOVERNMENT EMPLOYEES INSURANCE CO. Home Office- Washington 5, D.C. Vermont Ave & K St, N W, Washmgton 5, DC Senior mathematicians for Space Technology Leadership

REPRINTS OF THE DRAWINGS CREATED FOR THIS SERIES, SUITABLE FOR FRAMING, ARE AVAII.ABLE ON REQUEST.

Through the centuries, the dynamical top, a toy of their youth, fascinated such men as Euler, D'Aiembert, Lagrange, and Maxwell. It aided them in discovering laws regarding the rotation of bodies and in relating the phenomenon to mechanics. Today these laws serve as a springboard for the mathematician concerned with the advancement of space technology. The senior mathematician whose abilities exceed the challenge of his present assignment will find at Space Technology Laboratories, Inc. opportunity to explore and develop new areas of Space Technology Leadership. Your inquiry will be welcomed and will receive our meticulous attention.

SPACE TECHNOLOGY LABORATORIES, INC. P.O. BOX 95005TU, LOS ANGELES 45, CALIFORNIA a subsidiary of Thompson Ramo Wooldridge Inc. A Los Angeles • Santa Maria • Cape Canaveral • Washington, D. C. v, Boston • Dayton • Huntsville • ·Edwards AFB • Canoga Park • Hawaii Not necessarily in words and syllables but in electronic impulses, relayed by orbital, lunar and interplanetary communications systems. Scientific advancement depends upon the ability of man to overcome the limitations of current knowledge. These limitations are being pushed aside at the Amherst Laboratories by new concepts for aerospace communication and instrumentation systems. The role that microwave and millimeter wave technology will assume in future aerospace systems is being shaped through state-of-art advances achieved by our technical staff. PROFESSIONAL STAFF APPOINTMENTS in Microwave Communication Systems m·e available for Electrm1ic Engineers, Physicists a11d Mathematiciatls with advanced degrees a11d creative desire. You at·e invited to direct inquiries it1 confidence to Mr. E. Schlichter, Mattager, lHicrowave and Aerospace Systems Department, Amherst Laboratories . .. or call NF 3-8315 for ill/ormation. All inquiries will be ackt1owledged promptly.

SYLVANIA ELECTRONIC SYSTEMS A Division of SYLVANIA Subsidiary of GENERAL~ TELEPHONE & ELECTRONICS ~

1186 Wehrle Drive, Williamsville~ New York ABORATORIES

Shrinking the Universe ... through Communications HAVE YOU COMPLETED Associate THE QUESTIONNAIRE of the Director MATHEMATICAL For Applied SCIENCES SECTION of the Mathematics NATIONAL REGISTER OF SCIENTIFIC AND TECHNICAL PERSONNEL? Westinghouse-Baltimore is seeking a man to lead the mathematics section of the Electronics Division. Require­ At the request of the National Sci­ ments for this highly responsible post ence Foundation, the American Math­ are: a Ph.D. in mathematics, proven ematical Society has mailed the 1960 scientific ability with extensive experi­ reporting forms to its section in the ence in an applied mathematical en­ deavor, and demonstrated leadership National Register. The main objective qualities. of the Register is to provide up-to-date This man will head a staff of fifteen information on the scientific manpower engaged in research mathematicians resources of the United States. It is also programs relating to the military elec­ tronics field and providing mathemati­ increasingly valuable to our profession cal services to the Electronics Division. as a source of statistical information. The types of problems encountered are We are counting again on the splen those which support the conception did cooperation accorded to the previ and design of military electronics sys­ tems and devices. Types of mathe­ ous cycle of the Register by most of matics typically employed include the mathematicians and mathematical classical mathematics with emphasis scientists. on solutions and systems of linear and If you received a National Register non-linear differential equations, prob­ ability and statistics, and modern questionnaire from the Society last algebra with special reference to com­ month, won't you please fill it in now, binatorial problems. so that the information can be included Send resume to: Mr. J. F. Caldwell, in the final analysis of the 1960 Dept. 388 National Register data. Please return the completed ques­ Westinghouse tionnaire to the Headquarters Offices of the Society at 190 Hope Street, IW\ ~~7~Tl~m~~Y~ Providence 6, Rhode Island. \!:::!:) AIR ARM • ELECTRONICS • ORDNANCE BROOKHAVEN NATIONAL LABORATORY

APPLIED MATHEMATICS DIVISION

There are openings in this division at various levels for Scien­ tists with an interest in the use of large scale Digital Computing machines for the implementation of scientific research. These appointments allow the individual freedom to follow his own directions of research, but he must, in addition, accept respon­ sibility for the obligations of this division to research programs in other departments of the laboratory. These include inves­ tigations in high energy physics, solid state physics, reactor theory, chemistry, biology, and medicine.

The terms of these appointments are comparable to those in the best universities and research laboratories in the country. Interested persons should address inquiries to:

DR. JOHN W. CALKIN HEAD, APPLIED MATHEMATICS DIVISION BROOKHAVEN NATIONAL LABORATORY

ASSOCIATED UNIVERSITIES, INC.

UPTON, LONG ISLAND, NEW YORK • LUNAR • PLANETARY • INTERPLANETARY

The National Aeronautics and Space Administration has assigned the Jet Propulsion laboratory the re· sponsibility for the Nation's Lunar, Planetary and Interplanetary unmanned exploration programs.

J p L ... where qualified mathematicians, engineers and scientists have the opportunity to be leaders in SPACE VEHICLE Systems Analysis . . . . where you will have the opportunity to work for a leading research laboratory with advance programs in lunar, Planetary and Interplanetary Space flights.

Advanced degrees in Mathematics, Physics, Engineer· ing Sciences, or Astronomy, and three years experi· ence required in one or more of the following areas: Astrodynamics • Analysis of Guidance Systems • Trajectory Studies • Orbit Determination • Error Analysis and Correlation • Computations for Guidance Commands and Equations • Stochastic Analysis of Tracking Data • Midcourse, Terminal Guidance and Maneuvering Parameter Studies • Study and Develop· ment of Methods for Correcting Observable Data for Physical Phenomena, e.g., Refraction, Abbera· tion, etc. Qualified persons are invited to forward their resumes and salary requirements to:

CALIFORNIA INSTITUTE OF TECHNOLOGY JET PROPULSION LABORATORY 4800 OAK GROVE DRIVE PASADENA, CALIFORNIA • • Selective program expansion - and commencement of applied research projects in new and promising areas of nuclear reactor and power plant technology - have generated new openings of unusual potentiaL C/;) A "DIFFERENT" The Laboratory can offer iinmediate placement to: ~ MATHEMATICIANS - BS or MS in Math, with experience in hydraulic KIND OF STAFF and thermodynamic analysis. To develop and recommend systems and com­ t3 puter programs applicable to reactor problems. Will direct programming • c:-0> personnel. EXPANSION AT STATISTICIAN- PhD in Statistics. To perform studies leading to devel­ ~ opment of statistical techniques. Formulate and solve problems originating • c:-0> in Laboratory having wide application to reactor programs .

"+-) NUMERICAL ANALYSTS- BS or MS degree in Math or Physics. Work THE KNOLLS with engineers and physicists in formulating and programming problems t3 arising in the development of nuclear reactors and power plants. ATOMIC POWER U.S. Citizenship Required ~ ~rward your resume in confidence to Mr. F. W. Snell, Dept. 123-MD ~ LABORATORY ...s:: "+-) r~K~A~.~YL~ t3 GENERAL. ELECTRIC ~ Schenectady, New York MASTERS DEGREE PROGRAM- KAPL is now considering recent graduates in ME, M•t. MetE, ChE, Physics, EE, and Nuclear E for its masters degree program in nuclear engineering in conjunction with Rensselaer Po}ytechriic Institute. Applicants should have strong interest in the nuclear field and must have graduated in upper 10% of their class. Selection of candidates will be completed by early Spring; classes begin September, 1961. Write for further details . • •

PROFESSIONAL TRAINING IN MATHEMATICS

by

F. A. FicKEN AND C. C. MAcDuFFEE and a Selected List of Available Scholarships and Stipends in Mathematics

Reprinted from Special Issue Assistantships and Fellowships in Mathematics in 1961-1962; Volume 7, Number 7, Part II, Issue No. 51, December 1960.

25 pp 25¢

AMERICAN MATHEMATICAL SOCIETY

190 HOPE STREET, PROVIDENCE 6, RHODE ISLAND INDEX TO ADVERTISERS

Name Page Name Page

Academic Press ••...... 180 Ohio State University ...... 183

American Mathematical Society .. ·189, 190 Operations Evaluation Group (U. S. Navy) ...... ·.····· 181 Boeing Airplane Company ...... 182 Pergamon Press ...... 118 Brookhaven National Laboratory ...... 187 Philco Corporation ...... 183 Cushing-Malloy, Inc •...... ···· 132 Rocketdyne - Division of North Edwards Brothers, Inc •...... ··· 132 American Aviation .... ·········· 181

General Electric Company ...... · · · 189 Space Technology Laboratories, Inc .. 184

Government Employees Insurance Co. · · 183 Sylvania Electronic Systems - Amherst Laboratories· · · · · · · · · · · 185 Jet Propulsion Laboratory· · · · · · · · · · · 188 Vitro Laboratories ...... 93 National Register of the American Mathematical Society ...... · . · · · · 186 Westinghouse Electric Corporation .. 186

RUSSIAN -ENGLISH DICTIONARY OF MATHEMATICAL TERMS

Compiled by A. J. LoHWATER in collaboration with S. H. Gould

COMPLETE COVERAGE OF ALL BRANCHES OF MATHEMATICS

228 pages 1500 definitions short grammar

To be published late Summer

On publication list price will not be less than $7.70. For pre-publication orders received before June 1. 1961 copies may be reserved at the List price of $7.00, at Members' price of $5.25.

AMERICAN MATHEMATICAL SOCIETY 190 HOPE STREET. PROVIDENCE 6, RHODE ISLAND NEW YORK, NEW YORK :MEETING UNIVERSITY OF WASHINGTON AT THE HOTEL NEW YORKER SEATTLE, WASHINGTON

April 5-8, 1961 June 13-16, 1961

RESERVATION FORM RESERVATION FORM The New Yorker is the official headquarters hotel. FOR Your room reservations should be sent directly DORMITORY ACCOMODATIONS to Hotel New Yorker, Reservation Department, 34th Street and 8th Avenue, New York, 1, New York. Fill in the form below and send it to: PLEASE TEAR OFF THE ATTACHED COUPON AND USE AS YOUR ROOM RESERVATION BLANK. Professor S. P. Avann Be sure to specify the names of persons for whom Department of Mathematics reservations are being made. BE SURE TO BRING University of Washington THE CONFIRMATION SLIP WITH YOU AS PROOF SeatUe 5, Washington OF YOUR RESERVATION. Unless otherwise re- quested, the hotel will hold reservations only to The reservation form should be mailed 6:00 P.M. of the day of your arrival. Check-out NOT LATER THAN May 19, 1961 time is 3:00 P.M. HOTEL NEW YORKER-RATES Room and Bath Double bed Twin bed for one Room with bath Room with bath (Please send Hotel Reservations directly to the Hotel.) per day For two per day For two per day $8.00 $ ll. 50 $ 11.50 $13.00 $15.50 $16.50 10.00 12.50 13.50 15.00 17.00 18.00 13.00 14.50 15.50 18.00 18.50 20.00

Note: All rates subject to New York City 5 per cent hotel tax ------Please detach on this line Please detach on this line

TO: Hotel New Yorker Name 7.(l-a-~~)------~~~ir-s7.~--~W~l-ea_s_e_p_r~in~~ Reservation Department 34th Street and 8th Avenue Institution.______New York 1, New York _____y.m. a.m. Please enter my reservation for Arrival date.______Departure date______a.m.. Single Room Twin Bed ______p.m.

Double Bed Suite Dormitory accommodations desired for____ men

Rate. Approximately s to $ lt-----women'------children

(J) (We) plan to arrive th or without breakfast? ______

and remain until Please give ages and relationships of members of our party. If you wish to share a room with a olleague, please give his name.) Signature

Address

City and State

Please print below additional names of persons in Number of banquet reservations required______your group:

191 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE 190 Hope Street PAID AT Providence 6, Rhode Island ANN ARBOR, MICHIGAN

Return Postage Guaranteed

SOVIET MATHEMATICS DOKLADY

A Translation of all the Pure Mathematics Sections of Doklady Akademii Nauk SSSR

The total number of pages of the Russian journal to be translated in 1961 will be about 1600. All branches of Pure Mathematics are covered in the DOKLADY in short articles which provide a comprehensive, up-to-date report of what is going on in Soviet mathematics.

Six Issues a year Domestic Subscriptions $17.50 Foreign Subscriptions 20.00 Single Issues 5.00

Send Orders to

American Mathematical Society 190 Hope Street Providence 6, Rhode Island