OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 11, NUMBER 6 ISSUE NO. 77 OCTOBER 1964

OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. Walker

CONTENTS

MEETINGS Calendar of Meetings ••••••••••••••••••••••••••••••••••••• 632 Program of the October Meeting in Garden City, New York. , , • , • , , , • , , 633 Abstracts for the Meeting- Pages 662-667 PRELIMINARY ANNOUNCEMENTS OF MEETINGS, •• ,.,,.,, •• ,,,,,,,. 636

~I:JE 1964 SUMMER RESEARCH INSTITUTE ON ALGEBRAIC GEOMETRY, •• , • 639

NEWS ITEMS AND ANNOUNCEMENTS ••••••••••••••••••••••••• 638~ 641 THE ANNUAL SALARY SURVEY •••• ,.,, •• , ..... , •• ,.,,.,,.,., ••• 643 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D••••••• , •••• , 646 PERSONAL ITEMS •...... ••...... , • . • . • . • • . . . . • . . . • 647 NEW AMS PUBLICATIONS ••••••••••••••••••••••••••••••• , •• , • 655

SUPPLEMENTARY PROGRAM - Number 2 7 ••• , , ••••••• ~ , , , ••••••• , 658 MEMORANDA TO MEMBERS Postal Rates . . . • . . . . • ...... • • ...... • • • . . . . . • . . . • 661 ABSTRACTS OF CONTINUED PAPERS • • • • • • • • • • • • • • • • • • • • • • • • • • • • 662 ERR AT A - Volume 11 ...... • ...... • • . . • . . . • ...... • . . . . . • . . • 693 INDEX TO ADVERTISERS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 707 RESERVATION FORM •••••••••••••••••••••• -, • • • • • • • • • • • • • • .. 707 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 cNotiaiJ 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*

616 November 14, 1964 Los Angeles, California Sept. 30 617 November 21, 1964 Athens, Georgia Sept. 30 618 November 27-28, 1964 Evanston, illinois Sept. 30 619 January 26-30, 1.965 (71st Annual Meeting) Denve:c, Colorado Dec.. 2 April 9-10, 1965 Chicago, illinois April 12-15, 1965 New York, New York April 24, 1965 Stanford, California

June 19, 1965 Eugene, Oregon August 30-September 3, 1965 (70th Summer Meeting) Ithaca, New York November 12-13, 1965 Lexington, Kentucky January 24-28, 1966 (72nd Annual Meeting) Chicago, illinois August 29-September 2, 1966 (7lst Summer Meeting) New Brunswick, New Jersey January 24-28, 1967 {73rd Annual Meeting) Houston, Texas August 28-September 1, 1967 ( 72nd Summer Meeting) Toronto, Ontario, Canada August 26-30, 1968 (73rd Summer Meeting) Madison, Wisconsin * The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for by title abstracts are September 23, 1964, and November 25, 1964.

The cNotiaiJ of the American Mathematical Society is published by the Society in January, February, April, June, August, October and November. Price per annual volume is $7.00. Price per copy $2.00. Special price for copies sold at registration desks of meetings of the society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, 190HopeStreet, Providence, Rhode Island 02906. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34, 21, P. L. and R.). Accepted for mailing at the special rate of Postage provided for in section 34,40, paragraph (d).

Copyright~, 1964 by the American Mathematical Society Printed in the United States of America

632 Six Hundred Fifteenth Meeting Adelphi University Garden City, New York October 24, 1964

PROGRAM

The six hundred fifteenth meeting Blodgett Hall and will be open from 9:00 of the American Mathematical Society A.M. till 3:30 P.M. will be held at Adelphi University in Adelphi University is located one­ Garden City, New York on Saturday, third mile east of the Nassau Boulevard October 24. station of the Long Island Railroad (Hemp­ By invitation of the Committee to stead Branch), 40 minutes by rail from Select Speakers for Eastern Sectional Pennsylvania Station in New York City. Meetings, Professor Jun-lchi Igusa of the Those arriving by air at Kennedy or Johns Hopkins University will address the La Guardia airport may take the inter­ Society on "Modular functions and theta airport limousine to the Jamaica station functions" at 2:00 P.M. in the Waldorf of the Long Island Railroad and proceed School Auditorium. This building is located by rail. at the southern (Cambridge Avenue) en­ Those traveling by auto should enter trance to the campus. the campus at the Cambridge A venue gate Sessions for ten minute contributed and park in the adjacent parking lot. To papers will be held at 10:00 A.M. and at reach the campus, take one of the follow­ 3:15 P.M. in Blodgett Hall, which is in the ing routes: Northern State Parkway or center of the campus. Provision will be Long Island Expressway to the New Hyde made for a limited number of late papers. Park Road exit, then south on New Hyde Abstracts of contributed papers are Park Road, east on Stewart Avenue, south presented on pages 662 to 667 of these on Nassau Boulevard, east on Cambridge NOTICES. The abstract of a particular Avenue to campus; or: southern State paper in the program of the sessions may Parkway to Hempstead Avenue exit, then be located through the cross reference northeast on Hempstead Avenue, north on numbers which follow the identification of Nassau Boulevard, east on Cambridge the contributor. Avenue. The registration desk will be in

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. The papers are scheduled at fifteen minute intervals so that listeners can circulate easily between sessions. To maintain the schedule, time limits will be strictly en­ forced.

633 SATURDAY, 10:00 A.M. Session on Analysis and Applied Mathematics, Blodgett Hall, Room 204 10:00 - 10:10 (1) On the mean integral and a mean double integral Mr. C. B. Murray, The University of Texas (615-5) (Introduced by Professor H. S. Wall) 10:15 - 10:25 (2) On Ingham's summation method Professor S. L. Segal, University of Rochester ( 615-1 7) 10:30 - 10:40 (3) On a converse to a theorem of Fatou Dr. E. J. Beltrami* and Mr. M. R. Wohlers, Grumman Aircraft Engineer­ ing Corporation, Bethpage, Long Island, New York (615-4) 10:45 - 10:55 (4) Representations of tensor products of symmetric Banach algebras Professor H. A. Smith, Drexel Institute of Technology and University of Pennsylvania (615-10) 11:00 - 11:10 (5) A generalization of Holmgren's uniqueness theorem Professor G. W. Hedstrom, University of Michigan (615-16) 11:15- 11:25 (6) A Rodrigue's formula for the Laguerre polynomials Professor L. R. Bragg, Case Institute of Technology (615-7) 11:30- 11:40 (7) On Lagrange-Hermite interpolation Dr. J. F. Traub, Bell Telephone Laboratories, Murray Hill, New Jersey (615-13) 11:45 - 11:55 (8) Generalization of a Lax-Morawetz-Phillips time decay theorem Professor James Radlow, Purdue University (615-12)

SATURDAY, 2:00P.M. Invited Address, Waldorf School Auditorium Modular functions and theta functions Professor Jun-Ichi Igusa, The Johns Hopkins University

SATURDAY, 3:15P.M. Session on Algebra and Logic, Blodgett Hall, Room 204 3:15 - 3:25 (9) The rim of a locally nilpotent group Professor R. A. MeHaffey, University of Massachusetts (615-9) 3:30 - 3:40 ( 10) Some homomorphism theorems for a class of semigroups Professor K. D. Magill, Jr., State University of New York at Buffalo (615-2) 3:45 - 3:55 ( 11) Axiomatic theory of additive relations Professor J. B. Leicht, University of Toronto (615-1) 4:00 - 4:10 (12) A calculus of antinomies Dr. F. G. Asenjo, University of Pittsburgh (615-6)

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

634 SATURDAY, 3:15P.M. Session on Geometry and Topology, Blodgett Hall, Room 108 3:15 ~ 3:25 (13) Compact totally geodesic hypersurfaces Professor Robert Hermann, University of California, Berkeley (615~1) 3:30 ~ 3:40 (14) A nontopological 1~1 mapping onto E 3 Mr. Kenneth Whyburn, Cornell University (615-8) (Introduced by Professor G. A. Hunt) 3:45 ~ 3:55 (15) Extensions of Dehn's Lemma and the Loop Theorem Mr. D. W. Henderson, The Institute for Advanced Study (615-11) 4:00 - 4:10 (16) Some approximation theorems for rings of unbounded functions ProfessorS. G. Mrowka, The Pennsylvania State University (615-15) 4:15- 4:25 (17) The stellated forms of the sixteen~cell Professor B. L. Chilton, State University of New York at Buffalo (615~3) Everett Pitcher Bethlehem, Pennsylvania Associate Secretary

635 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Six Hundred Sixteenth Meeting University of Southern California Los Angeles, California November 14, 1964

The six hundred sixteenth meeting There are numerous hotels and of the American Mathematical Society will motels in the Los Angeles area. The be held on Saturday, November 14, 1964at nearest ones to the campus are the Vaga­ the University of Southern California in bond Motor Hotel, 3101 South Figueroa Los Angeles, California. Street, telephone (area Zl3) 746-1531, and By invitation of the Committee to the Coliseum Hotel, 457 West Santa Bar­ Select Hour Speakers for Far Western bara A venue. Sectional Meetings, an hour address will Luncheon will be available at the be presented by Professor Charles W. Commons Cafeteria. A section will be re­ Curtis of the University of Oregon. The served for people attending the meeting. title of the talk by Professor Curtis is Parking will be available on cam­ "Abstract groups of Lie type". This ad­ pus for a fee of fifty cents. Cars should dress will be given at 11:00 A.M. in enter at the main entrance which is Room 133 Founders Hall. There will be located at the corner of Exposition Boule­ sessions for contributed papers at 9:30 vard and Hoover Street. The guard at the A.M. and at Z:OO P.M. in Founders Hall. gate will direct motorists to the proper Registration for the meeting will parking area. begin at 9:00 A.M. The Registration Desk R. S. Pierce will be located outside Room 133 Found­ Associate Secretary ers Hall. Seattle, WaBhington

Six Hundred Seventeenth Meeting University of Georgia Athens, Georgia November 20-21, 1964

The six hundred and seventeenth There will be sessions for contrib­ meeting of the American Mathematical uted papers at 3:30 P.M. on Friday, Society will be held at the University of November ZO and at 10:00 A.M. on Satur­ Georgia on November ZO and Z1, 1964. day, November Zl. All sessions will be in the Center for The registration desk will be in the Continuing Education. main lob:Qy of the Center for Continuing By invitation of the Committee to Education; there will be a registration fee Select Hour Speakers for Southeastern of $Z.oo. Rooms will be available in the Sectional Meetings, Professor John R. Center at the rate of $7.00 single and Isbell of Tulane University of Louisiana $10.00 double; requests for reservations will speak on "Structure of Categories" should be addressed to the Registration at Z:OO P.M., Friday, November ZO. Desk, Center for Continuing Education,

636 University of Georgia, Athens, Georgia. night. Tickets are $1.50, and may be Since the number of single rooms is purchased at the time of registration. limited, it will be appreciated if double rooms will be utilized whenever possible. Morton L. Curtis There will be a beer party Friday Houston, Texas Associate Secretary

Six Hundred Eighteenth Meeting Northwestern University Evanston, Illinois November 27-28, 1964

The six hundred eighteenth meeting use the reservation form at the backof of the American Mathematical Society will the NOTICES so as to make themselves be held at Northwestern University on Fri­ eligible for these special rates. day and Saturday, November 27-28, 1964. By invitation of the Committee to Meeting headquarters will be at the North Select Hour Speakers for Western Sec­ Shore Hotel (1611 Chicago, Evanston) tional Meetings, Professor James Serrin though registration and all the scientific of the University of Minnesota will address sessions of the meeting will be held at the Society on Multiple integral problems Northwestern University. in the calculus ofvariations and Professor The hotel has guaranteed single I. N. Her stein of the University of Chicago rooms to the Society at $8.00 and doubles will chair a collection of 20-minute talks at $11.00. Consequently, those who wish to on Recent developments in ring theory. stay where most of the members of the Society will be housed and wish to avail Seymour Sherman themselves of these special rates should Detroit, Michigan Associate Secretary

Seventy-First Annual Meeting Denver, Colorado January 26-30, 1965

SPECIAL SESSIONS AND CONTRIBUTED PAPERS

There will be a number of special Topics presently under consideration for sessions at the Annual Meeting at Denver these sessions are papers, similar to those for twenty-minute Function Algebras preceding Annual Meetings. hald at the two Differential Topology presented at these sessions The papers Differential Geometry by invitation, and partly will be partly Ordinary Differential Equations papers submitted drawn from ten-minute Rings of Operators and Group Rep- authors of those selected for the meeting- resentations will have the opportunity of expanding their presentations to twenty minutes. Those contributing papers to the

637 Annual Meeting and who feel that they for contributed papers. would be appropriate to one of these ses- sions should submit hi·s abstract a week earlier than the ordinary deadline, to allow Los Angeles, john W. Green time for the additional handling- that is, California Secretary by November 25. As at recent Annual Meetings there Seattle, R. S. Pierce will be at most two hundred ten-minute Washington Associate Secretary papers accepted for the regular sessions

NEWS ITEMS AND ANNOUNCEMENTS

BELFER GRADUATE SCHOOL OF SCIENCE

The third Annual Science Confer­ Robert R. Wilson and Sam Treiman are ence sponsored by the Belfer Graduate chairmen of the sessions. School of Science will be held on Novem­ The schedule for the Tuesday, No­ ber 16 and 17, 19 64 at the Hotel Astor in vember 17, sessions on mathematics is City. The sessions are open free New York Nathan Jacobson 9:30- 10:30 A.M. to all who are interested in at­ of charge Harish-Chandra 11 :OO- 12:00 A.M. tending. The annual award for disgin­ R. L. Wilder service to science will be pre­ guished Chairman for morning session. the Conference by the Belfer sented at Louis Nirenberg 2:00 - 3:00 P.M. Graduate School of Science. I. M. Singer 3:30 - 4:30 P.M. At the Monday, November 16, ses­ Deane Montgomery on physics the speakers are: sions Chairman for afternoon session. William A. Fowler, Paul A. M. Dirac, Charles H. Townes, Robert Serber,

DYNAMIGAL SYSTEMS RESEARCH CENTER AT BROWN UNIVERSITY

In September, 1964, the faculty of to be more closely associated with a Brown University was joined by nine math­ strong graduate program in mathematics. ematicians who form the staff of the new The members of the group are Solomon Dynamical Systems Research Center with­ Lefschetz, Joseph P. LaSalle, Jack K. Hale, in the Division of Applied Mathematics. joseph j. Florentin, Henry G. Hermes, All nine members of the group engage Harold J. Kushner, Mauricio M. Peixoto, both in teaching and in research in their Leonard Weiss, and Walter M. Wonham. field of specialty, non-linear differential Three postdoctoral fellows have equations and applications to dynamical also joined the group. They are Kenneth systems. All of the mathematicians have Meyer, University of Cincinnati; Emilio been associated at one time or another 0. Roxin, University of Buenos Aires; and with RIAS in Baltimore; they are joining Frank W. Wilson, Jr., University of Mary­ the faculty of Brown because of a desire land.

638 The 1964 Summer Research Institute on Algebraic Geometry

The Eleventh Summer Research Scientific Research, each attented the In­ Institute conducted by the American Math­ stitute for about one week: ematical Society was held from July 6 to Scientific Program July 31, 1964 at Woods Hole, Massachu­ setts, with the generous financial support The topic under study at the Insti­ of the Air Force Office of Scientific Re~ tute, algebraic geometry, is a currently search, the Office of Naval Research, and active field that has attracted the interest the Unites States Steel Foundation. These of many young research mathematicians. agencies deserve our deep gratitude for Thus the program was planned to feature making the Institute possible. Dr. Ralph new developments and current research M. Kraus, Assistant Program Director work, which were presented in formal for the Mathematical Sciences Division of lectures, informal talks, and seminars. the National Science Foundation, and Dr. The speakers and their· topics are listed Robert H. Pohrer, Chief of the Mathemat­ below in the complete scientific program ics Division of the Air Force Office of of the Institute. The early part of the Summer Institute (July 6 - July 15) was devoted to formal lectures by invited speakers. There were fifteen formal lectures during the first week on four general topics as follows: I. Theory of singularities S. Abhyankar - Current status of the resolution problem. H. Hironaka - Equivalences and deformations of isolated singularities. 0. Zariski - Equisingularity and related questions of classification of singu­ larities. II. Classification of surfaces and moduli K. Kodaira - On the structure of compact complex anaLytic surfaces. T. Matsusaka - Deformations and varieties of moduli. D. Mumford - The boundary points of moduli schemes. M. Nagata - Invariants of a group in an affine ring. M. Rosenlicht - Transformation spaces, quotient spaces, and some classifi­ cation problems. J, lgusa - On the Siegel modular variety. III. Grothendieck cohomology M. Artin - Etale cohomology of schemes. J, L. Verdier - A duality theorem in the etale cohomology of schemes. J, Tate - Algebraic cohomology classes. IV. Zeta-functions and arithmetic of abelian varieties. J. W. S, Cassels- The arithmetic of elliptic curves and abelian varieties. B. M. Dwork - Same remarks concerning the Zeta-function of an algebraic variety over a finite field. G. Shimura - The Zeta-function of an algebraic variety and automorphic func­ tions, J, P. Serre - L-Series of schemes. During the three weeks July 14-July 31 the activities of the Summer Institute were centered in a number of seminars. The following is a list of these seminars and their speakers:

639 1. Theory of singularities. Speakers: Abhyankar, Hironaka, Zariski. Z. Moduli questions. Speakers: Rauch, Kodaira, Mayer, Ehrenpreis. 3. Cohomology of number fields. Speakers: Artin, Verdier. 4. Elliptic curves and formal groups. Speakers: Serre, Tate, Lubin. 5. Hyperbolic varieties and informal groups. Speakers: Shimura, Kuga. 6. Commutative algebra. Speakers: Samuel, Lichtenbaum, Schlessinger, Auslander, Rim. 7. Etale cohomology. Speakers: Hartshorne, Kleiman, Quillen. B. The Woods Hole duality theorem. Speakers: Atiyah, Bott. In addition, informal talks on self-contained topics were given by the following participants: Barsotti: Analytic groups (3 lectures) Cassels and Mattuck: Manin's proof of Mordell's conjecture over function fields. Ehrenpreis: Geometric theory of polynomial ideals. Fogarty: Hilbert schemes and Chow schemes. Hartshorne: Ample vector bundles and complete intersections. Lang: Diophantine approximations. Ogg: Wild ramification and elliptic curves. Rim: Koszul complexes. Lecture Notes the National Academy of Sciences. The Estate has a large informal summer Mimeographed copies of the fifteen residence converted to accommodate con­ formal lectures are collected in a paper­ ferences and offices. There is also a boat­ bound volume of the proceedings of .the house, a private beach and extensive Institute and in addition, brief summaries gardens, which were open to the partict­ of the seminars are being collected in a pants and their families. second and smaller volume. Copies of the lecture notes have been sent to the sup­ Acknowledgements porting agencies, to the participants, and The success of the 1964 Institute to about 90 other persons interested in was the result of the intelligent planning algebraic geometry, but there are none and hard work of many persons and of the available for further distribution. No for­ office staff of the American Mathematical mal publication of the proceedings of the Society, headed by Dr. Gordon L. Walker. Institute is planned although some indivi­ Executive Director. The scientific pro~ dual authors may arrange for publication gram was planned and directed by the of their papers through the usual channels Joint Invitations and Organizing Commit­ of scientific publication. tee, whose members were Oscar Zariski Participants (Chairman), Harvard University; W. L. Chow, Johns Hopkins University; Maxwell Attending the Institute were eighty.. Rosenlicht, University of California at· three mathematicians, along with forty­ Berkeley; D. C. Spencer, Stanford Univer­ eight wives and seventy children. Twelve sity; and John Tate, Harvard University. of the participants were from foreign countries, namely England, France, Italy, Japan, Mexico, and the Netherlands. Living Accommodations and Services The Institute was held at the Whit­ Oscar Zariski, Chairman ney Estate, which is located on the South­ joint Invitations and west tip of Cape Cod in the town of Wood's Organizing Committee Hole, Massachusetts, and is operated by 1964 Institute.

640 NEWS ITEMS AND ANNOUNCEMENTS

VISITING LECTURERS IN STATISTICS one in analysis by Professor Lynn H. Loomis of Harvard University and the With the financial support of the other in algebra by Professor Israel N. National Science Foundation, a Visiting Herstein of the University of Chicago. Lecturer Program in Statistics is being Stipends of $1,800 plus travel allowances sponsored jointly by the principal statisti­ will be awarded to 30 members of mathe­ cal organizations of the country, the Amer­ matics departments in institutions which ican Statistical Association, Biometric offer an undergraduate major but not a Society, and Institute of Mathematical Ph. D. in mathematics. Further informa­ Statistics. Leading teachers and research tion and application blanks can be obtained workers in statistics have agreed to par­ by writing the Director of the Seminar, ticipate as lecturers. The lecturers for the E. A. Cameron, Department of Mathema­ 1964-1965 academic year are:R. L. Ander­ tics, University of North Carolina, Chapel son, T. W. Anderson, R. E. Bechhofer, Hill, North Carolina. Z. W. Birnbaum, J. R. Blum, R. A. Brad­ ley, D. H. B,runk, J. M. Cameron, D. G. Chapman, H. Chernoff, H. T. David, A, P. Dempster, C. Derman, M. Dwass, S. Ehrenfeld, B. Epstein, T. S. Ferguson, F. A. Graybill, S. S. Gupta, W. J. Hall, H. 0. Hartley, L. Katz, A. W. Kimball, INSTITUTE IN HOMOLOGICAL C. H. Kraft, L. M. LeCam, L. E. Moses, ALGEBRA AND ITS APPLICATIONS J. Neyman, I. Olkin, H. E. Robbins, Joan Tentative Announcement R. Rosenblatt, J. Rosenblatt, J. Sacks, I. R. Savage, D. L. Wallace, G. S. Watson, In 1965 Bowdoin College plans to 0. Wesler, J. Wolfowitz. hold an eight-week summer institute com­ The Organizing Committee for these bining a research program for postdoc­ lectures consists of R. L. Anderson, D. J. toral mathematicians with advanced in­ Bogue, R. A. Bradley, G. J. Lieberman, struction for graduate students. For the and J. Kiefer, Chairman. For further in­ graduate students the central formal offer­ formation write to Professor J. Kiefer, ing will be a c~urse in Homological Alge­ Department of Mathematics, Cornell Uni­ bra given by Professor Ernst Snapper of versity, Ithaca, New York. Dartmouth College. The research program will center on a Colloquium in Homologi­ cal Algebra and Its Applications at which will appear a sequence of distinguished visiting speakers. The realization of these plans is 1965 COOPERATIVE SUMMER contingent upon a grant by the National SEMINAR Science Foundation. A decision is ex­ pected about November 15. If a grant is The Mathematical Association of made, announcements will appear con­ America will conduct a second Cooperative cerning the selection of members, sti­ Summer Seminar, in 1965, to be held at pends, exact dates, and the Colloquium Bowdoin College with approximate dates speakers. Department chairmen at Ph.D. June Z1 to August 13. Grants toward the granting institutions will be invited to financial support of this Seminar have nominate appropriate graduate students. been received from the Research Corpo­ Colloquium speakers will be asked to ration and the Alfred P. Sloan Foundation. nominate appropriate postdoctoral stu­ There will be two main series oflectures, dents. Individual queries will bewelcome.

641 STATISTICS AT THE NATIONAL BUREAU OF STANDARDS JOHNS HOPKINS UNIVERSITY POSTDOCTORAL

The Department of Statistics at the Twenty Postdoctoral Associate­ Johns Hopkins University offers a program ships have been awarded for 1964-1965 leading to the Ph.D. degree in mathemati­ to young physicists, chemists and mathe­ cal statistics. Normally an applicant for maticians to give them an opportunity to admission will hold a bachelor's degree perform advanced research under the in mathematics and preference is given guidance of scientists in the NBS labora­ to those applicants who also have some tories. The one appointee in mathematics scientific knowledge. Financial support is Richard A. Brualdi, Ph.D., Syracuse for students is available in the form of University, who will study the Kronecker NDEA fellowships, USPHS traineeships product and other combinatorial topics and research assistantships. A few post­ under Dr. Morris Newman. doctoral fellowships are also available. Early in the fall of each year the The Department of Biostatistics National Academy of Sciences, which ad­ offers graduate training in statistics, ministers the program, announces the mathematics and the medical sciences competition to all universities which grant leading to the Sc.M., Sc.D. and Ph.D. Ph.D. degrees in the physical sciences. degrees. Applicants for admission should Application may be made by United States have some background in mathematics and citizens who expect to have completed all preferably one full year of general biology. the requirements for their doctoral de­ Support for premasters, predoctoral and grees by the time they are ready to begin postdoctoral students is provided by their appointments. Candidates may apply USPHS traineeships. directly to the Academy before February 1. Inquiries about either program Awards are announced in April and are should be addressed to the department made for one year with the possibility of chairman, 615 North Wolfe Street, Balti­ extension for a second year. more, Maryland Zl205.

THE BRITISH MATHEMATICAL COLLOQUIUM

The 17th Annual Meeting of the University of St. Andrews from April 6 to British Mathematical Colloquium will be April 10, 19 65. The following lectures held at Queen's College, Dundee, in the will be given: C. Chevalley (Paris): Recent advances in the theory of algebraic groups A. Erdelyi (Edinburgh): Non- standard analysis: an extended system of real numbers P. R. Halmos (Michigan): Some recent progress in Hilbert space Invited addresses will also be given by: and discussion. Accommodation will be P. D. Barry, D.B.A. Epstein, R. 0. Gandy, provided in University Halls ofresidence. J. A. Green, R. C. Lyndon, J. E. Reeve, Further information, and application H. Reiter, I. N. Sneddon, and S. Vajda. forms, may be obtained from the Secretary, In addition there will be meetings Dr. H. G. Anderson, Department of Mathe­ of "splinter groups" for shorter papers matics, Queen's College, Dundee, Scotland.

642 THE ANNUAL SALARY SURVEY

The Annual Salary Survey for 1964 shows a 9% overall increase for 1964-1965 in the num­ ber of mathematical staff members at academic institutions, and a general increase in the salaries received in every rank on a staff. Over the last two years, the size of the staffs of universities have grown by an average of Z1%. The largest increase, 31% in two years, has occurred in the number of assistant professors in mathematics, whereas the number of instructors has continued to decrease during recent years, with a 6% decrease expected in 1964-1965. One group of schools, however, will be hiring a greater number of instructors in the coming academic year; Group II, defined below, will employ 31% more instructors in 1964-1965 than in 1963-1964. The basis of the classification of institutions remains the same in the 1964 Survey as in the last two annual surveys. Institutions included in the Survey are divided into two classes, Institu­ tional Non-Members and Institutional Members. The latter are further grouped according to the volume of their mathematical publications in the years from 19 59 through 1961. Group I is composed of those institutions which during the three-year period sponsored 37 1/Z or more pages in journals published or subsidized by the Society. Group II is made up of those institutions which contributed fewer than 37 1/Z pages during the same period. Every institution submitted a minimum, median and maximum salary figure for each of the academic ranks. The data presented here in each of the categories of salary figures is the range of the middle 50% of all of the salary figures received for that category. For example, the data in the following report indicates that the minimum salary of an instructor, with a Ph.D. at an institution in Group I in 1964-1965 is less than $7,000 at Z5% of the institutions and greater than $7,800 at Z5% of the institutions. The salaries covered by the Survey are those given by an institution in one fiscal year for a full-time appointment of either nine or twelve months. Grants and contracts are included but sabbati­ cal payments and other part-time salaries are excluded. All salary figures are given in hundreds of dollars. The information for the 1964 Survey was compiled from usable returns received from Z99 institutions reporting on 3504 academic positions in 1963-1964 and 38Z7 positions predicted for 1964-1965. This Survey is the eighth in an annual series begun in May, 1957 by the Society's Com­ mittee on the Economic Status of Teachers.

643 INSTITUTIONAL MEMBERS OF THE SOCIETY, GROUP I Number of usable returns: 73 Total number on the staffs working full time on the campus RANK 1963-1964 1964-1965 Instructor 155 141 Assistant Professor 582 648 Associate Professor 443 497 Professor 571 628 TOTAL 1751 1914

Salary Survey RANK 1963-1964 1964-1965 Minimum Median Maximum Minimum Median Maximum Instructor 64- 74 70- 75 71- 79 70- 78 71- 80 70- 82 Assistant Professor 73- 81 80- 87 86-100 76- 85 83- 90 90-103 Associate Professor 85-100 95-112 110-125 90-·105 102-118 116-131 Professor 106-130 130-158 176-200 113-135 136-170 162-215

INSTITUTIONAL MEMBERS OF THE SOCIETY, GROUP II Number of usable returns: 100 Total number on the staffs working full time on the campus RANK 1963-1964 1964-1965 Instructor 16 21 Assistant Professor 395 457 Associate Professor 304 330 Professor 314 340 TOTAL 1029 1148

Salary Survey RANK 1963-1964 1964-1965 Minimum Median Maximum Minimum Median Maximum Instructor 56- 71 60- 70 67- 78 61- 76 66- 77 70- 84 Assistant Professor 66- 80 74- 85 80- 90 70- 84 80- 90 85- 95 Associate Professor 80- 96 88-101 92- 111 85-100 91-104 100-117 Professor 90-115 107- 130 110-140 98-120 111-140 123-155

644 INSTITUTIONS WHICH ARE NOT MEMBERS OF THE SOCIETY Number of usable returns: 126 Total number on the staffs working full time on the campus 1963-1964 1964-1965 Instructor 18 16 Assistant Professor 321 431 Associate Professor 199 253 Professor 186 244 TOTAL 7224 944

Salary Survey 1963-1964 1964-1965 Minimum Median Maximum Minimum Median Maximum Instructor 50- 64 55- 70 60- 78 50- 64 60- 70 63- 78 Assistant Professor 62- 74 68- 79 72- 86 65- 75 73- 82 75- 90 Associate Professor 72- 90 80- 94 82-105 75- 92 84- 98 86-108 Professor 88-109 91-110 100-130 91-114 98-123 105-140

SUMMARY OF ALL INSTITUTIONS SURVEYED Number of usable returns: 299 Total number on the staffs working full time on the campus 1963-1964 1964-1965 Instructor 189 178 Assistant Professor 1298 1462 Associate Professor 946 1018 Professor 1071 1169 TOTAL 3504 3827

Salary Survey 1963-1964 1964-1965 Minimum Median Maximum Minimum Median Maximum Instructor 60- 70 62- 74 68- 78 60- 72 66- 76 70- 81 Assistant Professor 66- 80 72- 85 77- 91 69- 82 76- 87 82- 96 Associate Professor 8·0- 95 85-100 91-116 84- 99 90-108 96-120 Professor 94-118 102-140 ll0-159 100-125 110-143 ll8-175

645 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D.

This survey was compiled from questionaires sent to individuals receiving their Ph.D. in mathematics during 1963. Z13 usable returns were received. Academic institutions attracted by far the largest proportion of new Ph.D.'s .in mathematics with 7Zo/o of the individuals reporting. Of these, 59% were teaching primarily, ZZ% were doing research primarily, and 18% received fellowship grants. Industry, even with its comparatively higher salaries, attracted only 17% of the new mathematicians with doctorates. 5% went to research institutes, and 6% were employed by the government. In all categories the North East attracted the greatest percentages of mathematicians, 38% of the total. The Far West was next in popularity, with Z1%. The Midwest attracted 19%, and the South 11%. 1% were employed abroad. 6Z% of the mathematicians reporting had more than 1 year of previous professional ex­ perience. 15% had between six months and one year of experience and 16% had less than six months of experience.

UNIVERSITIES, COLLEGES· AND TECHNICAL INSTITUTES (Nine Month Salary) TEACHING RESEARC.H Year Minimum Median Maximum Minimum Median Maximum 1960 $4,900 $6,500 $8,000 $5,ZOO $6,500 $8,000 1961 4,500 6,300 8,ZOO 4,800 6,500 9,000 196Z 4,300 7,000 9,ZOO 4,500 6,500 9,000 1963 4,500 7,ZOO 9,500 4,500 6,800 9,800 1964 4,100 7,900 11,000 6,000 7,ZOO 10,500

FELLOWSHIP (Yearly Stipend) Year Minimum Salary Median Salary Maximum Salary 1963 $4,500 $6,500 $9,000 1964 4,000 6,000 8,500 INDUSTRY (Tweive Month Salary) Minimum Salary Median Salary Maximum Salary 1960 $7,800 $11,000 $15,000 1961 8, 700 11,000 17,400 196Z 9,000 11,500 16,ZOO 1963 10,500 1Z,OOO 18,500 1964 10,400 13,ZOO 16,800

RESEARCH INSTITUTES (Twelve Month Salary) Year Minimum Salary Median Salary Maximum Salary 1960 $9,700 $10,500 $14,000 1961 8,400 11,000 14,ZOO 196Z 6,000 10,000 14,500 1963 5,500 11,700 13,500 1964 9,000 11,800 17,000

GOVERNMENT (Twelve Month Salary)

Minimum Salary Median Salary ~aximum Salary 1960 $7,200 $9,300 $13,000 1961 7,800 8,900 16,000 196Z 8,800 10,700 14,300 1963 10,100 1l,ZOO 15,000 1964 7,000 9,900 16,700

646 PERSONAL ITEMS

Mr. M. A. AL-BASSAM of Texas Mr. j. A. BERTON of Indiana State Technological College has been appointed College has been appointed to an associate to a visiting professorship at the Ameri­ professorship at Ripon College. can University of Beirut, Beirut, Lebanon. Mr. R. E. BLOCK of the California Dr. W. L. ALLEN of the Louisiana Institute of Technology has been appointed State University in New Orleans has been to an associate professorship at the Uni­ appointed to an associate professorship at versity of illinois. Lamar State College of Technology. Professor L. M. BLUMENTHAL of Mr. D. R. ANDERSON of the Massa­ the University of Missouri has received chusetts Institute of Technology has been the Distinguished Faculty Award. appointed to an assistant professorship at Dr. W. M.BOGDANOWICZofGeorge­ the University of Wyoming. town University has been appointed to an Dr. M. A. ARKOWITZ of Princeton associate professorship at the Catholic University has been appointed to an assist­ University of America. ant professorship at Dartmouth College. Mr. S. E. BOHN of Bowling Green Professor M. G. ARSOVE of the Uni­ State University has been appointed to an versity of Washington has been awarded associate professorship at Miami Univer­ a NATO Fellowship and will be at the sity. University of Hamburg. Mr. j. D. BUCKHOLTZoftheUniver­ Professor RAFAEL ARTZY of Rut­ sity of North Carolina has been appointed gers, The State University has been ap­ to an associate professorship at the Uni­ pointed a Visitor at the Institute for Ad­ versity of Kentucky. vanced Study for the fall term. Mr. R. C. BUEKER of the Iowa State Professor F. V. ATKINSON of the University has been appointed to an assist­ University of Toronto has been appointed ant professorship at the University of to a visiting professorship at the Univer­ Wyoming. sity of Kentucky for the academic year Professor R. T. BUMBY of Rutgers, 1964-1965. The State University has been awarded Mr. H. R. ATKINSON of Queen's Uni­ a Rutgers Research Grant. He will spend versity has been appointed a Lecturer at the academic year 1964-1965 at the Uni­ the University of Windsor, Windsor, On­ versity of Michigan as a Visiting Scholar. tario, Canada. Mr. R. D. BYRDoftheLouisianaState Professor E. W.AVERILLofParsons University in New Orleans has been College has been appointed to a profess·or­ awarded a National Science Foundation ship at Clarion State College. Science Faculty Fellowship and will study Professor JOSE BARROS-NETO of at the Australian National University. Brandeis University has been appointed Professor EUGENIO CALABI of the to an associate professorship at the Uni­ University of Minnesota has been ap­ versity of Montreal. pointed to a professorship at The Univer­ Professor P. T. BATEMAN of the sity of Pennsylvania. University of illinois has been appointed Mr. j. C. CANTWELL of the Institute to a visiting professorship at the City for Advanced Study has been appointed to University of New York. an assistant professorship at the State Dr. j. C. BEIDELMAN of Pennsylvania University of Iowa. State University has been appointed to an ProfessorS. S.CHERN ofthe Univer­ assistant professorship at the University of sity of California, Berkeley has been ap­ Kentucky. pointed a Member of the Institute for Ad­ Dr. I. D. BERG ofYaleUniversityhas vanced Study for the fall semester 1964. been appointed to an assistant professor­ Professor j. B. CHICCARELLI of ship at the University of lllinois. Stonehill College has been appointed to an

647 associate professorship at Bridgewater of Technology has been appointed to an State College. assistant professorship at Smith College. Mr. D. R. COMSTOCK of Oregon Mr. H. M. FARKAS of the lllinois State University has been appointed to an Institute of Technology has been appointed assistant professorship at the Central to an assistant professorship at Kansas Washington State College. State University. Mr. R. M. CONKLING of the New Dr. V.I. FILIPPENKO of the Univer­ Mexico State University has been appointed sity of California, Berkeley has accepted Director of the Computing Center at the a position as Senior Research Mathemati­ New Mexico Highlands University. cian with the Defense Research Laboratory Dr. E. H. CONNELL ofBrandeisUni­ of General Motors Corporation, Santa versity has been appointed to an associate Barbara, California. professorship at Rice University. Mr. E. A. FISER of Miami University Miss G. A. COON of the University of has accepted a position as an Associate Connecticut has been appointed to a pro­ Engineer with the Electronics and Ord­ fessorship at Goucher College. nance Division of Auco Corporation. Dr. W. E. COPPAGE of Indiana State Professor PHILIP FRANKLIN of the College has been appointed to an assist­ Massachusetts Institute of Technology has ant professorship at the Dayton Campus retired with the title of Professor Emeri­ of Ohio State and Miami Universities. tus. Professor H. H. CORSON of the Uni­ Dr. J. B. FUGATE of the University versity of Washington has been awarded of Iowa has been appointed to an assistant a Sloan Foundation Fellowship. professorship at the University of Ken­ Mr. T. M. CREESE of the University tucky. of California, Berkeley, has been appointed Mr. E. G. P.GERLACHoftheUniver­ to an assistant professorship at the Uni­ sity of Kansas has been appointed to an versity of Kansas. assistant professorship at the University Dr. R. T. DAMES of Thompson Ramo­ of British Columbia. Wooldridge, Incorporated has accepted a Mr. F. M. GLASER of the University position as an Associate Manager of of Chicago has been appointed an Assist­ Engineering Programming with the Scien­ ant Professor of Physics at the Bowling tific Data Systems. Green State University. Mr. W. J. DAVIS of Case Institute of Mr. R. P. GOBLIRSCH of the Univer­ Technology has been appointed to an assist­ sity of Colorado has been appointed to an ant professorship at Ohio State University. associate professorship at the College of Mr. R. A. DIBRELL, JR. of the St. Thomas. Hughes Aircraft Company has become Professor G. W. GOES of the Univer­ President of a new company, the American sity of Kansas has been appointed to an Research Corporation. associate professorship at the lllinois In­ Mr. J. D. DIXON of the California stitute of Technology. Institute of Technology has been appointed Mr. C. A. GREATHOUSE of the Uni­ a Senior Lecturer at the University of versity of Tennessee has been appointed New South Wales. to an assistant professorship at Vanderbilt Professor P. L. DUREN of the Uni­ University. versity of Michigan, on leave as a Sloan Professor SIMON GREEN of the Ari­ Fellow for the academic year 1964-1965, zona State University has been appointed will spend the fall at Imperial College, to a senior associate professorship at the University of London. California State Polytechnic College. Mr. L. A. EDISON of Stanford Uni­ Professor M. J. GREENBERG of the versity has been appointed to an assistant University of California, Berkeley has professorship at Reed College. been appointed a National Science Founda­ Mr. H. B. ENDER TON of the Massa­ tion Postdoctoral Fellow at Harvard Uni­ chusetts Institute of Technology has been versity. appointed to an assistant professorship at Dr. H. B. GRIFFITHS of New York the University of California, Berkeley. University has been appointed to a pro­ Mr. R. J. FABIAN of Case Institute fessorship at The University,Southampton,

648 England. appointed to a professorship at the Univer­ Professor P. E. GUENTHER of Case sity of Illinois. Institute of Technology has been appointed Mr. J. E. JOHNSON of St. Mary's a Theodore M. Focke Professorial Fellow. College has been appointed to an assistant Mr. M. R. HAGAN of Oklahoma State professorship at Tarkio College. University has been appointed to an assist­ Mr. D. W. JONAH of Tufts University ant professorship at Stephen F. Austin has been appointed to an assistant pro­ State College. fessorship at Wayne State University. Mr. G. W. HAGGSTROM of the Univer­ Professor B. W. JONES of the Univer­ sity of illinois has been appointed to an sity of Colorado is on leave to serve as an assistant professorship at the University Adviser, on Mathematics curricula, to of Chicago. Central American universities. Professor 0. G. HARROLD, JR. of Mr. D. S. KAHN of the University of the University of Tennessee has been ap­ Chicago has been appointed to an assist­ pointed to a professorship at Florida ant professorship at Northwestern Univer­ State University. sity. Mr. N. P. HERZBERG of the Massa­ Mr. D. W. KAHN of Columbia Univer­ chusett.s Institute of Technology has been sity has been appointed to an assistant appqinted to an assistant professorship at professorship at the University of Minne­ Ohio State University. sota. Mr. D. W. HENDERSON of the Univer­ Mr. M. W. KATZ of the University of sity of Wisconsin has been appointed a Wisconsin--Milwaukee has been appointed Member of the Institute for Advanced Study. to a visiting associate professorship at Mr. J. K. HIGHTOWER of Claremont Wayne State University. Men's College has been appointed to an Professor J.E.KELLEYofMarquette assistant professorship at the University University has been appointed to an asso­ College, University of Richmond. ciate professorship at the University of Professor F. E. HOHN of the Univer­ South Florida. sity of Illinois, on sabbatical leave for the Dr. C. N. KELLOGG of Louisiana academic year 1964-1965, has received a State University has been appointed to an Fulbright grant to lecture at the Technical assistant professorship at the University University of Delft, Delft, Netherlands. of Kentucky. Professor L. I. HOLDER of San Jose Mr. C. F. KOCH of the University of State College has been appointed to a pro­ Minnesota has been appointed to an assist­ fessorship at Gettysburg College. ant professorship at Kansas State Univer­ Mr. CHORNG-SHI HOUGH of the Uni­ sity. versity of Florida has been appointed to Mr. YUKIHIRO KODAMA of the De­ an assistant professorship at the Univer­ fense Academy has been appointed to an sity of Manitoba. associate professorship at the Tokyo Uni­ Dr. J. T. HOWSON, JR. of the West­ versity of Education. inghouse Electric Corporation has been Dr. I. I. KOLODNER of the University appointed to an assistant professorship at of New Mexico has been appointed Headof Rose Polytechnic Institute. the Department of Mathematics at the Dr. R. E. HUGHS of the Sandia Cor­ Carnegie Institute of Technology. poration has been appointed to an assistant Mr. ADAM KORANYI of the University professorship at Carleton College. of California, Berkeley has been appointed Mr. T AQDIR HUSAIN of the University to a visiting assistant professorship at of Ottawa has been appointed to an asso­ Princeton University. ciate professorship at McMaster Univer­ Professor KLAUS KRICKEBERG of sity. the University of Heidelberg has been ap­ Dr. ALEXANDRA IONESCU TULCEA pointed to a visiting professorship in the of the University of Pennsylvania has been Department of Mathematical Statistics, appointed to an associate professorship at Columbia University. the University of Illinois. Dr. RUDOLF KURTH of Michigan Dr. CASSIUS IONESCU TULCEA of State University has been appointed to a the University of Pennsylvania has been professorship at Georgia Institute of Tech-

649 nology. Dr. J. D. McKNIGHT, JR. of the Dr. L. C. KURTZ of the Universityof General Electric Corporation has been Utah has been appointed to an assistant appointed to an associate professorship at professorship at the University of Ken­ the University of Miami. tucky. Mrs. F. J. MacWILLIAMS of the Bell Mr. WILLEM KUYK of the University Telephone Laboratories has been appointed of Ottawa has been appointed to an assist­ a Senior Fellow for Mathematical Research ant professorship at McGill University. at the University of Cambridge, England Mr. SAM LACHTERMAN ofWashing­ for the academic year 1964-1965. ton University has been appointed to an Mr. FUMI-YUKI MAEDA, on leave assistant professorship at St. Louis Uni­ from Hiroshima University has been ap­ versity. pointed a Research Associate at the Uni­ Dr. R. P. LANGLANDSofthelnstitute versity of illinois for the academic year for Advanced Study has been awarded a 1964-1965. fellowship at the Miller Institute for Mr. P. H. MASERICK of the Univer­ Basic Research, University of California, sity of Wisconsin has been appointed to an Berkeley. assistant professorship at Pennsylvania Mr. L. H. LANIER, JR. of the Ohio State University. State University has been appointed to an Mr. C. K. MEGIBBEN of Texas Tech­ assistant professorship at the University nological College has been awarded. an of Virginia. ONR Postdoctoral Research Associateship Mr. F. W. LA WVERE of Reed College at the University of Washington. has been appointed a NATO Postdoctoral Mrs. B. F. MEDINA of New York Fellow at the Swiss Federal Institute of University has accepted a position as a Technology, Zurich, Switzerland. Mathematical Analyst with the American Dr. J. A. LEAVITT of the University Power Jet Company. of Pisa has been appointed to an assistant Professor R. A. MELTER of the Uni­ professorship at the University of Minne­ versity of Rhode Island has been appointed sota. to an assistant professorship at the Uni­ Mr. YU-LEE LEE of the University versity of Massachusetts. of Oregon has been appointed to an assist­ Mr. CHING-HWA MENG, on sabbati­ ant professorship at the University of cal leave from Sacramento State College, Connecticut. is a Visiting Associate Professor at the Dr. E. H. LEHMAN, JR. has accepted University of California, Berkeley for the a position as Systems Engineer-Mathe­ academic year 1964-1965. matical with the Telecomputing Services, Mr. R. B. MERKEL of Sacramento Incorporated, Panorama City, California. State College has been appointed a Mathe­ Mr. G. W. LOFQUISToftheLouisiana matician at the University of California, State University in New Orleans has been Lawrence Radiation Laboratory. awarded a National Science Foundation Professor P.R. MEYER of St. John's Science Faculty Fellowship and will study University has been appointed to an at the Louisiana State University in Baton assistant professorship at Hunter College. Rouge. Dr. J. H. MICHAEL of the University Mr. R. E. LYNCH of the General of Adelaide has been appointed to a visit­ Motors Research Laboratories has been ing professorship at Purdue University. appointed to an assistant professorship Professor K. S. MILLER of New York at the Computation Laboratory, University University has accepted a position as of Texas. Senior Staff Scientist with the Columbia Mr. M. H. McANDREW of the Inter­ University Electronics Research Labora­ national Business Machines Corporation tories. has been appointed to an assistant pro­ Dr. DON MITTLEMAN oftheNational fessorship at the University of Washington. Bureau of Standards has been appointed Mr. R. M. McCONNEL of the Univer­ Director of the Computing Center at the sity of Arizona has been appointed to an University of Notre Dame. assistant professorship at the University Mr. G. S. MONK of the University of of Tennessee. Minnesota has been appointed to an assist-

650 ant professorship at the University of versity has been appointed to an assistant Washington. professorship at the University of Ken­ Professor C. C. MOORE of the Uni­ tucky. versity of California, Berkeley has been Mr. HWANG-WEN PU of the Ohio State appointed a member of the Institute for University has been appointed to an assist­ Advanced Study for the academic year ant professorship at Wayne S~ate Univer­ 1964-1965. sity. Mr. L. j. MORDELL of Cambridge Mr. HEYDAR RADJAVI of Pahlavi University has beP.n appointed to a visiting University has been appointed to an assist­ professorship at the University of Illinois. ant professorship at the University of Dr. A. 0. MORRIS of the University lllinois. College of Wales has been appointed to a Professor W. T. REID of the State Uni­ visiting assistant professorship at the Uni­ versity of Iowa has been appointed to a versity of lllinois. professorship at the University of Okla­ Dr. T. W. MULLIKIN of the Rand homa. Corporation has been appointed to a _pro­ Professor ERIC REISSNER of the fessorship at Purdue University. Massachusetts Institute of Technologyhas Mr. KUNIO MURASUGI of Hosei Uni­ been awarded an Honorary Doctor of En­ versity has been appointed to ·an assist­ gineering Degree from Hannover Institute ant professorship at the University of of Technology, Germany and the Theodore Toronto. von Karman Medal of the American Society Mr. JOHN MYHILL of the Institute of Civil Engineers. for Advanced Study has been appointed to Dr. j. R. RICE of the General Motors a visiting professorship at the University Corporation has been appointed to a pro­ of lllinois. fessorship at Purdue University. Dr. L. P. NEUWIRTH of the Institute Dr. R. F. RINEHART has returned to for Defense Analyses has been appointed Case Institute of Technology as a professor a· visiting Lecturer at the Graduate after a two-year leave of absence as Mathematics Center of the City University Director of Research with the Department of New York. of Defense, Weapons Systems, Evaluation Dr. IV AN NIVEN of the University of Group. Oregon has been appointed to a visiting Dr. j. B. ROSEN of Stanford Univer­ professorship at the University of Cali­ sity has been appointed to a professorship fornia, Berkeley for the academic year at the University of Wisconsin. 1964-1965. Mr. MILTON ROSENBERG of the U.S~ Dr. E. T. PARKER of the Rand Cor­ Naval Weapons Laboratory has been ap­ poration has been appointed to an associate pointed to an assistant professorship at professorship at the University of lllinois. the University of Tulsa. W. j. PER YIN of Pennsylvania State Professor MURRAY ROSENBLATT of University has been appointed to an asso­ Brown University has been appointed to a ciate professorship at the University of professorship at the University of Califor­ Wisconsin-Milwaukee. nia, San Diego. Dr. G. 0. PETERS of the Mitre Cor­ Professor K. A. ROSS of the Univer­ poration has accepted a position as a sity of Rochester has been appointed to a Mathematician with the General Electric visiting assistant professorship at Yale Company, Philadelphia, Pennsylvania. University Dr. W. F. POHL of Stanford Univer­ Dr. M. I. ROTHENBERG of the Uni­ sity has been appointed to an assistant versity of Chicago has been appointed professorship at the University of Minne­ Visiting Assistant Professor and Assist­ sota. ant Research Mathematician at the Uni­ Dr. H. 0. POLLAK of the Bell Tele­ versity of California, Berkeley for the phone Laboratories has been appointed fall semester 1964. a visiting Lecturer at the Graduate Mathe­ Professor LEON RUTLAND of the matics Center of the City University of University of Colorado has been appointed New York. to a professorship at Virginia Polytechnic Dr. D. L. PRULLAGE of Purdue Uni- Institute.

651 Professor A. A. SAGLE of Syracuse associate professorship at Stanford Uni­ University has been appointed to an assist­ versity. ant professorship at the University of Dr. HAROLD SHULMAN of Republic California, Los Angeles. Aviation Corporation has been appointed Mr. G.B.SAKSENA of North American to an assistant professorship at Hunter Aviation, Incorporated has accepted a po­ College of the City University of New York. sition as Director of Central Actuarial Professor H. A. SIMMONS of North­ Services with Ostheimer and Company, western University is a Visiting Professor Incorporated. at Wartburg College for the academic year . Dr. DIRAN SARAF Y AN of Michigan 1964-1965 • State University has been appointed to an Mr. SEYMOUR SINGER ofthe Univer­ associate professorship at the Louisiana sity of California, Berkeley has been ap­ State University in New Orleans. pointed to an assistant professorship and Dr. D. E. SARA SON has been appointed Director of the Computer Center at San to an assistant professorship at the Uni­ Francisco State College. versity of California, Berkeley. Mr. RAJINDER SINGH of Punjab Uni­ Mr. C. T. SCARBOROUGH, JR. of versity has been appointed to a visiting Tulane University has been appointed to assistant professorship at the University an assistant professorship at Wayne State of illinois. University. Mr. M. S. SKAFF of Douglas Aircraft Mr. R. A. SCHAUFELE of Stanford Company has accepted a position as a University has been appointed to an assist­ Mathematician with the Hughes Aircraft ant professorship at Columbia University, Company, El Segundo, California. the Department of Statistics. Professor W. A. SMALL of the State Dr. J. F. SCHELL ofthe Florida State University of New York College at Geneseo niversity has been appointed to an asso­ has been appointed a Fulbright-Hays Lec­ ciate professorship at Charlotte College. turer at the University of Aleppo, Aleppo, Dr. MORRIS SCHREIBER has been ap­ Syrian Arab Republic. pointed a Lecturer at the Rockefeller In­ Mr. JULIUS SMITH of the University stitute. of illinois has been appointed to an assis't­ Dr. R. G. SEGERS of General Pre­ ant professorship at the University of cision, Incorporated has accepted a posi­ Tennessee. tion as a Member of the Applied Mathe­ Professor L. B. SMITH, JR. of Salem matics Group, Esso Engineering Center. College has been appointed to an associate Professor V. L. SHAPIRO of the Uni­ professorship at Gaston College. versity of Oregon has been appointed to a Professor JOHN STALLINGS, on leave professorship at the University of Califor­ from Princeton University, will be a nia, Riverside. Visiting Research Professor at Rice Uni­ Mr. S. S. SHATZ of Stanford Univer­ versity under a grant received from the sity has been appointed to an assistant Alfred P. Sloane Foundation. professorship at the University of Penn­ Dr. N. R. STANLEY of the Perkin­ sylvania. Elmer Corporation has accepted a position Mr. HARRY SHERMAN of the System as a Senior Mathematician with the Vitro Development Corporation has accepted a Laboratories, West Orange, New Jersey. position as Member of the Technical Staff, Professor NORMAN STEIN of the with the ITT Communications System, State University of New York at Stony Incorporated. Brook has been appointed to an associate Professor SEYMOUR SHERMAN of professorship at Haverford College. Wayne State University has been appointed Mr. E. F. STORM of Harvard Univer­ to a professorship at Indiana University. sity has been appointed to an assistant Dr. T. L. SHERMAN of the University professorship at the University of Virginia. of Wisconsin has been appointed to an Dr. SRINIVASASWAMINATHANofthe assistant professorship at the Arizona University of Madras has been appointed State University. to an assistant professorship at the Indian Mr. J. R. SHOENFIELD of Duke Uni­ Institute of Technology, Kanpur, India. versity has been appointed to a visiting Dr. L. W. SWANSON of Arthur Ander-

652 sen and Company has been appointed a cian at the University of California, Berke­ Professor of Quantitative Methods and ley. Managerial Economics at the School of Mr. D. R. WEIDMAN ofthe University Business, Northwestern University. of Notre Dame has been appointed to an Dr. R. P. TEW ARSON of the Honey­ assistant professorship at Boston College. well Company has been appointed to an Professor ALEXANDER WEINSTEIN assistant professorship with the Depart­ of the University of Maryland has been ment of Engineering Analysis at the State elected a Foreign Member to the Accade­ University of New York at Stony Brook. mia Nazionale dei Lincei in Rome. P:::ofessor R. C. THOMPSON of the Mr. R. J. WHITLEY of New Mexico University of British Columbia has been State University has been appointed to an appointed to an assistant professorship at assistant professorship at the University the University of California, Santa Bar­ of Maryland. bara. Mr. C. 0, WILDE of the University of Dr. R. G. THOMPSON of the Univer­ Illinois ·has been appointed to an assistant sity of Washington has been appointed to a professorship at the University of Minne­ professorship at Eastern Washington State sota. College. Mr. ANDREW WILSON of the General Mr. J. K. THURBER of Adelphi Uni­ Dynamics/Electronics has accepted a po­ versity has accepted a position as Asso­ sition as an Associate Mathematician with ciate Mathematician with the Brookhaven the Xerox Corporation. National Laboratories. Mr. F. W. WILSON, JR. of the Uni­ Mr. T. W. TING of the Courant Insti­ versity of Maryland has been appointed a tute of Mathematical Sciences, New York R.esearch Associate at Brown University. University has been appointed to an asso­ Dr. H. K. WILSON of the University ciate professorship at the North Carolina of Minnesota has been appointed to an State of the University of North Carolina assistant professorship at the Georgia at Raleigh. Institute of Technology. Dr. I. M. TRA WINSKI of Keuka Col­ Professor LECH WLODARSKi of the lege has been appointed to an associate University of Lodz, Poland has been ap­ professorship at the Louisiana State Uni­ pointed to a visiting associate professor­ versity in New Orleans. ship at the University of Kentucky for the Mr. C. E. TSAI of the lllinois Insti­ academic year 1964-1965, tute of Technology has been appointed to Professor P. K. WONG of Lehigh an assistant professorship at Michigan University has been appointed to an assist­ State University. ant professorship at Michigan State Uni;, Professor MINORU URABE of Hiro­ versity. shima University has been appointed to a Dr. W. J. WONG df the University of professorship at Kyushu University, Fuk­ Otago has been appointed to an associate uoka, Japan. professorship at the University of Notre Dr. D. E. VARBERG of Hamline Uni­ Dame. versity has been appointed a National Mr. M. C. WUNDERLICH of the Uni­ Science Foundation Postdoctoral Fellow versity of Colorado has been appointed at the Institute for Advanced Study for to an assistant professorship at the State the academic year 1964-1965. University of New York at Buffalo. Mr. MICHAEL VOICHICK of Dart­ mouth College has been appointed to an The following promotions are announced: assistant professorship at the University of Wisconsin. W. G. BADE, UniversityofCalifornia, Dr. Y. D. WADHW A of the Indian Berkeley to a professorship. Institute of Technology has been appointed J, H. BARRETT, University of Ten­ to a visiting associate professorship at nessee, to Acting Head of the Department Iowa State University. of Mathematics. Dr. F. W. WARNER, Ill, has been ap­ T. N. BHARGA VA, Kent State Univer­ pointed to an acting assistant professor­ sity, to an associate professorship. ship and Assistant Research Mathemati- B. L. BRECHNER, Louisiana State

653 University in New Orleans, to an assist­ P. A. TUCKER, University of illinois, ant professorship, to an assistant professorship. W. E. BRIGGS, University of Colorado, K. E. WHIPPLE, Auburn University, to a Professor and Dean of the College of to an assistant professorship. Arts and Sciences. J. A. WOLF, UniversityofCalifornia, R. B. BROWN, University of Califor­ Berkeley, to an as.sociate professorship. nia, Berkeley, to an assistant professor-;. ship. The following appointments to Instructor­ JACOB FELDMAN, University of ships are announced: California, Berkeley, to a professorship. J. M.G. FELL, University of Washing­ Brooklyn College: WOLFE SNOW; ton, to a professorship. University of California, Berkeley: B. A. D, M. FRIEDLEN, Georgia Institute of BARNES, R. J. FAUDREE, JR., ALFRED Technology, to an associate professorship. GRAY, J. B. LEWIS, R. T. MOORE, R. K. GETOOR, University of Wash­ S.M. NEWBERGER, C. C. PUGH, FRED­ ington, to a professorship. ERICK ROWBOTTOM, ALAN SCHUMIT­ M. W. HIRSCH, University of Califor­ ZKY; University of Chicago: C. E. WElL; nia, Berkeley, to a professorship. Haverford College: R. E. GOEGHAN, J. N. CHARLES HOBBY, University of SIEGEL; University of Illinois: S. B. BANK, Washington, to an associate professorship. J. M. BECK, C. F. OSGOOD, P. D. W. R. HOOVER, Computer Sciences ZVENGROWSKI; University of Kentucky: Corporation, to Vice President. W. M. PRIESTLEY; Louisiana State Uni­ T. W. HUNGERFORD, University of versity in New Orleans: H. D. KAHN; Washingto.n, to an assistant professorship. University of Michigan: H. M. STARK; D. H. HUSEMOLLER, Haverford Col­ Phillips Exeter Academy: W. H. BURGIN, lege, to an associate professorship. JR.; Smith College: T. L. McFARLAND. J.P. JANS, UniversityofWashington, to a professorship. Deaths: H. J. JOHNSON, University of Wash­ ington, to an associate professorship. Professor M. K. FORT, JR, of the N. D. KAZARINOFF, University of University of Georgia died on August 2, Michigan, to a professorship. 1964 at the age of 43. He was a member of A. A. KOSINSKI, University of Cali­ the Society for 22 years. fornia, Berkeley, to an associate profes­ Professor A. S. GALE of the Univer­ sorship. sity of Rochester died on July 6, 1964 at SAMUEL KOTZ, University of the age of 87. He was a member of the Toronto, to an associate professorship. Society for 65 years. VISVANATHA KRISHNAMURTHY, Mr. J. W. PORTER of Baltimore,. University of illinois, to an assistant Maryland died' on May 24, 1964 at the age professorship. of 52. He was a member of the Society for J, D. NEFF, Georgia Institute of Tech­ 22 years. nology, to an associate professorship. Professor W. W. ROGOSINSKI of the E. A. NORDHAUS, Michigan State University of Durham, England died on University, to a professorship. July 23, 1964 at the age of 69. He was a C. W. PATTY, University of North member of the Society for 14 years. Carolina, to an associate professorship. Professor C. R. STOREY, JR. of D. A. ROBINSON, Georgia Institute Florida State University died on July 28, of Technology, to an associate professor­ 19 64 at the age of 32. He was a member ship. of the Society for 9 years. G. E. SACKS, Cornell University, to an associate professorship. J.D. STASHEFF, University of Notre Dame, to an associate professorship. S. J. TAYLOR, Westfield College, University of London, to a professorship.

654 NEW AMS PUBLICATIONS

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 8 THEORY OF NUMBERS. Edited by Albert Leon Whiteman About 200 pages; Prepublication prices Forms; III. Analytic Number Theory; IV. valid until November 10, 1964; List Price Analytic Number Theory and Modular $6.00; Member Price $4.50. After publica­ Functions. Each session consisted of a tion, List Price at least $6.60. one-hour invited address, followed by several fifteen-minute talks. The speakers The Symposium on Recent Develop­ for the invited addresses were Professors ments in the Theory of Numbers was held Selberg, Iwasawa, Birch and Carlitz. In all, on November 21 and 22, 1963 in conjunc­ the present volume contains twenty-two tion with the six hundred sixth meeting of papers. the American Mathematical Society. There The volume is dedicated to the were four sessions, with the topics: I. memory of Professor Morgan Ward, whose Diophantine Analysis and Algebraic Num­ untimely death on June 26, 1963 kept him her Theory; II. Matrices and Quadratic from delivering one of the invited lectures.

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS Number 16 STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND ENGINEERING Edited by Richard Bellman 326 pages; List Price $7.60; Member complexity and uncertainty has forced Price $5.70. the more frequent use of probabilitistic concepts. Interest in the theory of stochastic The processes takes its rise from the begin­ fifteen papers in this book ning of the Twentieth Century with the show how the theory of stochastic proc­ classic article of Bachelier (1900) on the esses can be applied to a wide range of problems "Brownian motion" of the stock market in such diverse fields as mod~ ern control and with Einstein's immediately subse­ theory, physics and astronomy, dynamic programming, quent work on the same sort of motion for and the theory of learning actual particles. As the century has pro­ processes in mathematical psy­ chology. gressed, the shift of interest from classi­ Some of the papers also give a careful, cal determinism to stochastic description detailed explanation of the ab­ stract foundations of phenomena has become more and more of the theory and of its inner-mathematical pronounced, largely as a consequence of the applications to such fundamentally continuing challenges to the mathemati­ important topics as the distribution cian to guide research in engineering, of solutions of a stochastic differential economics, biology, medicine, and oper­ equation. ations research, where a combination of

655 MEMOIRS Number 47 PERIODIC SOLUTIONS OF PERTURBED SECOND-ORDER AUTONOMOUS EQUATIONS By W. S. Loud 134 pages; List Price $2.10; Member x" + g(x,x') = 0. The analysis is carried Price $1.58. one step further than usual, in that an exhaustive study is made of all cases Implicit function methods, neces­ that can be resolved by a knowledge of sarily involving a study of singular cases the partial derivatives of the first three of the implicit function theorem, are used orders of certain functions. In particular, in this Memoir to construct periodic solu­ the methods are applied to the case that tions X = XQ(t) + ~Xl(t) + 0(~) for the per­ f is independent of t, which means that the turbed equation x" = g(x,x') = d(t,x,x',~), period of the solution will itself vary where x0(t) is a periodic solution of with ~.

Number 48 EXTENSION OF COMPACT OPERATORS By J. Lindenstrauss 112 pages; List Price $2.10; Member study of norm-preserving extensions and Price $1.58. some related geometrical topics, Exam­ ples of such geometrical topics are in­ The purpose of this paper is to tersection properties of cells, charac­ study the connection between various ex­ terizations of Banach spaces whose con­ tension properties for compact linear jugates are L 1 spaces and characteriza­ operators, and to characterize the Banach tions of finite-dimensional spaces whose spaces which have these properties. The unit cells are polyhedra. main part of the paper is devoted to the

Number 49 COTORSION MODULES By E. Matlis

66 pages; List Price $1.80; Member older concepts of the theory of Abelian Price $1.35. groups, particularly torsion-free, divi­ sible, and reduced groups. The completion In this Memoir the concept of a of a module in the topology determined Abelian group, recently intro­ cotorsion by the ideals of the underlying integral and R. Nunke, duced by D. K. Harrison J. domain is shown to be a cotorsion module theory of mo­ is built into the genei·al and a thorough study is made of the com­ integral domain. dules over an arbitrary pletion of modules. Homological properties examines the T() this end, the author of cotorsion modules are investigated in universal properties of cotorsion mo­ detail. dules and their relationships with the

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656 Volume 41. A. M. ll'in and 0. A. Oleinik, S.D. Eidel'­ man, A. A. Dezin, G. E. Silov, M. S. 320 pages; List Price $4.40; Member Sneerson, A. F. Filippov, N. V. Azbelev Price $3.30. and z. B. Caljuk, and V. A. Jakubovic. Four papers on partial differential equations, by S. D. Eidel'man, L. N. Volume 43. Slobodeckii, Czou Jui-lin', and A. A. 310 pages; List Price $4.40; Member Dezin. Price $3.30. Volume 42. Fifteen papers on series and func­ tions of complex variables, by Ju. V. 292 pages; List Price $4.20; Member Linnik, P. L. Ul'janov, N. K. Bari, A, A. Price $3.15. Talaljan, 0, A. Ziza, A. M. Olevski1, Fifteen papers on differential equa­ V. S, Fedorov, I. A. Morev, I. I. Ibragimov, N. U, Arakeljan, V. S, Vladimirov, tions, by A. N, Tihonov and A. A. Samar­ S. Ja, Havinson, M. M. Dragilev, Ju. skii, 0. A. Oleinik, B. L. Rozdestvenskii, E. Alenicyn, and I. I. Pjateckii-Sapiro.

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657 SUPPLEMENTARY PROGRAM-Number 27

During the interval from June 27, 1964 through September 3, 1964 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Con­ tributed papers in this issue of these NOTICES. One abstract presented by title may be accepted per person per issue of the NOTICES. 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) On uniform structures (9) Finitely generated free subsemi­ Professor Alexander Abian, The groups of a free semigroup Ohio State University (64T-438) Professor E. K. Blum, Wesleyan (Z) Convergence properties of generali- University (64T-471) zed splines (10) General spline functions and their Dr. J. H. Ahlberg, United Aircraft mjnimum properties Corporation, East Hartford, Con­ Mr. Carl de Boor and Mr. R. E. necticut and Dr. E. N. Nilson, Lynch, General Motors Research Pratt and Whitney Aircraft, ·East Laboratories, Warren, Michigan Hartford, Connecticut (64T-485) (64T-456) (3) Fundamental properties of generali- (11) Almost locally flat imbeddings of zed splines manifolds Dr. J. H. Ahlberg, United Aircraft Professor J. C. Cantrell and Pro­ Corporation, East Hartford, Con­ fessor C. H. Edwards, Jr., Univer­ necticut; Dr. E. N. Nilson, Pratt sity of Georgia (64T-466) and Whitney Aircraft Corporation, (lZ) A note on multiple exponential sums East Hartford Connecticut and Pro­ Professor Leonard Garlitz, Duke fessor J. L. Walsh, Harvard Uni­ University (64T-435) versity (64T-451) (13) Two refinements of Morley's method (4) Set functions and zero sets. Prelim­ on omitting types of elements inary report Professor C. C. Chang, University Professor W. D. L. Appling, North of California, Los Angeles (64T- Texas State University (64T-44Z) 449) (5) A transplantation theorem for ultra- (14) WITHDRAWN. spherical series ( 15) The relative mean value of the Euler Professor Richard Askey, Univer• function sity of Wisconsin and Professor Professor Eckford Cohen, The Uni­ Stephen Wainger, Cornell Univer­ versity of Tennessee (64T-436) sity ( 64T-418) ( 16) Affine images of certain sets of (6) Nilpotent matrices measures. II Professor C. E. Aull, Kent State Professor H. S. Collins, Louisiana University ( 64T-490) State University, Baton Rouge (64T- (7) A hierarchy in the theory of implicit 416) definability (17) The ring C(X) determines the cate- Mr. G. M. Benson, University of gory of X California, Berkeley (64T-480) Professor W. W. Comfort and Mr. (8) On the subgroups of SL(3,q). IV. Stelios Negrepontis, University of Preliminary report Rochester (64T-460) Professor D. M. Bloom, Brooklyn (18) Lower bounds to holonomy College (64T-453) Mr. L. W. Conlon and Reverend

658 A. P. Whitman, Loyola University (31) Equivalence of connectivity maps and (64T-468) peripherally continuous transforma­ ( 19) A model for finite hyperbolic panes tions in GF(qZ), q odd Mr. M. R. Hagan, Oklahoma State Professor D. W. Crowe, University University (64T-434) of Wisconsin (64T-467) (3Z) Hanf numbers for some generaliza­ (Introduced by Professor Robert tions of first-order language Turner) Mr. Martin Helling, University of (ZO) Convolution transforms whose in- California, Berkeley (64T-448) version functions have complex roots (33) Some theorems on relative recur­ Mr. John Dauns and Professor siveness of higher type objects D. V. Widder, Harvard University Mr. P. G. Hinman, University of (64T-431) California, Berkeley (64T-443) (Z1) On a result of D. Gallarati concern­ (34) Toeplitz sections on groups ing semigroups Professor I. I. Hirschman, Jr., Professor D. W. Dawson, North Washington University ( 64 T-446) Texas State University (64T-419) (35) Oscillation criteria for even order (ZZ) Fundamentals of Finsler geometry differential equations Profes.sor john DeCicco and Pro­ Professor H. C. Howard, Univer­ fessor A. Z. Czarnecki, niinois sity of Maryland (64T-430) Institute of Technology (64T-440) (36) Classes of partial propositional cal­ (Z3) Pseudo-orthogonal nets of multi- culi and r .e.d,u.' s isothermal families Mrs. A. H. Ihrig, University of Professor john DeCicco and Pro­ illinois (64T-417) fessor john Synowiec, illinois In­ (37) Weakly compact sets stitute of Technology (64T-439) Professor R. C. James, Harvey (Z4) A not-e on almost periodic solutions Mudd College (64T-447) of ordinary differential equations (38) Converse of the Banach theorem in Mr. L. G. Deysach, Harvard Uni­ the case of one to one contracting versity and Mr. G. R. Sell, Univer­ mapping sity of Minnesota (64T-4ZO) Professor Ludvik Janos, The (Z5) Combinatorial functions of Dedekind George Washington University infinite cardinals (64T-469) Mr. Erik Ellentuck, Institute for (39) Threads in compact semigroups Advanced Study (64T-43Z) Professor R. J, Koch, Louisiana (Z6) A result in spectral synthesis State University, Baton Rouge Mr. R. J. Elliott, King's College (64T-4Z5) (64T-4ZZ) (40) Homeomorphisms on manifolds (Z7) A set of nonnormal numbers Professor Yu-Lee Lee, University Mr. Michel Mendes France, Uni­ of Connecticut and University of versity of California, Los Angeles Oregon (64T-486) (64T-454) (41) A characterization of analyticity (Introduced by Professor T. S.Motzkin) Professor K. 0. Leland, Ohio State (Z8) On generalized axially symmetric University (64T-413) potentials whose associates are dis­ (4Z) Extension of unitary operators tributions Mr. C. E. Linderholm, University Professor R. P. Gilbert, University of illinois (64T-433) of Maryland (64T-479) (43) The sum of two crumpled cubes is s3 (Z9) Eleven nonequivalent conditions on a if it is a 3-manifold commutative ring Mr. L. L. Lininger, University of Professor R. W. Gilmer, Jr., The Missouri (64T-445) Florida State University (64T-474) (44) Generalized Haar theorem (30) On Fourier-Stieltjes-sine-series Mr. H. L. Loeb, Aerospace Corpo­ Professor G. W. Goes, illinois In­ ration, El Segundo, California ( 64 T- stitute of Technology (64T-477) 481)

659 (45) Further remarks on a linear Dio- Mr. J, S. Ratti, University of Nevada phantine problem of Frobenius (64T-487) Dr. M. S. Lynn and Dr. B. R. Heap, (Introduced by Professor Malcom National Physical Laboratory, Ted­ Graham) dington, England (64T-458) (58) On the essential spectrum of the hy­ (46) Fatigue geometry and "tired light" drogen energy operator. Preliminary Professor M. A. McKiernan, Uni­ report versity of Waterloo, Ontario, Can­ Professor P. A. Rejto, New York ada (64T-476) University ( 64 T-428) (47) Regressive sets, supercohesion and (59) The degree of convergence for entire permutations functions Dr. T. G. McLaughlin, University Dr. J. R. Rice, General Motors of illinois (64T-441) Research Laboratories, Warren, (48) Bounds for solutions of a system of Michigan (64T-462) partial differential equations in a do­ (60) Representation of consumer choice main with Bergman-Silov boundary Professor M. K. Richter, Univer­ surface sity of Minnesota (64T-459) Professor Josephine Mitchell, ( 61) An extremal problem for functions Pennsylvania State University (64T- with positive real part 484) Professor M. S. Robertson, Rut­ (49) A condition equivalent to categoricity gers, The State University (64T- in uncountable powers 429) Professor M. D, Morley, The Uni­ ( 62) Propositional calculi which are clas­ versity of Wisconsin (64T-473) sical in implication and minimal in (50) Some new analogues to classical negation. II number-theoretic functions. Prelim­ Dr. T. T. Robinson, University of inary report illinois (64T-414) Mr. A. A. Mullin, University of ( 63) Antiflexible algebras which are not California, Livermore (64T-450) power-associative (51) Baire sets in realcompact spaces. Professor David Rodabaugh, Van­ Preliminary report derbilt University (64T-482) Mr. Stelios Negrepontis, University (64) Hyperbolic mixed problems. Prelim­ of Rochester (64T-461) inary report (52) On the lattice of filters of an impli­ Mr. Leonard Sarason, Stanford Uni­ cative semi-lattice versity (64T-452) Professor W. C. Nemitz, South­ (65) The span and principal functions in western at Memphis (64T-427) Riemannian spaces (53) Definition of the order types of the Professor Leo Sario and Mr. Moses arc and the real line without reference Glasner, University of California, to separability Los Angeles and Professor Mena­ Mr. E. D. Nix, Box 28, Norwich, hem Schiffer, Stanford University Vermont (64T-483) (64T-455) (54) A theorem of the Phragmen-Lindelof (66) Multiplier transforms in half spaces type Professor Eliahu Shamir, Univer .. Mr. J. K. Odds on, University of sity of California, Berkeley (64T- Maryland (64T-437) 464) (55) A uniqueness theorem for the n-di­ (67) Characterization of nearly flat 2- mensional Helmholtz equation manifolds in 3-manifolds and of Miss Nora Pernavs, Wayne State pseudo-half spaces University (64T-465) Mr. C. D. Sikkema, University of (56) On sets of completeness for families Michigan (64T-421) of Haar functions (68) On the group (2,3,7;9) Professor J. J. Price and Professor Mr. C. C. Sims, Massachusetts R. E. Zink, Purdue University Institute of Technology (64T-475) (64T-444) (69) Proper tensor products of commuta­ (57) .A Watson transform tive Banach algebras

660 Professor H. A. Smith, Drexel In­ vanced Study (64T-470) stitute of Technology (64T-415) (75) Maximal states of Jordan operator (70) Zero-free regions of ~(kl(s) algebras Mr. R. S. Spira, Duke University Dr, D. M. Topping, University of (64T-463) Chicago (64T-472) (71) Filling n-space with crosses (76) Harnack's inequalities on Cartan Professor S. K. Stein, University domains. II of California, Davis (64T-457) Professor S.-H. Tung, Miami Uni­ (72) On certain ring-like continua versity (64T-489) Mr. H. H. Stratton, University of (77) On compactness of mappings California, Riverside (64T-4Z4) Professor G. T. Whyburn, Univer­ (73)· Related pairs of Hasse diagrams sity of Virginia (64T-488) Mr. M. T. Stroot and Mr. R, M. (78) On the use of the successive over­ Grassl, University of Sa,nta Clara, relaxation method with several re­ Santa Clara, (64T-4Z6) laxation factors (Introduced by Professor A. P. Professor D. M. Young, Jr., Miss Hillman) M. F. Wheeler and Mr. J. A. (74) A note on the bordism algebra of in­ Downing, The University of Texas volutions (64T-478) Mr. J, C, Su, The Institute for Ad-

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Beginning October 1, 1964, a charge paid. This step is made necessary by an for postage will be added to invoices for increase in handling costs, which include all books ordered from the Society, except in particular the recent rise in postal in the case that an order is received pre- rates.

661 ABSTRACTS OF CONTRIBUTED PAPERS

The October Meeting in Garden City, New York October 24, 1964

615-1. J. B. LEICHT, University of Toronto, Toronto 5, Canada. Axiomatic theory of additive relations.

S. Mac Lane's theory of additive relations can be based upon the axioms ( :R): gf n h C g(f n g- 1h) and (\n): D(O) = 1 (and their duals) or, alternatively, upon (l}l), (2{ 1): fD(f) = f ·<""\ -1 and ('-'): D(f n g) = g f n 1. To make the theory complete one needs an axiom for direct products (as given by D. Puppe), which allows largely to express sums by intersections. An axiom requiring the existence of sub- and quotient-objects can be replaced by a construction. (\R), (2{ 1) and

(2{ 2): D(gf) C D(f) represent the "quasi-exact" case in the sense of D. Puppe, whose conjecture re­ garding its characterization by proper morphisms is verified. A similar characterization of S. Mac Lane's system can be given by postulating additionally (a) Equalizers, in the sense of B. Eckmann-P. Hilton; (b) Any two coinitial epimorphisms l,o such that oKerf and fKero are monic represent the upper completion of the (cofinal) maps Coker(f Ker o), Coker( oKer f). (Received June 1, 1964.)

615-2. K. D. MAGILL, JR., State University of New York at Buffalo, Buffalo, Ne.w York 14214. Some homomorphism theorems for a class of semigroups.

Let F(X) denote the semigroup of all functions mapping X into X where the binary operation is that of composition. An a.-semigroup a.(X) is any subsemigroup of F(X) which contains the identity i of F(X) and the ideal K(X) of all constant functions. An a.-homomorphism from a.(X) into a.(Y) is a homomorphism with the property that if tP(i) o g = g for g E K(Y), then g E [a.(X)]. Representation theorems are obtained for a.-homomorphisms, a.-monomorphisms and isomorphisms. An example is Theorem 1. A mapping c/J from a.(X) into a.(Y) is an a.-monomorphism if and only if there exists a function h from X into Y and a function k from Y into X such that (1) k o h = i and (2) (f) = h of o k for each f E a.(X). These results are applied to various semigroups of functions on topological spaces. For instance, it is shown that if X is a completely regular, arcwise connected space and Y is an arbitrary space, any non-trivial a.-homomorphism from S(X) (the semigroup of continuous functions on X) into S(Y) is an a.-monomorphism and thus every epimorphism is an isomorphism. (Received August 6, 1964.)

615-3. B. L. CHILTON, State University of New York at Buffalo, Buffalo, New York 14214. The stellated forms of the sixteen-cell.

The stellated forms of the regular polyhedra in E 3 have been known for many years; the last regular polyhedron to be examined was the icosahedron, whose stellateil forms were found in 1938 by H. S. M. Coxeter. The four-dimensional polytopes can be treated in an analogous manner: that is,

662 the cells can be extended until they meet again, in such a way as to preserve the rotational symmetry of the original figure. Investigation shows that there exist five stellated forms of the polytope { 4,3,3} (sometimes referred to as the 16-cell.) Four of these forms are "solid" in the sense that a ray from the center of symmetry of each must necessarily intersect a bounding surface. The fifth form may be called "center-exposing". Attention is given to the properties of these five figures, such as the a-rrangement of vertices. (Received August 17, 1964.)

615-4. E. J. BELTRAMI and M. R. WOHLERS, Grumman Aircraft Engineering Corporation, Research Department, Bethpage, Long Island, New York. On a converse to a theorem of Fatou.

A well-known theorem of P. Fatou (Acta Math., 30, (1906)) states that bounded holomorphic functions in a half plane have limiting values a.e. as we approach the boundary. In this note we show that if f(w) is a bounded measurable function then the condition Of= (1/7ri)f •pv(1/w) is both necessary and sufficient that f be the boundary value of a bounded holomorphic function; the condition is to be interpreted in the distribution sense of L. Schwartz (Thliorie des distributions, Paris, 1957-1959). This result provides us with a converse to the theorem of Fatou. (Received August 19, 1964.)

615-5• .c. B. MURRAY, The University of Texas, Box 7053, University Station, Austin, Texas 78712.. On the mean integral and a mean double integral.

Theorem 1. The following condition is necessary and sufficient for existence of the refinement type mean integral (M)J~fdg in case f is bounded on [a,b] and g is of bounded variation on [a,b]: if w > 0 and s > 0 there exists a partition a= t0 < t 1 < ••• < tn = b such that (1) Jbldgl - "n lg(t ) - g(t ) < s, and (Z) if Q is a collection of positive integer(s) q n such a "'-'p= 1 p p- 1 I ~ that the segment (tq_ 1, tq) contains numbers x and y with f(y) - f(x) ~ w, then LQig(tq) - g(tq_ 1) I < s. Theorem z. If (M) J~f dg exists there is a partition of [a ,b] such that if f is unbounded on an interval of it then g is unbounded or constant on that interval. Analogous theorems hold for a new mean double integral (M) J~J~F dG having approximating sums (1/4) L [_F(x,r) + F(q,y) + F(u,s) + F(p,v)] •G-i>q/r 5 taken over rectangular subintervals fp,q; r,s] of [a,b; c,d], with x and u in the segment (p,q) andy and v in the segment (r,s). They do not hold for the "vertex mean" integral (VM) J~J=F dG, which has approximating sums (l/4)L(F(p,r) + F(q,r) + F(q,s) + F(p,s)] •G-i>qfr 5• Some corollaries: if g is of bounded variation, (M)J~fdg exists only in case the interior integral (I)J~fdg exists; if G is of bounded variation, (M) J~J~F dG exists only in case four corresponding "edge" integrals exist. (Received August 2.4, 1964.)

615-6. F. G. ASENJO, University of Pittsburgh, Pittsburgh, Pennsylvania 152.13. A calculus of antinomies.

Let us assume that atomic propositions have either one or two truth values. Propositions (atomic or compound) will then be either true, false, or true and false. Let us call those propositions which are true and false "antinomies." Our purpose is to extend the classical propositional calculus to include operations with antinomies. Consider the following redefinition of the Sheffer stroke using

a 3 X 3 matrix in which 0, 1, 2. indicate true, false, and antinomic. First row 1, 0, Z; second row 0, 0, 0; third row 2., 0, 2.; with entries 0, 1, 2. for rows and columns. The remaining propositional

663 connectives are automatically defined outright, The resulting semantic definitions provide an inter­ pretation for inconsistent axiom systems. With these definitions, for example, Da Costa's system is inconsistent and incomplete, By a convenient extension, Russell's paradox can be produced, but

T E T ~ l(T E T) does not yield T E T and l(T E T). T E Tis antinomic and unprovable, and the same holds for l (T E T).

6I5-7. L. R, BRAGG, Case Institute of Technology, Cleveland 6, Ohio. A Rodrigue's formula for the Laguerre polynomials,

Let J.L = 2(u + 1), a real, and let ~J.Ldenote the generalized Laplacian operator in the radially symmetric case, that is ~/.!= D; + ((It- I)/r)Dr• By application of the Appell Transform and the time translation property of the source solution of the generalized heat equation ut(r,t) = ~!Lu(r,t), the () 2 · 2· 1 2 i _ 2 author develops the Rodrigue's type formula (*) Lj" (r ) = ((- I)J /2 J j!) t er • ~p·e r } • Applications of this result are also given. The method used also gives the usual Rodrigue's formula for the Hermite polynomials; (Received September 3, I964.)

6I5-8, KENNETH WHYBURN, 124 Triphammer Road, Ithaca, New York. A nontopological I - I mapping onto E 3,

In this paper a 1 - 1 mapping which is not a homeomorphism is exhibited from a connected, 3 locally connected and locally compact metric space onto an open polyhedral 3-cell embedded in E • This contradicts a result supposedly proven by V. V. Proizvolov [Dokl. Akad. Nauk SSSR. 151 (1963), 1286-1287. See Soviet Mathematics, Amer. Math, Soc. Trans!., Vol.4, no. 4] asserting that any I - 1 mapping from a connected paracompact space onto En is a homeomorphism. Furthermore the domain space of this example is simply a polyhedral subset of E 3 on which the mapping is piecewise linear. (Received September 8, 1964.)

6I5-9. R. A. MeHAFFEY, University of Massachusetts, Amherst, Massachusetts. The rim of a locally nilpotent group.

Let G be a group and R(G) the set {r E G: for all x E G, [r,x] = 1 iff (r,x) is cyclic}.

Theorem I, If G is torsion-free and locally nilpotent, then R(G) consists just of the identity; unless G is abelian of rank 1, in which case R(G) = G. Theorem 2. If G is a mixed locally nilpotent group, then R(G) is trivial. Theorem 3, Let G be a p-group with nontrivial center Z. If x E R(G), x 'I I, then xPi E Z, xPi 'I I (i = i(x)) and Z is either cyclic or quasicyclic, (Received September 3, 1964.)

615-10, H. A. SMITH, 4625 Larchwood Avenue, Philadelphia 43, Pennsylvania, Representations of tensor products of symmetric Banach algebras.

Let A 1, A 2 be symmetric Banach algebras with unity, A3 the completion in a cross-norm, P, of AI ® A2• Suppose A3 a symmetric Banach algebra with inherited involution. Let F\ be the norm­ alized positive functionals on A. with w* topology, P. the indecomposable ones, p. E P .. Every 1 1 1 1 pi® p 2 is extendable to a p 3 iff (1) v·;:; Sup IP 1 ® p 2 i. The mapping to this extension is a homeomor- phism of P 1 X Pz to P 3• The extension is in P3 iff {p l'p2) E P 1 X P 2• Let Ri be the set of equivalence

664 classes of irreducible representations of Ai" Give Ri the quotient topology induced by the natural mapping, 1ri' of Pi onto Ri. Let Ri be the classes of finite-dimensional representations and r 1 E Ci E Ri. The spaces Ri are T 1• Each r 1 @ r 2 is extendable to an r 3 iff (1). The 1ri commute with®; i.e. if ri E 1ri(Pi), the extension of r 1 Gil r 2 E1r3 (extension of p 1 ® p 2). The mapping to the extension is 1 - 1, continuous of R 1 X R 2 to R3• Its restriction to R't X Rz is a homeomorphism onto R3. This generalizes, for symmetric Banach algebras, a result of Gelbaum, Tomiyama, and Gil de Lamadrid. As a corollary one gets a known similar result on unitary representations of products of compact groups. (Received September 8, 1964.)

615-11. D. W. HENDERSON, The Institute for Advanced Study, Princeton, New Jersey 08540. Extensions of Dehn's Lemma and the Loop Theorem,

The problem involved is to take a given singular disk whose interior does not intersect the boundary and to change it into a nonsingular disk which has certain desired properties in common with the original singular disk. The main technique used here is that of 2-sheeted coverings as developed by A. Shapiro and J. H. C. Whitehead. The primary difference between the methods used here and the previous work in the field is that we shall pay close attention to the geometrical relationships between the given singular disk and the obtained nonsingular disk. At the same time, the Loop Theorem is extended in certain cases to 2-submanifolds of a 3-manifold. Using these techniques, we attack the general question proposed by R. H. Bing: Does every simple closed curve which can be shrunk to a point in its -own complement bound a disk 7 A previously known but unpub­ lished result says "yes, if the simple closed curve is tame." This paper also gives an affirmative answer when the simple closed curve has finite penetration index and only a finite number of wild points, and also in the case that it has penetration index 2 at every point and' the wild points form a tame 0-dimensional set. This leads to a new condition for tameness of a simple closed curve. (Received September 8, 1964.)

615- 12. JAMES RADLOW, Purdue University, Lafayette, Indiana 4 7907. Generalization of a Lax-Morawetz-Phillips time decay theorem.

An exterior initial-boundary problem for the wave equation is considered: All boundary points of the bounded three-dimensional obstacle or obstacles (not necessarily star-shap.ed) are accessible; the wave function and its first time-derivative vanish at t = 0 (but nonhomogeneous initial data are easily treated); Dirichlet, Neumann or mixed boundary conditions are imposed, while the boundary function ("incident wave") need not have finite energy in the whole space, Theorem. The total field approaches zero exponentially with increasing time, at any point exterior to the obstacle or obstacles. The basic idea of the proof: Construct the appropriate time-reduced Green's function as the spherically uniform limit of a sequence of meromorphic functions of the complex wave-number k. This function is then itself meromorphic in the k-plane, and the statement of the theorem follows at once. A time-decay theorem of Lax, Morawetz and Phillips (Bull. Amer. Math. Soc. 68 (1962), 593-595) is therewith generalized in several respects. (Received September 9, 1964.)

665 615-13. j. F. TRAUB, Bell Telephone Laboratories, Murray Hill, New jersey. On Lagrange­ Hermite interpolation.

Let the p(n + 1) numbers y~m), 0 ~ i ~· n, 0 ~ m ~ p - 1, be given. A classical problem in 1 interpolation theory is to find a formula for the unique polynomial of degree p(n + 1) - 1 such that P~~~(xi) = y~m), 0 ~ i ~ n, 0 ~ m ~ p- 1. The solution is well known for the cases p = 1, p = 2, and n = 0. We give a solution for arbitrary p and n. The solution, which is of surprising simplicity, depends upon Bell polynomials. (Received September 9, 1964.)

615-14. ROBERT HERMANN, 701 Grizzley Peak, Berkeley, California. Compact totally geodesic hypersurfaces.

Let N be a simply connected, compact, totally geodesic hypersurface of a Riemannian manifold of positive Ricci curvature. Then, there exist other imbeddings of N arbitrarily close to the given one which have less area. (Received Septem·ber 10, 1964.)

615-15. s. G. MROWKA, The Pennsylvania State University, 227 McAll,ister Hall, University Park, Pennsylvania. Some approximation theorems for rings of unbounded functions.

Let X be a completely regular space and let A be a uniformly closed subring of C(X) con­ taining all constant functions and satisfying the condition: (a) if f E A and f(p) # 0 for every p E X, then 1/f EA. We say that A separates Z~X if for every two disjoint Z-sets z 1 and z2 in X there is an f E A such that f[Z1] nf[Z2] = g (the bars indicate the closure in the reals R). Theorem 1. If A separates Z-sets in X, then A= C(X). Theorem 2. If X is Lindelof and A separates points and closed subsets of X, then A= C(X). Theorem 3. Let X= Y X Z and let A be the smallest uniformly closed subring of C(X) containing C(Y) U C(Z) and satisfying the condition (a). If X is Lindelof, then A= C(X). (Received September 10, 1964.)

615-16. G. W. HEDSTROM, University of Michigan, Ann Arbor, Michigan 48104. A generalization of Holmgren's uniqueness theorem.

Let 1/1. be a real valued function in c 20J,), n an open set in Rn. Let P(x,D) be a differential operator with analytic coefficients defined inn, having real coefficients in the principal part. Let x0 k be a point inn where gradt/l(x0) =No# 0, and Pm(x0 ,N0) = 0, but 2:1Pm(x0.N0)1 > 0. Here Pm denotes the principal part of P and P~ (x,~) = iJp m (x,~)/o~k. Assume that for some T > 0 the bicharacteristic defined by: dxk/dt = iJpm (x,~)/o~k' d~k/dt =- 8Pm (x,~)/oxk, x(O) = x0, ~(0) = N0, has either the image of the interval 0 < t < T or the image of - T < t < 0 contained in jx: t/l(x) > l/>(x0)}. Then there exists a neighborhood n' C n of x0 such that every u E ~OJ,) satisfying P(x,D)u = 0 and vanishing when 1/>(x) > t{>(x 0), x E n, must also vanish in n '· A characteristic hyper surface is con­ structed, and a modification of Hormander's method is used in the proof. (Received September 10, 1964.)

666 615-17. S. L. SEGAL, University of Rochester, Rochester, New York. On Ingham's summation method.

If {an} is a given sequence of real numbers define I(t) = Ln~t'ri£-]an where [x] is the greatest integer in x. The series :Ean is said to be (I) - summable to s if limt~ 00 I(t) = s. (See e.g., Ingham, j. London Math. Soc. (1945), 171-180; Hardy, Divergent series, Clarendon, Oxford, 1949, Appendix IV). Ingham showed that, although (I)-summability is not comparable with convergence, (C, - o) => (I) => (C, o) for arbitrary o >0 where (C,k) denotes summability by Cesaro means of order k. Tauberian conditions under which (I)-summability can be inferred from Abel-summability or (C,k)-summability for some k ~ 0 are studied. Particular examples: Theorem: If :Ean is Abel-summable and Ldln dad= 0(1), then :Ean is (I)-summable. Theorem: There exists a series :Ea which is (C,1)-summable and such that I(t) is bounded, but which is not (I)-summable. n Concerning the related question of "limitation theorems" for (I)-summability we have the Theorem: If :Ea:n_ is (I)-summable then Ln;>xan = o(log x). titeceived September 10, 1964.)

667 ABSTRACTS PRESENTED BY TITLE

64T-413. K. 0. LELAND, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210. A characterization of analyticity.

Let B and C be Banach spaces, N > 0, and F a family of maps on open subsets of B into C closed under the operations of addition, multiplication by a scalar, and linear translation, which satisfies a weakened form of Schwarz's Lemma: Let f E F, M > 0, r > 0, such that x E B, llxll < r implies X E domain f and llf(x>ll ~ M. Then X E B. llx II < r implies llf(x) - f(O) II ~ 2NMr -lllx n. Then the elements of F are infinitely differentiable in the sense of Fr~chet, and under slightly stronger conditions expandable in power series. The theory applies equally well to complex Fr~chet differen­ tiable maps and harmonic functions. The work of Porcelli and Connell in topological analysis (see Duke Math. J. (1961), 73-81) is contained as a special case. (Received June 5, 1964.)

64T-414. T. T. ROBINSON, 2107 Grange Drive, Urbllna, Illinois 61801. Propositional calculi which are classical in implication and minimal in negation. II.

The following short decision procedure for Pxr is a direct corollary of Gentzen's Cut Elimin­ ation Theorem for that system (Propositional calculi which are classical in implication and minimal in negation, I, Abstract 64T-391, these Notices 11 (1964)): Metatheorem: f-xr A iff: A is both a classi­ cal tautology, and a tautology with the classical truth-tables modified as follows: f =!. (if x is f) and ~ t = ~i =!_(if x is ~ ). (Received June 9, 1964.)

64T-415. H. A. SMITH, Drexel Institute of Technology, Philadelphia, Pennsylvania. Proper tensor products of commutative Banach algebras.

Let A3 be the completion with respect to a cross-norm, v, of the tensor product of two com­ mutative Banach algebras, A 1 and A2, and suppose A3 a Banach algebra. Let Xi be the characters of Ai with weak* topology and xi E Xi, A continuous extension of x 1 ® x2 to A3 is in x3• Call a mapping of (x 1,x2) to this extension canonical. Call A3 proper iff x1 X x2 is canonically homeomorphic with x3• The following statements are equivalent. (1) A3 is proper, (2) v <:; lx 1 ® x 2 1 for all x 1 ® x2, (3) every x 1 ® x2 has a continuous extension to A3, (4) the Shilov boundary of A3 is canonically homeomorphic with the product of those of A 1 and A2• Let Ai denote the dual of Ai. If A3 is proper and semi-simple then A3 is the weak* closure of Ai ® Az• If v= "Y, the greatest cross-norm, the converse holds. If a proper A3 is isometrically isomorphic with its Gelfand representation, A 1 and A2 also have this property and v = A, the least uniform cross-norm. (Received June 17, 1964.)

64T-416. H. S. COLLINS, Louisiana State University, Baton Rouge, Louisiana 70803, Affine images of certain sets of measures. II.

Notation and terminology are as in Abstract 64T-405, these Notices 11 (1964). Theorem A.

These are equivalent: (1) K E 5f. (2)(a) there is compact T C K and z E K such that E =

668 {at + (1 - a)z: a E C, Ia I = 1, t E T ~. (b) Lz(K,C) separates points of K and contains 10 = 1 on T, (c) K1 = (T) is a simplex, (d) p EK -->p =Rep +.ilmp- iz, where (unique) Rep,Imp EK2 = (T U (2z - T)), (e) if :Lajxj + i( DjYj) - iz E K, these finite sums, aj, bj E R, xj'yj E K2, all j, and Laj = 1 = Dj' then :Lajxj and Lbjyj E K2, (3)(a) Part (a) of (2) holds, (b) Lz(K,C) separates points of K and each continuous real (or complex) f on Tis extendable to f E Lz(K,C), As an applica­ tion, denote by 9•, Ye•, ~· (resp.) those K E .9, Ye, and ~ (resp.) which become affine topological semigroups via the F-equivalence map relating K to the appropriate convolution semigroup of meas­ ures on S (now assumed to be a compact semigroup). Theorem B. K E 9• +--> K E 9 and E is a topol.ogical semigroup, Theorem C. K EYe' +--> K E ~and the T (resp. z) of Theorem 2 is a topo­ logical semigroup (resp, zero for K). Theorem D. K E ~· +--> K E ~and the T (resp,z) of Theorem A is a topological semigroup (resp. zero for K). Other applications to measure algebras and semi­ groups are given. (Received June 22, 1964.)

64T-417. A, H. IHRIG, 509 South Fifth Street Champaign, Illinois. Classes of partial propositional calculi and r.e.d,u.'s.

Let N = {1,2,. •• , n~ and the power set of N, .9(N) = {s 1, ... ,s 2 n~· For r.e.d.u.'s, D1, ... , On, define ~(0 1 , ... , Dn) to be the set of degrees { UiEsl Di''"' UiEs2nDtf. Theorem 1: For any finite set of r.e.d,u.'s, 0 0, Dl' ... 'Dn, there exists a class of partial propositional calculi such that each member is of degree o0 U D for some D E ~(0 1 , ... , Dn>• and the problem of determining of an arbitrary member to which D it corresponds is of degree 0 0• Theorem 2: For any pair of r.e.d,u. s, 0 1 and 0 2 (D 1 i D2), there exists a class of partial propositional calculi such that the problem of determining of an arbitrary member whether or not its decision problem is of degree 0 2 is of degree 0 1• Theorem 3: For any finite set of r.e.d,u.'s, Dl'"''Dn, there exists a class of Thue systems (partial propositional calculi) such that each member has word (decision) problem of degree

D E 9(01, ... , Dn)• Further, the problem of determining of an arbitrary member to which such D it corresponds is of degree Ui ENDi. The proofs of these theorems require Theorems 1 and 2 of Abstract 64T-302, Theorems 1 and 2 of Abstract 64T-406, and Result A of Boone, Bull. Amer. Math, Soc. 68 (1962), 616-623, (Received June 22, 1964.)

64T-418. RICHARD ASKEY, University of Wisconsin, Madison, Wisconsin and STEPHEN WAINGER, Cornell University, Ithaca, New York. A transplantation theorem for ultraspherical series.

Let f((J) be a measurable function on (0,7r) such that lf(O)IP(sin O)a.p E LP, 1 < p < oo, - 1/p < a. <1 - 1/p.· Then if Pn (x) is the Legendre polynomial and if f((J) ~:Lances nO, define Trf((J) = "-'anr" n P n(cos O)(n + 1/2) 1/2 (sin (J ) 1/2• Then Trf((J)(sm. fl)a. E L P , Trf((J) converges to Tf((J) as r ----> 1, and IITf((J)(sin Btllp ::> A llf(8)(sin 0) a.llp· Conversely the transplantation is bounded if one starts with the Legendre series and maps to the cosine series, Among the theorems that follow from this is a new proof of Pollard's theorem on mean convergence for Legendre series. There is an analogous result for ultraspherical polynomials. (Received June 24, 1964,)

669 64T-419. D. F. DAWSON, North Texas State University, Denton, Texas. On a result of D. Gallarati concerning semigroups.

Gallarati (Boll. Un. Mat. !tal. (3) 18 (1963), 279-280) observed that if ~is a semigroup with zeroid element A such that for each a E ~. the solutions x, y E ~ of xa = Aand ay = A are unique, then

~ is a group. The following two theorems show that the uniqueness of y in the above statement can be

omitted. Theorem 1. Suppose ~ is a semigroup with zeroid element A. If the solution e E ~ of e A = A is unique, then ~· = {ea Ia E ~I is a subgroup of ::!:. Theorem 2. Suppose ~ is a semigroup with zeroid element A. If for each a E ~.the solution x E ~ of xa =A is unique, then ~ = 1:•. (Received June 24, 1964.)

64T-420. L. G. DEYSACH, Harvard University, 2 Divinity Avenue, Cambridge, Massachusetts 02138 and G. R. SELL, Institute of Technology, University of Minnesota, Minneapolis, Minnesota 55455. A note on almost periodic solutions of ordinary differential equations.

Let f(x,t) be continuous on Rn X R1 and periodic in t. Assume that (1) x' = f(x,t) satisfies some uniqueness condition. Theorem. If there is a solution t/J(t) of (1) which is (i) bounded for all t ~ T and (ii) positively Lyapunov stable, then (1) has an almost periodic solution. This extends results of J. L. Massera [The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457-475] and C. R. Putnam [Unilateral stability and almost periodicity, J. Math. Mech. 9 (1960), 915-918]. (Received June 29, 1964.)

64T-421. C. D. SIKKEMA, University of Michigan, Ann Arbor, Michigan. Characterization of nearly flat 2-manifolds in 3-manifolds and of pseudo-half spaces.

Theorem 1. Let M be an (unbounded) 2-manifold in a 3-manifold N such that M is locally flat except at m points. Then there is a locally flat 2-manifold M' in N, there are m disjoint arcs in N which intersect M' in one endpoint and which are locally flat except at the other endpoint and there is a pseudo-isotopy tPt of N onto itself which shrinks each arc to a point such that t/J1 (M') is a 2-manifold equivalent to M. Moreover, M' and the arcs are unique up to equivalence class. Definition. An n-manifold Mn is an n-pseudo-half space if Int Mn "" Rn and Bd Mn "" R n- 1• Let Bn be the unit ball in R n centered at the origin. Theorem 2. If Mn is an n-pseudo-half space, then Mn ""Bn - n, where a, is an arc in Bn which intersects the boundary of Bn at one endpoint and is locally flat in Rn except at the other endpoint. Moreover, Bn - a, is ann-pseudo-half space for any such arc a. Corollary. Forni 3, Mn ""R~. (Received June 29, 1964.)

64T-422. R. J. ELLIOTT, King's College, Cambridge, England. A result in spectral synthesis.

G denotes a general locally compact abelian group. We prove Theorem. Every closed transla­ tion invariant subspace (variety) of the space of continuous functions (resp. Lfoc' 1 < p < oo), on G, whose annihilator ideal in the space of measures with compact support (resp. the L q functions with compact support, 1/p + 1/q = 1), is a principal ideal, is generated by the (nonempty) set of exponential monomials it contains. (An exponential monomial is a product of powers of real representations of G multiplied by a generalized character of G.) The main steps in the proof are: (i) observing that a

670 compact subset of G is contained in a compactly generated subgroup of the form R m X zn X L (R the reals, Z the integers, and L compact); (ii) an extension of a division theorem of L. Ehrenpreis (Amer. J. Math. 77 (1955), 293-328); and (iii) an application of techniques due to B. Malgrange (Se'minaire P. Lelong 1958/1959, Analyse, Facuitl! des Sciences de Paris, 1959). (Received June 30, 1964.)

64T-423. WITHDRAWN.

64T-424. H. H. STRATTON, University of California, Riverside, California. On certain ring-like continua.

Let M denote a compact connected metric space. If x is a point of M, let Lx denote the point set which contains x and all points of M at which M is not aposyndetic with respect to x. If whenever Lx n Ly = !il (x,y EM), there exist irreducible subcontinua H and K between Lx and Ly such that IntH n Int K = !il but M = H U. Kx U K U Ly' then M is said to be ring-like. If M is ring-like (e.g., when M is a simple closed curve), then the closure of the complement of a continuum is connected. If M is not the union of four indecomposable continua, then the converse of this theorem is true. In addition to sharpening this converse, other properties of such spaces are studied. (Received JUly 1, 1964.)

64T-425. R. J. KOCH, Louisiana State University, Baton Rouge, Louisiana. Threads in compact semigroups.

Theorem. If S is a compact connected Hausdorff topological semigroup with unit, and if each subgroup of S is totally disconnected, then S contains a standard thread which meets the minimal ideal and contains the unit. Here, a standard thread is a semigroup on an arc (not assumed

671 metrizable) in which the endpoints act as zero and unit, The theorem was known (with the additional hypothesis that S be normal (Sa= aS, each a E S)) to Hunter (Duke Math J, Z7 (1960), Z83-Z89); it can be used to furnish a simple proof of the fact (Hunter, Trans, Amer, Math. Soc. 93 (1959), 356-358) that if S is a one-dimensional acyclic continuum which is a semigroup with unit, then S is arcwise connected, The class !fi of semigroups with unit on a continuum, in which each subgroup is totally disconnected, is closed under continuous homomorphic images and cartesian products; also each member of .5ff is arcwise connected, acyclic, and in the metric case contractible, (Received July Z, 1964.)

64T-4Z6, M, T. STROOT and R, M, GRASSL, University of Santa Clara, Santa Clara, California, Related pairs of Hasse Diagrams,

Let {x1, ... ,xs~ be partially ordered by a Hasse Diagram D satisfying x 1 < xi for i > Z, x 1 1. Xz• Xz < xj for some j > z, and such that if~< xi for a fixed k and all i for which Xz < xi then k = 1 or z. Let D' result from D when x 1 1 Xz is replaced by x 1 < Xz and let f(n) and f'(n) be the numbers of realizations of D and D' by families of s subsets of a set of n elements. Using extensions of methods in Hillman, On the number of realizations of a Hasse Diagram by finite sets, Proc, Amer.

Math. Soc, 6 (1955), 54Z-548, it is shown that Lf,;0lcn,if'(i) is f(n) or Zf(n) depending on whether or not there is a j > Z with xz I. xj. (Received July Z, 1964.)

64T-4Z7, WILLIAM NEMITZ, Southwestern at Memphis, Memphis, Tennessee 3811Z, On the lattice of filters of an implicative semi-lattice,

Let (L, ~, 11, •) be an implicative semi-lattice, Let F(L) be the lattice of filters of L, ordered by containment, with intersection as infimum and span as supremum, For J and K filters

of L, let J o K = {x E L: p(x) n J C K ~. where p(x) is the principal filter of x. Then "o" is an impli­ cation in F(L), and hence F(L) is a complete, dual atomic, implicative lattice, Let L and M he im­

plicative semi-lattices. Let f: L --> M be onto. Let f': F(L) --> F(M) be such that for J E F(L), f'(J) = sp(f(J)), where sp(S) denotes the smallest filter containing a subsetS of M. Let F 0 (L) = L, f0 = f, and p0(x) = x, for x E L. For n a positive integer, let Fn(L) = F(Fn- 1(L)), fn = (fn- 1)•, and pn(x) = p(pri- 1(x)). So fl: Fn(L) --> Fn(M). Theorem, Iff is a homomorphism of L onto M with kernel A, then for each even positive integer n, fn is a homomorphism of Fn(L) onto Fn(M), the kernel of which is pn(A) E pn+1(L), If, for any odd positive integer n, f1 is a homomorphism, then f1 is a homomor­ phism for every positive integer n. This is the case iff A is a complemented element of F(L). In

0 0 this case, for n odd, the kernel of f1 is pn(A ), where A =A 0 {1~. (Received March 9, 1964,)

64T-4Z8, P. A. REJTO, New York University, Courant Institute of Mathematical Sciences, Z5 Waverly Place, New York 3, New York. On the e·ssential spectrum of the hydrogen energy operator, Preliminary report,

F. Wolf and F. Agudo formulated criteria for the compactness of a potential with reference to

~. the Laplacian (Rend. Acad, Lincei, (8) Z4 (1958),Nov,), In this report, his criteria are extended to include the potential r- 1• More specifically, Theorem 1, Suppose that the function p(x) is square in­

tegrable for x in any bounded region of E3, and it tends to 0.!! x --> oo. Then the operator p ~-

672 pact with respect to a. If square integrability is replaced by integrability of the fourth power then the conclusion holds in En for arbitrary n. Let A be an elliptic operator in free space such that its -coefficients tend to the corresponding coefficients of a, and their derivatives tend to 0 as x -> oo. Then using that the operator D is bounded by a, Theorem 1 yields the more general Theorem 2 • .!!_pis as before, then the operator p is compact with respect to A. Following an argument of Wolf (Indag. Math. 21. No. 2, 1959), Theorem 2 can be extended to an operator A acting in the exterior of a bounded region. If square integrability is replaced by integrability with power JJ., JJ. > n/2, then the conclusion holds in En. (Received July 3, 1964.)

64T-429. M. S, ROBERTSON, Rutgers, The State University, New Brunswick, New Jersey. An extremal problem for functions with positive real part,

Let 9 be the class of regular functions P(z) in E {z: lz I < 1} with P (0) = 1 and ~ P(z) > 0, Let F(w) denote a nonconstant function that is analytic in the convex domain D which is the image of E by the mapping w = £ 1 J 0((1 + t)/(1 - t))dt, This domain D lies in the half-strip given by the inequalities 1Yw I < 1r, ~w > 2 log 2 - 1. Then, for each r < 1, the minimum mF(r) = minPE!?'minlzl=r~F(z- 1 f0 P(t)dt)occurs for a function of the form P(z) = (1 + fz){1- fz)-1, where

E is an arbitrary complex constant of absolute value one, and for no other functions. The proof is by the method of subordination. The author obtained a second proof by variational techniques, An appli­ cation is made to the class ~a. of analytic functions f(z), f(O) = O,f' (0) = exp ia. for which ~f' (z) > 0 in E, (Received July 3, 1964.)

64T-430. H. C. HOWARD, Fluid Dynamics Institute, University of Maryland, College Park, Maryland, Oscillation criteria for even order differential equations,

Consider the equation (*) y<2n)(x) + f(y(x))p(x) = 0, with p(x) > 0 and continuous for 0 < x < oo,

f3 ~ f(y)/y ~ a. ({3 ~ 1 ~ a. > 0), f(y) E C for - oo .c y < oo, f(y) odd (an assumption that may be relaxed at the expense of extra computation) and such that yf(y) > 0 if y i 0. The following theorems hold, Theorem 1, If y is a solution of (*) existing for 0 < x < oo such that y(x) > 0 for all x sufficiently oo 2n-2 large, then f a x p(x)dx < oo, a > 0, Theorem 2, If y is a solution of (*) existing for 0 < x < oo, if there exists a positive function g E c• for 0 < x < oo such that J';:'Cr/g)dt = oo and J;:'

there exist integrable functions fi (i = 1,2, ... , 2n - 1) such that IY(i) I ;> f1 as x -. oo, if J~ ~n- 1 lpi lfidx < oo, if for some a. > 0, f3 > 0, Jpgdt =o(x) as x ---+ oo where g(t) = J! L~n- 1 lp 1 lfids, and if f~pdt = oo, then y is an oscillatory solution. (Received July 3, 1964,)

64T-431. JOHN DAUNS and D, V, WIDDER, Harvard University, Cambridge, Massachusetts. Convolution transforms whose inversion functions have complex roots,

00 -2 Let {a(k)} be complex numbers with :E 1 la(k) I < oo; let G(t) = (1/27r) J_~(exp(iyt)) flc;x> [1 + y 2a(k) - 2f 1dy, Let 1/l(t), - oo < t < oo, be locally absolutely integrable

673 and satisfy a certain order condition at oo. Inversion Theorem: Assume for some n z -z f > 0, larg a(k) I ;:;; 7r/4 - E for all k. Then limn ~oolll (1 - (d/dx) a(k) ](G • t/>)(x) = t/>(x) almost e.verywhere. This gen~ralizes a special case of a pre.vious inversion formula of I. I. Hirschman and D. V. Widder (Pacific j. Math. 1 (1951), Zll-ZZS) proved under the more restrictive hypothesis that limk_.00arga(k) = 0 very rapidly, i.e. l:'flarga(k)I-Z < oo. The proof involves estimating n[ z -z Gzn(t) = n1 1 - (d/dx) a(k) 1G(t) with respect ton and t by deforming the path of integration. 4/3"oo -z Under the additional restriction that limn ~oo la(n + 1) I "-'n+ 1 1a(k) I = oo, the asymptotic formula oo "'oo -z -1/Z f _ 00 IGzn (t) ldt - (cos arg .L.n+ 1a(k) ] ((n-oo) is proved and used to give an alternative proof of inversion. The above is to be published in the Pacific j. Math. (Received july 7, 1964.)

64T-43Z. ERIK ELLENTUCK, The Institute for Advanced Study, Princeton, New jersey. Combinatorial functions of Dedekind infinite cardinals.

For terminology see Abstract 64T-41Z. Every unary function f mapping the non-negative inte­ gers into itself can be represented as f(x) =Diof language ! let Cond(t/>) consist of a specification containing assertions of the form T(f) = w or T(f) = (v(n), v(m)) (where v(x) is the numeral for x) for every f occurring in t/>, no collisions occurring. Theorem 3. For every formula 1/>(x) of language !, and specification Cond(t/>), there is a formula "?i.,a.,{J) of language 18, such that if s0 is consistent then (\fx)(x E r- A -->t/>(x)) is a theorem of sO + Cond(t/>) iff (\fa.,{J)(1 ;:;; a. ;:;; fJ ;:;; W-->~(a,{J)) is true. (Received june Z4, 1964.)

64T-433. C. E. LINDERHOLM, University of Illinois, Urbana, Illinois. Extension of -qnitary operators.

Kakutani (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961, Vol. II) showed that if U is a unitary operator on a separable Hilbert space then there exists a measure space (X, ./tp.) isomorphic to the unit interval [0,1) with Lebesgue measure, a measure-preserving transformation Ton X, and a subspace til of yz(X) invariant under the uni­ tary operator UT induced by T, such that the restriction of UT to til is unitarily equivalent to U. Kakutani's proof involves a nontrivial argument based on properties of Gauss functions. The theorem is proved here in an elementary way using the spectral theorem for unitary operators. (Received July 7, 1964.)

64T-434. M. R. HAGAN, Oklahoma State University, Stillwater, Oklahoma. Equivalence of connectivity maps and peripherally continuous transformations.

j. Stallings has shown (Fund. Math. 57 (1959), Z49-Z63) that a local connectivity mapping of a locally peripherally connected polyhedron into a regular Hausdorff space is peripherally continuous. Theorem. If f is a peripherally continuous mapping of a locally peripherally connected space S having

674 Brouwer Property II into a space T such that S X T is completely normal, then f is a connectivity map. Thus, on an n-cell, n ;;;; 2, into itself, connectivity maps and peripherally continuous transformations are equivalent. (Received June 29, 1964.)

64T-435. LEONARD GARLITZ, Duke University~ Durham, North Carolina. A note on multiple exponential sums.

Mordell (Calcutta Mathematical Society Golden Jubilee Commemoration Volume (1958-1959), Part I, pp. 29-32) defined the double sum S(c) = I;e(x + y + cx'y'), where e(x) = exp(21ri/p) and xx' = 1 (mod p). He conjectured that S(c) = O(p). It is easy to prove the estimate S(c) = O(p312). This is improved to S(c) = O(p5/ 4). (Received June 29, 1964.)

64T-436. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee 37916. The relative mean value of the Euler function.

Let c/J(n) denote the Euler c/J-function and place a,r(x) = LcP(n)/n, where the summation is over

all n ;:;; x, n =a (mod r). Define {3 by {3(a,r) = limx.--.oo a,r(x)/(x), (x) = <1>0 , 1 (x). The function {3(a,r) is shown to be an~· primitive function of a (mod r) and three canonical evaluations of {3(a,r) are deduced. The extreme values of {3 (as a ranges over a set of residues (mod r)) are discussed. (Received June 24, 1964.)

64T-437. J. K. ODDSON, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. A theorem of the Phragmen- Lindelof type.

Let D be an open set within the cylindrical section 0 ;:;; llx II < ro, YO < y < y 1• where X= (Xp···•xn) and llxll = Oanddiff.e·rentiabla. Choose a.,f3,g(y), h 1(y), hz(y) 3aijX.iX.j;;;; a.IIX.II 2 • 0 < {3 <1rj2r0 , 2 -1 g ;:;; {3 (no.+ bixi)- c, 2ah1 ;;;; b- ({311xll tan f311xll)aixi ;;; 2ah2 and assume (n- 1)a. + bixi;;;; 0. 2 2 Suppose 3no and differentiable k 1(y), k 2(y) i1': 0 3 g- 2h1k 1 - k 1 - ak! i1': 0, g + 2h2k2 - k 2 + akz 2 ;;;; a. 0 >0, and I 1(y) = J~ 0 k 1 (t)/a(t)dt, I2(y) = J§0k2(t)/a(t)dt exist for Yo;:;; y < y 1• Let u(x,y) E c< )(D), continuous on compact subsets of D U 8, and satisfy Lu = f in D. Assume 31£(Y) ;;;; 12 (y) 3 luI ;:;; Ue- IL for (x,y) E 8 and if! ;;;Fe-lL for (x,y) ED. If u = o{exp(I1(y))}, uniformly in x as y ---.yl' (x,y) E 0, then u = 0 {exp(- 12 (y))}. uniformly in x as y ----> y 1, (x,y) ED U B. Special cases include the classical Phragmen-Lindeliif theorems for a sector and a strip. (Received July 15, 1964.)

64T-438. ALEXANDER ASIAN, Ohio State University, Columbus 10, Ohio. On uniform structures.

Let A be a topological space and I an index set. For every i E I, let ~ be a collection of open sets of A and Ai an element of J4{. Moreover, for every i E I and x E A, let Ai (x) be an element of ~such that x E ~ (x). Furthermore, for every i E I and x E A, let Si(x) = U Ai(x) E..Wi Ai (x). Definition. The family (J¥'i)iEI is called a uniform structure of the topo­ logical space A if (i) for every open set V of A with x E V, there exists an i E I such that

675 U yESi (x)Si (y) C V. (II) For every two elements i and e of I, (a) every element Ai of -l¥f is a subset of some element Ae of ~. or, (b) every element Ae of ~ is a subset of some element

Ai of~. (III) UJ&Ij_ = A1 for every i E I. Based on this definition it is possible to state and prove some classical theorems concerning uniformity. (Received July 15, 1964,)

64T-439, JOHN DeCICCO and J, A. SYNOWIEC, Illinois Institute of Technology, Chicago 16, Illinois, Pseudo-orthogonal nets of multi-isothermal families.

In a pseudo-conformal space ~Zn• of real dimension 2n ;?; 4, the tangent directions of all curves C, pseudo-orthogonal to a family of co' manifolds Szn-1' namely F(x;y) = U, obey the single Pfaffian equation (- BF ;al)dxk + (BF ;axk)dy~ = 0, A characterization of a multi-isothermal family is that this family F(x;y) = U, is multi-isothermal if and only if the given Pfaffian equation is integrable, The associated pseudo-orthogonal curves C generate a conjugate multi-isothermal family of co' manifolds, namely, G(x;y) = V. Thus F(x;y) = U and G(x;y) = V, are conjugate pseudo­ orthogonal multi-isothermal families, There is a direct holomorphic function w = (x;y) + i 1/;(x;y), with Bw;Bzk, k = 1,.,.,n, not all identically zero, such that U = U() = F(x;y) and V = V(l/;) = G(x;y). Thus these two families are obtained by setting the real and imaginary parts of such a direct holo­ morphic function equal to constants, i.e,,

64T-440. JOHN DeCICCO, and A. Z. CZARNECKI, Illinois Institute of Technology, Technology Center, 330 South Federal Street, Chicago, Illinois 60616, Fundamentals of Finsler geometry.

Let En be a Finsler space of n dimensions, in which the arc length of any admissible curve

C: x1 = xi(t), t 1 ::;; t ::;; t 2, with at most a finite number of corners is given by the line integral I(C lt) = It~ F(x;x)dt, where F = F(x;i) of 0 is positively linear homogeneous in the x' S, It is not as­ .sumed that F is positive definite. If the metric tensor gij = g dx;x) is not singular, there is obtained the characteristic function G(x;p) of En, which with its derived relations induces a duality on En• The contravariant and covariant component.$ of an absolute vector are defined on the basis of this duality, The inner product (,\,JL) of two absolute vectors ,\and JL in En is in general, neither symmetric, nor positive definite, It is symmetric if and only if En is a Riemannian space, The transversality law in En is equivalent to the vanishing of the inner product, For a curve C the curvature vector K is always transversal to the tangent vector dx/ds, These two vectors span the two-dimensional osculating extremal manifold of the curve C. Finally, applications are given to the Kasner Plane where F = x 2/y. (Received July 15, 1964,)

64T-441, T. G. McLAUGHLIN, University of Illinois, Urbana, Illinois, Regressive sets, supercohesion, and permutations,

Greek letters denote sets of natural numbers; "N" denotes the set of.!!lnatural numbers,

Def. a is supercohesive ~a is infinite and cannot be split by any regressive set, ~ 1, Every infinite a has a supercohesive subset. (Our proof of this uses the Axiom of Choice.) Lemma 2,

676 ( 3 a.)( a. is retrace able and a: is immune). (Our proof of Lemma Z, developed out of a suggestion of P. R. Young, is by a moveable-markers argument.) Prop. 1. Z No of the cohesive sets are non- N supercohesive. Prop. Z, There are Z 0 pairs of disjoint supercohesive sets a, /3 such that (3 i)(a. ~ "'i ~if> but 'Y regressive = $ 'Y V 'Y $ no Prop. 3 (Strengthening a theorem of c. F. Kent). Let ~ be any countable lattice of subsets of N containing the Boolean algebra generated by the finite X sets (ordinary U and n the operations in ~). Then there are Z 0 permutations, p, of N such that both p and p - 1 preserve membership in ~ and also preserve retraceability and regressiveness. Prop. 4. There are zNo permutations, p, of N such that (i) p preserves the property of having an infinite regressive subset, and (11) (3a.)(a. is retraceable & p(a.) is indecomposable). (Received july 16, 1964,)

64T-44Z. W. D. L. APPLING, North Texas State University, Denton, Texas. Set functions and zero sets. Preliminary report.

Suppose that U is the set of all real-nonnegative-valued finitely additive functions on the field S of subsets of the set R. All integrals considered are Hellinger-type limits of the appropriate sums. If A is a binary operation on the set of all nonnegative numbers and each of r and s is in U, then, provided the desired integrals exist, we let A(r,s) denote the function g on S defined by g(V) = fvA(r(I), s(I)); and if each of P and Q is a subset of U, then we let A(P,Q) denote the set of all f of the form A(t,w) fort in P and w in Q. For each fin U, we let Z(f) denote the set of all r in U such that JR min {r(I), f(I)} = 0, Theorem. If each off and g is in U and 0 < p < 1, then min {Z(f), Z(g) I = Z(f) n Z(g) = Z(max{f,g}) = Z(f +g)= Z(f)PZ(g)1-P, and max{Z(f), Z(g)} = Z(f) + Z(g) = Z(min{f,g}): z(lg1-p). (Received July 16, 1964.)

64T-443, P, G. HINMAN, University of California, Berkeley, California. Some theorems on relative recursiveness of higher type objects,

Following the terminology of Kleene's fundamental papers on recursive objects of higher type (Trans, Amer. Math. Soc, 91 (1959), 1-5Z; 108 (1963), l06-14Z) we call an object a.n of type n reducible iff, for some m < n and object {3m of type m,dg(a. t; = dg(/3m). Theorem 1. For any subsets A, B of NN such that A is nonempty but countable and B is meager (i.e. of first category) if B is recursive in A then for any /3 E B there is an a. E A which is recursive tn fJ, Corollary. For any a., /3 in NN, if {/3} is recursive in {a.} then a is recursive in {3. Let r+ZE be the representing func- r+ 1 r r+ 1 r t N { t tiona! of ta. : 31J (a. ('I ) = O)r. Since,for any a inN ,(i) a.r is irreducible and (ii) taf ls recursive in ZE iff a. is, it follows that (a) there are te0 irreducible Zdegrees less than dg(!E). Lemma 1 of Tugue (Comment. Math. Univ. St. Paul 8 (1960), 99) implies: (b) for any nonrecurslve a,

1 sc( { a.p is just the set of recursive functions. This and the corollary yield: (c) there exist re­ cursively incomparable irreducible type-Z objects with the same 1section. Results (a), (b), and (c) settle, respectively, the three open questions of Kleene, op. cit., §ll.Z1. The open problem of §ll.Z6 is settled by: Theorem Z, For any r > 0 not every :E~+ l n fi~+ l predicate of objects of types < r + Z is recursive in r+ZE. (Received july 17, 1964.)

677 64T-444, J. J. PRICE and R. E. ZINK, Purdue University, Division of Mathematical Sciences, Lafayette, Indiana. On sets of completeness for families of Haar functions.

Talalyan has shown that if any finite collection of functions be discarded from a complete set in L 2 (0,1), then to every positive number E there corresponds a measurable set of measure 1 - E on which the remaining set of functions is complete, Moreover, in giving a simple proof of Talalyan's result, Goffman and Waterman have observed that the same situation obtains if certain infinite collections be discarded, However, one may discard so many functions that the remaining family is complete on no set of positive measure. This is the case, for example, if one deletes from the Walsh system all functions save the Rademacher functions, Thus, it is of interest to investigate the relation between the set of functions discarded and the sets of completeness for the remaining family. Such an investigation is begun in this article with a study of the Haar system. Here one is able to deter­ mine precisely those sets on which a given subsystem is complete. There follow some interesting corollaries. On the one hand, no lacunary subset of the Haar system can be complete on any set of positive measure. On the other hand, the Haar system can be decomposed into an infinite number of subsystems, each of density zero with respect to the entire family, and each complete on a fixed set of measure 1 - t, (Received July 20, 1964.)

64T-445, L. L. LININGER, University of Missouri, Columbia, Missouri•. The sum of two crumpled cubes is s3 if it is a 3-manifold.

A compact continuum C is a crumpled cube if and only if C is homeomorphic to the union of a 2-sphere and its interior in E 3• Theorem 1. The sum of two crumpled cubes is connected and simply connected. Theorem 2, If the sum of two crumpled cubes is a 3-manifold then it is homeo­ morphic to s3• (Received July 20, 1964.)

64T-446, I. I. HIRSCHMAN, JR., Washington University, Box 146, St, Louis, Missouri 63130, Toeplitz sections on groups.

Let G be a locally compact (but not compact) Abelian group and r its dual. We assume that there exists a distinguished measurable order relation < on r compatible with its group structure. Let dx be Haar measure on rand let (x,t) be the value of the character x at t. We define A. to be the

Banach algebra of functions f0 (t), t E G, which have representations of the form f0 (t) = fr 0. Define E-(x) similarly. For c E.!J..; define W~f = E+(O)cf for all f E E+ (O)A *, and W-f = E- (O)cf for all f E E- (O)A *. We say that c E WH'A *) if w+ and We- both - c - ~ c have bounded inverses. If c E WH(A *) then there exists a number z ?:. 0 and a constant A both depend- ing only on c suGh that if x ?:. z and if W~(x)f = E+(O)E (x)cf, f E E+(O)E-(x)A* then llfll ~ AIIW~(x)fll and the range of W~(x) is E+(O)E-(x),a_*, etc, Starting from this result we can construct a theory of generalized Szegii functions which extends to the locally compact case the results of the author's paper Szegii polynomials on a compact group with ordered dual (to appear). Also as in this paper we may consider more general algebras. (Received July 20, 1964.)

678 64T-447. R. c. JAMES, Harvey Mudd College, Claremont, California. Weakly compact sets.

Some new (and some generalizations of known) characterizations of w-compact sets are given. E~g., with {xn} denoting a sequence in E, and for fn a continuous linear functional, nee. and suff. conditions for a bounded w-closed subset E of a complete LCS to be w-compact are: (1) For each

{xnl• 0 E cl[Uf(lin{x1, ••• ,xn }- conv{xn+l'""" p]. (2) For each {xn} and equicontinuous {fn }, inf{fn(xk): n< k} ~ sup{fn(xk): n >k}. (3) For no 8 >0, {xn}. and equicontinuous {fn~'is fn (xk) > 8 if n ~ k and fn (xk) = 0 if n > k. (4) For each { xn~ and f, the existence of lim f(xn) implies an x in E with lim f(xn) = f(x). (5) If K is closed and convex and E n K = {1, then 0 ¢. cl [E - K]. (6) Each w-continuous functional onE is bounded. If E also is convex, then fn(xk) > 8 can be replaced by fn (xk) = 8 in (3) and also: (7) For each {xn ~. 0 E cl [ U':'(lin{x1, ••• ,xn ~ - flat {zn+l'""" }>]. (8) Each affine continuous map of a nonempty closed convex subset of E into itself has a fixed point.

(9) If { ~ inf{f(y): y EY~. (Received July 24, 1964.)

64T-448. MAR TIN HELLING, University of California, Berkeley 4, California. Hanf numbers for some generalizations of first-order language.

Let K be an infinite cardinal. M(K) is the least cardinal X such that, for any 1st-order theory

T having at most K symbols, and any set l: of formulas with one free variable in the language ofT, if T has a model of power X in which no element satisfies all formulas of 2:, then T has arbitrarily .::r large such models. Put ::La.= N0 + ~) v(v

then M(K) = ::t ~· Let QK be the language obtained by adding the quantifier "there exist at least K11 to the first-order language. The Hanf number, H(K), of QK is the least X such that for any set T of at most K sentences of QK, if T has a model of pot.rer X, then 'T has arbitrarily large models. Using results of Keisler and Fuhrken (see Fuhrken, Languages with added quantifier: "there exist at least

Na.•" op. cit.), the following is obtained. Corollary. If K is a ::ta., cf(K) = N0 and K < the 1st meas­ urable cardinal, then H(K) = ::t +" (Received July 28, 1964.) K

64T-449. C. C. CHANG, University of California, Los Angeles, California 90024. Two refine­ ments of Morley's method on omitting types of elements.

For notation see preceding abstract [64T-448]. N(K) is the least cardinal X such that for any

first-order theory T having at most K symbols, and any collectionS= { 2:~: ~ < «~ of sets l:~of for­ mulas with one free variable in the language of T, if T has a model A of power X in which, for every 1: inS, no element of A satisfies a:ll formulas of ·2:, then T has arbitrarily large such models. O(«) is defined exactly as above except that in S the cardinal « is replaced by 2«. This of course amounts to placing no restriction on the number of such 2: 's. Refinements of Morley's method yield: Theorem 1.

If «is a ::::> and cf(K) = N0, then N(«) = ::J +" Theorem 2. In general, ~ + ~O(K) ~ :::J K)+" Theorem ll K K (2 is a slight improvement of Helling's main result in the preceding abstract , and in fact, as Helling pointed out, it can be derived as a consequence of his main result without further use of Morley's method. (Received July 28, 1964.) 679 64T-450 •• A. MULLIN, Lawrence Radaboratory, University of California, Box 808, Livermore, California 94551. Some new analogues to classical number-theoretic functions. Preliminary report.

For terminology see, e.g., Proc. Nat; Acad, Sci, U.S.A. 50 (1963), 604-606, and a forthcoming (1964) paper in Notre Dame J, of Formal Logic, This note shows that within the class of generalized multiplicative functions there are precise and concrete analogues to Mobius' function p. and Liouville's function A. Definitions. Let p.* (modified Mobius' function) be defined as follows: p.*(l) = 1; p.*(n) = 0, if the (unique) mosaic of natural number n > 1 has any prime repeated; and p.*(n) = (- l)m, if the mosaic of n > 1 has no prime repeated, where m is the number of (distinct) primes in the mosaic of n, Let A* (modified Liouville's function) be defined as follows: A*(l) = 1; and A*(n) = (- l)m if the mosaic of n > 1 has m primes counting repetitions according to their multiplicities, if any. By analogy to Euler's totient l/>, let l/>* (modified Euler's function) be defined as follows: l/>*(n) is the number of natural numbers ;;;; n whose mosaics have no prime in common with the mosaic of n. E.g., Theorem, Whereas J1. is multiplicative but !!.Q!_ completely so, p.* is generalized multiplicative but E.2!_ multiplicative; whereas A is completely multiplicative, A* is multiplicative but.!!£! completely so; and l/>* is dominated by if>. (Received July 31, 1964.)

64T-451, J, H. AHLBERG, E, N. NILSON and J. L, WALSH, Division of United Aircraft Corporation, Pratt and Whitney Aircraft, East Hartford, Connecticut. Fundamental properties of generalized splines. Given the linear differential operator L = L:~=Opk(x)dk/dxk, pk(x) E Cn(O,l] and a subdivision

6: 0 = x0 < x 1 < ... < xN = 1 of the unit interval, a generalized spline SN(x) satisfies L * L(SN) = 0 (L* = adjL) on each subinterval of (0,1] and E c 2n- 2 [0,l] overall. The fundamental integral identity (c.f. Walsh, Ahlberg, Nilson, J, Math. Mech, 11 (1962), 229), j~(Lf/dx = j~(LSN/dx + J~[L(f- SN)]2dx for f(x) E c 2n- 2{0,1) and SN(x) the generalized spline of interpolation to f(x), is valid for the periodic case, and for the nonperiodic case provided s<~-k)(O) = f(n-k)(O) or L:~ 1 (- l)i{pk-jL(SN)}~;J> = O(k = 1.2,... ,n- 1) with a similar condition at x= 1, Existence and uniqueness theorems, and best approximation, minimum norm and orthogonality properties are immediate consequences for generalized splines as well as are the extensions to higher dimensions. (Received August 3, 1964.)

64T-452, LEONARD SARASON, Stanford University, Stanford, California. Hyperbolic mixed problems, Preliminary report,

Lu(t,y,x) = ut - Aux - Buy= f is a strictly hyperbolic system with A, B real constant matrices and lA- 1 1 < oo. In the terminology of Hyperbolic mixed problems, I (here, June 1964), consider the mixed problem Lu = f(t,y ,x) in x > 0, t > 0; Pu(t,y,O) = g(t,y), u(O,y,x) = h(y,x), where P real obeys

(*), and h = 0 unless (#): If ~is a root of M(r,ll) with Im11 = Re ~ = 0, then the multiplicity of ~is ~ 2, Define fx>O e -p.t iw 12 dtdydx = llw 11;. fx=O e-p.t lw 12 dtdy = (w)~, and .ft=o iw 12 dydx = [w }2• Set 2 2 2 -1 2 2 2 r. 2 . [f,g,h)Jl. =(g) + [h]p. + p. llfllw Theorem, If J1. >0, then (u)f.L + 11Bu lip.~ c~,g,hJ,.., where cis mdepen- dent of p.. The proof uses Fourier-Laplace transformation in y and t, and Puiseux expansions of the eigenvalues and eigenvectors of M near their multiplicities. Correction: In I, A, B and P should be real. and in Theorems 1,2, assume(#). (Received August 3, 1964.)

680 64T-453. D. M. BLOOM, 195 Claremont Avenue, New York, New York 10027. On the subgrou-es

~SL(3,q). IV. Preliminary report.

Using generating relations, it can be shown that if a is even then PSL(3,5a) has a subgroup isomorphic to the alternating group on seven letters. This answers one of the unresolved questions in the preceding abstract (64T-317, these Notices 11 (1964), 461). (Received August 3, 1964.)

64T-454. MICHEL MENDES FRANCE, University of California, Los Angeles, California. A set of nonnormal numbers.

Let E be the set of all x E (0,1) such that for some real polynomial

64T-455. LEO SARlO, 521 Georgina Avenue, Santa Monica, California, MENAHEM SCHIFFER and MOSES GLASNER, University of California, Los Angeles, California. The span and -erincipal functions in Riemannian spaces.

Let V 1 be the complement of a regular subregion with border a 1 of a Riemannian space V. · Let a E C on V 1, a E H (harmonic) on V 1 and fa1• da = 0; let L be a normal operator for V 1• Then there exists a principal function p E H on V, piV 1 = a+ L(p- a). For a regular region n with border a U fJ and for f E C, there exist functions uo, u 1 on 0, ui = Lif such that the tangential component of •duo vanishes on fJ and uri.B = canst., J{j•du1 = 0. Set ux = (1 - X)u0 + Xu 1• Among functions u with 2 2 ula = f, f.a. du = 0, ux has the property B(u) + (2X- 1)A(u) = X A(u1)- (1- X )A(u0) + D(u- ux).

Here B(u) = JfJu A •du, A(u) = fau A • du and D(u) = J0 du 1\ •du. For noncompact bordered regions V 1 c V the limiting functions as 0 --> V 1 exist, ux has the same extremal property and ux = Lx f defines a normal operator Lx. For an arbitrary Riemannian space let a,b E V and P p.+X = {PIPE H in V- a- b, piCa= (p.+ X)sa + e, piCb =- (p.+ X)sb + f; e,f EH inCa, Cb, f(b) = 0} where sz is the fundamental solution relative to z in a region Cz bordered by a level surface of sz• Take

pi E P 1' pi = LiP i' and denote by hi the function e corresponding to pi' i = 1,2. Then Pp.X = P.Po + Xp 1 . 2 2 has the following property m PP.+X: B(p) +(X- p.)e(a) = X h 1(a)- p. h0(a) + D(p- Pp.X), As an immediate corollary the span h 1{a)- h0 (a) of Vis equal to D(p0 - p 1). {Received August 6, 1964.)

64T-456, CARL de BOOR and R, E, LYNCH, Research Laboratories, General Motors Technical Center, 12 Mile and Mound Roads, Warren, Michigan 48090, General spline functions and their minimum propertief!,

Fork i1:: 1 and b >a, let F {k) = {f(x) If E ck- 1 (a,b], t

f (k) E L 2 [a,b]}, let M be a kth order linear differential operator in normal form with coefficients in F(k), let M* be its adjoint, let a= x 0 ;> x 1 < ... < xn ;> xn+1 = b, n i1:: k, and let Li denote linear func­ tionals on F(k) defined by Li(f) = f(x1), f E F(k), i E [1,n], Then F(k) is a Hilbert space with respect to the inner product (f,g) = J~M[f]M[g]dx + L:t. 1L1(f)Li{g), and {Li}~ is a set of bounded linear function­ ala on F(k). The subspace T, spanned by the representing elements {

681 . 2k-2r. r, defimtion, of spline functions, i.e., functions inC La,b J which in each interval ~i'xi+ 1 J, i = O,,.,,n, are equal to an element in the annihilator of M*M. This definition coincides with that of odd-degree polynomial splines (c,f. Schoenberg, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148) in case M = dk/d~. Moreover, the various minimum properties of polynomial splines as stated in ~choen­ berg,op. cit.] are enjoyed by the elements in T with respect to the more general inner product, since these properties can be shown to follow in fact from simple properties of representing elements of bounded linear functionals on a Hilbert space, More generally, it is not necessary to insist that the Li be point functionals. Finally, these considerations lead to a useful definition of spline functions in more than one variable other than with the aid of tensor products. (Received August 6, 1964.)

64T-457, S, K. STEIN, University of California, Davis, California. Filling n-space with crosses.

A (k,n)-cross is a set of 2kn + 1 unit cubes inn-space, consisting of a central cube and at each of the 2n faces of this cube an arm of k cubes, A (k,n)-semicross consists of kn + 1 unit cubes re­ maining from a (k,n)-cross after one arm is deleted in each of then directions. Tesellation of n-space by translates of these figures is related to the existence of subsets A and B of Sm, the semigroup of integers mod m, such that every nonzero element of Sm is uniquely expressible in the form ab, a E A, b E B. Such factorizations impinge on the work of Hajos, DeBruijn, and Sands in groups and of Kummer and Mills in characters with prescribed values. Such algebraic considerations give information on those k and n for which tesellations of n-space by (k,n)-crosses or semicrosses exist.

Also, elementary purely geometrical proofs provide these restrictions: (1) If n ~ 2 and k > 2n - 2, then n-space can not be tesellated by (k,n)-crosses; (2) If n-space can be tesellated by (k,n)-crosses, then 2n-space can be tesellated by (k,2n)-semicrosses, (Received August 7, 1964.)

64T- 458, M. S. LYNN and B. R. HEAP, National Physical Laboratory, Teddington, Middlesex, England. Further remarks on a linear Diophantine problem of Frobenius.

We refer the reader to our earlier Abstract 64T-259 (these Notices, Vol. 11 (1964), 389) in which we stated that a certain Frobenius. problem (of finding the largest integer g(p1, ... ,pk) not representable as a linear combination with non-negative coefficients of a given set of k relatively prime numbers 0 < p 1 < p 2 < ... < Pk) may be replaced by one of computing the index of pri-mitivity, -y1, of a well-defined matrix, A 1, of order pk + pk_ 1 - 1. We have since shown, again using graph theoretic techniques, that if A2 = (aij] is the matrix of order pk defined by: ai,i+1 = 1 (i = 1, ... ,pk- 1), aij = 1 if i- j = Ps- 1 for some 1 ;;; s ~ k, aij = 0 otherwise, then g(p 1, ... ,pk) = ')'2 - Ik , where -y2 is the index of primitivity of A2• Since the order is only pk and since 'Y2 = 1'1 - (pk - 1) < 'Yl' the amount of computation is thus considerably reduced. (Received August 7, 1964.)

64T-459, M. K. RICHTER, Department of Economics, University of Minnesota, Minneapolis, Minnesota 55455, Representation of consumer choice.

Let B be the nonnegative orthant of Euclidean n-space En' A= {(p,m): p E En & 0 ~ m E E 1 p 1 > 0 & ... & Pn > 0 ~, and b(p,m) = {x: x E B & p•x :> m ~ for all (p,m) EA. A consumer is any func-

682 tion h from A to the class of nonempty subsets of B satisfying: x E h(p,m) => p·x ~ m. his rational if there is on B a total, strongly reflexive, transitive relation G ("at least as good as") such that

h(p,m) = {x: x E b(p,m) & \fyyEb(p,mfGyf for all (p,m) EA. For consumer h define binary relations M and Non B by: xMy ¢=:>3(p,m)(p,m)EAx E h(p,m) & p•y ~ m; N is the smallest transitive relation on B including M. his congruous if h(p,m) = {x: x E b(p,m) &.vyyEb(p,m)xNy} for all (p,m} E A

(i.e., x E h(p,m) & p•y ~ m & yNx => y E h(p,m)). h is representable if there is a function g on B to E 1 such that h(p,m) = {x: x E b(p,m) &VyyEb(p,m)g(x) ~ g(y)} for all (p,m) EA. For consumer h let D = Ujh(p,m): (p,m) E A}. Theorem 1. A consumer is congruous iff it is rationat Theorem 2. For any consumer h, if jx: x ED & 3y b & ,, & Vt (O )tx + {1 - t)y ¢ D} is countable and yE XlVlY 'oyNx tE ,1 {x• 3(p,m)(p,m)EAx E Closure (h(p,m)) ~h(p,m)} is countable and his congruous, then his representable. In Theorem 2 the last condition is necessary for representability; counterexamples are known omitting either of the first two conditions. (Received August 7, 1964.)

64T-460. W. W. COMFORT and STELIOS NEGREPONTIS, University of Rochester, Rochester, New York. The ring C(X) determines the category of X.

A topological property of the completely regular Hausdorff space X is paired with an algebraic property of the ring C(X) of real-valued continuous functions on X as follows: X is of category I iff 00 the collection _,It of real maximal ideals in C(X) may be written in the form__/(= Un=l _/(n' where for each none has (a) n( ../(\_/tn) = {0} and (b) forM Ef ../(n there exists f EC(X) such that f E n~ and f Ef M. Corollary: X is of category I iff the Hewitt completion (realcompactification) .JX is of category I. More generally, let (P) be either of the properties "is of category II," "is a Then (.JX ) has (P) iff X has (P). Corollary (partially duplicating Oxtoby, Baire. space." IT a. a. IT a.a Cartesian products of Baire spaces, Fund. Math. 49 (1961), 157-166): If each Xa. is pseudocompact, (Received August 7, 1964.) then IT a. X a. is Baire.

64T-461. STELIOS NEGREPONTIS, University of Rochester, Rochester, New York 14627. Baire sets in realcompact spaces. Preliminary report.

Let X be a Hausdorff completely regular topological space. For notation and terminology consult Gillman and jerison, Rings of continuous functions. Definition: A subset of X is Baire in X iff it belongs to the o--field generated by the zero-sets of X. Theorem: Each Baire set in a realcom­ pact space is realcompact. Corollary (E. R. Lorch): v X= n {A IX C A C (3X and A is Batre in (3X}. Corollary: If X is Baire in fJX then X is realcompact. Corollary: X is realcompact iff given p E (3X - X there is B a Baire set in (3X such that p E B and B C (3X - X. Corollary: X is pseudo­ compact iff every nonempty B a ire set of (3X meets X. (Received August 7, 1964.)

64T-462. ]. R. RICE, General Motors Research Laboratories, 12 Mile and Mound Roads, Warren, Michigan 48090. The degree of convergence for entire functions.

Consider functions f(z) for z in the complex plane. Let C be a closed and bounded point set whose complement is connected and regular [see ]. L. Walsh, Interpolation and approximation, Amer.

683 Math. Soc., 1960]. Theorem 1. Let f(z) be entire of order p > 0. Then there exists a sequence of polynomials Pn (z) of degree n and a constant K such that (*) maxzEC lf(z) - Pn (z) ll/n ~ Kn-l/P, Theorem z. Assume there exists a sequence P n (z) and constant K such that (*) holds for all n. Then f(z) is entire of order p. Let f(z) have an isolated singularity at z 0, set SE " { z I lz - z 0 I" E~ and M(E)" max lf(z)l, z E SE• The~ pof f(z) at z 0 is then defined by P" lim E~O~oglogM(f)lflogE. Corollary. Assume f(z) has a finite number of isolated singularities of maximum order P > 0, none of which lie in C. Then there exists a sequence of rational functions Rn (z) of total degree n and a constant K such that maxzEclf(z)- Rn(z)l1/n ~ Kn- 1/P, (Received August 10, 1964.)

64T-463, R, s. SPIRA, Duke University, Durham, North Carolina. Zero-free regions of r(k) (s).

It is shown that the kth derivative of the Riemann zeta function is 'I 0 for u :l:; 2k + 1 and also for the region u ::> uk and t ;<; tk• Sharper bounds are obtained for particular cases. A root is found for r"(s) off the real axis in the left half plane. The behavior of the roots of r• and r" is described, (Received August 10, 1964.)

64T-464. ELIAHU SHAMIR, University of California, Berkeley 4, California. Multiplier transforms in half spaces. n Let (x,y) E R , where x" (xl'... ,xn_ 1), y " xn• Points of the dual space are denoted by (~.!]). Y + is the characteristic function of the half space R!" {(x,y) IY "'- 0~. M(~,!J) is a m X m matrix of homogeneous functions, which is continuous and invertible for (~,!]),FO. The operator M is defmed. for u E (L 2 (R+))n m by Mu " Y+-"'-a-1 M2·u,= where -"'-=is the Fourier transform. The !-dimensional operator .MJ; is similarly defined in (L 2 (R!))m with the multiplier M(~,!J), ~fixed,

Lemma: The estimate llu II ~ C IIMu II for all u E (L 2 (R !»m holds if and only if llv II ~ C 1 IIM~v 11. v E' (L2(R!))m is true for 1~1" 1 uniformly. Solving M~ v" w is equivalent to Riemann-Hilbert problem +(!J)" M(~,17) cl>-(17) + W(!J) with ± sought in (H;(R))m, the space of Fourier transforms of functions supported in R!. The scalar case (m " 1) was treated by Widom (Trans. Amer. Math. Soc. 97, (1960), 131-160). We- have then, Theorem: Let (2r) -1/.00_ 00 d!]argM(t'l~ = k + (J, k integer, - 1/2 < (J ;;:; 1/2. If (J ¥ 1/2, M has a closed range, and is injective if k > 0, surjective if k < 0. (Received August 17, 1964.)

64T-465, NORA PERNAVS, Wayne State University, Detroit, Michigan. A uniqueness theorem for the n-dimensional Helmholtz equation.

Theorem. Let u(xl' x2, .. .,xn) be an everywhere twice continuously differentiable solution of the Helmholtz equation "'n~/J 2 u(x)/uAk"--2 + u" 0, n ;<; 2, and satisfy the integral condition limR-oofniJ~u(x)r(n- 2 )/ 2 drlctn "0, where n is the surface of then-dimensional unit sphere and x =· (x 1, x 2 .... ,xn) " rr (r being the spherical distance and T the unit vector). Then u = 0 throughout the entire (x 1, x2, ... ,xn)-space. This result generalizes a recent result obtained for the case n" 2 by o. G. Owens [Duke Math. j. 31 (1964), 91-98]. Since any twice continuously differentiable solution of thtl Helmholtz equation is an analytic function of its arguments it suffices to show that u together

684 with all its derivatives vanishes at the origin. This is accomplished by an analysis which utilizes the invariance of the equation under rotations, the integral condition, and the mean value equation n/2 n/2 r (27r) t Jn;2(t)u(O) = lr:;; tu(x)dx, where Jn/2 (t) is a Bessel function and dx = dx1dx2 ••• dxn. (Received August 17, 1964.)

64T-466. J, C. CANTRELL and C. H. EDWARDS, JR., University of Georgia, Athens, Georgia. Almost locally flat imbeddings of manifolds.

The following results are concerned with the imbedding of k-manifolds in n-manifolds in the

so-called "trivial" range of dimensions, n !1; 2k + 2. Lemma. If h is an imbedding of a combinatorial k-ball B in Euclidean n-space En (n ?; 2k + 2) which is locally piecewise linear except at a single point p E Int B, then h is locally flat at p. Theorem. Let M be a topological (not necessarily triangulated) k-manifold imbedded in a topological n-manifold N (n -.:: 2k + 2). Then the set E of points at which M fails to be locally flat can not contain an isolated point, so E must contain un­ countably many points if it is nonempty. Corollary, Let M be a closed combinatorial k-manifold

topologically imbedded in the combinatorial n-manifold N (n !1; 2k + 2). If M is locally flat except possibly at countably many points, then M is t-tame for every t > 0. The lemma is proved by con­ structing a k-sphere S which is locally flat except possibly at h(p), and such that S n h(B) contains a neighborhood (in h(B)) of h(p). A result of Stallings then applies. The theorem is obtained from the lemma by use of Gluck's modification of Homma's theorem. (Received August 17, 1964.)

64T-467. D. W. CROWE, University of Wisconsin, Madison, Wisconsin. A model for finite 2 hyperbolic planes in GF(q ), q odd.

In a finite field of order q2 (q an odd prime power) there are natural definitions of (square of) distance and orthogonality, The circles orthogonal to a given circle, C, either intersect C in two points or fail to intersect C. These circles yield a new model of a finite hyperbolic plane in the

sense of Ostrom (Amer. Math. Monthly (1962), 899-891). The proof is s~milar to the case q even, recently treated by the present author (Mathematika (1964)). However, in contrast to that case, not all lines have the same number of points. (Received August 17, 1964.)

64T-468. L. W. CONLON and A. P. WHITMAN, Loyola University, 6363 St. Charles Avenue, New Orleans 18, Louisiana. Lower bounds to holonomy.

Let M be a C00 manifold, K a Lie group, and P-> M a principal K-bundle. Let O(M) be the C00 loop space suitably topologized. Any connection form w on P defines a continuous map

hw: O(M)-> K. Proposition. Let A be an abelian group. !f.hw•: Hr~l(M); A) ->Hr(K;A) is not zero, then P admits no connection with holonomy group H of dimension < r. For the principal K-bundle G -> G/K corresponding to the compact symmetric pair (G,K) the map hw is explicitly computed in low dimensions using Nomizu's canonical connection. In particular the following lower bounds r(G,K) to the dimension of holonomy are found for G -> G/K: r(S0(2n), U(n)) = [n/2)(2 [n/2] - 1); r(U(2n), 0(2n) ) = 2n - 1; r(Sp(n), U(n)) = n(n + 1)/2 for n odd and n(n - 1)/2 for n even; 1 r(E7, E 6 X T ) =· 27; r(E 7, A7) = 27. (Received August 13, 1964.)

685 64T-469, LUDVIK JANOS, Monroe Hall 422, The George Washington University, Washington, D. C. Converse of the Banach theorem in the case of one to one contracting mapping,

Let S be a compact metrisable topological space and let c/>(x) be a continuous mapping of S into itself and such that the intersection of all iterated images of S: n c/>n(S) is one point set. Then for every a. E.(O,l) there exists a distance function p(x,y) (generating the given topology) such that P{'c/>(x),cp(y)) ;;; a.p(x,y) for x, yES, Let now c/>(x) be one to one and let us define a function a(x,y) as follows: a(x,y) = a.np(¢-n(x), c/>-n(y)) where n is the maximal integer for which both proimages 1/1-n(x), ¢-n(y) exist, In terms of this function we define a new distance function p*(x,y):

p*(x,y) = inf[L;f= 1a(xi,xi+l)] where the infimum is taken over all finite sequences x 1, x 2, ... ,xn+l for which x 1 = x and xn+l = y. Theorem: (1) p*(x,y) is a distance function topologically equivalent to P(x,y). {2) The mapping c/J(x) is with respect to p*(x,y) "pure" contraction: p*(cp(x), c/>(y)) = a.p(x,y) for x,y E; s, (Received August 20, 1964,)

64T-470, J, C. SU, The Institute for Advanced Study, Princeton, New Jersey. A note on the bordism algebra of involutions,

We consider the bordism group 9l0 ('Zz.) of involutions (P. E. Conner and E. E. Floyd, Differen­ tiable periodic maps, Springer Verlag, 1964), Tensor product of involutions and diagonal map of a classifying space of z 2 induce respectively a multiplication and a comultiplication in 9l 0 (Z2), making it a Hopf algebra over the Thorn bordism algebra 9l 0 , Theorem, There is a canonical 9l 0 -basis xn, n = 0, 1, 2, ... , in 9lo(Z2) so that 9l0 (Z2) is the exterior algebra over 9l0 generated by x 2n, n = 0,1,2, .. ., and the comultiplication is given by the formula A(xn) = Lf=Oxi 0 xn-i• (Received August 20, 1964,)

64T-471. E. K, BLUM, Wesleyan University, Middletown, Connecticut, Finitely generated free subsemigroups of a free semigroup.

Let l: be an alphabet and F(2:) the free semigroup of words over 2:, including the empty word e. Let E = {wl'"''wmf'• m ~ 1, be a set of distinct nonempty words of F(2:) and let E 00 be the subsemi­ group generated by E. An E-chain is a sequence, {t0, t 1, ... ,tn+l }. tiE F(2:) and tn+l = e, such that for 1 ;;; i ;;; n there exists wi E E and vi E'E00 satisfying the equation wi =\Viti+ I and (fori= 0)

either there exists w0 E E and v0 E E 00 satisfying w0 = t0v0t 1 or there exists v0 E E 00 satisfying t0 = v0 t 1• A distinct pair { wi, w j} C E is an E-couple if wi = w jtO and there exists an E-chain starting with t0• Theorem 1. Let E be a finite subset of nonempty words of F(2:), E 00 is a free sub­ semigroup of F(2:) withE as its unique irreducible generating set if and only if E contains no E-couple, Theorem 2, Let E be a finite subset of nonempty words of F(2:), There is an effective procedure for deciding whether or not E 00 is a free subsemigroup of F(2:) withE as its unique irreducible generating set, (Received August 21, .1964.)

64T-472. D, M. TOPPING, University of Chicago, Chicago, Illinois 60637. Maximal states of Jordan operator algebras.

A JC-algebra is a uniformly closed Jordan algebra of self-adjoint operators containing the identity. Otherwise the terminology follows Abstract 608-64, these Notices 11 {1964), 73,

686 Theorem 1. Each norm-closed quadratic ideal of a JC-algebra is the intersection of all maximal quadratic ideals containing it. Let .9 be the set of pure states, X its w*-closure (the "pure state space") and 1 the set of all maximal states (the latter are pure). Theorem 2. In any JC-algebra A, 1 is a boundary for A (for each a E A there is an w E 1 with lw(a) I = !Ia II), X is the w*-closure. of 1 and the state space of A is the w*-closed convex hull of 1. The set .9-1 is w*-nowhere dense and if A is finite-dimensional, then .9 = 1 .. For a state W-With kernel K and null space N, let H(I<) = ja E A: for each f > 0, there is an at E K with - ( f +a f) ~a ~ t+ at}. Proposition. A state w is maximal if and only if H(K) = N. Several other criteria for maximality are given. Theorem 3. If A is a JC-algebra and I is a norm-closed Jordan ideal, then A/I is isometrically isomorphic to a JC-algebra. (Received August 24, 1964.)

64T-473. M. D. MORLEY, 213 Van Vleck Hall, University of Wisconsin, Madison, Wisconsin 53706. A condition equivalent to categoricity in uncountable powers.

T is a complete theory in a countable first order language. B is a prime elementary extension of a model A if: (i) B is a proper elementary extension of A and (ii) if C is any proper elementary extension of A then there is an elementary map of B into C which is the identity on A. Our first theorem is a consequence of Vaught's two-cardinal theorem. Theorem 1. Suppose A is a countable model of T, S(A) t!ie space of elementary types of single elements with respect to A and p the only limit point in some neighborhood of A. If A has a proper elementary extension having no elements of type p then A has an uncountable elementary extension having no elements of type p. Theorem 2.

Tis M1-categorical if and only if every countable model of T has a prime elementary extension. Theorem 3. If T is }((categorical then the prime elementary extension of each countable model is also a minimal elementary extension and is therefore unique. (Received August 26, 1964.)

64T-474. R. W. GILMER, JR., Florida State University, Tallahassee, Florida 32306. Eleven nonequivalent conditions on a commutative rins;.

Let R be a commutative ring. This paper considers relations between the following conditions on R: (A) R has an identity, (B) R is generated by idempotent elements, (C) if A is a nonzero ideal of R such that VA f. R, then R/ A contains an identity, (D) A = RA for each ideal A of R, (E) if A is a proper ideal of R, vA f. R, (F) R = R 2, (G) an ideal A such that VA is maximal, is primary, (H) if p is a nonzero prime ideal of R, R/p has an identity, (J) maximal ideals of R are prime, (K) each proper ideal of R is contained in a maximal ideal, and (L) if A and B are proper comaximal ideals of R then A n B = AB. The following are all the simple implications which exist between these properties (A) __,(B) -->(D) --->(E) -->(F) ---> (J); (A) -->(C) --> (H); (C)--> (G); (D) -->(H); (D) --. (L); (E) --->(G). (Received August 26, 1964.)

64T-475. C. C. SIMS, Massachusetts Institute of Technology, Cambridge, Massachusetts. On the group (2,3, 7;9)

Coxeter (Trans. Amer. Math. Soc. 45 (1939), 73-150] has assigned the symbol (,£,,m,n;q) to the group defined by J = Sm = (RSf = (R -IS -IRS) q= 1. The groups (2,3, 7;q) have been shown to

be finite for q ~ 8 [Leech and Mennicke, Proc. Glasgow Math. Assoc. 5 (1961), 25-29). Two permu-

687 tations of the integers have been found which satisfy the relations for (2,3, 7; 9) and generate a transi­ tive group. Thus (2,3, 7;9) is infinite. The representation is obtained by numbering the co sets of the subgroup generated by RSR - 1s- 1 and (S- 1R - 1SR)4S. This can be done in such a way that both permu­ tations satisfy f(n + 56) = f(n) + 56, (Received August 26, 1964.)

64T-476. M. A. McKIERNAN, University of Waterloo, Waterloo, Ontario, Canada. Fatigue geometry and "tired light".

In a Riemann-Fatigue geometry (see 64T-263 and 64T-351, these Notices) the equations of geodesic and auto-parallel curves may be written in the form (1) /iii /Ot = dgiaa,\ga!3xiJ- BxgaJ3x.ax_i3X:1;2} for£=- 1 and£= 0 respectively. If the space is conformal in,\, that is, if gij(xa., ,\) = f(xa.,,\)gij(xa.,O) then (1) becomes llii/llt = (£/2)gia.a.\ga.{JxiJ. Further, if ,\(xa.) is a solution of the Hamilton-Jacobi equation then the corresponding characteristics (geodesics) satisfy g!~ = pii where p = 8,\ga.flaa..\8!},\. This suggests that in a Fatigue space generalized to nonsymmetric gij equation (1) becomes the classical equations of motion of a charged particle in an electro-magnetic field where f = e/m, ratio of charge to mass. The case £ = 0 becomes classical relativity. If radiation damping is approximated by the use of the linear theory, the equation of motion of a charge becomes llii/ot = £{gia.(Fa.{J + !LTa.{3)xfl- (F a.f3 + ILTa.iJ)ia.xflxi} where quadratic terms are absorbed in the intrinsic derivative, and T a.{J is the stress-energy tensor. Comparison with (l) suggests 8,\ga.{3 "" F a.f3 + !LT a.{J" The results thus far suggest the equation oxi /ot = Bxga.f3x_a.x.flxt/2 for photons, a type of "tired light". (Received August 27, 1964.)

64T-477. G. W. GOES, Illinois Institute of Technology, Chicago 60616, Illinois, On Fourier­ Stieltjes-sine-series.

The following theorem is proved: A Fourier-Stieltjes-sine-series wit!). only finitely many distinct coefficients, is a polynomial. For the proof a theorem of Helson [Proc. Amer. Math, Soc. 6 (1955), 235-242] is used, (The erratum on page 600 of this volume in connection with Abstract 64T- 332 is superfluous, i.e. the original abstract is correct.) (Received August 27, 1964.)

64T-478. D. M. YOUNG, JR., M. F. WHEELER, and J. A. DOWNING, The University of Texas, Computation Center, Austin, Texas. On the use of the successive overrelaxation method with several relaxation factors.

The use of the successive overre1axation (S.O.R.) method for solving the system Au+ d = 0 involves a relaxation factor w for each iteration. Rather than keeping w fixed we consider the use of m factors wl' w2 , ••• , wm in a cyclic order. It is shown that if A can be partitioned into a 2 X 2 block form with square diagonal matrices as diagonal blocks and if the eigenvalues of the matrix associated with Jacobi iteration are real and less than one in magnitude, then one cannot improve upon the repeated use of wb, the best single w, However, there are other sets besides w1 = w 2 = ... "Wm = wb which are equally effective in the sense of requiring no more iterations. This is so even when some of the wi are complex. Of course in such cases substantially more arithmetic operations would be required. If any wi = 1, then all of the eigenvalues of the matrix associated with the S. 0. R. method are real. One can accelerate the convergence using semi-iterative methods.

688 However, it is shown that this is less effective than the use of the S,O,R, method with a fixed "'b· (Received August 27, 1964.)

64T-479. R, P. GILBERT, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, On generalized axially symmetric potentials whose associates are distributions,

In this paper Bergman's operator method [Integral operators in the theory of linear differen­ tial equations, Ergeb, Math. u, Grenz. 23, Springer] is extended to the case where the associate is given as a distribution on the real axis, Using methods given by Bremermann and Durand (On analytic continuation, multiplication, and Fourier transformations of Schwartz distributions, J, Math. Physics 12 (1961), 240-258), Results are obtained concerning the analytic properties of solutions to the generalized axially symmetric potential equation in terms of the support of the associate, These re~ suits are analogous to earlier ones obtained by the author for analytic associates [Bergman's integral operator method in generalized axially symmetric potential theory, J, Math, Physics, July 1964], (Received August 27, 1964.)

64T-480. G. M. BENSON, 121 Overhill Road, Orinda, California. A hierarchy in the theory of implicit definability.

Paraphrasing Addison's definition of the effectively Borel subsets of NN (Ph.D, thesis, Wisconsin, 1954), let ( ff.l=-) be the applied effectively infinitary propositional language for number theory with one parameter (ranging over NN) defined inductively ~ (and recursively (!=)]by: (O) if P E NN and n EN then (o, Pin) E ff (and (if a. E NN) I= a. (o, 13ln) iff a. In= Pin]; (1)if c/JEffthen (1,c/J) Eff(and j:::a.(1,c/J)iffnot Fa.cp); (2)if -yE_¥N nL:~withgodel #ethen

( 2,e ) E ff [and I= a. ( 2 ,e ) iff for any n J=. a.-y(n)). For any ordinal v let _¥•v = {c/J: cp is obtainable from some ( 0, Pin) by v applications of (1) -followed-by- (2)~. Then it is known that for any {J in NN, {13} is effectively Borel (i.e., {P~ is explicitly definable in (Y,i=- )) iff pis implicitly definable in (Y, F)

(i.e,, for some c/J in 3'{3 = La. ) iff for some v < w , {3 is implicitly definable in iff Pis Fa. 1 (_~;v 1=-) L~ in some a. implicitly definable in (ffp F-) iff {3 E .6.~. Using Cohen's forcing-generic technique, Feferman (Berkeley Symposium, 1963) found a Pin NN n ~~which is not implicitly arithmeti'cally definable. This result is refined and generalized by: Theorem, For any constructive ordinal v > 0 there is a {3 in NN implicitly definable in (.¥;, J=.) but not in (.¥;.1=) for any tJ < v, For v = 3 this settles the main question of Kuznecov-Trahtenbrot in Doklady Akad, Nauk SSSR 105 (1955), 890-900, (Received May 8, 1964,)

64T-481, H. L. LOEB, 9508 Jellico, Northridge, California, Generalized Haar theorem,

Let {g0, ... ,~, ho, .. .,hm~ be a fixed set of n + m + 2 continuous real-valued functions over [0,1]. Let R = {r "'L~o ai~/L~obiht: L:~obiht > 0 on I!J,1]; where stand bt are real numbers~­ We are interested in approximating f E C[0,1] in the Chebyshev sense by members of R. We confine ourselves in the theorem only to members of C[0,1] which have Chebyshev approximations. Let

H{yi: i = 1, ... , k} designate the convex hull of set of vectors {y1, ... ,yk}· Theorem. The Chebyshev approximation is unique for every f iff. the following is always true: If there exists r E R; a finite

689 set of points (xl'''''xk) cf9,1] where k ;;> n + m + 2; a set of real numbers (>-. 1, ••• ,;\k) each of magnitude one; and a functiOn. m(x) of form "-'i=Oaigi"n + r("-'i=Obihi)"m· such that 0 E H 1;\i(g 0 (xi), ... ,gn(xi), r(xi)h 0(xi),.,., r(xi)hm (xi)); i = 1,.,., k} and m(x) has zeros at x 1, ... ,xk; then m(x) is identically zero, This theorem can also be stated using algebraic rather than geometric conditions. This theorem generalizes Haar's linear theorem. The main tools used in proving the theorem are some results of E. W. Cheney and author which will appear in J, Soc, Indust. Appl, Math, (Received August 28, 1964,)

64T-482, DAVID RODABAUGH, Vanderbilt University, Nashville, Tennessee 37203. Antiflexible algebras which are not power-associative.

For each char. p '12, a simple nodal p-dim. algebra satisfying the antiflexible law ((x,y,z) = (z,y,x)) is constructed which satisfies the condition that xaxb = xa+b for a+ b < p but xaxp-a 'I xp-axa. For p 'I 2,3, all of the structure theorems which have been proved for power-associative antiflexible .algebras can be proved for antiflexible algebras that satisfy (x,x,x) = 0 and contain idempotents. (Received August 31, 1964.)

64T-483, E. D. NIX, Box 28, Norwich, Vermont, Definition of the order types of the arc and the real line without reference to separability.

It is proven that the order type of the real line can be uniquely defined as the order type of a nonempty ordered setS, such that: (1) S is continuous; (2) S has no end elements; (3) any nonempty subset of S with the ordering derived from S which is continuous and without end elements is of the same order type asS. (Received August 31, 1964.)

64T-484. JOSEPHINE MITCHELL, Pennsylvania State University, 223 McAllister Hall, University Park, Pennsylvania 16801, Bounds for solutions of a system of partial differential equations in a domain with Bergman-Silov boundary surface.

A real solution of the system if!f;!8zj8zj= Fj(zj,z~*((j=l,2), Fj entire functions of the independent complex variables zj, zj, in some neighborhood of the,,origin; is given by 1/;(z 1,zl'z2 ,z2) = 2 Re[w1 + w2), where 'I}= g 1(zl'z2) + fo 1Tll(zl'z1, ~1 )g 1 (f1 ,z 2 )d~1 + J~ T 12 (z 2 ,z 2 ,l2 )g 1 (zl's2 )d~2 + 1 J~ J;t~? dzj'zj' 9g1 { ~1 • ~2 )d ~~ d~2 • I~ I ;;> lz jl' and w2 depends similarly on T2j and g 2 [Bergman, Ergebnisse der Math, u. Grenzgebiete, N. F, 23 (1961)]. Here zj is the complex conjugate of zj, Tkj are entire and gj arbitrary holomorphic functions of the indicated variables. If (i) gj are holomorphic on a domain 9Jl4 with a Bergman-Silov boundary surface ll'2 and continuous on \lli4 , (ii) 1/;(zl'O,z2,0), 3 4 1/;(z 1,o,o,z2) omit values ej 1, e j 2 respectively on the boundary m of 9Jl and are 0(1) ~m certain one-dimensional sets b1 cm3, (iii) 212 is an analytic surface meeting m3 in a closed curve a 1 but ll' 2 h a1 = jil, then from the Schottky inequality and maximum modulus theorem for one complex ,/, "2 Il2 [ ~ bl 212 crn4 ~ variable ., is bounded by "-'k=l j=ll + Tkj(lzjl)] lzjiBJc (go,r,ekl'ek2 'bk( )) on n "J' where Tkj are bounds of Tkj on IDl4 and r is related to (iii). Bounds are also obtained if a1 meets ll'2 in a finite number of points. (Received August 3.1, 1964.)

690 64T-485. J. M. AHLBERG and E. N. NILSON, United Aircraft Research Laboratories and Pratt and Whitney Aircraft, East Hartford, Connecticut. Convergence properties of generalized splines.

Given the linear differential operator L = Lk=Opk(x)dk/dxk, PJc(x) E cn[o,l], and a partition

A: 0 = x0 ••• xN = 1 of the unit interval, a generalized spline S(x) satisfies L* L(S) = 0 (L* = adj. L) on each subinterval formed by A and E c 2n-2 [0,1]. Let {Ak I be a sequence of partitions of [0,1] with 1 1 IAk I = maxi (xk,i+ 1 - xk,i) -> 0 as k -> oo and f E cn- [0,1] with f(n- ) absolutely continuous and f(n) E L 2 [0,1]. If Sk(x) represents a generalized spline of interpolation to f on Ak satisfying any of the standard end conditions, then (S~p)- f(p)) is O(IAkln-p- 1/), p = 0, 1, ••• , n- 1, and J;[L(Sk- f)fdx

-> 0 as k -> oo. If f E c 2n [0 ,1] and the ratio of IAk I/ mini (xk,i+ 1 - xk,i) is bounded, then (S~)- f(p)) is O(i~i 2 n-p- 1 ), p = 0,1, ••• , 2n- 2. These convergence properties carry over to multi­ dimensional generalized splines, thus constituting the extension of the results for conventional splines (these Notices vol. 11 (4) (1964), Abstract 64T-339). (Received August 31, 1964.)

64T-486. YU-LEE LEE, University of Connecticut, Storrs, Connecticut. Homeomorphisms on manifolds.

Let H(X, %') be the class of all homeomorphisms of a topological space (X,%') onto itself. Theorem 1. Let A and U be two open sets in ann-manifold (X,%') and p E Bndy(A) n U. Let R be an arc with p as an end point. Then there exists F E H(X, %') such that F is the identity outside U and p E Cl(R n F(A)). Theorem 2. Let A and U be two open sets in ann-manifold (X,%') and p E Bndy(A) n U. Then there exists a finite family of homeomorphisms {GpG2, ••• ,Gm} in H(X,%') such that Gi is fixed at p and outside U for each i and U {Gi(A): i = 1,2, ... ,m} U {P} is a neighborhood of p. (Received August 31, 1964.)

64T-487. j. S. RATTI, University of Nevada, Southern Regional Divion, Las Vegas, Nevada. A Watson transform.

Let K(ay) = G q,p (ayial, ..... ,ap• 2 -a1•· .. •••• 2-aP) with max .:: <- R(bj) < -1/2 < 2p,2q b1, ..... ,bq• 2-b1•·······2-bq 1=J=q min 1 ~j~pR(1 - aj), where K(ay) represents the Meijer G-function involving p + q parameters. It is shown that K(ay)/y is a Watson Kernel satisfying JQX>(K(ay)k(by)/i)dy = min(a,b) and f8g(x)dx = 2 J:(K(ay)/y)f(x)dx, J:oaf(x)dx = J 0txt,K(ay)/y)g(y)dy, JQX>ifi 2dx = J:lgi dy. The above results are also used to evaluate many new infinite integrals. (Received August 17, 1964.)

64T-488. G. T. WHYBURN, University of Virginia, Charlottesville, Virginia. On compactness of mappings.

Let f:X ------> Y be a monotone mapping where X andY are Hausdorff spaces: (1) if X is peri­ pherially compact, Y is locally connected and f is onto, f is closed if and only if connectedness is invariant under f- 1; (2) if X is connected and Y is a line, connectedness is invariant under f -l; (3) if X is peripherially compact and connected andY is the line, f is compact relative to f(X);

(4) if X is connected and locally connected and Y is the line, f is a homeomorphism if it is 1-1; (5) if X and Y are locally connected generalized continua, f is onto, and if the set T of singular points of f in X has a nonempty compact component, Y is multicoherent (T is the inverse of the set S of

691 points y of Y no closed neighborhood of which has a compact inverse); (6) if the restricted mapping of T onto S as in (5) is compact, so also is the whole mapping f whenever Y is unicoherent; (71· if f is 1-1 but not topological, there exists a topological ray in X which maps onto a simple closed curve under f. (Received September 3, 1964.)

64T-489. S-H. TUNG, Miami University, Oxford, Ohio 45056. Harnack's inequalities on Cartan domains. II.

Harnack's inequalities on Cartan domains which are called the classical domains by (*) L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc,, 1963, are obtained for ~k' k = 1,2,3 (see(*) for definition of .9i!'k] as application and extension of the results of (11) S. H. Tung, Harnack's inequality and theorems on matrix spaces, Proc. Amer. Math. Soc, 15 (1964), 375-381. Harnack's inequality for .9i!'4 is obtained independently of results in (l). The inequalities are derived by evaluating upper and lower bounds of the Poisson kernels Pk(z,u) in (*)as follows: For fixed z E ~k and u ranging over the characteristic manifold Sfk' (1/Vk >IIJ~J1((1 - rk)/(1 + rk))c ~ Pk(z,u) ~ (l/Vk)ITJt,J1((1 + rk)/(1 - rk))c where each non-negative rk < 1 is determined by the given z, and b = m, c = n for 9&'1 = .9i!'1 (m,n), b = n, c = (n + 1)/2 for

..9&'2 = ..9&'2 (n), b = n/2_, c = n- (1 + (- lf)/2 for .9&'3 (n) and b = 2, c = n/2 for ..9&'4 (l,n). (Received September 3, 1964.)

64T-490. C. E. AULL, Kent State University, Kent, Ohio. Nilpotent matrices.

Each nongroup .ot:class of all n X n matrices for fixed n, with elements in the field of real or complex numbers (see Abstract 614-51, these Notices (1964), 542) contains at least one element a such that a 2 is idempotent. We define the g-rank of an element b to be the rank of bn if bn is a group element. If k is the g-rank of b, there exist elements of the .M'-class of b with each of the g-ranks 0, 1, ... , k- 1. In particular,.there are nilpotent elements in every nongroup *class of the above matrices. Every element in an .M:class is nilpotent iff the square of some matrix in the cla~s is the zero matrix. (Received September 3, 1964.)

692 ERRATA-Volume 11

LEO SARlO and G. G. WEILL. Normal linear operators and some self­ adjoint elliptic linear partial differential equations. Page 336, Abstract 611-65. Line 1. The parenthetical remark should read "used thus far for Riemann surfaces only".

ERWIN ENGRLER. Ultrastructures in first-order model theory, Page 465, Abstract 64T-331. Lines 8-9. Change "58 is a normal ultrastructure of 2£" to "58 is a normal ultrastructure of a normal ultrastructure of 2l "·

YEHUDA RA V. On the representation of rational numbers of a sum of a fixed number of unit fractions. Pages 545-546, Abstract 614-63. Line 5. Instead of "Theorem" read "Conjecture".

ELIAHU SHAMIR. Asymptotic expansions for mixed elliptic problems in 2 dimensions. Page 576, Abstract 614-159. The expansion described is valid for homogeneous operators with con­ stant coefficients. A similar expansion holds for a general mixed problem with C00 coefficients, where the expansion terms are local null solutions with the same behavior near the origin.

A. BERIN. Convergent Gamma function formulae. Preliminary report. Page 551, Abstract 614-79. Line 13. "- 2 < x < + 2" should read "x < + 2".

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696 In preparation: FUNDAMENTAL CONCEPTS IN G. M. Fichtenholz Differential- und lntegralrechnung THE DESIGN OF EXPERIMENTS (Differential and Integral calculus) Charles R. Hicks, Purdue University College books for mathematics, Vol. 61, In concise and clear chapters, Professor Hicks 62,63 presents basic concepts in the design of experi­ Translation from the Russian ments using numerical examples. He empha­ Vol. I: abt. 5g0 pages, 16g illustrations, size go, sizes the distinction between the experiment, leatheroid, Price abt. OM 29.00 the design, and the analysis. This book is a Vol. II: abt. goo pages, 64 illustrations, size go, practical as well as theoretical approach to the leatheroid, Price abt. OM 34.00 design of experiments using statistical models. Vol. Ill: abt 650 pages, 145 illustrations, size go, It is intended primarily for the experimenter leatheroid, Price abt. OM 30.00 who has some background in basic statistical D. K. Faddejew and W. N. Faddejewa methods. Included: bibliography, 40 line draw­ Numerische Methoden der linearen ings, summary, glossary of terms, tables, and Algebra answers to questions. (Numerical methods of linear algebra) January, 1964 304 pp. $8.00 Mathematics in the natural sciences and engineering, Vol. 10 Translation from the Russian Latest Additions to the About 630 pages, 34 illustrations, size 8°, ATHENA SERIES ... leatheroid, Price abt. DM 60.00 Brief but brilliant studies covering special­ Publishers: ized areas of mathematics not found in VEB DEUTSCHER VERLAG DER conventional texts. WISSENSCHAFTEN • BERLIN W 8 THE GAMMA FUNCTION German Democratic Republic Emil Artin, late of the University of Hamburg Translated by Michael Butler A first-rate translation of a hard-to-find mathematical classic, incorporating cor­ MEMOIRS rections by the author. Number 48 August, 1964 48 pp. $1.75 (tent.) Other 1964 additions .•. EXTENSION OF RINGS AND HOMOLOGY COMPACT OPERATORS James P. ]ans, University by J. Lindenstrauss of Washington February 96 pp. $3.25 The purpose of this paper is to study the PERTURBATION TECHNIQUES IN MATH­ connection between various extension proper­ EMATICS, PHYSICS AND ENGINEERING ties for compact linear operators, and to Richard Bellman, Rand Corporation characterize the Banach spaces which have January 128 pp. $3.75 these properties. The main part of the paper is devoted to the study. of norm preserving ex­ COMBINATORIAL GEOMETRY IN THE tensions and some related geometrical topics. PLANE Examples of such geometrical topics are Hugo Hadwiger and Hans Debrunner, inter­ University of Berne section properties of cells, characterizations Translated by Victor Klee, University of Banach spaces whose conjugates are L1 of Washington spaces and characterizations of finite­ January 120 pp. $3.75 dimensional spaces whose unit cells are polyhedra. 112 pages $2.10 25% discount to members Order from American Mathematical Society 190 Hope Street, Providence, Rhode Island 02906

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INTRODUCTION TO MODERN ALGEBRA Neal H. McCoy, Smith College 304 pp. list $7.95 A popular text that presents the basic ideas of abstract algebra and features a clear exposition of the concepts of modern algebra. Adopted by approximately 250 colleges.

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698 NEW REPRINTS ...... ----Books ® ----...Journals N. H. ABEL Now Available Oeuvres Completes BOLLETTINO DI BIBLIOGRAFIA (Nouvelle Edition) Available Fall1964 E DI STORIA DELLE SCIENZE Cloth bound in 2 vols ...... $22.50 Oslo 1881, 964 pp., 1964 Reprint Niels Henrik Abel, one of the great men of the MATEMATICHE E FISICHE Norwegian mathematical school, was born in 1802 and Edited by B. Boncompagni (1821-1894) died in 1829, only 27 years old. Few scholars have had so brilliant a career in so short a space of time, nor Sources of Science No. 10 can it be predicted to what summits this spirit en­ Vols. 1-20 (All published). Rome 1868-1887 dowed with a genius for pure mathematics might have soared if he had been granted a longer existence. (with a General index to the 20 volumes in Volume XXl Cloth bound set ------$450.00 G. W. HILL Paper bound set ------405.00 Collected Mathematical Works Per volume, paper bound ------21.00 Available Falll964 Published by the Carnegie Institution of MATHEMATICAL Washington as their Publication No. 9 Cloth bound in 4 vols ...... $125.00 GAZETTE Washington 1905-1907, 1,788 pp., 1964 Reprint (Mathematical Association) "He knew all parts of celestial mechanics, yet the Now Available core of his work, that which made him immortal, is his Lunar Theory. He was, in this, not only a skillful artist Nos. 1-6. London 1894-1895 and an inquisitive researcher, but, truly, an original New Series inventor."- Henri Poincare Vols. 1-15. 1896-1931 (Including Index, 1894-1931) L. V. LORENZ Cloth bound set ------$320.00 Oeuvres Scientifrques Paper bound set ------275.00 Available Fall1964 Nos. 1-6 Cloth bound in 2 vols ...... $25.00 Paper bound in one volume ------7.50 Copenhagen, 1896-1908, 1,112.pp., Reprint 1964 "His work is nearly all mathematical as well as New Series. Vols. 1-15 physical; at times it is the mathematical and at other Per volume, paper bound ------20.00 times the physical side which predominates, but his Index 1894-1931, paper bound______10.00 ideas on physics were almost always worked out mathe­ matically. His main talent was his brilliant aptitude for the invention of methods, combined with a very fine MATHEMATICAL SOCIETY sense of the value of approximate results, even when he was unable to put any limit to the errors committed." -From the Preface OF JAPAN: JOURNAL Available Winter 1964 THE MATHEMATICAL Vols. 1-13. Tokyo 1948-1961 WORKS OF ISAAC NEWTON Cloth bound set ------$230.00 Paper bound set ------200.00 Vol. I. Assembled with an introduction Per volume, paper bound ------16.00 by Dr. Derek T. Whiteside Now Available MICHIGAN MATHEMATICAL Cloth bound ...... $14.50 New York, 1964,164 pp. JOURNAL Vol. II ... In preparation The first volume of Newton's Mathematical Works Now Available is concerned with the three works on which Newton's Vols. 1-7. Ann Arbor 1952-1960 renown as an inventor in calculus has historically depended. Their texts are largely self-explanatory and, Cloth bound set ------$125.00 though often dense in argument and subject matter, Paper bound set ------105.00 are rarely obscure. Per volume, paper bound ------15.00 JOHNSON REPRINT CORPORATION JOHNSON REPRINT COMPANY LTD. 111 Fifth Avenue, New York, N.Y. 10003 Berkeley Square House, London W.1, England

699 TOPOLOGICAL NEW FROM McGRAW-HILL

HANDBOOK OF MATHEMATICAL TABLES AND ANALYSIS FORMULAS/Fourth Edition By RICHARD S. BURINGTON, Chief Mathematician, Completely revised edition Bureau of Naval Weapons, Navy Deportment, Washington, D. C. 384 pages, $4.50 (Text Edition By also available). A major revision of on outstanding handbook designed to meet the needs of students and workers in mathematics, engineering, physics, chemistry, G. T. WHYBURN science, and other fields.

INTRODUCTION TO EXPERIMENTAL STATISTICS By C. C. Ll, University of Pittsburgh. McGraw-Hi// 134 pages $5.00 Series in Probability and Statistics. 460 pages, $11.50. Provides a basic working knowledge in the design and analysis of experiments for people in the biological, medical, and social science fields. Princeton ~\iathematical Series, # 23 INTRODUCTION TO FORTRAN: A Program for Self-Instruction By STEPHEN C. PLUMB, IBM. 192 pages, $5.50 PRIJ\"CETON UNIVERSITY PRESS (cloth), $3.50 (soft cover). In on overage of 16 hours a student con learn the basic information and skills needed to write com­ puter programs, using the FORTRAN II system de· signed for the 704/709!7090 family of computers. Neuerscheinungen! MATHEMATICAL METHODS IN RELIABILITY P. S. Alexandroff ENGINEERING Einfuhrung in die Mengenlehre und die By NORMAN H. ROBERTS, University of Washington. Theorie der reellen Funktionen 300 pages, $12.50. Provides the fundamental mathematical and analy­ tjochschulbucher fur Mathematik, Band 23 tical tools which ore the basis for reliability engin­ Ubersetzung aus dem Russischen eering. 2., univeranderte Auflage XXj279 Seiten, 25 Abb., gr. go, Leinen, CONVEX SETS 1g,-DM By FREDERICK A. VALENTINE, University of Calif­ ornia, Los Angeles. McGraw-Hi// Series in Higher Mathematics. 232 pages, $12.00. D. K. Faddejew und W. N. Faddejewa Develops the fundamental theory of convexity for and modern infinite Numerische Methoden der linearen both classical Euclidean geometry Algebra dimensional spaces. INTRODUCTION TO GENERAL RELATIVITY Mathematik fur Naturwissenschaft und Technik, Band 10 By RONALD ADLER, Stanford Univeristy; MAURICE BAZIN, Princeton University; and MENAHEN Russischen Ubersetzung aus dem SCHIFFER, Stanford University. International Series 772 Seiten, 35 Abb., g9 Tab., gr. go, in Pure and Applied Physics. Available in December. Kunstleder, g5,- DM An introduction to the fundamental mathematical and Vertriebsbeschrankung fur Westberlin, West­ physical concepts of the general theory of relativity. deutschland, Osterreich und die Schweiz Examination copies available on request VEB DEUTSCHER VERLAG DER McGRAW-HILL BOOK COMPANY WISSENSCHAFTEN · BERLIN W 8 330 West 42nd Street/New York, N. Y. 10036

700 These new and recent mathematics texts reflect the latest thinking in the field PROJECTIVE AND RELATED GEOMETRIES by Harry Levy, University of Illinois Intended for a year's course in geometry for the undergraduate mathematics major or the be­ ginning graduate student with a minimal preparation in geometry. It adopts Klein's formula­ tion of geometry as the invariant theory of a given set under a given group of transformations and develops this point of view consistently and systematically. Allendoerfer Advanced Series, 1964, 450 pages, $11.00, Sent on 30-day approval.* ABSTRACT ALGEBRA by W. E. Deskins, Michigan State University Makes deep-lying concepts and results of modern and classical algebra available to the student whose education may not have proceeded beyond college algebra. The text features the central theme of factorization, utilizes a number-theoretic viewpoint, proceeds from the familiar to the abstract, stresses the importance of analogy in the development of math~matics and in­ cludes numerous examples and heuristic discussions. Allendoerfer Advanced Senes, 1964, approx. 638 pages, $9.g5, sent on 30-day approval.* TOPICS IN HIGHER ANALYSIS by Harold K. Crowder and S. W. McCuskey, Case Institute of Technology The organization of definitions, theorems, and examples reflects the author's many years of ex­ perience in teaching- advanced calculus and advanced engineering mathematics. Complex num­ bers and the associated calculus, as well as vectors and the associated calculus, are introduced in early chapters. The student may therefore observe the similarity between the real and com­ plex planes. The result is unusual flexibility in the treatment of applied problems. For science and engineering students whose backgrounds include beginning differential equations. Allen­ doerfer Advanced Series, 1964, 561 pages, $10.00 INTERMEDIATE DIFFERENTIAL EQUATIONS, Second Edition by Earl D. Rainville, The University of Michigan Designed for a one-term course in differential equations at the advanced undergraduate or beginning graduate level, this book presents a broad selection of topics in classical analysis at a level beyond the introductory courses and below the highly-advanced treatises. It can also serve as a reference sourse for those who find use for more than elementary differential equations in their work or who wish to venture beyond their college training in this field. 1964, 307 pages, $9.50, sent on 30-day approval.* INTRODUCTION TO PROBABILITY THEORY by James R. McCord, III, Massachusetts Institute of Technology, and Richard M. Moroney, Jr., Consulting Mathematician Here is a truly brief introduction to probability theory that rapidly develops the basic con­ cepts. The authors stress the distinction between intuitive ideas and mathematical results and give special attention to sample spaces, sums of random variables, and limit theorems. They provide a sound elementary discussion of the law of large numbers and the central limit theorem for both discrete and continuous distributions, thus giving the student an understanding of those results which contain the essence of the concept of probability. Allendoerfer Mathematics Series, 1964, 232pages, $6.50. REGULAR POLYTOPES, Second Edition by H. S. M. Coxeter, University of Toronto This is a new edition of a masterpiece in the field of geometry by one of the world's major geometers. Self-contained, the book is designed for the student with a background in elementary algebra, geometry, and trigonometry. All of the geometry of the first six chapters is ordinary solid geometry; however, the topics have been carefully selected and organized to form a firm grounding for the subsequent developments. Macmillan Mathematics Paperbacks, 1963, 321 pages, $4.50, sent on 30-day approval.* THE THEORY OF GROUPS by Marshall Hall, Jr., California Institute of Technology An advanced text for students of considerable mathematical maturity. It provides both the fundamentals of the theory of groups and a broad selection from the most recent and active areas of research in group theory. 1959, 403 pages, $9.50. You'll want to evaluate these important titles. Write to Judith Wight for complimentary or approval copies.

*UTUUr Macmillan"s new "30-day approval plan,'" a book is billed only if you decide not to adopt it, but wish to keep a personal copy. THE MACMILLAN COMPANY 60 Fifth Avenue, New York 10011

701 Mathematics and IDA

Washington is the decision-making center of the free world. In that center, IDA functions as a scientific adviser to the Department of Defense. Our working environment is the gray area of those major national problems where too little is known and too much is at risk to hazard an intuitive decision. IDA provides responsible DOD decision makers with the scientific/ technical input required to eliminate or lessen the areas of uncertainty. Mathematics is applied in two principal fields at IDA: ..~ .... in the study of the technical feasibility of weapons systems . and in operations research to find the optimum choice IDA among competing weapons systems~ In a world in which the complexitit;s and exigencies of California Institute our nation's defense and foreign policy continue to in­ of Technology Case Institute crease and grow more critical, IDA's programs must also of Technology continue to expand. We invite qualified mathematicians to University of Chicago investigate both short term (two to three years) and per­ Columbia University manent appointments in our Weapons Systems Evaluation University of Illinois Massachnsetts Institute Division and our Research and Engineering Support Di­ of Technology vision. Respondents should preferably. have an advanced University of Michigan degree and be knowledgeable in at least one of the fol­ Pennsylvania State lowing: electromagnetic wave theory, information theory, Universit'J{ stochastic processes, automatic control theory, statistics, Princeton University queuing theory, numerical analysis, probability theory or Stanford University Tulane University the development of computer routines for problem solving or simulation. A career at IDA can present a challenge of satisfying proportions and provide a reward of substance. Write us; we may have mutual interests. Institute for Defense Analyses, 1666 Connecticut Ave­ nue, N.W., Washington 9, D. C. An equal opportunity employer.

702 More than a calculator-Almost a computer ... and now with automatic input-$4350

THE WYLE SCIENTIFIC IS THE FIRST DESK-TOP COMPUTATIONAL CENTER designed specifically for the solution of complex scientific and engineering problems. Almost as simple to operate as an adding machine, it provides arithmetic capability, efficiency and speed never before available in a calculator. WITH AUTOMATED DATA ENTRY, it eliminates all the tedious, wasted time of multi-step repetitive arithmetic problem solving. You enter only the variables manually. All the repetitive procedures are run off automatically from a prepared program library of simple punchcards fed into the calculator reader. And you don't have to be a programmer or need additional equipment to prepare your own input library. You simply punch in your instruc­ tions by hand on a Wyle stored-program card, which has the calculator keyboard reproduced on it. With this auto­ mated input added to the Wyle Scientific versatility, you will solve complex problems at speed approaching that of a computer. NEVER BEFORE CAPABILITIES LIKE THESE seen as they are entered and can be veri­ Its operation can be learned in minutes, The contents of all registers are displayed, fied before use. and it functions with the speed, quiet, and on an eight-inch cathode ray tube, as indi­ Transcription errors are eliminated reliability of its solid state design. cated in the following diagram. through complete versatility of transfer These capabilities, combined with from any register to any other without loss automatic entry, for the first time fill the of desired data. technical and economic gap between cal­ Multiplier-Quotient Register All registers handle 24-digit numbers. culators and computers. Entry Register Accumulator Rea;ister Decimal points are entered the same $3950 for basic calculator as digits, using an eleventh key, and all (You can add automatic Storage Register I input and answers are correctly aligned input later} Stor<§ge Register 2 with decimal point on the output display. Storage Register 3 $4350 complete with automatic input Automatic square root is provided, as For further information, write Dept. L, is single entry squaring and multiple sub­ Products Division, Wyle Laboratories, All parts of a problem are visible. The totals. El Segundo, California. Or telephone (213} contents not only of the three active arith­ The calculator has plug-in compatibility ORegon 8-4251. metic registers, but also of the three stor­ with auxiliary input-output devices includ­ age registers are displayed at all times. ing printers, paper tape equipment, and Numbers entered from the keyboard are other EDP equipment. WYLE LABORATORIES

703 Applied Mathematicians Operations Analysts Applied Physicists

CAREER APPOINTMENTS

PROFILE OF A CNA PROFESSIONAL A CN A analyst is a professional of superior competence. He may be a mathematician, a physical scientist, an economist, or a research engineer. He is a member of the Center for Naval Analyses of The Franklin Institute. CNA is a private scien­ tific organization engaged in opera­ tions research, systems evaluation, and broad-based studies for the New Directions United States Navy. CN A professionals work on current operational problems with the Operations Evaluation Group; on New directions in research are lead­ problems of cost effectiveness and ing to new dimensions in achieve­ force requirements of the mid­ ment at Booz•AIIen Applied Research. range future with the Naval War­ Our assignments-diverse, complex fare Analysis Group; on studies of and non-routine-are realistic tech· naval problems of the long-range nical problems brought to us by future with the Institute of Naval government and industrial clients. Studies; or on parametric studies Right now, for instance, our profes­ sional staff is treading unbeaten or development of new methodolo­ paths in astronautics ... CBR war­ gies in CN A's Research Group. fare ... communications ... com­ The CNA analyst has unusual ana­ puter technology . . . mathematic.s lytical ability. His imagination is and statistics ... meteorology ... operations research . . . reliability tempered by reality. He is capable . . . transportation . . . undersea of independent effort, but is amen­ warfare. The list is alphabetical and able to inter-disciplinary research. non~inclusive. He wants to apply his talents and Your career growth at Booz•AIIen knowledge to the nation's security. Applied Research can be exception· ally swift; potential is limited only A few CNA staff appointments are by the bounds of your own talents. available. For additional informa­ Creative, perceptive scientists and tion, write: engineers are invited to explore sev­ eral current career appointments. Director Please send your resume to Mr. CENTER FOR NAVAL ANALYSES Robert Flint, Director of Profes­ Dept. AM sional Appointments. 1401 Wilson Blvd., Arlington 9, Va. BOOZ•ALLEN APPLIED RESEARCH Inc. 4815 Rugby Avenue Bethesda, Maryland 20014 Washington • Cleveland CENTER FOR NAVAL ANALYSES Chicago • Los Angeles OF THE FRANKLIN INSTITUTE An equal opportunity employer OEG • OPERATIONS EVALUATION GROUP INS • INSTITUTE OF NAVAL STUDIES NAVWAG • NAVAL WARFARE ANALYSIS GROUP An equal opportunity employer

704 UNUSUAL CAREER

OPPORTUNITIES FOR Research Mathem.aticians The Applied Mathematics Laboratory of the David Taylor Model Basin-an advanced facility for fun­ damental and applied research into submarine, surface ship, aircraft, and missile design concepts -has important staff openings for research mathe­ maticians in • Stochastic Processes and Information Theory • Advance Programming • Numerical Solutions of Differential Equations • Critical Path Methods for Management Control Doctoral degrees in Mathematics, Statistics, or Physics are preferred, but those persons with a Master's Degree who have made significant contri­ butions to their fields will be considered. The Model Basin is actually a complex of four lab­ oratories-Hydromechanics, Aerodynamics, Struc­ tural Mechanics, as well as Applied Mathematics -and occupies 186 acres of suburban countryside along the Potomac River 12 miles northwest ofW ash­ ington, D. C. The living and working conditions are ideal, and these positions offer the added benefits of Career Civil Service. Please direct inquiries to Dr. Harry Polachek, Associ­ ate Director for Applied Mathematics (Code 800). David Taylor Model Basin DEPARTMENT OF THE NAVY Washington, D. C. 20007

An Equal Opportunity Employer Positions Filled Under PRNC Announcement 226B

70.') MONASH UNIVERSITY Clayton, Melbourne, Australia PROFESSOR OF PURE MATHEMATICS CUSlllNG-MAUOY, INC. -Applications are invited from mathematicians whose principal interests preferably lie in one or 1350 N. Main St., P. 0. Box 1187 more of the following fields: algebra, topology, Ann Arbor, Michigan functional analysis, mathematical logic. An ener· getic postgraduate programme is under way and there is opportunity to put into practice new thinking about undergraduate teaching. PROFESSOR OF APPLIED MATHE­ MATICS-Applications are invited from mathe· maticians whose principal interests preferably lie in one or more of the fields of partial differential equations, fluid mechanics, plasma physics, and astronomy. Other interests of members of the IJTHOPRINTERS Department are modern algebra, particularly semigroups, probability, and stochastic processes. Printen of the NOTICES A CDC 3200 computer is about to be installed in the University. Monash llniversity received its first students in 1960 and has at present established five full pro· fessorships in the Department of Mathematics which include the two mentioned above. Enquiries about the department may be addressed to the Chairman, Department of Mathematics. Known for Salary: £A4,600 per annum (under review). Superannuation on F.S.S.U. basis. QUALITY- ECONOMY- SERVICE Closing Date: 30th November 1964. Information on conditions of appointment and application Let us quote on your next printing procedure is available from the Registrar of the University.

TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY

The TRANSACTIONS is a cover-to-cover English translation of Trudy Moskovskogo Obscestva, which contains the results of original research in pure mathematics by many of the best mathematicians in the Soviet Union, as well as by some non-Soviet mathematicians. The papers are written in a survey style which renders them accessible to readers who might otherwise have difficulty studying papers on the same level. The translation will be prepared by the Ameri­ can Mathematical Society and the London Mathematical Society, with the support of the National Science Foundation. Each annual volume of the TRANSACTIONS will contain about 12 papers and approximately 500 pages and will be published as a hard-cover book. The trans­ lation begins with Volume 12 of the Russian journal.

Volume 12 496pages List Price $5.30 Member Price $3.98 Order from AMERICAN MATHEMATICAL SOCIETY 190 Hope Street, Providence, Rhode Island 02906

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Academic Press Inc, ••••• , •••••••••••• , • • • 694, 695 Holt, Rinehart and Winston, Inc ••••••••••.• , • • • 697

Allyn and Bacon, Inc ••••••••••••••••••••••••••• 698 Institute for Defense Analyses ••••••••••••• , •••• 702

American Mathematical Society. . • • • • • • • • • • • 697, 706 Johnson Reprint Corporation. • • • • • • • • • • • • • • • • • • 699

Booz•Allen Applied Research Inc...... 704 McGraw-Hill Book Company .••••••••••••••••.• 700

Center for Naval Analyses...... 704 The Macmillan Company ...... 696, 701

Cushing-Malloy, Inc••••••••• , ••.•••••••••••••• 706 Monash University •••••••••••.•••.•••••••••••• 706

David Taylor Model Basin, U.S. Navy•. , • . • • • • • • . 705 Princeton University Press. • • • • • • • • • • • • • • • • • . • 700

Deutscher Verlag der Wissenschaften ••.••••• 697, 700 Wyle Laboratories •••••••••••••••••••••••••••• 703

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PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Theory of Numbers Edited by Albert Leon Whiteman

The Symposium on Recent Developments in the Theory of Numbers was held on November 21 and 22, 1963 in conjunction with the six hundred sixth meeting of the American Mathematical Society. There were four sessions, with the topics: I. Diophantine Analysis and Algebraic Number Theory; II. Matrices and Quadratic Forms; III. Analytic Number Theory; IV. Anal­ ytic Number Theory and Modular Functions. Each session consisted of a one-hour invited address, followed by several fifteen-minute talks. The speakers for the invited addresses were Professors Selberg, Iwasawa, Birch and Carlitz. In all, the present volume contains twenty-two papers. The volume is dedicated to the·memory of Professor Morgan Ward, whose untimely death on June 26, 1963 kept him from delivering one of the invited lectures. Volume 8 about 200 pages Pre-publication List Price Member Price before November 10, 1964 before November 10, 1964 $6.00 $4.50 After November 10 not less After November 10 not less than$6.60 than $4.95

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