OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 14, NUMBER 2 ISSUE No. 96 FEBRUARY, 1967

OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by Everett Pitcher and Gordon L. Walker

CONTENTS

MEETINGS

Calendar of Meetings • • • • • • • • • • • • • • • • • • • • • . • • • • • . • • • • • • • • 204 Program of the February Meeting in New York ••••••••.•••••••••• 205 Abstracts for the Meeting - 241-249

PRELIMINARY ANNOUNCEMENTS OF MEETINGS. • • • • • . . • . • . • • . • • . • • • 208 ACTIVITIES OF OTHER ASSOCIATIONS. • • • • • • • . • • • • . • • • . . • . • • • • • • . 212 LETTERS TO THE EDITOR. • • • • • • • • • • • • • • • . • • • . • • . • • . . • • • • • • . • 213 SUMMER INSTITUTES FOR COLLEGE TEACHERS • • • • • • • • • • . • . • • • • • • . 214 NEWS ITEMS AND ANNOUNCEMENTS ••••••.••••••..•••••.•••• 215, 233 SUMMER INSTITUTES AND GRADUATE COURSES. . . • • . . • . • . . • • • . • • . • . 218

PERSONAL ITEMS • • • • • • • • • • • . • • • • • • • • • • • • • • • . • • . • • . • • • • • • • • 229 MEMORANDA TO MEMBERS Backlog of Mathematical Journals .••••.••••••••• , •••••••••••• 234

SUPPLEMENTARY PROGRAM-Number 44 •••••••••••••••••..••.••..• 235 ABSTRACTS OF CONTRIBUTED PAPERS. • • • • • • • • • • • . • . • • • • • • • • • • • • 241 RESERVATION FORMS...... 310 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 ctfoticeiJ 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•

644 April 5-8, 1967 New York, New York Feb. 20

645 April 14-15, 1967 Chicago, Illinois Feb. 20 646 April 22, 1967 San jose, California Feb. 20

647 june 17, 1967 Missoula, Montana May 4 August 28-September 1, 1967 Toronto, Canada (72nd Summer Meeting) january 23-27, 1968 San Francisco, California (74th Annual Meeting) August 26-30, 1968 Madison, Wisconsin (73rd Sum mer Meeting) january, 1969 New Orleans, Louisiana (75th Annual Meeting) August 25-29, 1969 Eugene, Oregon (74th Summer Meeting) January 22-26, 1970 Miami, Florida (76th Annual Meeting)

*The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next deadline dates for the by title abstracts are February 13, and April 27, 1967.

The ctfoticeiJ of the American Mathematical Society is published by the Society in January, February, April, June, August, October, November and December. Price per annual volume is $12.00. 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 mailing at the special rate of Postage provided for in section 34,40, paragraph (d).

Copyright © 1967 by the American Mathematical Society Printed in the United States of America

204 Six Hundred Forty-Third Meeting City College New York, New York February 25, 196 7

PROGRAM

The six hundred forty-third meeting Shepard Hall is two blocks east of of the American Mathematical Society will the !37th Street Station of the IRT subway be held at City College on Saturday, Febru­ (the Broadway-7th Avenue Line, not the ary 25, 1967, All sessions will be in 7th Avenue Line). Also it is one block west Shepard Hall. and five blocks south of the 145th Street By invitation of the Committee to Station of the IND subway (8th Avenue Select Hour Speakers for Eastern Sectional Line "A" train or 6th A venue Line "D" Meetings there will be an address by Pro­ train from mid-town New York). fessor Monroe D. Donsker of New York Buses marked "Broadway-230th University in Room 306 at 2:00 p.m. The Street" or "F art George" may be taken from title of his lecture is "Asymptotic evalua­ the !25th Street Station of the New Haven tion of function space integrals," or New York Central Railroad. Riders There will be sessions for contribu­ should get off at !38th Street and walk east ted papers at 10:00 a.m. and at 3:15p.m. one block to Convent Avenue. The registration desk will be located Persons who expect to travel by in Shepard Hall at Convent A venue and automobile may write to the Department of !39th Street. It will be open from 9:00 a.m. Mathematics, City College, New York, New till 3:30p.m. York 10031, before the meeting date, re­ Lunch will be available in a college questing a one-day campus parking permit cafeteria. This may be almost the only and directions to the parking area, place in the immediate vicinity for lunch.

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

SATURDAY, 10:00 A.M. Session on Analysis I, 306 Shepard Hall 10:00-10:10 ( 1) Geometry of Banach algebras Mr. I. N. Spatz, Polytechnic Institute of Brooklyn (643-15) 10:15-10:25 (2) On the continuity and measurability of certain transformation groups Professor R. E. Atalla, Ohio University (643-9)

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

205 10;30-10:40 (3) Trace class for an arbitrary H*-algebra Professor P. P. Saworotnow*, The Catholic University of America, and Professor j. C. Friedell, Loras College (643-17) 10:45-10:55 (4) Approximation theory on SU(2) with applications to Fourier analysis. Prelim­ inary report Mr. D. L. Ragozin, Harvard University (643-5) 11:00-11:10 ( 5) A functional calculus in Hilbert space based on operator valued analytic func­ tions Mr. P. A. Fuhrmann, Columbia University (643-19) 11:15-11:25 (6) Sums of deficiencies of entire functions Professor Albert Edrei, Syracuse University (643-23)

SATURDAY, 10:00 A.M.

Session on Logic, Algebra and Combinatorics, 315 Shepard Hall 10:00-10:10 (7) Simplified tree proofs in modal logic. Preliminary report Professor F. B. Fitch, Yale University (643-22) 10:15-10:25 (8) Some strengthenings of the Ulam nonmeasurability condition Professor Stanislaw Mr6wka, Pennsylvania State University (643-20) 10:30-10:40 ( 9) On semigroups embeddable in their endomorphism semigroup Professor L. M. Chawla* and Professor F. A. Smith, University of Florida (643-14) 10:45-10:55 ( 1 O) Inequalities in the formula of inclusion and exclusion for an additive set-function Professor H. J. Cohen, City University of New York, City College (643-18) 11:00-11:10 ( 11) Partially direct sum of inequality groups Professor Y. Kuo, The University of Tennessee (643-21) 11:15-11:25 (12) Some remarks on the Vander Waerden conjecture Dr. P. J. Eberlein, University of Rochester (643-3)

SATURDAY, 2:00 P.M.

Invited Address, 306 Shepard Hall Asymptotic evaluation of function space integrals Professor Monroe D. Donsker, New York University

SATURDAY, 3:15P.M.

Session on Analysis II, 306 Shepard Hall 3:15-3:25 ( 13) Periodic and almost periodic solutions of parabolic equations Professor Arnold Stokes, Georgetown UniversitY. and Mr. Carl Kallina*, Howard University (643-11) 3:30-3:40 ( 14) The first initial boundary value problem for quasilinear parabolic equations Dr. N. S. Trudinger, New York University (643-2)

206 3:45-3:55 ( 15) The range of the (n + 1) st moment for distributions on I o,l) Dr. Morris Skibinsky, Brookhaven National Laboratory, Upton, New York (643-12) 4:00-4:10 ( 16) Integration of operator-valued functions with respect to orthogonal volumes in Hilbert spaces Professor Witold Bogdanowicz and Professor john Welch*, The Catholic University of America (643-4) 4:15-4:25 ( 17) Integral representation of complete integral seminorms Professor Witold Bogdanowicz, The Catholic University of America (643-10) 4:30-4:40 ( 18) Derivatives of solutions of linear differential equations Mr. L. H. Haines, University of California, Berkeley (643-24) (Introduced by R. M •. Solovay)

SATURDAY, 3:15P.M.

Session on Topology, 315 Shepard Hall 3:15-3:25 ( 19) On monomorphisms in homotopy theory Professor Tudor Ganea, University of Washington (643-6) 3:30-3:40 (20) Metric dimension and equivalent metrics Professor J, H. Roberts* and Dr. F. G. Slaughter, Jr., Duke University ( 643 -16) 3:45-3:55 (21) Note on metric-dependent dimension functions Dr. R. E. Hodel, Duke University (643-13) 4:00-4:10 (22) Function spaces and the Aleksandrov-Urysohn conjecture Professor P. R. Meyer, City University of New York, Hunter College (643-8) 4:15-4:25 (23) The radical of topological abelian groups Professor K. C. Ha, University of South Florida (643-7) 4:30-4:40 (24) The proximal relation in coset transformation groups Professor H. B. Keynes, University of California, Santa Barbara (643-1) Herbert Federer Providence, Rhode Island Associate Secretary

207 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Six Hundred Forty-Fourth Meeting Americana of New York New York, New York April 5·8, 1967

The six hundred forty-fourth meet­ matical Society and the Society for Indus­ ing of the American Mathematical Society trial and Applied Mathematics, whose mem­ will be held at the Americana of New York bership at the time consisted of Professors in New York City on April 5-8, 1967. V. Bargmann, Garrett Birkhoff, G. E. For­ By invitation of the Committee to sythe, C. C. Lin, A. H. Taub (Chairman), Select Hour Speakers for Eastern Sectional and H. S. Wilf. The Joint Committee ap­ Meetings, Professor Harold Widom of Cor­ pointed the Organizing Committee consist­ nell University will give a lecture entitled ing of Professors Garrett Birkhoff (Chair­ "Eigenvalue asymptotics" on Friday, April man), R. E. Bellman, H. Grad, M. Krook, 7, at 2:00p.m., and Professor W. H. Tutte Dr. J. E. Moyal, and Professor T. W. of the University of Waterloo will address Mullikin. Financial support has been sup­ the Society on "The enumeration of planar plied by the Air Force Office of Scientific graphs" on Saturday, April 8, at 2:00 p.m. Research and the U. S. Army Research Contributed papers will be scheduled on the Office (Durham) under Contr.#44620-67-C- afternoon of Friday, April 7, and on Satur­ 0100. day, April 8. All sessions and invited ad­ Some of the speakers, with the titles dresses will be in Albert Hall. of their lectures, are as follows: M. Krook Abstracts of contributed papers and G. Rybicki, "Methods in radiative trans­ should be sent to the American Mathemati­ port theory"; C. Cercignani, "Boundary cal Society, Box 6Z48, Providence, Rhode value problems in linearized kinetic theory"; Island 02904, so as to arrive prior to the H. Grad, "Singularities of solutions of the deadline of February 20, 1967. Boltzmann equation"; E. Frieman; K. M. The Council of the Society will meet Case, "Linear transport problems"; T. W. at 5:00 p.m. on Friday, April 7. Mullikin, "Vector transport equations"; E. H. Bareiss; N. Corngold; R. Bellman, "Computational aspects of invariant em­ SYMPOSlUM ON TRANSPORT THEORY bedding and transport theory"; E. Gel bard, "The solution of discrete ordinate and Following is the present schedule spherical harmonics equations", M. Wing, of the symposium: "Mathematical methods suggested by trans­ Session A, April 5, 2:00 p.m. port theory"; G. I. Bell, "Stochastic for­ Kinetic theory and plasma transport mulations of neutron transport"; J. D. Session B, April 6, 9:30a.m. Moyal, "The theory of first passage multi­ Analytical neutron transport plicative processes." Session C, April 6, 2:00p.m. Numerical neutron transport (physical results) REGISTRATION Session D, April 7, 9:30 a.m. The registration desk will be in Stochastic aspects Albert Hall. It will be open from noon to The subject of the symposium was 5:00 p.m. on Wednesday, April 5, and from selected by the Joint Committee on Ap­ 9:00 a.m. to 5:00 p.m. on Thursday through plied Mathematics of the American Mathe- Saturday, April 6-8.

208 EXHIBITS of this issue of the cJioticeiJ. It will not appear again in the April issue. A joint book exhibit will be on dis­ play in Albert Hall from noon to 5:00 p.m. MAIL ADDRESS on Wednesday, April 5, and from 9:00a.m. to 5:00p.m. on Thursday through Saturday, Registrants at the meeting may re­ April 6-8. ceive mail addressed in care of the Ameri­ can Mathematical Society, Americana of ROOM RESERVATIONS New York, 7th Avenue and 52nd Street, New York, New York. Persons intending to stay at the Americana should make their own reserva­ Herbert Federer tions with the hotel. A reservation blank Associate Secretary and a listing of room rates are on page 310 Providence, Rhode Island

The Seventy Second Summer Meeting University of Toronto Toronto, Canada August 28 ·September 1, 1967

A full announcement of arrange­ The committee has also reserved 100 rooms ments for the Toronto meeting, including in private homes at Stratford for the nights information on dormitory accommodations, of August 25 and 26 and 50 rooms for the entertainment, and travel, will appear in night of August 2 7. Stratford, Ontario, is the April issue of.these ·cJiotiuiJ. However, approximately 90 miles southwest of Tor­ the Committee on Local Arrangements for onto, and is the site of the Shakespeare the Toronto meeting have made some special Festival. The program of the festival arrangements which necessitate this pre­ has not yet been announced, but the com­ announcement. mittee advises early reservation of these The Committee on Local Arrange­ rooms as accommodations may be difficult ments has reserved rooms at Loyola Col­ to obtain later in the season. lege in Montreal for 50 persons for each Those interested in making reserva­ of the following nights, August 25-27 and tions at either Loyola or Stratford should September 1-3. This will provide accom­ write as soon as possible to: Reservations, modations for mathematicians and their Department of Mathematics, University of families who want to visit the World's Toronto, Toronto 5, Ontario, Canada. Fair, either before or after the meeting. The rates will be $7 per person for a double Herbert Federer room and $8 per person for a single room. Associate Secretary Early reservation of these rooms is advised. Providence, Rhode Island

209 Six Hundred Forty-Fifth Meeting University of Chicago Center for Continuing Education Chicago, Illinois April 14-15, 1967

The six hundred forty-fifth meeting session has been arranged by Professor of the American Mathematical Society will Gerald R. MacLane of Purdue University. be held at the University of Chicago on The speakers will be joseph L. Doob, April 14-15, 1967. Registration and all ses­ Arthur j. Lohwater, john E. McMillan, sions will be held at the University of Makoto Ohtsuka, and Gail S. Young, Jr. Chicago Center for Continuing Education, Sessions for the presentation of con­ which is located at 1307 East 60th Street, tributed ten-minute papers will be held at Chicago, Illinois 60637. 3:15 p.m. on Friday, April 14, and at By invitation of the Committee to 9:00 a.m. and 3:15 p.m. on Saturday, April Select Hour Speakers for Midwestern Sec­ 15. The deadline for the receipt of abstracts tional Meetings, there will be three hour in the Society office is February 20. addresses. Professor Michio Suzuki of the Rooms will be available at the Center University of Illinois will speak on Friday, for Continuing Education at the rate of April 14, at 2:00p.m. Professor George j. $12 per single room and $8 per person in Minty of Indiana University will address a twin-bedded double. A form for requesting the Society on Saturday, April 15, at 11:00 accommodations may be found on page 310 a.m. Professor J. Frank Adams of the Uni­ of these cNotiaiJ. Those desiring accommo­ versity of Manchester will speak on Satur­ dations should complete the reservation day, April 15, at 2:00 p.m. Professor form or a reasonable facsimile thereof and Adams' talk will be entitled "Novikov's send it to the Reservations Department of work on complex cobordism." The titles the Center for Continuing Education. In the of the other two hour addresses will be event of an overflow, the Center will under­ announced later. All three lectures will be take to place people at nearby hotels. The held in the Assembly of the Conference hotel in question will in that case confirm Center. . the reservation. The official at the Center By invitation of the Committee to in charge of the meeting is Mrs. Lucy Ann Select Hour Speakers for Midwestern Sec­ Marx. tional Meetings, there will be a special session of 20-minute papers on Cluster Paul T. Bateman Sets on Friday, April 14, at 3:15 p.m. in Associate Secretary the Assembly of the Center. This special Urbana, Illinois

210 Six Hundred Forty-Sixth Meeting San Jose State College San Jose,California April 22, 1967

The six hundred forty-sixth meet­ Sainte Claire Hotel ing of the American Mathematical Society Market and San Carlos, San Jose will be held on Saturday, April 22, 1967, Single $12,50 at San Jose State College in San Jose, Double 13,00 California. San Jose Inn By invitation of the Committee to 1860 The Alameda, San Jose Select Hour Speakers for Far Western Single $10,00 Sectional Meetings, there will be an ad­ Double 13,50 dress by Professor Abraham Robinson of the University of California, Los Angeles, A list of nearby restaurants and their at 11:00 a.m. in Science 142. The title of locations will be available at the registra­ Professor Robinson's lecture is "Nonstan­ tion desk. dard arithmetic." The invited address will San Jose is served by Pacific Air­ be preceded by a brief welcome from Presi­ lines, Pacific Southwest Airlines, and by dent Robert D. Clark of San Jose State the San Francisco and Oakland Helicopter College. Service. Limousine service is available There will be sessions for contribu­ from the San Francisco Airport. There is ted papers during Saturday morning and frequent Greyhound Bus service from San afternoon. Registration for the meeting will Francisco, Persons driving to the meeting begin at 8:30 a.m. The Registration Desk from San Francisco should take the First will be located in the foyer of MacQuarrie Street exit when they reach San Jose, Hall. This building faces San Carlos Street. then proceed south on First Street ~o San MacQuarrie Hall is the second building Carlos where they can turn left and proceed west of the intersection of San Carlos to MacQuarrie Hall just beyond Seventh Street and Seventh Street. Maps of the San Street. Persons coming from the Berkeley Jose State College campus will be available area on the Nimitz Freeway are advised to at the Registration Desk. take the Thirteenth Street exit into San The following hotels and motels are Jose, and turn right onto San Carlos Street. near the campus of the college. Persons who There are several parking lots near the wish reservations should write to the hotel intersection of Seventh and San Carlos. or motel of their choice. These will be open free of charge to City Center Motel persons attending the meeting. 45 E. Reed Street, San Jose Coffee and doughnuts will be served Single or Double $8,00 in the registration area between 8:30 a.m. and 9:30 a.m. Hyatt House Hotel 1740 N. 1st Street, San Jose R. S, Pierce Single $12,00 $15,00 Associate Secretary Double 16,00 19,00 Seattle, Washington

211 ACTIVITIES OF OTHER ASSOCIATIONS

CANADIAN MATHEMATICAL CONGRESS ROYAL SOCIETY MEETING

A meeting of the Royal Society of to Professor G. F. D. Duff, Chairman--Re­ Canada will be held onJune4-7,1967,at the search Committee, Department of Mathe­ Carleton University, Ottawa. Papers will matics, University of Toronto, Toronto, be presented at the meeting by members of Ontario. the Canadian Mathematical Congress; titles The Council of the Congress will also of papers to be submitted should be sent hold a meeting at this time.

1967 SEMINAR AND CONGRESS The Biennial Seminar and Congress H. W. Wielandt, Universities of Tubin­ of the Canadian Mathematical Congress will gen and Wisconsin, Permutation groups. be held at the Glendon Hall Campus, York University, Toronto. Graduate Lectures The Seminar will be held on August 6-26, 1967. The Program Committee con­ M. Wonenburger, UniversityofBuffalo, sists of Professors P. Scherk, University The classical groups of Toronto; D. Russell, York University; H. Schwerdtfeger, McGill University, J. M. A. Maranda, University of Montreal; Topics in group theory. and H. Schwerdtfeger, McGill University. The program is not yet complete, but the L. LeBlanc, Universite de Montreal, following arrangements have been made: (Title not decided). A. Daigneault, Universite de Montreal, Research Lectures (Title not decided). A. W. Goldie, Leeds University, Non­ R. Ree, University of British Columbia, commutative noetherian rings Lie groups. J. Lehner, University of Maryland, The Biennial Meeting of the Congress Modular function theory in arithmetic and will be held from August 28 to September 2, analysis 1967. The Congress is organizing a joint P, Samuel, Universite de Sorbonne. program with the MAA for the first half Groupes de classes de diviseurs of this week.

SEMINAR ON COMPLEX ANALYSIS

The sixth session of the international speakers. Registrants may make applica­ Seminaire de Mathematiques Superieures tion for financial assistance to cover travel­ will be held from June 26 to July 28 under ing and living expenses. Requests for the sponsorship of the Canadian Mathemati­ registration forms and further information cal Congress and the Universite de Montreal. concerning the program should be addressed The subject of the seminar will be complex to Seminaire de Mathematiques Superieures, analysis. The program will consist of six Universite de Montreal, Case Postale 6128, main courses given by invited professors Montreal 3, Quebec. and a number of lectures given by guest

212 SESSION ON SYMBOLIC MATHEMATICS AT SPRING JOINT COMPUTER CONFERENCE

A session of the Special Interest puter aided symbolic mathematics," The Committee on Symbolic and Algebraic Mani­ speakers will describe their work at Project pulation of the ACM, will be held in conjunc­ MAC in the development of programming tion with the Spring joint Computer Con­ technology necessary for making the com­ ference in Atlantic City on April 21, 1967. puter an effective assistant in the solution The session will take place at 9:30-11:30 of nonnumerical mathematics. Computer a.m. in the Benjamin West Room (13th solution of problems will be demonstrated floor) of Chalfonte-Haddon Hall, head­ during the course of their presentation. quarters hotel for the conference, Further details may be obtained by The session will consist of a joint writing to the session chairman, Dr. james presentation by William A. Martin and H. Griesmer, Thomas J, Watson Research joel Moses of Project MAC, entitled "The Center, P, 0, Box 218, Yorktown Heights, mathematical laboratory--a study in com- New York 10598.

LETTERS TO THE EDITOR

Editor, the cJ/oticei) by indicating how it is related to previous work and to other mathematical topics, The letter by Irving Segal in the Published at the beginning of a paper such january c}/otiai) raises a number of issues an abstract would provide understandable that are of vital importance and should be information and assist in deciding whether further discussed. I am not sufficiently in­ to read intensively. With such abstracts formed to judge his specific comments on available, it should be a relatively easy Mathematical Reviews, but I agree that matter to prepare abstracts for prompt prompt publication of descriptive abstracts publication in a special journal, ordinarily would be very useful. As a virtually cost­ to appear before the paper itself. It would less step in this direction, I suggest that also be feasible to publish only the ab­ journals require authors to submit brief stracts of many papers whose complete abstracts with their manuscripts. These version might remain unpublished but abstracts should include a concise state­ available to any individual who wished it. ment of the main results and also place the work in its historical and logical context Kenneth 0, May

213 SUMMER INSTITUTES FOR COLLEGE TEACHERS

Leonard Gillman, Chairman, CUPM Panel on College Teacher Preparation Don Thomsen, Chairman, MAA Committee on Institutes Among various types of programs for 4. There have been several. institutes continuing the education of college mathe­ devoted to recent research with the objec- matics teachers, one which has natural tive of stimulating active interest in re­ logistic advantages is the summer institute. search and encouraging the development of Through its Committee on Institutes, the seminars and other scholarly activities at MAA has been directly involved in con­ the home institutions of the participants. ducting several institutes for experienced Other types of institutes would also college professors. Moreover, a study of serve important purposes. For instance, existing summer programs for college college teachers could be brought together teachers has been an item of high priority to collaborate in the preparation of instruc­ on the agenda of CUPM's newly formed tional materials or in the reorganization Panel on College Teacher Preparation. of courses and curricula to take into account This panel has conferred with a number recent trends in mathematics education. of persons who have directed two or more The range of possible programs such programs and is now planning further within each type of institute is wide, and study which may lead to some specific re­ any particular program may combine fea­ commendations for summer institute tures of several types. The purposes above courses. are all extremely worthwhile, and the Institutes held in recent years have panel is convinced that a greatly expanded pursued various objectives: effort in the direction of summer institutes 1. One kind of institute is directed of all types is urgently needed. In the pro­ at training teachers of precollege teachers grams for which we hava data, applications and is particularly concerned with curri­ have outnumbered acceptances by more than cular reforms and experiments in the teach­ six to one. Even if summer institutes are ing of precollege mathematics. to teach only ten percent of all college 2. Another provides fundamental in­ mathematics teachers each summer, the struction for college teachers not prepared number of institutes will need to be approxi­ for their current or forthcoming teaching mately tripled. assignments. At this kind of institute the Accordingly, the CUPM Panel on subject matter is often undergraduate math­ College Teacher Preparation and the Asso­ ematics presented from a higher standpoint. ciation's Committee on Institutes strongly 3. For teachers who have adequate urge that many more institutions submit general background but need special study proposals for summer institutes for college to introduce a new course, there have been teachers, that supporting agencies extend institutes devoted to specific topics in their support of such institutes to the level mathematics such as linear algebra, topol­ just described, and that directors of the ogy, probability and statistics, or applied institutes publicize their programs as widely mathematics. as possible to eligible college teachers.

214 NEWS ITEMS AND ANNOUNCEMENTS

1967 NSF NATIONAL REGISTER OF and Harold Widom. The proceedings of the SCIENTIFIC AND TECHNICAL conference will be published, PERSONNEL Support for the conference has been provided by a grant from the National Sci­ The 19 66 data on scientists who ence . Foundation. Though sufficient funds responded to the NSF National Register are not available to defray expenses of all of Scientific and Technical Personnel is participants, some limited support may be now available in "Salaries and Selected available. Characteristics of U. S, Scientists, 1966," All interested mathematicians are Reviews of Data on Science Resources, invited to attend. In order to be assured of No. 11. This bulletin includes current data hotel accommodations and transportation on mathematicians in the U, S, Charac­ from St. Louis, individuals should notify the teristics such as the median salary of director of plans to attend at an early date. mathematicians according to highest de­ Inquiries should be addressed to the direc­ grees, types of employers, work activities, tor of the conference, Professor Deborah median age, and academic rank are con­ Tepper Haimo, F acuity of Mathematical tained in the National Register report. The Studies, Southern Illinois University, geographic distribution of U, S, mathe­ Edwardsville, Illinois 62025. maticians is also presented. Other information concerning the relative position of mat~ematicians among all scientists is included in the report. For instance, 9o/o of all scientists reporting were mathematicians; among self-employed BOWDOIN SUMMER SEMINAR scientists, mathematicians reported the IN ALGEBRAIC GEOMETRY highest median salary ( $20,500); the youngest scientists reporting (median age Bowdoin College will conduct a third 34) were mathematicians and physicists. advanced summer seminar, with support Further details of these and other from the National Science Foundation, from data are reported in the National Register June 20 to August 10, 1967. For graduate bulletin. This bulletin is available from the students the central formal offering will be Mathematical and Statistical Sciences Sec­ a course in algebraic geometry based on tion of the National Register of Scientific lectures by Professor Arthur Mattuck of the and Technical Personnel, Box 6248, Provi­ Massachusetts Institute of Technology. The dence, Rhode Island 02904, lectures will be supported by work sessions. A research program for postdoctoral mathe­ maticians will focus on a colloquium at CONFERENCE AT which will appear a sequence of distin­ SOUTHERN ILLINOIS UNIVERSITY guished speakers including John Tate, David A conference on New Directions in Harrison, Gerard W ashnitzer, Steven Klei­ Orthogonal Expansions and their Continuous man, Jonathan Lubin, and David Mumford, Analogues will be held onApril27-29, 1967, Stipends are available for forty-eight in celebration of the dedication of the new graduate students and ten postdoctorals, and Edwardsville Campus of Southern Illinois a few small subsistence grants may be made University. A tentative schedule of speakers to mathematicians with their own principal included Richard Askey, F. M. Cholewinski, support. A brochure giving further infor­ Adriano Garsia, D. T. Haimo, Sigurdur mation may be obtained from Professor Dan Helgason, Edwin Hewitt, I. I, Hirschman, E. Christie, Chairman, Department of Math­ Jr., Samuel Karlin, Dan Rider, Victor ematics, Bowdoin College, Brunswick, Shapiro, Gabor Szego, David V. Widder, Maine 040 11 •

215 PANEL OF VOLUNTEERS FOR Benefit Life Insurance Company; J, P, CAREER INFORMATION Russell, Polytechnic Institute of Brooklyn; A. J, V. Sade, Pertuis, France; I. R. The American Mathematical Society Savage, Florida State University; Harvey and the many students who request career Sigal, International Business Machines information from the Society are deeply Corporation; L. W. Small, University of indebted to the volunteers who have, with Chicago; R. M. Smith, Huntingdon, Ten­ impressive care and thoughtfulness, en­ nessee; T. H. Southard, California State couraged students in mathematics by an­ College; R. A. Spong, General Dynamics; swering their letters. Nancy Tapper, Ithaca, New York; C. J, During 1966, these volunteers offered Thorne, Point Mugu, California; M. E, their services in answering studentletters: White, Stevens Institute of Technology; R. A. Alo, Carnegie Institute of Technology; J, W. Young, National Cash Register Com­ R. V. Andree, University of Oklahoma; pany. W. F. Atchison, Georgia Institute of Tech­ Every month, over 300 requests for nology; E, G. Begle, Stanford University; information on mathematics and careers B. H. Bissinger, Lebanon Valley College; in mathematics are received by the Ameri­ Margaret Butler; R. C. Carson, University can Mathematical Society. Many of these of Akron; Daniel Clock, Northern Michigan requests can be answered with a form letter College; L. J. Cohen, Applied Data Research, and pamphlets available for distribution Inc.; R, J, Cormier, Northern Illinois Uni­ by the Society. However, many others show versity; C. H. Cunkle, Clarkson College of a real interest in mathematics, and these Technology; J, M. Danskin, Center ofNaval are sent to the panel of volunteers to be Analyses of the Franklin Institute; R, C. answered. DiPrima, Rensselaer Polytechnic Institute; Anyone who would be willing to be a J, H. Engel, Center for Naval Analyses of part of this service is invited to send his the Franklin Institute; H. A. Gindler, Uni­ name, address, and field of interest to the versity of Pittsburgh; H, A. Gehman, Mathe­ Providence Office. matical Association of America; D. T, Haimo, Southern Illinois University; Frank­ lin Haimo, Washington University; R. G. Helsel, Ohio State University; Julius Hlavaty, College Board Commission; H. L. Hunzeker, University of Omaha; Julius Kane, University of Rhode Island; L. D. GRADUATE PROGRAM IN Kovach, George Pepperdine College; D. M, COMPUTER SCIENCE Krabill, Bowling Green State University; G. R, Kuhn, Northwestern Michigan College; The Department of Computer Sci­ J, B. Lane, University of South Carolina; ence of the University of Illinois has estab­ W. J, LeVeque, University of Michigan; lished a graduate program in Computer W. F. Lucas, Princeton University; M, E, Science leading to the degrees of Master Mahowald, Northwestern University; Ber­ of Science and Doctor of Philosophy in nard McGovern, Great Valley Laboratory; Computer Science. Research areas include, R. A. Melter, University of Massachusetts; but are not restricted to, digital computer W, F, Miller, Stanford University; Abraham arithmetic, switching and automata theory, Nemeth, Detroit, Michigan; L. W. Neustadt, circuit design, computer organization, com­ University of Southern California; Sam puter applications in physics, software Newman, National Aviation Facility Experi­ systems and languages, numerical analysis, mental Center; M. W. Oliphant, George­ pattern recognition, and information re­ town University; Otway Pardee, Syracuse trieval. University; George P iranian, University A limited number of assistantships of Michigan; L. E. Pursell, Grinnel Col­ and fellowships are available. Details of lege; Gordon Raisbeck, Arthur D, Little, this program may be obtained from Pro­ Inc.; S, M, Robinson, Union College; E. Y. fessor John R. Pasta, Head, Department Rodin, Wyle Laboratories; Alex Rosenberg, of Computer Science, University oflllinois, Cornell University; Paul Rotter, The Mutual Urbana, Illinois 61801.

216 GRADUATEPROGRAMrn tions to the mathematical theories of com­ BIOLOGICAL SYSTEMS ANALYSIS munications and information processing. John W. Milnor, Princeton University, re­ The State University of New York at ceived the award in the mathematical sci­ Buffalo is presently establishing a special ences for new approaches to topology. One program of graduate-level courses dealing of four awards given in the physical sciences with Biological Systems Analysis. This new was received by John H. van Vleck for his research discipline is concerned with the contributions to the development of the problems underlying the regulation, con­ theory of molecular structure and the theory trol, and organization of biological systems of the magnetic and dielectric properties of at all levels and includes investigations materials. into automata theory, information storage and handling in organized systems, adaptive systems, and optimality. EIGHTH ANNUAL SYMPOSIUM ON The program is sponsored jointly by SWITCHING AND AUTOMATA THEORY the Department of Mathematics and the De­ partment of Biophysics, with the cooperation The eighth annual symposium on of the Computing Center and the Center for Switching and Automata Theory, sponsored Theoretical Biology. At present, the pro­ by The University of Texas and the Switch­ gram comprises semester courses in neural ing and Automata Theory Committee of the nets, adaptive systems and artificial intel­ IEEE Computer Group, will be held at the ligence, regulation and control in biological University of Texas, Austin, Texas, on systems, biochemical automata, relational October 18, 19, and 20, 1967. Papers de­ biology, and special topics (reliability, scribing original research results in the self-reproduction, synthetic biological sys­ general. areas of automata theory, switching tems, etc.). It is envisagedthattheprogram theory, theory of computation, and theoreti­ will evolve into an independent and self­ cal aspects of logical design are being contained graduate program leading to the sought. Typical (but not exclusive) topics Ph.D. of interest include abstract languages~ Further information concerning this adaptive logic, asynchronous circuits, auto­ program may be obtained from Robert mata theory, computational complexity, Rosen, Department of Mathematics, State iterative circuits, minimization techniques, University of New York at Buffalo, Buffalo, reliability and fault diagnosis, sequential New York 14214. machines, theory of parallel computation, theory of programming languages, thresh­ old logic, and Turing machines. NATIONAL MEDAL OF SCIENCE Authors are requested to send six copies of detailed abstracts (no word limit) President Johnson announced the to Dr. Raymond Miller, IBM Thomas J. names of the eleven recipients of the Na­ Watson Research Center, P. 0. Box 218, tional Medal of Science in December 1966. Yorktown Heights, New York 10598, by Established in 19 59, the National Medal of May 1. Science is awarded by the President to Local arrangements are being han­ individuals who have made outstanding con­ dled by Professor C. L. Coates, Room 520, tributions to knowledge in the mathematical, Engineering Science Building, The Univer­ biological, physical, or engineering sci­ sity of Texas, Austin, Texas 78712. ences. The awards are made on the basis ~ of recommendations from the President's Errata committee on the National Medal of Science, headed by H. E. Carter of the University On Page 892 of the December 19 66 of Illinois. issue of the Notices (Volume 13, Number 8) Three members of the AMS received Professor Marvin Marcus was incorrectly awards this year. In the field of engineering, listed as Vice Chairman of the Department Claude E. Shannon, Massachusetts Institute of Mathematics at the University of Cali­ of Technology, received one of two awards fornia, Santa Barbara. Professor Marcus in the engineering sciences for contribu- is Chairman of the Department.

217 SUMMER INSTITUTES AND GRADUATE COURSES

The following is a list of graduate courses, seminars and institutes in mathematics being offered in the summer of 1967 for graduate students and college teachers of mathematics. The list was compiled from information received from grad1,1ate schools in the United States and Canada. Graduate Courses ALABAMA AUBURN UNIVERSITY, Auburn, Alabama 36830 june 12-August 22 Application deadline: May 24 Information: L, P. Burton, Head, Department of Mathematics MH610-Special Functions MH658-Algebraic Topology II MH628-Advanced Theory of Differential MH691-Directed Reading in Algebra Equations MH692-Directed Reading in Analysis MH634-Theory of Rings MH653-Dimension Theory

SAMFORD UNIVERSITY, Birmingham, Alabama 35209 Application deadline: May 15 Information: R, E. Wheeler, Department of Mathematics june 5-July 14 june 5-August 4 Ma 509-Modern Concepts of Mathematics Ma 508-Theory of Matrices for Secondary School Teachers ARIZONA NORTHERN ARIZONA UNIVERSITY, Flagstaff, Arizona 86003 Application deadline: june 12 Information: Dean of Graduate School, Department of Mathematics june 12-july 15 july 17-August 19 Introduction to Higher Algebra Introduction to Higher Algebra Theory of Functions of a Real Variable Analytic Geometry of Space Modern Mathematics for Teachers (Elementary) Six semester hours credit allowed in one term.

UNIVERSITY OF ARIZONA, Tucson, Arizona 85721 Application deadline: Prior to registration Information: Harvey Cohn, Head, Department of Mathematics or Graduate College, University of Arizona june 12-july 15 july 17-August 19 Math. 230-Matrix Analysis Math. 283-Elements of Complex Variables Math. 232-Linear Algebra Math. 231-lntroduction to Modern Algebra Math. 280a-lntroduction to Analysis CALIFORNIA CALIFORNIA STATE COLLEGE, Hayward, California 94542 june 20-September Application deadline: june 9 Information: john E. Weidlich, Acting Chairman, Department of Mathematics 4 Units: Math, 6331-Theory of Ordinary 3 Units: Math. 6960-Selected Topics in Differential Equations Graduate Point Set Topology 4 Units: Math. 6340-Introduction to Advanced Complex Analysis

218 STANFORD UNIVERSITY, Stanford, California 94305 June 26-August 19 Application deadline: June 1 Information: Department of Statistics Statistics 110-Statistical Methods in Engineer­ Statistics 21 7a,b-Introduction to Stochastic ing and the Physical Sciences Processes Statistics 116-Theory of Probability Statistics 300-Advanced Topics in Statistics Statistics 119-120-Statistical Inference Statistics 328-Topics on Rank and Nonparame­ tric Procedures Statistics 50 (Elementary Statistics) also offered during the Summer

UNIVERSITY OF CALIFORNIA, Riverside, California 92502 June 26-August 19 Application deadline: June 5 Information: Director of the Summer Session Math, S260 Seminar - Section 1, Continuous Section 2. Geometric Topology Functions

UNIVERSITY OF CALIFORNIA, Santa Barbara, California 93106 June 26-August 4 Application deadline: May 1 Information: Director, Summer Sessions Math, 260F -Seminar in Analysis Math, 260Q-Seminar in Algebra COLORADO COLORADO STATE COLLEGE, Greeley, Colorado 80631 June 12-August 18 Application deadline: May 15 Information: Forest N. Fisch, Chairman, Department of Mathematics Advanced Calculus Matrix Algebra Modern Algebra I and II Computer Mathematics Topology Theory of Rings Modern Geometry I and II History of Mathematics (Advanced) Linear Algebra Seminar in Mathematics The Department offers an Ed, D. in Mathematics Education and an Ed. D. in Mathematics

COLORADO STATE UNIVERSITY, Fort Collins, Colorado 80521 June 19-August 25 Application deadline: June 12 Information: M. L, Madison, Head, Department of Mathematics and Statistics Theory of Numbers Fourier Series and Boundary Value Problems Introduction to Complex Variables Matrices and Determinants Introduction to Modern Algebra Nonparametric Statistics Numerical Analysis Linear Algebra Vector Analysis FLORIDA

STETSON UNIVERSITY, DeLand, Florida 32720 June 12-August 4 Information: Richard B. Morland, Director of Graduate Studies Ms 405-406-Modern Algebra Ms 595-Seminar (Algebra)

219 GEIICIA UNIVERSITY OF GEORGIA, Athens, Georgia 30601 June 12-July 20 Application deadline: May 23 Information: B. J, Ball, Head, Department of Mathematics Elementary Set Theory Probability Theory Linear Algebra Seminar in Topology Advanced Calculus Seminar in Algebra ILLINOIS EASTERN ILLINOIS UNIVERSITY, Charleston, Illinois 61920 June 19-August 11 Application deadline: May 15 Information: Dean Lavern Hamand Advanced Calculus Digital Computer Techniques Statistics Modern Algebra Mathematics of Finance Topology History of Mathematics

ILLINOIS INSTITUTE OF TECHNOLOGY, Chicago, Illinois 60616 June 26-August 18 Application deadline: June l Information: Haim Reingold, Chairman, Department of Mathematics 507-Modern Algebra I 521-Topology 513-Real Variables I

ILLINOIS STATE UNIVERSITY, Normal, Illinois 61761 June 19-August 11 Application deadline: June 12 Information: C. T. McCormick, Department of Mathematics Math. 422-Topics in Geometry for Teachers Math, 480-Foundations of Mathematics Math. 455-Stochastic Processes

NORTHWESTERN UNIVERSITY, Evanston, Illinois 60201 June 27-August 19 Application deadline: May 19 Information: R. P. Boas, Chairman, Department of Mathematics Seminar in Algebra Seminar in Hydrodynamics Seminar in Analysis

SOUTHERN ILLINOIS UNIVERSITY, Edwardsville, Illinois 62025 June 19-August 11 Application deadline: May 15 Information: R.N. Pendergrass, Department of Mathematics Number Theory Topology

UNIVERSITY OF CHICAGO, Chicago,. Illinois 60637 June 19-September l Application deadline: No official deadline, but the applications should allow time for processing Information: Depa_rtment of Mathematics Theory of Functi"ons of a Complex Variable II Topics in Ring Theory Categorical Foundations of Set Theory Algebraic Numbers Interpolation Theory Selected Topics in Functor Theory There will be special seminars and visitors in the field of category theory

220 UNIVERSITY OF ILLINOIS, Urbana, Illinois 61801 June 19-August 12 Application deadline: May 20 Information: H. J. Miles, Department of Mathematics Courses for Advanced Undergraduate and Beginning Graduate Students Sets and Real Number System Complex Variables and Applications Topics in Geometry Introduction to Higher Analysis: Real Variables Selected Mathematical Topics for Secondary Introduction to Higher Analysis: Complex Teachers Variables Linear Transformations and Matrices Topics in Pure Mathematics Introduction to Higher Algebra I, II Elementary Theory of Numbers Introduction to Set Theory and Topology Advanced Statistics Advanced Calculus Introduction to Numerical Analysis Differential Equations and Orthogonal Mathematical Methods in Engineering and Functions Science Purely Graduate Courses Second Course in Abstract Algebra I, II Theory of Functions of a Complex Variable Advanced Topics in Abstract Algebra Real Analysis, I, II Algebraic Geometry Partial Differential Equations Elementary Geometry from a Modern View­ Hilbert Spaces point Mathematical Methods of Physics General Topology INDIANA BALL STATE UNIVERSITY, Muncie, Indiana 47306 Application deadline: June 1 Information: Earl H. McKinney, Head, Department of Mathematics June 12-July 14 July 17-August 17 Modern Algebra I Modern Algebra II Foundations of Algebra Theory of Numbers Non-Euclidean Geometry Higher Geometry Numerical Analysis History of Mathematics Advanced Calculus I Advanced Calculus II A maximum of nine quarter hours may be carried in each summer session

INDIANA UNIVERSITY, Bloomington, Indiana 47401 June 21-August 10 Application deadline: April 15 Information: S. G. Ghurye, Chairman, Department of Mathematics M404-Introduction to Modern Algebra II M511-Real Variables I M413-Introduction to Analysis I M514-Complex Variables II M439-Non-Euclidean Geometry IOWA IOWA STATE UNIVERSITY, Ames, Iowa 50012 Application deadline: Spring Information: Department of Statistics June 6-July 15 July 17-August 25 Non-parametric Inference Stochastic Processes Experimental Design Survey Design

221 KANSAS KANSAS STATE TEACHERS COLLEGE, Emporia, Kansas 66801 Application deadline: June 1 Information: Marion P. Emerson, Department of Mathematics June 5-August 4 August 5-August 25 Probability and Statistics Numerical Analysis Mathematical Programming for Projective Geometry Topology the Computer Non-Euclidean Geometry Group Theory Matrix Theory First Course in Abstract Real Variable Number Theory Algebra Advanced Calculus I

UNIVERSITY OF KANSAS, Lawrence, Kansas 66044 June 8-August 5 Application deadline: June 1 Information: G. Baley Price, Department of Mathematics Vector Analysis and Complex Variables Lie Algebras Applied Partial Differential Equations Summability Theory Programming for a Digital Computer MASSACHUSETTS

UNIVERSITY OF MASSACHUSETTS, Amherst, Massachusetts 01002 Application deadline: May 24 Information: Esayas G. Kundert, Mathematics Department

June 13-July 21 July 25-September 700-Topics in Advanced Mathematics 700-Topics in Advanced Mathematics 811-Advanced Algebra 812-Advanced Algebra

MICHIGAN EASTERN MICHIGAN UNIVERSITY, Ypsilanti, Michigan 48197 June 25-August 4 Application deadline: June 25 Information: Dean of the Graduate School 515-Elements of Set Theory 561-Modern Mathematics Content Sr. HS 531-Modern Mathematics Content Jr. HS 583-Modern Mathematical Methods Jr. HS 542-Non-Euclidean Geometry 586-Teaching of Modern HS Mathematics 544-Introduction to General Topology 500-Modern Mathematics Content K-6 547-Geometry for High School Teachers 501-Modern Mathematics K-8

WESTERN MICHIGAN UNIVERSITY, Kalamazoo, Michigan 49001 June 19-August 11 Application deadline: June 1 Information: School of Graduate Studies Programming for Computers, 506, 3 cr. Introduction to Analysis, I, 570, 3 cr. Introduction to Topology, 520, 3 cr. Abstract Algebra, I, 630, 3 cr. Linear Algebra, I, 530, 3 cr. Real Analysis, I, 670, 3 cr. MISSOURI UNIVERSITY OF MISSOURI, Columbia, Missouri 65201 June 12-August 4 Application deadline: June 1 Information: Director of Admissions Differential Equations Survey of Mathematics Applied Mathematics I Complex Function Theory Advanced Calculus I, II Foundations of Geometry Matrix Theory Dimension Theory Theory of Numbers

222 NEBRASKA CREIGHTON UNIVERSITY, Omaha, Nebraska 68131 June 12-August 4 Application deadline: May 1 Information: D .. J. H. Fuller, Acting Chairman, Department of Mathematics Modern Mathematics for High School Linear Algebra Teachers Complex Variables Numerical Analysis Ordinary Differential Equations NEW YORK FORDHAM UNIVERSITY, Bronx, New York 10458 July 3-August 11 Application deadline: June 30 Information: Paul Levack, Dean of Summer Session Fundamental Concepts of Mathematics Combinatorial Analysis Point Set Topology Conformal Mapping Introduction to Number Theory Abelian Groups Mathematical Logic

POLYTECHNIC INSTITUTE OF BROOKLYN, Brooklyn, New York 11201 Application deadline: June 1 Informati~n: Registrar

June 19-JulyE ~ugust ?-September 14 Advanced Calculus I Advanced Calculus I Real Variables I Real Variables I Elements of Complex Variables Modern Algebra II Modern Algebra I Linear Algebra II Linear Algebra I Numerical Analysis II Numerical Analysis I Partial Differential Equations of Math. Partial Differential Equations of Math. Physics II Physics I Linear Algebra and Differential Equations

SYRACUSE UNIVERSITY, Syracuse, New York 13210 Information: Summer Sessions

June 26-August 4 ~ugust ?-September 6 Intermediate Seminar Intermediate Seminar Functions of a Real Variable Modern Algebra (continuation) Functions of a Complex Variables Fundamentals of Analysis (continuation) Fundamentals of Analysis Modern Algebra The more advanced courses at the Master's degree level NORTH CAROLINA UNIVERSITY OF NORTH CAROLINA, Chapel Hill, North Carolina 27514 Information: Graduate Office, South Building June 9-July 15 July 17-August 2.4 Elementary Differential Equations Advanced Calculus Linear Algebra Introduction to Modern Algebra (9 wk. term) Elementary General Topology (9 wk. term) Topics in Analysis Elementary Theory of Number I Topics in Analytic Function Theory

223 OHIO BOWLING GREEN STATE UNIVERSITY, Bowling Green, Ohio 43402 Application deadlines: June 19 and July 24 Information: Director of Admissions June 19-Ju!y 21 July 24-August 25 Modern Geometry Advanced Calculus Modern Algebra Foundations of Mathematics Intermediate Analysis Intermediate Analysis Topology

MIAMI UNIVERSITY, Oxford, Ohio 45056 Information: S. E. Bohn, Chairman, Department of Mathematics April 24-June 14 June 20-August 9 Linear Algebra Linear Algebra Modern Algebra Real Analysis Complex Variables Geometry Probability and Statistics Modern Algebra Probability and Statistics II Probability and Statistics Topology Topics in Algebra Topics in Analysis

PENNSYLVANIA PENNSYLVANIA STATE UNIVERSITY, University Park, Pennsylvania !6802 June IS-September 3 Information: P. C. Hammer, Head, Computer Science Department CMPSC 401-Principles of Programming with CMPSC 490-Special Topics in Computer Science Physical Science Applications CMPSC 540-Information Processing Systems CMPSC 402-Principles of Programming with CMPSC 591-Special Topics in Computer Science Social and Biological Science Appli­ CMPSC 592-Special Topics in Foundations cations CMPSC 593-Special Topics in Languages and CMPSC 403-Principles of Programming with Systems Business Applications CMPSC 594-Special Topics in Nonnumerical CMPSC 410-Introduction to Computer Systems and Self-improving Systems CMPSC 454-Matrix Computation CMPSC 600-Research Problems in Computer Science

UNIVERSITY OF .PITTSBURGH, Pittsburgh, Pennsylvania 15213 Application dealines: April I and May 15 Information: Graduate Students Advisor, Department of Mathematics April_26- June 15 June 22-August 12 Math. 222-Analysis III Math. 220-Analysis I Math. 238-Functional Analysis Math. 239-Functional Analysis Math. 270-Topology I Math. 250-A!gebra I Math. 277-Laplace Transforms Math. 271-Topology II

WEST CHESTER STATE COLLEGE, West Chester, Pennsylvania 19380 June 26-August 4 Application deadline: June 15 Information: Director, Graduate Program Complex Variables Theory of Numbers Modern Geometry

224 PUERTO RICO UNIVERSITY OF PUERTO RICO, Rio Piedras, Puerto Rico 00931 June 10-July 25 Application deadline: March 15 Information: Francisco Garriga, Chairman, Department of Mathematics Advanced Calculus Differential Geometry Analytic Theory of Numbers RHODE ISLAND UNIVERSITY OF RHODE ISLAND, Kingston, Rhode Island 02881 Application deadline: Up to 4:00 p.m. on the first day of classes Information: Dean of the Summer Sessions June 19-July 25 July 17-28 Math. 291-Special Problems: "Stability by Math. 282S (Special two week Math. course) Lyapunov' s Second Method" "Difference Differential Equations" Math. 292-Special Problems: "Distribution Theory" TEXAS STEPHEN F. AUSTIN STATE COLLEGE, Nacogdoches, Texas 75961 Application deadline: See Graduate "Bulletin Information: Dean of the Graduate School June 5-July 14 July 17-August 25 411 Modern Algebra 412 Modern Algebra 509 Higher Geometry 510 Higher Geometry

TEXAS TECHNOLOGICAL COLLEGE, Lubbock, Texas 79409 Application deadline: June 1 Information: Dean of Graduate School or Head of Mathematics Department June 6-July 15 July 16-August 26 Topology Matrix Theory Mathematical Statistics I Mathematical Statistics II Advanced Mathematical Statistics I Advanced Mathematical Statistics II Numerical Analysis Distribution Theory II Distribution Theory I Approximation Theory II Approximation Theory I Intermediate Analysis II Intermediate Analysis I Methods of Applied Mathematics II Methods of Applied Mathematics I WASHINGTON UNIVERSITY OF WASHINGTON, Seattle, Washington 98105 June 19-July 19 and Application deadline: May 15 July 20-August 18 Information: R. S. Pierce, Department of Mathematics Invariant theory--old and new (Graduate lectures by Professor Jean Dieudonne)

WASHINGTON STATE UNIVERSITY, Pullman, Washington 99163 June 19-August 11 Application deadline: June 1 Information: D. Bushaw, Acting Chairman, Department of Mathematics· 551-Topological Groups 582-Advanced Topics in Matrix Theory WISCONSIN MARQUETTE UNIVERSITY, Milwaukee, Wisconsin 53233 Application deadline: May 10 Information: L. W. Friedrich, Dean, Graduate School Differential Equations Statistical Inference and Design of Experi­ Advanced Calculus ments Mathematical Statistics Topics in Abstract Analysis Abstract Algebra

225 UNIVERSITY OF WISCONSIN, Madison, Wisconsin 53706 June 20-August 12 Information: Registrar Computational Linguistics Artificial Intelligence Automata Theory

UNIVERSITY OF WISCONSIN, Milwaukee, Wisconsin 53201 June 19-August 12 Application deadline: May 1 Information: Robert Norris, Department of Mathematics Advanced Undergraduate Courses (may be taken for graduate credit) Modern Algebra I Projective Geometry Modern Algebra II Advanced Calculus I Graduate Courses Topics in Applied Mathematics Advanced Topics in Algebra Partial Differential Equations Rings and Fields CANADA DALHOUSIE UNIVERSITY, Halifax, Nova Scotia, Canada July 3-August 14 Application deadline: June 15 Information: Executive Secretary, Canadian Mathematical Congress, 985 Sherbrooke St. W, Montreal Tensor Analysis Dr. R. Blum, University of Saskatchewan Number Theory Dr. H. Abbott, Memorial University These courses are given under the supervision of the Canadian Mathematical Congress, with financial assistance from The Atlantic Provinces Inter-University Committee on the Sciences

UNIVERSITY OF BRITISH COLUMBIA, Vancouver 8, B. C. Canada June 26-August 18 Application deadline: April 15 Information: C. A. Swanson, Department of Mathematics 1. Permutation Representations of Finite Groups 2. Fourier and Hilbert Transformations 3. Asymptotic Methods Travel grants and tuition are paid to qualified applicants

Summer Institutes

CALIFORNIA UNIVERSITY OF CALIFORNIA, Santa Barbara, California 93106 PEPPERDINE COLLEGE, Los Angeles, Cali- LINEAR ALGEBRA CONFERENCE fornia 90044 Dates: July 31-August 24 SEMINAR IN INTERMEDIATE ANALYSIS Sponsoring Agency: National Science Foundation Dates: June 26-August 4 Subjects Covered: Topics in linear and multi- Subjects Covered: Most of the content of "Inter­ linear algebra mediate Analysis" by Olmsted Requirements for Admission: This is the third Requirements for Admission: Upper division year of a 3 -year course and participants are standing in mathematics enrolled on a continuing basis Application Deadline: June 26 Information: Marvin Marcus, Chairman, Depart­ Information: John S. tv1oore ment of Mathematics

226 COLORADO Information: Harry Pollard, Recitation Bldg. COLORADO STATE UNIVERSITY, Fort Collins, Colorado 80521 SYMPOSIUM ON TOPOLOGICAL DYNAMICS MAINE Dates: August 7-August 11 Sponsoring Agency: Air Force Office of Scienti­ BOWDOIN COLLEGE, Brunswick, Maine 04011 fic Research ADVANCED SEMINAR IN ALGEBRAIC GEOME- Other Information: Program will consist of a TRY (for Graduate and Postdoctoral Students) number of hour and half hour invited ad­ Dates: June 20-August 10 dresses. Provision will be made for contrib­ Sponsoring Agency: National Science Foundation uted papers, of duration at least twenty Subjects Covered: Course of lectures in algebraic minutes geometry. Research Colloquium in algebraic Information: Abstracts of contributed papers geometry and related topics. Miscellaneous should be submitted to joseph Auslander, seminars. Department of Mathematics, University of Requirements for Admission: Endorsement by Maryland, College Park, Maryland 20740 be­ graduate department fore May 15, 1967 Application Deadline: March 8 Other Information: Some stipends are available UNIVERSITY OF COLORADO, Boulder, Colo­ at various levels rado Information: Dan E. Christie, Department of ARMU ROCKY MOUNTAIN REGIONAL SUMMER Mathematics MATHEMATICS SYMPOSIUM Dates: June 19-August 11 Sponsoring Agency: Associated Rocky Mountain Universities, Inc. MICHIGAN Subjects Covered: Focus will be on ordinary differential equations MICHIGAN STATE UNIVERSITY, East Lansing, Requirements for Admission: Predoctoral (either Michigan 48823 graduate students or college teachers), post­ APPLIED MATHEMATICS AND MECHANICS FOR doctoral and senior faculty COLLEGE TEACHERS OF ENGINEERING Application Deadline: February 25 MATHEMATICS AND PHYSICS Other Information: Travel grants and stipends Dates: june 26-August 18 available, some being reserved for students Sponsoring Agency: Sponsored by the Department and faculty members at colleges and univer­ of Metallurgy, Mechanics and Materials Sci­ sities in the Rocky Mountain states ence of Michigan State University with the sup­ Information: Robert McKelvey, ARMUC, 68 South port of the National Science Foundation Main Street, Salt Lake City, Utah 84101 Subjects Covered: Vector analysis, differential equations, tensors, matrices, variational GEORGIA methods, and integral-transform methods Other Information: Prof·~ssor Ian N. Sneddon of UNIVERSITY OF GEORGIA, Athens, Georgia Glasgow, Scotland will be guest lecturer 30601 Information: Robert Wm. Little, Director INSTITUTE FOR COLLEGE TEACHERS OF MATHEMATICS Dates: June 12-August 18 Sponsoring Agency: National Science Foundation NEW JERSEY Subjects Covered: Elementary real .analysis, PRINCETON UNIVERSITY, Princeton, New Jer­ linear algebra, multivariable calculus sey 08540 Requirements for Admission: Equivalent of bache­ SIXTH SYMPOSIUM ON MATHEMATICAL PRO­ lor's or master's degree in mathematics; pre­ GRAMMING sently teaching mathematics at the college Dates: August 14-August 18 level Subjects Covered: Theory and applications of the Application Deadline: February 15 mathematics of constrained optimization, in­ Other Information: Late applications may be con­ cluding linear programming and its exten­ sidered sions to convex programming, general non­ Information: B. ]. Ball, Head, Department of linear programming, integer programming, Mathematics dynamic programming and network flow INDIANA Other Information: An international symposium which is a continuation of a series of re­ PURDUE UNIVERSITY, Lafayette, Indiana 4 7907 search conferences begun in 1949 at Chicago STABILITY IN CELESTIAL MECHANICS and last held in London in 1964 Dates: June 19-july 14 Information: Mathematical Programming Sympo­ Sponsoring Agency: National Science Foundation sium, Fine Hall, P. 0. Box 708

227 NEW YORK Information: Information on administration to Gordon L. Walker, Executive Director. Mathe­ STATE UNIVERSITY OF NEW YORK AT STONY maticians wishing to participate should write BROOK, Stony Brook, New York 11790 to A. Robinson, Department of Mathematics, WORKSHOP IN COMPUTING SCIENCES University of California at Los Angeles, Cali­ Dates: june 5- june 8 fornia 900Z4 Subjects Covered: Graduate Programs and rela­ ted research programs in computing sciences Information: S, Broder, Computing Center BROWN UNIVERSITY, Providence, Rhode Island PENNSYLVANIA OZ91Z PENNSYLVANIA STATE UNIVERSITY, Univer- SOLID MECHANICS SEMINAR sity Park, Pennsylvania 1680Z Dates: june 8-june 17 MATHEMATICAL SYSTEMS THEORY Sponsoring Agency: National Science Foundation Dates: September 11-September 15 Subjects Covered: Continuum Mechanics Subjects Covered: Fundamental systems theory, Requirements for Admission: To be a doctoral computer linguistics, extended topology candidate in final years of study, or recent Information: P, C. Hammer, 4Z6 McAllister Ph. D. employed in academic, governmental Bldg, or industrial concern Other Information: Request forms by March 1, PUERTO RICO Tuition fee for industrial participants only $ZOO,OO, Academic personnel can request UNIVERSITY OF PUERTO RICO, Rio Piedras, travel and subsistence funds Puerto Rico 00931 Information: jay B. Weidler, Assistant Seminar HOMOLOGICAL ALGEBRA Director, Division of Engineering Dates: june 10-july Z5 Requirements for Admission: Graduate standing Application Deadline: March 15 Information: Francisco Garriga, Box ZZ15Z TENNESSEE RHODE ISLAND VANDERBILT UNIVERSITY,Nashville, Tennessee 37203 AMERICAN MATHEMATICAL SOCIETY, P. 0, SUMMER INSTITUTE FOR COLLEGE TEACHERS Box 6Z48, Providence, Rhode Island OZ904 OF MATHEMATICS MATHEMATICS OF THE DECISION SCIENCES :es: june 5- july 14 Dates: july 10-August 10 Sponsoring Agency: National Science Foundation Sponsoring Agency: AFOSR, ARO(Durham), AEC, Subjects Covered: Linear algebra and matrix NSF, ONR, Small Business Agency, National theory, elementary topology Institute of Health Requirements for Admission: Any person teach­ Subjects Covered: Operations research, mathe­ ing mathematics at a four year college or matical economics, mathematical psychology, junior college may apply computer science Application Deadline: February 15 Requirements for Admission: Suitable graduate Other Information: Preference will be given to work, interest in the fields under discussion, teachers who lack the Ph, D. degree and/or recommendation from professor are relatively inexperienced in linear algebra Application Deadline: March Z7 and topology Information: Gordon L. Walker, Executive Di­ Information: james R. Wesson, Box 1595 rector

AMERICAN MATHEMATICAL SOCIETY and ASSOCIATION FOR SYMBOLIC LOGIC, P. 0, CANADA Box 6Z48, Providence, Rhode Island OZ904 AXIOMATIC SET THEORY UNIVERSITE DE MONTREAL, Montreal 3, P .Q., Dates: july 1 0-August 5 Canada Sponsoring Agency: National Science Foundation SEMINAIRE DE MA THEMA TIQUES SU- Subject Covered: Axiomatic set theory P ERIEURES (Sixth Session) - Requirements for Admission: Must be a qualified Dates: june 26-July 29 mathematician interested and active in the Sponsoring Agency: Universite de Montreal et field under study Societe Mathematique du Canada Additional Information: A few graduate students Subjects Covered: Analyse complexe and younger mathematicians will be invited Application Deadline: April 1 upon the recommendation of a senior pro­ Information: Seminaire de mathematiques su­ fessor perieures, Universite de Montreal

228 PERSONAL ITEMS

Professor TAKEO AKASAKI of Rutgers, Mr. RAYMOND BALBES of the Univer­ The State University, has been appointed sity of California, Los Angeles, has been to an assistant professorship at the Univer­ appointed to an assistant professorship at sity of California, Irvine. the University of Missouri, St. Louis Dr. YASUO AKIZUKI of Tokyo Univer­ Campus. sity of Education, Japan, has been appointed Professor E. J. BARBEAU of the Uni­ to a visiting professorship at the University versity of Western Ontario is in residence of Massachusetts. as a Postdoctoral Research Fellow at Yale Professor Emeritus B. E. ALLEN of University. Wayne State University has been appointed Professor F. W. BARBER of Wayne to a professorship at the Detroit Institute State University has been appointed to an of Technology. assistant professorship at Florida Atlantic Dr. R. A. ALO of Pennsylvania State University. University has been appointed to an assis­ Mr. R. W. BARBIERI of the Wolf Re­ tant professorship at Carnegie Institute of search and Development Corporation, Col­ Technology. lege Park, Maryland, has accepted a posi­ Dr. W. A. AL-SALAM of the University tion as a Mathematician with the National of Calgary has been appointed to an asso­ Aeronautical and Space Administration, ciate professorship at the University of Greenbelt, Maryland. Alberta. Professor R. E. BARNHILL, on leave Professor D. E. AMOS of the Univer­ from the University of Utah, has been ap­ sity of Missouri has accepted a position as pointed to a visiting assistant professorship a Staff Member with the Sandia Corpora­ in the Division of Applied Mathematics at tion, Albuquerque, New Mexico. Brown University. Professor J. M. ANDERSON of Iowa Dr. J. Y. BARRY of the Institute for State University has been appointed to a Defense Analyses, Arlington, Virginia, has visiting professorship at the University of returned after a leave of absence at the Puerto Rico, Mayaguez. University of Lancaster, England. Dr. LARRY ARMIJO of the National Dr. J. C. BECKER of Princeton Uni­ Engineering Science Company, Houston, versity has been appointed to an assistant Texas, has accepted a position as a Staff professorship at the University of Massa­ Analyst with Thompson Ramo Woolridge chusetts. Systems, Houston, Texas. Dr. STOUGHTON BELL~ fueSan~a Dr. W. W. ARMSTRONG of the Uni­ Corporation, Albuquerque, New Mexico, has versity of British Columbia has accepted been appointed Associate Professor and a position as a Member of the Technical Director of the Computer Centeratthe Uni­ Staff at Bell Telephone Laboratories, versity of New Mexico. Holmdel, New Jersey. Dr. ADI BEN-ISRAEL of the University Professor E. F. ASSMUS, Jr., of Wes­ of Illinois, at Chicago Circlt? has been ap­ leyan University has been appointed to an pointed Associate Professor of the Engi­ associate professorship at Lehigh Univer­ neering Sciences Department at North­ sity. western University. Professor W. F. ATCHINSON of the Professor H. L. BENTLEY of Rens­ Georgia Institute of Technology has been selaer Polytechnic Institute has been ap­ appointed Professor and Director of the pointed to an assistant professorship at the Computer Science Center at the University University of New Mexico. of Maryland. Professor MICHAEL BERNKOPF of Dr. A. H. BAARTMANS of Michigan Fairleigh Dickinson University has been State University has been appointed to an appointed to an associate professorship at assistant professorship at Bucknell Uni­ Pace College. versity. Dr. BARRY BERNSTEIN of Purdue

229 University has been appointed to a pro­ Hong Kong has been appointed a Senior fessorship at the Illinois Institute of Tech­ Lecturer at the University of Auckland, nology. New Zealand. Professor MANUEL BERRI of Tulane Professor J. CHIDAMBARASWAMY of University has been appointed to a visiting the University of Kansas has been appointed associate professorship at the Louisiana to an associate professorship at the Uni­ State University in New Orleans. versity of Toledo. Dr. H. J. BIESTERFELDT, JR., of the Dr. STEPHEN CHING of Philadelphia, University of Wisconsin,Milwaukee Campus, Pennsylvania, has accepted a position as has been appointed to a visiting assistant a Senior Engineer at the Research Center professorship at the University of Massa­ of the Burroughs Corporation, Philadelphia, chusetts. Pennsylvania. Dr. GAVIN BJORK of Seattle Univer­ Dr. TAE HO CHOE of the University of sity has been appointed to an assistant Florida has been appointed to an assistant professorship at Portland State College. professorship at the University of Massa­ Professor J. H. BLAU, on leave from chusetts. Antioch College, has been awarded a Na­ Dr. S. C. CHU of Bellcomm Incorpo­ tional Science Foundation Faculty Fellow­ rated, Washington, D.C., has been appointed ship and is at Stanford University for the to an assistant professorship at the Uni­ academic year 1966-1967. versity of Delaware. Mr. T. W. BOATMAN of Stephen F. Dr. J. R. CLAY of Washington, D. C., Austin State College has accepted a posi­ has been appointed to an assistant profes­ tion as an Aerosystems Engineer with sorship at the University of Arizona. General Dynamics, Ft. Worth, Texas. Dr. G. E. COLLINS of the International Dr. J. T. BORREGO, JR., of the Uni­ Business Machines Corporation, Yorktown versity of Florida has been appointed to an Heights, New York, has been appointed to assistant professorship at the University an associate professorship in the Computer of Massachusetts. Scien~es Department at the University of Professor JOHN BRACE of the Univer­ Wisconsin. sity of Maryland is on sabbatical leave at Dr. R. W. COTTLE, on leave from the Cambridge University, England, for the Bell Telephone Laboratories, Holmdel, New academic year 1966-1967. Jersey, has been appointed to an acting Professor C. A. BRYAN of the Arizona assistant professorship in the Department State University has been appointed to an of Industrial Engineering at Stanford Uni­ assistant professorship at the University of versity. Montana. Professor R. C. COURTER of the Professor T. A. BUR TON of the Uni­ University of Windsor has been appointed versity of Alberta has been appointed to an to an associate professorship at Wayne associate professorship at the Southern State University. Illinois University. Dr. A. S. COVER of the University of Mr. M. J. CAMBERN of Santiago, Chile, Arizona has been appointed to an assistant has been appointed a Lecturer at the Uni­ professorship at Clemson University. versity of California, Santa Barbara. Professor A. Z. CZARNECKI of the Dr. BUCHANAN CARGAL of the Lear Illinois Institute of Technology has been Siegler Corporation, Santa Monica, Cali­ appointed to an assistant professorship at fornia, has been appointed to a professor­ the Illinois Teachers College, Chicago­ ship at Prescott College. North. Professor W. M. CAUSEY of the Uni­ Dr. E. B. DAVIS of Stanford Univer­ versity of Kansas has been appointed to an sity has been appointed to an assistant assistant professorship at Mississippi State professorship at Lawrence University. University. Professor M. H. De GROOT of Car­ Dr. NATHANIEL CHAFFEE of Brown negie Institute of Technology has been ap­ University has been appointed to an assis­ pointed Professor of Mathematical Statis­ tant professorship at the University of tics and Head of the Department of Statis­ Michigan, Ann Arbor. tics. Dr. C. P. CHANG of the University of Professor C. N. DeSILVA of the

230 University of Minnesota has been appointed Stanford University for the academic year Professor and Chairman of the Department 1966-1967. of Engineering Mechanics at Wayne State Professor K. K. GOROWARA of the Uni­ University. versity of Montana has been appointed to an Dr. J. E. DUEMMEL of the University associate professorship at Ohio State Uni­ of Montana has been appointed to an asso­ versity and Miami University. ciate professorship at the Western Wash­ Professor L. D. GOULD of St. Augus­ ington State College. tine's College has been appointed to an Mr. V. H. DYSON of Miles College has associate professorship at Shaw University. been appointed to an assistant professor­ Professor J, W. GRAY of the Univer­ ship at the University of Illinois at Chicago sity of Illino!s has been awarded a National Circle. Science Foundation Senior Postdoctoral Mr. J. M. ERDMAN of Idaho State Uni­ Fellowship and is at the Forschungsinstitut versity has been appointed to an assistant fur Mathematik, Eidegenl:lssische Tech­ professorship at Portland State College. nische Hochschule, Ziirich, Switzerland, Professor R. A. ESTES of the Univer­ for 1966-1967. sity of Michigan has been appointed to an Mr. M.D. GREEN of the University of associate professorship at Gorham State Cincinnati has been appointed to an assis­ College. tant professorship at George Washington Professor H. M. FARKAS of Kansas University. State University has been appointed to an Professor R. J, GRIEGO of the Uni­ associate professorship at Indiana State versity of California, Riverside, has been University, Terre Haute. appointed to an assistant professorship at Professor HENRYK FAST of the Uni­ the University of New Mexico. versity of Notre Dame has been appointed Professor J, R. HATTEMER of Prince­ to an associate professorship at Wayne ton University has been appointed to an State University. assistant professorship at Southern Illinois Dr. M. L. FAULKNER of the Univer­ University, Edwardsville Campus. sity of Alberta has been appointed to an Dr. AKIO HATTORI of the University assistant professorship at Western Wash­ of Tokyo has been appointed a Research ington State College. Associate at Yale University for the aca­ Professor W. T. FISHBACK of Ohio demic year 1966-1967. University has been appointed to a pro­ Professor LOUISE HAY, on leave from fessorship at Earlham College. Mount Holyoke College, has a Postdoctoral Dr. J. R. FOOTE, formerly Director Fellowship at the Massachusetts Institute at the Holloman Graduate Center and Pro­ of Technology. fessor of Mathematics at the University of Professor R, B. HORA of Purdue Uni­ New Mexico, has been appointed Professor versity, Indianapolis Center, has been ap­ of Applied Mathematics at the University of pointed to an assistant professorship at Missouri, Rolla Campus. the University of Oklahoma. Professor HENRY FRANDSEN of Clark Professor T. C. HU ofthelnternational University has been appointed to an asso­ Business Machines Corporation, Yorktown ciate professorship at Wheaton College. Heights, New York, has been appointed to an Professor K. M. GARG of the Univer­ associate professorship at the University of sity of Calgary has been appainted to an Wisconsin. assistant professorship atthe University of Dr, V. S. HUZURBAZAR, Professor Alberta, and Head of the Department ofMathematics Professor JACOB GOLDHABER of the and Statistics, at the University of Poona, University of Maryland has received a India, has been elected President of the National Science Foundation Science Faculty Statistics Section of the Indian Science Fellowship and is at the University of Lon­ Congress Association for the year 1966- don, England, for the academic year 1966- 1967, 1967. Professor M. F. JANOWITZ of the Professor R. A. GOOD, on leave from University of New Mexico has been ap­ the University of Maryland, is on the School pointed to an associate professorship at Mathematics Study Group writing team at Western Michigan University.

231 Dr. B. A. JENSEN of Wesleyan Uni­ Professor J. E. L. PECK of the Univer­ versity has been appointed to an associate sity of Calgary is on leave of absence for professorship at Portland State College. one year at the Mathematisch Centrum, Dr. W. J. JONSSON of The University, Amsterdam, Netherlands. Birmingham, England, has been appointed Dr. V. V. RAO of the University of to an assistant professorship at McGill Hawaii has been appointed to an associate University. professorship at the University of Calgary. Dr. S. K. KA UL of the University of Dr. E. E. REED of the University of Saskatchewan has been appointed to an Colorado has been appointed to an assistant assistant professorship at the University professorship at the University of Massa­ of Calgary. chusetts. Professor H. B. KEYNES of Wesleyan Professor D. G. RIDER of the Massa­ University has been appointed to an assis­ chusetts Institute of Technology has been tant professorship at the University of Cali­ appointed to an assistant professorship at fornia, Santa Barbara. Yale University. Professor W. E. KIRWAN of the Uni­ Professor D. L. ROD of the University versity of Maryland is on leave at Hollo­ of Wisconsin has been appointed to an way College of the University of London, assistant professorship at the University England, as a visiting Lecturer for the of Calgary. academic year 1966-1967. Professor G. B. SELIGMAN of Yale Dr. V. LAKSHMIKANTHAM of the Uni­ University will be on leave and will reside versity of Calgary has been appointed in New Haven. Professor and Chairman of the Department Dr. L. H. SESHU of the University of of Mathematics at the University of Rhode Illinois has been appointed to an associate Island. professorship at Portland State College. Dr. PETER LANCASTER of the Cali­ Professor MOSHE SHIMRAT of the Uni­ fornia Institute of Technology has been ap­ versity of Calgary is on leave for one year pointed to an associate professorship at the at York University, Toronto. University of Calgary. Dr. J. L. SICKS of the University of Professor G. J. MALTESE of the Uni­ Pennsylvania has been appointed to an versity of Maryland is on leave as a visiting assistant professorship at the University Associate Professor at the University of of Massachusetts. Frankfurt, Germany, for the academic year Professor D. A. SPRECHER of Syra­ 1966-1967. cuse University has been appointed to an Professor G. D. MOSTOW of Yale Uni­ associate professorship at the University versity is on leave at the Institut des of California, Santa Barbara. Hautes Etudes Scientifiques, Paris, France, Professor AARON STRAUSS of the for the first term and will be at the Hebrew University of Maryland has been awarded University, Jerusalem, Israel, for the sec­ a National Science Foundation Postdoctoral ond term. Fellowship and is at the University of Dr. M. N. L. NARASIMHAN of the Uni­ Florence, Italy, for the academic year versity of Calgary has been appointed to a 1966-1967. professorship at Oregon State University. Dr. JIN CHEN SU of the Institute for Dr. N. NOB USA W A of the University of Advanced Study has been appointed to an Calgary has been appointed to an associate associate professorship at the University professorship at the University of Rhode of Massachusetts. Island. Professor TAKA YUKI TAMURA, on Professor D. S. PASSMAN of the Uni­ leave from the University of California, versity of California, Los Angeles, has Davis, is in Japan to give lectures and been appointed to an assistant professor­ seminars through June 1967. ship at Yale University. Mr. A. I. THALER of the Johns Hop­ Dr. G. P. PATIL, Professor of Mathe­ kins University has been appointed a Lec­ matical Statistics at the Pennsylvania State turer at the University of Maryland. University, has been elected an ordinary Professor R. J. VENTI of Albuquerque, member of the International Statistical In­ New Mexico, has been appointed to an stitute. assistant professorship at the University of

232 California, Santa Barbara. of Illinois: J. W. GRAY; University of Mary­ Mrs. G. B. WAGNER of The City land: JOSEPH AUSLANDER, LEON GREEN­ University of New York, Hunter College, BERG, C. R. KARP, R YSZARD SYSKI; has been appointed a Lecturer at the Uni­ Ohio State University: RANKO BOJANIC; versity of Maryland. Portland State College: R. L. STANLEY. Dr. D. R. WESTBROOK of the Courant Institute of Mathematical Sciences, New To Associate Professor: University of York University, has been appointed to an California, Santa Barbara: T. K. BOEHME; assistant professorship at the University of University of Maryland: G. J. MALTESE; Calgary. Portland State College: F. S. CATER; Dr. G. W. WHAPLES of the University Western Washington State College: ROBIN of Pennsylvania and Notre Dame University CHANEY,N.R.GRAY,D. F. SANDERSON. has been appointed to a professorship at the University of Massachusetts. To Assistant Professor: University Dr. FREDERICK YOUNG of Western of Massachusetts: M. K. BENNETT. Washington State College has been appointed to a professorship at Oregon State Univer­ To Associate Research Mathematician: sity. University of Michigan: ERGUN AR. Professor J. M. ZELMANOWITZ of the University of Wisconsin has been appointed To Instructor: Columbia University: to an assistant professorship at the Univer­ PETER BLUM; University of Maryland: sity of California, Santa Barbara. E. A. TIMSANS; University of Pennsylvania: F. J. FLANIGAN; Polytechnic Institute of PROMOTIONS Brooklyn: GERALD BIENMAN; Yale Uni­ versity: J. F. McCLENDON, M. E. SHAUCK, To Professor: University of California, S. J. SIDNEY. Santa Barbara: G. J. CULLER; University

NEWS ITEM

POINT SET TOPOLOGY CONFERENCE

A point set topology conference will connected continua. be held on the campus of Arizona State Uni­ Several hour lectures will be given versity in Tempe, Arizona, March 21-25, by invited participants. There will also be 1967. The conference will be particularly other sessions including shorter talks. concerned with ( l) metrization and unifor­ Inquiries concerning the conference mities, (2) classification of spaces with should be addressed to E. E. Grace or structures similar to uniformities, (3) con­ R. W. Heath at Arizona State University, tinuous curves, and (4) strongly nonlocally Department of Mathematics.

233 MEMORANDATOMEMBERS 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}fo!iai), with the kind co­ Editorial Department of the American Mathemati­ operation of the respective editorial boards, cal Society in the case of the Society's journals) It is important that the reader should in­ and based on these factors; manuscripts accepted, terpret the data with full allowance for the wide manuscripts received and under consideration, and sometimes meaningless fluctuations which are manuscripts in galley, and rate of publication. characteristic of them. Waiting times in particu­ There is no fixed formula. lar are affected by many transient effects, which Column 5. The first quartile ( Q1) 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 of location. 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}fo!iai). 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 issue 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 I 3 4 5 Observed waiting ! Est. time for paper time in latest No. Approx. No.I BACKLOG submitted published issue JOURNAL currently issues pages per 12/30/66 6/30/66 to be published (in months) per year year I year year (in months) Q1 Med. Q3 American J. Math. 4 1000 810 NR* 10-12 14 16 18 Annals of Math. 6 1200 1100 NR* 12 9 10 14 Annals of Math. Stat. 6 1900 0 0 5-8 6 8 9 Arch. Hist. Exact Sciences not fixed 450 0 0 5 8 *** Arch. Rational Mech. Anal. not fixed 1350 0 0 4 5 7 Canad. J. Math. 6 1350 500 1050 14 I·~·11 15 17 Duke Math. J. 4 750-800 304 326 12-18 11 12 14 Illinois J. Math. 4 720 510 300 17 17 17 18 J. Math. Analyses and Appl. 12 2300 1800 1200 9 ** ** ** J. Math. and Mech. 12 1400 970 700 8 10 12 12 J. Math. and Phys. 4 NR* NR* NR* NR* 15 17 18 J. Mathematical Physics 12 NR* NR* 0 NR* 9 10 12 J. Symbolic Logic 4 650# 300 NR* 11 16 18 22 Math. Comp 4 700 0 0 8 6 7 9 Michigan Math. J. 4 NR* NR* 130 NR* 9 10 12 Pacific J. Math. 12 2250 NR* NR* 15 16 17 20 Proceedings of AMS 6 1152 340 0 13 9 10 14 Quarterly of Appl. Math. 4 400 NR* 100 NR* 9 10 12 SIAM J. on Appl. Math. 6 1500 360 340 8-10 10 11 12 SIAM J. on Control 4 750 0 40 6 6 8 11 SIAM J. on Numer. Anal. 4 700 130 110 6-9 9 11 12 SIAM Review 4 700 50 0 6-9 8 9 10 Transactions of the AMS 12 1650 400 0 14 10 13 15

* NR means that no response was received to a request for information ** Dates of receipt of manuscripts not indicated in this journal *** The latest issue of this journal consisted of two articles # Contributed papers; 150 pages of reviews 234 SUPPLEMENTARY PROGRAM-Number 44

During the interval from November 26 to January 5, 1967, 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 Contributed Papers in this issue of these cJioticeiJ. One abstract presented by title may be accepted per person per issue of these cNoticeiJ. 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 new example of a nonmeasurable set Professor C. R. Borges, University Professor Alexander Abian, The of California, Davis (67T-124) Ohio State University (67T-139) (11) Commuting functions with no common ( 2) Permutations without rising or falling fixed point w-sequence Mr. W. M. Boyce, NASA Manned Mr. Morton Abramson and Profes­ Spacecraft Center and Tulane Uni­ sor Vf .O,J. Moser, McGill Univer­ versity of Louisiana (67T-218) sity (67T-232) (12) WITHDRAWN (3) Three theorems on summable set functions Professor W,D.L, Appling, North Texas State University (67T-159) ( 13) On certain finite-dimensional divisi- (4) On the interval of convergence of ble semigroups Picard's iteration Professor D. R. Brown and Pro­ Mr. P. B. Bailey, Sandia Corpora­ fessor Michael Friedberg, Univer­ tion, Albuquerque, New Mexico sity of Tennessee (67T-146) (67T-245) ( 14) On locally finite semigroups ( 5) Simultaneous extensions from dis- Professor T. C. Brown, Simon crete subspaces. Preliminary report Fraser University (67T-182) Professor HowardBanilower, Loui­ ( 15) The solution of a functional equation siana State University, Baton Rouge Professor R. C. Buck, University (67T-253) of Wisconsin (67T-213) (6) Extending decompositions of E 3 to ( 16) Distribution free tests for symmetry make them "nice". Preliminary re­ based on the number of positive sums. port Preliminary report Professor R. J. Bean, Universityof Mr. D. L. Burdick, University of Tennessee (67T-161) New Mexico (67T-l49) (7) Nonparametric tests for several hy­ ( 17) On certain isomorphisms between potheses. Preliminary report maximal nondetermining subalgebras Professor C. B. Bell, Case Institute of group algebras of Technology (67T-223) Mr. S. H. Butt, Tulane University (8) On the geometry of subspaces in (67T-160) Euclidean n-spaces ( 18) Perfect hypercyclic rings Professor Adi Ben-Israel, North­ Professor W. H. Caldwell, Univer­ western Uniyersity (67T-152) sity of South Carolina (67T-196) ( 9) On the coefficients of the cyclotomic ( 19) Sets with proportional brightness and polynomials. Preliminary report girth Professor D. M. Bloom, Brooklyn Professor G. D. Chakerian, Univer­ College (67T-233) sity of California, Davis (67T-l32) ( 10) An extension of Dugundji' s extension (20) Existence of segments in a topological theorem space with betweeness

235 Professor J, M. Cibulskis, Illinois p < 1, Preliminary report Teachers College, Chicago-North Professor Peter Duren, University (67T-185) of Michigan (67T-167) (21) The amalgamation property for clas­ ( 32) Groups occuring as automorphism ses of Cylindric algebras. Prelimin­ groups of semigroups ary report Professor P. L. Dussere, Idaho Mr. S. D. Comer, University of State University, and Professor D. Colorado (67T-251) W. Miller, University of Nebraska (22) On some function spaces related to (67T-215) quasi-homogeneous Bessel potentials ( 33) Dendritic extensions of clans Professor Mischa Cotler and Pro­ Professor Carl Eberhart, Univer­ fessor Cora Sadosky, University of sity of Kentucky (67T-174) the Republic of Montevideo, Uru­ (34) Estimation of the PSI-function for guay (67T-235) primes in arithmetic progression (23) Neighborhoods of polyhedra in 3- Dr. M. L. Faulkner, Western Wash­ manifolds ington State College (67T-255) Mr. Robert Craggs, The Institute (35) New formulae for the calculation of a for Advanced Study (67T-229) definite integral occuring in the statis­ (24) On the category oflattices with resid­ tical treatment of white noise uated mappings Mr. H. E. Pettis, Aerospace Re­ Mr. G. D. Crown, Western Michigan search Laboratories, Wright-Pat• University (67T-175) terson Air Force Base, Ohio (67T- (Introduced by Professor M. F. Janowitz) 133) (25) On surfaces of limit type ( 36) Algebraic geography: the set of struc­ Professor Ubiratan D'Ambrosio, ture constants for associative alge­ University of Rhode Island (67T~ bras. Preliminary report 169) Dr. F. J, Flanigan, University of (26) On absolutely independent group Pennsylvania (67T-191) axioms {37) Zeros of polynomials in a half plane. Professor D. F. Dawson, North Preliminary report Texas State University (67T-178) Professo! J, S. Frame, Michigan (27) Velocity and natural families in a State University (67T-138) Riemannian space V • ( 38) Irreducible continua and a paper of Professor John DeCicco, Illinois Thomas Institute of Technology, and Profes­ Mr. J. B. Fugate, University of sor R. V. Anderson, Illinois Teach­ Kentucky (67T-244) ers College, Chicago-South (67T- ( 39) Extensions of uniformly continuous 212) pseudometrics (28) Applications of the Hahn-Banach Mr. T. E. Gantner, University of Theorem in approximation theory Dayton (67T-134) Professor F. R. Deutsch and Pro­ (40) Isomorphisms of multiplier algebras fessor P. H. Maserick, The Penn­ Mr. G. I. Gaudry, Institut Henri sylvania State University (67T-164) Poincare, Paris (67T-242) (29) Algebraic models for measure pre- (41) Sequences of p-points of meromorphic serving transformations functions Professor Nicolae Dinculeanu, Mr. Paul Gauthier, Wayne State Uni­ Queen's University and Professor versity (67T-l42) C. Foias, University of Budapest, (Introduced by Professor W. Seidel) Romania (67T-194) (42) Nonstandard logic ( 30) Growth of entire functions and prob­ Dr. J, R. Geiser, Dartmouth Col­ lem of singular support of distribu­ lege (67T-136) tions (43) Spectral multiplicity of self-adjoint Dr. M.A. Dostal, New York Univer­ dilations sity ( 6 7T- 1 55) Professor R. C. Gilbert, California (Introduced by Professor Leon Ehrenpreis) State College atFullerton(67T-254) ( 31) Linear functionals on Hp spaces with (44) Contracted ideals with respect to

236 integral extensions. Preliminary re­ University of Kansas (67T-2.01) port {57) Curvature and characteristic classes Professor R.W. Gilmer, Jr., Florida of compact Riemannian manifolds State University (67T-183) Professor C. C. Hsiung, Lehigh (45) Rings of formal power series over a University, and Mr. Y. K. Cheung, Krull domain. Preliminary report Drexel Institute of Technology ( 67T- Professor R. W. Gilmer and Mr. 2.36) W. J. Heinzer, Florida State Uni­ {58) Conformal transformations of a com- versity (67T-2.07) pact Riemannian manifold (46) A note on the derivatives of functions Professor C. C. Hsiung and Mr. positive in a half-plane J. D. Liu, Lehigh University (67T- Professor J. L. Goldberg and Pro­ 2.09) fessor J. L. Ullman, The University (59) Isometries of compact submanifolds of Michigan (67T-2.16) on a Riemannian manifold ( 4 7) A Riesz representation theorem in Professor C. C. Hsiung and Mr. the setting of locally convex spaces B. H. Rhodes, Lehigh University Professor R. K. Goodrich, Univer­ (67T-192.) sity of Colorado (67T-l 72.) . ( 60) Two counterexamples to a conjecture (48) The p-primary components of the on commuting continuous functions of homotopy groups of spheres for p odd the closed unit interval. Dr. Brayton Gray, University of Mr. J. P. Huneke, Wesleyan Uni­ Illinois, Chicago (67T-2.10) versity (67T-2.31) (49) Hyper-irreducibility in an orthomodu- ( 61) Ramsey cardinals and the general lar lattice continuum hypothesis Professor R. J. Greechie, Univer­ Mr. R. B. Jensen, Universitat sity of Massachusetts, Boston (67T- Bonn, West Germany (67T-130) 168) {Introduced by Professor G. E. Hasen­ (50) Paths in graphs. Preliminary report jager) Mr. W. Gustin and Professor Sey­ (62.) A three cardinal theorem for w-logic mour Sherman, Indiana University Professor H. J. Keisler, University (67T-176) of Wisconsin ( 67T-140) (51) Laws of sines and cosines on a (63) Well-founded extensions of models of bounded domain in en with a constant set theory sectional curvature. Preliminary re­ Professor H. J. Keisler and Mr. port J. H. Silver, University of Wiscon­ Professor K. T. Hahn, Pennsylvania sin (67T-141) State University (67T-170) ( 64) Strongly representable functions. II (52.) Finite time stability of sets Professor C. F. Kent, Case Institute Professor T. G. Hallam and Pro­ of Technology (67T-12.9) fessor V. Komkov, Florida State (65) A note on generalized inverses for University (67T-2.2.5) matrices (53) A generalized Hausdorff summability Mr. J. B. Kim, Michigan State Uni­ method. Preliminary report versity (67T-2.06) Professor C. E. Harrell, Bowling (66) kth-order automata Green State University (67T-186) Mr. R. A. Knoebel, New Mexico (54) Tensor sequences of Abelian groups State University (67T-2.40) Professor T. J. Head, Universityof (67) Improved constructions for the Bose­ Alaska (67T-157) Nelson sorting problem. Preliminary (55) On compact divisible abelian semi- report groups Professor D. E. Knuth, California Professor J. A. Hildebrant, Louisi­ Institute of Technology, and Pro­ ana State University, Baton Rouge fessor R. W. Floyd, Carnegie Insti­ (67T-166) tute of Technology (67T-2.2.8) (56) P seudo-metrizability of quotient (68) The hyperplane sections through a spaces normal point of an algebraic variety Professor Charles Himmelberg, Professor Wei-eihn Kuan, Michigan

237 State University ( 67T-165) (82) Completion of a symmetrical unitary (69) Some remarks on uniform distribution matrix in compact topological groups Mr. R. F. Mathis, Ohio State Uni­ Professor Lawrence Kuipers, versity (67T-145) Southern Illinois University ( 67T- (83) Essential and strictly essential near- 187) ring modules (70) Finite images of polycyclic groups Professor C. J. Maxson, State Uni­ Dr. Joan Landman, Columbia Uni­ versity College at Fredonia, New versity (67T-222) York (67T-199) (71) The Brown-McCoyradicalsofahemi­ (84) Monte! subalgebras of the plane. Pre­ ring liminary report Professor D. R. LaTorre, Univer­ Mr. William Meyers, Tulane Uni­ sity of Tennessee (67T-163) versity (67T-158) (72) Some structure theorems of locally (85) On the resolvent of a linear operator compact groups. Preliminary report associated with a well-posed Cauchy Mr. D. H. Lee, Tulane University problem (67T-219) Dr. John Miller, University of ( 73) Consistency of the generalized Sous­ Massachusetts, Boston (67T-153) lin hypothesis. Preliminary report (86) Subspaces of continuous function Miss Elinor Lerner, University of spaces Rochester (67T-171) Professor P. D. Morris, Pennsyl­ ( 74) A complete axiomatization for weak vania State University (67T-135) second-order logic (87) Graph topologies for function spaces. Professor E. G. K. Lopez-Escobar, II University of Maryland (67T-128) Professor S. A. Naimpally and Mr. (75) Topological spaces which admitcom­ C. M. P areek, University of Alberta, plex-valued unisolvent systems Canada (67T-239) Professor J. A. Lutts, University (88) Retracts of compactly generated lat­ of Massachusetts, Boston (67T-137) tices ( 76) On spaces with certain hereditary Professor T. G. Newman, Southern properties Methodist University (67T-224) Professor Byron McCandless, Kent (89) Spectral interpolation in Lp spaces State University (67T-226) Professor K. K. Oberai, Queen's ( 77) Topological collapsing and piercing University (67T-193) points (90) Two splitting theorems for metare­ Professor D. E. McMillan, Jr., cursively enumerable sets University of Wisconsin (67T-188) Professor J. C. Owings, University (78) Rotations of the image domains of of Maryland ( 67T-1 77) analytic functions (91) Intersections of elementary substruc- Mr. T. H. MacGregor, Lafayette tures. I College (67T-217) Mr. D. M. R. Park, Programming ( 79) Polynomial bases for compact sets Research Group, Oxford, England in the plane. Preliminary report (67T-122) Dr. Victor Manjarrez, Catholic Uni­ (92) Homogeneous subgroups of 2Q versity of America (67T-241) Mr. J. T. Parr, University of Illin­ (80) Local behavior ofGaussianprocesses ois (67T-252) with stationary increments (83) Some remarks on the vector sub- Dr. M. B. Marcus, The RAND Cor­ spaces of a finite field poration, Santa Monica, California Professor R. L. Pele and Professor (67T-247) W. J. Leahey, University of Hawaii (81) Explicit form for the Fourier-Plan­ (67T-131) cherel transform over locally com­ ( 94) The integral in topological spaces. pact abelian groups Preliminary report Professor P. Masani, Catholic Uni­ Professor W. F.Pfeffer, University versity of America (67T-189) of California, Davis (67T-184)

238 (95) An answer to a question of Rudin Professor N. H. Schlomiuk, McGill Professor Mary Powderly, Fairfield University (67T-148) University (67T-143) (110) Multiple complementation in the lat­ (96) On decompositions and homotopy tice of topologies Professor T. M. Price, University Mr. P. S. Schnare, Louisiana State of Iowa (67T-197) University, New Orleans (67T-243) (97) Hausdorff m'omentproblem for opera­ ( 111) Summability by Dirichlet convolutions tors in locally convex spaces Mr. S. L. Segal, University of Professor M. S. Ramanujan, Uni­ Rochester ( 67T-246) versity of Michigan (67T-147) (112) Convergence of successive approxi­ (98) Automorphisms of Banach algebras mations and existence of examples of Mr. R.C.Reth,Polytechnic Institute Mueller type of Brooklyn (67T-250) Mr. ]. P. Shanahan, Boston College ( 99) Principal series representations of (67T-154) GL(n,F) ( 113) One sided approximation by cubic Professor Lewis Robertson, Uni­ splines versity of Washington (67T-195) Mr. A. Sharma and Professor Am­ ( 1 00) The initial value problem for a third ram Meir, University of Alberta, order equation Canada (67T-230) Dr. E. L. Roetman, Stevens Insti­ ( 114) A weak projection of C onto a Eucli­ tute of Technology (67T-205) dean subspace ( 101) A metric characterization of the 3- Professor Edward Silverman, Pur­ cell due University (67T-150) Mr. Dale Rolfs en, University of ( 115) Topologies on sets of relations Wisconsin (67T-220) Professor R. E. Smithson, Univer­ ( 102) Multitape finite automata with rewind sity of Florida ( 67T- 248) instructions (116) Nonrecursive real numbers. Part II: Dr. A. L. Rosenberg, IBM Watson Reducibility and productiveness Research Center, Yorktown Heights, Mr. R.I. Soare, Cornell University New York (67T-181) (67T-127) ( 1 03) Representations of groups of order 32 ( 11 7) Maximal chains in solvable groups. Mr. P. G. RuudandDr.E.R.Keown, Preliminary report Texas A & M University (67T-221) Professor A. E. Spencer, Western ( 1 04) Quasigroupes de Cardoso et pseudo­ Michigan University (67T-203) groupes de Aelmer ( 118) Thread actions. Preliminary report Professor Albert Sade, Marseille, Professor David Stadtlander, Uni­ France (67T-234) versity of Florida (67T-190) (105) On the Hellinger integrals and their ( 119) On a new inequality for the eigen­ applications to q-variate stochastic values of compact operators processes Mr. William Stenger, University of Professor Habib Salehi, Michigan Maryland (67T-126) State University ( 6 7T-162) (120) Subdirect product of semigroup and ( 106) A note on finite hyperbolic planes. rectangular band Preliminary report Professor Takayuki Tamura, Uni­ Professor S. C. Saxena, Northern versity of California, Davis (67T­ Illinois University (67T-204) l 79) ( 107) On intermediate extensions. I ( 121) A note on the growth of functions in HP Professor Martin Schechter, Belfer Mr. G. D. Taylor, Michigan State Graduate School of Science, Yeshiva University (67T-237) University (67T-200) (122) Gibbs phenomenon for the Hausdorff (108) A characterization of the category of means of double sequences topological spaces Mr. Fred Ustina, Univen·ity of Mrs. Dana Schlomiuk, McGill Uni­ Alberta, Canada (67T-238) versity (67T-180) ( 123) Differences, convolutions, primes x ( 109) Principal co-fibre bundles in the Mr. Benjamin Yolk, Yeshiva Uni­ category of simplicial groups versity (67T-120)

239 ( 124) Nonisomorphic abelian groups with ( 129) On entire functions with negative isomorphic Ulm factors zeros Mr. R. B. Warfield, Jr., Harvard Mr. Jack Williamson, University of University (67T-202) Wisconsin (67T-198) (125) L-bisimple semigroups (Introduced by Professor Simon Heller stein) Professor R. J. Warne, West Vir­ (130) Extension of a result of Dieudonne ginia University (67T-121) Professor J. M. Worrell, Jr., and (126) Weakly converging sequences of Dr. H. H. Wicke, Sandia Corpora­ measures. Preliminary report tion, Albuquerque, New Mexico Mr. B. B. Wells, Jr., University of (67T-125) California, Berkeley (67T-151) (131) Polynomial approximation on nested (127) Concerning an arc theorem regions Dr. H. H. Wicke, Sandia Corpora­ Professor Mishael Zedek, Univer­ tion, Albuquerque, New Mexico sity of Maryland (67T-227) (67T-249) ( 132) On choosing subsets of n-element ( 128) Topics in functional analysis sets. Preliminary report Professor Albert Wilansky, Lehigh Mr. M. M. Zuckerman, New York University (67T-156) University (67T-123)

240 ABSTRACTS OF CONTRIBUTED PAPERS

The February Meeting in NewYork February 25, 1967

643-1. H. B. KEYNES, University of California, Santa Barbara, California 93106. The proximal relation in coset transformation groups.

Let G be a topological group and H a syndetic subgroup of G (HK = G for some compact subset K of G). If H '\G = { Hg lg E G l. consider the coset transformation group (H \G, G). Let P be the proximal relation of (H\G,G) and set K(H) = n{H UHIU a neighborhood of the identity of Gl. Theorem. Let f,g E G, then (Hf,Hg) E P _!!!_gf- 1 E K(H). Using this result, one is able to completely characterize the various properties of P in terms of algebraic and topological properties of K(H). ln addition, it follows that if a single proximal cell is closed, then P is closed. If 'Y is an ordinal number, a

decreasing subnormal chain for H of order 'Y is a family (Hala ;:;; 'Y) of subgroups such that (l) H0 = G, H-y = H, (2) If fl;:;; 'Y and fj is a limit ordinal, then Hfj = n{Hala < fj f, (3) If {:f < -y, Hfj+l

643-2. N. S. TRUDINGER, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10003. The first initial boundary value problem for quasilinear parabolic equations.

Let n. be a bounded domain in En and Q a cylinder in En+ 1 given by Q = !J X {t; 0 < t ;:;; T l. ' 'C '' 2 T > 0. In Q consider the equations P uu = L 1.]=na1l (x,t,u, au/ axi) a u/ a xi axj + ua(x,t,u, au/ axi}

-au; at= 0, u E [0,1]. The coefficients aij(x,t,u,p}, a(x,t,u,p),p = (p 1, •.• ,pn) are assumed to be locally Holder continuous in Q X En+ 1 and to satisfy for any M > 0 and V(x,t,u,p) E Q X(- M, M} X En the inequalities (i) vl~l 2 ~ aii(x,t,u,p)~.t;:;; Jtl~l 2 v~ EEn, (ii) a(x,t,u,p);:;; JtWhere Jtand varepositive I J . 0- constants. Theorem. Let an satisfy an exterior sphere property and

first initial boundary value problem P 1 u = 0, u =

E = lu; Puu = O,u = u

IV; v = au; axi. i ;:;; n, u E: E I is bounded in C(l(K) for some (1 < 0 and any compact K c Q. The same result holds but with more general Pu if (ii) is extended to (ii)' a(x,t,u,p) ;:;; Jt(l + lpl)2. The analogous results for elliptic equations are also proved. The proofs depend on the application of a topological fixed point theorem in an appropriate locally convex space. (Received November 14, 1966.)

241 643-3. P. J. EBERLEIN, University of Rochester, Rochester, New York. Some remarks of the Vander Waerden conjecture.

The Van der Waerden conjecture is: let A be a doubly stochastic matrix of order n; then permanent (A) ~ n!/nn with equality holding if and only if A is the matrix having all elements equal to 1/n. This conjecture is proved for n = 3, (new proof) n = 4 and n = 5, and some special cases. The key tool is a theorem of J. Keilson on symmetric functions. (Received November 28, 1966.)

643-4. WITOLD BOGDANOWICZ and JOHN WELCH, The Catholic University of America, Washington, D. C. Integration of operator-valued functions with respect to orthogonal volumes in Hilbert spaces.

Let (X, V,v) be a volume space and L p(v, Y) be the Lp space of Bochner summable functions from X into a Banach spaceY. (See Bogdanowicz, Bull. Acad. Polan. Sci. 13 (1965), 793-800.)

A function Jl from the prering V into a Hilbert space H is called an orthogonal volume if (JJ.(A), JJ.(B)) =

0 for A, B E V such that A n B = 0, and Jl is countab1y additive on V. Denote by L(H) the B * algebra of linear continuous operators from H into H. We say that F, a B* subalgebra of L(H), preserves the orthogonality of 1J if (yJJ.(A), JJ.(B)) = 0 for ally E F whenever A n B = 0 and A,B E V. Theorem 1. If 1J is an orthogonal volume from V into H and F is a B subalgebra of L(H) preserving the ortho­ gonality of Jl then: (1) The function v mapping V into the real numbers is a volume, where v is de­ fined by the formula v(A) = IJL(A)I 2 for all A E V.

(3) The operator T defined by the formula Tf = Ju(f,dJL) is linear and continuous from L 2(v,F) into H and ITfl ~ llfll 2 for all f E L 2(v,F). (For the definition of the space M2u(H) and the integral Ju(f,dJL) and their properties see Bogdanowicz, Bull. Acad. Polan. Sci. 13 (1965), 801-808.) (Received November 21, 1966.)

643-5. D. L. RAGOZIN, Harvard University, 2 Divinity Avenue, Cambridge, Massachusetts. Approximation theory on SU(2) with applications to Fourier analysis. Preliminary report.

Let G = SU(2) and Xnbe the character of the irreducible representation of G of dimension n. 2 Call 3'~ = L (G) • L~k Xk the trigonometric polynomials of degree n, and set En (f) = inf lllf - T n 11 00: T E Y In terms of the metric p, which G inherits from c 4 , define the function classes Lip u. n n 1,. 1 11 Also, set M&"(h) = Jt(a- gah)da and define the symmetric Lipschitz classes, H = It: IIMgf- fll00 :'i Kp11 (g, 1), for some K j. Several direct and converse theorems of the Jackson and Bernstein type are proved. Samples are Theorem 1. If Dk is a product of k left-invariant derivations, then E k -k-a f C (G) and Dkf E Lip a, 0 < a < l, for all Dk iff En (f) = O(n ). Theorem 2. Theorem 1 with

H11 , 0 < a < 2, in place of Lip u. The proofs use positive convolution operators for k = 0, extending a method of Korovkin for U (1), and an estimate connecting Mg and the Laplace operator on G for k > 0. Corollaries. H u <:;; C 1 (G), a > 1. Lip a= H 11, a < 1. Let Snf = L 7f •k X k' the nth partial sum of the Fourier series for f. The direct theorems lead to Theorem 3. (i) f E C 1 (G) implies S f ---> f n 11 uniformly. (ii) f E H , a > 3/2, implies L~ II£• k Xkll 00 converges. (R. Mayer, unpublished, obtained 3i by different methods.) (Received November 15, 1966.)

242 643-6. TUDOR GANEA, University of Washington, Seattle, Washington 98105. On monomor­ phisms in homotopy theory.

A map h: Y ----> Z is monomorphic if, for any space X and any maps f,g: X ----> Y, h o f ~ h o g implies f~ g. Let h: Sq(m+l)-l ----> KPm be the Hopf map, where Sn is then-sphere, KPm is the projective m-space over the field K of real numbers, complex numbers, or quaternions, and q = l,Z, or 4. If q = 1 or Z, then h is monomorphic if and only if m is odd; if q = 4, then his mono­ morphic if and only if m =- 1 (mod Z4). Finally, the Hopf map S 15 ----> s8 is not monomorphic. (Received November Z5, 1966.)

643-7. KWANG CHUL HA, University of South Florida, Tampa, Florida. The radical of topological abelian groups.

F. B. Wright has defined the radical in a general topological abelian group as the intersection of what he has called the "residual subgroups". However, his definition suggests a more direct definition of the radical. The radical T(G) of a topological abelian group G is defined as the com­ plementary set of C(G), the conservative of G1 in G. Theorem 1. Let G be a topological abelian group and H be a subgroup of G. If G/H and H are both radical groups, then so is G. Theorem z. Let each Ga. (a. E A) be a topological abelian group, and let G = I1a.eAGa. be the cartesian product group of Ga. (a E A). Then T(G) = I1a.EAT(G J• where T(Ga.) (a E A) is the radical of Ga.• Theorem 3.

The radical T('G) of 1 G, the limit of an inverse system, is the intersection of 'TA (A E A), where

TA = { x: x E 1 G; x A E T(G A)}. Theorem 4. Let G be a connected locally compact abelian group. Then G is a torsion group if and only if G = (0). (Received December 1, 1966.)

643-8. P.R. MEYER, Hunter College of the City University of New York, 695 Park Avenue, New York, New York lOOZ 1. Function spaces and the Aleksandrov-Urysohn conjecture.

This paper gives an equivalent reformulation for the Aleksandrov- Urysohn conjecture that every first countable compact Hausdorff space has cardinality ~ c, the cardinality of the continuum.

Let X be a compact Hausdorff space; call the given topology the {j-topology and let C(j = C(X,R). The set of all bounded Baire functions generated by C{j induces a weak topology on X which is called the L-topology. The conjecture is equivalent to the following: If (X,{3) is first countable, then (X,t) is c-Lindelof. (A topology is said to be c- Lindelof if every open covering has a subcovering of cardinality ~ c.) This reformulation might be more tractable since many conditions equivalent to the ordinary LindelOf property for (X,L) are known. (See Abstract 64T-Z 19, these .cNOticetJ 11 (1964), 376; also Duke Math. j. 33 (1966), 33-40, and references there.) (Received November 16, 1966.)

643-9. R. E. ATALLA, Ohio University, Athens, Ohio. On the continuity and measurability of certain transformation groups.

Let G be a topological group, C(G) the bounded complex valued functions on G, and A a trans­ lation invariant self-adjoint subalgebra of C(G) which separates points and closed sets. Let aG be the compactification of G such that A is isomorphic to C(aG). If pEG, let Tp be the corresponding translation mapping, lifted to act on aG, and let 1r: G X aG ----> aG be defined by 1r(p, w) = TPw.

243 Theorem· l. 1r is continuous iff each element of A is uniformly continuous. Theorem 2, The map v:R X fJR----> {JR is discontinuous in the first variable. Specifically, there exist wE {JR\R, a sequence pn --> 0, and an f E C({JR) such that f(Tpfiw) + f(w). Theorem 3. The map v: R X {JR. --> fJR is not Baire measurable. Specifically, there is an f E C({JR) and a TP-invariant probability measure m on {JR such that the map p----. f • TP E L 1(m) is not continuous, from which fact nonmeasurability of f • v: R X {JR --> R can be proved directly, Theorem l makes it possible to generalize certain results on invariant means on the integers to invariant means on uniformly continuous functions, e.g., Thearem 3 of M. jerison, The set of all generalized limits of bounded seguences, Canad. J. Math. 9 (1957), 79-89. (Received November 30, 1966.)

643-10. WITOLD BOGDANOWICZ, The Catholic University of America, Washington, D. C. 20017. Integral representation of complete integral seminorms.

A nonnegative finite-valued functional J is called an integral seminorm over the space X if its domain j+ consists of functions from X into R + = < O,oo) and : t,s E R + and f,g E J+ implies tf + sg E J+ and J(tf + sg) = t]f + sjg; f,g E J +implies f U g E J+ and f n l E J +; if f(x) ~ g(x) on X and f,g E J+ then g - f E J+. The integral seminorm is called upper complete if, for every increasing sequence fn E j+, converging at every point of the space X to a finite-valued function f, for which the sequence of numbers jfn is bounded, we have f E J+ and Jfn--> ]f. An integral seminorm is called complete if in addition it satisfies the following condition: 0 ~ g ~ f E J+ and jf = 0 implies g E J+. Theorem 1. For every complete integral seminorm J there exists a volume v such that J+ = L +(v,R) and ]f = Jf dv for all f E J+, where L +(v,R) and Jdenote the space of nonnegative v-summable functions and the integral developed in the paper: Bogdanowicz, A generalization of the Lebesgue-Bochner­ Stieltjes integral and a new approach to the theory of integration, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 492-498). (Received November 21, 1966.)

643-ll. ARNOLD STOKES, Georgetown University, Washington, D. C., and CARL KALLINA, Howard University, Washington, D. C. Periodic and almost periodic solutions of parabolic equations.

Let Q = (0 < x < X, - oo < t < oo). Let S denote the lateral sides of Q. Consider in Q the equations (l) Uxx = A(x,t)Ut + B(x,t) Ut + C(x,t) U + F(x,t) and (2) Uxx = (l + tU) Ut+ B(x,t,U) Ux +

C(x,t,U) U t F(x,t). On S we specify the values (3) UIS = 1/t(x,t). Assuming all the given functions in (l) and (3) are periodic in t of period T, one can prove the existence of a unique periodic solution of

(l) and (3) in Q which is exponentially stable. Using this result one shows that there is a value t 0 such that for each £, 0 ;;;! t ~ t 0 , isolated periodic solutions U(x,t,t) of (2), (3) in Q can be found. This last result depends upon a new maximum principle for parabolic equations containing a small parameter. Under the assumption that the given functions in (l), (3) are almost periodic in t uniformly in llJ,x], one can extend the results of Amerio (Ann. Math, Pura Appl. 39 (1955), 97-119) to prove the existence of a unique solution U(x,t) of (1), (3) in Q which is almost periodic. Finally, the divergent form (4) Ut = <1/iJx(A(x,t,U,t) • Ux) + B(x,t,U,Ux) is considered. Using the Leray-Schauder theory, one can prove the existence of periodic solutions of (4), (3) when 0 ~ f ~ t 0 for some t0• This last result can easily be extended to the case of n space variables. (Received December l, 1966.)

244 643-12. MORRIS SKIBINSKY, Brookhaven National Laboratories, Applied Mathematics Depart­ ment, Upton, Long Island, New York 11973. The range of the (n + l)st moment for distributions on [0, 1].

Let P denote the class of all probability measures defined on the Borel subsets of the unit interval I= [0,1]. For each positive integer n, take Mn = j(c 1,c2, .. .,cn): ci = JixictP(x), i = 1,2, •.• ,n; P E Pl. It is known that Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve l v-. Define r on M by r = fin p .( 1 - p .), interior to M , and equal zero on its n n n n n J= 1 J J n boundary. Theorem. Let n be an arbitrary positive integer. Then everywhere on Mn, "~+l - "~+l = rn. Theorem 2.5, p.80 (Karlin and Shapley, Geometry of moment spaces, (Mem. Amer. Math. Soc. No. 12, Amer. Math. Soc., Providence, R. 1., 1953) paraphrased below may be viewed as a corollary. Theorem (Karlin and Shapley). The width in the en direction of Mn (i.e. the maximum range of the nth moment) is 2-2n+2. (Received December 19, 1966.)

643-13. R. E. HODEL, Duke University, Durham, North Carolina. Note on metric-dependent dimension functions.

Let (X,p) be a metric space, let dim X be the covering dimension of X, and let J1. dim(X,p) be the metric dimension of X. Let d 2, d 3, and d4 denote the dimension functions for metric spaces introduced by Nagami and Roberts in their paper Study of metric-dependent dimension functions, submitted to Trans. Amer. Math. Soc. A summary of the relation among these various dimension functions is as follows: d 2(X,p) ~ d 3(X,p) ~ JJ.dim(X,p) ~ d4 (X,p) =dim X ~ ZJJ.dim(X,p). In this paper the author continues the study of dimension functions for metric spaces which by their definition ap­ pear to depend upon the particular metric. A new metric-dependent dimension function d 5 is introduced, and it is shown that for any metric space (X,p), d 3(X,p) ~ d 5(X,p) ~ JJ..dim (X,p) and if X is separable d 5(X,p) = Jl. dim (X,p). The main result of the paper is the following. Theorem. Let (X,p) be a metric space. Then dim X ~ Zd3 (X,p}. (Received November 15, 1966.)

643-14. L. M. CHAWLA and F. A. SMITH, University of Florida, Gainesville, Florida. On semigroups embeddable in their endomorphism semigroup.

Let Si, i = 1,2, ... , n be any n additive abelian semigroups with zeros. Theorem 1. The direct sum of any semigroups Hij = Hom (Si' Si)' i even and j odd, can be embedded isomorphically in the multiplicative semigroup E(S) of e ndomorphisms of S = L~ si. Let z+ be the additive semigroup of 0 1 nonnegative integers. Theorem 2. If S is the direct sum of m ~ 4 copies z+ and an arbitrary additive abelian semigroup V with 0, then S can be embedded isomorphically in E(S). (Received December 22, 1966.)

245 643-15. I. N. SPATZ, Polytechnic Institute of Brooklyn, 202 Avenue F, Brooklyn, New York, 11218. Geometry of Banach algebras.

Let X be a real (complex) Banach algebra with identity e. lie II= 1. Let G(x,y) be the Gateaux differential of y at x; i.e., the limit as t approaches zero through the reals of ( llx + ty II - llx 11)/t. Theorem. If G(e,s) exists for each singular element s of X, then X is a division algebra. Corollary. If X is smooth, then X is the real numbers, the real quaternions, or the complex numbers (the complex numbers). This generalizes a result of L. lngelstam (Bull. Amer. Math. Soc. 69 (1963}, 794-796) and M. F. Smiley (Proc. Amer. Math. Soc. 16 (1965), 440-441) for Hilbert algebras. The proof of the theorem is both simple and elementary. An example of a nontrivial X which is uniformly convex is given. (Received December 15, 1966.)

643-16. J. H. ROBERTS and F. G. SLAUGHTER, JR., Duke University, Durham, North Carolina. Metric dimension and equivalent metrics.

Given a metric space (X,P}, the metric dimension of (X,p), indicated IL dim(X,P) is the smallest integer m such that for all f > 0 there exist an open cover %-f of X such that (i) p-mesh %-f < f and

(ii) ord %-f < m + I. It is trivial that IL dim (X,P) ;;; dim X, where dim is covering dimension. In the other direction, Katl!tov has shown that 2!Ldim (X, ~ dim X. Nagami and Roberts have given examples (for all n >I) of spaces (Xn,p) with dim Xn =nand IL dim(Xn,p) = [(n + 1}/2] (the biggest integer in (n + 1)/2). Theorem. Let (X,p) be a metric space with JJ. dim (X,p) = m and dim X = n. Then for any integer k such that m;;; k & n, there exists a metric pk for X such that (i}pk is topo­ logically equivalent to P and (ii) p. dim (X,pk) = k. (Received November 25, 1966.)

643-17. P. P. SAWOROTNOW, The Catholic University of America, Washington, D. C., and J. C. FRIEDELL, Loras College, Dubuque, Iowa. Trace class for an arbitrary H*-algebra.

For each proper H*-algebra ~the set T( ~) = !xyjx,y E ~~is a Banach algebra with respect to some norm T( ) such that T(a*a) = lla11 2 for each a E '11; T( ~)is a trace algebra in the sense that there is a trace t defined on it such that t(ab) = t(ba} = (a,b*) for all a,b E 21, t(a) = Ln(aea.,en) for each a E T(~) and each maximal family lenl of doubly orthogonal selfadjoint idempotents, and, for

all a E: ~. T(a) = t[a] where [a] is some positive (([a]x,x) ~ 0 for all x E ~)selfadjoint member of ~ such that [a]2 = a*a. This is a generalization of work of R. Schatten (Normed ideals of complet~ continuous operators, 1960). Conversely let T,T be a Banach algebra with an involution and a trace t

such that T(a *a)= t(a *a) for all a E T. Then Tis dense in some H*-algebra ~ with the scalar product

(x,y) which coincides with t(y*x) if x,y E T. Assume further that T is an ideal in ~and has the property that T(Ua) = T(u) for each a E T and each unitary operator U on '11 such that U(xy) = (Ux)y for

all x,y E ~. Then T = T( ~ ). (Received January 9, 1967.}

643-18. H. J. COHEN, City College of the City University of New York, New York, New York 10031. Inequalities in the formula of inclusion and exclusion for an additive set-function.

Let f be a real-valued, finitely-additive set-function on an arbitrary set T. Let A1, A2 ... , An be subsets ofT. For r = 1,2, ••. , n, define W(r) = Lf(Ai n Ai n ... n Ai ), where the summation is I 2 r taken over all the r-subsets of (1,2, ••• ,n). Let W(O) = f(T). Define Em as the set of all points of T

246 belonging to exactly m of the Ai' m = 0, l, •.• ,n. The formula of inclusion and exclusion states:

(1) f(E ) = LN-Um (- l)i(C . )W(m + i). Theorem l. Iff ~ 0, then the second half of the W(r), m 1= mt1,m r = 1, , .•. ,n, are monotone decreasing. That is, if [n/2] ::;; r ::;; n - 1, then W(r) ~ W(r + 1). Theorem 2,

Iff £; 0, then the partial sums of the right member in formula (1) alternately overestimate and under­ estimate f(Em). The proof of Theorem 1 uses the following: Lemma. LetS be a set of n elements, and suppose [n/2] ::;; r ::;; n - 1. Then the collection of all the (r + 1)- subsets of S can be mapped one­ one into the collection of all the r-subsets of Sin such a way that each (r + 1)-subset contains its image. (Received january 10, 1967.)

643- 19, P. A. FUHRMANN, Columbia University, New York, New York 10027. A functional calculus in Hilbert space based on operator valued analytic functions.

Let N be a Hilbert space, H2 (N) the 2-Hardy class of N-vector valued analytic functions in the unit disc. Let K be a left invariant subspace of H2 (N) and T* the left shift restricted to K. P the orthogonal projection of H2 (N) onto K. Let OH 00 (N) be the algebra of all bounded N-operator valued analytic functions in the unit disc, Definition. For each F inK, A in OH 00 (N) let A(T)F = P(AF). A(T) is a bounded operator in K whose norm is majorized by sup IIA(z) 11. lz I < 1. The map A ----> A(T) becomes an algebra homomorphism on the subalgebra of OH00 (N) consisting of all operator valued functions leaving K-'- invariant. By the Beurling- Lax theorem K-'- = SH 2 (N), S an operator valued func­ tion with partial isometries as its boundary values a.e. Theorem. Assume A in OH 00 (N), K-'- = SH2 (N) and Sinner then A leaves K-'- invariant if and only if there exists A1 in OH00 (N) such that AS= SA1. Let K = H2 (N) e SH2 (N) with S(z) = S(z)*. T* the left shift restricted to K. A such that it leaves K

invariant. A' 1 (z) = A1 (z)*. Theorem. A1 leaves K-'- invariant and A(T)* is unitarily equivalent to A' 1 (T). Theorem. A in OH00 (N) and AS= SA1 then 0 is in the point spectrum of A(T) if and only if A1,s have a common nontrivial right inner factor. Theorem. Assume N is finite dimensional. A(T) has a bounded inverse if and only if there exists a o > 0 such that infllxll=llliA(z)*xll + IIS(z)*x~j ~ oand

inf llx 11 =1IIIA 1 (z)x II + liS (z)x Ill ~ o for all z in the open unit disc, (Received january 10, 196 7.)

643-20. STANISLAW MROWKA, Pennsylvania State University, McAllister Building, University Park, Pennsylvania 16802. Some strengthenings of the Ulam nonmeasurability condition.

Let Xm denote a set of cardinality m; consider the following three conditions. S(m): there exists a class IK~: ~ EEl of families of subsets of Xm such that (a) card E ::;; m, card K~ < m for

every ~ E E. (b) for every finitely additive Ulam measure JL on Xm the equality JL( UK~)= sup I JL(A): A E K~ l fails for at least one ~ E ;;;. R(m): a product of m discrete spaces each of cardinal­

ity < m contains a closed discrete subspace of cardinality m. R0 (m): a product like above has a family I A ( ~ E E l of closed subsets such that card E = m, n I A~:~ E MIt- !il for every M c E with card M < m, niA~:~ EEl= !il. Theorem. Conditions S(m), R(m), Ro(m) are equivalent. (Note: each of the above conditions implies that m is "strongly incompact"; but these conditions do not express strongest properties of cardinals; seeS. Mrowka, Abstract 632-56, these c#otiai) 13 (1966), 342. It is likely that these conditions are equivalent to "strong incompactness".) (Received january 10, 1967.)

247 643-21. Y. KUO, The University of Tennessee, Knoxville, Tennessee 37916. Partially direct sum of inequality groups.

For every i in I = 11,2, ••. , r), let F i be an inequality group with order di of a linear program Li where di is the (integer) product of all preceding pivots [see Ralph E. Gomory, "An algorithm for integer solutions to linear programs," in Recent advances in mathematical programming, edited by

R. L. Graves and P. Wolf, pp. 269-302]. The notation (a0 .Jdi' al./di, ••• ,a0 .Jdi) (mod 1) is used for an 1 1 1 element of Fi where ak. is an integer for every k. in IO.,l., ••• ,n.), and ak jd. can be replaced by 1 1 11 1 i1 a'k./di' if ak· /di - a'k./di is an integer (i.e. ak./di = a'k /d1. (mod 1)). If d; II. 1d. has no repeated 1 1 1 1 i 1E 1 primes in its factorization, then there exists an element (ao/di, a 1/di'"""' a 0 /di) (mod I) in Fi which generates F. for every i in I, and

643-22. F. B. FITCH, Yale University, 1834 Yale Station, New Haven, Connecticut 06520. Simplified tree proofs in modal logic. Preliminary report.

A method is given for constructing tree proofs in modal logic similar to that described by the author in J. Symbolic Logic 31 (1966), 152, but simpler since no symbols for universes are needed. We say A is an mjp (modal disjunctive part) of B if A is a jp (disjunctive part in the usual sense) of B, or of some C such that oC is an mjp of B. A formula A is an axiom of some atomic formula B is such that B and ~ B are jps of the same mjp of A. The rules of deduction for the deontic system OM in­ clude the usual tree proof rules for nonmodal connectives (Boolean connectives and quantifiers) but with the concept mjp replacing jp, and with the further rule:(~ DA v .•• v o( ~A VB v ••• ))

--> ( "--' oA V •.• v o(B v ••• )), where (A) is any formula having A as an mjp. To get the alethic system M, add the alethic rule: ( ~ DA V ~A)--> ( ~ oA). Appropriate rules can be added to give S4-, S5- and 8 -type alethic or deontic systems. In any of these systems, if A is not provable, the attempt to construct a tree proof of A leads directly to a counterexample to A. Hence this method gives a simple way of proving the completeness of the various systems considered, and of proving Lowenheim-Skolem theorems for them. (Received January 10, 1967.)

643-23. ALBERT EDREI, Syracuse University, Syracuse, New York 13210. Sums of deficiencies of entire functions.

Let f(z) be an entire function of finite lower order ~(no assumption is made concerning the order of f(z), which may be + oo). Let ll(r,f) denote the deficiency, in the sense of R. Nevanlinna, of the value r(finite or oo) of the function f{z), and let .1-(f); Lro(r,f) be the total deficiency. If

1/2 ;:;; ~;:;; 1, it is known that (i) d(f) ;:;; 2 - sin -rr~ and that this inequality is "best possible" for every

~in the interval (1/2, 1]. The author proves: if ~ is equal to any one of the numbers l/2 + 1/2q (q; 1,2,3, .•• ) and if equality holds in (i), then f(z) has exactly two deficient values. The restrictions concerning the values of~ seem due to the method of proof and suggest the possibility of extending the result to all~ in the interval 1/2 < ~ ;> 1. (Received January 12, 1967.)

248 643-24. L. H. HAINES, University of California at Berkeley, Berkeley, California 94720. Derivatives of solutions of linear differential equations.

Theorem. Let (1) y(n)(t) + :E(r 1gi(t)y(il(t) = 0, jgi(t)l ;:!! C V t. Then given any f > 0 and any p (1 ;:;; p ;:!! oo) 3 constant K = K(C, n, f , p) 3 : for each solution f of (1) on any interval I of length at least f, llf(i) liP ;:;; K llf llp, ( 1 ;:!! i ;:;; n), where the Lp norms are evaluated over I. An immediate corollary of this theorem is a result proved by E. Landau, Math. Ann. 102 (1929), 177-188. Corollary. Iff is a bounded solution of (1) on the entire real line then the f(i) are also bounded (1 ;:;; i ;:;; n). (Received January 12, 1967.)

249 ABSTRACTS PRESENTED BY TITLE

67T-120. BENJAMIN VOLK, 1315 Dickens Street, Far Rockaway, New York 11691. Differences, convolutions, primes, X.

Definition. Let the two sequences of integers S ~ s 1, s 2 , s 3,. .• , and M ~ m 1, m 2 , m 3,. .. be given, where for all i and i, mi is relatively prime to mi. Define the kth order difference quotient of the sequenceS relative to the sequence M by induction as S(M; m 1, m 2 , .• .,mk_ 2 ,mk-l'mk) ~

(S(M; ml'm2, ... ,mk_ 2, mk)- S(M; ml'm2 , ... ,mk_ 2, mk_ 1))/(mm- mk_ 1)(modulo mk), whereS(M; m 1 )~ s 1(modulo m 1). Define now the Newton series of S relative toM as F(x) ~ S(M; m 1) + S(M; m 1,m2) •(x- m 1) + S(M; m 1,m2,m3 )(x- m 1)(x- m 2 ) + •••• Say that Sis continuous at the point mi of M if sk (modulo mk) ~ si (modulo mi) whenever mk - mi ~ !(modulo mk). Say also that the sequence S is differentiable at the point mi if the sequence S(M; mi,m 1), S(M; mi,m2 ), S(M; mi,m3),. .• is the same whenever mk- mi ~ !(modulo mk). Theorem 10. F(mk) ~ sk(modulo mk). Credit. The proof of the Chinese Remainder Theorem by a Lagrange Interpolation Formula (D. Shanks, Number theory) was our starting point here. Conjecture 9. If F(x) is a differentiable (with the underlying convergence being in the p-adic norm) rational-valued function of a rational-valued variable defined on (0,1), then so is the derivative of F(x). (Received April 4, 1966.)

67T- 121. R. J. WARNE, West Virginia University, Morgantown, West Virginia. L-bisimple semigroups.

If S is a semigroup, ES will denote its collection of idempotents. ES is lexicographically ordered if ES ~ l eij: i,j E I0 , the nonnegative integers l under the order. Suppose further that eij < eks if i > k or i ~ k and j > s. A bisimple semigroup S such that Es is lexicographically ordered is called an L-bisimple semigroup. Let C be the bicyclic semigroup and C 0 C be the Bruck product of C by C [Abstract 65T-336, these cNOticeiJ 12 (1965), 614]. Theorem. Sis an L-bisimple semi­ group if and only if S ~ G X (C o C) where G is a group under the multiplication -k 1 11 (g,(n,m), (k,l))(h,(n1,m 1), (k 1,1 1)) ~ (t, (n,m),(k,l)) ((n 1,m 1), (k 1,I 1)) where t ~ g((z (ha)z ) m-n 1-l 1 k 1 n 1-m-l k 1 k 1-o 1-0 •a {3 )), ((z- (ga)z )a {3 )h, (g{3 h{3 ) according to whether m > n 1, n 1 > m, or 0 n 1 ~ m, where a,{3 are endomorphisms of Ga ,{3° denoting the identity automorphlsms of G, and z E G such that {3a aC (xC ~ zxz -l for x E G) and 0 ~ min (k ,I) and juxtaposition denotes multiplication ~ z z 1 in G and Co C. This result is given for arbitrary finite dimensions. (Received June 23, 1966.)

67T-122. D. M. R. PARK, Programming Research Group, 43 Banbury Road, Oxford, England. Intersections of elementary substructures. I.

Theorem. Let ~ ~ \8 be first-order structures. The following conditions are equivalent:

(i) For each formula F(x0 ,x1, .. .,xn) and a 1, a 2 ,. •• , an E= I~ 1. {bib IE I~ I anct\Bf= Fj!>,a 1, ... ,anJl is either empty or infinite. (ii) There exist ~l > \8 and ~i < ~l' IE I s.t. ~ ~ ni 2li. (iii) There exist ~ 2 > \8 and an elementary embedding k: \8 --+ ~ 2 s.t. 21 ~ \8 n k( \8) and k ~ ~ ~ identity. (iv) There exist ~ 3 > \8 and an elementary self-embedding j: ~ 3 ---> ~ 3 s.t. 2l ~ n :o/(~ 3) and

250 j ~ ~ = identity, (v) There exist ~ 4 > \8 and an automorphism 0, \8 and an automorphism 1/;: ~ 5 ~ ~ 5 s, t. for all m > 0, c E I~ I iff 1/;m(c) = c. (N. B, Examples show that no ~ n may be replaced by \8 in (ii) - (vi),)

Definition. An elementary class has the .lt'Y-i.p, if, whenever ~ ,\8 satisfy one of (i)-(vi) and \8 E K, then ~ E K. Applying (vi) to Corollary 2 of Abstract 67T-3, these c){oticeiJ 14 (1967), 127- 128, we get: Corollary. K is closed under descending intersections iff K has the _l:f.sf-i.p. and is closed under ascending unions. (Received july 29, 1966,)

67T-123. M. M. ZUCKERMAN, New York University, Courant Institute of Mathematical Sciences, New York, New York. On choosing subsets of n-element sets, Preliminary report,

For n ~ 2 and for N (f- (1) s;; 11,2, .. ., n - 1}, let S(n; N) denote the statement: For every set X of n-element sets, there is a function f on X such that for each A EX, f(A) C A and the cardinality of f(A) is inN. S(n; It}) is obviously equivalent to Mostowski's axiom ljl] (Fund, Math, 33 (1945), 137-

168). For n ~ 3, let DC be the statement: For every set X of n-element sets, there is a function f on X such that for each A E X, f(A) is a (nontrivial) decomposition of A satisfying a certain condi- tion C. The interdependence of these axioms, for various subsets N and decomposition conditions C, is discussed, Some of the independence results are obtained by means of a generalization of Mostowski's condition (K). Typical results are the following: (1) For n > 4 composite and for N = ll,2,,.,,n - t}, [n] is independent of S(n; N). (2) If C is the decomposition condition, "for each A E X there is at least one c E f(A) of cardinality ~ 2," then ([3,5,13]--> DC(8))---> ([3,5,13]--> [15]). (Received August 22, 1966.)-

67T-124, C. R. BORGES, University of California, Davis, California, An extension of Dugundji's extension theorem.

For each positive integer n and topological space X, let P n denote the unit simplex in Euclidean n-space and xn the n-fold cartesian product of X. Definition. X is said to have a pseudo-convex structure if there exist functions hn: Pn X Xn--> X such that (a) hn((t1, ... ,ti-l' 0, ti+l'""tn), (xl'""xi-l' 0, xi+l'""xn)) = hn_ 1((t1, ... ,ti-l' ti+l'"''tn), (x1, ... ,xi-l' xi+l""'xn)); (b) the map t --> h (x,t) from P to X is continuous, for each x E Xn; (c) for each x E X and neighborhood U of x, n n there exists open V C U such that x E V and u: 1hn (P n X Vn) C U, Theorem. Every topological space X with a pseudo-convex structure is an absolute extensor space for the class of stratifiable (hence, the class of metrizable) spaces, Some results of Trans, Amer, Math, Soc, 115 (1965), 43-53 are immediate consequences of the preceding theorem. We also develop geodesic structures on x 2 which induce pseudo-convex structures on X. (Received October 3, 1966.)

67T-125, J, M. WORRELL, Jr., and H. H. WICKE, Sandia Corporation, Sandia Base, Albuquerque, New Mexico. Extension of a result of Dieudonno!.

Dieudonno! showed that although the space of countable ordinals with respect to the order topology possesses a unique uniformity, it is not complete with respect to that uniformity, [Kelley, General topology, Exercise 6E, p. 204]. The authors show that the example of Abstract 66T-411 [these cJ{oticfi) 13 (1966), 644], although a uniform space, is nowhere locally a complete uniform

251 space. This result may be of interest in view of the strong completeness properties of the example and the fact that the example is an open continuous image of a complete metric space [Abstract 66T-159, these cJ{oticei) 13, (1966), 265]. (Received September 22, 1966.)

67T- 126. WILLIAM STENGER, University of Maryland, College Park, Maryland. On a new inequality for the eigenvalues of compact operators.

Let A be a compact, symmetric, nonpositive operator on a Hilbert space H. Let H 1 be a closed subspace of Hand let H" = H 8 H 1 • Denote by P 1 and P" the projections onto H 1 and H" respectively.

and A~1 ~ >.z. ~ ... be the eigenvalues of the operators A, P 1 A P 1 , and P" A P" respectively. Using the minimum-maximum principle one has the seemingly new inequality >. 1 + Ai+j ~ ,\l + AJ 1 ' i, j = 1,2, ..•• This inequality is in a sense complementary to the inequality A1 + A11 < A. . , i, j = 1,2, ... given by Aronszajn [Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 1 J ~ l+J- l 476]. The first inequality can be extended to unbounded operators provided the lower parts of the spectra of A, P 1 A P 1 , and P" A P" consist of isolated eigenvalues each having finite multiplicity. (Received October 17, 1966.)

67T-l27. R. I. SOARE, 117 Oak Avenue, Ithaca, New York. Nonrecursive real numbers. Part II: Reducibility and productiveness.

For A a subset of the natural numbers N, let L(A) be the lower cut determined by A, as defined in Abstract 67T- 52, these cJ{oticei) 14 (1967), 144. Theorem 1. If L(A) is a nonrecursive cut and L(A) is many-one reducible to L(B), then L(B) is Turing reducible to L(A). It is easy to show that there is a cut of every truth-table degree. Corollary 1. No cut is many-one complete even with respect to r.e. cuts. Corollary 2. No cut is productive or even quasi-productive (as defined by

J. R. Shoenfield). It is then natural to ask about weaker types of productiveness such as semiproduc­ tiveness (Dekker). Theorem 2. Any cut which is truth-table productive or weakly truth-table pro­ ductive (Friedberg and Rogers) is semiproductive. Corollary 3. If a cut is truth-table complete its complement is semiproductive. Corollary 2 is a consequence also of results of C. G. Jockusch, Jr. since every cut is a semirecursive set. (See Jockusch, Abstract 65T-457, these c){oticei) 12 (1965), 816.) On the other hand, all the above results may be generalized by replacing "cut" by "semire­ cursive set". Theorem l may be further generalized by replacing "many-one reducible" by "positive reducible" or "norm-1 reducible" in the sense of jockusch. (Received October 25, 1966.)

67T-l28. E. G. K. LOPEZ-ESCOBAR, University of Maryland, College Park, Maryland. A complete axiomatization for weak second-order logic.

A complete axiomatization (i.e. such that the valid formulas coincide with the provable formulas) is obtained for the weak second-order logic. All rules of inference except one have finitely many premises and the one with infinitely many premises is a natural analogue of the w-rule. Furthermore imposing the condition that the infinitary rule be applied only when there exists a recursive function giving Godel numbers for the proofs of the premises does not change the set of provable formulas. (Received November 3, 1966.)

252 67T-l29. C. F. KENT, Case Institute of Technology, Cleveland, Ohio 44106. Strongly representable functions. II.

Refer to Abstract 67T-97, these c}/otiai) 14 (1967), 157. If-< is a primitive recursive well ordering of the natural numbers for which complete induction, CI-<, is a permissible rule of inference inS, then so is Ciw-<• for the natural, lexicographical, exponentiation w-<. Corollary (Kreisel, 1952). The s.r. functions of S are those ordinal recursive of finite type. Using the basic result of Abstract 67T-97, we obtain: Corollary. Fischer's properties P.l-P.5. are theorems of S. This verifies the suspicion that results of Rogers, Kent and Fischer hold within number theory. Typical new result: In the Boolean algebra of s.r. sets of natural numbers, the Dedekind finite sets (those which cannot be proved infinite) form a left segment which generates the Boolean algebra under pairwise union. (Received November 14, 1966.)

67T-130. R. B. JENSEN, Universitat Bonn, Lennestrasse 33a, West Germany. Ramsey cardinals and the general continuum hypothesis.

Let M be a model of ZF and AC. Then M can be imbedded in a Boolean-valued model 9Jl of ZF t AC t GCH such that: (i) The ordinals of M are the ordinals of 9Jl. (ii) Each measurable cardinal of M is measurable in 9Jl. (iii) Each Ramsey cardinal of M is a Ramsey cardin~! in 9Jl.

In particular follows: If ZF t AC + "there is a Ramsey cardinal" is consistant, then it remains so upon adjoining GCH. (Received November 25, 1966.)

67T-13l. R. L. PELE and W. ]. LEAHEY, University of Hawaii, Honolulu, Hawaii. Some remarks on the vector subspaces of a finite field.

Let F be a finite field of q elements and E an extension of F of degree n. Consider E as a vector space over F. It is shown that for every subspace V of E there exists a unique polynomial r r-1 r-2 f(X) = Xq + a 1 Xq + a 2Xq + ... + arX whose roots are the elements of V, and there exists a unique polynomial g(X) of the same form, but of degree qn-r, such that g(E) = V, where r is the dimension of V. Furthermore f(X) and g(X) split completely in E, and f(g(X)) = g(f(X)) = Xqn - X. (Received November 21, 1966.)

67T-132. G. D. CHAKERIAN, University of California, Davis, California. Sets with proportional brightness and girth.

If K is a convex body in 3-dimensional Euclidean space E 3 , let u(K,u) and A(K,u) be the area and perimeter respectively of the orthogonal projection of K onto a plane perpendicular to the direction u. Theorem. Let K and E be smooth convex bodies in E 3, withE= -E. Suppose there exist positive constants a, {3 such that u(K,u) = au(E,u), and >..(K,u) = {3.X.(E,u), for all directions u. Then K is homothetic to E. This generalizes the well-known result that the only smooth convex body of constant width and constant brightness is the sphere. (Received November 17, 1966.)

253 67T-l33. H. E. FETT!S, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio 45433. New formulae for the calculation of a definite integral occurring in the statistical treat­ ment of white noise.

The integral In (b) = (2/lr) J~ j(sin ~x l n cos bx dx occurs in many statistical problems. Although the integral can be evaluated in closed form as a succession of polynomials in b of order (n-1), the use of this form has not proved practical, particularly when n and b are large. Alternative schemes employing recursion relations have been proposed, but these are not satisfactory as a means of generating the function directly [e.g. by means of a subroutine]. The present paper expresses In (b) by means of rapidly converging trigonometric series which work equally well for moderately large and large values of n and b, and which allow the integral to be computed directly for any given values of these parameters. (Received November 18, 1966.)

67T-134. T. E. GANTNER, University of Dayton, Dayton 9, Ohio. Extensions of uniformly continuous pseudometrics.

A pseudometric d on a set X is 1'- separable (I' an infinite cardinal number) if the associated topological space (X, T d) has a dense subset of cardinality ;;; ')'. LetS be a uniform subspace of a uni­ form space X, and let"/ be an infinite cardinal number. A modification of an argument used by H. L. Shapiro (Extensions of pseudometrics, Canad. ]. Math. 18 (1966), 984) shows that every uniformly continuous )'-separable pseudometric on S has a continuous )'-separable pseudometric extension to X. Moreover, if the generalized continuum hypothesis is assumed, then every bounded uniformly continu­ ous )'-separable pseudometric on S has a bounded uniformly continuous )'-separable pseudometric extension to X. In fact, we prove this result with an assumption that is formally weaker than the generalized continuum hypothesis. In this form, a result of ]. R. Isbell (On finite-dimensional uniform spaces, Pacific J. Math. 9 (1959), 110) follows as an immediate corollary. Finally, the generalized continuum hypothesis is not needed in case 'Y = N0, and this leads to a new proof of a result of M. Kat~tov (On real-valued functions in topological spaces, Fund. Math. 38 (1951), 90). (Received November 16, 1966.)

67T-135. P. D. MORRIS, Pennsylvania State University, University Park, Pennsylvania. Subspaces of continuous function spaces.

It is a result of Banach's that C([O,l]) is a universal space for separable Banach spaces. This is no longer true if [0,1] is replaced by an arbitrary infinite compact Hausdorff space X. From some results of A. Pelczynski and Z. Semadeni (Studia Math. 18 (1959), 211-222), C(X) is universal for separable Banach spaces iffi X contains a nonvoid perfect subset. With the aid of this result, the following can be shown. Theorem. Let X be compact and Hausdorff. Then C (X) is universal for separable Banach spaces iffi C(X) contains an isomorphic isometric copy of 2-dimensional Euclidean space. Concerning those Banach spaces which are embeddable in every C (X), X infinite, compact, and Hausdorff, the following holds. Theorem. For E a Banach space, the following are equivalent: (i) E is embeddable in every C (X), for X infinite, compact, and Hausdorff; (ii) E is embeddable in (c); (iii) the weak*-closure of the set of extreme points of the unit ball of the dual E* is of the form z u (- Z), where z is countable and has at most one weak* limit point, and (Z n (- Z)) \ jo l = jil. (Received November 18, 1966.)

254 67T-136. J. R. GEISER, Dartmouth College, Hanover, New Hampshire 03755. Nonstandard

Let Y = ( l:, 9Jl,f--, f=) be a 1st order predicate logic. Assume that Y is strongly complete. Let D be a nonprincipal, non w-complete ultra filter over an infinite set I. For each f E 2: 1, define 1'= {glgEI1 and {ilg(i)=f(i)fE Dl. Definer= lflfE2:1f. lf.\i2liE9Jl 1,let IT2l/Ddenotethe ultra product of .\i 2.{i. Let 9Jl* = {f1~/DI.\i 2li E 9Y.If. Define ~· = {(f,g)l{ilf(i)~g(i)l E Df and define I=* = I ( f12.1 i /D,f) II il2l iF f(i)l E D}. Conjunction, negation and quantification are defined in 2:* and may be iterated infinitely often. Y• = : ( 2;*, 9Jl*,}-- *,1= *) is called a nonstandard (1st order predicate) logic. I. Y* is sound, complete, not compact and not strongly complete. S 1 C 2:* is called admissible if 3F:I -> {SISCl:j suchthatS 1 = {tl{ilf(i) E F(i)fE Dj; if {iiF(i)finitejE: D then S 1 is called p. finite. II. If S 1 is admissible then S 11= •f implies S 1f-•f. III. For ~countable, I.D may be chosen so that for any admissible S l' if every p. finite subset of S 1 has an ultra power model then s1 has an ultra power model. IV. Let 9Jl have the usual topology. I,D may be chosen so that for every Borel set 5e C 9Jl, 3 f E ~· such that 5e= I 2.1 E 9Jl I~ ~I= •fl. Nonstandard logic has been studied by A. Robinson 1!-Jagoya Math. J. 22 (1963), 83·117] and the author !)Joctoral Thesis, Massachusetts Institute of Technology, 1966]. (Received November 21, 1966.)

67T-137. J. A. LUTTS, University of Massachusetts at Boston, Boston, Massachusetts. Topological spaces which admit complex-valued unisolvent systems.

Let .lf(X,C) be the family of all continuous mappings of a topological space X into the complex numbers C. A set F of n distinct functions in ~(X,C) is called a complex-valued unisolvent system of order non X (or, more briefly, an n-C-U system on X) if each nonzero linear combination of the elements of F over C has at most n - 1 zeros in X. If n ~ 2 and X has at least n points, it has been conjectured that X admits of an n-C-U system if and only if there is an injective map in :t'(X,C). (Cf. Schoenberg and Yang, Ann. Mat. Pura Appl. Ser. IV. 54 (1961), 1-12 where die conjecture is proved for X a compact polyhedron.) The conjecture proves true for the following compact spaces:

(a) X not connected, (b) X connected and locally connected, (c) X connected and dim X = 2 (in general dim X~ 2 for such spaces), and (d) X connected and having a cut point. (Received November 22, 1966.)

67T-138. J. S. FRAME, 136 Oakland Drive, East Lansing, Michigan 48823. Zeros of polynomials in a half plane. Preliminary report.

The Routh-Hurwitz methods for counting zeros .\j of a real polynomial with negative, zero or positive real parts, and the corresponding methods of Frank and Wall for a complex polynomial, require special treatment in exceptional cases when any of the Hurwitz determinants vanish. Our

modified method avoids these breakdowns. A given complex polynomial a0 P(.\) of degree n = n0 with leading coefficient a 0 is written uniquely as a 0P 0 + a0 a 1 P 1, where the ratio a 1P /P 0 is a so-called J-function that maps the imaginary axis J into itself, and where Pk denotes a monic polynomial of degree nk. A sequence of decreasing degrees n k and constants ak is determined by a Euclidean

algorithm construction P k-l = QkP k + akak+ 1P k+ 1, that terminates with the highest common factor n -nk Prof P 0 and P 1. Here the leading term .\ k-1 /ak of Qk/ak is a J-function. The product IT~= 1 ( .\nk-1 -nk + a .j is shown to have the same right and left half plane distribution of zeros as

255 P (\ )/P r (,\). Nonlinear quotients Qk cause the breakdowns in earlier methods. Replacement of

P r by P r + P ~ decreases by 1 the multiplicities of pure imaginary zeros, moving these into the left half plane for counting. (Received November 14, 1966.)

67T-139. ALEXANDER ABIAN, The Ohio State University, Columbus, Ohio 43210. A new example of a nonmeasurable set.

Definition. A nonempty set S of real numbers is called linearly independent if no finite linear combination of the distinct elements of S with rational coefficients not all zero is equal to zero. Lemma. If a linearly independent set of real numbers is Lebesgue measurable then its measure is equal to zero. Theorem. There exists a linearly independent set A of real numbers such that A has at least one point in common with every nondenumerable closed subset of the real line. The set A is Lebesgue nonmeasurable. (Received November 28, 1966.)

67T-l40. H. j. KEISLER, University of Wisconsin, Madison, Wisconsin 53705. A three cardinal theorem for w-logic.

Consider models 1'1 = (A, u 21 ,v~, R 21 , •.• ) of a countable first order logic, where u 21 and v21 are unary relations. 211u 21 is the submodel of 21 whose set of elements is U~. For cardinals K and 2 ordinals a., define 2(K,a.) inductively by 2(K,0) = K, 2(K,a.) = kf3

Theorem l. Suppose w ;;; IV 21 1 < IU~ I « lA 1. Then for any cardinal ,\ ;;:: w 1 there exist models sa.~ such that ~ and ~ are elementary extensions of sa' IB I= w' Vsa = v~ ' and IV~ I= w' IU ~ I= w1, IC I= ,\. From now on suppose Tis a theory either in first order logic, w -logic, or weak second order logic. Corollary. If T has a model 21 with w ;;, IV~ I < IU 21 I« lA 1. then for any

X > w 1, T has a model~ with IV~ I= w, IU~ I= w 1, I~ I= .\. This adds to a theorem of Vaught and a theorem of Morley in Theory of models, Proc. of the 1963 Berkeley Symp., Amsterdam, 1965.

T is eventually homogeneous in power K if for all models ~ of T with K = IU ~ I« lA I, ~ 1u 21 is homogeneous. Assume hereafter that T is eventually homogeneous in power w1• Theorem 2. Tis eventually homogeneous in every power > w1• T is eventually categorical in power K if for all models 21' sa ofT with K = IU 21 I« lA I and K = IU sa I« IB I. 21IU21 ~ sa IUsa. Theorem 3. If Tis eventually categorical in power w1, then Tis eventually categorical in every power > w1• (Received November 22, 1966.)

67T- 141. H. J. KEISLER and j. H. SILVER, University of Wisconsin, Madison, Wisconsin 53 706. Well-founded extensions of models of set theory.

Consider models 21 = (A,E) where E is binary. Let Ord( 21) = (set of ordinals of 21, E). A p-extension of 21 is an extension sa of 21 preserving all formulas with < p alternations of quanti­ fiers. 'B is an end ex tens ion of ~ if Ord( 21) is a proper initial segment of Ord ('B). Theorem l.

Let 21 be a well-founded model of ZFC (Zermelo- Fraenkel with choice) such that, Ord( ~) is not cofinal with w. If 21 has an end p-extension which is a model of ZFC, then 21 has a well-founded end p-extension. Corollary. If 01 is the first strongly inaccessible cardinal > w, then (. R(O 1), k) has no end elementary extension. Theorem 2. Suppose ,\ is weakly inaccessible. Then for every cardinal a., w < a. < .\, there is a p < w and a well-founded model 21 of ZFC of power a. which has

256 no end p-extension. Theorem 3. Let 21 be a well-founded model of ZFC which "contains" all of its countable subsets. If 21 has an w -standard p-extension \B such that \B is a model of ZFC and Ord (~) is not cofinal in Ord (\B), then 21 has a proper well-founded p-extension. There are generalizations to models of ZFC with extra predicates and to "a.-well-founded" models. (Received November 28, 1966.)

67T-l42. PAUL GAUTHIER, Wayne State University, Detroit, Michigan 48202. Sequences of p-points of meromorphic functions.

Let f(z) be a function meromorphic in the unit disk \z \ < 1. If there is a sequence of non­ Euclidean disks jonl with non-Euclidean centers jznl and non-Euclidean radii jrnl , where rn tends to zero, such that for each n, the image of On under f(z) covers every point of the sphere with

the possible exception of two sets Kn and K~ whose diameters tend to zero with n; then jznl is called a sequence of p-points for the function f(z). (See Lange, Surles cercles de remplissage non-Euclidiens, Ann. Sci. Ecole Norm. Sup. 77 (1960), 257-280.) It is shown that a sequence jzn l of points of the unit disk is a sequence of p-points for a meromorphic function f(z) if and only if iznl is a sequence of P-points for f(z) in the sense of Gavrilov (On the distribution of values of nonnormal meromorphic functions in the unit disk (Russian), Mat. Sbornik 109 (1965), 408-427). It follows from Gavrilov's criterion for normalcy that a meromorphic function f(z) is normal in the unit disk if and

only if it does not possess a sequence of P-points. (Received November 28, 1966.)

67T-l43. MARY POWDERLY, Fairfield University, Fairfield, Connecticut. An answer to a question of Rudin.

M. E. Rudin in the Proc. Amer. Math. Soc. 16 (1965), 1320-1323, proposed the following

problem: Let R be the set of all points (x,y) in the plane such that 0 ~ x ~ l and 0 ~ y ~ x. Let

T = R - j(O,O)I. Let F be the set of all continuous real valued functions whose domain is the set of

all positive numbers < l and whose graph lies in T. There is a natural partial ordering for the ele­

ments of F: If f,g E F, define f < g if there exists an f > 0 such that for all x < f, f(x) < g(x). The question is: If the word "continuous" is deleted from the definition ofF, is there a totally ordered (by < ) subset of F of cardinality greater than that of the continuum? The answer is negative. (Received November 28, 1966.)

67T-l44. WITHDRAWN. 67T-145. R. F. MATHIS, The Ohio State University, Columbus, Ohio 43210. Completion of a

symmetrical unitary matrix.

Theorem. Given a square matrix A and a rectangular matrix B with the same number of rows as A, the rectangular matrix (A B) can be extended to a symmetrical unitary matrix if and only if A= AT and AA * + BB * = I. It is necessary to find a symmetrical matrix X such that (B T X) is the

rectangular matrix with the rows necessary to complete (A B). Let B l' X l' and I 1 denote the first columns of B, X, and I, respectively. It is always possible to find a column matrix a such that

BB*a = B 1. Then x 1 = BT(ii- Aa)- I 1• (Received january 17, 1967.)

67T-146. D. R. BROWN and MICHAEL FRIEDBERG, University of Tennessee, Knoxville, Tennessee 37916. On certain finite-dimensional divisible semigroups.

LetS be a commutative uniquely divisible metric clan with E(S) = lo,Ij and H(l) = III. Theorem. If S has dimension n, then S is iseomorphic to the one-point compactification of closed positive cone in En. In particular, S is (topologically) ann-cell and contains n algebraically indepen­ dent usual threads. This is a topological analogue to some results in (Brown- LaTorre, Pacific j. Math. 18 (1966), 57-60) and an extension of results of]. Hildebrant (to appear). (Received November 15, 1966.)

67T-147. M. S. RAMANUJAN, University of Michigan, Ann Arbor, Michigan 48104. Hausdorff moment problems for operators in locally convex spaces.

For notation and terminology, see M. S. Ramanujan [Math. Ann. 159 (1965), 365-373] and

K. Swong [Math. Ann. 155 (1964), 270-291]. Given a sequence ~~nl of continuous linear operators on a Fr~chet space E into a locally convex space F, define the Hausdorff matrix H = (H,~n) by Hnk = Cn,k.!ln-k~k (n ~ k) and= 0 (n < k). Define ~~nl to be a weak moment sequence if there exists avec­ tor measure 1/;: 2;---> L(E,F~) such that (a) for each z' E F', the set function z'---> 1/;z, is continuous for o(F', F) and u(M(2":,E'), C(I,E)); (b) 1/;z' is of bounded p-variation for each seminorm p generating the topology onE and (c) for n = O,l, ••• ,(~nx,z') = Jf(tllx, dt/;z,> for xE E and z' E F', where I = [0, 1]. The main result of the paper is that I ~nl is a weak moment sequence if and only if the corresponding Hausdorff matrix defines a convergence preserving transformation on E into F, where F has the weak topology u(F,F') and is weakly sequentially complete. The proof is via Swong's theorem for the representation of continuous linear operators on C(I,E) into F and the author's earlier results on Kojima matrices of operators. (Received November 16, 1966.)

67T-148. N. H. SCHLOMIUK, McGill University, Montreal 2, Canada. Principal co-fibre bundles in the category of simplicial groups.

We develop a theory of principal co-fibre bundles which is dual in the sense of Eckmann-Hilton to the theory of principal fibre bundles with a group G. This is done in the category of simplicial groups. We prove that any principal co-fibre bundle is a twisted free product between the co-fibre FK and the co-base A. This structure theorem is used to obtain a classification theorem which states that any principal co-fibre bundle of co-fibre FK is induced from the universal bundle with co-base GK and total space GCK. Two bundles are equivalent if the inducing maps are homotopic. (Received November 21, 1966.)

258 67T-149. D. L. BURDICK, University of New Mexico, Albuquerque, New Mexico. Distribution free tests for symmetry based on the number of positive sums. Preliminary report.

Let X 1, .•• ,Xn be independent random variables having a common, continuous, symmetric dis­ tribution function F and let K be a subset of the power set of (l,Z, •.• ,n j. From K a random variable N(K) is constructed by defining, for V C (l,Z, •.• ,nj, S(V) = LiEVXi then N(K) is the number of posi­ tive sums among (~(V): V E Kj. It has been proved that N(K) is independent ofF, that is distribution free, for: (a)IK = j(1j, l1,2j, •.•• 11,2, .•. ,njl by Sparre-Andersen, (b>!K =(all nonempty subsets of (l,2, •.• ,njj by Kraft and van Eeden, (c) K =any subset of the power set of 11,2, •.• ,nj which contain sets with only one or two elements by L. H. Koopmans, M. Katz, N. Friedman, (d) K = I all nonempty subsets of 1!,2, ••• ,nj except (1,2, ..• ,njj by L. H. Koopmans. A unified proof of the proceeding theorems is given by using geometric arguments. As a consequence of the proof, new examples of sets K for which N(K) is distribution free are constructed. The most interesting examples are: K = I all nonempty subsets of (1, 2, .•• ,nj with an even number of elements j and K = IIll, i 1,2}, ..• , ll,2, ..• ,n),ln),(n,n- lj ..... ln,n- 1, ••• ,2!). (ReceivedNovember2l,l966.)

67T- 150. EDWARD SILVERMAN, Purdue University, Lafayette, Indiana. A weak projection of C onto a Euclidean subspace.

Let B be a normed vector space. If bl'''"'bm E B then 11\b I = lb 1/\b zl\ ... /\bm I = sup(det[fi(b.)]if 1, ..• ,fm are linear functionals over Beach having norm one). If B is Euclidean, J then 11\bl is the volume of the parallelpiped spanned by b 1, •.• ,bm. Let C be the space of continuous functions on the (n - !)-sphere. Theorem. There exists En C C and a projection P of C onto En such that IPb/IPb/'1 .•. /\Pbkl ::;o 1/\bl whenever k > l. The main tool is, essentially, the following inequality: Let fi(x) = ab/2 + ,LJa~ coshx + b~sinhx], i = 1,2. Then la~b~- a~b~l ::;o max[f 1(x)f2 (y)­ f 1(y)f2 (x)]. A Kirzbraun-type theorem is then proved for C to show that a generalization of Lebesgue area, previously known to agree with Lebesgue area only on surfaces for which.,. lower area agreed with Lebesgue area, is, in fact, an extension of Lebesgue area. (Received November 23, 1966.)

67T-15l. B. B. WELLS, JR., 730 Blossom Way, no. 32, Hayward, California 94541. Weakly converging sequences of measures. Preliminary report.

Let X be an arbitrary compact Hausdorff space. An open set U is called regular if U = int 0. Theorem. If a sequence of regular Borel measures converges on each regular open set of X then it is convergent for the u(M(X), M*(X)) topology. This theorem with regular open replaced by open has been proved in the case of X metric by Dieudonn~ (see Ann, Acad, Brasil Cien. 20 (1951), 277-282) and by Grothendieck (see Canad, j. Math, 5 (1953), 129-173) in the case of X arbitrary compact Hausdorff. Our result is a generalization to arbitrary compact Hausdorff spaces of a theorem of Grothendieck (ibid.) which states that on an extremely disconnected space a sequence of regular Borel measures converges for the u(M(X), M*(X)) iff it converges for the u(M(X), C (X)) topology. In an extremely disconnected space the regular open sets are precisely the open closed sets, and a sequence of measures converges for each open closed set iff it converges for the u(M(X), C(X)) topology. Grothendieck's theorem is equivalent to a lemma of R. S. Phillips (see Trans. Amer. Math. Soc. 48 (1940), 516-541), and thus our result also gives a new proof of Phillips' lemma. (Received

November 18, 1966.)

259 67T-152. ADI BEN-ISRAEL, Northwestern University, Evanston, Illinois 60201. On the geometry of subspaces in Euclidean n-spaces.

The intersections of linear manifolds in En and the inclination between subspaces in En are studied. Sample results: Theorem. Let L, M be subspaces in En. Then L nM = PL x- PL(PL +PM)+ (PL x- PMy), x,y, range over En. Theorem. If the matrix c• =(A*, B*) is . - 1 + - 1 + +) s s b nonsmgular then C =(A ,0) + PN(A)(PN(A) + PN(B)) (-A , B . Theorem. Let 1, 2 e two subspaces in En, and A 1, A 2 matrices satisfying Sj = R(Aj), j = 1,2. Then the nonzero eigenvalues of A 1 A~A 2 A1 are the squares of the cosines of angles of inclination between s 1 and s2 in tire sense of Zassenhaus, [Amer. Math. Monthly 71 (1964), 218-219]. (Received December 1, 1966.)

67T-153. JOHN MILLER, University of Massachusetts, Boston, Massachusetts. On the resolvent of a linear operator associated with a well-posed Cauchy problem.

Theorem. If A is any m X m matrix satisfying j(zi - A)- 1 j ~ C/Re z, Re z > 0, then there exists a partition of its eigenvalues into cells C 1 ••.. ,Cr such that any cell Ci either contains a single point on the line Re z = 0 or is surrounded by a contour Y in the half plane Re z < 0 on which I 2 j(zi- A)- 1 j $ (KC)/o. where o. is the distance of C. from Rez = 0 and K = (12m)m +Zm (Received - I I I December 1, 1966.)

67T-154. J.P. SHANAHAN, Boston College, Chestnut Hill 67, Massachusetts. Convergence of successive approximations and existence of examples of Mueller type.

Consider the initial value problem y' = f(x,y), y(x0 ) = y0 and define (Ty) (x) = y0 + J~~f(s,y(s))ds, We are concerned with the convergence or divergence of the Picard successive approximations. It is well known that if f satisfies a Lipschitz condition in y then T is a contraction under the sup norm provided the interval of x values is sufficiently small. If the norm is changed a result in the large can be obtained as in Bielecki [Bull. Acad. Polan. Sci. 4 (1956), 261-268]. It is here shown that the same result can be obtained, while retaining the sup norm, by showing that Tn, for sufficiently large n, is a contraction. The Mueller construction is an example of an initial value problem for which (1) f(x,y) is nonincreasing in y for fixed x and (2) T 2 has nonunique fixed points and for which (3) the Picard successive approximations diverge. If (2) is changed to (2') T 2 has a unique fixed point then (3) is correspondingly changed to (3') the Picard successive approximations converge. A simple argument shows that there are no examples of Mueller type for a class of functions f(x,y) satisfying well-known conditions ensuring uniqueness of solutions. There is a con­ nection with the analogue of Kamke's theorem. (Received November 21, 1966.)

67T-155. M.A. DOSTAL, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012. Growth of entire functions and problem of singular support of distributions.

Let

260 on the line only, we find a condition on ;j; (= Fourier transform of cJ>) which is sufficient for valldity of the above equality with an arbitrary >¥ E ~· (R 1). The condition consists in some regularity of growth of the entire function .j; on a family of logarithmic curves. The proof is based on a theorem analogous to a theorem of Vladimir Bernstein. (Received November 15, 1966.)

67T-156. ALBERT WILANSKY, Lehigh University, Bethlehem, Pennsylvania 18015. Topics in functional analysis.

Results obtained in a seminar at Lehigh: The intersection of two vector topologies, T, T' need not be a vector topology. If T, T' are convex, TnT' is a vector topology iff it is convex. A W­ barreled space (= t property, Wilansky: Functional analysis p. 224) is ultrabarreled but not con­ versely. ]. D. Weston's comparison of topologies (J. London Math. Soc., 1957) is extended to almost open maps; f 0 g need not be almost open even iff is an almost open homeomorphism and g open, 1-1, onto; or if f,g are both almost open, linear, 1-1, onto. (The second example uses the exotic topologies of Klee.) A topology is maximal with respect to Weston's comparison iff S is open when­ ever S C (S)i; such topologies are irresolvable and have only trivial convergence of sequences.

T A T' is separated iff the identity has closed graph; this is a special case of separation of inductive limit iff quotient map from direct sum has closed kernel. The inductive limit of n spaces is sepa­ rated if n = 2 and the range has a T 2 topology T making the maps continuous, but not if n = 3 unless, in addition T has closed graph addition. A bornological space may have a sequential neighborhood of 0 which is not a neighborhood, (not convex, of course.) (Received December 2, 1966.)

67T-157. T. ]. HEAD, University of Alaska, College, Alaska. Tensor sequences of Abelian

~- a Let p be a prime, G an abelian group, and (*)0-> A-4 B ->C -+0 an exact sequence of abelian groups. We are concerned with (**) 0 -• G 0 A-> G 0 B-+ G 0 C _, 0. Definition: (*) is p-subpure

if a= pnb (a E im a, bE B) implies pka = pn+ka 1 (k some positive integer, a 1 E im a). Theorem 2. (**)is exact iff for each summand of G of the form Z(pn) we have im an pnB = pnim a and, for each

summand of G of the form Z(p00 ), (*) is p-subpure. Theorem 3. For G torsion, (**) is exact and splits iff (**) is exact and pure iff for each p-primary component P of T: (1) if P has an unbounded basic subgroup then im a n pnB = pnim a for all positive n and (2) if n is the largest positive integer for which P contains a cyclic summand of order pn then im an pkB = pkim a (l ;;; k ~ n) and if P is not reduced then (*)is p-subpure. Theorem 2 can be proved by applying known special cases to two canonical exact sequences. Alternatively, a proof can be based on: Theorem 1. Each abelian group G is the direct limit of the inclusion directed family of its subgroups of the form: finitely generated free abelian EB finite number of indecomposable primary groups where each of the indecomposable primary groups is isomorphic to a summand of G. (Received December 2, 1966.)

261 67T-l58. WILLIAM MEYERS, Tulane University, New Orleans, Louisiana 70118. Mantel subalgebras of the plane. Preliminary report.

An F -algebra in which every bounded subset is relatively compact is called Mantel. Let G be a polynomially convex domain in a::; it is known that the closure P{Gn) in the compact-open topology of polynomials on the subset Gn of Cn is precisely the algebra O{Gn) of holomorphic functions on Gn and thus Mantel. This fact together with the following theorem provides infinitely many nonisomorphic Mantel subalgebras, each containing O{G), of the F-algebra C{G) of continuous complex-valued func­ tions on G. Theorem. If D is any domain in C and n is a positive integer, then there is a subalgebra A of C{D) containing P{D) which is algebraically and topologically isomorphic to P{Dn). On the other hand, if A is a Mantel subalgebra of C{G) containing O{G) and having continuous homomorphism space homeomorphic to G, then A= O{G). {Received December 2, 1966.)

67T-l59. W. D. L. APPLING, North Texas State University, Denton, Texas. Three theorems on summable set functions.

Suppose F is a field of subsets of a set U, R is the set of all real-valued functions defined on F, R + is the set of all nonnegative-valued elements of R, R 1 is the set of all finitely additive elements of R +, and for each q in R ~, C q is the set of all p in R: absolutely continuous with respect to q, W ~ elements of R, W = W* R+, and s is the "q-snmmability operator" is the set of all "q-summable" q q n q defined on w; {see Abstract 65T-l60, these c}/oticei) 12 {1965), 365). Theorem l. Suppose m in R :. H in R, I H.l 00 a sequence of elements of w• and r in R + n C such that; {l) sm (IHn i){V) s r{V) 1 1= 1 m A m - for all V in F and all n, and {2) If 0 < min jc,d I• then there is a positive integer N such that if n is a positive integer ;?; N, then there is a subdivision On of U such that if E is a refinement of On' then

L:E.m{I) < d, where E* = !III in E, IH {I)- H{I)i !?; cl. Then His in w• and s 0, then Wq is a proper subset of R +, and {2) If each of m and p is in R: and W m <;;; W p' then p is in Cm. {Received December 2, 1966.)

67T-l60. S-S. H. BUTT, c/o San Joaquin General Hospital, Box 42, French Camp, California. On certain isomorphisms between maximal nondetermining subalgebras of group algebras.

A subalgebra A C L 1(G) is called maximal nondetermining if it is maximal relative to the property that A is not uniformly dense in c 0 {f), where G is a LCA group and r = G. Theorem. Let G be a discrete LCA group. Let Ai be a maximal nondetermining point-separating subalgebra of

L 1 {G) such that the largest closed ideal I. of L 1 {G) contained in A. has hull E., i = 1,2. If further, 1 1 1 the restriction of the uniform closure of i\ to Ei is a maximal function algebra on Ei for each i, then A 1 is isomorphic to A2 implies that A /I 1 is isomorphic to A2/I2" The proof uses the induced mapping theorem and the fact that an essential maximal function algebra is an integral domain.

Corollary. Let Ai and Ei be as in the Theorem. Then A 1 is isomorphic to A2 implies that E 1 is homeomorphic to E 2• Application. Let G = Z 00 , the direct sum of countably many copies of integers. Then L 1 {G) has a maximal and nondetermining subalgebra A, namely, the inverse image of the Fourier transform of a maximal function algebra on G = Tw that is essential on certain arc of Tw.

262 Hence A is not isomorphic to any maximal and nondetermining subalgebra B of L 1 (G) such that B- is an essential maximal function algebra when restricted to a Cantor set or a closed subgroup of Tw. (Received November 25, 1966,)

67T-l6l. R. j. BEAN, The University of Tennessee, Knoxville, Tennessee, Extending decompositions of E 3 to make them "nice". Preliminary report.

Bing has announced the following result (Decompositions of E3 , Topolo~y of 3- Manifolds, M. K. Fort, jr., (ed.), Prentice Hall, 1962). If G is a finite collection of disjoint nonseparating continua in E 3 then there is an upper semicontinuous decomposition G' of E 3 such that the members of G are members of G' and the decomposition space is E 3• We extend this result to any collection G of nonseparating continua which form (along with the points not covered by G) an upper semi­ continuous decomposition of E 3 which is definable by manifolds-with-boundary. Also we prove that we may fix up certain decompositions (satisfying stronger conditions) in the following way. (We state this theorem for a particular decomposition.) Let G be the dogbane decomposition. Then there is an upper semicontinuous decomposition G' such that the nondegenerate elements of G' miss those of G, the decomposition space of G' is E 3 and the decomposition space of G' t G is E 3. The decomposition G' untangles the nondegenerate elements in G without shrinking them. (Received November 25, 1966.)

67T-162. HABIB SALEHI, Michigan State University, East Lansing, Michigan 48823, On the Hellinger integrals and their applications to q-variate stochastic processes.

Let (xkF be a q-variate stationary SP with the spectral distribution measure F defined on -oo Jt will denote the subspace spanned by (xk)00 • For each the Borel family~ of subsets of (- 1r, 1r]. oo -oo

subset K of integers JtK will denote the subspace spanned by xk, k E K, and .A'K = A:-00 n1~c· Definition. (xk)~00 is called (i) interpolable if .A'K = I 0} for ;ach bounded K, (ii) minimal if 1k} f. I 0} for each k. For any matrix-valued measure M on~; J_7r((dMdM*)/dF) will denote the Hellinger integral of M with respect to F if it exists (this notion is properly introduced and studied). Theorem 1. (xk)~00 is interpolable iff for any trig-polynomial P, either f_:((dMpdMf,)/dF) = 0 or 7r ·o f_7r((dMpdMf,)/dF) does not exist, where for each B E ~ , Mp(B) = JBP(e1 )dO. Theorem 2, Let z 0 be the orthogonal projection of x0 onto -1 o}· Then (a) (zO'zO) = [l/27r J: ((dM JdMJ)/dF)r, where J is the projection operator on the space of q-tuples of complex numbers onto the range of (z0 ,z0 ) and [ T is the generalized inverse of [ ], (b) (xk)~00 is minimal iff I-:((dMJdMJ)/dF) f. 0. The Hellinger integrals also have interesting applications in the theory of detection of a signal in the presence of several stationary Gaussian noises. (Received November 16, 1966,)

67T-l63. D. R. LaTORRE, University of Tennessee, Knoxville, Tennessee 37916. The Brown- McCoy radicals of a hem iring.

The concept of the F-radica1 of a ring, and a specialization thereof known as the Brown-McCoy

radical, are studied for hemirings (additively commutative semirings with 0). The k-ideals, h-ideals, and hemirings of type (H) discussed in, On h-ideals and k-ideals in hemirings, Publ. Math. Debrecen 12 (1965), 219-226, are used, The main results concern the H-radical R of a hemiring S of type (H).

263 In caseS is a ring, R is just the Brown-McCoy radical. It is shown that R is the intersection of all h-ideals M of S such that a factor bemiring modulo M is a simple ring with identity, and that the jacobson radical J of S is contained in R. If S is additively regular or additively periodic, and satisfies the d.c.c. for right h-ideals, then J = R. An analogue of the Wedderburn-Artin theorem is given which states that if S satisfies the d.c.c. for h-ideals, then R = 0 if and only if S is semi­ isomorphic (homomorphic with kernel 0) to a direct sum of finitely many simple rings with identity elements. It is also shown that if S has an identity, then S is of type (H) if and only if the bemiring Sn of all n X n matrices over S is of type (H); and in this case, the H-radical of Sn is just Rn. (Received November 16, 1966.)

67T-l64. F. R. DEUTSCH and P. H. MASERICK, The Pennsylvania State University, 230 McAllister Building, University Park, Pennsylvania 16802. Applications of the Hahn-Banach theorem in approximation theory.

It is shown that a large number of results in the general theory of approximation in normed linear spaces can be deduced from a single geometric implication of the Hahn- Banach theorem: Lemma. If K is a convex subset of the normed linear space X and x EE K, then there exists a continuous linear functional f EX* such that llfll = 1 and infyEKIIx- Yll = Ref(x)- sup Ref(K). In geometric lan­ guage, there is a hyperplane H separating K from x whose distance from x is the same as the distance from K to x. This result is the main tool in proving: (!) a characterization of best approximations in K to x, (2) necessary and sufficient conditions that each x EX have at most one best approximation in K, (3) duality theorems for determining the distance from a point to a convex set. Results of Ivan Singer, A. L. Garkavi, and others are recovered. (Received November 17, 1966.)

67T-!65. WEI-EIHN KUAN, Michigan State University, East Lansing, Michigan. The hyper- plane sections through a normal point of an algebraic variety.

Let V be an affine variety (in Wei!' s sense) of dimension > 2 defined over a field k and let P be a k(P)-normal point on V. The following theorems are proved: Theorem I. Let V and P be as above. If k is of characteristic 0, then almost all hyperplane sections of V through P are irreducible, absolutely irreducible and k(P)-normal at P. Theorem 2, Let V and P be as above. If characteristic of k is p i 0 and k is infinite, and assume that not all tangent hyperplanes of V pass through P, then almost all hyperplane sections of V through P are irreducible, absolutely irreducible and k(P)­ normal at P. (Received November 18, 1966.)

67T-!66. J. A. HILDEBRANT, Louisiana State University, Baton Rouge, Louisiana. On compact divisible abelian semigroups.

A semigroup S is said to be [uniquely] divisible if for each x E S and each positive integer n, there exists an [unique] element y E S such that yn = x. Let N denote the set of all positive integers. Theorem. Let S be a compact divisible abelian semigroup. For each n E N, let Sn = S and define Tm: S ---+ S by Tm(z) = zm.'/ n.I for m > n. Then s = proj lim(S , Tm ,N) is a compact uniquely n m n n 0 n n divisible abelian semigroup. Moreover, if S is uniquely divisible, then s0 is iseomorphic to S. Corollary 1. Each compact divisible abelian semigroup is the continuous homomorphic image of a

264 compact uniquely divisible abelian semigroup. Corollary 2. Let S be a compact abelian semigroup whose subgroups are totally disconnected. Then S is divisible if and only if each nonidempotent of S lies on a usual or Calabi semigroup inS. Moreover, S is uniquely divisible if and only if each non­ idempotent of S lies on an unique usual semigroup inS. (Received November 18, 1966.)

67T-l67. PETER DUREN, University of Michigan, Ann Arbor, Michigan. Linear functionals on Hp spaces with p < l. Preliminary report.

Although Hp is not a Banach space if 0 < p < l, bounded linear functionals can be defined in the usual way. S. S. Walters [pacific J. Math. l (1951), 455-471] gave a unique representation for the general functional t/> E (HP)•. This may be expressed in the form t/>(f) = limr -d J~1Tf(reill)g(eill)dll, where g(z) is analytic in lz I < l and continuous in lz I ;;: l. The following theorem extends Walters' results. Suppose 1/(n + l) < p < 1/n (n = 1,2, ••• ). If g(z) is associated as above with a functional tf> E (HP)•, then g(n-l)(z) is continuous in lz I ;;: l and g(n-l)(eill) belongs to the Lipschitz class

A , a= 1/p - n. Conversely, if g(z) is any function analytic in lz I < l, with g(n-l)(z) continuous a in lz I ~ l and g(n-l)(eill) E A a for some a > 1/p - n, then the above limit exists for each f E Hp and defines a bounded linear functional on HP. (Received November 22, 1966.)

67T-l68. R. J. GREECHIE, University of Massachusetts at Boston, 100 Arlington Street, Boston, Massachusetts. Hyper-irreducibility in an orthomodular lattice.

Let L be an orthomodular lattice. A block in L is a maximal Boolean sub-orthomodular lattice of L. It is shown that D. E. Catlin's hyper-irreducibility (H-1) condition (cf. Abstract 642-56, these c/'foticei) 14 (1967), 77) is equivalent to the statement that the blocks separate the elements of L. Since H-I reduces to the atomic bisection property in an atomic (OM) lattice, the conjecture that H-I is equivalent to the condition "The blocks separate the atoms of (an atomic) L" seems reasonable. We exhibit a counterexample to this conjecture. (Received November 25, 1966.)

67T- 169. UBIRAT AN D'AMBROSIO, University of Rhode Island, Kingston, Rhode Island 02881. On surfaces of limit type.

Let U be the unit square f9,1] X [0,1], N an integer and a a real number. Then we consider surfaces in Rn of the type (T,U,N,a): wE U----> Lr=l k-a.pk(w), wherew being (u,v), pk(w) isthevec­ tor (x~(u).y~(v); i = l,2, .•. ,n), with x~,y~. i = 1,2, ..• ,n, all k, real valued continuous functions on [0,1).

Now consider limits of such surfaces, i.e., (T,U,a) = limN_, 00 (T,U,N,a). Then we have the following i i Theorem. Suppose the xk' yk' i = l,2, ••• ,n, all k, have equally bounded total variation on [0,1], and a > 2. Then (T, U, a) is a rectifiable surface. (Received November 25, 1966.)

67T-l70. K. T. HAHN, The Pennsylvania State University, University Park, Pennsylvania. Laws of sines and cosines on a bounded domain in en with a constant sectional curvature. Preliminary

report.

Let u(ua), v(va) be two vectors at a point z(z a) in a bounded domain D of en (n ~ 2) with the Bergman metric M(D):ds~= Tapdzadzf'l, Taj'i = iJ 2logKD;azaiJz/3, a,/3 = 1,2, .•. ,n, where KD is

265 the Bergman kernel of D. Here the summation convention is used. The analytic angle F between u and vis defined by cos F = I ~.v] 1/lullvl, [u,v] = Ta.f/o.vf:J, lul2 = ~.u], while the corresponding ordinary angle fll is given by cos @ = Re [u, v ]/ lu llv 1. A noneuclidean triangle in (D,M) consists of three sides given by the shortest geodesic and three angles given by the analytic angle. The length of a side is measured by M(D). The author proves: Theorem. Let ei be the length of a side of a noneuclidean triangle 6. in the hyper sphere H in en and F i the angle of 6. opposite to ei (i = 1,2,3). 2 2 1 2 (l) The law of sines holds: sinh(e/(n + 1) 11 )/ sin F 1 = sinh(e2/(n + 1)-1 )/sin F 2 =sinh (e3/(n + l) 1 )-t­ 1 2 1 2 1 2 sin F 3• (2) The law of cosines: cosh (e3/(n + 1) 1 ) =cosh (e 1/(n + l) / )cosh (e2/(n + 1) / )- 1 2 1 2 sinh (e/(n + 1) / ) sinh (e2/(n + 1) 1 ). cos F 3 holds if and only if F 3 =6 3• In particular, if F 3 is the right angle, ~h~n we obtain the no1~~uclidean Pythagorean theorem: cosh (e/(n + 1) 112) = cosh (e 1/(n + 1) I ) cosh (e2/(n + l) ) (see Duke Math. j. 33 (1966), 523). It follows from a result by K. H. Look that the same theorem holds on a bounded domain with a constant sectional curvature where the Bergman metric is complete. (Received November 25, 1966.)

67T- 171. ELINOR LERNER, University of Rochester, Rochester, New York 14627. Consistency of the generalized Souslin hypothesis. Preliminary report.

A tree, T, is a partially ordered set such that if x ~ z and y ~ z, then x and y are comparable. A chain in T is a linearly ordered subset. An anti-chain in T is a set of pairwise incomparable ele­ ments. An Ko. -Souslin tree is a tree of power X a. such that every chain and every anti-chain is of power less than xo.. Souslin's hypothesis is: there are no X1-Souslin trees. Tennenbaum and Solovay have shown, by means of an iterated Cohen-type extension, that it is consistent with contemporary set theory to assume that there are no x 1-Souslin trees. We use their method to show that, for Xo. regular, it is consistent with set theory to assume that there are no Xo.-Souslin trees. (For informa­ tion about the generalized Souslin problem see Gillman, Ann. of Math., val, 56, p. 446; Frayne, Abstract 623-47, these cifotiaiJ 12 (1965), 448.) (Received November 25, 1966.)

67T-172. R. K. GOODRICH, 1760 Athens Street, No. 4, Boulder, Colorado 80302. A Riesz representation theorem in the setting of locally convex spaces.

Let H be a compact Hausdorff space, and let E and F be locally convex topological vector spaces over the real or complex field where F is a Hausdorff space. Let C(H,E) be the space of continuous functions from H into E with the topology of uniform convergence. If T is a continuous linear transformation from C(H,E) into F then the problem is to find an integral representation for T. An additive set function K is defined on the set of differences of closed G-deltas having its values in a space of continuous linear transformations from E into F+, see Tucker's paper (Proc. Amer. Math. Soc. 16 (1965), 946-953). Then using only the differences of closed G-deltas in the definition of inte­ gral it is found that JdKf converges in the E00 topology (see Robertson and Robertson, .:!:£Eological vector spaces, p. 71) for every fin C(H,E) and T(f) = JdKf. (Received November 14, 1966.)

266 67T-l73. WITHDRAWN

67T-l74. CARL EBERHART, University of Kentucky, Lexington, Kentucky 40506. Dendritic. extensions of clans.

A clan is a continuum on which there is given a continuous associative operation with identity I. Let K be a clan contained in a metric continuum X. Then X is called a dendritic extension of K pro­ vided: (i) each component C of X\ K is open with a one point boundary and C • is a tree, (ii) the boundary of X\ K is totally disconnected, (iii) some component of X\ K has 1 as its boundary. Thea­

~ I. Every dendritic extension of a clan admits a clan structure. A theorem of A. D. Wallace states that every !-dimensional locally connected metric clan contains at most one simple closed curve. The converse of Wallace's theorem is a corollary to Theorem l. (Received November 15, 1966.)

67T-175, G, D. CROWN, Western Michigan University, Kalamazoo, Michigan 4900 I. On the category of lattices with residuated mappings.

A mapping from a poset P to a poset Q is said to be residuated if the inverse image of a principal ideal in Q is a principal ideal in P. The class of posets with residuated mappings is a category. Let !zf be the category of lattices with 0 and 1 with residuated mappings as the morphisms. Then !zf is a semiadditive balanced category with enough kernels, cokernels and finite biproducts. Let Y be the category of complete lattices with residuated mappings. Then Y is a semiadditive balanced category with enough kernels, cokernels, biproducts, injectives and projectives, A lattice L is injective in Y, or !zf, if and only if its dual is projective in il, or !zf. If L is a completely dis­ tributive lattice in Y then L is injective and hence projective, If L is an absolute subretract in Y, or !zf, then L is a distributive lattice and dually. Corollary. If L is a finite lattice, then L is injective in Y if and only if L is distributive and dually. (Received November 15, 1966,)

67T-176. W. GUSTIN and SEYMOUR SHERMAN,Indiana University, Bloomington, Indiana 4740 I. Paths in graphs. Preliminary report.

Consider a graph G with n vertices. Let V j be the number of j-stepped paths in G. Thus V 1 is twice the number of edges in G. Let B j = (n- 1v j//j. Theorem • ..!!. j ~ k and k is even, then B j ~ Bk. A counterexample shows that the condition k even is necessary, The corresponding Theorem for

267 closed walks is true and obviously fails for k odd. Conjecture. If D is a directed graph such that for each vertex the number of edges leading in is equal to the number of edges leading out, then the cor­ responding Theorem is true. The Theorem was motivated by a lower bound for a counting problem in statistical mechanics. (Received November 21, 1966.)

67T-l77. J. C. OWINGS, University of Maryland, College Park, Maryland 20408. Two splitting theorems for metarecursively enumerable sets.

For definitions of undefined terms, see G. Kreisel and G. E. Sacks, Metarecursive sets, J. Symbolic Logic 30 (1965), 318-338. L is the set of recursive ordinals; N is the set of nonnegative integers. A :;;, MB means A is metarecursive in B; A :;;, wB means A is weakly metarecursive in B. Sacks calls a set A <; L completely regular if, for all B <; L, B :;;, wA implies B is regular. If A is

a I1 1. set, then A is simple if there is no infinite rr. set C such that C <; N - A. Theorem l. If A is a ------1 nonmetarecursive regular meta-r.e. set, there exist disjoint completely regular meta-r.e. sets B(O), B(l) such that A i wB(O), A { wB(l) and A= B(O) UB(l). Corollary. Every nonmetarecursive meta-r.e. metadegree is the least upper bound of two incomparable completely regular meta-r.e. metadegrees. Theorem 2. If A is a simple~ set there exist disjoint rri sets B (0), B (l) such that A= B(O)UB(l), B(O):;;, MA, B(l):;;, MA but A~ wB(O), A iwB(l). Corollary (with Sacks and G. Driscoll). The ili metadegrees are dense. (Received November 28, 1966.)

67T-l78. D. F. DAWSON, North Texas State University, Denton, Texas. On absolutely independent group axioms.

Harary (Amer. Math. Monthly 68 (1961), 159-162) introduced the notions of "very" and "absolutely" independent axiom systems. Jacobson and Yocom (Amer. Math. Monthly 72 (1965), 756-758) gave a set of absolutely independent group axioms, which Morgado (Gaz. de Mat. 27 (1966), 8-10) improved in a sense. In this paper three sets of absolutely independent group axioms are given. One of these improves Morgado's result. Jacobson and Yocom defined the statement that an axiom system is absolutely independent (mod n). The systems of Jacobson-Yocom and Morgado

and the first two systems given in this paper are absolutely independent (mod M0 ), while the third system has the property that there is no k such that the axioms are absolutely independent (mod k). The axioms of this system are as follows (G is a groupoid). (l) There exists e E G such that if x E G - leI· then xe = x. (2) There exists b E G such that if c E G, then there exists at most one s E G such that cs = b, there exists at most one t E G such that tc = b, and if c f b, then there exists x E G such that ex= b. (3) If x E (G- lbiJ U lei andy, z E G- lei, then x(yz) = (xy)z. The paper concludes with two axioms which define a group and which are very independent (mod n) for every integer n > l. (Received December l, 1966.)

67T-179. TAKA YUKI TAMURA, University of California, Davis, California. Subdirect product of semigroup and rectangular band.

The results presented in Abstract 639-8, these cNOticeiJ 13 (1966), 831 can be generalized as follows: Theorem 1. LetS be a semigroup and B be a rectangular band. Every subdirect product of Sand B is s-indecomposable if and only if Sis s-indecomposable. Theorem 2. LetS be a semi-

268 group and B be a rectangular band of order > 1. The only direct product of S and B is a subdirect product of S and B if and only if one of the following conditions is satisfied: (1) S is right simple hut not left simple and B is a left zero semigroup. (2} S is left simple but not right simple and B is a right zero semigroup. (3} S is a group and B is an arbitrary rectangular band. (Received December 2, 1966.)

67T-180. DANA SCHLOMIUK, McGill University, Montreal 2, Canada. A characterization of the category of topological spaces.

We characterize the category of topological spaces in the spirit of Lawvere's characterization of the category of sets. We give a finite number of elementary axioms and one nonelementary axiom--the axiom of completeness--and we prove that every model of the system is equivalent in the sense of category theory to the category of topological spaces. An axiom which plays an im­ portant role, since it enables us to define "open subs paces", states the existence of an object with exactly three maps into itself, two of which are constant. In the category of topological spaces any space with two points and three open sets satisfies this axiom. (Received December 5, 1966.)

67T-181. A. L. ROSENBERG, IBM Watson Research Center, P. 0. Box 218, Yorktown Heights, New York 10598. Multitape finite automata with rewind instructions.

gn (resp. _A/n) is the class of sets of n-tuples of tapes defined by deterministic (resp. non­ deterministic) one-way n-tape finite automata (see Abstract 64T-343, these cJ{oticei) 11 (1964}, 469).

Let ~n Le the aualogous class for n-tape finite automata which are one-way except that, at any point in their computations, they can simultaneously rewind all n tapes to the beginning. Lemma. Given any rewind n-tape FA, one can effectively find an equivalent one which halts on all inputs.

The main results are: Theorem. For all n, ~n is the Boolean closure of .91n. Theorem. For n > 1,

(a) g{'n - _A/n is nonempty; (b) _A/n - ~n is nonempty. Part (a) is proved by exhibiting a set in g{'n whose projection on any coordinate is not context-free (and hence not regular). Part (b) is proved by exhibiting sets in gn whose concatenation is not in _q{>n· As one would readily expect, most interesting questions about g{'n are recursively unsolvable for n > 1. (Received December 5, 1966.)

67T-182. T. C. BROWN, Simon Fraser University, Burnaby 2, B. C., Canada. On locally finite semigroups.

The theorem below, a generalization of a theorem of Shevrin (Dokl. Akad. Nauk SSSR 162 (1965), 770-773 = Soviet Math. Dokl. 6 (1965), 769), is the exact analogue for semigroups of the theorem of Schmidt on extensions of locally finite groups by locally finite groups. A complete proof will appear in the Ukrainian journal of Mathematics. Theorem. Let be a homomorphism of the semigroup S upon the locally finite semigroup T such that -l (e) is a locally finite subsemigroup of S for each idempotent e ofT. Then S is locally finite. (Received December 5, 1966.)

269 67T-183. R. W. GILMER, JR, Florida State University, Tallahassee, Florida. Contracted ideals with respect to integral extensions. Preliminary report.

Let R be a subring of the commutative ringS. Then R has property (C) with respect to S pro­ vided A~ Aens for each ideal A of R; here Ae denotes the ideal of S generated by A. The following conjecture is false, even in case L/K is a finite-dimensional Galois extension: If R is an integrally closed domain with quotient field K and if S is the integral closure of R in L, an algebraic extension field of K, then R has property (C) with respect to S. Theorem 1. With notation as in the conjecture, i between 1 and n, (C .) xi E Ai. Corollary. if [L: K] ~ n < oo, then for x E A en Rand any integer n,1 - With notation as in the conjecture, if R is a Prufer domain, R has property (C) with respect to S. Theorem 2. With notation as in the conjecture, if S has an integral basis over R, R has property (C) with respect to S. (Received December 5, 1966.)

67T- 184. W. F. PFEFFER, University of California, Davis, California. The integral in topological spaces. Preliminary report.

In Abstract 625-37, these cNoticeiJ 12 (1965), 555, an integral in a locally compact Hausdorff space was defined. The following modification of the definition of a major function leads to the integral in any topological space which preserves many properties of the original integral. Let P be a topological space and let u be a prering of subsets of P on which a nonnegative additive function G is defined. With every x E P U (oo) associate a family Kx of nets {B a l C u. For A E u, A- denotes the closure of A in P and uA ~ {B E u: B C A j. Let A E u, x E P U (oo) and let F be a function defined on uA. We denote #F(x) ~ inf~im inf F(Ba)] and .F(x) ~ inf~im inf F(Ba)/G(Ba)], where infimum is taken over all nets {sal C uA for which {Bal E Kx· A superadditive function M on uA is said to be a major function of a function f on A- whenever there is a countable set Z C A- such that (i) x E Z = #(- G)(x) ;;;;; 0, (ii) x E Z U(oo) = #M(x) ;;;;; 0, (iii) x E A-- Z =- oo t • M(x) ;;;;; f(x). (Received December 7, 1966.)

67T-185. j. M. CIBULSKIS, Illinois Teachers College, Chicago-North, Chicago, Illinois 60625. Existence of segments in a topological space with betweeness,

Let X be a topological space with a ternary relation B. If (x,y,z) E B, we write simply xyz.

For x t z, define B(x,z) to be the set of all y E X for which x ~ y or xyz or y ~ z. B shall be called a betweeness on X provided that it satisfies the conditions: (1) For no x, y E X do we have xyx. (2) wxy and wyz are equivalent to xyz and wxz. (3) If x t y, then xyz for at least one z E X. (4) If x t y, then B(x,y) is closed. We define a segment from x toy to be an arc f[a,b] with the properties that: (1) .f(a) ~ x, f(b) ~ y, and (2) a ~ p < q < r ~ b implies that f(p) f(q) f(r). Lemma. Let K be an infinite chain with first and last elements and which is topologized in such a manner that the following conditions are satisfied: (a) each closed interval in K is closed and compact; (b) K is (topologically) separable. Then, K is homeomorphic to [9,1] with preservation of order. Theorem. If B(x,y) is compact and second countable, then a segment from x toy exists. (Received December 2, 1966.)

270 67T-186. C. E. HARRELL, 1075 Varsity West, Bowling Green, Ohio. A generalized Hausdorff summability method. Preliminary report.

Suppose 0 < b < 1 and B = (l,b,b2 , .•• ). The sequence X= (x0 ,xl'x2 , •.• ) is said to be B-conver­ 2 gent if and only if the sequence Y = (x0 ,x 1/b,x2;b , .•. ) converges. If X is B-convergent the limit of Y will be called the B-limit of X. An infinite triangular matrix will be called B-regular if and only if it is B-limit preserving. Suppose H(B,D) is the infinite triangular matrix defined by H(B,D) = np bn-pH(D) , where H(D) is the Hausdorff matrix for the moment sequence D. Theorem 1. The np following two statements are equivalent: (1) H(B,D) is regular and B-regular, and (2) Dis a regular moment sequence having mass function a with a(l) - a(l-) = 1. If (l) is true and X is a 1:Jounded se­ quence then H(B,D)X converges or diverges according as X converges or diverges. Theorem 2. If a is a nondecreasing mass function, continuous at (1, a(l)) and Jol/[1- IJ da---> Las x---> 1 and D is the moment sequence for a, then H(B,D) transforms each sequence which has a sum into a sequence which has a sum. (Received November 25, 1966.)

67T- 187. LAWRENCE KUIPERS, Southern Illinois University, Carbondale, Illinois 62901. Some remarks on uniform distribution in compact topological groups.

Given a sequence an (n = 1,2, ••• ) of real numbers the behaviour of which with respect to uniform distribution mod 1 is known, one can consider transforms of an into sequences which have different (or similar) distribution mod 1 properties. Some results of this type are known in the theory of uni­ form distribution in compact topological groups. We prove a.o.: If G and Hare compact topological groups, if a is a "fixed pointfree" (with respect to all nontrivial representations of G) element of G, if b is an arbitrary element of H, then the sequence (a,b}, (a,b2}, (a2 ,b), (a 2,b2}, (a,b3}, (a3,b), •.• , belonging to the direct product F = G X H, is uniformly distributed in F. (Received November 14, 1966.)

67T-188. D. R. McMILLAN, JR., University of Wisconsin, Madison, Wisconsin 53706. Topological collapsing and piercing points.

Suppose that K is a finite complex, that L is a subcomplex of K, and that K collapses to L.

Let h: K ---> Int Mn be a homeomorphism, where M n is a piecewise-linear n-manifold. Theorem 1 • .!f.n i 4 and if h(K) is cellular in Mn, then h(L) is cellular in Mn. The proof uses the author's "cellularity criterion" in Ann. of Math. 79 (1964}, 327-337, and Theorem 1 is a generalization of Theorem 6 of that paper. Theorem 1 can be used to prove the following: Theorem 2. Let D be a 2-cell topologically embedded in s3, and suppose X C Int D is a cellular subset of s3• Then, with the possible exception of one point, D can be pierced with a tame arc at each point of X. A corollary of Theorem 2: Theorem 3. LetS be a 2-sphere topologically embedded in s3 in such a way that each component of s3 - S is an open 3-cell. Then, with the possible exception of two points, S can be pierced with a tame arc at each of its points. An important tool in proving Theorems 2 and 3 is the result (due to Gillman and Martin) that a cellular arc cannot fail to be tame only at its endpoints. (Received December 1, 1966.)

271 67T-189. P. MASANI, Catholic University of America, Washington, D. C. 20017. Explicit form for the Fourier-Plancherel transform over locally compact abelian groups.

Let ~ = L 2(X, 18,#£), where X is a locally compact, abelian group with Borel family \l!: and Haar measure #£, and let 180 = {B : 8 E 18 & #£(8) < 00 I. Let ~ = L2 (X, sa.il>. where X, 18.~. s8 0 are the dual entities. For x E X, a EX, [x,a] will denote the value of the character a at x. For x E X and

B E 18 0, define ~(B)(x) = JB[x,a],l(da). We assert that ~(ih E ~ and (~(A), ~(B))~ = ,i(A n8), and that IS~ = ~ (cf. Abstract 67T-119, these cJVoticeiJ-14 (1967), 164). Thus ~is an orthogonal differential basis for ~ defined on sBo· Itfollows at once that the correspondence: i ---> f xf

spect to Haar measure, we conclude (cf. cited abstract) that VfE~. 1(a) = limN -> faj (f, ~ (f:ViL(l\la))~, ~ A (l a.e. (ji}, where Na is a symmetric neighborhood of a EX. The limit can be brought inside the inner product when and only when ii{ aj > 0, i.e. since jL is invariant, when P, is wholly atomic. Our formula then yields f(a) = (f, ~{aj/il{al = (f,[•,a]), i.e. the well-known expression for f when #t(X) < oo, i.e. X is compact and X discrete. When X = (- oo, oo) our general formula yields the familiar f(a) = 1 2 limb _.0 (l/(27r) / )J~00 f(x)(sin hx/hx)e -ixadx. (Received December 21, 1966.)

67T-190. DAVID STADTLANDER, 3502 N. W. 17 Terrace, Gainesville, Florida. Thread actions. Preliminary report.

Let X be a Hausdorff space and T a topological semigroup. An act is a continuous function u: T X X ---> X which is onto and satisfies u(ts,x) = u(t,u(s,x)). A K-space X is a compact metric space containing 0 EX, 8 C X and 2: = [(O,b]: bE B] satisfying (1) for each b E 8, there is a unique arc

[O,b] in 2:, (2) X =U ~. (3) for b 1, b2 E 8 with b 1 1- b 2, [O,b 1] n (O,b 2] is a proper subarc of each, and (4) letting [O,x] denote the subarc of any member of~ containing x if xn ---> x, then [O,xn] ---> [O,x]. These are the first four properties used by Koch and McAuley to define a ruled continuum (Fund, Math. 56 (1964), 1-8). Theorem 1. LetT be a metric thread and X a compact metric space. Then T acts on X with 0 acting as a constant mapping iff X is a K-space. Theorem 2. Let T be a metric clan. If T acts on the compact metric space X with some element of T acting as a constant mapping, there is a K-space Y and a monotone mapping a: X--+ Y such that for each x E X, a -la(x) is the under­ lying space of an abelian topological group. If X is one dimensional, X is a K- space. (Received November 28, 1966.)

67T-191. F. J. FLANIGAN, University of Pennsylvania, Philadelphia, Pennsylvania 19104. Algebraic geography: the set of structure constants for associative algebras. Preliminary report.

The author studies the global family of n-dimensional associative algebras by examining the

'geography' of the algebraic set !ff of structure constants. A point (cijk), i,j,k = 1, ••• , n, of affine n3-space is in ~iff the multiplication x .x . = I:... c ..,_x k is associative. The set ~ is thus the zero 1 l " lJ"- locus of homogeneous polynomials in n3-space obtained from the associativity condition and is, therefore, a finite union of Zariski-closed cones, There is a natural 'change-of-basis' action of

GL(n) on ~. The dimensions of certain orbits and components are calculated. The author then utilizes the deformation theory of Gerstenhaber (Annals of Mathematics Studies (1964)) and Nijenhuis-

272 Richardson (Bull. Amer. Math. Soc. 72 (1966), 1-29) to treat the question 'Does every component of ~carry the (Zariski-open) orbit of a rigid algebra?' It is shown that every deformation of an algebra is equivalent to a constrained deformation of its radical. This is used to show that every component of .l:f·carries a Zariski-open subset which is either the orbit of a rigid algebra or an infinite union of orbits of algebras all with isomorphic semisimple parts, isomorphic actions of their semisimple parts on their radicals, and radicals which are deformations of each other. Examples are given. (Received November 28, 1966.)

67T-192. c. C. HSIUNG and B. H. RHODES, Lehigh University, Bethlehem, Pennsylvania 18015. Isometries of compact submanifolds on a Riemannian manifold.

Let R be a Riemannian manifold which admits an infinitesimal conformal transformation ~; let x,x* : M ----+ R be two immersed compact submanifolds on R, and f: x(M) ----+ x*(M) a volume­ preserving diffeomorphism. Conditions are found for f to be an isometry. These conditions were obtained by S. S. Chern and C. C. Hsiung in a joint paper [Math. Ann. 149 (1963), 278-285], when R

is Euclidean and ~is generated by the position vector field with respect to a fixed point 0 in R. (Received December 1, 1966.)

67T-193. K. K. OBERAI, Queen's University, Kingston, Ontario, Canada. Spectral interpolation in Lp spaces.

Let (X,2:,#) be a finite measure space. Let 1 ;> r ;> p ;> s ;> oo. Let T be a continuous linear

operator on Lr ( =Lr(X,2:,~£)) which leaves Ls invariant and let T /Ls be also continuous. Also, let T and T/Ls be spectral operators on Lr and Ls' respectively. Then T and T/Ls have the same spectrum and if E(•) is the spectral measure ofT, then Ls is invariant under E(·) and E(•)/Ls is the spectral measure of T/Ls. Also Lp is invariant under T and E(•) and T/Lp is a spectral operator with E(•)/Lp as the corresponding spectral measure. An example shows that there exist spectral

operators on L2 (0, 1) which leave Lp(O, 1) invariant 1 ;> p ;> oo but T /Lp(O, 1) is not spectral for p 1 2. (Received November 25, 1966.)

67T-194. NICOLAE DINCULEANU, Queen's University, Kingston, Ontario, Canada, and C. FOIAS, University of Budapest, Romania. Algebraic models for measure preserving transformations,

( r, U,f/l) is an algebraic ergodic system (a.e.s.) if r is an abelian group, U is an automorphism of r and 4» is a function of positive type such that 4» • U = 4». Example. Tan invertible measure preserving (i.m.p.) transformation on a probability measure space (X,2:,1'), r(p) the set of the equi­

valence classes f E L 00 (~t) with jfj = 1, UTf = f o T for f E L 2 (~t), fjll'(f) = jfd~t; then (r(~t), UT' fjll') is an a.e.s. An a.e.s. (r,U,fjl) is an algebraic model forT if there exists an injective homomorphism

j: r----+ r<~t> with UTj = jU and 4» = «PI' 0 j, such that jr generates L 2 (~t). Every T has an algebraic model. Every a.e.s. is an algebraic model for some T. Conjugacy can be characterized by means of algebraic models. An a.e.s. (r,U,f/l) is discrete if r contains the circle group C and fjl(-y) = 'Y f/lc('Y) for 'Y E r. The i. m.p. transformations with discrete algebraic models can be characterized by

273 means of the (f, U) part of algebraic models ( r, U,4>). These transformations contain the ergodic transformations with discrete spectrum and those with quasi-discrete spectrum. (Received November 25, 1966.)

67T- 195. LEWIS ROBERTSON, University of Washington, Seattle, Washington 98105. Principal series representations of GL(n,F).

Let F be an arbitrary nondiscrete locally compact field, and GL(n,F) the group of all invertible linear transformations on Fn. Then principal series representations can be defined as analogous of the Gelfand-Naimark principal series (induced) representations of SL(n,C). There are principal series representations of GL(n,R) which are irreducible when the analogous representations of SL(n,R) are reducible. General results on induced representations of semidirect products can be used to establish irreducibility for a large class of representations which includes all nondegenerate and many degenerate principal series representations of GL(n, F). These results give irreducibility proofs for representations of SL(n,C) and SL(2n + 1, R). Irreducibility of nondegenerate principal series representations of SL(2n + 1, R) follows immediately from the existence of t 1/n when t is real and n is odd. (Received November 14, 1966.)

67T-196. W. H. CALDWELL, University of South Carolina, Columbia, South Carolina. Perfect hypercyclic rings.

Definition. A ring R is called hypercyclic provided every cyclic right R-module has a cyclic injective hull. It is known that uniserial rings are characterized as being left or right artinian or right noetherian hypercyclic rings (Faith, Kothe rings, Math. Ann. 164 (1966)). Lemma l. If R is left perfect and right self injective, then any injective cyclic right R-module is projective. Lemma 2. If R is hypercyclic and left perfect, then annihilator left ideals are principal left ideals. Theorem. R is hypercyclic and left perfect if and only if R is uniserial. (Received November 18, 1966.)

67T-197. T. M. PRICE, University of Iowa, Iowa City, Iowa 52240. On decompositions and homotopy groups.

Let X be a metric space. Let G be an upper semicontinuous decomposition of X. Let Y denote the decomposition space, X/G, and let f denote the natural map of X onto Y. The map f is called semi n-connected iff for every y E Y and each open subset, U, of X with f- 1(y) <;; U, there exists an open set V such that f-l (y) <;; V <;; U and each map of the r-sphere into V, 0 ~ r ~ n, can be extended to take the (r + 1)-cell into U. Let K be a finite k-complex, k ~ n, and let g: K ----> Y be a map. Let t > 0. Iff is semi n- connected, then there exists a map h: K ----> X such that f o h is homotopic to g under an t-homotopy. An easy corollary is that if W is an open subset of Y then f induces an isomor­ phism between ilk(f- 1(W),x) and ilk(W ,f(x)) for 0 ;;! k ;;; n and a monomorphism fork = n + 1. These results can also be proven under the assumption that X and Y are metric spaces and f is a compact map (inverse images of compact sets are compact) onto Y. Hence it yields another example (S. Smale, Proc. Amer. Math. Soc. 8 (1957), 604-610) of a Vietoris mapping theorem for homotopy. (Received November 30, 1966.)

274 67T-198. JACK WILLIAMSON, University of Wisconsin, Madison, Wisconsin 53705. On entire functions with negative zeros.

Lemma. Let f(z) be a canonical product of nonintegral finite order p, having only negative zeros •. If we set C (r) = {8 E [0, 1r]: log lf(rei 8) I <:: 0 j and q = [p], then there exist points a 1 = a 1 (r), ••• , aq+l = aq+l (r) with ((j- (1/2))/(q + 1)) 1r < ai < ((j -(1/2))/q) 1r, j = l, ••• ,q; ((q + (1/2))/(q + l))1r < <1q+l :;; lrSUCh that C(r) = u\~~l)/l(a2i-~C12i), if q is odd and C(r) = u~J[a2i' C12i+lJ, ao= 0, if q is even. This lemma is then used to prove Nevanlinna's theorem for entire functions of genus one, namely: Theorem. Let f(z) be an entire function of genus one, order P, with negative zeros. Then lim supr__,00N(r,O)/T(r,f) <:; lsin1rPI/(l +!sin lrP!),l :i!p:!! 3/2; limsupr_,00N(r,O)/T(r,f) <:: lsinlrPI/2, 3/2 :!! P < 2. (Received December 1, 1966.)

67T-199. C. J. MAXSON, State University College, Fredonia, New York 14063. Essential and strictly essential near-ring modules.

Let M be a unitary near-ring module over a near-ring N with identity. A submodule A of M is said to be essential (strictly essential) in M if for every nonzero submodule (N-subgroup) K of M, K n A i (0). If A is strictly essential in M then A is essential in M. Examples show that the converse is not true. Theorem. A necessary and sufficient condition that an essential submodule A of M be strictly essential is that for every N- subgroup S of M, S n A = (O) =;. N ( S) n A = (0) where N(S) is the submodule of M generated by S. We call x EM a singular element if there is a strictly essential left ideal A of N such that Ax= (0). Contrary to ring theory the set S(M) of singular elements of M is not a submodule of M although S(M) is an N-subset. Theorem. A necessary and sufficient condition that the set S(M) be a submodule of M is: Vx, y E S(M), Vm E M, Vn EN, 3 strictly essential left ideals Bl' B2, B 3 of N such that for any strictly essential left ideal A of N B 1(x- y) UB2(m + x- m) U B 3(n(x + m)- nm) c:; Ax. (Received December 1, 1966.)

67T-200. MARTIN SCHECHTER, Belfer Graduate School of Science, Yeshiva University, New York, New York 10033. On intermediate extensions. I.

Let X, Y be Banach spaces and let Ao be a preclosed linear operator from X to Y with D(A0 ) dense in X. Then D(A0) is weakly • dense In Y*. Let S be a linear manifold in D(A()) which is also weakly • dense in Y*. Consider the class .lf(S) of closed extensions A of A0 such that D(A*) 2 s. The closure A of Ao is the smallest extension in .lf(S) and is called the minimal extension. There is a largest extension A in .lf(S). D(A) consists of those u E X for which there is an f E Y satisfying

(u, Ai) v) = (f, v) V vE S. Au = f. A is called the maximal extension of A0 relative to S. Let W be a Banach space containing D(A0 ) and continuously embedded in X, and let Z be a Banach space contain­ ing S and continuously embedded in Y*. An operator A from X to Y will be called an intermediate

extension of A0 relative to W and Z (or a W-Z extension of A0) if (a) There Is a continuous bilinear form a(u,v) on W X Z such that a (u,v) = (A0u, v) on D(Ao) X Z (b) u E D(A) and Au= f iff u E W, fEY and a(u, v) = (f, v) V vE Z. (Received December 8, 1966.)

275 67T-201. CHARLES HIMMELBERG, University of Kansas, Lawrence, Kansas 66044. Pseudo­ metrizability of quotient spaces.

Theorem. Let f be a map from a pseudo-metrizable space X onto a topological space Y such that Y has the quotient topology relative to f. Then Y is pseudo-metrizable if and only if there exists a pseudo-metric space (Z, o) containing X as a subspace and a continuous extension g: Z __, Y off such that the function ,\ defined by .\(y, w) = o(g- 1 (y), g- 1 (w)), if y, w E Y, is a pseudo-metric compati­ ble with the topology of Y. If D is a pseudo-metric compatible with the topology of X in the theorem above, then (Z, 0) can be required to contain (X,d) isometrically, if either (X,d) is bounded or o is allowed to take extended values. (Received December 8, 1966.)

67T-202. R. B. WARFIELD, JR., Harvard University, Cambridge, Massachusetts. Nonisomorphic abelian groups with isomorphic Ulm factors.

If G is an abelian p-group we define subgroups G where G0 = G, and inductively, G/3+l is the set of elements of infinite height in G/3 (i.e. divisible by all powers of p), and if a is a limit ordinal, Ga is the intersection of the G/3 for /3 < a. The Ulm factors are the quotient groups G = Ga/Ga+l. a If Ga = 0 and G/3 f. 0 for /3 < a, then G is of Ulm type a. If G is of type 2 and if G0 and G1 are countable then Ulm' s theorem implies that G is completely determined (up to isomorphism) by G0

and G1. On the other hand, Kulikov has given a special example of two p-groups A, B for which he showed that there existed at least two nonisomorphic groups G of type 2 with G0 ~ A, G1 ~ B. We extend this result as follows. Theorem. If A, B are nonzero p-groups with no elements of infinite height, such that the cardinality of A is c (the power of the continuum), the cardinality of B is at most c, and A has a countable basic subgroup (i.e. the p-adic metric makes A into a separable metric

space) then there are exactly 2c nonisomorphic groups G of type 2 with G0 ~A, G1 ~B. Similarly, Theorem. If G is a reduced group of infinite Ulm type with a countable basic subgroup, then there are 2c nonisomorphic groups having the same Ulm factors as G. (Received December 8, 1966.)

67T-203. A. E. SPENCER, Western Michigan University, Kalamazoo, Michigan. Maximal chains in solvable groups. Preliminary report.

Let G be a finite solvable group. Define h(G) = n if (l) Every upper chain of length n, G =

G0 > G1 > G2 ..• > Gn' with Gi maximal in Gi-l, contains at least one proper subnormal entry and (2) There is at least one upper chain of length (n - l) which contains no proper subnormal entry.

(Note. h(G) = l if and only if G is nilpotent.) Using classical techniques, the following principal results are obtained: Theorem. If h(G) ~ n then the nilpotent length of G is less than or equal to n.

Theorem. Suppose jGj is divisible by exactly m distinct primes. Then: (l) If h(G) ;:i (m + l), G has

a Sylow tower. (2) If h(G) ~ (m - l) then h(G) = l, i.e. G is nilpotent. (3) If h(G) = m ;;; 2 then G is a towered A- group. In fact the nonnormal Sylow subgroups of G are cyclic, and the normal Sylow sub­ groups are cyclic or elementary abelian. (Received December 8, 1966.)

276 67T-204. S.C. SAXENA, Northern Illinois University, Dekalb, Illinois 60115. A note on finite hyperbolic planes. Preliminary report.

A finite hyperbolic plane is obtained from the postulates of a finite-affine plane except that the Playfair axiom is replaced by a postulate which says, "Through a point P not lying on a line l there pass m (m > l) lines which do not have any point in common with I". m cannot always be arbitrary. Some results are discussed in this paper. (Received November 25, 1966.)

67T-205. E. L. ROETMAN, Stevens Institute of Technology, Hoboken, New Jersey 07030. The initial value problem for a third order equation.

Consider the third order partial differential equation 0 with initial data u(x,O+) = f(x). Iff is CBV with compact support then u(x,t) = t-l/3 JAi(t-l/3 (x- y))f(y)dy is the solution of the equation which converges as t-' 0 to f(x) for x in the interior of the support.

Let f be as above except that it has a jump discontinuity at a point x0. Then this representation has the interesting property that u(x,t) -' (l/3)f(x0 - 0) + (2/3)f(x0 + 0) as t --> 0. (Received December l, 1966.)

67T-206. JIN BAI KIM, Michigan State University, East Lansing, Michigan 48823. A note on generalized inverses for matrices.

The following theorem is a modification of a theorem of Penrose [?roc. Cambridge Philos. set of all n by m matrices over a finite Soc. 51 (1955), 406-413]. Theorem. Let M n,m (F q ) be the field F q of order q. p(A) denotes the rank of a matrix A in Mn, m (F q). Then the number of all solu-

tions X (E Mm,n (F q)) of the equations AXA = A and XAX = X is given by qp(A) (m + n - 2 P(A)). (Received November 30, 1966.)

67T-207. R. W. GILMER, JR. and W. J. HEINZER, Florida State University, Tallahassee, Florida 32306. Rings of formal power series over a Krull domain. Preliminary report.

Let D be an integral domain with identity which can be expressed as the unio~ of a family ~ of D, where satisfies these three conditions: (l) Each D in ~ is a Krull l D a I a E A of subrings 11 a domain and 11 is a directed set under c::;;, (2) For any pair Da' D13 of elements of 11 such that Da c:;;DfJ and for any minimal prime P of DfJ• either PnDa = (0) or PnDa is a minimal prime of Da• and (3) If Da and DfJ are as in (2), and if M is any minimal prime of Da, there is a minimal prime of DfJ containing M. Theorem. In order that D be a Krull domain, it is necessary and sufficient that the following condition hold: for any a in A and any minimal prime P of D a' there is a fJ in A such

that (a) Da c::;; DfJ, (b) there are only finitely many minimal primes Q1, Q 2 , ••• , Qm of DfJ lying over P, and (c) for any )' in A such that DfJ c::;; D-y and for any Qi' there is a unique minimal prime Hi of D·y lying over Qi and Qi g; H}2 >. Corollary. If J is a Krull domain, then the ring J[[lx{JIJJ of formal power series in any set I XfJ I of indeterminates over J is again a Krull domain. (Received December 9, 1966.)

277 67T-208. WITHDRAWN.

67T-209. C. C. HSIUNG and]. D. LIU, Lehigh University, Bethlehem, Pennsylvania 18015. Conformal transformations of a compact Riemannian manifold.

Let Rhijk' Rij (h,i,j,k = l, .•• ,n) be respectively the Riemann and Ricci tensors of a compact Riemannian manifold M of dimension n > 2 with constant scalar curvature R, C (M) (respectively I(M)) the group of conformal transformations (respectively isometries) of M, and C 0 (M) (respectively I0 (M)) hi .k the connected component of the identity of C(M) (respectively I(M)). Suppose that P = R J Rhijk and Q = RiiR .. satisfy a 2P + b(2a + nb)Q =canst., where a and bare constants such that 2a + nb i 0. lJ If (*) C(M) i I(M), then it is shown that M is isometric to a sphere. This result was obtained by C. Barbance (C. R. Acad. Sci. Paris 260 (1965), 1547-1549) for a= 0, by A. Lichnerowicz (C. R. Acad.

Sci. Paris 259 (1964), 697-700) for a= 0 and (*)replaced by (**) c 0 (M) i I0 (M), and by C. C. Hsiung (Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1509-1513) forb= 0 and(**) instead of(*). (Received December 2, 1966.)

67T-210. BRAYTON GRAY, University of Illinois, Circle Campus, Chicago, Illinois. The p-primary components of the homotopy groups of spheres for p odd.

Theorem. There is a spectral sequence converging to the p-primary components of the homotopy groups of s2n+ 1, in which E 2 is the p-primary component of the homotopy groups of s2n- 1 and p-primary components of the homotopy groups of low stems on s2np-l and s2np+ l. The differen­ tials can be computed in terms of Toda brackets and Toda bracket formulae are available for all new generators. With this theorem the 3-component is calculated unstably in stems ::;; 45. These results are in agreement with the independent stable results of]. Cohen. Many patterns appear, especially in stems 2k(p - I)- 2, which are too complicated to state here. (Received November 30, 1966.)

67T-2ll. WITHDRAWN.

67T-2!2. JOHN DeCICCO, Illinois Institute of Technology, 3300 South Federal Street, Chicago, Illinois, and R. V. ANDERSON, Illinois Teachers College (South), 6800 South Stewart Avenue, Chicago,

Illinois. Velocity and natural families in a Riemannian space V0 •

This theory was originally developed by Kasner and further results were obtained by Kasner and DeCicco. In this article it is proved that the property (A) characterizes a velocity family in a n-1 Riemannian space V n· This states that at every admissible point x, the corresponding oo centers of geodesic curvature describe a flat space En to the Riemannian space Vn at the given point x. A natural family is a special type of a velocity family. It is composed of the oo 2n- 2 extremals C of the variation problem: JieJI.ds = minimum, of the Riemannian space V n· A velocity family is a natural family if and only if the related field of force in the Riemannian space V n is conservative. In relation to the above results an extension of Hamilton's Principle for dynamics has been obtained. Also there have been found applications of the generalized Hamiltonian H to physical systems Sk of trajectories in Vn and to optics in V n• (Received December !2, 1966.)

278 67T-213. R. C. BUCK, University of Wisconsin, Madison, Wisconsin 53705. The solution of a functional equation.

The equation in question is: (*) t/>(x) - K(x)tf>({j(x)) = u(x) where K, u, {j are inC [0,1) and 0 ~ {j(x) ~ 1. One seeks a solution t/> E C [0,1]. Let Ak = {all x E [0,1] with {jk (x) = x }. Suppose that for some m, Am= A2m = ••• ,and that for any x E Am' IT~IK({jj(x))l < 1. Then,(*) has a (unique) solution for any choice of u(x). The proof is based on a special technique for solving functional equations introduced in an earlier abstract [Abstract 608-164, these cN"otiaiJ 11 (1964), 104] and an observation made by j. B. Diaz and S. C. Chu [Atti Accad. Sci. Torino 99 (1964/1965), 351-363]. If

T(f)(x) = K(x) f(/3(x)), one seeks an operator S such that liS TmS- 1 11 < 1. The existence of S depends upon finding approximate solutions of the related equation (**) h(x) - h('Y(x)) = v(x) for a specific v, with 'Y = pk. This in turn is converted, as in the earlier abstract, into a problem of showing that a specific function F lies in the uniform closure of a space of functions of two variables on a planar compact set r. (Received November 25, 1966.)

67T-214. WITHDRAWN.

67T-215. P. L. DUSSERE Idaho State University, Pocatello, Idaho 83201, and D. W. MILLER, University of Nebraska, Lincoln, Nebraska 68508. Groups occurring as automorphism groups of semigroups.

For each group G let K(G) denote the class of all semigroups whose automorphism group is isomorphic to G. Theorem 1. Let G be a group of order less than or equal to the cardinality of the real numbers. Then K(G) is nonempty. Moreover: (I) If G has infinite order M then there is an S in

K(G) whose order is 2M. (II) If G is finite, there is a finite S in K(G). (III) If G is finite abelian, there is a finite commutative S in K(G). (IV) If G is finitely generated abelian, there is a countable com­ mutative S in K(G). Theorem 2. Let G 1, G~···• Gn be groups such that K(Gi) is nonempty for i = 1,2, ••• ,n. Let G be the direct product of Gl' G2, ••. ,Gn. Then K(G) is nonempty. Furthermore, if for each i = 1,2, ••• , n there is a commutative Si in K(Gi), then there is a commutative S in K(G). (Received November 25, 1966.)

67T-216. j. L. GOLDBERG and j. L. ULLMAN, The University of Michigan, 347 West Engineering, Ann Arbor, Michigan 48104. A note on the derivatives of functions positive in a half-plane.

Suppose f is single-valued and analytic in the right half-plane and has positive real part there. For each such f, f(z) ->A ;;; 0 as z = x > 0-> oo. Then f - Az is a normalized positive function. We have proved: Theorem. Suppose f is a normalized positive function with the property that at some z0 (x0 > 0) there exists an integer N ~ 1 such that the complex numbers, f(N)(z0 ), f(N+l)(z0 ), •.• ,.!!!_ lie in the half-plane H = {wl¢' ~ argw ~ ¢'+ 1r, 0 ~ ¢'< 21r}. Then these numbers all lie on the line through the origin with slope, tan ¢'. A number of consequences of this result are also established. The proof of the above theorem is an application of the relationship between positive functions and functions with real part positive in the unit circle. This latter class has bounded Taylor coefficients and this fact together with the linear fractional transformations which relate the two classes of func­ tions from the basis of the proof. (Received December 2, 1966.)

279 67T- 217, T. H. MACGREGOR, Lafayette College, Easton, Pennsylvania 18042, Rotations of the image domains of analytic functions.

1 Let f(z) = zn + an+ 1zn+ + ... be analytic for lz I < 1 and let D denote the image of lz I < 1 under f(z). It is known that if D + b no= ~then lb I ~ 7r/2 and also if aD + b no= ~. where Ia I= 1, then lbl ~ 1 [Proc. Amer, Mach. Soc, 16 (1965), 1280-1286]. We now prove that if ei0o + bnD = ~.

0 < 0 ;:;i 1r, then (*) lbl ~ (sin 0/2)7r/0. This result implies the two theorems just quoted, Let F(O) be the class of functions f(z) = L :oanzn analytic in lz I < 1 and mapping onto D such that ei0o n D = ~. where 0 < 0 ~ 1r, The following theorems hold iff E F(O): (1) lf(z) I ~ lf(O) I [(1 + lz l>/(1 - lz I>J 0/1r; -1 (2) lanl :S.,C(0)(7r- 0) , 0 < 0 < 1r, n ~ 1; (3) an---> 0 if 0 < 0 < 7r/2. These and other results can be obtained from (*), the principle of subordination, and properties of univalent functions. (Received November 14, 1966.)

67T-218. W. M. BOYCE, 2824 Briarhurst, Apartment 3, Houston, Texas 77027, Commuting functions with no common fixed point,

In 1954 Dyer raised the following question. Iff and g map the unit interval continuously into itself and commute under functional composition, must they have a common fixed point? A negative answer is given by the construction through a limit process of a pair of functions which commute but have no fixed points in common. The functions were discovered as the result of a computer- aided search based in part on necessary conditions derived by Baxter (Proc. Amer, Math. Soc. 15 (1964), 851-855). (Received November 14, 1966,)

67T-219. D. H. LEE, Tulane University, New Orleans, Louisiana 70118. Some structure theorems of locally compact groups. Preliminary report.

Let G be a locally compact topological group and G0 the component containing the identity of G. We shall be concerned with the class [c] of all locally compact topological groups G such that G/Go is compact, We prove the following global structure theorem for groups in [C]. Theorem I. Let

G E [c]. Then G contains a compact, totally disconnected subgroup K such that G = G 0• K. This theorem enables us to prove the following: Theorem 2, Let G E [cj. Then the following two state­

ments are equivalent: (i) G is a nilpotent group, (ii) G0 , G/G0 are nilpotent and Zi(G0 ) = Zi(G) nG0 , for 1 ~ i ~ r, where Zi(G) (resp, Zi(G0 )) are the characteristic subgroups in the ascending central series for G (resp. G0), and r is the length of nilpotency of G0• Among the immediately available corollaries to these two theorems, the following are typical: Corollary 1. Let G E [C] and assume

that G0 is homeomorphic to a finite dimensional euclidean space, then G splits over G0 • Corollary 2, Let G be a locally compact, nilpotent group. If G/G0 is cyclic, then G splits over G0. Corollary 3, Let G E [c] and assume that G is nilpotent. If G0 is abelian, then G0 is central in G. (Received December 1, 1966,)

67T-220. DALE ROLFSEN, University of Wisconsin, Madison, Wisconsin 53706, A metric characterization of the 3-cell.

Bing and Borsuk have conjectured the following: A compact a-dimensional space is a topologi­ cal cell if it may be metrized in such a way that each pair of points has a unique midpoint and when-

280 ever x,y, and z are distinct points, the midpoint of x and z differs from that of y and z, This has been proved recently for n < 3 by Lelek and Nitka, This paper shows that the conjecture is true also for the case n = 3, If we should assume in addition that the space is a topological manifold, a similar argument verifies the case n > 5, using Connell's proof of the topological Poincar~ hypothesis in dimensions greater than four. (Received December 12, 1966.)

67T-221, P. G. RUUD and E. R. KEOWN, Texas A and M University, College Station, Texas, Representations of groups of order 32,

In groups of order 2n, n ;;;;; 6, the task of finding irreducible representations has been reduced to a single method, This method is inductive, starting with the representations of a group of order 2 to build those of a group of order 22 and so on to 2n, n ;;; 6, This method requires that the Cayley Table of the group be such that an ascending chain of subgroups of index two always exists, Let H be a subgroup of index 2 in the group G and T' be any irreducible representation of H. Then if T' is self-conjugate it gives two irreducible representations of G. When T' is conjugate to some T", then T' and T" combine to give one irreducible representation of G which has twice the degree of T' and T". In order to facilitate calculations, the procedure was programmed on the IBM 7094. One element from each class of equivalent irreducible representations of groups of order 2n, n ;;; 6, has been calculated. It is the conjecture of the authors that with slight modification, the method will work equally well for groups of order pn, p a prime, (Received December 5, 1966.)

67T-222, jOAN LANDMAN, Columbia University, New York, New York 10027, Finite images of polycyclic groups.

Let G, H be polycyclic groups such that every finite epimorphic image of G is an epimorphic image of H and conversely. K. A. Hirsch has raised the question: can one conclude that G, H are iso- morphic? Theorem. There exist nonisomorphic polycyclic groups with the same collection of finite images. Sufficient conditions are obtained for the class of nilpotent groups to insure isomorphism, (Received December 6, 1966.)

67T-223, C. B. BELL, Case Institute of Technology, Cleveland, Ohio. Nonparametric tests for several hypotheses, Preliminary report,

For the appropriate hypotheses, let Y = l s} and !§be, resp., the permutation transformation group under which the LF 's (likelihood functions) and the null hypothesis class, resp,, are invariant;

R(v(z)) = l::dv(z)- v(s(z))}. where E is the 0-1 distribution and vis a B-Pitman function, i.e. v(z) t v(s(z)) for a,e, z and sf. e. For the !-factor, 2-sample, independence and randomness hypo­ theses: (I) each NP statistic T is a function of some R(v), and has a discrete distribution with proba­ bilities k (.5/)- 1; (II) T is a rank statistic iff T is invariant under :!§; (III) for a specified simple alternative, the MP test is based on R(v), where v is a monotone function of the LF; (IV) for any preassigned v, R(v) is the statistic of the MP test against a Koopman-Pitman class generated by v; (V) for arbitrary preassigned F and v, there exists a randomized NP statistic with H-distribution F, and based on a randomized transformation which preserves R(v)-information. (Received December 5,

1966.)

281 67T-224. T. G. NEWMAN, Southern Methodist University, Dallas, Texas. Retracts of compactly generated lattices.

In this paper several equivalent characterizations of those lattices which are retracts of com­ pactly generated lattices will be given. Let us define a lattice L to be meet-continuous iff L is com­ plete and for every a E L and for every ideal A of L, an ( U A) = U (a nA). For any lattice L let

I(L) be the lattice of all ideals of L (including the void set ~). and for a E L let (a] denote the principal ideal generated by a. Theorem. Let f be a homomorphism of a lattice M into a compactly generated lattice L. There exists a unique join-complete homomorphism f* from I(M) into L such that f*(~) =

0 and f*{{a]) = f{a) for all a EM. Theorem. For any lattice L the following conditions are equivalent: (I) L is a retract of a compactly generated lattice, (2) L is a retract of I(L), {3) L is meet-continu­ ous, {4) for any lattice M and any homomorphism f from M into L, there exists a homomorphism f* from I(M) into L such that f*{{a]) = f{a) for all a EM. (Received December 5, 1966.)

67T-225. T. G. HALLAM and V. KOMKOV, Florida State University, Tallahassee, Florida. Finite time stability of sets.

Finite time stability with respect to the origin of a differential system x = f{t,x) {1) has been defined by Weiss and Infante [Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 44-48). This concept is general­ ized to a finite time stability with respect to an arbitrary closed set. Following some extensions of the results of Weiss and Infante in the case when r is compact, this paper proves Theorem 1. The system (1) is unstable with respect to (a, {3,t0 , T,p,[ ), where p = p(x, f) is the metric distance between

X and the set r; if there exists a function V{t,x) E c 1, t E [to,T], p{k,r) ~ {3 with the properties:

(i) V(t,x) ~ b(p(x,f)); (ii) Given a > 0 and a t0 , there exists an x0 such that p(x0 ,r) < a and V(t0 ,x0 ) > 0; (iii) V{t,x{t,t0 ,x0 )) ;;; c(p(x, f)) where b{p), c(p) are monotone nondecreasing functions 1 of p(x,f), b{O) = 0, c(O) = 0; (iv) b{{3) < c[!l- (V(t0 ,x0 ))]{T- t0 ). Also thoerems are given for the finite time stability, quasicontractive stability, and contractive stability with respect to an arbitrary closed set f. (Received December I, 1966.)

67T-226. BYRON McCANDLESS, Kent State University, Kent, Ohio. On spaces with certain hereditary properties.

Let X and Y be topological spaces, A closed in X, and f: A --> Y a continuous map. Let Z be the adjunction space obtained by adjoining X to Y by means of f. We have previously shown (see Abstract 635-16, these c}/oticei) 13 {1966), 477) that Z preserves the properties:-· hereditary normal­ ity, the hereditary Lindelof property, hereditary paracompactness, and hereditary countable para­ compactness. In addition, we show that Z preserves hereditary collectionwise normality. Therefore, a space Y is an ANR for one of these classes if and only if Y is an ANE for that class. Theorem. Every metric ANR (hereditarily paracompact) is an absolute 0 0• Corollary I. Let Y be a metric space which is an ANR {hereditarily collectionwise normal), an ANR (hereditarily countably paracompact normal) or an ANR (totally normal). Then Y is an absolute G0• An example is given to show that a metric ANR (hereditarily Lindelof) need not be an absolute G0• Corollary 2. Every metric ANR (hereditarily paracompact) is an ANR (paracompact). (Received December 1, 1966.}

282 67T-227, MISHAEL ZEDEK, University of Maryland, College Park, Maryland 20740. Polynomial approximation on nested regions,

Let D and D 1 be Jordan regions whose boundaries are star shaped with respect to the same point z 0 and D 1 CD. Let f(z) be analytic in D, continuous in 5 and map D and D 1 respectively onto G and G1• Then by a theorem of M. Thompson and J, L. Walsh (J, Math. Mech. 13 (1964), 1015-10 19) there exists a sequence of polynomials IPn(z) I converging uniformly to f(z) on i5 and such that Pn(z) maps i5 into G. In this paper it is shown that Pn (z) can be further restricted to map 5 1 into G1• A generalization to n( > 2) nested regions which can be transformed by a univalent and analytic function onto nested starshaped Jordan regions (with common center) is carried out, A modification of these results yields the following application: If f(z) is analytic in lz I < R and f(z) 'I 0 in lz - a I < r ( Ia I + r ;;; R) then there exists a sequence of polynomials Pn (z) 'I 0 in lz - a I :;;;. r con­ verging uniformly to f(z) in every lz I ;;; R 1 < R and mapping lz I ;;; R into the image of lz I < R by the mapping w = f(z). (Received December 2, 1966,)

67T-228. D. E. KNUTH, California Institute of Technology, Pasadena, California, and R. W. FLOYD, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, Improved constructions for the Bose-Nelson sorting problem. Preliminary report,

For I ;;; k ~ m,let ak,bk be integers with I ;;; ak < bk;;; n, Given real numbers x 1, ... ,xn define x(i,O) =xi; x(ak,k) = min(x[ak,k- 1], x(bk,k- I]); x[bk,k] = max(x(ak,k- 1), x[bk,k- 1]); and xu,kJ = xO,k- I]; for I ;;; k;;; m; I & i, j & n; j 'I ak; j 'I bk' The sequence of pairs (a1,b1) ... (am,bm) is said to be a shuttle sort of order n if x [1, m] ;;; x [2, m] ;;; .. • ;;; x [n, m] for all x l'"' ,xn; it is said to be a

shuttle merge of order r if n = 2r and if x[l,m] ;;; ... ,;:;; x[n,m] whenever x 1 & ... ;;; x r andxr+l ;;; ... :£ xn. Let S(n) (resp. M(n)) be the minimum m for which a shuttle sort (resp. shuttle merge) of order n exists. In J, Assoc. Comput, Mach. 9 (1962), 282-296, R. C. Bose and R, J, Nelson construct shuttle sorts and merges for which in particular S(2k), S(2k + 1), M(2k) are shown to be respectively at most 3k - 2k, 3k, 3k; and they further conjecture that their construction gives S(n) exactly for all n, In the

present paper we show that (a) S(mn) ;;; mS(n) + M(n}S(m) and M(mn) ::;! M(n)M(m);

(b) S(9) ;;; 25, S(25) ;;; 144, S(27) ::;! 163, and similar constructions give upper bounds when n is any prime power; (c) S(l6) ;;; 64; (d) M(n) ::;;; n + 2M(k) + M(n - I - k) for 0 ;;; k < n; (e) as a consequence of (a)-(d), the Bose-Nelson construction, which is of order n log23, is nonoptimal for all n > 8;

(f) S(n) = o(n I+ E) for all E > 0; (g) the Bose-Nelson construction is optimal for n ::;! 8. (Received December 16, 1966.)

67T-229. ROBERT CRAGGS, The Institute for Advanced Study, Princeton, New Jersey 08540, Neighborhoods of polyhedra in 3-mapifolds,

Suppose that finite polyhedron K collapses to subpolyhedron L by an elementary collapse. A retraction r: K ---> L is induced by the collapse if r(K - L) is contained in the interior of the pwl cell Cl(K - L) n L. Suppose that finite polyhedron K collapses to sub-polyhedron L by a finite sequence of elementary collapses. A retraction r: K---> L is induced by the collapse if it is the com­ position of retractions induced by the elementary collapses which make up the collapse of K to L. Say that K E-collapses to L if there is a retraction r: K---> L which is induced by the collapse such that

283 the preimage of each point of L under r-l has diameter less than E. The following theorem extends recent results of McMillan. Theorem. Suppose that M is a combinatorial 3-manifold, X is a subset of M which is homeomorphic to a finite polyhedron, and E is a positive number. There is a polyhedron K in M which is homeomorphically within E.£! X, there is a regular neighborhood N(K)..£! Kin M which E-collapses to K, and there is a finite collection {Hi I of mutually exclusive polyhedral cubes-with­ handles each of which has diameter less than E and intersects N(K) in exactly a disk on Bd(N(K)) such that N(K) U ( U Hi) contains a neighborhood of X in M. (Received December 15, 1966.)

67T-230. A. SHARMA and AMRAN MEIR, University of Alberta, Edmonton, Alberta, Canada.

One sided approximation by cubic splines.

Estimates for the degree of one sided approximation by polynomials in the L 1-norm have been obtained by G. Freud (Acta Sci. Math. 16 (1955), 12-28) and have been used to refine the remainder in a Tauberian theorem. We have obtained the following analogue of Freud's result for cubic splines on 2 equi-distant nodes. Theorem. Let f(x) E c [0,1] and let f"(x) be the integral of f 3(x), f 3 E BV [0,1]. Then there exist cubic splines sn(x) and Sn (x) with nodes xk = k/n (k = O,l, •.. ,n) such that s (x) ~ f(x) ~ S (x) and liS (x)- s (x)IIL ~ A·v ;n4 where A= constant ~ l/12 independent of n n n--n n 1 3 --- and V 3 = J~ ldf3 (x) 1. (Received December 16, 1966.)

67T-23l. J.P. HUNEKE, Wesleyan University, Middletown, Connecticut. Two counterexamples to a conjecture on com muting continuous functions of the closed unit interval.

Conjecture. Two continuous functions from the closed unit interval to the closed unit interval which commute (under composition) must have a common fixed point. One counterexample can !Je defined as the limits of uniformly convergent sequences of piecewise linear functions (fn/n EN),

(gn/n E N) which have the property: fn • gn+ 1 = gn o fn+ 1 (for n EN). The second counterexample can be defined more explicitly as follows. Notation. For each real function k, k*(x) = l - k(l - x). Pick b E (0,1/2); lets= (3 - 2b + (6- 4b)112)/(l - 2b); define three homeomorphisms of the reals, 1 1 1 h 1(x) = sx- sb + b, h2 (x) = 2- h 1(x), h3 (x) =- h2 (x); and let x 1 = hj_ (l), x2 = h3 (0), x 3 = h3 (l- b), -1 -1 - l *-1 * x4=hj (l), x 5 = hz (h2 (O)).x6 = h 1 (0), and c = the fixed point of h 2• Then the defining relations for continuous functions f, g can be stated: g(x) = b for x E [O,b], g(x) = h 1 (x) for ~ E [b,x 1], g(x) = 1 h 2(x) for x E [xl'x2], g(x) = h 3(x) for x E [x 2,x3], g(x) = hr- [g(hj(x)] for x E [x 3,x4], g(x) = 1 1 h~- 1 [g(h2(x))] for x E [x4,,x5]. g(x) = h2.- [g(h2.(x))] for x E [x 5,x6], g(x) = hr [g(hr(x))] for x E ~ 6 ,1 - b], g(x) = c for x E [1 - b, 1], and f = g*. The functions f,g of this second counterexample are both differentiable on a dense open subset of [0,1], and satisfy the Lipschitz condition: lf(x) - f(y) I ~ s lx - y 1. and lg(x) - g(y) I ~ s lx - y 1. for all x, yin [0,1]. (Received December 20, 1966.)

67T-232. MORTON ABRAMSON and W. 0. ]. MOSER, McGill University, Montreal 2, Quebec, Canada. Permutations without rising or falling w-seguences.

A permutation contains a rising (falling) w-sequence if it has w consecutive entries which are consecutive increasing (decreasing) integers. Explicit expressions are obtained for the number of permutations containing exactly r rising (falling; rising and/or falling) w- sequences in the straight

284 line case and the "circular" case (where l and n are considered consecutive). This generalizes results contained in I. Kaplansky (Bull. Amer. Math. Soc. 50 (1944), 906-914). ]. Riordan (ibid. 51 (1945), 745-748) and others. (Received December 20, 1966.)

67T- 233. D. M. BLOOM, 135 Amersfort Place, Brooklyn, New York 11210. On the coefficients of the cyclotomic polynomials. Preliminary report.

Let F n(x) = I:akxk be the nth cyclotomic polynomial. An algorithm is known for computing the coefficients ak using only operations defined in the ring Z [x], Z denoting the integers. Using this algorithm, the following results are obtained: (l) If n has exactly three distinct prime factors p < q < r, then lak I ;::;; p - 1 for all k; if p = 5 then lak I ;::;; 3. (2) If n has exactly four distinct odd prime factors p < q < r < s, then lak I ;::;; p(p - l)(pq - l) for all k. The proof of (2) uses the result of L. Garlitz, Amer. Math. Monthly 73 (1966), 979-981. (Received December 20, 1966.)

67T-234. ALBERT SADE, 42, Bd du Jardin Zoologiq, Marseille 4°,13, France. Quasigroupes de Cardoso et psE:udogroupes de Zelmer.

Tout quasigroupe soustractif, Q = E( ) est selfadjoint et r

(x ----> x- 1), et Q est un quasigroupe de Cardoso. Pour qu'un pseudogroupe de Zelmer, Z, so it soustractif

if faut et il suffit qu'il soit unipotent; alors Z est isotope de(C3)n. Tout gropoide Q, avec neutre et inverse 1l droite, satisfaisant a(bc) = ll(ab)c I .•• c )c = (ab)cm' a pour loi de composition xy = x·ym, oll G = E( •) est un groupe, et Q = G(l,q,l), oil q = (xm--> x) E ®E est un automorphisme de Get ry- 1) dans S. (Received December 21, 1966.)

67T-235, MISCHA COTLAR and CORA SADOSKY, Universidad de la Republica, Montevideo, Uruguay. On some function spaces related to quasi-homogeneous Bessel potentials. a al an Let a= (a 1, ••• ,an) be a fixed n-tuple of rationals ~ 1, >. = (A , •.• ,A ), m the least integer such that m/Zai is an integer, i = l,. • .,n. A function f is quasi-homogeneous of degree r if f(Aax) = ;\rf(x), A > 0. Fixing the quasi-homogeneous "norm" given by [x) =

285 67T-236. C. C. HSIUNG, Lehigh University, Bethlehem, Pennsylvania andY. K. CHEUNG, Drexel Institute of Technology, Philadelphia, Pennsylvania. Curvature and characteristic classes of compact Riemannian manifolds.

Curvature conditions are found for a compact orientable Riemannian manifold to have vanishing

Pontr jagin classes or Euler- Poincar~ characteristic. These conditions are weaker than those obtained by S. S. Chern (Abh. Math. Sem. Univ. Hamburg 20 (1956), 117- 126) and J. A. Thorpe (Ann. of Math. 80 (1964), 429-443). (Received December 2, 1966.)

67T-237. G. D. TAYLOR, Michigan State University, East Lansing, Michigan 48823. A note on the growth of functions in HP.

It is first shown that f E HP implies lf(z) I = o {(I - lz If l/p I where HP is the usual Hardy space with I ~ p < oo. This is then shown to be the best possible result in the sense that given any real valued function c/>(r) such that 0 for 0 ~ r < I and c/>(r) ---->0 as r ----> l-, one can exhibit a function f(z) E Hp such that lf(z)l f. o{ 0, there is a iO I -a function g(z) and a sequence { r n I with 0 < r n < I and rn T I such that Jg(r n e ) ~ (I /8)( I - rn) for each n. (Received December 12, 1966.)

67T-238. FRED USTINA, University of Alberta, Edmonton, Alberta, Canada. Gibbs phenomenon for the Hausdorff means of double sequences.

0. Szasz [Trans. Amer. Math. Soc. 69 (1950), 440-456] proved that if 1/;(t) = 0, t = 0;

1/;(t) = (7r- t)/2, 0 < t < 27r; and 1/;(t + 2k7r) = 1/;(t), k =±!,±2, ... , and if hm(if;;t) denotes the mth Hausdorff transform of the sequence of partial sums of the Fourier series of 1/;(t), relative to the regular weight function g(u), then taking the limit superior as m ----> m and t ----> 0, lim sup hm (1/;;t) = max T >O J6 {1 - g(u) I (sin Tu)/u du. The result is extended to two dimensions in the following manner. Let c/>(s,t) = 1/;(s) Y.(t), and let hm,n((s,t), relative to the regular weight function g(u,v). Then hm,n<>t-;s) = hm(if;;s) lhm,n(l/;;t) = hn(.J-;t)j where hm(>t-;s) {hn(>t-;t)j denotes the one dimensional transform relative to the regular weight function g(u, I){g(l,v)j. Taking the limit superior as m, n ----> 0, and s, t ----> 0, lim sup hm,n; x,y) = maxrl' T2>o I6:6 ((sin Tlu)/u)((sin T2v)/v)g(l,l; u,v) dudv where g(l,l;u,v) = g(l, I)- g(l,v)- g(u, I)+ g(u,v). (Received December 21, 1966.)

67T- 239. S. A. NAIMPALLY and C. M. P AREEK, University of Alberta, Edmonton, Alberta, Canada. Graph topologies for function spaces. II.

A collection %'(f) of open subsets of X X Y is a clover off EyX iff G(f) n U i for each U E ~(f) and G(f) C 1%-(f)l = U {UIU E %' (f)j, where G(f) is the graph of f. Following Marjanovit! (Publ. Inst. Math. 6 (20) (1966), 125-!30) a general method of constructing several graph topologies on yX Is explained, which include as special cases, r 1 constructed by Poppe (in a paper to be pub­ lished) and T2 constructed by the first author (Trans. Amer. Math. Soc. 123 (1966), 267-272). Topologies f 3 and f 4 generated by finite and locally finite clovers respectively are also considered.

286 It is shown that ri C ri+ 1 (i = 1,2,3) and examples are constructed to show strict inclusion. Poppe has discussed the relations between r 1 and r 2• In this paper it is proved that r 2 = r 3 if and only if X is T 1 and that r 3 f. r 4 even on the subspace of continuous functions, (X, Y being "nice" spaces). Following Stallings (Fund. Math. 47 (1959), 249-263) ri-almost continuous functions are defined and their elementary properties studied. When X, Yare closed intervals, r 4-almost continuous functions reduce to continuous functions. Comparisons are made between the graph topologies and the usual function space topologies; e.g. if X is compact Hausdorff and Y a uniform space then on the subspace of continuous functions, r 4 is larger than the u.c. topology. Separation axioms T 1, T 2, T 3, complete regularity and metrizability are discuss.ed; e.g. if X X Y is T 3 and 9 = If E yX jG(f) is a para compact subset of X xY}, then (9,r4) is T 3• (Received December 22, 1966.)

67T-240. R. A. KNOEBEL, New Mexico State University, Box AM, Las Cruces, New Mexico 88001. kth-order automata.

We solve the problem posed by C. L. Liu (IEEE Trans. on Electronic Computers, Vol. 12, Oct. 1963, pp. 470-475) of when a given automaton is equivalent to some kth-order automaton. An automaton is a map M: S X 2:--> S(2:, S f. (1). For a positive integer k, a kth-order automaton is a map

M: U X 2:--> S where (1 f. U ~ sk, where for all u = (u0, ... ,uk- 1} E U and u E 2:, (u 1, .. .,uk-l' M(u,u)) E U, and where for all s E S there is a u E U and an i < k such that s = ui. The automaton

M: S X 2: ->S is equivalent to the kth-order automaton M 1: U X 2:--> s 1 if there is a one-to-one map f: U--> S such that for all u = (u0, ... ,uk_ 1) E U and uE 2:, f(u 1, ... ,uk-l' M 1(u,u)) = M(f(u), u). We define for any relation on Sa doubly infinite sequence, •.• ~ R_ 2 ~ R_ 1 ~ R ~ R 1 ~ R 2 ~ ••• , of re­ lations which are given as Rn = (U~=OR[i]{wJ where R[i~ = I

R -n = n ~=0 R[-i] where R [- i] = I (s,t) : for all x, y E 2: 1 • M(s,x)RM(t,y) j. Here (w) is transitive closure, i.e., Q (w) = Q U (Q 0 Q) U (Q o Q) U (Q o Q o Q) lJ •.. . Theorem. An automaton M: S X 2:---> S is equivalent to some kth-order automaton iff (Ik_ 1) l-k =I (where I is the identity relation on S). Corollary. There is a natural correspondence between kth-order automata equivalent to a given automaton M:S X 2:-->s and relations Ron S such that R1_k =I. (Received December 22, 1966.)

67T-241. VICTOR MANJARREZ, Catholic University, Washington, D. C. 20017. Polynomial bases for compact sets in the plane. Preliminary report.

Let E be a compact set in the plane whose complement is connected and possesses a Green's function with pole at infinity. Let (an) be a sequence of points of E such that, for any function f analytic on E, the sequence of interpolation polynomials for f in the points (an) converges uniformly to f on E. Let w0 (z) = 1, wn(z) = (z - a 1) .•. (z - an). Let Pn be a complex polynomial of degree n, and let wn(z) = L~=Oqnkpk(z). Let f be analytic onE with interpolation polynomials LJ=Ocjwf A necessary and sufficient condition on the (pn)• that bk = L :ocnqnk exist and :Ef:obkpk converge uniformly to f on E, is given. The condition is shown to be independent of the (an). B. Cannon and J. M. Whittaker established the condition for the case E = the closed unit disk, an= 0. The polynomial convergence is shown to be maximal convergence, as defined by J. L. Walsh. Ostrowski's theorem that overconvergence of the Taylor series outside the unit disk implies lacunary structure of the Taylor series, is shown to hold for the series L :obkpk on E. Certain classical polynomials, such as Faber

287 polynomials and all kinds of orthogonal polynomials, are shown to satisfy the condition mentioned. (Received December 2 7, 1966.)

67T-242. G. I. GAUDR Y, Institut Henri Poincar~. 11 rue Pierre-Curie, Paris (Ve) France. Isomorphisms of multiplier algebras.

Let G 1 and G2 be locally compact Hausdorff groups, and suppose i ~ p < oo. Let LP(Gi) (i = 1,2) be the usual Lebesgue space over Gi formed relative to left Haar measure. By a right multiplier of LP(G) we mean a continuous endomorphism of Lp(G) which commutes with right trans­ lations. Denote by m (G) the Banach algebra of all right multipliers of Lp(G.) (with operator norm p I I and composition multiplication). Theorem I. Suppose that Tis a norm-preserving isomorphism of

mp(G 1) onto mp(G2 ), I ~ p < oo, p f. 2. Then G1 and G2 are isomorphic topological groups. The theorem is easily seen to be false when p = 2. LetT be an isomorphism of mp(G 1) onto mp(G2). We say that Tis bipositive if Tm ;;; 0 iff m ;;; 0. (m ); 0 if m(f) ;;; 0 when f ;;; 0.) Theorem 2. Suppose that

Tis a bipositive isomorphism of mp(G 1) onto mp(G2), I ~ p < oo. Then G1 and G2 are isomorphic topological groups. (Received December 27, 1966.)

67T-243. P. S. SCHNARE, 4350 Stemway Drive, New Orleans, Louisiana 70126. Multiple complementation in the lattice of topologies.

For terminology and notation see Steiner, Trans. Amer. Math. Soc. 122 (1966), 379-398. A question of Berri, Fund. Math. 58 (1966), 159-162 is answered affirmatively in Theorem 6.

Theorem 1. If IX I ;;; 3, then there exists T E_2: without unique complement. Proof: Let x,y,z EX be distinct and letT=- S(x, ~(y)). Then, T' = liil. I x \. X\ and T" = l iiJ, lx 1. I x,z I. X I are distinct principal complements for r. Q.E.D. Theorem 1 is known but this extremely simple proof establishes simul- taneously results of Hartmanis (finite case) and Berri (infinite case) and, moreover, has as an im­

mediate: Corollary (Steiner). If IX I ;;; 3, then~ is not modular. Theorem 2. Every ultraspace on an infinite set has infinitely many complements. Theorem 3. An ultraspace is principal iff it has a

maximal (principal-) complement. Theorem 4. AT 1 topology without isolated points has no maximal (principal-) complement. Theorem 5. Every nondiscrete T 1 topology has infinitely many principal complements. Theorem 6. If IX I ;;; 3, then every proper topology on X has at least two principal complements. (Received December 30, 1966.)

67T-244. j. B. FUGATE, University of Kentucky, Lexington, Kentucky. Irreducible continua and a paper of Thomas.

For definitions of the terms used, see E. F. Thomas, jr.1 Monotone decompositions of

irreducible continua, Rozprawy Matematyczne. A compact metric continuum M is~ A iff there is a monotone u.s.c. decomposition G of M such that the decomposition space M/G is an arc. Theorem 1. A continuum M is of type A iff l W: W is a subcontinuum of M, W = w0 l is uncountable. Theorem 2. A continuum M is of type A iff I K: K is a minimal separating continuum of M \is un­ countable. (K is a separating continuum of Miff M - K is not connected.) Theorem 3. If M is of type A, then the elements of the minimal admissible decomposition of M are continua K such that ( 1) M - K has at most two components, and each component is continuum- wise connected, (2) K is

288 irreducible with respect to 1. Theorem 4, A continuum M of type A is tree-chainable iff there is an admissible decomposition G of M such that each element of G is tree-chainable. (Received December 23, 1966.)

67T-245, P. B. BAILEY, Sandia Base Albuquerque, New Mexico, On the interval of conver- gence of Picard's iteration.

It is shown that "best possible" existence and uniqueness results for boundary value problems of the form y"(t) + f(t,y(t), y'(t)) = 0, y(a) =A, y(b) = B, with f(t,y,y') continuous and satisfying the

Lipschitz conditions lf(t,y,y')- f(t,x,x')l ~ Kly- xl + Lly'- x'l, cannot be obtained by Picard's itera­ tion. The iteration sequence for the problem u"(t) + Lu'(t) + K u(t) = 1, u(O) = 0 = u(1) will diverge if

L > 27r and K ~ !, (Received December 23, 1966,)

67T-246. S, L. SEGAL, University of Rochester, Rochester, New York. Summability by Dirichlet convolutions,

Let h(n) be a function on the positive integers with h(l) = I. A series a will be called L: n (."2,h(n))-summable to the sums if lim (1/x)~ < n~dl adh(n/d) = s, ( 0!, 1/n)-summability is x---+oo .£...tn=x .L....J n the method of Ingham and Wintner, and is closely connected with the prime number theorem, The relationships between various subclasses of ( .01',h(n))-methods and Cesll.ro means is examined, Among other results: No ( .s?',h(n))- method is regular; and various sets of conditions, some necessary and sufficient, connected with the inclusions (C, - I) =(9,h(n)) and (.s?',h(n)) =(C,l) are studied, Results of a Tauberian nature are also considered, as well as the influence of subjecting h(n) itself to a summability condition, One particular subclass arising in the above investigations is closely connected with the approximation of integrals by weighted sums. (Received November 30, !966,)

67T-247, M. B. MARCUS, The RAND Corporation, 1700 Main Street, Santa Monica, California 90406. Local behavior of Gaussian processes with stationary increments,

Analogues to the well-known results for Brownian motion, the law of the iterated logarithm and Paul Levy's uniform Holder condition are found for a wide class of real valued, separable Gaussian processes with stationary increments (denoted by X(t)), for which E I 0. Define g(s) = - log u(e -s). If 1/g'(log 1/h) < a log 1/h (a< I) and if 1/g'(s) is concave for s sufficiently large then the following event has probability one: 2 1 2 c 0 ~ lim suph-•O IX(t +h)- X(t)l/(2u (h)f(h)) / ~ C 1 where f(h) = max(loglog 1/h, 1/g'(log !/h)) and 2 c 0 and C 1 are constants, As an example, if u (h) = exp 1- (log 1/h)vj (0 < v < !) then f(h) = (1/v)(log 1/h)l- v. If 1/g'(log 1/h) = o(log !/h) the following event has probability one: 2 1 2 I-E~ limsuplt-t'l=h__.O,O:?t,t;;;IIX(t')- X(t)l/(2u (h) log l/h) 1 ::;; I+ f for arbitrary f > 0. In many cases, it is possible to sharpen these inequalities. (Received November 14, 1966.) 67T-248. R. E. SMITHSON, University of Florida, Gainesville, Florida. Topologies on sets of relations.

Let X be a set and Y a topological space, and denote the set of all relations on X into Y by YmX. The set YmX is topologized by requiring that the projections (which are now multi-functions) be upper semicontinuous, lower semicontinuous or continuous. Then if §is a set of multi-functions on X into Y, we can topologize 5' by relativization. We call the topology obtained by requiring that the projections be continuous the pointwise topology. Further, compact open topologies on a set of multi­ functio.1S are defined by extension of the definition for single valued functions. Results analogous to the results in Kelley, General topology, Chapter 6, are obtained for these topologies. Some typical results are: Proposition. If Y is a regular, T 2 space and if 5' is a set of point closed functions, then the pointwise topology is T 2. Also, Theorem. Let Y be a regular, Hausdorff space, and let 5' be a family of po·int closed functions from X into Y. Then 5' is compact with the pointwise topology if and only if (a) it is closed in the pointwise topology and (b) the sets ff[x) = IY: y E F(x) for some FEffj have compact closure. (Received January 3, 1967.)

67T- 249. H. H. WICKE, Sandia Corporation, Sandia Base, Albuquerque, New Mexico 8 7115. Concerning an arc theorem.

Let C denote the class of Hausdorff open continuous images of complete metric spaces. The class C has been shown by J. M. Worrell, Jr. and the author to be the class of T 1 locally monotonically complete spaces having bases closurewise of countable order [Abstract 66T-331, these cNoticeiJ 13 (1966), 510]. Theorem. A connected and locally connected member of C is arc-wise connected, The class C includes the class R of spaces for which Aronszajn proved a similar theorem [Fund. Math. !5 (1930), 228-234] and R includes the class of spaces satisfying axioms 0 and 1 of R. L. Moore [Amer. Math. Soc, Colloq. Pub!., Vol. 13, Amer. Math. Soc., Providence Rhode Island , 1932]. The author has also proved that the class R is the class of spaces for which Heath proved an arc theorem (Pacific J, Math. 12 (1964), 1315]. A notable feature of the present theorem is that the members of C are not necessarily regular. (Received January 3, 196 7.)

67T-250. R. C. RETH, 93-06 95 Street, Woodhaven,New York 11421. Automorphisms of Banach algebras,

Automorphisms of nonarchimedean Banach algebras over fields with nonarchimedean rank one valuation are studied. Typical of some of the theorems proved are (!) If W = V, all automorphisms have norm one, where W = I x EX : lx(M) I ~ I all M l and V = I x E X: llx II ~ lj, and (2) If X is semisimple and regular the map G --> G is 1-1 where G is the automorphism group of X over F, G is the automorphism group of W /H over v /p, and the map is given by --> <1> 1 where 1(x + H)= (x) + H. Various topologies are discussed for X and G. One of them, the automorphism topology in

which the open sets are given by X( ¢1,. .. ,n) = I x EX: ; (x) = x some i = l, ... ,nj where ; E G, is adopted to this more general setting from the topology introduced by Soundararajan (A topology for extension fields and Galois theory, Indag. Math. 27 (1965), 136-140) for fields. It is shown that closed sets correspond with subgroups of the automorphism group of X via a Galois type correspondence: V = I x E X: (x) = x whenever (y) = y for all y E V l = JI(V) where I(V) = I E G: (x) = x all x E V l

290 and J(H) = I x EX: cP(x) = x all cP E H}. An analogous topology is introduced on G and conditions are discussed for when IJ(H) = H. (Received January 4, 1967.)

67T-25l. S.D. COMER, 2031 Grandview, Apartment M, Boulder, Colorado 80302. The amalgamation property for classes of cylindric algebras. Preliminary report.

Denote by (RCAa) CAa the class of (representable) a-dimensional cylindric algebras.

Property (A) means the amalgamation property. Theorem. (i) The class CA1 has Property (A); (ii) For l < a < UJ, Property (A) fails for the classes CAa and RCA a. The argument for (i) is essen­ tially due to Daigneault (On automorphisms of polyadic algebras, Trans. Amer. Math. Soc. 112 (1964). 84-130). The theorem also holds for the classes of (representable) polyadic algebras (with equality) of dimension a and the second part holds for the class of (representable) relation algebras. This provides a large number of finitely axiomatizable equational classes of BA's with operators for which

Property (A) fails. Let d(a X a)= I: (K,A) E aXa,.,ldla dKA." Theorem. (i) For a= 2,3,4 every CAa which satisfies the equation d(a X a) = 1 is representable; (ii) For 4 < a < w, RCAa contains the class of

CA 's which satisfy the equation d(4 X 4) = l. Using this theorem, its proof and a logical argument a involving reducts we can prove the following. Theorem. (i) For a= 2,3,4 Property (A) holds for the class of CAa's which satisfy d(a X a)= 1; (ii) For 4

67T- 252, J. T. PARR, University of Illinois, Urbana, Illinois 6180 l. Homogeneous subgroups of 2Q.

Let H be an indecomposable homogeneous rank two torsion-free abelian group. Then H is embedded essentially uniquely in the additive group of a ring A = n{Ap: p a prime} where Ap is either the ring of p-adic integers or Z/pnZ for some n. Invariants which determine H are its type; its cotype, which is the~ of Beaumont and Pierce (Mem. Amer. Math. Soc., No. 38, Amer. Math. Soc., Providence, R. I., 1961); and the embedding ratio, an element of A which generalizes Kuro1l'' p-adic number invariant (Ann. of Math. 38 (1937), 175). Among the results are the following: His

K 0 T where K is of type zero and T has rank one. The endomorphism ring of H is a subring of the image of H in A; it has rank two iff the embedding ratio is quadratic over Z in A, and is explicitly determined in terms of the embedding ratio. If H is locally free, its endomorphism ring is Z or 2Z.

Any rank two integral domain with indecomposable additive group is commutative and is the endo­

morphism ring of its additive group. The Krull cSchmidt theorem holds for homogeneous subgroups

if 3Q. (Received January 4, 1967.)

67T-253. HOWARD BANILOWER, Louisiana State University, Baton Rouge, Louisiana.

Simultaneous extensions from discrete subspaces. Preliminary report.

For background see W. W. Comfort (Trans. Amer. Math. Soc. 114 (1965), 1-9] and J. B.

Conway [Proc. Amer. Math. Soc. 17 (1966), 843]. A simultaneous extension is a bounded, linear transformation which extends functions. Let N be a closed, C*-embedded, denumerable, discrete subspace of a locally compact space X. Consider the following statements: (A) there exists a simultaneous extension from C({JN - N) into C({JX - X); (B) there exists a simultaneous extension

291 from C(flX - X) into C(/'lX) (i.e. X has the projection property); (C) there exists a simultaneous ex­ tension from C*(N) into C*(X) taking C0 (N) into C0 (X). Then (C) implies (A) and (A) and (B) are incompatible. If N is C-embedded in X or if X is basically disconnected then (C) holds. If /'lN - N is a retract of flX - X then (A) holds. In particular, if flN - N = i'lX - X then X doesn't have the projec­ tion property. If Y is an infinite, basically disconnected, completely regular space, then (m) is em­ bedded inC *(Y). Normality of Y is not necessary. See D. W. Goodner [Pro c. A mer. Math. Soc. 16 (1965), 933). (Received January 4, 1967.)

67T-254. R. C. GILBERT, California State College at Fullerton, 800 N. State College Boulevard, Fullerton, California 92631. Spectral multiplicity of selfadjoint dilations.

Theorem. Let A be a closed symmetric operator in a Hilbert space H. Suppose that A has deficiency indices (1,1). If p.0 is a point of regular type for A, then there is a neighborhood of p.0 in which every minimal selfadjoint dilation of A has spectral multiplicity not exceeding 1. Proof.

According to M. G. Krein, any generalized resolvent R(,\) of A can be written R(,\) = R0 (,\) - 1 [r(,\) + Q 1 (A)r ( • ,g(~))g(,\), where I,\ > 0, A0 is a selfadjoint extension of A in H, R0 (,\) is the resolvent of A0 , Q 1 (,\) and g(,\) are certain functions depending on A0 , and r(,\) is a function deter­ mined by R(,\). Substituting into the Stieltjes inversion formula, we obtain ([1!12\jE(/'l) + E( fl + 0) I -

ll/2\jE(a) + E(a + 0) IJ f,h) = J~ (f,g(u))(g(u), h)dp(u), where Pis a certain nondecreasing function,

E(p.) is the spectral function corresponding to R(X), and [a,l'l] is any interval in the resolvent set of A0• The theorem follows from this expression and the fact that there exists an Ao for which p. 0 is in the resolvent set. (Received january 5, 1967.)

67T-255. M. L. FAULKNER, Western Washington State College, Bellingham, Washington 98225. Estimation of the PSI-function for primes in arithmetic progression.

P. Tschebyschef obtained nontrivial estimates for the function 1/;(x) = a log p by using an L p 5_x elementary sieve argument on the expression (i) 1/;(x) = Ln~:>f(n)T(x/n), where p. is the Mobius func- tion, and (ii) T(x) = Lm~}og m = Ln~xl/;(x/n). Suppose, now, that a and b are integers with (a,b) = l, l ~ b ~ a - 1. Let 1/;(x; a,b) = L' a< log p, where L' indicates that the sum is over primes p such . p ~X that pa = b(mod a), and let T(x; a,b) = L' m·~xlog m, where L' indicates that the sum is over positive integers m = b (mod a). Formulae analogous to (i) and (ii) are developed for the functions 1/;(x; a,b) and T(x; a,b), and these formulae, together with an elementary sieve argument, provide a means of estimating 1/;(x; a,b) for particular a and b. As an illustration, it is shown that .40x < 1/;(x; 4,b) <. 59x for b = 1,3 and x ;?; 43. (Received December l, 1966.)

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CALCULUS, Volume 1: One-variable calculus with an introdudion to linear algebra. Tom M. Apostol, California Institute of Technology The second edition has been divided into smaller chapters, each centering on an important concept. Several sedions have been rewritten and reorganized to provide better motivation and to improve the flow of ideas. Proofs now follow immediately after discussion of the theorems. The last third of Volume I provides a natural blending of algebra and analysis and helps pave the way for the transition from one-variable calculus to multi-variable calculus discussed in Volume II. In press

NUMERICAL INTEGRATION Philip J. Davis, Brown University Philip Rabinowitz, Weizmann Institute, Israel This monograph provides a balance between pradical applications and theoretical topics which underlie numerical integration. The coverage is broad, comprehensive, and computer oriented. The Appendix includes FORTRAN programs, a bibliography of ALGOL PROCEDURES, and a reprodudion of "On the Pradical Evolution of Integrals" by M. Abramowitz. In press

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299 New From Macmillan Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations By Arthur W. Babister, University of Glasgow Textbooks in Applied Mathematics, C. C. Lin, Editor This book presents the first comprehensive treat­ are related to well known transcendental equations. ment of the properties of solutions of nonhomo­ Many of the results appear in book form for the geneous linear differential equations, together with first time. a study of certain nonhomogeneous equations that 1967, 432 pages, $14.95 Algebra By Saunders Mac Lane, The University of Chicago, and Garrett Birkhoff, Harvard University Here is a fresh presentation of algebra for under­ cessive chapters on the matrix representation of graduate or graduate courses. From the idea of linear transformations, on similarity and eigen­ functions and composition of functions and set values, on determinants and tensor products, and theory the concept of universality is introduced. It on quadratic form. Special topics are covered in is further developed in the next chapter on the following chapters. A bibliography, lists of symbols, systems of integers. A discussion of the basic types and an index are included. of algebraic systems follows. On this background linear algebra is developed systematically, in sue- 1967, approx. 672 pages, prob. $11.95 Applications of Undergraduate Mathematics in Engineering Written and edited by Dr. Ben Noble, Mathematics Research Center, U.S. Army, The University of Wisconsin, A Publication of the Mathematical Association of America Now, for the first time, students of engineering and Professor Noble, and a considerable amount of applied mathematics have a text that clearly and supplementary material was added. The text has five simply explains various ways in which elementary sections: elementary mathematics, ordinary differ­ mathematics can be used to solve significant engi­ ential equations, field problems, linear algebra, and neering problems. Examples selected by a panel of probability theory. This is not a problem book- it members of the Committee on the Undergraduate is a fascinating study of important uses of elemen­ Program in Mathematics (M.A.A.) and the Commis­ tary mathematics in engineering. sion on Engineering Education were edited by 1967, approx. 400 pages, prob. $9.00 Boundary Value Problems of Mathematical Physics Volume I By lvar Stakgold, Northwestern University Macmillan Series in Advanced Mathematics and Theoretical Physics, Mark Kac, Editor This is the first of a two-volume, systematic treat­ Hilbert space, integral equations, extremal methods, ment of linear boundary value problems. It uses two and the spectral theory of differential operators. principal approaches, based respectively on eigen­ There are numerous examples and exercises. The function expansions and Green's functions. The book is intended as a text for graduate students in necessary mathematical apparatus is developed, in­ applied mathematics, engineering, and the physical cluding such topics as the theory of distributions, sciences. generalized solutions of differential equations, 1967, approx. 320 pages, prob. $12.95 A Survey of Modern Algebra Third Edition By Garrett Birkhoff, Harvard University, and Saunders Mac Lane, The University of Chicago " ... still probably the best introduction to the sub- examples and exercises." ject because it never loses sight of the concrete -Science Books, May, 1966 origins of the abstract ideas or of their applications in other fields, and is clearly written with many 1965, 437 pages, $8.50

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302 FUNDAMENTALS OF ABSTRACT ANALYSIS INTRODUCTION TO DIOPHANTINE By ANDREW M. GLEASON, Harvard University. APPROXIMATIONS By SERGE LANG, Columbia University. This text is designed for use in the first course in real variable theory, at the advanced undergraduate­ The aim of this book is to illustrate by significant graduate level. special examples three aspects from the theory of An important feature of this text is the explicit diophantine approximations: first, the formal relation­ formulation of the set-theoretic approach to abstract ships which exist .between various counting processes mathematics. This point of view is maintained faith­ and functions entering into the theory; second, the fully throughout. Although the foundations are not determination of these functions for numbers which given axiomatically, the book is entirely consistent are given as classical numbers; and third, certain with the Hilbert-Bernays-Godel treatment of set asymptotic estimates holding almost everywhere theory. The axiom of choice and the ideas behind ( e.g. the Khintchine theorems and the Leveque­ axiomatic set theory are carefully explained in con­ Erdos-Schmidt theorems). junction with the theory of cardinals. 83pp, $6.75 404 pp, 25 illus, $13.75

INTRODUCTION TO TOPOLOGICAL VECTOR SPACES AND DISTRIBUTIONS, VOLUME I TRANSCENDENTAL NUMBERS By SERGE LANG, Columbia University. By JoHN HoRVATH, University of Maryland The theory of transcendental numbers consists in This book is an elementary introduction to topological determining the transcendence and algebraic inde­ vector spaces and their most important application: pendence of numbers obtained as values of classical the theory of distributions of Laurent Schwartz. The functions, suitably normalized. This advanced text text is intended for use in junior-senior-graduate examines all of the several variations of the one main courses in linear topological spaces and distributions, method of this theory. Applications range from a very and as a supplement in courses in partial differential equations. Prerequisites include advanced calculus elementary setting ( concerning the function et), to and a minimum of abstract algebra, metric space rather sophisticated contexts involving abelian func­ topology, and complex function theory. tions and automorphic functions. 105 pp, $8.95 449 pp, $12.75

ELEMENTARY TOPO·LOGY A FIRST COURSE IN ABSTRACT ALGEBRA By MICHAEL C. GEMIGNANI, State University of By JoHN B. FRALEIGH, University of Rhode Island. New York, Buffalo. Aimed at the average mathematics major, this text This text is written at the appropriate level for an has the primary objective of achieving maximum undergraduate course in topology, although under depth and comprehension in a first course. The book certain circumstances, it might also be used for a is designed to teach material of a level previously beginning graduate course. If the student has com­ available only in texts intended for honors students. pleted at least three semesters of a calculus and The coverage constitutes a basic introduction to analytic geometry sequence, he should have sufficient modern algebra, exclusive of linear algebra. Content background to understand this book. However, in areas are: group theory and applications to topology, .order to gain a deeper appreciation of its contents, he introduction to ring theory, integral domains and should also have had a course in real analysis or its unique factorization domains, and field theory up to equivalent. and including Galois theory. In Press In Press

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LECTURES ON ELLIPTIC LECTURES ON QUASICONFOR­ BOUNDARY VALUE PROBLEMS MAL MAPPINGS 10 by Lars B. 2 by Shmuel Agmon, Hebrew Univer­ Ahlfors, Harvard University. 1966; sity of Jerusalem. 1965; 289 pages; 146 pages; $2.75 (paper). $3.95 (paper). SIMPLICIAL OBJECTS IN ALGE­ NOTES ON DIFFERENTIAL BRAIC TOPOLOGY 11 by J. Peter GEOMETRY 3 by Noel J. Hicks, May, Yale University. Just Published. The University of Michigan. 1965; 183 pages; $2.95 (paper). TOPICS IN THE THEORY OF FUNCTIONS ON ONE COMPLEX TOPOLOGY AND ORDER 4 by VARIABLE 12 by Wolfgang Fuchs, Leopolda Nachbin, University of Cornell University, in collaboration Rochester and Instituto de Matema­ with A. Schumitsky. Just Published. tica Pura e Aplicada, Brazil. 1965; 128 pages; $2.50 (paper). SELECTED PROBLEMS IN EX­ CEPTIONAL SETS 13 by Lennart NOTES ON SPECTRAL THEORY Carleson, University of Uppsala. Avail­ 5 by Sterling Berberian, University of able March, 1967. About 160 pages; Iowa. 1966; 121 pages; $2.50 (paper). approximately $2.95.

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304 Current Highlights from Holt, Rinehart and Winston CALCULUS AND ANALYTIC GEOMETRY, Second Edition ABRAHAM SCHWARTZ, The City College of the City University of New York As in the first edition, this text for the introductory calculus course begins with chapters on the differential and integral calculus which rest on an intuitive basis rather than an abstract one. In this second edition, t~e definition of "function" at the beginning of the book has been rewritten in more precise terms, and the first, intuitive definition for integrals in Chapter Two has been improved. A completely new feature is the addi­ tion of a chapter on differential equations. March 1967 1056 pp. 811.95 tent, A FIRST COURSE IN ABSTRACT ALGEBRA HIRAM PALEY and PAUL M. WEICHSEL, both of the University of Illinois Designed for the undergraduate course in abstract algebra, this book includes naive set theory, elementary number theory with modular arithmetic, and the elementary theory of groups and rings. 1966 334 pp. 88.95 TOPOLOGICAL STRUCTURES WOLFGANG J. THRON, University of Colorado Discusses a number of advanced subjects: lattice theory aspects of point set topology; detailed accounts of both net theory and filter theory and the relationship between these theories; and proximity spaces. 1966 240 pp. 89.50 A FIRST COURSE IN INTEGRATION EDGAR ASPLUND, University of Stockholm, LUTZ BUNG ART, University of California, Berkeley A basic text for a one-semester course in the theory of Lebesgue inte­ gration. Develops the technique of integrating functions as a means of approaching the Lebesgue theory earlier than usual in the course. 1966 489 pp. 810.95 Athena Series A SEMINAR ON GRAPH THEORY FRANK HARARY, The University of Michigan April1967 128 pp. 86.00 tent. APPROXIMATION OF FUNCTIONS G. G. LORENTZ, Syracuse University 1966 200 pp. 85.50 SQUARE SUMMABLE POWER SERIES LOUIS DeBRANGES and JAMES L. ROVNYAK, both of Purdue University 1966 112 pp. $4.50 A SHORT COURSE IN AUTOMORPHIC FUNCTIONS JOSEPH LEHNER, University of Maryland 1966 144 pp. 85.50 Holt, Rineharl • and Winslon. Inc. 383 Mad1son Avenue. New York. New York 10017

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Weapon Systems This lecture-note volume is intended for a first-year graduate course in algebraic topology and may be used as a supplement for a course in differential topology. It presents the basic material a graduate Analysis student should know for the Ph.D. qualifying exam­ ination in algebraic topology. The first part of the book treats elementary homotopy theory, and the In addition to performing compre­ second deals with singular homology theory. Part Ill is concerned h_ensive systems analysis of missiles, with the orientability and duality properties of manifolds, and the fourth part pre­ a1rcraft and other weapon systems, sents the theory of products (KUnneth formula, Booz•AIIen Applied Research is in­ cross and slant products) and applies it to prove the creasingly involved in military weap­ Lefschetz Fixed Point Theorem for compact oriented ons planning for the day after tomor­ manifolds. Emphasis throughout is on the functorial row ... the 1970 to 1980 period. viewpoint which has begun to dominate all of This demands knowledge of the state modern mathematics. of the art in related sciences plus the ability to derive development objec­ CONTENTS: Elementary Homotopy Theory. Intro­ tives and detailed performance pro­ duction. Homotopy of Paths. Homotopy of Maps. Fundamental Group of the jections from long-range Circle. Covering Spaces. technologi­ A Lifting Criterion. Loop Spaces and Higher cal forecasts. Homotopy Groups. Singular Homology Theory. Today's work rests on a strong Homology Functors. Affine Preliminaries. Singular foundation of achievement for all Theory. Homotopy I nvariance Theorem. Relation three military services ... achieve­ Between rr, and H1• Relative Homology. The Exact ment in problem definition, systems Homology Sequence. The Excision Theorem. Fur­ and operations analysis, planning ther Applications to Spheres. Mayer-Vietoris and control, computer simulation Sequence. The Jordan-Brouwer Separation Theorem. mathematical modeling, and cost: Construction of Spaces: Spherical Complexes. effectiveness analysis. Orientation and Duality on Manifolds. Orientation If you wish to share our future of Manifolds. Singular Cohomology. Cup and Cap achievements in weapon system Products. Algebraic Limits. Poincare Duality. Abso­ lute Neighborhood Retracts. Alexander Duality.· analysis and can bring us pertinent Lefschetz Duality. Products and the Lefschetz Fixed technical skills, please write Mr. Point Theorem. Products. Thorn Class and Lefschetz Robert C. Flint, Director of Profes­ Theorem. Bibliography. Table of Symbols. List of sional Appointments. Definitions.

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307 F THE CENTER -....,.... FOR NAVAL ANALYSES of the Franklin Institute 0 Would you welcome providing scientific Rapport sur Ia assistance to fteet and force commanders in the Cohomologie U. S. and overseas? We have openings for qualified scien­ tists at the Center for Naval Analyses des Groupes of the Franklin Institute. For example, as a CNA Analyst, you may be assigned Serge Lang, Columbia University. on rotation for about one year, serve 250 Pages (1966) afloat and ashore - conduct on-the­ scene studies of urgent problems, assist Regular Price: $3.95 Paper; $8.00 Cloth. in determining and improving Fleet ca­ Prepaid Price: $3.16 Paper; $6.40 Cloth.* pabilities by helping to plan and analyze Fleet operations. These advanced notes, in French, present some IM_ME~IATE OPENINGS FOR: Physical previously unpublished basic results in the coho­ Sc1ent1sts, Mathematicians and Statisti­ mology of groups to graduate students and mathe­ cians, Systems Analysts, Operations maticians in homological algebra and number Research Analysts, Research Engineers theory. Lang's book is a detailed introduction, Social Scientists. CNA offers an excel: missing until now, to the Artin-Tate notes on class lent salary and fringe benefits - plus field theory and other topics such as the coho­ an opportunity to serve your country as mology of Abelian varieties or elliptic curves. a scientist. If you're interested - arid Necessary background includes graduate-level if you're a qualified scientist with a PhD algebra and some knowledge of the terminology of or MA degree and could bring us scien­ homological algebra. tific imagination and insight, we'd like CONTENTS: Existence et unicite: Theoremes d'un­ to hear from you. icite abstraits; Notations et theoremes d'unicite What we do at CNA - CNA investigates dans mod(G); Existence; Formules explicites; problems of future force requirements Groupes cycliques. Relations avec les sous-groupes: and allocations, the cost effectiveness Morphismes varies; Groupes de Sylow; Represen­ of proposed Naval systems, the evalua­ tations induites; Doubles cosets. Trivialite coho­ tion of new weapons and sensors tech­ mologique: Le theoreme des jumeaux; Le theoreme nical aspects of strategic planning and des triplets; Splitting module et le theoreme de the correlation of research and develop­ Tate. Cup produits: Effa~abilite et unicite; Exist­ ence; Relations avec les sous-groupes; Le theoreme ment programs with Navy and Marine des triplets; Anneau de cohomologie et dualite; Corps needs. Such investigations are Periodicite; Les theoremes de Tate-Nakayama; ~ade !n all of the major warfare fields, Nakayama maps explicites. Produits augmentes: mcludmg undersea, surface, air anti-air Definition; Existence. Suites Spectrales: Defini­ amphibious and space. ' ' tions; Suite spectrale de Hochschild-Serre; Suites Send resume and letter to: spectrales et cup produits. Groupes de type Galois James M. Hibarger (article non-publie de Tate): Definitions et proprie­ tes elementaires; Cohomologie; Dimension cohomo­ CENTER FOR NAVAL ANALYSES logique; Dimension cohomologique 1; Theoreme 1401 Wilson Blvd. de Ia Tour; Groupes de type Galois sur un corps. Arlington, Va. 22209 Extensions des groupes: Morphismes d'extensions; Commutateurs et transfert dans une extension; La CN/l..... ______deflation. Formation de classes: Definitions; L'homo­ morphisme de reciprocite; Groupes de Weil. CENTER FOR NAVAL ANALYSES OF THE FRANKLIN INSTITUTE *A discount of 20o/o off the regular price INS • Institute of Naval Studies is granted on all prepaid orders. SEG • Systems Evaluation Group OEG • Operations Evaluation Group NAVWAG • Naval Warfare Analysis Group MCOAG • Marine Corps Operations Analysis W. A. Benjamin, Inc. Group No. 1 Park Avenue, New York, N.Y. 10016 ~n equal opportunity employ~

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This new journal will publish original research papers in EDITORIAL BOARD: A. BEURLING TOSIO KATO all branches of science in which functional analysis plays A. P. CALDERON M.G. KREIN J. DIXMIER PETER LAX an essential role. The material presented will include new LARS GARDING J. L. LIONS developments in functional analysis itself, applications I. M. GELFAND EDWARD NELSON GILBERT A. HUNT JOHN WERMER or examples in other parts of mathematical science, as Volume 1, 1967 (Quarterly) $16.00 well as novel problems or conceptual challenges within Personal Subscription: $1 0.00* the field of functional analysis. Plus $1.00 postage outside U.S.A. • Valid only on orders placed directly with the Pub­ lishers certifying that the subscription Is paid for by the subscriber /or his personal use.

Two New Volumes In MATHEMATICS IN SCIENCE AND ENGINEERING edited by Richard Bellman GRAPHS, DYNAMIC PROGRAMMING DYNAMIC PROGRAMMING AND FINITE GAMES SEQUENTIAL SCIENTIFIC MANAGEMENT by A. KAUFMANN by A. KAUFMANN and R. CRUON Contents: Graphs. Dynamic Programming. The A thorough treatment of dynamic programming in Theory of Games of Strategy. The Principal Prop­ finite Markov chains is given, including the "de­ erties of Graphs. Mathematical Properties of Dy­ composable" case introduced by the authors, which namic Programming. Mathematical Properties of should prove very useful in practical applications. Games of Strategy. March 1967, 278 pp., $12.00 May 1967, about 500 pp., $14.50