OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by Everett Pitcher and Gordon L. Walker

CONTENTS MEETINGS

Calendar of Meetings ••••• , •••••..••.•••.••.••••..••••.•• 1002 Program for the November Meeting in Baton Rouge, Louisiana • • • . • . • . 1003 Abstracts for the Meeting- Pages 1043-1057 Program for the November Meeting in Claremont, California • • • • • • . . • 1009 Abstracts for the Meeting- Pages 1058-1061 Program for the November Meeting in Ann Arbor, Michigan 1012 Abstracts for the Meeting- Pages 1061-1075

PRELIMINARY ANNOUNCEMENTS OF MEETINGS ..••..••.•. 1018

STARTING SALARIES FOR MATHEMATICIANS WITH A Ph. D ..•••••.••.•. 1026

DISMISSAL OF DO~TOR DUBINSKY .•.•••.•.•...•.••..••••••••... 1027 NEWS ITEMS AND ANNOUNCEMENTS •.•...•...... •.. 1008, 101I, 1017,1031

PERSO:--IAL ITEMS . • . • . • . • . . . . . • . . . • . • . • • . . . . . • . • • . . • • • • . . • • 1032 NEW AMS PUBLICATIO="'S •...••.....•...••••....•••.•••• 1033 MEMORANDA TO MSMBERS 1969 Summer Institute on Number Theory~ • . • . . • • . • . • • • . • • • • . . 1039 Abstracts . . . . • . . • • . • • . • • • . • • . • • • • • • • . • . . • • • • • • • • . • . • 1039 Computing and Mathematics •.••.••..•.••••.•.••..•.•••.•.• 1040 Additional Audio Recording of Mathematical Lectures Available •..•..•• 1040 ACTIVITIES OF OTHER ASSOCIATIONS ...... •...•.•••.•.•••.... 1040 VISITING MATHEMATICIANS ••..•.•.• 104I ABSTI\.ACTS OF CONTRIBUTED PAPERS . 1043 ERRATA .••••.•....•••.•••..• . 1095 INDEX TO ADVERTISERS ' 1107 RESERVATI0:--1 FORM ...... •...•••..••...... •••.•.••.•• 1108 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 c}/otiaiJ 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•

672 january 22-26, 1970 San Antonio, Texas Nov. 6, 1969 673 March 25-28, 1970 New York, New York jan. 28, 1970 674 April 14-18, 1970 Madison, Wisconsin Feb. 27, 1970 675 April 2 5, 1970 Davis, California Feb. 27, 1970 August 24-28, 1970 Laramie, Wyoming (75th Summer Meeting) january 21-25, 1971 Atlantic City, New jersey (77th Annual Meeting)

*The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadlines for by-title abstracts will be January 21, 1970· and February 2.0, 1970.

OTHER EVENTS

D·~cemher 2 7, 1969 Symposium on Some Mathematical Questions Boston, Massachusetts in Biology

The cJVofiai) of the American Mathematical Society is published by the Society in January, February, April, June, August, October, November and December. Price per annual volume is $10.00. Price per copy $3.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices.

Copyright© , 1969 by the American Mathematical Society PriJited in the United States of America

1002 The Six Hundred Sixty-Ninth Meeting Louisiana State University Baton Rouge, Louisiana November 21-22, 1969

The six hundred and sixty-ninth meet­ Lockett Hall. A beer party at the Capitol ing of the American Mathematical Society House Hotel is planned for Friday evening. will be held at Louisiana State University Tickets for this may be purchased at the at Baton Rouge, Louisiana, November 21- time of registration. 22, 1969. Pleasant Hall is a university-owned Sessions for contributed papers will hotel located on campus within a five-minute be held in the designated rooms in Lockett walk of Lockett Hall. The accommodations Hall. The three invited addresses are sche­ are adequate but not luxurious. There are duled to be held in the auditorium in Lock­ about 95 rooms with private baths and 45 ett Hall. rooms with hall baths available. Rates are By invitation of the Committee to as follows: Select Hour Speakers, Professor Marvin With private bath $ 7.00 1 person Rosenblum of the University of Virginia, 10.00 2 persons Professor Swarupchand M. Shah of the Uni­ 12.00 3persons versity of Kentucky and Professor Nickolas Heerema of Florida State University will With hall bath 4.00 1 person present hour talks. The title of Professor 7.00 2 persons Shah 1 s talk will be "Univaient functions 9.00 3 persons with univalent derivatives"; the title of Make reservations at Pleasant Hall Professor Heerema 1 s talk will be "Higher Reservation Desk, Louisiana State Univer­ derivations and automorphisms ofcomplete sity, Baton Rouge, Louisiana 70803. local rings"; the title of Professor Rosen­ Additional accommodations, con­ blum 1 s talk will be "Shifts and Hilbert venient to the University, with approximate space factorization problems." driving times to the university are given The registration desk will be located below: in the basement of the Mathematics Build­ JACK TAR CAPITOL HOUSE HOTEL ing, Lockett Hall, where all sessions will ( 10 minutes) be held. Registration hours will be 9 a.m. Lafayette at Convention, Baton Rouge, La. to 5 p.m. Friday, November 21 and 9 a.m. 70821 to 12 noon Saturday, November 22. Single $ 9.00 to $15.00 Baton Rouge, which is approximately Double 12.00 to 19.00 75 miles northwest of New Orleans, is lo­ (two persons, 1 bed) cated on U.S. 61 and U.S. 190. It is served by Twin 14.00 to 19.00 Delta, Southern, and Trans-Texas Air­ (two persons, 2 beds) lines, and by Greyhound and Trailways Bus Companies. Passenger train service is PRINCE MURAT INN (5 minutes) extremely limited. From the New Orleans 1480 Nicholson Drive, Baton Rouge, La. Airport to the Baton Rouge campus is a 70821 drive of less than an hour and a half. Single $10.00 up Several persons might wish to rent a car Double 14.00 up (two beds) jointly at the airport and drive down. Meals and snacks will be available The following motels might not have at campus cafeterias and off-campus estab­ accommodations available for Saturday lishments. Coffee and doughnuts will be night because of a home football game, but served each morning in the basement of they do expect to have facilities available

1003 for Friday night. HOLIDAY INN-SOUTH (15 minutes) (Inter­ section of I-12 and U.S. 61) BATON ROUGE TRAVELODGE MOTEL 9940 Airline Highway, Baton Rouge, La. (10 minutes) 70821 427 Lafayette Street, Baton Rouge, La. Single $11,00 70821 Double 15,00 (two double beds) Single $11.50 ( $2.00 each additional person) Double 13,50 Reservations should be made directly (two persons, 1 bed) with Pleasant Hall, the hotel, or one of the Double 15,50 motels. It is suggested that reservations be (two persons, 2 beds) made as early as is practical.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is 10 minutes. The contributed papers are scheduled at 15 minute intervals. To maintain this schedule, the time limit will be strictly enforced.

FRIDAY, 1:00 P.M. Invited Address, Lockett Hall Auditorium (Room B2) Shifts and Hilbert space factorization problems Professor Marvin Rosenblum, University of Virginia

FRIDAY, 2:30P.M. Session on Algebra, Room Bl5, Lockett Hall 2:30-2:40 (1) Domains in which every ideal is a finite product of semiprime ideals. Preliminary report Professor Nick H. Vaughan, North Texas State University (669-24) 2:45-2:55 (2) R-endomorphisms of R[[X]} Professor Matthew O'Malley*, NASA Manned Spacecraft Center, Houston, and Professor Craig A. Wood, Oklahoma State University (669-22) 3:00-3:10 (3) The field of constants of an integral derivation on a p-adic field Professor Henry W. Thwing, Stetson University (669-17) (Introduced by Professor William A. LaBach) 3:15-3:25 (4) Higher derivations and inseparable Galois theory Professor Richard L. Davis, Louisiana State University (669-16) 3:30-3:40 ( 5) Quotient over rings of integral domains Professor William J, Heinzer, Louisiana State University (669-14) 3:45-3:55 (6) R-automorphisms of R [[X]] Professor Robert Gilmer, Florida State University (669-2)

FRIDAY, 2:30 P.M. Session on Analysis, Room B5, Lockett Hall 2:30-2:40 ( 7) A note on power bounded operators Mr. Richard H. Bouldin, University of Georgia (669-7) *For papers with more than one author, an asterisk follows the name of the author who plans to presentthe paper at the meeting.

1004 2:45-2:55 (8) An extension of the Hausdorff-Young theorem. Preliminary report Professor Charles N. Kellogg, University of Kentucky (669-29) 3;00-3:10 (9) A functional integral for vector measures Mr. Daniel R. Lewis, Louisiana State University (669-36) (Introduced by Professor james R. Dorroh) 3;15-3:25 ( 1 0) Algebras of analytic functions. Preliminary report Professor Kenneth 0. Leland, Illinois Institute of Technology ( 669-40) 3;30-3:40 ( 11) Analyticity and quasi-analyticity of trajectories of semigroups of bounded linear transformations Professor John W. Neuberger, Emory University (669-41) 3:45-3:55 (12) A lattice of complete inner product spaces. Preliminary report Professor J. S. MacNerney, University of Houston (669-47)

FRIDAY, 2:30 P.M. Session on Topology, Room B6, Lockett Hall 2;30-2;40 (13) Cellular mappings on three manifolds Mr. William E. Haver, State University of New York at Binghamton (669-39) (Introduced by Professor Louis F. McAuley) 2:45-2:55 ( 14) Strongly acyclic map between simply connected manifolds Professor R. Christopher Lacher, Florida State University ( 669-37) 3:00-3:10 (15) Absolute Z-sets Miss Jean Pollard, Louisiana State University (669-32) (Introduced by Professor Richard D. Anderson) 3;15-3:25 (16) Embeddings of !-dimensional compacta in En Professor John L. Bryant, Florida State University (669-38) 3;30-3:40 (17) A note on cyclic subelement theory -- reducibility of local connectedness and local simple connectedness Professor Louis F. McAuley*, and Professor Byron L. McAllister, State University of New York at Binghamton (669-50) 3:45-3:55 (18) Homotopy groups of PL embedding spaces Professor Lawrence S. Husch, Virginia Polytechnic Institute (669-33)

FRIDAY, 2:30P.M. General Session, Room Bl6, Lockett Hall 2:30-2:40 ( 19) A geometric characterization of the line graph of a finite affine plane Professor Renu Laskar, Clemson University (669-11) 2:45-2:55 (20) Optimal excitation theory for the p.d.e. systems of symmetric hyperbolic type, or of elastic vibration type Professor Vadim Komkov, Texas Technological University (669-46) 3:00-3:10 (21) Nonsolvability theorems for second order two point boundary value problems Professor J. B. Garner, Louisiana Polytechnic Institute (669-26)

1005 3:15-3:25 (22) A property of the Rayleigh function Professor Fredric T. Howard, Wake Forest University (669-23) 3:30-3:40 (23) A characterization of subsets of Radon partitions. Preliminary report Professor William R. Hare*, and Professor John W. Kenelly, Clemson University (669-48)

FRIDAY, 4:15P.M. Invited Address, Lockett Hall, Room 82 Higher derivations and automorphisms of complete local rings Professor Nickolas Heerema, Florida State University

SATURDAY, 9:00A.M. Invited Address, Lockett Hall, Room 82 Univalent functions with univalent derivatives Professor S. M. Shah, University of Kentucky

SATURDAY, 10:15 A.M. Session on Algebra, Room 85, Lockett Hall 10:15-10:25 ( 24) The covering theorem for upper basic subgroups Professor Paul D. Hill, Florida State University ( 669-43) 10:30-10:40 (25) A generalization of the concept of a ring of quotients Professor John K. Luedeman, Clemson University (669-12) 10:45-10:55 (26) On the algebraic independence of symmetric polynomials Mr. Guenter K. Haeuslein, Oak Ridge National Laboratory, Oak Ridge, Tennessee (669-4) 11:00-11:10 (27) Abstract contents. I Professor Joseph Diestel, West Georgia College (669-15) 11:15-11:25 (28) Abelian torsion groups with artinian primary components and their automor­ phisms Professor Jutta Hausen, University of Houston ( 669-49) 11:30-11:40 (29) Schreier varieties of semigroups Professor Trevor Evans, Emory University (669-6) 11:45-11:55 (30) Density theorems for some arithmetic functions. Preliminary report Mr. Charles R. Wall, University of Tennessee (669-44)

SATURDAY, 10:15 A.M. Session on Analysis, Room 815, Lockett Hall 10:15-10:25 ( 31) Periodic solutions for perturbed nonlinear differential equations Professor Thomas G. Proctor, Clemson University (669-19) 10:30-10:40 (32) A product integral solution of a Stieltjes-Volterra integral equation Professor James A. Reneke, Clemson University (669-21)

1006 10:45-10:55 (33) One dimensional point derivation spaces in Banach algebras Professor Richard M. Crownover, University of Missouri-Columbia ( 669-28) 11:00-11:10 (34) In what spaces is every closed normal cone regular? Professor Charles W. McArthur, Florida State University ( 669- 30) 11:15-11:25 (35) Periodic solutions of functional differential equations. Preliminary report Professor Robert E. Fennell, Clemson Unive:r;sity (669-35) 11:30-11:40 (36) An arc in Hilbert space Professor Gordon G. Johnson, Virginia Polytechnic Institute (669-25)

SATURDAY, 10:15 A.M. Session on Topology, Room B6, Lockett Hall 10:15-10:25 (37) Notes on separation by continuous functions Professor Charles E. Aull, Virginia Polytechnic Institute (669-5) 10:30-10:40 (38) Weakly normal spaces. Preliminary report Professor Eugene S. Ball, Auburn University and Tennessee Technolo­ gical University (669-8) 10:45-10:55 (39) On compactifications with continua as remainders Professor Jack W. Rogers, Emory University (669-9) 11:00-11:10 (40) Locally connected continua embeddable in a torus. Preliminary report Professor Ralph B. Bennett, Auburn University (669-27) 11:15-11:25 (41) Dimension of the space of ho1omorphic cross-sections of a complex line bundle Professor Margaret M. LaSalle,University of Southwestern Louisiana (669-18) 11:30-11:40 (42) Infinite deficiency in Frechet manifolds Mr. Thomas A. Chapman, Louisiana State University (669-34) (Introduced by Professor Richard D. Anderson) 11:45-11:55 (43) Real compactifications with projective spectra Professor Phillip L. Zenor, Auburn University (669-31)

SATURDAY, 10:15 A.M. Session on General Topology and Topological Algebra, Room B2, Lockett Hall 10:15-10:25 (44) Natural extensions of compact semigroups Professor Charles E. Clark*, University of Tennessee, and M:r. JosephS. Starr, University of Missouri ( 669-42) 10:30-10:40 (45) Homological study of purity in locally compact groups Professor Ronald 0. Fulp, North Carolina State University (669-13) 10:45-10:55 (46) Semigroups that are unions of groups Mr. Reginald Mazeres, Tennessee Technological University (669-45) (Introduced by Richard Ball)

1007 11:00-11:10 ( 47) The existence of Irr( X) Mr. Michael W. Mislove, University of Tennessee (669-1) 11:15-11:25 ( 48) Semigroups and semilattices on generalized trees Professor W. Wiley Williams, University of Louisville ( 669-20) 11 : 30-11 :40 (49) The compactness of countably compact spaces Professor Philip Bacon, University of Florida ( 669-1 0) 11:45-11:55 (50) Countable subspaces. Preliminary report Mr. James M. Boyte, Virginia Polytechnic Institute (669-3) 0. G. Harrold Tallahassee, Florida Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

SPECIAL YEAR IN NUMERICAL Financial support for this symposium (and ANALYSIS some other aspects of the special year) by the Office of Naval Research is gratefully Purdue University has designated acknowledged. For further informa.tion 1969-1970 as a special year of emphasis about the symposium write: John R. Rice, on numerical analysis. There are a post­ 428 Math Sciences Building, Purdue Uni­ doctoral fellowship and several speciai versity, Lafayette, Indiana 4 7907. graduate research positions associated with this special year. In addition to regular seminars and visiting speakers, three spe­ cial sets of lectures have been arranged. On November 10 and 11 there will 'be lectures by A. Householder, J. Ortega, W. Rheinboldt and J. Traub in the general SIXTH INTERNATIONAL CONGRESS area of the solution of nonlinear equations. ON CYBERNETICS On December 4 and 5 there will be lectures by R. Barnhill, P. Davis, J. Lyness and The International Association for A. Stroud in the general are of quadratures. Cybernetics is organizing the Sixth Inter­ In early May of 1970 there will be lectures national Congress on Cybernetics in Namur, by G. Birkhoff, M. Schultz and R. Varga in Belgium, from September 7 through Sept­ the general area of projection methods for ember 11, 19 70. The subjects dealt with the solution of operator equations. Visitors will be divided into the following groups: are welcome to attend these lectures. 1) principles and methods of cybernetics, Also in conjunction with this special 2) semantic machines, 3) automation, year is a three-day symposium (April 1,2, 4) cybernetics and huma11 sciences, and 3) on "Mathematical Software." Approxi­ 5) cybernetics and life. For further infor­ mately twenty papers (some invited and mation write to: Secretariat of the Interna­ some contributed) will be presented. This tional Association for Cybernetics, Palais symposium is co-sponsored by the Associa­ des Expositions, Place A. Rijckmans, Na­ tion for Computing Machinery and SIGNUM. mur, Belgium.

1008 The Six Hundred Seventieth Meeting Claremont Graduate School Claremont, California November 22, 1969

The six hundred seventieth meeting HOWARD JOHNSON'S MOTOR HOTEL of the American Mathematical Society will 721 S. Indian Hill, Claremont be held at the Claremont Graduate School, Single $10.50 up Claremont, California, on November 22, Double $13.65 up 1969. By invitation of the Committee to UPLANDER MOTEL Select Hour Speakers for Far Western Foothill and Euclid, Upland Sectional Meetings, there will be two in­ Single $12.00 up vited hour addresses at this meeting. Pro­ Double $16.00 up fessor Solomon Feferman of Stanford University will lecture at 11:00 a.m. on BONITA MOTEL Saturday. The title of his talk is "Systems 151 E. Bonita, Pomona of ordinal functions and functionals."Pro­ Single $ 8.50 up fessor R. D. Richtmyer of the University Double $10.50 up of Colorado will address the Society at 2:00 p.m. on Saturday. His lecture is en­ TWILIGHT MOTEL titled "L2 -spaces of distributions." 433 E. Foothill, Pomona There will be sessions for con­ Single $ 6.30 up tributed papers at 9:30 a.m. and 3:30p.m. Double $ 8.40 up on Saturday. Late papers may be added to the program. Information concerning late The first two motels listed above are a papers will be a vail able at the registration long walk from the Claremont Graduate desk. All sessions of the meeting will be in School. Persons who stay in the other Bauer Center. motels will need auxiliary transportation. All reservations should be made directly REGISTRATION with the preferred motel.

Registration for the meeting will MEALS begin at 8:30a.m. on Saturday. The regis­ tration desk will be located in Bauer Cen­ Luncheon will be available at Collins ter at 9th Street and Mills Avenue. Hall of Claremont Men's College. The price for this meal is $2.00. Tickets for 1un­ ACCOMMODATIONS cheon will be sold at the registration desk.

The list of motels near the Clare­ TRAVEL mont Graduate School includes the follow­ ing. The Ontario International Airport, which is about ten miles from Claremont, GRISWOLD MOTEL is serviced by direct flights from San 555 W. Foothill, Claremont Diego, San Francisco, and Oakland, as Single $12.00 up well as commuter service from the Los Double $14.00 up Angeles International Airport. Taxi fare

1009 from the airport is approximately $5,00, Avenue, then north and watch for signs to Persons driving to the meeting on the AMS meeting after crossing 6th Street. the San Bernardino Freeway (Interstate 5) If coming along Foothill Boulevard, should exit at Indian Hill Boulevard in drivers should turn south on Mills Avenue Claremont, turn north to the first signal and watch for signs directing them to the (San Jose), turn east and go to Mills meeting.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is 10 minutes. The papers are scheduled at 15 minute intervals in order that lis­ teners can circulate among the sessions. To maintain the schedule, the time limit will be strictly enforced.

General Session, Room 25, Bauer Center 9:30-9:40 (1) Some homotopy groups of Thorn complexes Professor Patrick J, Ledden, University of California, San Diego (670-1 0) 9:45-9:55 (2) B-set of a family of sets Professor Sherman K. Stein, University of California, Davis (670-1) 10:00-10:10 ( 3) On unitary polarities Mr. Alan R. Hoffer, University of Montana (670-4) 10:15-10:25 (4) Dirichlet L-functions and character power sums Professor Tom M. Apostol, California Institute of Technology (670-6)

SATURDAY, 11:00 A.M. Invited Address, Lecture Hall, Bauer Center Systems of ordinal functions and functionals Professor Solomon Feferman, Stanford University

SATURDAY, 2:00P.M. Invited Address, Lecture Hall, Bauer Cente:I; L2 -spaces of distribitions Professor R. D. Richtmyer, University of Colorado

SATURDAY, 3:30 P.M. Session on Analysis and Topology, Room 24, Bauer Center 3:30-3:40 (5) On the Darboux-Neugebauer property of uxy and a problem of S. Marcus Professor John M. Bownds, University of Arizona (670-9) 3:45-3:55 ( 6) Uniform approximation of rational functions by polynomials with integral co­ efficients Professor LeBaron 0. Ferguson, University of California, Riverside (670-2) 4:00-4:10 ( 7) Bounding sets in various ways Professor Raymond B. Killgrove*, California State College at Los Angeles, Professor Henry G. Bray, San Diego State College, and Mr. Gordon Whitnall, University of California, San Diego (670-8) ------. *For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting. 1010 4:15-4:25 (8) On hereditarily arc-wise connected plane continua Dr. Charles L. Hagopian, Sacramento State College (670-3)

SATURDAY, 3:30P.M. Session on Algebra, Logic and Foundations, Room 25, Bauer Center 3:30-3:40 (9) Modules over Priifer domains Dr. S.M. Fakhruddin, University of Manitoba (670-5) (Introduced by Professor R. Venataraman) 3:45-3:55 (10) Congruences on N-semigroups Professor Takayuki Tamura, University of California, Davis (670-11) 4:00-4:10 ( 11) Boolean algebras of logics of higher order. Preliminary report Mr. Mohamed A. Amer*, Cairo University, and Dr. William P. Hanf, University of California, Berkeley (670-7) R. S. Pierce Seattle, Washington Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

C. L. E. MOORE INSTRUCTORSHIPS INSTITUTE FOR ADVANCED STUDY MEMBERSHIPS The Department of Mathematics at the Massachusetts Institute of Technology The School of Mathematics will grant announces that the C.L.E. Moore Instruc­ a limited number of memberships, some torships in Mathematics are open to young with financial support, for research in mathematicians with doctorates who show mathematics at the Institute during the aca­ definite promise in research. The base demic year 1970-71. Candidates must have salary will be $10,600, and the teaching given evidence of ability in research com­ load is six hours per week in one semes­ parable at least with that expected for the ter and three hours per week in the other. Ph. D. degree. Application blanks may be The salary can be supplemented by sum­ obtained from the secretary of the School mer work supported by a research grant of Mathematics, Institute for Advanced sponsored by the Air Force Office of Study, Princeton, New Jersey 08540. Com­ Scientific Research, or by teaching in the pleted applications should be returned ( whe­ summer session. The appointments are ther or not funds are expected from some annual but renewable for one additional other source) by January 15, 1970, or as year. Applications should be filed not later soon thereafter as possible. then January 5, on forms obtained from the Department.

1011 The Six Hundred Seventy-First Meeting University of Michigan Ann Arbor, Michigan November 29, 1969

The six hundred seventy-first Joel A. Smoller, and W. Gilbert Strang. meeting of the American Mathematical There will also be four sessions of con­ Society will be held at the University of tributed ten-minute papers. Michigan, Ann Arbor, Michigan, on Satur­ day, November 29, 1969. All sessions of REGISTRATION the meeting will be held in the Auditorium Unit of James B. Angell Hall. Angell Hall The registration desk will be is a large building on the east side of located in the lobby connecting Mason State Street; the Auditorium Unit is on the Hall, Haven Hall, and the Auditorium Unit ground floor on the far side from the of Angell Hall. This lobby faces the main street. University library and is known locally as By invitation of the Committee "The Goldfish Bowl." The desk will be to Select Hour Speakers for Western Sec­ open from 9:00 a.m. to 4:p.m. on Friday, tional Meetings there will be two one-hour November 28 and from 8:30 a.m. to 3:30 addresses. Dr. Alan Baker of Cambridge p.m. on Saturday, November 29. University and the Universities of Michi­ gan and Colorado will address the Society ACCOMMODATIONS at 11:00 a.m. His subject will be "A sur­ vey of recent results in the theory of The hotel headquarters for the diophantine equations." Professor Avner meeting will be the Michigan Union, which Friedman of Northwestern University will is located on the west side of State Street. speak at 1:45 p.m. on the topic "Free The Union will have sleeping accommoda­ boundary problems for parabolic equa­ tions for two hundred guests. The rates tions." are $10.50 for a single room, $12.50 for a By invitation of the same com­ double-bedded room, and $13.00 to $15.00 mittee there will be two special sessions for a twin-bedded room. All these rooms of selected twenty-minute papers, each have private baths and television. In ad­ of which will meet both morning and after­ dition a limited number of single rooms noon. Professor Donald J. Lewis of the with semi-private baths at $6.50 are avail­ University of Michigan has arranged one able, and also dormitory-style accom­ such session on the subject of Number modations at $4.50 per bed. Please direct Theory; the speakers will be James B. Ax, all requests for room reservations to Peter E. Blanksby, Harold G. Diamond, Room Reservations, November Mathema­ Patrick X. Gallagher, William J. Ellison, tics Meeting, Michigan Union, 530 South Takashi Ono, Carl Riehm, Wolfgang M. State Street, Ann Arbor, Michigan 48104. Schmidt, J. Roderick Smart, Harold M. It is important to mention the A. M. S. Stark, and George Whaples. The other meeting in making reservations. special session has been arranged by Professor James B. Serrin of the Univer­ FOOD SERVICE sity of Minnesota on the subject of Partial Differential Equations; the speakers will The Michigan Union will pro­ be Felix E. Browder, Todd Dupont, Jr., vide breakfast at nominal cost to those Eugene B. Fabes, Robert Finn, Keith staying there. In fact for planning purposes Miller, Louis Nirenberg, Johannes C. C. the Union will assume that those staying Nitsche, Ralph S.Phillips, James B. Serrin, in the Union will want breakfast unless the

1012 contrary is mentioned in the request for tinuation of State Street. a room reservation. A light lunch will also Visitors may park in the Thomp­ be available in the Union, but dinner defini­ son Street Multistory Parking Structure tely will not be. However, there are many one block east of the Union at a cost of at satisfactory restaurants within a mile of most one dollar per day. Theoretically it the campus. is necessary to pick up a special guest parking permit at the hotel desk of the TRAVEL AND LOCAL INFORMATION Union or at the registration desk for the meeting before parking, but there is a Those coming by air should use reasonable chance that the magic words Detroit Metropolitan Airport, which also "mathematics meeting" will persuade the serves Ann Arbor. Frequent bus and attendant at the structure to let cars limousine connections are available to the enter directly. Michigan Union. Ann Arbor can be reached Ann Arbor is on Eastern Stan­ from New York or Chicago by the New dard Time throughout the year. York Central Railroad; in fact sleeping accommodations are available from New SYMPOSIUM York. Ann Arbor is also served by Grey­ hound and Shortway Bus Lines. On Friday, November 28, 1969, Two major expressways inter­ the day before the meeting itself, the Uni­ sect just southeast of the city: U.S. 23 is versity of Michigan will sponsor a Sym­ a north-south route passing just east of posium on Function Algebras and Rational Ann Arbor, while 1-94 is an east-west Approximation, with the anticipated sup­ Interstate Highway passing just south of port of the National Science Foundation. A Ann Arbor. Northbound drivers are ad­ detailed announcement may be obtained by vised to exit from U. S. 23 onto Washtenaw writing to Professor Allen L. Shields, De­ Avenue West. Those using 1-94 should use partment of Mathematics, University of the State Road exit; State Road is a con- Michigan, Ann Arbor, Michigan 48104.

PROGRAM OF THE SESSIONS

The time limit for each contributed paper is 10 minutes. The contributed papers are sche­ duled at 15 minute intervals. To maintain this schedule, the time limit will be strictly enforced.

SATURDAY, 8:45A.M. Special Session on Number Theory, Auditorium A 8:45-9:05 On Schanuel' s conjectures Professor James B. Ax, State University of New York at Stony Brook (671-39) (Introduced by Professor Donald J. Lewis) 9:10-9:30 Waring's problem for fields Dr. William J. Ellison, University of Michigan (671-42) (Introduced by Professor Donald J. Lewis) 9:35-9:55 On some arithmetic questions on forms Professor Takashi Ono, Johns Hopkins University (671-43) 10:00-10:20 Some theorems in Galois cohomology with applications to quadratic forms and simple algebras over local fields Professor Carl R. Riehm, University of Notre Dame (671-44)

1013 10:25-10:45 Two remarks on class field theory Professor George Whaples, University of Massachusetts and University of Notre Dame (671-48)

SATURDAY, 8:45 A.M.

Special Session on Partial Differential Equations, Auditorium B 8:45-9:05 Uniqueness and comparison theorems for nonlinear elliptic boundary value problems Professor Todd Dupont, University of Chicago (671-37) 9:10-9:30 Barriers and boundary pathology for the nondivergence equation Professor Keith Miller, University of California, Berkeley (671-30) 9:35-9:55 On solvability of linear partial differential equations Professor Louis Nirenberg,* Courant Institute of Mathematical Sciences, New York University, and Professor J. F. Treves, Purdue University (671-38) 10:00-10:20 On the boundary behavior of minimal surfaces Professor Johannes C. C. Nitsche, University of Minnesota (671-31) 10:25-10:45 Elliptic-parabolic equations of the second order Professor RalphS. Phillips*, Stanford University, and Professor Leonard Sarason, University of Washington (671-34)

SATURDAY, 9:00A.M.

Session on Analysis, Auditorium C 9:00-9:10 ( 1) Zeros of analytic functions with infinitely differentiable boundary values Professor James G. Caughran, University of Kentucky (671-13) 9:15-9:25 (2) p-unique families of meromorphic functions Professor Benjamin Lepson, United States Naval Research Laboratory, Washington, D.C. and Catholic University of America (671-27) 9:30-9:40 ( 3) Approximation by polynomials whose zeros lie on nonanalytic curves Professor Jacob Korevaar, Claremont Graduate School ( 671-19) 9:45-9:55 ( 4) An existence theorem for bounded vector-valued functions Professor Samuel Zaidman, Universite de Montreal (671-2) 10:00-10:10 (5) On Mo(G) and the quotient M(G)/Mo(G) of a measure algebra Mr. Colin C. Graham, Northwestern University (671-17) 10:15-10:25 (6) Stable potentials. I Professor A. Leonard and Professor Seymour Sherman*, Indiana Univer­ sity (671-5) 10:30-10:40 ( 7) Approximation of vector-valued continuous functions Professor Alan H. Shuchat, University of Toledo (671-21)

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

1014 SATURDAY, 9:00A.M.

Session on Applied Mathematics and Probability, Auditorium D 9:00-9:10 (B) Mathematical aspects in relativistic fluid dynamics in non vacuum Professor Mari Z. v. Krzywoblocki, Michigan State University ( 671-3) 9:15-9:25 ( 9) The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3 Professor HowardS. G. Swann, Antioch College (671-6) 9:30-9:40 ( 10) Odd order boundary value problems Mr. T. Gifford, Carnegie-Mellon University (671-24) (Introduced by Professor Roger N. Pederson) 9:45-9:55 ( 11) On the computation of rigorous bounds for the solutions of linear integral equations with the aid of interval arithmetic Professor Colin W. Cryer, University of Wisconsin (671-12) 10:00-10:10 ( 12) Smoothest moving average interpolation formulas Professor Thomas N. E. Greville, University of Wisconsin (671-25) 10:15-10:25 ( 13) A result on the Sn/n optimal stopping problem Mrs. Mary E. Thompson, University of Waterloo (671-28)

SATURDAY, 11:00 A.M. Invited Address, Auditorium A A survey of recent results in the theory of diophantine equations Dr. Alan Baker, Cambridge University and the Universities of Michigan and Colorado

SATURDAY, 1:45 P.M. Invited Address, Auditorium A Free boundary problems for parabolic equations Professor Avner Friedman, Nowthwestern University

SATURDAY, 3:00P.M. Special Session on Partial Differential Equations II, Auditorium A 3:00-3:20 Existence theorems for nonlinear elliptic boundary value problems without coercivity hypotheses Professor Felix E. Browder, University of Chicago (671-33) 3:25-3:45 An existence theorem in the initial-value problem for the equations of Navier­ Stokes Professor Eugene B. Fabes*, and Professor Nestor M. Riviere, Univer­ sity of Minnesota (671-32) 3:50-4:10 On the behavior of a capillary surface in a wedge Dr. Paul Concus, University of California, Berkeley, and Professor Robert Finn*, Stanford University (671-35) 4:15-4:35 Existence theorems for two-by-two hyperbolic systems Professor Joel A. Smaller, University of Michigan (671-36)

1015 4:40-5:00 Puiseux expansions for Cauchy problems with multiple characteristics Professor Gilbert Strang, Massachusetts Institute of Technology (671-29) 5:05-5:25 An a priori bound for surfaces of constant mean curvature Professor James B. Serrin, University of Minnesota (671-49)

SATURDAY, 3:00P.M. Special Session on Number Theory II, Auditorium B 3:00-3:20 On a problem of Schinzel and Zassenhaus Dr. Peter E. Blanksby*, Ohio State University, and Mr. Hugh L. Mont­ gomery, University of Cambridge (671-40) 3:25-3:45 An elementary proof of the prime number theorem with remainder term Professor Harold G. Diamond, University of Illinois (671-41) 3:50-4:10 On Fogels' density theorem Professor Patrick X. Gallagher, Barnard College, Columbia University (671-26) 4:15-4:35 Dirichlet's theorem on Diophantine approximation Professor H. Davenport, University of Cambridge and University of Colorado, and Professor Wolfgang M. Schmidt*, University of Colorado (671-45) 4:40-5:00 On the values of the Epstein zeta function Professor J. Roderick Smart, University of Wisconsin (671-46) 5:05-5:25 Sign changes of some number theoretic functions Professor Harold M. Stark, Massachusetts Institute of Technology and University of Michigan (671-47)

SATURDAY, 3:00P.M. Session on Algebra and Lattice Theory, Auditorium C 3:00-3:10 (14) Linear equations over cones with interior: A solvability theorem with applica­ tions to matrix theory Professor Abraham Berman* and Professor Adi Ben-Israel, Northwestern University (671-1) 3:15-3:25 ( 15) Orthogonal similarity in a finite field Professor A. Duane Porter, University of Wyoming (671-16) 3:30-3:40 (16) Infinitely many nonisomorphic nilpotent algebras Professor Chong-Yun Chao, University of Pittsburgh (671-4) 3:45-3:55 (17) On the cohomology of Jordan algebras of a symmetric bilinear form Professor Roger A. Avelsgaard, Bemidji State College (671-20) 4:00-4:10 (18) Abstract contents. II Mr. Joseph Diestel, West Georgia College (671-7) 4:15-4:25 ( 19) Discrete linear lattices Mr. Ronald G. Mosier* and Professor Hidegoro Nakano, Wayne State Uni­ versity (671-8)

1016 4:30-4:40 (20) Semicontinuous linear lattices Mr. Bernard C. Anderson* and Professor Hidegoro Nakano, Wayne State University (671-9) 4:45-4:55 (·21) Cluster lattices Professor Hidegoro Nakano and Mr. Stephen Romberger*, Wayne State University (671-10)

SATURDAY, 3:00P.M. Session on Geometry and Topology, Auditorium D 3:00-3:10 (22·) Connexions on submanifolds. Preliminary report Mr. Joe W. Knickmeyer, University of Oklahoma (671-11) 3:15-3:25 ( 23) Restricted homotopies of curves Professor George K. Francis, University of Illinois (671-15) 3:30-3:40 (24) Homotopy groups of spaces of PL embeddings Mr. Ewing L. Lusl<; University of Chicago (671-22) 3:45-3:55 (25) Block bundles and embeddings Professor John E. Connett, Northern Illinois University (671-23) (Introduced by Professor Rodney Angotti) 4:00-4:10 (26) Regular-closed spaces and proximities Professor Douglas Harris, Marquette University ( 671-18) 4:15-4:25 (27) Uniform spaces and antinomies Professor D. V. Thampuran, State University of New York at Stony Brook (671-14) Paul T. Bateman Urbana, Illinois Associate Secretary

NEWS ITEM

SEVENTH ANNUAL HOLIDAY mathematicians from the Southwest are MATHEMATICS SYMPOSIUM AT being invited as guest participants. Some NEW MEXICO STATE UNIVERSITY informal seasonal social events will also be held. State University The New Mexico All interested mathematicians are Department of Mathematical Sciences will invited to attend the Symposium. While Holiday Symposium hold its Seventh Annual the University cannot support travel ex­ December 27-31, 1969, during the period penses, there is a small amount of money With partial support by the National Science available to helpdefraylivingexpenses. The Foundation and the Conference Board ofthe Department will assist in making motel Mathematical Sciences, the Symposium will reservations and will provide auto trans­ be centered around a series of lectures on portation to and from El Paso-, Texas, which topology by Professor George W. Whitehead is the nearest airport. of theMassachusetts Institute of Technology. Additional information may be ob­ In addition to the main lectures by tained from the Department of Mathematical Professor Whitehead, there will be an op­ Sciences, New Mexico State University, Las portunity for informal discussions and ses­ Cruces, New Mexico 88001. sions for contributed papers; twenty-five

1017 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Winter Meeting Moved from Miami to San Antonio

The Mathematical Association of In moving the meeting, the govern­ America and the Society learned early in ing bodies were highly conscious of the in­ October that the hotels in Miami, where convenience to members whose personal the meeting of January 1970 was originally plans are disrupted by the change. In announced, were having difficulty deliver­ making their decision, they gave greater ing the number of rooms needed for the weight to the factors of excellent meeting meeting. Department Chairmen and others space all under one roof and of nearby were notified at once of the difficulty. Our hotel space at low cost, factors that con­ needs were larger than estimated when the tribute to pleasant and successful scientific meeting was planned several years ago. sessions. Those who are specially disap­ Successive efforts to find more rooms pointed in the change should take cheer yielded chiefly rooms at considerable dis­ from the fact that the average cost of hotel tance from Miami (for example, at the far rooms in Miami was just about double that end of Miami Beach) and at very high in San Antonio, without counting the cost prices. Finally, some rooms already and the inconvenience of shuttle bus ser­ promised artd some meeting space were vice at the earlier location. Further, the withdrawn to fill needs of the owners of transportation problem to San Antonio one of the hotels. seems much simpler. Fortunately, it happened that an A copy of the new reservation form, excellent convention center and many near­ to be mailed to the San Antonio Housing by hotel rooms were available in San Bureau, is incorporated in this issue as is Antonio, a very rare occurrence when one all of the pertinent information on the San considers that convention calendars are Antonio facilities. The deadline for re­ frequently filled several years in advance. ceipt of the reservations has been extended At the height of the crisis, the space was to January 2, 1970. being inspected by Society officers for Please let mathematicians in your possible use in some future year. immediate vicinity know of this change. The Miami Convention Bureau Please inform your friends and your asso­ kindly released us from our commitment ciates who are traveling and who may not and, by action of the Board of Governors receive the c}/oticei). of the Association and the Executive Com­ mittee of the Society, the meeting has been relocated in San Antonio on the same Everett Pitcher dates as originally planned. Secretary

1018 The Seventy-Sixth Annual Meeting San Antonio Convention Center San Antonio, Texas January 22-25, 1970

The seventy-sixth annual meeting East Market Street directly across from of the American Mathematical Society will the Hilton Hotel. The desk will be open be held at the San Antonio Convention from 2:00p.m. to 8:00p.m. on Wednesday, Center in San Antonio, Texas. The meeting January 21; from 8:00 a.m. to 5:00p.m. is being held in conjunction with the Mathe­ on Thursday, January 22; from 9:00 a.m. matical Association of America. Hour ad­ to 5:00 p.m. on Friday through Sunday, dresses will be given by Professor James January 23-25; and from 9:00 a.m. i:o B. Ax of the State University of New York 3:00 p.m. on Monday, January 26. at Stony Brook and Professor Morris W. The registration fees for the meet­ Hirsch of the University of California, ings are as follows: Berkeley. The Josiah Willard Gibbs Lec­ Member $ 3,00 be given by Professor Walter H. ture will Member's family 0,50 Munk of the University of California, San Student no fee Diego. Nonmember 7,50 The general program is outlined as Nonmember's family 0,50 follows: January 22-25: AMS contributedpapers EMPLOYMENT REGISTER and invited addresses January 24-26: MAA sessions The Mathematical Sciences Em­ ployment Register will be maintained from The Council of the American Math­ 9:00 a.m. to 5:00p.m. from Friday through Society has set no limit on the ematical Sunday, January 23-25, in The Gallery of number of ten-minute papers that will be the Convention Center. accepted for presentation in the regular sessions for contributed papers at the EXHIBITS annual meeting. However, only the first 720 abstracts received will be assigned to The book and educational media day sessions because of space limita­ exhibits will be displayed in the east end tions, Abstracts received after the first of the Exhibit Hall of the Convention Cen­ 720 will be assigned to general evening ter. They will be open from Friday through sessions. The deadline for abstracts to be Sunday, January 23-25, from 9:00 a.m. to received in the Providence office is No­ 5:00p.m. vember 6, 1969. Because of the short period of time that is a vail able to prepare BOOK SALE the program for the annual meeting, the Providence office will not be able to accept Books published by the Society will changes in abstracts. be sold for cash prices somewhat below Authors are requested to notify the the usual prices when these same books Providence office of papers to be withdrawn. are sold by mail.

REGISTRATION ACCOMMODATIONS

The registration desk for this meet­ Accommodations for the meeting ing will be in the west end of the Exhibit will be handled by the City of San Antonio Hall o£ the Convention Center, located on Convention Bureau. A form for requesting

1019 CENTRAL SAN ANTONIO

1. Alamo Travelodge 8. Hilton Palacio del Rio 15. St. Mary's Travelodge 2. Blue Bonnet Hotel 9. Holiday Inn-Downtown 16. Travis-Plaza Hotel 3. Crockett Hotel & Motor Inn 10. La Posada Motor Hotel 17. Wayfarer Motel 4. Downtowner Motel 11. La Quinta Inn 18. YMCA 5. El Tropicano Motor Hotel 12. Menger Hotel 6. Granada Inn 13. Rode way Inn # 103 7. Gunter Hotel 14. St. Anthony Hotel

1020 accommodations will be found on page 1108 Suites--1 Bedroom and Parlor of these Notices. Persons desiring accom­ $35.00 to $40.00 modations should complete this reserva­ Suites--2 Bedrooms and Parlor tion form or a reasonable facsimile and $55.00 to $90.00 send it to the Mathematical Meetings Housing Bureau, City of San Antonio Con­ HILTON PALACIO del RIO vention Bureau, P .O.Box 2277, San An­ Singles $17.00 to $21.00 tonio, Texas 78206. Reservations will be Doubles $22.00 to $26.00 made in accordance with preferences Twins $22.00 to $26.00 indicated on the reservation form, insofar Suites--1 Bedroom and Parlor as this is possible, and all reservations $34.00 to $38.00 will be confirmed. A deposit will not be Suites--2 Bedrooms and Parlor required. REQUESTS FOR RESERVA­ $100.00 to $125.00 TIONS SHOULD ARRIVE IN SAN ANTO­ NIO NO LATER THAN JANUARY 2, 1970. HOLIDAY INN, DOWNTOWN Singles $12.00 CENTRAL SAN ANTONIO Doubles $17.00 Twins $19.00 ALAMO TRA VELODGE Singles $10.00 to $13.00 LA POSADA MOTOR HOTEL Doubles $12.00 to $15.00 Singles $14.00 to $20.00 Twins $14.00 to $17.00 Doubles $18.00 to $24.00

BLUE BONNET HOTEL LA QUINT A INN Singles $ 9.50 to $13.00 Singles $10.00 Doubles $11.50 to $17.50 Doubles $13.00 Twins $13.50 to $18.00 Twins $15.00

CROCKETT HOTEL & MOTOR INN MENGER HOTEL Singles $ 9.00 to $12.00 Singles $13.00 Doubles $13.00 to $15.00 Doubles $16.00 Twins $15.00 Twins $15.00 to $17.00 Suites--1 Bedroom and Parlor DOWNTOWNER MOTEL $30.00 to $46.00 Singles $ 9.00 to $10.00 Suites- -2 Bedrooms and Parlor Doubles $11.00to $12.00 .$44.00 to $70.00 Twins $13 .oo to $14.00 RODEWA Y INN-No. 103 EL TROPICANO MOTOR HOTEL Singles $ 9 .oo Singles $14.00 to $22.00 Doubles $12.00 Doubles $16.00 to $24.50 Twins $15.00 Twins $15.50 to $24.50 Suites--1 Bedroom and Parlor ST. ANTHONY HOTEL $25.00 Singles $13.00, $14.00, $17.00 Suites--2 Bedrooms and Parlor Doubles $16.00, $17 .oo, $20.00 $40.00 to $150.00 Twins $17.00, $18.00, $20.00, $25.00 and up GRANADA INN Suites--1 Bedroom and Parlor Singles $10.00 $35.00 to $55.00 Doubles $14.00 Suites--2 Bedrooms and Parlor Twins $14.00 $55.00 to $85.00

GUNTER HOTEL ST. MARY'S TRAVELODGE Singles $10.50 to $16.50 Singles $ 9.00 Doubles $12.50 to $18.50 Doubles $12.00 to $14.00 Twins $15.50to $21.50 Twins $13.00 to $16.00

1021 SU AITGIIO J•l AND l•o I .. OIUS 11011$ IU U()to.(O .. YINTIQH (101111 -----"--COI,OI•II n '!",~· "·\.,'\..,11'""""

1. Albert Pick Hotel & Motel 10o Rodeway Inn-Pan Am Expressway 20 Aloha Inn llo Rodeway Inn-Wonderland 3o Holiday lnn-N oW o Expressway 12o San Antonio Inn 4o Holiday Inn-Pan Am Expressway 13o Sheraton San Antonio 5o La Quinta Motor Inn-Loop 410 14o Town & Cotu1try Lodge 6o Ramada Inn-Austin Highway 15o Western Sun 7o Ramada Inn-Military Drive 16o Westerner Motor Hotel So Rio Best Western 17o Howard Johnson's 9o Rodeway Inn-Austin Highway

1022 TRAVIS-PLAZA HOTEL RAMADA INN, MILITARY DRIVE Singles $ 8,00 to $10,00 Singles $10,00 to $11,00 Doubles $10.00 to $12,00 Doubles $13,00 to $14.00 Twins $12.00 to $14.00 Twins $14,00 to $15,00

WAYFARER MOTEL RIO BEST WESTERN Singles $ 9.50 Singles $ 9,00 Doubles $10,50 to $13.50 Doubles $ 9,00 to $11.00 Twins $12,50 to $14.50 Twins $11.00 to $18.00

YMCA* RODEWAY INN, AUSTIN HIGHWAY Singles $ 3,25 Singles $ 8,00 to $14,00 *One night's deposit is required by the Doubles $11,00 to $14,00 YMCA Twins $12.00 to $16,00

RODEWAY INN, PAN AM EXPRESSWAY SAN ANTONIO Singles $ 9,00 Doubles $10.00 to $11.00 ALBERT PICK HOTEL&MOTEL Twins $12,00 to $14,00 Singles $11,00 Doubles $15,00 RODEWAYINN, WONDERLAND Twins $17,00 Singles $ 9.00 to $ 9.50 Doubles $10,00 to $11.00 ALOHA INN Twins $12,00 to $14,00 Singles $ 8,50 to $12,00 Doubles $10,50 to $14,00 SAN ANTONIO INN Twins $12,50 to $14.00 Singles $10,00 Doubles $14,00 HOLIDAY INN, NORTHWEST EXPRESS­ Twins $16.00 WAY Singles $12,00 SHERATON SAN ANTONIO Doubles $17,00 Singles $14.00 to $20.00 Twins $19.00 Doubles $18.00 to $24,00 Twins $18,00 to $24.00 HOLIDAY INN, PAN AM EXPRESSWAY Singles $12,00 TOWN & COUNTRY LODGE Doubles $17,00 Singles $ 9.50 to $11,50 Twins $19.00 Doubles $12.00 to $14.00 Twins $13.00 to $16.00 HOWARD JOHNSON'S Singles $ 8.50 to $9.50 WESTERNER MOTOR HOTEL Doubles $10,50 to $11.50 Singles $ 6,00 Twins $10,50 to $11,50 Doubles $ 8,00 Each additional person $ 2,00 Twins $ 9.00

WESTERN SUN LA QUINTA MOTOR INN, LOOP 410 Singles $ 7,00 Singles $11,00 Doubles $ 8.00 Doubles $13.00 Twins $10,00 to $12.00 Twins $16,00 to $17,00 NATIONAL SCIENCE FOUNDATION INFORMATION CENTER RAMADA INN, AUSTIN HIGHWAY NSF staff members will be avail­ Singles $ 8,00 to $ 9.50 able to provide counsel and information Doubles $10,50 to $11.50 on all NSF programs of interest to mathe­ Twins $12.00 to $13.00 maticians from 9:00 a.m. to 5:00 p.m. on

1023 January 23, 24, and 25, 1970, in Room 12 the same area to receive incoming calls for of the Convention Center. all members in attendance. The center will be open from January 22 through January ENTERTAINMENT 26 between 9:00 a.m. and 5:00 p.m. Messages will be recorded, and the name The Paseo del Rio, an arm of the of any member for whom a message has San Antonio River, extends for about two been received will be posted until the and one-half miles through the center of message has been picked up at the mes­ the city. Located on the river are several sage center. Members are advised to of the major hotels as well as restaurants, leave the following numbers with anyone shops, craftsmen, and art galleries. Small who might want to reach them at the meet­ river taxis ply the river, and one may go ing: (512) 222-8061 or 222-8062. from one end of the river to the other for $0.50. The banks of the Paseo del Rio are TRAVEL beautifully landscaped, and a walkway known as the "River Walk" extends the In winter, San Antonio is on Central length of the river with frequent egresses Standard Time. There is regular airline to hotels and shops away from the river. service to the San Antonio International The Convention Center is at one end of the Airport by the following airlines: Ameri­ Paseo del Rio. Among the many places of can, Braniff, Continental, Eastern, and interest are The Alamo, including the Texas International Airlines. Southern Alamo Museum; Brackenridge Park, which Pacific and Missouri Pacific have daily has a Chinese Sunken Garden and one­ passenger service to San Antonio. Southern fifth scale model of a diesel train; the Pacific has one train daily from New Hertzberg Circus Collection; La Villita and Orleans and one from Los Angeles; Mis­ the Arneson River Theater, a small his­ souri Pacific has daily service from St. toric Mexican village and an open air Louis. theater; the four Missions of San Antonio founded between 1720 and 1731; the Spanish WEATHER Governor's Palace; and the San Antonio Zoo ranked as one of the finest zoos in the The location of San Antonio on the world. edge of the Gulf Coastal Plains results in San Antonio has a large number o, a modified subtropical climate, predomi­ excellent restaurants, including many open nately continental during the winter air and sidewalk cafes. While the cuisine months. Normal mean temperature during is slanted toward the Mexican, restaurants the month of January is 52°. Mild weather serving the food of continental Europe are prevails during the winter months with in abundance. below-freezing temperatures occurring on an average of about 20 days each year. MAIL AND MESSAGE CENTER For the past 30 years, the average daily maximum temperature during the month All mail and telegrams for persons of January has been 62.3 ° and the average attending the meeting should be addressed daily minimum temperature has been in care of Mathematical Meetings, San 41.6°. The possibility of showers exists; Antonio Convention Bureau, Post Office however, in general January can be ex­ Box 22 77, San Antonio, Texas 78206. Mail pected to be sunny and clear. and telegrams so addressed may be picked up at the Mail and Information Desk located 0. G. Harrold in the west end of the Exhibit Hall in the Associate Secretary Convention Center. A message center will be located in Tallahassee, Florida

1024 Some Mathematical Questions in Biology Boston, Massachusetts December 27, 1969

The fourth annual symposium on with some mathematical background and Some Mathematical Questions in Biology mathematicians. Most of the &peakers are will be held on December 27, 1969, in the biologists who will address themselves to State Suite of the Sheraton Plaza Hotel in questions which are primarily of biological Boston, Massachusetts. This symposium interest, but in which some mathematical is cosponsored by the American Mathe­ analysis is involved. The morning session matical Society and the Society for Indus­ will be devoted to models of developing trial and Applied Mathematics, and it is organisms and the afternoon to models of being held in cooperation with Section A the brain. (Mathematics) of the American Association The program will consist of six for the Advancement of Science. The sym­ lectures, and it was arranged by the AMS­ posium will be supported by a grant from SIAM Joint Committee on Mathematics in the Institute for Defense Analyses. Regis­ the Life Sciences. The members of this tration and hotel arrangements will be committee are Murray Gerstenhaber announced in SCIENCE. (chairman), Hans Bremermann, Robert This is the fourth in a series of MacArthur, Alston S. Householder, and annual symposia whose purpose is to R. C. Lewontin. stimulate direct contact between biologists

PROGRAM 9:00 a.m Chairman: Murray Gerstenhaber, University of Pennsylvania Periodic wave propagation and pattern formation: a model Morrel H. Cohen, Director, James Franck Institute, University of Chicago Periodic wave propagation and pattern formation: application to problems in development Brian C. Goodwin, Reader, University of Sussex, Brighton, Sussex,. England. Metabolic stability, epigenesis, and self-replication in randomly construc­ ted macromolecular systems Stuart Kauffman, Assistant Professor of Mathematical Biology, Uni­ versity of Chicago

2:00 p.m. Chairman: Jack D. Cowan, chairman, Committee on Mathematical Biology, University of Chicago Point processes and neural ensembles George L. Gerstein, Professor of Physiology and Biophysics, Univer­ sity of Pennsylvania Neuronal periodicity and frequency discrimination Vernon Mountcastle, Professor of Physiology, Johns Hopkins University Neurophysiological insight provided by mathematical boundary value prob­ lems, Wilfred Rall, Research Physicist, National Institutes of Health Theory of nervous action Jerom.~ Y. Lettvin, Professor of Experimental Epistemology, Massa­ chusetts Institute of Technology

Murray Gerstenhaber Philadelphia, Pennsylvania Chairm1n 1025 Starting Salary for Mathematicians with a Ph. D.

This survey was compiled from questionnaires sent to 11 56 individuals who received the Ph.D. in mathematics during the academic year 1968-1969. There were 577 usable returns. The academic life attracted 85o/o of those reporting; of these, 61 o/o were primarily engaged in teaching, 11 o/o in research, and 10o/o in a combination of teaching and research. Industry attracted only 9o/o of those reporting; 3o/o were connected with research institutes and 3o/o by government. Geographically 33o/o accepted jobs in the northeast, 21 o/o in the south, 23o/o in the midwest, 6o/o in the mountain states, 14o/o in the far west and 3o/o in Canada. The greater majority of Ph.D.'s had experience in their fields prior to receiving their doctorates, 52o/o had more than a year of experience, 16o/o had between 1/2 and a year, and 32o/o had less than one year of experience. It should be noted that the first category listed below (teaching, nine-month salary) repre­ sents 57o/o of all positions reported in the survey. Salaries reported on a nine-month basis represent 74o/o of those responding; of these, two-thirds accepted salaries in the range of $9,800 to $11,500, with approximately an equal number accepting salaries lower and higher than these amounts. Salaries are listed in hundreds of dollars. Dashes indicate that not enough returns were received to warrant including the salary figures.

TEACHING RESEARCH FELLOWSHIP (Nine-Month Salary) (Nine-Month Salary) (Yearly Stipend) Year Min. Median Max. Year Min. Median Max. Year Min. Median Max. 1966 60 89 120 1966 72 84 96 1966 60 81 90 *1967 70 93 120 *1967 78 93 108 *1967 40 80 120 **1967 65 96 140 **1967 70 93 103 **1967 60 85 100 1968 72 102 170 1968 78 100 115 1968 1969 80 105 165 1969 63 105 125 1969

TEACHING AND RESEARCH RESEARCH TEACHING (Nine-Month Salary) (Twelve-Month Salary) (Twelve-Month Salary) Year Min. Median Max. Year Min. Median Max. Year Min. Median Max. 1966 73 89 120 1966 75 90 120 1966 80 105 129 *1967 75 90 105 *1967 80 101 130 *1967 85 112 150 **1967 **1967 80 105 132 **1967 83 122 200 1968 78 100 130 1968 80 100 134 1968 95 120 180 1969 95 po 138 1969 78 149 180 1969 75 128 168

INDUSTRY RESEARCH INSTITUTES GOVERNMENT (Twelve-Month Salary). (Twelve-Month Salary) (Twelve-Month Salary) Min. Median Max. year Min. Median Max. Year Min. ~ Max. Year 150 1966 103 137 170 1966 68 135 190 1966 78 109 111 161 *1967 125 145 200 *1967 60 120 150 *1967 84 **1967 97 151 204 **1967 60 135 215 **1967 170 1968 110 156 248 1968 120 157 192 1968 85 134 1969 125 168 250 1969 75 156 235 1969 82 138 192

*These figures represent the survey compiled from returns sent in by individuals who received their doctorates in 1966. **These figures represent the survey compiled from returns sent in by individuals who received their doctorates during the first six months of 1967.

1026 DISMISSAL OF DR. DUBINSKY

Effective September 4, 1969, the Board was understood to have handled the Board of Administrators of the Tulane case, not with the substance of the case that Educational Fund dismissed Dr. Edward came before the Senate Committee on Aca­ L. Dubinsky, a member of the Society, demic Freedom, Tenure, and Responsibili­ from his post of Associate Professor of ties of Tulane University. The Secretary Mathematics at Tulane University. did send the telegram to Mr. Harry B. The problem, from the viewpoint of Kelleher, President of the Board, and to the Council, can be divided into two parts, President Herbert E. Longenecker. The One is the actual offenses with which Dr. telegram in each case was followed by a Dubinsky was charged, The formal charges letter of confirmation, are epitomized in the phrase "willful dis­ The Business Meeting of the Society ruption of regularly scheduled class and in Eugene on August 28, 1969, passed the academic activities," Several brief ver­ following motion: sions of the "facts" have appeared, They do not fully agree, chiefly in the matter of "Be it resolved that this Business emphasis and interpretation, No state­ Meeting of the American Mathematical So­ ment purporting to be a summary of the ciety supports the Council's resolution facts will be given in this article. It is concerning the dismissal of Professor Ed understood that the transcript of the hear­ Dubinsky from Tulane," ing amounts to about eight hundred pages. Even the Decision on the Appeal consists of The Senate position was subsequent­ twenty-four pages; it is already a summary ly stated in a letter from Professor Leo­ and could hardly be further summarized. nard Reissman, Secretary of the Hearing The other part is a procedural mat­ Committee, to Mr. Kelleher, dated Septem­ ter. Here the Council of the Society has ber 10, 1969, and later releasedby Pro­ entered a protest. Specifically, at its fessor Reissman for publication, It reads meeting of August 26, 1969, the Council as follows: passed the following resolution: Dear Mr. Kelleher: "Whereas Edward L. Dubinsky, a "It was with considerable dismay tenured Associate Professor of Mathemat­ that I read the Board's decision to termi­ ics at Tulane University, is being dis­ nate Professor Dubinsky's appointment at missed after a faculty committee investi­ Tulane. Specifically, I was discouraged by gated charges against him at the request two actions of the Board: ( 1) their decision of the President and recommended Pro­ not to refer the case with their recommen­ fessor Dubinsky's retention, but with re­ dations back to the Senate Committee that primand and warning; Therefore be it re­ heard the evidence in the Dubinsky case, solved that the Council of the American and (2) their decision to impose a much Mathematical Society in Eugene, Oregon harsher penalty than the Senate Committee on 26 August 1969 deplores the decision had recommended. of the Board of Administrators of Tulane "In the first instance, the members University to go against the advice of its of the Senate Committee were given to faculty committee and urges reconsidera­ understand that the Board intended to refer tion of this action, and further be it re­ its decision to them for discussion and solved that the Secretary send by telegram comment. Indeed, such was the purpose of a copy of this resolution to the President Professor Oppenheim's work in arranging and Board of Administrators of Tulane for a meeting oftheCommitteeonSeptem­ University." ber 8. Yet, the Board's letter of August 23 requested almost immediate action and the It should be noted that this resolution Board remained deaf to the request (my is concerned only with the way that the letter of August 29) for a delay. I know of

1027 no good purpose that was served by such expectation that their decision will be speedy action, and in fact, I believe that the viewed as "strong," may very well create Board faulted in this instance their other­ the unwanted reaction that the decision was wise careful procedures. As a member of boldly unjust. the Senate Committee, I felt I was being "Obviously, something has been dismissed without the Board's customary learned from the entire matter but differ­ courtesy; receiving no reply to our letter ent things have been learned by different and learning of the decision in the morn­ people and in the coming months we proba­ ing's newspaper. bly will find that those differences are "More importantly, however, the critical for the University." Board's decision to impose the harsher Sincerely yours, penalty on Professor Dubinsky was, I be­ lieve, a mistake. I do not dispute the IS/Leonard Reissman Board's findings as to guilt; necessarily, perhaps, two different courts can conclude differently on the basis of the evidence. The position of the Board of Ad­ (However, I do not subscribe to the Board's ministrators is stated in the reply of Mr. implication that the Faculty committee's Kelleher to the telegram that the Secretary recommendations were somehow less valid was instructed to send. Mr. Ashton Phelps, than those of the Board because we "heard" the vice-chairman of the Board, has the testimony while the Board "read" it.) kindly agreed to the publication of Mr. "The penalty itself, however, is Kelleher's letter. another matter that must be placed in broader context. Compared with the vio­ Dear Mr. Pitcher: lence and take-overs of buildings that have "I acknowledge with thanks your marked other campuses, Professor Dubin­ letter of August 27, 1969, transmitting to sky's actions last spring can hardly be con­ me the substance of the telegram with the sidered in the same light. Yet, the recom­ resolution of the American Mathematical mendations by the Board and President Society concerning Dr. Edward L. Dubin­ Longenecker for Dubinsky's dismissal are sky. I am enclosing herewith for your and extremely severe; in fact, so severe that the Society's reference and information a I am convinced that the purpose of punish­ copy of the decision of the Board of Ad­ ment to discourage such rebellious actions ministrators of the Tulane Educational will work in reverse. The severity of the Fund in the matter of Dr. Edward L. Du­ punishment is more than likely to attract binsky. more followers to the "cause" of Dubinsky "The substance of the resolution than would the firm--but less severe-­ contained in the Society's telegram and recommendation suggested by the Senate letter was considered by the Board of Committee. I expect a visit by the AAUP Administrators at its meeting held on in the matter, and !further expect that they September 4, 19 69. Upon conclusion of the will concur with these views in the light discussion of said resolution, my col­ of other activities on other campuses. leagues on the 8 oard requested that I write "Unfortunately, the Dubinsky case you, confirming that it is the position of will not remain closed, much as we all the Board of Administrators of Tulane Uni­ would wish that it did. Repercussions are versity that its decision, effective Septem­ bound to develop during the coming aca­ ber 4, 1969, terminating Dr. Dubinsky's demic year, and necessarily, the Univer­ employment by Tulane University was sity will become polarized around issues based solely on the Board's considered that are not immediately related to this conclusion that Dr. Dubinsky's conduct case. I suspect that faculty and students constituted a deliberate and calculated dis­ might well interpret the Board's action as ruption of regularly scheduled academic "another example" of unjust control or of functions on the Tulane campus. This con­ a failure to consider them in their deci­ duct on Dr. Dubinsky's part was not only sions. I think that this would be unfortu­ deliberate, but was persisted in by him nate, even as I believe that it will be un­ notwithstanding the warnings and remon- avoidable. The Board's and the President's

1028 strances given him in advance ofthis mis­ "I am sure you will understand that conduct. this unfortunate controversy was a matter "In the view of the Board of Admini­ of great regret and concern to the Board str a tors, Dr. Dubinsky stands adjudged of Administrators, but it is the conviction guilty of misconduct in that he violated the of the Board thatthe maintenance and even­ principle to which Tulane University stands handed enforcement of the established committed that the free exercise of in­ policies is the only alternative to chaos in herent and fundamental constitutional the governance of the affairs of Tulane Uni­ rights carries with it a concomitant stan­ versity. dard of responsibility to respect that "I am, with best wishes, right in others. The principle at issue Sincerely, was succinctly and clearly stated in Is I Harry B. Kelleher" President Longenecker's letter to the Tulane University community of july Z5, 1968, as follows: Harry B. Kelleher 'At Tulane we respect the constitu­ Chairman tional rights of individuals and we believe that these rights are sustained by accep­ tance of the responsibilities that make The action of the Council and of the them possible. Civil disobedience is not Business Meeting are much milder than a accepted as a right. Tulane permits ex­ statement that has been circulating for pression of views by peaceful picketing and signature of individual mathematicians and demonstrations on campus under specific not as a Society action. The statement rules as to where, when and how. However, notes at the bottom that it is intended for we will not tolerate interruption of the Mr. Kelleher, Mr. Longenecker, andPro­ normal processes of the University, such fessor Paul S. Mostert, Chairman of the as academic functions, administrative Department of Mathematics at Tulane. functions, business functions, and recrea­ Its text is as follows: tional activities on campus. We intend to safeguard the right of each studentto par­ "Ed Dubinsky should be restored to take of any University program or service, his Associate Professorship at Tulane Uni­ especially the attendance of classes. No versity. No professor should ever be fired individual or group can be given the right for legitimate political expression. This by demonstration, picketing, sit-in or case is particularly clear-cut in that Pro­ other device to impede regularly scheduled fessor Dubinsky had tenure, and in that the University ft'mctions. We will not permit appropriate faculty committee found there dissent to deprive students of their rights were no grounds for dismissal. If Pro­ to the educational opportunities that fessor Dubinsky is not reinstated, mathe­ brought them here. Should problems arise maticians should boycott Tulane in all which cannot be solved through normal ways. In particular, channels of communication I shall make 1. The American Mathematical So­ myself available for prompt consultation.' ciety should withdraw institutional mem­ "Neither Dr. Dubinsky's profession­ bership from Tulane; al competence nor his political views were Z. Mathematicians should refuse ever at issue at any stage of the disciplin­ employment at Tulane; ary proceedings against him. On the con­ 3. Students should be advised not to trary, the sole issue was whether Dr. study at Tulane; Dubinsky's conduct amounted to disruption 4. Mathematicians should refuse to of regularly scheduled official University give colloquia or seminars at Tulane; functions. As clearly appears in the de­ 5. NSF should be called upon to cision of the Board of Administrators in cancel research grants at Tulane." the Appeal of Edward L. Dubinsky, the Board concluded Dr. Dubinsky's conduct In declining himself to subscribe to did constitute serious disruptive activity the statement, Professor Mostert wrote to in violation of the established policies of Professor Chandler Davis and asked that Tulane University. his reply be published. It reads

1029 Dear Professor Davis: "It would seem, then, unfair to penalize this department for actions by "Thank you for sending me the copies the administration which this department of the resolution signed by (so far) some so stubbornly fought and is still fighting. 69 mathematicians interested in the Dubin­ The battle will go on, and there will be sky case at Tulane. We appreciate the those who will wish to stay and fight. They interest that other mathematicians have will need and deserve the support of the shown in this case and the support that we mathematics community, not the condem­ have received. I would, however, have pre­ nation." ferred a statement that could receive the support of a substantial number of out­ Sincerely yours, standing mathematicians, such as the one /Sl Paul S. Mostert adopted by the Council. I could not sign a document like the one distributed by MAG PaulS. Mostert [Mathematics Action Group] whether Tu­ University Chairman lane or some other university were in­ volved. The issues are not as clearcut as this statement infers. The Faculty Com­ A group of members of the Tulane mittee did find that there were serious Department of Mathematics proposed the violations involved but felt that they were following resolution to the Faculty of the not sufficiently serious to warrant dis­ College of Arts and Science: this time. The mathematicians missal at "The undersigned members [Editor­ this department worked very hard to in ial Note: 17 signatures] of the Department try to convince the administration of the of Mathematics propose the adoption of folly of the particular action that was taken. the following resolution by the F acuity of This included a petition signed by all but Arts and Sciences: three members of the department asking to find some solution less the President "Whereas Edward L. Dubinsky, a than dismissal, a letter to the Board of tenured Associate Professor of Mathe­ Administrators pointing out the possible matics, has been dismissed after the Sen­ on the Department of Mathematics, effects ate Committee on Academic Freedom, second letter from me to the Chairman a Tenure, and Responsibility investigated of the Board, a letter from Gail Young to charges against him at the request of the the Chairman of the Board, a two-hour President and recommended Professor of the President and Deans with meeting Dubinsky's retention, but with reprimand the full professors of the Department and and warning, therefore, be a resolved that similar meeting with the Chairman of a the F acuity of the College of Arts and Board. Gail and I also solicited a the Sciences deplores the decision of the letter from Edwin Moise to the Chairman Board of Administrators to go against the the Board and I worked with faculty of advice of its faculty committee and urges members from other parts of the Univer­ reconsideration of this action." sity trying to effect a compromise solution. the mathematicians of this De­ Finally, The F acuity passed the resolution. introduced a resolution to the partment It is understood that in other parts of the of the College of Arts and Sci­ faculties University F acuity there are contrary ences and Newcomb College (copy en­ views. was adopted by a 74-43 vote closed) which The incident concerning Tulane Uni­ the first and is still pending in the sec­ by versity and Professor Dubinsky raise ond. That resolution contains the signa­ larger issues, pointed up in a letter from of all but three members of the math­ tures Professor Thomas I. Seidman, most of faculty in residence at the time ematics which is quoted: and the votes of two others who arrived in time for the faculty meetings. You will "Two of the resolutions put forward note that it is a very close paraphrase of at the recent Business Meeting of the So­ the resolution adopted by the Council of the ciety [in Eugene on August28,1969]raise, American Mathematical Society. in connection with the specific situation in-

1030 volving Professor Dubinsky at Tulane, an establishing a "strike fund," and so on, issue whose larger implications I should (c) the Society encourage the formation of like to see considered independently: an independent organization for these pur­ Should the American Mathematical Society poses." take on some of the functions of a "union" for mathematicians? In fact, a Committee of the Council "A committee might be appointed to is considering the financial and legal im­ examine and report to the membership on plications of some of these points of view, the relevant considerations involved in to advise the Executive Committee and the such alternatives as (a) the Society ignore Trustees. The collected views will go be­ the whole issue, (b) the Society take on fore the Council in January. such functions as negotiating with and per­ haps invoking sanctions against such or­ Everett Pitcher ganizations as universities, NSF, etc., Secretary

NEWS ITEMS AND ANNOUNCEMENTS

SOCIETY FOR NATURAL PHILOSOPHY and summary should be received by Janu­ ary 15, 1970. Authors will be notified of The Society for Natural Philosophy acceptance by February2,1970.Allmanu­ will have its tenth annual meeting in Ann scripts are to be submitted to Professor Arbor, November 30--December 1, 1969, S. C. Schwartz or Professor J.D. Ullman, on the subject: Optimal Control and the Cal­ Engineering Quadrangle, Princeton Uni­ culus of Variations. The meeting will be versity, Princeton, New Jersey 08540. held in the Chrysler Center of the North Campus of the University of Michigan.

FOURTH ANNUAL THE COLLECTED WORKS OF PRINCETON CONFERENCE ON NORBERT WIENER INFORMATION SCIENCES AND SYSTEMS March 26-27, 1970 The MIT Press is in the process of publishing the Collected Works ofNorbert Authors are invited to submit pa­ Wiener. They will form the first of a series pers describing new advances, applications of publications of the collected works of and ideas in the areas of computer science, eminent contemporary mathematicians un­ communications, control systems, and cir­ der the general direction of Dr. Gian­ cuits. Two kinds of papers are solicited. Carlo Rota. The first will be standard papers requiring The publishers earnestly solicit approximately thirty minutes for presenta­ assistance from the mathematical com­ tion; these will be reproduced in full in munity by way of information on Wiener's the conference Proceedings. The second works, such as titles of papers not listed will be short papers suitable for presen­ in the Bibliography which appeared in the tation in 10-15 minutes; 250 word sum­ Wiener Memorial Volume of the Bun: in aries will be published in the Proceed­ Amer. Math. Soc. 72 (1966), no. 1, partll, ings. 1-145; errata within papers; and other For regular papers, a title, 50 word comments. Such information should please abstract and summary are to be submitted be mailed as soon as possible to Professor before January 2, 1970.Summaries should P. R. Masani, Department of Mathematics, be of sufficient detail and length to permit Indiana University, Bloomington, Indiana careful reviewing. For short papers, a title 47401, who is the editor of these works.

1031 PERSONAL ITEMS

Professor F. G. ASEN JO of the Uni­ Professor DALE W. LICK of the Uni­ versity of Pittsburgh has been awarded a versity of Tennessee has been appointed to Fulbright Faculty Fellowship and will be at an associate professorship at the Drexel the University of Lisbon, Portugal, for the Institute of Technology. second semester of the academic year 1969- Professor BERTRAM MONO of the La 1970. Trobe University, Melbourne, Australia, Dr. MICHAEL BRAND of Purdue Uni­ has been appointed Editor of the Journal of versity has been appointed to an assistant the Australian Mathematical Society. professorship at Oakland University. Professor RICHARD A. MOORE of Professor T. F. BRIDGLAND, JR. of Carnegie-Mellon University has been a­ Florida State University has been appointed warded the 1969 William H. and Frances S. to a professorship at the Drexel Institute Ryan Award for Meritorious Teaching. of Technology. Dr. CHRIS RORRES of the Courant Professor ROBERT C. BUSBY of Oak­ Institute of Mathematical Sciences, New land University has been appointed to an York University has been appointed to an assistant professorship at the Drexel In­ assistant professorship at the Drexel In­ stitute of Technology. stitute of Technology. Dr. J. CURTIS CHIPMAN of Dart­ Mr. WILLIAM A. VEECH of the Institute mouth College has been appointed to an for Advanced Study has been appointed to assistant professorship at Oakland Uni­ an associate professorship at Rice Uni­ versity. versity. Mrs. ELAINE C. ENSIGN of James­ PROMOTIONS town College has been appointed a Lec­ To Professor. Rice University: B. F. turer at the University of Alaska. JONES, JR. Professor HERMAN E. GOLLWITZER of the University of Tenessee has been ap­ To Associate Professor. Rice Univer­ pointed to an assistant professorship at sity: J. P. HEMPEL; R. 0. WELLS, JR. the Drexel Institute of Technology. Dr. WILLIAM C. HOFFMAN of Oregon INSTRUCTORSHIPS State University has been appointed to an assistant professorship at Oakland Univer­ Oakland University: S. T AKIF F. sity. DEATHS Dr. ROBERT W. JOHNSON of Bowdoin College has been appointed to an assistant Professor SHU-TEH C. MOY of Santa professorship at Lehigh University. Barbara, California, died on August 12, Dr. BARBARA A. LANDO has been ap­ 1969, at the age of 49. She was a member pointed to an assistant professorship at of the Society for 20 years. the University of Alaska. Mr. T. 0. WALTON of Kalamazoo, Dr. CLIFTON A. LANDO has been ap­ Michigan, died on November 11, 1969, at pointed to an assistant professorship at the age of 78. He was a member of the the University of Alaska. Society for 47 years.

1032 NEW AMS PUBLICATIONS

MEMOIRS OF THE AMERICAN in some sense and to demonstrate limi­ MATHEMATICAL SOCIETY tations of the general integral. A few open questions are mentioned. Number 83 REPRESENTATION OF RINGS BY SEC­ TIONS By J. Dauns and K. H. Hofmann Number 87 192 pages; List Price $2.20; Member Price $1.65 STUDIES IN ABSTRACT FAMILIES OF LANGUAGES The theory of representing rings By Ginsburg, Greibach and Hopcroft (or other algebraic structures) as rings (or corresponding substructures) of sec­ 56 pages; List Price $1.60; Member Price tions in sheaves has received increasing $1.20. attention in recent years. In this Memoir The Memoir consists of three papers the authors introduce a general theory in devoted to the study of families of languages which sheaves are replaced by more gen­ accepted by arbitrary families of one-way eral structures especially suited for the nondeterministic acceptors. These families representation of topological rings by con­ of languages, called AFL, provide a unifying tinuous sections. These structures (called structure for results in automata and formal fields) share with bundles the property of language theory. AF L are characterized as having non-discrete stalks and with the families of languages closed under (i) sheaves the property of allowing continu­ union, (ii) concatenation, (iii) Kleene clo­ ously variable stalks and having an abun­ sure, (iv) intersection with regular sets, dance of local sections. (v) nondecreasing homomorphism, and (vi) As principal applications the au­ inverse homomorphism. The independence thors prove a Gelfand-Nai:mark Theorem of these six operations is examined. Various for non-commutative C*-algebras andre­ properties of AFL are determined. Finally, presentation theorems for certain classes certain families of languages, called pre­ of discrete rings including biregular and AF L, which become AF L when closed under some types of function rings. nondecreasing homomorphisms, are con­ sidered. Number 85 DENJOY INTEGRATION IN ABSTRACT SPACES Number 88 By D. W. Solomon A RIEMANN-TYPE INTEGRAL THATIN­ 72 pages; List Price $1.90; Member CLUDES LEBESGUE-STIELT JES, BOCH­ Price $1.43 NER AND STOCHASTIC INTEGRALS This paper is concerned with the By E. J. McShane definition and properties of vector-valued 56 pages; List Price $1.50; Member Denjoy-Perron type integrals on the funda­ Price $1.13 mental sets of a Romanovski space (cf. c}/otiai) of the A.M.S., 13 (1965), Abstract The Riemann integral is defined by 630-45, p. 69). General constructive and first using finite sums to define a func­ descriptive methods of defining such inte­ tional of partitions of the domain of inte­ grals are presented. Several special cases gration and then taking the limit of that are studied. Examples are given to show functional under a familiar limit process. that results presented are best possible By isolating the essential elements of the

1033 processes of forming the finite sums and 168 pages; List Price $2.20; Member Price passing to the limit, a type of abstract $1.65. Riemann integral is formed that includes Building on algebraic foundations among its special cases the Riemann, laid by Frobenius, G. W. Mackey in the Riemann-Stieltjes, Lebesgue and Lebes­ 1950's developed a powerful method for gue-Stieltjes integrals, the extension of studying the unitary representations of the latter to locally compact spaces with a separable locally compact group extensions. (finitely-additive content, the Bochner Using techniques of Loomis, Glimm, and integral and Bogdanowicz' s generalization Blattner, one can to a large extent remove of it, Henstock's "Riemann-complete" in­ tegral, the author's "belated" stochastic the assumption of separability. In the pre­ integral, and Ito's stochastic integral. sent Memoir, this nonseparable version of Mackey's method is developed in the more general context of Banach *-algebraic bun­ Number 89 dles. These include locally compact group FORMALIZED RECURSIVE FUNCTION­ extensions as a special case. They also in­ ALS AND FORMALIZED REALIZABILITY clude quite different kinds of objects, such By S.C. Kleene as the "convariance algebras" which have recently found application in quantum phy­ 108 pages; List Price $2.60; Member sics. In Part I of this Memoir, we present Price $1.95 the general properties of Banach *-alge­ In Part I, the theory of general and braic bundles and prove the Imprimitivity partial recursive functions of type 0 Theory for them, and develop the Mackey­ (natural number) and type 1 (one-place Blattner analysis of their representations. number-theoretic function) variables is formalized in the author's basic formal Number 92 system for intuitionistic and classical DOUBLY TIME LIKE SURF ACES analysis (Chapter I of the 1965 monograph By J. K. Beem and P. Woo of Kleene and Vesley). The- treatment in­ cludes the informal theory, in a version 116 pages; List Price $1.90; Member Price chosen as a result of experimentation $1.45~ (intermittently since 1959), to make the This Memoir gives a synthetic purely formalization as manageable as possible. geometric treatment of two dimensional in­ The version starts from ~-recursiveness, definite metric spaces. A doubly timelike and utilizes a representation of any finite surface is a two dimensional general G­ selection of values of a partial function space having exactly two null directions at argument. The formalization is intended to each point. Locally there are two partial provide basic machinery for investigations orderings determined and the timelike ine­ such as those in Part II, where two for­ quality holds for each particle ordering. malized notions of realizability for intui­ General G-spaces are the extension of tionistic analysis are used to establish Busemann's G-spaces to the indefinite case. some conjectures of the author ( 1964, Consequently, many of the methods and re­ 1965, 1967); e.g. that in intuitionistic analy­ sults are similar to those of ordinary G­ sis, if a closed formula vx 3yA(x,y) is spaces. The minkowski spaces are a natural provable, then Vx 3y[T1 (e,x,y) & A(x,U(y))] extension of perpendicularity in pseudo­ is provable for some numeral e, where euclidean spaces. A synthetic theory of dif­ T1 and U formalize the T1 and U of the ferentiability is developed and used in author's normal form theorem (in the characterizing the spaces which are locally version used in Part I). exterior hyperbolic.

Number 93 Number 90 HITTING PROBABILITIES OF SINGLE AN EXTENSION OF MACKEY'S METHOD POINTS FOR PROCESSES WITH STATION­ TO BANACH *-ALGEBRAIC BUNDLES ARYINDEPENDENTINCREMENTS By J. M. G. Fell By Harry Kesten

1034 132 pages; List Price $2.10; MemberPrice Semigroups given by identities with $1.58, distinguished elements by A. P. Birjukov; Asymptotic behavior and ergodic proper­ It is investigated what processes ties of solutions of the generalized Hardy­ {Xt }, t ;;;; 0, with stationary independent in­ Littlewood equation by B. M. Bredihin and crements hit points with positive probability, Ju. V. Linnik; Finite groups whoseproper i.e. when h(r) = P{Xt = r or Xt- = r for some subgroups all admit nilpotent partitions t > OJ is strictly positive. For various by V. M. Busarkin and A. I. Starostin; On special cases (all of them stable processes) systems of congruences by A. A. Karacuba; this was determined by Levy, Erdos, Kac, A functional equation for Dirichlet L­ Port and Stone. Here it is determined for a series and the problem of divisors in general one-dimen_sional right continuous arithmetic progressions by A. F. Lavrik; process with stationary independent incre­ On the representation of numbers by ments when h(r) > 0 and when h(r) = 0, It positive binary diagonal quadratic forms turns out that for most processes either by G. A. Lomadze; Identical relations on h(r) > 0 for all r or h(r) = 0 for all r. For varieties of quasi-groups by A. I. Mal'cev; "honestly" higher dimensional processes Some finiteness conditions in the theory of one always has h(r) 0, The one dimen­ = semigroups by L. N. Sevrin; Upper bounds sional result is used to answer a question and numerical calculation of the number of of Chung's, namely to show that one can ideal classes of real quadratic fields by find a Borel measure W on [O,oo) which I. S. Slavutski~; Sets of nonnegative integers satisfied Joss:srO'(r - s) W (ds) = 1 for .i!ll not containing an arithmetic progression r > o whenever o-( s) ,1. o as s ..... oo, a right of length p by Ju, T. Tka~enko; Groups continuous and J~o-(s) ds < oo. with a normalizer condition for closed sub­ groups by V. I. Usakov; On a theorem of infinite dimensionality of an associative algebra by E. B. Vinberg; On extension to the left halfplane of the scalar product by TRANSLA liONS-SERIES II Heeke L-series with magnitude characters by A. I. Vinogradov; On the density conjec­ ture for Dirichlet L- series by A. I. Vino­ Volume 81 gradov; An estimate for a certain sum FOUR PAPERS ON FUNCTIONS OF REAL extended over the primes of an arithmeti­ VARIABLES cal progression by I. M. Vinogradov; In­ variant measures on Boolean algebras 284 pages; List Price $14.20; Member by D. A. Vladimirov. Price $10,65

On equivalent norms for fractional spaces by K. K. Golovkin; Some inequali­ Volume 83 ties in function spaces and their applica­ tion to the study of the convergence of EIGHTEEN PAPERS ON LOGIC AND variational processes of V. P. 11' in; On THEORY OF FUNCTIONS some properties of differentiable func­ 288 pages; List Price $12.80; Member tions of several variables by V. P. ll'in Price $9.60, and V. A. Solonnikov; The properties of some classes of differentiable functions A uniqueness theorem for harmonic of several variables in an n-dimensional functions in a halfspace by I. S. ArS'on and region by V. P. 11 'in. M. A. Pak; On the reduction to single­ valued form of certain functions analytic in domains of finite connectivity by L. E. Volume 82 DunduC:enko; On approximation of functions SIXTEEN PAPERS ON NUMBER THEORY continuous on Jordan arcs by L .I. Kolesnik; AND ALGEBRA Some problems in the structure of measur­ able functions by S. G. Kozlovcev; On dif­ 264 pages; List Price $13,40; Member ferential properties of measurable func­ Price $10,05 tions by S. G. Kozlovcev; Estimates from

1035 below for entire functions of finite order by COllOQUIUM PUBLICATIONS I. F. Krasi~kov; Some problems of the theory of classes of models by A. I. Mal' cev; On the decrease of functions analytic in a Volume 39 halfplane by Ju. I. Masljukov; On reduction STRUCTURE AND REPRESENTATIONS OF of the decision problem of recursively en­ JORDAN ALGEBRAS umerable sets to the separability problem By Nathan Jacobson by A. A. Mu~nik; Boundary properties of functions defined on a region with angular 455 pages; List Price $10.80; Member points. I, II and III by S.M. Nikol'skii; Ex­ Price $8.10. tension of functions of several variables The purpose of this book is to give a preserving differential properties by S. M. comprehensive account of the structure and Nikol 'skil; On best approximation of func­ representation theory of Jordan algebras tions of class Zzk by certain linear poly­ over a field of characteristic not two. The nomial operators by I. M. Petrov; Extremal author indicates the limits he has set him­ properties of certain classes of univalent self and what lies immediatelybeyondthese functions by V. A. Pohilevil::; On simul­ limits. In the first place, a substantial part taneous orthogonality of functions on a do­ of the theory carries over to algebras over main and on its boundary by E. A. Sinev; On a commutative ring with 1 containing an ele­ some extremal properties of r regular func­ ment ~ such that ~ + ~ = 1. A reader who tions bounded in the disk lz <. 1 by V. A. I has need of a result of this generality will Turkovski1; Some remarks on the Baire have no difficulty in ascertaining whether or property. I by Yang Zong-pan (Yang Tsung­ not the corresponding result in the field case P'an). carries over. A more serious and significant extension of the theory, which can now be made, is the passage from a linear theory to a quadratic one. An important step in the transition to a quadratic theory was taken by the author in a paper which appeared in Volume 84 1966, in which he gave an Artinian-like structure theory for Jordan algebras founded TWELVE PAPERS ON ALGEBRA, ALGE­ on axioms on quadratic ideals. This theory BRAIC GEOMETRY AND TOPOLOGY is considered in detail in Chapter IV. A very 280 pages; List Price $14.00; Member important area of applications of the Jordan Price $10.50. theory, especially of exceptional Jordan algebras, is to exceptional Lie groups and Finiteness conditions in the general algebras and related geometries. This is theory of groups by S. N. Cernikov; Some discussed in part in Chapter IX. Indications problems of Burnside type by E. S. Golod; of additional results and the or-iginal papers On polynomials with small prime factors. containing these are given in the notes at II by V. I. Hmyrova; On birational forms of the end of the book entitled" Further Results rational surfaces by V. A. Iskovskih; Topo­ and Open Questions." logical imbeddings in a manifold and pseudo­ isotopy by L. Keldy~; Rational surfaces over perfect fields. I by Ju. I. Manin; Invariant measure of a compact ring group by V. G. TRANSACTIONS OF THE Paljutkin; A generalization of the Jacobian MOSCOW MATHEMATICAL SOCIETY variety by A. N. Par~in; On the dimension of increments by bicompact extensions of Volume 17 proximity spaces and topological spaces. I and II by Ju. M. Smirnov; On multipli­ 388 pages; List Price $21. 70; Member Price $16.28. cative and spectral properties of spatial matrices with nonnegative elements by Existence of a phase transition for N. P. Sokolov; Measurable realization of the lattice gas with interpa~ticle attraction, continuous automorphism groups of a uni­ F. A. Berezin and Ja. G. Sind; Some new tary ring by A. M'. Ver~ik. results on first order phase transitions in

1036 ~attice gas models, R. A. Minlos and ja. G. The Milne problem consists of deter­ Sinai; On an integral transform and its ap­ mining the diffusion of monochromatic ra­ plication to the estimation of the number diation through a material layer of semi­ of eigenvalues of certain integral operators, finite thickness. The present monograph M. A. Evgrafov; Some concepts and applica­ presents the solutions of the relevant inte­ tions of nonabelian cohomology theory, gral equations under much more general A. L. Oniscik; Scales of Banach structures laws of scattering than the isotropic case ••i measurable functions, S. G. Krein, ju. I. extensively studied up to now; particular Petunin, and E. M. Semonov; Expansion in attention is given to the asymptotic proper­ eigenfunctions of the Laplace operator on ties of the solutions at large optical depths. the fundamental domain of a discrete group on the Labacevski! plane, L. D. Faddeev; Representation of operators by means of functions, F. A. Berezin; Free T-sums of multioperator fields, I. S. Ivanov; Some existence conditions for K-decompositions for special flows; B. M. GureviC':; Thermo­ PROCEEDINGS OF SYMPOSIA dynamic limit for entropy, R. A. Minlos IN PURE MATHEMATICS and A. ja. Povzner. Volume 12 Volume 18 NUMBER THEORY 296 pages; List Price $19.80; Member Edited by William j. LeVeque and Ernst Price $14.85. G. Straus Discretev subgroups of Lie groups, 112 pages; List Price $5.60; Member I. I. P jateckiT-Sapiro; Integral limit theo­ Price $4.20 rems for certain classes of additive arith­ metic functions, V. V. Levin and A. S. At the seventy-third annual meet­ F ainleib; Algebraic independence of the ing of the American Mathematical Society, values of certain hypergeometric E-func­ Houston, Texas, january 24-28, 1967, a tions, A. B. Sidlovski1; An elliptic analog Special Session on recent advances in the of an inequality of A. 0. Gel 'fond, N. I. theory of numbers was held. The Program Fel'dman; On an asymptotic formula, A. A. Committee for the Special Session con­ Karacuba; On extremal problem, M. B. sisted of E. G. Straus and W. j. LeVeque. Bar ban and P. P. Ulhov; An approximate The articles in this volume comprise all equation for the Dirichlet L-function, A. F. but one of the papers presented. All of the Lavrik; An analog of the canonical product talks were of 20 minutes duration except for entire functions of several complex that of Professor H. M. Stark, which was variables, L. I. Ronkin; On the special an hour address. In several cases, what theory of nonautonomous linear systems of was only sketched in the talk, because of differential equations, V. M. Millionscikov; lack of time, is presented in this volume Extension of variational problems, A. D. in detail. Ioffe and V. M. Tihomirov; Integration of continuous bounded functionals on a function space, L. S. Grinblat. TRANSLATIONS OF MATHEMATICAL MONOGRAPHS PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Volume 25 Number 97 MAHLER'S PROBLEM IN METRIC NUM­ THE MILNE PROBLEM WITH ANISTROPIC BER THEORY SCATTERING By V. G. Sprindluk Edited by M. V. Maslennikov 200 pages; List Price $12.70; Member 168 pages; List Price $11.40; Member Price Price $9.52 $8.55. This book deals with the solution of

1037 a group of questions related both to the Volume 26 general theory of transcendental numbers GEOMETRIC THEORY OF FUNCTIONS and to the metrical theory of diophantine OF A COMPLEX VARIABLE (and also algebraic) approximations. The By G. M. Goluzin in this field has been fundamental problem 684 pages; List Price $35.50; Member known in the literature since 19 32 as Price $26.62 Mahler's conjecture. The main result of this book is a proof of Mahler's conjecture The first edition of Goluzin' s mono­ and some analogous theorems. graph was published in 19 52, shortly after In Part I, the "Classical" case of the author's death. In the last decade, an Mahler's conjecture, dealing with real and extensive literature has appeared on complex numbers, is considered. This part themes closely related to the content of should be comprehensible to anyone who this monograph, and many of these results knows the elements of measure theory and were obtained in the works of Goluzin' s possesses sufficient perseverance in over­ pupils. A survey of this literature is given coming purely logical difficulties. Part II in a special supplement. is concerned with locally compact fields The text of the book has undergone with nonarchimedean valuation. This part only slight modifications. Three biblio­ requires a general familiarity with the graphic lists have been added. One of these structure of fields with nonarchimedean corresponds to references made in the valuation. All the necessary information main text, another to the supplement. In is given in the text with references to the addition, a complete list is given of sources. Goluzin's works.

PROCEEDINGS OF THE SYMPOSIUM ON FINITE GROUPS

94 pages; List Price $2.20; Member Journal of Mathematics from whose March Price $1.65 1969 issue it is reprinted. The book pre­ sents a composite of six lectures given at Proceedings of the Symposium on the symposium of Finite Groups held at the Theory of Finite Groups has been pub­ Urbana on November 24, 1967, and spon­ lished by the American Mathematical So­ sored by the University of Illinois in con­ ciety with the cooperation of the Illinois junction with their centennial celebration.

1038 MEMORANDA TO MEMBERS 1969 SUMMER INSTITUTE ON NUMBER THEORY: ANALYTIC NUMBER THEORY, DIOPHANTINE PROBLEMS, AND ALGEBRAIC NUMBER THEORY

The sixteenth summer institute of lems. There were 16 such series, ranging the American Mathematical Society was from 2 to 14 lectures, and the lecturers held from July 7 to August l, 1969, at the were James Ax, Alan Baker, B. J. Birch, State University of New York at Stony Kenkichi Iwasawa, Yukiyosi Kawada, Tomio Brook. The subject of the institute was Kubota, Kurt Mahler, Albrecht Pfister, selected by the Committee on Summer H. E. Richert, Julia Robinson, Andrzej Institutes consisting of Jim Douglas, Jr., Schinzel, Atle Selberg, Wolfgang Schmidt, Murray Gerstenhaber, I. N. Herstein, Louis H. M. Stark, H. P. F. Swinnerton-Dyer, N_irenberg, I. M. Singer (chairman), and Paul Turan, and Eduard Wirsing. In addi­ E. H. Spanier. The Organizing Committee tion to the lectures, a program of seminars was composed of James Ax, Paul T. Bate­ was arranged at which the individual parti­ man, Kenkichi Iwasawa, D. J. Lewis (chair­ cipants reported on their research. The man), and Atle Selberg. lecture notes and certain of the reports on Since the last number theory institute seminars will be published by the Society a decade ago, a large number of old prob­ in a volume in the series, Proceedings of lems in number theory have been solved, Symposia in Pure Mathematics. some by refinements of old methods, others The institute was attended by 139 by the introduction of entirely new methods. mathematicians, who were accompanied by One of the prime purposes of the meeting 51 wives or husbands and 54 children. Of was to acquaint the participants from the the participants, 23 were from foreign various areas of number theory with the countries. important results and methods developed The institute was supported finan­ in recent years in other areas. To achieve cially by the National Science Foundation this end, the Organizing Committee invited and the Research Foundation of State Uni­ a number of mathematicians to give a series versity of New York. of lectures on particular areas and prob-

ABSTRACTS

A new form for presenting abstracts All abstracts for future meetings to the Society was approved at the Council should be submitted on the new form. Meeting in August at Eugene, Oregon. The Please destroy al!_ old abstract !orms. principal change consists of an enlarged Copies of the new form may be obtained subject classification. Authors will be re­ by writing to the Editorial Department, quired from now on to classify abstracts by American Mathematical Society, P. 0. Box subject. 6248, Providence, Rhode Island 02904. The printed abstracts in the cNoficei) It would be appreciated if authors will be grouped by the author's classifica­ would check abstracts very carefully before tion for both those presented in person sending them toP rovidence. There is so lit­ and those presented by title; this will take tle time between receipt of manuscript and effect as early in 1970 as is practical. The sending of reproduction proof to the printer Associate Secretaries may find the clas­ that the Providence office cannot be re­ sification useful, but they cannot promise sponsible for omissions (a common occur­ to follow it in grouping papers on the rence), illegible handwriting, or question­ program. able notation.

1039 COMPUTING AND MATHEMATICS

The winter meetingin San Antonio G. E. Forsythe Stanford University will hear a panel discussion on Computer W. Givens Argonne National Science and Mathematics. The panel is Laboratories sponsored by the Conference Board of H. Greenberg University of Denver the Mathematical Sciences. The session R. M. Thrall Rice University is being planned by a committee consisting R. F. Traub, Chairman of: Bell Telephone Laboratories

ADDITIONAL AUDIO RECORDING OF MATHEMATICAL LECTURES AVAILABLE

As announced in the October issue sity,· a Colloquium lecture delivered at of c#otit:ei), the Society is issuing a series the August 1969 meeting in Eugene, Oregon. of taped lectures entitled Audio Recordings The Audio Recordings of Mathe­ of Mathematical Lectures. This series in­ matical Lectures may be purchased for $6, cludes tape recordings, With accompanying and additional copies of the manual may manual, of Gibbs Lectures, Colloquium be ordered for $0.30 each. Standing orders Lectures, and invited hour addresses. In for the entire series of lectures may be addition to the four listed in last rrionth' s placed. There will be approximately 37 issue, No. 5 is now ready for distribution. lectures recorded during the coming year. It is "On the periodicity theorem for the Orders should be sent to the American classical groups and some of its appli­ Mathematical Society, P .0. Box 6248, cations" by Raoul Bott of Harvard Univer- Providence, Rhode Island 02904.

' 2EZQ

ACTIVITIES OF OTHER ASSOCIATIONS

ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC

The Annual Meeting of the Associa­ tures as well as meetings for contributed tion for Symbolic Logic will take place on papers. December 27-28, 1969, at the Waldorf­ Professor Jon Barwise, Department Astoria in New York City in conjunction of Mathematics, Yale University, New with a meeting of the Eastern Division of Haven, Connecticut 06520, U.S.A. is Pro­ the American Philosophical Association. gram Chairman. There will be three invited one-hour lee-

1040 VISITING MATHEMATICIANS

SUPPLEMENTARY LIST

The list of visiting mathematicians is being expanded this year to include both foreign mathemati• visiting in the United States and Canada, and Americans visiting abroad. Note that there are two separate li1

FOREIGN MATHEMATICIANS VISITING IN THE UNITED STATES AND CANADA

Name and Home Countr:l': Host Institution Field of SEecial Interest Period of Vi! Barratt, Michael G. (England) Northwestern University Algebraic Topology 9/69-6/70 Basu, Debabrata (India) University of New Mexico Statistics 9/68-2/70 Bombiere, Enrico (Italy) New York University, Computer Science 2/70-6/70 Courant Institute Ching, Wai-Mee (China) New York University, Functional analysis 9/69-6/70 Courant Institute Christiansen, Peter (Denmark) New York University, Diffraction Theory 9/69-6/70 Courant Institute Currie, Peter (England) Simon Fraser University Continuum Mechanics 9/69-8/70 Dubuc, Eduardo J. (Argentina) University of Illinois Category Theory 1/70-6/70 Enquist, Bjorn E. (Sweden) New York University, Computer Science 9/69-6/70 Courant Institute Farooqui, A.S. (India) Simon Fraser University Dynamic Plasticity 1/69-8/70 Fujiwara, Daisuke (Japan) New York University Partial differential 9/69-6/70 Courant Institute equations Griffiths, David F. (Wales, U.K.) New York University Numerical Analysis 9/69-6/70 Courant Institute G runbaum, Branko (Israel) Michigan State University Geometry, Convexity 9/69-8/70 Jager, Willi (Germany) New York University, Analysis 9/69-6/70 Courant Institute Jarvis, RichardJ. (Wales,U.K.) New York University, Water Waves 9/69-6/70 Courant Institute Kaneyuki, Soji (Japan) Washington University Differential Geometry 9/69-9/70 Kolodner, Ignace Izaak (Poland) New York University, 9/69-2/70 Courant Institute Kucera, Jan E. (Czechoslovakia) University of New Mexico Theory of Distribution 9/69-6/70 Leviatan, Dany (Israel) University of Illinois Summability 9/69-8/70 Liron, Nadav (Israel) New York University, Computer Science 9/69-6/70 Courant Institute Marik, Jan (Czechoslovakia) Michigan State University Real Analysis 9/69-8/70 Marti, Jurg T. (Switzerland) University of lllinois Functional Analysis 9/69-8/70 McMullen, Peter (England) Michigan State University Geometry, Convexity 9/69-8/70 Meinardus, Gunter (Germany) Michigan State University Approximation Theory 9/69-12/69 Meyer, Wolfgang (Germany) SUNY at Stony Brook Riemannian Geometry 9/69-8/70 Mimura, Mamoru (Japan) Michigan State University Topology 9/69-8/70 Miyake, Katsuya (Japan) New York University, Algebraic Geometry 9/69-6/70 Courant Institute N'Uesch, Peter E. (Switzerland) Johns Hopkins University Statistics 9/68-5/70 Olivier, Reinhard M; (Germany) New York University, Differential Geometry 9/69-6/70 Courant Institute and Topology

1041 Name and Home Countr;r Host Institution Field of SEecial Interest Period of Visit Quinn, FrankS. (Cuba) New York University Topology 9/69-6-70 Courant Institute Robinson, Peter D. (England) University of Wisconsin, Applications in 9/69-1/70 Mathematics Research Center Quantum Mechanics Rojas, Basilio (Mexico) Iowa State University Probability and Statistics 1/70-12/71 Rottenberg, Reuven (Israel) Michigan State University Geometry 9/69-8/70 Sarndal, Carl-Eric (Sweden) New York University, Statistics 9/69-6/70 Courant Institute Shephard, Geoffrey C. (England) Michigan State University Geometry, Convexity 9/69-8/70 Sirao, Tunekiti (Japan) New York University Probability 9/69-6/70 Courant Institute Som, M.M. (India) Simon Fraser University Relativity 11/68-12/69 Soward, Andrew (England) New York University, 9/69-6/70 Courant Institute Srivastava, Hari Mohan (India) University of Montana Classified Applied 9/69-6-70 Mathematics Takeuchi, Kei (Japan) New York University, Statistics 9/69-6/70 Courant Institute Toda, Hirosi (Japan) Northwestern University Algebraic Topology 9/69-6/70 Villegas, Cesareo (Uruguay) University of Rochester Statistics 9/68-9/70 Viswanath, Kasturi (India) University of Illinois Hilbert Spaces 9/69-6/70 Wood, Geoffrey V. (England) University o{ Illinois Functional Analysis 9/69-6/70 Zaks, Abraham (Israel) Northwestern University Algebra 9/69-6/70 Zwas, Gideon (Israel) New York University, Numerical Analysis 9/69-6/70 Courant Institute

AMERICAN MATHEMATICIANS VISITING ABROAD

Blum, J.R. (U.S.A.) Israel Institute of Technology Ergodic Theory 9/69-9/70 D'Alarcao, Hugo (U.S.A.) University of Chile Semigroups 7/69-1/70 DeMarr, Ralph (U.S.A.) University of Lenigrad, Functional Analysis 9/69-7/70 Russia Griego, Richard (U.S.A.) Cer.tro de Investigacion y Probabilistic and 9/69-6/70 de Estudios, Mexico Potential Theory Mohr, Richard (U.S.A.) University of Vienna Mathematical Statistics, 9/69-6/70 Philosophy of Science Wendroff, Burton (U.S.A.) Uppsala University, Numerical Analysis 9/69-6/70 Sweden

1042 ABSTRACTS OF CONTRIBUTED PAPERS

The November Meeting in Baton Rouge, Louisiana November 21-22, 1969

669-1. MICHAEL W. MISLOVE, University of Tennessee, Knoxville, Tennessee 37916. The existence of Irr(X).

Definition. Let S be a compact connected semigroup with identity. S is said to be irreducible if there is no compact connected subsemigroup of S which contains the identity of S and meets the minimal ideal of S. Theorem. Let X be a compact totally ordered space. There is an irreducible semigroup Irr(X) with idempotents X which satisfies: If S is any irreducible semigroup with idem­ potents X, there is an idempotent separating surmorphism of Irr(X) onto S. Moreover, the Clifford­ Miller endomorphism of Irr(X) is an injection when restricted to any 1{-class. Hofmann and Mostert attempted to construct Irr(X) in "Elements of compact semigroups," Charles Merrill, {1966), but there were errors in the proof. Ours is an existence proof, and it is at best questionable that an explicit description of Irr(X) can be found. (Received July 28, 1969.)

669-2. ROBERT GILMER, Florida State University, Tallahassee, Florida 32306. R-automorphisms of R[[X]].

If R is a commutative ring with identity, an endomorphism of the power series ring R[[x]] which is the identity on R is called an R -endomorphism of R [[X]] . If rp is an R- automorphism of oo · oo n R ([X)) such that tp{X) = Li=ObiX1 , where n n=lbOR = (0), then O'Malley (see Abstract 68T-l2l, these cJVoticri) l 5 ( 1968), 22 7) has shown that R is complete in the (b J- adic topology and that If' is uniquely determined by tp{X). O'Malley leaves open the question as to whethern : 1 b~R must be zero if there exists an R- automorphism If' of R [[X) J such that ~(X) has constant term b0 . Theorem l. If P is a fixed element of R [(X] J, these conditions are equivalent. (l) There is an R·automorphism of R[[XJ] mapping X onto p. {2) The mapping r ~ r + ({3) of R into R([X)]/(~) is bijective. (3) R [[x]] = R ~ (,). Theorem 2. Suppose that there is an R-automorphism of R [[x]J mapping X onto ~ = :E g::\Xi. Then these conditions are equivalent. {l) n : 1bgR = (0). (2) ({J) is closed in the (b0 )-adic topology on R[[X]J. (3) (~)is closed in the (X)-adic topology on R ([X]). Theorem 3. If S is a ring containing an elements such that s n : 1snS # n :l sns, then the ring R = S [[Y)] /(s - Y) is such that R [ [x]) admits an R-automorphism sending X onto an element with constant term r 0 , where n :{~R # (0). (Received August 4, 1969.)

669-3. JAMES M. BOYTE, Virginia Polytechnic Institute, Blacksburg, Virginia 24060. Countable subs paces. Preliminary report.

Theorem 1. If (X, T) is a first countable topological space such that each countable subspace is T 3 then (X, T) is T 3. Corollary l. Let (X, T) be a separable space. Then each countable subspace is metrizable if and only if (X, T) is first countable and T 3" Theorem 2. If Q is a property such that

1043 every metric space satisfies Q and Q implies T 3 , then the following for a given topological space (X, T) are equivalent. (l) If (X, T) is first countable and T 3 then (X, T) has property Q. (2) If (X, T) is first countable then (X, T) has property Q if and only if each countable subspace has property Q.

Corollary 2. Every first countable, separable, T 3 space has for each pair of disjoint countable separated sets A and B disjoint open sets V(A) and W(B) containing A and B respectively.

Remark. If (R, T) is the half open interval topology on the reals then the product topology R X R is not normal but the countable product topology of R satisfies the property of Corollary 2.

Corollary 3. If (X,T) is first countable, hereditary Lindelof, and each countable subspace is T 3 then (X,T) is T 5 . (Received August 12, 1969.)

669-4. GUENTER K. HAEUSLEIN, Oak Ridge National Laboratory, Oak Ridge, Tennessee 3 7831. On the algebraic independence of symmetric polynomials.

Let h 1, ... ,hn be homogeneous symmetric polynomials in x 1, ... ,xn with complex coefficients;

hk is of degree k. Theorem. h 1, ... ,hn are algebraically independent if hk(wk' w~ •...• c{. 0, ... , 0) i 0 fork= 1, ...• n, where b.k is a primitive kth root of unity. Application. Let P 2k = (.tx, ±x ,± ... ±x )2 k, k = 1, ...• n. Theorem P P P are algebrai·cally 1"ndependent This ~ ± 2 n • 2' 4•···• 2n • solves a problem of L. Flatto (Bull. Amer. Math. Soc. 74 (1968), 730-734). (Received August 22, 1969.)

669-5. CHARLES E. AULL, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. Notes on separation by continuous functions.

Referring to the separation axioms of Van Est and Freudenthal [w. T. Van Est and

H. Freudenthal, "Trennung durch Stetigkeit Functionen in topologischen Raumen", Indag. Math. 15 satisfying P'T q satisfies P'T A. (1951), 359-368 J the following results are proved. A Lindelof space s s A countably paracompact (countably compact or semiregular and countably H- closed) p 'Ts A space is regular (completely regular). A countably compact A'T sB space is normal. (Received August 25, 1969.)

669-6. TREVOR EVANS, Emory University, Atlanta, Georgia 30322. Schreier varieties of

semigroups.

It is shown that the only varieties V of semigroups which have the Schreier property that any nontrivial subsemigroup of a free V-semigroup is again a free V-semigroup are (i) the variety of

left-zero semigroups defined by xy = x, (ii) the variety of right-zero semigroups defined by xy = y, (iii) the variety of constant semigroups defined by xy = zt, (iv) the varieties of abelian groups

satisfying xP = l, p prime. The corresponding result for groups was given by P. M. Neumann and ]. Wiegold, Math. Z. 85 (1964), 392-400. The proof of the semigroup theorem consists of first dis-

posing of the case where the variety satisfies only consequences of the commutative law and then considering the various possibilities in the other case where the variety satisfies a law xm = xm+n

For m = 1, n prime, the group theorem is required. The only other nontrivial cases are

m = l, 2; n = l. (Received August 29, 1969.)

1044 669-7. RICHARD H. BOULDIN, University of Georgia, Athens, Georgia 30601. A note on power bounded operators.

The work of Wermer, Rota, Sz-Nagy and Foi~s, Gilfeather, and others give rise to the hypothesis that the bounded linear operator T on the complex Hilbert space H be "power bounded" i.e. the sequence (litk 111 is uniformly bounded. Theorem l. It is necessary and sufficient for the

operator T to be power bounded that either r(T) < l or else r(T) = 1 and IITk lll/k = 1 + 0(1/k).

_!.emma. Let T be the weighted shift defined by Tek = tkek+ 1 where (tk1 is a mo•1otoae decreasin.s sequence of positive numbers. If ak = (t 1t 2 ... tk)l/k then the following are true: (1) the sequence (ak) is decreasing and determines the operator T, (2) r(T) = inf faiJ. (3) IIR(z)lf = lzl- 20 00 lzl- 2k(ak) 2k for any z such that lzl > r(T). Theorem 2. The power bounded operators are k=O a proper subset of the operators satisfying the two conditions (i) r(T) ~ l and (ii) IIR(z) II~ c/(lz 1- 1) for lzl > l. (Received September 3, 1969.)

669-8. EUGENE S. BALL, Auburn University, Auburn, Alabama 3 6830 and Tennessee Technological University, Cookeville, Tennessee 38501. Weakly normal spaces. Preliminary report.

Definition (Zenor). A topological spaceS is weakly normal provided that if (Hi)~ 1 is a monotonically decreasing sequence of closed sets inS with no common part and His a closed set in

S not intersecting H 1, then there exist a positive integer N and an open set D such that HN c D and cl(D) does not intersect H. Theorem. There is a weakly normal, countably paracompact, T 2 space which is not normal. C. H. Dowker (Canad. J. Math. 3 (1951), 219-224) proved that countable para- compactness and countable pointwise paracompactness are equivalent in a normal topological space.

Theorem. If S is weakly normal, then S is countably paracompact if and only if S is countably point­ wise paracompact. R. E. Hodel (Proc. Amer. Math. Soc. 17 (1966), 462-465) defined the following axioms for topological spaces: Axiom 1. If every open subset of X has property P, then every subset of X has property P. Axiom 2. If X has property P, then every Fa subset has property P.

Axiom 3. If (Vu; u in A} is a locally finite open cover of X such that for all u in A, cl(V u> has property P, then X has property P. It is well known that if property P is replaced by normality then each axiom is true. _Iheorem. If property P is replaced by weak normality, then (i) Axiom is

~true; (ii) Axiom 2 is false; (iii) Axiom 3 is false. (Received September 8, 1969.)

669-9. JACK W. ROGERS, Emory University, Atlanta, Georgia 30322. On compactifications with continua as remainders.

If K is a continuum (compact connected metric space) we say that a space X can be compactified

A by K if and only if there is a compact Hausdorff space X with a dense subspace X' homeomorphic to X

A such that X - X' is homeomorphic to K. J. M. Aarts and P. van Em de Boas have shown that any locally compact, noncompact, separable metric space can be compactified by any continuum. K. D. Magill has shown that any locally compact nonpseudocompact Hausdorff space can be compactified by any Peano continuum. Recently A. K. Steiner and E. F. Steiner have observed [Fund. Math. 63 ( 1968), 221- 223] that the methods used in obtaining the former result are applicable to the latter: We show that in fact these methods yield a generalization of both results, namely that any locally compact nonpseudo­ compact Hausdorff space can be compactified by any continuum. A partial converse is given, together with an example of a locally compact pseudocompact Hausdorff space which can be compactified by any continuum. (Received September 8, 1969.) 1045 669-10. PHILIP BACON, University of Florida, Gainesville, Florida 32601, The compactness of countably compact spaces.

A topological space is said to be countably compact if every countable open cover of it contains a finite subcover. We say a space is isocompact if every closed count ably compact subset of it is compact. Theorems. A space is isocompact if it is a countable union of closed isocompact subspaces. The product of an isocompact space X and an isocompact space Y is isocompact if every point of Y has a closed and compact neighborhood or if Y is hereditarily isocompact. The product of any collection of hereditarily isocompact spaces is isocompact. Every regular almost realcompact space (for a definition see Z. Frolik, "A generalization of realcompact spaces," Czechoslovak Math. J, 13 (88) (1963), 127-137) is isocompact. The paper will appear in the Pacific Journal of Mathematics, (Received September li, I969.)

669-Il, RENU LASKAR, Clemson University, Clemson, South Carolina 2963I. A geometric characterization of the line graph of a finite affine plane.

Let G be a graph, d(x, y), the distance and t, (x, y), the number of vertices adjacent to both x andy, x, y (: V(G). Let G satisfy the following properties: (I) IV(G)i = n2 (n + 1), (2) deg x = 2n­ for x IE V(G), (3) if d(x, y) = I, t,(x, y) = n- 2 or n- 1, (4) if d(x, y) = 2, t,(x, y) = I, It is shown here that any graph G with no loops and no multiple edges satisfying the above properties (1)-- (4) must be the line graph of an affine plane of order n, if n > 2. A characterization in terms of the eigenvalues of the adjacency matrix was given by A. J, Hoffman and D. K. Ray-Chaudhuri (Canad. J, Math. 17 (1965), 687-694). (Received September 15, 1969.)

669-12. JOHN K. LUEDEMAN, Clemson University, Clemson, South Carolina 29631. A generalization of the concept of a ring of quotients.

A collection 1: of left ideals of a ring is a I:-set iff (1) A c: B, A E !:, ., B E I;;

(2) A E I:, r E R => Ar- 1 E I:; (3) J £ :t. Kx -l E I: Vx E J ., K E I:. Sanderson (Canad. Math. Bull. 8 (I965), 505-513) defined the concepts of !:,-injective and I;-essential and proved the existence of a I:-injective hull for any R module when 1 E R. In this paper, we remove the need for 1 E R. The maximal Utumi 1:-quotient ring UI: of R is defined and is shown to be its own UI:-maximal quotient ring. The !:-singular ideal ZI:(R) is defined and when ZI:(R) = 0, UI: is shown to be UI: -injective and is the 1:-injective hull of R. The Johnson maximal !:-quotient ring is defined and is compared with the Gabriel %:-quotient ring. Bourbaki, Vol. 27, Alg~bra commutative,. Hermann, Paris, 1961. (Received September 12, 1969.)

669-13. RONALD 0. FULP, North Carolina State University, Raleigh, North Carolina 27607. Homologicaf study of purity in locally compact groups.

Let ~ denote the category of locally compact abelian groups with continuous homomorphisms as morphisms. A morphism lfJ: A - B of ~ is called a proper morphism iff 1fJ (U) is open in 1fJ (A) whenever U is open in A. An exact sequence A !1! 8 .2 C is proper pure exact iff lfJ and 9 are proper morphisms of .e., lfJ (A) is pure in 8, and 9(8) is pure in C. We show that a short exact sequence is

1046 proper pure exact iff its Pontryagin dual is proper pure exact. We show that the set of all n-fold proper pure exact sequences forms a group. If Pexf (A, B) denotes this group for A and B in.!., it is shown that Pextn is a functor and that for proper pure exact sequences A ,.... B..,, C in.!. the Pextn measure the inexactitude of Hom as in the discrete case. Contrary to the discrete case Pext2 does not vanish identically. The pure projectives and pure injectives of various subcategories of .!. are obtained. If Q is pure injective in .!. , then Q = lli.n $ (11!./Z) rr $ Q* where Q* is a totally disconnected pure injective of .!. . Moreover if any one of the compact open subgroups of Q* is pure in Q*, then Q* = A a'! DX (aX} where A is discrete and algebraically compact and (a>..) is finite cyclic (fi(aA) has the product topology). (Received September 16, 1969.)

669-14. WILLIAM]. HEINZER, Louisiana State University, Baton Rouge, Louisiana 70803. Quotient over rings of integral domains.

If D is an integral domain with quotient field K, then D is called a QR-domain if each ring between D and K has the form Ds for some multiplicative system S of D; and D is called a GQR-domain if each ring between D and K has the form D.;= {x ~ KlxA c:: D for some A ~ ./,. Here ./denotes a multiplicative family of subsets of D. A Prilfer domain with torsion class group is a QR -domain. However, the converse is false, as is here shown by the construction of a QR-domain not having torsion class group. Also the problem of characterizing GQR -domains is studied. If D is integrally closed, then Dis a GQR-domain if and only if Dis Priifer; but in general a GQR-domain need not be integrally closed. Toward characterizing GQR-domains the following partial result is obtained. Theorem. Let D be a GQR-domain such that the integral closureD* of Dis a finite D-module. Then D* is Priifer. Moreover, if Dis quasi-local and not integrally closed, then D* is the unique minimal overring of D in the sense that each subring of K properly containing D contains D*. (Received September 19, 1969.)

669- 15. JOSEPH DIES TEL, West Georgia College, Carrollton, Georgia 30117. Abstract contents. I.

Let X be a set, let L 1, L2 be lattices of subsets of X. We call Ll' L2 well-placed whenever A E L 1 implies (i) there is B E L 2: A c Band (ii) A c B 1 U B2 , Bi !! L 2 yields Al' A2 E L 1 such that A= A1 U A2, \ c Bi. Let L 1, L2 be well-placed. Examples include: (a) L 1 =ideal of subsets of an algebra of sets L2; (b) L 1 = (compact sets in locally compact, T2 space Xl. L 2 = (open sets in X) or L 1 = (compact Go's}, L 2 = (open F o-'s); (c) L 1 = (zero sets in completely regular space X}, L2 = (cozero sets in X}. Let Y be a Dedekind complete vector lattice with positive cone y+. Let 1.1: L 1 ~ y+ be finitely additive, bounded. Define {l: L2 ~ y+ by ~(B)= sup (!J(A): A c: B, A E L 1 }. Theorem. jl is finitely additive on L2 with same bounds as 1.1· Remark. In most of the above cases, ~ is actually countably additive on L 2. (Received September 19, 1969.)

669-16. RICHARD L. DAVIS, Louisiana State University, Baton Rouge, Louisiana 70803. Higher derivations and inseparable Galois theory.

Let K;k be a finite dimensional, purely inseparable, modular field extension having exponent n + 1 and characteristic p. Denote the group of all rank pn higher derivations of K by H(K) and the

1047 group of rank pn higher derivations of K trivial on k by H(K/k). Necessary and sufficient conditions that a subgroup G of H(K) be of the form H(K/k) are given. This characterization involves technical requirements on the upper central series for G and certain groups derived from G. This extends, to arbitrary exponent, the Galois theory of the author's "A Galois theory for a class of purely inseparable exponent two field extensions," Bull. A mer. Math. Soc. 75 ( 1969), 255-257. (Received September 18, 1969.)

669- 17. HENRY W. THWING, Stetson University, DeLand, Florida 32 720. The field of constants of an integral derivation on a p-adic field.

When is K0 , a subfield of p- adic field K, the field of constants of an integral derivation on K? Necessary conditions are that K0 be (l) p-adic, (2) algebraically closed in K, and (3) contain the inertial subfield of K. If the residue fields ko c k satisfy (4) there exists a transcendency basis T for k/ko such that k is of bounded exponent over ko(T), then conditions ( 1), (2), and (4) are sufficient. An example proves (4) not necessary and shows that no condition on the residue fields replacing

(4) can be both necessary and sufficient. All p-adic subfields Ko c K having k0 as residue field are algebraically closed in K iff ko is algebraically closed ink. At least one such K0 is algebraically closed in K if k0 is separably algebraically closed ink, trans. deg. k/kQ > 0 and k n k 0p -l is finitely generated over k 0 . Suppose k is finitely generated over k0 and trans. deg. k/k0 > 0. Then some p-adic subfield K0 c K with residue field k 0 is the field of constants of an integral derivation on K iff k 0 is separably algebraically closed ink. All such K0 have this property iff k 0 is algebraically closed ink. Techniques of proof are primarily constructive using results from S, Mac Lane [Ann. of Math. 40 (1939), 423-442j, (Received September 15, 1969.)

669-18. MARGARET M. LaSALLE, University of Southwestern Louisiana, Lafayette, Louisiana 70501. Dimension of the space of holomorphic cross sections of a complex line bundle.

The characteristic on a compact Riemann surface is a constant which is determined by the Riemann-Roch Theorem to be 1 - g. Dimension of the space of holomorphic cross section of a complex line bundle will be classified by use of the Riemann-Roch Theorem and Serre! Duality Theorem. To effect this, a sheaf of rings over a Riemann surface is introduced, Topological con­ sideration for genus, g, arises from inferences for an exact sequence of sheaves and corresponding exact cohomology sequence. (Received September 22, 1969.)

669-19. THOMAS G. PROCTOR, Clemson University, Clemson, South Carolina 29631. Periodic solutions for perturbed nonlinear differential equations.

Let D c R n, let f(t,x) and g(t,x) be continuous functions from R X D into R n with period P in t and let x(t,y) denote a solution of x = f(t,x), x(O) = ')'. Iff is C', f(t,O) = 0, the function 'Y - x(P, y) is invertible on some set K, g satisfies a certain geometric condition and is small enough, then there i~ a periodic solution of x = f(t,x) + g(t,x). This generalizes a well-known result for perturbed linear systems. (Received September 26, 1969.)

1048 669-20. W. WILEY WILLIAMS, University of Louisville, Louisville, Kentucky 40204. Semi- groups and semilattices on generalized trees.

A ~eneralized tree is an arcwise connected hereditarily unicoherent continuum X containing a point 0 so that [0, xa] - [0, x] (in lim suplim inf convergence) whenever xa - x. The quadrant of x E X, Q(x), is the component of x in X \(OJ. Theorem l. Let X be a semigroup on a generalized tree. Suppose there exists a net (ea) - l so that [0, ea] l'l (0, l] - {0}. Then for each x E X, (0, eax] n [0, 1] - {o}. Corollary. With X as in Theorem 1, X is not locally connected at any x E Q (1). If X is a semilattice with identity, similar results hold if l is replaced by any a E (0, 1) and X is replaced by Xa. Theorem 2. Let X be a generalized tree with zero 0 and let l E X. Suppose there exists a quadrant Q(x) 1- Q(l) such that Q(l) is the only quadrant whose closure meets

Q(x). Suppose also there exists a sequence (Q,}? of quadrants such that Q = Q(x), Qn = Q(1), and l 1= 1 1 Qj' 11 q+ 1 1- 0 for each i. Then X does not admit the structure of a semilattice with zero 0 and identity l. Two examples are given of generalized trees not admitting a semilattice with identity. (Received September 26, 1969.)

669-21. JAMES A. RENEKE, Clemson University, Clemson, South Carolina 29631. A product integral solution of a Stieltjes-Volterra integral equation.

Lets be a compact number interval [a, b], {X, I• n a Banach space, F a function from s X s into the continuous linear transformations of X, and k a nondecreasing function from S such that

F [, t] is quasicontinuous and IIF(t, u)- F(t, v)ll ~·ik(u) - k(v)l for all t, u, and v inS, where 11•11 is the usual operator norm. Let F be the Banach space of quasicontinuous functions from S into X and

V the function from S X S into the continuous linear transformations of F defined by [V(u, v)g](t) = (L)J~dF (t, ]g. Let W be the function from S X S defined by W(u, v) = u nv [1 + V], where 1 is the identity transformation on F and the product integral exists as a limit with respect to the usual operator norm. Theorem. Iff is in F and h(t) = [W(t, a)f](t) then h is the only function in F such that h(t) = f(t) t (L)Jt dF [t, ]h, for all tinS. The same methods are also shown to hold for a nonlinear a equation. (Received September 29, 1969.)

669-22. MATHEW O'MALLEY, NASA, Manned Spacecraft Center, Mail Code ED-13, Houston, Texas 77058 and CRAIG A. WOOD, Oklahoma State University, Stillwater, Oklahoma 74074. R-endomorphisms of R[[X]].

Let R be a commutative ring with identity, X an indeterminate over R, and S = R[[x]] the formal power series ring. If p E S, then the problem of determining n.a.s.c. conditions for the existence of an R-endomorphism If> of S mapping X onto {l is considered. In particular, n.a.s.c. are given in order that there exist such a one -to-one and/or onto R -endomorphism of S. If n : 1 (B)n = (0), n.a.s.c. are given without regard to the one-to-one or onto properties, and moreover, in this case

(n 00 (S)n = (0)), if there exists an R-endomorphism I() of S such that !f>(X) = fJ, then I() is unique. n=1 The following is also shown: Theorem. Let fJ = E~ 0 aiXi E S and suppose that there exists an R- endomorphism I() of S such that If> (X) = P. Then: (i) lfl is onto if and only if a 1 is a unit of R; (ii) if

If> is onto, then I() is one-to-one; (iii) qJ is an automorphism if and only if a 1 is a unit of R. (Received September 29, 1969.)

1049 669-23. FREDRIC T. HOWARD, Wake Forest University, Winston-Salem, North Carolina 27106. A property of the Rayleigh function.

The Rayleigh function of order 2n, cr zn(v), is defined by Kishore in (Proc. A mer_ Math. Soc. 14

( 1963 ), 52 7- 53 3). Let v be a rational number a/b, a odd and b even, b = (2k + l)2t. By using induction on one of the recurrence formulas for cr2n (a/b), we determine the exponent of the highest power of 2 dividing the denominator of cr2n(a/b) to be 2n + l + (l - 2n)t - m, where m is the number of terms in the base 2 expansion of 2n. Congruences (mod 4) and (mod 8) can be determined from the same formula. For example, if 2n = zs 1 + ... + zsm. si- SI+l > l fori= l, ...• m- I, then, letting x = Zn + I+ (1 - 2n)t- m, we have zxcr2n(a/b) "'(- l)m+l(2k + I)/a (mod 4). (Received September 30, I969.)

669-24. NICK H. VAUGHAN, North Texas State University, Denton, Texas 76203. Domains in which every ideal is a finite product of semiprime ideals. Preliminary report.

Let D be a domain with l t- 0 and quotient field K such that every proper ideal A of D is a finite product of semiprime ideals, i.e., A= n ns. such that JS. = S. (e.g., a Dedekind domain is such a 1 1 1 1 domain). Theorem l. Every primary ideal of Dis a prime power and conversely. Theorem 2. Every ideal of D contains a power of its radical. Theorem 3. If the ascending chain condition for primes holds in D, then D is an almost Dedekind domain. In addition we study the prime ideal structure of D. In particular we have: Theorem 4. If P > P * are prime ideals of D and there are no primes properly between them, then n Pn = P or n Pn = P *. Corollary. If PI > P 2 are prime ideals of D, then

P ~ > P 2 . (Received October I, 1969.)

669-25. GORDON G. JOHNSON, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. An arc in Hilbert space.

Theorem. There is an arc Hilbert space such that its closed convex hull is the Hilbert Cube. (Received October l, 1969.)

669-26. j. B. GARNER, Louisiana Polytechnic Institute, Ruston, Louisiana 71270. Nonsolvability theorems for second order two point boundary value problems.

Conditions are given for f(t, y, y') under which there exist real numbers A and B such that the boundary value problem y'' = f(t, y, y'), y(a) = A, y(b) = B does not have a solution. These results are used to study the sharpness of known solvability results for this problem. (Received October l, 1969.)

669-27. RALPH B. BENNETT, Auburn University, Auburn, Alabama 36830. Locally connected continua embeddable in a torus. Preliminary report.

Theorem. There is a finite class S of locally connected continua such that a locally connected metric continuum K is embeddable in a torus if and only if .K does not contain a subcontinuum homeo­ morphic to a member of S. Unfortunately, there are many members of S and this characterization is not easy to apply. (Received October 3, 1969.)

1050 669-28. RICHARD M. CROWNOVER, University of Missouri, Columbia, Missouri 65201. , One dimensional point derivation spaces in Banach algebras.

Let A be a commutative semisimple Banach algebra with identity, and let M(A) be the set of complex homomorphisms on A. A point derivation D at a point cp E M(A) is a linear functional such -1 n that D(gh) =cp ( g)D(h) + cp(h)D(g), for all g, h EA. Let Acp = tp (0), let Acp be the ideal generated by products of n elements in A and let A~ = n;: lAcp. Theorem. If the point derivation space at cp is one dimensional, and cp is not norm isolated in M(A), then (i) each An is closed, (ii) dim An I A n+l = 1, cp cp cp and (iii) A/ A~ is isomorphic to a Banach algebra of power series. Two sufficient conditions are given for an analytic disk to pass through cp. Theorem. If !!ll is a two-manifold embedded in M(A) with the norm topology, and the point derivation space at each point in !In is one dimensional,

then !!ll can be made into a Riemann surface on which the Gelfand transforms of elements of A are analytic. (Received October 3, 1969.)

669-29. CHARLES N. KELLOGG, University of Kentucky, Lexington, Kentucky 40506. An extension of the Hausdorff-Young theorem. Preliminary report.

For fl. = { >..(n)) E t00 and l ~ p, q < oo, let 1\ A. I\ p,q = {:E:=-oo (~n E I(m) I >..(n) lp)q/p) 1/q where I(m) = (n: 2m-l ~ n <2m) for m > 0, I(O) = (0) and I(m) = - I(- m) form < 0. If p or q is infinite replace the corresponding sum by a supremum. Denote by Lp'q, the space of all A for which 1\ A 1\ < oo. If A and B are sequence spaces, the space of multipliers from A to B, denoted by (A, B), p,q is the set of all A E ( 0 such that Aa = [\(n)a(n)} E B for all a EA. Theorem 1, Let p,q,r,s E [1,00] and define a and fj by 1/a = 1/r- 1/p if p > r, a= oo if p ~ r, 1/~ = 1/s - 1/q if q > s, and$= oo if q ~ s. Then (Lp,q• Lr,s) = L"• fl. Repeated application of Theorem 1 and some results of]. H. Hedlund ("Multipliers of HP spaces,"]. Math. Mech. 18 (1969), 1067-1074) yield the following extension of the classical theorem of Hausdorff and Young. Theorem 2. Suppose that 1 < p l!! 2. p " p' 2 " (a) There exists a constant, Ap' such that, iff E L , then f E L ' and 1\f 1\p' , 2 ~,.Ap llf 1\p ~here 1/p + 1/p' = 1. (b) There exists a constant, Bp' such that, if A E LP'2 the series !:A(n)emx converges p' " in L to a function f such that f = A and llf 1\ , ~ B IP·II 2 where again 1/p + 1/p' = 1. This gives an p p p, I extension of the classical theorem because if. p i 2, L p' •2 is a proper subset of tP and tP is a proper subset of LP•2 . (Received October 2, 1969.)

669-30. CHARLES W. McARTHUR, Florida State University, Tallahassee, Florida 32306. In what spaces is every closed normal cone regular?

A cone K in a real topological vector space E is normal if there is a family of seminorms P = (p} which generates the topology of E and which has the property that if p E P and x,y E K with

x :& y then p(x) ~ p(y). A cone Kin a real topological vector space E is regular if each nondecreasing sequence in K which is majorized by an element of K converges. Theorem 1. Each closed normal

cone of a Frechet space E is regul~r if and only if E has no subspace isomorphic to (c0 ). Theorem 2. Let E be a Banach space whose topological dual space E' is separable with the strong topology. Then with respect to the strong topology each closed normal cone in E' is fully regular

and hence regular. (Received October 2, 1969.)

1051 669-31. PHILLIP L. ZENOR, Auburn University, Auburn, Alabama 36830. Real compactifica­ tions with projective spectra.

Theorem. The limit spaceS~ X of the countable projective spectrum, SeX' of the normal and countably paracompact space X is homeomorphic to the Hewitt real compactification, vX, of X. (Received October 3, 1969.)

669-32. JEAN POLLARD, Louisiana State University, Baton Rouge, Louisiana 70803. Absolute z-sets.

All spaces in this paper are assumed to be separable and metric. A Z- set K in a space X is a closed subset of X such that for each nonempty homotopically trivial open set U, U\K is nonempty and homotopically trivial. Z-sets have been used frequently in infinite dimensional topology. For each i > 0, let I. = [0, l] and r? = (0, 1). Define !00 = n.> 1. and s = n. r0 ; thus, s c 100 • It is known I I I 0 I 1> 0 I that any topologically complete [or compact] space can be embedded in s [or 100] as a Z- set. The question then arises as to which spaces admit closed embeddings into s [or 100] only as Z-sets. An absolute Z-~~ is a space M such that for any closed embedding h of M into s, h(M) is a Z-set in s.

An absolute compact Z-~ is a space M such that for any embedding h of Minto 100 , h(M) is a Z-set in 100 . The main result in this paper is the following: Theorem. A space M is an absolute Z-set iff M is topologically complete and

669-33. LAWRENCE HUSCH, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. Homotopy groups of P L embedding spaces.

Let PL(N, M) and C(N, M) denote the spaces of piecewise linear embeddings and continuous mappings, respectively, of the piecewise linear manifold N into the piecewise linear manifold M. Theorem. Let N be a compact PL-n-manifold with k-spine, k < n, and let M be a PL-m-manifold without boundary. If m ~ n t k + s t 1, the homomorphism i#: 1Ts(PL(N, M))- 1Ts(C(N, M)) induced by inclusion is an isomorphism; if m ~ n t k + s, il is onto. (Received October 6, 1969.)

669-34. THOMAS A. CHAPMAN, Louisiana State University, Baton Rouge, Louisiana 70803. Infinite deficiency in Frechet manifolds.

A Frechet manifold (orE-manifold) is a separable metric space having an open cover by sets homeomorphic to s, the countable infinite product of lines. A set Kin a space X is a Z-set if K is closed and for each nonnull homotopically trivial open set U in X, U \l( is nonnull and homotopically trivial. A subset K of s has infinite deficiency (or infinite codimension) provided that for each of infinitely many different coordinate directions, K projects onto a single point. The factor theorem of Anderson and Schori (Bull. Amer. Math. Soc. 75 (1969), 53-56}, which says that any F-manifold X is homeomorphic to X X s, is used to prove: If X is an F- manifold, then a closed subset K of X is a

Z- set iff there exist a homeomorphism h of X onto X X s such that 1Ts • h(K) has infinite deficiency.

1052 Anderson (Mich. Math. j. 14 (1967), 365-383) had earlier proved a similar theorem for X= s. This theorem and the theorem of Henderson (Bull. Amer. Math. Soc. 75 (1969), 759-762), which says that every F-manifold can be embedded as an open subset of s, are used in proving: If X is an F-manifold K c X is a Z-set, then there exists an open embedding h: X- s such that h(K) is closed ins. (Received October 6, 1969.)

669-35. ROBERT E. FENNELL, Clemson University, Clemson, South Carolina 29631. Periodic solutions of functional differential equations. Preliminary report.

Let {Ch, 11·11} denote the Banach space of continuous functions mapping [- h, OJ into R n with the supremum norm. For a given function x with domain [- h, b), b > 0, and 0 ,; t < b define xt on [- h, OJ by xt(9) = x(t + 9). Let L, f [0, ro) XC h- Rn be continuous, let L(t,

T-periodic in t for fixed cD, and let L(t, rD) be linear in cD for fixed t. The Schauder fixed point theorem is used to prove the following result. Theorem. Suppose f maps closed bounded sets into bounded sets and jf(t,

669-36. DANIEL RALPH LEWIS, Louisiana State University, Baton Rouge, Louisiana 70803. A functional integral for vector measures.

Let 'r be a semitribe (o-ring) of subsets of S, E be a real or complex locally convex space and

/): 'r- E be a measure. A scalar valued function f on S is ~~-integrable iff is (IJ, x') -integrable for

each x' E E', and if, for each set A locally in 1; the linear form (the integral over A) fA f(t) (IJ. (dt), •) is o(E', E) continuous. Theorem. If E is sequentially complete, then the dominated convergence theorem holds in -r(E,E'). The gauge of a circled, convex zero neighborhood U is denoted by Pu• and vu(IJ., •) is the extended real valued measure whose value at A E -ris the supremum over finite parti­ tions of A of the sums :E . ., p (1J (A.)). A measure IJ. is of finite variation if v (1J., •) is real valued l=n 0 1 0 for each U. Theorem. For a space E the following are equivalent: (I) Every subseries summable

series in E is absolutely summable. (2) If IJ is any E-valued measure, then a wintegrable function is v JJJ• •)-integrable for each U. (3) Every E-valued measure defined in a semitribe is of finite variation. (Received October 6, 1969.)

669-37. R. CHRISTOPHER LACHER, Florida State University, Tallahassee, Florida 32306. Strongly acyclic map between simply connected manifolds.

Let M and N be topological n-manifolds without boundary, fa proper, strongly acyclic map of M onto N. Define Cf to be { y E N If 1 (y) is not cellular in M}. Remark. When M is not simply connected, Cf can be a polyhedron of any dimension k, - 1 "'k ~ n- 3. Wright [Abstract 69T-G!2!, these cJiotiai.J 16 ( 1969), 8 53] showed that Cf is locally finite when n = 3. Thus, if M = S 3, Cf is empty. For higher

dimensions, a weaker theorem can be proved: Theorem. If M is simply connected, and n ~ 5, then Cf contains no isolated points. The proof uses some naive surgery and Van Kampen to deduce UV 1;

then a kind of Hurewicz theorem for UV- properties deduces UV 00 , which implies cellularity. A simi­ lar Hurewicz-type theorem is used to prove previously announced [October 1969) results on "Cellularity criteria for maps." (Received October 6, 1969.)

1053 669-38. JOHN L. BRYANT, Florida State University, Tallahassee, Florida 32306. Embeddings of !-dimensional compacta in En.

Theorem 1. Suppose that xk is a k-dimensional compactum in En, n ~ 5 and n - k !!: 3, such

that En - X is 1· ULC. Then for each £ > 0 there exists a covering N 1, ...• Nr of X by compact P L n- manifolds satisfying (a) diam Ni < £ (i = l, .•. ,r), (b) ( N 1, ... ,Nr) has order ~ k, and (c) each of BdNi and each Ni ,., Nj is d-connected, where d = min {n - k - 2, [(n- 1)/2]- 1}. Theorem 2. Suppose

that X is a !-dimensional compactum in En (n ~ 5) such that En -X is 1-ULC. Then for each f > 0, there exists a !-dimensional polyhedron P in En such that X can be engulfed by an arbitrary neighbor­ hood of P via an (-push of (En, X). Theorem 3. Suppose that X is a 1- dimensional compactum and

that f0, f 1 : X- En(n!!: 5) are embeddings such that En- fi (X) is 1-ULC (i = 0, 1) and d(f0 , f 1) roc. A mer. Math. Soc. 23 (1969), 46- 51] concerning cmbeddings of k-dimensional compacta in En for n ~ 6. (Received October 6, 1969.)

669-39. WILLIAM E. HAVER, State University of New York, Binghamton, New York 13901. Cellular mappings on three manifolds.

Let Mn be a manifold without boundary. Theorem l. Let f: Mn - Mn be an onto map, n "'3. Then f can be approximated uniformly by homeomorphisms iff it is compact and cellular. The suffi­ ciency of the conditions is due to Armentrout. We then have the following Corollary. Let f and g be

compact cellular maps from Mn on,to Mn, n l!! 3. Then g • f is compact and cellular. Let H(N) denote the space of maps from M onto M which can be approximated uniformly by homeomorphisms and Ce(M) be the compact cellular maps of M onto M. Both spaces are considered with the compact open topology. Using the techniques of Edwards and Kirby it is shown that H(Mn) is locally contractible if Mn is compact or is the interior of a compact manifold. In particular, we have Theorem 2. Suppose Mn is compact or the interior of a compact manifold, n ~ 3. Then Ce(Mn) is locally contractible. (Received October 6, 1969.)

669-40. KENNETH 0. LELAND, Illinois Institute of Technology, Chicago, Illinois 60616. Algebras of analytic functions. Preliminary report.

The results of "A characterization of analyticity." III, (J. Math. and Mech. 18 (1968), 109-123), announced. in Abstract 648-144, these cJVotiai) 14 (1967), 672, are extended to the case of an algebra of functions F on open subsets of an Euclidean space En into areal Banach algebra B, closed under multiplication by real scalars, translations of the form x:- rx + x0 , r > 0, and uniform limits on suitable domains. Under suitable conditi.ons on B, either the elements of F are analytic, or there exist e0 E En' A0 e B, such that F contains all functions of the form x- g[(x, e 0)]A0 , where g is an arbitrary map on the reals R into R. In the simplest case B can be taken as the complex numbers considered as

a Banach algebra over R. More generally B can be taken as simple. Set B0 = Uf E F range f. B0 is called strongly semi simple (SSS) if the intersection of all proper maximal two sided ideals of B0 is ( 0). It suffices to require that B be finite dimensional and (1) commutative and semisimple; or (2) that B0 be SSS. Conditions ill' the infinite dimensional case involving SSS algebras and semisimple annihilator rings are given. The method of proof centers on complexifications of En and B, generated by suitably chosen subalgebras ofF, which are then shown to be analytic in the classical sense. (Received October 7, 1969.) 1054 669-41. JOHN W. NEUBERGER, Emory University, Atlanta, Georgia 30322. Analyticity and quasi-analyticity of trajectories of semigroups of bounded linear transformations.

Suppose that Tis a strongly continuous one-parameter semigroup of bounded linear transforma­ tions on a Banach space and T has generator A. Theorem. If lim supx-oiT(x)- II< 2 then AT(x) is bounded for all x > 0. Suppose ( liqJ ';; 1 is a sequence of positive numbers convergent to zero and each of N(q) = fnqJ ~l' q = l, 2, ... is an increasing sequence of positive integers. Denote by Q the collection consisting of (l) all real analytic functions on (0, oo) and (2) all h on (0, oo) for which there is a Banach spaceS, a member p of S, a member f of s• and a strongly continuous semigroup L of bounded linear transformations so that h(x) = fiL(x)pl for all x > 0 where L satisfies

lim sup ( EN( )) IL( 6 /n .) - I I < 2. Theorem. No two members of Q agree on an open subset n -oo n q q q,J of (O,oo). (Received October 7, 1969.)

669-42. CHARLES CLARK, University of Tennessee, Knoxville, Tennessee 37916 and JOSEPHS. STARR, University of Missouri, Columbia, Missouri 65201. Natural extensions of compact semigroups.

If S is a compact semigroup with zero and T a compact semigroup with a closed ideal I, then T will be called a natural extension of I by S if (i) T/I is isomorphic to S, and (ii) the closed subsemi­ group generated by (f '11)- 1(S\ (0)) coincides with T, where f: T1 -Sis the isomorphism given by

(i), and 11: T- T/1 is the natural homomorphism. (The algebraic theory of extensions (where only (i) is assumed) is treated in "The algebraic theory of semigroups," Vol.I, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R.I., 1961, by A. H. Clifford and G. B. Preston, and references to the original work in this area are given there.) GivenS and I, a method is given for constructing certain natural extensions of I by S and the existence of a maximal natural extension is established. More precisely,

if S is a compact semigroup with zero, then there exists a compact semigroup To with a closed ideal

I 0 such that T 0 is a natural extension of I0 by S, and if (T,I) is any other pair with these properties, then there is a surmorphism h: T 0 - T such that h(I0) .=I and an appropriate commutative diagram is obtained. In fact, T 0 is a natural extension of I0 by T/I. If Tis a natural extension of I by S, conditions are given under which Tis connected (abelian) if Sis connected (abelian). (Received October 7, 1969.)

669-43. PAUL D. HILL, Florida State University, Tallahassee, Florida 32306. The covering

theorem for upper basic subgE.9~~

A basic subgroup B of a primary abelian group G is called an upper basic subgroup of G if the rank of G/B is less than or equal to the rank of G/ A for every basic subgroup A of G. Theorem. If A is a basic subgroup of G, then there exists a decomposition G = H + K of G such that K is a direct sum

of cyclic groups and such that A= B + (A 1"1 K) where B is an upper basic subgroup of H.

~orollary. Any basic subgroup is contained in an upper basic subgroup. (Received October 7, 1969.)

669-44. CHARLES R. WALL, University of Tennessee, Knoxville, Tennessee 3 7916. Density theorems for some arithmetic functions. Preliminary report.

A divisor d of n is a unitary divisor if (d, n/d) = l. Let cr(n) be the sum of divisors of n, cr*(n)

the sum of unitary divisors of n, and ~·(n) = n n(l + p- 1) with the product over primes p dividing n. For x real, let A(x) [respectively B(x), C(x)J be the asymptotic density of integers n such that

1055 cr (n) [resp. 1/J (n), d'(n)] exceeds xn. It is known that for all x, A(x) exists and is continuous; a similar result is obtained for B(x) and C(x). From n ~ d'(n) ~ [t:r(n) + d'(n)}/2 "'1/J(n) ~ cr(n) it follows that

C(x) ~ B(x) ,; A(x) ;[ B((x + l)/2). It can be shown that B(l + x) ,; l - x for 0 ~ x ~ 0.28; it is a corollary that if F(x) denotes the density of odd integers n with cr(n) '!': xn, then the existence of F'(2) is a sufficient condition for the nonexistence of odd perfect numbers. (Received October 7, 1969.)

669-45. REGINALD MAZERES, Tennessee Technological University, Cookeville, Tennessee 3 8 50 l. Semigroups that are unions of groups.

If a semigroup S is the union of four disjoint groups its idempotents need not form a band. An example is given showing this. Suschkewitsch ("Theory of generalized groups," Gos. Wauk Tekh. Izd. Ukranii, Kharkow, 1937) showed that in a semigroup S that is the union of two groups, the idempotents form a band. This shows that if a semigroup S is the union of three groups, then the idempotents form

a band. Also if a semigroup is the union of four groups, either the idempotents form a band or they have the structure of the given example. (Received October 7, 1969.)

669-46. VADIM KOMKOV, Texas Tech University, Lubbock, Texas 79409. Optimal excitation theory for the p.d.e. ~stems of symmetric hyperbolic type, or of elastic vibration type.

Some results in optimal control theory of vibrating elastic systems are easily extended to cover the optimal excitation case. However, some of the most important theorems of optimal control theory are false when "control" is changed to "excitation", i.e. when maximum energy level is desired. In particular the uniqueness of the finite state can no longer be proved, and the basic con­ vexity lemma is clearly false. In fact, we can prove complete nonconvexity of optimal excitations: if

1/J 1, 1/J 2 are optimal excitations then 'llr is not optimal if 1/J = A¢1 + (l - /\) ¢2 , 0 < A < l. Because of this shortcoming the Pontryagin's principle which takes a form analogous to the optimal control case is largely meaningless. The author proves the corresponding maximum principle for instantly optimal excitation (as defined in Abstract 650-11, these c/'.fotictiJ 14 (1967), 919) and shows that it conveys meaningful information. (Received October 8, 1969.)

669-4.7. J. S. MAC NERNEY, University of Houston, Houston, Texas 77004, A lattice of complete inner product spaces. Preliminary report.

The inner product space (S 1, Q 1 Jis continuously situated in the inner product space (s2, Q2 1 [these cJ.fotictiJ 16 (1969), 677] in case s 1 is a linear subspace of s2 and the identity function on s 1 is continuous from (S 1, Q 1 } into ( s2 , Q2 1. Suppose K is a collection of complete inner product spaces, for each two of which there is a third in which both are continuously situated: if each of {S 1, Q1) and

( s2 , Q2) is inK then s1 s2 denotes the common part of s1 and s2 and s1 al s2 denotes the linear space to which z belongs only in case z = x + y for some x inS 1 andy in s2 . Theorem. If (S 1, Q 1 1 and

( s2 , Q2 ) are in K then (l) if S 1 is a subset of s2 then (S 1, Q1) is continuously situated in ( s2 , Qz), (2) Q1 + Q2 is an inner product for S 1 s2 with respect to which S 1s 2 is complete,

(3) (s 1s2 , Q1 + Q2 } is continuously situated both in (s 1, Q1 J and in (s 2 , Q2), and each inner product space which is continuously situated both in (s 1, Q1 } and in (S 2 , Q2 ) is also continuously situated in

(s 1s2, Q1 + Q2), and (4) there is an inner product Q3 for s1 al s2 such that (s 1 E>l s2 , Q3 ) is com-

1056 plete, both (S 1, Q 1 J and (S 2, Q2 j are continuously situated in (S 1 $ S 2, Q3 ), and (S 1 $ S2, Q3 ) is continuously situated in every complete inner product space in which both (S 1, Q1 ) and (S 2 , Q2 J are continuously situated. (Received October 8, 1969.)

669-48. WILLIAM R. HARE, and JOHN W. KENELLY, Clemson University, Clemson South Carolina 29631. A characterization of subsets of Radon partitions. Preliminary report.

Let pc Rd be a set of n !!! d + 2 points in general position. By Radon's theorem P admits a

partition A U B = P such that conv A n conv B 'f ~. If S c P, IS I = k, the question arises: Is S one "half" of a Radon partition 7 (I.e., does there exist a Radon partition P = A U B with S c A or S c B 7)

Theorem. S is in one part of a Radon partition of P iff either (i) k ;!il n - d - 1 or (ii) if k !; n - d, then conv S n aff (P - S) 'f ~- This becomes a theorem. of Proskuryakov [Uspehi Mat. Nauk 14 (1959) 215.-222] and Kosmak (Spisy PZ:irod. Fak. Univ. Brno (1963),223-225) in the special case of n = d + 2

and k = 2. (Received October 8, 1969.)

.669-49. JUTT A HAUSEN, University of Houston, Houston, Texas 77004. Abelian torsion groups with artinian primary components and their automorphisms.

A group is called artinian if it satisfies the minimum condition for subgroups. If r is contained in a group A, the centralizer of r in A is denoted by E./1 r. Theorem. Let G be an abelian torsion group and A(G) its group of automorphisms. Then every primary component of G is artinian if and only if A(G) is residually finite and A(G)/.£A(Gl is finite for every primary normal subgroup r of A(G). (Received October 8, 1969.)

669-50. LOUIS F. McAULEY and BYRON L. McALLISTER, State University of New York, Binghamton, New York 13901. A note on cyclic subelement theory--reducibility of local connectedness and local simple connectedness.

The principal purpose of this note is to show that the property of being a plane locally simply connected Peano continuum is subelement reducible. An example is given in E 3 of a locally simply connected Peano continuum with a subelement that fails to be locally simply connected. Furthermore, in the plane, a Peano continuum exists with a nonlocally connected subelement. It is also shown that two forms of local simple connectedness (weak and strong) are equivalent for plane continua. In general, they are not equivalent. (Received October 8, 1969.)

1057 The November Meeting in Claremont, California November 22, 1969

670-l. SHERMAN K. STEIN, University of California, Davis, California 95616. B-set of a family of sets.

Theorem. Let a regular graph of degree three on the surface of the ordinary sphere partition the surface into simply-connected regions. Then the graph has a Hamiltonian if and only if the regions can be labelled A orB in such a way that no cycle of length three or more is all A or all B. This result is part of an investigation of "B-set of a family of sets," that is, a set that meets each member of the family and yet contains no member of the family. (Received July 28, 1969.)

670-2. LeBARON 0. FERGUSON, University of California, Riverside, California 92502. Uniform approximation of rational functions by polynomials with integral coefficients.

In this note we give a necessary and sufficient condition in order that a rational function be uniformly approximable on a class of compact subsets of the complex plane by polynomials whose coefficients are, in a certain sense, integers. Indeed, let A be my discrete subring of the complex numbers with rank 2 and unique factorization (for example, the Gaussian integers). Let X be a compact subset of the open unit disk with connected complement in the plane having the origin in its interior. Then a rational function f is uniformly approximable on X by polynomials with coefficients in A iff it can be represented in the form f = p/g where the coefficients of p and g lie in A, g(O) is a unit of A, and the roots of g lie outside of X. (Received September 15, 1969.)

670-3. CHARLES L. HAGOPIAN, Sacramento State College, Sacramento, California 95819. On hereditarily arc-wise connected plane continua.

A continuum M is said to be aposyndetic at a point p of M with respect to a set N in M - {p} if there exist an open set U and a continuum H in M such that p E U c H c. M - N. In this paper it is proved that if a compact plane continuum M contains a finite set of points F such that for each point x in M - F, M is not aposyndetic at x with respect to F and M is aposyndetic at x with respect to each point of M - (F U ( x}), then M is hereditarily arc-wise connected. (Received September 15, 1969 .)

670-4. ALAN R. HOFFER, University of Montana, Missoula, Montana 59801. On unitary polarities.

Let 1T be a finite projective plane which admits a unitary polarity li. Theorem. If rr contains three noncollinear points which are centers of elations that commute with li, then If is either a Desarguesian plane or in the Lenz -Barlotti Class I-1. This yields the following answer to a question posed by Professor M. Suzuki. Theorem. If each absolute point of 1T is the center of an elation which commutes with li and if 1T contains one transitivity center, then 1T is a Des argues ian plane and the

1058 group of collineations which is generated by these elations contains the three-dimensional projective unitary group. (Received September 30, 1969.)

670-5. S. M. F AKHR UDDIN, University of Manitoba, Winnipeg 19, Manitoba, Canada. Modules over Prufer domains.

Let R be a Priifer domain with field of quotients K. Let R • be another Priifer domain containing

R, with field of quotients K •, such that R • n K = R. Let '!JlR denote the category of all torsion free modules over R. Let C- be the category whose objects are of the form 0- M*-1 M* ~ • K* [ V- 0 where (1) M* E ~R •, and V E Cat(k) with Rank (M*) = dim(V). (2) :>- is the canonical injection

(3) 'Tis K-monomorphism with certain properties. The morphism in C- is obvious. Theorem l. ~ is equivalent to C-. From now on R is a valuation ring. Applications: Theorem 2. (Baer-Kulikov­

Kaplansky). M E. 'Ill R. If M is completely decomposable, so is any direct summand of M. Theorem 3. (Matlis). R is maximal iff every M E !IDR of rank two is completely decomposable. We also calculate a complete set of invariants for !DlR, when R is almost maximal. (Received September 29, 1969.)

670-6. TOM M. APOSTOL, California Institute of Technology, Pas adena, California 91109. Dirichlet L -functions and character power sums.

A new representation for Dirichlet L-functions L(s, X), valid for primitive characters X modulo k and all complex s, is given in terms of the function F(x,s) defined for real x and R(s) > 1 by the series 'L: ~ 1 e 2 n!Tix/ns. Evaluation of L(s,)() for negative integers leads to a class 1 of identities relating mth power moments !: ~: 1x(r)rm with finite cotangent power sums. Special emphasis is given to the quadratic character x.(n) = (n lp). p an odd prime. A new proof of the func­ tiona! equation for L-functions is also given. (Received October 2, 1969.)

670-7. MOHAMED A. AMER, Cairo University, United Arab Republic and University of California, Berkeley, California 94720, and WILLIAM P. HANF, University of California, Berkeley, California 94720. Boolean algebras of logics of higher order. Preliminary report.

Let L be a second-order language, T L the L-theory consisting of all valid L-sentences, and L T(T L) the Boolean algebra of equivalence classes of sentences of L. The cardinality of L is the cardinality of the set of all of its nonlogical constants. Lis relational if all of its nonlogical constants correspond to relations on individuals, and it is trivial if all of these relations are 0-ary or unary, otherwise it is nontrivial. First, if L is denumerable, L T(T L) is completely characterized by being denumerable and atomless. Second, if Lis relational, is finite and trivial, LT(TL) is completely characterized by being denumerable, atomistic an such that for each a E L T(T L)' if a is greater than infinitely many atoms, then there is b E LT(T L) such that each of ab and a - b is greater than infinitely many atoms. Third, if L is relational, is finite, and nontrivial the structure of L T(T L) depends on the meta -theory. If the axiom of construct ability is assumed, L T(T L) is completely characterized as in the second case. If the nonexistence of a set-theoretically definable well-ordering ofw is assumed, LT(TL) is not atomistic. The same results are also obtained for logics of order n,

where 3 ~ n ~ w. (Received October 6, 1969.)

1059 670-8. RAYMOND B. KILLGROVE, California State College, Los Angeles, California, HENRY G. BRAY, San Diego State College, San Diego, California 92115 and GORDON WHITNALL, University of California, Santa Barbara, California 93106. Bounding sets in various ways.

Let HBT abbreviate "compact sets are exactly those which are closed and bounded". In a metric space, let a usual-bounded set be called sphere bounded. In any space, a set is B-bounded if contained in some member of base B (Abstract 666-23, these cJ{oticei) 16 (1969), 557) and topologicall_y bounded if contained in some compact set (Kimber, "Two extended Balzano-Weierstrass theorems", Math. Monthly 72 (1965), 1011). Busemann ("Local metric geometry," Trans. Amer. Math. Soc. 56 (1944), 205) has shown necc. and suff. cond. for existence of metric so that HBT holds using sphere bounded are for the space to be Hausdorff, locally compact, and second countable. Suff. cond. for existence of a base B so that HBT holds using B -bounded are for the space to be locally compact and KC (compact sets are closed, Wilansky, "Between Tl and T2" Math. Monthly 74 (1967), 262). Trivially necc. and suff. cond. for HBT holding using topologically bounded is KC. Also KC is necc. and suff. for existence of base B so that (1) every compact set is closed and B-bounded. Locally compact is not necc. cond. for converse to (1). To see this, use infinite product of this space: open sets are-- all reals, empty, all left open rays. Open question: is there a KC, not locally compaCt space with base so that HBT holds? (Received October 6, 1969.)

670-9. JOHN M. BOWNDS, University of Arizona, Tucson, Arizona 85721. On the Darboux­ Neugebauer property of uxy and a problem of S. Marcus.

The Darboux property for the derivative uxy ·as defined by Neugebauer is established as a corollary to a more general theorem which states that intermediate values for this derivative are actually attained on an infinite set. The corollary answers in the affirmative a question posed by S. Marcus. This affirmative answer is also obtainable using results in recent years of K. Bogel and

L. Mi~ik. (Received October 6, 1969.)

670-10. PATRICK]. LEDDEN, University of California, San Diego, California 92037. Some homotopy groups of Thorn complexes.

The computation of the groups 1Tn+k(MO(n)) fork< n is classical. (R. Thorn, 1954). The groups

1T2n(MO(n)) can be computed by the same method: Theorem. If n is an even integer, 1T2n(MO(n)) = z + r • z2 ; if n is an odd integer, ~n (MO(n)) = s • z2 . (Here r and s are integers which depend upon the dyadic expansion of n.) The Postnikov invariants are also obtained (they are, in general, nonzero)

and they yield interesting geometric results. Similar computations determine 1T2n+l (!:MO(n)),

1T 2n+2(0 MO(n)), etc. In each case theorems about homology classes in manifolds result: e.g. 2 Corollary. A necessary condition that u E Hn(M n: z2) be realized by a submanifold with nonvanishing normal vector field is that the Poincare dual class, v = D- 1u, have v 2 = 0. (Received October 6, 1969.)

670-11. TAKAYUKI TAMURA, University of California, Davis, California 95616. Congruences on N-semigroups.

An N-semigroup S is a commutative archimedean cancellative semigroup without idempotent. S is faithfully represented in terms of a abelian group G and an ..J -function I: G X G- [0,1,2, ... l

1060 satisfying some conditions. S is finitely generated if and only if G is finite. Let cp : G- (1,2, ... } be defined by cp(a) = !: «E G I(a, ( ). Then S is determined by G and cp • An N- congruence on S is a congruence 0' such that S/0' is an N-semigroup. This paper determines all N-congruences on S in caseS is finitely generated. Let p be a congruence on G satisfying ap fJ => cp(~) "'((!(~) (mod. IGI).

Now 0' is defined on S as follows: (m,a)cr(m + (cp (a)- cp($ )/ IGI, M if cp(a) ~ cp({J );

(m + (tp ({J)- tp(a))/IGI, a) cr(m, ft) if cp(a) < tp(B). Then(] is anN-congruence on S. All N-congruences on S are obtained in this manner. (Received October 8, 1969.)

The November Meeting in Ann Arbor, Michigan November 29, 1969

671-l. ABRAHAM BERMAN and AD! BEN-ISRAEL, Department of Engineering Sciences, Northwestern University, Evanston, Illinois 60201. Linear equations over cones with interior: A solvability theorem with applications to matrix theory.

The solvability of the system (l) Tx = b, x E int S where T E Cmxn, b ~ em and S a closed

convex cone in en is characterized as follows: (1) is consistent iff [ 0 'f THy E S • => Re(b,y) > OJ. Corollaries and special cases of this theorem include Lyapunov's characterization of stable matrices (by takingS the cone of positive semidefinite matrices in the real space of hermitian matrices,

T(x) = AHX + XA, b = - I) and a generalization of Stiemke's theorem of the alternative for complex linear inequalities (by taking b = 0). (Received June 23, 1969.)

671-2. SAMUEL ZAIDMAN, Universite de Montreal, Montreal, Qu.fbec, Canada. An existence theorem for bounded vector-valued functions.

Let X be a reflexive Banach space; Tt : t !!:: 0 - L(X,X) a strongly con.tinuous semigroup of

linear bounded operators; f(t) an almost-periodic function; - oo < t < oo -X. Let u(t); t ~ 0 -X be a bounded continuous function admitting representation u(t) = Ttu(O) + Jci Tt- Cf(l;)dl;:, t ~ 0. Then, there exists a bounded continuous function v(t), - oo < t < oo - X admitting representation v(t) = Tt-to v(t0 ) + J~ 0 Tt_,f(l;)dl;:, '\ft;,; t0 . (Received August 21, 1969.)

671-3. MARl Z. v. KRZYWOBLOCKI, Division of Engineering Research, Michigan State University, East Lansing, Michigan 48823. Mathematical aspects in relativistic fluid dynamics in nonvacuum.

In 1 'l07, 2 years after his historical paper on the special theory of relativity, Albert Einstein published his second paper on the special theory of relativity, in which he included the effect of the gravitational field upon the velocity of the propagation of light. In 1948 A. Taub published his paper on the fundamentals of the relativistic fluid dynamics. In his approach Taub assumes the conditions in

~acuo. In the present work the author assumes more realistic environmental conditions of a non­ vacuum, i.e., the existence of a gravitational field. Under these assumptions the velocity of the propagation of light is effected by the gravitational field. The fundamentals of fluid dynamics, as proposed by Taub, are revised due to these effects. The following mathematical aspects appearing

1061 in the work are of interest: The metric of the four-dimensional space-time is a Riemannian one, whereas in the original approach it is an Euclidean one. This causes some inconveniences in trans­ formations. The case of the propagation of one-dimensional waves of finite amplitude in the Euclidean space-time leads to homogeneous equations, in the Riemannian one to nonhomogeneous, which, according to the author's knowledge, were not solved, as yet. (Received August 25, 1969.)

671-4. CHONG YUN CHAO, University of Pittsburgh, Pittsburgh, Pennsylvania 15213. Infinitely many nonisomorphic nilpotent algebras.

Theorem 1. Let K be a field whose characteristic is not 2 and whose cardinality is ~ 0 (greater

than N0 ). Then there exist countably infinitely (uncountably) many nonisomorphic nilpotent Lie alge­ bras of Class 3 (L =:> L2 = [L,L] =:> [L,L 2] = 0) over K for any given dimension 1!; 10. Theorem 2. Let K be the same as in Theorem 1. Then there exist countably infinitely (uncountably) many nonisomorphic solvable not nilpotent Lie algebras of index 3 (L =:> L 2 =:> L 3 = [L 2 , L 2] =:> [L 3, L 3] 0) for any given dimension :;; 11. These are generalizations of the results in the author's article [ Proc. Amer. Math. Soc. 13(1962), 903]. In Suprunenko and Tyshkevich's book "Commutative

matrices", it is shown that for any integer n ~ 6, there exist infinitely many nonisomorphic commuta­ tive nilpotent associative algebras of class 3(A =:> A2 =:> A3 = 0) and of dimension n over an infinite field F. Definition. A field R is said to be an s-field (s'-field) if R contains a set S(S') of countably infinitely (uncountably) many elements which are algebraically independent over a subfield Q of R. Theorem 3. Let F be an s -field (s '-field). Then there exist countably (uncountably) many nonisomor-

phic noncommutative nilpotent associative algebras of class 3 over F for any given dimension ~ 5. (Received September 8, 1969.)

671-5. A. LENARD and SEYMOUR SHERMAN, Indiana University, Bloomington, Indiana 47401. Stable potentials. I.

If lfJ is an even, real valued, measurable function on a topological group, G, such that

~ <" • "' lfJ (x. - x .) ~ 0 for all n ~ 1 and all x , ...• x in G, then tp is called a stable potential. 1 ~l,J _ n 1 J 1 n David Ruelle ["Statistical Mechanics," Benjamin, New York, 1969] observed that the sum of an even, measurable, nonnegative function and an even, continuous, positive semidefinite function is a stable potential and raised the question whether every stable potential can be decomposed in this way. For

G = z2k+ 1' k ~ 2, examples are given of stable potentials not admitting such a decomposition. For the physical case G = R3 or even G = R the problem is still open. To appear in "Communications in Mathematical Physics" (work of S. Sherman supported by NSFGP 7469). (Received September 15, 1969.

671-6. HOWARDS. G. SWANN, Antioch College, Yellow Springs, Ohio 45387. The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R 3 .

It is shown here that a unique solution to the N avier-Stokes equations exists in R 3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge

uniformly to a function that is a solution to the equations for ideal flow in R 3 . The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed-point methods with estimates derived from using analytic semigroup theory. (Received September 15, 1969.) 1062 671-7. JOSEPH DIESTEL, West Georgia College, Carrollton, Georgia 30117. Abstract contents. II.

Let X be a set, L 1, L 2 be well-placed lattices of subsets of X. Let Y be a Dedekind com­ plete vector lattice with positive coney+. For any 1J.: L 1 ~ y+- that is finitely additive, bounded, let ~ : L 2 ~ y+ be the "lifting" of to L2 . Definition. If ). •IJ: L 1 ~ y+ is finitely additive, bounded, we say). < IJ. whenever ). (A)= 0 for any A E L 1 : IJ. (A)= 0;).. << IJ whenever given f E y+ 10, :!! 6 E y+ IO: IJ. (A) < 6 implies ).(A) < r. Theorem. Let IJ.•).: L 1 ~ y+ be as above. Then if a.~: L 2 ~ y+- denote the "liftings II of IJ• " to L 2 respectively. we have (i) ). < ,.,. if and only if 1 < ~ and (ii) ). << IJ. if and only if X << ,1. (Received September 18, 1969.)

671-8. RONALD G. MOSIER and HIDEGORO NAKANO, Wayne State University, Detroit, Michigan 48202. Discrete linear lattices.

A linear lattice V of functions on a space R is called a vector lattice on R if V contains the characteristic function of each single point of R. The purpose of this paper is to characterize vector lattices. An x f. 0 in a linear lattice L is called discrete if y E L and lx I != IY I imply the existence of a real number a such that x = ay. A linear lattice L is called discrete if 0 f. y E L implies the existence of a discrete x E L such that IY I !':: lx 1. In this paper it is proven that a linear lattice is isomorphic to a vector lattice if and only if it is discrete and archimedean. Then different kinds of discrete linear lattices are investigated. Finally, discrete linear lattices are characterized by their proper spaces. (Received September 24, 1969.)

671-9. BERNARD C, ANDERSON and HIDEGORO NAKANO, Wayne State University, Detroit, Michigan 48202. Semicontinuous linear lattices.

A linear lattice L is called semicontinuous if L"= (x} J.J. Ell {x}J. for each x E L; and is called complete for uniform convergence if each uniform Cauchy sequence in L is order convergent.· A sequence (x#J.} is said to be a uniform Cauchy sequence if there is 0 ;!; I £ L such that, for each r > 0, there exists IJ(r) for which ''• 1J. != Jt(£) implies lxv- xl'l;!; ( I. In [H. Nakano, "Linear Lattices", Wayne State Univ. Press, Detroit, Mich., 1966] it is shown that every sequentially continuous linear lattice is semicontinuous and complete for uniform convergence. By using spectral theory for linear lattices one can prove the converse. Theorem l. If a linear lattice L is semicontinuous and complete for uniform convergence then L is sequentially continuous. The following theorem is also established. Theorem 2. Every Banach lattice is complete for uniform convergence. One can then apply Theorems l, 2 to obtain Theorem 3. A Banach lattice is sequentially continuous if and only if it is semi­ continuous. (Received September 30, 1969.)

671-10. HIDEGORO NAKANO and STEPHEN ROMBERGER, Wayne State University, Detroit, Michigan 48202. Cluster lattices.

Let L be a linear lattice and M a manifold of L. MJ. is the set of all x E L such that lxi/\IYI = 0 for each y E M. M is called a cluster if M = MJ.J.. In this paper it is proved that the clusters of L form a complete Boolean algebra under the inclusion order. (Received September 29, 1969.)

1063 671-11, JOE W. KNICKMEYER, University of Oklahoma, Norman, Oklahoma 73069. Connexions on submanifolds. Preliminary report.

Let M be a C 00 d-manifold, N a C 00 e-submanifold of M, and D a linear connexion on M. Classically, a rigging of N (which may not exist globally) has been used to induce a connexion on N from D. Let A be a function which assigns to each m EN a subspace Am of M m such that

.Am ~Nm = Mm, and such that this decomposition splits vector fields in a C 00 manner. Then A in­ duces a connexion on N by decomposition. Call such functions .A-congruences. Every submanifold

of a paracompact, T2 manifold supports a .A-congruence, If N is a closed submanifold with the sub­

space topology of a paracompact, T 2 manifold M, each A-congruence on N is the normal distribution to N re some Riemannian metric on M (the proof proceeds by gluing local normalizations together with a C 00 partition of unity). A-congruences permit a pleasant, coordinate-free analysis of union curves and related concepts. A A-congruence may be represented as a cross section of a C 00 fibre bundle with fibre a certain open subm an if old of the Grassmann manifold of (d-e)- planes in Rd. (Received September 29, 1969.)

671- 12. COLIN W. CRYER, University of Wisconsin, Madison, Wisconsin 53 706. On the computation of rigorous bounds for the solutions of linear integral equations with the aid of interval arithmetic.

A method is given for approximately solving linear Fredholm integral equations of the second kind with nonnegative kernels. The basis of the method is the construction of piecewise-polynomial degenerate kernels which bound the given kernel. The method is a generalization of a method sug- gested by Gerberich. When implemented on a computer, interval arithmetic is used so that rigorous bounds for the solution of the integral equations are obtained. The method is applied to two problems: the equation considered by Gerberich; and the equation of Love which arises in connection with the problem of determining the capacity of a circular plate condenser. (Received September 30, 1969.)

671-13. JAMES G. CAUGHRAN, University of Kentucky, Lexington, Kentucky 40506. Zeros of analytic functions with infinitely differentiable boundary values. iB A necessary and sufficient condition is proved that a set ofyoints (rne nJ in the unit disk be the set of zeros of an analytic function with infinitely differentiable boundary values for every choice

of (rnJ, 0 < rn < 1 and 'B(l - rn) < oo. (Received October 1, 1969.)

671-14. D. V. THAMPURAN, State University of New York, Stony Brook, New York 11790. Uniform spaces and antinomies.

It is possible to construct Hausdorff uniform spaces, using subsets of ordinals and reals, whose completions are unique. Taking completions in two ways and identifying these antinomies can be obtained one of which is similar to that of Burali-Forti. (Received October 1, 1969.)

1064 671-15. GEORGE K. FRANCIS, University of Illinois, Urbana, Illinois 61801. Restricted homotopies .of curves.

A class of restricted homotopies for curves, discovered by C. J. Titus (unpublished) is used, together with results of S. ]. Blank (Seminaire Bourbaki, 1967/68, no. 342), to prove the following. Let g be an immersion of the circle into the plane whose image curve ji] lies in general position. Let N(g) denote the number, possibly zero, of topologically inequivalent extensions of g to an immer­ sion of the disc. (1) Titus' homotopies preserve N(g). (2) The pair of points on the circle correspond­ ing to a self intersection of 18-J determines two circular arcs. If all these arcs are pairwise disjoint

or nested, then N(g) ~ l. Equality holds according to Titus' criterion on the intersection sequence

(W. Kaplan~. "Lectures on functions of a complex variable," University of Michigan Press, Ann Arbor, 1955, p. 435). (Received October 2, 1969.)

671-16. A. DUANE PORTER, University of Wyoming, Laramie, Wyoming 82070. Orthogonal similarity in a finite field.

Let F = GF(q) be the finite field of q = pr elements, p odd. A number of the well-known theorems concerning orthogonal similarity of symmetric, skew, and normal matrices, which are valid for the real field, do not hold over F. The usual proofs fail for one or more of the following reasons: (1) the sum of nonzero squares of F may equal zero, (2) for p t- 2, exactly one half of the elements of F do not have square roots in F, (3) the eigenvalues of a matrix over F may lie in an extension field of F of degree greater than 2. Through a consideration of these reasons as well as the properties of F, we obtain several canonical forms, valid over F, for certain classes of the above noted matrices. (Received October 3, 1969.)

671-17. COLIN C. GRAHAM, Northwestern University, Evanston, Illinois 60201. On M0(G) and the quotient M(G)/M0(G) of a measure algebra.

The L-ideal M0(G) of measures on the LCA group G whose Fourier-Stieltjes transforms vanish at infinity is studied. It is shown that the orthogonal complement of M0 (G) contains positive measures y whose product lies in M0(G) and that the Silov boundary of the quotient algebra M(G); M0 (G) is not all of the maximal ideal space of M(G)/M0(G). (Received October 3, 1969.)

671-18. DOUGLAS HARRIS, Marquette University, Milwaukee, Wisconsin 53233. Regular­ closed spaces and proximities.

A topological space is regular-closed if every regular filter has a cluster point. A topological space is RC-regular if it can be embedded as a dense subspace of a regular-closed space. An R -proximity on a set X is a relation p on 2 X satisfying: (P 1) !tl non- p A for any A c: X, (P2) A p A for any A ex with At- !tl, (P3) Ap(B U C) if and only if ApB or ApC. Defining A< B to mean A non-p(X - B), (P4) If (pJ

1065 A > B. Theorem A. The operator u: zX ~ zX given by uA = (x: (x} p A} is a topological closure operator for X if p is an R -proximity. Theorem B. A topological space is regular if and only if its topology is given by the topological closure operator of an R- proximity. Theorem C. A topological space is RC-regular if and only if its topology is given by the topological closure operator of an RC -proximity. (Received October 6, 1969.)

671-19. JACOB KOREVAAR, Claremont Graduate School, Claremont, California 91711. Approximation by polynomials whose zeros lie on nonanalytic curves.

Let D be the union of two domains D 1 and D2 bounded by mutually exterior rectifiable Jordan curves C 1 and C 2 . Denote the harmonic measure of C 1 relative to the exterior of D by w(z) and set C~.~(oo) = 11. For analytic curves Cj, Korevaar and Chui (Abstract 662-5, these cN"oticeiJ 15 (1968), 1031; Proc. Conf. Approximation Theory in Oberwolfach, (1968)] obtained the following results on the approximability of zero-free holomorphic functions F(z) in D by polynomials whose zeros lie on o D. When 11 is irrational, every F(z) is uniformly approximable on every compact subset of D. However, when n is rational, only special F(z) are approximable. In the present paper it is shown that when at least one of the curves Cj fails to be analytic at one or more points, all F(z) are approximable whether n is irrational or not. (Received October 6, 1969.)

671-20. ROGER A. AVELSGAARD, Bemidji State College, Bemidji, Minnesota 56601. On the cohomology of Jordan algebras of a symmetric bilinear form.

Let A be an algebra over a field ~ in a class r of algebras defined by a set of identities S. M. Gerstenhaber (Proc. Nat!. Acad. Sci. U.S.A. 52 (1964), 626-629) and N. D. Glassman (dissertation, Yale University, 1968) have defined cohomology functors Hn(A, ) from the category of A-bimodules to the category of ~-spaces. In Glassman's theory there is a choice for H 1(A, ) so that for any A-bimodule M, the space Hn(A,M) coincide with the classical cohomology spaces when A is associative, Lie, associative with involution, Lie with involution, or a Jordan algebra of m X m matrices, m ;, 4. Approaching the problem from the point of view of projective resolutions, we show that there is a

"natural" choice of a bimodule C and bimodules Xi and maps X2 ~ X 1 - x0 ~ C ~ 0 which extends the resolution of Gerstenhaber to a projective complex of C, for any r. It is shown for the classes of associative, Lie, and Jordan algebras of a symmetric bilinear form, that the complex is a projec- tive resolution of C, that the cohomology of the resolution is Hn(A, and therefore that for these alge- bras the cohomology functors are the derived functors of Hom (C, ). (Received October 6, 1969.)

671-21. ALAN H. SHUCHAT, University of Toledo, Toledo, Ohio 43606. Approximation of vector- valued continuous functions.

The following results are useful w proving Riesz-type representation theorems for spaces of continuous functions with values in a topological vector space (TVS) that is not locally convex. Let Q be a locally compact Hausdorff space, E a TVS and C(Q,E) the space of continuous functions from Q to E with compact support. When E is the scalar field we write C(Q) for C(Q,E). If a E C(Q) and

x E E the function q ~ a(q)x, denoted by a ® x, belongs to C(Q,E) and the linear span of these functions is denoted by C(Q) ®E. When E is a Hausdorff space C(Q) ® E consists of the functions with finite-

1066 dimensional range. It is well known that when E is locally convex C(Q) ® E is dense in C(Q,E) for the topology of uniform convergence (the density property). Theorem l. If Q is a locally compact sub­ space of some Rn and E Is any TVS then C(Q,E) has the density property. Theorem 2. If Q is any locally compact Hausdorff space and E is an F-space (complete metrizable TVS) with a basis then C(Q,E) has the density property. Theorem lis contained in a result of Turpin and Waelbroeck (C. R. Acad. Sci. Paris (Ser. A- B 267 (1968), 94- 97) on differentiable functions but our proof is much simpler for this special case. (Received October 6, 1969.)

671-22. EWING L. LUSK, University of Chicago, Chicago, Illinois 60637. Homotopy groups of spaces of P L embeddings.

A recent theorem of Claude Morlet enables us to use known embedding theorems to immediately obtain information about spaces of PL embeddings. Let (M c Q) denote the space of PL embeddings of a closed PL m-manifold Min a PL q-manifold Q, considered as a semisimplicial complex in the usual way. Let f be an embedding of M in Q to be used as base point. Suppose oQ = Jil and m + i "§' q- 3. Theorem l. Let M be (2m - q t i)-connected and suppose that 1Tr(Q) = 0 for r !!<2m - q tit l and for i ~ r "it m. Then 1Ti((M c Q), f)= 0. Theorem 2. Suppose that 1Tr(Q) = 0 fori~ r" it m and that f is (2m - q tit I)-connected. Then 1Ti ((M c Q), f)= 0. Zeeman's sphere unknotting Theorem [110 (sm c Sq) = 0 if m ~ q - 3] thus generalizes to: Corollary. 1T. (Sm c S q) = 0 if m t i ~ q - 3. Theorem 2 is the result of combining Morlet' s theorem on the 1 equivalence of generalized concordances and generalized isotopies in the given dimension range with an embedding theorem of Browder, Sullivan, and Casson. Theorem l can be considered as a corollary of Theorem 2 or can be proved independently using Irwin's embedding theorem. (Received October 6, 1969.)

671-23. JOHN E. CONNETT, Northern Illinois University, DeKalb, Illinois 60115. Block bundles and embeddings.

"Fiber homotopy type" and "Thorn space" are defined for block bundles, and the following proved: Theorem. Let Mn be a closed simply connected P.L.-n-manifold, n '!!: 5, n "f 4K t 2. Suppose (l/2)n t 3/2 and (i) there exists a E 1T (T(~)) such that ~q is a block bundle over Mn such that q ~ ntq h(a) =

October 6, 1969.)

671-24. T. GIFFORD, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. Odd order

boundary value problems.

Let P(X,D) be a partial differential operator of order 2m+ 1, and let P 2mtl(x,O be the symbol of the principal part of p. If n is a domain whose boundary is sufficiently smooth and noncharacteristic for P then the following theorem holds: If lfJ is a sufficiently regular function and

'V rp 'V cp (x, ~) ~ I t:.1 2 m for x E n, ~ E R n then there exists well posed problems for P(X,D)u = f. x ., 2m+l

1067 Examples of operators satisfying the above are P 3 (X,D) = ~ + D~ + D~ and QZp+l (X,D) = (0~=laiDxi)t,P, The above condition on PZm+l(x,O, properly interpreted, is also a necessary condi­ tion. The method of proof involves the use of weighted Sobolev spaces and extensions of theorems originally due to Friedrichs, Interior and boundary regularity theorems are also proved and a partial algebraic characterization of these directionally elliptic operators is offered. (Received October 6, 1969.

671-25. THOMAS N. E. GREVILLE, University of Wisconsin, Madison, Wisconsin 53706.

An interpolation formula of the form vx =!:;:=-co L(x - v)yv with the support of L(x) contained in (a, a+ n) (with nan integer) is called exact for degree r if vx = p(x) for all x when Yv = p(v) for all v for every polynomial p of degree r. It is called reproducing if L(v) = 60 v for every integer v. 2 For given a, n, r, m there is a unique L E cm-l such that crL = J~00 (L(rn) (x)) dx is smallest. It is a piecewise polynomial of degree at least r and at most max(Zn - 1, r) with knots only at a + v (for integers v) and is the only such function for which/:; r+ 1 L(x) is given in (a, a + n - r - 1) by a single polynomial. There is also a unique reproducing formula such that cr Lis smallest. For this formula L is a piecewise polynomial with degree and knots as before, but with additional discontinuities in L (Z m- l) (x) at the integers in (a, a + n) if a is not an integer. It is the only such function for which 6r+lL(x) is given in (a, a+ n -· r - l) by a spline of degree 2m - 1 with simple knots. (Received October 8, 1969.)

671-26. PATRICK X. GALLAGHER, Barnard College, New York, New York 10027, On Fogels' density theorem.

On Fogels' Density Theorem. The proof of an estimate, due to Linnik and Fogels (Acta Arith. ll (1965), 67-96) for the density of zeros of Dirichlet L-functions near cr= lis simplified, using the following lemma: Let S(t) = 'f;c( v) • e(vt) be an absolutely convergent Dirichlet series with real 2 2 frequencies v. Put C(x) =!; lc(v)l (x < v 1" x + 1). Then J61S(t)l dt << J~00(C(x)) dx. A refinement of the lemma gives another proof of Montgomery's (Invent. Math,, to appear) "hybrid" sieve inequality. (Received October 8, 1969.)

671-27. BENJAMIN LEPSON, United States Naval Research Laboratory, Washington, D. C. 20390 and Catholic University of America, Washington, D. C. 20017. p-unique families of merornorphic functions.

We start with the following definition: A family ;; of functions merornorphic on a set E is called p-unique on E if no two distinct members of ;; are equal at p or more points of E. Clearly, a family of polynomials of degrees not exceeding p - l is p-unique on any set E, and a subfamily of a p-unique family is alsop-unique. It is then possible to establish the following results, which extend a basic property concerning the distributions of zeros and a-points of polynomials to the larger class of finite linear combinations of merom orphic functions: Theorem. Let f1 , f 2, ... , fn be any finite set of functions meromorphic on a compact set K. Let ;; be the family of nonconstant functions of the form !)~ c.f., where the c are arbitrary complex numbers, and .J be the family of all functions of 1= 1 1 1 j the same form. Then there is an integer p such that ;; and .J are p-unique, and such that no member of ;; has more than p - 1 zeros, poles, or a-points (for any value of a) in K. (Received October 8, 1969.) 1068 671-28. MARY E. THOMPSON, University of Waterloo, Waterloo, Canada. A result on the Sn/n optimal stopping problem.

Let X 1, x 2, ... be independent and identically distributed random variables with mean 0, and let S n.= ~ ~= 1xi. The problem is to maximize if possible E(St/t) over finite -valued stopping times t. If there exist constants C > 0, T > 0 and v >I such that the truncated varianceU(X) = J~xy 2 dP(X 1 "!' y) satisfies U(tx)/U(x) "!' Ct2 - 11 for all x ~ l, t ~ T, then a maximizing stopping time t0 exists. In parti­ cular, if for some sequence (an),Sn/an converges in distribution to a random variable Y which is stable

with exponent greater than one, then t0 = min[n ~ 1: Sn 1!:: P(n)] where fJ(n) increases with n, and ~ (n) -can for some positive finite c. (Received October 8, 1969.)

671-29. GILBERT STRANG, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Puiseux expansions for Cauchy problems with multiple characteristics.

The constant-coefficient equation P(Dt, Dx)u = 0 has solutions of the form

exp i(x~ t t!:ak~-k/Pj, where the sum is the Puiseux expansion of a root of P(A,() = 0. We carry out a corresponding construction, under suitable hypotheses, for a variable-coefficient equation P(Dt, Ox, t, x)u = 0; the coefficients tak are replaced by lk(t,x). Such constructions have been used to establish incorrectness in the Cauchy problem, assuming that the principal part of P has the form m 1 m 1 y ... y , where y. = o/a t- r_ o/a x and the !'. are distinct. With simple characteristics 1 1 1 1 1 (mi = 1) the series has integer powers (p = 1), and P. Lax (Duke Math. j. 24 (1957), 627-646) proved incorrectness unless the r; are real. The author introduced fractional exponents (Arch. Rational Mech. Anal. 33 (1969), 358-373) to prove also the necessity of the Levi-Lax conditions, with real

characteristics of multiplicity m i ~ 3. We report on subsequent work with H. Flaschka, eliminating this last restriction. (Received October 8, 1969.)

671-30. KEITH MILLER, University of California, Berkeley, California 94720. Barriers and boundary pathology for the nondivergence equation.

We survey some of our work on the uniformly elliptic equation in nondivergence form, (1) Lu = ~ na .. u = 0, where L E rl ; i.e. (aij(x)) is symmetric, measurable, and has its eigen- 1 1J xixj a values in [a, 1], 0 < a < 1. We show that any boundary point x 0 of the domain S1 having an exterior

cone is regular for all L in rl . In fact, we construct barriers for rl on T , , of the form r Af(9), a a .,1 where r = lx - x0 1. 9 is the angle to the polar axis, and Tl/! is the right circular cone of aperature 1/J•

0 < 1/J < TT /2. However, the regular boundary points for (1) are not the same as those for Laplace's equation, even though this equivalence is the case for the divergence equation ~~(aijux.>x. = 0 by 1 l results of Littman, Stampacchia, and Weinberger. This nonequivalence can occur even with coefficients continuous at x0 • The barriers rAf(9) yield Phragman-Lindelof theorems on unbounded S1 cTI/J for solutions with growth rate o(r). ). The optimal growth condition o(r) (1/J, a)) is determined by means of extremal barriers, solutions of M (rAf(9)) = 0, where M (u)., max(Lu: L E rl ) defines Pucci's a a a maximizing operator. The growth condition for Laplace's equation, o(rA (1/J 'l)), suffices only when

1/> = TT/2; it may even fail to suffice with coefficients continuous at infinity. (Received October 8, 1969.)

1069 671-31. JOHANNES C. C. NITSCHE, University of Minnesota, Minneapolis, Minnesota 55455. On the boundary behavior of minimal surfaces.

(1) LetS= (:; =i(u,v); (u,v)E P}, where P = (u,v; u 2 + v 2 ~ 1 J. be a solution of Plateau's prob­ lem for a given jordan curve r. The regularity (up to the boundary) of the position vector x(u,v) depends on the regularity of f. In 1951 H. Lewy proved that x(u,v) is analytic in P if r is a regular analytic curve. Recently further results were obtained by various authors. Using potential theoretic methods now a simple proof of the following final result is presented: "If r is regular and of class cm,a, where m "' 1 and 0

671-32. EUGENE B. FABES and NESTOR M. RIVIERE, University of Minnesota, Minneapolis, Minnesota 55455. An existence theorem in the initial-value problem for the equations of Navier-Stokes.

Let ST = R nx(O, T) and consider the initial-value problem for the equations of Navier-Stokes, i.e. find a vector u(x,t) = (u 1(x,t), ... , un(x,t)) and associated scalar pressure p(x,t) such that

Dtu - 6u + u\7u + 17p = 0 in sT. \7 X • u = 0 in sT. u(x,O) = g(x) where g(x) is a prescribed initial value (u\7u is the vector with ith component=~ nk_luk(x,t)D u. (x,t)). Say that the vector g(x) is weakly - xk 1 divergence free if g(x) = g 1(x) + g 2 (x) where gi(x) E Lp~Rn) for some pi~ l, i = 1,2, and JRng(x) • \71/)(x)dx = 0 for all scalar functions cp(x) E C.~(Rn). A vector u(x,t) is called a weak solution to the above problem if (l) for each t E (0, T) u(x,t) is weakly divergence free, (2) u(x,t) E LP(ST) for some p "'l, and (3) for all divergence free vectors ¢>(x,t) I' C~(Rnx[O, T)), J J" fRnu •IA

671-33. FELIX E. BROWDER, University of Chicago, Chicago, Illinois 60637. Existence theorems for nonlinear elliptic boundary value problems without coercivity !Jypotheses.

Let A= "Biul"'m (- l)luiDuAu(x,u, .•. ,Dmu) be a quasi-linear strongly elliptic operator of

order 2m in generalized divergence form, a(u,v) = ~I I (A (x,u), Da.v) the corresponding a. ~ m a generalized Dirichlet form which we assume to be defined on a closed subspace V of ·the Sobolev space Wm'p(G) for a bounded, smoothly bounded open subset G of Rn. A new existence theorem is given for the variational boundary value problem: a(u, v) =(f, v) for all v in V, where f is a given

1070 element in V* and u is the desired solution in V, under the weakest hypotheses imposed previously in the coercive case but with the coercivity assumption (a(u,u)llullm,p- oo as Hullm,p -oo) replaced by the following hypothesis: There exists a one-parameter family of problems, at (u,v) where a 1 is the given problem and a0 corresponds to an odd operator A0, such that for all solutions of the problem at(u,v) = (f,v) for each fin V* and each tin [0,1], Hull~ lf'(lifli) for a suitable (but arbitrary) function If'. This existence theorem corresponds to an abstract extension of the Borsuk- Ulam theorem for pseudo-monotone mappings T of a reflexive separable Banach space V into its conjugate space V* and uses a new characterization of pseudo-monotone mappings as the uniform limits on bounded sets of mappings T which satisfy the stronger condition (S) introduced by the writer in the study of nonlinear eigenvalue problems. (Received October 8, 1969.)

671-34. RALPHS. PHILLIPS, Stanford University, Stanford, California 94305 and LEONARD SARASON, University of Washington, Seattle, Washington 98105. Elliptic-parabolic equations of the second order.

The purpose of this paper is to study the uniqueness of the weak solution of the Dirichlet boundary-value problem for the second order elliptic-parabolic partial differential equation Lu = ,,iiuij + Bi ui + "Yu = f, aij t i t j ~ 0, in a bounded domain G in R m with smooth boundary I:; here subscripts denote differentiation and the usual summation convention is employed. The investigation into this class of problems was initiated by Fichera in 1956 who established the existence of weak solutions and in the process obtained certain estimates on the solutions. More recently Oleinik proved for a large class of problems that the weak solution is unique. In the present paper we con­ sider a slightly different class of weak solutions and as a consequence are able to establish the uniqueness for a larger class of problems than that treated by Oleinik. For example we do not require, as does Oleinik, that the aii be smoothly continuable to a larger domain (including G) in which aij~. ~. ~ 0. However we require of the weak solution that both u 2 and aiiu.u. be integrable 1 . l 1 l in G whereas the comparable requirement of Oleinik's is that u3 be integrable in G. (Received October 8, 1969.)

671-35. PAUL CONCUS, University of California, Lawrence Radiation Laboratory, Berkeley, California 94720 and ROBERT FINN, Stanford University, Stanford, California 94305. On the behavior of a capillary surface in a wedge.

Estimates above and below are obtained for the height of the equilibrium free surface of a liquid, when the liquid partially fills a cylindrical container whose cross section contains a corner with interior angle 2a. The surface is characterized by the condition that its mean curvature be proportional to its height above a reference plane (or, in the case of zero gravity, that its mean cur­ vature be constant), and by the requirement that it meet the container wall with prescribed contact angle y. It turns out that the qualitative behavior of such a surface changes markedly, according as

a+ 'Y

1071 surfaces; here the interest centers on the fact that it is the mean curvature of an (n - I)-dimensional boundary element that controls the local behavior of the n-dimensional solution surface. (Received October 8, 1969.)

671-36. JOEL A. SMOLLER, University of Michigan, Ann Arbor, Michigan 48104. Existence theorems for two-by-two hyperbolic systems.

Consider the 2 X 2 hyperbolic system in a single space variable Ut + F(U)x = 0, where the mapping F = (f(u,v), g(u,v)) from R 2 into R 2 is smooth, fvgu > 0 and tid2 F(rj,rj) > 0, i,j = 1,2, where ti and rj are the left and right eigenvectors of dF, suitably normalized. Conditions are given in order to assure that the Riemann problem is solvable for an arbitrary jump. Under the stability conditions, which state that a shock is formed by the intersection of characteristics from exactly one family, the solution is unique in the class of centered shock and rarefaction waves. Using Glimm's difference scheme, the existence of a solution for the problem of elementary interactions is proved. Here the initial data consists of three constant states separated by two waves which move in such a way as to intersect and destroy the middle state after a finite time. Observations of certain patterns in the behavior of computer programmed solutions led to the proof. The author was assisted in this by C. Moler. (Received October 8, 1969.)

671-3 7. TODD DUPONT, University of Chi_cago, Chicago, Illinois 6063 7. Uniqueness and comparison theorems for nonlinear elliptic boundary value problems.

Jim Douglas, Jr., James Serrin, and I have recently established the uniqueness of smooth solu­ tions of the Dirichlet problem for nonlinear elliptic equations of the form div c7(x,u,Du) = /3 (x,u). We assume only that c7 satisfies mild differentiability conditions and that I!J is a nondecreasing function of its second argument. When contrasted with N. G. Meyers' example of nonuniqueness for a Dirichlet problem for an elliptic equation of the form !:ai,j(x,u)DiDju = 0, our result points out an important difference between elliptic equations with divergence structure and those without it. The proof of the uniqueness theorem can be extended to give comparison theorems for smooth solutions of elliptic inequalities which satisfy a wide variety of boundary and degeneracy conditions. (Received October 8, 1969.)

671-38. LOUIS NIRENBERG, Courant Institute, New York University, New York, New York

10012 and J. F. Treves, Purdue University, Lafayette, Indiana 47907. On solvability of linear partial differential equations.

This is a report on the problem of local solvability of a differential equation of order m:

(1) Pu = f; ( 1) is said to be locally solvable at a point x if x has a neighborhood V such that for every f E C~(V) there is a distribution u in V satisfying (1). Only operators of principal type are considered, i.e., those for which the leading symbol,of P, Pm(x.~). has only simple real zeros, i.e., if ~O 'f 0 is a real vector such that at some point x0 , Pm (x0 , ~ 0 ) = 0 then grad t, Pm (x0 , ~ ~ 'f 0. Necessary conditions for local solvability are obtained, and in case the coefficients of pm are analytic these conditions are also proved to be sufficient. (Received October 8, 1969.)

1072 671-39. JAMES B. AX, State University of New York, Stony Brook, New York 11790. On Schanuel' s conjectures.

Conjecture. Let y1 , ... ,yn E

671-40. PETER E. BLANKSBY, Ohio State University, Columbus, Ohio 43210 and HUGH L. MONTGOMERY, University of Cambridge, Cambridge, England. On a problem of Schinzel and Zassenhaus.

This paper is an extension of earlier work by Schinzel and Zassenhaus, and Cassels. The following and related results are discussed: if a is an algebraic integer of degree n > 1, with conjugates a 1 , •.• ,an such that max 1 ~j;!!n laj I < 1 t 1/lO(n log 2n)2 , then a is a root of unity. (Received October 6, 1969.)

671-41. HAROLD G. DIAMOND, University of Illinois, Urbana, Illinois 61801. An elementary proof of the prime number theorem with remainder term.

An elementary proof of the prime number theorem is given which improves on the. error term achieved by elementary methods by Bombieri (Riv. Mat. Univ. Parma (2) 3(1962), 393-440) and Wirsing (J. Reine Angew. Math. 211(1962), 205-214; 214(1964), 1-18). The starting point of the investigation is the establishment of a sequence of formulas tr'nl:o generalizing Selberg's formula (which is Fo ). The error term in each formula is estimated explicitly. From each formula rp(x) - x is estimated by a tauberian theorem, and for each x the optimal choice of n is made. (Received October 6, 1969.)

671-42. WILLIAM J. ELLISON, University of Michigan, Ann Arbor, Michigan 48104. Waring's problem for fields.

Waring's problem is generalized to arbitrary fields. It is shown that in a very wide class of fields that representations of the form a=L:;r=1 x~ are possible, with N;!! g(k, K), a number depending only on k and the field K. Estimates for g(k,K) are given. (Received October 6, 1969.)

671-43. TAKASHI ONO, Johns Hopkins University, Baltimore, Maryland 21218. On some arithmetic questions on forms.

Let k be an A-field. In the arithmetical questions of polynomial mapping f: kn - km, the inte­ gral transform Gf!p( ~) =J knfP(x) X((f(x), ())dx plays a basic role. We discuss the corresponding Eisenstein-Siegel series an'l!, when m = 1, zeta-functions. (Received October 6, 1969.)

671-44. CARL R. RIEHM, University of Notre Dame, Notre Dame, Indiana 46556. Some theo­ rems in Galois cohomology with applications to quadratic forms and simple algebras over local fields.

Let G be a group and H a subgroup of finite index. A simple generalization to the non-Abelian

1073 case is given of the corestriction map of Galois cohomology, involving a G-group, an H-group, and their cohomologies, A different notion of corestriction is then considered, namely the notion dual (in a suitable category) to that of restriction in algebraic geometry. Finally the foregoing is applied to objects over local fields: a theorem of Milnor concerning quadratic forms is proved, along with an analogous theorem about simple algebras, Both theorems ultimately rely on the injectivity of the usual corestriction between Brauer groups of local fields, (Received October 6, I969.)

671-45. H, DAVENPORT, University of Cambridge, Cambridge, England and WOLFGANG M. SCHMIDT, University of Colorado, Boulder, Colorado 80302. Dirichlet's theorem on Diophantine approximation,

There are two forms of Dirichlet's theorem: (a) For any positive integer N there exist inte­

gers xp ... ,xn•Y• not all zero, satisfying j11 1x1+... + llzrn + yj < N"n and max(jxii• ... ,jxnl> ;! N. (b) For any positive integer N there exist integers xl, ... ,xn•Y• not all zero, with max(j11IY- xlj, ... ,jllnY- xnl>

exist integers xl' ... 'xn•Y, not all zero satisfying j11 1 x 1 + ...+ 11nx n + y I < !JN-n and max (jxll•· ... lxnl> ;! !JN. The meaning of the phrase" (b) can be improved" is analogous. Theorem 1,

For any n-tuple (111, ... ,a.4 ), form (a) of Dirichlet's theorem can be improved if and only if form (b) can be improved, Theorem 2. For almost every n-tuple (11I,. .. ,11n), neither form of Dirichlet's theorem can be improved. The proofs involve the study of certain one-parameter families of point lattices. Further results can be obtained when n = 1. (Received October 6, I969.)

671-46, J. RODERICK SMART, University of Wisconsin, Madison, Wisconsin 53706. On the values of the Epstein zeta function.

Let Q(m,n) be a positive definite binary quadratic form. This paper is concerned with the values of 1:: q(k), k l!< 2, k integral, the associated Epstein zeta function. Taking a clue from Kronecker's first limit theorem, a formula is derived which in this case involves modular forms of positive dimen­ sion with period polynomials and their derivatives. The period polynomials associated with the generators of the modular group are determined. Finally a formula for approximating l::q(k) is given. (Received October 6, I969.)

671-47, HAROLD M. STARK, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and University of Michigan, Ann Arbor, Michigan 48104. Sign changes of some number theoretic functions.

Ingham proved a Tauberian theorem that allowed Haselgrove to disprove the Polya conjecture

(L(x) =I:n:sxX(n) ~ 0 if x ~2), In fact, computation on a high speed (then) computer enabled Haselgrove to find a c-;;nstant c > 0 such that lim stipx.... oox- 112L(x) >c. It is possible to generalize Ingham's theorem so as to attack other problems similar to the Polya conjecture (the generalization is too long to be conveniently stated here), In particular, we can apply the theorem to the problem of changes of sign of 11(x,k,a)- 11(x,k,b). The method is computational and so will yield results for k,a,b only on an individual basis, (Received October 6, 1'969.)

1074 671-48. GEORGE WHAPLES, University of Massachusetts, Amherst, Massachusetts 01003 and University of Notre Dame, Notre Dame, Indiana 46556. Two remarks on class field theory.

I. One can study axiomatically the "field crossing" method used by E. Artin in his original proof of his Reciprocity Law and prove a Theorem. A norm residue symbol can be uniquely extended from one system of field extensions to a larger system whenever the first system contains enough cyclic extensions. Examples. Going from the unramified extensions to the abelian extensions in the local case, or from the cyclotomic extension to the abelian extensions in the global case. II. Call a field K of algebraic functions of one variable with k as field of constants a "function field over k". Such a K is always a product formula field for the set of all valuations of K which are trivial on k. If k is quasi-finite the usual local class field theory holds for the completion of K at each of these valuations. Call a quasi-finite k !!:ood if global class field theory holds for every function field over k. Known results. Every quasi-finite field with no nontrivial valuation is good. The field of formal power series in one variable over ~ is not good. Related result. If k is quasi-finite and has a nontrivial valuation there is a function field of genus 1 over k in which the Dirichlet-Hasse-Chevalley unit theorem does not hold. (Received October 6, 1969.)

671-49. JA!VIES B. SERRIN, University of Minnesota, Minneapolis, Minnesota 55455. A!l a priori bound for surfaces o!_ constant mean curvature.

We obtain an a pri_c:>,!_! limitation on the region in which a surface of constant mean curvature may lie, in terms of the diameter of its boundary set. Specifically, letS be a surface of constant mean curvature H > 0, having no self-intersections and spanning a curve r in E3 . Then the following Theorem holds: If r is contained in a ball of radius R about the origin 0, then S is contained in a ball of radius R + 2H · 1 about 0. The value R + ZH- 1 is best possible. The proof makes use of Bonnet's associated surface of constant mean curvature, together with certain techniques of A. D. Alexandrov. ln joint work with A. Aeppli, the above result has been extended to apply to surfaces of constant positive Gaussian curvature, and other elliptic Weingarten surfaces, i.e., surfaces satisfying a condition W(kl' k2) = 0 with Wk W k > 0. In these cases, the value R + ZH - 1 in the conclusion of the 1 2 theorem must be replaced by R + 2Jk J- 1, where k is the unique root of W(k, k) = 0. Generalizations to higher dimensional surfaces can also be given. (Received October 6, 1969.)

1075 ABSTRACTS PRESENTED TO THE SOCIETY

The next deadline for Abstracts will be January 31, 1970. The papers printed below were accepted by the American Mathematical Society for presentation by title. The abstracts are grouped according to subjects chosen by the author from categories listed on the abstract form. The miscel­ laneous group includes all abstracts for which the authors did not indicate a category. One abstract presented by title may be accepted per person per issue of these cNotiaiJ. 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. Algebra and Theory of Numbers

69T -A190. GEORGE A. GRATZER and J. P-!:.ONKA, University of Manitoba, Winnipeg 19, Manitoba, Canada. On the number of polynomials of an idempotent algebra. II.

1 heorem 1. Let 21 be an idempotent algebra and let n be the smallest integer with Pn( 21) ! 0.

!1.1 ) ( !1.1) for all m ~ n. Theorem 2. Let M be an idempotent algebra having a If n > 2, then Pm( 1 < pm+ 1 binary polynomial • satisfying the identity x 1 • (x2 • ( .•• (xn-l • xn) •.. )) = x 1 • (x2 • ( ... (xn • xn-l ) ... )) for some integer n. Then pin (~I) < Pm+ 1 ( !1.1) for all m "' 2. (Received September 4, 1969.)

69T -Al9l. AF LRED HORN, University of California, Los Angeles, California 90024. Injective hulls and projective covers of semilattices.

Let K be the category of meet semilattices (A, • ).. If ai E A for all i, the sequence (

called distributive if "tai exists and for all a E A, :Eaai =a- :E ai. A nonempty subset 1 of A is called

a d-ideal if 1 is hereditary and closed under sums of distributive sequences. Let A0 be the result of adding a smallest element to A if A does not have one. Theorem. If A E K, the K injective hull of A is the lattice of all d-ideals of Ao. Definition. If A E K, then a K projective cover of A is an epimor­

phism a: P ~A, where pis K projective and for any homomorphism f: B ... P, B E K, such that af is onto, f is onto. P is unique up to isomorphism. Theorem. Let A E K and let M be the set of most irre­ ducible elements of A. Then A has a projective cover if and only if M generates A and every principal filter of M is finite. In this case, the projective cover of A is the semilattice "freely generated" by Mas a partially ordered set. (Received September 15, 1969.)

69T -Al92. GRANT A. FRASER and ALFRED HORN, University of California, Los Angeles, California 90024. Congruence relations in direct products.

Let C(A) be the lattice of congruence relations of A. Definition. A1 X A2 has property P if for every 9 E C(At X .AQ ), there exist fli E C(AiJ such that e = 9 1 X 82. Theorem. Let K be an equa­ tional class of finitary algebras. Then A1 X A2 has property P for all A 1 , A 2 E K if and only if K

satisfies the following Malcev type condition. For some n ~ 1, there exist k(i)-ary polynomials

Pi• 0 'l!' i < n, polynomials ~j(xo, x1 ) and rij(Yo• y1 , y), 0;;; i < n, 0 ""j < k(i)- 1, and integers <(i) which are 0 or 1, 0 ~ i < n, such that the following identities hold inK: xo =

Po(xf(O)' qoo

Po!Yf(O)' r oo!Yo • Yt• y), ..• ) =pn-1 (Yt-f(n)' lll-t,o (y0 , y1 , y), .•• ), and for 0,. i < n- 1 Pi(xl-((i)' qiO(xo, xl), ••• ) =pi+l(xf(i+1)qi+I,o(xo• xl), .•• ), Pi(Yt-£(i) • riO(Yo• Yt• y), ..• ) =

Pi+l (x((i+l)' ri+t,o!Yo•Y['Y), ..• ). This is a solution of Problem 40 in Gratzer's "Universal algebra",

1076 Van Nostrand, !968. Examples are given. If C(A1 X A2 ) is distributive, then A 1 X A2 has property P (remark due to Alfred Hales). If for every a E C(!T Ail• there exist ai E C(Aj_) such that a= IT ai' then almost all Ai are trivial. (Received September !5, !969.)

69T-A!93. JONATHAN S. GOLAN,. The Hebrew University, Jerusalem, Israel. Quasi-perfect modules.

A unitary module M is quasi-perfect iff it is quasi-projective, has small Jacobson radical, and satisfies the property that U + V = M implies the existence of a submodule V' of V minimal with respect to the property U + V' = M. This generalizes the concept of semiperfect modules introduced by Mares (Math. Z. 82 (!963), 347-360). The structure of quasi-perfect modules and their endomor- phism rings is investigated. Theorem. The following are equivalent for an associative ring with I: (a) R is left perfect [resp. semiperfect]. (b) Every [finitely-generated] quasi-projective left R-module is quasi-perfect. (c) Every [finitely-generated] left R-module has a quasi-perfect cover. (d) Every [finitely-generated] quasi-perfect left R-module has a semiperfect cover, (Received September !5, !969.)

69T-A!94. RONSON J. WARNE, West Virginia University, Morgantown, West Virginia 26505. E -regular semigroups. Part !.

An E-regular semigroup is a regular semigroup whose idempotents form a naturally ordered band [R. J. Warne, Abstract 69T-Al69, these cJ{oticei) 16 (!969), 960]. Let I 0(N) denote the non- negative integers (natural numbers). Theorem. S is a simple E -regular semigroup if and only if 0 * 0 d-1 0 2 S "'((I X [0)) X (Go X Ko ))U {(! XN) X{Go xK0 ))U(U io:I ((I ) X (Gi x Ki ))),where dis a positive integer, d-1 * dF1 ( Gi) i=O (K 0 , {Kili=O) is a collection of disjoint groups (sets), under the multiplication ((n,k), (g1, p)) ((r,s),(hj, 0 q)) = ((n,k)(r,s), gicpi+kd,t~j(jlj+rd,t• x) where gi E Gi' hj E Gj' where form, n E I , m

IPm,n = lt'miPm+! • • · 'Pn-1' where fork E I , 'Pk ='Pk(mod d) and f/li(O ~ i j + rd or i + kd ~ j +rd. Yo is a homomorphism of Gd-! into GK 0 and 'Y! {I ~ i ~ d- I) is a homomorphism of Gi-l into GKi where GKi {0 ~ i !fl d- 1) is the full transformation group on Ki. As special cases, we obtain the structure theorems for simple regular w-semigroups [W. D. Munn, Glasgow Math, J. 9 {!968), 46-66] and E-bisimple semigroups lR. J. Warne, above reference]. (Received September 15,1969.)

69T-A195. C. Y. TANG, University of Waterloo, Waterloo, Ontario, Canada. On Frattini sub­ groups of generalized free products with cyclic amalgamations.

Let P = (P) = ( 4> (-<\) n N; i (- I), where N is the maximal normal subgroup (possibly trivial) of H normal in p. Theorem 2. If H is an infinite cyclic subgroup such that for any strictly descending sequence of sub- groups H = Ko :::> Kr :::> ••• :::> Kn :::> ••• , (~_ 1 , K;) : Kp , where Kp is the normal closure of K in P, r n-1 n n then ot·(P) =I. (Received August 22, !969.)

1077 69T -Al96. RAYMOND BALBES, University of Illinois, Chicago, Illinois 60680. The category of Post algebras. Preliminary report.

Let It be the category of finite chains with more than one element and isotone maps which preserve o, 1; and 13 the category of Boolean algebras and Boolean homomorphisms. For a Post algebra p, denote by p1 the Boolean algebra of complemented elements of P. Theorem. The cate­ gory II of Post algebras and post homomorphisms is equivalent with C X B. Corollary 1. The in­ jectives in (} are exactly those objects p such that Pt is complete. Corollary 2. Q is an essential

injective extension of a Post algebra P of order n if and only if Q has order n and Q 1 is a MacNeille completion of p1 • Corollary 4. If p is countable then it is projective. (Received September 25, 1969.)

69T -Al97. JOHN DAUNS, Tulane University, New Orleans, Louisana 70118. Multiplier rings and primitive ideals.

Let A be a C*-algebra, 1 ~ A; M(A) its double centralizer, R = center M(A), Z = center A, and A c: R + A c: M(A). The primitive ideal space of M(A) is described in terms of a retraction 9: Prim M(A) ... Prim (R + A) which is a homeomorphism on Prim A c: Prim M(A) in the hull-kernel topologies. Proposition. The complete regularization tp: Prim A ... M contains Prim Z c: M as an

open subset. A result of R. C. Busby, j. Functional Analysis 1 (1967), 370-3771 is a Co~ol~!J· If

Prim A is Hausdorff, Prim A contains Prim Z as an open subset. Let A 2 be a closed ideal in

A c: A2 c: R + A and M, M 2 the complete regularizations of Prim A, Prim A2• There is a monic map i: M ... M2 • Let M(l) = {mE MIZ 1 mJ, M(2) = {mE MIZ ~=: m, A2 C: At m.n RJ, M(3) = {mE MIZ ~=:m, A2 ciA+ mt:) RJ. For m 2 (' M2 'd(M), Az!m2 e. C, the complexes. Form I' M(l) U M(2). A/i(m)!!!! A/m, while form E M(3). A/i(m) e. C X A/m. (Received September 26, 1969.)

69T -A198. SIN-MIN LEE, University of Manitoba, Winnipeg 19, Manitoba, Canada. On a class of semirings. Preliminary report.

Let R: be a semiring. R is called '0-semiring if and only if it can be represented as the sum (in the sense of J. P!onka, Fund. Math. 41 (1967), 183-189) of the join direct system of associative rings. In this note we give a characterization of 'E-semirings by a unary operation satisfying five regular equationl3. This result implies immediately that the class of all U-semirings form an equa­ tional class. An example is given to show that this class is a proper subclass of the class of additively commutative and additively regular semirings. (Received September 29, 1969.} (Author introduced by George A. Gratzer.)

69T-Al99. ROBERT GILMER, Florida State University, Tallahassee, Florida 32306. An embedding theorem for HCF -rings.

Let D be an integral domain with identity on which the v-operation is endlich arithmetisch brauchbar, and let Dv be the Kronecker function ring of D with respect to the v-operation. Then D is weakly inertly embedded in Dv ( in the sense of p. M. Cohn [Proc. Cambridge Philos. Soc. 64(1968), 251-264]) if and only if Dis an HCF-ring. Moreover, if Dis a unique factorization domain, then D is weakly inertly embedded in D v, a principal ideal domain. If D is a Krull domain, then D is inertly embedded in Dv (in the sense of P. Samuel [Proc. Cambridge Philos. Soc. 64(1968), 249-250]), and Dv is a principal ideal domain. Further, Dv is a flat averring of D if Dis a Priifer v-multiplication

1078 ring: in particular ,Dv is a flat averring of D if D is a Krull domain. These results complement certain embedding theorems of Cohn [!bid.] and Samuel (ibid.]. (Received September 29, 1969.)

69T-A200. GORDON H. BRADLEY, Administrative Sciences Department, Yale University, New Haven, Connecticut 06520. Algorithm and bound for the greatest common divisor of n integers.

A new version of the Euclidean algorithm for finding the greatest common divisor of n integers ai and multipliers Xi such that GCD = x 1 a 1 + ••• +· xnan is developed, The number of arithmetic opera­ tions is 5n plus five times the number of iterations of the Euclidean algorithm. The theorem of Lame bounding the number of iterations of the Euclidean algorithm (nonnegative remainder version) for two integers is extended to the n integer case. Theo_re~. The number of iterations of the Euclidean algorithm for n integers is never greater than n - 2 plus five times the number of digits J.n the smallest number. s;orollary. The number of iterations of the Euclidean algorithm for n integers is less than n- I plus the logarithm of the smallest number to the base !.6. (Received September 29, 1969.) (Author introduced by Dr. Leopold B. Willner.)

Analysis

69T-B202. C. J, MOZZOCHl, Yale University, New Haven, Connecticut 06520, On the point-

~~~': convergence of Fourier series. Preliminary report.

This monograph is a detailed, (essentially) self contained treatment of the work of Carleson, Hunt, and others needed to establish the Theorem. lf f E Lp(-'Tr, 'Tr) p > l. then Sn(x) converges to f(x) a.e. where Sn(x) is the nth partial sum of the Fourier series for f. (Received May 7, !969.)

69T-B203. MITSURU NAKAI, University of California, Los Angeles, California 90024 and Nagoya University, Nagoya, Japan. On the principal function problem.

Let R be an open Riemann surface (or noncompact sm::>oth Riemannian manifold) and let A be an ideal boundary neighborhood such that u = ?JA is smooth. An admissible linear subspace F(u) of

C(u) is a subspace such that there exists a region BF containing R -A with F(u) :::> [uJuJu E H(BF )J. An imitation operator L is a linear operator of an admissible F(u) into H(A) n C(A) such that (L.l) (Lf)Ju = f; (L.2) if [unl c::H(BF) converges uniformly to zero on each compact subset of BF, then [L(unJu)J converges to zero uniformly on each compact subset of A; (L.3) no u E H(R) satisfies L(uju) = uJA unless it is constant; (L.4) if Ll f- 1, then Jli * d(Lf) = 0 across the ideal boundary li of R. For a given function s E H(A) n C(A), a function p E H(R) with L(pju - sJu) = p(A - s, if it exists, is called an (L,s)-principal function ..T!_leorem. If Ll f- 1, then there exists a unique (L,s)-principal function p for any s. If Ll = 1, then there exists an (L,s)-principal function, unique up to an additive constant, if and only if J li *ds = 0. (Received June 30, !969.)

69T-B204. PAUL SCHAEFER, State University College, Geneseo, New York 14454. Perfect infinite matrices. Preliminary report.

If A =(a nl<) is an infinite matrix and x = [xn} is a sequence, Ax denotes the sequence

["0kankXk}. If Y is an FK space, then YA = [x: Ax E Y} is an FK space. Suppose that F E Y', the space of all continuous linear functionals on Y. The matrices A and B are said to be F -consistent if F(Ax) = F(Bx) for all x E Y An '8 . If X c:: YA then A is an X - Y matrix •. The X - Y matrix A is

1079 perfect if X is dense m YA. Theorem 1. if A and B are X - Y matrices, if YA c Y8 , if A is perfect, and if for some F E Y', F(Ax); F(Bx) for all x EX, then B is F-consistent with A. Theorem 2. Let

A be an X - Y matrix and let F E Y'. Suppose that to each f E YA there corresponds a matrix B ; B(f)

with y8 ::> YA so that f(x) ; F(Bx). If for each such matrix the condition F(Bx) ; F(Ax) for x EX

implies that B is F-consistent with A, then A is perfect. The matrix A is Y-reversible if to each

y E y there corresponds exactly one x E YA so that Ax; y. A matrix A is of type My if (1) the columns

of A are elements of y and (2) G E Y' vanishing on the columns of A implies that G is the zero func

tiona!. Theorem 3. Let A be a Y-reversible matrix and suppose that X c YA has a Schauder basis 1 2 ( e , e , .•. } in the relative topology of Y A. Then A is a perfect X - Y matrix if and only if A is of type My. (Received July 14, 1969.)

69T-B205. SAMUEL ZAIDMAN, Universite de Montreal, Montreal, Quebec, Canada. A variational property for a family of vector-valued functions.

Let H be a Hilbert space; G(t) a family of linear operators H - H, - OJ

V x E H, G(t)x is continuous Vx E H. Let h(t), h(B) ; 8, be continuous, - oo < t

U- the class of functions admitting representation u(t) ; G(t)x + h(t), U B the subclass of bounded

(on- OJ< t

Then we have Theorem. Suppose 0

'lx E H, vt E (- OJ, OJ). Then, every minimizing sequence in l£ is uniformly convergent on- OJ < t < OJ is strong H-topology. (Received August 21, 1969.)

69T-B206. STEWART P. HASTINGS; Case Western Reserve University, University Circle,

Cleveland, Ohio 44106. A boundary value problem related to the Falkner-Skan equation.

The following theorem extends a result announced in Abstract 69T-Bl29, these cJiotiaiJ 16 (1969), 678-679. The proof is essentially the same. Theorem. For any a > 0 there is a >.. 0 < 0 ------. 2 such that for 0 >). > >.. 0 the boundary value problem f'" + ff" + ).(h- f' ) ; 0, h" t fh' ; 0, f(O); f'(O); 0, h(O); a; f'(OJ); h(OJ) = 1 has at least one solution. (Received August 21, 1969.)

69T-B207. SYED M. MAZHAR, Aligarh Muslim University, Aligarh, India, and R. S. KHAN,

Department of Applied Science, College of Engineering and Technology, Aligarh Muslim University, Aligarh, India. On Fourier coefficients of positive functions.

In this note a number of theorems have been proved which generalize certain results due to

Askey and Boas [Math. Z. 100 (1967), 373-379]. A typical theorem is as follows: Theorem. Let

G(x) l on (0,'11'), G bounded below and JgxdG(x) finite so that dG has generalized sine coefficients bn

(2/'11') J~sin nxdG(x). If 1 < p < OJ, ).(x) is a positive function such that (i) x 1+ 0 ).(x) is decreasing p+l-O 1/p p for some small 0 > 0, (ii) x A(x) J t OJ for some small c5 > 0, as x- OJ, then (). (n)bnJ E 1 1/p -1-2/p t p if and only if). ('11'/t)t J 0 x dG(x) E L . (Received August 18, 1969.)

69T-B208. E. WARD CHENEY and KENNETH HUGH PRICE, University of Texas, Austin, Texas 78712. Projections into subspaces of finite codimension.

Theor~. If Y is a closed subspace of codimension n in a normed space X, then for each f > 0 there exists a linear projection of norm at most n t 1 t f from X onto Y. If Y is an 8- space 1080 (i.e., each point in X has at least one nearest point in Y) then (can be taken to be 0. lt is an open question whether the infimum of (IJPIJ: P projects X onto Yl is necessarily attained when Y is an c'J -space of finite codimension in X. (Received August 4, 1969.)

69T-B209. U. KRENGEL, Ohio State 'University, Columbus, Ohio 43210. On the existence of !_inite strong generators in a given exhaustive subalgebra.

Theorem. Let (11, F, ~) be a probability space for which ~ is countab1y generated mod ~, and let T be a nonsingular invertible transformation in n without T -invariant probability measure k u 0 < < ~· If ~ is an exhaustive increasing sub-~:r-algebra of£; (i.e. TQ ::> g, U T g generates !:; mod~), then the system of sets E E Q which are strong generators ( {E, TE, .-f E , ••• ) generates

£, mod ~) is dense in Q; in particular strong finite generators for~ exist in g. (Received September 5, 1969.) (Author introduced by Professor D. K. Ray-Chaudhuri.)

69T-B210. JOHN WILBUR, Pacific Union College, Angwin, California 94508. Direct and inverse limits of Banach spaces.

Let D be a directed set such that any countable subset has an upper bound. To each a. E D associate a normed space Ba. and to each pair a. ~ fJ a linear operator f~: Ba. - BfJ with IJf!IJ ~ 1. Let the maps fe satisfy f~ = 1 and f~ • f~., f: for a. ~ fl ~,. The resulting system {B a., f~ ) is called a direct system of normed spaces. If instead we write a,. and retain all conditions but reverse the maps the system (8,., fe lis called an inverse system of normed spaces.The direct limit, lim_ (Ba., f ~ )• is defined by taking the usual equivalence classes and algebraic operations and defining the norm of an equivalence class to be the infimum of the norms of its members. The inve:J;Se limit, lim_ {Ba.• fUl. is defined by taking the usual subset of the product space and algebraic operations and defining the norm to be the supremum of the norms of the components. If all the Ba.'s (or Ba.'s) are Banach, reflexive, or pre-Hilbert, the limit is shown to be the same. The topology of the direct limit is the topology of the corresponding inductive limit, while the topology of the inverse limit is the bornological topology associated with the corresponding projective limit. The duality of direct and inverse systems is discussed and examples are given in the Lp spaces. (Received September 10, 1969.)

69T -B211. DAVID ZEITLIN, 1650 Vincent Avenue North, Minneapolis, Minnesota 55411. On moment identities for the coefficients of cyclotomic polynomials of order P1P2 ••• Pn·

Let 3 ~ p1 < p2 < ..• < Pn be any set of n distinct odd primes, and for n ~ 2, set P(n) = p 1p2 ••• Pn and d = !p(P(n)) in the cyclotomic polynomial DrjP(n)(xr - l)~(P(n)/r) = d - d k i F(x, P(n)) :C0 + c1 x + .•. + Cdx • Let M(i,a) = 'i:k=O a k Ck, with M(i,1) sM(i). Theorem 1. 6M(2) = n d(A - 3B), where A = (2p1 - 1)"Dj=2(1 + Pjl• and with an empty product defined as 1, B =

!:~_= 2 !p(P1P 2 ••• Pk_ 1lnJ=k+1(1 + pj); M(1) = M(1,- 1) = d/2, 2M(2,- 1) = d(A- 3B- d), 4M(3) = d2(A - 3B - d), and 4M(3, - 1) = d2(3A- 98 - 4d). An explicit formula for M(4) is omitted. The ' s 4 4 3 2 2 3 2 3'2 Slmplest case Ol::k=O k Ck =- 3S +(9pq t 10)s t ((p t q) - 2)s - (p t p t q t q t pq- 1)s,

where s = lfl(pq). Theorem 2. 2M(4,- 1) = 30 M(4)- 3d3(A- 3B)- d2(A- 3B)2t 5d\ 12 M(5) = 30dM(4) + 6d5 - 5d4 (A - 3B), and 4M(5, - 1) = 150d M(4) - 20d4(A - 3B) - 5;f (A - 3B)2 t 32d:,. Special cases of Theorem 1 for n = 2 and n = 3 appear in the author's paper, "On coefficient identities

1081 for cyclotomic polynomials Fpq (x)," A mer. Math. Monthly 75 (1968), 976-980. Application of Lemma 6 (p. 979) in the cited paper gives Theorem 3, For n = 3, 4, ... , F(x,P(n)) is a factor of 2 xa(b-l/ 2 Ub-I(f(xa)), where a= 1 for n = 3, a= p4p5 ... Pn• n ~ 4. b = p3p2PJ.. f(x) = (1 + x)/(2{x l/ )), and Um(Y) is the Chebyshev polynomial of the second kind, (Received September 15, 1969.)

69T-B212. JOSEPH L. WALSH, University of Maryland, College Park, Maryland 20742. Mean approximation by polynomials on a Jordan curve.

Let C be an analytic Jordan curve containing 0 in its interior, H2(k,a.), 0

classes, Then if f(z) E L2(k,a.), 0 < a. ~ 1 on C, we have f(z) "'f1(z) + f2(z) on C, where f1(z) E H2 (k,a.) and f2(z) E G2(k,a.) on C. The latter functions are uniquely determined by f(z), (Received September 12, 1969.)

69T-B213. LOUIS PIGNO, State University of New York, Stony Brook, New York 11790. Some multiplier problems in Fourier analysis on groups.

Let G be a LCA group with dual a. Let H and F be Banach spaces contained in Lp 1 (G) and Lp2(G) respectively where 1 ~ p 1 ~ 2 and 1 ~ p2 ~ 2. Suppose Eisa Haar measurable subset of G

and 0 is a complex-valued function defined on E. Suppose further, that E is contained in some a-compact subset of G." Definition. 0 is a multiplier of type (H,F ,E) if, given h E H, there corres- ponds some f E F such that ((JhA = /'t.f a,e, onE, 1'\hand "f being the Fourier transforms of hand f respec-

tively, The Banach spaces L 1(G), Lp1(G) n Lp2 (G), L 1(G) nC(G), L 1(G) n C0(G), and L 1(G) n B(G) are defined in Rudin r· Fourier analysis on groups," lnterscience, New York, 1962].

Theorem I. (Lp L 1 n B, E)= (L1 n B)" JE, Theorem 2. (Ll' Lp1 n Lp 2• E)= (LPr n Lp2 (' JE,

1 ~ p1 ~ 2 and 1

69T -B214. HARI M.· SR IV AST AVA, University of Victoria, Victoria, British Columbia, Canada, A class of bilateral generating functions for the Jacobi polynomial.

put(*) G~,y) = !;p+q~ n (- n) c Jl yq, where n.l is the Pochhammer symbol and the c... q p,q=O p+q p,q m I!• are arbitrary constants. Making use of the specialized forms of some of his earlier results [H. M, Srivastava," Infinite series of certain products involving Appell's double hypergeometric functions," Glasnik Mat, Ser. lll, 4 (24) (1969), 67-73] the author derives here bilateral generating .oo [ functions of the type (**l'Bn=O (X) n2F1 p- n, a.; X+ p; x] G[y,zJn t /n~, where p, X and a. are arbitrary complex numbers. In particular, it is shown that when G[y,z] is a double hypergeometric polynomial, the right-hand side of(**) belongs to a class of general triple hypergeometric functions introduced by the author [H. M. Srivastava, "Generalized Neumann expansions involving hypergeometric functions," Proc, Cambridge Philos, Soc, 63 (1967), 425-429]. An interesting special case of(**) when p =- m,

1082 m being a nonnegative integer, yields a class of bilateral generating functions for the Jacobi poly­ nomials ~o.·fJ 1(x)ln = 0, I, 2, ••• } in the form (***)I;00 (m~n )Jn-n,p-n)(x)G[Y,z]tn /n~, which n · n=0 m+n proves a unification of several known results. Further extensions of(**) and(***) with G(y,z]

replaced by an analogous multiple sum H(y 1, ••• ,y m] are also discussed. (Received September 18,1969.)

69T-B215. ELLEN M. TORRANCE, Mount Holyoke College, South Hadley, Massachusetts 01075.

Adjoints of ope.~tors. Preliminary report,

Let X be a complex Banach space, and H the set of hermitian operators on X, where T is hermitian iff III+ io.TII = 1 + o(o.) as a. .. 0, a. E II!., Define (T +iS)*= T-iS, forT, S, in H. If X= lp(n), 1 ~ p ~ oo, p F 2, then Tin B(X) has an adjoint iff T has a complex diagonal matrix, and (hij)* = {hijl· If Ko is (locally) compact, X =Co (l

69T-B216. ROBERT E. ATALLA, Ohio University, Athens, Ohio 45701. On the multiplicative behavior of Toeplitz matrices.

Let X be compact T 2 , T a bounded linear operator on C(X) such that Tl = 1. For p E X, let Kp be the support of the Borel measure representing f ... Tf(p), and K =closure U ( !)>: p E X}. Let MT = (g E C(X): T(fg) = TfTg for all f E C(X)l, and CT = fg E C(X): Tg =constant}. We show that g E MT iff g is constant on each i)>, and g E MT f'l CT iff g is constant on K. If T!!; O, then g E Mr iff (Tg)2 = Tl. These ideas apply to matrix summability, via the representation given in (Proc. Amer, Math, Soc, 21 (1969), 36-42, with J. Bustoz), For instance if T!!; 0, then Tis multipli­

cative in the sense of Henrikson (Math, Z. 71 (1959), 427-435) iff CT c ~. (Cf, Hill and Sledd, Canad, J, Math, 20 (1968), 410-418.) Also, we give an example of a positive regular multiplicative matrix such that the corresponding set K (see Henrikson, op. cit,, page 432) is nowhere dense in {JN\N. (Received September 24, 1969,)

69T-B217. VlJENDRA SINGH, Aligarh M•J.slim University, Aligarh, India, On a relation be­ tween harmonic summability and Riemann-Ceshro summability.

Let I:a be a given infinite series with s as its nth partial.sum. We denote by Sk the nth n n n Ceshro sum of order k of this series, A series ~an is said to be harmonic summable to the sums, if T /log n .. s, as n .. oo, where T ='tn p s and p = 1/(n + 1). The series !:a is said to n n v=O n-v v n n be summable by Riemann-Ces~ro method of order a. and index 1 or briefly summable (R, 1, a.) to the 1 sum s, 1·f the senes· c-a. t a.+I~ooL.Jn=l sna. sm· nt1 nt converges to a f unction F (t) m· some interval 0 < t

1083 69T-B218. VOJ!SLAV S. MAR!C, Institute of Physics and Mathematics, University of Novi

Sad, Yugoslavia. !'~ymptotic _behavi~-~~~olutions of a nonlinear different~_al equation of th_:_ __.~~::st order.

The asym;>totic behavior at infinity of solutions of the equation p.' = P(u,t)/Q(u,t) is studied. P, Q are p:>lynomials in p. whose coefficients are functions oft and belong to the Hardy class 'II. The latter being defined as the set of all real valued functions defined, for large t, by a finite combination of ordinary algebraic, exp and log operations acting on the real variable t and on real constants. It is proved that for any continuously differentiable solution l.l (t) there exists one or the other of the asymptotic formulae l.l (t) - h(t), In u(t) - h(t), within the class 'II, i.e. h(t) E 'II . As the main tool for the proof it is first shown that any (real) solution y(t) of an algebraic equation whose coefficients are elements of 'JI behaves at infinity again as an element of 'JI. (Received Septem':Jer 19, 1969.)

69T-B219. JAMES D. BAKER, Texas Instruments, Inc., 13500 North Central Expressway, Dallas, Texas 75222 • .!_'!:obability representations with Hight Cauchy integrals.

While the Right Cauchy refinement integral is not a measure integral, it is related to the Young refinem.:mt integral by Theorem 1. Suppose his of bounded variation on [ a,b] and that h(a+) = h(a). Then there is a function g of bounded variation on [ a,bl such that RC rb fdg = rbfdh for · ,J a J a each quasi-continuous function f. Theo_:_em 2. If one of the integrator or integrand functions is a probability distribution and the other is quasi-continuous on [a,b] , then the Right Cauchy and Young integrals are equal. Theor_em 3. Suppose t is a real number, and ,. 1 and T 2 are independent random variables with distribution functions F 1 and F 2• If gt(x) is continuous except at a finite number of points on (-co, too), and ifF (g) is quasi-continuous, then P(T :s; g (T )) = RC +ooF (g (x))dF (x). 1 t 1 - t 2 •r -co 1 t 2 An example is given which shows that this representation, which is a generalization of the convolution integral, does not hold for many other refinement integrals. (Received September 25, 1969.)

69·~·-B220. DOUGLAS W. WILLETT, University of Utah, Salt Lake City, Utah 84112. A necessary and sufficient condition for the oscillation of some linear second order differential equation.

t Assume that p E C [ a,oo) and P is an averaged integral of p. Thus, P(t) = C - Ja P for some constant C, and C = r00p if this integral exists. Let E(t, s) = exp (2jsP) and Q(t) = r00P 2 (s)E(t,s)ds. oa t .Jt The equation y" + p(t)y = 0 is disconjugate on [a,oo), if and only if, L)~= Qk(a) < co, where ~(t) = Q(t), co... 2 oo n-2 ° . Q1(t)= J th(t,s)Q (s)ds, Qn(t) = ft E(t,s) Qn_ 1(s) [Qn_ 1(s) t 2:Bk=OQk(s)] ds, n = 2, 3, •••• (Rece1ved September 29, 1969.)

69T-B22l. NORBERTO ANGEL FAVA, University of Minnesota, Minneapolis, Minnesota.

55455. ~ea~_!:ype inequalities for iterated operators.

Let (n, :;, l.l) be a a -finite measure space. We denote by Rk the class of all functions f, such that Jn·lil/t (log+ lfl/t)kdj.l is finite for every t > 0. Theorem. Let Mr. •.. ,Mk be positive sublinear operators of weak type (1,1), defined on L1 t L00 , such that 0 ~ f ;!! g implies Miif !!1 Mig• and

IIMi Ill 00 :o; llfll 00 • Under these conditions, we have: (1 °) If f is a nonnegative function in Rk-l' then the operation Mk···Ml f is well defined and jJ. ( Mk •.. M 1f > 4t) ~ Con st. J (f/t) (log+ f/t)k-l dj.l (t > 0); (2°) Iff is in Rk, then Mk ••. M1f is integrable over every set of finite measure, and conse­ quently it belongs to L 1 + L 00 • By using this result, it is possible to extend the validity of the individual.

1084 ergodic theorems of Dunford and Schwartz fork semigroups of operators to any function fin Rk-I• The theorem also yields a simple proof of the strong differentiation theorem of Jessen, Marcinkiewicz and Zygmund. (Received September 29, I969.)

69T-B222. RICHARD STAUM, Polytechnic Institute of Brooklyn, Brooklyn, New York II20I.

_03__1l_Pherical completeness and sep_!!ability in non-Archim~~an normed spaces. Preliminary report.

Let X be a non-Archimedean normed space over a valued field F, with IIX II = IF j. Let V = (x EX: II xll ~ l}; for 0 < d ~ 1, let Pd = (x EX: llxll

space over F properly extending X yields a proper extension of R1 or of IIX 11. This extends a result given by I. Kaplansky for non-A-rchimedean valued fields [Duke Math. j. 9(1942), 303-32I]. (Received September 29, 1969.) (Author introduced by Professor Harry Hochstadt.)

Applied Mathematics

69T-C34. ALAN D. SOLOMON, Department of Applied Mathematics, Tel Aviv University; Ramat

Aviv, 1 el Aviv, Israel. ~~aximum principle for an ini~ially parabolic, hyperbolic equation. Pre­ _liminary report.

We consider the equation (*) ut + tutt = uxx' - oo < x < oo, t ;;; 0 which for t > 0 is of hyper- bolic type and for t = 0 reduces to the heat equation. The characteristics of (*) are the parabolas t = ( I/4) (x - c )2 having horizontal tangents for t = 0, two of which pass through each point of the

upper half-plane. Let u(x, t) be a solution of (*) satisfying the initial condition u(x,O) = f(x), and necessarily ut(x,O) = r• (x), for f(x) a given twice continuously differentiable function on - oo < x < oo. Let P be any point of the upper half-plane; suppose the two characteristics passing through P inter- sect the x-axis at the points A, B. Using standard techniques of partial differential equations we prove the following _:I'heorem. If f(x) is not constant on (A,B) then min f(x)

Geometry

69T-D22. DONALD A. EISENMAN, University of California, Berkeley, California 94720. Holomorphic mappings into tight manifolds.

If M and N are complex manifolds the set of holomorphic mappings from M to N, A(M,N), is a topological space under the topology of uniform convergence on compact sets. For U an open subset of M define iU: A(M,N) ... A(U ,N) to be the restriction to U of functions in A(M,N). Theorem. Let M be tight (Wu) (same as hyperbolic (Kobayashi)). Then for any complex manifold M iU: A(M,N) ... A(U, N) is a homeomorphism into. Theorem. Let N be tight, p EM, and f: M ... M holomorphic

1085 with f(p) = p. Then: (i) Jdet dfpl ~ 1, (ii) dfp =identity matrix iff f = idM' and (iii) Jdet dfpJ = 1 iff f is an automorphism of M. (This extends a theorem of Wu.) Theorem. Let M be tight, U open and relatively compact in M, and f: M - M holomorphic such that f(U) = U and iu(f) is an automorphis1.1 of U. Then f is an automorphism of M. (Received June 13, 1969.) (Author introduced by Professor Shiing-Shen Chun,)

69T-D23. PATRICK BARRY EBERLEIN, University of California, Los Angeles, California 90024. Negative curvature manifolds admitting no constant negative curvature metric,

Let M be a compact n-dimensional Riemannian manifold of negative sectional curvature. lf M admits a Riemannian metric of constant negative curvature, its Pontryagin classes are zero and consequently, if M is orientable and of dimension 4k, its index in the sense of Hirzebruch is zero, For every positive integer k, there exist compact complex analytic manifolds of real dimension 4k, negative sectional curvature and arbitrarily large index. Such manifolds can admit no Riemannian metric of constant negative curvature. (Received September 18, 1969.)

Logic and Foundations

69T-E81. SAHARON SHELAH, The Hebrew University, Jerusalem, Israel. On infinitary languages with nonhomogeneous strings of quantifiers.

This notice presents a solution for a problem presented in an article by Keisler in" The syntax and semantics of infinitary languages." Let the set of predicates be fixed. Let LX •X be a

language with the usual logical operations and with conjunction on

Theorem 1. Every formula in L~,). has an equivalent formula in Lx•).' Let L(oo) be the union of all the Lx 'X. and L 1(oo) will have in addition quantifications where the quantifiers are ordered in any linear ordering. Theorem 2. Every formula in L 1(co) has an equivalent formula in L(oo). For example

[ ... (Vxn)(:i!yn) ... (VxoH:!rYol] rp+-+(Vxo ... xn ... )(:i!yo ... Yn ... l/'.n

69T-E82. MURRAY JORGENSEN, University of British Columbia, Canada. An equivalent form of Levy's axiom schema, Preliminary report.

Levy's schema (Levy, Pacific J. Math. 10 (1960), 227) is (1) Every normal function defined for all ordinals (d,f,a.o,) has at least one inaccessible number in its range. Con,;ider the statement: (2) Every normal function d.f.a.o. has at least one regular number in its range. From Levy, p. 228, we see that ( 1) implies (2) in ZF. ln ZF + AC (axiom of choice) the two statements are equivalent as (2) then implies the existence of arbitrarily great inaccessible numbers, Definition. Iff is an ordinal valued function d,f,a,o. (not necessarily normal!) an ordinal a is inaccessible (f) if /3 f3. In particular in

1086 Z F t AC one can consider the instance of (3) where f( y) = iY and thereby recognize Levy's schema as a natural strengthening of Tar ski's axiom of inaccessible cardinals. (Received September 5, 1969·.) (Author introduced by professor David W. Bressler.)

69T-E83. RICHARD J. LAVER, University of California, Berkeley, California 94720 and University of Bristol, England. Partitioning order types.

If (L, ~ J is a linear ordering, let tp(L, ~ L) be the order type (isomorphism type) of (L, ~ L).

For order types ~ and 1/J• ~ ~ 1/J <--+ dfll' is embeddable in rP• and tp "' 1/• <--+ df ~ ~ 1/J and 1/J ~ lfJ • Let 17 be

the order type of the rationals. Theorem 1. If ~-is a countable order type= tp(L, ~ L)' then there is

an integer n such that if L is partitioned into any finite number of subsets L1 , L 2 •···• Lk' there will be a ~ n-element set I ~ {1,2, ... , k} with ~ ~ tp( UiEl L 1, ~ L). If n can be taken to be I in this theorem, 1fJ is said to be strongly indecomposable. Let .P 0 =the set of order types (0,1), and let tp E .Pat!<--+ tp = (tp 1 + tp 2 + ••• + 'Pn + ••• )or tp = ( ... + 'l'n + ... + ~ 2 + 'P!) for order types 'Pn E .Pa , n < w, where each ~i ~~~'it!' and let .PX = Up<>.. .P fl for X a limit ordinal. Theorem 2. If

lfJ is a countable order type, then lfJ is strongly indecomposable <--+ ttJ "17 or 'il ~· E Ua

69T-E84. ERIC MENDELSOHN, Universite de Montreal, C. P. 6128, Montreal, Quebec, Canada.

Let p be the category of partly ordered sets ( ~) and monotone maps. A set of elementary axioms on a category is given such that every categor.y which is a model of this set of axioms, and also complete as a category, is naturally equivalent toP. (Received September !8, 1969.)

69T-E85. A. L. RUBIN and JEAN E. RUBIN, Purdue University, West Lafayette, Indiana 47907. Accumulation functions on the ordinals.

In J. Doner and A. Tar ski ("An extended arithmetic of ordinal numbers," to appear in Fund.

Math.) higher operations, 0 , on the ordinals are defined as follows: a00 fJ =a.+ f3, and if y > O,aO B ~ . y U1!<{3•~<.y (aOy?j )0 ~a.. Now define functions =y on the ordinals as follows: (!) e 0(0) = "'O ( 1 ) = 0. ( 2) If y > 0, "'y( 0) = "'y (I ) = I . ( 3) If a > 0, "''Y( a. + I) = "' ~a. )0 y a. ( 4) If a. = Ua f 0, =y(a.) = U B<" '=y(fJ). The values of •o are the triangular numbers and = 1 is the factorial function. In this paper we describe these accumulation functions, '"y, and their fixed points, for all ordinals y. (Received Septem!Jer 22, !969.)

69T-E86. ALISTAIR H. LACHLAN and J. T. BALDWIN, Simon Fraser University, Burnaby 2,

British Columbia, Canada. On countable theories categorical in power ~ 1 but not in power ~o.

Theorem I. Let a be a model of such a theory then some inessential expansion of a contains

a strongly m;.nimal set. Theorem 2. Any such theory has ~O distinct isomorphism types of count­ able models. The main tools are the methods of Marsh (Ph. D. Thesis, Dartmouth !966) and Keisler's theorem that such theories do not possess the finite cover property. (Received September 26, !969.)

1087 69T-E87. DOUGLASS B. MORRIS, University of Wisconsin, Madison, Wisconsin 53706.

Ch~ice and cofinal well-ordered subsets. Preliminary report.

Let MAT ="Every simply ordered set has a cofinal subset which is well-ordered by the in­ duced ordering." 0 ="Every set is simply orderable," OE ="Every partial ordering can be extended to a simple ordering."' At the AMS Institute held at U. C. L. A., in the summer of 1967 A. Mathias gave a model of ZF + 0 t , OE and asked whether ZF t MAT+ , AC was consistent.

Theorem 1. Con(ZF) ... Con(ZF + MAT + 0 + ~ OE ). The proof involves adding a set C of Cohen reals

whose labels are in wx (1.1 and symmetrically well-ordering every subset of C whose labels are in m X 14,

for each m. Let X(X) be the least aleph not l!i X. Let a W-set be a set obtained from well-ordered sets by iterating the taking of unions of well-ordered collections. The class W was first studied by Keisler (Ann. of Math." Logic," first issue, forthcoming). Theorem 2_ ZF +MAT 1- (a) Each simply ordered W-set is well-orderable. (+b) If for some nR(a.) is not well-orderable, the least

such n is a successor ordinal.(+ c) 1f N ( N+ is regular a, N+ l!i 2 N). (+ d) Every simply ordered set has a maximal well-orderable initial segment. (+ e) No infinite Dedekind finite set is simply

orderable. (+ f) For every simply ordered X, X(X) is a successor cardinal. From (a) and (b) we obtain ZF t MAT+ V = WI-AC. Theorem 3. ZF t MATt OE \- AC. (Received September 29, 1969.)

69T-E88. JAMES R. GEISER, Dartmouth College, Hanover, New Hampshire 03755. Complete­ ness of the nonstandard reals.

Terminology from E. Zakon, "'Remarks on the Nonstandard Reals," in" Applications of Model Theory •.. ."' Using ultra limits and continued fractions we establish: I. There is an enlargement *A" of A in which *E 1, the nonstandard reals, is metrizable. ll. In an enlargement, if *E 1 is metrizable then it is not cauchy complete. Zakon raised the possibility that comprehensiveness and nonmetri­ zability might be equivalent. This is answered negatively by: III. There is a comprehensive en­ largement in which *E 1 is not metrizable. II is a consequence of a more general result: IV. If

~: "'A- *A" is a monomorphism and *N, the nonstandard integers, satisfies the following conditionS, then *E 1 is not complete in its uniform topology (see Kelley's" General Topology," p. 192). Condition S: There is a segment S c *N, bounded above but with no least upper bound and a map p: S ... *N whose range is cofinal in *N and for all n E S, pfSeg(n) is internal. V, If w1 is cofinal with *Nand *A is N0-comprehensive then condition S holds and *E1 is not co~plete in its uniform topology. Results II and V give partial answers to the question; is *E1 ever com:;>lete in its uniform topology? (Received September 29, 1969.)

69T-E89. E. M. KLEINBERG, Massachusetts Institute of Technology, Cambridge, Massachu­ setts 02139. Somewhat homogeneous sets. II. Preliminary report.

Let x > X> 6 be cardinals, x uncountable, and let ')' and 11 be limit ordinals less than x. Then X ... (')')~ ,o is the partition relation for each partition of the 11-sequel!._c~~!".£!!1 x into X pieces, there exists a type-y subset of x the set of whose 11-seguences meets at m~st II of these pieces.

Theorem 1. ZF + ACI-w~ 11 and II

1088 w+w _!heorem_4. ZF r If x- (1<) 2 , 1 then xis measurable (under the same measure as in Theorem 2). Theorem 5. ZF 1- x- (xl~', 1 is inconsistent with x-well-ordered choice. (Received September 30,1969.)

69T-E90. ALEXANDER ABIAN, Iowa State University, Ames, Iowa 50010. Consist~cy of ~H via the method of synergistic models.

Using definitions and results of Abstract 691 -E48, these cNoliCe.O 16 ( 1969), 686 and Abstract 69T-E64, these cNoliCe.O 16(1969), 844,let (K, £)be a model for ZF t C. But then between every two denumerable well ordered sets e and g of (K, £)it is possible to assert the existence of a well defined equipollence E(e,g) in (K, £). Consider the Synergistic model (S, £)corresponding to the finite in- 1 n . stances F (x,y), ... ,F (x,y) of R deflned as follows: Ao = (O,l, } •.• ,IL' and Aut! =Au U Pu us uUFuUl ··• n U Fu U.:U UE uU( UEu) U(IJUEu) and Au = Uv

69T-E91. ROBERT I. SOARE, University of Illinois at Chicago Circle, Chicago, Illinois 60680. Automorphisms of the lattice of r. e. sets modulo finite sets. Preliminary report.

Let a;"J denote the lattice of r .e. sets modulo finite sets as in Hartley Rogers," Theory of recursive functions and effective com;Jutability," McGraw-Hill, New York, 1967. The following results resolve questions posed by Rogers on pp. 228-229. Theorem 1. Every autom:>rphism of a j"J is induced by some permutation f of N, such that f restricted to a is an automorphism 3f a. (A. H. Lachlan has shown that there are z~O automorphisms of a;'J, so in general f will not be a recursive permutation.) Corollary. Every automorphism of &j"J is induced by some automorphism of a. _!heorem 2. Turing degree is not invariant under automorphisms of a;'J. Theorem 3. Every non­ recursive element of a; "J may be taken by some automorphism into an element of different one-one degree. Corollary (D. A. Martin). Creativeness is not invariant under autom:>rphisms of a;"J. (The proof of Theorem 1 is direct; those of Theorems 2 and 3 use priority arguments and a device of D. A. Martin which he used to prove that hypersim?licity and creativeness are not invariant (unpublished). (Received October 1, 1969.)

Statistics and Probability

69T-F21. ALBERT A.MULLIN, USATACOM, Building, 200A, Warren Michigan 48090. A new class of queueing problems.

Consider a queueing situation in which a customer Pnters a commodity-dispensing facility

1089 (CDF) containing.!!. items located at positions equally spaced along a single straight-line goods -counter (G.C.). One wishes to place !!l items located in consecutive bins into a container and make payment for them at a single servicing facility (S. F.). Assuming that one can enter (but then not leave) the servicing queue (SQ) at~ time after entering the CDF, that the spatial length of the S. Q. is negligible compared to the length of the G. C., that the S. F. is negligibly distant from the reference end of the G. C., that the customer can transfer only one item at a time from the G. C. to his container, and that the customers outside the S. Q. move with uniform speed, an estimate of the optimum time of entry into the S. Q. is given to minimize the total time spent in the CDF. The existence of an optimum time to enter the S. Q. is clear, since if one enters the S. Q. judiciously before collecting all!!! items, then during the waiting period in the S. Q. and S. F., one can collect the remaining items from the G. C. However, miscalculation results in leaving the CDF without all .m_ items or results in an unnecessarily long stay in the S. Q. (Received August 18, 1969.)

691-F22. JOHN E. DUTT, Ayerst Laboratories, Medical Department, New York, New York

10017. New expansions for the m ·~ltivariate probability integral.

By making use of certain integral transformations on existing series expansions, new rapidly convergent expansions are found for the normal probability integral for dimensions 2, 3, 4 etc. Numerical comparisons are made to existing tables. (Received September 25, 1969.)

69T- F23. WALTER J. HENDRICKS, Case Western Reserve University, Cleveland, Ohio 44106. Lower envelopes for processes with stable components.

A type of Markov process X(t) in R d with stationary independent increments is defined by

considering X(t) "' (X1 (t), ... ,Xn(t)), where, for 1 "'i ~ n, Xi (t) is a stable process of index ai in

Euclidean space of dimension di and d = d1 t ... t dn. We let 0 < an < ... < a. 1 ~ 2 and call X(t) a process with stable components. By use of some potential theory, estimates are made upon the capacity and delayed hitting probability of certain spheres with respect to certain types of X(t). Integral tests are developed for determining lower and upper envelopes of functions with respect to X(t) as defined above as t approaches zero or infinity. (Received September 26, 1969.)

Topology

69T-G151. F. THOMAS FARRELL and JOHN B. WAGONER, University of California, Berkeley, California 94720. An alg.,braic criterion for a map to be a proper homotopy equivalence.

Preliminary report.

Let f: K ... L be a proper map between finite dimensional, connected, locally finite CW- i i . i i complexes with finitely many stable ends E' k and E'£ (1 = 1 •···~n) such that f( E'k) c: E't (i.e., given 1 1 neighborhoods Vi of the ££, there are neighborhoods Ui of E'k with f(q ) c Vi). Suppose the induced 1 1 - homomorphisms '7Tl ( E'k) ... '7T1 (K) and '7Tl (E'£) ... '7Tl (L) are monomorphisms. If X is a space, let X= universal covering space of X and Hc(X)* =cohomology with compact support of X. Theorem l. f is a

proper homotopy equivalence iff (l) f: 11 (K) ... 111 (L) and f: 111(E'ki) ... 11 (E'~) are isomorphisms, - - * *jl 1 *- ,If 1 G and (2) f*: H*(K) ... 1-\. (L) and f : Hc(L) ... He (K) are isomorphisms. This criterion is a generalization i i of the Whitehead Theorem on homotopy equivalences. When 11 1 (E' k) ... 111 (K) and '7T1 (E'.t') ... 111 (L) are

1090 not monomorphisms, there is a result (useful for applications) in which condition (2) is replaced by one which is more geometric, Theorem 2. Iff: X .. Y is a proper map between connected, locally finite, finite dimensional CW -complexes which are simply connected and simply connected at infinity, * * * then f is a proper homotopy equivalence iff f*: H*(X) .. H*(Y) and f : Hc(Y) .. Hc(X) are isomorphlsms, L. C. Siebenmann has informed us that he has some results in this area, (Received July 22, 1969.)

69T-Gl52, CHARLES L. HAGOPIAN, Sacramento State College, Sacramento, California 95819. Concerning the cyclic connectivity of plane continua,

Suppose that a compact metric continuum M is semilocally-connected at all except a finite number of its points and is such that for each point x in M, M is either not semilocally-connected or not aposyndetic at x, M is known to be arc-wise connected (See Abstract 69T-G75, these cJfoticei) 16 (1969), 694), In this paper it is proved that if p and q are distinct points of M and no point cuts M weakly between p and q, then there exists a simple closed curve in M which contains p and q. Also an example is given which indicates that the theorem does not remain true if the word 'finite' (in the first sentence of this abstract) is replaced by the word 'countable'. (Received July 24, 1969.)

69T-G153. OFELIA T, ALAS, Universidade de Sao Paulo, Brasil. Topological groups and uniform continuity. III.

Let m be an uncountable regular cardinal number, (G, al be the product topological group of a family ((G , a )) S of topological groups, with lSI 2: m, Let 'f' denote the topology on G generated S S SE' - . by the class of the subsets nseS vs of G such that l[s E SIVs F Gs 11 < m, Theorem. Put H =

[(xs lses E Gl[s E Sl Xs F e 8 )is finite}, where es is the neutral element of Gs for any s E S. The '!'-subgroup H has the property K if and only if IGsl < m, for any s E S. Theorem. If for any s E S, Gs has a dense subset if cardinality less than m, then G has the property K( a, '1'). Furthermore, if

m = ~~, (G, a) is nonpseudo-compact and (Gs, as) is metrizable for any s E S, then 'f' is the smallest topology on G such that G has the property K( a,'!'). Theorem. If for any s E' S, Xs is a Hausdorff space with a dense subset of cardinality less than m, then for any continuous map g offiseS Xs into

R, there is a subset T of S, with ITI < m, such that if x,y E fisES ~ and xs = Ys for any s E T, then g(x) = g(y). Other results are proved, (Received July 28, 1969.)

69T -Gl54, ROGER W. HANSELL, University of Conecticut, Storrs, Connecticut 06268. On a -discrete decompositions and Borel isomorphism,

A, H. Stone ("On a-discreteness and Borel isomorphism," Amer. J. Math. 85 (1963), 655-666) has shown that .for absolute Borel spaces the property of If-discreteness (for sets) is invariant under Borel isomorphism. We show that this does not continue to hold for the corresponding property for .collections of sets, even in very special cases, However, it is shown that the slightly weaker property of being " a-discretely decomposable" is an invariant for suitably restricted domains. A collection Cl of sets is said to be C1-discretely decomposable in a space X if each A E Cl can be written as a countable union of sets An (n = 1 ,2, ... ) such that {An : AE Cl ) is discrete in X for each n. Basic Theorem. A disjoint collection of analytic sets in a complete metric space X has the property that the union of every subcollection is analytic in X iff it is a -discretely decomposable. The invariance

1091 property mentioned above is a consequence of the Corollary. Iff is a Borel measurable mapping from an absolutely analytic (metric) space X to a space Y and a is a a-discretely decomposable collection in Y, then (f- 1(A): A E Cl) is r:J.discretely decomposable in X. (Received September 24,1969.)

69T-Gl55. EUGENE S. BALL, Auburn University, Auburn, Alabama 36830 and Tennessee Technological University, Cookeville, Tennessee 38501. Weakly normal spaces and regular spaces. Preliminary report.

00 Definition (Zenor). A topological spaceS is weakly normal provided that if (l\1i=l is a monotonically decreasing sequence of closed sets in S with no common part and H is a closed set in S not intersecting H1 , then there exist a positive integer N and an open set D such that HN c: D and cl(D) does not intersect H. Theorem. There is a weakly normal, 12 , point Gil space which is not regular. J. Mack in a paper to appear entitled" Countable paracompactness and weak normality properties'' showed that if a space is ll-normal and either countable or satisfies the first axiom of countability, then it is regular. Theorem. If Sis weakly normal, then S is ll-normal. Corollary. If S is weakly normal and either countable or satisfies the first axiom of countability then S is regular. Carlos J. R. Borgers (Canad. J. Math. 20(1968), 795-804) defined para-Lind.elof. !heorem. If Sis weakly normal, 12 and para-Lindeli:if, then Sis regular. C. W. Proctor (Abstract 660-13, these cN0tiai) 15( 1968), 1011) defined pseudonormal. Theorem. If S is weakly normal and regular, then S is pseudonormal. ~·heorem. If S is weakly normal, T2 and Lindelof, then S is normal. Theo­ :em. If S is regular and the boundary of each open set is Lindelof, then S is normal. (Received September 8, 1969.)

69T-GI56. ALLAN M. KIRCH, Macalester College, St. Paul, Minnesota 55101. Connectedness

-~nd local connectedness of restricted products of topological spaces.

A topological space A is an Sa-space if B is a subspace of A and A is the union of nonvoid dis­ joint sets U and V such that U (1 a and V n B are open in a and U n V <:: a, the bar denoting closure m A. We call A a C a-space if it is not an Sa-space. A neighborhood (nbhd) N in A is an Sa - or

Ca -nbhd according as the subspace N is an Sa- or Ca -space. Let I be a set, J a subset of I, ~ a J­ ideal of subsets of I; for each i E J let Ai be a topological space, and for i E I - J let ai be a sub­ space of Ai; let X be the ~-restricted product of the Ai with respect to the ai (cf. '"Generalized restricted direct products," Duke Math. J., to appear shortly!. Theorem_ I. The space X is connected if and only if each Ai is connected and for each i E I - J, Ai is a Csi -space. Theorem 2. The space X is locally connected (6c) if and only if (1) Ai is ..t'c for each i € J; (2) a i is .t'c for each i E I _ J; (3) lf i E I - J, xi (the ith component of x E X) is in Ai - ai, and N i is any nbhd of xi in Ai, then there exists an open connected CBj_ -nbhd Qi such that xi E Qi <:: Ni; (4) For any K E ~, Ai is connected for all but finitely many i € K; (5) Bi is connected for all but finitely many i E I - J. (Received September 15, 1969.)

69T-GI57. MAX K. AGOSTON, Wesleyan University, Middletown, Connecticut 06457. arowder­ _Novikov theory for maps of degree d > 1. I.

This paper extends the results of Browder (polloquium on Algebraic Topology, Aarhus (1962), 42-46] and Novikov [Amer. Math. Soc. Trans!. 2(48) (1965), 271-396] to the case of maps of arbitrary

1092 n nonzero degree. Let ~ be a q-disk bundle over an oriented closed manifold M with q;; n t 3, Defi- nition. Let x E l!ntqT(~), deg x = d ! 0. (T( ~) is the Thorn complex of ~ and deg: 1T ntqT( ~) -1\tq T( ~) = 7i. is the Hurewicz homomorphism.) x is said to split completely if x can be represented by a map F: Sntq- T( ~).transverse regular on M, such that F-l (M) has the homotopy type of the jdj-fold con­ nected sum of M. Definition. If G is an abelian group and Gk is the direct sum of k copies of G, let

lli: Gk- G be the projection onto the ith factor. Definel'l: Gk- G by l'l(g) = 11 1(g) + •.. +llk(g) and set l'lk(G) =kernel of 1'1. Finally, let TT~ = limt 1TttkSt denote the k'th stable homotopy group of the sphere.

Theorem. Suppose Mn is !-connected with n ~ 6, and let x I' 1T T( ~ ), deg x = d ! 0. Assume that ------ntq either jdj = 1 and n i 2 (mod 4) or n is odd. Then x splits completely if and only if a sequence of ob- . s structions o.(x) E tf (M;l'll !( 11 . )), l

69T -G !58. ER NES1 p. LANE, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. Insertion of continuous functions between normal semicontinuous functions.

Normal semicontinuous functions are defined in R. p. Dilworth's paper, "The normal completion of the lattice of continuous functions" (Trans. Amer. Math. Soc, 68(1950), 427-438). All functions con­ sidered are real valued. Theorem. The following are equivalent in a topological space X: (i) Iff is a normal lower semicontinuous function on X and g is a normal upper semicontinuous function on X and if g(x) "' f(x) for all x in X, then there exists a continuous function h on X such that g(x) ~ h(x) ~ f(x) for all x in X. (ii) If A and B are disjoint regular closed sets then there exist disjoint open sets that contain A and B. (iii) If h is a bounded continuous function defined on a regular closed subset of X, then h extends to a bounded continuous function defined on X. (iv) If A and B are disjoint regular

closed sets, there exists a continuous function h on X such that h(A) = 0 and h(B) = l. (Received September 22, 1969.)

69T -Gl59. GARY D. JONES, Murray State University, Murray, Kentucky 42071. The embed­ ding of homeomorphisms of the plane in continuous flows.

A systematic study of fundamental regions of the plane under a fixed point free, orientation preserving, self-homeomorphism f is made [S. A. Andrea, Bull. Amer. Math, Soc. 71 (1965), 381- 3 83] under the conditions that there exists exactly n fundamental regions Ri under f; if x E R i -

Int Ri , then x E C c Ri - Int ~ where C is homeomorphic to the real line and f(C) = C; and if x 1 ,

x 2 E Int R 1. then there exists an arc K c:: lnt R. joining x and x such that ~(K) - oo as n - ± oo. 1 1 2 The following embedding theorems are established, Theorem I. Under the above hypothesis, if R 1 and Ri are not separated fori= 2, ... ,n, then R 1 is open, fjR 1 can be embedded in a continuous flow, and if fi'R 1 can be embedded in a continuous flow, then f can be embedded in a continuous flow. Theorem 2. Let f be an orientation preserving self-homeomorphism of the plane p with one fixed point x • Suppose if x E P - {x } there is a nbhd U of x such that fn(U) - w or x as n - ± oo, and 0 0 0 there is a curve C c P - {x0} such that C is homeomorphic to the reals and f(C) = C. Then f can be embedded in a continuous flow. (This work is part of the author's University of Missouri Ph. D. thesis under the direction of Professor W. R. Utz.) (Received September 26, 1969.)

69T-GI60. JAMES W. CANNON, University of Wisconsin, Madison, Wisconsin 53706. *-taming sets for crumpled cubes.

A compact set X in s3 is said to be a *-taming set if the following condition is satisfied: If C

1093 3 is a crumpled cube in S , X c: C, and Bd C is locaHy tame at each point of Bd C - X, then the com- 3 plementary crumpled cube C* = S - lnt C is a 3-cell. If one requires only that the condition be satis- fied when X c: Bd C, then one has the definition of taming set (see (J. W. Cannon, Abstract 658-163, these cJVotiaiJ 15 (1968), 768)). Theorem 1. A compact subset X of s3 is a *-taming set if any of the following conditions is satisfied: (i) X lies on a tame 2-sphere and has no degenerate components; 3 (ii) ((Burgess, Amer. j. Math. 88 (1966), 309-313]) X is a crumpled cube and X*= S - Int X is a 3 -cell; (iii) X is a locally peripherally unknotted arc; (iv) X has no degenerate components and can be described uniformly by sequentially 1-ULC cubes; (v) X is a countable union of *-taming sets. 3 3 Theorem 2, If X is a *-taming set, C is a crumpled cube in S containing X, and h : C .. S is an embedding, then the following are equivalent: (i) h(X) is a *-taming set; (ii) X and h(X) are equi- 3 3 valently embedded inS ; (iii) h(C)*- h(X) is 1-ULC (where h(C)* = S - Int h(C)). (Received September 22, 1969.) Miscellaneous Fields

69T-H54. N. S. MENDELSOHN, University of Manitoba, Winnipeg 19, Manitoba, Canada. Orthogonal Steiner systems.

Two Steiner triple systems on the same elements are said to be orthogonal if (1) they have no triple in common and (2) if the two triples determined by two pairs of elements of one system intersect then the triples determined by the same pairs of elements of the other systems do not intersect. It is shown that if p is a prime p 11 1 mod 3, then a pair of orthogonal Steiner triple sys­ tems of order p can be constructed. If p " 2 mod 3, the same construction can be carried out for 2 order p . The notion of orthogonality is extended to other types of block designs. (Received September 8, 1969.)

69T-HS5. GEORGE F. CLEMENTS,.University of Colorado, Boulder, Colorado 80302. Another application of the generalized Macaulay theorem. Preliminary report.

Let .t' and n be fixed positive integers, ..,t!!! 2n and let A~ denote an L -element subset of the lexicographically ordered set of n-tuples of O's and 1's. A connection (x,y) = (y,x) is a pair of n­ tuples, x andy, which disagree at exactly one place. The connection (x,y) is !!! A£ iff x EAt! and y E Ae• Let Se be the set of...! smallest n-tuples, and let C(At') denote the number of connections in

Al' The papers of Harper @lAM J, Appl. Math, 12(1964), 131-135] and Bernstein @lAM j. Appl. Math.

15 (1967), 1485-1489] contain (properly): Theorem. maxA C(At') = C(~). j. B. Kruskal points out e [J. Combinatorial Theory 6(1969), 86-89] that this theorem seems to be related to his theorem r· Mathematical optimization techniques," Univ. of California Press, Berkeley and Los Angeles, California, 1963, pp. 251-278]. In this paper it is shown that the above theorem is equivalent to the k = 2 special case of the Lindstrom -Zetterstrom theorem [Proc. A mer. Math. Soc. 18(1967), I 66-170] which is a corollary of Kruskal' s theorem. All of these results are corollaries of the generalization of Macaulay's theorem due to the author and B. Lindstrom [Abstract 663-297, these cJVotiai) 16(1969), 171]. (Received September 29, 1969.)

1094 SABBAGH, 69T-H56. L. HADDAD, 127 avenue Philippe Auguste, Paris 13, France and GABRIEL of Erdos and Yale University, New Haven, Connecticut 06520. Ramsey-like numbers and a conjecture Moser.

427-489) m and n are positive integers. Erdos and Rado (Bull. Amer. Math. Soc. 62 (1956), Mat. Kutat6 have introduced Ramsey-like numbers .t'0(m,n). Erdos and Moser (Magyar Tud. Akad. tournament lnt. Kozl. 9("1964), 125-132) have denoted by f(m) the largest integer n such that every a simple proof with m vertices contains a transitive subtournament with n vertices. One can find indicated in by induction of the existence of t'0(m,n) which yields better upper bounds than those f(m) 1!; n if and Erdos and Rado's paper. Another result is that~(5,2) is at most equal to 15. Since mentioned by Erdos and only ift'o(n,2) ;!; m, one obtains f(l5) = 5, which disproves a conjecture 1Moser. (Received September 29, 1969.)

ERRATA Volume 15

655-23, H. 0. F ATTORINI. Differential equations in linear topological spaces. III. Abstract Page 472. In Theorem 2 "L 2 " should read rrrf ". Theorem 5 is incorrect and should be deleted.

FRED GALVIN. Partition theorem for the real line, Abstract 68T-530, Page 660. Line 6: Replace "for all finite rand n" by "all finite nand r ~ 2".

Volume 16 683. BARUCH GERSHUNI. Continuous sequences of totalities, Abstract 69T-E37, Page Lines 2 and 3: Replace "Such a sequence is called a continuous sequence of operations." by "The sequence of operants belonging to the sequence of operations in question is called a continuous sequence of totalities."

BARUCH GERSHUNI. Systematics of the totalities, Abstract 69T-E52, Page 841. Line 7: Replace "abc ... ;" by "abc ... ;".

of Abstract 665-73. Application of singular integral equation methods to static problems by Guillamo Miranda. nonsmooth elastic bodies, Page 646, was incorrectly stated as having been written The author's correct name is Guillermo Miranda.

KARL E. GUSTAFSON. Toeplitz-Hausdorff Theorem in Banach space, Abstract 69T-Bl89, Page 973. The stated result is false.

simplexes, jOHN N. McDONALD. On certain compact convex sets which are similar to Choquet Abstract 664-34, Page 510. The example is incorrect.

Page 849. MILTON ULMER. C-embedded 0-spaces. Preliminary report. Abstract 69T-Gl05, Title: Replace "0-spaces" by "!:-spaces".

1095 New and Recent Wiley-lnterscience Books for Mathematicians

Ordinary Differential Equations By JACK K. HALE, Brown University. Vol. 21 in the Wiley-lnterscience Pure and Applied Mathematics Series, edited by R. Courant, Courant Institute of Mathematical Sciences; L. Bers, Columbia University; and J. J. Stoker, Cou­ rant Institute of Mathematical Sciences. The emphasis is on the theory of nonlinear equations with considerable attention given to special analytical methods used in applications. Much of the material presented has never ap­ peared in a text before. The topics are ordered in such a way as to introduce new concepts earlier than usual, making the sub­ ject stimulating as well as permitting the reader to begin think­ ing independently as soon as possible. 1969 Approx. 368 pages $14.95

Introduction to Potential Theory By L. L. HELMS, University of Illinois. Vol. 22 in the Wiley-lnterscience Pure and Applied Mathematics Series, edited by R. Courant, Courant Institute of Mathematical Sciences; L. Bers, Columbia University; and J. J. Stoker, Cou­ rant Institute of Mathematical Sciences. This book contains a complete treatment of harmonic and superharmonic functions, Green functions, Green potentials, the Dirichlet problem, Chequet's theory of capacities, and the Mattin boundary. It provides the background necessary for modern probability theory and axiomatic potential theory. Most of the important aspects of Laplace's theory are treated in this book. 1969 304 pages $14.95

Topics in Complex Function Theory Volume 1 Elliptical Functions and Uniformization Theory By CARL L. SIEGEL, University of Gottigen, West Germany. This is the first of three new volumes by a famous mathema­ tician. The only prerequisite is a basic knowledge of complex variables. The higher parts of function theory are developed in a unifield, lucid presentation, starting with elliptical integrals and functions and uniformization theory. It continues with auto­ morphic functions and the theory of abelian integrals and ends with the theory of functions and modular functions in several variables. The last topic owes its existence to the author and has not yet been presented in any other book. 1969 186 pages $9.95

1096 The Chi-Squared Distribution By H. 0. LANCASTER, University of Sydney. This volume describes the theory and practice of the distribution of X2 as introduced by Karl Pearson and devel­ oped by R. A. Fisher and others, bringing out where possible analogies with the analysis of variance. Special attention is paid to the problems of approximation to discrete distri­ bution, the use of orthagonal functions, the canonical de­ scription of distribution, contingency tables, and interference. Ample historical references are given and the book con­ cludes with a bibliography of over 1200 entries, indexed by subject. 1969 356 pages $14.95

Fourier Analysis in S:weral Complex Variables By LEON EHRENPREIS, Courant Institute of Mathematical Sciences, New York University. Vol. 17 in the Wiley-lnterscience Pure and Applied Mathe­ matics Series, edited by R. Courant, Courant Institute of Mathematical Sciences; L. Bers, Columbia University; and J. J. Stoker, Courant Institute of Mathematical Sciences. This volume represents a way of looking at the central problem of Fourier analysis. It is concerned with linear prop­ erties which tend to lead to abstract thinking, although the classical content of the work is also emphasized. Each chapter begins with a detailed summary and ends, when the occasion permits, with general remarks, bibliographical re­ marks, and problems for further study. 1969 Approx. 528 pages $14.95 Generalized Integral Transformations By ARMEN H. ZEMANIAN, State University of New York at Stony Brook. Vol. 18 in the Wiley-lnterscience Pure and Applied Mathe­ matics Series, edited by R. Courant, Courant Institute of Mathematical Sciences; L. Bers, Columbia University; and J. J. Stoker, Courant Institute of Mathematical Sciences. This volume modernizes the subject of integral transfor­ mations by basing it on the theory of generalized functions. It gives detailed descriptions, a variety of applications to boundary-value problems and system theory, worked-out examples, and problems. The generalized functions have be­ come powerful tools and can be used to solve a variety of problems that could not be attacked with the classical transformations. 1968 300 pages $16.00 Numerical Control By GLENN ERTELL, formerly Head of the Digital Laboratory, National Radio Astronomy Observatory. "Numerical control, as treated here, is defined as that part of an electromechanical system that utilizes digital logic circuits to cause the system to respond to instructions re­ ceived from digitally coded tapes."- from the Preface. T:-Jis book was written for engineers and technicians who must apply and maintain numerical controls. It develops the techniques that are common to all numerical controls with emphasis on logic functions. 1969 160 pages $9.50 Continued on next page

1097 Crack Problems in the Classical Theory of Elasticity By IAN N. SNEDDON, The University of Glasgow, Scotland; and MORTON LOWENGRUB, Indiana University. The calculation of the distribution of stress in the neigh­ borhood of a crack in an elastic solid is the subject of this volume. An account is given of calculations in the mathe­ matical theory of elasticity relating to Griffith cracks and their three-dimensional analogues and having some relev­ ance for the theory of brittle fractures. Research results obtained by the authors and others during the last 20 years are collected in the volume. A brief account is given of solu­ tions to dynamical crack problems. Throughout, the mathe­ matical aspects of the problem are emphasized and trans­ form techniques are considered in great detail. Numerous illustrations and tables as well as an extensive bibliography are included. 1969 232 pages $14.95 Mathematical and Theoretical Physics Volumes 1 and 2 By the late EGIL A. HYLLERAAS, formerly at the University of Oslo, Norway. This is one of the most clearly written works covering all areas of classical physics. It also presents the necessary mathematical apparatus. Four basic themes are expounded: classical mechanics, theory of heat, electricity and magnet­ ism, and atomic theory. These themes encompass a wide scope. Classical mechanics, for example, means not only the traditional Lagrange- Hamilton-Jacobi transformation theory for systems with a finite number of degrees of free­ dom, but also the mechanics of continua. Nothing but a certain experimental background is presupposed. 1969 Volume 1. 512 pages $15.00 Volume 2. 528 pages $15.00

Modern Mathematical Methods in Engineering, Volume 1 By STEFAN FENYO, Department of Mathematics, and THOMAS FREY, Computer Center, Hungarian Academy, Budapest. A Volume in the North Holland {lnterscience) Series in Applied Mathematics and Mechanics. This volume acquaints the reader with mathematical disciplines which are important from the viewpoint of recent applications, but which are not yet taught in many colleges and universities. Consideration is given to the mathematician who is interested in the latest fields of application. The reader is only required to have a good grasp of elementary algebra, geometry, and analysis. 1969 In press $21.50

Parabolic Systems By SAMUIL D. EYDEL'MAN. Translated from the Russian, this volume is devoted to systems that are parabolic in the sense of Petrovskiy. It dis­ cusses fundamental matrices of the solutions of linear para­ bolic equations, properties of the solutions of parabolic sys­ tems, the cauchy problem, and the mixed problem for linear parabolic systems. A North Holland {lnterscience) book. 1969 480 pages $21.00

1098 Linear Operators By NELSON DUNFORD, formerly the James E. English Prof­ essor of Mathematics, Yale University; and JACOB T. SCHWARTZ, Courant Institute of Mathematical Sciences, New York University; with the assistance of Robert G. Bartle, University of Illinois, and William G. Bade, University of California, Berkeley. Volume 7 in the Wiley-lnterscience Pure and Applied Mathe­ matics Series, edited by R. Courant, Courant Institute of Mathematical Sciences; L. Bers, Columbia University; and J. J. Stoker, Courant Institute of Mathematical Sciences.

Part 3 Spectral Operators in Banach Spaces This volume completes the treatise on linear operators by Dunford and Schwartz. It is devoted to the spectral theory of non-selfadjoined operators, and also discusses the simi­ larity theory of operators due to Freidricks and Kato. CON­ TENTS: Spectral Operators. Spectral Operators in Sufficient Conditions. Algebras of Spectral Operators. Perterbations of Spectral Operators with Discrete Spectra. Spectral Oper­ ators with Continuous Spectra. Applications of the General Theory. 1969 In press

Part 2 Spectral Theory, Self-Adjoint Operators in Hilbert Space CONTENTS: B-Aigebras. Bounded Normal Operators in Hilbert Space. Miscellaneous Applications. Unbounded Oper­ ators in Hilbert Space. Ordinary Differential Operators. Linear Partial Differential Equations and Operators. 1963 1072 pages $35.00 Part 1 General Theory CONTENTS: Preliminary Concepts. Three Basic Principles of Linear Analysis. Integrations and Set Functions. Special Spaces. Convex Sets and Weak Topologies. Operators and the Adjoints. General Spectral Theory. Applications. 1958 872 pages $23.00 Representations of Groups: With Special Consideration for the Needs of Modern Physics Second Edition By HERMANN BOERNER, University of Giessen, Germany. This volume is of importance to mathematicians,physicists, and theoretical chemists. Though the book's contents are mathematical, the method of presentation is designed to satisfy the requirements of physicists. The subject matter is arranged so that a reader interested in a particular topic need not read more than is necessary. The principle concern of the book is to give the representations and character of the most important groups. 1969 In press $16.00

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1099 Mathematics: Its Content, Methods, and Meaning, Second Edition edited by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev translated by S. H. Gould, K. A. Hirsch, and T. Bartha translation edited by S. H. Gould Published in cooperation with the American Mathematical Society These volumes have achieved the status of a standard work and are now available in paper editions. They cover the whole sweep of the development of mathematics from earliest times to the present, and cover the whole range of its content from the theory of numbers to the geometry of colors, and thus fulfill their purpose of providing a kind of diving bell, a means of access for the seriously interested layman into mathematical depths usually con­ sidered hopelessly over his head. " ... nothing less than a major contribution to the scientific culture of this world in the mid-20th century." -Harry Schwartz The New York Times Book Review "This is a masterful English translation of a stupendous and formidable mathematical masterpiece originally published in Russian ... one reads actual mathematics in these three volumes, not just about mathematics ... Anyone interested in becoming familiar with various aspects of elementary and advanced mathematics will find this unique and impressive work more than valuable." -Social Science Three-volume boxed set. paperback, $12.00; hardcover, $30.00 Survey of Applicable Mathematics edited by Karel Rektorys A team of Czech applied mathematicians, all experts in their own fields, have combined to produce this unique survey of applicable mathematics. While ·the mathematics is always sound, accurate and comprehensive, it is the applications of the theory that have at all times been in the forefront of the writers' minds. All proofs, which in general only interest mathematicians, have been omitted, but every theorem or formula is carefully and pre­ cisely stated, so that a non-mathematician knows exactly when and under what circumstances it can be used. This is a mine of information-the table of solutions of differential equations, for instance, is the most complete ever published in English; it is a book for any user of mathematics to keep, not on his bookshelves but within arm's length on his desk. $16.95 Number Words and Number Symbols: A Cultural History of Numbers by Karl Menninger Since all cultures have evolved or inherited number systems of some sort, a comparison of their likenesses and differences yield significant clues as to universals of language and culture and a measure of the actual (or possible) extent of their divergence. This is the immense work undertaken by the distinguished German scholar, Karl Menninger, in Number Words and Number Symbols. Fortunately, the author is a natural story-teller, and his story has an intrinsic fascination he brings out by casting much of it in anecdotal form. Although his story is disciplined by the scholarly imperatives of a number of diverse fields-linguistics, anthropology, historiography, mathematics, conceptual psychology among them-Menninger wears his learning as lightly as is consistent with the seriousness of his purpose and the thoroughness of his presentation. $15.00 The Mathematical Sciences: A Collection of Essays edited by COSRIMS "This absolutely first-rate collection of essays has been written by a group of the most outstanding mathematicians, economists, biologists, and mathematical physicists in this country in an attempt to acquaint the non-professional reader with the latest developments in the most ancient of the exact sciences ... Most of them are self-contained, and all are written with such lucidity that a reader who is willing to take the pains will come away with an under­ standing of why this field has flourished with such vigor since the dawn of recorded intellectual history." -The New Yorker ' ..... Cb .paperback, $3.95; hardcover, $8.95 ~lllo....-v;,;~ TheMITPress ~ ~ Massachusetts Institute of Technology . ... ~ ~,.... ,...._ .. ~~ Cambridge, Massachusetts 02142 :V v.,V ~ ,~V ~~~~~~v c; ~~~~ S>G ~

·1100 From Distinguished Authors: BOOKS OF VITAL INTEREST ON STO CDASTIC PROCESSES AND OPTIMAL CONTROL Andreyev: CORRELATION THEORY OF STATISTICALLY OPTIMAL SYSTEMS In this new work on stochastic control, several distinguished Soviet scientists consider the design of systems to produce optimum outputs according to statistical criteria. The authors discuss criteria used for comparing systems. Discussions are based on the calculus of varia­ tions, probability theory, game theory, statistical decision theory, and nonlinear programming. The material is organized for practical utilization by mathematically-oriented engineers working in the field of automatic control and for study by advanced undergraduates in mathematics as well as graduate students in engineering. By N. I. Andreyev. 370 pp. illustrated. $14.50. 5/69. White & Tauber: SYSTEMS ANALYSIS Here is a unique book providing the fundamentals of systems analysis and developing a unified treatment for handling the mathematically analogous systems occuring in various fields of engineering and science. The book emphasizes linear algebra, variational methods, and dynami­ cal systems. It embodies an extensive array of applications drawn from branches of modern engineering, physical science, and mathematics. This is an excellent text for advanced under­ graduate and graduate students in engineering, applied mathematics, and applied science and as a reference for practicing engineers. By Harry J. White and Selma Tauber, Portland State University. 499 pp. 116 illustrations. $14.00. 5/69. Young: Lectures on the CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY This graduate-level volume can help prepare its reader for contemporary research in optimal control. The book covers Hamiltonian theory on a global basis; it is one of few books which treats unicity; it provides a highly detailed introduction to Morse theory. Generalized curves, originally conceived and developed by the author, are fully discussed. The treatment of the classical calculus of variations includes discussions of the Huygens Principle, Lagrange brackets, and the geometrical form of exactness. A complete account of the sufficiency theory of optimal control is presented. By L. C. Young, Univ. of Wisconsin. 331 pp. 175 illustrations. $15.00. 5/69. Gikhman & Skorokhod: INTRODUCTION TO THE THEORY OF RANDOM PROCESSES Starting with fundamentals [including measure theory, Lebesgue integration, and axiomatic probability theory), this book goes on to a penetrating study of specialized aspects of random processes. Much of the authors' original work on limit theorems is also included, as well as--a unique treatment of processes with independent increments. The book is eminently suited for graduate-level students, mathematicians, and mathematically-oriented engineers. By I. I. Gikh­ man and A. V. Skorokhod, Kiev State University. 516 pages. About $17.50. Just Ready. W. B. SAUNDERS COMPANY West Washington Square, Philadelphia, Pa. 19105 Please send on 30-day approval and bill me: D For my personal library D For possible adoption as a classroom text 0 Andreyev: CORRELATION THEORY $14.50 0 White & Tauber: SYSTEMS ANALYSIS $14.00 0 Young: CALCULUS OF VARIATIONS $15.00 0 Gikhman & Skorokhod: RANDOM PROCESSES About $17.50

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FUNDAMENTAL CONCEPTS OF There are a substantial number of new, MODERN MATHEMATICS more challenging exercises added to By Max D. Larsen, problem sets and many of the routine University of Nebraska. problems have been changed. Many of the sections have been extensively re­ This- book is designed as a textbook written in the interest of improved expo­ for an introductory course in symbolic sition. Sample copies will be available logic and the theory of sets. Chapter by the end of November. One is an introduction to symbolic logic. In Chapter Two the theory of sets In press (1970) is introduced and the major concepts COLLEGE CALCULUS WITH of set theory needed for a course in ANAL VTIC GEOMETRY, modern algebra or geometry are dis­ cussed. The remaining chapters pro­ Second Edition vide an opportunity for the student to By Murray H. Protter and Charles B. use this material, and many exercises Morrey, Jr., University of California. are included to strengthen the student's This is the second .edition of a well­ understanding of the concepts involved. known text in calculus and analytic In press (1970) geometry designed for a three-semes­ ter course. In addition to the material CALCULUS WITH ANAL VTIC covered in the authors' First Course, GEOMETRY: A FIRST COURSE, the text includes linear algebra in order Second Edition to prepare the student for further study in multivariate calculus. Considerable By Murray H. Protter and Charles B. attention has been devoted to elemen­ Morrey, Jr., University of California. tary topics, illustrative examples, and The revision of the text for a full year intuitive explanations of many of the course in calculus of one variable and fundamental concepts, thus making the analytic geometry has maintained the book attractive both to students of sci­ successful approach of the first edition ence and engineering and to students in tone and emphasis. Set notation has of pure mathematics. been introduced throughout the text. In press (1970)

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1102 NEW MATHEMATICAL TITLES FROM GORDON AND BREACH NON-LINEAR SAMPLED-DATA SYSTEMS By PIERRE VIDAL, Faculle des Sciences de Litle 1969 384 pp. $24.50/Prepaid $19.60 CONTENTS: Fundamental Elements of the Calculus of Finite Differences. The z-transform Method. The Discrete Phase Plane Method. Method Using Signal-Flow Graphs. Stability of Non-linear Sampled-data Systems. Transient Response and Periodic Oscillations. Pulse-width Modulated Systems. Quantized Sampled-data Systems. Author Index. Subject Index. ' CODE NUMBER: 3023 REAL ANALYSIS By JOHNSTON A. ANDERSON, University ot Nottingham 1969 356 pp. Reference $28.00/Prepaid $22.40 Text $11.00 A comprehensive account of the material covered in a course on real analysis, this volume is recommended as the basis for such a course. To help the reader acquire a thorough under­ standing of both the concepts and methods of theorem proofs used in real analysis, numerous illustrative examples are given on the material covered. Over 180 illustrative examples and worked exercises, with nearly 100 diagrams and more than 200 problems, reinforce the con­ cepts developed in the definitions and theorems, motivating the user to proceed from concrete situations to abstract ideas. The treatment of the subject is along fairly traditional lines. Thus, when the reader subsequently encounters the more abstract analysis of metric and topological spaces, he is already thoroughly acquainted with a particular realization of these topics and is better equipped to appreciate their generality. CODE NUMBER: 6146 POCKET MATHEMATICAL LIBRARY Edited by JACOB T. SCHWARTZ, Courant Institute of Mathematical Sciences A series of primers, workbooks and courses for both classroom and self-study. These books provide short, readable and concise treatment of specific mathematical subjects and fulfill a need for explicit terse textbooks in well-defined subject areas. Primers LEARN LIMITS THROUGH PROBLEMS! THE COORDINATE METHOD By S. I. Gelfand et al. By I. M. Gelfand, E. G. Glagoleva 1969 78 pp. Reference $6.50/Prepaid & Text $5.00 and A. A. Kirillov CODE NUMBER: 2072 1969 80 pp. Reference $7.50/Prepaid & Text $5.00 MATHEMATICAL PROBLEMS: AN ANTHOLOGY CODE NUMBER: 2064 By E. B. Dynkin et al. FUNCTIONS AND GRAPHS 1969 78 pp. Reference: $7.50/Prepaid & Text $5.00 By I. M. Gelfand, E. G. Glagoleva CODE NUMBER: 2071 and E. E. Shnol Courses 1969 110 pp. Reference $7.50/Prepaid & Text $5.00 LIMITS AND CONTINUITY CODE NUMBER: 2069 By P. P. Korovkin Workbooks 1969 134 pp. Reference $8.50/Prepaid & Text $5.00 SEQUENCES AND COMBINATORIAL CODE NUMBER: 2074 PROBLEMS DIFFERENTIATION By S. I. Gelfand et al. By P. P. Korovkin 1969 93 pp. Reference $7.50/Prepaid & Text $5.00 1969 94 pp. Reference $7.50/Prepaid & Text $5.00 G CODE NUMBER: 2073 CODE NUMBER: 2075 ------ORDER FORM ------B GORDON AND BREACH, SCIENCE PUBLISHERS, INC. 150 FIFTH AVENUE • NEW YORK, NEW YORK 10011 Code # Author Edition Quantity Payment Enclosed

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ADVANCED CALCULUS: An Introduction to Analysis Second Edition By WATSON FULKS, University of Colorado. This highly respected textbook emphasizes the careful development of theory, and covers in depth those results that are useful to further work in mathematics and immediate use in science and engineering applications. A Solutions Manual will be available. 1969 Approx. 616 pages $11.95

LINEAR ALGEBRA AND MATRIX THEORY Second Edition By EVAR D. NERING, Arizona State University. A revision of the popular linear algebra text, designed for the junior-senior level course. Changes in the new edition include rewriting of some passages for in­ creased clarity, updating of notation, and new problems and applications. 1969 In press

INTRODUCTION TO GEOMETRY Second Edition By H. S. M. COXETER, University of Toronto. Improvements made in the second edition add to the usefulness of this book. Each of the 22 chapters is reasonably self-contained. There are exercises at the end of almost every section, the hardest ones having hints for their solution. 1969 Approx. 496 pages $10.95

PROJECTIVE AND EUCLIDEAN GEOMETRY Second Edition By W. T. FISHBACK, Earlham College. This volume develops projective geometry first from Euclidean geometry, then from an axiomatic viewpoint. Projective geometry is then used as a starting point for the development of the beginnings of affine, Euclidean, hyperbolic, and ellip­ tic geometries. 1969 Approx. 320 pages $10.95

ELEMENTARY DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS Second Edition By WILLIAM F. BOYCE and RICHARD DiPRIMA, both of Rensselaer Polytechnic Institute. Already established as a superior textbook, Boyce and DiPrima is now even clearer, more accurate, and timely. Among the features are the balance between theory and application, careful motivation, and outstanding treatment of the many topics. 1969 533 pages $10.95

AN INTRODUCTION TO PROBABILITY THEORY AND STATISTICAL INFERENCE By HAROLD J. LARSON, Naval Postgraduate School, Monterey, California. This book offers a pedagogically and mathematically sound introduction to the subject. It is clearly written and mathematically concise. There are over 400 prob­ lems with answers, and more than 250 worked examples selected from general science. 1969 387 pages $10.95

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1104 UNIFORM ALGEBRAS Theodore W. Gamelin, University of Calif., Los Angeles, California Dr. Gamelin offers an up-to-date and comprehensive treatment of the field of uniform algebras. He empha- sizes the interplay between the techniques of func- tional analysis which are used to study uniform algebras, and the theory of analytic functions of one and several complex variables. Topics covered include commutative Banach algebras, methods of several complex variables, Hardy spaces, invariant subspace theory, embedding of analytic structure in maximal ideal spaces, generalized analytic functions, and analytic capacity and rational approximation. A series of exercises is included in each chapter. Also included is a bibliography with short notes to bibliographical references. October 1969, approx. 288 pp., $11.00, (93780-5)

For further information, write Box 903 PRENTICE-HALL, ENGLEWOOD CLIFFS, N. J., 07632 CLASSICAL MODERN ALGEBRA Seth Warner, Duke University, Durham, North Carolina A one volume revision of the first eight chapters of Seth Warner's MODERN ALGEBRA. This new text adds a com- plete discussion of the Sylow theorems. Exposition has been simplified for greater ease in teaching. Independ- ence of the sections is noted, allowing the instructor to make many different selections of material. Binary opera- tions are introduced first. An axiomatic development of the natural numbers is followed by the construction of the integers and rational numbers, abstract algebraic systems-groups, rings, and fields are introduced as needed to aid in the discussion. The ordered field of real numbers is characterized by the least upperbound axiom and is constructed by means of Cauchy se­ quences. Discussion of polynomials and fields is ap- plied to classical geometric problems, preceded by a chapter containing the needed linear algebra. The final chapter on Galois theory includes a proof of the alge- braic closure of the complex numbers and culminates with examples of fifth-degree polynomials with integral coefficients not solvable by radicals. April 1970, 448 pp., $13.50, (13606-9)

1105 Cut r field down to size. r------, The size of a mini-catalogue. A mini-catalogue of reprints of the world's foremost journals in the field of mathematics. Journals such as American Journal of Mathematics, Michigan Mathematical Journal, Bibliotheca Mathematica, and Mathematische Annalen. These reprints-like all Johnson reprints-are identical or similar in format to the originals. In addition to volume numbers, years, and prices for individual volumes and cloth and paper bound sets, the mini-catalogue con­ tains precis describing the editorial contents and objectives of each journal. For a free copy, simply mail the coupon. 82

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1108 LINEAR ALGEBRA John dePillis, University of California, Riverside Suitable for freshman or sophomore courses, this book discusses concrete vector spaces; linear transformations and their matrices, linear equations, and determinants; the structure of operators. It introduces ideas of inner product, diagonalization, and the spectral decomposition theorems for self-adjoint operators and hermitian matrices. Each chapter contains examples, exercises, and a summary of the material discussed. April1969l528 pages I $8.95 COLLEGE GEOMETRY David C. Kay, University of Oklahoma The author begins with a set of eleven intuitively reasonable axioms that provide a foundation for spherical, Euclidean, and hyperbolic geometry. The material is devel­ oped in a gradual style, with many examples and counter-examples for clarification. The usual ideas regarding points, lines, distance, segments, rays, angles, angle-meas­ ure, triangles, and congruence are developed, including a brief chapter on the quadrilateral, polygon, and circle. Then the three geometries are studied both axio­ matically and from models, with an extensive development of non-Euclidean trigo­ nometry and its consequences. Solutions Manual. July 1969 1384 pages I $9.50 TRIGONOMETRY: A Programmed Text Mervin L. Keedy, Purdue University, and Marvin L. Bittinger, Indiana University, Indianapolis Instead of the usual "frame-by-frame" format encountered in most programmed texts, this book incorporates a significant amount of reading material with frequent ques­ tions, demanding student response and participation in the learning process. Diag­ nostic tests, a final examination, and several summaries are included. April 1969 I 272 pages I $5.95 paper TRIC:Of\0\/\ETRY: A Furctional Aooroach Bill Rice and Joe Dorsett,bcth of St. Petersb~rg Junor College All the major trigonometric topics are developed through a distinctive classroom approach. The authors ask the student questions during the reading, require him to give reasons for steps in the proof, ask him to make conjectures, and introduce impor­ tant concepts in the exercises. May 19691320 pages I $7.95 PLANE TRIC:Of\0\/\ETRY: Third Editon Fff'k A. Rickey and J. ~ry Cole both of louisk:lna Stan::: University The authors present a concise, analytical development of plane trigonometry that provides the trigonometric background for modern courses in analytic geometry and the calculus of functions of real variables. The third edition includes increased stress on general results with reduced memorizing of formulas and more consistent use of the language and symbolism of sets for the making of succinct statements. The exer­ Holt cises are almost entirely new. April 1969 1272 pages I $7.50 Rinehart LINEAR ALGEBRA AND GEOV'\ETRY and James A. Murtha and EariRWil6rd, both of Marietta College Winston, The authors' basic theme is the unification of linear algebra with classical geometry. In their presentation, they motivate each new concept by appealing to the student's Inc. intuition and to his knowledge of high school geometry. Throughout the text, there 383 Madison Ave. is a consistent emphasis on maps which acknowledge the algebraic or geometric New York, N.Y. 10017 structure present in their domains. June 1969 I 240 pages I $8.95 SECOND-CLASS POSTAGE AMERICAN MATHEMATICAL SOCIETY PAID AT P.O. Box 6248 PROVIDENCE, RHODE ISLAND AND Providence, Rhode Island 02904 ADDITIONAL MAILING OFFICES

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recent progress in combinatorics edited by W. T. TUTTE, umverslfy of Waterloo. Waterloo, Ontario. Canada This \'olume, based on the lectures gi\'en at the Third Waterloo Conference on Combinatorics describes the recent research done in the field by the contributors. Many combinatorial results are pre­ sented for the first lime and coloring problems are discussed, especially the Four Cc.1lor Conjecture. CONTRIBUTORS: E. R. IJERLEKAMP. FRANK IIARARY. R. G. STANTON, J G KALHFLEISCH and P. H. WEILANIJ. CLA!JilE BERGE. N. G. d., BR!IIJN, ROBERTO FR!JNCHT. JAY COI.Ili-1AN .1nd Glt\N-CARLO ROT/1. BRANKO CRIINil:\IIM. R. HALIN. A. j. HOFFI-1:\N. LEROY M. KELLY. N. S. MENDEL­ SOHN. C. ST. j. A. NASH-WII.I.IA~1S. R. RAllO. RONALD C. READ. HORST S:\C:IIS. J J SEIDEL. W T. TUTTE. II. 1.. AllHOTT and ll. CARIJNEI'. RUTH !L\Rl. J 1-1. CANIJ!II. AI.I.AN GEWIRTZ and LOlliS V. QUINTAS. RM>1 PRAKASH C!IPTA. RIC!IARIJ K. CIIY and STEFAN ZNAM. ll. A. !IICCS. ARTHliR 1-1. HOBBS. MICHEL JEAN. J C. Kt\I.HFLEISC:II. II. S. R. MlmTY. OYSTEIN ORE and MICHAEL IJ. PLII~1MER. J 1.1 S. SII-1EOS PEREIRA. I·RI-:Il S ROllERTS. j. SCHONHEIM. llAVIIJ I' SI11.1NER. MARK E. WATKINS "'"I IJ ll YOIINGER. 1!11>'1. :;r 1'1'. S!l;oo 0 proof techniques in graph theory edited by FRANK HARARY, University of Mic:higCJn, ,\nn Arbor, Michigcm This book will stand as the most important and stimulating of symposium books on graph theory to date for the high quality of the contributions and for its definitive and useful bibliography on the subject. The title of the book reflects the diversity of methods of proof used by the individual contributors, all active specialists in this rapidly growing field of pure and applied mathematics. The up-to-date bibliography compiled by James Turner, with both alphabetical and key word indexes, facilitates the location of individual articles and of particular topics in the total literature. 1969, :J30 pp., $14.50