Notices of the American Mathematical Society

OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 13, NUMBER 4 ISSUE N0.90 JUNE, 1966

cAfotiaiJ OF THE

AMERICAN MATHEMATICAL SOCIETY

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

CONTENTS

MEETINGS Calendar of Meetings • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 420 Program of the June Meeting in British Columbia...... • • . • • • • • • • • 421 Abstracts for the Meeting- Pages 4720478 PRELIMINARY ANNOUNCEMENT OF MEETING. • • • • • • . • • • • • • • • • • • • • 425 DOCTORATES CONFERRED IN 1965...... 431 RECENT POLICIES CONCERNING FEDERAL RESEARCH GRANTS. • • • • • • • • 450 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••.••• , • 424, 451, 452,468 MEMORANDA TO MEMBERS Request for Visiting Foreign Mathematicians • • • • • • • . • • • • • • • • • . • • 456 The Employment Register. • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • 456 PERSONAL ITEMS. • • • • • • • • • • • • • • • • • • • • • • • • • • • . . • • • • • • • • • • • 457 ACTIVITIES OF OTHER ASSOCIATIONS..... • • • • • • • • • . • • • • • • • • • • • • 459 NEW AMS PUBLICATIONS.. • • • • • • • • • • • • • • • • • • . • . • • • • • • • • • • • • • 460 SUPPLEMENTARY PROGRAM- Number 39...... 464 ABSTRACTS OF CONTRIBUTED PAPERS...... 469

ERRATA. • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • . • • • • • • • • • • • . • • • 513 INDEX TO ADVERTISERS • • • • • • • • • • • • • • • • • • • • . . • • • • . • • • • • • • • • 525 RESERVATION FORMS...... 526 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 cJforicei) 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·*

636 August 2-9-September 2, 1966 New Brunswick, New Jersey July 8 13 637 October 29, 1966 Philadelphia, Pennsylvania Sept. Sept. 28 638 November 11' 1966 Tampa, Florida 28 639 November 19, 1966 Los Angeles, California Sept. 28 640 November 25-26, 1966 Mexico City, Mexico Sept. January 23-27, 1967 Houston, Texas August 28-September 1, 1967 Toronto, Ontario, Canada January, 1968 San Francisco, California

*The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for the by title abstracts are .July 1, and September 6, 1966.

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

Copyright©. 1966 by the American Mathematical Society Printed in the United States of America

420 Six Hundred Thirty-Fifth Meeting University ofVictoria and Canadian Services College, Royal Roads June 18,1966

PROGRAM

The six hundred thirty-fifth meeting ROOM RESERVATIONS of the American Mathematical Society will be held at the University of Victoria in A limited amount of residence ac­ Victoria, Br~tish Columbia, on June 18, commodation will be available on campus 1966, in conjunction with meetings of the for the nights of June 16, 17, 18. The daily Pacific Northwest Section of the Mathemati­ adult rates for rooms without meals are cal Association of America and the Society $5.00 for single $4.00 per person in a for Industrial and Applied Mathematics. double room; children under age 10, $1.00. The Society will meet on Saturday, and the Reservations for residence accommodation Association and SIAM will hold their regu­ should be sent as soon as possible to Pro­ lar sessions on Friday, June 17. There will fessor Phoebe Noble, Department of Mathe­ be special sessions of the Association for matics, University of Victoria, Victoria, members of mathematics departments of B.C. Requests should include the number two year colleges, junior colleges, and and names of adults and children, dates community colleges during the morning and and times of arrival and departure, and afternoon of Saturday, June 18. check or money order for pre-payment of By invitation of the Committee to room. (If payment is made in American Select Hour Speakers for Far Western money, rates are $4.75, $3.80, $0.95.} Sectional Meetings, the Society will be ad­ As a major tourist center, Victoria dressed at 11:00 A.M. on Saturday by Pro­ abounds in motels and motor hotels. For a fessor P. Emery Thomas of the University list of motels or further information con­ of California, Berkeley. The title of the cerning motels or ferry timetables write talk by Professor Thomas is "Vectorfields to Phoebe Noble at the given address. on manifolds." It will be given in Elliott Many hotels are available in Victoria 168. although none are located close to the Sessions for contributed papers will Gordon Head Campus of the University. be held on Saturday at 9:00A.M. and 2:00 Below is a partial list of downtown hotels. P.M. Late papers may be added to these (Minimum rates are quoted.) sessions. A supplementary program listing Anyone who wishes to stay in a hotel changes and late papers will be available should write directly to the hotel for re­ at the registration desk. All sessions of servations. These hotels are approximately the meeting will be held in the Lecture 3 miles from the campus. Room annex of the Elliott Building. The Dominion Hotel 759 Yates Street REGISTRATION Single Double Twin $7.00-10.00 $7.00-11.00 $10.00-15.00 The registration desk will be in the lobby of this building, and it will be open The Douglas Hotel from 8:30A.M. to 5:30 P.M. on Friday and 1450 Douglas Street Saturday, June 17, and 18. $6.00-8.00 $8.00-10.00 $10.00-12.00

421 The Empress Hotel 5:00P.M. on Saturday, June 18, at Canadian 721 Government Street Services College, Royal Roads. Tea will $13.00 $19.00 $19.00 be served, and there will be an opportunity The Strathcona Hotel to explore the gardens and grounds of the 919 Douglas Street College. All persons attending the meet­ $8.00 $9.50 $10.50 ings are invited to this reception. Trans­ portation will be provided from the Gordon ENTERTAINMENT Head Campus of the University of Victoria to Royal Roads for those who need it. A beer party will be held in the Invitations should be picked up at the time Student Union Building at 8:30 P.M. on the of registration. evening of June 17. All persons attending the meetings are invited as guests of the Victoria is served by Air Canada departments of mathematics at the Univer­ from Seattle to Vancouver. There is also sity of Victoria and Canadian Services a direct bus service from downtown Van­ College, Royal Roads. A complimentary couver to downtown Victoria via the B.C. luncheon, through the courtesy of the British Government Ferries and a bus service from Columbia Centennial Committee, will be downtown Seattle to Port Angeles, Wash­ held in the cafeteria on the Gordon Head ington, via the Hood Canal Bridge which Campus of the University of Victoria be­ connects directly with the Port Angeles­ tween 12:00 noon and 2:00P.M. on Saturday, Victoria ferry. Members who are able to June 18. There will be an informal recep­ do so are advised to come to Victoria by tion from noon until 12:30 P.M. in the cafe­ private car since only in this way can one teria, followed by the luncheon. Tickets take full advantage of the scenic beauties for this luncheon may be picked up at the of the Victoria area. There is an excellent registration desk. Since it may be necessary car ferry service from Tsawwassen to to limit attendance at this luncheon, those Swartz Bay, twenty miles north ofVictoria; persons who wish to attend are urged to from Anacortes, Washington, to Sidney, obtain their complimentary tickets when British Columbia; and from Port Angeles, they register. A reception will be held at Washington, to downtown Victoria.

PROGRAM OF THE SESSIONS

The time limit for each contributed paper is ten minutes. To maintain the schedule the time limit will be strictly enforced. • --

SATURDAY, 9:00 A.M. General Session, Room 060, Elliott Building 9:00-9:10 ( 1) Totally positive matrices whose entries are permanental functions of a given matrix Dr. D. W.Sasser and Dr.M.L. Slater*, Sandia C-orporation, Albuquerque, New Mexico (635-7) 9:15-9:25 (2) On error bounds for generalized inverses Professor Adi Ben-Israel, University of Illinois at Chicago (635-6) 9:30-9:40 (3) An example of a Julia line which is not a Milloux line Professor W. J. Schneider, Syracuse University (635-14)

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

422 9:45-9:55 (4) Reduction of Fredholm integral equations with Green's function kernels to Volterra equations Mr. S. K. Aalto, Oregon State University (635-15) 10:00-10:10 ( 5) Singular perturbations of a nonlinear system of ordinary differential equations Professor J. W. Macki, University of Alberta (635-8) 10:15-10:25 (6) On the behavior of the solutions of some nonlinear differential equations Professor D. W. Willett and Professor J. S. W. Wong*, University of Alberta (635-4) 10:30-10:40 ( 7) Some convergence theorems connected with the transformation of integrals in measure space Professor Robin Chaney, Western Washington State College (635-19)

SATURDAY, 11:00 A.M. Invited Address, Room 168, Elliott Building Vector fields on manifolds Professor P. Emery Thomas, University of California, Berkeley

SATURDAY, 2:00 P.M. Session on Analysis and Topology, Room 060, Elliott Building 2:00-2:10 (8) Multiplicative invariant means on subalgebras of m(S) Professor E. E. Granirer, University of British Columbia (635-9) 2:15-2:25 (9) Certain isomorphisms between quotients of a group algebra Dr. 0. C. McGehee, University of California, Berkeley (635-10) 2:30-2:40 (10) Ring of generalized continuous functions (g.c.f.) Professor Chien Wenjen, California State College at Long Beach ( 635-12) 2:45-2:55 ( 11) On the spectral sequence of the generalized homotopy suspension Professor Theodor Ganea, University of Washington (635-11) 3:00-3:10 ( 12) On the adjunction space and hereditary properties Professor Byron McCandless, Western Washington State College ( 635-16) 3:15-3:25 ( 13) Proximate homotopy Professor A. L. Yandl, Western Washington State Call ege (635-17)

SATURDAY, 2:00 P.M. Session on Algebra, Logic and Number Theory, Room 061, Elliott Building 2:00-2:10 ( 14) The ultraspace theorem is equivalent to the Boolean prime ideal theorem. Preliminary report Dr. P. S. Schnare, Louisiana State University in New Orleans (635-3) 2:15-2:25 ( 15) A natural transfinite hierarchy of hyperdegrees ProfessorS. K. Thomason, Simon Fraser University (635-1) 2:30-2:40 (16) Addition theorems for sets of integers Professor C. T. Long, Washington State University (635-5)

423 2:45-2:55 (17) Primary decompositions of ideals in gamma-rings Professor W. E. Barnes, Washington State University (635-13) 3:00-3:10 (18) Endomorphism rings of p-primary Abelian groups Professor R. W. Stringall, University of California, Davis (635-Z) 3:15-3:25 ( 19) On generalizations of the Gel 'fand-Mazur representation theory Professor D. F. Sanderson, Western Washington State College (635-18) R. S. Pierce Seattle, Washington Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

MATHEMATICS LEARNING CENTER MATHEMATICS IN BIOLOGY OPENS IN SEATTLE A task committee representing the A regional Mathematics Learning Committee on the Undergraduate Program Center for the Pacific Northwest, to serve in Mathematics (CUPM) and the Committee as a model for the nation, recently opened on Undergraduate Education in the Biologi­ in Seattle's Pacific Science Center. Planned cal Sciences (CUEBS) is engagedinaneffort as a facility for both the general public and to collect significant examples of the appli­ school groups, the Center encompasses a cation of mathematics in the solution of large exhibit area, a comprehensiv,"! mathe­ biological problems. This effort is stimu­ matics reference library, a film library lated by the fact that the quantity and quality and preview room, and a 36-seatclassroom. of mathematics studied and used by biolo­ The Center was made possible by grants gists is increasing at a striking rate. from the Carnegie Corporation of New York The immediate objective of this effort and the Alfred P. Sloan Foundation. Exhibits is to place before the scientific community were designed by Charles· and Ray Eames a collection of examples which ( 1) will and sponsored by the International Business add meaning and motivation to mathematics Machines Corporation. courses attended by biologists, and (2) will Planning for the Mathematics Center be cross-referenced according to the field was carried out by an Advisory Committee of biology involved, for use in biology in­ headed by Carl Allendoerfer, University of atruction. The stimulation of two-way re­ Washington. Other members of the com­ search interchanges between mathemati­ mittee are: George Swift, Washington State cians and biologJ.sts is one of the long range University, Pullman; Theodore S. Chihara, objectives. Seattle University; Nick Massey, Seattle Members of the mathematical com­ School District; Robert Willson, Seattle munity who wish to be informed of these School District; John Trivett, American In­ developments are requested to write to the stitute of Development, Bellevue, Washing­ Committee on the Undergraduate Program ton; Edward Burke, architect, Seattle; Elden in Mathematics, P .0. Box 1024, Berkeley, B. Egbers, Office of State Superintendent California 94 701. The Committee welcomes of Public Instruction, Olympia, Washington; names of other persons who are interested Thomas Hanna, King County Schools, in this information. Seattle; Carol Otterson, Tacoma Public Schools; Ronald Smallman, Pacific Science Center.

424 PRELIMINARY ANNOUNCEMENT OF MEETING

Seventy-First Summer Meeting Rutgers, The State University New Brunswick, New Jersey August 30-September 2, 1966

The American Mathematical Society number of papers contributed to a summer will hold its seventy-first summer meeting meeting has never exceeded the limit. How­ at New Brunswick, New jersey from Tues­ ever, .if necessary, suitable contributed day through Friday, August 3D--September papers will be accepted for the program in 2, 1966. order of their receipt, with random choice All sessions will be held in lecture among the finallotofpapers simultaneously rooms and classrooms of Rutgers, The received. In any event, the deadline for the State University, on the College Avenue receipt of papers to be placed on the pro­ Campus in New Brunswick. Times are gram is july 8, 1966. Abstracts of con­ EASTERN DAYLIGHT TIME throughout. tributed papers should be sent to the Ameri­ There will be no colloquium, follow­ can Mathematical Society, P .0. Box 6248, ing the established pattern for years in Providence, Rhode Island 02904. Abstract which there is an International Congress. blanks can be obtained on request from the The Committee to Select Hour Speak­ same address. ers for Summer and Annual Meetings issued There will be no provision for late invitations to the following persons, whose papers. choice of title is noted: Several organizations will cooperate in holding meetings or council meetings on Professor Nachman Aronszajn, Univer­ the same campus and at approximately the sity of Kansas same time as the Society. These include (Title to be announced) the Mathematical Association of America, Professor Alex Heller, City University The Institute of Mathematical Statistics, of New York the Society for Industrial and Applied Math­ Stable homotopy categories ematics, Pi Mu Epsilon, and Mu Alpha Professor jerome Keisler, University of Theta. The Society for Industrial and Ap­ Wisconsin plied Mathematics will present Professor Recent developments in model theory Eugene P. Wigner of Princeton University Professor Karl Menger, Illinois Institute as the john von Neumann Lecturer. His of Technology address is titled "Statistical theory of Ideas on complex functions spectra." The Mathematical Association Professor V. S. Varadarajan, University of America will present Professor Nathan of California at Los Angeles j. Fine of the Pennsylvania State University Representations of semisimple Lie as the Earle Raymond Hedrick Lecturer. groups His set of lectures is entitled "Basic hy­ There will be sessions for contributed pergeometric series and applications." papers on Tuesday afternoon, August 30, on Wednesday morning, August 31, and on COUNCIL AND BUSINESS MEETINGS Thursday and Friday, September 1 and 2, both morning and afternoon. The number of The .Council of the Society will meet contributed papers which can be accepted at 5:00 P.M. on Thursday, September 1. is limited by action of the Council. The The Business Meeting of the Society limit for the Rutgers meeting is 180. The will be held on Friday, September 2, at

425 11:00 A.M. The preferred procedure is to regis­ ter in advance (see section ADVANCE ADVANCE REGISTRATION REGISTRATION). One then completes the process by picking up a registration packet The advance registration procedures at the registration desk. It is desirable to will be used. On Page 414 of the April issue have one's local address already established and on Page 526 of this issue of these when completing registration, as this infor­ dfoti.cW is a registration form. The form will mation will be recorded at the registration not appear in the August issue. The form desk _fo_r the visual index. This will be easy provides for registration and payment of for those occupying dormitory rooms since registration fees, parking permits, and re­ room registration will be accomplished at servation of dormitory rooms. It allows The Ledge also. for the preparation in advance of badges and an information packet for the registrant EMPLOYMENT REGISTER and his party. Copies of the advance registration The Mathematical Sciences Employ­ form may also be obtained by writing to the ment Register will be in Frelinghuysen Society at the address already given for the Hall, Level A, on George Street east of submission of abstracts. The Ledge. It will be open Tuesday through Thursday, August 30 through September 1, REGISTRATION from 9:00A.M. to 5:00 P.M. on each of the three days. Attention is invited to the an­ The registration desk will be in nouncement of the Employment Register on The Ledge on George Street in the River page 456, in particular to the deadline dates Dormitory area of the College A venue Cam­ for application to the register and to the pus. It will be open on Sunday, August 28, necessity for prompt registration at the from 2:00 P.M. till 8:00 P.M.; on Monday, Employment Register desk by both appli­ August 29, from 8:00 A.M. till 5:00P.M.; cants and employers. OI! Tuesday through Thursday, August 30 through September 1, from 9:00 A.M. EXHIBITS through 5:00 P.M.; and on Friday, Septem­ ber 2, from 9:00 A.M. till 1:00 P.M. Book exhibits and exhibits of educa­ The registration fees will be as tional media will be displayed in Records follows: Hall on Tuesday through Thursday. Records Member $3.00 Hall is across George Street from The Member's family $0.50 Ledge. for the first such registration and no charge BOOK SALE for additional registrations

Student no fee Books published by the Society will Others $6.00 be sold for cash prices somewhat below the usual prices when these same books are The registration fee will be used to pay sold by mail on invoice. part of the costs incurred by the Society and by the host institution in holding the meet­ DORMITORY HOUSING ing. (Note: Partial payment of costs of the host institution has been handled in various Dormitory rooms will be available. ways in the past, including a separate re­ Guests with children will be assigned to the gistration fee, a share in the general regis­ Bishop Campus Halls on George Street, one tration fee, an increment to dormitory rents, block from The Ledge. Others will be a fixed fee for service, rental of meeting housed in the Riverside Halls on George rooms, and itemized service bills. The Street. These dormitories are provided with matter is mentioned here, not because it is elevators. They are not air-conditioned. unusual this year, but because of the im­ Reservations for dormitory rooms pression, wide spread in the membership; should be made in advance, using the form that meetings at universities are "free.")

426 provided. See the section on ADVANCE FOOD SERVICE REGISTRATION. Advance reservations will be acknowledged by mail and a local map The air-conditioned University Com­ will be enclosed. mons will be open for breakfast on Sunday, Although it is possible that rooms August 28, through breakfast, Saturday, will still be available to persons who have September 3. The service will be cafeteria not registered in advance, this is not style, on a cash basis, and the hours for guaranteed. meals follow: Rooms will be available from 10:00 Breakfast 7:00 A.M. - 9:00 A.M. A.M., Saturday, August 27, to 10:00 A.M., Luncheon 11:00 A.M. - 1:30 P.M. Saturday, September 3. Bed linen, towels, Dinner 5:00 P.M. - 7:00 P.M. soap, and room cleaning service are pro­ vided. The rates are $5.00 per day, single, The Rathskeller of the Commons will and $3.50 per day, per person, double. be open on Sunday through Friday from Extra beds will be furnished for older chil­ 7:00A.M. to 7:30P.M. dren (not adults) in double rooms at a charge The Snack Bar in The Ledge will be of $2.00 per day. Extra beds will be avail­ open on Saturday, August 27, from 8:30 able by advance reservation only. A.M. to 11:00 P.M. and, if the service is There are no special provisions or required, the same hours will continue furniture available for babies or young through Friday, September 2. On Saturday, children. Persons who wish to rent cribs September 3, it will be open from 8:30A.M. for babies or small children should com­ through 12:00 noon. municate directly with Bud Rentals, P .0. A list of nearby restaurants will be Box 595, Stelton Road and Washington available at the Registration Headquarters Avenue, Piscataway, New Jersey 08854 in The Ledge. (Phone: (201) 752-8500) or Miller Rental Service, 218 Sandford Street, New Bruns­ MOTELS AND HOTELS wick, New Jersey 08903, (Phone: (201) 247-8888). Cribs rent for $4.00 to $5.00 There are a number of motels and per week. hotels in the area. Some of them are listed Upon arrival on campus all guests below, with coded information which is should check in at The Ledge on George explained at the end of the list. Street to register for dormitory rooms and to secure keys and other housing informa­ LIST OF HOTELS AND MOTELS tion. Student bellhops will be available at The Ledge to guide guests to residence halls THE ARCH MOTEL and parking areas. They will accept tips; U.S.Highway 22, East bound, Somerville however, it is not necessary to use their 42 Rooms (E) Single $9- $14; Double $10-$16 services. Extra per. $2.00. Code: RT-FP-SP-TV-AC Guests are urged to arrive on the 25 minutes drive from The Ledge campus during the normal registration Tel: (201) 722-3555 at The Ledge. Those who must ar­ hours BRUNSWICK INN rive at unusual hours should go to The Exit 9, N.j. Turnpike and Route 18, East Ledge, where a member of the Housing Brunswick Staff will be on duty 24 hours a day to issue 200 Rooms (E) Single $10- $13; Double $14- dormitory keys for advance reservations and $18 rooms for late arrivals. assign Code: RT-AC-TV-SP-CL Guests may pay for their rooms when 15-20 minutes drive from The Ledge campus or when they they arrive on the Tel: (201) 846-1400 check out of their rooms. A member of the University Cashier's Office will be on duty EDISON MOTOR LODGE at the Registration Headquarters in The U.S. Highway 1 and Wilson Ave. (P .O.Edison) Ledge during the hours indicated in the sec­ 50 Rooms (E) Single $7- $9; Double W- $12 tion titled REGISTRATION and on Saturday, Extra Per. $2. Code: FP-TV-AC September 3, from 9:30 A.M. through 12:00 20-25 minutes drive from The Ledge noon. Tel: (201) 985-6000

427 HOLIDAY INN CODE U.S.Highway 1 and Aqueduct Rd., Princeton R T-Restaurant SF-Swimming Pool 104 Rooms (E) Single $10- $12; Double $14- CL-Cocktail Lounge TV-Television $18 FP-Free Parking AC-Air Conditioned Extra Per.$2.Code: RT-CL-FP-SP-TV-AC E -European Plan 30 minutes drive from The Ledge Tel: (609) 452-9100 CAMPING HOST WAYS MOTEL There are no suitable camping sites 247 State Highway 18, East Brunswick located near the Rutgers New Brunswick 60 Rooms (E) Single $9; Double $11 Campuses. Extra Per. $2. Code: RT-SP-TV-AC 15-20 minutes drive from The Ledge ENTERTAINMENT Tel: (201) 257-8700 The tentative plan calls for the HOWARD JOHNSON'S MOTOR LODGE following events: U.S. Highway 1, New Brunswick 119 Rooms (E) Single $10- $12; Double $14- Monday evening- SIAM Beer Party at 8:30 $20 P.M. in the Field House. Tickets on Extra Per. $3. Code RT-CL-FP-SP-TV-AC sale at the registration desk for $1.50. 15-20 minutes drive from The Ledge (Ticket sale will extend beyond the Tel: (201) 249-8000 Monday closing of registration.) Tuesday evening - Chicken Fry at 6:00P.M. LAKE-WOOD COURT MOTEL in the Field House. Tickets on sale at U.S. Highway 130, Deans the registration desk.--Square Dance 14 Rooms (E) Single $7.50-$8.50 at 8:00 P.M. in the Field House. No Double $9.50-$11.50 Extra Per. $2 charge. Code: FP-TV-AC Wednesday afternoon - Presidential Tea at 25 minutes drive from The Ledge 4:30 P.M. Tel: (20 1) 29 7-1036 Wednesday evening- IMS Mixer at9:00 P.M. PARK HOTEL (following the von Neumann Lecture) 123 W. Seventh St., Plainfield at the University Commons. Tickets on 104 Rooms (E) Single $9.50- $15; Double $12- sale at the registration desk. $20 Thursday - Excursion to Island Beach State Extra Per. $3. Code: RT-CL-FP-TV-AC Park. 25-30 minutes drive from The Ledge The park features one of the most Tel: (201) 756-3400 attractive beaches on the New Jersey shore PRINCETON MOTOR LODGE line. Bath houses and refreshments are U.S. Highway 1 and Meadow Rd., Princeton available in the vicinity of the beach area. 28 Rooms (E) Single $8- $10; Double $10-$16 Air-conditioned buses will leave at Extra Per. $2. Code: FP-SP-TV-AC about 10:00 A.M. and, if sufficient demand Tel: (609) 452-2100 exists, a bus can be provided at 1:00 P.M. Tickets for the transportation at $2.25 each ROGER SMITH MOTOR HOTEL will be available at the registration desk 10 Livingston Avenue, New Brunswick all day Monday, August 29, and Tuesday 100 Rooms (E) Single $7- $12.50; .Double morning, August 30. Transportation tickets $10-$18.50 will be available after Tuesday afternoon if Extra Per. $3. Code: RT-CL-FP-TV-AC there are seats remaining in the chartered 5 minutes drive from The Ledge buses. Maps will also be available for those Tel: (201) 247-6000 planning to travel privately. The beach is SILVER CREST MOTOR LODGE about 60 miles from New Brunswick. Travel U.S. Highway 130, North Brunswick time is about one hour and twenty minutes. 50 Rooms (E) Single $7-$9; Double $8-$12 A box lunch may be prearranged at Extra Per. $2 Code: FP-SP-TV-AC-RT the time of registration for the trip at a 25 minutes drive from The Ledge cost of about $1. 75. Tel: (20 1) 297-2100 In the event of inclement weather,

428 the alternative arrangeme:at is for the to New Brunswick is by taxi or bus to the buses to go to a couple of museums in New Pennsylvania Railroad Station in Newark, York City, traveling part way by ferry to and by train from there to New Brunswick. lower New York, In the April issue of these cNoticeiJ The 18-hole Rutgers Golf Course, the announcement of air taxi service from located on the University Heights Campus, Kennedy Airport directly to New Brunswick will be available, The greens fees are appears to be INCORRECT. daily $3,00, and Saturday andSunday, $5.00, Tennis courts are available without PARKING charge on the firstcome, first served basis. The Rutgers Swimming Pool in the The Campus Parking Security and Gymnasium on College A venue will be Transportation Office will issue, gratis, available to the conferees and their families, parking stickers which will permit the use providing planned repairs are completed of campus facilities. The stickers will be prior to this conference, Swimmers must made available to those who indicate on the bring their own suits and towels. Lifeguards advance registration form that they will will be provided. There will be no charge drive to the meeting. for this service. Johnson Park offers excellent facili­ WEATHER ties for picnics: tables, benches, fire­ places, drinking water, sanitary and recrea­ The climate of central New Jersey, tion facilities, and a small wildlife exhibit, as exemplified by New Brunswick, is charac­ It is located on the opposite side of the terized by great seasonal variety. The Raritan River from the George Street Dor­ general climatic type is often referred to mitories. as modified continental. For example, great extremes in weather conditions are some­ TRAVEL what modified by the proximity of the region to the Atlantic Ocean. The moder­ Labor Day weekend follows this ation of climate is at best only partial meeting. Persons who need return trav­ mainly because the prevailing winds are el reservations are advised to make from westerly quadrants- -northwest in them in advance even though there will winter; southwest in summer. be a travel desk at The Ledge. In the summer the main system New Brunswick, located in central effecting the weather of central New Jersey New Jersey, is on the main line of the is the "Bermuda High." This quasi-perman­ Pennsylvania Railroad, 30 miles southwest ent anticyclonic-pressure system is gen­ of New York City, aad 60 miles northeast erally located over an area extending from of Philadelphia. The daily train service is the Sargasso Sea region of the Atlantic very good. Ocean to the eastern Gulf of Mexico. The Express bus service {one hour) is clockwise movement of winds around this available from New York City at the Port system serves to pump warm moist air Authority Bus Terminal, 40th Street and from the Gulf of Mexico over much of the 8th A venue. The following long distance bus eastern third of the United States. This lines also stop in New Brunswick: Continen­ results in rather frequent hot muggy sum­ tal Trailways, Eastern Greyhound Lines, mer days in central New Jersey. Instability Safeway Trails, and Suburban Transit. in this south to north moving flow of air By automobile, New Brunswick is may yield thunder- showers that bring relief, easily accessible from Exit 9 of the New occasionally only temporarily, to the region. Jersey Turnpike and U.S. Highway1.0ther Under such conditions temperatures of nearly main highways are the Garden State 1 00 F 0 or more have been reached. (See Parkway, and U, S. Highway 22, 130, 202, Shulman, M.D., 1965. The Climate of New 206, 287 and N. J, Highway 18. Brunswick, New Jersey, N. J, Agricultural The nearest airport in Newark, N.J. Experiment Station, College of Agriculture, {25 miles) provides facilities for air travel Rutgers, The State University, New Bruns­ via most of the major airlines. The most wick, N. J,, pp. 1-14). economical way to travel from the airport Facts on the climate of New Bruns-

429 wick for the week of August 26-.-Septem­ vide out-patient clinic service, Monday ber 2: through Friday, from 9:00A.M. to 5:00P.M. Average maximum temperature: 82°F or After these hours and on Saturday and 28°C. Sunday, for emergency illnesses and ac­ Average minimum temperature: 61 °F or cidents, one may phone CHarter 7-1766, 16°C. Extension 6211 (Campus Parking Security and Transportation) for assistance. There is one chance in 10 thatthe tempera­ ture during this week will exceed 95°F or ADDRESS FOR MAIL AND TELKGRAMS 35°C. There is one chance in 10 that the tempera­ Individuals may be addressed at ture during this week will fall below 44°F or AMS - Summer Meeting -The Ledge, 7°C. Rutgers, The State University, New Bruns­ The average precipitation for this period wick, New jersey 08903. is about 1 inch or 25.4 mm normally oc­ curring from thunderstorm activity. Sunrise during this period is at 6:25 local COMMITTEE time (Eastern Daylight) and sunset is at 19:35. The meeting arrangements have been made through the good offices of Mr. BOOKSTORE Robert Collett, Director of the Bureau of Special Services of Rutgers, The State The University Bookstore (air-con­ University. ditioned) is adjacent to the University Com­ The Committee consists of the fol­ mons and will be open Monday through lowing persons: Friday from 9:00 A.M. to 5:00 P.M. In ad­ H. L. Alder Katherine E. Hazard dition to textbooks, it sells University joshua Barlaz L. F. McAuley, Chairman souvenirs and articles for personal use. john Bender Mrs. Barbara L. Osofsky Saul Blumenthal Everett Pitcher LIBRARY F. E. Clark M. S. Robertson R. M. Cohn G. L. Walker The University Library (air-condi­ j. H. Griesmer K. G. Wolfson tioned) is located on College Avenue, and will be op·~n on Monday through Friday Everett Pitcher from 9:00 A.M. to 5:00 P.M. Associate Secretary Bethlehem, Pennsylvania MEDICAL SERVICES

The Student Health Center located on Bishop Place, Bishop Campus, will pro-

Six Hundred Fortieth Meeting Mexico City, Mexico November 25-26, 1966

There will be a joint meeting of the Torre de Ciencias at the University of Sociedad Mathematica Mexicana and the Mexico. American Mathematical Society on. Friday Everett Pitcher and Saturday, Nbvember 25-26, 1966, in Associate Secretary Mexico City. The sessions will be at the Bethlehem, Pennsylvania

430 DOCTORATES CONFERRED IN 1965

The following are among those who received doctorates in the mathematical sciences and related subjects from universities in the United States and Canada during 1965. After each university is given the number of doctorates listed; note that for some universities this includes both the depart­ ment of mathematics and related departments. Each entry includes the dissertation title and nonmathe­ matical minor if applicable. 95 universities are listed with a total of 716 names.

ADELPHI UNIVERSITY (3) Stevens, Robert Ray Probabilistic metrics and probability meas­ Cannonito, Frank Benjamin ures on metrics Hierarchies of computable groups and the Swartz, Charles Wayne word problem An operational calculus based on the Laplace Eisen, Dennis transform Difference schemes with singular coefficients Munick, Herman AUBURN UNIVERSITY (4) On nonlinear optimal control problems with Anders, Edward Benoit control appearing linearly An extension of Romberg integration pro­ UNIVERSITY OF ALABAMA (1) cedures to N-variables Dixon, Edward Dale Gray, William Jesse Metric polynomials which are higher com­ Topological transformation groups with a mutators fixed end-point Fulp, Ronald Owen UNIVERSITY OF ALBERTA, Edmonton (4) Characters of commutative semigroups Plemmons, Robert James Abbott, Harvey Leslie Ideals in direct products of semigroups Some problems in combinatorial analysis Chorneyko, Ihor Zinovie BOSTON UNIVERSITY (4) Unbiased estimation, simplicity and optimal­ Chand, Donald R. ity in certain binomial sampling plans Approximation of curves by piece-wise con­ Muldowney, James Stephen tinuous functions Analyticity and asymptotics of Jost functions Chandy, A. John Stenger, Frank Rings generated by the inner automorphisms Error bounds for solutions of differential of non-Abelian groups equations Koval, Daniel UNIVERSITY OF ARIZONA (8) A perturbation solution of linear elliptic equations Bronder, Joseph Bertram Peterson, Richard M. Minor: Geophysics Some minimum time transfer problems On the number of critical configurations of charges on an m-torus. UNIVERSITY OF BRITISH COLUMBIA (5) Conway, William Edward Booth, Raymond Sidney The two-dimensional jet under gravity from Sequentially localizable functional . an aperture in the lower of two horizontal Chew, Kim Lin planes which bound a liquid Extensions of rings and modules Haskell. Charles Thomson Fraga, Robert J. Multi-restricted and rowed partitions A non-divisorial variety Klassen, Vuryl Jess Robert, J. Pierre A free boundary problem for the flow of a On some non-normed linear spaces liquid under gravity through a partially Willmott, Richard Curtis obstructed orifice Hausdorff measures in topological spaces Leonard, Jon Norman A model theoretic characterization of the BROWN UNIVERSITY (16) consistent sentences of the theory of total ordering with K unary predicates Howard, Alan Nymann, James Eugene On the homotopy groups of an affine alge­ Ideal structure of relative quadratic fields braic hypersurface arising from fixed points of the Hilbert O'Neill, Bernard Vincent, Jr. modular group Parts and one-dimensional analytic spaces

431 Ramanujam, Srinivasa Flaherty, Francis Joseph An application of Morse theory to certain The cut locus of a submanifold symmetric spaces Freeman, Michael Benton Richardson, Henry Russell III Some conditions for uniform approximation Regression analysis when the least-squares on a manifold estimate is not asymptotically efficient Gardner, Robert Brown Differential geometric methods in partial Department of Applied Mathematics differential equations Anderson, Charles Alfred Grosser, Siegfried Karl On the meaning of concentrated load solutions Compactness conditions in structure theory in the plastic theory of structures topological groups Carlson, Donald Earle Gupta, Haragauri Narayan Second and higher orders effects in a class Contributions to the axiomatic foundations of of problems in plane finite elasticity geometry Carroll, Michael Mary Howard, Charles Malone Electro-magneto-optical effects An approach to algebraic logic Fine, Arthur Davis Knoebel, Robert Arthur Second order effects in the propagation of Functionally complete algebras elastic waves Kraines, David Paul Gordon, William John Higher order cohomology operations Interactions in closed queueing systems Kraines, Vivian Yoh Leitman, Marshall Justin Topology of quaternionic manifolds Some results on variational principles for Lopez-Escobar, Edgar George Kenneth linear initial-value and initial-history Infinitely long formulas with countable quan­ problems tifier degrees Nadkarni, Mahendra Ganpatrao Maurer, Ward Douglas Vector valued weakly stationary stochastic On minimal decompositions of group mach­ processes and factorization of matrix­ ines valued functions and strongly mixing and Meissner, Loren Phillip uniformly ergodic Gaussian processes Nonlinear resonance for Duffing's differen­ Perella, Carlos tial equation Periodic solutions of ordinary differential Peterson, David Carl equations with and without time lag Under sums Shirely, Leland Kimble Pohlmann, Henry John On the stability of free convection boundary On the zeta-function of an Abelian variety layers of complex multiplication type Salathe, Eric Paul Riddell, Ronald Cameron Two dimensional magnetohydrodynamics Spectral concentration for self adjoint opera­ Taylor, Albion Dennis tors Water waves at the shoreline--a singular Smoke, William Henry boundary value problem Differential operators on homogeneous Valentin, Richard Alan spaces Applications of functional analysis to optimal Zame, Alan numerical approximation for analytic func­ On the distribution of the fractional parts of tions certain sequences UNIVERSITY OF CALIFORNIA, BERKELEY (34) Group in Logic and Methodology of Science Alves, Jose Ubyrajara Barnes, Robert Featherstone, Jr. Detection of a map by stable operations The classification of the closed-open and the Bradford, Robert Earl recursive sets of number-theoretic func­ Cardinal addition and the axiom of choice tions Brown, Benson Samuel Chuaqui, Rolando Basim Some results in mod C homotopy theory A definition of probability based on equal Charnow, Allen Kenneth likelihood Parameter specialization in a free module Hanna, Joseph Ford over a polynomial ring The methodology of the testing of learning Crandall, Michael Grain models, with applications to a new stimulus Two families of periodic solutions of the discrimination model of two-choice be­ plane four-body problem havior De Pillis, John Linear operators preserving positive semi­ Department of Statistics definite n· X n matrices Afifi, Abdelmonem Abdelaziz Douglas, Roy Rene Missing observations in multivariate sta­ Homotopy-commutativity in H-spaces tistical analysis

432 Deo, Chandrakant Mahadeorao in the theory of transcendental numbers Prediction theory of non stationary processes Warner, Kenneth Knapp Doksum, Kjell Andreas Some meromorphic differential operators Asymptotically minimax distribution-free UNIVERSITY OF CALIFORNIA, RIVERSIDE (3) procedures Good, Phillip Frey Boswell, Marilyn Tuley Limiting conditional behaviour of some tran­ Estimating and testing trend in a stochastic sient chains process of Poisson type Lawton, William Harvey Neal, Scotty Ray Concentration of random quotients Measures in lattices Marcus, Allan Harvey Lick, Dale Wesley A stochastic model of the formation and A Quasi-linear singular Cauchy problem survival of lunar craters Strauch, Ralph Eugene UNIVERSITY OF CALIFORNIA, SANTA BAR­ Negative dynamic programming BARA (3) UNIVERSITY OF CALIFORNIA, LOS ANGELES Botta, Earle Peter (21) Linear transformations on matrices: The Balsam, Richard Em invariance of a class of general matrix Normal differential operators functions Bernstein, Allen Richard Gordon, William Robert Invariant subspaces for linear operators Inequalities for generalized matrix functions Caradus, Selwyn Ross Williamson, Stanley Gill Contributions to the theory of operators with Some applications of tensor algebraic tech­ meromorphic resolvents niques to classical problems in matrix Dyer, James theory Automorphisms of groups Dyer, Michael Neal CALIFORNIA INSTITUTE OF TECHNOLOGY (8) Two term conditions in 1r exact couples Baez-Duarte, Luis Gold, Edward Meyer Pointwise Abelian ergodic theorems Models of goal-seeking and learning Baumert, Leonard Daniel Gold, Robert Extreme copositive quadratic forms On the structure of linear binary sequences Chase, Phillip John Halpern, Benjamin Rigler Sublattices of partition lattices Fixed point theorems for outward maps Derzko, Nicholas Anthony Hilliker, David Lee Minor: Physics On analytic functions which have algebraic Wave limits and generalized Hilbert trans­ val11es at a convergent sequence of points forms H urw it2;, Alexander Holbrook, John Arthur Rankin Matrices of zeros and ones all of whose The Egoroff property and related properties pairs of rows are isomorphic in the theory of Riesz spaces Kendig, Keith Milo Keller, Gordon Ernest Algebraic geometry over Dedekind domains Groups with only the identity fixing three Lambert, William Miles letters Effectiveness, elementary definability and Wilkinson, John Fergus prime polynomial ideals Minor: Economics Loeb, Henry Leon A coloring problem related to Konig's theo­ Contributions to rational approximation rem Maltz, Robert Wong, James Sai Wing The nullity of the curvature operator Generalizations to the converse of contrac­ Rabkin, Eugene Elliot tion mapping principle Characterizations of star shaped sets Robertson, Lewis Chandlee CARNEGIE INSTITUTE OF TECHNOLOGY (4) Homogeneous dual pairs of locally compact Abelian groups Chew, Milton Carl Russo, Bernard Some optimal search procedures Linear mappings and unitary structure of Dempster, Michael Alan Howarth operator algebras On stochastic programming Senge, Hans Georg Ryder, Gerald Hastings On closed sets of algebraic numbers Boundary value problems for a class of Smith, Stoddart nonlinear differential equations Classification of Riemanian spaces Tal, Abraham Wadleigh, Francis Rawle Eigenfunctions for a class of nonlinear dif­ Algebraic independence of entire functions ferential equations

433 CASE INSTITUTE OF TECHNOLOGY (14) Holman, Herbert Harlan Minor: Physics Bartlett, Noel Sloane On aspects of harmonic analysis of non­ Ranking and selection procedures for con­ stationary stGchastic processes tinuous and certain discrete populations Kolmer, Sister Mary Kenneth Bittner, Edward Clarence Minor: Philosophy Numerical analysis of Laplace's equation Generalization of the symplectic group with nonlinear boundary conditions Savary, Lot;.is Michael Davis, William Jay Minor: Psychology Solution representations for linear initial A theory of spherical probability distributions value problems Fabian, Robert John UNIVERSITY OF CHICAGO (23) Hierarchies of general recursive functions Allan, Nelo da Silva and ordinal recursion Maximality of some arithmetic groups Hopper, Allen Taylor Fabes, Eugene Barry Expansion theorems for solutions of two Parabolic partial differential equations and generalizations of the heat equation singular integrals Johnston, Robert Laurence Gedwiser, Samuel Mayer Block generalizations of some Gerschgorin­ Similarity of matrices with elements in com­ type theorems m':.tative rings with unit Joshi, Madhukar Vishvanath Gray, Brayton I. Testing statistical hypotheses for the loca­ Operations and a problem of Heller tion and scale parameters of the chi­ Handel, David distribution ( 1 D .F.) Vector bundles over real projective spaces. McGuinness, David Joseph Kirby, Robion Cromwell Differential equations with second order Smoothing locally flat imbeddings turning points Krishnarao, Gade Venkata Price, Harvey Simon Unstable homotopy of O(n) Monotone and oscillation matrices applied to Kuo, Yung- Yung Lu finite difference approximations Simple lie algebras of rank three Division of Organizational Sciences Opera­ Larson, Richard G. tions Research Group Hopf algebras and group algebras Greaney, William A. Levin, Gerson Louis Interaction of goals in economic organiza­ Homology of local rings tions MacDonald, John Lauchlin Guha, Dilip Group derived functors and relative repre­ An operational theory of oligopoly , sentability Hain, Robert C. McNamara, James Nevin A computable theory of strategies in differ­ Bundle theory for quasi-topological spaces entiated oligopolies Sadosky, Cora Susanna Marshall, Wayne S. Parabolic singular integral operators A simulation model of human behavior in Schafer, James Arthur communication networks experiments The homology ring of an Abelian group Schrady, David A. Slater, Michael Individual choice behavior and Markov re­ Weakly prime alternative rings newal processes Small, Lance W. Studies on non-commutative Noetherian rings Spitznagel, Edward Lawrence THE CATHOLIC UNIVERSITY OF AMERICA (6) On the Hall subgroups of certain families of Andersen, Gerald Ralph finite groups Minor: Physics Tsang, Hwa On some almost sure convergence theorems Gauss's Lemma in probability theory Vagi, Stephen Cada, Brother Lawrence John On multipliers and singular integrals in Minor: Philosophy LP spaces of vector valued functions The structure and classification of nilpotent Wheeden, Richard Lee algebras On trigonometric series associated with Gilroy, James Francis hypersingular integrals Minor: Biology Correction factor:;; for estimates of para­ Department of Statistics meters obtained from grouped sample data Abrahamson, Innis Gillian in the case of a probability distribution on On the stochastic comparison of tests of a circle hypotheses

434 Scott, Alastair john Beckman, Frank Samuel Allocation of effort in the design of selection The behavior of weak solutions of the Dirich­ procedures let problem for strongly elliptic equations Trumbo, Bruce Edward in domains with some lower dimensional Sufficient conditions for the weak conver­ bounding surfaces gence of conditional probability distribu­ Goldberg, Estelle Maxine tions in a metric space Relations in categories Kocan, Daniel, Jr. UNIVERSITY OF CINCINNATI (2) Spectral manifolds for a class of operators Green, Marvin D. McAlpin, john Harris Some measures on groups Infinite dimensional manifolds and Morse Hicks, Troy Lee theory Some results concerning the theory of locally Tierney, Myles convex topological linear spaces On the classifying spaces for K-theory mod p Smith, Harry Foster, Jr. UNIVERSITY OF COLORADO (9) Numerical development of harmonic series Birnbaum, Sidney for the coordinates of the moon Spectral theory for a pair of closed linear Strauss, Phyllis Rubin transformations acting between two Banach Topologies and Borel structures associated spaces with Banach algebras of functions Burling, james Perkins Department of Statistics On coloring problems of families of proto­ Bramblett, jerry Earl types Some approximations to optimal stopping Davis, Henry Werner procedures periodic functions on groups and Almost Engelberg, Ora (Mrs. jerome Percus) mean values their Applications of combinatorial methods to Eldridge, Klaus E. random walk and ballot problems chain condition rings with cyclic Descending Pickands, james III groups quasi-regular Maxima of stationary Gaussian processes Keiser, Victor Hugo, Jr. Starr, Norman projective and affine planes with Finite On the sequential estimation of the mean of collineation groups primitive a normal population with unknown variance Morez, Nicholas Sommerville Generators for the semi-topogeneous orders of Csarzar Shotwell, David Alan CORNELL UNIVERSITY (12) Boundary problems for the differential equa­ Benson, Clark Tabor tion Lu = 2uu and associated eigenfunction Generalized quadrangles and (B,N) pairs expansions Driscoll, Graham Cameron, Jr. Vaidya, Arunkumar Madhusudan Contributions to metarecursive theory Contributions to the theory of squarefree Hay, Louise Schmir numbers Topics in recursion theory, I. The co-simple Department of Applied Mathematics isols, II, Creative sets Poole, Michael George Herzog, Marcel Laplace transforms of rapidly increasing On finite groups with a subgroup containing functions with applications to initial­ the normalizer of each of its nonunit boundary value problems elements Kaufman, Sol COLORADO STATE UNIVERSITY (4) Asymptotic efficiency of the maximum likeli­ Barr, Donald Roy hood estimator Sequential decision rules for a multiple Manaster, Alfred Berry choice theory Higher order indecomposable isols Owen, William Burrage McConnell, Alan General confidence intervals Studies on Amitsur cohomology Rounding, Robert Campbell Mineka, john Crawford A two-step sample size problem Existence and uniqueness of positive solu­ Urquhart, Norman Scott tions to the Wiener-Hop£ equation with On the permutation distribution of a multi­ positive kernel variate test statistic Pressman, Irwin Samuel A comparison of two functors in homological COLUMBIA UNIVERSITY (12) algebra Appelgate, Harry Wesley Schroer, David Edward Acyclic models and resolvent functors The Church-Rosser theorem

435 Schumitzky, Alan Howell, Leonard R. Jr. Wiman- Valiron theory for entire functions A tameness condition for 3-cells of several complex variables Rubin, Leonard Whyburn, Kenneth Gordon The product of an unusual decomposition Operational calculus and related topics space with E 1 yields E4 Russo, John Peter DARTMOUTH COLLEGE (1) Monotone and E-Schauder bases of subspaces Curtis, Ellwood Count Wishart, Edward F. Potential theory for supermartingales and Higher derivations on w-adic field applications of Choquet's theorem to Mar­ GEORGE PEABODY COLLEGE FOR TEACHERS tin representation theory (3) DUKE UNIVERSITY (9) Flatt, James LaVerne Al-Salam, Nadhla Abdul-Halim Minor: Fundamentals of Education Minor: Physics The relative effectiveness of three types of A class of hypergeometric polynomials follow-up when teaching remedial mathe­ Church, Charles Alexander, Jr. matics by television Minor: Philosophy Kerce, Robert H. Some permutations with restricted positions Minor: Psychology Fray, Robert Dutton A study of the interaction between class size Minor: Philosophy and the teaching method in Freshman Congruence properties and generating func­ college mathematics instruction tions Lindsay, Charles M. Long, Andrew F., Jr. Minor: Psychology Minor: Philosophy An experimental investigation of two methods Factorable polynomials in several indeter­ used in the in-service education of teachers minates over a finite field of arithmetic Menon, Korandatti Venugopalan GEORGE WASHINGTON UNIVERSITY (2) Minor: Philosophy Inequalities for symmetric functions Department of Statistics Roselle, David Paul Khairat, Mahmoud Khairat Ahmed Minor: Philosophy Some results for random elements taking Generalized Eulerian functions and a problem values in a linear space in partitions Shumay, Robert Herman Vaughan, Jerry Eugene An information theoretic approach to the Minor: Philosophy problem of discriminating between two A modification of Morita's characterization stochastic processes of dimension Wells, Charles GEORGETOWN UNIVERSITY (1) Minor: Philosophy Permutation polynomials over finite fields Maloney, John P. Winslow, Leon Edward On certain nonlinear operators with applica­ Minor: Physics tions to ordinary and partial differential Jordan structures for bounded linear trans­ equations formations in Hilbert space UNIVERSITY OF FLORIDA (3) UNIVERSITY OF GEORGIA (6) Bevis, Jean Harwell Bryant, John Logan Minor: Physics Taming polyhedra in the trivial range Quantifiers and dimension equivalence rela­ Durham, Harvey Ralph tions on orthomodular lattices Taming arcs and Cantor sets in Euclidean Catlin, Donald Edward spaces Minor: Engineering Mechanics Gentry, Karl Ray Implicativity and irreducibility in orthomod­ Inverse limit spaces; periodicity and trans­ ular lattices formation groups Lin, Shwu- Yeng Tzeng Hodge, James Edgar Imbeddability of a topological space into a Certain subsets of locally connected plane generalized arc continua Pittman, Chatty Roger FLORIDA STATE UNIVERSITY (5) Paracompactness and ordered spaces Fleming, Richard Joseph Rhee, Choon- Jai Weak-sequential closure of subspaces of Homotopy functors determined by set valued conjugate spaces maps

436 GEORGIA INSTITUTE OF TECHNOLOGY (2) Contributions to the nonresponse problem in sample surveys Cain, George Lee, Jr. Switzer, Paul Compact and related mappings Sampling of random set processes in the Stiles, Wilbur Janes plane On closest-point maps in Banach spaces UNIVERSITY OF IDAHO (1) HARVARD UNIVERSITY (20) Wang, Ya Yen (Mrs. C. W.) Bolker, Ethan David Minor: Chemistry Functions resembling quotients of measures On the variational problems of curves under Halpern, Herbert P au! different groups of transformations Decomposition of centrally reducible and of reducible functionals UNIVERSITY OF ILLINOIS (22) Jones, William Burton Bavel, Zamir Types of completeness in locally convex Minor: Electrical Engineering spaces On the structure and automorphism s of finite Kleiman, Steven Lawrence automata Toward a numerical theory of ampleness Brosamler, Gunnar A. Labute, John Paul Minor: Physics Classification of Demushkin groups Potential theoretic analysis of a certain in­ Landweber, Peter Steven tegral equation Kiinneth formulas for bordism theories Davis, Harvey Samuel Lipman, Joseph Minor: Philosophy Quasi-ordinary singularities of embedded Manifolds with local codimension one surfaces Donaldson, James Ashley Schultz, Martin Harvey Integral representations of the extended airy The convergence of difference approxima­ integral type for the modified Bessel func­ tions to Cauchy problems in the space of tion tempered distributions Inselberg, Alfred Minor: Physics Committee on Applied Mathematics On classification and superposition principles Comstock, Craig for nonlinear operators A three fluid model ionosphere Kelleher, James Joseph Lindzen, Richard Minor: Electrical Engineering Radiative and photochemical processes in Rings of meromorphic functions strato- and mesospheric dynamics LaBach, William Anderson Division of Engineering and Applied Physics Minor: Philosophy Berger, Robert Manifolds with transverse fields in Euclidean The undecidability of the domino problem space Cole, Stephen Nisson Loustaunau, Joaquin Octavio Real-time computation by iterative arrays Minor: Astronomy of finite-state machines On the state diagram of a linear operator and Friedman, Joyce its adjoint in locally convex sets A new decision procedure in logic with a Low, Marc Edmon computer realization Minor: Philosophy Hooper, Philip Kuehne Real zeros of the Dedekind zeta function of The undecidability of the Turing machine an imaginary quadratic field immortality problem Mayoh, Brian Henry Minor: Philosophy ~cy_g_f_fu;_ience~artment Solvable and unsolvable problems in Merzbach, Uta Caecilia the theory of computable numbers Quantity of structure: Development of modern Nelson, Wayne Bryce algebraic concepts from Leibniz to Dede­ Minor: Physics kind Minimax solution of statistical decision prob­ Department of Statistics lems by iteration Carlson, Roger Allan P aliouras, John Demetrios A quadratic model for multivariate predic­ Topological aspects of stability of set tra­ tion jectories under a continuous motion Davis, Miles Pu, Arthur Ta-Shiang Comparison of sequential experiments for Minor: Electrical Engineering estimating the median lethal dose Generalized decomposition of incomplete Owen, Joel finite automata An investor in the stock market Sanders, William Mack Rao, Poduri Sree Rango Sambasiva Perpendicularity in Hilbert planes

437 Shao, Tzu-Sien Mirbagheri, Ahmad Minor: Aeronautical Engineering Finiteness in radical algebras On the Brachistochronic flight of a rocket Nagaraj, Muppinaiya in the interplanetary space Minor: Physics Stone, Alexander Paul Special cases of Einstein-Maxwell fields in Minor: Electrical Engineering presence of matter and pressure The existence of conservation laws on analy­ Phillips, Joan Ruth tic manifolds Inclusion-transfer theorem of class field Taylor, Bert Alan theory Minor: Astronomy Salehi, Habib Duality and entire functions The prediction theory of multivariate sto­ Taylor, Doug1as Henley chastic processes with continuous time Minor: Philosophy Sicks, Jon Lewis Circumscribing circles of hyperellipsoids Minimal sequences of modifications in Euclidean n- space UNIVERSITY OF IOWA (8) Thornton, Melvin Chandler Minor: Philosophy Christiansen, Robert Alvin Singularly fibered manifolds Uniformly recurrent transformation groups Totten, Donald Fred Gregorac, Robert John Minor: Electrical Engineering On families of normal subgroups of certain Vitali-Denjoy theorem in the product space generalized free products of two abstract measure spaces Mestecky, Frank Joseph Wong, Bing Kuen Varieties over two-dimensional local rings Penetration of set-trajectories into pre­ Otto, Albert Dean assigned regions in a discrete dynamical Certain sets of p-power automorphisms system Wheaton, Burdette Carl Yntema, Mary Katherine On groups having the same group characters Minor: Philosophy Zipoy, Rogert Thomas A generalized regular notation used to define Baer-like invariants of algebras some families of context-free languages Department of Statistics and to obtain their closure properties and Bulgren, William Gerald containment relations Computer sampling study of type I and type II ILLINOIS INSTITUTE OF TECHNOLOGY (2) error rates for certain multiple compari­ son procedures Senechalle, Marjorie Lee Hewett, John Earl Approximate functional equations and proba­ Some joint distributions of order statistics bilistic inner product spaces of combined samples Krajkiewicz, Paul M. IOWA STATE UNIVERSITY (14) Some theorems on polygenic functions Anderson, Gary Lytle UNIVERSITY OF INDIANA (li) Minor: Physics Ahluwalia, Daljit Singh Baffling of fluid sloshing in cylindrical tanks Minor: Physics Boessenroth, Theodore Ulrich integral trans­ Plastic flow and fracture of a thick plate On convergence-preserving under uniform stress formations Di Franco, Roland B. Jensen, Jens Allen ordinary dif­ On extending a norm residue symbol to in­ Linear operator methods for trunca­ separable extensions ferential equations which minimize Hou, Roger Hsiang-Dah tion and propagated errors An axiomatic treatment of cohomology of Kemper, Gene Allen systems finite groups Some results in delay-differential Isaak, Samuel Rue, James Sandvik linear operators Minor: Analysis and Mechanics Some stability properties of Reduction of unknowns in Einstein-Maxwell in Banach spaces field equations Rutter, Edgar Andrews of quasi-Frobenius rings Leitzel, James Robert Charles Characterizations On the divisibility of the group of divisor Department of Statistics classes of degree zero of an algebraic Easson, Rodney Peter curve On unbiased estimation in variance compon­ Mielke, Marvin Victor ent models Minor: Physics Cochran, Robert Smith Generalized modifications and cobounding Theory and application of multiple frame manifolds surveys

438 Doerfler, Thorn as E. LOUISIANA STATE UNIVERSITY (6) Size and power of some tests under experi­ Conway, john Bligh mental randomization Strict topologie and their applications El Mawaziny, Ahmed Hassan Estes, Dennis Ray Chi- square distribution theory with applica­ Classes of binary quadratic forms over poly­ tions to reliability problems nomial rings Lawing, William Dennis, jr. Friedberg, Michael Multiple decision sequential procedures Measures and affine semigroups Roseberry, Thomas Dean Hardy, john Thomas, Jr. The utilization of concomitant information Modules and rings of integers in the Cagley in sequential procedures for the comparison algebra of two treatments Mooney, Michael Cannon Tetreault, Florence Gertrude Structures in analysis A statistical outlier methodology for observed Smith, William Walker points and lines Priifer rings Thomas, David R. Asymptotic value distributions for certain UNIVERSITY OF MARYLAND (7) 2 X n games and n-stage games of perfect Dupre, Arthur Mason III information Real Borel cohomology of locally compact THE JOHNS HOPKINS UNIVERSITY (9) groups Fusaro, Bernard Alexander Hammond, William Franklin On a singular, inhomogeneous Cauchy prob­ The modular groups of Hilbert and Siegel lem with variable coefficients Pugh, Charles Chapman Gustafson, Karl Edwin The closing lemma for dimensions two and A priori bounds with applications to inte­ three gl:odifferential boundary problems Sauve, james Willard Henney, Dagmar R. A reduction of the three body problems by Theory of additive set-valued functions on means of transformation groups base cones in Banach spaces Tramer, Henry john Oddson, john Keith The cohomology ring of pseudo-projective Maximum principles and Phragrnen-LindelOf spaces theorems for second order differential Department of Biostatistics equations Archer, Philip Gordon P alrner, Daniel john Minor: Chronic Diseases On the structure of nilpotent algebras A mathematical framework for the investi­ Sigillito, Vincent George gation of precision in cytological screen­ A priori inequalities and interior pointwise ing studies bounds for solutions of certain parabolic Levy, Paul Samuel and elliptic partial differential equations The combination of unbiased estimators Mellits, Edwin David MASSACHUSETTS INSTITUTE OF TECHNOLOGY Estimation and design for intersecting re­ (27) gressions Priore, Roger Lawrence Billik, Martin A method for analysing tumor response to Minor: Physics chemotherapy Structure theorems for Reynolds operators Walker, Strother Holland Brodsky, Alan Robert Minor: Chronic Diseases Asymptotic decay of solutions to the rela­ Estimation of the probability of occurrence tivistic wave equation and the existence of of a disease as a function of several scattering for certain nonlinear hyperbolic variables equations Chadarn, LEHIGH UNIVERSITY (3) john Martin Minor: Physics Shahin, Jamal Khalil Unitary implementability of propagators of Affine differential geometry of closed hyper­ perturbed Klein-Gordon equations surfaces Dennett, john Roy Snyder, Andrew Kagey Minor: Philosophy Surnrnability of continuous functions on count­ A functorial description of 1r L 2K(A,q) able spaces; A classification of FK-spaces Dym, Harry n Snyder, Herbert Howard Minor: Electrical Engineering A hypercornplex function-theory associated Stationary measures for the flow of a linear with Laplace's equation differential equation driven by white noise

439 Freeman, Richard Don, Jr. Steen, Lynn Arthur Minor: Information Theory Minor: Philosophy Atom: Approximations toward optimal mem­ Uniform approximation by rational functions ory Spiegel, Eugene Haines, Leonard Harold On the structure and automorphisms of Generation and recognition of formal lan­ arbitrary dimensional unitary groups guages Sweedler, Moss Eisenberg Herzberg, Norman Peter Cocommutative Hopf algebras with antipode Minor: Music Tharp, Leslie Howard On the solutions of systems of quadratic Constructibility in impredicative set theory forms over the p-adic integers Wan, Frederic Yui-Ming Julian, William Harold Twisting and stretching of helicoidal shells Minor: Electrical Engineering McGILL UNIVERSITY (5) On the enhancement of random velocities of stars in disk-like galaxies Armstrong, Kenneth William Keenan, Edward Milton Topological f- rings Minor: Modern Languages Bhargava, Mira On the automorphisms of classicalgroups over Some problems associated with Ricatti-type local rings differential equations Kopperman, Ralph David Harrow, Martin Application of infinitary languages to analy­ Stability of solutions of certain third and sis fourth order differential equations Kuzmack, Arnold Mark Henniger, James Perry Minor: Political Science Uniformly almost periodic weighted se­ Statistical estimation for imperfectly-ob­ quences served stochastic processes Rawlinson, Elizabeth Lavine, Richard Bengt On algebraic equations with prescribed Minor: Philosophy Galois groups The Weyl transform-Fourier analysis of operators in L 2 -spaces UNIVERSITY OF MICHIGAN (23) Marcus, Michael Barry Local behavior of stationary Gaussian pro­ Agard, Stephen Browning cesses Topic in the theory of quasiconformal map­ Miller, john James Henry pings Minor: Physics Anderson, Glen Douglas An improvement to tl;le Kreiss matrix theo­ Extremal rings and quasiconformal map­ rem pings in 3-space Poritz, Alan Bram Beineke, Lowell Wayne Minor: Biology On the decomposition of complete graphs into Parallelism on p-surfaces in Riemannian planar and other subgraphs manifolds Bickel, Thomas Fulcher Putz, Herbert Byrne Some properties of groups with (B,N)-struc­ Minor: Physics ture The triangulation of differentiable fibre Boudreau, Paul Emile bundles Functions of finite Markov chains and ex­ Rehm, Ronald George ponential type processes Minor: Physics Cohen, Marshall Meyer On the calculation of spiral structure in a Transversely cellular mappings of combina­ disk galaxy torial manifolds Sattinger, David Harvey Courtney, Cameron Budd Minor: Physics A history and critical study of the function Solutions of the neutron transport equation concept and of certain elementary functions Segal, Robert with an evaluation of their diverse defini­ Minor: Literature tions and of the methods of developing their An application of Bernoulli polynomials to properties the theory of cyclotomic fields Dickey, James Mills Simmonds, James Gordon Bayesian statistical inference ror multi­ Minor: Structural Mechanics variate location parameters Asymptotic solutions for circular cylindrical Dudley, Underwood thin elastic shells The distribution modulo of oscillating Singleterry, Ann Marie functions Minor: Germanic Literature Falk, James Edward Degrees of unsolvability in the hyperarith­ A constrained Lagrangian approach to non­ metical hierarchy linear programming

440 Fossum, Robert Merle UNIVERSITY OF MINNESOTA (7) The Noetherian differences of projective Colwell, Peter orders On the boundary behavior of functions Heisler, Joseph Patrick bounded and analytic in the unit disk Diophantine problems for matrix rings, rings Grimm, Louis john of functions, and other rings Minor: Physical Chemistry Hilding, Stephen Reinhold General solutions of nonlinear difference Commutative Banach algebras: Fundamental equations theory and selected topics Kallman, Ralph Arthur Hollister, Herbert Allen On using the llstou integral to solve a cer­ Contributions to the theory of partially tain partial differential equation ordered groups Kuelbs, James D. Lipkin, Leonard Jay Integration on spaces of continuous functions Free boundary problems in the calculus of Pitchaiah, Vadlamudi variations Finite difference methods in the solution of Locker, john Stewart Laplace's equation with mixed boundary An existence analysis for nonlinear non­ conditions on a rectangular domain self-adjoint boundary value problems of Spicer, Donald Zane ordinary differential equations Minor: Physics Palmer, Edgar Milan Group algebras of vector-valued functions Graphical enumeration and the power group deValcourt, john Taylor Parrott, Stephen Kinsley Minor: Education Weighted translation operators A study of axial symmetry of planar convex Rosenbaum, jordan Tobias sets Interpolation in Hilbert spaces of analytic functions UNIVERSITY OF MISSOURI (Z) Sikkema, Carl Dean Hallam, Thomas Guy A "duality" between certain spheres and Asymptotic behavior and stability problems arcs in 3 s in ordinary differential equations Kimeldorf, George Williams, Robert Elvin Applications of Bayesian statistics to actu­ Simple near-rings and their associated rings arial graduation Taylor, Gerald Duncan MONTANA STATE UNIVERSITY (Z) Contributions to" the theory of multipliers Engle, Robert Dean Williams, James Patrick Bilinear forms on vector spaces of countable Spectral sets and finite-dimensional opera­ dimension in the case characteristic of K tors equals two \UCHIGAN STATE UNIVERSITY (8) Loftsgaarden, Don Owen Nonparametric classification and nonpara­ Behzad, Mehdi metric density function estimator Graphs and their chromatic numbers Gantos, Richard Laham UNIVERSITE de MONTREAL (3) On finite commutative semi-groups having Cleroux, Robert a group-like property Sur la robustesse du test F dans l'analyse Osborne, Richard Paul intrabloc du schema a blocs incomplets Embedding Cantor sets in manifolds partiellement equilibre et a deux classes Polimeni, Albert David associees. A study of the automorphisms and chains of Lemieux, Louis a finite group Etudes de certaines classes d'equivalence Wenger, Ronald Harold dans les demi-groupes Some semigroups having quasi-Frobenius Venne, Marc algebras Langues, theories et algebres polyadiques Department of Statistics and Probability d'ordres superieurs McCloskey, john William UNIVERSITY OF NEBRASKA (3) A model for the distribution of individuals by species in an environment Cobb, Ernest Benton Srivastava, Ramesh Chandra The character.i.zation of the solution sets for Estimation of the parameter in the stochastic generalized reduced moment problems and model for phage attachment to bacteria applications Zemach, Rita Heimes, Kenneth Anthony On orthogonal arrays of strength four and Boundary value problems for ordinary second their applications order systems

441 Mathsen, Ronald Melvin Logemann, George Wahl Subfunctions for third order ordinary differ­ Existence and uniqueness of rarefaction waves ential equations McCarthy, Donald John On just-infinite groups UNIVERSITY OF NEW BRUNSWICK (2) Morgan, Richard Chester Rahman, Mizanur Some mathematical problems in the pheno­ Correlation functions in ionized gases menological theory of multimode surface Viswanathan, Bhaskaran wave diffraction Generating functions for ultraspherical func­ Nowosad, Pedro tions On the variational problems of optimal allo­ cations of mass and its applications UNIVERSITY OF NEW MEXICO (4) Pincus, Martin Mark Steiner, Anne Kercheval Gaussian processes and Hammerstein inte­ The topological complementation problem gral equations Tarwater, Jan Dalton Roseman, Joseph Jacob Galois theory of Abelian groups On the principle of Saint- Venant Van Domelon, Sarah Rosenbaum, Hannah Leona Wolfson The structure of commutative bounded semi­ Analytic representation of L(X,l) for certain simple cones ray class characters X Wagner, Edwin Ferris Ross, George Gray On the convergence of Mikusinski operators Some calculations bearing on the Bieberbach conjecture Schilder, Michael NEW YORK UNIVERSITY (30) A Laplace asymptotic formula for Wiener Balch, Michael Slavo integrals A generalized Laurent expansion for solu­ Sormani, George Andrew tions of elliptic partial differential equa­ Three singular integral equations containing tions Jacobi elliptic functions Bernkopf, Michael Stern, Frederick The development of function spaces with Some results involving Wiener and related particular reference to their origins in Gaussian singular integrals integral equations theory Stevens, Donald Charles Bleistein, Norman The average and variance of the number of Asymptotic methods for the solution of initial­ real zeros of random functions boundary value problems for dispersive Strait, Peggy Tang hyperbolic equations Sample function regularity for Gaussian Dickey, Ronald Wayne processes with the parameter in a Hilbert Dynamic deformation of axially symmetric space elastic surfaces Swenson, Eva Valencia Dupras, Andre Alain Numerical computation of hypersonic flow Similarity of certain Volterra operators past a two-dimensional blunt body Fattorini, Hector Osvaldo Wells, Raymond O'Neil, Jr. Controllability in finite time of differential On the local holomorphic hull of a real sub­ equations in Banach space manifold in several complex variables Flack, Thomas Rudolph Wolfe, Peter Scattering by a semi-infinite dielectric cy­ Detraction and refraction of scalar waves linder Troue, Jacques Granoff, Barry The orders of the Postnikov invariants of Asymptotic solution of first order hyperbolic the Thorn spectrum MSO systems of partial differential equations UNIVERSITY OF NORTH CAROLINA (5) Goldfarb, Edwin F. The finite dimensional irreducible repre­ Deal, Albert Leonard III sentations of the complex connected simple Green's matrices for differential and differ­ Lie group of type G2 ence systems Handelsman, Richard Allen Ness, Sandra Gordon The asymptotic theory of Cerenkov radia­ On some Conti-type theorems tion Showalter, David Watts Kadish, Abraham A Perron-type existence theorem with sev­ Stability criteria and the initial value prob­ eral related theorems lem for collisionless, guiding center plasma Tonne, Philip Charles Landman, Joan Gottlieb Power-series versus sequences Polycyclic groups and residual finiteness Vegh, Emanuel Lieberman, Sidney Moses On the minimum distance determined by the Asymptotic methods for the inverse scatter­ roots of an algebraic equation and a con­ ing problem jecture of Lee Moser

442 NORTH CAROLINA STATE UNIVERSITY (10) Sehgal, Surinder K. On the existence of Cartan subgroups of Fulton, John David finite groups Tactical configurations induced by matrices Thie, Paul R. over the Galois domain for the integer m The Lelong number Marlin, Joe Alton Minor: Electrical Engineering OHIO STATE UNIVERSITY (4) Solutions of perturbed Hamiltonian systems Aggarwal, Satish Teague, David Boyce Diophantine inequalities in fields of formal Minor: Physics - Engineering Mechanics Laurent power series A Boussinesq problem for a finite cylinder Breiter, Thomas Department of Experimental Statistics On the Lebesque property and related topics Braaten, Melvin Ole Hans, Rajinder Minor: Genetics. Some results on lattice packings and cover­ The union of partial diallel mating designs ings and incomplete block environmental de­ Santos, Eugene signs Mathematical foundations of the theory of Gun, Atindramohan thin shells The use of a preliminary test for interactions in the estimation of factorial means Harvey, James Raymond OKLAHOMA STATE UNIVERSITY (9) Fractional moments of a quadratic form in Blondeau, Ernest Eugene, Jr. noncentral normal random variables Some properties of the Cartesian product Kanofsky, Paul Benjamin of two sets of points Parametric confidence bands on cumulative Glynn, William Allen distribution functions Strictly convex metrics and natural delta Nasoetion, Andi Hakim functions Minor: Genetics Bee, David Elliott An evaluation of two procedures to estimate Estimation of variance components in in­ genetic and environmental parameters in complete block design a simultaneous selfing and partial diallel Blankenship, John Hamilton test-crossing design Use of order statistics for graphical estima­ Nixon, David Eugene tion A study of the distributions of two test statis­ Faulkenberry, G. David tics for periodicity in variance Uniformly most accurate tolerance limits Nuri, W alid Abdul Hamid Gurley, William Raymond Fourier methods in the study of variance Optimal symmetrical designs fluctuations in time series analysis Mirham, George Arthur NORTHWESTERN UNIVERSITY (3) Bivariate warming-time/failure-time/dis­ tributions Larmore, Lawrence Louis, Jr. Pierce, Donald Alan On the classification of liftings to AK(G,N) A method for determining the sensitivity of bundle and oriented vector bundles over a a Bayes procedure to the prior distribu­ finite complex tion Mullins, Robert Emmet Pulley, Paul Eugene, Jr. Some results on algebras of functions On the goodness of estimated parameters in Ostrand, Phillip Arthur the ratio model Some new results in the dimension theory of metric spaces UNIVERSITY OF OREGON (8) UNIVERSITY OF NOTRE DAME (6) DeMeyer, Frank Raymond Gemignani, Michael Caesar Galois theory in rings and algebras Topological geometries and a new charac­ Fein, Burton Ira terization of R m The field of characters of a finite group Johnson, Arnold A. Fischler, Roger Morris Integral representations of Hermitian forms Mixing as a generalization of independence over local fields Harper, Lawrence Hueston McGrath, Gary Lee Capacities of sets and harmonic analysis on Normal families of nonnegative divisors on the group zw polycylinders Ingrham, Edward Chisholm Oppelt, John Andrew On the existence of inertial sub-algebras Abelian groups which are a direct sum of Janusz, Gerald Joseph p-mixed groups Separable algebras over commutative rings

443 Kennel, Ethan Finley PRINCETON UNIVERSITY (16) Class field theory in dimension greater than Bloom, Thomas one Local fiberings of complex analytic variety Woodroofe, Michael Barrett Brown, Morton Bertram Statistical properties of the number of posi­ A secondarily Bayes approach to the two­ tive sums means problem UNIVERSITY OF PENNSYLVANIA (5) Cardenas Trigos, Humberto The cohomology alfebra of the symmetric Alter, Ronald group of degree p On a necessary condition for the validity of Faris, William Guignard the Rie~ann hypothesis for functions that Perturbation formulas for linear propagators generalize the Riemann zeta function Goldman, Jay Robert Kolman, Bernard Stochastic point processes limit theorems On the lattice of subalgebras of a Lie alge­ and infinite divisibility bra Knapp, Anthony William Leahy, John V. Distal function on Abelian groups Deformation of filtered and graded objects Prill, David Donald in an abelian category Some local properties of analytic varieties Sellers, Peter Hoadley Rayna, Gerhard Algebraic complexes which characterize On the classification of modules over finite chemical networks groups Moore School of Electrical Engineering Robbin, Joel William Wexelblat, Richard Lee Subrecursive hierarchies The development and mechanization of a Roskies, Ralph problem-solving facility R-matrix approach to complex angular mo­ mentum and to three-particle interaction PENNSYLVANIA STATE UNIVERSITY (6) Siebermann, Laurence Carl Alo, Richard Anthony The obstruction to finding a boundary for an Order topologies open manifold of dimension greater than Blefko, Robert Lamar five On E-compact spaces Silberstein, Martin Louis Gerber, Harvey Many particle processes Normal functions- and ordinal notations Smythe, Neville Francis Hager, Anthony Wood Isotopy invariants of links On the tensor product of function rings Sondow, Jonathan David Levitz, Hilbert Disproof of the Hauptvermutung for manifold On the ordinal notations of Schutte and the pairs ordinal diagrams of Takeuti Thompson, James Robert Moore, Robert Clyde A shrinking technique for modifying minim urn Some multiplicative problems in C(X) variance unbiased linear estimators Weiss, Benjamin UNIVERSITY OF PITTSBURGH (6) Vibrating systems and positivity preserving Freeman, Louis Barton semigroups Singular perturbations of boundary value problems arising in the study of chemical reactors PURDUE UNIVERSITY (13) Hall, Charles Allan Poly-functions on S-algebras Diem, John Edwin Hiergeist, Franz Xavier A radical for lattice-ordered rings Some generalizations and extensions of uni­ Friedman, Eugene Myron formly distributed sequences A-convergence: A generalization of the notion Itkin, Ivan of order convergence in a vector lattice Approximate methods for solving linear and Gorman, William Joseph nonlinear multidimensional integral equa­ Minor: Physics tions and boundary value problems On the effects of homeomorphic transforma­ Moritz, Roger Homer tions upon certain nontopological properties Results pertaining to the Lototsky method of of real functions and of sets summability Khazanie, Ramakant Salter, James William On a Mendelien Markov process with multi­ Real representatil5n of complex three-space nomial transition probabilities in Argand six-space with respect to linear Lane, Ernest Paul varieties in these spaces Bitopological spaces

444 Masterson, john joseph UNIVERSITY OF ROCHESTER (4) An extension of the concept of the order dual Connolly, Francis Xavier of a vector lattice Natural transformations of singular homo­ Perry, Clifford Raymond logy theory Theory of solomanifolds Itzkowitz, Gerald Lee Ryan, Francis Daniel Extensions of Haar measure for compact On the uniform closure of the linear space connected abelian groups generated by the nonnegative upper semi­ Negrepontis, Stelios Amvrossiou continuous functions A homology theory for realcompact spaces Shapiro, Harvey Lee Willard, Stephen Whitman Extensions of pseudometrics Absolute Borel sets and their Stone-Cech Smith, Frank Englebert Anthony compactifications A structure theory for a class of lattice ordered semirings RUTGERS, THE STATE UNIVERSITY (5) Henry Ellis, Jr. White, Barback, joseph The approximations of one-one measurable Contributions to the theory of isols by diffeomorphisms transformations Bentsen, Irving Otman Department of Statistics Existence of solutions of abstract partial Deely, john joseph difference polynomials Multiple dedision procedures from an em­ Mancuso, Vincent pirical Bayes approach Retracts and extension spaces for certain McLean, Robert Allen classes of normal spaces integral of a function defined on a semi­ The Statistics Center Markov process Hayton, Thomas QUEEN'S UNIVERSITY (4) A synoptic theory of inference Walker, William Franklin, Jr. Atkinson, Harold Russell An application of the decomposition principle Semi-normed linear lattices with duals of to convex programming problems integral type Dawkins, Brian P. SAINT LOUIS UNIVERSITY (l) isomorphism of certain approximate The Keezer, Sister james Miriam finite geometries Minor: Philosophy Griffin, Malcolm P. An axiom system for plane Euclidean geo­ Rings of Krull type metry McDonald, Robert Ian Banach spaces of operators UNIVERSITY OF SASKATCHEWAN, SASKATOON (2) RENSSELAER POLYTECHNIC INSTITUTE (4) Chakravartty, lswar Chandra Bentley, Herschel Lamar The prime number summation formula and Topology without the union axiom its applications Gross, Arthur Gerald Pearson, Terrance Laverne Numerical investigation of the stability of Almost elliptic functions Couette flow UNIVERSITY OF SOUTH CAROLINA (5) Haines, Charles Wills An analysis of stochastic eigenvalue prob­ Fritz, Mrs. Carol Stone lems The dependence of initial structure on uni­ O'Neil, Peter Vincent verses and forgetful" functors On tree counting in networks Leonard, William Wilson Some results on small modules Maddox, Billy Hoyte Absolutely pure modules RICE UNIVERSITY (3) Pleasant, James Carroll Certain relations between objects and mor­ Bryant, jack Douglas phisms in categories and module cate­ Theorems relating convolution and Fourier gories series Thomas, John Pelham Hall, Robert Lester Maximal topological spaces Asymptotic behavior of functions holomor­ phic in the unit disc Ryan, Frank Beall STANFORD UNIVERSITY (28) A characterization of the set of asymptotic Diamond, Harold George values of a function holomorphic in the Asymptotic distribution of generalized inte­ unit disc gers and prime numbers

445 Edison, Larry Alvin Parent, Elias Alphonse, Jr. On the spectrum of generalized almost per­ Sequential ranking procedures iodic functions Shorack, Galen Jain, Naresh Chandra Nonparametric tests and estimation of scale Some limit theorems for a general Markov in the two-sample problem process Sternin, Chaim Loeb, Peter Albert Statistical sequencing of archaeological data An axiomatic treatment of pairs of elliptic Stewart, Leland Taylor differential equations Continuous sampling by variables under de­ Lowenthal, Franklin structive testing On some subgroups and subsemigroups of Swain, Donald Derek the projective group on the real line and Bounds and rates of convergence for the the Moebius group on the plane extended compound estimation problem in Meeker, Loren David the sequence case Some generalized Hardy spaces Taylor, Howard Milton III Moler, Cleve Barry Statistical control Finite difference methods for the eigenvalues STATE UNIVERSITY OF NEW YORK, COLLEGE of Laplace's operator AT BUFFALO (I) Perko, Lawrence Marion Asymptotic matching in the restricted three Lowerre, James Myrick body problem Some stochastic birth chain theorems Rosenthal, Haskell P au! STEVENS INSTITUTE OF TECHNOLOGY ( 1) Projections onto translation-invariant sub­ spaces of LP( G) Martin, John Francis Patrick Rudin, Bernard David On the exponential representation of solutions Convex polynomial approximation of systems of linear differential equations Shenk, Norman Alexander II and an extension of Erugin' s theorem Spectral repr-esentations and scattering the­ SYRACUSE UNIVERSITY (6) ory for the wave equation Smith, Donald Ray Abramson, Paul On the exterior stationary problem for the An analysis of topological characterizations Navier-Stokes equations in 2-dimensions of the dimension concepts Snow, Donald Ray Brown, David Theodore Reachable regions and optimal controls On some aspects of the Arens-Hoffman Switzer, Robert Massey, Jr. extension of Banach algebras Homology with local coefficients and applica­ Humm, Margaret Ann tions to fiber spaces On a class of right alternative rings with an Thoe, Dale Wesley idempotent not the identity Scattering theory for the wave equation with O'Brien, Thomas Vincent a potential On the modified Postnikov tower Troutman, John Lewis Shea, Daniel Francis, Jr. Eigenvalues and eigenfunctions of a class On the relations between growth of a mero­ of potential operators in the plane morphic function and the angular distribu­ Ziegler, Zvi tion of two of its values Generalized convexity cones Vosburg, Albert Chapman Some topics on E-entropy ,Graduate Division-Special Program Grace, Donald Wayne UNIVERSITY OF TENNESSEE (1) Minor: Computational analysis Hildebrant, John A. Computer search for non-isomorphic convex Minor: Philosophy polyhedra On compact uniquely divisible semigroups Department of Statistics UNIVERSITY OF TEXAS (11) Arnold, Barry Charles A correlated urn-scheme for a continuum of Baker, Blanch Jo Anne Monger responses Minor: Physics Fromov.itz, Stanley Concerning uncountable collections of triods A class of inventory models Baker, Harvey Lee Hollander, Myles Minor: Physics Rank tests for randomized blocks when the Complete amonotomic collections alternatives have an a priori ordering Borm, Alfred Ervin Norman, Maurice Frank Minor: Physics A probabilistic model for free-responding On the lattice of continuous functions on a

446 completely regular G li-space UNIVERSITY OF UTAH (11) Elliott, Roger William Bryan, Robert Neff Minor: English A linear differential system A model for a fact retrieval system Cannon, Lawrence Orson Hudson, Francis Marion Sums of solid horned spheres Minor: Education Gill, Gurcharan Singh A generalization of the finite Fourier trans­ Conformal mapping and retraction formations Hansen, Richard Allan Maher, Laurence Paul, Jr. Round-off procedures in the numerical treat­ A generalization of the real-numbers axioms ment of differential equations Pratt, Terrence Wendall Irwin, Ronald Lee Minor: Psychology Absolute Hardy-Bohr factors Syntax-directed translation for experimental Jamison, Ronald Dee programming languages Systems of differential equations Reed, Dennis Keith Komkov, Vadim Minor: Physics Minor: Electrical Engineering Upper semi-continuous collection of finite Applications of homogeneous contact trans­ point sets formations Roach, Everett D. Kurtz, J. Cleo Minor: Education Summability factors An integral Loveland, Lowell Duane Tulloch, Lynn H. Tame subsets of spheres in E 3 Minor: English Peterson, Gerald E. Operational solution of partial derivative Cardinal and ordinal numbers equations of hyperbolic and parabolic type Runckel, Hans Joachim Young, Sam Wayne On the number of zeros of entire functions of Minor: Engineering finite order Uniform completeness of sets generated by a single function

UNIVERSITY OF TORONTO (9) VANDERBILT UNIVERSITY (3) Brown, Margaret Gillian Bailey, Donald Forest Metric differential geometry Minor: Physics Csima, Joseph On contractive and expansive mappings Investigations on a time-table problem Nelson, Oscar Tivis, Jr. Davison, Thomas Mathew Kerr Minor: Physics Arithmetical convolutions and generalized Subdirect products of finite lattices prime number theorems Williams, Richard Kelso Derome, Jean-Robert Minor: Physics Foundations of Racah algebra for a general Expansive mappings group Heinig, Hans Paul UNIVERSITY OF VIRGINIA (4) The Laplace transform and generalized Hp Bobo, Edwin Ray spaces Automorphism groups of Jordan algebras Maag, Richard Urs Duke, Richard Alter Contributions to the design of experiments Minimal imbeddings and open mappings of Norminton, Edward Joseph graphs in manifolds Diffusion of heat and circulation in known Rice, Barbara Louise Slyder convection fields A definition of a stochastic integral Tam, Kwok Kuen Stroud, Junius Brutus On two singular perturbation problems in Simple Jordan algebras of characteristic two fluid mechanics Williams, Kenneth Stuart VIRGINIA POLYTECHNIC INSTITUTE (8) Arithmetical theory of forms Hanson, John Robert TULANE UNIVERSITY (2) Connections between binary systems and admissible topologies Penney, David Emory Klassen, Vyron M, On algorithm for establishing isomorphism Complexes with invert points between tame prime knots Persinger, Carl Allan Wilken, Donald Rayl On the embedding of subsets of N -books in Local•behaviour in function algebras E3

447 Department of Statistics UNIVERSITY OF WATERLOO (3) Barnett, Frederic Charles The problem of classifying members of a Cowan, Donald Douglas population on a continuous scale The numerical solution of unilateral equations Kanjo, Anis Ismail Lawson, John Douglas -Recovery of interblock information Some one-step methods for the numerical Mendel, Jack Marvin solution of systems of ordinary differential The statistical analysis of nonlinear response equations functions with application to the genetics Mullin, Ronald Cleveland of poultry growth An enumerative survey of triangular maps Norman, James E., Jr. WAYNE STATE UNIVERSITY (3) Restricted ranking Wolock, Fred W. Derderian, Jean-Claude Cyclic designs Residuated mappings Stebbins, John LaForest UNIVERSITY OF WASHINGTON (11) Spiral asymptotic values of functions mero­ Beldin, Donald Ross morphic in the unit disk On some problems in the harmonic analysis Brown, Linda Marie of compact groups Span spaces Colby, Robert Ray UNIVERSITY OF WESTERN ONTARIO (2) On indecomposable modules over rings with minimum condition Hosking, Roger John Denzel, George Eugene Resistive and Hall instabilities A characterization of Markov processes in Saleh, A. K, Md. Ehsanes terms of their hitting characteristics On the estimation of the parameters of ex­ Ferguson, Le Baron Otis ponential distribution based on optimum Uniform approximations by polynomials order statistics in censored samples with coefficients in discrete subrings of C WESTERN RESERVE UNIVERSITY (5) Kim, Choo Whan Uniform approximation and almost periodi­ Geiger, William city of doubly stochastic operators Finite-dimensional subsets of infinite- Mansfield, Larry Everett dimensional metric spaces A generalization of the Cartan-Kahler theo­ Hopper, Maia Leetmaa rem On stability and boundedness of solutions Mines, Ray III of a system x = X(x,t) A family of functors defined on generalized P au!, Jerome Larson primary groups Approximation and Schoenflies extension of Roush, Galen James cm-diffeomorphisms (m ?, 0} Algebras of bounded measurable functions Skidmore, Alexandra Samanich Shih, Chung Tuo Two ordinary differential equations; fourth Markov processes whose hitting distribu­ order and second order tions are dominated by those of a given Tomastik, Edmond Charles of n dependent process Singular quadratic functionals Stringall, Robert William variables Endomorphism rings of P -primary Abelian UNIVERSITY OF WISCONSIN (21) groups Yandl, Andre Louis Bronikowski, Thomas On 6-continuity proximate retracts, fixed­ Minor: Physics points and homotopy On systems of integrodifferential equations occurring in reactor dynamics WASHINGTON UNIVERSITY (4) Brown, Gerald Leonard Egan, Howard Leslie Minor: Engineering Mechanics The Frattini subgroup and its relation to The inverse reflection problems for electric certain classes of generalized nilpotent waves on non-uniform transmission lines groups Dunk!, Charles Frank Hunt, Richard Allen Harmonic analysis on spheres Operators acting on Lorentz spaces Foote, Stuart Ashby Machtinger, Lawrence Arnold Universal control extensions of groups Rings with a not necessarily transitive order Glauberman, George relation Fixed point sub-groups that contain central­ Zwart, Philip Bernard izers of involutions Compact homogeneous spaces possessing Ganser, Carl Coleman invariant contact, symplectic, or cosym­ Infinitely differentiable positive functions plectic structures and integrability of ultra spherical series

448 Graham, George Department of Statistics On dimension subgroups-- some ordering Bacon, David Walter properties of torsion free locally nil- Seasonal time series potent groups Hall, James Emerson YALE UNIVERSITY (9} Differential equations on product spaces Hoppensteadt, Frank Allen, Harry Prince of Lyapunov' s second method An application Jordan algebras and Lie algebras of type D4 to singular perturbation problems Frampton, John Noel Liu, Shih-Chao Smooth partitions of unity on Banach mani­ On many-one degrees folds Mattson, Don Arthur Kaufman, Robert Philip Structures in extended topology Commutative semi-groups Moschovakis, Joan Rand Klotz, Eugene Arthur Disjunction, existence, and A -eliminability Isomorphisms of simple Lie rings in formalized intuitionistic analysis McCrimmon, Kevin Mor Mullikin, Albert Norms and noncommutative Jordan algebras Some topics in differential equations Merrill, Samuel III Reiter, Allen HP spaces derived from function algebras On topological equivalences Rickert, Neil William Roberts, Arthur Wayne Amenable groups and groups with the fixed Classes of differentiable mappings point property Sastry, Tadepalli N arazimhs Rovnyak, Virginia Garrett Turning point problems for certain systems Self reciprocal functions of linear differential equations Winter, David John Weinstein, Joseph M. On automorphisms of Lie algebras and alge­ First order properties preserved by direct braic groups product West, Donald Christian Transitive lifting functions in fiber spaces YESHIVA UNIVERSITY (2) Williams, Francis Homotopy-commutativity of H-spaces Farkas, Hershel Mark Department of Computer Sciences Special divisors, theta nulls and analytic Keller, Sister Mary Kenneth subloci of teichmueller space Inductive interence on computer generated Lebowitz, Aaron patterns Degeneration of Riemann surfaces

449 RECENT POLICIES CONCERNING FEDERAL RESEARCH GRANTS

In recent years congressional in­ of the Budget issued guidelines to each of terest in the entire pattern of the Federal the agencies involved in setting procedures support of research in universities has to be followed in issuing grants made after intensified, partly because of the greatly March 1, 1966. increased funds which have been made Because of the large number of available for this purpose since 1946. As a mathematicians who are affected by the consequence, there has been a general policies of the National Science Foundation tightening of procedures by agencies on the it may be useful to summarize some of their one hand and aroused concern by university procedures here (a fuller description is scientists and administrators on the other. contained in the NSF Important Notice of The results of lengthy discussions within January 24, 1966): the government have recently led to a series (a) Cost-sharing is not required on grants of recommendations by the Bureau of the made solely for the support of conferences, Budget for improvements and can be ex­ travel, equipment, and publication ofmanu­ pected to have a substantial influence with­ scripts. in agencies in the administration of their (b) A relatively higher level of cost­ research grants programs. This report, sharing is required on grants which provide "The Administration of Government Sup­ for a significant reimbursement of aca­ ported Research at Universities" is dis­ demic-year fact1lty salaries than those cussed briefly in the April 29, 1966, issue grants which provide a substantial fraction of Science. of support for major equipment items. The universities and the government (c) For institutions on a calendar year have often approached questions of Federal basis the institution must contribute to the support from two points of view. Partly cost of the project an amount which is at reflecting the rationale for support which least half of the NSF support of faculty existed during World War II, many univer­ salaries (including applicable indirect sities have held that their relationship with costs). For institutions on an academic the government was based upon a purchaser­ year basis the institution must match that vendor concept. On the other hand, the part of NSF support provided for faculty government has often held that, in the case salaries during the academic year (including of basic research at universities, this applicable indirect costs). Those research support represented a grant-in-aid for work projects where the proposed institutional which the university faculty wanted to ac­ contribution, through faculty salary match­ complish. For several years this latter ing vr otherwise, does not constitute more principle was reaffirmed by Congress in than a token amount will be subject to setting a statutory limitation of 20o/o on the negotiation with respect to cost-sharing. indirect costs of grants awarded by the (d) Budgets for all research proposals National Science Foundation. In effect, this should include the following: acted to insure a level of cost participation by the institution which accepted these ( 1) The amount requested from NSF grants. for direct cost of each of the cost elements The appropriation acts in fiscal year of the research proposed. Where applicable, 1966 removed this limitation on the pay­ faculty salaries should be separately allo­ ment of full indirect costs of research cated between academic year and summer sponsored by Federal grants. At the same salaries, unless salaries are claimed on a time Federal agencies were prohibited from calendar year basis. paying any recipient of a grant an amount (2) The amount requested for indirect equal to as much as the entire cost of such costs on appropriate direct cost items. a project. In its Circular A-74 the Bureau (3) The total requested from NSF.

450 (4) The amount proposed as the insti­ ment the very substantial degree of cost­ tution's cost-sharing contribution from non­ sharing which institutions had been pro­ Federal sources, itemized by the same viding and thereby emphasize to Congress categories of cost as above. the fact that research support has been a joint enterprise of the institutions and the (e) Finally, procedures for establishing Foundation. It appears, however, that many approved indirect cost rates are set forth institutions feel it to be advantageous to in this Notice. show no more than token participation and As may be evident, the introduction it is obvious that several interpretations of these procedures has led to differences can be given to the meaning of "token." of interpretations of many kinds. Generally, This discussion may suggest some of program officials within the Foundation the complex factors which surround grant feel that these procedures can help docu- negotiations at this time.

NEWS ITEMS AND ANNOUNCEMENTS

SYMPOSIUM ON BIOMATHEMATICS FIFTH INTERNATIONAL CONGRESS ON AND COMPUTER SCIENCE IN CYBERNETICS THE LIFE SCIENCES The Fifth International Congress on The Division of Continuing Education Cybernetics will be held in Namur, Belgium, of The University of Texas Graduate School from September 11 to September 15, 1967, of Biomedical Sciences at Houston will con­ under the auspices of the International duct the Fifth Annual Symposium on Bio­ Association for Cybernetics. The Congress mathematics and Computer Science in the will coincide with the tenth anniversary of Life Sciences, March 1967 at the Shamrock the founding of the Association. Hilton Hotel in Houston, Texas. Further information about the Con­ The theme of the symposium is gress may be obtained from the Secretariat "Medical Uses of Man-Machine Systems." of the International Association for Cyber­ Topics include the following: the engineer­ netics, Palais des Expositions, Place A. ing and mathematical stimulation of arti­ Rijckmans, Namur, Be.lgium. ficial organs; computer data editing display and graphic systems; patient monitoring; radiation treatment planning; research NATO SUMMER SCHOOL ON laboratory automation; medical applications CONTEMPORARY METHODS OF of computer hardware and software sys­ DISCRETE MATHEMATICS tems; training program in biostatistics, biomathematics, computer sciences and A NATO summer school on Con­ bioengineering. temporary Methods of Discrete Mathemat­ Those interested in submitting ab­ ics will be held in Varenna, Italy, August stracts should contact the Office of the 1-13, 1966. The co-directors of the sum­ Dean; Division of Continuing Education, mer school are Dr. Bernard Roy of Paris The University of Texas Graduate School and Dr. Frank Harary, Department of Math­ of Biomedical Sciences at Houston, 102. ematics, University of Michigan, Ann Arbor, Jesse Jones Library Building, Texas Medi­ Michigan, 48104. North American mathe­ cal Center, Houston, Texas 7702.5. Abstracts maticians interested in attending the sum­ will not be reviewed after the deadline, mer school should write to Dr. Harary for November Z.O, 1966. further information.

451 NEWS ITEMS AND ANNOUNCEMENTS

1966 INTERNATIONAL SUMMER SCHOOL Aldridge Bousfield, topology, Brandeis Uni­ ON LOCAL FIELDS versity; Spencer Dickson, algebra, Univer­ sity of Oregon; David Knudson, algebra, The Netherlands Universities Foun­ Cornell University; John Polking, analysis, dation for International Cooperation (NUF­ Brandeis University; Donald Ramirez, anal­ FIC) will conduct an international summer ysis, University of Washington; Wolmer school on local fields, July 25-August 6, Vasoncelos, algebra, Cornell University, 1966, Sponsored by NATO, the school will be held at Driebergen, The Netherlands. program consists of four sub­ The COMMITTEE ON SCIENTIFIC AND the general topic. ject areas related to TECHNICAL COMMUNICATION speakers are as follows: Subject areas and ESTABLISHED (l) p-adic analysis/Theory ofp-adic functions: J, Tate, Harvard University; The National Academy of Sciences H. Grauert, University of Gottin&"en; R. and the National Academy of Engineering Remmert, University of Gottingen; M. P. have jointly established the Committee on Lazard, University of Poitiers; A. F. Monna, Scientific and Technical Communication, at University of Utrecht. the request of the National Science Founda­ (2) Algebraic groups over loc.al tion. The committee will provide a means fields: F, Bruhat, University of Nancy; whereby scientists and engineers, through J, Tits, University of Bonn. their societies, can participate in the con­ ( 3) Elliptic curves and abelian vari­ sideration of plans for a national network of eties in connection with local fields: J, W. S, information systems in science and tech­ Cassels, Trinity College, University of nology, as proposed by the Committee on Cambridge; A, Neron, Universite de Paris. Scientific and Technical Information (4) Rationality of Z- and L- functions: (COSATI) of the Federal Council for Sci­ B. M. Dwork, Princeton University. ence and Technology. In its study of the present status and In addition,· J. P. Serre, College de future requirements of the national scientific has been invited to lecture, and France, and engineering communities with respect Artin, Massachusetts Institute of Tech­ M. to the flow and transfer of information, the and J, L. Verdier, University of nology, committee expects to work closely with have provisionally agreed to Strasbourg, COSA TI, the Office of Science Information lecture on etale cohomology and rationality Service of the National Science Foundation, and L-functions, of Z- the Office of Science and Technology in the Executive Office of the President, and with the professional groups that perform in­ ONR RESEARCH ASSOCIATESHIP~ formation services, The committee will give special at­ The Office of Naval Research ( ONR) tention to information activities and policies has announced the award of six postdoctoral of private groups and organizations, both at research associateships in mathematics for home and abroad, and to the interaction and the academic year 1966-1967. These awards interrelation of the Federal Government were made upon the recommendation of the with these groups. Committee on Mathematics Advisory ~o the The following topics will be of par­ ONR, a committee of the Division of Mathe­ ticular concern to the committee. matics, National Academy of Sciences-­ l, Methods of promoting more effective National Research Council, The recipients, relationships between information systems their fields of research, and the institutions and the principal producers and users of making the appointments are as follows: scientific and technical information

452 z. Techniques and systems for improv­ ment of Terrestrial Magnetism, Carnegie ing information transfer. Institution of Washington,; Paul Weiss, Uni­ 3, New means of providing greater versity Professor, Graduate School of Bio­ selectivity and consolidation in information medical Sciences, University of Texas; transfer. W, B. Wiley, president, john Wiley and The committee will make recom­ Sons, Inc.; and VanZandt Williams, director, mendations both to private organizations American Institute of Physics. and to Federal agencies on courses of action required to maintain effective communica­ tion within and among fields of science and SYMPOSIUM ON PROSPECTS FOR technology. SIMULATION AND SIMULATORS OF Dr. Robert W. Cairns, director of DYNAMIC SYSTEMS research of the Hercules Powder Company, Wilmington, Delaware, is chairman of the A symposium on Prospects for Simu­ committee. A former Deputy Assistant lation and Simulators of Dynamics, spon­ Secretary of Defense and past chairman of sored by the Air Force Office of Scientific the Division of Chemistry and Chemical Research and Westinghouse Electric Cor­ Technology of the National Research Coun­ po~ation, will be held at the Westinghouse cil, Dr. Cairns is presently a director of Defense and Space Center, Baltimore, the American Chemical Society. Maryland, on September Z6 and Z7, 1966, F, Joachim Weyl named The meeting will be divided into four ses­ Special Assistant sions. Two of the sessions will be devoted to problems which can only be solved by When announcing the formation of the simulation; the other two will be concerned committee, Dr. Frederick Seitz, president with the concepts of simulators and novel of the National Academy of Sciences, also organizations which may be useful in the announced the appointment of Dr, F. Joachim solution of such problems. Further infor­ Weyl to the executive staff of the Academy mation may be obtained from S. Burik, as a special assistant to the president. Symposium Secretary, MS 45ZA, Westing­ Dr. Weyl, who has resigned his position house Electric Corporation, P. 0, Box 746, as Chief Scientist of the Office of Naval Baltimore, Maryland Z1Z03, Research, Department of the Navy, will serve as executive secretary of the Com­ mittee on Scientific and Technical Com­ munication, SEMINAR ON GRAPH THEORY Members of the committee are as follows: Robert W. Cairns, chairman; The dates for the seminar on Graph George E. Holbrook, vice president, E. I. Theory, to be held in Rome, which were du Pont de Nemours and Company, Inc.; given in the February issue of these cJ{ofir.ti) J, C. R. Licklider, consultant to the direc­ as july 8-1Z, 1966, have now been changed tor of research, International Business to july 5-9, 1966, The seminar, conducted Machines Corporation; Clarence H. Linder, under the auspices of the International Com­ vice president and group executive (retired), putation Centre (ICC), is now to be held General Electric Company; H. W. Magoun, simultaneously with the International Sym­ Dean, Graduate Division, and Professor of posium on Mathematical and Computational Physiology, University of California, Los Methods in the Social Sciences (July 4-8, Angeles; Nathan M, Newmark, Head, De­ 1966,) partment of Civil Engineering, University The provisional list of lectures for of Illinois; W, H, Pickering, director, Jet this seminar is as follows: C. Picard, Propulsion Laboratory, California Institute G. Haj6s, J, M. Hammersley, V, V, Menon, of Technology; Byron Riegel, director of P. Erdos, P. Camion, H. W. Kuhn, P, W. chemical research, G, D. Searle and Com­ Kasteleyn, J, Edmonds, J. Nash-Williams, pany; William Steere, director, The New G, Minty. York Botanical Garden; john W. Tukey, Travel and accommodations may be Professor of Mathematics, Princeton Uni­ arranged through the Wagons-Lite Cook versity; Merle A, Tuve, director, Depart- agencies.

453 Requests for further information R, Boudon. Persons interested in submitting should be addressed to ICC, 23, viale della a paper should have sent an abstract to the Civilta del Lavoro, Rome-EUR, Italy. program committee before the end of March, 1966. Abstracts should be addressed to the Secretary General of the symposium, 8. Jaulin, Centre de Calcul de la Maison des SYMPOSIUM ON GENERAL TOPOLOGY Sciences de l'Homme, 14, rue Monsieur le AND ITS RELATIONS TO MODERN Prince, Paris 5e, France. ANALYSIS AND ALGEBRA The Proceeding of the Congress, which will contain invited lectures, the con­ The Czechoslovak Academy of Sci­ tributions to the symposia, and the program will conduct the second symposium ences of the Congress, will be available by on General Topology and its Relations to October 1966. Only papers which have been Analysis and Algebra on August 30 Modern submitted to the program committee in their through September 4, 1966, in Prague. This final version by May 1966 will be included. to the one of the same symposium is related The official languages of the sym­ held five years ago. The aim of this name posium are French and English with simul­ is to provide interchange of in­ symposium taneous translation. on new achievements and on formation The participation fee for member­ problems of general topology as actual ship in the Congress is $25. its applications in other fields of well as A number of social events will take Special questions regarding mathematics. place during the symposium, including ex­ algebraic topology will probably not be in­ cursions to the Mediterranean beaches and in the program. Persons interested cluded archaeological sites in the Rome area. should note that the in this symposium This symposium will be followed by a sem­ dates follow the conclusion of the Inter­ inar on Graph Theory. national Congress of Mathematicians to be held in Moscow from August 16-26, 1966. This will enable attendants of the Moscow Congress to participate in the Prague sym­ NSF AWARDS OVER posium, 2000 FELLOWSHIPS

The National Science Foundation re­ cently announced award of over 2000 fellow­ SYMPOSIUM ON 1966 INTERNATIONAL ships given for postdoctoral work in the MATHEMAT~ALANDCOMPUTATIONAL sciences, for advanced science graduate METHODS IN THE SOCIAL SCIENCES study, and for summer study and research. One hundred and seventy awards symposium on Math­ An international were given to individuals with doctoral de­ and Computational Methods in the ematical grees, or the equivalent, in the mathemati­ will be held in Rome from Social Sciences cal, physical, engineering and life sciences 8, 1966 under the auspices July 4 to july for the purpose of aiding continued research. the International Computation Centre, of Each of the awards includes an annual sti­ individuals will be Honorary The following pend of $5,500, an allowance for dependents, the symposium: F. Braudel, Members of and an allowance to help defray travel costs B. de Finetti, University College de France; to the fellowship institution, Fell ow ship Guilbaud, Ecole Pratique of Rome; G.~ Th, recipients in this program will study or P. F. Lazarsfeld, Col­ des Hautes Etudes; carry on research at institutions in the Levi-Strauss, College umbia University; c. United States and abroad. The following Piaget, University of Geneva; de France; J, mathematicians were among those who re­ Lelong, Paris University; C. M. Lerici, P. ceived the awards: William Adl

454 David Hayes, University of Tennessee; NAS ELECTS NEW MEMBERS Arnold Johnson, University of Notre Dame; Robert Latzer, Washington University, Mis­ The National Academy of Sciences souri; Fred Linton, Wesleyan University; recently elected 42 new members in recog• Jon Luke, California Institute of Technology; nition of their distinguished and continuing Nelson Max, Harvard University; Robert achievements in original research. The Robinson, Cornell University; James Rovn­ election took place April 26, 1966, during yak, Purdue University;Maxwell Shauck, Jr., the 103rd Annual Meeting of the Academy Tulane University; Lawrence Smith, Yale at its headquarters in Washington, D. C. University; Jonathan Sundow ,Institut Hautes Election to membership in the NAS Etudes; John Wagoner, Princeton Univer­ is considered to be one of the highest honors sity; Aaron Wyner, Bell Telephone Labora­ that can be accorded to an American scien­ tories, New Jersey. tist or engineer. Up to 42 members may be Graduate fellowships were awarded elected each year. Those elected to date to 1,998 individuals in the sciences, mathe­ bring the total to 745. matics, and engineering ior the academic Three mathematicians were among year 1966-1967. These awards provide basic those elected. They are Andrew M. Gleason, stipends (for 12 months) of $2,400 for the professor of mathematics, Harvard Univer­ first year level of graduate study, $2,600 sity; Irving Kaplansky, professor ofmathe­ for study at the intermediate level, and matics, University of Chicago; Stanislaw $2,800 for the terminal year. Additional Ulam, research advisor, director's office, allowances for dependents and for limited Los Alamos Scientific Laboratory. travel are also included. In lieu of tuition and fees, United States institutions will receive a standardized $2,500 cost-of­ education allowance for each Fell ow. At its own discretion each fellowship institution SOVIET PHYSICIST A WARDED may augment the stipend by not more than DANNIE HEINEMAN PRIZE $1,000 per year. Of the 1,998 fellowships, 433 were in mathematics, as compared to 279 in engineering; 626 in the physical The 1966 Dannie Heineman Prize for sciences; 321 in the biological sciences; Mathematical Physics was recently awarded and 339 in the social sciences and psycho­ to Academician Nikolay Nikolayevich Bogo­ logy. lyubov of the Soviet Union. The award, NSF awarded 910 Summer Fellow­ which includes $2500, is presentedannually ships for Graduate Teaching Assistants under the auspices of The American Institute for study and research in the summer of of Physics and The American Physical 1966. Of these awards, 156 were made in Society. mathematics; 88 in engineering; 322 in the Dr. Bogolyubov, who was unable to physical sciences, including a number in attend the ceremonial presentation, was interdisciplinary fields; 139 in the life cited "for several outstanding achievements sciences; and 205 in the social sciences in bringing the resources of modern math­ and psychology. Awardees will receive sti­ ematics to bear upon fundamental problems pends ranging from $50 to $85 for each in physics and, in particular, for the first week of their Summer Fellowships, the rigorous proof of dispersion relations for exact amount to be determined by each the nonforward scattering of elementary fellowship institution according to local particles." Dr. Bogolyubov is presently conditions. The Foundation will also pay director of the Laboratory of Theoretical the fellowship institutions the tuition and Physics of the Joint Institute for Nuclear fees incurred by these Fellows. Research in Dubna.

455 MEMORANDA TO MEMBERS

REQUEST FOR NAMES OF employers is applicable. Provision will be VISITING FOREIGN MATHEMATICIANS made for anonymity of applicants upon re­ quest and upon payment of $5,00 to defray The editors of these d'foficti) would the cost involved in handling anonymous appreciate receiving the names and ad­ listings. dresses of foreign mathematicians who will Applicants and employers who wish be visiting the United States in 1966-1967, to be listed should write to the Employment The information then will be published in Register, P. 0. Box 6Z48, Providence, the August issue of these c/'{otiui), in time Rhode Island OZ904 for applicant forms or for mathematics departments to extend position description forms. These forms speaking invitations for the coming .aca­ must be completed and returned to Provi­ demic year. The regular annual list of dence no later than July 15, 1966, in order foreign mathematicians will appear in the to be included in the August Lists. Those November issue, forms, which are received after the dead- line date and too late for inclusion in the printed lists, will be displayed at the meet­ THE EMPLOYMENT REGISTER ing. Employers who enter listings after the deadline dates are charged a fee of $5,00 The Mathematical Sciences Employ­ for each late listing. The printed lists will ment Register, established by the American be mailed to subscribers the first week in Mathematical Society, the Mathematical August and will be available for distribution Association of America, and the Societyfor both during and after the meeting. Industrial and Applied Mathematics, will be A subscription, which includes both maintained at the Summer Meeting at the Lists of Applicants and Positions pub­ Rutgers; The State University in New Bruns­ lished in January, May, and August, is wick, New Jersey from Tuesday, August 30, $15,00; a single issue of the List of Appli­ through Thursday, September 1, 1966, The cants and Positions may be purchased Register will be open from 9:00 A.M~ to for $7 ,50; and a List of Positions only is 5:00 P.M. in Level A, Frelinghuysen Hall. $3,00, Persons wishing either a year's It is important that applicants and employers subscription or individual issues should register as soon as possible after the make checks payable to the American Math­ Register opens to facilitate the arrange­ ematical Society and send them to the ment of appointments. Mathematical Sciences Employment Regis• There is no charge for registration, ter in care of the American Mathematical either to job applicants or to employers, Society, P ,0, Box 6Z48, Providence, Rhode except when the late registration fee for Island OZ904,

456 PERSONAL ITEMS

Professor B. A. AMIRA of the Hebrew and Operations Research, starting Septem­ University in Jerusalem, Israel, has been ber 1966. designated Chevalier de la Legion d'Honneur Professor L. K. DURST of Rice Uni­ by decree of the President of the French versity has accepted a position as Asso­ Republic in recognition of his contribution ciate Executive Director of the Committee to international cultural relations. for Undergraduate Programs in Mathemat­ Professor J. J. BENEDETTO of New ics, Berkeley, California. York University has been appointed to an Professor HERBERT FEDERER of assistant professorship at the University Brown University has been appointed to a of Maryland. Florence Pirce Grant University professor­ Mr. R. J. BENICE of the International ship. Business Machines Corporation, Federal Professor H. R. FISCHER of the Uni­ Systems Division, Rockville, Maryland re­ versity of Oklahoma has been appointed to ceived an award as co-author of the best a professorship at the University of Massa­ technica·l paper _published by the Institute of chusetts. Electronics Engineers during the last year. Professor H. T. FREITAG of Hollins Dr. G. M. BROWN of the University of College was appointed to a visiting pro­ Toronto has been appointed a Lecturer at fessorship at the University of Tennessee the University of Canterbury, Christchurch, for the academic year 1965-1966. New Zealand. Professor HARRY FURSTENBERG of Professor R. G. BUSCHMAN of SUNY the University of Minnesota has been ap­ at Buffalo has been appointed to a profes­ pointed to a professorship at the Hebrew sorship at the University of Wyoming. University, Jerusalem, Israel. Dr. F. B. CANN ONITO of the Hughes Dr. MARGARET GURNEY of the Bu­ Aircraft Company, Fullerton, California reau of the Census was honored by the has been appointed to an assistant pro­ United States Government for "meritorious fessorship at the University of California, Federal service." She was presented a U.S. Irvine. Department of Commerce Silver Medal Dr. D. L. CAUSEY of the University Award at ceremonies held in Washington. of Alabama Research Institute has accepted Dr. W. R. G. HAKEN of the Institute a position as Manager of Scientific Pro­ for Advanced Study has been appointed to gramming and Analysis with the Computer a professorship at the University of Illinois. Sciences Corporation, Richland, Washing­ Dr. T. E. HARRIS of The Rand Cor­ ton. poration, Santa Monica, California has been Professor FREDERICK CHONG of the appointed to a professorship at the Univer­ University of Auckland, New Zealand has sity of Southern California. been appointed to a professorship at Professor SIN HITOTUMATU of St. Macquarie University, Eastwood, Aus­ Paul's University, Tokyo, Japan will spend tralia. one year in the Applied Mathematics De­ Professor R. li. CORNELL of SUNY partment at the Brookhaven Natiom' T,abo­ at Stony Brook has been appointed to an ratory, Upton, New York starting St.. :em­ assistant professorship at Wellesley Col­ ber 1966. lege. Dr. R. E. HUGHS of Carleton College Professor Emeritus RICHARD COU­ has been appointed to an associate pro­ RANT of New York University, has been fessorship at Southern Illinois University elected a member of the Academy of at Edwardsville. Sciences of the U.S.S.R. Professor D. G. JAMES of the Univer­ Professor G. B. DANTZIG of the Uni­ sity of Auckland, New Zealand has been versity of California, Berkeley has been appointed to an assistant professorship at appointed a Professor of Computer Science Pennsylvania State University.

457 Dr. A. W. KIMBALL of Johns Hopkins Dr. D. P. ROSELLE of Duke Univer­ University has been named Dean of the sity has been appointed to an assistant Faculty of Arts and Sciences at the Univer­ professorship at the University of Mary­ sity's Homewood campus. land. Dr. R. D. KOPPERMAN of the Massa­ Professor JEANNETTE ROSENST ARK chusetts Institute of Technology has been of the City College (CUNY) has been ap­ appointed to an assistant professorship at pointed to an assistant professorship at the University of Rhode Island. Fair leigh Dickinson University, Rutherford. Professor K. 0. KORTANEK of the Mr. R. D. SECREST of the University University of Chicago has been appointed of Tulsa has accepted a position as Systems Associate Professor of the Department of Programmer/ Analyst with North American Industrial Engineering at Cornell Univer­ Aviation Incorporated, Tulsa, Oklahoma. sity. Professor D. M. SILBERGER ·of Idaho Dr. M. D. KUDLICK of the Massachu­ State UniverBity has been appointed to an setts Institute of Technology has accepted associate professorship at Butler Univer­ a position as Mathematician with the Shell sity. Development Company, Emeryville, Cali­ Mr. D. R. SMITH of Stanford Univer­ fornia. sity has been appointed a Visiting Member Dr. P. S. LANDWEBER of Harvard at the Courant Institute of Mathematical University has been appointed to an assis­ Sciences, New York University. tant professorship at the University of Dr. R. R. STEVENS of the University Virginia. of Arizona has been appointed to an assis­ Dr. CHIA-CHIAO LIN of the Massa­ tant professorship at the California State chusetts Institute of Technology has been College at Hayward. appointed to the distinguished rank of In­ Professor H. A. STILL of Alfred Uni­ stitute Professor. He is on leave this year, versity has been appointed to a visiting dividing his time between the Institute for professorship at Queen's University. Advanced Study and Harvard University. Dr. A. C. SUGAR of Los Angeles, Professor H. F. J. LOWIG of the Uni­ California has been appointed to a pro­ versity of Alberta will be on Sabbatical fessorship at Northern Michigan Univer­ leave during the academic year 1966-1967 sity. as a Visiting Fellow at the Australian Na­ Professor MASAHIRO SUGA WARA of tional University, Camberra, Australia. Kyoto University, Japan has been appointed Professor L. N. MANN of the Univer­ to a professorship at Hiroshima University, sity of Virginia has been appointed to an Hiroshima, Japan. associate professorship at the University ProfessorS. K. THOMASON ofCornell of Massachusetts. University has been appointed to an assis­ Professor TORSTEN NORVIG of the tant professorship at Simon Fraser Uni­ University of Massachusetts has been ap­ versity. pointed to an assistant professorship at Professor F. G. TRICOMI of the Uni­ Wellesley College. versity of Turin, Italy has been appointed Professor R. E. PRATHER of San Jose to a visiting professorship at the Institute State College has been appointed Associate for Fluid Dynamics and Applied Mathemat­ Professor of the Department of Electrical ics, University of Maryland. Engineering at Syracuse University. Professor D. F. WEI of McGill Univer­ Professor W. H. PURCELL, JR. of sity has been appointed to an assistant pro­ Fort Worth, Texas has been appointed to an fessorship at Marianopolis College. assistant professorship at the Georgia In­ Mr. J. B. WILLIAMSON of Indiana stitute of Technology. University has accepted a position as Assis­ Dr. A. S. REHM of North American tant, Data AnalysisJ with the National Aero­ Aviation Incorporated has been appointed nautical and Space Administration, Manned to an assistant professorship at Clarkson Spacecraft Center, Houston, Texas. College. Professor J. W. WYMAN of Pasadena Dr. E. D. ROACH of the University College has been awarded a National Sci­ of Texas has been appointed to an assistant ence Foundation Science Faculty Fellow­ professorship at McMurry College. ship for a 15 month period to complete his

458 Ph.D. in mathematics at the University of To Instructor: Southern California. Professor MIYUKI YAMADA of Sacra­ Benedict College: D. F. MONCK; Univer­ mento State College has been appointed to sity of California, Berkeley: DAVID HAN­ an assistant professorship at Shimane Uni­ DEL; University of California, Irvine: S. H. versity, Matsue, Japan. LACHENBRUCH; University of Chicago: Professor F. R. YETT of the Univer­ R. A. HUNT; De Paul University: C. L. sity of Texas has been appointed to Pro­ DEVITO; Haverford College: J. T. WOOD; fessor and Chairman of the Mathematics Lafayette College: ZAFARULLAH SHEIKH; Department at the University of South Lamar State College of Technology: D. R. Alabama. READ; Lehigh University: M. R. KIRCH; Professor E. C. ZACHMANOGLQU of Manhattan College: C. H. STOLZE; Mar­ Purdue University was awarded a Fulbright quette University: SHU-CHUNG KOO; Uni­ Research Grant at the University of Rome, versity of Maryland: G. E. LINDAMOOD; Italy for the academic year 1965-1966. University of Miami: P. J. O'HARA, JR; Modesto Junior College: WEI HSUAN; Naz­ The following promotions are announced: areth College of Rochester: IRA GEDAN; Northern Michigan University: S. F. ROD­ To Professor: FONG; Northwestern University: G. A. LORDEN; University of Portland: H. G. City College (CUNY): SOLOMON HURWITZ; LAWRENCE; Purdue University, Calumet Duke University: S. L. WARNER; University Campus: J. D. KADASHAW; University of of Pittsburgh: F. G. ASENJO. Richmond: S. W. SKIP PER; Trinity College: J. W. MARVIN; Wellesley College: R. L. To Associate Professor: PIERCE; Wichita State University: H. H. WATHAN. University of California, Berkeley: R. E. BARLOW;Oberlin College: G. H. ANDREWS; Deaths: Pennsylvania State University: E. H. UM­ BERGER;Stanford University: G. H. GOLUB. Professor M. B. KELLY of Wells Col­ lege died March 30, 1966 at the age of 62. To Assistant Professor: She was a member of the Society for 29 years. Brooklyn College: H. L. ZUCKERBERG; Dr. H. P. THIELMAN of Alexandria, University of California, Los Angeles: Virginia died February 17, 19 66 at the age P. A. LOEB; Wellesley College: B. L. of 64. He was a member of the Society for AUSLANDER. 37 years.

ACTIVITIES OF OTHER ASSOCIATIONS

The Northwest Computing Associa­ papers are solicited, Abstr~cts for con·­ tion will conduct its ninth annual computing tributed papers must be received by June 1, conference August 11-13, 1966, in Seattle, 1966. Additional information may be ob­ Washington. The theme for this year's con­ tained by writing to P. 0. Box 9394, Seattle, ference is time sharing, and contributed Washington 98009.

459 NEW AMS PUBLICATIONS

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Volume 16 THE TOPOLOGY OF THE CALCULUS OF VARIATIONS IN THE LARGE By L.A. Ljusternik 110 pages; List Price $9.30; Member Price calculus of variations in the large; applica­ $6.98. tions to nonlinear integral equations; criti­ cal sets of geodesic lines; the function This book presents the results ob­ space R of curves with common end points tained by applying cohomology theory, and on a sphere; closed geodesics on manifolds in particular the duality principle and the homeomorphic to then-dimensional sphere. Pontrjagin removing theorem, to problems of analysis in the large. The five chapter headings are: general principles of the

SELECTED TRANSLATIONS, Series II Volume 50 Volume 51 320 pages; List Price $9.60; Member Price 336 pages; List Price $10.10; Member Price $7 .20. $7 .58. Fifteen papers on Algebra by N. I. Eleven papers on Differential Equa­ Ahiezer, S. D. Berman, K. K. Billevic, tions, Functional Analysis and Measure I. V. Bogacenko, C. E. Dididze, D. B. Fuks, Theory, by I. A. Bahtin, S. V. Fomin, I. C. V. E. Govorov, P.M. Gudivok,A.H.Livsic, Gohberg, V. V. Grusin, V. E. Ljance, M. V. Ju. I. Manin, A. L. Oniscki, A. V. Ruko­ Keldys, M.A. Krasmosel'skii, M.G. Krein, lalne, A. B. Sidlovskii, S. P. Demuskin. L. H. Liberman, S.M. Nikol'skii, A. Povz­ ner, P. E. Sobolevskii, V. G. Sprindzuk.

Volume 52 Volume 53 280 pages; List Price $8.40; Member Price 288 pages; List Price $8. 70; Member Price $6.30. $6.52. Twelve papers on Topology, Algebra, Eleven papers on Analysis, by I. I. and Number Theory, by M. M. Postnikov, Rjabcev, M.S. Birman, R. M. Martirosjan, V. Kuz'minov, L. V. Keldys, Ju.A.Kaz'min, M. G. Gasymov, B. M. Levitan, D. M. E!dus, I. M. Gel'fand, G. E. Silov, V. B. Demranov, Ju. A. Dubinski!, V.I. Arnol'd, V.P.Havin, V. A. Andrunakievic, V. P. Elizarov, B. I. V. K. Dzjadyk. Plotkin, I. C. Gohberg, A. S. Markus, E. V. Novoselov.

460 MEMOIRS

Number 60 TORSION IN SU-BORDISM By P. E. Conner and E. E. Floyd 76 pages; List Price $1,50; Member Price torsion consists entirely of elements of or­ $1.13. der 2 and the 2-rank is as in the preceding case. There is no torsion in any other di­ In this paper the authors compute the mensions. Chapter I is an exposition of stable homotopy of the MSU-spectrum, thus stable structures on bundles, especially we determine additively the coefficient ring tangent bundles. Chapter II sets up a cer­ for the generalized homology functor su­ tain chain complex and computes its hom­ bordism. Denoting the bordism groups of a ology and in Chapter III it is shown that this point by f n we show that if n = Bk + 1 then is the crucial step in computing SU -bordism. rn consists entirely of elements of order 2 For completeness, we have included a good and that its 2-rank is equal to the number deal of expository material, of partitions of k, For n = Bk + 2 the 2-

Number 61 A NEW FORMULATION OF PARTICLE MECHANICS By Reese T. Prosser 64 pages; List Price $1.50; Member Price particles as special cases. Questions of $1.13, classification, representation and inter­ pretation are taken up in turn. In particular, This paper presents an abstract it is shown that under this formulation every formulation of particle mechanics and de• system of mechanics is a suitable com­ rives its most elementary properties. The bination of the two special cases, so that formulation is based on a suitable generali­ no essentially i:lifferent systems are pos­ zation of the moment problem, and is de­ sible, signed to include both the classical and the quantum mechanics of systems of point

Recent Reprints

COLLOQUIUM PUBLICATIONS Volume 20 INTERPOLATION AND APPROXIMATION BY RATIONAL FUNCTIONS IN THE COMPLEX DOMAIN By J, L. Wal.sh 416 pages; List Price $19.10; Member Price is relevant in view of the present day in­ $14,33, terest not only in approximation by func­ tions of this type, but also in such generali­ Since its first appearance more than zations as approximations by bounded analy· thirty years ago, the standard treatise of tic functions. Walsh has continued to provide a basis for In addition to the original treatise, extensive research in interpolation and this edition contains an appendix on recent approximation. This fourth edition of the work in approximation by rational functions 1935 Colloquium volume, which deals chiefly and the references have been expanded to with approximation by polynomials and by include later works. rational functions with preassigned poles,

461 Volume 9 DYNAMICAL SYSTEMS By G. D. Birkhoff 3Z4 pages; List Price $6.00; Member Price tion contains a special preface by Marston $4.50, Morse. Jiirgen Moser has written an intro­ duction and added new footnotes, an adden­ In the 40 years since its first pub­ dum, and a bibliography of pertinentlitera­ lication, G. D. Birkhoff's book on dynamical ture which has appeared since 19 Z 7. systems has been a source of inspiration to many mathematicians. The present edi-

MEMOIRS

Number 16 PRODUITS TENSORIELS TOPOLOGIQUES ET ESPACES NUCLEAIRES By A. Grothendieck 340 pages; List Price $4.90; Member Price will form the natural framework for such $3,68. theories as those of distributions and analy­ tic functionals. The concept of a "nuclear space" An appendix has been added to this is the result of the author's desire to create reprint of the 1955 edition. a special class of locally convex spaces that

Number Z6 PRINCIPAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN THE COMPLEX DOMAIN By W. Strodt llZ pages; List Price $3,00; Member Price P is a polynomial in y,y' , ... ,y(n) with co­ $2,Z5. efficients analytic in x, In the present In this Memoir, as in his earlier Memoir ,reprinted with no essential changes from the 1955 edition, the analysis offirst­ one (No, 13, 1954), the author studies the order equations is made more precise and minimal rate of growth for large lx I of the efficient, and exact solutions of minimal solution of a nonlinea; diffl~)ntial equation rate of growth are obtained for a broad of the form P(x,y,y , ... ,y ) = O, where class of equations of arbitrary finite order n,

Number Z7 THE GENERALIZED PONTRJAGIN COHOMOLOGY OPERATIONS AND RINGS WITH DIVIDED POWERS By Emery Thomas 84 pages; List Price $1,80; Member Price states, for a space X and its cohomology $1.35, with coefficients in a p-cyclic ring with divided powers, that the cohomology ring The generalized Pontrjagin coho­ of the space is itself a (bigraded) ring with mology operation Pt(t = 0,1,Z, ...) is identi­ divided powers. The second theorem gives cal for t = Z with the Pontrjagin square certain properties of the model operations, cohomology operation, The present memoir and, consequently of the Pt themselves, gives the proofs, by means of subsidiary leading to information not obtained from functions Pt called "model operatio.ns," of other known cohomology invariants. two "main theorems," The first theorem

462 Number 33

COHOMOLOGY THEORY AND ALGEBRAIC CORRESPONDENCES By Ernst Snapper

100 pages; List Price $2.00; Member Price complex Ki has as cohomology groups $1.50. those of Xi with values in Oi. We can then prove, by analyzing K by means of spectral Let X1 and X2 be two irreducible sequences, that if T has no fundamental normal projective varieties defined over an points on x , then h'(X ,o ) ~ h'(X ,o ) arbitrary fixed ground field k and let T be 2 1 1 2 2 and if T is birational and r is the common an irreducible algebraic correspondence dimension of x and x , then hr(x ,o ) between them. Denote the sheaf of local 1 2 1 1 ~ hr(x2 ,o 2). For general Tit is possible rings of Xi by Oi and consider a double to extend a large part of Serre's theory of complex K whose cohomology groups are "faisceaux algebrigues coherents" to alge­ those of x3 (where x3 denotes the graph braic correspondences. of T) with values in 03, and whose edge

MATHEMATICAL SURVEYS Volume 2 THE THEORY OF RINGS By N. jacobson

160 pages; List Price $5. 70; Member Price for ideals (except in the lastchapter,where $4.28. rings of the type of a maximal order in an This is a reprint of the volume 2 algebra are considered). The central idea edition of MATHEMATICAL SURVEYS, pub­ consists of representing rings as rings of lished in 1943. The book is mainly con­ endomorphisms of an additive group, which cerned with the theory of rings in which can be achieved by means of the regular both maximal and minimal conditions hold representation.

Volume 3

GEOMETRY OF THE ZEROS OF A POLYNOMIAL IN A COMPLEX VARIABLE By Morris Marden

251 pages; List Price $11.00; Member of Cauchy and Gauss. Thus, the number of Price $8.25. entries in the bibliography of this edition had to be increased from about 300 to about During the seventeen years since the 600 and the book enlarged by one third. It first edition of this well-known monograph now includes a more extensive treatment appeared, the subject (the geometry of the of Hurwitz polynomials and other topics. zeros of a complex polynomial} has con­ The new material on infrapolynomiais, tinued to display the same outstanding abstract polynomials, and matrix methods vitality as it did in the first 150 years of is of particular interest. its history, beginning with the contributions

463 SUPPLEMENTARY PROGRAM-Number39

During the interval from February 12, 1966 to April 28, 1966 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Contributed Papers in this issue of these cJioriceiJ. One abstract presented by title may be accepted per person per issue of these c}/olit;ci). Joint authors are treated as a separate category; thus in addition to abstracts from two authors individually one joint abstract by them may be accepted for a particular issue. ( 1) Some recursively undecidable prob­ University (66T-235) lems in automat,l theory and quantifi­ ( 1 O) The injective envelope in the category cation theory of semigroup representations Mr. S. 0. Aanderaa, Harvard Uni­ Professor Pierre Berthiaume, Uni­ versity (66T-327) versity of Ottawa (66T-307) (Introduced by Professor Hao Wang) ( 11) Existence and characteristic proper­ (2) Aleph limit points of sets of trans­ ties of maximal and minimal volume finite dyadic sequences spaces generating the same integration Professor Alexander Abian, The as a given volume Ohio State University (66T-322) Professor Witold Bogdanowicz, The (3) No r .c. set is the union of finitely Catholic University of America (66T- many immune retrac·~able sets 236) Professor K. I. Appel, University of ( 12) A note on a problem of Mr6wka Illtnois (66T-261) Professor C. R. Borges, University (4) A transformation with no invariant of California, Davis (66T-305) measure ( 13) A theorem on regular graphs Mr. L. K. Arnold, Brown University Professor R. C. Bose, University of (66T-271) North Carolina (66T-316) ( 5) A necessary condition for 4-coloring ( 14) A characterization of tetrahedral a planar graph graphs Professor Glen Baxter, Purdue Uni­ Professor R. C. Bose and Dr. Renu versity (66T-252) Laskar, University of North Caro­ ( 6) A polynomial identity for symmetric lina at Chapel Hill ( 66T- 31 7) elements and commutativity. Prelim­ ( 15) Undecidability of the elementary the­ inary .report ory of Dedekind cardinal addition. Professor W. E. Baxter, University Professor Robert Bradford, Univer­ of Delaware (66T-282) sity of Southern California (66T-296) (7) Characteristic polynomials of sym- ( 16) Extension to nonconvex, closed sets of metric matrices Motzkin' s theorem on convexity Mr. E. A. Ben\fer, California Insti­ Dr. Lorenzo Calabi and Dr. W. E. tute of Technology and Institute for Hartnett, Parke Mathematical Labo­ Defense Analyses (66T-262) ratories, Inc., Carlisle, Massachu­ (8) Remarks on the method of interpola­ setts (66T-293) tion by functional quasinorms ( 17) Integration over fields with a non­ Mr. C. A. Berenstein, University of archimedean valuation Buenos Aires, Argentina (66T-3ll) Mr. V. R. CeLeste, Polytechnic In­ (Introduced by Mr. Mischa Cotlar) stitute of Brooklyn (66T-272) (9) Special classes of subordinate func­ ( 18) On boundary behavior of Bergman's tions kernel function in the theory of two ProfessorS. D. Bernardi, New York complex variables

464 Mr. B. L. Chalmers, Stanford Uni­ (32) A representation theorem for additive versity (66T-285) functionals (Introduced by Professor Karel de Leeuw) Professor N. A. Friedman and Mr. ( 19) Transformations preserving tame sets M. L. Katz, University of New Mexico Mr. H. J. Charlton, Virginia Poly­ (66T-312) technic Institute (66T-324) (33) On the Kato-Taussky- Wielandt com­ (20) Picard-Lefschetz theorem for families mutator relation. Preliminary report of algebraic varieties acquiring cer­ Mr. Fergus Gaines, Californiainsti­ tain singularities tute of Technology (66T:294) Mr. C. H. Clemens, Jr., University (34) An exact description of the concepts of California, Berkeley (66T-244) class and set (21) The Baer radical of the standard Dr. B. Germansky-Gershuni, Am­ wreath product sterdam, Netherlands (66T-313) Mr. C. R. Combrink, The University (35) One-way stack automata. II of Kansas (66T-256) Dr. Seymour Ginsburg, Systems De­ (22) Almost sure convergence of spatial velopment Corporation,Santa Monica, averages for a branching process of California, ProfessorS. A. Greibach, Brownian particles Harvard University and Professor Professor H. E. Conner·, University M. A. Harrison, University of Cali­ of Wisconsin, Madison (66T-242) fornia, Berkeley (66T-233) (23) Locally cyclic continua. II (36) The bracket function, q-binoml~l co­ Mr. A. C. Connor, University of efficients, and some new Stirling num­ Georgia (66T-240) ber formulas (24) On a new class of functions Professor H. W. Gould, West Vir­ Professor J. T. Darwin, Jr., Auburn ginia University (66T-241) University (66T-301) (37) State diagrams for Hilbert spa<.'e op­ (25) On merom orphic functions of two com­ erators plex variables Dr. Karl Gustafson, Universita de Mr. E. B. Davis, Stanford University Roma, Italy (66T-274) (66T-276) (38) Generallzation of the Schwarz-Pick (26) Ideal decompositions in r-rings. Pre­ lemma to invariant volume in a Kahler liminary report manifold Miss Marjorie A. Enneking, Wash­ Professor K. T. Hahn and Professor ington State University (66T-263) Josephine Mitchell, Pennsylvania (27) A generalization of the Radon-Nikodym State University (66T-315) theorem (39) A maximum principle for a mathemati­ Mr. Charles Fefferman, University cal programming problem in infinite of Maryland (66T-24"8) dimensional space (28) Fixed point theorems and Galerkin Professor Hubert Halkin, University approximations of California, San Diego (66T-265) Mr. D. G. de Figueiredo, University (40) A property of Hilbert spaces. Prelim­ of Chicago (66T-292) inary report (29) Infi.1ite recursively enumerable clas­ Mr. T. L. Hayden, University of ses with any finite number of infi 1~te Kentucky (66T-300) recursively enumerable subclasses (41) Markov processes. I Mr. J. B. Florence, Simon Fraser Professor Naresh Jain and Professor University (66T-281) Benton Jamison, University of Min­ ( 30) Error bounds on numerical solutions of nesota (66T-247) Dirichlet problems for quasilinear el­ (42) Continuity of solution of a generalized liptic equations diffusion equation Mr. T. G. Frank, Los Alamos Sci­ Professor M. P. Jones, Southwestern entific Laboratory, Los Alamos, New at Memphis (66T-267) Mexico (66T-287) (43) Varieties of groups of nilpotency three ( 31) Strength of certain set theories Professor Bjarni Jonsson, Univer­ Mr. Harvey Friedman, Massachu­ sity of Minnesota (66T-258) setts Institute of Technology ( 66T- (44) Spectral theory of Wiener-Hop£ type 280) integral operators on generalized

465 Sobolev spaces versity of Washington(66T-237) Professor j. S. Kalme, Drexel Insti­ (57) S)P spaces of harmonic functions tute of Technology (66T-329) Professor Linda Lumer-Naim, Uni­ (45) A convergence criterion for multiple versity of Washington (66T-237) ratios (Introduced by Mr. Ronald Getoor) Dr. Tatsuji Kambayashi, Universita (58) On the enumeration of finite topologies de Pisa, Italy (66T-277) Mr. M. S. Lynn, Professor Frank (46) Intermediate spaces of n + 1 quasi­ Harary and Mr. j. W. Evans, Inter­ normed spaces national Business Machines,Houston, Mr. N. L. Kerzman, University of Texas (66T-260) Buenos Aires, Argentina (66T-310) (59) Sets of uniqueness and sets of multi­ (Introduced by Mr. Mischa Cotlar) plicity ( 47) Riemann functions for a linear system Mr. 0. C. McGehee, University of of "hyperbolic like" form California, Berkeley (66T-319) Dr. S. R. Kraft, Negev Institute, (60) Splitting and decomposition by re­ Israel (66T-308) gressive sets. II (48) Measures in function spaces Professor T. G. McLaughlin, Uni­ Professor j. D. Kuelbs, University versity of Illinois (66T-264) of Wisconsin, Madison (66T-250) (61) Spaces determined by left ideals of (49) A new interpretation of the method of semigroups stationary phase Professor K. D. Magill, Jr., SUNY Professor Ali Kyrala, Arizona State at Buffalo (66T-326) University (66T-246) (62) Locally connected images of ordered (50) Nonaxiamatizability of the elementary compacta are locally peripherally met­ theory of standard enumerations of ric recursiv~ly enumerable sets Professor Sibe Mardesic, University Mr. A. H. Lachlan, Simon Fraser of Zagreb, Yugoslavia and Univer­ University (66T-284) sity of Washington (66T-320) (51) Disjointing infinite sums in Boolean (63) On an analogue of the Ramanujan sum algebras Mr. A. A. Mullin, University of Cali­ Mr. Robert LaGrange, University fornia, Livermore (66T-279) of Colorado (66T-291) (64) On kth power coset representatives and (Introduced by Professor j. D. Monk) diagonal congruences modulo a prime (52) A topological property of Bing's de­ power. Preliminary report comp'.)Sition of E into points and tame Mr. K. K. Norton, University of arcs Illinois (66T-314) Mr. H. W. Lambert, University of (65) Representing measures and hypo­ Utah (66T-321) Dirichlet algebras (53) A characterization of cubic lattice Professor B. V. O'Neill, New York graphs University (66T-295) Dr. Renu Laskar, University of (66) Some results on lengths of proofs North Carolina at Chapel Hill (66T- Mr. Rohit Parikh, University of 318) of Bristol, England (66T-255) (Introduced by Professor R. C. Bose) (67) Radial symmetrization and capacities (54) Topological spaces with large uni- in space. Preliminary report formities Professor J. A. Pfaltzgraff, Indiana Professor Yu-Lee Lee, University University (66T-328) of Florida and Mr. C. J. Mozzochi, ( 68) Orbits of analytic functions University of Connecticut (66T-269) Professor W. A. Pranger, Jr., De­ (55) Maximal subgroups of the kernel of a Paul University (66T-325) semigroup with left zeroids ( 69) A knotted cell pair with knot group Z Professor C. W. Leininger, Arling­ Professor T. M. Price, University ton State College (66T-304) of Iowa (66T-249) (56) A polynomial ap.tJr••ach to topological ( 70) Extrema concerning asymmetric analysis graphs Professor K. 0. Leland, University Professor L. V. Quintas, St. John's

466 University, New York (66T-243) Mr. Brian Smyth, Brown University ( 71) Bisimple inverse semigroups (66T-238) Professor N. R. Reilly, Tulane Uni­ (85) Bi-arc boundary functions versity, New Orleans (66T-299) Professor L. E. Snyder, University ( 72) On the stability of some algebras of of Virginia (66T-259) absolutely convergent Fourier series (86) A minimal hull theorem for operators Mr. N. M. RiviereandMr. Y.Sagher, Professor J. G. Stampfli, New York University of Chicago (66T-270) University ( 66T-234) (73) A generalization of the Bieberbachco­ (87) Multiplication in Grothendieck rings efficient problem fgr univalent func­ of integral group rings tions Mr. D. L. Stanci, University of Professor M. S. Robertson, Rutgers, Illinois ( 66T- 302) The State University (66T-306) (88) On the ratio ergodic theorem (74) Holomorphy of Bol loops Professor Louis Sucheston, Ohio Professor D. A. Robinson, Georgia State University (66T-273) Institute of Technology (66T-309) ( 89) The factorization of the ad jugate matrix (75) A remark on the double equichordal Dr. Olga Taussky, California Insti­ point problem. Preliminary report tute of Technology (66T-275) Professor E. E. Robkin, University (90) Uniform continuity and general topology of Massachusetts (66T-332) Professor D. V. Thampuran, SUNY (76) Real-time definable languages at Stony Brook (66T-245) Dr. A. L. Rosenberg, IBM Watson (91) On the stability of midpoint smoothing Research Center, Yorktown Heights, with Legendre polynomials New York (66T-286) Professor W. F. Trench, Drexel (Introduced by Dr. Richard Goldber·g) Institute of Technology (66T-289) (77) Second structure theorem for com­ (92) Differences, convolutions, primes. V plemented algebras. Preliminary re­ Mr. Benjamin Yolk, Yeshiva Uni­ port versity (66T-232) Professor P. P. Saworotnow, The (93) Symmetric addition Catholic University of America ( 66T- Mr. Ralph Yolk, Far Rockaway High 268) School (66T-298) (78) On fractional integration operators (Introduced by Mr. Benjamin Yolk) Dr. R. K. Saxena, McGill University (94) Congruences on 1-bisimple semi­ (66T-330) groups. I (79) On Hs-bounded operators Professor R. J. Warne, West Vir­ Professor Martin Schechter, Insti­ ginia University (66T-231) tute for Advanced Study (66T-278) (95) Open continuous mappings satisfying a (80) Nilpotent automorphisms completeness condition. Preliminary Professor Eugene Schenkman, Pur­ report due University (66T-303) Dr. H. H. Wicke, Sandia Corporation, (81) The maximal T 0 (respectively T 1) Albuquerque, New Mexico (66T-333) subspace lemma is equivalent to the (96) Polynomials associated with meas­ axiom of choice ures in the complr:!x: plane Mr. P. S. Schnare, Louisiana State Professor Harold Widom, University University in New Orleans (66T-290) of California, Berkeley (66T-257) (82) Symmetry of complex involutory Ba­ (97) Boundedness and unboundedness re­ nach algebra. Preliminary report sults for some nonlinear differential Mr. Satish Shirali, Harvard Univer­ equations sity (66T-239) Professor D. W. Willett and Pro­ (83) Simple rings with involution. Prelim- fessor J. S. W. Wong, University of inary report Alberta (66T-266) Mrs. Lynne Small, 2525 Le Conte (98) Local completeness and a variation of Avenue, B,~rkeley, California (66T- Arhangels'kii's base of countable or­ 323) der concept. Preliminary report (8,1) Differential geometry of complex hy­ Dr. J. M. Worrell, Jr., Sandia Cor­ persurfaces poration, Albuquerque, New Mexico

467 (66T-331) ( 1 00.) A distributional convolution transfor­ (99) Restrictions of uniformly monotoni­ matidn cally complete open continuous map­ Professor A. H. Zemanian, SUNY at pings. Preliminary report Stony Brook (66T-251) Dr. j. M. Worrell, Jr. and Dr.H.H. (101) Some theorems on the axiom of choice Wicke, Sandia Corporation, Albu­ for finite s~ts. II querque. New Mexico (66T-334) Mr. M. M. Zuckerman, New York Unive"['sity (66T-254)

NEWS ITEMS AND ANNOUNCEMENfS

EIGHTEENTH BRITISH Further information concerning the MATHEMATICAL COLLOQUIUM Symposium may be obtained from the Or­ ganizing Committee, Matematicky tistav The Eighteenth British Mathematical GSA V, Zitna 25, Praha 1, Czechoslovakia. Colloquium was held at Imperial College, London, from March 29 to April 2, 1966, THIRD INTERNATIONAL CONGRESS under the chairmanship of Professor j. G. FOR LOGIC, METHODOLOGY AND Clunie, with Dr. T. Kovari as secretary. PHILOSOPHY OF SCIENCE About four hundred mathematicians attended the colloquium. The Netherlands Society for Logic There were three hour lectures and and Philosophy o{ Science announces that nine forty-minute lectures. Speakers in­ the Third International Congress for Logic, cluded B. Segre, R. V. Kadison, I. Kaplan­ Methodology and Philosophy of Science will ·sky, j.F.P.Hudson,R.H.Bott,G. A. Dirac, be held in Amsterdam from August 25 to C. Pommerenke, j. F. C. Kingman, D. A. September 2, 1967. The Congress will be Burgess, A. L. S. Corner, J. E. Roseblade held under the auspices of the Division of and D. G. Northcott. In addition, informal Logic, Methodology and Philosophy of Sci­ splinter groups were organized for short ence of the International Union of History lectures and discussions. and Philosophy of Science. Arrangements were made to conduct The Congress will be organized into the Nineteenth Colloquium at the University the following sections: mathematical logic, of Swansea, from April 4 to April &, 1967. foundations of mathematical theories, auto­ mata and programming languages, philos­ ophy of language and mathematics, general problems of methodology and philosophy of science, methodology and philosophy of physical sciences, methodology and philos­ SECOND SYMPOSIUM ON GENERAL ophy of biological sciences, methodology TOPOLOGY AND ITS RELATIONS TO and philosophy of psychological sciences, MODERN ANALYSIS AND ALGEBRA methodology and philosophy of social sci­ ences, methodology and philosophy of lin­ The Second Symposium on General guistics, and history of logic, methodology Topology and its Relations to Modern Analy­ and philosophy of science. sis and Algebra will be held in Prague from Additional information may be ob­ August 30 to September 4, 1966. The Sym­ tained from the Secretariat, in care of the posium will be organized by the Czecho­ Holland Organizing Centre, 16 Lange Voor­ slovak Academy of Sciences. hout, The Hague, the Netherlands.

468 ABSTRACTS OF CONTRIBUTED PAPERS The April Meeting inNew York April4-7, 1966

632-72, G. T. McALLISTER, Lehigh University, Bethlehem, Pennsylvania. A priori bounds on difference quotients of discrete potential functions.

Let>! be a disc. Then PiE >lh iff Pi = (xi'yi), (xi+ h,yi), (xi- h,yi' (xi'yi- h), and (xi' yi + h) En. Let U be the vector such that ~h Ui = 0 for Pi E >lh and Uj = v j for P j E an h. Let D v, D v, ... ,D( )v denote difference quotients of v along 8>1h and let D(k)u denote any kth s ss ps 2 order partial difference quotient of U. Then max0 hiD(l)UI ~ 50M1, max0 hiD( )UI ~ 42M2 , and 3 max0 hiD( )Ui ~6M 3 where Mp = maxanhiD(ps)vl. We have continued the Dirichlet data into a neighborhood of the 8n so that Mp is independent of h. Such estimates are important in solving quasilinear problems. (Received February 21, 1966.)

632-73. A. M. DuPRE, Howard University, Washington, D. C. Continuously split extensions of locally compact groups.

If E : 0 ---> H-->i G--> jG/H ---> 0 is an algebraically exact sequence of separable locally compact groups, with i and j continuous, E is said to be an exact sequence of locally compact separable groups. E is said to split continuously if there is a continuous section s: G/H ---> G for j: G ---> G/H.

The major theorem proved is Theorem. If E: 0 ---> H---> iG ___,jG/H ---> 0 is an extension of locally compact separable groups, and if H is central and isomorphic to IRn X Tm with G/H abelian, then E splits continuously. The proof uses machinery involving spectral sequences in the cohomology theory of locally compact groups, as developed further in the author's dissertation and first intro­ duced by C. C. Moore. (Received February 19, 1966.)

632-74. MORTEZA ANVARI, University of British Columbia, Vancouver, B. C., Canada, A topology on a measure space.

Let (X,2:,p.) be a measure space on which no topology is presupposed. If a covering theorem of Vitali type exists for X (or A C X), then it is possible to topologize X (or A) by constructing a family of closed sets out of the covering family. This topology is T 1• (Received March 9, 1966.)

632-75, J. F. KENNISON, Clark University, Worcester, Massachusetts. Generalized injectivity and reflective subcategories.

Let C be a category. Maranda [Trans, Amer. Math. Soc. 110 (1964), 98-135) defines an object

X to be injective w. r. t. a morphism f: A ---> B if every g: A---> X factors so that g = hf, If S is a class of morphisms of C then clo(S) is the class of all objects which are injective w. r. t. every member of S. If Q is a class of objects of C then V(Q) is the class of all morphisms w. r. t. which every member of Q is injective. (S,Q) is an injective structure on C if Q = clo(S), S = '1/(Q) and if

469 every object X admits a completing morphism f: X ----> Y for which f E S and Y E Q. The structure (S,Q) is regular if Q is the class of objects of a full reflective subcategory of C. Using Freyd's theorem on reflective subcategories, as formulated by Isbell {?acific ]. Math. 14 (1964), 1276], and related bicategory techniques, we obtain an explicit description of when certain injective structures associated with the Jacobson-Chevalley and Brown-McCoy radicals are regular. This answers, in part, a question raised by Maranda [op. cit.]. The same techniques yield a new condition on full sub­ categories necessary for reflectivity. We also obtain necessary and sufficient conditions for a set of morphisms to be a subset of the class of all universal junctions associated with a full subcategory. (Received March 17, 1966.)

The April Meeting in Honolulu April9, 1966

633-33. R. P. DILWORTH, 1201 East California Boulevard, Pas adena, California. Contiguous mappings of partially ordered sets.

A mapping f of a finite partially ordered set P into itself is contiguous if the images of any pair of covering elements are either equal or a covering pair. More generally, f is k-contiguous if the images are joined by a covering sequence of at most length k. Several theorems on families of k-contiguous maps are proved. A simple example is the following: Any commuting family of conti­ guous mappings of a finite totally ordered set into itself has a common generalized fixed point, A generalized fixed point is either an ordinary fixed point or a pair of covering elements which are interchanged by the mapping. (Received February 21, 1966.)

633-34. H. M. LIEBERSTEIN, Indiana University, Bloomington, Indiana 47401. Numerical studies of the steady state equations for a Hodgkin-Huxley model. Preliminary report.

It has been found (see Abstract, Amer. Math. Monthly) that effects of line inductance and cap­ acitance can be included in the Hodgkin-Huxley model (J. Physiol. 117 (1952), 500-544) for develop- ment and propagation of voltage impulses on nerve fibers without increasing the number of para­ meters utilized. Using a system of partial differential equations, it was shown that from a -15mv square or sawtoothed wave impulse applied at a cut end of the axon a -lOOmv "steady state" wave developed. The wave agreed well with those obtained from previous computations, and the extreme sensitivity to propagation rate was removed. The steady state which developes at some distance from the cut end was found to satisfy the wave equation, and this gave four first order ordinary differ­ ential equations describing the steady state. A forcing term that provides an initial sawtoothed impulse is here added to the voltage equation. This brings the voltage to threshold starting from an initial zero voltage, and thus allows the computation of initial conductances, Runge- Kutta computations of the propagated action potential then give none of the former "splitting" difficulties (R. Fitzhugh and H. A. Antosiewicz, J. Soc. Indust. Appl. Math. vol. 7, no. 4, p. 456). (Received February 21, 1966.)

470 633-35. E. G. STRAUS, University of California, Los Angeles, California 90024. On a problem of P. Erdos in combinatorial number theory.

P. Erdos has asked for the maximal number k of integers·a 1, ••• ,ak; 1 ~ ai ~ n so that two

sums, ai 1 + ••• + ail = aj 1 + ••• + ajm of distinct elements from {a l'"""'akJ are equal only if J = m. He conjectured that the maximal k is attained by choosing the ai at the upper end of the interval (l,n). This would yield k -(2JI2. It is proved that k < c(n) 1/ 2 by showing that the number of distinct sums of £-tuples from fa 1, ••• ,ak} is minimal when the ai are in arithmetic progression. (Received February 21, 1966.)

633-36. D. S. CARTER, Oregon State University, Corvallis, Oregon 97331. The exponential of a measure space.

Let exp X denote the class of all unordered sequences of arbitrary length n ~ 0 in a set X.

Let the concatenation of two unordered sequences a= a 1 ••• am, b = b 1 ••• bn be defined by ab = a 1 ••• amb1 ... bn' the symmetric product of two subsets A,B of exp X be defined by AB = [ ab: a E A, b E B}, and the exponential of the subset A be the union of symmetric powers, exp A= LAn. Then the exponential law holds in two forms: (1) (exp A) (exp B)= exp(A + B). (2) Let the pair A,B be unlinked, in the sense that there exist disjoint subsets Y,Z of X such that A C Y, B C z. Then the direct product (exp A) X (exp B) is isomorphic, under concatenation, to exp(A + B). Now let J1 be a u-algebra of subsets of X, and 11. be a u-finite measure on df. Then there is a corresponding u-algebra exp cJ'f induced naturally on exp X, and a u-finite measure exp 11. on exp cJf-, such that the t-wo exponential laws hold, with subsets of exp X replaced by subspaces of the measure space (exp X, exp cl'f, exp II.). The same is true for the completion of this space, if direct products are replaced by their completions. The second exponential law is useful in statistical applications, by invoking the standard theory of product measures whenever the space X is decom­ posed into a disjoint union. (Received February 21, 1966.)

633-37. ADOLF MADER, University of Hawaii, Honolulu, Hawaii 96822. A note on direct and semi-direct products of groups.

G a group, N a normal subgroup of G, f: G --+ G/N the natural homomorphism, h E Hom(G/N,N). The map 1 + fh: G --+ G: g(1 + fh) = g(gfh) Is one-to-one and onto. Theorem. Let G = N X M. Then there is a one-to-one correspondence between all semidi'rect complements of N in G and Hom(G/N,N). Such a correspondence is given by h--+ M(1 + fh) where f, h are as above. M(l + fh) is a direct com­ plement of N if and only if h E Hom(G/N, center N). (Received April 13, 1966.)

471 The June Meeting in Victoria, British Columbia June 18, 1966

635-1. S. K. THOMASON, Simon Fraser University, Burnaby 2, British Columbia, Canada. A natural transfinite hierarchy of hyperdegrees.

Define a hierarchy of hyperdegrees Jl(a.) for certain countable ordinals a. by: .!J.(O) = (hyperdegree of N), h(a + 1) (hyper jump of !!(a.)), h(A) = lub [h

It is shown that h.(a.) is defined for all a. less than the limit 110 of the sequence o0 = w1, 0 n+ 1 = w ~(on>.

Let 111 be the least a. such that ll(a.) is not defined. The only ordinals w~ < 111 are the ordinals w~(a.). w~(a.) < w~ iff !!(a.+ 1) ~ (hyperdegree of A). If >.. < 110 is a limit ordinal, then lim [w~(a.)la. < AJ is A not of the form w·1 • (Received February 7, 1966.)

635-2. R. W. STRlNGALL, University of California, Davis, California 95616. Endomorphism rings of p-primary abelian groups.

Let E(G) denote the endomorphism ring of the Abelian p-group G, and define H(G) = {a E E (G): x E G, px = 0 and height x < oo imply height a(x) >height x J. Then H(G) is an ideal in E(G) which contains the jacobson radical of E(G). It is known that the factor ring E(G)/H(G) is naturally isomorphic to a subring R of a direct product n~ 1 En with L: 1 En contained in R and where each En is the ring of all linear transformations of a Zp space whose dimension is equal to the n - 1 Ulm invariant of G. Theorem. Let R be any countable subring of flv Z which contains "0 p the identity and LxoZP. Then there exists a p-group G with a standard basic subgroup and containing no elements of infinite height such that E(G)/H(G) is isomorphic to R. Moreover, G can be chosen without proper isomorphic subgroups; in this case, H(G) is the Jacobson radical of E(G). (Received February 15, 1966.)

635-3. P. S. SCHNARE, Louisiana State University, New Orleans, Louisiana 70122. The ultra­ space theorem is equivalent to the Boolean prime ideal theorem. Preliminary report.

A topology on a set E is proper iff it is nondiscrete. A proper topology Ton E is an ultraspace iff the only topology on E which properly contains Tis the discrete topology. The ultraspace theorem (U) states that every proper topology is contained in an ultraspace. The Boolean prime ideal theorem (B) states that every proper ideal in a Boolean algebra is contained in a prime ideal. Theorem.

U ~B. The proof uses the fact that B is equivalent to the statement that every filter is contained in an ultrafilter, implicit in A. Tar ski, Prime ideal theorems for set algebras and the axiom of choice.

Preliminary report, Bull. Amer. Math. Soc. 60 (1954), 391. (That B implies U is implicit in 0. Frohlich, Das Halbordnungssystem der topologischen Raume auf einer Menge, Math. Ann. 156

(1964), 79-95.) It follows as a corollary of the theorem that several strong and interesting topologi­ cal theorems are equivalent to U. (See H. Rubin and D. Scott, Some topological theorems equivalent to the Boolean prime ideal theorem. Preliminary report,.Bull. Amer. Math. Soc. 60 (1954), 389. (Received March 7, 1966.)

472 635-4, D. W. WILLETT and J, S, W. WONG, University of Alberta, Edmonton, Alberta, On the behavior of the solutions of some nonlinear differential equations,

We prove the following three theorems concerning solutions of the differential equation

[p(t)x']' + q(t)f(x) = 0, where all functions are at least continuous and p > 0: Theorem 1. Let q > 0, -l/2. Jx (pq)' ?; 0, (pq) p' -----+ 0 as t ---+oo, J±oo f = oo, and f ::;; vxf(x) for some constant v. If there exists + foo-l -1 b(t) > 0 such that b' ?; 0, b = oo, and lim inf (pq) (pq)' b > 0, then each solution x(t) of (*) can be extended as a solution tot ?; 0 and (x')2.q- 1p + x2. ->0 as t ->oo; Theorem 2., If q ::;; 0, j 00p -l = 1 x(t)-> 0 or oo as t-> t - , where t is the right end J00 0 for x f 0, then 1 1 point of the right maximal interval of existence of x; Theorem 3, Let q > 0, pq-l be bounded, xf(x) ?; 0, and j±00f = oo. If either (pq) ?; 0 and pq is bounded, or (pq)' ::;; 0 and (pq) -l is bounded, then each solution x of (*) can be extended as a solution to t ?; 0, x and x' are bounded, and lim sup lx(t) I > 0 as t ~ oo. Theorem l generalizes a theorem of the authors [Abstract 66T-9l, these cNotiaiJ 13 (1966), 2.33] for the linear case, and Theorems 2. and 3 generalize some results mentioned in Utz (Monatsh.

Math, 69 (1965), 353-361]. (Received March 7, 1966.)

635-5. C. T. LONG, Washington State University, Pullman, Washington, Addition theorems for sets of integers.

Let C be a set of integers. Two sets A and B are said to be complementing subsets of C in case A C C, B C C, and every integer c E C is uniquely represented in the sum C = A + B = i x lx = a + b, a E A, b E B}. It is shown that A and B are infinite complementing subsets of N, the set of nonnegative integers, if and only if there exists an infinite sequence of integers [ mt5 with

m. ?; Z.'fi such that A and B are the sets of all finite sums of the forms a= ,Lx MZ. and b = 1 Z.i i ,Lx2.i+ 1 MZ.i+ 1 respectively where M0 = l, M1 = TI~= 1 mj and 0 ::;; xi-l < mi for i ?; l. If one of A and B is finite the result is similar. For example, if A is finite there exists a fixed integer k

such that A = [a Ia = :L~:~~i ~iJ and B = [b lb = ,Lf;dxZ.i+ 1 MZ.i+ 1} and the only restriction on

x2.k- 1 is that it be positive, These results are also extended to complementing k-tuples of subsets of N. If C = Nn = [xiO ::;; x < nJ, the results are again similar. In this case A and B are defined by finite sums as in the case of complementing subsets of N with A or B finite and here

0 ::;; xi < m!+ 1 for 0 ~ i ::;; k for some k. The results in case C = I where I is the set of all inte- gers are again similar but, in this case, it is only shown that the conditions are sufficient, (Received

February 2.8, 1966,)

635-6. ADI BEN-ISRAEL, University of Illinois at Chicago Circle, P.O. Box 4348, Chicago, Illinois 60680, On error bounds for generalized inverses,

Error bounds for inverses and solutions of linear equations are generalized to the singular case. For a matrix A let A •• A+. IIA n. R(A) denote respectively the conjugate transpose, generalized inverse, spectral norm and range space of A. llx II is the Euclidean norm of the vector x. Pr(L) is the projector on the subspace L. Some results: (l) !!All < l and La subspace containing R(A) implies

!I(Pr(L) + A)+ II ::;; (l - IIA II>- 1 and !I(Pr(L) + A)+ - Pr(L + R(A *))II ::;; IIA 11/(l - IIA II). (2.) If dA satisfies AA+dAA+A = dA and IIA+IIIIdAII < l then (A+ dA)+- A+= :L~ 1 (- l)~A+dA)kA+ and!I(A + dA)+- A+ll ~ IIA +liZ !IdA 11/.(l -II A+ II !IctAl!). (3) For A,dA as in (2.), Ax = b solvable, x = A +b and (x + dx) =

473 (A+ dA)+(b +db) one has: (a) lldxll ~ IIAIIIIA+IIIIdbllllxll/llbll if dA = 0, (b) lldxll ~ IIA+IIIIdAIIIIxll +­ (1- IIA+IIIIdAII) if db= 0 and (c) lldxll ~ IIA+II(IIdAIIIIA+IIIIbll + lldbll)/(1- IIA+IIIIdAII). (Received April 4, 1966.)

635-7. D. W. SASSER and M. L. SLATER, Sandia Corporation, Albuquerque, New Mexico 87115. Totally positive matrices whose entries are permanental functions of a given matrix.

For an n X n matrix A define P 0 (A) to be 1, and for l ~ k ~ n, define P k (A) to be the sum of the permanents of all k X k submatrices. Then set Pfj(A) = I;P i (Ak)P j(Ak) for 0 ~ i, j ~ k (where the sum is taken over all c-;,k k X k submatrices of A), Pk(A) = matrix [Prj(A)j, 2 and E = (epq>• epq =l,p,q = 1, ••• , n. Theorem. For 1 ~ k ;> n - 1, if the Cn,k dimensional vectors fP i(Ak)j, i = 0, •.• , k, are linearly independent, then for 0 < IJ < Ek = Ek(A) the matrix Pk(IJA + (1 - 8)E) is totally positive. The proof is based on the following identity: Set T = L::il;Bapq' 2 p,q = l, ••• ,n, and Frj(A) = [(k - i)!(k - j)!) P~/A). Then T F~j(A) = F~_ 1 ,lA) + F ~.j- 1(A). Remarks. (l) If in pk(A), k = 1, ••• ,n - 1, the [~:j submatrix has a nonnegative determinant for all nonnegative A, the van der Waerden conjecture follows. (2) There exist matrices A with positive entries for which Pk(A) is not totally nonnegative; however, in the cases examined numerically, when A has had non­ negative entries, all the 2 X 2 submatrices of Pk(A) have had nonnegative determinants. (Received April 5, 1966.)

635-8. J. W. MACKI, University of Alberta, Edmonton, Alberta, Canada. Singular perturbations of a nonlinear system of ordinary differential equations.

The nonlinear singular perturbation problem x' = f(x,y,t,E), Ey' = g(x,y,t,E), a 1x(O, E)+ a 2y(O, E)=

o.(E), b 1x(l, E)+ b 2y(l,E); {3(E) is examined under the assumption that the degenerate problem x' = f(x,y,t,O), 0 = g(x,y,t,O), b 1x(1) + b 2y(1) = {3(0), has a C 1 solution, x0 , y0• The major assumptions made on f and g are that they are expandable to first order in E about E = 0 (for real positive epsilon),

twice differentiable with respect to x andy (including mixed partials), with gy(x0 (t), y0 (t), t, E)= k(t) + O(E), with k(t) of constant sign on [0,1], and with f , g satisfying certain inequalities in an yy yy appropriate neighborhood of the degenerate solution. It is shown that the problem has a solution of

the form x(t,~) = x 0 (t) + fp(t, ~) + tp(t,E), y(t,t) = y0 (t,E) + t'q(t,E) + r(t,<), where p and q are 0(1)

while p and T exhibit, in general, a nonuniformity at one of the end points of the interval. (Received

April 7, 1966.)

635-9. E. E. GRANIRER, University of British Columbia, Vancouver, 8, B. C. Canadi!. Multiplicative invariant means on subalgebras of m(S).

LetS be a semtgp. m the bded real fens. on S. f = supjf(s)j, fa(s) = f(as), fa(s) = f(sa), Va, s E S, f Em. S is (ELA)LA if there is a (multiplicative) left invariant mean on m. S is ERS if V f E m~a; a E Sj contains a net converging pointwise to a canst. fen. K(f) is the set of all such consts. H = lL~ fj(gj- (gj)at fj,gj Em, aj E S, 0 < n < oo]. Thm: The following are equivalent: (l) Sis ELA. (2) Sis ERS and in this case K(f) = [(f); a multiplicative left inv. mean} (compare with Mitchell (Trans. Amer. Math. Soc.) 119 (1965), 244-261. (3). Vh E H, 3 a E S s.t.h(a) = 0

474 (4) inf {Ill- hil; hE H} = 1.(5). sup{h(s); s E sj 6 0 Vh EH. (6). His not dense in m. (7), Va, bE S, 3c E S s.t. ac = be= c. (3) and (7) do not have analogues for the LA case. A construction of a

"huge" class of left cancellation (lc)ELA semigps. is given. Any lc. semigp. is a subsemigp. of some lc. ELA semigp. The only right cancellation ELA semigp. is S = {ej with e 2 =e. Refine­ ments of the above conditions to subalge~ras A C m and applications to top. gps. (not necessarily lac. comp.) are given. (Received April 13, 1966.)

635-10. 0. C. McGEHEE, University of California, Berkeley, California 94720. Certain isomorphisms between quotients of a group algebra.

We consider closed, perfect sets E, of zero measure, contained in the circle group T. Let A be the algebra of functions on T which have absolutely convergent Fourier series, and let A(E) be the algebra of restrictions to E of functions in A. DeLeeuw and Katznelson (On certain homomor­ phisms of quotients of group algebras, Israel j. Math. 2 (1964), 120-126) have shown that whenever the norm of an isomorphism of A(E) into A(F) is equal to one, it must arise from a map c/>: F ---> E which is the restriction to F of a character composed with a rotation. We present a procedure for constructing pairs of sets E and F, which are not Helson sets, such that a map cl>: F--> E induces an isomorphism of A(E) into A(F); we obtain cases of an isomorphism of norm one, where cJ> is the

restriction to F of a discontinuous character. In general, our cJ> is such a restriction at least on a dense subset of F, with the norm of the isomorphism not necessarily equal to one. (Received

April 14, 1966.)

635-11. THEODOR GANEA, University of Washington, Seattle, Washington 98105. On the spectral se11uence of the generalized homotopy suspension.

Let F -tiE ---> Ps be a fibration and let A ---" dX ____,he be a cofibration with a retraction r: X --->A; let H be the fibre of h with projection j: H -->X. Using the associativity of the operation

F X riB ---> F one proves the Lemma. If there is a map f: X ----> F with if ~ ifdr, then fj ~ fdrj. Now,

let A be any space and define a sequence of cofibrations A---> ekF k----. hkck as follows: F 0 and c 0 are the cone and the suspension of A, Fk Is the fibre of hk-l with projection ik: Fk---> Fk-l, ek is the can­ onical lift of ek-l' and Ck is the cofibre of ek; define fk,£= iJ+l ••• ik: Fk---> F i· The cocategory of A is the least k 6 0 (if any) for which there is a retraction r: F k --->A [T. Ganea, Comment. Math. Helv. 39 (1965), 295-322]. Induction on £and use of the Lemma yields: If cocat A ;;; k, then fm, £. ~ e p,rfm,k for m 6 i + k. This implies: dr = 0 for r >co cat A in the spectral sequence of homotopy groups arising from the factorization A ---> ... ---. Fk---> ... ---> F0 • This result is dual to that of M. Ginsburg [Ann. of Math. 77 (1963), 538-551]; the proofs are much shorter and more con­ ceptual for both category and cocategory. In the study of the above spectral sequence, the following fact which improves results of the author [Comment. Math. Helv. 39 (1965), 295-322] is useful: If

A --> dx ---+ he is a cofibration with H the fibre of h and e: A ---> H the lift of d, then the suspension of the cofibre of e has the homotopy type of the join A • nc. (Received April 28, 1966.)

475 635-12, CHIEN WENJEN, 12191 Reagan Street, Los Alamitos, California 90720, Ring of generalized continuous functions (g.c.f,)

(1) A g.c.f. on a topological space T is defined as a continuous mapping from T to the extended real number system R. Stone's theorem is generalized and Gelfand- Kolmogoroff theorem [Gillman­ Jerison: Rings of continuous functions, p. 102) is a particular case of generalized Stone's theorem, (2) Let C(X) be the set of all real continuous (unbounded) functions on a completely regular space X. By the generalized spectrum 2: of C(X) we mean the set of all nonzero homomorphisms from C(X) to R. 2: endowed with Stone topology is proved to be the Stone-Cech compactif!cation of X by ring theoretic treatment, parallel to the one given by Gelfand-Silov [Uber Verschiedene Methoden, Mat, Sbornik N. S. (1941)] for C*(X). (3) A subring K of C(T) on a topological space Tis closed if (1) K is a lattice containing 1 and (2) any real continuous function f on T that is the limit of a monotonic sequence of functions in C*(T) belongs to K, To each closed subring K of C(T), corresponds the Hewitt's realcompactification Q' of some continuous image T' ofT so that K is isomorphic to C(Q'), and conversely. (Received May 2, 1966,)

63 5-13, W. E. BARNES, Washington State University, Pullman, Washington 99163. Primary decompositions of ideals in gamma-rings.

N. Nobusawa (On a generalization of the ring theory, Osaka J, Math. 1 (1964), 81-89) introduced the notion of a gamma-ring, and extended the Wedderburn theorems to this case. Here we define prime and primary ideals, m-systems, and the radical of an ideal in a gamma-ring M. If M satis­ fies the maximum condition for ideals, the radical of a primary ideal is a prime ideal, an ideal Q 'I M is P-primary iff pn ~ Q <;; P for some n, and AB ~ Q, A 1= P implies B ~ Q, and if an ideal has a primary decomposition, then the usual uniqueness theorems hold, (Received May 2, 1966.)

635-14, W. J, SCHNEIDER, Syracuse, University, Syracuse, New York 13201, An example of a Julia line which is not a Milloux line,

In his book, Lecons sur les familles normales (p. 92) Monte! posed the question whether or not such an example existed, To construct an entire function f(z) for which the negative real axis is a Julia line, but not a Milloux line we first construct a sequence of harmonic functions hn (z) as follows: hn (z) will solve the Dirichlet problem in the half plane to the left of Re z = n for the boundary func­ tion ~ (n + iy) which has the properties: (i) o.n is continuous and for points near the real axis o.n is large and negative. (ii) As one moves past the point n + isn (sn > 0), o.n slowly becomes positive and large and then after one passes the point n + itn(tn > > sn), ~ slowly goes to minus infinity, (iii) For sn < Bn < y < tn, o.(n - iy) = - a(n + iy) and as y ---+ ± oo, ~(n + iy)----> - oo. Let gn(z) = exp(hn + ihri). Since gn(z) is continuous at infinity by the method described in my Abstract 632-54 frhese c)fotiuiJ 13 (1966), 342], we can find an entire function fn(z) which approximates gn(z) within (l/2)n in the domain where gn(z) is defined, If the sn (> > tn_ 1) grow fast enough and the o.n are chosen carefully (using the Gross-Iverson Theorem) then I:: 1 fn(z) (= f(z)) converges and f(z) can be shown to have the desired properties, (Received May 2, 1966,)

476 635- 15. S. K. AAL TO, Oregon State University, Corvallis, Oregon 97331. Reduction of Fredholm integral equations with Green's function kernels to Volterra equations.

G. F. Drukarev has given a method for solving the Fredholm equations which arise in the study of collisions between electrons and atoms. He transforms the Fredholm equations into Volterra equa­ tions plus finite algebraic systems. H. Brysk observes that Drukarev's method applies generally to a Fredholm integral equation (I - ,\ G)u = h with a Green's function kernel. In this thesis connections between the Drukarev transformation and boundary value problems for ordinary differential equations are investigated. In particular, it is shown that the induced Volterra operator is independent of the boundary conditions. The resolvent operator can be expressed in terms of the Volterra operator for regular A. The characteristic values of G satisfy a certain transcendental equation. The Neumann expansion provides a means for approximating this resolvent and the characteristic values. To illus­ trate the theory several classical boundary value problems are solved by this method. Also included is an appendix which relates the resolvent operator mentioned above and the Fredholm resolvent operator. (Received May 3, 1966.)

635-16. BYRON McCANDLESS, Western Washington State College, Bellingham, Washington. On the adjunction space and hereditary properties.

Let X and Y be spaces, A closed in X, and f: A-+ Y a map. Let Z be the adjunction space obtained by adjoining X to Y by means of the map f. Suppose Z' is a subspace of the space z. Under certain conditions, Z' can be considered as the adjunction space obtained by adjoining a space X' to a spaceY' by means of a map f', where X' andY' are subspaces of X andY respectively. This is true, for example, if Z' is open (closed) in z. Using this, we can prove that the adjunction space pre­ serves certain hereditary properties, e.g., hereditary normality, the hereditary Lindelof property, and hereditary paracompactness (countable paracompactness). (Received May 4, 1966.)

635-17. A. L. Y ANDL, Western Washington State College, Bellingham, Washington. Proximate homotopy.

Let X be a space, M a metric space and f: X -+ M, g: X ·-+ M mappings not necessarily continu­ ous. Then f and g are said to be o-proxhomotopic if there exists a /)-continuous mapping h: X X I---+ M such that h(x,O) = f(x) and h(x,1) = g(x) for all x in X. A mapping f: X ---+ M is said to be /)-null prox­ homotopic if it is o-proxhomotopic to a constant mapping. Theorem: If X is an ANR then for each f > 0 there exists a positive of such that each (if-null proxhomotopic mapping of X into itself leaves a point f-invariant (see Abstract 630-134, these cNoticeiJ 13 (1965).97). (Received May 4, 1966.)

635-18. D. F. SANDERSON, Western Washington State College, Bellingham, Washington. On generalizations of the Gel'fand-Mazur representation theory.

Let M be an extension module of a module N and J1 be a set of submodules of M. Further, let PI be the natural map from N to Mil, where I E rl, and E be the equalizer of the family £N,Mjl,P I' J J. If we define qi:M -+ M/1-+ E, we may define a map f from M to a collection of maps

477 from ci toE by setting f(m) = m', where m'(l) = q1(m). We explore this construction for topological modules with special regard to generalizations of spectral theory. (Received May 4, 1966.)

635-19. ROBIN CHANEY, Western Washington State College, Bellingham, Washington. Some convergence theorems connected with the transformation of integrals in measure space.

Assume f~,J!,~t J and t~•h• v} are u-finite measure spaces having completions [~.M. ~t} and {~.N. vJ. Let T be a function from .S. onto X for which T- 1fl C M and finally assume that Q. is a sub­ family of M such that TQ. C ,N. An extended real valued function 'W having domain X X M is a signed weighting function for Tin case 'W(.,G) = 0 a. e. v off TG for each Gin Q, 'W(.,M) = L'W(.,Mi) a.e.

11 on ~whenever M is the union of countably many pairwise disjoint sets Mi in M, and each function

'W(.,M) is ,N-measurable. If each 'W(.,M) is also in L 1(v) then a function fin L1 (It) is a gage for 'W in case, for ,N-measurable functions H, one has JM(H 0 T)fd~t= J~H'W(.,M)dv whenever these inte­ grals exist. A sequence of signed weighting functions 'W n for T with gages fn converges to a signed weighting function in mean uniformly with respect toM iff f\t] converges in mean. This result and some like i.t can be used to show that certain function spaces which arise in connection with the trans­ formation of integrals in measure space are Banach spaces. (Received May 4, 1966.)

478 ABSTRACTS PRESENTED BY TITLE

66T-231, R. J, WARNE, 428 Cedar Street, Morgantown, West Virginia, Congruences on I-bisimp1e semigroups, I.

LetS= (G,a) be an I-bisimple semigroup (Abstract 66T-81, these c){oticeiJ 13 (1966), 230). Theorem. Let e be the identity of G. Let N = {_g E G/gan = e for some n E I0 (the nonnegative integers)}· Define (xN)8 = (xa)N. Let g ~ g be the natural homomorphism of G onto G/N. The maximal group homomorphic image S/u of S is isomorphic to G/N X I under the following definitions of equality and multiplication: (g,b - a)= (ii,d - c), a,b,c,d E IO if there exist x, y E IO such that x + b = y + d, x + a= y + c, and gfJX = lifJY. (g,b - a)(ii,d - c)= (gfJCjj(Jb, (b +d) - (a + c)). The homo­ morphism of S onto Sju Is given by (g,a + I, b + i).P = (g,b - a), i E I, a,b E IO. The group congruences p of S (S/p is a group) are uniquely determined by the normal subgroups of S/u, u n H and uvH are given, The determination of the maximal group homomorphic image of an I-bisimple semigroup de­ pends heavily on a result of the author (Trans, Amer. Math, Soc. 106 (1963), 427-435; Theorem 2,1). (Received November 3, 1965,)

66T-232, BENJAMIN YOLK, 1315 Dickens Street, Far Rockaway, New York 11691, Differences, convolutions, primes, V.

Conjectures 1 and 3 are false, The Lagrange interpolation formula for F(z) taken at the n distinct points a,b,c,.,,,s, is (F(t)l(t- z) (t;a,b,c, .. .,s))l(1l(t- z) (t;a,b,c, ... ,s)). There exists a func­ tion analytic in the closed unit disc D' (with differentiability understood in the set-theoretic sense of approach by means of points of the set) whose first derivative is not analytic in D'. Theorem 5, Let Z = (z(n)) be any set of points having a finite limit point, p. Let W = (w(n)) be any other sequence of complex numbers. Then W can be interpolated at the points of Z by an entire function w = w(z) if

and only if lw(z;z(O), z(1), z(2), ... ,Z(n))l ~ (R(n))I(R(n)- lz(O)I) (R(n)- lz(l)I)(R(n)- lz(2)i) ... (R(n) - lz(n) I> for every natural number ri, and for some sequence of positive real numbers (R(n)) increasing to infinity. Furthermore, W can be interpolated at the points of Z by a function w = w(z), analytic for lz - pI less than R and having uniformly bounded coefficients in its Taylor series expan­ sion about p if and only if lw(z; a,b,c, ... ,t)l ;:;; MI(R- Ia- pi)(R- lb- pi)(R- lc- pi) ... (R- It- pi) whenever a,b,c, ... ,t, are a finite number of distinct points of Z, (Received February 24, 1966.)

66T-233, SEYMOUR GINSBURG, Systems Development Corporation, 2500 Colorado, Santa Monica, California, S, A. GREIBACH, Harvard University, Cambridge, Massachusetts and M, A. HARRISON, University of California, Berkeley, California, One-way stack automata, II.

A 1-sa A= (K,:E,¢,$,f,o,q0 ,z0,F) is deterministic if (1) ois a function from K X (l: Uf¢,$j) X f Into {0,1} X K X [- 1,0,1j X r•, and (2) (d,q',e,y) E o(q,a,Z0) implies y = z0 w for some win r•. L Is a D-language if there is some deterministic 1-sa A such that L = T(A). For each r,e, set R, there

exists ,a homomorphism h and deterministic languages L 1 and L 2 such that R = h(L 1 n L2). The

479 D-languages are closed under (i) intersection with regular sets, (ii) inverse gsrn mappings, and (iii) complementation. The D-languages are not closed under union, intersection, homomorphism, and left product with regular sets. It is recursively unsolvable to determine whether T(A) is a D-language for an arbitrary 1-sa. There are D-languages which are not context free. Every D-language is accepted by some deterministic linear bounded automation. (Received January 17, 1966.)

66T-234. J. G. STAMPFLI, New York University, New York 53, New York. A minimal hull

theorem for operators.

Let T be an invertible operator on Hilbert space. Designate the numerical range of T by

W(T) and the convex hull of the spectrum ofT by ~(T). Assume both a(T) and a(T- 1) lie on convex curves. For a wide class of such curves, it is shown that Tis normal if and only if W(T) C E(T) and

W(T- 1) C l(T- 1). Examples are provided which indicate that this is the widest class of curves for

which the result holds. (Received January 26, 1966.)

66T-235. S. D. BERNARDI, New York University, University Heights, Bronx, New York.

Special classes of subordinate functions.

Let (K) denote the class of functions f(z) ~ " 00 a zn regular, univalent and mapping lz I < L...1 n onto a convex domain, SN(z) ~ L~anzn, FN(z) ~ cz-cJ~tc-lsN(t)dt, F 00 (z) ~ cz-cJ~tc-lf(t)dt. Theorem 1. Let f(z) E (K), 0 ;;; B ~ 1, c > 0, N ;;; a ~ N + 1, N ~ 1,2,3,... . Then g(z) ~ L~ [(c + Bn)/(c + n)J (1- n/a)anzn-< f(z). The subordinate function g(z) may be written as g(z) ~ PSn (z) - QzSN(z) + (1 - P)F N(z), where P " (B - c(l - B)/ a], Q ~ (B I a). Corollaries.

(B ~ 0): (1) (N ->en): Bf(z) + (1- B)F 00(z)-< f(z), (2) (B ~ 1): SN(z)- (z/a)SN(z)-< f(z), (3)

(1 + c/a.)F N(z) - (c/a.)Sn(z)-< f(z), (4) (B ~ 0, c ~ a.): ZFN(z) - SN(z)-< f(z), (5) (B ~ c/(a. +c)):

F N(z) - [c/a.(a. + c)]zSN(z) -< f(z). Theorem 2. Let f(z) E (K), then (a) Re(f(z)/zf'(z)) ~ (l/2)Re(l/f'(z)),

(b) Re[(f(z)- Sn(z))/zf'(z)) ~ (- 1/a.)Re[sN(z)/f'(z)]. These and additional results will appear in the

March 1966 issue of the Duke Math. j. (Received February 14, 1966.)

66T-236. WITOLD BOGDANOWICZ, The Catholic University of America, Washington, D. C.

20017. Existence and characteristic properties of maximal and minimal volume spaces generating

the same integration as a given volume.

A family V of subsets of a space X is called a pre-ring if A, B E V implies An B E V and

A\ B ~ C 1 U •.• U Ck, where Cj E V are disjoint. If in addition AU B E; V then V is a ring. A non­ negative function v is called a volume if it is defined on a pre-ring and is countably additive. Given

2 volumes v 1,v2 on pre-rings v 1,v2 , respectively, define the order relation v 1 C v 2 by the condition:

V 1 C V 2 and v 1 is the restriction of v2 to V 1. A volume v is called complete if its domain V is a ring, every v-null-set belongs to V, and if An E V is an increasing sequence such that the sequence v(An)

is bounded then U An E V. A volume v is called discrete if its domain V consists of disjoint sets. Let Y, R.be a 8-space and the space of reals. Let L(v,Y), II llv be the space of Lebesgue-Bochner

summable functions and the seminorm generated by v. (For construction of the space and other

notions see: Bogdanowicz, Proc. Nat. Acad. Sci. USA 53 (1965), 492-498.) Denote by D the family of

all volumes w such that L(w,Y) ~ L(v,Y), and II llw ~ II llv for every B-space Y. Put v0 (A) ~

480 JXAdv if XA E L(v,R). Theorem 1. The function v0 is the greatest volume in Do Theorem 2. The volume v generating D is the greatest in D iff v is complete. Theorem 3, The family D has a minimal volume iff it contains a discrete volume. Theorem 4. The volume v generating D is a minimal volume in D iff vis discrete. (Received March 24, 1966.)

66T-237. LINDA LUMER-NAIM, University of Washington, Seattle, Washington 98105. S)P spaces of harmonic functions.

In the context of Brelot's axiomatic theory of harmonic functions, S) denotes the class of complex-valued harmonic functions in the basic space rl. Definition: S)P, 1 ;;;; p ;;;; + oo, is the class of functions in rl, whose :rY -norms, with respect to the harmonic measures pwtxo relative to a fixed point x0 E rl and the open relatively compact subsets w, of rl which contain x0 , are uniformly bounded in ~. Main results. The functions in S)P are the solutions of Dirichlet problems with the minimal bound­ ary ~l of rl, the fine filters ·in rl, and boundary functions in Lp (~ 1 ), for l < p ;::;, + oo, the integrals of finite complex measures on ~l' for p = 1. Every function in S)P, 1 ;;;; p ~ + oo, has a finite fine

limit E LP(L\) at almost every point of Ll1. Similar results hold for classes &), defined by replac­ ing tP, p > l, by a positive convex increasing function in [0, + oo G and for classes S)~, ~ ~. of real subharmonic functions in 11, defined like the S)P, ~<1>. Application_. Extension of a result of Garding-Hormander, giving in particular: Iff E S) 1, and if q,-l(lfl) is subharmonic for some as

above, with limt~+oo (t)/t = + oo, (in particular if log If I is subharmonic), then f is the integral of

a measure on ~ 1 • absolutely continuous with respect to the harmonic measure at the point x0• (Received February 14, 1966.)

66T-238. BRIAN SMYTH, Brown University, Providence, Rhode Island. Differential geometry of complex hypersurfaces.

Let M be a complex analytic submanlfold (of complex dimension n <::. 2) of M, where M is any one of the complex space forms pn+l(C) (Fubini-Study metric), cn+1 (complex Euclidean metric), Dn+l (Bergman metric). M is a Kahler manifold in the metric induced from M and all metric properties

refer to this metric. When M is Einstein, the Ricci tensor S equals Pg, where g is the metric on M induced from M and pis some constant. Theorem._!! M is Einstein it is locally symmetric • ..!!._ M= pn+ 1(C) and M is a complete Einstein manifold, then M is either Pn(C) or the complex quadric Qn. ___!! M = Dn+l (resp. cn+l) and M is a simply-connected complete Einstein manifold, then M is on

(resp. en). If M is complete and Einstein with p > 0, then by the theorems of Myers [Duke Math. j.

8 (1941), 401-404] and Kobayashi [Ann. of Math. 74 (1961), 570-574) M is compact and simply c.on­

nected. This accounts for bests result being obtained when M = pn+l(C). The restricted holonomy group of M is computed for all possible p and used together with Cartan's list of irreducible symmet­

ric spaces to obtain the classification in the above theorem. (Received February 15, 1966.)

481 66T-239. SATISH SHIRALI, 320 Pine Street, Fall River, Massachusetts 02722. Symmetry of complex involutory Banach algebras. Preliminary report.

A complex Banach algebra with a continuous involution such that every hermitian element has a real spectrum must be symmetric. This answers a question raised by Kaplansky (Duke Math. J. 16 (1949), 399-418). (Received February 17, 1966.)

66T-240, A. C. CONNOR, University of Georgia, Athens, Georgia. Locally cyclic continua. II.

In this paper, which will also be sub.mitted to Fundamenta, the following additional results on locally cyclic continua are obtained. Theorem 1. A necessary and sufficient condition for a locally cyclic continuum M not to be unicoherent is that there exist a monotone open mapping f of M onto a simple closed curve. Theorem 2. Every locally cyclic continuum is a web. (A continuum M is said to be a web if there exist monotone mappings f and g of M onto dentrites H and K such that if p E H and q E K, then f- 1(p) nf- 1(q) 'f (1.) Theorem 3. A necessary and sufficient condition for a plane continuum M to be a disk is that for any two points p, q, of M there exist a monotone open mapping f of M onto (0,1] such that C 1(0): p, r 1(1): q and if 0 < t < 1 then C 1(t) is locally connected. This theorem remains true if the condition that C 1(t) be locally connected is replaced by the requirement that f- 1(t) be hereditarily indecomposable. (Received February 17, 1966.)

66T-241. H. W. GOULD, West Virginia University, Morgantown, West Virginia 26506.

The bracket function, q-binomial coefficients, and some new Stirling number formulas.

We prove the following main inversion Theorem. For two sequences f,g, f(n,k,q): where: Rk(j,q): Lk:,;:,;;_j._n g(n,j,q)R k (j,q) if and only if g(n,k,q): Lk,_j:S__ n f(n,j,q)Ak(j,q), " qd-k[c ]1-10/d) and A (J q) = " (- l)j-mk[c 1 qk(qn-J+ 1 - 1)/(qJ- 1). It is shown that integer ~ n/k, and g(n,k,q) = [C k] satisfy the general theorem. Various f(n,k,q) = [!1/k]: greatest n, combinatorial results and other expansions follow. This extends previous work of the author in Fibonacci Quart. 2 (1964), 241-260. (Received February 17, 1966.)

66T-242. H. E. CONNER, University of Wisconsin, Madison, Wisconsin. Almost sure convergence of spatial averages for a branching process of Brownian particles.

The results of A. V. Skorohod (Theor. Prob. Appl. 9) give a Markov branching process of particles which have common independent laws of motion on (a,b) determined by the equation (1/2)(d 2;dx Z.,IL(x) - V(x)!L(x) = >q.t(x),!L(a) = !L(b) = 0, >. > 0, a < x < b, and have common independent

laws of branching bk(x), k 'I 1, a ~ x ~b. Let Nt(E) denote the random number of particles at time t in a Borel set E of (a,b). The existence is established of a random variable TJ such that, for all bound-Bore! f on (a,b), the random average entJ~f(y)Nt(dy) converges a.s. to 1JJ~f(y)(y)dy, where (y) is the unique positive eigenfunction for the maximum eigenvalue a of the equation (A);

(l/2)(d2 /dx2)1L- V(x) [Lkt-1kbk(x) - ~ 11 = A!L,IL(a): 1-!(b) = 0, >.. > 0. The proof is made by first show­ ing such convergence for averages having the eigenfunctions of (A) as weights and then extending by approximation to general weights f. In particular, entNt(E) converges a.s. to TJJ E(y)dy and, subject

482 to no extinction, Nt(E)/Nt(F) converges a, s, to JsJ>(y)dy / JFtb(y)dy, The restriction to particles in (a,b) is convenient, not necessary, and the extension to particles in higher dimension sets with smooth boundaries is clear and direct, (Received February 17, 1966.)

66T- 243, L. V. QUINT AS, St. John's University, Jamaica, New York 11432, Extrema concerning asymmetric graphs,

By a ~we mean a finite undirected graph without loops and without multiple edges, The automorphism group of a graph consists of those permutations of the vertex set of the graph which preserve adjacency relations, A graph is called asymmetric if its automorphism group consists only of the identity automorphism (cf, P. Erdos and A. R~nyi, Asymmetric graphs, Acta Math, Acad, Sci, Hungar, 14 (1963), 295-315), We establish the extreme values of p and q for which there exist asymmetric graphs having p vertices and q edges, This is done for the following four classes of graphs: all graphs, connected graphs, graphs having no vertices of degree 2 (topological graphs), and connected topological graphs. An asymmetric graph K has asymmetry equal to 1 if it is possible by either adjoining one edge to K or deleting one edge of K to obtain a graph which is not asymmetric, For the above mentioned four classes of graphs, we resolve the following problem posed by P. Erdos and A. Rc!nyi (ibid): Determine the least value G(p,l) for which there exists a graph with asymmetry equal to 1, having p vertices, and G(p,l) edges, (Received February 14, 1966.)

66T-244, C. H, CLEMENS, JR., 1512 North Street, Berkeley, California 94703, Picard­ Lefschetz theorem for families of algebraic varieties acquiring certain singularities,

Let B be a complex analytic line bundle on the nonsingular complex projective variety X£, and let s 0 and s 00 be linearly independent holomorphic sections such that S00 = (zeros of s 00 ) is a nonsingular variety. Let St = (zeros of (so+ ts00)). Let a::s = [t: St singular]. Assume

0 Ea::s' -y: [0,1] -->(fl:- fl:s) is a loop around 0 beginning and ending at 1. Let FIJ :S1 --> S-y(IJ) be the diffeomorphism induced as 'Y(IJ) = t moves, and let F = F 1• Assume the singular locus cr of s0 is a nonsingular double r-variety. Then there exists a "vanishing bundle" C 1 ~ S 1 such that C 1 is an ( j,- r - 1)- sphere bundle over C. Let ~ be the associated bundle of coefficients, and 1/1.: Hk(C;~) --> Hq(C 1) the map in the corresponding Gysin sequence, where q = k + (£- r-1). Let {31, .. , f3s be a basis for Hk(C; 31 ® Q); let(') denote duals under intersection. For a E Hq (S 1; Q), F *(a) - a = (- 1) hLi(a •1/IJ. f3i)) 1/1.( {3i) where h = k( £ - r - 1) + (1/2)( J,- r)( ,2 - r + 1),

(Formula must be modified slightly in case C n S00 "I j1,) Further a formula is derived for the case of a family of algebraic surfaces acquiring double curves with triple points, (Received February 18, 1966.)

66T-245, D. V. THAMPURAN, State University of New York at Stony Brook, Stony Brook, New York 11790, Uniform continuity and general topology.

In 1937 Well succeeded in generalizing some of the uniformity concepts (such as uniform con­ tinuity) of metric spaces, to completely regular spaces; in the present paper these concepts are now generalized to arbitrary topological spaces and the "expected" theore-ms are proved, Two of the new ideas here introduced are those of u-metric and u-continuity. The following are a few of the

483 results obtained. Theorem. A topological space is homeomorphic to a subspace of a product of u-metric spaces. Theorem. Each quasi-uniform structure for a set S is generated by the family of all u-metrics which are u-continuous on S X S. Theo.rem. An arbitrary topological space is complete iff each family of closed sets which has the finite intersection property and contains small sets has a nonempty intersection. (Received February 21, 1966.)

66T- 246. ALI KYRALA, 801 Lemon Street, Tempe, Arizona. A new interpretation of the method of stationary phase.

The method of stationary phase is an asymptotic method of calculation of integrals of the form Je iwu(t)dt which exploits the cancellation of contributions over ranges outside the neighborhoods of points of stationary phase for which u' (t) = 0. The method gains considerably in analytical clarity and interest when viewed as a problem in functional inversion and Fourier synthesis. To do this one need only consider that the change of variable s = u(t) corresponding also to the inverse transforma­ tion t = w(s) makes the integral a Fourier transform in which w'(s) (according to the relationship between the derivatives of a function and its inverse u'(t)w'(s) = 1) assumes its largest values in the neighborhoods of points of stationary phase for which u'(t) = 0. This associates such points precisely with the most important spectral ranges in the Fourier development Je iwsw' (s)ds. (Received February 21, 1966.)

66T-247. NARESH JAIN and BENTON JAMISON, University of Minnesota, Minneapolis, Minnesota. Markov processes. I.

Let P(x,B) be a transition probability function on the measurable space (X,.9!i'). For termin­ ology and notation, see Chung, z., Wahrscheinlichkeitstheorie 2 (1964), 230-254; Harris, Third Berkeley Symposium on Math. Statistics and Probability, vol. II, 113-124; and Doeblin, Ann. Sci. Ecole Norm. Sup. (3) 57 (1940), 61-111. Theorem 1. If .!18 is separable, and X is indecomposable and absolutely essential, then X = H + I, where H is a stochastically closed set on which the process is recurrent in the sense of Harris, and where I is either inessential or improperly essential. Corollary. Under the hypotheses of Theorem 1, the process is normal. An example of Blackwell, Bull. Amer. Math. Soc. 51 (1945), 465-468, shows that the assumption that t?g is separable cannot be removed. Theorem 2. If X is a final set, then the process is recurrent in the sense of Harris. (Received February 21, 1966.)

66T-248. CHARLES FEFFERMAN, 9006 Linton Street, Silver Spring, Maryland 20901. A generalization of the Radon-Nikodym theorem.

Suppose that 2: is a field of subsets of the set S. Let J.L be a bounded complex-valued finitely

additive set function on~, and let 'Y be a finite complex-valued finitely additive set function on L, absolutely continuous with respect to J.L. Theorem 1. There is a sequence {f nl of ~~-simple functions, by a pure such that limn~ 00 JEfn (s )J.L(ds) = -y(E) uniformly for E f= ~. Theorem l is established existence proof and gives no method of construction for fn• A result which provides more information on the nature of {fn) is sketched below. A partition of S is a finite collection of sets Ei belonging to L, such that Sis the disjoint union of the Ei, and J.L(Ei) t 0 fori= l, .•• ,n. The set of partitions of S

484 is denoted by 9. 9 is a directed set under the following order: P 1 < P 2 if and only if for every

E E P 1 there are sets F 1, ... ,Fr E P 2 (r may depend on E) such that E = F 1U ... UFr. If Pis a parti­ tion, define f p = LEEP(-y(E)/ f.l(E))XE, where XE is the characteristic function of E. fp is obviously a f.l-simple function. Theorem 2, If f.l is positive, then limPE9 JEfp(s) p(ds) =')'(E), uniformly forE E ~.where 9is directed as explained above. (Received February 28, 1966.)

66T-249. T. M. PRICE, University of Iowa, Iowa City, Iowa 52240, A knotted cell pair with knot group z.

For n ~ 6, examples of locally flat cell pairs (C,C ') of type (n,n - 2) are constructed so that

IT1 (C - C ') = Z but n1(Bd C - Bd C ') f. Z. They are constructed by spinning and twist spinning, see E. C. Zeeman, Trans, Amer. Math. Soc. 115 (1965), 485. (Received February 22, 1966.)

66T-250, J, D. KUELBS, University of Wisconsin, Madison, Wisconsin 53706. Measures in function spaces.

Let X = TI~= 1 [0,1] and C(X) the space of real-valued continuous functions on X with the uniform topology. If F(p 1••"•Pk), p. EX, is a consistent system of finite-dimensional distributions such that 1 2 for all p 1,p2 in X we have J~J~IY- xladF(pl'p )(x,y);;; C[d(p 1,p2)]a where a> 0, C > 0, a> n are constants independent of p 1,p2 then there exists a probability measure m on the Borel subsets of C(X) whose finite-dimensional distributions agree with those given, If the initial finite-dimensional distributions are Gaussian the conclusions follow with a= 2 and a > 0. In fact, under suitable cir- cumstances the argument used can be extended to the situation where X is a compact metric space (p1, ... ,pk) provided the distance function d( , ) can be suitably chosen, a ~ 1, and the F , pi EX, are Gaussian, Furthermore, if X= [0,1] X [0,1] and {mkl is a sequence of probability measures on C(X) whose finite-dimensional distributions converge to limit distributions then it can be shown that f mk 1 converges weakly to a measure m provided each mk satisfies suitable conditions, Moreover, the finite-dimensional distributions of m are the limit distributions. (Received February 21, 1966.)

66T-251. A. H. ZEMANIAN, State University of New York at Stony Brook, New York, A distributional convolution transformation.

The convolution transformation of Hirschman and Widder can be extended to certain classes of distributions. Through various changes of variables, this provides a distributional generalization and inversion theory for quite a variety of integral transformations, Given a nonfinite Class I, II, or III kernel G(t), a testing function space L is defined which contains G (k)(x - t) for every x and every nonnegative integer k, The dual L' of L consists of distributions f which possess convolution

transforms F(x), which are defined by F(x) = (f(t), G(x- t) ). This transform is inverted by the Hirschman- Widder real inversion formula when the limit process in that formula is taken in a dis­ tributional sense, More extensive results of this nature are obtained when G(t) is either a Class II

of III nonfinite kernel. (Received February 22, 1966,)

485 66T-252. GLEN BAXTER, Purdue University, Lafayette, Indiana 47907. A necessary condition for 4-coloring a elanar graph.

Let G be a finite graph in' the plane, regular of degree 3 (exactly three edges at each vertex), with no loops or multiple edges, and such that edges intersect only at vertices. Let the faces of the graph be labeled with numbers 1, 2, 3, 4 in such a way that no two faces with a common edge have the same label. A vertex v will be called clockwise if the labels on the three faces incident to v when read clockwise have the order 123, 124, 134, 234. A vertex which is not clockwise is called counter-clockwise, i.e., the labels would have the above order when read counter-clockwise. Let A i denote the number of clockwise vertices whose three incident faces omit the label i minus the number of counter-clockwise vertices whose three incident faces omit the label i. Theorem. lA il is an integer independent of i. ~· A 1 = A3 = - A 2 = - A4• The result can be generalized to include loops and multiple edges, etc. (Received February 21, 1966.)

66T-253. WITHDRAWN

66T-254. M. M. ZUCKERMAN, New York University, Bronx, New York 10453. Some theorems on the axiom of choice for finite sets. II. Preliminary report.

Let I= zinteger ~ 2j. I*= 1 U[lJ. and 9: fprimesj. US £I and n E I, S(n) will denote

1x: xES & x ~ nj. Mostowski showed (M) to be necessary for the implication [Z]----> [n] (for termin­ ology see Fund. Math. 33 (1945), 137-168). Theorem. For p E .9 and n E I, (M) is a NASC for the implication [9(p)J --> [n). Definition. Let m i(Z) be the ith smallest integer satisfying [mtJ ----> [Z] i E I*. Theorem. For n E I, m 1 is prime iff n is a prime or 4; mi<{n }> is composite for i E I. Theorem. A sufficient condition that m 1(Z) = £cm(Z) is that each element of Z is square-free. For such sets Z, mi(Z) = i •icm(Z) for all i E I*. All the mi( [n}). i EI*, are determined for 2 ~ n ~ 19.

For 10 ~ n ~ 50, m 1 ( fn}) = n. Conjecture. m 1( {n}) = n for n ~ 10. Definition. An element m E I and a finite subset Z C I satisfy (M') iff for each n E Z, fm} and n satisfy (M). Theorem. (M') is necessary for the implication [m] ----> [z] (M') is a NASC for this implication in each of the following

486 cases: (l) m < 40, (2) m is a prime power, (3) Z = I(n)' n E I, and (4) Z = 9(n)' n E I. Theorem.

Let m, n E I. Let p(n) be the largest prime ~ n. The following are equivalent: (l) m is a multiple of every prime ~ p(n). (2) [m} ____, [9(n)). (3) [m] ----> [I(n)]. (Received March 11, 1966.)

66T-255. ROHIT j. PARIKH, University of Bristol, Bristol 8, England. Some results on lengths of proofs.

Let T be a formal system, formalised in the language of the first order functional calculus including equality, individual, predicate and function constants, and axiomatised by a finite number of axioms and axiom schemata (either unrestricted or with restrictions of the "provided t is free for x in A" type) with a finite number of schematic rules of inference like modes ponens, Gentzen type rules, etc. Assume that "S" and "0" are among the constants and let "n" denote "0" preceded by n occurrences of "S". Let "f-kA" mean "A is provable in Tin k steps". Then Theorem 1. For a given formula A(x) and fixed k the set [n I f-kA(u) J is a union of a finite set with a finite number of arithmetic progressions, obtainable effectively from k and A(x). Theorem 2. If T is Peano arithme­ tic then ('v'n) 1-kA(!!) implies f-('v'x)A(x). (Note. Dan Richardson has obtained similar results by a somewhat different method.) (Received February 21, 1966.)

66T-256. C. R. COMBRINK, The University of Kansas, Lawrence, Kansas 66044. The Baer radical of the standard wreath product.

Definition. A group G is a Baer n!lgroup if every cyclic subgroup of G is subnormal. Every group G has a maximum normal Baer nil-subgroup R(G), called the Baer radical of G (see Nilgruppen, Math. z. 62 (1955), 402). If A and Bare groups, W = AWrB (W = AwrB) will denote the standard unrestricted (restricted) wreath product of A and B. Let F = AB and F = A(B) be the cartesian power and direct power of A. Then W = FB and W = FB. Lemma. If G is a solvable p-group of bounded exponent, then G is a Baer nilgroup. Theorem 1. Let W = AWrB (W = AwrB) where A 1 E 1 B. Then W (W) is a Baer nilgroup iff A is a solvable p-group of bounded exponent and B is a Baer nil p-group. Corollary. If A is a solvable p-group of bounded exponent and B is a Baer nil p-group, then every extension of A by B is a Baer n!lgroup. Theorem 2. Let W = AWrB and W = AwrB. (l) If A

is a solvable p-group of bounded exponent and B 1 is the Sylow p-subgroup of R(B), then R(W) = FB 1

and R(W) = FB 1• (2) If A is not a solvable p-group of bounded exponent, then R(W) = R(F) and R(W) = R(F). (Received February 28, 1966.)

66T-257. HAROLD WIDOM, University of California, Berkeley, California. Polynomials associated with measures in the complex plane.

Associated with a finite positive measure p. on a closed bounded set E in the plane, and a nonnegative function satisfying certain conditions, there is a sequence {P n(z)j of monic nth degree polynomials: P n minimizes J( IP i)dp. over all monic nth degree polynomials. (For (t) = t 2 we obtain the orthogonal polynomials associated with p..) With appropriate assumptions on J.L asymptotic formu­ las are found for Pn(z) and mn• the minimum in question. (Roughly it is assumed J.L is thick near the outer boundary of E; it suffices that harmonic measure '>n E be a. c. with respect to J.L.) If G(z) is the

Green's function with pole at oo for the unbounded component r! of the complement of E and C(E) is

487 the capacity of E then n- 1 logiP (z)l--> G(z) +log C(E) uniformly on any closed set disjoint from the n convex hull of E; a slightly more complicated formula, which takes into account possible zeros of P n• holds for an arbitrary closed subset of rl. In addition mn = ci>((C(E) + o(l))n). As an application, let ,\N be the smallest eigenvalue of the matrix with m,n entry JzmzndJ.I (m,n = O, •.. ,N). Then under certain conditions N-l log 1/,\N approaches twice the maximum of G(z) on the unit circle. (Received February 28, I966.)

66T-258. BJARNI JONSSON, University of Minnesota, Minneapolis, Minnesota 55455. Varieties of groups of nilpotency three.

. - I - 1 Wnte (x0 , x 1) = x 0 x 1 x0xi and (x0 ,xi, ••. ,xk+I) = ((x0 ,x1, •.• ,xk), xk+I). A variety (equational class) U of groups is said to be n!l-n iff every member of U satisfies the identity (x0 ,xl' ... ,xn) = I. Theorem. For every nil-3 variety U of groups there exist unique natural numbers m, n, p, and q such that n•gcd(Z,m)lm.pln,qlp, q•gcd(6,m)lm, and pl3q, and such that U is defined by the identities xm =I, (y,x)n = 1, (y,x,z)P = I, (y,x,z)q = I, (y,x,z,u) = I. Corollary. The lattice of all nil-3 varieties of groups is distributive. (Received March 3, I966.)

66T-259. L. E. SNYDER, University of Virginia, Charlottesv!lle, Virginia. Bi-arc boundary functions.

Let :R X (O,oo)---> R and assume that f: R--> R is a boundary function of 4> such that for x E R there are two simple arcs in the upper half-plane terminating at (x,O) along which the boundary limit of 4> exists and is equal to f(x). In this case f is said to be a bl-are boundar:y function of 4>. Theorem. Iff: R __, R is a bl-are boundary function of 4>: R X (O,oo) --> R, then f is an honorary function of the second Baire class, i.e., there is a function g: R --> R such that g is in the first Baire class and g(x) = f(x) except on a countable set. (Received November 26, I965.)

66T-260. M.S. LYNN, F. HARARY and]. W. EVANS, International Business Machines, 6900 Fannin Street, Houston, Texas 77025. On the enumeration of finite topologies.

We consider the problem of enumerating the set, Tn of topologies on a finite number, n, of points, where we regard the points as being distinct (the labeled case), and also the related problem of counting homeomorphism classes (the unlabeled case). If T E Tn' the relation R =

488 66T-26i. K. I. APPEL, University of Illinois, Urbana, Illinois, No r,e, set is the union of finitely many immune retraceable sets,

The proof follows by induction. If RS 1 U s2 U ... 0 Sn with R r,e. and each s1 immune retrace­ able, it follows by induction that an infinite r.e. subset R 1 of R lies in the intersection of the domains of the retracing functions of the St. Next, by a combinatorial argument if can be shown that an infinite r,e. subset R" of R' is disjoint from some s1•. Again using the induction hypothesis it is shown that some Sj, j f i 0 is r.e. (hence of course recursive), T. G. McLaughlin has recently observed that an r.e. set is the union of finitely many immune regressive sets only if it has a recursive subset which is the union of finitely many immune retraceable sets, Hence, no r.e. set is the union of finitely many immune regressive sets, (Received March 2, 1966,)

66T-262, E. A. BENDER, Institute for Defense Analysis, Princeton, New jersey, Characteristic polynomials of symmetric matrices,

Let F be a field and pan F-polynomial, IfF is formally real, we say that pis F-real if and only if the splitting field of p over F is formally real; if F is not formally real, we always say that p is F -real. Theorem, 1!! F be an algebraic number field and p a monic F-polynomial with an odd de­ gree factor over F. ~p !!_F-real if and only if it is the characteristic polynomial of a symmetric F-~. The proof is a generalization of methods mentioned in Abstract 66T-120, these c}foticeiJ 13 (1966), 242, Let p = p 1p2 ... be the prime decomposition of p over F, It suffices to show that L:x~ is a direct sum of quadratic forms of the type D 1xj trace (X~i~j) where X EG = F[x]/(pk(x)) and ~1'~ 2 , ... is a basis for Gover F. Special forms of this type are constructed as noted in 66T-120 and their direct sum is shown to be equivalent to ,L:xf. (Received February 25, 1966.)

66T-Z63. M. A. ENNEKING, Washington State University, Pullman, Washington 99163, ~ decompositions in r-rings. Preliminary report,

W. E. Barnes, (On the r-rin&s of Nobusawa, to appear in the Pacific J, Math.), by modifying the definition of the r-rings of Nobusawa, (On a generalization of the ring theory, Osaka J, Math, 1 (1964), 81-89), has defined and obtained the standard results for residue class rings, the radical of an ideal, prime and primary ideals, and primary decompositions of an ideal, Defining tertiary and primal ideals and radicals as usual', we obtain the usual existence and uniqueness theorems for ter- n n ni tiary and primal decompositions, The unique primary decomposition A = i= 1(A + Pi )p 1 of an ideal A having a primary decomposition (W. E. Barnes and W, M. Cunrtea, Ideal decompositions in Noetherian rings, Canad, J, Math 17 (1965), 178-184) is also obtained, as well as the known necessary and sufficient conditions for the existence of a primary decomposition in ordinary rings, (Received February 24; 1966,)

66T-264, T, Q. McLAUGHLIN, University of Illinois, Urbana, Illinois, Splitting and decomposi­ tion by regreuive sets, II.

K. I. Appel has recently shown: any finite union of immune retraceable 11ets is immune. More­ over, the following more general result is obtainable by an easy reduction to Appel's theorem: any

489 finite union of immune regressive sets is immune. The latter result enables us to strengthen greatly a pair of theorems proven in McLaughlin, Splitting and decomposition by regressive sets, Michigan Math. J. 12 (1965), 499-505. For definitions the reader is referred to that paper. Theorem A. There exists a cohesive set a of numbers such that (1) a is split by a retraceable set of degree ~ 0 1, and (ii) a is not R •-decomposable. Theorem B. There exists a set a of numbers such that (1) a is sequen­ tially decomposable, and (ii) a is not R *-decomposable by a collection each of whose members is either recursive or immune. We take the opportunity of remarking that the use of the axiom of choice in Lemma 2 of the cited paper is unnecessary and occurred as the result of a simple oversight. (That axiom is likewise not required for the proofs of Theorems A and B above.) (Received March 7, 1966.)

66T- 265. HUBERT HALKIN, University of California, San Diego, La Jolla, California 92038. A maximum principle for a mathematical programming problem in infinite dimensional space.

O:tven a set L, a mapping f = (f1 , ••• ,fk): L -> Ek and an integer m with 1 ~ m & k, find an ~ E L which maximizes f 1 (ll.) subject to the constraints fi(x) <; 0 if i = 2, ••• ,m and fi(x) = 0 if i = m + 1, ••• ,k. To simplify the notation we assume that x = 0. We assume also that L is a subset of a linear space X and that there is a set M C X and a mapping h = (h 1, ••• ,hk): X --> Ek such that (i) M is convex and x E M, (ii) hi is convex for i = 1, ••• , m and linear for i = m + 1, ••• ,k, (iii) for any setS= cofx,x1•···•x£J C M there is a mapping r: M-> L such that tor is continuous overS (with respect to the usual finite topology on S) and lim 0~o; o>ol(f(r 0, ( {3) \ <; 0 for i = 1,2, ••• ,m, ('Y) hi (x) & hi (x) for all x E X and all i = 1, ... , m) (o) :Lt, 1 ,;\ihi (x) ;:; Lf= 1 ;\ihi (x) for all x EM where iii= hi if i = m + 1, ... ,k. (Received March 21, 1966.)

66T-266. D. W. WILLETT and J. s. W. WONG, University of Alberta, Edmonton, Alberta. Boundedness and unboundedness results for some nonlinear differential e9uations.

Theorem. For each positive integer n, there exists a continuous function a(t) such that a(t) -> oo as t _, oo and the differential equation x" + a(t)x2n- 1 = 0 (*)has an unbounded solution. On .t, -A , -1 the other hand, suppose an equation of the type (*) is given. Let 'l'(t) = a (t), where /\ = (n + 1) , 1/1"' = max (0, '~/;"' ), and ¢! = 1/;nft 1/;"'. Theorem. If n = 1 and q, ~ Z, or if n > 1 and c/> is bounded, + n 0 + 1 n then all solutions of (*) are bounded. Theorem. If a(t) ---> oo as t _, oo and c/>n (t) --+0 as t --+ oo, then all solutions of (*) tend to zero as t --> oo. (Received March 7, 1966.)

66T-267. M.P. JONES, Southwestern at Memphis, Memphis, Tennessee 38ll2. Continuity of solution of a generalized diffusion equation.

For M > 0 and T = [a,b), let IJJM denote the set of all real valued stochastic processes y satisfying: (i) [y(t), iJt, a ~ t & b] is a martingale; (11) E [(y(a))2} < oo; (iii) there is an isotone function F such that if s < t, E t

490 1953; p, 277), the equation x y(t,w)- xY(a,w) = ftm(s,xy(s,w))ds + ru(s,xy(s,w))dy(s,w) has an a a essentially unique dtdP-measurable solution x Y, In the topology 1 (defined by E. J, McShane in A weak topology for stochastic processes, Bull. Math. Soc, Sao Paulo, 14 (1959), 86-90), x y depends continuously on Y over Q~. (Received March 10, 1966,)

66T-268. P. P. SAWOROTNOW, The Catholic University of America, Washington, D. C, 20017, Second structure theorem for complemented algebras, Preliminary report,

A complemented algebra is a Banach algebra with a Hilbert space norm such that the orthogonal complement of an ideal is an ideal of the same kind, The author proves the following generalization of a result of Smiley {Proc, Amer, Math, Soc, 4 (1953), 1-4], For each simple complemented alge- bra A there exists a set J and a complex-valued Hermitian matrix {aijf on J X J such that A is isomor­ phic to the algebra of all complex matrices {x ij ~ on J X J such that Lij xij a.Jkxik < oo, The matrix [ aij} has the property that L:i !xi 12 :;> Lijxi aijxj for each x E L 2(J), (Received March 11, 1966,)

66T-269, YU-LEE LEE and C, J, MOZZOCHI, University of Florida, Gainesville, Florida, Topological spaces with large uniformities,

Theorem, U (X, 1) is a locally compact u-compact T 2 space then (X,1) admits a uniformity U for. X whose uniform topology is 1 and U is the family of all neighborhoods of the diagonal in X X X with the product topology, (Received March 11, 1966,)

66T-270, N, M. RIVIERE andY, SAGHER, University of Chicago, Chicago, Illinois 60637, On the stability of some algebras of absolutely convergent Fourier Series,

The notation is that used in Abstract 66T-180, these c}fofiai), 13 (1966), 262, Theorem, Let F (x) be defined in the open interval I, and such that if the domain of F contains the range of

f(x), f(x) E As (0 < s :;> 1), then F(f(x)) E Us 1

66T-271. L, K, ARNOLD, Brown University, Providence, Rhode Island 02912, A transformation with no invariant measure.

Assume 0

Define c/J on [o,x0) by c/J(x) = ax + x 0 and on [xk,xk+ 1) by c/J(x) = (a-k)(x - xk) + x_ (k+ 1)• for k = 0,1,2,. .. , Then by deleting a set of Lebesgue measure 0 from [0,1) we obtain a set X such that the restriction of

c/J to X is a bijective, measurability preserving, nonsingular transformation with respect to Lebesgue measure on X. Moreover, there is no u-finite measure iJ. which is equivalent to Lebesgue measure on X and is such that iJ.(c/J(B)) = iJ.(B) for all measurable subsets of X. (Received March 14, 1966,)

66T-272, V. R, CELESTE, Polytechnic Institute of Brooklyn, 333 Jay Street, Brooklyn, New York, 11201. Integration over fields with a nonarchimedean valuation,

A theory of integration for set functions from a measure space to an arbitrary field with a nonarchimedean valuation has been developed, A function f is strongly measurable (!J) on (X,S) if

491 there is a sequence of elementary functions, each measurable (when regarded as transformations), and the sequence converges a.e. [IlL I] (i.e., a.e. with respect to the total variation of !L). Theorem. f is strongly measurable (!L) iff there is a measurable function g with separable range and g = f a.e. [l~tl]. A function f is called integrable (!L) if there is a sequence {fnJ of measurable elementary functions Jlf - f I] is functions which converge to f .a,e. [ l~tl] and if the sequence of real valued l n m l~tl-mean convergent to zero and each lfn I is IILl-integrable, Iff is integrable (!L) then Jfd~t is defined as Limn Jfnd!L, the integral of an elementary function being defined as the sum of the products of the values of the function times the measure of the set upon which the function takes this value, Most of the classical results of integration theory are obtained, such as estimates of integrals and a dominated convergence theorem, but the nonarchimedean nature of the valuation is heavily used to obtain a Fubini type theorem and a countably additive integral. (Received March 15, 1966.)

66T-273. LOUIS SUCHESTON, Ohio State University, Columbus, Ohio 43210, On the ratio ergodic theorem.

Let (X,Uf,p) be a probability space and letT be a positive linear operator on L 1 "' L 1(x,cfil,p).

Assume that for each f E L 1 the sequence JITnfldp is bounded. Then the space X decomposes uniquely (p-modulo zero) into two measurable sets, Y and Z, with the following properties, The Chacon-Ornstein theorem holds on Y : if supp f = {x: f(x) 'I 0}, f E L 1' g E L 1' g ~ 0, then the ratio (f + Tf + ... + Tn- 1f)/(g + Tg + ... + Tn-lg) converges to a finite limit on the set YnSupp(g + Tg + ... ). Z is the largest set which "disappears" under Tin the sense that n- 1(JTllzdP + JTj+l1zdP + ... + JTl+n- 11Zdp) converges to zero uniformly in j (1 Z is the indicator function of Z), Z is closed: if supp f C Z, then supp Tf c Z. (Received March 17, 1966.)

66T-274. KARL GUSTAFSON, Istituto Matematico, Universita di Roma, Rome, Italy. State diagrams for Hilbert space operators.

State diagrams (see the terminology and results of Seymour Goldberg, Linear operators and their conjugates, Pacific J, Math, 9 (1959), 69-79) are obtained for the following important classes of (generally unbounded) operators: symmetric, essentially self-adjoint, normal Cayley Transforms, normally solvable. These diagrams generalize, and related characterizations of these and other operators are also obtained. (Received March 14, 1966,)

66T-275, OLGA TAUSSKY, California Institute of Technology, Pasadena, California,~ factorization of the adjugate matrix.

Various factorizations are being examined. In particular for A a nonsingular n X n matrix with n n-1 characteristic polynomial x + a 1x + ... + an_ 1x +an the following relation is obtained: adj(A) = (- l)n- 1 Tiy-~A- A1I), where At are the zeros of xn- 1 + a 1xn-2 + ... + an+1• Each factor has detA as its determinant, If A has only one characteristic root then adj(A) can be expressed as the (n-1-st power of a matrix with the same characteristic root, (Received March 16, 1966,)

492 66T-2.76, E. B. DAVIS, Stanford University, Box 2.966, Stanford, California 94305, On mero­ morphic functions of two complex variables.

For each fixed value of a parameter s, a surface F(s) C c 2 is constructed. Uaing an approach similar to that in Bergman, Ann. Acad. Sci. Fenn. A. I. 336 12. (1963), 5-9, an analogue of Nevanlinna's First Fundamental Theorem is obtained with F(s) assuming the role that the circle assume& in the one-dimenllional case. Next, an analogue of the Second Fundamental Theorem is obtained for G = u:s F(s), by using the results in one complex variable applied to each of a series of domains h(s,6) C Cg, which are holomorphic images of a circle in an auxiliary complex plane. By construction,

F(s) = a( U#:0h(s,6)), so that aspecu of Nevanlinna's theory can be extended from the h(s,6) to F(s), and thence to G. After the error terms are shown to be sufficiently small, the deficiency, ~(a), and the index of multiplicity, O(a), of a meromorphic function with respect to a complex number a, and to G are introduced, The following theorem is proved: Let f be a nonconatant meromorphic function defined in C. Iff satisfies certain conditions (mainly imposed on its behavior in the plane z 1 = 0), then for any finite set a 1, ... ,aq' of distinct complex numbers (including perhaps oo), 2:~ .. 1(6'(ak) + B'

66T-Z77, TATSUJI KAMBAYASHI, University of Pisa, Pisa, Italy, A convergence criterion for multiple ratios.

Let .25. be a point on a finite-dimensional complex (reap. real) projective apace J!• and let I, be a projective transformation on !: repreaented by a complex (resp. real) linear transformation

T with det(T)"' 0. Then, the sequence of points f:!:n~} converges in !!_as n -+ oo for every~ if and only if no two diatinct eigenvalue& of T have the same absolute value, The theorem has certain application& in mathematical economics, (Received March 18, 1966.)

66T-Z78. MARTIN SCHECHTER, Institute for Advanced Study, Princeton, New Jersey. On H -bounded operator&,

For a complex valued function q(x) defined on n-dimensional Euclidean space En, a. >- n, p ~ 1,

let Ma.,p(q) = aupxflx-yl<11q(y)IPa.(x- y)dy, where a.(z) = lzla., a.< O,l/l0(z) = 1- loglzl, tpJz) = 1, a. > 0. Let Ma.,p denote the aet of those functions q(x) for which Ma.,p(q) < oo. For u E Cg'(En) (the aet of infinitely differentiable function& with compact aupports in En), let ii(~) denote its Fourier tranaform. For a real set llu II! = J(1 + lslz)s lii(s) lzd~ and let Hs be the completion of Cg'(En) with respect to tllia norm. Theorem. If a > 0 and q E M zs-n,2.' then for each t auch that 0 ~ t ~ s one !las llqullz < const. M (q)llull2 for all u E Hs-t. (Received March 18, 1966.) -t - 2 a-n, 2 '11-t

66T-Z79. A. A. MULLIN, University of California, Box 808, Livermore, California 94551. On an analogue of the Ramanujan aum.

Define R • (modified Ramanujan sum of tile first kind) and R ** (modified Ramanujan sum of

the aecond kind) 111 follows: put R*(m,n) = LM(x,n)=ll' exp Z11"imx/n, where M(x,n) =!I indicates the condition that the mosaics of x and n have no prime in common, and where, in addition, the sum is over a reduced reaidue system~ n; put R **(m,n) = 1 - IR *(m,n)l. Lemma 1, R *(O,n) ;;; 1/l(n) for

493 every natural number n (with equality if n is prime), where f is Euler's totient. Lemma z. IR *(l,n)l;;; jl'(n) I + (1/>(n) - 1/>*(n)), for every natural number n (with equality if n is prime), where It is Mobius' function and 1/>* is defined in Abatxact 64T-450, these c}/oticeiJ 11 (1964), 680. Lemma 3, Lm1iinl¢"(m)- !L(m)l + Lm1iin p.•(m)- IL*(m)l = Lm;;;nlX(m)- !L(m)l ~ Lm1iinR**(l,m) + Lmo1nll/:l(m) - c/l*(m)l for every natural number n, where ~is Liouville's function, and "*• I"* are defined loc. cit, (Received March ;u, 1966,)

66T-Z80, HARVEY FRIEDMAN, 50 Massachusetts Avenue, Apartment Z07, Cambridge, Massachusetts, Strength of certain set theories,

We define ZF* as in Strength of certain set theo:dea. Let M be an interpretation of set theory. Let D(M) be the submodel of M consisting of the objects definable in M. Let L(M) be the submodel of M consisting of the objects satisfying the predicate L in M. Theorem. U M F ZF • then L(M) n D(M)!=ZF + V = L. Corollary. U ZF* has an E-model thenZF and ZF* has the same minimal tralfsitive E-model, Now let A be the set of all analytic seta of integers. Let A(L) be the set of all

sets of integers analytic in L. Theorem, A(L) < AnL< LnP(w)! Corollary. U all analytic sets are constructible, then the analytic seta form a model of analysis, In fact, they form an elementary sub­ model of P(w)nL'. Theorem, A sentence of number theory is provable in ZF* if and only it it is provable in ZF, M =(A,R.)~ ZF is said to be an w-model for ZF it the integers in Mare well-ordered by R, M is said to be an w1-model of ZF if the countable ordinals in M are well-ordered by R. Theorem. Every .w1-model of ZF satisfies the sentence "there exists an w-model for ZF." (Received March Z3, 1966.)

66T-Z81, J, B. FLORENCE, Simon Fraser University, Burnaby Z, B. c. Canada, Infinite recursively enumerable claaaea with any finite number of infinite recursively enumerable aubclaues.

There exists, for any natural number m, an infinite recursively enumerable class with exactly m proper infinite recursively enumerable subclaaaea, This answers a question of P. R, Young [Abstract 66T-49, these cJVoticti) 13 (1966), 141]. (Received March Zl, 1966.)

66T-Z8Z. W. E. BAXTER, University of Delaware, Newark, Delaware 19711. A polynomial identity for symmetric elements and commutativity, Preliminary report.

The following theorems are proved: Theorem 1, Let R be a simple ring, with involution J, such that either Z = (8) or (R: Z) > 4, then it an= s, fixed n, all a E S(J) = {a E R lsj = s J, R is com­ mutative, Theorem la, Let R be a division ring with involution, J, and such that sn(s) = a,n(s) ~ Z, for all s E S, then R is commutative, Theorem lb. Let Fn be the n X n matrices over F, n <:; 3, then there exists a E S(J) such that am;. a for all m <:; z. These results depend upon results of Martindale, Pacific J, Math, Vol. 11, pp. 1431-1441, and the observation that Martindale's conditions of algebraic algebra can be weakened under the above hypothesis, The main tool is the proof that every nonzero left ideal contains an idempotent. (Received March Z4, 1966,)

494 66T- 283. K. 0. LELAND, University of Virginia, Charlottesv!lle, Virginia, A polynomial approach to topological analysis.

The machinery of topological analysis of Whyburn, Porcelli, and Connell is developed for the case of twice continuously differentiable functions without use of algebraic topology, using only elementary methods and the Stone-Weierstrass Theorem (SWT). The central lemma states that if Pis a polynomial whose real part is bounded byE > 0 on B = !lxl =I}, then IP(x)l ~ 2Eixl/(l- lxl) for all x E U = /lx I < 1 J. For a real valued function

66T-284, A. H. LACHLAN, Simon Fraser University, Burnaby 2, B. C. Canada, Nonaxiomatiz- ability of the elementary theory_ of standard enumerations of recursively enumerable sets.

Let W be a binary relation on the natural numbers such that iW J is a standard enumeration of the r.e. sets, where Wi = {x!W(i,x)j. Denote by SE the elementary theory of W. Theorem, SE is not axiomatizable, (Received March 2!, 1966.)

66T-285, B. L. CHALMERS, Stanford University, Stanford, California, On boundary behavior of Bergman's kernel function in the theory of two complex variables.

2 In the space of two complex variables consider, for p > 0, the domains R ~ Rp = [iz 1 ! /P

+ iz 2 12 < 1] introduced as domains of comparison in the theory of the Bergman kernel function (Mat. Sb, (N.S) 1 (43), {1936), 79-96). 8R has an "edge" s 1 which is homeomorphic to the circumference of a 1 circle, Let P = ( 11 = 0, 12 = I) E s • At this point the kernel function KR (z l'z 2; z 1,z 2) becomes infinite of order n = {p + 2}, when approaching in a certain sense, A. In (*) (Crell's journal, 169 (1933), l-42, and 172 (1934), 89-128) Bergman obtained sufficient conditions for a domain B so that the order n = 2, 3 or 4, Generalizing the above approach and using certain transforms of the R- domains as domains of comparison, the author proves: Let, in the neighborhood of the origin Q, 1 2 B = [2x 1 - 1/;(y 1,!z2 1 /p) > 0], where zk = xk + iyk' k = 1,2, and 1/;(s,t) = as + u~ + c/J(s,t}, where u > 0 and lim ¢(s,t)(s 2 + t 2)- 1 = 0. If for a given (arbitrarily small) £ > 0, there exists an r > 0 s,t- 0 such that [Bn(z 1 = Y)] C [lz2 1 <£]for 11'1 ~ r, then, when approaching in the sense A, lim( ) (z + :Z )P+2K (z ,z2,z ,z ) =uP (p + 1)/rr 2. Further results concerning the boundary z 1,z 2 __, Q 1 r B 1 1 2 behavior at Q of Bergman's invariant metric and other functionals (see (*), 98-10 1) are obtained.

(Received March 18, 1966.)

66T-286. A. L. ROSENBERG, P. 0. Box 2!8, Yorktown Heights, New York. Real-time definable

languages.

The closure properties of the class 1R of languages defined by real-time on-line multi-tape Turing machines are studied. Theorem, The class R is a Boolean algebra, Corollary. (a) IR

495 contains languages which are not context-free, (b) lR contains inherently ambig).lous languages. Theorem. IR is closed under inverse generalized sequential machine (gsm) mapping. Theorem, If A is in R, and if B is regular, then AB is in IR. Theorem, IR is not closed under the operations of (a) concatenation, even with a regular set; (b) (unary concatenation) closure; and (c) reversal. Corollary. There is a language which is definable in twice real time, but not in real time. Theorem. (a) R is not closed under gsm mapping, (b) The derivative of a set in 1R with respect to a regular set need not be real-time definable. (c) The quotient of a set in IR by a regular set need not be real­ time definable. It is further shown that these results are invariant under certain modifications to the model used to define IR, (Received March 25, 1966.)

66T-287, T, G. FRANK, P. 0. Box 1663, Los Alamos, New Mexico. Error bounds on numerical solutions of Dirichlet problems for quasilinear elliptic equations.

Let R be a closed, bounded, simply connected region in the plane. Let P denote the Dirichlet problem Auxx + 2Buxy + Cuyy = G on R in which A, B, C, G depend on x,y,u,ux,uy• It is assumed that A,B,C satisfy a uniform ellipticity condition and a condition (see L, Bers, F, john, and M. Scheichter, Partial differential equations, Intersclence, New York, 1964, pp. 262-264) which enables uniqueness of the solution of P to be established by means of a maximum principle; also it is assumed that R and the coefficient functions are such that u E c 4 on R. Several finite difference analogues of P are studied which use, essentially, central differences except near the boundary. One such scheme uses the method of J, H. Bramble and B. E. Hubbard, Contributions to differential equations 2, (1963), 319-340, to treat the term 2Buxy' It is shown that the solutions of the finite difference ana­ logues converge, with decreasing mesh width h, to the solution of P. Moreover, the error is O(hP) with p either one or two depending on which particular combination of difference equations in the interior and at the boundary of R is used, (Received March 25, 1966.)

66T-288, K. G. JOHNSON, Virginia Polytechnic Institute, Blacksburg, Virginia, Functions of bounded variation and change of variable in a Lebesgue integral.

An affirmative answer to each Of two questions of Varberg in Amer. Math. Monthly 72 (1965), 841, is contained in the following result. Theorem 1. Let 1P be real and of bounded variation on a closed interval [tl't2] and let M be a measurable subset for which m [IP(M)) = 0 where m is Lebesgue measure. Then !j)'(t) = 0 almost everywhere on M. An application is in proving the following general change of variable formula, other versions of which require tfJ either to be absolutely continuous or monotone continuous, Theorem 2, Let f be extended real valued and summable over [a,b]. Let F(X) =

J:f(u)du, Let 1P be of bounded variation on a closed interval [1_,t2] such that cf>((! 1,t2]) C [a,b], and such that F(f/>) is absolutely continuous on [t1 ;t2]. Let c/>' be a function equal almost everywhere to the derivative of lj>, Then f(cp)cJ>' is summable over [t1 ,t2) and J:\~~~f(u)du = J~~ f(c/>(t)>IP' (t)dt = F (lj>(t2)) - F (!P(t1 )). (Received March 24, 1966.)

496 66T-289. W. F. TRENCH, Drexel Institute of Technology, Philadelphia, Pennsylvania. On the stability of midpoint smoothing with Legendre polynomials.

Let f be square integrable on [- 1,1], and let p be the projection off on the subspace of poly­ nomials of degree :'i 2k + 1, relative to the usual inner product (f lg) : J-~f(x)g(x)dx. It is known

(see reference cited below) that p(O): .[iw(x)f(x)dx, where w(x): (k + 1/2)P2k(O)P2k+ 1 (x)/x, P2k and

P 2k+ 1 being Legendre polynomials. Let C(IJ) = j_~w(x) cosxDdx. Theorem. IC(ll)l < 1 if 8i 0, and C(O): 1. The theorem is established by exhibiting a rational function E(O) of the form E({J): {J Zk+2 "2k+ 1b o2s)-1 ("k IJ2r) · h d b b · h · 1 - (L..s=O s L..r:Qar , Wlt a 0 , •.• ,ak an o•···• Zkt1 positlve, whic interpolates C 2 (0) at the critical points of C(IJ). Using a different method of proof, Wilf [The stability of smooth­ ing by least squares, Proc. Amer. Math. Soc. 15 (1964), 933-937] has previously shown that there is an integer ko such that the present theorem holds if k ~ k 0 , and conjectured that ko : 0. The present proof is valid for all k ~ 0. (Received March 25, 1966.)

66T-290. P. S. SCHNARE, Louisiana State University in New Orleans, New Orleans, Louisiana

70122. The maximal T0 (respectively T 1) subspace lemma is equivalent to the axiom of choice.

The following problem is considered. Given some topological property P. Is this statement true: (MP) every topological space (X,r) has a subspace (Y,riY) (where riY is the relativization of T to Y) with property P, which is maximal (with respect to inclusion) 7 (The blanket topological refer­ ence is]. L. Kelley, General topology, Van Nostrand, Princeton, 1955.) Gaifman proved that the axiom of choice (AC) implies MT0 and noted that AC implies MT 1 in an unpublished paper. By prov­ ing the converses the apparently new: Theorem A. AC iff MT0 iff MT 1 is established. A source of counterexamples is provided by: Theorem B. Let P be possessed by every finite discrete space but not by a countably infinite minimal T 1 space. Then MP is false. Corollary. MP is false for P: T 2, regular, completely regular, normal, T 3, T 4, paracompact, metrizable, totally disconnected or dis­ crete. (Received March 28, 1966.)

66T-291. ROBERT LAGRANGE, University of Colorao, Boulder, Colorado. Disjointing infinite sums in Boolean algebars.

Let (Jt be a Boolean algebra, and suppose a is the supremum in Mot an indexed set {at: t E T ~. We say that {at: t E T} can be disjointed if there is an indexed set fbt: t E T} of elements of ()f_ satisfying (i) a is the supremum of {bt: t E T J. (ii) for each t E T, bt :'i at and (iii) if t i t' then bt • bt' : 0. Also fat: t E T} can be weakly-disjointed if the indexed set {cf: f E F} can be disjointed, where F is the set of all finite, nonempty subsets of T, and for each f, cf is the supremum of all at such that t Ef. The following result holds: If tYi satisfies the 911-chain condition, and T has power ~ 911, then (1) fat: t E T} can be weakly-disjointed and (2) If fat: t E Tj is totally ordered by the Boolean inclusion then {at: t E Tj can be disjointed. An example of a Boolean algebra rJ£, and an in­ dexed set [at: t E T} with supremum 1, which can be neither disjointed nor weakly-disjointed is con­ structed. This solves Problem 1 of Sikorski, A few problems on Boolean algebras, Colloq. Math. XI. Assume the Generalized Continuum Hypothesis. Let ()i be an 911-complete Boolean algebra which is representable as an 911-homomorphic image of a 2 911 -complete field of sets. If Ot.-is weakly 911-distributive, then ()(is 911-distributive. (Received March 8, 1966.)

497 66T-292. D. G. deFIGUEIREDO, University of Chicago, Chicago, Illinois 60637. Fixed point theorems and Galerkin approximations.

Let E be a Banach space with property (rr1), i.e. (i) there exists a directed set (Fa.) of finite dimensional subspaces of E whose union is dense in E, and (ii) for each a. there is a projection P a. of norm 1 and range Fa.' Hilbert spaces and all Banach spaces used in applications have this property.

Let C be a bounded closed convex set in E. A (nonlinear) operator T: C ----> E is a G-operator (or a Galerkin approximable operator) if(!) PaT: Fa nc __,Fa is continuous, and (ii) the solvability of Pa Tx = x in Fa implies the solvability of Tx = x in E. Weakly continuous, strongly continuous, com­ pletely continuous, P-compact and nonexpans!ve mappings are examples of G-operators. Theorem.

Let E be a reflexive Banach space with property (rr1). Let T: C __, E be a G-operator. Suppose

(Tx, Jx) ~ Jlx Jill( Jlx Jl) for all x on the boundary of C. (J is the duality mapping with gauge function llJ Then T has a fixed point. This theorem contains fixed point theorems of Schauder, Altmen, Rothe, Petryshyn, Browder and others. (Received March 30, 1966.)

66T-293. LORENZO CALABI and W. E. HARTNETT, Parks Mathematical Laboratories, Inc., River Road, Carlisle, Massachusetts 01741. Extension to nonconvex, closed sets of Motzkin's theorem on convexity.

In the Euclidean plane we shall say that a closed ball B is a ball of support for the set A if there are points of A on the boundary of B but not in its interior. To every closed set A we then associate its skeleton S, viz. the set of the centers of the maximal balls of support for A. For x E S let q(x) denote the distance from x to A. The pair (S,q) is called the skeletal pair of A. Finally if C is the closed convex hull of A, we call C\A the convex deficiency of A. Theorem. Two closed sets have equal convex deficiencies iff they have equal skeletal pairs. This theorem includes Motzkin' s, which may be stated: a closed set is convex iff its skeleton is empty. Corollary. A closed set is uniquely determined by its closed convex hull and its skeletal pair. Corollary. If a closed set A has empty interior but its convex hull does not, then A is uniquely determined by its skeletal pair. The notion of skeletal pair was introduced during a study of the Blum model for the neuro-physiological process of recognition of "line figures" (Symposium on models for perception, M.I. T. Press, forth­ coming). (Received March 31, 1966.)

66T-294. FERGUS GAINES, California Institute of Technology, Pasadena, California. On the Kato-Taussky- Wielandt commutator relation. Preliminary report,

Let ..::!'be the algebra of linear transformations on an n-dimens!onal vector space V over a field _y-and let A, BEY. Let Ai+ 1 = A1B- BAi, i = 0,1,2,. .. , where Ao =A and let fk(A,B;b) =

A2K+ 1 - li 1A2K_ 1+ li 2A2K_ 3 - ... + (- l)KiiKA1 whereK=k(k- 1)/2 and o= (/i 1,1i2 , ... ,1iK) with Iii E-St: Taussky and W!elandt (Proc. Amer. Math. Soc. 13 (1962), 732-735] proved fn(A,B; ) = 0 if Iii is the 2 ith elementary symmetric function of ( /3r - (3 s ) , 1 ::? r < s ~ n where (Jr are the characteristic roots of B. Theorem. Let A and B generate Y and suppose the characteristic of Y does not divide n. If there exists li so that fk(A,B;o) = 0 where 1 ~ k < n, then the characteristic roots (Jr of B belong to 2 2 the splitting field of gk(x;li) = x K+l- o1x K-l + ... + (- l)K/iKx over§. The relation f 1(A,B;/i) ~ 0 is commutativity which implies that A and B have property L [cf. Motzk!n and Taussky, Trans. Amer.

498 Math. Soc. 73 (19 52), 108- 114]. A version of the theorem above is used (when k = 2) to prove a result which implies A and B have a generalized property L. (Received March 21, 1966.)

66T-295. B. V. O'NEILL, New York University, 251 Mercer Street, New York, New York 10012. Representing measures and hypo- Dirichlet algebras.

Let A be a function algebra on a compact Hausdorff space X and let be a complex homomor­ phism of A. Assume further that A is hypo-Dirichlet on X. (J. Wermer, Analytic disks in maximal ideal spaces, Amer. ]. Math. january 1964). Theorem. Let m be the unique Arens-Singer measure representing

respect to p, Corollary. Let P be a Gleason part of the maximal ideal space of A containing more than one point. Let T1 and T 2 be two points in P. Let p1 and Pz be representing measures for

T 1, T2 resp. Then p 1 and Pz are mutually absolutely continuous. The result is a partial converse to a theorem of Ahern and Sarason (P. Ahern, D. Sarason, Representing measures for function algebras, Abstract 630-128, these cJiotiaiJ 13 (1966), 95, and the proof makes use of their work. (Received April 4, 1966.)

66T-296. ROBERT BRADFORD, University of Southern California, Los Angeles, California 90007. Undecidability of the elementary theory of Dedekind cardinal addition.

Two cardinals a and {3 are comparable (in symbols C(a,{J)) iff a ;:i {3 or !3 ;:i a. A cardinal a is called a Dedekind cardinal iff a i a+ l. The axiom of choice implies that any two cardinals are com­ parable and all Dedek!nd cardinals are finite. Let ZF denote Zermelo- Fraenkel set theory without the axiom of choice. It is assumed that ZF includes special variables for cardinals and Dedekind cardinals as well as the nonlogical constants C, ;:i, +, =. ZF(C), or ZF'(C), denotes the class of theo- rems in ZF which contain no nonlogical constants other than C and no variables other than variables for cardinals, or Dedekind cardinals, respectively. ZF( ;:i), ZF'(;:i ), ZF(+, =), ZF'(+,=) are defined analogously. Theorem • ...!!_ZF is consistent, then ZF(C) and ZF'(C) (and therefore also ZF( ;:i ),

ZF'( ;:i), ZF(+,=)~ ZF'(+,=)) are hereditarily undecidable. Since ZF'(+,~) is hereditarily undecidable it follows that the elementary theory of commutative semigroups with the refinement property is undecidable. This settles a problem raised by Tarski and extends his result on the undecidability of the elementary theory of commutative semigroups [Abstract 591-29, these cJ{otiai) 9 (1962), 205]. The decision problems for ZF(+,=) and ZF'(+,=) were also posed by Tarski. (Received April 4, 1966.)

66T-297. WITHDRAWN

499 66T- 298. RALPH YOLK, I3I5 Dickens Street, Far Rockaway, New York II69l. Symmetric addition.

Definitions, The symmetric sum of two fractions is (p/q) +' (m/n) = (pn + qm)/(pm + qn). The natural product of two real quantities is then nx'r = ((I + r)n - (I - r)n)/((1 + r)n + (1 - r)n). The nth inverse constant is the number c(n) such that the n-fold iterate of T defined by T(z) = z + 'c(n) is the identity. Theorems. p + 'q = q + 'p, p+'(q + 'r) = (p + 'q) + 'r, p + '(- p) = 0, px'(l/p) = 1, p + '0 = 0 + 'p = p, 1x'p = p, p + '1 = px'I = 1, p + 'q = r implies q = (- p) + 'r, px'q =rim­ plies q = (I/p)x'r, px'(q + 'r) = (px 'q) +' (px'r), (p + q)x'r = (px'r) +' (qx'r) so (1/(1- r))x'a =a +'(rx'a) +' (rx'rx'a) +' ... , ((I- u)/(1 + u)) +' ((1- v)/(1 + v)) = ((I- uv)/(1 + uv)), tanh'(p + 'q) = tanh'p + tanh'q and tanh'(nx'r) = n tanh'r where tanh' denotes the inverse hyperbolic tangent, p + 'q = (1/p) +' (1/q) and p +'(1/q) = (I/p) + 'q = 1/(p + 'q) since c(n) = itan(7rk/n) with k between 0 and n - 1. So consider Y = X +' itan (7r/cos w) and notice that in XYZ coordinates (Z = cos w) this corresponds to a twisting of the plane Y = X, and that Y is real just when X is iff Y = X, (1/X). (Received April 4, 1966.)

66T-299. N. R. REILLY, Newcomb College, Tulane University, New Orleans, Louisiana 70118. Bisim"Jle inverse semigroups.

This paper extends results of Clifford's (A class of d-simple semigroups, Amer. J. Math. 75 (1953), 547-556) to bisimple inverse semigroups without identity. A right partial semigroup Sis a set S with a partial binary operation satisfying the condition: if a,b and c are elements of S such that a(bc) is defined then (ab)c is defined and a(bc) = (ab)c. An RP-system (R,P) is a right partial semigroup R with a subsemigroup P such that: ab is defined if and only if a E P; R has a left identity contained in P; ac = be implies that a= b, for all a, b E P, c E R; for all a, b E R, Pan Pb = Pc for some c E' R. The equivalence .!£'• is defined on R by (a,b) E.!£'' if and only if Pa = Pb. A representa­ tive is chosen from each .!£'•-class and the representative from the .Y'i-class containing c, where

Pan Pb = Pc, is denoted by a V b. For a, b E;: R, (a *b) is defined by (a • b)b = a Vb. Then R X R under the multiplication (a,b)(c,d) = ((c * b)a, (b • c)d) is a bisimple inverse semigroup, denoted by

R - 1 o R. Conversely, for e 2 = e E S, a bisim:;:>le inverse semigroup, let P e = Re nese. Then (Re, P e> is an RP-system such that S ~ R~ I " Re• (Received April 6, I966.)

66T-300. T.L. HAYDEN, University of Kentucky, Lexington, Kentucky 4(f506. A property of

Hilbert spaces. Preliminary report.

We say that two Banach spaces U and V have the extension property provided that: if S and T are subspaces of U and V respectively and f is a bounded bilinear functional on S X T, then f can be extended to U XV with the same norm. Theorem. The extension property holds if U and V are Hilbert spaces. Theorem. If U is a Banach space and the extension property holds for each Hilbert space. (Received April I I, 1966.)

500 66T- 301. j. T. DAR WIN, JR., Auburn University, Auburn, Alabama. On a new class of functions.

The class of quasi-absolutely continuous (q.a.c.) functions on [0,1] = j is defined, and some

properties of it shown. "The function f is q.a.c." means if f > 0, 3 a finite set E and (J > 0 such that, if {[ri,sJ~ is a countable collection of nonoverlapping intervals, each in j- E, such that Li (si - r i) < o, then Li if (si) - f (r i) I < f. This class forms a closed linear manifold in the Banach space of functions of bounded variation, with usual norm. (Received April 11, 1966.)

66T-302. D. L. STANGL, 1005 S. Sixth Street, Champaign, Illinois 61820. Multiplication in Grothendieck rings of integral group rings.

Let G be an abelian group, R the ring of algebraic integers in a number field which splits G, and denote by K0 (RG) the Grothendieck ring of the group ring RG. Then multiplication in K0 (RG) can be explicitly determined. Theorem 1. Let G be elementary abelian of exponent p, and take R = Z [IJP], where IJP is a primitive pth root of unity. Then there exists a ring monomorphism from K0 (ZG) into K0 (RG), and hence multiplication in K0 (ZG) is known. Next, let Gr denote a cyclic group of order r, Zr the ZGr -module Z [Or), where IJr is a primitive rth root of unity, and Ar a Z [IJr]-ideal in Q(Or). Theorem 2. Let G be a cyclic group, and let r,s be divisors of the order of G. Then multiplication in K0 (ZG) is determined by the formula [ZGr]([As]- [Zs]) = Ld {[Ns/s'(As)Zd]- [zdJ}' where

s' = s/(r,s), Ns/s' denotes the norm from Q(l:ls) to Q(l:ls 1), and d ranges over the divisors of [r,s] such that ([r,s]/d,s') = I. (Received April 12, 1966.)

66T-303. EUGENE SCHENKMAN, Purdue University, Lafayette, Indiana. Nilpotent automor­ phisms.

We shall say that an automorphism a is nilpotent or acts nllpotently on a group G if in the holomorph H = [G](n) of G with a, a is a bounded left Engel element; that is [H, ku] = 1 for some natural number k. Here [H, kn] means [H,(k - l)u] with [H, On] denoting H. We generalize a theorem

of Blackburn, j. of Algebra 3 (1966), 28-29 as follows. Theorem I. Let F be a subgroup of a nil­

potent group G such that F contains its centralizer CG(F). If n is an automorphism of G which maps F into F and which is nilpotent on F, then n is nilpotent on G. Theorem 2. Let G be a nilpotent group and suppose that for each prime p and fixed j, the Sylow p- subgroup Sp of G has exponent at most pl. Let F be an abelian subgroup of G so that F contains the elements of order 4, the torsion­ free elements, and the elements of order p for odd p of CG(F). If a is an automorphism of G which maps F into F and which is nilpotent on F, then a is nilpott:nt on G. (Received April 12, 1966.)

66T-304. C. W. LEININGER, Arlington State College, Arlington, Texas 76010. Maximal subgroups of the kernel of a semigroup with left zeroids.

Let K denote the kernel of a semigroup S which contains a left zeroid. If L denotes the set of left zeroids of S, then L = K (Clifford and Preston, The algebraic theory of semigroups, vol. I, Amer. Math. Soc., Providence, R. I., 1961, p. 70). Theorem 1. If S contains a left zeroid and an idempotent element e, then the following statements are mutually equivalent: (i) e is a left zeroid of

501 S; (ii) eS is a maximal subgroup of Se; (iii) Se = K. Theorem 2. If e and f are distinct idempotent left zeroids of S, then eS and fS are isomorphic and disjoint. Theorem 3. If e is an idempotent left zeroid of Sand Se = {x: x E S, xe E eS J. then Se is a sub-semigroup of Sand eS is the subgroup of (two-sided) zeroids of Se• Theorem 4. If e is an idempotent left zeroid of Sand P = {eS: e E KJ, then the following statements are equivalent: (i) K ~ UeSe; (ii) P is a partition of K. (Received April 12, 1966.)

66T-305. C. R. BORGES, University of California, Davis, California 95616. A note on a problem of Mr6wka.

Recently Mr6wka [On nearly-LindelOf spaces, Abstract 622-63, these cJiotiaiJ 12 (1965), 344] defined nearly-Lindelof spaces (A topological space X is nearly-Lindeli:if provided that every un­ countable strictly decreasing family (indexed by a well-ordered set) of nonempty closed subsets of X has nonempty intersection) and asked the following question: Is every nearly- Lindel6f space a LindelOf space? We will answer this question positively. Furthermore we will extend our results, by generalizing the concepts of Lindelofness and near- Lindel6fness in terms of cardinality (these results have been accepted for publication in the Israel J. Math.). (Received April 12, 1966.)

66T-306. M.S. ROBERTSON, 361 Lenox Avenue, Milltown, New Jersey 08850. A generalization of the Bieberbach coefficient problem for univalent functions.

Let f(z) = L~anzn be regular and univalent in E (z: lz I< 1), and map E on a domain which is convex in one direction. Let n and m be nonnegative integers. Then (1) In jan 1- m lam I I ~ !n2 - m 2 1 2 ·la 1 j where a 0 = 0. Equality signs are attained for the function a 1z(1- fz)- where jfj = 1. In parti- cular,(- 3 t 2/n)ja 1 1 ~jan!- lan_ 1 1 ~ (2- l/n)ja 1 !. If f(z) = L:fanzn is regular, univalent and close-to-convex in E and n- m is an even integer then (1) again holds and(- 6 + 8/n)ja 1 1 ~ lanl

- lan_ 2 1 ~ (4- 4/n)ja 1 !. Furthermore, for close-to-convex f(z) (l) holds when n = 3, m = 2. (Received April 14, 1966.)

66T-307. PIERRE BERTHIAUME, University of Ottawa, Ottawa, Canada. The injective envelope in the category of semigroup.

All semigroups are noted multiplicatively. AnS-set A is a set with a semigroup S of operators, i.e., it is essentially a semigroup representation. An extension A' of A is an essential extension iff every congruence relation on A' not the identity is not the identity on A. A characterization in terms of elements can be given. Theorem. The injective envelope (i.e. a minimal injective extension) of an S-set always exists: it is a maximal essential extension of A and is unique up to isomorphism over A. Example. LetS be a dense-in-itself chain with smallest and greatest elements. It is a semi­ group if the product of two elements is taken to be their inf., and thus an S-set. Then the injective envelope of S is its dedekind- MacNeil completion. In particular, the extended reals can be regarded as the injective envelope of the extended system of rationals. (Received April 15, 1966.)

502 66T-308. S. R. KRAFT, Negev Institute; P. 0. Box 1025, Beersheva, Israel. Riemann functions for a linear system of "hyperbolic like" form.

Riemann functions are defined for systems equivalent to L.(U) ~a ui + "N A .. ui = 0, 1 Xi L.,J= l 1] i = l, ••• ,N, where Cauchy data is given on the initial surface H: "N x. = 1. The value of ui is ex- .L..1= 1 1 pressed as a quadrature of the Cauchy data and the Riemann functions over subsets of H. The Rie- mann functions satisfy the adjoint equations of systems obtained from Li(U) = 0, by setting certain combinations of the ui equal to zero. Initial data for these functions involve the Aif (Received April 15, 1966.)

66T-309. D. A. ROBINSON, Georgia Institute of Technology, Atlanta, Georgia 30332. Holomorphy of B ol loops.

Let (G, •) be a loop and, for each x in G, let x- 1 denote the right inverse of x. The loop (G, ·) is a B ol loop if and only if (xy • z)y = x(yz • y) for all x,y, z in G. (See Robinson, B ol loops, Trans. Amer. Math. Soc., to appear.) If A is a group of automorphisms of (G, • }, construct the A-holomorph of (G, •) in the usual manner. Theorem. The A- holomorph of (G, ·)is a Bol loop if and only if (i) (G, ·) is a Bol loop and (ii) each a in A is right nuclear in the sense that x -l • xa is in the right nucleus of (G, ·)for each x in G. (Received April 18, 1966.)

66T-310. N. L. KERZMAN, Facultad de Ciencias Exactas y Naturales, University de Buenos Aires, Pen! 272, Buenos Aires, Argentina. Intermediate spaces of n + l quasinormed spaces.

A quasinorm is a function U•ll such that llxll ~ 0, x = 0 if llxll = 0; II Axil= IAI llxll (A escalar); and there is c > 0 that llx + y II ~ c(llx II + IIY Ill. Given n + l quasinormed spaces (F j)j=O, •.• ,n which are continuously included in a Hausdorff linear topological space, the intermediate spaces [(F.).] (J can be J J ,p defined in the same way as Lions-Peetre's spaces (F0 ,F 1)(J,p· Let K be the convex hull of the origin and the canonical basis of Rn; it is shown that ni~aFi---> i[(Fjlj](J,p___,hLJ=O Fj holds, the inclusions j and h being continuous in the case 0 < p ~ oo and fJ belonging to the interior of K; if p = oo this also holds for all B belonging to K. Moreover if 0 ¢ K then [(F j) JJIJ,p = f 0 }. Several properties of (F O, F l )e, p are extended to these spaces such as the- interpolation theorem, some characterisations due to P. Krt!e and a convexity inequality•for an interpolated operator. (Received April 18, 1966.)

66T-3ll. C. A. BERENSTEIN, Facultad de Ciencias Exactas y Naturales, University de Buenos Aires, Pen! 272, Buenos Aires, Argentina. Remarks on the method of interpolation by functional quasinorms.

It is known that if (Aj)j=O, 1 are Banach spaces and 0 < 0 < l, l ~ 'p ,; oo, then the intermediate spaces (A0 ,A 1)(J,p are com;:>lete. It is shown that this is also true if (Aj)j=O, 1, are quasinormed complete spaces and the intermediate spaces are obtained by Peetre-Krt!e' s method of functional quasinorms. The same is true of Gagliardo's method is used. Using Krt!e's extension of Calder6n's identities it follows that all the Lorentz spaces Lpq' 0 < p, q ~ oo, are complete (this is known if l < p ~ oo, l ;;; q ;;; oo). (Received April 18, 1966.)

503 66T-312. N. A. FRIEDMAN and M. L. KATZ, University of New Mexico,,ALlbuquerque, New Mexico. A representation theorem for additive functionals.

Let S be a compact Hausdorff space and let C(S) denote the class of real-valued continuous functions defined on S with llfll = supsESif(s)l. We consider functionals ¢defined on C(S) such that (l) Continuity. For each E > 0 and b > 0 there exists o= o(E ,b) such that llf II :'£ b, llg U :'£ b, and ilf- gil~ o imply I(f)- ¢(g) I :'£E. {2) Boundedness. For each b > 0 there exists B = B(b) such that

llfll :'£ b Implies 1{f)l < B. {3) Additivity. If f 1 and f2 have disjoint support then (f 1 + f 2) = (f 1) + ¢{f2). Theorem. Conditions (l)- (3) are necessary and sufficient in order that a functional defined on C(S) can be represented in the form ¢(f)= J8 K(f(s),s)J.L(ds), where (i) J.L is a regular measure of finite variation defined on the Borel sets of S, (ii) K{x,s) is a measurable function of s for each x,

(iii) K(x,s) is a continuous function in x for J.L- a. a. s, (iv) for each b > 0 there exists H = H{b) such that 11c I :'£ b implies IK(x, s) I ;2; H for J.L- a.a. s. This extends our previous result, which appeared in Arch. Rational Mech. Anal. 21, 1, {1966), 49-57, where S was assumed to be a compact metric space. (Received April 20, 1966.)

66T-313. B. GERMANSKY, Singe! 270, Amsterdam, Netherlands. An exact description of the concepts class and set.

A totality contains its building-stones either directly or by mediation of uniting them to a one.

In the first case It is called a class, its building-stones s 1 ,s2 .•• its members and it is written in the form s 1,s2 , .•• in the second case it is a set, its building-stones--its elements and it is written In the form (sl's2, ••. ) or fsl's 2 , ••• }. Totality is a common name for class and set; building-stone--for member and element. A building-stone may also be an Individual, i.e. an indivisible entity. A class having one member is called a singular; having more than one member It is called a plural. A totality which Is a general set, may simultaneously also be grasped as a class having one member, i.e. it is a singular. Any set may be obtained by comprehending (uniting) a class, i.e. symbolically by including the class in brackets. The following axioms may be accepted: I. Only singulars {and indi­ viduals) are building- stones. II. A totality having one building- stone is equal to this building-stone. III. An application of the union-operator U a suitable finite number n of times to a totality t trans­ forms it in a totality having only individuals lli2 .•• as building-stones: unt = il,i2'""" or= (il,l2, •.• ). (Received April 21, 1966.)

66T-314. K. K. NORTON,l830A Orchard Place, Urbana, Illinois 6180I • ..Q!!_kth power coset representatives and diagonal congruences modulo a prime power. Preliminary report.

Let p be an odd prime and k, w be positive integers. Write k = pfk0 (where pfk0 ), a= w a-1 min(w,f + I), and let o = (k0 ,p- 1) and o(w) = (k,(p )) = p o. Let Hk be the subgroup of kth powers in the group of residue classes mod pw which are prime top. Let I= c0 < c 1 < c 2 < ••• < co(w)-l be the smallest positive representatives of the o(w) distinct cosets of Hk. Then for each r, c :'£ 1 + (27rf 1apa-l/2 (log p){d/(o(w) - r))1/ 2 (l + O{log log p/log p)), where 0 implies an absolute r 1 4 constant. Also, If w = 1, then c 0_ 1 ~ B I {k,f)p / H for each f > 0. If w > 1 and f = 0, then 3 8 c o(w)- 1 ~ B 2 {k, f )p 1 + f for each E > 0. These results may be compared with those of Davenport and Erdlis (Publ. Math. Debrecen 2 {1952), 252-265) and Jordan (Pacific]. Math. 16 {1966), 77-85).

504 As an application of inequalities of the first type above, we have r•{k) < {3/{2 log 2) + f)k log k for a large class of odd k > k{t) {for notation, see Abstract 65T-474, these cNoticeiJ 12 {1965), 821). Com­ putations show that r•(11) = 45; f•(l3) =53 or 66; r•(15) = 61, 76, or 91; f•(17) =52, 69, 86, or 103; r• {19) =58, 77, 96, or 115; f*(21) = 106, 127, or 148; T*(23) = 116, 139, or 162; f*(25) = 101, 126, 151, or 176. {Received April 21, 1966.)

66T-315. K. T. HAHN and JOSEPHINE MITCHELL, Pennsylvania State University, McAllister Building, University Park, Pennsylvania 16802. Generalization of the Schwarz- Pick lemma to invariant volume in a Kahler manifold.

Let M be a Kahler manifold C en with metric given by du~ = ga.lfdwa.dw/3 (w local co­ ordinates on M) such that (i)- ra.puOU/3 ~ 0 for every vector (ua.) and {ii) det (- ra.p> ~ gM(w,w) = det (ga.i§) - r a.P being components of the Ricci curvature tensor of M. Then: Theorem 1. _!!_ D ~ bounded homogeneous star-shaped domain in Cn with the Bergman metric given by ds~ = T a.,Bdza.dz 13, which can be mapped biholomorphically by w = w(z) into M, then gM(w, w) I B(w)/ 8{z) 12

~ TD(z,z) for all g E D(T D(z,z) = det(T a.iJ»· __!! {ii) is replaced by det {- r a.ii) ~ T D(z0,z0)gM(z,z), where Dis starshaped with respect to the point z 0 , then the inequality becomes gM(w,w)IB(w)/a(z)l2 ;<;:TD(z,z)ITD{z0 ,z0}1. Theorem 2. Let G CD and w: G---> w(G) CM. Under the same hypotheses as Theorem 1 the invariant volume of G satisfies the inequality VM{w{G)) = fw(G)gM{w,w)dww ~ JG T D{z,z")dwz =- V D{G) (dw Euclidean volume element). _g_ w maps D into D and duD = dsD, then

VD{w{G)) ~ VD{G) and equality holds if the mapping is onto. The results are applied to the classical Cartan domains. The Corollary is an extension of a result due to Dinghas in case D is the unit hyper­ sphere. (Received April 21, 1966.)

66T-316. R. C. BOSE, University of North Carolina, Chapel Hill, North Carolina 27514. A theorem on regular graphs.

Finite unoriented regular graphs without loops are considered. A clique K of G is a set of vertices any two of which are adjacent. A clique K is maximal if no vertex A of G can be found such that K UA is a clique. Given any two distinct vertices A and B of G, A{A,B) denotes the number of vertices adjacent to both A and B. Let r ~ 1, x ~ 0, y ~ 0 be integers such that ry <: 2x. Let 'Y(r,x) = 1 + {r - 1)x, p(r,x,y) = 1 + y + (2r- 1)x, p(r,x,y) = 1 + {l/2)(r + 1)(ry - 2x). It is then proved Theorem. If G is a finite connected graph without loops satisfying the conditions (i) G is regular of degree r(k - 1), {ii) For any two distinct vertices A and B, A{A,B) = (k - 2) + x, if A and B are adjacent, and A(A,B) ~ 1 + y if A and Bare nonadjacent (iii) k > max[p(r,x,y), p(r,x,y)] then there exists in G a set ~of maximal cliques K1,k2, ... ,kb such that {a) Each vertex of G is contained in exactly r cliques of ~. (b) Each adjacent pair of di·stinct vertices of G is contained in exactly one clique of ~.{c) IKi I ~ k + 1 - ')'(r,x,), i = 1,2, •.. , b. This generalizes a previous result due to the author (Pacific J. Math. 13 (1963), 389-419). (Received April 22, 1966.)

66T-317. R. C. BOSE and RENU LASKAR, University of North Carolina, Chapel Hill, North Carolina 27514. A characterization of tetrahedral graphs.

An unoriented finite graph G without loops is defined to be tetrahedral with characteristic

505 n if the vertices of G can be identified with the unordered triplets on n symbols, such that two ver­ tices of G are adjacent if and only if the corresponding triplets have two symbols in common, Clearly such a graph has the following properties (i) the number of vertices in G is n(n - 1)(n - 2/6), (11) G is connected and regular of degree 3 (n - 3), (iii) for any two vertices x andy the distance d(x,y) ;e 3, (iv) the number of vertices adjacent to both x and y to be denoted by Ll(x,y) = n - 2,4 or 0 according as d (x,y) = 1,2 or 3 respectively, (v) If d(x,y) = 2, there exists a vertex Z adjacent to X such that d(Z,y) = 3, Now one may ask the following question, Is a graph G with the above properties (i)- (v) a tetrehedral graph with characteristic n? It is shown that for n > 16, the answer is in the affirmative, so that the properties characterize the graph. The question is open for n < 16. This is an extension of some results due to Connor and Hoffman (Ann. Math, Statist, 29 (1958), 262-266, 31 (1960), 492-497), (Received April 22, 1966,)

66T-318. RENU LASKAR, University of North Carolina, Chapel Hill, North Carolina 27514, A characterization of cubic lattice graphs,

An unoriented finite graph G without loops is defined to be a cubic lattice graph with character­ istic n if the vertices of G can be identified with the ordered triplets on n symbols, such that two vertices of G are adjacent if and only if the corresponding triplets have two common symbols in the same positions, Clearly such a graph has the following properties (!) the number of vertices in G is n3 , (ii) G is connected and regular of degree 3(n - 1), (iii) for any two vertices x andy the distance d(x,y) :':i 3, (iv) the number of vertices adjacent to both x and y to be denoted by D.(x,y) = n - 2,2 or 0 according as d(x,y) = 1,2, or 3 respectively, (v) If d(x,y) = 2 there exist exactly (n- 1) vertices zi (i = 1,2, ... ,n - 1) adjacent to x such that d(y ,zi) = 3, Now one may ask the following question. Is a graph G with the above properties (i) - (v) a cubic lattice graph with characteristic n? It is shown that for n > 7, the answer is in the affirmative, so that the properties characterize the graph. The question is open for n < 7, This is an extension of some results due to Shrikhande (Ann, Math, Statist, 30 (1959), 781-798), (Received April 22, 1966.)

66T-319, 0. C. McGEHEE, University of California at Berkeley, Berkeley, California 94709, Sets of uniqueness an sets of multiplicity.

See chapters I, V, an VI of Kahane an Salem, Ensembles parfaits et s~ries trigonom~trigues, Hermann, Paris, 1963, for definitions and backgFound, Theorem 1, Let E be a perfect set of trans­ lation constructed in the interval [0,1] by a sequence of dissections of types (H(k), u 1 ,k'"''uH(k),k'~k), k = 1,2,.,,. Let dr = ~ 1 ~ 2 ... ;r. If b > 4 and lim infr~oodrexp[(Iog br) L~= 1 (H(k)- 1)] = 0, then the measure set E is a set of uniqueness in the broad sense (a u0 - set), In particular, if E has Hausdorff zero with respect to the function h(t) = (log t- 1f 1, then E is a LtJ -set, Theorem 2, There exists a translation set which is a set of multiplicity in the strict sense (an M0 -set) and has Hausdorff measure zero with respect to all the functions fb(t) = exp[- (log t-l)b], 0 < b < 1. Remarks. Theorem 1, obtained by the use of elementary Diophantine approximation, throws some light on the construction and Hausdorff of arbitrarily thin M0 -sets by Iva~ev- Musatov (translated by Allen Shields, M-sets measure, Soviet Math, Dokl. 3 (1962), 213-216,), That Theorem I is close to being sharp follows from the existence proofs of Salem, which we extend to obtain Theorem 2, (Received April 25, 1966.)

506 66T-320, SIBE MARDESIC, University of Washington, Seattle, Washington, Locally connected images of ordered compacta are locally peripherally metric.

Let Sf denote the class of Hausdorff compact spaces X which can be obtained as images of

(linearly) ordered compact spaces K under (continuous) maps f: K --> X onto X. Theorem 1, If X E Sf and G is an open Fu- set in X such that Cl G is connected, then Fr G is metrizable, As a corol­ lary one obtains L. B. Treybig's result that a separable continuum X E Sf is always metrizable [Duke Math, J, 32 (1965), 417-422]. Another consequence is Theorem 2. If X E: Sf is locally connected, then it is locally peripherally metric, i.e. admits a basis of open sets with metrizable frontiers. The proof of Theorem 1 uses this metrization theorem, A Hausdorff compact space Y is metrizable if and only if it has the following two properties Jl and u, Property f.J., If F C Y is a closed subset of Y, then the space of components Z(F) of F is metrizable, Property u. There exists a countable family \B of open sets V C Y such that any two disjoint closed subsets M, N C Y can be separated in Y by some member V E \B, (Received April 25, 1966,)

66T-32l. H. W. LAMBERT, University of Utah, Salt Lake City, Utah 84112. A topological property of Bing's decomposition of E3 into points and tame arcs.

Let G be the decomposition in Bing's paper, A decomposition of E 3 into points and tame arcs 3 such that the decomposition space is topologically different from E (Ann, of Math, 65 (1957), 484-500), Using the notation of Bing's paper, let A be the solid double torus which is the first stage of the de­ fining sequence for the arcs of G, P the projection map of E 3 onto E 3 /G, and Ao the sum of the non­ degenerate elements of G. Bing showed that there is no homeomorphism between E 3 and E 3 /G and asked whether a topological property of E 3 /G could be found which would distinguish it from E 3• Armentrout announced (Abstract 612-61, these cJVotiaiJ 11 (1964), 369-370) that certain points of E 3 /G have no small simply connected open neighborhoods, Another property of E 3 /G which distin­ guishes it from E 3 is the following. Theorem, For each point x of P (A0 ), there is no neighborhood of x in P(A) that has a 2-sphere as a boundary. (Received April 25, 1966.)

66T-322. ALEXANDER ABIAN, The Ohio State University, Columbus, Ohio 43210, Aleph limit points of sets of transfinite dyadic sequences,

For convenience, a transfinite sequence made up of 0 and 1 is called a dyadic sequence. For

;'i v the sequence (si)i< j is called the j-segment of a sequence (si)i< v where i,j and v are ordinal

numbers, Definition. A dyadic sequence (s. )1.< is called an X -limit point of a set S of dyadic 1 WA (l sequences of type wA if for every ordinal j < wA the power of the subset of S of sequences with j-segment equal to (si)i

X -limit point for any a. Theorem 2, Let a and A be ordinals and let wT be the smallest ordinal with a K ~ which wA is confinal. If MT ,; Na ,; 2 A then there exists a set S of dyadic sequences with S ; N a such that S has an Ka-limit point, Theorem 3, Let a and A be ordinal numbers such that (i) wa is confinal with no ordinal less than or equal to z'tl·f3 for any {3 < A and (ii) w is confinal with no ordinal less (l than or equal to wA. Then a set S of dyadic sequences of type wA contains an Ka-limlt point if and only if S ;c; Xa. (Received April 25, 1966.)

507 66T-323, LYNNE SMALL, 2525 LeConte Avenue, Berkeley, California 94709. Simple rings with involution,

Let R be a simple ring with involution •, characteristic i 2, the center, Z, equal to 0 or dim R/Z > 4 ( > 16 for Theorem 3), LetS= [x E Rix* = xJ and K = [x f Rlx* =- xJ. Theorem l, If 1/> is an additive map from R to some ring such that the ring generated by the image of R is semi­

prime with char i 2 and 1/>(xx*) = 1/>(x)~(x*) for all x E R, then

This extends results of Her stein which cover the cases T C S and T c K. Again using Theorem 2, part of a result of Jacobson (Prop. 19.4, Structure of alternative and Jordan bimodules, Osaka J, Math, 6 (1954), 1-71) on characterizing irreducible Jordan representations can be extended from finite dimensional central simple algebras with involution, char i 2, to rings R as above. A counter­ example shows that the entire result can not be extended. (Received April 26, 1966.)

66T-324, H. J, CHARLTON, Virginia Polytechnic Institute, Blacksburg, Virginia 24061, Transformations preserving tame sets,

In answer to the question, "What continuous functions of E 3 into E 3 (En into En) or s3 into s 3 (Sn into Sn) take tame sets onto tame sets?", the following corollary and theorem provide a partial but strong answer, If X is a triangulated topological space then a polyhedron of X with respect to this triangulation is a subset of X corresponding to a finite subcomplex. A homeomorph P of a polyhedron of X with respect to a triangulation is tame if there is a homeomorphism of X onto a polyhedron of X with respect to some triangulation of X. Corollary. Let f be a function from a triangulated n- manifold

M 1 into a triangulated n-manifold M2,n > 1. Iff takes each tame arc onto a connected tame set con­ taining no triod then f is a homeomorphism if f(M 1) is not an arc, point or simple closed curve, Theorem, Let K bean-dimensional complex and letS be the unit square [0,1] X [0,1], There exists a map f of K onto S such that the image of every tame set is tame, (Received April 25, 1966,)

66T-325, W, A. PRANGER, JR., DePaul University, Chicago, Illinois 60604. Orbits of analytic functions.

If Tis holomorphic on the unit disk U and takes values in U, define the iterates ofT by Tn = TTn-l, n = 2,3, ... ; T = T 1, For z and win U define z equivalent tow if both have the same image under a pair of iterates of T, This equivalence relation defines the orbits of T. If T lacks fixed points the sequence [Tkl converges continuously to a constant function of value r. Ill= 1. The angular derivative a of T at f exists and 0 < a ;:;;; !, If a < 1 I call T hyperbolic, Theorem, If T is hyperbolic the orbits topologized with the quotient topology has the natural structure of a Riemann surface, (Received April 25, 1966,)

508 66T-326. K. D. MAGILL, JR., State University of New York at Buffalo, Buffalo, New York 14214. Spaces determined by left ideals of semigroups.

Z denotes a left ideal of the semigroup T. A C T X Z is a bond if for every finite subset

{(t. ,z. )l ~ of A, the system of equations ft. x = z.l N has a common solution x in Z. A bond which 1 1 J 1= 1 1 1 J 1= 1 is not properly contained in any other bond is called an ultrabond. Let _57 (T,Z) denote the family of all ultrabonds and for each (t,z) E T X Z, let H(t,z) = fA E 3(T,Z): (t,z) E A]. The topological space obtained by taking {H(t,z): (t,z) E T X Z} as a subbasis for the closed subsets of 3\T, Z) is referred to as the structure space of the pair (T,Z). If the pair (T,Z) satisfies certain conditions, the ideal Z corresponds in a natural way to a subspace .9t'(T,Z) of §(T,Z) and it is shown that 3\T,Z) is a Hausdorff compactification of .9i!'(T, Z). Furthermore, every element in T determines in a natural

way a function which maps 9i.'(T,Z) into ~(T,Z), It is shown that every such function is continuous and moreover has a continuous extension to a function mapping Y(T,Z) into 3'(T,Z). These results are used to show that for certain X, the structure space of the pair (S(X), Z(X)) is the Stone-Cech compactification of X where S(X) is the semigroup (under composition) of all continuous functions mapping X into X and Z(X) is the kernel of S(X). (Received April 26, 1966.)

66T-327, S. 0. AANDERAA, The Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138, Some recursively undecidable problems in automata theory and quantification

theory.

One-way infinite one-dimensional iterative arrays have been defined and dealt with in several papers, The iterative arrays dealt with here are two-way infinite one-dimensional, and the machines have a halting state. The problems we consider here are to decide (l) if there exists an assignment of states at time 0 such that none of the machines will ever halt, and (2) the same problem under the restriction that the assignment be periodic. An iterative array is called signal preserving iff signals are passed unchanged in both directions with a uniform one unit delay, Theorem, Both problems (l) and (2) are recursively unsolvable. This remains true under a signal preserving array. Corollaries, The problems of deciding satisfiabil!ty and finite satisfiability of the formulas of the form ('

66T-328,J, A. P F AL TZGRAF F, Indiana University, Bloomington, Indiana 4 7401, Radial sym-

metrization and capacities in space, Preliminary report,

Let DC En be an open set containing the origin, 0. For each p ~ l a symmetrization process is defined. The symmetrized set D* is starlike (with respect to 0), and intersects any ray from 0 in

a segment of length f 0, small), where Jx J denotes the norm of the point x E En and the integration is on the linear set consisting of D inter­ sected with the given ray. Symmetrization of rings and functions is defined, Let R C En be a ring and define the capacity r p(R) = inf f JR j'Vu JP dm: u E C 1(R). u = 0,1 on oRJ. Theorem. r p(R) ~ r p(R *). Additional symmetrization methods are developed. These methods replace starlike domains by star­ like domains with m-fold symmetry about a line. Again, capacity is not increased by symmetrization,

509 The results of this paper are a generalization of methods due to G. Szego [Ann. Mat, Pures Appl. (4) 40 (1955), 113-119], and M. Marcus [Pacific J, Math, 14 (1964), 613-626]. The proofs employ techniques of Hayman [Multivalent functions, Cambridge Univ, Press, 1958) and Gehring [Trans, Amer, Math. Soc. 101 (1961), 499-519]. (Received April 26, 1966.)

66T-329, J, S. KALME, 5807 North Camac Street, Philadelphia, Pennsylvania 19141. Spectral theory of Wiener-Hopf type integral operators on generalized Sololev spaces,

Consider the Wiener-Hopf type integral operators (T¢)(A) = J~00K{A - !J-)¢{1'-)dll over the spaces W~·P (R), where s i':: 0, 1 < p < oo. W~'P(R) consists of functions ¢(11}, 11 E: (- oo,oo}, such that .- T)¢11 s p • Theorem 2, Suppose o- 1(A,T) has a total of m f' 0 simple W~·P(R) " w.• (R) ---- poles in the strips - 1 - a< ImT < - 1 if a<. 1/p, and m ~ 0 poles in the strips - 1 - 1/p < ImT < -

if a > 1/p, where 1 = O,p, ... , [s]. Then A - Tis a 1-1 operator, the range of A- Tis closed and of codimension m in W~·P(R). The results are extended to systems of operators. {These results con­ stitute a portion of a Ph.D. thesis written at the University of Pennsylvania under the direction of Professor Martin Schechter of the Courant Institute.) (Received .\pril 26, 1966.)

66T-330, R. K. SAXENA, McGill University, Montreal 2, Canada, On fractional integration operators,

We define the fractional integration operators by means of the following relations: 3' {a.,,B;y,m)f(x) =(x_,_ 1/{I'{1- a.)))j~(a.,,B-m;,B;t/x)t~'f(t)dt, m(a.,fJ;y,m)f(x) =(x1 /(l'(1- a.))) ·J~F(a.,{J- m;,B;x/t)t-'Y- 1f{t)dt where m = 0,1,2,. .. ; a.,f3,'Y are complex parameters and F(a,b;c;z) denotes the ordinary hypergeometric function, Kober's operators of fractional integration can be ob­ tained from above by taking m = 0 (Quart. j. Math, Oxford Ser, 11 {1940), 193-211). Three theorems are established for the above defined operators. The first two theorems give the Mellin transforms of these operators under appropriate conditions and third is the Parseval property satisfied by these operators. It is expected that the study of these operators might be useful in the solution of dual integral equations and other allied topics. (Received April 27, 1966.)

66T-331, J. M. WORRELL, JR., Sandia Corporation, Sandia Base, Albuquerque, New Mexico, Local completeness and a variation of Arhangels'kiT•s base of countable order concept. Preliminary report.

To say here that a base B for a space is closurewise of countable order means that if P is a point common to the closures of the elements of a perfectly decreasing [Canad, J, Math, 17 (1965),

510 SZ 1] subcollection K of 8, then every open set containing P includes some element of K. A space is said to be monotonically complete if and only if it has a monotonically complete base [Abstract

66T-159, these cNOtice0 13 (1966), Z55]. Theorem. If (1) the locally monotonically complete T 1 space R has a base closurewlse of countable order and (Z) for each n,Sn is a monotonically complete sub­ space of R, then (1) if R' is a subspace of R in which a point P of R He·s if and only if P lies in each

Sn' R 1 is locally monotonically complete and (Z) if each Sn is dense in R, there is an R 1 as in (1) and

R 1 is dense in R. Remarks. (1). All subspaces of R have bases closurewise of countable order, (Z), Locally monotonically complete spaces are monotonically complete. (3), H. H. Wicke and the author have shown that the class C of Hausdorff open continuous images of complete metric spaces is the class of spaces R as above, (4). All spaces complete in the sense of Cech having bases of countable order belong to C. (Received Aprll Z7, 1966.)

66T-33Z. E. E. ROBKIN, University of Massachusetts, Amherst, Massachusetts. A remark on the double equichordal point problem. Preliminary report.

Suppose there exists a compact convex set C inEz with two equichordal points. Then C is sym­ metric [G. Dirac, Ovals with equichordal points, J, London Math, Soc. Z7 (195Z), 4Z9-437J and the boundary of C is analytic [E. Wirsing, Zur Analyzitiit der Doppelspeichen Kurven, Arch. Math, 9 (19 58), 300-307]. Adopt the polar coordinate system of the latter with the equichordal points at (0,0) and (Ze,1r), Then the boundary of C is given by r "' 1 - e cos llg(ll) and g(ll)"' .L::odzn (sin II)Zn. There is an angle 1/tsuch that (1 + ecosl/lg(l/t))sin~: (1- ecosll (li))sinll and (1 + ecosl/tg(l/t))cosl/1- Ze"' (1 - e cos llg(ll))cos 8 and sin 1/t"' sin II.L::oazn(sin 11ln. The first four coefficients of g(fJ) are 2 2 d0 "' 1; dz"' e 2 /1 + e 2; d4 "' e 4(e4 + se2 + 3)/(1 + e ) (e4 + 6e2 + 1); d 6 "' e 6(e 10 + Z4e8 + 194e6 + 4 Z Z3 4 Z 6 4 Z . . . Z64e + 133e + Z4)/(1 + e ) (e + 6e + 1)(e + 15e + 15e + 1), which does not agree w1th W1rsmg . ZN Z N Z [Ibid. p. 305]. Conjecture: dzN "' e RzN/(1 + e ) Q2N where RzN is some polynomial in e and Q2N(e2)"' f1:, 0 [(! + e)Zm + (1 - e)2m]. Let (sin 1/l)j "' (sin IJ)jLKij(sin II)Zi-Z(Cauchy product). 2 Let cos II"' .L::obzn(sin 8) n and let P 2N"' .L::,ObZN-ZmdZm and B2N"' [b2N- ed2N + ed2N_ 2]. Conjecture: ePzN"' a 2N(1- e)- e .L:Wt, 1PzN+Z-zmKm,ZN+3-zm and [bzN + ed2N- ed2N_ 2]"' .L;Wt, 1B2N+Z-zmKm,ZN+Z-Zm' (Received April 28, 1966.)

66T-333. H. H. WICKE, Sandia Laboratory, Sandia Base, Albuquerque, New M:oxico 87115, Open continuous mappings satisfying a completeness condition. Preliminary report.

A mapping M of a space S onto a T 1- space R is said to be uniformly monotonically complete if and only if there exists a base 8 for S such that if P is a point of R and K is a monotonic sub- -1 -1 collection of 8 each element of which intersects M (P), then there is a point of M (P) common to the closures of the elements of K. If S is rnetrizable, M satisfies the above condition if and only if there exists a topology-preserving metric d for S such that for every P in R, M- 1 (P) is complete with respect to d. The following complements Theorem 1 of Abstract 66T-159, these cNoticeiJ 13 (1966), Z55. Theorem 1. Every T 1-space having a base of countable order is an open continuous uniformly monotonically complete image of a metrizable space, This, with a result submitted by

Worrell and Wicke (Abstract 66T-334) yields the following. Theorem z. ~ T 1-space has a base of countable order if and only if it is an open continuous uniformly monotonically complete image of a

511 metrizable space, A theorem of Michael [Duke Math. J, 27 (1959)] follows from Theorem 2 and a theorem of Arhangel'skH [Uspehi Mat. Nauk 18 (1963)]: Iff is an open continuous mapping of a metric space E onto a T 2 paracompact space F such that C 1(y) is complete for every y in F, then F is met­ rizable, (Received April 28, 1966,)

66T-334, J, M. WORRELL, JR. and H. H. WICKE, Sandia Laboratory, Sandia Base, Albuquerque, New Mexico 87115, Restrictions of uniformly monotonically complete open continuous mappings. Preliminary report.

For definitions, see Abstracts 66T-331 and 66T- 333, The notion of a uniformly monotonically complete mapping is quite general. For example, if¢ is a continuous mapping with a metrizable domain and a T 1 range having a base of countable order, then 4> is necessarily a restriction of a range-preserving uniformly monotonically complete mapping whose domain is metrizable, Also every bicompact continuous mapping whose range is T 1 is uniformly monotonically complete. Here open uniformly monotonically complete mappings are considered. Theorem 1, Suppose 4> is an open continuous mapping of a subspace S' of a locally monotonically complete T 1-space Shaving a base closurewise of countable order onto a T 1- space R. The following conditions are equivalent: (1) R has a base of countable order. (2) cJ> is a restriction of an open continuous uniformly monotonically complete mapping of a subspace of S ~R. Example. There exists an open LindeHifian mapping cJ> of a metrizable space S onto a nonparacompact T 2 space such that the elements of the decomposition induced by 4>are absolute G0's. (Received April28, 1966.)

512 ERRATA Volume 12

J. E. DONER. Decidability of the weak second-order theory of two successors, Abstract 65T-468, Page 819. Line 4: Replace'the fourth sentence by the following: "A proof of the de­ cidability of WMT ( \)1 1) was first found by Ehrenfeucht, but was not pub­ lished by him; see Robinson, Proc. Amer. Math. Soc. 9 (1958), and Fe­ ferman and Vaught, Fund. Math. 47 (1959). Subsequent proofs were found independently by Buehl and Elgot, Abstract 553-12, these cJVotiai) 5 (1958), 834, and published by them; Buehl, z. Math. Logik 6 (1960), and Elgot, Trans. Amer. Math. Soc. 87 (1961)."

Volume l:i

G. J. CHAITIN. On the length of programs for computing finite binary sequences by bounded-transfer Turing machines. II. Abstract 631-6, Page 229. Line 5: Replace "P" by "L ".

DANIEL RICHARDSON. Some unsolvable problems involving functions of a real variable. Abstract 66T-31, Page 135. Line 7: "f~(x)dx" should read "J~g(x)dx."

HABIB SALEHI. On prediction theory with continuous time. Preliminary report. Abstract 66T-108, Page 239. Line 2: Replace "log det F"' by "log det F'(X)/(1 + >. 2)".

D. TOPPING. An isomorphism invariant for spin factors. Abstract 630-60, Pages 73-74. Line 6: "Consequently, all operator topologies coincide on a spin factor" should read "In a spin factor, the strong operator topology and the norm topology coincide."

S. K. THOMASON. Lattice-embeddings in the degrees and hyperdegrees. Ab­ stract 66T-35, Page 136. Line 6: Replace the corollary by "Corollary. Any partial ordering of car­ dinality at most c in which each element has only finitely many prede­ cessors is embeddable on the hyperdegrees in a way which preserves finite meets."

R. E. PEINADO. On the commutativity of rings, Abstract 66T-170, Page 259. Line 3: Replace " .•• n is the largest integer ••• "by " ••• n is the smallest positive integer ••• "

513 NEW FROM Available August 1966: QDCONSULTANTS BUREAU 9? PLENUM PRESS International Congress AN IMPORTANT SOVIET JOURNAL IN COVER·TO·COVER TRANSLATION: of Mathematicians. SIBERIAN MATHEMATICAL JOURNAL of the Academy of Sciences of the USSR, Novosibirsk Clothbound volumes: SIBERSKII MATEMATICHESKII ZHURNAL SERIES 1: Editors-in-Chief: A. V. Bitsadze, Institute of Mathematics and Computation Center, Novosibirsk and the Steklov Mathe­ 1st, matical Institute, and Yu. G. Reshetnyak Zurich, 1897 (Verhandlungen). $16.00 Available for the first time in comflete and authoritative translation, this bimonthly journal o mathematics is under 2nd, the editonal direction of internationally eminent Soviet mathematicians. The journal is published by the Academy of Paris, 1900 (Compte-rendu). $16.00 Sciences in Moscow and is issued by the Siberian Section of the Academy with headquarters in Akademgorod ("Science 3rd, City"), the dramatic new educational and research center near Novosibirsk. The Consultants Bureau translation begins Heidelberg, 1904 (Verhandlungen). $26.00 with the 1966 issues, Volume 7. 4th, SUBSCRIPTION: 6 ISSUES $150.00 Rome, 1908 (Atti). 3 parts in 2 vols. $45.00 AN IMPORTANT CHINESE JOURNAL IN COVER· TO-COVER TRANSLATION: 5th, Cambridge (England), 1912 (Pro· ACTA MECHANICA SINICA ceedings). 2 vols. $45.00 Advisory Editor: Professor Hsuan Yeh, Director, Towne School of Civil and Mechanical Engineering, University of Pennsylvania, Philadelphia, Pa. SERIES II: An outstanding journal of mechanics devoted to research in applied mechanics (elasticity, plasticity1 automatic control 1st, theory, fluid mechanics, etc.) and appliea mathematics from the Institute of Mechanics in Peking. Available for the first Strasbourg, 1920 (Comptes-rendus). $26.00 time in complete and authoritative English translation begin­ ning with the 1966 issues, Volume IX. 2nd, SUBSCRIPTION: 4 ISSUES $135.00 Toronto, 1924 (Proceedings). 2 vols. $75.00 SAMPLE COPIES AVAILABLE UPON REQUEST. 3rd, Bologna, 1928 (Atti). 6 vols. bound in 3. $99.00 MATRIX ALGEBRA FOR PHYSICISTS 4th, By R. K. Eisenschitz, Emeritus Professor of Theoretical Physics, Queen Mary College, London Zurich, 1932 (Verhandlungen). A valuable intermediate text for college physics students. It briefly but thoroughly sets forth the fundamental concepts 2 vols. in 1. $26.00 and techniques of matrix algebra and then shows how they are applied in areas of theoretical physics including oscilla­ 5th, tions, mvariance laws, statistical mechanics, and quantum phySICS. Oslo, 1936 (Comptes-rendus). 136 PAGES PP SUMMER 1966 $4.95 2 vols. in 1. $26.00 6th, CORRELATION EQUATIONS FOR Cambridge, Mass., 1950 (Pro· ceedings). 2 vols. $50.00 STATISTICAL COMPUTATIONS 7th, By A. K. Mitropol'skii Amsterdam, 1954 (Proceedings). Translated from Russian by Edwin S. Spiegelthal 3 vols. $50.00 Completely revised and updated by the author for the English edition. A concise treatment of the various methods for computing correlation equations, this work is a useful contribution to applied mathematics, statistics, and all scientific and engi­ neering disciplines. It describes the use of the Chebyshev method for fitting polynomials to various data requiring the use of desk calculators for accurate computation. KRAUS REPRINT 103 PAGES CB MARCH 1966 $9.50 CORPORATION CONSULTANTS BUREAU/PLENUM PRESS 16 East 46th Street A Division of Plenum Publishing Corporation New York, N.Y. 10017 227 W. 17th ST., NEW YORK, N.Y. 10011

514 Devoted to the publication of original mathematical articles dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science ... JOURNAL OF COMBINATORIAL THEORY A NEW Journal which fully explores the potential of the long neglected field of combinatorial analysis. In the Journal will be found the efforts of scientists working in many independent disciplines forging common links between separate studies in combinatorics.

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515 Previously published:

1. NUMBERS: RATIONAL AND IRRA­ TIONAL by Ivan Niven 2. WHAT Is CALCULUS ABOUT? by W. THE NEW W. Sawyer 3. AN INTRODUCTION TO INEQUALITIES by E. Beckenbach and R. Bellman 4. GEOMETRIC INEQUALITIES by N. D. MATHEMATICAL Kazarinoff 5. THE CONTEST PROBLEM BOOK I: High School Contest Problems of the Mathematical Association of America, 1950-1980 LIBRARY ~Jl 6. THE LoRE OF LARGE NUMBERS by I A unique serJes, sponsored by Philip J. Davis the School Mathematics Study ~ 7. UsEs oF INFINITY by Leo Zippin Group, that makes important 8. GEOMETRIC TRANSFORMATIONS by J. mathematical ideas understand- M. Yaglom al>le to interested laymen and D. students. Paperbound, $1.95 each. 9. g1;:INUED FRACTIONS by c. 10. GRAPHS AND THEIR UsEs by Oystein Ore Two new titles: 11. HuNGARIAN PROBLEM BooK II, based on the Elltvlls Contests, 1894-1905 FROM PYTHAGORAS TO EINSTEIN 12. HUNGARIAN PROBLEM Bomc II, based By K. 0. FRIEDRICHS. Traces the Pythago- on the Eiltvos Contests, 1906-1928 rean Theorem from itS elementary form to its 13. EPISODES FROM THE EARLY HISTORY i,l,i:,=;, role in Relativity. #16 in the series. oF MATHEMATics by Asger Aaboe : 14. GROUP$ AND THEIR GRAPHS by :: THE MAA PROBLEM BOOK II: ~~::~.Grossman and Wilhelm 1961 1965 c il d b CHARL s SAL- 15. MATHEMATICS OF CHOICE (How to r::,:-: - · omp e Y E T. Count Without Counting) by Ivan :-:=· KIND. #17 in the series. 1 :::l Now at your bookstore. Published by RA: ;~M HOUSE, INC. cia! edition for high school teachers and students published by L. W. Singer Co., West Erie Blvd., Syracuse 2, N. Y., 99¢ each. Reinforced cloth binding available $2.95 through Random House School & Library Service, 457 Madison Ave., N. Y. 10022 ~=-¥.~:::::::::::::::::::::::::::;:::;:::::::::::::::::::::::::::::::::::=-:.. ··::~:::::::::::::::::;:;::;::::::::::::::::::~::::::::::::::::::::;::::::::::::~·=:<:~:·~:·Wllllllllllllll118111181111811.

REPRINTS-now available: LEaURES II APPLIED IllTHEliA TICS Volumes 5, 6, 1 BULLETIN DES SCIENCES SPACE MATHEMATICS MATHEMATIQUES. edited by J. Barkley Rosser Series I, v.1-11. Paris, 1870-1876 These volumes consist of the lectures presented Series II, v.1-85. Paris, 1877-1961 US$1700.00 at the AMS 1963 Summer Seminar in Applied Individual volumes, paperbound US$ 20.00 Mathematics, plus some notes from the two CZECHOSLOVAK MATHEMATICAL previous institutes. Articles indicate (I) the current state of research on behavior of non­ JOURNAL. v.1-14. Praha, 1951-1964 us$ 259.00 propulsive vehicles; (2) unsolved problems; Individual volumes, paperbound us) 18.50 (3) examples of useful mathematical techniques. Besides presenting advanced material, these ENSEIGNEMENT MATHEMATIQUE. volumes provide readers with basic information Series I, v.1-40; Series II, v.1-8. which can be applied to space problems in Geneva, 1899-1961 US$ 475.00 their own fields. Approximately 200 pages per volume QUARTERLY JOURNAL OF PURE AND APPLIED MATHEMATICS. SPACE MATHEMATICS, Part 1, Vol. 5 Availallle July 30 Ust Price $18.20 MMiber Price $7.65 v.1-50. London, 1855-1927 us $1175.00 SPACE MATHEMATICS, Part II, Vol. 6 Available M8 30 Individual volumes: Ust Price $18.20 Member Price $1.65 v.1-35 each US $ 25.00 SPACE "MATHEMATICS, Parrill, Vol. 7 Available July 30 v.36-50 each US $ 20.00 Usl Price $10.00 (approx.) MMiber Price $1.50 (approx.)

Please order from SWETS & ZEITLINGER N.Y. Keizersgracht 471 555 Woodside Ave. AMERICAN MATHEMATICAL SOCIETY Amsterdam-Holland Berwyn, Pa. USA P. 0. Box 6248, Providence, Rhode Island 02904

516 Academic Press NEW MATHEMATICS TEXTBOOKS

Introduction to Ordinary Functional Analysis Differential Equations by George Bachman and Lawrence Narici This textbook is intended for either a one­ by Albert L. Rabenstein, Maca lester College or two-semester introductory course at the This text is designed for use in a one­ advanced undergraduate or first year grad­ semester, introductory and intermediate­ uate level. The book meets the demand for level course. It presents a thorough intro­ an in-depth, axiomatic and hi~hly detailed duction to the elementary topics of ordinary treatment of functional analysis in its pure differential equations. The book also covers aspects. Its extensive applications to sepa­ the more advanced theories which are of rate studies have been confined to numerous interest to mathematicians, as well as engi­ references and examples at the ends of neers and physicists who require a knowl­ the chapters. The main topics covered are: edge of the applications to their own fields. normed and Banach spaces along with metric spaces; Hilbert spaces and inner CONTENTS: Linear Differential Equations. Addi­ tional Properties of Linear Differential Equations. product spaces; and spectral theory. The Complex Variables. Series Solutions. Bessel text is continuously related to examples and Functions. Orthogonal Polynomials. Eigenvalue problems illustrating each new concept and Problems. Fourier Series. Systems of Differential Equations. Laplace Transforms. Partial Differen­ extending the material covered. tial. Equations. Appendix. INDEX. Special attention has been given to provide Apri I 1966, 431 pp., $9.95 an elementary orientation which facilitates progress into inter-disciplinary applications m physics, electrical engineering, and A fortran Introduction to related subjects. INDEX. Programming and Computers: April1966, 543 pp., $14.50 Including fortran IV A first Course in by Marvin L. Stein and William D. Munro, Stochastic Processes University of Minnesota by Samuel Karlin CONTENTS: Elements of Stochastic Processes. Intended for a one-semester introductory Markov Chains. The Basic Limit Theorem of course in programming at the undergrad­ Markov Chains and Applications. Algebraic Meth­ uate level, this text can also be used for ods in Markov Chains. Ratio Theorems of Transi­ tion Probabilities and Applications. Sums of a course of self-study by graduate students, Independent Random Variables as a Markov professors and researchers. Chain. Classical Examples of Continuous Time CONTENTS: Introduction. Elements of Fortran. Markov Chains. Continuous Time Markov Chains. Input and Output. Program Control and Organi­ Order statistics, Poisson Processes and Applica­ zation. Extended Modes of Arithmetic. Some Non­ tions. Brownian Motion. Branching Processes. arithmetic Aspects of Fortran. Program Check· Compoundinlj Stochastic Processes. Deterministic ing and Execution: Miscellaneous. Modern Digital and StochastiC Genetic and Ecological Processes. Computing Systems. INDEX. Queuing Processes. INDEX. 1966, 502 pp., $10.95 March 1966, 123 pp., $3.95 January Theoretical Elementary Numerical Analysis Differential Geometry by Burton Wendroff by Barrett O'Neill, CONTENTS: Interpolation and Quadrature. Ap­ proximation. Ordinary Differential E~uations. Solu­ University of California, Los Angeles f~BE~ Equations. Partial Differential Equations. Assuming as prerequisites only elementary calculus and linear algebra, and written March 1966, 241 pp., $10.95 specifically for the one- or two-semester junior-level course, this new text provides Number Theory an up-to-date introduction to the geometry by Z. I. Borevich and 1. R. Shafarevich of curves and surfaces. translated from the Russian by CONTENTS: Calculus on Euclidean Space. Frame Newcomb Greenleaf Fields. Euclidean Geometry. Calculus on a Sur­ CONTENTS: Congruences. Representation of face .. Shape. Operators. Geometry of Surfaces in Integers by Reducible Forms Theory of Divisi­ E3• Rtemanntan Geometry. Answers to Exercises. bility. Local. Methods. Analytic Methods. Appen­ INDEX. dix. Some Concepts from Algebra. Tables. INDEX. May 1966, 410 pp., $9.95 May 19661 about 431 pp., $12.95

WNEW YORK AND LONDON ACADEMIC PRESS~ 111 FIFTH AVENUE, NEW YORK, N.Y. 10003

517 Mathematics Journals

APPLIED MATHEMATICS AND MECHANICS Translation Editor: GEORGE HERMANN, Evans­ ton. A cover-to-cover translation of the Soviet Journal Prikladnaia Matematika i Mekhanika. AUTOMATICA Bi-monthly $75.00 (The International Journal of Automatic Con­ On the Syllogism trol and Automation) Executive Editors: J. A. AsELTINE, Houston; and Other Logical Writings H. M. PAYNTER, Cambridge, Mass. et al. This journal provides a medium for the multilingual by Augustus De Morgan publication of theoretical and experimental research work and original development in control systems, edited by Peter Heath including all facets of automatic control theory, de­ sign and characterization of components, practical Augustus De Morgan, distinguished 19th-century Brit­ applications, etc. Quarterly $30.00 INFORMATION STORAGE ish mathematician, is chiefly AND RETRIEVAL, including remembered today as mechanical translation one of the founders of modern mathematical logic. Editor-in-Chief: J. FARRADANE, London. Nevertheless, his writings, apart from Formal Logic, Original work on the theory and techniques of in­ formation storage and retrieval, have for the most part lain buried in inaccessible with emphasis on scientific information and the intellectual problems periodicals. In this volume his five memoirs on the involved. Quarterly $30.00 syllogism are assembled for the first time, together PROBLEMS OF CYBERNETICS with more condensed accounts of his researches Translation Editor: RICHARD GOODMAN, Brighton. given in his syllabus of a Proposed System of Logic This is the first journal to deal with the whole and in his article on logic contributed to the English field of cybernetics and related activities in the Cyclopedia. $10.00 Soviet Union. Irregular $60.00 SANKHYA, INDIAN JOURNAL OF STATISTICS (Distributed on behalf of the Indian Statisti­ cal Institute) Disquisitiones Arithmeticae Editor: P. C. MAHALANOBIS, Calcutta. This journal is published in co-operation with the by Carl Friedrich Gauss Indian Statistical Institute in two separate series. Series A is for papers and notes on new methods translated by Arthur A. and techniques. Series B is mainly for original data, records and applications of new methods. Quarterly Clarke, S. J. Series A: $20.00; Series B: $20.00; Series A and B combined: $30.00 The forst translation into English of the standard TEACHING ARITHMETIC work on the theory of numbers by one of the great­ (A Joumal of Elementary Mathematics) est masters of modern mathematical analysis. First Editor: G. PRICE, Chorley. Created specially for all teachers of elementary published in 1801 in Latin, this classic has continued mathematics, this journal caters to teachers of to be important to mathematicians as the source of varying age groups, student-teachers, etc. Editorial the ideas from which number theory was developed. coverage includes discussions of structural appar­ atus, psychological findings as they influence class­ "Whatever set of values is adopted, [this book] room practice and the relationship of mathematics surely belongs among the greatest mathematical to other subjects. 3 times per year $3.00 treatises of all fields and periods."-Asger Aaboe. TOPOLOGY (An International Journal of Mathematics Cloth $12.50; paper $2.95 founded by J. H. C. Whitehead, Oxford) Editorial Board: R. BoTT, Cambridge, Mass .. et a!. A journal of high quality, publishing papers in math­ available through your bookstore ematics with special emphasis on topology and its Yale University Press Dr:! related subjects using topological techniques such as homological algebra, Lie groups, differential ge­ New Haven and London IIIli ometry and algebraic geometry. Quarterly $30.00 in Canada: McGill All orders and inquiries should be directed University Press to your subscription agent, or to MR. WILLIAM BIRNEY Director, Subscription Sales PERGAMON PRESS 44-01 21st Street, long Island City, N. Y. 11101

518 Select Publications for Current Problems

INTRODUCTION TO CYBERNETICS by V. M. Glushkov Translated from the Russian by Hansjorg Oser, National Bureau of Standards Written by a leading Russian authority, this book concerning the theory of Boolean functions, the contains discussions of contemporary cyber- abstract and structural theory of discrete auto- netics such as the theory of algorithms, digital mata, the theory of discrete self-organizing sys- automata, digital self-organizing systems, and terns, and the universal algorithmic language and mathematical logic. It relates fundamental data foundations of the restricted predicate calculus. August 1966, about 335 pp., approx. $12.50

VOLUME 21: PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks LINEAR DIFFERENTIAL EQUATIONS AND FUNCTION SPACES by J. L. Massera and J. J. Schaffer Presents in systematic and detailed form, the between properties of the nonhomogeneous recent studies of the authors concerning linear equation and the behavior of the solutions of ordinary differential equations in the real do- the homogeneous equation are thoroughly dis- main. Contributing to the deveiopment of an cussed. Methods of functional analysis and the idea put forth by 0. Perron in 1930, relations use of function spaces are emphasized. June 1966,404pp.,$16.00

VOLUME 22: PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks AN INTRODUCTION TO NONASSOCIATIVE ALGEBRAS by Richard D. Sc:haler In elementary terms, this book presents topics text to illustrate important techniques and alter­ which will be of value to graduate students native algebras are considered in detail. Jordan encountering nonassociative algebras for the first algebras and their relationships to the excep­ time. A number of proofs are included in the tional Lie algebras are described. August 1966, about 175 pp., approx. $6.50 FUNCTIONAL ANALYSIS AND NUMERICAL MATHEMATICS by L. Collatz Translated from the German Highlighting the dramtic changes occurring in spaces, the theory of Frechet derivatives, and numerical mathematics during the past twenty topological fixed-point theorems. Problems con­ years, the text describes those parts of functional cerning iterative methods, differential and inte­ analysis which have proven useful for numerical gral equations and approximation theory are applications including Hilbert space theory, Ban­ used to illustrate the applications. ach spaces, metric spaces and pseudometric July 1966, about 475 pp., $18.50

ADVANCE ANNOUNCEMENT: TOPOLOGICAL VECTOR SPACES, DISTRIBUTIONS, AND KERNELS by F. Treves November 1966, about 500 pp., approx. $18.00

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519 ENGINEERS· PHYSICISTS· MATHEMATICIANS

NAVAL PROBLEM SOLVING

The Applied Physics Laboratory, The Johns Hopkins University now offers both senior and associate staff appointments in naval operations analysis covering broadly defined areas of submarine, surface, and air operations ... present and future.

Primary requirements include ( 1) interest in solving complex problems of conflict that go far beyond the limitations of Game Theory, and (2) ability to abstract operational models from the complexity of real or proposed systems. Background in physics, engineering or mathematics is most applicable; knowledge of operations research techniques and modern computer usage in combat simulation is useful but not mandatory.

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

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

The Applied Physics Laboratory · The Johns Hopkins University

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

520 ~~fessor Wilson is lecturing in 14 cities. He's going to do it via Tele-Lecture (two-way amplified voice transmission system) with the help of electronic blackboards. He will be able to transmit notes, graphs and even artwork while he is lecturing. And after the lecture, a discussion can take place. In this way, teachers, researchers and others in allied fields can keep up with the latest information available. There's a man available to help you. He's a Communications Consultant in the field of education, and he knows your problems. Bell System American Telephone & Telegraph ® and Associated Companies

521 Are you a PhD in math, strong on statistics and probability, with a Renaissance man's capability to absorb other disciplines?

Astropower, a research laboratory of the Douglas Aircraft Company, is conducting basic research in the field of artificial intelligence. Over eight different technologies are involved and you will assimilate something of each. Experience or training in these areas is an asset. As for your normal profession, you must be a PhD in math, strong on statistics and probability. And you should have a working knowledge of adaptive or learning systems. If you did your dissertation on that subject, it is sufficient. Your primary responsibility will be basic research in the field of artificial intelligence systems. Currently, we are seeking to develop learning systems capable of recognizing specific patterns or objects of interest in complex photographic imagery such as that received from the Tiros or Ranger scientific satellites. If your professional qualifications meet our requirements, and if you have the capability and interest to absorb other disciplines, please send your professional biography to Mr. F. L. MacDonald, 2750 Ocean Park Blvd., Santa Monica, Calif. DOUGLAS A II equal OJIJIOI'illllily employer, males & je111ai£w may apply. ~WiHILE & SPACE SYSTE~IS DJVISIOX

522 A MESSAGE OF CAREER INTEREST TO THE ADVANCED, CREATIVE MATHEMATIC/AN FROM National Security Agency

A unique organization operating within the structure of the Federal Government, the National Security Agency is responsible for developing "secure" (i.e., invulnerable) communications systems to transmit and receive vital information.

Mathematicians are key members of NSA's scientific fraternity. And the Agency's advanced career opportunities for senior mathematicians have never been more exciting and attractive than they are today ... in their challenge, their scope and their potential.

Theoretical Research Opportunities There are presently a number of NSA groups engaged in theoretical mathematical research programs. Many other branches of mathematics which might fruitfully advance the Agency's work simply await the capabilities and interest of new mathematicians. In addition, there is considerable opportunity to make substantive scientific contributions in bridging gaps between the theoretical and the practical. Computer software and applications comprise still another vital area for potential contributions.

NSA mathematicians are frequently called in to help define new problems. Observing their origins, characteristics and the trends of any data associated with them, the mathematician must determine whether the problem and data are susceptible to mathematical solution. If so, he may then help define and determine the method of that solution.

Advanced Facilities, Educational Programs NSA will encourage you to pursue advanced mathematical study at any of several area universities, including the highly regarded University of Maryland. Educational assistance ranges from tuition payment for part time study-to full time, full salary fellowships.

As a professional staff member of NSA, you enjoy all the benefits of Federal employment without the necessity of Civil Service certi­ fication. The Agency is located between Washington and Baltimore, its site p~rmitting your choice of city, suburban or country living and offering a wide range of nearby nsa recreational/cu ltu ral/h istorical attractions. NATIONAL SECURITY AGENCY Suite 10, 4435 Wisconsin Ave., N.W. Washington, D.C. 20016 For further career information, please An equal opportunity employer (M & F) send us your resume now. . .. where imagination is the essential qualification

523 ·~-~~~~~!!~ ,.!/' '~~ •1 1• TRANSLATORS II II RUSSIAN TO ENGLISH II II II II SCIENTISTS & ENGINEERS II II You can keep abreast of the latest Soviet research in your field while supplementing your II II income by translating in your own home on a II II part-time basis. In the expanding Consultants Bureau publishing program, we guarantee a II II continuous flow of translation in your specialty. 1 will II If you have a native command of English, a These scientists good knowledge of Russian, and experience 1 see the results of 11 and academic training in a scientific or engi­ neering discipline, you may be qualified for our 1 their work in action 11 program. Immediate openings are available in On the blackboard before these men is II the following fields: physics, instrumentation, I a part of the mathematical model they automation, quality control, mathematics, engi­ are developing for the Center for Naval 1 neering, chemistry, biology, and earth sciences. I Analyses of The Franklin Institute to 1 help the Navy coordinate sea, air, under- 1 sea, and amphibious forces. Further information may be obtained from I The Navy uses this and other CNA I Translations Editor studies as part of its decision-making I I apparatus for force and systems re- CONSULTANTS BUREAU quirements and disposition, tactics, stra- I A Division of Plenum Publishing Corporation I tegy, weapons evaluation, and logistics. 227 West 17 Street, New York, N.Y. 10011 1 A private scientific organization engaged I in operations research and systems I analysis for the Navy and Marine Corps, I CNA needs graduate mathematicians, I statisticians, physical scientists, sys- I I terns analysts, research engineers, and I operations research analysts. CUSHING-MALLOY, I CNA staff members must have the ability 1 to apply the scientific method in new INC. 1 contexts . .. to aid in the formulation 1 of new operational requirements . .. to 1350 North Main St., P. 0. Box 632 I develop and test mathematical models. I CNA provides for professional growth; I I the opportunity for stimulating work im- Ann Arbor, Michigan 48107 1 portant to national security, and compe- 1 titive salaries with the customary fringe LITHOPRINTERS benefits attached to responsible posi- 1 I tions. I Send resume and letter to: I • I James M. Hibarger I CENTER FOR NAVAL ANALYSES Known for 1 1401 Wilson Blvd., Arlington, Va. 22209 II QUALITY- ECONOMY II I CNA______SERVICE I CENTERFORNAVALANALYSES II OF THE FRANKLIN INSTITUTE I INS • Institute of Naval Studies II • SEG • Systems Evaluation Group I OEG • Operations Evaluation Group II NAVWAG • Naval Warfare Analysis Group We specialize in reproducing out-of­ I MCOAG • Marine Corps Operations Analysis II Group print books, as well as typewriter com­ a:' An equal opportunity employer /...... , . position. Let us quote on your next ·------··'- printing.

524 AUSTRALIA UNIVERSITY OF QUEENSLAND, St. Lucia, Brisbane. Department of Mathematics During 1966 the Department hopes to make appointments at the level of lecturer ($ Aust. 4800-6200 p.a.), senior Lecturer ($ Aust. 6400-7600 p.a.) and reader ($ Aust. 8000-8600 p.a.). Applications for two new chairs ($ Aust. 10,400) in the Department are at present being considered. The present staff establishment (lecturer and above) is 25 and it is likely that further appointments will be made in 1967. Efforts are being made to expand the research activities of the Department considerably. The University has at present a student population of 14,500 and is expanding rapidly. Further details of the Department and of study leave, travelling and removal expenses, superannuation, etc. may be obtained from the Registrar, who will be happy to receive applications at any time.

C. J. CONNELL Registrar

Conference on the Advanced Placement Program in Mathematics June 23-25, 1966, Harvey Mudd College Those wanting additional details should write to, Professor R. C. James Harvey Mudd College, Claremont, California

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526 Mathematical Physics Monograph Series

SYMMETRY GROUPS IN NUCLEAR AND THE COMPLEX j-PLANE, Roger G. Newton, PARTICLE PHYSICS, Freeman J. Dyson, In­ Indiana University. The mathematical tech­ stitute for Advanced Study. This collection of nique of complex angular momentum is de­ thirty-two key journal articles which appeared scribed in detail in this new monograph. The during the years 1937 to 1965 applies group author has limited this discussion of angular theory to nuclear and particle physics. It pro­ momentum to those areas in which mathema­ vides the mathematical physicist with a readily tical proofs are based on well-established dy­ accessible compendium of essential literature. namics. For example, a complete review of the Professor Dyson begins the text with an intro­ application of the complex angular momentum duction emphasizing the growing similarity be­ technique to the Schrodinger theory is given; tween the use of symmetry groups in nuclear this application is based on the particle dynam­ and particle physics, both in the observed data ics expressed in the Schrodinger differential and in theoretical analysis. The reprints begin equation. In addition to proofs and their discus­ with the classical papers of Wigner and Barg­ sion, the book contains all the important nu­ man which set up the basic mathematical merical results of various applications and framework of relativistic symmetry groups. examples. A complete bibliography is included. These are followed by recent papers applying 244 Pages; $9.00 Cloth, $5.95 Paper. symmetry groups in a more or less speculative way to current problems of particle physics. PCT, SPIN&: STATISTICS, AND ALL THAT, Included in this last group of papers are at­ Raymond F. Streater, Imperial College, and tempts to apply groups such as SU(6) and Arthur S. Wightman, Princeton University. SU(l2) to problems of particle physics, as well Recent advances in the foundations of relativ­ as a criticism of these attempts. 332 Pages; istic quantum field theory are presented in this $9.00 Cloth, $4.95 Paper. monograph, together with the mathematical tools required for understanding them. These advances give simple, satisfying, but nontrivial explanations of the origin of such physical regularities as PCT symmetry, and the connec­ tions of spin with statistics. Results that can be rigorously proven are presented in a style which makes them digestible. The book also LIE GROUPS FOR PHYSICISTS, Robert Her· serves as a rather self-contained introduction mann, Argonne National Laboratory and to axiomatic quantum field theory. 189 Pages; Northwestern University. These lecture notes $9.00 Cloth, $4.95 Paper. are directed to the mathematically inclined physicist and provide a broad survey of the MATHEMATICAL FOUNDATIONS OF important ideas in recently developed parts of QUANTUM MECHANICS, George W. Mack­ the theory of Lie groups. Special emphasis has ey, Harvard University. The aim of the course been given to the theory of homogeneous on which these notes are based was to explain spaces, particularly the theory of symmetric quantum mechanics and certain parts of classi­ spaces, and the possible application of Lie cal physics from a point of view more congenial groups to recent progress in elementary par­ to pure mathematics than that commonly en­ ticle physics is indicated. In addition to de­ countered in physics texts. Accordingly, the scribing the relation between geometry of ho­ emphasis in this volume is on careful formu­ mogeneous vector bundles, representations of lation rather than on the technique of solving noncompact Lie groups, and application to problems. In the interests of making the work quantum mechanics, Dr. Hermann presents a accessible to mathematicians with little or no mathematical treatment of the ideas of con­ knowledge of physics, the author defines physi­ traction and limit of Lie groups and representa­ cal concepts in the terms of pure mathematics tion. Complete references are included. 202 and of space and time. 147 Pages; Cloth $8.00, Pages; $12.50. Paper $3.95.

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THE GEOMETRY OF New and INCIDENCE Current Mathematics By HAROLD DORWART, Trinity College 1, Connecticut. Dr. Dorwart begins with a synthetic introduction of ideal points of the Euclidean plane. He follows Books from with analytical discussions on homogeneous co­ ordinates, duality, and the classic configuration theorems of Pappus, Desargues, and Pascal. Postu­ Prentice Hall lates for an abstract projective plane are presented, and Dr. Dorwart shows how a projective plane may ORDINARY DIFFERENTIAL be converted into analytic geometry by introduction EQUATIONS of coordinates. He then introduces finite projective planes and related combinatoric problems such as By 1. G. PETROVSKI, Moscow State University. A self­ Squares. January 1966, 156 pp. $5.95 contained introduction to the theory of ordinary Latin differential equations, this new volume provides an exceptionally lucid development of basic general MODERN ALGEBRA, theory. A good background in advanced calculus or introductory mathematical analysis is a prerequisite. Volumes I & II Some knowledge of the elements of linear algebra By SETH WARNER, Duke University. Introduces the is helpful but not essential. A book suited for both student to the rigors of abstract mathematical course use and independent study by mathematically thought, while the coverage is thorough enough to inclined engineers and physical scientists. (In the provide extensive treatment of almost every aspect Selected Russian Publication Series in the Mathe­ of modern algebra. Dr. Warner has provided over matical Sciences, translated and edited by RICHARD I ,300 carefully selected exercises, ranging from the A. SILVERMAN.) May 1966, approx. 256 pp. $795. routine to the most advanced. Both volumes have been class tested for three years. UNIVERSITY ALGEBRA VOLUME I: Relates concepts arising in the study By RICHARD E. JOHNSON, Montana State University. and solid The basic theme is to define abstract concepts in as of vector spaces to concepts of plane and then carry them analytic geometry. Contains much material unavail­ simple a setting as possible, able in texts at this level such as the Density over to more complex systems. Thus, cartesian Theorem on products, homomorphisms, and quotient structures Theorem and the Wedderburn-Artin groups and extended to simple rings with minimal condition. Introduces the are first defined for Abelian language of set theory and the notion of a com­ commutative rings. Contains a complete set of exer­ pp. $12.95 cises from routine ones to difficult ones as well as position or binary operation. 1965, 4457 special theoretical projects. March 1966, 271 pp. $.50. VOLUME II: Discusses in detail the classical geo­ $7.95. Selected answers to problems, metric problems of constructing figures by ruler and compass; includes Gauss's theorem on the con­ MATHEMATICAL struction of regular polygons. Contains material INTRODUCTION TO unavailable in texts at this level: The Euler-La­ grange Theorem on real-closed fields; the "Funda­ CELESTIAL MECHANICS mental Theorem of Algebra"; Frobenius's Theorem By HARRY POLLARD, P'urdue University. The main on real division algebras; The Axiom of Choice; purpose of this book is to interest mathematicians Zorn's Lemma and Steinitz's theorems on algebraic to take a fresh look at the field; it follows the closures of fields. /965, 347 pp. $11.50 recommendations of the CUPM course on celestial mechanics. Features an account of the n-Body (Prices shown are £or student use) Problem, including new material on the growth of For approval copies, write: Box 903 the systems, new derivations of the planetary equa­ tions based on vector methods, and a simplified PRENTICE-HALL account of the Hamilton-Jacobi theory. January 1966, Ill pp., $4.95. Englewood Cliffs, N.j. 07632