OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 11, NUMBER 4 ISSUE NO. 75 JUNE 1964

OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by John \V. Green and Gordon L. 'Valker

CONTENTS

MEETINGS Calendar of Meetings ••.•••••••••••••••••••••.•.•••••.••• 408 Program of the june Meeting in Pullman, Washington •••••• , •••••••• 409 Abstracts for the Meeting - Pages 442-444 PRELIMINARY ANNOUNCEMENT OF MEETING ••••••••.••••••••.•••• 411 MEMORANDA TO MEMBER:> The Employment Register •••••••••••••••••••.••.•••••• , , •. 414 Announcement of Changes in the Combined Membership List , • , ••••••• 414 DOCTORATES CONFERRED IN 1963 •••••••••••••.•••••••••••••••• 415

LETTERS TO THE EDITOR ••••••••••••• o • ,· •••••••••••••••••• , • 433

PERSONAL ITEMS ••••••••••••••• o •••••••••••••••••••••••••• 435 NEWS ITEMS AND ANNOUNCEMENTS ••••••••••••••••••••••••••••• 437 SUPPLEMENTARY PROGRAM - Number 25 ••••••••••••••••••••••••• 438 ACTIVITIES OF OTHER ASSOCIATIONS •••••••••••••••••••••••••••• 441 ABSTRACTS OF CONTRIBUTED PAPERS..... • • • • • • • • • • • • • • • • • • • • • 442 ERRATA ...... 470 INDEX TO ADVERTISERS • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 475 RESERVATION FORM •••••••••••••••••••••••••.•••••• ,...... 475 MEETINGS

Calendar of Meetings

NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change. This is particularly true of the meetings to which no numbers have yet been assigned. Meet Deadline ing Date Place for No. Abstracts* 614 August 24-28, 1964 (69th Summer Meeting) Amherst, Massachusetts July 3 615 October 24, 1964 Garden City, New York Sept. 10 616 November 14, 1964 Los Angeles, California Sept. 30 617 November 21, 1964 Athens, Georgia Sept. 30 618 November 27-28, 1964 Evanston, Illinois Sept. 30 January 26-30, 1965 (71st Annual Meeting) Denver, Colorado April 9-10, 1965 Chicago, illinois April 12-15, 1965 New York, New York April 24, 1965 Stanford, California August 30-September 3, 1965 (70th Summer Meeting) Ithaca, New York November 12-13, 1965 Lexington, Kentucky January 24-28, 1966 (72nd Annual Meeting) Chicago, Illinois August 29-September 2, 1966 (71st Summer Meeting) New Brunswick, New Jersey January 24-28, 1967 (73rd Annual Meeting) Houston, Texas August 28-September 1, 1967 (72nd Summer Meeting) Toronto, Ontario, Canada August 26-30, 1968 ( 7 3rd Summer Meeting) Madison, Wisconsin *The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for by title abstracts are June 26, 1964 and September 3, 1964.

The NOTICES of the American Mathematical Society is published by the Society in January, February, April, June, August, October and November. Price per annual volume is $7.00. Price per copy $2.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, 190 Hope Street, Providence, Rhode Island 02906. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August 4, 1947 (Sec. 34,21, P. L.andR.). Accepted for mailing at the special rate of postage provided for in section 34.40, paragraph (d). Copyright© 1964 by the American Mathematical Society Printed in the United States of America

408 Six Hundred Thirteenth Meeting Washington State University Pullman, Washington June 20, 1964

PROGRAM

The six hundred thirteenth meeting provided by his parents. Reservations for of the American Mathematical Society will dormitory housing should be sent to Pro­ be held at Washington State University in fessor Wilfred E. Barnes, Department of Pullman, Washington, in conjunction with a Mathematics, Washington State University, joint meeting of the Pacific Northwest Sec­ Pullman, Washington. tion of the Mathematical Association of Rates in Pullman motels are as America and the Society for Industrial and follows (all prices include tax). Applied Mathematics. The Society will Al Kircher's Hilltop Motel and Steak meet on Saturday, June 20, 1964, and the House (Colfax Highway) Association and SIAM will hold their Single $7.28 sessions on Friday, June 19. Double (double bed) 8.84 By invitation of the Committee to Double (twin beds) 10.40 - $12.48 Select Hour Speakers for Far Western Sectional Meetings, there will be an hour Hillside Motel (Lewiston Highway) address by Professor H. G. Eggleston of Single $5.20 the University of London and the Univer­ Double (double bed) 8.32 sity of Washington. Professor Eggleston Double (twin beds) 9.36 will speak at 11:00 P.M. on Saturday, Manor Lodge Motel (505 Paradise) June 20 in Room 175, Sloan Hall. The title Single $6.24 of his talk is "Convex sets of constant Double (double bed) 8.32 width." Double (twin beds) 9.88 Sessions for contributed papers will be held at 9:30 A.M. and at 2:00 P.M. on Saturday. All sessions will be in Room 175 Anyone who wishes to stay in a motel Sloan Hall. Abstracts of the papers to be should write directly and as soon as pos­ presented at these sessions appear on sible to the motel for reservations. pages 442-444 of these NOTICES. There are Breakfast and lunch will be available cross references to the abstracts in the at the Compton Union Building. A no host program. Late papers may be added to the dinner is being arranged for the evening program. Information concerning late pa­ of Friday, June 19. Coffee will be served pers will be available at the Registration on Friday and Saturday in the foyer of Desk, located in the foyer of Sloan Hall. Sloan Hall. Dormitory space will be available Pullman is approximately 80 miles on campus for the nights of June 18, 19 south of Spokane on U. S. Highway 195. and 20. The rates per night are $3.00 for It is served by the Northern Pacific Rail­ a single room and $Z.OO each for a double. way, West Coast Airlines, and the Grey­ No extra charge is made for a child whose hound Bus Company. Ample parking is bedding (e.g. sleeping bag or crib) is available· in college parking areas.

409 PROGRAM OF THE SESSIONS The time li.r!it for each contributed paper is ten minute.:, The contributed paper!:! are scheduled at 15 minute iutervals. _I~ __ .!_l1aintain the S£_hed~ule, th~ time limit will be st:tic.:tly en­ for.;;ed.

SATURDAY, 9:30A.M. General_§ession, Room 175, Sloan Hall 9:30 - 9:40 ( 1) Characters of locally compact abelian groups. Preliminary report Professor M. Rajogopalar., Lehigh University (613-6) 9:45 - 9:55 (2) Boundedly divergent sequences in FK-spaces Professor Gunther Go'"s. University of Kansas (613-8) 10:00 - 10:10 (3) Determination of associate functions for Bergman integral operators yielding transonic flow patterns of certain types Professor J. M. Stark, Lamar State College of Technology (613-9) 10:15 - 10:25 (4) Existence theorems for difference and q-difference equations Professor Selmo Tauber, Portland State College (613-7) 10:30 - 10:40 {5) A trunking problem for three channels Dr. S. F. Neustadter, Sylvania Electronic Systems, Waltham, Massachu­ setts (613-10)

SATURDAY, 11:00 A.M. lnvited Address, Room 175 Sloan Hall Convex sets of constant width Professor H •. G. Eggleston, University of London and University of Wash­ ington

SATURDAY, 2:00P.M. General Sessl011, Room 175, Sloan Hall 2:00 - 2:10 (6) Convex bodies witll preasGigned sum of principal radii of curvature. Prelim­ inary report Professor W, J, Firey, Oregon State University (613-4) 2:15 - 2:25 {7) A new proof of the Bonnicc:-Klee theorem Profc::ssor J, R, Reay, Western Washington State College (613-3) 2:30 - 2:40 {8) Some natural topologies in algebra. Preliminary report P .rofessor D. F. Sander: son, Western Washington State College ( 613-5) 2:45 - 2.:55 (9) The divisibility of Gaussian integers by large Gaussian primes Professor j. H. jordan, Washington State University (613-2) 3:00 - 3:10 {10) Application of Goodstein's solution in solving some diophantine equations Mr. D. R. Rao, Sjtafalmandi, Secunderabad, India (613-1) R. S. Pierce Seattle, Washington Associate Secretary

410 PRELIMINARY ANNOUNCE!\J.ENTS OF l\liEETINGS

Sixty-Ninth Summer _Meeting and Forty-Second. ColJoquiotnn University of J\lassachusetts Amherst, _Massachusetts August 25-28, 1964

The A;:nerican Mathematical. Society day morning ant~ afternoon, and Friday will hold its sixty-ninth summer meeti.ng mo:rnJLg auc dternoon. THE NUMBER OF and will present its fort:y·-second Collo­ P II.PER.S T (l BE ACCEPTED WILL NOT quium at the University of Massachusetts F XCEE!J 11:!0, AhAtracts of papers which in Amherst, Massachusetts f:rorn Tuenday ''leet the crj_tcria. of Article X, Section 5 through Friday, August 25-28, 1964. These of the By-Laws will be accepted in order are the same dates aA announced in the of the date of receipt at the office of the April issue of these NOTICES hut a:re ONE Society, of which the address is The Amer­ WEEK EARLIER than dates listed several ic'ln Mathematical Society, 190 Hope months ago in the calendar of meeting,;. Stre~t. Providence, Rhode Island 02906. All sessions will bf' held in lectu.re Tne deadline beyond which abstracts will rooms and classrooms on the campus of not be considered is july 3, 1964. There the University of Massachusetts. wiH be no provision for late papers, Professor Charles B. Morrey of the University of California will present the Colloquium Lectu-res. The title of the lec­ ActivJties of Other Organizations tures is "Multiple integrals in the calculus of variations." The first lecture will be on Several organizations will cooper­ Tuesday, August25, at2;00 P.M. in Bo·Nker ate in hol ctjr·p ''leir meetings on the same Auditorium (Stockbridge HaJl). The three campus and at approximately the same remaining lectures will be presented on tinw c.e the Svdety. The arrangements consecutive days in the sarne han at commiltee to date is aware of the following: 9:00A.M. Mathematical Ascociation of America By invitation of the Committee to Select Hour Speakers for Summer and Monday through Wednesday, August 24- Annual Meetings there will be two add.res­ 26, with Board of Governors meeting ses. Sunday, August 23 at 10:00 A.M. Professor Art.hur Sar.d of Queens Pi Mu Epsilon College of the University of the City of New York wiJl speak on "Function spaces" Tuesday and Wednesday, August 25-26, at 2:00 P.M. on Thursday, August ?..? in with dinner and speaker Tuesday even­ Bowker Auditorium (Stockbridge Hall), ing in recognition of the fiftieth anni­ Professor John Stallings of Prince­ versary, breakfast Wednesday morn­ ton University will speak on "The founda­ ing, Council meeting Wednesday after­ tions of polyhedral topology'' at 2:00 P.M. noon, and sessions for papers Tues­ on Friday, AuguAt 28 in Bowker Auditor­ day and Wednesday qfternoons. ium (Stockbridge Hall). Mu Alpha Theta Governing Council There will be sessions for contrib­ uted papers on Tuesday afternoon, 'Jhnr!'- Wednesday, August 26, 9:00A.M.

411 Socie:t:.L!.2.!.Jndustrial and AJ2.12lied Mathe­ 25-27, from 9:00 A.M. to 5:00 P.M. on matics each of these three days. Attention is in­ Wednesday and Thursday, August26-27, vited to the announcement of the Employ­ with a party Wednesday evening. ment Register on page 414, in particular to the deadline dates for application to the Institute of Mathematical Statistics register and to the necessity for prompt Wednesday through Saturday, August 26- registration at the Employment Register 29, with council and business meetings Desk by both applicants and employers. on Thursday and Friday evenings. Room and Board These statements should not be taken as definitive inasmuch as each organization There will be rooms available in officially notifies its membership and University dormitories at the rates of governing bodies. $4.00 per day for a single room and $3.50 Council and Business Meetings per day for half of a double room. Meals will be available in University The Council of the Society will meet dining rooms on a cafeteria basis. One on Tuesday evening at 5:00 P.M. in the may contract separately for room and for Council Chamber on the second floor of the board. It is intended that one who eats in Student Union. There will be an inter­ the cafeterias shall contract for all of his mission for dinner in the Student Union. meals there. Most persons will find it This schedule is subject to revision if the desirable to contract for meals since there agenda promises to be brief. are few good places to eat in the area. The Business Meeting of the Society Hours and prices of the meals are be held in Bowker Auditorium (Stock­ will Breakfast 7:00 - 8:30 $0.75 Hall) on Wednesday, August 26 at bridge Lunch 11:30 - 1:30 1.25 10:30 A.M. Dinner 5:00 - 6:30 1.50

Registration On page 475 of these NOTICES (the inside back cover) is a reservation form The Registration Desk will be in the for room and board. Such a form will NOT Ballroom on the First Floor of the Student appear in the August issue, because all Union. It will be open on Sunday, August 23, reservations must be in the hands of Mr. from 2:00 P.M. till 8:00 P.M.; on Monday, Harold C. Durgin, Conference Office, The August 24, from 8:00 A.M. till 5:00 P.M.; University of Massachusetts, Amherst, on Tuesday through Friday, August 25-28, Massachusetts by Saturday, August 15, from 9:00 A.M. till 5:00 P.M.; and on Sat­ 1964, which may be earlier than the de­ urday, August 29 from 9:00 A.M. till livery of the August issue. Rooms will be 2:00P.M. available as early as Saturday, August 22 The registration fees will be as and may be occupied through 3:00 P.M. follows: on Sunday, August 30. Member $2.00 The arrangements committee has Member's family 0.50 made block reservations at hotels and for the first such registrant and no charge motels in and near Amherst. The names, for additional registrants addresses, rates and distance from the campus of these hostelries are as follows: Student No fee Others $5.00 Lord Jeffery Inn, 30 BoltwoodAvenue, Am­ herst, Massachusetts. 23 single rooms Employment Register at $9.00; 8 double rooms at $14.00- $16.00 per room. One and one half The Mathematical Sciences Em­ miles from the campus. Register will be in Rooms E 13 ployment Amherst Motel, Amherst, Massachusetts. of Machmer Hall, across Ellis and E 15 3 single rooms at $10.00- $12.00; 18 from the Student Union. It will be Drive double rooms at $15.00 per room (will open Tuesday through Thursday, August hold 3 or 4 persons at $17.00 for 3 and

412 $19.00 for 4). Twomilesfromthecam­ train. pus on Route 9. Massachusetts State Highways 9 and 116 go through Amherst. The East-West Hotel Northampton, 36 King Street, North­ Massachusetts Turnpike connecting with ampton, Massachusetts. 75 rooms; New York State Thruway lies less than single rooms at $9.00 and $10.00 and 25 miles south of Amherst. Use West double rooms at $14.00 and $15.00. Springfield or Chicopee exits. Seven miles from the campus. The New York Central Railroad and There are at least seven campsites the New York, New Haven and Hartford from ten to forty miles from the campus, Railroad serve the area. Passengers de­ with more than two hundred fifty individual train at Springfield, Massachusetts, and sites. take Peter Pan or Trailways Bus for Am­ herst. ENTERTAINMENT Bradley Field, Connecticut (Hart­ ford-Springfield) is the airport which Three features of the area near serves Amherst. It is located 21 miles Amherst are highlighted. Old Sturbridge south of Springfield. Commercial lines village at Sturbridge, Massachusetts (about coming into Bradley Field include Alle­ 50 miles from Amherst) is "a center of gheny, American, Eastern, Northeast, living history which brings to life the way Mohawk, United and TWA. Airfield Service people lived, worked and traded during the Taxi Company will provide limousine and first fifty years of American independ­ bus service from Bradley Field to Am­ ence." An all day excursion by chartered herst through Springfield, with travel time bus to Old Sturbridge Village is planned. about an hour and a half. Fare from Brad­ Reservations will be taken at the time of ley Field to Amherst is $3.50 per person registration. one way. On advance registration, please Tanglewood at Lenox in Berkshire indicate date of arrival, airline, and flight County (about 60 miles from Amherst) is number. the "summer home of the Boston Sym­ Greyhound, Trailways, Interstate, phony Orchestra and its eight week Berk­ Blue Lines, Peter Pan, and Vermont shire Festival and Berkshire Music Cen­ Transit bus lines come into Springfield. ter." The last performance is on August Peter Pan and Trailways serve Amherst 23. directly. Travel time by bus from Boston There is a Summer Theater at is about three and one half hours. Mountain Park about 15 miles from the A traveler whose flight plans re­ University. quire a change of planes to reach Bradley Each of these enterprises charges Field might inquire whether he can go admission. directly to Boston. He would have a longer bus trip but no change of planes. Direct SIAM will conduct a party in its plane service to Bradley Field is available traditional manner on Wednesday evening from many places, for example Miami and at a time, place, and price to be announced. Chicago. The University tennis courts and the bowling alleys in the Student Union WEATHER will be available. The town of Amherst has an out­ The mean temperature at this time door swimming pool, open to the public of year is 72°. Temperatures as high as at an admission charge of twenty-five 90° are not unknown. There is a possibility cents per person. of rain. Average humidity is 72.8%.

TRAVEL MAIL AND TELEGRAMS The address which individuals may Amherst is located in the Connecti­ use for mail and telegrams is American cut Valley, just 92 miles west of Boston Mathematical Society, c/o Conference Of­ and 27 miles north of Springfield. It can fice, University of Massachusetts ,Amherst easily be reached by highway, air, bus, or Massachusetts.

413 COMMITTEE ON ARRANGEMENTS

H. L. Alder T. Norvig A. E. Andersen E. Pitcher R. H. Breusch R. W. Wagner H. G. Jacob, Jr. G. L. Walker L. C. Lavallee Everett Pitcher Bethlehem. Pennsylvania Associate Secretary

MEMORANDA TO MEMBERS

THE EMPLOYMENT REGISTER will be available for distribution ~o~l-t during and after the meeting. The Mathematical Sciences Em­ It is essential that applicants and ployment Register, established by the employers register at the Employment American Mathematical Society, the Register Desk promptly upon arrival at the Mathematical Association of America, and meeting to facilitate the arrangement of the Society for Industrial and Applied appointments. Mathematics will be maintained at the Summer Meeting at the University of ANNOUNCEMENTOFCHANGESW Massachusetts, Amherst, Massachusetts THE COMBINED MEMBERSHIP LIST on August 25, 26, and 27,1964. The Regis­ ter will be conducted from 9:00 A.M. to The format used in listing members 5:00 P.M. on each of these three days. of AMS, MAA, and SIAM will differ in the There is no charge for registration, next issue of the Combined Membership either to job applicants or to employers, List, which will go to press in October, except when the late registration fee for 1964. All the listings, however, may not be employers is applicable. Provision will changed before the next issue. New addr:.>s­ be made for anonymity of applicants upon ses and changes of address are being request and upon payment of $3.00 to de­ processed in the new manner as they are fray the cost involved in handling anony­ received. The changes are: mous listings. (1) The entry for a member willlisthis Job applicants and employers who name, title, and place of employment, wish to be listed will please write to the and only one address, the mailing ad­ Employment Register, 190 Hope Street, dress. Both the home and the business Providence, Rhode Island 02906 for appli­ addresses will no longer be included. cation forms or for position description (2) Certain standard abbreviations will forms. These forms must be completed be introduced; e.g., the names of and returned to Providence not later than states in all state colleges and uni­ July 15, 1964, in order to be included in versities will be abbreviated, the listings at the Summer Meeting in (3) ZIP code numbers will be added to Amherst. Position Description Forms the addresses. which arrive after this closing date, but before August 10 will be included in the These changes are being undertaken tore­ register at the meeting for a late regis­ duce the size and production cost of the tration fee of $3.00. The printed listings Combined Membership List.

414 DOCTORATES CONFERRED IN 1963

The following are among those who received doctorates in the matematical sciences and related subjects from universities in the United States and Canada during 1963. In each case, when available, the university, minor subjects (other than mathematical), and title of dissertation are given. 579 names are listed. ADELPHI UNIVERSITY Fosdick, Roger Lee Some results in the theory of small elastic MacHenry, Trueman deformations superimposed on large elas­ Metabelian D1r-groups: A construction tic deformations UNIVERSITY OF ALABAMA Roberts, Howard Mahlon On the dual Pontrjagin classes of a manifold Smith, Gaston On the (f,dn) method of summability Shen, Mei-Chang Minor - Fluids UNIVERSITY OF ALBERTA Problems in the theory of surf Trollope, John Reginald Sun, Tze-Chien Exact solutions of the Einstein-Maxwell A central limit theorem for non•linear equations that determine null fields functions of a normal stationary process Tyndall, William Felker UNIVERSITY OF ARIZONA A duality theorem for a class of continuous Bryan, Charles Allen linear programming problems On an iterative technique for algebraic systems CALIFORNIA INSTITUTE OF TECHNOLOGY Irwin, Robert Cook Knuth, Donald Ervin Generalized quaternion algebras over alge­ Finite semifields and projective planes braic number fields Willett, Douglas Warren Solutions of nonlinear integral equations AUBURN UNIVERSITY and their application to singular pertur­ Ford, Ralph Montgomery bation problems Basis properties in dimension theory Megibben, Charles Kimbrough, III UNIVERSITY OF CALIFORNIA, Berkeley On algebraically compact and primary abelian groups Aigner, Dennis John A study of recursive proQ"ramming:Dynamic BOSTON UNIVERSITY supply programming Glazer, Harold Astromoff, Andrew B. Comparison of the student two- sample Functionally complete algebras and their t test and the Wilcoxon-Mann-Whitney generalizations test for normal distributions with un­ Balslev, Erik equal variances Perturbation of differential operators BRANDEIS UNIVERSITY Bickel, Peter John Asymptotically nonparametric statistical Williamson, Susan inference in the multivariate case Crossed products and hereditary orders Borgman, Leon Emry UNIVERSITY OF BRITISH COLUMBIA Stationarity in a Markov chain approxima­ tion to the Neyman-Scott clustering model Ahuja, Jagdish Chand for populations Minor - Economics Statistical problems in the study of growth Borrelli, Robert Louis A priori estimates for elliptic boundary BROWN UNIVERSITY problems Breuer, Shlomo Brown, Barry Wilson Some general results in linear viscoelas­ Dynamic programming in finite state dis­ ticity crete time Markov chains

415 Brown, Robert Dillon Seever, Galen Lathrop Reflection laws of fourth order elliptic Measure on F-spaces differential equations in two independent Sills, William Hall variables Arens multiplication and spectral theory Bucy, Richard Snowden Evans Sinha, Samirendra Mohan Recurrent events for transient Markov chains Stochastic programming Cambern, Michael James Swimmer, Alvin Analytic range functions Valuation ideals in polynomial rings over do Carmo, Manfredo Perdigao arbitrary ground fields The cohomology ring of certain Kahlerian Wigley, Neil Marchand manifolds Asymptotic expansions at a corner of solu­ Fabius, Jaap tions of mixed boundary value problems Asymptotic behavior of Bayes' estimates Williamson, Robert Emmett, Jr. Gamelin, Theodore William A Thorn theorem for the combinatorial The extension problem for restrictions of cobordism group functions in a subspace of C(X) Wirth, Niklaus Emil Hanerfeld, Harold A generalization of Algol Conditions for the discreteness of the spec­ Wittenberg, Helen trum of singular elliptic operators Limiting distributions of random sums of Hanf, William Porter independent random variables Some fundamental problems concerning Yahav, Joseph Aharon languages with infinitely iong expressions On optimal stopping Hoyland, Arnljot Yee, Kane Shee-Gong Some problems in robust point estimation Boundary-value problems for Maxwell's Kallin, Eva Marianne equations A non-local function algebra UNIVERSITY OF CALIFORNIA, LOS ANGELES Kelley, Allen Frederick, Jr. Change of variable techniques in ordinary Biriuk, George differential equations Extension of formally normal operators in a larger Hilbert space Kodama, Laura Stith Ketchum Some problems in the theory of uniform Clark, Virginia Ann Leader approximation Minor - Epidemiology Choice of levels for the several factor Labrousse, Jean-Philippe Marcel design problem Homotopy invariants associated with ellip­ tic partial differential operators Fraenkel, Aviezri* Rational approximations to algebraic num­ Maitra, Ashok Prasad bers Dynamic programming for countable state systems Franklin, Stanley Phillip Concerning continuous relations Miller, Willard, Jr. On a class of vector-valued functions co-. Fuller, Derek Joseph Haggard variant under the classical groups, with Mappings into abstract Riemann surfaces applications to physics Goldstein, Myron Purves, Roger Alexander L- and K-kernels on an arbitrary Riemann Bimeasurable functions surface Resnikoff, Howard Leonard Hutchison, Gerald Andrew On fundamental domains for discrete groups Meromorphic minimal surfaces Rosenberg, Harold William McLaughlin, Thomas Graham Inequalities for periodic surfaces of vector Decompositions of recursively enumerable fields sets, relative to the post-Dekker-Uspen­ skij classification scheme Roth, Richard Lewis Collineation groups of finite projective Rosenfeld, Melvin planes Commutative F-algebras

1961 degree not reported in the list of doctorates published in these NOTICES in 196Z.

416 Sallin, Edward Arthur Elsey, Sister Margaret Grace Bounds for iterates, inverses and spectral Minor - Physics variation of non-normal matrices Some convergence theorems for normal Ritt series Sato, Daihachiro Integer valued entire functions UNIVERSITY OF CHICAGO Stiel, Edsel Ford Deliyannis, Platon Constantine Isometric immersions of Riemannian mani­ Measure bundles and group representations folds Hungerford, Thomas William CARNEGIE INSTITUTE OF TECHNOLOGY Bockstein spectra F ederowicz, Alexander John Linderholm, Carl Eric A generalized algorithm for solution of a On some questions in ergodic theory class of non-convex programming prob­ Martinez, Hugo Murua lems Studies in stochastic learning theory Goyal, Vinod Bhushan Minkus, Jerome Bernard Bounds for solutions of certain classes of On embedding highly connected manifolds nonlinear partial differential equations in Euclidean space Smith, Frank Beckley Richman, Fred A Bayesian procedure for detection of Generalized quotient rings and zero­ slippage divisor-preserving modules Voytuk, James Anthony Tsutakawa, Robert K. Singularities for piecewise analytic bound­ Up-and-down method in bio-assay ary conditions and boundary curves Venter, Johannes Hendrik Winter, David Lawrence On stochastic approximation methods Finite groups with a faithful representation of degree less than (2.p + l)/3 UNIVERSITY OF CINCINNATI CASE INSTITUTE OF TECHNOLOGY Cerimele, Benito Joseph The conformal representation of a perfor­ Mirasol, Noel Mario ated strip upon a canonical domain A systems approach to logistics Mond, Bertram Roudebush, William Henry On the direct sum and tensor product of Analysis of discretization errors for dif­ linear and non-linear programs ferential equations with discontinuous coefficients Mushenheim, Harold George Applications of interpolation series to Sanders, Jerry function theory Minor - Operations Research The application of a theory of multiload Rolwing, Raymond systems to optimization problems On a system of quadratic equations and its integral equation Seidman, David Richard Traffic congestion on upgrades of two-lane Sjoberg, John Calvin highways A family of associated functions Sprague, Charles Fremont, III UNIVERSITY OF COLORADO Self-organizational aspects of the general systems theory models of organizations Bierstedt, Ronald Gale Some problems on the distribution of kth Stewart, Donald I. power residues modulo a prime Minor - Electrical Engineering Overall mathematical models of replace­ Segal, Sanford Leonard ment and maintenance The error term in the formula for the average value of the Euler ¢ - function Wolff, Ronald William Statistical inference in queuing theory Thompson, Robert Glenn Classification of quadratic forms

CATHOLIC UNIVERSITY OF AMERICA COLUMBIA UNIVERSITY Austing, Richard Henry Abramson, Lee Richard Minor - Physics Sequential design of experiments with two Groups of circulant matrices random variables

417 Cohen, Arthur CORNELL UNIVERSITY Some questions of predicting from a re­ Atkinson, Glenn Francis gression function and a hybrid problem Designs for sequences of treatments with on the exponential density carry-over effects Comer, John Preston, Jr.** Berger, Charles Arnold Some stochastic approximation procedures Normal dilations for use in process control Brown, Gordon Elliott Feldman, Edgar On commutators in certain Lie algebras The differential geometry of immersions Djorup, Frans Martin, Jr. Freiman, Charles Visvald* On the operator equation AX = XB Major - Engineering Science Protective block codes for asymmetric Newman, David Stanley binary channels Minor - Physics On the exact calculation of the partition Gastwirth, Joseph Lewis functions of one-dimensional systems On some problems in the theory of particle counting and queues with infinitely many Porter, Gerald Joseph servers Higher order Whitehead products Lawvere, Francis William, Jr. DUKE UNIVERSITY Functorial semantics of algebraic theories Cavior, Stephan Robert Linhart, Peter Bernard Minor - Philosophy Overflow traffic from a trunk group with Equivalence classes of functions over a balking finite field Marshall, Clifford Wall ace* Hayes, David Ryan Contributions to the statistics of multiple Minor - Philosophy correspondences The distribution of irreducibles in GF [q,x) May, Warren Lee King, Lunsford Richardson Binary forms over number fields Minor - Philosophy Rieffel, Marc On <1>-compactifications A characterization of commutative group Rosenstein, George Morris, Jr. algebras and measure algebras Minor - Philosophy Samuel, Ester* Generalizations and an extension of On the compound decision problem in the Lebesgue's covering theorem nonsequential and the sequential case Van Meter, Robert Guy Schanuel, Stephen Minor - Philosophy Heights in number fields The number of solutions of certain systems of equations in a finite field Sigler, Laurence Edward On the real asymptotic theory of the fac­ torization of ordinary differential opera­ tors FLORIDA STATE UNIVERSITY

St~rmer, Erling Chandler, Richard Edward Point measures in two-sided non-commu­ Cellular subcomplexes of piecewise linear tative integration theory manifolds Thorpe, John Alden Greathouse, Charles Alfred High order sectional curvatures Locally Hat, locally tame and tame em­ beddings Wagner, Eric Gerhardt Uniformly reflexive structures: toward an Neggers, Joe abstract theory of computability Derivations on p-adic fields Welch, Peter Dunbar Retherford, James Ronald Some contributions to the theory of priority Basic sequences, bases and weak*-bases queues in Banach spaces

* See footnote on page 416 **1962 degree not reported in the list of doctorates in these NOTICES in 1963,

418 Rice, Peter Milton Denham, Walter Frank Homotopically homogeneous spaces and Steepest-ascent solution of optimal pro­ manifolds gramming problems Ruckle, William Henry Denton, John Stanton, Jr. Schauder decompositions and bases Applications of the Herbrand theorem Elashoff, Robert Myles UNIVERSITY OF FLORIDA Multivariate two-sample problems with Hatfield, Francis discrete and continuous variables Minor - Philosophy Ericson, William Arnold Root-powering of polynomial equations Minor - Economics Optimum stratified sampling using prior GEORGE PEABODY COLLEGE information Hodges, John Raymond Even, Shimon Minor - Psychology On information loss less automata A study of the ability of a group of eighth grade students to learn and use certain Germeles, Apostolos Evangelos mathematical concepts Minor - Electricity and Magnetism Laminar channel-flow of conducting fluids UNIVERSITY OF GEORGIA under an applied transverse magnetic field Bastida, Julio Rafael Group homomorphisms associated with the Griebach, Shiela Adele H-equivalence in certain semigroups Inverses of phase structure generators Bell, Curtis Porter Hayden, Seymour An isotopy theorem On finite linear groups whose order con­ tains a prime larger than the degree Rao, Pejover Vishwamber The effect of truncation on the analysis of Hirschfield, Judith Barbara variance test for PBIB designs The curves of genus 3 defined over z;zZ Selden, John Jacobs, Stanley Joel Theorems on topological semigroups and Low Rossby number flow over bottom semirings topography Johnson, Nicolas HARVARD UNIVERSITY Numerical methods in communication en­ Ackerberg, Robert Cyril gineering The viscous incompressible flow inside a cane Kennison, John Frederick Natural functors in topology Adorno, David Samuel Stochastic optimization in the theory of Kirchner, Roger Burr control Elementary moduli for finite Riemann sur­ faces Anderson, Donald Gordon Numerical experiments in kinetic theory Kitchen, Joseph Weston, Jr. Barcilon, Victor Almost periodic functionals in the conjugate Thermally driven motion of a stably strati­ space of a Banach algebra fied fluid in a rotating annulus Lave, Lester Bernard Bloom, David Michael Measurements of technological change in On the subgroups of SL(3,q) American agriculture Bossert, William Hines Lawler, Eugene Leighton Minor - Biology Some aspects of discrete mathematical Character displacement in animals programming Brandl, John Edward Lechner, Robert Joseph The early development of econometric Affine equivalence of switching functions method Lubin, Jonathan Darby Choi, Keewhan One-parameter formal Lie groups over Analysis of multi-factor experiments with p-adic integer rings prior knowledge Conlon, Lawrence William Lubkin, Saul Spaces of paths on a symmetric space Theory of covering spaces

419 Lynch, Robert Emmett Ifram, Adnan Farhan Error bounds of a numerical solution of the Minor - Philosophy quasi-linear wave equation On the asymptotic behavior of densities with applications to several sequential Manjarrez, Victor Manuel tests of composite hypotheses Polynomial bases for compact sets in the plane Miech, Ronald Joseph Almost primes generated by a polynomial Rockafellar, Ralph Tyrrell Convex functions and dual extremum prob­ P atnaik, Surendranath lems Minor - Astronomy Some mapping theorems with applications Schwartz, Lawrence Elliot to fixed points of multiple-valued trans­ From core to ring: An econometric study formations of the relocation of the central office and its workers Resh, James Albert On networks and hi-complete graphs Sims, Charles Coffin Enumerating p-groups Rosenkrantz, Walter Abraham Minor - Philosop)ly Taylor, Lester Dean Probability and Fourier series Minor - Economics The principal-components-instrumental­ Rublein, George Thomas variable approach to the estimation of Minor - Physics systems of simultaneous equations Representations of generalized homology and cohomology theories Turyn, Richard Joseph Character sums and difference sets Schubert, Sanford Roy On a new class of generalized analytic Zahl, Samuel functions Bounds for the central limit theorem error Skeath, James Edward Zilber, Joseph Abraham Minor - Philosophy Categories in homotopy theory An extension of the Denjoy-Carleman­ Ahlfors theorem in subharmonic form UNIVERSITY OF IDAHO Tang, Francis Chi-yu Fisher, Newman Hersch Minor - Physics Minor - Physics Generalized direct products with amalga­ Stability and periodicity properties of the mated subgroups space sled Tung, Hsue Chong Minor - Aeronautical Engineering UNIVERSITY OF ILLINOIS Generalized functions and their Weierstrass transforms Blakney, Simmie Samuel Minor - Education Vayo, Harris Westcott Lusin's theorem in metric theory Minor - Biophysics The determination of electrical parameters Embree, Earl Owen of a mammalian visceral muscle system Minor - Philosophy A class of linear differential equations Vogt, Robert Martin involving distributions Minor - German Finiteness conditions in group theory: Foulk, Clinton Ross Schmidt's problem Minor - Electrical Engineering Cyclic spot-error-correcting codes Welker, Russell Arthur Minor - Electrical Engineering Grossman, Marvin Willard Difference equations in fields of charac­ Choquet and Silov boundaries teristic p Gylys, Vytautas Minor - Astronomy ILLINOIS INSTITUTE OF TECHNOLOGY Deterministic machines with apparently Flaxman, Abraham John random output Representation of a quadratic integer as a Harris, John Henry binary quadratic form Minor - Electrical Engineering An abstract integral defined for all real­ Rodabaugh, David J, A generalization of the flexible law valued measurable functions

420 Senechalle, Lester john Grosvenor, Dale David On uniform operators Minor - Economics Uses of integer programming in problems Stueben, Edwin Frank of optimization and their computational Ordinal invariants of quadratic forms over aspects algebraic number fields Hemmerle, William Jacob Witthoft, William G. Minor - Industrial Engineering A class of nilstable algebras The logical structure of analysis of vari­ ance and its implementation on digital INDIANA UNIVERSITY computers Fregoso, Arturo Urbina Hinkelmann, Klaus Heinrich Minor - History and Logic of Science Minor - Genetics Local categories Design and analysis of multi-way genetic Geissinger, Ladnor Dale experiments A generalization of local class field theory Leaverton, Paul Emmett Gillman, Albert Franklin, III Minor - Economics Minor - History and Logic of Science Statistical procedures applicable in the Noncommutative schemes analysis of bioassays when the usual assumptions are not fulfilled Glass, Kenneth William Henry On extending a norm residue symbol Lee, llbok Minor - Economics Grant, Kenneth Louis Bioassay with quantal response observed Minor - Physics at different times Abstract class formations Lewish, William Thomas Howe, Wanda Jane Minor - Chemical Engineering A new approach to Witt vectors and Witt's Linear estimation in convex parameter reciprocity law spaces Miller, john David Mordeson, john Nelson Minor - Physics Minor - Philosophy Modifications and cobounding manifolds Coefficient integral domains in commutative Rao, Uppuluri V. Ramamohana algebras I. Maximum likelihood estimation in finite Mueller, Elsie Christine stationary ergodic Markov chains, Minor - Philosophy II. A strong converse to the coding theorem Generating sets for uncleft algebras for continuous memoryless channels Richards, Dale Owen Sato, Kenkichi Incompletely specified models in life test­ Local triangulation of real analytic varia­ ing ties Robinson, Thomas john Schmidt, Bernd Siegfried Minor - Philosophy Minor - Physics Mappings with small point-inverses The gravitational tensor as a concomitant of a given electromagnetic tensor in the Sanderson, Donovan Forest null case Minor - Philosophy Galois completely primary rings IOWA STATE UNIVERSITY Walling, Derald Dee Booker, Aaron Hicks Minor - Statistical Education Nonlinear estimation Numerical methods for non-linear least squares curve fitting problem Byrne, George Dennis Minor - Aerospace Engineering Multi-point Runge-Kutta integration Van Aarde, Izak Mauritz Rohm Minor - Genetics Deckert, Kenneth Laurel Covariances of relatives in random mating Minor - Physics populations with linkage Solutions for nonlinear diffusion equations with integral type boundary conditions White, Robert F, Graham, john Elwood Minor - Dairy Husbandry Rotation designs for sampling on successive Randomization analysis of the general occasions experiment

421 STATE UNIVERSITY OF lOW A Vobach, Arnold Raleigh On two-dimensional continua structured Luther, Norman Yeomans by finite families of simple closed curves Quadratic forms in normally distributed variables McGILL UNIVERSITY Shekoury, Raymond Nejib Connell, Ian G. The L-system and Postnikov invariants of On the group ring a CW -complex Csorgo, Miklos Tanis, Elliot Alan Some Kolmogorov-Smirnov-Renyi type theo­ Linear forms in the order statistics from rems of probability an exponential distribution Fortin, Jacques G. JOHNS HOPKINS UNIVERSITY On the algebraic invariance of the dimen­ sion of a manifold Shanahan, John Peter On some questions on hyperbolic differen­ Lorimer, Peter James tial equations A study of T2 -groups Williams, George Trevor McMASTER UNIVERSITY Stochastic model for incubation periods in infectious diseases Kerr-Lawson, Angus Carmichael A filter description for the homomor­ UNIVERSITY OF KANSAS phisms of the algebra of bounded analy­ tic functions on the unit disc Deeter, Charles Raymond Minor - Physics UNIVERSITY OF MARYLAND Discrete harmonic kernels Kezlan, Thomas Phillip Bordelon, Derrill J. Topics in orthogonal polynomials in sev­ Minor - Physics eral variables Rings with nil commutator ideals Chu, Sherwood C. Mulla, Fuad Shakir On mixed boundary value problems for a Minor - Physics linear third order partial differential On spaces of potentials connected with equation of hyperbolic type LP-classes Lagnese, John Edward Stuth, Charles James Self-adjoint differential operators of the Minor - Physics pure wave type A generalization of the Cartan-Brauer-Hua theorem

UNIVERSITY OF KENTUCKY MASSACHUSETTS INSTITUTE OF TECHNOLOGY Pfaltzgraff, John Andrew Abrahams, Paul William Minor - Physics Minor - Electrical Engineering Extremal problems for analytic functions Machine verification of mathematical proof defined by a Stieltjes integral Arbib, Michael Anthony LOUISIANA STATE UNIVERSITY Minor - Electrical Engineering Hitting and Martingale characteristics of Brooks, Robert Morgan one-dimensional diffusions A ring of analytic functions Becker, Ronald Isadore Brown, Dennison Robert Periodic solutions of linear and quasi­ Topological semigroupl!l of non-negative linear parabolic differential equations matrices Brown, Edward Martin Decell, Henry Planchard, Jr. Hauptvermutung for 3-complexes Measure algebras over a compact topo­ logical semigroup Clark, Alfred, Jr. Minor - Physics Leland, Kenneth Owen On the hydrodynamics of superfluidhelium Topological analysis of analytic functions Dawson, Donald Andrew Mott, Joe Leonard Minor - Physics On invertible ideals in a commutative ring Construction of diffusions with specified Phillips, Richard Carlisle mean hitting times and hitting proba­ Almost Dedekind domains bilities

422 Evans, Thomas George UNIVERSITY OF MICHIGAN Minor - Electrical Engineering Heuristic program to solve geometric­ Andrews, George Harold analogy problems Periodograms of graduation and subtabula­ tion operators Freeman, John Markham Minor - Modern Languages Arnold, Richard Fair banks Perturbations of Volterra operators Group methods in combinatorial switching theory Fristedt, Bert Elmer Auslander, Bernice Rita Liberman Minor - Physics Reflexive modules over integrally closed Generalized variations, continuity, and noetherian domains Hausdorff measures of sample functions of stochastic processes with independent Battle, Joseph increments Imbedding of graphs in orientable 2-mani­ folds Goodman, Roe William Borges, Carlos Alberto De Buarque Minor - Modern Languages Periodic solutions of nonlinear differential Causal S-operators and domains of depen­ equations: Existence and error bounds dence for hyperbolic equations Cicchinelli, Alexander Louis Hackman, Morton Matthew The composite of two Gaussian distribu­ Minor - Modern Languages tions as a model for blood pressure de­ Abstract time-dependent Cauchy problem composition in a man Claborn, Luther E. James, Donald Gordon On the theory of E rings Minor - Electrical Engineering Extension of isometries of sublattices over Foulser, David Arthur local fields On finite affine planes and their collinea­ tion groups Leibowitz, Gerald Martin Minor - Modern Languages Graves, Glenn William On generalized analytic functions A complete constructive algorithm for the general mixed linear programming prob­ Luckham, David Compton lem that does not require augmentation Minor - Philosophy or perturbation Topics in the theory of constructive hier­ Harrison, Michael A. archies Combinatorial problems in Boolean alge­ McKissick, Robert Jones bras and applications to the theory of Existence of a non-trivial normal function switching algebra Kelingos, John Alexander Silver, Edward Allan Contributions to the theory of quasiconfor­ Minor - Hydraulics mal mapping Bayesian estimation in Markov processes Kuzma, Jan Waldemar A statistical study of various aspects of a Smith, John Howard battery of clinical neurologic tests Minor - Modern Languages Krom, Melven Robert Certain imbedding problems for algebraic Separation principles in the hierarchy the­ number fields ory of pure first-order logic Tauer, Sister Rita Jean Lewkowicz, Raymond Edward Minor - Physics A characterization of the analytic operator Maximal abelian subalgebras in finite fac­ among the Loewner-Benson operators tors of Type II and on M-dimensional area of continuous Warner, Frank Wilson, III mappings Minor - Physics Lim, Yu Chin Conjugate locus of a Riemannian manifold Minor - Physics and Electrical Engineering Wu, Hung-Hsi Linear theory of beam-plasma interaction On the deRahm decomposition theorem Lucas, William Franklin On solutions to n-person games in parti­ Young, Paul Rue! tion function form Minor - Philosophy On the structure of recursively enumerable Luh, Jiang sets On the centralizer of a Galois ring

423 Moriguchi, Chikashi Wrobleski, William Joseph Manufacturers' short-term production de• Extensions of the Dwyer-Macphail matrix cisions under the different phases of derivative calculus with applications to business cycles as shown by selected estimation problems involving errors­ industries in-variable and errors-in-equations Nashed, M. Zuhair z. MICHIGAN STATE UNIVERSITY Iterative methods for the solutions of non­ Fisk, Donald LeRoy linear operator equations in Hilbert Quasi-martingales and stochastic integrals space Hemminger, Robert Louis Pillai, Ramanpillai Krishna On the Wedderburn principle theorem for Some aspects of statistical inference for commutative power-associative algebras Mth order Markov processes Kay, David Clifford PatH, Venkutai Hanamant Ptolemaic metric spaces and the charac­ The Behrens-Fisher distributions terization of geodesics by vanishing Ruehr, Otto George metric curvature Nonlinear modeling functions of a special Oodaira, Hiroshi type The equivalence of Gaussian stochastic Sarason, Donald Erik processes Minor - Physics Turner, Walter William The Hp -spaces of annuli A method of generating integral represen­ Schork, Michael Anthony tations A multivariate investigation of the effects UNIVERSITY OF MINNESOT-{\. of inbreeding on size and growth in man Ahern, Patrick Robert Schuette, Donald Richard Minor - Physics Some extensions in multiple decrement Tauberian methods in the theory of primes theory with applications to mortality tables analyzed by cause of death Alam, Khursheed A two-stage sampling procedure for esti­ Schuur, Jerry Dee mating a common mean Upper and lower bounds for the norms of solutions of systems of differential equa­ Antelman, Gordon Randolph tions Bayes decision theory: lnsensitivitytonon­ optimal design Seiken, Arnold Minor - Physics Avishalom, Dov Issachar Connexions in principal fibre bundles with Minor - Philosophy group GL(nm,R) The axiomatic processes in the foundations of geometry and their translation into Sparrow, Frederick Tom~inson reality and education A queuing theory model of marketequilib· rium Beekman, John Alfred Solutions to generalized Schroedinger Swong, Khyson equations via Feynman integrals con­ A representation theory of continuous lin­ nected with Gaussian Markov stochastic ear maps processes Tracy, Derrick Shannon Burgstahler, Sylvan,Duane Finite moment formulae and products of Minor - Physics generalized K-statistics with a generali­ Contributions to the study of heat conduc­ zation of Fisher's combinatorial method tion in composite systems and to heat potential theory Trauth, Charles Arthur, Jr. Gaalswyk, Arie On the connectedness of directed graphs Minor - Mechanics and Materials under binary operations An investigation of the existence and quali­ Walker, Thomas Patrick, Jr. tative behavior of one-dimensional com­ Two metric theorems in diophantine ap­ bustion shock layers proximation Jorgensen, Roy Alfred, Jr. Minor - Physics Walsh, Bertram John Construction and variation of canonical Structure of spectral measures on locally conformal maps of multiply-connecte.d convex spaces domains

424 Knatterud, Genell Lavonne UNIVERSITY OF NEBRASKA Minor - Psychology Gross, Mildred Foreman (Mrs.) An evaluation of certain data from the Sieve methods and some applications cancer detection center Herzog, John Orlando Krishnaiah, P arachuri Ram a Phragmen-Lindelof theorems for elliptic Simultaneous tests and efficiency of gen­ partial differential equations eralized balanced incomplete block de­ signs Heuer, Charles Vernon An extension problem for cancellative Mielke, Paul William, Jr. semigroups Minor - Public Health Variance component estimation and pre­ Peinado, Rolando E. cision for the general two-way classifi­ The generalized module type of a ring cation UNIVERSITY OF NEW MEXICO Rizvi, Mohammad Haseeb Ranking and selection problems of normal Entringer, Roger Charles populations using the absolute values of Minor - Engineering their means: fixed sample size case Some properties of certain sets of coprime integers Sather, Duane Paul Minor - Physics Kent, Darrell Conley Maximum properties for Cauchy's problem Minor - Physics on n-dimensional space-time Convergence functions and their related topologies Sather, Jerome Odell Minor - Physics NEW SCHOOL FOR SOCIAL RESEARCH The initial boundary-value problem for the Navier-Stokes equations in regions with Sardy, Hyman moving boundaries Minor - Sociology The determinants of corporate savings Schmaedeke, Wayne William Optimal control theory for nonlinear vector NEW YORK UNIVERSITY differential equations with measure co­ efficients Carroll, William J. Application of an inspection scheme for Spielberg, Stephen Esreal attributes Solutions to certain non-homogeneous sec­ ond order partial differential equations Bachmuth, Seymour expressed in terms of Wiener integrals Automorphisms of free groups and free metabelian groups Stackelberg, Olaf Patrick Probabilistic limit theorems arising in Chen, Yung-Ming arithmetic analysis Diffraction by a smooth transparent object Walczak, Hubert Ronald Fithian, James Hampton Distortion theorems for quasiconformal An investigation of a certain singular inte­ mappings using the complex dilatation gral equation and its solution by an ap­ proximation method Weckwerth, Vernon Erwin Minor - Public Health Ginsberg, Felix Elias Imputation of non-response values with The Cauchy problem for the one dimen­ special emphasis on hospital and health sional heat equation surveys Kahane, Charles Williamson, John Alexander Analyticity of solutions of mildly singular Some renewal theorems for positive inde­ integral equations. Extension of a theo­ pendent random variables rem of Lewy Yen, Elizabeth Yu- Yin Hsi Kazdan, Jerry Lawrence On two-stage non-parametric estimation Minor - Physics A boundary value problem ansmg in the theory of univalent functions Kuby, George Application of nonlinear membrane theory UNIVERSITY OF MISSOURI to pressure vessels Steiner, Eugene Francis Laska, Eugene M. Lattice of topologies on linear spaces A general theory of robustness

425 Leavitt, Jay A. UNIVERSITY OF NORTH CAROLINA A power series solution for compressible Cox, Raymond Howard flow past a conical shock wave Integral equations in certain topological Levine, David Alan rings An implicit finite difference solution for Das Gupta, Somesh symmetric hyperbolic quasilinear dif­ Minor - Psychology ferential equations Some problems in classification Machover, Maurice Kendell, Peter James Generalized eigenvectors and separation Estimation of the mean life of the exponen­ of variables tial distribution from grouped data when Morse, Burton the sample is censored-with application Diffraction by polygonal cylinders to life-testing Orkand, DonaldS. Leadbetter, Malcolm Ross Minor - Economics On the nonparametric estimation ofprbba­ Some statistical techniques for the man­ bility densities agement of incentive contracts Mudholkar, Govind Shrikrishna Reed, Kennard Some contributions to the theory of uni­ An abstract approach to a problem in neu­ variate and multivariate analysis tron transport theory Novick, Melvin Robert Resende, Eliseu Minor - Psychology Propagation reflection and diffraction of A Bayesian indifference procedure elastic waves Painter, Richard James Rosenstark, Jeannette Salkind Extensions of theorems of Ostrowski on The group of stable vector bundles the zeros of cert.ain classes of poly­ nomials Rushfield, Norman Inner factors and Blaschke products Portman, Ruth Melba Minor - Animal Physiology Segal, Martin Estimation of time, age, and cohort effects Stability regions for systems of linear differential equations with periodic co­ Ray, Sudhindra Narayan efficients Some sequential Bayes procedures for comparing two binomial parameters when Sibner, Robert J. observations are taken in pairs Uniformization of symmetric Riemann surfaces Roberts, Charles DeWitt An asymptotically optimal sequential de­ Siegel, Carol F. sign for comparing several experimental Contributions to the theory of queues with categories with a standard or control an absorbing barrier Searls, Donald Turner Solomon, Alan David Minor - Agricultural Economics The minimal surface of many sheets On the "large" observation problem Steinberg, Arthur Warren, William George On free quotient images of single relator Minor - Forestry groups Contributions to the study of spatial point Turner, Robert Edward Lee processes Perturbation of compact operators Witz, Klaus Gerhard Weinbaum, Carl Applications of compactification for Plane graphs and the word problem in some bounded operator semigroups groups Yniguez, Amador D'Bayan Zauderer, Erich Minor - Econometrics Wave propagation around a smooth object Some estimation problems of a nonlinear supply model implied by a minimax de­ NORTH CAROLINA STATE COLLEGE OF cision rule AGRICULTURE AND ENGINEERING NORTHWESTERN UNIVERSITY Reutlinger, Shlomo Minor - Economics Lin, Bor-Luh Evaluation of some uncertainty hypothesis Two topological problems concerning in­ for predicting supply finite dimensional Banach spaces

426 Unni, Kodakara Raman Bain, Lee J. Hankel transforms and weighted entire Tolerance limits for some life testing functions distributions Johnson, Charles Henry UNIVERSITY OF NOTRE DAME Some properties of the method of steepest Derwent, John Edwin ascent On the realization of the Morse inequalities Williams, Donald Ray as equalities A new approach to factorial experimenta­ Lappan, Peter Ambrose, Jr. tion Some sequential properties of normal and non-normal functions with applications UNIVERSITY OF OREGON to automorphic functions Jewett, Robert Israel Michaels, Sister Elizabeth Louise Partial differentiation on abelian groups A study of simple groups of even order Stafney, James Dean Riedl, John 0. Arens multiplication and convolution Partially ordered locally convex vector spaces and extensions of positive con­ OREGON STATE UNIVERSITY tinuous linear mappings Kohfeld, John Jacob Rueve, Father Charles R. Minor - Physics A geometrical approach to Markoff's min­ Stability of numerical solutions of ordinary ima problem of indefinite ternary quad­ differential equations ratic forms Olive, Gloria Sulski, Leonard C. Generalized powers On a statically determined problem in Roetr.:J.an, Ernest Levane shell theory Minor - Mechanical Engineering Vibrations of elastic bars OHIO STATE UNIVERSITY Soni, Raj Pal Coppage, William Eugene Minor - Physics Peirce decomposition in simple lie-admis­ The diffraction of cylindrical waves by two sible power-associative algebras parallel half planes Laffer, Walter Ball, II Tucker, Richard Ray Decomposition theorems for sets of ele­ Error analysis, convergence, divergence ments in finite abelian groups and sets and the acceleration of convergence of non-negative integers Wirshup, Arthur 0. McWorter, William Andrew Application of the Puiseux polygon to the j6 algebras solution of nonlinear integral equations Norris, Donald Oliver A topology for Mikusinski operators UNIVERSITY OF OTTAWA Outcalt, David Lewis Suhasini, Ravipati (Mrs.) A class of weakly alternative rings which Some properties of the distribution func­ are not power-associative tions of the index numbers of simple continued fractions UNIVERSITY OF OKLAHOMA PENNSYLVANIA STATE UNIVERSITY Assenza, Joseph Robert Biesterfeldt, Herman John, Jr. Minor - Sanitary Engineering Extensions of the Cauchy-Riemann equa­ Use of multiple regression techniques for tions estimating municipal sewage treatment costs Cima, Joseph Anthony Some properties of harmonic functions Brandt, Edward Newman, Jr. generated by the Bergman-Whittaker An application of multivariate homogeneity operator of variance tests to the electrocardio­ gram in myocardial infarction Konhauser, Joseph Daniel Edward Biorthogonal polynomials and some of their OKLAHOMA STATE UNIVERSITY properties Allen, John Ed Lercher, Bruce Leon Starlike sets, inverse starlike sets, and a Strong reduction and recursion in combina­ generalization of convexity tory logic

427 Lu, Chin-Pi Park, Richard Miller The ring of formal power series in a Representation of complex three-space in countably infinite number of indetermin­ Argand six-space ates Rigler, A. Kellam Magill, Kenneth Derwood, Jr. On solution of biharmonic type problems On rings of continuous functions by finite differences Sieber, James Leo Schaefer, Paul Theodore Generalizations of some topological notions On summability methods proposed by to syntopogenic spaces Fekete Srivastava, Om Prakash Spangler, Charles Bishop Asymptotic behavior of certain functions Summability and the core of a generalized of order statistics bounded sequence on a Banach space Stringer, Loren Frank UNIVERSITY OF PENNSYLVANIA On the mathematical theory of the rolling Gabai, Hyman of metals at elevated temperatures On the discrepancy of certain sequences Taylor, Floyd Heckman Mod. 1 The development of a procedure for the Grabois, Neil Robert utilization of social security records in Pure extensions of algebras the study of mobility and mortality as experienced by a cohort of chromate Kosobud, Richard Frank workers Forecasting relationships for selected U.S.A. government expenditures andre­ Terrell, James Clinton venues A criterion for choosing between alterna­ tive sample designs when the primary Lui, Teng-sun interest is in relation Vanishing algebras and invariant subspaces of some function spaces PRINCETON UNIVERSITY UNIVERSITY OF PITTSBURGH Ash, Michael Edward The Neumann problem on stronglypseudo­ Allen, Stephen Ives convex complex multifoliate manifolds Representation of algebraic functions of a complex variable by two-dimensional Baum, Paul Frank surfaces in four-space, and application Cohomology of homogeneous spaces to Riemann surface theory Brumer, Armand Angotti, Rodney The structure of hereditary orders A pseudo-euclidean geometry in the pro­ Hartshorne, Robert Cope jective three-space Connectedness of the Hilbert scheme Davis, Budmon Roy Hensel, Gustav Bernard Affine transformation in complex two­ On the notational independence of various space interpreted in Argand four-space hierarchies of degrees of unsolvability Dillon, Thaddeus Michael Hsiang, Wu-Chung Statistical hypothesis modification Obstructions to sectioning fibre bundles Dudziak, Donald John Mjolsness, Raymond Minor - Nuclear Engineering A study of the stability of a relativistic Stability of a multiplicative stochastic particle beam passing through a plasma process with linear feedback Nemenyi, Peter Bjorn Ghaznavi, Mohammed Ovais Hanif Khan Distribution-free multiple comparisons Affine null spaces of linear sets Richter, Wayne Harvey Guilinger, Willis Herbert, Jr. Systems of notations for the "constructively The Peaceman-Rachford method for small accessible" ordinals mesh increments McDermot, Robert Gibson Robinson, Thomas Thacher The circle and related methods of sum­ Interpretations of Kleene' s metamathemat- mability for double series ical predicate riA in intuitionistic arithmetic N Ortner, Gene Merle Equiconvergence of Riesz and Cesaro trans­ Rosen, Michael Ira forms of infinite series Representations of twisted group rings

428 Sand, Francis Michael Howson, Joseph Thacher, Jr. Investigations in residual analysis and a Orthogonality in linear systems modification of the least multiple square Tsao, Sherman method for multiple regression Stability of flow between arbitrarily spaced Scheinberg, Stephen concentric cylindrical surfaces and re­ Hardy spaces and boundary problems in lated mathematical problems one complex variable Walowit, Jed Schorr, Herbert The stability of Couette flow between ro­ Towards the automatic analysis and syn­ tating cylinders in the presence of a thesis of digital systems radial temperature gradient Veech, William Austin RICE UNIVERSITY Almost automorphic functions Batten, George Washington, Jr. Wonnacott, Thomas Herbert Iterative solution of integral equations of A Monte-Carlo method for obtaining the first kind with applications to continua­ power of certain tests of location when tion problems sampling from non-normal distributions Deckard, Donald PURDUE UNIVERSITY Complete sets of unitary invariants for compact and trace-class operators Alexander, John Ralph, Jr. Minor - Physics Miller, Charles Keith On the theory of asymptotic density Three circle theorems in partial differen­ tial equations and applications to im­ Berens, Alan Paul properly posed problems Minor - Engineering On an equilibrium life model for multi­ UNIVERSITY OF ROCHESTER component systems Sarma, Vemuri L. N. ** Boes, Duane Charles Eberlein measure and mechanical quadra­ On the estimation of mixing distributions ture formulae Cline, Randall Eugene Warner, C. Robert** Minor - Industrial Engineering The Banach algebra I.} (G) n Lz (G) Representations for the generalized in­ verse of matrices with applications in linear programming

Lynn, Irvin Leon RUTGERS, THE STATE UNIVERSITY Linearly orderable spaces Ferguson, Donald Carvel Smoller, Joel Allan Infinite products of non-negative integers Translation- invariant functionals Gewirtzman, Leonard Tremel, Jerome G. Anti-isomorphisms of the endomorphism Minor - Education and Physics rings of classes of torsion-free modules A study of the relationships among basic ability factors and the learning of se­ Hogben, David lected operations on the set of integers Some properties of Tukey's test for non­ additivity Venkataraman, Valavanur Krishnaswamy On commutators and absolutely continuous Langford, Eric Siddon operators in a Hilbert space Separation in topological vector lattices and the Radon-Nikodym theorem Wei!, Clifford Edward Properties of derivatives Livingston, Albert Edward P -valent close-to-convex functions RENSSELAER POLYTECHNIC INSTITUTE Lodato, Michael William Goodwin, Bruce Edward On topologically induced proximity rela­ On the theory and application of Fredholm tions integral equations and on integral equa­ Zuckerman, Israel tions whose kernels depend on the eigen­ A new measure of a partial differential value parameter field extension Gustavson, Fred Gehrung The stability of periodic response of an on-off control system to sinusoidal inputs ** See footnote on page 418

429 ST. LOUIS UNIVERSITY STEVENS INSTITUTE OF TECHNOLOGY Cronin, Daniel Dennis Helm, Harry Arthur Minor - Philosophy On exponential type solutions of systems Proximity spaces, their products and their of linear differential equations relation to topological groups Hill, David Byrne Systems of linear differential equations UNIVERSITY OF SOUTHERN CALIFORNIA and generalized exponentials Bergquist, James William Macy, James Richard Minor - Physics and Electrical Engineering Theory of serial codes Difference sets and congruences modulo a product of primes Nichols, Leonard Frank The application of Pontryagin's maximum Datko, Richard Frank principle to the minimum time problem Minor - Physics for relay control systems Some results in nonlinear control theory UNIVERSITY OF TENNESSEE STANFORD UNIVERSITY Davis, Kenneth joseph Bhargava, Raghunandan Prasad Minor - Physics Multivariate tests of hypotheses with in­ A generalization of the Dirichlet product complete data of arithmetical functions Edwards, James Eugene Hinton, Don Barker On the existence of solutions of the steady­ Minor - Electrical Engineering state Navier-Stokes equations for a class Two Stieltjes-Volterra integral equations of non-smooth boundary data Stewart, Donald George Fortier, Jean Jacques Minor - Physics Contributions to item selection Cellular subsets of the 3-sphere Gieser, Leon Jay The comparison of multivariate tests of UNIVERSITY OF TEXAS hypothesis by means Bahadur efficiency Atchison, Thomas Andrew Halkin, Hubert Rene Minor - Psychology On the necessary condition for optimal A class of Riemann surfaces control of nonlinear systems Calder, James Richard Kraemer, Helena Chmura Minor - Educational Psychology Point estimation in learning models Concerning completely convex sets Moore, Ramon Edgar Clinger, Barbara Ann Interval arithmetic and automatic error Structure theorems for a class of groups analysis in digital computing Coleman, William Paul Thiess, Frank Bartlett A study of the structure of linear spaces An analytic representation of the invariant Darwin, James Thomas, Jr. distributions of quantum field theory Minor - Physics Zweig, Hans J. Some theorems on functional analysis Counter models and applications to de­ Ehrlich, Louis William tection theory The block symmetric successive over­ relaxation method STATE UNIVERSITY OF NEW YORK AT BUFFALO Falbo, Clement Earl Minor - Educational Psychology Rozycki, Eugene Paul Some generalizations of Green's function On Egoroff' s theorem Jolly, Robert Franklin Uschold, Richard Louis Concerning a functional inequality Minor - Physics Topological C-spaces Leininger, Charles William Concerning a mean integral Walbesser, William John Minor - Electrical Engineering Perryman, John Keith The structure of a partial ordering of A class of integral transforms for func­ channels in information theory tions which are O(ebt2) as t ->oo

430 Stokes, Russell Aubrey UNIVERSITY OF VIRGINIA An integral of a function with respect to a Haley, Joseph Arnold function in a rectangular interval On a certain spectral sequence UNIVERSITY OF TORONTO Krall, Allan Morton Inhomogeneous solutions and eigenfunction Brown, William Gordon expansions in £2 concerning a second Enumerational problems of linear graph order ordinary differential equation theory Lemmon, Charles Walker Kakita, Takao Power and exponential maps of Lie groups Theory of hyperbolic distributions Puttaswamaiah, B. M. VIRGINIA POLYTECHNIC INSTITUTE Group representations Dunn, James Eldon Singh, Jarnail Minor - Mathematical Chemistry Statistical theory of selection Power characteristics of Kramer's method for analysis of variance of a two-way White, George Kerr classification with disproportionate sub­ Algebra and the theory of numbers class numbers TULANE UNIVERSITY Gibbons, Jean Dickinson The small-sample power of some non­ Alford, William Robert parametric tests Some wild embeddings of the one and two dimensional spheres in the three sphere Smith, Armand Verne, Jr. Comparison of two drugs in multistage Haque, Mohammad Rashidul sampling using Bayesian decision theory Cech homology and cohomology groups of compact, len-spaces WASHINGTON UNIVERSITY Hightower, Collin James Davis, Jeffrey Robert On the minima of real indefinite binary The asymptotic behavior of extreme eigen­ quadratic forms values of Toeplitz operators of a Hankel Hudson, Sigmund Nyrop type Topological loops with invariant uniform­ Lachterman, Sam ities Exponentially convex functions on a cone Schneider, Walter Jan in a Lie group Some theorems in constructive function Mahoney, Richard Thomas John theory On Howarth endomorphisms UNIVERSITY OF UTAH UNIVERSITY OF WASHINGTON Gillette, Richard Morris Alvarez de Araya Munoz, Jorge Enrique The space of oo-smooth homeomorphisms Invariant measures on compactifications of of the 2-cell the integers Nath, Radha Gobinda Argabright, Loren Neil An asymptotic formula for partitions Invariant means on topological semigroups Sherman, Thomas Lawrence Craswell, Keith Jamie Properties of solutions of nth order linear Density estimation and estimation of a differential equations selection function Froderberg, Albert John A class of convolution algebras VANDERBILT UNIVERSITY Mochizuki, Horace Yomishi Depew, Rex Denwil Finitistic homological dimension and dual­ Minor - Education ity theory for rings Lagrange resolvent transformations of finite ordered point sets in the complex Reay, John Robert plane Generalizations of a theorem of Caratheo­ dory Jones, William Branham Minor - Physics Schaufele, Ronald Alfred Contributions to the theory of Thron con­ Stationary measures and potential theory tinued fractions for Markov renewal processes

431 WAYNE STATE UNIVERSITY Paine, Dwight Milton Some grouplike properties of partitioning Janowitz, Melvin Fiva systems Quantifiers on quasi-orthomodular lattices Pilgrim, Donald Harry Vollmer, John Edward Engel conditions on groups Semi-cubical homotopy theory of Kann Triads I Van Buskirk, James Morton Braid groups of compact 2-manifolds with UNIVERSITY OF WISCONSIN elements of finite order Bechtell, Homer Franklin, Jr. Vermes, Robert Elementary groups On the location of the zeros of linear com­ binations of polynomials Black, Richard Holland A method of pattern recognition by machine Walkup, David William Lie rings satisfying Engel conditions Brown, Robert Freeman Generalized n-plane bundles with applica­ Weston, Kenneth William tions to path fields on manifolds Minor - Foundations Z-A groups which satisfy the mth Engel Carlson, David Hilding conditions Rank and inertia theorems for matrices: The semidefinite case Day, James Thomas On the approximate solution of differential equations Greenwood, Priscilla Irene (Mrs. Donald) The convolution equation over a compact real interval for some special kernels YALE UNIVERSITY Hanna, Martin Slafter Barnes, Richard Thomas A coerciveness inequality for second-order On derivation algebras and Lie algebras elliptic operators in polyhedra of prime characteristic Krause, Eugene Franklin Glennie, Charles Milne On the restricted Burnside problem and Identities in Jordan algebras theorems like Sanov's Greenblatt, Robert Losey, Nora Lazier (Mrs. Gerald) Homology with local coefficients and Simple commutative nonassociative alge­ characteristic classes of vector bundles bras satisfying a polynomial identity of degree 5 Greenleaf, Frederick Paul Characterization of group algebras in terms Lynch, William Charles of their translation operations Ambiguities in Backus normal form lan­ guages Gulick, Sidney Lewis, III The bidual of a locally multiplicatively­ Moore, Mack Arthur convex algebra Minor- Law The role of temporary help services in the McCord, Michael Campbell clerical labor market Inverse limit systems Moschovakis, John Nicolas Rajagopalan, Minakshisundaram Recursive analysis IY -spaces and convolution in locally com­ pact groups Moursund, David Garvin Optimal approximation of functions: Rovnyak, James Leo Chebyshev-type approximations Some Hilbert spaces of analytic functions Nielsen, George Marius Smith, David Alexander A determination of the minimal right On Chevalley's method in the theory of ideals in the enveloping algebra of a Lie Lie algebras and linear groups of prime algebra of classical type characteristic Nirschl, Nicholas Edward Watkins, Mark E. Applications of norms, Eigenvalue locali­ A characterization of the planar geodetic zation theorems and field of values graph and some geodetic properties of theorems in matrix theory non-planar graphs

432 YESHIVA UNIVERSITY The three dimensional Dirichlet problem associated with a plane lamina Bagnato, Robert Anthony Gevrey classes and hypoellipticity Zuckerberg, Hyam Leiter Solution of the Dirichlet problem in paral­ Senter, Harvey Z. lel slit domains

LETTERS TO THE EDITOR

Editor, The NOTICES the Conference Board, from which the The letter in the April issue of the NOTICES summary was taken, Halmos NOTICES, signed by Accola etal, assert­ talks about this at length, and with great ing that there is a lack of good doctoral wisdom. I believe his talk should be read students, and hence no need to create new by anyone interested in Ph.D. production. "centers of excellence", is most surpris­ Second, the shortage of qualified ing. college teachers is so great, and will be­ First, the "surplus of people who come greater, that anyone who can get are qualified to supervise Ph.D. level re­ anything called a Ph.D. is sure of a good search". I have made a study of the entire lifetime job. Again, taking the class of 1951 class of mathematics Ph.D's., which 1951 as a basis, one can extrapolate that will appear in the Monthly. Of that class, from the 1949-1960 classes, there are one-fifth had published at the rate of a about 500 Ph.D's. teaching in the 900 col­ paper a year, a rate of publication which leges that have undergraduate majors in I would regard as just about minimal for mathematics but no doctoral programs. a Ph.D. supervisor (I admit the existence Mathematics enrollments are rising rap­ of exceptions). In 1949-19 60, there were idly. We are now producing as manyunder­ almost exactly 2500 Ph.D's. produced. graduate majors as all the physical sci­ There is some reason to think the 19 51 ences combined; in 1960 we had a third of class was outstanding, but if we assume their faculties. Lindquist has estimated that one-fifth of the entire 2500 is compe­ that by 1970, one first degree out of every tent to supervise theses, this would give 19 will be in mathematics. The "Manpower" 500 supervisors from the post-war crop. conferees want us to prepare for the prob­ How many more can one take from the lem. I do not believe we have enough pre-war Ph.D's.? Perhaps 300? If so, this "centers of excellence" now. But if we do means there are 800 people in the country for the moment, we certainly do not have really qualified to supervise theses. We enough to train competent faculties for the are now actually producing around 400 near future. What do the signers of this Ph.D's. a year. Of course, there are un­ letter propose to do about the teaching qualified people supervising theses, at shortage? I myself would like to make sure schools unqualified to give Ph.D's. The that there are enough good graduate de­ figure of 400 does not mean that the 800 partments to come near meeting the de­ qualified supervisers are each producing mands. I want to make clear that I am not a student every two years. proposing a dilution of the Ph.D. More The fact is, thesis supervision is a Ph.D's. will be produced. I want them to be most difficult form of teaching, I think the competent. most difficult. In the original report of the Third, I am simply baffled by the conference on "Manpower Problems in the statement that good students are in short Training of Mathematicians", published by supply. My own school is rejecting students

433 we would have jumped at two years ago. reasons that everybody knows but them. I know certainly that half of our admissions this year would have been accepted by al­ I regarded the complete report of the conference as an unusually thorough most any department ten years ago. I do the need for creation of not believe that we have suddenly gotten documentation of confused with Harvard or Chicago. But more strong graduate departments. I students that these schools would have wonder if the signers of this letter all have accepted ten years ago now don't get in, read the report. The conferees deserve a and we get some of them. I have confirmed detailed, serious, responsible answer to this impression with graduate schools of their documentation, pointing out in what about our standing, and find that this is the respect their analyses are wrong, if they general phenomenon. There are depart­ are wrong. ments with brilliant faculties that have difficulty getting appropriately able stu­ dents, sometimes for reasons that seem Gail S. Young to me simply inexplicable, sometimes for

434 PERSONAL ITEMS

Dr. E. Z. ANDALAFTE of Southwest versity. Missouri State College has been appointed Dr. D. L. FISK of Knox College has to an associate professorship at the Uni­ been appointed to an assistant professor­ versity of Missouri, St. Louis Campus. ship at Kent State University. Dr. D. W. ANDERSON of the Univer­ Dr. GEORGE GIOUMOUSIS of the Shell sity of California, Berkeley, has been Development Company has accepted a posi­ awarded an NAS-NRC Postdoctoral Re­ tion as a Research Scientist with the Lock­ search Fellowship. heed Missiles and Space Company. Mr. P. B. ANDREWS of Princeton Dr. H. S. HAYASHI of Northrup Cor­ University has been appointed to an assist­ poration has accepted a position as. Staff ant professorship at the Carnegie Institute Consultant to Control Data Corporation's of Technology. System Sciences Division of Los Angeles. Dr. LARRY ARMIJO of Rice Univer­ Miss R. I. HOFFMAN of the Denver sity has accepted a position as a Research Public Schools has been appointed to an Scientist with the Martin Company. associate professorship at the University Professor LOUIS AUSLANDER of the of Denver. University of California, Berkeley has been Mr. R. A. HOLT of the Mitre Corpora­ appointed to a professorship at the Belfer tion has accepted a position as Senior Pro­ Graduate School of Science of Yeshiva grammer with the New England Merchants University. National Bank. Professor LIPMAN BERS of New Professor FRITZ JOHN of New York York University has been elected to the University has been elected to the National National Academy of Sciences. Academy of Sciences. Dr. B. R. BHONSLE of the Maulana Professor JACOB KOREV AAR of the Azad College of Technology has been ap­ University of Wisconsin has been appointed pointed to a professorship at the Govern­ to a professorship at the University of ment Engineering College, Jabal pur, India. California, San Diego. Professor R. H. BOTT of Harvard Mr. P. H. KRATZ of the National Sci­ University has been elected to the National ence Foundation has accepted a position as Academy of Sciences. Special Assistant to the Deputy Director Mr. L. D. BROWN of Cornell Univer­ at the National Bureau of Standards. sity has been awarded an NAS-NRC Post­ Mr. V. LAKSHMIKANTHAM of the doctoral Research Fellowship. University of Alberta has been appointed Mr. BRUCE CHANDLER of Fairleigh to a professorship at the Marathwada Uni­ Dickinson University has been appointed to versity, Maharashta, India. an assistant professorship at New York Professor HANS LEWYofthe Univer­ University. sity of California, Berkeley has been Mr. C. W. CUR TIS of the University elected to the National Academy of Sci­ of Wisconsin has been appointed to a pro­ ences. fessorship at the University of Oregon. Mr. YOU-FENG LIN oftheUniversity Mr. D. A. DAWSON of Massachusetts of Florida has been appointed to an assist­ Institute of Technology has been appointed ant professorship at the University of South to an assistant professorship at McGill Florida. University. Mr. C. K. MILLER ofRiceUniversity Professor ELDON DYER of the Uni­ has been awarded an NAS-NRC Post­ versity of Chicago has been appointed to doctoral Research Fellowship. a professorship at Rice University. Mr. E. E. S. MOYERS of Rice Univer­ Mr. N. J. FINE of the University of sity has accepted a position as Research Pennsylvania has been appointed to a pro­ Specialist with the Tidewater Oil Company. fessorship at the Pennsylvania State Uni- Dr. JUN-ITI NAGATA of Osaka City

435 University has been appointed to a pro­ R. A. CLARK, Case Institute of Tech­ fessorship at Texas Christian University. nology, to a professorship. Dr. H. R. PITT of the University of C. V. HOLMES, San Diego State Col­ Nottingham has been appointed Vice­ lege, to a professorship. Chancellor of Reading University ,England. C. S. HOUGH, University of Florida, Mr. M. RAJAGOPALAN of Yale Uni­ to an assistant professorship. versity has been appointed to an assistant J. A. HUMMEL, University of Mary­ professorship at Lehigh University. land, to a professorship. Mr. H. P. RAWLINGS of the University C. F. KENT, Case Institute of Tech­ of Maryland has been appointed to an nology, to an associate professorship. assistant professorship at Morgan State TOMIO KUBOTA, Nagoya University, College. to a professorship. Dr. T.E.REGGE of the University of R. G. LAHA, Catholic University, to an Turin has been awarded the Dannie Heine­ associate professorship. man Prize, given by the American Physical M. W. MILLIGAN, Albion College, to a Society and the American Institute of Phy­ professorship. sics, for outstanding achievement in math­ AKIHIKO MORIMOTO, Nagoya Uni­ ematical physics. versity, to a professorship. Mr. NORMAN RUSHFIELD of the GLORIA OLIVE, Anderson College, to New York University, Courant Institute of a professorship. Mathematical Sciences, has accepted a A. M. PEISER, Socony Mobil Oil Com­ position as Programmer with the Compu­ pany, to an Engineering Consultant. ter Applications, Incorporated. DANIEL SALTZ, San Diego State Professor PETER SCHERK of the College, to an associate professorship. University of Toronto has been appointed F. J. SANSONE, Case Institute of a Visiting Lecturer at the University of Technology, to an assistant professorship. Colorado for the summer term. SEYMOUR SHERMAN, Republic Avia­ Dr. T. I. SEIDMAN of the Boeing tion Corporation, Research Division, to Scientific Research Laboratories has been Chief of the Applied Mathematics Sub­ appointed to an associate professorship at division. Wayne State University. N. B. SMITH, San Diego State College, Dr. HERMANN SIMON of McGill Uni­ to a professorship. versity has been appointed to an assistant D. T. WALKER, Memphis State Uni­ professorship at the University of Miami. versity, to an associate professorship. Mr. SEYMOUR SINGER of the Univer­ L. J. WARREN, San Diego State Col­ sity of California, Berkeley has been ap­ lege to a professorship. pointed to an assistant professorship at the San Francisco State College. Deaths: Professor STEPHEN SMALE of Col­ umbia University has been appointed to a Professor H. L. AGARD of Williams professorship at the University of Califor­ College died on February 23, 1964 at the nia, Berkeley. He has also been elected to age of 81. He was a member of the Society a corresponding membership at the Aca­ for over 50 years. demia Brasileira de Ciencias, Rio de Professor J. W. GADDUM of the Uni­ Janeiro, Brazil. versity of Florida died on February 14, Dr. F. J. WEYL of the OfficeofNaval 1964 at the age of 39. He was a member of Research has been chosen to receive one the Society for 18 years. of the ten 1964 Career Service Awards, Professor CORINNE HATTAN of the given by the National Civil Service League. University of Illinois died on February 8, 1964 at the age of 60. She was a member The following promotions are announced: of the Society for 22 years. Professor P. M. NASTUCOFF of the C. B. BELL, JR., San Diego State University of Notre Dame died on AprilS, College, to a professorship. 1963 at the age of 66. He was a member of C. C. CHANG, Seattle University, to the Society for 14 years. an assistant professorship. Mr. ARTHUR RADIN of Norwich, Con-

436 necticut, died on January 27, 1963 at the a member of the Society for 46 years. age of 40. He was a member of the Society Professor H. M. SHEFFER of Harvard for 17 years. University died on March 17, 1964 at the Professor NORBERT WIENER of the age of 80. He was a member of the Society Massachusetts Institute of Technology died for 35 years. on March 19, 1964 at the age of 69. He was

NEWS ITEMS AND ANNOUNCEMENTS

SUMMER INSTITUTE ON CONFERENCE ON ALGEBRAIC THEORY OF DISTRIBUTIONS NUMBER THEORY Lisbon, Sept. 7-18, 1964 University of Sussex September 1-17, 1965 Lecturers at the international Sum­ mer Institute on Theory of Distributions An instructional conference on Al­ will be J. B. Diaz, Maryland; J. L. Lions, gebraic Number Theory will be held from Paris; A. Martineau, Montpellier; E. T. September 1-17, 1965 at the University of Poulsen, Aarhus; and J. Sebastiao E Silva, Sussex, Brighton, England, under the aus­ Lisbon. The institute will be sponsored by pices of the London Mathematical Society. NATO, the Gulbenkian Foundation, and the Introductory lectures on the cohomology University of Lisbon. theory of groups and on local and global Interested persons are invited to fields will take up the first part of the write for information and applications to conference. More advanced courses will the Secretariat, Centro de Ca1culo Cientr­ follow on class field theory, on L­ fico, Rua D. Joao V, 30, Lisbon 2, Portugal. series and the Dedekind zeta function. Accommodations and board will be provided by the University of Sussex.

AID CONTRACT PROGRAM IN RESEARCH AND ANALYSIS

The Agency for International De­ velopment, Department of State, Washing­ ton, D. C., has published a pamphlet on CORNING BOOK PRIZE their program of support of research projects and analytical studies. The objec­ On March 3, the Corning Glass tives of the program are outlined in the Works and Little, Brown & Company pamphlet by F. Joachim Weyl, Science opened the competition for the 1965 Corn­ Advisor to the Administrator of the AID. ing Science Prize. In the program for support of sys­ Author of thebestbook-lengthman­ tems and operations analyses, research uscript submitted in the field of natural groups are being encouraged to investigate or physical sciences for the general problems of their own devising within the reader will win $12,500. following outline: Systems Analysis: Only English-language manu- (1) Economic analysis; (2) Analytical aids scripts, typewritten and doublespaced, in development decisions; (3) Strategic will be accepted. Entries are offered to analyses; (4) Data needs and handling; Little, Brown for publication throughout Operations Analysis: (1) Policy and pro­ the world, on terms to be arranged by gram coordination problems; (2) Country author and publisher. There is no entry studies; (3) Analysis offoreign assistance blank. Entries should be submitted by modes and channels; (4) Pilot design and July 1, 1965, to Corning Science Prize evaluation of projects; (5) Administrative Editor, Little, Brown & Company, 34 analyses. Beacon Street, Boston 6, Massachusetts.

437 SUPPLEMENTARY PROGRAM-No. 25

During the interval from February Z8, 1964throughApril 30, 1964 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Contributed Papers in this issue of these NOTICES. For a new policy on "by title" abstracts see page Z94 of the April issue of these NOTICES.

(1) Orthogonality properties of the spline Institute of Technology (64T-336) function (Introduced by Professor Dr. J, H. Ahlberg, United Aircraft A. M. Garsia) Corporation, East Hartford, Con­ (7) On semi-simple Lie algebras of prime necticut; Dr. E. N. Nilson, Pratt characteristic. Preliminary report and Whitney Aircraft,EastHartford, Professor R. T. Barnes, The Ohio Connecticut and Professor J. L. State University (64T-311) Walsh, Harvard University (64T- (8) A general ergodic theorem with 338) weighted averages (Z) Extremal, orthogonality and conver­ Professor G. E. Baxter, University gence properties of multi-dimen­ of Minnesota (64T-3Z6) sional splines (9) Error bounds for spline fits Dr. J. H. Ahlberg, United Aircraft Professor Garrett Birkhoff, Har­ Corporation, East Hartford, Con­ vard University (64T-Z96) necticut; Dr. E. N. Nilson, Pratt (10) The subgroups of SL(3,q). II. Prelim­ and Whitney Aircraft, East Hart­ inary report ford, Connecticut and Professor Professor D. M. Bloom, University j. L. Walsh, Harvard University of Massachusetts (64T-314) (64T-339) (11) The subgroups of SL(3,q).III.Prelim­ (3) The nine point hyperbolic circle C n inary report of a triangle T in the Minkowski plane Professor D. M. Bloom, University Mz of Massachusetts (64T-317) Mr. R. V. Anderson, illinois Insti­ (lZ) On a linear continuous operator in the tute of Technology (64T-Z76) space of continuous functions (Introduced by Professor John Professor Witold Bogdanowicz, DeCicco) Georgetown University (64T-Z80) (4) The circumscribed and tangent hyper­ (13) On the existence of almost periodic bolic circles of a triangle in the Min­ and asymptotically almost periodic kowski plane solutions for nonlinear partial-differ­ Mr. R. V. Anderson, Illinois Insti­ ential equations tute of Technology ( 64T-Z77) Professor Witold Bogdanowicz, (Introduced by Professor John Georgetown University (64T-Z81) DeCicco) ( 14) On the existence of periodic solutions (5) Set functions and an extremal prob­ of nonlinear partial differential equa­ lem in absolute continuity. Prelimin­ tions ary report Professor Witold Bogdanowicz, Professor W. D. L. Appling, North Georgetown University (64T-Z8Z) Texas State University (64T-310) (15) Finite sets and axioms of limitation (6) An ergodic theorem of Abelian type. Mr. K. R. Brown and Professor Hao Preliminary report Wang,Harvard University(64T-300) Mr. Luis Baez-Duarte, California (16) Short definitions of the class of

438 ordinals (28) On a class of elliptic partial differen- Mr. K. R. Brown and Professor Hao tial equations Wang, Harvard University (64T- Professor R. P. Gilbert and Pro­ 299) fessor H. C. Howard, University of ( 17) Zeros of partial sums of power series Maryland (64T-304) for certain entire functions (29) Propositional calculi with decision Professor J.D. Buckholtz, Univer­ problems of any required recursively sity of North Carolina (64T-295) enumerable degree of unsolvability (18) Problems in linked operators. II. Mr. M.D. Gladstone, University of Preliminary report Bristol, England (64T-294) Professor R. W. Carroll, Rutgers, (Introduced by ProfessorW.W.Boone) The State University (64T-340) (30) Classes of conjugate gap Fourier ( 19) Approximation by generalized ra- series tional functions Professor G. W. Goes, The Univer­ Professor E. W. Cheney, Univer­ sity of Kansas (64T-332) sity of California, Los Angeles (31) Cellularity at the boundary of a mani­ ( 64T- 319) fold. Preliminary report (20) On certain convex functions Professor Charles Greathouse, Mr. T. W. Cusick, University of University of Tennessee (64T-269) illinois (64T-272) (32) A formal reduction theorem for a (Introduced by Professor G. R. Blakley) system of differential equations with (21) Combinatorial manifolds havinglocal a turning point co-dimension one. Preliminary re­ Mr. R.J, Hanson, University of Wis­ port consin (64T-337) Mr. H. S. Davis, University of (Introduced by Professor Wolfgang illinois (64T-327) Was ow) (22) Applications of analytic number the­ (33) Locally nice embeddings of manifolds ory to the study of type sets of torsion Dr. J. P. Hempel and Professor free Abelian groups. I D. R. McMillan, Jr., The Institute Professor D. W. Dubois, The Uni­ for Advanced Study (64T-285) versity of New Mexico (64T-301) (34) Locally nice embeddings of mani- (23) Ultrastructures in first-order model folds. II theory Dr. J. P. Hempel and Professor Dr. Erwin Engeler, IBM Research D. R. McMillan, Jr., Institute for Laboratories, Zurich, Switzerland Advanced Study (64T-309) ( 64 T- 331) (35) Post-lineal theorems for r.e.d.u. (24) The solvability of the halting prob- Mrs. A. H. Ihrig, University of lem of a 2-state Post machine illinois (64T-302) Professor P. C. Fischer and Mr. ( 36) Coincidence of partial propositional Stal Aanderaa, Harvard University calculi (64T-291) Mrs. A. H. Ihrig, University of (25) Perturbation of a Sturm-Liouville illinois (64T-303) operator by a finite function ( 37) On standard ideals Professor R. C. Gilbert,.California Professor M. F. Janowitz, Univer­ State College at Fullerton (64T-324) sity of New Mexico (64T-286) (26) Multivalued harmonic functions in (38) On Baer semigroups. II four variables Professor M. F. Janowitz, Univer­ Professor R. P. Gilbert, University sity of New Mexico (64T-313) of Maryland (64T-306) ( 39) A characterization of the lattice of (2 7) On solutions of the generalized axially real-valued continuous functions on a symmetric wave equation represented real compact space by Bergman operators Miss G. A. Jensen, Purdue Univer­ Professor R. P. Gilbert and Pro­ sity (64T-270) fessor H. C. Howard, University of (40) Non-Desarguesian planes of order Maryland (64T-304) 22m+l

439 Professor D. E. Knuth, California (52) Outer automorphisms of nilpotent Institute of Technology (64T-268) p-groups (41) On some applications ofP.J.Cohen's Dr. M. F. Newman, Australian Na­ method in mathematical analysis tional University and New York Uni­ Professor Motokiti Kondo, Tokyo versity (64T-283) Metropolitan University, Tokyo, (53) On absolutelyconvergenttrigonomet­ Japan (64T-325) ric series. Preliminary report (42) The Cauchy functional equation and Mr. R. E. Pippert, University of scalar product in vector spaces Kansas (64T-292) Professor Svetozar Kurepa, Uni­ (54) On the space of homeomorphisms of a versity of Zagreb, Yugoslavia ( 64 T- multiply punctured surface of genus g 287) Professor L. V. Quintas, St. John's (43) Quadratic and sesquilinear function- University (64T-278) ala (55) A note on semi-homogeneous func­ Professor Svetozar Kurepa, Uni­ tions versity of Zagreb, Yugoslavia ( 64T- Professor L. V. Quintas, St. John's 293) University (64T-289) (44) Characterization of rational trigo- (56) A lattice-point estimate nometric Chebyshev approximation Dr. B. S. Randol, Yale University Mr. H. L. Loeb, Aerospace Corpo­ (64T-279) ration, El Segundo, California (64T- (57) Global representation of bounded 320) linear functionals on Orlicz spaces (45) More pointlike decompositions of E 3 Professor M. M. Rao, Carnegie Professor L. F. McAuley, Rutgers, Institute of Technology (64T-288) The State University (64T-315) (58) On n-tape finite state acceptors (46) Taming Cantor sets in En Mr. A. L. Rosenberg, Harvard Uni­ Professor D. R. McMillan, Jr., versity (64T-343) Institute for Advanced Study (64T- (Introduced by Professor P. C. 316) Fischer) (47) Isomorphisms of semigroups of con- (59) Hyperbolic mixed problems. I tinuous functions Professor Leonard Sarason, Stan­ Professor K. D. Magill, Jr., State ford University (64T-266) University of New York at Buffalo (60) Hyperbolic mixed problems. II (64T-333) Professor Leonard Sarason, Stan­ (48) Subsocles which do not support pure ford University (64T-267) subgroups (61) Homeomorphic isomorphic Abelian Professor C. K. Megibben, Texas groups Technological College (64T-322) Mr. Stephen Scheinberg, Princeton (49) Some results on investigations of in­ University (64T-328) jective and projective spaces sug­ (62) Modular pairs in an orthomodular gested by N achbin.Preliminary report lattice Mr. Nick Metas, IBM Corporation, Professor E. A. Schreiner, West­ Yorktown Heights, New York (64T- ern Michigan University (64T-318) 290) (63) Homology and central series of groups (50) An analogue for projective spaces of Professor J. R. Stallings, Prince­ a theorem of Sobczyk. Preliminary ton University (64T-334) report (64) Local limit theorems for asymptoti­ Mr. Nick Metas, IBM Corporation, cally stable distribution functions Yorktown Heights, New York (64T- Professor Charles Stone, Cornell 308) University (64T-329) (51) Approximation in the sense of a (65) Circumscribing cubes of hyperellip­ deviator integral soids. Preliminary report Professor T. S. Motzkin, Univer­ Mr. D. H. Taylor, University of sity of California, Los Angeles lllinois (64T-330) (64T-Z84) (66) A characterization of certain P-se-

440 quences. Preliminary report (73) Continuous decompositions of para- Professor Dmitri Thoro, San Jose compact spaces State College (64T-323) Dr. J, M. Worrell, Jr., Sandia Cor­ (67) Polynomially convex disks poration, Albuquerque, New Mexico Professor John Wermer, Brown (64T-298) University (64T-321) (74) Upper semicontinuous decomposi- (68) An extension of a theorem by Mar­ tions of paracompact spaces shall Hall. Preliminary report Dr. J. M. Worrell, Jr., Sandia Cor­ Professor J. R. Wesson, Vanderbilt poration, Albuquerque, New Mexico University (64T-271) ( 64 T- 335) ( 69) Generalizations of even and odd func­ (75) Weakly exact categories tions Professor Oswald Wyler, Univer­ Professor M. E. White, Stevens sity of New Mexico (64T-307) Institute of Technology (64T-275) (76) Representation of functions ofn vari­ (70) Invariant means and the Stone-Cech ables by superposition of functions of compactification one variable and addition Mr. C. 0. Wilde and Dr. K. G. Witz, Dr. J. Z. Yao, Illinois Institute of University of Illinois ( 64 T- 342) Technology (64T-341) (71) Homotopy type of almost differen­ {77) On the structure of quasi-open maps tiable manifold Professor W. P. Ziemer, Indiana Mr. Y,-F. Wong, Atlanta University University (64T-274) (64T-312) (78) On semicontinuous functions and (72) Upper semicontinuous decomposi- Baire functions tions of metacompact spaces Professor R. E. Zink, Purdue Uni­ Dr. J. M. Worrell, Jr.,SandiaCor­ versity (64T-273) poration, Albuquerque, New Mexico (64T-297)

ACTIVITIES OF OTHER ASSOCIATIONS

SOCIETY OF ENGINEERING SCIENCE SCHEDULES SECOND TECHNICAL MEETING

The Society of Engineering Science, papers. Scientists and engineers actively Incorporated which was founded last year engaged in serious theoretical and/or for "the advancement of interdisciplinary experimental research on interdiscipli­ research and the establishment of a bridge nary problems are invited to submit manu­ between science and engineering, "Will have scripts for presentation at this meeting its second technical meeting on November and for subsequent publication in the Pro­ 2-4, 1964 at the Michigan State University, ceedings of the Society. Deadline for the East Lansing, Michigan. The first two days submission of manuscripts is August 15, of the three-day meeting will be devoted 1964. Three copies of each manuscript entirely to tutorial lectures on modern should be sent to: topics in mathematical techniques, mate­ Professor A. C. Eringen rials science, fluid mechanics, and special School of Aeronautical and Engineering physical and chemical phenomena. Out­ Sciences standing scientists and engineers will pre­ Purdue University sent discussions on varied aspects of and West Lafayette, Indiana 47907 different approaches to the specific topic chosen for each session. Inquiries about the meeting and other The last day of the meeting is re­ matters pertaining to the Society may also served for the presentation of contributed be addressed to Dr. Eringen.

441 ABSTRACTS OF CONTRIBUTED PAPERS

The June Meeting in Pullman, Washington June 20, 1964

613-1. D. R. RAO, 11-1-243, Sitafalmandi, Secunderabad A. P. India. Application of Goodstein's solution in solving some Diophantine equations. 2 2 Certain elementary Diophantine equations are solved by using a solution of U 1 V 1 (U 1 - V 1) = 2 2 U2V2(U2- V2) by Goodstein in Mathematical Gazette 23 (1939), p. 266. Ordinary identities were taken and from these identities advantage has been taken from the resulting expressions of the 2 2 2 2 form V 1 (U 1 - V 1) = V 2 (U 2 - V 2) to arrive at the solutions of interesting equations of the type n 1 n2 n3 nk N1 N2 N3 Nk xl + x2 + X3 + ••• + xn = X1 + X2 + X3 + ••• + XN. (Received March 6, 1964.)

613-2. J. H. JORDAN, Washington State University, 1820 D Street, Pullman, Washington. The divisibility of Gaussian integers by large Gaussian primes.

For 0 < a < 1 and x positive let P(a,x) be the set of Gaussian primes such that pin P(a,x) implies xa ;:> IP I ::> x. Let G(a,x) be the set of Gaussian integers such that g in G(a,x) implies (i) lg I < x and (ii) there is a p in P (a,x) that divides g. Let t/;cf.a,x) be the cardinality of G(a,x). 2 -1 -1 It is shown in this paper that limit tf;G(a,x) (JC1r) = p(a ) where p is N. C. de Bruijn's function. (Received March 17, 1964.)

613-3. J. R. REAY,Western Washington State College, Bellingham, Washington. A new proof of the Bonnice-Klee theorem.

The following theorem with appropriate definitions appears in these Notices, Abstract 601-4, June 1963. Theorem 1. Each positive basis B for n-dimensional linear space E admits a partition

B = B 1 U ••• U Bk (1 ::> k ::> n) into pairwise disjoint subsets such that card Bi 01:; card Bi+l 01:; ?. (i = 1, ••. , k - 1), and Pos(B 1 U ••• U Bj) is a linear subspace of E of dimension [<2:1= 1 card Bi) - j), for j = l, ••• ,k. This theorem may be used to give an easy new proof of the following theorem due to Bonnice-Klee. [See these Notices, Abstract 606-4, November 1963, for references and definitions.]

Theorem 2. If X is a subset of n-dimensionallinear space E, 0 ::> d ~ n, and p E intdcon X, then p E intdcon U for some subset U of X with cardinality at most max(n + 1,2d). (Received April 29, 1964.)

442 613-4. W. J. FIREY, Oregon State University, Corvallis, Oregon. Convex bodies with preassigned sum of principal radii of curvature. Preliminary report.

5e is the class of convex bodies in 3-space whose support functions have Hessian matrices of (maximum) rank 2 over the unit spherical surface 81;. Consider Ki E ~ i = 1,2, with associated Hessian matrices Mi and let u(Mi'u) be the sum of the 2nd order principal minors of Mi at u E BE. Then (i) u > 0 over BE, (ii) the centroid of mass distribution specified on aE by u is the centre of sphere E. We define

613-5. D. F. SANDERSON,Western Washington State College, Bellingham, Washington. Some natural topologies in algebra. Preliminary report.

Let A and B be sets and let 27 be a set of functions from A to B. If S is a subset of A, define

C 1 (S) = {xI there is an f E 27 and a y E S such that f(x) = f(y)} and c2 (S) = {xI there is a y E S such that for every f E ff. f(x) = f(y) f. One readily verifies that the operations are legitimate closure operations in Kuratowski's sense, i.e. they satisfy Kuratowski's axioms. The resulting topologies seem to be natural in algebra. For example: Theorem. If L is a left ideal in a ring R and 27 consists of all those right multiplications of elements of R in the right annihilator of L, then c 2 (L) is the left annihilator of the right annihilator of L. A similar result holds for commuting subrings of a ring. (Received April 30, 1964.)

613-6. M. RAJAGOPALAN, Lehigh University, Bethlehem, Pennsylvania. Characters of locally compact abelian groups. Preliminary report.

Let G be an abelian group and let T1 and T2 be two locally compact topologies on G which make G a locally compact group. Let T1 be strictly stronger than T2• It is proved in this paper that there are at least zc continuous characters of G in the T1 topology which are discontinuous in the T2 topology. This solves a conjecture of Edwin Hewitt proposed in his paper A remark on the charac­ ters of a locally compact abelian group (Fund. Math. 53 (1963), 55-64). (Received May 1, 1964.)

613-7. SELMO TAUBER, Portland State College, Portland, Oregon 97201. Existence theorems for difference and q -difference equations.

Let E and A be the operators such that Ef(t) = f(t + h), h ;;:;; 1, and Af(t) = f(qt), q > 1. The vector equations Ex= q;'(x,t,h), and, Ax= lj(i',t,q), include all difference and q-difference equations of any order as well as systems of simultaneous equations. Existence and uniqueness theorems for

443 solutions of such equations have been proved assuming that for t 0 ::;; t ::;; t 0 + h (difference equations), and, to ::;; t ::;; qto (q-difference equations), x = x0(t), where x0 (t) is a given vector. These results are refined by assuming simply that fort= t 0, x = x0 (t0), a constant vector, and proving that then t 0 ::;; t ::;; t0 + h, or, t0 ::;; t ::;; qt0, x = x0 (t), so that the conditions of the known existence and uniqueness theorems are satisfied, (Received May 4, 1964.)

613-8. GUNTHER GOES, The University of Kansas, Lawrence, Kansas. Boundedly divergent sequences in FK-spaces.

Let X be the FK-space of all complex sequences x = { xk} with seminorms qi(x) = !xi 1. E a Banach space, T = {Tn} a sequence of linear and continuous mappings Tn of X into E with the property that the spaces ETB = {x EX: supniiTnxiiE < oo} and ETN = {x E X:IITmx- TnxiiE----> 0 (m > n----> oo)} are BK-spaces under the norm llxll = supniiTnxiiE' Let furthermore F be an FK-space and T 00 a linear continuous operator which maps some FK-space containing all elements ofF U ETN into E and with the property that liT nx - T 00x liE --> 0 (n --> oo) for every x E ETN' Then the following theorem holds: If F n ETN is not closed in the F-topology and if ETB f. ETN' then F n (ETB - ETN) f. 10 and F rj. ETB' Applications to Fourier series are given. (Received May 4, 1964.)

613-9. J, M. STARK, P.O. Box 6126, Stanford University, Stanford, California 94305. Determination of associate functions for Bergman integral operators yielding transonic flow patterns of certain types.

For solutions 1/; = 1/;(A, JJ) of the compressibility equation L(l/;) = 0 in a transonic region of the (A,JJ) plane, following Bergman (A mer. J. Math. 70 (1948), 888), we define X1 (!?) = limM ~ 1 1/;, Xz (JJ) = lim M ~ 1 a'if;/ aM, where M = Mach number, A= A(M), and 1J defines direction of velocity vector q. Let P(f) be the Bergman integral operator generating solution if;= P(f) of L(,P) = 0 from associate function f = f( .\). Associate functions f<::)

Fourier coefficients of corresponding X1 ( JJ), X2 ( !?). Associate functions are found by this method for both subsonic and supersonic regions. Similar results for Taylor developments also are given. (Received May 5, 1964.)

613-10. S. F. NEUSTADTER, Applied Resea:tch Laboratory, Sylvania Electronic Systems, 40 Sylvan Road, Waltham 54, Massachusetts. A trunking problem for three channels.

Calls of fixed duration are initiated between two points connected by three parallel channels and assigned to any one of the channels which may be free. The analysis covers both the cases of exponential and of arbitrary interarrival times for the calls. For exponential interarrival times there is obtained the chance of not losing a call in any given time interval as well as the average time that elapses before a call is lost. (Received May 7, 1964.)

444 ABSTRACTS PRESENTED BY TITLE

64T-266. LEONARD SARASON, Stanford University, Stanford, California. Hyperbolic mixed problems. I.

Let Lu(t,y ,x) = ut - Aux - Buy= 0 be a strictly hyperbolic system, with A,B constant matrices and A nonsingular. For real1/ and all T, define M(T,1/) = A- 1(rt inB). For ReT >0, let M± be M restricted to its eigenspaces whose e-values ~i have± real part. Let P ±be projectors such that

P± M = MP ± = M±. On ReT= 0, P ±M± are defined a.e. by continuity. Let M0 , P 0 correspond to Re ~i = 0. Consider the mixed problem: Lu = 0 on x > 0, t > 0; u(O,y,x) = 0 and Pu(t,y,O) =

g(t,y) (t > 0) where P is a projector such that K = (P + P-rJ- 1P exists for Re T > 0. Define z rT f oo z z r x z . . llv(x)IIT= Jodt _00 lv(t,y,x)l dy, and lllviiiT,x = Jo llv(x')IITdx'. Conditions:(*) K 1s bounded on

Re T = 0; (**) Ko = P 0K is bounded on Re T = 0. Theorem l. Under (*), llu(x) 11 00 ::'i c llg 11 00 • If L is symmetric hyperbolic, P is strictly dissipative = (*). Theorem 2. Under (**), if, for each 1/,K is unbounded along ReT= 0 only at simple poles T = iTj11' then u(r,y,O) = K 1g + Lici J_~(ag(t',y- ri(t- t'))/ay)dt', where K 1 is a bounded operator in Lz and Ci are constant matrices. Hence llu(O)IIT ::'i c{llgiiT + Tllag;ayiiT}. Also, lllv(x)uiiiT,oo ::'i cTIIgll. (Received February 28, 1964.)

64T-267. LEONARD SARASON, Stanford University, Stanford, California. Hyperbolic mixed problems. II.

Using the terminology and definitions of I, consider the mixed problem Lu = h(r,y,x) in x > 0, t > 0; u(O,y,x) = f(y,x), Pu(r,y,O) = 0, where f,h E c00 with bounded support. Let u, h be the Laplace-Fourier transforms of u,h in t,y, i.e., u(T,1/,X) = (1/211") fadtdye-tT-ihyu(r,y,x), and let f be the y-Fourier transform of f. Theorem: With k( T,1/,X) = h - A - 1!, and for ReT > 0, u(T,1/,X) = J000 eM(x-~)(H(x- OP_- H(~- x)P+) A- 1 k(T,1/,0d~- eMxP_(P + P+) 1 fo 00 P+e-M~k(T,1/,0d~. The first term represents the solution of a pure Cauchy problem, and the second contains the effects of the boundary condition. Theorem: If L is symmetric hyperbolic and (*) obtains, then 2 2 llu liT ::'i cT( llf II + T llh II ). (Received February 28, 1964.) ,oo T,oo

64T-268. D. E. KNUTH, California Institute of Technology, Pasadena, California. . 2m+1 Non-desargueslan planes of order 2

The planes described in these Notices, June 1962, p. 218 can be generalized to a construction of planes of order 2mn, where n ~ 3 is odd, and mn > 3. Let q =2m, and let f: GF(qn) ->GF(q) be the unique function satisfying f(x + b + bq) = x for x E GF(q), b E GF(qn). Then operations o,

*are defined as follows. a o b = ab + (f(a)b + f(b)a)2; (1 0 a)* (1 o b)= a o b. The system (GF (qn), +,,.) is a nonassociative division algebra. All autotopisms of this algebra, (a • b)H = aF * bG,

are given by the following rule: Let aL = 1 0 a; then aH =au, aF = ((aL- 1)ux)L, aG = ((aL - 1)ux - 1)L,

445 where u is an automorphism of GF(qn) and 0 1 x E GF(q). Therefore the collineation group of the plane has order mn23mn(2m - 1), (Received March 2, 1964.)

64T-269. CHARLES GREATHOUSE, University of Tennessee, Knoxville 16, Tennessee, Cellularity at the boundary of a manifold. Preliminary report.

A subset X of ann-manifold Mn is said to be cellular at the boundary of Mn if there is a nt n r, n n-1 n-1 n-1 sequence {Bi 1 of closed n-cells satisfying: Bi n LBd(M )] = Bi , Bi+ 1 C Int(Bi ), . [ n rrll n n . n Bi+n 1 n Int(M )J ClntLt\ n Int(M ) J , n Bi = X. Theorem. Let X be a subset of an n-mamfold M . n n n { }k such that X 1s cellular at the boundary of M . Then M /X = M • Corollary. Let Xi i= 1 be a finite collection of disjoint subsets of an n-manifold Mn such that each Xi is either cellular in Mn or cellular at the boundary of Mn. Then Mn/ U Xi = Mn. A closed set X in a noncompact n-manifold Mn is said to be openly cellular in Mn provided there is a sequence { C~} satisfying n n-1 rt _n n n n n Ci = R X L0,1), ~i+ 1 C Int(C 1), Ci = X. Theorem. Let X be a subset of an n-manifold M • Then X is cellular at the boundary of Mn if and only if both: (1) X n Bd(Mn) is cellular in Bd(Mn) and (2) X n Int(Mn) is openly cellular in Int(Mn). (Received February 28, 1964.)

64T-270. G. A. JENSEN, Purdue University, Lafayette, Indiana. A characterization of the lattice of real-valued continuous functions on a realcompact space.

Let C(X) be the lattice of continuous functions on a space X into the reals Rand let R be the extended reals. If a lattice L contains (a copy of) R, then a homomorphism ¢from L into R is an R-homomorphism if ¢(r) = r for all r E R, and if L is a lattice of functions on a space X, then ¢is fixed in X if there is an x EX such that

64T-271. J, R. WESSON, Box 1595, Vanderbilt University, Nashville 5, Tennessee. An extension of a theorem by Marshall Hall. Preliminary report.

Let R be a ternary ring (Hall, Theory of Groups, Macmillan, 1959, p. 355) with operation xyz and with or without a unit, Let ¢, 1/1 be fixed nonzero elements of R. Take as points (0,¢,0), (¢,y,O), (x,y,¢), where x,y may be any elements of R, and take as lines z = 0, x!f(zcO) = 0, yt/1 [xb(zcO)) = 0, where b,c may be any elements of R. A point (x,y,z) is said to be incident with a line if and only if its coordinates satisfy the equation of the line. The points and lines

446 so defined form a projective plane 7r isomorphic to the plane induced by R in the usual way [Hall,

Theory of Groups, p. 3 56]. 7r is independent of the choice of and f. In case R has no unit, then a new operation on the elements of R is defined so as to give a ternary ring R 1 with a unit, and the plane constructed in the same way from R' is isomorphic to 7r, The proof of all of the preceding when the ternary ring is specialized to a Veblen-Wedderburn system is already known [Hall, Projective Planes, Trans. Amer, Math. Soc. 54 (1943), 253]. The proof of the present extension is analogous. (Received March 4, 1964.)

64T-272. T. W. CUSICK, University of Illinois, Urbana, Illinois. On certain convex functions.

Theorem 1. If B is a finite set of fixed points of Euclidean n-space Rn and d(u, {3) is the distance between the points u and f3 of Rn, the generalized ellipsoid E = { u ERn: Lf3EBd(u,{3) ;<;; c, c a positive constant} is convex. From this the following generalization of a result of]. I. Nasser and R. W. Wheeling (Abstract 609-5, Notices Amer. Math. Soc. 11 (1964), 206) is easily obtained.

Theorem 2. Let A= ( ApJ• 1 ;<;; p ;<;; k, 1 ;';; q ;<;; n, be the k by n matrix whose rth row is the point

Ar = ( An• Ar 2 , ... , Arn) for each r, 1 ;';; r ;';; k. If gij is a convex and nondecreasing function of a nonnegative argument for each i, j, 1 ;<;; i ;';; k, 1 ;';; j ;';; k, and F (A)= Li,jgij(d(\ ,Aj )), the function F is convex in the matrix A. The result cited above follows from this by setting n = 2 and holding some rows of A fixed while the others vary. (Received March 4, 1964.)

64T-273. R. E. ZINK, Purdue University, Lafayette, Indiana. On semicontinuous functions and Baire functions.

According to a classical theorem, every semicontinuous real-valued function of a real variable can be obtained as the limit of a sequence of continuous functions. The corresponding proposition need not hold for the semicontinuous functions defined on an arbitrary topological space; indeed, it is known that the theorem holds in the general case if and only if the topology is perfectly normal. For some purposes, it is important to know when a topological measure space has the property that each semicontinuous real-valued function defined thereon is almost every­ where equal to a function of the first Baire class. In the present article this question has been resolved when the topology is completely regular. A most interesting by-product of the considera­ tion of this problem is the discovery that every Lebesgue measurable function is equivalent to the limit of a sequence of approximately continuous ones. This fact enables one to answer a question posed in an earlier work (Abstract 603-157, these Notices, August 1963), concerning the existence of topological measure spaces in which every bounded measurable function is equivalent to a Baire function of the first class. (Received November 26, 1963.)

64T-274. W. P. ZIEMER, Indiana University, Bloomington, Indiana. On the structure of quasi-open maps.

Suppose M and N are compact 2-manifolds without boundary and f: M -> N is a continuous map. If y E N,D is an open 2-cell containing y, V is a component off - 1(D), then f induces a homomorphism

on the Cech cohomology groups f*: H2 (clos D, bdry D) ---> H2(clos V, bdry V). A component C of C 1(y)

447 is called essential iff* is nontrivial whenever y E D and C C V. Let K(y) be the number of components of f- 1(y) and let S(y) be the number of essential components. f is called quasi-open provided that for any pointy EN and any open set U containing a component of f- 1(y),y is interior to f(U), Theorem, f is quasi-open iff K(y) = S(y) for all but countably many y EN. From this it follows that the middle space off is the closure of the union of countably many proper cyclic elements. If S(y) < oo on an uncountable set, then the middle space consists only of a finite number of proper cyclic elements. Hence, in this case there exists a closed set A C M such that f(A) is a finite set and f is either locally sense-preserving or locally sense-reversing on each component of M - A. In this regard see [c. ]. Titus and G. S. Young, An extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329-340], (Received March 4, 1964.)

64T-275, M, E. WHITE, Stevens Institute of Technology, Castle Point Station, Hoboken, New Jersey, Generalizations of even and odd functions.

The familiar separation of a real function f(x) into even and odd components is generalized in two ways: (1) f(z) is congruent to k modulo m iff f(rz) = rkf(z), where r is a primitive mth root of unity and k = 0, 1, 2,..,, m - 1. Then an arbitrary function f(z) can be decomposed into m com- "m-1 "m-1 jk m-j ponents modulo m: mf(z) = L...k=O fk(z), where fk(z) = L...j=O r f(r z) is congruent to k modulo m.

In case m = 2 and z is real, f0 (x) and f 1(x) are the even and odd components of 2f(x), (2) A function 4F(x,y) can be expressed as the sum of four components given by 4Fij = F(x,y) + (- l)iF(- x,y) + . "+" (- 1)1F(x,- y) + (- 1)1 1F(- x,y) (i = 0, 1; j = 0,1), where Fij is congruent to i modulo 2 as a function of x and congruent to j modulo 2 as a function of y. (Received March 5, 1964,)

64T-276, R, V, ANDERSON, Illinois Institute of Technology, 3300 S, Federal Street, Chicago

16, Illinois. The nine point hyperbolic circle CN of a triangle Tin the Minkowski plane M2•

The three midpoints of the sides of a triangle Tin Mz, the feet of the Minkowski altitudes of T, and the three midpoints of the three segments with the Minkowski orthocenter H as the common initial point, and the three vertices A, B, C, ofT, are all on one hyperbolic circle, called the nine point hyperbolic circle CN of T, Its center N is on the Euler line OH and bisects the segment OH, Thus the circumcenter 0, the nine point center N, the centroid G, and the orthocenter H are on the

Euler line. The nine point radius RN is one half of the circumradius R 0 • The barycentric equations of the inscribed, the three escribed hyperbolic circles of a modular or pseudomodular triangle T, are found, If T is modular, pseudo modular or indefinite, the equation of the nine point hyperbolic circle is obtained in barycentric coordinates, When Tis modular or pseudomodular, CN is tangent to the inscribed hyperbolic circle CI, and to each of the three escribed hyperbolic circles, This is an analogue of Feuerbach's theorem, Finally a point P is on the hyperbolic circumcircle of a triangle T, if and only if the three feet of the M2-projections on the three sides ofT from the point P, are on a line L. This is the Simpson line L relative to P, of the triangle T in M2• (Received March 6, 1964,)

448 64T-277. R. V. ANDERSON, Illinois Institute of Technology, 3300 S. Federal Street, Chicago 16, Illinois. The circumscribed and tangent hyperbolic circles of a triangle in the Minkowski plane.

Let T be a triangle in the Minkowski plane M2, whose vertices are given by the position vectors z 1, zz, z3. If u = (N23 + N31 + N 1z)/2, a= u- N23• f3 = u- N31, 'Y= u- N12, the circum­ center 0 of T is z 0 = [a(f) + 'Y)z 1 + f3('Y + a)z2 + 'Y(a + {3)z3 l/2(a{3 + f3'Y + 'Ya). The circum radius R for a modular or pseudomodular triangle T is given by 2R = a/sinh lA I = b/sinh IB I = c/sinh IC 1. The incenter I of a modular or pseudomodular triangle T is z1 = (az 1 + bzz + cz3)/2s. The excenter 11 of a modular or pseudomodular triangle Tis z 1 = (- az 1 + bz2 + cz3)/Z(s- a). The other two excenters 12 , and 13, are obtained by cyclic permutations. The inradius r is r = k/s = 1 2 [-(s- a)(s - b)(s- c)/s] / . Similarly an exradius r 1 is given by r 1 = k/(s- a)= [- s(s - b)(s - c)/(s - ad/2 • The other two exradii r 2 and r 3 are given by similar expressions. Analogues of the Gergonne points for the tangent hyperbolic circles of T are obtained. If T is either modular or pseudomodu1ar or indefinite, the centroid G, the Minkowski circumcenter 0, and the Minkowski orthocenter H, determine the Euler line. Moreover G divides the segment OH in the ratio 1:2. In general the tangent hyperbolic circles of an indefinite triangle T do not exist. (Received March 6, 1964.)

64T-Z78. L. V. QUINT AS, St. John's University, Jamaica, New York 11432. On the space of homeomorphisms of a multiply punctured surface of genus g.

Let Hk denote the space obtained from a closed orientable surface of genus g by deleting k g G(Hk) the identity component of the space of all homeomorphisms of Hk onto itself points and g g topologized by the compact-open topology. In this paper, using a result of M.-E. Hamstrom (The space of homeomorphisms on a torus, these Notices 10 (1963), 275) and homeotopy sequence methods of G. S. McCarty (Homeotopy groups, Trans. Amer. Math. Soc. 106 (1963), 293-304), the following theorems concerning the homotopy groups Trn[G(H~l) of G(H~) are proven. Theorem 1.

If 1 ~ k < oo, then Trn[G(Hk1)] = 0 for all n ;;; 0. I.e., G(H~) is n-connected for all n ;;:; 0. Theorem 2. If n f:: 0, 1 ~ k < oo, and g f:: 2, then TTn[G(H~)] ""JJn[G(H~)J. It is conjectured by those familiar with the problems considered here that, if g ;;; 2, then G(H~) is n-connected for all n ;;; 0. Theorem 2 may

be useful in verifying this. In any case the conjecture now is, if 0 ~ k < oo and g ;;; 2, then G(H~) is n-connected for all n ;;; 0. (Received March 6, 1964.)

64T-279. B. S. RANDOL, 12 Hillhouse Avenue, New Haven, Connecticut. A lattice-point estimate.

Let Y = (y l' ... ,y n) be a typical point in En. Suppose T(Y) is a positive, smooth function in En - {o} such that T(rY) = r 2T(Y) (r > 0). Define c = {Y IT(Y) ~ 1 }. ac = {Y IT(Y) = 1}. and

suppose ac has the following property: there exists 0 < (l ~ (n -1)/2 such that for any function f(Y) which is smooth in a neighborhood of ac, there is a positive constant M(f) such that 27ri(X Y) -a I f.acf{Y)e • dsy I < M(f) IX I for X i o. Moreover, M(f) depends only on bounds for f(Y) and a fixed, finite number of its derivatives in a neighborhood of ac. (Here X= (xl•···•xn), lXI =(xi+ .•. + x~) 1 1 2 , and dsy is the area element on ac.) Let V be the volume of C, and

denote by N(x) the number of integral lattice-points N, such that T(N) ~ x. Theorem. 2 N(x) = Vxn/2 + O(x(n -n-n.a.)/2(n-a)). (Received March 5, 1964.)

449 64T-280. WITOLD BOGDANOWICZ, Georgetown University, Washington, D. C. 20007. On a linear continuous operator in the space of continuous functions.

Let Q be a compact metric space and let h be a continuous mapping from the space Q into itself. Define compositions hn+l = hn o h, for n = 1,2, •••• Let C be the space of all continuous complex valued functions on the space Q. Assuming a E C, define a linear continuous operator g = H(f) from the space C into itself by the formula g(x) = a(x)f(h(x)) for x E Q, f E C. Theorem. If all the functions hn have exactly one fixed pointy and they converge uniformly toy on the set Q, then the spectrum of the operator H is the set { P: IPI ;:;; ia(y) I}. The proof of the theorem makes use of some construction in the space of convergent sequences. This type of operator one meets in cer­ tain boundary problems for partial differential equations. (Received March 9, 1964.)

64T-281. WITOLD BOGDANOWICZ, Georgetown University, Washington, D. C. 20007. On the existence of almost periodic and asymptotically almost periodic solutions for nonlinear partial differential equations.

Let Y be a complex Banach space and I be either the additive group R of reals or the additive semigroup R+ of non-negative reals. Let A= A 1P 1 -A2P 2, where the Pi are complementary projective operators, PiY = Y i (Y i may be also equal to the whole space Y or { 0 }>. and Ai is a closed linear operator with dense domain in the space Y i with resolvent Ri (p) satisfying the condition

IIRf(p) II ;:;; M(p- J..l.) -n for p > J.l., n = 1,2,. •• and some M and J.1. < 0. Let f(x,y) from I X Y into the domain of A satisfy Lipschitz's condition with respect toy with a constant L. In the case when

I= R+, let the function f(x,y0) be asymptotically almost periodic for any fixed Yo in Y and P 2 = 0. In the case when I= R, let the function f(x,y0) be almost periodic for any Yo E Y. Then there exists a constant C depending only on the operator A such that if L < C then the differential equation y' + Ay = f(x,y) has an asymptotically almost periodic or almost periodic solution, respectively. The proof makes use of Hille-Yosida and contraction mapping theorems. (Received March 9, 1964.)

64T-282. WITOLD BOGDANOWICZ, Georgetown University, Washington, D. C. 20007. On the existence of periodic solutions of nonlinear partial differential equations.

Let Y be a complex Banach space and R the space of reals. Let A= A1P 1 - A2P2, where Pi are complementary projective operators, Pi Y = Y i (Yi may be also equal to the whole space Y or

{ 0} ), and A1 is a closed linear operator with dense domain in the space Y i with resolvent Ri (p) satisfying the condition IIRf(p) II ;:;; M(p - J..!) -n for p > J.l., n = 1,2, ••• and some M and J.1. < O. Theorem 1. Let f(x,y) be uniformly continuous from R X Y into the domain of A, have a period T in x and map R X Y into a compact set. Then the differential equation y' + Ay = f(x,y) has a solution with period T. Let S denote a sphere in Y with the center at the origin. Theorem 2. Iff satisfies similar conditions as in Theorem 1 and maps R X S into a compact set then there exists a constant C such that if llf(x,y) II ;:;; C for x E R, y E S then the differential equation has a solution with period T. The proof makes use of Hille-Yosida and Schauder theorems. If Y is a space of functions the operator A of the above form may be obtained in some cases as a closure of a partial differential operator in the space Y. (Received March 9, 1964.)

450 64T-283. M. F. NEWMAN, New York University, Courant Institute of Mathematical Sciences, 25 Waverly Place, New York, New York. Outer automorphisms of nilpotent p-groups.

Theorem l. Every infinite nilpotent p-group has an outer automorphism. Theorem 2. Let G be a finite p-group and let N be a normal subgroup of G such that G/N is elementary abelian of order p 2• If the second centre of G does not lie in N, then G has an outer automorphism of p-power order stabilizing N. Theorem 3. A nonsimple finite p-group G of nilpotency class at most 3 has an outer automorphism of p-power order which stabilizes a normal subgroup N with G/N elementary abelian of order p 2• See also: Godino (these Notices 8 (1961), 368; 9 (1962), 117), Huval (ibid. 7 (1960), 723), Liebeck (in publication), Ree (Proc. Amer. Math. Soc. 7 (1956), 962-964; 9 (1958), 105-109), and Schenkman (ibid. 6 (1955), 6-11). (Received March 9, 1964.)

64T-284. T. S. MOTZKIN, University of California, Los Angeles 24, California. Approximation in the sense of a deviator integral.

For a function a.(x), x inn-space, where a. belongs to a (nonlinear) family A, let v(a.) ~

JT(x,a.(x))d'Y(x) for given T and 'Y, be the (nonmetric) deviation of a. from 0. Conditions are given under which the variation ov of v(a.) equals I aT(x, a.(x)) oa.(x)d'Y(x)/ aa., where a; aa. if one- sided is right or left as oa.(x) is > 0 or < 0; this entails suborthogonality and orthogonality conditions ov > 0, ;;; 0, = 0 for various kinds of minima of ·v. (If A is linear and T(x,y) is convex in y there follow simple conditions for uniqueness of the minimizing a..) Other results are strong oscillation of the best approximator and conversely, best approximation of 0 by any strongly oscillating a. in the sense of a suitable weight-related deviation JT(x,a.(x))w(x)d'Y(x). Here oscillation refers to a family of sets inn-space on which the family of variations oa. is oscillable. (Received March 9, 1964.)

64T-285. J.P. HEMPEL and D. R. McMILLAN, The Institute for Advanced Study, Princeton, New Jersey. Locally nice embeddings of manifolds.

Let Nk be a compact pw£ (piecewise-linear) k-manifold (possibly with boundary) topologically embedded in Int Mn, where Mn is a pw£ n-manifold and k ;:;; n - l. This embedding is called locally nice if Mn- Nk is 1-LC (locally simply-connected) at each point of Nk (if k f. n- 2) or if Mn- Nk has "uniformly abelian local fundamental groups" (in case k = n - 2) in the sense of Harrold (Trans. Amer. Math. Soc. 67 (1949), 120-129 and Duke Math. J. 17 (1950), 269-272). Theorem. Let

Nk C Int Nf be a locally nice embedding. Let n ;;; 5. Then, each compact absolute retract in Nk ~ cellular with respect to pw£ cells in Mn. The proof uses "A criterion for cellularity in a manifold" (abstract in these Notices). (Received March 11, 1964.)

64T-286. M. F. JANOWITZ, University of New Mexico Albuquerque, New Mexico. On standard ideals.

The notation follows Gratzer and Schmidt (Acta Math. A cad. Sci. Hungar. 12 (1961), 17- 86). Theorem 1. Let L be a section complemented lattice with 0 or a relatively atomic lattice such that every nonzero element can be expressed as the join of a finite number of atoms. An ideal J of L is

standard if and only if a E J, aVx ;;; bvx, a Ax~ bAx ~ 0 for some x E L implies bE J. An ideal of a

451 relatively complemented lattice with 0 is standard if and only if it is closed under perspectivity. Theorem 2. Every homomorphism kernel of the lattice of closed subspaces of a Hilbert space H is neutral if and only if H is finite dimensional. Corollary. The lattice of all ideals of a weakly modular lattice need not be weakly modular. Theorem 3. Let L be a lattice with 0 such that every homomorphism kernel is a principal ideal. Every congruence is determined by its kernel and every

homomorphism kernel is a standard ideal if and only if a > b ;<;; d implies the existence of an element

c such that a ;<;; c > d and a/b---> c/d. (Received March 12, 1964.)

64T-287. SVETOZAR KUREPA, University of Zagreb, Marulicev TRG IS-I, Zagreb, Yugoslavia. The Cauchy functional equation and scalar product in vector spaces.

A real-valued functional n(x) on a real vector space X is quadratic if (i) n(x + y) + n(x - y) = 2n(x) + 2n(y) holds through X. If (ii) n(tx) = t 2n(x) holds for all x E X and every real t, then in an

algebraic basic set {ek} of X one has: n L: n(tx + y) and the question raised by I. Halperin, is solved in negative. Details will be published in Glasnik Mat. Fiz. Astr. Zagreb. (Received March 11, 1964.)

64T-288. M. M. RAO, Carnegie Institute of Technology, Schenley Park, Pittsburgh, Pennsylvania 15213. Global representation of bounded linear functionals on Orlicz spaces.

Let cf>, 'II be Young's functions and Lcf> , JJ" be the Orlicz spaces on (G, 2:, JJ.) with the property:

A E 2:, JL(A) > 0 ~ 3 B E ~. '3 B CA and 0 < JL(B) < oo. Defs. (1) An additive scalar set function

G on 2:, G < < J.l., is of - bounded variation, if {Ai} C 2: is a finite disjoint collection of sets with

0 < JL(Ai) < oo, then Icf>(G) = sup 2: cf>(G(Ai)/JJ.(Ai)) ~t(Ai) exists (sup is over all such collections { Ai} in 2:). (2) Acf>(!L) is the class of additive set functions G of (1), such that Icf>(G/K) < oo for some K > O. (IV and A\llare defined similarly.) The norm on Acf>(IL) is IIGIIcf>= inf {K > 0, Icf>(G/K) ;;; 1} for G EAcf>(IL) and that on Lcf> is lifllcf> =sup Jnlfgld~t, f E Lcf> and all g with Jn'{J(g)dJL;;; 1. Results: Theorem 1. If cf>, '11, and Lei>, Av(JL) are as above, then for every x* E (Lcf>)*, the conjugate space of L cf>, there is a unique G E Av(IL), such that (*) x*(f) = JnfdG, f E L cf>, and IIGIIv;;; llx*ll;;; SIIGII'IIi" Theorem 2. Let cf> be continuous, Lei>, Av(IL) be as above, and Mci> be the closed subspace determined by J.l. simple functions in Lcf>. If x* E (Mci>)*, then there is a subspace { C A~JJ.), of countably additive set functions, such that(**) x*(f) = JnfdG, f EM«<>, G €' 4 and llx*ll = IIGI'v· Theorem 3. If cf>, \It are continuous, and Mci> =Lei> ,-M'~' = L'~', then JY (L'~') is reflexive. (Received March 11, 1964.)

64T-289. L. V. QUINT AS, St. John's University, jamaica, New York 11432. A note on semi­ homogeneous functions.

Let f and p denote functions on the reals to the reals with p product preserving and not necessarily continuous. The function f is called semi-homogeneous with respect top and a set A, if f(ax) = p(a)f(x) is satisfied for all x in the domain off and for all a in A. For semi-homogeneous f,

452 it has been noted (§5, L. V. Quintas and F. Supnick, Semi-homogeneous functions, Proc. Amer. Math. Soc. 14 {1963), 620-625) that, (i) if A has positive Lebesgue measure and contains at least one negative number, then f(x) = f(l)p(x) and (ii) there exist sets A having measure zero and such that f(x) = f(l)p(x). It is noted here that (iii) there exist sets A having the property that for every open interval (a,b) ~ A n (a,b) has outer measure b - a and inner measure zero, and such that f(x) = f(l)p(x). If A denotes the additive subgroup of the reals generated by any Hamel basis, then the measure property of A is the statement of a theorem by P. Erdos (Theorem 1, On some properties of Hamel bases, Colloq. Math. 10 (1963), 267-269). The set A - jo} generates the multiplicative group of nonzero reals and this implies f(x) = f(l)p(x). (Received March 13, 1964.)

64T-290. NICK METAS, Thomas J. Watson Research Center, IBM Corporation, P.O. Box 218, Yorktown Heights, New York. Some results on investigations of injective and projective spaces suggested by Nachbin. Preliminary report.

Definition 1: A B-space I is injective if for every B-space X, every closed subspace Y of X, and every bounded linear operator T: Y --+ I, there exists a bounded linear operator T: X -+ I which extends T. A B-space Pis projective if for every B-space X, every closed subspace x 0 of X, and every bounded linear operator T: P --> X/X0, there exists a bounded linear operator T:P -+X such that QT = T where Q is the quotient map from X onto x;x0• Definition 2: A B-space I is dually injective (projective) if in Definition 1 we restrict our B-spaces X to those which are congruent to dual spaces. A B-space is £00-injective (C(S)-injective) if in Definition 1 we restrict our B-spaces

X to those which are congruent to £00(S) for some setS (to those which are congruent to C(S), S compact Hausdorff). A B-space P is .£1-projective if in Definition 1 we restrict our B-spaces X to those which are congruent to l 1 (S) for some set S. Theorem: ~B-space is injective iff it is dually injective iff it is lao-injective iff it is C(S)-injective. ~B-space is projective iff it is dually

projective iff it is £1-projective. This theorem answers some of the questions whose investigation was suggested by L. Nachbin (Some Problems in Extending and Lifting Continuous Linear Transfor­ mations in the Proceedings of the International Symposium on Linear Spaces). (Received March 13, 1964.)

64T-291. P. C. FISCHER and STAL AANDERAA, Harvard University, Computation Laboratory,

33 Oxford Street, Cambridge 38, Massachusetts. The solvability of the halting problem of a 2-~ Post machine.

Turing's original formulation of a Turing machine as a set of quintuples and Post's formulation as a set of quadruples (cf. Davis, Computability and Unsolvability, p. 5) have been used more or less interchangeably in the development of computability theory, since any partial recursive function is realizable by either variety of machine. Shannon has shown that there exists a universal 2-state Turing machine ("A universal Turing machine with two internal states," Automata Studies, pp. 157- 166). In the same paper Shannon shows that no !-state Turing machine can generate the binary expansion of an irrational number, but he does not show that the halting problem of a !-state Turing machine is decidable. The latter result has been proved independently by R. J. Abbott and Richard Boyd and also follows immediately from the following: Theorem. The halting problem for a 2-state

453 Post machine is recursively solvable. Thus, by a theorem of Davis, there is no universal 2-state Post machine. (It is quite easy, however, to construct a 3-state universal Post machine.) (Received March 13, 1964.)

64T-292. R. E. PIPPERT, University of Kansas, Lawrence, Kansas 66045. On absolutely convergent trigonometric series. Preliminary report.

The following two theorems are proved. Theorem 1. If (i) the sequence {an} satisfies lan+ 1 1 ~ K ian I for some constant K, and (ii) either L ian cos nx0 I < oo for some x0 or

L ian sin nxo I < 00 for some Xo "' 0 (mod 11"), then L lan I < oo. Theorem 2. Write ao/2 + ,L(an cos nx + bn sin nx) = ,LAn(x), and write Pn = (a~+ b~) 1 1 2 . If the sequence {Pn} satisfies Pn+ 1 ~ Kpn for some constant K, and if L IAn(x) I < oo for two distinct values of x, say x 1 and x2, with lx 1 - x 2 1 < 11", then LPn < oo. (Received March 13, 1964.)

64T-293. SVETOZAR KUREPA, University of Zagreb, MaruU~ev trg 19, Zagreb, Yugoslavia. Quadratic and sesquilinear functionals.

Theorem. Let X be a complex vector space and n(x) a complex-valued functional such that (i) n(x + y~ + n(x - y) = 2n(x) + 2n(y) and (ii) n(Ax) = I X1 2n(x) hold for all x,y EX and all complex numbers .\. Under these conditions the functional B(x,y) = !P-

64T-294. M. D. GLADSTONE, Queen's Building, University Walk, Bristol 8, England. Propositional calculi with decision problems of any required recursively enumerable degree of unsolvability.

Let S be any set of logical connectives adequate to express implication, and let ---+ be an expression of implication in terms of s. Let A be any set of tautologies built up from S. Then P (S,A, --->) is to denote the propositional calculus with connectives S, axioms A, and rules of inference (i) substitution, and (ii) a, a---> b f- b. We abbreviate recursively enumerable degree of unsolvability to r.e.d.u., and decision problem to d.p. Theorem 1: GivenS and ---> , we can choose A (finite) so that the d.p. of P (S,A, ---> ) is of any required r.e.d.u. (I learn that this theorem, with others, has been proved recently for the case when S consists of 1 and ::::J, and ---+ = ::::J, by Mrs. A. H. Ihrig, and also by W. E. Singletary, both of the University of Illinois.) Theorem 2: Given P(S,A, --->) and any connective ~having ~ 1 argument-place and not E S, we can choose B (finite, if S is finite) so that the d.p. of P(S + ~. A+ B, --->) is of any required r.e.d.u. (Received March 16, 1964.)

64T-295. J. D. BUCKHOLTZ, 360 Phillips Hall, University of North Carolina, Chapel Hill, North Carolina. Zeros of partial sums of power series for certain entire functions.

Let f(z) = ,LapzP = exp {g(z) }, where g(z) = _Lbkzk is an entire function which is not a poly­ nomial. For each positive integer n let rn denote the least modulus of a zero of L~=O~zP, the nth partial sum of f(z). For each n define Bn to be the positive number which satisfies the equation

454 k 2 n = maxlbkiBn, l ,?; k ,?; n. Theorem l, rn ~ Bn as n ---> oo. Furthermore, Bn/2e < rn,?; Bn for every positive integer n, Theorem 2, If the order of g is finite, then rn ~ M- 1 (n), where M-l de,;otes the inverse of the maximum modulus of g. In both theorems the proof depends on obtaining for rna lower bound which is asymptotic to Bn. This lower bound is the maximum value of r exp{- 2Gn(r)/n}, 0 < r < oo, where Gn(r)= L~=llbklrk. (Received March 16, 1964.)

64T-296. GARRETT BIRKHOFF, Harvard University, 2 Divinity Avenue, Cambridge, Massa­ chusetts 02138. Error bounds for spline fits.

Let f(x) be a function having an absolutely continuous third derivative on a finite interval a ,?; x ,?; b. Let s7r(x) be the interpolating spline function to f(x) associated with a given partition 1r: a= x0 < x 1 < ... < xn-l < xn =band endpoint conditions, Let M(7r) = max(l

true if 1/2 <

64T-297. J, M. WORRELL, JR., Division 5421, Sandia Corporation, Sandia Base, Albuquerque, New Mexico 87115. Upper semicontinuous decompositions of metacompact spaces,

A topological space is defined much as in J, L. Kelley's General topology. Let~ denote a

topological space, let G denote an upper semicontinuous decomposition of ~. and let I denote the

decomposition space, Theorem 1, 1!. ~ is metacompact then I is metacompact, Theorem 2. 1!_ ~ ~ paracompact and no element of G has a nonbicompact boundary, then I is paracompact. The proof for Theorem 2 is a corollary to that for Theorem 1. E. Michael's theorem that the closed, continuous images of paracompact Hausdorff spaces. are paracompact [Another note on paracompact spaces, Proc, Amer. Math. Soc, 8 (1957), 822-828] is implicit in the proof for Theorem 1. (Received March 16, 1964.)

64T-298. J. M. WORRELL, JR., Division 5421, Sandia Corporation, Sandia Base, Albuquerque, New Mexico 87115, Continuous decompositions of paracompact spaces,

A topological space is defined much as in J, L. Kelley's General topology. While E. Michael has shown that the closed, continuous images of paracompact Hausdorff spaces are paracompact, the stipulation that space be Hausdorff is not superfluous, Moreover there exist open, continuous images of metrizable spaces that are not paracompact [see, for example, A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math, Soc. 7 (1956), 690-700]. Theorem, Continuous decomposi­ tions of paracompact topological spaces yield paracompact spaces. (Received March 16, 1964.)

64T-299. K. R, BROWN and HAO WANG, 33 Oxford Street, Cambridge 38, Massachusetts. Short definitions of the class of ordinals,

Four alternative short definitions are given of the class On of ordinals as construed by

Zermelo and von Neumann. First, On(x) iff for all X, x EX if (w) (~w ~ w ~X ~w EX). Second,

On(x) iff for all u, x E u if (w) [(~w ~ w ~ u A w ~x) ~ w E u ]. Third, replace w ~ x by w E x' in the

455 second definition. Fourth, On(x) iff for all u, x E u only if (Ew) (w E u A ~w <:;;; w 11 w n u = A). In each

case, it is shown that every transitive set of ordinals is an ordinal and that On <:;;; F if (w)(~w <:;;; w <:;;; F

::J Fw), The axioms of set theory required in developing the four definitions are different. The third and the fourth require only extensionality, (Ey) (y = x U {x} ), and the schema (Ey)(x)(x E y = (x E z A Fx)). For the second definition, add (Ey)(x)(x E y = x <:;;; z). The first definition can be developed on different bases depending on whether we assume impredicative classes. (Received March 17, 1964.)

64T-300. K. R. BROWN and HAO WANG, 33 Oxford Street, Cambridge 38, Massachusetts. Finite sets and axioms of limitation.

An axiom system F of finite sets is given as an analogue for the Peano axioms. AI. Exten­ sionality corresponds to the two axioms that 0 is not a successor and no two numbers have the same successor. A2. (Ey)(x)(x E y = xi x) (0 is a number). A3. (Ez)(z = x U {y }> (successor of a number is a number). A4. {H A A (u)(v) [(Hu A Hv) ::J H(u U { v} )] } ::J Hx (induction). This is more satisfactory than a related system G* (Math. Ann. 126 (1953), 391) in which A4 is replaced by Aussonderung and an axiom of limitation using natural numbers explicitly. Either through G* or directly, all axioms of set theory (with axiom of infinity negated) are derivable in F. A related development is to use the power set ?rX as the successor x' of x in developing arithmetic. Thus, Nx iff for every y, x E y if (1) 0 E y, and (2) v' E y if v E y and v' <:;;; x. Then we have the system P as follows. B 1. extensionality. B2. (Ey)(z)(z E y = (z <:;;; x A Hz)). B3. (x)(Ey)(Ny Ax E y). (Received March 17, 1964.)

64T-301. D. W. DUBOIS, The University of New Mexico, Albuquerque, New Mexico. Applications of analytic number theory to the study of type sets of torsion free Abelian groups. I.

The problem studied is that of giving necessary and sufficient conditions that a set of types be the type set of a torsion free Abelian group, assumed, in this first paper, to be of rank two. A necessary condition, a sufficient but not necessary condition, and .examples are given, which improve the results of §7. 7-7.11, and answer questions (2) (a) (no), (2)(b) (no), and (2)(c) (yes), of R. A. Beaumont and R. S. Pierce, Torsion free groups of rank two, Mem. Amer. Math. Soc., No. 38. The proofs involve a simple group construction scheme, the prime number theorem, and a corollary of the latter. (Received March 19, 1964.)

64T-302. A. H. IHRIG, 509 South Fifth Street, Champaign, Illinois. Post-lineal theorems for r.e.d.u.

Abbreviations; r.e.d.u., recursively enumerable degree of unsolvability; p.p.c., partial propositional calculus; s-T.s., semi-Thue system. Theorem 1. For any s-T.s., T, whose operation rules include empty words, one can construct a s-T.s., T*, such that (i) the operation rules ofT* include only nonempty words, (ii) T and T* have the same r.e.d.u. Theorem 2. Given any s-T.s.,

T, one can construct a p.p.c. (connectives ~and ::J), PT, such that the decision problem for PT reduces (by unbounded truth-tables) to that of T. Theorem 3. For every r.e.d.u., D, there exists a p.p.c., P T(D)' whose decision problem is of degree D. (D.C. Shepherdson wrote to W. W. Boone

456 that M. D, Gladstone of Bristol proved a stronger theorem, :::J the unique connective, in the summer of 1963. Subsequently, W. E. Singletary of Illinois proved the theorem.) Theorem 4. For every

r e d U D there exists a class of p p c .•. " ' • ' • , C P(D) , such that the problem of determining of an arbitrary member whether or not it is complete is of degree D. Theorem 5. For every r.e.d.u., D, there exists a class of p.p.c., CP(D)' such that the problem of determining of an arbitrary member whether or not its axiom set is independent is of degree D. (Received March 20, 1964.)

64T-303. A. H. IHRIG, 509 South Fifth Street, Champaign, Illinois. Coincidence of partial propositional calculi.

Let T be a Thue system with unsolvable word problem of degre D. Let {w i' Wif be the class

of word pairs on T recursively enumerated. From the ith pair construct Ti as done by Boone (J. Symbolic Logic 27 (1962),375). From each Ti construct T[ as in Th. 1 of the pr.eceding abstract Post-lineal Theorems for r.e.d.u. From each T": construct PT* as in Th. 2 of same. Lemma: The - 1 i following are equivalent: (i) Wk +--> Wk in T, k = i,j; (ii) Tk' k = i,j, trivial; (iii) Ti""' Tj; • (iv) T k' k = i,j, decidable; (v) T! = Tj'; (vi) PTk' k = i,j, decidable; (vii) PTi coincides with PT*" J Theorem 1. For every r.e.d.u., D, there exists a class of p.p.c., C P(Ti,DY such that the problem of determining of an arbitrary member whether or not it is decidable is of degree D. Theorem 2. For every r.e.d,u., D, there exists a class of p.p.c., CP(T[,D)' such that the problem of determining of an arbitrary pair whether or not they coincide is of degree D. (Received March 20, 1964.)

64T-304. R. P. GILBERT and H. C. HOWARD, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. On solutions of the generalized axially symmetric wave equation represented by Bergman operators.

In this paper we use the integral operator method to investigate solutions to the partial differential equation a 2u;ax2 + 82u;ay2 + 2v/y + au;ay + k 2u = 0. An operator which maps analytic functions onto solutions is given, along with its inverse. The location of singularities is investigated and criteria established for a point to be a singularity. The growth of entire solutions is described in terms of the Bessel-Gegenbauer coefficients and upper and lower bounds for the maximum modulus of the solution. (Received March 30, 1964.)

64T-305. R. P. GILBERT and H. C. HOWARD, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. On a class of elliptic partial differential equations.

In this paper the integral operator method of S. Bergman (Integral operators in the theory of linear partial differential equations, Ergeb. Math. u. Grenzzeb., vol. 23, Springer, Berlin, 1960) is

used to investigate the (p + 2)-dimensional, elliptic equations, T p+2 [o/] = a2 'lr/ axiL axiL+ 2 2 A(r )Xu a>~r;8xJ.L + C(r ) = O. (Repeated indices indicate summation from 0 top+ 2, p ~ 0, A(r 2), and 2 C (r ) are analytic functions of i.) An operator l:lp+2 [f] is constructed which maps holomorphic functions of (p + 1) complex variables onto solutions of Tp+2 (o/] = 0 by making use of Bergman's operator Bp+2 [f] for harmonic functions of (p + 2) variables. Results are obtained concerning location of singularities, growth of entire solutions, and line integrals of vector solutions. (Received March 30, 1964.)

457 64T-306. R. P. GILBERT, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland. Multivalued harmonic functions in four variables.

In this paper solutions of Laplace's equation in four variables H(X) are investigated by

means of the integral operator B4 [f] =-(l/4'11"1J, 2 Jf(u,11,~)(d11/11)(d~/~), u= x 1(1 + 1/11~) + ixz(l- 1/11~) + x 3(1/~ - 1/17) + ix 4(1/~ + 1/11), w!ere r(Z) is a 2-cycle, and H(X) = B4 [f}. It is shown that in the case where f(u,17, ~) is a holomorphic function defined on an algebraic surface the solutions can be represented in terms of the periods of Weierstrass integrands of the 1st, 2nd, and 3rd kind over a suitable Riemann surface. Line integrals of harmonic vectors are also evaluated in this manner. (Received March 30, 1964.)

64T-307. OSWALD WYLER, University of New Mexico, Albuquerque 6, New Mexico. Weakly exact categories.

The basic diagram lemmas and theorems of homology theory are proved for a class of categories including the category of groups. The axioms are self-dual. Ker, Coker, Im, Coim are defined in the usual way. a E C is called normal if ker coker a = im a, conormal if coker ker a = coim a. One distinguishing axiom: If e is epimorphic and conormal, m monomorphic and normal, then em is normal and conormal. Other axioms are also needed; a complete list is too long for an abstract. One distinguishing feature of the theory: Exact (im a = ker b) and co exact (coim b = coker a) pairs have to be distinguished; the two dual properties are not equivalent. (Received November 29, 1963.)

64T-308. NICK METAS, Thomas J. Watson Research Center, IBM Corporation, Post Office Box 218, Yorktown Heights, New York. An analogue for projective spaces of a theorem of Sobczyk. Preliminary report.

Definition 1: A Banach space P is projective if for every Banach space X, every closed subspace Y of X, and every bounded linear transformation T from P to X/Y, there exists a bounded linear transformation T from P to X such that QT = T, where Q is the quotient map from X onto X/Y.

Definition 2: Let A ~ 1. A projective Banach space ·P is said to be in class IA if the bounded linear transformation Tin Definition 1 can always be chosen so that IITII & AliT II. Theorem: If a Banach

~ P is projective, then P is in class IA for some A. This theorem is the analogue for projective Banach spaces of a theorem of Sobczyk for injective Banach spaces (Extension Properties of Banach Spaces in Bull. Amer. Math. Soc., May, 1962). (Received April 1, 1964.)

64T-309. JOHN HEMPEL and D. R. McMILLAN, JR., Institute for Advanced Study, Princeton, New Jersey. Locally nice embeddings of manifolds. II.

The embedding of a compact piecewise-linear manifold Nk (possibly with boundary) in the interior of a pwl manifold Mn (k & n - 1) is said to be locally nice if Mn - Nk is 1 - LC (locally simply connected) or 1 - ALG (has abelian local fundamental groups) at each point of Nk according as k 1 n - 2 or k = n - 2. The authors have shown (Locally nice embeddings of manifolds, abstract in these Notices) that if Nk C lnt Mn (n ~ 5) is a locally nice embedding then each absolute retract in ~

458 is cellular in Mn. It is shown here that if Nk C Int Mn and Pr C Int Qs are embeddings of compact pwl manifolds then the embedding of Nk X Pr in Mn X Qs is locally nice in each of the following cases:

(a) r ;;; s - 1 and k ;;; n - 2; (b) r = s,k ;;; n - 1, and the embedding of N in M is locally nice; (c) r = s - 1, k = n - 1 and at least one of the embeddings is locally nice. By suspending a knotted n-3 n-1 n-1 n-3 sphere pair S C S for which 1r1 (S - S ) = Z (such examples have been given by Neuwirth and Hirsch and by C, T. C. Wall), we obtain piecewise-linear embeddings Sn-2 C Sn (n ;;; 6) which are locally nice but not locally flat, (Received April 2, 1964.)

64T-310, W. D. L. APPLING, North Texas State University, Denton, Texas, Set functions and an extremal problem in absolute continuity, Preliminary report,

All integrals discussed are Hellinger type limits of the appropriate sums. Suppose S is a field of subsets of the set R, U is the set of all real-valued, finitely additive functions on S having bounded variation, m is a nonnegative-valued element of U, and W is the set of all elements of U absolutely continuous with respect to m. Suppose his in U, for each V in S, g(V) = l,u,b, fvmin jlh(I) j, Km(I)} for 0 < K, B (V) = 1 if 0 ;;; h\ V), and B (V) = - 1 if h(V) < 0, Suppose further that for each V in S, g*(V) = JVB (I)g(I). Theorem: If u is in W and u is not g*, then JRjh(I)- u(I)j > fRjh(I)- g*(I)j. (Received April 2, 1964.)

64T- 311, R. T. BARNES, The Ohio State University, Columbus, Ohio, On semi-simple Lie algebras of prime characteristic, Preliminary report.

Let denote a field of prime characteristic different from 2 or 3, and {2 the algebraic closure of , An extension field P of is called a splitting field for a Lie algebra !£'over if lip is of classical type in the sense of Seligman and Mills. Seligman has shown that if is alge­ braically closed and of characteristic p > 7, and if Y over is restricted, containing no abelian ideals and possessing a restricted representation with nondegenerate trace form, then !:;tis of classical type. Theorem. Every semi-simple Lie algebra over with nondegenerate Killing form

has a separable splitting field, This result permits an alternate proof of the fact that if ~ is the split Cayley algebra or rhe split exceptional Jordan algebra over and !£'is a Lie algebra over

such that ~ is isomorphic to the derivation algebra of '21: 11 , then Yis isomorphic to the derivation algebra of 18 where 18 is an algebra over and 1811 is isomorphic to ~n. (Received April 6, 1964.)

64T-312, YUEN-FAT WONG, Box 216, Atlanta University, Atlanta 14, Georgia. Homotopy type of almost differentiable manifold,

Theorem, Let n be an~ integer greater than 5 and X be a connected finite polyhedron, with

1r1(X) = 0, Suppose the following two conditions are satisfied: (i) X satisfies the Poincare duality; i.e. Hn(X) = Z and if g is a generator of Bu(X) then the cap product ,..,.g:Hi(X) -->~-i(X) is an isomorphism for all i. (ii) There is an oriented k-vector bundle ~over X such that (g) E Hn+k(T( \)) is spherical, Where is the Thorn's isomorphism Hn(X) --> Hn+k(T(l)) and T(l) is the Thorn complex. Then there is a combinatorial manifold Nn with the same homotopy type of X and N admits a differen­ tiable structure except possibly on one n-dimensional cell. For n = 6 or n = 14, Nn is a differentiable

459 manifold, Remark, W. Browder's results for n = 2m- 1 and n = 4q can be found in Colloquium on algebraic topology, Aarhus Universitet, Denmark, 1962. (Received April 6, 1964.)

64T-313, M. F. JANOWITZ, University of New Mexico, Albuquerque, New Mexico. On Baer semigroups, II.

The terminology is that of Abstract 608-21. Theorem. The set of right annihilators of elements of a Baer semigroup forms a lattice with 0 and 1. Every lattice with 0 and 1 is isomorphic to the lattice of right annihilators of a suitable Baer semigroup. (Received April 7, 1964.)

64T-314. D. M. BLOOM, 1023 Lincoln Apartments, Amherst, Massachusetts. The subgroups of SL(3,q). II. Preliminary report.

Let p be an odd prime; let q = pa be a power of p with q ~ 1 (mod 3). Let G be a subgroup of SL(3,q). Then one of the following six cases occurs: (1) G has a nontrivial normal elementary­ abelian subgroup; (2) G is conjugate to the subgroup SL(3,pb) obtained by restricting all matrix coefficients to a subfield; (3) G is conjugate to the unitary subgroup U(3,pb), Zb ja, corresponding to a subfield; (4) G is isomorphic to PSL(Z,pb) or PGL(Z,pb) where b Ia and pb f. 3; (5) G is isomor­ phic to PSL(Z, 5) and q = ± 1 (mod 10); (6) G is isomorphic to PSL(Z, 7) and q3 = 1 (mod 7). In cases (4), (5), and (6), there exists exactly one conjugacy class of subgroups of each type mentioned, and representatives of these classes are given explicitly. The above strengthens the results of a preceding abstract (63T-18). (Received April 13, 1964.)

64T-315. L. F. McAULEY, Rutgers, The State University, New Brunswick, New Jersey. More pointlike decompositions of E 3•

An example is given of an upper semicontinuous decomposition G of E 3 into points and straight-line intervals such that the decomposition space is not homeomorphic to E 3• The proof provided relies on some fundamental theorems about E 3 established by R. H. Bing. (Received March 6, 1964.)

64T-316, D. R. McMILLAN, JR., Institute for Advanced Study, Princeton, New Jersey. Taming Cantor sets in En.

A Cantor set C in En is said to be~ if there is a homeomorphism of En onto En throwing C into one of the coordinate axes. Wild (i.e., nontame) Cantor sets in En exist for each n <:: 3 (Blankenship, Ann, of Math. (1951)). Homma and Bing have shown (independently) that a Cantor set C in E 3 is tame if and only if E 3 -Cis 1 - ULC. Theorem. Let C CEn be a Cantor set (n f. 4). Then, C is tame if and only if En - C is 1 - ULC. (If K is a compact set in En, En - K is said to be

1 - ULC if for each f > 0 there is a o > 0 such that each loop in En - K of diameter less than ois null-homotopic in En- K on a set of diameter less than £,) (Received April 16, 1964.)

460 64T-317. D. M. BLOOM, 10Z3 Lincoln Apartments, Amherst, Massachusetts. The subgroups

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

Let q = pa be a power of an odd prime p. Then all simple subgroups of PSL(3,q) are of known types. Specifically, if G is such a simple subgroup, then G is isomorphic to one of the following: (1) the simple groups of orders 60, 168, 360; (2) PSL(2,pb), PSL(3,pb), or PU(3,pb), where b divides a; (3) the alternating group on seven letters, with p = 5; (4) SL(3,3), with p = 13; (5) U(3,3}, with p = 7. It is not yet certain whether case (3), (4) or (5) actually occurs. The above strengthens results in the two preceding abstracts (The subgroups of SL(3,q). I, Abstract 63T-18, these Notices (1963); II, Abstract 64T-314, ibid. (1964)). (Received April 17, 1964,)

64T-318. E. A. SCHREINER, Western Michigan University, Kalamazoo, Michigan. Modular pairs in an orthomodular lattice.

L always represents an orthomodular lattice. >'he pair (a,b) is called a modular pair, written

M(a,b), if e ~ b implies e v(aAb) = (eva)Ab, Say that a commutes with b, written aCb, if a= (aAb)v(aAb'). Theorem. Let a,b,e E L. Then M(a,b), aCe and bCe implies (i) M(ave, bve), (ii) M(aAe, bAe), (iii) M(ave,b) and (iv) M(a,bAe). The pairs (aAe,b) and (a,bVe) need not be modular. Theorem. Let L be atomic and complete. Then (i) L satisfies the Atomic Exchange Property, [i.e., a,b atoms, a ~ b ve, and a f, e implies b ~ aVe for all e E L) is equivalent to (ii) M(e,f) iff M(f,e) for all e,f E L. Theorem. Let L be atomic and complete. Then (*) M(a,b) iff M(b',a') for all a,b E L is equivalent to the two conditions (i) M(a,b) iff M(b,a) for all a,b E L and

(ii) M(a,b), aAb = 0, x ~ avb, and a < avx implies (avx)Ab i 0. If L does satisfy(*), then M(a,b) and x ~ avb imply M(avx,b), M(a,bVx) and M(avx, bvx). (Received April 20, 1964.)

64T-319. E. W. CHENEY, University of California, Los Angeles, California 90024. Approximation by generalized rational functions.

P and Q are linear subspaces of C [a,b] of dimensions n + 1 and m + 1. R consists of all p/q with p E P, q E Q, and q > 0 on [a,b]. Results akin to those of Rice are given for approximations

in R. If r E.R let v(r) + 1 be the cardinal of a maximal Tchebycheff system in P e rQ. Let ~-t(r) be

the l.u.b. of the number of roots of nonzero elements in P 8j rQ, counting doubly a root in (a,b) where the function does not change sign. Say "f alternates k times" if points xi exist with i a~ x 1 < ... < xk ~ b, f(xi) = (- 1) A, and IA.I = llfll. Let r E Rand f Ec. Let Tf be the (possibly void) set of best approximations in R to f. Theorems. If f - r alternates 1-'(r) + 2 times then r E Tf.

If r E Tf then f - r alternates 11 (r) + 2 times. If r E Tf and v(r) = J.L(r) then r is alone in Tf. If r E Tf

and v(r) = n + m then r is "strongly" unique: a > 0 exists so that if r 1 E R, llf - r 1 11 ~ llf- r II + allr - r 1 11. If r E Tf and v(r) = n + m then Tg is nonvoid for g near f, and Tis continuous at f; i.e., llr - r 1 11 < fj llf - g II if r 1 E Tg. If P and Q consist of polynomials of degrees ~ n and ~ m resp., and if r = pjq E R with p/q irreducible, then ~-t(r) = v(r) = max{n + aq, m + ap }. The proofs depend on results of H. L. Loeb and the author to appear in J, Soc, Indust. Appl. Math. (Received April 20, 1964.)

461 64T-320. H. L. LOEB, 9508 jellico, Northridge, California. Characterization of rational trigonometric Chebyshev approximation.

Let Rn = jT/B: Tis a trigonometric polynomial of maximum order nand B is a trigonometric m polynomial of maximum order m}. Let f be any continuous function defined on [- 1r, r] with period 21r. It has been shown by E. W. Cheney and author, in a paper to appear in j. Soc. Indust. Appl. Math., that a Chebyshev approximation from R~ to f exists with a positive denominator. It can be shown that T /B E Rn with B > 0 can be reduced to an equivalent form T* /B * E R n where algebraic poly- m m nomials P*(z) and Q*(z) defined by following relationship are relatively prime: P*(e iB) = ein'BT*(8); Q*(ei6) = eim'BB*(8) where order T* = n' and order B* = m'. Theorem. A T*/B* of the above type is a Chebyshev approximation to f from R~ over [- 1r,1r) iff f - T* /B * induces 42 max (n + m', m + n') + 2} maximum residuals in absolute value which alternate in sign. This Cheby­ shev approximation is unique if the denominator is restricted to have norm one. By definition, the order of the zero trigonometric polynomial is - oo and the zero algebraic polynomial is only relatively prime to one. The main tools used in proving this theorem are a geometric characterization theorem of E. W. Cheney and author for "Generalized Rational Functions" and some results ofT. Rivlin which will appear in the journal referred to above. (Received April 20, 1964.)

64T-321. JOHN WERMER, Freudenbergstrasse 144, Zurich 6, Switzerland. Polynomially convex disks.

Let D be a closed disk in the z-plane and f a smooth complex-valued function defined on D.

Assume the image in C 2 of D under the map: z ----> (z,f(z)) is polynomially convex. Write (z,f] for the class of all functions on D which are uniform limits of polynomials in & and f. Denote by E the set of zeros of fz and by R(E) the space of functions on E which are uniform limits of rational functions without poles on E. Theorem: [z,f] consists of those continuous functions on D whose restriction to

E lies in R(E). Corollary: ..!!.f:z "f 0 everywhere (or almost everywhere),~ [z,f] = C(D). (Received April 24, 1964.)

64T-322. C. K. MEGIBBEN, Texas Technological College, Lubbock, Texas. Subsocles which do not support pure subgroups.

If G is a p-primary abelian group, call a subgroup S a subsocle of G is S ~ G [p ]. A subsocle S of G is said to be semi-dense if Sis dense (relative to the p-adic topology on G) in (pnG)jp] for some n. A subsocle S of G is said to support a pure subgroup if there is a pure subgroup H of G such that H[p] = S. The author (Bull. Soc. Math. France 91 (1963), 453-454) has shown that there exists a p-group G with G1 = 0 containing a subsocle which does not support a pure subgroup. A more general result involving a simpler construction is the following: Theorem. If B is an unbounded closed p-group with a countable basic subgroup and if Sis a subsocle of B which is not semi-dense and has cardinality of the continuum, then there is a pure subgroup G of B such that (i) B/G is divisible, (ii) S ~ G and (iii) S does not su!'port a pure subgroup of G. (If S were either countable or semi-dense, no such G could exist.) (Received April 24, 1964.)

462 64T-323, DMITRI THORO, San Jose State College, San Jose, California 95114. A characteriza- tion of certain P-sequences, Preliminary report.

Let S(x,n) designate the sequence of consecutive positive integers {x, x + 1, ... , x + n - 1 f. S(x,n) is a P-sequence if it contains an integer which is prime to all the rest; otherwise S(x,n) will be called a P'-sequence. Previous results (these Notices 10 (1963}, 512, Abstract 63T-311) are ex­ tended. Theorem. If 17 "' n "' 6000, S(x,n) is a P- sequence for infinitely many integers x. Theorem.

S(x, 17) is a P'-sequence iff x ~ 2184 or x ~ 27830 (mod 30030}. Similar results have been established for other values of n. (Received April 24, 1964.}

64T-324. R. C. GILBERT, California State College at Fullerton, 800 N. State College Boulevard, Fullerton, California. Perturbation of a Sturm- Liouville operator by a finite function.

Theorem. Let q be a real-valued continuous function on [O,oo). Let T be a self-adjoint operator in L 2 [O,oo) defined by L = - D2 + q and the boundary condition u' (0) = 0. (If L is in the limit circle case at infinity, a boundary condition at infinity is also included.} Let p(x) be a real-valued continuous function on [O,oo) which vanishes for x >A. If T is bounded below, IIVR(iT) liz = O(T - 3/ 4) as T---> oo, where V is the operator of multiplication by p, the norm is the Schmidt norm and R(z)

is the resolvent ofT. ~· If 1/;(x,A) is the solution of Lu = Au, u(O) = 1, u'(O) = 0, then = [1/;(x + t, A) + 1/l(x - t, A)] + (1/2) jx+tl/;(s, A)w(x,t,s}ds, where w is constructed 1/;(x, A) cos-v).t (1/2) x-t · from the Riemann function of a hyperbolic equation. Multiplying both sides by e -atg(t}, where g is suitably chosen, integrating from 0 to 1, and using the isometry of L 2 [9,oo) onto L 2(p}, where pis . . JOO f1 .A -at 2 sx+l 2 the spectral functiOn ofT, one obtams 0 11/;(x,,\) 0 cosv,\te g(t}dtl dP(A) = 0 IF(x,s,a)l ds. The result then follows by estimation of both sides of this equation. The situation in which T is defined by more general boundary conditions or L has singularities at both ends of"the interval can be handled similarly. (Received April 2 7, 1964,)

64T-325, MOTOKITI KONDO, Tokyo Metropolitan University, Fukazawa-Cho, Setagaya, Tokyo, Japan, On some applications of P. J. Cohen's method in mathematical analysis.

In using the results of S. Ferferman and A. Lfivy (Amer. Math. Soc. Notices 10 (1963}, No. 6), we can prove the existence of a denumerable complete model .A/of ZF with a property (a) there is a 11"~ set Q(n, a.}, where a. is a variable on NN, such that ( V n)( E a.)Q (n, a.), and that ll Q(n,g'n) is finite for all one valued function g which is a sub-set of N X NN, from that of a denumerable complete model 1 of ZF + V = L. However, (a) can be replaced by a property (b) there is a not finite sub-set E of NN such that S(g'n E E) is finite for all one valued function g which is a sub-set of N X NN and for which n i m implies g'n i g'm. From these results, we can deduce several ones in mathematical analysis. In fact, many problems on infinite sets, derived sets or continuity of functions, proposed in a famous paper of W. Sierpinski on the axiom of choice (Bull. Internat. Acad. Sc. Cracovie (1918}, §1, §4, §6) can be solved. (Received April 29, 1964.}

463 64T-326. G. E. BAXTER, University of Minnesota, Minneapolis, Minnesota 55455, A general ergodic theorem with weighted averages.

Let (X,~.~) be a a-finite measure space, let L(X) denote the class of real-valued functions f(x)

integrable over X, and letT be a positive linear operator from L(X) to L(X) such that 11Tfii 1 ~ llfll 1 for all fin L(X). Let {wk• 1 ~ k} be non-negative numbers whose sum is one, and let {uk, 0 ~ k} be defined by un = w 1un_ 1 + ... + wnuo, uo = 1. Finally, for any pair of functions fin L(X), pin L(X), p ~ 0, set Qn(f,p) = zn(f)/zn(p), zn(g) = :Lg- 1ukTkg. Theorem, The ratios Qn(f,p) have a finite limit almost everywhere on the set where p > 0. In case w 1 = 1, wk = 0, k 1;; 2, the theorem reduces to that of Chacon-Ornstein. The proof is based on a very general maximal ergodic theorem and a weak form of up-crossings inequality. (Received April 27, 1964,)

64T-327. H. S. DAVIS, University of Illinois, Urbana, Illinois. Combinatorial manifolds having local co-dimension one. Preliminary report.

An n-dimensional combinatorial manifold Mn embedded in ann + k dimensional Euclidean space En+k is said to have local co-dimension p (LCD p) iff for every vertex v of Mn,

D(St(v)) - dim(St(v)) ~ p, where D(X) denotes the dimension of the linear subspace generated by X. ~· Mn has LCD 1 iff there is a collection of disjoint closed subcomplexes Kj of Mn containing all the vertices of Mn and whose stars cover Mn such that for all j D(St(Kj)) ~ n + 1, and, for all i, j, D(St(Ki) n St(Kj)) ;;;;; n. Each St(Kj) is then differentiably approximated using a theorem of Cairns (S. S. Cairns, The manifold smoothing problem, Bull. Amer. Math. Soc. 67 (1961), 237-238) and that approximation is made to coincide with St(Kj) outside a closed neighborhood of Kj by well­ known methods, The Union of these is a differentiable approximation to Mn which gives us: Theorem. An n-dimensional combinatorial manifold having local co-dimension one in an n + k dimensional Euclidean space can be differentiably approximated. (Received April 27, 1964.)

64T-328. STEPHEN SCHEINBERG, Princeton University, Princeton, New Jersey 08540. Homeomorphic isomorphic Abelian groups.

Let ni be a sequence of integers tending to infinity. Let G be the complete direct sum of cyclic groups of orders pni, p a prime. Let T be the subgroup of G consisting of elements

x = (xpx2, ... ) such that the largest power of p dividing xi tends to infinity. For any cardinal m let R(m) be the complete direct sum of m copies of the group of p-adic integers. The techniques of

Kaplansky (Infinite Abelian Groups) yield the Theorem. G ~TEll R(Xo). Corollary. G ~ G ED R(m)

if m ~ N0• The compact groups G (f) R(m) are all distinct as topological groups because of the

different number of Z(p00 ) summands in their Pontrjagin duals but are homeomorphic (to the Cantor set), as well as isomorphic. It is well known that for compact connected Abelian groups homeomor­ phism implies topological isomorphism (the first Cech cohomology group with integer coefficients is the Pontrjagin dual). By an elementary homology argument (T. Petrie, oral communication) this can be extended to locally compact connected Abelian groups. (Received April 27, 1964.)

464 64T-329. CHARLES STONE, White Hall, Cornell University, Ithaca, New York. Local limit theorems for asymptotically stable distribution functions.

Let F 1 be a non-lattice distribution function in the domain of attraction of a nondegenerate

stable distribution function F of order a., 0 < a. ~ 2. Let An and Bn be constants such that

F n (Bn (x + An)) ----> F (x) as n ----> oo for - oo < x < oo, where F n denotes the n-fold convolution of F 1 with itself. Theorem l. (Fn(Bn(x + h +An))- Fn(Bn(x +An)))- (F(x +h)- F(x)) = on(1) (h + B~ 1 ),

where on (1) ----> 0 as n ----> oo uniformly in x and h. Theorem 2. If lim sup It I ~ 00Jf 1 (t) I < l, where f 1 denotes the characteristic function of F 1, then for any v > 0 the error term of Theorem 1 can be replaced by on(l) (h + n- v). (Received April 27, 1964.)

64T-330. D. H. TAYLOR, F -40-B Stadium Terrace, Champaign, Illinois. Circumscribing

cubes of hyperellipsoids. Preliminary report,

2 This study considers E = { (X1, X2, .•• , Xn): L~O<:i /ai) = l}, a hyperellipsoid in euclidean n space, and its space of circumscribing cubes. Theorem l. Every circumscribing cube of E has edge length equal to 2((1/n) I:~ a~) l/2• There is a one-to-one correspondence between sets of conjugate

diameters of E and orthonormal frames. If {d1, .... dn} is a set of conjugate diameters, where d1 is

of length 2 o1, then the corresponding orthonormal frame determines a circumscribing box of dimensions 2o1 X 2o2 X ... X 2on· Since I:~ of= L:~af-(Dresden, Arnold, Solid analytical geometry and determinants, Chapter 10, Theorem 5), if {d1, ... ,dn} determines a circumscribing cube of edge 2 length 2o then no2 = L:~af and 2o = 2((1/n)L~ a~ l/ • Theorem 2. If n = 3 and the a 1•s are distinct then the space of circumscribing cubes is homeomorphic to S'. The intersection of the 2-sphere of

radius ((1/3) z::ia~) 1 / 2 and Eisa pair of topological 1-spheres, s•1 and S2_. The space of circum­

scribing cubes is the orbit space Sl/z3 , which is homeomorphic to S', i = l or 2. It has been conjectured and verified in some special cases that the space of circumscribing cubes of the hyper­ ellipsoid in euclidean n space is homeomorphic to SO(n- 2). (Received April 27, 1964.)

64T-331. ERWIN ENGELER, International Business Machines Corporation, Research Laboratory Zurich, Riischlikon Zurich, Switzerland. Ultrastructures in first-order model theory.

Let ~ = ( A,R) , R <:;;;A X A, be a relational structure, B any nonempty set and P a family of partial mappings of B into A such that for any finite subset E <:;;; B the set PE= { p E P: E <:;;; dom p

A p fE one-one} is nonempty. Let D be an ultrafilter on P which contains all PE. Define the relation E S iff {pEP{ }: (px, py) E R} ED. Then = (B,S) is called an ultra- S <:;;;B XB by: (x,y) x,y 5B structure of ~. If, moreover, there exists a set F of Skolem-functions for ~with the property that An Bn for any f E F n A there exists g E B such that for any x 1, ... , xn E B we have

{P E P j' ( )}: f(px 1, ... ,px ) = p(g(x1, ... ,x ))} E D, then 5B is called a normal ultra- x1,. .. ,x0, g x 1,. .. ,xn n n structure of ~. Theorem: Two relational structures ~. 5B are elementarily equivalent iff 5B is

a normal ultrastructure of ~. Ultrastructures may be used to derive the familiar theorems of first-order model theory, in the way ultrapowers are used. The construction of ultra structures generalizes to stronger languages. (Received April 29, 1964.)

465 64T-33Z. G. W. GOES, The University of Kansas, Lawrence, Kansas. Classes of conjugate gap Fourier series.

Let dV be the linear space of complex sequences >. = {>.k} (- oo < k < oo) of Fourier-Stieltjes coefficients (i.e. - i L: (>.kjk)eikt is the Fourier. series of a function of bounded variation). Let furthermore E be any linear space of complex sequences c = {ck} (- oo < k < oo) of Fourier coeffi­ cients with the property that >.c = {>.kck} E E for every c E E if and only if >. E dV. Then thlil follow­ ing is true: For any subsequence {nk} of the sequence of integers {k} (- oo < k < oo) there exist c E E such that {i(sign nk)cnk! ¢ E, i.e. the sequence of coefficients of the conjugate gap series does not belong to E. The well-known spaces L, V, L 00 and C, but also the space of Fourier coefficients whose Fourier series has boundedly convergent Cesaro partial sums, can be taken for E. The theorem follows mainly from a theorem of H. Helson [?roc. Amer. Math. Soc. 6 (1955), Z35-Z4Z]. (Received April Z9, 1964.)

64T-333. K. D. MAGILL, JR., State University of New York at Buffalo, Buffalo, New York 14Z14. Isomorphisms of semigroups of continuous functions.

All spaces here are Hausdorff. A space X is defined to be an S-space if it consists of one point or each point x E X has a basis Bx such that for each G E Bx• there is a continuous function f mapping cl G into X with the properties f(x) i x and f(y) = y for each y E cl G - G. The class of S-spaces includes 0-dimensional spaces and locally Euclidean spaces. Examples of spaces which are not S-spaces are also discussed. For a space X, let S(X) denote the semigroup of continuous functions which map X into itself where f o g is defined by (f o g)(x) = f(g(x)) for f and g in S(X) and x EX. Theorem. Let X andY be S-spaces. A mapping 4> from S(X) onto S(Y) is an isomorphism if and only if there exists a homeomorphism h from X onto Y such that t/J(f) = h Of o h~ for each f E S(X). (Received April Z9, 1964.)

64T-334. J. R. STALLINGS, Princeton University, Fine Hall, P.O. Box 708, Princeton, New Jersey. Homology and central series of groups.

Theorem 1. Let S be a normal subgroup of G with quotient group Q. Then there is an exact sequence: Hz(G) ->Hz(Q) -->S/[G,S] -+H1(G) -+H1(Q) -+0. Theorem Z. Let f:K--> L be a map of complexes inducing an isomorphism on H 1 and an epimorphism on Hz. Then f induces isomorphisms on the quotient groups of 1r1 by terms in their lower central series. Theorem 3. Let f: A --> B be a homomorphism of nilpotent groups inducing an isomorphism on H 1 and an epimorphism on Hz. Then f is an isomorphism. Theorem 4. Let K be a complex and F a field; let Hz(K;F) = 0.

Let {x11 } be elements of 1r1(K) which are linearly independent in H 1(K;F). Then {x11} is the basis of a free subgroup of 1r1(K). These results all follow easily from the exact sequence in Theorem 1, which is the exact sequence for terms of low degree in the spectral sequence Hp(Q;Hq(S)) =>p Hn(G). (Received April Z9, 1964.)

466 64T-335. J. M. WORRELL, JR, Division 5421, Sandia Co:rporation, Sandia Base, Albuquerque,

New Mexico 8 7115. YJ>P~~_:;-~mic.£!~£inuous decompositions of i?!rac<.!!!!E!!-~t:...:'!la ce~.

ToiJ_ological spa co: and ~Pp~emico~tinuous decompositio1~ are defined much as in J. L.

Kelley's _s.'-._::~~~!..!£?....?..!.'?.£".1:> But the stipulation in DieudC>nue'~ definition that ~"" ~, be Hausdorff in the definition of paracompactness is not r<>quired here. A cr.rtain prcp-er~y is defined. The following two results are corollaries to theorems proved concerning this propsrty. Let S d<.mote a parac.ompact topological space and let I denote a topological space obtained from an upper semicontinuous paracompact. Theorem 2. If it is true for S that every closed point set is an inner limiting set and I satisfief!__t.!!_e firs!~xiom~~<2UPE'-_J:>l~.!!:.Y.!_0.~-~ I ~.Pa.r.~c..:,,mP<·'~t, (Rec.,o'ved Aoril 29, 1964.)

64T-336. LUIS BAEZ·DUARTE, Ca.lifornia Institute of Technology, Pasadena, California.

An ergodic_theor~ of Abeli~J:~ FreU.ml.nary report.

This work was motivated by an ergodic theorem with weighted averag~s d•.1e to G. E. Baxter (these Notices ll (1964), Abstract 64T-l92.), Let (X,S,m) be a m<: <>pe.-:e, an::i T a positive linear transformation of L 1 (X,S,m) with norm not greater than one. Assume jtn: n ?: q is a Hequence of non-negative numbers whose sum is one. Define recurrently a sequence {un: n ?: 0} as follows:

u 0 = l, and ''n; f 1un_ 1 + f 2un .. 2 l- ... + fG

lim(I;~untnTnf)(~ountnTnp) -l as t ---> 1 exists and is finite a.e. in X. The proof uses a combina­ tion of the techniques in the papers of E. Hopf (On the ergodic theorem for _positive operators,

Grelle's Journal 205-206 (1960}, 101) and G. C. Rota (On the maximal ergo<~ic._.,~~-eorem for Abel­ limits, Proc. Amer. Math. Soc. 14 (1963}, 722}. (Received April 30, 19'>4.\

64T-337. R. J. HANSON, University of Wisconsin, 1\hclism, Wi'"'"c""i!l .\.formal reduction

theorem for a system of differential equations with a turn~.J?.2int.

Studied are nth order systems of differential equatkn~ wit'!. 8. turning p:)int: (*) f(dy/dz} = A(z,f)y,where the matrix A(z,f) <.omits a formal <>xpansion: A\z.f) ~'[~ 0 Ar(z)E;:, f __, IJ, larg E I < o, with all matrices Ar(z) holomorphic for lzl ;;; z0 !z0 > 0). Without loss of generality we m!l.y assume det(Ao(O)- .X.n = {- l)n ,)-,nand tbo~ A 0 (~) is in Jordan form. Three hypotb~ses 9.re then made: (l) Ao(O) has one elementary divisor. (2) For the •.::haracteristic polynomial

(*"'}det(Ao(z)- .\I)=(- 1)n[An- ... - a 0 _ 1(z) .\- an(z)), an(z) has at least a second order zero while an-l (z) has exactly a first order zero at z = 0. (3) n ?: 3. The system (*) is then formally reduced by a. nonsingular transfon'.:tadcn y =(L~o F r(z}·E~}y* to E(dy*/dz) = [~ 0 (z) + EB(E))y', with a.n P r(z) holomorphic for lz I ;;; z 0• Here A0 (z) is the companion matrix associated with (**}, while the only nonzero entry for B(E) is a nroperly cbosen constant formal series in nonnegative powers of f in the (n, l)et positinn. By a change. of th10 independent variable we may suppose an-l (z) "" z. This generalizes a theorem of K. Okubo,, Proc. Japan A-::ad. 39 (1961), 544-549. (Received April 30, 1964.)

467 64T.-338. J. H. AHLBERG, Research Laboratory, United Aircraft Corporation, East Hartford, Connecticut, E. N. NILSON, Pratt and Whitney Aircraft Corporation, East Hartford, Connecticut and ]. L. WALSH, 474 Widener Library, Cambridge 38, Massachusetts. Orthogonality properties of the spline function.

Given two subdivisions .:land a-:::> .:l of the unit interval, splines u and 'il of degree 2n + 1 on .:land :i, respectively, u"' 0 on .:l, are orthogonal in the sense of the pseudo inner product (U, u) "' Jlu(n+ 1) u(n+l)dx if periodic, also if nonperiodic provided standard end conditions are satisfied. Complete sequences of orthogonal splines for the space cn+l [0,1] are introduced. Under suitable conditions, c 2n[O,l] is a direct sum of linear spaces generated by such splines. If f(z) is analytic in lz I < 1, E c 2 in lz I ;;; 1, f(t) is approximated by a periodic cubic spline gn(t) of interpolation on 1 ltl"' 1, and Gn(z) = (l/27ri). Jltl= 1 gn(t)(t- z)- dt, then as ll.:lnll ---->0, G~k)(z)----> f(k)(z) (k = 0,1) uniformly in lz I ~ 1, and also for k = 2 if f"(t) satisfies a Holder condition and the mesh becomes asymptotically uniform. (Received April 30, 1964.)

64T-339. j. H. AHLBERG, Research Laboratory, United Aircraft Corporation, East Hartford, Connecticut, E. N. NILSON, Pratt and Whitney Aircraft Corporation, East Hartford, Connecticut and ]. L. WALSH, 474 Widener Library, Cambridge, Massachusetts. Extremal, orthogonality and convergence properties of multi-dimensional splines.

Existence and uniqueness of piecewise bicubics are established for a number of important sets of boundary conditions, thus extending the work of DeBoor. Minimal curvature, best approximation, and convergence properties valid for cubic splines are shown to be valid here as well. Extensions to higher odd degrees as well as higher dimensions are established. It is shown that under the pseudo­ norm fvlan(k+l)f;(ax1 )k+l ••• (aXu)k+ll2dx1 , •• dxn the class of functions such that an(k+l)f;(ax1)k+l •.• (axn)k+l exists and is continuous, as a pre-Hilbert space, has an orthogonal bas is of spline functions. The representations arising from this basis converge pointwise and are termwise differentiable. (Received April 30, 1964.)

64T-340. R. W. CARROLL, Rutgers, The State University, New Brunswick, New jersey. Problems in linked operators. II. Preliminary report.

Let E and H be separable Hilbert spaces with L: E ----> E, A: H ----> H closed densely defined linear operators. One considers various realizations S of L 0 I + I 0 A in F = E 0u H (cf. (1): R. Carroll, Math. Annalen 151 (1963), 272-282). Under the hypotheses of (1) it is shown that S = Lo + Ao equals S* = (Li'J + Ai'J)* and applicatiO"Ils are given. Under different kinds of hypotheses (with L and A accretive) the equation Su = f is considered by spectral techniques. It is shown for -1 n -1 example that if the Banach algebra generated by a = A 0 , 1- = L 0 , and the identity satisfies llxll ~ clxl00, then g(z) = z 1z 2;z1 + z 2 bounded on the joint spectrum u(a,£) implies that an operator G E L(F) inverting Scan be found (here z 1 ~a and z 2 ~ £ are spectral variables). Various general examples are studied where this holds. The results of (R. Carroll, Proc. Amer. Math. Soc. 15 (1964), 225-230) are adapted to various S as above and the structures of some abstract Green's kernels are determined in a calculable way. (Received April 30, 1964.)

468 64T-341. j. Z. YAO, Illinois Institute of Technology, Technology Center, Chicago, Illinois. Representation of functions of n variables by superposition of functions of 1 variable and addition.

Let R be the real line and n an integer > 1. Theorem 1. There exist n functions gi: R ----. R, i = 1,. •• , n such that every function F: Rn----. R is representable in the form F(xp ••• ,xn) = f(g 1 (x1) + ••• + gn (xn)), where f: R --> R depends on F, but can be chosen to be a left inverse of all gi, i = 1,.,,, n. ~· Let H be a Hamel basis and H = H 1 U ••• U Hn be a partition of H into n sets of equal cardinality. Hence there exist n 1-1 correspondences gi: R -->Hi' i = 1, ... , n. Given any -1 F: R n ----> R, one defines f: R ---> R as follows: (1) f(x) = gi (x) if x E Hi for some i (2) f(x) =

F(x1, ... ,xn), if x = g 1 (x1) + ... + gn(xn) and f(x) is arbitrary, if otherwise. By the linear independence of H, f is consistently defined. By (1), f is a left inverse of all gi, i = 1, .. ,, n. By (2), always

F(x1, ... ,xn) = f(g1 (x) + ... + gn(xn)). q.e.d. By a similar method one can prove Theorem 2. There exists a fixed function g: R ----> R such that every function F: Rn----> R, symmetric in all the variables xi, i = 1, ... , n, can be represented in the form F(x1,. .. ,xn) = f(g(x1) + ... + g(xn)) where f: R----. R depends on F, but can be chosen to be a left inverse of g. (Received April 30, 1964.

64T-342. c. 0. WILDE, 612 W. California, Urbana, Illinois and K. G. WITZ, 412 Hessel Boulevard, Champaign, Illinois. Invariant means and the Stone-Cech compactification.

Let S be a discrete left amenable semigroup. This paper (a) simplifies and extends a number of known results on amenability of subsemigroups and ideals of S, and (b) studies relationships between invariant means and supporting sets in {3S. Sample theorems: (a) Theorem. If S has two­ sided cancellation, then a subsemigroup A of S is left amenable iff each two right ideals of A intersect,

(b) Theorem. If J1. is a left invariant mean on S, then there exists a smallest closed subset, .5-'tJL), of

{3S such that JL EK(..Y'(JL)) (=the w*-closed convex hull of 5/(JI.)), and SltJL) = ri{A-: A ~Sand

JL E K(A)}. Theorem. In caseS also has cancellation, Y"(JL) is a left ideal of {3S, under the Arens multiplication. An application is that if I is a left ideal of {3S, then K(I) contains an extreme point of the set of left invariant means. (Received April 30, 1964.)

64T-343. A. L. ROSENBERG, Computation Laboratory, 33 Oxford Street, Cambridge 38, Massachusetts. On n-tape finite state acceptors.

~n(~)(.3';;_) is the class of sets of n-tuples of tapes defined by deterministic (non-determin­ istic)(two-way) n-tape automata. Theorem. -.4ri is closed under union, catenation, closure and

reversal. For n > 1, it is not closed under complementation or intersection. Theorem. ~n is closed under complementation. For n > 1 it is not closed under union, intersection, catenation, closure, or reversal. For n = 2, the first half of this theorem is a result of Rabin and Scott (IBM J. Res. Dev. 3,2 (1959), 114-125). Theorem. Yn is a Boolean algebra, For n > 1, it is

closed under reversal but not under catenation or closure. Theorem. If A E ~·the projection of

A on any co-ordinate is a regular set. The same is true if A E ~n (generalizing the result of Rabin

and Scott). If A E ~. the projection of A on any co-ordinate need not be a context-free language. Let 2: be a finite alphabet, and A= {A}. the set containing only the null tape. Let Sn = 2: X An- 1 U A Xl:X An-2 U ... U An--1 X:!:. Ann-regular expression is a regular expression over Sn• Theorem. --{ is exactly the class of n-regular expressions. (Received April 28, 1964.)

469 ERRATA- Volume 11

G. G. BILODEAU. Summability of Laguerre series. Preliminary report. Page 63, Abstract 608-31. Line 8. Add to the hypotheses of the theorem the statement "and let t-3/ 4f{t} E L(O,R) for somP. finite and positive R".

::'FYMOUR SHERMAN. Flnctuation and periodicity. Page 74, Abstract 608-117.

M. K. FORT, JR. and M. C. McCORD. Approximation of maps of inverse limit sp8.<:ef' hy induced maps. Page 99, Abstract 608-146.

Line l. Chang"' "polyhedra" to "connected polyhedra".

470 Mathematics and IDA

Washington is the decision-making center of the free world. In that center, IDA functions as a scientific adviser to the Department of Defense. Our working environment is the gray area of those major national problems where too little is known and too much is at risk to hazard an intuitive decision. IDA provides responsible DOD decision makers with the scientific/ technical input required to eliminate or lessen the areas of uncertainty. Mathematics is applied in two principal fields at IDA: in the study of the technical feasibility of weapons systems and in operations research to find the optimum choice a~nong corrrpeting 'Neapons systems. In a world in which the complexities and exigencies of California Imtit1

471 MATHEMATICIAN

An interesting and challenging opportunity exists at ITT Federal Laboratories, in San Fernando, California, for a PhD Mathematician specializing in probability and decision theory. Immediate projects involve the application of deci­ sion theory techniques to the fields of optical com­ munications, bio-physical models for neural systems,· and space guidance and control systems. In addition to a high degree of analytical ability, candidates should have some background and apti­ tude in the physical sciences. INTERCEPTION This unusual position affords an excellent opportun­ ity to perform advanced state-of-the-art assignments TACTICS in a variety of fields, along with stable growth in a highly professional working environment that is Problem: Given a Combat Air ideally suited to the needs of the creative scientist. Patrol (CAP) of interceptor air­ Please Submit Resume, in Confidence, to craft having known performance, Professional Staff Relations carrying weapons of known types and radar of known range, deter­ mm fed~ral mine the most effective and econom­ I ical combination of these aircraft .1...1. LABORATORIES and their weapons and sensors to A DIVISION OF INTERNATIONAL TELEPHONE AND TELEGRAPH CORPORATION provide the greatest degree of pro­ tection 15151 BLEDSOE STREET, SAN FERNANDO, CALIFORNIA for a fleet. An Equal Opportunity Employer This is an example of the challeng­ ing tasks assigned to the Center for Naval Analyses of The Franklin Institute. CN A analysts devised a graphic technique for examining various combinations of the factors. It CUSHING-MALLOY, INC. incorporated such data as aircraft 1350 N. Main St., P. 0. Box ll87 speeds and altitudes available, num­ Ann Arbor, Michigan ber and locations of CAP stations, and the types of weapons involved. In addition to its initial purpose, the method can be used to help determine the composition and dis­ position of task forces, to compare effectiveness of different intercept tactics, and to find the location from which a given CAP is best able to LITHOPRINTERS defend a task force at sea. Printers of the NOTICES CAREER OPPORTUNITIES with CNA are now available for Operations Analysts, Mathematicians, Physi­ cists, and Engineers. For additional information, write: Director CENTER FOR NAVAL ANALYSES Dept. AM Known for 1710 H St., N. W., Washington, D. C. QUAUTY- ECONOMY- SERVICE Let us quote on your next printing c CENTER FOR NAVAL ANALYSES OF THE FRANKLIN INSTITUTE

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473 New references cmd texts for pure and applied mathematicians .... MECHANISMS FOR THE GENERATION OF PUNf CURVES by I. I. Artobolevskii, U.S.S.R. Academy of Sciences A highly useful general study for mechanical design engineers. Based on material developed in the author's researches and long practical experience, the book deals with the generation ~,f pl;<.ne eurve2 using various kinematic linkages, and comprises principles of kinematic synthesis and theoretical a.dvances in geometric and alge­ braic mathematics. CHAPTER HEADINGS: Basic Theory of the Generation of Curves by Mechanisms with Higher Pairs of the Fourth Class. FuHdauwntals of the Theory of Generation of Curves Using Mechanisms with Lower Pairs of Class V. Mechanical Generation of Straight Lines and Circles. Mechanical Generation of Ellipses, Hyperbolas and Parabolas. The Mechanical Generation of Curves of the Third Degree. Mechanical Generation of Curves of the Fourth Degree. Mechanical Generation of Certain Alge­ braic Curves of Higher Degree, and of Transcendental Curves. 296 pages A Pergamon Press Book $12.00

A GUlDE-BOOK TO !IJ!ATH~MAT!!;S fOR TECHNOLOGISTS AND ENGINEERS by I. N. Bronshtein and K. A. Semc:I>.dyayev, (I,-tternational Series of Monographs on Pure and Applied Mathematics, Vol. 36) Covers all essential mathematics for students and practicing engineers. CONTENTS: Tables and Graphs. Elementary 1.-fathematics: Approximate Computa­ tions, Alg:obra., Geometry, Tiigor10Iilttry. A11alytic a11d Differential Geometry: Ana­ lytic Geometry, Differential Geometry. Foundations of Mathematical Analysis: Introduction to Analysis, Differential Calculus, Integral Calculus, Differential Equations. Supplementary Chapters on Analysis: Complex Numbers and Functions of a Complex Variable, Vector Calculus, the Calculus of Variations, Integral Equa­ tions, Fourier Series. Interpretation of Experimental Results: Foundations of the Theory of Probability and the Theory of Errors, Empirical Formulas and Inter­ polation. Index. 783 p!!ges A. Pergamon Press Book $20.00

THEORY OF APPROXIMATION OF FUNCTIONS INTRODUCTION TO HIGHER ALGEBRA OF A REAL VARIABU. by A. Mostowski, Member of the Polish Academy of Ly A. F. Th~.ta..::.. Sciences, and M, Stark, Mathematical Institute of the Polish Academy of Sciences (International Series of (International Series of Monographs Otl Pure and Ap­ Monographs on Pure a11dApplied Mathematics, Vol. 35) plied Mathematics, Vol. 34) An authoritative and remarkably lucid introduction to A complere developmental a-::cOI.tll< of tile theory of ap­ all phases of higher algebra. proximations of a real variable. The work discusses the origins of the theory in the researches uf Chebyshev and CONTENTS: Introduction. Some Combinatoriai Prob­ 'Veierst1·ass, and thoroughly elaborates recent "xten­ lems. Complex Numbers. Determinants. Vector Spaces sions and refinements. and Linear Equations. Polynomials in One Variable. Rings of Real and Complex Polynomials. Ring of Rational CONlENTS: Weierstrass's Theorem. The Best Approxi­ Polynomials, Algebraic and Transcendental Numbers. mation. Some Compact Classes of Functions and Their Polynorr.ials in Several Variables and Symmetric Func­ Structural Characteristics. Some Properties of Alge­ tions. The Theory of Elimination. Quadratic and Hermi­ braic Polynomials and Transcendental Integral FunC·· tian Forms. Appendix: Some Properties of Matrices and tions of Exponential Type. Direct Theorems of the Quadratic Forms. Index. Constructive Theory of Functions. Converse Theorems, 474 pages A Pergamon Press Book $6.50 Constructive Characteristics of Some Classes of Func­ tions. Further Theorems Connecting the Best Approxi­ mations of Functions with Their Structural Properties. Linear Processes of Approximation of Functions by Poly­ nomials and Some Estll:nates Connected with Them. Some Results from the Theory of Functions and Func­ THE MACMILLAN COMPANY tional Analysis. Index. 631 pages A Pergamon Press Book $14.50 60 Fifth Avenue, New York 10011

474 INDEX TO ADVERTISERS

Name Page Name Page American Mathematical Society •..•••• 476 ITT Federal Laboratories •••••••.••••. 472 Center for Naval Analyses ••.....••.•. 472 The Macmillan Company ••..••.•.•••••• 474 Cushing-Malloy, Inc •...•..•.•••..... 472 Princeton University Press •••••..••... 473 Deutscher Verlag der Wissenschaften .• 473 The University of Wisconsin Press ...•. 473 Institute for Defense Analyses ...... •.. 4 71

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MATHEMATICAL SURVEYS Volume 8 DISCONTINUOUS GROUPS AND AUTOMORPHIC FUNCTIONS by Joseph Lehner

Much has been written on the theory of discontinuous groups and auto­ morphic functions since 1880, when the subject received its first formulation. The purpose of this book is to bring together in one volume both the classical and the modern aspects of the theory, and to present them clearly and in modern language and notation. The emphasis in this book is on the funda­ mental parts of the subject.

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