OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 13, NUMBER 2 ISSUE NO. 88 FEBRUARY, 1966

cNotiaiJ OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. \\' alker

CONTENTS

MEETINGS Calendar of Meetings • • • • • . • . . . • • • . • . • . • • . . • . . • • • . . • • • • . . • 18 8 Program of the Meeting in New York .••••.•.••.•••• , ...•••.•.• 189 Abstracts for the Meeting - Pages 227-229. PRELIMINARY ANNOUNCEMENTS OF MEETINGS...... 191 ACTIVITIES OF OTHER ASSOCIATIONS .•.•.•• , .•••..•••.••• , . • • . . 195 LETTERS TO THE EDITOR •• ·• . • • • • • • • • • . • • . • • . • . . . • . . • . . • . • . • 196 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••• 190, 195, 197, 199,202,223 MORE ON FLIGHT ARRANGEMENTS. • • . • • • • • • • . . • . • . • • . • • . . • • • • . 198 MEMORANDA TO MEMBERS Other Journals Available to Members at Reduced Rates • . • . • . • • . • • • • 200 Backlog of Mathematical Research Journals •••••... , ... , . • • • • • • • 201 Corporate Members and Institutional Associates , ••••••.••••. , .•. , 202 SUMMER INSTITUTES AND GRADUATE COURSES ••••• , • • • • . • • . . . • • • • 203 PERSONAL ITEMS ..•..••.•...•.•.••••• , •.•••..•.. , • . • • . . • . 214 SUPPLEMENTARY PROGRAM- Number 37 .•• , •.. , ••.•• , ...•••. ,.. 219 ABSTRACTS OF CONTRIBUTED PAPERS • . . . • . . . • • . • . . • • . • . . . • . • . 224 ERRATA •.•.•.••••.••.••••••••••...••..••...•.•.•••••.•. 263 INDEX TO ADVERTISERS ••...••.••••.••.•.••• , . . • • • . . . • . • . • • 277 RESERVATION FORMS ••••.• , • • • . • . • • • • • • • • • • • . • • • • . • • • . . • • • 278 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 cifot:i.tYV 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*

632 April 4-7, 1966 New York City Feb. 18 633 April 9, 1966 Honolulu, Hawaii Feb. 18 634 April 20-23, 1966 Chicago, Illinois Feb. 18 635 june 18, 1966 Victoria, British Columbia May 4 August 29 - September 2, 1966 (71st Summer Meeting) New Brunswick, New jersey January 24-28, 1967 (73rd Annual Meeting) Houston, Texas August 28 - September 1, 1967 (72nd Summer Meeting} Toronto, Ontario, Canada january, 1968 (74th Annual Meeting) San Francisco, California *The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society iri Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for the by title abstracts are February 11, and April 21. 1966.

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The cJioticeiJ of the American Mathematical Society is published by the Society in January, February, Apn1, June, August, October, November and December. Price per annual volume is $7.00. Price per copy $2.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34,21, P. L. and R.). Accepted for mru1ing at the special rate of Postage provided for in section 34,40, paragraph (d).

Copyright«:;>, 1966 by the American Mathematical Society Printed in the United States of America

188 Six Hundred Thirty-First Meeting· New York UDiversity Washington Square Campus New York, New York February 26, 1966

PROGRAM

The six hundred thirty-first meeting the south side of Washington Square but of the American Mathematical Society will the street signs on the stretch along the be held on the Washington Square Campus square read "Washington Square South." of New York University at the Courant Subway and bus transportation may Institute of Mathematical Sciences. All conveniently be used as follows: Lexington Sessions will be in Warren Weaver Hall. A venue (Interborough) Subway (IR T)--Local By invitation of the Committee to to Astor Place Station. Walk west on Astor Select Hour Speakers for Eastern Sectional Place to Broadway, then south on Broadway Meetings, Professor W. H. J. Fuchs of to Fourth Street and west to Mercer Street. Cornell University will give an hour ad­ Seventh Avenue (Interborough) Sub­ dress in Room 109 of Warren Weaver Hall way (IRT)--Local to Sheridan Square Sta­ at Z:OO P.M. His title is "Developments in tion. Walk east on Waverly Place to Wash­ the classical Nevanlinna theory of mero­ ington Square. morphic functions." Broadway(Brooklyn-Manhattan) Sub­ There will be sessions for contrib­ way (BMT)--Brighton local or Fourth Ave­ uted papers in Room 109 of Warren Weaver nue local to Eighth Street Station. Walk Hall. There will be provision for a limited south on Broadway to Fourth Street and number of late papers. west to Mercer Street. The registration desk will be set up Sixth or Eighth A venue (Independent) in the lobby at the entrance to Warren Subway (IND)--Express to West Fourth Weaver Hall and will be open at 9:00 A.M. Street--Washington Square Station. Walk A list of restaurants near Washing­ east on West Fourth Street to Washington ton Square will be available atthe registra­ Square. tion desk. Fifth Avenue Bus--Busses numbered Warren Weaver Hall is at 251 Mer­ 3, and some numbered 5, to University cer Street, one block east of the southeast Place. Walk south and cross the square to corner of Washington Square between Third Washington Square South. Walk east to and Fourth Streets. Fourth Street borders Mercer Street.

PROGRAM OF THE SESSIONS

SATURDAY, 10:00 A.M. General Session, Room 109 Warren Weaver Hall 10:00 - 10:10 ( 1) Triples versus theories. Preliminary report Professor F. E. j. Linton, Wesleyan University (631-1) 10:15 - 10:25 (Z) Sample function regularity for Gaussian processes with the parameter in a Hilbert space Professor P. T. Strait, Queens College (631-Z)

189 10:30 - 10:40 (3) Extensions of I-bisimple semigroups . Professor R. j. Warne, West Virginia University (631-3) 10:45 - 10:55 (4) Orthogonal conjugacies in finite groups Professor E. j. Taft, Rutgers, The State University (631-4) 11:00 - 11:10 (5) An analytic function with limit zero along some path tangent to each radius Professor W. j. Schneider, Syracuse University (631-5) 11:15 - 11:25 (6) On the length of programs for computing finite binary sequences by bounded­ transfer Turing machines. II Mr. G. J. Chaitin, The City Coliege of the City University of New York (631-6) (Introduced by Professor Martin Davis) 11:30- 11:40 ( 7) On groups of automorphisms of Lie algebras Mr. D. j. Winter, Yale University (631-7) 11:45- 11:55 (8) A direct proof that S0(3) is homomorphic to SU(Z) Mr. F. P .·Callahan, General Electrrc Company, Philadelphia, P ennsyl­ vania (631-8)

.SATURDAY, Z:OO P.M.

Invited Address, Room 109 Warren Weaver Hall Developments in the classical Nevanlinna theory of meromorphic functions Professor W. H. j. Fuchs, Cornell University Everett Pitcher Bethlehem, Pennsylvania Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

NEW DOCTORAL PROGRAMS NEWSLETTER FOR NUMERICAL ESTABLISHED ANALYSTS AT UNIVERSITY OF TOLEDO The ACM Special Interest Committee New doctoral programs in Mathe­ on Numerical Mathematics (SICNUM) will matics,· Applied Math,ematics, and Physics begin publication of a newsletter in order have been established at the University of to provide numerical analysts with a fast Toledo. The new programs, supported by means of communication below the journal local funds, grants from Owens-lllinoisCo.; level. The newsletter is free and will be and state funds, were approved by the Board sent upon request. The newsletter will ap­ of Regents of the State of Ohio in August, pear as frequently as there is sufficient 1965 and by the North Central Association material. in November, 1965. Areas of research in Requests to receive the newsletter, Mathematics include Algebra, Logic and material for the newsletter, and requests Foundations, Applied Mathematics· and for additional information should be sent Mathematical Physics, and Meas.tire Theory. to the Chairman of SlCNUM, Dr. J. F. Traub, Research are.as in Physics include Astro­ Bell Telephone Laboratories, Inc., Murray physics, Lunar Luminescence, Solid State Hill, New jersey. Physics, and Low Energy Nuclear Physics.

190 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Six Hundred Thirty-Second Meeting Waldorf-Astoria Hotel New York, New York April4-7, 1966

The six hundred thirty-second meet­ SYMPOSIUM ON MATHEMATICAL ing of the American Mathematical Society ASPECTS OF COMPUTER SCIENCE will be held at the Waldorf-Astoria Hotelin New York on April 4-7, 1966. All sessions The Symposium will be scheduled in will be held in public rooms of the hotel. four sessions on the afternoon of Tuesday, By invitation of the Committee to April 5, on Wednesday, April 6, and on the Select Hour Speakers for Eastern Sectional morning of Thursday April 7, probably as Meetings, there will be two addresses. follows: Feit of Yale University Professor Walter Session I. .Computation with symbolic April4, at 2:00P.M. will speak on Monday, and algebraic data. The title of his lecture is in the Sert Room. Session II. Numerical methods for com­ of finite groups." "Modular representations puters. Tamagawa of Yale l:Jni­ Professor Tsuneo Session III. Software systems; mechani­ the Society on Tues­ versity will address cal linguistics; computer analysis of 5, at 11:00 A.M. in the Sert day, April language. His lecture is entitled "Z -functions Room. Session IV. Theory of automata. of simple algebras." The Sert Room is at the northwest corner of the main floor The subject of the Symposium was (the corner nearest to Park A venue and chosen by the Committee on Applied Math·­ Fiftieth Street) and is near the registration ematics, which consisted of A. H. Taub area. (Chairman), V. Bargmann, G. E. Forsythe, There will be sessions for contrib­ C. C. Lin, Alfred Schild, and H. S. Wilf. uted papers on Monday morning and after­ Financial support comes from the Air noon and on Tuesday morning, April 4 and Force Office of Scientific Research, the 5. Abstracts should be submitted to the Institute for Defense Analyses, and the American Mathematical Society, P .0. Box U.S. Army Research Office--Durham. The 6248, Providence, Rhode Island 02904, so Association for Computing Machinery and as to arrive prior to the deadline of Feb­ the Association for Symbolic Logic are ruary 18, 1966. co-sponsoring the Symposium. The Invitations Committee, respon­ sible for the planning of the program and THE ASSOCIATION FOR the choice of speakers, consists of Jack SYMBOLIC LOGIC Schwartz (Chairman), Courant Institute of The Association for Symbolic Logic Mathematical Sciences; Martin Davis, will meet in the same hotel, also on Courant Institute of Mathematical Sciences; April 4. Their call for papers has been H. H. Goldstine, IBM Research Center; issued. Their program includes an invited D. H. Lehmer, University of California; address by Professor Hilary Putnam of John Todd, California Institute of Tech­ Harvard University. The program chairman nology; H. S. Wilf, University of Pennsyl­ is Professor Martin D. Davis, Courant In­ vania; Calvin C. Elgot, IBM Research Cen­ stitute of Mathematical Sciences, 251 Mer­ ter; Saul Gorn, University of Pennsylvania; cer Street, New York, New York 10012. Harry Huskey, University of California;

191 and Anthony G. Oettinger, Harvard Univer­ take the I.R.T. Lexington Avenue· local sub­ sity. way to the 51st Street stop. Some of the speakers., with the Those arriving by bus may take the tentative titles of their lectures, are as Independent Subway System (E or F cars) follows: from the west side bus terminal. There is a shuttle bus service from La Guardia and R. W. Floyd: An axiomatic approach to Kennedy Airports to the East Side Terminal the analysis of context-free gram­ with a transfer bus to Grand Central Station. mars (It is suggested that those arriving in a Juris Hartmanis: Theory of automata. group of three or more may find it as Susumu Kuno: Computer analysis of economical to take a taxi directly to the natural languages hotel.) Mark Mahowol.d: The use of computers Those arriving· at Newark Airport in topological calculations can take a shuttle bus to the west side ter­ Michael Rabin: (Title not yet specified) minal and use the Independent Subway Sys­ j. A. Robinson: A review of automatic tem (E or F cars) to the 53rd Street stop. theorem proving Those arriving by car will find many H.P .F. Swinnerton-Dyer: The use of com­ parking facil~ties in the neighborhood in ad­ puters in the theory of numbers dition to those at the hotel. Pickup and J. F. Traub: Tl:i.e calculation of poly­ delivery service can be arranged through nomial zeros the hotel at a cost of $3.25 for a 24-hour period, plus $1.25 for each pickup-delivery. REGISTRATION ROOM RESERVATIONS The registration desk for attendance at the meeting will be in the Terrace Court Persons intending to stay at- the on the main floor near the west end of the Waldorf-Astoria should make their own re­ hotel (the Park A venue entrance). It will servations with the hotel. A reservation be open Monday through Wednesday, April blank and a listing of room rates are on 4-6, from 9:00 A.M. to 5:00 P.M. and on page Z78 of this issue of the .cJiotiaiJ, It Thursday, April 7, from 9:00A.M. till noon. will not appear again in the April issue. TRAVEL MAIL ADDRESS The Waldorf-Astoria occupies an -entire city block on the east side of New Registrants at the meeting may re­ York City, from 49th to 50th Street and c-eive mail addressed in care of the Ameri­ from Lexington to Park Avenues. can Mathematical Society, The Waldorf­ Those arriving by train at Pennsyl­ Astoria, 301 Park Avenue, Nevr York, New vania Station may take the Independent Sub­ York lOOZZ. way System (E or F cars) to the 53rd Street Everett Pitcher and Lexington Avenue stop, a short walk Associate Secretary from the hotel. Bethlehem, Pennsylvania From Grand Central Station onemay

192 Six Hundred Thirty-Third Meeting University ofHawaii Honolulu, Hawaii April9, 1966

The six hundred thirty-third meeting tion area at coffee breaks in the morning of the American Mathematical Society will and afternoon. Luncheon will be available be held on Saturday, April 9, 1966 at the in the cafeteria in Jefferson Hall at the University of Hawaii in Honolulu, Hawaii. East-West Center near Kuykendall Hall, By invitation of the Committee to The University of Hawaii is located Select Hour Speakers for Far Western Sec­ about three miles from downtown Honolulu, tional Meetings, there will be hour ad­ .and about two miles from Waikiki. The cam­ dresses by Professor Branko Griinbaum of pus can be reached by bus, but it is advisable Michigan State University and the Hebrew to use taxi service, since the bus trip re­ University, Jerusalem, and by Professor quires two transfers. The cost of a taxi trip Jakob Korevaar of the University of Cali­ from Waikiki is about $1.75, with reduced fornia, San Diego. Professor Griinbaum will rates if two or more persons share a cab. speak at 11:00 A.M. on "Polytopes, graphs, Honolulu is served by several air­ and complexes." Professor Korevaar's lec­ lines, including Northwest, Pan American, ture on "Distributions" will begin at 1:30 and United Airlines. The round trip thrift P.M. Both of these addresses will be given class fare from Los Angeles and San Fran­ in the Kuykendall Hall Auditorium. cisco is approximat~ly $200. From Portland There will be sessions for contribu­ and Seattle the fare is about $220. Group ted papers at 9:30 A.M. and at 3:30 P.M. travel from the West Coast to this meeting in Kuykendall Hall. The deadline for con­ will not be practical. However, one free tributed papers is February 18. Late pa­ ticket is given with block reservations of pers will be accepted, and if necessary a sixteen or more. Limousine service is special session for late papers will be available between the airport and hotels in scheduled. downtown Honolulu and Waikiki. The Registration. Desk will be located There are numerous hotels in Hono­ on the Lanai of Kuykendall Hall, and it will lulu. All travel agents are happy to provide be open from 8:30 A.M. Since this meeting complete information on available accom­ is being held at the Conference Center of modations and rates. It is recommended that the University, the Council of the American hotel reservations be made early. Mathematical Society has authorized a registration fee to cover costs of the meet­ R. S. Pierce ing. A fee of $3.00 will be collected from Associate Secretary members upon registration. Coffee will be served in the registra- Seattle, Washington

193 Six Hundred Thirty-Fourth Meeting University of Chicago Center for Continuing Education Chicago, Illinois April20-23, 1966

The six hundred thirty-fourth meet­ Brandeis University; and Antoni Zygmund, ing of the American Mathematical Society University of Chicago. will be held at the UnivE!rsity of Chicago The principal topics to be discussed Center for Continuing Education in Chicago, are: Illinois on April ZO-Z3, 1966. Singular Integrals and Real Variables Contributed papers and invited ad­ Singular Integrals and Partial Differen­ dresses will be scheduled on Friday, April tial Equations ZZ, and Saturday, April Z3. There will be a Singular Integrals on Manifolds Symposium on Singular Integrals on Wed­ New and Miscellaneous Aspects of the nesday afternoon, April ZO, Thursday, Theory of Singular Integrals April Z1, and Friday morning, April ZZ. The invited addresses include: Glen Bax­ Approximately 17 persons will be invited ter, Purdue University, "Some aspects of to participate as speakers or discussion the Ising model"; Hans Grauert, Go"ttingen leaders. There will be four hour lectures and Notre Dame, "Nonarchimedean analy­ and a series of shorter papers presenting sis"; and R. G. Swan, University of Chicago, individual attainments in the field of Singu­ "Modules over finite groups." lar Integrals. The subject of the Symposium was The deadline for receipt of abstracts chosen by the Committee to Select Hour for the sessions of ten minute papers on Speakers for Western Sectional Meetings. Friday afternoon is February 18, 1966. The Committee consisted of Seymour Sher­ Abstracts should be sent as usual to the man (Chairman), Felix Browder, and Irving American Mathematical Society, P .0. Box Reiner. Financial support comes from the 6Z48, Providence, Rhode Island 0Z904 so National Science Foundation and the Uni­ as to arrive prior to the deadline. versity of Chicago. A reservation blank is on the last The Invitations Committee respon­ page of this issue of the c}/otiui). sible for the planning of the program and the choice of speakers consists of: Alberto s·~ Sherman P. Calder6n (Chairman), University of Chi­ Associate Secretary cago; K. 0. Friedrichs, Courant Institute of Mathematical Sciences; Robert T. Seeley, Bloomiiigton, Indiana

194 ACTIVITIES OF OTHER ASSOCIATIONS

SIAM ANNOUNCES 1966 FALL MEETING

"Applications of Generalized Func­ of papers will be pre-printed and circu­ tions to System Theory" is the theme of the lated at the meeting. Deadline for the sub­ 1966 Fall Meeting of SIAM to be held at mission of abstracts for contributedpapers the State University of New York, Stony is july 11, 1966. Abstracts should not ex­ Brook, New York, on September lZ-14. The ceed ZOO words. Persons desiring to present meeting will provide an opportunity for the papers at the meeting who are not mem­ discussion of research trends and open bers of SIAM must be introducedbya!1lem­ problems and will allow participants to ac­ ber of SIAM or a member of the Program quaint themselves with the tools and tech­ Committee, which consists of E. J. Betrami, niques of this rapidly growing field of math­ M. Dresden, W. C. Fox, I. Gerst, R. Woh­ ematics. lers, and A. H. Zemanian, Chairman. Contributed papers on all fields of All Abstracts and inquiries should be applied mathematics are solicited. In par­ addressed to: Professor A. H. Zemanian, ticular, for this meeting, contributed pa­ Department of Applied Analysis, State Uni­ pers are being solicited which concern the versity of New York, Stony Brook, New applications of generalized function theory York 11790. to all types of physical models. Abstracts

NEWS ITEMS AND ANNOUNCEMENTS

CONFERENCE ON GENERALIZED FUNC­ or application forms should be addressed TIONS TO BE HELD IN POLAND to: Dr. Piotr Antosik, ul. jaworowa Z, Siemianowice, Silesia, Poland. The Mathematical Institute of the Polish Academy of Sciences will sponsor a Conference on Generalized Functions, August ,30 through September 4, 1966, fol­ SY:MPOSIUM TO BE HELD AT lowing the International Congress in Mos­ U.S. NAVAL ACADEMY cow. Participants in the Moscow Congress may reach the Polish Conference by plane A Symposium on Current Trends in or. train from Moscow. Lattice Theory and Related Topics, spon­ The Conference will be held, for the sored jointly by the Department of Mathe­ most part, in Katowice, Poland's industrial matics of the United States Naval Academy center. However, from September Z, the and the Office of Naval Research, will be Conference will take place from a mountain held April Z, 1966 at the United States resort area 50 miles south of Katowice. Naval Academy, Annapolis, Maryland. Accommodations for all participants The program includes the following will be reserved. Pensions are available at speakers: S. Holland, Boston College; the rate of about eight to ten dollars per G. Birkhoff, Harvard University; G. Rota, day per person. Rockefeller University; and G. Gratzer, Requests for further information and/ Pennsylvania State University.

195 LETTERS TO THE EDITOR

Editor, the c/{otiJ:RV these grants will still suffice. The National Science Foundation, through its research I am writing to urge all interested grants, plans to provide about the same members of the Society to write to the Na­ number of round-trip jet economy :glaces tional Science Foundation asking them to as it did for Stockholm. Funds available for change their policy regarding the use of foreign travel are limited, and the use of travel allowances on NSF grants for the such funds is kept under surveillance by purpose of attendance at the forthcoming Congress and others; under these conditions International Congress in Moscow. Con­ it. is to be expected that competition for sidering the small number of direct grants these funds is becoming much keener. available for travel to the Congress, it seems urgent to find other means of allow­ John Green ing as many other American Mathemati­ cians as possible to attend the Congress. One way which requires no additional allot­ ment of funds is to permit travel money already budgeted for domestic travel to be us~d to attend the Congress. This has been past NSF policy. To change it now, or even Editor, the .c#Oti.ceJ) to limit the permission to a few grant­ On the morning of Moolday, Decem­ holders seems short-sighted. Certainly ber Z9, 195Z, John von Neumann gave a many would prefer to skip several domes­ lecture on Game Theory at a meeting in tic meetings in order to be able to attend St. Louis of the American Association for the International Congress. Surely there the Advancement of Science. The American should be a choice. 1 need not detail the Mathematical Society was also meeting gains to be derived from attendance at the there and his talk had auditors from both Congress. No small savings in tourist meetings. Someone in the audience--identity dollars generated by present policy can unknown--tape recorded this talk. Can you compensate for the loss of these gains. 1 help us locate this tape? urge everyone to write to the NSF in sup­ The Committee on Educational Media port of these views. of the Mathematical Association of America Earl Taft is preparing a film on the life and achieve­ ments of John von Neumann. We are trying Editor's CQmment: to trace down this recording and other recordings, and gather pictures and other The National Research Council is information about him. Photographs are making approximately the same number of especially wanted; all such will be care­ direct grants (with funds from NSF) to at­ fully handled and returned to the sender. tend the Moscow Congress as were made If you can help us, please write or call for that in Stockholm. The grants are (collect) Miss Patricia Powell, 344 West smaller in amount, but it is hoped that with 1Zth Street, New York, New York 10014-­ greater use of supplementary funds from (212) 243-5318, or Dr. A. N. Feldzamen, other sources, as universities, for the Committee on Educational Media, P .0. Box domestic portion of the travel, and with 2310, San Francisco, California 94126-­ wide use of charter and group flights, as (415) 362-7582.. arranged by many university student and faculty groups, as well as by the Society, A. N. Feldzamen

196 NEWS ITEMS AND ANNOUNCEMENTS

SYMPOSIUM ON NUMERICAL SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS

A Symposium on the topic Numeri­ A detailed program of the Symposium cal Solution of Nonlinear Differential Equa­ and information on registration and ac­ tions will be held May 9-11, 1966, spon­ commodations will be available about March sored by the M~thematics Jlesearch Cen­ 1, 1966. Requests for the program and all ter, U. S. Army. The first two days of the related inquiries should be directed to meeting, May 9-10, will be heldinMadison, Professor Donald Greenspan, Symposium Wisconsin, and will consist ~of fourteen Chairman, Mathematics Research Center, half-hour lectures by invited speakers. The U.S. Army, The University of Wisconsin, third day of the Symposium, May 11, will Madison, Wisconsin 53706. be sponsored jointly by the Mathematics Persons interested in other aspects Research Center and SIAM. This session of numerical analysis should note that the will be held at the University of Iowa in jointly sponsored May 11 session on non­ Iowa City and will include an address by linear differential equations will initiate the the president of SIAM, lectures by visiting annual SIAM meeting. This meeting will Russian mathematicians, andthe presenta­ continue in Iowa City through May 14 and will tion of contributed papers. Persons inter­ also consider numerical methods related to ested in presenting a contributed paper matrix computatipns and to integral and should send· an abstract to Professor George integra-differential equations. For further Seifert, Department of Mathematics, Iowa information write to Dr. W. J. Jameson, State University, Ames, Iowa 50010, no SIAM Secretary, Collins Radio Company, later than February 15, 1966.Abstractsare Cedar Rapids, Iowa 52406. limited to 200 words.

MATHEMATICS CONFERENCE TO BE PART OF WEAVER HALL DEDICATION AT COURANT INSTITUTE

The Courant Institute of Mathematical include M. Atiyah, Oxford University, En­ Sciences, New York University will conduct gland; L. Bers, Columbia University; S. a Mathematics Conference, March 21-23, Chandrasekhar, University of Chicago; 1966, in its new building, Warren Weaver P. Cohen, Stanford University; M. Kac, Hall, 251 Mercer Street, New York City. Rockefeller University; M. Lighthill, 1m­ The Conference is part of the Courant In- perial College, London; and J. Mil_nor, stitute's dedication of Warren Weaver Hall. Princeton University. The speakers at the Conference will

MUNICH SITE FOR NATO'S INTERNATIONAL SCHOOL OF NONLINEAR MATHEMATICS AND PHYSICS

Persons interested in the Interna­ been decided that it will be held at the Max tional School of Nonlinear Mathematics and Planck Institute for Physics and Astrophys­ Physics, described in the previous issue of ics in Munich, Germany. Deadline for the c}fotiuiJ, should note that it has now receipt of applications is March 1, 1966.

197 MORE ON FLIGHT ARRANGEMENTS

To date, the status of travel arrangements made by the Society for the 1966 International Congress is as follows:

I. CHARTER FLIGHTS A. August 14: New York-Moscow (SAS) August 29: Moscow-New York (SAS) There is a fixed price for this 138 passenger jet, and the price for each individual will be prorated to the number of passengers particip~ting in the flight, Plans and arrange­ ments will be based on a minimum of 120 persons for which the approximate price would be $434. However, as many seats as possible up to the 138 limit will be sold, and in that case, individuals will be entitled to a :refund, for example the cost might be re­ duced to approximately $378.

B. August 14: New York-Moscow (PAA) August 30: Moscow-New York (P AA) The approximate fare, based on: the minimum of 120 passengers, is $443, As many seats as possible up to a limit of 154 will be sold and an approximate reduced cost might be $347.

C. The following charter flight is available for those applicants who desire trans­ portation from Western Europe to Moscow. August 15: Paris-Moscow (LOT) September 1: Moscow-Paris (LOT) The approximate fare for· this flight, based on a minimum of 70 passengers, is $228. As many seats as possible up to 87 will be sold and an approximate reduced cost might be $182. Ail of the fares stated above include the cost of insurance for loss incurred by an applicant who, by reason of injury or sickness, is prevented from actually parti­ cipating in the flight.

II. GROUP FLIGHTS The following flights, all nonstop jets, are available for groups consisting of a minimum of 2.5 people. No.1. june: New York-London (TWA) Fare: $300 September 13: London-New York (TWA) No.2. june 7: New York-Paris (TWA) Fare: $331 September 7: Paris-New York (TWA) No.3, june 21: New York-Paris (PAA) Fare: $331 September 12: Paris-New York (PAA) No.4, june 28: New York-Paris (P AA) Fare: $331 September 13: Paris-New York (PAA) No.5, july 20: New York-Paris (P AA) Fare: $331 September 15: Paris-New York (P AA)

198 No.6. July 28: New York-Paris (Air France) Fare: $545 August 12: Paris-Moscow (Aeroflot) August 31: Moscow-Paris (Aeroflot) September 5: Paris-New York (Air France) No.7. August 4: New York-Paris (Air Fra:p.ce) Fare $545 August 12.: Paris-Moscow (Aeroflot) August 31: Moscow-Paris (Aeroflot) September 5: Paris-New York (Air France)

INFORMATION CONCERNING AMS FLIGHTS

1. Deadline for all applications April 1. 2. $50 deposit for applicant and for each dependent accompanying him must be included in applications. Deadline for payment of balance is April 15. 3. Arrangements for one-way flights (available only on charters) are contingent upon the number of requests that can be matched up to fill round trip seats. 4. Only a limited number of seats are still available on any of the charters. Therefore anyone applying for a charter flight subsequent to this announcement should list at least one of the group flights as an alternate choice. 5. All group flights are contingent upon a minimum number of applicants. We must have at least 25 for a group before their arrangement is definite. 6. Participants must be members of the Society (or their families living in the same household) six months prior to the time of departure. 7. Regular commercial flights from New York to Moscow and return are available as follows: 1st Class - $1,109.50 Economy - $815.50 Persons interested in commercial flights should make private arrangements with travel agents.

NEWS ITEMS AND ANNOUNCEMENTS

POLISH ACADEMY TO HOLD SEMINAR ON CONFERENCE ON ANALYTIC FUNCTIONS SPECTRAL ANALYSIS OF TIME SERIES

The Institute of Mathematics of the An advanced seminar on Spectral Polish Academy of Sciences will conduct Analysis of Time Series will be held at the a Conference on Analytic Functions in Mathematics Research Center, U.S. Army, ~6di from September 1 through September University of Wisconsin, Madison, Wiscon­ 7, 1966. The Conference will consist of sin, during October 3-5,1966. The proceed­ deliveredlectures, ten-minute communica­ ings will be published. tions and three seminars on the subjects of Tentatively scheduled as participants extremal problems, quasi-conformal map­ are Professors E. Parzen, S. K. Zaremba, pings, and functions of several complex M. Rosenblatt, L. J, Tick, G.J.MacDonald, variables. All correspondence relative to H. Panofsky, and M.D. Godfrey. the Conference should be directed to the For further information please write Organizing Committee, Institute of Mathe­ to Professor Bernard Harris, Mathematics matics Conference, al. Kosciuszki 21, Research Center, U, S, Army, University ~6dz, Poland. of Wisconsin, Madison, Wisconsin, 53706,

199 MEMORANDA TO MEMBERS

OTHER JOURNALS AVAILABLE TO MEMBERS AT REDUCED RATES

The Mathematical journals li-sted but sent directly to the journals. Subscrip­ below, some of which receive grants from tion prices are per volume in 1966. the American Mathematical Society, are The special rates are offered only to available to members of the Society at re­ individual members of the Society, not to duced subscription rates. Orders for sub­ institutions. scriptions should not be sent to the Society, AMS Name of .Journal List Members American Journal of Mathematics The Johns Hopkins Press Baltimore, Maryland Zl218 $11.00 $ 8.80* Annals of Mathematics (2 volumes per year) Princeton University P .0. Box 231, Princeton, New Jersey 08540 18.00 9.00 Canadian Journal of Mathematics University of Toronto Toronto, Ontario, Canada 12.00 6.00 Illinois Journal of Mathematics University of Illinois Urbana, lllinois--61803 9 .oo 5.00

Journal D' Analyse Mathematique 13 Abrabanel Street Jerusalem, Israel $20.00** $12.00** Journal of Mathematics and Physics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 8.00 5.00 Michigan Mathematical Journal University of Michigan Ann Arbor, Michigan 48104 12.00 4.00 Pacific Journal of Mathematics 103 Highland Boulevard Berkeley, California 94 708 32.00 16.00 Journal of the Society for Industrial and Applied Mathematics 33 South 17th Street Philadelphia, Pennsylvania 19103 20.00 16.00 Theory of Probability and its Applications Society for Industrial and Applied Mathematics 33 South 17th Street Philadelphia, Pennsylvania 19103 28.50 Z0.60

*Postage per volume: 30 cents Canada; 60 cents foreign. **Two volumes per year; prices per volume.

200 Backlog of Mathematical Research Journals

Information on this important matter .is received but not yet accepted are being ignored.) being published twice a year, in the February and Column 4. Estimated by the editors (or the August issues of the c/{otiui), with the kind Editorial Department of the American Mathemati­ cooperation of the respective editorial boards. cal Society in the case of the Society's journals) It is important that the reader should inter­ and based on these factors; manuscripts accepted, pret the data with full allowance for the wide and manuscripts received and under consideration, sometimes meaningless fluctuations which are manuscripts in galley, and rate of publication. characteristic of them. Waiting times in particular There is no fixed formula. are affected by many transient effects, which Column 5. The first quartile ( Q1l and the arise in part from the refereeing system. Ex­ third quartile (Q3) are presented to give a meas­ treme waiting times as observed from the pub­ ure of the dispersion which will not be too much lished dates of receipt of manuscripts may be distorted by meaningless extreme values. The very misleading, and for that reason, no data on median (Med.) is used as the measure oflocation. extremes are presented in the table at the bottom The observations were made from the latest issue of this page. received in the Headquarters Offices before the Some of the columns in the table are not deadline date for this issue of the c/{otiai) .. The quite self-explanatory, and here are some further waiting times were measured by counting the details on how the figures were computed. months from receipt of manuscript in final re­ Column 2. These numbers are rounded off vised form, to month in which the is!!lue was re­ to the nearest 50. ceived at the Headquarters Offices. It should be Column 3. ·For each journal, this is the noted that when a paper is revised, the waiting estimate as of the indicated dates, of the total time between receipt by editors of the final number of printed pages which will have been ac­ revision and its publication may be much shorter cepted by the next time that manuscripts are to than is the case for a paper which is not revised, be sent to the printer, but which nevertheless will so these figures are to that extent distorted on the not be sent to the printer at that time. (Pages low side.

1 2 3 4 5 Est. time for paper Observed waiting time No. f\pprox.no. BACKLOG submitted currently in latest published JOURNAL issues pages per 12/31/65 6/30/65 to be published issue (in months) per year year pages pages (in months) Q1 Med. Q3 American J. Math. 4 NR* NR* NR* NR* 14 14 16 Annals of Math. 6 1200 1195 1217 12 10 12 16 Annals of Math. Statist. 6 1900 0 0 5-8 9 9 17 Arch. Hist. Exact Sciences not fixed 500 0 Nll* 5-6 7 8 9 Arch Rational Mech. Anal. not fixed 1300 0 NR* 5 8 8 9 Canad. J. Math. 6 1320 500 500 14 15 16 17 Duke Math. J. 4 680-700 420 661 15-18 18 18 19 Illinois J. Math. 4 700 300 510 20 20 20 21 J. Math. Analyses and Appl. 6 NR* NR* NR* NR* *" *" *" J. Math. and Mech, 12 1000 600 1000 7 *" *" *" J. Math. and Phys. 4 NR* NR* 400 NR* 13 13 15 J. Mathematical Physics 12 6400 255 25 6 9 11 12 J. Symbolic Logic NR* NR* NR* NR* NR* 12 14 19 Math. C<1mp. 4 725 0 22 8 7 9 9 .Michigan Math. J. 4 500 100 50 12 9 10 13 Pacific J. Math. 12 2250 1500 1050 12 20 22 24 Proceedings of the AMS 6 1584 75 350 8 15 16. 16 Quarterly of Appl. Math. 4 NR* NR* NR* NR* 15 15 16 SIAM Journal 6 1500 350 200 9-12 15 17 20 SIAM Journal on Control 4 680 0 0 6-9 10 13 15 SIAM J. Qn Numer. Anal. 4 680 70 0 6-9 9 10 16 SIAM Review 4 700 0 0 6-9 10 11 20 Transactions of the AMS 12 2750 50 500 11 17 19 21

* NR means that no response was rece1ved to a request for information. *·* Dates of receipt of manuscript not indicated in this journal.

201 CORPORATE MEMBERS AND INSTITUTIONAL ASSOCIATES.

We are pleased to announce that, as Radio Corporation of America of December 23, 1965; the following com­ Shell Development Company panies and corporations are supporting the Socony Mobil Oil Company, Incorporated Society through Corporate Membership. Standard Oil Company Incorporated in New Jersey Academic Press, Incorporated TRW Systems, Incorporated Bell Telephone Laboratories, Incorpo- United Gas Corporation rated The Boeing Company The following corporations and companies Corning Glass Works Foundation are supporting the Society as Institutional E. I. duPont de Nemours and Company, Associates. Incorporated Eastman Kodak Company Ford Motor Company Chelsea Publishing Company General Dynamics Corporation Consultants Bureau Enterprises, ln­ General Motors Corporation co:r:po.rated Hughes Aircraft Company Dover Publications International Business Machines Cor­ McGraw-Hill Book Company, Incorpo­ poration rated Lockheed Missiles and Space Company Springer-Verlag Marathon Oil Company Union Oil Company of California

NEWS ITEMS AND ANNOUNCEMENTS

UNIVERSITY OF ARIZONA OFFERS PH.D. AIR FORCE TO SPONSOR PROGRAM IN SYSTEMS ENGINEERING BIONICS SYMPOSIUM

The University of Arizona is now The_ Air Force Systems Command's offering a Ph.D. program in Systems En­ Aerospace Medical Division and Research gineering. Students holding an M.S. degree and Technology Division will jointly sponsor may be admitted to this program by passing a Bionics Symposium May 3-5, 1966 at the a qualifying -examination which requires a Sheraton Hilton Hotel in Dayton, Ohio. The broad background in probability and statis­ Symposium will consist of invited and con­ tics, engineering mathematics, numerical tributed papers, including invited review analysis and computers, optimization and lectures covering the status of selected operations research, human factors, and areas. The scope of the Symposium is the systems theory. Students entering with a entire field of bionics, but main emphasis Bachelo:t"'s degree may participate in a will be on those areas which are related to Master's program which insures their pro­ cybernetics. Topics of special interest in­ ficiency in the six above mentioned areas. clude: biological control and information­ Thesis and dissertation subjects are chosen processing systems, neu.romime networks, according to the interest of the student in artificial intelligence, pattern recognition, one of these fields of knowledge. self-organizing and learning systems, auto­ Faculty members of the Department mata theory and related areas. includeS. R. Browning, L. Duckstein, J. W. Further information may be obtained Perry, J. L. Sanders, E. W. Titt, H. Tucker, by writing to: Bionics Symposium 1966, R. j. Weldon, A. W. Wymore, Chairman. Station D, P .0. Box 51, Dayton, Ohio 45410 Additional information concerning or by contacting Dr. H. L. Oestreicher, graduate curricula in Systems Engineering Aerospace Medical Research Laboratorie-s, can be obtained by writing to the Depart­ Wright-Patterson Air Force Base, Ohio ment. 45433.

202 SUMMER INSTITUTES AND GRADUATE COURSES

The following is a list of graduate courses, seminars and institutes in mathematics being offered in the summer of 1966 for graduate students and college teachers of mathematics. The list was compiled from information received from graduate schools in this country.

GRADUATE COURSES

ANDREWS UNIVERSITY, Berrien Springs, Michigan 49104 June 13-August 18 · Application deadline: May 13 . Information: Edward J. Specht, Chairman, Department of Mathematics Algebraic structures

UNIVERSITY OF ARIZONA, Tucson, Arizona 85721 Application deadlines: June 10 and July 15 Information: J. F. Foster, Department of Mathematics June 12 -July 15 July 17 -August 19 205a Advanced Analysis for Engineers 205b Advanced Analysis for Engineers 213a Foundations of Geometry 213b Foundations of Geometry 230 Matrix Analysis 203 Elements of Complex Variables 231 Introduction to Modern Algebra

ARIZONA STATE COLLEGE, Flagstaff, Arizona 86003 Information: Registrar June 13-July 16 July 18-August 19 540 Finite Dim. Vector SPaces Theory of Numbers Advanced Calculus Advanced Calculus Theory of Groups

ARIZONA STATE UNIVERSITY, Tempe, Arizona 85281 · Application deadlines: June 13 and July 18 Information: Dr. Evar D. Nering, Chairman, Department of Mathematics June 13-July 16 Julx18-Ausust 20 404 Projective Geometry 46P .1\pplied Real Analysis 460 Applied Real Analysis 461 Applied Complex Analysis 461 Applied Complex Analysis 462 Partial.Differential Equations 462 Partial Differential Equations 471 Continuation of 470 470 Foundations of Analysis 442 Vector Spaces, Matrix Theory

AUBURN UNIVERSITY, Auburn, Alabama 36830 June 13-August 22 Application deadline: May 25 Information: Professor L. P. Burton, Head, Department of Mathematics Topics in Applied Mathematics Dimension Theory Real Variables I Homological Algebra Complex Variables IT Functional Analysis ill Advanced Theory of Differenttal Eq11ations

BOSTON. UNIVERSITY, Boston, Massachusetts 02215 Informat1uu: .r LvLw~-'r Scheid, Department of Mathematics, College of Liberal Arts MA 841S Applied Math. Seminar MA 842S Applied Math •. Seminar MA 843S Algebra Seminar

UNIVERSITY OF BRITISH COLUMBIA, Vancouver 13, B. C., Canada June 27-August 19 Application deadline: April 15 Information: C. A. Swanson, Department of Mathematics Linear Topological Spaces Topic to be announced Homological Algebra Travel grants and tuition are paid to qualified applicants.

c. W. POST, Greenvale, New York 11548 Information: Dr. Sylvan Wallach, Department of Mathematics June 27-July 29 August !-September 2 501 Mathematics for Elementary School Teachers 502 Mathematics for Elementary School Teachers 511 Set Theory 512 Mathematical Logic 522 Number Theory 524 History of Mathematics 617 Abstract Algebra 618 Abstract Algebra 661 Geometry 662 Geometry 674 Applications of Mathematics in the 675 Applications of Mathematics in the Physical and Biological Sciences Physical and Biological Sciences

203 UNIVERSITY OF CALIFORNIA, Berkeley, California 94720 Application deadlines: May 19 and July 5 Information: H. Helson, Chairman, Mathematics Department June 20 -July 29 August !-September 9 S202 Foundations of Analysis S203 Measure and Integration S206 Linear Spaces S205A Theory of Functions of a Complex Variable S215A Algebraic Topology S250A Groups, Rings, and Fields

UNIVERSITY OF CALIFORNIA, Berkeley, California 94720 Application deadline: May 18 Information: Henry Scheff~, Chairman, Department of Statistics June 20-July 29 August !-September 9 202A Theory of Probability and Statistics 202B Theory of Probability and Statistics (Statistics) (Probability) · 202D Laboratory course for 202B 202C Laboratory course for 202A

UNIVERSITY OF CALIFORNIA, Los Angeles, California 90024 Information: University of California Extension, 405 Hilgard Avenue University extension has a tentative list of 80 courses; each course is of two weeks or less duration. These courses are in engineering, physics and chemistry, many of which have a mathematical emphasis.

UNIVERSITY OF CALIFORNIA, Santa Barbara, California 93106 June 20-July 30 Application deadline: June 20 Information: Office of the Su=er Sessions, Administration Building Topics in Field Theory CENTRAL MICHIGAN UNIVERSITY, Mount Pleasant, Michigan 48858 June 20-July 29 Application deadline: June 15 Information: Professor Edward H. Whitmore, Chairman, Department of Mathematics Differential Equations Theory of Numbers History of Mathematics

UNIVERSITY OF CHATTANOOGA, Chattanooga, Tennessee 37403 Information: Winston L. Massey, Chairman, Department of Mathematics June 8-July 15 June 11-August 18 (Evening classes) Fundamentals of Modern Mathematics for Introduction to Analysis Teachers Mathematical Statistics

COLORADO STATE COLLEGE, Greeley, Colorado 80630 June 13-August 19 Application deadline: May 1 Information: Dr. William D. Popejoy Algebra Set Theory Geometry Logic Advanced Calculus Structure of Numbers Educational Statistics Computer Mathematics

COLUMBIA UNIVERSITY, TEACHERS COLLEGE, New York, New York 10027 July 1.,-August 12 Application deadline: May 1 Dr. Howard F. Fehr, Chairman, Department of Mathematical Education TX4451 Mathematics Applied to Computers TX5455 Abstract Algebra TX4455 Secondary Mathematics from an Advanced TX5459 Theory of Functions of a Complex Variable Viewpoint TX5450 Mathematical Logic

CREIGHTON UNIVERSITY, Omaha, Nebraska 68131 June 14-August 5 Application deadline: May 15 Information: G. A. Hutchison, Chairman, Department of Mathematics Fourier Series Topology Group Theory

EAST CAROLINA COLLEGE, Greenville, North Carolina 27834 Application deadlines: June 6 and July 13 Information: Dr. Tullio J. Pignani, Chairman, Department of Mathematics June 6-July 12 July 13 -August 18 MA345G Non-Euclidean Geometry MA365G Theory of Numbers MA371G Theory of Equations MA369G Historical Development of Mathematics MA381G Modern Mathematics for Elementary MA405 Modern Mathematics "for Secondary Teachers Teachers MA419 Infinite Series · . · MA391G Complex Variables

204 EAST TENNESSEE STATE UNIVERSITY, Johnson City, Tennessee 37601 Application deadlines: March 20 and April 22 Information: Mr. William M. Beasley, Dean of Admissions, or Dr. RichardS. Stevenson, Dean, School of Graduate Studies June 20-July 22 July 25-August 26 Mathematical Statistics Mathematical Statistics Introduction to Modern Algebra Introduction to Modern Algebra Introduction to Modern Geometry Introduction to Modern Geometry Foundations and Structure of Mathematics Foundations and Structure of Mathematics Theory of Matrices Theory of Numbers Modern Algebra Modern Algebra Introduction to Research

EASTERN ILLINOIS UNIVERSITY, Charleston, illinois 61920 June 20-August 12 Application deadline: May 20 Information: Lawrence A. Ringenberg, Chairman, Department of Mathematics 530 Real Variables I 471 Statistics II 461 Advanced Calculus II 480 Finance 507 Digital Computer Techniques 540 Problems in Teaching of Mathematics (K-6) 4 07 Statistics I 570 Problems in the Teaching of Mathematics (9-12)

EASTERN MICHIGAN UNIVERSITY, Ypsilanti, Michigan 48197 June 27-August 5 Application deadline: June 27 Information: James H. Glasgow, Dean of the Graduate School , Topics in Modern Mathematics-Senior High School Elements of Set Theory Selected Topics in Modern Mathematics -Junior Non-Euclidean Geometry High School Geometry for High School Teachers

EMORY UNIVERSITY, Atlanta, Georgia 30322 Application deadlines: May 17 and June 29 Information: Henry Sharp, Jr., Associate Professor of Mathematics, Department of Mathematics June 18-July 29 July 30-August 20 260 Probability and Statistics 421 Complex Analysis 411 Foundations of Geometry 421 Complex Analysis FLORIDA STATE UNIVERSITY, Tallahassee, Florida 32306 Application deadlines: March 21 and May 13 Information: Dr. Ralph A. Bradley, Head, Department of Statistics May 2-June 18 June 23-August 13 Ss 405 Statistical Procedures in the Natural Ss 406 Statistical Procedures in the Social Sciences Sciences Ss 412 Discrete Probability II Ss 407 Statistical Methods & Design Ss 523 Limit Theory of Statistics Ss 411 Discrete Probability I Ss 599 & 699 Thesis & Dissertation Ss 522 Statistical. Inference II Ss 526 Advanced Biometry Ss 624 Multivariate Analysis II Ss 599 & 699 Thesis & Dissertation

FORT HAYS KANSAS STATE COLLEGE, Hays, Kansas 67601 June 7 -August 5 Application deadline: June 7 Informatiom Professor W. Toalson, Department of Mathematics Elementary Probability Advanced Calculus I Differential Equations Thesis Scientific Computer Programming Laboratory

GEORGE WASIDNGTON UNIVERSITY, Washington, D. C. 20006 June 13-July 20 Application deadline: May 1 Information: D. Nelson, Chairman, Department of Mathematics Topics in Projective Differential GeomE)try

UNIVERSITY OF GEORGIA, Athens, Georgia 30601 June 13-July 21 Application deadline: May 24 Information: Dr. B. J. Ball, Head, Department of Mathematics Theory of Numbers Fundamental Ideas of Algebra Higher Algebra College Geometry Advanced Calculus Elementary Set Theory Projective Geometry Basic Ideas of Calculus Basic Ideas of Arithmetic Seminar in Algebra

GEORGIA STATE COLLEGE, Atlanta, Georgia 30303 June 10-August 19 Information: Dr. G. L. Tiller, Head, Department of Mathematics 642 Modern Algebra II (prerequisite-Math 641) 841 General Topology I 661 Advanced Calculus I 880 Topics in Mathematics 832 Complex Variables II (prerequisite...:Math 831)

205 HARVARD UNIVERSITY, Cambridge, Massachusetts 02138 July 5-August 26 Application deadline: June 20 Jnfor.mation: Mr. T. E. Crooks, 626 Holyoke Center, 75 Mount Auburn Street, Cambridge, Massachusetts Introduction to Mathematical Methods I (advanced calculus) Introduction to Mathematical Methods II (differential equations) Prerequisites: One year of calculus with some differential equations

HENDERSON STATE TEACHERS COLLEGE, Arkadelphia, Arkansas 71923 Information: Dr. C. W. Thomasson, Dean of Graduate Division, .Box 672, Henderson State Teachers College June 6-July 9 · · July 11-August 13 P~bability and Statistics Theory of Numbers Fundamental Algebraic Concepts Foundations of Mathematics Elementary Mathematics ill* Elementary Mathematics I* :tVIathematics Workshop* Elementary Mathematics II* . *Elementary Mathematics and Mathematics Workshop are opeil to teachers ofEiementaiy School Mathematics only. All eight courses (including workshop) carry th:~ree semester hours credit.

ILLINOIS INSTITUTE OF TECHNOLOGY, Chicago, illinois 60616 June 27-August 19 Application deadline: June 20 Information: Dr. Haim Reingold, Chairman, Department of Mathematics ·507 Modern Algebra I 521 Topology 513 Real Variables I

INDIANA UNIVERSITY OF PENNSYLVANIA, India:rul, Pennsylvania 15701 Application deadlines: May 15 Pre-session, June 15 Main Session, July 20 Post-session Information: Dr. I. L. Stright, Dean, Graduate School June 9-June 22 · June 28-August 4 Pre-Session Main-Bession 510 Seminar I 501 Fortran 521 Fundllmental Concept: E 1 542 Elementary Arithmetic 529 Differential Equations I of Analysis August a-August 19 532 Advanced Calculus II 531 Advanced Calculus I 539 Infinite Series I 533 Functions of a Post-session 552 Number Theory 511 Seminar II Complex Variable 521 Basic Concepts 563 Linear Algebra I 562 Modern Algebra II 581 Advanced Statistics 571 Modern Geometry I

IOWA STATE UNIVERSITY, Ames, Iowa 50010 Registration deadlines: June 7 and July 18 Information: George Seifert, Chairman, Department of Mathematics June· 8-July 15 · · July 19-August 25 Linear Algebraic Computations Complex Variables Real Variables I Real Variables II Applied Mathematics I Applied Mathematics II

KANSAS STATE COLLEGE, Pittsburg, Kansas 66764 June 7-August 5 Application deadline: May 1 Information: Professor R. G. Smith, Chairman, Department of Mathematics 503 Modern Mathematics for Elementary 613 Modern Algebra I Teachers G5;1 Functions of cl}mplex Variables 510.Basic Concepts of Algebra 690 Research and Thesis 530 Basic Concepts of Geometry KANSAS STATE TEACHERS COLLEGE, Emporia, Kansas 66802 Application deadlines: June 6 and July 18 Information: Dr. Marion P. Emerson, Chairman, Department of Mathematics June 6-July 15 July 18-August 26 Programming Computer for Mathematics Projective Geometry Probability and Statistics Abstract Algebra II Abstract Algebra I Advanced Calculus II Differential Equations Mathelilatical StatiStics Advanced Calculus I Introduction to Mathematical Logic Topology Complex Variables Differential Geometry Theory of Matrices

KANSAS STATE UNIVERSITY, Manhattan, Kansas 66504 June 13-August 5 Application deadline: May 1 Information: Professor R. G. Sanger, Head, Department of Mathematics Numerical Methods Non-Euclidean Geometry Cal.culus of Variations

KUTZTOWN STATE COLLEGE, ~tztown, Pennsylvania 19530 June·27-August 5 Application deadline: June 27 Information: Dr. Edward W. Evans, Chairman, Department of Mathematics Group Theory Foundations of Analysis

206 MARQUETTE UNIVERSITY, Milwaukee, Wisconsin 53233 June 22-August 3 Application deadline: May 11 Information: L. W. Friedrich, S.J., Dean, Graduate School Introductory Topology Theory of Games and Linear Programming Linear Algebra Advanced Mathematical Statistics* College Geometry Advanced Calculus Probability Topics in Abstract Analysis* Topics in Abstract Algebra* Starred courses carry graduate credit only; others carry both undergraduate and graduate credit.

UNIVERSITY OF MIAMI, Coral Gables, Florida 33124 Application deadlines: May 31 and June 30 Information: E. F. Low, Jr., Chairman, Department of Mathematics June 16-July 22 July 25-August 30 504 College Geometry 511 Advanced Calculus II 510 Advanced Calculus I 671 Topics in Modern Mathematics 671 Topics in Modern Mathematics

MIAMI UNIVERSITY, Oxford, Ohio 45056 Information: S. E. Bohn, Chairman, Department of Mathematics April 24-June 14 June 19-August 9 Linear Algebra Linear Algebra Topics in Applied Analysis Geometry Probability and Statistics (II) Introduction to Modern Algebra Topology Probability and Statistics (I) Topics in Advanced Mathematics Topics in Advanced Mathematics

MICHIGAN STATE UNIVERSITY, East Lansing, Michigan 48823 June 20-September 1 Application deadline: June 1 Information: Professor M. L. Tomber, Department of Mathematics Set Theory and Abstract Spaces Boundary Value Problems Advanced Analysis Combinations Matrices and Groups

MICHIGAN STATE UNIVERSITY, East Lansing, Michigan 48823 June 22-8eptember 3 Application deadline: May30 Information, K. J. Arnold, Chairman, Department of Statistics and Probability Statistics 864 Stochastic Models in Biology

MONTCLAffi STATE COLLEGE, Upper Montclair, New Jersey 07043 June 27-August 5 Application deadline: June 22 Information: Paul C. Clifford, Chairman, Department of Mathematics Algebra Advanced Calculus Geometry Computer Programming Probability Applications

MOORHEAD STATE COLLEGE, Moorhead, Minnesota 56560 Application deadlines: May 15 and June 21 Information: Marion V. Smith, Chairman, Department of Mathematics June 15-July 20 July 21-August 24 Foundations of Mathematics History of Mathematics Modern Algebra Linear Algebra Computer Programming Problems in Mathematics Numerical Analysis Independent Study Problems in Mathematics Independent Study

UNIVERSITY OF NEBRASKA, Lincoln, Nebraska 68503 June 13-August 5 Registration: June 10 Information: Department of Mathematics Algebra Topology Geometry Statistics Number Theory

UNIVERSITY OF NEW MEXICO, Albuquerque, New Mexico 87106 June 27-August 19 Application deadline: June 20 Information: Professor J. Mayer, Assistant Chairman, Department of Mathematics Universal Algebra

207 NEW YORK UNIVERSITY, New York, New York 10012 Application deadline for both sessions: May 16 Information: Ruth Shor, New York University, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012 June 20-July 29 August !-September 9 Numerical Methods Linear Algebra Linear Algebra Special Functions Probability Statistical Inference

UNIVERSITY OF NORTH CAROLINA, Chapel Hill, North Carolina 27515 Application deadlines: May 21 and July 5 Information: Professor C. Wayne Patty, Department of Mathematics, Phillips Hall June 9-July 16 July 18-August 25 Math 181 Elementary Theory of Numbers Math 179 Summability Math 175 Topics in Analysis Math 194 Mathematical Analysis Math 193 Mathematical Analysis Math 226 Foundations of Geometry Math 281 General Topology I

NORTH CAROLINA STATE UNIVERSITY, Raleigh, North Carolina 27607 June 6-August 8 Application deadline: May 1 Information: Dr. Jack Suberman, Director, Summer Sessions MA 532 Theory of Ordinary Differential Equations MA 541 Theory of Probability I MA 622 Linear Algebra MA 625 Differential Geometry MA 632 Operational Mathematics I

NORTH TEXAS STATE UNIVERSITY, Denton, Texas 76203 Application deadlines: June 9 and July 21 Information: John T. Mohat, Chairman, Department of Mathematics June 4-July 13 July 16-August 24 Modern Algebra Modern Algebra Matrix Theory Geometry Calculus of Variations Analysis Analysis Complex Variables Complex Variables Functional Analysis Topology

NORTHEAST LOUISIANA STATE COLLEGE, Monroe, Louisiana June 6-August 5 Application deadline: May 15 Information: Dale R. Bedgood, Head, Department of Mathematics College Geometry Introduction to Modern Algebra Fundamental Concepts of Modern Mathematics Topology (Elementary Teachers) Modern Algebra I Foundations of Mathematics Introduction to Analysis I Linear Algebra

NORTHEAST MISSOURI STATE TEACHERS COLLEGE, June 6-August 12 Kirksville , Missouri 635 01 Application deadline: June 6 Information: Dale Woods, Chairman, Division of Mathematics Advanced Calculus Probability & Statistics Linear Algebra Modern Geometry Abstract Algebra Readings in Mathematics

NORTHERN ILLINOIS UNIVERSITY, DeKalb, illinois 60115 June 13-August 5 Information: Admissions Office or Mathematics Department 407 Theory of Numbers I 451 Teaching Practices in High School Mathematics II 420 Differential Equations I 502 The Teaching of Arithmetic in Elementary School 425 Introduction Topics in Modern Algebra 515 Advanced Analytic Geometry 430 Advanced Calculus I 532 Introduction to Complex Variables and 435 Foundations of Geometry Applications I 450 Teaching Practices in High School 534 Real Variables I Mathematics I 535 Foundations of Mathematics

NORTHERN MICIDGAN UNIVERSITY, Marquette, Michigan June 27-August 5 Application deadline: June 1 Information: Harry Rajala, Office of Admissions Linear Algebra Mathematical Logic (Elementary Education) Modern Geometry Theories of the Teaching of Arithmetic Probability & Statistics Inference I Seminar in Elementary Mathematics

208 NORTHWESTERN UNIVERSITY, Evanston, illinois 60201 June 20-August 13 Application deadline: May 20 Information: Helen M. Clark, Lunt 218D C02 Probability and Statistics C41 Foundations of Algebra for Teachers C05 Complex Variables for Applications C42 Foundations of Geometry for Teachers C10-1 Advanced Calculus C43 The History of Mathematics I C 14 Ordinary Differential Equations C45 Advanced Geometry for Teachers C19-1 Digital Computer Programming and C46-1 Foundations of Calculus for Teachers Error Analysis C46-2 Foundations of Calculus for Teachers C30 Probability E 10 Analysis Seminar C34-1 Vectors, Matrices, Quadratic Forms E 12 Topology and Geometry Seminar C35 Introduction to Theory of Numbers E90 Research for Ph.D Dissertation

UNIVERSITY OF OKLAHOMA, Norman, Oklahoma 73069 June 8-August 4 Information: Dr. Richard V. Andree, Mathematics Service Committee Introduction to Computer Science Theory of Numbers Introduction to Abstract Algebra Honors Readings Arithmetic for Elementary Teachers Advanced Calculus I Engineering Mathematics (4 hrs.) Higher Algebra I Engineering Mathematics (3 hrs.) Functions of a Complex Variable I Basic Concepts of Calculus Functions of Real Variables Foundations of Analysis Special Problems in Mathematics College Geometry Seminar: Analysis Foundations of Geometry Seminar: Algebra Numerical Analysis Seminar: Geometry Principles of Mathematical Statistics I Seminar: Applied Mathematics Elementary Differential Equations Research for Master's Thesis Honors Reading Research for Doctor's Dissertation Tensor and Vector Analysis Differential Geometry Algebra for Secondary Teachers Functional Analysis Analytic Projective Geometry Calculus of Variations Theory of Probability

PEPPERDINE COLLEGE, Los Angeles, California 90044 June 20-July 29 Application deadline: June 20 Information: Ladis D. Kovach, Chairman, Department of Mathematics and Physics, Pepperdine College, So. Vermont Avenue at 79th Street Complex Analysis

UNIVERSITY OF PITTSBURGH, Pittsburgh, Pennsylvania 15213 June 16-August 5 Application deadline: June 1 Information: Dr. George Laush, Department of Mathematics Mathematics 270 Topology I Mathematics 280 Geometry I

POLYTECHNIC INSTITUTE OF BROOKLYN, Brooklyn, New York 11201 Application deadlines: June 9 and July 20 Information: Andrew J. Terzuoli, Department of Mathematics June 13-July 21 August 1-September 8 Calculus Mathematical Statistics Differential Equations Vector Analysis Probability Applied Mathematics Vector Analysis Advanced Calculus Applied Mathematics Complex Variables Computer Techniques Numerical Analysis Advanced Calculus Partial Differential Equations Complex Variables Linear Algebra Numerical Analysis Finite Differences Partial Differential Equations Linear Algebra PURDUE UNIVERSITY, West Lafayette, Indiana 47!;107 Information: Associate Professor Robert E. Zink, Division of Mathematical Sciences CS 514 Numerical Analysis Ma 547 Analysis for Teachers I Ma 519 Introductory Applied Probability Ma 548 Analysis for Teachers ll Ma 530 Functions of a Complex Variable Ma 550 Algebra for Teachers I Ma 544 Principles of Analysis I (real variables) Ma 551 Algebra for Teachers ll Ma 546 Introduction to Functional Analysis Ma 560 Fundamental Concepts of Geometry Ma 554 Linear Algebra Ma 563 Advanced Geometry Ma 561 Projective Geometry Ma 583 History of Elementary Mathematics Ma 571 Elementary Topology

QUEENS COLLEGE, Flushing, New York 11367 July 5-August 12 Application deadline: June 1 Information: Dr. Howard Knag, Director Probability

209 RHODE ISLAND COLLEGE, Providence, Rhode Island 02908 June 27-August 5 Application deadline: June 24 Information: Graduate Office Topology

RUTGERS, THE STATE UNIVERSITY, New Brunswick, New Jersey 08903 July 5-August 12 Information: Dr. Angus Austen, Director of the Summer Session Introductory Theory of Flinctions of a Complex Introductory Abstract Algebra Variable

ST. JOHN'S UNIVERSITY, Jamaica, New York 11432 July 5-August 13 Registration for non-matriculated students June 29-30 Information: Dr. Richard C. Morgan, Chairman, Department of Mathematics Partial Differential Equations Projective Geometry Request Summer Schoof brochure from Miss Margaret B. Kelly, Coordinator of Curricula, St. John's University, Grand Central & Utopia Parkways, Jamaica, New York 11432

ST. MARY'S UNIVERSITY, San Antonio,. Texas 78228 Application deadlines: June 11 and July 22 Information: Brother Fraxik Gutting, S.M. , Chairman, Department of Mathematics June 8-July 15 July 19-August 26 Mt401G Introduction to Modern Algebra Mt402 Linear Algebra Mt403G Modern Geometry* Mt404 Modern Geometry* Mt431G Probability Mt432 Mathematical Statistics Mt607 Analysis I* Mt608 Analysis ll* Mt633 Complex Variables Mt704 Modern Geometry** Mt703 Modern Geometry** Mt733 Stochastic Processes *Offered to teachers of High School Mathematics. **Doubtful about offering this sequence; offered to teachers of High School Mathematics.

SOUTHERN ILLINOIS UNIVERSITY, Carbondale, illinois 62901 June 13-August 19 Application deadline: March 26 Information: Professor John M. H. Olmsted, Chairman, Department of Mathematics Linear Algebra Seminar Course in Algebra; Geometry; Analysis; Probability Probability and Statistics (probably two of these Finite Mathematics four)

SOUTHERN ILLINOIS UNIVERSITY, Edwardsville, lllinois 62025 June 13-August 5 Application deadline: June 1 Information: Dr. R. N. Pendergrass, Director, Southern illinois University Theory of Numbers Theory of Point Sets

UNIVERSITY OF SOUTHERN MISSISSIPPI, Hattiesburg, Mississippi39401 June 6-August 6 Application deadline: February 15 Information: Dr. Ed Kelly, Jr., Chairman, Department of Mathematics Modern Algebra Foundations of Mathematics Geometry Probability and Statistics Applicant must be teaching mathematics in a secondary school, with at least three years experience, and have a degree from a credited institution.

SOUTHWEST TEXAS STATE COLLEGE, San Marcos, Texas 78666 Information: Don Cude, Chairman, Department of Mathematics June 1-July 19 July 12-August 20 Functions of Complex Variables Partial Differential Equations Theory of Functions of Real Variables Theory of Determinants Introduction to Modern Algebra Probability and Statistics Advanced Calculus

STATE UNIVERSITY OF NEW YORK, COLLEGE AT GENESEO, June 27-August 5 Geneseo, New York 14454 Information: W. A. Small, Chairman, Department of Mathematics Theory of Functions of Real Variables I Introduction to Modern Algebra I* Abstract Algebra I Higher Geometry* Advanced Calculus I* *Sometimes carry graduate credit. SUNY AT ALBANY, Albany, New York 12203 July 5-August 19 Application deadline: June 1 Information: Office of Graduate Studies, State University of New York, Albany, New York 12203 Mathematical Logic Abstract Algebra Projective Geometry Groups, Rings and Fields Theory of Numbers Functions of a Complex Variable Topology

210 SUNY AT BUFFALO, Buffalo, New York Application deadlines: June 6 and June 27 or between May 14-20 Information: Professor D. Tamari, Chairman, Department of Mathematics June 6-July 15 June 27-August 5 July 18-August 26 S319 Introduction to Higher Algebra S425 Introduction to Complex S320 Higher Algebra S421 Introduction to Real Variables Variables S422 Real Variables S499 Mathematical Logic I S427 Introduction to Topology S527 Non-Euclidean Geometry S537 Operational Calculus S539 Theory of Matrices S551 Theory of Sets S657 Integral Equations

UNIVERSITY OF TENNESSEE, Knoxville, Tennessee 37916 June 13-August 25 Registration date: June 13 Information: Professor John H. Barrett, Head, Department of Mathematics Matrix Differential Equations Advanced Calculus Foundations of Analysis Credit Seminars: Differential Equations, Algebra, Partial Differential Equations Topology, Number Theory, Complex Variables Linear Algebra Some of these may be available for period June 12-July 19 only

TRENTON STATE COLLEGE, Trenton, New Jersey 08625 June 27-August 5 Application deadline: June 27 Information: William Hausdoerffer, Chairman, Department of Mathematics Mathematical Logic Seminar in Mathematics Education

TULANE UNIVERSITY, New Orleans, Louisiana 70118 Application deadlines: June 13 and July 25 Information: Professor G. S. Young, Chairman, Department of Mathematics June 13-July 22 July 25-September 2 601 Probability 605 Probability

VANDERBILT UNIVERSITY, Nashville, Tennessee 37203 June 13-August 27 Information: James R. Wesson, Box 1595, Vanderbilt University Topics in Advanced Calculus Intermediate Differential Equations Theory of Functions of a Complex Variable

VILLANOVA UNIVERSITY, Villanova, Pennsylvania 19085 July 11-August 12 Application deadline: June 11 Information: Mr. Emil Amelotti, Chairman, Department of Mathematics 535 Intermediate Analysis I 524 Sets, Relations and Functions 53 6 Intermediate Analysis II 526 Real Number System 529 Advanced Calculus 534 Linear Algebra 527 Modern Algebra 520 Introduction to Mathematical Logic and 537 Topology Foundations of Mathematics 544 Modern Geometry 601 Logic, Sets, Relations* 602 Analysis I* 603 Algebraic Structures* 604 Analysis II 605 Geometric Foundations* 606 Analysis ill* *For junior and senior high school teachers of mathematics.

UNIVERSITY OF VIRGINIA, Charlottesville, Virginia 22901 June 20-August 13 Application deadline: June 1 Information: Miss Anne P. Brydon, Registrar, University of Virginia Summer Session, Garrett Hall, Charlottesville, Virginia 107 Foundations of Algebra 114 Introductory Analysis 111 Advanced Calculus 216 Advanced Calculus

WAGNER COLLEGE, Staten Island, New York 10301 June 13-August 5 Information: Robert J. Dwyer, Chairman, Department of Mathematics 141 Theory of Numbers

UNIVERSITY OF WASHINGTON, Seattle, Washington 98105 Application deadline for both sessions: May 15 Information: Graduate Secretary, Department of Mathematics June 20-July 20 July 21-August 19 507 Foundations of Mathematics 508 Foundations of Mathematics 531 Special Topics in Analysis 532 Special Topics in Analysis

WAYNE STATE UNIVERSITY, Detroit, Michigan 48202 Information: Karl W. Folley, Chairman, Department of Mathematics Complex Analysis (Second Course) Advanced Topics in Algebra Topology (Introductory) Advanced Topics in Functional Analysis

211 WEST CHESTER STATE COLLEGE, West Chester, Pennsylvania 19380 Jime 27 -August 5 Application deadline: June 15 Information: Dr. Albert E. Filano, Chairman, Department of Mathematics History of Mathematics Theory of Sets

WEST TEXAS STATE UNIVERSITY, Canyon, Texas 79015 June 6-July 15 Application deadline: June 6 Information: Dr. H. L. Cook, Head, Department of Mathematics Linear Algebra Partial Differential Equations

WEST VffiGINIA UNIVERSITY, Morgantown, West Virginia June 13-August 20 Application deadline: June 1 Information: Dr. Laddie R. Bell, Associate Professor of Education 235 Introduction to Analysis and Topology 366 Algebraic Plane Curves 237 Introduction to Linear Algebra 378 Functional Analysis 257 Applied Mathematics I 264, 265 Foundations of Algebra 260 Advanced Real Calculus 268, 269 Probability and Statistics 309 Group Theory 271 PR Structure of Number Systems* *From June 13-July 2, for elementary teachers

WESTERN ILLINOIS UNIVERSITY, Macomb, lllinois 61455 June 20-August 12 Application deadline: June 6 Information: Joseph Stipanowich, Head, Department of Mathematics Logic and Sets* History of Mathematics* Foundations of Geometry* Complex Variables Point Set Topology* Modern Algebra Real Number System* Teaching of Secondary Mathematics Linear Algebra* Elementary Mathematics and Methodology *For advanced undergraduates as well.

WESTERN MICillGAN UNIVERSITY, Kalamazoo, Michigan 49001 June 20-August 12 Application deadline: June 1 Information: James H. Powell, Head, Department of Mathematics Programming for Computers Mathematical Statistics Linear Algebra Advanced Calculus Algebraic Geometry Complex Analysis

WESTERN WASHINGTON STATE COLLEGE, Bellingham, Washington 98225 June 20-August 19 Information: Dr. William Abel, Department of Mathematics Advanced Abstract Algebra Theory of Functions of a Real Variable Theory of Numbers Complex Variables Topology Mathematical Logic and Sets Analysis Theory of Algorithms

UNIVERSITY OF WISCONSIN, Madison, Wisconsin 53706 June 20-August 13 Information: Mathematics Department Linear Transformations in Hilbert Space Advanced Topics in Algebra Group Theory Advanced Topics in Point Set Topology Introductory Topology Advanced Topics in Algebraic Topology Advanced Topics in Real Analysis Advanced Topics in Number Theory UNIVERSITY OF WYOMING, Laramie, Wyoming 82071 Application deadlines: June 13 and July 18 Information: Dr. W. Norman Smith, Box 3036, University Station, University of Wyoming June 13-July 15 July 18-August 19 Fundamental Concepts of Mathematics Partial Differential Equations, Fourier Series and Theory of Probability Boundary Value Problems Point Set Topology Numerical Analysis Theory of Functions of a Complex Variable Theory of Numbers Elementary Matrix Theory and Linear Algebra Advanced Calculus Introduction to Higher Algebra-Advanced Calculus College Geometry Point Set Topology Theory of Functions of a Complex Variable

XAVIER UNIVERSITY, Cincinnati, Ohio 45207 June 20-July 29 Application deadline: June 17 Information: Dr. Charles Wheeler, Dean, Summer School Mathematical Logic Real Analysis Seminar in Abstract Algebra

212 SUMMER INSTITUTES AND SEMINARS

BOWDOIN COLLEGE, Brunswick, Maine 04011 June 21-August 11 ADVANCED SEMINAR IN ALGEBRAIC NUMBER THEORY AND CLASS FIELD THEORY Requirements for admission: Suitable previous graduate work, specific interest and opportunity to continue work in these fields, endorsements Other information: Stipends open to graduate and postdoctoral students; dislocation allotments for mathematicians having other research support Application deadline: March 1 Information: Professor Dan E. Christie, Chairman, Department of Mathematics

UNIVERSITY OF OKLAHOMA, Norman, Oklahoma 73069 June 8-August 17 RESEARCH PARTICIPATION FOR COLLEGE TEACHERS Sponsoring Agency: National Science Foundation Subjects covered: Research projects in any area, particularly in abstract algebra, geometry, calculus of variations, differential equations, functional analysis, bio-statistics, computer science, mathematical logic, applied mathematics, advanced programming and numerical analysis. Each applicant's summary is read and rated by faculty members interested in that area. Participants are selected largely on this basis. Requirements for admission: Predoctoral applicants should hold master's degree. All applicants should be full-time faculty members teaching in a college or university Other information: First round of awards will be made in February. Five predoctoral and ten postdoctoral stipends at $750 and $1000 will be made; dependency and travel allowances will be awarded. Each participant will enroll in an advanced course or seminar in mathematics or computer science and will contribute two lectures (one elementary, one advanced) to the summer colloquium series. It is hoped that each participant will prepare at least one research paper during the summer. Advanced seminars will be conducted for small groups with common interests Application deadline: Applications accepted anytime Information: Dr. Richard V. Andree, Mathematics Service Committee

RUTGERS, THE STATE UNIVERSITY, New Brunswick, New Jersey 08903 June 20-August 12 SUMMER INSTITUTE FOR COLLEGE TEACHERS Sponsoring Agency: National Science Foundation Subjects covered: Foundations of analysis, advanced topics in calculus, mathematical statistics, introductory abstract algebra, functions of a real variable Requirements for admission: For college teachers of mathematics who do not have Ph, D. Application deadline: May 15 (Deadline for stipend consideration was February 15) Information: Joshua Barlaz, Department of Mathematics

SUNY AT BUFFALO, Buffalo, New York 14214 June 27-August 5 SEMINARS IN ALGEBRA, ANALYSIS AND LOGIC Requirements for admission: Permission of instructor Information: Dov Tamari, Chairman, Department of Mathematics, 208 Michael Hall

VANDERBILT UNIVERSITY, Nashville, Tennessee 37203 June 6-July 1 CONFERENCE IN LINEAR ALGEBRA AND TOPOLOGY Sponsoring Agency: National Science Foundation Subjects covered: Linear algebra (Finite-dimensional vector spaces, matrix algebra, linear transformations); elementary topology (fundamental ideas in the setting of general spaces and metric spaces) Requirements for admission: Must be teaching mathematics at college or junior college level Other information: Preference will be given to teachers who lack the Ph. D. degree and are relatively inexperienced in linear algebra and topology Application deadline: March 15 Information: James R. Wesson, Box 1595, Vanderbilt University

213 PERSONAL ITEMS

Professor N. D. ALLAN of the Uni­ SW AMY of the University of California, versity of Chicago has been appointed to an Berkeley has been appointed to an assistant assistant professorship at De Paul Univer­ professorship at the University of Kansas. sity. Professor W. E. CLARK of the Cali­ Professor E. S. ANDERSEN of the Uni­ fornia Institute of Technology has been ap­ versity of Aarhus, Aarhus, Denmark has pointed to an assistant professorship at the been appointed to a professorship at the University of Florida. University of Copenhagen, Copenhagen, Professor E. D. CONWAY of the Cour­ Denmark. ant Institute of Mathematical Sciences, Professor C. E. AULL of Kent State New York University has been appointed to University has been appointed to an asso­ an assistant professorship at the University ciate professorship at the Virginia Poly­ of California, San Diego. technical Institute. Professor W. E. CONWAY of the Uni­ Professor D. F. BAILEY of Vanderbilt versity of Arizona has been appointed to an University has been appointed to an assis­ associate professorship at West Texas tant professorship at East Carolina College. State University. Professor R. G. BIERSTEDT of the Uni­ Professor H. H. CRAPO of North­ versity of California, Berkeley has been eastern University has been appointed to an appointed to an assistant professorship at associate professorship at the University the University of New Mexico. of Waterloo. Dr. E. R. BOBO of the University of Professor H. K. CROWDER of the Case Virginia has been appointed to an assis­ Institute of Technology has been appointed tant professorship at the Georgetown Uni­ to an associate professorship at the Cleve­ versity. land State University. Mr. C. E. BOYNDON, IV of Columbia Professor K. J. DAVIS of the United University has accepted a position as Pub­ States Army has been appointed to an asso­ lications Editor of the College Entrance ciate professorship at Old Dominion Col­ Examination Board, New York, New York. lege. Professor D. T. BROWN of Syracuse Dr. W. E. DAVIS of the Hercules Pow­ University has been appointed to an assis­ der Company, Wilmington, Delaware has tant professorship at Hiram College. been appointed to an associate professor­ Professor J. R. CANNON of the Brook­ ship at the University of Delaware. haven National Laboratory, Upton, New Professor R. A. DE VILLIERS of York has been appointed to an associate Allen University has been appointed to an professorship at Purdue University. associate professorship at the Sacred Heart Dr. P. J. CHASE of the California University. Institute of Technology has been appointed Professor R. E. DOWDS of Butler to an assistant professorship at the College University has been appointed to an asso­ of Wooster. ciate professorship at the State University Professor KUO-TSAI CHEN of Rutgers, of New York, College at Fredonia. The State University has been appointed to Professor D. M. EAVES of the Uni­ a professorship at the State University of versity of Washington has been appointed New York at Buffalo. to an assistant professorship at the Simon Professor Y. M. CHEN of Purdue Fraser University. University has been appointed to an asso­ Professor H. L. EGAN of the Washing­ ciate professorship in the Department of ton University has been appointed to an Engineering Science and Mechanics at the assistant professorship at the University University of Florida. of Maryland. Professor JA YANTHI CHIDAMBARA- Professor MURRAY EISENBERG of

214 Wesleyan University has been appointed to Dr. JURIS HAR TMANIS of the General an assistant professorship atthe University Electric Research Laboratory, Schenectady, of Massachusetts. New York has been appointed Professor and Miss J. E. H. ELLIOTT of the Univer­ Chairman of the Computer Science Depart­ sity of Miami has been appointed to an as­ ment at Cornell University. sistant professorship at the University of Professor L. F. HEATH of the Univer­ Florida. sity of Kansas has been appointed to an Professor JACOB ENGELHARDT of assistant professorship at Arlington State Washington State University has accepted a College. position as a Mathematician with the Hudson Professor H. P. HEINIG of the Univer­ Laboratories of Columbia University. sity of Toronto has been appointed to an Professor M. A. FELDSTEIN of the assistant professorship at McMaster Uni­ University of California, Los Angeles has versity. b~en appointed to an assistant professorship Dr. D. R. HENNEY of the University of at Brown University. Maryland has been appointed to an assistant Professor A. I. FINE of the Illinois professorship at the George Washington Institute of Technology has been appointed University. to an assistant professorship at the Univer­ Professor H. H. W. HERDA of Wayne sity of Illinois. State University has been appointed to an Professor R. L. GANTOS of Michigan assistant professorship at Salem State Col­ State University has been appointed to an lege. assistant professorship at the University of 1st Lt. C. J. HIGHTOWER of the Wisconsin, Milwaukee. United States Army has been appointed to Dr. K. R. GENTRY of the University an assistant professorship atthe University of Georgia has been appointed to an assis­ of Colorado. tant professorship at the University of Mrs. S. N. HILT of the University of North Carolina at Greensboro. North Carolina has been appointed to an Professor W. 0. GORDON has retired assistant professorship at the Colorado from the Department of Mathematics at College. Pennsylvania State University with the rank Miss NOBUO HITOTSUYANAGI of the of Emeritus. Hiroshima Women's Junior College, Hiro­ Professor J. W. GRAY of the Univer­ shima, Japan has been appointed to an sity of Illinois will be visiting at the assistant professorship at the Kagoshima Eidgenossiche Technische Hochschule, University, Kagoshima, Japan. Zurich, Switzerland from February 1966 to Mr. LOUIS HODES of the International September 1967. Business Machines Corporation, Yorktown Professor R. J. GREECHIE of the Uni­ Heights, New York has been appointed a versity of Florida has been appointed to an Visiting Member of the Courant Institute of assistant professorship at the University of Mathematical Sciences at New York Uni­ Massachusetts, Boston. versity. Dr. LEON GREENBERG of the Univer­ Mr. J. M. HORNER of the General sity of Maryland has accepted a position as Motors Research Laboratories, Huntsville, a Mathematician at the Institute for Basic Alabama has been appointed to an assistant Standards with the National Bureau of professorship at the University of Alabama, Standards, Washington, D. C. Huntsville. Professor V. B. GYLYS of the Univer­ Professor J. T. HOWSON, JR. of the sity of Illinois has been appointed to an Rose Polytechnic Institute has been ap­ assistant professorship at the Illinois In­ pointed to an assistant professorship at the stitute of Technology. University of Cincinnati. Professor M. R. HAGAN of the Stephen Dr. W. L. HOYT of Indiana University F. Austin State College has been appointed has been appointed to an associate pro­ to an assistant professors-hip at North Texas fessorship at Rutgers, The State University. State University. Professor J. W. HURST of Montana Professor R. T. HARRIS of Duke Uni­ State University, Bozeman has been ap­ versity has been appointed to an associate pointed to a professorship at the State Uni­ professorship at the University of Mary­ versity of New York, College at Fredonia. land. Professor TERUO IKEBE of the Uni-

215 versity of Tokyo, Tokyo, Japan has been town University has been appointed to an appointed to an assistant professorship at associate professorship at the University the University of Kyoto, Kyoto, Japan. of Toronto. Professor Emeritus SHIKAO IKEHARA Mrs. M. P. LEE of Brown University of the Tokyo Institute of Technology, Tokyo, has been appointed to an assistant professor­ Japan has been appointed to a professorship ship at Ohio State University. at the Tokyo Electrical Engineering Col­ Mr. ARTHUR LIB EN SON of the Ray­ lege, Tokyo, Japan. theon Company, Sudbury, Massachusetts Dr. ALFRED INSELBERG of the Uni­ has accepted a position on the Technical versity of Illinois has been appointed Re­ Staff of the Mitre Corporation, Bedford, search Assistant Professor at the Biologi­ Massachusetts. cal Computer Laboratory at the University Mr. JOSEPH LIPMAN of Queen's Uni­ of Illinois. versity has been appointed to an assistant Professor J. W. JAWOROWSKI of Cor­ professorship at Purdue University. nell University has been appointed to a Dr. R. G. LONG of Wesleyan University professorship at Indiana University. has accepted a position as Associate Exec­ Professor 0. T. JONES of Florida utive Director of the Committee on Educa­ State University has been appointed to an tional Media at the Mathematical Associa­ assistant professorship at Stetson Univer­ tion of America, San Francisco, California. sity. Professor E. R. LORCH of Columbia Mr. ABRAHAM KAREN of New York University will be on sabbatical leave until University has accepted a position as Sys­ july 1966 as a Visiting Professor at the tems Designer with Univac/Whippany, Han­ Middle East Technical University, Ankara, over, New jersey. Turkey. Dr. E. G. KIMME of the Bell Telephone Dr. EUGENE LUKACS, Director of the Laboratories at Murray Hill, New jersey Statistical Laboratory at Catholic Univer­ has been appointed Head of the Applied sity is on leave during the academic year Science Department, Research and Develop­ 1965-1966. He is spending part of the time ment Division of the Collins Radio Com­ at the University of Vienna and part at the pany, Newport Beach. California. Sorbonne. Professor HANS-HEINRICH KOERLE Professor W. D. Me INTOSH of the of ·the University of Utah has been appointed University of Kansas has been appointed to to an associate professorship at Eastern an assistant professorship at the Univer­ Michigan University. sity of Missouri. Professor H. S. KONIJN of City Col­ Dr. j. W. MACK! of the Los Alamos lege, City University of New York has been Scientific Laboratory, Los Alamos, New appointed to a professorship at the Tel Mexico has been appointed to an assistant Aviv University, Ramat Aviv, Israel. professorship at the University of Alberta. Dr. L. H. KOOPMANS of the Sandia Professor ROBERT MALTZ of San Corporation, Albuquerque, New Mexico has Diego State College has been appointed to been appointed to an associate professor­ an assistant professorship at the University ship at the University of New Mexico. of California, Irvine. Professor CHARLES KRAFT of the Dr. J. F. P. MAR TIN of Mendham, University of Minnesota has been appointed New jersey has accepted a position as to a professorship at the University of Principal Mathematician with the Cornell Montreal. Aeronautical Laboratory, Buffalo, New Mr. ANDREW KRAUS of the General York. Electric Company, New Hartford, New York Dr. H. J. MISER of the Mitre Cor­ has accepted a position as an Electronics poration has accepted a position as Vice Engineer with the Bell Aerosystems Com­ President and Director of the Department pany, Tucson, Arizona. of Environmental Systems Management Dr. R. J. LARSEN of Yale University Studies with the Travelers Research Cen­ has been appointed to an assistant profes­ ter Incorporated, Hartford, Connecticut. sorship at Cowell College of the University Professor WILLIAM NACHBAR of of California, Santa Cruz. Stanford University has been appointed a Professor j. E. LE BEL of George- Professor in the Department of the Aero-

216 space and Mechanical Engineering Sciences Professor CHARLES SWARTZ of Ari­ at the University of California, San Diego. zona University has been appointed to an Professor 0. T. NELSON, JR. of Van­ assistant professorship at New Mexico derbilt University has been appointed to an State University. assistant professorship at Emory Univer­ Professor OSAMU T AKENOUCHI of sity. Okayama University, Okayama, japan has Professor A. B. J. NOVIKOFF of the been appointed to a professorship at the Stanford Research Institute has been ap­ Osaka University, Osaka, japan. pointed to an associate professorship at Dr. F. C. Y. TANG of the Illinois Insti­ the Courant Institute of Mathematical Sci­ tute of Technology has been appointed to an ences at New York University. associate professorship at the University Professor T. E. OBERBECK of the of Waterloo. United States Naval Postgraduate School Professor G. I. TARGONSKI of Ford­ has accepted a position as Advisor for ham University is spending the academic Research Operations at the Office of Re­ year 1965-1966 at the Research Institute search Analyses, Holloman Air Force Base, for Mathematics of the Swiss Federal In­ New Mexico. stitute of Technology, Zurich, Switzerland. Professor E. T. ONAT of Brown Uni­ Professor R. L. TENNISON of the East versity has been appointed a Professor in Central State College has been appointed the Department of Engineering and Applied to an assistant professorship at Arlington Science at Yale University. State College. Professor DANIEL PEDOE of Purdue Professor W. L. TERWILLIGER of University has been appointed to a pro­ Washington State University has been ap­ fessorship at the University of Minnesota. pointed to an assistant professorship at Dr. C. R. PITTMAN of the University the Bowling Green State University. of Georgia has been appointed to an assis­ Mr. J. W. THATCHER of the Watson tant professorship at West Georgia College. Research Center of the International Bu­ Mr. CHOON-JAI RHEE of the Univer­ siness Machines Corporation, Yorktown sity of Georgia has been appointed to an Heights, New York has been appointed to assistant professorship at Randolph-Macon a visiting assistant professorship at Fisk Woman's College. University. Mr. j. A. RILEY of the Parke Mathe­ Mr. MILES TIERNEY of Columbia matical Laboratories, Incorporated, Car­ University has been appointed to an assis­ lisle, Massachusetts has been appointed tant professorship at Rice University. to an associate professorship at the Lowell Professor R. j. TROYER of Indiana Technological Institute. University has been appointed to an assis­ Dr. 0. S. ROTHAUS of the Institute tant professorship at the University of for Defense Analyses, Princeton, New jer­ North Carolina, Chapel Hill. sey has been appointed to a visiting pro­ Dr. E. F. WAGNER of the University fessorship at Yale University. of New Mexico has been appointed to an Dr. LEONARD SARASON of Stanford assistant professorship at the University of University has been appointed to an assis­ Nevada. tant professorship at the University of Dr. C. H. WARLICK of the General Washington. Electric Company, Cincinnati, Ohio has Professor KENITI SA TO of the Tokyo been appointed Assistant Director of the Metropolitan University, Tokyo, Japan has Computation Center at the University of been appointed to an associate professor­ Texas. ship at the Tokyo University of Education, Professor A. I. WEINZWEIG of North­ Tokyo, japan. western University has been appointed to Dr. SAMUEL SCHECHTER of New an associate professorship at the Univer­ York University has been appointed to a sity of Illinois, Chicago Circle. visiting associate professorship at Stan­ Dr. L. R. WELCH of the Institute for ford University. Defense Analyses, Princeton, New jersey Miss L. R. SONS of Cornell University has been appointed Visiting Associate Pro­ has been appointed to an assistant pro­ fessor of Electrical Engineering at the fessorship at Northern Illinois University. University of Southern California.

217 Professor C. F. WELLS of Duke Uni­ To Associate Professor: versity has been appointed to an assistant University of Connecticut: HELENE RES­ professorship at We stern Reserve Univer­ CHOVSKY; Fairleigh Dickinson University: sity. MAURICE MACHOVER; University of Illi­ Dr. BURTON WENDROFF on leave nois: R. L. BISHOP; UniversityofKansas: of absence from the Los Alamos Scientific C. J. HIMMELBERG; University of Min­ Laboratory, Los Alamos, New Mexico has nesota: W. F. POHL; Texas Technical been appointed to a visiting associate pro­ College: T. A. ATCHISON. fessorship at the University of New Mexico. Professor B. C. WHEATON of the To Assistant Professor: Western Illinois University has been ap­ pointed to an assistant professorship at Illinois Institute of Technology: A. Z. CZAR­ Mankato State College. NECKI; University of Illinois: CHIN-PI L U; Professor J. H. M. WHITFIELD of University of Wisconsin: J. E. HALL. Case Institute of Technology has been ap­ pointed to an assistant professorship at The following appointments are announced: Lakehead University. Dr. E. F. WISHART of Florida State To Instructor: University has been appointed to an assis­ Columbia University: R. L. HALL; Cornell tant professorship at the University of University: SIEGFRIED GROSSER; Harvard Nevada. University: J. R. GOLDMAN; University of Dr. Y. K. WONG of Princeton, New Illinois: MARCEL HERZOG; University of Jersey has been appointed to a professor­ Illinois, Chicago Circle: N. D. GOODMAN; ship at the State University of New York University of Minnesota: G. H. KNIGHTLY; at Albany. New Haven College: E. H. BIRD; State Professor L. E. T. WU of the Western University of New York at Stony Brook: Washington State College has been appointed CHRISTOPHER WASIUTYNSKI; Northwest­ to an assistant professorship at Cowell ern University: E. L. SPITZNAGEL, Jr.; College of the University of California, Ohio University: W. T. FISHBACK; Univer­ Santa Cruz. sity of Pennsylvania: R. L. FABER; Prince­ ton University: M.L. SILVERSTEIN; Wash­ The following promotions are announced: ington University: D. R. BELDIN; Yale University: W. RICKERT, DA YA-NAND To Associate Vice President of Aca­ VERMA. demic Affairs: University of Notre Dame, T.E.STEWART. Deaths:

To Director of Research Services: Professor G. E. F. SHERWOOD of the University of California, Los Angeles died Chesapeake and Ohio Railway: H. N. LADEN. on December 16, 1965 at the age of BZ. He was a member of the Society for 44 To Professor: years. Birla Institute of Technology and Science: Mr. R. M. SMITH of Bethel College VISVANATHA KRISHNAMURTHY; Kyoto died September Z7, 1965 at the age of Z6. University: YUKIO KUSUNOKI; Southwest Professor S. D. ZELDIN of the Massa­ Center for Advanced Studies: WOLFGANG chusetts Institute of Technology died No­ RINDLER; University of Wisconsin: E. H. vember Z, 1965 at the age of 71. He was FELLER. a member of the Society for 51 years.

218 SUPPLEMENTARY PROGRAM-Number37

During the interval from November 29, 1965 through January 6, 1966 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Con­ tributed Papers in this issue of these cJVotirrV~ One abstract presented by title may be accepted per person per issue of the cJioticrV, Joint authors are treated as a separate category; thus in addition to abstracts from two authors individually one joint abstract by them may be accepted for a particular issue. ( 1) A reduction method for special cases (9) On compatibility of extensions of of the decision problem semiregular integral transformations Mr. S. 0. Aanderaa, Harvard Univer­ Professor Nachman Aronszajn and sity (66T-176) Professor Pawel Szeptycki, Univer­ (Introduced by Professor Burton Dreben) sity of Kansas (66T-116) (2) On the minimal length of sequences ( 1 O) Generating submodules in simple representing simply ordered sets rings with involution Professor Alexander Abian and Mr. Professor W. E. Baxter and Mr. E. David Deever, The Ohio State Uni­ F. Haeussler, Univevsity of Delaware versity ( 66T-162) (66T-106) (3) Complex cubic splines (11) The minimum dimension of symmet­ Professor J. H. Ahlberg, United ric matrices with a given root Aircraft,East Hartford, Connecticut, Mr. E. A. Bender, California Insti­ Dr. E. N. Nilson, Pratt and Whitney tute of Technology (66T-120) Aircraft, East Hartford, Connecticut, (12) Two nonisomorphic (v,k, A) designs and Professor J. L. Walsh, Univer­ with similar incidence matrices sity of Maryland (66T-84) Professor C. T. Benson, University (4) The dual space of an operator algebra. of Manitoba (66T-141) Preliminary report ( 13) An oscillation theorem Mr. C. A. Akemann, University of Professor N. P. Bhatia, Western California, Berkeley (66T-109) Reserve University (66T-122) (5) Cantor sets in Euclidean three space ( 14) Slice algebras of bounded analytic Dr. W. R. Alford, F. J. Seiler Re­ functions. Preliminary report search Laboratory, USAF Academy, Professor F. T. Birtel, Tulane Uni­ Colorado (66T-148) versity (66T-93) (6) Fractional q-integration and q-differ­ ( 15) On polynomials of best one sided ap- entiation proximation Dr. W. A. Al-Salam, University of Professor Ranko Bojanic and Mr. Alberta (66T-121) R. H. DeVore, Ohio State University (7) On quasi-homogeneous potential op- (66T-147) erators ( 16) Finite dimensional perturbations of Mr. P. J. Aranda, Mr. E. P. Catta­ spectral systems neo, Universidad del Litoral, and Professor R. D. Brown and Professor Mr. Cora Sadosky, Universidad de Nachman Aronszajn, University of Buenos Aires, Argentina (66T-177) Kansas (66T-112) (8) Extension of unbounded operators in ( 1 7) Every fully indecomposable matrix a Hilbert space if:! diagonally equivalent to a doubly Professor Nachman Aronszajn, Uni­ stochastic matrix versity of Kansas (66T-107) Professor R. A. Brualdi, Professor

219 S. V. Parter and Professor Hans versal algebras Schneider, University of Wisconsin Professor.George Gratzer, The Penn­ (66T-102) sylvania State University (66T-92) (18) Local weak norms and almost every­ (32) On the structure of finite orthomodu­ where convergence lar lattices Professor D. L. Burkholder, Univer­ Mr. R. j. Greechie, University of sity of Illinois (66T-179) Massachusetts at Boston (66T-123) (19) The univalence of an integral. II (33) On the generalized inverse of a ma- Mr. W. M. Causey, University of trix product Kansas (66T-85) Dr. T. N. E. Greville, Mathematics (20) On continuous functions commuting Research Center, U. S. Army and functions and fixed points University of Wisconsin ( 66T-160) Dr. S.C. Chu, Bellcom, Inc., Wash­ (34) Factorization of periodic functions ington, D. C. and Professor R. D. Dr. Fred Gross, Bellcom, Inc., Moyer, Pennsylvania State University Washington, D. C. (66T-145) (66T-149) (35) A property of the Riesz function (21) On the sum of the squares oftwocon- Professor Emil Grosswald, Univer­ secutive numbers sity of Pennsylvania (66T-124) Mr. E. L. Cohen, The MITRE Cor­ (36) A modified Monte-Carlo quadrature poration, Bedford, Massachusetts Dr. Seymour Haber, National Bureau (66T-130) of Standards, Washington, D. C. (22) Subresultants and reduced polynomial (66T-125) remainder sequences (37) Isotonic spaces in convexity. I Dr. G. E. Collins, lnternationalBus­ Professor P. C. Hammer, The Penn­ iness Machines, Yorktown Heights sylvania State University (66T-99) (66T-138) (38) On systems of differential equations (23) Quasi-open maps in uniform algebras with special types of singularities Professor R. M. Crownover, Univer­ Professor R. j. Hanson, University sity of Missouri (66T-168) of Southern California (66T-158) (24) Generalized R-uniformities (39) On operators with a Fredholm spec- Mr. D. W. Curtis andProfessorj.C. trum. Preliminary report Mathews, Iowa State University(66T- Dr. M. A. Kaashoek, University of 151) California, Los Angeles (66T-153) (25) On subgroups of semigroups (40) On the characterization of weakly Professor D. F. Dawson, North Texas representable Boolean algebras State University (66T-167) Professor Carol Karp, University of (26) Group representations and cardinal Maryland (66T-174) algebras (41) A characterization of semiperfect Professor P. C. Deliyannis, Athens, modules Greece (66T-105) Dr. Friedrich Kasch, University of (27) Decomposition of modules. II. Rings Munich, Munich, West Germany, and without chain conditions Dr. E. A. Mares, Swarthmore Col­ Professor S. E. Dickson, University lege and HRB-Singer, Inc., State of Nebraska (66T-146) College, Pennsylvania (66T-156) (28) Model theory of ranks and orders (42f Products of zero-one matrices Mr. H. M. Friedman, Massachusetts Professor j. B. Kelly, Arizona State Institute of Technology (66T-178) University (66T-lll) (29) On representations of the Weyl group (43) Primitive idempotents in a regular D. A. Gay, Dartmouth College (66T- semigroup with 0 95) Dr. j. B. Kim, Michigan State Uni­ (30) Periodic and semiperiodic sequences versity (66T-131) and Fourier-Stieltjes transforms of (Introduced by Dr. S. H. Gould) discrete measures (44) Compatibility of uncertainty in 3- Professor Gunther Goes, Illinois In­ space with determinism in space­ stitute of Technology (66T-133) time (31) On direct and inverse limits of uni- Professor Ali Kyrala, Arizona State

220 University (66T-144) Mr. Lawrence Narici, Polytechnic (45) A property of universal sets Institute of Brooklyn (66T-8) Professor A. H. Lachlan, Simon (Introduced by Dr. George Backman) Fraser University (66T-83) (59) Homotopy properties of Fredholm (46) A generalized Sheffer function operators in Banach spaces Mr. Samuel LaMacchia and Professor Dr. G. J. Neubauer, University of Alexander Abian, The Ohio State Uni­ Notre Dame (66T-98) versity (66T-88) {60) A uniform boundedness theorem ( 47) On the radius of univalence of certain Professor R. M. Nielsen, University analytic functions of Delaware, and Professor J. W. Professor A. E. Livingston, Lafayette Brace, University of Maryland(66T- College (66T-126) 114) (48) A topological property of Euclidean (61) On the stability of the index of certain 3-space mod a certain wild arc singular integral operators Professor F. D. Lonergan, Sylvania Professor Joseph Nieto, University Electronic Systems, Needham, Mas­ of Maryland (66T-100) sachusetts (66T-118) (62) Inequalities for the permanent and (49) Certain averages of univalent func­ determinant tions Mr. P. J. Nikolai, Aerospace Re­ Professor T. H. MacGregor, Lafay­ search Laboratories, Wright-Patter­ ette College (66T-119) son AFB, Ohio (66T-101) (50) Semigroup structures for families of (63) A cohomology theory of Harrison for functions. I connected rings Professor K. D. Magill, Jr., SUNY Professor Morris Orzech, Cornell at Buffalo (66T-161) University (66T-165) (51) Upper semicontinuous decompositions (64) On the commutativity of rings of irreducible continua Professor R. E. Peinado, University Professor W. S. Mahavier, Emory of Puerto Rico at Mayaquez (66T- University (66T-173) 170) (52) Completely regular mappings and the (65) On certain discrete inequalities and slicing structure properties their continuous analogues Professor L. F. McAuley, Rutgers, Mr. A. M. Pfeffer, California Insti­ The State University (66T-80) tute of Technology (66T-166) (53) Singular homology of n-cell-like ( 66) Infinite products of substochastic spaces matrices Professor M. C. McCord, University Professor N. J. Pullman, University of Georgia (66T-164) of Alberta, Canada (66T-134) (54) A calculus for a certain class of word (67) On certain algebras of analytic func­ problems in groups tions. II Professor N. S. Mendelsohn and Pro­ Mr. K. V. R. Rao, Purdue University fessor C. T. Benson, University of (66T-87) Manitoba (66T-137) ( 68) Module extensions and blocks (55) Coset enumeration used to solve a Professor Irving Reiner, University word problem in groups of Illinois (66T-171) Dr. W. 0. J. Moser, McGill Univer­ (69) Projective representations in arbi­ sity (66T-127) trary fields (56) An analogue of Ramanujan's tau­ Professor W. F. Reynolds, Tufts function. Preliminary report University (66T-97) Mr. A. A. Mullin, University of (70) Symbolic calculus of kernels with California, Livermore (66T-82) mixed homogeneity (57) On factoring matrix valued functions Professor N. M. Riviere, University in the Bohr group of Chicago and De Paul University Dr. M. G. Nadkarni, Washington (66T-132) University (66T-136) (71) The converse of Wiener-Levi-Mar­ {58) On nonarchimedian linear spaces cinkiewicz theorem

221 Professor N. M. Riviere and Mr. (84) Remarks on minimal topological Y. Sagher, University of Chicago spaces {66T-180) Mr. R. M. Stephenson, Jr., Tulane (72) Quasigroupes isotopes d'un quasi­ University (66T-163) groups demi-symetrique {Introduced by Professor M. P. Berri) Professor A. j. V. Sade, Marseille, (85) Ratio limit theorems for random France (66T-143) walks on groups (73) On prediction theory with continuous Professor Charles Stone, University time. Preliminary report of California, Los Angeles (66T-l 72) Dr. Habib Salehi, Michigan State (86) A remark on nonlinear elliptic equa­ University (66T-108) tions {74) Turning point problems for certain Professor W. A. Strauss, Stanford systems of linear differential equa­ University (66T-139) tions (87) M•lltiple points for the sample paths Dr. T. V. Sastry, Illinois Institute of of the symmetric stable processes Technology (66T-150) Professor S. J. Taylor, University {75) An inversion for a kernel of Michigan (66T-96) Dr. R. K. Saxena, McGill University (88) The exact Hausdorff measure of the ( 66T-152) zero set of a stable process (Introduced by Professor William Moser) Professor S. J. Taylor and Profes­ (76) Minimal metric spaces sor J. G. Wendel, University of Professor C. T. Scarborough, Jr., Michigan (66T-89) Wayne State University (66T-128) {89) Measurable cardinals and Bernays' {77) Estimates for boundary operators set theory Professor Martin Schechter, The Professor L. H. Tharp, Massachu­ Institute for Advanced Study {66T- setts Institute of Technology (66T- 117) 90) (78) Measures on F-spaces (Introduced by Dr. S. H. Gould) Professor G. L. Seever, California (90) A note on the Herglotz-Noerther theo­ Institute of Technology {66T-115) rem for rigid motion in relativity (79) Mixed estimates for systems of sin­ theory gular integral operators and applica­ Dr. A. H. Thompson, University of tions to elliptic problems Pittsburgh (66T-113) Professor Eliahu Shamir, North­ (91) The cohomology of the classifying western University (66T-175) space for K-theory mod p {80) A free boundary problem for the heat Professor Myles Tierney, Rice Uni­ equation with heat input at a melting versity (66T-140) interface (92) Immersions of a circle in the plane Dr. Bernard Sherman, Rocketdyne, Professor C. J. Titus, University of A Division of North American Avia­ Michigan (66T-129) tion, Inc., Canoga Park, California (93) On Gaussian measures equivalent to (66T-154) Wiener measure. II {81) Construction of Markov processes Professor D. E. Varberg, Hamline from hitting probabilities. Prelimi­ University ( 66T-155) nary report (94) Differences, convolutions, primes. III Dr. C. T. Shih, Cornell University Mr. Benjamin Volk, Yeshiva Univer­ (66T-94) sity (66T-79) (82) Permutation grammars. Preliminary {95) 1-bisimple semigroups report Professor R. J. Warne, West Vir­ Mr. Walter Sillars, The Pennsyl­ ginia University (66T-81) vania State University (66T-169) (96) On the behavior of solutions of Max- {Introduced by Professor W. E. Singletary) well equations for small frequencies (83) On a question of Rowbottom Professor Peter Werner, Mathema­ Mr. Jack Silver, University of Cali­ tics Research Center, U. S. Army, fornia, Berkeley (66T-103) University of Wisconsin (66T-142)

222 (97) Riesz operators Alberta, Canada (66T-110) Professor T. T. West, University of (100) On the asymptotic behaviour of solu­ California, Los Angeles (66T-157) tions to certain second order linear (Introduced by Professor L. J. Paige) ordinary differential equations (98) Open continuous images of complete Dr. D. W. Willett and Dr. J. S. W. metric spaces Wong, University of Alberta, Canada Dr. H. H. Wicke, Sandia Corporation, (66T-91) Albuquerque, New Mexico, (66T-159) ( 101) Continuous automorphisms on locally (99) On an example in second orderlinear compact groups ordinary differential equations Professor T. S. Wu, University of Dr. D. W. Willett, University of Massachusetts (66T-104)

NEWS ITEMS AND ANNOUNCEMENTS

FLORIDA STATE UNIVERSITYHOSTS GRAPH THEORY SEMINAR ACM--SIAM SYMPOSIUM TO BE HELD IN ROME

A Symposium on The Recommended A Seminar on Graph Theory will be Impact of Computer Applications on The held in Rome, from July 8-12, 1966, under Undergraduate Mathematics Curriculum the auspices of the International Computa­ will be conducted as a joint open meeting of tion Centre. The Seminar will consist of the Southeastern Section of SIAM and the invited lectures as well as contributed pa­ Northwest Florida Chapter of ACM at pers. The Seminar follows immediately B'lorida State University, March 14, 1966. after the International Symposium on Mathe­ Dr. Knox Millsaps, University of matical and Computational Methods in Social florida, will preside over the Symposium. Sciences (Rome, July 4-8, 1966). Speakers include W. F. Atchison, Georgia Persons who wish to present a paper Institute of Technology; W. Givens, North­ should submit abstracts not exceeding two western University; H. H. Goldstine, Na­ pages. Abstracts may be written in French tional Research Council; N. Macon, Institute or English and should be sent before for Defense Analyses; F. J. Murray, Duke March 1, 1966. Authors should submit ab­ University and Army Research Office, Dur­ stracts to the Chairman of the Program ham; G. Young, Tulane University. Committee: Dr. P. C. Gilmore, Thomas This Symposium will be a part ofthe J. Watson Research Center, Box218, York­ 6th Annual Computing Center Conference at town Heights, New York. In addition, Euro­ Florida State University, to be held during pean lecturers must also forward abstracts the period March 12-19,1966. The complete to: Professor Claude Berge, Director of the program of this conference, "The Impactof International Computation Centre, 23, Viale the Computer on Society," may be obtained Civilta del Lavoro, Rome E.U.R., Italy. by writing to the General Chairman, E. P. The participation fee for participants who Miles, Jr., Computing Center, Florida State are not giving lectures is $20.00. The official University, Tallahassee,· Florida 32306, languages will be French and English. after January 15, 1966.

223 ABSTRACTS OF CONTRIBUTED PAPERS The Meeting in Berkeley, California December 29, 1965

629-31, G. J, LANDRY, Louisiana State University iJl New Orleans, Lake Front, New Orleans, Louisiana 70122, A new family of 5th Order Runge-Kutta type formulas,

Using a modified version of a method originated by Sarafyan, the author has established a new family of 5th Order Runge-Kutta type formulas that can be used for the numerical solution of the ordinary differential equation y' = f(x,y). Certain formulas in this family, because of their simplicity, yield better approximations than the classical 5th Order formula of Nystrom. Also, the results com­ pare favorably with those of the 6th Order formulas of Butcher and Huta, When lfy I > 1, the author's simple formulas, in all studied cases, gave better results than the 6th Order formulas mentioned above, The simplicity of the author's formulas is mainly attributable to the exactness and smallness of the values assigned to the 21 parameters of the 5th Order system, For instance, the value zero is assigned four times and exact values such as 1/8, 3/16, 1/4, 1/2, 3/4, and 1 are also used, Fur­ thermore, these formulas involve the assignment of 17 nonzero values to the parameters as compared to 18 in Nystrom's formula, 24 in Butcher's formula, and 35 in Huta's formula. This facilitates their programming. A specific formula from the family will be distributed for examination and practical use, (Received November 22, 1965,)

629-32. DIRAN SARAFY AN, Louisiana State University in New Orleans, Lake Front, New Orleans, Louisiana 70122, Runge-Kutta formulas in pseudo-iterative form.

Concerning Runge-Kutta formulas Milne in his Numerical solution of differential equations (P. 74) writes: "The process does not contain in itself any simple means for estimating the error in detecting arithmetical mistakes," This objection appears to be universally accepted, but is un­ founded. In fact, any Runge-Kutta type formula provides, by quantities that appear directly in the computation, lst and 2nd order approximations to the true value y(x0 + h). For instance a 3rd order formula at its lst, 2nd and 3rd stages, provides 1st, 2nd and 3rd order approximations, respectively. Comparing these consecutively improved approximations one obtains valuable information, particularly in regard to their accuracy, This internal property of Runge-Kutta formulas is further exploited with the derivation of sets of a family of formulas of order n > 4 in which are incorporated or im­ bedded formulas of order 1, 2 and 3 or 4. In all investigated cases these formulas yielded better error estimates than other known methods including Richardson's. It seemed appropriate to refer to these formulas when n > 4 as to pseudo-iterative formulas, One such formula has been presented at the 625th meeting of the So·ciety. Families of pseudo-iterative formulas will be made available to participants, (Received November 22, 1965.)

224 629-33. KEITH MILLER, University of California, Berkeley, California. Barriers on cones for elliptic operators with measurable coefficients.

In real n-dimensional space let ~be the class of linear elliptic operators of the form Lu = l:~aijuxixj' where (aij(x)) is measurable with all eigenvalues in [a,l], 0 0 on Tp - { 01. and Lu ~ 0 in T3 for every L E .Ya· Theorem. Given fJ < 1r, there exists a constant }.. > 0 and a barrier for .sf on Tp of the form lxl}l.f(8). The supremum of such constants,\, a- for given n,a, and {J, is related to the first zero of the solution of a certain extremal ordinary differ- ential equation. There exist analogous "singular barriers" of the form lx 1- Af(8), >.. > 0. Applications include the Dirichlet problem, regularity of solutions at the boundary, Phragmen-Lindelof Theorems, removable singularities at a boundary point, and pathological examples. (Received December 2, 1965.)

629-34. E. A. HERMA~, Grinnell College, Grinnell, Iowa. The symbol of the algebra of singular inte&ral· operators.

A The symbol a of the B *-algebra tJtof singular integral operators in Rn is well known to have the properties: (a) a is a *-homomorphism from rJt onto a B*-algebra of continuous complex-valued -'\ functions, and the kernel of a is the set ([ of compact operators; (b) llaK 11 0 ~ IlK II for all K E ()[,

(e.g.: H. O; Cordes, The algebra of singular integral op~rators in R, J. Math. Mech., to appear). These results are obtained rather quickly and in a readily generalizable fashion by using the theory

of Banach algebras to prove the following theorem: Let £t-1 be the B •-algebra of multiplication operators where multiplication is by continuous functions on Rn with limits at infinity. Let rJt,2 be the B*-algebra generated by the Riesz operators. fJt is tbe B*-algebra generated by iJt 1, ~·and ([. fJt, is defined to be the quotient algebra if;([ . Let M1, M2, M be the maximal ideal spaces of £% 1 ,~, &respectively. Theorem. M is homeomorphic to M 1 X M2 under the mapping

,(9)1) = (J (9)1), 1r1 (9)1)) (for ID1 EM) where 1r1! is the associated dual map of 1r1. and 1r. is the restriction 1 2 "' 1 to 0£i of the natural projection a-.(){,, i = 1,2. (Received November 26, 1965.)

629-35. c. C. TORRANCE, U. S. Navar Postgraduate School, Monterey, California. Real vs. apparent. time.

The Lorentz transformation is here derived solely from considerations of symmetry, without reference to the speed of light or the principles of mechanics. This puts the Lorentz transformation at a more fundamental level, i.e., as an abstract axiomatic formulation, so that experiments relating to this transformation are not verifications of a theory, but searches for possible interpretations of the undefined terms in the formulation. The term "the same" is defined operationally; this leads to some implications contrary to common usage. One implication is that "clocks" measure only apparent time, so that only "proper time" is physically real. (Thus Lorentz-Fitzgerald "contractions" are only foreshortening&.) The (relativized) Hamilton equations and Maxwell equations can be similarly regarded as abstract axiomatic formulations of principles of symmetry, so that such difficult-to­ define concepts as force, mass, and electric charge play only the role of possible interpretations of the undefined terms in these equations. Thus familiar relations, such as "Newton's law", result

225 from sp~cialized approximate interpretations of thes~ und~fin~d terms, such as by th~ id~ntification of r~al and appar~nt tim~. From th~ pr~c~ding point of vi~w, th~ conc~pt and rol~ of op~rational

"d~finitions" of physical quantiti~s ar~ clarifi~d. (R~c~iv~d D~c~mb~r 6, 1965.)

629-36. R. A. KNOEBEL, N~w M~xico Stat~ Univ~rsity, Box AM, Univ~rsity Park, N~w M~xico.

A class of ord~r-pr~s~rving algebras which are categorical and precomplete.

lf R is a relation on a set A and f is an n-ary function on A, then f preserves R iff whenever a, b E: An and aR (n)b then faR fb. Let F A,R be the set of all finitary functions on A which preserve R.

For a universal algebra ()t = (A, f 0, f 1, ••. ,fv•··· )• (Jt_# is the set of all finitary functions generated by composition of the f0 , f 1, ••• ,fv•···. Theorem. _g_ ot= (A, f0 , f 1, ••. ,fv•···) is a finite finitary algebra with at least two elements, R is a lattice-ordering on A, and F A,R ~ tJt#, then tJt is cat~gori­ cal i.~ •• ev~ry alg~bra satisfying the identities of iJt is a subdirect power of tY£. It can be shown that any algebra satisfying the hypotheses of the theqrem is either primal, i.e., all finitary functions on A are in r:Jt~ or is precomplete, i.e., F A,R = r:Jt#. The theorem has been proven in the first case by A. L. Foster, Generalized 'Boolean' theory of universal algebras, Math. Z. 58 (1953), 306-336. The second case is new. (Received December 6, 1965.)

226 The Meeting in New York City February 26, 1966

631-1. F. E. J, LINTON, Wesleyan University, Middletown, Connect.icut 06457. Triples versus theories. Preliminary report.

Let .JI-be a category. Extending earlier definitions of Lawvere and Linton, a theory over .ft is a category I' (having the same object class as .Jl) and contravariant functors i: .Jrf __, lr {identity on the object class), t: .I' ->J'l, adjoint to each other on the right. A I'-algebra is a configuration

(X,A,71), where 11 is a natural equivalence between .Jf(- ,A) and the composition of i with X: I' --+ {sets}. It is well known that the functor ti can be made part of a dual standard construction (Godement-Huber) or triple (Beck-Eilenberg-Moore) in .Jf, and that each triple arises in this way (Kleisli). Theorem. With suitable definition of what the morphisms are, the constructions alluded to above set up an equi­ valence between the category of theories over Jtf and that of triples in .Jl; moreover, if, under the equivalence, theory I' and triple T correspond to each other, the category of ]:'-algebras is canonically equivalent with Beck's category ofT-algebras, (Received November 24, 1965,)

631-2, PEGGY TANG STRAIT, 1270 Fifth Avenue, New York, New York 10029. Sample function regularity for Gaussian processes with the parameter in a Hilbert space.

Gaussian processes f?t: t E H} where H is the Hilbert space l 2 are considered. It is shown that if Tis a compact subset of the set [(t1,t2, ••• ,tn''''): an;:;;; tn;:;;; an+ (l/2)n, (al'a2, ... ,an••··> E HJ and there exist constants D >0 and K-:>- 0 such that E

constants a > 0 and K such that E(lfft- ~si 2 ) ;:;; K lit- s 11 11 for all t, s in H, then "almost all" sample functions of the process are Lipschitz-,8 continuous on T for 0 < {1< a/2. The phrase "almost all" is defined in the paper, but has the usual meaning if the process is separable and is separated by the set of dyadic numbers in H. Several examples of processes satisfying the conditions for continuity or Lipschitz-.8 continuity are given. In particular, it is shown that the Brownian process in a Hilbert

space defined by Paul Levy satisfies the latter condition for a= 1. (Received Decembe-r 2, 1965,)

631-3, R. j. WARNE, 428 Cedar Street, Morgantown, West Virginia. Extensions of 1-bisimple semigroups.

Let S = (G,a) be an 1-bisimple semigroup. {Abstract 66T-2). The multiplication in S, the translational hull of S, is completely described. If T = M0 (H;K, A,P) is a completely 0-simple semi­ group, the partial homomorphisms of T\0 into S are given in terms of mappings of K into I, A into I,

K into G, A into G and a homomorphism of H into G. A simpler determination is given in the case T is a Brandt semigroup. In either case, if T has proper divisors of zero, every extension of S by T is given by a partial homomorphism. We also give an explicit determination of the extension of S

by T ~n the case T has no proper divisors of zero. If S is a weakly reductive semigroup and T is a

227 0-bisimple semigroup having proper divisors of zero, every extension of S by T is given by a partial homomorphism provided S = S or S\S is a semigroup. (Received December 22, 1965.)

631-4, E. J. TAFT, 14 Vandeventer Avenue, Princeton, New Jersey 08540, Orthogonal conjugacies in finite groups.

Let A be a finite group, G a group of automorphisms and antiautomorphisms of A whose order is relatively prime to that of A. For a fixed prime p, let S and T be two p-Sylow subgroups of A left fixed setwise by G(such subgroups exist). Then S and T are G-orthogonally conjugate, i.e., via an element of A which is a fixed point of the automorphisms in G and which is inverted by the antiautomorphisms in G. This result is essentially due to Wielandt and Glauberman (who considered only automorphisms), but the techniques used here are analogous to those used by the author in investigating certain group-invariant decompositions of associative and Lie algebras into the radical and a semisimple subalgebra (cf. Orthogonal conjugacies in associative and Lie algebras, Trans. Amer. Math. Soc. 113 (1964), 18-29). This leads to certain natural conjectures where A is an alge­ braic group, and the relative primeness condition is replaced by certain cohomology conditions on G and certain subgroups of A. (Received December 29, 1965.)

631-5. W. J, SCHNEIDER, Syracuse University, Syracuse, New York. An analytic function with limit zero along some path tangent to each radius.

A well-known result of Lusin and Privalow states that there are no nonconstant analytic functions on lz I < 1 with radial limit zero on a set of radii whose end points are metrically dense and of 2nd category on an arc of lz I = 1. However, Theorem, There exists a nonconstant analytic function on lz I < 1 which has limit zero along some path tangent to each radius. The method of proof is quite similar to a method used in Noshiro (Cluster sets, pp. 76-78, Springer, Berlin, 1960) except that a generalization of Bagemihl and Seidel's notion of "tress," which we shall call a "jointed tress," is needed, An example of the kind of "jointed tress" used is: let s1 = [i}, s2 = f- 1/2 t 1/24 t i/22 , ... , - 1/2 t z 4;z 4 t i/22 J, ... , Sn = [- 1/2 t 1/2 Zn t i/2 n, ... ,- 1/2 t z2n ;z2n + i/2 n~ .... and let S consist of the line segments joining each point of Sn to one of the points of S n- 1 nearest it, for n = 2,3,... . As far as the author knows it was not even known previously whether there existed nonconstant analytic functions with zero as an asymptotic value at all points of lz I = 1 (for meromorphic functions it was known). (Received December 2, 1965,)

631-6. G. J, CHAITIN, 819 Madison Avenue, New York, New York 10021, On the length of programs for computing finite binary sequences by bounded-transfer Turing machines. II.

Refer to Abstract 66T-26, these cNoticeiJ 13 (1966), 133. There it is proposed that elements of Cn may be considered patternless or random. This is applied; some properties of the L function are derived by using what may be termed the simple normality (Borel) of these binary sequences.

Note that (4) L(S*S') ~ L(S) + L(S'), where S and S' are finite binary sequences and • is the concate­

nation operation. Hence L(Cn+m) ~ L(Cn) + L(Cm). With (l) this subadditivity property yields

(5) an ~ L(Cn) (actually, subadditivity is used in the proof of (1)). Also, (6) for any natural number k, if an element of en is partnioned into successive subsequences of length k, then each of the 2k pos-

228 sible subsequences will occur~ 2-k(n/k) times. (6) follows from (1) and a generalizatiOn of (3). (4), (5) and (6) give immediately (7) an ;;; 2-n :LL(S), where the summation is over binary sequences S of length n. Denote the binary sequence of length n consisting entirely of zeros by on. As L(On) =

O(log n), for n sufficiently large L(Cn) > 2-n LL(S) ~ an, or (8) an< L(Cn). For each k it follows from (4) and (6) that for s sufficiently large L(Cs) = L(S) = P(S' • Ok • S"), where S' • Ok • S" = S E Cs, so that L(Cs) :;;; L(Cn) + L(Cm) + L(Ok) (n + m = s - k). This last inequality yields (9) (L(Cn)- an) is unbounded. (Received January 6, 1966.)

631-7. D. J. WlNTNER, Yale University, New Haven, Connecticut. On groups of automorphisms of Lle algebras.

Let L be a finite dimensional Lie algebra over an arbitrary field F. Let u be an automorphism of L. The following theorems generalize known results at characteristic 0 to arbitrary fields. Theorem 1. If L is not solvable, u has a nonzero fixed point in L. Theorem 2. If L is not solvable and u is semisimple, then u ha.s a fixed point x in L such that ad x is not nilpotent. Theorem 3. Let L be solvable and let G be a finite group of semisimple automorphisms of G. Then L has a Cartan subalgebra which is stable under G. Definition. L is w-restricted if either characteristic F is 0 or characteristic F is p and for each x in L, there exists y in L such that (ad x)p = ad y. Theorem 4. Suppose that L is w-restricted. Let G be a group of automorphisms of L having a normal subgroup T such that T is a torus and G/T is finite and supersolvable. Then L has a Cartan subalgebra which is stable under G. (Received January 6, 1966.)

631-8. F. P. CALLAHAN, General Electric Company, P. 0. Box 8661, Philadelphia, Pannsylvania. A direct proof that S0(3) is homomorphic to SU(2).

Expressing the 3 X 3 proper orthogonal group, S0(3), in terms of axes that diagonalize rotations about the z-axis leads to a rational parametrization of S0(3), in terms of one complex number and one angle, which is useful in the study of rigid body motion (1), and to the usual stereographic projection association of complex numbers with real unit vectors. The mapping of these complex numbers in- duced by real rotations is now easily shown to be a fractional linear transformation associated with a 2 X 2 unitary matrix having determinant unity (member of SU(2)), or to its negative, thus establish­ ing the well-known homomorphism of SU(2) and S0(3), (2,3). References. (1) F. P. Callahan, Approx­

imate pl~nar rigid body motion, Jour. Rat. Mech. and Anal. 4 (1955); (2) F. Klein, The icosahedron, Dover, New York, 1956; (3) E. T. Whittaker, Analytical dynamics, Dover, New York, 1944. ·(Received January 14, 1966.

229 ABSTRACTS PRESENTED BY TITLE

66T-79. BENJAMIN YOLK, 1315 Dickens Street, Far Rockaway, New York 11691. Differences, convolutions, primes. Ill.

Theorem 3. L(f) = f(a,b,c, ••• ,t) is a linear operator on the space of complex valued functions of one variable defined at the finite set of distinct complex values a,b,c, ••• ,t. Furthermore f(z) is pro­ portional to a function absolutely monotonic in (0,1) iff (f( (a(, (b (, (c (, ••• , (t () l is monotonic in (a(, and (f(a,b,c, ••• ,t) l is not greater than (f((a (,(b (, (c (, ••• ,(t I> I for every finite set of distinct points in the open unit disc D: a,b,c, ••• ,t. Here f(a,b,c, ••• ,t) is the Nth order difference quotient off taken at the N + 1 points a,b,c, ••• ,t. Credit. Theorem 3 is due to conversations with Professors H. E. Rauch and D. J. Newman. Corollary 1. If f(z) is defined and bounded by M on a set S dense in D, then f(z) has an ex­ tension analytic in D iff (1): {1- lxl)(l- lyl)(l- lzl)lf(x,y,z)l is not greater than M when x,y,z are three distinct points of S. Furthermore if S = D then condition (1) may be relaxed to (2): (1 - lz I> •(1- lz + hl)(l- lz + khl)lf(z,z + h, z + kh)l is not greater than M whenever z, z + h, z + kh are three distinct points in D, h is real, k is a fourth root of unity. Conjecture 3. The factor (1 - lx I> •(1 - IY 1)(1 - lz I> of (1) cannot be replaced by ((1 - Max(lx l.ly j.jz !)) to the power 3 - r) for any positive r. (Received October 4, 1965.)

66T-80. L. F. McAULEY, Rutgers, The State University, New Brunswick, New Jersey. Completely regular mappings and the slicing structure properties.

A mapping p: T--> B has the generalized slicing structure property (c){h J denoted by (GSSP(c)) {GSSP(h)} iff for each point bin B, there is an open set Ub containing band a mapping t/lub: p -l(Ub) --+K such that (a) t/lubiP -l(b) is a (continuous mapping) [homeomorphism} of p -l(b) onto K and; (b) the collection [ tPuJ is continuous. Theorem. Suppose that p: T --+B is a completely regular mapping (Dyer-Hamstrom), Tis a complete metric space, and B is a metric space with covering dim B ;;; n + 1. Furthermore, there is a metric space K, an open covering W of B such that -1 -1 -1 -1 if bE w EW, then the space Gb of all mappings f of p (w) into p (b) such that fjp (b) maps p (b) (onto)fhomeomorphically ontoJ ~dentically onto] p -l(b) is locally n-connected (LCn). Then p has the (GSSP(c)) {GSSP(h)J [SSP] with respect to K ~{p- 1 (b)J. (Received September 27, 1965.)

66T- 81. R. J. WARNE, 428 Cedar Street, Morgantown, West Virginia. I-bisimple semigroups.

A semigroup I) is called I-bisimple if S is bisimple and Es the set of idempotent& of S is order isomorphic to I, the integers, under the reverse of the usual order. Theorem. S is a I-bisimple semi­ group iff S ~ G xI xI under the multiplication (g,n,m){h,k,t) = (gTk-rhTm-r, n + k - r, m + t - r) where r. = min(m,k) and T is an endomorphism of G, T0 denoting the identity transformation of G or equi­ valently (g,n,m) (h,k,t) = (gTs-m-khTs-t, n + k, s) where s = max (m + k, t)., Conditions for two such semigroups to be isomorphic are given. (Received October 14. 1965.)

230 66T-8Z, A. A. MULLIN, Lawrence Radiation Laboratories, Box 808, Livermore, California 94551, An analogue of Ramanujan's tau-function. Preliminary report,

A few years ago L. J, Mordell showed that S. Ramanujan's tau-function is multiplicative, This note considers an analogue of it, Lemma 1. Put r• = r(l/t(•)), where 1/tis defined(Bull. Amer,

Math. Soc, 69 (1963), 446~ Then r• is generalized multiplicative, but..!!.!!! multiplicative. Lemma Z, T•(n) 'I 0 for every natural number n if and only if r(m) 'I 0 for every natural number m. Lemma 3,

The least zeros, if any, of T and r• coincide. Using results of D. H. Lehmer,{Bull, Amer. Math, Soc, 43 (1948), 663~ and properties of the arithmetic function 1/t' (defined as follows: 1/t•(n) = min [m: l/l(m) = nJ> comparative estimates of possible zeros ofT and T• are given. (Received October Z5, 1965.)

66T-83. A. H. LACHLAN, Simon Fraser University, Burnaby Z, B. C. Canada, A property of univex:sal sets,

A set is called universal if every r,e, set is many-one reducible to it, Theorem, If AX B is universal and one of A, B is r,e, then one of A, B is universal. (Received November 8, 1965,)

66T-84. J, H. AHLBERG, United Aircraft Research Laboratories, East Hartford, Connecticut, E. N. NILSON, Pratt and Whitney Aircraft, East Hartford, Connecticut, and J, L. WALSH, University of Maryland, College Park, Maryland. Complex cubic splines,

Let r denote a smooth jordan curve, R its interior, and [ llkJ a sequence of subdivisions of r with llllkll--->0 as k--HXl, For a given f(t) defined on r, qk(f;t) is the spline in the complex variable t interpolating to f(t) on the points of Ilk (piecewise cubic, of class cZ on f). We define an "analytic spline" in R by Sk(f;t) = qk(f;t)/Z + (1/Z7ri) Jr[qk(f;r)/( T- t))dr, Convergence properties announced (Abstract 66T-47, these c}/oficei) 13 (1966), 140) for real cubic splines carry over to compl:ex splines. If f(z) is analytic within R and if f(q)(t) (q = 0,1,Z or 3) satisfies on r a Holder condition of order o. (0 < o. ~ 1), then for any o.', 0 < o.' < o., we have f(p)(z)- sf:>>(z)= O(il~kllq+o.'-p)inR (p = 0,1, ... ,q) provided ( • ) llllk 11/(minimum chord length of Ak) ~ /3 < oo when q = 0 or 3, If f(q+ 1)(t) is continuous on f (q = 0, 1,Z), then f(p)(z) - S~)(z) = O(i!llkll q-1) in R without ( • ). On any closed subset of R, f(p)(z) - s~>(z) = O(i!llkllq+o.') (p = 0,1, ... ) with the designated mesh conditions. It is further shown N that s~V (f; z) = L:i=k1 (uk,j+1 - uk,j)/(tk,j - z) where tk,j are the mesh po~nts of ~ and qic" (f; t) = uk,j on the interval tk,j- 1, tk,j' This sum is related directly to (1/Z7ri) frf1 (t)dt/(t- z). (Received November 8, 1965,)

66T-85. W. M. CAUSEY, University of Kansas, Lawrence, Kansas 66045. The univalence of an integral. II.

Let f and g be regular in lz I < 1 and normalized by f(O) = 0, f'(O) = 1. Suppose F(z) =

J;~(t)/trdt and G(z) = J~[g(t)/trdt. Iff is close-to-convex with respect tog for lzl < 1, then G is convex and F is close-to-convex with respect to G for lz I < 1 and 0 ~ E ~ 1, (Received November 17, 1965,)

231 66T-86. LAWRENCE NARICI, Polytechnic Institute of Brooklyn, 333 Jay Street, Brooklyn, New York. On nonarchimedian linear spaces,

Following A. Monna, we define the index of the nonarchimedian normed space X over a field F with a nonarchimedian valuation I I. in the case where t IPI: p E F J = { IIX II: X E XJ' as the order of the additive group V' = [x E X: llx II ~ lJ modulo the subgroup P' = [ x: llx 0· < 1}. For the field F itself this is just the cardinality of the residue class field. In the case when these additive groups are finite, various relations between the (finite) dimension of X over F, dim X IF = r, and the dimension f of V'/P' over V/P and the indices, Typical of these results are: (1) In the finite-dimensional case, if the valuation on F is discrete then f = r, and the norm on X is a maximum norm. (2) If r is finite and the norm on X is a maximum norm then f = r. (3) If F is locally compact and f is finite then the indices are finite and m ~ nr where m is the index of X and n is the index of F. The proofs are modi­ fications of the known proofs in the case where X is a field extension of F and the norm is a valuation extending the given one of F. (Received November 22, 1965.)

66T-87, K. V. R. RAO, Purdue University, West Lafayette, Indiana, On certain algebras of analytic functions. II.

Let W be a hyperbolic subregion of a compact Riemann surface S, and let g be a nonconst,ant meromorphic function on S, Denote by A = A(W,S,g) the algebra of all those holomorphic functions f on W which, together with df/dg, are of bounded characteristic on W. Let A' be an algebra similarly defined relative to· W', S' and g'. Theorem 1. J!.. A: A ->A' is an. algebra isomorphism, then there exists a conformal homeomorphism. cf>: W' -> W such that A(f) = f iJ c/J, f EA. Theorem 2, The con­ formal structure of a hyperbolic plane region is determined by the algebra of all those holomorphic functions f, all of whose derivatives f(n)(n = 0, 1,2, ... oo) are of bounded characteristic, The proofs are based on the results of the author's paper (Michigan IVIath, J; 11 (1964), 231-235) and the following observations: (i) the algebra A is quotient-closed; (ii) every meromorphic function of S which is holomorphic on W belongs to A. (Received November 26, 1965,)

66T-88, SAMUEL LaMACCHIA and ALEXANDER ABIAN, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43214. A generalized Sheffer function.

Denoting by a natural number n ~ 3 the set of n elements 0,1, ... , n - 1 and by nk the k-fold cartesian product of n fork·= 1,2,3, .. , it is established: Theorem. For every natural number n ~ 3 2 the function p n (x,y) from n onto n given by p n (n - 2,0) = 0, p n (n - 1, 0) = n - 1 and otherwise p n (x,y) = 1 + x (mod n) is such that every function from nk into n for k = 1,2,3, ... is obtained in terms of super- positions of Pn (x,y) where none of the constants 0, 1,. .. , n - 1 appears in any superposition. Clearly, for n-valued logic with n ~ ~ the function pn(x,y) can serve as a primitive binary connective in the same sense that the Sheffer stroke is a primitive binary connective for 2-valued logic, (Received November 29, 1965.)

232 66T-89. S. j. TAYLOR and j. G. WENDEL, University of Michigan, Ann Arbor, Michigan 48104. The exact Hausdorff measure of the zero set of a stable process.

Let x(t) be a suitably normalized stable process of index a > 1, and Z its set of zeros. Blumen­ thal and Getoor [Illinois j. Math. 6 (1962), 308-316] showed that the Hausdorff dimension of Z is almost surely f3 = 1- 1/a. We refine this as follows. Let- m(•) be the Hausdorff measure induced by (h)= h/3 (log log 1/h)l/a. Let A(t) be the local time at zero for the process x(t), in the sense of Blumenthal and Getoor [Z. Wahr. 3 (1964), 50-74]. Theorem 1. Almost surely, tj>- m(Z n[O,t]) = A(t) for 0 < t < oo. The main tool in the proof is the fact, essentially contained in the last reference, that the function inverse to A(t) is r(t), the stable subordinator of index {3. We prove a local law of the iterated logarithm for T. Theorem 2, Almost surely, lim inft~ at -l//3 (log log(l/t)f l+l/J1 T(t) = 1, Let Q denote the range ofT, Theorem 3. P[cb- m(Q n[O,r(t)]) = t, all tJ = 1. Theorem 1 then follows. (Received November 22, 1965.)

66T-90. L. H. THARP, Massachusetts Institute of Technology, Cambridge, Massachusetts. Measurable cardinals and Bernays' set theory.

(1) Let B be the system of set theory given by Bernays in Zur Frage der Unendlichkeitsschemata in der Axiomatischen Mengenlehre. It can be proven in B that there are cardinals which are not strongly incompact. (2) In Z F with choice it can be proven that if there is a measurable cardinal, then there is a set which is a model for B and V = L. The proof uses an ultrapower construction and the notion of a nameable model, that is, a model each of whose members is definable in the model. (3) The conjunction of (1) and (2) gives an alternative proof that Po is not sufficient for strong incom­ pactness (see Tarski, Some problems and results relevant to the foundations of set theory), and indi­ cates that the weaker type of compactness is compatible with V = L. (Received November 29, 1965.)

66T-91. D. W. WILLETT, and j. S. W. WONG, University of Alberta, Edmonton, Alberta, Canada. On the asymptotic behaviour of solutions to certain second order linear ordinary differential equations.

Let a(t) be a positive, nondecreasing continuously differentiable function such that a(t)----. oo. Let b(t) be any positive continuously differentiable function such that J00dt/b(t) = oo and lim inf b(t)b- 1 (t)a- l/2 (t) ~ 0 as t--> oo. Theorem. A sufficient condition to imply that all solutions x = x(t) of the equation x + a(t)x = 0 are such that x(t)--> 0 as t--> oo is that lim inf a(t)b(t)a- 1 (t) > 0 as t-->oo. Interesting corollaries are obtained by taking b(t) = t, tInt, a- 1/ 2 (t)fJa112, and a(t)a- 1(t). The case b(t) = a(t) produces the result of Sansone (Scritti mat. offerti a Luigi Berzolari, Pavia, 1936, 385-403). (Received November 29, 1965.)

66T-92. GEORGE GRATZER, The Pennsylvania State University, McAllister Building, University Park, Pennsylvania. On direct and inverse limits of universal algebras.

A direct limit (inverse limit) system ()[is defined as a collection of algebras ~. i E I, (I; ;<;)

is a directed partially ordered set and for i ~ j (i,j E I) there is a homomorphism t/>ij: ~ --> Aj (for inverse limit system: a homomorphism i: ~ --J~). The direct (inverse) limit will be denoted

233 by lim 0£-(Iim.ot}. Let I = v (I Ja E P}, such that ( P; ~) is also directed, if a ;> /3then I ~ I/3 and - - . Q a. each (Ia; 2) is directed. Let .J1a denote the direct (inverse) limit of L9t- restricted to I a; then if a 2 {3, there is a natural homomorphism

. Let K be a class of algebras closed under isomorphism. _!::(K), ~(K}, ~w (K}, ,!:-_ w (K) stand for the class of direct, inverse, well-ordered direct ((I; 2 ) is well-ordered}, well-ordered inverse· limits formed from algebras in K. If we combine the theorem with a well-known result of T. Iwamura on directed sets we get: Corollary. If .!:: w (K) c;;;; K then .!::(K) c;;;; K, (if b_w (K) c;;;; K, then ,!:-_(K) c;;;;K). (Received November 22, 1965.)

66T-93. F. T. BIRTEL, Tulane University, New Orleans, Louisiana 70118. Slice algebras of bounded analytic functions. Preliminary report.

Let H00 be the algebra of bounded analytic functions on the disc D and ..A'H its maximal 00 ideal space. By S denote the function algebra in C(..A'H X ..A'H ) such that f E S implies f(•,N) E H 1\ 00 00 00 and f(M ,·) E H". We have the following results: If Tis the unit circle, H (TXT) (I L ~h L 00 00 OOAOO 1 1 ~ S ~ .<£'K(L ; H 1, H ). the space of compact operators from L /H0 into H where L 1;H 1 has H as 1 ooo 00 0 00 dual. Since S ~H 00 (D X D), ..A'H X AtH C 1 f~H • The restriction map 1r:A'H (D D) oo oo S oo(D X D) 11 oo X -> ..A'H x _,/{H is onto, continuous and an homeomorphism on D X D. Every f E H00 (D X D)" which ro oo is constant on 1T- 1 (M,N) where (M,N) E 1 H X 1 H is a continuous extension of f E S JD X D. 00 00 (Received November 18, 1965.)

66T-94. C. T. SHIH, Cornell University, Ithaca, New York 14850. Construction of Markov processes from hitting probabilities. Preliminary report.

Suppose K is a compact metric space with a distinguished point 6. Let 'l8 denote its Borel field and~ be a base of the topology closed w.r.t. finite union and intersection. Let 9J = [DID= (K- U) U {A} for some U E ~}. Suppose HD(x,•), x E K, D E ~.are probability measures on 'l8 satisfying (1) HD(•,B) is measurable, (2) H0 (x,D) = 1, (3) H0 (x,{xJ) = 1 if xED, and (4) if DC D' then H0 (x,B) = JH0 (x,dy}H0 (y,B). Define holding points, excessive functions and then the fine topo­ logy of K in an obvious manner. Let ..'6'0 be the class of continuous real-valued functions on K vanishing at 6. Theorem. Suppose (A} if f E ..'6'0 then H 0f E ..'6'0 where H0 f(x) = JH 0 (x,dy}f(y),

(B) there is a non-negative g E '!f0 satisfying grb (x) = g(x) - H0 g(x) > 0 if x t/:- D, (C) for each x and fine neighborhood A of x there is E > 0 such that if A C U, U open, then sup ig0 (x)JK - U CD E ~} > E, and (D) fx Jx = 6 or x is a holding point J is closed. Then there exists a Hunt process on K with A as absorbing point satisfying: (1} H0 (x, •) is its hitting distribution of D starting at x ~ D, and (2) the expected time of reaching 6, starting at x, equals g(x). This result generalizes that of Dawson (Illinois J. Math. 8 (1964), 657-684}; the general approach follows Knight and Orey (J. Math. Mech. 13 (1964), 857-873}. (Received November 23, 1965.}

234 66T-95. D. A. GAY, Dartmouth College, Hanover, New Hampshire. On representations of the Weyl group.

Let~ be a semisimple complex Lie algebra and 91 C ~ a Cartan subalgebra. Let 1/; be the highest root of~ relative to a simple system of roots. If m is a positive integer, denote by Vm the simple ~-module with highest weight ml/;. The Weyl group W of~ acts on ffi*, the dual of 91. Thus a W-module structure is induced in Sm( 91), the homogeneous polynomials of degree m over· 91. More­ over, a ~-module V with nontrivial zero-weight space Zy induces a W-module structure in Zy.

Let k,£ be positive integers. Define P n (k) to be the number of integer m-tuples (i 1, ••• ,i ) with m,~ m 1 ~ ij ~ .f and L~ 1ni n= k. Now let ~ = AJ. Theorem 1. Zy m and Sm(ffi) are isomorphic as W-modules. Theorem 2. The multiplicity of k as a generalized exponent (see B. Kostant, Amer. ]. Math. 85 (1963), 391-399) of V m is given by P m,.£ (k). (Received November 19, 1965.)

66T-96. S. ]. TAYLOR, University of Michigan, Ann Arbor, Michigan. Multiple points for the sample paths of the symmetric stable processes.

The sample path of a symmetric stable process of index a (0 < a ~ 2) in R n visits a preassigned point with probability zero if 0 < a ~ n. A sample path X(t,w), t ;:;, 0, is said to have maximum multi­ plicity k = k(w) (k a positive integer) if there are some points in R n which are visited k times by the path but no point is visited (k + 1) times. In the present paper it is proved that for almost all sample paths: (i)·if n ;:;, 4, then k = 1; (ii) if n = 3, k = 1 for 0 n((r - l)r), the set Lr of the points on the sample path which have multiplicity r has Hausdorff dimensional num­ ber ra- n(r- 1) with probability one. (Received November 19, 1965.)

66T-97. W. F. REYNOLDS, Tufts University, Medford, Massachusetts 02155. Projective representations in arbitrary fields.

Let K be an arbitrary field of characteristic p. Let r be a twisted group algebra of a finite group G over K: f has a K -basis B {(g): g E G} and multiplication (g)(h) E h(gh), 0 i E E K. = = g, g,h Let w be the number of nonisomorphic irreducible f-modules. It is shown that w is equal to the num- ber of those K-conjugate classes [Curtis and Reiner, Representation theory of finite groups, p. 306] of G which consist o! p-regular elements, are special with respect to the factor set E [Osima, Math. ]. Okayama Univ. 1 (1952), 33-61, §11], and satisfy an additional condition too complicated to state here. The proof depends on the following fact. Let L be a suitably large finite normal extension of K. Then the Galois group of L over K has a canonical representation by L-linear transformations of L 0K f, which are monomial with respect to the L-basis B, and whose restrictions to the center of L 0K r are automorphisms. (For a special case of this fact, see p. 317 of Burnside, Theory of groups of finite order, 2d edition.) (Received November 18, 1965.)

235 66T-98. G. J. NEUBAUER, University of Notre Dame, Notre Dame, Indiana 46556. Homotopy properties of Fredholm operators in Banach spaces.

Let X. be a compact topological space, [X,L] the set of homotopy classes of continuous maps from X into L, Y a Banach space, F(Y) the metric space of closed semi-Fredholm operators with finite index and dense domain in Y (G. Neubauer, Math. Alin. 160 (1965), 93-130), FB(Y) the subspace of all bounded operators in F (Y), A(Y) the automorphism group in Y, K(X) the Grothendieck group for the vector bundles over X. (1) One can find an "index" i(f) in K(X) associated with each continuous map­ ping f from X into F(Y) such that i induces a mapping 1 from [X,F(Y)] into K(X). For Y separable tis one-to-one and onto. Some similar but partially weaker results can be obtained if .one allows infinite index or nondense domains. (2) Imposing some restrictions on Y (as for example that Y = Z al J P for some p) one can establish: 0----> (X,A(Y)]----> [X,FB(Y)]----> K(X)----> 0 is exact. This extends results pre­ viously obtained by K. Janich for Hilbert space. (Received November 24, 1965.)

66T-99. P. C. HAMMER, Pennsylvania State University, 426 McAllister Building, University Park, Pennsylvania 16802. Isotonic spaces in convexity. I.

Let V be a flat in a. real linear vector space M with null set N and let A be any convex set maximal with respect to exclusion of V. Then A is called a demispace with vertex set V. Let M have dimension n. Then one of each pair of nonempty complementary convex sets is a demispace. Let p E ukX for 1 ;:;! k < n, provided X·nY f. N for every demispace Y which has some (k - I)-dimensional vertex set V through p. Then ~ is an isotonic set-valued set-function which maps each subset X of M into the k-interior of its convex hull. In this framework, the appropriate class of demispaces is shown to be the minimum neighborhood base for p and a topological type interpretation is given to Theorems of Caratheodory, Steinitz and Bonnice-Klee. (Received November 22, 1965.)

66T-100. JOSEPH NIETO, University of Maryland, College Park, Maryland. On the stability of the index of certain singular integral operators.

Let r be a simple closed plane curve of class c 2 (this assumption can be weakened), r and s continuous functions on r satisfying the condition ~(z) = r 2 (z) - s 2(z) + 0 for all z E r. Further, let H be the operator Hc(l(z) =(1/'II"~Jr 1/>(5)/(5- z) d.f (z E r) and K an arbitrary compact operator in LP(r), 1 < p = riP+ sH + K, is a bounded and Fredholm operator in Lp (r) for every p, 1 < p < co. One shows that if T is perturbed by any bounded operator C with IIC liP < (MaxzEr lr(z)/~(z) I + MaxzEr ls(z)/~(z) I•IIS llp>-l where S is the operator S!/J(t) = ~/211") J~1r(s) cotan(s - t)/2ds, then the index of T + C coincides with index of T. Furthermore, liS liz = 1 and S is of weak type (1,1) so that using Marcinkiewicz's interpolation theorem one finds estimates for liS lip = liS llq; whenever 1/p + 1/q = 1. (Received November 17, 1965.)

66T-101 •. P .J. NIKOLAI, Aerospa·ce Research Laborato+ies, Wright-Patterson AFB, Ohio 45433. Inequalities for the permanent and determinant.

Let A= (aij} be ann-square positive semidefinite hermitian matrix with proper values

Xz ~ ••• ?; Xn; per A= LuES n:=laiu(i); det A =:LuES (- l)sgn(u)llf=laiu(i)" Let Aw• n · n

236 w = l, ••• ,cn,k, denote the cn,k principal k-square minor matrices of A. {1) Let A have row sums 2 r 1, r 2, ... ,rn' Lf=l ri = r f 0. Then (a) Lw per Aw ~ 1Pk(r1, ... ,rn)i k!/rk and (b) trPk(A) ~ 2 lqk (r 1 , ... ,rn) i k!/rk where P k (A) is the kth power matrix of A and tr is the trace; I\c (r 1, ... ,r n) and qk(r 1, ... ,rn) denote the kth weighted elementary symmetric function and kth weighted completely symmetric function, respectively, of r l'''''rn. Let A be regular. Then (2) per ~AT} ~ m Pi[Ati2 h ~m ~m s (t s)/(t ) rri=lper ( J• were T = L...i=lPh• pi> 0, L...i=lpi = 1. (3) per (A)~ per - -r (Ar) (s- r)/(t- r). t ·per (A ), if 0 < r < s < t. Let K = K(A) denote the condition number ;\I An cor;responding to the spectral norm of A. Then for K > 1, (4) per A ~ (Kn/(Kn-l) det A)/(e log Kn/(«n.: 1>), and

(5) per A ~ det A+ A~ {

66T-102. R. A. BRUALDI, S. V. PARTER and HANS SCHNEIDER, 213 Van Vleck Hall, University of Wisconsin, Madison, Wisconsin 53706. Every fully indecomposable matrix is diagonally equivalent to a doubly stochastic matrix.

An n X n nonnegative matrix is fully indecomposable provided there does not exist an r X s block of zeros with r + s = n, The following result constitutes a generalization of a theorem of Sinkhorn for positive matrices. Theorem. !!.._A is a nonnegative fully indecomposable matrix, then there exists diagonal matrices D 1 and D2 with positive elements on the diagonal such that D 1 AD2 = S is doubly stochastic. The doubly stochastic matrix S is uniquely determined while D 1 and D2 ~ uniquely determined up to scalar multiplication, The proof is accomplished by showing that a certain

nonlinear homogeneous operator on the cone [ x = (xl' ••• ,xn); x 1 ~ 0 J (defined by M. V. Menon for positive matrices in a forthcoming paper) has maximum eigenvalue one with corresponding unique eigenvector lying in the interior of the cone. (Received November 22, 1965.)

66T-103. JACK SILVER, University of California, Berkeley, International House, Berkeley 4, California. On a question of Rowbottom,

For terminology see my Abstract 65T-444, these cJioticeiJ 12 (1965), 723. Theorem. If (No) 0, then a. ->(NJ 0 in the model L of constructible sets. (This answers a question

of Frederick Row bottom, who had shown in his thesis, University of Wisconsin, 1964, that a.-----+( N 1) < N 0, then

a. -->(N 0 ) subsets of a. with two equivalence classes, put

G E s 0 (E) iff G is a finite subset of a. homogeneous with respect to E (i.e. any two finite subsets of G of the same size are E-equivalent), G E S iJ+l (E) iff G has some proper extension in SiJ(E), and if !3 = UiJ > 0, set Sil(E) = n 'Y(o) where 0 is a countable ordinal in the sense of 1 (and where o refers to the order type of the homogeneous set in the obvious way). (Received December 2, 1965.)

237 66T-104. T. S. WU, University of Massachusetts, Amherst, Massachusetts. Continuous auto­ morphisms on locally compact groups.

Theorem. Any continuous automorphisms on a noncompact, connected locally compact topo­ logical group is not ergodic with respect to the left Haar measure. (Received November 22, 1965.)

66T-105. P. C. DELIYANNIS, 7, Pafou Str.,Athens 219, Greece. Group representations and cardinal algebras.

The general theory of von Neumann algebras has been studied by Kaplansky and Loomis within the abstract frames of ring and lattice theory respectively; in a sense this also covers group re­ presentations, as Mackey has pointed out. It appears, however, to be a somewhat awkward approach, since the basic idea of direct sums is not immediately available. The starting point in this work is the observation that the operation of forming discrete direct sums of representations satisfies a suitable version of Tarski's axioms for cardinal algebras; accordingly, the global theory of represen­ tations is completely developed within this abstract frame. All concepts and many arguments are obtained by straight-forward translation, but several basic theorems have to be reproved by different techniques. (Received December 2, 1965.)

66T-l06. W. E. BAXTER and E. F. HAEUSSLER, University of Delaware, Newark, Delaware 19711. Generating submodules in simple rings with involution.

In several papers concerning a simple associative ring, A, with involution relationships between Jordan and Lie products of the sets S and K and their commutator subgroups have been established. The main result of this paper is Theorem 1: If A is of characteristic not 2 or 3 and either Z, the center, : [OJ or (i) dimension of A over Z Is greater than 16, then (K,K) o (K,K] : S, ([K,S), (K,S]] :

[K,K] and [K,K] "' [K,S] : K; (ii) dimension of A over Z is greater than 64, then [K,S] o [K,S] : S. The argument In (i) follows rather easily, while the argument in (ii) is one involving the use of the density theory for primitive rings and the results of M. Slater (J. of Algebra, Vol. I). (Received November 19, 1965.)

66T-107. NACHMAN ARONSZAJN, University of Kansas, Lawrence, Kansas. Extension of unbounded operators in a Hilbert space.

In a Hilbert space ~ conside·r an unbounded closed linear operator T with dense domain '.D. We denote by ~T the numerical range ofT, i.e. [X:3u E '.D, llull: 1, X: (Tu,u)j. Twill be called standard if the complement of §T is in the resolvent set ofT. We consider the problem of extending a nonstandard T to a standard f without changing the closure of Its numerical range: 3T : 3T. Theorem. (a) _!!_ 3T Is neither a half plane nor a parallel strip there exists such a canonical exten­ sion f (generalization of Friedrich's extension for symmetric operators bounded below); (b) if -- - T 3 is a half plane there always exists such an extension (noncanonical); (c) .!:!.._ 3T Is a parallel strip, such an extension exists if and only if for ,\ in the two compleme~tary half planes of 3T the deficiency is the same. This theorem (especially cases (a) and (c)) can be applied in several types of linear differential problems. (Received November 24, 1965.)

238 66T-108, HABIB SALEHI, Michigan State University, East Lansing, Michigan 48823, On prediction theory with continuous time. Preliminary report,

It is known that a nonnegative, hermitian q X q matrix-valued function F' on (- oo, oo) such that • fat. iA.t F' and logdet F' E L 1 (- oo,oo) can be factored in the form F =«<>«<>a,e,, cl>(>..) = 0l,;(t)e dt E L2(- oo, oo), where cl> is an outer function in the upper half-plane, It was announced by the author in the Abstract 65T-246, these cNot:iuiJ 12 (1965), 469, that cl>can be determined by an iterative procedure in the 2 case F'(>..) = 2[I + M(>..)}/(1 + A ), where J.L= ess,l,u,b._ 00 is determinable under a weaker condition on F ', namely there exist nonnegative measurable functions g and h on(- oo, oo) such that g(>..)I ~ F' ~ h(X)I a,e,, where (h/g) E L00 (- oo,oo). In general cl> is given by an expression containing infinitely many terms, However in the case g,h,F are rational functions and the poles of fl/(X + i)j (g +h) (F'f 1 are simple, our algorithm is considerably simplified and a closed-form expression for the factor cl> is available. (Received November 22, 1965,)

66T-109. c. A. AKEMANN, 1843 Channing Way, Berkeley 3, California. The dual space of an operator algebra. Preliminary report,

If M is a W*-algebra, it is well known that there exists a unique Banach space F such that F* = M. Sakai (jllinois J, Math. 9 (1965), 236] conjectured that the Mackey topology of M (as the dual of F) agrees with the strong* topology on the unit sphere of M. This is proved in the present paper by a modification of the Sakai proof for the finite case [ibid). An application of the above is the following extension of a result of Dunford (Canad, J, Math, 7 (1955), 289-305], Theorem. A bounded subset K of M* is weakly relatively compact iff there exists positive f in M* such that given e1 > 0 there is fJ > 0 such that if f(p) < fJ, then lg(p) I < <1 for each g in K, where p is a projection in M, The following theorem is a generalization of a classical result of Banach, Theorem, If M has the property that every projection in M dominates a nonzero projection of finite rank in M, then weak and norm convergence coincide in the positive cone of F. (Received Novembe-. 22, 1965,)

66T-110. D. W. WILLETT, University of Alberta, Edmonton, Alberta, Canada, On an example in second order linear ordinary differential equations.

Let b(t), t !1; 0, be any given positive nondecreasing function. Theorem. There exist positive nondecreasing infinitely differentiable functions a(t), t !1; 0, such that a(t) !1; b(t), and there are solutions x = x(t) of x.. + a(t)x = 0 for which lim sup x 2() t > 0 as t ---> oo. This generalizes the example given by Galbraith, McShane, and Parrish (Proc. Nat. Acad, Sci, 53 (1965), 247-249). (Received December 2, 1965,)

66T-lll, J, B. KELLY, Arizona State University, Tempe, Arizona 85281. Products of zero-one matrices.

Let B be an n X n symmetric matrix with nonnegative integral elements, B is said to be realizable if there exists an n X p matrix A with elements 0 and 1 such that AAT = B. If B is realizable, the minimum value of p taken over all 0-1 matrices A with AA T = B is called the content of B, The equation AA T = B is given a customary combinatorial interpretation and the determination of the content of B is reduced to a problem in integral linear programming, Consideration of the

239 dual problem leads one to study n X n matrices with the property that the sums of the elements in all principal submatrices are ;:;! 1. Necessary and sufficient conditions for realizability are given for n ;:;! 4 and the content of B is found to be the maximum of a certain finite number of linear func­ tions of the elements of B. The content of matrices of the form (k - A)In + AJn is determined in a number of cases but not in general. Related problems for nonsymmetric matrices are discussed and a connection is made with the Hasse-Minkowski congruence theory. (Received November 19, 1965.)

66T-112. R. D. BROWN and NACHMAN ARONSZAJN, University of Kansas, Lawrence, Kansas. Finite dimensiox.al perturbations of spectral systems.

Let (I): (V,W,H,G] be a spectral system; i.e., let V,W be Banach spaces and G,H be bounded linear transformations of V into W. The finite part RI of the quasi-resolvent of (I) consists of those complex >. such that the range of A). = G - >-H is closed and the nullity o.(A >,) and deficiency {J(A >,) are finite. By theorems bf Gokhbe;g, Krein, and Kato, RI is the countable union of connected open components, in each of which o.(A>, ), {J(A >,) remain constant except at a countable number of isolated points, the isolated eigenvalues of (I) in RI• Corresponding to each eigenvalue >. there is a finite dimensional eigenspace which can be decomposed into finitely many irreducible subspaces called elementary divisors of (I) at >.. If (II): [V,W,Hl'Gl] is another spectral system such that H = H1 and G = 0 1 on a closed subspace of finite docodimension in V, then RII = RI' and a matrix can be explicitly constructed from which can be determined the number and dimensions of the elementary divisors of (II) at >. E RII knowing those of (I). One can apply this result to extend approximation methods used for self adjoint eigenvalue problems to non self adjoint problems. (Received November 24, 1965.)

66T-113. A. H. THOMPSON, University of Pittsburgh, Pittsburgh, Pennsylvania. A note on the Herglotz-Noerther theorem for rigid motion in relativity theory.

An analysis of rigid motion in general relativity is given in terms of an anholonomic subspace v! of space-time (J. A. Schouten, Ricci calculus, Chapter V). Identities in the v! are obtained by projecting the Bianchi Identities of space-time on to the anholonomic subspace. These are used to give a set of sufficient conditions for the validity of the Herglotz-Noerther Theorem (F. Noerther, Ann. Phys. Lpz. 31 {1910), p. 919) in general relativity. A conformal generalisation of the Herglotz­ Noerther theorem is conjectured. (Received November 26, 1965.)

66T-114. R. M. NIELSEN, University of Delaware, Newark Delaware an1 j. W. BRACE, University of Maryland, College Park, Maryland. A uniform boundedness theorem.

Let E be a linear topological space, F a locally convex linear space, and H a family of con­ tinuous linear transformations with domain E and range in F. Definition. H is bounded on a sequence fxnJ C E if for each closed, convex neighborhood of Oin F, there is a real number r > 0 with the property that for each f in H and positive integer N there exists an n > N such that xn is in rr 1 (V). Theorem. Let B be a bounded convex subset of E. Then, H is uniformly bounded on B if and only if H is bounded on every Cauchy sequence in B. Note that the set B is not assumed to be sequentially closed or sequentially complete as in the usual uniform boundedness theorem. Using similar ideas

240 a definition is formulated for a sub.set A of E to absorb a sequence of elements from E. An analogue to the classical absorption theorem of functional analysis is obtained with the usual category require­ ment removed. (Received November 24, 1965.)

66T-115. G. L. SEEVER, California Institute of Technology, Pasadena, California 91109. Measures on F-spaces.

An F-space is a compact Hausdorff space in which disjoint open Fu sets have disjoint closures. A partially ordered set (P, ;;;;) has the property (1.) iff for any pair of sequences {xnJnEw and r y ~ in P such that x ;;;; y for all n, m E w, there exists x E P such that x ;;;; x ;;;; y for all l n5nEw n m n m n, m E w, Theorem 1. A compact Hausdorff space X is an F-space iff C(X) has the property (1).

For ~ a Boolean algebra, ba( ~) is the set of bounded, additive, real-valued functions on ~and is equipped with the total variation norm. Theorem 2. Let ~ be a Boolean algebra which has the property (1). Then K C ba(~) is bounded iff sup {IIL(E)I: 11. E K J < oo for all E E ~. Theorem 3. Let X be an F-space. Then u(C(X)*, C(:X:))-convergent sequences are u(C(X)*, C(X)**)-convergent,

Theorem 4. Let PA be a Boolean algebra which has the property (1), IL E ba(~). and let [!Lnf n cw

be a sequence in ba(~) such that t_!l.n(E) }nEw is convergent for all E E~, and each !l.n is absolutely

continuous with respect to ·,II; Then f!LnJ nEW is u(ba(9J), ba(~)*)-convergent, and the !l.n are uniformly absolutely continuous with respect to 11.. (Received November 22, 1965.)

66T- 116. NACHMAN ARONSZAJN and P AWEL SZEPTYCKI, University of Kansas, Lawrence, Kansas. On compatibility of extensions of semiregular integral transformations.

Let (X,IL)(Y,v) be !I.-finite measure spaces, 9.n, 91-linear spaces of measurable, finite a,e. scalar valued functions on X and Y, provided with complete metric topologies of convergence in measure on all sets of finite measure. A is an (F)-subspace of 9.n if A is a complete linear metric space continuously contained in 9.n; A is an (FL)-subspace if, in addition, u E A, v E 9.n,

IY(x) I 1! lu(x)l a. e. => v EA. Let K(x,y) be a scalar valued measurable function on (X X Y, 11. X 11) and define 'IlK= [u E 9.n: JIK(x,y)llu(x)ld~(x) < oo a.e. Pj, (Ku)(y) = JKK(x,y)u(x)d!l.(x), u E 'IlK. Assume that there is u E 1)K' u(x) > 0 a. e .. If A is an (F) subspace of 9Jl then K is A-semiregular (A-s.r.) if (i) 'I)K n A is dense in A, (ii) K: 'I)K n A----> 91 is continuous. For an A-s.r.K, KA denotes its continl!ous extension to A. Consider the following compatibility condition: (*)If K is A and B - s.r., u E A n B then K Au= KBu a.e.v. Theorem I. (*)holds for all couples of (F)-subspaces of 9Jl iff the transformation K: 9Jl ----> 91 is closable. Theorem II. There exists an (FL)-subspace 1) of 9Jl such that (i) K is f>- s.r., (ii) if K is A-s.r. and A is an (FL) - subspace of 9Jl then A C 1) and Kl)u = K Au for u EA. Thus(*) holds for all couples of (FL)-subspaces of 9.n. (Received November 19, 1965.)

66T-117. MARTIN SCHECHTER, The Institute for Advanced Study, Princeton, New jersey. Estimates for boundary operators,

Let G be a bounded domain in En with smooth boundary oG and let A be a properly elliptic operator of order m with coefficients smooth in G (for definitions cf. Comm. Pure Appl. Math. 12 (1959), 457-486). Let 'Yj denote the normal derivative on oG of order j, and denote the norms of the

241 spaces Hs,p(G) and ws•P(aG) by 11•11 and ( •) , respectively (cf. Ricerche di Matematica 13 s,p s,p (1964), 19Z-Z06). Theorem. For each real~ and each 1 < p < oo there is a constant K such that s,p "'m- 1 ('YjU) j 1/ ~K

66T-118. F. D. LONERGAN, Sylvania Electronic Systems, Needham, Massachusetts OZ194. A topological property of Euclidean 3-space mod a certain,wild arc.

Let fJ be the wild arc of example 1.3 in Some wild cells and spheres in 3-dimensional euclidean space, R. H. Fox and E. Artin, Ann. of Math. 49 (1948), 979-990. Let X be the quotient space obtained by collapsing fJ, embedded in the 3-sphere, s3, to a point and let x (point) be the image of fJ under the natural projection p: s3 ___, X. Then, Theorem. X does not have a contractible basis at x. The concept of a contractible basis was defined previously in Abstract 608-134, these cJ.foficei) 11 (1964), 96, wherein it was stated that a basis G x = fu xT of open neighborhoods of a point x in the space X forms a contractible basis at x if and only if each Ux is contractible (relative to some point) and further­ more Ux - xis pathwise connected and simply connected, (Received December 3, 1965,)

66T-119. T. H. MacGREGOR, Lafayette College, Easton, Pennsylvania. Certain averages of univalent functions.

Suppose that f(z) is analytic for lzl < 1, and let g(z,a) = (1/a)J~f(zei 8)d8, where 0 0, and f'(O) =I 0, then f(z) satisfies (1). (Received November ll, 1965,)

66T-1ZO. E. A. BENDER, California Institute of Technology, Pasadena, California. The minimum dimension of symmetric matrices with a given root.

Let k be an algebraic number field and ()algebraic of degree n over k. Krakowski (Comment, Math, Helv. 3Z (1958), 2Z4-Z40) has shown that 8is the root of a symmetric k-matrix if and only if k(8) is totally real if k is formally real. Theorem. If such a symmetric matrix exists, it can be chosen to have dimension n or n + 1, whichever is odd, Generalization of a method of Sapiro (Sibiriskii Math. j. 3 (196Z), Z80-Z91) reduces the problem to finding for each local spot p on k a >.p in the p-adic algebra corresponding to k(8) and a basis a 1..... ,an over kp such that the quadratic form I:x C'j tract (X\la.1 aj) has the same Hasse invariant over kll as L:x~ and, for odd n, square dis­ criminant, The local problem is easily solved if ll is prime to l; otherwise, several cases must be considered. This dimension is best possible when n is odd or when n = Z(m6d 4) and - 1 ft. Norm(k(8)/k) • k 2• (Received December 6, 1965.)

242 66T-121. W. A. AI-SALAM, University of Alberta at Calgary, Calgary, Alberta, Canada. Fractional q-integration and q-differentiation.

Let 0 < q < 1, [n] = (qn - 1)/(q - 1), [n)! = [n)[n - 1] ••• [1] If n is a positive integer and [0]! = 1. The q-derivative of a function f is Df(x) = (f(qx) - f(x))/x(q - 1). The inverse operation is called q-integration. Two definite q-integrals are xD- 1f(t) = J~f(t)d(q,t) = x(l - q)L:oqkf(xqk) and xif (x) = j~f(t)d(q,t) = x(l- q)L~ 1q-jf(xq-j) •. We prove the following q-analog of Cauchy's formula -n - 1 r. ] for multiple integrals. Theorem 1. xD f(x) = xD (x - qt)n_ 1f(t)fLn - 1 ! where (x - qt)n = 2 n-1 n r. ] -1 1-n (x - qt)(x- q t) ••• (x - q t). We also have Theorem 2. xi f(x) = (Ln - 1 !) xi(t- x)n_ 1f(tq ). By means of these two theorems we then define the fractional integrals xD-"t(x) =

1 qt) _f(t)d(q,t) and K 11f(x) = 2 1 x)n_ f(tq 1- 11)d(q,t) where (fq(n))- J~(x- 11 q-n(n-l)/ (f 2 (n))- j~(t- 1 r q(n) is the q-analog of the gamma function due to Ja.ckson, n arbitrary real number, and (t - x>n-l = t 11 - 1 IJ~ 0 (1 + xt -lqj)/(1 - xt -lqn+j-l). These operations satisfy the semigroup property: Theorem 3. We have Dllo ,8f(x) = on+,8f(x) and K11K(jf(x) = K n+,8f(x). Various other properties and applications will be given. (Received November 19, 1965.)

66T-122. N. P. BHATIA, Western Reserve University, Cleveland, Ohio 44106. An oscillation theorem.

Consider the second order nonlinear differential equation (l(t)x')' + m(t)g(x) = 0, where g(x)

is defined and continuously differentiable for all x, xg(x) > 0 for x 'I 0, dg(x)/dx ~ 0, l(t), m(t) are defined and continuous on [O,oo) and the conditions J+00 1/l(r)dT = J+00m(r)d7 = + oo, l(t) > 0 on [O,oo) hold. Then all solutions x = u(t) of the differential equation are oscillatory. The result is unusual

as m(t) ~ 0 is not assumed. This generalizes the results of Leighton and Wintner when the equation is linear, and that of Waltman, when g(x) = x 2n-l, n positive integer. References: W. Leighton, The detection of the oscillation of solutions of a second order linear equation, Duke Math. J. 17 (1950), 57-51; P. Waltman, An oscillation criterion for a nonlinear second order equation, J. Math. Anal. Appl. 10 (1965), 439-441; A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115-117. (Received November 26, 1965.)

66T-123. R. J. GREECHIE, University of Massachusetts, 100 Arlington Street, Boston, Massachusetts 02116. On the structure of finite orthomodular lattices.

Let (L, ~ ,')be an orthomodular lattice (OML). For a, b E L and M C L, let M(a,b) = fx EM: a ~ x ~ b}, let Sa= L(a,l) U L(O,a'), and let ~L be the set of all maximal Boolean sub-OML's of L. Theorem. Let S 1 and s 2 be proper sub-complete sub-OML 's of the disjoint complete OML 's

L 1 and L 2, reap., let 8: s 1 ~ s 2 be a join-meet ortho-isomorphism, and let there exist Mi C: Li such

that si = U{sa: a E Mi} (i = 1,2). Then 8 determines an equivalence relation "'on L 1 U L2 and (L1 U L 2)/"' is a complete OML. If L 1 and L2 are both Boolean lattices, then in order to obtain the conclusion it is necessary that si = Sa for some ai E Li (i = 1,2). (The internal point of view is i more perspicuous: If L is a complete OML and L = B 1 U B2 where B l' B2 E ~L' then ~L = {B 1,B2J and B 1 n B2 = Se' for some eEL.) That any finite OML may be constructed by "pasting together" Boolean lattices in a similar fashion is shown in the following Theorem. Let (L, &. , ')be a finite

OML. Let Bl' B2 E ~Land let el' ••• ,en be the (distinct) atoms of B 1 n B2• Then, for 1 ~ k ~n + 1,

243 k-1 n . k B 2 (0, Vi= 1 e i) U B 1 (0, V i=k ei) determmes an element of ~L which we denote by B Moreover B 1 = Bl' Bn+l = B2, andfor 1 ~k ~ n, Bk n Bk+l = Se n (Bk UBk+l). (Received December 1, 1965.) k

66T-124. EMIL GROSSW ALD, University of Pennsylvania, Philadelphia, Pennsylvania 19104. A property of the Riesz function.

Two functions, P (z) and Q(z), whose Maclaurin expansions contain only real coefficients are said to form a KL-couple, if they satisfy the following conditions: (1) they have only real zeros; (2) the zeros of P separate those of Q and reciprocally; (3) Q'(z)P(z)- Q(z)P'(z) > 0 for real z; (4) the indicator diagrams of P and Q are identical. It is trivial to verify that P = cos z, Q = sin z are a KL-couple. By a theorem of Krein and Levine (Levine, Nullstellenverteilung ganzer Funktionen, Berlin, 1962, p. 488), fP,Qf E KL ~P(z) + iQ(z) = G(z) is of class P, i.e. G(z) 'I 0 for -2 -z/m2 lm z < 0 and ha(- 7r./2) - ha(11'/2) ~ 0. Let F (z) = z L:=f(m)m e be Riesz' function and set G(z) = F(- iz)/(- iz) = P(z) + iQ(z). Then P(z) = Lm=l(!L(m)/m2>cos (z/m2) = "oo 2 oo 2 2 oo 2 n L...n- =O ((- z )")/(2n)! r<4n + 2), Q(z) • Lm= 1 (!l(m)/m ) sin (z/m ) = z L n =O(- z ) )/(2n + 1)! r<4n + 4).

It may be shown that G(z) 'I 0 for Im z ~ 0 and it is known (Wilf, Illinois J. Math. 8 (1964), 639-641) that hF (IJ}) = max (0, - cos IJ). It follows that G(z) E P, so that [P ,Q} E KL. Many other KL-couples are obtained by applying the Krein- Levine theorem to generalizations of the Riesz function (see Grosswald, Publicationes Mathematicae, Debrecen, 1964). (Received November 29, 1965.)

66T-125. SEYMOUR HABER, National Bureau of Standards, Washington, D. C. A modified Monte-Carlo quadrature.

A Monte-Carlo method for numerical quadrature, which incorporates completely automatic forms of stratified sampling and of the "method of antithetic variates", is described, The method applies to integration over an s-dimensional hyper-rectangle, and certain other regions. Where the simple Monte-Carlo method produces an estimate whose standard deviation is O(N- 1/ 2), the estimates by 2 forms of the present method have standard deviations which are asymptotically O(N-1/2-1/s) and O(N- 1/2-2/s). In experimental calculations, the asymptotic estimates have been found to represent the standard deviations quite well, for practical values of N. (Received November 26, 1965.)

66T-126. A. E. LIVINGSTON, Lafayette College, Easton, Pennsylvania. On the radius of univalence of certain analytic functions,

Let C denote the class of functions f, regular and univalent in E = {z I lz I < 1 }, which satisfy f(O) = 0 and f' (0) = 1 and which are close-to-convex in E. Let .5e and Sf'* denote the subfamilies of 't/:, made up of functions which are convex and starlike in E, respectively. Recently, Libera lProc. Amer. Math. Soc. 16 (1965), 755-758] has shown that iff is a member of 5t; SA< or -'i:f. then the function F(z) = (2/z)J~f(t)dt is also a member of X, Y. or Sf. We consider the converse question. In particular, letting f(z) = [1/2] [zF(z))', we prove the following theorems. Theorem 1. IfF is a member of 't!:, .Y'• or .Jt' then f is respectively close-to-convex, starlike or convex for lz I < 1/2. In each case the constant 1/2 is the best possible. Theorem 2. If Re [F'(z)] > 0 for z in E, then Re~'(z)] > 0 for lzl < (\/(5)- 1)/2. The resultisthebestpossibleresult. (Received December 6, 1965,)

244 66T-127. W. 0. J. MOSER, McGill University, Montreal, Quebec, Canada. Coset enumeration used to solve a word problem in groups.

Let G be a group generated by a finite number of generators g 1, ..• ,gm and defined by a finite -1 number of relations Ri (g)= E, i = 1, ••. ,n. (Ri(g) denotes a word in the gi and gi .• ) Let Wi = Wi (g), i = 1, ••• ,s be s elements of G, H the subgroup generated by them and suppose [G: H)= d is finite. In this paper we describe an algorithmic method to find a set of representatives hi= hi(g), i = 1, .•• ,d, of the cosets of H and then for any X= X(g) E G to find hi and U E H such that X= Uhi with U expressed as a word in the W i and W i - 1 • A partial .solution has been described by Leech (Pro c. Glasgow Math. Assoc. 5(1962), 166- 175) and a solution with U expressed as a word in the Schreier generators of H has been described by Mendelsohn (Canad. J. Math. 16 (1964), 509- 516). In all cases the main tool is the Todd-Coxeter coset enumeration method (see Coxeter and Moser, Generators and relations for discrete groups, Springer, 1965). Some applications of the method are described. (Received November 29, 1965.)

66T-128. C. T. SCARBOROUGH, JR., Wayne State University, Detroit, Michigan. Minimal metric spaces.

A topological space (X, T) is metrically closed if T is a metric topology on X and (X, T) is closed in every metric space in which it is topologically embedded. The topological space (X, T) is minimal metric (minimal completely Hausdorff) if T is a metric (completely Hausdorff) topology on X and no metric (completely Hausdorff) topology on X is properly contained in T. Theorem .1. If X is metrically closed or minimal metric, then X is compact. Theorem 2. If X is minimal completely Hausdorff, then X is compact. (Received November 30, 1965.)

66T-129. C. J. TITUS, University of Michigan, 363 West Engineering, Ann Arbor, Michigan 48104. Immersions of a circle in the plane.

All mappings are C 00 • Given an immersion f of the oriented circle S in the Euclidean plane (= regular, closed, parametrized curve) there is associated the Gauss mapping of S into S (= unit normal mapping); let r(f) be the degree of the Gauss mapping (= tangent winding number). Define the equivalence relation: f ~ g if and only if there is a sense-preserving diffeomorphism h of S onto S such that f 0 h and g have the same Gauss mapping. This equivalence relation may be considered the natural one associated with the classical Schwartz-Christoffel mapping function. Theorem. Given any immersion f there exists an immersion g, with g ~f, such that the number of self intersections of g is exactly I jr(f)j- 11. (Received November 29, 1965.)

66T- 130. E. L. COHEN, The MITRE Corporation, P. 0. Box 208, Bedford, Massachusetts. On the sum of the squares of two consecutive numbers.

In The theory of algebraic numbers (Carus Monograph No. 9), H. Pollard proves (p. 17)- -The Gaussian primes fall into the following three classes: (I) all positive rational primes of the form 4m + 3 and their associates in G (the Gaussian integers); (2) the number 1 + i and its associates (of no concern here); (3) all integers associated with either x + iy or x - iy where x > 0, y > 0, xis

245 even, and x 2 + y 2 Is a rational prime of the form 4m + 1. Let q, q 1' q 2'•·••qs be odd primes. n 2 + (n + 1)2 = [n + (n + l}i] (n - (n + 1)1] = qk, k positive. lf q = 3(mod 4}, then since q is a Gaussian prime q l(n, n + 1). Only 1 can do this, a contradiction. Now we look at q = 1 (mod 4). 0 2 + 1 2 = 1l, 12 + 2 2 = 5, 32 + 4 2 = 52 , and 42 +52 = 41. Excluding these possibilities, we have Theorem. n2 + (n + 1)2 t qk for all primes ~ 109 except for 13, 29, and 61. We are still looking for proofs when q = 13, 29, and61. In thesecases22 + 3 2 = 13, 52 + 6 2 = 61,202 + 21 2 = 292 , 1192 + 1202 = 134 , and there may be other cases which we haven't come up with. The theorem was proved for q = 5 by C. Engelman, Volume IT-7, Number 1, jan. 1961. As a Corollary to the above discussion: _ q rz rs . . _ 2 2 £ l lf e- q 1 q 2 ••• qs contams any pnme ql'q2 , ... ,q 8 = 3 (mod 4}, then n + (n + 1) "I e for any (positive). (Received December 7, 1965.)

66T-131. jiN BAI KIM, 701-208 Cherry Lane, Michigan State University, East Lansing, Michigan. Primitive idempotent& In a regular semigroup with 0.

Let S be a regular semigroup with zero 0, and let a E s\o. Define E a = [e E S: e = e 2, ea = a J, Fa= ff E S: f = f 2, af = aJ. lf T <;; S, E(T) = feET: e = eeJ, V(a) =)xES: axa =a, xax = xJ. Theorem. The following conditions on a regular semlgroup S with 0 are equivalent (Lallement and Petrich, Bull. Amer. Math. Soc. 70 (1964}, 777). (i) All its nonzero ldempotents are primitive.

(ii) For all a, x in S, axa = a "I 0 implies xax = x. (iil} For a, b, x and y in S, xa = xb t 0 and ay = by "I 0 implies a= b. (lv) Every nonzero principal left (and right) ideal of S is 0-minimal. (v) S Is a mutually annihilating sum of completely 0-simple semigroup. (vi) For all a, b in S, aS n Sb contains at most one nonzero idempotent. (vii) For all a, b and c in S, aS n Sb contains at most one element of V(c). (viii) For all a in S\0, Ea n Fa contains at most one element. (ix) For all e in E(S\0), Ee and Fe are right and left zero semlgroups withEe n Fe= £e f· (x) For all e E E(S\0), there exists a set I of nonzero idempotents of S such that ei and Ie are right and left zero subsemi­ groups of S, respectively, ei n Ie = !e} and e(E(S\I)) = )OJ= E(S\I)e. (Received December 6, 1965.)

66T-132. N. M. RIVIERE, University of Chicago, Box 39, Chicago, Illinois 60637. Symbolic calculus of kernels with mixed homogeneity.

Let X E En, X= (xl ..... xn). a= (al, .... an); ai ~ 1, al = 1; call Ia I= L~tai and xax =

(Xa 1x 1, ... ,>.anxn). Consider the transformation x =Pax', where lx'l = 1, dx = plal-lj()dpdx'. Let k(x,y) defined in Enx(En- fol ), such that(!) k(x, ,\ay) = >.-lalk(x,y); (il) Jlyl=lk(x,y)J(}du(y) = 0; (iii) For every (3, I< a; ay)(3k(x,y) I ~ C(3 in E l1x Ln where Ln = [ x,lx I = 1]. Under these conditions:

K(f) = b(x)f(x) + lim t~Q JP(x-y)?.t k(x,x- y)f(y)dy, satisfies IIK(f)llp & C pllfllp (this result is part of a paper submitted to Studia Mathematica by Fabes and Riviere). Consider the operator Pp. defined for f E cg>(En) by p;(f} = PJI.(x)f(x), p. > 0. Define K E: C~ -l, ,\ > 0, if k satisfies (!}, (il}, (lli) and for )'. ~ Xa.- 1, 1' integers: (1) l.a.- 1] is the integer part of 1 1 1 ,\L...n 1 1 Aa.- 1; l(a;ax)i3k(x,y)- (a;ax)i3k(x + h,y)l ~ C"':l 1hl. ai- /3i in EnxL ; (1) and (2) must also hold 1 L1= 1 1 n for b(x). Following a technique similar to that used by Calder6n-Zygmund in Singular integral opera- tors and differential equations, Amer. j. Math. 79 (1957), the symbol of these operators can be de­ fined; using the same nomenclature: Theorem. lf p.

246 (Z) If lu(K 1)(x,y)l > f > 0 in EnxLn then there exists K; u(K) = u(K1) and K has a two sided inverse, (Received December 7, 1965.)

66T-133, GUNTHER GOES, Illinois Institute of Technology, Chicago, Illinois 60616, Periodic and semiperiodic sequences and Fourier-Stieltjes transforms of discrete measures,

With notations as in Rudin's book, Fourier analysis on groups, let Mdr(T) be the closed sub­ algebra of discrete measures in M(T) (T = additive group of reals modulo 1) concentrated in rational points, Let Bdr(Z) be the space of Fourier-Stieltjes transforms of measures in Mdr(T). It is shown: (i) Bdr(Z) is the closure with respect to the norm 11!111 = IIJLII in B(Z) of the set of periodic sequences, (ii) Bdr(Z) = B(Z) n Q where Q is the space of semiperiodic complex sequences, (iii) IfF is a complex valued analytic function in a region which includes the uniform closure of the range of Bdr(Z) then also IF(a(k))} E Bdr(Z). (iv) The /3-dual Kothe space of Bdr(Z), i.e. the space of complex se­ quences c = [c kl ~00 such that I;~p(k) converges for every ft E Bdr(Z), is the Banach space of c for which sn(r) = Lk=-nckeZ1rikr converges for n -+00 boundedly in the set ot rational numbers rET. Here llcll = supn supxETisn(x)l. (Received November Z6, 1965,)

66T-134, N. J. PULLMAN, University of Alberta, Edmonton, Alberta, Canada, Infinite products of substochastic matrices,

A real matrix (aij) none of whose entries is negative is substochastic iff Ljaij ::> 1 for all i, and it is stochastic iff I;jaij = 1 for all i. A sequence of nonnegative real numbers is substochastic if

its sum doesn't exceed 1. Theorem, If {An} is a sequence of oo X oo substochastic matrices then there exists a nonempty set E of substochastic sequences each of which, except perhaps the zero sequence, is the componentwise limit of a sequence of rows, one from each of the partial products

AnAn_ 1 ... A 1; and any sequence fPnt of rows, one from each of the partial products AnAn_ 1 ... A1, can be approximated by a sequence fcnJ of convex combinations of elements of E (i.e, [Pn- cnJ

converges componentwise to 0,) Corollary. If fAn' is a sequence of r X r stochastic matrices then

there is an m X r stochastic matrix A with 1 ;;; m :i r and permutation matrices Qn such that if

m < r then limn~ 00 (AnAn_ 1 ... A 1 - QnCA,D) = 0 where Dn = CnA for some (r- m) Xm stochastic

matrices en and; if m = r then limn~ 00{AnAn_ 1 ... A1 - QnAj = 0. Some results on products of the form A1A2 ... An ... are also obtained, (Received December 6, 1965,)

66T-135, WITHDRAWN.

247 66T-136. M. G. NADKARNI, Washington University, St. Louis, Missouri 63130. On factoring matrix valued functions in the Bohr group.

Let R denote the real line with discrete topology and B = R be its compact dual. Let H be a Hilbert space and f be a function on B taking as values bounded positive definite Hermitian operators on H. Suppose (i) (f(x)a,a) is summable for every a E H; (ii) log II (f(x))- 1 11 is summable, then f = cf>.P• where c/> is an operator valued function on B such that Jc/>(x)Xt (x)dx = 0 for t > 0. Here summable means summable with respect to the Haar measure on B and Xt is the character in B corresponding to the real number t. This generalizes a portion of a theorem of A. Devinatz proved in Ann. of Math. (Z) 73 (1961), 458-495. The theorem for the groups does not seem to be obtainable by Devinatz' method. We adopt a method due to D. Lowdenslager, Ann. of Math. (Z) 78 (1963), 450-454. (Received December 3, 1965.)

66T-137. N. S. MENDELSOHN and C. T. BENSON, University of Manitoba, Winipeg 19, Canada. A calculus for a certain class of word problems in groups.

This paper continues the work of Mendelsohn, An algorithmic solution for a word problem in group theory, Canad. J. Math. vol. 16, pp. 509-516. The following problem is solved in an algorithmic manner. Let G be a finitely presented group with generators gl' gz•···•gt. Let H be a subgroup of finite index generated by hl' hz•···•hs where each h1 is a word in the gj's. A practical algorithm is given for writing a word in the generators of G in the form Wu where W is a word in the generators i of H and ui is a coset representative. Several applications are given. (Received December 3, 1965.)

66T-138. G. E. COLLINS, Thomas J. Watson Research Center, P. 0. Box Zl8, Yorktown Heights, New York 10598. Subresultants and reduced polynomial remainder sequences.

Let Y be an integral domain, 9r,f) the ring of polynomials over Y. Let P, Q E 9(.7) with m = deg (P) ~ n = deg(Q) > 0. Let M be the matrix whose determinant defines the resultant of P and Q. Let Mij be the submatrix of M obtained by deleting the last j rows of P coefficients, the last j rows of Q coefficients, and the last Zj + 1 columns, excepting column n 1 + nz - I - j (0 ;:> i ;:> j < nz). The polynomial Rj(x) = Li=odet(Mij)x1 is the jth subresultant of P and Q, R0 being the resultant. If b = .st'tQ), the leading coefficient of Q, there exist uniquely R, S E _9(..JF) such that bm -n+ 1P = QS + R with deg (R) < n; define !JR(P, Q) = R. Define 'P1 E 9(.¥}, Y the fraction field of J. inductively; - -- ..., --- -- ~i-r+l . - P 1 = P, Pz = Q, P 3 = .=(Pp Pz)• Pi+z = !}R(Pi'Pi+l)/c for 1 ~ z and ni+l > 0, where ci = .sf(Pi), ni = deg(Pi) and q1 = n1 - ni+l• P1,Pz, ••• ,Pk is a reduced polynomial remainder sequence; it is regular in case ~~ = l for Z ;:> I < k. Theorem • If k ~ 3 and nk ;:> j < n k-l' then - nk-r-J-lp- - nk-2(-~i-1 (~i-1))_ ~k-z(nk-1-j-1) R C 11 I (l) -p - nk-Z(- ~i-1(~1-1)) ck k - ± i=2 ci ck-1 j" oro ar es. k - ± i=Z cl - E /L'J - n nk-1- 5t-1 (~c 1) - - - Rnk_ 1_1" (Z)Pk .7'(JI). (3)lfnk=O,thenPk k-1=± i=z(ci )R0 • (4) lfPl'Pz, ... ,Pk is regular, then Pk = + Rnk_ 1_ 1; If also nk = 0 then Pk = ± R0 • These results provide improved algo­ rithms for computing resultants, greatest common divisors and Sturm Sequences. (Received December Z, 1965.)

248 66T-139. W. A. STRAUSS, Stanford University, Stanford, California. A remark on nonlinear elliptic equations.

A theorem of Minty, Browder, and Leray and Lions is generalized to nonreflexive spaces. Let X be any Banach space (of dimension > 1) with dual space X'. Theorem. A mapping A of X' into X is surjective under the following conditions. (1) A is weakly continuous on finite-dimensional sets. (2) j(Au,u)l/lluU tends to+ oo with !lull (norm in X'). (3) There exists a mapping A• of X' XX' into X, weakly continuous on linea in ita second variable, such that Au = A •(u, u) for u E X'· (4) Re(Au - A•(u,v), u- v) <': 0 for u, vEX'. (5) U a net un tends to u weakly• in X', then for all vEX', limsup(A•(un,v), un- v) <': 0. (6) U un tends to u weakly• in X' and if Re(Aun- A•(un,u), un- u) tends to zero, then lim sup Re(A•(un,v), un- v) <':Re(A•(u,v), u- v)for all vEX'. (Conditions (3)-(6) are satisfied if A is itself monotonic.) The proof is minimally different from that of the above-men­ tioned authors. (Received November 26, 1965.)

66T-140. MYLES TIERNEY, Rice University, Houston, Texas 77001. The cohomology of the classifying space forK-theory mod p. Let p: S I_, s 1 be of prime degree p. A Puppe sequence for p gives a sequence .lfp: sl-.2, sl ___, L(p,2)---> s 2 ..2. s 2 ••.• ('ifp, BSU) ~ _lf'(p,BSU) and one obtains a fibration BU---> BP---> SU induced by p = (p, BSU): SU---> SU from the universal fibration BU---> E---> SU. BP = (L(p, 2), BSU) is the classifying space for complex K-theory mod p. To compute H•(uBP; Zp) one applies n to the above fibration, computes (Op)•: H•(BU;Zp) ---> H'03 U;Z J• and uses the known rings, H•(U;Zp) and H•(BU;Zp). This gives H•(nBP; Zp) ~ (P p ~ .. llzj ••• ])/(••• g.~j ••• ) ®

Ep(••• g,2i-l ••• ), i ¢ o(p) for p odd. A more complicated argument gives H•(nB2; Z2) ""E2 (••• g,2j ••• ) ® (f2 G•• g4i+1 .. ~)/( ••• g:i+l ••• ). Using these results and two theorems of W. Browder, one may compute H•(BP; Zp) as an algebra. The result is: for any prime p, H•(BP; ~) ""Ep(••• 112i+l ••• ) ® P p G•• 112i ...], i ¢ o(p). (Received November 26, 1965.)

66T-141. C. T. BENSON, University of Manitoba, Winnipeg, Canada. Two nonisomorphic (v ,k,A) designs with similar incidence matrices.

Two nonisomorphic (v,k, A) designs with v = (1 + q) (1 + q '?, k = 1 + q + q2, A= 1 + q with q an odd prime power are constructed such that the incidence matrices can be arranged to be similar. One of the designs is projective 3-space over the field of q elements with points as objects and planes as blocks. Moreover, the incidence matrices of the two designs are symmetric and for q = 3 corres­ pond to rationally equivalent quadratic forms. Finally, for arbitrary q the collineation groups for the two designs are isomorphic. (Received December 1, 1965.)

66T-142. PETER WERNER, Mathematics Research Center, U.S. Army, University. of Wisconsin, Madison, Wisconsin. On the behavior of solutions of Maxwell equations for small frequencies.

The calculation of a stationary electromagnetic field with frequency w which is produced by the reflection of a given incoming field at n perfectly conducting obstacles with surfaces s 1 , ... ,Sn, involves finding vector fields E,H such that (11) V X E - tw H = 0, V X H + LW E = 0 in the exterior

249 De of S = s1 + ... + Sn' (/3) n XE =conS, (-y) (ajar- tw) E = o(r- 1) as r ---->oo, It is known that for every w > 0 a uniquely determined solution (E,H) exists and that (E,H) depends continuously (analytically) on w if the boundary data c likewise depend continuously (analytically) on w (J. Math. Anal. Appl. 7 (1963), 348-396. In the present paper the behavior of E and H as w---->0 is discussed. It is shown that E and H tend, for a large class of boundary data, to limit fields which can be charac­ terized as solutions of boundary value problems for harmonic vector fields, The study of this limit relationship is of particular interest since the mathematical structure of the limit problems differs essentially from that of the electromagnetic problem formulated above, For example, the theory of harmonic vector fields depends to a large extent on the topological structure of the boundaries whereas the theory of the original boundary value problem for w > 0 is independent of topological properties, (Received November 19, 1965,)

66T-143, A. J, V. SADE, 14 Bd du Jardin Zoologique, Marseille 4 BDR, France, Quasigroupes isotopes d'un quasigroupe demi-sym~trique.

Soit un quasigroupe Q = E( ), isotope de son transpos~ par la distorsion (l,/,,F), F d'ordre 3m dans @;E; Si !'ensemble des ~~ments de E constituant les cycles de longueur 3i de F et gi le nombre de ces cycles, Alors! (i), chaque sous-ensemble Ei = s0 + s 1 + ... + Si est un sous-quasigroupe de Q, isotope de son transpos~ par la distorsion (/, J,,F1) induite par F sur Ei' (ii) L'ordre de Q est divisible par 3, ou bien tous les gi sont multiples de 3, (Hi) Tout quas!groupe isotope de son trans­ pas~. dont l'ordre est premier avec 3, est isotope d'un quasigroupe demi-sym~trique, sauf peut-etre si Q est isotope d'un quasigroupe admettant une autotopie gauche distorsive (l, l,F) d'ordre 3m, pour laquelle le nombre des cycles non monomes de meme longueur dans F est divisible par 3, quelle que so it cette longueur, Les quasigroupes de Bruck, d' Artzy et de Cardoso fournissent des exemples,

L'auteur a construit deux quasigroupes, d'ordres 10 et 12, qui sont isotopes de leur transpos~ par des distorsions r~guli~res du 3 !~me ordre et qui ne sont isotopes d'aucun quasigroupe demi-sym~trique, Chaque solution en fournit un grand nombre d'autres par isomE!rie, (Received December 8, 1965,)

66T-144, ALI KYRALA, Arizona State University, 801 Lemon Street, Tempe, Arizona, Compatab!l!ty of uncertainty in 3-space with determinism in space-time,

It has been previously shown that the laws of special relativistic mechanics in the absence of electromagnetic fields are implied by § e dt - p • dR = 0 where e is mass-energy density, p is momentum density, t time, R position (see: A. Kyrala; Acta Phys. Austriaca, 1961, Bd, XIV, 448-459). The path of integration in the above is any closed one in space-time and the condition is simply that of unique integrability of the 4-dimensional action, The old Bohr-Sommerfeld quantization conditions are concerned with 3-dimens!onal and 1-d!mens!onal projections of this after a volume integration. f Edt= nh = fp • dR implies and is implied by the uncertainty principle with a return to determinism as h approaches zero. It is clearly possible to have determinism in 4-dimensional space-time together with uncertainty in the lower dimensional projections thereof provided the latter B/S integrals cancel, In the presence of nonpropagating electromagnetic fields the original uniqueness condition in 4-space does ~hold and the conclusion is uncertainty in 4-space also. (Received December 8, 1965.)

250 66T-145. FRED GROSS, Bellcomm, Incorporated, Washington, D. C. Factorization of periodic functions.

Theorem 1. Let f(z) and g(z) be two entire functions with f(z) periodic and g(z) nonlinear.

If f(g(z)) is of finite lower order, then it cannot be periodic. Theorem 2. Let g(z) be entire non­ periodic and not a polynomial of degree ;;; 2. If f(g) is of finite lower order, then it cannot be periodic. Theorem 3. If f is entire of lower order < 1/2 and g is not periodic, then f(g) cannot be periodic. Definition. An entire function f is said to be pseudo-prime if it cannot be expressed as f = g(h) with g and h transcendental entire. Theorem 4. Every periodic function of exponential type is pseudo­ prime. Furthermore, iff= g(p) where g is transcendental then p is a polynomial of degree at most 2 while f = p(g) is only possible if p is linear. Some analogous results are proved for meromorphic periodic functions while for elliptic functions the conclusions are somewhat stronger. (Received November 24, 1965.)

66T-146. S. E. DICKSON, University of Nebraska, Lincoln, Nebraska 68508. Decomposition of modules. II. Rings without chain conditions.

Call a ring R with unit a T-ring, if the primary decomposition theorem holds for torsion left R -modules (see Math. z. 90 (1965), 9-13 for the torsion theory in question). Theorem. If each localizing subcategory of left R-modules is closed under taking injective envelopes, then R is a T-ring. Theorem. If R is a commutative ring, then R is a T-ring if and only if for any collection .Li(i E I) of maximal ideals of R, the ideal J'l= fr E Rjr E .Li for all but finitely many i E IJ is not maximal. Examples are given of: 1. A noncommutative, left-Noetherian, non-T-ring. (2) A non­ Noetherian T-ring. (3) A commutative non-T-ring. (Received December 1, 1965.)

66T-147. RANKO BOJANIC and R. H. DeVORE, Ohio State University, Columbus, Ohio 43210. On polynomials of best one sided approximation.

Let f be an extended real valued function bounded from below on [a,b], w a weight function

and A (f) the supremum of Jbw(x)P(x)dx over all polynomials P of degree ;;; n such that P ;;! f on n a [a,b). If Q is such a polynomial and An(f) = J~w(x)Q(x)dx then Q is a polynomial of best one sided approximation to f of degree ;;; n. The existence of a polynomial of best one sided approximation to f of degree ;;; n is proved assuming that f is either L-integrable on [a,b) or finite on a set

fx0 , ••• ,xn} of n + 1 points of [a,b) which have the property that for any polynomial R of degree ;;; n, J~w(x)R(x)dx =~=oo AkR(xk) with Ak > 0, k = O, ••• ,n. As far as the uniqueness of polynomials of best one sided approximation is concerned, in contrast to the theory of best L 1-approximation, it is shown by a counterexample that continuity of f is not sufficient and it is proved that a differentiable function has a unique polynomial of best one sided approximation. Polynomials of best one sided approximation of degree ;;; n - 1 to functions whose nth derivative is of constant sign on (a,b) are determined explicitly as are polynomials of best one sided approximation of degree ;;; n on [- 1,1] to functions of the form h(x2) assuming that the [(l/2)n] + lth derivative of h is of constant sign on (0,1). (Received December 3, 1965.)

251 66T-148. W. R. ALFORD, F. j. Seiler Research Laboratory, USAF Academy, Colorado 80840. Cantor sets in Euclidean three space.

There are many embeddings of the Cantor set in E 3; however, they all share the following: Theorem. Let X be a Cantor set in E 3• Then there is a sequence of coverings \Uif ~X such that: (1) mesh fU il --> 0 _!!!! i--> co, (2) each Ui is the sum of a finite number of adjoint tame cubes with handles, (3) lnt Ui+l is a refinement of Int Ui' (4) i: Ui+l CUi is null homotopic. The techniques of the proof of the above theorem are new; however, if we use the above theorem and the W. B. Raymond Lickorish homeomorphism as described by john Hempel in Topology of 3-manifolds, pp. 207-212, then we get: Theorem. Let X be a Cantor set in E 3• Then there is a countable number of disjoint tame solid unknotted tori [Til~ E 3 and a homeomorphism such that (1) any finite subcollection of

{Ti} is splittable, (2) h: E3 - Lint Ti --> E3 - L~t Ti' (3) h(Bd Ti) = Bd T 1, (4) X • (LTt) = 0 and h(X) is tame. (Received December 3, 1965.)

66T-149. S. C. CHU, Bellcomm, Incorporated, 1100 17th Street N. W. Washington, D. C. and R. D. MOYER, Pennsylvania State University, University Park, Pennsylvania. On continuous functions, commuting functions, and fixed points.

Let f and g be two continuous functions which map the interval [0,1] into itself. It has been conjectured that if f and g commute, then they possess a common fixed point. The conjecture is trivially verified if one of the two functions, say f, has the property that it has a fixed point in any nonempty closed subset G (of I!J,1]) which is mapped into itself by f, or that repeated application off to any point x in the interval produces a convergent sequence. ln an attempt to bring these conditions to a more concrete form, one is led to a theorem which is of interest in itself. Let fl(x) = f(x), fk(x) = f(fk-l(x)) for k = 2,3, ••• , and x in (a,b]. Theorem. Let f be a continuous mapping of the interval [a,b] into itself. Then the following conditions are equivalent: (i.a) for each x E [a,b), f(x) t x => i (x) t x; (i.b) for each x E (a,b], f(x) > x (f(x) < x) => f2 (x) > x (f2(x) < x); (ii) If G is any nonempty closed subset of ~,b] mapped into itself by f, then f has a fixed point in G; (iii.a) for each x E [a,b], f(x) i x => rk(x) i x for every k > 1; (iii.b) for each x E (a,b], f(x) > x (f(x) < x) => rk(x) > x(rk(x) < x) for all k > 1; (iv) [rk}: 1 is a convergent sequence for every x in [a,b]. (Received November 18, 1965.)

66T-150. T.V. SASTRY, lllinois Institute of Technology, Chicago, lllinois 60616. Turning point problems for certain systems of linear differential equations.

The work is concerned with the asymptotic nature of the solutions of certain vectorial differen­ tial equations of the form Eh(du/dt) = A(t, E)u as the parameter E --> 0. Here h is a nonnegative integer and A(t,E) is an (n X n) matrix, holomorphic in the complex variables t and E in a region of the

(t,E)-space defined by the inequalities ltl ~to· 0 < (E( ~ Eo· (arg E( ~ocr At f = 0 the matrix A(t,E) may have a singularity of such a nature that a uniform asymptotic expansion A(t, E) ::::: L:oAr(t)Er is valid as E --> 0 in the above mentioned region. The coefficients A r(t) are supposed to be holomor­

phic for (t I ~ t0• The coefficients of the formal power series solutions become singular in the neighborhood of a turning point, but a different type of series expansions can be derived by first subjecting the differential equation to stretching and shearing transformations. It is shown here

252 that the expansions are not only true in larger regions than claimed heretofore but also the regions

do overlap for arbitrary small f, (Received December 1, 1965.)

66T-151. D. W. CURTIS and J. C. MATHEWS, Iowa State University, Ames, Iowa 50010. Generalized R-uniformities.

Let R be a relation in X X Y with dom(R) = X and ran(R) = Y. We say U C X X Y is R-enlarged

iff [(x,y) E U, (a,y) E R, and (a,y') E R together imply (x,y') E U] and ~x,y) E U, (x,b) E R, and

(x' ,b) E R together imply (x' ,y) E U]. For U, V C X X Y we define the operations U o V and U -l

relative toR, and consider R-uniformities ~R as nonvoid collections of R-enlarged subsets of X XY

on which are imposed the usual uniformity conditions, with R taking the place of the diagonal ~. Generalized R-uniformities ~R on X X Y are obtained by relaxing some of these conditions in the manner as A. S. Davis (Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886-891). The uniform topologies on X and Y are defined in the usual way, and for given spaces X and Y the relations R for which such uniform structures exist are character­ ized. Uniform continuity, generating families of metric functions on X X Y, and various compactness conditions on X and Y are considered with respect to generalized R-uniformities. (Received December 10, 1965.)

66T-15Z. R. K. SAXENA, McGill University, Montreal 2, Canada. An inversion for a kernel.

In this paper we solve the integral equation (1) F(x) = 00 R (xu)h(u)du, where R (x) has a J0 p,q p,q Mellin-Barnes type integral representation, F(x) is given and h(u) is an unknown function to be determined. It has been shown by means of fractional integration theory developed by Erdelyi (Rend. Sem. Mat. Torino 10 (1940), 217-234) that (1) can be reduced to classical Laplace transform, which can be solved by known methods. A result quite recently given by Fox (Proc. Cambridge Philos. Soc. 61 (1965), 466) is included in our result. On account of the general character of the kernel Rp,q (x), inversion formulae for various integral transforms possessing Bessel functions, Whittaker functions and Meijer's G-function as kernels, can be derived as particular cases of the inversion theorem of the paper. (Received December 10, 1965.)

66T-153. M. A. KAASHOEK, University of California, Los Angeles, California 90024. On operators with a Fredholm spectrum. Preliminary report.

The following theorems extend some results of I. C. Gohberg and M. G. Krern [Amer. Math. Soc. Trans!. (2) 13 (1960), 185-265), 185-265]. Theorem. Let X andY be Banach spaces, and letT and S be bounded linear operators from X into Y. Suppose that S is a homeomorphism, and that

T + XS is a Fredholm operator for each complex value of >.. Then dim X ~ dim Y < oo. Corollary. Let T be a closed linear operator with domain D(T) and range in the Banach space X. Suppose that D(T) is closed, and that T has a Fredholm spectrum. Then dim X is finite. The condition D(T) is closed and the preceding corollary is satisfied if, e.g., the quotient space X/D(T) is finite dimensional or even more generally, if there exists a closed subspace complementary to D(T) in X. (Received December Z, 1965.)

253 66T-154. BERNARD SHERMAN, Rocketdyne, A Division of North American Aviation, Incorporated, Canoga Park, California. A free boundary problem for the heat equation with heat Input at a melting interface.

A slab of heat conducting material initially occupying the interval 0 ~ x ~ a is insulated at x : 0 and has heat Input Q(t) per unit area per unit time at the opposite face. U melted material is formed we assume it is removed immediately; the heat input is always at the melting interface. We prove an existence and uniqueness theorem for small t for the free boundary s(t) using the contraction mapping principle, i.e., a contracting mapping of a complete metric space into itself has a unique fixed point. (Received November 26, 1965.)

66T-155. D. E. V ARB ERG, Hamline University, Saint Paul, Minnesota. On Gaussian measures equivalent to measure. II.

We continue a study initiated in Trans. Amer. Math. Soc. 113 (1964), 262-273. The most important new result that we obtain is that sufficient conditions for a general Gaussian measure determined by a covariance function r(s,t) and mean function zero to be equivalent to Wiener measure are that (i) r(O,t): 0, (ii) (1 21f(s,t) -min (s,t))/iltils exists, is square Integrable and has smallest eigenvalue greater than - 1. Recent results of Hiroshi Sato (private correspondence) make it appear plausible that our conditions are also necessary. Our method is to examine the circumstances under which a Gaussian process y(t) may be represented as a linear transformation of the Wiener process x(t) of the form y(t): x(t) + Jgnf(u,s)dudx(s) (stochastic integral) and then apply a generalization of D. A. Woodward's theorem, Trans. Amer. Math. Soc. 100 (1961), 459-480. In the process, we find it necessary to solve the integral equation 2K(s,t): F(s,t)- sgK(s,u)K(t,u)du where F(s,t) Is the kernel in (i!) above and K(s,t) is the unknown function. This we accomplish both by the method of successive approximations and by expanding in terms of the eigenfunctions of F(s,t). (Received November 29, 1965.)

66T-156. FRIEDRICH KASCH, Mathematisches Institut der Universitlit Munchen, Schell!ng­ strasse 2-8, West Germany and E. A. MARES, Swarthmore College, Swarthmore, Pennsylvania. A characterization of sem!perfect modules.

A projective cover of a module M is an exact sequence P L M ----<0 where P Is projective and Ke(f) is small in P. A projective module is called semiperfect if every epimorphic image of it has a projective cover. E. Mares has shown (Semi-perfect modules, Math. z. 82 (1963)) that a projective module P is semiperfect if and only if the radical Ra(P) of P is small in P, P : P /Ra(P) is semi­ simple, and every direct decomposition of j5 can be raised to P. This result is used to give another characterization of a semiperfect module. Theorem. A projective module is semiperfect If and only if It is complemented. M is said to be complemented if for every submodule U ~ M there exists a minimal submodule V with respect to the property M: U + V. By dualizing the proof of the theorem, one obtains a new proof for the existence of the injective hull in which the use of Zorn's lemma is made only to get the complement of a submodule with respect to Intersection. (Received December 3, 1965.)

254 66T-157, T. T. WEST, University of California, Los Angeles, California, Riesz operators,

Riesz operators in a Hilbert space are bounded linear operators with the spectral properties of compact operators. They have been characterized by Ruston. The class of Riesz operators is not closed under addition, multiplication or in the uniform topology. A result due to Stampfli stating that every bounded linear operator is the sum of eight projections shows that the algebra generated by the Riesz operators is the whole algebra of bounded linear operators. Also Ringrose's super­ diagonalization technique may be used to prove that every Riesz operator is the sum of a compact and a quasi-nilpotent operator, (Received November 26, 1965,)

66T-158, R. J, HANSON, University of Southern California, Los Angeles, California 90007, On systems of differential equations with special types of singularities.

The. system B(z)(dy/dz) = A(z)y + f(z) is studied where A and B are n X n matrices holomorphic at z = 0 with det B (z) =' 0 for all z. By using the fact that the rank of a matrix of functions holomor­ phic in a domain changes only at discrete points, it is possible, provided A satisfies a certain condi­ tion of compatibility, to reduce the given system to a system of linear differential equations with a singularity of finite rank at z = 0 and a system of linear algebraic equations, (Received December 8, 1965.)

66T-159. H. H. WICKE, Sandia Corporation, Division 5261, Sandia Base, Albuquerque, New Mexico. Open continuous images of complete metric spaces,

By a monotonically complete base for a space R is meant a base B for R such that the closures of the elements of any monotonic subcollection of B have a point in common. Recall that a base of countable order for a space R is a base B for R such that if P is a point belonging to each element of a perfectly decreasing subcollection K of B, then K is a base for R at P [Abstract 64T-40 1, these c}foticei) 11 {1964), 595]. Theorem, A regular To -space having a monotonically complete base of countable order is an open continuous image of a complete metric space. From this result and a theorem of Worrell and Wicke [Abstract 628-4, these c}foticei) 12 {1965), 803lthe following theorem is obtained, Theorem. A regular T 0- space has a monotonically complete base of countable order if and only if it is an open continuous image of a complete metric space. Remark. The class of

;regular T0 -spaces having monotonically complete bases of countable order is equivalent to the class of spaces introduced axiomatically by Aronszajn in Fund, Math, 15 (1930), 228-241. This equivalence is contained in the paper reported on in Abstract 628-4. (Received November 26, 1965.)

66T-160, T. N. E, GREVILLE, Mathematics Research Center, U. S. Army, Madison, Wisconsin, On the generalized inverse of a matrix product.

Let A+ denote the Moore-Penrose generalized inverse of an arbitrary complex matrix A, and A* the conjugate transpose. Each of the following is necessary and sufficient for (AB )+ = B +A+: {1) A+ABB*A* = BB*A* and BB+A*AB = A*AB; (2) both A+ABB* and A*ABB+ are Hermitian; (3) A+ABB*A*ABB+ = BB*A*A; (4) A*A = 0 and B B* = 0, where A = ABB+, A =A- A , B = 21 21 1 2 11 A+AB, B 2 = B- B 1• (Received December 21, 1965,)

255 66T-161. K. D. MAGILL, JR., State University of New York at Buffalo, Buffalo, New York 14214. Semlgroup structures for families of functions. I.

Let f denote a function from a nonempty set Y into a nonempty set X and let 0' denote a family

of functions with domains contained in X and ranges contained in Y such that f o f o g E 0' whenever f, g E 0'. Define the product fg of two elements f and g of 3' by fg = f o f o g. lJ Is a semlgroup with multiplication defined in this manner. It is referred to as a '5-semlgroup and is denoted by (5 (X, Y, f) if it separates points in X and has the additional property that for each x in X and y In Y, it contains a function whose domain contains x and whose range consists of the point y. Results are obtained concerning homomorphisms from one such semigroup into another. An example Is the following theorem. Let f and g be surjections from Y onto X and V onto U respectively. Then a bijection c/> from ®(X, Y. f) onto ®(U, V, g ) is an isomorphism If and only if there exist bijections

~ and t from X onto U and Y onto V respectively such that ~ 0 f = g o t and c/>(f) = to f o ~ for each f in ®(X, Y, f). These results have applications to various families of functions with domains con­ tained In one topological space and ranges contained in another. (Received December 21, 1965.)

66T-162. ALEXANDER ABIAN and DAVID DEEVER, Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210. On the minimal length of sequences representing simply ordered sets.

Based on the definitions of "weak-denseness" and "order by strong first differences" as introduced respectively in Abstract 65T-428, these c){oticei) 12 (1965), 718 and Abstract 65T-456, these c){oticei) 12 (1965),815, the following are established. Theorem 1. Let (S, ~)be a simply

ordered set with a weakly dense subset of power ~ K).. Then every subset of S contains a weakly

dense subset of power ~ Kx· Theorem 2. Let (S.~) be a set of sequences of 0 and 1 of type w X simply ordered by strong first differences. Then there exists a set S' of sequences of 0 and 1 of type wX simply ordered by strong first differences such that S C S', S = S' and S' has a weakly dense sub­

set of power ;;:; XX. Theorem 3. Let (S, ~ ) be a simply ordered set with a weakly dense subset of power Xx such that S has no weakly dense subset of power < Kx· Then Sis Isomorphic to a set of sequences of 0 and 1 of type wX ordered by strong first differences and Is not isomorphic to any set of such sequences of type smaller than wX. (Received December 21, 1965.)

66T-163. R. M. STEPHENSON, JR, Tulane University, New Orleans, Louisiana 70118. Remarks on minimal topological spaces.

Each space considered is assumed to be a Hausdorff space. Let X be a set, and let :r be a topology on X which has a property P. (X, :;t) is said to be a minimal P -space provided that for each P-topology :r• on X, :;t• C :r implies :r= :r•. (X, :r) is said to be P-closed provided that X is closed in each P-space in which it can be embedded. Theorem 1. Let P denote any one of metric, paracompact, or completely normal, and suppose that (X, :r) is a P-space. Then the following are equivalent: (a) (X, :r) is a minimal P-space; (b) (X, :r) Is P-closed; (c) (X, :r) is compact. Theorem 2. The product of an absolutely closed Urysohn space and a Urysohn-closed space is Urysohn-closed; the product of a compact space and a minimal Urysohn space is a minimal Urysohn space. (Received December 21, 1965.)

256 66T-l64. M. C. McCORD, University of Georgia, Athens, Georgia 3060 l. Singular homology £!. n-cell-like spaces.

A space is n-cell-like if it can be represented as the limit of an inverse sequence of (closed) n-cells, with all bonding maps onto. Such spaces are of course compact, connected, metrizable, and of dimension ~ n. Although they are acyclic in Cech homology, their singular homology can be interesting. Theorem. For each n ~ 3 there exists ann-cell-like continuum X whose rational singular homology groups Hq(X;Q) are uncountable for.!!.!_ q ~ 0. The construction uses a result due to M. G. Barratt and John Milnor, Proc. Amer. Math. Soc. 13 (1962), 293-297. On the other hand, it can be seen that if X is a 1-cell-like space (a chainable or snake-like continuum), then

Hq(X;G) = 0 for each q ~ l and each coefficient group G. This raises the following questions: If X is a 2-cell-like space and G is any coefficient group, must Hq(X;G) = 0 for allq ~ 3? In fact must

Hq(X;G) = 0 for all q ~ 2? (Received December 21, 1965.)

66T-l65. MORRIS ORZECH, Cornell University, Ithaca, New York 14850. A cohomology theory of Harrison for connected rings.

Let R be a commutative ring, S a faithfully flat commutative R-algebra, and J a finite cyclic group. Let Sn = S ®R ••• ®R S (n times). A bicomplex C(R,J,S) is constructed with cP•q = U(Sp+Iuq+l)), where U(T) ={units of TJ. The horizontal coboundary operators are those introduced by Harrison (Abelian extensions of arbitrary fields, Trans. Amer. Math. Soc. 52 (1965), l-14), the vertical ones arise from Amitsur's complex. Let T(J,R) denote the group of isomorphism classes of Galois extensions of R with Galois group J (Harrison, Abelian extensions of commutative rings, Mem. Amer. Math. Soc. No. 52, pp. 1-14). Associated with C(R,J,S) is a total complex with cohomology groups Hn(R,J,S). Then H 1(R,J,S) , (A E T(J,R)IS ®RA has a normal basis over sj. Passing to a direct limit over all S of the form S = L Ell Rxi' with I:xi = I, we get a cohomology group !j 1(R,J) ~ T(J,R). If R is a field, !:! 1(R,J) is isomorphic to Harrison's H2 (R,J). If R is connected and 1r is the fundamental group of R, then· Chase (Abelian extensions and a cohomology theory of Harrison, Proceedings of the La Jolla Conference on Categorical Algebra 1965)has shown T(J ,R), Home (1r,J). This and the result above generalize Harrison's isomorphism of the first-mentioned paper to connected rings. (Received December 9, 1965.)

66T-166. A. M. PFEFFER, California Institute of Technology, Pasadena, California. On certain discrete inequalities and their continuous analogues.

In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequali­ ties. Their techniques are generalized to mixed and higher derivatives and inequalities with weight functions in the integrals. An analogue of an inequality of Muller and Redheffer is used to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the kth derivative of some periodic function and the second number is the square norm of the mth derivative of the same periodic function. (Part of Doctoral Dissertation submitted to the California Institute of Technology. Supervisor Professor John Todd.) (Received December 13, 1965.)

257 66T-167. D. F. DAWSON, North Texas State University, Denton, Texas. On subgroups of semigroups.

Throughout, S denotes a semigroup with a left zeroid (Clifford and Miller, Amer. J. Math. 70 (1948), 117-125) and L denotes the set of all left zeroids of S. Each of the following conditions is sufficient for L to be a group: (1) L contains a regular element JJ. of S such that the element x E S for which JJ. = J.l.X!J- is unique, (2) L contains a normal element, i.e., there exists JJ. E L such that

J

66T-168. R. M. CROWNOVER, University of Missouri, Columbia, Missouri. Quasi-open maps In uniform algebras.

Let A be a uniform algebra on a compact Hausdorff space X. That is, A is a uniformly closed subalgebra of C(X) which separates points and contains constants. Let M [A] and r [A) be the carrier space and Silov boundary, respectively. Definition. A mapping f from a topological space S into the plane is said to be quasi-open provided that for any y in the plane, and any open set U InS containing a compact component of r 1(y), then y is interior (relative to the plane) to f(U). Theorem. If M[A] = X, then the restriction of each f E A to X - r[A] Is a quasi-open map. The proof of this theorem employs Theorem 3.3.23 of C. E. Rickart's General theory of Banach algebras, and results In H. Rossi's paper: The local maximum modulus principle, Ann. of Math. 72 (1960), 1-11. (Received December 16, 1965.)

66T-169. WALTER SILLARS, Pennsylvania State University, McAllister Building, University Park, Pennsylvania. Permutation grammars. Preliminary report.

A permutation grammar Is a context sensitive grammar as defined by S. Ginsburg (The mathe­ matical theory of context free languages, not yet published) with the restriction that every production is context free or of the form a {3--> {3a for variables a and {3. Theorem 1. Not every permutation language can be generated by a context free grammar. Theorem 2. The language L(G) generated by a permutation grammar G and that generated by its associated context free grammar of (obtained by dropping the permutation rules) are both empty, both finite, or both infinite. Lemma. If a context free language L contains infinitely many words of the form anbncn then for some word W in L the number of times a,b, and c occur in W are not all equal. Theorem 3. Not every context sensitive language can be generated by a permutation grammar. An example of one which cannot is the set L of all words of the form anbncn for n ;;:: 1, because if L = L(G) for some permutation grammar G, then by Theorem 2, L(of) Is infinite, and since L(of) ~ L(G), the lemma is contradicted. (Received December 13, 1965.)

258 66T-170. R. E. PEINADO, University or Puerto Rico, Mayaguez, Puerto Rico 00709. On the commutativity of rings.

A ring is said to be a P-ancestral ring if all proper nonzero subrings of R have property P. A subring of R of the form nR = { nR: r in R,n a fixed integer} is called a multiple of R. nR is called

S- maximal if n is the largest integer for which nR ~ S. For S a sub ring of R we consider the follow­ ing property: P: S contains an S-maximal multiple subring of R. Our main result is: If R is a P -ancestral ring then R is a commutative ring. These results are a generalization of a paper by F. Szasz in Pub!. Math. Debrecen 4 (1956), 237-238. (Received December 23, 1965.)

66T-171. IRVING REINER, University of Illinois, Urbana, Illinois 61803. Module extensions and blocks.

Notation. G: finite group, R: alg. int. /KJ, P : prime ideal in R, R p: P-adic completion of R, Kp : quotient field of Rp• Theorem 1. Let U,V be KG-modules such that the set of block idempotents of Rp G, to which the composition factors of K pU belong, is disjoint from the corres­ ponding set for Kp V. Then for any pair of RG-modules M,N for which KM: U, KN = V, the P-primary component of Ex~G(M,N) is zero. Theorem 2. Let e be a block idempotent of RpG which is primi­ tive in Rp G, and let X, Y be distinct irreducible Kp G-modules both of which belong to e. Then there exist Rp G-modules M,N such that Kp M = X, KpN = Y, Exti G (M,N) f. 0. These results generalize p the followihg recent theorem of Berman and Lichtman: Let G be nilpotent, G: G1 X ••• X Gr• where the iGiJ are the Sylow subgroups of G. Let U= u 1# ••. #Ur, V: v1 # ••• #Vr be distinct irreducible QG-modules, where for each i, 1 ;> i ;:;; r, Ui and Vi are irreducible QGi-modules. In order that there should exist ZG-modules M,N satisfying QM = U, QN = V, ExtiG(M,N) f. 0, it is necessary and suffi­ cient that the { Ui} differ from the [Vi j for exactly one value of the subscript i. (Received December 22, 1965.)

66T-172. CHARLES STONE, University of California, Los Angeles, California 90024. Ratio limit theorems for random walks on groups.

Let G be a locally compact Abelian group and let I I denote Haar measure on G. Let 11. be a regular probability measure on G with supportS and suppose that (1) lim sup (1J.(n)(C)) 1/n: 1 n~ 00 for some compact set C, and (2) G has no proper closed subgroups containing S - S. Let ~denote the collection of all Borel subsets of G having compact closure and positive Haar measure. Let $

denote the collection of those sets in ~ whose boundary has zero Haar measure. Theorem 1. For

A and B in .YI and n0 an integer limll-'001J.(n+no) (x t A)/11. (n) (y t B) : lA I/ IB I uniformly for x and y in compact sets. Theorem 2. If some IJ.(n) has a nontrivial absolutely continuous component with respect to Haar measure, then Jf can be replaced by ~in Theorem 1. Theorem 3. If for n sufficiently large n lim p(n+no)(x)/p(n)(y): It (n) has a continuous bounded density p(n) then for any integer 0 n .---.. oo uniformly for x andy in compact sets. (Received December 22, 1965.)

259 66T-173. W. S. MAHAVIER, Emory University Atlanta, Georgia 30322. Upper semicontinuous decompositions of irreducible continua.

It is known that there is no continuous collection of decomposable continua such that the sum of the members is a compact metric irreducible continuum; and that there is an upper semicontinuous collection of arcs which is an arc with respect to its elements and such that the union of its elements is a compact metric irreducible continuum. In this note we consider an upper semicontinuous col- lection G having the following property. (A) If g E G, each point of g is a limit point of the union of the members of each component of G - g. Theorem. If G is an upper semicontinuous collection of mutually exclusive continua such that (I) G has property A, (2) G is an arc with respect to its ele­ ments, and (3) each member of G is nondegenerate, hereditarily decomposable, and hereditarily irreducible, then a• is not a compact irreducible metric continuum. (Received December 22, 1965.)

66T-174. CAROL KARP, University of Maryland, College Park, Maryland 20742. On the characterization of weakly representable Boolean algebras.

This is a report on the algebraic consequences of some recent work in infinitary logic. Present proofs are metamathematical, but algebraic proofs are in preparation. Let K be an infinite cardinal and call a K-complete Boolean algebra (K- BA) weakly K-representable (K-WRBA) if it is isomorphic to a K-complete field of sets modulo a K-complete ideal. The following extension of Loomis' Theorem on K-BA's was found by the author in 1958. Theorem. If Kis a strong limit number of cofinality w, then a K-BA is a K-WRBA iff it is 'Y-distributive for all -y

Corollary. Assuming the generalized continuum hypothesis, if K is an infinite cardinal then cf(K) = w iff the K-WRBA's can be characterized by a single equation. This answers a question posed by

Prof. Tar ski. Corollary. For any infinite cardinal K, there is a 2K -complete BA which is K-distribu­ tive, but not a 2K- WRBA. (Received December 21, 1965.)

66T-175. ELIAHU SHAMIR, Northwestern University, Evanston, Illinois. Mixed estimates for systems of singular integral operators and applications to elliptic problems.

Let (x,y) denote points of Rn, with x E Rn- 1, y E R. R~[R~) is the half-spacey;;:; 0 (Y;;; 0). Hs,p is the space of distributions u in Rn for which llull = IIF- 1(1 + lsl2 + l>Full p < oo. Here F s,p L is Fourier transform, ($>q) dual to (x,y) and 1 < p < oo. H~·P is the subspace of elements supported in Rn. HS•P(Rn) is the quotient H8 •P;Hs,p, Y the canonical map onto the quotient and IIY ull the - + - + + s,p quotient norm. H s,p(R~) and Y _ are similarly defined. The definitions extend to vector-valued u componentwise. Let M(s,IJ) be aN X N matrix of functions, positively homogeneous Of degree 0,

C00 on ltl2 + 11 2 = 1. The operator Mu = F- 1 M(~,IJ)Fu is bounded in Hs,p, invertible if det [M(~,IJ)] t 0. Theorem. The operator T: u--> (Y _u, Y +Mu) has a closed range in HS•P(R~)XHS•P(R~) for every s except for at most N exceptional values of s(mod 1). There exists s 1 ;;:; 0 such that for s ;;:; s 1 non­ exceptional: llulls,p;;; C[IIY_ullx,p + IIY+Mulls,J• for all u E HS•P;L±IIY±V±II-s,p;;; cnv_ + MV+II-s,p• for all V ± E H- s,p(R~). The 1st estimate implies that T is 1-1, the 2nd is used to determine its

260 range. We proved them before for n = 1 and general M, or M = I and general n. They settle com­ pletely the problem of a priori LP estimates for elliptic problems in (n + 1) dimensions with piece­ wise smooth boundary operators (so called mixed problems), (Received December 2, 1965.)

66T-176. S, 0. AANDERAA, The Computation Laboratory, Harvard University, Cambridge, Massachusetts, A reduction method for special cases of the decision problem,

In Solvable Surinyi subclasses, Ann. Comp. Lab. Harvard University 31 (1962), 32, Dreben

showed that the class K 1 of wffs with prefixes of the form (y 1 )(y 2)(3 z)(y3) and with matrices containing no elementary part in which y 3 occurs either with z or withy 1 is solvable by showing that K' is finitely controllable. This can also be proved by reducing K' to the class G with prefixes of the form

(y 1)(y2)(3z) in the following way. Let the matrix M be written in conjunctive normal form C 1 & c 2 & ... & em' where Ci is the ith clause. Let A 1, A 2, ... ,Am be new distinct monadic predicate letters, Let Bi be the subclause of Ci which contains all elementary parts in which y 3 occurs; and let Di be the subclause of Ci containing the remaining elementary parts, Let M' be the result of

substituting y 1 for y 3 in (,A1y 2 vB 1) & (A 1y 2 v D1) & (1A2y 2 v B 2 ) & (A2y 2 vD2) & ... & (-rAmy 2 vBm) & (Amy2 vDm). Theorem, (y 1)(y2)(3z)(y3) M is satisfiable in a domain D iff (y 1)(y2)(3z)M' is satisfiable in the same domain; 1(y 1 )(y 2)(3 z)(y3)M has property C of order p iff 1 (y 1 )(y 2)(3 z) M' has property C of the same order, A generalization of this reduction method may be used to reduce other classes of wffs to classes which are easier to solve. (Received January 3, 1966,)

66T-177, P. J. ARANDA and E. P. CATTANEO, Universidad del Litoral, Argentina, CORA SADOSKY, Universidad de Buenos Aires, Argentina, On quasi-homogeneous potential operators.

Let a = (a 1, ... ,an) be a fixed n-tuple of positive real numbers and Ia I= a 1 + ... an' Ia I m = a 1 + ... am form< n. Let En denote then-dimensional Euclidean space and Em= fx = (x 1, ... ,xn): xi= 0, m < i;:; nj. A function f is quasi-homogeneous of degree r if f(Aa 1 x 1, ... , Aanxn) = Arf(x), A >0. If p(x) > 0 is a real-valued sublinear function, quasi-homogeneous of degree 1, it defines a distance " m/a 1/m . m. n E • It may be considered p(x) = (L..Ixi I i) , m bemg the least integer such that every ai divides m/2. Definition: H'Y is a quasi-homogeneous potential operator If H-yf(x) = p(x?-lal*f(x), 0 < 'Y < Ia 1. These operators satisfy similar properties as held by the Riesz potentials, to which

they reduce for a 1 = ... = 1, Theorem 1. For a. > 0, H-y is a continuous operator fro Ill. LP(En,p(x)p,8dx) to Lq(Em,p(x)(/3-a.)qdx) for all p;:; q such that 1/p- (lalm/1al)(1/q) = (-y- a.)/lal and

1 < P < lal/(a.- J3) if J3 < 0 or lal/(lal- /3) < p < lalh if J3 > 0. The weak type of H-y for p = 1 holds too. The proof given are simpler than the usual for the Riesz potentials. Theorem 2. H'Y is of vec­ torial type (P,Q) (for definition see A. Benedek and R. Panzone, Duke Math, J, 28 (1961), 301), P =

(p, ... p), R = (r1,. .. rn) and 1/p- 1/ri = 1'/nai for 1 < p < (n/1') min ai' (Received January 3, 1966,)

66T-178, H. M. FRIEDMAN, 50 Massachusetts Avenue, Apt. 207, Cambridge, Massachusetts, Model theory of ranks and orders.

Theorem. If (R0 , tR )!=Z.F. and if the union U of the ordinals< Odefinable in (Ro,tR ) is 0 . 0 < O,thenletRu<::;;Ra.<::;;R0 • Thenthereare Oordinals !3<0 with (Ra.,tRa.) = (RJ3,tRJ3)· For

261 be~ow we let i denote the cardinal of x, and if xis constructible let lxl be the constructible cardinal (the cardinal with respect to the constructible sets) of x. Theorem. U (Ma.' EMa.) < (MfJ, EMfJ) and a.< lfJI < IJ, then (M , EM )I=Z.F. The following is intended as an analogue to the well-known result a. a. that if 8 is inaccessible then (R8 , ER 8 ) F Z. F. Theorem. If 8 > w, then if for any a. < 8 definable in ( R8 , ER ) we have both IJ not cofinal with a., and~ < 8, then (R8, ER )I=Z.F. We let wi be con- 8 8 structible w1• Theorem. Let A 2 Mw.i be a constructibl~ set, Then the union of the ordinals < w~ definable in (A, EA) = the order type of the set of ordinals < wi definable in (A, EA). Theorem. lfxlx = Th((Ma.,EMa. )) for some a.JI = wi· lf V = L, then {x(x- Th((Ma.,EMa.)) some a.J= x lx = Th( (R , ER ) ) some a. J = w • Using the methods of P. J. Cohen, one can show Theorem. It is l a. a. 1 consistent with Z. F. + Choice to assume that {x 1x = Th( (R ,ER )) for some a. is countable, or ·that a. a. J it is zxO, or neither. (Received January 3, 1966.)

66T-179. D. L. BURKHOLDER, University of lllinois, Urbana, Illinois 61803, Local weak norms and almost everywhere convergence.

Let G be a locally compact Hausdorff group, p. a left Haar measure on G, and 1 ;:;o p ;:;o 2, For each positive integer n, let T n be a bounded linear operator in L p(G,p.) commuting with left [right] translations. U B is a measurable set, let M 8 be the smallest extended real number M such that if X> 0 and llfllp ;;; 1, then the measure of the set of all x in B satisfying ITnf(x)l >>.for some n is no greater than M/.\. P, Theorem. lf M8 = oo for some bounded measurable set B, then. for every u-flnite set C there is an f in Lp such that supn IT nf(x) I = oo for almost all x in C. The theorem is not true if the boundedness of B is dropped, A similar theorem holds for homogeneous spaces possessing a relatively invariant measure. These theorems extend some results of E. M. Stein who considered the compact case [Ann. of Math, 74 (1961), 140-170). Theorems for measure spaces with no group structure are also obtained, Using these results, several almost everywhere convergence problems are settled, (Received January 6, 1966.)

66T-180, N. M. RIVIERE and Y. SAGHER. University of Chicago, Box 39, Chicago, Illinois 6063 7. The converse of Wiener- LE!vy- Marcinkiewicz theorem,

A real variable function F(x) defined in the open interval I; F E Gs (0 < s ;> 1) if and only if: For every I' closed contained in I, IF(n)(x)l ;> snnn/s, x E I', B depending only on I', where F(n) denotes the nth derivative of F(x). Define: Yf = {f,f(x) = L: 00 a einx; such that ("00 Ia lp)l/p = p -oo n £..,_ 00 n A (f) < oo) Marcinkiewicz (Sur la convergence absolue des series de Fourier--Collected papers-- p Wars·aw (1964), 588-594) proved that if: (i) The domain of F(x) contains the range of f(x). (ii) f(x) E ..Y/-s'

F(x) E ~s· Then F(f(x)) E J¢1• A. Zygmund has pointed out, that the proof of Marcinkiewicz can readily be extended to show that actually F(f(x)) E ~· Theorem. Let F(x) be defined in the open interval I and such that U the domain of F(x) contains the range of f(x), f E ~ (O < s ;> 1); F(f(x)) E.)lZP, p < 2 (p depending on f) then F E Gs. The result can be extended to infinite compact abelian groups. (Received January 5, 1966.)

262 ERRATA Volume 12

R. ]. COCH. Some open questions concerning topological semi-groups. Page 801, Abstract 627-49. The author's name is misspelled; it should be Koch.

R. L. HEMMINGER. The lexicographic product of graphs. Page 787, Abstract 627-6.

Line 9. Add "(c)(x1,xa,) EE(X) for all a. EO [lJ."

P. G. HINMAN. Generalizations of some standard theorem on recursive functions. Page 406. Abstract 65T-238. Lines 10-11. S. K. Thomason has pointed out that Corollary 2 does not follow directly from Theorem 2. For arithmetic comparability the Corollary fol­ lows as stated, but the result concerning hyperarithmetic comparability, although true, seems to require other methods for its proof (see Abstract 624-2, these Notices 12 (1965), 449.

T. S. MOTZKIN. Extension of the Minkowski-Carath~odory theorem on convex hulls. Page 705, Abstract 65T-385.

Line 2. Replace "F 1n ••• nFk"by "F 1u ... UFk".

PETER KUGEL. Two theorems about !-predictable sets of tapes. Page 8\)9, Abstract 629-11.

Line 3. Replace " ~ 11 by 11 ?; ".

Volume 13

T. TAMURA and.R. LEVIN. Locally cyclic semigroups. Page 149, Abstract 66T-75. Line 4. "··· a holomorphic image T of the semigroup R of all positive rational ••• " should read "··· a holomorphic image T of a semigroup R if positive rational ••• "

A. C. MOREL. Remarks on absolutely convex subgroups. Preliminary report. Page 100, Abstract 630-142. Line 2. "preceding abstract" should be replaced by "Abstract 66T-48 on page 48 of this issue".

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The APL location in suburban Washington affords a choice of city, suburban or country living and offers many benefits including complete service and recreational facilities and superior public and private schools. Programs leading to advanced degrees from The Johns Hopkins University may be pursued at APL. In addition, staff members may continue their education at six other area universities with APLfinancial support.

Direct your inquiry to: Mr. W. S. Kirby, Professional Staffing

The Applied Physics Laboratory · The johns Hopkins University

8638 Georgia Avenue, Silver Spring, Maryland (Suburb of Washington, D. C.) An equal opportunity employer

273 Mathematicians and Statisticians looking for an opportunity to apply your special skills and your analytical abilities? The Center for Naval Analyses of The Franklin Institute is inter­ ested in men who can derive mathematical bases for new 7lte~uest tactical and operational proce­ dures for the U.S. Navy. &tlie Your ability to develop and test mathematical models will be Gommitment valuable in formulating opera· tiona! requirements for equip· The Age of Space is also the Age of ment under evaluation. Land and Sea. At Lockheed there are CNA is a private scientific no environmental limits to techno­ organization engaged in opera· logical exploration and progress. tions research and systems On land: highly advanced vehicle analysis for the U.S. Navy and systems for missions of the future. Marine Corps. It offers a prom· In the sea: deep submersibles to ising career with good salary probe the ocean depths, Poseidon and the accepted concomi· and Polaris to keep the peace. In tants of a position that de­ space: A gena, most versatile mands responsibility and vehicle system of the age. returns satisfaction. Engineers and scientists are invited to write Mr. K. R. Kiddoo, For an interview, write, Professional Placement Manager, enclosing resume, to: Sunnyvale, California. An Equal James M. Hibarger Opportunity Employer. CENTER FOR NAVAL ANALYSES LOCKHEED 1401 Wilson Boulevard MIBBILW$ • SI"AC• COM-NY Arlington, Virginia 22209

CIV/'a.-w---~ .. CENTER FOR NAVAL ANALYSES OF THE FRANKLIN INSTITUTE INS • Institute of Naval Studies SEG • Systems Evaluation Group OEG • Operations Evaluation Group NAVWAG • Naval Warfare Analysis Group MCOAG • Marine Corps Operations Analysis Group An equal opportunity employer

274 strategy: the scLence and art of interrelating political, economic, psychological and military forces of a nation to afford maximum support for its adopted policies

The Institute for Defense Analyses contributes to our nation's deliberations on strategy by advising on the weapons systems, processes, and economics of defense. In this activity, IDA's guiding belief is that meaningful advice can best be developed by bringing together highly qualified experts from a variety of fields, and providing stimulating guidance and a creative environment. IDA conducts studies for elements of the Department of Defense such as the Joint Chiefs of Staff and the Director of Defense Research and Engineering, as well as other executive agencies of the Government. To enhance our response to their requests, we wish now to add as professional staff members a few outstanding people at the level of Ph.D., with backgrounds in physics, engineering, operations research and systems analysis. Write toT. J. Shirhall, Institute for Defense Analyses, 400 Army-Navy Drive, Arlington, Virginia (near the Pentagon). An equal opportunity employer sponsored by twelve of the nation's leading universities. 4Ab IDA

275 Math.-Statistical Applied Mathematicians Marketing Consultant Operations Analysts Applied Physicists A. T. Kearney & Company, Inc., national firm, CAREER APPOINTMENTS general management consultant ACME member, seeks a marketing oriented individual trained in the field of operations research, statistics, and mathematical techniques. Industrial or distribution experience preferred. Depth of experience not important, but must have practical approach as well as theoretical knowledge of simulation, linear programming, queuing theory, probability theory, sampling tech­ niques, etc. Desirable to be conversant with EDP operations. Advanced degree required. Age 28-33. Moderate travel involved, home all weekends. Chicago location. Commen­ surate salary, plus later bonus and profit­ sharing. Please submit full details, in confidence to D. J. Parsons, A. T. Kearney & Company, Inc., 135 South LaSalle Street, Response Chicago, Illinois, 60603.

AN EQUAL OPPORTUNITY EMPWYER

Booz•AIIen Applied Research today serves many of the nation's top gov­ ernmental, military and private or­ THE UNIVERSITY OF ADELAIDE ganizations. Responding swiftly to their most complex technical prob­ AT BEDFORD PARK lems, we have worked to solve them SCHOOL OF PHYSICAL SCIENCES with a blend of interdisciplinary tal­ ents, modern analytical techniques, READER IN MATHEMATICS and wide-ranging research experi­ ence. Applications are invited from candidates for the This has proven a most success­ position of Reader in Mathematics in the School of ful combination, both for our clients Physical Sciences in the University of Adelaide at and for BAARINC. A good indicator Bedford Park. our success is our current need of Preference will be given to a candidate who is actively for new talent to respond to still research in Pure Mathematics. The duties of greater challenges ... in astronau­ engaged in the super­ tics, communications, transports~ the post will include research, lecturing and tion, computer technology and naval vision of graduate students. There is an ample provision warfare, to cite just a few. Creative, of staff, secretarial aid and librory facilities, and a high perceptive scientists and engineers degree of co-operation with the University of Adelaide are invited to investigate these new at North Terrace in graduate teaching and research avenues of career opportunity. work. Apart from this, graduate students in Mathematics Please send your resume to Mr. will be enrolled at Bedford Park from 1966. Robert H. Flint, Director of Profes­ sional Appointments. The salary for this post is £A4,300 ($A8,600) per annum. Superannuation is on the f.S.S.U. pattern. BOOZ•ALLEN The successful applicant should be able to take up APPLIED RESEARCH Inc. duty in 1966 but the date will be arranged with him 135 South LaSalle Street after appointment. Chicago, Illinois 60603 A statement giving information about the develop­ New York • Washington • Cleveland Chicago • Los Angeles ment of Bedford Park and setting out conditions of An equal opportunity employer appointment may be obtained from the Secretary, University of Adelaide at Bedford Park, Bedford Park, South Australia with whom applications in duplicate should be lodged not later than 28th february, 1966.

276 ·--.... THE UNIVERSITY OF AUCKLAND ----· New Zealand

The University invites applications for permanent positions in the Department of Mathematics. Present vacancies include a full professorship and at least one other post. Candidates may be of any nationality but PEER MEASURING should have a working knowledge of English. Applicants Occasionally, the best of us need to look through a window for a full professorship should have a substantial record at our peers-in order to judge ourselves. For the typical of publication, preferably in some branch of modern person, this is an extremely difficult task. An individual may analysis, topology or geometry. not have all the facts nor is he able to remain objective in The University is the largest in .New Zealand and is self evaluation. EDP is unique in this regard. We are able to expanding rapidly; total enrolment in 1965 was approxi­ "peer gaze" for you. Through our extensive field exposure mately 5,500. Teaching loads in mathematics allow time to the scientific and mathematical computer field, we will for research, and there is a well-established mathematical assist you in evaluating your progress and professional research library. standing against that of your peers. Peer gazing may lead you into another world of greater challenge and compensa­ Further particulars may be obtained from the Registrar tion. • Write today for our free scientific computer oppor­ or the Head of the Mathematics Department, Box 2175, tunities bulletin. For immediate consideration, please for­ Auckland, New Zealand. Applications should if possible ward a detailed experience resume. All inquiries are held be received by the Registrar by April 15th 1966. in strict confidence. Further posts are expected to become available shortly due to the expansion of the Department. Also some visiting and temporary appointments may be avail­ inc. able. Enquiries concerning these are invited at any time. ~dp p;;;~~';;l, ~ EXCLUSIVELY DATA PROCESSING J. A. KIRKNESS REGISTRAR 100 S. Wacker Drive, Chicago, Illinois 60606, (312) 782-0857 CLIENT COMPANIES ASSUME OUR CHARGES

PROGRAMMERS, SENIOR An intensive short course TEXACO INC. Theory and Techniques of Linear Programming SYSTEMS ANALYSTS OPERATIONS RESEARCH ANALYSTS March 14-25, 1966, San Francisco mM SYSTEM/34i0,-709411, Lecturers include George B. Dantzig and Alan Manne 7010, 1'10, 1401, 1440, 1620 Submit Resuines toR. C. PINKEI'lTON For brochure write: Marvin Chachere, University 135Eaat42St.,NewYork,N. Y. An EqUDl Opportunity EmpiDyer (M!F) of California Extension, 2223 Fulton St., Berkeley, California 94720.

INDEX TO l [)VERTISERS Academic Press Inc . . 265 The Johns Hopkins University, The The University of Adelaide . 276 Applied Physics Laboratory .... 273 Allyn and Bacon, Inc . . . . . 264 A. T. Kearney & Company, Inc ... 276 The University of Auckland 277 Lockheed Missiles & Space Company. 274 W. A. Benjamin, Inc ..... 266 McGraw-Hill Book Blaisdell Publishing Company . 268 Company ...... inside back cover Booz ·Allen Applied Research Inc 276 Mathematical Algorithms . . . 266 University of California Extension 277 Mechanical Enterprises, Inc . 270 Center for Naval Analyses . . . . 274 Charles E. Merrill Books, Inc. 270 The Chemical Rubber Company. 269 National Security Agency . . . 272 Cushing-Malloy, Inc ..... 270 Prentice-Hall, Inc ... outside back cover EDP Personnel, Inc...... 277 Springer-Verlag New York, Inc . 267 Holt, Rinehart and Winston, Inc . 266 Texaco, Inc ...... 277 Institute for Defense Analyses . . . 275 D. VanNostrand Company, Inc. 271

277 RESERVATION FORMS

UNIVERSITY OF CHICAGO NEW YORK MEETING CENTER FOR CONTINUING EDUCATION AT THE WALDORF-ASTORIA April 20-23. 196'6 April 4-7, 1966

The Center for Continuing Education will be the The Waldorf-Astoria is the official headquarters headquarters for the meeting. Room reservations hotel. Your sleeping room reservations should be should be sent to: sent directly to Front Office Manager, The Wal­ dorf-Astoria, 301 Park Avenue, New York, New Reservations York 10022. PLEASE TEAR OFF THE ATTACHED University of Chicago COUPON AND USE AS YOUR ROOM RESERVA­ Center for Continuing Education TION BLANK, Be sure to specify the n(lmes of 1307 East 60th Street persons for whom reservations are being made. Chicago, Illinois 60637 BE SURE TO BRING THE CONFIRMATION SLIP PLEASE TEAR OFF THE ATTACHED COUPON WITH YOU AS PROOF OF YOURRESERVATION. AND USE AS YOUR ROOM RESERVATION Unless otherwise requested, the hotel will hold BLANK. The Center will send you a confirmation. reservations only to 6:00 P.M. of the day of your BE SURE TO BRING THE CONFIRMATION SLIP arrival. Check-out time is 3:00 P.M. In order to WITH YOU AS PROOF OF YOUR RESERVATION. secure the special rates listed below, you must Unless otherwise requested, the Center will hold use the form below. THE DEADLINE FOR ALL reservations until 6:00 P.M. Reservations must RESERVATIONS IS MARCH 21. reach the Center by April 7, 1966. If reservation THE WALDORF-ASTORIA RATES,AMS MEETING is accompanied by one-half the room rate, reser­ April 4-7, 1966 vations will be accepted beyond the April 1 dead­ line Room and Bath for one per day $12.00 RATES Double Bed Room w-ith Bath for two per day 18,00 Twin Room with Bath for two per day 18.00 Single occupancy $10.00 Suites $35,00 and up - Double occupancy 14.00 Note: All rates subject to New York City 5 per cent hotel tax.

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Mail this form to: To: Front Office Manager Reservations The Waldorf-Astoria University of Chicago 301 Park A venue Center for Continuing Education New York, New York 10022 1307 East 60th Street Chicago, lllinois 60637 During the Meeting of the American Mathematical Society, please enter my reservation for: (Please Print) ___Single Room ___Twin Bed Reserve: Single(s) ___Double Bed Suite Double(s) A.M. I wlll arrive at Rate:$ P.M. (date) I wlll depart at A.M. (I) {We) plan to arrive on (date) P.M. and remain until Occupants: List names below and type of room requested, (Signature) 1. ------2. ______(Address) Send confirmation to: (City and State) Name ______Please print below additional names of persons for Institution ______whom only you are making reservations, Street Address------City and State.______

278 Outstanding Mathematics Texts MATHEMATICS OF PHYSICS AND MODERN ENGINEERING, Second Edition By I. S. SOKOLNIKOFF and R. M. REDHEFFER, both of University of California, Los Angeles. Available in April. .. Like it's predecessors, this edition provides a sound, comprehensive, and stimulating presentation of fundamental mathematics for engineers and applied scientists. Its scope is considerably broader than that of other engineering math texts, and much of the material included cannot be found in comparable books. ADVANCED ENGINEERING MATHEMATICS, Third Edition By C. R. WYLIE, Jr., University of Utah. Available in April. .. This highly successful text has been extensively rewritten to achieve greater clarity. This edition includes 40% more exercises than before, for a total of 1380 carefully graded problems. INTRODUCTION TO LINEAR ALGEBRA By ALLAN D. MARTIN, late of Carnegie Institute of Technology; and VICTOR J. MIZEL, Carnegie Institute of Technology. Off press ... This unusually clear presentation covers: (1) the algebraic and geometric fundamentals of real finite dimensional vector spaces; (2) the fundamentals of the theory of linear operators on finite dimensional vector spaces; and (3) the spectral theorem for symmetric operators. PLANE TRIGONOMETRY with Tables, Third Edition By GORDON FULLER, Texas Technological College. 350 pages, $6.50... Analytic trigo­ nometry is emphasized and brought into proper balance with numerical trigonometry. Major changes in this edition include the expansion of introductory material, the addition of an appendix on polar coordinates, and the use of second color for emphasis and clarification. MODERN ALGEBRA AND TRIGONOMETRY By J. VINCENT ROBISON, Oklahoma State University. 400 pages, $7.50 ... A modern integrated text, developing college algebra and trigonometry through the use of set theory. The concept of function as a special kind of relation provides a unifying thread that connects various types of functions. FINITE MATHEMATICS: With Applications in the Social and Management Sciences By LOUIS 0. KATTSOFF and ALBERT J. SIMONE, Boston College. 407 pages, $8.95 ... A mathematics text designed for the modern student in schools of business, departments of mathematics, or liberal arts colleges. Presented in the rigorous and precise language of the mathematician. McGraw-Hill Book Company 330 WEST 42ND STREET/ NEW YORK, N. Y. 10036 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE PAID AT P.O. Box 6248 PROVIDENCE, RHODE ISLAND Providence, Rhode Island 02904 AND ADDITIONAL MAILING OFFICES

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Pl'esenting Pl'entice-Hall's New and Fol'thcoming Mathematics Publications

MATHEMATICAL INTRODUCTION TO CELESTIAL MECHANICS by Harry Pollard, Purdue University. Here is a new look at a classical subject that can serve well as a basic text in departments of astronomy, mathematics, and aeronautics. Junior­ Senior graduate level. January 1966, approx. 128 pp., $4.95 ORDINARY DIFFERENTIAL EQUATIONS by I. G. Petrovski, Rector of Moscow State University. A self-contained introduction to the theory of ordinary differential equations which provides an exceptionally lucid develop­ ment of the basic general theory. (In the Selected Russian Publications in the Mathemati­ cal Sciences Series translated and edited by Richard A. Silverman.) May 1966, approx. 256 pp., $7.95

INTEGRAL, MEASURE AND DERIVATIVE: A Unified Approach by G. W. Shilov, Moscow State University and B. L. Gurevich, noted Russian author. Out­ standing features of this book include: Unified systematic exploitation of the Daniell ap­ proach, lucid expository style, an abundance of carefully selected problems, and an up-to­ date bibliography. (In the Selected Russian Publications in the Mathematical Sciences Series translated and edited by Richard A. Silverman.) April 1966, approx. 224 pp., $8.50

for approval copies, write: Box 903

PRENTICE-HALL, Englewood Cliffs, N. d. 07832 (Prices shown are for student use).